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arXiv:1611.04732v1 [math.AC] 15 Nov 2016 BETTI NUMBERS OF CERTAIN SUM IDEALS JOYDIP SAHA, INDRANATH SENGUPTA, AND GAURAB TRIPATHI A BSTRACT. In this paper we compute the Betti numbers for ideals of the form I1 (XY ) + J, where X and Y are matrices and J is the ideal generated by the 2 × 2 minors of the matrix consisting of any two rows of X. 1. I NTRODUCTION Let K be a field and {xij ; 1 ≤ i, j ≤ n}, {yj ; 1 ≤ j ≤ n} be indeterminates over K; n ≥ 2. Let R := K[xij , yj ] denote the polynomial algebra over K. Let X denote an n × n matrix such that its entries belong to the ideal h{xij ; 1 ≤ i ≤ n, 1 ≤ j ≤ n}i. Let Y = (yj )n×1 be the n × 1 column Pnmatrix. Let I1 (XY ) denote the ideal generated by the polynomials gj = i=1 xji yi , j = 1, . . . , n, which are the 1 × 1 minors or entries of the n × n matrix XY . The primality, primary decomposition and Betti numbers of ideals of the form I1 (XY ) have been studied in [9] and [10], with the help of Gröbner bases for I1 (XY ). Ideals of the form I1 (XY ) + J are particularly interesting because they occur in several geometric considerations like linkage and generic residual intersection of polynomial ideals, especially in the context of syzygies. Bruns-Kustin-Miller [1] resolved the ideal I1 (XY )+Imin(m,n) (X), where X is a generic m×n matrix and Y is a generic n×1 matrix. Johnson-McLoud [5] proved certain properties for the ideals of the form I1 (XY ) + I2 (X), where X is a generic symmetric matrix and Y is either generic or generic alternating. We say that I and J intersect transversally if I ∩J = IJ. Suppose that F· resolves R/I and G· resolves R/J minimally. It is interesting to note that if I and J intersect transversally, then the tensor product complex F ⊗R G 2010 Mathematics Subject Classification. Primary 13D02; Secondary 13C40, 13P10, 13D07. Key words and phrases. Gröbner basis, Betti numbers, transversal intersection, mapping cone. The first author thanks UGC for the Senior Research Fellowship. The second author is the corresponding author, who is supported by the the research project EMR/2015/000776 sponsored by the SERB, Government of India. The third author thanks CSIR for the Senior Research Fellowship. 1 2 JOYDIP SAHA, INDRANATH SENGUPTA, AND GAURAB TRIPATHI resolves R/I + J minimally; see Lemma 3.7. Therefore, it is useful to know if two ideals intersect transversally, especially when one is trying to compute minimal free resolutions and Betti numbers for ideals of the form I + J, through iterated techniques; see [4]. 2. N OTATION M AIN T HEOREM xi1 xi2 · · · xin e • If X is generic and i < j; let Xij = . xj1 xj2 · · · xjn • If X is generic symmetric and i < j; let x1i · · · xii · · · xij · · · xin e Xij = . x1j · · · xij · · · xjj · · · xjn AND THE eij . • Let Gij denote the set of all 2 × 2 minors of X eij ) denote the ideal generated Gij . • I2 ( X Our aim in this paper is to prove the following theorem: Theorem 2.1. Let X = (xij ) be either the generic or the generic symmetric matrix of order n. Let 1 ≤ i < j ≤ n. eij ) + hgi , gj i are given by (1) The total Betti numbers for theideal I2 (X n n n b0 = 1, b1 = 2 + 2, b2 = 2 3 + n, bi+1 = i i+1 + (i − 2) ni , for 2 ≤ i ≤ n − 2 and bn = n − 2. (2) Let 1 ≤ k ≤ n−2. Let βk,p denote the p-th total Betti number for the eij )+hgi , gj , gl1 , . . . , gl i, such that 1 ≤ l1 < . . . < lk ≤ n ideal I2 (X k and lt is the smallest in the set {1, 2, . . . , n} \ {i, j, l1 , . . . , lt−1 }, for every 1 ≤ t ≤ k. They are given by βk,0 = 1, βk,p = βk−1,p−1 + βk−1,p for 1 ≤ p ≤ n + k − 1 and βk,n+k = n − 2. eij ) are In particular, the total Betti numbers for the ideal I1 (XY ) + I2 (X βn−2,0 , βn−2,1 , . . . , βn−2,2n−2 . 3. P RELIMINARIES 3.1. Determinantal Ideals. We recall some useful results on determinantal ideals pertaining to our work. We refer to [2], [3], [8] for detailed discussions on these. Theorem 3.1. Let K be a field and let xij : 1 ≤ i ≤ m, 1 ≤ j ≤ n be indeterminates over K. Let A = (xij ) be the m×n matrix of indeterminates and Im (A) denotes the ideal generated by the maximal minors of A. The set of maximal minors of A is a universal Gröbner basis for the ideal Im (A). SUM IDEALS 3 Proof. See [2]. The Eagon-Northcott Complex. We present the relevant portion from the book [3] here. Let F = Rf and G = Rg be free modules of finite rank over the polynomial ring R. The Eagon-Northcott complex of a map α : F −→ G (or that of a matrix A representing α) is a complex df −g df −g+1 EN(α) : 0 → (Symf −g G)∗ ⊗ ∧f F −→ (Symf −g−1 G)∗ ⊗ ∧f −1 F −→ d d2 g ∧g α 3 · · · −→ (Sym2G)∗ ⊗ ∧g+2 F −→ G∗ ⊗ ∧g+1 F ∧ F −→ ∧g G. Here Symk G is the k-th symmetric power of G and M ∗ = HomR (M, R). The map dj are defined as follows. First we define a diagonal map (Symk G)∗ → G∗ ⊗ (Symk−1G)∗ X ′ ′′ ui ⊗ ui u 7→ i as the dual of the multiplication map G ⊗ Symk−1G −→ Symk G in the symmetric algebra of G. Next we define an analogous diagonal map ∧k F −→ F ⊗ ∧k−1 F X ′ ′′ v 7→ vi ⊗ vi i as the dual of the multiplication in the exterior algebra of F ∗ . Theorem 3.2 (Eagon-Northcott). The Eagon-Northcott complex is a free resolution of R/Ig (α) iff grade(Ig (α)) = f − g + 1 where Ig (α) denotes the g × g minors of the matrix A representing α. Proof. See [3]. 3.2. Mapping Cone. We present the relevant portion from the book [8] ′ ′ here. Let R be the polynomial ring. Let φ· : (U· , d· ) → (U· , d· ) be a map of complexes of finitely generated R-modules. The mapping cone of φ· is the ′ complex W· with differential δ· defined as follows. Let Wi = Ui−1 ⊕ Ui , ′ ′ ′ ′ with δ\|Ui−1 = −d + φ : Ui−1 −→ Ui−2 ⊕ Ui−1 and δ\|U ′ = d : Ui → Ui−1 i for each i. Theorem 3.3. Let M be an ideal minimally generated by the polynomials f1 , . . . , fr . Set Mi = hf1 , . . . , fi i, for 1 ≤ i ≤ r. Thus, M = Mr . For each i ≥ 1, we have the short exact sequence fi+1 0 −→ S/(Mi : fi+1 ) −→ S/Mi −→ S/Mi+1 −→ 0. If resolutions of S/Mi and S/(Mi : fi+1 ) are known then we can construct a resolution of S/Mi+1 by the mapping cone construction. 4 JOYDIP SAHA, INDRANATH SENGUPTA, AND GAURAB TRIPATHI Proof. See Construction 27.3 in [8]. 3.3. Gröbner basis and transversal intersection of Ideals. Lemma 3.4. Let h1 , h2 · · · , hn ∈ R be such that with respect to a suitable monomial order on R, the leading terms of them are mutually coprime. Then, h1 , h2 · · · , hn is a regular sequence in R. Proof. . See Lemma 4.3 in [9]. Lemma 3.5. Suppose that X is either generic or generic symmetric. The eij ), with respect a suitable set Gij is a Gröbner basis for the ideal I2 (X monomial order. Proof. We choose the lexicographic monomial order given by the following ′ ′ ordering among the variables: xst > xs′ t′ if (s , t ) > (s, t) and yn > yn−1 > · · · , y1 > xst for all s, t. We now apply Lemma 4.2 in [9] for the matrix X t and for k = 2. Definition 1. Let T ⊂ R be a set of monomials. We define supp(T ) = {(i, j, 0) \| xij divides m for some m ∈ T } ∪ {(0, 0, k) \| yk divides m for some m ∈ T }. If T = {m}, then we write supp(m) instead of supp({m}). Lemma 3.6. Let > be a monomial ordering on R. Let I and J be ideals in R, such that m(I) and m(J) denote unique minimal generating sets for their leading ideals Lt(I) and Lt(J) respectively. Then, I ∩ J = IJ if supp(m(I))∩supp(m(J)) = ∅. In other words, the ideals I and J intersect transversally if the set of variables occurring in the set m(I) is disjointed from the the set of variables occurring in the set m(J). Proof. Let f ∈ I ∩ J; we show that f ∈ IJ. Now f ∈ I ∩ J implies that f ∈ I and therefore Lt(f ) ∈ Lt(I). Hence, mi ∈ m(I) such that mi \| Lt(f ). Similarly, there exists monomial mj ∈ m(J) such that mj \| Lt(f ). Given that mi and mj are of disjoint support, we have mi mj \| Lt(f ) and this (f ) proves that Lt(f ) ∈ Lt(IJ). We replace f by f − Lt . Now the proof mi mj Lt(f ) < Lt(f ). follows by induction since, Lt f − m i mj SUM IDEALS 5 3.4. Homological Lemmas. Lemma 3.7. Let I and J be graded ideals in a graded ring R, such that I ∩ J = I · J. Suppose that F and G are minimal free resolutions of I and J respectively. Then F ⊗ G is a minimal free resolution for the graded ideal I + J. Proof. Consider the short exact sequence 0 −→ I −→ R −→ R/I −→ 0 and tensor it with R/J over R. We get the exact sequence 0 −→ Tor1 (R/I, R/J) −→ I/I · J −→ R/J −→ R/I + J −→ 0. The terms on the left are 0 since R is a flat R module. Moreover, the kernel of the map from I/I · J −→ R/J is I ∩ J/I · J. Therefore Tor1 (R/I, R/J) = 0 if and only if I ∩J = I ·J. By the corollary 1 of theorem 3 proved in [6], Tor1 (R/I, R/J) = 0 implies that Tori (R/I, R/J) = 0 for all i ≥ 1. Therefore, Hi (F ⊗ G ) ≃ Tori (R/I, R/J) ≃ 0 for all i ≥ 1 and H0 (F ⊗ G ) ≃ R/I + J. This proves that F ⊗ G resolves I + J. The resolution is minimal since F and G are minimal. Lemma 3.8. Let A A 1 2 Ra1 −→ Ra2 −→ Ra3 be an exact sequence of free modules. Let Q1 , Q2 , Q3 be invertible matrices of sizes a1 , a2 , a3 respectively. Then, Ra1 Q−1 2 A1 Q 1 Ra2 −→ Q−1 3 A2 Q 2 −→ Ra3 is also an exact sequence of free modules. Proof. The following diagram is a commutative diagram is free modules and the vertical maps are isomorphisms: A1 ROa1 / RaO 2 Q1 Q−1 2 A1 Q 1 Ra1 Therefore, Ra1 Ra3 is exact. Q−1 2 A1 Q 1 −→ Ra2 / R Q−1 3 A2 Q 2 −→ A2 / RaO 3 Q2 Q3 Q−1 A2 Q 2 3 a2 / a3 R A A 1 2 Ra3 is exact since Ra1 −→ Ra2 −→ Corollary 3.9. Let C B A Ra1 −→ Ra2 −→ Ra3 −→ Ra4 be an exact sequence of free modules. Let P1 , P2 , P3 be invertible matrices of sizes a1 , a2 , a3 respectively. Then, P −1 CP1 BP AP −1 3 Ra1 2−→ Ra2 −→2 Ra3 −→ Ra4 6 JOYDIP SAHA, INDRANATH SENGUPTA, AND GAURAB TRIPATHI is also an exact sequence of free modules. C B Proof. Consider the sequence Ra1 −→ Ra2 −→ Ra3 . If we take Q1 = P1 , Q2 = P2 and Q3 = I and apply Lemma ??, we get that the sequence Ra1 P2−1 CP1 −→ BP Ra2 −→2 Ra3 is exact. We further note that the entire se- P −1 CP1 BP A An+1 A quence Ra1 2−→ Ra2 −→2 Ra3 −→ Ra4 is exact as well, since Im(B) = BP Im(BP2 ) and P2 is invertible. Let us now consider the sequence Ra2 −→2 A Ra3 −→ Ra4 . We take Q1 = Q3 = I, Q2 = P3−1 and apply Lemma 2.3 to arrive at our conclusion. Lemma 3.10. Let An−1 n · · · −→ Rβn+1 −→ Rβn −→ Rβn−1 −→ Rβn−2 −→ · · · be an exact sequence of free R modules. Let aij denote the (i, j)-th entry of An . Suppose that alm = ±1 for some l and m, ali = 0 for i 6= m and ′ ajm = 0 for j 6= l. Let An+1 be the matrix obtained by deleting the m-th ′ row from An+1 , An−1 the matrix obtained by deleting the l-th column from ′ An−1 and An the matrix obtained by deleting the l-th row and m-th column from An . Then, the sequence ′ · · · −→ R βn+1 An+1 −→ R ′ ′ βn −1 An −→ R βn−1 −1 An−1 −→ Rβn−2 −→ · · · is exact. Proof. The fact that the latter sequence is a complex is self evident. We need to prove its exactness. By the previous lemma we may assume that l = m = 1, for we choose elementary matrices to permute rows and columns and these matrices are always invertible. Now, due to exactness of the first complex we have An−1 An = 0. This implies that the first col′ umn of An−1 = 0, which implies that Im(An−1 ) = Im(An−1 ). Therefore, the right exactness of An+1 is preserved. By a similar argument we can ′ prove that the left exactness of An+1 is preserved. ′ from R. If (x) ∈ ker(A Let (x) denote a tuple with entries n ), then (0, x) ∈ ker(An ). There exists y ∈ Rβn+1 such that An−1 y = (0, x). ′ ′ It follows that An−1 y = (x), proving the left exactness of An . By a ′ similar argument we can prove the right exactness of An . Lemma 3.11. Let A be q × p matrix over R with Aij = ±1, for some i and j. Let C be a p × s matrix and B a r × q matrix over R. There exist an invertible q × q matrix X and an invertible p × p matrix Y , such that SUM IDEALS 7 (i) (XAY )kj = δki and (XAY )ik = δjk , that is   ··· 0 ···  ··· 0 ···    ..  .. .  . .  .  .   XAY = 0 · · · 0 1 0 · · · 0 ; 1 at the (i, j)−th spot.  . ..  ..  .. .  .    ··· 0 ···  ··· 0 ··· P (ii) (Y −1 C)kl = Ckl for k 6= j and (Y −1 C)jl = Cjl + t6=i (ait )Ctl P (iii) (BX −1 )kl = Bkl for l 6= i and (BX −1 )ki = Bki + t6=i (atj )Bkt . Proof. (i) We prove for aij = 1. The other case is similar. We take Y = Πk6=j Ejk (−aik ) and X = Πk6=i Eki (−akj ), where Ekl (α) denotes the matrix E with Ekl = α, Ett = 1 and Eut = 0 for u 6= t and (u, t) 6= (k, l). (ii) and (iii) are easy to verify. Lemma 3.12. Let A be q × p matrix, C be a p × s matrix and B a r × q matrix over R. The matrices A, B and C satisfy property Pij if they satisfy the following conditions: • Aij = 1 , Aik ∈ m for k 6= j and Akj ∈ m for k 6= i; • Bki ∈ m, for 1 ≤ k ≤ r; • Cjl ∈ m, for 1 ≤ l ≤ s. The matrices XAY , BX −1 and Y −1 C satisfy property Pij , if A, B, C satisfy property Pij . Proof. This follows from the above lemma since aik and akj belong to m. 4. M INIMAL eij ) + hgi , gj i FREE RESOLUTION OF I2 (X Lemma 4.1. Let X be generic or generic symmetric. Let i < j. eij )) = n − 1. (i) ht(I2 (X fij ). (ii) The Eagon-Northcott complex minimally resolves the ideal I2 (X Proof. (i) We show that f1 , . . . , fn−1 , given by fk = xik xj,k+1 − xjk xi,k+1, 1 ≤ k ≤ n − 1 form a regular sequence. Let us first assume that X is generic. We take the lexicographic monomial order induced by the following ordering among the variables: xi1 > e and xi2 > · · · > xin > xj2 > xj3 > · · · > xjn > xj1 not appearing in X the variables yk are smaller than xj1 . Then, Lt(fk ) = xik xj,k+1 and hence 8 JOYDIP SAHA, INDRANATH SENGUPTA, AND GAURAB TRIPATHI gcd(Lt(fk ), Lt(fl )) = 1 for every k 6= l. Therefore, f1 , . . . , fn−1 is a regular e ≥ n − 1. On the other hand, sequence by Lemma 3.4 and hence ht(I2 (X)) e ≤ n − 1, by Theorem [13.10] in [7]. Hence, ht(I2 (X)) e ≤ n − 1. ht(I2 (X)) If X is generic symmetric, then we have to choose the lexicographic monomial order induced by xii > xij > x1i > x2i > · · · > xi−1,i > xi,i+1 > · · · > xin > xjj > · · · > xc ij > · · · > xj−1,j > xj,j+1 > · · · > xjn fij and the variables yp are smaller than and variables xkl not appearing in X xjn . fij ) is n − 1, which is the maximum. Hence, the (ii) The height of I2 (X fij ). Eagon-Northcott complex minimally resolves the ideal I2 (X Lemma 4.2. Let X be generic or generic symmetric. Let i < j. Then, eij ) ∩ hgi i = I2 (X eij ) · hgi i, that is, the ideals I2 (X eij ) and hgi i intersect I2 ( X transversally. Proof. Let X be generic. We choose the lexicographic monomial order ′ ′ given by the following ordering among the variables: xst > xs′ t′ if (s , t ) > (s, t) and yn > yn−1 > · · · > y1 > xst for all s, t. Then, by Lemma 3.5 eij ). the set of all 2 × 2 minors forms a Gröbner basis for the ideal I2 (X eij ))) doesn’t involve the inClearly, the minimal generating set m(Lt(I2 (X determinates xin and yn , whereas Lt(gi ) = xin yn . Hence, the supports of eij ))) and m(Lt(gi )) are disjoint. Therefore, by Lemma 3.6 we m(Lt(I2 (X are done. Let X be generic symmetric. Once we choose the correct monomial order, the rest of the proof is similar to the generic case. Suppose that (i, j) = (n − 1, n). We choose the lexicographic monomial order given by the following ordering among the variables: y1 > yn > yn−1 > · · · > y2 > > > > xn−1,n−1 > xn−1,n > x1,n−1 > x2,n−1 > · · · > xn−2,n−1 xnn x1n > · · · > xn−2,n xst for all other s, t. Suppose that (i, j) 6= (n − 1, n). We choose the lexicographic monomial order given by the following ordering among the variables: yn > yn−1 > · · · > y1 > xii > xij > x1i > x2i > · · · > xi−1,i > xi,i+1 > · · · > xin > xjj > · · · > xj−1,j > xj,j+1 > · · · > xjn > xst for all other s, t. SUM IDEALS 9 eij ) + hgi i : gj ) = Lemma 4.3. Let X be generic and i < j. Then, (I2 (X eij ) + hgi i : hxi1 , . . . , xin i. If X is generic symmetric and i < j, then (I2 (X gj ) = hx1i , . . . , xi−1,i , xii , . . . , xin i. P Proof. X be generic. We have xit gj = xjt gi + nk=1 (xit xjk − xik xjt )yk . fij ) + hgi i : gj ). Moreover, I2 (X eij ) + hgi i ⊆ Hence, hxi1 , · · · , xin i ⊆ (I2 (X hxi1 , · · · , xin i and gj ∈ / hxi1 , · · · , xin i. The ideal hxi1 , · · · , xin i being a eij ) + hgi i : gj ). The prime ideal, it follows that hxi1 , · · · , xin i ⊇ (I2 (X proof for the generic symmetric case is similar. 5. R ESOLUTION OF THE SUM IDEALS eij ) + Our aim is to construct a minimal free resolution for the ideal I2 (X eij ) and hgi i intersect hg1, . . . , gn i. We have proved that the ideals I2 (X eij ) + hgi i can therefore be resolved transversally; see 4.2. The ideal I2 (X eij ) + minimally by Theorem 3.7. We have also proved that the ideal I2 (X hgi i and the ideal hgj i have linear quotient; see 4.3. Therefore, the ideal eij ) + hgi , gj i can be resolved by the mapping cone construction. A I2 ( X minimal free resolution can then be extracted from this resolution by apeij ) + hgi , gj i plying Lemma 3.12. Next, we will show that the ideal I2 (X intersects transversally with the ideal hgl1 i, if l1 is the minimum in the eij ) + set {1, 2, . . . , n} \ {i, j}; see Lemma 5.4. Therefore, the ideal I2 (X hgi , gj , gl1 i can be resolved minimally by Theorem 3.7. Proceeding in this eij )+hgi , gj , gl1 , . . . , gl i manner, we will be able to show that the ideals I2 (X k and hglk+1 i intersect transversally, if 1 ≤ l1 < . . . < lk < lk+1 ≤ n and lk+1 is the smallest in the set {1, 2, . . . , n} \ {i, j, l1 , . . . , lk }; see Lemma eij ) + 5.4. This finally gives us a minimal free resolution for the ideal I2 (X hgi , gj , gl1 , . . . , gln−2 i, with 1 ≤ l1 < . . . < ln−2 ≤ n and lt ∈ / {i, j} for every t. Let us assume that X is generic and i = 1 and j = 2. The proofs for the general i and j, with i < j would be similar according to the aforesaid scheme. The proofs in the case when X is generic symmetric would be similar as well. Comments for general i < j and the symmetric case have been made whenever necessary. 10 JOYDIP SAHA, INDRANATH SENGUPTA, AND GAURAB TRIPATHI e12 ) + hg1 , g2 i. The minimal free 5.1. A minimal free resolution for I2 (X e12 ) is given by the Eagon-Northcott complex, which is resolution of I2 (X the following: δk e −→ 0 Ek−1 −→ · · · E1 −→ E0 −→ R/I2 (X) E : 0 −→ En−1 −→ · · · Ek −→ n where E0 ∼ = R1 , Ek = Rk(k+1) and for each k = 0, . . . , n − 2, the map δk : Ek → Ek−1 is defined as δk (ei1 ∧ · · · ∧ eik+1 ) ⊗ v2k−1 δk (ei1 ∧ · · · ∧ eik+1 ) ⊗ v1k−1 δk (ei1 ∧ · · · ∧ eik+1 ⊗ v1j v2k−j−1) = k+1 X x2s (ei1 ∧ · · · ∧ eˆs ∧ · · · ∧ eik+1 ) ⊗ v2k−2 = k+1 X (−1)s+1 x1s (ei1 ∧ · · · ∧ eˆs ∧ · · · ∧ eik+1 ) ⊗ v1k−2 s=1 s=1 = k+1 X (−1)s+1 x1s (ei1 ∧ · · · ∧ eˆs ∧ · · · ∧ eik+1 ) ⊗ v1j−1v2k−j−1 s=1 + k+1 X x2s (ei1 ∧ · · · ∧ eˆs ∧ · · · ∧ eik+1 ) ⊗ v1j v2k−j−2 s=1 for every ordered k + 1 tuple (i1 , i2 , · · · , ik+1 ), with 1 ≤ i1 < · · · < ik+1 ≤ n and for every j = 1, 2, · · · , k − 2. A minimal resolution of hg1 i is given by: g1 G : 0 −→ R −→ R −→ R/hg1 i −→ 0. e12 ) and hg1 i intersect transversally, by Lemma 4.2. ThereThe ideals I2 (X e12 ) + hg1 i is given fore, by Lemma 3.7, a minimal free resolution for I2 (X by the tensor product complex ψk+1 e12 )+hg1 i → 0 E ⊗G : 0 → En−1 → · · · → Ek+1 ⊕Ek −→ Ek ⊕Ek−1 → · · · → E0 → R/I2 (X such that ψk : Ek ⊕ Ek−1 −→ Ek−1 ⊕ Ek−2 is the map defined as j k−j−1 j k−j−1 ψk ei1 ∧ · · · ∧ eik+1 ⊗ v1 v2 = δk (ei1 ∧ · · · ∧ eik+1 ) ⊗ v1 v2 ψk (ei1 ∧ · · · ∧ eik ) ⊗ v1j v2k−j−2 = (−1)k−1g1 (ei1 ∧ · · · ∧ eik ) ⊗ v1j v2k−j−2 j k−j−2 . + δk−1 (ei1 ∧ · · · ∧ eik ) ⊗ v1 v2 e12 ) + hg1, g2 i by mapping Now we find a minimal free resolution for I2 (X e12 ) + cone. Let Ck := (E· ⊗ G· )k . We have proved in Lemma 4.3 that hI2 (X hg1i : g2 i = hx11 , x12 , · · · , x1n i; which is minimally resolved by the Koszul SUM IDEALS 11 complex. Let us denote the Koszul Complex by (F· ; σk ), where σk is the kth differential. We first construct the connecting map τ· : F· → E· ⊗ G· . Let n n n us write Fk := R(k ) and Ck := Rk(k+1) ⊕R(k−1)( k ) . The map τk : Fk → Ck is defined as: X τk (ei1 ∧ · · · ∧ eik ) = yj (ei1 ∧· · ·∧eik ∧ej )⊗v1k−1 −(ei1 ∧· · ·∧eik )⊗v1k−2 . j Let us choose the lexicographic ordering among the k tuples (i1 , . . . , ik ), n such that 1 ≤ i1 < · · · < ik ≤ n in order to write an ordered basis for R(k ) . We define lexicographic ordering among the tuples (i1 , . . . , ik+1 , k − j, j), for j = 0, . . . , k and k = 1, . . . , n − 1 to order the basis elements for n n n Rk(k+1) . Moreover, in the free module Ck = Rk(k+1) ⊕ R(k−1)( k ) , we order n the basis elements in such a way that those for Rk(k+1) appear first. The matrix representation of τk with respect to the chosen ordered bases is the following:   Ak( n )×(n) 0k( n )×(n) k k+1 k k+1          .  −I(n)×(n)   k k    0    (k−1)(nk)×(nk)   0(k−2)(n)×(n) k k Theorem 5.1. The following diagram commutes for every k = 1, . . . , n−1: τk FO k / σk+1 CO k ψk+1 Fk+1 τk+1 / Ck+1 Proof. It suffices to prove the statement for a basis element (ei1 ∧· · ·∧eik+1 ) of Fk+1 . Without loss of generality we consider (e1 ∧ · · · ∧ ek+1 ). We first compute (τk ◦ σk+1 )(e1 ∧ · · · ∧ ek+1 ). σk+1 (e1 ∧ · · · ∧ ek+1 ) 7−→ k+1 X (−1)j+1 x1j (e1 ∧ · · · ∧ eˆj ∧ · · · ∧ ek+1 ) j=1 τk 7−→ k+1 X (−1) j=1 − k+1 X j=1 j+1 n X x1j [ ys (e1 ∧ e2 ∧ · · · ∧ eˆj ∧ · · · ∧ ek+1 ∧ es ) ⊗ v1k−1] s=1 (−1)j+1 x1j (e1 ∧ e2 ∧ · · · ∧ eˆj ∧ · · · ∧ ek+1 ) ⊗ v1k−2 . 12 JOYDIP SAHA, INDRANATH SENGUPTA, AND GAURAB TRIPATHI We now compute (ψk+1 ◦ τk+1 )(e1 ∧ · · · ∧ ek+1 ). n τk+1 X ys (e1 ∧ · · · ∧ ek+1 ∧ es ) ⊗ v1k (e1 ∧ · · · ∧ ek+1 ) 7−→ s=1 −(e1 ∧ · · · ∧ ek+1 ) ⊗ v1k−1 n X ψk+1 X 7−→ [ (−1)j+1 ys x1j (e1 ∧ · · · ∧ eˆj ∧ · · · ∧ ek+1 ∧ es ) ⊗ v1k−1 s=1 j=1,2,··· ,k+1,s −(−1)k g1 (e1 ∧ · · · ∧ eˆj ∧ · · · ∧ ek+1 ∧ ej ) ⊗ v1k−1 − k+1 X (−1)j+1 x1j (e1 ∧ · · · ∧ eˆj ∧ · · · ∧ ek+1 ∧ ej ) ⊗ v1k−2 j=1 = n k+1 X X [x1j ys (−1)j+1 (e1 ∧ · · · ∧ eˆj ∧ · · · ∧ ek+1 ∧ es ) ⊗ v1k−1] j=1 s=1 n X (−1)s+1 ys x1s (e1 ∧ · · · ∧ eˆs ∧ · · · ∧ ek+1 ∧ es ) ⊗ v1k−1 + s=1 −(−1)k g1 (e1 ∧ · · · ∧ eˆj ∧ · · · ∧ ek+1 ∧ ej ) ⊗ v1k−1 − k+1 X (−1)j+1 x1j (e1 ∧ · · · ∧ eˆj ∧ · · · ∧ ek+1 ∧ ej ) ⊗ v1k−2 j=1 = k+1 X n X [x1j ys (−1)j+1 (e1 ∧ · · · ∧ eˆj ∧ · · · ∧ ek+1 ∧ es ) ⊗ v1k−1 ] j=1 s=1 n X (−1)s+1 (−1)k+1−s ys x1s (e1 ∧ · · · ∧ eˆs ∧ · · · ∧ ek+1 ∧ es ) ⊗ v1k−1 + s=1 −(−1)k g1 (e1 ∧ · · · ∧ eˆj ∧ · · · ∧ ek+1 ∧ ej ) ⊗ v1k−1 − k+1 X (−1)j+1 x1j (e1 ∧ · · · ∧ eˆj ∧ · · · ∧ ek+1 ∧ ej ) ⊗ v1k−2 j=1 = k+1 X n X [x1j ys (−1)j+1 (e1 ∧ · · · ∧ eˆj ∧ · · · ∧ ek+1 ∧ es ) ⊗ v1k−1 ] j=1 s=1 +(−1)k g1 (e1 ∧ · · · ∧ eˆj ∧ · · · ∧ ek+1 ∧ ej ) ⊗ v1k−1 −(−1)k g1 (e1 ∧ · · · ∧ eˆj ∧ · · · ∧ ek+1 ∧ ej ) ⊗ v1k−1 − k+1 X (−1)j+1 x1j (e1 ∧ e2 ∧ · · · ∧ eˆj ∧ · · · ∧ ek+1 ∧ ej ) ⊗ v1k−2 j=1 = k+1 X n X [x1j ys (−1)j+1 (e1 ∧ · · · ∧ eˆj ∧ · · · ∧ ek+1 ∧ es ) ⊗ v1k−1 ] j=1 s=1 − k+1 X j=1 (−1)j+1 x1j (e1 ∧ · · · ∧ eˆj ∧ · · · ∧ ek+1 ∧ ej ) ⊗ v1k−2. SUM IDEALS 13 e12 )+ Hence the mapping cone M(E· ⊗G· ; F· ) gives us the resolution for I2 (X hg1, g2 i as described in 3.2. However, this resolution is not minimal. We now construct a minimal free resolution from M(E· ⊗ G· ; F· ). e12 ) + hg1 , g2 i has been constructed in A free resolution for the ideal I2 (X 3.1, which is given by dn+2 dk+1 d k Dk−1 · · · −→ D1 −→ D0 −→ 0, 0 −→ Dn+2 −→ Dn+1 · · · −→ Dk −→ n n n such that Dk = Fk−1 ⊕ Ck = R(k−1) ⊕ (Rk(k+1) ⊕ R(k−1)( k ) ) and dk = (−σk−1 + τk−1 , ψk ). Let us recall that the map ψ is the differential in the e12 ) + hg1 i, the map σ is the differential in the Koszul free resolution for I2 (X resolution for hx11 , x12 , . . . , x1n i and τ is the connecting homomorphism between the complexes defined in 3.1. Let us order bases for Fk−1 and Ck with respect to the lexicographic ordering. Finally we order basis for Dk in such a way that the basis elements for Fk−1 appear first, followed by the basis elements for Ck . Therefore, the matrix representation for the differential map dk is given by   −σk−1    τk−1 −σk−1        0 A     =         ψk τk−1 =     0   0 0       −I         0         .   ψk      The entries in the matrices representing σk−1 and ψk can only belong to the maximal ideal hxij , yj i, since both are differentials of minimal free resolutions. The block matrix A has also elements in the maximal ideal hxij , yj i. The only block which has elements outside the maximal ideal hxij , yj i is in the identity block appearing in τk−1 . Therefore, it is clear from the matrix representation of the map dk that we can apply Lemma 3.12 repeatedly to get rid of non-minimality. Hence, we get a minimal free resolution and the 14 JOYDIP SAHA, INDRANATH SENGUPTA, AND GAURAB TRIPATHI e12 ) + hg1 , g2 i are total Betti numbers for the ideal I2 (X b0 = 1, n + 2, b1 = 2 n + n, b2 = 2 · 3 n n n n n − − + + (k − 1) bk+1 = k k k−1 k−1 k k+1 n n , for 2 ≤ k ≤ n − 1, + (k − 2) = k k k+1 bn = n − 2. eij ) + hg1 , . . . , gn i. 5.2. A minimal free resolution for I2 (X Lemma 5.2. Let Gk = G12 ∪ {g1 , g2 , . . . , gk }, 1 ≤ k ≤ n, where G12 is the e12 defined in the list of notations in section 2. The set of all 2×2 minors of X e12 ) + hg1 , . . . , gk i with respect set Gk is a Gröbner basis for the ideal I2 (X to a suitable monomial order. Proof. We take the lexicographic monomial ordering in R induced by the following ordering among the indeterminates: xnn > · · · > xtt > · · · > x33 > y1 > · · · > yn > x11 > · · · > x1n > x21 > · · · > x2n > xst for other s, t. Then, we observe that for every s ≥ 3, Lt(gs ) is coprime with Lt(gt ) for every 1 ≤ t ≤ k; t 6= s and also coprime with Lt(h) for every h ∈ G. e12 ). Therefore, Moreover, by Lemma 3.5, G12 is a Gröbner basis for I2 (X we only have test the S-polynomials S(g1, g2 ), S(g1 , h) and S(g2 , h), for h ∈ G. Pn We can write S(g1 , g2 ) = k=1 [12\|1k]yk and note that Lt([12\|1k]) ≤ Lt(S(g1 , g2 ) for every 1 ≤ k ≤ n. Hence, S(g1 , g2 ) →G ′ 0. We note that, if i 6= 1 then the leading terms of g1 and [12\|st] are mutually coprime and therefore P S(g1 , [12\|st]) →Gk 0. Next, the expression S(g1 , [12\|1t]) = x1t g2 + s6=t [12\|st]ys shows that S(g1 , [12\|1t]) →Gk 0. Similarly, if s 6= 1 SUM IDEALS 15 then the leading terms of g2 and [12\|st] are mutually coprime and therefore S(g2 , [12\|st]) →Gk 0. The proof for S(g2 , [12\|1t]) is similar to that of S(g1, [12\|st]). Remark. The corresponding result for i < j in general would be the following: Lemma 5.3. Let Gi,j,k = Gij ∪ {gi , gj , gl1 , . . . , glk−2 }, 1 ≤ k ≤ n, 1 ≤ l1 < · · · < lk−2 ≤ n and lt is the smallest in the set {1, 2, . . . , n} \ eij defined {i, j, l1 , . . . , lt−1 }; Gij denotes the set of all 2 × 2 minors of X in the list of notations in section 2. The set Gi,j,k is a Gröbner basis for the eij ) + hg1 , . . . , gk i with respect to a suitable monomial order. ideal I2 (X Proof. While proving this statement with i < j arbitrary, we have to choose the following monomial orders. The rest of the proof remains similar. Suppose that X is generic, we choose the lexicographic monomial ordering in R induced by the following ordering among the indeterminates: xnn > · · · > xc cii > · · · > x11 > jj > · · · > x > > > y1 > · · · > yn xi1 > · · · > xin xj1 > · · · > xjn xst for all other s, t. If X is generic symmetric, we choose the lexicographic monomial ordering in R induced by the following ordering among the indeterminates: xnn > · · · > xc cii > · · · > x11 > y1 > · · · > yn jj > · · · > x > xii > xij > x1i > x2i > · · · > xi−1,i > xi,i+1 > · · · > xin > xjj > · · · > xj−1,j > xj,j+1 > · · · > xjn > xst for all other s, t. e12 )+hg1 , . . . , gk i and hgk+1i intersect transverLemma 5.4. The ideals I2 (X sally, for every 2 ≤ k ≤ n − 1. e12 ) + hg1 , . . . , gk i such Proof. Suppose not, then, there exists hk+1 ∈ / I2 ( X e that hk+1 gk+1 ∈ I2 (X12 ) + hg1, . . . , gk i. Let us choose the same monomial order on R as defined in Lemma 5.2. Upon division by elements of Gk , we may further assume that Lt(h) ∤ Lt(hk+1 ) for every h ∈ Gk , since Gk e12 ) + hg1, . . . , gk i by Lemma 5.2. On is a Gröbner basis for the ideal I2 (X e12 ) + hg1 , . . . , gk i and therefore Lt(h) \| the other hand hk+1 gk+1 ∈ I2 (X Lt(hk+1 ), for some h ∈ Gk , since Lt(h) and Lt(gk+1 ) are mutually coprime, - a contradiction. 16 JOYDIP SAHA, INDRANATH SENGUPTA, AND GAURAB TRIPATHI Remark. The corresponding result for i < j in general would be the eij ) + hgi , gj , gl1 , . . . , gl i and hgl i intersect following: The ideals I2 (X k k+1 transversally, if 1 ≤ l1 < . . . < lk < lk+1 ≤ n and lk+1 is the smallest in the set {1, 2, . . . , n} \ {i, j, l1 , . . . , lk }, for every 1 ≤ k ≤ n − 3. The proof is essentially the same as above after we use the Lemma 5.3. Proof of Theorem 2.1. Part (1) of the theorem has been proved in 5.1. We now prove part (2) under the assumption i = 1, j = 2. Let the minimal e12 ) + hg1 , g2 , · · · , gk i be (L , δ ). By Lemma 5.4 free resolution of I2 (X e12 ) + hg1 , . . . , gk+1i is and Lemma 3.7, the minimal free resolution of I2 (X gk+1 given by the tensor product of (L , δ ) and 0 −→ R −→ R −→ 0, and that is precisely (K , ∆ ), with Kp = Lp ⊕Lp−1 and ∆p = (λp , (−1)p gk+1 + λp−1 ). Let βk,p , 0 ≤ p ≤ n + k, denote the p-th total Betti number for the ideal e12 ) + hg1 , . . . , gk i. Then, the total Betti numbers βk+1,p , 0 ≤ p ≤ I2 ( X e12 ) + hg1 , . . . , gk+1i are given by βk+1,0 = 1, n + k + 1 for the ideal I2 (X βk+1,p = βk,p−1 + βk,p for 1 ≤ p ≤ n + k and βk+1,n+k+1 = n − 2. The proof for general i < j follows similarly according to the strategy discussed in the beginning of section 5. eij )+hg1 , . . . , gn i In particular, the total Betti number βn−2,p for the ideal I2 (X are given by βn−2,0 = 1, βn−2,p = βn−3,p−1 + βn−3,p for 1 ≤ p ≤ 2n − 3 and βn−2,2n−2 = n − 2. Example. We show the Betti numbers at each stage for n = 4 and n = 5. 1 6 1 7 n=4: 1 8 1 9 1 10 1 1 1 n=5: 1 1 1 10 11 12 13 14 15 8 3 14 11 3 12 7 2 20 19 9 2 29 39 28 11 2 20 5 4 30 25 9 4 25 25 14 3 37 50 39 17 3 50 87 89 56 20 3 64 137 176 145 76 23 3 R EFERENCES [1] W. Bruns, A.R. Kustin, M. Miller, The Resolution of the Generic Residual Intersection of a Complete Intersection, Journal of Algebra 128 (1990) 214-239. [2] A. Conca, Emanuela De Negri, Elisa Gorla, Universal Gröbner bases for Maximal Minors, International Mathematics Research Notices 11(2015) 3245-3262. SUM IDEALS 17 [3] D. Eisenbud, Geometry of Syzygies, Springer-Verlag, NY, 2005. [4] P. Gimenez, I. Sengupta and H. Srinivasan, Minimal graded free resolution for monomial curves defined by arithmetic sequences, Journal of Algebra 388 (2013) 294-310. [5] M.R., Johnson, J. McLoud-Mann, On equations defining Veronese Rings, Arch. Math. (Basel) 86(3)(2006) 205-210. [6] S.Lichtenbaum, On the vanishing of Tor in regular local rings, Illinois J.Math. 10: 220- 226,1966. [7] H. Matsumura, Commutative Ring Theory, Cambridge University Press, NY, 1986. [8] I. Peeva, Graded Syzygies, Springer-Verlag London Limited, 2011. [9] J. Saha, I. Sengupta, G. Tripathi, Ideals of the form I1 (XY ), arXiv:1609.02765 [math.AC] 2016. [10] J. Saha, I. Sengupta, G. Tripathi, Primality of certain Determinanatal ideals, arXiv:1610.00926 [math.AC] 2016. Department of Mathematics, RKM Vivekananda University, Belur Math, Howrah 711202, India. E-mail address: [email protected] Discipline of Mathematics, IIT Gandhinagar, Palaj, Gandhinagar, Gujarat 382355, INDIA. E-mail address: [email protected] Department of Mathematics, Jadavpur University, Kolkata, WB 700 032, India. E-mail address: [email protected]	0
arXiv:1711.07737v1 [math.GT] 21 Nov 2017 SUPERRIGIDITY OF ACTIONS ON FINITE RANK MEDIAN SPACES ELIA FIORAVANTI Abstract. Finite rank median spaces are a simultaneous generalisation of finite dimensional CAT(0) cube complexes and real trees. If Γ is an irreducible lattice in a product of rank one simple Lie groups, we show that every action of Γ on a complete, finite rank median space has a global fixed point. This is in sharp contrast with the behaviour of actions on infinite rank median spaces. The fixed point property is obtained as corollary to a superrigidity result; the latter holds for irreducible lattices in arbitrary products of compactly generated groups. We exploit Roller compactifications of median spaces; these were introduced in [Fio17a] and generalise a well-known construction in the case of cube complexes. We provide a reduced 1-cohomology class that detects group actions with a finite orbit in the Roller compactification. Even for CAT(0) cube complexes, only second bounded cohomology classes were known with this property, due to [CFI16]. As a corollary, we observe that, in Gromov’s density model, random groups at low density do not have Shalom’s property HF D . Contents 1. Introduction. 2. Preliminaries. 2.1. Median spaces and median algebras. 2.2. Bridges. 2.3. The Haagerup class. 3. Haagerup class and elementarity of actions. 3.1. The main statement. 3.2. Elementarity and Shalom’s property HF D . 4. Superrigidity. 4.1. The superrigidity result. 4.2. Homomorphisms to coarse median groups. Appendix A. Structure of UBS’s. References 1 2 9 9 20 21 23 23 26 27 27 34 35 41 2 ELIA FIORAVANTI 1. Introduction. A metric space X is median if, for any three points x1 , x2 , x3 ∈ X, there exists a unique point m ∈ X such that d(xi , m) + d(m, xj ) = d(xi , xj ) for all 1 ≤ i < j ≤ 3. Simple examples are provided by real trees and Rn with the `1 metric. Under a finite dimensionality assumption, to each connected median space b The spaces X and X b are biX corresponds a canonical CAT(0) space X. b Lipschitz equivalent and isometries of X induce isometries of X [Bow16]. For instance, to Rn with the `1 metric we associate Rn with its euclidean distance. More elaborate examples of median spaces are provided by simply connected cube complexes satisfying Gromov’s link condition; in this case we b obtain a median space X by endowing each cube with the `1 metric and X is the corresponding CAT(0) cube complex. Median spaces generally display wilder features than cube complexes, as, like real trees, they can be essentially non-discrete objects. Note in this regard that the class of median spaces is closed under ultralimits; these also preserve a notion of dimension, usually called rank (see Section 2.1 for a precise definition). Despite non-discreteness, finite rank median spaces retain many of the good combinatorial properties of cube complexes. In addition to the CAT(0) metric, there is a notion of boundary compatible with the median property [Fio17a] and many groups of isometries contain free non-abelian subgroups [CS11, Fio17b]. We would expect many known results for CAT(0) cube complexes to extend to finite rank median spaces without significant complications, for instance [BCG+ 09, GH10, NS13, Fer15, CFI16, KS16] to name a few. There is however a notable exception to this pattern: general group actions on median spaces do not have a clear connection with the existence of codimension one subgroups [Sag95, Ger97, Fio17b]. Such close similarities between cube complexes and general median spaces can be ascribed to the existence of a collection W of walls. These encode the geometry of the space in the same way as hyperplanes do in CAT(0) cube complexes. The set W should not be thought of as discrete and needs to be endowed with a measure µ that encodes the “thickness” of sets of walls [CDH10, Fio17a]. Indeed, the concept of median space is in a certain sense dual to the notion of space with measured walls [CMV04, dCTV08, CDH10]; these extend spaces with walls [HP98], which are the dual viewpoint on CAT(0) cube complexes. Our main theorem is a superrigidity result for irreducible lattices Γ in products G = G1 × ... × G` of locally compact, compactly generated groups. Namely, under weak assumptions of non-elementarity, every action of Γ on a finite rank median space X essentially arises from continuous actions Gi y Yi on median spaces of lower rank. For cube complexes, this was known due to [CFI16]. SUPERRIGIDITY OF ACTIONS ON FINITE RANK MEDIAN SPACES 3 In the more general context of CAT(0) spaces, similar results were obtained long ago in [Mon06, CM09]. Unfortunately, applying these to a median space X only provides actions of the factors Gi on CAT(0) subspaces b these subspaces might bear no relation to the median structure on Zi ⊆ X; X. This might seem like an irrelevant subtlety, but it is, on the contrary, key to the fixed point properties that this paper provides. As an illustration of this, consider an irreducible lattice Γ < SL2 R×SL2 R. It acts properly and cocompactly on the CAT(0) space H2 ×H2 . In particular, it is quasi-isometric to a product of surface groups, which can easily be recognised as cubulated. It was shown in [CD17] that Γ acts properly and coboundedly on an infinite rank median space. The group Γ is moreover coarse median in the sense of [Bow13a]. Thus, it should appear particularly striking that every action of Γ on a connected finite rank median space fixes a point; this follows from our superrigidity result, see Corollary D below. The proof of our superrigidity theorem follows a very similar outline to Monod’s [Mon06]. This is mostly hidden in our application of Theorem 4.1 from [Sha00], thus we believe it is important to highlight the analogy here. We have already mentioned that, to each finite rank median space X, one b However, one can also can associate a finite dimensional CAT(0) space X. consider the infinite dimensional CAT(0) spaces L2 (W , µ) and L2 (H , νb); here (H , νb) is a close relative of (W , µ) and will be introduced below. Retracing the proof of Monod Superrigidity in our context, we would first induce a continuous action of G on the infinite dimensional CAT(0) space b (see [Mon06] for a definition). Then, we would prove that a L2 (G/Γ, X) b splits as a product Z1 × ... × Zk , where the action subspace of L2 (G/Γ, X) of G on the i-th factor only depends on the projection to Gi . Finally, we b the information gained about Γ y L2 (G/Γ, X). b would carry back to Γ y X Our application of Shalom’s machinery instead constructs a continuous action of G on the infinite dimensional CAT(0) space L2 (G/Γ, L2 (H , νb)). Again, one then proves a splitting theorem for this action and carries the gained insight back to Γ y L2 (H , νb). Our main contribution lies in transferring information back and forth between the actions Γ y L2 (H , νb) and Γ y X. Indeed, Shalom’s machinery can only be set in motion once we have a nonvanishing reduced cohomology class for Γ y L2 (H , νb); similarly, once we have a superrigidity statement for Γ y L2 (H , νb), we need to translate it back to Γ y X. We now describe our results in greater detail. 1.1. A cohomological characterisation of elementary actions. Each median space X has a distinguished collection H of subsets called halfspaces; every wall gives rise to two halfspaces and each halfspace arises from a wall. The collection H is equipped with a measure νb, see [Fio17a]. In the case of cube complexes, one recovers the usual notion of halfspace and νb is simply the counting measure. 4 ELIA FIORAVANTI Given a topological group G and an isometric action G y X, one naturally obtains a unitary representation ρ : G → U (L2 (H , νb)) and a cocycle b : G → L2 (H , νb) which will be referred to as the Haagerup cocycle. This construction is well-known and appears for instance in [CMV04, dCTV08, CDH10, FV16]. If G y X has continuous orbits, ρ and b are continuous; thus b induces a reduced continuous cohomology class [b] ∈ Hc1 (G, ρ). In [Fio17b], we introduced a notion of elementarity for actions on median spaces; namely, we say that G y X is Roller elementary if G has at least one finite orbit within a certain compactification X, the Roller compactification. Roller elementarity implies the existence of a finite orbit in the visual b compactification of the CAT(0) space X. If G y X is an isometric action with continuous orbits, Roller elementarity can be described in terms of the reduced cohomology class [b]. Theorem A. Let X be a complete, finite rank median space. The Haagerup class [b] ∈ Hc1 (G, ρ) vanishes if and only if G y X is Roller elementary. Theorem A extends various known results. In the case of simplicial trees, it appears in [FV16]. For CAT(0) cube complexes, the implication “Roller nonelementary ⇒ [b] 6= 0” is implicit in [DP16]. In [CFI16], the authors construct a family of bounded cohomology classes detecting Roller elementarity in CAT(0) cube complexes. We remark that Theorem A equally holds if we replace L2 (H , νb) with any Lp (H , νb), 1 ≤ p < +∞, although it is slightly simpler to exploit the richer structure of Hilbert spaces in its proof. Our superrigidity result only relies on the implication of Theorem A that yields [b] 6= 0, but we believe the full statement of Theorem A to be of independent interest. The proof of the other implication turns out to be quite technical and requires a careful study of the structure of UBS’s in median spaces; these are a generalisation of the simplices in Hagen’s simplicial boundary of a CAT(0) cube complex [Hag13]. Most of these details will be relegated to the appendix. 1.2. Superrigidity of actions. Once we have a nontrivial reduced cohomology class, as provided by Theorem A, we can apply well-established machinery (namely Theorem 4.1 in [Sha00]) to obtain superrigidity results. Let X be a complete, finite rank median space. Its Roller compactification X is partitioned into components [Fio17a]. The subset X ⊆ X forms a full component; every other component is a complete median space of strictly lower rank. This aspect of X shares some similarities with refined boundaries of CAT(0) spaces [Lee00, Cap09]. Given a component Z ⊆ X, a median subalgebra of Z is a subset Y ⊆ Z that is itself a median space with the restriction of the median metric of Z; equivalently, the median map m : Z 3 → Z takes Y 3 into Y . We are now ready to state our main superrigidity result. SUPERRIGIDITY OF ACTIONS ON FINITE RANK MEDIAN SPACES 5 Theorem B. Let X be a complete, finite rank median space. Let Γ be a uniform, irreducible lattice in a product G = G1 × ... × G` of compactly generated, locally compact groups with ` ≥ 2. Suppose Γ y X is a Roller nonelementary action. There exist a finite index subgroup Γ0 ≤ Γ, a Γ0 invariant component Z ⊆ X and a Γ0 -invariant closed median subalgebra Y ⊆ Z where the action Γ0 y Y extends to a continuous action G0 y Y , for some open finite index subgroup G0 ≤ G. Remarks. (1) Theorem B also applies to nonuniform lattices, as long as they are square-integrable; this is a well-known technical condition that implies finite generation and ensures that Theorem 4.1 in [Sha00] still holds. Irreducible lattices in G1 × ... × G` are square-integrable if each Gi is the group of ki -rational points of a semisimple, almost ki simple, ki -isotropic linear algebraic group defined over some local field ki [Sha00]. Further examples of nonuniform square-integrable lattices include minimal Kac-Moody groups over sufficiently large finite ground fields; these can be regarded as irreducible lattices in the product of the closed automorphism groups of the associated buildings [Rém99, Rém05]. (2) Theorem B should be compared to Shalom’s superrigidity result for actions on simplicial trees, Theorem 0.7 in [Sha00]. If X is a simplicial tree, Y is always a subcomplex of X and Γ0 = Γ, G0 = G. The complications in the statement of Theorem B reflect phenomena that do not happen in the world of trees. However, as soon as we leave the context of rank one median spaces, our result is optimal even if one restricts to CAT(0) square complexes; see Examples 4.7 and 4.8. We remark that, when X is a general CAT(0) square complex, the median algebra Y might not be a subcomplex of X or Z. (3) We can take Γ0 = Γ, G0 = G and Z = X as long as X does not split as a nontrivial product of median spaces and G has no finite b see Theorem 4.4 below. orbits in the visual compactification of X; However, even in this case, the action in general extends only to a proper median subalgebra of X. (4) For CAT(0) cube complexes, the superrigidity result of [CFI16] is slightly more general than Theorem B as it applies to all nonuniform lattices. This is due to the use of bounded cohomology (namely Theorem 16 in [BM02]) rather than reduced cohomology. Our strategy of proof was hinted at on page 9 of [Sha00]. 1.3. Fixed point properties for irreducible lattices. Unlike automorphism groups of CAT(0) cube complexes, the isometry group of a median space needs not be totally disconnected. Still, it is possible to exploit Theorem B to derive a fixed point property for irreducible lattices in connected groups. 6 ELIA FIORAVANTI Given a locally compact topological group Q, we denote the connected component of the identity by Q0 . We say that Q satisfies condition (∗) if Q/Q0 is amenable or has Shalom’s property HF D (see Section 1.5 for a definition). In particular, all almost-connected and connected-by-(T) groups satisfy condition (∗). Theorem C. Let X be a complete, finite rank median space. Let Γ be a square-integrable, irreducible lattice in a product G1 × ... × G` with ` ≥ 2. Suppose that every Gi is compactly generated and satisfies condition (∗). (1) Every action Γ y X is Roller elementary. (2) If Γ does not virtually map onto Z, every action Γ y X has a finite orbit within X. If moreover X is connected, every action Γ y X has a global fixed point. When X is a real tree, Theorem C also follows from Theorem 6 in [Mon06]. We remark that every group that virtually maps onto Z admits a Roller elementary action on Rn with unbounded orbits, for some n ≥ 1. Corollary D. Let X be a complete, connected, finite rank median space. Let Γ be an irreducible lattice in a connected, higher rank, semisimple Lie group G. Every action Γ y X fixes a point. An analogous result for CAT(0) cube complexes was proved in [CFI16]. If each simple factor of G has rank at least two, then Γ has property (T) and Corollary D follows from Theorem 1.2 in [CDH10]. The assumption that X have finite rank is essential for Corollary D to hold. If at least one simple factor Gi < G is locally isomorphic to O(n, 1) or U (n, 1), n ≥ 2, the lattice Γ admits an action on an infinite rank median space with unbounded orbits [CDH10]. Moreover, if all Gi ’s are locally isomorphic to O(ni , 1), ni ≥ 2, then Γ even admits a proper and cobounded action on an infinite rank median space [CD17]. 1.4. Homomorphisms to coarse median groups. Coarse median spaces were introduced in [Bow13a] as an attempt to formulate a coarse notion of nonpositive curvature. They have recently received a lot of attention [Bow13b, Hae16a, Zei16, SW17, ANWZ17, NWZ17] and proved instrumental to striking results such as [Hae16b, BHS17b]. A group is said to be coarse median if its Cayley graphs are coarse median spaces. Examples of finite rank coarse median groups include hyperbolic groups, cubulated groups [HS16], fundamental groups of closed irreducible 3-manifolds not modelled on Nil or Sol, mapping class groups and, more generally, all groups that are HHS [Bow15, BHS17a, BHS15]. We will be mainly interested in equivariantly coarse median groups. If we view coarse median groups as a generalisation of groups that are HHS, equivariantly coarse median groups generalise hierarchically hyperbolic groups (HHG). In particular, hyperbolic groups, cubulated groups and mapping class groups also are equivariantly coarse median of finite rank. SUPERRIGIDITY OF ACTIONS ON FINITE RANK MEDIAN SPACES 7 More precisely, we say that a group H is equivariantly coarse median if it is equipped with a finite generating set S ⊆ H and a coarse median µ : H 3 → H [Bow13a] such that dS (µ(ha, hb, hc), hµ(a, b, c)) ≤ C < +∞ for all elements h, a, b, c ∈ H; here dS denotes the word metric induced by S. Note that this definition does not depend on the choice of S. Equivariantly coarse median groups have already been considered in [Zei16] under a different name. If H is a coarse median group of finite rank, every asymptotic cone of H is endowed with a bi-Lipschitz equivalent median metric [Zei16]. As an example, in asymptotic cones of mapping class groups the median geodesics are limits of hierarchy paths [BDS11b, BDS11a]. When H is equivariantly coarse median, the median metric on each asymptotic cone is preserved by the action of the ultrapower of H. Given a group Γ and an infinite sequence of pairwise non-conjugate homomorphisms Γ → H, we can apply the Bestvina-Paulin construction [Bes88, Pau91]. The result is an isometric action on a median space X with unbounded orbits; this is obtained as the canonical median space bi-Lipschitz equivalent to an asymptotic cone of H. Along with Theorem C, this implies the following. Corollary E. Let H be an equivariantly coarse median group of finite rank. Let Γ be as in the second part of Theorem C. There exist only finitely many pairwise non-conjugate homomorphisms Γ → H. Corollary E applies in particular to the case when Γ is an irreducible lattice in a connected, higher rank, semisimple Lie group. When Γ has property (T), this result already appears in [Zei16] and, if in addition H is HHG, a much stronger statement is provided by Corollary D of [Hae16b]. Note that Haettel’s method cannot be applied in our context as lattices in products of rank one groups do admit nonelementary actions on hyperbolic spaces. Also compare Theorem 4.1 in [BF14] for the case of hyperbolic H and an arbitrary product G of locally compact, second countable groups. We remark that a stronger conclusion can be reached if H acts freely on a complete, finite rank median space. Indeed, the following is an immediate consequence of Theorem F in [Fio17b]. Proposition F. Let H be a group admitting a free action on a complete, finite rank median space X. Suppose that every action Γ y X is Roller elementary. Then every homomorphism Γ → H factors through a virtually abelian subgroup of H. Proposition F applies for instance to the case when Γ has no non-abelian free subgroups, has property HF D or satisfies the hypotheses of the first part of Theorem C. In particular, if Γ is an irreducible lattice in a connected, higher rank, semisimple Lie group, every homomorphism Γ → H has finite image. This should motivate a certain interest in groups acting freely on complete, finite rank median spaces. If a group acts freely on a finite dimensional 8 ELIA FIORAVANTI CAT(0) cube complex, it clearly falls into this class; however, it is unclear at this stage whether these are the only finitely generated examples. See [CRK15] for partial results in this direction. Note that the infinitely generated group Q even admits a proper action on a rank two median space, which splits as a product of a simplicial tree and the real line; see Example II.7.13 in [BH99]. However, since Q is a divisible group, all its elements must act elliptically on any (possibly infinite-dimensional) CAT(0) cube complex [Hag07]. Even within finitely generated groups, actions on median spaces tend to be more flexible than actions on CAT(0) cube complexes. For every group H, we can consider dimf m H, i.e. the minimum rank of a complete median space X admitting a free action of H; if H does not act freely on any complete median space, we set dimf m H := −1. Restricting to CAT(0) cube complexes, we can similarly define dimf c H and, if we only consider (metrically) proper actions, we obtain dimpm H and dimpc H. Thus, dimpc Q = dimf c Q = −1, while dimpm Q = 2 and dimf m Q = 1. We remark that 1 ≤ dimf m H < dimcm H and 1 ≤ dimpm H < dimpm H for many finitely generated groups H. For instance, dimf c H = 1 if and only if H is free. On the other hand, by work of E. Rips, dimf m H = 1 if and only if H is a free product of free abelian and surface groups (excluding a few nonorientable surfaces); see e.g. Theorem 9.8 in [BF95]. One can use the same observation to construct free actions of various RAAGs on median spaces of rank strictly lower than the dimension of the Salvetti complex. Considering more general actions, we mention that there exist finitely generated groups admitting actions on real trees with unbounded orbits, but whose actions on simplicial trees (and in fact even finite dimensional CAT(0) cube complexes) must have a global fixed point [Min16]. 1.5. Shalom’s property HF D and random groups. Theorem A also allows us to prove that various (non-amenable) groups do not have property HF D . The latter was introduced in [Sha04]: a topological group G has property HF D if every unitary representation π with Hc1 (G, π) 6= 0 has a finite dimensional subrepresentation. Property HF D is trivially satisfied by every locally compact group with property (T) [Del77], but also, at the opposite end of the universe of groups, by a large class of amenable groups. This includes polycyclic groups, lamplighter groups and all connected, locally compact, amenable groups [Sha04, Mar06]. An example of an amenable group without HF D is provided by the wreath product Z o Z [Sha04]. We prove the following; see Proposition 3.7 below for a more general result. Corollary G. Let Γ be a discrete group with property HF D . If Γ acts freely and cocompactly on a CAT(0) cube complex X, then Γ is virtually abelian. Property HF D has been studied almost exclusively within the class of amenable groups, where it happens to be a quasi-isometry invariant [Sha04]. It was a key ingredient (implicitely, or explicitely) in recent more elementary SUPERRIGIDITY OF ACTIONS ON FINITE RANK MEDIAN SPACES 9 proofs of Gromov’s theorem on groups of polynomial growth [Kle10, Oza16]. It has moreover interesting applications to the study of quasi-isometric embeddings into Hilbert spaces [dCTV07]. Property HF D is inherited by uniform lattices and it is stable under direct products and central extensions [Sha04]. Being satisfied by groups that fall into two extremely different classes, namely amenable and Kazhdan groups, it is reasonable to expect a wide variety of groups with property HF D . However, it seems that no answer is known to the following question. Question. Does every finitely generated group with property HF D virtually split as a direct product of an amenable group and finitely many groups with property (T)? Does every word hyperbolic group with property HF D also satisfy property (T)? Corollary G and the results of [OW11] imply that random groups at low density do not satisfy HF D . Corollary H. With overwhelming probability, random groups at density d < 16 in Gromov’s density model do not have property HF D . Note however that, at density d > 13 , random groups are Kazhdan [Ż03, KK13], hence satisfy property HF D . Acknowledgements. The author warmly thanks Brian Bowditch, Pierre-Emmanuel Caprace, Indira Chatterji, Yves Cornulier, Thomas Delzant, Mark Hagen, Masato Mimura, Narutaka Ozawa, Romain Tessera, Pierre Pansu, Alain Valette for helpful conversations. The author expresses special gratitude to Cornelia Druţu and Talia Fernós for contributing many of the ideas of this paper. This work was undertaken at the Mathematical Sciences Research Institute in Berkeley during the Fall 2016 program in Geometric Group Theory, where the author was supported by the National Science Foundation under Grant no. DMS-1440140 and by the GEAR Network. Part of this work was also carried out at the Isaac Newton Institute for Mathematical Sciences, Cambridge, during the programme “Non-positive curvature, group actions and cohomology” and was supported by EPSRC grant no. EP/K032208/1. The author was also supported by the Clarendon Fund and the Merton Moussouris Scholarship. 2. Preliminaries. 2.1. Median spaces and median algebras. Let X be a metric space. Given points x, y ∈ X, the interval I(x, y) is the set of points z ∈ X that lie between x and y, i.e. that satisfy d(x, y) = d(x, z)+d(z, y). We say that X is a median space if for all x, y, z ∈ X there exists a unique point m(x, y, z) that lies in I(x, y)∩I(y, z)∩I(z, x). The median map m : X 3 → X that we obtain this way endows X with a structure of median algebra. Most definitions in the theory of median spaces can also be given for arbitrary median algebras; 10 ELIA FIORAVANTI we will follow this approach in introducing the necessary notions. The reader can consult e.g. [Rol98, Nic08, CDH10, Bow13a, Bow16, Fio17a, Fio17b] for more background on median spaces and algebras. In a median space, I(x, y) = {z ∈ I(x, y) \| z = m(x, y, z)}; this can be taken as a definition of intervals in general median algebras. If (M, m) is a median algebra, we say that a subset C ⊆ M is convex if I(x, y) ⊆ C whenever x, y ∈ C. The intersection of a finite family of pairwise intersecting convex sets is always nonempty; this is known as Helly’s Theorem, see Theorem 2.2 in [Rol98]. A subset h ⊆ M is a halfspace if both h and h∗ := M \ h are convex; we will denote the set of halfspaces of M by H (M ), or simply by H when there is no ambiguity. Halfspaces h, k are said to be transverse if no two distinct elements of the set {h, h∗ , k, k∗ } are comparable in the poset (H , ⊆). Equivalently, the intersections h ∩ k, h ∩ k∗ , h∗ ∩ k, h∗ ∩ k∗ are all nonempty. Given A ⊆ H , we write A∗ for {h ∈ H \| h∗ ∈ A}. A subset σ ⊆ H is said to be an ultrafilter if any two halfspaces in σ intersect and H = σ t σ ∗ . For instance, for each x ∈ M the set σx := {h ∈ H \| x ∈ h} is an ultrafilter. Given subsets A, B ⊆ M , we write H (A\|B) := {h ∈ H \| B ⊆ h, A ⊆ h∗ } and σA := H (∅\|A); we refer to sets of the form H (x\|y), x, y ∈ M , as halfspace intervals. If C, C 0 ⊆ M are disjoint and convex, the set H (C\|C 0 ) is nonempty, see Theorem 2.7 in [Rol98]. In particular σx = σy if and only if the points x, y ∈ M coincide. A subset Ω ⊆ H is inseparable if, whenever j ∈ H satisfies h ⊆ j ⊆ k for h, k ∈ Ω, we have j ∈ Ω. Given a subset A ⊆ H , its inseparable closure is the smallest inseparable subset of H that contains A; it coincides with the union of the sets H (k∗ \|h), for h, k ∈ A. A wall is a set of the form w = {h, h∗ }, with h ∈ H ; we say that h and ∗ h are the sides of w. The wall w separates subsets A, B ⊆ M if either h or h∗ lies in H (A\|B); we denote by W (A\|B) = W (B\|A) the set of walls separating A and B and by W (M ), or simply W , the set of all walls of the median algebra M . A wall is contained in a halfspace k if one of its sides is; a wall w is contained in disjoint halfspaces k1 , k2 if and only if k2 = k∗1 and w = {k1 , k2 }. If a side of the wall w1 is transverse to a side of the wall w2 , we say that w1 and w2 are transverse. The rank of the median algebra M is the maximum cardinality of a set of pairwise transverse walls; various alternative (and equivalent) definitions of the rank can be found in Proposition 6.2 of [Bow13a]. We remark that M has rank zero if and only if it consists of a single point. If X is a median space of finite rank r, the topological dimension of every locally compact subset of X is bounded above by r, see Theorem 2.2 and Lemma 7.6 in [Bow13a]. If moreover X is complete and connected, X is b [Bow16]. The visual bi-Lipschitz equivalent to a canonical CAT(0) space X b is finite dimensional by Proposition 2.1 in [CL10]. boundary of X SUPERRIGIDITY OF ACTIONS ON FINITE RANK MEDIAN SPACES 11 b yielding a homomorEvery isometry of X extends to an isometry of X b Every convex subset of X is also convex in X; b phism Isom X ,→ Isom X. the converse is not true: the euclidean convex hull of the points (1, 0, 0), (0, 1, 0) and (1, 1, 1) in [0, 1]3 is not even a median subalgebra. Halfspaces in finite rank median spaces are fairly well-behaved [Fio17a]: Proposition 2.1. Let X be a complete median space of finite rank r. Every halfspace is either open or closed (possibly both). Moreover, if h1 ) ... ) hk is a chain of halfspaces with h∗1 ∩ hk 6= ∅, we have k ≤ 2r. The following is a simple but extremely useful observation: given ultrafilters σ1 , σ2 ⊆ H (M ) and h, k ∈ σ1 \ σ2 , we either have h ⊆ k, or k ⊆ h, or h and k are transverse. Along with Dilworth’s Theorem [Dil50] this yields the following. Lemma 2.2. Let M be a median algebra of finite rank r and let σ1 , σ2 ⊆ H be ultrafilters. (1) We can decompose σ1 \ σ2 = C1 t ... t Ck , where k ≤ r and each Ci is nonempty and totally ordered by inclusion. (2) Every infinite subset of σ1 \ σ2 contains an infinite subset that is totally ordered by inclusion. If C ⊆ M is a subset and x ∈ M , a gate for (x, C) is a point y ∈ C such that y ∈ I(x, z) for every z ∈ C; gates are unique when they exist. If a gate exists for every point of M , we say that C is gate-convex ; in this case we can define a gate-projection πC : M → C by associating to each point of M the unique gate. The gate-projection to C is always a morphism of median algebras and satisfies W (x\|C) = W (x\|πC (x)) for every x ∈ M . Gate-convex subsets are always convex, but the converse is not always true. For every y, z ∈ M , the interval I(y, z) is gate-convex with gateprojection x 7→ m(x, y, z). A proof of the following statements can be found in [Fio17a]. Proposition 2.3. Let C, C 0 ⊆ M be gate-convex. −1 (1) The sets {h ∈ H (M ) \| h∩C 6= ∅, h∗ ∩C 6= ∅}, {πC (h) \| h ∈ H (C)} and H (C) are all naturally in bijection. (2) There exists a pair of gates, i.e. a pair (x, x0 ) of points x ∈ C and x0 ∈ C 0 such that πC (x0 ) = x and πC 0 (x) = x0 . In particular, we have H (x\|x0 ) = H (C\|C 0 ). (3) The set πC (C 0 ) is gate-convex with gate-projection πC ◦ πC 0 . Moreover, πC ◦ πC 0 ◦ πC = πC ◦ πC 0 . (4) If C ∩ C 0 6= ∅, we have πC (C 0 ) = C ∩ C 0 and πC ◦ πC 0 = πC 0 ◦ πC . In particular, if C 0 ⊆ C, we have πC 0 = πC 0 ◦ πC . A median algebra (M, m) endowed with a Hausdorff topology is said to be a topological median algebra if the median map m : M 3 → M is continuous; here we equip M 3 with the product topology. Median spaces always provide 12 ELIA FIORAVANTI topological median algebras; indeed, the median map m is 1-Lipschitz in that case. In compact median algebras and complete median spaces, a subset is gateconvex if and only if it is closed and convex; moreover, gate-projections are continuous. In median spaces, gate-projections are even 1-Lipschitz. Let X be a complete, finite rank median space. In [Fio17a] we endowed b and a measure νbX (usually denoted just νb). the set H with a σ-algebra B b as measurable sets. Unlike [Fio17a], here we simply refer to the elements of B The map ∗ : H → H sending each halfspace to its complement is measure preserving. Every inseparable subset of H is measurable; in particular, all ultrafilters are measurable and, for all x, y ∈ X, we have νb(σx 4σy ) = d(x, y). Almost every halfspace h ∈ H is thick, i.e. both h and h∗ have nonemtpy b and νb differ from their counterparts in interior. Note that in general B [CDH10]. Proposition 2.4. Let X be a complete, finite rank median space and σ ⊆ H an ultrafilter such that νb(σ4σx ) < +∞ for some x ∈ X. There exists y ∈ X such that νb(σ4σy ) = 0. Thus X can be equivalently described as the collection of all ultrafilters on H that satisfy νb(σ4σx ) < +∞ for some x ∈ X; we identify ultrafilters whose symmetric difference is νb-null. Considering the space of all ultrafilters on H we obtain a set X in which X embeds. A structure of median algebra can be defined on X by setting m(σ1 , σ2 , σ3 ) := (σ1 ∩ σ2 ) ∪ (σ2 ∩ σ3 ) ∪ (σ3 ∩ σ1 ). We endow X with a topology such that ultrafilters σn ⊆ H converge to σ ⊆ H if and only if lim sup(σn 4σ) is νb-null. We refer to X as the Roller compactification of X [Fio17a]. Proposition 2.5. The Roller compactification X is a compact topological median algebra. The inclusion X ,→ X is a continuous morphism of median algebras with dense, convex image. In general, X is not open in X and the inclusion X ,→ X is not a homeomorphism onto its image. The Roller boundary is defined as ∂X := X \ X. A point of X can in general be represented by several distinct ultrafilters with null symmetric differences. However, for each ξ ∈ X there is a unique preferred ultrafilter σξ representing ξ [Fio17a]; this should be seen as a generalisation of the ultrafilters σx when x ∈ X. We can extend each halfspace h of X to a halfspace e h of X such that e e h ∩ X = h; indeed, it suffices to define h := {ξ ∈ X \| h ∈ σξ }. When ξ, η ∈ X, we save the notation H (ξ\|η) for the set ση \ σξ ⊆ H (X), instead of the analogous subset of H (X). If Y ⊆ X is a closed median subalgebra, the restriction of the metric of X turns Y into a complete median space with rank(Y ) ≤ rank(X); moreover: SUPERRIGIDITY OF ACTIONS ON FINITE RANK MEDIAN SPACES 13 Lemma 2.6. There is a canonical morphism of median algebras ιY : Y ,→ X. Proof. We write HY := {h ∈ H (X) \| h ∩ Y 6= ∅, h∗ ∩ Y 6= ∅}; intersecting with Y gives a map p : HY → H (Y ). Lemma 6.5 in [Bow13a] implies that p is surjective. Thus, for every ultrafilter σ ⊆ H (Y ), there is a unique ultrafilter σ 0 ⊆ H (X) such that σY ⊆ σ 0 and p(σ 0 ∩ HY ) = σ. Applying this to canonical ultrafilters yields the required embedding. Given ultrafilters σ1 , σ2 ⊆ H , we set d(σ1 , σ2 ) := νb(σ1 4σ2 ). We refer to d as the extended metric on X as it satisfies all the axioms of a metric, even though the value +∞ is allowed. Note that for points of X this is the same as the original median metric on X. A component Z ⊆ X is a maximal set of points having finite pairwise distances; components are convex subsets of X. One component always coincides with X ⊆ X; all other components are contained in ∂X. The following appears in [Fio17a]. Proposition 2.7. Let X be a complete median space of finite rank r. Let Z ⊆ ∂X be a component and let d denote the extended metric on X. (1) The metric space (Z, d) is a complete median space of rank at most r − 1. (2) Every thick halfspace of Z is of the form e h ∩ Z for a unique h ∈ H . If C ⊆ X is closed and convex, the closure of C in X is gate-convex and naturally identified with the Roller compactification of C; thus the notation C is not ambiguous. We denote by πC : X → C the corresponding gateprojection; it extends the usual gate-projection X → C. If σ ⊆ H (X) is an ultrafilter representing the point ξ ∈ X, the set σ ∩ H (C) is an ultrafilter on H (C) and represents πC (ξ). Similarly, if Z ⊆ ∂X is a component, the closure of Z in X is gateconvex and naturally identified with the Roller compactification Z. The gate-projection πZ : X → Z satisfies πZ (X) ⊆ Z. In terms of ultrafilters, πZ takes the point of X represented by σ ⊆ H (X) to the point of Z represented by σ ∩ H (Z) ⊆ H (Z). The intersection makes sense as, by part 2 of Proposition 2.7, almost every halfspace of Z arises from a halfspace of X. Let Γ be a group. An isometric action Γ y X is said to be Roller elementary if there exists a finite orbit within X. The action is Roller minimal if rank(X) ≥ 1 and Γ does not preserve any proper, closed, convex subset C ⊆ X. Roller elementarity implies – but is in general much stronger than – the b When X is existence of a finite orbit in the visual compactification of X. a CAT(0) cube complex, an action is Roller minimal if and only if, in the terminology of [CS11], it is essential and does not fix any point in the visual b boundary of X. Neither Roller elementarity, nor Roller minimality implies the other one. However, Roller minimal actions naturally arise from Roller nonelementary ones [Fio17b]: 14 ELIA FIORAVANTI Proposition 2.8. Let X be a complete, finite rank median space with an isometric action Γ y X. Either Γ y X fixes a point or there exist a Γinvariant component Z ⊆ X and a Γ-invariant, closed, convex subset C ⊆ Z such that Γ y C is Roller minimal. A Γ-invariant, closed, convex subset C ⊆ Z ⊆ X always gives rise to a ∗ ); here we introduced the measurable decomposition H = HC t (σC ∪ σC sets HC := {h ∈ H \| C ∩ e h 6= ∅, C ∩ e h∗ 6= ∅} and σC := {h ∈ H \| C ⊆ e h}. Note that, by part 2 of Proposition 2.7, the measure spaces (HC , νbX ) and (H (C), νbC ) are isomorphic. We say that an action Γ y X is without wall inversions if there do not exist g ∈ Γ and h ∈ H such that gh = h∗ . By Proposition 2.1, any action of a connected, complete, finite rank median space is without wall inversions. The following was proved in [Fio17b]; compare with [CS11]. Proposition 2.9. Let X be a complete, finite rank median space with thick halfspaces h ⊆ k. Let Γ y X be a Roller minimal action without wall inversions. (1) There exists g ∈ Γ such that gh∗ ( h and d (gh∗ , h∗ ) > 0. (2) There exists g ∈ Γ such that gk ( h ⊆ k and d(gk, h∗ ) > 0. When G is a topological group, all isometric actions G y X will be implicitely required to have continuous orbit maps. Equivalently, the homomorphism G → Isom X is continuous, where we endow Isom X with the topology of pointwise convergence. We remark that Isom X is a Hausdorff, sequentially complete topological group as soon as X is complete. If (M1 , m1 ) and (M2 , m2 ) are median algebras, the product median algebra is defined as (M1 × M2 , m), where m = (m1 ◦ p1 , m2 ◦ p2 ); here pi denotes the projection onto the i-th factor. If (X1 , d1 ) and (X2 , d2 ) are median spaces, we endow the product X1 ×X2 with the `1 metric, namely d = d1 ◦p1 +d2 ◦p2 . The median algebra associated to the median space (X1 × X2 , d) is just the product median algebra arising from X1 and X2 . A median space X is said to be irreducible if it is not isometric to any nontrivial product X1 × X2 . Proposition 2.10. Let X1 , ..., Xk be irreducible, complete, finite rank median spaces; consider the product X = X1 × ... × Xk . (1) We have a measurable partition H (X) = H1 t ... t Hk , where each Hi is canonically identified with H (Xi ). If h ∈ Hi and k ∈ Hj with i 6= j, the halfspaces h and k are transverse. (2) Every isometry of X permutes the members of the partition. The product Isom X1 × ... × Isom Xk sits inside Isom X as an open, finite index subgroup. (3) Every closed, convex subset C ⊆ X is of the form C1 × ... × Ck , where each Ci is a closed convex subset of Xi . (4) The Roller compactification X is naturally identified with the product median algebra X1 × ... × Xk with the product topology. SUPERRIGIDITY OF ACTIONS ON FINITE RANK MEDIAN SPACES 15 Proof. For part 1 and the first half of part 2, see Corollary 2.9 and Proposition 2.11 in [Fio17b]. We conclude the proof of part 2 by showing that Isom X1 × ... × Isom Xk is open in Isom X. Choose points xi , yi ∈ Xi for each i and a real number > 0 such that 2 < d(xi , yi ) for all i; denote by pi : X → Xi the projection onto the i-th factor. Let x ∈ X be the point with coordinates xi ; we also consider the points zi ∈ X such that pj (zi ) = xj for all i 6= j and pi (zi ) = yi . Suppose that F ∈ Isom X is such that d(F (x), x) < and d(F (zi ), zi ) < for all i. We claim that F (Hi ) = Hi for all i; this implies that the product of the isometry groups of the factors is open in Isom X. Suppose for the sake of contradiction that, for indices i 6= j, we have F (Hi ) = Hj . This induces an isometry f : Xi → Xj such that pj ◦ F = f ◦ pi , see [Fio17a]. In particular, we have d(xj , f (xi )) ≤ d(x, F (x)) < and d(xj , f (yi ) ≤ d(zi , F (zi )) < ; thus, d(xi , yi ) = d(f (xi ), f (yi )) < 2, a contradiction. Irreducibility of the factors plays no role in part 3, so it suffices to consider the case k = 2. Let C1 and C2 be the projections of C to X1 and X2 . If x ∈ C1 and y ∈ C2 , there exist u ∈ X1 and v ∈ X2 such that the points x := (x, v) and y := (u, y) lie in C. It is immediate to observe that the point (x, y) lies in I(x, y) ⊆ C. Finally, part 4 is Lemma 2.10 in [Fio17b]. Halfspaces h1 , ..., hn form a facing n-tuple, n ≥ 2, if they are pairwise disjoint. We say that h, k ∈ H are strongly separated if h ∩ k = ∅ and no j ∈ H is transverse to both h and k. See [Fio17b] for the following. Proposition 2.11. Let X be an irreducible, complete, finite rank median space; let h be a thick halfspace. (1) If X admits a Roller minimal action without wall inversions, there exist thick halfspaces h0 ⊆ h ⊆ h00 such that h0 and h00∗ are strongly separated. (2) If X admits a Roller nonelementary, Roller minimal action without walls inversions, h is part of a facing n-tuple of thick halfspaces for every n ≥ 2. Every complete, finite rank median space can be isometrically embedded into its barycentric subdivision X 0 . This is a complete median space of the same rank, see [Fio17b]; when X is the 0-skeleton of a CAT(0) cube complex with the `1 metric, the space X 0 is given by the 0-skeleton of the customary barycentric subdivision. We have a natural homomorphism Isom X ,→ Isom X 0 . Given any isometric action Γ y X, the induced action Γ y X 0 is without wall inversions. We write (H 0 , ν 0 ) instead of (H (X 0 ), νX 0 ). There is an inclusion preserving map p : H 0 → H ; it is surjective, (Isom X)-equivariant and its fibres have cardinality at most two. We have #(p−1 (h)) = 2 if and only if {h} is an atom for νb; in this case, we refer to each element of p−1 (h) as a hemiatom. See [Fio17b] for the following lemma. 16 ELIA FIORAVANTI Lemma 2.12. Let X be a complete, finite rank median space. An action Γ y X is Roller elementary if and only if the induced action Γ y X 0 is. The sets {−1, 1} and {−1, 0, 1} inherit a median algebra structure from the median space R; in particular, we can consider the product median algebras {−1, 1}k and {−1, 0, 1}k for every k ≥ 0. For every point x ∈ X 0 \ X, b there exists a canonical, gate-convex subset C(x) ⊆ X 0 ; it is isomorphic k to {−1, 0, 1} , for some k ≥ 1, via an isomorphism that takes x to the b centre (0, ..., 0). The intersection C(x) := C(x) ∩ X is gate-convex in X k and corresponds to the subset {−1, 1} ⊆ {−1, 0, 1}k . For x ∈ X, we set b C(x) = C(x) = {x}. See [Fio17b] for more details. Lemma 2.13. Let X be a complete, finite rank median space. Every infinite, convex subset C ⊆ X 0 intersects X. Proof. Let x ∈ C be a point minimising R := rank(C(x)); if R = 0 we have b x ∈ C ∩ X. Otherwise, there exists a point y ∈ C that does not lie in C(x); b b the gate-projection z of y to C(x) lies in C(x) \ {x}. In particular C(z) is contained in a face of C(x) and has strictly lower rank, a contradiction since z ∈ I(x, y) ⊆ C. In particular, we can obtain the following extension of Lemma 2.12. Lemma 2.14. If Γ y X is Roller nonelementary and Roller minimal, so is the action Γ y X 0 . Proof. Suppose for the sake of contradiction that C ⊆ X 0 is a nonempty, Γ-invariant, closed, convex subset. By Corollary 4.31 in [Fio17a] there exists a Γ-invariant component W ⊆ X 0 with W ∩ C 6= ∅; note that C ∩ W is unbounded by Corollary 2.16 in [Fio17b], since Γ y X 0 is Roller nonelementary by Lemma 2.12. The component W is the barycentric subdivision of a component Z ⊆ X and Lemma 2.13 implies that C ∩ Z 6= ∅. Since Γ y X is Roller minimal, we must have Z = X and C ∩ X = X; hence X 0 ⊆ C by part 2 of Proposition 2.14 in [Fio17b]. We conclude that C = X 0 . Let us now fix a basepoint x ∈ X, where X is a complete finite rank median space; the following discussion is independent of our choice of x. A diverging chain of halfspaces is a sequence (hn )n≥0 such that d(x, hn ) → +∞ and hn+1 ⊆ hn for each n ≥ 0; we use the same terminology for the set {hn \| n ≥ 0}. Given ξ ∈ ∂X, a UBS for ξ is an inseparable subset Ω ⊆ σξ \σx containing a diverging chain of halfspaces. Given UBS’s Ω1 , Ω2 ⊆ σξ \ σx , we say that Ω1 is almost contained in Ω2 if the halfspaces in Ω1 \ Ω2 lie at uniformly bounded distance from x; this is denoted by Ω1 Ω2 . If Ω1 Ω2 and Ω2 Ω1 , the UBS’s are equivalent and we write Ω1 ∼ Ω2 . We denote the equivalence class of Ω ⊆ H by [Ω] and the set of all equivalence classes of UBS’s for ξ by U(ξ); the relation descends to a partial order on U(ξ). A UBS Ω is said to be minimal if SUPERRIGIDITY OF ACTIONS ON FINITE RANK MEDIAN SPACES 17 [Ω] is a minimal element of (U(ξ), ). A minimal UBS is equivalent to the inseparable closure of any diverging chain that it contains. We define a directed graph G(ξ) as follows [Hag17, Fio17b]. The vertex set of G(ξ) is identified with the set of minimal elements of (U(ξ), ). Given diverging chains (hm )m≥0 and (kn )n≥0 in Ω1 and Ω2 respectively, we draw an oriented edge from [Ω1 ] to [Ω2 ] if almost every hm is transverse to almost every kn , but not vice versa; this is independent of the choices involved. A subset A ⊆ G(ξ)(0) is said to be inseparable if every directed path between vertices in A only crosses vertices in A. The following can be found in [Fio17b]. Proposition 2.15. Let X be a complete median space of finite rank r and ξ ∈ ∂X. (1) The graph G(ξ) has at most r vertices and contains no directed cycles. (2) The poset (U(ξ), ) is isomorphic to the poset of inseparable subsets of G(ξ)(0) , ordered by inclusion. The isomorphism maps [Ω] ∈ U(ξ) to the set of equivalence classes of minimal UBS’s almost contained in Ω. In particular, the set U(ξ) is finite. (3) Given a UBS Ω and a set {Ω1 , ..., Ωk } of representatives of all equivalence classes of minimal UBS’s almost contained in Ω, we have sup d(x, h) < +∞. h∈Ω4(Ω1 ∪...∪Ωk ) If Ω is a minimal UBS, we denote by Ω0 the subset of halfspaces k ∈ Ω that are not transverse to any diverging chain of halfspaces in Ω. We say that Ω is reduced if Ω = Ω0 . We say that Ω is strongly reduced if we can write Ω = C1 t ... t Ck for some k ≥ 1, where each Ci is totally ordered by inclusion and contains a diverging chain of halfspaces. Consider the median spaces in Figures 1 and 2; both are subsets of R2 with the restriction of the `1 metric. In both cases, the UBS Ω = σξ \ σx is minimal. Figure 1 shows that Ω can be reduced, but not strongly reduced. In Figure 2 the UBS Ω is strongly reduced and exhibits how the decomposition Ω = C1 t ... t Ck can require k ≥ 2. Lemma 2.16. Let X be a complete, finite rank median space. Consider x ∈ X and ξ ∈ ∂X; let Ω ⊆ σξ \ σx be a minimal UBS. (1) The subset Ω0 is a reduced UBS equivalent to Ω. (2) There exists a strongly reduced UBS contained in Ω; all its sub-UBS’s are strongly reduced. (3) If Ω is strongly reduced, it is reduced. Proof. If h ⊆ k lie in σξ \ σx and k is transverse to a diverging chain of halfspaces, then h is transverse to an infinite subchain. This implies that Ω0 is inseparable; moreover, Ω \ Ω0 cannot contain a diverging chain or Ω would contain two inequivalent UBS’s. This proves part 1. To obtain part 2, we decompose Ω = C1 t ... t Ck as in Lemma 2.2; let A be the union of the sets Ci that do not contain a diverging chain. 18 ELIA FIORAVANTI Figure 1 Figure 2 SUPERRIGIDITY OF ACTIONS ON FINITE RANK MEDIAN SPACES 19 There exists D < +∞ such that d(x, h) ≤ D for every h ∈ A. The set {h ∈ Ω \| d(x, h) > D} is a strongly reduced UBS and so are its sub-UBS’s. Regarding part 3, decompose Ω = C1 t ... t Ck , where each Ci is totally ordered by inclusion and contains a diverging chain. If there existed k ∈ Ω \ Ω0 , we would have k ∈ Ci for some i; in particular, k would not be transverse to a diverging chain in Ci . Since Ω is minimal, k would not be transverse to any diverging chain in Ω, a contradiction. Given ξ ∈ ∂X, we denote by Isomξ X the group of isometries that fix ξ. Let Kξ be the kernel of the action of Isomξ X on U(ξ); it is a finite-index subgroup of Isomξ X. If Ω ⊆ σξ \ σx is a UBS, we denote by KΩ the subgroup of Isomξ X that fixes [Ω]. We can define χΩ : KΩ → R by the for a transfer−1character −1 mula χΩ (g) = νb g Ω \ Ω − νb Ω \ g Ω . This is a homomorphism and only depends on the equivalence class [Ω]. If Ω1 , ..., Ωk are a set of representatives for G(ξ)(0) , we can also consider the full transfer homomorphism χξ = (χΩ1 , ..., χΩk ) : Kξ → Rk . Proposition 2.17. Let X be a complete, finite rank median space; consider ξ ∈ ∂X. The subgroup Kξ is open in Isomξ X and the full transfer homomorphism χξ : Kξ → Rk is continuous. Every finitely generated subgroup of ker χξ has a finite orbit in X; if X is connected, every finitely generated subgroup of ker χξ fixes a point. Before proving the proposition, we will need to obtain the following lemma. Note that, for every point ξ ∈ ∂X and every halfspace h ∈ σξ , the set H (h∗ \|ξ) = {k ∈ σξ \| k ⊆ h} is a UBS. Lemma 2.18. For every thick halfspace h ∈ σξ and every > 0, there exists a neighbourhood U of the identity in Isomξ X such that H (h∗ \|ξ) ∼ H (gh∗ \|ξ) and νb (H (h∗ \|ξ)4H (gh∗ \|ξ)) < for all g ∈ U . Proof. Pick a point x ∈ X with d(x, h) > 0; in a neighbourhood of the identity of Isomξ X, we have x ∈ gh∗ . If k ∈ H (h∗ \|ξ) \ H (gh∗ \|ξ) and y is the gate-projection of x to k, we have y ∈ gh∗ by part 4 of Proposition 2.3, since k ∩ gh∗ 6= ∅ and x ∈ gh∗ ; thus d(y, g −1 y) ≥ d(k, h∗ ). We conclude that, for every k ∈ H (h∗ \|ξ) with d(k, h∗ ) > 0, there exists a neighbourhood Vk of the identity in Isomξ X such that k ∈ H (gh∗ \|ξ) for all g ∈ Vk . Decompose H (h∗ \|ξ) = C1 t ... t Ck as in Lemma 2.2. Let ki be the union of all k ∈ Ci with d(h∗ , k) ≥ 2k ; if nonempty, it is a halfspace and d(h∗ , ki ) ≥ 2k . ∗ Elements of H (h \|ξ) not contained in any k form a subset of measure at i P most νb(H (h∗ \|ki )) ≤ 2 . Let V be the intersection of the Vki for ki 6= ∅; if g ∈ V , the set H (h∗ \|ξ) \ H (gh∗ \|ξ) has measure at most 2 and consists of halfspaces at uniformly bounded distance from x ∈ X. It now suffices to consider U := V ∩ V −1 . Proof of Proposition 2.17. We only need to prove that Kξ is open and that χξ is continuous; the rest of the statement is contained in Theorem F of [Fio17b]. 20 ELIA FIORAVANTI For every v ∈ G(ξ)(0) , there exists hv such that the vertices w ∈ G(ξ)(0) with w [H (h∗v \|ξ)] are precisely v and those that are at the other end of an incoming edge at v. Indeed, given a diverging chain in a UBS representing v, almost every halfspace in the chain can be chosen as hv . Let Ai be the set of vertices v ∈ G(ξ)(0) such that there exists no directed path of length ≥ i in G(ξ) that ends at v. Note that, by Proposition 2.15, we have ∅ = A0 ( A1 ( ... ( Ak = G(ξ)(0) for some k ≤ r. We will show that the subgroup Ki ≤ Isomξ X that fixes Ai ⊆ G(ξ)(0) pointwise is open in Isomξ X and that, for every [Ω] ∈ Ai , the homomorphism χΩ : Ki → R is continuous. We proceed by induction on i, setting K0 := Isomξ X. The base step is trivial. If i ≥ 1, let Ai \ Ai−1 = {v1 , ..., vs } and consider the halfspaces hv1 , ..., hvs . Setting Ξj := H (h∗vj \|ξ), Lemma 2.18 provides a neighbourhood U of id in Isomξ X such that gΞj ∼ Ξj for all j and all g ∈ U . A minimal UBS almost contained in Ξj projects to an element of Ai−1 or to vj ; hence, we have gvj = vj for every g ∈ U ∩ Ki−1 . Since, by the inductive hypothesis, Ki−1 is open, so is Ki . Continuity of the transfer characters is obtained with a similar argument. 2.2. Bridges. Let a median algebra (M, m) and two gate-convex subsets C1 , C2 be fixed throughout this section. All the following results have analogues in Section 2.G of [CFI16]. Denote by πi : M → Ci the gate-projection to Ci . We will refer to the sets S1 := {x1 ∈ C1 \| ∃x2 ∈ C2 s.t. (x1 , x2 ) are gates for (C1 , C2 )} , S2 := {x2 ∈ C2 \| ∃x1 ∈ C1 s.t. (x1 , x2 ) are gates for (C1 , C2 )} , as the shores of C1 and C2 , respectively. By part 3 of Proposition 2.3, these coincide with π1 (C2 ) and π2 (C1 ), hence they are gate-convex. The map π2 \|S1 : S1 → S2 is a bijection with inverse π1 \|S2 ; if M arises from a median space X, this is an isometry as gate-projections are 1-Lipschitz. The bridge is the set G G (∗) B := I (x1 , π2 (x1 )) = I (π1 (x2 ), x2 ) . x1 ∈S1 x2 ∈S2 The union is disjoint because, if (x1 , x2 ) is a pair of gates for (C1 , C2 ), we have πi (I(x1 , x2 )) = {xi } for i = 1, 2; this follows from part 4 of Proposition 2.3 and the observation that I(x1 , x2 ) ∩ Ci = {xi }. Proposition 2.19. The bridge B is gate-convex and W (B) = (W (C1 ) ∩ W (C2 )) t W (C1 \|C2 ). For any pair of gates (x1 , x2 ), the bridge is canonically isomorphic to the product S1 × I(x1 , x2 ); this is an isometry if M arises from a median space. Proof. Pick a pair of gates (x1 , x2 ), set I := I(x1 , x2 ) and consider the morphism of median algebras φ := πS1 × πI : M → S1 × I. If (x01 , x02 ) is another SUPERRIGIDITY OF ACTIONS ON FINITE RANK MEDIAN SPACES 21 pair of gates, the projection πI provides an isomorphism I(x01 , x02 ) → I mapping each x0i to xi . This observation and the decomposition (∗) above imply that the restriction φ\|B is bijective. The map πB = (φ\|B )−1 ◦φ : M → B is surjective and it is a gate-projection by Proposition 2.1 in [Fio17a]. By part 1 of Proposition 2.3 and the discussion above, every wall of B arises either from a wall of M cutting S1 or from a wall of M cutting I; the latter correspond to W (C1 \|C2 ) by part 2 of Proposition 2.3, so we are left to show that W (S1 ) = W (C1 ) ∩ W (C2 ). This follows from the fact that S1 = π1 (C2 ). When M arises from a median space X, the measure νb induces a measure µ b on the set W [Fio17a]. In this case, the fact that φ\|B is an isometry follows from the decomposition of W (B) above and the observation that µ b(W (x\|y)) = d(x, y) for all x, y ∈ X. We can extend the notion of strong separation to arbitrary gate-convex subsets of median algebras. We say that C1 and C2 are strongly separated if they are disjoint and W (C1 ) ∩ W (C2 ) = ∅. Note that the condition W (C1 ) ∩ W (C2 ) = ∅ alone already implies that C1 ∩ C2 consists of at most one point. In a median space, two halfspaces are strongly separated in the sense of Section 2.1 if and only if their closures are strongly separated according to the definition above; see Lemma 2.21 below for a stronger result. Proposition 2.19 implies that two disjoint, gate-convex sets are strongly separated if and only if their shores are singletons; this yields the following result. Corollary 2.20. Let C1 , C2 ⊆ M be strongly separated. There exists a unique pair of gates (x1 , x2 ) and π1 (C2 ) = {x1 }, π2 (C1 ) = {x2 }. We will also need the following: Lemma 2.21. Let h, k ∈ H be strongly separated as halfspaces. The closures H, K of e h, ek in X are strongly separated as subsets of X. Proof. Since h, k have disjoint closures in X, the sets H and K are disjoint (see e.g. the proof of Theorem 5.1 in [Fio17b]). By Proposition 2.19, it then suffices to prove that the shore S ⊆ H is a singleton. Suppose for the sake of contradiction that S contains distinct points ξ, η and let ξ 0 , η 0 ∈ K be their projections to K; in particular, ση \ σξ = ση0 \ σξ0 . Given j ∈ ση \ σξ , the argument at the beginning of the proof shows that the closures of j and h inside X intersect nontrivially; by Lemma 3.6 in [Fio17a], almost every j ∈ ση \ σξ intersects h. Considering complements, we conclude that almost every j ∈ ση \σξ is transverse to h and, similarly, to k. Since d(ξ, η) > 0, there exists such a j, contradicting the fact that h, k are strongly separated. 2.3. The Haagerup class. Let X be a median space, G a topological group and p ∈ [1, +∞). Given a Banach space E, we denote by U (E) the group of linear isometries of E. 22 ELIA FIORAVANTI An isometric action G y X corresponds to a measure preserving action G y (H , νb). We obtain a continuous representation ρp : G → U (Lp (H , νb)); when p = 2, we simply write ρ := ρ2 . We will use interchangeably the notations Hc1 (G, ρp ) and Hc1 (G, Lp (H , νb)) to denote continuous cohomology. Given x ∈ X, we can consider the continuous 1-cocycle bx : G → Lp (H , νb) defined by bx (g) := g · 1σx − 1σx ; it satisfies kbx (g)kp = (2 · d(x, gx))1/p . We will refer to bx as a Haagerup cocycle. The cohomology class [bx ] ∈ Hc1 (G, ρp ) does not depend on the point x and we will simply denote it by [b]. The action G y X has bounded orbits if and only if the affine action G y Lp (H , νb) induced by bx fixes a point; this follows for instance from the Ryll-Nardzewski Theorem for p > 1 and from Theorem A in [BGM12] in the case p = 1. Thus, we have [b] = 0 if and only if the action G y X has bounded orbits. We can also consider the projection [b] of [b] to reduced continuous cohomology, which carries more interesting geometrical information (see Theorem A in the introduction). We will refer to [b] ∈ Hc1 (G, ρ) as the Haagerup class of G y X. The choice of p = 2 here is not particularly relevant and the same discussion could be equally carried out for any other p ∈ [1, +∞) with few complications; see Remark 3.6. We conclude this section by collecting a few straightforward lemmata for later use. Let H be a Hilbert space and G → U (H) a continuous unitary representation. Lemma 2.22. If H ≤ G is open and finite-index, functoriality induces an injective map Hc1 (G, H) ,→ Hc1 (H, H). Lemma 2.23. Given a G-invariant decomposition H = H1 ⊥ H2 , the projections onto the two factors induce Hc1 (G, H) ' Hc1 (G, H1 ) ⊕ Hc1 (G, H2 ). Recall that we denote by X 0 the barycentric subdivision of X and by i : X ,→ X 0 the standard inclusion; we write (H 0 , ν 0 ) instead of (H (X 0 ), νbX 0 ). Every isometric action G y X also induces a continuous representation ρ0 : G → U (L2 (H 0 , ν 0 )). Lemma 2.24. Let X be complete and finite rank and let G y X be an isometric action. The projection p : H 0 → H induces an isometric embedding p∗ : L2 (H , νb) ,→ L2 (H 0 , ν 0 ) and a monomorphism p∗ : Hc1 (G, ρ) ,→ Hc1 (G, ρ0 ) taking the Haagerup class of G y X to the Haagerup class of G y X 0 . Proof. The fact that p∗ is an isometric embedding follows from the observation that p∗ ν 0 = νb. Injectivity of p∗ follows from Lemma 2.23 applied to L2 (H , νb) and its orthogonal complement. Finally, if x ∈ X and bx is the corresponding Haagerup cocycle for G y X, the cocycle p∗ ◦ bx is the Haagerup cocycle for G y X 0 relative to the point i(x) ∈ X 0 . SUPERRIGIDITY OF ACTIONS ON FINITE RANK MEDIAN SPACES 23 3. Haagerup class and elementarity of actions. 3.1. The main statement. Let X be a complete, finite rank median space and let G y X be an isometric action of a topological group G. The goal of this section is to prove Theorem A. By Lemmata 2.12 and 2.24, it suffices to consider the case when G y X is without wall inversions; this will be a standing assumption throughout the rest of the section. Lemma 3.1. Suppose that X is irreducible and that G y X is Roller minimal and Roller nonelementary. There exists a non-abelian free subgroup H ≤ G such that ρ has no H-almost-invariant vectors and H y X has unbounded orbits. Proof. Part 2 of Proposition 6.4 in [Fio17b] provides a non-abelian free subF group H ≤ G and a measurable, ∗-invariant partition H = h∈H Hh with gHh = Hgh for all g, h ∈ H. It is immediate from the construction of H that it acts on X with unbounded orbits. If there existed a sequence of almost invariant vectors (Fn )n≥0 in L2 (H , νb), say with kFn k2 = 1, we could define functions fn ∈ `2 (H) by fn (h) := kFn 1Hh k2 . It is immediate to check that kfn k2 = 1 for every n ≥ 0 and, for every g ∈ H, 2 X kgfn − fn k22 = kFn 1Hg−1 h k2 − kFn 1Hh k2 = h∈H = X k(gFn ) 1Hh k2 − kFn 1Hh k2 2 ≤ h∈H ≤ X k(gFn ) 1Hh − Fn 1Hh k22 = kgFn − Fn k22 −−−−−→ 0. h∈H n→+∞ Thus, the regular representation of H would contain almost-invariant vectors, implying amenability of H (see e.g. Theorem G.3.2 in [BdlHV08]). This is a contradiction. We can already prove the “only if” half of Theorem A. Proposition 3.2. If G y X is Roller nonelementary, we have [b] 6= 0. Proof. Note that, by functoriality of reduced cohomology, it suffices to consider the case when G has the discrete topology; thus, we do not need to worry about continuity issues. We proceed by induction on r = rank(X). When r = 0, all actions are Roller elementary; assume for the rest of the proof that r ≥ 1. We can also assume that G y X is Roller minimal. Indeed, if C ⊆ Z ⊆ X are the subsets provided by Proposition 2.8, the action G y C is again without wall inversions and rank(C) ≤ r, by Proposition 2.7. The G-equivariant, ∗ ) induces an orthogonal decommeasurable partition H = HC t (σC ∪ σC 2 position of L (H , νb) and a G-equivariant splitting ∗ Hc1 G, L2 (H , νb) = Hc1 G, L2 (H (C), νbC ) ⊕ Hc1 G, L2 (σC ∪ σC , νb) . 24 ELIA FIORAVANTI If p1 and p2 are the orthogonal projections of L2 (H , νb) onto the two direct summands, we can write [bx ] = [p1 bx ] + [p2 bx ] for every x ∈ X. Note that the gate-projection πC : X → C maps x to a point ξ ∈ C. The cocycle p1 bx is precisely the Haagerup cocycle bξ for the action G y C, by part 2 of Proposition 2.7. In particular, if [bξ ] 6= 0 we can conclude that [b] 6= 0. We thus assume in the rest of the proof that X = C. If X is irreducible, Lemma 3.1 provides i : H ,→ G such that H y X has unbounded orbits and ρ has no H-almost-invariant vectors. The first condition implies that i∗ [b] 6= 0; the second condition and Theorem 1 in [Gui72] thus yield i∗ [b] 6= 0. In particular, [b] 6= 0. If instead X splits as a nontrivial product X1 × X2 and j : G0 ,→ G is the finite-index subgroup preserving this decomposition, it suffices to show that j ∗ [b] 6= 0. Writing νbi instead of νbXi , Proposition 2.10 and Lemma 2.23 imply: Hc1 G0 , L2 (H , νb) = Hc1 G0 , L2 (H (X1 ), νb1 ) ⊕ Hc1 G0 , L2 (H (X2 ), νb2 ) ; hence, if x = (x1 , x2 ), we have [bx ] = [bxX11 ] + [bxX22 ]. The action G0 y X is Roller nonelementary, since G0 is finite-index in G; thus, up to exchanging the two factors, G0 y X1 is Roller nonelementary. Since rank(X1 ) < r, the inductive hypothesis guarantees that [bxX11 ] 6= 0 and this concludes the proof. Before proving the rest of Theorem A we need to obtain a few more results. Lemma 3.3. If the G-orbit of ξ ∈ X is finite, the G-stabiliser of ξ is open. Proof. Suppose that Gξ = {ξ} t {ξ1 , ..., ξk } and d(ξ, ξi ) ≥ > 0 for all i, where d is the extended metric on X. By Proposition 4.24 in [Fio17a], we can find xi , yi ∈ X such that, denoting by πi the projection to Ii := I(xi , yi ), we have d(πi (ξ), πi (ξi )) > 34 . In a neighbourhood U ⊆ G of the identity element we have d(gxi , xi ) < 41 , d(gyi , yi ) < 41 and d(gπi (ξ), πi (ξ)) < 41 , for all i. If g ∈ U , we have d(πgIi (ξi ), πi (ξi )) ≤ d(gxi , xi ) + d(gyi , yi ) < 12 ; if in addition we had gξ = ξi , we would have πgIi (ξi ) = gπi (ξ). As a consequence, d(πi (ξ), πi (ξi )) ≤ d(πi (ξ), gπi (ξ)) + d(gπi (ξ), πi (ξi )) < 34 , which would contradict our choice of Ii . We conclude that U is contained in the stabiliser of ξ, which must be open in G. The proof of the following fact is rather lengthy and technical; it will be carried out in Appendix A. Proposition 3.4. Let ξ ∈ ∂X and K ⊆ Isomξ X be a compact set of isometries acting trivially on U(ξ). There exists a point xK ∈ X such that σξ \σxK coincides, up to a null set, with ΩK := Ω1K t ... t ΩkK , where • each ΩiK is a strongly reduced, minimal UBS; SUPERRIGIDITY OF ACTIONS ON FINITE RANK MEDIAN SPACES 25 • if g ∈ K we have gΩiK ⊆ ΩiK whenever χΩi (g) ≥ 0 and gΩiK ⊇ ΩiK K whenever χΩi (g) ≤ 0; K • if i 6= j and g ∈ K, we have ΩiK ∩ gΩjK = ∅. Given points ξ ∈ ∂X, x ∈ X and a UBS Ω ⊆ σξ \ σx , we can define a function αΩ : Ω → R as αΩ (h) := νb (H (x\|h) ∩ Ω). The dependence on the point x is not particularly relevant, so we do not record it in our notation. We can consider the sets Ωc := {h ∈ Ω \| αΩ (h) ≤ c}. In Appendix A we will obtain the following result (see Lemma A.9). Proposition 3.5. Suppose that Ω is minimal and strongly reduced. Let K ⊆ Isomξ X be a compact set of isometries such that gΩ ⊆ Ω for all g ∈ K. As c → +∞, the functions i h αΩ · 1Ωc (g − id) · − 1 − c converge to 1Ω\gΩ in L2 (H , νb), uniformly in g ∈ K. If instead gΩ ⊇ Ω for all g ∈ K, they converge to the function −1gΩ\Ω . We are finally ready to complete the proof of Theorem A. Proof of Theorem A. By Proposition 3.2, it suffices to consider the case when G has a finite orbit in X and, by Lemmata 2.22 and 3.3, we can actually assume that G fixes a point ξ ∈ X. If ξ ∈ X, we have [b] = 0; suppose instead that ξ ∈ ∂X. By Proposition 2.17, an open, finite-index subgroup G0 ≤ G acts trivially on U(ξ); by Lemma 2.22, it suffices to consider the case G = G0 . Fix x ∈ X. For every > 0 and every compact subset K ⊆ G, we need to construct a function ψ ∈ L2 (H , νb) such that kbx (g) − (gψ − ψ)k2 < for all g ∈ K. Considering the point xK ∈ X provided by Proposition 3.4, it suffices to find a function φ ∈ L2 (H , νb) such that kbxK (g) − (gφ − φ)k2 < for all g ∈ K and then set ψ := φ + 1σx − 1σxK . If g ∈ K, considering all equalities up to null sets, we have σgxK \ σxK = [(σgxK \ σxK ) ∩ σξ ] t (σgxK \ σxK ) ∩ σξ∗ = = [(σξ \ σxK ) \ (σξ \ σgxK )] t [(σξ \ σgxK ) \ (σξ \ σxK )]∗ = = (ΩK \ gΩK ) t (gΩK \ ΩK )∗ . In particular, since by construction ΩiK ∩ gΩjK = ∅ whenever i 6= j, σgxK \ σxK = k G ∗ ΩiK \ gΩiK t gΩiK \ ΩiK . i=1 Introducing the notation 2A := 1A − 1A∗ for subsets A ⊆ H , we can rewrite X X bxK (g) = 2σgxK \σxK = 2Ωi \gΩi − 2gΩi \Ωi . K χΩi (g)>0 K K K χΩi (g)<0 K K 26 ELIA FIORAVANTI For c > 0, consider the function Gc := k X i=1 αΩi K 2(ΩiK ) . − 1− c c Proposition 3.5 shows that it suffices to take φ = Gc for large c. Remark 3.6. Theorem A also holds for the analogous class in Hc1 (G, ρp ), for every p ∈ [1, +∞). Indeed, Lemma 2.23 applies to any decomposition of a Banach space into closed subspaces. In the proof of Lemma 2.24, a closed complement to Lp (H , νb) within Lp (H 0 , ν 0 ) is always provided by the subspace of functions on H 0 that take opposite values on hemiatoms. Theorem 1 in [Gui72] holds for representations in general Banach spaces. Finally, if F is a non-abelian free group, `p (F ) has no F -almost-invariant vectors for every p ∈ [1, +∞). The value of p also has little importance for most of the material in Appendix A. Note however that Proposition 3.5 fails for p = 1; one needs to consider functions with a quicker decay in that case. 3.2. Elementarity and Shalom’s property HF D . Let X be a complete, finite rank median space and let G be a topological group. Our main result on property HF D is the following. Proposition 3.7. If G has property HF D , every isometric action G y X is Roller elementary. We will need the following lemma. Lemma 3.8. Suppose that X is irreducible and let G y X be a Roller nonelementary, Roller minimal action. Let E ⊆ H be a measurable subset such that νb(gE4E) = 0 for all g ∈ G. Then νb(E) is either 0 or +∞. Proof. Without loss of generality we can assume that G y X is without wall inversions, as Lemma 2.14 allows us to pass to the barycentric subdivision X 0 if necessary. Now, suppose that E is such a set and 0 < νb(E) < +∞. Since X has finite rank, we can find a thick halfspace h such that, replacing E with E ∗ if necessary, the set Eh := E ∩ {k ∈ H \| k ⊆ h} satisfies a := νb(Eh ) > 0. By part 2 of Proposition 2.11, the halfspace h is part of a facing n-tuple with n > a1 · νb(E). By Proposition 2.9, there exist g2 , ..., gn ∈ G such that h, g2 h, ..., gn h are a facing n-tuple. The sets Eh , g2 Eh , ..., gn Eh are pairwise disjoint and contained in E, up to null sets. However, their union has measure na > νb(E), a contradiction. Proof of Proposition 3.7. Suppose for the sake of contradiction that G y X is Roller nonelementary. Without loss of generality, we can assume that X has minimal rank r among complete median spaces admitting Roller nonelementary actions of G. In particular, X must be irreducible, see the proof of Proposition 3.2. By Proposition 2.8, we can also assume that G y X is Roller minimal. Theorem A guarantees that Hc1 (G, ρ) 6= {0} and, since SUPERRIGIDITY OF ACTIONS ON FINITE RANK MEDIAN SPACES 27 G has property HF D , there exists a finite dimensional subrepresentation V < L2 (H , νb). We will construct a measurable G-invariant subset of H with positive, finite measure, thus violating Lemma 3.8. Let f1 , ..., fk be measurable functions on H whose equivalence classes in L2 (H , νb) form an orthonormal basis of V . Define, for c > 0, n o Ec := h ∈ H \| ∃α = (α1 , ..., αk ) ∈ Sk−1 s.t. \|(α1 f1 + ... + αk fk )(h)\| > c . Since in the definition of Ec it suffices to look at α’s lying in a countable dense subset of Sk−1 , we conclude that Ec is measurable. If h ∈ Ec , we must have \|fi (h)\| > kc for some i, hence νb(Ec ) < +∞; if c is sufficiently small, we have νb(Ec ) > 0. Given g ∈ G, there exist real numbers P αij , 1 ≤ i, j ≤ k, such that, outside a measure zero set, we have fi = j αij (gfj ) for every i. P If h ∈ Ec \ gEc , we must have fi (h) 6= j αij (gfj )(h) for some i; we conclude that νb(gEc 4Ec ) = 0 for all g ∈ G. Corollary 3.9. Let Γ be a discrete group with property HF D . If Γ acts freely and cocompactly on a CAT(0) cube complex X, then Γ is virtually abelian. Proof. Cocompactness of the action implies that X is finite dimensional. By Propositions 2.17 and 3.7, there exists a finite-index subgroup Γ0 ≤ Γ and a normal subgroup N C Γ0 consisting of elliptic elements, such that Γ0 /N is abelian. Since Γ acts freely, N is trivial. Recall that, in Gromov’s density model, random groups at density d < 12 are nonelementary hyperbolic with overwhelming probability [Gro93, Oll04]. Together with Theorem 10.4 in [OW11], Corollary 3.9 then immediately implies the following result. Corollary 3.10. With overwhelming probability, random groups at density d < 61 in Gromov’s density model do not have property HF D . 4. Superrigidity. 4.1. The superrigidity result. Let X be a complete median space of finite rank r and Γ y X an action by isometries of a discrete group Γ. Lemma 4.1. Suppose h1 , h2 , h3 ∈ H form a facing triple; let ki ∈ H and h∗i be strongly separated for i = 1, 2, 3. There exists a point z ∈ X such that m(ξ1 , ξ2 , ξ3 ) = z whenever ξi ∈ kei . Proof. Let C be the intersection of the closures of e h∗i inside X; it is nonempty, closed and convex. Given points ξi ∈ kei , set m := m(ξ1 , ξ2 , ξ3 ). By convexity we have I(ξ2 , ξ3 ) ⊆ e h∗1 , hence m ∈ e h∗1 ; permuting the indices, we obtain m ∈ C. In particular, denoting by πC the gate projection X → C, we have m = πC m(ξ1 , ξ2 , ξ3 ) = m(πC ξ1 , πC ξ2 , πC ξ3 ). The closures of e h∗i and eki in X are strongly separated by Lemma 2.21; let {xi } be the shore of h∗i and set z := m(x1 , x2 , x3 ). By Corollary 2.20, 28 ELIA FIORAVANTI we have πC (ξi ) = xi , hence m = z no matter which points ξi ∈ kei we have chosen. Lemma 4.2. Suppose that X is irreducible and assume that Γ y X is Roller nonelementary and Roller minimal. Given 0 6= f ∈ L2 (H , νb), consider N S (f ) := (gn ) ∈ Γ \| gn g · f −−−−−→ g · f, ∀g ∈ Γ . n→+∞ There exists z ∈ X such that, for every (gn ) ∈ S (f ), there exists N ≥ 0 such that gn · z = z for all n ≥ N . Proof. By part 1 of Proposition 2.10, the barycentric subdivision X 0 of X is an irreducible median space of the same rank. The action Γ y X 0 is without wall inversions, Roller nonelementary and Roller minimal by Lemma 2.14. As usual, we write (H 0 , ν 0 ) for (H (X 0 ), νbX 0 ). The function f ∈ L2 (H , νb) induces f 0 ∈ L2 (H 0 , ν 0 ) with S (f ) ⊆ S (f 0 ). We approximate f 0 by a linear combination F of characteristic functions of halfspace intervals H 0 (x1 \|y1 ), ..., H 0 (xk \|yk ) such that kF − f 0 k < 31 kf 0 k. Proposition 2.9 implies that ν 0 (H 0 ) = +∞; all halfspace intervals have finite measure, so there exists h0 ∈ H 0 such that xi , yi ∈ h0 for all i. Propositions 2.9 and 2.11 provide a thick halfspace h ∈ H 0 such that h∗ and h0 are strongly separated; in particular, h contains every wall in the set W 0 (x1 \|y1 ) ∪ ... ∪ W 0 (xk \|yk ). Propositions 2.9 and 2.11 also provide γ1 ∈ Γ such that h and γ1 h are strongly separated and γ2 ∈ Γ such that γ1 h∗ and γ2 h are strongly separated. We can assume without loss of generality that d(γ1 h∗ , γ2 h) ≥ 1, as by Proposition 2.1 the quantity d(γ1 h∗ , γ2n h) diverges as n goes to infinity. Thus, we can choose elements γm ∈ Γ such that h∗ ) γ1 h ) γ2 h ) ... and, for all i ≥ 1, the halfspaces γi h∗ and γi+1 h are strongly separated and at distance at least 1. We denote by S ⊆ H 0 the support of the function F and by P the set of all points xi , yi . Let D be the maximum distance from h∗ of a point of P and d := dD + 2e. Let us fix an integer m > d and g ∈ Γ such that kgγm f 0 − γm f 0 k < 31 kf 0 k and kgf 0 − f 0 k < 13 kf 0 k. We prove that gγm h ⊆ γm−d h. A straightforward repeated application of the triangle inequality yields kgF − F k < 2kF k and kgγm F − γm F k < 2kF k; thus, νb(S ∩ gS) > 0 and νb(γm S ∩ gγm S) > 0. Let w be a wall corresponding to a halfspace in γm S ∩ gγm S. Since w is contained in both γm h and gγm h and Γ y X 0 is without wall inversions, we conclude that γm h ∩ gγm h 6= ∅. A similar argument shows that h ∩ gh 6= ∅. Let u be the wall corresponding to gγm h. If u is contained in γm−1 h, we either have gγm h ⊆ γm−1 h or γm−1 h∗ ∩ gγm h∗ = ∅. The former case immediately yields gγm h ⊆ γm−d h, while the latter leads to a contradiction as h ⊆ γm−1 h∗ and gh ⊆ gγm h∗ intersect. SUPERRIGIDITY OF ACTIONS ON FINITE RANK MEDIAN SPACES 29 If instead u is not contained in γm−1 h, it is contained in γm h∗ , by strong separation. Let 1 ≤ l ≤ m be minimum such that γl h∗ contains u. We have γl h ⊆ gγm h, since γl h ∩ gγm h ⊇ γm h ∩ gγm h 6= ∅. Let k be the side of w that is contained in γm h. Since either k or k∗ lies in gγm S, there exists q ∈ P such that gγm q ∈ k ⊆ γm h. Hence, m − l ≤ d(γm h, γl h∗ ) ≤ d(gγm q, gγm h∗ ) = d(q, h∗ ) ≤ D and m − l + 2 ≤ D + 2 ≤ d. By strong separation and minimality of l, the wall u is contained in γl−2 h ⊆ γm−d h. Hence gγm h ⊆ γm−d h, since otherwise gγm h∗ ∩ γm−d h∗ = ∅ would again violate the fact that h ∩ gh 6= ∅. Now consider the intersection C of the closures in X 0 of the halfspaces γm e h. It consists of a single point ξ since any j ∈ H 0 with ej ∩ C 6= ∅ and ej∗ ∩ C 6= ∅ would have to be transverse to almost all γm h, violating strong separation. Strong separation also implies that ξ actually lies in X ⊆ X 0 . Given (gn ) ∈ S (f 0 ) we can assume, removing a finite number of elements if necessary, that kgn f 0 − f 0 k < 13 kf 0 k for all n. Let N (m) be a natural number such that kgn γm f 0 − γm f 0 k < 13 kf 0 k for all n ≥ N (m). When n ≥ N (m), we have shown that gn γm h ⊆ γm−d h; thus we have gn ξ ∈ gn γme h ⊆ γm−de h. In this case, strong separation implies that σgn ξ 4σξ consists only of halfspaces whose corresponding walls are contained in γm−d−1 h. This shows that lim sup σgn ξ 4σξ = ∅; we conclude that gn ξ → ξ in the topology of X, for every (gn ) ∈ S (f 0 ). We finally construct the point z ∈ X. Let j, m be thick halfspaces of X 0 so that m∗ and j are strongly separated and ξ ∈ ej. Part 2 of Proposition 2.11 provides a facing triple consisting of m, m1 , m2 . We choose thick halfspaces j1 , j2 ∈ H 0 such that m∗i and ji are strongly separated for i = 1, 2; by Proposition 2.9 we can find hi ∈ Γ such that hi j ⊆ ji . Let z ∈ X 0 be the point provided by Lemma 4.1 applied to j, j1 , j2 and m, m1 , m2 ; in particular, we have z = m(ξ, h1 ξ, h2 ξ), hence z ∈ X. Since the set S (f 0 ) is closed under conjugation by elements of Γ, we have gn hi ξ → hi ξ for all (gn ) ∈ S (f 0 ). Hence, given (gn ) ∈ S (f 0 ), there exists N ≥ 0 such that for every n ≥ N we have gn ξ ∈ ej and gn hi ξ ∈ eji , for i = 1, 2. Thus, gn z = gn m(ξ, h1 ξ, h2 ξ) = m(gn ξ, gn h1 ξ, gn h2 ξ) = z. In the rest of the section, we consider a locally compact group G and a lattice Γ < G. Any Borel fundamental domain U ⊆ G defines a cocycle α : G × U → Γ such that gu ∈ α(g, u) · U . We say that Γ is square-integrable if Γ is finitely generated and U can be chosen so that Z \|α(g, u)\|2S du < +∞, ∀g ∈ G; U here du is the Haar measure on U and \| · \|S denotes the word length with respect to a finite generating set S ⊆ Γ. Integrability does not depend 30 ELIA FIORAVANTI on the choice of S. Uniform lattices are always square-integrable and a few nonuniform examples were mentioned in the introduction; see [Sha00, Rém99, Rém05, CR09, CR10] for more details and examples. We assume that G splits as a product G1 × ... × G` , where each Gi is compactly generated and ` ≥ 2. We also require the lattice Γ < G to be irreducible, i.e. to project densely into each factor Gi . Consider a unitary representation π : Γ → U (H); we denote by H0 the subspace of invariant vectors. Let c : Γ → H be a 1-cocycle for π. We will make use of the following result of Y. Shalom in an essential way; see page 14 and Theorem 4.1 in [Sha00] for a proof. Theorem 4.3. Suppose that Γ is square-integrable and that H0 = {0}. There exist Γ-invariant closed subspaces Hi ≤ H, i = 1, ..., `, where the restriction πi : Γ → U (Hi ) extends to a continuous representation πi : G → U (Hi ) that factors through the projection pi : G → Gi . Furthermore, there are cocycles ci : Γ → Hi , i = 1, ..., `, such that c and c1 + ... + c` represent the same class in H 1 (Γ, π). The following is a version of our Theorem B under stronger hypotheses. Theorem 4.4. Suppose that Γ is square-integrable and X is irreducible; let Γ y X be Roller nonelementary and Roller minimal. There exists a Γ-invariant, closed median subalgebra Y ⊆ X where Γ y Y extends to a continuous action G y Y . Moreover, G y Y factors through a projection p i : G → Gi . Proof. We have H 1 (Γ, ρ) 6= {0} by Theorem A. Lemma 3.8 implies that ρ has no nonzero invariant vectors; thus, Theorem 4.3 provides a Γ-invariant subspace {0} = 6 Hi ⊆ L2 (H , νb) where the action of Γ extends to a continuous action of G factoring through a projection pi : G → Gi . Pick any 0 6= f ∈ Hi and consider the set S (f ) introduced in Lemma 4.2. Any sequence (gn ) ∈ ΓN with pi (gn ) → id lies in S (f ). Thus, Lemma 4.2 implies that the Γ-invariant set n o Y := x ∈ X \| ∀(gn ) ∈ ΓN s.t. pi (gn ) → id, we have gn x → x is nonempty. Note that Y is a median subalgebra of X, thus the restriction of the metric of X gives Y a structure of median space. Since Y is a closed subset of X, it is a complete median space. Finally, Proposition 4.3 in [Sha00] provides a continuous extension to G of Γ y Y and it factors through pi . The assumption that Γ y X be Roller minimal and Roller nonelementary can be replaced with the (stronger) requirement that Γ have no finite orbit in b see Proposition 5.2 in [Fio17b]. the visual boundary of the CAT(0) space X; The homomorphism φ : G → Isom Y provided by Theorem 4.4 is continuous with respect to the topology of pointwise convergence. We remark however that φ remains continuous even if we endow Isom Y with the topology mentioned in Remark 4.5 below; this will be a key point in our proof of Theorem C. SUPERRIGIDITY OF ACTIONS ON FINITE RANK MEDIAN SPACES 31 Remark 4.5. In the proof of Theorem 4.4, Lemma 4.2 actually yields that the smaller set n o Y0 := x ∈ X \| ∀(gn ) ∈ ΓN s.t. pi (gn ) → id, ∃N ≥ 0 s.t. gn x = x, ∀n ≥ N is nonempty. Thus, φ : G → Isom Y is continuous with respect to the topology on Isom Y that is generated by stabilisers of points of Y0 . In the statement of Theorem 4.4, we can always take Y to be the closure of Y0 in X. This topology on Isom Y might seem a lot finer than the topology of pointwise convergence. To clarify this phenomenon, we mention the following fact, without proof. Let W be an irreducible, complete, finite rank median space admitting a Roller nonelementary, Roller minimal action; then there exists a dense, convex subset C ⊆ W such that, for every x ∈ C, the stabiliser of x is open for the topology of pointwise convergence on Isom W . This essentially follows from Lemma 4.1. Relaxing the hypotheses of Theorem 4.4, we obtain Theorem B for all square-integrable lattices: Corollary 4.6. Suppose that Γ is square-integrable; let Γ y X be Roller nonelementary. There exist a finite index subgroup Γ0 ≤ Γ, a Γ0 -invariant component Z ⊆ X and a Γ0 -invariant closed median subalgebra Y ⊆ Z where the action Γ0 y Y extends to a continuous action G0 y Y , for an open finite index subgroup G0 ≤ G. Proof. We proceed by induction on rank(X); when the rank is zero there is nothing to prove, so we assume that the statement holds for all median spaces of rank at most r − 1. By Proposition 2.8, there exists a Γ-invariant, closed, convex subset D of a component W ⊆ X such that Γ y D is Roller minimal and Roller nonelementary. If W ⊆ ∂X, we have rank(D) < r and we conclude by the inductive hypothesis; thus we can assume that W = X. Let D = D1 × ... × Dk be the splitting of D into irreducible factors; if k = 1, the result follows from Theorem 4.4. If k ≥ 2, let Γ1 ≤ Γ be a finite index subgroup preserving the splitting of D; up to permuting the factors, we can assume that Γ1 y Di is Roller nonelementary for 1 ≤ i ≤ s and Roller elementary for i > s. A further finite index subgroup Γ2 ≤ Γ1 fixes a point ξi ∈ Di for each i > s; we denote by Zi ⊆ Di the component containing ξi . Note that Γ2 is a square-integrable, irreducible lattice in an open, finite index subgroup of G. Since rank(Di ) < r, for each i ≤ s the inductive hypothesis yields a finite index subgroup Γ0i ≤ Γ2 , an open finite index subgroup G0i ≤ G and a Γ0i -invariant, closed median subalgebra Yi of a component Zi ⊆ Di where the action of Γ0i extends to a continuous action of G0i . Let Γ0 be the intersection of all Γ0i and G0 be the intersection of all G0i , for i ≤ s. The set Y := Y1 × ... × Ys × {ξs+1 } × ... × {ξk } ⊆ D is a closed median subalgebra of Z1 × ... × Zk , which is a component of D; in particular, Z1 × ... × Zk is a closed, convex subset of a component Z ⊆ X. The action Γ0 y Y trivially extends to a continuous action G0 y Y . 32 ELIA FIORAVANTI Figure 3 We now describe two examples that illustrate how: • in Theorem 4.4 the space Y cannot be taken to coincide with X, nor with a convex subset (Example 4.7); • in Corollary 4.6 it cannot be avoided to pass to the finite index subgroup Γ0 , even when the action is Roller minimal (Example 4.8). The actions that we consider are actually on CAT(0) square complexes. Since Burger-Mozes groups play an important role in the construction of the two examples, we briefly recall a few facts regarding their construction. Given an integer n ≥ 3, we denote by Tn the n-regular tree and by An the group of even permutations on n elements. We fix a legal colouring on Tn , i.e. a way of associating an integer in {1, ..., n} to every edge of Tn so that we see all n integers around each vertex; in particular, we have a bijection iv : lk(v) → {1, ..., n} for every vertex v. Let U (An ) ≤ Isom Tn be the subgroup of isometries g such that igv ◦ g ◦ i−1 v ∈ An for every vertex v of Tn ; we denote by U (An )+ the intersection of U (An ) with the subgroup of Isom Tn generated by edge stabilisers. If n ≥ 4, the subgroup U (An )+ has index 2 in U (An ), see Proposition 3.2.1 in [BM00a]. The subgroup U (An ) is closed in Isom Tn ; in particular, it is locally compact, second countable and compactly generated (e.g. by Theorem 4.C.5 and Proposition 5.B.5 in [CdlH16]). By Theorem 6.3 in [BM00b], there exists a uniform irreducible lattice Λ ≤ U (A2k ) × U (A2k ) for every integer k ≥ 19. For the next two examples, we fix such a lattice Λ. Let p1 , p2 : Λ → U (A2k ) + be the projections into the two factors and set Λ0 := p−1 1 (U (A2k ) ); this is + an irreducible lattice in the open, index 2 subgroup U (A2k ) × U (A2k ) of U (A2k ) × U (A2k ). Let τ : Λ → Z/2Z be the homomorphism with kernel Λ0 . Example 4.7. Given any tree T , we can blow up every edge to a square as in Figure 3, thus obtaining a “tree of squares” T. Adjacent squares only share a vertex; if T has no leaves, each square has a pair of opposite vertices that are shared with other squares and a pair of opposite vertices that are not shared. The space T is a complete, rank two median space in which T embeds as a median subalgebra; edges of T correspond to diagonals joining shared pairs of vertices of a square. SUPERRIGIDITY OF ACTIONS ON FINITE RANK MEDIAN SPACES 33 We can embed Isom T ,→ Isom T by extending each isometry of T so that the restriction to each square is orientation preserving. Let σ ∈ Isom T be the isometry that fixes pointwise the image of the embedding T ,→ T and acts on each square as a reflection in a diagonal; we have σ 2 = id and Isom T × hσi ,→ Isom T. Viewing p2 : Λ → U (A2k ) as a homomorphism into Isom T2k we can define a homomorphism Λ → Isom T2k × hσi by λ 7→ (p2 (λ), τ (λ)). We denote by ψ the composition of this map with the embedding Isom T2k × hσi ,→ Isom T2k . The action Λ y T2k induced by ψ is Roller nonelementary and Roller minimal since the action Λ y T2k induced by p2 is. As T2k is irreducible, Theorem 4.4 guarantees a continuous extension of Λ y Y to U (A2k ), for some Λ-invariant median subalgebra of T2k . Indeed, one can take Y to be the image of T2k ,→ T2k . However, Y cannot be taken to be a convex subspace (or even a subcomplex) of T2k . Indeed, Y would be forced to be the whole T2k , as this is the only Λ-invariant convex subset of T2k . The action Λ y T2k does not extend to U (A2k ) × U (A2k ) by factoring via p1 ; this is because, whenever elements gn ∈ Λ satisfy p1 (gn ) → id, the sequence (p2 (gn )) must diverge. However, Λ y T2k also does not extend by factoring through p2 : we have p2 (Λ0 ) = p2 (Λ) = U (A2k ), but ψ(Λ0 ) is contained in the closed subgroup Isom T2k < Isom T2k and ψ(Λ) is not. Example 4.8. Choose an element g ∈ Λ \ Λ0 and consider the action Λ0 y T2k × T2k given by λ · (x, y) = (p2 (λ) · x, p2 (g −1 λg) · y). Since the action U (A2k ) y T2k does not preserve any proper closed subtree, the same holds for the action of p2 (Λ0 ). Part 3 of Proposition 2.10 then implies that Λ0 does not leave any proper, closed, convex subset of T2k × T2k invariant. Note that no component of ∂(T2k × T2k ) = (∂T2k × T2k ) ∪ (T2k × ∂T2k ) is preserved by Λ0 , as this would correspond to a fixed point for p2 (Λ0 ) y T2k , hence to a fixed point for U (A2k ) y T2k . We conclude that Λ0 y T2k × T2k is Roller minimal and the same argument also shows that it is Roller nonelementary. One can easily check that Λ0 y T2k × T2k can be extended to an action of the whole Λ by setting λg · (x, y) = (p2 (λg 2 ) · y, p2 (g −1 λg) · x) for all λ ∈ Λ0 . This action also is Roller minimal and Roller nonelementary. We will show, however, that there exists no Λ-equivariant isometric embedding j : Y ,→ T2k × T2k of a median space Y such that the action on Y extends continuously to U (A2k ) × U (A2k ) by factoring through one of the factors. Let j : Y ,→ T2k × T2k be a Λ-equivariant embedding; note that j(Y ) is entirely contained in a Λ-invariant component Z ⊆ T2k × T2k . In particular, the previous discussion shows that Z = T2k × T2k . By Lemma 6.5 in [Bow13a], each wall of j(Y ) arises from a wall of T2k ×T2k , i.e. a wall of one of the two factors, see part 1 of Proposition 2.10. Since the two factors are exchanged by g ∈ Λ, we conclude that Y splits as Y1 × Y2 , with Λ0 y Y preserving this decomposition and g exchanging Y1 and Y2 . 34 ELIA FIORAVANTI Suppose for the sake of contradiction that Λ y Y extends to an action of U (A2k ) × U (A2k ) by factoring through one of the two factors. As in Example 4.7, we see that the extension cannot factor via p1 . However, since p2 (Λ0 ) is dense in U (A2k ) and Λ0 preserves the splitting Y = Y1 × Y2 , part 2 of Proposition 2.10 implies that an extension factoring through p2 would also preserve the splitting Y = Y1 × Y2 . This contradicts the fact that g exchanges Y1 and Y2 . We conclude the section by proving Theorem C. Proof of Theorem C. We begin by observing that part 2 follows from part 1 and Proposition 2.17. Now, suppose for the sake of contradiction that Γ admits a Roller nonelementary action on X. As in the proof of Proposition 3.7, we can assume that X is irreducible and that Γ y X is Roller minimal. Theorem 4.4 then yields a factor Gi , a closed median subalgebra Y ⊆ X and actions Gi y Y and Γ y Y . Without loss of generality, we can assume that Y is the closure of Y0 inside X, as in Remark 4.5. Stabilisers of points of Y0 are open in Gi , thus the identity component G0i must fix Y0 pointwise. As Y0 is dense in Y , the entire action G0i y Y vanishes and Gi y Y descends to an action of the group Gi /G0i . Since Gi satisfies condition (∗), Proposition 3.7 above and Corollary 6.5 in [Fio17b] imply that the action Gi /G0i y Y is Roller elementary. However, by Lemma 2.6, the actions Γ y Y and Gi y Y are Roller nonelementary, a contradiction. 4.2. Homomorphisms to coarse median groups. We defined equivariantly coarse median groups in the introduction. Here we simply prove Corollary E. Proof of Corollary E. Fix a non-principal ultrafilter ω on N and let Hω be the corresponding ultrapower of H. We endow H with a word metric dS arising from a finite generating set S ⊆ H. Given λ = (λn ) ∈ RN + , we denote by Coneω (H, λ) the asymptotic cone obtained by taking all basepoints at the identity and λ as sequence of scaling factors. Let d denote the metric that dS induces on Coneω (H, λ); it is a geodesic metric and it is preserved by the natural action Hω y Coneω (H, λ). If λn → +∞, the coarse median on H induces a structure of finite rank median algebra on Coneω (H, λ), see Section 9 in [Bow13a]; we denote by m the corresponding median map. The action Hω y Coneω (H, λ) is by automorphisms of the median algebra structure. By Propositions 3.3 and 5.1 in [Zei16], we can endow Coneω (H, λ) with a median metric dm that is biLipschitz equivalent to d and preserved by the Hω -action; furthermore, the median algebra structure associated to dm is given by the map m. Now suppose for the sake of contradiction that there exist pairwise nonconjugate homomorphisms φn : Γ → H, for n ≥ 0; these correspond to a homomorphism Γ → Hω , hence to an action on every asymptotic cone of H that preserves the median metric dm . The Bestvina-Paulin construction SUPERRIGIDITY OF ACTIONS ON FINITE RANK MEDIAN SPACES 35 [Bes88, Pau91] provides us with a sequence µn → +∞ such that, modifying each φn within its conjugacy class if necessary, the induced action Γ y Coneω (H, µ) has no global fixed point. This, however, contradicts Theorem C. Appendix A. Structure of UBS’s. Let X be a complete median space of finite rank r. We fix points x ∈ X and ξ ∈ ∂X. Let Ω ⊆ σξ \ σx be a minimal, reduced UBS. Lemma A.1. Let T g ∈ Isomξ X be an isometry satisfying gΩ ∼ Ω. Consider the UBS Ω1 := 0≤i≤r g −i Ω ⊆ Ω. (1) If χΩ (g) ≥ 0, we have g r! Ω1 ⊆ Ω1 and g r! h ⊆ h for all h ∈ Ω1 . (2) If χΩ (g) = 0, we have g r! Ω1 = Ω1 and g r! h = h for all h ∈ Ω1 . Proof. Observe that Ω1 ∼ Ω; we fix a diverging chain (kn )n≥0 in Ω1 . If h ∈ Ω1 , the halfspaces g i h with 0 ≤ i ≤ r all lie in Ω ⊆ σξ \ σx and cannot be pairwise transverse; thus, there exists 0 ≤ i ≤ r such that either g i h ⊆ h, or g i h ⊇ h. Hence, for every h ∈ Ω1 we either have g r! h ⊆ h or g r! h ⊇ h. Suppose that g r! h0 ( h0 for some h0 ∈ Ω1 ; set hk := g kr! h0 . For each k ≥ 0, a cofinite subchain of {g −kr! kn }n≥0 is a diverging chain in Ω1 , as gΩ1 ∼ Ω1 . Hence, for each k ≥ 0 we have h0 ⊇ g −kr! kn if n is sufficiently large, since Ω is reduced; in particular h0 ⊇ hk ⊇ kn . We conclude that each hk lies in Ω1 ; Proposition 2.1 guarantees that (hk )k≥0 is a diverging chain. Let Ξ ⊆ Ω1 be the inseparable closure of {hk }k≥0 ; it is a UBS equivalent to Ω and it satisfies g r! Ξ ⊆ Ξ. Observe that, for each k ≥ 0, kr! · χΩ (g) = χΩ (g kr! ) = χΞ (g kr! ) = νb(Ξ \ g kr! Ξ) ≥ d(hk , h∗0 ); since d(hk , h∗0 ) > 0 for some k ≥ 0, we conclude that χΩ (g) > 0. The same argument applied to g −1 shows that χΩ (g) < 0 if there exists h0 ∈ Ω1 with g r! h0 ) h0 . This proves part 2 and shows that g r! h ⊆ h for all h ∈ Ω1 if χΩ (g) > 0. In the latter case, for every h ∈ Ω1 we have h ⊇ g r! h ⊇ kn for sufficiently large n, since Ω is reduced; thus, g r! h ∈ Ω1 and g r! Ω1 ⊆ Ω1 . Corollary A.2. Let g ∈ Isomξ X be an isometry satisfyingTgΩ ∼ Ω. Define Ω1 as in the previous lemma and consider the UBS Ω2 := 1≤i≤r! g i−1 Ω1 . (1) If χΩ (g) ≥ 0, we have gΩ2 ⊆ Ω2 . (2) If χΩ (g) = 0, we have gΩ2 = Ω2 . In the rest of the appendix, we also consider a compact subset K ⊆ Isomξ X such that gΩ ∼ Ω for every g ∈ K. Lemma A.3. (1) There exists a constant C1 = C1 (Ω, K) such that every h ∈ Ω with d(x, h) > C1 lies in gΩ for every g ∈ K. (2) If Ξ ⊆ σξ \ σx is a minimal, reduced UBS such that Ξ 6∼ Ω and gΞ ∼ Ξ for all g ∈ K, there exists a UBS Ω ⊆ Ω that is disjoint from gΞ for all g ∈ K. 36 ELIA FIORAVANTI Proof. Let (hn )n≥0 be a diverging chain in Ω with h0 thick. For every g ∈ K, a cofinite subchain of (g −1 hn )n≥0 is contained in Ω, as gΩ ∼ Ω; since Ω is reduced, there exists n(g) ≥ 0 so that g −1 hn(g) ⊆ h0 . By Proposition 2.1 we can assume that νb(H (gh∗0 \|hn(g) )) > 0 and Lemma 2.18 provides a neighbourhood U (g) of g in K such that H (γh∗0 \|ξ)∩H (gh∗0 \|hn(g) ) 6= ∅ for all γ ∈ U (g); in particular, hn(g) ⊆ γh0 for all γ ∈ U (g). There exist g1 , ..., gk ∈ K such that K = U (g1 ) ∪ ... ∪ U (gk ); if N is the maximum of the n(gi ), we have hN ⊆ gh0 for all g ∈ K. Let ΩN be the inseparable closure of {hn }n≥N . If m ≥ N and g ∈ K, we have gh0 ⊇ hm ⊇ ghn for every sufficiently large n, since Ω is reduced. This shows that ΩN is contained in gΩ for all g ∈ K. Since ΩN ∼ Ω, there exists a constant C1 such that every h ∈ Ω with d(x, h) > C1 lies in ΩN . To prove part 2, let C be the supremum of distances d(x, h) for h ∈ Ω ∩ Ξ; we have C < +∞ since Ω 6∼ Ξ. Let M be the maximum distance d(x, gx) for g ∈ K and consider C 0 := max{C, C1 (Ξ, K −1 ) + M }; we define Ω to be the set of h ∈ Ω with d(x, h) > C 0 . If there existed h ∈ Ω ∩ gΞ for some g ∈ K, we would have g −1 h ∈ Ξ and d(x, g −1 h) > C1 (Ξ, K −1 ); thus, part 1 implies that g −1 h ∈ g −1 Ξ, i.e. h ∈ Ξ, and this contradicts the fact that h ∈ Ω and d(x, h) > C. Recall that we have introduced the function αΩ : H → R, defined by the formula αΩ (h) := νb (H (x\|h) ∩ Ω) and the sets Ωc := {h ∈ Ω \| αΩ (h) ≤ c}. Observe that αΩ (k) ≤ αΩ (h) whenever h ⊆ k; in particular αΩ is measurable. We have αΩ (h) ≤ νb(H (x\|h)) = d(x, h) for all h ∈ H . We say that Ω is small if νb(Ω) < +∞; otherwise, Ω is large. If X is a CAT(0) cube complex, every UBS is large; an example of a small UBS in a rank two median space appears in Figure 3 of [Fio17a]. If Ω is small, we have χΩ (h) = 0 for every isometry h fixing [Ω]. Note that the supremum of αΩ is precisely νb(Ω). Lemma A.4. Let hn ∈ Ω be halfspaces with hn+1 ⊆ hn for all n ≥ 0. Then, αΩ (hn ) → νb(Ω) if and only if (hn )n≥0 is a diverging chain of halfspaces. Proof. The fact that αΩ (hn ) → νb(Ω) if (hn )n≥0 is a diverging chain follows from the fact that Ω is reduced. For the other implication, let (km )m≥0 be a diverging chain in Ω. Since αΩ (h) ≤ d(x, h), it suffices to consider the case when Ω is small. For every m ≥ 0, the set {j ∈ Ω \| j ⊆ km } has measure am > 0 by Proposition 2.1. For large n we have αΩ (hn ) > νb(Ω) − am , hence there exists j ⊆ km such that j ∈ H (x\|hn ); in particular, hn ⊆ km . Since m is arbitrary, this shows that (hn )n≥0 is a diverging chain. Lemma A.5. (1) For every 0 ≤ c < νb(Ω), the set Ω \ Ωc is a UBS. (2) For all c ≥ 0, we have νb(Ωc ) ≤ rc. Proof. Since Ω is reduced, any h ∈ Ω \ Ωc contains almost every halfspace in any diverging chain in Ω; this provides a diverging chain in Ω \ Ωc . Inseparability follows from the monotonicity of αΩ . SUPERRIGIDITY OF ACTIONS ON FINITE RANK MEDIAN SPACES 37 To prove part 2, we decompose Ωc = C1 t ... t Ck as in Lemma 2.2. If h, k ∈ Ci and h ⊆ k, we have c ≥ αΩ (h) ≥ νb (H (k∗ \|h)). Hence, Lemma 2.27 in [Fio17a] implies that the inseparable closure of Ci has measure at most c. We conclude that νb(Ωc ) ≤ kc ≤ rc. Lemma A.6. Assume that gΩ ⊆ Ω for all g ∈ K. (1) For all h ∈ Ω and g ∈ K, we have −d(x, gx) − χΩ (g) ≤ αΩ (g −1 h) − αΩ (h) ≤ d(x, gx); in particular kgαΩ − αΩ k∞ ≤ C2 for some constant C2 = C2 (Ω, K). (2) For every c > 0, we have gΩc ⊆ Ωc+C2 . Proof. Indeed, αΩ (g −1 h) ≤ νb H (x\|g −1 x) + νb H (g −1 x\|g −1 h) ∩ Ω = = d(x, g −1 x) + νb (H (x\|h) ∩ gΩ) ≤ d(x, gx) + αΩ (h), and αΩ (h) ≤ νb (H (x\|gx)) + νb (H (gx\|h) ∩ Ω) = = d(x, gx) + νb H (x\|g −1 h) ∩ g −1 Ω ≤ ≤ d(x, gx) + νb H (x\|g −1 h) ∩ Ω + νb g −1 Ω \ Ω = = d(x, gx) + αΩ (g −1 h) + χΩ (g). We then take C2 to be the maximum of d(x, gx) + χΩ (g) for g ∈ K; this exists due to Proposition 2.17. Regarding part 2, observe that, if h ∈ Ωc , αΩ (gh) = αΩ (h) + (αΩ (gh) − gαΩ (gh)) ≤ c + C2 . Lemma A.7. Assume that Ω is strongly reduced. (1) For every d ≥ 0, there exists a constant C3 = C3 (Ω, d) such that h ⊆ k for all h, k ∈ Ω with d(x, h) > C3 and d(x, k) ≤ d. (2) If Ξ ⊆ Ω is a UBS and d(x, k) ≤ d for all k ∈ Ω \ Ξ, we have αΩ (h) − αΞ (h) = νb(Ω \ Ξ) for all h ∈ Ω with d(x, h) > C3 (Ω, d). Proof. Decompose Ω = C1 t ... t Ck , where each Ci is totally ordered by inclusion and contains a diverging chain. Pick halfspaces ki ∈ Ci with d(x, ki ) > d. By part 3 of Lemma 2.16, the halfspace ki cannot be transverse to a diverging chain in Ω; thus, part 2 of Lemma 2.2 guarantees that the halfspaces of Ω that are transverse to ki lie at uniformly bounded distance from x. We conclude that there exists C3 such that every h ∈ Ω with d(x, h) > C3 is contained in each ki , hence also in each k ∈ Ω with d(x, k) ≤ d. This proves part 1; part 2 is an immediate consequence. In the rest of the section, we assume that Ω is minimal and strongly reduced. Lemma A.8. Suppose that χΩ (g) ≥ 0 for all g ∈ K. 38 ELIA FIORAVANTI (1) There exists a constant C4 = C4 (Ω, K) such that, for every halfspace h ∈ Ω with d(x, h) > C4 and every g ∈ K, we have αΩ (gh) ≥ αΩ (h). (2) There exists a constant C5 = C5 (Ω, K) < νb(Ω) such that we have g(Ω \ Ωc ) ⊆ Ω \ Ωc and Ωc \ gΩc = Ω \ gΩ whenever c > C5 and g ∈ K. Proof. We first observe that, given a minimal UBS Ξ ⊆ σξ \ σx and an isometry g ∈ Isomξ X with gΞ ⊆ Ξ, we have αΞ (gh) − αΞ (h) = νb H (g −1 x\|h) ∩ g −1 Ξ − νb (H (x\|h) ∩ Ξ) = = νb H (g −1 x\|h) ∩ Ξ − νb (H (x\|h) ∩ Ξ) + νb H (g −1 x\|h) ∩ (g −1 Ξ \ Ξ) = = −b ν H (x\|g −1 x, h) ∩ Ξ + νb H (g −1 x\|h) ∩ (g −1 Ξ \ Ξ) , since H (g −1 x\|x) ∩ Ξ = ∅, as Ξ ⊆ σξ \ σx . Note that H (x\|g −1 x) ∩ gΞ = g −1 H (gx\|x) ∩ g 2 Ξ ⊆ g −1 (H (gx\|x) ∩ Ξ) = ∅. Thus αΞ (gh) − αΞ (h) equals − νb H (x\|g −1 x, h) ∩ (Ξ \ gΞ) + νb H (g −1 x\|h) ∩ (g −1 Ξ \ Ξ) ≥ ≥ −b ν (σx∗ ∩ (Ξ \ gΞ)) + νb (H (x\|gh) ∩ (Ξ \ gΞ)) ≥ ≥ −b ν ((σx∗ \ H (x\|gh)) ∩ (Ξ \ gΞ)) ≥ −b ν ({k ∈ Ξ \ gΞ \| gh 6⊆ k}) . T Now, consider for each g ∈ K the set Ω2 (g) := −r+1≤i≤r! g i−1 Ω as in Corollary A.2; we have gΩ2 (g) ⊆ Ω2 (g) and, by part 1 of Lemma A.3, there exists a constant C1 such that each h ∈ Ω with d(x, h) > C1 lies in gΩ2 (g) for every g ∈ K. By part 1 of Lemma A.7, if d(x, gh) > C3 (Ω, C1 ) we have αΩ2 (g) (gh) − αΩ2 (g) (h) ≥ −b ν ({k ∈ Ω2 (g) \ gΩ2 (g) \| gh 6⊆ k}) = 0. By part 2 of Lemma A.7, if d(x, h) > C3 (Ω, C1 ) and d(x, gh) > C3 (Ω, C1 ), we have αΩ (gh) − αΩ (h) = αΩ2 (g) (gh) − αΩ2 (g) (h) ≥ 0. We conclude that αΩ (gh) ≥ αΩ (h) whenever d(x, h) > C4 := C3 (Ω, C1 ) + M , where M is the maximum of the distances d(x, gx) for g ∈ K. We now prove part 2. Let C4 be as in part 1. Part 1 of Lemma A.3 provides a constant C10 such that each h ∈ Ω with d(x, h) > C10 lies in gΩ ∩ g −1 Ω for every g ∈ K. Let C5 be the supremum of the values αΩ (h) for h ∈ Ω with d(x, h) ≤ max{C4 , C10 }; by Lemma A.4 we have C5 < νb(Ω). If c > C5 , any h ∈ Ω \ Ωc satisfies d(x, h) > max{C4 , C10 }, hence αΩ (gh) ≥ αΩ (h) > c and gh ∈ Ω; in particular g(Ω \ Ωc ) ⊆ Ω \ Ωc . Observe that (Ω \ gΩ) ∩ gΩc ⊆ g(Ωc \ Ω) = ∅ and Ω \ gΩ ⊆ Ωc , by our choice of the constant C10 ; thus, Ω \ gΩ ⊆ Ωc \ gΩc . Conversely, it is clear that Ωc \ gΩc ⊆ Ω and (Ωc \ gΩc ) ∩ gΩ = Ωc ∩ g (Ω \ Ωc ) ⊆ Ωc ∩ (Ω \ Ωc ) = ∅. Consider now, for c > 0, the functions Fc,Ω := − 1 − αΩ c 1Ωc in L2 (H , νb). SUPERRIGIDITY OF ACTIONS ON FINITE RANK MEDIAN SPACES 39 Lemma A.9. Assume that gΩ ⊆ Ω for all g ∈ K. For every , there exists a constant C = C (Ω, K) < +∞ such that k(g − id)Fc,Ω − 1Ω\gΩ k2 < for all g ∈ K and all c ≥ C . If instead gΩ ⊇ Ω for all g ∈ K, we have k(g − id)Fc,Ω + 1gΩ\Ω k2 < for c ≥ C . Proof. Observe that gαΩ αΩ (g − id)Fc,Ω = − 1 − 1gΩc + 1 − 1Ωc = c c gαΩ αΩ gαΩ − αΩ =− 1− 1gΩc \Ωc + 1 − 1Ωc \gΩc + 1gΩc ∩Ωc . c c c We will analyse the three summands separately. By Lemma A.6, we have gαΩ (h) αΩ (h) − C2 C2 αΩ (h) + C2 2C2 ≤ 1− ≤ 1− < ≤ 1− , − c c c c c for each h ∈ gΩc \ Ωc ⊆ Ωc+C2 \ Ωc . By part 2 of Lemma A.5, gαΩ 2C2 (r(c + C2 ))1/2 2C2 1− 1gΩc \Ωc ≤ ·b ν (gΩc \Ωc )1/2 ≤ −−−−→ 0. c→+∞ c c c 2 By part 1 of Lemma A.3, there exists a constant C1 such that each h ∈ Ω with d(x, h) > C1 lies in gΩ for every g ∈ K; in particular, if k ∈ Ω \ gΩ for some g ∈ K, then αΩ (k) ≤ d(x, k) ≤ C1 . By part 2 of Lemma A.8, if c ≥ C5 we have Ωc \ gΩc = Ω \ gΩ and αΩ αΩ 1Ωc \gΩc − 1Ω\gΩ = 1Ωc \gΩc − 1Ω\gΩ − 1 = 1− c c Ωc \gΩc 2 2 αΩ C1 M 1/2 C1 = 1Ω\gΩ ≤ · νb (Ω \ gΩ)1/2 ≤ −−−−→ 0, c→+∞ c c c 2 where M is the maximum of χΩ (g) for g ∈ K, which exists by Proposition 2.17. Finally, by part 2 of Lemma A.5 and part 1 of Lemma A.6, gαΩ − αΩ 1gΩc ∩Ωc c ≤ 2 C2 C2 (cr)1/2 · νb(Ωc )1/2 ≤ −−−−→ 0. c→+∞ c c If instead gΩ ⊇ Ω for all g ∈ K, the previous discussion shows that, for large c, we have k(g −1 − id)Fc,Ω − 1Ω\g−1 Ω k2 < ; the conclusion follows by applying g. Lemma A.10. Let Ξ ⊆ σξ \ σx be a UBS and let Ξ1 , ..., Ξk ⊆ Ξ be pairwise inequivalent, reduced UBS’s representing all minimal equivalence classes of UBS’s almost contained in Ξ; for every i, set σi := νb(Ξi ). There exist (i) increasing sequences (cn )n≥0 such that (i) • cn → σi for all i =1, ..., k; 1 1 • Ξ \ Ξ (1) ∪ ... ∪ Ξk \ Ξk(k) is inseparable for all n ≥ 0. cn cn Proof. We proceed by induction on k. If k = 1, the lemma is immediate. Suppose that k ≥ 2; without loss of generality, we can assume that Ξk corresponds to a vertex with no incoming edges in the full subgraph of G(ξ) 40 ELIA FIORAVANTI with vertices [Ξ1 ], ..., [Ξk ]. Fix > 0; we will construct c(i) > σi − satisfying the inseparability condition. (i) (k) Pick a diverging chain (hn )n≥0 in each Ξi ; up to replacing (hn )n≥0 with a cofinite subchain, we can assume that, for each i ≤ k − 1 and each m ≥ 0, (k) (i) the halfspace hm is transverse to hn for almost every n. By Lemma A.4, there exists c(k) > σk − such that Ξk \ Ξkc(k) is contained in the inseparable (k) closure of {hn }n≥0 . As a consequence, for every h ∈ Ξk \ Ξkc(k) and every (i) i ≤ k − 1, the halfspaces h and hn are transverse for almost every n. Halfspaces lying in the inseparable closure of Ξ1 ∪ ... ∪ Ξk , but in neither of the Ξi , are at uniformly bounded distance from x, by part 3 of Proposition 2.15; say that these distances are bounded above by M < +∞. Enlarge M so that all j ∈ Ξk with d(x, j) > M lie in Ξk \ Ξkc(k) . By part 4 of Proposition 2.15 there exists a UBS contained in Ξ such that [Ξ1 ], ..., [Ξk−1 ] are all the equivalence classes of minimal UBS’s almost contained in Ξ. The inductive hypothesis and Lemma A.4 imply that we can find c(i) > σi − so that Ξ1 \ Ξ1c(1) ∪ ... ∪ Ξk−1 \ Ξk−1 is inseparable and d(x, h) > M for c(k−1) all h ∈ Ξi \ Ξic(i) with i ≤ k − 1. Now, if Ξ1 \ Ξ1c(1) ∪ ... ∪ Ξk \ Ξkc(k) were not inseparable, there would exist j ∈ H such that ku ⊆ j ⊆ kv , for halfspaces ku ∈ Ξu \ Ξuc(u) and kv ∈ Ξv \ Ξvc(v) , but j would not lie in any of the Ξi \ Ξic(i) . In particular, u 6= v and k ∈ {u, v}; observe that v 6= k, otherwise kv ∈ Ξk \ Ξkc(k) would not be transverse to diverging chains in Ξu . Thus, u = k; moreover, for all (i) i ≤ k − 1, the halfspace j must be transverse to hn for almost every n, since v ≤ k − 1 and j does not lie in Ξ1 \ Ξ1c(1) ∪ ... ∪ Ξk−1 \ Ξck−1 (k−1) , which is inseparable. Since d(x, j) ≥ d(x, kv ) > M , we have j ∈ Ξ1 ∪ ... ∪ Ξk ; the fact that each Ξi is reduced implies that j ∈ Ξk . By our choice of M , we have j ∈ Ξk \ Ξkc(k) , a contradiction. Lemma A.11. There exists a UBS Ξξ ⊆ σξ \σx such that every UBS Ξ ⊆ Ξξ with Ξ ∼ Ξξ is of the form σξ \ σy for some y ∈ X, up to a null set. Proof. Let Ξ1 , ..., Ξk ⊆ σξ \σx be pairwise inequivalent minimal UBS’s representing all minimal elements of U(ξ); we can assume that they are all reduced by Lemma 2.16. Halfspaces in σξ \σx that are transverse to a diverging chain in each Ξi lie at uniformly bounded distance from x, by part 3 of Proposition 2.15; say that these distances are bounded above by M < +∞. Let Ξξ consist of all h ∈ σξ \ σx with d(x, h) > M ; it is a UBS equivalent to σξ \ σx . Observe that, if Ξ ⊆ Ξξ is a UBS equivalent to Ξξ and ξ ∈ e h for halfspaces h ⊆ k ∈ Ξ, then h ∈ Ξ. Indeed, otherwise h would not contain any halfspace of Ξ, by inseparability, and it would therefore be transverse to a diverging chain in each of the Ξi ; hence d(x, k) ≤ d(x, h) ≤ M , a contradiction. SUPERRIGIDITY OF ACTIONS ON FINITE RANK MEDIAN SPACES 41 Now, given Ξ ⊆ Ξξ , consider the set σ := (σξ \ Ξ) t Ξ∗ ⊆ H . Observe that σ is an ultrafilter. Indeed, since σξ \ Ξ ⊆ σξ and Ξ∗ ⊆ σx , it suffices to check that h ∩ k 6= ∅ whenever h ∈ σξ \ Ξ and k ∈ Ξ∗ . If such halfspaces were disjoint, we would have ξ ∈ e h and h ⊆ k∗ ∈ Ξ, contradicting the observation we made above since h 6∈ Ξ. Finally, by Lemma 4.3 in [Fio17b], we can decompose σξ \ σx = Ξ t Σ for some set Σ with νb(Σ) < +∞; in particular, σx \ σξ = Ξ∗ t Σ∗ . Since Ξ and σx are disjoint, we have σx \ σ = σx \ (σξ ∪ Ξ∗ ) = Σ∗ , hence νb(σx 4σ) < +∞. Proposition 2.4 implies that there exists y ∈ X such that σ4σy is null; thus, up to measure zero, σξ \ σy = σξ \ σ = Ξ. We are now ready to prove Proposition 3.4. Proof of Proposition 3.4. Let Ξ1 , ..., Ξk ⊆ σξ \ σx be pairwise inequivalent UBS’s representing all minimal elements of the poset U(ξ), ; by Lemma 2.16, we can assume that each Ξi is strongly reduced. Up to replacing each Ξi with a smaller UBS, part 2 of Lemma A.3 guarantees that we can assume that Ξi ∩ Ξj = ∅ and Ξi ∩ gΞj for all i 6= j and g ∈ K. By part 2 of Lemma A.8, we have g(Ξi \ Ξic ) ⊆ Ξi \ Ξic for all C5 (Ξi , K) ≤ c < νb(Ξi ) and all g ∈ K with χΞi (g) ≥ 0, while we have g(Ξi \ Ξic ) ⊇ Ξi \ Ξic if χΞi (g) ≤ 0. Lemma A.10 provides constants C5 (Ξi , K) ≤ ci < νb(Ξi ) such that ΩK := Ξ1 \ Ξ1c1 ∪ ... ∪ Ξk \ Ξkck is inseparable. Thus ΩK is a UBS equivalent to σξ \ σx by part 3 of Proposition 2.15. We conclude by Lemma A.11, enlarging the constants ci if necessary, so that ΩK ⊆ Ξξ ; this is possible by Lemma A.4. References [ANWZ17] Goulnara Arzhantseva, Graham A. Niblo, Nick Wright, and Jiawen Zhang. A characterization for asymptotic dimension growth. arXiv:1612.06638v2, 2017. 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Massive MIMO Performance Comparison of Beamforming and Multiplexing in the Terahertz Band Sayed Amir Hoseini∗† , Ming Ding† and Mahbub Hassan∗† arXiv:1710.09031v2 [cs.IT] 26 Oct 2017 ∗ School of Computer Science and Engineering, University of New South Wales, Sydney, Australia † Data61, CSIRO, Sydney, Australia Email: [email protected], [email protected], [email protected] Abstract—In this paper, we compare the performance of two main MIMO techniques, beamforming and multiplexing, in the Terahertz (THz) band. The main problem with the THz band is its huge propagation loss, which is caused by the tremendous signal attenuation due to molecule absorption of the electromagnetic wave. To overcome the path loss issue, massive MIMO has been suggested to be employed in the network and is expected to provide Tbps for a distance within a few meters. In this context, beamforming is studied recently as the main technique to take advantage of MIMO in THz and overcome the very high path loss with the assumption that the THz communication channel is Line-of-Sight (LoS) and there are not significant multipath rays. On the other hand, recent studies also showed that the well-known absorbed energy by molecules can be reradiated immediately in the same frequency. Such re-radiated signal is correlated with the main signal and can provide rich scattering paths for the communication channel. This means that a significant MIMO multiplexing gain can be achieved even in a LoS scenario for the THz band. Our simulation results reveal a surprising observation that the MIMO multiplexing could be a better choice than the MIMO beamforming under certain conditions in THz communications. I. I NTRODUCTION To respond to the huge increasing demand for the wireless data traffic, recently the terahertz (THz) band (0.1-10 THz) is envisioned to make Tbps wireless link feasible [1]. In spite of the wide unused bandwidth in this spectrum, the high propagation loss is the main issue of using such spectrum. Thus, the potential applications of the THz link are limited to short range communications such as nanosensors [2], wireless on-chip communications and wireless personal area networks [3]. Moreover, part of the radio signal attenuation at the THz frequencies is due to molecular absorption which is frequency selective and increases the total loss to more than 200 dB for some frequencies at 10-meters distance. Basically, to overcome the very high path loss the transmit power could be largely increased. Unfortunately, this is not feasible with the current technology and it is limited to a few of mW [3]. Alternately, channel gain can be significantly improved by means of the multi-antenna beamforming technique. Indeed, Due to the very small footprint of a large number of antennas at the THz band, beamforming using very large scale Multiple Input Multiple Output (MIMO) systems has been considered in the field as a practical solution which can provide up to 55 dB channel gain at 1 THz [1]. However, beamforming comes at the cost of system complexity and signaling overhead where the transmitter should receive the channel state information continuously and align the beam to the receiver. On the other hand, to achieve a significant MIMO beamforming gain in high frequency spectrum the beam would become very narrow which is sometimes described as a pencil beam. This makes beamforming vulnerable to any transmitter/receiver mobility because it is difficult to perform beam re-alignment in a very short time interval. Another approach to take advantage of MIMO is the MIMO multiplexing technique. While the beamforming technique strives to focus the transmission energy and achieve a large channel gain in a specific direction, the multiplexing technique builds it strength on creating parallel information channels. However, the multiplexing gain is significant only when there are enough non-negligible multipath signal components in a rich scattering environment. Because of the huge path loss, THz communication is usually assumed to be applied in as a Line-of-Sight (LoS) dominant channel and thus, the research focus has been on beamforming rather than multiplexing. However, recent studies show that in the channel medium, molecules absorb and re-radiate the electromagnetic energy in THz band [3–6], which transforms the LoS channel into a rich-scattering environment. The re-radiation is usually considered as noise but the theorical model shows it is highly correlated to main signal [6]. In this paper, we will theoretically investigate the THz channel capacity for both cases of beamforming and multiplexing in a MIMO set-up. We find that the multiplexing technique can provide a considerable capacity gain in comparison with the beamforming technique on certain conditions. Also, in some other conditions where the beamforming yields a higher capacity, the multiplexing technique is still preferable choice due to its easier implementation. Note that in this work we assume a multiplexing technique using a blind precoding scheme without channel state information (CSI). In contrast, the beamforming technique always requires accurate CSI to smartly direct its energy in the spatial domain. The rest of the paper is structured as follows. In Section II, we present the molecular absorption model for the calculation of channel transfer function, Section III analyzes the MIMO channel model considering the molecular re-radiation, followed by simulation results in Section IV. Finally, we conclude the paper in Section V. II. C HANNEL MODEL AND MIMO CAPACITY The molecular absorption model defines how different species of molecules in a communication channel absorb energy from the electromagnetic signals and how they re-radiate them back to the environment. This section first explains the concept of absorption coefficient used to characterize the absorption capacity of a given molecule species, followed by the attenuation and re-radiation models that are built upon this coefficient. A. Molecular absorption coefficient The medium absorption coefficient, k(f ), at frequency f is a weighted sum of the molecular absorption coefficients in the medium [5], which can be formulated as k(f ) = N X mi ki (f ), (1) i=1 where ki (f ) is the molecular absorption coefficient of species Si on condition of temperature T and and pressure P. ki (f ) can be obtained from HITRAN [7]. In this work, to get the values of k(f ), we will use some predefined standard atmosphere conditions and their corresponding ratio of molecules in the air, which are tabulated in [7]. B. Attenuation of radio signal The attenuation of the radio signal at the THz frequencies is due to spreading and molecular absorption. In more detail, the spreading attenuation is given by C. Molecular re-radiation The existing molecules in communication medium will be excited by electromagnetic waves at specific frequencies. The excitement is temporary and the vibrational-rotational energy level of molecules will come back to a steady state and the absorbed energy will be re-radiated in the same frequency. These re-radiated waves are usually considered as noise in the literature [3]. Molecular absorption is not white and its power spectral density (PSD) is not flat because of the different resonant frequencies of various species of molecules. The PSD of the molecular absorption noise that affects the transmission B of a signal, SNabs , is contributed by the atmospheric noise SN X and the self-induced noise SN as addressed in [5]: B X SNabs (f, d) = SN (f, d) + SN (f, d), c 2 B , SN (f, d) = limd→∞ (kB T0 (1 − e−k(f )d )) √ 4πf c 2 X , SN (f, d) = Pt (f )(1 − e−k(f )d ) 4πf d (4) (5) (6) where k(f ) is the absorption coefficient of the medium at frequency f , T0 is the reference temperature (296K), kB is the Boltzmann constant, Pt (f ) is the power spectral density of the transmitted signal and c is the speed of light. The first term in (4), which is called sky noise and defined in (5) is independent of the signal wave. However, the self-induced noise in (6) is highly correlated with the signal wave [6], and can be considered as a distorted copy of the signal wave. Thus, equation (6) can be revised as the received power of the reradiated signal by molecules at the receiver by c 2 Pr,a (f, d) = Pt (f )(1 − e−k(f )d ) . (7) 4πf d Since the phase of the re-radiated wave depends on the phase of molecular vibration, which varies from molecules to molecules [8], the received power in this case is affected by a large number of phase-independent re-radiated photons. Thus, we assume a uniformly distributed random phase for the received signal, with its power given by (7). D. Channel Transfer Function (3) The channel transfer function for a single LoS channel is given by s 2 d c e−k(f )×d × ej2π λ h̃LoS (f, d) = 4πf d (8) d c j2π −k(f )× d 2 × e λ. = e 4πf d where k(f ) is the absorption coefficient of the medium at frequency f . Thus, the line-of-sight (LoS) received power at the receiver becomes 2 c Pr,LoS (f, d) = Pt (f ) × × e−k(f )×d . 4πf d Then, the partial channel transfer function resulted from the molecular absorption and excluding the LoS component can be represented by s c 2 × ej2πβrandom h̃a (f, d) = (1 − e−k(f )d ) 4πf d (9) c j2πβrandom −k(f )d 21 ×e . = (1 − e ) 4πf d Aspread (f, d) = 4πf d c 2 , (2) where c is the speed of light. The attenuation due to molecular absorption is characterized as Aabs (f, d) = ek(f )×d , Hence, the total channel transfer function is the superposition of the partial channel transfer functions, which is written as h̃(f, d) = h̃LoS (f, d) + h̃a (f, d), h̃(f, d) = c 4πf d (10) d d e−k(f )× 2 × ej2π λ c 1 × ej2πβrandom . (11) +(1 − e−k(f )d ) 2 4πf d E. MIMO channel model and capacity In this paper, we consider a MIMO system that is consisted of nt transmitting antennas and nr receiving ones. The received signal vector y at nr receiving antennas can be formulated as y = H̃x + n, (12) where x is the transmitted signal vector form nt transmitting antennas, and n is an nr ×1 vector with zero-mean independent noises with variance σ 2 . H̃ is the channel matrix where each of its elements, h̃ij , is a complex value denoting the transfer coefficient associated with the jth transmitter antenna and the ith receiver antenna. Note that h̃ij can be obtained from (11) for frequency f and distance dij . The capacity of MIMO channel can be written as C = log2 det(Inr + P H̃H̃† ), nt σ 2 (13) where P is total transmitting power, and I is the identity matrix Since the determinant of (Inr + ntPσ2 H̃H̃† ) can be computed by the product of the eigenvalues of the matrix H̃H̃† , the MIMO capacity can thus be written in the form of a product of non-zero eigenvalues as [9] C= κ X i=1 log2 (1 + P λ2i ), κσ 2 (14) where λi denotes singular values of the matrix H̃, and hence the squared singular values λ2i denotes the eigenvalues of the matrix H̃H̃† . Each of the λ2i characterize an equivalent P λ2 information channel where kσ2i is the corresponding signalto-noise ratio (SNR) of the channel at the receiver. Note that κ denotes the number of non-zero λ2i , which for beamforming technique it is equal to one and in multiplexing technique it could be the rank of H̃ with κ ≤ min(nr , nt ) [9]. However, because we use blind precoding and uniform power allocation for multiplexing technique κ = nt . Therefore, equation (14) is valid for uniform power allocation at the transmitter. P λ2 Furthermore, the equivalent channel SNR, κσ2i , should meet a minimum receiver threshold to be reliably detectable by the receiver. In this paper, we assumed 0 dB as the SNR threshold and uniform power allocation at the transmitter. The main difference between beamforming and multiplexing techniques is how to tune or exploit the eigenvalue distribution. In more details, beamforming technique aims to maximize λ1 to improve the channel SNR for a single data stream while in the multiplexing technique, a uniform eigenvalue distribution is preferable. In this way, multiplexing technique can utilize parallel data streams through MIMO and maximize the data rate. The complexity of beamforming comes from eigenvalues tuning because it means the channel state information (CSI) should be measured and sent back to the transmitter periodically for optimum precoding. This also results in a protocol overhead in the channel. On the other hand, multiplexing gain can take advantage of eigenvalue value distribution even with a blind precoding. This is more beneficial when there is a rich scattering environment in the channel. In next section, we will discuss how the re-radiation can provide a rich scattering environment. III. A NALYSIS ON THE CHANNEL WITH MOLECULAR ABSORPTION To analyze the MIMO channel capacity and characterize the scattering richness of channel quantitatively, lets decompose and normalize channel transfer function H̃ as H(f, d) = r K HLoS (f, d) + K +1 r 1 Ha (f, d), (15) K +1 where H, HLoS and Ha are normalized with corresponding channel gain. Because of uniformly distributed random phase of received re-radiated signal, elements of Ha are independent and identically distributed (i.i.d) complex Gaussian random variables with zero mean and unit magnitude variance. K is the ratio of powers of the LoS signal and the re-radiated components and if we assume the channel distance is much longer than antenna space, it can be obtained by K= e−k(f )d Pr,LoS (f, d) . = Pr,a (f, d) 1 − e−k(f )d (16) This is same as the well-known Rician channel model where the K is called Rician K-factor. Equivalently, K-factor shows how much channel is rich in term of scattering and multipath rays. Equation (16) shows K is a function of absorption coefficient of channel medium k(f ) and the distance between transmitter and receiver d so that a longer distance and a higher absorption result smaller K, as shown in Figure 1a. The capacity of MIMO channel considering Rician K-factor is studied in several works [10, 11]. Authors in [11] showed the lower bound of Rician channel expected capacity for large number of antennas is the expected capacity of channel considering only NLoS component, r 1 (17) E(C(H)) ≥ E(C( Ha )), K +1 p =⇒ E(C(H)) ≥ E(C( 1 − e−k(f )d Ha )), (18) where E(.) denotes the expectation. It is clear that the lower band is a increasing function of absorption coefficient, ∀f1 , f2 such that k(f2 ) ≥ k(f1 ), Emin (C(f2 )) ≥ Emin (C(f1 )). (a) K-factor (b) MIMO capacity using beamforming (c) MIMO capacity using multiplexing Fig. 1: K-factor is an increasing function of distance and absorption coefficient. For both multiplexing and beamforming techniques, the performance gain is affected by K-factor. The capacity is calculated for 225x225 MIMO system. IV. S IMULATION AND DISCUSSION A. Simulation set-up In this section, to evaluate the molecular absorption impact on THz MIMO capacity, we consider a simple n × n MIMO system with a square uniform Arrays, where at both transmitter and receiver, the inter-element spacing s is equal to half of the wavelength and the channel distance is d. Moreover, we consider uniform power allocation to transmitter arrays operating in an open-space LoS scenario. The default values of the parameters are listed in Table I, and different values will be explained when necessary. Since we apply random phases on NLoS components created by molecular re-radiation, we conduct the evaluation of the MIMO capacity with molecular re-radiation for 1000 times and show the average result. We use the online browsing and plotting tools1 , which is based on HITRAN databases [7] to generate absorption coefficients for different single gas or some predefined standard gas mixture of the atmosphere at sea level, as shown in Table II. Since the water molecules play main roles in a normal air environment at THz bands we use the highest and lowest water ratio in Table II, i.e., the ”USA model, high latitude, winter” and ”USA model, tropics”. The corresponding absorption coefficients in THz bands have been shown in Figure 2a for an ambient temperature of 273 K and a sea level pressure of 1 atm. For a tropic atmosphere, the water ratio is higher than that of the winter atmosphere, and thus we can see a significant increase in the absorption coefficient among these two gas mixtures. In our simulation, we assume a constant transmit power over the entire frequency spectrum and display the MIMO capacity in bps/Hz for THz bands. We consider a MIMO set-up with 225 antennas at each side in a uniform square planar array. Our aim is to compare the beamforming and multiplexing 1 http://hitran.iao.ru/gasmixture/simlaunch TABLE I: Simulation parameters Transmitter and receiver distance (d) Inter-element spacing (s) Transmitter arrays angle (φ) Receiver arrays angle (θ) Number of arrays on each side (n) Transmit power Noise power 0.1, 1, 10 m 0.5λ (wave length) 90◦ 90◦ 225 0, 10 dBm −80 dBm techniques in different channel conditions. First, we calculate the channel capacity for beamforming while the re-radiation is totally ignored in the channel. Next, the beamforming capacity is re-calculated when the re-radiation is taken into account. Finally, the multiplexing gain is calculated with and without the consideration of re-radiation. In all scenarios, capacity is obtained by 14. In the first step, the simulation is run at 500 GHz with the practical range of absorption coefficient (10−5 ∼ 10+3 ) over the THz spectrum, as shown in Figure 1. It should be noted that the actual value of absorption coefficient at 500 GHz is shown in Figure 2a. The beamforming and multiplexing techniques capacity is calculated for a range of 0.1∼10 m distance and a 1 mW transmit power. Secondly, the channel is simulated for two different transmit power and three distances with realistic absorption coefficients. Our assumption on the transmit power is based on current technology [3] and a previous work on THz massive MIMO [1]. Furthermore, distances have been chosen to cover various application scenarios. For example, THz nanosensors are considered to communicate in a very short distance in the order of 0.1-10 cm or less, while THz communications are also nominated to provide terabit per second ultra high video communication link at around 1 m distance for home entrainment devices like TV or virtual reality (VR). In addition, longer distances to a few meters characterize wireless personal or local networks. Simulation results are presented in Figure 2. B. The MIMO Capacity vs. the K-factor Figure 1 illustrates how the channel is transformed from a LoS dominant channel to a Rayleigh channel and how it effects on the MIMO beamforming and multiplexing capacity gain. As can be seen in Figure 1b, the beamforming gain is decreasing when the absorption coefficient increases which is because in the very high absorption, the channel is not LoS dominant anymore and there is significant NLoS signal component generated by molecule re-radiation or equivalently lower K-factor. In contrast, Figure 1c shows the multiplexing technique takes advantage of higher absorption to reach a huge data rate. However, the low SNR limit the multiplexing gain in longer distances so that it drops sharply to zero beyond 2 m. In Figure 2, more results for the THz spectrum with realistic absorption coefficients will be presented. TABLE II: Atmosphere standard gas mixture ratio in percentage for different climates [7] USA USA USA USA USA model, model, model, model, model, mean latitude, summer, H=0 mean latitude, winter, H=0 high latitude, summer, H=0 high latitude, winter, H=0 tropics, H=0 H2O: H2O: H2O: H2O: H2O: 1.860000 0.432000 1.190000 0.141000 2.590000 CO2: CO2: CO2: CO2: CO2: 0.033000 0.033000 0.033000 0.033000 0.033000 O3: O3: O3: O3: O3: 0.000003 0.000003 0.000002 0.000002 0.000003 C. The MIMO Capacity vs. the Transmit Power and Distance The channel attenuation including molecular attenuation in (3) and spreading attenuation in (2) is illustrated in Figure 2b. While the spreading attenuation is increasing linearly in dB with distance and frequency, the molecular attenuation is also increasing with distance but is frequency selective. For example, while the total loss at 10m is 107 dB for 500 GHz, the total attenuation at 550 GHz is 86 dB at 1 m and it grows to 220 dB at 10 m which is mostly because of very high absorption of water molecules in the channel medium at this frequency. Note that the channel atmosphere for this case is from tropic data where the ratio of water molecules in the air is more than 0.02, as shown in Table II. Figure 2c and 2d illustrate the capacity of the investigated transmission techniques for a 10 cm distance. The transmit power is increased from 1 mW in Figure 2c to 10 mW in Figure 2d. It can be seen that a huge performance difference exists between multiplexing and beamforming, thanks to the tremendous multiplexing gain provided by the rich scattering environment due to molecule re-radiation. Furthermore, in very high absorption frequencies which existing studies consider as infeasible windows for THz communications, a significant capacity improvement can be observed. This is because more absorption leads to more re-radiation, which transforms a LoS dominant channel to Rayleigh channel. The details can be found in Section III, where we have discussed about how the re-radiation decreases the K-factor and creates a rich scattering environment. To sum up, the re-radiation improves the multiplexing gain which is fundamentally supported by a better eigenvalue distribution and channel matrix rank in mathematical analysis. In Figure 2e and 2f, the distance is increased to 1 m. With a relatively large distance for THz communications, it can be seen the beamforming gain is comparable with the multiplexing gain. However, we can see the multiplexing gain in high absorption windows, such as 540-580 GHz, is significantly higher than the rest of spectrum for a 10 mW transmit power. It is a different story for a 1 mW transmit power where the capacity drops to zero in high P λ2i kσ2 absorption windows because the equivalent SNR, ( ), of most parallel channels created by the multiplexing technique is less than 0 dB and practically such parallel channels are useless because the receiver can not reliably detect the received signals. Such results are not surprising since it has been shown in several works on conventional communication band [12] that the multiplexing performance drops dramatically in low SNR. However, considering the implementation challenges of beamforming, the multiplexing technique might still be a preferable choice for frequency up to 1 THz. For example, it can be observed in Figure 2e at 0.9 THz, the capacity is 4 and N2O: N2O: N2O: N2O: N2O: 0.000032 0.000032 0.000031 0.000032 0.000032 CO: CO: CO: CO: CO: 0.000015 0.000015 0.000015 0.000015 0.000015 CH4: CH4: CH4: CH4: CH4: 0.000170 0.000170 0.000170 0.000170 0.000170 O2: O2: O2: O2: O2: 20.900001 20.900001 20.900001 20.900001 20.900001 N2: N2: N2: N2: N2: 77.206000 78.634779 77.876781 78.925780 76.476779 11.7 bps/Hz for the multiplexing and beamforming techniques, respectively. Finally, Figures 2g and 2h present the results for a 10 m distance. For such a distance, path loss leads to a very low reception SNR and thus the beamforming performance is significantly better than the multiplexing performance. It is wellknown that beamforming technique is not very effective where there are strong multipath rays [12]. Thus, it is observed that in very high absorption frequency windows, the beamforming performance drops sharply. It is not only because of receiving strong NLoS rays caused by molecule re-radiation but also due to LoS signal attenuation. Note that the multiplexing technique can take advantage of same windows in high SNR as we discussed above for Figure 2f. V. C ONCLUSION In this paper, we compared the beam forming and multiplexing techniques of MIMO in the terahertz band. We showed in high SNR, high transmit power or lower distance, the multiplexing technique can provide a considerable capacity gain compared with beamforming. However, for beyond a few meters such as 10 meters, there should be enough transmitting power possibility to use multiplexing technique, otherwise the capacity drops to zero where the beamforming technique can still provide effective spectrum efficiency at the cost of complexity and protocol overhead. Our theoretical model also showed re-radiation of molecules in the THz band can be helpful for massive MIMO system to improve the channel performance using multiplexing technique. The re-radiation can provide significantly strong multipath components to achieve a full spatial multiplexing gain where the receiver is in an enough SNR coverage. It means some very high absorption frequency windows which have been formerly pointed as not feasible for communication might be more preferable choices for MIMO in some certain applications. R EFERENCES [1] I. F. Akyildiz and J. M. Jornet, “Realizing ultra-massive mimo (10241024) communication in the (0.0610) terahertz band,” Nano Communication Networks, vol. 8, pp. 46 – 54, 2016. [2] E. Zarepour, M. Hassan, C. T. Chou, and A. A. Adesina, “Semon: Sensorless event monitoring in self-powered wireless nanosensor networks,” ACM Transactions on Sensor Networks (TOSN), vol. 13, no. 2, p. 15, 2017. [3] I. F. Akyildiz, J. M. Jornet, and C. Han, “Terahertz band: Next frontier for wireless communications,” Physical Communication, vol. 12, pp. 16 – 32, 2014. [4] J. Kokkoniemi, J. Lehtomäki, and M. Juntti, “A discussion on molecular absorption noise in the terahertz band,” Nano Communication Networks, 2015. [5] J. M. Jornet and I. F. Akyildiz, “Femtosecond-Long Pulse-Based Modulation for Terahertz Band Communication in Nanonetworks,” IEEE Transactions on Communications, vol. 62, no. 5, pp. 1742–1754, May 2014. 10 4 US Model tropic - High H 2 O Absorption Coefficient (m -1 ) US Model High latitude winter - low H 2 O 10 2 10 0 10 -2 10 -4 10 -6 10 2 10 3 10 4 Frequency (GHz) (a) absorption coefficient, T= 273 K, P= 1 atm (b) Signal Attenuation 1200 600 Beamforming w re-radiation Beamforming w/o re-radiation Multiplexing - w re-radiation Multiplexing - w/o re-radiation Beamforming w re-radiation Beamforming w/o re-radiation Multiplexing - w re-radiation Multiplexing - w/o re-radiation 1000 400 800 Capacity (bps/Hz) Capacity (bps/Hz) 500 300 200 600 400 100 200 0 10 2 10 3 0 10 2 10 4 Frequency (GHz) 10 3 10 4 Frequency (GHz) (c) distance=0.1m, transmit power=1mW (d) distance=0.1m, transmit power=10mW 20 250 Beamforming w re-radiation Beamforming w/o re-radiation Multiplexing - w re-radiation Multiplexing - w/o re-radiation 18 16 Beamforming w re-radiation Beamforming w/o re-radiation Multiplexing - w re-radiation Multiplexing - w/o re-radiation 200 Capacity (bps/Hz) Capacity (bps/Hz) 14 12 10 8 150 100 6 4 50 2 0 10 2 10 3 0 10 2 10 4 Frequency (GHz) (e) distance=1m, transmit power=1mW 10 4 (f) distance=1m, transmit power=10mW 12 15 Beamforming w re-radiation Beamforming w/o re-radiation Multiplexing - w re-radiation Multiplexing - w/o re-radiation 10 Beamforming w re-radiation Beamforming w/o re-radiation Multiplexing - w re-radiation Multiplexing - w/o re-radiation 8 Capacity (bps/Hz) Capacity (bps/Hz) 10 3 Frequency (GHz) 6 4 10 5 2 0 10 2 10 3 10 4 Frequency (GHz) (g) distance=10m, transmit power=1mW 0 10 2 10 3 10 4 Frequency (GHz) (h) distance=10m, transmit power=10mW Fig. 2: 225x225 MIMO channel performance in tropic atmosphere. [6] J. M. Jornet Montana, “Fundamentals of electromagnetic nanonetworks in the terahertz band,” Ph.D. dissertation, Georgia Institute of Technology, 2013. [7] L. Rothman, I. Gordon, Y. Babikov, A. Barbe et al., “The hitran2012 molecular spectroscopic database,” Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 130, pp. 4 – 50, 2013. [8] L. D. Barron, Molecular light scattering and optical activity. Cambridge University Press, 2004. [9] D. Tse and P. Viswanath, “Chapter 07: MIMO I : spatial multiplexing and channel modeling,” Fundamentals of Wireless Communication, pp. 290–331, 2005. [10] F. R. Farrokhi, G. J. Foschini, A. Lozano, and R. A. Valenzuela, “Link-optimal space-time processing with multiple transmit and receive antennas,” IEEE Communications Letters, vol. 5, no. 3, pp. 85–87, 2001. [11] G. Lebrun, M. Faulkner, M. Shafi, and P. J. Smith, “Mimo ricean channel capacity: an asymptotic analysis,” IEEE Transactions on Wireless Communications, vol. 5, no. 6, pp. 1343–1350, 2006. [12] D. Gesbert, M. Shafi, D.-s. Shiu, P. J. Smith et al., “From theory to practice: An overview of mimo space-time coded wireless systems,” IEEE Journal on selected areas in Communications, vol. 21, no. 3, pp. 281–302, 2003.	7
An Emptiness Algorithm for Regular Types with Set Operators arXiv:cs/9811015v1 [cs.LO] 11 Nov 1998 Lunjin Lu and John G. Cleary Department of Computer Science University of Waikato Hamilton, New Zealand Phone: +64-838-4627/4378 {lunjin,jcleary}@cs.waikato.ac.nz Abstract. An algorithm to decide the emptiness of a regular type expression with set operators given a set of parameterised type definitions is presented. The algorithm can also be used to decide the equivalence of two regular type expressions and the inclusion of one regular type expression in another. The algorithm strictly generalises previous work in that tuple distributivity is not assumed and set operators are permitted in type expressions. Keywords: type, emptiness, prescriptive type 1 Introduction Types play an important role in programming languages [6]. They make programs easier to understand and help detect errors. Types have been introduced into logic programming in the forms of type checking and inference [5,9,12,26,32] or type analysis [25,33,17,19,13,22,7,23] or typed languages [16,21,28,31]. Recent logic programming systems allow the programmer to declare types for predicates and type errors are then detected either at compile time or at run time. The reader is referred to [27] for more details on types in logic programming. A type is a possibly infinite set of ground terms with a finite representation. An integral part of any type system is its type language that specifies which sets of ground terms are types. To be useful, types should be closed under intersection, union and complement operations. The decision problems such as the emptiness of a type, inclusion of a type in another and equivalence of two types should be decidable. Regular term languages [14,8], called regular types, satisfy these conditions and have been used widely used as types [29,25,33,9,17,21,28,31,12,32,19,13,22,7,23]. Most type systems use tuple distributive regular types which are strictly less powerful than regular types [29,25,33,17,21,28,31,12,32,19,13,22,7,23]. Tuple distributive regular types are regular types closed under tuple distributive closure. Intuitively, the tuple distributive closure of a set of terms is the set of all terms constructed recursively by permuting each argument position among all terms that have the same function symbol [32]. This paper gives an algorithm to decide if a type expression denotes an empty set of terms. The correctness of the algorithm is proved and its complexity is analysed. The algorithm works on prescriptive types [28]. By prescriptive types, we mean that the meaning of a type is determined by a given set of type definitions. We allow parametric and overloading polymorphism in type definitions. Prescriptive types are useful both in compilers and other program manipulation tools such as debuggers because they are easy to understand for programmers. Type expressions may contain set operators with their usual interpretations. Thus, the algorithm can be used to decide the equivalence of two type expressions and the inclusion of one type expression in another. The introduction of set operators into type expressions allows concise and intuitive representation of regular types. Though using regular term languages as types allow us to make use of theoretical results in the field of tree automata [14], algorithms for testing the emptiness of tree automata cannot be applied directly as type definitions may be parameterised. For instance, in order to decide the emptiness of a type expression given a set of type definitions, it would be necessary to construct a tree automaton from the type expression and the set of type definitions before an algorithm for determining the emptiness of an tree automaton can be used. When type definitions are parameterised, this would make it necessary to construct a different automaton each time the emptiness of a type expression is tested. Thus, an algorithm that works directly with type definitions is desirable as it avoids this repeated construction of automata. Attempts have been made in the past to find algorithms for regular types [25,12,32,33,31,10,9]. To our knowledge, Dart and Zobel’s work [10] is the only one to present decision algorithms for emptiness and inclusion problems for prescriptive regular types without the tuple distributive restriction. Unfortunately, their decision algorithm for the inclusion problem is incorrect for regular types in general. See [24] for a counterexample. Moreover, the type language of Dart and Zobel is less expressive than that considered in this paper since it doesn’t allow set operators and parameterised type definitions. Set constraint solving has also been used in type checking and type inference [3,2,20,18,11]. However, set constraint solving methods are intended to infer descriptive types [28] rather than for testing emptiness of prescriptive types [28]. Therefore, they are useful in different settings from the algorithm presented in this paper. Moreover, algorithms proposed for set constraint solving [3,4,2,1] are not applicable to the emptiness problem we considered in this paper as they don’t take type definitions into account. The remainder of this paper is organised as follows. Section 2 describes our language of type expressions and type definitions. Section 3 presents our algorithm for testing if a type expression denotes an empty set of terms. Section 4 addresses the of the algorithm. Section 5 presents the complexity of the algorithm and section 6 concludes the paper. Some lemmas are presented in the appendix. 2 Type Language Let Σ be a fixed ranked alphabet. Each symbol in Σ is called a function symbol and has a fixed arity. It is assumed that Σ contains at least one constant that is a function symbol of arity 0. The arity of a symbol f is denoted as arity(f ). Σ may be considered as the set of function symbols in a program. Let T (Φ) be the set of all terms over Φ. T (Σ) is the set of all possible values that a program variable can take. We shall use regular term languages over Σ as types. A type is represented by a ground term constructed from another ranked alphabet Π and {⊓, ⊔, ∼, 1, 0}, called type constructors. It is assumed that (Π ∪ {⊓, ⊔, ∼, 1, 0}) ∩ Σ = ∅. Thus, a type expression is a term in T (Π ∪ {⊓, ⊔, ∼, 1, 0}). The denotations of type constructors in Π are determined by type definitions whilst ⊓, ⊔, ∼, 1 and 0 have fixed denotations that will be given soon. Several equivalent formalisms such as tree automata [14,8], regular term grammars [14,10,8] and regular unary logic programs [32] have been used to define regular types. We define types by type rules. A type rule is a production rule of the form c(ζ1 , · · · , ζm ) → τ where c ∈ Π, ζ1 , · · · , ζm are different type parameters and τ ∈ T (Σ ∪ Π ∪ Ξm ) where Ξm = {ζ1 , · · · , ζm }. The restriction that every type parameter in the righthand side of a type rule must occur in the lefthand side of the type rule is often referred to as type preserving [30] and has been used in all the type definition formalisms. Note that overloading of function symbols is permitted as a function symbol can appear in the righthand sides of many type rules. We def S denote by ∆ the set of all type rules and define Ξ = c∈Π Ξarity(c) . hΠ, Σ, ∆i is a restricted form of context-free term grammar. Example 1. Let Σ = {0, s(), nil, cons(, )} and Π = {Nat, Even, List()}. ∆ defines natural numbers, even numbers, and lists where    Nat   → 0 \| s(Nat),  ∆ = Even → 0 \| s(s(Even)),    List(ζ) → nil \| cons(ζ, List(ζ))  where, for instance, Nat → 0 \| s(Nat) is an abbreviation of two rules Nat → 0 and Nat → s(Nat). ∆ is called simplified if τ in each production rule c(ζ1 , · · · , ζm ) → τ is of the form f (τ1 , · · · , τn ) such that each τj , for 1 ≤ j ≤ n, is either in Ξm or of the form d(ζ1′ , · · · , ζk′ ) and ζ1′ , · · · , ζk′ ∈ Ξm . We shall assume that ∆ is simplified. There is no loss of generality to use a simplified set of type rules since every set of type rules can be simplified by introducing new type constructors and rewriting and adding type rules in the spirit of [10]. Example 2. The following is the simplified version of the set of type rules in example 1. Σ = {0, s(), nil, cons(, )}, Π = {Nat, Even, Odd, List()} and ∆= ( Nat → 0 \| s(Nat), Even → 0 \| s(Odd), Odd → s(Even), List(ζ) → nil \| cons(ζ, List(ζ)) ) A type valuation φ is a mapping from Ξ to T (Π ∪{⊓, ⊔, ∼, 1, 0}). The instance φ(R) of a production rule R under φ is obtained by replacing each occurrence of each type parameter ζ in R with φ(ζ). E.g., List(Nat⊓(∼Even)) → cons(Nat⊓(∼Even), List(Nat⊓(∼Even))) is the instance of List(ζ) → cons(ζ, List(ζ)) under a type valuation that maps ζ to Nat⊓(∼Even). Let def ground(∆) = {φ(R) \| R ∈ ∆ ∧ φ ∈ (Ξ 7→ T (Π ∪ {⊓, ⊔, ∼, 1, 0}))} ∪ {1 7→ f (1, · · · , 1) \| f ∈ Σ} ground(∆) is the set of all ground instances of grammar rules in ∆ plus rules of the form 1 → f (1, · · · , 1) for every f ∈ Σ. Given a set ∆ of type definitions, the type denoted by a type expression is determined by the following meaning function. def [1]]∆ = T (Σ) def [0]]∆ = ∅ def [E1 ⊓E2]∆ = [E1]∆ ∩ [E2]∆ def [E1 ⊔E2]∆ = [E1]∆ ∪ [E2]∆ def [∼E]]∆ = T (Σ) − [E]]∆ def [ω]]∆ = [ {f (t1 , · · · , tn ) \| ∀1 ≤ i ≤ n. ti ∈ [Ei]∆ } (ω→f (E1 ,···,En ))∈ground(∆) [·]]∆ gives fixed denotations to ⊓, ⊔, ∼, 1 and 0. ⊓, ⊔ and ∼ are interpreted by [·]]∆ as set intersection, set union and set complement with respect to T (Σ). 1 denotes T (Σ) and 0 the empty set. Example 3. Let ∆ be that in example 2. We have [Nat]]∆ = {0, s(0), s(s(0)), · · ·} [Even]]∆ = {0, s(s(0)), s(s(s(s(0)))), · · ·} [Nat⊓∼Even]]∆ = {s(0), s(s(s(0))), s(s(s(s(s(0))))), · · ·} [List(Nat⊓∼Even)]]∆ = {cons(s(0), nil), cons(s(s(s(0))), nil), · · ·} The lemma 5 in the appendix states that every type expression denotes a regular term language, that is, a regular type. We extend [·]]∆ to sequences θ of type expressions as follows. def [ǫ]]∆ = {ǫ} def [hEi • θ′]∆ = [E]]∆ × [θ′]∆ where ǫ is the empty sequence, • is the infix sequence concatenation operator, hEi is the sequence consisting of the type expression E and × is the Cartesian product operator. As a sequence of type expressions, ǫ can be thought of consisting of zero instance of 1. We use Λ to denote the sequence consisting of zero instance of 0 and define [Λ]]∆ = ∅. We shall call a sequence of type expressions simply a sequence. A sequence expression is an expression consisting of sequences of the same length and ⊓, ⊔ and ∼. The length of the sequences in a sequence expression θ is called the dimension of θ and is denoted by kθk. Let θ, θ1 and θ2 be sequence expressions of the same length. def [θ1 ⊓θ2]∆ = [θ1]∆ ∩ [θ2]∆ def [θ1 ⊔θ2]∆ = [θ1]∆ ∪ [θ2]∆ def [∼θ]]∆ = T (Σ) × · · · × T (Σ) −[[θ]]∆ \| {z kθk times } A conjunctive sequence expression is a sequence expression of the form γ1 ∧ · · · ∧ γm where γi for, 1 ≤ i ≤ m, are sequences. 3 Emptiness Algorithm This section presents an algorithm that decides if a type expression denotes the empty set with respect to a given set of type definitions. The algorithm can also be used to decide if (the denotation of) one type expression is included in (the denotation of) another because E1 is included in E2 iff E1 ⊓∼E2 is empty. We first introduce some terminology and notations. A type atom is a type expression of which the principal type constructor is not a set operator. A type literal is either a type atom or the complement of a type atom. A conjunctive type expression C is of the form ⊓i∈I li with li being a type literal. Let α be a type atom. F (α) defined below is the set of the principal function symbols of the terms in [α]]∆ . def F (α) = {f ∈ Σ \| ∃ζ1 · · · ζk .((α → f (ζ1, · · · , ζk )) ∈ ground(∆))} Let f ∈ Σ. Define def Afα = {hα1 , · · · , αk i \| (α → f (α1 , · · · , αk )) ∈ ground(∆)} We have [Afα]∆ = {ht1 , · · · , tk i \| f (t1 , · · · , tk ) ∈ [α]]∆ }. Both F (α) and Afα are finite even though ground(∆)) is usually not finite. The algorithm repeatedly reduces the emptiness problem of a type expression to the emptiness problems of sequence expressions and then reduces the emptiness problem of a sequence expression to the emptiness problems of type expressions. Tabulation is used to break down any possible loop and to ensure termination. Let O be a type def expression or a sequence expression. Define empty(O) = ([[O]]∆ = ∅). 3.1 Two Reduction Rules We shall first sketch the two reduction rules and then add tabulation to form an algorithm. Initially the algorithm is to decide the validity of a formula of the form empty(E) (1) where E is a type expression. The first reduction rule rewrites a formula of the form (1) into a conjunction of formulae of the following form. Reduction Rule One. empty(σ) (2) where σ is a sequence expression where ∼ is applied to type expressions but not to any sequence expression. It is obvious that a type expression has a unique (modulo equivalence of denotation) disjunctive normal form. Let DNF(E) be the disjunctive normal form of E. empty(E) can written into ∧C∈DNF(E) empty(C). Each C is a conjunctive type expression. We assume that C contains at least one positive type literal. This doesn’t cause any loss of generality as [1⊓C]]∆ = [C]]∆ for any conjunctive type expression C. We also assume that C doesn’t contain repeated occurrences of the same type literal. Let C = ⊓1≤i≤m ωi ⊓ ⊓1≤j≤n ∼τj where ωi and τj are type atoms. def The set of positive type literals in C is denoted as pos(C) = {ωi \| 1 ≤ i ≤ m} while the set of complemented type atoms are denoted as def neg(C) = {τj \| 1 ≤ j ≤ n}. lit(C) denotes the set of literals occurring in C. By lemma 3 in the appendix, empty(C) is equivalent to ∀f ∈ ∩α∈pos(C) F (α). (3) empty((⊓ω∈pos(C) (⊔Afω ))⊓(⊓τ ∈neg(C) ∼(⊔Afτ ))) The intuition behind the equivalence is as follows. [C]]∆ is empty iff, for every function symbol f , the set of the sequences ht1 , · · · , tk i of terms such that f (t1 , · · · , tk ) ∈ [C]]∆ is empty. Only the function symbols in ∩α∈pos(C) F (α) need to be considered. We note the following two special cases of the formula (3). (a) If ∩α∈pos(C) F (α) = ∅ then the formula (3) is true because ∧∅ = true. In particular, F (0) = ∅. Thus, if 0 ∈ pos(C) then ∩α∈pos(C) F (α) = ∅ and hence the formula (3) is true. (b) If Afτ = ∅ for some τ ∈ neg(C) then ⊔Afτ = h0, · · · , 0i and ∼(⊔Afτ ) = h1, · · · , 1i. Thus, τ has no effect on the subformula for f when Afτ = ∅. In order to get rid of complement operators over sequence subexpressions, the complement operator in ∼(⊔Afτ ) is pushed inwards by the function push defined in the following. def push(∼(⊔i∈I γi )) = ⊓i∈I push(∼γi ) def push(∼hE1 , E2 , · · · , Ek i) = ⊔1≤l≤k h 1, · · · , 1, ∼El , 1, · · · , 1 i def push(∼ǫ) = Λ \| {z l−1 } \| {z k−l for k ≥ 1 } It follows from De Morgan’s law and the definition of [·]]∆ that [push(∼(⊔Afτ ))]]∆ = [∼(⊔Afτ )]]∆ . Substituting push(∼(⊔Afτ )) for ∼(⊔Afτ ) in the formula (3) gives rise to a formula of the form (2). The second reduction rule rewrites a formula of the form 2 to a conjunction of disjunctions of formulae of the form 1. Formula 2 is written into a disjunction of formulae of the form. empty(Γ ) Reduction Rule Two. where Γ be a conjunctive sequence expression. In the case kΓ k = 0, by lemma 4 in the appendix, empty(Γ ) can be decided without further reduction. If Λ ∈ Γ then empty(Γ ) is true because [Λ]]∆ = ∅. Otherwise, empty(Γ ) is false because [Γ ]∆ = {ǫ}. In the case kΓ k = 6 0, empty(Γ ) is equivalent to ∨1≤j≤kΓ k empty(Γ↓j) def where, letting Γ = γ1 ⊓ · · · ⊓γk , Γ↓j = ⊓1≤i≤k γij with γij being the j th component of γi . Note that Γ↓j is a type expression and empty(Γ↓j) is of the form 1. 3.2 Algorithm The two reduction rules in the previous section form the core of the algorithm. However, they alone cannot be used as an algorithm as a formula empty(E) may reduce to a formula containing empty(E) as a sub-formula, leading to nontermination. Suppose Σ = {f (), a}, Π = {Null} and ∆ = {Null → f (Null)}. Clearly, empty(Null) is true. However, by the first reduction rule, empty(Null) reduces to empty(hNulli) which then reduces to empty(Null) by the second reduction rule. This process will not terminate. The solution, inspired by [10], is to remember in a table a particular kind of formulae of which truth is being tested. When a formula of that kind is tested, the table is first looked up. If the formula is implied by any formula in the table, then it is determined as true. Otherwise, the formula is added into the table and then reduced by a reduction rule. The emptiness algorithm presented below remembers every conjunctive type expression of which emptiness is being tested. Thus the table is a set of conjunctive type expressions. Let C1 and C2 be def conjunctive type expressions. We define (C1 C2 ) = (lit(C1 ) ⊇ lit(C2 )). Since Ci = ⊓l∈lit(Ci ) l, C1 C2 implies [C1]∆ ⊆ [C2]∆ and hence (C1 C2 ) ∧ empty(C2 ) implies empty(C1 ). Adding tabulation to the two reduction rules, we obtain the following algorithm for testing the emptiness of prescriptive regular types. Let BCf = (⊓ω∈pos(C) (⊔Afω ))⊓(⊓τ ∈neg(C) push(∼(⊔Afτ ))). def etype(E) = etype(E, ∅) def etype(E, Ψ ) = ∀C ∈ DNF(E).etype conj(C, Ψ ) (4) (5) def etype conj(C, Ψ ) =  if pos(C) ∩ neg(C) 6= ∅,  true, true, if ∃C ′ ∈ Ψ.C C ′ ,  ∀f ∈ ∩ f otherwise. α∈pos(C) F(α).eseq(BC , Ψ ∪ {C}), def eseq(Θ, Ψ ) = ∀Γ ∈ DNF(Θ).eseq conj(Γ, Ψ ) def eseq conj(Γ, Ψ ) = ( true if kΓ k = 0 ∧ Λ ∈ Γ , false if kΓ k = 0 ∧ Λ 6∈ Γ , ∃1 ≤ j ≤ kΓ k.etype(Γ↓j, Ψ ) if kΓ k = 6 0. (6) (7) (8) Equation 4 initialises the table to the empty set. Equations 5 and 6 implement the first reduction rule while equations 7 and 8 implement the second reduction rule. etype(, ) and etype conj(, ) test the emptiness of an arbitrary type expression and that of a conjunctive type expression respectively. eseq(, ) tests emptiness of a sequence expression consisting of sequences and ⊓ and ⊔ operators while eseq conj(, ) tests the emptiness of a conjunctive sequence expression. The expression of which emptiness is to be tested is passed as the first argument to these functions. The table is passed as the second argument. It is used in etype conj(, ) to detect a conjunctive type expression of which emptiness is implied by the emptiness of a tabled conjunctive type expression. As we shall show later, this ensures the termination of the algorithm. Each of the four binary functions returns true iff the emptiness of the first argument is implied by the second argument and the set of type definitions. Tabling any other kind of expressions such as arbitrary type expressions can also ensure termination. However, tabling conjunctive type expressions makes it easier to detect the implication of the emptiness of one expression by that of another because lit(C) can be easily computed given a conjunctive type expression C. In an implementation, a conjunctive type expression C in the table can be represented as lit(C). The first two definitions for etype conj(C, Ψ ) in equation 6 terminates the algorithm when the emptiness of C can be decided by C and Ψ without using type definitions. The first definition also excludes from the table any conjunctive type expression that contains both a type atom and its complement. 3.3 Examples We now illustrate the algorithm with some examples. Example 4. Let type definitions be given as in example 2. The tree in figure 1 depicts the evaluation of etype(Nat⊓∼Even⊓∼Odd) by the algorithm. Nodes are labeled with function calls. We will identity a node with its label. Arcs from a node to its children are labeled with the number of the equation that is used to evaluate the node. Abbreviations used in the labels are defined in the legend to the right of the tree. Though [A]]∆ = [B]]∆ , A and B are syntactically different type expressions. The evaluation returns true, verifying [Nat⊓∼Even⊓∼Odd]]∆ = ∅. Consider etype conj(B, {A}). We have B A as lit(A) = lit(B). Thus, by equation 6, etype conj(B, {A}) = true. etype(A) 3 etype(A, ∅) 4 etype conj(A, ∅) ∧ 5 5 eseq(C, {A}) eseq(ǫ⊓Λ, {A}) 6 6 eseq conj(ǫ⊓Λ, {A}) eseq conj(C, {A}) 7 7 true etype(B, {A}) 4 etype conj(B, {A}) 5 true Legend: A = N at⊓∼Even⊓∼Odd B = N at⊓∼Odd⊓∼Even C = hN ati⊓h∼Oddi⊓h∼Eveni Fig. 1. Evaluation of etype(N at⊓∼Even⊓∼Odd)) Example 5. Let type definitions be given as in example 2. The tree in figure 2 depicts the evaluation of etype(List(Even⊓∼Nat)) by the algorithm. The evaluation returns false, verifying [List(Even⊓∼Nat)]]∆ 6= ∅. Indeed, [List(Even⊓∼Nat)]]∆ = {nil}. The rightmost node is not evaluated as its sibling returns false, which is enough to establish the falsity of their parent node. etype(A) (3) etype(A, ∅) (4) etype conj(A, ∅) ∧ (5)/nil (5)/cons(,) eseq(ǫ, {A}) eseq(hB, Ai, {A}) (6) eseq conj(ǫ, {A}) (7) false Legend: A = List(Even⊓∼N at) B = Even⊓∼N at Fig. 2. Evaluation of etype(List(Even⊓∼N at)) Example 6. The following is a simplified version of the type definitions that is used in [24] to show the incorrectness of the algorithm by Dart and Zobel for testing inclusion of one regular type in another [10]. Let Π = {α, β, θ, σ, ω, ζ, η}, Σ = {a, b, g(), h(, )} and ∆= ( α → g(ω), β → g(θ) \| g(σ), θ → a \| h(θ, ζ), σ → b \| h(σ, η), ω → a \| b \| h(ω, ζ) \| h(ω, η), ζ → a, η→b ) Let t = g(h(h(a, b), a)). t ∈ [α]]∆ and t 6∈ [β]]∆ , see example 3 in [24] for more details. So, [α]]∆ 6⊆ [β]]∆ . This is verified by our algorithm as follows. Let Ψ1 = {α⊓∼β} and Ψ2 = Ψ1 ∪ {ω⊓∼θ⊓∼σ}. By applying equations 4, 5, 6, 7, 8 and 5 in that order, we have etype(α⊓∼β) = etype conj(ω⊓∼θ⊓∼σ, Ψ1 ). By equation 6, we have etype(α⊓∼β) = eseq(ǫ⊓Λ⊓ǫ, Ψ2 ) ∧ eseq(ǫ⊓ǫ⊓Λ, Ψ2 ) ∧ eseq(Θ, Ψ2) where Θ = (hω, ζi⊔hω, ηi)⊓(h∼θ, 1i⊔h1, ∼ζi)⊓(h∼σ, 1i⊔h1, ∼ηi). We choose not to simplify expressions such as ǫ⊓ǫ⊓∼Λ so as to make the example easy to follow. By applying equations 7 and 8, we have both eseq(ǫ⊓Λ⊓ǫ, Ψ2 ) = true and eseq(ǫ⊓ǫ⊓Λ, Ψ2 ) = true. So, etype(α⊓∼β) = eseq(Θ, Ψ2). Let Γ = hω, ζi⊓h∼θ, 1i⊓h1, ∼ηi. To show etype(α⊓∼β) = false, it suffices to show eseq conj(Γ, Ψ2 ) = false by equation 7 because Γ ∈ DNF(Θ) and etype(α⊓∼β) = eseq(Θ, Ψ2). Figure 3 depicts the evaluation of eseq conj(Γ, Ψ2). The node that is linked to its parent by a dashed line is not evaluated because one of its siblings returns false, which is sufficient to establish the falsity of its parent. It is clear from the figure that etype conj(Θ, Ψ2) = false and hence etype(α⊓∼β) = false. 4 Correctness This section addresses the correctness of the algorithm. We shall first show that tabulation ensures the termination of the algorithm because the table can only be of finite size. We then establish the partial correctness of the algorithm. etype conj(Γ, Ψ2 ) ∨ 7 7 etype(ζ⊓∼η, Ψ2 ) etyp(ω⊓∼θ, Ψ2 ) 4 4 etyp conj(ω⊓∼θ, Ψ2 ) etype conj(ζ⊓∼η, Ψ2 ) ∧ 5/b 5/a 5/h(,) 5 eseq(ǫ⊓Λ, Ψ3 ) eseq(ǫ⊓ǫ, Ψ4 ) eseq(ǫ⊓ǫ, Ψ3 ) eseq(Θ1 , Ψ3 ) 6 6 6 eseq conj(ǫ⊓Λ, Ψ3 ) eseq conj(ǫ⊓ǫ, Ψ3 ) eseq conj(ǫ⊓ǫ, Ψ4 ) 7 7 7 true false false Legend: Θ1 = (hω, ζi⊔hω, ηi)⊓(h∼θ, 1i⊔h1, ∼ζi) Ψ3 = Ψ2 ∪ {ω⊓∼θ} Ψ4 = Ψ2 ∪ {ζ⊓∼η} Γ = hω, ζi⊓h∼θ, 1i⊓h1, ∼ηi Fig. 3. Evaluation of etype conj(Γ, Ψ2 ) 4.1 Termination Given a type expression E, a top-level type atom in E is a type atom in E that is not a sub-term of any type atom in E. The set of top-level type atoms in E is denoted by TLA(E). For instance, letting E = ∼List(Nat)⊔T ree(Nat⊓∼Even), TLA(E) = {List(Nat), T ree(Nat⊓∼Even)}. We extend TLA(·) to sequences def S by TLA(hE1 , E2 , · · · , Ek i) = 1≤i≤k TLA(Ei ). Given a type expression E0 , the evaluation tree for etype(E0 ) contains nodes of the form etype(E, Ψ ), etype conj(C, Ψ ), eseq(Θ, Ψ ) and eseq conj(Γ, Ψ ) in addition to the root that is etype(E0 ). Only nodes of the form etype conj(C, Ψ ) add conjunctive type expressions to the table. Other forms of nodes only pass the table around. Therefore, it suffices to show that the type atoms occurring in the first argument of the nodes are from a finite set because any conjunctive type expression added into the table is the first argument of a node of the form etype conj(C, Ψ ). The set RTA(E0 ) of type atoms relevant to a type expression E0 is the smallest set of type atoms satisfying – TLA(E0 ) ⊆ RTA(E0 ), and – if τ is in RTA(E0 ) and τ → f (τ1 , τ2 , · · · , τk ) is in ground(∆) then TLA(τi ) ⊆ RTA(E0 ) for 1 ≤ i ≤ k. The height of τi is no more than that of τ for any τ → f (τ1 , τ2 , · · · , τk ) in ground(∆). Thus, the height of any type atom in RTA(E0 ) is finite. There are only a finite number of type constructors in Π. Thus, RTA(E0 ) is of finite size. It follows by examining the algorithm that type atoms in the first argument of the nodes in the evaluation tree for etype(E0 ) are from RTA(E0 ) which is finite. Therefore, the algorithm terminates. 4.2 Partial Correctness The partial correctness of the algorithm is established by showing etype(E0 ) = true iff empty(E0 ). Let Ψ be a set of conjunctive type def expressions. Define ρΨ = ∧C∈Ψ empty(C). The following two lemmas form the core of our proof of the partial correctness of the algorithm. Lemma 1. Let Ψ be a set of conjunctive type expressions, E a type expression, C a conjunctive type expression, Θ a sequence expression and Γ a conjunctive sequence expression. (a) (b) (c) (d) If If If If ρΨ ρΨ ρΨ ρΨ \|= empty(C) \|= empty(E) \|= empty(Γ ) \|= empty(Θ) then then then then etype conj(C, Ψ ) = true, and etype(E, Ψ ) = true, and etype(Γ, Ψ ) = true, and etype(Θ, Ψ ) = true. Proof. The proof is done by induction on the size of the complement of Ψ with respect to the set of all possible conjunctive type expressions in which type atoms are from RTA(E0 ) where E0 is a type expression. Basis. The complement is empty. Ψ contains all possible conjunctive type expressions in which type atoms are from RTA(E0 ). We have C ∈ Ψ and hence etype conj(C, Ψ ) = true by equation 6. Therefore, (a) holds. (b) follows from (a) and equation 5. (c) follows from (b), equation 8 and lemma 4 in the appendix, and (d) follows from (c) and equation 7. Induction. By lemma 3 in the appendix, ρΨ \|= empty(C) implies ρΨ \|= empty(BCf ) for any f ∈ ∩α∈pos(C) F (α). Thus, ρΨ∪{C} \|= empty(BCf ). The complement of Ψ ∪ {C} is smaller than the complement of Ψ . By the induction hypothesis, we have eseq(BCf , Ψ ∪{C}) = true. By equation 6, etype conj(C, Ψ ) = true. Therefore, (a) holds. (b) follows from (a) and equation 5. (c) follows from (b), equation 8 and lemma 4 in the appendix and (d) follows from (c) and equation 7. This completes the proof of the lemma. Lemma 1 establishes the completeness of etype(, ), etype conj(, ), eseq(, ) and eseq conj(, ) while the following lemma establishes their soundness. Lemma 2. Let Ψ be a set of conjunctive type expressions, E a type expression, C a conjunctive type expression, Θ a sequence expression and Γ a conjunctive sequence expression. (a) (b) (c) (d) ρΨ ρΨ ρΨ ρΨ \|= empty(C) \|= empty(E) \|= empty(Γ ) \|= empty(Θ) if if if if etype conj(C, Ψ ) = true, and etype(E, Ψ ) = true, and etype(Γ, Ψ ) = true, and etype(Θ, Ψ ) = true. Proof. It suffices to prove (a) since (b),(c) and (d) follow from (a) as in lemma 1. The proof is done by induction on dp(C, Ψ ) the depth of the evaluation tree for etype conj(C, Ψ ). Basis. dp(C, Ψ ) = 1. etype conj(C, Ψ ) = true implies either (i) pos(C) ∩ neg(C) 6= ∅ or (ii) ∃C ′ ∈ Ψ.C C ′ . In case (i), empty(C) is true and ρΨ \|= empty(C). Consider case (ii). By the definition of and ρΨ , we have etype conj(C, Ψ ) = true implies ρΨ \|= empty(C). Induction. dp(C, Ψ ) > 1. Assume etype conj(C, Ψ ) = true and ρΨ \|= ¬empty(C). By lemma 3, there is f ∈ ∩α∈pos(C) F (α) such that ρΨ \|= ¬empty(BCf ). We have ρΨ∪{C} \|= ¬empty(BCf ). dp(BCf , Ψ ∪ {C}) < dp(C, Ψ ). By the induction hypothesis, we have etuple(BCf , Ψ ∪ {C}) = false for otherwise, ρΨ∪{C} \|= BCf . By equation 6, etype conj(C, Ψ ) = false which contradicts etype conj(C, Ψ ) = true. So, ρΨ \|= empty(C) if etype conj(C, Ψ ) = true. This completes the induction and the proof of the lemma. The following theorem is a corollary of lemmas 1 and 2. Theorem 1. For any type expression E, etype(E) = true iff empty(E). Proof. By equation 4, etype(E) = etype(E, ∅). By lemma 1.(b) and lemma 2.(b), we have etype(E, ∅) = true iff ρ∅ \|= empty(E). The result follows since ρ∅ = true. 5 Complexity We now address the issue of complexity of the algorithm. We only consider the worst-case time complexity of the algorithm. The time spent on evaluating etype(E0 ) for a given type expression E0 can be measured in terms of the number of nodes in the evaluation tree for etype(E0 ). The algorithm cycles through etype(, ), etype conj(, ), eseq(, ) and eseq conj(, ). Thus, children of a node of the form etype(E, Ψ ) can only be of the form etype conj(C, Ψ ), and so on. Let \|S\| be the number of elements in a given set S. The largest possible table in the evaluation of etype(E0 ) contains all the conjunctive type expressions of which type atoms are from RTA(E0 ). Therefore, the table can contain at most 2\|RTA(E0 )\| conjunctive type expressions. So, the height of the tree is bounded by O(2\|RTA(E0 )\| ). We now show that the branching factor of the tree is also bounded by O(2\|RTA(E0 )\| ). By equation 5, the number of children of etype(E, Ψ ) is bounded by two to the power of the number of type atoms in E which is bounded by \|RTA(E0 )\| because E can only contain type atoms from RTA(E0 ). By equation 6, the number of children of etype conj(C, Ψ ) is bounded by \|Σ\|. The largest number of children of a node eseq(Θ, Ψ ) is bounded by two to the power of the number of sequences in Θ where Θ = BCf . For each τ ∈ neg(C), \|push(∼(⊔Afτ ))\| is O(arity(f )) and \|C\| < \|RTA(E0 )\|. Thus, the number of sequences in Θ is O(arity(f ) ∗ \|RTA(E0 )\|) and hence the number of children of eseq(Θ, Ψ ) is O(2\|RTA(E0 )\| ) since arity(f ) is a constant. By equation 8, the number of children of eseq conj(Γ, Ψ ) is bounded by maxf ∈Σ arity(f ). Therefore, the branching factor of the tree is bounded by O(2\|RTA(E0 )\| ). The above discussion leads to the following conclusion. Proposition 1. The time complexity of the algorithm is O(2\|RTA(E0 )\| )). The fact that the algorithm is exponential in time is expected because the complexity coincides with the complexity of deciding the emptiness of any tree automaton constructed from the type expression and the type definitions. A deterministic frontier-to-root tree automaton recognising [E0]∆ will consist of 2\|RTA(E0 )\| states as observed in the proof of lemma 5. It is well-known that the decision of the emptiness of the language of a deterministic frontier-to-root tree automaton takes time polynomial in the number of the states of the tree automaton. Therefore, the worst-case complexity of the algorithm is the best we can expect from an algorithm for deciding the emptiness of regular types that contain set operators. 6 Conclusion We have presented an algorithm for deciding the emptiness of prescriptive regular types. Type expressions are constructed from type constructors and set operators. Type definitions prescribe the meaning of type expressions. The algorithm uses tabulation to ensure termination. Though the tabulation is inspired by Dart and Zobel [10], the decision problem we consider in this paper is more complex as type expressions may contain set operators. For that reason, the algorithm can also be used for inclusion and equivalence problems of regular types. The way we use tabulation leads to a correct algorithm for regular types while the Dart-Zobel algorithm has been proved incorrect for regular types [24] in general. To the best of our knowledge, our algorithm is the only correct algorithm for prescriptive regular types. In addition to correctness, our algorithm generalises the work of Dart and Zobel [10] in that type expressions can contain set operators and type definitions can be parameterised. Parameterised type definitions are more natural than monomorphic type definitions [12,26,32] while set operators makes type expressions concise. The combination of these two features allows more natural type declarations. For instance, the type of the logic program append can be declared or inferred as append(List(α), List(β), List(α⊔β)). The algorithm is exponential in time. This coincides with deciding the emptiness of the language recognised by a tree automaton constructed from the type expression and the type definitions. However, the algorithm avoids the construction of the tree automaton which cannot be constructed a priori when type definitions are parameterised. Another related field is set constraint solving [3,2,20,18,11]. However, set constraint solving methods are intended to infer descriptive types [28] rather than for testing the emptiness of a prescriptive type [28]. Therefore, they are useful in different settings from the al- gorithm presented in this paper. In addition, algorithms proposed for solving set constraints [3,4,2,1] are not applicable to the emptiness problem we considered in this paper. Take for example the constructor rule in [3,2] which states that emptiness of f (E1 , E2 , · · · , Em ) is equivalent to the emptiness of Ei for some 1 ≤ i ≤ m. However, empty(List(0)) is not equivalent to empty(0). The latter is true while the former is false since [List(0)]]∆ = {nil}. The constructor rule doesn’t apply because it deals with function symbols only but doesn’t take the type definitions into account. References 1. A. Aiken, D. Kozen, M. Vardi, and E. Wimmers. The complexity of set constraints. In Proceedings of 1993 Computer Science Logic Conference, pages 1–17, 1992. 2. A. Aiken and T.K. Lakshman. Directional type checking of logic programs. In B. Le Charlier, editor, Proceedings of the First International Static Analysis Symposium, pages 43–60. Springer-Verlag, 1994. 3. A. Aiken and E. Wimmers. Solving systems of set constraints. In Proceedings of the Seventh IEEE Symposium on Logic in Computer Science, pages 329–340. The IEEE Computer Society Press, 1992. 4. A. Aiken and E. Wimmers. Type inclusion constraints and type inference. In Proceedings of the 1993 Conference on Functional Programming Languages and Computer Architecture, pages 31–41, Copenhagen, Denmark, June 1993. 5. C. Beierle. Type inferencing for polymorphic order-sorted logic programs. In L. Sterling, editor, Proceedings of the Twelfth International Conference on Logic Programming, pages 765–779. The MIT Press, 1995. 6. L. Cardelli and P. Wegner. On understanding types, data abstraction, and polymorphism. ACM computing surveys, 17(4):471–522, 1985. 7. M. Codish and V. Lagoon. Type dependencies for logic programs using aciunification. In Proceedings of the 1996 Israeli Symposium on Theory of Computing and Systems, pages 136–145. IEEE Press, June 1996. 8. H. Comon, M. Dauchet, R. Gilleron, D. Lugiez, S. Tison, and M. Tommasi. Tree Automata Techniques and Applications. Draft, 1998. 9. P.W. Dart and J. Zobel. Efficient run-time type checking of typed logic programs. Journal of Logic Programming, 14(1-2):31–69, 1992. 10. P.W. Dart and J. Zobel. A regular type language for logic programs. In Frank Pfenning, editor, Types in Logic Programming, pages 157–189. The MIT Press, 1992. 11. P. Devienne, J-M. Talbot, and S. Tison. Co-definite set constraints with membership expressions. In J. Jaffar, editor, Proceedings of the 1998 Joint Conference and Symposium on Logic Programming, pages 25–39. The MIT Press, 1998. 12. T. Fruhwirth, E. Shapiro, M.Y. Vardi, and E. Yardeni. Logic programs as types for logic programs. In Proceedings of Sixth Annual IEEE Symposium on Logic in Computer Science, pages 300–309. The IEEE Computer Society Press, 1991. 13. J.P. Gallagher and D.A. de Waal. Fast and precise regular approximations of logic programs. In M. Bruynooghe, editor, Proceedings of the Eleventh International Conference on Logic Programming, pages 599–613. The MIT Press, 1994. 14. F. Gécseg and M. Steinby. Tree Automata. Akadémiai Kiadó, 1984. 15. F. Gécseg and M. Steinby. Tree languages. In G. Rozenberg and A. Salomma, editors, Handbook of Formal Languages, pages 1–68. Springer-Verlag, 1996. 16. M. Hanus. Horn clause programs with polymorphic types: semantics and resolution. Theoretical Computer Science, 89(1):63–106, 1991. 17. N. Heintze and J. Jaffar. A finite presentation theorem for approximating logic programs. In Proceedings of the seventh Annual ACM Symposium on Principles of Programming Languages, pages 197–209. The ACM Press, 1990. 18. N. Heintze and J. Jaffar. A decision procedure for a class of set constraints. Technical Report CMU-CS-91-110, Carnegie-Mellon University, February 1991. (Later version of a paper in Proc. 5th IEEE Symposium on LICS). 19. N. Heintze and J. Jaffar. Semantic types for logic programs. In Frank Pfenning, editor, Types in Logic Programming, pages 141–155. The MIT Press, 1992. 20. N. Heintze and J. Jaffar. Set constraints and set-based analysis. In Alan Borning, editor, Principles and Practice of Constraint Programming, volume 874 of Lecture Notes in Computer Science. Springer, May 1994. (PPCP’94: Second International Workshop, Orcas Island, Seattle, USA). 21. D. Jacobs. Type declarations as subtype constraints in logic programming. SIGPLAN Notices, 25(6):165–73, 1990. 22. L. Lu. Type analysis of logic programs in the presence of type definitions. In Proceedings of the 1995 ACM SIGPLAN Symposium on Partial Evaluation and Semantics-Based program manipulation, pages 241–252. The ACM Press, 1995. 23. L. Lu. A polymorphic type analysis in logic programs by abstract interpretation. Journal of Logic Programming, 36(1):1–54, 1998. 24. L. Lu and J. Cleary. On Dart-Zobel algorithm for testing regular type inclusion. Technical report, Department of Computer Science, The University of Waikato, October 1998. http://xxx.lanl.gov/ps/cs/9810001. 25. P. Mishra. Towards a theory of types in Prolog. In Proceedings of the IEEE international Symposium on Logic Programming, pages 289–298. The IEEE Computer Society Press, 1984. 26. A. Mycroft and R.A. O’Keefe. A polymorphic type system for Prolog. Artificial Intelligence, 23:295–307, 1984. 27. Frank Pfenning, editor. Types in logic programming. The MIT Press, Cambridge, Massachusetts, 1992. 28. U.S. Reddy. Types for logic programs. In S. Debray and M. Hermenegildo, editors, Logic Programming. Proceedings of the 1990 North American Conference, pages 836–40. The MIT Press, 1990. 29. M. Soloman. Type definitions with parameters. In Conference Record of the Fifth ACM Symposium on Principles of Programming Languages, pages 31–38, 1978. 30. J. Tiuryn. Type inference problems: A survey. In B. Roven, editor, Proceedings of the Fifteenth International Symposium on Mathematical Foundations of Computer Science, pages 105–120. Springer-Verlag, 1990. 31. E. Yardeni, T. Fruehwirth, and E. Shapiro. Polymorphically typed logic programs. In K. Furukawa, editor, Logic Programming. Proceedings of the Eighth International Conference, pages 379–93. The MIT Press, 1991. 32. E. Yardeni and E. Shapiro. A type system for logic programs. Journal of Logic Programming, 10(2):125–153, 1991. 33. J. Zobel. Derivation of polymorphic types for Prolog programs. In J.-L. Lassez, editor, Logic Programming: Proceedings of the fourth international conference, pages 817–838. The MIT Press, 1987. Appendix Lemma 3. Let C be a conjunctive type expression. empty(C) iff ∀f ∈ ∩α∈pos(C) F (α). empty((⊓ω∈pos(C) (⊔Afω ))⊓(⊓τ ∈neg(C) ∼(⊔Afτ ))) Proof. Let t be a sequence of terms and f a function symbol. By the definition of [·]]∆ , f (t) ∈ [C]]∆ iff f ∈ ∩α∈pos(C) F (α) and t ∈ [⊓ω∈pos(C) (⊔Afω ))]]∆ \ [(⊔τ ∈neg(C) (⊔Afτ ))]]∆ . t ∈ [⊓ω∈pos(C) (⊔Afω ))]]∆ \ [(⊔τ ∈neg(C) (⊔Afτ ))]]∆ iff t ∈ [(⊓ω∈pos(C) (⊔Afω ))⊓(⊓τ ∈neg(C) ∼(⊔Afτ ))]]∆ . Thus, empty(C) iff empty((⊓ω∈pos(C) (⊔Afω ))⊓(⊓τ ∈neg(C) ∼(⊔Afτ ))) for each f ∈ ∩α∈pos(C) F (α). Lemma 4. Let Γ be a conjunctive sequence expression. Then empty(Γ ) iff ⊔1≤jkΓ kempty(Γ↓j) Proof. Let kΓ k = Tn and Γ = γ1 ⊓γ2 ⊓ · · · ⊓γm with γi = hγi,1 , γi,2, · · · , γi,n i. We have [Γ ]∆ = 1≤j≤m [γj]∆ . We have Γ↓j = γ1,j ⊓γ2,j ⊓ · · · ⊓γm,j . T ∃1 ≤ j ≤ n.empty(Γ↓j) iff ∃1 ≤ j ≤ n. 1≤i≤m [γi,j]∆ = ∅ iff [Γ ]∆ = ∅ iff empty(Γ ). Lemma 5. [M]]∆ is a regular term language for any type expression M. Proof. The proof is done by constructing a regular term grammar for M [14]. We first consider the case M ∈ T (Π ∪ {1, 0}). Let R = hRTA(M), Σ, ∅, Υ, Mi with Υ = {(α → f (α1 , · · · , αk )) ∈ ground(∆) \| α ∈ RTA(M)} R is a regular term grammar. It now suffices to prove that t ∈ [M]]∆ iff M ⇒∗R t. – Sufficiency. Assume M ⇒∗R t. The proof is done by induction on derivation steps in M ⇒∗R t. • Basis. M ⇒R t. t must be a constant and M → t is in Υ which implies M → t is in ground(∆). By the definition of [·]]∆ . t ∈ [M]]∆ . (n−1) • Induction. Suppose M ⇒ f (M1 , · · · , Mk ) ⇒R t. Then ni t = f (t1 , · · · , tk ) and Mi ⇒R t with ni ≤ (n − 1). By the induction hypothesis, ti ∈ [Mi]∆ and hence t ∈ [M]]∆ by the definition of [·]]∆ . – Necessity. Assume t ∈ [M]]∆ . The proof is done by the height of t, denoted as height(t). • height(t) = 0 implies that t is a constant. t ∈ [M]]∆ implies that M → t is in ground(∆) and hence M → t is in Υ . Therefore, M ⇒R t. • Let height(t) = n. Then t = f (t1 , · · · , tk ). t ∈ [M]]∆ implies that (M → f (M1, · · · , Mk )) ∈ ground(∆) and ti ∈ [Mi]∆ . By the definition of Υ , we have (M → f (M1 , · · · , Mk )) ∈ Υ . By the definition of RTA(·), we have Mi ∈ RTA(M). By the induction hypothesis, Mi ⇒∗R ti . Therefore, M ⇒R f (M1 , · · · , Mk ) ⇒∗R f (t1 , · · · , tk ) = t. Now consider the case M ∈ T (Π ∪ {⊓, ⊔, ∼, 1, 0}). We complete the proof by induction on the height of M. – height(M) = 0. Then M doesn’t contain set operator. We have already proved that [M]]∆ is a regular term language. – Now suppose height(M) = n. If M doesn’t contain set operator then the lemma has already been proved. If the principal type constructor is one of set operators then the result follows immediately as regular term languages are closed under union, intersection and complement operators [14,15,8]. It now suffices to prove the case M = c(M1 , · · · , Ml ) with c ∈ Π. Let N = c(X1 , · · · , Xl ) where each Xj is a different new type constructor of arity 0. Let Π ′ = Π{X1, · · · , Xl }, Σ ′ = Σ ∪ {x1 , · · · , xl } and ∆′ = ∆ ∪ {Xj → xj \|1 ≤ j ≤ l}. [N ]∆′ is a regular term language on Σ ∪ {x1 , · · · , xl } because N doesn’t contain set operators. By the induction hypothesis, [Mj]∆ is a regular term language. By the definition of [·]]· , we have [M]]∆ = [N ]∆′ [x1 := [M1]∆ , · · · , xl := [Ml]∆ ] which is a regular term language [14,15,8]. S[y1 := Sy1 , · · · , ] is the set of terms each of which is obtained from a term in S by replacing each occurrence of yj with a (possibly different) term from Syj . This completes the induction and the proof. The proof also indicates that a non-deterministic frontier-to-root tree automaton that recognises [M]]∆ has \|RTA(M)\| states and that a deterministic frontier-to-root tree automaton that recognises [M]]∆ has O(2\|RTA(M)\| ) states.	6
1 A Study of the Allan Variance for Constant-Mean Non-Stationary Processes arXiv:1702.07795v3 [math.ST] 22 Jun 2017 Haotian Xu, Stéphane Guerrier, Roberto Molinari & Yuming Zhang Abstract—The Allan Variance (AV) is a widely used quantity in areas focusing on error measurement as well as in the general analysis of variance for autocorrelated processes in domains such as engineering and, more specifically, metrology. The form of this quantity is widely used to detect noise patterns and indications of stability within signals. However, the properties of this quantity are not known for commonly occurring processes whose covariance structure is non-stationary and, in these cases, an erroneous interpretation of the AV could lead to misleading conclusions. This paper generalizes the theoretical form of the AV to some non-stationary processes while at the same time being valid also for weakly stationary processes. Some simulation examples show how this new form can help to understand the processes for which the AV is able to distinguish these from the stationary cases and hence allow for a better interpretation of this quantity in applied cases. Index Terms—Metrology, Sensor Calibration, Bias-Instability, Longitudinal Studies, Haar Wavelet Variance, Heteroscedasticity. I. I NTRODUCTION The Allan Variance (AV) is a widely used quantity in areas going from engineering to physics where there is an interest in studying the stochastic stability of error measurements from various instruments such as, among others, clocks and oscillators. Its usefulness resides in the fact that it provides an extremely informative summary on the variance of time series or, more generally, of autocorrelated processes, especially when these are non-stationary and with infinite variance. Indeed, [8] underlined how the AV is a better measure of uncertainty compared to standard methods (e.g. moving average variance) for processes such as random walks and non-stationary Fractional Autoregressive-Moving Average (ARFIMA) models, while being considerably useful also for stationary processes. For these processes the AV has a well known form which can help detect the kind of process, for example, from the log-log plot of the AV of an observed signal. The behaviour and forms of the AV for stationary and some non-stationary processes was studied in [8] where the AV is used to detect and understand the process underlying a signal issued from different voltage measurements. However, there are many other applications for which the AV is of interest H. Xu is a PhD student, Geneva School of Economics and Management, University of Geneva, 1211, Switzerland (E-mail: [email protected]). S. Guerrier is Assistant Professor, Department of Statistics, Pennsylvania State University, PA, 16801, USA. R. Molinari is Visiting Assistant Professor, Department of Statistics & Applied Probability, University of California, Santa Barbara, CA, 93117, USA. Y. Zhang is a graduate student, Department of Statistics, Pennsylvania State University, PA, 16801, USA. such as the detection of noise terms characterising inertial sensors [see 2] and many others [see 5, for an overview]. However, although the AV is extremely useful in the above settings, it is not known how it behaves when in the presence of other types of processes and whether it is able to distinguish between them. In this paper we intend to investigate the form of the AV for a particular class of processes which includes all those processes that have a constant mean but have a time-varying variance-covariance structure such as, for example, the Generalized AutoRegressive Conditional Heteroscedasticity (GARCH) models [see 1], while processes with specific forms of mean or higher-order non-stationarity are not considered since they can either be dealt with through statistical regression techniques or simply cannot be detected by the AV. In particular, we focus on those processes which are characterized by a dependence structure by blocks since they are common in settings such as longitudinal studies or sensor calibration for navigation engineering. In the latter cases, the AV is often approximated by that of other known stationary processes such as, for example, the bias-instability process whose AV is often approximated by that of a firstorder autoregressive process [see, for example, 7]. Moreover, it is not clear whether the AV can actually help to distinguish between these processes and those processes for which its form is currently known. The latter aspect is of particular relevance since it could lead to an erroneous interpretation of the observed process, for example assuming stationarity when this is not the case and reaching false conclusions. In order to deal with the above mentioned non-stationary processes, this paper intends to study the theoretical form of the AV when the covariance structure is non-stationary. The consequent advantage of this study is that, by considering the varying covariance structure in the AV definition, it extends the applicability of those approaches that make use of the AV and raises awareness on its limitations, and inappropriate interpretation, in distinguishing and identifying these processes from stationary ones. With this in mind, Section II briefly defines the AV and describes its theoretical form for those processes which have been considered up to now. Section III introduces the new theoretical form of both the overlapping and non-overlapping AV for processes whose covariance structure is non-stationary and shows how the form of the AV for stationary processes is a special case of this new form. In the same section, three case studies are presented which highlight the importance of these findings in order to better interpret processes through the AV. Finally Section IV concludes. 2 II. OVERVIEW OF THE A LLAN VARIANCE γ(h) = cov(Xt+h , Xt ), mean to estimate different kind of processes [see for example 4]. However, there are many commonly encountered processes whose AV is not known and it is unclear to what point these can actually be distinguished from stationary processes. The next section delivers a more general form of the AV which includes these processes and studies if this quantity can actually be helpful in detecting them. which depends solely on h, the distance between observations, 2 with σX = γ(0) being the process variance. We can consequently define the autocorrelation function as III. A LLAN VARIANCE FOR C ONSTANT-M EAN N ON -S TATIONARY P ROCESSES To introduce the AV, let us first define (Xt )t=1,...,T as a weakly stationary, discrete time and regularly spaced stochastic process with a constant mean µ (i.e. E[Xt ] = µ) and an autocovariance function defined as follows ρ(h) = γ(h) . γ(0) We consider the AV computed at dyadic scales (τ ) starting from local averages of the process which can be denoted as n (n) X̄t ≡ 1X Xt−n+i , n i=1 (1) where n ∈ {x ∈ N : 1 ≤ x < log2 (T ) − 1} therefore determines the number of consecutive observations considered for the average. If the process has constant mean µ, this implies (n) that X̄t also has the same mean and, based on these averages and following [6], the maximum-overlapping AV (MOAV) T 2 1 X (n) (n) E X̄k − X̄k−n . (2) AVarn (Xt ) ≡ 2m? As underlined in the previous sections, the AV is particularly useful for measuring uncertainty in non-stationary processes, especially when these have infinite variance. Nevertheless, there are other forms of non-stationarity for which the properties of the AV are unknown and these consist in those processes (Xt ) with a constant mean µ (independent of time) but a non-stationary covariance structure. This implies that the covariance function between observations at distance h is also a function of time t and can therefore be denoted as γ(h, t). This type of process is very common in different areas going from engineering [see 2] to economics [see 3]. To study the theoretical form of the AV for this class of (n) processes, let us first define Xt as being the following vector of n consecutive observations starting at t − n + 1 (n) Xt T ≡ [Xt−n+1 · · · Xt ] . k=2n ? where m = T − 2n + 1 and whose corresponding estimator is given by [ n (Xt ) = AVar T 2 1 X (n) (n) x̄k − x̄k−n . ? 2m (3) k=2n (n) (n) where x̄t denotes the sample equivalent of X̄t based on a realization of the process (Xt ). Another version of the AV is the non-overlapping AV (NOAV) whose estimator however is not statistically as efficient as that of the MOAV (see App. B for more details). Keeping in mind the above definitions of the MOAV, [8] delivered a general theoretical form of this quantity when applied to weakly stationary processes which is given by 2 σX AVarn (Xt ) = n [1 − ρ(n)] (4) n2 n−1 X + i [2ρ(n − i) − ρ(i) − ρ(2n − i)] . i=1 Based on the above equation, the exact form of the AV for different stationary processes, such as the general class of AutoRegressive Moving Average (ARMA) models, can be derived. Moreover, [8] provided the theoretical AV for nonstationary processes such as the random walk and ARFIMA models for which the AV, as mentioned earlier, represents a better measure of uncertainty compared to other methods. Using the known theoretical forms of the AV, it is therefore possible to detect and distinguish different processes based on the pattern of their AV. Due to this, this quantity (or similar quantities such as the Haar wavelet variance) can be used as a which contains the observations used to build the average in (1). Using the above vector, for t = n, ..., T , we can then define the matrix (n) (n) Σt ≡ var Xt , (5) and, for t = 2n, ..., T , we define the matrix (n) (n) (n) , Γt ≡ cov Xt−n , Xt (6) These matrices represent the covariance matrices of the obser(n) vations contained in each consecutive average. Indeed, Σt represents the covariance matrix of the observations within the (n) average X̄t which is used in definitions (D-1) and (2) while (n) Γt represents the cross-covariance between these two sets of observations. A visual representation of these quantities is given in App. A. In this section we will only consider the non-stationary MOAV while the form of the non-stationary NOAV is given in App. B. Based on the above matrices, we can also define different quantities according to the matrix of reference and the lags between observations. More specifically, let us first consider the case in which we are interested in lags h such that 0 ≤ h < n. Because of the overlapping nature of the AV, the observations at these lags can belong to the sets of (n) (n) observations within both the matrix Σt and the matrix Γt (n) and, for these sets of observations within the matrix Σt , we can define the following quantity γ e(h) ≡ T n−h−1 X X 1 cov(Xt−n−s−h , Xt−n−s ) ? 2m (n − h) t=2n s=0 + cov(Xt−s−h , Xt−s ). 3 If, however, the observations at the considered lags are among (n) the set of observations only within the matrix Γt , we define the quantity below T h 1 XX γ e (h) ≡ ? cov (Xt−n+s−h , Xt−n+s ) . m h t=2n s=1 ∗ Finally, when considering lags h such that n ≤ h ≤ 2n − 1, the set of observations at these lags can only be considered (n) within the matrix Γt and, for this final case, we define the quantity γ e(h) ≡ T 2n−h−1 X X 1 cov (Xt−s−h , Xt−s ) . ? m (2n − h) t=2n s=0 The above definitions can be seen as generalized definitions of the autocovariance which consists in the average autocovariance for a given lag h. Because of this, it must be underlined that these definitions are not at all equivalent to the covariance function γ(h, t) but correspond to an average of this function over all times t. Having specified this, we can now provide the following lemma. L EMMA 1: The non-stationary MOAV is given by ( " Pn−1 1 ? ? e(0) + 2 e(h) AVarn = 2m? n2 2nm γ h=1 2m (n − h) γ #) −m? h γ e∗ (h) − P2n−1 h=n m? (2n − h) γ e(h) . The proof of this lemma is given in App. C. Considering this expression, an aspect that must be underlined is that the definitions of the functions γ e(h) and γ e∗ (h) given earlier simplify to the autocovariance function γ(h) when dealing with a weakly stationary process. In the latter case, the form of the non-stationary AV consequently reduces to the expression in (4) for which a more detailed discussion is given in App. ??. As a final note to this result it should also be highlighted that, in some of the considered non-stationary cases, the estimators of the MOAV defined in (3) and of the NOAV (see App. B) do not necessarily have the same expectation. Having underlined these points and with the general definition of the AV given in Lemma 1, we can now investigate its properties when assuming the process of interest is within the class of processes treated in this paper. The next sections report some simulation studies regarding some of these processes in attempt to understand also whether the AV is a useful quantity to detect them and distinguish them from weakly stationary processes. In all cases, the simulated process is of length T = 1, 000 (except for the bias-instability process where T = 2, 500) and is simulated 50 times. The estimated AV is represented in plots along with the theoretical stationary and non-stationary forms of the AV in order to understand its behaviour under these different assumptions. 1) Non-Stationary White Noise: The first process we study is the non-stationary white noise by which we intend, without loss of generality, all those zero-mean processes whose variance changes with time. The evolution of the variance in time can either be completely random or can follow a specific Fig. 1. Logarithm of the MOAV of the non-stationary white noise process for scales τ = 2n . Estimated MOAV (light-blue lines); theoretical non-stationary MOAV (black line with dots) and theoretical stationary AV based on the average variance (red line with triangles). parametric model such as, for example, a GARCH process. The goal of studying the AV for these types of processes is to understand whether it is able to detect such a structure in a time series and if it can be distinguished from a stationary white noise process. For this purpose, the true non-stationary process considered in the simulation study is generated from the following model Xt ∼ N (0, σt2 ), where σt2 = t, with t = 1, . . . , T . The theoretical stationary form is based on the average of the variances used to simulate the processes (i.e. σ̄ 2 = (1+T )/2 in this example). Fig. 1 represents the estimated AVs along with the theoretical forms (stationary and non-stationary). In this case, it can be seen how both theoretical forms correspond and the estimated AVs closely follow these quantities. This example confirms that the AV is therefore unable to distinguish between a stationary white noise process and a white noise process whose secondorder behaviour is non-stationary. 2) Bias-Instability: The bias-instability process is a commonly known process in the engineering domain, specifically for inertial sensor calibration for navigation. The characteristic of this process is that it consists in different concatenated sequences (blocks) where, within each block, the realization of a random variable is repeated (i.e. constant). More formally, let bi , i = 1, . . . , B, represent the set of time indices belong iid to the ith block within the time series, and let Ci ∼ N (0, σ 2 ). We can then define this process as Xt = ci if t ∈ bi . One realization of the bias-instability process is illustrated in the top panel Fig. 2 where the length of block bi is 10 for all i = 1, . . . , B, and B = 250. Since the theoretical form of the AV for this process is not known exactly, it is often approximated by the AV of a First-Order AutoRegressive (AR1) process. Although this approximation can be useful, it is nevertheless still an approximation and, using the form given in Lemma 1, we can now obtain a theoretical form for the AV of this process which is represented in the bottom panel of Fig. 4 Fig. 2. Top: Realization of the bias-instability process with σ 2 = 1 and the length of block bi = 10, ∀i = 1, . . . , B, and B = 250. Bottom: Logarithm of the MOAV of the bias-instability process for scales τ = 2n . Estimated MOAV (light-blue lines); theoretical non-stationary MOAV (black line with dots) and theoretical stationary MOAV of an AR1 approximating the biasinstability MOAV (red line with triangles). Fig. 3. Top: Realization of the block-structure autoregressive process with φ = 0.9, σ 2 = 1 and the length of block bi = 10, ∀i = 1, . . . , B, and B = 100. Bottom: Logarithm of the MOAV of the block-structure firstorder autoregressive process for scales τ = 2n . Estimated MOAV (light-blue lines); theoretical non-stationary MOAV (black line with dots) and theoretical stationary MOAV assuming no block structure (red line with triangles). process 2. Indeed the latter plot shows that the estimated AVs closely follow the theoretical non-stationary form given earlier. The red line represents the AV of a stationary AR1 process which is supposed to approximate the true AV of bias-instability. The latter is the result of the averaging of the theoretical AV for a stationary AR1 process estimated via maximum-likelihood on each of the simulated processes. It is clear how, although close over some scales, this approximation is not good enough when considering the logarithmic representation of the AV. Therefore, knowing the exact form of the AV for this process would allow to better interpret the signals characterised by bias-instability. 3) Block-Structure Autoregressive Processes: As a final example we consider a block-structure AR1 process. Similarly to the bias-instability process, within this paper, we define a block-structure process as a process whose parameters are fixed but is made by concatenated time periods (blocks) where observations within each block are generated independently from those in the other blocks. An example is given by the settings of longitudinal studies in which each subject can be measured over time and, although the subjects are independent from each other, these measurements can be explained by an autocorrelated process within each subject. To define this (i) process formally, let Xt ∼ Fθ denote the following AR1 (i) Xt (i) = φXt−1 + t with parameter vector θ = [φ σ 2 ]T where φ ∈ (−1, 1) and iid t ∼ N (0, σ 2 ). If again we let bi denote the ith block, then the block-structure AR1 process can be defined as (i) Xt = Xt (i) if t ∈ bi . (j) where Xt is independent of Xt , ∀ i 6= j. By defining φ = 0.9 and σ 2 = 1 for the simulation study, the top panel of Fig. 3 shows a realization of this process while the bottom panel of Fig. 3 illustrates the results of the simulations for this particular process. As for bias-instability, it can be observed how the stationary form of the AV (that does not consider the block structure) is not close to the estimated AVs while the non-stationary form provided in this paper adequately represents this process and can therefore allow to distinguish between a stationary autoregressive process and a block-structure one. IV. C ONCLUSIONS Within this paper we wanted to underline an issue concerning the AV which had not yet been studied. Indeed, the behaviour of the AV in commonly occurring settings where the covariance structure of the processes is non-stationary was 5 unknown and, in many cases, was either ignored or dealt with through approximations. The consequence of the latter approaches would probably consist in erroneous interpretations and conclusions drawn from an AV analysis. For this reason, this paper studied the form of the AV for this class of processes thereby generalizing its form also for weakly stationary processes. Based on this, several examples were provided in which the properties of the AV were studied, highlighting its ability to detect these processes and to eventually distinguish them from stationary ones, making researchers and practitioners more aware of issues related to the interpretation and use of this quantity in more general and common settings. Γi Γ 2n Σn Σ 2n Σ i− ΓT n ΣT Σi −n Γi Γ 2n ΣT ΓT trix ce Ma an i var Co R EFERENCES [1] Bollerslev, T.: Generalized autoregressive conditional heteroskedasticity. Journal of econometrics 31(3), 307–327 (1986) [2] El-Sheimy, N., Hou, H., Niu, X.: Analysis and modeling of inertial sensors using allan variance. IEEE Transactions on instrumentation and measurement 57(1), 140–149 (2008) [3] Gallegati, M., Semmler, W.: Wavelet applications in economics and finance, vol. 20. Springer (2014) [4] Guerrier, S., Skaloud, J., Stebler, Y., Victoria-Feser, M.P.: Wavelet-variance-based estimation for composite stochastic processes. Journal of the American Statistical Association 108(503), 1021–1030 (2013) [5] Percival, D.B.: A wavelet perspective on the allan variance. IEEE transactions on ultrasonics, ferroelectrics, and frequency control 63(4), 538–554 (2016) [6] Percival, D.B., Guttorp, P.: Long-memory processes, the allan variance and wavelets. Wavelets in geophysics 4, 325–344 (1994) [7] Unsal, D., Demirbas, K.: Estimation of deterministic and stochastic imu error parameters. In: Position Location and Navigation Symposium (PLANS), 2012 IEEE/ION, pp. 862–868. IEEE (2012) [8] Zhang, N.F.: Allan variance of time series models for measurement data. Metrologia 45(5), 549 (2008) A PPENDIX A G RAPHICAL ILLUSTRATION OF MOAV To graphically illustrate the quantities defined in Section III, Fig. 4 represents the true covariance matrix for a given process and highlights how the AV is related to this matrix by overlapping square matrices along the diagonal, each of which is composed of the quantities defined in Eq. (5) and Eq. (6). (n) Fig. 4. Graphical illustration of matrices Σt k=1 This estimator is less efficient than the MOAV, mainly because it is based on fewer averages and therefore on a smaller sample size. To define the theoretical form of the NOAV for the non(n) stationary processes of interest, we first define the vector Xj of n consecutive observations starting at (j − 1)n + 1, i.e. T (n) Xj ≡ X(j−1)n+1 · · · Xjn . Using the above, for k = 1, ..., m, we define the matrices (n) (n) (n) Σ2k , Σ2k−1 and Γk as follows: (n) (n) (n) (n) Σ2k ≡ var X2k , Σ2k−1 ≡ var X2k−1 (n) (n) (n) and Γk ≡ cov X2k−1 , X2k . As in Appendix A, the above matrices are graphically represented in Fig. 5 where, as opposed to the MOAV, these matrices do not overlap along the diagonal of the covariance (n) (n) (n) matrix of the process. We then let σ̄2k , σ̄2k−1 and γ̄k (n) (n) (n) denote the averages of the matrices Σ2k , Σ2k−1 and Γk , respectively, i.e. n n 1 X X (n) (n) σ̄2k ≡ 2 Σ2k , n i=1 j=1 i,j (n) A PPENDIX B T HEORETICAL FORM OF THE NOAV FOR N ON -S TATIONARY P ROCESSES k=1 (n) γ̄k (D-1) for the MOAV. T where m = b 2n c. The corresponding estimator for this quantity is given by m 2 X (n) (n) ] n (Xt ) = 1 AVar x̄2k − x̄2k−1 . 2m σ̄2k−1 ≡ The non-overlapping AV (NOAV) is defined as: m 2 1 X (n) (n) E X̄2k − X̄2k−1 , AVarn (Xt ) ≡ 2m (n) and Γt n n 1 X X (n) Σ2k−1 , n2 i=1 j=1 i,j n n 1 X X (n) ≡ 2 Γk . n i=1 j=1 i,j We further define σ̄ (n) and γ̄ (n) as follows: m m i 1 X h (n) 1 X (n) (n) σ̄ (n) ≡ σ̄2k + σ̄2k−1 and γ̄ (n) ≡ γ̄k . 2m m k=1 k=1 6 L EMMA 2: We define the Non-stationary NOAV as ( " Pn−1 1 AVarn = 2mn2 2mn γ e(h) e(0) + 2 h=1 2m(n − h) γ Γ1 Σ1 Σ2 Γ1 Σ 2i− Γi 1 Σ 2i Γi Σ 2m #) Γm −1 ∗ Σ 2m −mh γ e (h) − Γm ce h=n m(2n − h) γ e(h) . Proof of Lemma 2: The proof of Lemma 2 is direct from the above definitions. Indeed, we have 2 (n) (n) (n) (n) E X̄2k − X̄2k−1 = var X̄2k + var X̄2k−1 (n) (n) −2 cov X̄2k−1 , X̄2k trix an i var Co P2n−1 Ma (n) (n) (n) = σ̄2k + σ̄2k−1 − 2γ̄k . (n) (n) (n) Fig. 5. Graphical illustration of matrices Σ2k , Σ2k−1 and Γk NOAV. for the Then, using Eq. (D-1), we obtain 2 Pm (n) (n) 1 E X̄ − X̄ AVarn = 2m k=1 2k 2k−1 1 2m Pm (n) (n) (n) σ̄ + σ̄ − 2γ̄ 2k(n) 2k−1 (n) k 1 2m σ̄ − 2m γ̄ = ( 2m " Pn−1 1 2mn γ e(0) + 2 = 2mn e(h) 2 h=1 2m(n − h) γ = k=1 #) Based on the earlier defined matrices, as for the MOAV, we can also define different quantities according to the matrix of reference and the lags between observations. More specifically, let us first consider the case in which we are interested in lags h such that 0 ≤ h < n. The observations at these lags can belong to the sets of observations within the matrix (n) (n) (n) Σ2k , Σ2k−1 and Γk and, for these sets of observations (n) (n) within matrices Σ2k and Σ2k−1 , we can define the following quantity 2m n−h XX 1 γ e(h) ≡ cov X(k−1)n+s , X(k−1)n+s+h . 2m(n − h) s=1 k=1 If, however, the observations at the considered lags are among (n) the set of observations only within the matrix Γk , we define the quantity below m γ e∗ (h) ≡ h 1 XX cov X(2k−1)n+s−h , X(2k−1)n+s . mh s=1 k=1 Finally, when considering lags h such that n ≤ h ≤ 2n − 1, the set of observations at these lags can only be considered (n) within the matrix Γt and, for this final case, we define the quantity ∗ −mh γ e (h) − P2n−1 h=n m(2n − h) γ e(h) , which concludes the proof. A PPENDIX C (n) Proof of Lemma 1: In order to prove Lemma 1 let σ̄t (n) (n) (n) and γ̄t denote the averages of the matrices Σt and Γt , respectively, i.e. n n 1 X X (n) (n) Σt , σ̄t ≡ 2 n i=1 j=1 i,j (n) γ̄t n n 1 X X (n) ≡ 2 Γt . n i=1 j=1 i,j Further, we define σ̄ (n) and γ̄ (n) as follows: σ̄ (n) ≡ T X 1 (n) (n) σ̄ + σ̄t−n 2(T − 2n + 1) t=2n t γ̄ (n) ≡ T X 1 (n) γ̄ . T − 2n + 1 t=2n t m 2n−h X X 1 cov X(k−1)n+s , X(k−1)n+s+h . m(2n − h) Based on these definitions we have k=1 s=1 2 (n) (n) (n) (n) As for the MOAV case, the above definitions can be seen as E X̄k − X̄k−n = var X̄k + var X̄k−n generalized definitions of the autocovariance which consists in (n) (n) the average autocovariance for a given lag h. Using the above −2 cov X̄k , X̄k−n notations and definitions, we can provide the following result. (n) (n) (n) = σ̄k + σ̄k−n − 2γ̄k . γ e(h) ≡ 7 Then, using Eq. (2), we obtain AVarn 1 2m? = = 1 2m? = ( = 1 2m? n2 PT k=2n E (n) X̄k (n) k=2n σ̄k + ? (n) PT 1 2m∗ (n) − X̄k−n (n) 2 (n) σ̄k−n − 2γ̄k − 2m? γ̄ (n) 2m σ̄ " ? 2nm γ e(0) + 2 Pn−1 h=1 2m? (n − h) γ e(h) #) ? ? −m h γ e (h) − which concludes the proof. P2n−1 h=n ? m (2n − h) γ e(h) ,	10
ACCEPTED BY IEEE TRANSACTIONS ON CYBERNETICS 1 Learning Domain-Invariant Subspace using Domain Features and Independence Maximization arXiv:1603.04535v2 [cs.CV] 22 Jun 2017 Ke Yan, Lu Kou, and David Zhang, Fellow, IEEE Abstract—Domain adaptation algorithms are useful when the distributions of the training and the test data are different. In this paper, we focus on the problem of instrumental variation and time-varying drift in the field of sensors and measurement, which can be viewed as discrete and continuous distributional change in the feature space. We propose maximum independence domain adaptation (MIDA) and semi-supervised MIDA (SMIDA) to address this problem. Domain features are first defined to describe the background information of a sample, such as the device label and acquisition time. Then, MIDA learns a subspace which has maximum independence with the domain features, so as to reduce the inter-domain discrepancy in distributions. A feature augmentation strategy is also designed to project samples according to their backgrounds so as to improve the adaptation. The proposed algorithms are flexible and fast. Their effectiveness is verified by experiments on synthetic datasets and four realworld ones on sensors, measurement, and computer vision. They can greatly enhance the practicability of sensor systems, as well as extend the application scope of existing domain adaptation algorithms by uniformly handling different kinds of distributional change. Index Terms—Dimensionality reduction, domain adaptation, drift correction, Hilbert-Schmidt independence criterion, machine olfaction, transfer learning I. I NTRODUCTION I N many real-world machine learning problems, the labeled training data are from a source domain and the test ones are from a target domain. Samples of the two domains are collected under different conditions, thus have different distributions. Labeling samples in the target domain to develop new prediction models is often labor-intensive and time-consuming. Therefore, domain adaptation or transfer learning is needed to improve the performance in the target domain by leveraging unlabeled (and maybe a few labeled) target samples [1]. This topic is receiving increasing attention in recent years due to its broad applications such as computer vision [2], [3], [4] and text classification [5], [6]. It is also important in the field of sensors and measurement. Because of the variations in the The work is partially supported by the GRF fund from the HKSAR Government, the central fund from Hong Kong Polytechnic University, the NSFC fund (61332011, 61272292, 61271344), Shenzhen Fundamental Research fund (JCYJ20150403161923528, JCYJ20140508160910917), and Key Laboratory of Network Oriented Intelligent Computation, Shenzhen, China. K. Yan is with the Department of Electronic Engineering, Graduate School at Shenzhen, Tsinghua University, Shenzhen 518055, China (e-mail: [email protected]). L. Kou is with the Department of Computing, The Hong Kong Polytechnic University, Kowloon, Hong Kong (e-mail: [email protected]). D. Zhang is with the Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen 518055, China, and also with the Department of Computing, Biometrics Research Centre, The Hong Kong Polytechnic University, Kowloon, Hong Kong (e-mail: [email protected]). fabrication of sensors and devices, the responses to the same signal source may not be identical for different instruments, which is known as instrumental variation. Furthermore, the sensing characteristics of the sensors, the operating condition, or even the signal source itself, can change over time, which leads to complex time-varying drift. As a result, the prediction model trained with the samples from the initial device in an earlier time period (source domain) is not suitable for new devices or in a latter time (target domains). A typical application plagued by this problem is machine olfaction, which uses electronic noses (e-noses) and pattern recognition algorithms to predict the type and concentration of odors [7]. The applications of machine olfaction range from agriculture and food to environmental monitoring, robotics, biometrics, and disease analysis [8], [9], [10], [11]. However, owing to the nature of chemical sensors, many e-noses are prone to instrumental variation and time-varying drift mentioned above [12], [13], which greatly hamper their usage in real-world applications. Traditional methods dealing with these two kinds of drift (“drift correction” methods hereinafter) require a set of transfer samples, which are predefined gas samples needed to be collected with each device and in each time period [12], [10], [14], [15]. They are often used to learn regression models to map the features in the target domain to the source domain [10], [14]. Nevertheless, collecting transfer samples repeatedly is a demanding job especially for nonprofessional e-nose users. In such cases, domain adaptation techniques with unlabeled target samples are desirable. An intuitive idea is to reduce the inter-domain discrepancy in the feature level, i.e. to learn domain-invariant feature representation [5], [16], [17], [3], [18], [19], [20], [21], [22]. For example, Pan et al. [5] proposed transfer component analysis (TCA), which finds a latent feature space that minimizes the distributional difference of two domains in the sense of maximum mean discrepancy. More related methods will be introduced in Section II-A. When applied to drift correction, however, existing domain adaptation algorithms are faced with two difficulties. First, they are designed to handle discrete source and target domains. In time-varying drift, however, samples come in a stream, so the change in data distribution is often continuous. One solution is to split data into several batches, but it will lose the temporal order information. Second, because of the variation in the sensitivity of chemical sensors, the same signal in different conditions may indicate different concepts. In other words, the conditional probability P (Y \|X) may change for samples with different backgrounds, where “background” means when and with which device a sample was collected. Methods like ACCEPTED BY IEEE TRANSACTIONS ON CYBERNETICS TCA project all samples to a common subspace, hence the samples with similar appearance but different concepts cannot be distinguished. In this paper, we present a simple yet effective algorithm called maximum independence domain adaptation (MIDA). The algorithm first defines “domain features” for each sample to describe its background. Then, it finds a latent feature space in which the samples and their domain features are maximally independent in the sense of Hilbert-Schmidt independence criterion (HSIC) [23]. Thus, the discrete and continuous change in distribution can be handled uniformly. In order to project samples according to their backgrounds, feature augmentation is performed by concatenating the original feature vector with the domain features. We also propose semi-supervised MIDA (SMIDA) to exploit the label information with HSIC. MIDA and SMIDA are both very flexible. (1) They can be applied in situations with single or multiple source or target domains thanks to the use of domain features. In fact, the notion “domain” has been extended to “background” which is more informative. (2) Although they are designed for unsupervised domain adaptation problems (no labeled sample in target domains), the proposed methods naturally allow both unlabeled and labeled samples in any domains, thus can be applied in semi-supervised (both unlabeled and labeled samples in target domains) and supervised (only labeled samples in target domains) problems as well. (3) The label information can be either discrete (binary- or multi-class classification) or continuous (regression). To illustrate the effect of our algorithms, we first evaluate them on several synthetic datasets. Then, drift correction experiments are performed on two e-nose datasets and one spectroscopy dataset. Note that spectrometers suffer the same instrumental variation problem as e-noses [24]. Finally, a domain adaptation experiment is conducted on a well-known object recognition benchmark: Office+Caltech [25]. Results confirm the effectiveness of the proposed algorithms. The rest of the paper is organized as follows. Related work on unsupervised domain adaptation and HSIC is briefly reviewed in Section II. Section III describes domain features, MIDA, and SMIDA in detail. The experimental configurations and results are presented in Section IV, along with some discussions. Section V concludes the paper. 2 information between all samples and their binary domain labels, which can be viewed as a primitive version of the domain features used in this paper. They also minimized the negated mutual information between the target samples and their cluster labels to reduce the expected classification error. The low-rank transfer subspace learning (LTSL) algorithm presented in [19] is a reconstruction guided knowledge transfer method. It aligns source and target data by representing each target sample with some local combination of source samples in the projected subspace. The label and geometry information can be retained by embedding different subspace learning methods into LTSL. Another class of methods first project the source and the target data into separate subspaces, and then build connections between them [17], [25], [26], [3]. Fernando et al. [17] utilized a transformation matrix to map the source subspace to the target one, where a subspace was represented by eigenvectors of PCA. The geodesic flow kernel (GFK) method [25] measures the geometric distance between two different domains in a Grassmann manifold by constructing a geodesic flow. An infinite number of subspaces are combined along the flow in order to model a smooth change from the source to the target domain. Liu et al. [26] adapted GFK to correct timevarying drift of e-noses. A sample stream is first split into batches according to the acquisition time. The first and the latest batches (domains) are then connected through every intermediate batch using GFK. Another improvement of GFK is domain adaptation by shifting covariance (DASC) [3]. Observing that modeling one domain as a subspace is not sufficient to represent the difference of distributions, DASC characterizes domains as covariance matrices and interpolates them along the geodesic to bridge the domains. B. Hilbert-Schmidt Independence Criterion (HSIC) HSIC is used as a convenient method to measure the dependence between two sample sets X and Y . Let kx and ky be two kernel functions associated with RKHSs F and G, respectively. pxy is the joint distribution. HSIC is defined as the square of the Hilbert-Schmidt norm of the cross-covariance operator Cxy [23]: HSIC(pxy , F, G) = kCxy k2HS =Exx0 yy0 [kx (x, x0 )ky (y, y 0 )] + Exx0 [kx (x, x0 )]Eyy0 [ky (y, y 0 )] II. R ELATED W ORK A. Unsupervised Domain Adaptation Two good surveys on domain adaptation can be found in [1] and [2]. In this section, we focus on typical methods that extract domain-invariant features. In order to reduce the inter-domain discrepancy while preserving useful information, researchers have developed many strategies. Some algorithms project all samples to a common latent space [5], [16], [19]. Transfer component analysis (TCA) [5] tries to learn transfer components across domains in a reproducing kernel Hilbert space (RKHS) using maximum mean discrepancy. It is further extended to semi-supervised TCA (SSTCA) to encode label information and preserve local geometry of the manifold. Shi et al. [16] measured domain difference by the mutual − 2Exy [Ex0 [kx (x, x0 )]Ey0 [ky (y, y 0 )]]. Here Exx0 yy0 is the expectation over independent pairs (x, y) and (x0 , y 0 ) drawn from pxy . It can be proved that with characteristic kernels kx and ky , HSIC(pxy , F, G) is zero if and only if x and y are independent [27]. A large HSIC suggests strong dependence with respect to the choice of kernels. HSIC has a biased empirical estimate. Suppose Z = X × Y = {(x1 , y1 ), . . . , (xn , yn )}, Kx , Ky ∈ Rn×n are the kernel matrices of X and Y , respectively, then [23]: HSIC(Z, F, G) = (n − 1)−2 tr(Kx HKy H), (1) n×n where H = I − n−1 1n 1T is the centering matrix. n ∈R Due to its simplicity and power, HSIC has been adopted for feature extraction [28], [5], [29] and feature selection ACCEPTED BY IEEE TRANSACTIONS ON CYBERNETICS 3 [27]. Researchers typically use it to maximize the dependence between the extracted/selected features and the label. However, to our knowledge, it has not been utilized in domain adaptation to reduce the dependence between the extracted features and the domain features. III. P ROPOSED M ETHOD A. Domain Feature We aim to reduce the dependence between the extracted features and the background information. A sample’s background information should (1) naturally exist, thus can be easily obtained; (2) have different distributions in training and test samples; (3) correlate with the distribution of the original features. The domain label (which domain a sample belongs) in common domain adaptation problems is an example of such information. According to these characteristics, the information clearly interferes the testing performance of a prediction model. Thus, minimizing the aforementioned dependence is desirable. First, a group of new features need to be designed to describe the background information. The features are called “domain features”. From the perspective of drift correction, there are two main types of background information: the device label (with which device the sample was collected) and the acquisition time (when the sample was collected). We can actually encode more information such as the place of collection, the operation condition, and so on, which will be useful in other domain adaptation problems. Formally, if we only consider the instrumental variation, the following one-hot coding scheme can be used. Suppose there are ndev devices, which result in ndev different but related domains. The domain feature vector is thus d ∈ Rndev , where dp = 1 if the sample is from the pth device and 0 otherwise. If the time-varying drift is also considered, the acquisition time can be further added. If a sample was collected from the pth device at time t, then d ∈ R2ndev , where   1, q = 2p − 1, (2) dq = t, q = 2p,   0, otherwise. According to (1), the kernel matrix Kd of the domain features needs to be computed for HSIC. We apply the linear kernel. Suppose D = [d1 , . . . , dn ] ∈ Rmd ×n , md is the dimension of a domain feature vector. Then Kd = DT D. (3) Note that in traditional domain adaptation problems with several discrete domains, the one-hot coding scheme can be applied to construct domain features, because the problems are similar to instrumental variation. B. Feature Augmentation Feature augmentation is used in this paper to learn background-specific subspaces. In [30], the author proposed a feature augmentation strategy for domain adaptation by replicating the original features. However, this strategy requires that data lie in discrete domains and cannot deal with timevarying drift. We propose a more general and efficient feature augmentation strategy: concatenating the original features and the domain features, i.e. x x̂ = ∈ Rm+md . (4) d The role of this strategy can be demonstrated through a linear dimensionality reduction example. Suppose a projection matrix W ∈ R(m+md )×h has been learned for the augmented feature vector. h is the dimension of the subspace. W has two Wx parts: W = , Wx ∈ Rm×h , Wd ∈ Rmd ×h . The embedWd ding of x̂ can be expressed as W T x̂ = WxT x + WdT d ∈ Rh , which means that a background-specific bias (WdT d)i has been added to each dimension i of the embedding. From another perspective, the feature augmentation strategy maps the samples to an augmented space with higher dimension before projecting them to a subspace. It will be easier to find a projection direction in the augmented space to align the samples well in the subspace. Take machine olfaction for example, there are situations when the conditional probability P (Y \|X) changes along with the background. For instance, the sensitivity of chemical sensors often decays over time. A signal that indicates low concentration in an earlier time actually suggests high concentration in a later time. In such cases, feature augmentation is important, because it allows samples with similar appearance but different concepts to be treated differently by the background-specific bias. The strategy also helps to align the domains better in each projected dimension. Its effect will be illustrated on several synthetic datasets in Section IV-A and further analyzed in the complementary materials. C. Maximum Independence Domain Adaptation (MIDA) In this section, we introduce the formulation of MIDA in detail. Suppose X ∈ Rm×n is the matrix of n samples. The training and the test samples are pooled together. More importantly, we do not have to explicitly differentiate which domain a sample is from. The feature vectors have been augmented, but we use the notations X and m instead of X̂ and m + md for brevity. A linear or nonlinear mapping function Φ can be used to map X to a new space. Based on the kernel trick, we need not know the exact form of Φ, but the inner product of Φ(X) can be represented by the kernel matrix Kx = Φ(X)T Φ(X). Then, a projection matrix W̃ is applied to project Φ(X) to a subspace with dimension h, leading to the projected samples Z = W̃ T Φ(X) ∈ Rh×n . Similar to other kernel dimensionality reduction algorithms [31], [32], the key idea is to express each projection direction as a linear combination of all samples in the space, namely W̃ = Φ(X)W . W ∈ Rn×h is the projection matrix to be actually learned. Thus, the projected samples are Z = W T Φ(X)T Φ(X) = W T Kx . (5) Intuitively, if the projected features are independent of the domain features, then we cannot distinguish the background ACCEPTED BY IEEE TRANSACTIONS ON CYBERNETICS 4 of a sample by its projected features, suggesting that the interdomain discrepancy is diminished in the subspace. Therefore, after omitting the scaling factor in (1), we get the expression to be minimized: tr(Kz HKd H) = tr(Kx W W T Kx HKd H), where Kz is the kernel matrix of Z. In domain adaptation, the goal is not only minimizing the difference of distributions, but also preserving important properties of data, such as the variance [5]. It can be achieved by maximizing the trace of the covariance matrix of the project samples. The covariance matrix is cov(Z) = cov(W T Kx ) = W T Kx HKx W, (6) where H = I − n−1 1n 1T n is the same as that in (1). An orthonormal constraint is further added on W . The learning problem then becomes max W − tr(W T Kx HKd HKx W ) + µ tr(W T Kx HKx W ), s.t. W T W = I, (7) where µ > 0 is a trade-off hyper-parameter. Using the Lagrangian multiplier method, we can find that W is the eigenvectors of Kx (−HKd H + µH)Kx corresponding to the h largest eigenvalues. Note that a conventional constraint is requiring W̃ to be orthonormal as in [29], which will lead to a generalized eigenvector problem. However, we find that this strategy is inferior to the proposed one in both adaptation accuracy and training speed in practice, so it is not used. When computing Kx , a proper kernel function needs to be selected. Common kernel functions include linear (k(x, y) = xT y), polynomial (k(x, y) = (σxT y + 1)d ), Gaussian radial 2 basis function (RBF, k(x, y) = exp( kx−yk 2σ 2 )), and so on. Different kernels indicate different assumptions on the type of dependence in using HSIC [27]. According to [27], the polynomial and RBF kernels map the original features to a higher or infinite dimensional space, thus are able to detect more types of dependence. However, choosing a suitable kernel width parameter (σ) is also important for these more powerful kernels [27]. The maximum mean discrepancy (MMD) criterion is used in TCA [5] to measure the difference of two distributions. Song et al. [27] showed that when HSIC and MMD are both applied to measure the dependence between features and labels in a binary-class classification problem, they are identical up to a constant factor if the label kernel matrix in HSIC is properly designed. However, TCA is feasible only when there are two discrete domains. On the other hand, MIDA can deal with a variety of situations including multiple domains and continuous distributional change. The stationary subspace analysis (SSA) algorithm [33] is able to identify temporally stationary components in multivariate time series. However, SSA only ensures that the mean and covariance of the components are stationary, while they may not be suitable for preserving important properties in data. Concept drift adaptation algorithms [34] are able to correct continuous time-varying drift. However, most of them rely on newly arrived labeled data to update the prediction models, while MIDA works unsupervisedly. D. Semi-supervised MIDA (SMIDA) MIDA aligns samples with different backgrounds without considering the label information. However, if the labels of some samples are known, they can be incorporated into the subspace learning process, which may be beneficial to prediction. Therefore, we extend MIDA to semi-supervised MIDA (SMIDA). Since we do not explicitly differentiate the domain labels of the samples, both unlabeled and labeled samples can exist in any domain. Similar to [28], [5], [29], [27], HSIC is adopted to maximize the dependence between the projected features and the labels. The biggest advantage of this strategy is that all types of labels can be exploited, such as the discrete labels in classification and the continuous ones in regression. The label matrix Y is defined as follows. For c-class classification problems, the one-hot coding scheme can be used, i.e. Y ∈ Rc×n , yi,j = 1 if xi is labeled and belongs to the jth class; 0 otherwise. For regression problems, the target values can be centered first. Then, Y ∈ R1×n , yi equals to the target value of xi if it is labeled; 0 otherwise. The linear kernel function is chosen for the label kernel matrix, i.e. Ky = Y T Y. (8) The objective of SMIDA is max W s.t. tr(W T Kx (−HKd H + µH + γHKy H)Kx W ), W T W = I, (9) where γ > 0 is a trade-off hyper-parameter. Its solution is the eigenvectors of Kx (−HKd H + µH + γHKy H)Kx corresponding to the h largest eigenvalues. The outline of MIDA and SMIDA is summarized in Algorithm III.1. The statements in brackets correspond to those specialized for SMIDA. Algorithm III.1 MIDA [or SMIDA] Input: The matrix of all samples X and their background information; [the labels of some samples]; the kernel function for X; h, µ, [and γ]. Output: The projected samples Z. 1: Construct the domain features according to the background information, e.g. Section III-A. 2: Augment the original features with domain features (4). 3: Compute the kernel matrices Kx , Kd (3), [and Ky (8)]. 4: Obtain W , namely the eigenvectors of Kx (−HKd H + µH)Kx [or Kx (−HKd H + µH + γHKy H)Kx ] corresponding to the h largest eigenvalues. 5: Z = W T Kx . Besides variance and label dependence, another useful property of data is the geometry structure, which can be preserved by manifold regularization (MR) [35]. MR can be conveniently incorporated into SMIDA. In our experiments, adding MR generally increases the accuracy slightly with the cost of three more hyper-parameters. Consequently, it is not adopted in this paper. ACCEPTED BY IEEE TRANSACTIONS ON CYBERNETICS IV. E XPERIMENTS In this section, we first conduct experiments on some synthetic datasets to verify the effect of the proposed methods. Then, drift correction experiments are performed on two enose datasets and a spectroscopy dataset. To show the universality of the proposed methods, we further evaluate them on a visual object recognition dataset. Comparison is made between them and recent unsupervised domain adaptation algorithms that learn domain-invariant features. A. Synthetic Dataset In Fig. 1, TCA [5] and MIDA are compared on a 2D dataset with two discrete domains. The domain labels were used to construct the domain features in MIDA according to the one-hot coding scheme introduced in Section III-A. The similar definition was used in synthetic datasets 3 and 4. For both methods, the linear kernel was used on the original features and the hyper-parameter µ was set to 1. In order to quantitatively assess the effect of domain adaptation, logistic regression models were trained on the labeled source data and tested on the target data. The accuracies are displayed in the caption, showing that the order of performance is MIDA > TCA > original feature. TCA aligns the two domains only on the first projected dimension. However, the two classes have large overlap on that dimension, because the direction for alignment is different from that for discrimination. Incorporating the label information of the source domain (SSTCA) did no help. On the contrary, MIDA can align the two domains well in both projected dimensions, in which the domainspecific bias on the second dimension brought by feature augmentation played a key role. A 3D explanation is included in the supplementary materials. Thus, good accuracy can be obtained by using the two dimensions for classification. In Fig. 2, SSA [33] and MIDA are compared on a 2D dataset with continuous distributional change, which resembles timevarying drift in machine olfaction. Samples in both classes drift to the upper right. The chronological order of the samples was used to construct the domain features in MIDA, i.e. d = 1 for the first sample, d = 2 for the second sample, etc. The parameter setting of MIDA was the same with that in Fig. 1, whereas the number of stationary components in SSA was set to 1. The classification accuracies were obtained by training a logistic regression model on the first halves of the data in both classes, and testing them on the last halves. SSA succeeds in finding a direction (z1 ) that is free from time-varying drift. However, the two classes cannot be well separated in that direction. In plot (c), the randomly scattered colors suggest that the time-varying drift is totally removed in the subspace. MIDA first mapped the 2D data into a 3D space with the third dimension being time, then projected them to a 2D plane orthogonal to the direction of drift in the 3D space. No label information was used in the last two experiments. If keeping the label dependence in the subspace is a priority, SMIDA can be adopted instead of MIDA. In the 3D synthetic dataset in Fig. 3, the best direction (x3 ) to align the two domains also mixes the two classes, which results in the output of MIDA in plot (b). The labels in the source domain 5 were used when learning the subspace. From plot (c), we can observe that the classes are separated. In fact, class separation can still be found in the third dimension of the space learned by MIDA. However, for the purpose of dimensionality reduction, we generally hope to keep the important information in the first few dimensions. Nonlinear kernels are often applied in machine learning algorithms when data is not linearly separable. Besides, they are also useful in domain adaptation when domains are not linearly “alignable”, as shown in Fig. 4. In plot (a), the inter-domain changes in distributions are different for the two classes. Hence, it is difficult to find a linear projection direction to align the two domains, even with the domainspecific biases of MIDA. Actually, domain-specific rotation matrices are needed. Since the target labels are not available, the rotation matrices cannot be obtained accurately. However, a nonlinear kernel can be used to map the original features to a space with higher dimensions, in which the domains may be linearly alignable. We applied an RBF kernel with width σ = 10. Although the domains are not perfectly aligned in plot (c), the classification model trained in the source domain can be better adapted to the target domain. A comparison on different kernel and kernel parameters on two synthetic datasets is included in the supplementary materials. B. Gas Sensor Array Drift Dataset The gas sensor array drift dataset1 collected by Vergara et al. [36] is dedicated to research in drift correction. A total of 13910 samples were collected by an e-nose with 16 gas sensors over a course of 36 months. There are six different kinds of gases at different concentrations. They were split into 10 batches by the authors according to their acquisition time. Table A in the supplementary material details the dataset. We aim to classify the type of gases, despite their concentrations. Similar to [36], [26], we took the samples in batch 1 as labeled training samples, whereas those in batches 2–10 are unlabeled test ones. This evaluation strategy resembles the situation in real-world applications. In the dataset, each sample is represented by 128 features extracted from the sensors’ response curves [36]. Each feature was first normalized to zero mean and unit variance within each batch. The timevarying drift of the preprocessed features across batches can be visually inspected in Fig. 5. It is obvious that samples in different batches have different distributions. Next, the labeled samples in batch 1 were adopted as the source domain and the unlabeled ones in batch b (b = 2, . . . , 10) as the target domain. The proposed algorithms together with several recent ones were used to learn domain-invariant features based on these samples. Then, a logistic regression model was trained on the source domain and tested on each target one. For multiclass classification, the one-vs-all strategy was utilized. As displayed in Table I, the compared methods include kernel PCA (KPCA), transfer component analysis (TCA), semi-supervised TCA (SSTCA) [5], subspace alignment (SA) 1 http://archive.ics.uci.edu/ml/datasets/ Gas+Sensor+Array+Drift+Dataset+at+Different+Concentrations ACCEPTED BY IEEE TRANSACTIONS ON CYBERNETICS 2 6 Pos. source data 15 Neg. source data Pos. target data 10 Neg. target data 1.5 1 2 1.5 1 5 0.5 0.5 x z2 z2 2 0 0 0 −5 −0.5 −0.5 −10 −1 −1.5 −15 (a) −2 −2 −1 0 x1 1 2 −1 −1.5 (b) −20 −10 −5 0 z1 5 (c) −2 −10 10 −5 0 z1 5 10 Fig. 1. Comparison of TCA and MIDA in a 2D synthetic dataset. Plots (a)-(c) show data in the original space and projected spaces of TCA and MIDA, respectively. The classification accuracies are 53%, 70% (only using the first projected dimension z1 ), and 88%. 16 10 20 14 15 8 12 10 10 6 5 z2 z2 x 2 8 4 0 6 4 −5 2 Pos. old data Neg. old data Pos. new data Neg. new data 2 0 (a) −2 −2 0 2 x1 4 −10 0 −15 (b) −2 −4 6 −2 0 z1 2 (c) −20 −80 4 −60 −40 −20 z1 0 20 40 Fig. 2. Comparison of SSA and MIDA in a 2D synthetic dataset. Plots (a)-(c) show data in the original space, projected spaces of SSA and MIDA, respectively. The chronological order of a sample is indicated by color. The classification accuracies are 55%, 74% (only using the first projected dimension z1 ), and 90%. 3 3 Pos. source data Neg. source data Pos. target data 2 Neg. target data (a) 2 1 5 4 (b) (c) 3 1 0 z2 z2 x3 2 0 0 −1 −1 1 −1 −2 −2 −2 0 2 1.5 2 −0.5 0 0.5 x1 1 x −2 −3 −10 −1 −5 0 −3 −4 5 −2 0 z1 2 4 6 z1 Fig. 3. Comparison of MIDA and SMIDA in a 3D synthetic dataset. Plots (a)-(c) show data in the original space and projected spaces of MIDA and SMIDA, respectively. The classification accuracies are 50%, 55%, and 82%. 4 Pos. source data Neg. source data Pos. target data Neg. target data 3 5 1.9 4 1.8 3 1.7 2 1.6 1 1.5 z2 x 2 1 0 z2 2 0 1.4 −1 1.3 −1 −2 −2 −3 −4 −3 (a) −2 0 2 x 1 4 6 −4 −60 1.2 1.1 (b) −40 −20 0 z 1 20 40 60 1 (c) 9 9.5 10 10.5 z 1 Fig. 4. Comparison of different kernels in a 2D synthetic dataset. Plots (a)-(c) show data in the original space and projected spaces of MIDA with linear and RBF kernels, respectively. The classification accuracies are 50%, 57%, and 87%. ACCEPTED BY IEEE TRANSACTIONS ON CYBERNETICS 7 75 Average classification accuracy (%) 5 PC2 0 −5 Batch 1 Batch 3 Batch 5 Batch 7 Batch 9 −10 −15 −30 −20 −10 0 10 70 65 60 55 50 20 PC1 Fig. 5. Scatter of ethanol (dots) and acetone (plus signs) samples in batches 1,3,5,7,9 in the gas sensor array drift dataset. Samples are projected to a 2D subspace using PCA. Different colors indicate different batches. [17], geodesic flow kernel (GFK) [25], manifold regularization with combination GFK (ML-comGFK) [26], informationtheoretical learning (ITL) [16], structural correspondence learning (SCL) [20], and marginalized stacked denoising autoencoder (mSDA) [21]. For all methods, the hyper-parameters were tuned for the best accuracy. In KPCA, TCA, SSTCA, and the proposed MIDA and SMIDA, the polynomial kernel with degree 2 was used. KPCA learned a subspace based on the union of source and target data. In TCA, SSTCA, MIDA, and SMIDA, eigenvalue decomposition needs to be done on kernel matrices. In order to reduce the computational burden, we randomly chose at most nt samples in each target domain when using these methods, with nt being twice the number of the samples in the source domain. GFK used PCA to generate the subspaces in both source and target domains. The subspace dimension of GFK was determined according to the subspace disagreement measure in [25]. The results of ML-comGFK are copied from [26]. In SCL, the pivot features were binarized before training pivot predictors using logistic regression. We also compared several variants of our methods. In Table I, the notation “(discrete)” means that two discrete domains (source and target) were used in MIDA and SMIDA, which is similar to other compared methods. The domain feature vector of a sample was thus [1, 0]T if it was from the source domain and [0, 1]T if it was from the target. However, this strategy cannot make use of the samples in intermediate batches. An intuitive assumption is that the distributions of adjacent batches should be similar. When adapting the information from batch 1 to b, taking samples from batches 2 to b − 1 into consideration may improve the generalization ability of the learned subspace. Concretely, nt samples were randomly selected from batches 2 to b instead of batch b alone. For each sample, the domain feature was defined as its batch index, which can be viewed as a proxy of its acquisition time. MIDA and SMIDA then maximized the independence between KPCA TCA SSTCA SA ITL MIDA (continuous) SMIDA (continuous) 0 20 40 60 # Projected dimensions (h) 80 100 Fig. 6. Performance comparison on the gas sensor array drift dataset with respect to the subspace dimension h. the learned subspace and the batch indices. The results are labeled as “(continuous)” in Table I. Besides, the accuracies of continuous SMIDA without feature augmentation (no aug.) are also shown. From Table I, we can find that as the batch index increases, the accuracies of all methods generally degrade, which confirms the influence of the time-varying drift. Continuous SMIDA achieves the best average domain adaptation accuracy. The continuous versions of MIDA and SMIDA outperform the discrete versions, proving that the proposed methods can effectively exploit the chronological information of the samples. They also surpass ML-comGFK which uses the samples in intermediate batches to build connections between the source and the target batches. Feature augmentation is important in this dataset, since removing it in continuous SMIDA causes a drop of four percentage points in average accuracy. In Fig. 6, the average classification accuracies with varying subspace dimension are shown. MIDA and SMIDA are better than other methods when more than 30 features are extracted. C. Breath Analysis Dataset As a noninvasive approach, disease screening and monitoring with e-noses is attracting more and more attention [8], [11]. The concentration of some biomarkers in breath has been proved to be related to certain diseases, which makes it possible to analyze a person’s health state with an e-nose conveniently. For example, the concentration of acetone in diabetics’ breath is often higher than that in healthy people [11]. However, the instrumental variation and time-varying drift of e-noses hinder the popularization of this technology in real-world applications. Unsupervised domain adaptation algorithms can be applied to solve this problem. We have collected a breath analysis dataset in years 2014– 2015 using two e-noses of the same model [11]. In this paper, samples of five diseases were selected for experiments, including diabetes, chronical kidney disease (CKD), cardiopathy, lung cancer, and breast cancer. They have been proved to be related to certain breath biomarkers. We performed ACCEPTED BY IEEE TRANSACTIONS ON CYBERNETICS 8 TABLE I C LASSIFICATION ACCURACY (%) ON THE GAS SENSOR ARRAY DRIFT DATASET. B OLD VALUES INDICATE THE BEST RESULTS . Batch 2 3 4 5 6 7 8 9 10 Average 80.47 79.26 69.57 77.16 77.39 64.21 52.04 47.87 48.78 66.30 KPCA 75.88 69.04 49.07 57.87 62.65 52.26 37.07 47.66 49.97 55.72 TCA [5] 82.96 81.97 65.22 76.14 89.09 58.98 49.32 66.17 49.50 68.82 SSTCA [5] 84.57 80.90 80.12 75.63 87.26 66.37 54.76 61.28 54.44 71.70 SA [17] 80.79 80.01 71.43 75.63 78.35 64.68 52.04 48.51 49.58 66.78 GFK [25] 77.41 80.26 71.43 76.14 77.65 64.99 36.39 47.45 48.72 64.49 ML-comGFK [26] 80.25 74.99 78.79 67.41 77.82 71.68 49.96 50.79 53.79 67.28 ITL [16] 76.85 79.45 59.63 96.45 78.00 60.95 49.32 77.02 48.58 69.58 SCL [20] 77.57 82.03 68.32 82.74 77.22 65.18 53.74 48.51 48.08 67.04 mSDA [21] 73.87 79.19 65.84 80.20 76.39 65.90 51.70 48.51 48.92 65.61 MIDA (discrete) 81.03 85.62 60.25 75.63 87.61 62.44 48.30 67.87 48.36 68.57 SMIDA (discrete) 80.47 87.07 65.22 75.63 90.04 59.20 50.00 62.77 44.81 68.36 MIDA (continuous) 84.32 81.59 68.32 75.63 91.74 63.13 78.91 62.34 45.14 72.35 SMIDA (no aug.) 82.23 83.17 67.70 75.13 85.22 61.67 51.02 61.49 54.61 69.14 SMIDA (continuous) 83.68 82.28 73.91 75.63 93.00 63.49 79.25 62.34 45.50 73.23 1 0.8 (a) 0.6 0.4 0.2 0 0 Sensor response (Volt) five binary-class classification tasks to distinguish samples with one disease from the healthy samples. Each sample was represented by the steady state responses of nine gas sensors in the e-nose. When a gas sensor is used to sense a gas sample, its response will reach a steady state in a few minutes. The steady state response has a close relationship with the concentration of the measured gas. Therefore, the 9D feature vector contains most information needed for disease screening. To show the instrumental variation and time-varying drift in the dataset, we draw the steady state responses of two sensors of the CKD samples in Fig. 7. Each data point indicates a breath sample. In plot (a), the sensitivity of the sensor in both devices gradually decayed as time elapsed. In plot (b), the aging effect was so significant that we had to replace the sensors in the two devices with new ones on about day 200. In this case, a signal at 0.3 V will suggest low concentration on day 0 but high concentration on day 150. In addition, the responses in different devices are different (e.g. plot (b), after day 200). The numbers of samples in the six classes (healthy and the five diseases mentioned above) are 125, 431, 340, 97, 156, and 215, respectively. We chose the first 50 samples collected with device 1 in each class as labeled training samples. Among the other samples, 10 samples were randomly selected in each class for validation, the rest for testing. The hyper-parameters were tuned on the validation sets. Logistic regression was adopted as the classifier, with F-score as the accuracy criterion. Results are compared in Table II. In KPCA, TCA, SSTCA, MIDA, and SMIDA, the RBF kernel was used. Because methods other than stationary subspace analysis (SSA) [33], MIDA, and SMIDA are not capable of handling the chronological information, we simply regarded each device as a discrete domain and learned device-invariant features with them. The same strategy was used in discrete MIDA and SMIDA. In continuous MIDA and SMIDA, the Sensor response (Volt) Original feature 50 100 150 200 250 300 Acquisition time (day) 350 400 450 Device 1 Device 2 1.5 (b) 1 0.5 0 0 50 100 150 200 250 300 Acquisition time (day) 350 400 450 Fig. 7. Illustration of the instrumental variation and time-varying drift in the breath analysis dataset. Plots (a) and (b) show the steady state responses of the CKD samples of sensors 2 and 7, respectively. domain features were defined according to (4), where t was the exact acquisition time converted to years and the number of devices ndev = 2. SSA naturally considers the chronological information by treating the sample stream as a multivariate time series and identifying temporally stationary components. However, SSA cannot deal with time series with multiple sources, such as the multi-device case in this dataset. Thus, the samples were arranged in chronological order despite their device labels. From Table II, we can find that the improvement made by SSA is little, possibly because the stationary criterion is not suitable for preserving important properties in data. For example, the noise in data can also be stationary [5]. MIDA and SMIDA achieved obviously better results than other methods. They can address both instrumental variation and ACCEPTED BY IEEE TRANSACTIONS ON CYBERNETICS 9 TABLE II C LASSIFICATION ACCURACY (%) ON THE BREATH ANALYSIS DATASET. B OLD VALUES INDICATE THE BEST RESULTS . Task 1 2 3 4 5 Average Original feature 34.34 63.67 73.71 43.17 42.93 51.57 KPCA 58.05 72.58 84.78 44.95 42.60 60.59 TCA [5] 67.19 68.31 59.93 67.08 68.17 66.14 SSTCA [5] 67.01 68.06 74.14 68.31 67.36 68.97 SA [17] 29.95 72.42 72.74 42.19 44.54 52.37 GFK [25] 41.49 68.50 58.96 75.63 70.16 62.95 ITL [16] 68.59 66.53 74.75 66.67 68.03 68.91 SSA [33] 49.77 72.10 33.49 52.64 55.38 52.68 SCL [20] 32.52 61.16 75.43 35.35 51.86 51.26 mSDA [21] 36.86 69.51 76.69 35.51 50.49 53.81 MIDA (discrete) 62.17 71.74 84.21 67.05 67.06 70.45 SMIDA (discrete) 80.16 84.18 88.47 68.45 52.41 74.73 MIDA (continuous) 68.30 67.54 74.01 73.04 69.63 70.50 SMIDA (no aug.) 82.80 72.57 72.61 80.33 70.05 75.67 SMIDA (continuous) 85.29 80.18 91.67 74.28 66.55 79.59 time-varying drift. With the background-specific bias brought by feature augmentation, they can compensate for the change in conditional probability in this dataset. SMIDA is better than MIDA because the label information of the first 50 samples in each class was better kept. D. Corn Dataset Similar to e-noses, data collected with spectrometers are one-dimensional signals indicating the concentration of the analytes. Instrumental variation is also a problem for them [24]. In this section, we test our methods on the corn dataset2 . It is a spectroscopy dataset collected with three near-infrared spectrometers designated as m5, mp5, and mp6. The moisture, oil, protein, and starch contents of 80 corn samples were measured by each device, with ranges of the measured values as [9.377, 10.993], [3.088, 3.832], [7.654, 9.711], and [62.826, 66.472], respectively. Each sample is represented by a spectrum with 700 features. This dataset resembles traditional domain adaptation datasets because there is no time-varying drift. Three discrete domains can be defined based on the three devices. We adopt m5 as the source domain, mp5 and mp6 as the target ones. In each domain, samples 4, 8, . . . , 76, 80 were assigned as the test set, the rest as the training set. For hyper-parameter tuning, we applied a three-fold crossvalidation on the training sets of the three domains. After the best hyper-parameters were determined for each algorithm, a regression model was trained on the training set from the source domain and applied on the test set from the target domains. The regression algorithm was ridge regression with the L2 regularization parameter λ = 1. Table III displays the root mean square error (RMSE) of the four prediction tasks and their average on the two target domains. We also plot the overall average RMSE of 2 http://www.eigenvector.com/data/Corn/ the two domains with respect to the subspace dimension h in Fig. 8. ITL was not investigated because it is only applicable in classification problems. In KPCA, TCA, SSTCA, MIDA, and SMIDA, the RBF kernel was used. For the semisupervised methods SSTCA and SMIDA, the target values were normalized to zero mean and unit variance before subspace learning. The domain features were defined according to the device indices using the one-hot coding scheme. We can find that when no domain adaptation was done, the prediction error is large. All domain adaptation algorithms managed to significantly reduce the error. KPCA also has good performance, which is probably because the source and the target domains have similar principal directions, which also contain the most discriminative information. Therefore, source regression models can fit the target samples well. In this dataset, different domains have identical data composition. As a result, corresponding data can be aligned by subspaces alignment, which explains the small error of SA. However, this condition may not hold in other datasets. MIDA and SMIDA obtained the lowest average errors in both target domains. Aiming at exploring the prediction accuracy when there is no instrument variation, we further trained regression models on the training set of the two target domains and tested on the same domain. The results are listed as “train on target” in Table III. It can be found that SMIDA outperforms these results. This could be attributed to three reasons: (1) The inter-domain discrepancy in this dataset is relatively easy to correct; (2) The use of RBF kernel in SMIDA improves the accuracy; (3) SMIDA learned the subspace on the basis of both training and test samples. Although the test samples were unlabeled, they can provide some information about the distribution of the samples to make the learned subspace generalize better, which can be viewed as the merit of semi-supervised learning. To testify this assumption, we conducted another experiment with multiple target domains. The training samples from the source domain and the test ones from both target domains were leveraged together for subspace learning in MIDA and SMIDA. The average RMSE for the two target domains are 0.209 and 0.217 for MIDA, and 0.208 and 0.218 for SMIDA. Compared with the results in Table III with single target domain, the results have been further improved, showing that incorporating more unlabeled samples from target domains can be beneficial. E. Visual Object Recognition Dataset In [25], Gong et al. evaluated domain adaptation algorithms on four visual object recognition datasets, namely Amazon (A), Caltech-256 (C), DSLR (D), and Webcam (W). Ten common classes were selected from them, with 8 to 151 samples per class per domain, and 2533 images in total. Each image was encoded with an 800-bin histogram using SURF features. The normalized histograms were z-scored to have zero mean and unit variance in each dimension. Following the experimental setting provided in the sample code from the authors of [25], experiments were conducted in 20 random trials for each pair of domains. For each unsupervised trail, 20 (for A, C, W) or 8 (for D) labeled samples per class ACCEPTED BY IEEE TRANSACTIONS ON CYBERNETICS 10 TABLE III R EGRESSION RMSE ON THE CORN DATASET. B OLD VALUES INDICATE THE BEST RESULTS . Mp5 as target domain Mp6 as target domain Moisture Oil Moisture Oil Protein Starch Average Original feature 1.327 0.107 1.155 2.651 1.310 1.433 0.101 1.413 2.776 1.431 KPCA 0.477 0.165 0.215 0.315 0.293 0.396 0.164 0.238 0.290 0.272 TCA [5] 0.539 0.322 0.217 0.402 0.370 0.398 0.145 0.259 0.572 0.343 SSTCA [5] 0.343 0.093 0.140 0.366 0.235 0.367 0.088 0.186 0.318 0.240 SA [17] 0.302 0.094 0.186 0.351 0.233 0.324 0.079 0.158 0.390 0.238 GFK [25] 0.267 0.197 0.342 0.621 0.357 0.263 0.189 0.264 0.485 0.301 SCL [20] 0.283 0.115 0.249 0.619 0.316 0.311 0.108 0.257 0.683 0.340 mSDA [21] 0.264 0.107 0.211 0.446 0.257 0.285 0.097 0.198 0.471 0.263 MIDA 0.317 0.078 0.141 0.378 0.228 0.317 0.084 0.158 0.352 0.228 SMIDA 0.287 0.072 0.143 0.339 0.210 0.316 0.073 0.152 0.325 0.217 Train on target 0.176 0.094 0.201 0.388 0.215 0.182 0.108 0.206 0.414 0.228 0.5 KPCA TCA SSTCA SA MIDA SMIDA Average regression RMSE 0.45 0.4 0.35 0.3 0.25 0.2 0 10 Protein Starch Average 20 30 40 # Projected dimensions (h) 50 60 Fig. 8. Performance comparison on the corn dataset with respect to the subspace dimension h. were randomly chosen from the source domain as the training set (other samples were used unsupervisedly for domain adaptation), while all unlabeled samples in the target domain made up the test set. In semi-supervised trails, three labeled samples per class in the target domain were also assumed to be labeled. Averaged accuracies on each pair of domains as well as standard errors are listed in Tables IV and V. For GFK, low-rank transfer subspace learning (LTSL), domain adaptation by shifting covariance (DASC), and a recent method called integration of global and local metrics for domain adaptation (IGLDA), we copied the best results reported in the original papers [18], [19], [3], [22]. For other methods tested, the hyper-parameters were tuned for the best accuracy. Logistic regression was adopted as the classifier. The polynomial kernel with degree 2 was used in KPCA, TCA, SSTCA, MIDA, and SMIDA. The domain features were defined according to the domain labels using the onehot coding scheme. MIDA and SMIDA achieve the best average accuracies in both unsupervised and semi-supervised visual object recognition experiments. We observe that TCA and SSTCA have comparable performance with MIDA and SMIDA, which may be explained by the fact that the HSIC criterion used in MIDA and MMD used in TCA are identical under certain conditions when there are one source and one target domain [27]. Besides, the feature augmentation strategy in MIDA is not crucial in this dataset because there is no change in conditional probability. On the other hand, TCA and SSTCA can only handle one source and one target domains. SSTCA uses the manifold regularization strategy to preserve local geometry information, hence introduces three more hyper-parameters than SMIDA. Moreover, computing the data adjacency graph in SSTCA and the matrix inversion operation in TCA and SSTCA make them slower than MIDA and SMIDA. We compared their speed on the domain adaptation experiment C → A. They were run on a server with Intel Xeon 2.00 GHz CPU and 128 GB RAM. No parallel computing was used. The codes of the algorithms were written in Matlab R2014a. On average, the running times of each trial of MIDA, SMIDA, TCA, and SSTCA were 2.4 s, 2.5 s, 3.0 s, and 10.2 s, respectively. Therefore, MIDA and SMIDA are more practical to use than TCA and SSTCA. Besides, they were initially designed for drift correction. This dataset is used to show their universality. V. C ONCLUSION In this paper, we introduced maximum independence domain adaptation (MIDA) to learn domain-invariant features. The main idea of MIDA is to reduce the inter-domain discrepancy by maximizing the independence between the learned features and the domain features of the samples. The domain features describe the background information of each sample, such as the domain label in traditional domain adaptation problems. In the field of sensors and measurement, the device label and acquisition time of the each collected sample can be expressed by the domain features, so that unsupervised drift correction can be achieved by using MIDA. The feature augmentation strategy proposed in this paper adds domainspecific biases to the learned features, which helps MIDA to align domains. ACCEPTED BY IEEE TRANSACTIONS ON CYBERNETICS 11 TABLE IV U NSUPERVISED DOMAIN ADAPTATION ACCURACY (%) ON THE VISUAL OBJECT RECOGNITION DATASET. B OLD VALUES INDICATE THE BEST RESULTS . X → Y MEANS THAT X IS THE SOURCE DOMAIN AND Y IS THE TARGET ONE . C→A Ori. ft. D→A W→A A→C D→C W→C A→D C→D W→D A→W C→W D→W Average 43.2±2.2 34.9±1.1 36.8±0.6 38.5±1.6 31.7±1.2 32.7±0.9 37.3±3.1 40.5±3.6 80.6±2.0 37.5±2.9 37.1±3.6 76.7±2.0 43.97 KPCA 27.4±2.0 27.0±1.6 27.6±1.2 25.8±1.3 22.0±2.4 23.7±1.6 29.2±2.9 27.6±2.7 55.1±1.8 29.0±2.7 27.1±3.2 50.3±2.6 30.97 TCA [5] 49.8±2.7 38.6±1.4 39.2±1.0 42.8±2.1 35.3±1.5 35.7±0.8 42.8±3.2 45.9±3.8 83.9±1.7 41.7±3.3 42.8±5.4 81.5±2.1 48.35 SSTCA [5] 50.5±2.8 39.3±1.6 40.5±0.7 42.4±1.8 36.1±1.5 35.9±0.8 42.8±3.1 46.6±3.5 80.6±2.3 42.5±2.7 42.8±4.7 81.8±1.9 48.48 ITL [16] 41.2±3.0 35.7±2.0 38.4±1.1 34.8±1.8 28.7±1.5 31.4±1.6 34.5±3.0 32.3±3.8 67.0±2.7 31.8±3.6 36.4±4.2 71.1±3.2 40.27 SA [17] 48.4±2.9 36.3±2.6 37.3±1.7 38.5±2.1 33.4±1.9 35.1±0.9 35.0±3.6 39.7±5.0 63.2±2.8 36.7±4.4 41.3±5.5 70.3±2.5 42.93 GFK [18] 40.4±0.7 36.2±0.4 35.5±0.7 37.9±0.4 32.7±0.4 29.3±0.4 35.1±0.8 41.1±1.3 71.2±0.9 35.7±0.9 35.8±1.0 79.1±0.7 42.50 LTSL [19] 50.4±0.4 40.2±0.6 44.1±0.3 38.6±0.3 35.3±0.3 37.4±0.2 38.3±1.1 53.7±0.9 79.8±0.4 38.8±1.3 47.0±1.0 72.8±0.7 48.03 DASC [3] 39.1±0.3 39.3±0.8 37.7±0.7 49.8±0.4 48.5±0.8 45.4±0.9 36.5±0.3 35.6±0.3 88.3±0.4 36.3±0.4 33.3±0.3 79.8±0.9 47.47 SCL [20] 43.5±2.2 34.7±1.3 36.9±0.7 38.5±1.5 31.7±1.3 33.2±0.9 37.8±3.0 40.6±3.6 81.1±1.8 37.6±3.3 37.1±4.0 76.7±2.0 44.10 mSDA [21] 45.3±1.9 37.4±1.2 38.3±0.7 40.3±1.8 33.7±1.3 35.5±1.1 38.1±2.8 40.4±4.0 82.0±1.8 38.5±3.4 37.8±3.7 79.0±2.1 45.51 IGLDA [22] 51.0±2.7 38.4±1.9 38.6±1.2 41.5±1.8 36.4±2.2 34.2±1.5 38.9±2.5 45.1±2.4 82.6±1.8 40.0±3.4 42.2±3.6 82.4±2.4 47.61 MIDA 50.3±2.5 39.2±1.9 39.8±1.0 42.7±1.8 35.5±1.1 35.7±0.7 42.3±2.8 45.7±3.6 82.2±2.0 42.8±2.8 43.6±5.0 82.4±2.0 48.51 SMIDA 50.5±2.4 39.1±1.8 39.8±1.1 42.7±2.0 35.5±1.2 35.4±0.8 42.4±2.6 45.8±3.3 82.5±2.1 42.9±2.8 43.4±5.1 81.9±2.0 48.49 TABLE V S EMI - SUPERVISED DOMAIN ADAPTATION ACCURACY (%) ON THE VISUAL OBJECT RECOGNITION DATASET. B OLD VALUES INDICATE THE BEST RESULTS . C→A D→A W→A A→C D→C W→C A→D C→D W→D A→W C→W D→W Average Ori. ft. 48.8±1.8 44.5±1.6 43.5±1.4 41.6±1.9 36.7±2.1 37.3±1.4 48.1±4.2 49.3±3.2 81.7±2.4 51.0±3.4 50.9±4.4 80.5±2.2 51.17 KPCA 53.3±2.4 46.2±1.7 43.2±1.1 44.1±1.4 39.1±1.8 37.8±1.1 47.3±3.3 53.9±3.4 81.5±2.9 49.7±2.7 54.0±4.1 81.1±2.2 52.60 TCA [5] 55.3±2.2 48.6±1.8 45.7±1.4 46.1±2.0 40.3±2.1 39.7±1.4 52.1±3.0 56.3±4.5 83.7±2.9 55.4±3.5 58.3±4.5 84.2±1.9 55.46 SSTCA [5] 55.3±2.2 48.6±1.8 45.6±1.4 46.0±2.0 40.3±2.1 39.7±1.3 52.1±3.1 56.3±4.6 83.7±2.9 55.4±3.5 58.4±4.5 84.2±1.9 55.47 ITL [16] 51.5±3.1 47.7±2.5 44.1±2.1 40.0±2.2 36.8±3.2 36.6±2.2 44.4±4.1 48.2±4.0 59.7±2.6 51.5±4.5 54.9±3.8 68.5±3.3 48.65 SA [17] 51.8±2.3 47.6±2.8 44.7±1.7 42.6±1.7 36.9±2.9 36.8±2.0 45.3±4.2 49.0±3.9 71.0±2.9 47.2±2.9 49.0±3.6 76.1±2.5 49.82 GFK [18] 46.1±0.6 46.2±0.6 46.2±0.7 39.6±0.4 33.9±0.6 32.3±0.6 50.9±0.9 55.0±0.9 74.1±0.9 56.9±1.0 57.0±0.9 80.2±0.4 51.53 LTSL [19] 50.4±0.5 47.4±0.5 47.8±0.4 39.8±0.4 36.7±0.4 38.5±0.3 59.1±0.7 59.6±0.6 82.6±0.5 59.5±1.1 59.5±0.8 78.3±0.4 54.93 SCL [20] 48.8±1.7 45.0±1.4 43.4±1.3 41.3±1.8 36.3±2.2 37.4±1.4 48.9±4.4 49.3±3.5 81.8±2.5 51.9±3.7 52.0±4.4 81.0±2.2 51.42 mSDA [21] 50.4±2.2 48.1±1.6 45.6±1.6 43.6±1.9 38.9±2.2 39.3±1.5 48.9±4.5 49.2±4.9 82.3±2.5 52.9±3.9 52.1±4.4 81.6±2.1 52.75 MIDA 55.2±2.2 48.6±1.8 45.6±1.4 46.1±2.0 40.4±2.2 39.7±1.4 52.1±3.0 56.4±4.6 83.7±2.8 55.3±3.4 58.5±4.6 84.2±1.8 55.48 SMIDA 55.2±2.2 48.6±1.8 45.7±1.4 46.1±2.0 40.3±2.1 39.7±1.4 52.2±3.1 56.3±4.6 83.7±2.9 55.4±3.5 58.5±4.5 84.3±1.9 55.49 MIDA and SMIDA are flexible algorithms. 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IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, SUBMITTED 1 Large-Scale Low-Rank Matrix Learning with Nonconvex Regularizers arXiv:1708.00146v2 [cs.LG] 23 Sep 2017 Quanming Yao, Member IEEE, James T. Kwok, Fellow IEEE, Taifeng Wang, Member IEEE, and Tie-Yan Liu, Fellow IEEE Abstract—Low-rank modeling has many important applications in computer vision and machine learning. While the matrix rank is often approximated by the convex nuclear norm, the use of nonconvex low-rank regularizers has demonstrated better empirical performance. However, the resulting optimization problem is much more challenging. Recent state-of-the-art requires an expensive full SVD in each iteration. In this paper, we show that for many commonly-used nonconvex low-rank regularizers, the singular values obtained from the proximal operator can be automatically threshold. This allows the proximal operator to be efficiently approximated by the power method. We then develop a fast proximal algorithm and its accelerated variant with inexact proximal step. A convergence rate of O(1/T ), where T is the number of iterations, can be guaranteed. Furthermore, we show the proposed algorithm can be parallelized, and the resultant algorithm achieves nearly linear speedup w.r.t. the number of threads. Extensive experiments are performed on matrix completion and robust principal component analysis. Significant speedup over the state-of-the-art is observed. Index Terms—Low-rank matrix learning, Nonconvex regularization, Proximal algorithm, Parallel algorithm, Matrix completion, Robust principle component analysis F 1 I NTRODUCTION L OW -rank matrix learning is a central issue in many machine learning and computer vision problems. For example, matrix completion [1], which is one of the most successful approaches in collaborative filtering, assumes that the target rating matrix is low-rank. Besides collaborative filtering, matrix completion has also been used on tasks such as video and image processing [2], [3], [4], [5], [6]. Another important use of low-rank matrix learning is robust principal component analysis (RPCA) [7], which assumes that the target matrix is low-rank and also corrupted by sparse noise. RPCA has been popularly used in computer vision applications such as shadow removal, background modeling [7], [8], [9], and robust photometric stereo [10]. Besides, low-rank matrix learning has also been used in face recognition [11] and subspace clustering [12]. However, minimization of the matrix rank is NP-hard [1]. To alleviate this problem, a common approach is to use a convex surrogate such as the nuclear norm (which is the sum of singular values of the matrix). It is known that the nuclear norm is the tightest convex lower bound of the rank. Though the nuclear norm is non-smooth, the resultant optimization problem can be solved efficiently using modern tools such as the proximal algorithm [13], [14], [15], Frank-Wolfe algorithm [16], and active subspace selection method [17]. Despite the success of the nuclear norm, recently there have been numerous attempts to use nonconvex surrogates that better approximate the rank function. The key idea is • • Q. Yao and J. Kwok are with the Department of Computer Science and Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong. E-mails: {qyaoaa, jamesk}@cse.ust.hk T. Wang and T. Liu are with Microsoft Research Asia, Beijing, China 100010. E-mails: {taifengw, tyliu}@microsoft.com that the larger, and thus more informative, singular values should be less penalized. Example nonconvex low-rank regularizers include the capped-`1 penalty [18], log-sum penalty (LSP) [19], truncated nuclear norm (TNN) [3], [9], smoothly clipped absolute deviation (SCAD) [20], and minimax concave penalty (MCP) [21]. They have been applied on various computer vision tasks, such as image denoising [6] and background modeling [9]. Empirically, these nonconvex regularizers achieve better recovery performance than the convex nuclear norm regularizer. Recently, theoretical results have also been established [22]. However, the resultant nonconvex optimization problem is much more challenging. Most existing optimization algorithms that work with the nuclear norm cannot be applied. A general approach that can still be used is the concave-convex procedure [23], which decomposes the nonconvex regularizer into a difference of convex functions [3], [18]. However, a sequence of relaxed optimization problems have to be solved, and can be computationally expensive [24], [25]. A more efficient approach is the recently proposed iteratively re-weighted nuclear norm (IRNN) algorithm [5]. It is based on the observation that existing nonconvex regularizers are concave with non-increasing super-gradients. Each IRNN iteration only involves computing the supergradient of the regularizer and a singular value decomposition (SVD). However, performing SVD on a m × n matrix takes O(mn2 ) time (assuming m ≥ n), and can be expensive on large matrices. Recently, the proximal algorithm has been used for nonconvex low-rank matrix learning [3], [5], [9], [26]. However, it requires the full SVD to solve the proximal operator, which can be expensive. In this paper, we observe that for the commonly-used nonconvex low-rank regularizers [3], [9], [18], [19], [20], [21], the singular values obtained from the corresponding proximal operator can be automatically IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, SUBMITTED thresholded. One then only needs to find the leading singular values/vectors in order to generate the next iterate. Moreover, instead of computing the proximal operator on a large matrix, one only needs to use the matrix projected onto its leading subspace. The matrix size is significantly reduced and the proximal operator can be made much more efficient. Besides, by using the power method [27], a good approximation of this subspace can be efficiently obtained. While the proposed procedure can be readily used with the standard proximal algorithm, its convergence properties are not directly applicable as the proximal step here is only approximately solved. In the sequel, we will show that inexactness on the proximal step can be controlled, and a O(1/T ) convergence rate can still be guaranteed. Moreover, the algorithm can be further speeded up using acceleration. Effectiveness of the proposed algorithms is demonstrated on two popular low-rank matrix learning applications, namely matrix completion and robust principal component analysis (RPCA). For matrix completion, we show that additional speedup is possible by exploring the problem’s “sparse plus low-rank” structure; whereas for RPCA, we extend the proposed algorithm so that it can handle the two parameter blocks involved in the RPCA formulation. With the popularity of multicore shared-memory platforms, we parallelize the proposed algorithms so as to handle much larger data sets. We will show that they can achieve almost linear speedup w.r.t. the number of threads. Experiments are performed on both synthetic and realworld data sets. Results show that the proposed nonconvex low-rank matrix learning algorithms can be several orders faster than the state-of-the-art, and outperform other approaches including factorization and the use of nuclear norm regularization. Preliminary results of this paper have been reported in [28]. In this full version, we speed up the algorithm with acceleration, and demonstrate how it can be applied to two important instances of low-rank matrix learning problems, namely matrix completion and RPCA. Besides, we show how the proposed algorithms can be parallelized. More extensive empirical evaluations are also performed on both the sequential and parallel versions of the algorithms. Notation: In the sequel, vectors are denoted by lowercase boldface, matrices by uppercase boldface, and the transpose by the superscript (·)> . For a square matrix X, tr(X) is its trace. For a rectangle matrix X,P kXkF = tr(X> X) is its Frobenius norm, and kXk∗ = i σi (X), where σi (X) is the ith leading singular value of X, is the nuclear norm. Given x = [xi ] ∈ Rm , Diag(x) constructs a m × m diagonal matrix whose ith diagonal element is xi . I denotes the identity matrix. For a differentiable function f , we use ∇f for its gradient. For a nonsmooth function, we use ∂f for its subdifferential, i.e., ∂f (x) = s : f (y) ≥ f (x) + s> (y − x) . 2 2.1 where f is a smooth loss, r is a nonsmooth low-rank regularizer, and λ is a regularization parameter. We make the following assumptions on f . A1. A2. f is not necessarily convex, but is differentiable with ρ-Lipschitz continuous gradient, i.e., k∇f (X1 ) − ∇f (X2 )kF ≤ ρkX1 − X2 kF . Without loss of generality, we assume that ρ ≤ 1. f is bounded below, i.e., inf f (X) > −∞, and limkXkF →∞ f (X) = ∞. In recent years, the proximal algorithm [29] has been popularly used for solving (1). At iteration t, it produces Xt+1 = prox λ r (Xt − τ 1 ∇f (Xt )), τ (2) where τ > ρ is the stepsize, and 1 prox λ r (Z) ≡ arg min kX − Zk2F + λr(X) (3) τ X 2 is the proximal operator [29]. The proximal step in (2) can also be rewritten as Xt+1 = arg minY tr(∇f (Xt )> (Y − Xt )) + τ2 kY − Xt k2F + λr(Y). When f and r are convex, the proximal algorithm converges to the optimal solution at a rate of O(1/T ), where T is the number of iterations. This can be further accelerated to the rate of O(1/T 2 ), by replacing Xt in (2) with a proper linear combination of Xt and Xt−1 [30]. Recently, the accelerated proximal algorithm has been extended to problems where f or r may be nonconvex [25], [31]. The state-ofthe-art is the nonmonotone accelerated proximal gradient (nmAPG) algorithm [25] (Algorithm 1). Each iteration may perform two proximal steps (steps 4 and 8). Acceleration is performed in step 3. The objective is then checked to determine whether Xat+1 is accepted (step 5). As the problem is nonconvex, its convergence rate is still open. However, empirically it is much faster. Algorithm 1 Nonmonotone APG (nmAPG) [25]. Input: choose τ > ρ, δ > 0 and η ∈ [0, 1); 1: initialize X0 = X1 = Xa 1 = 0, α0 = α1 = 1, b1 = F (X1 ) and q1 = 1; 2: for t = 1, 2, . . . , T do −1 3: Yt = Xt + ααt−1 (Xat − Xt−1 ) + αt−1 (Xt − Xt−1 ); αt t 1 a 4: Xt+1 = prox λ r (Yt − τ ∇f (Yt )); τ 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: BACKGROUND 2 if F (Xat+1 ) ≤ bt − 2δ kXat+1 − Yt k2F then Xt+1 = Xat+1 ; else Xpt+1 = prox λ r (Xt − τ1 ∇f (Xt )); ( τ Xat+1 if F (Xat+1 ) ≤ F (Xpt+1 ) Xt+1 = ; Xpt+1 otherwise end if p αt+1 = 21 ( 4αt2 + 1 + 1); 1 bt+1 = qt+1 (ηqt bt + F (Xt+1 )) where qt+1 = ηqt + 1; end for return XT +1 . Proximal Algorithm In this paper, we consider low-rank matrix learning problems of the form min F (X) ≡ f (X) + λr(X), X (1) 2.2 Nonconvex Low-Rank Regularizers For the proximal algorithm to be successful, the proximal operator has to be efficient. The following shows that the IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, SUBMITTED proximal operator of the nuclear norm k · k∗ has a closedform solution. > Proposition 2.1 ([32]). proxµk·k∗ (X) = U (Σ − µI)+ V , where UΣV> is the SVD of X, and (A)+ = [max(Aij , 0)]. While the (convex) nuclear norm makes low-rank optimization easier, it may not be a good enough approximation of the matrix rank [3], [5], [6], [9], [26]. As mentioned in Section 1, a number of nonconvex surrogates have been recently proposed. In this paper, we make the following assumption on the low-rank regularizer r in (1), which is satisfied by all nonconvex low-rank regularizers in Table 1. A3. r is possibly non-smooth and nonconvex, and of the Pm form r(X) = i=1 r̂(σi (X)), where r̂(α) is concave and non-decreasing for α ≥ 0 and r̂(0) = 0. TABLE 1 r̂’s for some popular nonconvex low-rank regularizers. For the TNN regularizer, θ ∈ {1, . . . , n} is the number of leading singular values that are not penalized; for SCAD, θ > 2; and for the others, θ > 0. µr̂(σi (X)) capped-`1 [18] µ min(σi (X), θ) 1 LSP [19] (µ log θ σi (X) + 1 µσi (X) 0 TNN [3], [9] SCAD [20] MCP [21]   µσ (X)   i2 if i > θ otherwise −σi (X)+2θµσi (X)−µ2 2(θ−1)    (θ+1)µ2 2 ( σ 2 (X) µσi (X) − i2θ θµ2 2 if σi (X) ≤ µ if µ < σi (X) ≤ θµ otherwise if σi (X) ≤ θµ otherwise Recently, the iteratively reweighted nuclear norm (IRNN) algorithm [5] has been proposed to handle this nonconvex low-rank matrix optimization problem. In each iteration, it solves a subproblem in which the original nonconvex regularizer is approximated by a weighted version of the nuclear Pm norm kXkw = i=1 wi σi (X) and 0 ≤ w1 ≤ · · · ≤ wm . The subproblem has a closed-form solution, but SVD is needed which takes O(mn2 ) time. Other solvers that are designed for specific nonconvex low-rank regularizers include [8] (for capped-`1 ), [3], [9] (for TNN), and [21] (for MCP). All these (including IRNN) perform SVD in each iteration, which takes O(mn2 ) time and are slow. While the proximal algorithm has mostly been used on convex problems, recently it is also applied to nonconvex problems [3], [5], [6], [8], [9], [26]. The generalized proximal gradient (GPG) algorithm [26] is the first proximal algorithm which can handle all the above nonconvex regularizers. In particular, its proximal operator can be computed as follows. Proposition 2.2 (Generalized singular value thresholding (GSVT) [26]). For any r satisfying assumption A3, proxµr (Z) = UDiag(y∗ )V> , where UΣV> is the SVD of Z, and y∗ = [yi∗ ] with 1 2 (4) yi∗ ∈ arg min (yi − σi (Z)) + µr̂(yi ). yi ≥0 2 In [26], problem (4) is solved by fixed-point iteration. However, closed-form solutions indeed exist for regularizers in Table 1 [24]. Nevertheless, Proposition 2.2 still involves SVD, which takes O(mn2 ) time. 3 3 P ROPOSED A LGORITHM In this section, we show how the proximal algorithm can be made much faster by using approximate GSVT. 3.1 Automatic Thresholding of Singular Values The following Proposition shows that yi∗ in (4) becomes zero when σi (Z) is smaller than a regularizer-specific threshold. Proof can be found in Appendix C.1. Proposition 3.1. There exists a threshold γ > 0 such that yi∗ = 0 when σi (Z) ≤ γ . Together with Proposition 2.2, solving the proximal operator (3) only needs the leading singular values/vectors of Z. For the nonconvex regularizers in Table 1, simple closed-form solutions of γ can be obtained by examining the optimality conditions of (4). Proof can be found in Appendix C.2. Corollary 3.2. The γ values for the following regularizers are: √ • Capped-`1 : γ = min 2θµ, µ ; • LSP: γ = min µθ , θ ; • TNN: γ = max (µ, σθ+1 (Z)); • SCAD: γ =√µ; • MCP: γ = θµ if 0 < θ < 1, and µ otherwise. This can also be used with the nuclear norm. It can be shown that γ = λτ , and yi∗ = max σi (A) − λτ , 0 . However, since our focus is on nonconvex regularizers, the case for nuclear norm will not be further pursued in the sequel. 3.2 Approximate GSVT Proposition 2.2 computes the proximal operator by exact SVD. In this section, we show that one can use approximate SVD, which is more efficient. 3.2.1 Reducing the Size of SVD Assume that Z has k̂ singular values larger than γ , then we only need to a rank-k SVD on Z with k ≥ k̂ . Let the rank-k̂ SVD of Z be Uk̂ Σk̂ Vk̂> . The following Proposition shows that proxµr (Z) can be obtained from the proximal operator on a smaller matrix.1 The proof can be found in Appendix C.3. Proposition 3.3. Assume that Q ∈ Rm×k , where k ≥ k̂ , is orthogonal and span(Uk̂ ) ⊆ span(Q). Then, proxµr (Z) = Q proxµr (Q> Z). 3.2.2 Obtaining an Approximate GSVT To obtain such a Q, we use the power method [27] (Algorithm 2) which has been recently used to approximate the SVT in nuclear norm minimization [15], [17]. As in [17], we set the number of power iterations to 3. Warm-start can be used via matrix R in Algorithm 2. This is particularly useful because of the iterative nature of proximal algorithm. Obtaining an approximate Q using Algorithm 2 takes O(mnk) time. As in [14], [34], [35], the PROPACK 1. We noticed a similar result in [33] after the conference version of this paper [28] has been accepted. However, [33] only considers the case where r is the nuclear norm regularizer. IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, SUBMITTED 4 algorithm [36] can also be used to obtain Q in O(mnk) time. However, it finds the Q exactly and cannot benefit from warm-start. Hence, though it has the same time complexity as power method, empirically it is much less efficient [15]. Xt for the proximal gradient algorithm in (2)). An approximate proximal step solution X̃p is generated in step 4, and we try to ensure Algorithm 2 Powermethod(Z, R). (note that this is less stringent than where c1 = τ −ρ 4 the condition in Lemma 3.4). If (5) holds, we accept X̃p ; otherwise, we improve X̃p by using Ṽp−1 to warm-start the next iterate. The following Proposition shows convergence of Algorithm 4. The proof can be found in Appendix C.4. m×n n×k Input: Z ∈ R , R ∈ R and the number of power iterations J = 3. 1: Y1 = ZR; 2: for j = 1, 2, . . . , J do 3: Qj = QR(Yj ); // QR decomposition (returning only the Q matrix) 4: Yj+1 = Z(Z> Qj ); 5: end for 6: return QJ . The approximate GSVT procedure is shown in Algorithm 3. Step 1 uses the power method to efficiently obtain an orthogonal matrix Q that approximates span(Uk̂ ). Step 2 performs a small SVD. Though this SVD is still exact, Q> Z is much smaller than Z (k × n vs m × n), and SVD(Q> Z) takes only O(nk 2 ) time. In step 3, the singular values Σii ’s are thresholded using Corollary 3.2. Steps 6-8 obtains an (approximate) proxµr (Z) using Proposition 2.2. The time complexity for GSVT is reduced from O(mn2 ) to O(mnk). F (X̃p ) ≤ F (X) − c1 kX̃p − Xk2F , (5) Proposition 3.5. If k ≥ k̂A , where k̂A is the number of singular values in A larger than γ , then limp→∞ X̃p = prox λ r (A). τ Algorithm 4 Inexact proximal step: InexactPS(X, R). Input: X ∈ Rm×n , and R ∈ Rn×k for warm-start; 1: A = X − τ1 ∇f (X); 2: Ṽ0 = R; 3: for p = 1, 2, . . . do 4: [X̃p , Ṽp ] = ApproxGSVT (A, Ṽp−1 , λτ ); 5: if F (X̃p ) ≤ F (X) − c1 kX̃p − Xk2F then 6: break; 7: end if 8: end for 9: return X̃p . Algorithm 3 Approximate GSVT: ApproxGSVT(Z, R, µ). Input: Z ∈ Rm×n and R ∈ Rn×k for warm-start; 1: Q = PowerMethod(Z, R); 2: [U, Σ, V] = SVD(Q> Z); 3: a = number of Σii ’s that are > γ in Corollary 3.2; 4: Ua = a leading columns of U; 5: Va = a leading columns of V; 6: for i = 1, 2, . . . , a do 7: obtain yi∗ from (4); 8: end for 9: return low-rank components of X̃ (QUa , Diag([y1∗ , . . . , ya∗ ]) and Va> ), and V. 3.3 Inexact Proximal Step In this section, the proximal step will be inexact, and so it can utilize the approximate GSVT in Algorithm 3. Inexact proximal step has been considered in [37], [38]. However, r in (1) is assumed to be convex in [38]. Attouch et al. [37] considered nonconvex r, but they require a difficult and expensive condition to control inexactness (an example is provided in Appendix B). Let A = X − τ1 ∇f (X). The following shows that the objective F is always decreased (as τ > ρ) after an exact proximal step. Lemma 3.4 ( [24], [37]). F prox λ r (A) ≤ F (X) − τ −ρ (A) 2 kprox λ τr − Xk2F . τ Motivated by this Lemma, we propose to control the proximal step’s inexactness by Algorithm 4 (note that X = 3.4 The Complete Procedure The complete procedure for solving (1) is shown in Algorithm 5, and will be called FaNCL (Fast NonConvex Lowrank). Similar to [15], [17], we perform warm-start using the column spaces of the previous iterates (Vt and Vt−1 ). For further speedup, we employ a continuation strategy at step 3 as in [5], [14], [34]. Specifically, λt is initialized to a large value and then decreases gradually. Algorithm 5 FaNCL (Fast NonConvex Low-rank) algorithm. Input: choose τ > ρ, λ0 > λ and ν ∈ (0, 1); 1: initialize V0 , V1 ∈ Rn as random Gaussian matrices and X1 = 0; 2: for t = 1, 2, . . . T do 3: λt = (λt−1 − λ)ν t + λ; 4: Rt = QR([Vt , Vt−1 ]); // warm start 5: Xt+1 = InexactPS(Xt , Rt ); 6: end for 7: return XT +1 . Assume that evaluations of f and ∇f take O(mn) time, which is valid for many applications such as matrix completion and RPCA. Let rt be the rank of Xt at the tth iteration, and kt = rt + rt−1 . In Algorithm 5, step 4 takes O(nkt2 ) time; and step 5 takes O(mnpkt ) time as Rt has kt columns. The iteration time complexity is thus O(mnpkt ). In the experiment, we set p = 1, which is enough to guarantee (5) empirically. The iteration time complexity of Algorithm 5 thus reduces to O(mnkt ). In contrast, exact GSVT takes O(mn2 ) time, and is much slower as kt n. Besides, the space complexity of Algorithm 5 is O(mn). IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, SUBMITTED 5 Proposition 3.6. r can be decomposed as r̆ − r̃, where r̆ and r̃ are convex. O(mnkt ). Besides, its space complexity is O(mn), which is the same as Algorithm 5. There are several major differences between Algorithm 6 and nmAPG. First, the proximal step of Algorithm 6 is only inexact. To make the algorithm more robust, we do not allow nonmonotonous update (i.e., F (Xt+1 ) cannot be larger than F (Xt )). Moreover, we use a simpler acceleration scheme (step 4), in which only Xt and Xt−1 are involved. On matrix completion problems, this allows using the “sparse plus low-rank” structure [14], [15] to greatly reduce the iteration complexity (Section 4.1). Finally, we do not require extra comparison of the objective at step 10. This further reduces the iteration complexity. Based on this decomposition, we introduce the definition of critical point. Algorithm 6 Accelerated FaNCL algorithm (FaNCL-acc). 3.5 Convergence Analysis The inexact proximal algorithm is first considered in [38], which assumes r to be convex. The nonconvex extension is considered in [37]. However, as discussed in Section 3.3, they use an expensive condition to control inexactness of the proximal step. Thus, their analysis cannot be applied here. It is known that r̂ in Assumption A3 can be decomposed as a difference of convex functions [24]. The following Proposition shows that r also admits such a decomposition. The proof is in Appendix C.5. Definition 1 ([39]). If 0 ∈ ∇f (X)+λ (∂ r̆(X) − ∂ r̃(X)), then X is a critical point of F . The following Proposition shows that Algorithm 5 generates a bounded sequence. The proof is in Appendix C.6. Proposition 3.7. The sequence {Xt } generated Algorithm 5 is bounded, and has at least one limit point. from Let G λ r (Xt ) = Xt − prox λ r (Xt − τ1 ∇f (Xt )), which τ τ is known as the proximal mapping of F at Xt [29]. If G λ r (Xt ) = 0, Xt is a critical point of (1) [24], [37]. This τ motivates the use of kG λ r (Xt )k22 to measure convergence τ in [31]. However, kG λ r (Xt )k22 cannot be used here as r τ is nonconvex and the proximal step is inexact. As Proposition 3.7 guarantees the existence of limit points, we use kXt+1 − Xt k2F instead to measure convergence. If the proximal step is exact, kG λ r (Xt )k2F = kXt+1 − Xt k2F . The τ following Corollary shows convergence of Algorithm 5. Its proof can be found in Appendix C.7. Corollary 3.8. mint=1,...,T kXt+1 − Xt k2F ≤ F (X1 )−inf F c1 T . The following Theorem shows that any limit point is also a critical point. The proof is in Appendix C.8. Theorem 3.9. Assume that Algorithm 4 returns X only when X = prox λ r (X− τ1 ∇f (X)) (i.e., the input is returned as output τ only if it is a limit point). Let {Xtj } be a subsequence of {Xt } generated by Algorithm 5 such that limtj →∞ Xtj = X∗ . Then, X∗ is a critical point of (1). 3.6 Acceleration In convex optimization, acceleration has been commonly used to speed up convergence of proximal algorithms [30]. Recently, it has also been extended to nonconvex optimization [25], [31]. A state-of-the-art algorithm is the nmAPG [25] (Algorithm 1). In this section, we integrate nmAPG with FaNCL. The whole procedure is shown in Algorithm 6. The accelerated iterate is obtained in step 4. If the resultant inexact proximal step solution can achieve a sufficient decrease (step 7) as in (5), this iterate is accepted (step 8); otherwise, we choose the inexact proximal step solution obtained with the nonaccelerated iterate Xt (step 10). Note that step 10 is the same as step 5 of Algorithm 5. Thus, the iteration time complexity of Algorithm 6 is at most twice that of Algorithm 5, and still Input: choose τ > ρ, λ0 > λ, δ > 0 and ν ∈ (0, 1); 1: initialize V0 , V1 ∈ Rn as random Gaussian matrices, X0 = X1 = 0 and α0 = α1 = 1; 2: for t = 1, 2, . . . T do 3: λt = (λt−1 − λ)ν + λ; −1 4: Yt = Xt + αt−1 (Xt − Xt−1 ); αt 5: Rt = QR([Vt , Vt−1 ]); // warm start 6: Xat+1 = InexactPS(Yt , Rt ); 7: if F (Xat+1 ) ≤ F (Xt ) − 2δ kXat+1 − Yt k2F then 8: Xt+1 = Xat+1 ; 9: else 10: Xt+1 = InexactPS(Xt , Rt ); 11: end if p 12: αt+1 = 12 ( 4αt2 + 1 + 1); 13: end for 14: return XT +1 . The following Proposition shows that Algorithm 6 generates a bounded sequence. Proof can be found in Appendix C.9. Proposition 3.10. The sequence {Xt } generated Algorithm 6 is bounded, and has at least one limit point. from In Corollary 3.8, kXt+1 − Xt k2F is used to measure progress before and after the proximal step. In Algorithm 6, the proximal step may use the accelerated iterate Yt or the non-accelerated iterate Xt . Hence, we use kXt+1 − Ct k2F , where Ct = Yt if step 8 is performed, and Ct = Xt otherwise. Similar to Corollary 3.8, the following shows a O(1/T ) convergence rate. Proof can be found in Appendix C.10. Corollary 3.11. For Algorithm 6, mint=1,...,T kXt+1 −Ct k2F ≤ F (X1 )−inf F min(c1 ,δ/2)T . On nonconvex optimization problems, the optimal convergence rate for first-order methods is O(1/T ) [31], [40]. Thus, the convergence rate of Algorithm 6 (Corollary 3.11) cannot improve that of Algorithm 5 (Corollary 3.8). However, in practice, acceleration can still significantly reduce the number of iterations on nonconvex problems [25], [31]. On the other hand, as Algorithm 6 may need a second proximal step (step 10), its iteration time complexity can be higher than that of Algorithm 5. However, this is much compensated by the speedup in convergence. As will be demonstrated in Section 6.1, empirically Algorithm 6 is much faster. IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, SUBMITTED The following Theorem shows that any limit point of the iterates from Algorithm 6 is also a critical point. Proof can be found in Appendix C.11. Theorem 3.12. Let {Xtj } be a subsequence of {Xt } generated by Algorithm 6 such that limtj →∞ Xtj = X∗ . With the assumption in Theorem 3.9, X∗ is a critical point of (1). 4 A PPLICATIONS In this section, we consider two important instances of problem (1), namely, matrix completion [1] and robust principal component analysis (RPCA) [7]. As the accelerated FaNCL algorithm (Algorithm 6) is usually faster than its nonaccelerated variant, we will only consider the accelerated variant here. For matrix completion (Section 4.1), we will show that Algorithm 6 can be made even faster and require much less memory by using the “sparse plus low-rank” structure of the problem. In Section 4.2, we show how Algorithm 6 can be extended to deal with the two parameter blocks in RPCA. 4.1 Matrix Completion Matrix completion attempts to recover a low-rank matrix O ∈ Rm×n by observing only some of its elements [1]. Let the observed positions be indicated by Ω ∈ {0, 1}m×n , such that Ωij = 1 if Oij is observed, and 0 otherwise. Matrix completion can be formulated as an optimization problem in (1), with F (X) = 1 kPΩ (X − O)k2F + λr(X), 2 (6) where [PΩ (A)]ij = Aij if Ωij = 1 and 0 otherwise. In the following, we show that the time and space complexities of Algorithm 6 can be further reduced. 4.1.1 Utilizing the Problem Structure First, consider step 7, which checks the objectives. Computing F (Xt ) relies only on the observed positions in Ω and the singular values of Xt . Hence, instead of explicitly constructing Xt , we maintain the SVD Ut Σt Vt> of Xt and a sparse matrix PΩ (Xt ). Computing F (Xt ) then takes O(kΩk1 rt ) time. Computing F (Xat+1 ) takes O(kΩk1 kt ) time, as Rt has rank kt . Next, since Yt is a linear combination of Xt and Xt−1 in step 4, we can use the above SVD-factorized form and compute kXat+1 − Yt k2F in O((m + n)kt2 ) time. Thus, step 7 then takes O(kΩk1 kt + (m + n)kt2 ) time. Steps 6 and 10 perform inexact proximal step. For the first proximal step (step 6), Yt (defined in step 4) can be rewritten as (1+βt )Xt −βt Xt−1 , where βt = (αt−1 −1)/αt . When it calls InexactPS, step 1 of Algorithm 4 has 1 A = Yt + PΩ (Yt − O) τ = (1 + βt )Xt − βt Xt−1 + 1 PΩ (Yt − O). τ (7) The first two terms involve low-rank matrices, while the last term involves a sparse matrix. This special “sparse plus low-rank” structure [14], [15] can speed up matrix 6 multiplications. Specifically, for any V ∈ Rn×k , AV can be obtained as > AV =(1 + βt )Ut Σt (Vt> V) − βt Ut−1 Σt−1 (Vt−1 V) 1 + PΩ (O − Yt )V. (8) τ Similarly, for any U ∈ Rm×k , U> A can be obtained as > U> A =(1 + βt )(U> Ut )Σt Vt> −βt (U> Ut−1 )Σt−1 Vt−1 1 + U> PΩ (O − Yt ). (9) τ Both (8) and (9) take O((m + n)kt k + kΩk1 k) time (instead of O(mnk)). As Rt in step 5 of Algorithm 6 has kt columns, each call to approximate GSVT takes O((m+n)kt2 +kΩk1 kt ) time [15] (instead of O(mnkt )). Finally, step 5 in Algorithm 4 also takes O((m + n)kt2 + kΩk1 kt ) time. As a result, step 6 of Algorithm 6 takes a total of O((m + n)kt2 + kΩk1 kt ) time. Step 10 is slightly cheaper (as no Xt−1 is involved), and its time complexity is O((m + n)rt kt + kΩk1 rt ). Summarizing, the iteration time complexity of Algorithm 6 is O((m + n)kt2 + kΩk1 kt ). (10) Usually, kt n and kΩk1 mn [1], [14]. Thus, (10) is much cheaper than the O(mnkt ) complexity of standard FaNCL-acc (Section 3.6). The space complexity is also reduced. We only need to store the low-rank factorizations of Xt and Xt−1 , and the sparse matrices PΩ (Xt ) and PΩ (Xt−1 ). These take a total of O((m+n)kt +kΩk1 ) space (instead of O(mn) in Section 3.6). These techniques can also be used on Algorithm 5. It can be easily shown that its iteration time complexity is O((m + n)rt kt + kΩk1 rt ), and its space complexity is O((m + n)rt + kΩk1 ) (as no Xt−1 is involved). 4.1.2 Comparison with Existing Algorithms Table 2 compares the convergence rates iteration time complexities, and space complexities of various matrix completion algorithms that will be empirically compared in Section 6. Overall, the proposed algorithms (Algorithms 5 and 6) enjoy fast convergence, cheap iteration complexity and low memory cost. While Algorithms 5 and 6 have the same convergence rate, we will see in Section 6.1 that Algorithm 6 (which uses acceleration) is significantly faster. 4.2 Robust Principal Component Analysis (RPCA) Given a noisy data matrix O ∈ Rm×n , RPCA assumes that O can be approximated by the sum of a low-rank matrix X plus some sparse noise S [7]. Its optimization problem is: min F (X, S) ≡ f (X, S) + λr(X) + υg(S), X,S (11) where f (X, S) = 21 kX+S−Ok2F , r is a low-rank regularizer, and g is a sparsity-inducing regularizer. Here, we allow both r and g to be nonconvex and nonsmooth. Thus, (11) can be seen as a nonconvex extension of RPCA (which uses the nuclear norm regularizer for r and `1 -regularizer for g ). Some examples of nonconvex r are shown in Table 1, and examples of nonconvex g include the `1 -norm, capped-`1 norm [18] and log-sum-penalty [19]. IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, SUBMITTED 7 TABLE 2 Comparison of the iteration time complexities, convergence rates and space complexity of various matrix completion solvers. Here, kt = rt + rt−1 , ν ∈ (0, 1) and integer Ta > 0 are constants. For the active subspace selection method (active) [17], Ts is the number of inner iterations required. regularizer (convex) nuclear norm fixed-rank factorization nonconvex method APG [13], [34] active [17] AIS-Impute [15] LMaFit [41] ER1MP [42] IRNN [5] GPG [26] FaNCL FaNCL-acc convergence rate O(1/T 2 ) O(ν T −Ta ) O(1/T 2 ) — O(ν T ) — — O(1/T ) O(1/T ) While (11) involves two blocks of parameters (X and S), they are not coupled together. Thus, we can use the separable property of proximal operator [29]: proxλr+υg ([X, S]) = [proxλr (X), proxυg (S)]. For many popular sparsity-inducing regularizers, computing proxυg (S) takes only O(mn) time [24]. For P example, when g(S) = = i,j \|Sij \|, [proxυg (S)]ij sign(Sij ) max(\|Sij \| − υ, 0), where sign(x) is the sign of x. However, directly computing proxλr (X) requires O(mn2 ) time and is expensive. To alleviate this problem, Algorithm 5 can be easily extended to Algorithm 7. The iteration time complexity, which is dominated by the inexact proximal steps in steps 6 and 13, is reduced to O(mnkt ). Algorithm 7 FaNCL-acc algorithm for RPCA. Input: choose τ > ρ, λ0 > λ, δ > 0, ν ∈ (0, 1) and c1 = τ −ρ 4 ; 1: initialize V0 , V1 ∈ Rn as random Gaussian matrices, X0 = X1 = 0, S0 = S1 = 0 and α0 = α1 = 1; 2: for t = 1, 2, . . . T do 3: λ (λt−1 − λ)ν + λ ; t = YtX , YtS = Xt , St 4: −1 Xt , St − Xt−1 , St−1 ; + αt−1 αt 5: Rt = QR([Vt , Vt−1 ]); // warm start 6: Xat+1 = InexactPS(YtX , Rt ); 7: Sat+1 = prox υ g (YtS − τ1 ∇S f (YtX , YtS )); τ 8: ∆t = kXat+1 − YtS k2F + kSat+1 − YtS k2F ; 9: if F (Xat+1 , Sat+1 ) ≤ F (Xt , St ) − 2δ ∆t then 10: Xt+1 = Xat+1 ; 11: St+1 = Sat+1 ; 12: else 13: Xt+1 = InexactPS(Xt , Rt ); 14: St+1 = prox υ g (St − τ1 ∇S f (Xt , St )); τ 15: end if p 16: αt+1 = 12 ( 4αt2 + 1 + 1); 17: end for 18: return XT +1 and ST +1 . iteration time complexity O(mnrt ) O(kΩk1 kt Ts ) O(kΩk1 kt + (m + n)kt2 ) O(kΩk1 rt + (m + n)rt2 ) O(kΩk1 ) O(mn2 ) O(mn2 ) O (kΩk1 rt + (m + n)rt kt ) O kΩk1 kt + (m + n)kt2 space complexity O(mn) O((m + n)kt + kΩk1 ) O((m + n)kt + kΩk1 ) O((m + n)rt + kΩk1 ) O((m + n)rt + kΩk1 ) O(mn) O(mn) O((m + n)rt + kΩk1 ) O((m + n)kt + kΩk1 ) Theorem 4.3. Let [Xtj , Stj ] be a subsequence of {[Xt , St ]} generated by Algorithm 6 such that limtj →∞ Xtj = X∗ and limtj →∞ Stj = S∗ With the assumption in Theorem 3.9, [X∗ , S∗ ] is a critical point of (11). 5 PARALLEL FA NCL FOR M ATRIX C OMPLETION In this section, we show how the proposed algorithms can be parallelized. We will only consider the matrix completion problem in (6). Extension to other problems, such as RPCA in Section 4.2, can be similarly performed. Moreover, for simplicity of discussion, we focus on the simpler FaNCL algorithm (Algorithm 5). Its accelerated variant (Algorithm 6) can be similarly parallelized and is shown in Appendix A. Parallel algorithms for matrix completion have been proposed in [43], [44], [45]. However, they are based on stochastic gradient descent and matrix factorization, and cannot be directly used here. 5.1 Proposed Algorithm Convergence results in Section 3.6 can be easily extended to this RPCA problem. Proofs of the following can be found in Appendices C.12, C.13, and C.14. Operations on a matrix X are often of the form: (i) multiplications U> X and XV for some U, V (e.g., in (8), (9)); and (ii) element-wise operation (e.g., evaluation of F (X) in (5)). A popular scheme in parallel linear algebra is block distribution [46]. Assume that there are q threads for parallelization. Block distribution partitions the rows and columns of X into q parts, leading to a total of q 2 blocks. Figure 1 shows how computations of XV, U> X and element-wise operation can be easily parallelized. In Algorithm 5, the most important variables are the low-rank factorized form Ut Σt Vt> of Xt , and the sparse matrices PΩ (Xt ), PΩ (O). Using block distribution, they are thus partitioned as in Figure 2. The resultant parallelized version of FaNCL is shown in Algorithm 8. Steps that can be parallelized are marked with “B”. Two new subroutines are introduced, namely, IndeSpan-PL (step 6) which replaces QR factorization, and ApproxGSVT-PL (step 9) which is the parallelized version of Algorithm 3. They will be discussed in more detail in the following Sections. Note that Algorithm 8 is equivalent to Algorithm 5 except that it is parallelized. Thus, the convergence results in Section 3.5 still hold. Proposition 4.1. The sequence {[Xt , St ]} generated from Algorithm 7 is bounded, and has at least one limit point. Corollary 4.2. Let Ct = YtX , YtS if steps 10 and 11 are performed, and Ct = [Xt , St ] otherwise. Then, (X1 ,S1 )−inf F mint=1,...,T k[Xt+1 , St+1 ] − Ct k2F ≤ Fmin(c . 1 ,δ/2)T 5.1.1 Identifying the Span (Step 5) In step 4 of Algorithm 5, QR factorization is used to find the span of matrix [Vt , Vt−1 ]. This can be parallelized with the Householder transformation and Gaussian elimination [46], which however is typically very complex. The following IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, SUBMITTED 8 (a) U> X. (b) XV. (c) Element-wise operation. Fig. 1. Parallelization of different matrix operations. Here, the number of threads q is equal to 3. Each dotted path denotes operation of a thread. Moreover, though step 3 uses SVD it only takes O(k 3 ) time. Algorithm 9 Parallel algorithm to identify the span of A: IndeSpan-PL(A). (a) UΣV> . (b) O. Fig. 2. Partitioning of variables UΣV> and O, and three threads are used (q = 3). Input: matrix A ∈ Rn×k ; 1: B B = A> A; 2: [U, Σ, V] = SVD(B); 3: construct w as in Proposition 5.1; 4: V = VDiag(w); 5: B Q = AV; > > 6: return Q. // Q = [Q> 1 , . . . , Qp ] Algorithm 8 FaNCL in parallel: FaNCL-PL. Input: choose τ > ρ, λ0 > λ and ν ∈ (0, 1); 1: initialize V0 , V1 ∈ Rn as random Gaussian matrices and X1 = 0; 2: partition X1 , PΩ (X1 ) and PΩ (O); 3: start q threads for parallelization; 4: for t = 1, 2, . . . T do 5: λt = (λt−1 − λ)ν t + λ; 6: B Rt = IndeSpan-PL ([Vt , Vt−1 ]); 7: B At = Xt − τ1 PΩ (Xt − O); 8: for p = 1, 2, . . . do 9: B [X̃p , Rt ] = ApproxGSVT-PL(At , Rt , λτ ); 10: B ap = F (X̃p ); 11: B at = F (Xt ); 12: B aF = kX̃p − Xt k2F ; 13: if ap ≤ at − c1 aF then 14: break; 15: end if 16: end for 17: B Xt+1 = X̃p ; 18: end for 19: return XT +1 . Proposition proposes a simpler method to identify the span of a matrix. Proof can be found in Appendix C.15. Proposition 5.1. Given a matrix A, let the SVD of A> A be VΣV> , w = [wi ] where wi = Σii if Σii > 0 and 1 otherwise. −1 Then, AV (Diag(w)) 2 is orthogonal and contains span(A). The resultant parallel algorithm is shown in Algorithm 9. Its time complexity is O(( nq + q)k 2 + k 3 ). Algorithm 8 calls Algorithm 9 with input [Vt , Vt−1 ], and thus takes O(( nq + q)kt2 + kt3 ) time, where kt = rt + rt−1 . We do not parallelize steps 2-4, as only k × k matrices are involved and k is small. 5.1.2 Approximate GSVT (Step 8) The key steps in approximate GSVT (Algorithm 3) are the power method and SVD. The power method can be parallelized straightforwardly as in Algorithm 10, in which we also replace the QR subroutine with Algorithm 9. Algorithm 10 Parallel power method: PowermethodPL(Z, R). Input: matrix Z ∈ Rm×n , R ∈ Rn×k . 1: B Y1 = ZR; 2: for j = 1, 2, . . . , J do 3: B Qj = IndeSpan-PL(Yj ); 4: B Yj+1 = Z(Z> Qj ); 5: end for 6: return QJ . As for SVD, multiple QR factorizations are usually needed for parallelization [46], which are complex as discussed in Section 5.1.1. The following Proposition performs it in a simpler manner. Proof can be found in Appendix C.16. Proposition 5.2. Given a matrix B ∈ Rn×k , let P ∈ Rn×k be orthogonal and equals span(B), and the SVD of P> B be UΣV> . Then, the SVD of B is (PU)ΣV. The resultant parallelized procedure for approximate GSVT is shown in Algorithm 11. At step 5, a small SVD is performed (by a single thread) on the k × k matrix P> B. At step 8 of Algorithm 8, X̃p is returned from Algorithm 11, and we keep X̃p in its low-rank factorized form. Besides, when Algorithm 11 is called, Z = At and has the “sparse plus low-rank” structure mentioned earlier. Hence, (8) and (9) can be used to speed up matrix multiplications2 . As Rt has kt columns in Algorithm 8, 2. As no acceleration is used, βt is equal to 0 in these two equations. IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, SUBMITTED 2 PowerMethod-PL in step 1 takes O( kqt kΩk1 + m+n q kt ) time, n 2 3 steps 2-6 take O(( q + q)kt + kt ) time, and the rest takes O(kt ) time. The total time complexity for Algorithm 11 is 2 2 O( kqt kΩk1 + m+n q kt + (q + kt )kt ). Algorithm 11 Approximate GSVT in parallel: ApproxGSVTPL(Z, R, µ). Input: partitioned matrix Z ∈ Rm×n and R ∈ Rn×k ; 1: B Q = PowerMethod-PL(Z, R); 2: B B = Z> Q; // B ∈ Rn×k 3: B P = Iden-Span(B); 4: B A = P> B; 5: [U, Σ, V] = SVD(A); // U, Σ, V, A ∈ Rk×k 6: B U = PU; 7: a = number of Σii ’s that are > γ in Corollary 3.2; 8: B Ua = a leading columns of U; 9: B Va = a leading columns of V; 10: for i = 1, 2, . . . , a do 11: obtain yi∗ from (4); 12: end for 13: return the low-rank components of X̃ (QUa , Diag([y1∗ , . . . , ya∗ ]) and Va> ), and V. 5.1.3 Checking of Objectives (steps 9-11) As shown in Figures 1(c), computation of kPΩ (Xt − O)k2F in F (·) can be directly parallelized and takes O( p1 kΩk1 ) time. As r only relies on Σt , only one thread is needed to evaluate r(Xt ). Thus, computing F (Xt ) takes O( rqt kΩk1 ) time. Similarly, computing F (X̃p ) takes O( kqt kΩk1 ) time. > > As kX̃p − Xt k2F = tr(X̃> p X̃p − 2X̃p Xt − Xt Xt ), the lowrank factorized forms of X̃p and Xt can be utilized. Based 2 on Figures 1(a) and 1(b), it can be performed in O( m+n q kt ) time. Thus, the time complexity for steps 9-11 in Algorithm 8 2 is O( kqt kΩk1 + m+n q kt ). The iteration time complexity for Algorithm 8 is thus O( 1q ((m + n)kt2 + kΩk1 kt ) + (q + kt )kt2 ). Compared with (10), the speedup w.r.t. the number of threads q is almost linear. 6 E XPERIMENTS In this section, we perform experiments on matrix completion, RPCA and the parallelized variant of Algorithm 6 (Appendix A). Experiments are performed on a Windows server 2013 system with Intel Xeon E5-2695-v2 CPU (12 cores, 2.4GHz) and 256GB memory. All the algorithms in Sections 6.1 and 6.2 are implemented in Matlab. For Section 6.3, we use C++, the Intel-MKL package3 for matrix operations, and the standard thread library4 for multi-thread programming. 6.1 Matrix Completion We compare a number of low-rank matrix completion solvers, including models based on (i) the commonly used (convex) nuclear norm regularizer; (ii) fixed-rank factorization models [41], [42], which decompose the observed 3. https://software.intel.com/en-us/intel-mkl 4. http://www.cplusplus.com/reference/thread/thread/ 9 matrix O into a product of rank-k matrices U and V. Its optimization problem can be written as: minU,V 12 kPΩ (UV − O)k2F + λ2 (kUk2F + kVk2F ); and (iii) nonconvex regularizers, including √ the capped-`1 (with θ in Table 1 set to 2λ), LSP (with θ = λ), and TNN (with θ = 3). The nuclear norm minimization algorithms to be compared include: 1) 2) 3) Accelerated proximal gradient (APG) algorithm [13], [34], with the partial SVD by PROPACK [36]; AIS-Impute [14], an inexact and acceleration proximal algorithm. The “sparse plus low-rank” structure of the matrix iterate is utilized to speed up computation (Section 4.1); and Active subspace selection (denoted “active”) [17], which adds/removes rank-one subspaces from the active set in each iteration. The nuclear norm optimization problem is then reduced to a smaller problem defined only on this active set. We do not compare with the Frank-Wolfe algorithm [16] and stochastic gradient descent [47], as they have been shown to be less efficient [15], [17]. For the fixed-rank factorization models (where the rank is tuned by the validation set), we compare with the two state-of-the-art algorithms: 1) 2) Low-rank matrix fitting (LMaFit) algorithm [41]; and Economical rank-one matrix pursuit (ER1MP) [42], which pursues a rank-one basis in each iteration. We do not compare with the concave-convex procedure [3], [18], since it has been shown to be inferior to IRNN [24]. For models with nonconvex low-rank regularizers, we compare with the following solvers: 1) 2) 3) Iterative reweighted nuclear norm (IRNN) [5]; Generalized proximal gradient (GPG) algorithm [26], with the underlying problem (4) solved using the closed-form solutions in [24]; and The proposed FaNCL algorithm (Algorithm 5) and its accelerated variant FaNCL-acc (Algorithm 6). We set J = 3 and p = 1. All the algorithms are stopped when the difference in objective values between consecutive iterations becomes smaller than 10−5 . 6.1.1 Synthetic Data The observed m × m matrix is generated as O = UV + G, where the elements of U ∈ Rm×k , V ∈ Rk×m (with k = 5) are sampled i.i.d. from the standard normal distribution N (0, 1), and elements of G sampled from N (0, 0.1). A total of kΩk1 = 2mk log(m) random elements in O are observed. Half of them are used for training, and the rest as validation set for parameter tuning. Testing is performed on the unobserved elements. For performance evaluation, we use (i) the normalized mean squared error NMSE = kPΩ⊥ (X − UV)kF /kPΩ⊥ (UV)kF , where X is the recovered matrix and Ω⊥ denotes the unobserved positions; (ii) rank of X; and (iii) training CPU time. We vary m in the range {500, 1000, 2000}. Each experiment is repeated five times. IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, SUBMITTED 10 TABLE 3 Matrix completion performance on the synthetic data. Here, NMSE is scaled by 10−2 , CPU time is in seconds and the number in brackets is the data sparsity. The best results (according to the pairwise t-test with 95% confidence) are highlighted. m = 500 (12.43%) m = 1000 (6.91%) m = 2000 (3.80%) NMSE rank time NMSE rank time NMSE rank time nuclear norm APG 4.26±0.01 50 12.6±0.7 4.27±0.01 61 99.6±9.1 4.13±0.01 77 1177.5±134.2 AIS-Impute 4.11±0.01 55 5.8±2.9 4.01±0.03 57 37.9±2.9 3.50±0.01 65 338.1±54.1 active 5.37±0.03 53 12.5±1.0 6.63±0.03 69 66.4±3.3 6.44±0.10 85 547.3±91.6 fixed rank LMaFit 3.08±0.02 5 0.5±0.1 3.02±0.02 5 1.3±0.1 2.84±0.03 5 4.9±0.3 ER1MP 21.75±0.05 40 0.3±0.1 21.94±0.09 54 0.8±0.1 20.38±0.06 70 2.5±0.3 capped `1 IRNN 1.98±0.01 5 14.5±0.7 1.99±0.01 5 146.0±2.6 1.79±0.01 5 2759.9±252.8 GPG 1.98±0.01 5 14.8±0.9 1.99±0.01 5 144.6±3.1 1.79±0.01 5 2644.9±358.0 FaNCL 1.97±0.01 5 0.3±0.1 1.98±0.01 5 1.0±0.1 1.79±0.01 5 5.0±0.4 FaNCL-acc 1.97±0.01 5 0.1±0.1 1.95±0.01 5 0.5±0.1 1.78±0.01 2.3±0.2 LSP IRNN 1.96±0.01 5 16.8±0.6 1.89±0.01 5 196.1±3.9 1.79±0.01 5 2951.7±361.3 GPG 1.96±0.01 5 16.5±0.4 1.89±0.01 5 193.4±2.1 1.79±0.01 5 2908.9±358.0 FaNCL 1.96±0.01 5 0.4±0.1 1.89±0.01 5 1.3±0.1 1.79±0.01 5 5.5±0.4 FaNCL-acc 1.96±0.01 5 0.2±0.1 1.89±0.01 5 0.7±0.1 1.77±0.01 2.4±0.2 TNN IRNN 1.96±0.01 5 18.8±0.6 1.88±0.01 5 223.1±4.9 1.77±0.01 5 3220.3±379.7 GPG 1.96±0.01 5 18.0±0.6 1.88±0.01 5 220.9±4.5 1.77±0.01 5 3197.8±368.9 FaNCL 1.95±0.01 5 0.4±0.1 1.88±0.01 5 1.4±0.1 1.77±0.01 5 6.1±0.5 FaNCL-acc 1.96±0.01 5 0.2±0.1 1.88±0.01 5 0.8±0.1 1.77±0.01 2.9±0.2 Results are shown in Table 3. As can be seen, nonconvex regularization (capped-`1 , LSP and TNN) leads to much lower NMSE’s than convex nuclear norm regularization and fixed-rank factorization. Moreover, the nuclear norm and ER1MP output much higher ranks. In terms of speed among the nonconvex low-rank solvers, FaNCL is fast, and FaNCLacc is the fastest. The larger the matrix, the higher is the speedup of FaNCL and FaNCL-acc over GPG and IRNN. 6.1.2 MovieLens Experiment is performed on the popular MovieLens data set (Table 4), which contain ratings of different users on movies. We follow the setup in [42], and use 50% of the observed ratings for training, 25% for validation and the rest for testing. For performance evaluation, we use q the root mean squared (a) capped-`1 . error on the test set Ω̄: RMSE = kPΩ̄ (X − O)k2F /kΩ̄k1 , where X is the recovered matrix. The experiment is repeated five times. TABLE 4 Recommendation data sets used in the experiments. #users #movies #ratings MovieLens 100K 943 1,682 100,000 1M 6,040 3,449 999,714 10M 69,878 10,677 10,000,054 netflix 480,189 17,770 100,480,507 yahoo 249,012 296,111 62,551,438 (b) LSP. Fig. 3. Objective vs CPU time for the capped-`1 and LSP on MovieLens100K. The plot of TNN is similar and thus not shown. Results are shown in Table 5. Again, nonconvex regularizers lead to the lowest RMSE’s. Moreover, FaNCLacc is also the fastest among nonconvex low-rank solvers, even faster than the state-of-the-art GPG. In particular, FaNCL and its accelerated variant FaNCL-acc are the only solvers (for nonconvex regularization) that can be run on the MovieLens-1M and 10M data sets. Figure 3 compares the objectives vs CPU time for the nonconvex regularization solvers on MovieLens-100K. As can be seen, FaNCL and FaNCL-acc decrease the objective and RMSE much faster than the others. Figure 4 shows the testing RMSEs on MovieLens-10M. Again, FaNCL-acc is the fastest. use 50% of the observed ratings for training, 25% for validation and the rest for testing. Each experiment is repeated five times. Results are shown in Table 6. APG, GPG and IRNN cannot be run as the data set is large. AIS-Impute has similar running time as LMaFit but inferior performance, and thus is not compared. Again, the nonconvex regularizers converge faster, yield lower RMSE’s and solutions of much lower ranks. Figure 5 shows the objectives and RMSE vs time, and FaNCL-acc is the fastest.5 6.1.3 Netflix and Yahoo Next, we perform experiments on two very large recommendation data sets, Netflix and Yahoo (Table 4). We randomly 5. On these two data sets, ER1MP easily overfits as the rank increases. Hence, the validation set selects a smaller rank (relative to that obtained by the nuclear norm) and ER1MP stops earlier. However, as can be seen, its RMSE is much worse. IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, SUBMITTED 11 TABLE 5 Matrix completion results on the MovieLens data sets (time is in seconds). The best results (according to the pairwise t-test with 95% confidence) are highlighted. MovieLens-100K MovieLens-1M MovieLens-10M RMSE rank time RMSE rank time RMSE rank time nuclear APG 0.877±0.001 36 47.1±7.6 0.818±0.001 67 2174.4±117.3 — — > 105 norm AIS-Impute 0.878±0.002 36 1.2±0.1 0.819±0.001 67 12.4±0.9 0.813±0.001 100 540.6±107.9 active 0.878±0.001 36 2.7±0.3 0.820±0.001 67 82.6±5.2 0.814±0.001 100 2338.3±304.7 fixed LMaFit 0.865±0.002 2 1.2±0.2 0.806±0.003 6 24.8±1.3 0.792±0.001 9 501.7±95.2 rank ER1MP 0.917±0.003 5 0.1±0.1 0.853±0.001 13 1.3±0.1 0.852±0.002 22 62.7±17.8 capped `1 LSP TNN IRNN GPG FaNCL FaNCL-acc IRNN GPG FaNCL FaNCL-acc IRNN GPG FaNCL FaNCL-acc 0.854±0.003 0.855±0.002 0.855±0.003 0.860±0.009 0.856±0.001 0.856±0.001 0.856±0.001 0.853±0.001 0.854±0.004 0.853±0.005 0.865±0.016 0.861±0.009 3 3 3 3 2 2 2 2 3 3 3 3 289.9±60.6 223.6±49.7 0.5±0.2 0.3±0.1 286.6±48.1 277.5±56.9 0.5±0.1 0.2±0.1 133.8±30.7 591.3±127.2 1.2±0.2 0.4±0.1 — — 0.788±0.002 0.791±0.001 — — 0.786±0.001 0.787±0.001 — — 0.786±0.001 0.786±0.001 — — 5 5 — — 5 5 — — 5 5 > 104 > 104 16.8±2.9 4.7±0.5 > 104 > 104 24.6±1.5 5.4±0.5 > 104 > 104 23.8±0.9 5.4±0.3 — — 0.783±0.001 0.778±0.001 — — 0.779±0.001 0.779±0.001 — — 0.780±0.001 0.778±0.001 — — 8 8 — — 9 9 — — 8 9 > 105 > 105 341.6±58.5 98.4±14.7 > 105 > 105 641.4±82.6 209.3±42.5 > 105 > 105 712.0±142.6 207.6±56.9 TABLE 6 Results on the netflix and yahoo data sets (CPU time is in minutes). The best results (according to the pairwise t-test with 95% confidence) are highlighted. netflix yahoo RMSE rank time RMSE rank time fixed rank LMaFit 0.811±0.001 15 116.4±12.2 0.666±0.001 10 229.3±62.8 ER1MP 0.862±0.006 25 7.1±1.1 0.810±0.003 77 27.9±7.5 capped-`1 FaNCL 0.798±0.001 13 220.0±24.0 0.656±0.001 8 333.0±113.9 FaNCL-acc 0.795±0.001 13 92.8±13.8 0.651±0.001 8 90.2±16.2 LSP FaNCL 0.794±0.001 15 145.5±10.2 0.652±0.001 9 339.8±93.9 FaNCL-acc 0.792±0.001 15 69.6±21.1 0.650±0.001 8 76.1±33.2 TNN FaNCL 0.797±0.001 13 275.1±16.7 0.657±0.001 7 321.2±69.1 FaNCL-acc 0.795±0.001 13 104.3±17.1 0.650±0.001 7 72.9±16.1 (a) netflix. Fig. 4. RMSE vs CPU time on the MovieLens-10M data sets. 6.2 6.2.1 Robust Principal Component Analysis Synthetic Data In this section, we first perform experiments on a synthetic data set. The observed m × m matrix is generated as O = UV+ S̃+G, where elements of U ∈ Rm×k , V ∈ Rk×m (with k = 0.01m) are sampled i.i.d. from N (0, 1), and elements of G are sampled from N (0, 0.1). Matrix S̃ is sparse, with 1% of its elements randomly set to 5kUVk∞ or −5kUVk∞ with equal probabilities. The whole data set is then randomly split into training and test sets of equal size. The standard `1 regularizer is used as the sparsity regularizer g in (11), and different convex/nonconvex lowrank regularizers are used as r. Hyperparameters λ and υ in (11) are tuned using the training set. For performance evaluation, we use (i) NMSE = k(X + S) − (UV + S̃)kF /kUV + S̃kF , where X and S are the recovered low-rank and sparse components, respectively; (ii) (b) yahoo. Fig. 5. RMSE vs CPU time on the netflix and yahoo data sets. accuracy on locating the sparse support of S̃ (i.e., percentage of entries that S̃ij and Sij are nonzero or zero together); (iii) the recovered rank and (iv) CPU time. We vary m in {500, 1000, 2000}. Each experiment is repeated five times. Note that IRNN and active subspace selection cannot be IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, SUBMITTED 12 TABLE 7 RPCA performance on synthetic data. NMSE is scaled by 10−3 , and CPU time is in seconds. The best results (according to the pairwise t-test with 95% confidence) are highlighted. m = 500 m = 1000 m = 2000 NMSE rank time NMSE rank time NMSE rank time nuclear norm APG 4.88±0.17 5 4.3±0.2 3.31±0.06 10 24.5±1.0 2.40±0.05 20 281.2±26.7 capped-`1 GPG 4.51±0.16 5 8.5±2.6 2.93±0.07 10 42.9±6.6 2.16±0.05 20 614.1±64.7 FaNCL-acc 4.51±0.16 5 0.8±0.2 2.93±0.07 10 2.8±0.1 2.16±0.05 20 24.9±2.0 LSP GPG 4.51±0.16 5 8.3±2.3 2.93±0.07 10 42.6±5.9 2.16±0.05 20 638.8±72.6 FaNCL-acc 4.51±0.16 5 0.8±0.1 2.93±0.07 10 2.9±0.1 2.16±0.05 20 26.6±4.1 TNN GPG 4.51±0.16 5 8.5±2.4 2.93±0.07 10 43.2±5.8 2.16±0.05 20 640.7±59.1 FaNCL-acc 4.51±0.16 5 0.8±0.1 2.93±0.07 10 2.9±0.1 2.16±0.05 20 26.9±2.7 TABLE 8 PSNR (in dB) and CPU time (in seconds) on the video background removal experiment. The PSNRs for all the input videos are 16.47dB. bootstrap campus escalator hall PSNR time PSNR time PSNR time PSNR time nuclear norm APG 23.07±0.02 524±84 22.47±0.02 101±6 24.01±0.01 594±86 24.25±0.03 553±85 capped-`1 GPG 23.81±0.01 3122±284 23.21±0.02 691±43 24.62±0.02 5369±238 25.22±0.03 4841±255 FaNCL-acc 24.05±0.01 193±18 23.24±0.02 53±5 24.68±0.02 242±22 25.22±0.03 150±10 LSP GPG 23.93±0.03 1922±111 23.61±0.02 324±27 24.57±0.01 5053±369 25.37±0.03 2889±222 FaNCL-acc 24.30±0.02 189±15 23.99±0.02 69±8 24.56±0.01 168±15 25.37±0.03 144±9 TNN GPG 23.85±0.03 1296±203 23.12±0.02 671±21 24.60±0.01 4091±195 25.26±0.04 4709±367 FaNCL-acc 24.12±0.02 203±11 23.14±0.02 49±5 24.66±0.01 254±30 25.25±0.06 148±11 used here. Their objectives are of the form “smooth function plus low-rank regularizer”, but RPCA also has a nonsmooth `1 regularizer. Similarly, AIS-Impute is only for matrix completion. Moreover, FaNCL, which has been shown to be slower than FaNCL-acc, will not be compared. Results are shown in Table 7. The accuracies on locating the sparse support are always 100% for all methods, and thus are not shown. Moreover, while both convex and nonconvex regularizers can perfectly recover the matrix rank and sparse locations, the nonconvex regularizers have lower NMSE’s. As in matrix completion, FaNCL-acc is much faster; the larger the matrix, the higher the speedup. 6.2.2 1 [6]: PSNR = −10 log10 ( mn kX − Ok2F ) where X ∈ Rm×n is the recovered video, and O ∈ Rm×n is the ground-truth. Results are shown in Table 8. As can be seen, the nonconvex regularizers lead to better PSNR’s than the convex nuclear norm. Moreover, FaNCL-acc is much faster than GPG. Figure 7 shows PSNR vs CPU time on the bootstrap and campus data sets. Again, FaNCL-acc converges to higher PSNR much faster. Results on hall and escalator are similar. Background Removal on Videos In this section, we use RPCA for background removal in videos. Four benchmark videos in [7], [8] are used (Table 9), and example frames are shown in Figure 6. As in [7], the image background is considered low-rank, while the foreground moving objects contribute to the sparse component. TABLE 9 Videos used in the experiment. bootstrap campus escalator #pixels / frame 19,200 20,480 20,800 total #frames 9,165 4,317 10,251 (a) bootstrap. (b) campus. (c) escalator. (a) bootstrap. hall 25,344 10,752 (d) hall. Fig. 6. Example image frames in the videos. Given a video with n image frames, each m1 × m2 frame is first reshaped as a m-dimensional column vector (where m = m1 m2 ), and then all the frames are stacked together to form a m × n matrix. The pixel values are normalized to [0, 1], and Gaussian noise from N (0, 0.15) is added. The experiment is repeated five times. For performance evaluation, we use the commonly used peak signal-to-noise ratio (b) campus. Fig. 7. PSNR vs CPU time on the bootstrap and campus videos. 6.3 Parallel Matrix Completion In this section, we experiment with the proposed parallel algorithm in Section 5 on the Netflix and Yahoo data sets (Table 4). We do not compare with factorization-based IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, SUBMITTED algorithms [44], [45], as they have inferior performance (Section 6.1). The machine has 12 cores, and we use one thread for each core. As suggested in [44], we randomly shuffle all the matrix columns and rows √ before partitioning. We use the LSP penalty (with θ = λ) and fix the total number of iterations to 250. The hyperparameters are the same as in Section 6.1.3. Experiments are repeated five times. Convergence of the objective for a typical run is shown in Figure 8. As we have multiple threads running on a single CPU, we report the clock time instead of CPU time. As can be seen, the accelerated algorithms are much faster than the non-accelerated ones, and parallelization provides further speedup. (a) netflix. (b) yahoo. Fig. 8. Objective value vs clock time for the sequential/parallel versions of FaNCL on the netflix and yahoo data sets. Figure 9 shows the speedup with different numbers of threads. As can be seen, the parallelized variants scale well with the number of threads. In particular, scaling is better on yahoo. The observed entries in its partitioned data submatrices are distributed more evenly, which improves performance of parallel algorithms [48]. Another observation is that the speedup can be larger than one. As discussed in [49], in performing multiplications with a large sparse matrix, a significant amount of time is spent on indexing its nonzero elements. When the matrix is partitioned, each submatrix becomes smaller and easier to be indexed. Thus, the memory cache also becomes more effective. 7 C ONCLUSION In this paper, we considered the challenging problem of nonconvex low-rank matrix optimization. The key observations are that for the popular low-rank regularizers, the singular values obtained from the proximal operator can 13 Fig. 9. Speedup vs the number of threads for parallel FaNCL. The red dashed line indicates linear speedup. be automatically thresholded, and the proximal operator can be computed on a smaller matrix. This allows the proximal operator to be efficiently approximated by the power method. We extended the proximal algorithm in this nonconvex optimization setting with acceleration and inexact proximal step. We further parallelized the proposed algorithm, which scales well w.r.t. the number of threads. Extensive experiments on matrix completion and RPCA show that the proposed algorithm is much faster than the state-of-the-art. 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IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, SUBMITTED A PPENDIX A PARALLEL FA NCL- ACC Algorithm 12 shows the parallel version of FaNCL-acc. Acceleration is performed at step 6. The first inexact proximal step is performed at steps 8-18. Step 19 checks whether the accelerated iterate is accepted. If the condition fails, a second inexact proximal step is performed at steps 22-32. Note that the algorithm is equivalent to Algorithm 6, and thus the convergence analysis in Section 3.6 still holds. Algorithm 12 FaNCL-acc in parallel: FaNCL-acc-PL. Input: choose τ > ρ, λ0 > λ, δ > 0 and ν ∈ (0, 1); 1: initialize V0 , V1 ∈ Rn as random Gaussian matrices, X0 = X1 = 0 and α0 = α1 = 1; 2: partition X0 , X1 , PΩ (X0 ), PΩ (X1 ) and PΩ (O); 3: start q threads for parallelization; 4: for t = 1, 2, . . . T do 5: λt = (λt−1 − λ)ν t + λ; α −1 6: B Yt = Xt + t−1 (Xt − Xt−1 ); αt 7: B Rt = IndeSpan-PL ([Vt , Vt−1 ]); 8: B Aat = Yt − τ1 PΩ (Yt − O); 9: for p = 1, 2, . . . do 10: B [X̃p , Rt ] = ApproxGSVT-PL(Aat , Rt , λτ ); 11: B ap = F (X̃p ); 12: B at = F (Xt ); 13: B aF = kX̃p − Xt k2F ; 14: if ap ≤ at − c1 aF then 15: break; 16: end if 17: end for 18: B Xat+1 = X̃p ; 19: if F (Xat+1 ) ≤ F (Xt ) − 2δ kXat+1 − Yt k2F then 20: B Xt+1 = Xat+1 ; 21: else 22: B At = Xt − τ1 PΩ (Xt − O); 23: for p = 1, 2, . . . do 24: B [X̃p , Rt ] = ApproxGSVT-PL(At , Rt , λτ ); 25: B bp = F (X̃p ); 26: B bt = F (Xt ); 27: B bF = kX̃p − Xt k2F ; 28: if bp ≤ bt − c1 bF then 29: break; 30: end if 31: end for 32: B Xt+1 = X̃p ; 33: end if p 34: αt+1 = 12 ( 4αt2 + 1 + 1); 35: end for 36: return XT +1 . 15 example. Using Proposition 3.6, we can decompose r(X̃p ) as r̆(X̃p ) + r̃(X̃p ), where r̆(X̃p ) r̃(X̃p ) 1 kX̃p k∗ , θ " !# n X σi (X̃p ) σi (X̃p ) . = − log 1 + θ θ i=1 = Let the SVD of X̃p be UΣV> . Assume that X̃p has k singular values larger than 0. Let Uk (resp. Vk ) be the matrix containing the first k columns of U (resp. V). Then, 1 ∂ r̆(X̃p ) = Uk Vk> + B , θ where B ∈ {C : U> k C = 0, CVk = 0, and σ1 (C) ≤ 1}. Let c = [ci ] with ci = θ1 − σi (X1p )+θ . Then, ∂ r̃(X̃p ) = UDiag(c)V> . Thus, a full SVD on X̃p is needed, which is expensive and impractical for large matrices. A PPENDIX C P ROOFS C.1 Proposition 3.1 For simplicity of notations, we write σi (Z) as σi . First, we introduce the definition of super-gradient for a concave function and two lemmas. Definition 2 ( [51]). For a concave function f , its super-gradient ˆ ≡ ∂ (−f ). is given by g ∈ ∂f Lemma C.1 ( [51]). (i) inf g∈∂ˆr̂(y) g ≥ 0; (ii) Assume that yj ≥ yi ≥ 0. Then, supgj ∈∂ˆr̂(yj ) gj ≤ inf gi ∈∂ˆr̂(yi ) gi . Lemma C.2. (i) yi∗ − max (σi − µgi , 0) = 0, where gi ∈ ∂ˆr̂(yi∗ ); (ii) if yi∗ > 0, then yi∗ increases with σi . Proof. (Part (i)): Let gi ∈ ∂ˆr̂(yi∗ ). From the first-order optimality condition of (4), consider the two possibilities: (a) (b) σi + µgi ≤ 0: In other words, the optimal solution is achieved at the boundary, and yi∗ = 0. σi + µgi > 0: We have 0 = yi∗ − σi + µgi , and yi∗ > 0. Combining these two cases, the relationship between yi∗ and σi can be expressed as yi∗ = max (σi − µgi , 0) . (12) (Part (ii)): Assume that yi∗ > 0. Then, (12) becomes yi∗ = σi − µgi . (13) Let σi becomes larger as σj . according to (13), we have two possibilities for its corresponding yj∗ , i.e., A PPENDIX B T HE C HECKING C ONDITION IN [37] The condition in [37] to accept an approximate X̃p is: ∃A ∈ ∂ r̆(X̃p ) − ∂ r̃(X̃p ) where r̃ and r̆ are two convex functions, such that kA + ∇f (X̃p )k2F ≤ bkX̃p − Xk2F for some constant b > 0. Using the LSP regularizer as an • • yj∗ > yi∗ : Then, supgj ∈∂ˆr̂(y∗ ) gj ≤ inf gi ∈∂ˆr̂(y∗ ) gi from j i Lemma C.1. Together with the fact that σj > σi , there exists a yj∗ which is not smaller than yi∗ to make (12) hold. yj∗ ≤ yi∗ : Then, inf gj ∈∂ˆr̂(y∗ ) gj ≥ supgi ∈∂ˆr̂(y∗ ) gi from j i Lemma C.1. However, such a solution may not exist (e.g., when r̂(α) = α). IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, SUBMITTED Thus, while there can be multiple solutions to ensure (13), the first case must exist. We take the largest solution of all possible candidates. Thus, if σi gets larger, yi∗ also becomes larger. Proof of Proposition 3.1. From Lemma C.2, we have and   if 0 ≤ σi ≤ µ 0 p∗i = σi − µ if µ < σi ≤ µ + θ .   θ otherwise qi∗ 0 = yi∗ − max (σi − µgi , 0) . We can see that y ∗ = 0 once σi − µgi ≤ 0. However, if σi becomes smaller, σi − µgi will reach 0 before σi reaches zero. This comes from two facts. First, yi∗ becomes smaller as σi gets smaller (Lemma C.2), but inf gi ∈∂ˆr̂(y∗ ) gi i will not become smaller (Lemma C.1). Second, we have limy→0+ inf g∈∂ˆr̂(y) g > 0. An illustration of the relationships among σi , yi∗ and gi is shown in the following figure. Thus, there exists γ > 0 such that once σi ≤ γ , σi − µgi ≤ 0, and yi∗ becomes 0. 16 (14) Let qi∗ = arg minqi >θ h2 (qi ). As h2 is also quadratic and cannot be θ, we have ( no solution if 0 ≤ σi ≤ θ ∗ qi = . (15) σi otherwise Note that when σi ∈ [0, θ], there is no solution to qi∗ , as qi∗ can arbitrarily close to θ. Since h1 (θ) = h2 (θ), the possibility for θ = arg minyi ≥0 h(yi ) is covered by arg min0≤pi ≤θ h1 . Thus, (14) and (15) have covered all possibilities of yi∗ . Using them, we have 1) If θ ≤ µ, then  h1 (0)    min (h (0), h (σ )) 1 2 i min h = min (h1 (σi − µ), h2 (σi )) yi ≥0    min (h1 (θ), h2 (σi )) 0 ≤ σi ≤ θ θ < σi ≤ µ . µ < σi ≤ µ+θ σi > µ+θ In order to get yi∗ = 0, we need min (h1 (0), h2 (σi )) = h1 (0), which leads to p σi ≤ 2µθ. √ Thus, if 0 ≤ σi ≤ min 2µθ, µ , then yi∗ = 0. C.2 2) Corollary 3.2 In this section, we show how to derive the threshold γ for the capped-`1 penalty. Derivations for the other penalties can be obtained similarly. C.2.1 Capped-`1 Penalty Proof. Note that problem (4) considers each singular value separately. For simplicity of notations, let σi denote σi (Z). For the ith singular value, let h(yi ) ≡ 1 2 (yi − σi ) + µ min (yi , θ) . 2 Thus, Finally, combining the above two cases, we can con√ clude that√ once σi ≤ min( 2θµ, µ), then yi∗ = 0. Thus, γ = min( 2θµ, µ). Proposition 3.3 Proof. First, we introduce the following theorem. , where 1 (pi − σi )2 +µpi , 2 1 h2 (qi ) = (qi − σi )2 + µθ. 2 Note that h1 is quadratic. There are only three possibilities for p∗i = arg min0≤pi ≤θ h1 (pi ), i.e.,  1 2  if p∗i = 0  2 σi 1 2 min h1 (pi ) = µσi − 2 µ if p∗i = σi − µ ,  0≤pi ≤θ 1 2 ∗ 2 (θ − σi ) + µσi if pi = θ h1 (pi ) = 0 ≤ σi ≤ µ µ < σi ≤ θ . θ < σi ≤ µ+θ σi > µ+θ Thus, if 0 ≤ σi ≤ µ, then we have yi∗ = 0. C.3 ( arg min0≤yi ≤θ h1 (yi ) arg min h(yi ) = yi ≥0 arg minyi >θ h2 (yi ) If θ > µ, then   h1 (0)  h (σ − µ) 1 i min h =  yi ≥0 min (h1 (σi − µ), h2 (σi ))    min (h1 (θ), h2 (σi )) Theorem C.3 (Separation theorem [52]). Let A ∈ Rm×n and B ∈ Rm×r with B> B = I. Then σi B> A ≤ σi (A), for i = 1, . . . , min(r, n). Let the SVD of Z be UΣV> . Z can then be rewritten as Σk̂ Z = [Uk̂ ; U⊥ ] [Vk̂ ; V⊥ ]> , (16) Σ⊥ where Uk̂ contains the k̂ leading columns of U, and U⊥ the remaining columns. Similarly, Σk̂ (resp. Vk̂ ) contains the k̂ leading eigenvalues (resp. columns) of Σ (resp. V). Hence, σi Q> Z = max ũ> Q> Z ṽi . (17) i ũi ,ṽi IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, SUBMITTED Let ũi = Q> ui and ṽi = vi , (18) u> i Zvi σi (Z), > p > > Then, kBp B> p − Uk Uk kF ≤ η kB0 B0 − Uk Uk kF , where η ∈ (0, 1) is a constant. Proof. At the pth iteration of Algorithm 4, inside Algorithm 2 (step 3), since Z = A and R = Ṽp−1 , we have where ui (resp. vi ) is the ith column of U (resp. V). Q> Z ṽi = u> QQ> Zvi ũ> i i = = 17 Q1 = QR(AṼp−1 ) = Bp−1 . (19) (20) where (19) is due to span(Uk̂ ) ⊆ span(Q). From Theorem C.3, by substituting Q = B and A = Z, we have σi (Q> Z) ≤ σi (Z). Combining with (20), we obtain that (18) is the optimal solution of (17). Thus, the rank-k̂ SVD of Q> Z is (Q> Uk̂ )Σk̂ Vk̂> , with the corresponding left and right singular vectors contained in Q> Uk̂ and Vk̂ , respectively. Again, by Theorem C.3, we have σk̂+1 Q> Z ≤ σk̂+1 (Z) ≤ γ. Besides, using (16), > ṽi . σi Q> Z =max ũ> Q> Uk̂ Σk̂ Vk̂> +Q> U⊥ Σ⊥ V⊥ i Then, for Ṽp , inside Algorithm 3 (step 2), we have span(Ṽp ) = span(A> Q). Thus, span(AṼp ) = span(A(A> Q)) = span(A(A> QJ )) = span(YJ+1 ), = QR(AṼp ) = Bp . > > > kBp B> p − Uk Uk kF = kQJ+1 QJ+1 − Uk Uk kF > ≤ αJ kQ1 Q> 1 − Uk Uk kF > proxµr (Q Z) = > proxµr (Q Uk̂ Σk̂ Vk̂> + Q> U⊥ Σ⊥ V⊥ ) > > > > ) proxµr (Q Uk̂ Σk̂ Vk̂ ) + proxµr (Q U⊥ Σ⊥ V⊥ > > proxµr (Q Uk̂ Σk̂ Vk̂ ). > = ηkBp−1 B> p−1 − Uk Uk kF , > (22) where η = αJ ∈ (0, 1). Thus, > p > > kBp B> p − Uk Uk kF ≤ η kB0 B0 − Uk Uk kF . (23) where (22) follows from that Q> Uk̂ (resp. Vk̂ ) is orthogonal to QU⊥ (resp. V⊥ ). (21) shows that there are only k̂ singular values in Q> Z larger than γ . Thus, > proxµr (Q> U⊥ Σ⊥ V⊥ ) = 0 and we get (23). Finally, Qproxµr (Q> Z) = Q Q> Uk̂ proxµr (Σk̂ )Vk̂> = Uk̂ proxµr (Σk̂ )Vk̂> (24) = proxµr (Z), (25) where (24) comes from span(Uk̂ ) ⊆ span(Q); (25) comes from that rank-k̂ SVD of Z is Uk̂ Σk̂ Vk̂> and Z only has k̂ singular values larger than γ . C.4 (30) Note that C = Q1 in Lemma C.4. Together with (27) and (30), we have Then, = (29) QJ+1 = QR(YJ+1 ) ũ,ṽ = (28) where (28) comes from the fact that Q is returned after J iterations of Algorithm 2; and (29) from the definition of Yj+1 at step 4 in Algorithm 2. Thus, ũi ,ṽi The first k̂ singular values are from the term Q> Uk̂ Σk̂ Vk̂ . Hence, > ṽ ≤ γ. (21) σk̂+1 Q> Z = max ũ> Q> U⊥ Σ⊥ V⊥ (27) Proposition 3.5 Proof. First, we introduce the following Lemmas. Lemma C.4 ( [27], [53]). In Algorithm 2, let the SVD of Z be ŪΣ̄V̄> , and Ūk contain the first k columns of Ū. We have > j−1 kQj Q> kCC> − Ūk Ū> j − Ūk Ūk kF ≤ α k kF , where α = σk+1 (Z)/σk (Z) ∈ (0, 1) and C = QR(ZR). Lemma C.5. In Algorithm 4, let the rank-k SVD of A be Uk Σk Vk> , and Bp = QR(AṼp ). (26) (Proof of Proposition 3.5) For Bp in (26), we have limp→∞ Bp = Uk from Proposition C.5 where Uk comes from rank-k SVD of A. As k ≥ k̂A , span(Uk̂A ) ⊆ span(Uk ). Then, from Proposition 3.3, we have Uk prox λ r (U> k A) = prox λ r (A). τ τ Thus, limp→∞ X̃p = prox λ r (A). τ C.5 Proposition 3.6 Proof. First, we introduce Lemma C.6. Pm Lemma C.6 ( [54]). Let φ(X) = i=1 f (σi (X)). If f is convex, φ is also convex on X. For r̂ in Assumption A3, it can be rewritten as r̂(α) = r̂1 (α) − r̂2 (α), where r̂1 (α) = κα (for some constant κ) and r̂2 (α) = κα − r̂(α). Obviously, both r̂1 and r̂2 are convex. Define m m X X r̆(X) = r̂1 (σi (X)), and r̃(X) = r̂2 (σi (X)). i=1 i=1 From Lemma C.6, both r̆ and r̃ are convex. Thus, r can also be written as a difference of convex functions: r(X) = r̆(X) − r̃(X). IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, SUBMITTED C.6 C.9 Proposition 3.7 Proof. From step 5 of Algorithm 5 (which ensures (5)), we have F (Xt+1 ) ≤ F (Xt ) − c1 kXt+1 − Xt k2F . Proposition 3.10 Proof. Consider the two cases: 1) Step 8 in Algorithm 6 is performed: Then, 2) δ F (Xt+1 ) ≤ F (Xt ) − kXt+1 − Yt k2F . 2 Step 10 is performed: Then, Summing this from t = 1 to T , we have c1 T X kXt+1 − Xt k2F ≤ F (X1 ) − F (XT +1 ) F (Xt+1 ) ≤ F (Xt ) − c1 kXt+1 − Xt k2F . t=1 ≤ F (X1 ) − inf F. (31) As F is bounded from below (Assumption A2), a1 ≡ F (X1 ) − inf F is a positive constant. Let T → ∞, we have ∞ X kXt+1 − Xt k2F ≤ t=1 a1 . c1 (32) lim f (X) → ∞, F (X1 ) − F (XT +1 ) X δ X ≥ kXt+1 − Yt k2F + c1 kXt+1 − Xt k2F . 2 1 2 t∈ΩT Corollary 3.8 Proof. Combining (31) and (32), we have min kXt+1 − Xt k2F t=1,...,T ≤ T 1X kXt+1 − Xt k2F T t=1 ≤ ∞ 1X kXt+1 − Xt k2F T t=1 ≤ F (X1 ) − inf F . c1 T lim kXkF →∞ As {Xtj } is a subsequence of {Xt } with limit point X∗ , lim Xtj +1 = InexactPS lim Xtj , lim Rtj , (33) tj →∞ lim kXt+1 − Xt k2F = 0, X 3) Both \|Ω1∞ \| and \|Ω2∞ \| are infinite: As in the above two cases, {Xt } is bounded when either of \|Ω1∞ \| and \|Ω2∞ \| is infinite. Combining the above, {Xt } generated from Algorithm 6 is bounded and has at least one limit point. t→∞ which implies tj →∞ \|Ω1∞ \| is infinite but \|Ω2∞ \| is finite: For tj ∈ Ω1∞ , note that, F (Xtj +1 ) ≤ F (Xtj ) due to (35) and (36). From Assumption A2, then the sequence {F (Xtj )} is bounded. Again, from Assumption A2, we have (40), then {Xtj } is also bounded which has at least one limit point [51]. From (32), we have lim Xtj +1 = lim Xtj = X∗ , (40) inf F (X) > −∞, Lemma C.7 ( [24], [37]). If X = prox λ r (X − τ1 ∇f (X)), then τ X is a critical point of (1). tj →∞ f (X) = ∞, which indicates that maxtj =1,...,∞ kXtj kF < ∞. Together with (39), the sequence {Xt } is bounded, which has at least one limit point [51]. Proof. First, we introduce Lemma C.7. tj →∞ \|Ω1∞ \| is finite but \|Ω2∞ \| is infinite: For tj ∈ Ω2∞ , we have from (38) X a1 kXtj +1 − Xtj k2F ≤ . (39) c1 2 From Assumption A2, we also have Theorem 3.9 tj →∞ (38) t∈Ω∞ tj ∈Ω∞ 2) C.8 (37) where a1 = F (X1 ) − inf F > 0 is a constant. Consider the three cases: 1) C.7 (36) t∈ΩT t∈Ω∞ which implies that maxt=1,...,∞ kXt kt < ∞. Together with (32), {Xt } is a bounded sequence with at least one limit point [51]. (35) Partition the iterations {1, . . . , T } into two sets Ω1T and Ω2T , such that t ∈ Ω1T if step 8 is performed, and t ∈ Ω2T if step 10 is performed. Sum (35) and (36) from t = 1 to T , As F is bounded from below (Assumption A2), X Xδ kXt+1 − Yt k2F + c1 kXt+1 − Xt k2F ≤ a1 . 2 2 1 From Assumption A2, we also have kXkF →∞ 18 (34) for {Xtj }. Combining (33) and (34), we have lim X∗ = InexactPS lim Xtj , lim Rtj , tj →∞ tj →∞ tj →∞ = InexactPS X∗ , lim Rtj . C.10 Corollary 3.11 Proof. Let c2 = min(δ/2, c1 ). From (37), we have F (X1 ) − F (XT +1 ) X X ≥ c2 ( kXt+1 − Yt k2F + kXt+1 − Xt k2F ) t∈Ω1T tj →∞ Thus, X∗ = prox λ r (X∗ − τ1 ∇f (X∗ )) holds by the assumpτ tion. From Lemma C.7, X∗ is a critical point of (1). = c2 T X t=1 kXt+1 − Ct k2F . t∈Ω2T (41) IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, SUBMITTED Thus, min kXt+1 − Ct k2F ≤ T 1X kXt+1 − Ct k2F T t=1 ≤ ∞ 1X kXt+1 − Ct k2F T t=1 t=1,...,T Combining (45) and (46), we have lim Xtj +1 = InexactPS lim Ytj , lim Rtj tj →∞ tj →∞ tj →∞ = InexactPS X∗ , lim Rtj tj →∞ = X∗ . F (X1 ) − inf F ≤ , c2 T Thus, by the assumption, we also have 1 ∇f (X∗ )). τ From Lemma C.7, X∗ is also a critical point of (1). X∗ = prox λ r (X∗ − where the last inequity comes from (41). C.11 τ Theorem 3.12 Proof. Partition the iterations {1, . . . , ∞} into two sets Ω1∞ and Ω2∞ , such that t ∈ Ω1∞ if step 8 is performed, and t ∈ Ω2∞ if step 10 is performed. Consider the three cases: 1) \|Ω1∞ \| is finite but \|Ω2∞ \| is infinite: Let {Xtj } be a subsequence of {Xt } where t ∈ Ω2∞ , and limtj →∞ Xtj = X∗ . From (39), we have lim Xtj = lim Xtj +1 = X∗ . tj →∞ tj →∞ (42) Besides, lim Xtj +1 = InexactPS tj →∞ 3) C.12 1) Steps 10 and 11 are performed: Then, 2) F (Xt+1 , St+1 ) ≤ F (Xt , St ) (47) δ − (kXt+1 − YtX k2F + kSt+1 − YtS k2F ). 2 Steps 13 and 14 are performed: Then, (43) tj →∞ = X∗ . Proposition 4.1 Proof. Consider the two cases: tj →∞ Combining (42) and (43), we have lim Xtj +1 = InexactPS lim Xtj , lim Rtj tj →∞ tj →∞ tj →∞ = InexactPS X∗ , lim Rtj Both \|Ω1∞ \| and \|Ω2∞ \| are infinite: From the above two cases, we can see that the limit point X∗ is also a critical point of (1) when either \|Ω1∞ \| or \|Ω2∞ \| is infinite. Thus, limit points of {Xt } are also critical points of (1). lim Xtj , lim Rtj . tj →∞ F (Xt+1 , St+1 ) ≤ F (Xt , St ) −(c1 kXt+1 − ΘT = 1 X∗ = prox λ r (X∗ − ∇f (X∗ )). τ τ Xδ t∈Ω1T + From Lemma C.7, X∗ is also a critical point of (1). \|Ω1∞ \| is infinite but \|Ω2∞ \| is finite: Let {Xtj } be a subsequence of {Xt } where t ∈ Ω1∞ , and limtj →∞ Xtj = X∗ . From (37), we have X δ kXt+1 − Yt k2F < ∞ 2 1 2 X τ −ρ (c1 kXt+1 − Xt k2F + kSt+1 − St k2F ). 2 2 t∈ΩT Summing (47) and (48) from t = 1 to T , we have F (X1 , S1 ) − F (XT +1 , ST +1 ) ≥ ΘT . (49) As F is bounded from below (Assumption A2), we have Θ∞ ≤ a2 , which indicates (50) where a2 = F (X1 , S1 ) − inf F . We consider the three cases: lim Xtj +1 − Ytj = 0. tj →∞ (44) From (44), we have lim Ytj = lim Xtj +1 = X∗ . tj →∞ (45) Besides, lim Xtj +1 = InexactPS 1) \|Ω1∞ \| is finite but \|Ω2∞ \| is infinite; For tj ∈ Ω2∞ , X a2 kXtj +1 − Xtj k2F ≤ , c1 tj ∈Ω2∞ X 2a2 kStj +1 − Stj k2F ≤ . τ −ρ 2 tj ∈Ω∞ tj →∞ (48) −ρ 2 kSt+1 − St kF ). 2 (kXt+1 − YtX k2F + kSt+1 − YtS k2F ) t∈Ω∞ tj →∞ τ Xt k2F + Partition {1, . . . , T } into two sets as Ω1T and Ω2T , where t ∈ Ω1T if steps 10-11 are performed; otherwise, t ∈ Ω2T (and steps 13-14 are performed). Let Thus, by the assumption, we also have 2) 19 lim Ytj , lim Rtj . tj →∞ tj →∞ (46) Again from Assumption A2, we have lim kXkF →∞ or kSkF →∞ f (X, S) → ∞. (51) IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, SUBMITTED Thus, maxtj =1,...,∞ k[Xtj , Stj ]kF < ∞, and the sequence {[Xtj , Stj ]} is bounded with at least one limit point [51]. 2) 3) \|Ω1∞ \| is infinite but \|Ω2∞ \| is finite: For tj ∈ Ω1∞ , from (47) and (48), we have F (Xtj +1 , Stj +1 ) ≤ F (Xtj , Stj ). As, F is bounded from below (Assumption A2), the sequence {F ([Xtj , Stj ])} must be bounded. Besides, again from Assumption A2, we have (51), which indicates the sequence [Xtj , Stj ]) must be bounded with at least one limit point [51]. \|Ω1∞ \| 20 Partition {1, . . . , ∞} into two sets as Ω1∞ and Ω2∞ , where t ∈ Ω1∞ if steps 10-11 are performed; otherwise, t ∈ Ω2∞ (and steps 13-14 are performed), we consider three cases here. 1) \|Ω1∞ \| is finite but \|Ω2∞ \| is infinite: Let {[Xtj , Stj ]} be a subsequence of {[Xt , St ]} where t ∈ Ω2∞ , and lim Xtj , Stj = [X∗ , S∗ ] . tj →∞ From (49), we have ∞ X \|Ω2∞ \| Both and are infinite: As in the above two cases, {[Xt , St ]} is bounded with at least one limit point once \|Ω1∞ \| or \|Ω2∞ \| is infinite. kXt+1 − Xt k2F < ∞, t=1 ∞ X kSt+1 − St k2F < ∞. t=1 Thus, the sequence {[Xt , St ]} generated from Algorithm 6 is bounded and has at least one limit point. These indicate lim Xtj +1 − Xtj = 0, (53) lim Stj +1 − Stj = 0. (54) tj →∞ C.13 Corollary 4.2 tj →∞ Proof. Let c2 = min(δ/2, c1 ). First, we have Xδ t∈Ω1T + 2 (kXt+1 − YtX k2F + kSt+1 − From (53), we have YtS k2F ) lim Xtj +1 = lim Xtj = X∗ . tj →∞ X τ −ρ (c1 kXt+1 − Xt k2F + kSt+1 − St k2F ) 2 2 T X k[Xt+1 , St+1 ] − Ct k2F . (52) t=1 tj →∞ Together with (49) and (50), we have = X∗ . min k[Xt+1 , St+1 ] − Ct k2F Thus, t=1,...,T 1 ∇X f (X∗ , S∗ )) (56) τ holds by the assumption. Then, the proximal operator is always exact for S. Using (54), we have T 1X ≤ k[Xt+1 , St+1 ] − Ct k2F T t=1 ≤ ∞ X X∗ = prox λ r (X∗ − τ k[Xt+1 , St+1 ] − Ct k2F lim Stj +1 = lim prox µ g (Stj − t=1 tj →∞ F (X1 , S1 ) − inf F , ≤ c2 T τ Definition 3 ( [39]). If X and S satisfy 0 ∈ ∇X f (X, S)+λ (∂ r̆(X)−∂ r̃(X)) , 0 ∈ ∇S f (X, S) + λ (∂ğ(S) − ∂g̃(S)) , 1 ∇S f (Xtj , Stj )) τ 1 ∇S f (X∗ , S∗ )) τ (57) Combining with (56) and (57), [X∗ , S∗ ] is a critical point of (11) by using Lemma C.8. 2) \|Ω1∞ \| is infinite but \|Ω2∞ \| is finite: Let {[Xtj , Stj ]} be a subsequence of {[Xt , St ]} where t ∈ Ω1∞ , and lim Xtj , Stj = [X∗ , S∗ ] . tj →∞ From (49), we have X kXtj +1 − YtXj k2F ≤ ∞, tj ∈Ω2∞ then [X, S] is a critical point of F . X Lemma C.8 ( [37]). If X and S satisfy 1 X = prox λ r (X− ∇X f (X, S)), τ τ 1 S = prox ν g (S − ∇S f (X, S)), τ τ then [X, S] is a critical point of F . τ = S∗ Theorem 4.3 Proof. Let g = ğ + g̃ be the difference of convex decomposition of g . As two blocks of variables are involved, its critical points are defined as follows. tj →∞ = prox µ g (S∗ − where the last inequality comes from (52). C.14 (55) Combing (53) and (55), we have lim Xtj +1 = InexactPS lim Xtj , lim Rtj tj →∞ tj →∞ tj →∞ = InexactPS X∗ , lim Rtj t∈ΩT ≥ c2 tj →∞ kStj +1 − YtSj k2F ≤ ∞, tj ∈Ω2∞ and then lim Xtj +1 − YtXj = 0, (58) lim Stj +1 − YtSj = 0. (59) tj →∞ tj →∞ IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, SUBMITTED C.16 Thus, lim Xtj +1 = lim YtXj = X∗ . tj →∞ (60) tj →∞ Combing (58) and (60), we have lim Xtj +1 = InexactPS lim YtXj , lim Rtj tj →∞ tj →∞ tj →∞ = InexactPS X∗ , lim Rtj tj →∞ 1 ∇X f (X∗ , S∗ )) (61) τ τ holds by the assumption. Then, the proximal operator is always exact for S. Using (59), X∗ = prox λ r (X∗ − lim Stj +1 = lim prox µ g (YtSj τ t →∞ t →∞ j j = prox µ g (S∗ − τ 1 − ∇S f (YtXj , YtSj )) τ 1 ∇S f (X∗ , S∗ )) τ = S∗ (62) Combining with (61) and (62), [X∗ , S∗ ] is a critical point of (11) by using Lemma C.8. Both \|Ω1∞ \| and \|Ω2∞ \| are infinite: As above, either \|Ω1∞ \| or \|Ω2∞ \| is infinite, a limit point [X∗ , S∗ ] is a critical point of (11). Thus, the limit points of the sequence {[Xt , St ]} are also critical points of (11). Proposition 5.1 Proof. As the SVD of A> A is VΣV> , the SVD of A can 1 be written as UΣ 2 V> where U is an orthogonal matrix containing the span of A. From the construction of w, we have AV (Diag(w)) − 21 1 = UΣ 2 (V> V) (Diag(w)) 1 = UΣ 2 (Diag(w)) − 12 − 12 . Consider the two cases. 1) 2) Proof. Let the SVD of B be ŪΣ̄V̄> . Then, P> B = (P> Ū)Σ̄V̄> . Note that (P> Ū)> P> Ū = Ū> (PP> )Ū = Ū> (ŪŪ> )Ū = I, span(P) = span(Ū). Thus, C.15 Proposition 5.2 where the second equality comes from = X∗ . 3) 21 A is of full column rank: Then, 1 1 1 −1 UΣ 2 (Diag(w)) 2 = U Σ 2 Σ− 2 = U, which contains the span of A. Assume that A has k columns and its rank is k̄ < k : Then, 1 −1 UΣ 2 (Diag(w)) 2 1 1 2 = UDiag Σ11 , . . . , Σk̄2k̄ , 0, . . . , 0 −1 −1 Diag Σ112 ,. . ., Σk̄k̄2 , 1,. . ., 1 = [Uk̄ , 0] , where Uk̄ contains the first k̄ columns of U. As A is 1 −1 only of rank k̄ , UΣ 2 (Diag(w)) 2 again covers the span of A. The Proposition then follows. > > (63) Thus, the SVD of P B is (P Ū)Σ̄V̄. As a result, we have V = V̄, Σ = Σ̄. Finally, from U = P> Ū, we have PU = PP> Ū = Ū(Ū> Ū) = Ū, where the second equality again comes from (63).	2
When 3D-Aided 2D Face Recognition Meets Deep Learning: An extended UR2D for Pose-Invariant Face Recognition arXiv:1709.06532v1 [cs.CV] 19 Sep 2017 Xiang Xu, Pengfei Dou, Ha A. Le, Ioannis A. Kakadiaris∗ Computational Biomedicine Lab University of Houston 4800 Calhoun Rd. Houston, TX, USA Abstract Most of the face recognition works focus on specific modules or demonstrate a research idea. This paper presents a pose-invariant 3D-aided 2D face recognition system (UR2D) that is robust to pose variations as large as 90◦ by leveraging deep learning technology. The architecture and the interface of UR2D are described, and each module is introduced in detail. Extensive experiments are conducted on the UHDB31 and IJB-A, demonstrating that UR2D outperforms existing 2D face recognition systems such as VGG-Face, FaceNet, and a commercial off-the-shelf software (COTS) by at least 9% on the UHDB31 dataset and 3% on the IJB-A dataset on average in face identification tasks. UR2D also achieves state-of-the-art performance of 85% on the IJB-A dataset by comparing the Rank-1 accuracy score from template matching. It fills a gap by providing a 3D-aided 2D face recognition system that has compatible results with 2D face recognition systems using deep learning techniques. Keywords: Face Recognition, 3D-Aided 2D Face Recognition, Deep Learning, Pipeline 2010 MSC: 00-01, 99-00 Preprint submitted to SI on Biometrics in the wild, Image and Vision Computing September 20, 2017 Figure 1: Depiction of existing pose problem from selected samples. Distribution of yaw angles are from −90◦ to +90◦ in (T) constrained dataset UHDB31 [1] and in (B) the wild dataset IJB-A [2]. 1. Introduction Face recognition is an application in which the computer either classifies human identity according to the face (face identification) or verifies whether two images belong to the same subject (face verification). A common face recognition system has two steps: enrollment and matching. Specifically, in the enrollment stage, features are obtained from a facial image or a set of images to obtain a signature or a template for each subject. The enrollment usually has three steps: (i) face detection, (ii) face alignment, and (iii) signature generation. In the matching stage, these signatures are compared to obtain a distance for the identification or verification problem. Recently, face recognition technology has significantly advanced by the deployment of deep learning technology, especially using Convolutional Neural Networks (CNN). Pure 2D face recognition (2D-FR) systems have achieved human performance or even better. DeepFace, proposed by Taigman et al. [3], first reported performance on the Labeled Faces in the Wild (LFW) standard benchmark [4] that was better than human efforts. FaceNet, proposed by Schroff et al. [5], used triplet loss to train a deep neural ∗ Corresponding author Email address: [email protected] (Ioannis A. Kakadiaris) 2 network using 200 million labeled faces, and obtained a performance of 99.63% verification accuracy on the same dataset. The success of deep learning techniques in face recognition indeed relies on the following four aspects: (i) a large amount of data either from public datasets such as WebFace [6] and Ms-Celeb-1M [7], or private datasets, (ii) advanced network architecture such as VGG [8] and ResNet [9], (iii) discriminative learning approaches such as Triplet Loss [5], Center Loss [10], Range Loss [11], SphereFace [12], and (iv) regularization methods such as Noisy Softmax [13]. However, face recognition is still not a solved problem in real-world conditions. Some datasets, such as LFW, use Viola-Jones face detector, which is not designed to work in the whole pose distribution from −90◦ to +90◦ . In an unconstrained scenario, especially using surveillance camera, there is a plethora of images with large variations in head pose, expression, illumination, and occlusions. To overcome these challenges, a 3D face model can be applied to assist a 2D face recognition. A 3D facial model is intrinsically invariant to pose and illumination. To use a 3D face model, a model should is fitted on the facial images and a 3D-2D projection matrix is estimated. With the help of a projection matrix and fitted 3D model, it is easy to rotate the face out-plane and align the input images from any arbitrary large pose positions to the frontal position for the feature extraction and signature matching. In the last few years, researchers focused on the 2D face recognition from pure 2D image view and have developed numerous loss function approaches to learn the discriminative features from the different poses. A limited number of 3D-aided 2D face recognition systems (3D2D-FR) have been developed using the 3D model to help align 2D images. Kakadiaris et al. [14] proposed a pose and illumination invariant system which frontalized the face image using annotated face model (AFM). Hu et al. [15] proposed a unified 3D morphable model (U-3DMM) which has additional PCA subspace for perturbation. To address the problem mentioned above, this paper presents a 3D-aided 2D face recognition system called UR2D which significantly improves face recognition performance using the AFM and deep learning technology, especially in large pose scenarios. There is enormous demand [16] for pose-invariant face recognition systems because frontal face recognition can be considered as a solved problem. 3 UR2D consists of several independent modules: face detection, landmark detection, 3D model reconstruction, pose estimation, lifting texture, signature generation, and signature/template matching. Despite face detection methods, all other methods are developed in the Computational Biomedicine Lab. It provides sufficient tools and interfaces to use different sub-modules designed in the system. The core code is written in efficient C++, which provides bindings to Python. The system leverages several open-sourced libraries such as OpenCV [17], glog [18], gflags [19], pugixml [20], JSON for modern C++ [21], and Caffe [22]. In UR2D, after detecting the face and 2D landmarks from image, a 3D model is constructed from a 2D image or several 2D images. By estimating the 3D-2D projection matrix, the correspondence between the 3D model and 2D image can be computed. Then, a 3D model is used to help frontalize the face. The pose-robust features and occlusion encodings are extracted to represent the face. For matching, we use cosine similarity to compute the similarity between two signature vectors. In summary, this paper extends Xu et al. [23] and make the following contributions: • A brief survey of recent face recognition pipeline and each module are summarized; • A pose-invariant 3D-aided 2D face recognition system using deep learning is developed. The intrinsic value of a 3D model is explored to frontalize the face, and the pose-invariant features are extracted for representation. We demonstrate results that a 3D-aided 2D face recognition system exhibits a performance that is comparable to a 2D only FR system. Our face recognition results outperform the VGG-Face, FaceNet, and COTS by at least 9% on the UHDB31 dataset and 3% on the IJB-A dataset on average. In addition, we demonstrate that UR2D can generate template signatures from multiple images and achieve state-of-the-art performance of 85% on the IJB-A dataset. The rest of the paper is organized as follows: modern face recognition systems are reviewed in Sec. 2. In Sec. 3, the architecture of UR2D and its functionalities are discussed. In Sec. 4, each module separately is introduced in detail. Detailed evaluations on the indoor and in-the-wild datasets are reported in Sec. 5. 4 2. Related work We divide the current existing face-related work into two categories: In Sec. 2.1, we discuss detailed recent related work for each module in the common face recognition pipeline from an academic view. System level papers about the implementation are discussed in Sec. 2.2 . 2.1. Modules Face Detection: Face detection is the first step, as well as the most studied topic, in the face recognition domain. Zefeiriou et al. [24] presented a comprehensive survey on this topic. They divided the approaches into two categories: rigid template-based methods, and deformable-parts-models-based methods. In addition to the methods summarized in [24], the approaches of object detection under the regions with a convolutional neural network (R-CNN) framework [25] have been well developed. Some techniques can be directly integrated to face detection [26]. Li et al. [27] used a 3D mean face model and divided the face into ten parts. They joined face proposals into a single R-CNN model. The approach proposed by Hu and Ramanan [28] explored context and resolution of images to fine-tune the residual networks (ResNet) [29], which was demonstrated to detect a face as small as three pixels. Despite the two-stage face detectors above using proposal and classification technique, single-stage detectors have also been developed. SSD [30] and YOLO [31] classify a fixed grid of boxes and learn regression functions to map to the objects simultaneously. Lin et al. [32] address the issue that the performance of single-stage detectors are not as strong as two-stage detectors because of unbalanced positive and negative samples. With focal loss, they also trained state-of-the-art single-stage object detector. Very recently, SSH has been proposed by Najibi et al. [33] using multi-task loss for both classification and regression in the network. Face Alignment: Face alignment refers to aligning the face image to a specific position. Usually, researchers include landmark detection in this topic. Jin and Tan [34] summarized the categories of popular approaches for this task. Cascaded regression was a major trend in this topic and classification frameworks tend to be popular recently. Zhu et al. [35] searched for similar shapes from exemplars and regressed the 5 shapes by using SIFT features and updating the probability of shapes. An ensemble of random ferns [36] are used to learn the local binary discriminative features. Xu and Kakadiaris [37] proposed to jointly learn head pose estimation and face alignment tasks in a single framework (JFA) using global and local CNN features. Some researchers treat the face alignment task as a classification problem. KEPLER [38] joined CNN features from different layers and captured the response map to localize the landmarks. Wu et al. [39] proposed the GoDP algorithm to localize landmarks under a fully convolutional network (FCN) framework by exploring two-pathway information. Some recent works use generative adversarial networks (GAN) to frontalize the face [40, 41, 42]. Huang et al. [40] used two-pathway GAN (TP-GAN) for photorealistic frontal synthesis images, but kept identity and details of texture. Yin et al. [41] incorporated a 3D model with GAN to frontalize faces for large poses in the wild. DRGAN was proposed by Tran et al. [42] to generate the frontalized face from face images under different poses. They also demonstrated the usage of GAN in face recognition. Signature Generation: An emerging topic in face recognition research is generating a discriminative representation for a subject. When training with millions of face images using deep learning technology, many feature descriptors have been proposed recently. Parkhi et al. [8] proposed the VGG-Face descriptor within VGG-Very-Deep architectures. Triplet loss was proposed by Schroff et al. [5] to train a deep neural network using 200 million labeled faces from Google. Masi et al. [43] developed face recognition for unconstrained environments by fine-tuning the ResNet and VGG-Face on 500K 3D rendering images. In addition to frontalizing the face, they also rendered face images to half-profile 40◦ , and full-profile (75◦ ). Masi et al. [44] addressed the question of whether we need to collect millions of faces for training a face recognition system. They argued that we can use synthesized images instead of real images to train the model and still obtain comparable results. Despite triplet loss, many other loss functions have been proposed recently. Center loss was added by Wen et al. [10] alongside cross entropy loss to obtain more discriminative features for deep face recognition. Range loss [11] was designed by Zhang et al. to train deep neural networks with a long tail distribution. A-Softmax Loss [12] was used in SphereFace and demonstrated efficiency in learning the discriminative features. Marginal Loss [45] was proposed to 6 Name Category Core Detection Alignment Representation Matching OpenBR [46] 2D C++ X X X X FaceID1-3 [47] 2D - X X X DeepFace [3] 2D - X X X FaceNet [5] 2D - X X X VGG-Face [8] 2D - X X X OpenFace [48] 2D Torch X X X U-3DMM [15] 3D2D - X X X UR2D 3D2D C++ X X X X X X X Modern Active X Table 1: Comparison of recent existing 2D face recognition pipelines. We employ the same definition of modern and active made by Klontz et al. [46] (“-” means that this information is not provided in the paper). enhance the discriminative ability by maximizing the inter-class distances from large scale training data. 2.2. System OpenCV and OpenBR are some well known open-source computer vision and pattern recognition libraries. However, the eigenface algorithm in OpenCV is out-ofdate. OpenBR has not been updated since 9/29/2015. Both libraries only support nearly frontal face recognition, since the face detector can only detect the frontal face. OpenFace is an open-source implementation of FaceNet [5] by Amos et al. [48] using Python and Torch, which provides four demos for usage. OpenFace applied Dlib face detector and landmark detector to do the pre-processing, which is better than OpenBR. There is another official Tensorflow implementation of FaceNet in which the authors use MTCNN [49] to detect and align face, which boosts performance speed and detection accuracy. To the best of our knowledge, there is a limited amount of well-designed system papers. Most face-related papers focus on different sub-modules or the research of face representations. The comparison of recent existing 2D face recognition systems is presented in Tab. 1 including the research on face representation. 7 3. System Design UR2D is a 3D-aided 2D face recognition system designed for pose-invariant face recognition. Moreover, this system is suitable for face-related research, and can fast pre-process images, provide baselines, plot the results, and support further development. 3.1. System requirements UR2D is written in clean and efficient C++, which is developed on a Linux platform (Ubuntu system). It requires GCC 4.9 or above for compilation. It leverages a list of open-sourced libraries and tools such as CMake, Boost, OpenCV, gflags, glog, puxixml, JSON, and Caffe. Most of the dependencies are available in the Ubuntu repository except Caffe. Therefore, to install dependencies, it only requires installing Caffe manually. 3.2. Architecture Overview Figure 2 illustrates the architecture of UR2D, which explicitly illustrates modules and functionality. The blue blocks are external shared libraries. The other three components belong to our system. As a base of the software, green blocks provide the basic functions. The algorithm modules are constructed as high-level APIs. The applications and GUIs are the top of the software and are built by combining these APIs. The users can directly call these applications and obtain the results. The advantages of this architecture are that it is simple and well-structured. With full development of libraries, the system can use CPUs/GPU and other features easily. 3.3. Data Structures In UR2D, the basic element is File on the disk. All operations or algorithms are based on the files. The basic data structure is Data, which is a hash table with pairs of keys and values. Both keys and values are in string type. Unlike OpenBR [46], to avoid saving giant data in the memory, we only keep the file path in the memory. 8 GUI Applications Detection … File System External Core Caffe Utility Matching Logging Dataset Utility IO GLog Caffe GFlag Modules Internal QT OpenCV Apps Figure 2: Depiction of UR2D’s architecture. In addition to external libraries, it includes some other base libraries to process files, use CUDA, manage the data files, etc. Based on these basic libraries, high level APIs were implemented by calling function from each module. Based on UR2D’s SDK, it is easier to write various applications for different purposes. Also, we created the GUIs to demonstrate our UR2D. 3.4. Configuration We have two approaches to run UR2D. The first one is defining the configuration file (JSON format), which points out the datasets, input files, output directories, involving modules and their model locations, and evaluation. Attribute dataset contains the information of input dataset including the name and path. Attribute input contains the list of galleries and probes. Attribute output defines the output directories. Attribute pipelines defines the modules used in the pipeline. The pip command line application only accepts the argument of the configuration file, which will parse the configuration file, load the models, and run defined modules. The advantages of this approach are simplicity and flexibility. Unlike the OpenBR framework, it does not require a detailed understanding of the option or input long arguments in the command line. The users only need to change some values in the attributes dataset and 9 input (e.g., set dataset directory and file to enroll), and program pip will generate the output they defined in this configuration file. 3.5. Command Line Interface To make full use of SDK of UR2D, we created some corresponding applications to run each module. All applications accept the file list (text or csv file by default, which includes tag at the top line), a folder, or a single image. The IO system will load the data in the memory and process the data according to the data list. The arguments specify the location of the input file/directory and where the output should be saved. UR2D’s enrollment is executed and generates signatures to the output directory. The path of the signature is recorded in the Data. By calling the API from IO system, the list of Data will be written to the file (default is in .csv format). 4. Face Recognition Figure 3 depicts the overview of enrollment in the UR2D, which contains face detection, face alignment, 3D face reconstruction, pose estimation, texture lifting, and signature generation. 4.1. Face Detection A serious problem in OpenBR [46], OpenFace [48], and even the commercial offthe-shelf face recognition software (COTS) is the face detection rate. OpenBR only supports OpenCV frontal face detector. OpenFace also supports Dlib [50] face detector. However, in recent years, many face detection algorithms have been developed [51, 27, 28] with deep learning technology to support multi-view images. To detect the face in multi-view poses, some modern detectors such as Headhunter [52] and DDFD [51], and Dlib-DNN face detector are supported in our system. Mathias et al. [52] trained Headhunter by using multi-scale templates. DDFD face detector is proposed by Farfade et al. [51] by fine-tuning AlexNet [53] and using non-maximum suppression (NMS-max, NMS-avg). To support different face detectors for downstream modules, we perform the bounding box regression on detected bounding box to reduce the variations of the bounding 10 Generate Bounding box Face Detection Gallery Lift textures from original image according to 3D-2D correspondence Reconstruct 3D face model from a single image 3D Reconstruction Euclidean Distance Texture Lifting Refinement Landmark Detection Pose Estimation Representation Localize landmarks based on the response map Compute 3D-2D projection matrix Residual network to learn pose-invariant features Probe Cosine Similarity 𝑃"" 𝑃"# 𝑃"$ 𝑃"& 𝑃#" 𝑃## 𝑃#$ 𝑃#& 𝑃$" 𝑃$# 𝑃$$ 𝑃$& Feature & Occlusion Encoding Images Enrollment Matching Figure 3: Depiction of the whole pipeline (follow the arrow in the middle) of UR2D. The rounded rectangles represent different modules. Dashed arrows represent the workflow. The enrollment encompasses the modules listed. A face is first detected and then transferred to localize landmarks. A 3D model is constructed directly from a 2D image with a bounding box. With 2D landmarks and a 3D model, a 3D-2D projection matrix can be estimated. The frontalized image and occlusion map are generated according to the 3D model and projection matrix. The pose robust features are extracted from these images along with occlusion encoding. The matching step computes features from visible parts and outputs a similarity score. box. The first advantage of this approach is that we do not need to re-train or fine-tune the models for downstream modules after switching the face detector. The second advantage is that this approach provides a more robust bounding box for the landmark localization module. 4.2. Landmark Localization To detect face landmarks, we use GoDP proposed by Wu et al. [39], which is demonstrated to be robust to pose variations. GoDP landmark detector relies on confidence maps generated by a fully convolutional network. A confidence map is generated for each landmark to indicate the possibility of a landmark appearing at a specific location in the original image. The prediction is made by simply selecting the location that has the maximum response in the confidence map. This winner-take-all strategy helps to suppress false alarms generated by background regions and improves the robustness 11 of the algorithm under large head pose variations. Compared to other confidence-mapbased landmark detectors, the novel architecture of GoDP merges the information of the deep and shallow layers based on a new loss function, increases the resolution and discrimination of the confidence maps, and achieves state-of-the-art results on multiple challenging face alignment databases. 4.3. 3D Reconstruction of Facial Shape To reconstruct the 3D facial shape of the input 2D image, we integrate into our pipeline the E2FAR algorithm proposed by Dou et al. [54]. It uses a subspace model to represent a 3D AFM as a parameter vector and employs CNN to estimate the optimal parameter values from a single 2D image. To train the deep neural network, a large set of synthetic 2D and 3D data has been created using the 3D rendering of randomly generated AFMs. To improve the robustness to illumination variation, the deep neural network is pre-trained on real facial images and fine-tuned on the synthetic data. Compared with existing work, it is more efficient due to its end-to-end architecture, which requires a single feed-forward operation to predict the model parameters. Moreover, it only relies on face detection to localize the facial region of interest on the image. As a result, compared with landmark-based approaches, it is more robust to the pose variation that can degrade landmark detection accuracy. 4.4. Pose estimation Given 2D landmarks X2D obtained from landmark detection and 3D landmarks X3D obtained from a 3D model, the transformation matrix P can be estimated by solving a least-squares problem as follows: min \|\|X2D − P X3D \|\|22 + λ\|\|P \|\|22 . P (1) In our implementation, we use the Levenberg-Marquardt algorithm, also known as DLS, to solve this equation. 4.5. Texture Lifting Facial texture lifting is a technique first proposed by Kakadiaris et al. [14], which lifts the pixel values from the original 2D images to a UV map. Given the 3D-2D 12 projection matrix P , 3D AFM model M , and original image I, it first generates the geometry image G, each pixel of which captures the information of an existing or interpolated vertex on the 3D AFM surface. With G, a set of 2D coordinates referring to the pixels on an original 2D facial image is computed. In this way, the facial appearance is lifted and represented into a new texture image T . A 3D model M and Z-Buffer technique are used to estimate the occlusion status for each pixel. This process generates an occlusion mask Z. This module has the following two advantages: It generates the frontal normalized face images, which is convenient for feature extraction and comparison. Second, it generates occlusion masks, which identify the parts of the face images that are occluded, providing the evidence to exclude the face regions. 4.6. Signatures To improve the performance of face recognition in matching non-frontal facial images, we integrate into our pipeline the algorithm proposed by Dou et al. [55] for extracting Pose-Robust Face Signature (PRFS), a part-based face representation with discriminative local facial features and explicit pose and self-occlusion encoding. The facial texture T and the self-occlusion mask Z are first divided into multiple local patches. Then, on each local patch, discriminative features are extracted and selfocclusion encoding is computed. The ensemble of local features, each enhanced by the self-occlusion encoding, forms the pose-robust face signature. We use two types of local features, namely the DFD feature proposed by Lei et al.[56] and a deep feature we trained by following Wen et al.[10] using center loss. To train the DFD feature, we use a small subset of the FRGC2 database that consists of 907 frontal facial images of 109 subjects. We divide the facial texture into 64 non-overlapping patches and train a DFD feature extractor for each local patch separately. To train the deep feature, the CASIA WebFace dataset [6] is used as training data. We divide the facial texture into 8 partially-overlapping patches and train a deep neural network for each local patch separately. In this paper, we call the face signature with the DFD feature PRFS, and the face signature with the deep feature DPRFS. 13 5. Experiments In this section, we provide a systematical and numerical analysis on two challenging datasets in both constrained and in-the-wild scenarios. First, the datasets used to verify UR2D are introduced. Then, a fair comparison of UR2D with VGG face descriptor (VGG-Face) and a commercial face recognition software (COTS) on these two challenging datasets is conducted for the image matching. In the end, the template matching experiments on IJB-A dataset was performed. Dataset Images Subjects Environment UHDB31 IJB-A 24,255 77 Constrained 25,808 500 In-the-wild Poses Illuminations Usage 21 3 2D-2D, 3D-2D, 3D-3D face recognition Various Various 2D unconstrained face recognition Table 2: Comparison of datasets: UHDB31 [1] and IJB-A [2]. Both are challenging due to pose variations, illumination, and resolution. 5.1. Datasets UHDB31 [1] was created in a controlled lab environment, which allows facerelated research on pose and illumination issues. In addition to 2D images, it also provides the corresponding 3D model of subjects. An interesting fact of this dataset is that pose follows the uniform distribution on three dimensions: pitch, yaw, and roll. For each subject, a total of 21 high-resolution 2D images from different views and 3D data are collected at the same time. Then, a 3D model is registered from the 3D data from different poses to generate a specific 3D face model. In addition to three illuminations, the resolutions are downsampled to 128, 256, and 512 from the original size. IJB-A [2] is another challenging dataset which consists of images in the wild. This dataset was proposed by IARPA and is managed by NIST. This dataset merges images and frames together and provides evaluations on the template level. A template contains one or several images/frames of a subject. According to the IJB-A protocol, it splits galleries and probes into 10 folders. In our experiment, we modify this protocol to use it for close-set face identification. The details will be introduced in Sec. 5.4. A summary of these two datasets is presented in Tab. 2. Our system provides dataset utility to parse and load the data from these two datasets. 14 Yaw −90◦ −60◦ −30◦ 0◦ +30◦ +60◦ +90◦ 14/11/58/ 69/32/95/ 94/90/100/ 99/100/100/ 95/93/99/ 79/38/92/ 19/7/60/ 47/82 90/99 100/100 100/100 100/99 95/99 47/75 22/9/84/ 88/52/99/ 100/99/100/ 100/100/100/ 94/73/99/ 27/10/91/ 81/96 100/100 100/100 100/100 100/100 84/96 8/0/44/ 2/19/80/ 91/90/99/ 96/99/99/ 96/98/97/ 52/15/90/ 9/3/35/ 44/74 90/97 99/100 100/100 99/100 95/96 58/78 Pitch +30◦ 0◦ −30◦ - Table 3: Comparison of Rank-1 percentage of different systems on UHDB31.R128.I03. The methods are ordered as VGG-Face, COTS v1.9, FaceNet, UR2D-PRFS, and UR2D-DPRFS. The index of poses are ordered from the left to right and from the top to bottom (e.g., pose 3 is pitch −30◦ and yaw −90◦ , pose 11 is pitch 0◦ and yaw 0◦ ). The frontal face is gallery while the other poses are probes. In all cases, our system achieves the best performance compared with the state-of-the-art. 5.2. Baselines To perform a fair comparison with current state-of-the-art face recognition systems, we choose VGG-Face and COTS v1.9 as baselines. The VGG-Face descriptor was developed by Parkhi et al. [8]. The original release contains a Caffe model and a MATLAB example. We re-used their model, implemented their embedding method on multi-scaled images, and fused the features in C++. In our implementation, we tried different combinations of descriptor and matching methods. We found that embedding features with cosine similarity metrics works the best for the VGG-Face. In our experiment, we use VGG-Face to represent the embedding features with matching using a cosine similarity metric. As in the baseline module, UR2D provides API to obtain the features. The FaceNet algorithm was proposed by Schroff et al. [5]. We use a personal implemented FaceNet from GitHub 1 trained using WebFace [6] and MS-Celeb-1M [7]. They first use MTCNN [49] to align face and extract 128 dimensions features. They provide pre-trained models that achieves 99.20% ± 0.30% accuracy on the LFW dataset. The accuracy is a little bit lower than the original paper, but still can be con1 https://github.com/davidsandberg/facenet 15 sidered state-of-the-art. COTS is a commercial software developed for scalable face recognition. It provides SDK and applications which can be used directly. In our experiments, we used version 1.9 to compare with our system. This version is considered to be a significant boost compared with previous versions. In our experiment, we report the performance using both PRFS and DPRFS features. The summary of software configuration is reported in Tab. 4. We compute the Rank-1 identity accuracy from successfully enrolled signatures. System Features Dims Metric VGG-Face Embedding 4096 Cosine COTS v1.9 - - - FaceNet - 128 Cosine UR2D PRFS 64 × 1024 Cosine UR2D DPRFS 8 × 1024 Cosine Table 4: Comparison of systems configuration used in our experiments. 5.3. UHDB31: Pose-Invariant Face Recognition In this experiment, we chose a configuration from the UHDB31 dataset named UHDB31.R0128.I03. This is a subset in which all images are down-sampled to the size 153 × 128 in the neutral illumination. This subset was chosen to demonstrate that our system, UR2D, is robust to different poses. Therefore, we use this configuration to exclude the other variations such as illumination, expressions, etc, but only keep the pose variations. We treated the frontal face images (pose 11) as gallery and images from the other 20 poses (poses 1 − 10, 12 − 21) as probes, independently. Both the gallery and the probe contain 77 images, each of which belongs to a subject. The face identification experiment was performed using 20 pairs of sigsets. Table 3 depicts the comparison of Rank-1 accuracy among 20 poses (except pose 11, which is used for gallery), which indicates that UR2D is robust to the different poses 16 compared with other systems. We observed the VGG-Face and COTS v1.9 algorithms cannot generalize all pose distributions. FaceNet works better than VGG-Face and COTS v1.9 on the extreme poses. One possible answer is that this model is trained from the most available datasets using Ms-Celeb-1M and WebFace, which provide more extreme pose cases. However in cases such as pose 3 (−30◦ , −90◦ ) and pose 21 (−30◦ , −90◦ ) in Tab. 3, the performance of 2D only face recognition pipelines still has significant room for improvement. On the other hand, with the help of the 3D model, our system keeps the consistent and symmetric performance among the different poses. Even in the cases with yaw −90◦ or +90◦ , our system can tolerate the pose variations, and achieves around 80% Rank-1 identity accuracy with DPRFS features and around 50% Rank-1 identity accuracy with PRFS features on average. 5.4. IJB-A: In-the-Wild Face Recognition However, in a real-world case, a face recognition system does not suffer only from pose variations. In this experiment, we want to explore whether our system is can also be used in an in-the-wild environment. We designed a different protocol for face identification experiments based on the original 10 splits. Unlike the original templatelevel comparison, we conducted an image pairs comparison. First, we removed some samples in the IJB-A splits to make 10 close-set comparison pairs. Then, we cropped the face according to the annotations. Image thumbnails with resolution below 50 were up-sampled, while those with resolution larger than 1000 were down-sampled. Herein, we do not compare with FaceNet since there are overlapping samples between the training set and IJB-A dataset. Method Split-1 Split-2 Split-3 Split-4 Split-5 Split-6 Split-7 Split-8 Split-9 Split-10 Avg. VGG-Face 76.18 74.37 24.33 47.67 52.07 47.11 58.31 54.31 47.98 49.06 53.16 COTS v1.9 75.68 76.57 73.66 76.73 76.31 77.21 76.27 74.50 72.52 77.88 75.73 UR2D-PRFS 47.61 49.27 47.71 47.71 48.97 44.83 52.98 44.14 43.40 49.02 47.56 UR2D-DPRFS 78.20 76.97 77.31 79.00 78.01 79.00 81.15 78.40 74.97 78.57 78.16 Table 5: Comparison of Rank-1 percentage of different systems on 10 splits of IJB-A. UR2D achieves the best performance with DPRFS features on each split. Table 5 depicts the rank-1 identification rate with different methods on IJB-A 17 dataset. Our system UR2D with DPRFS reports better performance compared with VGG-Face and COTS v1.9. Also, our system results are consistent on 10 splits, which indicates that our system is robust. Why do PRFS features in our system not perform well on the IJB-A dataset? One possible answer is that PRFS features are trained on the FRGC dataset, which has notably fewer variations of pose, illumination, and resolution problems. The current PRFS features cannot generalize on these images with large variances. The corresponding solution is retraining the PRFS feature model on the in-the-wild dataset. Third, COTS performs well on this challenging dataset, since it is designed for the real scenario. By comparing the experiment in Sec. 5.3, we are left with the question why does our system perform only slightly better than baselines? We argue that in in-the-wild scenarios there are complicated combinations of pose variations, illumination, expression, and occlusions. A robust face recognition system should take all cases into consideration. In addition, COTS dropped hard samples and enrolled fewer signatures than ours, which would boost the performance to some extent. Method Rank-1 (%) OpenBR [46] 24.60 ± 1.10 Wang et al. [57] 82.20 ± 2.30 DCNN [58] 85.20 ± 1.80 PAM [43] 77.10 ± 1.60 DR-GAN [42] 85.50 ± 1.50 UR2D 85.65 ± 1.74 Table 6: Comparison of Rank-1 percentage of different systems on 10 splits of IJB-A. UR2D achieves the best performance with DPRFS features on each split. We extended UR2D to enroll the several images for a subject to generate a template. The template is an average of signatures computed by generating a unified 3D model from several 2D images. Here we use the results from [42] to do the comparison. Table 6 lists the average Rank-1 identification accuracy for each method. UR2D achieved 18 Method Split-1 Split-2 Split-3 Split-4 Split-5 Split-6 Split-7 Split-8 Split-9 Split-10 Avg. VGG-Face 74.44 74.26 70.68 73.96 69.60 72.64 72.91 70.03 72.25 71.78 72.25 UR2D(DPRFS) 87.22 86.82 83.68 86.05 83.52 88.22 85.14 83.59 84.86 87.38 85.65 Table 7: Detailed Rank-1 percentage of different systems on 10 splits of IJB-A. the best performance. The detailed comparison of Rank-1 identification accuracy with VGG-Face is summarized in Tab. 7 for 10 splits in the IJB-A dataset. 5.5. Memory Usage and Running Time We conducted the analysis of UR2D in terms of both memory and time. Cafferelated implementation runs on GPU (GTX TITAN X). COTS v1.9 makes full use of eight CPUs. Table 8 summarizes the system run-times for different systems. Some modules in our implementation or external libraries run on CPU, such as face detection, pose estimation, text-lifting, and PRFS feature extraction. Therefore, the time used by PRFS features takes 1.5 s more than DPRFS features. Due to loading several large models, DPRFS requires more memory. The user can define the suitable feature extractors according to their needs. Since we optimized Memory for DPRFS, it shares the memory block in the GPU. The memory cost is reduced to the same level as PRFS. We use DPRFS by default. System GPU Memory (GB) Time (s) VGG-Face Full 1.2 0.9 COTS v1.9 No 0.1 0.5 UR2D (PRFS) Partial 2.4 2.5 UR2D (DPRFS) Partial 2.4 1.0 Table 8: Comparison of system run-times. “Partial” in GPU column denotes part of the code does not support GPU acceleration. Time means the average enrollment time for a single image. 19 6. Conclusion In this paper, a well-designed 3D-aided 2D face recognition system (UR2D) that is robust to pose variations as large as 90◦ using deep learning technology has been presented. An overview of the architecture, interface, and each module in UR2D are introduced i detailed. Extensive experiments are conducted on UHDB31 and IJB-A to demonstrate that UR2D is robust to the pose variations, and it outperforms existing 2D-only face recognition systems such as VGG face descriptor, FaceNet, and a commercial face recognition software by at least 9% on UHDB31 dataset and 3% on IJB-A dataset in average. And the system achieves the state-of-the-art performance of 85% in template matching on IJB-A dataset. 7. Acknowledgment This material is based upon work supported by the U.S. Department of Homeland Security under Grant Award Number 2015-ST-061-BSH001. This grant is awarded to the Borders, Trade, and Immigration (BTI) Institute: A DHS Center of Excellence led by the University of Houston, and includes support for the project “Image and Video Person Identification in an Operational Environment” awarded to the University of Houston. 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arXiv:1611.05067v1 [math.AC] 15 Nov 2016 On S-coherence Driss Bennis1,a and Mohammed El Hajoui1,b 1. Department of Mathematics, Laboratory of Analysis, Algebra and Decision Support, Faculty of Sciences, B.P. 1014, Mohammed V University in Rabat, Rabat, Morocco a. [email protected]; driss [email protected] b. [email protected] Abstract. Recentely, Anderson and Dumitrescu’s S-finiteness has attracted the interest of several authors. In this paper, we introduce the notions of Sfinitely presented modules and then of S-coherent rings which are S-versions of finitely presented modules and coherent rings, respectively. Among other results, we give an S-version of the classical Chase’s characterization of coherent rings. We end the paper with a brief discussion on other S-versions of finitely presented modules and coherent rings. We prove that these last S-versions can be characterized in terms of localization. Key Words. S-finite, S-finitely presented, S-coherent modules, S-coherence rings. 2010 Mathematics Subject Classification. 13E99. 1 Introduction Throughout this paper all rings are commutative with identity; in particular, R denotes such a ring, and all modules are unitary. S will be a multiplicative subset of R. We use (I : a), for an ideal I and an element a ∈ R, to denote the quotient ideal {x ∈ R; xa ∈ I}. According to [1], an R module M is called S-finite if there exists a finitely generated submodule N of M such that sM ⊆ N for some s ∈ S. Also, from [1], an R-module 1 2 D. Bennis and M. El Hajoui M is called S-Noetherian if each submodule of M is S-finite. In particular, R is said to be an S-Noetherian ring, if it is S-Noetherian as an R-module; that is, every ideal of R is S-finite. It is clear that every Noetherian ring is S-Noetherian. The notions of S-finite modules and of S-Noetherian rings were introduced by Anderson and Dumitrescu motivated by the works done in [8] and [2]. They succeeded to generalize several well-known results on Noetherian rings including the classical Cohen’s result and Hilbert basis theorem under an additional condition. Since then the S-finiteness has attracted the interest of several authors (see for instance [6, 7, 10, 11, 12, 14]). Recentely, motivated by the work of Anderson and Dumitrescu, S-versions of some classical notions have been introduced (see for instance [6, 10]). In this paper we are inerested in S-versions of finitely presented modules and coherent rings. Actually, there are two possibilities which could be considered as S-versions of finitely presented modules which lead to two S-versions of coherent rings. We prove that the S-version of coherent rings defined by one of them has a characterization similar to the classical one given by Chase for coherent rings [3, Theorem 2.2]. This is why we adopt this notion as the suitable S-version of finitely presented modules. However, it seems not evident to characterize this notion in terms of localization. We prove that indeed it is the other S-version, which is briefly studied at the end of the paper, has a characterization in terms of localization. The organization of the paper is as follows: In Section 2, we introduce and study an S-version of finitely presented modules. We call it an S-finitely presented module (see Definition 2.1). Then, we study the behavior of S-finiteness in short exact sequences (see Theorem 2.5). We end Section 2 with some change of rings results (see Proposition 2.7 and Corollary 2.8). Section 3 is devoted to the S-version of coherent rings which are called S-coherent rings (see Definition 3.3). Our main result represents the S-counterpart of Chase’s result [3, Theorem 2.2] (see Theorem 3.8). Also an S-version of coherent modules is introduced (see Definition 3.1 and Proposition 3.2). We end the paper with a short section which presents the other Sversion of S-finiteness (see Definitions 4.1 and 4.4). We prove that these notions can be characterized in terms of localization (see Proposition 4.3 and Theorem 4.7). We end the paper with results which relate S-finiteness with the notion of S-saturation (see Propositions 4.9 and 4.8 and Corollary 4.10). 3 On S-coherence 2 S-finitely presented modules In this section, we introduce and investigate an S-version of the classical finitely presented modules. Other version is discuted in Section 4. Definition 2.1 An R-module M is called S-finitely presented, if there exists an exact sequence of R-modules 0 −→ K −→ F −→ M −→ 0, where K is S-finite and F is a finitely generated free R-modules. Clearly, every finitely presented module is S-finitely presented. However, the converse does not hold in general. For that, it suffices to note that when R is a nonNoetherian S-Noetherian ring, then there is an S-finite ideal I which is not finitely generated. Then, the R-module R/I is S-finitely presented but it is not finitely presented. Also, it is evident that every S-finitely presented module is finitely generated. To give an example of a finitely generated module which is not S-finitely presented, it suffices to consider an ideal I which is not S-finite and then use Proposition 2.4 given hereinafter. One could remark that in Definition 2.1 we assume that the free module F is finitely generated rather than S-finite. In fact, because of the following result both of notions coincide for free modules. Proposition 2.2 Every S-finite free R-module is finitely generated. Proof. Let M = L Rei be an S-finite free R-module, where (ei )i∈I is a basis of M i∈I and I is an index set. Then, there exist a finitely generated R-module N and an s ∈ S such that sM ⊆ N ⊆ M . Then, N = Rm1 + · · · + Rmn for some m1 , ..., mn ∈ M (n > 0 is an integer). For every k ∈ {1, ..., n}, there exists a finite subset Jk of I n S P Jk . Then, the finitely generated R-module λkj ej . Let J = such that mk = j∈Jk k=1 L Rej contains N . We show that M ′ = M . Deny. There exists an i0 ∈ I\J M′ = j∈J P ′ λj ej for some λ′j ∈ R. such that ei0 ∈ / M ′ . But sei0 ∈ N ⊆ M ′ and so sei0 = j∈J This is impossible since (ei )i∈I is a basis. Remark 2.3 Similarly to the proof of Proposition 2.2 above, one can prove that any S-finite torsion-free module cannot be decomposed into an infinite direct sum of non-zero modules. This shows that any S-finite projective module is countably 4 D. Bennis and M. El Hajoui generated by Kaplansky [9, Theorem 1]. Then, naturaly one would ask of the existence of S-finite projective module which is not finitely generated. For this, consider ∞ Q the Boolean ring R = ki , where ki is the field of two elements for every i ∈ N. i=1 Consider the projective ideal M = ∞ L ki and the element e = (1, 0, 0, ...) (see [4, i=1 Example 2.7]). Then, S = {1, e} is a multiplicative subset of R. Since eM = k1 is a finitely generated R-module, M is the desired example of S-finite projective module which is not finitely generated. However, determining rings over which every S-finite projective module is finitely generated could be of interest. It is worth noting that rings over which every projective module is a direct sum of finitely generated modules satisfy this condition. These rings were investigated in [13]. Next result shows that, as in the classical case [5, Lemma 2.1.1], an S-finitely presented module does not depend on one specific short exact sequence of the form given in Definition 2.1. Proposition 2.4 An R-module M is S-finitely presented if and only of M is finitely f generated and, for every surjective homomorphism of R-modules F −→ M −→ 0, where F is a finitely generated free R-module, ker f is S-finite. Proof. (⇐) Obvious. (⇒) Since M is S-finitely presented, there exists an exact sequence of R-modules 0 −→ K −→ F ′ −→ M −→ 0, where K is S-finite and F ′ is finitely generated and free. Then, by Schanuel’s lemma, K ⊕ F ∼ = ker f ⊕ F ′ , then ker f is S-finite. The following result represnts the behavior of S-finiteness in short exact sequences. It is a generalization of [5, Theorem 2.1.2] for modules with λ-dimension at most 1. Note that one can give an S-version of the classical λ-dimension (see [5, page]). However, here we prefer to focus on the notion of S-finitely presented modules, and a discussion on the suitable S-version of the λ-dimension could be the subject of a further work. f g Theorem 2.5 Let 0 −→ M ′ −→ M −→ M ′′ −→ 0 be an exact sequence of Rmodules. The following assertions hold: 1. If M ′ and M ′′ are S-finite, then M is S-finite. In particular, every finite direct sum of S-finite modules is S-finite. 2. If M ′ and M ′′ are S-finitely presented, then M is S-finitely presented. In particular, every finite direct sum of S-finitely presented modules is Sfinitely presented. 5 On S-coherence 3. If M is S-finite, then M ′′ is S-finite. In particular, a direct summand of an S-finite module is S-finite 4. If M ′ is S-finite and M is S-finitely presented, then M ′′ is S-finitely presented. 5. If M ′′ is S-finitely presented and M is S-finite, then M ′ is S-finite. Proof. 1. Since M ′′ is S-finite, there exist a finitely generated submodule N ′′ of n P M ′′ and an s ∈ S such that sM ′′ ⊆ N ′′ . Let N ′′ = Rei for some ei ∈ M ′′ and i=1 n ∈ N. Since g is surjective, there exists an mi ∈ M such that g(mi ) = ei for every i ∈ {1, ..., n, }. Let x ∈ M , so sx ∈ N = g −1 (N ′′ ). Then g(sx) ∈ g(N ) = N ′′ , and n n n n P P P P so g(sx) = αi ei = αi g(mi ) = g( αi mi ). Then, g(sx − αi mi ) = 0. Thus, (sx − n P i=1 i=1 i=1 i=1 αi mi ) ∈ ker g = Imf which is S-finite. So there exist a finitely generated i=1 submodule N ′ of Imf and an s′ ∈ S such that s′ Imf ⊆ N ′ . Then, s′ sx ∈ N ′ + and so s′ sM is a submodule of N ′ + n P n P Rmi i=1 Rmi which is a finitely generated submodule i=1 of M . Therefore, M is S-finite. 2. Since M ′ and M ′′ are S-finitely presented, there exist two shorts exacts sequences: 0 −→ K ′ −→ F ′ −→ M ′ −→ 0 and 0 −→ K ′′ −→ F ′′ −→ M ′′ −→ 0, with K ′ and K ′′ are S-finite R-modules and F ′ and F ′′ are finitely generated free R-modules. Then, by Horseshoe Lemma, we get the following diagram 0O 0O 0O 0 / M′ O /M O / M ′′ O /0 0 / F′ O / F ′ ⊕ F ′′ O / F ′′ O /0 0 / K′ O /K O / K ′′ O /0 0 0 0 By the first assertion, K is S-finite. Therefore, M is S-finitely presented. 3. Obvious. 6 D. Bennis and M. El Hajoui 4. Since M is S-finitely presented, there exists a short exact sequence of R-modules 0 −→ K −→ F −→ M −→ 0, where K is S-finite and F is a finitely generated free R-module. Consider the following pullback diagram 0O 0O 0 / M′ O /M O / M ′′ /0 0 /D O /F O / M ′′ /0 KO KO 0 0 By (1), D is S-finite. Therefore, M ′′ is S-finitely presented. 5. Since M ′′ is S-finitely presented, there exists a short exact sequence 0 −→ K −→ F −→ M ′′ −→ 0 where K is S-finite and F is a finitely generated free R-module. Consider the following pullback diagram 0O 0O 0 / M′ /M O / M ′′ O /0 0 / M′ /D O /F O /0 KO KO 0 0 Since F is free, D ∼ = M ′ ⊕ F , and so D is S-finite (since M ′ and F are S-finite). ′ Therefore, M is S-finite. As a simple consequence, we get the following result which extends [5, Corollary 2.1.3]. Corollary 2.6 Let N1 and N2 be two S-finitely presented submodules of an Rmodule. Then, N1 + N2 is S-finitely presented if only if N1 ∩ N2 is S-finite. 7 On S-coherence Proof. Use the short exact sequence of R-modules 0 −→ N1 ∩ N2 −→ N1 ⊕ N2 −→ N1 + N2 −→ 0. We end this section with the following change of rings results. The following result extends [5, Theorem 2.1.7]. Proposition 2.7 Let A and B be rings, let φ : A −→ B be a ring homomorphism making B a finitely generated A-module and let V be a multiplicative subset of A such that 0 6∈ φ(V ). Every B-module which is V -finitely presented as an A-module it is φ(V )-finitely presented as a B-module. Proof. Let M be a B-module which is V -finitely presented as an A-module. Then M is a finitely generated A-module. Then, M is a finitely generated B-module. Thus there is an exact sequence of B-modules 0 −→ K −→ B n −→ M −→ 0, where n > 0 is an integer. This sequence is also an exact sequence of A-modules. Since M is an V -finitely presented A-module and B n is a finitely generated A-module (since B is a finitely generated A-module), K is an V -finite A-module, and so K is a φ(V )-finite B-module. Therefore, M is a φ(V )-finitely presented B-module. The following result extends [5, Theorem 2.1.8 (2)]. Proposition 2.8 Let I be an ideal of R and let M be an R/I-module. Assume that I ∩ S = ∅ so that T := {s + I ∈ R/I; s ∈ S} is a multiplicative subset of R/I. Then, 1. M is an S-finite R-module if and only if M is a T -finite R/I-module. 2. If M is an S-finitely presented R-module, then M is a T -finitely presented R/I-module. The converse holds when I is an S-finite ideal of R. Proof. 1. Easy. 2. Use the canonical ring surjection R −→ R/I and Proposition 2.7. Conversely, if M is a T -finitely presented R/I-module. Then, there is an exact sequence of R/I-modules, and then of R-modules 0 −→ K −→ (R/I)n −→ M −→ 0, where n > 0 is an integer and K is a T -finite R/I-module. By the first assertion, K is also an S-finite R-module. And since I is an S-finite ideal of R, (R/I)n is an S-finitely presented R-module. Therefore, by Theorem 2.5 (4), M is an S-finitely presented R-module. 8 3 D. Bennis and M. El Hajoui S-coherent rings Before giving the definition of S-coherent rings, we give, following the calssical case, the definition of S-coherent modules. Definition 3.1 An R-module M is said to be S-coherent, if it is finitely generated and every finitely generated submodule of M is S-finitely presented. Clearly, every coherent module is S-coherent. However, using Proposition 3.2(1) below, one can show that, for an S-finite ideal I of R which is not finitely generated, the R-module R/I is S-coherent but it is not coherent. The reason of why we consider finitely generated submodules rather than S-finite submodules is explained in assertion (4) of Remark 3.4. The following result studies the behavior of S-coherence of modules in short exact sequences. It generalizes [5, Theorem 2.2.1]. f g Proposition 3.2 Let 0 −→ P −→ N −→ M −→ 0 be an exact sequence of Rmodules. The following assertions hold: 1. If P is S-finite and N is S-coherent, then M is S-coherent. 2. If M and P are S-coherent, then so is N . In particular, every finite direct sum of S-coherent modules is S-coherent. 3. If N is S-coherent and P is finitely generated, then P is S-coherent. Proof. 1. It is clear that M is finitely generated. Let M ′ be a finitely generated submodule of M . Then, f (P ) ⊆ g −1 (M ′ ), so there exist two shorts exacts sequences of R-modules 0 −→ K −→ Rn −→ P −→ 0 and 0 −→ K ′ −→ Rm −→ M ′ −→ 0, where n and m 9 On S-coherence are two positive integers. Then, by Horseshoe Lemma, we get the following diagram 0O 0O 0O 0 /P O / g −1 (M ′ ) O / M′ O /0 0 / Rn O / Rn+m O / Rm O /0 0 /K O / K ′′ O / K′ O /0 0 0 0 Since g −1 (M ′ ) is a finitely generated submodule of the S-coherent module N , g −1 (M ′ ) is S-finitely presented. Then, using Theorem 2.5 (5), K ′′ is S-finite, and so K ′ is S-finite. Therefore, M ′ is S-finitely presented. 2. Clearly N is finitely generated. Let N ′ be a finitely generated submodule of f g N . Consider the exact sequence 0 −→ Ker(g/N ′ ) −→ N ′ −→ g(N ′ ) −→ 0. Then, g(N ′ ) is a finitely generated submodule of the S-coherent module M . Then, g(N ′ ) is S-finitely presented. Then, Ker(g/N ′ ) is finitely generated by Theorem 2.5 (5), and since P is S-coherent, Ker(g/N ′ ) is S-finitely presented. Therefore, by (2) of Theorem 2.5, N ′ is S-finitely presented. 3. Evident since a submodule of P can be seen as a submodule of N . Now we set the definiton of S-coherent rings. Definition 3.3 A ring R is called S-coherent, if it is S-coherent as an R-module; that is, if every finitely generated ideal of R is S-finitely presented. Remark 3.4 1. Note that every S-Noetherian ring is S-coherent. Indeed, this follows from the fact that when R is S-Noetherian, every finitely generated free R-module is S-Noetherian (see the discussion before [1, Lemma 3]). Next, in Example 3.6, we give an example of an S-coherent ring which is not S-Noetherian. 2. Clearly, every coherent ring is S-coherent. The converse is not true in general. As an example of an S-coherent ring which is not coherent, we consider the trivial extension A = Z ⋉ (Z/2Z)(N) and the multiplicative set 10 D. Bennis and M. El Hajoui V = {(2, 0)n ; n ∈ N}. Since (0 : (2, 0)) = 0 ⋉ M is not finitely generated, T is not coherent. Now, for every ideal I of A, (2, 0)I is finitely generated; in fact, (2, 0)I = 2J ⋉ 0, where J = {a ∈ Z; ∃b ∈ (Z/2Z)(N) , (a, b) ∈ I}. Since J is an ideal of Z, J = aZ for some element a ∈ Z. Then, (2, 0)I = 2J ⋉ 0 = (2a, 0)A. This shows that A is V -Noetherian and so V -coherent. 3. It is easy to show that, if M is an S-finitely presented R-module, then MS is a finitely presented RS -module. Thus, if R is a S-coherent ring, RS is a coherent ring. However, it seems not evident to give a condition so that the converse holds, as done for S-Noetherian rings (see [1, Proposition 2 (f)]). In Section 4, we give another S-version of coherent rings which can be characterized in terms of localization. 4. One would propose for an S-version of coherent rings, the following condition “S-C: every S-finite ideal of R is S-finitely presented”. However, if R satisfies the condition S-C, then in particular, every S-finite ideal of R is finitely generated. So, every S-finite ideal of R is finitely presented; in particular, R is coherent. This means that the notion of rings with the condition S-C cannot be considered as an S-version of the classical coherence. Nevertheless, these rings could be of particular interest as a new class of rings between the class of coherent rings and the class of Noetherian rings. To give an example of a coherent ring which does not satisfy the condition ∞ Q S-C, one could consider the Boolean ring B = ki , where ki is the field of i=1 two elements for every i ∈ N, and the multiplicative subset V = {1, e} of B, ∞ L where e = (1, 0, 0, ...) ∈ B. Indeed, the ideal B = ki is V -finite but not i=1 finitely generated. Also, note that the following condition “S-c: every S-finite ideal of R is finitely generated” could be of interest. Indeed, clearly one can show the following equivalences: (a) A ring R satisfies the condition S-C if and only if R is coherent and satisfies the condition S-c. (b) A ring R is coherent if and only if R is S-coherent and satisfies the condition S-c. (c) A ring R is Noetherian if and only if R is S-Noetherian and satisfies the condition S-c. To give an example of an S-coherent ring which is not S-Noetherian, we use the following result. 11 On S-coherence Proposition 3.5 Let R = S = n Y n Y Ri be a direct product of rings Ri (n ∈ N) and i=1 Si be a cartesian product of multiplicative sets Si of Ri . Then, R is S- i=1 coherent if and only if Ri is Si -coherent for every i ∈ {1, ..., n}. Proof. The result is proved using standard arguments. Example 3.6 Consider the ring A given in Remark 3.4 (2). Let B be a coherent ring which has a multiplicative set W such that BV is not Noetherian. Then, A × B is V × W -coherent (by Proposition 3.5), but it is not V × W -Noetherian (by [1, Proposition 2 (f )]). Now, we give our main result. It is the S-counterpart of the classical Chase’s result [3, Theorem 2.2]. We mimic the proof of [5, Theorem 2.3.2]. So we use the following lemma. Lemma 3.7 ([5], Lemma 2.3.1) Let R be a ring, let I = (u1 , u2 , ..., un ) be a finitely generated ideal of R (n ∈ N) and let a ∈ R. Set J = I + Ra. Let F be f a free module on generators x1 , x2 , ..., xr+1 and let 0 −→ K −→ F −→ J −→ 0, be an exact sequence with f (xi ) = ui (1 ≤ i ≤ r) and f (xr+1 ) = a. Then there exists n L g an exact sequence 0 −→ K ∩ F ′ −→ K −→ (I : a) −→ 0, where F ′ = Rxi . i=1 Theorem 3.8 The following assertions are equivalent: 1. R is S-coherent. 2. Every S-finitely presented R-module is S-coherent. 3. Every finitely generated R-submodule of a free R-module is S-finitely presented. 4. (I : a) is an S-finite ideal of R, for every finitely generated ideal I of R and a ∈ R. 5. (0 : a) is an S-finite ideal of R for every a ∈ R and the intersection of two finitely generated ideals of R is an S-finite ideal of R. 12 D. Bennis and M. El Hajoui Proof. The proof is similar to that of [3, Theorem 2.2] (see also [5, Theorem 2.3.2]). However, for the sake of completeness we give its proof here. (1⇒2) Follows from Proposition 3.2 (1). (2⇒1) Obvious. (1⇒3) Let N be a finitely generated submodule of a free R-module F . Hence, there exists a finitely generated free submodule F ′ of F containing N . Then, by (1), F ′ is S-coherent. Therefore, N is S-finitely presented. (3⇒1) Trivial. (1⇒4) Let I be a finitely generated ideal of R. Then, I is S-finitely presented. Consider J = I + Ra, where a ∈ R. Then, J is finitely generated, and so it is S-finitely presented. Thus, there exists an exact sequence 0 −→ K −→ Rn+1 −→ J −→ 0, where K is S-finite. By Lemma 3.7, there exists a surjective homomorphism g : K −→ (I : a) which shows that (I : a) is S-finite. (4 ⇒ 1) This is proved by induction on n, the number of generators of a finitely generated ideal I of R. For n = 1, use assertion (4) and the exact sequence 0 −→ (0 : I) −→ R −→ I −→ 0. For n > 1, use assertion (4) and Lemma 3.7. (1 ⇒ 5) Since R is S-coherent, Proposition 2.4 applied on the exact sequence 0 −→ (0 : a) −→ R −→ aR −→ 0 shows that the ideal (0 : a) is S-finite. Now, Let I and J be two finitely generated ideals of R. Then, I + J is finitely generated and so Sfinitely presented. Then, applying Theorem 2.5 (5) on the short the exact sequence 0 −→ I ∩ J −→ I ⊕ J −→ I + J −→ 0, we get that I ∩ J is S-finite. (5 ⇒ 1) This is proved by induction on the number of generators of a finitely generated ideal I of R, using the two short exact sequences used in 1 ⇒ 5. It is worth noting that, in Chase’s paper [3], coherent rings were characterized using the notion of flat modules. Then, naturaly one can ask of an S-version of flatness that characterizes S-coherent rings similarly to the classical case. We leave it as an interesting open question. We end this section with some change of rings results. The following results extends [5, Theorem 2.4.1]. Proposition 3.9 Let I be an S-finite ideal of R. Assume that I ∩ S = ∅ so that T := {s + I ∈ R/I; s ∈ S} is a multiplicative subset of R/I. Then, an R/I-module M is T -coherent if only if it is an R-module S-coherent. In particular, the following assertions hold: 1. If R is an S-coherent ring, then R/I is a T -coherent ring. 2. If R/I is a T -coherent ring and I is an S-coherent R-module, then R is an S-coherent ring. On S-coherence 13 Proof. Use Proposition 2.8. Next result generalizes [5, Theorem 2.4.2]. It studies the transfer of S-coherence under localizations. Lemma 3.10 Let f : A → B be a ring homomorphism such that B is a flat Amodule, and let V be a multiplicative set of A. If an A-module M is V -finite (resp., a V -finitely presented), then M ⊗A B is an f (V )-finite (resp., f (V )-finitely presented) B-module. Proof. Follows using the fact that flatness preserves injectivity. Proposition 3.11 If R is S-coherent, then RT is an ST -coherent ring for every multiplicative set T of R. Proof. Let J be a finitely generated ideal of RT . Then, there is a finitely generated ideal I of R such that J = IT . Since R is S-coherent, I is S-finitely presented. Then, using Lemma 3.10, the ideal J = I ⊗R RT of RT is ST -finitely presented, as desired. 4 Other S-version of finiteness In this short section, we present another S-version of S-finiteness and we prove that this notion can be characterized in terms of localization. The following definition gives another S-version of finitely presented modules. Definition 4.1 An R module M is called c-S-finitely presented, if there exists a finitely presented submodule N of M such that sM ⊆ N ⊆ M for some s ∈ S. Remark 4.2 1. Clearly, every finitely presented module is c-S-finitely presented. However, the converse does not hold in general. For that it suffices to consider a coherent ring which has an S-finite module which is not finitely generated. An example of a such ring is given in Remark 3.4 (4). 2. The inclusions in Definition 4.1 complicate the study of the behavior of of cS-finitely presented modules in short exact sequences as done in Theorem 2.5. This is why we think that c-S-finitely presented modules will be mostly used by commutative rings theorists rather than researchers interested in notions of homological algebra. This is the reason behind the use of the letter “c” in “c-S-finitely presented”. 14 D. Bennis and M. El Hajoui 3. It seems that there is not any relation between the two notions of c-S-finitely presented and S-finitely presented modules. Nevertheless, we can deduce that in a c-S-coherent ring (defined below), every S-finitely presented ideal is c-Sfinitely presented. It is well-known that if, for an R-module M , MS is a finitely presented RS -module, then there is a finitely presented R-module N such that MS = NS . Nevertheless, what doest not make things work with respect to localization for S-finitely presented modules is the fact that the module N which satisfies MS = NS is not necessarily a submodule of M . For c-S-finitely presented modules we give the following result. Proposition 4.3 1. If an R-module M is c-S-finitely presented, then MS is a finitely presented RS -module. 2. A finitely generated R-module M is c-S-finitely presented if and only if there is a finitely presented submodule N of M such that MS = NS . Proof. 1. Obvious. 2. (⇒) Clear. (⇐) Since M is finitely and MS = NS , there is an s ∈ S such that sM ⊆ N , as desired. Now we define the other S-version of the classical coherence of rings. Definition 4.4 A ring R is called c-S-coherent, if every S-finite ideal of R is Sfinitely presented. Clearly, every coherent ring is c-S-coherent. The converse is not true in general. The ring given in Example 3.4 (2) can be used as an example of a c-S-coherent ring which is not coherent. Also, it is evident that every S-Noetherian ring is c-S-coherent. As done in Example 3.6, we use the following result to give an example of a c-S-coherent ring which is not S-Noetherian. Proposition 4.5 Let R = S = n Y n Y Ri be a direct product of rings Ri (n ∈ N) and i=1 Si be a cartesian product of multiplicative sets Si of Ri . Then, R is c- i=1 S-coherent if and only if Ri is c-Si -coherent for every i ∈ {1, ..., n}. Proof. The result is proved using standard arguments. On S-coherence 15 Example 4.6 Consider a c-S-coherent ring A which is not coherent. Let B be a coherent ring which has a multiplicative set W such that BV is not Noetherian. Then, A×B is c-V ×W -coherent (by Proposition 4.5), but it is not V ×W -Noetherian (by [1, Proposition 2 (f )]). The follwoing result characterizes c-S-coherent rings can be characterized in terms of localization. Theorem 4.7 The following assertions are equivalent: 1. R is c-S-coherent. 2. Every finitely generated ideal of R is c-S-finitely presented. 3. For every finitely generated ideal I of R, there is a finitely presented ideal J ⊆ I such that IS = JS . In particular, RS is a coherent ring. Proof. (1⇒ 2 ⇒ 3 ) Straightforward. (3⇒1) Let I be an S-finite ideal of R. Then, there exist an s ∈ S and a finitely generated ideal J of R such that sI ⊆ J ⊆ I. By assertion (3), there is a finitely presented ideal K ⊆ J such that KS = JS . Then, there is a t ∈ S such that tJ ⊆ K. Therefore, tsI ⊆ K ⊆ I, as desired. We end the paper with a result which relates c-S-coherent rings with the notion of S-saturation. In [1], the notion of S-saturation is used to characterize S-Noetherian rings. Assume that R is an integral domain. Let SatS (I) denotes the S-saturation of an ideal I of R; that is, SatS (I) := IRS ∩ R. In [1, Proposition 2 (b)], it is proved that if SatS (I) is S-finite, then I is S-finite and SatS (I) = (I : s) for some s ∈ S. This fact was used to prove that a ring R is S-Noetherian if and only if RS is Noetherian and, for every finitely generated ideal of R, SatS (I) = (I : s) for some s ∈ S (see [1, Proposition 2 (f)]). The following result shows that the implication of [1, Proposition 2 (b)] is in fact an equivalence in more general context. Consider N ⊆ M an inclusion of R-modules. Let f : M → MS be the canonical R-module homomorphism. Denote by f (N )RS the RS -submodule of MS generated by f (N ). We set SatS,M (N ) := f −1 (f (N )RS ) and (N :M s) := {m ∈ M ; sm ∈ N }. Proposition 4.8 Let N be an R-submodule of an R-module M . SatS,M (N ) is Sfinite if and only if N is S-finite and SatS,M (N ) = (N :M s) for some s ∈ S. 16 D. Bennis and M. El Hajoui Proof. (⇒) Set K = SatS,M (N ). Since K is S-finite, there exist an s ∈ S and a finitely generated R-module J such that sK ⊆ J ⊆ K. Thus, sN ⊆ sK ⊆ J. We can write J = Rx1 + Rx2 + · · · + Rxn for some x1 , x2 , ..., xn ∈ J. For each xi , there n Q exists a ti ∈ S such that ti xi ∈ N . We set t = ti . Then, tsN ⊆ tsK ⊆ tJ ⊆ N . i=1 Then, N is S-finite. On the other hand, since sK ⊆ tJ ⊆ N ⊆ K, K ⊆ (N :M s). Conversely, let x ∈ (N :M s). Then, sx ∈ N , so x ∈ K, as desired. (⇐) Since N is S-finite, there exist a t ∈ S and a finitely generated R-module J such that tN ⊆ J ⊆ N . On the other hand, since K = (N : s) for some s ∈ S, sK ⊆ N . Consequently, tsK ⊆ tN ⊆ J ⊆ N ⊆ K. Therefore, K is S-finite. The following result is proved similarly to the proof of Proposition 4.8. However, to guarantee the preservation of finitely presented modules when multiplying by elements of S, we assume that S does not contain any zero-divisor of R. Proposition 4.9 Assume that every element of S is regular. Let N be an Rsubmodule of an R-module M . SatS,M (N ) is c-S-finitely presented if and only if N is c-S-finitely presented and SatS,M (N ) = (N :M s) for some s ∈ S. Corollary 4.10 Assume that every element of S is regular. The following assertions are equivalent: 1. For every finitely generated ideal I of R, SatS (I) is c-S-finitely presented. 2. R is c-S-coherent and, for every finitely generated ideal I of R, SatS (I) = (I : s) for some s ∈ S. Acknowledgement. A part of this work was presented by the second author at the Scientific day of Algebra “JA GRAAF 2016” held in Faculty of Sciences of Rabat (May 17, 2016). The authors would like to thank Professor Zine El Abidine Abdelali for his helpful comments during the preparation of this paper. References [1] D. D. Anderson and T. Dumitrescu, S-Noetherian rings, Comm. Algebra, 30 (2002), 4407–4416. [2] D. D. Anderson, D. J. Kwak and M. Zafrullah, Agreeable domains, Comm. Algebra, 23 (1995), 4861–4883. On S-coherence 17 [3] S. U. Chase, Direct products of modules, Trans. Amer. Math. Soc. 97 (1960), 457–473. [4] D. Costa, Parameterizing families of non-Noetherian rings, Comm. Algebra 22 (1994), 3997–4011. [5] S. Glaz, Commutative coherent rings, Lecture Notes in Math., Springer-Verlag, Berlin, 1989. [6] A. Hamed and S. Hizem, Modules Satisfying the S-Noetherian Property and S-ACCR, Comm. Algebra 44 (2016), 1941–1951. [7] A. Hamed and S. Hizem, S-Noetherian rings of the forms A[X] and A[[X]], Comm. Algebra 43 (2015), 3848–3856. [8] E. Hamann, E. Houston and J. L. Johnson, Properties of uppers to zero in R[X], Com. Alg. 23 (1995), 4861–4883. [9] I. Kaplansky, Projective modules, Ann. Math. 68 (1958), 372–377. [10] H. Kim, M. O. Kim and J. W. Lim, On S-strong Mori domains, J. Algebra, 416 (2014), 314–332. [11] J. W. Lim and D. Y. Oh, S-Noetherian properties on amalgamated algebras along an ideal, J. Pure Appl. Algebra 218 (2014), 1075–1080. [12] J. W. Lim and D. Y. Oh, S-Noetherian properties of composite ring extensions, Comm. Algebra 43 (2015), 2820–2829. [13] W. W. McGovern, G. Puninski and P. Rothmaler, When every projective module is a direct sum of finitely generated modules, J. 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Fast construction of efficient composite likelihood arXiv:1709.03234v1 [math.ST] 11 Sep 2017 equations Zhendong Huang and Davide Ferrari ∗ School of Mathematics and Statistics, The University of Melbourne Abstract Growth in both size and complexity of modern data challenges the applicability of traditional likelihood-based inference. Composite likelihood (CL) methods address the difficulties related to model selection and computational intractability of the full likelihood by combining a number of low-dimensional likelihood objects into a single objective function used for inference. This paper introduces a procedure to combine partial likelihood objects from a large set of feasible candidates and simultaneously carry out parameter estimation. The new method constructs estimating equations balancing statistical efficiency and computing cost by minimizing an approximate distance from the full likelihood score subject to a `1 -norm penalty representing the available computing resources. This results in truncated CL equations containing only the most informative partial likelihood score terms. An asymptotic theory within a framework where both sample size and data dimension grow is developed and finite-sample properties are illustrated through numerical examples. Keywords: Composite likelihood estimation, likelihood truncation, `1 -penalty. ∗ Corresponding author: Davide Ferrari, School of Mathematics and Statistics, The University of Mel- bourne, Parkville, VIC 3010, Australia. E-mail: [email protected]. 1 1 Introduction Since the idea of likelihood was fully developed by Fisher (1922), likelihood-based inference has played a role of paramount importance in statistics. The complexity of modern data, however, poses nontrivial challenges to traditional likelihood methods. One issue is related to model selection, since the full likelihood function can be difficult or impossible to specify in complex multivariate problems. Another difficulty concerns computing and the necessity to obtain inferences quickly. These challenges have motivated the development of composite likelihood (CL) methods, which avoid intractable full likelihoods by compounding a set of low-dimensional likelihood objects. Besag (1974) pioneered CL inference in the context of spatial data; Lindsay (1988) developed CL inference in its generality. Due to its flexible framework and established theory, the CL framework has become a popular tool in many areas of applied statistics; see Varin et al. (2011) for an overview of CL inference and common applications. Consider n independent observations on the d × 1 random vector X = (X1 , . . . , Xd )T with pdf in the parametric family {f (x; θ), x ∈ X , θ ∈ Θ ⊆ Rp }, where θ∗ ∈ Θ denotes the true parameter. In this paper, we are mainly concerned with large data sets where both the data dimension d and the sample size n are large. Given i.i.d. observaP tions X (1) , . . . , X (n) on X, we write EFn (g) = n−1 i≤n g(X (i) ) for the empirical mean of P the function g, where Fn (x) = n−1 i≤n I(X (i) ≤ x) is the empirical cdf, and use E(g) to denote its expected value. The operator “∇” denotes differentiation with respect to θ. In the CL setting, the maximum likelihood score uM L (·, θ) = ∇ log f (·, θ) and the associated estimating equations EFn uM L (θ) = 0 are intractable due to difficulties in computing or specifying the full d-dimensional density f (·; θ). Suppose, however, that one can obtain m tractable pdfs f1 (s1 ; θ), . . . , fm (sm ; θ) for sub-vectors S1 , . . . , Sm of X, where each Sj has dimension much smaller than d. For example, S1 could represent a single element of X like X1 , a variable pair like (X1 , X2 ), or a conditional sub-vector like (X1 , X2 )\|X1 . Typically, the total number of sub-models m grows quickly with d; for instance, taking all variable pairs 2 in X results in m = d(d − 1)/2 candidate sub-likelihoods. The specific choice for the set of pdfs {fj , j = 1, . . . , m} is sometimes referred to as CL design (Lindsay et al., 2011) and is typically specified by the practitioner . For simplicity, here the CL design is treated as given, and we assume f1 = · · · = fm , as it is often the case in applications. b We focus on the maximum composite likelihood estimator (MCLE), θ(w), defined as the solution to the CL estimating equations 0 = EFn [u(θ, w)] = EFn [w1 u1 (θ) + · · · + wm um (θ)], (1) where uj (·, θ) = ∇ log{fj (·; θ)} is the jth partial score (sub-likelihood score) associated with the jth subset Sj of X. Here w ∈ Rm is a given vector of coefficients to be determined, which we refer to as composition rule. In addition to well-known computational advantages compared to MLE and flexible modeling, the MCLE enjoys first-order properties analogous to those of the maximum likelihood estimator (MLE). Since the partial scores commonly define unbiased estimating equations (i.e. Euj (θ) = 0 at θ = θ∗ , for all 1 ≤ j ≤ m), the CL b score u(θ, w) in (1) is also unbiased, a property leading to consistency of θ(w). Unfortunately, the MCLE does not have the same second-order properties as the MLE since the asymptotic b variance of θ(w) is generally different from the inverse of Fisher information −E[∇uM L (θ∗ )], with the two coinciding only in special families of models. The choice of the composition rule w is crucial in determining both efficiency and computb ing cost associated with θ(w). Established theory of unbiased estimating equations prescribes b to find w so to minimize the asymptotic variance of θ(w) (Heyde, 2008, Chapter 2), given by the inverse of the p × p Godambe information matrix G(θ, w) = E{∇u(θ, w)} var {u(θ, w)}−1 E{∇u(θ, w)}. (2) Although theoretically appealing, this is a notoriously difficult task due the well-known instability of common estimators of the term var {u(θ, w)} in G(θ, w) (Lindsay et al., 2011). 3 On the other hand, the common practice of retaining all terms in (1) by choosing fixed wj 6= 0 for all j ≥ 1 (e.g. wj = 1, j ≥ 1) is undesirable from both computational and statistical efficiency perspectives, especially when the partial scores uj exhibit pronounced correlation. Cox and Reid (2004) discuss the detrimental effect caused by the presence of b many correlated scores on the variance of θ(w) when n is small compared to m in pairwise likelihood estimation. In the most serious case where the correlation between scores is overwhelming, keeping all the terms in (1) may lead to lack of consistency for the implied b MCLE θ(w). Motivated by the above considerations, we introduce a new method called sparse composite likelihood estimation and selection (SCLE) consisting of two main steps: a truncation Step (T-Step) and an estimation Step (E-Step). In the T-Step, the composition rule w is obtained by minimizing an approximate distance between the unknown full likelihood score uM L (θ) and the CL score u(θ, w), subject to a `1 -norm constraint on w. This step may be viewed as maximizing statistical accuracy for given afforded computing. Alternatively it may be interpreted as minimizing the computing cost for given level of statistical efficiency. Due to the geometry of the `1 -norm, the resulting composition rule, say w, b contains a number of non-zero elements (see Lemma 3.1). While the most useful terms for improving MCLE’s statistical accuracy are retained, the noisy sub-likelihoods contributing little or no improvement are dropped. In the E-step, we solve the estimating equations (1) with w = w b and find b w). the final estimator θ( b Compared to traditional CL estimation, the main advantage of our approach is to reduce the computational burden, while retaining relatively high efficiency in large data sets. The reduced number of terms in the estimating equations (1) translates into fast computing and enhanced stability for the final estimator at a relatively small cost in terms of statistical efficiency. The remainder of this paper is organized as follows. In Section 2, we describe the main methodology for simultaneous likelihood truncation and parameter estimation. In Section 3, we study the properties of the truncated composition rule and for the implied estimator 4 within a framework where both the sample size n and the data dimension d are allowed to diverge. Section 4 illustrates the properties of our methodology in the context of estimation of location and scale for multivariate normal models. In Section 5, we study the trade-off between computational and statistical efficiency in finite samples through numerical simulations. Section 6 concludes with final remarks. Technical lemmas used in our main results are deferred to the appendix. 2 Main methodology Throughout the paper, we consider unbiased partial scores {uj (θ), 1 ≤ j ≤ m} satisfying Euj (θ) = 0, for all 1 ≤ j ≤ m, (3) when θ = θ∗ and assume that θ∗ is the unique solution for all the equations in (3). The approach described in this section is applicable to problems with arbitrary sample size n and data dimension d, but we are mainly concerned with the situation where the data dimension d (and number of available sub-likelihood objects m) is large compared to the sample size n. Although we focus on log-likelihood partial scores for concreteness, our methodology and the properties in Section 3 remain essentially unchanged if uj (θ) is any arbitrary unbiased M-estimating equation. For instance, when θ is a location parameter, a more appropriate choice in the presence of outliers may be the Huber-type partial score uj (θ) = ψ(sj − θ), where ψ(z) = −k if z ≤ k, ψ(z) = z if \|z\| ≤ k and ψ(z) = k if z ≥ k, with k > 0. Another suitable choice in the same setting is the Lq-likelihood estimating equation of Ferrari and Yang (2010) defined by uj (θ) = ∇ logq {fj (sj ; θ)}, where logq (z) = log(u) if q = 1, and logq (z) = (z 1−q − 1)/(1 − q) if q 6= 1. In the rest of the paper we use U (θ) to denote the p × m matrix with column vectors u1 (θ), . . . , um (θ) and define the m × m matrix S(θ) = U (θ)T U (θ) with (jk)th entry {S(θ)}j,k = uj (θ)T uk (θ). We write UA (θ) for the sub-matrix of U (θ) with columns corre5 sponding to A ⊆ {1, . . . , m}, while U\A (θ) denotes the sub-matrix containing the remaining columns. Accordingly, we define the \|A\| × \|A\| matrix SA (θ) = U (θ)TA UA (θ) and use wA to denote the sub-vector of w with elements {wj , j ∈ A}, while w\A represents the vector containing all the elements in w not in wA . 2.1 Sparse and efficient estimating equations Our main objective is to solve the CL estimating equations 0 = EFn u(θ, w) defined in (1) with respect to θ using coefficients w = wλ (θ) obtained by minimizing the ideal criterion m X 1 wj uj (θ) Qλ (θ, w) = E uM L (θ) − 2 j=1 2 +λ 2 m X αj \|wj \| , (4) j=1 where k · k2 denotes the Euclidean norm, λ ≥ 0 is a given constant, and the αj s are pre-set constants not depending on the data. For clarity of exposition, we set αj = 1 for all j ≥ 1 in the remainder of the paper. The optimal solution wλ (θ) is interpreted as one that maximizes the statistical accuracy of the implied CL estimator, subject to a given level of computing. Alternatively, wλ (θ) may be viewed as to minimize the complexity of the CL equations, subject to given efficiency compared to MLE. The tuning constant λ balances the trade-off between statistical efficiency and computational burden The first term in Qλ (θ, w) aims to obtain efficient estimating equations by finding a CL score close to the ML score. When λ = 0 and θ = θ∗ , the composition rule w0∗ = w0 (θ∗ ) is optimal in the sense that the score function u(θ, w0∗ ) is closest to the MLE score uM L (θ). Although this choice gives estimators with good statistical efficiency, it offers no control for the CL score complexity since all the partial likelihood scores are included in the final P estimating equation. The second term λ m j=1 αj \|wj \| in (4) is a penalty discouraging overly complex estimating equations. In Section 3.1, we show that typically this form of penalty implies a number of elements in wλ (θ) exactly zero for any λ > 0. For relatively large λ, many elements in wλ (θ) are exactly zero, thus simplifying considerably the CL estimating 6 equations 0 = EFn u(θ, wλ (θ)). When a very large fraction of such elements is zero, we say that wλ (θ) and the CL equations 0 = EFn u(θ, wλ (θ)) are sparse. Sparsity is a key advantage of our approach to reduce the computational burden when achievable without loosing much statistical efficiency. On the other hand, if λ is too large, one risks to miss the information in some useful data subsets which may otherwise improve statistical accuracy. 2.2 Empirical criterion and one-step estimation Obvious difficulties related to direct minimization of the ideal criterion Qλ (θ, w) are the presence of the intractable likelihood score uM L and the expectation depending on the unknown parameter θ∗ . To address these issues, first note that, up to a negligible term not depending on w, Criterion (4) can be written as m X 1 E wj uj (θ) 2 j=1 2 − 2 m X m X ML T wj E u (θ) uj (θ) + λ αj \|wj \| . j=1 (5) j=1 If θ = θ∗ , we have E[uM L (θ)uj (θ)T ] = E[uj (θ)uj (θ)T ]. To see this, recall that partial scores are unbiased and differentiate both sides of 0 = Euj (θ) under appropriate regularity conditions. This result is used to eliminate the explicit dependency on the score uM L . Finally, replacing expectations in (5) by empirical averages leads to the following empirical objective: m X 1 b Qλ (θ, w) = EFn wj uj (θ) 2 j=1 2 − 2 m X j=1 m X T wj EFn uj (θ) uj (θ) + λ αj \|wj \| . (6) j=1 Under appropriate regularity conditions, the empirical criterion (6) estimates consistently the population criterion (4) up to an irrelevant constant not depending on w, with the caveat that θ must be close to θ∗ . These considerations motivate the following estimation strategy: b compute the truncated 1) T-Step. Given a preliminary root-n consistent estimator θ, 7 composition rule w bλ by solving b w). bλ (θ, w bλ = argmin Q (7) w∈Rm 2) E-Step. Update the parameter estimator by the one-step Newton-Raphson iteration h i−1 b b b bw θλ = θ − EFn ∇u(θ, w bλ ) EFn u(θ, bλ ). (8) Theorem 3.2 shows that the convex minimization problem in the T-Step has unique solution. Particularly, let Eb ⊆ {1, . . . , m} is the subset of partial scores such that n o T b b EFn uj (θ) rj (θ, w bλ ) ≥ λ, (9) where rj is the pseudo-residual defined by rj (θ, w) = uj (θ) − u(θ, w) and and write \Eb for b Then the solution of the T-Step is the set {1, . . . m} \ E. o o−1 n n b − λ sign(w b bλ,Eb) , w bλ,\Eb = 0, diag{EFn SEb(θ)} w bλ,Eb = EFn SEb(θ) (10) b sign(w) is the vector where: SEb = UEbT UEb and UEb is a matrix with column vectors {uj , j ∈ E}; sign function with jth element taking values −1, 0 and 1 if wj < 0, wj = 0 and wj > 0, respectively; and diag(A) denotes the diagonal of the square matrix A. More insight on the meaning of (9) may be useful. Differentiating (5) in wj 6= 0 and then expanding around θ∗ under Conditions C.1 and C.2 in Section 3.1 gives n o b T rj (θ, b w) = E uj (θ∗ )T uM L (θ∗ ) − u(θ∗ , w) + op (1). EFn uj (θ) (11) Combining (9) and (11) highlights that the jth partial likelihood score uj (θ) is selected when it is sufficiently correlated with the residual difference uM L (θ) − u(θ, w). Hence, our 8 criterion retains only those uj s which are maximally useful to explain the gap between the full likelihood score uM L (θ) and the CL score u(θ, w), while it drops the remaining scores. When λ = 0, we have Eb = {1, . . . , m} meaning that the corresponding composition rule w b0 does not contain zero elements. From (10) for λ = 0 it is required that the empirical b is non-singular which is violated when n < m. covariance matrix for all partial scores EFn S(θ) b may be nearly singular due to the presence of largely Even for n > m, however, EFn S(θ) correlated partial scores. On the other hand, setting λ > 0 always gives a non-singular b and guarantees existence of w matrix EFn SEb(θ) bλ,Eb. The proposed approach requires an initial root-n consistent estimator, which is often easy to obtain when the partial scores are unbiased. One simple option entails solving EFn u(w, θ) = 0 with w = (1, . . . , 1)T . If m is large, one may choose w by the stochastic CL strategy of Dillon and Lebanon (2010), where the elements of w may be set as either 0 or 1 randomly according to some user-specified scheme. Although the initial estimator θb could be quite inefficient, the one-step update (8) improves upon this situation. Moreover, the estimator θbλ and coefficients w bλ can be refined by iterating the T-Step (with θb = θbλ ) and the E-Step a few times. 2.3 Computational aspects: LARS implementation and selection of λ The empirical composition rule w bλ in (7) cannot be computed using derivative-based apb w). To address this issue, we propose an bλ (θ, proaches due to non-differentiability of Q implementation based on the least-angle regression (LARS) algorithm of Efron et al. (2004) originally developed for sparse parameter estimation in the context of linear regression modb our implementation of LARS minimizes Q b w) by including one score bλ (θ, els. For given θ = θ, b at the time in the composite likelihood score u(θ, b w). In each step, the score with the uj (θ) b − u(θ, b w) is included, largest correlation with the currently available residual difference uj (θ) followed by an adjustment step on w. The numerical examples in Section 5, suggest that 9 our implementation of the LARS algorithm for CL selection is very fast. In at most m × p steps, it returns a path of estimated composition rules w bλ1 , . . . , w bλm , where λj here is the value of the tuning constant λ in (6) at which the jth partial score enters the CL estimating equation. Selection of λ is of practical importance since it balances the trade-off between statistical and computational efficiency. For a given budget on afforded computing, say λ∗ , we include one partial score at the time, for example using the LARS approach above, and stop when b = max{λ : φ(λ) > τ }, for some user-specified 0 < τ ≤ 1, where we reach λ φ(λ) = b tr{EFn Sλ (θ)} I(λ > λ∗ ). b tr{EFn S(θ)} (12) Here EFn Sλ = EFn UλT Uλ denotes the empirical covariance matrix for the selected partial scores indexed by the set Ebλ = {j : w bλ,j 6= 0}. The criterion φ(λ) can be viewed as the proportion of score variability explained by currently selected partial scores. In practice, we choose τ close to 1, such as τ = 0.9, 0.95 or 0.99. If the computing budget is reached, we set b = λ∗ . In analogy with principal component analysis, the selected combination of scores λ accounts for the largest variability in the collection of empirical scores. 3 Properties This section investigates the asymptotic behavior of the sparse composition rule w bλ and the corresponding SCLE θbλ defined in (8) within a setting where m – the number of candidate partial likelihoods – is allowed to grow with the sample size n. We use m∗ = EkuM L (θ∗ )k22 to denote the trace of Fisher information based on the full likelihood. Here m∗ may be interpreted as the maximum knowledge about θ if the full likelihood score uM L were available. Although m∗ can grow with m, reflecting the rather natural notion the one can learn more about the true model as the overall data size increases, it is not allowed to grow as fast as n; e.g., m∗ = o(log n). This is a rather common situation in CL estimation occuring, for 10 instance, when the sub-likelihood scores are substantially correlated or they are independent but with heterogeneous and increasing variances (see examples in Section 4.1). 3.1 Sparsity and optimality of the composition rule In this section, we give conditions ensuring uniqueness of the empirical composition rule w bλ and weak convergence to its population counterpart wλ∗ . To this end, we work θ within the root-n neighborhood of θ∗ , Θn = {θ : kθ − θ∗ k < c0 n−1/2 }, for some c0 > 0, and assume the following regularity conditions on S(θ): C.1 There exist positive constants c1 , c2 > 0 such that E{supθ∈Θn S(θ)j,k } < c1 , and V ar{supθ∈Θn S(θ)j,k } < c2 , for all j, k ≥ 1. C.2 Each element ES(θ)j,k is continuous with uniformly bounded first and second order derivatives on Θn . Our analysis begins by deriving the Karush-Kuhn-Tucker Condition (KKTC) (Kuhn, 2014) for the population objective Qλ (θ∗ , w) defined in (4). The KKTC characterizes the amount of sparsity – and, the computational complexity – associated with the selected estimating equations depending on the value of the tuning constant λ. Let c(θ, w) = diag(S(θ))−S(θ)w, where S(θ) = U (θ)U (θ)T is as defined in Section 2 Lemma 3.1 (KKTC). Under Condition C.1, the minimizer wλ∗ of Qλ (θ∗ , w) defined in (4) satisfies E\|{c(θ∗ , wλ∗ )j }\| = λ · γj , j = 1, . . . , m, ∗ ∗ ∗ where γj ∈ {1} if wλ,j > 0, γj ∈ {−1} if wλ,j < 0, and γj ∈ [−1, 1] if wλ,j = 0; c(·, ·)j is the jth element of vector c(·, ·). ∗ Proof. Let dj = −Ec(θ∗ , wλ∗ )j +λ·sign(wλ,j ) and note that the Tayor expansion of Qλ (θ∗ , wλ∗ ) ∗ around wλ,j 6= 0 is Qλ (θ∗ , wλ∗ + ) = Qλ (θ∗ , wλ∗ ) + dj + 11 2 tr{Ij (θ∗ )}, 2 (13) where = (0, . . . , j , . . . , 0)T , and Ij (θ∗ ) = E uj (θ∗ )uj (θ∗ )T is the p × p Fisher information matrix for the jth likelihood component, and tr{Ij (θ∗ )} < c1 by Condition C.1. ∗ 6= 0, we have dj = 0. Otherwise, if dj 6= 0, choosing j such that sign(j ) = If wλ,j −sign(dj ) and \|j \| < 2\|dj \|/c1 , implies Qλ (θ∗ , wλ∗ + ) < Qλ (θ∗ , wλ∗ ), but this is a contra∗ = 0, we need to show \|Ec(θ∗ , wλ∗ )j \| ≤ λ. diction since wλ∗ minimizes Qλ (θ∗ , ·). If wλ,j Assume \|Ec(θ∗ , wλ∗ )j \| > λ and take j such that sign(j ) = sign(Ec(θ∗ , wλ∗ )j ) and \|j \| < 2\|Ec(θ∗ , wλ∗ )j − λ\|/\|c1 \|. Then dj + 2 tr{I(θ∗ )}/2 < −\|j \|(\|Ec(θ∗ , wλ∗ )j \| − λ) + 2 c1 /2 < 0, which implies Qλ (θ∗ , wλ∗ +) < Qλ (θ∗ , wλ∗ ). But this is contradicted by wλ∗ being the minimizer of Qλ . Hence, Ec(θ∗ , wλ∗ )j = λ · γj , for all j = 1, . . . , m. An argument analogous to that used in the proof of Lemma 3.1 leads to the KKTC for b w). Specifically, for w bw b θ, w bλ , the minimizer of the empirical loss Q( bλ we have EFn c(θ, bλ )j = λ·γ bj , j = 1, . . . , m, where γ bj ∈ {1} if w bλ,j > 0, γ bj ∈ {−1} if w bλ,j < 0, and γ bj ∈ [−1, 1] if w bλ,j = 0. Lemma 3.1 has important implications in our current setting, since it relates λ to the size of the covariance between the jth sub-likelihood score uj (θ) and the residual difference uM L (θ) − u(θ, w) at θ = θ∗ . Particularly, if such a covariance is sufficiently small, i.e. λ > E{c(θ∗ , wλ∗ )j } = \|E {uj (θ∗ ) [u(θ∗ , wλ∗ ) − uj (θ∗ )]}\| = E uj (θ∗ ) u(θ∗ , wλ∗ ) − uM L (θ∗ ) , ∗ then the correspondent coefficient is wλ,j = 0. Thus, the tuning parameter λ controls the level of sparsity of the composite score u(θ∗ , wλ∗ ) by forcing the weights of those non-important score components with small pseudo-covariance c(θ∗ , wλ∗ )j to be exactly zero. For uniqueness of wλ∗ and w bλ , a simple condition is that the partial scores cannot replace each other, i.e. we require that the scores are in general position. Specifically, we say that the score components u1 , . . . , um are in general position if any affine subspace L ⊂ Rm of dimension l < m contains at most l + 1 elements of {±u1 , ..., ±um } excluding antipodal pairs of points. 12 C.3 The partial scores uj (x, θ), j ≥ 1, are continuous and in general position with probability 1 for all θ ∈ Θn . Theorem 3.2. Under Conditions C.1-C.3 the solution of the T-Step, w bλ , defined in (7) is unique and is given by (10) for any λ > 0. Moreover, w bλ contains at most np ∧ m non-zero elements. Proof. Let Eb = {j ∈ {1, . . . , m} : \|b γj \| = 1} to be the index set of non-zero elements of w bλ where where γj is as defined after Lemma 3.1. First note that the composite likelihood score bw b Tw b ·) defined in (6), due bλ (θ, u(θ, bλ ) = U (θ) bλ is unique for all solutions w bλ which minimize Q b w). Uniqueness of u(θ, bw bλ (θ, to strict convexity of Q bλ ) implies that γ b and the corresponding index set Eb are unique by Lemma 3.1. b we first note that the square matrix Next, to show uniqueness of w bλ,j for all j ∈ E, b has full rank. Otherwise, some row the matrix can be written as a linear b T U b(θ)] EFn [UEb(θ) E b b = P aj EFn [uj (θ) b T U b(θ)]. b T U b(θ)] b i.e. EFn [uk (θ) combination of other rows in the set E, k6=j E E P b 2 −P aj EFn uj (θ) b 2 = λb Then Lemma 3.1 implies also the event EFn uk (θ) γk − j6=k aj λb γj for j6=k the same set of coefficients aj s, which has probability equals to 0 since each uj is continuous b b(θ) b T ] has full rank, meaning that the size of Eb satisfies \|E\| b ≤ and random. Thus, E[UEb(θ)U E b T ] implies strict convexity of b b(θ) b full rank of E[U b(θ)U np ∧ m. For fixed wj = 0, j ∈ \E, E E b w b) where w b is the sub-vector of w containing elements indexed by E. bλ (θ, b Hence, w Q bλ,Eb λ,E λ,E is unique. The arguments in Theorem 3.2 go through essentially unchanged for the population composition rule wλ∗ by showing the full rank of E[UE (θ∗ )T UE (θ∗ )] using Condition C.3 and Lemma 3.1, where E is the index set of non-zero elements in wλ∗ . This implies also uniqueness of wλ∗ . Next, we turn to convergence of the empirical composition rule w bλ to wλ∗ , thus showing bλ (θ, w) (6) is a suitable replacement for the intractable criterion Qλ (θ, w) that the objective Q (4). Since criterion b w) = wT EFn {S(θ)}w/2 b b T w + λkwk1 bλ (θ, Q − EFn {diag(S(θ))} 13 is used as an approximation of the population criterion Qλ (θ∗ , w) defined in (4), clearly the b and ES(θ∗ ) affects the accuracy of such an approximation. Let distance between EFn S(θ) b and r1 = supθ∈Θn kEFn S(θ) − ES(θ∗ )k2 be the supreme variation between matrices EFn S(θ) ES(θ∗ ), where kAk2 is the matrix induced 2-norm for matrix A. As n → ∞, the rate at which r1 goes to 0 depends mainly on the number of partial scores m and the behavior of the random elements in S, which can vary considerably in different models. For example, when the elements of S(θ) are sub-Gaussian, one needs only log(m)/n = o(1) (Cai et al., 2010). In more general cases, m4 /n = o(1) suffices to ensure r1 = op (1) (Vershynin, 2012). Next we investigate how m and m∗ should increase compared to r1 when λ → 0 as n → ∞ to ensure a suitable behavior for w bλ . To obtain weak convergence of w bλ to wλ∗ , we introduce the additional requirement that the covariance matrix of the partial scores ES(θ∗ ) does not shrink to zero too fast. 2 C.4 There exists a sequence cn , such that cn r1λm∗2 → ∞ and xT ES(θ∗ )x ≥ cn kxk1 for any x ∈ Rm , as n → ∞. Condition C.4 is analogous to the compatibility condition in `1 -penalized least-squares estimation for regression (Bühlmann and Van De Geer, 2011), where it ensures a good behavior of the observed design matrix of regressors. Differently from the sparse regression setting, where Condition C.4 is applied to the set of true nonzero regression coefficients, here no sparsity assumption on the composition rule w is imposed. P Theorem 3.3. Under Conditions C.1-C.4, if r1 m∗ 2 λ−2 = op (1) then kw bλ − wλ∗ k1 → 0, as n → ∞. b w Proof. From Lemma 7.3, EFn kU (θ)( bλ − wλ∗ )k22 = op (1). Note that b w b w EFn kU (θ)( bλ − wλ∗ )k22 = (w bλ − wλ∗ )T EFn S(θ)( bλ − wλ∗ ) n o ∗ ∗ T ∗ ∗ ∗ T b =(w bλ − wλ ) ES(θ )(w bλ − wλ ) + (w bλ − wλ ) EFn S(θ) − ES(θ ) (w bλ − wλ∗ ), 14 and the second term of the last equality is Op (r1 m∗ 2 λ−2 ) by Lemma 7.2. Thus, (w bλ − P wλ∗ )T ES(θ∗ )(w bλ −wλ∗ ) = Op (r1 m∗ 2 λ−2 ), which implies kw bλ −wλ∗ k1 → 0 by Condition C.4. Corollary 3.4. Let λ be a sequence such that λ → 0 as n → ∞. Under Conditions C.1-C.4, if r1 m∗ 2 λ−2 = op (1), we have P sup EFn ku(θ, w bλ ) − u(θ, wλ∗ )k2 → 0, as n → ∞. (14) θ∈Θn bw b w∗ )k2 = op (1). The result follows by noting Proof. From Lemma 7.3, EFn ku(θ, bλ ) − u(θ, λ 2 bw b w∗ )k2 = that for any θ ∈ Θn , the difference EFn ku(θ, w bλ ) − u(θ, wλ∗ )k22 − EFn ku(θ, bλ ) − u(θ, λ 2 b w bλ − wλ∗ ) is op (1) according to Conditions C.1, C.2 and Theorem (w bλ − wλ∗ )T EFn [S(θ) − S(θ)]( 3.3 Corollary 3.4 states that the composite likelihood score u(θ, w bλ ) is a reasonable approximation to u(θ, wλ∗ ). Particularly, even for λ close to zero, the composite score u(θ, w bλ ) still b uses a fraction \|E\|/m of sub-likelihood components. At the same time u(θ, w bλ ) is near the optimal score u(θ, w0∗ ), where w0∗ is the composition rule yielding the closest CL score u(θ) to the maximum likelihood score uM L (θ). Moreover, the implied Godambe information G(θ, w bλ ) = E{∇u(θ, w bλ )}var{∇u(θ, w bλ )}−1 E∇{u(θ, w bλ )} is expected to be close to G(θ, w) with w = w0∗ . However, while the MCLE based on w0∗ (or other choice of wj 6= 0, j ≥ 1) may be unavailable or computationally intractable due to common difficulties in estimating var{∇u(θ, w0∗ )} (Lindsay et al., 2011; Varin et al., 2011), our truncated composition rule w bλ implies a more stable estimation of G(θ, w bλ ) by requiring only a fraction of scores. 15 3.2 Asymptotic behavior of the one-step SCLE In this section, we show consistency and give the asymptotic distribution for the SCLE bw θbλ = θ( bλ ) defined in the E-Step (8). One advantage of one-step estimation is that consistency and asymptotic normality are treated separately. The one-step estimator θbλ inherits b under standard rethe properties leading to consistency from the preliminary estimator θ, quirements on S(θ). For normality, additional conditions on the sub-likelihood scores are needed. Let H(θ) be the p×mp matrix obtained by stacking all the p×p sub-matrices ∇uj (θ). Let r2 = supθ∈Θn maxj,k \|EFn H(θ)j,k −EH(θ∗ )j,k \| be the maximum variation between the emb − Euj (θ∗ )k1 be pirical and the optimal Hessian matrices. Let r3 = supθ∈Θn maxj kEFn uj (θ) the supreme variation between empirical scores and their expected value around Θn . In the rest of this section, we use Jλ∗ = Cov [u(θ∗ , wλ∗ )] and Kλ∗ = −E∇u(θ∗ , wλ∗ ) to denote the population variability and sensitivity p × p matrices, respectively, both depending implicitly on n. We further assume: C.5 There exist positive constants c5 and c6 such that E[supθ∈Θn H(θ)j,k ] < c5 , and V ar[supθ∈Θn H(θ)j,k ] < c6 , for all j, k ≥ 1. C.6 Each element EH(θ)j,k , j, k ≥ 1 of the matrix EH(θ) is continuous with uniformly bounded first and second derivatives on θ ∈ Θn . Theorem 3.5. Suppose there exist N > 0 such that Kλ∗ is non-singular with all eigenvalues bounded away from 0 for all n > N . Under Conditions C.1 - C.6, if r1 m∗ 2 λ−2 = op (1), √ r2 m∗ λ−1 = op (1) and r3 m∗ λ−1 = op (1), then as n → ∞ we have P (i) kθbλ − θ∗ k1 → 0, and (ii) √ 1 D nJλ∗ − 2 Kλ∗ (θbλ − θ∗ ) → Np (0, I), where Jλ∗ = Cov [u(θ∗ , wλ∗ )] and Kλ∗ = −E∇u(θ∗ , wλ∗ ) denote p × p population variability and sensitivity matrices. 16 Proof. Without loss of generality, we only prove the case p = 1. Since p is fixed, the bλ = proof can be easily generalized to the case p > 1 without additional conditions. Let K bw −EFn ∇u(θ, bλ ) be the empirical sensitivity matrix. Then θbλ can be written as θbλ = θb + bw b −1 EFn u(θ, K bλ ), with θb being a consistent preliminary estimator. Note that Eu(θ∗ , wλ∗ ) = 0 λ and bw bw b w∗ )k1 kEFn u(θ, bλ ) − Eu(θ∗ , wλ∗ )k1 ≤ kEFn u(θ, bλ ) − EFn u(θ, λ (15) b w∗ ) − EFn u(θ∗ , w∗ )k1 + kEFn u(θ∗ , w∗ ) − Eu(θ∗ , w∗ )k1 . + kEFn u(θ, λ λ λ λ The first term on the right hand side of (15) is op (1) by Lemma 7.3. The second term is op (1) b − EFn uj (θ∗ )\|kw∗ k1 , which converges b w∗ ) − EFn u(θ∗ , w∗ )k1 ≤ maxj \|EFn uj (θ) since kEFn u(θ, λ λ λ to 0 by Theorem’s assumptions and Lemma 7.2. The last term of (15) is also op (1) by the P bw Law of Large Numbers. This shows that EFn u(θ, bλ ) → 0. Moreover, from Lemma 7.4, b λ − K ∗ k1 = op (1). Since K ∗ has all eigenvalues bounded away from 0 for large n, we have kK λ λ P P P bw bw b −1 EFn u(θ, b −1 EFn u(θ, K bλ ) → 0. Since θb → θ∗ , we have θbλ = θb + K bλ ) → θ∗ , which shows λ λ part (i) of the theorem. bw b −1 EFn u(θ, To show normality in (ii), re-arrange θbλ = θb + K bλ ) and obtain λ bw b λ (θbλ − θ∗ ) = K b λ (θb − θ∗ ) + EFn u(θ, bλ − K e λ ](θb − θ∗ ) K bλ ) = EFn u(θ∗ , w bλ ) + [K bλ − K e λ ](θb − θ∗ ), =EFn u(θ∗ , wλ∗ ) + [EFn u(θ∗ , w bλ ) − EFn u(θ∗ , wλ∗ )] + [K (16) ew e λ = −EFn ∇u(θ, where K bλ ) and θe is some value between θb and θ∗ . The second equality b For the first term in (16), we follows from the first-order expansion of EFn u(θ∗ , w) b at θ. √ 1 D have nJλ∗ − 2 EFn u(θ∗ , wλ∗ ) → Np (0, I), since the Lindeberg-Feller Central Limit Theorem applies to u(θ∗ , wλ∗ ) by Lemma 7.6. By Lemma 7.5 Jλ∗ = O(m∗ ), so the first term in (16) is p Op ( m∗ /n). The second term in (16) EFn u(θ∗ , w bλ ) − EFn u(θ∗ , wλ∗ ) = EFn U (θ∗ )T (w bλ − wλ∗ ) P is of smaller order compared to first term EFn u(θ∗ , wλ∗ ) = EFn U (θ∗ )T wλ∗ since kw bλ − wλ∗ k → 0 17 by Theorem 3.3. For the last term in (16), we have b − E∇uj (θ∗ ) bλ − K e λ )(θb − θ )\|1 ≤ max EFn ∇uj (θ) \|(K ∗ j ∗ e − E∇uj (θ ) + max EFn ∇uj (θ) j kw bλ k1 \|θb − θ∗ \| ≤2r2 kw bλ k1 \|θb − θ∗ \|. (17) and the last expression in (17) is op (r2 m∗ n−1/2 λ−1 ) by Lemma 7.2. Theorem’s assumption that r1 m∗ 2 λ−2 = op (1) implies that the last term in (16) is of smaller order compared to the b λ − K ∗ k1 = op (1) according to Lemma 7.4, Slutsky’s Theorem first term. Finally, since kK λ implies the desired result. Consistency and asymptotic normality for the one-step estimator θbλ follow mainly from w bλ converging in probability to the target composition rule wλ∗ . Since each sub-likelihood score is unbiased and asymptotically normal, their linear combination is also normally dis√ 1 tributed. The overall convergence rate is given by k nJλ∗ − 2 Kλ∗ k1 which is of order between √ √ n and nm. The actual order depends on the underlying correlation between partial √ scores u1 , . . . , um . While the optimal rate nm is achieved when the scores are perfectly independent, combining highly correlated scores into the final estimating equation will give √ rates closer to n. 4 Examples for special families of models In this section, we illustrate the SCLE through estimation of location and scale estimation for special multivariate normal models. 4.1 Estimation of common location for heterogeneous variates Let X ∼ Nm (θ1m , Σ), where the m × m covariance matrix Σ has off-diagonal elements σjk (j 6= k) and diagonal elements σk2 (j = k). Computing the MLE of θ requires Σ−1 and in 18 b = EFn X T X. When n < m, Σ b is singular and the MLE practice Σ is replaced by the MLE Σ of θ is not available in practice, whilst CL estimation is still feasible. The jth partial score is uj (θ) = (Xj − θ)/σj2 and the CL estimating equation (1) based on the sample X (1) , . . . , X (n) is 0 = EFn u(θ, w) = m n X wj X (i) (Xj − θ), 2 nσ j j=1 i=1 leading to the profiled MCLE b θ(w) = m X ! wj σj−2 X j j=1 / m X ! wk σj−2 , (18) j=1 which is a weighted average of marginal sample means X j = n−1 Pn i=1 (i) Xj , j ≥ 1. In this example, one can work out directly the optimal composition rule wλ∗ and no estimation is required. Particularly, it is useful to inspect the special case where X has independent components (σjk = 0 for all j 6= k). This corresponds to the fixed-effect meta-analysis model where estimators from m independent studies are combined to improve accuracy. Under independence, we have the explicit solution ∗ wλ,j = (1 − σj2 λ)I σj2 < λ−1 , 1 ≤ j ≤ m, which highlights that overly noisy data subsets with variance σj2 ≥ λ−1 are dropped and thus do not influence the final estimator (18). The number of non-zero elements in wλ∗ is Pm 2 −1 j=1 I(σj < λ ). Note that when λ = 0, we have uniform weights w0∗ = (1, . . . , 1)T and the corresponding b ∗) MCLE is the usual optimal meta-analysis solution. Although the implied estimator θ(w 0 has minimum variance, it offers no control for the overall computational cost since all m sub-scores are selected. On the other hand, choosing judiciously λ > 0 may lead to low computational burden with negligible loss for the resulting estimator. For instance, assuming 19 σj2 = j 2 , for θ ∈ Θn , a straightforward calculation shows E [u(θ, wλ∗ ) − u(θ, w0∗ )]2 ≤ λ2 X j∈E Since the number of the non-zero scores Pm j=1 j2 + X j −2 + o(1), (19) j ∈E / 1 I (j 2 < λ−1 ) = bλ− 2 c, the first term the mean squared difference between u(θ, wλ∗ ) and the optimal score u(θ, w0∗ ) is bounded by 1 1 λ2 λ−1 λ− 2 = λ 2 , up to a vanishing term. Thus, if λ = o(1), the composite score u(θ, wλ∗ ) converges to the optimal composite score u(θ, w0∗ ). Particularly, if λ decreases at a sufficiently slow rate, the truncated score u(θ, wλ∗ ) can still contain a relatively small number of terms, b ∗ ) is approximately equal the optimal estimator θ(w b ∗) while the correspondent estimator θ(w 0 λ in terms of statistical accuracy. If the elements of X are correlated (σjk 6= 0 for j 6= k), the partial scores contain overlapping information on θ. In this case, tossing away some highly correlated partial scores improves computing while maintaining satisfactory statistical efficiency for the final estimator. Figure 1 shows the solution path of wλ∗ and the asymptotic relative efficiency of b ∗ ) compared to MLE for different values of λ. When m is large the corresponding SCLE θ(w λ (e.g. m = 1000), the asymptotic relative efficiency drops gradually until a few scores are left. This example illustrates that a relatively high efficiency can be achieved by our truncated CL equations, when a few partial scores already contains the majority of information about θ. In such cases, the final SCLE with a sparse composition rule is expected to achieve a good trade-off between computational cost and statistical efficiency. 4.2 Location estimation in exchangeable normal variates In our second example we consider exchangeable variables with X ∼ Nm (θ1m , Σ) with Σ = (1 − ρ)Im + ρ1m 1Tm , 0 < ρ < 1. The marginal scores uj (θ) = Xj − θ are identically distributed and exchangeable with equal correlation. Differently from Example 4.1, the 20 ρ = 0, m = 20 Number of sub-likelihoods 1 200 0 −2 log(λ) 0 −7 −1 0.8 0.0 0.4 ARE 0.8 0 −5 ^ log(λ) 0.0 0.4 ARE 0.8 ^ log(λ) 0.0 −2 log(λ) −3 log(λ) 0.0 −4 ^ log(λ) −4 1 0.4 −2 log(λ) 10 wλ,j 0.0 wλ,j 0.0 wλ,j −4 ARE 3 1.0 11 ρ = 0.5, m = 1000 Number of sub-likelihoods 1.0 3 1.0 14 ρ = 0.5, m = 20 Number of sub-likelihoods −4 −2 log(λ) 0 −8 −4 log(λ) 0 Figure 1: Top Row: Solution paths for the minimizer wλ∗ of Criterion Q(θ∗ , w) in (4) for different values of λ with corresponding number of sub-likelihoods. Bottom Row: Asymptotic b ∗ ) compared to MLE. The vertical dashed lines on relative efficiency (ARE) of the SCLE θ(w λ b selected by Criterion (12) with τ = 0.9. Results correspond to the the bottom represent λ common location model X ∼ Nm (θ∗ 1m , Σ)√with jth diagonal element of Σ equal to j, and (jk)th off-diagonal element of Σ equal to ρ jk. 21 solution wλ∗ to Criterion (4) has equal elements ∗ = wλ,j 1−λ I(λ < 1), 1 ≤ j ≤ m, ρ(m − 1) + 1 b ∗ ) = Pm X j /m regardless of the value of λ. so the optimal parameter estimator is θ(w λ j=1 The first eigenvalue of ES(θ) is θ(m − 1) + 1, whilst the remaining m − 1 eigenvalues are all equal to 1 − θ, suggesting that the first score contains a relatively large information on b ∗ )} = θ compared to the other scores. When m is much larger than n, we have var{θ(w 0 [ρ2 (m − 1) + 1]/(mn) ρ2 /n. The trade-off between statistical and computational efficiency may be measured by the ratio of estimator’s variance with m = ∞ compared to that with m < ∞. This ratio is t(m) = ρ2 m/{ρ2 (m − 1) + 1}, which increases quickly for smaller m and much slower for larger m (e.g., t(5) = 0.83, t(9) = 0.90 and t(50) = 0.98, if ρ = 0.75). Thus, although all the elements in wλ∗ are nonzero, a few partial scores contain already the majority of the information on θ. This suggests that in practice taking a sufficiently large value for λ, so that the sparse empirical solution w bλ contains only a few of zero elements, bw already ensures a relatively high statistical efficiency for the corresponding MLCE θ( bλ ). 4.3 Exponentially decaying covariances Let X ∼ Nd (0, Σ(θ)), where the jkth element of Σ(θ) is σjk (θ) = exp{−θd(j, k)}. The quantity d(j, k) may be regarded as the distance between spatial locations j and k. Evaluating the ML score in this example is computationally expensive when d is large, since it requires computing the inverse of Σ(θ), a task involving O(d3 ) operations. On the other hand, the CL score is obtained by inverting 2 × 2 covariance matrices, thus requiring at most b O(d2 ) operations. Given i.i.d. observations X (1) , . . . , X (n) on X, the MCLE θ(w) solves the 22 equation 0= X wjk j<k n X (i) (i) ujk (θ, Xj , Xk ) i=1   (i) 2 (i) 2 (i) (i) n X σjk (θ){Xj + Xk − 2Xj Xk σjk (θ)}  σjk (θ)d(j, k)  = wjk 2 }2 {1 − σ (θ) jk i=1 j<k # " (i) (i) n X X σjk (θ) + Xj Xk − σjk (θ)d(j, k), wjk 1 − σjk (θ)2 i=1 j<k X where ujk corresponds to the score of a bivariate normal distribution for the pair (Xj , Xk ). Figure 2 shows the analytical solution path of the minimizer wλ∗ of Criterion (4) for b ∗ ) compared to different values of λ, and the asymptotic relative efficiency of the SCLE θ(w λ MLE. We consider a number of pairs ranging from m = 45 to m = 1225 for various choices of θ. When λ = 0, the SCLE has relatively high asymptotic efficiency. Interestingly, efficiency remains steady around 90% until only a few sub-likelihoods are left. This suggests again that a very small proportion of partial-likelihood components contains already the majority of the information about θ. In such cases, the SCLE reduces dramatically the computing burden while retaining satisfactory efficiency for the final estimator. 5 Numerical examples In this section, we study the finite-sample performance of the SCLE in terms by assessing its mean squared error and computing cost when the data dimension d increases. As a b preliminary estimator, we use the MCLE θ(w) with w = (1, . . . , 1)T , which is perhaps the most common choice for w in CL applications (Varin et al., 2011). 5.1 Example 1 We generate samples of size 50 from X ∼ Nm (θ1m , Σl ), l = 1, . . . , 4. We specify the following covariance structures: Σ1 = Im ; Σ2 is diagonal with kth diagonal elements σk2 = k; Σ3 has 23 458 200 wλ,j 0 −4 0 −5 −2 log(λ) 0 −1 −3 log(λ) −1 0.8 ARE 0.0 0.0 0.4 ARE 0.4 0.0 −4 −3 log(λ) ^ log(λ) 0.8 ^ log(λ) 0.8 ^ log(λ) −2 log(λ) 0.4 −2 log(λ) 60 −0.4 −0.4 0.2 wλ,j 0.2 −0.4 wλ,j −4 ARE 18 0.6 44 θ = 0.6, d = 50, m = 1225 Number of sub-likelihoods 0.6 18 0.6 44 θ = 0.6, d = 10, m = 45 Number of sub-likelihoods 0.2 θ = 0.4, d = 10, m = 45 Number of sub-likelihoods −4 −2 log(λ) 0 −5 Figure 2: Top Row: Solution paths for the minimizer wλ∗ of Criterion Q(θ∗ , w) defined in (4) for different values of λ with corresponding number of sub-likelihoods reported. Botb ∗ ) compared to MLE. The tom Row: Asymptotic relative efficiency (ARE) of SCLE θ(w λ b selected by Criterion (12) with vertical dashed lines on the bottom row correspond to λ τ = 0.9. Results correspond to the model X ∼ Nd (0, Σ(θ)) with (j, k)th element of Σ equal p to exp{−θ 2\|j − k\|}. 24 unit diagonal elements with the first 10 elements of X uncorrelated with any other element while the other elements in X have pairwise correlations 0.8\|j−k\| (10 < j < k < d); Σ4 has unit diagonal elements and a block diagonal structure with independent blocks of six elements each and within-block correlation of 0.6. Figure 3 (left), shows the relative mean squared error of the SCLE θbλ compared to that of the MLE for a moderate data dimension (d = m = 30). The points in the trajectories correspond to inclusion of a new sub-likelihood component according to the the least-angle algorithm described in Section 2.3. The SCLE θbλ achieves more than 90% efficiency compared to MLE for all the covariance structures considered, always before all the candidate partial likelihoods are included. The advantage of SCLE becomes evident when the sub-likelihood scores exhibit relatively strong correlation. For example, for Σ = Σ4 where sub-likelihoods are independent between blocks, the maximum efficiency is achieved when only a few representative partial scores are selected from each block. Figure 3 (right) shows the ratio between mean squared error of the SCLE compared and that of the MLE for a relatively large data dimension (d = m = 1000) compared to the sample size (n = 50). Although here the MLE is used as a theoretical benchmark, in practice such an estimator is not available as m is larger than the sample size n. Interestingly, when the sample size n is fixed, including all the sub-likelihoods eventually leads to substantial loss of efficiency. In this examples, selecting too many sub-likelihoods not only wastes computing resources but also implies estimators with larger errors . On the other hand, a proper choice of the tuning constant λ (corresponding to about 20 selected sub-likelihoods) can balance computational and statistical efficiency. 5.2 Example 2 In our second numerical example, we consider covariance estimation for the model X ∼ Nd (0, Σ(θ)) with Σ(θ)j,k = exp{−θ2(j − k)2 }. Here the covariance between components Xj and Xk in the random vector X decreases rapidly as the distance (j − k)2 between 25 Σ=Σ1 Σ=Σ2 Σ=Σ3 Σ=Σ4 0 5 MSEMLE MSESCLE MSEMLE MSESCLE 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 10 15 20 25 30 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 m=50 m=100 m=200 m=500 m=1000 0 Number of partial scores included 10 20 30 40 50 Number of partial scores included Figure 3: Monte-Carlo estimate of the mean square error of the MLE (MSEMLE ) divided by that of the SCLE (MSESCLE ), for the model X ∼ Nm (θ1m , Σ). Each trajectory is based on 1000 Monte-Carlo samples of size n = 50. Each point in the trajectories correspond to inclusion of a new sub-likelihood component based on the least-angle algorithm described in Section 2.3. Left: Different specifications for Σ detailed in Section 5 with m = 30. Right: Covariance Σ = Σ2 with m ranging from 50 to 1000. components of X increases. Figure 4 shows Monte-Carlo estimates for the mean square bw error of the SCLE θ( bλ ) compared to that of the MCLE with uniform composition rule (i.e. b θ(w) with w = (1, . . . , 1)T ), for θ = 0.2 and 0.4. Each point in the trajectories correspond to inclusion of a new sub-likelihood component using the least-angle algorithm described in Section 2.3. The SCLE is already more efficient than the uniform MCLE when a handful of partial scores are selected. For example if θ = 0.2 and m = 1035, selecting ten sub-likelihoods already ensures 1.5 times the accuracy of the uniform MCLE. Since the uniform MCLE uses all the m = 1035 pairs of sub-likelihoods, the SCLE obtains more accurate results at a much lower computing cost. 26 θ=0.2 m=45 m=435 m=1035 1.0 0.5 1.5 MSEUnif MSESCLE MSEUnif MSESCLE 1.5 θ=0.4 m=45 m=435 m=1035 1.0 0.5 0.0 0.0 0 10 20 30 40 50 60 Number of partial scores included 0 10 20 30 40 50 60 Number of partial scores included Figure 4: Monte-Carlo estimate of the mean square error of the MCLE with w = (1, . . . , 1)T (MSEUnif ) divided by that for the SCLE (MSESCLE ). Each point in the trajectories corresponds to the inclusion of a new sub-likelihood component based on the least-angle algorithm described in Section 2.3. Results are based on 1000 Monte Carlo samples of size n = 50 from the model X ∼ Nd (0, Σ(θ)) with Σ(θ)j,k = exp{−2θ(j − k)2 }. Trajectories correspond to θ = 0.2 (left) and θ = 0.4 (right) for different numbers of sub-likelihoods, m, ranging from 45 to 1035. 6 Conclusion and final remarks In recent years, inference for complex and large data sets has become one of the most active research areas in statistics. In this context, CL inference has played an important role in applications as a remedy to the drawbacks of traditional likelihood approaches. Despite the popularity of CL methods, how to address the trade-off between computational parsimony and statistical efficiency in CL inference from a methodological perspective remains a largely unanswered question. Motivated by this gap in the literature, we introduced a new likelihood selection methodology which is able truncate quickly overly complex CL equations potentially encompassing many terms, while attaining relatively low mean squared error for the implied estimator. This is achieved by selecting CL estimating equations satisfying a `1 -constraint on the CL complexity while minimizing an approximate `2 -distance from the full-likelihood score. Inference based on statistical objective functions with `1 -penalties on the parameter 27 θ is not new in the statistical literature (e.g., see Giraud (2014) for a book-length exposition on this topic). Note, however, that differently from existing approaches the main goal here is to reduce the computational complexity of the overall CL estimating equations regardless of the model parameter θ, which is viewed as fixed in size. Accordingly, our `1 -penalty involves only the composition rule w, but not the model parameter θ. In the future, developing approaches for simultaneous penalization on θ and w may be useful to deal with situations where both the data dimension and the size of the parameter space increase. Two main perks of the proposed approach make it an effective alternative to traditional CL estimation from practitioner’s perspective. The first advantage is that the SCLE methodology constructs CL equations and returns inferences very quickly. Theorem 3.2 shows that for any λ > 0 the empirical composition rule w bλ retains at most np ∧ m non-zero elements. This is an important feature of our method, which reduces – sometimes dramatically – the bw amount of computing needed to obtain the implied MCLE θ( bλ ) and its standard error. Lemma 3.1 highlights that the non-zero elements in w bλ correspond to partial scores maximally correlated with the residual difference r(θ, w) = uM L (θ) − u(θ, w). This means that our approach constructs estimators with relatively high efficiency by dropping only those uj s contributing the least in the CL equations for approximating uM L (θ). The second desirable feature of our method concerns model selection and the ability to reduce the complexity of large data sets. In essence, the truncation step (T-Step) described in (7) is a dimensionreduction step: starting from observations on a possibly large the d-dimensional vector X, our method generates a collection of lower-dimensional subsets Sλ = {Sj , j ∈ Ebλ , λ > 0} where Ebλ = {j : w bλ,j 6= 0}. While individually the selected data subsets in Sλ are of size much smaller than d, collectively they contain most of the information on θ for a given level of computing represented by λ. From a theoretical perspective, little work has been devoted to study the properties of CL estimators when the number of sub-likelihoods m diverges. Cox and Reid (2004) discuss estimators based CL equations with m = d2 + d terms by taking all pairwise and marginal 28 scores for the d-dimensional vector X. They take non-sparse and more rigid composition rules compared to ours with wjk = 1 for all pairs (j 6= k) and wjj = −a × d for all marginals (j = k), where a is a tuning constant used to increase efficiency. To our knowledge, the current paper is the first studying the behavior of more flexible sparsity-inducing composition rules and implied CL estimating equations in the setting where both m and n grow. Theorem 3.3 and Corollary 3.4 provide us with guidance on when the selected score u(θ, w bλ ) is a meaningful approximation to the unknown ML score in the sense of the objective (4). A first requirement is that the total information on θ available if the full likelihood were actually known, m∗ = kuM L (θ∗ )k22 , is not overwhelming compared to the sample size n. If X ∼ Nm (θ1m , Σ), we require m∗ /n = tr{Σ−1 }/n → 0. This condition is very mild when relatively few elements X contain a strong signal on θ, whilst the remaining elements are noisy and with heterogeneous variances. In Section 4.1, we illustrate this by taking Σ diagonal with increasing diagonal elements. A second requirement is that the tuning √ √ constant λ dominates asymptotically m∗ r1 , where r1 represents the convergence rate of the empirical covariance of scores EFn {S(θ)}. For instance, if the elements of S(θ) are subGaussian, we have r1 = op (log(m)/n), meaning that λ should be asymptotically larger than p m∗ log(m)/n. Finally, we show that statistical optimality and computationally parsimony can co-exhist within the same selection procedure when λ is judiciously selected. If λ → 0 at the rate described in Theorem 3.3, the truncated composition rule w bλ with \|Ebλ \| scores approximates the optimal composition rule w0∗ consisting of m nonzero terms. Accordingly, Corollary 3.4 suggests that the implied truncated CL score function u(θ, w bλ ) approximates the optimal score u(θ, w bλ ), uniformly on a neighborhood of θ∗ . Extending this type of result and developing further theoretical insight on the interplay between the type of penalty and the MCLE accuracy beyond the current i.i.d. setting would represent another exciting future research direction. For example, findings would be particularly valuable in spatial statistics where often the number of sub-likelihood components is overwhelming and poses serious challenges 29 to traditional CL methods. Appendix In this section, we show technical lemmas required by the main results in Section 3. Lemma 7.1. kw bλ k1 and kwλ∗ k1 are decreasing in λ. Proof. Denote the first term of Criterion Qλ (θ, w) defined in (4) (without the penalty term) by Q1 (θ, w). Suppose λ1 > λ2 , and let w1 , w2 be the minimizers of Qλ1 (θ∗ , w), Qλ2 (θ∗ , w) respectively. Then, Q1 (w1 )+λ1 kw1 k1 ≤ Q1 (w2 )+λ1 kw2 k1 and Q1 (w1 )+λ2 kw1 k1 ≥ Q1 (w2 )+ λ2 kw2 k1 . Subtracting the last two inequalities gives (λ1 − λ2 )kw1 k1 ≤ (λ1 − λ2 )kw2 k1 . Since λ1 > λ2 , we have kw1 k1 ≤ kw2 k1 . An analogous argument shows that kw bλ k1 is decreasing. Lemma 7.2. Under Conditions C.1 - C.3, if r1 /λ = op (1), then kwλ∗ k1 = O(m∗ /λ) and kw bλ k1 = Op (m∗ /λ). Proof. For wλ∗ , note that Qλ (θ∗ , wλ∗ ) = EkuM L (θ∗ ) − u(θ∗ , wλ∗ )k22 /2 + λkwλ∗ k1 ≤ Qλ (θ∗ , 0) = m∗ /2, or λkwλ∗ k1 ≤ m∗ /2. Hence, kwλ∗ k1 = O(m∗ /λ). bw b θ, For w bλ , we have Q( bλ ) = bw b Tw b 0) = 0. Since EuM L (θ∗ )T uj (θ∗ ) = b θ, w bλT EFn S(θ) bλ /2 − diag(EFn S(θ)) bλ + λkw bλ k1 ≤ Q( Euj (θ∗ )T uj (θ∗ ), we have b Tw b − ES(θ∗ ))T w λkw bλ k1 ≤ diag(EFn S(θ)) bλ = diag(EFn S(θ) bλ + EuM L (θ∗ )T u(θ∗ , w bλ ), b − ES(θ∗ ))T w b − ES(θ∗ ))T \|j,k kw with diag(EFn S(θ) bλ ≤ maxj,k \|EFn S(θ) bλ k1 ≤ r1 kw bλ k1 and EuM L (θ∗ )T u(θ∗ , w bλ ) = Op (m∗ ) Hence, kw bλ k1 = Op (m∗ /λ). 30 Lemma 7.3. Let λ → 0 as n → ∞. Under Conditions C.1-C.3, if r1 m∗ 2 λ−2 = op (1), we bw b w∗ ) have EFn u(θ, bλ ) − u(θ, λ P 2 → 0, as n → ∞, where θb is the preliminary root-n consistent estimator used to compute w bλ in the T-Step (7). Proof. Note that r1 m∗2 λ2 = op (1) implies r1 /λ = op (1). Therefore, we have kw bλ − wλ∗ k1 = bw b w∗ ) gives bλ (θ, bλ (θ, Op (m∗ /λ) by Lemma 7.2. Moreover, re-arranging Q bλ ) ≤ Q λ 1 b T (w b ∗ } ≤ EFn {U 2 (θ) bw bλ k1 + λkwλ∗ k1 . bλ − wλ∗ )} − λkw bλ − wλ∗T S(θ)w EF {w bT S(θ) λ 2 n λ b (θ) b T w∗ }T (w Subtracting EFn {U (θ)U bλ − wλ∗ ) from both sides gives λ 1 b T (w b w∗ )}T (w EFn kU (θ) bλ − wλ∗ )k22 ≤ EFn {c(θ, bλ − wλ∗ ) − λkw bλ k1 + λkwλ∗ k1 λ 2 h iT h iT b w∗ ) − Ec(θ∗ , w∗ ) w b w∗ ) − Ec(θ∗ , w∗ ) w∗ = EFn c(θ, bλ − EFn c(θ, λ λ λ λ λ + Ec(θ∗ , wλ∗ )T w bλ − λkw bλ k1 − Ec(θ∗ , wλ∗ )T wλ∗ − λkwλ∗ k1 h iT ∗ ∗ ∗ b ≤ EFn c(θ, wλ ) − Ec(θ , wλ ) (w bλ − wλ∗ ) h iT ∗ ∗ b b = diag EFn S(θ) − ES(θ ) − EFn S(θ)) − E(S(θ ) wλ∗ (w bλ − wλ∗ ), where the inequality is implied by Lemma 3.1. The last expression is op (1), since r1 m∗ 2 /λ2 = op (1) and kw bλ −wλ∗ k1 = Op (m∗ /λ) by Lemma 7.2, and the matrix maximum norm is bounded by matrix 2-norm. √ Lemma 7.4. If r1 m∗ 2 λ−2 = op (1) and r2 m∗ λ−1 = op (1), then under conditions C.1-C.6, b λ − K ∗ k1 = op (1). kK λ P b λ − K ∗ k1 ≤ r2 kw Proof. This is a direct result since kK b − w∗ k1 , r2 → 0 according to lemma λ P assumption and kw b − w∗ k1 → 0 by Theorem 3.3. Lemma 7.5. Under Conditions C.1-C.6, Eku(θ∗ , wλ∗ )k22 = O(m∗ ). Proof. Note that EkuM L (θ∗ )−u(θ∗ , wλ∗ )k22 ≤ EkuM L (θ∗ )−u(θ∗ , wλ∗ )k22 +λkwλ∗ k1 ≤ EkuM L (θ∗ )k22 = 31 m∗ . Expanding EkuM L (θ∗ ) − u(θ∗ , wλ∗ )k22 gives Eku(θ∗ , wλ∗ )k22 ≤ 2EuM L (θ∗ )T u(θ∗ , wλ∗ ) q √ q ≤ 2 EkuM L (θ∗ )k22 · Eku(θ∗ , wλ∗ )k22 = 2 m∗ Eku(θ∗ , wλ∗ )k22 . Re-arranging gives Eku(θ∗ , wλ∗ )k22 ≤ 4m∗ . Lemma 7.6. Assume Conditions C.1-C.6. For every > 0, we have n o p 1 X n ∗ ∗ 2 ∗ ∗ ∗ ) → 0, E u (θ , w ) I(\|u(θ , w )\| ≥ nJ i λ λ λ nJλ∗ i=1 where ui (θ, w) = Pm j=1 as n → ∞, (i) wj ∇ log fj (Xj ; θ) is the composite likelihood score corresponding to the ith observation. Proof. Without loss of generality, assume p = 1. Recall that Jλ∗ = Eu(θ∗ , wλ∗ )2 . For every > 0, and constants a, b > 1 such that 1/a + 1/b = 1 n o p 1 X n ∗ ∗ 2 ∗ ∗ ∗ E u (θ , w ) I(\|u(θ , w )\| ≥ nJ ) i λ λ λ nJλ∗ i=1 o p 1 n = ∗ E ui (θ∗ , wλ∗ )2 I(\|u(θ∗ , wλ∗ )\| ≥ nJλ∗ ) Jλ 1 1 1 b (J ∗ )a−1 ∗ ∗ 2a ≤ ∗ E \|ui (θ , wλ )\| · 2 = 2λ 1/b , Jλ n ( n) (20) where the inequality follows by applying Hölder’s and Chebyshev’s inequalities. By the assumption at the beginning of Section 3 that m∗ = o(log(n)), Lemma 7.5 implies Jλ∗ = o(log(n)). Hence, (20) converges to 0 as n → ∞, which proves the desired result. References J. Besag. Spatial interaction and the statistical analysis of lattice systems. Journal of the Royal Statistical Society. Series B (Methodological), pages 192–236, 1974. 32 P. Bühlmann and S. Van De Geer. Statistics for high-dimensional data: methods, theory and applications. Springer Science & Business Media, 2011. T. T. Cai, C.-H. Zhang, H. H. Zhou, et al. Optimal rates of convergence for covariance matrix estimation. 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Noname manuscript No. (will be inserted by the editor) Identifying Hazardousness of Sewer-Pipeline Gas-Mixture using Classification Methods A Comparative Study arXiv:1707.00561v1 [cs.NE] 16 May 2017 Varun Kumar Ojha · Parmartha Dutta · Atal Chaudhuri Received: date / Accepted: date Abstract In this work, we formulated a real-world problem related to sewerpipeline gas detection using the classification-based approaches. The primary goal of this work was to identify the hazardousness of sewer-pipeline to offer safe and non-hazardous access to sewer-pipeline workers so that the human fatalities, which occurs due to the toxic exposure of sewer gas components, can be avoided. The dataset acquired through laboratory tests, experiments, and various literature-sources were organized to design a predictive model that was able to identify/classify hazardous and non-hazardous situation of sewer-pipeline. To design such prediction model, several classification algorithms were used and their performances were evaluated and compared, both empirically and statistically, over the collected dataset. In addition, the performances of several ensemble methods were analyzed to understand the extent of improvement offered by these methods. The result of this comprehensive study showed that the instance-based-learning algorithm performed better than many other algorithms such as multi-layer perceptron, radial basis function network, support vector machine, reduced pruning tree, etc. Similarly, it was observed that multi-scheme ensemble approach enhanced the performance of base predictors. V. K. Ojha IT4Innovations, VŠB Technical University of Ostrava, Ostrava, Czech Republic and Dept. of Computer Science & Engineering, Jadavpur University, Kolkata, India E-mail: [email protected] P. Dutta Dept. of Computer & System Sciences, Visva-Bharati University, India E-mail: [email protected] A Chaudhuri Dept. of Computer Science & Engineering, Jadavpur University, Kolkata, India E-mail: [email protected] Neural Computing and Applications DOI: 10.1007/s00521-016-2443-0 2 Varun Kumar Ojha et al. Keywords Sewer gas detection · Neural network · Classification · KS test 1 Introduction This is in the view of providing a solution to a real-world problem using technology, where the human fatalities need to be avoided. Hence, the technology should be as simple as possible. In this work, we addressed a complex realworld problem related to sewer-pipeline gas detection, where sewer-pipeline safety detection (in terms of non-toxic environment) was required to allow maintenance and cleaning of the pipeline. The sewer gas detection is a highly complex problem because of the presence of several toxic gases in a mixture form, and a single gas detector may not offer reliable solution. Therefore, we studied the complexity of this problem in terms of gas mixture. The primary goal was to offer a simple solution with a high accuracy so that it was easy to categorize the hazardous situation in straightforward way such as “hazardous” or “non-hazardous.” To meet this simplicity, we formulated sewer-pipeline gas detection problem as a classification problem. Sewer-pipeline contains a mixture of several toxic gases such as hydrogen sulphide (H2 S), ammonia (NH3 ), methane (CH4 ), carbon dioxide (CO2 ), nitrogen oxides (NOx ), etc., [1, 2, 3]. Usually, this mixture is generated due to the biodegradation of the waste and the sewage into the sewer-pipeline. Such toxic gas-mixture is fatal for those who come to the proximity/exposure of these gases. Following this, an alarming number of human fatalities are reported each year by the newspapers and the other agencies [4, 5,6]. The authorities those are responsible for maintaining and cleaning of the sewer pipeline provides various electronic portable gas detectors available in the market to the employed persons so that they can determine the safeness of the sewer-environment before physically get involve into the maintenance work. However, the available electronic portable gas detectors are not providing satisfactory results. It is evident from the recent comments from the judiciary to these authorities. In a judgment to a civil appeal number 5322 of 2011, the Supreme Court of India stated, “the State and its agencies/instrumentalities cannot absolve themselves of the responsibility to put in place effective mechanism for ensuring safety of the workers employed for maintaining and cleaning the sewage system [7].” Similarly, in another judgment, the Supreme Court of India stated, “...entering sewer lines without safety gears should be made a crime even in emergency situations... [8, 9].” This motivated us to carry out our research in this domain and to come out with a simple solution so that without having the minimum knowledge of the technicalities of gas composition and safety limits, a person is able to understand the environment of a sewer system before entering. To ensure the simplicity in model, we collected and preprocessed data to realize sewer gas-detection as a binary class classification problem. However, in this work, apart from the objective of constructing a prediction model, we set a secondary objective, which was to analyze the performances of the classifiers, Identifying Hazardousness of Sewer-Pipeline 3 both empirically and statistically. To meet these objectives, we used 12 base predictors from four different categories such as neural network based classifiers, tree based classifiers, instance based classifiers, and rule based classifiers. The algorithms were applied over the collected dataset and the performance of the algorithms were collected in terms of the accuracy. The collected results were then used for analyzing the performance superiority of the one algorithm over another or the one category of algorithms over another. We observed that the performance of the algorithms were independent of the category they belong to. For example, the performance of instance based k-nearest neighbor, logistic model tree, and support vector machine came from three different categories, but they had a very competitive performance. However, we must consider the “No-free-lunch theorem” that suggests that some algorithms perform better on some problem and some on another [10]. Therefore, to find out which predictor performs best in this case, we used 12 base predictors and nine ensemble methods. Rest of the article is organized as follows. A background study is provided in Section 2.1, which leads to setting ground for describing our contribution to the sewer-pipeline gas detection. In Section 2.2, we provide a detailed description of the data collection and preprocessing mechanisms, which constitute the core and significant part for formulating gas detection problem as a binary classification problem. Section 2.3 deals with the brief descriptions of the classifiers/algorithms and methods used for constructing the prediction model. The design of comprehensive experiment set for the evaluation of the classifiers is reported in Section 3. Whereas, Section 3 describes empirical and statistical evaluation of the classifiers, discussions and conclusion are reported in Sections 4 and 5, respectively. 2 Methodology In this Section, we put together the background study, the data collection mechanisms, and the classification methods definitions. The background study describes the significance of the sewer-pipeline gas detection problem and the data collection mechanism describes the formulation of gas-detection as a classification problem. 2.1 Background Study Literature review was conducted in the perspective of electronic-nose (ENOSE) and gas-detection-system to cover a broad area of research in the field of gas detection and modeling using intelligent computing techniques/algorithms. Although not much work specifically on sewer gas-mixture-detection was reported in the past, few notable contributions were observed. Li et al. [11] reported a noticeable research work on the development and design of an electronic nose (E-NOSE) and gas detection system, where a neural network 4 Varun Kumar Ojha et al. (NN)-based mixed gas (NOx, and CO) measurement system was developed. On the other hand, Sirvastava et al. [12, 13] proposed a design of intelligent ENOSE system using backpropagation (BP) and neuro-genetic approach. Llobet et al. [14] presented a pattern recognition approach, based on the wallet transformation for gas mixture analysis using single tin-oxide sensor. Liu et al. [13] addressed a genetic-NN algorithm to recognize patterns of mixed gases (a mixture of three component gases) using infrared gas sensor. Lee et al. [15] illustrated uses of micro gas sensor array (GSA) combined with NN for recognizing combustible leakage gases. Ambard et al. [16] have demonstrated use of NN for gas discrimination using a tin-oxide GSA for the gases H2 , CO and CH4 . In [17], authors have illustrated a NN-based technique for developing a gas sensory system for sensing gases in a dynamic environment. Pan et al. [18] have shown several applications of E-NOSE. Wongchoosuka et al. [19] have proposed an E-NOSE detection system based on carbon nanotube-SnO2 gas sensors for detecting methanol. Zhang et al. [20] developed a knowledge-based genetic algorithm for detecting mixed gas in mines. Won et al. [21] proposed a system for estimation of hazardous gas release rate using optical sensor and NN-based technique. The following salient points came out of the above mentioned articles: – Mainly, BP and NN-based approaches were studied so far for detecting gas-mixtures. – Mostly, the E-Nose systems reported in the past were developed for the gas-mixtures of only two or three gases and the sensors of the gases used were less cross-sensitive to the other gases in mixtures. – Cross-sensitivity during sensing is an important factor in gas detection system, which was least reported in literature as yet. However, Ojha et al. [22, 23, 24, 25,26, 27] offered a few methods such as neuro-genetic, neuro-swarm, ant-colony-based, neuro-simulated annealing, etc., where cross-sensitivity factor has been addressed to some extent. However, these works were primarily related to regression modeling. – The impact of humidity and temperature on sensors remained ignored so far. – The gas detection system or E-Nose was viewed only in the framework of regression problems and not classification problem. Classification based approach led us to determine the hazardous and nonhazardous situation of a sewer-pipeline. In addition, the collection, organization, and the preprocessing of the collected data enabled us to address the cross-sensitivity issue firmly. The cross-sensitivity issue occurs because of the sensitivity of one gas-sensor towards multiple gases. So was our case, where a semiconductor-based GSA was designed using five gas-sensors. Each gas-sensor was typically meant for detecting its respective target gas. Hence, when the GSA was used for collecting data for a mixture of gases, the crosssensitivity in the sensed values (collected data) became inevitable. Therefore, rather than considering pure results of the respective gases, we registered the cross-sensitive results as a part of our-collected data. Since a computation- Identifying Hazardousness of Sewer-Pipeline 5 ally intelligent model learned from the data and also maintained the crosssensitivity patterns registered in terms of data values itself, a learned model accurately predicts an unknown gas mixture. 2.2 Equipment and Data Collection Mechanism (A/trained/Classifier/embedded/into/ electronic/(EEPROM)./ (Buzzer/light)/ Intelligent/unit./ Output/Unit (Figure/2) Gas/Sensor/ Array/ Before explaining the details of data collection and equipment, we need to explain the basic design and the purpose of our work, which is to offer an intelligent gas detection system (an electronic portable gas detector) that will be a result of embedding learned-predictor (trained-classifier) into an electronic system. The data flow into our developed intelligent system is shown in Fig. 1, which describes the entire process of the intelligent system design, which is divided into three phases: 1) The data acquisition unit, which consists of gas suction-motor chamber, GSA, and data acquisition-cum data-preprocessor block; 2) An intelligent unit (classifier unit), which receives data from dataacquisition unit and classifying the acquired data patterns; 3) The output unit, which prompts the result in terms of colored light and buzzer. Hence, our objective here was limited to only train a classifier using the collected data. We describe the data collection process as follows. Fig. 1 Block diagram of intelligent system design (real time data flow process) At first, we collected the data samples from the data-sheets, literature, and laboratories test of the collected gas mixture samples from sewer-pipelines. Second, we designed our own metal oxide semiconductor (MOS) gas sensors array (GSA) that was used for verifying the literature and laboratory data and for generating the data samples for the purpose experiments. Our designed GSA consists of five gas-sensors for sensing five different gases. They include hydrogen sulphide (H2 S), ammonia (NH3 ), methane (CH4 ), carbon dioxide (CO2 ), and nitrogen oxides (NOx ). Typically, MOS sensors are resistance-type electrical sensors, where responses are change in circuit resistance proportional to gas concentration, A resistance type sensor responds to change in resistance due to change in the concentration of gases. The change in resistance is given as δRs /R0 , where δRs is change in MOS sensor resistance and R0 is base resistance or the sensing resistance at a specifics gas concentration in clean air [19]. The R0 of the sensors MiCS - 4514, MQ - 7, MQ - 136, MQ - 135, and MQ - 4 is 0.25 ppm, 100 ppm, 10 ppm, 100 ppm and 1000 ppm, respectively. Here, ppm is the unit for measuring concentration of gas into air which is defined as follows: 1 ppm is equal to 1 volume of a gas into 106 volume of air. 6 Varun Kumar Ojha et al. A typical arrangement of a gas sensor array is shown in Fig. 2. The circuitry shown in Fig. 2 (left) was developed in our laboratory. Here, the fabricated and installed sensors were MiCS - 4514, MQ - 7, MQ - 136, MQ - 135, and MQ - 4 for gases NO2 , CO, H2 S, NH3 , and CH4 , respectively [28, 29]. The gas sensors used were sensitive to not only their target gases, but they were sensitive also to other gases in the gas-mixture [30, 31]. Hence, crosssensitivity effect over MOS sensors was confirmed [32]. It was moreover confirmed that the sensor responses were noisy and accordingly the pattern of such noise were considered and recorded as an instance into our dataset. Hence, a non-intelligent use of raw values of sensor response for hazardousness prediction may be misleading in operating (real-world) environment. Therefore, a training electronic portable gas detector may be used to predict sewer hazardousness, accurately. So was the effort in this work to provide a classifier. Data collection had vital role in training of a classifier. Data samples were collected as per the following steps. At first, several manhole samples collected from the Kolkata, India municipal area were tested in laboratory to identify the presence of several toxic gases such as nitrogen dioxide (NO2 ), carbon monoxide (CO), hydrogen sulphide (H2 S), ammonia (NH3 ), methane (CH4 ), and carbon dioxide (CO2 ). Secondly, gas sensors were identified for each of the respective gases. As a result we came out with the procurement of gas sensor MiCS - 4514, MQ - 7, MQ - 136, MQ - 135, and MQ - 4 for NO2 , CO, H2 S, NH3 , and CH4 , respectively. We collected data sheets form the companies for the respective sensors. In the third step, a laboratory was setup for the verification and collection of the sensor response of the respective gas sensors in certain range of their concentration. Specifically, the concentration range in ppm laid down in sensor manuals of sensors MiCS - 4514, MQ - 7, MQ - 136, MQ - 135, and MQ - 4 are [0.25 - 5], [20 - 1000], [1 - 100], [10 - 300] and [300 - 10000] of the gases NO2 , CO, H2 S, NH3 , and CH4 , respectively. In addition, the lab was setup (see Fig. 2 [right]), where gas cylinders were connected to a gas concentration measuring unit called mass flow controller (MFC), which was further connected to a gas chamber, where each gas was allowed to pass in a specific concentration over an array of gas sensor. More specifically, the behavior of each of the gas sensors was recorded. Fig. 2 Laboratory-scale gas sensor array (GSA) [28, 29] Identifying Hazardousness of Sewer-Pipeline 7 The following steps were used for preparing data sample for the classifiers’ training. First, hazardous (safety) limits of the component gases of manhole gas mixture were collected. Secondly, three different levels, (i) above safetylimit, (ii) at safety-limit, and (iii) below safety-limit for each manhole gas were recognized. Thirdly, gases were mixed in different combination to prepare several mixture sample that were used to pass over GSA. Table 1 indicates few examples of such mixture of gases in different combinations. For example, when we mix five gases each of which has three different recognized concentration levels, we get 243 different combinations (35 ). In addition, we considered the role of humidity and temperature to influence the sensor’s behavior. Accordingly, the data values were recorded. Hence, our collected dataset contained seven input features and an output class. Each sample was labeled with “0” for safe sample (if the responses of all five sensors were under the maximum safety limit) or “1” for unsafe sample (if the responses of any among the five sensors were above the maximum safety limit). The safety limits of the manhole gases are as follows: safety limit of NH3 is between 25 ppm and 40 ppm [33], CO is in between 35 ppm and 100 ppm [34], H2 S is in between 50 ppm and 100 ppm [35], CO2 is in between 5000 ppm and 8000 ppm [36] and CH4 is in between 5000 ppm and 10000 ppm [37]. Table 2 illustrates a fraction of the collected data samples. 2.3 Classification Based Approach We categorized the classifiers in the four different groups of classifiers. Each category of classifiers contains three classifiers. Table 1 Samples of gas-mixture in different concentration. Concentration of gases in ppm # Humidity Temperature NO2 CO H2S NH CH Class Status 1 2 3 : 7535 7536 7537 : 16036 16037 16038 65 65 65 20 20 20 0 0 0 10 10 10 10 10 10 20 20 20 2000 5000 10000 0 0 1 safe safe unsafe 65 65 65 30 30 30 0 0 0 10 10 10 10 10 10 20 20 20 2000 5000 10000 0 1 0 safe unsafe safe 75 75 75 50 50 50 20 20 20 50 50 50 50 50 50 50 100 100 10000 2000 5000 1 1 1 unsafe unsafe unsafe 8 Varun Kumar Ojha et al. 2.3.1 Network Based Classifiers Multi-layer perceptron (MLP) is a computational model that imitates human brain, and learn from environment, i.e., data. In our work, we used threelayered MLP, where layers are input layer, hidden layer, and output layer [38]. Radial Basis Function Network (RBF) is a special class of MLP, where inputs are mapped onto a hidden layer that consists of radial basis function, which does the non-linear mapping of input to a hidden layer [39]. Support vector machine (SVM) is a supervised learning computational model that maps input to a high dimension feature space using kernel trick. Hence, non-linear separable patterns in input space are linearly classified on a high dimensional feature space [40]. 2.3.2 Tree Based Classifiers Reduced pruning tree (REP) is a tree based classifier method, where a treelike structure is designed for predicting target class based on the input variables [41, 42]. More specifically, the leaves of tree offers decision of the class based on the conjunction of the input feature represented by the branches of the tree. REP tree is a decision tree, where the tree size is reduced by pruning inefficient branches [43]. Naive Bayes tree (NBT) is a special class of decision tree, where the leaf nodes of decision tree that offer decision on the class is replaced by a Naive Bayes classifier, which decides the class label, based on the features and learned threshold [44]. Table 2 Samples of calibrated sensor responses based on the knowledge gathered from literature, data-sheets, lab tests, and scaling process. Sensors response (δRs /R0 ) # Humidity Temperature InNO2 InCO InH2S InNH InCH Class 1 2 3 : 7535 7536 7537 : 16036 16037 16038 65 65 65 20 20 20 0.813 1.301 1.035 6.929 7.521 6.658 5.938 5.525 5.841 3.433 3.521 3.633 3.985 2.178 1.620 0 0 1 65 65 65 30 30 30 1.038 1.054 0.642 7.565 6.694 7.210 5.658 5.745 5.819 3.228 3.692 1.326 2.275 1.268 3.530 0 1 0 75 75 75 50 50 50 4.645 4.712 4.911 2.764 2.985 2.433 0.608 0.641 0.381 2.709 1.228 0.937 0.499 0.450 0.481 1 1 1 Identifying Hazardousness of Sewer-Pipeline 9 Logistic Model Trees (LMT) is similar to NBT that does the transformation of leaves of a decision tree into a logistic regression node. A logistic regression maps independent variables to categorical dependent variables using a logistic function [45, 46]. Hence, LMT is a simple idea, where nodes of a decision/classification tree are replaced by logistic regression model [47]. 2.3.3 Rule Based Classifiers Decision Table (DT) is a simple representation of data into a table based system, where the decision is made based on the features matching or searched into a decision table. On a successful search, the majority class label is returned, otherwise the majority class label of the entire dataset is returned as a decision for an unlabeled data [48]. PART is a rule based classification method based on partial decision tree that generates a list of rules, used subsequently for making prediction of unknown data instance. The rules are generated based on the partial decision tree, which splits dataset into subsets until the entire dataset gets exhausted to form nodes and leaf nodes of the tree [49]. Majority Predictor (Zero R) is the simplest possible form of classification method. It is based on the majority of class label into a dataset. In simple words, it always predicts the majority class. 2.3.4 Instance Based Classifiers Instance-Based Learning (IBK) provides the concept description which is the primary output of an IBK algorithm. It is a function that maps an instance to a category (class label). The concept description function is updated based on training procedure that involves two functions similarity and classification. The similarity function computes the similarity between the training instances and the pre-stored instances, and returns a numeric-value. Then, the classification function provides class label to the instances based on the results of similarity function. Accordingly, the concept description is updated [50]. K∗ (K Star) is an instance-based learner that uses an entropy-based similarity matching function for searching/matching test instances to the learned instances [51]. Locally Weighted Learning (LWL). In a locally weighted learning, the prediction models are allowed to create at local points in a dataset or the specific point of interest rather than creating model for entire dataset. Hence, a linear regression or naive Bayes classifier or any other classifier may be used to create local models. In this case, we use Decision Stamp, which is a single level decision tree model for prediction [52, 53]. 10 Varun Kumar Ojha et al. 2.3.5 Ensemble Methods In this work, we tried to exploit different method of making ensemble. For an ensemble to perform well, we need to take into account two things which are accuracy of predictors and diversity among the predictors [54]. For example, Bagging maintains diversity by bootstrapping dataset, AddBoost combines several weak predictors, Random Subspace maintains diversity by splitting feature space, Random committee maintains diversity by creating predictors using different random seeds, and Rotation forest maintains diversity by splitting and extracting feature subspace using principal component analysis. Similarly, in multi-scheme and voting scheme, we combine several predictors to maintain diversity. Here, we describe the ensemble methods as follows. Bagging. In Bagging, several copies of same predictor is created. Each copy of the predictor learns a different replicate of learning set created from the complete training set using bootstrapping. Finally, the predictor’s decision is combined using plurality voting method [55]. Adaptive Boosting (AdaBoost) is an ensemble technique that combines several weak predictors and inaccurate rules to create an accurate predictor [56]. Random Subspace (Random SUB). In random subspace ensemble method feature space is divided into several feature subset. Hence, predictors are constructed for each feature subset. Finally, the decision of each constructed predictors are combined using voting method [57]. Random Committee (Random COM): In a random committee ensemble, several predictors are constructed over similar dataset, but they use different random seeds to maintain diversity in the ensemble. Rotation Forest (Rotation FRST). In this approach, training set for the predictors are created by splitting feature set into K subsets, and Principal Component Analysis is applied to extract all the principle components [58]. Hence, diversity among the predictors are maintained by K axis rotation to form new feature set for training [58]. Ensemble Selection (Ensemble SEL). In the ensemble selection approach, the ensemble starts with an empty bag, and the predictors (chosen from a library of trained predictors) maximizing the performance of ensemble are added to the bag one by one to compute the decision of ensemble by using voting method [59]. Voting Scheme (Vote). The voting scheme combines probability distribution of several chosen predictors/classifiers (or predictors available in a bag for making ensemble) using majority voting combination method [60]. Identifying Hazardousness of Sewer-Pipeline 11 Multi-Scheme (Multi). The multi-scheme ensemble approach uses a bag of predictors and selects the output class by selecting a predictor from the bag of predictors based on cross-validation performance of the predictors [60]. Weighted Predictor Ensemble (WPE). In this scheme of ensemble, the weight of predictors were determined. Subsequently, the ensemble output of k many predictors were computed as follows: c y = arg max j=1 k X wj I (Pj = ωj ) , j=1 where c is the number of classes (here it is two), I (Pj = ωj ) is a function that returns value one for the predicted class ωj . 3 Experimental Framework and Results Our aim in the experiment design was to obtain a highly accurate model for predicting hazardousness of the environment in a sewer pipeline. The sewerpipeline environment was represented by the collected dataset. The second objective of the experiment design was to obtain results for analyzing the classifiers (predictors). Accordingly, the results of the classifiers were collected. Table 3 represents the parameter setting of the chosen classifiers. For the evaluation of the classifiers, we repeated our experiments 10 times. Finally, the results were compared based on empirical and statistical (Kolmogorov– Smirnov test) evaluation. We used WEKA [61] and MATLAB tools [62] for the purpose of our experiments. We organized the experimental results into three parts as reflected in Table 4. The first part in the table describes the category wise performance of classifier. Hence, the performance of the category of classifiers was evaluated. We represented the performance of the classifiers as per their training and test accuracy. An accuracy close to 1.0 indicates 100% classification accuracy. Accordingly, the standard deviation (std) of training and test accuracies were reported for understanding the consistency of the classifiers’ performance. In Table 4, the performance of the classifiers were arranged as follows. The category is arranged in the ascending order of their average accuracy over 10-fold CV test set, i.e., better performing classifier to the less performing classifier. The dataset was portioned into 10 equal sets and each time 9 sets were used for training and one set for testing. This process was repeated 10 times and each time a unique test set was used. In the second part, we organized the results according to rank of the classifiers’ performance over 10-fold test set. It may please be noted that for each classifier, we collected 10 instances of 10-fold CV training and test results. Hence, the results in Table 5 reflect averaged training and test accuracy of the classifiers. However, ranking the classifiers based only on the average results does not say much about the quality of the classifier. Hence, in the 12 Varun Kumar Ojha et al. Table 3 Parameter setting of different classifiers category Network-Based Classifiers (F1) Tree-Based Classifiers (F2) Instance-Based Classifiers (F3) Rule-Based Classifiers (F4) Ensemble Classifiers (E1) Ensemble Classifiers (E2) Classifiers Parameters MLP Learning rate: 0.3, momentum factor: 0.2, iteration: 500, nodes in hidden layer: 100 Kernel: Gaussian basis function. Kernel: Radial basis function Minimum no. of instance per leaf: 2, split proportion: 0.001 Leaf node: nave Bayes classifier. Node: logistic function, Number of instance per node for splitting: 15 Similarity function: linear nearest neighbor search, neighbor size: 1 Similarity function: entropy distance measure. Similarity function: linear nearest neighbor search, Weight function: Linear, Classifier: Decision Stamp. Evaluation metric: accuracy, Search method: best first Confidence threshold for pruning: 0.25 Ensemble size: 10. Classifier: REP Tree Ensemble size: 10. Classifier: Decision Stamp. Ensemble size: 10. Classifier: REP Tree Ensemble size: 10. Classifier: Random Tree Ensemble size: 10. Classifier: Random Tree Ensemble size: 10. Classifier: REP Tree Ensemble size: 12. Classifiers: F1, F2, F3 and F4 Ensemble size: 12. Classifiers: F1, F2, F3 and F4 Ensemble size: 12. Classifiers: F1, F2, F3 and F4 RBF SVM REP NBT LMT IBK K Star LWL DT PART Zero R Bagging AdaBoost Random SEL Random COM Rotation FRST Ensemble SEL Vote Multi Scheme WPE third part of the results, we used pairwise comparison of the classifiers using Kolmogorov–Smirnov (KS) test, which ascertains whether the supremacy of one classifier over the other is statistically significant or not. A comprehensive matrix of the pairwise KS test results are presented in Table 6. The KS Test is a non-parametric statistical test that determines the difference between the cumulative frequency distribution (cfd) of two samples. In other words, it indicates whether the empirical cfd of one sample is equal “=”, larger “”, or smaller “≺” than the other. It tells whether two dataset A and B are statistically similar “A=B”, dissimilar “A≺B”, where A being statistically dominated by B, or dissimilar “AB”, where A being statistically dominant over B. In our experiments, the KS test was evaluated with 5% significance level, i.e., with 95% confidence. 4 Discussions Since the developed electronic portable gas detector shall be used by naive persons who are engaged in maintaining sewer-pipeline, we are looking for binary answer. Hence, our objective is to search for classification accuracy and Identifying Hazardousness of Sewer-Pipeline 13 Table 4 Experimental Results of Classifiers over 10 fold cross validation error category NN-Based Classifiers Tree-Based Classifiers Instance-Based Classifiers Rule-Based Classifiers Ensemble Classifiers Classifiers Training avg. accuracy std Test avg. accuracy std SVM MLP RBF LMT REP Tree NB Tree IBK K Star LWL PART Decision Table Zero R Multi Rotation FRST Random COM Bagging WPE Ensemble SEL Vote Random SUB AdaBoostM1 0.9407 0.8681 0.8064 0.9697 0.9528 0.9064 1.0000 0.9997 0.7613 0.9275 0.8672 0.7613 1.0000 1.0000 1.0000 0.9728 0.9635 0.9577 0.9423 0.9160 0.7613 0.9340 0.8664 0.8051 0.9360 0.9265 0.8898 0.9671 0.9638 0.7613 0.9062 0.8553 0.7613 0.9672 0.9622 0.9549 0.9395 0.9356 0.9330 0.9214 0.8720 0.7613 0.0041 0.0081 0.0187 0.0023 0.0067 0.0418 0.0023 0.0036 0.0128 0.0103 0.0154 0.0091 0.0035 0.0036 0.0073 0.0077 0.0056 0.0077 0.0143 0.0165 0.0091 0.0008 0.0029 0.0090 0.0039 0.0025 0.0469 0.0000 0.0001 0.0014 0.0097 0.0049 0.0010 0.0000 0.0000 0.0000 0.0008 0.0006 0.0009 0.0128 0.0103 0.0010 Table 5 Ranking algorithms according to their performance on test set (10 Fold CV). Rank category Classifiers Training Test 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 E2 F3 F3 E1 E1 E1 E2 F2 E1 F1 F2 E2 F4 F2 F1 E1 F4 F1 F3 F4 E1 Multi IBK KStar Rotation FRST Random COM Bagging WPE LMT Ensemble SEL SVM REPTree Vote PART NBTree MLP Random SUB DT RBF LWL ZeroR AdaBoost 100.0000 100.0000 99.9653 100.0000 100.0000 97.2874 96.3564 96.7454 95.7173 94.0837 95.2469 93.9593 92.7331 89.4770 86.8566 90.9650 86.6874 80.7795 76.1285 76.1285 76.1302 96.8060 96.7945 96.4677 96.2725 95.7737 94.1025 93.9865 93.4674 93.3778 93.3647 92.4733 92.1314 90.8675 88.0795 86.6336 86.4397 85.4790 80.7571 76.1451 76.1451 76.1302 14 Varun Kumar Ojha et al. RBF REPTree SVM ZeroR Multi-Scheme Vote AdaBoost Bagging Ensemble SEL Random COM ≺ ≺ . . . . . . . . . . . . . . . . ≺ ≺ ≺ . . . . . . . . . . . . . . . ≺ ≺ ≺ ≺ . . . . . . . . . . . . . . ≺ . . . . . . . . . . . . . ≺ ≺ ≺ ≺ ≺ ≺ . . . . . . . . . . . . ≺ ≺ ≺ ≺ ≺ ≺ ≺ . . . . . . . . . . . = . . . . . . . . . . ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ . . . . . . . . . ≺ ≺ ≺ ≺ ≺ ≺ ≺ . . . . . . . . = = . . . . . . . ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ . . . . . . ≺ ≺ ≺ ≺ ≺ ≺ ≺ = ≺ ≺ ≺ . . . . . ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ . . . . WPE PART . . . . . . . . . . . . . . . . . Random SUB NBTree ≺ . . . . . . . . . . . . . . . . . . Rotation FRST MLP ≺ . . . . . . . . . . . . . . . . . . . LMT ≺ . . . . . . . . . . . . . . . . . . . . LWL . . . . . . . . . . . . . . . . . . . . . IBK DT IBK Kstar LMT LWL MLP NBTree PART RBF REPTree SVM ZeroR Multi-Scheme Vote AdaBoost Bagging Ensemble SEL Random COM Random SUB Rotation FRST WPE Kstar Classifiers DT Table 6 Ranking algorithms according to their performance on test set (10 Fold CV). ≺ ≺ = ≺ ≺ ≺ . . . ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ . . ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ . the model (weights) with the highest accuracy so that such a combination may be implemented into electronic portable gas detector form. Moreover, it is also a difficult task to be certain with the accuracy of an implemented electronic portable gas detector because the toxic exposure of a gas is also proportional to the time and not only its safety limit. However, with a real-time monitoring and requisite maintenance involved, the accuracy of detector may be relaxed and hence, we resorted to choose 90% accuracy as the accuracy for our developed detector. So, the classifier’s performance was compared with a threshold setting of 90% accuracy. First, let us discuss on the obtained results. For the classifiers belonging to network-based category F1, the classifier SVM performs better than its counterparts MLP and RBF both in terms of high accuracy (test accuracy 0.93403) and high consistency (std on test accuracy 0.0041). On the other hand, the performance of MLP was reported next to SVM with high consistency. The performance of the RBF was found to be inconsistent and poorer in comparison to its counterparts. In the tree based category F2, the performance of LMT and REPTree was comparable to whereas, NBTree has shown poor performance compared to its counterparts. Identifying Hazardousness of Sewer-Pipeline 15 In instance-base category F3, the performance of IBK and K Star was comparative with a high accuracy and high consistency. LWL performed poor with a very low accuracy. When it came to the category of rule based classifier F4, PART has outperformed others in its category, but the consistency was not as high as the consistency of the other well performing classifiers IBK, SVM, MLP, etc. The classifier ZeroR consistently performed poor in comparison to all other classifiers. In the ensemble category E1 and E2, the multi-scheme, Random COM, Rotation FRST, Bagging, WPE, and Ensemble SEL performed with high accuracies (over 90%) and consistency. However, the performance of the ensembles Random Forest, Vote, and AdaBoost were not as satisfactory as compared to the other ensembles. One of the reason behind poor performance of Random SUB was the usage of subset of the features. Therefore, the feature selection may not help in case of this dataset because of the high correlation maintained by each of the features with the output feature. Similarly, Voting used probability measures to combine the predictors and AddBoost combined weak predictors, whereas, the entirely better performing ensemble exploited the best predictors. Hence, they performed better in this scenario. Considering the assumption of 90% accuracy being a good predictor for implementation as gas detector, we can figure out from Table 5 that the classifiers belong to category F3 (exception of the classifier LWL) had performed better than the classifiers of other categories. However, the instance based classifier IBK is not suitable for the implementation as electronic gas detector since it required a large memory for its computation for saving all the instances of the training set. IBK prediction is computed based on all training samples. Hence, it takes long time to compute the output, which is unacceptable in real time. The next category whose performance was found close to IBK were the classifiers of category F2 (tree based classifier). Two classifies, LMT and REP Tree qualified the 90% accuracy threshold. On the contrary, two classifiers from each F1 and F4 had performed lower than 90% accuracy. However, SVM performed significantly well with a very high accuracy 93.36%. Similarly, classifier PART from category F4 had an accuracy of 90.86%. However, since the SVM produced less number of parameters than the tree based predictor and it robustly accommodates the noisy attributes, it was recommended from these experiments that SVM is a proper choice for the implementation of the proposed gas detector. 5 Conclusion In this work, we explored a real world problem in the context of classification, where we simplified the approach by offering binary decision to the problem. We explored the problem related to the detection of hazardousness of a sewer pipeline environment. This is very crucial problem since it is related to the safety of the persons who have to work under the toxic environment of 16 Varun Kumar Ojha et al. the sewer-pipeline. Usually, a sewer-pipeline environment contains mixture of toxic gases. Hence, we collected samples from sewer pipelines from different locations. Then we examined those samples to identify data samples for our experiments. We prepared a large dataset by collecting gas sensor responses from laboratory tests, literature and scaled the collected gas sensor responses to form a dataset where non-hazardous samples were labeled 0 and hazardous samples were labeled 1. Finally, we applied 21 different classifiers over the identified dataset and their empirical and statistical performance were evaluated. We discovered that for this problem, the instance based classifier performed best followed by the performance of tree based classifiers. 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The Generic Model of Computation Nachum Dershowitz School of Computer Science Tel Aviv University Tel Aviv, Israel [email protected] Over the past two decades, Yuri Gurevich and his colleagues have formulated axiomatic foundations for the notion of algorithm, be it classical, interactive, or parallel, and formalized them in the new generic framework of abstract state machines. This approach has recently been extended to suggest a formalization of the notion of effective computation over arbitrary countable domains. The central notions are summarized herein. 1 Background Abstract state machines (ASMs), invented by Yuri Gurevich [24], constitute a most general model of computation, one that can operate on any desired level of abstraction of data structures and native operations. All (ordinary) models of computation are instances of this one generic paradigm. Here, we give an overview of the foundational considerations underlying the model (cobbled together primarily from [18, 3, 12]).1 Programs (of the sequential, non-interactive variety) in this formalism are built from three components: • There are generalized assignments f (s1 , . . . , sn ) := t, where f is any function symbol (in the vocabulary of the program) and the si and t are arbitrary terms (in that vocabulary). • Statements may be prefaced by a conditional test, if C then P or if C then P else Q, where C is a propositional combination of equalities between terms. • Program statements may be composed in parallel, following the keyword do, short for do in parallel. An ASM program describes a single transition step; its statements are executed repeatedly, as a unit, until no assignments have their conditions enabled. (Additional constructs beyond these are needed for interaction and large-scale parallelism, which are not dealt with here.) As a simple example, consider the program shown as Algorithm 1, describing a version of selection sort, where F(0), . . . , F(n − 1) contain values to be sorted, F being a unary function symbol. Initially, n ≥ 1 is the quantity of values to be sorted, i is set to 0, and j to 1. The brackets indicate statements that are executed in parallel. The program proceeds by repeatedly modifying the values of i and j, as well as of locations in F, referring to terms F(i) and F( j). When all conditions fail, that is, when j = n and i + 1 = n, the values in F have been sorted vis-à-vis the black-box relation “>”. The program halts, as there is nothing left to do. (Declarations and initializations for program constants and variables are not shown.) This sorting program is not partial to any particular representation of the natural numbers 1, 2, etc., which are being used to index F. Whether an implementation uses natural language, or decimal numbers, 1 For a video lecture of Gurevich’s on this subject, see http://www.youtube.com/v/7XfA5EhH7Bc. E. Kashefi, J. Krivine, F. van Raamsdonk (Eds.) DCM 2011 EPTCS 88, 2012, pp. 59–71, doi:10.4204/EPTCS.88.5 c N. Dershowitz This work is licensed under the Creative Commons Attribution License. 60 Generic Model of Computation Algorithm 1 An abstract-state-machine program for sorting.   i := i + 1 if j = n then if i + 1 6= n then do  j := i + 2     F(i) := F( j)    if F(i) > F( j) then do  F( j) := F(i) else do     j := j + 1 Algorithm 2 An abstract-state-machine program for bisection search.   if sgn f ((a + b)/2) = sgn f (a) then a := (a + b)/2 if \|b − a\| > ε then do  if sgn f ((a + b)/2) = sgn f (b) then b := (a + b)/2 or binary strings is immaterial, as long as addition behaves as expected (and equality and disequality, too). Furthermore, the program will work regardless of the domain from which the values of F are drawn (be they integers, reals, strings, or what not), so long as means are provided for evaluating the inequality (>) relation. Another simple ASM program is shown in Algorithm 2. This is a standard bisection search for the root of a function, as described in [22, Algorithm #4]. The point is that this abstract formulation is, as the author of [22] wrote, “applicable to any continuous function” over the reals—including ones that cannot be programmed. What is remarkable about ASMs is that this very simple model of computation suffices to precisely capture the behavior of the whole class of ordinary algorithms over any domain. The reason is that, by virtue of the abstract state machine (ASM) representation theorem of [25] (Theorem 2 below), any algorithm that satisfies three very natural “Sequential Postulates” can be step-by-step, state-for-state emulated by an ASM. Those postulates, articulated in Section 2, formalize the following intuitions: (I) an algorithm is a state-transition system; (II) given the algorithm, state information determines future transitions and can be captured by a logical structure; and (III) state transitions are governed by the values of a finite and input-independent set of terms. The significance of the Sequential Postulates lies in their comprehensiveness. They formalize which features exactly characterize a classical algorithm in its most abstract and generic manifestation. Programs of all models of effective, sequential computation satisfy the postulates, as do idealized algorithms for computing with real numbers (e.g. Algorithm 2), or for geometric constructions with compass and straightedge (see [34] for examples of the latter). Abstract state machines are a computational model that is not wedded to any particular data representation, in the way, say, that Turing machines manipulate strings using a small set of tape operations. The Representation Theorem, restated in Section 3, establishes that ASMs can express and precisely emulate any and all algorithms satisfying the premises captured by the postulates. For any such algorithm, there is an ASM program that describes precisely the same state-transition function, state after state, as does the algorithm. In this sense, ASMs subsume all other computational models. It may be informative to note the similarity between the form of an ASM, namely, a single repeated loop of a set of generalized assignments nested within conditionals with the “folk theorem” to the effect N. Dershowitz 61 that any flowchart program can be converted to a single loop composed of conditionals, sequencing, and assignments, with the aid of some auxiliary variables (see [29]). Parallel composition gives ASMs the ability to perform multiple actions sans extra variables, and to capture all that transpires in a single step of any algorithm. This versatility of ASMs is what makes them so ideal for both specification and prototyping. Indeed, ASMs have been used to model all manner of programming applications, systems, and languages, each on the precise intended level of abstraction. See [13] and the ASM website (http://www.eecs.umich. edu/gasm) for numerous exemplars. ASMs provide a complete means of describing algorithms, whether or not they can be implemented effectively. On account of their abstractness, one can express generic algorithms, like our bisection search for arbitrary continuous real-valued functions, or like Gaussian elimination, even when the field over which it is applied is left unspecified. AsmL [26], an executable specification language based on the ASM framework, has been used in industry, in particular for the behavioral specification of interfaces (see, for example, [1]). Church’s Thesis asserts that the recursive functions are the only numeric functions that can be effectively computed. Similarly, Turing’s Thesis stakes the claim that any function on strings that can be mechanically computed can be computed, in particular, by a Turing machine. More generally, one additional natural hypothesis regarding the describability of initial states of algorithms, as explained in Section 5, characterizes the effectiveness of any model of computation, operating over any (countable) data domain (Theorem 4). On account of the ability of ASMs to precisely capture single steps of any algorithm, one can infer absolute bounds on the complexity of algorithms under arbitrary effective models of computation, as will be seen (Theorem 6) at the end of Section 5. 2 Sequential Algorithms The Sequential Postulates of [25] regarding algorithmic behavior are based on the following key observations: • A state should contain all the relevant information, apart from the algorithm itself, needed to determine the next steps. For example, the “instantaneous description” of a Turing machine computation is just what is needed to pick up a machine’s computation from where it has been left off; see [38]. Similarly, the “continuation” of a Lisp program contains all the state information needed to resume its computation. First-order structures suffice to model all salient features of states. Compare [32, pp. 420–429]. • The values of programming variables, in and of themselves, are meaningless to an algorithm, which is implementation independent. Rather, it is relationships between values that matter to the algorithm. It follows that an algorithm should work equally well in isomorphic worlds. Compare [19, p. 128]. An algorithm can—indeed, can only—determine relations between values stored in a state via terms in its vocabulary and equalities (and disequalities) between their values. • Algorithms are expressed by means of finite texts, making reference to only finitely many terms and relations among them. See, for example, [31, p. 493]. The three postulates given below (from [25], modified slightly as in [4, 5, 6, 3]) assert that a classical algorithm is a state-transition system operating over first-order structures in a way that is invariant under isomorphisms. An algorithm is a prescription for updating states, that is, for changing some of the interpretations given to symbols by states. The essential idea is that there is a fixed finite set of terms Generic Model of Computation 62 that refer (possibly indirectly) to locations within a state and which suffice to determine how the state changes during any transition. 2.1 Sequential Time To begin with, algorithms are deterministic state-transition systems. Postulate I (Sequential Time) An algorithm determines the following: • A nonempty set2 S of states and a nonempty subset S0 ⊆ S of initial states. • A partial next-state transition function τ : S ⇀ S . Terminal states S‡ ⊆ S are those states X for which no transition τ (X ) is defined. Having the transition depend only on the state means that states must store all the information needed to determine subsequent behavior. Prior history is unavailable to the algorithm unless stored in the current state. State-transitions are deterministic. Classical algorithms in fact never leave room for choices, nor do they involve any sort of interaction with the environment to determine the next step. To incorporate nondeterministic choice, probabilistic choice, or interaction with the environment, one would need to modify the above notion of transition. This postulate is meant to exclude formalisms, such as [21, 33], in which the result of a computation—or the continuation of a computation—may depend on (the limit of) an infinite sequence of preceding (finite or infinitesimal) steps. Likewise, processes in which states evolve continuously (as in analog processes, like the position of a bouncing ball), rather than discretely, are eschewed. Though it may appear at first glance that a recursive function does not fit under the rubric of a state-transition system, in fact the definition of a traditional recursive function comes together with a computation rule for evaluating it. As Rogers [36, p. 7] writes, “We obtain the computation uniquely by working from the inside out and from left to right”. 2.2 Abstract State Algorithm states are comprehensive: they incorporate all the relevant data (including any “program counter”) that, when coupled with the program, completely determine the future of a computation. States may be regarded as structures with (finitely many) functions, relations, and constants. To simplify matters, relations will be treated as truth-valued functions and constants as nullary functions. So, each state consists of a domain (base set, universe, carrier) and interpretations for its symbols. All relevant information about a state is given explicitly in the state by means of its interpretation of the symbols appearing in the vocabulary of the structure. The specific details of the implementation of the data types used by the algorithm cannot matter. In this sense states are “abstract”. This crucial consideration leads to the second postulate. Postulate II (Abstract State) The states S of an algorithm are (first-order) structures over a finite vocabulary F , such that the following hold: • If X is a state of the algorithm, then any structure Y that is isomorphic to X is also a state, and Y is initial or terminal if X is initial or terminal, respectively. • Transitions preserve the domain; that is, Dom τ (X ) = Dom X for every non-terminal state X . 2 Or class; the distinction is irrelevant for our purposes. N. Dershowitz 63 • Transitions respect isomorphisms, so, if ζ : X ∼ = Y is an isomorphism of non-terminal states X ,Y , then also ζ : τ (X ) ∼ = τ (Y ). State structures are endowed with Boolean truth values and standard Boolean operations, and vocabularies include symbols for these. As a structure, a state interprets each of the function symbols in its vocabulary. For every k-ary symbol f in the vocabulary of a state X and values a1 , . . . , ak in its domain, some domain value b is assigned to the location f (a1 , . . . , ak ), for which we write f (ā) 7→ b. In this way, X assigns a value [[t]]X in Dom X to (ground) terms t. Vocabularies are finite, since an algorithm must be describable in finite terms, so can only refer explicitly to finitely many operations. Hence, an algorithm can not, for instance, involve all of Knuth’s arrow operations, ↑, ↑↑, ↑↑↑, etc. Instead one could employ a ternary operation λ x, y, z. x ↑z y. This postulate is justified by the vast experience of mathematicians and scientists who have faithfully and transparently presented every kind of static mathematical or scientific reality as a logical structure. In restricting structures to be “first-order”, we are limiting the syntax to be first-order. This precludes states with infinitary operations, like the supremum of infinitely many objects, which would not make sense from an algorithmic point of view. This does not, however, limit the semantics of algorithms to first-order notions. The domain of states may have sequences, or sets, or other higher-order objects, in which case, the state would also need to provide operations for dealing with those objects. Closure under isomorphism ensures that the algorithm can operate on the chosen level of abstraction. The states’ internal representation of data is invisible and immaterial to the program. This means that the behavior of an algorithm, in contradistinction with its “implementation” as a C program—cannot, for example, depend on the memory address of some variable. If an algorithm does depend on such matters, then its full description must also include specifics of memory allocation. It is possible to liberalize this postulate somewhat to allow the domain to grow or shrink, or for the vocabulary to be infinite or extensible, but such “enhancements” do not materially change the notion of algorithm. An extension to structures with partial operations is given in [3]; see Section 4. 2.3 Effective Transitions The actions taken by a transition are describable in terms of updates of the form f (ā) 7→ b, meaning that b is the new interpretation to be given by the next state to the function symbol f for values ā. To program such an update, one can use an assignment f (s̄) := t such that [[s̄]]X = ā and [[t]]X = b. We view a state X as a collection of the graphs of its operations, each point of which is a location-value pair also denoted f (ā) 7→ b. Thus, we can define the update set ∆(X ) as the changed points, τ (X ) \ X . When X is a terminal state and τ (X ) is undefined, we indicate that by setting ∆(X ) = ⊥. The point is that ∆ encapsulates the state-transition relation τ of an algorithm by providing all the information necessary to update the interpretation given by the current state. But to produce ∆(X ) for a particular state X , the algorithm needs to evaluate some terms with the help of the information stored in X . The next postulate will ensure that ∆ has a finite representation and its updates can be determined and performed by means of only a finite amount of work. Simply stated, there is a fixed, finite set of ground terms that determines the stepwise behavior of an algorithm. Postulate III (Effective Transitions) 3 For every algorithm, there is a finite set T of (ground) critical terms over the state vocabulary, such that states that agree on the values of the terms in T also share the same update sets. That is, ∆(X ) = ∆(Y ), for any two states X ,Y such that [[t]]X = [[t]]Y for all t ∈ T . In particular, if one of X and Y is terminal, so is the other. 3 Or Bounded Exploration. Generic Model of Computation 64 The intuition is that an algorithm must base its actions on the values contained at locations in the current state. Unless all states undergo the same updates unconditionally, an algorithm must explore one or more values at some accessible locations in the current state before determining how to proceed. The only means that an algorithm has with which to reference locations is via terms, since the values themselves are abstract entities. If every referenced location has the same value in two states, then the behavior of the algorithm must be the same for both of those states. This postulate—with its fixed, finite set of critical terms—precludes programs of infinite size (like an infinite table lookup) or which are input-dependent. A careful analysis of the notion of algorithm in [25] and an examination of the intent of the founders of the field of computability in [18] demonstrate that the Sequential Postulates are in fact true of all ordinary, sequential algorithms, the (only) kind envisioned by the pioneers of the field. In other words, all classical algorithms satisfy Postulates I, II, and III. In this sense, the traditional notion of algorithm is precisely captured by these axioms. Definition 1 (Classical Algorithm) An object satisfying Postulates I, II, and III shall be called a classical algorithm. 2.4 Equivalent Algorithms It makes sense to say that two algorithms have the same behavior, or are behaviorally equivalent, if they operate over the same states and have the same transition function. Two algorithms are syntactically equivalent if their states are the same up to renaming of symbols (α -conversion) in their vocabularies, and if transitions are the same after renaming. For a wide-ranging discussion of algorithm equivalence, see [2]. 3 Abstract State Machines Abstract state machines (ASMs) are an all-powerful description language for the classical algorithms we have been characterizing. 3.1 Programs The semantics of the ASM statements, assignment, parallel composition, and conditionals, are as expected, and are formalized below. The program, as such, defines a single step, which is repeated forever or until there is no next state. For convenience, we show only a simple form of ASMs. Bear in mind, however, that much richer languages for ASMs are given in [24] and are used in practice [27]. Programs are expressed in terms of some vocabulary. By convention, ASM programs always include symbols for the Boolean values (true and false), undef for a default, “undefined” value, standard Boolean operations (¬, ∧, ∨), and equality (=, 6=). The vocabulary of the sorting program, for instance, contains F = {1, 2, +, >, F, n, i, j} in addition to the standard symbols. Suppose that its states have integers and the three standard values for their domain. The nullary symbols 0 and n are fixed programming constants and serve as bounds of F. The nullary symbols i and j are programming “variables” and are used as array indices. All its states interpret the symbols 1, 2, +, >, as well as the standard symbols, as usual. Unlike i, j, and F, these are static; their interpretation will never be changed by the program. Initial states have n ≥ 0, i = 0, j = 1, some integer values for F(0), . . . , F(n − 1), plus undef for all other points N. Dershowitz 65 States X such that Update set ∆(X ) 0 [[ j]] = [[n]] = [[i]] + 1 ⊥ 1 [[ j]] = [[n]] 6= [[i]] + 1 i 7→ [[i]] + 1, j 7→ [[i]] + 2 2 [[ j]] 6= [[n]] , [[F(i)]] > [[F( j)]] F([[i]]) 7→ [[F( j)]] , F([[ j]]) 7→ [[F(i)]] , j 7→ [[ j]] + 1 3 [[ j]] 6= [[n]] , [[F(i)]] 6> [[F( j)]] j 7→ [[ j]] + 1 Table 1: Update sets for sorting program. of F. This program always terminates successfully, with j = n = i + 1 and with the first n elements of F in nondecreasing order. There are no hidden variables in ASMs. If some steps of an algorithm are intended to be executed in sequence, say, then the ASM will need to keep explicit track of where in the sequence it is up to. 3.2 Semantics Unlike algorithms, which are observed to either change the value of a location in the current state, or not, an ASM might “update” a location in a trivial way, giving it the same value it already has. Also, an ASM might designate two conflicting updates for the same location, what is called a clash, in which case the standard ASM semantics are to cause the run to fail (just as real-world programs might abort). An alternative semantics is to imagine a nondeterministic choice between the competing values. (Both were considered in [24].) Here, we prefer to ignore both nondeterminism and implicit failure, and tacitly presume that an ASM never involves clashes, albeit this is an undecidable property. To take the various possibilities into account, a proposed update set ∆+ P (X ) (cf. [4]) for an ASM P may be defined in the following manner: ∆+f (s1 ,...,sn ):=t (X ) = { f ([[s1 ]]X , . . . , [[sn ]]X ) 7→ [[t]]X } + ∆+ (X ) ∪ · · · ∪ ∆+ Pn (X ) do {P1 ···Pn } (X ) = ∆ (P1 + ∆P (X ) if X \|= C ∆+ (X ) = if C then P else Q ∆+ (X ) otherwise ( Q ∆+ P (X ) if X \|= C ∆+ (X ) = if C then P ∅ otherwise . Here X \|= C means, of course, that Boolean condition C holds true in X . When the condition C of a conditional statement does not evaluate to true, the statement does not contribute any updates. When ∆+ (X ) = ∅ for ASM P, its execution halts with success, in terminal state X . (Since no confusion will arise, we are dropping the subscript P.) Otherwise, the updates are applied to X to yield the next state by replacing the values of all locations in X that are referred to in ∆+ (X ). So, if the latter contains only trivial updates, P will loop forever. For terminal states X , the update set ∆(X ) is ⊥, to signify that there is no next state. For non-terminal X , ∆(X ) is the set of non-trivial updates in ∆+ (X ). The update sets for the sorting program (Algorithm 1) are shown in Table 1, with the subscript in [[·]]X omitted. For example, if state X is such that n = 2, i = 0, Generic Model of Computation 66 j = 1, F(0) = 1, and F(1) = 0, then (per row 2) ∆+ (X ) = {F(0) 7→ 0, F(1) 7→ 1, j 7→ 2}. For this X , ∆(X ) = ∆+ (X ), and the next state X ′ = τ (X ) has i = 0 (as before), j = 2, F(0) = 0 and F(1) = 1. After one more step (per row 1), in which F is unchanged, the algorithm reaches a terminal state, X ′′ = τ (X ′ ), with j = n = i + 1 = 2. Then (by row 0), ∆+ (X ′′ ) = ∅ and ∆(X ′′ ) = ⊥. 4 The Representation Theorem Abstract state machines clearly satisfy the three Sequential Postulates: ASMs define a state-transition function; they operate over abstract states; and they depend critically on the values of a finite set of terms appearing in the program (and on the unchanging values of parts of the state not modified by the program). For example, the critical terms for our sorting ASM are all the terms appearing in it, except for the left-hand sides of assignments, which contribute their proper subterms instead. These are j 6= n, ( j = n) ∧ (i + 1 6= n), F(i) > F( j), i + 2, j + 1, and their subterms. Only the values of these affect the computation. Thus, any ASM describes a classical algorithm over structures with the same vocabulary (similarity type). The converse is of greater significance: Theorem 2 (Representation [25, Theorem 6.13]) Every classical algorithm, in the sense of Definition 1, has a behaviorally equivalent ASM, with the exact same states and state-transition function. The proof of this representation theorem constructs an ASM that contains conditions involving equalities and disequalities between critical terms. Closure under isomorphisms is an essential ingredient for making it possible to express any algorithm in the language of terms. A typical ASM models partial functions (like division or tangent) by using the special value, undef, denoting that the argument is outside the function’s domain of definition, and arranging that most operations be strict, so a term involving an undefined subterm is likewise undefined. The state of such an ASM would return true when asked to evaluate an expression c/0 = undef, and it can, therefore, be programmed to work properly, despite the partiality of division. In [3], the analysis and representation theorem have been refined for algorithms employing truly partial operations, operations that cause an algorithm to hang when an operation is attempted outside its domain of definition (rather than return undef). The point is that there is a behaviorally equivalent ASM that never attempts to access locations in the state that are not also accessed by the given algorithm. Such partial operations are required in the next section. 5 Effective Algorithms The Church-Turing Thesis [30, Thesis I† ] asserts that standard models capture effective computation. Specifically: All effectively computable numeric (partial) functions are (partial) recursive. All (partial) string functions can be computed by a Turing machine. We say that an algorithm computes a partial function f : Dk ⇀ D if there are input states I ⊆ S0 , with particular locations for input values, such that running the algorithm results in the correct output values of f . Specifically: • The domain of each input state is D. There are k terms such that their values in input states cover all tuples in Dk . Other than that, input states all agree on the values of all other terms. N. Dershowitz 67 • For all input values ā, the corresponding input state leads, via a sequence of transitions τ , to a terminal state in which the value of a designated term t (in the vocabulary of the algorithm) is f (ā) whenever the latter is defined, and leads to an infinite computation whenever it is not. To capture what it is that makes a sequential algorithm mechanically computable, we need for input states to be finitely representable. Accordingly, we insist that they harbor no information beyond the means to reach domain values, plus anything that can be derived therefrom. We say that function symbols C construct domain D in state X if X assigns each value in D to exactly one term over C , so restricting X to C gives a free Herbrand algebra. For example, the domain of the sorting algorithm, consisting of integers and Booleans, can be constructed from 0, true, false, undef, and a “successor” function (call it c) that takes non-negative integers (n) to the predecessor of their negation (−n − 1) and negative integers (−n) to their absolute value (n). Postulate III ensures that the transition function is describable by a finite text, and—in particular–by the text of ASM. For an algorithm to be effective, its states must also be finitely describable. Definition 3 (Effectiveness) 1. A state is effective if it includes constructors for its domain, plus operations that are almost everywhere the same, meaning that all but finitely-many locations (these can hold input values) have the same default value (such as undef ). 2. A classical algorithm is effective if its initial states are. 3. Moreover, effective algorithms can be bootstrapped: A state is effective also if its vocabulary can be enriched to C ⊎ G so that C constructs its domain, while every (total or partial) operation in G is computed by an effective algorithm over those constructors. 4. A model (of computation), that is, a set of algorithms with shared domain(s), is effective if all its algorithms are, via the same constructors. This effectiveness postulate excludes algorithms with ineffective oracles, such as the halting function. Having only free constructors at the foundation precludes the hiding of potentially uncomputable information by means of equalities between distinct representations of the same domain element. This is the approach to effectiveness advocated in [11], extended to include partial functions in states, as in [3]. For each n ≥ 1, our sorting algorithm is effective in this sense, since addition (+) of the natural numbers and comparisons (>) of integers, operations that reside in its initial states, can be programmed from the above-mentioned constructors (0, true, false, undef, c). In particular, partial-recursion for natural numbers and Turing machines for strings form effective models [11]. Furthermore, it is shown in [12] that three prima facie different definitions of effectiveness over arbitrary domains, as proposed in [11, 18, 35], respectively, comprise exactly the same functions, strengthening the conviction that the essence of the underlying notion of computability has in fact been captured. Theorem 4 (Church-Turing Thesis [11]) For every effective model, there is a representation of its domain values as strings, such that its algorithms are each simulated by some Turing machine. Call an effective computational model maximal if adding any function to those that it computes results in a set of functions that cannot be simulated by any effective model. Remarkably (or perhaps not), there is exactly one such model: Theorem 5 (Effectiveness [12, Theorem 4]) The set of partial recursive functions (and likewise the set of Turing-computable string functions) is the unique maximal effective model, up to isomorphism, over any countable domain. Generic Model of Computation 68 We have recently extended the proof of the Church-Turing Thesis and demonstrated the validity of the widely believed Extended Church-Turing Thesis: Theorem 6 (Extended Church-Turing Thesis [17]) Every effective algorithm can be polynomially simulated by a Turing machine. 6 Conclusion We have dealt herein with the classical type of algorithms, that is to say, with the “small-step” (meaning, only bounded parallelism) “sequential-time” (deterministic, no intra-step interaction with the outside world) case. Abstract state machines can faithfully emulate any algorithm in this class, as we have seen in Theorem 2. Furthermore, we have characterized the distinction between effective algorithms and their more abstract siblings in Theorem 4. There are various “declarative” styles of programming for which the state-transition relation is implicit, rather than explicit as it is for our notion of algorithm. For such programs to be algorithms in the sense of Definition 1, they would have to be equipped with a specific execution mechanism, like the one for recursion mentioned above. For Prolog, for example, the mechanism of unification and the mode of search would need to be specified [14]. The abstract-state-machine paradigm can be extended to handle more modern notions: • When desired, an algorithm can make an explicit distinction between successful and failing terminal states by storing particular values in specific locations of the final state. Alternatively, one may declare failure when there is a conflict between two or more enabled assignments. See [24]. • There is no difficulty in allowing for nondeterminism, that is, for a multivalued transition function. If the semantics are such that a choice is made between clashing assignment statements, then transitions are indeed nondeterministic. See [24, 28]. • More general forms of nondeterminism can be obtained by adding a choice command of some sort to the language. See [24]. • Nothing needs to be added to the syntax of ASMs to apply to cases for the environment provides input incrementally. One need only imagine that the environment is allowed to modify the values of some (specified) set of locations in the state between machine steps. See [24]. • In [4, 5, 6], the analysis of algorithms was extended to the case when an algorithm interacts with the outside environment during a step, and execution waits until all queries of the environment have been responded to. • In [8, 9], all forms of interaction are handled. • In [7], the analysis was extended to massively parallel algorithms. • Distributed algorithms are handled in [24, 20]. • The fact that ASMs can emulate algorithms step-for-step facilitates reasoning about the complexity of algorithms, as for Theorem 6 above. Parallel ASMs have been used for studying the complexity of algorithms over unordered structures. See [10, 37]. • Quantum algorithms have been modeled by ASMs in [23]. • Current research includes an extension of the framework for hybrid systems, combining discrete (sequential steps) and analog (evolving over time) behaviors [15, 16]. N. Dershowitz 69 Acknowledgements I thank Yuri Gurevich and Nikolaj Bjørner for their perspicacious suggestions, the referees for their questions, and Evgenia Falkovich for her help. References [1] Mike Barnett & Wolfram Schulte (2001): The ABCs of Specification: AsmL, Behavior, and Components. Informatica (Slovenia) 25(4), pp. 517–526. Available at http://research.microsoft.com/pubs/73061/ TheABCsOfSpecification(Informatica2001).pdf (viewed June 7, 2009). [2] Andreas Blass, Nachum Dershowitz & Yuri Gurevich (2009): When are Two Algorithms the Same? Bulletin of Symbolic Logic 15(2), pp. 145–168, doi:10.2178/bsl/1243948484. 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Commonsense L OCATED N EAR Relation Extraction arXiv:1711.04204v2 [cs.CL] 16 Nov 2017 Frank F. Xu∗, Bill Y. Lin∗ and Kenny Q. Zhu {frankxu, yuchenlin}@sjtu.edu.cn, [email protected] Department of Computer Science and Engineering Shanghai Jiao Tong University, Shanghai, China 1 Introduction Artificial Intelligent systems can benefit from incorporating commonsense knowledge as background, such as ice is cold (H AS P ROPERTY), chewing is a sub-event of eating (H AS S UBEVENT), chair and table are typically found near each other (L OCATED N EAR), etc. This kind of commonsense facts have been utilized in many downstream tasks, such as textual entailment [4, 1] and visual recognition tasks [29]. The commonsense knowledge is often represented as relation triples in commonsense knowledge bases, such as ConceptNet by MIT [20], one of the largest commonsense knowledge graph available today. However, this kind of commonsense knowledge bases are usually manually curated or crowd-sourced by community efforts and thus do not scale well. This paper aims at automatically extracting the commonsense L OCATED N EAR relation between physical objects from textual corpora, which is defined as two objects typically found near each other in real life. We focus on L OCATED N EAR relation for these reasons: (i) L OCATED N EAR facts are helpful prior knowledge for object detection in complex image scenes; Figure 1 illustrates two motivating examples; (ii) such commonsense knowledge can potentially benefit general reasoning in reading comprehension, question answering as well as many other AI tasks; (iii) existing knowledge bases have very few facts for this relation (ConceptNet 5 has only 49 triples of L OCATED N EAR). Figure 1: L OCATED N EAR relation facts assist the detection of vague objects: in a dimly lit room with settings shown in the left sub-figure, if a bright laptop is present on a table, one may guess that a lamp, a photo frame or books maybe nearby. Similarly in the right sub-figure, if a set of knife, fork and plate is on the table, one may believe there could be a glass beside based on the commonsense, even though these objects are hardly visible due to low light. We propose two novel tasks in extracting L OCATED N EAR relation from textual corpora. One is a binary relation classification problem which judges whether or not a sentence is describing two objects physically close by. The other task is to produce a ranked list of L OCATED N EAR facts with the given classified results of large number of sentences. We believe both two tasks can help the community further automatically complete and populate existing commonsense knowledge bases. ∗ The first two authors contribute equally. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. Additionally, we also create two benchmark datasets for evaluating L OCATED N EAR relation extraction systems on the two tasks: one is 5,000 sentences each describing a scene of two physical objects and with a label indicating if the two objects are co-located in the scene; the other consists of 500 pairs of objects with human-annotated scores indicating confidences that a certain pair of objects are commonly located near in the real life. We propose several methods to solve the tasks including feature-based and LSTM-based neural architecture. The proposed neural architecture compares favorably with the current state-of-the-art method for general-purpose relation classification problem. From our relatively smaller proposed datasets, we extract in total 2,067 new L OCATED N EAR triples that are not in ConceptNet. 2 Sentence-level L OCATED N EAR Relation Classification Given a sentence s mentioning a pair of physical objects <ei , ej >, we call <s, ei , ej > an instance. In this section, we aim to determine whether ei and ej are located near each other in a physical scene described in the sentence s. For example, suppose ei is “dog", ej is “cat”, and s = “The King puts his dog and cat on the table.”. As it is true that the two objects are located near in this sentence, a successful classification model is expected to label this instance as True. While if s2 = “My dog is older than her cat.”, then the answer to the instance <s2 , ei , ej > is False, for s2 is just talking about a general comparison. In the following subsections, we present two different kinds of baseline methods for this binary classification task: feature-based methods and LSTM-based neural architectures. 2.1 Feature-based Methods Our first baseline is an SVM classifier based on following features. We claim that such semantic and syntactic features are widely utilized among existing relation classification models [2, 6, 28, 17]. Note that we put special focus on adverbs and prepositions based on the assumption that these lexical units describing directions and positions in physical world will help identify L OCATED N EAR relations. Proposed features: - Bag of Words (BW) The set of words that ever appeared in the sentence. - Bag of Path Words (BPW) The set of words that appeared on the shortest dependency path between objects ei and ej in the dependency tree of the sentence s, plus the words in the two subtrees rooted at ei and ej in the parse tree. - Bag of Adverbs and Prepositions (BAP) The existence of adverbs and prepositions in the sentence as binary features. - Global Features (GF) The length of the sentence, the number of nouns, verbs, adverbs, adjectives, determiners, prepositions and punctuations in the whole sentence. - Shortest Dependency Path Features (SDP) From the dependency parse tree of the sentence and the shortest path between the two objects ei and ej . - Semantic Similarity Features (SS) The cosine similarity between the pre-trained GloVe word embeddings [16] of the two object words. Obtaining such features for every instances, we then feed processed data into a SVM classifier. We evaluate linear and RBF kernels with different parameter settings, and the RBF kernel with {C = 100, γ = 10−3 } performs the best overall. 2.2 LSTM-based Neural Architectures Long Short Term Memory based recurrent neural architectures (LSTMs) [8] are widely used in relation classification [19, 5, 22, 24]. We observe that the existence of L OCATED N EAR relation in an instance <s,e1 ,e2 > depends on two major information sources: one is from the semantic and syntactical features of sentence s and the other is from the object pair <e1 ,e2 >. By this intuition, we design our LSTM-based model with two parts, shown in Figure 2. The left part is for encoding the syntactical and semantic information of the sentence s, while the right part is encoding the semantic similarity between the pre-trained word embeddings of e1 and e2 . 2 Output confidence σ LSTM Dense Layer Token Vector Representation Position Normalized Sequence Original Sentence DT lead#s lead DT 𝐸1 into PR JJ 𝐸2 -4 -3 -2 -1 0 1 2 3 4 -8 -7 -6 -5 -4 -3 -2 -1 0 The king Token Pre-trained word vectors of 𝐸1 and 𝐸2 Position led the dog into his nice garden . dog garden Figure 2: The proposed LSTM-based model 2.2.1 Sentence Normalization Using the original word sequence as of a sentence s as input has two problems: (i) the irrelevant words in the sentence can take noise into model; (ii) the large vocabulary of original words induce too many parameters, which may cause over-fitting. For example, given two sentences “The king led the dog into his nice garden.” and “A criminal led the dog into a poor garden.”. The object pair is <dog, garden> in both sentences. The two words “lead” and “into” are essential for determining whether the object pair is located near, but they are not given more bias than other words. Also, the semantic differences between irrelevant words, such as “king” and “criminal”, “beautiful” and “poor”, are not useful to the co-location relation between the “dog” and “garden”, and thus tends to act as noise. Level Objects Lemma Dependency Role POS Tag Examples E1 , E2 open, lead, into, ... open#s, open#o, into#o, ... DT, PR, CC, JJ, ... Table 1: Examples of four types of tokens during sentence normalization. (#s represents the subject of given verb or preposition, and #o represents the object) Considering above problems, we propose utilizing POS (Part-of-Speech) tags instead to capture more syntactical information and reduce the vocabulary size. However, solely doing this loses too much semantic dependency between the words. Thus, we propose a normalized sentence representation method merging the three most important and relevant kinds of information about each instance: lemma, POS tags and dependency role 2 . We first replace the two nouns in the object pair as E1 and E2 , keep the lemmatized form of the original words for all the verbs, adverbs and prepositions, which are highly relevant to describing physical scenes. Then, we replace the subjects and direct objects of the verbs and prepositions (nsubj, dobj for verbs and case for prepositions in dependency parse tree) with special tokens indicating their dependency roles. For the remaining words, we simply use their POS tags to replace the originals. The four kinds of tokens are illustrated in Table 1. Table 2 is a real example of our normalized sentence representation, where the object pair of interest is <dog, garden>. The DT king open#s opened open the DT door open#o and CC led lead the DT dog E1 into into his PR Table 2: Sentence Normalization Example 2 We utilize Stanford CoreNLP tool: https://stanfordnlp.github.io/CoreNLP/ 3 nice JJ garden. E2 . 2.2.2 Model Training As shown in Figure 2, the bottom of the figure shows the original sentence, which is transformed to normalized sequence described above. Apart from the normalized tokens of the original sequence, to capture more structural information, we also encode the distance from each token to E1 and E2 . Such word position embeddings (position/distance features) are proposed by [27] with the intuition that information needed to determine the relation between two target nouns normally comes from words which are close to the target nouns. Then, we leverage LSTM to encode the whole sequence of the tokens of normalized representation plus position embedding. In the meantime, two pretrained GloVe word embeddings [16] of the original two physical object words are fed into a hidden dense layer. Finally, we concatenate both outputs and then use sigmoid activation function to obtain the final prediction. We choose to use the widely-used standard binary cross-entropy as our loss function, and RMSProp [7] is used as optimizer. Following [26], we add 0.5 dropout in LSTM as well as embedding layer, and utilize batch normalization [10, 3] for overfitting problem due to relatively small dataset. 3 L OCATED N EAR Relation Extraction Figure 3 shows the overall workflow of our automatic framework to mine LocatedNear relations from raw text. We first construct a vocabulary of physical objects and generate all candidate instances. For each sentence in the corpus, if a pair of physical objects ei and ej appear as nouns in a sentence s, then we apply our L OCATED N EAR relation classifier on this instance. The relation classifier yields a probabilistic score s indicating the confidence of the existence of L OCATED N EAR relation. Finally, all scores of <s,ei ,ej > instances from the corpus are grouped by the object pairs and aggregated, where each object pair is associated with a final score. Such mined physical pairs with scores can easily be integrated into existing commonsense knowledge base. More specifically, for each object pair <ei , ej >, we find all the m sentences in our corpus mentioning both objects. We classify the m instances with the sentence-level relation classifier and get confidences for each instance, feed them into a function f to obtain the final score of the object pair. There are five variants of the scoring functions: m m X 1 X f0 = m, f1 = conf(sk , ei , ej ), f2 = conf(sk , ei , ej ) m k=1 f3 = m X k=1 m 1 X f4 = 1{conf(sk ,ei ,ej )>0.5} m 1{conf(sk ,ei ,ej )>0.5} , k=1 Object Pairs < 𝑠𝑚 , 𝑒𝑖 , 𝑒𝑗 > ... 𝑓 ... 𝑐𝑜𝑛𝑓 < 𝑠𝑚 , 𝑒𝑖 , 𝑒𝑗 > ... 𝑐𝑜𝑛𝑓 < 𝑠1 , 𝑒𝑖 , 𝑒𝑗 > Object co-location classifier ... Corpus Classification Confidence .. .. .. .. < 𝑒𝑖 , 𝑒𝑗 > ... ... < 𝑠1 , 𝑒𝑖 , 𝑒𝑗 > k=1 𝑠𝑐𝑜𝑟𝑒 < 𝑒𝑖 , 𝑒𝑗 > LocatedNear Relation Scores Figure 3: Computing the L OCATED N EAR scores of object pairs 4 Datasets Our proposed vocabulary of single-word physical objects is constructed by the intersection of all entities that belong to “physical object” class in Wikidata and all ConceptNet concepts. We then manually filtered out some words that have the meaning of an abstract concept, which results in 1169 physical objects in total. Afterwards, we utilize a cleaned subset of the Project Gutenberg corpus [11], which contains 3,036 English books written by 142 authors. An assumption here is that sentences in fictions are more 4 Acc. P R F1 Random 0.500 0.551 0.500 0.524 Majority 0.551 0.551 1.000 0.710 SVM 0.584 0.606 0.702 0.650 SVM(-BW) 0.577 0.579 0.675 0.623 SVM(-BPW) 0.556 0.567 0.681 0.619 SVM(-BAP) 0.563 0.573 0.811 0.672 Acc. P R F1 SVM(-SDP) 0.579 0.597 0.728 0.656 SVM(-SS) 0.584 0.605 0.708 0.652 DRNN [22] 0.635 0.658 0.702 0.679 LSTM+Word 0.637 0.635 0.800 0.708 LSTM+POS 0.641 0.650 0.751 0.697 LSTM+Norm 0.653 0.654 0.784 0.713 SVM(-GF) 0.605 0.616 0.751 0.677 Table 3: Performance of baselines on co-location classification task with ablation. (Acc.=Accuracy, P=Precision, R=Recall, “-” means without certain feature) likely to describe real life scenes. We sample and investigate the density of L OCATED N EAR relations in Gutenberg with other widely used corpora, namely Wikipedia, used by Mintz et al. (2009) and New York Times corpus, created by Riedel et al. (2010) and used by Lin et al. (2016), Hoffmann et al. (2011), Surdeanu et al. (2012). In the English Wikipedia dump, out of all sentences which mentions at least two physical objects, 32.4% turn out to be positive. In the New York Times corpus, the percentage of positive sentences is only 25.1%. In contrast, that percentage in the Gutenberg corpus is 55.1%, much higher than the other two corpora, making it a good choice for L OCATED N EAR relation extraction. From this corpus, we identify 15,193 pairs that co-occur in more than 10 sentences. Among these pairs, we randomly select 500 object pairs and 10 sentences with respect to each pair for annotators to label their commonsense L OCATED N EAR. Each instance is labeled by at least three annotators who are college students and proficient with English. The final truth label of a sentence is decided by a majority vote from the four annotators. The Cohen’s Kappa among the three annotators is 0.711 which suggests substantial agreement. We randomly choose 4000 instances as the training set and 1000 as the test set for evaluating the first sentence-level relation classification task. For the second task, we further ask the annotators to label whether each pair of objects are likely to locate near each other in the real world. Majority votes determine the final truth labels. The inter-annotator agreement here is 0.703. Both datasets are made publicly available.3 5 Evaluation 5.1 Sentence-level L OCATED N EAR Relation Classification We evaluate the proposed methods against the state-of-the-art general domain relation classification model (DRNN) [23]. The results are shown in Table 3. For feature-based SVM, we do feature ablation on each of the 6 feature types (Section 2.1). For LSTM-based model, we experiment on variants of input sequence of original sentence. “LSTM+Word” uses the original words as the input tokens, while “LSTM+POS” uses just the POS tag sequence as the input tokens. “LSTM+Norm” uses the tokens of sequence after sentence normalization. 4 From the results, we find that the SVM model without the Global Features performs best, which indicates that bag-of-word features benefit more in shortest dependency paths than on the whole sentence. We find that DRNN performs best (0.658) on precision but not significantly higher than LSTM+Norm (0.654). The experiment also shows that LSTM+Word enjoys the highest recall score. In terms of the overall performance, LSTM+Norm is the best one. One possible reason is that our proposed the normalization representation reduces input sequences’ token vocabulary size, while preserving important syntactical and semantic information. While LSTM+POS also reduces the vocabulary size, it loses too much information. Another reason is that L OCATED N EAR relation are described in sentence mostly with the prepositions/adverbs decorating them, which are the descendants of object word in the dependency tree, other than words merely along the shortest dependency path. Thus, DRNN cannot capture the information from the words belonging to the descendants of the two object words in the tree, while this 3 https://adapt.seiee.sjtu.edu.cn/~frank/location_relation_data.zip Besides, we added two naive baselines: “Random” baseline classifies the instances into two classes with equal probability; “Majority” baseline considers all the instances to be positive. 4 5 f f0 f1 f2 f3 f4 MAP 0.42 0.58 0.48 0.59 0.56 P@50 0.40 0.70 0.56 0.68 0.40 P@100 0.44 0.60 0.52 0.63 0.48 P@200 0.42 0.53 0.49 0.55 0.50 P@300 0.38 0.44 0.42 0.44 0.42 Table 4: Ranking performances of the 5 scoring methods. information is captured by LSTM+Norm. For the rest of the experiments, we will use LSTM+Norm as the classifier of our choice. 5.2 L OCATED N EAR Relation Extraction Once we have classified the sentences using LSTM+Norm, we can extract L OCATED N EAR relation using the four scoring functions in Section 3. We first present the quantitative results. We use each of the scoring functions to rank the 500 commonsense L OCATED N EAR object pairs described in Section 3. Table 4 shows the ranking results using Mean Average Precision (MAP) and Precision at K as metric. Accumulative scores (f1 and f3 ) generally do better. (door, room) (ship, sea) (fire, wood) (fire, smoke) (book, table) (boy, girl) (house, garden) (house, fire) (door, hall) (fruit, tree) (cup, tea) (arm, leg) (horse, saddle) (door, street) (table, chair) Table 5: Top object pairs returned by best performing scoring function f3 Qualitatively, we show 15 object pairs with some of the highest f3 scores in Table 5. Setting a threshold of 40.0 for f3 , which is the minimum non-zero f3 score for all true object pairs in the L OCATED N EAR object pairs data set (500 pairs), we obtain a total of 2,067 L OCATED N EAR relations, with a precision of 68% by human inspection. 6 Related Work Classifying relations between entities in a certain sentence plays a key role in NLP applications and thus has been a hot research topic recently. Feature-based methods [6] and neural network techniques [19, 5] are most common. Xu et al. (2015) introduce multi-channel SDP-based LSTM model to classify relations incooperating several different kinds of information of a sentence improved by Xu et al. (2016), which performed best on SemEval-2010 Task 8 and is one of our baseline methods. The most related work to ours is the extraction of visual commonsense knowledge by Yatskar et al. (2016). This work learns the textual representation of seven types of fine-grained visual relations using textual caption for the image in MS-COCO dataset [13]. Another important related work is from Li et al. (2016), which enriches several popular relations in ConceptNet with little textual information from real large corpora. However, L OCATED N EAR relation was not studied in this work, while this relation is extremely scarce in ConceptNet and has its own distinctiveness. 7 Conclusion We presented a novel study on enriching L OCATED N EAR relationship from textual corpora. Based on our two newly-collected benchmark datasets, we proposed several methods to solve the sentence-level relation classification problem. We showed that existing methods do not work as well on this task and discovered that LSTM-based model does not have significant edge over simpler feature-based model. Whereas, our multi-level sentence normalization turns out to be useful. Future directions include: 1) better utilizing distant supervision, 2) incorporating knowledge graph embedding techniques, 3) applying the L OCATED N EAR knowledge into downstream applications of Computer Vision and Natural Language Processing. 6 References [1] S. R. Bowman, G. Angeli, C. Potts, and C. D. Manning. A large annotated corpus for learning natural language inference. arXiv preprint arXiv:1508.05326, 2015. [2] R. C. Bunescu and R. J. Mooney. A shortest path dependency kernel for relation extraction. In HLT/EMNLP, 2005. [3] T. Cooijmans, N. Ballas, C. Laurent, Ç. Gülçehre, and A. Courville. Recurrent batch normalization. arXiv preprint arXiv:1603.09025, 2016. [4] I. Dagan, B. Dolan, B. Magnini, and D. Roth. Recognizing textual entailment: Rational, evaluation and approaches. Natural Language Engineering, 15(4):i–xvii, 2009. [5] J. Ebrahimi and D. Dou. Chain based rnn for relation classification. In HLT-NAACL, pages 1244–1249, 2015. [6] I. Hendrickx, S. N. Kim, Z. 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Continuous-time GARCH process driven by semi-Lévy process arXiv:1803.00733v1 [math.ST] 2 Mar 2018 M. Mohammadi ∗ S. Rezakhah∗ N. Modarresi† March 5, 2018 Abstract In this paper we study the simple semi-Lévy driven continuous-time generalized autoregressive conditionally heteroscedastic (SS-COGARCH) process. The statistical properties of this process are characterized. This process has the potential to approximate any semi-Lévy driven COGARCH processes. We show that the state representation of such SS-COGARCH process can be described by a random recurrence equation with periodic random coefficients. The almost sure absolute convergence of the state process is proved. The periodically stationary solution of the state process is shown which cause the volatility to be periodically stationary under some suitable conditions. Also it is shown that the increments with constant length of such SS-COGARCH process is itself a periodically correlated (PC) process. Finally, we apply some test to investigate the PC behavior of the increments (with constant length) of the simulated samples of proposed SS-COGARCH process. Keywords: Continuous-time GARCH process; Semi-Lévy process; Periodically correlated; Periodically stationary. 1 Introduction Many financial data and indices have heteroscedastic structure. Examples of this kind are stocks returns, network traffic and natural data, see [4, 18, 16]. Popular model for these data are autoregressive conditionally heteroscedastic (ARCH) model proposed by Engle [13] and generalized ARCH (GARCH), Bollerslev [3]. The GARCH type processes have become the most popular tools to model heteroscedasticity in discrete time. ∗ Faculty of Mathematics and Computer Science, Amirkabir University of Technology, 424 Hafez Avenue, Tehran 15914, Iran. E-mail: [email protected](M. Mohammadi) and [email protected](S. Rezakhah). † Department of Mathematics and computer science, Allameh Tabatabai University, Tehran, Iran. E-mail: [email protected](N. Modarresi). 1 In practice, for various reasons such as high-frequency data, many time series are irregularly spaced and this has created a demand for continuous-time models, [8]. For the first time, Kluppelberg et al. [17] introduced a continuous-time version of the GARCH(1,1) (COGARCH(1,1)) process, which preserves the essential features of the discrete-time GARCH(1,1) processes. They replaced the noise of the discrete-time GARCH(1,1) process with the increments of some Lévy process. The volatility of this process satisfies a stochastic differential equation. They proved the stationarity property and also second order properties under some regularity conditions on the corresponding Lévy process. Brockwell et al. [8] generalized the Lévy driven COGARCH(1,1) process to the Lévy driven COGARCH(p, q) process for q ≥ p ≥ 1 when its volatility is a continuous-time ARMA (CARMA) process [7]. They showed that the state representation of the volatility can be expressed as a stochastic recurrence equation with random coefficients. Periodic behavior is common in many real-world time series such as power market prices, car accident claims for an insurance company and sales with seasonal interest. The term periodically correlated (PC) was introduced by Gladyshev [14], but the same property was introduced by Bennett [1] who called them cyclostationary ([15]). Properties of PC processes are studies by Hurd and Miamee [15]. Bibi and Lescheb [2] studied the class of bilinear processes with periodic time-varying coefficients of periodic ARMA and periodic GARCH models. Lévy processes introduced by Lévy have stationary and independent increments and right continuous paths with left limits [21]. Such processes have potential to be applied to financial data following stochastic volatility structure. A generalization of Lévy process is semi-Lévy process, that has periodically stationary increments, studied by Maejima and sato [19]. We considered this process as the underlying process in CARMA [7] and COGARCH [8, 17] processes that can be applied when there is evident that the underlying process has PC increments. The observations of such processes have significant dependency to the ones of previous periods. So semi-Lévy process are more prominent than Lévy processes in such cases. In this paper we introduce a COGARCH process driven by some simple semi-Lévy process, which we call SS-COGARCH process. The simple semi-Levy process is defined as a compound Poisson process with periodic time-varying intensity with period τ . This process enables us to provide the statistical properties of the SS-COGARCH process. Moreover, we find a random recurrence equation with periodic random coefficients for the state representation of such process. By some regularity condition we show the absolute convergence of the state equation. We also show that the volatility of the SS-COGARCH process is strictly periodically stationary. The increments of the SS-COGARCH process with constant length h = τ /% where % is some integer is a discrete-time PC process with period %. Such SS-COGARCH process has the potential to provide an approximation for every semi-Lévy driven COGARCH process. Finally, we investigate the theoretical results concerning PC structure of the increment process by simulation. We show that the increments of the SS-COGARCH process with length h is PC with some period % and the support of the squared coherence statistics consists of lines parallel to the main 2 diagonal and having spacing of 2π/%. This paper is organized as follows. In section 2 we introduce the simple semi-Levy driven COGARCH processes. For this, we present the simple semi-Levy process and obtain the characteristic function of it. Section 3 is devoted to some sufficient conditions which make the volatility process strictly periodically stationary. We obtain the mean, covariance function of the state process and volatility process in section 4. We also investigate second order properties of the squared increments of the COGARCH process in this section. In section 5 we illustrate the results with simulations. All proofs are contained in Section 6. 2 Simple semi-Lévy driven COGARCH processes In this section we study the preliminaries such as the additive processes and their characteristic functions and semi-Lévy process in subsection 2.1. We also describe the structure of simple semi-Lévy process and characteristics it in subsection 2.2. Then we introduce the simple semi-Lévy driven COGARCH (SS-COGARCH) process in subsection 2.3. 2.1 Preliminaries Let (Ω, F, (Ft )t≥0 , P) be a filtered probability space, where Ft is the smallest rightcontinuous filtration such that F0 contains all the P-null sets of F. A process (Xt )t≥0 defined on the probability space (Ω, F, (Ft )t≥0 , P) is called an additive process if X0 = 0 a.s., it is stochastically continuous, it has independent increments and its sample paths are right-continuous and have left limits in t > 0. Further, if (Xt )t≥0 has stationary increments, it is a Lévy process [11, 21]. The characteristic function of the additive process (Xt )t≥0 has a following Lévy-Khinchin representation [21, Theorems 9.1-9.8]. Theorem 2.1 Let (Xt )t≥0 be an additive process on Rd . Then (Xt )t≥0 has infinitely divisible distribution for t ≥ 0. The law of (Xt )t≥0 is uniquely determined by its spot characteristic triplet (Γt , Πt , ψt )t≥0 E[ei<w,Xt > ] = eϕt (w) , 1 ϕt (w) = i < w, Γt > − < w, Πt w > + 2 Z w ∈ Rd , (ei<w,x> − 1 − i < w, x > I{\|\|x\|\|≤1} )ψt (dx). Rd where < ·, · > is inner product and \|\| · \|\|Ris Euclidean vector norm. The spot Lévy measure ψt satisfies the integrability condition Rd min{1, \|\|x\|\|2 }ψt (dx) < ∞ for t ≥ 0. Remark 2.1 By [11, p.458-459], the spot characteristic triplet (Γt , Πt , ψt )t∈[0,T ] can be 3 defined by Z t Γt = γs ds 0 Z t Πt = σs2 ds Z0 t υs (B)ds, ψt (B) = ∀B ∈ Rd , 0 where Rd is σ−field on the Rd . The triplet (γt , σt2 , υt )t∈[0,T ] is called the local characteristic triplet of (Xt )0≤t≤T which satisfy the following conditions: • γt : [0, T ] → Rd is a deterministic function with finite variation. • σt : [0, T ] → Md×d (R) is a symmetric, continuous and matrix valued function which RT verifies 0 σt2 dt < ∞. • (υt )t∈[0,T ] is a family of Lévy measures which verifies Z TZ min{1, \|\|x\|\|2 }υt (dx)dt < ∞. 0 Rd As an extension of Lévy process, we present the definition of semi-Lévy processes [19]. Definition 2.1 A subclass of additive processes (Xt )t≥0 is called semi-Lévy process with period τ > 0, if for any 0 ≤ s ≤ t, d Xt − Xs = Xt+τ − Xs+τ d where = denotes the equality in distributions. 2.2 Structure of simple semi-Lévy process For describing the structure of the simple semi-Lévy process, we define the general structure of the intensities function of the Poisson process with periodically stationary increments. We also characterize this pure jump process by representation the characteristic function and introduce the corresponding semi-Lévy measure. Definition 2.2 : Poisson process with periodically stationary increment A process N (t) t≥0 is a Poisson process with periodically stationary increment where E N (t) = Λ(t), Z t Λ(t) = λ(u)du (2.1) 0 and the intensity λ(·) is a periodic non-negative function with some period τ > 0, so λ(t) = λ(t + kτ ) for t ≥ 0, k ∈ N. 4 Definition 2.3 : Simple compound Poisson process Let 0 = t0 < t1 < · · · be a partition of positive real line. Also assume that Aj = Pl [tj−1 , tj ), j ∈ N, and \|Aj \| = \|Aj+l \| for some integer l ∈ N and τ = j=1 \|Aj \|. Let N (t) t≥0 be a Poisson process which has periodically stationary increments with period τ > 0 and intensity function Λ(t) defined by (2.1). Then the simple compound Poisson process (St )t≥0 is defined as S t = Dt + N (t) X Zn (2.2) n=1 P S where Zn = lj=1 Znj I{Υn ∈Dj } , Υn is the arrival time of nth jump Zn , Dj = ∞ A and R 2 k=0 j+kl j Zn are independent and have distribution Fj , j = 1, · · · , l, such that R z Fj (dz) < ∞ for j = 1, · · · , l. Also Dt , t > 0, is a deterministic drift function with period τ , say Dt = Dt+τ , and D0 = 0. One can easily verify that (St )t≥0 has independent increment. Now we find characteristic function of the simple compound Poisson process (St )t≥0 by the following Lemma. Lemma 2.2 Let (N (t))t≥0 be a Poisson process with periodically stationary increment and mean Λ(t), defined by (2.1). Then the process (St )t≥0 defined by (2.2) has the following characteristic function for t ≥ 0 E[eiwSt ] = eϕt (w) , Z ϕt (w) = iwΓt + (eiwz − 1 − iwz I{\|z \|≤1} )ψt (dz ), R where Γt = Dt + m−1 l Z XX k=0 r=1 + j−1 Z X Zr=1 z Λ(tkl+r ) − Λ(tkl+r−1 ) Fr (dz) \|z\|≤1 z Λ(tml+r ) − Λ(tml+r−1 ) Fr (dz) \|z\|≤1 + z Λ(t) − Λ(tml+j−1 ) Fj (dz), \|z\|≤1 and ψt (dz) = m−1 l XX Λ(tkl+r ) − Λ(tkl+r−1 ) Fr (dz) k=0 r=1 j−1 + X Λ(tml+r ) − Λ(tml+r−1 ) Fr (dz) r=1 + Λ(t) − Λ(tml+j−1 ) Fj (dz), where m = [ τt ] and (t − mτ ) ∈ Aj for some j = 1, · · · , l. 5 Proof: see Appendix, P1. Remark 2.2 By Remark 2.1, Lemma 2.2 and (2.1), the spot characteristic triplet of process (St )0≤t≤T , (Γt , 0, ψt )t∈[0,T ] , have the local characteristic triplet (γs , 0, υs )s∈[0,T ] which has the following form m−1 l Z XX γs = dDs + zλ(s)I[tkl+r−1 ,tkl+r ) (s)Fr (dz) \|z\|≤1 k=0 r=1 + j−1 Z X r=1 zλ(s)I[tml+r−1 ,tml+r ) (s)Fr (dz) \|z\|≤1 Z zλ(s)I[tml+j−1 ,t] (s)Fj (dz), + \|z\|≤1 and υs (dz) = m−1 l XX λ(s)I[tkl+r−1 ,tkl+r ) (s)Fr (dz) k=0 r=1 j−1 + X λ(s)I[tml+r−1 ,tml+r ) (s)Fr (dz) r=1 + λ(s)I[tml+j−1 ,t] (s)Fj (dz). (2.3) It follows from definition 2.3 and Remark 2.2 that the family (υs )s∈[0,T ] of semi-Lévy measures verify Z TZ \|z\|2 υs (dz)ds < ∞. (2.4) 0 R This implies that (St )t≥0 is semi-martingale, so it has Lévy-Ito decomposition and has quadratic variation process [11, p.459-460]. Corollary 2.3 By lemma 2.2, the stochastic process (St )t≥0 defined by (2.2) is a semiLévy process with period τ. Proof: see Appendix, P2. 2.3 Structure of simple semi-Lévy driven COGARCH process Let (St )t≥0 be a simple semi-Lévy process with period τ defined by (2.2). Process (Gt )t≥0 with parameters α0 > 0, α1 , · · · , αp ∈ R, β1 , · · · , βq ∈ R, αp 6= 0, βq 6= 0, and αp+1 = · · · αq = 0 is a simple semi- Lévy √ driven COGARCH(p,q) process (SS-COGARCH(p,q)), q ≥ p ≥ 1, defined by dGt = Vt dSt or equivalently Z tp Gt = Vu dSu , t > 0, G0 = 0, (2.5) 0 6 in which the left-continuous volatility process (Vt )t≥0 is defined by Vt = α0 + a0 Yt− , t > 0, V0 = α0 + a0 Y0 , (2.6) where the state process (Yt )t≥0 is the unique càdlàg solution of the stochastic differential equation dYt = BYt− dt + e(α0 + a0 Yt− )d[S, S]t , t > 0, (2.7) d denotes differentiation with respect to t. The initial value Y0 is F0 -measurable and independent of the driving semi-Lévy process (St )t≥0 , and       0 1 0 ··· 0 0 α1  0  0 0 1 · · · 0   α2     ..    . . . . . . . . . B= . (2.8)  , a =  ..  , e =  ..  . . . . .     .    0  0 0 ··· 1 0 αq −βq −βq−1 −βq−2 · · · −β1 1 3 Periodic stationarity conditions In this section we provide some conditions to prove that the volatility process (Vt )t≥0 defined by (2.6) is strictly periodically stationary with period τ . As a result of main theorem, we prove that the increments with constant length of process (Gt )t≥0 is itself a periodically correlated (PC) process which is the mian aim of this paper. We also give a sufficient an necessary condition by which we can determine the volatility is non-negative. In the following theorem in (b) a Lr −matrix norm of the (q × q)-matrix C is defined as kCkr = kCckr . c∈Cq \{0} kckr sup Theorem 3.1 (a) Let (Yt )t≥0 be the state process of the SS-COGARCH(p,q) process with parameters B, a and α0 defined by (2.5). Suppose that (St )t≥0 be a simple semi-Lévy process defined by (2.2). Then for all 0 ≤ s ≤ t Yt = Js,t Ys + Ks,t , (3.1) where (Js,t , Ks,t )0≤s≤t is a family of random (q × q)−matrix Js,t and random vector Ks,t in q R . In addition, Js+kτ,t+kτ , Ks+kτ,t+kτ k∈N0 are independent and identically distributed. (b) Let ηi , i = 1, · · · , q, be the eigenvalues of invertible matrix B which have strictly negative real parts. Also suppose that exists one r ∈ [1, ∞] such that Z 1 log 1 + \|\|P −1 ea0 P \|\|r z 2 dνt (z) < − νt (R)ητ, ∀t ∈ [0, τ ), (3.2) Λ(t + τ ) − Λ(t) R 7 where P is a matrix in which P −1 BP is diagonal and η := maxi=1,··· ,q ηi and (νt )t≥0 is semi-Lévy measure defined by (2.3). Then Yt+mτ converges in distribution to a finite random vector U(t) for fixed t ∈ [0, τ ), as m goes to infinity. The distribution of the vector U(t) is the unique solution of the random equation d where U(t) U(t) = Jt,t+τ U(t) + Kt,t+τ , is independent of Jt,t+τ , Kt,t+τ . (3.3) d (c) Let the conditions of (b) hold and Y0 = U(0) , Then (Yt )t≥0 and (Vt )t≥0 are strictly periodically stationary with period τ . In the other hands, for any s1 , s2 , · · · , sn ≥ 0 and Borel sets E1 , E2 , · · · , En of Rd and Borel sets J1 , J2 , · · · , Jn of R and k ∈ N, P Ys1 ∈ E1 , Ys2 ∈ E2 , · · · , Ysn ∈ En = P Ys1 +kτ ∈ E1 , Ys2 +kτ ∈ E2 , · · · , Ysn +kτ ∈ En , and P Vs1 ∈ J1 , Vs2 ∈ J2 , · · · , Vsn ∈ Jn = P Vs1 +kτ ∈ J1 , Vs2 +kτ ∈ J2 , · · · , Vsn +kτ ∈ Jn . Proof: see Appendix P3. In the following remark we describe the non-negativity of the Lyapunov exponent which leads to the absolutely convergence of the state process (Yt )t≥0 in Theorem 3.1. Remark 3.1 (a) The proof of Theorem 3.1 will be based on the use of the general theory of multivariate random recurrence equations, as discussed by Bougerol and Picard [5], Brandt [6] and Vervaat [22] in the one dimensional case. The state vector (Yt )t≥0 defined by (2.7) satisfies multivariate random recurrence equation. (b) The condition (3.2) which provides the stability of the model based on the existence of a vector norm \|\| · \|\|r such that Jt,t+τ and Kt,t+τ for all t ∈ [0, τ ) satisfy the conditions E log + \|\|Kt,t+τ \|\|r < ∞ (3.4) E log\|\|Jt,t+τ \|\|r < 0, where log + (x) = log(max{1, x}). E log\|\|Jt,t+τ \|\| < 0 is equivalent to the assertion that r the Lyapunov exponent of the Jt+kτ,t+(k+1)τ k∈N0 is strictly negative almost surely. i.e. 1 lim sup log\|\|Jt,t+τ · · · Jt+kτ,t+(k+1)τ \|\|r < 0, k−→∞ k a.s. (c) The conditions of Theorem 3.1 imply (3.4) with the natural matrix norm \|\|A\|\|B,r = \|\|P −1 AP \|\|r , for some matrix A, which corresponds to the following the natural vector norm \|\|c\|\|B,r := \|\|P −1 c\|\|r , where P is a matrix in which P −1 AP is diagonal. 8 c ∈ Cq Corollary 3.2 If (Vt )t≥0 is a strictly periodically stationary process with period τ , then increments with constant length of the process (Gt )t≥0 make a PC process. In the other words, for any t ≥ 0 and h ≥ p > 0 and k ∈ N, (p) (p) E Gt = E Gt+kτ , (p) (p) (p) (p) cov Gt , Gt+h = cov Gt+kτ , Gt+h+kτ , (p) where Gt := R t+p √ t Vs dSs . Proof: see Appendix P4. Theorem 3.3 Let (Yt )t≥0 be the state process of the SS-COGARCH(p,q) process (Gt )t≥0 with parameters B, a and α0 > 0. Suppose that γ ≥ −α0 is a real constant and the following two conditions hold: a0 eBt e ≥ 0 ∀t ≥ 0, a0 eBt Y0 ≥ γ a.s. ∀t ≥ 0. (3.5) (3.6) Then, with probability one, V t ≥ α0 + γ ≥ 0 ∀t ≥ 0. Conversely, if either (3.6) fails, or (3.6) holds with γ > −α0 and (3.5) fails, then there exists a simple semi-Lévy process (St )t≥0 and t0 ≥ 0 such that P (Vt0 < 0) > 0. The proof of the non-negativity volatility process (Vt )t≥0 is similar to the proof of Theorem 5.1 in [8] for Lévy process. 4 Characterization of the state process The aim of this section is to study expected value and covariance function of the state process {Yt : t ≥ 0} and volatility process {Vt : t ≥ 0}. First, we prove that by some sufficient conditions the expected value and covariance Yt exist. Then, by presenting the first and second moments of the random vector U (0) , we find the expected value and covariance function of the state process. Furthermore, a closed form for square increments of the COGARCH process is characterized. c Lemma 4.1 Let the assumptios of Theorem 3.1 hold. If E \|\|Y0 \|\|r < ∞, for c = 1, 2, then (a) If E(St2 ) < ∞, then E(Yt ) < ∞ and E(U(0) ) < ∞. (b) If E(St4 ) < ∞, then cov(Yt ) < ∞ and cov(U(0) ) < ∞. where {St : t ≥ 0} is the simple semi-Levy process. 9 Proof: see Appendix P5. Remark 4.1 By Theorem 3.1(b), (a) we find that E(U(0) ) is the solution of the following random equation I − E(J0,τ ) E U(0) = E(K0,τ ), and (b) E (U(0) )(U(0) )0 is the solution of the following equation Iq2 − E(J0,τ ⊗ J0,τ ) vec E (U(0) )(U(0) )0 = E(K0,τ ⊗ J0,τ ) + E(J0,τ ⊗ K0,τ ) E(U(0) ) + vec E(K0,τ K00,τ ) , where ⊗ is the Kronecker product of two matrices and for a matrix C, vec(C) is the 2 column vector in Cq which is constructed by stacking the columns of matrix A in a vector. The following lemmas establish the the mean and covariance function of the state process. Lemma 4.2 Suppose that {Yt : t ≥ 0} be the state process and the conditions of Theorem 0 3.1 and lemma 4.1 hold. Then for t, h ≥ 0, there exists m, n ∈ N and t1 , t2 ∈ [0, τ ) such that t ∈ mτ, (m + 1)τ , t + h ∈ nτ, (n + 1)τ , t = t1 + mτ and t + h = t2 + nτ and E(Yt ) = E(J0,t1 )E(U) + E(K0,t1 ), (4.1) m−n−1 h cov(Yt , Yt+h ) = E(J0,t2 ) E(J0,τ ) E(J0,τ E(UU0 )J00,t1 ) − E(J0,τ )E(U)E(U0 )E(J00,t1 ) + E(J0,τ E(U)K00,t1 ) − E(J0,τ )E(U)E(K00,t1 ) 0 + E(K0,τ E(U0 )J00,t1 ) − E(K0,τ )E(U )E(J00,t1 ) i 0 0 + E(K0,τ K0,t1 ) − E(K0,τ )E(K0,t1 ) . (4.2) Proof: see Appendix P6. Corollary 4.3 Let {Vt : t ≥ 0} be the volatility process. Then for t, h ≥ 0, expected value and covariance function of Vt have the following forms. E(Vt ) = α0 + a0 E(Yt ) cov(Vt , Vt+h ) = a0 cov(Yt , Yt+h )a. Proof: see Appendix P7. In financial time series, the returns have negligible correlation while the squared returns are significantly correlated, therefore we investigate the behavior of the second-order properties of the increments of the COGARCH process. We assume that volatility process is strictly periodically stationary and non-negative. (p) Now we present the first and second orders of the increment process Gt that in defined in Corollary 4.2. 10 Proposition 4.4 Let G be a zero mean simple semi-Levy driven COGARCH process. Then for t ≥ 0 and h ≥ p > 0, (a) (p) E(Gt = 0, (p) (p) Gt , Gt+h cov (4.3) = 0. (4.4) (b) There exist m, m0 ∈ N0 where t ∈ Aml+i , i = 1, · · · , l and t + p ∈ A(m+m0 )l+i0 , i0 = i, · · · , l, then Z Z t+p tml+i (p) 2 2 2 E Vs λ(s)ds E Vs λ(s)ds + E (Zi0 ) E (Gt ) = E (Zi ) t(m+m0 )l+i0 −1 t m0 l+i0 −2 + X 2 E (Zr+1 ) Z tml+r+1 E Vs λ(s)ds. (4.5) tml+r r=i Moreover, there exist n, n0 ∈ N0 (n ≥ m and n0 ≥ m0 ) where t + h ∈ Anl+j , j = 1, · · · , l and t + h + p ∈ A(n+n0 )l+j 0 , j 0 = j, · · · , l, then cov (p) (p) (Gt )2 , (Gt+h )2 2 0 Z tnl+j = E (Zj ) a (p) λ(s)E(Jt+p,s− )cov (Gt )2 , Yt+p ds t+h + E (Zj 0 )2 a0 Z t+h+p (p) λ(s)E(Jt+p,s− )cov (Gt )2 , Yt+p ds t(n+n0 )l+j 0 −1 n0 l+j 0 −2 + X 2 0 Z tnl+r+1 E (Zr+1 ) a (p) λ(s)E(Jt+p,s− )cov (Gt )2 , Yt+p ds. tnl+r r=j (4.6) Proof: see Appendix P8. Remark 4.2 For s ≥ 0, if we assume that s ≥ 0, R R z 3 νs (dz) = 0, Then (p) cov (Gt )2 , Yt+p = 2E(It+p Yt+p ) − 2E(Jt,t+p )E(It Yt ) Z t+p − cov(Yt+p ) + cov(Yt+p , Yt ) − cov(Yt+p , Ys− )dsB 0 e, t+ where It := Rt 0 √ Gs− Vs dSs and Z E(It Yt ) = B t Z tZ E(Is Ys )ds + α0 0 0 11 R E(Is Ys )z 2 νs (dz)ds. 5 Simulation In this section we simulate the simple semi-Lévy process defined by (2.2). This process is a compound Poisson process with time-varying arrival rate Λ(t) defined by (2.1). Then we verify the theoretical results concerning PC structure of the increments of the SSCOGARCH(p,q) process (Gt )t≥0 defined by (2.5) by simulation. For this we simulate the state process (Yt )t≥0 defined by (2.7) at jump time points and non-jump time points using its random recurrence equation (3.1). Then we evaluate the discretized version of the volatility process (Vt )t≥0 defined by (2.6) and corresponding the SS-COGARCH(p,q) process (Gt )t≥0 . Finally, we verify the PC structure of the increments of the SS-COGARCH process by following the method of [12]. For simulating the simple semi-Lévy process defined by (2.2) with the underlying Poisson process N (t) t≥0 , we consider T1 as the time of the first jump and Tn , n = 2, 3, · · · the P time intervals between the (n − 1)th and nth jumps. Then Υn = nj=1 Tj , n ∈ N, are the arrival times and Υ0 = 0. Therefore for j = 1, 2, · · · FTsn (x) := P Tn ≤ x Υn−1 = s = 1 − P N (s + x) − N (s) = 0 = 1 − e−Λ(s+x)+Λ(s) , (5.1) where Λ(t) defined by (2.1). The arrival times Υ1 , Υ2 , · · · are generated by the following algorithm. 1. Generate the independent and identically distributed (iid) sequence U1 , U2 , · · · from Uniform (0,1). Then by (5.1) as Υ0 = 0 the first arrival time Υ1 = T1 has distribution FΥ0 1 (x) = 1 − e−Λ(x) . Therefore d Λ(Υ1 ) = −ln(1 − U ), where U denotes a Uniform (0,1). So by generating U1 , Υ1 = Λ−1 − ln(1 − U1 ) can be considered as a generated sample for the first arrival time. If Υn−1 , n = 2, 3, · · · , Υ is the (n − 1)th evaluated arrival time, then by (5.1) Tn has distribution FTnn−1 (x) = 1 − e−Λ(x+Υn−1 )+Λ(Υn−1 ) . Therefore d Λ(Υn ) = Λ(Υn−1 ) − ln(1 − U ). So by generating Un , Υn = Λ−1 Λ(Υn−1 ) − ln(1 − Un ) is a generated sample for the nth arravial time. Thus applying the iid sample U1 , U2 , · · · we can evaluate successively the nth arrival time by the (5.1), for the details see [10, p.99]. So by having the periodic intensity function λ(u) in (2.1), one can evaluate Λ−1 (·) by available software. 2. Consider some periodic drift function Ht and as the successive jump size Zn generate independently and has distribution Fn (·) if corresponding arrival time belongs to Dj = S∞ A , j = 1, 2, · · · , l. Now evaluate the simple semi-Lévy process (St )t≥0 from (2.2) k=0 j+kl 12 as St = Ht + N (t) l X X Znj I{Υn ∈Dj } . (5.2) n=1 j=1 Now we consider the following steps for the simulation of the SS-COGARH(p,q) process defined by (2.5)-(2.7). 1. Consider p and q as some integer such that q ≥ p ≥ 1. 2. Choose real parameters β1 , · · · , βq and α1 , · · · , αp and α0 > 0 such that the eigenvalues of the matrix B defined by (2.8) have strictly negative real parts and conditions (3.2), (3.5) and (3.6) are satisfied. 3. Having evaluated arrival times Υn by the above algorithm, generate the state process (YΥn )n∈N by the following the recurrence equation after assuming some initial value for YΥ0 2 B Υn −Υn−1 0 B Υn −Υn−1 YΥn = e YΥn−1 + e α0 + a e YΥn−1 Zn , n ∈ N. This recurrence equation obtained by replacing s = Υn−1 and t = Υn in (3.1). The jump size Zn can be simulated by (5.2) for predefined distributions µ1 , µ2 , · · · , µl . 4. As the simple semi-Lévy process (St )t≥0 (2.2) has no jump over [Υn−1 , Υ− n ], n ∈ N, − ]. Therefore for t ∈ [Υ it follows from (2.7) that dYt = BYt dt for t ∈ [Υn−1 , Υ− n−1 , Υn ] n e−Bt dYt = Be−Bt Yt dt so that d e−Bt Yt = 0. From this follows that for t ∈ [Υn−1 , Υ− n] Z t d e−Bu Yu = 0 Υn−1 hence Yt = eB(t−Υn−1 ) YΥn−1 . (5.3) By (2.6) and (5.3), the discrete-time version of the process (Vt )t≥0 is as VΥn = α0 + a0 YΥ−n = α0 + a0 eB(Υn −Υn−1 ) YΥn−1 , 13 (5.4) and using that the process (St )t≥0 (2.2) has one jump at time Υn over [Υn−1 , Υn ] it follows from (2.5) that Z Υn p Z Υn−1 p GΥn − GΥn−1 = Vu dSu − Vu dSu 0 0 Z Υn p Vu dSu = Υn−1 p = VΥn Zn . (5.5) 5. Having evaluated values of the process (YΥn )n∈N and G0 = 0, generate the process (VΥn )n∈N by (5.4) and corresponding the process (GΥn )n∈N by (5.5). 6. Finally, using the values of VΥn and GΥn provided by previous step, evaluate the sampled processes (Vih )i∈N and (Gih )i∈N for some h > 0 by the followings: (i) Suppose that ih ∈ [Υn−1 , Υn ), for i, n ∈ N. Since the simple semi-Lévy process (St )t≥0 (2.2) has no jump over [Υn−1 , Υn ), it follows from (2.6) and (5.3) that for ih ∈ [Υn−1 , Υn ) Vih = α0 + a0 Yih− = α0 + a0 eB(ih−Υn−1 ) YΥn−1 , note that if ih = Υn−1 , then it follows from step 4 that Vih = α0 + a0 YΥ−n−1 = α0 + a0 eB(Υn−1 −Υn−2 ) YΥn−2 . (ii) Using that the process (St )t≥0 (2.2) has no jump over [Υn−1 , ih] it follows from (2.5) that Z ih Gih − GΥn−1 = Z p Vu dSu − 0 Z Υn−1 p Vu dSu 0 ih = p Vu dSu = 0 Υn−1 hence Gih = GΥn−1 . 5.1 Test for the PC Structure of the increments process To detect the PC structure of a process, Hurd and Miamee [15] and Dudek et al. [12] showed that their proposed spectral coherence can be used to test whether a discrete-time 14 process is PC. Their method is based on the fact that the support of the spectral coherence of a PC process with period % is contained in the subset of parallel lines λs = λr + 2jπ/% for j = −(% − 1), · · · , −1, 0, 1, · · · , (% − 1). The squared coherence statistic for the series X1 , X2 , · · · , XN is computed as follows PM −1 X̃N (λr−M/2+m )X̃N (λs−M/2+m )\|2 \|γ̂(λr , λs , M )\| = PM −1 PM −1 2 2 m=1 \|X̃N (λr−M/2+m )\| m=1 \|X̃N (λs−M/2+m )\| 2 \| m=1 P −iλj k where X̃N (λj ) = N is discrete Fourier transform of Xk for j = 0, 1, · · · , N −1, k=1 Xk e λj = 2πj/N and λj ∈ (0, 2π]. This statistic satisfies 0 ≤ \|γ̂(λr , λs , M )\|2 ≤ 1. Under the null hypothesis that X̃N (λj ) are complex Gaussian with uncorrelated real and imaginary parts for each j, squared coherence statistic has probability density, [15] p(\|γ\|2 ) = (M − 1) 1 − \|γ\|2 M −2 , 0 ≤ \|γ\|2 ≤ 1. For type I error α, the squared coherence α-threshold is determined, [15] xα := \|γ\|2α = 1 − elog(α)/(M −1) . The values of statistic \|γ̂(λr , λs , M )\|2 are computed for all r and s that pair (λr , λs ) ∈ [0, 2π) × [0, 2π). By plotting the values of statistic that exceed the α−threshold, if there are some significant values of statistic that lie along the parallel equally spaced diagonal lines, then Xk is PC. The graph of these significant values indicates the presence of the subset of parallel lines s = r + jN/% for j = −(% − 1), · · · , −1, 0, 1, · · · , (% − 1). To ensure that periodic structure of the series Xk is not a consequence of a periodic mean, it is recommended to remove the periodic mean from this series first. Example 5.1 Let (St )t≥0 be a simple semi-Lévy process by the rate function λ(t) = −cos( π6 t) +4, for t ≥ 0. Furthermore, τ = 12, l = 5 and the length of the successive partitions of each period intervals are 2, 2, 2, 3, 3. Moreover, the distribution of jumps size on these subintervals are assumed to be N (3, 1), N (0, 1), N (1.25, 1.25), N (4, 1), and N (0, 1.5), where N (µ, σ 2 ) denotes a Normal distribution with mean µ and variance σ 2 . In this example we consider SS-COGARCH(1,3) process with parameters of α0 = 1, α1 = 0.03, β1 = 5, β2 = 9 and β3 = 5. Thus, the matrix B is   0 1 0 0 1 B= 0 −5 −9 −5 and conditions (3.2), (3.5) and (3.6) are satisfy. For such the SS-COGARCH process we simulate GΥn for the duration of 40 period intervals with the parameters specified above, Y0 = (8.3580, 2.3377, 0.9040)0 and G0 = 0. Then, using step 6, we sample from this process in equally space partition with distance one (h=1). So we get 480 discretized 15 samples of this 40 period intervals. Then we follow to verify that the increments of the sampled process are a PC process. Figure 1: Top: the increments of the simulated process {Gi : i ∈ N} of size 480; bottom left: the sample autocorrelation (1) plot of Gt ; bottom right: the significant values of the sample spectral coherence with α = 0.05. In figure 1 graph of the increments of the sampled process of size 480 (top) with the sample autocorrelation graph of this process (bottom left) are presented. The bottom right graph shows the sample coherent statistics values for a specified collection of pairs (λr , λs ) ∈ [0, 2π) × [0, 2π) and M = 240 that exceed the threshold corresponding to α = 0.05. The parallel lines for the sample spectral coherence confirm the increments of the sampled process are PC. Also in this graph, the significant off-diagonal is at \|r−s\| = 40 which verifies the first peak at 40 and shows that there is a second order periodic structure with period % = 480/40 = 12. 16 Table 1: Some different values of \|γ̂(λr , λs , M )\|2 for some values r, s with α = 0.05 and xα = 0.0125. The some different values of sample coherence statistics for the test that the increments of the sampled process have period 12 are presented in Table 1. As the corresponding α = 0.05 threshold shows the test is significant on the corresponding parallel lines of Figure 1. 6 Appendix P1: Proof of Lemma 2.2 For any t ≥ 0 there exist j = 1, · · · , l and m ∈ N0 such that t ∈ Aj+ml . Thus, using the Definition 2.2, Definition 2.3 and the fact that (St )t≥0 has independence increments we have PN (t) PN (t) iw Dt + n=1 Zn iwSt = eiwDt E eiw n=1 Zn E e =E e PN (t) PN (t1 ) 1 PN (t ) j 2 iw n=N (t Zn iw n=N2 (t )+1 Zn ml+j−1 )+1 1 = eiwDt E eiw n=1 Zn × e × ··· × e iwDt =e l m−1 YY iw E e k=0 r=1 PN (t) iw n=N (t × E e PN (tkl+r ) n=N (tkl+r−1 )+1 j Zn ml+j−1 )+1 r Zn × j−1 Y r=1 . 17 iw E e PN (tml+r ) n=N (tml+r−1 )+1 r Zn Since for r = 1, · · · , l, Znr are independent and have distribution Fr , it follows from Definition 2.2 and conditional expected value for k = 0, · · · , m and r = 1, · · · , l that PN (t ) PN (t )−N (tkl+r−1 ) r iw n=Nkl+r Zr Zn iw n=1kl+r (tkl+r−1 )+1 n \|N (tkl+r ) − N (tkl+r−1 ) = N E e =E E e ∞ X = E eiw PN n=1 r Zn P N (tkl+r ) − N (tkl+r−1 ) = N N =0 ∞ X = N =0 = e− N e r N Λ(tkl+r ) − Λ(tkl+r−1 ) E(eiwZn ) N! hR iN iwz X ∞ Λ(t ) − Λ(t ) F (dz) e kl+r kl+r−1 r R Λ(tkl+r )−Λ(tkl+r−1 ) N! N =0 R =e − Λ(tkl+r )−Λ(tkl+r−1 ) iwz −1) R (e Λ(tkl+r )−Λ(tkl+r−1 ) Fr (dz) Therefore R E e iwSt iwDt =e e h iwz −1) R (e iw Dt + =e h R \|z\|≤1 z + R ×e r=1 k=0 + Pm−1 Pl Pj−1 Pm−1 Pl r=1 k=0 r=1 i Λ(tml+r )−Λ(tml+r−1 ) Fr (dz)+ Λ(t)−Λ(tml+j−1 ) Fj (dz) r=1 Pj−1 Λ(tkl+r )−Λ(tkl+r−1 ) Fr (dz) Λ(tkl+r )−Λ(tkl+r−1 ) Fr (dz) i Λ(tml+r )−Λ(tml+r−1 ) Fr (dz)+ Λ(t)−Λ(tml+j−1 ) Fj (dz) h iwz −1−iwzI {\|z \|≤1} ) R (e Pm−1 Pl r=1 k=0 + Pj−1 r=1 Λ(tkl+r )−Λ(tkl+r−1 ) Fr (dz) i Λ(tml+r )−Λ(tml+r−1 ) Fr (dz)+ Λ(t)−Λ(tml+j−1 ) Fj (dz) . P2: Proof of Corollary 2.3 It is sufficient to prove that for any 0 ≤ s < t and K ∈ N, d St − Ss = St+Kτ − Ss+Kτ . For any 0 ≤ s < t there exist m, m0 ∈ N0 and j, j 0 = 1, · · · , l such that s ∈ Aj+ml and t ∈ Aj 0 +(m+m0 )l . Thus PN (t) PN (t) PN (s) iw St −Ss iw Dt −Ds + n=1 Zn − n=1 Zn iw Dt −Ds E e =E e =e E eiw n=N (s)+1 Zn PN (tml+j ) j PN (t) j0 iw n=N (t Zn iw n=N (s)+1 Zn iw Dt −Ds 0 )l+j 0 −1 )+1 (m+m =e E e × ··· × e 18 iw(Dt −Ds ) =e ×E e PN (tml+j ) iw N (s)+1 j Zn × l Y E e iw PN (tml+r ) n=N (tml+r−1 )+1 r Zn r=j+1 m+m0 −1 × l Y Y iw PN (tkl+r ) E e n=N (tkl+r−1 )+1 r Zn k=m+1 r=1 × 0 −1 jY iw E e PN (t(m+m0 )l+r ) n=N (t(m+m0 )l+r−1 )+1 r Zn ×E e iw 0 PN (t) n=N (t(m+m0 )l+j 0 −1 )+1 j Zn . r=1 By similar method in the Proof of Lemma 2.2, we have h R iwz (e −1) Λ(t )−Λ(s) Fj (dz)+···+ j+ml R E eiw St −Ss = eiw Dt −Ds e P 0 P 0 + m −1 k=1 Λ(t(m+k)l+1 )−Λ(t(m+k)l ) F1 (dz)+···+ m −1 k=1 Λ(t(m+1)l )−Λ(t(m+1)l−1 ) Fl (dz) Λ(t(m+k+1)l )−Λ(t(m+k+1)l−1 ) Fl (dz) i + Λ(t(m+m0 )l+1 )−Λ(t(m+m0 )l )F1 (dz) +···+ Λ(t)−Λ(t(m+m0 )l+j 0 −1 )Fj 0 (dz) . Since s + Kτ ∈ Aj+(m+K)l and t + Kτ ∈ Aj 0 +(m+m0 +K)l , it follows the same method used in the computation of the characteristic function of St − Ss that h R iwz (e −1) Λ(tj+(m+K)l )−Λ(s+Kτ ) Fj (dz)+··· R iw Dt+Kτ −Ds+Kτ iw St+Kτ −Ss+Kτ e =e E e + Λ(t(m+K+1)l )−Λ(t(m+K+1)l−1 ) Fl (dz) + Pm0 −1 + Pm0 −1 k=1 k=1 Λ(t(m+K+k)l+1 )−Λ(t(m+K+k)l ) F1 (dz)+··· Λ(t(m+K+k+1)l )−Λ(t(m+K+k+1)l−1 ) Fl (dz) + Λ(t(m+m0 +K)l+1 )−Λ(t(m+m0 +K)l )F1 (dz) +··· i + Λ(t+Kτ )−Λ(t(m+m0 +K)l+j 0 −1 )Fj 0 (dz) . By Definition 2.2 and Definition 2.3, we have for partition 0 ≤ ti < tj and K ∈ N Λ(tj ) − Λ(ti ) = Λ(tj + Kτ ) − Λ(ti + Kτ ) = Λ(tj+Kl ) − Λ(ti+Kl ) (6.1) and Dt = Dt+Kτ for t ≥ 0. Thus iw St −Ss E e =E e iw St+Kτ −Ss+Kτ . P3: Proof of Theorem 3.1 Proof a : Let (S)t≥0 be the simple semi-Lévy process defined by (2.2) and Zi be ith jump 19 size. Furthermore T1 denote the time that the first jump occurs P and Tj , j = 2, 3, · · · , be th th the time intervals between the (j − 1) and j jumps and Υn := nj=1 Tj , for n ∈ N, and Υ0 = 0. For n ∈ N Qn := I + (Zn )2 ea0 eB(Υn −Υn−1 ) , Rn := α0 (Zn )2 e. It follows from [8] that Yt satisfies in Yt = Js,t Ys + Ks,t , 0≤s≤t (6.2) where, Js,t = eB(t−ΥN (t) ) QN (t) · · · QN (s)+2 I + (ZN (s)+1 )2 ea0 eB(ΥN (s)+1 −s) , Ks,t = eB(t−ΥN (t) ) RN (t) + QN (t) RN (t)−1 + · · · + QN (t) · · · QN (s)+2 RN (s)+1 . In order to prove that the sequence Js+kτ,t+kτ , Ks+kτ,t+kτ k≥0 is independently identically distributed, let mτ ≤ s ≤ t ≤ (m + 1)τ such that m ∈ N0 . We define (m+1) := ΥN (s)+1 − s, Υ1 (m+1) Z1 := ZN (s)+1 , .. . (m+1) ΥN (t)−N (s) := ΥN (t) − s, (m+1) ZN (t)−N (s) := ZN (t) . (6.3) (m+1) (m+1) (m+1) , , · · · , ΥN (t)−N (s) , Z1 Therefore, Js,t , Ks,t is function of random vector N (t)−N (s), Υ1 (m+1) · · · , ZN (t)−N (s) . Using the fact that increments of the Poisson process {N (t) : t ≥ 0} are independent and the density function of random vector (X1 , · · · , Xn ) can be computed as follows: P x < X ≤ x + δ , · · · , x < X ≤ x + δ 1 1 1 1 n n n n (x1 , · · · , xn ) = f lim , X1 ,··· ,Xn δ1 ,··· ,δn →∞ δ1 · · · δn (m+1) (m+1) , · · · , Υn \|N (t) − N (s) = n as follows: we can give the conditional density of Υ1 f ) (s1 , · · · , sn ) = n!λ(s1 ) × · · · × λ(s n n , Λ(t) − Λ(s) (m+1) (m+1) Υ1 ,··· ,Υn \|N (t)−N (s)=n 0 < s1 < · · · < sn < τ. Since the increment process N (t) − N (s) is a Poisson process with mean Λ(t) − Λ(s) such that Λ(t) − Λ(s) = Λ(t + kτ ) − Λ(s + kτ ), for all k ∈ N0 , it follows from Definition 2.1 and conditional density that d Js,t , Ks,t = Js+kτ,t+kτ , Ks+kτ,t+kτ . 20 The independence of the sequence Js+kτ,t+kτ , Ks+kτ,t+kτ is clear, since Js+kτ,t+kτ and Ks+kτ,t+kτ are constructed only from the segment Su , (s + kτ ) ≤ u ≤ (t + kτ ), of the semi-Levy process S. If 0 ≤ s≤ t, there are n, m ∈ N0 such that s ∈ nτ, (n + 1)τ and t ∈ (n + m)τ, (n + m + 1)τ . By iterating (3.1) we obtain h Yt = J(n+m)τ,t J(n+m−1)τ,(n+m)τ · · · J(n+1)τ,(n+2)τ Js,(n+1)τ Ys + K(n+m)τ,t i + J(n+m)τ,t K(n+m−1)τ,(n+m)τ + · · · + J(n+m)τ,t J(n+m−1)τ,(n+m)τ · · · J(n+1)τ,(n+2)τ Ks,(n+1)τ . It follows from [8] and (6.2) that Js,t = J(n+m)τ,t J(n+m−1)τ,(n+m)τ · · · J(n+1)τ,(n+2)τ Js,(n+1)τ Ks,t = K(n+m)τ,t + J(n+m)τ,t K(n+m−1)τ,(n+m)τ + · · · + J(n+m)τ,t · · · J(n+1)τ,(n+2)τ Ks,(n+1)τ , therefore, for all k ∈ N0 , d Js,t , Ks,t = Js+kτ,t+kτ , Ks+kτ,t+kτ . (b) By iterating (3.1) we obtain h Yt+mτ = Jt+(m−1)τ,t+mτ · · · Jt,t+τ Yt + Kt+(m−1)τ,t+mτ + Jt+(m−1)τ,t+mτ Kt+(m−2)τ,t+(m−1)τ i + · · · + Jt+(m−1)τ,t+mτ · · · Jt+τ,t+2τ Kt,t+τ . Since Jt+(m−1)τ,t+mτ , Kt+(m−1)τ,t+mτ follows immediately d Yt+mτ = m Y m∈N , are independent and identically distributed it Jt+(k−1)τ,t+kτ Yt + Kt,t+τ + m−1 X Jt,t+τ · · · Jt+(k−1)τ,t+kτ Kt+kτ,t+(k+1)τ . k=1 k=1 Note that the Kt,t+τ + infinite series (t) U Pm−1 k=1 Jt,t+τ · · · Jt+(k−1)τ,t+kτ Kt+kτ,t+(k+1)τ is the partial sums of the := Kt,t+τ + ∞ X Jt,t+τ · · · Jt+(k−1)τ,t+kτ Kt+kτ,t+(k+1)τ . (6.4) k=1 Thus, using the general theory of random recurrence equations (see Bougerol and Picard [5], Brandt [6] and Vervaat [22]) and condition (3.2), we prove the almost sure absolute convergence of the series (6.4). Let P be such that ∆ := P −1 BP is diagonal. Then we have for t ≥ 0, \|\|eBt \|\|B,r = \|\|P e∆t P −1 \|\|B,r = \|\|P −1 P e∆t P −1 P \|\|B,r = \|\|e∆t \|\|r = eηt . 21 Using (6.2), (6.3) and condition (3.2) show (2) (2) E log\|\|Jt,t+τ \|\|B,r ≤ E ητ + log 1 + (ZN (t+τ )−N (τ ) )2 \|\|ea0 \|\|B,r + · · · + log 1 + (Z1 )2 \|\|ea0 \|\|B,r (1) (1) + log 1 + (ZN (τ )−N (t) )2 \|\|ea0 \|\|B,r + · · · + log 1 + (Z1 )2 \|\|ea0 \|\|B,r < 0 and it follows from [8], (6.2) and (2.4) E log\|\|Kt,t+τ \|\|B,r )−N (t) h N (t+τ i X ≤E log(1 + (ZN (t+τ )−j+1 )2 \|\|ea0 \|\|B,r ) + (ZN (t+τ )−j+1 )2 j=1 + log(α0 \|\|e\|\|B,r ) < ∞. Hence, the strong law of large numbers yield k 1 X log\|\|Jt+(j−1)τ,t+jτ \|\|B,r + log\|\|Kt+kτ,t+(k+1)τ \|\|B,r < 0 a.s., lim sup k→∞ k j=1 i.e. k1 lim sup \|\|Jt,t+τ · · · Jt+(k−1)τ,t+kτ Kt+kτ,t+(k+1)τ \|\|B,r < 1 a.s. k→∞ From Cauchy’s root criterion follows that series (6.4) is almost sure absolute convergence. Since the state process Y has cadlag paths, it follows that \|\|Yt \|\|B,r is almost surely finite. Therefore (t) \|\|Yt+mτ − U \|\|B,r ≤ \|\| m Y \|\| Jt+(k−1)τ,t+kτ \|\|B,r \|\|Yt − U(t) m B,r −→ 0 a.s. k=1 P (t) where Um := Kt+mτ,t+(m+1)τ + ∞ k=m Jt+mτ,t+(m+1)τ · · · Jt+kτ,t+(k+1)τ Kt+(k+1)τ,t+(k+2)τ . It follows from [9] that Yt+mτ converges in distribution to U(t) , for fixed t ∈ [0, τ ). That U(t) satisfies (3.3) and is the unique solution is clear by the general theory of random recurrence equations. (c) It suffices to show that for any s1 , s2 , · · · , sn and k ∈ N0 d Ys1 , Ys2 , · · · , Ysn = Ys1 +kτ , Ys2 +kτ , · · · , Ysn +kτ . Using the recursion equation (6.2) and analysis is used in (a) we obtain above relation. We give the proof for s1 ∈ [0, τ ) and s2 ∈ [τ, 2τ ). The general case is similar. Therefore Ys2 = Jτ,s2 Js1 ,τ Ys1 + Kτ,s2 + Jτ,s2 Ks1 ,τ , and Ys1 = J0,s1 Y0 + K0,s1 . 22 The random vector Ys1 , Ys2 is function from J0,s1 , K0,s1 , Js1 ,τ , Ks1 ,τ , Jτ,s2 , Kτ,s2 , Y0 and with similar argument also shows that the random vector Ys1 +kτ , Ys2 +kτ is function from Jkτ,s1 +kτ , Kkτ,s1 +kτ , Js1 +kτ,(k+1)τ , Ks1 +kτ,(k+1)τ , J(k+1)τ,s2 +kτ , K(k+1)τ,s2 +kτ , Ykτ . Using d d (a) and assumption Y0 = U(0) , it follows that Ys1 , Ys2 = Ys1 +kτ , Ys2 +kτ . P4: Proof of Corollary 3.2 Since for all s ∈ [t, t + p], the process dSs is independent of Fs , it follows from (2.5) and Corollary 3. that Z t+p p (p) E Gt = E( Vs )E(Ss+ds − Ss ) t Z t+p p d (p) E( Vs+kτ )E(Ss+ds+kτ − Ss+kτ ) = E Gt+kτ . = t (p) In order to prove that the covariance function of Gt is periodic, it suffices to show that (p) (p) (p) (p) E Gt Gt+h = E Gt+mτ Gt+h+mτ . Let Et+p denote conditional expectation with respect to the σ-algebra Ft+p . Since the increments of S on the interval (t + h, t + h + p] are independent of Ft+p and the increment (p) process Gt is measurable Ft+p , we have (p) (p) (p) (p) E Gt Gt+h = E Gt Et+p Gt+h Z t+h+p Z t+p p p Vs Vu dSu E dSs . = E t+h t √ √ Since Vs Vu dSu is function of Ju+du,s− , Ku+du,s− , Ju,u+du , Ku,u+du , Ju− ,u , Ku− ,u , Jt,u− , Kt,u− , − +kτ , Ju+kτ,u+du+kτ , Yt and this vector has the same distribution with Ju+du+kτ,s− +kτ , Ku+du+kτ,s − − − − Ku+kτ,u+du+kτ , Ju +kτ,u+kτ , Ku +kτ,u+kτ , Jt+kτ,u +kτ , Kt+kτ,u +kτ , Yt+kτ , it follows that (p) (p) (p) (p) cov Gt , Gt+h = cov Gt+kτ , Gt+h+kτ . P5: Proof of Lemma 4.1 (a) Let Ỹt be state process of semi Levy driven COGARCH(1,1) process. Then Ỹt = J̃0,t Ỹ0 + K̃0,t , where J̃0,t = exp ηt + N (t) X log(1 + \|\|ea0 \|\|B,r Zi2 ) , i=1 23 (6.5) K̃0,t = α0 \|\|e\|\|B,r × exp N (t) X log(1 + \|\|ea 0 N (t) X \|\|B,r Zi2 ) i=1 Zi2 . i=1 It follows from [8] that for all t ≥ 0 \|\|J0,t \|\|B,r ≤ exp ηt + N (t) X log(1 + \|\|ea 0 \|\|B,r Zi2 ) , (6.6) i=1 \|\|K0,t \|\|B,r ≤ α0 \|\|e\|\|B,r × exp N (t) X log(1 + \|\|ea 0 \|\|B,r Zi2 ) N (t) X i=1 Zi2 . (6.7) i=1 Now define a cadlag process {Xt : t ≥ 0} by Xt = −ηt − N (t) X log(1 + \|\|ea0 \|\|B,r Zi2 ), t ≥ 0. i=1 Then, Xt is a negative simple pure jump semi Levy procress. It follows from Definition 2.1 and Remark 3.2 that E e −cXt = E exp cηt + c N (t) X log(1 + \|\|ea0 \|\|B,r Zi2 ) i=1 Z tZ = exp cηt + 0 (1 + \|\|ea0 \|\|B,r z 2 )c − 1 νs (dz)ds . R Using a similar analysis is used in proof of Proposition 3.2 [17], it follows from [8] that Z t h i 0 −1 −Xt K̃0,t = \|\|ea \|\|B,r α0 \|\|e\|\|B,r e −η e−(Xt −Xu ) du − 1 . 0 It follows from (6.5), (6.6) and (6.7) that \|\|Yt \|\|B,r < Ỹt , for all t ≥ 0. Thus E(St2 ) < ∞ and E(St4 ) < ∞ imply E(Yt ) < ∞ and cov(Yt ) < ∞, respectively. In the proof of Theorem 3.1(a) we have seen that (3.2) implies that the sequence Ỹmτ converges in distribution to a finite random vector Ũ which of the vector Ũ is the unique solution of the random equation d Ũ = J̃0 Ũ + K̃0 , d where J̃0 , K̃0 = J̃0,τ , K̃0,τ and Ũ is independent of J̃0 , K̃0 . It follows from (6.4), (6.6) and (6.7) that U ≤ Ũ. Thus E(St2 ) < ∞ and E(St4 ) < ∞ imply E(U) < ∞ and cov(U) < ∞, respectively. 24 P6: Proof of Lemma 4.2 Using (5.1) and independence Ymτ and (Jmτ,s1 +mτ , Kmτ,s1 +mτ ), we obtain E(Yt ) = E(Jmτ,s1 +mτ )E(Ymτ ) + E(Kmτ,s1 +mτ ) = E(J0,s1 )E(U) + E(K0,s1 ) d where the last equality follows from that (Jmτ,s1 +mτ , Kmτ,s1 +mτ ) = (J0,s1 , K0,s1 ) and assumption of section (c) of the Theorem 3.1. For computing cov(Yt , Yt+h ) it is sufficient to obtain E(Yt+h Yt0 ). It will therefore be followed from recursion equations which used in the proof of Theorem 4.1 that Yt = Jmτ,t Ymτ + Kmτ,t , Yt+h = Jnτ,t+h J(n−1)τ,nτ · · · Jmτ,(m+1)τ Ymτ h + Knτ,t+h + Jnτ,t+h K(n−1)τ,nτ + n−m−1 X Jnτ,t+h · · · J(n−i)τ,(n−i+1)τ K(n−i−1)τ,(n−i)τ i i=1 The relation (4.2) follows from independence the sequence (Jkτ,s+kτ , Kkτ,s+kτ ) for any k ∈ N0 and s ∈ [0, τ ] and also independence Ymτ from this sequence for any k ≥ m. P7: Proof of Corollary 4.3 Since for fixed t, almost surely Vt = Vt+ = α0 + a0 Yt , we have the expected value and covariance function volatility process from (2.6). P8: Proof of Proposition 4.4 (a) We imitate the proof of Theorem 6.1 of Brockwell, Chadraa, and Lindner [8]. Since S is a martingale with zero mean, we have (4.3). It follows from Ito isometry for square integrable martingales as integrators (e.g. [20], IV 27) that Z t+h+p (p) (p) Vs I[t,t+p) (s)I[t+h,t+h+p (s)d[S, S]s = 0, E Gt Gt+h = E 0 and hence (4.4) follows. (b) It follows from partial integration that Z t+p (p) 2 (Gt ) = 2 Gs− dGs + [G, G]t+p t+ t+ Z t+p X p =2 Gs− Vs dSs + Vs (∆Ss )2 . t t<s≤t+p By similar analysis is used in (a), the compensation formula and [11] we have Z t+p Z X (p) 2 2 E (Gt ) = E Vs (∆Ss ) = E(Vs )z 2 νs (dz)ds t t<s≤t+p 25 R (6.8) From Remark 3.2 the relation (4.5) follows. For proof of (4.6), Since the increments of S on the interval (t, t + p] are independent of Ft+p and S has expectation 0, it follows that Z t+p p Et+p Gs− Vs dSs = 0. t Thus it follows from the compensation formula and (6.2) that X (p) Et+p (Gt+h )2 = Et+p α0 + a0 Jt+p,s− Yt+p + a0 Kt+p,s− (∆Ss )2 t+h<s≤t+h+p Z t+h+p Z = t+h α0 + a0 E(Jt+p,s− )Yt+p + a0 E(Kt+p,s− ) z 2 νs (dz)ds, R therefore (p) (p) (p) (p) (p) (p) cov (Gt )2 , (Gt+h )2 = E (Gt )2 Et+p (Gt+h )2 − E (Gt )2 E (Gt+h )2 , (p) and by Remark 3.2 and (2.6) we have (4.6). To calculate cov Yt+p , (Gt )2 , partial integration (6.8) to get Z t+p Z t+p p (p) 2 Vs d[S, S]s . cov Yt+p , (Gt ) = 2cov Yt+p , Gs− Vs dS̄s + cov Yt+p , t+ t Rt √ To calculate the first term, let It := 0 Gs− Vs dSs . We know E(It ) = 0 for all t ≥ 0. Therefore Z t+p p cov Yt+p , Gs− Vs dSs = E It+p Yt+p − E(Jt,t+p )E It Yt − E(It )E(Kt,t+p ). t From [8], partial integration and substituting dVt+ = a0 BYt dt + α0 Vt d[S, S]t it follows that Z t Z t E(It Vt+ ) = E Is− dVs+ + E Vs dIs + E [V+ , I]t 0 0 Z t Z tZ 0 =aB E(Is− Ys )ds + αq E(Is− Vs )z 2 νs (dz)ds 0 Z0 t ZR t p p +E Gs− Vs Vs dSs + αq E Gs− Vs Vs dMs , 0 0 where Ms := )3 is a locally integrable martingale, with mean zero as a 0<u≤s (∆SsR result of assumption that R z 3 νs (dz) = 0, for all s ≥ 0. Thus, using the fact that Rt √ E 0 Gs− Vs Vs dSs = 0, E(It Vt+ ) = a0 E(It Yt ) and that Is Ys = Is− Ys = Is− Ys− almost surely for fixed s, so we have Z t Z tZ 0 0 0 a E(It Yt ) = a B E(Is Ys )ds + α0 a E(Is Ys )z 2 νs (dz)ds, P 0 0 26 R The equality holds for any vector a, hence Z t Z tZ E(It Yt ) = B E(Is Ys )ds + α0 E(Is Ys )z 2 νs (dz)ds. 0 0 R To calculate the second term of the covariance, it follows from [8] and (2.7) that Z t+p Z t+p 0 0 Ys− dsB 0 )e cov Yt+p , Vs d[S, S]s = cov Yt+p , (Yt+p − Yt − t+ t+ Z t+p = cov(Yt+p ) − cov(Yt+p , Yt ) − cov(Yt+p , Ys− )dsB 0 e. t+ References [1] W. R. Bennett (1958). Statistics of regenerative digital transmission. Bell System Technical Journal, 37, 1501-1542. [2] A. Bibi and I. Lescheb (2012). On general periodic time-varying bilinear processes. Economics Letters, 114, 353357. [3] T. Bollerslev (1986). Generalized Autoregressive Conditional Heteroskedasticity. Journal of Econometrics, 31, 307327. [4] T. Bollerslev, A.J. Patton and W. 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AdaDNNs: Adaptive Ensemble of Deep Neural Networks for Scene Text Recognition Chun Yang† , Xu-Cheng Yin†∗ , Zejun Li† , Jianwei Wu† , arXiv:1710.03425v1 [cs.CV] 10 Oct 2017 † Chunchao Guo‡ , Hongfa Wang‡ , and Lei Xiao‡ Department of Computer Science and Technology, University of Science and Technology Beijing, Beijing, China ‡ TEG, Tencent Co. LTD, Shenzhen, China ∗ Corresponding author: [email protected] Abstract Recognizing text in the wild is a really challenging task because of complex backgrounds, various illuminations and diverse distortions, even with deep neural networks (convolutional neural networks and recurrent neural networks). In the end-to-end training procedure for scene text recognition, the outputs of deep neural networks at different iterations are always demonstrated with diversity and complementarity for the target object (text). Here, a simple but effective deep learning method, an adaptive ensemble of deep neural networks (AdaDNNs), is proposed to simply select and adaptively combine classifier components at different iterations from the whole learning system. Furthermore, the ensemble is formulated as a Bayesian framework for classifier weighting and combination. A variety of experiments on several typical acknowledged benchmarks, i.e., ICDAR Robust Reading Competition (Challenge 1, 2 and 4) datasets, verify the surprised improvement from the baseline DNNs, and the effectiveness of AdaDNNs compared with the recent state-ofthe-art methods. Scene text is widely used as visual indicators for navigation and notification, and text recognition from scene images and videos is one key factor for a variety of practical applications with reading in the wild (Ye and Doermann 2015; Yin et al. 2016; Tian et al. 2017), such as assisting for visually impaired people (Goto and Tanaka 2009; Sanketi, Shen, and Coughlan 2011), real-time translation (Shi and Xu 2005; Fragoso et al. 2011), user navigation (Minetto et al. 2011), driving assistance systems (Wu, Chen, and Yang 2005), and autonomous mobile robots (Létourneau et al. 2003). Scene text (cropped word) recognition methods can be generally grouped into segmentation-based word recognition and holistic word recognition. Typical segmentationbased approaches over-segment the word image into small segments, combine adjacent segments into candidate characters, classify them using convolutional neural networks (CNNs) or gradient feature-based classifiers, and find an approximately optimal word recognition result (Bissacco et al. 2013; Jaderberg, Vedaldi, and Zisserman 2014). Because of complex backgrounds and diverse distortions, character segmentation is another more challenging task. Thereby, holistic word recognition approaches with deep neural networks are more impressive for text reading in the wild. Copyright c 2017-2018, All rights reserved. Word spotting, the direct holistic approach, usually calculates a similarity measure between the candidate word image and a query word (Jaderberg et al. 2016; Gordo 2015). Sequence matching, the indirect holistic approach, recognizes the whole word image by embedding hidden segmentation strategies. For example, Shi et al. constructed an endto-end training deep neural network for image-based sequence recognition (scene text recognition) (Shi, Bai, and Yao 2017). However, there are a variety of grand challenges for scene text recognition (see samples in Fig. 1), even with recent deep neural networks (DNNs), where additional characters will be probably identified for text distortions and complex backgrounds, some characters are wrongly recognized for changing illuminations and complex noises, and characters are sometimes missed for low resolutions and diverse distortions. Figure 1: Some challenging examples (from 2015 ICDAR Robust Reading Competition Challenge 4 dataset) of scene text images which are incorrectly recognized by the baseline DNNs (see related descriptions in Experiments). The captions show the recognized text (left) versus the ground truth (right): additional characters, wrong characters and missing characters in target words. Stochastic Gradient Descent (SGD) (Bottou 2010) and its variants have become the defacto techniques for optimizing DNNs, where SGD always leads to local minima, even though the popularity of SGD can be attributed to its ability to avoid spurious saddle-points and local minima (Dauphin et al. 2014). There are a plenty number of (more than million) possible local minima in DNNs (Kawaguchi 2016), and local minima with flat basins are supposed to generalize better in the learning system (Keskar et al. 2017). As a result, although different local minima often have similar error rates, the corresponding neural networks in DNNs tend to make different mistakes. This diversity and complementarity can be exploited via classifier ensemble (Huang et al. 2017). There are two major ways for ensemble of deep neural networks. On the one hand, different learning systems with DNNs are first trained independently, and then the final system is a trivial ensemble of these different deep learning architectures via majority voting or averaging. For example, most high profile competitions in ImageNet 1 and Kaggle 2 are won by such ensemble techniques. Because of the huge computation complexity, this ensemble becomes uneconomical and impossible for most researchers in the universities and even in the small companies. On the other hand, one learning system with DNNs is first trained, and then the final ensemble selects and combines neural network components 3 in this only one system without incurring any additional training cost. Huang et al. proposed such an ensemble technique, called as Snapshot Ensembling, where a specific optimization strategy is designed to train DNNs and “model snapshots” (neural network components) in all cycles are combined for the final ensemble in the learning procedure (Huang et al. 2017). However, how to design the specific and effective optimization algorithms for DNNs is also a challenge. In this paper, we propose a new and adaptive ensemble of deep neural networks (AdaDNNs) in the most simplest way, i.e., given trained neural networks (of all iterations) from a learned DNNs system 4 , a subset of neural network components are simply selected and adaptively combined to perform the final predictions. And the ensemble is formally formulated as a Bayesian framework for classifier weighting and combination. We argue that because of the diversity and complementarity in DNNs with SGD, AdaDNNs via ensembling with diversity can improve robust performance of the final learning system. On the same time, because of the high accuracy of components in DNNs, AdaDNNs via combination with accurate neural network components can improve precision performance of the final classification system. A variety of experiments on several acknowledged benchmarks, i.e., ICDAR Robust Reading Competition (Challenge 1, 2 and 4) datasets, have shown that the simple but effective AdaDNNs improves largely from the baseline DNNs. Moreover, our proposed approach has the 1 www.image-net.org. www.kaggle.com. 3 The neural network component means the resulting DNN of each iteration in the whole training procedure. 4 Here, the DNNs system can be trained with conventional optimization algorithms (Bottou, Curtis, and Nocedal 2016), or even with the specific algorithms, e.g., Snapshot Ensembling (Huang et al. 2017). 2 top performance compared with the latest state-of-the-art methods. Related Work Recognizing text in scene videos attracts more and more interests in the fields of document analysis and recognition, computer vision, and machine learning. The existing methods for scene text (cropped word) recognition can be grouped into segmentation-based word recognition and holistic word recognition. In general, segmentation-based word recognition methods integrate character segmentation and character recognition with language priors using optimization techniques, such as Markov models (Weinman et al. 2014) and CRFs (Mishra, Alahari, and Jawahar 2012; Shi et al. 2013). In recent years, the mainstream segmentationbased word recognition techniques usually over-segment the word image into small segments, combine adjacent segments into candidate characters and classify them using CNNs or gradient feature-based classifiers, and find an approximately optimal word recognition result using beam search (Bissacco et al. 2013), Hidden Markov Models (Alsharif and Pineau 2014), or dynamic programming (Jaderberg, Vedaldi, and Zisserman 2014). Word spotting (Manmatha, Han, and Riseman 1996), a direct holistic word recognition approach, is to identify specific words in scene images without character segmentation, given a lexicon of words (Wang and Belongie 2010). Word spotting methods usually calculate a similarity measure between the candidate word image and a query word. Impressively, some recent methods design a proper CNN architecture and train CNNs directly on the holistic word images (Jaderberg et al. 2014; Jaderberg et al. 2016), or use label embedding techniques to enrich relations between word images and text strings (Almazan et al. 2014; Gordo 2015). Sequence matching, an indirect holistic word recognition approach, recognizes the whole word image by embedding hidden segmentation strategies. Shi et al. constructed an endto-end train deep neural network for image-based sequence recognition (scene text recognition), where a convolutional recurrent neural networks framework (CRNN) is designed and utilized (Shi, Bai, and Yao 2017). In this paper, a similar CRNN architecture is used in AdaDNNs for recognizing scene text sequently and holistically. Classifier ensemble can be mainly divided into two categories. The first one aims at learning multiple classifiers at the feature level, where multiple classifiers are trained and combined in the learning process, e.g., Boosting (Freund and Schapire 1997), Bagging (Breiman 1996), and Rotation Forest (Rodriguez, Kuncheva, and Alonso 2006). The second tries to combine classifiers at the output level, where the results of multiple available classifiers are combined to solve the targeted problem, e.g., multiple classifier systems (classifier combination) (Zhou 2012; Yin et al. 2014). AdaDNNs in this paper follows the second one. Namely, given multiple classifiers (neural network components sequently learned in DNNs), AdaDNNs is constructed by combining intelligently these component classifiers within a Bayesian-based formulation framework. Adaptive Ensemble of Deep Neural Networks As we have known, both SGD and batch optimization can lead to different local minima in DNNs, and neural network components are always with diversity and complementarity. Conventionally, there are tens of thousands of iterations and also neural network components in the learning system of DNNs. Considering the acceptable computation complexity in the testing procedure, one thing is to quickly select a small subset of neural network components in different training iterations. At the same time, considering the high accuracy requirement, another thing is to adaptively combine this subset of neural network components and construct a final classification system. In the following, the unified framework of AdaDNNs is first formulated. Next, the detail procedure of AdaDNNs is then described. There are two key issues for optimizing Eq. 3. The first one is the calculation of W (y, hi (x)). As mentioned above, P (y\|hi , x) is the distribution of describing the correlation between decision y and hi (x). Thus, W (y, hi (x)) can be derived from y, hi (x) and the distance between y and hi (x). Here, W (y, hi (x)) is assumed to be computed as W (y, hi (x)) = I(y = hi (x)) + U (y) ∗ V (y, hi (x)) (4) where both U (∗) and V (∗, •) are functions. I(y = hi (x)) returns 1 when y = hi (x); otherwise, I(y = hi (x)) = 0. For the scene text recognition task, on the one hand, with a given dictionary 5 , U (y) can be calculated as U (y) = { 1 0 y ∈ Dict y∈ / Dict (5) Unified Framework To formulate the ensemble decision, the individual classifier decisions can be combine by majority voting, which sums the votes for each class and selects the class that receives most of the votes. While the majority voting is the most popular combination rule, a major limitation of majority voting is that only the decision of each model is taken into account without considering the distribution of decisions. In particular, all the possible models in the hypothesis space could be exploited by considering their individual decisions and the correlations with other hypotheses. Here, we use a Bayesian-based framework to combine classifiers. Given a sample x and a set H of independent classifiers, the probability of label y can be estimated by a Bayesian Model (BM) as X P (y\|H, x) = P (y\|hi , x)P (hi \|x) (1) On the other hand, the correlation between y and hi (x) can be assumed by the function V of Cost Levenshtein Distance (CLD). In the traditional Levenshtein Distance, the cost of any two different characters is always 1. However, in spelling correction, the cost of two characters with similar shape tends to have a smaller distance. In this paper, we statistics the frequencies of different character pairs at the same location from the label and the hypothesis on the validation set (bootstrapped from the training set in Experiments), and calculate the cost of two different characters (a and b) as cost(a, b) = 1 − P (a\|b) (6) Note that if both y and hi (x) are from the given dictionary, then they will have a competitive relationship with each other. Thus, V (y, hi (x)) can be calculated with hi ∈H where P (y\|hi , x) is the distribution of describing the correlation between decision y and hi (x), and P (hi \|x) denotes the posterior probability of model hi . The posterior P (hi \|x) can be computed as P (hi \|x) = P P (D\|hi )P (hi ) hi ∈H P (D\|hi )P (hi ) (2) where P (hi ) is the prior probability of classifier hi and P (D\|hi ) isP the model likelihood on the training set D. Here, P (hi ) and hi ∈H P (D\|hi )P (hi ) are assumed to be a constant in Eq. 2. Therefore, BM assigns the optimal label y to y ∗ according to the following decision rule, i.e., P (y ∗ ) = argmaxy PP(y\|H, x) = argmaxy Phi ∈H P (y\|hi , x)P (hi \|x) = argmaxy P hi ∈H P (y\|hi , x)P (D\|hi ) = argmaxy 2P hi ∈H P (y\|hi , x)P (D\|hi ) − P (D) = argmaxy Phi ∈H (2P (y\|hi , x) − 1)P (D\|hi ) = argmaxy hi ∈H W (y, hi (x))P (D\|hi ) (3) where W (y, hi (x)) is a function of y and hi (x). By multiplying the scaling factor λ > 0, W (y, hi (x)) can have a different range in R. hi (x) ∈ Dict hi (x) ∈ / Dict (7) where F is a function of the CLD between y and hi (x). By a heuristic approach, the values of F can be empirically assigned at the multiple integral points, and the values at other points can be calculated by the piecewise linear interpolation. An example of F is shown in Fig. 2. In general, F has a small range, e.g., [1.5, 1.5] in Fig. ??. So, the obtained weights from Eq. 4 are convenient for linear combination of classifiers. The second issue is about generating voting candidates (more probable labels of the hypotheses). Obviously, the ground truth doesn’t always appear in the decisions made by H. It is necessary to find an effective way to generate good candidates from all the decisions, i.e., to find a more probable label yi (x) from the existed initial label yi0 (x) of hypothesis hi . Generally speaking, a good candidate means it has a small edit distance with most of the hypotheses. Following this idea, we propose an algorithm to semantically generate voting candidates (see Algorithm 1). V (y, hi (x)) = { F (−CLD(y, hi (x))) F (CLD(y, hi (x))) 5 In our experiments, a 90k word dictionary from (Jaderberg et al. 2016) is used as the given dictionary. batch orderings will converge to different solutions. Those snapshots often have the similar error rates, but make different mistakes. This diversity can be exploited by ensembling, in which multiple snapnots are average sampling and then combined with majority voting. Focusing on scene text recognition, the CRNN model (Shi, Bai, and Yao 2017) is used to generate base classifiers (neural network components) as our text recognizer. CRNN uses CTC (Graves et al. 2006) as its output layer, which estimates the sequence probability conditioned on the input image, i.e. P (h\|x), where x is the input image and h represents a character sequence. Figure 2: An example of F of describing the relationship between V (in Y-axis) and the Cost Levenshtein Distance (in X-axis). Algorithm 1: Generating Voting Candidates. Input: H = {h1 , h2 , ..., hL }: the base classifier set, \|H\| = L. Y0 : the initial decisions made by H. ED: the measurement function of the pairwise distance. θ: the upper bound of the distance between the candidate and the hypothesis. Output: Y : the voting candidate set. Parameter: H ? : a subset of H, ∀h?i , h?j ∈ H ? , ED(h?i , h?j ) ≤ 2θ. Procedure: 1: Y = ∅. 2: For each H ? ⊂ H; 3: For each y ∈ Y0 : 4: If maxh?i ∈H ? ED(y, h? (x)) ≤ θ: 5: Y = Y ∪ {y}. 6: End 7: End In Algorithm 1, the searching process of H ? is an implicit computational way for P (D\|hi ). In our experiments, a special simple case of algorithm 1 is used, where during the voting candidates generation process, Y0 is initialized only by H, the upper bound is set from θ to inf, and P (D\|hi ) is assumed to be a constant. AdaDNNs Algorithm Within the above framework, the procedure of AdaDNNs for scene text recognition includes three major steps, i.e., base classifiers generation, classifier combination, and ensemble pruning. Base Classifiers Generation Ensembles work best if the base models have high accuracy and do not overlap in the set of examples they misclassify. Deep neural networks (DNNs) are naturally used as a base classifier generator for ensembles. On the one hand, DNNs have dramatically improved the state-of-the-art in many domains, such as speech recognition, visual object recognition and object detection, by being composed of multiple processing layers to learn representations of data with multiple levels of abstraction. On the other hand, during the training phase of one individual deep neural network, two snapshots with different mini- Classifier Combination for AdaDNNs The core of AdaDNNs is to calculate y ∗ (by Eq. 3), i.e., the calculation of F , which is a function of distance between y and hi (x). Here, F is represented by the set of values at the multiple integral points. These values are assigned with the highest recognition rate on the validation set. The detail procedure of the AdaDNNs ensemble is shown in Algorithm 2. Algorithm 2: AdaDNNs (classifier combination). Input: H = {h1 , h2 , ..., hL }: the base classifier set, \|H\| = L. Dict: the given dictionary. F : a function of distance between y and hi (x). Parameter: Y : the voting candidates set generated by Algorithm 1. Output: y ∗ : the label of prediction. Procedure: 1: Initialize Y by H and Dict. 3: For y ∈ Y : 4: Calculate P (y\|H, x) through Eq. 2. 5: End 6: Calculate y ∗ through Eq. 3. AdaDNNs Pruning In classifier ensemble, pruning can generally improve the ensemble performance. Here, we use Genetic Algorithm (GA) to pruning the ensemble. GA is a meta heuristic inspired by the process of natural selection that belongs to the larger class of evolutionary algorithms. GAs are commonly used to generate high-quality solutions for optimization and search problems by relying on bioinspired operators such as mutation, crossover and selection. In AdaDNNs pruning, firstly, a population of binary weight vectors is randomly generated, where 1 means the classifier is remained. Secondly, the population is iteratively evolve where the fitness of a vector w is measured on the V validation set V , i.e., f (w) = Rw (R stands for the recognition rate). Finally, the ensemble is correspondingly pruned by the evolved best weight vector w∗ . Experiments To evaluate the effectiveness of the proposed AdaDNNs method, a variety of experiments for text (cropped word) recognition are conducted on acknowledged benchmark datasets. We first focused on the most challenging task, i.e., incidental scene text recognition (ICDAR Robust Reading Competition Challenge 4), trained our AdaDNNs learning system (on both the synthetic dataset from (Jaderberg et al. 2016) and the training set of Challenge 4), and performed comparative experiments. Then, we also conducted experiments of this learned AdaDNNs on other text recognition tasks, i.e., focused scene text recognition and borndigital text recognition (ICDAR Robust Reading Competition Challenge 1 and 2), and checked the generalization of AdaDNNs. Here, the baseline DNNs model, CRNN, is same to the one in (Shi, Bai, and Yao 2017). The official metrics in ICDAR 2011/2013/2015 Robust Reading Competition (Shahab, Shafait, and Dengel 2011; Karatzas et al. 2013; Karatzas et al. 2015) are used. Figure 3: Challenging samples of scene text from COCO-text which are correctly recognized (with C.R.W. upper) by AdaDNNs: “GEMS”, “mgennisgal”, “RGAO”, “RAILROAD”, “UNITED”, “Kappa”, “XMAS”, “ZOOM”, “YouTube”, “YORK”, “WALK”, “WPRD”, “WHEN”, “YEAR”, and “WISCONSIN”. Experiments with Incidental Scene Text Recognition The ICDAR 2015 Robust Reading Competition Challenge 4 database (Karatzas et al. 2015) is a widely used and highly competitive benchmark database for scene text recognition within complex situations in the recent 3 years. The public dataset includes a training set of 1, 000 images and a test set of 500, with more than 10, 000 annotated text regions (cropped words). Because of complex backgrounds, various illuminations and diverse distortions, this incidental scene text recognition topic is a very challenging task. In our experiments, a variety of methods are conducted and compared, i.e., the baseline DNNs, AdaDNNs, AdaDNNs pruning, the winning participation method in the official competition (marked as bold words), and the latest top submissions of the Robust Reading Competition (RRC) website 6 in 2017 (marked as italic words). Table 1: Comparative results on 2015 ICDAR Challenge 4 dataset (incidental scene text recognition), where the comparative results are from the RRC website. Date Method T.E.D 2017/7/6 2017/7/6 2017/6/29 2015/4/1 - Baidu IDL v3 HIK OCR v3 HKU-VisionLab MAPS Baseline DNNs AdaDNNs AdaDNNs Pruning 211.59 191.25 258.59 1,128.01 384.76 251.98 224.7 C.R.W (%) 80.02 78.29 72.03 32.93 60.18 76.31 79.78 T.E.D. (upper) 171.15 158.84 212.17 1,068.72 303.77 185.36 147.11 C.R.W. (upper) 82.33 80.12 74.19 33.90 64.90 80.55 84.21 As can be seen from Table 1, our proposed AdaDNNs is much better than the baseline DNNs. For example, for the measure of “C.R.W (upper)”, AdaDNNs has a surprised improvement, i.e., from 64.90% to 80.55%. That is to say, the adaptive ensemble of DNNs in a simple but effective strategy can largely improved the performance from the original baseline DNNs. Moreover, compared with the latest top submissions (e.g., “Baidu IDL v3” and “HIK OCR v3”), our method, AdaDNNs Pruning, has the best performance with “C.R.W (upper)”, i.e., 84.21%. We also perform experiments on the COCO-text dataset (Veit et al. 2016), a similar challenging but largescale incidental scene text dataset. Images in this dataset are from the MS COCO dataset that contain text (63, 686 images with 173, 589 text regions). ICDAR2017 Robust Reading Challenge on COCO-Text is holding and will be released 6 http://rrc.cvc.uab.es. in ICDAR 2017. So, the comparative results of AdaDNNs, AdaDNNs Pruning and the baseline DNNs are only on the validation set; they are 58.08%, 66.07%, and 66.27%, respectively. Some scene text recognition samples for COCOtext are shown in Fig. 3. Experiments with Focused Scene Text Recognition and Born-Digital Text Recognition In order to investigate the generalization of AdaDNNs, we directly use the trained AdaDNNs system above (for 2015 ICDAR Challenge 4), and perform experiments on 2013 ICDAR Challenge 2 (cropped word recognition) dataset. The Challenge 2 dataset contains 1, 015 ground truths cropped word images. In our experiments, a variety of methods are conducted and compared, i.e., the baseline DNNs, AdaDNNs, AdaDNNs pruning, the winning participation method in the official competition (marked as bold words), the top three results in published papers, and the latest top submissions of the RRC website in 2017 (marked as italic words). Table 2: Comparative results on 2013 ICDAR Challenge 2 dataset (focused scene text recognition), where the comparative results without publications are from the RRC website. Date Method T.E.D 2017/8/14 2017/7/28 2017/2/24 2016 2016 2017 2013/4/6 - TencentAILab Tencent Youtu HIK OCR CNN (Jaderberg et al. 2016) RARE (Shi et al. 2016) CRNN (Shi, Bai, and Yao 2017) PhotoOCR Baseline DNNs AdaDNNs AdaDNNs Pruning 42 48.12 64.95 – – – 122.75 306.43 193.51 182.7 C.R.W (%) 95.07 92.42 90.78 – – – 82.83 75.34 83.20 85.21 T.E.D. (upper) 39.35 40.37 42.31 – – – 109.9 282.35 170.13 164.38 C.R.W. (upper) 95.34 93.42 93.33 90.8 88.6 89.6 85.30 78.63 86.67 88.13 Similarly, AdaDNNs is much better than the baseline DNNs, e.g., the measure of “C.R.W (upper)” increases from 78.63% to 86.67%. Surprisedly, only trained for another task (Challenge 4), the AdaDNNs (AdaDNNs Pruning) has a competitive performance on a new dataset (Challenge 2 dataset), compared with the recent published methods (e.g., CRNN (Shi, Bai, and Yao 2017)), and even with the latest submission results. Apart from the above experiments on text recognition from scene images (ICDAR Robust Reading Competition Challenge 2 and 4), we also directly perform the learned AdaDNNs on the born-digital images track (Challenge 1). Though born-digital images are not scene images, they have similar challenging issues for text recognition, e.g., complex backgrounds, low resolution and various colors. We also compare AdaDNNs (AdaDNNs Pruning) with the baseline DNNs, the winning participation method in the official competition (marked as bold words), and the latest top submissions of the RRC website (marked as italic words). The similar conclusions are drawn. Firstly, AdaDNNs improves largely compared with the baseline DNNs (from 84.50 to 92.22 for “C.R.W (upper)”). Secondly, AdaDNNs has a comparative performance with the latest submission results (e.g., “Dahua OCR v1” with 92.49% in 2017/9/1). Table 3: Comparative results on ICDAR Challenge 1 dataset (born-digital text recognition), where the comparative results are from the RRC website. Date Method T.E.D 2017/8/22 2017/7/21 2017/9/1 2013/4/6 - Tecent Youtu TecentAILab Dahua OCR v1 PhotoOCR Baseline DNNs AdaDNNs AdaDNNs Pruning 17.51 18.77 57.47 103.41 87.7 55.44 55.64 C.R.W (%) 96.80 96.18 91.31 82.21 82.42 89.58 89.53 T.E.D. (upper) 13.67 12.91 42.87 87.19 72.32 39.61 39.81 C.R.W. (upper) 97.29 97.22 92.49 85.41 84.50 92.22 92.17 We fully believe that if AdaDNNs (AdaDNNs Pruning) performs re-training on ICDAR Challenge 1 and Challenge 2 datasets, the performance will correspondingly be improved and obtain a more impressive results compared with the latest submission systems. This is also a near issue for our future work. Conclusion and Discussion A variety of DNNs based methods have been proposed and are still being investigated in the literature for scene text recognition because of the grand challenges, e.g., complex backgrounds, various illuminations and diverse distortions. In order to fully take advantage of the complementary diversity and the high accuracy of neural network components in DNNs, an adaptive ensemble of deep neural networks (AdaDNNs) is proposed to simply select and adaptively combine neural networks in the whole training procedure. Comparative experiments of scene text (cropped word) recognition showed that AdaDNNs achieves a remarkable increase in the final performance (more than 10%) compared with the baseline DNNs. Note that the DNNs methods have dramatically improved the state-of-the-art in object detection, object recognition, speech recognition and many other domains. 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Yang et al. BMC Systems Biology 2014, 8(Suppl 4):S7 http://www.biomedcentral.com/1752-0509/8/S4/S7 RESEARCH Open Access Microbial community pattern detection in human body habitats via ensemble clustering framework Peng Yang1, Xiaoquan Su2, Le Ou-Yang3, Hon-Nian Chua1, Xiao-Li Li1, Kang Ning2* From Asia Pacific Bioinformatics Network (APBioNet) Thirteenth International Conference on Bioinformatics (InCoB2014) Sydney, Australia. 31 July - 2 August 2014 Abstract Background: The human habitat is a host where microbial species evolve, function, and continue to evolve. Elucidating how microbial communities respond to human habitats is a fundamental and critical task, as establishing baselines of human microbiome is essential in understanding its role in human disease and health. Recent studies on healthy human microbiome focus on particular body habitats, assuming that microbiome develop similar structural patterns to perform similar ecosystem function under same environmental conditions. However, current studies usually overlook a complex and interconnected landscape of human microbiome and limit the ability in particular body habitats with learning models of specific criterion. Therefore, these methods could not capture the real-world underlying microbial patterns effectively. Results: To obtain a comprehensive view, we propose a novel ensemble clustering framework to mine the structure of microbial community pattern on large-scale metagenomic data. Particularly, we first build a microbial similarity network via integrating 1920 metagenomic samples from three body habitats of healthy adults. Then a novel symmetric Nonnegative Matrix Factorization (NMF) based ensemble model is proposed and applied onto the network to detect clustering pattern. Extensive experiments are conducted to evaluate the effectiveness of our model on deriving microbial community with respect to body habitat and host gender. From clustering results, we observed that body habitat exhibits a strong bound but non-unique microbial structural pattern. Meanwhile, human microbiome reveals different degree of structural variations over body habitat and host gender. Conclusions: In summary, our ensemble clustering framework could efficiently explore integrated clustering results to accurately identify microbial communities, and provide a comprehensive view for a set of microbial communities. The clustering results indicate that structure of human microbiome is varied systematically across body habitats and host genders. Such trends depict an integrated biography of microbial communities, which offer a new insight towards uncovering pathogenic model of human microbiome. Background Metagenomic background The human body is a content that complex microbial communities are living inside and on. This microbiome occupies body habitats and endows us with ecosystem functions, such as nutrition, pathogen resistance and * Correspondence: [email protected] 2 Computational Biology Group of Single Cell Center, Shandong Key Laboratory of Energy Genetics and CAS Key Laboratory of Biofuels, Qingdao Institute of Bioenergy and Bioprocess Technology, Chinese Academy of Science, Qingdao 266101, China Full list of author information is available at the end of the article immune system development [1,2], to help maintain our health. Hence systematically defining the “normal” states of human microbiome is an important step towards understanding role of microbiota in pathogenesis [3]. However, the majority of microbiomes have been poorly investigated. To understand the principle of human microbiome, prior research concentrated on particular body habitats [3-8]. For example, Turnbaugh et al. [9] investigated the gut microbiome in obese and lean twins to address how host, environmental condition and diet influence the © 2014 Yang et al.; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The Creative Commons Public Domain Dedication waiver (http:// creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated. Yang et al. BMC Systems Biology 2014, 8(Suppl 4):S7 http://www.biomedcentral.com/1752-0509/8/S4/S7 microbial components. Grice et al. [10] targeted human skin microbiome to characterize its topological and personal variations within multiple sites. Bik et al. [11]’s research indicated the distinctness of microbial structure on oral cavity and tongue. However, human microbial habitats are not isolated with one another; instead they reveal community structure correlation across body habitats [12]. In this case, ensemble of different habitat samples could bring global and full-scale insights into microbiome. Recent studies had aggregated microbial samples from different body habitats to perform a comprehensive study. Costello et al. [13] surveyed the microbiomes that were gathered from 27 body habitats of nine adults. Mitreva [12] carried out the extensive sampling on 18 body habitats from 242 individuals. In order to establish a global insight of human microbiome, they built a “whole-body” microbial similarity network where the nodes were consisted of metagenomic samples from multiple human body sites and the edges as pair-wise phylogenetic similarity of samples were measured in terms of their shared evolutionary history. Clustering approaches [14] had been applied on this large-scale similarity network to group samples that shared more similar phylogenetic structures with each other within the clusters than other ones. From these clusters, researchers could infer how microbial patterns were affected by body habitat, host gender and environmental condition with time. Costello et al. [13] proposed a hierarchical clustering algorithm on a microbial community network and found out personal microbiota relatively stable within habitats over time. Turnbaugh et al. [9] identified two distinct functional modules on gut microbiome via principal components analysis (PCA) and hierarchical clustering algorithm, and experimental results disclosed that microbiome within same clusters carried out similar ecosystem-level functions. Mitreva [12] adopted a centroid-based clustering algorithm and discovered the covariation and co-exclusion of microbiome between different habitats. Current Limitations Clustering approach aims to group metagenomic samples with similar phylogenetic patterns. It can be achieved by various algorithms that differ significantly in terms of computational principles and measures, by which each generated clustering results can be viewed as taking a different “look” through data (as shown in Table 1). However, most of prior studies employ one particular clustering approach, by which the clustering outputs tend to be specific towards the criterion of the proposed approach. For example, density-based clustering algorithm groups samples that are densely connected in similarity network. However, true microbial Page 2 of 12 communities are not limited to densely connected structures; samples with sparsely microbial structure widely exist in the lake [15]. Graph partition-based clustering such as MCL [16] and K-means clustering [17] explores the best partition of a network. But these algorithms do not allow the overlaps between clusters. Therefore, they are unable to discover shared microbe between two communities, such as some species that could adapt in multi-environmental conditions like microbial mats and biofilms. Hierarchical clustering algorithm [18] learns the hierarchical structure of a network, which has been used in [13], but hierarchical structure is determined by local optimization criterion as such there is no global objective function, which might lead to small clusters with only part of similar samples. Distribution-based clustering approach, like expectation-maximization (EM) [19], identifies the clusters that follow statistical Condorcet criteria. But statistical model for microbial community remains rarely known and therefore it is difficult to evaluate reliability of the results. Advantage of proposed Ensemble clustering framework Ideally, a clustering algorithm should be able to exploit clustering patterns as comprehensive as possible. However, as we have mentioned above, few algorithms are capable of taking into consideration all factors. Different clustering algorithms may produce different partitions of the network. Given multiple clustering results, we need to explore their information and output more robust results that can exploit the complementary nature of these patterns. Ensemble clustering was proposed recently which has been successfully used to solve many community detection problems [20-23]. Thus, we use ensemble clustering framework to integrate the various kinds of clusters (here we call them base clustering results) and output more comprehensive results. In this study, we first construct a consensus matrix which measures similarity of samples based on co-occurrence of samples in base clustering results [24]. Next we apply Symmetric Nonnegative Matrix Factorization (NMF) [25] on the consensus matrix to derive clusters. Symmetric NMF provides a lower rank approximation of a nonnegative matrix, which could be easily related to the clustering of the nonnegative data. As mentioned in [25], the factorization of the consensus matrix will generate a clustering assignment matrix that could capture the cluster structure inherent in the network. Unlike prior researches that applied single cluster algorithm on particular habitat microbiome, our framework assembled clustering algorithms of different human microbiome in different body habitats. We carried out our experiments to demonstrate its capability in capturing the microbial community. Experimental Yang et al. BMC Systems Biology 2014, 8(Suppl 4):S7 http://www.biomedcentral.com/1752-0509/8/S4/S7 Page 3 of 12 Table 1 Summary of four particular clustering approaches Clustering Approaches Density-based clustering Characteristics Limitations on microbial pattern Clusters are defined as connected dense regions in the network True microbial community are not limited to densely connected structures; sparsely microbial structure still exists Graph partition- Clusters are generated via graph partitioning techniques based clustering Partition based algorithms do not allow the overlaps between clusters. Therefore, they are unable to discover shared microbe among clusters, such as some species that could adapt in multi-environmental conditions like microbial mats and biofilms Hierarchical clustering Clusters are built based on an agglomerative clustering model that shows relations between the members and groups Hierarchical structure is determined by local optimization criterion as such there is no global objective function, which might lead to small clusters with only part of similar samples Distributionbased clustering Clusters are modelled using statistical distributions Statistical models of microbial communities are still unknown and need to be further explored results showed that predicted clusters were capable of revealing the spatial and gender roles of human microbiota and eventually elaborated human microbiome biogeography, which provided new insights about disease pathogenesis of human microbiome [9,12,13]. Material and methods In this section, we first briefly introduced the experimental data, the similarity measurements of metagenomic samples and GPU based fast similarity matrix computing. Then we described the schema of ensemble clustering framework and its phases to structure microbial community. Experimental data In this work, we used 1920 metagenomic samples from the project “Moving pictures of human microbiome” [26] to build the microbial matrix and similarity network (refer to section “Similarity measurements of metagenomic samples” for details). A sample of metagenomic matrix and network were illustrated in Figure 1 and the similarity matrices of all datasets were shown in Additional file 2: Table S1. GPU-Meta-Storms [27] were performed to measure structural similarity of metagenomic samples (Efficiency of GPU-Meta-Storms is shown in Additional file 2: Figure S1). Metagenomic samples were annotated by two meta-labels: Habitat = {gut, skin, oral cavity} defined human body habitat the samples live in, while Gender = {male, female} defined the gender of host the samples inhabit. Combining the two meta-labels, each sample was partition into one of six meta-classes, they were {male & gut, male & skin, male & oral cavity, female & gut, female & skin, female & oral cavity}. Table 2 summarized the distribution of 1920 metagenomic samples on three body habitats and two host genders. Similarity measurements of metagenomic samples The scoring function of Meta-Storms [27] compared two microbial samples’ structure by calculating the maximum common component of their common phylogenetic tree Figure 1 An example of (A) similarity matrix and (B) its similarity network. In the matrix, each tile indicates a similarity value between 2 samples by colour gradient from red (high) to green (low). In the network, each node represents a sample, and edges represent similarity values in the matrix. Yang et al. BMC Systems Biology 2014, 8(Suppl 4):S7 http://www.biomedcentral.com/1752-0509/8/S4/S7 Page 4 of 12 Table 2 1920 microbial samples on six human body habitats Gut Skin Oral 1395 Male 331 698 366 Female 130 262 133 525 Total 461 960 499 1920 Total considering the b-diversity, phylogenetic distance and abundance of each species (Formula 1). The scoring function first evaluated the common abundance of each species on the leaf node, which was considered as the smaller abundance value in two samples. These abundance values were propagated to their ancestors iteratively, and the accumulative common abundance values at the root node reflected the overall similarity between the two metagenomic samples, which could be computed using Similarity (Root) defined in Formula 1. ⎧ ⎪ ⎪ Common Abundance(X) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ Similarity(X) = If X is a leaf node Common Abundance(X) If X is an internal node ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ +Similarity(X.Left) ⎪ ⎪ ⎩ +Similarity(X.Right) (1) Then we constructed the similarity matrix based on the pair-wised similarity among all sample pair (Figure 1 (A)). Exploiting the multi-core architecture of the GPU [28], Formula 1 could be invoked in parallel using a large number of threads to compute similarity between different pairs of metagenomic samples. To compute the pair-wise similarity matrix for N samples, we spawned N * N threads in the GPU such that each similarity value in the matrix was processed by an independent thread. Figure 2 Overview of the GPU based similarity matrix computing. Figure 2 illustrated the GPU computing workflow: to build the common phylogenetic tree, we first loaded and initialize abundant specie data from the file system to main memory; this data was then reloaded to the GPU for computing. When all threads of the GPU kernel had been completed (Figure 1, step 3, the key step), these values were returned back to RAM to populate the similarity matrix, which was then stored in the file system. Ensemble clustering framework In this subsection, we proposed a novel ensemble clustering framework, namely Meta-EC, to perform microbial community pattern detection. The framework consisted of two stages: a generation phase where a consensus matrix was constructed based on base clustering results and an identification phase in which a symmetric NMF-based clustering was used to detect reliable clusters from the consensus matrix. The schema of our Meta-EC algorithm was presented in Figure 3. Terminology: After computing the pair-wise similarity matrix of the metagenomic samples, we used it to construct the microbial similarity network that was reformatted as a simple undirected graph G = (V, E), where V defined a vertex set which containeed \|V\| = N vertices, and E an edge set. A vertex v ∈ V represented a metagenomic sample and a weighted edge e ∈ E represented the polygenetic structure similarity of two samples (Figure 1(B)). A cluster Ci = Vci , Eic was a subnetwork of G such that Vci ⊂ V and Eic was the set of edges induced by Vci from G. A microbial community of G was a set of predicted microbial clusters, defined as {C1,...,Cm}. Generation phase: When the similarity network was ready, a set of base clustering results were calculated by Yang et al. BMC Systems Biology 2014, 8(Suppl 4):S7 http://www.biomedcentral.com/1752-0509/8/S4/S7 Page 5 of 12 Figure 3 The schema of Meta-EC algorithm. applying four clustering algorithms (base clustering algorithms) on the similarity network with different initializations, as shown in Figure 3(A). The base clustering algorithms included EM algorithm, K-mean clustering, hierarchical clustering and density-based clustering, as present in Table 1 and Additional file 1: Section 1. A consensus matrix W was introduced to measure the co-occurrence of samples in clusters of the base clustering results. Each Wij indicated the number of base clustering results in which sample i and sample j were assigned to the same cluster, divided by the total number of base clustering results. Therefore, matrix W took into consideration all generated clusters and reflected the co-clusters similarity between each pair of samples based on different clustering criterions. The higher the value of Wij, the more likely sample i and sample j belonged to the same cluster. Identification phase: When the consensus matrix was constructed, we applied a symmetric NMF-based clustering algorithm on this matrix to derive the clusters. The flowchart of this algorithm was shown in Figure 3 (B). The main idea of this algorithm was outlined as follows: The symmetric NMF defined in Equation (2) was suitable for network clustering based on similarity matrix W: minH≥0 D W\|HHT (2) Here D(W\|HH T ) was a predefined cost function K and K was the predefined number of clusters. H was a cluster indicator matrix in which each entry hi,k denoted the real-valued membership of sample i belonging to cluster k. So we could easily infer the clustering assignment of sample i from the i-th row of H. In this study, we used Kullback-Leibler (KL) divergence [28] as the cost function, which could be represented as: D W HHT = DKL W HHT = N i,j=1 Wij Wij log HHT ij − Wij + HHT ij (3) We chose KL-divergence as the cost function since it was free of noise parameter and had been widely used in NMF. A sample may belong to more than one cluster, but it seldom belonged to all clusters. Thus, the cluster indicator matrix H should be sparse. To achieve sparsity of Yang et al. BMC Systems Biology 2014, 8(Suppl 4):S7 http://www.biomedcentral.com/1752-0509/8/S4/S7 Page 6 of 12 the solution of H, a L1-norm regularization for H was integrated. Neglecting constants and adding the L1norm regularization for H, the modified formulation was as follows: min − H≥0 N N i=1 j=1 Wi,j log HHT i,j − HHT i,j + N K i=1 z=1 βhi,z (4) where the hyper-parameter b > 0 controlled the sparsity of H, and H ≥ 0 was the cluster indicator matrix. Solution to NMF-based Ensemble Clustering: Minimization of the cost function in equation (4) with constraints formed a constrained nonlinear optimization problem. Similar to [29,30], we adopted the multiplicative update rule [31] to estimate H, which was widely accepted as a useful algorithm in solving nonnegative matrix factorization problem. By the multiplicative update rule, we obtained the following update rules for hi,z: \|V\| j=1 hi,z ← Wi,j hj,z K l=1 hi,l hj,l hi,z 1 + hi,z \|V\| 2 2 j=1 hj,z + 0.5β (5) We iteratively updated H according to the updating rule (5) until they satisfied a stopping criterion. Let Hl be the cluster indicator matrix at iteration time l (l > 1). The algorithm was stopped whenever \|\|Hl - Hl-1\|\|1 < r, where r was a predefined tolerance parameter. Here we set r = 10-6 as the default value of tolerance parameter. In addition, the maximum of iteration time was limited to 200 iterations if the stopping criteria r was unsatisfied. In order to avoid local minimum, for random initialization, we repeated the algorithm 10 times with random initial conditions and chose the results with lowest value of the cost function (4). From cluster indicator matrix to microbial clusters: Similar to [32], we obtained microbial clusters from cluster indicator matrix H by taking the threshold τ to assign a sample to a cluster when its weight for the cluster exceeded τ. In this way, we obtained theresultant samplecluster membership matrix H∗ = h∗i,z , where h∗i,z = 1 if hi,z ≥ τ and h∗i,z = 0 if hi,z < τ . Here, h∗i,z = 1 mean sample i was assigned to detected cluster z and h* mean the final output of h. After completing these steps, we obtained the refined clusters EK that satisfied the following conditions: EK = {C1 , ..., CK } : vi ∈ CZ , if .h∗i,z = 1 (6) where i = 1,...,N and z = 1,...,K. We summarized the whole algorithm in Figure 4. Results In this section, we focused on evaluating the effectiveness of Meta-EC algorithm. Before presenting the experimental results, we first introduced our experiment design: evaluation metrics and experimental settings in Figure 4 The algorithm of Meta-EC for microbial community pattern detection. our study. Then we conducted experimental comparison between Meta-EC and base clustering approaches, and comparison between constructed consensus network and original metagenomic similarity network. Finally, from clustering results, we investigated how human microbial community was influenced by body habitat and host gender. Evaluation metrics In this work, we evaluated the effectiveness of clustering algorithms by observing how well detected clusters corresponded to the sampling information of habitats and genders (six meta-classes, refer to subsection Terminology for details). Since the true number of cluster patterns for habitat and gender was unknown, and there were no literature references to clearly mention how to determine the number of cluster patterns in either body habitat or host gender, we empirically defined reference clusters based on six meta-classes. Assuming that metagenomic samples with identical meta-classes were likely to have similar microbial structures [13], we bring the metagenomic samples with identical meta-classes into one reference cluster. Typically, the quality of the predicted clusters could be evaluated by following three quantity measures, f-measure [33], PR metrics and F-score, which could measure how well the detected clusters corresponded to reference clusters. Among these three measures, f-measure which was the harmonic mean of Precision and Recall, aimed at assessing how well the detected clusters matched reference Yang et al. BMC Systems Biology 2014, 8(Suppl 4):S7 http://www.biomedcentral.com/1752-0509/8/S4/S7 Page 7 of 12 ones at cluster level (Precision measured what fraction of the detected clusters were matched with reference ones and Recall measured what fraction of reference clusters were matched to detected clusters). PR-based metric took into account the overlap between detected and reference clusters. F-score focused on measuring whether samples within identical habitats were grouped together in the detected clusters. The value of each measure varied from 0 to 1, and the higher value indicated better match. For more details of f-measure, PR metrics and F-score, please refer to Additional file 1: Section 2. And the parameter setting in the experiments is introduced in Additional file 1: Section 3. Evaluation of clustering results generated by Meta-EC algorithm In this subsection, to evaluate the performance of MetaEC algorithm, we presented performance comparison of proposed Meta-EC algorithm with base clustering approaches and comparison of constructed consensus matrix with original microbial similarity matrix. Comparison against four base clustering approaches: To evaluate the performance of ensemble clustering approach, the accuracy of the clustering results derived from our proposed approach was compared with the ones derived from these base clustering algorithms. Figure 5 illustrated the performance of different clustering algorithms in terms of three metrics (PR, f-measure and F-score) with respect to the reference clusters. From Figure 5, we could observe that our ensemble-based approach had competitive performance compared with the base clustering algorithms as regard to all three measures. Among the base clustering algorithms, K-means with cluster number set to 6 had better performance in terms of PR, while K-means and Density-based clustering with cluster number set to 6 had better performance in terms of f-measure, and Hierarchical clustering with cluster number set to 9 and 10 had comparable performance with K-means and Density-based clustering with cluster number set to 6 in terms of F-score. But none of them could have superior performance than others as regard to all three measures. However, our ensemble-based approach obtained the best performance in terms of all the three measures. This may be owing to the fast that ensemble-based approach could make use of clusters derived from different base clustering algorithms and extract more reliable results. In addition, we conducted sensitivity study of phylogenetic structure similarity on microbial network. We ran algorithm Meta-EC with threshold value of metagenomic similarity in matrix tuning from 0.7 to 0.9 with 0.1 as step size, the results in Additional file 2: Figure S4 showed Meta-EC outperformed other state-of-art clustering techniques in the wide range of edge threshold, indicating that our Figure 5 Performance comparison of ensemble clustering framework to base clustering algorithms with respect to fmeasure, PR and F-score. Note that ensemble-based approach with random initialization is denoted as “Ensemble_random”, while ensemble-based approach with a base clustering result as initial input is denoted as “Ensemble_initial”. The result of “Ensemble_random” is obtained with β = 1. algorithm is robust and insensitive to the similarity network noisy and data coverage. In addition, we have compared the computational time with base clustering approaches in Table 3 and results show that Meta-EC Yang et al. BMC Systems Biology 2014, 8(Suppl 4):S7 http://www.biomedcentral.com/1752-0509/8/S4/S7 Page 8 of 12 Table 3 Comparison with bases clustering approaches on computational time Method EM Time(S) 221.17 K-Means Hierarchical Density-based Meta-EC 11.57 55.04 12.24 72.8 exactly spend more time than that by K-mean and Hierarchical clustering but less than EM clustering, so the total time cost of MetaEC is the sum of all base clustering algorithms plus 72.8 seconds. With rapid development of computational capability, we could improve the time efficiency on large amount of operations. Comparison of constructed consensus network with original similarity network: To demonstrate the benefits of combining different base clustering results, we applied symmetric NMF on original metagenomic similarity network and evaluated its performance. To be fair, the results of symmetric NMF on original metagenomic similarity network were obtained over the best tuned parameter. The comparison of the two tested similarity network is present in Figure 6 as regard to F-measure, PR and f-measure. The results in Figure 6 showed that applying symmetric NMF on consensus matrix achieved better performance than that on the original similarity network. These results demonstrated the benefits of combining different base clustering results. If the similarity matrix was well constructed (each element reflected the cocluster similarity), the factorization of the similarity matrix would generate a clustering assignment matrix Figure 6 Performance comparison of Bayesian NMF based clustering algorithm applied on ensemble clustering similarity network and original microbial similarity network. Additive values of three measures are present for each data source. For random initialization case, the value of β is set to 1 and the result corresponds to “Original_random”. We also choose the base clustering results which presents the best performance as the initial input of symmetric NMF and the result corresponds to “Original_initial”. that could well capture the cluster structure inherent in the network representation. However, the original network weighted the interaction via measuring the phylogenetic structure of samples. In this way, metagenomic samples with higher phylogenetic similarity were more likely to be involved in one cluster. If the actual microbial pattern was uncorrelated with phylogenetic similarity, the community detected by symmetric NMF may be unreliable. In ensemble clustering framework, we generated a consensus matrix that integrated the clustering results derived from different clustering algorithms. Each element in consensus matrix indicated the frequency of the corresponding sample pair being clustered together in these base clustering results. Thus, applying symmetric NMF on consensus matrix could take into consideration the co-cluster strength of multiple clustering patterns and output a more comprehensive and robust result. Interpretation of Microbial community patterns on human body habitats based on clustering results Recall that metagenomic samples were clustered in terms of co-occurrence frequency in base clustering results. Hence the final output clusters assembled samples to represent unique microbial patterns that are the consensus from base clustering approaches. Next, from the clustering results, we infer how microbial pattern was influenced by body habitats and host genders. Structural variation across body habitats: Through analyzing the enrichment of body habitat and host gender over six predicted clusters, the results in Figure 7 revealed a stronger coherence by body habitat than host gender. These clusters dominated by particular body habitats inferred that these body habitats harboured distinctive microbial patterns, which was also observed in base clustering results in Additional file 2: Figure S3. Although four base clustering algorithms generate clustering patterns with different criterions, most clusters in Additional file 2: Table S2 were enriched with particular habitats. Meanwhile, we observed that microbial communities at different body habitat exhibited different degree of compositional structure variation. Figure 7 showed that microbial structure remained relatively stable in oral cavity, compared with diverse microbial structures harboured in skin. It was biologically reasonable to detect diverse patterns on skin, since there were quite different places where skin microbial communities could be sampled. Different extend of habitat structural variation were also observed in base clustering results. In Additional file 2: Figure S2, gut and oral cavity microbial community patterns were only fit with one clustering criterion, gut consistent with K-means and oral cavity with Yang et al. BMC Systems Biology 2014, 8(Suppl 4):S7 http://www.biomedcentral.com/1752-0509/8/S4/S7 Page 9 of 12 Figure 7 Sample distribution on predicted clusters with respect to body habitat and host gender. hierarchical clustering. Contrary to gut and oral cavity, skin-enriched cluster could be recognized by four clustering criterions in all experimental settings, inferring skin samples have many cluster patterns with diverse microbial structures. Note that the proposed Meta-EC generates a more comprehensive community patterns with respect to meta-data since our result is an agreement by consensus of multiple base clustering approaches. For example, compared to Hierarchical and EM clustering results in Additional file 2: Figure S3 that only capture male-gut cluster, ensemble clustering is able to uncover femalegut specific clusters (shown in Figure 7), indicating that Meta-EC could reveal degree of structural variation over body habitat more comprehensively than base clustering results. Structural variation across host gender: We further assessed microbial structure variation with respect to host gender. Meta-Storm [27] was used to measure similarity of two metagenomic samples. The results in Figure 8 indicated that over all habitats, variation was significantly less within same gender samples than between opposite gender samples. However, these habitats perform different degree of structural variation with respect to host gender. Oral cavity microbiome exhibited a stable structure both among same and opposite gender individuals (both above 92% phylogenetic structure similarity). And skin communities had no unique structural variation patterns regarding to host gender. Gut community structure was highly variable between samples from opposite gender hosts (less than 90% similarity value for opposite gender samples of gut cluster 3), but exhibited strong coherence to same gender hosts. On the other hand, the enrichment study in Figure 7 showed that two gut clusters were distinct with host gender, indicating that opposite sexual individuals may exhibit a distinct microbial composition in gut. Microbial interconnection over habitats: Although microbial communities reflected unique structures (distributions) over body habitats, the interconnected microbial components among the body habitats were still observed in the clustering results. For example, cluster 1 in Figure 7 contained 10 skin samples that shared similar microbial compositions with oral cavity communities, while skin cluster 2, 4 and 6 harboured 6, 15 and 2 oral cavity samples respectively. Since skin microbial pattern was closely associated with external Yang et al. BMC Systems Biology 2014, 8(Suppl 4):S7 http://www.biomedcentral.com/1752-0509/8/S4/S7 Figure 8 Structural variation over host gender in oral cavity, gut and skin-dominated clusters. environment [34] and oral cavity was an open system where microbiome from external environment was imported by breathing, eating food and drinking water [35], oral cavity and skin would respond to outside environmental conditions, and gradually evolve similar microbiomes. Conclusions and discussions The human microbiomes are microbiomes that are hosted in gut, oral mucosa and multi-layer of skin etc. These organisms perform ecosystem-level functions that are useful for human host to maintain healthy, yet detailed factors that attribute the microbial community structures in human body habitats and host gender remain poorly conceptualized. To fully understand the roles of human microbiome in disease and health, prior studies focus on particular body habitats of health individuals with specific clustering approaches, based on the assumption that metagenomic samples of same body habitats would develop similar microbial structure patterns. However, human habitats are not isolated; they are interacted and correlated to form an integrated and complex system. And identified structures might be unsuccessful due to noisy sample similarity and specific topological structure within metagenomic network. Hence, single clustering algorithm rarely achieves optimal outcome. To uncover a global and comprehensive landscape of human microbiome, we perform an ensemble clustering framework Meta-EC on large- Page 10 of 12 scale metagenomic samples. In this study, our proposed Meta-EC algorithm has four main advantages on microbial pattern detection: (1) Meta-EC could effectively identify more reliable microbial communities via integrating many base clustering results, (2) As regard to the modularity of microbial communities, defined as the clustering of microbial communities (modularity) according to the effects of their related environments or treatments (meta-data), the consensus clustering network is much clearer at showing such modularity property (such as how environments (or meta-data) shape microbial communities in body habitat, which is critical to healthcare and diognosis) than the original metagenomic similarity network [25], (3) Ensemble framework is robust for the coverage of metagenomic similarity network (as shown in Additional file 2: Figure S4), and (4) Compared to base clustering results in Additional file 2: Figure S3, Meta-EC algorithm could reveal the spatial and gender patterns of microbiome (as shown in Figure 7) more comprehensively, as the ensemble clustering result is a general agreement by multiple base clustering approaches. Nevertheless, it should be acknowledged that the performance of our algorithm depends on the base clustering results and quality of original metagenomic similarity network. If all these base results were generated by poor clustering algorithms, the ensemble outputs would be far from real microbial community similarity patterns. If the original similarity network is unreliable to capture the modularity of metagenomic samples, none of clustering approaches could work. To address this problem, we have to integrate more base clustering approaches with diverse optimization criterions and pattern assumptions, to reduce the bias generated by base approaches. We assume these algorithms can capture a wide variety of clustering patterns in similarity network to alleviate the effect of unreliable clustering results. On the other hand, the proposed NMF based mode, which could be used in association study of bioinformatics domain [36-39], is a more complex method to implement, and convergence could be slow, as shown in Table 3. With rapid development of computational capability, we could improve the time efficiency on large amount of operations. And the nonnegative constraints on cluster indicator matrix H may be an insufficient condition for achieving sparseness in some cases [20]. Then one may set appropriate thresholds to enforce sparseness. In summary, Meta-EC is an ensemble clustering framework for large-scale metagenomic data analysis and microbial community pattern detection. In the future, NMF based model could be exploited to offer potential applications on bipartite model of drug-target association [40] and disease gene prediction [41]. Yang et al. BMC Systems Biology 2014, 8(Suppl 4):S7 http://www.biomedcentral.com/1752-0509/8/S4/S7 Availability The data sets and supporting experimental results of this article are available for download from http:// datam.i2r.a-star.edu.sg/MetaEC. Page 11 of 12 5. 6. 7. Additional material Additional file 1: Experimental Design. The file show the experimental design in this paper, including: (1) introductory of four base clustering approaches; (2) evaluation of microbial clusters; (3) parameter setting. Additional file 2: Supplementary Material. The file presents several figures, tables and additional experimental results mentioned in this paper, including: (1) the efficiency of GPU-meta-storm algorithm; (2) evaluation of four base clustering results; (3) sensitivity study of phylogenetic structure similarity on microbial network. 8. 9. 10. 11. Competing interests The authors declare that they have no competing interests. Authors’ contributions Conceptualized and designed the method and drafted manuscript: PY KN. Responsible for the implementation: PY XS LOY. Provided raw data: KN XS. Participated in discussion and improved the method as well as revised the draft: XS LOY H-NC X-LL. Read and approved the manuscript: PY XS LOY HNC X-LL KN. Acknowledgements This work is supported in part by Chinese Academy of Sciences’ e-Science grant INFO-115-D01-Z006, Ministry of Science and Technology’s high-tech (863) grant 2009AA02Z310 and 2014AA21502, as well as National Science Foundation of China grant 61103167 and 31271410. Declarations Publication costs for this article were partially funded by Chinese Academy of Sciences’ e-Science grant INFO-115-D01-Z006, Ministry of Science and Technology’s high-tech (863) grant 2009AA02Z310 and 2014AA21502, as well as National Science Foundation of China grant 61103167 and 31271410 and by the Institute for Infocomm Research, Agency for Science, Technology & Research (ASTAR), Singapore. This article has been published as part of BMC Systems Biology Volume 8 Supplement 4, 2014: Thirteenth International Conference on Bioinformatics (InCoB2014): Systems Biology. The full contents of the supplement are available online at http://www.biomedcentral.com/bmcsystbiol/supplements/ 8/S4. Authors’ details Institute for Infocomm Research, Agency for Science, Technology & Research (ASTAR), Singapore, 138632, Singapore. 2Computational Biology Group of Single Cell Center, Shandong Key Laboratory of Energy Genetics and CAS Key Laboratory of Biofuels, Qingdao Institute of Bioenergy and Bioprocess Technology, Chinese Academy of Science, Qingdao 266101, China. 3Center for Computer Vision and Department of Mathematics, Sun Yat-Sen University, Guangzhou, 510275, China. 1 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. Published: 8 December 2014 References 1. Wilson M: Bacteriology of humans: an ecological perspective. John Wiley & Sons 2009. 2. Dethlefsen L, McFall-Ngai M, Relman DA: An ecological and evolutionary perspective on human-microbe mutualism and disease. Nature 2007, 449(7164):811-818. 3. Turnbaugh PJ, Ley RE, Hamady M, Fraser-Liggett CM, Knight R, Gordon JI: The human microbiome project: exploring the microbial part of ourselves in a changing world. 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Dynamic Loop Parallelisation Adrian Jackson∗ and Orestis Agathokleous∗ ∗ EPCC, arXiv:1205.2367v1 [cs.PL] 10 May 2012 The University of Edinburgh, Kings Buildings, Mayfield Road, Edinburgh, EH9 3JZ, UK Abstract—Regions of nested loops are a common feature of High Performance Computing (HPC) codes. In shared memory programming models, such as OpenMP, these structure are the most common source of parallelism. Parallelising these structures requires the programmers to make a static decision on how parallelism should be applied. However, depending on the parameters of the problem and the nature of the code, static decisions on which loop to parallelise may not be optimal, especially as they do not enable the exploitation of any runtime characteristics of the execution. Changes to the iterations of the loop which is chosen to be parallelised might limit the amount of processors that can be utilised. We have developed a system that allows a code to make a dynamic choice, at runtime, of what parallelism is applied to nested loops. The system works using a source to source compiler, which we have created, to perform transformations to user’s code automatically, through a directive based approach (similar to OpenMP). This approach requires the programmer to specify how the loops of the region can be parallelised and our runtime library is then responsible for making the decisions dynamically during the execution of the code. Our method for providing dynamic decisions on which loop to parallelise significantly outperforms the standard methods for achieving this through OpenMP (using if clauses) and further optimisations were possible with our system when addressing simulations where the number of iterations of the loops change during the runtime of the program or loops are not perfectly nested. I. I NTRODUCTION High Performance Computing (HPC) codes, and in particular scientific codes, require parallel execution in order to achieve a large amount of performance increase. Depending on the underlying parallel platform which is used, programmers use different programming models in order to achieve parallel execution. In distributed memory systems, the message passing programming model is the most commonly used approach for applying parallelism in the codes. In shared memory systems however, an attractive choice for parallel programming is through OpenMP[16]. The parallelisation of codes with OpenMP is often achieved with loop parallelisation. As long as the iterations of a loop are independent, they can be distributed to the available processors of the system in order to execute them in parallel. A programmer is required to specify a loop that can be parallelised by placing compiler directives before the loop, resolving any dependency issues between the iterations beforehand. HPC codes often consist of regions with nested loops of multiple levels. In order to parallelise these regions, a choice must be made on how parallelism should be applied on the loops. Even though OpenMP supports a variety of strategies for parallelising nested loops, only a single one can be used to parallelise the code. A static choice however, cannot exploit any runtime characteristics during the execution of the program. Changes in the input parameters of the executable which affect the iterations of the loops may render the parallelisation decision suboptimal. In addition to this, the iterations of a loop can change at runtime due to the nature of the code. A common feature of HPC codes is to organise the data into hierarchies, for example blocks of multi-dimensional arrays. Depending on the problem, the blocks can have different shapes and sizes. These parameters affect the loops that are responsible for accessing this data. In some situations, a static decision has the potential to impose a limitation on the amount of processors that can be used for the parallel execution of the loops. With the current trend of chip manufactures to increase the number of cores in the processors in each generation leading to larger and larger shared memory system being readily available to computational scientists on the desktop and beyond, a more dynamic approach must be considered for taking such decisions. This report outlines our investigations into various strategies that can be applied at runtime in order to make a dynamic decision on how to parallelise a region with nested loops. Our approach is to try to automatically perform modifications to users code before compilation in order to enable the code to make these decisions dynamically at runtime. Specifically, we investigated the possibility of having multiple versions of a loop within a region of nested loops in order to make a dynamic choice on whether a loop should be execute sequentially or in parallel. II. O PEN MP OpenMP [1] is, arguably, the dominant parallel programming model currently used for writing parallel programs for used on shared memory parallel systems. Now at version 3.1, and supported by C and FORTRAN, OpenMP operates using compiler directives. The programmer annotates their code specifying how it should be parallelised. The compiler then transforms the original code into a parallel version when the code is compiled. By providing this higher level of abstraction, OpenMP codes tend to be easier to develop, debug and maintain. Moreover, with OpenMP it is very easy TABLE I S TRATEGIES FOR PARALLELISING NESTED LOOP REGIONS Name Description Outermost Inner Loop Nested Loop Parallelisation of the outermost loop Parallelisation of one of the inner loops Parallelisation of multiple loops with nested parallel regions Collapsing the loops into a single big loop Loop Collapsing Loop Selection Runtime loop selection using if clauses to develop the parallel version of a serial code without any major modifications. Whilst there are a number of different mechanisms that OpenMP provides for adding parallel functionality to programs, the one that is generally used most often is loop parallelisation. This involves taking independent iterations of loops and distributing them to a group of threads that perform these sets of independent operations in parallel. Since each of the threads can access shared data, it is generally straightforward to parallelise any loop with no structural changes to the program. III. N ESTED L OOPS HPC codes, and particularly scientific codes, deal with numerical computations based on mathematical formulas. These formulas are often expressed in the form of nested loops, where a set of computations is applied to a large amount of data (generally stored in arrays) and parallelisation can be applied to each loop individually. The arrays often consist of multiple dimensions and the access on the data is achieved with the presence of nested loops. Furthermore it is not uncommon that the arrangement of the data is done in multiple hierarchies, most commonly in blocks with multidimensional arrays, where additional loops are require in order to traverse all the data. When such code is presented, a choice must be made on which loop level to parallelise (where the parallelisation should occur) [2]. A summary of the available strategies is presented in Table I A. Outermost loop The most commonly used approach is to parallelise the outermost loop of a nested loop region, as shown in Listing 1. Using this strategy, the iterations of the loop are distributed to the members of the thread team. The threads operate in parallel by executing the portion of iterations they are assigned to them individually. The nested loops of the parallel region are executed in a sequential manner. 1 2 3 4 5 6 #pragma omp parallel for private (j) for(i = 0; i < I; i++){ for(j = 0; j < J; j++){ work(); } } Listing 1. Outer loop parallelisation of a nested loop region Parallelising the outermost loop is often a good choice, as it minimises the parallel overheads of the OpenMP implementation (such as the initialisation of the parallel region, the scheduling of loop iterations to threads and the synchronisation which takes place at the end of the Parallel loops). More extensive work on the overheads of various OpenMP directives can be found in [3]. Despite the advantages of the Outermost Loop parallelisation strategy in this context, there are drawbacks of this choice. The maximum amount of available parallelism is limited by the number of iterations of the outerloop loop. Considering the example code in Listing 1, it is only possible to have I tasks being executed in parallel. This restricts the number of threads the code can utilise upon execution, and therefore the number of processors or cores that can be exploited. B. Inner loop This is a variant on the outermost loop strategy, with the difference that one of the inner loops of the region is chosen to be parallelised. This approach will only be required or beneficial if the outer loop does not have enough iterations to parallelise efficiently as this variant on the parallelisation strategy introduces parallelisation overheads by requiring the parallelisation to be performed for each loop of the outerloop rather than once for all the loops (as shown in Listing 2). Further nesting of the parallelisation (at deeper loop levels) will further increase the performance problems; the parallel overheads appear a lot more times, whereas the amount of work of each iteration becomes finer. 1 2 3 4 5 6 for(i = 0; i < I; i++){ #pragma omp parallel for shared (i) for(j = 0; j < J; j++){ work(); } } Listing 2. Inner loop parallelisation of a nested loop region Another issue with this strategy is the scenario where loops are not perfectly nested. In this situation, when there are computations in-between the loops, as shown in Listing 3, parallelising a loop of a deeper level will result in sequential execution of that work. Depending on the amount of the execution time which is now serialised, this approach has the potential to increase the execution time of the code. 1 2 3 4 5 6 for(i = 0; i < I; i++){ somework(); for(j = 0; j < J; j++){ otherwork(); } } Listing 3. Poorly nested loop region example C. Nested The Nested parallelisation strategy exploits the fact that more than one loop can be executed in parallel. By opening multiple nested parallel regions at different levels of loops, as presented in Listing 4, more threads can be utilised during the parallel execution of the code. Unlike the Outermost Loop and the Inner Loop approaches, which can only utilise as many threads as the iterations of the loop with the biggest number of iterations, this strategy can exploit further parallelisation opportunities. Other studies have shown that nested parallelism can give good results on systems with a large number of processors [4] [5]. 1 2 3 4 5 6 7 #pragma omp parallel for private (j) for(i = 0; i < I; i++){ #pragma omp parallel for shared (i) for(j = 0; j < J; j++){ work(); } } Listing 4. Nested loop parallelisation of a nested loop region D. Loop Collapsing The loop collapsing strategy takes a different approach for exposing additional parallelism within nested loop regions. By performing code transformations, multiple nested loops are combined, or collapsed, into a single loop. The newly created loop has a larger amount of iterations, which can be distributed to the threads. As of version 3.0, OpenMP supports loop collapsing by using the COLLAPSE clause in the Loop Construct, requiring the programmer to provide the number of loop levels to collapse. To be able to use the COLLAPSE clause the loops have to be perfectly nested (i.e. no code between the loops) and the number of loop iterations (when multiplied together) need to be able to be regularly divided. Loop collapsing can produce better results than both the inner loop and nested loop strategies, since the parallel overheads are minimal, however it is not always available, either because not all compilers support OpenMP version 3.0, or because the conditions outlined above cannot be met. 1 2 3 4 5 6 7 8 #pragma omp parallel for collapse (3) for(i = 0; i < I; i++){ for(j = 0; j < J; j++){ for(k = 0; k < K; k++){ work(); } } } parallel region is always created in either case. The presence of the if clause only affects the number of threads that get assigned to the parallel region. When sequential execution is triggered, the code is only executed by the master thread, for parallel execution all threads execute the code. Furthermore, with the if clause, programmers are still required to manually write code which makes the decision, construct sensible scalar-expressions to be evaluated, and manually parallelise each loop that is a potential target for parallelisation. IV. DYNAMIC LOOP One of the motivators for this work was a parallelisation that was undertaken of a finite-volume cell-centred structured Navier-Stokes code for undertaking Computational Fluid Dynamics (CFD) simulation. It is a structured mesh, multigrid, code which works with multiblock grids, and includes a range of CFD solvers including; steady state, time-domain dual time-stepping, frequency-domain harmonic balance, and timedomain Runge-Kutta. The general pattern for the computations within the code is shown in Listing 6. Whilst this type of computational pattern is not uncommon for scientific codes one of the challenges in the parallelisation is that as the code can use a range of different methods, as previously outlined, the range of these loops can vary. For instance when performing a time domain simulation the harmonic loop has a single iteration. However, when performing a harmonic balance simulation it can have a range of values, generally between 2 and 16. Furthermore, it is not uncommon to run large simulations with a single block, or a small number of blocks, meaning that the block loop has a very small number of iterations. Finally, each block in the simulation can have different values for its dimensions. In theory, the loop collapsing strategy would be ideal for this type of simulation code as this would enable parallelisation without having to deal with the varying sizes of the nested loops. However, it cannot be guaranteed that for all input datasets the loop iterations can be regularly divided, and there are also particular areas of the code where the loops are not perfectly nested. 1 2 3 4 5 Listing 5. Parallelisation of a nested loop region with loop collapsing 6 7 8 E. Loop Selection OpenMP already provides a way of forcing a parallel region to execute sequentially with the use of the if clause on OpenMP directives. The if clause, of the following form, if (scalar − expression), is used to determine at runtime whether the code enclosed in the parallel region should execute sequentially or in parallel. When the scalar expression of the clause evaluates to 0, the region is executed sequentially. Any other value will result in parallel execution. However, a new PARALLELISATION 9 10 11 for(iter = 0; iter < n_iters; iter++){ for(block = 0; block < n_blocks; block++){ for(harmonics = 0; harmonics < n_harmonics; harmonics++){ for(j_cell = 0; j_cell < n_cells_j; j_cell++){ for(i_cell = 0; i_cell < n_cells_i; i_cell++){ perform computations; } } } } } Listing 6. Example scientific code loops Given the different techniques that can be used to parallelise nested loops, the occurrence of nested loops in many scientific simulation codes, and the fact that the loop iterations of nested loops can change for different input datasets of a code or when performing different functions with a code, we wanted a system that enabled the selection of different parallelisation choices to be available to code at runtime when the specific ranges of the nested loops are known. Our strategy for providing this functionality is to create code, based on the provided user code, that can perform a parallelisation of any of the nested loops and add decision making algorithms to dynamically choose, at runtime, which parallelisation is used. Specifically, we have created tools that create multiple versions of a loop within a region of nested loops in order to make a dynamic choice on whether a loop should execute sequentially or in parallel In general, code duplication is considered bad programming practice as it can, amongst other issues, lead to update anomalies (where not all instances of the functionality are modified when modifications occur) and thus damage the maintainability of the code. However, if the duplicate code (in our instance the serial and parallel versions of each loop in the nested loop structure) can be generated automatically for standard user code then it will not adversely affect the maintainability of the user program. We created a source-to-source compiler that recognises compiler directives within user’s source code and uses them to pre-process the source code and generate a program that has the alternative parallelisation strategies encapsulated within it. By exposing a simple interface to the programmers through compiler directives, which are similar to the already familiar OpenMP compiler directives, we can automatically provide the dynamic parallelisation functionality for users without requiring significant changes to the original source code. Furthermore, this approach provides the users the choice of enabling or disabling our functionality with minimum effort. To complement the code duplication we have also implemented functionality (in a small runtime library) that produces the code which is responsible for deciding what parallelisation to perform automatically. The decision functionality considers the number of iterations of a loop in order to chose a parallelisation strategy that makes best use of the processors or cores available. Our implementation is currently limited to parallelising a single loop of a nested loop region, taking advantage only the Outermost and Inner loop strategies. Other authors [2] have already taken a similar approach by modifying the OpenMP runtime library in order to make these decisions dynamically. However, applying this logic in the OpenMP runtime library would have limited the implementation to a specific compiler. Using our source-to-source compile approach we are aiming to transfer the logic in user code in order to maintain the portability of our solution. In addition to simple heuristics, we also explored the idea of a profile-based approach at runtime in order to detect the best possible parallelisation strategy with time measurements. A heuristics based approach alone cannot capture any information on the amount of the actual computations when making a decision on parallelising a loop. Whilst this is generally irrelevant for perfectly nested loops (as all the work is in the lowest loop), it may have more of an impact where there is work between the different loops as well. There may also Fig. 1. Compilation process using the source-to-source compiler be situations where a different inner loop has slightly more iterations than an outer loop so could be chosen by a simple heuristic as the place where the parallelisation occurs but the overheads associated with parallelising that inner loop actually make this a suboptimal choice. Providing a profiling based decision mechanism may help with both these scenarios, and enable us to identify situations where, for instance, using less threads to parallelise an outer loop might provide a better execution time. The idea of an auto tuning code has already been proposed by other compiler-related researches [6] [7] for producing optimised code, we apply similar logic. V. S OURCE - TO - SOURCE COMPILER Our source-to-source compiler acts as a preprocessor to C code which can contain OpenMP directives, as well as our own directives. The compiler parses the code, and creates an internal representation of the code in the form of an Abstract Syntax Tree (AST). The regions of the input code that contain our directives are translated into the semantics of the C programming language and OpenMP directives during the parse phase, and appropriate nodes for these regions are placed in the AST. The created AST is then translated back to C code with OpenMP directives. This generated code is then compiled using a standard, OpenMP enabled, C compiler to produce a parallel executable (this process is illustrated in Figure 1. Our compiler, implemented using the Lua [8] programming language along with the Lpeg [9] parsing library, recognises a number of our own bespoke compiler directives of the form #pragma preomp. A loop that is preceded by a #pragma preomp f or directive is considered by our compiler as a suitable candidate for applying parallelisation. When such a loop is found, our compiler performs the necessary code transformations so that a decision can be made at runtime whether the loop should run sequentially or in parallel (and to ensure that both the sequential and parallel versions of the loop are available in the executable at runtime). In addition to this, a simple analysis of the loop is performed in order to facilitate the computation of a loops iterations during the making of the decision. An example of such a code is presented in Listing 7. 1 2 3 4 5 6 7 #pragma preomp parallel for private(j) for(i=0; i<I; i++){ #pragma preomp parallel for shared(i) for(j=0; j<J; j++){ work(); } } Listing 7. A nested loop region with preomp Furthermore we also extend the grammar to support an additional clause, the parallel threshold(expression) clause. This is optional, and when it is not present the compiler will assume a default value of 1.0. This clause is used to allow control over when a loop is parallelised, and will be discussed further in Section VI. A. Code Duplication The main function of the source-to-source compiler is to take the original user code and duplicate the loops to be parallelised so that there are both serial and parallel versions of those loops that can be selected at runtime. As previously mentioned our system only allows one loop to be parallelised at any given time (although which loop is parallelised can change over the runtime of a program as the parameters of the loop change), but both the serial and parallel versions of all the loops to be parallelised must appear in the executable to enable a selection at runtime to take place. When a loop is preceded by a #pragma preomp f or directive, the loop is duplicated and wrapped in a normal if − else statement which evaluates a decision function from our runtime library and selects the if or else branch based on the outcome of the evaluation. B. OpenMP if As a comparison to our code duplication approach we also implemented the same functionality uses the existing if clause of the OpenMP Parallel Construct. Our custom directive is translated into an OpenMP Parallel For directive, with an attached if clause in order to decide whether to execute the loop in parallel or not (rather than a serial and parallel version of the loop). The expression of the if clause consists of a call to a decision function of our runtime library, which takes the evaluated expressions of the loops information in order to make a decision. This functionality was included to allow a comparison of our approach to the standard method that developers could currently use to provide dynamic selection of parallelism with OpenMP. However, a major drawback of this approach (and the reason we do not uses it for our functionality) is that a parallel region will be created regardless of whether a loop is parallelised or not. Considering the example in Figure 2, parallelising the outer loop of two nested loops with two threads will result in three parallel regions. Each thread of the outer region will Fig. 2. An example of using the if clause to parallelise (a) the outer and (b) the inner loop of two nested loops with two threads create a new parallel region and become its master. In the case of the inner loop being parallelised, two parallel regions are created. For nested regions with a larger number of loops this method has the potential to produce excessive parallel overheads. VI. D ECISION FUNCTIONS AND THE RUNTIME LIBRARY The runtime library implements the logic for deciding which version of a loop is chosen during execution. Once a code has been processed by the source-to-source compiler it must then be linked with our runtime library to enable this functionality to be used. A. Decision Based On Heuristics Here we use heuristics, based on information collected at runtime, to decide whether a loop should execute sequentially or in parallel. The idea of this approach is to look for the first loop that has enough iterations to utilise all of the available threads, based on the assumption that parallelising outer loops is more efficient than parallelising inner loops as the amount of parallel overheads should be lower (as the OpenMP parallel regions are encountered less frequently). Before the execution of a loop, the decider checks whether a loop of an outer level is already running in parallel. If this condition is met, then the loop is serialised. In the case that no outer loop is running in parallel the number of iterations of the loop is calculated and it is divided with the available number of threads. If this results in a value that is greater than or equal to a specified threshold, then the parallel version of a loop is chosen, otherwise the loop is serialised. As discussed in Section V, the default value of the threshold is 1 (there must be no idle threads) although this can be controlled by the user. The calculations of the iterations is based on the parameters of the loop which are extracted by the source to source compiler and are provided as arguments to the decision function. In the case that the original code of the loop uses variables for its boundaries, any change in their value will also be captured by the decision function during the calculation. This design allows constant monitoring of any changes in the iterations of the loops which also results in dynamic adaptation of the parallelisation strategy during the execution of the program. The algorithm is very simple and with minimum overheads. Moreover, there is no need to maintain any state for the loops. However, the logic which is used by the function is of a program the profiling overhead will only be imposed in the first few iterations of the program. Figure 3 outlines this with an example of three nested loops. VII. P ERFORMANCE E VALUATION Fig. 3. An example of the Heuristics With Profiling Decider on three loops based on optimism. It only considers the amount of parallelism exposed by the loop regardless of whether the amount of work of the loop is big enough to justify any overheads of the parallelisation or whether there is any work between loops. To evaluate the performance of our new functionality we aimed to benchmark it against standard, static, OpenMP parallelisations with a range of different configurations. In particular, we focussed on varying the number of loop iterations, the amount of work between and within loops, and the number of changes that occur to loop bounds during execution to evaluate whether and when our approach is beneficial compared to a static parallelisation. To undertake these benchmarks we used two different codes. The first is a synthetic, configurable, benchmark C code, shown in Listing 8, which we constructed for this evaluation. The number of iterations of each loop can be configured, as can the amount of work that is simulated (by calling the delay function) between the second and third loops, and within the third loop. 1 B. Decision Based On Heuristics With Profiling 2 To address the potential issue with the basic decision based on heuristics previously discussed we also implemented a more complex decision function based on both the size of loops and some evaluation of the work in the loops. In the same manner as the heuristics decider, it uses the same information extracted by the source to source compiler in order to determine whether the loop should be parallelised or not. However, if a loop does not meet the conditions, then the function reverts to a profiling mode in order to decide which version of the loop, serial or parallel, to choose from based on timings. The first time a loop is executed, the heuristics decider determines if the loop should be parallelised. If the conditions are not met, the sequential version of the loop is chosen and profiling is enabled for this loop. At the next execution of the loop, the evaluation of the heuristics is still performed. If the conditions are still not met (for example there where no changes in the iterations of the loop), the loop is now parallelised since at this point we only have timing information for the serial version. Consecutive executions of the loop will first check the heuristics conditions, falling back to profiling mode if the condition is not satisfied. However, the function will detect that timings for both versions are available and utilise the information gathered from profiling to decide what loop to parallelise (providing the number of iterations of the loop have not changed), with the fastest version chosen as the final decision. In contrast to this, if the amount of work is not the same (i.e. the number of loop iterations has changed) the timings get invalidated, and profiling is re-initiated. To implement this functionality requires additional code, when compared to the basic heuristic decision function. This will impose an extra overhead to the produced program, although if the loop iterations are static throughout the run 4 3 5 6 7 8 for(i=0; i<num_iters; i++){ for(j=0; j<outer_iters; j++){ delay(outer_delayreps); for(k=0; k<inner_iters; k++){ delay(inner_delayreps); } } } Listing 8. Synthetic benchmark code The second benchmark code was an extract from the CFD code outlined in Listing 6. This code is more complex than the synthetic benchmark and more representative of realistic scientific simulation codes. This code is used to explore the performance of our solution when the loop iterations vary and when the bounds of loops are dynamic during the course of the execution of the benchmark (i.e. one or more loops change their loop bound as the outer loops are progressed). A. Benchmark Environment The platform used to evaluate the dynamic loop parallelisation functionality was Ness [10], at EPCC. The system is composed by two parts, a front-end for development and job submission and a back-end for job execution. The management of the two parts is handled by the Sun Grid Engine which allows submission of jobs from the front-end that must be executed on the back-end nodes in isolation. The back-end part of the system is composed by two SUN X4600 Shared Memory nodes. The central processing unit (CPU) of each node is an AMD Opteron processor of 16 2.6GHz processing cores and 32 GB or main memory. Each core has 64K of L1 cache for data and 64K L1 cache for instructions. In addition there is also 1 MB of L2 available to each core (combined for data and instructions). We used the Portland Group (PGI) C compiler for the majority of the benchmarks, with the following compiler flags: -O4,-c99,-mp. For the benchmarking involving the OpenMP if functionality we used the GNC C compiler instead as the version of the PGI compiler we used does not support a thread team of a nested parallel region to have more than one threads when an outer region is serialised with the if clause (this seems contrary to the OpenMP specification where the if clause only affects the number of threads that get assigned to a particular parallel region, not the thread teams of its nested regions). When using the GNU C compiler we used the following compiler flags: -O3,-stf=c99,-fopenmp. Timing information was collected using the omp get wtime() function, with each benchmark executed three times and the worst time taken (since this is the limiting factor for the execution time). (a) Outer Loop Work : 0s (b) Outer Loop Work : 0.022s (c) Outer Loop Work : 0.079s (d) Outer Loop Work : 0.15s B. Synthetic benchmark results If we consider the example code in Listing 8, the execution time of the code of the two internal nested loops when only the outer loop is parallelised with a certain amount of threads (outer threads) can be calculated as shown in Equation 1. TpOuter is the execution time when parallelising the outer loop, Touter work is the time needed for the work in-between the loops and Tinner work is the time needed for the amount of work within the innermost loop. basicstyle= Tp Outer = outer_iters ∗ (Touter_work + (inner_iters ∗ Tinner_work )) (1) outer_threads In a similar fashion, when parallelising the inner loop using inner threads, the execution time of the loops is shown in Equation 2. basicstyle= Tp Inner = outer_iters ∗ (Touter_work + inner_iters ∗ Tinner_work ) (2) inner_threads If we want to have a reduction in the overall execution time by parallelising the inner loop, the constraint TpInner < TpOuter must be satisfied. Solving this constraint in terms of Touter work we can get the maximum allowed threshold of the execution time for the work of the outer loop as shown in Equation 3. It is worth mentioning that this model is an ideal performance model, where the work is evenly distributed to the threads. In reality, the time of Touter work might be affected by the presence of parallel overheads. basicstyle= Touter_work < 1 1 inner_iters ∗ Tinner_work ∗ ( − ) outer_threads inner_threads 1 1− outer_threads (3) In order to test our hypothesis, we measured the amount of time which is required by the delay function for various values, with the results shown in Figure 4. The graphs in Figure 4 show the performance of four different parallelisation strategies. OpenM P Outer(1) and OpenM P Inner(2) are the results from manual, static, parallelisations of the individual loops in the benchmark. Heuristics are the results from our basic decision function using a value of one (i.e. only parallelise the loop if there are more iterations than threads available), and Heuristic P rof iler are the results from our system using the profiling functionality where appropriate. Fig. 4. Synthetic Benchmark results with varying levels of work between the loops From the results it is evident that when the loops are perfectly nested, and regular (i.e. the loop bounds are not changing), then there is no benefit from using the profiling functionality. The basic heuristics will choose the optimal loop to parallelise apart from when we are using 6 threads. The variation in outcomes for 6 threads is is a consequence of the number of loop iterations chosen for the benchmark (8 iterations of the outer loop and 16 iterations of the inner loop). The distribution of 8 iterations to 6 threads results in all of the threads to get assigned 1 iteration of the outer loop each, and 2 of the threads get and extra iteration. The total execution time in this case is limited by the slowest threads, which is the time of 32 iterations; 2 iterations of the outer loop multiplied by 16 iterations of the inner one. Parallelising the inner loop with 6 threads however, 2 of the thread get from 2 iterations whereas the rest the threads get 3 iterations each. In this case, the total execution time of the parallel loops is the amount of time required for 24 iterations; 3 iterations of the inner loop multiplied by 8 iterations of the outer loop. Since both decision functions only utilise the heuristics decision (when the number of threads is less than the number of iterations) they cannot exploit this opportunity as no profiling is actually performed in this case. This could be altered by setting the decision heuristic to a value other than 1 (i.e setting the heuristic to 1.5). From the graphs we can observe that our threshold value calculations hold. For the parameters we used for this benchmark the calculated threshold value is approximately Touter work < 0.0468 seconds. When the work of the outer work is less than the calculated threshold (Figures 4a 4b) parallelising the inner loop with 16 threads is still faster than parallelising the outer loop with 8 threads. As the amount of work increases, the impact on the execution time when parallelising the inner loop TABLE II L OOP PARAMETERS USED FOR THE CFD CODE BENCHMARKING Parameter Value iters n cell j n cell i 500 2496 or 8 8 or 2496 is increased, since more work is now being serialised. In these cases, the heuristics decider makes the wrong choice (Figures 4c and 4d) since its decision only concerns the amount of iterations of the loops and the available threads. In contrast to this, when profiling is used in the decision function, it correctly detected that the fastest execution time is achieved by not parallelising the inner loop. In the case, where the amount of work of the outer loop exceeds the calculated threshold, parallelising the inner loop, even with 16 threads, increases the total execution time. The benefit from using 16 threads to parallelise the inner loop is not enough to justify the work that is serialised. C. CFD benchmarking results The first benchmark that we performed using the extract from the CFD code was to compare the OpenMP if clause with our basic heuristic functionality. We used, as a reference, the timings of the manually parallelised the n blocks, n harmonics and n cell j loops and compare the execution time of the heuristics decision function for the two code generation modes of our compiler. In order to avoid cases of the iterations not being evenly distributed to the threads, we only consider cases of 2, 4, 8, 12 and 16 threads. The parameters used for the loop iterations are shown in Table II, with varying amount of work in the inner loop. We also consider cases where blocks do not have the same shape by altering the values of the n cell j and n cell i loops. No alterations indicate that all of the blocks have a grid shape of 2496x8(j cell x i cell). An alteration of 2 means that the first and third blocks have a grid shape of 8x2496 whereas the second and fourth blocks have a shape of 2496x8. The performance results shown in Figure 5 highlight the fact that there is a significant difference between our implemented functionality and that provided by OpenMP (the if clause). Not only is the if clause slower than the basic OpenMP parallelisation, but it also increases the overall execution time of the code. For Figure 5a, where 2 and 4 threads are available, only the loop of the outer level is parallelised in both code generation modes. However, the if clause mode produces a slower execution time than the code duplication mode. When more than 4 threads are used, the parallelisation is applied on the n cell j loop. In contrast to the code duplication mode which produces an execution time similar to the case of statically parallelising the loop, the if clause mode is still slower. A similar performance pattern is seen at 16 threads. Moreover, in the presence of alterations in the shape of the blocks, as shown in Figures 5b and 5d, the if clause mode produces an even slower execution time. On the other hand, (a) Small work, no alterations (b) Small work, 2 alterations (c) Large work, no alterations (d) Large work, 2 alterations Fig. 5. CFD benchmark with n blocks = 4 and n harmonics = 4 with varied alterations in the i and j cell loops and varied amount of work in the inner loop the code duplication mode can exploit this opportunity in order to utilise all of the available threads by applying parallelism on the n cell i loop. Increasing the amount of work in the core calculation has a positive effect on the if clause code generation mode. We can observe from Figure 5c that compared to Figure 5a the difference between using the if clause and the static parallelisation is not as large for small numbers of threads. This is likely to be because the performance cost of executing the if clause is proportionally smaller compared to the overall execution time. However, the same performance degradation is still observed when increasing the number of threads. The execution times of the code using the OpenMP if clause raised some concerns over whether the code was operating correctly. After extensive testing and verification we ascertained that both versions of the code (the if clause and code duplication) were correct and producing the same behaviour. Therefore, we investigated the parallel overheads of the OpenMP runtime library of the GCC compiler. Other authors [11] have already studied the overheads of nested parallelism on various compilers, including a more recent version of the GCC compiler than the one used in this work. Their findings suggest that the implementation of nested parallel regions of the GCC compiler has significant overheads. What is not presented in their work is whether or not the use of the if clause on nested parallel regions produces the same overheads. In order to ensure that the behaviour we observed in our results is the cause of nested parallel regions and not the presence of the if clause, we have constructed a simple micro benchmark. TABLE III M ICRO BENCHMARK RESULTS OF GNU’ S C COMPILER ’ S IMPLEMENTATION OF NESTED PARALLELISM Parallel loop Execution time (seconds) Outer Inner Nested (with if clause) Nested (with num threads clause) 38.845619 153.06809 163.05681 162.85479 (a) n blocks = 4, n harmonic = 8,(b) n blocks = 4, n harmonic = 8, no alterations 2 alterations D. Nested parallel micro benchmark We created four versions of a benchmark code with three nested loops and the delay function of the EPCC Microbenchmark Suite in the block of the innermost loop. The first version of the benchmark creates a parallel region on the loop of the second level. The second version performs the same operation on the innermost loop. The third version uses the if clause on both loops by serialising the outer loop with a value of 0 and parallelising the inner loop with a value of 1. Finally, the last version creates a parallel region on both of these loops, however we force the number of threads on the thread team of the outer loop to 1 using the num threads clause. Through this we manage to reproduce the same behaviour as with the if clause code case when the inner loop is parallelised. The number of iterations of the parallel loops are the same as the number of available threads. Table III presents the execution times of each case. We can see that parallelising the inner loop with nested parallel regions takes 10 seconds longer than parallelising the inner loop manually, even for this small and simple benchmark. Moreover, the two versions that contain nested parallel regions achieve very similar execution times. From this test we can concluded that it is likely that the behaviour we observed from the if clause code generation mode is affected by the overheads of the implementation of the GCC compiler for nested parallel regions. E. Decision function benchmarking Finally, we investigated the performance of our profiling decision functionality for the CFD extract code. This code is perfectly nested, so the basic heuristic decision function should be optimal here as it should chose the best loop to parallelise with very little overheads, whereas the profiling function has extra functionality and therefore imposes extra overheads on the performance of the code. The results from our experiments are shown in Figure 6. We can observe from Figures 6a that both the decision functions make the correct choice of parallelisation strategy up to 12 threads. However, the overheads of the profiling functionality have a negative impact on the overall execution time. Even when profiling is not actually being performed, the functions which are inserted before and after the execution of each loop to count the amount of work performed at each loop level increase the overall time. Moreover, we can observe that at 16 threads the profiler actually chooses to parallelise the harmonics loop, whereas the heuristics decider produces (c) n blocks = 8, n harmonic = 4,(d) n blocks = 8, n harmonic = 4, no alterations 4 alterations Fig. 6. CFD benchmark with varied alterations in the i and j cell loops and a large amount of work in the inner loop the correct behaviour of profiling the n cell loop. The timings which are performed for each loop version during the profiling mode are sensitive to the presence of any overheads which ultimately affect the decision of the function (such as the overhead of taking the timings). When alterations are present in the shape of the loops, as shown in Figures 6b and 6d, the heuristics decider manages to adapt its behaviour, parallelising the innermost loop in order to utilise more threads, and can significantly out perform the static parallelisation. In all of the test cases, the decision function which is based on profiling provides slower execution times than the decision function which is based on heuristics. Moreover, the additional logic which is included in the decision function with profiling caused a suboptimal decision to be made in some situations. VIII. I MPROVED PROFILING DECISIONS The results from the previous benchmarks lead to considerations of the reasons behind the poor execution of the decision function which performs profiling. Comparing the functionality of this function with the simple case of the heuristics decision function there are two sources of additional overheads. The first one is the logic of profiling each version of a loop. In order to make a choice between the two versions of a loop, the slow version must also be executed. However, if an actual simulation code runs for a significant amount of time this overhead should be negligible (providing the loop bounds do not alter and trigger the profiling functionality too many times) as it should only be incurred infrequently. (a) Small work, n blocks = n harmonic = 4, 2 alterations 4,(b) Large work, n blocks = n harmonic = 4, 4 alterations 8, Fig. 7. CFD benchmark with varied alterations in the i and j cell loops and a large amount of work in the inner loop The second source of overheads is the inclusion of additional function calls before and after each loop in order to measure the time of the execution and count the amount of work performed. The elimination of the functionality for taking the slow path is not possible since this is the essence of profiling. Both versions of a loop must be executed in order to make a comparison between their execution time. However, we can relax the conditions on the validity of the timings. If we only consider the number of the iterations of the specific loop which is being profiled, then we can eliminate all the logic that performs the counting of the work for the internal loops. When the decision function decides that a version of a loop should be profiled (after the failure of the heuristics conditions) the number of the iterations of the version of the loop that is going to be executed is saved in the state of the loop at that point. This way, the code of the function calls which are placed before and after each loop remains simple, only adjusting the loop level counter of each thread as well as marking the starting and ending times of the execution of a loop which is being profiled, rather than counting the iterations of internal loops as the initial profiling functionality does. In order to test our theory we have created a new version of the runtime library which includes the above modifications, called the relaxed profiler. From the graphs in Figure 7 we can see that the removal of the additional logic which performs the counting benefits the decision function with profiling. When no profiling is performed (2 and 4 threads), the relaxed version of the decision function is faster than the accurate version, and the same performance pattern holds when the profiling is performed (8 threads and more for Figure 7a and 12 threads and more for Figure 7b). Comparing the execution time of the new version of the decision function with profiling to the execution time of the heuristics decision function, the latter still produces a faster execution time, however the difference is not large. This behaviour is expected, since the presence of profiling introduces additional computations within the code itself from the functions which are placed before and after each loop. Moreover, in the cases where the parallelisation is applied on a nested loop, the decision function must execute both versions of a loop, one of them being the slow version, in order to make a decision. Finally, we can see that the relaxed decision function rectifies the problem with the original profiling decision function of it choosing the wrong option in some cases. For Figure 7a we can see that at 12 and 16 threads the relaxed profiler makes the correct choice, and the same for Figure 7b at 16 threads (where the performance of the relaxed profile decision function is comparable to the heuristics decision function). IX. C ONCLUSION The main focus of this work was to investigate the possibility of dynamically choosing at runtime the best loop of a nested loop region which best utilises the available threads. We have successfully created a source-to-source compiler and a runtime library in order to automatically allow a dynamic choice to be made at runtime. As our solution uses a directives based approach, similar to OpenMP, we requires minimum effort and code change from the users point of view. We have discovered that the current mechanism users can exploit to perform this, the OpenMP if clause, does not perform efficiently (at least for the implementation we tested). Despite the fact that this behaviour is the result of the inefficient implementation of the GCC compiler which was used in this work, the same compiler with the code duplication mode was able to provide additional speedup in the execution time of the code. From this we conclude that by relying on the OpenMP runtime library to perform loop nesting, the execution time is limited by the compilers implementation of nested parallel regions. Although code duplication is considered to be a bad programming practice, when it is done automatically, it can eliminate unnecessary parallel overheads. 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Avoiding Your Teacher’s Mistakes: Training Neural Networks with Controlled Weak Supervision Mostafa Dehghani1 Aliaksei Severyn2 Sascha Rothe2 Jaap Kamps1 1 University of Amsterdam 2 Google Research [email protected], [email protected], [email protected], [email protected] arXiv:1711.00313v2 [cs.LG] 7 Dec 2017 Abstract In this paper, we propose a semi-supervised learning method where we train two neural networks in a multi-task fashion: a target network and a confidence network. The target network is optimized to perform a given task and is trained using a large set of unlabeled data that are weakly annotated. We propose to weight the gradient updates to the target network using the scores provided by the second confidence network, which is trained on a small amount of supervised data. Thus we avoid that the weight updates computed from noisy labels harm the quality of the target network model. We evaluate our learning strategy on two different tasks: document ranking and sentiment classification. The results demonstrate that our approach not only enhances the performance compared to the baselines but also speeds up the learning process from weak labels. 1 Introduction Deep neural networks have shown impressive results in a lot of tasks in computer vision, natural language processing, and information retrieval. However, their success is conditioned on the availability of exhaustive amounts of labeled data, while for many tasks such a data is not available. Hence, unsupervised and semi-supervised methods are becoming increasingly attractive. Using weak or noisy supervision is a straightforward approach to increase the size of the training data. For instance in web search, for the task of ranking, the ideal training data would be rankings of documents ordered by relevance for a large set of queries. However, it is not practical to collect such a data in large scale and only a small set of judged query-document pairs is available. However, for this task, the output of heuristic methods (Dehghani et al., 2017c) or clickthrough logs (Joachims, 2002) can be used as weak or noisy signals along with a small amount of labeled data to train learning to rank models. This is usually done by pre-training the network on weak data and fine-tuning it with true labels (Dehghani et al., 2017c; Severyn and Moschitti, 2015a). However, these two independent stages do not leverage the full capacity of information from true labels. For instance, in the pet-raining stage there is no handle to control the extent to which the data with weak labels contribute in the learning process, while they can be of different quality. In this paper, we propose a semi-supervised method that leverages a small amount of data with true labels along with a large amount of data with weak labels. Our proposed method has three main components: A weak annotator, which can be a heuristic model, a weak classifier, or even human via crowdsourcing and it is employed to annotate massive amount of unlabeled data, a target network which uses a large set of weakly annotated instances by weak annotator to learn the main task, and a confidence network which is trained on a small human-labeled set to estimate confidence scores for instances annotated by weak annotator. We train target network and confidence network in a multi-task fashion. In a joint learning process, target network and confidence network try to learn a suitable representation of the data and this layer is shared between them as a two-way communication channel. The target network tries to learn to predict the label of the given input under the supervision of the weak annotator. In the same time, the output of confidence network, which are the confidence scores, define the magnitude of the weight updates to the target network with respect to the loss computed based on labels from weak annotator, during the back- propagation phase of the target network. This way, confidence network helps target network to avoid mistakes of her teacher, i.e. weak annotator, by down-weighting the weight updates from weak labels that do not look reliable to confidence network. From a meta-learning perspective (Dehghani et al., 2017b), the goal of the confidence network trained jointly with the target network is to calibrate the learning rate for each instance in the batch. I.e., the weights w of the target network fw at step t+1 are updated as follows: wt − w t+1 =w lt b wt ) (1) ∑cθ (τi ,ỹi )∇L(fwt (τi ),y˜i )+∇R(w b i=1 where lt is the global learning rate, b is the batch size, L(⋅) the loss of predicting ŷ = fw (τ ) for an input τ when the target label is ỹ; cθ (⋅) is a scoring function learned by the confidence network taking input instance τ i and its noisy label ỹi and R(.) is the regularization term. Thus, we can effectively control the contribution to the parameter updates for the target network from weakly labeled instances based on how reliable their labels are according to the confidence network (learned on a small supervised data). Our setup requires running a weak annotator to label a large amount of unlabeled data, which is done at pre-processing time. For many tasks, it is possible to use a simple heuristic, or implicit human feedback to generate weak labels. This set is then used to train the target network. In contrast, a small expert-labeled set is used to train the confidence network, which estimates how good the weak annotations are, i.e. controls the effect of weak labels on updating the parameters of the target network. Our method allows learning different types of neural architectures and different tasks, where a meaningful weak annotator is available. In this paper, we study the performance of our proposed model by focusing on two applications in information retrieval and natural language processing: document ranking and sentiment classification. Whilst these two applications differ considerably, as do the exact operationalization of our model to these cases, there are also some clear similarities. First, in both cases the human gold standard data is based on a cognitively complex, or subjective, judgments causing high interrater variation, increasing both the cost of obtaining labels as the need for larger sets of labels. Second, also in both cases, the weak supervision signal is more systemic or objective, which facilitates the learning of the data representation. Our experimental results suggest that the proposed method is more effective in leveraging large amounts of weakly labeled data compared to traditional fine-tuning in both tasks. We also show that explicitly controlling the weight updates in the target network with the confidence network leads to faster convergence since the filtered supervision signals are more solid and less noisy. In the following, in Section 2, we introduce the general architecture of our model and explain the training process. Then, we describe the details of the applications to which we apply our model in Section 3. In Section 4 we present the experimental setups for each of the tasks along with its results and analysis. We then review related works and conclude the paper. 2 The Proposed Method In the following, we describe our recipe for semi-supervised learning of neural networks, in a scenario where along with a small human-labeled training set a large set of weakly labeled instances is leveraged. Formally, given a set of unlabeled training instances, we run a weak annotator to generate weak labels. This gives us the training set U . It consists of tuples of training instances τi and their weak labels ỹi , i.e. U ={(τi ,ỹi ),...}. For a small set of training instances with true labels, we also apply the weak annotator to generate weak labels. This creates the training set V , consisting of triplets of training instances τj , their weak labels ỹj , and their true labels yj , i.e. V = {(τj , ỹj ,yj ),...}. We can generate a large amount of training data U at almost no cost using the weak annotator. In contrast, we have only a limited amount of data with true labels, i.e. ∣V ∣<<∣U ∣. 2.1 General Architecture In our proposed framework we train a multi-task neural network that jointly learns the confidence score of weak training instances and the main task using controlled supervised signals. The high-level representation of the model is shown in Figure 1: it comprises a weak annotator and two neural networks, namely the confidence network and the target network. The goal of the weak annotator is to provide weak labels ỹi for all the instances τi ∈ U ∪V . We have this assumption that ỹi provided by the weak annotator are imperfect estimates of true labels yi , where yi are available for set V , but not for set U . Prediction loss wrt. the weak labels Supervision Layer Prediction loss wrt. the weak labels Confidence Network Supervision Layer Goodness of instances Representation Learning Weak Annotator Goodness of instances Representation Learning True Labels (a) Full Supervision Mode: Training on batches of data with true labels. Confidence Network Weak Annotator True Labels (b) Weak Supervision Mode: Training on batches of data with weak labels. Figure 1: Learning from controlled weak supervision: Our proposed multi-task network for learning a target task in a semi-supervised fashion, using a large amount of weakly labeled data and a small amount of data with true labels. Faded parts of the network are disabled during the training in the corresponding mode. Red-dotted arrows show gradient propagation. Parameters of the parts of the network in red frames get updated in the backward pass, while parameters of the network in blue frames are fixed during the training. The goal of the confidence network is to estimate the confidence score c̃j of training instances. It is learned on triplets from training set V : input τj , its weak label ỹj , and its true label yj . The score c̃j is then used to control the effect of weakly annotated training instances on updating the parameters of the target network in its backward pass during backpropagation. The target network is in charge of handling the main task we want to learn, or in other words, approximating the underlying function that predicts the correct labels. Given the data instance, τi and its weak label ỹi from the training set U , the target network aims to predict the label ŷi . The target network parameter updates are based on noisy labels assigned by the weak annotator, but the magnitude of the gradient update is based on the output of the confidence network. Both networks are trained in a multi-task fashion alternating between the full supervision and the weak supervision mode. In the full supervision mode, the parameters of the confidence network get updated using batches of instances from training set V . As depicted in Figure 1b, each training instance is passed through the representation layer mapping inputs to vectors. These vectors are concatenated with their corresponding weak labels ỹj generated by the weak annotator. The confidence network then estimates c̃j , which is the probability of taking data instance j into account for training the target network. In the weak supervision, mode the parameters of the target network are updated using training set U . As shown in Figure 1a, each training instance is passed through the same representation learning layer and is then processed by the supervision layer which is a part of the target network predicting the label for the main task. We also pass the learned representation of each training instance along with its corresponding label generated by the weak annotator to the confidence network to estimate the confidence score of the training instance, i.e. c̃i . The confidence score is computed for each instance from set U . These confidence scores are used to weight the gradient updating target network parameters or in other words the step size during back-propagation. It is noteworthy that the representation layer is shared between both networks, so besides the regularization effect of layer sharing which leads to better generalization, sharing this layer lays the ground for the confidence network to benefit from the largeness of set U and the target network to utilize the quality of set V . 2.2 Model Training Our optimization objective is composed of two terms: (1) the confidence network loss Lc , which captures the quality of the output from the confidence network and (2) the target network loss Lt , which expresses the quality for the main task. Both networks are trained by alternating between the weak supervision and the full supervision mode. In the full supervision mode, the parameters of the confidence network are updated using training instance drawn from training set V . We use cross-entropy loss function for the confidence network to capture the difference between the predicted confidence score of instance j, i.e. c̃j and the target score cj : Ranker Lc = ∑ −cj log(c̃j )−(1−cj )log(1−c̃j ), (2) j∈V The target score cj is calculated based on the difference of the true and weak labels with respect to the main task. In the weak supervision mode, the parameters of the target network are updated using training instances from U . We use a weighted loss function, Lt , to capture the difference between the predicted label ŷi by the target network and target label ỹi : Lt = ∑ c̃i Li , Compositionality Embedding Weights (3) i∈U where Li is the task-specific loss on training instance i and c̃i is the confidence score of the weakly annotated instance i, estimated by the confidence network. Note that c̃i is treated as a constant during the weak supervision mode and there is no gradient propagation to the confidence network in the backward pass (as depicted in Figure 1a). We minimize two loss functions jointly by randomly alternating between full and weak supervision modes (for example, using a 1:10 ratio). During training and based on the chosen supervision mode, we sample a batch of training instances from V with replacement or from U without replacement (since we can generate as much train data for set U ). Since in our setups usually ∣U ∣ >> ∣V ∣, the training process oversamples the instance from V . The key point here is that the “main task” and “confidence scoring” task are always defined to be close tasks and sharing representation will benefit the confidence network as an implicit data augmentation to compensate the small amount of data with true labels. Besides, we noticed that updating the representation layer with respect to the loss of the other network acts as a regularization for each of these networks and helps generalization for both target and confidence network since we try to capture all tasks (which are related tasks) and less chance for overfitting. We also investigated other possible setups or training scenarios. For instance, we tried updating the parameters of the supervision layer of the target network using also data with true labels. Or instead of using alternating sampling, we tried training the target network using controlled weak supervision signals after the confidence network is fully trained. As shown in the experiments the architecture and training strategy described above provide the best performance. Figure 2: The target network for the document ranking. 3 Applications In this section, we apply our semi-supervised method to two different tasks: document ranking and sentiment classification. For each task, we start with an introduction of the task, followed by the setup of the target network, i.e. description of the representation learning layer and the supervision layer. 3.1 Document Ranking This task is the core information retrieval problem which is challenging as it needs to capture the notion of relevance between query and documents. We employ a state-of-the-art pairwise neural ranker architecture as target network (Dehghani et al., 2017c). In this setting, each training instance τ consists of a query q, and two documents d+ and d− . The labels, ỹ and y, are scalar values indicating the probability of d+ being ranked higher than d− with respect to q. The general schema of the target network is illustrated in Figure 2. The Representation Learning Layer is a setup proposed in (Dehghani et al., 2017c). This layer is a function ψ, which learns the representation of the input data instances, i.e. (q,d+ ,d− ), and consists of three components: (1) an embedding function ε ∶ V → Rm (where V denotes the vocabulary set and m is the number of embedding dimensions), (2) a weighting function ω ∶ V → R, and (3) a compositionality function ⊙ ∶ (Rm , R)n → Rm . More formally, the function ψ is defined as: ∣q∣ ψ(q,d+ ,d− )=[⊙i=1 (ε(tqi ),ω(tqi )) ∣∣ ∣d+ ∣ + + ∣d− ∣ − − ⊙i=1 (ε(tdi ),ω(tdi )) ∣∣ ⊙i=1 (ε(tdi ),ω(tdi )) ], (4) where tqi and tdi denote the ith term in query q respectively document d. The embedding function ε maps each term to a dense m- dimensional real value vector, which is learned during the training phase. The weighting function ω assigns a weight to each term in the vocabulary. The compositionality function ⊙ projects a set of n embedding-weighting pairs to an m- dimensional representation, independent from the value of n: n ⊙(ε(ti ),ω(ti ))= i=1 ∑ni=1 exp(ω(ti ))⋅ε(ti ) , ∑nj=1 exp(ω(tj )) (5) which is in fact the normalized weighted elementwise summation of the terms’ embedding vectors. It has been shown that having global term weighting function along with embedding function improves the performance of ranking as it simulates the effect of inverse document frequency (IDF), which is an important feature in information retrieval (Dehghani et al., 2017c). In our experiments, we initialize the embedding function ε with word2vec embeddings (Mikolov et al., 2013) pre-trained on Google News and the weighting function ω with IDF. The Supervision Layer receives the vector representation of the inputs processed by the representation learning layer and outputs a prediction ỹ. We opt for a simple fully connected feed-forward network with l hidden layers followed by a softmax. Each hidden layer zk in this network computes zk = α(Wk zk−1 +bk ), where Wk and bk denote the weight matrix and the bias term corresponding to the k th hidden layer and α(.) is the non-linearity. These layers follow a sigmoid output. We employ the weighted cross entropy loss: Lt = ∑ c̃i [−ỹi log(ŷi )−(1− ỹi )log(1− ŷi )], (6) i∈BU where BU is a batch of instances from U , and c̃i is the confidence score of the weakly annotated instance i, estimated by the confidence network. The Weak Annotator is BM25 (Robertson et al., 2009) which is a well-performing unsupervised retrieval method. In the pairwise documents ranking setup, ỹi for a given instance τj =(q,d+ ,d− ) is the probability of document d+ being ranked higher than d− , based on the scores obtained from the annotator: ỹi =Pq,d+ ,d− = sq,d+ , sq,d+ +sq,d− (7) whereas sq,d is the score obtained from the weak annotator. To train the confidence network, the target label cj is calculated using the absolute difference of the true label and the weak label: cj =1−∣yj − ỹj ∣, where yj is calculated similar to ỹi , but sq,d comes from true labels created by humans. 3.2 Sentiment Classification This task aims to identify the sentiment (e.g., positive, negative, or neutral) underlying an individual sentence. Our target network is a convolutional model similar to (Deriu et al., 2017; Severyn and Moschitti, 2015a,b; Deriu et al., 2016). Each training instance τ consists of a sentence s and its sentiment label ỹ. The architecture of the target network is illustrated in Figure 3 The Representation Learning Layer learns a representation for the input sentence s and is shared between the target network and confidence network. It consists of an embedding function ε ∶ V → Rm , where V denotes the vocabulary set and m is the number of embedding dimensions. This function maps the sentence to a matrix S ∈ Rm×∣s∣ , where each column represents the embedding of a word at the corresponding position in the sentence. Matrix S is passed through a convolution layer. In this layer, a set of f filters is applied to a sliding window of length h over S to generate a feature map matrix O. Each feature map oi for a given filter F is generated by oi = ∑k,j S[i ∶ i+h]k,j Fk,j , where S[i∶i+h] denotes the concatenation of word vectors from position i to i+h. The concatenation of all oi produces a feature vector o∈R∣s∣−h+1 . The vectors o are then aggregated over all f filters into a feature map matrix O ∈Rf ×(∣s∣−h+1) . We also add a bias vector b ∈ Rf to the result of a convolution. Each convolutional layer is followed by a non-linear activation function (we use ReLU(Nair and Hinton, 2010)) which is applied element-wise. Afterward, the output is passed to the max pooling layer which operates on columns of the feature map matrix O returning the largest value: pool(oi ) ∶ R1×(∣s∣−h+1) → R (see Figure 3). This architecture is similar to the state-of-the-art model for Twitter sentiment classification from Semeval 2015 and 2016 (Severyn and Moschitti, 2015b; Deriu et al., 2016). We initialize the embedding matrix with word2vec embeddings (Mikolov et al., 2013) pretrained on a collection of 50M tweets. The Supervision Layer is a feed-forward neural Classifier Pooled Repr. Conv. Feature Map Embedding Embedding Figure 3: The target network for the sentiment classification. network similar to the supervision layer in the ranking task (with different width and depth) but with softmax instead of sigmoid as the output layer which returns ŷi , the probability distribution over all three classes. We employ the weighted cross entropy loss: Lt = ∑ c̃i ∑ −ỹik log(ŷik ), i∈BU (8) k∈K where BU is a batch of instances from U , and c̃i is the confidence score of the weakly annotated instance i, and K is a set of classes. The Weak Annotator for the sentiment classification task is a simple unsupervised lexicon-based method (Hamdan et al., 2013; Kiritchenko et al., 2014). We use SentiWordNet03 (Baccianella et al., 2010) to assign probabilities (positive, negative and neutral) for each token in set U . Then a sentencelevel distribution is derived by simply averaging the distributions of the terms, yielding a noisy label ỹi ∈ R∣K∣ , where ∣K∣ is the number of classes, i.e. ∣K∣=3. We empirically found that using soft labels from the weak annotator works better than assigning a single hard label. The target label cj for the confidence network is calculated by using the mean absolute difference of the true label and the weak 1 label: cj =1− ∣K∣ ∑k∈K ∣yjk − ỹjk ∣, where yj is the onehot encoding of the sentence label over all classes. 4 Experiments and Results Here we first describe baselines. Afterward, we present the experimental setups for each of our tasks along with their results and analysis. 4.1 Baselines and General Setups For both tasks, we evaluate the performance of our method compared to the following baselines: • (1.WA) Weak Annotator, i.e. the unsupervised method that we used for annotating the unlabeled data. • (2.WSO) Weak Supervision Only, i.e. the target network trained only on weakly labeled data. • (3.FSO) Full Supervision Only, i.e. the target network trained only on true labeled data. • (4.WS+FT) Weak Supervision + Fine Tuning, i.e. the target network trained on the weakly labeled data and fine-tuned on true labeled data. • (5.WS+SFT) Weak Supervision + Supervision Layer Fine-Tuning, i.e. the target network trained only on weakly labeled data and the supervision layer is fine-tuned on true labeled data while the representation learning layer is kept fixed. • (6.WS+RFT) Weak Supervision + Representation Fine Tuning, i.e. WS+SFT, except the supervision layer is kept fixed during fine tuning. • (7.NLI) New Label Inference (Veit et al., 2017) is similar to our proposed neural architecture inspired by the teacher-student paradigm (Hinton et al., 2015; Romero et al., 2014), but instead of having the confidence network to predict the “confidence score” of the training instance, there is a label generator network which is trained on set V to map the weak labels of the instances in U to the new labels. The new labels are then used as the target for training the target network. • (8.CWSJT ) Controlled Weak Supervision with Joint Training is our proposed neural architecture in which we jointly train the target network and the confidence network by alternating batches drawn from sets V and U (as explained in Section 2.2). • (9.CWSJT+ ) Controlled Weak Supervision + Full Supervision with Joint Training is the same as CWSJT , except that parameters of the supervision layer in target network are also updated using batches from V , with regards to the true labels. Additionally, we compare the performance of CWSJT , with other possible training setups: • (a.CWSST ) Separate Training, i.e. we consider the confidence network as a separate network, without sharing the representation learning layer, and train it on set V . We then train the target network on the controlled weak supervision signals. • (b.CWSCT ) Circular Training, i.e. we train the target network on set U . Then the confidence network is trained on data with true labels, and the target network is trained again but on controlled weak supervision signals. • (c.CWSPT ) Progressive Training is the mixture of the two previous baselines. Inspired by (Rusu et al., 2016), we transfer the learned information from the converged target network to the confidence network using progressive training. We then train the target network again on the controlled weak supervision signals. The proposed architectures are implemented in TensorFlow (Tang, 2016; Abadi et al., 2015). We use the Adam optimizer (Kingma and Ba, 2014) and the back-propagation algorithm. Furthermore, to prevent feature co-adaptation, we use dropout (Srivastava et al., 2014) as a regularization technique in all models. In our setup, the confidence network to predict c̃j is a fully connected feed forward network. Given that the confidence network is learned only from a small set of true labels and to speed up training we initialize the representation learning layer with pre-trained parameters, i.e., pre-trained word embeddings. We use ReLU (Nair and Hinton, 2010) as a non-linear activation function α in both target network and confidence network. In the following, we describe task-specific setups and the experimental results. 4.2 Document Ranking Setup & Results Collections. We use two standard TREC collections for the task of ad-hoc retrieval: The first collection (Robust04) consists of 500k news articles from different news agencies as a homogeneous collection. The second collection (ClueWeb) is ClueWeb09 Category B, a large-scale web collection with over 50 million English documents, which is considered as a heterogeneous collection. Spam documents were filtered out using the Waterloo spam scorer 1 (Cormack et al., 2011) with the default threshold 70%. Data with true labels. We take query sets that contain human-labeled judgments: a set of 250 queries (TREC topics 301–450 and 601–700) for the Robust04 collection and a set of 200 queries (topics 1-200) for the experiments on the ClueWeb collection. For each query, we take all documents judged as relevant plus the same number of documents judged as non-relevant and form pairwise combinations among them. Data with weak labels. We create a query set Q using the unique queries appearing in the AOL 1 http://plg.uwaterloo.ca/˜gvcormac/clueweb09spam/ query logs (Pass et al., 2006). This query set contains web queries initiated by real users in the AOL search engine that were sampled from a three-month period from March 2006 to May 2006. We applied standard pre-processing (Dehghani et al., 2017c,a) on the queries. We filtered out a large volume of navigational queries containing URL substrings (“http”, “www.”, “.com”, “.net”, “.org”, “.edu”). We also removed all non-alphanumeric characters from the queries. For each dataset, we took queries that have at least ten hits in the target corpus using our weak annotator method. Applying all these steps, We collect 6.15 million queries to train on in Robust04 and 6.87 million queries for ClueWeb. To prepare the weakly labeled training set U , we take the top 1000 retrieved documents using BM25 for each query from training query set Q, which in total leads to ∼∣Q∣×106 training instances. Parameters and Settings. We conducted a nested 3-fold cross validation with 80/20 training/validation split in each fold. All hyperparameters of all models and baselines were tuned individually on the validation set using batched GP bandits with an expected improvement acquisition function (Desautels et al., 2014). The size and number of hidden layers for the ranker and the confidence network were separately selected from {64, 128, 256, 512} and {1, 2, 3, 4}, respectively. The initial learning rate and the dropout parameter were selected from {10−3 ,10−5 } and {0.0,0.2,0.5}, respectively. We considered embedding sizes of {300,500}. The batch size in our experiments was set to 128. In all experiments, the parameters of the network are optimized employing the Adam optimizer (Kingma and Ba, 2014) and using the computed gradient of the loss to perform the back-propagation algorithm. At inference time, for each query, we take the top 2000 retrieved documents using BM25 as candidate documents and re-rank them using the trained models. We use the Indri2 implementation of BM25 with default parameters (i.e., k1 =1.2, b=0.75, and k3 =1000). Results and Discussions. We evaluate on set V and report two standard evaluation metrics: mean average precision (MAP) of the top-ranked 1000 documents and normalized discounted cumulative gain calculated for the top 20 retrieved documents (nDCG@20). Statistical significant differences of MAP and nDCG@20 values are determined using 2 https://www.lemurproject.org/indri.php Table 1: Performance of the proposed method and baseline models on different datasets. (Ĳ or Ź indicates that the improvements or degradations are statistically significant, at the 0.05 level using the paired two-tailed t-test. For all model, the improvement/degradations is with respect to the “weak supervision only” baseline (WSO). For CWSJT , the improvement over all baselines is considered and the Bonferroni correction is applied on the significant tests.) Method 1 WABM25 2 3 WSO 4 5 6 WS+FT 7 8 9 NLI FSO WS+SFT WS+RFT CWSJT CWS+JT Robust04 Table 2: Performance of the variants of the proposed method on different datasets. (Ĳ orŹ indicates that the improvements or degradations are statistically significant, at the 0.05 level using the paired two-tailed t-test. For all model, the improvement/degradations is with respect to the “weak supervision only” baseline (WSO on Table 1) . For CWSJT , the improvement over all baselines is considered and the Bonferroni correction is applied on the significant tests.) ClueWeb MAP nDCG@20 MAP nDCG@20 0.2503 0.4102 0.1021 0.2070 0.2702 0.1790Ź 0.4290 0.3519Ź 0.1297 0.0782Ź 0.2201 0.1730Ź 0.2830Ĳ 0.2711 0.2810Ĳ 0.4355Ĳ 0.4203 0.4316 0.1346Ĳ 0.1002Ź 0.1286 0.2346Ĳ 0.1940Ź 0.2240 0.2421Ź 0.3024Ĳ 0.2786Ĳ 0.4092Ź 0.4507Ĳ 0.4367Ĳ 0.1010Ź 0.1372Ĳ 0.1310 0.2004Ź 0.2453Ĳ 0.2244 the two-tailed paired t-test with p value < 0.05, with Bonferroni correction. Table 1 shows the performance on both datasets. Based on the results, CW S JT provides a significant boost on the performance over all datasets. There are two interesting points we want to highlight. First, among the fine-tuning experiments, updating all parameters of the target network is the best fine tuning strategy. Updating only the parameters of the representation layer based on the true labels works better than updating only parameters of the supervision layer. This supports our designed choice of a shared embedding layer which gets updated on set V . Second, while it seems reasonable to make use of true labels for updating all parameters of the target network, CWS+JT achieves no better results than CWSJT . It also performs mostly even worse than WS+FT. This is because during training, the direction of the parameter optimization is highly affected by the type of supervision signal and while we control the magnitude of the gradients, we do not change their directions, so alternating between two sets with different label qualities (different supervision signal types, i.e. weak and string) confuses the supervision layer of the target network. In fine tinning, we don not have this problem since we optimize the parameters with respect to the supervision from these two sets in two separate stages. It is noteworthy that we have also tried CWS+JT with another objective function for the target network taking both weak and true labels into account which was slightly better, but gives no Method a b c CWSST CWSCT CWSPT CWSJT Robust04 ClueWeb MAP nDCG@20 MAP nDCG@20 0.2716 0.2961Ĳ 0.2784Ĳ 0.3024Ĳ 0.4237 0.4440Ĳ 0.4292 0.4507Ĳ 0.1320 0.1378Ĳ 0.1314 0.1372Ĳ 0.2213 0.2431Ĳ 0.2207 0.2453Ĳ improvement over CWSJT . In the ranking task, the target network is designed in particular to be trained on weak annotations (Dehghani et al., 2017c), hence training the network only on weak supervision performs better than FSO. This is due to the fact that ranking is a complex task requiring many training instances, while relatively few true labels are available. The performance of NLI is worse than CWSJT as learning a mapping from imperfect labels to accurate labels and training the target network on new labels is essentially harder than learning to filter out the noisy labels, hence needs a lot of supervised data. The reason is that for the ranking, due to a few training instances with regards to the task complexity, NLI fails to generate better new labels, hence it directly misleads the target network and completely fails to improve the performance. Table 2 shows the performance of different training strategies. As shown, CWSJT and CWSCT perform better than other strategies. CWSCT is to let the confidence network to be trained separately, while still being able to enjoy shared learned information from the target network. However, it is less efficient as we need two rounds of training on weakly labeled data. CWSST performs poorly since the training data V is too small to train a high-quality confidence network without taking advantage of the vast amount of weakly annotated data in U . We also noticed that this strategy leads to a slow convergence compared to WSO. Also transferring learned information from target network to confidence network via progressive training, i.e. CWSPT , performs no better than full sharing of the representation learning layer. Table 3: Performance of the baseline models as well as the proposed method on different datasets. (Ĳ orŹ indicates that the improvements or degradations are statistically significant, at the 0.05 level using the paired two-tailed t-test. For all model, the improvement/degradations is with respect to the “weak supervision only” baseline (WSO). For CWSJT , the improvement over all baselines is considered and the Bonferroni correction is applied on the significant tests.) Method SemEval-14 SemEval-15 1 WALexicon 0.5141 0.4471 2 3 WSO 0.6719 0.6307 0.5606 0.5811 4 5 6 WS+FT 0.7080Ĳ 0.6875 0.6932 0.6441Ĳ 0.6193Ĳ 0.6102Ĳ 7 8 9 NLI 0.7113Ĳ 0.7362Ĳ 0.7310Ĳ 0.7162Ĳ 0.6433Ĳ 0.6626Ĳ 0.6551Ĳ 0.6618Ĳ FSO WS+SFT WS+RFT CWSJT CWS+JT SemEval1th Table 4: Performance of the variants of the proposed method for sentiment classification task, on different datasets. (Ĳ orŹ indicates that the improvements or degradations are statistically significant, at the 0.05 level using the paired two-tailed t-test. For all model, the improvement/degradations is with respect to the “weak supervision only” baseline (WSO on Table 3) . For CWSJT , the improvement over all baselines is considered and the Bonferroni correction is applied on the significant tests.) Method a b c CWSST CWSCT CWSPT CWSJT 4.3 SemEval-14 Ĳ 0.7183 0.7363Ĳ 0.7009Ĳ 0.7362Ĳ SemEval-15 0.6501Ĳ 0.6667Ĳ 0.6118Ĳ 0.6626Ĳ Sentiment Classification Setup & Results Collections. We test our model on the twitter message-level sentiment classification of SemEval-15 Task 10B (Rosenthal et al., 2015). Datasets of SemEval-15 subsume the test sets from previous editions of SemEval, i.e. SemEval-13 and SemEval-14. Each tweet was preprocessed so that URLs and usernames are masked. Data with true labels. We use train (9,728 tweets) and development (1,654 tweets) data from SemEval13 for training and SemEval-13-test (3,813 tweets) for validation. To make our results comparable to the official runs on SemEval we use SemEval-14 (1,853 tweets) and SemEval-15 (2,390 tweets) as test sets (Rosenthal et al., 2015; Nakov et al., 2016). Data with weak labels. We use a large corpus containing 50M tweets collected during two months for both, training the word embeddings and creating the weakly annotated set U using the lexicon-based method explained in Section 3.2. Parameters and Settings. Similar to the docu- ment ranking task, we tuned the hyper-parameters for each model, including baselines, separately with respect to the true labels of the validation set using batched GP bandits with an expected improvement acquisition function (Desautels et al., 2014). The size and number of hidden layers for the classifier and the confidence network were separately selected from {32,64,128} and {1,2,3}, respectively. We tested the model with both, 1 and 2 convolutional layers. The number of convolutional feature maps and the filter width is selected from {200, 300} and {3, 4, 5}, respectively. The initial learning rate and the dropout parameter were selected from {1E − 3,1E − 5} and {0.0,0.2,0.5}, respectively. We considered embedding sizes of {100,200} and the batch size in these experiments was set to 64. Results and Discussion. We report the performance of our model and the baseline models in terms of official SemEval metric, Macro-F1, in Table 3. We have also report statistical significance of F1 improvements using two-tailed paired t-test with p value < 0.05, with Bonferroni correction. Our method is the best performing among all the baselines. Unlike the ranking task, training the network only on data with true labels, i.e. TSO, performs rather good. In the sentiment classification task, learning representation of input which is a sentence (tweet) is simpler than the ranking task in which we try to learn representation for query and long documents. Consequently, we need fewer data to be able to learn a suitable representation and with the amount of available data with true labels, we can already capture a rather good representation without helps of weak data, while it was impossible in the ranking task. However, as the results suggest, we can still gain improvement using fine-tuning. In this task, behaviors of different fine-tuning experiments are similar to the ranking task. Furthermore, updating parameters of the supervision layer, with respect to the true labels, i.e. CWS+JT model, does not perform better than CWSJT , which again supports our choice of updating just the representation learning layer with respect to the signals from data with true labels. In the sentiment classification task, the performance of NLI is acceptable compared to the ranking task. This is first of all because generating new classification labels is essentially simpler. Secondly, in this task, we need to learn to represent a simpler input, and learn a simpler function to predict the labels, but a relatively bigger set of supervised data which helps to generate new labels. However, the performance of NLI is still lower than CWSJT . We can argue that CWSJT is a more conservative approach. It is in fact equipped with a soft filter that decreases the effect of noisy training examples from set U on parameter updates during training. This is a smoother action as we just down-weight the gradient, while NLI might change the direction of the gradient by generating a completely new label and consequently it is prone to more errors, especially when there is not enough high-quality training data to learn to generate better labels. In the sentiment classification task, besides the general baselines, we also report the best performing systems, which are also convolution-based models (Rouvier and Favre (2016) on SemEval-14; Deriu et al. (2016) on SemEval-15). Our proposed model outperforms the best system on both datasets. Table 4 also presents the results of different training strategies for the sentiment classification task. As shown, similar to the ranking task, CWSJT and CWSCT perform better than other strategies. Although CWSCT is slightly better (not statistically significant) in terms of effectiveness compared to CWSJT , it is not as efficient as CWSJT during training. Compared to the ranking task, for sentiment classification, it is easier to estimate the confidence score of instances with respect to the amount of available supervised data. Therefore, CWSST is able to improve the performance over WSO significantly. Moreover, CWSPT fails compared to the strategies where the representation learning layer is shared between the target network and the confidence network. 4.4 Faster Learning Pace Controlling the effect of supervision to train neural networks not only improves the performance, but also provides the network with more solid signals which speeds up the learning process. Figure 4 illustrates the training/validation loss for both networks, compared to the loss of training the target network with weak supervision, along with their performance on test sets, with respect to different amounts of training data for the sentiment classification task3 . As shown, in the training, the loss of the target network in our model, i.e. Lt is higher than the loss of the network which is trained only on weakly 3 We have observed similar speed-up in the learning process of the ranking task, however we skip bringing its plots due to space limit since we have nested cross-validation for the ranking task and a set of plots for each fold. Figure 4: Loss of the target network (Lt ) and the confidence network (Lc ) compared to the loss of WSO (LWSO ) on training/validation set and performance of CWS, WSO, and WA on test sets with respect to different amount of training data on sentiment classification. supervised data, i.e. LWSO . However, since these losses are calculated with respect to the weak labels (not true labels), having very low training loss can be an indication of overfitting to the imperfection in the weak labels. In other words, regardless of the general problem of lack of generalization due to overfitting, in the setup of learning from weak labels, predicting labels that are similar to train labels (very low training loss) is not necessarily a desirable incident. In the validation set, however, Lt decreases faster than LWSO , which supports the fact that LWSO overfits to the imperfection of weak labels, while our setup helps the target network to escape from this imperfection and do a good job on the validation set. In terms of the performance, compared to WSO, the performance of CWS on both test sets increases very quickly and CWS is able to pass the performance of the weak annotator by seeing much fewer instances annotated by the weak annotator. 5 Related Work Learning from weak or noisy labels has been studied in the literature (Frénay and Verleysen, 2014). We briefly review research most relevant to our work. There are semiSemi-supervised learning. supervised learning algorithms (Zhu, 2005) developed to utilize weakly or even unlabeled data. Self-training (Rosenberg et al., 2005) or pseudo-labeling (Lee, 2013) tries to predict labels of unlabeled data. This unlabeled data is provided additionally. In particular for neural networks, methods use greedy layer-wise pre-training of weights using unlabeled data alone followed by supervised fine-tuning (Deriu et al., 2017; Severyn and Moschitti, 2015b,a; Go et al., 2009). Other methods learn unsupervised encodings at multiple levels of the architecture jointly with a supervised signal (Ororbia II et al., 2015; Weston et al., 2012). Meta-learning. From the meta-learning perspective, our approach is similar to Andrychowicz et al. (2016) where a separate recurrent neural network called optimizer learns to predict an optimal update rule for updating parameters of the target network. The optimizer receives a gradient from the target network and outputs the adjusted gradient matrix. As the number of parameters in modern neural networks is typically on the order of millions the gradient matrix becomes too large to feed into the optimizer, so the approach of Andrychowicz et al. (2016) is applied to very small models. In contrast, our approach leverages additional weakly labeled data where we use the confidence network to predict per-instance scores that calibrate gradient updates for the target network. Direct learning with weak/noisy labels. Many studies tried to address learning in the condition of imperfect labels. Some noise cleansing methods were proposed to remove or correct mislabeled instances (Brodley and Friedl, 1999). Other studies showed that weak or noisy labels can be leveraged by employing a particular architecture or defining a proper loss function to avoid overfitting the training data imperfection (Dehghani et al., 2017c; Patrini et al., 2016; Beigman and Klebanov, 2009; Zeng et al., 2015; Bunescu and Mooney, 2007). Modeling imperfection. There is also research trying to model the pattern of the noise or weakness in the labels. Some methods leverage generative models to denoise weak supervision sources that a discriminative model can learn from (Ratner et al., 2016; Rekatsinas et al., 2017; Varma et al., 2017). Other methods aim to capture the pattern of the noise by inserting an extra layer or a separated module (Sukhbaatar et al., 2014; Veit et al., 2017), infer better labels from noisy labels and use them to supervise the training of the network. This is inspired by the teacher-student paradigm (Hinton et al., 2015; Romero et al., 2014; Xiao et al., 2015) in which the teacher generates a new label given the training instance with its corresponding weak or noisy label. However, as we show in our experiments, this approach is not sufficient when the amount of supervised data is not enough to generate better labels. 6 Conclusion and Future Directions Training neural networks using large amounts of weakly annotated data is an attractive approach in scenarios where an adequate amount of data with true labels is not available. In this paper, we propose a multi-task neural network architecture that unifies learning to estimate the confidence score of weak annotations and training neural networks to learn a target task with controlled weak supervision, i.e. using weak labels to updating the parameters, but taking their estimated confidence scores into account. This helps to alleviate updates from instances with unreliable labels that may harm the performance. We applied the model to two tasks, document ranking and sentiment classification, and empirically verified that the proposed model speeds up the training process and obtains more accurate results. As a promising future direction, we are going to understand to which extent using weak annotations has the potential of training high-quality models with neural networks and understand the exact conditions under which our proposed method works. References Martı́n Abadi et al. 2015. TensorFlow: Large-scale machine learning on heterogeneous systems. Software available from tensorflow.org. http://tensorflow.org/. Marcin Andrychowicz, Misha Denil, Sergio Gomez, Matthew W Hoffman, David Pfau, Tom Schaul, and Nando de Freitas. 2016. Learning to learn by gradient descent by gradient descent. In Advances in Neural Information Processing Systems. pages 3981–3989. 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Semi-supervised learning literature survey .	9
Machine Learning Application in the Life Time of Materials Xiaojiao Yu Abstract: Materials design and development typically takes several decades from the initial discovery to commercialization with the traditional trial and error development approach. With the accumulation of data from both experimental and computational results, data based machine learning becomes an emerging field in materials discovery, design and property prediction. This manuscript reviews the history of materials science as a disciplinary the most common machine learning method used in materials science, and specifically how they are used in materials discovery, design, synthesis and even failure detection and analysis after materials are deployed in real application. Finally, the limitations of machine learning for application in materials science and challenges in this emerging field is discussed. Keywords: Machine learning, Materials discovery and design, Materials synthesis, Failure detection 1. Introduction Materials science has a long history that can date back to the Bronze age 1. However, only until the 16th century, first book on metallurgy was published, marking the beginning of systematic studies in materials science 2. Researches in materials science were purely empirical until theoretical models were developed. With the advent of computers in the last century, numerical methods to solve theoretical models became available, ranging from DFT (density functional theory) based quantum mechanical modeling of electronic structure for optoelectronic properties calculation, to continuum based finite element modeling for mechanical properties 3-4. Multiscale modeling that bridge various time and spatial scales were also developed in the materials science to better simulate the real complex system 5. Even so, it takes several decades from materials discovery to development and commercialization 6-7 . Even though physical modeling can reduce the amount of time by guiding experiment work. The limitation is also obvious. DFT are only used for functional materials optoelectronic property calculation, and that is only limited to materials without defect 8 . The assumption itself is far off from reality. New concept such as multiscale modeling is still far away from large scale real industrial application. Traditional ways of materials development are impeding the progress in this field and relevant technological industry. With the large amount of complex data generated by experiment, especially simulation results from both published and archived data including materials property value, processing conditions, and microstructural images, analyzing them all becoming increasingly challenging for researchers. Inspired by the human genome initiative, Obama Government launched a Materials Genome Initiative hoping to reduce current materials development time to half 9. With the increase of computing power and the development of machine learning algorithms, materials informatics has increasingly become another paradigm in the field. Researchers are already using machine learning method for materials property prediction and discovery. Machine learning forward model are used for materials property prediction after trained on data from experiments and physical simulations. Bhadeshia et al. applied neural network(NN) technique to model creep property and phase structure in steel 10-11. Crystal structure prediction is another area of study for machine learning thanks to the large amount of structural data in crystallographic database. K -nearest- neighbor’s method was used to identify materials’ structure type based on its neighbors’ structure types 12-13 . Machine learning is also applied for materials discovery by searching compositional, structural space for desired properties, which is essentially solving a constrained optimization problem. Baerns et al. was able to find an effective multicomponent catalyst for low-temperature oxidation of lowconcentration propane with a genetic algorithm and neural network 14. There are a few reviews on machine learning application in materials science already. Dane Morgan and Gerbrand Ceder reviewed the data mining methods in materials development 15. Tim Mueller, Aaron Gilad Kusne, and Rampi Ramprasad also reviewed the progress and application of machine learning in materials science, more specifically in phase diagram, crystal structural and property prediction 16. However, their reviews are mostly based on applications in fundamental of materials science. Here, we are taking a more practical approach of reviewing machine learning application in material design, development and stages after deployment. We first discuss data problems specifically in materials science. Then, machine learning concept and most widely used methods are introduced. Up-to-date reviews on machine leaning application in materials discovery, design, development, deployment and recall is conducted. The relation between data driven research and traditional experimental and physical modeling is discussed afterwards. Finally, challenges and future endeavors of machine learning based materials science research is pointed out for researchers in this niche area. 2.1 Data Problem in Materials Science The successful application of informatics in biology, astronomy and business has inspired similar application in materials science. However, materials science differs from other subjects due to its unique characteristics. Some researchers are debating whether there is a big data problem in materials science, after all the size of materials data is nothing comparable to biology data. The largest existing database based on experimental results from materials has 5x105 data records 17. However, the rapid progress in computational science and microscopy techniques is resulting in enormous amounts of output data 18 . Furthermore, Materials science data tends to be complex and heterogeneous in terms of their sources and types ranging from discrete numerical values to qualitative descriptions of materials behavior and imaging data 19. Data in materials science also exhibit the Veracity characteristics of big data problem, by that we acknowledge the practical reality of data missing and uncertainties with the data 19 . According to the 4V (volume, variety, velocity, veracity) characteristics of big data, materials science does have a big data problem 19 . With the emergence of this big data in materials science, how to extract hidden information from the complex data and interpret resulted information is becoming increasingly important for materials design and development. 2.2 Machine Learning Methods Machine learning, a branch of artificial intelligence, is about computer learning from existing data without being explicitly programmed and make predictions on new data by building a model from input samples. Depending on the assigned task, machine learning can be classified into three categories: supervised learning, machine learning algorithms are trained with a set of input value and labeled output value first, then they are used to predict output values for corresponding unseen input values; unsupervised learning, where there is no labelled output value for training data and machine learning algorithm is used to discover patterns in the input value; reinforcement learning (program interact with environment dynamically to maximize accumulated rewards). Reinforcement learning is not used in materials science field; hence it is not introduced in detail in this manuscript. Supervised learning can either be a classification problem or a regression problem depends on the whether the output value is discrete or continuous. 2.3 Method Workflow Machine learning method typically comprise several steps including raw data collection, data preprocessing (filling in missing data, handling outliers, data transformation), feature engineering for feature selection and extraction (principle component analysis), model selection, training, validations and testing. A detailed workflow is presented in Fig 1. To select the best algorithm for a particular task, model evaluation is important. Different algorithms are evaluated with different metrics. For instance, a classifier’s evaluation metrics include confusion matrix, AUC (area under curve), precision recall, F measure, Kolomogorov Smirnov chart (K-S). Confusion matrix is a 2X2 matrix with four elements: true positive (TP), true negative(TN), false positive (FP), false negati ve(FN) 20. Other accuracy measures are 𝑇𝑃 𝑇𝑁 sensitivity (True Positive Rate= ), specificity (True negative rate= ). AUC is the area under 𝑇𝑃+𝐹𝑁 𝑇𝑁+𝐹𝑃 ROC curve, which consider the relation between sensitivity and specificity. The greater the area under the curve, the more accurate is the model. Precision is 𝑇𝑃 , recall is the true positive rate defined as 𝑇𝑃+𝐹𝑃 above. Precision-recall shows the fraction of predictions that are false positive 21. F measure is also a measure of the model accuracy and is defined as the weighted harmonic mean of the precision and recall of the test. F is the balance between precision and recall 22 . K-S evaluate how the model separates between the positive and negative distributions. Higher KS value means better separation 23 . For regression algorithms, evaluation metric includes mean absolute error, (root) mean squared error (RMSE = √∑𝑁 ̂𝑖 ) 2 𝑖=1(𝑦𝑖 −𝑦 𝑟𝑒𝑔𝑟𝑒𝑠𝑠𝑖𝑜𝑛 𝑣𝑎𝑟𝑖𝑎𝑡𝑖𝑜𝑛 𝑁 𝑡𝑜𝑡𝑎𝑙 𝑣𝑎𝑟𝑖𝑎𝑡𝑖𝑜𝑛 ), coefficient of determination (𝑅 2 ). 𝑅 2 = = ̂ ̅ 2 ∑𝑁 𝑖=1(𝑌𝑖−𝑌) ∑𝑁 (𝑌−𝑌̅)2 measures the percent of total variability that is explained by the regression model 24. Fig. 1. Flowchart of a typical machine learning method 2.4 Method Comparison 𝑖=1 Some of the most common machine learning algorithms are SVM (support vector machine), ANN (artificial neural network), logistic regression, decision trees. Support vector machine algorithms are used to find the hyperplane that separate different classes with highest margin 25. The advantage of SVM is that the solution is global and unique. Computation complexity of SVM does not depend on the dimension of the input space and is less prone to overfitting 26-27 . However, SVM does not work well on unbalanced data 26 . Artificial neural network is inspired by biological brain, where artificial neurons are connected to mimic the connection of neurons in the brain 28. Multiple hidden layers and neurons can add to the complexity of the neuron network architecture. The strength of ANN is that they are flexible and can represent any nonlinear and linear function. However, it needs large amount of training data and is prone to overfitting. Hyperparameter tuning is tedious and troublesome for ANN. Decision tree is another commonly used basis classification algorithm, which comprises a root node, internal node, branch, leaf node, and depth 29. Decision tree progressively splits the tested data based on input feature value, decision process follows the branch, which is the collection between an internal node and its parent node, until it reaches a leaf node 30. Ensemble methods such as random forest and adaboost which are based on constructing a large number of trees with bootstrap samples and iteratively build an ensemble of weak learners, in an attempt to generate a strong overall model. Ensemble methods usually perform better than basic machine learning algorithms in terms of reducing variance and bias 31. 3.1 Machine Learning Application in Materials Discovery and Design An important concept in materials science field is structural-property-performance relationship 32. Developing materials that meet the required performance and property goes back to control processing conditions, structural and compositions of the materials. Hence, understanding how processing condition, structural and compositions affect materials property and performance is the first step towards materials design. Traditionally, controlled experiments are conducted to isolate the effect of one variable. However, variables often are correlated with each other. It is infeasible to isolate some variable for experimental testing 33. Data mining can help revealing hidden relations between large amount of materials parameters, processing conditions and their re lations with dependent materials properties 33. Traditional ways of materials development can be disrupted and reshaped by making the use of available data. 3.1.1 Materials Property Prediction Materials design first of all requires understanding of how de sired properties such as materials’ yield strength, toughness, ultimate tensile strength, and fatigue life etc. are affected by intrinsic microstructure, chemical composition, crystal structure, and external processing, loading conditions and temperatures. Machine learning algorithm can derive the quantitative relation between the independent and dependent variables and hence make prediction with enough training data when physical model does not exist or is too complicated to apply 33. Neural network algorithm has been used in ferritic steel welds toughness prediction due to their ability to handle complex models 33. Toughness was studied as a function of chemical composition, microstructure, welding process and testing temperature. Their influence on toughness was shown in Fig 2. The interaction between different variables can also be predicted with neural network algorithm as shown in Fig 3. The cross of the two toughness curves as a function of temperature and manganese compositions indicates at higher temperatures the influence of manganese on toughness was not only reduced but also negative. Fig 2. Bar chart showing a measure of the model-perceived significance of each of the input variable in influencing toughness. 33 Fig 3. Variation in the normalized toughness as a function of the manganese concentration and the test temperature. 33 ANN can also be used to predict constitutive relations. For instance, the constitutive flow behavior of 42CrMo Steel is predicted with strain, log strain rate and temperature as input and flow stress as output. Predicted results show good correlation with experimental value, indicating excellent capacity of the developed model in predicting flow stress, Fig 4 34. Austenite Stainless Steel grade 304L and 316L ultimate tensile strength, yield strength, tensile elongation rate, strain hardening exponent and strength coefficient were also able to be predicted by ANN with a function of temperature and strain rate. The optimum architecture is [2-6-5] for ASS 304L and [2-17-5] for ASS 316L using feed forward back propagation learning. Model accuracy is verified with correlation coefficient, average absolute error and its standard deviation 35 . Fatigue properties has always been among the most difficult ones to predict due to the high cost and long time for fatigue testing and the prevalence of structural failure caused by fatigue 36-37. Existing physical models are either lacking of generality or fail to give quantitative indications 38. Agrawal et al. predicted the fatigue strength of steel using data from the Japan National Institute of Materials Science (NIMS) MatNavi database 39-40. They used 12 regression-based predictive model, among them, neural network, decision tree and multivariate polynomial regression were able to achieve a high R 2 value of >0.97. Fig 4. Comparison between experimental value and predicted flow stress of 42CrMo steel using BP ANN. 34 (a) Predicted training data (b) Predicted testing data 3.1.2 Inversed Design of Materials Understanding how mechanical properties are influenced by materials internal and external factors help reducing searching space in the inversed materials design task. However, the inverse problem is more challenging because of the possibility of multiple solutions and the enormous structural dimension 41 . Machine learning application has shown promise in inversed materials discovery and design by reducing searching path and searching region. Ruoqian Liu et al. developed a machine learning method for the inverse design of Fe-Ge alloy microstructure with enhanced elastic, plastic and magnetostrictive properties 41. A systematic approach consisting of random data generation, feature selection, and classification was developed. Firstly, features that can quantitatively describe microstructures and properties were developed. Then, randomly generated structural and properties pairs were simulated to form the most desired and least desired classes. Two crucial steps, search path refinement and search space reduction are conducted prior to the actual searching to find the most efficient orders of features in search and the most promising search regions of features. This method was validated with five design problems, which involves identification of microstructures that satisfy both linear and nonlinear property constraints. This framework shows supremacy comparing with traditional optimization methods in reducing as much as 80% of running time and achieving optimality that would not be attained. 3.2 Machine learning application in materials processing and synthesis Design of materials can be facilitated with the data driven machine learning approach, however the commercialization of materials is still impeded by the availability to synthesize them. To disrupt the trial and error synthesis methods, Olivetti group in MIT is working on creating a predictive synthesis system for advanced materials processing. They are building a curated database of solid state materials and their synthesis methods compiled from thousands of materials synthesis journal articles. The database also contains algorithms developed through machine learning approaches, which are capable of predicting synthesis routes for novel materials based on chemical formulae and other known physical input data.  Even failed experiments can be used by the machine learning algorithm for materials discovery and synthesis which truly shows the power of data mining and machine learning. After all, onl y a small amount of information is published in the research work, most of the data are archived and not been used to its full potential. Paul Raccuglia, et al. trained a machine learning model based on failed hydrothermal syntheses data to predict reaction outcomes under different conditions such as temperature, concentration, reactant quantity and acidity 42 . The model was validated and tested with previously untested data and shown better performance than human researchers who have 10 years’ experience. It was able to predict conditions for new organically templated inorganic product formation with a success rate of 89%. 3.3 Machine learning application in microstructure recognition and failure analysis Microstructure damage and failure pre-detection is another area that machine learning find its applications. Traditionally, materials scientist examines the SEM, OPM images of samples for failure analysis similar to medical doctors analyze X-Ray images of patients. With the increasing penetration of machine learning methods in medical imaging analysis, the same kind of application in materials imaging is expect to happen as well 43. In fact, there are already reports on machine learning and computer vision researches on materials microstructure automatic recognition. Aritra et al. applied computer vision methods to identify images that contain dendritic morphology and then classify whether the dendrites are al ong the longitudinal direction or traverse direction if they do exist in the image. To extract features and reduce feature dimensions, they used visual bag of words, texture and shape statistics, and pre -trained convolutional neural network. Classification was conducted using support vector machine, nearest neighbors and random forest models 44. It was shown that pre-trained convolutional neural network performs best in terms of micrograph recognition and feature extraction, which confirmed with other reports 45-46. Classification methods were able to reach great accuracy for both task. Another example is the automatic measurement of ferrite volume fraction from the ferrite-austenite binary phase structures based on GPF (Graph Processing Framework) algorithm developed by Hafiz Muhammad Tanveer, et al 47. Machine learning algorithm can also be used in failure detection by examining microstructure images. Matthias Demant et al. introduced an enhanced machine learning algorithm for crack detection in photoluminescence (PL) images of as-cut wafers. The detection algorithm is based on a classification of cracks due to the comparison of the crack descriptions with previous trained crack data. Crack centers are identified by detecting features appearing as star or line-like structure. Grain boundary information is extracted from additional images in the visible range to avoid false detections. Support vector machine is used to train labelled data for crack and non-crack structures classification 48. The algorithm is able to achieve a high precision of 91.1% and sensitivity of 80.4% for crack length greater than 3 mm. Elaheh Rabiei et al. developed a dynamic Bayesian network(DBN) based on the variation of modulus of elasticity to estimate damages from a prognostic approach when crack is not observable yet. Various sources of information were taken into account to reduce uncertainties. DBN was applied to relate the variables and their causal or correlation relationship. Degradation model parameters are learned with joint particle filtering technique. Support vector regression models was applied to define unknown nonparametric and nonlinear correlation between the input variables. More precise damage estimation and crack initiation prediction in a metallic alloy under fatigue was confirmed by experimental observations 49. This method is different from traditional empirical damage models (Paris law) since direct damage indicators such as crack is not required to predict damage stage. Thus, underling damages can be monitored at an earlier stage. It is easy to imagine manufacturing companies such as GE can monitor their jet engine data to predict whether it needs inspection or maintenance. Fig. 5. Overview of the crack detection algorithm 48 . 4. limitations of machine learning in materials science applications Although machine learning has been widely used in a lot of fields and increasingly been used in materials science, machine learning is by no means a panacea. Without understanding its limitations and blindly apply it to every possible area can lead to wrongful predictions and a waste of time and effort. First of all, machine learning system are opaque, making them very hard to debug. Machine learning prediction heavily relies on training data. Machine learning often have overfitting or overfitting problems that needs to be concerned when taking their prediction results into consideration. Input data quality needs to be ensured. Interpolation and extrapolation can lead to problems when training data is not sufficient in the interpolated or extrapolated regime or when training data is noisy. Hence, error bar prediction is needed for evaluating prediction accuracy. Machine learning does not explain the results from the physics point of view. Materials scientists often are interested in understanding the mechanism of certain phenomena. Machine learnin g cannot elucidate the mechanism since it works on data driven model training and prediction. Interpretation of the machine learning results needs domain knowledge. Without understanding the underline physics, nonsense predictions can’t be recognized. Even in the process of feature selection, a good understanding of the causal relationship between these variable and dependent properties can be helpful for selecting most effective features and build less complicated models. Machine learning is also inseparable from experiment and physical simulation. 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SELF-LEARNING TO DETECT AND SEGMENT CYSTS IN LUNG CT IMAGES WITHOUT MANUAL ANNOTATION Ling Zhang1 , Vissagan Gopalakrishnan2 , Le Lu1 , Ronald M. Summers1 , Joel Moss2 , Jianhua Yao1 arXiv:1801.08486v1 [cs.CV] 25 Jan 2018 1 Radiology and Imaging Sciences Department, 2 Cardiovascular and Pulmonary Branch, NHLBI, National Institutes of Health (NIH), Bethesda MD ABSTRACT Image segmentation is a fundamental problem in medical image analysis. In recent years, deep neural networks achieve impressive performances on many medical image segmentation tasks by supervised learning on large manually annotated data. However, expert annotations on big medical datasets are tedious, expensive or sometimes unavailable. Weakly supervised learning could reduce the effort for annotation but still required certain amounts of expertise. Recently, deep learning shows a potential to produce more accurate predictions than the original erroneous labels. Inspired by this, we introduce a very weakly supervised learning method, for cystic lesion detection and segmentation in lung CT images, without any manual annotation. Our method works in a self-learning manner, where segmentation generated in previous steps (first by unsupervised segmentation then by neural networks) is used as ground truth for the next level of network learning. Experiments on a cystic lung lesion dataset show that the deep learning could perform better than the initial unsupervised annotation, and progressively improve itself after self-learning. Index Terms— Convolutional neural networks, weakly supervised learning, medical image segmentation, graph cuts 1. INTRODUCTION Image segmentation is a fundamental problem in medical image analysis. Classic segmentation algorithms [1] are usually formulated as optimization problems relying on cues from low-level image features. In recent years, deep learning has made much progress on image segmentation tasks (e.g., FCN [2], HED [3]), achieved dominant performances on many medical image segmentation benchmarks, e.g., UNet [4] is competitive enough for many applications. The success of deep learning based segmentation requires supervised learning on large manually annotated data. However, expert annotations on big medical datasets are expensive to obtain This research was supported - in part - by the Intramural Research Program of the National Institutes of Health Clinical Center. The authors thank Dr. Li Zhang from Beijing Institute of Big Data Research for his inspiring discussion, and Nvidia for the TITAN X Pascal GPU donation. Mild Moderate Severe Fig. 1. Examples of the cystic lung lesions with different severity levels in CT image and their manual annotation (red). or even unavailable. For example, manual annotation of hundreds of cysts in CT volume dataset (examples shown in Fig. 1) is not feasible for a recent large-sized clinical study of lymphangioleiomyomatosis (LAM) [5]. To alleviate the annotation burden, researchers exploit weakly supervised methods for deep learning based segmentation. One direction is to reduce the effort (e.g., time, expertise) for annotation. By combining FCN and active learning, 50% training data is needed to train a model with comparable performance as training on all data [6]. Another direction applies image-level annotation by incorporating FCN in a multiple instance learning framework [7]. However, expertise from physicians are still needed, such as assigning imagelevel annotations and estimating the lesion size. Recently, deep learning has shown a potential to beat the teacher (i.e., perform better than the training data labels) [8, 9] or even self-learn to be an expert without human knowledge in AlphaGo Zero [10]. Specifically, for some classification [8] and semantic segmentation [9] tasks, when provided with data labels with certain amount of errors, deep learning could produce lower errors than the original erroneous labels. In addition, with assisting by domain-specific algorithm (e.g., Monte Carlo tree search in Go game [10]; GrabCut in image segmentation [9]), training samples/labels can be generated to iteratively or recursively update the neural network parameters to achieve better performance. transfer Unsupervised Segmentation Segmentation Net – level 1 transfer Segmentation Net – level 2 … Segmentation Net – level n Data stream Annotation stream Fig. 2. Learning to segment medical images without manual annotation. Segmentation networks (level 1 – level n) are recursively trained with the previous network segmentation as training labels. In this paper, we propose a very weakly supervised approach for LAM cyst detection and segmentation. As shown in Fig. 1, the detection and segmentation of cysts is a challenging task due to the large number of cysts, greatly variation of cyst sizes, severe touching of cysts, inconsistent image quality, and image noise and motion artifact, etc. Moreover, it is infeasible to obtain manual segmentation on LAM studies. Our method, differs from weakly supervised methods, can automatically learn from medical images without any manual pixel- [6], sparse- [9], or image-level [7] annotation and without pre-training a segmentation network on other labeled datasets [9]. Starting from classic segmentation techniques, specifically unsupervised K-means clustering with spatial information followed by graph cuts [11] refinement, the initial annotation is generated and serves as labels for a segmentation network (UNet [4] in this paper) learning. New networks are then recursively trained with the previous network predictions as training labels. An improved segmentation network could be trained under two hypotheses: 1) deep learning might generate better predictions than the training data labels [8], and 2) better training data labels produce better predictions [10]. Note that the value of K in K-means clustering is the only value provided to the framework. 2. METHODS Given a medical image dataset without manual annotation, our method works in a self-learning manner (Fig. 2), where the previously generated (first by unsupervised segmentation then by segmentation networks) pixel-level annotations serve as inputs for the next level of network learning. 2.1. Unsupervised Segmentation K-means clustering is an unsupervised segmentation approach. By involving pixel intensity, average and median pixel intensities of a local window into a feature space, a spatial K-means [12] classifies the image by grouping similar pixels in the feature space into clusters. The number of clus- ters K needs to be manually set in different applications. For the cyst segmentation in CT images, we set K = 3 to obtain three clusters. c1 , c2 , and c3 are the cluster centers, indicating cyst, lung tissue, and others, respectively. With c1 , c2 , and c3 , we construct a three-terminal graph with the energy function consisting of a data term and a pixel continuity term as in [11]. The data term is assigned as the squared intensity differences between pixels and the cluster centers; The pixel continuity term is 0 when two neighboring pixels values are the same, and δ otherwise (δ = 0.003 through empirical evaluation on our data). Then, max-flow algorithm [11] is used to optimize the energy function, and the global optimal pixel labels are obtained. 2.2. Segmentation Network After obtaining the initial annotation for all the images in the dataset by using spatial K-means graph cuts, UNet is used as the network architecture to learn a better segmentor because of its efficiency and accuracy for medical image segmentation [4]. UNet is constituted of four layers of contraction (pooling) and four layers of expansion (up-convolution). Skip connections from contracting path to expansive path strengthen context information in higher resolution layers. During UNet training, the inputs are raw CT images with original resolution, and the outputs are 1-channel annotations (cross-entropy loss is utilized). The training focuses on distinguishing between cysts and lung tissues and ignoring background labels. One critical problem in training UNet for medical images is that the label/class distribution can be highly imbalanced, e.g., much more positive samples than negative or vice versa. In our experiments, we use the distribution of cysts and lung tissues in the image to balance the positive and negative classes in loss function as in [3]. We also avoid sampling empty CT slices (no cyst in the slice) in the training. 2.3. Recursive Learning The trained UNet will become its own teacher – it is applied to segment all the CT images in training set to generate a new set of pixel-level cyst labels, which will be used as the new ground truth to train a next level UNet. The network parameters of the previous UNet are transferred to initialize the next network, and a lower learning rate is used to train the next network. The self-learning terminates when the similarity between successive segmentation is larger than a threshold. 3. EXPERIMENTAL METHODS In this study, we evaluated our method on a LAM dataset. A total of 183 CT volumes from patients with LAM in a natural history protocol were studied. High resolution CT scans of the chest were obtained. The scans contained 9-13 slices and the slice thickness ranged from 1 to 1.25 mm at 3-cm intervals. Each CT slice is with 512×512 pixels. The UNet is implemented using Caffe [13]. We train the UNet model from scratch. Three UNet models are trained progressively in the recursive framework, named as UNetlevel1, UNet-level2, and UNet-level3, respectively. The initial learning rate is 1×10−7 for UNet-level1 and decreases by a factor of 10 for every next level thanks to transfer learning from previous level. Each UNet-level is trained for 50k iterations. Mini-batch of 1 image since it provides better performance (than 5, 10, etc.) in a preliminary experiment. The proposed method is tested on a DELL TOWER 7910 workstation with 2.40 GHz Xeon E5-2620 v3 CPU, 32 GB RAM, and a Nvidia TITAN X Pascal GPU of 12 GB of memory. Our model is trained on 166 CT volumes. The remaining 17 volumes including 5 mild, 6 moderate, and 6 severe cases are left out as unseen testing data. To evaluate the segmentation performance, a medical student manually detect and segment one slice from each of the 17 testing volumes. The manual segmentation was tedious that it took 4 working days. Quantification metrics include Dice coefficient and absolute difference of cyst scores (ADCS). Cyst score is defined as the percentage of lung region occupied by cysts, which is a critical clinical factor in LAM assessment [5]. It’s worth mentioning that differing from traditional concept of training set, our model does not learn from any manual annotation from the 166 training data (which is not available), therefore, these data can also be seen as testing data for performance evaluation. Six images (from 6 CT volumes) with large ADCS between unsupervised segmentation results and UNet results are additionally selected from the 166 dataset. Manual segmentation is then conducted on these slices for evaluation of the progressive improvement of our framework. In addition, we compare our method with the cyst segmentation method in [5] where semi-automated thresholding followed by some postprocessing techniques were used. 4. RESULTS Table 1 shows the performance on unseen images. 15 out of the 17 images are with good image quality while 2 are noisy. Table 1. Performance comparison on 17 unseen CT images. SK-GC: spatial K-means graph cuts; ADCS: absolute difference of cyst scores. Bold indicates the best results. Dice (%) ADCS (%) Semi-automated [5] 62.64 8.34 SK-GC (teacher) 74.67 3.71 UNet-LV.1 (student) 75.41 3.65 UNet-LV.2 (student) 75.87 3.38 UNet-LV.3 (student) 74.94 4.56 Table 2. Performance comparison on 6 CT images with large ADCS between SK-GC and UNet from learning set. SK-GC: spatial K-means graph cuts; ADCS: absolute difference of cyst scores. Bold indicates the best results. Dice (%) ADCS (%) Semi-automated [5] 79.05 5.19 SK-GC (teacher) 70.39 11.25 UNet-LV.1 (student) 82.25 2.89 UNet-LV.2 (student) 82.65 1.98 UNet-LV.3 (student) 81.93 3.42 Student (i.e., UNet) learning could achieve higher segmentation accuracy than its teacher (i.e., spatial K-means graph cut, SK-GC), but the self-improvement seems to stop at level 3. The same trends could be observed in Table 2, where the performance on images from the learning set is shown. In these 6 CT images with large ADCS between SK-GC and UNet, compared to manual annotation, UNet learning performs substantially better than SK-GC. The lower Dice of UNet in Table 1 compared to which in Table 2 is mainly caused by the lower Dice values from the 5 mild cases, where both SK-GC and UNet have Dice values around 60%. Our proposed self-learning method is also more accurate than the semi-automated method [5]. Three examples in Fig. 3 show how the proposed strategy recursively improves the segmentation performance itself. Given inaccurate segmentation provided by SK-GC, one level of UNet learning (UNet-LV.1) can already correct most oversegmentation and undersegmentation of cysts, thus achieve both higher sensitivity and higher specificity. Higher levels of UNet tend to obtain more accurate cyst boundaries especially for the overlapping cysts. The whole training process takes about 17 hours and testing is 0.13 sec./slice. 5. CONCLUSIONS We report the first results of very weakly supervised learning to detect and segment cysts in lung CT images without manual annotation. By first learning from classic unsupervised segmentation, deep learning shows its potential to perform even better after a few levels of self-learning. In future work, we will extend this method to segment other medical images. CT slice SK-GC UNet-LV.1 UNet-LV.2 Manual annotation Fig. 3. 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Zhang, and D.Z. Chen, “Suggestive annotation: A deep active learning framework for biomedical image segmentation,” in MICCAI, 2017. [7] Z. Jia, X. Huang, E. I. Chang, and Y. Xu, “Constrained deep weak supervision for histopathology image segmentation,” IEEE TMI, 2017. [8] M.Y. Guan, V. Gulshan, A.M. Dai, and G.E. Hinton, “Who said what: Modeling individual labelers improves classification,” arXiv preprint arXiv:1703.08774, 2017. [9] A. Khoreva, R. Benenson, J. Hosang, M. Hein, and B. Schiele, “Simple does it: Weakly supervised instance and semantic segmentation,” in CVPR, 2017. [10] D. Silver, J. Schrittwieser, K. Simonyan, I. Antonoglou, A. Huang, A. Guez, T. Hubert, L. Baker, M. Lai, A. Bolton, Y. Chen, T. Lillicrap, F. Hui, L. Sifre, G. Van Den Driessche, T. Graepel, and D. Hassabis, “Mastering the game of Go without human knowledge,” Nature, vol. 550, pp. 354–359, 2017. [11] Y. Boykov and V. 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arXiv:1801.07743v1 [cs.IR] 23 Jan 2018 Entity Retrieval and Text Mining for Online Reputation Monitoring Pedro dos Santos Saleiro da Cruz Departamento de Engenharia Informática Faculdade de Engenharia da Universidade do Porto In partial fulfillment of requirements for the degree of Doctor in Informatics Engineering FEUP 2017 Supervisor: Dr. Carlos Soares Co-Supervisor: Dr. Eduarda Mendes Rodrigues Faculdade de Engenharia Universidade do Porto Rua Dr. Roberto Frias, s/n 4200-465 Porto, Portugal Copyright © 2017 by Pedro Saleiro Doctoral Committee: Dr. Eugénio Oliveira, Full Professor at FEUP, University of Porto Dr. Mark Carman, Senior Lecturer at Monash University Dr. Bruno Martins, Assistant Professor at IST, University of Lisbon Dr. Luís Torgo, Associate Professor at FCUP, University of Porto Dr. Carlos Soares, Associate Professor at FEUP, University of Porto Esta tese é dedicada à minha Mãe, Maria de Lurdes, pelo seu amor e dedicação constantes. Acknowledgements First, I would like to thank everybody that contributed somehow to this work, from co-authors to reviewers, colleagues and faculty staff. I will probably forget to mention someone in particular and I sincerely apologize for that. This work was funded by SAPO Labs, FCT and Microsoft Research. Without their financial support, I would not be able to conclude this thesis. I am deeply grateful to my supervisor, Carlos Soares. Although I was not working in meta-learning :) Carlos always showed a genuine enthusiasm about this work. Thank you for giving me the freedom to grow independently as a researcher and to pursuit my own ideas, even in the moments I was not delivering at the rhythm you expected. I believe you made me more pragmatic after all. We had really interesting and thorough discussions during the last 5 years. We have not tried all the cool ideas but I hope we will do it someday. Last, I would like to thank you for all the support and protection, as well as, for believing that I should expand my horizons to the US! I will send you a postcard from Chicago! I am also thankful to Eduarda Mendes Rodrigues, co-supervisor of this work. It all started with you. Thank you for receiving me at FEUP back in October 2012. I still remember the day we drew the first draft of our framework for ORM. You always have encouraged me and your positive feedback was a source of inspiration and motivation. Even at distance you have always been available when I needed. I must also mention your decisive role in helping me pursuing a Summer internship in a top notch place such as Microsoft Research. Being a graduate student is an opportunity to collaborate with new and inspirational people and that was what happened when I had the chance to start working with Natasa Milic-Frayling at Microsoft Research. When we are around Natasa we believe we can make things happen. Thank you for your patience, support and motivation. I hope we can keep our collaboration for many years. I show my gratitude to Prof. Eugénio Oliveira who helped me with a smooth transition to LIACC and always had the door open to discuss my issues. Furthermore, Prof. Eugénio was very enthusiastic about my work even not being my supervisor and I iv sincerely appreciate that. Thank you for your advices and I will miss our conversations about the past and future of AI. I wish to thank Luís Sarmento for introducing me to the world of Data Science, and Text Mining in particular. I just regret we did not had the chance to collaborate more often. I must thank two special friends that are also graduate students, Jorge Teixeira and Damião Rodrigues. Jorge, you were my “brother in arms” throughout this journey and I will always be thankful. Damião, my friend and colleague for more than 10 years, is the friend I searched for sharing the ups and downs of being graduate student. Thank you for your support and motivation. I would like to thank Cristina Ribeiro for the administrative support regarding my funding throughout these years. I must also address Rosaldo Rossetti for believing in my abilities and for starting our productive collaboration, and of course, for being a really enjoyable and funny colleague. Arian Pasquali also deserves a personal mention for all the support, as well as, Luís Gomes, Luís Rei, Sílvio Amir, Tiago Cunha and Gustavo Laboreiro. I would like also to thank all the members of the POPSTAR project, specially Pedro Magalhães. Nesta hora não poderia deixar de estender os agradecimentos aos amigos e à família. Em especial queria referir o grupo do Rumo ao Penta onde todos os passarões me proporcionam grandes momentos de boa disposição sem os quais seria impossível abstrair-me dos problemas do dia a dia. Um grande abraço para o André, João Miguel, Jorge e Márcio. Queria também deixar aqui uma palavra para a minha prima Xana pela amizade desde sempre. Como é óbvio tenho imenso a agradecer à minha Mãe, a quem dedico esta tese. Obrigado pelo amor, dedicação e pela liberdade que sempre me deste em todas as minhas escolhas. Como não deixaria de ser, sempre foste uma entusiasta deste meu desafio que agora chega ao fim. Parabéns a ti! Deixo também um beijinho à minha irmã Guida, ao querido Tomás e ao bebé Diogo que ainda não conheci mas a vida de emigrante tem destas coisas. E por fim, deixo o meu sentido agradecimento à minha namorada, Maria. Foste crucial nesta caminhada. Ter-te ao meu lado deu-me força todos os dias para avançar mais um pouco. Sei que este trabalho comprometeu muito o nosso tempo a dois mas agradeço-te a compreensão e os incentivos constantes para levar isto até ao fim. Prometo compensar no futuro :) Ah e claro tenho que agradecer ao Bobby pela companhia que me fez durante a escrita e por me conseguir fazer sorrir mesmo quando a vida é madrasta. Abstract Online Reputation Monitoring (ORM) is concerned with the use of computational tools to measure the reputation of entities online, such as politicians or companies. In practice, current ORM methods are constrained to the generation of data analytics reports, which aggregate statistics of popularity and sentiment on social media. We argue that this format is too restrictive as end users often like to have the flexibility to search for entity-centric information that is not available in predefined charts. As such, we propose the inclusion of entity retrieval capabilities as a first step towards the extension of current ORM capabilities. However, an entity’s reputation is also influenced by the entity’s relationships with other entities. Therefore, we address the problem of Entity-Relationship (E-R) retrieval in which the goal is to search for multiple connected entities. This is a challenging problem which traditional entity search systems cannot cope with. Besides E-R retrieval we also believe ORM would benefit of text-based entity-centric prediction capabilities, such as predicting entity popularity on social media based on news events or the outcome of political surveys. However, none of these tasks can provide useful results if there is no effective entity disambiguation and sentiment analysis tailored to the context of ORM. Consequently, this thesis address two computational problems in Online Reputation Monitoring: Entity Retrieval and Text Mining. We researched and developed methods to extract, retrieve and predict entity-centric information spread across the Web. We proposed a new probabilistic modeling of the problem of E-R retrieval together with two fusion-based design patterns for creating representations of both entities and relationships. Furthermore, we propose the Entity-Relationship Dependence Model, a novel early-fusion supervised model based on the Markov Random Field framework for Retrieval. Together with a new semi-automatic method to create test collections for E-R retrieval, we released a new test collection for that purpose that will foster research in this area. We performed experiments at scale with results showing that it is possible to perform E-R retrieval without using fix and pre-defined entity and relationship types, enabling a wide range of queries to be addressed. vi We tackled Entity Filtering and Financial Sentiment Analysis using a supervised learning approach and studied several possible features for that purpose. We participated in two well known external competitions on both tasks, obtaining state-of-the-art performance. Moreover, we performed analysis of the predictive power of a wide set of signals extracted from online news to predict the popularity of entities on Twitter. We also studied several sentiment aggregate functions on Twitter to study the feasibility of using entity-centric sentiment on social media to predict political opinion polls. Finally, we created and released an adaptable Entity Retrieval and Text Mining framework that puts together all the building blocks necessary to perform ORM and can be reused in multiple application scenarios, from computational journalism to politics and finance. This framework is able to collect texts from online media, identify entities of interest, perform entity and E-R retrieval as well as classify sentiment polarity and intensity. It supports multiple data aggregation methods together with visualization and modeling techniques that can be used for both descriptive and predictive analytics. Resumo A Monitorização da Reputação Online (MRO) consiste na utilização de ferramentas computacionais para medir a reputação de entidades online, como por exemplo, políticos ou empresas. Na prática, os métodos actuais de MRO estão restringidos à produção de relatórios constituídos por análises de dados, tais como estatísticas agregadas da popularidade e do sentimento nos media sociais. Consideramos que esta prática é demasiado restritiva uma vez que os utilizadores finais das plataformas MRO desejam frequentemente ter a flexibilidade que lhes permita pesquisar por informação centrada nas entidades que vai além da disponibilizada nos gráficos pré-definidos. Por conseguinte, propomos a inclusão da capacidade de recuperação de entidades como um primeiro passo no sentido de estender as o estado atual das ferramentas de MRO. No entanto, a reputação de uma dada entidade também é influenciada pelas relações desta com outras entidades. Neste sentido, propomo-nos a tratar do problema de recuperação de entidade-relações (E-R) onde o objectivo consiste na pesquisa por múltiplas entidades relacionadas entre si. Trata-se de um desafio que os sitemas tradicionais de recuperação de entidades ainda não são capazes de lidar. Para além da recuperação E-R, também acreditamos que a MRO iria beneficiar da capacidade de efectuar previsões baseadas em texto e centradas nas entidades, como por exemplo a previsão da popularidade de entidades nos media sociais utilizando eventos retratados nas notícias ou o resultado de sondagens. No entanto, nenhuma destas tarefas terá sucesso e utilidade se não houver a capacidade efetiva de desambiguar entidades mencionadas nos textos, assim como uma análise de sentimento específica para o contexto da MRO. Consequentemente, esta tese trata dois problemas computacionais da Monitorização da Reputação Online: Recuperação de Entidades e Prospeção de Texto. Investigámos e desenvolvemos métodos para extrair, recuperar e prever informação centrada em entidades e espalhada pela Internet. Propomos um novo modelo probabilístico do problema de recuperação E-R conjuntamente com dois padrões de desenho baseados em fusão de texto para criar representações de entidades e relações. Propomos também o Modelo de Dependência viii Entitdade-Relação (MDER), um novo modelo supervisionado de fusão antecipada baseado no Campo Aleatório de Markov para a Recuperação de Informação. Conjutamente com um novo método semi-automático de geração de coleções de teste para recuperação E-R, lançamos uma nova coleção de teste com esse propósito que irá fomentar a investigação nesta área. Efetuamos experiências de grande escala e os resultados mostram que é possível realizar recuperação E-R sem utilizar tipos fixos e pré-definidos de entidades e relações, o que permite atuar sobre o conjunto alargado de pesquisas. Tratamos também das tarefas de Filtragem de Entidades e Análise de Sentimento Financeiro utilizando uma abordagem de aprendizagem supervisionada em que estudamos várias características para esse fim. Participámos em duas competições exterrnas em ambas as tarefas, atingindo resultados ao nível do estado da arte. Além disso, realizámos uma análise do poder preditivo de um grande conjunto de sinais extraídos das notícias online para parever a popularidade de entidades no Twitter. Assim como, um estudo de várias funções de agregação de sentimento do Twitter para estudar a praticabilidade de utilizar informação de sentimento nos media sociais para prever sondagens eleitorais. Finalmente, criámos e disponibilizámos uma plataforma de recuperação de entidades e prospeção de texto que conjuga todos os blocos necessários para a realização de MRO. Pode ser reutilizada em diversos cenários de aplicação, desde o jornalismo computacional à política e finança. Esta plataforma é capaz de recolher textos dos media online, identificar entidades alvo, efectuar recuperação de entidades e relaçãos, assim como classificar sentimento e intensidade associada. Suporta vários métodos de agregação de dados e juntamente com métodos de visualização e previsão pode ser utilizada tanto para análises descritivas como preditivas. Table of contents List of figures xiii List of tables xv 1 Introduction 1.1 Thesis Statement . . . . . . . . 1.2 Objectives . . . . . . . . . . . . 1.3 Research Methodology . . . . . 1.4 Contributions and Applications 1.5 Foundations . . . . . . . . . . . 1.6 Thesis Outline . . . . . . . . . . . . . . . . 1 2 5 7 8 10 12 . . . . . . . . . . . 13 13 14 15 19 20 22 23 24 26 28 29 . . . . . . . . . . . . . . . . . . 2 Background and Related Work 2.1 Online Reputation Monitoring . . . . 2.1.1 Related Frameworks . . . . . 2.2 Entity Retrieval and Semantic Search 2.2.1 Markov Random Field for IR 2.2.2 Sequential Dependence Model 2.2.3 MRF for Entity Retrieval . . 2.3 Named Entity Disambiguation . . . . 2.4 Sentiment Analysis . . . . . . . . . . 2.5 Word Embeddings . . . . . . . . . . 2.6 Predicting Collective Attention . . . 2.7 Political Data Science . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Entity Retrieval for Online Reputation 3.1 Entity-Relationship Retrieval . . . . . 3.1.1 E-R Queries . . . . . . . . . . . 3.1.2 Modeling E-R Retrieval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Monitoring 33 . . . . . . . . . . . . . . . . . . 34 . . . . . . . . . . . . . . . . . . 35 . . . . . . . . . . . . . . . . . . 36 Table of contents x 3.2 3.3 3.4 Design Patterns for Entity-Relationship 3.2.1 Early Fusion . . . . . . . . . . . 3.2.2 Association Weights . . . . . . 3.2.3 Early Fusion Example . . . . . 3.2.4 Late Fusion . . . . . . . . . . . 3.2.5 Late Fusion Example . . . . . . 3.2.6 Implementation . . . . . . . . . Entity-Relationship Dependence Model 3.3.1 Graph Structures . . . . . . . . 3.3.2 Feature Functions . . . . . . . . 3.3.3 Ranking . . . . . . . . . . . . . 3.3.4 Discussion . . . . . . . . . . . . Summary of the Contributions . . . . . Retrieval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Entity-Relationship Retrieval over a Web Corpus 4.1 RELink Query Collection . . . . . . . . . . . . . . 4.1.1 Tabular Data and Entity Relationships . . . 4.1.2 Selection of Tables . . . . . . . . . . . . . . 4.1.3 Formulation of Queries . . . . . . . . . . . . 4.1.4 Collection Statistics . . . . . . . . . . . . . . 4.2 Experimental Setup . . . . . . . . . . . . . . . . . . 4.2.1 Data and Indexing . . . . . . . . . . . . . . 4.2.2 Retrieval Method and Parameter Tuning . . 4.2.3 Test Collections . . . . . . . . . . . . . . . . 4.3 Results and Analysis . . . . . . . . . . . . . . . . . 4.4 Summary of the Contributions . . . . . . . . . . . . 5 Entity Filtering and Financial Sentiment 5.1 Entity Filtering . . . . . . . . . . . . . . 5.1.1 Task Overview . . . . . . . . . . 5.1.2 Pre-processing . . . . . . . . . . . 5.1.3 Features . . . . . . . . . . . . . . 5.1.4 Experimental Setup . . . . . . . . 5.1.5 Results . . . . . . . . . . . . . . . 5.2 Financial Sentiment Analysis . . . . . . 5.2.1 Task Overview . . . . . . . . . . 5.2.2 Financial Word Embeddings . . . Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 41 43 44 46 48 51 52 53 55 60 61 62 . . . . . . . . . . . 63 64 65 65 67 67 69 69 70 71 72 77 . . . . . . . . . 79 80 81 81 81 82 84 86 87 87 Table of contents 5.3 5.2.3 Approach . . . . . . . 5.2.4 Experimental Setup . . 5.2.5 Results and Analysis . 5.2.6 Concluding Remarks . Summary of the Contributions xi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Text-based Entity-centric Prediction 6.1 Exploring Online News for Reputation Monitoring 6.1.1 Approach . . . . . . . . . . . . . . . . . . 6.1.2 Experimental Setup . . . . . . . . . . . . . 6.1.3 Results and Discussion . . . . . . . . . . . 6.2 Predicting Political Polls using Twitter Sentiment . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Methodology . . . . . . . . . . . . . . . . 6.2.2 Data . . . . . . . . . . . . . . . . . . . . . 6.2.3 Experimental Setup . . . . . . . . . . . . . 6.2.4 Results and Discussion . . . . . . . . . . . 6.2.5 Feature Importance . . . . . . . . . . . . . 6.2.6 Outlook . . . . . . . . . . . . . . . . . . . 6.3 Summary of the Contributions . . . . . . . . . . . 7 A Framework for Online Reputation Monitoring 7.1 Framework Overview . . . . . . . . . . . . . . . . 7.1.1 RELink . . . . . . . . . . . . . . . . . . . 7.1.2 TexRep . . . . . . . . . . . . . . . . . . . 7.2 RELink Use Case . . . . . . . . . . . . . . . . . . 7.2.1 News Processing Pipeline . . . . . . . . . . 7.2.2 Demonstration . . . . . . . . . . . . . . . 7.3 TexRep Use Case . . . . . . . . . . . . . . . . . . 7.3.1 Data Aggregation . . . . . . . . . . . . . . 7.3.2 Visualization . . . . . . . . . . . . . . . . 7.4 Learning Word Embeddings for ORM . . . . . . . 7.4.1 Neural Word Embedding Model . . . . . . 7.4.2 Experimental Setup . . . . . . . . . . . . . 7.4.3 Results and Analysis . . . . . . . . . . . . 7.4.4 Concluding Remarks . . . . . . . . . . . . 7.5 Summary of the Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 90 90 93 93 on Twitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 96 97 101 103 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 107 110 112 113 116 117 118 . . . . . . . . . . . . . . . 119 119 120 122 127 128 129 130 132 134 134 136 137 140 144 145 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii Table of contents 8 Conclusions 147 8.1 Summary and Main Contributions . . . . . . . . . . . . . . . . . . . . . 147 8.2 Limitations and Future Work . . . . . . . . . . . . . . . . . . . . . . . 154 References 157 List of figures 1.1 Entity Retrieval and Text Mining as computational problems of ORM. 3 2.1 Markov Random Field document and term dependencies. . . . . . . . . 19 3.1 3.2 3.3 Bayesian networks for E-R Retrieval with queries of different lengths. . Markov Random Field dependencies for E-R retrieval, \|Q\| = 3. . . . . . Markov Random Field dependencies for E-R retrieval, \|Q\| = 5. . . . . . 39 53 54 4.1 4.2 4.3 4.4 Example of Wikipedia table row. . . . . . . . . . . Example of metadata provided to editors. . . . . . Illustration of E-R indexing from a web corpus. . . ′ ′ Values of λ for ERDM: (a) all λ, (b) λE , (c) λR . obtained using sum normalization. . . . . . . . . . . . . . . . . . . . . . were . . . . 67 67 69 5.1 Results grouped by entity’s category using Run 2. . . . . . . . . . . . . 85 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 Daily popularity on Twitter of entities under study. . . . . . . . . . . Training and testing sliding window - first 2 iterations. . . . . . . . . Individual feature type F1 score for tp = 12 at k = 0.5. . . . . . . . . Negatives share (berminghamsovn) of political leaders in Twitter. . . Representation of the monthly poll results of each political candidate Error predictions for polls results. . . . . . . . . . . . . . . . . . . . . Error predictions for polls results variation. . . . . . . . . . . . . . . . Mean absolute error buzz vs sentiment. . . . . . . . . . . . . . . . . . Aggregate functions importance in the Random Forests models. . . . . . . . . . . . . 102 102 105 110 112 114 114 115 117 7.1 7.2 7.3 7.4 High-level overview on the ORM framework. . . . . . . RELink Framework architecture overview. . . . . . . . Architecture and data flows of the TexRep framework. News processing pipeline. . . . . . . . . . . . . . . . . . . . . . 120 121 124 128 . . . . . . . . . . . . . . . (b) and . . . . . . . . . . . . . . . . . . . . . . . (c) . . . . . . . . . . . . . . . . . . . . . . 76 xiv 7.5 7.6 7.7 List of figures Cristiano Ronaldo egocentric network. . . . . . . . . . . . . . . . . . . 129 Twitter buzz share of political leaders. . . . . . . . . . . . . . . . . . . 133 Continuous line represents loss in the training data while dashed line represents loss in the validation data. Left side: effect of increasing \|V \| using 100% of training data. Right side: effect of varying the amount of training data used with \|V \| = 32768. . . . . . . . . . . . . . . . . . . . 142 List of tables 3.1 3.2 3.3 3.4 3.5 E-R retrieval definitions. . . . . . . . . . . . . . . . . . . . . . . . . . Illustrative example of the entity index in Early Fusion. . . . . . . . . Illustrative example of the relationship index in Early Fusion. . . . . Illustrative example of the document index in Late Fusion. . . . . . . Clique sets and associated feature functions by type and input nodes. . . . . . 34 44 46 49 56 4.1 4.2 4.3 4.4 4.5 4.6 Examples of query annotations. . . . . . . . . . . . . . . . RELink collection statistics. . . . . . . . . . . . . . . . . . ClueWeb09-B extractions statistics. . . . . . . . . . . . . . Description of query sets used for evaluation. . . . . . . . . Early Fusion and ERDM comparison using LM and BM25. Results of ERDM compared with three baselines. . . . . . . . . . . . . . . . . . . . . . . . 68 68 70 71 73 74 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 RepLab 2013 Filtering Task dataset description. . . . . . . . . . . . Entity filtering versions description. . . . . . . . . . . . . . . . . . . Official results for each version plus our validation set accuracy. . . Training set examples for both sub-tasks. . . . . . . . . . . . . . . . Microblog results with all features on validation and test sets. . . . Features performance breakdown on test set using RF. . . . . . . . News Headlines results with all features on validation and test sets. Features performance breakdown on test set using MLP. . . . . . . . . . . . . . . . . . . . . . . 83 84 84 87 90 91 92 92 6.1 6.2 Summary of the four type of features we consider. . . . . . . . . . . . . 100 F1 score of popularity high as function of tp and k equal to 0.5, 0.65 and 0.8 respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Distribution of positive, negative and neutral mentions per political party110 6.3 7.1 . . . . . . . . . . . . . . . . . . . . . . . . Number of 5-grams available for training for different sizes of target vocabulary \|V \| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 xvi List of tables Overall statistics for 12 combinations of models learned varying \|V \| and volume of training data. Results observed after 40 training epochs. . . 141 7.3 Evaluation of resulting embeddings using Class Membership, Class Distinction and Word Equivalence tests for different thresholds of cosine similarity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 7.2 Chapter 1 Introduction Nowadays, people have pervasive access to connected devices, applications and services that enable them to obtain and share information almost instantly, on a 24/7 basis. With Social Media growing at an astonishing speed, user opinions about people, companies and products quickly spread over large communities. Consequently, companies and personalities are under thorough scrutiny, with every event and every statement potentially observed and evaluated by a global audience, which reflects one’s perceived reputation. Van Riel and Fombrun [1] define reputation as the “overall assessment of organizations by their stakeholders.” The authors use the term organization in the definition, but it may as well apply to individuals (e.g. politicians) or products (e.g. mobile phone brands). A stakeholder is someone who has some relationship with the organization, such as employees, customers or shareholders. This definition and other similar ones [2], focus on the perspective that reputation represents perceptions that others have on the target entity. However, the rise of Social Media and online news publishing has brought about wider public awareness about the entities’ activities, influencing people’s perceptions about their reputation. While traditional reputation analysis is mostly manual and focused on particular entities, with online media it is possible to automate much of the process of collecting, preparing and understanding large streams of content, to identify facts and opinions about a much wider set of entities. Online Reputation Monitoring (ORM) addresses this challenge: the use of computational tools to measure the reputation of entities from online media content. Early ORM started with counting occurrences of a brand name in Social Media as a channel to estimate the knowledge/reach of a brand. Introduction 2 There are several challenges to collect, process and mine online media data for these purposes [3]. Social Media texts are short, informal, with many abbreviations, slang, jargon and idioms. Often, the users do not care about the correct use of grammar and therefore the text tends to have misspellings, incomplete and unstructured sentences. Furthermore, the lack of context poses a very difficult problem for tasks relevant in the context of Text Mining, such as Named Entity Disambiguation or Sentiment Analysis. Once we classify the sentiment polarity of a given document (e.g. tweet or news title), it is necessary to aggregate several document scores to create meaningful hourly/daily indicators. These tasks are technically complex for most of the people interested in tracking entities on the web. For this reason, most research has focused on investigating parts of this problem leading to the development of tools that only address sub-tasks of this endeavor. Text data usually includes a large number of entities and relationships between them. We broadly define an entity to be a thing or concept that exists in the world, such as a person, a company, organization, an event or a film. Entities exist as mentions across documents and in external knowledge resources. In recent years, entities have gained increased importance as the basic unit of information to answer particular information needs, instead of entire documents or text snippets [4, 5]. The volume of entity-centric data is rapidly increasing on the Web, including RDF and Linked Data, Schema.org, Facebook’s Open Graph, and Google’s Knowledge Graph, describing entities (e.g., footballers and coaches) and relationships between them (e.g., “manages”). These developments have a great impact in Online Reputation Monitoring as it is mainly focused on entities. More specifically, the ORM process consists in searching and tracking an entity of interest: the personality, the company, organization or brand/product under analysis. On the other hand, news stories, topics and events discussed in the news or Social Media usually contain mentions of entities or concepts represented in a Knowledge Base. Thus, we can say that entities are the gravitational force that drives the Online Reputation Monitoring process. 1.1 Thesis Statement The ultimate goal of ORM is to track everything that is said on the Web about a given target entity and consequently, to assess/predict the impact on its reputation. From our perspective, this goal is very hard to achieve for two reasons. The first reason has to do with the difficulty of computationally processing, interpreting and accessing the huge amount of information published online everyday. The second 1.1 Thesis Statement 3 reason is inherent to the definition of reputation as being intangible but having tangible outcomes. More specifically, Fombrun and Van Riel [6] and later Stacks [7] found a correlation between several indicators, such as reputation or trust, and financial indicators, such as sales or profits. However, this finding does not imply causality, as financial indicators can be influenced by many factors, besides stakeholders’ perceived reputation. In conclusion, there is no consensus on how to measure reputation, neither intrinsically nor extrinsically. To the best of our knowledge, current ORM is still very limited and naive. The most standard approach consists in counting mentions of entity names and applying sentiment analysis to produce descriptive reports of aggregated entity popularity and overall sentiment. We propose to make progress in ORM by tackling two computational problems: Entity Retrieval and Text Mining (Figure 1.1). Online Reputation Monitoring Text and Entities Entity Retrieval Text Mining Fig. 1.1 Entity Retrieval and Text Mining as computational problems of ORM. We believe that a ORM platform, besides providing aggregated statistics and trends about entity popularity and sentiment on the news and social media, would benefit from providing entity retrieval capabilities. End users often like to have the flexibility to search for specific information that is not available in predefined charts. However, ORM has some specificities that traditional entity search systems cannot cope with. More specifically, an entity’s reputation is also influenced by the entity’s relationships with other entities. 4 Introduction For instance, the reputation of Apple Inc. was severely damaged with the so called “Apple Foxconn scandal”. Foxconn was one of the several contractor companies in Apple’s supply chain that was accused of exploiting Chinese workers. Although the facts were not directly concerned with Apple itself, its relationship with Foxconn triggered bad public opinion about Apple. The same happened recently with the “Weinstein sex scandal”, as accusations of sexual harassment aimed at Harvey Weinstein created a wave of damage to companies and personalities associated with the disgraced Hollywood producer. Therefore, a ORM platform should provide entity-relationship search capabilities. Entity-Relationship (E-R) Retrieval is a complex case of entity retrieval where the goal is to search for multiple unknown entities and relationships connecting them. Contrary to traditional entity queries, E-R queries expect tuples of connected entities as answers. For instance, “US technology companies contracts Chinese electronics manufacturers" can be answered by tuples <Apple, Foxconn>, while “Companies founded by disgraced Hollywood producer" is expecting tuples <Miramax, Harvey Weinstein>. In essence, an E-R query can be decomposed into a set of sub-queries that specify types of entities and types of relationships between entities. On the other hand, ORM requires accurate and robust text processing and data analysis methods. Text Mining plays an essential enabling role in developing better ORM. There are several challenges with collecting and extracting relevant entity-centric information from raw text data. It is necessary to filter noisy data otherwise downstream processing tasks, such as sentiment analysis, will be compromised. More specifically, it is essential to develop named entity disambiguation approaches that can distinguish relevant text passages from non-relevant. Named entities are often ambiguous, for example, the word “bush” is a surface form for two former U.S. presidents, a music band and a shrub. The ambiguity of named entities is particularly problematic in social media texts, where users often mention entities using a single term. ORM platforms would be even more useful if they would be able to predict if social media users will talk a lot about the target entities or not. For instance, on April 4th 2016, the UK Prime-minister, David Cameron, was mentioned on the news regarding the Panama Papers story. He did not acknowledge the story in detail on that day. However, the news cycle kept mentioning him about this topic in the following days and his mentions on social media kept very high. He had to publicly address the issue on April 9th, when his reputation had already been severely damaged, blaming himself for not providing further details earlier. Thus we also want to study the feasibility of 1.2 Objectives 5 using entity-centric knowledge extracted from Social Media and online news to predict real world surveys results, such as political polls. 1.2 Objectives The work reported on this dissertation aimed to understand, formalize and explore the scientific challenges inherent to the problem of using unstructured text data from different Web sources for Online Reputation Monitoring. We now describe the specific research challenges we proposed to overcome. Entity-Relationship Retrieval: Existing strategies for entity search can be divided in IR-centric and Semantic-Web-based approaches. The former usually rely on statistical language models to match and rank co-occurring terms in the proximity of the target entity [8]. The latter consists in creating a SPARQL query and using it over a structured knowledge base to retrieve relevant RDF triples [9]. Neither of these paradigms provide good support for entity-relationship (E-R) retrieval. Recent work in Semantic-Web search tackled E-R retrieval by extending SPARQL to support joins of multiple query results and creating an extended knowledge graph [10]. Extracted entities and relationships are typically stored in a knowledge graph. However, it is not always convenient to rely on a structured knowledge graph with predefined and constraining entity types. In particular, ORM is interested in transient information sources, such as online news or social media. General purpose knowledge graphs are usually fed with more stable and reliable data sources (e.g. Wikipedia). Furthermore, predefining and constraining entity and relationship types, such as in Semantic Web-based approaches, reduces the range of queries that can be answered and therefore limits the usefulness of entity search, particularly when one wants to leverage free-text. To the best of our knowledge, E-R retrieval using IR-centric approaches is a new and unexplored research problem within the Information Retrieval research community. One of the objectives of our research is to explore to what degree we can leverage the textual context of entities and relationships, i.e., co-occurring terminology, to relax the notion of an entity or relationship type. Instead of being characterized by a fixed type, e.g., person, country, place, the entity would be characterized by any contextual term. The same applies to the relationships. Traditional knowledge graphs have fixed schema of relationships, e.g. child of, created by, works for while our approach relies on contextual terms in the text proximity of 6 Introduction every two co-occurring entities in a raw document. Relationships descriptions such as “criticizes”, “hits back”, “meets” or “interested in” would be possible to search for. This is expected to significantly reduce the limitations which structured approaches suffer from, enabling a wider range of queries to be addressed. Entity Filtering and Sentiment Analysis: Entity Filtering is a sub-problem of Named Entity Disambiguation (NED) in which we have a named entity mention and we want to classify it as related or not related with the given target entity. This is a relatively easy problem in well formed texts such as news articles. However, social media texts pose several problems to this task. We are particularly interested in Entity Filtering of tweets and we aim to study a large set of features that can be generated to describe the relationship between a given target entity and a tweet, as well as exploring different learning algorithms to create supervised models for this task. Sentiment Analysis has been thoroughly studied in the last decade [11]. There have been several PhD thesis entirely dedicated to this subject. It is a broad problem with several ramifications depending on the text source and specific application. Within the context of ORM, we will focus in a particular domain: finance. Sentiment Analysis on financial texts has received increased attention in recent years [12]. Neverthless, there are some challenges yet to overcome [13]. Financial texts, such as microblogs or newswire, usually contain highly technical and specific vocabulary or jargon, making the development of specific lexical and machine learning approaches necessary. Text-based Entity-centric Prediction: We hypothesize that for entities that are frequently mentioned on the news (e.g. politicians) it is possible to establish a predictive link between online news and popularity on social media. We cast the problem as a supervised learning classification approach: to decide whether popularity will be high or low based on features extracted from the news cycle. We aim to assess if online news are valuable as source of information to effectively predict entity popularity on Twitter. More specifically, we want to find if online news carry different predictive power based on the nature of the entity under study and how predictive performance varies with different times of prediction. We propose to explore different text-based features and how particular ones affect the overall predictive power and specific entities in particular. On the other hand, we will study if it is possible to use knowledge extracted from social media texts to predict the outcome of public opinion surveys. The automatic content analysis of mass media in the social sciences has become necessary and possible 1.3 Research Methodology 7 with the rise of social media and computational power. One particularly promising avenue of research concerns the use of sentiment analysis in microblog streams. However, one of the main challenges consists in aggregating sentiment polarity in a timely fashion that can be fed to the prediction method. A Framework for ORM: The majority of the work in ORM consists in ad-hoc studies where researchers collect data from a given social network and produce their specific analysis or predictions, often unreproducible. The availability of open source platforms in this area is scarse. Researchers typically use specific APIs and software modules to produce their studies. However, there has been some effort among the research community to address these issues through open source research platforms. We therefore aim to create an adaptable text mining framework specifically tailored for ORM that can be reused in multiple application scenarios, from politics to finance. This framework is able to collect texts from online media, such as Twitter, and identify entities of interest and classify sentiment polarity and intensity. The framework supports multiple data aggregation methods, as well as visualization and modeling techniques that can be used for both descriptive analytics, such as analyze how political polls evolve over time, and predictive analytics, such as predict elections. 1.3 Research Methodology We adopted distinct research methodologies in the process of developing the research work described in this thesis. The origin of this work was the POPSTAR project. POPSTAR (Public Opinion and Sentiment Tracking, Analysis, and Research) was a project that developed methods for the collection, measurement and aggregation of political opinions voiced in microblogs (Twitter), in blogs and online news. A first prototype of the framework for ORM was implemented and served as the backend of the POPSTAR website (http://www.popstar.pt/). The ground work concerned with the development of a framework for ORM was carried in the scope of the project. Therefore, the POPSTAR website served as use case for validating the effectiveness and adaptability of the framework. The Entity Filtering and Sentiment Analysis modules of the framework were evaluated using well known external benchmarks resulting in state-of-the-art performance. We participated in RepLab 2013 Filtering Task and evaluated our Entity Filtering method using the dataset created for the competition. One of our submissions obtained the first place at the competition. We also participated in SemEval 2017 Task 5: Introduction 8 Fine-grained Sentiment Analysis on Financial Microblogs and News. We were ranked 4th using one of the metrics at the sub-task 5.1 Microblogs. We performed two experiments regarding the text-based entity centric predictions. For predicting entity popularity on Twitter based on the news cycle we collected tweets and news articles from Portugal using the SocialBus twitter collector and online news from 51 different news outlets collected by SAPO. We used the number of entity mentions on Twitter as target variable and we extracted text-based features from the news datasets. Both datasets were aligned in time. We used the same Twitter dataset for studying different sentiment aggregate functions to serve as features for predicting political polls of a private opinion studies company, Eurosondagem. Improvements of Entity-Relationship (E-R) retrieval techniques have been hampered by a lack of test collections, particularly for complex queries involving multiple entities and relationships. We created a method for generating E-R test queries to support comprehensive E-R search experiments. Queries and relevance judgments were created from content that exists in a tabular form where columns represent entity types and the table structure implies one or more relationships among the entities. Editorial work involved creating natural language queries based on relationships represented by the entries in the table. We have publicly released the RELink test collection comprising 600 queries and relevance judgments obtained from a sample of Wikipedia List-of-lists-of-lists tables. We evaluated the new methods proposed for E-R retrieval using the RELink query collection together with two other smaller query collections created by research work in Semantic Web-based E-R retrieval. We used a large web corpus, the ClueWeb-09B containing 50 million web pages for creating E-R retrieval tailored indexes for running our experiments. Moreover, we implemented a demo using a large news collection of 12 million Portuguese news articles, resulting in the best demo award at ECIR 2016. 1.4 Contributions and Applications This work resulted in the following contributions: 1. A Text Mining framework that puts together all the building blocks required to perform ORM. The framework is adaptable and can be reused in different application scenarios, such as finance and politics. The framework provides entityspecific Text Mining functionalities that enable the collection, disambiguation, sentiment analysis, aggregation, prediction and visualization of entity-centric information from heterogeneous Web data sources. Furthermore, given that it is 1.4 Contributions and Applications 9 built using a modular architecture providing abstraction layers and well defined interfaces, new functionalities can easily be integrated. 2. Generalization of the problem of entity-relationship search to cover entity types and relationships represented by any attribute and predicate, respectively, rather than a pre-defined set. 3. A general probabilistic model for E-R retrieval using Bayesian Networks. 4. Proposal of two design patterns that support retrieval approaches using the E-R model. 5. Proposal of a Entity-Relationship Dependence model that builds on the basic Sequential Dependence Model (SDM) to provide extensible entity-relationship representations and dependencies, suitable for complex, multi-relations queries. 6. An Entity-relationship indexing and retrieval approach including learning to rank/data fusion methods that can handle entity and relationships ranking and merging of results. 7. The proposal of a method and strategy for automatically obtaining relevance judgments for entity-relationship queries. 8. We make publicly available queries and relevance judgments for the previous task. 9. Entity Filtering and Financial Sentiment Analysis methods tailored for Twitter that is able to cope with short informal texts constraints. 10. Analysis of the predictive power of online news regarding entity-centric metrics on Twitter, such as popularity or sentiment. 11. Analysis of how to combine entity-centric knowledge obtained from heterogeneous sources for survey-like prediction tasks. We believe this work can be useful in a wide range of applications from which we highlight six: Reputation Management is concerned with influencing and controlling company or individual reputation and consequently tracking what is said about entities online is one of the main concerns of this area. For instance, knowing if a given news article will have a negative impact on entity’s reputation would be crucial for damage control. Introduction 10 Digital Libraries are special libraries comprising a collection of digital objects (e.g. text or images) stored in a electronic media format. They are ubiquitous nowadays, from academic repositories, to biomedical databases, law enforcement repositories, etc. We believe the contributions we make to the Entity-Relationship Retrieval research problem can be applied to any digital library enabling a new wide range of search capabilities. Fraud Detection and inside trading detection is an area where information about entities (individuals and companies) and relationships between entities is very useful to discover hidden relationships and contexts of entities that might represent conflicts of interests or even fraud. Journalism, or more specifically, computational journalism would benefit of a powerful entity-relationship search tool in which journalists could investigate how entities were previously mentioned on the Web, including online news through time, as well as relationships among entities and their semantics. Political Science has given a lot of attention to Social Media in recent years due to the sheer amount of people reactions and opinions regarding politically relevant events. Being able to analyze the interplay between online news and Social Media from a political entity perspective can be very interesting for political scientists. On the other hand, it is becoming increasingly difficult to obtain pollsresponses via telephone and it is necessary to start testing alternative approaches. Social Media Marketing focuses on communicating through social networks with company potential and effective customers. Evaluating the success of a given campaign is a key aspect of this area. Therefore assessing the volume and polarity of mentions of a given company before and after a campaign would be very useful. 1.5 Foundations Most of the material of this thesis was previously published in journal, conference and workshop publications: • P.Saleiro, E. M. Rodrigues, C. Soares, E. Oliveira, “TexRep: A Text Mining Framework for Online Reputation Monitoring”, New Generation Computing, Volume 35, Number 4 2017 [14] 1.5 Foundations 11 • P. Saleiro, N. Milic-Frayling, E. M. Rodrigues, C. Soares, “RELink: A Research Framework and Test Collection for Entity-Relationship Retrieval”, 40th International ACM SIGIR Conference on Research and Development in Information Retrieval (SIGIR 2017) [15] • P. Saleiro, N. Milic-Frayling, E. M. Rodrigues, C. Soares, “Early Fusion Strategy for Entity-Relationship Retrieval”, The First Workshop on Knowledge Graphs and Semantics for Text Retrieval and Analysis (KG4IR@SIGIR 2017) [16] • P. Saleiro, L. Sarmento, E. M. Rodrigues, C. Soares, E. Oliveira, “Learning Word Embeddings from the Portuguese Twitter Stream: A Study of some Practical Aspects”, Progress in Artificial Intelligence (EPIA 2017) [17] • P. Saleiro, E. M. Rodrigues, C. Soares, E. Oliveira, “FEUP at SemEval-2017 Task 5: Predicting Sentiment Polarity and Intensity with Financial Word Embeddings”, International Workshop on Semantic Evaluation (SemEval@ACL 2017) [18] • P. Saleiro and C. Soares, “Learning from the News: Predicting Entity Popularity on Twitter” in Advances in Intelligent Data Analysis XV (IDA 2016) [19] • P. Saleiro, J. Teixeira, C. Soares, E. Oliveira, “TimeMachine: Entity-centric Search and Visualization of News Archives” in Advances in Information Retrieval: 38th European Conference on IR Research (ECIR 2016) [20] • P. Saleiro, L. Gomes, C. Soares, “Sentiment Aggregate Functions for Political Opinion Polling using Microblog Streams” in International C* Conference on Computer Science and Software Engineering (C3S2E 2016) [21] • P. Saleiro, S. Amir, M. J. Silva, C. Soares , “POPmine: Tracking Political Opinion on the Web” in IEEE International Conference on Computer and Information Technology; Ubiquitous Computing and Communications; Dependable, Autonomic and Secure Computing; Pervasive Intelligence and Computing (IUCC 2015) [22] • P. Saleiro, L. Rei, A. Pasquali, C. Soares, et al., “POPSTAR at RepLab 2013: Name ambiguity resolution on Twitter” in Fourth International Conference of the CLEF initiative (CLEF 2013) [23] Introduction 12 1.6 Thesis Outline In Chapter 2 we discuss related work to this thesis. In Chapter 3 we present a formalization of the problem of E-R retrieval using a IR-centric approach. We provide two design patterns for fusion-based E-R retrieval: Early Fusion and Late Fusion. We end the chapter by introducing a new supervised early fusion-based Entity Relationship Dependence Model (ERDM) that can be seen as an extension of the MRF framework for retrieval adapted to E-R retrieval. In Chapter 4 we describe a set of experiments on E-R retrieval over a Web corpus. First we introduce a new query collection, RELink QC, specifically tailored to this problem. We developed a semi-automatic approach to collect relevance judgments from tabular data and the editorial work consisted in creating E-R queries answered by those relevance judgments. We run experiments using the ClueWeb09-B as dataset and provide evaluation results for the new proposed methods for E-R retrieval. Chapter 5 is dedicated to Entity Filtering and Financial Sentiment Analysis. We evaluate our approaches using well known external benchmarks, namely, RepLab 2013 and SemEval 2017. In Chapter 6, we present two experiments of text-based entity-centric predictions. In the first experiment, we try to predict the popularity of entities on social media using solely features extracted from the news cycle. On the second experiment, we try to assess which sentiment aggregate functions are useful in predicting political polls results. In Chapter 7, we present an unified framework of ORM. The framework is divided in two major containers: RELink (Entity Retrieval) and TexRep (Text Mining). We present the data flow within the framework and how it can be used as a reference open source framework for researching in ORM. We also present some case studies of using this framework. We end this thesis with Chapter 8 which is dedicated to the conclusions. Chapter 2 Background and Related Work This chapter introduces an overview of the background concepts and previous research work on the tasks addressed in this dissertation. We start by presenting a brief description of the task of Online Reputation Monitoring (ORM), including related frameworks for ORM. We then survey previous research work in Entity Retrieval and Semantic Search, including a detailed explanation of the Markov Random Field model for retrieval and its variations. We describe the tasks of Named Entity Disambiguation, Sentiment Analysis and previous work on training word embeddings. We end this chapter by providing an overview of related work on text-based predictions, including predicting social media attention or the outcome of political elections. 2.1 Online Reputation Monitoring The reputation of a company is important for the company itself but as well for the stakeholders. More specifically, stakeholders make decisions about the company and its products faster if they are aware of the image of the company [24]. From the company perspective, reputation is an asset as it attracts stakeholders and it can represent economic profit at the end [25, 6] In 2001, Newell and Goldsmith used questionnaire and survey methodologies to introduce the first standardized and reliable measure of credibility of companies from a consumer perspective [26]. There have been also studies that find a correlation between company indicators such as reputation, trust and credibility, and financial indicators, such as sales and profits [6, 7]. These studies found that although reputations are intangible they influence tangible assets. Following this reasoning, Fombrum created a very successful measurement framework, named RepTrak [27]. Background and Related Work 14 A different methodology compared to questionnaires is media analysis (news, TV and radio broadcasts). Typically, the analysis involves consuming and categorizing media according to stakeholder and polarity (positive, negative) towards the company. Recently, Social Media analysis is becoming an important proxy of people opinion, originating the field of Online Reputation Monitoring [28]. While traditional reputation monitoring is mostly manual, online media pose the opportunity to process, understand and aggregate large streams of facts about about a company or individual. ORM requires some level of continuous monitoring [29]. It is crucial to detect early the changes in the perception of a company or personality conveyed in Social Media. Online buzz may be good or bad and consequently, companies must react and address negative trends [30, 31]. It also creates an opportunity to monitor the reputation of competitors. In this context, Text Mining plays a key, enabling role as it offers methods for deriving high-quality information from textual content [32]. For instance, Gonzalo [31] identifies 5 different Text Mining research areas relevant to ORM: entity filtering, topic tracking, reputation priority detection, user profiling and automatic reporting/summarization. Social Media as a new way of communication and collaboration is an influence for every stakeholder of society, such as personalities, companies or individuals [33]. Social Media users share every aspect of their lives and that includes information about events, news stories, politicians, brands or organizations. Companies have access to all this sharing which opens new horizons for obtaining insights that can be valuable to them and their online reputation. Companies also invest a big share of their public relations on Social Media. Building a strong reputation can take long time and effort but destroying it can take place overnight. Therefore, as the importance of Social Media increased, so did the importance of having powerful tools that deal with this enormous amount of data. 2.1.1 Related Frameworks The great majority of work in ORM consists in ad-hoc studies and platforms for ORM are usually developed by private companies that do not share internal information. However, there are some open source research projects that can be considered as related frameworks to this work. Trendminer [34] is one of such platforms that enables real time analysis of Twitter data, but has a very simple sentiment analysis using word counts and lacks flexibility in order to support entity-centric data processing. A framework for ORM should be 2.2 Entity Retrieval and Semantic Search 15 entity-centric, i.e., collect, process and aggregate texts and information extracted from those texts in relation to the entities being monitored. conTEXT [35] addresses adaptability and reusability by allowing a modular interface and allowing plugin components to extend their framework, specially from the perspective of the data sources and text analysis modules. For instance, it does not support Sentiment Analysis module by default but it could be plugged in. Neverthless, conTEXT does not support the plugin of aggregation and prediction modules which makes it not suitable for ORM. The FORA framework [30] is specifically tailored for ORM. It creates an ontology based on fuzzy clustering of texts but it is only concerned with extracting relevant linguistic units regarding the target entities and does not include automatic sentiment analysis and it does not allow the plugin of new modules. POPmine [36] was the first version of our Text Mining framework for ORM and it was developed specifically in the context of a project in political data science. It comprises a richer set of modules, including cross media data collection (Twitter, blog posts and online news) and real-time trend analysis based on entity filtering and sentiment analysis modules. In fact, our current version of TexRep, our Text Mining framework for ORM, can be seen as an extension of the POPmine architecture by creating a more general purpose framework for ORM which is not restricted to political analysis. While it would be possible to adapt POPmine’s entity disambiguation and sentiment analysis modules, its aggregations are specific to the political scenarios. On the other hand, TexRep supports users to define and plug custom-specific aggregate functions. Moreover, POPmine has limited user configurations (e.g. lacks support for pre-trained word embeddings) and does not include predictive capabilities. 2.2 Entity Retrieval and Semantic Search Information Retrieval deals with the “search for information”. It is defined as the activity of finding relevant information resources (usually documents) that meet an information need (usually a query), from within a large collection of resources of an unstructured nature (usually text) [37]. In early boolean retrieval systems, documents were retrieved if the exact query term was present and they were represented as a list of terms [37]. With the introduction of the Vector Space Model, each term represents a dimension in a multi-dimensional space, and consequently, each document and query are represented as vectors [38]. Values of each dimension of the document vector correspond to the term frequency 16 Background and Related Work (TF) of the term in the document. Therefore, the ranking list of documents is produced based on their spatial distance to the query vector. The concept of inverse document frequency (IDF) was later introduced to limit the effect of common terms in a collection [39]. A term that occurs in many documents of the collection has a lower IDF than terms that occur less often. The combination TF-IDF and variants, such as BM25 [40], became commonly used weighting statistics for Vector Space Model. Recently, it has been observed that when people have focused information needs, entities better satisfy those queries than a list of documents or large text snippets [5]. This type of retrieval is called Entity Retrieval or Entity-oriented retrieval and includes extra Information Extraction tasks for processing documents, such as Named Entity Recognition (NER) and Named Entity Disambiguation (NED). Entity Retrieval is closely connected with Question answering (QA) though, QA systems focus on understanding the semantic intent of a natural language query and deciding which sentences represent the answer to the user. Considering the query “British politicians in Panama papers”, the expected result would be a list of names rather than documents related to British politics and the “Panama Papers” news story. There are two search patterns related to Entity Retrieval [4]. First, the user knows the existence of a certain entity and aims to find related information about it. For example, a user searching for product related information. Second, the user defines a predicate that constrains the search to a certain type of entities, e.g. searching for movies of a certain genre. Online Reputation Monitoring systems usually focus on reporting statistical insights based on information extracted from Social Media and online news mentioning the target entity. However, this kind of interaction limits the possibility of users to explore all the knowledge extracted about the target entity. We believe Entity Retrieval could enhance Online Reputation Monitoring by allowing free text search over all mentions of the target entity and, consequently, allow users to discover information that descriptive statistical insights might not be able to identify. Entity Retrieval differs from traditional document retrieval in the retrieval unit. While document retrieval considers a document as the atomic response to a query, in Entity Retrieval document boundaries are not so important and entities need to be identified based on occurrence in documents [41]. The focus level is more granular as the objective is to search and rank entities among documents. However, traditional Entity Retrieval systems does not exploit semantic relationships between terms in the 2.2 Entity Retrieval and Semantic Search 17 query and in the collection of documents, i.e. if there is no match between query terms and terms describing the entity, relevant entities tend to be missed. Entity Retrieval has been an active research topic in the last decade, including various specialized tracks, such as Expert finding track [42], INEX entity ranking track [43], TREC entity track [44] and SIGIR EOS workshop [45]. Previous research faced two major challenges: entity representation and entity ranking. Entities are complex objects composed by a different number of properties and are mentioned in a variety of contexts through time. Consequently, there is no single definition of the atomic unit (entity) to be retrieved. Additionally, it is a challenge to devise entity rankings that use various entity representations approaches and tackle different information needs. There are two main approaches for tackling Entity Retrieval: “profile based approach” and “voting approach” [46]). The “profile based approach” starts by applying NER and NED in the collection in order to extract all entity occurrences. Then, for each entity identified, a meta-document is created by concatenating every passage in which the entity occurs. An index of entity meta-documents is created and a standard document ranking method (e.g. BM25) is applied to rank meta-documents with respect to a given query [47, 48]. One of the main challenges of this approach is the transformation of original text documents to an entity-centric meta-document index, including pre-processing the collection in order to extract all entities and their context. In the “voting approach”, the query is processed as typical document retrieval to obtain an initial list of documents [46, 49]. Entities are extracted from these documents using NER and NED techniques. Then, score functions are calculated to estimate the relation of entities captured and the initial query. For instance, counting the frequency of occurrence of the entity in the top documents combined with each document score (relevance to the query) [46]. Another approach consists in taking into account the distance between the entity mention and the query terms in the documents [50]. Recently, there is an increasing research interest in Entity Search over Linked Data, also referred as Semantic Search, due to the availability of structured information about entities and relations in the form of Knowledge Bases [51–53]. Semantic Search exploits rich structured entity related in machine readable RDF format, expressed as a triple (entity, predicate, object). There are two types of search: keyword-based and natural language based search [54, 55]. Regardless of the search type, the objective is to interpret the semantic structure of queries and translate it to the underlying schema of the target Knowledge Base. Most of the research focus is on interpreting the query intent [54, 55] while others focus on how to devise a ranking framework that deals with 18 Background and Related Work similarities between different attributes of the entity entry in the KB and the query terms [53] Relationship Queries: Li et al. [56] were the first to study relationship queries for structured querying entities over Wikipedia text with multiple predicates. This work used a query language with typed variables, for both entities and entity pairs, that integrates text conditions. First it computes individual predicates and then aggregates multiple predicate scores into a result score. The proposed method to score predicates relies on redundant co-occurrence contexts. Yahya et al. [10] defined relationship queries as SPARQL-like subject-predicateobject (SPO) queries joined by one or more relationships. The authors cast this problem into a structured query language (SPARQL) and extended it to support textual phrases for each of the SPO arguments. Therefore it allows to combine both structured SPARQL-like triples and text simultaneously. It extended the YAGO knowledge base with triples extracted from ClueWeb using an Open Information Extraction approach [57]. In the scope of relational databases, keyword-based graph search has been widely studied, including ranking [58]. However, these approaches do not consider full documents of graph nodes and are limited to structured data. While searching over structured data is precise it can be limited in various respects. In order to increase the recall when no results are returned and enable prioritization of results when there are too many, Elbassuoni et al. [59] propose a language-model for ranking results. Similarly, the models like EntityRank by Cheng et al. [60] and Shallow Semantic Queries by Li et al. [56], relax the predicate definitions in the structured queries and, instead, implement proximity operators to bind the instances across entity types. Yahya et al. [10] propose algorithms for application of a set of relaxation rules that yield higher recall. Entity Retrieval and proximity: Web documents contain term information that can be used to apply pattern heuristics and statistical analysis often used to infer entities as investigated by Conrad and Utt [61], Petkova and Croft [50], Rennie and Jaakkola [62]. In fact, early work by Conrad and Utt [61] demonstrates a method that retrieves entities located in the proximity of a given keyword. They show that using a fixed-size window around proper-names can be effective for supporting search for people and finding relationship among entities. Similar considerations of the co-occurrence statistics have been used to identify salient terminology, i.e. keyword to include in the document index [50]. 2.2 Entity Retrieval and Semantic Search 2.2.1 19 Markov Random Field for IR In this section we detail the generic Markov Random Field (MRF) model for retrieval and its variation, the Sequential Dependence Model (SDM). As we later show, this model is the basis for our entity-relationship retrieval model. The Markov Random Field (MRF) model for retrieval was first proposed by Metzler and Croft [63] to model query term and document dependencies. In the context of retrieval, the objective is to rank documents by computing the posterior P (D\|Q), given a document D and a query Q: P (D\|Q) = P (Q, D) P (Q) (2.1) For that purpose, a MRF is constructed from a graph G, which follows the local Markov property: every random variable in G is independent of its non-neighbors given observed values for its neighbors. Therefore, different edge configurations imply different independence assumptions. Fig. 2.1 Markov Random Field document and term dependencies. Metzler and Croft [63] defined that G consists of query term nodes qi and a document node D, as depicted in Figure 2.1. The joint probability mass function over the random variables in G is defined by: PG,Λ (Q, D) = 1 Y ψ(c; Λ) ZΛ c∈C(G) (2.2) where Q = q1 , ...qn are the query term nodes, D is the document node, C(G) is the set of maximal cliques in G, and ψ(c; Λ) is a non-negative potential function over clique P Q configurations. The parameter ZΛ = Q,D c∈C(G) ψ(c; Λ) is the partition function that normalizes the distribution. It is generally unfeasible to compute ZΛ , due to the exponential number of terms in the summation, and it is ignored as it does not influence ranking. Background and Related Work 20 The potential functions are defined as compatibility functions between nodes in a clique. For instance, a tf-idf score can be measured to reflect the “aboutness” between a query term qi and a document D. Metzler and Croft [63] propose to associate one or more real valued feature function with each clique in the graph. The non-negative potential functions are defined using an exponential form ψ(c; Λ) = exp[λc f (c)], where λc is a feature weight, which is a free parameter in the model, associated with feature function f (c). The model allows parameter and feature functions sharing across cliques of the same configuration, i.e. same size and type of nodes (e.g. 2-cliques of one query term node and one document node). For each query Q, we construct a graph representing the query term dependencies, define a set of non-negative potential functions over the cliques of this graph and rank documents in descending order of PΛ (D\|Q): rank PΛ (D\|Q) = log PΛ (D\|Q) rank = log PΛ (Q, D) − log PΛ (Q) rank = X log ψ(c; Λ) (2.3) c∈C(G) rank = X log exp[λc f (c)] c∈C(G) rank = X λc f (c) c∈C(G) (2.4) Metzler and Croft concluded that given its general form, the MRF can emulate most of the retrieval and dependence models, such as language models [64]. 2.2.2 Sequential Dependence Model The Sequential Dependence Model (SDM) is the most popular variant of the MRF retrieval model [63]. It defines two clique configurations represented in the following potential functions ψ(qi , D; Λ) and ψ(qi , qi+1 , D; Λ). Basically, it considers sequential dependency between adjacent query terms and the document node. The potential function of the 2-cliques containing a query term node and a document node is represented as ψ(qi , D; Λ) = exp[λT fT (qi , D)]. The clique configuration containing contiguous query terms and a document node is represented by two real valued functions. The first considers exact ordered matches of the 2.2 Entity Retrieval and Semantic Search 21 two query terms in the document, while the second aims to capture unordered matches within N fixed window sizes. Consequently, the second potential function is ψ(qi , qi+1 , D; Λ) = exp[λO fO (qi , qi+1 , D) + λU fU (qi , qi+1 , D)]. Replacing ψ(c; Λ) by these potential functions in Equation 3.38 and factoring out the parameters λ, the SDM can be represented as a mixture model computed over term, phrase and proximity feature classes: rank X P (D\|Q) = λT fT (qi , D) + qi ∈Q X λO fO (qi , qi+1 , D) + qi ,qi+1 ∈Q X λU fU (qi , qi+1 , D) qi ,qi+1 ∈Q where the free parameters λ must follow the constraint λT +λO +λU = 1. Coordinate Ascent was chosen to learn the optimal λ values that maximize mean average precision using training data [65]. Considering tf the frequency of the term(s) in the document D, cf the frequency of the term(s) in the entire collection C, the feature functions in SDM are set as: " # cfq tfqi ,D +µ \|C\|i fT (qi , D) = log (2.5) \|D\|+µ  fO (qi , qi+1 , D) = log  tf#1(qi ,qi+1 ),D +µ \|D\|+µ  fU (qi , qi+1 , D) = log  cf#1(q ,q i i+1) \|C\| tf#uwN (qi ,qi+1 ),D +µ cf#uwN (q ,q i i+1) \|C\| \|D\|+µ   (2.6)   (2.7) where µ is the Dirichlet prior for smoothing, #1(qi , qi+1 ) is a function that searches for exact matches of the phrase “qi qi+1 ” and #uwN (qi , qi+1 ) is a function that searches for co-occurrences of qi and qi+1 within a window of fixed-N terms (usually 8 terms) across document D. SDM has shown state-of-the-art performance in ad-hoc document retrieval when compared with several bigram dependence models and standard bag-ofwords retrieval models, across short and long queries [66]. Background and Related Work 22 2.2.3 MRF for Entity Retrieval The current state-of-the-art methods in ad-hoc entity retrieval from knowledge graphs are based on MRF [53, 67]. The Fielded Sequential Dependence Model (FSDM) [53] extends SDM for structured document retrieval and it is applied to entity retrieval from knowledge graphs. In this context, entity documents are composed by fields representing metadata about the entity. Each entity document has five fields: names, attributes, categories, similar entity names and related entity names. FSDM builds individual language models for each field in the knowledge base. This corresponds to replacing SDM feature functions with those of the Mixture of Language Models [68]. The feature functions of FSDM are defined as: f˜T (qi , D) = log F X   tfqi ,Dj + µj wjT  j f˜O (qi , qi+1 , D) = log F X f˜U (qi , qi+1 , D) = log \|Dj \| + µj  tf#1(qi ,qi+1 ),Dj + µj O wj   (2.8)   cf#1(qi ,qi+1 ),j \|Cj \| \|Dj \| + µj j F X cfqi ,j \|Cj \|  tf#uwN (qi ,qi+1 ),Dj + µj U  wj  cf#uwN (qi ,qi+1 ),j \|Dj \| + µj j \|Cj \|    (2.9)    (2.10) where µj are the Dirichlet priors for each field and wj are the weights for each field P and must be non-negative with constraint Fj wj = 1. Coordinate Ascent was used in two stages to learn wj and λ values [53]. The Parameterized Fielded Sequential Dependence Model (PFSDM) [67] extends the FSDM by dynamically calculating the field weights wj to different query terms. Part-of-speech features are applied to capture the relevance of query terms to specific fields of entity documents. For instance, NNP feature is positive if query terms are proper nouns, therefore the query terms should be mapped to the names field. Therefore, the field weight contribution of a given query term qi and a query bigram qi ,qi+1 in a field j are a linear weighted combination of features: wqi ,j = X U αj,k ϕk (qi , j) (2.11) k wqi ,qi+1 ,j = X k B αj,k ϕk (qi , qi+1 , j) (2.12) 2.3 Named Entity Disambiguation 23 U where ϕk (qi , j) is the k feature function of a query unigram for the field j and αj,k is its respective weight. For bigrams, ϕk (qi , qi+1 , j) is the k feature function of a query B bigram for the field j and αj,k is its respective weight. Consequently, PFSDM has F ∗ U + F ∗ B + 3 total parameters, where F is the number of fields, U is the number of field mapping features for unigrams, B is the number of field mapping features for bigrams, plus the three λ parameters. Their estimation is performed in a two stage optimization. First α parameters are learned separately for unigrams and then bigrams. This is achieved by setting to zero the corresponding λ parameters. In the second stage, the λ parameters are learned. Coordinate Ascent is used in both stages. The ELR model exploits entity mentions in queries by defining a dependency between entity documents and entity links in the query [69]. 2.3 Named Entity Disambiguation Given a mention in a document, Named Entity Disambiguation (NED) or Entity Linking aims to predict the entity in a reference knowledge base that the string refers to, or NIL if no such entity is available. Usually the reference knowledge base (KB) includes a set of documents, where each document describes one specific entity. Wikipedia is by far the most popular reference KB [70]. Previous research typically performs three steps to link an entity mention to a KB: 1) representation of the mention, i.e. extend the entity mention with relevant knowledge from the background document, 2) candidate generation, i.e. find all possible KB entries that the mention might refer to and their representation 3) disambiguation, by computing the similarity between the represented mention and the candidate entities. Entity Filtering, or targeted entity disambiguation, is a special case of NED in which there is only one candidate entity, i.e. the entity that is being monitored. There is an increasing interest in developing Entity Filtering methods for Social Media texts, considering its specificities and limitations [71, 72]. These approaches focus on finding relevant keywords for positive and negative cases using co-occurrence, web and collection based features. Another line of work creates topic-centric entity extraction systems where entities belong to a certain topic and are used as evidence to disambiguate the short message given its topic [73]. Similarly, Hangya et al. [74] create features representing topic distributions over tweets using Latent Dirichlet Allocation (LDA). The majority of research work in NED is usually applied to disambiguate entities in reasonably long texts as news or blog posts. In recent years, there has been an increasing interest in developing NED methods for Social Media texts and its specificities and Background and Related Work 24 limitations [75–78]. A survey and evaluation of state-of-the-art NER and NED for Tweets concluded that current approaches do not perform robustly on “ill-formed, terse, and linguistically compressed” microblog texts [79]. Some Twitter-specific methods reach F1 measures of over 80%, but are still behind the state-of-the-art results obtained on well-formed news texts. Social Media texts are too short to provide sufficient information to calculate context similarity accurately [76, 80, 78, 77, 81]. In addition, most of state-of-theart approaches leverage on neighboring entities in the documents but, once again, tweets are short and do not have more than one or two entities mentioned. Most of them [82, 77, 81] extract information obtained from other tweets, and disambiguate entity mentions in these tweets collectively. The assumption is that Twitter users are content generators and tend to scatter their interests over many different messages they broadcast, which is not necessarily true [83]. Entity Filtering has also been studied in the context of real-time classification. Davis et al. [81] propose a pipeline containing three stages. Clearly positive examples are exploited to create filtering rules comprising collocations, users and hashtags. The remaining examples are classified using a Expectation-Maximization (EM) model trained using the clearly positive examples. Recently, Habib et al. [84] proposed an hybrid approach where authors first query Google to retrieve a set of possible candidate homepages and then enrich the candidate list with text from the Wikipedia. They extract a set of features for each candidate, namely, a language model and overlapping terms between tweet and document, as well as URL length and mention-URL string similarity. In addition, a prior probability of the mention corresponding to a certain entity on the YAGO [85] knowledge base is also used. Recent work in NED or Entity Linking includes graph based algorithms for collective entity disambiguation, such as TagMe[86], Babelfy [87] and WAT [88]. Word and entity embeddings have been also used for entity disambiguation [89–91]. More specifically, Fang [90] and Moreno [91] propose to learn an embedding space for both entities and words and then compute similarity features based on the combined representations. 2.4 Sentiment Analysis In the last decade, the automatic processing of subjective and emotive text, commonly known as Sentiment Analysis, has triggered huge interest from the Text Mining research community [92]. A typical task in Sentiment Analysis is text polarity classification and in the context of this work can be formalized as follows: given a text span that mentions 2.4 Sentiment Analysis 25 a target entity, decide whether it conveys positive, negative or neutral sentiment towards the target. With the rise of Social Media, research on Sentiment Analysis shifted towards Twitter. New challenges have risen, including slang, misspelling, emoticons, poor grammatical structure [92]. A number of competitions were organized, such as SemEval [93], leading to the creation of resources for research [94]. There are two main approaches to sentiment polarity classification: lexicon-based using a dictionary of terms and phrases with annotated polarity – or supervised learning – building a model of the differences in language associated with each polarity, based on training examples. In the supervised learning approach, a classifier is specifically trained for a particular type of text (e.g. tweets about politics). Consequently, it is possible to capture peculiarities of the language used in that context. As expected, this reduces the generality of the model, as it is biased towards a specific domain. Supervised learning approaches require training data. In Twitter, most of previous work obtained training data by assuming that emoticons represent the tweet polarity (positive, negative, neutral) [95], or by using third party software, such as the Stanford Sentiment Analyzer [96]. Lexicon-based approaches have shown to work effectively on conventional text [97] but tend to be ill suited for Twitter data. With the purpose of overcoming this limitation, an algorithm that uses a human-coded lexicon specifically tailored to Social Media text was introduced [98]. SentiStrength has become a reference in recent years due to its relatively good performance and consistent performance on polarity classification of Social Media texts. Nevertheless, it is confined to a fixed set of words and it is context independent. The recent interest in deep learning led to approaches that use deep learned word embeddings as features in a variety of Text Mining tasks [99, 100]. In Sentiment Analysis, recent work integrated polarity information of text into the word embedding by extending the probabilistic document model obtained from Latent Dirichlet Allocation [101]. While others learned task-specific embeddings from an existing embedding and sentences with annotated polarity [102]. Or learning polarity specific word embeddings from tweets collected using emoticons [103] and directly incorporating the supervision from sentiment polarity in the loss functions of neural networks [104]. Background and Related Work 26 2.5 Word Embeddings The most popular and simple way to model and represent text data is the Vector Space Model [105]. A vector of features in a multi-dimensional feature space represents each lexical item (e.g. a word) in a document and each item is independent of other items in the document. This allows to compute geometric operations over vectors of lexical items using well established algebraic methods. However, the Vector Space Model faces some limitations. For instance, the same word can express different meanings in different contexts - the polysymy problem - or different words may be used to describe the same meaning - the synonymy problem. Since 2000, a variety of different methods (e.g. LDA [106]) and resources (e.g. DBpedia [107]) have been developed to try to assign semantics, or meaning, to concepts and parts of text. Word embedding methods aim to represent words as real valued continuous vectors in a much lower dimensional space when compared to traditional bag-of-words models. Moreover, this low dimensional space is able to capture lexical and semantic properties of words. Co-occurrence statistics are the fundamental information that allows creating such representations. Two approaches exist for building word embeddings. One creates a low rank approximation of the word co-occurrence matrix, such as in the case of Latent Semantic Analysis [108] and GloVe [109]. The other approach consists in extracting internal representations from neural network models of text [110, 111, 100]. Levy and Goldberg [112] showed that the two approaches are closely related. Although, word embedding research goes back several decades, it was the recent developments of Deep Learning and the word2vec framework [100] that captured the attention of the NLP community. Moreover, Mikolov et al. [113] showed that embeddings trained using word2vec models (CBOW and Skip-gram) exhibit linear structure, allowing analogy questions of the form “man:woman::king:??.” and can boost performance of several text classification tasks. In this context, the objective is to maximize the likelihood that words are predicted given their context. word2vec has two models for learning word embeddings, the skip-gram model (SG) and the continuous-bag-of-word model (CBOW). Here we focus on CBOW. More formally, every word is mapped to a unique vector represented by a column in a projection matrix W ∈ Rd×V with d as embedding dimension and V as the total number of words in the vocabulary. Given a sequence of words w−2 , w−1 , wt , w1 , w2 , ..., wT , the objective is to maximize the average log probability: V X 1X log P (wt \|wt+j ) T t=1 −c≤j≤c,j̸=0 (2.13) 2.5 Word Embeddings 27 where c is the size of the context window and wt+j is a word in the context window of the center word wt . The context vector is obtained by averaging the embeddings of each word w−c≤j≤c,j̸=0 and the prediction of the center word wt is performed using a softmax multiclass classifier over all vocabulary V : eywt P (wt \|wt+j ) = P yw e i (2.14) Each of yi is un-normalized log-probability for each output word i. After training, a low dimensionality embedding matrix E encapsulating information about each word in the vocabulary and its surrounding contexts is learned, transforming a one-hot sparse representation of words into a compact real valued embedding vector of size d × 1. This matrix can then be used as input to other learning algorithms tailored for specific tasks to further enhance performance. For large vocabularies it is unfeasible to compute the partition function (normalizer) of softmax therefore Mikolov [100] proposes to use the hierarchical softmax objective function or to approximate the partition function using a technique called negative sampling. Stochastic gradient descent is usually applied for training the softmax where the gradient is obtained via backpropagation. There are several approaches to generating word embeddings. One can build models that explicitly aim at generating word embeddings, such as Word2Vec or GloVe [100, 109], or one can extract such embeddings as by-products of more general models, which implicitly compute such word embeddings in the process of solving other language tasks. One of the issues of recent work in training word embeddings is the variability of experimental setups reported. For instance, in the paper describing GloVe [109] the authors trained their model on five corpora of different sizes and built a vocabulary of 400K most frequent words. Mikolov et al. [113] trained with 82K vocabulary while Mikolov et al. [100] was trained with 3M vocabulary. Recently, Arora et al. [114] proposed a generative model for learning embeddings that tries to explain some theoretical justification for nonlinear models (e.g. word2vec and GloVe) and some hyper parameter choices. The authors evaluated their model using 68K vocabulary. SemEval 2016-Task 4: Sentiment Analysis in Twitter organizers report that participants either used general purpose pre-trained word embeddings, or trained from Tweet 2016 dataset or “from some sort of dataset” [115]. However, participants neither report the size of vocabulary used neither the possible effect it might have on the task specific results. 28 Background and Related Work Recently, Rodrigues et al. [116] created and distributed the first general purpose embeddings for Portuguese. Word2vec gensim implementation was used and authors report results with different values for the parameters of the framework. Furthermore, authors used experts to translate well established word embeddings test sets for Portuguese language, which they also made publicly available and we use some of those in this work. 2.6 Predicting Collective Attention Online Reputation Monitoring systems would be even more useful if they would be able to know in advance if social media users will talk a lot about the target entities or not. In recent years, a number of research works have studied the relationship and predictive behavior of user response to the publication of online media items, such as, commenting news articles, playing Youtube videos, sharing URLs or retweeting patterns [117–120]. The first attempt to predict the volume of user comments for online news articles used both metadata from the news articles and linguistic features [119]. The prediction was divided in two binary classification problems: if an article would get any comments and if it would be high or low number of comments. Similarly, other studies found that shallow linguistic features (e.g. TF-IDF or sentiment) and named entities have good predictive power [121, 122]. Research work more in line with ours, tries to predict the popularity of news articles shares (url sharing) on Twitter based on content features [117]. The authors considered the news source, the article’s category, the article’s author, the subjectivity of the language in the article, and number of named entities in the article as features. Recently, there was a large study of the life cycle of news articles in terms of distribution of visits, tweets and shares over time across different sections of the publisher [123]. Their work was able to improve, for some content type, the prediction of web visits using data from social media after ten to twenty minutes of publication. Other lines of work, focused on temporal patterns of user activities and have consistently identified broad classes of temporal patterns based on the presence of a clear peak of activity [124–126, 118]. Classes differentiate by the specific amount and duration of activity before and after the peak. Crane and Sornette [124] define endogenous or exogenous origin of events based on being triggered by internal aspects of the social network or external, respectively. They find that hashtag popularity is mostly influenced by exogenous factors instead of epidemic spreading. Other work [125] extend these classes by creating distinct clusters of activity based on the distributions 2.7 Political Data Science 29 in different periods (before, during and after the peak) that can be interpreted based on semantics of hashtags. Consequently, the authors applied text mining techniques to semantically describe hashtag classes. Yang and Leskovec [118] propose a new measure of time series similarity and clustering. The authors obtain six classes of temporal shapes of popularity of a given phrase (meme) associated with a recent event, as well as the ordering of media sources contribution to its popularity. Recently, Tsytsarau et al. [127] studied the time series of news events and their relation to changes of sentiment time series expressed on related topics on social media. The authors proposed a novel framework using time series convolution between the importance of events and media response function, specific to media and event type. Their framework is able to predict time and duration of events as well as shape through time. 2.7 Political Data Science Content analysis of mass media has an established tradition in the social sciences, particularly in the study of effects of media messages, encompassing topics as diverse as those addressed in seminal studies of newspaper editorials [128], media agenda-setting [129], or the uses of political rhetoric [130], among many others. By 1997, Riffe and Freitag [131], reported an increase in the use of content analysis in communication research and suggested that digital text and computerized means for its extraction and analysis would reinforce such a trend. Their expectation has been fulfilled: the use of automated content analysis has by now surpassed the use of hand coding [132]. The increase in the digital sources of text, on the one hand, and current advances in computation power and design, on the other, are making this development both necessary and possible, while also raising awareness about the inferential pitfalls involved [133, 134]. One avenue of research that has been explored in recent years concerns the use of social media to predict present and future political events, namely electoral results [135– 143]. Although there is no consensus about methods and their consistency [144, 145]. Gayo-Avello [146] summarizes the differences between studies conducted so far by stating that they vary about period and method of data collection, data cleansing and pre-processing techniques, prediction approach and performance evaluation. One particular challenge when using sentiment is how to aggregate opinions in a timely fashion that can be fed to the prediction method. Two main strategies have been used to predict elections: buzz, i.e., number of tweets mentioning a given candidate or 30 Background and Related Work party and the use of sentiment polarity. Different computational approaches have been explored to process sentiment in text, namely machine learning and linguistic based methods [147–149]. In practice, algorithms often combine both strategies. Johnson et al. [150] concluded that more than predicting elections, social media can be used to gauge sentiment about specific events, such as political news or speeches. Defending the same idea, Diakopoulos el al. [151] studied the global sentiment variation based on Twitter messages of an Obama vs McCain political TV debate while it was still happening. Tumasjan et al. [140] used Twitter data to predict the 2009 Federal Election in Germany. They stated that “the mere number of party mentions accurately reflects the election result”. Bermingham et al. [135] correctly predicted the 2011 Irish General Elections also using Twitter data. Gayo-Avello et al. [145] also tested the share of volume as predictor in the 2010 US Senate special election in Massachusetts. On the other hand, several other studies use sentiment as a polls result indicator. Connor et al. [142] used a sentiment aggregate function to study the relationship between the sentiment extracted from Twitter messages and polls results. They defined the sentiment aggregate function as the ratio between the positive and negative messages referring an specific political target. They used the sentiment aggregate function as predictive feature in the regression model, achieving a correlation of 0.80 between the results and the poll results, capturing the important large-scale trends. Bermingham et al. [135] also included in their regression model sentiment features. Bermingham et al. introduced two novel sentiment aggregate functions. For inter-party sentiment, they modified the share of volume function to represent the share of positive and negative volume. For intra-party sentiment , they used a log ratio between the number of positive and negative mentions of a given party. Moreover, they concluded that the inclusion of sentiment features augmented the effectiveness of their model. Gayo-Avello et al. [145] introduced a different aggregate function. In a two-party race, all negative messages on party c2 are interpreted as positive on party c1, and vice-versa. In summary, suggestions for potentially independent or in other words predictive metrics appear in a wide variety of forms: the mention share that a party received within all party mentions during a given time-span [135, 152–155, 140], the mention share of political candidates [156–159, 153], the share of positive mentions a party received [135, 160], the positive mention share of candidates [142, 161, 158], the share of users commenting on a candidate or party [155], the share of mentions for a candidate followed by a word indicative of electoral success or failure [162], the relative increase of positive mentions of a candidate [163] or simply a collection of various potentially 2.7 Political Data Science 31 politically relevant words identified by their statistical relationship with polls or political actors in the past [164–167]. Suggestions for the dependent variable, metrics of political success, show a similar variety. They include the vote share that a party received on election day [135, 163, 152– 154], the vote share of a party adjusted to include votes only for parties included in the analysis [140], the vote share of candidates on election day [157–159, 162, 153], campaign tracking polls [164, 165, 158, 166, 142, 161, 160], politicians’ job approval ratings [167, 142], and the number of seats in parliament that a party received after the election [155]. Chapter 3 Entity Retrieval for Online Reputation Monitoring We start by presenting a formal definition of E-R queries and how can we model the E-R retrieval problem from a probabilistic perspective. We assume that a E-R query can be formulated as a sequence of individual sub-queries each targeting a specific entity or relationship. If we create specific representations for entities (e.g. context terms) as well as for pairs of entities, i.e. relationships then we can create a graph of probabilistic dependencies between sub-queries and entity/relationship representations. We show that these dependencies can be depicted in a probabilistic graphical model, i.e. a Bayesian network. Therefore, answering an E-R query can be reduced to a computation of factorized conditional probabilities over a graph of sub-queries and entity/relationship documents. However, it is not possible to compute these conditional probabilities directly from raw documents in a collection. Such as with traditional entity retrieval, documents serve as proxies to entities (and relationships) representations. It is necessary to fuse information spread across multiple documents. We propose two design patterns inspired from Model 1 and Model 2 of Balog et al. [46] to create entity/relationship centric and document centric representations. The first design pattern - Early Fusion - consists in aggregating context terms of entity and relationship occurrences to create two dedicated indexes, the entity index and the relationship index. Then it is possible to use any retrieval method to compute the relevance score of entity and relationship documents given the E-R sub-queries. The second design pattern - Late Fusion - can be applied on top of a standard document index alongside a set of entity occurrences in each document. First we compute the relevance score of documents given a E-R sub-query, then based on Entity Retrieval for Online Reputation Monitoring 34 the entity occurrences of the top k results we compute individual entity or relationship scores. Once again any retrieval method can be used to score documents. When combined with traditional retrieval methods (e.g. Language Models or BM25) these design patterns can be used to create unsupervised baselines for E-R retrieval. Finally, we follow a recent research line in entity retrieval [53, 69, 67] which exploits term dependencies using the Markov Random Field (MRF) framework for retrieval[63]. We introduce the Entity-Relationship Dependence Model (ERDM), a novel supervised Early Fusion-based model for E-R retrieval that creates a MRF to compute term dependencies of E-R queries and entity/relationship documents. 3.1 Entity-Relationship Retrieval E-R retrieval is a complex case of entity retrieval. E-R queries expect tuples of related entities as results instead of a single ranked list of entities as it happens with general entity queries. For instance, the E-R query “Ethnic groups by country" is expecting a ranked list of tuples <ethnic group, country> as results. The goal is to search for multiple unknown entities and relationships connecting them. Table 3.1 E-R retrieval definitions. Q QEi QRi−1,i DEi DRi−1,i QE QR DE DR \|Q\| TE E-R query (e.g. “congresswoman hits back at US president”). Entity sub-query in Q (e.g. “congresswoman”). Relationship sub-query in Q (e.g. “hits back at”). Term-based representation of an entity (e.g. <Frederica Wilson> = {representative, congresswoman}). We use the terminology representation and document interchangeably. Term-based representation of a relationship (e.g. <Frederica Wilson, Donald Trump> = {hits,back}). We use the terminology representation and document interchangeably. The set of entity sub-queries in a E-R query (e.g. {“congresswoman”,“US president” }). The set of relationship sub-queries in a E-R query. The set of entity documents to be retrieved by a E-R query. The set of relationship documents to be retrieved by a E-R query. E-R query length corresponding to the number of entity and relationship sub-queries. The entity tuple to be retrieved (e.g. <Frederica Wilson, Donald Trump>). 3.1 Entity-Relationship Retrieval 35 In this section, we present a definition of E-R queries and a probabilistic formulation of the E-R retrieval problem from an Information Retrieval perspective. Table 3.1 presents several definitions that will be used throughout this chapter. 3.1.1 E-R Queries E-R queries aim to obtain a ordered list of entity tuples TE = <E1 , E2 , ..., En > as a result. Contrary to entity search queries where the expected result is a ranked list of single entities, results of E-R queries should contain two or more entities. For instance, the complex information need “Silicon Valley companies founded by Harvard graduates” expects entity-pairs (2-tuples) <company, founder> as results. In turn, “European football clubs in which a Brazilian player won a trophy" expects triples (3-tuples) <club, player, trophy> as results. Each pair of entities Ei−1 , Ei in an entity tuple is connected with a relationship R(Ei−1 , Ei ). A complex information need can be expressed in a relational format, which is decomposed into a set of sub-queries that specify types of entities E and types of relationships R(Ei−1 , Ei ) between entities. For each relationship sub-query there must be two sub-queries, one for each of the entities involved in the relationship. Thus a E-R query Q that expects 2-tuples, is mapped into a triple of sub-queries Q = {QE1 , QR1,2 , QE2 }, where QE1 and QE2 are the entity attributes queried for E1 and E2 respectively, and QR1,2 is a relationship attribute describing R(Ei , Ei+1 ). If we consider a E-R query as a chain of entity and relationship sub-queries Q = {QE1 , QR1,2 , QE2 , ..., QEn−1 ,QRn−1,n , QEn } and we define the length of a E-R query \|Q\| as the number of sub-queries, then the number of entity sub-queries must be \|Q\|+1 and the number of relationship sub-queries equal to \|Q\|−1 . Consequently, the 2 2 size of each entity tuple TE to be retrieved must be equal to the number of entity sub-queries. For instance, the E-R query “soccer players who dated a top model” with answers such as <Cristiano Ronaldo, Irina Shayk>) is represented as three sub-queries QE1 = {soccer players}, QR1,2 = {dated}, QE2 = {top model}. Automatic mapping of terms from a E-R query Q to sub-queries QEi or QRi−1,i is out of the scope of this work and can be seen as a problem of query understanding [168, 54, 169]. We assume that the information needs are decomposed into constituent entity and relationship sub-queries using Natural Language Processing techniques or by user input through an interface that enforces the structure Q = {QE1 , QR1,2 , QE2 , ..., QEn−1 ,QRn−1,n , QEn }. 36 3.1.2 Entity Retrieval for Online Reputation Monitoring Modeling E-R Retrieval Our approach to E-R retrieval assumes that we have a raw document collection (e.g. news articles) and each document Dj is associated with one or more entities Ei . In other words, documents contain mentions to one or more entities that can be related between them. Since our goal is to retrieve tuples of related entities given a E-R query that expresses entity attributes and relationship attributes, we need to create term-based representations for both entities and relationships. We denote a representation of an entity Ei as DEi . In E-R retrieval we are interested in retrieving tuples of entities TE = <E1 , E2 , ..., En > as a result. The number of entities in each tuple can be two, three or more depending on the structure of the particular E-R query. When a E-R query aims to get tuples of more than two entities, we assume it is possible to combine tuples of length two. For instance, we can associate two tuples of length two that share the same entity to retrieve a tuple of length three. Therefore we create representations of relationships as pairs of entities. We denote a representation of a relationship R(Ei−1 , Ei ) as DRi−1,i . Considering the example query “Which spiritual leader won the same award as a US vice president?” it can be formulated in the relational format as QE1 = {spiritual leader}, QR1,2 = {won}, QE2 = {award}, QR2,3 = {won}, QE3 = {US vice president}. Associating the tuples of length two <Dalai Lama, Nobel Peace Prize> and <Nobel Peace Prize, Al Gore> would result in the expected 3-tuple <Dalai Lama, Nobel Peace Prize, Al Gore>. For the sake of clarity we now consider an example E-R query with three sub-queries (\|Q\| = 3). This query aims to retrieve a tuple of length two, i.e. a pair of entities connected by a relationship. Based on the definition of a E-R query, each entity in the resulting tuple must be relevant to the corresponding entity sub-queries QE . Moreover, the relationship between the two entities must also be relevant to the relationship sub-queries QR . Instead of calculating a simple posterior P (D\|Q) as with traditional information retrieval, in E-R retrieval the objective is to rank tuples based on a joint posterior of multiple entity and relationship representations given a E-R query, such as P (DE2 , DE1 , DR1,2 \|Q) when \|Q\| = 3. E-R queries can be seen as chains of interleaved entity and relationship subqueries. We take advantage of the chain rule to formulate the joint probability P (DE2 , DE1 , DR1,2 , Q) as a product of conditional probabilities. Formally, we want to rank entity and relationship candidates in descending order of the joint posterior P (DE2 , DE1 , DR1,2 \|Q) as: 3.1 Entity-Relationship Retrieval 37 P (DE2 , DE1 , DR1,2 , Q) P (Q) E2 E1 R1,2 , Q).P (DE1 \|DR1,2 , Q).P (DR1,2 \|Q).P (Q) rank P (D \|D , D = P (Q) rank P (DE2 , DE1 , DR1,2 \|Q) = rank = P (DE2 \|DR1,2 , Q).P (DE1 \|DR1,2 , Q).P (DR1,2 \|Q) rank ∝ P (DE2 \|DR1,2 , QE2 ).P (DE1 \|DR1,2 , QE1 ).P (DR1,2 \|QR1,2 ) (3.1) (3.2) We consider conditional independence between entity representations within the joint posterior, i.e., the probability of a given entity representation DEi being relevant given a E-R query is independent of knowing that entity DEi+1 is relevant as well. As an example, consider the query “action movies starring a British actor”. Retrieving entity representations for “action movies” is independent of knowing that <Tom Hardy> is relevant to the sub-query “British actor”. However, it is not independent of knowing the set of relevant relationships for sub-query “starring”. If a given action movie is not in the set of relevant entity-pairs for “starring” it does not make sense to consider it as relevant. Consequently, P (DE2 \|DE1 , DR1,2 , Q) = P (DE2 \|DR1,2 , Q). Since E-R queries can be decomposed in constituent entity and relationship subqueries, ranking candidate tuples using the joint posterior P (DE2 , DE1 , DR1,2 \|Q) is rank proportional to the product of conditional probabilities on the corresponding entity and relationship sub-queries QE2 , QE1 and QR1,2 . We now consider a longer E-R query aiming to retrieve a triple of connected entities. This query has three entity sub-queries and two relationship sub-queries, thus \|Q\| = 5. As we previously explained, when there are more than one relationship sub-queries we need to join entity-pairs relevant to each relationship sub-query that have one entity in common. From a probabilistic point of view this can be seen as conditional dependence from the entity-pairs retrieved from the previous relationship sub-query, i.e. P (DR2,3 \|DR1,2 , Q) ̸= P (DR2,3 \|Q). To rank entity and relationship candidates we need to calculate the following joint posterior: 38 Entity Retrieval for Online Reputation Monitoring rank P (DE3 , DE2 ,DE1 , DR2,3 , DR1,2 \|Q) = P (DE3 \|DE2 , DE1 , DR2,3 , DR1,2 , Q). P (DE3 \|DE2 , DR2,3 , DR1,2 , Q).P (DE1 \|DR2,3 , DR1,2 , Q). P (DR2,3 \|DR1,2 , Q).P (DR1,2 \|Q) rank = P (DE3 \|DR2,3 , Q).P (DE2 \|DR2,3 , DR1,2 , Q). P (DE1 \|DR1,2 , Q).P (DR2,3 \|DR1,2 , Q).P (DR1,2 \|Q) rank ∝ P (DE3 \|DR2,3 , QE3 ).P (DE2 \|DR2,3 , DR1,2 , QE2 ). P (DE1 \|DR1,2 , QE1 ).P (DR2,3 \|DR1,2 , QR2,3 ).P (DR1,2 \|QR1,2 ) (3.3) (3.4) When compared to the previous example, the joint posterior for \|Q\| = 5 shows that entity candidates for DE2 are conditional dependent of both DR2,3 and DR1,2 . In other words, entity candidates for DE2 must belong to entity-pairs candidates for both relationships representations that are connected with E2 , i.e. DR2,3 and DR1,2 . We are now able to make a generalization of E-R retrieval as a factorization of conditional probabilities of a joint probability of entity representations DEi , relationship representations DRi−1,i , entity sub-queries QEi and relationship sub-queries QRi−1,i . These set of random variables and their conditional dependencies can be easily represented in a probabilistic directed acyclic graph,i.e. a Bayesian network [170]. In Bayesian networks, nodes represent random variables while edges represent conditional dependencies. Every other nodes that point to a given node are considered parents. Bayesian networks define the joint probability of a set of random variables as a factorization of the conditional probability of each random variable conditioned Q on its parents. Formally, P (X1 , . . . , Xn ) = ni=1 P (Xi \|pai ), where pai represents all parent nodes of Xi . Figure 3.1 depicts the representation of E-R retrieval for different query lengths \|Q\| using Bayesian networks. We easily conclude that graphical representation contributes to establish a few guidelines for modeling E-R retrieval. First, each sub-query points to the respective document node. Second, relationship document nodes always point to the contiguous entity representations. Last, when there are more than one relationship sub-query, relationship documents also point to the subsequent relationship document. Once we draw the graph structure for the number of sub-queries in Q we are able to compute a product of conditional probabilities of each node given its parents. Adapting 3.1 Entity-Relationship Retrieval 39 (a) \|Q\| = 3 (b) \|Q\| = 5 (c) \|Q\| = 7 Fig. 3.1 Bayesian networks for E-R Retrieval with queries of different lengths. the general joint probability formulation of Bayesian networks to E-R retrieval we come up with the following generalization: \|Q\|−1 2 \|Q\|+1 2 rank Y P (DE , DR \|Q) = P (DEi \|DRi−1,i , DRi,i+1 , QEi ) i=1 P (DRi,i+1 \|DRi−1,i , QRi,i+1 ) Y i=1 (3.5) We denote D as the set of all candidate relationship documents in the graph and DE the set of all candidate entity documents in the graph. In Information Retrieval is often convenient to work in the log-space as it does not affect ranking and transforms the product of conditional probabilities in a summation, as follows: R rank P (DE , DR \|Q) = log P (DE , DR \|Q) \|Q\|+1 2 rank X = i=1 \|Q\|−1 2 logP (DEi \|DRi−1,i , DRi,i+1 , QEi ) + X logP (DRi,i+1 \|DRi−1,i , QRi,i+1 ) i=1 (3.6) (3.7) 40 Entity Retrieval for Online Reputation Monitoring We now present two design patterns to compute each conditional probability for every entity and relationship candidate documents. 3.2 Design Patterns for Entity-Relationship Retrieval Traditional ad-hoc document retrieval approaches create direct term-based representations of raw documents. A retrieval model (e.g. Language Models) is then used to match the information need, expressed as a keyword query, against those representations. However, E-R retrieval requires collecting evidence for both entities and relationships that can be spread across multiple documents. It is not possible to create direct term-based representations. Raw documents serve as proxy to connect queries with entities and relationships. Abstractly speaking, entity retrieval can be seen as a problem of object retrieval in which the search process is about fusing information about a given object, such as in the case of verticals (e.g. Google Finance). Recently, Zhang and Balog [171] presented two design patterns for fusion-based object retrieval. The first design pattern – Early Fusion – is an object-centric approach where a termbased representation of objects is created earlier in the retrieval process. First, it creates meta-documents by aggregating term counts across the documents associated with the objects. Later, it matches queries against these meta-documents using standard retrieval methods. The second design pattern - Late Fusion - is a document-centric approach where relevant documents to the query are retrieved first and then later in the retrieval process, it ranks objects associated with top documents. These design patterns represent a generalization of Balog’s Model 1 and Model 2 for expertise retrieval [46]. In essence, E-R retrieval is an extension, or a more complex case, of object-retrieval where besides ranking objects we need to rank tuples of objects that satisfy the relationship expressed in the E-R query. This requires creating representations of both entities and relationships by fusing information spread across multiple raw documents. We propose novel fusion-based design patterns for E-R retrieval that are inspired from the design patterns presented by Zhang and Balog [171] for single object-retrieval. We extend those design patterns to accommodate the specificities of E-R retrieval. We hypothesize that it should be possible to generalize the term dependence models to represent entity-relationships and achieve effective E-R retrieval without entity or relationship type restrictions (e.g. categories) as it happens with the Semantic Web based approaches. 3.2 Design Patterns for Entity-Relationship Retrieval 3.2.1 41 Early Fusion The Early Fusion strategy presented by Zhang and Balog [171] consists in creating a term-based representation for each object under retrieval, i.e., a meta-document containing all terms in the proximity of every object mention across a document collection. As described in previous section, E-R queries can be formulated as a sequence of multiple entity queries QE and relationship queries QR . In a Early Fusion approach, each of these queries should match against a previously created term-based representation. Since there are two types of queries, we propose to create two types of term-based representations, one for entities and other for relationships. Our Early Fusion design pattern is similar to Model 1 of Balog et al. [46]. It can be thought as creating two types of meta-documents DE and DR . A meta-document DEi is created by aggregating the context terms of the occurrences of Ei across the raw document collection. On the other hand, for each each pair of entities Ei−1 and Ei that co-occur close together across the raw document collection we aggregate context terms that describe the relationship to create a meta-document DRi−1,i In our approach we focus on sentence level information about entities and relationships although the design pattern can be applied to more complex segmentations of text (e.g. dependency parsing). We rely on Entity Linking methods for disambiguating and assigning unique identifiers to entity mentions on raw documents D. We collect entity contexts across the raw document collection and index them in the entity index. The same is done by collecting and indexing entity pair contexts in the relationship index. We define the (pseudo) frequency of a term t for an entity meta-document DEi as follows: f (t, DEi ) = n X f (t, Ei , Dj )w(Ei , Dj ) (3.8) j=1 where n is the total number of raw documents in the collection, f (t, Ei , Dj ) is the term frequency in the context of the entity Ei in a raw document Dj . w(Ei , Dj ) is the entity-document association weight that corresponds to the weight of the document Dj in the mentions of the entity Ei across the raw document collection. Similarly, the term (pseudo) frequency of a term t for a relationship meta-document DRi−1,i is defined as follows: f (t, D Ri−1,i )= n X j=1 f (t, Ri−1,i , Dj )w(Ri−1,i , Dj ) (3.9) 42 Entity Retrieval for Online Reputation Monitoring where f (t, Ri−1,i , Dj is the term frequency in the context of the pair of entity mentions corresponding to the relationship Ri−1,i in a raw document Dj and w(Ri−1,i , Dj ) is the relationship-document association weight. In this work we use binary associations weights indicating the presence/absence of an entity mention in a raw document, as well as for a relationship. However, other weight methods can be used. The relevance score for an entity tuple TE can then be calculated using the posterior P (DE , DR \|Q) defined in previous section (equation 3.6). We calculate the individual conditional probabilities as a product of a retrieval score with an association weight. Formally we consider: logP (DEi \|DRi−1,i , DRi,i+1 , QEi ) = score(DEi , QEi )w(Ei , Ri−1,i , Ri,i+1 ) logP (DRi,i+1 \|DRi−1,i , QRi,i+1 ) = score(DRi,i+1 , QRi,i+1 )w(Ri,i+1 , Ri−1,i ) (3.10) (3.11) where score(DRi,i+1 , QRi,i+1 ) represents the retrieval score resulting of the match of the query terms of a relationship sub-query QRi,i+1 and a relationship meta-document DRi,i+1 . The same applies to the retrieval score score(DEi , QEi ) which corresponds to the result of the match of an entity sub-query QEi with a entity meta-document DEi . For computing both score(DRi,i+1 , QRi,i+1 ) and score(DEi , QEi ) any retrieval model can be used. Different scoring functions will be introduced below. We use a binary association weight for w(Ei , Ri−1,i , Ri,i+1 ) which represents the presence of a relevant entity Ei to a sub-query QEi in its contiguous relationships in the Bayesian network, i.e. Ri−1,i and Ri,i+1 which must be relevant to the sub-queries QRi−1,i and QRi,i+1 . This entity-relationship association weight is the building block that guarantees that two entities relevant to sub-queries QE that are also part of a relationship relevant to a sub-query QR will be ranked higher than tuples where just one or none of the entities are relevant to the entity sub-queries QE . On the other hand, the entity-relationship association weight w(Ri,i+1 , Ri−1,i ) guarantees that consecutive relationships share one entity between them in order to create triples or 4-tuples of entities for longer E-R queries (\|Q\| > 3). The relevance score of an entity tuple TE given a query Q is calculated by summing individual relationship and entity relevance scores for each QRi−1,i and QEi in Q. We define the score for a tuple TE given a query Q as follows: 3.2 Design Patterns for Entity-Relationship Retrieval E R rank P (D , D \|Q) = 43 \|Q\|+1 2 score(DEi , QEi )w(Ei , Ri−1,i , Ri,i+1 )+ X i=1 (3.12) \|Q\|−1 2 X score(DRi,i+1 , QRi,i+1 )w(Ri,i+1 , Ri−1,i ) i=1 (3.13) Considering Dirichlet smoothing unigram Language Models (LM) the constituent retrieval scores can be computed as follows:  scoreLM (DRi,i+1 , QRi,i+1 ) = log  t∈DRi,i+1 ∩QRi,i+1  scoreLM (DEi , QEi ) = X t∈DEi ∩QEi  f (t,C R ) R µ  \|C R \|  + µR Ri,i+1 )+  f (t, D X \|DRi,i+1 \|  f (t,C E ) E  f (t, D ) + \|C E \| µ  log   \|DEi \| + µE Ei (3.14) (3.15) where t is a term of a sub-query QEi or QRi,i+1 , f (t, DEi ) and f (t, DRi,i+1 ) are the (pseudo) frequencies defined in equations 3.8 and 3.9. The collection frequencies f (t, C E ), f (t, C R ) represent the frequency of the term t in either the entity index C E or in the relationship index C R . \|DEi \| and\|DRi,i+1 \| represent the total number of terms in a meta-document while \|C R \| and \|C E \| represent the total number of terms in a collection of meta-documents. Finally, µE and µR are the Dirichlet prior for smoothing which generally corresponds to the average document length in a collection. 3.2.2 Association Weights Both Early Fusion and Late Fusion share three components: w(Ri,i+1 , Dj ), w(Ei , Dj ) and w(Ei , Ri,i+1 ). The first two represent document associations which determine the weight a given raw document contributes to the relevance score of a particular entity tuple TE . The last one is the entity-relationship association which indicates the strength of the connection of a given entity Ei within a relationship Ri,i+1 . In our work we only consider binary association weights but other methods could be used. According to the binary method we define the weights as follows: w(Ri,i+1 , Dj ) = 1 if R(Ei , Ei+1 ) ∈ Dj , 0 otherwise (3.16) Entity Retrieval for Online Reputation Monitoring 44 w(Ei , Dj ) = 1 if Ei ∈ Dj , 0 otherwise (3.17) w(Ei , Ri−1,i , Ri,i+1 ) = 1 if Ei ∈ DRi−1,i and Ei ∈ DRi,i+1 , 0 otherwise (3.18) w(Ri,i+1 , Ri−1,i ) = 1 if Ei ∈ DRi−1,i and Ei ∈ DRi,i+1 , 0 otherwise (3.19) Under this approach the weight of a given association is independent of the number of times an entity or a relationship occurs in a document. A more general approach would be to assign real numbers to the association weights depending on the strength of the association [8]. For instance, uniform weighting would be proportional to the inverse of the number of documents where a given entity or relationship occurs. Other option could be a TF-IDF approach. 3.2.3 Early Fusion Example Let us consider an illustrative example of the Early Fusion design pattern for E-R retrieval using unigram Language Models and the E-R query Q = {soccer players who dated a top model}. This query can be decomposed in three sub-queries, QEi = { soccer players}, QEi+1 = { top model} and QRi,i+1 = { dated }. The first two sub-queries target the entity index and the last targets the relationship index. Table 3.2 presents a toy entity index with 3 entities as example for each of the two entity sub-queries, including the term frequency f (t, DEi ) for each sub-query term. Table 3.2 Illustrative example of the entity index in Early Fusion. Ei <Tom Brady> <Cristiano Ronaldo> <Lionel Messi> <Luís Figo> <Gisele Bundchen> <Irina Shayik> <Helen Svedin> ... f (t, DEi ) soccer:0 player:600 soccer:800 player:800 soccer:700 player:700 soccer:200 player:200 top:400 model:400 top:300 model:300 top:150 model:150 ... \|DEi \| 3000 5000 4000 800 3000 2000 600 ... 3.2 Design Patterns for Entity-Relationship Retrieval 45 Considering the remaining variables required to calculate the scoreLM (DEi , QEi ): \|C E \| = 100000 \|µE \| = 1500 f (soccer, C E ) = 3000 f (player, C E ) = 8000 f (top, C E ) = 8000 f (model, C E ) = 4000 We calculate the scoreLM (DEi , QEi ) for the respective entities and sub-queries. For the first entity query – “soccer players” – the ranked list of relevant entities and the respective LM score would be the following: 1. <Lionel Messi>: -1.6947 2. <Cristiano Ronaldo>: -1.7351 3. <Luís Figo>: -1.8291 4. <Tom Brady>: -2.7958 For the second entity query – “top models”: 1. <Gisele Bundchen>: -1.6295 2. <Irina Shayik>: -1.7093 3. <Helen Svedin>: -1.9698 Table 3.3 shows 3 relationships, i.e. entity pairs, relevant to the sub-query “dated” and the respective term frequency f (t, DRi,i+1 ). Considering the remaining variables required to calculate the scoreLM (DRi,i+1 , QRi,i+1 ): \|C R \| = 20000 \|µR \| = 500 f (dated, C R ) = 5000 Entity Retrieval for Online Reputation Monitoring 46 Table 3.3 Illustrative example of the relationship index in Early Fusion. Ri,i+1 <Gisele Bundchen, Tom Brady> <Irina Shayik, Cristiano Ronaldo> <Helen Svedin, Luís Figo> ... f (t, DRi,i+1 ) dated:500 dated:300 dated:100 ... \|DRi,i+1 \| 800 600 200 ... We calculate the scoreLM (DRi,i+1 , QRi,i+1 )for the respective relationship and the sub-query and we obtain the following ranked list: 1. <Gisele Bundchen, Tom Brady>: -0.3180 2. <Irina Shayik, Cristiano Ronaldo>: -0.4130 3. <Helen Svedin, Luís Figo>: -0.4929 We can now sum up individual scores for each sub-query and calculate the final score for the early fusion design pattern score(TE , Q) using the equation 3.12. The final ranked list of tuples is the following: 1. <Irina Shayik, Cristiano Ronaldo>: -3.8575 2. <Helen Svedin, Luís Figo>: -4.2919 3. <Gisele Bundchen, Tom Brady>: -4.6977 The entity tuple <Irina Shayik, Cristiano Ronaldo> is the most relevant to the query “soccer players who dated a top model”. Although <Gisele Bundchen, Tom Brady> has higher individual scores in two sub-queries (“top model” and “dated”) it ranks last due to the poor relevance of Tom Brady to the sub-query “soccer player”. The entity <Lionel Messi> is the most relevant entity to the sub-query “soccer player” but it is not relevant to the relationship sub-query, therefore it is excluded from the final ranked list of entity tuples. 3.2.4 Late Fusion The Late Fusion design pattern presented by Zhang and Balog [171] is a documentcentric strategy, i.e. first we query raw individual documents then we aggregate the associated objects with the relevant documents. Instead of creating term-based representations of entities and relationships (pairs of entities), in late fusion we use the 3.2 Design Patterns for Entity-Relationship Retrieval 47 raw documents as hidden variables, separating the E-R query from the relevant entity tuples to be retrieved. Our vision of ORM implies processing raw documents to detect entities occurrences and extract sentence level information that will be used in downstream Entity Retrieval and Text Mining tasks. Therefore, we are not interested in applying a Late Fusion strategy in this work. However, we believe it makes sense to present a theoretical formulation of a Late Fusion design pattern for E-R retrieval. We leave the practical experiments with Late Fusion for future work in the context of generic E-R retrieval. The process of retrieving entity tuples using our late fusion strategy consists in processing each sub-query independently, as in the early fusion strategy, but in this case, we use a single index comprising a term based representation of the collection of raw documents. A retrieval model is used to calculate a relevance score of each individual raw document and a given sub-query. Once we have the relevant documents we use entity linking to extract the entities that are mentioned in each relevant raw document. Following this strategy we calculate aggregated counts of entity occurrences weighted by the individual relevance score of the individual raw documents. At the end, we join the results of each sub-query and calculate the overall relevance score of the entity tuples. Formally, we define the relevance score of an entity tuple TE given a query Q as follows: rank P (DE , DR \|Q) = \|Q\|+1 2 n X X score(Dj , QEi )w(Ei , Dj )w(Ei , Ri−1,i , Ri,i+1 )+ i=1 j=1 (3.20) \|Q\|−1 2 n X X score(Dj , QRi,i+1 )w(Ri,i+1 , Dj )w(Ri,i+1 , Ri−1,i ) i=1 j=1 (3.21) where score(Dj , QRi,i+1 ) represents the retrieval score resulting of the match of the query terms of a relationship sub-query QRi,i+1 and a raw document Dj . The same applies to the retrieval score score(Dj , QEi ) which corresponds to the result of the match of an entity sub-query QEi with a raw document Dj . The weights w(Ri,i+1 , Dj ) and w(Ei , Dj ) represent association weights between relationships and raw documents, and entities and raw documents, respectively. We use binary association weights in this work but other weights can be used. We also use a binary association weight Entity Retrieval for Online Reputation Monitoring 48 for w(Ei , Ri−1,i , Ri,i+1 ) and w(Ri,i+1 , Ri−1,i ) which represent the entity-relationship association weights, similarly to what happens with the case of Early Fusion. For computing both score(Dj , QRi,i+1 ) and score(Dj , QEi ) any retrieval model can be used. Considering BM25 the scores can be computed as follows: scoreBM 25 (Dj , QRi,i+1 ) = X log t∈Dj ∩QRi,i+1 N − n(t) + 0.5 f (t, Dj )(K1 + 1) . \|Dj \| n(t) + 0.5 f (t, Dj ) + K1 (1 − b + b avg(\|D\|) ) (3.22) f (t, Dj )(K1 + 1) N − n(t) + 0.5 . \|Dj \| n(t) + 0.5 ) f (t, Dj ) + K1 (1 − b + b avg(\|D\|) t∈Dj ∩QEi (3.23) Ei Ri,i+1 where t is a term of a sub-query Q or Q and f (t, Dj ) is the query term frequency in a raw document Dj . The inverse document frequency, IDF (t), is computed −n(t)+0.5 as log Nn(t)+0.5 with N as the number of documents on the collection and n(t) the number of documents where the term occurs.\|Dj \| is the total number of terms in a raw document Dj and avg(\|D\|) is the average document length. K1 and b are free parameters usually chosen as 1.2 and 0.75, in the absence of specific optimization. scoreBM 25 (Dj , QEi ) = 3.2.5 X log Late Fusion Example Considering the same toy example query introduced in the previous sub-section, we now have a single index, the document index, as illustrated in Table 3.4. The remaining parameters required for calculating the scoreBM 25 (Dj , QEi ) and scoreBM 25 (Dj , QRi,i+1 ) are the following: • N=2000 • n(soccer)=100 • n(player)=130 • n(dated)=60 • n(top)=250 • n(model)=80 • avg(\|D\|)=120 3.2 Design Patterns for Entity-Relationship Retrieval 49 Table 3.4 Illustrative example of the document index in Late Fusion. Dj docid-1 docid-2 docid-3 docid-4 docid-5 docid-6 docid-7 docid-8 docid-9 ... f (t, Dj ) soccer:10 player:10 soccer:5 player:5 soccer:5 player:5 top:4 model:4 \|Dj \| dated:5 top:6 model:6 model:4 dated:2 player:2 dated:3 top:2 model:2 dated:2 soccer:2 player:2 ... 200 Ei <Cristiano Ronaldo> <Lionel Messi> 150 <Cristiano Ronaldo> 100 <Luís Figo> 150 <Gisele Bundchen> 80 <Gisele Bundchen> <Tom Brady> 100 <Irina Shayik> 100 120 <Gisele Bundchen> <Adriana Lima> <Tom Brady> <Irina Shayik> <Cristiano Ronaldo> 150 <Luís Figo> <Helen Svedin> ... ... For the first entity sub-query, “soccer players”, the relevant documents ranked by the scoreBM 25 (Dj , QEi ) are the following: 1. docid-1 (<Cristiano Ronaldo>, <Lionel Messi>): 4.7606 2. docid-3 (<Luís Figo>): 4.6426 3. docid-2 (<Cristiano Ronaldo>): 4.3716 4. docid-9 (<Luís Figo>, <Helen Svedin>): 3.2803 5. docid-7 (<Gisele Bundchen>, <Adriana Lima>, <Tom Brady>): 1.8418 For the second entity sub-query, “top model”: 1. docid-6 (<Irina Shayik>): 3.1618 50 Entity Retrieval for Online Reputation Monitoring 2. docid-4 (<Gisele Bundchen>): 2.7393 3. docid-9 (<Luís Figo>, <Helen Svedin>): 2.1694 4. docid-7 (<Gisele Bundchen>, <Adriana Lima>, <Tom Brady>): 1.4714 For the relationship sub-query, “dated”: 1. docid-5 (<Gisele Bundchen>, <Tom Brady>): 2.8081 2. docid-8 (<Irina Shayik>, <Cristiano Ronaldo>): 2.3668 3. docid-7 (<Gisele Bundchen>, <Adriana Lima>, <Tom Brady>): 2.1728 4. docid-9 (<Luís Figo>, <Helen Svedin>): 1.9349 Since in Late Fusion there is no relationship meta-documents that could be used directly as entity tuples, we need to extract the candidate tuples from the raw documents retrieved using the relationship sub-query. When there are more than two entity associations in a relevant document we combine entities to create tuples. For instance, docid-7 has three entity associations therefore we extract three candidate tuples: <Gisele Bundchen, Tom Brady>, <Gisele Bundchen, Adriana Lima> and <Adriana Lima, Tom Brady>. For each candidate tuple we sum up scoreBM 25(Dj , QRi,i+1 )w(Ri,i+1 , Dj ) over every relevant document Dj for the relationship sub-query that is associated with each entity tuple. The same applies to individual entities from the candidate tuples that are associated with relevant documents for each entity sub-query. For instance, for the entity sub-query “soccer players” we sum score(Dj , QEi )w(Ei , Dj )w(Ei , Ri,i+1 ) over the relevant documents that mentioned an entity that belongs to a candidate tuple. When both entities of the candidate tuple are mentioned in relevant documents for both entity sub-queries, e.g. <Helen Svedin, Luís Figo>, we assign each entity to the sub-query that maximizes the final score score(TE , Q), i.e., we use the scores of the entity sub-query “soccer player” for <Luís Figo> and the entity sub-query “top model” for <Helen Svedin>. The final ranked list of entity tuples is the following: 1. <Irina Shayik, Cristiano Ronaldo>: 14.0443 2. <Helen Svedin, Luís Figo>: 12.5970 3. <Gisele Bundchen, Tom Brady>: 10.9784 3.2 Design Patterns for Entity-Relationship Retrieval 51 4. <Gisele Bundchen, Adriana Lima>: 9.3459 5. <Adriana Lima, Tom Brady>: 6.9245 Once again <Lionel Messi> is excluded from the final ranked list of entity tuples because he is not associated with any document relevant to the relationship sub-query “dated”. On the other hand, <Adriana Lima> is included in the final ranking although it is not true that she has dated either <Tom Brady> or <Gisele Bundchen>. In this example, the top three entity tuples are ranked in the same order as in the Early Fusion strategy example. 3.2.6 Implementation In this section we proposed two design patterns for E-R retrieval: Early Fusion (EF) and Late Fusion (LF). Both can be seen as a flexible framework for ranking tuples of entities given a E-R query expressed as a sequence of entity and relationship sub-queries. This framework is flexible enough to allow using any retrieval method to compute individual retrieval scores between document and query nodes in a E-R graph structure. When using Language Models (LM) or BM25 as scoring functions, these design patterns can be used to create unsupervised baseline methods for E-R retrieval (e.g. EF-LM, EF-BM25, LF-LM, LF-BM25, etc.). In the case of Early Fusion there is some overhead over traditional document search, since we need to create two E-R dedicated indexes that will store entity and relationship meta-documents. The entity index is created by harvesting the context terms in the proximity of every occurrence of a given entity across the raw document collection. This process must be carried for every entity in the raw document collection. A similar process is applied to create the relationship index. For every two entities occurring close together in a raw document we extract the text between both occurrences as a term-based representation of the relationship between the two. Once again, this process must be carried for every pair of co-occurring entities in sentences across the raw document collection. Late Fusion requires less overhead and can be implemented on top of a web search engine with reduced effort. We only need to have a list of entity occurrences alongside each document. Therefore there is no need to create a separate index(es). On the other hand, it requires more processing on query time since we need to first rank raw documents for each sub-query and then aggregate entity occurrences at the top k documents retrieved. Moreover, it does not contain any proximity-based information 52 Entity Retrieval for Online Reputation Monitoring on the entity occurrences, so two entities occurring very far in the text might be considered as relationship candidates. It might be prone to a higher false positive rate. One advantage of Early Fusing lies in its flexibility as we need to create two separate indexes for E-R retrieval it is possible to combine data from multiple sources in seamless way. For instance, one could use a well established knowledge base (e.g. DBpedia) as entity index and use a specific collection, such as a news collection or a social media stream, for harvesting relationships having a more transient nature. Common to both design patterns is a challenge inherent to the problem of E-R retrieval: the size of the search space. Although the E-R problem is formulated as a sequence of independent sub-queries, the results of those sub-queries must be joined together. Consequently, we have a multi-dimensional search space in which we need to join results based on shared entities. This problem becomes particularly hard when sub-queries are short and contain very popular terms. Let us consider “actor” as QEi , there will be many results to this sub-query, probably thousands. There is a high probability that will need to process thousands of sub-results before finding one entity that is also relevant to the relationship sub-query QRi−1,i . If at the same time we have computational power constraints, we will probably apply a strategy of just considering top k results for each sub-query which can lead to reduced recall in the case of short sub-queries with popular terms. 3.3 Entity-Relationship Dependence Model In this section we present the Entity-Relationship Dependence Model (ERDM), a novel supervised Early Fusion-based model for E-R retrieval. Recent approaches to entity retrieval [53, 67, 69] have demonstrated that using models based on Markov Random Field (MRF) framework for retrieval [63] to incorporate term dependencies can improve entity search performance. This suggests that MRF could be used to model E-R query term dependencies among entities and relationships documents. One of the advantages of the MRF framework for retrieval is its flexibility, as we only need to construct a graph G representing dependencies to model, define a set of non-negative potential functions ψ over the cliques of G and to learn the parameter vector Λ to score each document D by its unique and unnormalized joint probability with Q under the MRF [63]. The non-negative potential functions are defined using an exponential form ψ(c; Λ) = exp[λc f (c)], where λc is a feature weight, which is a free parameter in the model, 3.3 Entity-Relationship Dependence Model 53 associated with feature function f (c). Learning to rank is then used to learn the feature weights that minimize the loss function. The model allows parameter and feature functions sharing across cliques of the same configuration, i.e. same size and type of nodes (e.g. 2-cliques of one query term node and one document node). 3.3.1 Graph Structures The Entity-Relationship Dependence Model (ERDM) creates a MRF for modeling implicit dependencies between sub-query terms, entities and relationships. Each entity and each relationship are modeled as document nodes within the graph and edges reflect term dependencies. Contrary to traditional ad-hoc retrieval using MRF (e.g. SDM), where the objective is to compute the posterior of a single document given a query, the ERDM allows the computation of a joint posterior of multiple documents (entities and relationships) given a E-R query which consists also of multiple sub-queries. Fig. 3.2 Markov Random Field dependencies for E-R retrieval, \|Q\| = 3. The graph structures of the ERDM for two E-R queries, one with \|Q\| = 3 and other with \|Q\| = 3 are depicted in Figure 3.2 and Figure 3.3, respectively. Both graph structures contain two different types of query nodes and document nodes: entity query and relationship query nodes, QE and QR , plus entity and relationship document nodes, DE and DR . Within the MRF framework, DE and DR are considered “documents” but they are not actual real documents but rather objects representing an entity or a relationship between two entities. Unlike real documents, these objects do not have direct and explicit term-based representations. Usually, it is necessary to gather evidence across multiple real documents that mention the given object, in order to be able to match them against keyword queries. Therefore, ERDM can be seen as Early Fusion-based retrieval model. The existence of two different types of documents implies two different indexes: the entity index and the relationship index. 54 Entity Retrieval for Online Reputation Monitoring Fig. 3.3 Markov Random Field dependencies for E-R retrieval, \|Q\| = 5. The relationship-specific dependencies of ERDM are found in the 2-cliques formed by one entity document and one relationship document: DEi−1 - DRi−1,i , DEi - DRi−1,i and for \|Q\| = 5, DEi - DRi,i+1 and DRi−1,i - DRi,i+1 . The graph structure does not need to assume any explicit dependence between entity documents given a relationship document. They have an implicit connection through the dependencies with the relationship document. The likelihood of observing an entity document DEi given a relationship document DRi−1,i is not affected by the observation of any other entity document. Explicit dependence between the two entity documents could be used to represent the direction of the relationship between the two entities. To support this dependence, relationship documents would need to account the following constraint: R(Ei−1 , Ei ) ̸= R(Ei , Ei−1 ), ∀ DRi−1,i ∈ C R , with C R representing the relationship index. Then, we would compute an ordered feature function between entities in a relationship, similar to the ordered bigram feature function in SDM. In this work, we do not explicitly model asymmetric relationships. For instance, if a user searches for the relationship entity A “criticized” entity B but was in fact entity B who criticized entity A we assume that the entity tuple <entity A, entity B> is still relevant for the information need expressed in the E-R query. ERDM follows the SDM [63] dependencies between query terms and documents due to its proved effectiveness in multiple contexts. Therefore, ERDM assumes a dependence between neighboring sub-query terms: Ei Ei P (qjEi \|DEi , qj̸E=i l ) = P (qjEi \|DEi , qj−1 , qj+1 ) (3.24) 3.3 Entity-Relationship Dependence Model R R 55 R R R i−1,i i−1,i i−1,i Ei P (qj i−1,i \|DRi−1,i , qj̸=i−1,i \|DRi−1,i , qj−1 , qj+1 ) l , D ) = P (qj (3.25) MRF for retrieval requires the definition of the sets of cliques (maximal or nonmaximal) within the graph that one or more feature functions is to be applied to. The set of cliques in ERDM containing at least one document are the following: • T E - set of 2-cliques containing an entity document node and exactly one term in a entity sub-query. • OE - set of 3-cliques containing an entity document node and two ordered terms in a entity sub-query. • T R - set of 2-cliques containing a relationship document node and exactly one term in a relationship sub-query. • OR - set of 3-cliques containing a relationship document node and two ordered terms in a relationship sub-query. • S ER - set of 2-cliques containing one entity document node and one relationship document node. • S RER - set of 3-cliques containing one entity document node and two consecutive relationship document nodes. The joint probability mass function of the MRF is computed using the set of potential functions over the configurations of the maximal cliques in the graph [63]. Non-negative potential functions are constructed from one or more real valued feature functions associated with the respective feature weights using an exponential form. 3.3.2 Feature Functions ERDM has two types of feature functions: textual and non-textual. Textual feature functions measure the textual similarity between one or more sub-query terms and a document node. Non-textual feature functions measure compatibility between entity and relationship documents, i.e., if they share a given entity. Table 3.5 presents an overview of the feature functions associated with clique sets and the type of input nodes. Although we could define a wide set of different feature functions, we decided to adapt SDM textual feature functions to ERDM clique configurations. Therefore we define unigram based feature functions fTE and fTR to Entity Retrieval for Online Reputation Monitoring 56 Table 3.5 Clique sets and associated feature functions by type and input nodes. Clique Set TE OE TR OR S ER S RER Feature Functions fTE fOE and fUE fTR fOR and fUE fSER fSRER Type Textual Textual Textual Textual Non-textual Non-textual Input Nodes {qjEi , DEi } Ei {qjEi , qj+1 , DEi } Ri−1,i {qj , DRi−1,i } R Ri−1,i {qj i−1,i , qj+1 , DRi−1,i } {DEi , DRi−1,i } {DEi , DRi−1,i , DRi,i+1 } 2-cliques containing a single sub-query term and a entity or relationship document node. For 3-cliques containing consecutive sub-query terms and a document node, we define two feature functions. One considers consecutive sub-query terms and matches ordered bigrams with entity or relationship documents. This feature function is denoted as fOE and fOR , depending if the clique is OE or OR . The second feature function matches bigrams with documents using an unordered window of 8 terms (uw8), i.e., it matches bigrams with documents if the two terms of the bigram occur with a maximum of 6 other terms between each other. This feature function is denoted as fUE and fUR , depending if the clique is OE or OR . For each textual feature function we decided to use two variants: Dirichlet smoothing Language Models (LM) and BM25. We now present the summary of the textual feature functions used in this work. LM-T-E  E (qjEi , DEi ) = log  fT,LM  E f (q i ,C E ) E f (qj i ,DEi )+ j E µE \|C \| \|DEi \|+µE    LM-O-E E E f#1 (q i ,q i ,C E ) E Ei j j+1 f#1 (qj i ,qj+1 µE ,DEi )+ \|C E \| \|DEi \|+µE  E E f#uw8 (q i ,q i ,C E ) E Ei j j+1 µE f#uw8 (qj i ,qj+1 ,DEi )+ E \|C \| E E \|D i \|+µ   Ei E fO,LM (qjEi , qj+1 , DEi ) = log     LM-U-E  Ei E fU,LM (qjEi , qj+1 , DEi ) = log     3.3 Entity-Relationship Dependence Model 57 LM-T-R  Ri−1,i  f (qj R R (qj i−1,i , DRi−1,i ) = log  fT,LM Ri−1,i R ,C ) j µR \|C R \| Ri−1,i \|D \|+µR ,DRi−1,i )+ f (q    LM-O-R Ri−1,i Ri−1,i R f#1 (q ,q ,C ) j j+1 µR \|C R \| Ri−1,i \|D \|+µR  Ri−1,i Ri−1,i R f#uw8 (q ,q ,C ) Ri−1,i R j j+1 ,DRi−1,i )+ f#uw8 (qj i−1,i ,qj+1 µR \|C R \| Ri−1,i R   R Ri−1,i  f#1 (qj R i−1,i R (qj i−1,i , qj+1 , DRi−1,i ) = log  fO,LM R i−1,i ,qj+1 ,DRi−1,i )+   LM-U-R  R R i−1,i R fU,LM (qj i−1,i , qj+1 , DRi−1,i ) = log   \|D \|+µ R Here, f (qjEi , DEi ) and f (qj i−1,i , DRi−1,i ) represent the sub-query term frequencies in a entity document and relationship document, respectively. The collection frequencies R f (qjEi , C E ), f (qj i−1,i , C R ) represent the frequency of sub-query term in either the entity index C E or in the relationship index C R . The variants of these functions f#1 and f#uw8 represent ordered and unordered bigram matching frequency. \|DEi \| and\|DRi,i+1 \| represent the total number of terms in a meta-document while \|C R \| and \|C E \| represent the total number of terms in a collection of meta-documents. Finally, µE and µR are the Dirichlet prior for smoothing which generally corresponds to the average document length in a collection. BM25-T-E E Ei E Ei fT,BM 25 (qj , D ) BM25-O-E = log N E −n(qj i )+0.5 E n(qj i )+0.5 E . f (qj i ,DEi )(K1 +1) E f (qj i ,DEi )+K1 (1−b+b \|D Ei \| ) avg(\|D E \|)   Entity Retrieval for Online Reputation Monitoring 58 Ei Ei E Ei fO,BM 25 (qj , qj+1 , D ) =log Ei N E − n#1 (qjEi , qj+1 ) + 0.5 · Ei Ei n#1 (qj , qj+1 ) + 0.5 (3.26) Ei f#1 (qjEi , qj+1 , DEi )(K1 + 1) Ei \|D \| Ei f#1 (qjEi , qj+1 , DEi ) + K1 (1 − b + b avg(\|D E \|) ) (3.27) BM25-U-E Ei E Ei fU,BM 25 (qj , D ) =log Ei N E − n#uw8 (qjEi , qj+1 ) + 0.5 · Ei Ei n#uw8 (qj , qj+1 ) + 0.5 (3.28) Ei , DEi )(K1 + 1) f#uw8 (qjEi , qj+1 Ei \|D \| Ei , DEi ) + K1 (1 − b + b avg(\|D f#uw8 (qjEi , qj+1 E \|) ) (3.29) BM25-T-R R Ri−1,i R fT,BM , DRi−1,i ) 25 (qj =log N R − n(qj i−1,i ) + 0.5 R n(qj i−1,i ) + 0.5 · (3.30) R f (qj i−1,i , DRi−1,i )(K1 + 1) R Ri−1,i \| \|D f (qj i−1,i , DRi−1,i ) + K1 (1 − b + b avg(\|D R \|) ) (3.31) BM25-O-R 3.3 Entity-Relationship Dependence Model 59 R R R i−1,i i−1,i R fO,BM , qj+1 , DRi−1,i ) =log 25 (qj R i−1,i N R − n#1 (qj i−1,i , qj+1 ) + 0.5 R R i−1,i n#1 (qj i−1,i , qj+1 ) + 0.5 R · R i−1,i f#1 (qj i−1,i , qj+1 , DRi−1,i )(K1 + 1) R Ri−1,i R \|D \| i−1,i f#1 (qj i−1,i , qj+1 , DRi−1,i ) + K1 (1 − b + b avg(\|D R \|) ) (3.32) (3.33) BM25-U-R R R Ri−1,i Ri−1,i R fU,BM , qj+1 , DRi−1,i ) 25 (qj =log i−1,i ) + 0.5 N R − n#uw8 (qj i−1,i , qj+1 R R i−1,i ) + 0.5 n#uw8 (qj i−1,i , qj+1 R · R i−1,i , DRi−1,i )(K1 + 1) f#uw8 (qj i−1,i , qj+1 R Ri−1,i R \| \|D i−1,i , DRi−1,i ) + K1 (1 − b + b avg(\|D f#uw8 (qj i−1,i , qj+1 R \|) ) (3.34) (3.35) Here, N E and N R represent the total number of documents in the entity index and relationship index, respectively. The document frequency of unigrams and bigrams is represented using n(),n#1 () and n#uw8 (). \|DEi \| and \|DRi−1,i \| are the total number of terms in a entity or relationship document while avg(\|DE \|) and avg(\|DR \|) are the average entity or relationship document length. K1 and b are free parameters usually chosen as 1.2 and 0.75, in the absence of specific optimization. We define two non-textual features in ERDM. The first one, fTER is assigned to 2-cliques composed by one entity document and one relationship document and it is inspired in the feature function fE of Hasibi and Balog’s ELR model [69]. It is defined as follows: h i) fSER (DEi , DRi−1,i ) = (1 − α)f (DEi , DRi−1,i ) + α n(E NR i (3.36) where the linear interpolation implements the Jelinek-Mercer smoothing method with α ∈ [0, 1] and f (DEi , DRi−1,i ) = {0, 1} which measures if the entity Ei represented Entity Retrieval for Online Reputation Monitoring 60 in DEi belongs to the relationship R(Ei−1 , Ei ) represented in DRi−1,i . The background model employs the notion of entity popularity within the collection of relationship documents. n(DEi ) represents the number of relationship documents DR that contain the entity Ei and N R represents the total number of relationship documents in the relationship index. For E-R queries with more than one relationship sub-query, we draw an edge between consecutive relationship documents within the ERDM graph. This edge creates a 3-clique containing two relationship documents and one entity document. The feature function fSRER measures if a given entity Ei is shared between consecutive relationship documents within the graph. We opted to define a simple binary function: fSER (DEi , DRi−1,i , DRi,i+1 ) = 1 if Ei ∈ DEi ∩ DRi−1,i ∩ DRi,i+1 , 0 otherwise (3.37) In summary, we described the set of feature functions associated with each clique configuration within the ERDM graph. We leave for future work the possibility of exploring other type of features to describe textual similarity and compatibility between different nodes in the ERDM graph, such as neural language models. 3.3.3 Ranking We have defined the set of clique configurations and the real valued feature functions that constitute the non-negative potential functions over the cliques in the graph of ERDM. We can now formulate the calculation of the posterior P (DE , DR \|Q using the probability mass function of the MRF, as follows: 3.3 Entity-Relationship Dependence Model rank PΛ (DE , DR \|Q) = X 61 λc f (c) c∈C(G) rank E = λT XX λE O XX λE U XX λR T X X fTE (qjEi , DEi )+ E QEi Ei fOE (qjEi , qj+1 , DEi )+ E QEi Ei , DEi )+ fUE (qjEi , qj+1 E QEi R fTR (qj i−1,i , DRi−1,i )+ (3.38) R QRi,j λR O X X λR U X X R R R R i−1,i , DRi−1,i )+ fOR (qj i−1,i , qj+1 R QRi,j i−1,i , DRi−1,i )+ fUR (qj i−1,i , qj+1 R QRi,j λER S XX R λRER S fSER (DEi , DRi−1,i )+ E XX R fSER (DEi , DRi−1,i , DRi,i+1 ) E (3.39) In essence, E-R retrieval using the ERDM corresponds to ranking candidate entity tuples using a linear weighted sum of the feature functions over the cliques in the graph. Therefore, we can apply any linear learning to rank algorithm to optimize the ranking with respect to the vector of feature weights Λ. Given a training set T composed by relevance judgments, a ranking of entity tuples RΛ and an evaluation function E(RΛ ; T ) that produces a real valued output, our objective is to find the values of the vector Λ that maximizes E. As explained in [65], we require E to only consider the ranking produced and not individual scores. This is the standard characteristic among information retrieval evaluation metrics (e.g. MAP or NDCG). 3.3.4 Discussion In this section we introduced the Entity-Relationship Dependence Model (ERDM), a novel supervised Early Fusion-based model for E-R retrieval. Inspired by recent work in entity retrieval we believe that modeling term dependencies between sub-queries and entity/relationship documents can increase search performance. Entity Retrieval for Online Reputation Monitoring 62 ERDM can be seen as an extension of the SDM model [63] for ad-hoc document retrieval in a way that besides modeling query term dependencies we create graph structures that depict dependencies between entity and relationship documents. Consequently, instead of computing a single posterior P (D\|Q) we propose to use the MRF for retrieval for computing a joint posterior of multiple entity and relationship documents given a E-R query, P (DE , DR \|Q). Moreover, since ERDM is a supervised model, we believe that tuning weights of feature functions, besides optimizing search performance, can also help to explain the inter-dependencies between sub-query terms and the respective documents, but also how entity documents and relationship documents contribute to the overall relevance of entity tuples given a E-R query. 3.4 Summary of the Contributions In this chapter we present several contributions to the problem of entity-relationship retrieval from a IR perspective: • Generalization of the problem of entity-relationship search to cover entity types and relationships represented by any attribute and predicate, respectively, rather than a predefined set. • A general probabilistic model for E-R retrieval using Bayesian Networks. • Proposal of two design patterns that support retrieval approaches using the E-R model. • Proposal of a Entity-Relationship Dependence model that builds on the basic Sequential Dependence Model (SDM) to provide extensible entity-relationship representations and dependencies, suitable for complex, multi-relations queries. Chapter 4 Entity-Relationship Retrieval over a Web Corpus We start this chapter by presenting a new semi-automatic method for generating E-R test collections together with a new E-R test collection, the RELink Query Collection comprising 600 E-R queries. We leverage web tabular data containing entities and relationships among them as they share the same row in a table. We exploit the Wikipedia Lists-of-lists-of-lists tree of articles containing lists of Entities in the form of tables. We developed a table parser that extracts tuples of entities from these tables together with associated metadata. This information is then provided to editors that create E-R queries fulfilled by the extracted tuples. We then report a set of evaluations of the ERDM model using four different query sets. In order to leverage information about entities and relations in a corpus, it is necessary to create a representation of entity related information that is amenable to ER search. In our approach we focus on sentence level information about entities although the method can be applied to more complex segmentation of text. Our experiments are based on the ClueWeb-09-B data set with FACC1 text annotation that refer to entities found in the text, including the variances of their surface forms. Each entity is designated by its unique ID and for each unique entity instance we created ’entity documents’ comprising a collection of sentences that contain the entity. These context documents are indexed, comprising the entity index. The same is done by creating entity pair documents and the entity pair index. These two indexes enable us to execute E-R queries using different retrieval models, including the ERDM that models the dependence between entities. Entity-Relationship Retrieval over a Web Corpus 64 4.1 RELink Query Collection 1 Improvements of entity-relationship (E-R) search techniques have been hampered by a lack of test collections, particularly for complex queries involving multiple entities and relationships. In this section we describe a method for generating E-R test queries to support comprehensive E-R search experiments. Queries and relevance judgments are created from content that exists in a tabular form where columns represent entity types and the table structure implies one or more relationships among the entities. Editorial work involves creating natural language queries based on relationships represented by the entries in the table. We have publicly released the RELink test collection comprising 600 queries and relevance judgments obtained from a sample of Wikipedia List-of-lists-of-lists tables. The latter comprise tuples of entities that are extracted from columns and labelled by corresponding entity types and relationships they represent. Improvement of methods for both extraction and search is hampered by a lack of query sets and relevance judgments, i.e., gold standards that could be used to compare effectiveness of different methods. In this section we introduce: 1. A low-effort semi-automatic method for acquiring instances of entities and entity relationships from tabular data. 2. RELink Query Collection (QC) of 600 E-R queries with corresponding relevance judgments Essential to our approach is the observation that tabular data typically includes entity types as columns and entity instances as rows. The table structure implies a relationship among table columns and enables us to create E-R queries that are answered by the entity tuples across columns. Following this approach, we prepared and released the RELink QC comprising 600 E-R queries and relevance judgments based on a sample of Wikipedia List-of-lists-of-lists tables. The query collection and the research framework are publicly available2 , enabling the community to expand the RELink Framework with additional document collections and alternative indexing and search methods. It is important to maintain and enhance the RELink QC by providing updates to the existing entity types and creating new queries and relevant instances from additional tabular data. 1 The material contained in this section was published in P. Saleiro, N. Milic-Frayling, E. M. Rodrigues, C. Soares, “RELink: A Research Framework and Test Collection for Entity-Relationship Retrieval”[15]. 2 https://sigirelink.github.io/RELink/ 4.1 RELink Query Collection 4.1.1 65 Tabular Data and Entity Relationships Information that satisfies complex E-R queries is likely to involve instances of entities and their relationships dispersed across Web documents. Sometimes such information is collected and published within a single document, such as a Wikipedia page. In such cases, traditional search engines can provide excellent search results without applying special E-R techniques or considering entity and relationship types. Indeed, the data collection, aggregation, and tabularization has been done by a Wikipedia editor. That also means that a tabular Wikipedia content, comprising various entities, can be considered as representing a specific information need, i.e., the need that motivated editors to create the page in the first place. Such content can, in fact, satisfy many different information needs. We focus on exploiting tabular data for exhaustive search for pre-specified E-R types. In order to specify E-R queries, we can use column headings as entity types. All the column entries are then relevance judgments for the entity query. Similarly, for a given pair of columns that correspond to distinct entities, we formulate the implied relationship. For example the pair <car, manufacturing plant> could refer to “is made in” or “is manufactured in” relationships. The instances of entity pairs in the table then serve as evidence for the specific relationship. This can be generalized to more complex information needs that involve multiple entity types and relationships. Automated creation of E-R queries from tabular content is an interesting research problem. For now we asked human editors to provide natural language and structured E-R queries for specific entity types. Once we collect sufficient amounts of data from human editors we will be able to automate the query creation process with machine learning techniques. For the RELink QC we compiled a set of 600 queries with E-R relevance judgments from Wikipedia lists about 9 topic areas. 4.1.2 Selection of Tables Wikipedia contains a dynamic index “The Lists of lists of lists”3 which represents the root of a tree that spans curated lists of entities in various domains. We used a Wikipedia snapshot from October 2016 to traverse “The Lists of lists of lists” tree starting from the root page and following every hyperlink of type “List of ” and their children. This resulted in a collection of 95,569 list pages. While most of the pages contain tabular data, only 18,903 include tables with consistent column and row structure. As in [172], we restrict content extraction to wikitable HTML class that 3 http://en.wikipedia.org/wiki/List_of_lists_of_lists 66 Entity-Relationship Retrieval over a Web Corpus typically denotes data tables in Wikipedia. We ignore other types of tables such as infoboxes. In this first instance, we focus on relational tables, i.e., the tables that have a key column, referring to the main entity in the table [173]. For instance, the ”List of books about skepticism” contains a table “Books” with columns “Author”, “Category” and “Title”, among others. In this case, the key column is “Title” which contains titles of books about skepticism. We require that any relationship specified for the entity types in the table must contain the “Title” type, i.e., involve the “Title” column. In order to detect key columns we created a Table Parser that uses the set of heuristics adopted by Lehmberg et al. [173], e.g., the ratio of unique cells in the column or text length. Once the key column is identified, the parser creates entity pairs consisting of the key column and one other column in the table. The content of the column cells then constitutes the set of relevant judgments for the relationship specified by the pair of entities. For the sake of simplicity we consider only those Wikipedia lists that contain a single relational table. Furthermore, our goal is to create queries that have verifiable entity and entity pair instances. Therefore, we selected only those relational tables for which the key column and at least one more column have cell content linked to Wikipedia articles. With these requirements, we collected 1795 tables. In the final step, we selected 600 tables by performing stratified sampling across semantic domains covered by Wikipedia lists. For each new table, we calcuated the Jaccard similarity scores between the title of the corresponding Wikipedia page and the titles of pages associated with tables already in the pool. By setting the maximum similarity threshold to 0.7 we obtained a set of 600 tables. The process of creating RELink queries involves two steps: (1) automatic selection of tables and columns within tables and (2) manual specification of information needs. For example, in the table “Grammy Award for Album of the Year” the columns “winner”, “work” were automatically selected to serve as entity types in the E-R query (Figure 4.1). The relationship among these entities is suggested by the title and we let a human annotator to formulate the query. The RELink query set was created by 6 annotators. We provided the annotators with access to the full table, metadata (e.g., table title or the first paragraph of the page) and entity pairs or triples to be used to specify the query (Figure 4.2). For each entity pair or triple the annotators created a natural language information need and an E-R query in the relational format Q = {QEi−1 , QRi−1,i , QEi }, as shown in Table 4.1. 4.1 RELink Query Collection 67 Fig. 4.1 Example of Wikipedia table row. Fig. 4.2 Example of metadata provided to editors. 4.1.3 Formulation of Queries The relational query format is introduced to support a variety of experiments with E-R queries. In essence, a complex information need is decomposed into a set of subqueries that specify types of entities E and types of relationships R(Ei−1 , Ei ) between entities. For each relationship query there is one query for each entity involved in the relationship. Thus a query Q that expects a pair of entities for a given relationship, is mapped into three sub-queries (QEi−1 , QRi−1,i , QEi ), where QEi−1 and QEi are the entity types for Ei−1 and Ei respectively, and QRi−1,i is a relationship type describing R(Ei−1 , Ei ). 4.1.4 Collection Statistics RELink QC covers 9 thematic areas from the Lists-of-Lists-of-Lists in Wikipedia: Mathematics and Logic, Religion and Belief Systems, Technology and Applied Sciences, Miscellaneous, People, Geography and Places, Natural and Physical Sciences, General Reference and Culture and the Arts. The most common thematic areas are Culture and the Arts with 70 queries and Geography and Places with 67 queries. In Table 4.2 we show the characteristics of the natural language and relational queries. Among 600 E-R queries, 381 refer to entity pairs and 219 to entity triples. As Entity-Relationship Retrieval over a Web Corpus 68 Table 4.1 Examples of query annotations. ID NL Query Relational Format RELink_P_164 What are the regiments held by the Indian Army? {regiment, held by, Indian Army} RELink_T_071 In which seasons NHL players scored more than 50 goals and the team they represented? {NHL season, scored more than 50 goals in, NHL player, played for, NHL team } Table 4.2 RELink collection statistics. Total queries Avg. queries length Avg. QE length Avg. QR length # uniq. entity attributes (QE ) # uniq. relationships (QR ) Avg. # relevant judgments 2-entity 381 56.5 20.9 11.8 679 145 67.9 3-entity 219 83.8 20.9 12.6 592 205 41.8 All 600 66.5 20.9 12.3 1251 317 58.5 expected, natural language descriptions of 3-entity queries are longer (on average 83.8 characters) compared to 2-entity queries (56.5 characters). We further analyze the structure of relational queries and their components, i.e., entity queries QE that specify the entity type and relationship queries QR that specify the relationship type. Across 600 queries, there are 1251 unique entity types QE (out of total 1419 occurrences). They are rather unique across queries: only 65 entity types occur in more than one E-R query and 44 occur in exactly 2 queries. The most commonly shared entity type is “country”, present in 9 E-R queries. In the case of relationships, there are 317 unique relationship types QR (out of 817 occurrences) with a dominant type “located in” that occurs in 140 queries. This is not surprising since in many domains the key entity is tied to a location that is included in one of the columns. Nevertheless, there are only 44 relationship types QR occurring more than once implying that RELink QC is a diverse set of queries, including 273 relationship types occurring only once. 4.2 Experimental Setup 4.2 69 Experimental Setup In this section we detail how we conducted our experiments in E-R retrieval. Since we only have access to test collections comprising general purpose E-R queries we decided to use a Web corpus as dataset, more precisely ClueWeb-09-B4 .The ClueWeb09 dataset was created to support research on information retrieval and related human language technologies and contains 1 billion web pages. The part B is a subset of the most popular 50 million English web pages, including the Wikipedia. Part B was created as a resource for research groups without processing power for processing the all ClueWeb09 collection. We used the ClueWeb-09-B Web collection with FACC1 text span annotations linked to Wikipedia entities to show how RELink can be used for E-R retrieval over Web content. We developed our prototype using Apache Lucene for indexing and search. We used a specific Python library (PyLucene) that allowed our customized implementation tailored for E-R retrieval. 4.2.1 Data and Indexing E1 E3 E4 E3 E2 E1E3 E3E4 E3 E1 E2 E3E4 D1 D2 D3 D4 E1 E1 E2 E2 E1E3 E3 E3 E3 E3E4 E3E4 E4 Entity Index Relationship Index Fig. 4.3 Illustration of E-R indexing from a web corpus. As a text corpus, we use ClueWeb-09-B combined with FACC1 text span annotations with links to Wikipedia entities (via Freebase). The entity linking precision and recall in FACC1 is estimated to be 80-85% and 70-85%, respectively [174]. For our experiments we created two main indexes: one for entity extractions and one for entity pairs 4 https://lemurproject.org/clueweb09/ Entity-Relationship Retrieval over a Web Corpus 70 (relationships) extractions. We extract entity and pairs occurrences using an Open Information Extraction method like OLLIE [57] over the annotated ClueWeb-09-B corpus as follows. For each entity annotation, we extract the sentence where it occurred as an entity context. For pairs of entities, we look for co-occurring entities in the same sentence and we extract the separating string, i.e., the context of the relationship connecting them. Figure 4.3 illustrates the indexing process adopted in this work. We obtained 476 million entity extractions and 418 million entity pairs extractions, as described in Table 4.3. In order to compute \|DEi \| and \|DRi−1,i \| we incrementally updated two auxiliary indices, containing the number of terms per entity and per entity pair, respectively. We ran our experiments using Apache Lucene and made use of GroupingSearch for grouping extractions by entity and entity pair at query time. To get the statistics for ordered and unordered bigrams we made use of SpanNearQuery. Table 4.3 ClueWeb09-B extractions statistics. Entities Entity pairs 4.2.2 Total 476,985,936 418,079,378 Unique 1,712,010 71,660,094 Avg. doc. len. 9977 138 Retrieval Method and Parameter Tuning For experiments using ERDM we adopted a three stage retrieval method. First, queries QEi−1 ,QEi are submitted against the entity index and QRi−1,i is submitted against the entity-pair index. Initial sets of top 20000 results grouped by entity or entity-pairs, respectively, are retrieved using Lucene’s default search settings. Second, the feature functions of the specific retrieval model are calculated for each set, using an in-house implementation. This process is easily parallelized. The final ranking score for each entity-pair is then computed using the learned λ weights. Evaluation scores are reported on the top 100 entity-pair results. Parameter tuning for ERDM and baselines was directly optimized with respect to the Mean Average Precision (MAP). We make use of the RankLib’s implementation of the coordinate ascent algorithm under the sum normalization and non-negativity constraints with 3 random restarts. Coordinate ascent is a commonly used optimization technique [65] that iteratively optimizes a single parameter while holding all other parameters fixed. Parameters are estimated using 5-fold cross validation for each of the 4 query sets separately. To be able to use the same train and test folds throughout all experiments, 4.2 Experimental Setup 71 we first randomly create fixed train and test folds from the initial result set, for each query set. All reported evaluation metrics were macro-averaged over 5 folds. We do not optimize the Dirichlet priors µE and µR in language models and set them equal to the traditional average document length, i.e., the average entity and entity pairs extractions length, respectively. The unordered window size N for fUE and fUR is set to be 8, as suggested in [63]. 4.2.3 Test Collections We ran experiments with a total of 548 E-R queries. We decided to just perform experiments using queries aiming 2-tuples of entities. We leave for future work the evaluation of queries aiming at triples. Besides RELink QC we used other 3 relationshipcentric query sets, with pairs of Wikipedia entities as answers, i.e., relevance judgments. The query sets cover a wide range of domains as described in Table 4.4. Query sets for entity-relationship retrieval are scarce. Generally entity retrieval query sets are not relationship-centric [10]. Table 4.4 Description of query sets used for evaluation. Query Set QALD-2 Count 79 Domains Geography and places, Politics and society, Culture and the Arts, Technology and science ERQ 28 COMPLEX 60 RELink 381 Award, City, Club, Company, Film, Novel, Person, Player, Song, University Cinema, Music, Books, Sports, Computing, Military conflicts General Reference, Culture and the Arts, Geography and places, Mathematics and logic, Natural and physical Sciences, People, Religion and belief systems, Society and social sciences, Technology and applied science Total 548 One exception is the QALD-2 query set used in the DBpedia-entity collection [175]. It contains a subset of relational queries, e.g.“Who designed the Brooklyn Bridge?”. Most of relational queries in QALD-2 have a fixed relevant entity, e.g., “Brooklyn Bridge” and can be easily transformed from single entity relevance judgments into pairs. From the 79 relational queries in QALD-2, we identified 6 with no fixed relevant entity 72 Entity-Relationship Retrieval over a Web Corpus in the query (e.g. “Give me the capitals of all countries in Africa.”). In these cases, for provided single entity relevance judgment we needed to annotate the missing entity manually to create a pair. For instance, given a capital city in Africa we identified the corresponding African country. In addition, we used two benchmarks created in previous work using SemanticWeb-based approaches: ERQ [56] and COMPLEX [10]. Neither ERQ nor COMPLEX provide complete relevance judgments and consequently, we manually evaluated each answer in our experiments. ERQ consists of 28 queries that were adapted from INEX17 and OWN28 [56]. However, 22 of the queries have a given fixed entity in the query (e.g. “Find Eagles songs”). Only 6 queries are asking for pairs of unknown entities, such as “Find films starring Robert De Niro and please tell directors of these films.”. COMPLEX queries were created with a semi-automatic approach [10]. It contains 70 queries from which we removed 10 that expect 3-tuples of entities. This query set consists of pure relationship-centric queries for unknown pairs of entities, such as “Currency of the country whose president is James Mancham “Kings of the city which led the Peloponnesian League.” and “Who starred in a movie directed by Hal Ashby?”. We used four different retrieval metrics, Mean Average Precision at 100 results (MAP), precision at 10 (P@10), mean reciprocal rank (MRR) and normalized discounted cumulative gain at 20 (NDCG@20). 4.3 Results and Analysis We start by performing a simple experiment for comparing Early Fusion and ERDM using both Language Models (LM) and BM25 as retrieval functions. Since we are only interested in comparing relative performance we opted to scale down our experimental setup. Instead of computing the term frequency for every extraction for a given entity or relationship we cap to 200 the number for each group of documents retrieved in the first passage. We tried several different values and for values below 200 extraction the performance reduced significantly. For 200, while the performance reduces it is not dramatic. This setup reduces the experimental runtime and since we had limited resources this proved to be useful. Table 4.5 depicts the results for this comparative evaluation. We decided to only use the three test collections specifically tailored for relationship retrieval. As we can see the results are very similar between EF and ERDM for both LM and BM25 variants. In the three test collections ERDM presents slightly better performance than the corresponding EF variant (e.g. BM25). However when performing statistical 4.3 Results and Analysis 73 significance tests we obtained p-values above 0.05 when comparing EF and ERDM. This is very interesting as it shows that for general purpose E-R evaluation the overhead of computing sequential dependencies does not carry significant improvements. Table 4.5 Early Fusion and ERDM comparison using LM and BM25. EF-LM EF-BM25 ERDM-LM ERDM-BM25 EF-LM EF-BM25 ERDM-LM ERDM-BM25 EF-LM EF-BM25 ERDM-LM ERDM-BM25 ERQ MAP P@10 MRR NDCG@20 0.251 0.15 0.3408 0.3508 0.1939 0.1423 0.1783 0.2861 0.2611 0.1615 0.3151 0.3589 0.2106 0.1462 0.2839 0.3257 COMPLEX MAP P@10 MRR NDCG@20 0.1703 0.0596 0.1839 0.2141 0.1855 0.0719 0.1907 0.2454 0.1719 0.0789 0.2466 0.2492 0.1955 0.0772 0.2257 0.248 RELink(381 queries) MAP P@10 MRR NDCG@20 0.0186 0.0063 0.0192 0.0249 0.0203 0.0071 0.0227 0.0259 0.0213 0.0058 0.0273 0.0255 0.0213 0.0061 0.0265 0.0275 On the other hand, we detect sensitivity to the retrieval function used. In ERQ, both ERDM-LM and EF-LM outperform BM25 but the opposite happens for COMPLEX and RELink. This sensitivity means that we cannot generalize the assumption that one of the retrieval functions is more adequate for E-R retrieval. Another important observation has to do with the overall lower results on the RELink test collection in comparison with ERQ and COMPLEX. Contrary to our expectations ClueWeb-09B has very low coverage of entity tuples relevant to the RELink test collection. We now present the results of comparing ERDM with three baselines using sequential dependence to evaluate the impact of modeling dependencies between query terms. The first baseline method, BaseEE, consists in submitting two queries against the entity index: QEi−1 + QRi−1,i and QRi−1,i + QEi . Entity-pairs are created by cross product of the two entity results set retrieved by each query. For each method we compute the Sequential Dependence Model(SDM) [63] scores. Entity-Relationship Retrieval over a Web Corpus 74 The second baseline method, BaseE, consists in submitting again a single query Q towards the entity index used in ERDM. Entity-pairs are created by cross product of the entity results set with itself. The third baseline method, BaseR, consists in submitting a single query Q towards an entity-pair index. This index is created using the full sentence for each entity-pair co-occurrence in ClueWeb-09-B, instead of just the separating string as in ERDM. This approach aims to capture any entity context that might be present in a sentence. ERDM relies on the entity index for that purpose. In this evaluation we decided to not cap the number of extractions to compute term frequencies inside each group of results returned from the first passage with Lucene GroupingSearch. Due to the low coverage of ClueWeb for the entire RELink collection, we decided to just perform the evaluation using the top 100 queries with highest number of relevance judgments in our indexes. We also include results for the adapted QALD-2 test collection. Table 4.6 Results of ERDM compared with three baselines. BaseEE BaseE BaseR ERDM MAP 0.0087 0.0306 0.0872 0.1520 BaseEE BaseE BaseR ERDM MAP 0.0085 0.0469 0.1041 0.3107 BaseEE BaseE BaseR ERDM MAP 0.0035 0.0264 0.0585 0.2879 BaseEE BaseE BaseR ERDM MAP 0.03 0.0395 0.0451 0.1249 QALD-2 P@10 MRR NDCG@20 0.0027 0.0093 0.0055 0.004684 0.0324 0.0363 0.01678 0.0922 0.0904 0.0405 0.1780 0.1661 ERQ P@10 MRR NDCG@20 0.004 0.00730 0.0030 0.01086 0.0489 0.038 0.05086 0.1089 0.1104 0.1903 0.37613 0.3175 COMPLEX P@10 MRR NDCG@20 0 0.00430 0 0.005 0.03182 0.1223 0.01836 0.0748 0.0778 0.1417 0.32959 0.3323 RELink(100 queries) P@10 MRR NDCG@20 0.01 0.0407 0.02946 0.019 0.0679 0.03948 0.021 0.0663 0.07258 0.048 0.1726 0.1426 4.3 Results and Analysis 75 Table 4.6 presents the results of our experiments on each query set. We start by comparing the three baselines among each other. As follows from Table 4.6, BaseR baseline outperforms BaseEE and BaseE on all query sets, while BaseEE is the worst performing baseline. The BaseR retrieval is the only relationship-centric approach from the three baselines, as its document collection comprises entity-pairs that co-occurred in ClueWeb-09-B corpus. BaseEE and BaseE retrieve entity pairs that are created in a post-processing step which reduces the probability of retrieving relevant results. This results shows the need for a relationship-centric document collection when aiming to answer entity-relationship queries. ERDM significantly outperform all baselines on all query sets. We performed statistical significance testing of MAP using ERDM against each baseline obtaining p-values below 0.05 on all the query sets. This results show that our Early Fusion approach using two indexes (one for entities and other for relationships) is adequate and promising. We believe this approach can become a reference for future research in E-R retrieval from an IR-centric perspective. Nevertheless, based on the absolute results obtained on each evaluation metric and for each query set we can conclude that E-R retrieval is still very far from being a solved problem. There is room to explore new feature functions and retrieval approaches. This is a very difficult problem and the methods we proposed are still far from optimal performance. Queries such as “Find world war II flying aces and their services” or “Which mountain is the highest after Annnapurna?” are examples of queries with zero relevant judgments returned. On the other hand, ERDM exhibits interesting performance in some queries with high complexity, such as “Computer scientists who are professors at the university where Frederick Terman was a professor.” We speculate about some aspects that might influence performance. One aspect has to do with the lack of query relaxation in our experimental setup. The relevant entity tuples might be in our indexes but if the query terms used to search for entity tuples do not match the query terms harvested from ClueWeb-09B it is not possible to retrieve those relevant judgments. Query relaxation approaches should be tried in future work. More specifically, with the recent advances in word embeddings it is possible to expand queries with alternative query terms that are in the indexes. On the other hand, we adopted a very simple approach for extracting entities and relationships. The use of dependency parsing and more complex methods of relation extraction would allow to filter out noisy terms. We also leave this for future work. Moreover, to further assess the influence of the extraction method we propose to use Entity-Relationship Retrieval over a Web Corpus 76 selective text passages containing the target entity pairs and the query terms associated as well. Then different extraction methods could be tried and straightforward evaluation of their impact. (a) (b) (c) ′ ′ Fig. 4.4 Values of λ for ERDM: (a) all λ, (b) λE , (c) λR . (b) and (c) were obtained using sum normalization. To understand how much importance is attributed to the different types of clique sets, we plot the values of the lambda parameters: λE parameters represent the feature importance of the set of functions targeting the dependence between entity query terms and the entity documents in overall ranking score for entity-pairs; λR represent the importance of the feature functions of the relationship type queries and finally, the value for λER which is assigned to the feature function that evaluates if each entity retrieved from both entity type queries belongs to the entity-pair retrieved from the relationship type query. 4.4 Summary of the Contributions 77 We plot the feature weights learned on each query set, as depicted in Figure 4.4. We see that λER and λE T (weight for the unigram language model in the entity type queries) dominate the ranking function. We further evaluated the relative weights for each one of the three SDM-like functions using a sum normalization of the three weights for both entity documents and entity-pair documents. We observe that λE T dominates on R every query set, however the same does not happen with λT . For relationship type queries the bigram features have higher values for COMPLEX and RELink. 4.4 Summary of the Contributions In this chapter we presented the following contributions to the E-R retrieval research area: 1. Indexing method that supports generalization of entity types and entity-relationships to any attribute and predicate, respectively 2. A semi-automatic method for generating E-R test collections, which resulted in the RELink Query Collection comprising 600 E-R queries. 3. Results of experiments at scale, with a comprehensive set of queries and corpora. Chapter 5 Entity Filtering and Financial Sentiment Analysis In this chapter we present the work developed to tackle two fundamental Text Mining problems in ORM: Entity Filtering and Sentiment Analysis. We start by describing our participation at the Filtering task of RepLab 2013 [32]. We developed a supervised method to classify tweets as relevant or non-relevant to given target entity. This method obtained the first place at the competition. Entity Filtering can be seen as target based Named Entity Disambiguation (NED). Given a target entity under study, we need to develop a binary classifier to filter out tweets that are not talking about the target entity. This task is fundamental in ORM as downstream tasks such as Sentiment Analysis or entity-centric predictions would produce misleading results if noisy signals were used. Sentiment Analysis has been widely studied over the last decade. It is a research area with several ramifications as it is dependent on the type of texts and the objective of the analysis. We decided to focus our efforts in a not so well explored sub-area of Sentiment Analysis. SemEval 2017 Task 5 focused on fine-grained sentiment analysis of financial news and microblogs. As one of the use cases of ORM is to track the online reputation of companies and try to assess its impact on the stock market we decided it was a specific task within Sentiment Analysis in which we could make a contribution. We obtained the fourth place in the Microblogs sub-task using one of the evaluation metrics. The task consisted in predicting a real continuous variable from -1.0 to +1.0 representing the polarity and intensity of sentiment concerning companies/stocks mentioned in short texts. We modeled it as a regression analysis problem. Entity Filtering and Financial Sentiment Analysis 80 5.1 Entity Filtering1 The relationship between people and public entities has changed with the rise of social media. Online users of social networks, blogs and micro-blogs are able to directly express and spread opinions about public entities, such as politicians, artists, companies or products. Online Reputation Monitoring (ORM) aims to automatically process online information about public entities. Some of the common tasks within ORM consist in collecting, processing and aggregating social network messages to extract opinion trends about such entities. Twitter, one of the most used online social networks, provides a search system that allows users to query for tweets containing a set of keywords. ORM systems often use Twitter as a source of information when monitoring a given entity. However, search results are not necessarily relevant to that entity because keywords can be ambiguous. For instance, a tweet containing the word “columbia” can be related with several entities, such as a federal state, a city or a university. Furthermore, tweets are short which results in a reduced context for entity disambiguation. When monitoring the reputation of a given entity on Twitter, it is first necessary to guarantee that all tweets are relevant to that entity. Consequently, other processing tasks, such as sentiment analysis will benefit from filtering out noise in the data stream. In this work, we tackle the aforementioned problem by applying a supervised learning approach. Given a set of entities E = {e1 , e2 , ..., ei , ...}, a stream of texts S = {s1 , s2 , ..., si , ...} (e.g. tweets), we are interested in monitoring the mentions of an entity ei on the stream S, i.e. the discrete function fm (ei , S). We cast the prediction of fm as a supervised learning classification problem, in which we want to infer the target variable fˆm (ei , S) ∈ {0, 1} We implemented a large set of features that can be generated to describe the relationship between an entity representation and a text mention. We use metadata (e.g. entity names, category) provided in the user configurations, text represented with TF-IDF, similarity between texts and Wikipedia, Freebase entities disambiguation, feature selection of terms based on frequency and feature matrix transformation using SVD. The learning algorithms from scikit-learn Python library that were tested for Entity Filtering include Naive Bayes, SVM, Random Forests, Logistic Regression and MultiLayer Perceptron. 1 Most of the material contained in this section was published in P.Saleiro, E. M. Rodrigues, C. Soares, E. Oliveira, “TexRep: A Text Mining Framework for Online Reputation Monitoring” [14] 5.1 Entity Filtering 5.1.1 81 Task Overview RepLab 2013 [32] focused on monitoring the online reputation of entities on Twitter. The Filtering task consisted in determining which tweets are relevant to each entity. The corpus consists of a collection of tweets obtained by querying the Twitter Search API with 61 entity names during the period from the June 2012 until the December 2012. The corpus contains tweets both in English and Spanish. The balance between both languages varies for each entity. Tweets were manually annotated as “Related” or “Unrelated” to the respective target entity. The data provided to participants consists in tweets and a list of 61 entities. For each tweet in the corpus we have the target entity id, the language of the tweet, the timestamp and the tweet id. The content of each URL in the tweets is also provided. Due to Twitter’s terms of service, the participants were responsible to download the tweets using the respective id. The data related with entities contain the query used to collect the tweets (e.g. “BMW”), the official name of the entity (e.g. “Bayerische Motoren Werke AG”), the category of the entity (e.g. “automotive”), the content of its homepage and both Wikipedia articles in English and Spanish. 5.1.2 Pre-processing The Entity Filtering module includes methods to normalize texts by removing all punctuation, converting text to lower case, removing accents and converting non-ASCII characters to their ASCII equivalent. Lists of stop words for several languages are also available, which are used to filter out non relevant words. We rely on the Natural Language Toolkit (NLTK) to provide those lists. Contrary to other types of online texts (e.g. news or blog posts) tweets contain informal and non-standard language including emoticons, spelling errors, wrong letter casing, unusual punctuation and abbreviations. Therefore, when dealing with tweets, the Entity Filtering module uses a tokenizer [176] optimized for segmenting words in tweets. After tokenization we extract user mentions and URLS and hashtags textual content. 5.1.3 Features Many different types of features can be used to optimize relevance classification, including language models, keyword similarities between tweets and entities as well as external resources projections. We implemented a large number of those. We assume that future users of our framework for ORM will provide entity-specific data (e.g. 82 Entity Filtering and Financial Sentiment Analysis homepage/Wikipedia content) prior to training and configuring the Entity Filtering module. Language Model: text is encapsulated in a single feature to avoid high dimensionality issues when adding other features. A TF-IDF representation of unigrams, bigrams and trigrams for training a text classifier which calculates the probability of a text being related to the expected entity. The output probabilities of the classifier are used as a feature. Keyword similarity: similarity scores between metadata and the texts, obtained by calculating the ratio of the number of common terms in the texts and the terms of query and entity name. Similarities at character level are also available in order to include possible spelling errors in the text. Web similarity: similarity between the text and the normalized content of the entity’s homepage and normalized Wikipedia articles are also available. The similarity value is the number of common terms multiplied by logarithm of the number of terms in tweet. Freebase: For each keyword of the entity’s query that exists in the text, two bigrams are created, containing the keyword and the previous/subsequent word. These bi-grams are submitted to the Freebase Search API and the list of retrieved entities are compared with the id of the target entity on Freebase. A Freebase score is computed by using the inverse position of the target entity in the list of results retrieved. If the target entity is the first result, the score is 1, if it is the second, the score is 0.5, and so on. If the target entity is not in the results list, the score is zero. The feature corresponds to the maximum score of the extracted bigrams of each text. Category classifier: a sentence category classifier is created using the Wikipedia articles of each entity. Each sentence of the Wikipedia articles is annotated with the category of the corresponding entity. TF-IDF for unigrams, bigrams and trigrams are calculated and a multi-class classifier (SVM) is trained to classify each text. The feature is the probability of the text being relevant to its target class. 5.1.4 Experimental Setup The dataset used for the competition consists of a collection of tweets both in English and Spanish, possibly relevant to 61 entities from four domains: automotive, banking, 5.1 Entity Filtering 83 Dataset Related Unrelated Total Training Development Validation Test 33,193 26,534 6,659 75,470 10,389 8,307 2,082 21,378 43,582 34,841 8,741 96,848 Table 5.1 RepLab 2013 Filtering Task dataset description. universities and music.The dataset consists of a collection of tweets obtained by querying the Twitter Search API with 61 entity names during the period from the June 2012 until the December 2012. The balance between both languages varies for each entity. The complementary data about each target entity is the following: • query used to collect the tweets (e.g. “BMW”) • official name of the entity (e.g. “Bayerische Motoren Werke AG”) • category of the entity (e.g. “automotive”) • content of entity homepage • Wikipedia article both in English and Spanish Tweets were manually annotated as “Related” or “Unrelated” to the respective target entity. The dataset is divided in training, test and development (Table 5.1). The training set consists in a total of 45,671 tweets from which we were able to download 43,582. Approximately 75% of tweets in the training set are labeled as “Related”. We split the training dataset into a development set and a validation set, containing 80% and 20% of the original, respectively. We adopted a randomly stratified split approach per entity, i.e., we group tweets of each target entity and randomly split them preserving the balance of “Related”/“Unrelated” tweets. The test dataset consists of 90,356 tweets from which we were able to download 88,934. We used the development set for trying new features and test algorithms. We divided the development set in 10 folds generated with the randomly stratified approach. We used the validation set to validate the results obtained in the development set. The purpose of this validation step is to evaluate how well the Entity Filtering classifier generalizes from its training data to the validation data and thus estimate how well it will generalize to the test set. It allows us to spot overfitting. After validation, we trained the classifier using all of the data in the training dataset and evaluated in the test set. Entity Filtering and Financial Sentiment Analysis 84 5.1.5 Results We created different classifier runs using different learners, features and we also created entity specific models as explained in Table 5.2, [177]. We applied selection of features based on frequency and transformation of content representation using SVD. The learners tested include Naive Bayes (NB), SVM, Random Forests (RF), Logistic Regression (LR) and MultiLayer Perceptron (MLP). The evaluation measures used are accuracy and the official metric of the competition, F-measure which is the harmonic mean of Reliability and Sensitivity [178]. We present results for the top 4 models regarding the F-measure. We replicated the best system at RepLab 2013 in the run 1. Run Learner Features No. of models 1 2 3 SVM RF RF All All All global global per entity Table 5.2 Entity filtering versions description. Table 5.3 shows the results of top performing runs and the official baseline of the competition. This baseline classifies each tweet with the label of the most similar tweet of target entity in the training set using Jaccard similarity coefficient. The baseline results were obtained using 99.5% of the test set. Run Acc. (Val. Set) 1 0.944 2 0.945 3 0.948 Official Baseline Best RepLab - Acc. R S F-measure 0.906 0.908 0.902 0.8714 0.908 0.759 0.729 0.589 0.4902 0.729 0.428 0.470 0.451 0.488 0.444 0.448 0.3199 0.3255 0.451 0.488 Table 5.3 Official results for each version plus our validation set accuracy. Based on the results achieved we are able to conclude that the models of our classifier are able to generalize successfully. Results obtained in the validation set are similar to those obtained in the test set. During development, solutions based on one model per entity were consistently outperformed by solutions based on global models. We also noticed during development that language specific models (English and Spanish) did not exhibit improvements in global accuracy, therefore we opted to use language as a feature. Results show that the best model uses the Random Forests 5.1 Entity Filtering 85 classifier with 500 estimators for training a global model. Though, the Language Modeling feature encapsulates text by using a specific model trained just with TF-IDF of n-grams of tweets. We performed a “break down” analysis for each one of the four categories of RepLab 2013 using Run 2 model, as depicted in Figure 5.1. We observe that University, Banking and Automotive categories exhibit similar average F-measure results, all above 0.50. In contrast, results for Music shows it is a rather difficult category of entities to disambiguate (achieving F-measure of 0.39). In fact, some of the entity names of this category contain very ambiguous tokens, such as “Alicia Keys”, “U2”, “The Wanted” or “The Script”. Fig. 5.1 Results grouped by entity’s category using Run 2. The main goal of this task was to classify tweets as relevant or not to a given target entity. We have explored several types of features, namely similarity between keywords, language models and we have also explored external resources such as Freebase and Wikipedia. Results show that it is possible to achieve an Accuracy over 0.90 and an F-measure of 0.48 in a test set containing more than 90000 tweets of 61 entities. In future work, we expect to include the possibility of using entity-specific embedding to learn a joint embedding space of entities and words, similar to [91]. Entity Filtering and Financial Sentiment Analysis 86 5.2 Financial Sentiment Analysis2 Sentiment Analysis on financial texts has received increased attention in recent years [12]. Nevertheless, there are some challenges yet to overcome [13]. Financial texts, such as microblogs or newswire, usually contain highly technical and specific vocabulary or jargon, making the development of specific lexical and machine learning approaches necessary. Most of the research in Sentiment Analysis in the financial domain has focused in analyzing subjective text, labeled with explicitly expressed sentiment. However, it is also common to express financial sentiment in an implicit way. Business news stories often refer to events that might indicate a positive or negative impact, such as in the news title “company X will cut 1000 jobs”. Economic indicators, such as unemployment and change over time such as drop or increase can also provide clues on the implicit sentiment [179]. Contrary to explicit expressions (subjective utterances), factual text types often contain objective statements that convey a desirable or undesirable fact [92]. Recent work proposes to consider all types of implicit sentiment expressions [180]. The authors created a fine grained sentiment annotation procedure to identify polar expressions (implicit and explicit expressions of positive and negative sentiment). A target (company of interest) is identified in each polar expression to identify the sentiment expressions that are relevant. The annotation procedure also collected information about the polarity and the intensity of the sentiment expressed towards the target. However, there is still no automatic approach, either lexical-based or machine learning based, that tries to model this annotation scheme. In this work, we propose to tackle the aforementioned problem by taking advantage of unsupervised learning of word embeddings in financial tweets and financial news headlines to construct a domain-specific syntactic and semantic representation of words. We combine bag-of-embeddings with traditional approaches, such as pre-processing techniques, bag-of-words and financial lexical-based features to train a regressor for sentiment polarity and intensity. We study how different regression algorithms perform using all features in two different sub-tasks at SemEval-2017 Task 5: microblogs and news headlines mentioning companies/stocks. Moreover, we compare how different combinations of features perform in both sub-tasks. The system source code and word embeddings developed for the competition are publicly available.3 2 The material contained in this section was published in P. Saleiro, E. M. Rodrigues, C. Soares, E. Oliveira, “FEUP at SemEval-2017 Task 5: Predicting Sentiment Polarity and Intensity with Financial Word Embeddings” [18] 3 https://github.com/saleiro/Financial-Sentiment-Analysis 5.2 Financial Sentiment Analysis 5.2.1 87 Task Overview The task 5 of SemEval 2017 [181] consisted of fine-grained sentiment analysis of financial short texts and it was divided in two sub-tasks based on the type of text. Sub-task 5.1 – Microblogs – consisted of stocktwits and tweets focusing on stock market events and assessments from investors and traders. Companies/stocks were identified using stock symbols, the so called cashtags, e.g.“$AMZN” for the company Amazon.com, Inc. Sub-task 5.2 – News Headlines – consisted of sentences extracted from Yahoo Finance and other financial news sources on the Internet. In this case, companies/stocks were identified using their canonical name and were previously annotated by the task organizers. Sub-task 5.1 - Microblogs 5.2 - Headlines Company JPMorgan Glencore Text Span Sentiment Score “its time to sell banks" -0.763 “Glencore’s annual results +0.900 beat forecasts" Table 5.4 Training set examples for both sub-tasks. The goal of both sub-tasks was the following: predict the sentiment polarity and intensity for each of the companies/stocks mentioned in a short text instance (microblog message or news sentence). The sentiment score is a real continuous variable in the range of -1.0 (very negative/bearish) to +1.0 (very positive/bullish), with 0.0 designating neutral sentiment. Table 5.4 presents two examples from the training set. Task organizers provided 1700 microblog messages for training and 800 messages for testing in sub-task 5.1, while in sub-task 5.2, 1142 news sentences were provided for training and 491 for testing. Submissions were evaluated using the cosine similarity [181]. 5.2.2 Financial Word Embeddings Mikolov et al. [182] created word2vec, a computationally efficient method to learn distributed representation of words, where each word is represented by a distribution of weights (embeddings) across a fixed set of dimensions. Furthermore, Mikolov et al. [100] showed that this representation is able to encode syntactic and semantic similarities in the embedding space. The training objective of the skip-gram model, defined by Mikolov et al. [100], is to learn the target word representation (embeddings) that maximize the prediction of its surrounding words in a context window. Given the wt word in a vocabulary the objective is to maximize the average log probability: Entity Filtering and Financial Sentiment Analysis 88 T X 1X log P (wt+j \|wt ) T t=1 −c≤j≤c,j̸=0 (5.1) where c is the size of the context window, T is the total number of words in the vocabulary and wt+j is a word in the context window of wt . After training, a low dimensionality embedding matrix E encapsulates information about each word in the vocabulary and its use (surrounding contexts). We used word2vec to learn word embeddings in the context of financial texts using unlabeled tweets and news headlines mentioning companies/stocks from S&P 500. Tweets were collected using the Twitter streaming API with cashtags of stocks titles serving as request parameters. Yahoo Finance API was used for requesting financial news feeds by querying the canonical name of companies/stocks. The datasets comprise a total of 1.7M tweets and 626K news titles. We learned separate word embeddings for tweets and news headlines using the skip-gram model. We tried several configurations of word2vec hyperparameters. The setup resulting in the best performance in both sub-tasks was skip-gram with 50 dimensions, removing words occurring less than 5 times, using a context window of 5 words and 25 negative samples per positive example. Even though the text collections for training embeddings were relatively small, the resulting embedding space exhibited the ability to capture semantic word similarities in the financial context. We performed simple algebraic operations to capture semantic relations between words, as described in Mikolov et al. [113]. For instance, the skip-gram model trained on tweets shows that vector (“bearish”) - vector(“loss”) + vector(“gain”) results in vector (“bullish”) as most similar word representation. 5.2.3 Approach In this section we describe the implementation details of the proposed approach. Pre-Processing A set of pre-processing operations are applied to every microblog message and news sentence in the training/test sets of sub-tasks 5.1 and 5.2, as well as in the external collections for training word embeddings: • Character encoding and stopwords: every message and headline was encoded in UTF-8. Standard english stopword removal is also applied. 5.2 Financial Sentiment Analysis 89 • Company/stock and cash obfuscation: both cashtags and canonical company names strings were replaced by the string _company_. Dollar or Euro signs followed by numbers were replaced by the string _cash_amount_. • Mapping numbers and signs: numbers were mapped to strings using bins (0-10, 10-20, 20-50, 50-100, >100). Minus and plus signs were coverted to minus and plus, “B” and “M” to billions and millions, respectively. The % symbol was converted to percent. Question and exclamation marks were also converted to strings. • Tokenization, punctuation, lowercasing: tokenization was performed using Twokenizer [183], the remaining punctuation was removed and all characters were converted to lowercase. Features We combined three different group of features: bag-of-words, lexical-based features and bag-of-embeddings. • Bag-of-words: we apply standard bag-of-words as features. We tried unigrams, bi-grams and tri-grams with unigrams proving to obtain higher cosine similarity in both sub-tasks. • Sentiment lexicon features: we incorporate knowledge from manually curated sentiment lexicons for generic Sentiment Analysis as well as lexicons tailored for the financial domain. The Laughran-Mcdonald financial sentiment dictionary [184] has several types of word classes: positive, negative, constraining, litigious, uncertain and modal. For each word class we create a binary feature for the match with a word in a microblog/headline and a polarity score feature (positive - negative normalized by the text span length). As a general-purpose sentiment lexicon we use MPQA [185] and created binary features for positive, negative and neutral words, as well as, the polarity score feature. • Bag-of-Embeddings: we create bag-of-embeddings by taking the average of word vectors for each word in a text span. We used the corresponding embedding matrix trained on external Twitter and Yahoo Finance collections for sub-task 5.1 and sub-task 5.2, respectively. Entity Filtering and Financial Sentiment Analysis 90 5.2.4 Experimental Setup In order to avoid overfitting we created a validation set from the original training datasets provided by the organizers. We used a 80%-20% split and sampled the validation set using the same distribution as the original training set. We sorted the examples in the training set by the target variable values and skipped every 5 examples. Results are evaluated using Cosine similarity [181] and Mean Average Error (MAE). The former gives more importance to differences in the polarity of the predicted sentiment while the latter is concerned with how well the system predicts the intensity of the sentiment. We opted to model both sub-tasks as single regression problems. Three different regressors were applied: Random Forests (RF), Support Vector Machines (SVM) and MultiLayer Perceptron (MLP). Parameter tuning was carried using 10 fold cross validation on the training sets. 5.2.5 Results and Analysis In this section we present the experimental results obtained in both sub-tasks. We provide comparison of different learning algorithms using all features, as well as, a comparison of different subsets of features, to understand the information contained in each of them and also how they complement each other. Task 5.1 - Microblogs Table 5.5 presents the results obtained using all features in both validation set and test sets. Results in the test set are worse than in the validation set with the exception of MLP. The official score obtained in sub-task 5.1 was 0.6948 using Random Forests (RF), which is the regressor that achieves higher cosine similarity and lower MAE in both training and validation set. Regressor Set RF Val RF Test SVR Val SVR Test MLP Val MLP Test Table 5.5 Microblog results with all Cosine MAE 0.7960 0.1483 0.6948 0.1886 0.7147 0.1944 0.6227 0.2526 0.6720 0.2370 0.6789 0.2132 features on validation and test sets. 5.2 Financial Sentiment Analysis 91 We compared the results obtained with different subsets of features using the best regressor, RF, as depicted in Table 5.6. Interestingly, bag-of-words (BoW) and bag-of-embeddings (BoE) complement each other, obtaining better cosine similarity than the system using all features. Financial word embeddings (BoE) capture relevant information regarding the target variables. As a single group of features it achieves a cosine similarity of 0.6118 and MAE of 0.2322. It is also able to boost the overall performance of BoW with gains of more than 0.06 in cosine similarity and reducing MAE more than 0.03. The individual group of features with best performance is Bag-of-words while the worst is a system trained using Lex (only lexical-based features). While Lex alone exhibits poor performance, having some value but marginal, when combined with another group of features, it improves the results of the latter, as in the case of BoE + Lex and BoW + Lex. Features Cosine MAE Lex 0.3156 0.3712 BoE 0.6118 0.2322 BoW 0.6386 0.2175 BoE + Lex 0.6454 0.2210 Bow + Lex 0.6618 0.2019 Bow + BoE 0.7023 0.1902 All 0.6948 0.1886 Table 5.6 Features performance breakdown on test set using RF. Task 5.2 - News Headlines Results obtained in news headlines are very different from the ones of the previous sub-task, proving that predicting sentiment polarity and intensity in news headlines is a completely different problem compared to microblogs. Table 5.7 shows that MLP obtains the best results in the test set using both metrics while SVR obtains the best performance in the validation set. The best regressor of sub-task 5.1, RF is outperformed by both SVR and MLP. The official result obtained at sub-task 5.2 was a cosine similarity of 0.68 using MLP. Table 5.8 shows the results of the different groups of features in sub-task 5.2 for MLP regressor. The most evident observation is that word embeddings are not effective in this scenario. On the other hand, lexical based features have significantly better performance in news headlines than in microblogs. Despite this, the best results are obtained using all features. Entity Filtering and Financial Sentiment Analysis 92 Regressor Set RF Val RF Test SVR Val SVR Test MLP Val MLP Test Table 5.7 News Headlines results with Features BoE Lex BoW BoE + Lex BoW + Lex BoW + BoE All Table 5.8 Features performance Cosine MAE 0.5316 0.2539 0.6562 0.2258 0.6397 0.2422 0.6621 0.2424 0.6176 0.2398 0.6800 0.2271 all features on validation and test sets. Cosine MAE 0.0383 0.3537 0.5538 0.2788 0.6420 0.2364 0.5495 0.2830 0.6733 0.2269 0.6417 0.2389 0.6800 0.2271 breakdown on test set using MLP. Analysis Financial word embeddings were able to encapsulate valuable information in sub-task 5.1 - Microblogs but not so much in the case of sub-task 5.2 - News Headlines. We hypothesize that as we had access to a much smaller dataset (∼ 600K) for training financial word embeddings for news headlines, this resulted in reduced ability to capture semantic similarities in the financial domain. Other related works in Sentiment Analysis usually take advantage of a much larger dataset for training word embeddings [186]. On the other hand, lexical features showed poor performance in microblog texts but seem to be very useful using news headlines. The fact that microblogs have poor grammar, slang and informal language reveals that financial lexicons created using well written and formal financial reports, result better in news headlines rather than in microblog texts. After inspecting microblog texts and headlines in which our models showed poor performance we believe it would be important to also encapsulate syntactic and semantic dependencies in our models. For instance, our model predicted a sentiment score of -0.467 for the microblog message “was right to reject the offer” while the true value is 0.076. Similar examples include “Glencore shares in record crash as profit fears grow” and “I would rather be a buyer at these levels then trying to sell”, in which our models 5.3 Summary of the Contributions 93 has absolute errors around 0.5. Other type of errors have to do with intensity of the sentiment in which our model correctly predicts the polarity but still has a large error. 5.2.6 Concluding Remarks Work reported here reported is concerned with the problem of predicting sentiment polarity and intensity of financial short texts. Previous work showed that sentiment is often depicted in an implicit way in this domain. We created financial-specific continuous word representations in order to obtain domain specific syntactic and semantic relations between words. We combined traditional bag-of-words and lexicalbased features with bag-of-embeddings to train a regressor of both sentiment polarity and intensity. Results show that different combination of features attained different performances on each sub-task. Future work will consist on collecting larger external datasets for training financial word embeddings of both microblogs and news headlines. We also have planned to perform the regression analysis using Deep Neural Networks. 5.3 Summary of the Contributions In this chapter we present some contributions to two fundamental Text Mining problems in ORM. • A supervised learning approach for Entity Filtering on tweets, achieving state-ofthe-art performance using a relatively small training set. • Created and made available word embeddings trained from financial texts. • A supervised learning approach for fine-grained sentiment analysis of financial texts. Chapter 6 Text-based Entity-centric Prediction In this chapter we explore the predictive power of entity-centric information in online news and social media in the context of ORM. We address two different predictive tasks. The first is concerned with predicting entity popularity on Twitter based on signals extracted from the news cycle. We aim to study different sets of signals extracted from online news mentioning specific entities that could influence or at least are correlated with future popularity of those entities on Twitter. We know that entity popularity on social media can be influenced by several factors but we are only interested in exploring the interplay between online news and social media for entities that are frequently mentioned on the news cycle such as politicians or footballers. This could be particularly interesting for anticipating public relations damage control once a polemic news article is published. Or even for editorial purposes to maximize buzz on social media. The second predictive task consists in using entity-centric sentiment polarity extracted from tweets to predict political polls. There has been several research work trying to assess the predictive power of social media to predict the outcome of political opinion surveys or elections. However, each study proposes its own method of aggregating polarity scores over time, however, there is not a consensus on which sentiment aggregate function is the most adequate for this problem. We propose to use and contrast several sentiment aggregate functions reported in the literature, by assessing their predictive power on a specific case comprising data collected during the Portuguese bailout (2011-2013). Text-based Entity-centric Prediction 96 6.1 Exploring Online News for Reputation Monitoring on Twitter 1 Online publication of news articles has become a standard behavior of news outlets, while the public joined the movement either using desktop or mobile terminals. The resulting setup consists of a cooperative dialog between news outlets and the public at large. Latest events are covered and commented by both parties in a continuous basis through the social media, such as Twitter. When sharing or commenting news on social media, users tend to mention the most predominant entities mentioned in the news story. Therefore, entities, such as public figures, organizations, companies or geographic locations, can act as latent connections between online news and social media. Online Reputation Monitoring (ORM) focuses on continuously tracking what is being said about entities on social media and online news. Automatic collection and processing of comments and opinions on social media is now crucial to understand the reputation of individuals and organizations and therefore to manage their public relations. However, ORM systems would be even more useful if they would be able to know in advance if social media users will talk a lot about the target entities or not. We hypothesize that for entities that are frequently mentioned on the news (e.g. politicians) it is possible to establish a predictive link between online news and popularity on social media. We cast the problem as a supervised learning classification approach: to decide whether popularity will be high or low based on features extracted from the news cycle. We define four set of features: signal, textual, sentiment and semantic. We aim to respond to the following research questions: • Is online news a valuable source of information to effectively predict entity popularity on Twitter? • Do online news carry different predictive power based on the nature of the entity under study? • How do different thresholds for defining high and low popularity affect the effectiveness of our approach? • Does the performance remain stable for different prediction times? • What is the most important feature set for predicting entity popularity on Twitter based on the news cycle? 1 The material contained in this section was published in P. Saleiro and C. Soares, “Learning from the News: Predicting Entity Popularity on Twitter” [19] 6.1 Exploring Online News for Reputation Monitoring on Twitter 97 • Do individual sets of features exhibit different importance for different entities? 6.1.1 Approach The starting point of our hypothesis is that for entities that are frequently mentioned on the news (e.g. politicians) it is possible to predict popularity on social media using signals extracted from the news cycle. The first step towards a solution requires the definition of entity popularity on social media. Entity Popularity There are different ways of expressing the notion of popularity on social media. For example, the classical way of defining it is through the number of followers of a Twitter account or the number of likes in a Facebook page. Another notion of popularity, associated with entities, consists on the number of retweets or replies on Twitter and post likes and comments on Facebook. We define entity popularity based on named entity mentions in social media messages. Mentions consist of specific surface forms of an entity name. For example, “Cristiano Ronaldo” might be mentioned also using just “Ronaldo” or “#CR7”. Given an set of entities E = {e1 , e2 , ..., ei , ...}, a daily stream of social media messages S = {s1 , s2 , ..., si , ...} and a daily stream of online news articles N = {n1 , n2 , ..., ni , ...} we are interested in monitoring the mentions of an entity ei on the social media stream St , i.e. the discrete function fm (ei , St ). Let T be a daily time frame T = [tp , tp+h ], where the time tp is the time of prediction and tp+h is the prediction horizon time. We want to learn a target popularity function fp on social media stream S as a function of the given entity ei , the online news stream N and the time frame T : t=tp+h fp (ei , N, T ) = X fm (ei , St ) t=tp which corresponds to integrating fm (ei , S) over T . Given a day di , a time of prediction tp , we extract features from the news stream N until tp and predict fp until the prediction horizon tp + h. We measure popularity on a daily basis, and consequently, we adopted tp+h as 23:59:59 everyday. For example, if tp equals to 8 a.m, we extract features from N until 07:59:59 and predict fp in the interval 08:00 - 23:59:59 on day di . In the case of tp equals to midnight, we extract Text-based Entity-centric Prediction 98 features from N on the 24 hours of previous day di−1 to predict fp for the 24 hours of di . We cast the prediction of fp (ei , N, T ) as a supervised learning classification problem, in which we want to infer the target variable fˆp (ei , N, T ) ∈ {0, 1} defined as: fˆp =   0(low), if P (fp (ei , N, T ) ≤ δ) = k  1(high), if P (fp (ei , N, T ) > δ) = 1 − k where δ is the inverse of cumulative distribution function at k of fp (ei , N, T ) as measured in the training set, a similar approach to Tsagkias et al. [119]. For instance, k = 0.5 corresponds to the median of fp (ei , N, T ) in the training set and higher values of k mean that fp (ei , N, T ) has to be higher than k examples on the training set to consider fˆp = 1, resulting in a reduced number of training examples of the positive class high. News Features Previous work has focused on the influence of characteristics of the social media stream S in the adoption and popularity of memes and hashtags [126]. In contrast, the main goal of this work is to investigate the predictive power of the online news stream N . Therefore we extract four types of features from N which we label: (i) signal, (ii) textual, (iii) sentiment and (iv) semantic, as depicted in Table 6.1. One important issue is how can we filter relevant news items to ei . There is no consensus on how to link a news stream N with a social media stream S. Some works use URLs from N , shared on S, to filter simultaneously relevant news articles and social media messages [117]. As our work is entity oriented, we select news articles with mentions of ei as our relevant N . Signal Features - This type of features depict the “signal” of the news cycle mentioning ei and we include a set of counting variables as features, focusing on the total number of news mentioning ei in specific time intervals, mentions on news titles, the average length of news articles, the different number of news outlets that published news mentioning ei as well as, features specific to the day of the week to capture any seasonal trend on the popularity. The idea is to capture the dynamics of news events, for instance, if ei has a sudden peak of mentions on N , a relevant event might have happened which may influence fp . Textual features - To collect textual features we build a daily profile of the news cycle by aggregating all titles of online news articles mentioning ei for the daily time frame [0, tp ] in di . We select the top 10,000 most frequent terms (unigrams and bi-grams) in 6.1 Exploring Online News for Reputation Monitoring on Twitter 99 the training set and create a document-term matrix R. Two distinct methods were applied to capture textual features. The first method is to apply TF-IDF weighting to R. We employ Singular Value Decomposition (SVD) to capture similarity between terms and reduce dimensionality. It computes a low-dimensional linear approximation σ. The final set of features for training and testing is the TF-IDF weighted term-document matrix R combined with σR which produces 10 real valued latent features. When testing, the system uses the same 10,000 terms from the training data and calculates TF-IDF using the IDF from the training data, as well as, σ for applying SVD on test data. The second method consists in applying Latent Dirichlet allocation (LDA) to generate a topic model of 10 topics (features). The system learns a topic-document distribution θ and a word distribution over topics φ using the training data for a given entity ei . When testing, the system extracts the word distribution of the news title vector r on a test day d′i . Then, by using φ learned on training data, it calculates the probability of r belonging to one of the 10 topics learned before. The objective of extracting this set of features is to create a characterization of the news stream that mentions ei , namely, which are the most salient terms and phrases on each day di as well as the latent topics associated with ei . By learning our classifier we hope to obtain correlations between certain terms and topics and fp . Sentiment features - We include several types of word level sentiment features. The assumption here is that subjective words on the news will result in more reactions on social media, as exposed in [187]. Once again we extract features from the titles of news mentioning ei for the daily time frame [0, tp ]. We use a sentiment lexicon as SentiWordNet to extract subjective terms from the titles daily profile and label them as positive, neutral or negative polarity. We compute count features for number of positive, negative, neutral terms as well as difference and ratio of positive and negatives terms. Similar to textual features we create a TFIDF weighted term-document matrix R using the subjective terms from the title and apply SVD to compute 10 real valued sentiment latent features. Semantic features - We use the number of different named entities recognized in N on day di until tp, as well as, the number of distinct news category tags extracted from the news feeds metadata. These tags, common in news articles, consist of author annotated terms and phrases that describe a sort of semantic hierarchy of news categories, topics and news stories (e.g. “european debt crisis”). We create a TF-IDF weighted entity-document and TF-IDF tag-document matrices and applied SVD to each of them to reduce dimensionality to 10. The idea is to capture interesting entity Text-based Entity-centric Prediction 100 Table 6.1 Summary of the four type of features we consider. Number Signal 1 2 3 4 5 6 7 8 Textual 9-18 19-28 Sentiment 29 30 31 32 33 34 35-44 Semantic 45 46 47-56 57-66 Feature Description news news di−1 news total di−1 news titles avg content sources weekday is weekend number of news mentions of ei in [0, tp ] in di number of news mentions of ei in [0, tp ] in di−1 number of news mentions of ei in [0, 24[ in di−1 number of title mentions in news of ei in [0, tp ] in di average content length of news of ei in [0, tp ] in di number of different news sources of ei in [0, tp ] in di day of week true if weekend, false otherwise tfidf titles LDA titles TF-IDF of news titles [0, tp ] in di LDA-10 of news titles [0, tp ] in di pos neg neu ratio diff subjectivity tfidf subj number of positive words in news titles [0, tp ] in di number of negative words in news titles [0, tp ] in di number of neutral words in news titles [0, tp ] in di positive/negative positive − negative P (positive + negative + neutral)/ words TF-IDF of subjective words (pos, neg and neu) entities tags tfidf entities tfidf tags number of entities in news [0, tp ] in di number of tags in news [0, tp ] in di TF-IDF of entities in news [0, tp ] in di TF-IDF of news tags [0, tp ] in di co-occurrences as well as, news stories that are less transient in time and might be able to trigger popularity on Twitter. Learning Framework Let x be the feature vector extracted from the online news stream N on day di until tp. We want to learn the probability P (fˆp = 1\|X = x). This can be done using the inner product between x and a weighting parameter vector w ∈ R, w⊤ x. Using logistic regression and for binary classification one can unify the definition of ˆ p(fp = 1\|x) and p(fˆp = 0\|x) with 6.1 Exploring Online News for Reputation Monitoring on Twitter p(fˆp \|x) = 101 1 1+ e−fˆp w⊤ x Given a set of z instance-label pairs (xi ,fˆp i ), with i = 1, ..., z and fˆp i ∈ {0, 1} we solve the binary class L2 penalized logistic regression optimization problem, where C>0 n X 1 ˆ ⊤ min w⊤ w + C log(1 + e−fp i w xi ) w 2 i=1 We apply this approach following an entity specific basis, i.e. we train an individual model for each entity. Given a set of entities E to which we want to apply our approach and a training set of example days D = {d1 , d2 , ..., di , ...}, we extract a feature vector xi for each entity ei on each training day di . Therefore, we are able to learn a model of w for each ei . The assumption is that popularity on social media fp is dependent of the entity ei and consequently we extract entity specific features from the news stream N . For instance, the top 10,000 words of the news titles mentioning ei are not the same for ej . 6.1.2 Experimental Setup This work uses Portuguese news feeds and tweets collected from January 1, 2013 to January 1, 2016, consisting of over 150 million tweets and 5 million online news articles2 . To collect and process raw Twitter data, we use a crawler, which recognizes and disambiguates named entities on Twitter [188]. News data is provided by a Portuguese online news aggregator3 . This service handles online news from over 60 Portuguese news outlets and it is able to recognize entities mentioned on the news. We choose the two most common news categories: politics and football and select the 3 entities with highest number of mentions on the news for both categories. The politicians are two former Prime-ministers, José Sócrates and Pedro Passos Coelho and the incumbent, António Costa. The football entities are two coaches, Jorge Jesus and José Mourinho, and the most famous Portuguese football player, Cristiano Ronaldo. Figure 7.5 depicts the behavior of daily popularity of the six entities on the selected community stream of Twitter users for each day from July 2014 until July 2015. As expected, it is easily observable that in some days the popularity on Twitter exhibits 2 3 Dataset is available for research purposes. Access requests via e-mail. http://www.sapo.pt Text-based Entity-centric Prediction 102 6000 5000 Pedro Passos Coelho José Sócrates António Costa Jorge Jesus Cristiano Ronaldo José Mourinho 4000 3000 2000 1000 0 Aug Sep Oct Nov Dec Jan 2015 Feb Mar Apr May Jun Fig. 6.1 Daily popularity on Twitter of entities under study. Training Iteration 1 Jan 2013 Feb 2013 Test ... Dec 2014 Jan 2015 Training Iteration 2 Jan 2013 Feb 2013 Mar 2013 Feb 2015 Test ... Jan 2015 Feb 2015 Fig. 6.2 Training and testing sliding window - first 2 iterations. bursty patterns. For instance, when José Sócrates was arrested in November 21st 2014 or when Cristiano Ronaldo won the FIFA Ballon d’Or in January 12th 2015. We defined the years of 2013 and 2014 as training set and the whole year of 2015 as test set. We applied a monthly sliding window setting in which we start by predicting entity popularity for every day of January 2015 (i.e. the test set) using a model trained on the previous 24 months, 730 days (i.e. the training set). Then, we use February 2015 as the test set, using a new model trained on the previous 24 months. Then March and so on, as depicted in Figure 6.2. We perform this evaluation process, rolling the training and test set until December 2015, resulting in 365 days under evaluation. 6.1 Exploring Online News for Reputation Monitoring on Twitter 103 The process is applied for each one of the six entities, for different time of predictions tp and for different values of the decision boundary k. We test tp = 0, 4, 8, 12, 16, 20 and k = 0.5, 0.65, 0.8. Therefore, we report results in Section 6.2.4 for 18 different experimental settings, for each one of the six entities. The goal is to understand how useful the news cycle is for predicting entity popularity on Twitter for different entities, at different hours of the 24 hours cycle and with different thresholds for considering popularity as high or low. 6.1.3 Results and Discussion Results are depicted in Table 6.2. We report F1 on positive class since in online reputation monitoring is more valuable to be able to predict high popularity than low. Nevertheless, we also calculated overall Accuracy results, which were better than the F1 reported here. Consequently, this means that our system is fairly capable of predicting low popularity. We organize this section based on the research questions we presented in the beginning of this section. Is online news valuable as source of information to effectively predict entity popularity on Twitter? Do online news carry different predictive power based on the nature of the entity under study? Results show that performance varies with each target entity ei . In general, results are better in the case of predicting popularity of politicians. In the case of football public figures, Jorge Jesus exhibits similar results with the three politicians but José Mourinho and especially Cristiano Ronaldo represent the worst results in our setting. For instance, when Cristiano Ronaldo scores three goals in a match, the burst on popularity is almost immediate and not possible to predict in advance. Further analysis showed that online news failed to be informative of popularity in the case of live events covered by other media, such as TV. Interviews and debates on one hand, and live football games on the other, consist of events with unpredictable effects on popularity. Cristiano Ronaldo can be considered a special case in our experiments. He is by far the most famous entity in our experiments and in addition, he is also an active Twitter user with more than 40M followers. This work focus on assessing the predictive power of online news and its limitations. We assume that for Cristiano Ronaldo, endogenous features from the Twitter itself would be necessary to obtain better results. Text-based Entity-centric Prediction 104 Table 6.2 F1 score of popularity high as function of tp and k equal to 0.5, 0.65 and 0.8 respectively. Entity \tp (hour) k = 0.50 António Costa José Sócrates Pedro Passos Coelho Cristiano Ronaldo Jorge Jesus José Mourinho k = 0.65 António Costa José Sócrates Pedro Passos Coelho Cristiano Ronaldo Jorge Jesus José Mourinho k = 0.80 António Costa José Sócrates Pedro Passos Coelho Cristiano Ronaldo Jorge Jesus José Mourinho 0 4 8 12 16 20 0,76 0,77 0,72 0,35 0,73 0,62 0,67 0,66 0,63 0,41 0,68 0,46 0,74 0,73 0,70 0,45 0,69 0,51 0,77 0,75 0,70 0,37 0,68 0,56 0,75 0,75 0,74 0,35 0,69 0,55 0,72 0,75 0,71 0,32 0,70 0,45 0,61 0,63 0,58 0,29 0,63 0,56 0,60 0,57 0,57 0,35 0,61 0,39 0,66 0,62 0,65 0,42 0,63 0,48 0,64 0,66 0,67 0,41 0,59 0,56 0,60 0,64 0,67 0,36 0,62 0,47 0,60 0,62 0,65 0,30 0,64 0,38 0,48 0,48 0,47 0,14 0,50 0,32 0,51 0,42 0,46 0,29 0,48 0,32 0,55 0,47 0,56 0,31 0,51 0,36 0,53 0,53 0,56 0,26 0,48 0,41 0,44 0,47 0,52 0,20 0,57 0,41 0,49 0,35 0,54 0,21 0,56 0,36 How do different thresholds for defining high and low popularity affect the effectiveness of our approach? Our system exhibits top performance with k = 0.5, which corresponds to balanced training sets, with the same number of high and low popularity examples on each training set. Political entities exhibit F1 scores above 0.70 with k = 0.5. On the other hand, as we increase k, performance deteriorates. We observe that for k = 0.8, the system predicts a very high number of false positives. It is very difficult to predict extreme values of popularity on social media before they happen. We plan to tackle this problem in the future by also including features about the target variable in the current and previous hours, i.e., time-series auto-regressive components. Does performance remain stable for different time of predictions? Results show that time of prediction affects the performance of the system, specially for the political entities. In their case, F1 is higher when time of prediction is noon 6.1 Exploring Online News for Reputation Monitoring on Twitter 105 Fig. 6.3 Individual feature type F1 score for tp = 12 at k = 0.5. and 4 p.m. which is an evidence that in politics, most of the news events that trigger popularity on social media are broadcast by news outlets in the morning. It is very interesting to compare results for midnight and 4 a.m./8 a.m. The former use the news articles from the previous day, as explained in Section 6.1.1, while the latter use news articles from the first 4/8 hours of the day under prediction. In some examples, Twitter popularity was triggered by events depicted on the news from the previous day and not from the current day. What is the most important feature set for predicting entity popularity on Twitter based on the news cycle? Do individual set of features exhibit different importance for different entities? Figure 6.3 tries to answer these two questions. The first observation is that the combination of all groups of features does not lead to substantial improvements. Semantic features alone achieve almost the same F1 score as the combination of all features. However in the case of Mourinho and Ronaldo, the combination of all features lead to worse F1 results than the semantic set alone. Text-based Entity-centric Prediction 106 Sentiment features are the second most important for all entities except José Mourinho. Signal and Textual features are less important and this was somehow a surprise. Signal features represent the surface behavior of news articles, such as the volume of news mentions of ei before tp and we were expecting an higher importance. Regarding Textual features, we believe that news articles often refer to terms and phrases that explain past events in order to contextualize a news article. In future work, we consider alternative approaches for predicting future popularity of entities that do not occur everyday on the news, but do have social media public accounts, such as musicians or actors. In opposition, entities that occur often on the news, such as economics ministers and the like, but do not often occur in the social media pose also a different problem. 6.2 Predicting Political Polls using Twitter Sentiment4 Surveys and polls using the telephone are widely used to provide information of what people think about parties or political entities [150]. Surveys randomly select the electorate sample, avoiding selection bias, and are designed to collect the perception of a population regarding some subject, such as in politics or marketing. However this method is expensive and time consuming [150]. Furthermore, over the years it is becoming more difficult to contact people and persuade them to participate in these surveys [189]. On the other hand, the rise of social media, namely Twitter and Facebook, has changed the way people interact with news. This way, people are able to react and comment any news in real time [135]. One challenge that several research works have been trying to solve is to understand how opinions expressed on social media, and their sentiment, can be a leading indicator of public opinion. However, at the same time there might exist simultaneously positive, negative and neutral opinions regarding the same subject. Thus, we need to obtain a value that reflects the general image of each political target in social media, for a given time period. To that end, we use sentiment aggregate functions. In summary, a sentiment aggregate function calculates a global value based on the number of positive, negative, and neutral mentions of each political target, in a given period. We conducted an exhaustive study and collected and implemented several sentiment aggregate functions from the state of the art [135–143]. 4 The material contained in this section was published in P. Saleiro, L. Gomes, C. Soares, “Sentiment Aggregate Functions for Political Opinion Polling using Microblog Streams” [21] 6.2 Predicting Political Polls using Twitter Sentiment 107 Thus, the main objective of our work is to study and define a methodology capable of successfully estimating the poll results, based on opinions expressed on social media, represented by sentiment aggregators. We applied this problem to the Portuguese bailout case study, using Tweets from a sample of the Portuguese Tweetosphere and Portuguese polls as gold standard. Given the monthly periodicity of polls, we needed to aggregate the data by month. This approach allows each aggregate value to represent the monthly sentiment for each political party. Due to the absence of a general sentiment aggregate function suitable for different case studies, we decided to include all aggregate functions as features of the regression model. Therefore the learning algorithm is able to adapt to the most informative aggregate functions through time. 6.2.1 Methodology To collect and process raw Twitter data, we use an online reputation monitoring platform [36] which can be extended by researchers interested in tracking political opinion on the web. It collects tweets from a predefined sample of users, applies named entity disambiguation [177] and generates indicators of both frequency of mention and polarity (positivity/negativity) [190] of mentions of entities over time. In our case, tweets are collected from the stream of 100 thousand different users, representing a sample of the Portuguese community on Twitter. This sample was obtained by expanding a manually annotated seed set of 1000 users using heuristics such as, as language of posts, language of followers posts or geo-location [188]. The platform automatically classifies each tweet according to its sentiment polarity. If a message expresses a positive, negative or neutral opinion regarding an entity (e.g. politicians), it is classified as positive, negative or neutral mention, respectively. The sentiment classifier uses a corpus of 1500 annotated tweets as training set and it has achieved an accuracy over 80% using 10-fold cross validation. These 1500 tweets were manually annotated by 3 political science students. Mentions of entities and respective polarity are aggregated by counting positive, negative, neutral and total mentions for each entity in a given period. Sentiment aggregate functions use these cumulative numbers as input to generate a new value for each specific time period. Since we want to use sentiment aggregate functions as features of a regression model to produce an estimate of the political opinion, we decided to use traditional poll results as gold standard. Text-based Entity-centric Prediction 108 Sentiment Aggregate Functions Let Mei be a mention on Twitter of an entity ei , then Me+i , Me∗i and Me−i are positive, neutral and negative classified mentions of entity ei on Twitter. Therefore, given a time frame T (e.g. a month), sentiment aggregate functions applied to the aggregated data between polls are the following: P • entitybuzz: T Mei , the sum of the number of mentions (buzz) of a given entity in the time frame T . • entitypositives: T Me+i , sum of the positively classified mentions of a given entity in the time frame T . P • entityneutrals: T Me∗i , the sum of the neutral classified mentions of a given entity in a time frame T . P • entitynegatives: T Me−i , the sum of the negatively classified mentions of a given entity in a time frame T . P M + +M − P ei T P ei • entitysubjectivity: , the ratio of positive and negative classified M e i T mentions of entity ei over its buzz in a time frame T . M+ P • entitypolarity: PT Me−i , the ratio of positive over negative classified mentions in ei T a time frame T . P M− ei T • berminghamsovn: P P , the ratio of the negative classified mentions of Me−i T E entity ei over the total number of negative mentions of all entities in time frame T. P Me+i +1 T P P log10 Me− +1 - bermingham [135]: T - berminghamsovp [135]: T E Me+i P M+ PT e−i - connor [191]: T P Mei Me−j Ej̸=i + Mei +Me−i E Me+i + P T P P - gayo [141]: T - polarity: E P Me+i T P P P T Me+i − - polarityON eutral: P Me−i P Me+i − T Me−i P Me0i T T T P i 6.2 Predicting Political Polls using Twitter Sentiment P - polarityOT otal: P P T P - subjOT otal: - subjN euv: T M ei Me+i + T Me−i P M ei T P T P 109 Me−i T Me+i − T Me+i + T Me−i P Me0i T P P + P − E ei ei M + M - subjSoV : PT Pei M +T+Me−i T - subjV ol: P T - share [135]: Me+i + Me−i P M ei T P P T E M ei - shareOf N egDistribution: P − M P T ei Mei T P P Me− T i P E T in the poll , where n is the number of political entities M ei + P M - normalized_positive: PT Meei i T - normalized_negative: P M− PT e−i T - normalized_neutral: Mei P M0 PT ei T M ei - normalized_bermingham: log10 - normalized_connor: normalized_positives+1 normalized_negatives+1 normalized_positives normalized_negatives - normalized_gayo: normalized_positives+normalized_others_negatives normalized_total_positives+normalized_total_negatives - normalized_polarity: normalized_positives − normalized_negatives The sentiment aggregate functions are used as features in the regression models. Text-based Entity-centric Prediction 110 Fig. 6.4 Negatives share (berminghamsovn) of political leaders in Twitter. 6.2.2 Data The data used in this work consists of tweets mentioning Portuguese political party leaders and polls from August 2011 to December 2013. This period corresponds to the Portuguese bailout when several austerity measures were adopted by the incumbent right wing governmental coalition of the PSD and CDS parties. Twitter Table 6.3 Distribution of positive, negative and neutral mentions per political party PSD PS CDS CDU BE Negative 69 723 28 660 41 935 2 445 9 603 Positive 121 225 51 79 306 Neutral 37 133 15 326 17 554 5 604 4 214 Total Mentions 106 977 44 211 59 540 8 128 14 123 The Twitter data set contains 232,979 classified messages, collected from a network of 100 thousand different users classified as Portuguese. Table 6.3 presents the distribution of positive, negative, and neutral mentions of the political leaders of the 5 most voted political parties in Portugal (PSD, PS, CDS, PCP and BE). The negative mentions represent the majority of the total mentions, except for CDU where the number of negative mentions is smaller than the neutral ones. The positive mentions represent less than 1% of the total mentions of each party, except for BE where they represent 6.2 Predicting Political Polls using Twitter Sentiment 111 2% of the total mentions. The most mentioned parties are PS, PSD and CDS. The total mentions of these three parties represent 90% of the data sample total mentions. Figure 6.4 depicts the time series of the berminghamsovn (negatives share) sentiment aggregate function. The higher the value of the function the higher is the percentage of negative tweets mention a given political entity in comparison with the other entities. As expected, Pedro Passos Coelho (PSD) as prime-minister is the leader with the higher score throughout the whole time period under study. Paulo Portas (CDS) leader of the other party of the coalition, and also member of the government is the second most negatively mentioned in the period, while António José Seguro (PS) is in some periods the second higher. PSD and CDS are the incumbent parties while PS is the main opposition party in the time frame under study. PSD and CDS as government parties were raising taxes and cutting salaries. PS was the incumbent government during the years that led to the bailout and a fraction of the population considered responsible for the financial crisis. The bailout and the consequent austerity measures could explain the overwhelming percentage of negative mentions although we verified that in other time periods the high percentage of negatives mentions remains. We can say that Twitter users of this sample when mentioning political leaders on their tweets tend to criticize them. Political Opinion Polls The polling was performed by Eurosondagem, a Portuguese private company which collects public opinion. This data set contains the monthly polls results of the five main Portuguese parties, from June 2011 to December 2013. Figure 6.5 represents the evolution of Portuguese polls results. We can see two main party groups: The first group, where both PSD and PS are included, has a higher value of vote intention (above 23%). PSD despite starting as the preferred party in vote intention, has a downtrend along the time, losing the leadership for PS in September 2012. On the other hand, PS has in general an uptrend. The second group, composed by CDS, PCP and BE, has a vote intention range from 5% to 15%. While CDS has a downtrend in public opinion, PCP has an ascendant one. Although the constant tendencies (up and down trends), we noticed that the maximum variation observed between two consecutive months is 3%. In June 2013 there was political crises in the government when CDS threaten to leave the government coalition due to the austerity measures being implemented and corresponds to the moment when PS takes the lead in the polls. Text-based Entity-centric Prediction 112 Fig. 6.5 Representation of the monthly poll results of each political candidate 6.2.3 Experimental Setup We defined the period of 2011 to December 2012 as training set and the whole year of 2013 as test set. We applied a sliding window setting in which we predict the poll results of a given month using the previous 16 months as training set: • Training set – containing the monthly values of the aggregators (both sentiment and buzz aggregator) for 16 months prior the month intended to be predicted. • Test set - containing the values of the aggregators (both sentiment and buzz aggregator) of the month intended to be predicted. We start by predicting the poll results of January 2013 using the previous 16 months as training set: 1. We select the values of the aggregators of the 16 months prior January 2013 (September 2011 to December 2012). 2. We use that data to train our regression model. 3. Then we input the aggregators’ values of January 2013 - the first record of the test set - in the the trained model, to obtain the poll results prediction. 6.2 Predicting Political Polls using Twitter Sentiment 113 4. We select the next month of the test set and repeat the process until all months are predicted. The models are created using two regression algorithms: a linear regression algorithm (Ordinary Least Squares - OLS) and a non-linear regression algorithm (Random Forests - RF). We also run an experiment using the derivative of the polls time series as gold standard, i.e., poll results variations from poll to poll. Thus, we also calculate the variations of the aggregate functions from month to month as features. Furthermore, we repeat each experiment including and excluding the lagged self of the polls, i.e., the last result of the poll for a given candidate (yt−1 ) or the last polls result variation (∆yt−1 ) when predicting polls variations. We use Mean Absolute Error (MAE) as evaluation measure, to determine the absolute error of each prediction. Then, we calculate the average of the twelve MAE’s so we could know the global prediction error of our model. Pn \|fi − yi \| (6.1) n n is the number of forecasts, fi is the model’s forecast and yi the real outcome. M AE = 6.2.4 i=1 Results and Discussion In this section we explain in detail the experiments and their results. We perform two different experiments: (1) using absolute values and (2) using monthly variations. Predicting Polls Results In this experiment, the sentiment aggregators take absolute values in order to predict the absolute values of polls results. Mathematically speaking, this experiment can be seen as: y ← {yt−1 , buzzAggregators, sentimentAggregators}. In Figure 6.6 we see the global errors we obtained. The results show that we obtain a MAE for the 5 parties poll results over 12 months of 6.55% using Ordinary Least Squares and 3.1% using Random Forests. The lagged self of the polls, i.e., assuming the last known poll result as prediction results in a MAE of 0.61 which was expectable since the polls exhibit slight changes from month to month. This experiment shows that the inclusion of the lagged self (yt−1 ) produces average errors similar to the lagged self. Text-based Entity-centric Prediction 114 Fig. 6.6 Error predictions for polls results. Fig. 6.7 Error predictions for polls results variation. Predicting Polls Results Variation According to our exploratory data analysis, the polls results have a small variation between two consecutive months. Thus, instead of predicting the absolute value of poll results, we tried to predict the variation, ∆y ← {∆(yt−1 ), ∆buzzAggregators, ∆sentimentAggregators} In this particular experiment, the inclusion of the ∆yt−1 as feature in the regression model has not a determinant role (Figure 6.7). Including that feature we could not obtain lower MAE than excluding it. It means that the real monthly poll variation is not constant over the year. In general, using a non-linear regression algorithm we obtain lower MAE. The results show that when leading with polls results with slight 6.2 Predicting Political Polls using Twitter Sentiment 115 Fig. 6.8 Mean absolute error buzz vs sentiment. changes from poll to poll it makes sense to transform the dataset by taking differences between consecutive time-steps. Buzz and Sentiment Several studies state that the buzz has predictive power and reflects correctly the public opinion on social media. Following that premise, we trained our models with buzz and sentiment aggregators separately to predict polls variations: • ∆y ← {∆(yt−1 ), ∆buzzAggregators} • ∆y ← {∆(yt−1 ), ∆sentimentAggregators} This experiment allowed us to compare the behavior of buzz and sentiment aggregators. According to Figure 6.8, buzz and sentiment aggregators have similar results. Although the OLS algorithm combined only with buzz aggregators has a slightly lower error than the other models, it is not a significant improvement. These results also show that Random Forests algorithm performs the best when combined only with sentiment aggregators. Feature Selection One of the main goals of our work is to understand which aggregator (or group of aggregators) better suits our case study. According to the previous experiments, we can achieve lower prediction errors when training our model with buzz and sentiment aggregators separately. However, when training our model with these two kinds of aggregators separately, we are implicitly performing feature selection. We only have Text-based Entity-centric Prediction 116 two buzz features (share and total_mentions). Due to that small amount of features, it was not necessary to perform any feature selection technique within buzz features. Thus, we decided to apply a feature selection technique to the sentiment aggregators, in order to select the most informative ones to predict the monthly polls results variation. We use univariate feature selection, selecting 10% of the sentiment features (total of 3 features). Using this technique, the Random Forests’ global error rose from 0.65 to 0.73. However, OLS presents an MAE drop from 0.72 to 0.67. Another important fact to notice is that if we perform univariate feature selection to all aggregators (buzz and sentiment), we will achieve the same MAE value that when applied only to sentiment aggregators. It means that buzz aggregators are discarded by the feature selection technique. We try a different approach and perform a recursive feature elimination technique. In this technique, features are eliminated recursively according to a initial score given by the external estimator. This method allows us to determine the number of features to select. Thus, also selecting 3 features, the OLS’ MAE drops to 0.63. Once again, none of the buzz features were selected. Furthermore, both feature selection techniques select different features for each monthly prediction. 6.2.5 Feature Importance We select the Random Forest model of monthly variations to study the features importance as depicted in Figure 6.9. The higher the score, the more important the feature is. The importance of a feature is computed as the (normalized) total reduction of the criterion brought by that feature. It is also known as the Gini importance. Values correspond to the average of the Gini importance over the different models trained in the experiments. The single most important feature is the bermingham aggregate function, followed by neutrals. It is important to notice that when combining all the aggregate functions as features in a single regression model, the buzz does not comprise a high Gini importance, even though when used as a single feature it produces similar results to the sentiment aggregate functions. In general, the standard deviation of the Gini importance is relatively high. This has to do with our experimental setup, as the values depicted in the bar chart correspond to the average of the Gini importance over 12 different models (12 months of testing set). Therefore, feature importances vary over time while the MAE tends to remain unchanged. We can say that different features have different informative value over time and consequently it is useful to combine all the sentiment aggregation functions as features of the regression models over time. 6.2 Predicting Political Polls using Twitter Sentiment 117 Fig. 6.9 Aggregate functions importance in the Random Forests models. 6.2.6 Outlook We studied a large set of sentiment aggregate functions to use as features in a regression model to predict political opinion poll results. The results show that we can estimate the polls results with low prediction error, using sentiment and buzz aggregators based on the opinions expressed on social media. We introduced a strong baseline for comparison, the lagged self of the polls. In our study, we built a model where we achieve the lowest MAE using the linear algorithm (OLS), combined only with buzz aggregators, using monthly variations. The model has an MAE of 0.63%. We performed two feature selection techniques: (1) univariate feature selection and (2) recursive feature elimination. Applying the recursive technique to the sentiment features, we can achieve an MAE of 0.63, matching our best model. Furthermore, the chosen features are not the same in every prediction. Regarding feature importance analysis 118 Text-based Entity-centric Prediction our experiments showed that bermingham aggregate function represents the highest Gini importance in the Random Forests model. 6.3 Summary of the Contributions In this chapter we presented research work about entity-centric text-based prediction for ORM, making the following contributions: • Analysis of the predictive power of online news regarding entity popularity on Twitter for entities that are frequently mentioned on the news. • Analysis of how to combine different sentiment aggregate functions to serve as features for predicting political polls. Chapter 7 A Framework for Online Reputation Monitoring In this chapter, we present a framework that puts together all the building blocks required to perform ORM. The framework is divided in two distinct components, one is dedicated to Entity Retrieval and the other to Text Mining. In practice these two components can act as two separate frameworks. Both are adaptable and can be reused in different application scenarios, from computational journalism to finance or politics. We start with a framework overview description and then we focus specifically on each of the two components. The first component is RELink, a research framework for E-R retrieval. We carried the experiments on E-R retrieval, described in Chapter 4 using RELink. Furthermore, since we did not have access to training data based on news articles, we describe a case study of using RELink for entity retrieval from a large news collection. We then describe the TexRep framework which is responsible for Text Mining related tasks for ORM, such as Entity Filtering, Sentiment Analysis or Predictive tasks. The experiments described in both Chapter 5 and Chapter 6 were carried out using TexRep. We also provide further detail how TexRep was used as backend of the POPSTAR project. Finally, we perform an independent study of practical aspects of general purpose word embeddings from the Twitter stream to serve as resource for future users of TexRep. 7.1 Framework Overview The framework provides Entity Retrieval and Text Mining functionalities that enable the collection, disambiguation, retrieval of entities and relationships, sentiment analysis, data aggregation, prediction and visualization of entity-centric information from A Framework for Online Reputation Monitoring 120 heterogeneous Web data sources. Furthermore, given that both components are built using modular architectures providing abstraction layers and well defined interfaces, new functionalities or methods can be easily integrated. The framework is divided in two components: RELink and TexRep. Both can work as independently dedicated frameworks using specific data sources or can be put together in a unifying setup for ORM. As depicted in Figure 7.1, when working together, RELink and TexRep are connected through the Entity Occurrences Warehouse. This is the central module of our framework for ORM. The Entity Occurrences Warehouse contains extractions from occurrences of the entities of interest across the Web data sources. RELink (Entity Retrieval) ENTITY OCCURRENCES WAREHOUSE TexRep (Text Mining) Fig. 7.1 High-level overview on the ORM framework. The data flow starts with TexRep collecting data from Web text data sources, extraction of text passages containing entity mentions and disambiguation. Entitycentric text passages are then stored in the Entity Occurrences Warehouse. This data can then be used for E-R retrieval indexing using RELink or for downstream Text Mining tasks (e.g. Sentiment Analysis) using other modules of TexRep. We now describe RELink and TexRep architectures and internal data flow. 7.1.1 RELink The RELink framework is designed to facilitate experiments with E-R Retrieval query collections. The formulation of E-R queries in natural language and relational format (QEi−1 , QRi−1,i , QEi ) provide opportunities to define and explore a range of query formulations and search algorithms. Although, RELink provides support for Late Fusion design patterns, it is mostly tailored for Early Fusion approaches where it is necessary to create entity and relationship representations at indexing time. A typical Early Fusion E-R retrieval experimental setup would involve search over a free-text collection to extract relevant instances of entity tuples and then verify their correctness against the relevance judgments. The key enabling components therefore are: (1) test collections of documents with annotated entity instances that could be 7.1 Framework Overview 121 extracted during E-R search, (2) an indexing facility, and (3) a retrieval module to process queries and rank results. Fig. 7.2 RELink Framework architecture overview. Figure 7.2 depicts the architecture of RELink used in the experiments described in Chapter 4. We include the modules responsible for deriving relevance judgments from Wikipedia. The Table Parser module is described in Section 4.1.2 in Chapter 4. Currently, the RELink Framework includes the ClueWeb-09-B1 collection combined with FACC1[174] text span annotations with links to Wikipedia entities (via Freebase). The entity linking precision and recall in FACC1 are estimated at 85% and 70-85%, respectively [174]. The RELink Extractor, part of E-R Indexer, applies an Open Information Extraction method [57] over the annotated ClueWeb-09-B corpus. The two additional components are Corpus E-R Index and E-R Retrieval, both depicted in Figure 7.2. The implementation of all modules in E-R Retrieval and the Indexer 1 http://www.lemurproject.org/clueweb09/ A Framework for Online Reputation Monitoring 122 module in Corpus E-R Index are based on Apache Lucene and the Letor module serves as a wrapper for RankLib2 . Indexing and Retrieval Based on the ClueWeb-09-B collection we create two essential resources: entity index and entity pair relationship index for the entities that occur in the corpus. For a given entity instance, the ER Indexer identifies co-occuring terms within the same sentence and considers them as entity types for the observed entity instance. Similarly, for a given pair of entities, the ER Indexer verifies whether they occur in the same sentence and extracts the separating string. That string is considered a context term for the entity pair that describes their relationship type. We obtain 476M entity and 418M entity pair extractions with corresponding sentences that are processed by the Indexer. Once the inverted index (ER Index) is created, any instance of an entity or entity pair can be retrieved in response to the contextual terms, i.e., entity types and relationship types, specified by the users. Search Process The E-R retrieval process is managed by the RELinker module (Figure 7.2). The Query Analyzer module processes information requests and passes queries in the structured format to the Retriever. Query search is performed in stages to allow for experimentation with different methods and parameter settings. First, the Retriever provides an initial set of results using Lucene’s default search settings and groups them by entity or entity pairs on query time using the Lucene’s GroupingSearch. The Scorer then generates and applies feature functions of specific retrieval models with required statistics. Currently, the Scorer has implementations for Early Fusion variants EF-LM, EF-BM25 and ERDM. The RELinker is responsible for re-ranking and providing final results based on the scores provided by the Scorer and the parameter weights learned by Letor. 7.1.2 TexRep TexRep is a research framework that implements Text Mining techniques to perform Online Reputation Monitoring (ORM) in various application domains, such as computational social sciences, political data science, computational journalism, computational finance or online marketing. 2 http://www.lemurproject.org/ranklib.php 7.1 Framework Overview 123 TexRep was designed with two main challenges in mind: 1) it should be able to cope with the Text Mining problems underlying ORM and 2) it should be flexible, adaptable and reusable in order to support the specificities of different application scenarios. We define that a Text Mining based system for Online Reputation Monitoring must follow a set of technical and operational requirements: • Batch and real-time operation: such a system must naturally be able to operate in real-time, i.e. collecting data as it is generated, processing it and updating indicators. However, it is also important to be able to operate in batch mode, in which it collects specific data from a period indicated by the user, if available, and then processes it. The system should use a distributed approach to deal with great volumes of data, (e.g. Hadoop). It should also be able to operate autonomously for long periods of time, measured in months. • Adaptability: the system should be able to adapt its models (e.g. polarity classification) through time as well as across different applications. Updating models often requires manually annotated data (e.g. NED). Therefore the system should provide a flexible annotation interface. • Modularity: researchers should be able to plug in specific modules, such as a new data source and respective crawler or a different visualization. The system interfaces should use REST APIs and JSON data format, which allow users to add new modules that interact with other data sources (e.g. Wikipedia or Facebook). • Reusability: the system should enable repeatability of all experiments to allow the research community to obtain equal results. We will make the software package of a prototype publicly available as well as the data sources and configuration parameters used in experiments. • Language independence: each component of the system should apply a statistical language modeling completely agnostic to the language of the texts. We decompose the use of Text Mining for ORM into four distinct but interconnected tasks: Data Collection, Entity Filtering, Sentiment Analysis and Analytics. Each task is accomplished by one or more software modules. For instance, Analytics tasks usually involves the use of the Aggregation, Prediction and Visualization modules. Figure 7.3 presents the TexRep architecture, including the data flow between modules. A Framework for Online Reputation Monitoring 124 PIPELINE MANAGER SENTIMENT ANALYSIS CONFIGURATIONS TRAINING DATA DATA COLLECTION SERVER DATA COLLECTION CLIENT VISUALIZATION DATA COLLECTION CLIENT ENTITY OCCURRENCES WAREHOUSE ENTITY FILTERING AGGREGATION PREDICTION TEXREP DATA SOURCES KNOWLEDGE BASE Fig. 7.3 Architecture and data flows of the TexRep framework. Entity Filtering and Sentiment Analysis represent the most challenging Text Mining problems tackled in the TexRep framework. When tracking what is being said online about the target entities it is necessary to disambiguate mentions. When this is done incorrectly, the knowledge obtained by the other modules is negatively affected. Consequently, other Text Mining tasks, such as Sentiment Analysis, will benefit from filtering non relevant texts. The current implementation of the Entity Filtering module uses the scikit-learn Python library as the Machine Learning library interface, providing access to TexRep users to the most suitable learning algorithm and parameter tuning for their specific needs. We studied a large set of features that describe the relationship between the target entity representation and a given text and we tried several different supervised learning algorithms that are available through the framework, such as Support Vector Machines (SVM) and Random Forests (RF). The Sentiment Analysis module also uses scikit-learn implementation of supervised learning algorithms in order to predict sentiment polarity and intensity in short texts 7.1 Framework Overview 125 using regression analysis. We use unsupervised learning of word embeddings [182] in short texts to construct syntactic and semantic representations of words. The Sentiment Analysis module combines word embeddings with traditional approaches, such as pre-processing techniques, bag-of-words and lexical-based features to train a classifier for sentiment polarity and a regressor for sentiment intensity. Analytics modules include Aggregation, Visualization and Prediction. These modules are application specific and depend on user configurations. For instance, in the political domain it is common to create aggregate functions that represent relative popularity indicators between political parties or candidates. These indicators are then used to predict elections. On the other hand, if we consider the financial domain, due to its high volatility, aggregation is usually performed with lower granularity (minutes instead of days) and target prediction variables are individual stock prices or variations. TexRep implements various aggregation functions and allows custom plug-in of tailored prediction models based on each application. Therefore, TexRep is able to adapt itself to the specificities of different application scenarios by implementing a modular and flexible design through user configurations and abstraction layers. Data Collection depends on the specified data sources, thus TexRep decouples client-side implementations from the data collection process management using a REST API. If a user needs a different Data Collection Client from the ones provided by default, she is able to implement a specific client that is easily integrated into the framework. The same applies to the Analytics modules which are extensible by loading user-implemented methods through an abstraction layer. Furthermore, if users wish to extend TexRep with Topic Modeling, they only need to plug-in the new module and write topic assignments through the Entity Occurrences Warehouse. New aggregation functions could be implemented that use the topic of each mention as input in order to create entity-centric topic trends visualizations. The framework can be fully configured using configuration files that are processed in the Pipeline Manager, which is the module responsible for forwarding specific parameterization to the other modules. It is possible to specify the entities of interest, data sources, aggregate functions and prediction time windows. Module specific configurations are also specified in this module, such as which training data should be used by the modules that rely on machine learning. As explained, TexRep addresses the two aforementioned challenges of developing a Text Mining framework for ORM. The current version of the framework is implemented in Python, uses MongoDB as NoSQL database and implements the MapReduce paradigm for aggregations. The external and pluggable resources used are the scikit- 126 A Framework for Online Reputation Monitoring learn library and the matplotlib for visualization, though users can replace these two resources by others of their preference. We provide the implementations of each module that we believe are the most generic as possible within the context of ORM. Nevertheless, users are also able to extend each module with the methods they see fit, such as, new features or data pre-processing steps. We now describe in detail how the different modules interact with each other, as well as, a detailed explanation of the current implementation of the Entity Filtering, Sentiment Analysis and Analytics modules. Data Flow TexRep collects data continuously and performs mini-batch processing and analytics tasks. The standard data flow is organized as follows. First the user defines the entities of interest in the configurations files, including canonical and alternative names. These configurations are processed by the Pipeline Manager and forwarded to the Data Collection clients to search for texts (e.g. news articles and tweets) using entity names as queries on each data source-specific API. The Data Collection Clients implement source-specific API clients, such as the case of Twitter and Yahoo Finance, for instance. If the user is interested in collecting RSS feeds of news outlets, then the Data Collection Client can be adapted to subscribe to those feeds and process them accordingly. Once collected, texts are stored in the Entity Occurrences Warehouse. Entity Filtering classifies each text as relevant or not for each target entity using a supervised learning approach. A knowledge base (e.g. Freebase) is used to extract target entity representations and to compute similarity features with extracted mentions contexts. Once the non-relevant texts are filtered, Sentiment Analysis takes place. The framework implements both polarity classification and sentiment regression for sentiment intensity detection. Then, Analytics modules are able to aggregate and create visualizations of trends in data or predictions of application specific dependent variables. Data Collection The Data Collection Server communicates with each Data collection Client using a REST API and therefore it allows modularity and a plugin approach for adapting to specific data sources. The task of data collection is based on user-defined entity configurations containing the list of entities under study. Each data source has specific web interfaces (e.g. RSS feeds, Yahoo Finance API or Twitter API). The Data Collection Server manages the Data Collection Clients through specific interfaces (plugins) that are adequate for the corresponding source. For instance, collecting data 7.2 RELink Use Case 127 from Twitter poses some challenges, namely due to the limits on the amount of data collected. We opted to create by default a Data Collection Client for SocialBus [192], a distributed Twitter client that enables researchers to continuously collect data from particular user communities or topics, while respecting the established limits. Some data sources allow query by topics (e.g. entity names) while others do not (e.g. RSS feeds). Moreover, in the case of Twitter, we might be interested in continuously monitoring a fixed group of Twitter users (e.g., the accounts of the entities of interest). In such cases, when we cannot search directly by entity name in the specific data source, we use the list of entity names to process collected texts that might be relevant. The Data Collection Server applies a sequential classification approach using a prefix tree to detect mentions. This method can be seen as first step of filtering but it is still prone to noisy mentions. For instance, a tweet with the word “Cameron” can be relative to several entities, such as a former UK prime minister, a filmmaker or a company. Consequently, this problem is later tackled by the Entity Filtering module. Collected texts (e.g. news or tweets) are stored in a centralized document-oriented NoSQL database (e.g. MongoDB), the Entities Occurrence Warehouse. This setup provides modularity and flexibility, allowing the possibility of developing specific data collection components tailored to specific data sources and is completely agnostic to the data format retrieved from each data source. The Data Collection Server annotates each text with the target entity which will be used by the Entity Filtering module to validate that annotation. 7.2 RELink Use Case In this section we present a use case of the RELink framework in the context of ORM applied to computational journalism. Never before has computation been so tightly connected with the practice of journalism. In recent years, the computer science community has researched [193–196, 36, 197–199] and developed3 new ways of processing and exploring news archives to help journalists perceiving news content with an enhanced perspective. We created a demo the TimeMachine, that brings together a set of Natural Language Processing, Text Mining and Information Retrieval technologies to automatically extract and index entity related knowledge from the news articles [177, 200, 36, 201, 197–199]. It allows users to issue queries containing keywords and phrases about news stories or events, and retrieves the most relevant entities mentioned in the news articles through 3 NewsExplorer (IBM Watson): http://ibm.co/1OsBO1a 128 A Framework for Online Reputation Monitoring time. TimeMachine provides readable and user-friendly insights and a temporal perspective of news stories and mentioned entities. It visually represents relationships among public figures co-mentioned in news articles as a social network graph, using a force atlas algorithm layout [202] for the interactive and real-time clustering of entities. 7.2.1 News Processing Pipeline The news processing pipeline, depicted in Figure 7.4, starts with a news cleaning module which performs the boilerplate removal from the raw news files (HTML/XML). Once the news content is processed we apply the NERD module which recognizes entity mentions and disambiguates each mention to an entity using a set of heuristics tailored for news, such as job descriptors (e.g. “Barack Obama, president of USA”) and linguistic patterns well defined for the journalistic text style. We use a bootstrap approach to train the NER system [201]. Our method starts by annotating entity names on a dataset of 50,000 news items. This is performed using a simple dictionarybased approach. Using such training set we build a classification model based on Conditional Random Fields (CRF). We then use the inferred classification model to perform additional annotations of the initial seed corpus, which is then used for training a new classification model. This cycle is repeated until the NER model stabilizes. The Fig. 7.4 News processing pipeline. entity snippet extraction consists of collecting sentences containing mentions to a given entity. All snippets are concatenated generating an entity document, which is then indexed in the entity index. The entity index represents the frequency of co-occurrence of each entity with each term that it occurs with in the news. Therefore, by relying on the redundancy of news terms and phrases associated with an entity we are able to retrieve the most relevant entity to a given input keyword or phrase query. As we also index the snippet datetime it is possible to filter query results based on a time span. For instance, the keyword “corruption” might retrieve a different entity list results in different time periods. Quotations are typically short and very informative sentences, which may directly or indirectly quote a given entity. Quotations are automatically 7.2 RELink Use Case 129 extracted (refer to "Quotations Extraction" module) using linguistic patterns, thus enriching the information extracted for each entity. Finally, once we have all mentioned entities in a given news articles we extract entity tuples representing co-occurrences of entities in a given news article and update the entity graph by incrementing the number of occurrences of a node (entity) and creating/incrementing the number of occurrences of the edge (relation) between any two mentions. 7.2.2 Demonstration The setup for demonstration uses a news archive of Portuguese news. It comprises two different datasets: a repository from the main Portuguese news agency (1990-2010), and a stream of online articles provided by the main web portal in Portugal (SAPO) which aggregates news articles from 50 online newspapers. The total number of news articles used in this demonstration comprises over 12 million news articles. The system is working on a daily basis, processing articles as they are collected from the news stream. TimeMachine allows users to explore its news archive through an entity search box or by selecting a specific date. Both options are available on the website homepage and in the top bar on every page. There are a set of “stories” recommendations on the homepage suited for first time visitors. The entity search box is designed to be the main entry point to the website as it is connected to the entity retrieval module of TimeMachine. Fig. 7.5 Cristiano Ronaldo egocentric network. Users may search for surface names of entities (e.g. “Cristiano Ronaldo”) if they know which entities they are interested to explore in the news, although the most 130 A Framework for Online Reputation Monitoring powerful queries are the ones containing keywords or phrases describing topics or news stories, such as “Eurozone crisis” or “Ballon d’Or nominees”. When selecting an entity from the ranked list of results, users access the entity profile page which contains a set of automatically extracted entity specific data: name, profession, a set of news articles, quotations from the entity and related entities. An entity timeline is also provided to allow users to navigate entity specific data through time. By selecting a specific period, different news articles, quotations and related entities are retrieved. Furthermore, users have the option of “view network” which consists in a interactive network depicting connections among entities co-mentioned in news articles for the selected time span. An example of such visualization is depicted in Figure 7.5, and it is implemented using the graph drawing library Sigma JS, together with "Force Atlas" algorithm for the clustered layout of entities. Nodes consist of entities and edges represent a co-occurrence of mentioned entities in the same news articles. The size of the nodes and the width of edges is proportional to the number of mentions and co-occurrences, respectively. Different node colors represent specific news topics where entities were mentioned. By selecting a date interval on the homepage, instead of issuing a query, users get a global interactive network of mentions and co-occurrences of the most frequent entities mentioned in the news articles for the selected period of time. 7.3 TexRep Use Case This section describes the design and implementation of the POPmine system, an use case of the proposed framework, developed in the scope of the POPSTAR project. It is an open source platform which can be used and extended by researchers interested in tracking reputation of political entities on the Web. POPmine operates either in batch or online mode and is able: to collect texts from web-based conventional media (news items in mainstream media sites) and social media (blogs and Twitter); to process those texts, recognizing topics and political entities; to analyze relevant linguistic units; to generate indicators of both frequency of mention and polarity (positivity/negativity) of mentions to political entities across sources, types of sources, and across time. As a proof of concept we present these indicators in a web application tailored for tracking political opinion in Portugal, the POPSTAR website. The system is available as an open source software package that can be used by other researchers from social sciences but also from any other area that is interested in tracking public opinion on the web. 7.3 TexRep Use Case 131 We opted to use data from news articles, tweets and blog posts and each of these data sources requires its specific crawler. News articles and blog posts are collected using RSS feeds which eases the implementations of a specific crawler. Collecting data from Twitter poses some challenges. The need for large amounts of data, coupled with Twitter’s imposed limits demand for a distributed system. We opted to use SocialBus4 which enables researchers to continuously collect data from particular user communities, while respecting Twitter’s imposed limits. The data collection components crawl data from specific data sources which implement specific web interfaces (e.g. RSS feeds, Twitter API). Each data source must have its own data collection module which in turn connects to the POPmine system using REST services. POPmine stores data collected in a document oriented NoSQL database (MongoDB). This configuration allows modularity and flexibility, allowing the possibility of developing specific data collection components tailored to specific data sources. The default setting of data collection modules comprise the following components: • News: Data from online news are provided by the service Verbetes e Notícias from Labs Sapo. This service handles online news from over 60 Portuguese news sources and is able to recognize entities mentioned in the news. • Blogs: Blog posts are provided by the blogs’ monitoring system from Labs Sapo, which includes all blogs with domain sapo.pt, blogspot.pt (Blogger) and Wordpress (blogs written in Portuguese). • Twitter: Tweets are collected using the platform SocialBus, responsible for the compilation of messages from 100.000 Portuguese users of Twitter. Tweets are collected in real time and submitted to a language classification. In our experiments we opted to collect the tweets written in Portuguese. The information extraction component comprises a knowledge base containing metadata about entities, e.g., names or jobs. Using a knowledge base is crucial to filter relevant data mentioning politicians, such as news, tweets and blog posts. In our application scenario, we opted to use Verbetes, a knowledge base which comprises names, alternative names, and professions of Portuguese people mentioned often in news articles. The Information Extraction components address two tasks: Named Entity Recognition and Named Entity Disambiguation. We envision an application scenario where we 4 http://reaction.fe.up.pt/socialbus/ A Framework for Online Reputation Monitoring 132 need to track political entities. Usually this type of entities are well known therefore we opted to use a knowledge base to provide metadata about the target entities, namely the most common surface forms of their names. Once we had the list of surface forms to search for we applied a sequential classification approach using a prefix tree to detect mentions. This method is very effective on news articles and blog posts but can result in noisy mentions when applied to Twitter. For instance, a tweet containing the word “Cameron” can be related with several entities, such as the former UK prime minister, a filmmaker or a company. Furthermore, tweets are short which results in a reduced context for entity disambiguation. We then apply the Entity Filtering approach of TexRep. The opinions warehouse contains the messages filtered by the information extraction component and applies polarity classification to those messages using an external resource - the Opinionizer classifier [203]. One of the requirements of the Opinionizer is to use manually labeled data to train the classifier. We developed an online annotation tool for that effect. We create opinion and polls indicators using the aggregator which is responsible to apply aggregation functions and smoothing techniques. Once we obtain the aggregated data we make available a set of web services that can be consumed by different applications such as the POPSTAR website or other research experiences, such as polls predictions using social media opinions. 7.3.1 Data Aggregation Buzz is the daily frequency with which political leaders are mentioned by Twitter users, bloggers and online media news. We use two types of indicators. The first type is the relative frequency with which party leaders are mentioned by each medium (Twitter, Blogs and News), on each day. This indicator is expressed, for each leader of each party, as a percentage relative to the total number of mentions to all party leaders. The second indicator is the absolute frequency of mentions, a simple count of citations for each political leader. To estimate trends in Buzz, we use the Kalman Filter. We allow users to choose the smoothing degree for each estimated trend. Users can choose between three alternatives: a fairly reactive one, where trend is highly volatile, allowing close monitoring of dayby-day variations; a very smooth one, ideal to capture long term trends; and an intermediate option, displayed by default. After identifying the polarity in each of the tweets, there are several ways to quantify the overall sentiment regarding political leaders. We can, for instance, look at each 7.3 TexRep Use Case 133 target independently or in relative terms, compare positive with negative references or simply look at one side of the polarity, or look at daily, weekly or monthly data records. In this first prototype we opted to present two separate indicators and their evolution across time, using in both cases the day as reference period. The fist indicator is the logarithm of the ratio of positive and negative tweets by political leader (party leaders and the president). In other words, a positive sign means that the political leader under consideration received more positive than negative tweets that day, while a negative result means that he received more negative than positive tweets. In mathematical notation: logsentimenti = log( positivesi + 1 ) negativesi + 1 The second approach is to simply look at the negative tweets (the vast majority of tweets in our base classifier) and calculate their relative frequency for each leader. In this way it is possible to follow each day which party leaders were, in relative terms, more or less subject to tweets with negative polarity. In mathematical notation: negativesi,d negativesshare = P negativesd Fig. 7.6 Twitter buzz share of political leaders. A Framework for Online Reputation Monitoring 134 7.3.2 Visualization We created a website5 to allow interactive visualization of the data collected and processed in real time by the POPmine platform. The site was developed within the scope of the POPSTAR project (Public Opinion and Sentiment Tracking, Analysis, and Research) and presents the following data: a) mentions to Portuguese party leaders in Twitter, in the blogosphere and in online news; b) sentiment conveyed through tweets regarding party leaders, c) voting intentions for the main political parties, measured by traditional polls; and d) evaluation of the performance of said party leaders, measured by polls. An example chart is depicted in Figure 7.6. Besides providing our indicators in the form of charts, the website also has a dashboard offering a more compact view of trends across indicators for all politicians. 7.4 Learning Word Embeddings for ORM6 Word embeddings have great practical importance since they can be used as precomputed high-density features to ML models, significantly reducing the amount of training data required in a variety of Text Mining tasks. We aim to provide general purpose pre-trained word embeddings for the Text Mining tasks in ORM. We are particularly interested in learning word embeddings from the Twitter stream due to the specificities of user generated content. It is relatively easy to get access to word embeddings trained from well formed texts such as Wikipedia or online news. However, to the best of our knowledge there are no publicly available word embeddings learned from the Portuguese Twitter stream. There are several inter-related challenges with computing and consistently distributing word embeddings concerning the: • intrinsic properties of the embeddings. How many dimensions do we actually need to store all the “useful" semantic information? How big should the embedded vocabulary be to have practical value? How do these two factors interplay? • type of model used for generating the embeddings. There are multiple possible models and it is not obvious which one is the “best", both in general or in the context of a specific type of applications. 5 6 http://www.popstar.pt The material contained in this section was published in P. Saleiro, L. Sarmento, E. M. Rodrigues, C. Soares, E. Oliveira, “Learning Word Embeddings from the Portuguese Twitter Stream: A Study of some Practical Aspects” [17]. 7.4 Learning Word Embeddings for ORM 135 • the size and properties of training data: What is the minimum amount of training data needed? Should we include out of vocabulary words in the training? • optimization techniques to be used, model hyperparameter and training parameters. Not only the space of possibilities for each of these aspects is large, there are also challenges in performing a consistent large-scale evaluation of the resulting embeddings [204]. This makes systematic experimentation of alternative word-embedding configurations extremely difficult. In this work, we make progress in trying to find good combinations of some of the previous parameters. We focus specifically in the task of computing word embeddings for processing the Portuguese Twitter stream. User-generated content (such as twitter messages) tends to be populated by words that are specific to the medium, and that are constantly being added by users. These dynamics pose challenges to NLP systems, which have difficulties in dealing with out of vocabulary words. Therefore, learning a semantic representation for those words directly from the user-generated stream - and as the words arise - would allow us to keep up with the dynamics of the medium and reduce the cases for which we have no information about the words. Starting from our own implementation of a neural word embedding model, which should be seen as a flexible baseline model for further experimentation, our research tries to answer the following practical questions: • how large is the vocabulary the one can realistically embed given the level of resources that most organizations can afford to buy and to manage (as opposed to large clusters of GPU’s only available to a few organizations)? • how much data, as a function of the size of the vocabulary we wish to embed, is enough for training meaningful embeddings? • how can we evaluate embeddings in automatic and consistent way so that a reasonably detailed systematic exploration of the previously describe space of possibilities can be performed? By answering these questions based on a reasonably small sample of Twitter data (5M), we hope to find the best way to proceed and train embeddings for Twitter vocabulary using the much larger amount of Twitter data available (300M), but for which parameter experimentation would be unfeasible. This work can thus be seen as a preparatory study for a subsequent attempt to produce and distribute a large-scale database of embeddings for processing Portuguese Twitter data. 136 7.4.1 A Framework for Online Reputation Monitoring Neural Word Embedding Model The neural word embedding model we use is the Continuous Bag-of-words (CBOW) [182]. Given a sequence of 5 words - wi−2 wi−1 wi wi+1 wi+2 , the task the model tries to perform is that of predicting the middle word, wi , based on the two words on the left - wi−2 wi−1 - and the two words on the right - wi+1 wi+2 : P (wi \|wi−2 , wi−1 , wi+1 , wi+2 ). This should produce embeddings that closely capture distributional similarity, so that words that belong to the same semantic class, or which are synonyms and antonyms of each other, will be embedded in “close” regions of the embedding hyper-space. The neural model is composed of the following layers: • a Input Word Embedding Layer, that maps each of the 4 input words represented by a 1-hot vectors with \|V \| dimensions (e.g. 32k) into a low dimension space (64 bits). The projections matrix - Winput - is shared across the 4 inputs. This is not be the embedding matrix that we wish to produce. • a Merge Layer that concatenates the 4 previous embeddings into a single vector holding all the context information. The concatenation operation ensures that the rest of the model has explicit information about the relative position of the input words. Using an additive merge operation instead would preserve information only about the presence of the words, not their sequence. • a Intermediate Context Embedding Dense Layer that maps the preceding representation of 4 words into a lower dimension space, still representing the entire context. We have fixed this context representation to 64 dimensions. This ultimately determines the dimension of the resulting embeddings. This intermediate layer is important from the point of view of performance because it isolates the still relatively high-dimensional input space (4 x 64 bits input word embeddings) from the very high-dimensional output space. • a final Output Dense Layer that maps the takes the previous 64-bit representation of the entire input context and produces a vector with the dimensionality of the word output space (\|V \| dimensions). This matrix - Woutput - is the one that stores the word embeddings we are interested in. • A Softmax Activation Layer to produces the final prediction over the word space, that is the P (wi \|wi−2 , wi−1 , wi+1 , wi+2 ) distribution 7.4 Learning Word Embeddings for ORM 137 All neural activations in the model are sigmoid functions. The model was implemented using the Syntagma7 library which relies on Keras [205] for model development, and we train the model using the built-in ADAM [206] optimizer with the default parameters. 7.4.2 Experimental Setup We are interested in assessing two aspects of the word embedding process. On one hand, we wish to evaluate the semantic quality of the produced embeddings. On the other, we want to quantify how much computational power and training data are required to train the embedding model as a function of the size of the vocabulary \|V \| we try to embed. These aspects have fundamental practical importance for deciding how we should attempt to produce the large-scale database of embeddings we will provide in the future. All resources developed in this work are publicly available8 . Apart from the size of the vocabulary to be processed (\|V \|), the hyperparamaters of the model that we could potentially explore are i) the dimensionality of the input word embeddings and ii) the dimensionality of the output word embeddings. As mentioned before, we set both to 64 bits after performing some quick manual experimentation. Full hyperparameter exploration is left for future work. Our experimental testbed comprises a desktop with a nvidia TITAN X (Pascal), Intel Core Quad i7 3770K 3.5Ghz, 32 GB DDR3 RAM and a 180GB SSD drive. Training Data We randomly sampled 5M tweets from a corpus of 300M tweets collected from the Portuguese Twitter community [192]. The 5M comprise a total of 61.4M words (approx. 12 words per tweets in average). From those 5M tweets we generated a database containing 18.9M distinct 5-grams, along with their frequency counts. In this process, all text was down-cased. To help anonymizing the n-gram information, we substituted all the twitter handles by an artificial token “T_HANDLE". We also substituted all HTTP links by the token “LINK". We prepended two special tokens to complete the 5-grams generated from the first two words of the tweet, and we correspondingly appended two other special tokens to complete 5-grams centered around the two last tokens of the tweet. Tokenization was perform by trivially separating tokens by blank space. No linguistic pre-processing, such as for example separating punctuation from words, was made. We 7 8 https://github.com/sarmento/syntagma https://github.com/saleiro/embedpt A Framework for Online Reputation Monitoring 138 Table 7.1 Number of 5-grams available for training for different sizes of target vocabulary \|V \| \|V \| # 5-grams 2048 2,496,830 8192 6,114,640 32768 10,899,570 opted for not doing any pre-processing for not introducing any linguistic bias from another tool (tokenization of user generated content is not a trivial problem). The most direct consequence of not performing any linguistic pre-processing is that of increasing the vocabulary size and diluting token counts. However, in principle, and given enough data, the embedding model should be able to learn the correct embeddings for both actual words (e.g. “ronaldo") and the words that have punctuation attached (e.g. “ronaldo!"). In practice, we believe that this can actually be an advantage for the downstream consumers of the embeddings, since they can also relax the requirements of their own tokenization stage. Overall, the dictionary thus produced contains approximately 1.3M distinct entries. Our dictionary was sorted by frequency, so the words with lowest index correspond to the most common words in the corpus. We used the information from the 5-gram database to generate all training data used in the experiments. For a fixed size \|V \| of the target vocabulary to be embedded (e.g. \|V \| = 2048), we scanned the database to obtain all possible 5-grams for which all tokens were among the top \|V \| words of the dictionary (i.e. the top \|V \| most frequent words in the corpus). Depending on \|V \|, different numbers of valid training 5-grams were found in the database: the larger \|V \| the more valid 5-grams would pass the filter. The number of examples collected for each of the values of \|V \| is shown in Table 7.1. Since one of the goals of our experiments is to understand the impact of using different amounts of training data, for each size of vocabulary to be embedded \|V \| we will run experiments training the models using 25%, 50%, 75% and 100% of the data available. Metrics Related with the Learning Process We tracked metrics related to the learning process itself, as a function of the vocabulary size to be embedded \|V \| and of the fraction of training data used (25%, 50%, 75% 7.4 Learning Word Embeddings for ORM 139 and 100%). For all possible configurations, we recorded the values of the training and validation loss (cross entropy) after each epoch. Tracking these metrics serves as a minimalistic sanity check: if the model is not able to solve the word prediction task with some degree of success (e.g. if we observe no substantial decay in the losses) then one should not expect the embeddings to capture any of the distributional information they are supposed to capture. Tests and Gold-Standard Data for Intrinsic Evaluation Using the gold standard data (described below), we performed three types of tests: • Class Membership Tests: embeddings corresponding to members of the same semantic class (e.g. “Months of the Year", “Portuguese Cities", “Smileys") should be close, since they are supposed to be found in mostly the same contexts. • Class Distinction Test: this is the reciprocal of the previous Class Membership test. Embeddings of elements of different classes should be different, since words of different classes ere expected to be found in significantly different contexts. • Word Equivalence Test: embeddings corresponding to synonyms, antonyms, abbreviations (e.g. “porque" abbreviated by “pq") and partial references (e.g. “slb and benfica") should be almost equal, since both alternatives are supposed to be used be interchangeable in all contexts (either maintaining or inverting the meaning). Therefore, in our tests, two words are considered: • distinct if the cosine of the corresponding embeddings is lower than 0.70 (or 0.80). • to belong to the same class if the cosine of their embeddings is higher than 0.70 (or 0.80). • equivalent if the cosine of the embeddings is higher that 0.85 (or 0.95). We report results using different thresholds of cosine similarity as we noticed that cosine similarity is skewed to higher values in the embedding space, as observed in related work [207, 208]. We used the following sources of data for testing Class Membership: • AP+Battig data. This data was collected from the evaluation data provided by [116]. These correspond to 29 semantic classes. 140 A Framework for Online Reputation Monitoring • Twitter-Class - collected manually by the authors by checking top most frequent words in the dictionary and then expanding the classes. These include the following 6 sets (number of elements in brackets): smileys (13), months (12), countries (6), names (19), surnames (14) Portuguese cities (9). For the Class Distinction test, we pair each element of each of the gold standard classes, with all the other elements from other classes (removing duplicate pairs since ordering does not matter), and we generate pairs of words which are supposed belong to different classes. For Word Equivalence test, we manually collected equivalente pairs, focusing on abbreviations that are popular in Twitters (e.g. “qt" ≃ “quanto" or “lx" ≃ “lisboa" and on frequent acronyms (e.g. “slb" ≃ “benfica"). In total, we compiled 48 equivalence pairs. For all these tests we computed a coverage metric. Our embeddings do not necessarily contain information for all the words contained in each of these tests. So, for all tests, we compute a coverage metric that measures the fraction of the goldstandard pairs that could actually be tested using the different embeddings produced. Then, for all the test pairs actually covered, we obtain the success metrics for each of the 3 tests by computing the ratio of pairs we were able to correctly classified as i) being distinct (cosine < 0.7 or 0.8), ii) belonging to the same class (cosine > 0.7 or 0.8), and iii) being equivalent (cosine > 0.85 or 0.95). It is worth making a final comment about the gold standard data. Although we do not expect this gold standard data to be sufficient for a wide-spectrum evaluation of the resulting embeddings, it should be enough for providing us clues regarding areas where the embedding process is capturing enough semantics, and where it is not. These should still provide valuable indications for planning how to produce the much larger database of word embeddings. 7.4.3 Results and Analysis We run the training process and performed the corresponding evaluation for 12 combinations of size of vocabulary to be embedded, and the volume of training data available that has been used. Table 7.2 presents some overall statistics after training for 40 epochs. The average time per epoch increases first with the size of the vocabulary to embed \|V \| (because the model will have more parameters), and then, for each \|V \|, with the volume of training data. Using our testbed (Section 7.4.2), the total time of learning in our experiments varied from a minimum of 160 seconds, with \|V \| = 2048 and 25% 7.4 Learning Word Embeddings for ORM 141 Table 7.2 Overall statistics for 12 combinations of models learned varying \|V \| and volume of training data. Results observed after 40 training epochs. Embeddings # Training Data Tuples Avg secs/epoch Training loss Validation loss \|V \| = 2048 561,786 (25% data) 4 3.2564 3.5932 \|V \| = 2048 1,123,573 (50% data) 9 3.2234 3.4474 \|V \| = 2048 1,685,359 (75% data) 13 3.2138 3.3657 \|V \| = 2048 2,496,830 (100% data) 18 3.2075 3.3074 \|V \| = 8192 1,375,794 (25% data) 63 3.6329 4.286 \|V \| = 8192 2,751,588 (50% data) 151 3.6917 4.0664 \|V \| = 8192 4,127,382 (75% data) 187 3.7019 3.9323 \|V \| = 8192 6,114,640 (100% data) 276 3.7072 3.8565 \|V \| = 32768 2,452,402 (25% data) 388 3.7417 5.2768 \|V \| = 32768 4,904,806 (50% data) 956 3.9885 4.8409 \|V \| = 32768 7,357,209 (75% data) 1418 4.0649 4.6 \|V \| = 32768 10,899,570 (100% data) 2028 4.107 4.4491 of data, to a maximum of 22.5 hours, with \|V \| = 32768 and using 100% of the training data available (extracted from 5M tweets). These numbers give us an approximate figure of how time consuming it would be to train embeddings from the complete Twitter corpus we have, consisting of 300M tweets. We now analyze the learning process itself. We plot the training set loss and validation set loss for the different values of \|V \| (Figure 7.7 left) with 40 epochs and using all the available data. As expected, the loss is reducing after each epoch, with validation loss, although being slightly higher, following the same trend. When using 100% we see no model overfitting. We can also observe that the higher is \|V \| the higher are the absolute values of the loss sets. This is not surprising because as the number of words to predict becomes higher the problem will tend to become harder. Also, because we keep the dimensionality of the embedding space constant (64 dimensions), it becomes increasingly hard to represent and differentiate larger vocabularies in the same hyper-volume. We believe this is a specially valuable indication for future experiments and for deciding the dimensionality of the final embeddings to distribute. On the right side of Figure 7.7 we show how the number of training (and validation) examples affects the loss. For a fixed \|V \| = 32768 we varied the amount of data used for training from 25% to 100%. Three trends are apparent. As we train with more data, we obtain better validation losses. This was expected. The second trend is that by using less than 50% of the data available the model tends to overfit the data, as indicated by the consistent increase in the validation loss after about 15 epochs (check 142 A Framework for Online Reputation Monitoring Fig. 7.7 Continuous line represents loss in the training data while dashed line represents loss in the validation data. Left side: effect of increasing \|V \| using 100% of training data. Right side: effect of varying the amount of training data used with \|V \| = 32768. dashed lines in right side of Figure 7.7). This suggests that for the future we should not try any drastic reduction of the training data to save training time. Finally, when not overfitting, the validation loss seems to stabilize after around 20 epochs. We observed no phase-transition effects (the model seems simple enough for not showing that type of behavior). This indicates we have a practical way of safely deciding when to stop training the model. Intrinsic Evaluation Table 7.3 presents results for the three different tests described in Section 7.4.2. The first (expected) result is that the coverage metrics increase with the size of the vocabulary being embedded, i.e., \|V \|. Because the Word Equivalence test set was specifically created for evaluating Twitter-based embedding, when embedding \|V \| = 32768 words we achieve almost 90% test coverage. On the other hand, for the Class Distinction test set - which was created by taking the cross product of the test cases of each class in Class Membership test set - we obtain very low coverage figures. This indicates that it is not always possible to re-use previously compiled gold-standard data, and that it will be important to compile gold-standard data directly from Twitter content if we want to perform a more precise evaluation. The effect of varying the cosine similarity decision threshold from 0.70 to 0.80 for Class Membership test shows that the percentage of test cases that are classified as correct drops significantly. However, the drop is more accentuated when training with 7.4 Learning Word Embeddings for ORM 143 only a portion of the available data. The differences of using two alternative thresholds values is even higher in the Word Equivalence test. The Word Equivalence test, in which we consider two words equivalent word if the cosine of the embedding vectors is higher than 0.95, revealed to be an extremely demanding test. Nevertheless, for \|V \| = 32768 the results are far superior, and for a much larger coverage, than for lower \|V \|. The same happens with the Class Membership test. On the other hand, the Class Distinction test shows a different trend for larger values of \|V \| = 32768 but the coverage for other values of \|V \| is so low that it would not make sense to hypothesize about the reduced values of True Negatives (TN) percentage obtained for the largest \|V \|. It would be necessary to confirm this behavior with even larger values of \|V \|. One might hypothesize that the ability to distinguish between classes requires larger thresholds when \|V \| is large. Also, we can speculate about the need of increasing the number of dimensions to be able to encapsulate different semantic information for so many words. Table 7.3 Evaluation of resulting embeddings using Class Membership, Class Distinction and Word Equivalence tests for different thresholds of cosine similarity. Embeddings \|V \|, %data Class Membership coverage 2048, 25% 2048, 50% 12.32% Acc. Acc. @0.70 Class Distinction Word Equivalence TN TN @0.80 @0.70 30.71% 4.94% 29.13% 12.69% coverage 1.20% Acc. Acc. @0.80 @0.85 @0.95 100% 100% 26.67% 2.94% 100% 100% coverage 31.25% 26.67% 2.94% 2048, 75% 29.13% 18.12% 100% 100% 33.33% 2.94% 2048, 100% 32.28% 26.77% 100% 100% 33.33% 6.67% 8192, 25% 14.17% 4.94% 100% 100% 14.71% 2.94% 99% 100% 8192, 50% 29.60% 22.41% 12.69% 6.54% 70.83% 20.59% 2.94% 8192, 75% 27.51% 18.12% 99% 100% 20.59% 2.94% 8192, 100% 33.77% 21.91% 97% 100% 29.41% 5.88% 32768, 25% 17.73% 5.13% 98% 100% 16.28% 2.33% 83% 98% 32768, 50% 47.79% 52.30% 21.06% 18.31% 89.58% 34.88% 9.30% 32768, 75% 85.15% 49.41% 44% 88% 58.14% 23.26% 32768, 100% 95.59% 74.80% 13% 57% 72.09% 34.88% Further Analysis regarding Evaluation Metrics Despite already providing interesting practical clues for our goal of trying to embed a larger vocabulary using more of the training data we have available, these results also 144 A Framework for Online Reputation Monitoring revealed that the intrinsic evaluation metrics we are using are overly sensitive to their corresponding cosine similarity thresholds. This sensitivity poses serious challenges for further systematic exploration of word embedding architectures and their corresponding hyper-parameters, which was also observed in other recent works [208]. By using these absolute thresholds as criteria for deciding the similarity of words, we create a dependency between the evaluation metrics and the geometry of the embedded data. If we see the embedding data as a graph, this means that metrics will change if we apply scaling operations to certain parts of the graph, even if its structure (i.e. relative position of the embedded words) does not change. For most practical purposes (including training downstream ML models) absolute distances have little meaning. What is fundamental is that the resulting embeddings are able to capture topological information: similar words should be closer to each other than they are to words that are dissimilar to them (under the various criteria of similarity we care about), independently of the absolute distances involved. It is now clear that a key aspect for future work will be developing additional performance metrics based on topological properties. We are in line with recent work [209], proposing to shift evaluation from absolute values to more exploratory evaluations focusing on weaknesses and strengths of the embeddings and not so much in generic scores. For example, one metric could consist in checking whether for any given word, all words that are known to belong to the same class are closer than any words belonging to different classes, independently of the actual cosine. Future work will necessarily include developing this type of metrics. 7.4.4 Concluding Remarks Producing word embeddings from tweets is challenging due to the specificities of the vocabulary in the medium. We implemented a neural word embedding model that embeds words based on n-gram information extracted from a sample of the Portuguese Twitter stream, and which can be seen as a flexible baseline for further experiments in the field. Work reported in this paper is a preliminary study of trying to find parameters for training word embeddings from Twitter and adequate evaluation tests and gold-standard data. Results show that using less than 50% of the available training examples for each vocabulary size might result in overfitting. The resulting embeddings obtain reasonable performance on intrinsic evaluation tests when trained a vocabulary containing the 32768 most frequent words in a Twitter sample of relatively small size. Nevertheless, results exhibit a skewness in the cosine similarity scores that should be further explored 7.5 Summary of the Contributions 145 in future work. More specifically, the Class Distinction test set revealed to be challenging and opens the door to evaluation of not only similarity between words but also dissimilarities between words of different semantic classes without using absolute score values. Therefore, a key area of future exploration has to do with better evaluation resources and metrics. We made some initial effort in this front. However, we believe that developing new intrinsic tests, agnostic to absolute values of metrics and concerned with topological aspects of the embedding space, and expanding gold-standard data with cases tailored for user-generated content, is of fundamental importance for the progress of this line of work. Furthermore, we plan to make public available word embeddings trained from a large sample of 300M tweets collected from the Portuguese Twitter stream. This will require experimenting with and producing embeddings with higher dimensionality (to avoid the cosine skewness effect) and training with even larger vocabularies. Also, there is room for experimenting with some of the hyper-parameters of the model itself (e.g. activation functions, dimensions of the layers), which we know have impact on final results. 7.5 Summary of the Contributions The work reported in this chapter makes the following contributions: • A framework that supports research in Entity Retrieval and Text Mining tasks in the context of Online Reputation Monitoring. This framework is composed by two major components that can act as independent frameworks: RELink and TexRep. • The RELink framework that supports comprehensive research work in E-R retrieval, supporting the semi-automatic creating of test queries, as well as, Early Fusion based approaches for E-R retrieval. • The TexRep framework that is able to collect texts from online media, such as Twitter or online news, and identify entities of interest, classify sentiment polarity and intensity. The framework supports multiple data aggregation methods, as well as visualization and modeling techniques that can be used for both descriptive analytics, such as analyze how political polls evolve over time, and predictive analytics, such as predict elections. 146 A Framework for Online Reputation Monitoring • A study of some practical aspects, namely vocabulary size, training data size and intrinsic evaluation for the training and publishing word embeddings from the Portuguese Twitter stream that can be later used for ORM related tasks. Chapter 8 Conclusions In this thesis we have addressed two computational problems in Online Reputation Monitoring: Entity Retrieval and Text Mining. Entities are the gravitational force that drives the ORM process and consequently the work reported in this thesis gravitates around entities and their occurrences across the Web. We researched and developed methods for text-based extraction, entity-relationship retrieval, analysis and prediction of entity-centric information spread across the Web. The main objectives of this thesis were achieved resulting in several contributions to the problem of Online Reputation Monitoring. Several competitive baselines were developed which we believe represent significant progress in a research area where open source work is scarce. However, there are still many issues to be addressed in the future. Recent developments in Deep Neural Networks create opportunities to improve performance in several tasks we addressed in this thesis. Once we have access to larger quantities of training data it will be possible to easily adapt our research framework to include these techniques. 8.1 Summary and Main Contributions Entity-Relationship Retrieval We have established that ORM benefits from entity retrieval capabilities and should not be constrained to classic data analytics reports. Users ought to be able to search for entity-centric information from Social Media and online news. Furthermore, reputation is not an isolated asset and depends also of the reputation of “neighboring” entities. We studied the problem of Entity-Relationship Retrieval using a IR-centric perspective and we made several contributions to this line of research: 148 Conclusions • Generalization of the problem of entity-relationship search to cover entity types and relationships represented by any attribute and predicate, respectively, rather than a predefined set. • A general probabilistic model for E-R retrieval using Bayesian Networks. • Proposal of two design patterns that support retrieval approaches using the E-R model. • Proposal of a Entity-Relationship Dependence model that builds on the basic Sequential Dependence Model (SDM) to provide extensible entity-relationship representations and dependencies, suitable for complex, multi-relations queries. • Proposal of an indexing method that supports a retrieval approach to the above problem. • A semi-automatic method for generating E-R test collections, which resulted in the RELink Query Collection comprising 600 E-R queries. • Results of experiments at scale, with a comprehensive set of queries and corpora. Entity-Relationship (E-R) Retrieval is a complex case of Entity Retrieval where the goal is to search for multiple unknown entities and relationships connecting them. Contrary to entity retrieval from structured knowledge graphs, IR-centric approaches to E-R retrieval are more adequate in the context of ORM. This happens due to the dynamic nature of the data sources which are much more transient than other more stable sources of information (e.g Wikipedia) used in general Entity Retrieval. Consequently, we developed E-R retrieval methods that do not rely on fixed and predefined entity types and relationships, enabling a wider range of queries compared to Semantic Web-based approaches. We started by presenting a formal definition of E-R queries where we assume that a E-R query can be decomposed as a sequence of sub-queries each containing keywords related to a specific entity or relationship. Then we adopted a probabilistic formulation of the E-R retrieval problem. When creating specific representations for entities (e.g. context terms) and for pairs of entities (i.e. relationships) it is possible to create a graph of probabilistic dependencies between sub-queries and entity plus relationship representations. We use a Bayesian network to depict these dependencies in a probabilistic graphical model. To the best of our knowledge this represents the first probabilistic model of E-R retrieval. 8.1 Summary and Main Contributions 149 However, these conditional probabilities cannot be computed directly from raw documents in a collection. In fact, this is a condition inherent to the problem of Entity Retrieval. Documents serve as proxies to entities and relationship representations and consequently, we need to fuse information spread across multiple documents to be able to create those representations. We proposed two design patterns, Early Fusion and Late Fusion, inspired from Model 1 and Model 2 of Balog et al. [46]. However, in the context of ORM, we are only interested in Early Fusion. Early Fusion aggregates context terms of entity and relationship occurrences to create two dedicated indexes, the entity index and the relationship index. Once we have the two indexes it is possible to apply any retrieval method to compute the relevance scores of entity and relationship documents (i.e. representations) given the E-R sub-queries. The joint probability to retrieve the final entity tuples is computed using a factorization of the conditional probabilities, i.e., the individual relevance scores. On the other hand, Late Fusion consists in matching the E-R sub-queries directly on a standard document index alongside a set of entity occurrence in each document. Once we compute the individual relevance scores of each document given a E-R sub-query, we then aggregate the entity occurrences of the top k results to compute the final joint probability. When using traditional retrieval models, such as Language Models or BM25, these design patterns can be used to create unsupervised baselines for E-R retrieval. Since our objective was to explore an Early Fusion approach to E-R retrieval we developed a novel supervised Early Fusion-based model for E-R retrieval, the EntityRelationship Dependence Model (ERDM). It uses Markov Random Field to model term dependencies of E-R sub-queries and entity/relationship documents. ERDM can be seen as an extension of the Sequential Dependence Model (SDM) [63] for ad-hoc document retrieval in a way that it relies on query term dependencies but creates a more complex graph structure that connects terms of multiple (sub-)queries and multiple documents to compute the probability mass function under the MRF. One of the difficulties we faced while researching E-R retrieval was the lack of test collections. We therefore decided to contribute to this research problem by creating a semi-automatic method for creating test collections. We realized that web tabular data often include implicit relationships between entities that belong to the same row in a table. We developed a table parser that extracts tuples of related entities from Wikipedia Lists-of-lists-of-lists tables. We then extract metadata, such as table title or column name, and provide it to editors, together with the list of entity tuples. We 150 Conclusions asked editors to create E-R queries in which the list of entity tuples could serve as relevance judgments. This process resulted in the creation and publication of the RELink Query Collection comprising 600 E-R queries. We believe RELink QC will foster research work in E-R retrieval. We performed experiments at scale using the ClueWeb-09B Web corpus from which we extracted and indexed more than 850 million entity and relationship occurrences. We evaluated our methods using four different query sets comprising a total of 548 E-R queries. As far as we know, this is the largest experiment in E-R retrieval, considering the size of the query set and the data collection. Results show consistently better performance of the ERDM model over three proposed baselines. When comparing Language Models and BM25 as feature functions we observed variance on the performance depending on the query set. Furthermore, using unsupervised Early Fusion proved to be very competitive when compared to ERDM, suggesting that it can be used in some application scenarios where the overhead of computing sequential dependencies might be unfeasible. Entity Filtering and Sentiment Analysis Entity Filtering and Sentiment Analysis are two fundamental Text Mining problems in ORM. We participated in two well known external benchmark competitions in both tasks resulting in state-of-the-art performance. We made the following contributions to these two problems: • A supervised learning approach for Entity Filtering on tweets, achieving state-ofthe-art performance using a relatively small training set. • Created and made available word embeddings trained from financial texts. • A supervised learning approach for fine-grained sentiment analysis of financial texts. Entity Filtering can be seen as targeted named entity disambiguation. We developed a supervised method that classifies tweets as relevant or non-relevant to a given target entity. This task is fundamental in ORM as downstream tasks, such as prediction, can be highly affected by noisy input data. We implemented a large set of features that can be generated to describe the relationship between a tweet mentioning a entity and a reference entity representation. 8.1 Summary and Main Contributions 151 We relied on metadata, such as entity categories, text represented with TF-IDF, similarity between tweets and Wikipedia entity articles, Freebase entities disambiguation, feature selection of terms based on frequency and feature matrix transformation using SVD. Although our approach can be perceived as relatively simple and low cost, we achieved first place with an Accuracy over 0.90 at the Filtering Task of RepLab 2013, in a test set containing more than 90 thousand tweets and 61 different target entities. Regarding Sentiment Analysis, we decided to focus our efforts in a not so well explored sub-area, namely financial texts. We participated in SemEval 2017 Task 5 which focused on fine-grained sentiment analysis of financial news and microblogs. The task consisted in predicting a real continuous variable from -1.0 to +1.0 representing the polarity and intensity of sentiment concerning companies/stocks mentioned in short texts. We modeled it as a regression analysis problem. Previous work in this domain showed that financial sentiment is often depicted in an implicit way. We created financial-specific word embeddings in order to obtain domain specific syntactic and semantic relations between words in this context. We combined traditional bag-of-words, lexical-based features and bag-of-embeddings to train a regressor of both sentiment and intensity. Results showed that different combination of features attained different performances on each sub-tasks. Nevertheless, we were able to obtain cosine similarities above 0.65 in both sub-tasks and mean average errors below 0.2 in a scale range of 2.0, representing less than 10% of the maximum possible error. Text-based Entity-centric Prediction We explored two text-based prediction problems in the context of ORM, performing analysis of the predictive power of entity-centric information on the news to predict entity popularity on Twitter, as well, as a study of sentiment aggregate functions to predict political opinion. We made the following contribution in this research area: • Analysis of the predictive power of online news regarding entity popularity on Twitter for entities that are frequently mentioned on the news. • Analysis of how to combine different sentiment aggregate functions to serve as features for predicting political polls. We are aware that entity popularity on social media can be influenced by endogenous and exogenous factors but we are only interested in exploring the interplay between 152 Conclusions online news and social media reactions. This could be useful for anticipating public relations damage control or even for editorial purposes to maximize attention and consequently revenue. We explored different sets of signal extracted from online news mentioning entities that are frequently mentioned on the news such as politicians of footballers. These signals could influence or at least are correlated with future popularity of those entities on Twitter. Results show that performance varies depending on the target entity. In general, results are better in the case of predicting popularity of politicians, due to the high unpredictability of live events associated with sports. This is a general conclusion of this study as online news do not have predictive power for live events as Twitter reactions happen quickly than the publication of the news for such cases. Results also show that the time of prediction affects the performance of the models. For instance, in the case of politicians F1 score is higher when time of prediction occurs after lunch time, which is an evidence that in politics most of the news events that trigger social media reactions are reported in the morning news. The second predictive studied we carried out consisted in using entity-centric sentiment polarity extracted from tweets to predict political polls. There is no consensus on previous research work on what sentiment aggregate functions is more adequate to predict political results. We explored several sentiment aggregate functions described in the literature to assess which one or combination would be more effective on predicting polls during the Portuguese bailout (2011-2013). In our study, we achieved the lowest mean average error using a combination of buzz aggregation functions to predict monthly poll variations instead of absolute values. On the other hand, the most important individual feature was an aggregate function consisting on the logarithm of the ration positive and negative classified tweets. A Framework for ORM We also created a framework specifically tailored for ORM that puts together the sub-tasks we tackled throughout this thesis. We believe this framework represents a significant contribution and paves the way to future research in the computational problems inherent to the process of monitoring reputation online. More precisely we make the following contributions: • A framework that supports research in Entity Retrieval and Text Mining tasks in the context of Online Reputation Monitoring. This framework is composed by two major components that can act as independent frameworks: RELink and TexRep. 8.1 Summary and Main Contributions 153 • The RELink framework that supports comprehensive research work in E-R retrieval, supporting the semi-automatic creating of test queries, as well as, Early Fusion based approaches for E-R retrieval. • The TexRep framework that is able to collect texts from online media, such as Twitter or online news, and identify entities of interest, classify sentiment polarity and intensity. The framework supports multiple data aggregation methods, as well as visualization and modeling techniques that can be used for both descriptive analytics, such as analyze how political polls evolve over time, and predictive analytics, such as predict elections. • A study of some practical aspects, namely vocabulary size, training data size and intrinsic evaluation for the training and publishing word embeddings from the Portuguese Twitter stream that can be later used for ORM related tasks. The framework is divided in two distinct components, one is dedicated to Entity Retrieval and the other to Text Mining. In practice these two components can act as two separate frameworks. Both are adaptable and can be reused in different application scenarios, from computational journalism to finance or politics. RELink framework is designed to facilitate experiments with E-R Retrieval query collections. TexRep was designed with two main challenges in mind: 1) it should be able to cope with the Text Mining problems underlying ORM and 2) it should be flexible, adaptable and reusable in order to support the specificities of different application scenarios. We also presented two use cases of our framework for ORM. In the first we use RELink in the context of computational journalism while in the second we described the design and the implementation of the POPmine system, an use case of the proposed framework in the scope of the POPSTAR project. Furthermore, we presented a study of the practical aspects of learning word embeddings from the Twitter stream. Our goal was to try to assess the feasibility of producing and publishing general purpose word embeddings for ORM. Results showed that using less than 50% of the available training examples for each vocabulary size might result in over-fitting. We obtained interesting performance on intrinsic evaluation when trained a vocabulary containing 32768 most frequent words in a Twitter sample of relatively small size. We proposed a set of gold standard data for intrinsic evaluation of word embeddings from user generated content. Nevertheless, we realized that evaluation metrics using absolute values as thresholds might not be suitable due to the cosine skewness effect on large dimensional embedding spaces. We propose to develop topological intrinsic evaluation metrics in future work. Conclusions 154 8.2 Limitations and Future Work One of the major obstacles we faced during the course of this thesis was the limited availability of labeled data for training and evaluation of the different tasks we tackled. This is a common limitation in the scope of Online Reputation Monitoring. Due to this obstacle we did not have the chance to perform extensive experimentation using more than one data source and language for each task. This aspect reduces the generalization of the results obtained since they might be biased towards the available datasets we had access to. Therefore, we leave for future work experimentation on each task with multiple datasets using different data sources and languages to perform comparable evaluations. We also recognize that we tried to address many different tasks which reduced our capability of addressing every task with the same level of depth. Nevertheless, we believe that exploring several new tasks in the scope of ORM constitutes a strong contribution to foster future research work in this area. During the course of this thesis, we did not have the possibility of performing user studies to assess the global usefulness of our framework for ORM. We would like to leave that as future work. While we had the objective of applying E-R retrieval in online news and social media which represent the natural data sources for ORM, it was not possible to evaluate our approaches using these type of data sources. Research work in E-R retrieval is still in its early stages and we believed it was necessary to first contribute to general E-R retrieval and leave for future work specific evaluation in the context of ORM. We implemented and created a demo of the Early Fusion approach since it is unsupervised. However, it was not possible to apply ERDM to online news due to the lack of training queries and relevance judgments for parameter tuning. In either cases, we aim to conduct an user experience in a near future to collect queries and relevance judgments in the context of ORM. Recent work in Deep Neural Networks makes the opportunity to beat the baselines we created in this thesis however, most of the tasks we addressed do not have enough labeled data to use these techniques. One of the most interesting avenues we would like to explore would be the use of neural networks as feature functions of the ERDM model. Since we have a dataset of more than 850 million entity and relationship extractions this represents an ideal scenario for Deep Learning. We propose to use a window based prediction task similar to the CBOW model for training word embeddings. Given a fixed window size, one would learn a neural network that would provide a ranked list of entities/relationships given an input query. We believe this approach would reduce 8.2 Limitations and Future Work 155 the computational costs of the current ERDM feature functions since we would not need to keep two huge indexes at query time. We would like also to explore different priors in entity and relationship documents within ERDM. For instance, creating source and time sensitive rankings would be useful when using transient information sources. 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International Journal of Computing and Business Research (IJCBR) ISSN (Online) : 2229-6166 Volume 3 Issue 3 September 2012 TIME EFFICIENT APPROACH TO OFFLINE HAND WRITTEN CHARACTER RECOGNITION USING ASSOCIATIVE MEMORY NET Tirtharaj Dash B.Tech Final Year Student, Department of Information Technology National Institute of Science and Technology Berhampur-761008, India Abstract: In this paper, an efficient Offline Hand Written Character Recognition algorithm is proposed based on Associative Memory Net (AMN). The AMN used in this work is basically auto associative. The implementation is carried out completely in ‘C’ language. To make the system perform to its best with minimal computation time, a Parallel algorithm is also developed using an API package OpenMP. Characters are mainly English alphabets (Small (26), Capital (26)) collected from system (52) and from different persons (52). The characters collected from system are used to train the AMN and characters collected from different persons are used for testing the recognition ability of the net. The detailed analysis showed that the network recognizes the hand written characters with recognition rate of 72.20% in average case. However, in best case, it recognizes the collected hand written characters with 88.5%. The developed network consumes 3.57 sec (average) in Serial implementation and 1.16 sec (average) in Parallel implementation using OpenMP. Keywords: Offline; Hand Written Character; Associative Memory Net; OpenMP; Serial; Parallel. 1. Introduction In the recent years, Hand Written Character Recognition has been a challenging and interesting research area in the field of pattern recognition and image processing (Impedovo et al., 1991; Mori et al., 1992). It contributes mainly to the Human-Computer interaction and improves the interface between the two (Pradeep et al., 2011). Other International Journal of Computing and Business Research (IJCBR) ISSN (Online) : 2229-6166 Volume 3 Issue 3 September 2012 human cognition methods viz. face, speech, thumb print recognitions are also being great area of research (Imtiaz and Fattah, 2011; Khurana and Singh, 2011; Kurian and Balakriahnan, 2012). Generally, character recognition can be broadly characterized into two types (i) Offline and (ii) Online. In offline method, the pattern is captured as an image and taken for testing purpose. But in case of online approach, each point of the pattern is a function of time, pressure, slant, strokes etc. Both the methods are best based on their application in the field. Yielding best accuracy with minimal cost of time is a crucial precondition for pattern recognition system. Therefore, hand written character recognition is continuously being a broad area of research. In this work, an approach for offline character recognition has been proposed using Associative Memory Network (AMN). In fact, to make it time efficient, a parallel algorithm has been developed for the implementation of AMN using OpenMP (Open Multiprocessing) (www.openmp.org). AMN is a neural network which can store patterns as memories. When the network is being tested with a key pattern, it corresponds by producing one of the stored patterns, which closely resembles to key pattern. Based on the testing pattern, AMN can be of two types (i) auto-associative memory net or (ii) hereto-associative memory net. Both the networks contains two layers (a) input layer and (b) output layer. In case of auto-associative memory net, the input and target pattern are same (Sivanandam and Deepa, 2011). But, in case of hetero-associative memory net the two patterns are different. This work uses the auto-AMN, as the character to be tested is same as the stored character. The characters considered in this work are English alphabets (both small and capital letters). This paper is organized as follows. Section 1 presented a general introduction to the character recognition systems and methods. Section 2 gives a brief literature review of some methods proposed for character recognition. Section 3 describes the proposed methodology of this work. Section 4 is a result and discussion section which gives a detailed analysis of the work. The paper is concluded in section 5 with a note to future works. 2. Literature Review International Journal of Computing and Business Research (IJCBR) ISSN (Online) : 2229-6166 Volume 3 Issue 3 September 2012 Available literatures convey that various algorithms and techniques have been used in order to accomplish the task of character recognition. Some studies are described below. Source of the literature are Google scholar, Scopus and IEEE library. Neural Network (NN) has been a backend of character classification in most of the methods. This is due to its faster and reliable computation. The methods used in front end could be (a) statistical approaches (b) kernel methods (c) support methods or (d) hybrid of fuzzy logic controllers. Multilayer Perceptron (MLP) was used for ‘Bangla’ alphabet recognition by Basu et al., 2005. The accuracy achieved in this work was 86.46% and 75.05% on the samples of training and testing respectively. Manivannan and Neil, 2010 proposed and demonstrated an optical correlator-neural network architecture for pattern recognition. English alphabet used as patterns for the training and testing process. (Pal and Singh, 2010) proposed NN based English character recognition system. In this work, MLP with one hidden layer was used. About 500 testing were carried out to test the performance of the design. The best case accuracy obtained in this work was 94%. (Perwej and Chaturvedi, 2011) worked on English alphabet recognition using NN. In this work, binary pixels of the alphabets were used train the NN. The accuracy achieved was found to be 82.5%. (Pal et al., 2007) proposed a modified quadratic classifier approach for handwritten numerals of six popular Indian scripts with high level of recognition accuracy. (Dinesh et al., 2007) used horizontal and vertical strokes and end points as feature for handwritten numerals. This method reported an accuracy rate of 90.5% in best case. However, this method used a thinning method resulting in loss of features. Yanhua and Chuanjun, 2009 recommended a novel Chinese character recognition algorithm which was based on minimum distance classifier. The algorithm attempted to work with two classes of feature extraction-structure and statistics. The statistic feature decided the primary class and the structure feature used to identify Chinese characters. A good method of character recognition was proposed by (Huiqin et al, 2011). In this work, they proposed a distribution based algorithm based on image segmentation and International Journal of Computing and Business Research (IJCBR) ISSN (Online) : 2229-6166 Volume 3 Issue 3 September 2012 distribution of pixels. Deflection Correction method was adopted for flexibility as well as reduction of matching error. This work avoided the burden of extracting the skeleton from the character. The method gave excellent result and was robust. 3. Methodology A step-wise methodology has been proposed which is demonstrated in Figure 1. Figure 1: Proposed Methodology Step-1: Collection of English alphabets (both small and capital) from (i) system and (ii) persons (Hand Written) Step-2: Extraction of pixels from the characters Step-3: Implementation of auto AMN: (i) Training and (ii) Testing using both (a) serial and (b) parallel algorithms. Step-4: Comparison of results from serial and parallel processing with respect to time of execution 3.1 Generation of English alphabets English alphabets (both small and capital) are designed in the system using MS Paint version 6.1 in Arial font size-28 (No Bold) in BMP file format. The dimension of the bmp file is 31×39, with bit depth of 4. Some alphabets are given in Figure 2. International Journal of Computing and Business Research (IJCBR) ISSN (Online) : 2229-6166 Volume 3 Issue 3 September 2012 Figure 2: English alphabets of the system Hand written English alphabets are collected, each one from different persons. The characters are given in Figure 3. Figure 3: English alphabets collected from different persons 3.2 Extraction of Pixel from the characters Pixels are extracted from the character images (bitmap files) using a standard image function of MATLAB version 10. The function is imread(‘filename.bmp’). The function extracts the decimal values associated with each pixel. The pixels are then stored in a text (.txt) file for the experiment purpose. 3.3 Auto-associative memory net (Auto-AMN) implementation 3.3.1 Serial algorithm INITIALIZE weight (W) to 0 SET the target pattern as the system’s pattern International Journal of Computing and Business Research (IJCBR) ISSN (Online) : 2229-6166 Volume 3 Issue 3 September 2012 INPUT the Handwritten pattern to the first layer of AMN FOR i=1 to n DO FOR j=1 to n DO CALCULATE the weight as Wij(new)=W ij (old) + INPUTi×TARGETj END END FOR i=1 to n DO FOR j=1 to n DO CALCULATE the net input to each output node as, n Yinj= ∑ x i Wij i =1 IF(Yinj>0) Yj=+1; ELSE Yj=-1; END END 3.3.2 Parallel algorithm INITIALIZE weight (W) to 0 SET the target pattern as the system’s pattern INPUT the Handwritten pattern to the first layer of AMN #pragma omp paralle shared(W,Yin,chunk,p) private(tid,i,j) DO #pragma omp for schedule (static, chunk) FOR i=1 to n DO FOR j=1 to n DO CALCULATE the weight as Wij(new)=W ij (old) + INPUTi×TARGETj END END International Journal of Computing and Business Research (IJCBR) ISSN (Online) : 2229-6166 Volume 3 Issue 3 September 2012 #pragma omp for schedule (static, chunk) FOR i=1 to n DO FOR j=1 to n DO CALCULATE the net input to each output node n Yinj= ∑ x i Wij i =1 IF(Yinj>0) Yj=+1; ELSE Yj=-1; END END END 3.3.3 System Specification A computer system having 1 GB RAM and Four processors is used for the complete work. The operating system is Ubuntu 10.04 (Linux). However, for auto optimization by the compiler, ‘–g’ tag is used in the compilation command. 4. Results and Discussion The contribution of this work is a detailed analysis on the recognition accuracy for all the handwritten English alphabets (Total 52). Time of computation has been noted for both serial and parallel algorithm to compare the decision making speed. 4.1 Recognition accuracy Table 1 shows a result of the testing the developed AMN for a set of hand written characters. It should be noted that the network is trained with the machine’s alphabets and tested with the hand written alphabet. However, for reliability issue, a hand written character is checked for 5 times and the matching percentage is the average of the 5 results. Table 1: Recognition accuracy of AMN for offline Hand written character recognition International Journal of Computing and Business Research (IJCBR) ISSN (Online) : 2229-6166 Volume 3 Issue 3 September 2012 System’s Alphabet A B C D E F G H I J K L M N O P Q R S T U V W X Y Z a b c d e f g h i j Hand Written Alphabet for which highest match is achieved A B C O F F G H I J K L H H O P O R S T U V U X Y Z o b e d e p y b i j Recognition Accuracy (%) 66.56 56.80 67.19 60.88 62.85 68.34 58.78 70.73 85.74 86.28 64.85 85.16 70.38 71.13 64.17 67.74 61.42 61.16 63.20 80.18 71.46 73.69 73.13 73.39 76.67 64.01 69.49 65.35 73.96 67.26 67.47 73.54 69.36 70.52 83.62 88.50 Time of Computation (sec.) Serial Parallel 3.00 0.99 3.11 0.98 2.98 0.74 4.60 1.55 3.77 2.01 4.65 2.32 3.12 1.87 3.67 0.99 4.01 1.03 4.32 1.93 4.17 1.05 3.04 0.87 4.12 1.21 4.61 1.35 3.02 0.76 4.00 1.04 2.98 0.83 3.41 0.99 4.04 1.12 2.87 0.77 2.76 0.76 2.78 0.87 3.05 1.00 3.83 1.43 4.05 1.45 4.00 1.45 3.77 1.88 3.78 1.43 3.17 1.42 3.77 1.54 3.18 1.31 3.17 1.22 3.18 1.03 4.01 1.12 3.19 1.27 4.01 1.23 International Journal of Computing and Business Research (IJCBR) ISSN (Online) : 2229-6166 Volume 3 Issue 3 September 2012 k l m n o p q r s t u v w x y z K l m n o p q r s t u v w x y z 69.44 83.60 62.62 72.97 70.03 68.61 65.45 82.90 68.66 79.92 78.54 82.52 67.16 78.55 78.27 70.08 3.66 3.75 2.76 2.89 2.75 3.78 3.89 4.05 3.00 2.01 4.89 4.02 3.96 2.95 3.81 3.99 1.44 1.03 1.05 0.75 0.76 0.94 0.94 1.11 0.77 0.52 1.33 1.04 1.21 0.76 0.98 1.23 Table-1 can be viewed as a detailed analysis of performance of the developed auto AMN for offline hand written English alphabet recognition. The network recognizes the handwritten character ‘j’ with highest matching of 88.50%. However, the network doesn’t recognize some alphabets like ‘D’, ‘E’, ‘M’, ‘N’, ‘Q’, ‘W’, ‘a’, ‘c’, ‘f’, ‘g’, ‘h’, ‘j’ and ‘k’ and these alphabets are recognized as ‘O’, ‘F’, ‘H’,’H’, ‘O’, ‘U’, ‘o’, ‘e’, ‘p’, ‘y’, ‘b’, ‘I’ and ‘K’ respectively with some matching error. 4.2 Level of matching of each alphabet A plot has been given in Figure 4 to view the level up to which each English alphabet is matched by the AMN. The alphabets which are not recognized are awarded with 0% matching. International Journal of Computing and Business Research (IJCBR) ISSN (Online) : 2229-6166 Volume 3 Issue 3 September 2012 90 80 Level of Matching (%) 70 60 50 40 30 20 10 0 A B C D E F G H I J K L M N O P Q R S T U VW X Y Z a b c d e f g h i j k l m n o p q r s t u v w x y z English Alphabet Figure 4 This plot shows level of matching of each alphabet 4.3 Time Efficiency As it is already mentioned that the network is developed with two algorithms, (i) serial and (ii) parallel; it will be a good idea to check the timing variation in both the cases. A plot given in Figure 5, shows speed up after achieved after the execution of parallel algorithm. International Journal of Computing and Business Research (IJCBR) ISSN (Online) : 2229-6166 Volume 3 Issue 3 September 2012 5 Serial Parallel 4.5 Decision making speed (sec.) 4 3.5 3 2.5 2 1.5 1 0.5 0 5 10 15 20 25 30 35 Alphabet Serial Number (1-52) 40 45 50 Figure 5 Decision making speed by Serial and Parallel algorithm 5. Conclusion In this paper, an offline English character recognition system has been proposed. The system is developed using auto Associative Memory Net. To make the developed system faster and reliable, a parallel algorithm has been developed and tested successfully. Experimental study showed that, the system recognizes characters with average recognition rate of 72.20%. Character ‘j’ is recognized with highest accuracy rate of 88.5%. The average time required by the serial algorithm to recognize a character is 3.57 sec where as the parallel algorithm takes only 1.16 sec on an average. However, automatic checking of a sequence of character by the network will play a great role in the world of character recognition. The author is currently working on this issue. References 1. Basu, S. et al. (2005) “Handwritten ‘Bangla’ alphabet recognition using an MLP based classifier,” Proceeding of 2nd National Conference on Computer Processing of Bangla, pp. 285-291. International Journal of Computing and Business Research (IJCBR) ISSN (Online) : 2229-6166 Volume 3 Issue 3 September 2012 2. Dinesh, A. U. et al. (2007) “Isolated handwritten Kannada numeral recognition using structural feature and K-means cluster,” IISN, pp. 125-129. 3. Huiqin L. et al.(2011) “The Research of Algorithm for Handwritten Character Recognition in Correcting Assignment System,” Proceeding of 6th International Conference on Image and Graphics (ICIG), pp.456-460. 4. Impedovo, S. et al. (1991) “Optical character recognition,” International Journal of Pattern Recognition and Artificial Intelligence, Vol. 5(1-2), pp. 1-24. 5. Imtiaz, H. and Fattah, S. A. (2011) “A wavelet-domain local dominant feature selection scheme for face recognition,” International Journal of Computing and Business Research, Vol.3, Issue.2. 6. Khurana, P. and Singh, V. (2011) “A model for human cognition,” International Journal of Computing and Business Research, Vol.2, Issue.3. 7. Kurian, C. and Balakriahnan, K. (2012) “Continuous speech recognition system for Malayalam language using PLP cepstral coefficient,” Journal of Computing and Business Research, Vol.3, Issue.1. 8. Manivannan, N. and Neil, M.A.A. (2010) “Optical correlator-neural network hybrid system for many patterns recognition,” Proceeding of 8th International Symposium on Intelligent Systems and Informatics (SISY), pp.249-251. 9. Mori, S. et al. (1992) “Historical review of OCR research and development,” Proceedings of IEEE, vol. 80, pp. 1029-1058. 10. Pal, U. et al. (2007) “Handwritten numeral recognition of six popular scripts,” 9th International conference on Document analysis Recognition, vol. 2, pp. 749-753. 11. Pal, A. and Singh, D. (2010) “Handwritten English Character Recognition Using Neural Network,” International Journal of Computer Science and Communication, Vol.1, No.2, pp. 141-144. 12. Perwej, Y. and Chaturvedi, A. (2011) “Neural Networks for Handwritten English Alphabet Recognition,” International Journal of Computer Applications, Vol. 20, No. 7, pp. 1-5. 13. Pradeep J. et al. (2011) “Diagonal based feature extraction for handwritten alphabet recognition system using neural network,” International Journal of Computer Science and Information Technology, Vol.3, No.1, pp. 27-38. 14. Sivanandam, S.N and Deepa, S.N. (2011) “Principles of Soft Computing,” Wiley-India publisher, 2nd edition. 15. Yanhua, M. and Chuanjun, L. (2009) “A Recognition Algorithm for Chinese Character Based on Minimum Distance Classifier,” Proceeding of 2nd International Workshop on Computer Science and Engineering, vol.2, pp.246-249.	9
arXiv:1512.03987v3 [math.ST] 8 Oct 2016 On the Finite-Sample Analysis of Θ-estimators Yiyuan She Department of Statistics Florida State University, Tallahassee, FL 32306 Abstract In large-scale modern data analysis, first-order optimization methods are usually favored to obtain sparse estimators in high dimensions. This paper performs theoretical analysis of a class of iterative thresholding based estimators defined in this way. Oracle inequalities are built to show the nearly minimax rate optimality of such estimators under a new type of regularity conditions. Moreover, the sequence of iterates is found to be able to approach the statistical truth within the best statistical accuracy geometrically fast. Our results also reveal different benefits brought by convex and nonconvex types of shrinkage. 1 Introduction Big data naturally arising in machine learning, biology, signal processing, and many other areas, call for the need of scalable optimization in computation. Although for low-dimensional problems, Newton or quasi-Newton methods converge fast and have efficient implementations, they typically do not scale well to high dimensional data. In contrast, first-order optimization methods have recently attracted a great deal of attention from researchers in statistics, computer science and engineering. They iterate based on the gradient (or a subgradient) of the objective function, and have each iteration step being cost-effective. In high dimensional statistics, a first-order algorithm typically proceeds in the following manner β (t+1) = P ◦ (β (t) − α∇l(β (t) )), 1 (1) where P is an operator that is easy to compute, ∇l denotes the gradient of the loss function l, and α gives the stepsize. Such a simple iterative procedure is suitable for large-scale optimization, and converges in arbitrarily high dimensions provided α is properly small. P can be motivated from the perspective of statistical shrinkage or regularization and is necessary to achieve good accuracy when the dimensionality is moderate or high. For example, a proximity operator (Parikh and Boyd, 2014) is associated with a convex penalty function. But the problems of interest may not always be convex. Quite often, P is taken as a certain thresholding rule Θ in statistical learning, such as SCAD (Fan and Li, 2001). The resulting computation-driven estimators, which we call Θ-estimators, are fixed points of β = Θ(β − ∇l(β); λ). To study the non-asymptotic behavior of Θ-estimators (regardless of the sample size and dimensionality), we will establish some oracle inequalities. During the last decade, people have performed rigorous finite-sample analysis of many high-dimensional estimators defined as globally optimal solutions to some convex or nonconvex problems—see Bunea et al. (2007), Zhang and Huang (2008), Bickel et al. (2009), Lounici et al. (2011), Zhang and Zhang (2012), She (2014), among many others. Θ-estimators pose some new questions. First, although nicely, an associated optimization criterion can be constructed for any given Θ-estimator, the objective may not be convex, and the estimator may not correspond to any functional local (or global) minimum. Second, there are various types of Θ-estimators due to the abundant choices of Θ, but a comparative study regarding their statistical performance in high dimensions is lacking in the literature. Third, Θ-estimators are usually computed in an inexact way on big datasets. Indeed, most practitioners (have to) terminate (1) before full computational convergence. These disconnects between theory and practice when using iterative thresholdings motivate our work. The rest of the paper is organized as follows. Section 2 introduces the Θestimators, the associated iterative algorithm–TISP, and some necessary notation. Section 3 presents the main results, including some oracle inequalities, and sequential analysis of the iterates generated by TISP. Section 4 provides proof details. 2 2 Background and Notation 2.1 Thresholding functions Definition 1 (Thresholding function). A thresholding function is a real valued function Θ(t; λ) defined for −∞ < t < ∞ and 0 ≤ λ < ∞ such that (i) Θ(−t; λ) = −Θ(t; λ); (ii) Θ(t; λ) ≤ Θ(t′ ; λ) for t ≤ t′ ; (iii) limt→∞ Θ(t; λ) = ∞; (iv) 0 ≤ Θ(t; λ) ≤ t for 0 ≤ t < ∞. A vector version of Θ (still denoted by Θ) is defined componentwise if either t or λ is replaced by a vector. From the definition, Θ−1 (u; λ) := sup{t : Θ(t; λ) ≤ u}, ∀u > 0 (2) must be monotonically non-decreasing and so its derivative is defined almost everywhere on (0, ∞). Given Θ, a critical number LΘ ≤ 1 can be introduced such that dΘ−1 (u; λ)/ du ≥ 1 − LΘ for almost every u ≥ 0, or LΘ := 1 − ess inf{ dΘ−1 (u; λ)/ du : u ≥ 0}, (3) where ess inf is the essential infimum. For the perhaps most popular soft-thresholding and hard-thresholding functions ΘS (t; λ) = sgn(t)(\|t\| − λ)+ , ΘH (t; λ) = t1\|t\|≥λ , LΘ equals 0 and 1, respectively. For any arbitrarily given Θ, we construct a penalty function PΘ (t; λ) as follows PΘ (t; λ) = Z 0 \|t\| −1 (Θ (u; λ) − u) du = Z 0 \|t\| (sup{s : Θ(s; λ) ≤ u} − u) du (4) for any t ∈ R. This penalty will be used to make a proper objective function for Θ-estimators. The threshold τ (λ) := Θ−1 (0; λ) may not equal λ in general. For ease in notation, in writing Θ(·; λ), we always assume that λ is the threshold parameter, i.e., λ = τ (λ), unless otherwise specified. Then an important fact is that given λ, any thresholding rule Θ satisfies Θ(t; λ) ≤ ΘH (t; λ), ∀t ≥ 0, due to property (iv), from which it follows that PΘ (t; λ) ≥ PH (t; λ), 3 (5) where PH (t; λ) = Z 0 \|t\| 2 2 (Θ−1 H (u; λ) − u) du = (−t /2 + λ\|t\|)1\|t\|<λ + (λ /2)1\|t\|≥λ . (6) 2 In particular, PH (t; λ) ≤ P0 (t; λ) := λ2 1t6=0 and PH (t; λ) ≤ P1 (t; λ) := λ\|t\|. When Θ has discontinuities, such as t = ±λ in ΘH (t; λ), ambiguity may arise in definition. To avoid the issue, we assume the quantity to be thresholded never corresponds to any discontinuity of Θ. This assumption is mild because practically used thresholding rules have few discontinuity points and such discontinuities rarely occur in real applications. 2.2 Θ-estimators We assume a model y = Xβ ∗ + ǫ, (7) where X is an n × p design matrix, y is a response vector in Rn , β ∗ is the unknown coefficient vector, and ǫ is a sub-Gaussian random vector with mean zero and scale bounded by σ, cf. Definition 2 in Section 4 for more detail. Then a Θ-estimator β̂, driven by the computational procedure (1), is defined as a solution to the Θ-equation ρβ = Θ(ρβ + X T y/ρ − X T Xβ/ρ; λ), (8) where ρ, the scaling parameter, does not depend on β. Having ρ appropriately large is crucial to guarantee the convergence of the computational procedure. All popularly used penalty functions are associated with thresholdings, such as the ℓr (0 < r ≤ 1), ℓ2 , SCAD (Fan and Li, 2001), MCP (Zhang, 2010a), capped ℓ1 (Zhang, 2010b), ℓ0 , elastic net (Zou and Hastie, 2005), Berhu (Owen, 2007; He et al., 2013), ℓ0 + ℓ2 (She, 2009), to name a few. Table 1 lists some examples. From a shrinkage perspective, thresholding rules usually suffice in statistical learning. Equation (8) can be re-written in terms of the scaled deign X̃ = X/ρ and the corresponding coefficient vector β̃ = ρβ β̃ = Θ(β̃ + X̃ T y − X̃ T X̃ β̃; λ). 4 (9) We will show that the λ in the scaled form does not have to adjust for the sample size, which is advantageous in regularization parameter tuning. A simple iterative procedure can be defined based on (8) or (9): β̃ (t+1) = Θ(β̃ (t) + X̃ T y − X̃ T X̃ β̃ (t) ; λ), β (t+1) = β̃ (t+1) /ρ, (10) which is called the Thresholding-based Iterative Selection Procedure (TISP) (She, 2009). From Theorem 2.1 of She (2012), given an arbitrary Θ, TISP ensures the kXk2 following function-value descent property when ρ ≥ 2−L : Θ f (β (t+1) ; λ) ≤ f (β (t) ; λ). (11) Here, the energy function (objective function) is constructed as p X 1 f (β; λ) = kXβ − yk22 + P (ρ\|βj \|; λ), 2 j=1 (12) where the penalty P can be PΘ as defined in (4), or more generally, P (t; λ) = PΘ (t; λ) + q(t; λ), (13) with q an arbitrary function satisfying q(t, λ) ≥ 0, ∀t ∈ R and q(t; λ) = 0 if t = Θ(s; λ) for some s ∈ R. Furthermore, we can show that when ρ > kXk2 /(2 − LΘ ), any limit point of β (t) is necessarily a fixed point of (8), and thus a Θ-estimator. See She (2012) for more detail. Therefore, f is not necessarily unique when Θ has discontinuities—for example, penalties like the capped ℓ1 , 2 P0 (t; λ) = λ2 1t6=0 and PH are all associated with the same ΘH . Because of the many-to-one mapping from penalty functions to thresholding functions, iterating (1) with a well-designed thresholding rule is perhaps more convenient than solving a nonconvex penalized optimization problem. Indeed, some penalties (like SCAD) are designed from the thresholding viewpoint. The following theorem shows thatPthe set of Θ-estimators include all locally optimal solutions of 12 kXβ − yk22 + pj=1 PΘ (\|βj \|; λ) =: fΘ (β). Theorem 1. Let β̂ be a local minimum point (or a coordinate-wise minimum point) of fΘ (·). If Θ is continuous at β̂ + X T y − X T X β̂, β̂ must satisfy β = Θ(β + X T y − X T Xβ; λ). The converse is not necessarily true. Namely, Θ-estimators may not guarantee functional local optimality, let alone global optimality. This raises difficulties in statistical analysis. We will give a novel and unified treatment which can yield nearly optimal error rate for various thresholdings. 5 Table 1: Some examples of thresholding functions and their associated quantities. Θ LΘ soft (t − λsgn(t))1\|t\|>λ 0 ridge −η PΘ λ\|t\| η 2 t 2 hard t1\|t\|>λ 1 ( − 21 t2 + λ\|t\|, 1 2 λ , 2 t 1+η if \|t\| < λ if \|t\| ≥ λ 2 P elastic net (η ≥ 0) berhu (η ≥ 0)  if \|t\| < λ 0  t − λsgn(t) if λ ≤ \|t\| ≤ λ + λ/η   t if \|t\| > λ + λ/η 1+η 0 ( λ\|t\| if \|t\| ≤ λ/η min(λ\|t\|, λ2 ) (‘capped ℓ1 ’), hard-ridge (η ≥ 0) λ2 1 2 t6=0 Θ t−λsgn(t) 1\|t\|≥λ 1+η LΘ −η PΘ λ\|t\| + 21 ηt2 P (‘ℓ0 + ℓ2 ’) scad (a > 2) mcp (γ ≥ 1)   0, if \|t\| ≤ λ   if \|t\| < λ   t − λ sgn(t), 0, if λ < \|t\| ≤ 2λ t−λsgn(t) , if λ ≤ \|t\| < γλ (a−1)t−aλ sgn(t) 1−1/γ   , if 2λ < \|t\| ≤ aλ   a−2   t, if \|t\| ≥ γλ t, if \|t\| > aλ 1/(a − 1) 1/γ  ( 2 if \|t\| ≤ λ λ sgn(t), t  + λ\|t\|, if \|t\| < γλ − 2γ aλ sgn(t)−t dP = = γ1 PH (t; γλ) , if λ < \|t\| ≤ aλ dt a−1 γλ2   , if \|t\| ≥ γλ 2 0, if \|t\| > aλ lr (0 < r < 1, ζ ≥ 0) ( 0, if \|t\| ≤ ζ 1/(2−r) (2 − r)(2 − 2r)(r−1)/(2−r) sgn(t) max{ζ 1/(2−r) [r(1 − r)]1/(2−r) ≤ θ ≤ \|t\| : θ + ζrθ r−1 = \|t\|}, otherwise.(The set is a singleton.) 1 ζ\|t\|r Θ LΘ PΘ Θ LΘ P ηt2 2 + 2 λ 2η if \|t\| > λ/η. t 1 1+η \|t\|>λ 1 ( − 21 t2 + λ\|t\|, 2 1 2 λ ηt + 12 1+η , 2 η 2 1 λ2 1 + 2t 2 1+η t6=0 if \|t\| < if \|t\| ≥ λ 1+η λ . 1+η 3 Main Results To address the problems in arbitrary dimensions (with possibly large p and/or n), we aim to establish non-asymptotic oracle inequalities (Donoho and Johnstone, 1994). For any β = [β1 , . . . , βp ]T , define J (β) = {j : βj 6= 0}, J(β) = \|J (β)\| = kβk0 . 2 (14) Recall P1 (t; λ) = λ\|t\|, P0 (t; λ) = λ2 1t6=0 , PH (t; λ) = (−t2 /2 + λ\|t\|)1\|t\|<λ + (λ2 /2)1\|t\|≥λ. For convenience, we use P1 (β; λ) to denote λkβk1 when there is no ambiguity. P0 (β; λ) and PH (β; λ) are used similarly. We denote by . an inequality that holds up to a multiplicative constant. Unless otherwise specified, we study scaled Θ-estimators satisfying equation 6 (9), where β̃ = ρβ, X̃ = X/ρ, and ρ ≥ kXk2 (and so kX̃k2 ≤ 1). By abuse of notation, we still write β for β̃, and X for X̃. As mentioned previously, we always assume that Θ is continuous at β̂ + X T y − X T X β̂ in Sections 3.1 & 3.2; similarly, Section 3.3 assumes that Θ is continuous at β (t) + X T y − X T Xβ (t) . The past works on the lasso show that a certain incoherence requirement must be assumed to obtain sharp error rates. In most theorems, we also need to make similar assumptions to prevent the design matrix from being too collinear. We will state a new type of regularity conditions, which are called comparison regularity conditions, under which oracle inequalities and sequential statistical error bounds can be obtained for any Θ. 3.1 PΘ -type oracle inequalities under R0 In this subsection, we use PΘ to make a bound of the prediction error of Θestimators. Our regularity condition is stated as follows. A SSUMPTION R0 (δ, ϑ, K, β, λ) Given X, Θ, β, λ, there exist δ > 0, ϑ > 0, K ≥ 0 such that the following inequality holds for any β ′ ∈ Rp ϑPH (β ′ − β; λ) + ≤ LΘ ′ kβ − βk22 2 2−δ kX(β ′ − β)k22 + PΘ (β ′ ; λ) + KPΘ (β; λ). 2 (15) Roughly, (15) means that 2kX(β ′ − β)k22 can dominate LΘ kβ ′ − βk22 with the help from PΘ (β ′ ; λ) and KPΘ (β; λ) for some K > 0. T T Theorem 2. Let p β̂ be any Θ-estimator satisfying β = Θ(β + X y − X Xβ; λ) with λ = Aσ log(ep) and A a constant. Then for any sufficiently large A, the following oracle inequality holds for β ∈ Rp E[kX β̂ − Xβ ∗ k22 ] . kXβ − Xβ ∗ k22 + PΘ (β; λ) + σ 2 , (16) provided R0 (δ, ϑ, K, β, λ) is satisfied for some constants δ > 0, ϑ > 0, K ≥ 0. Theorem 2 is applicable to any Θ. Let’s examine two specific cases. First, consider LΘ ≤ 0, which indicates that PΘ is convex. Because PH ≤ PΘ and PH is sub-additive: PH (t + s) ≤ PH (t) + PH (s) due to its concavity (Zhang and Zhang, 2012), R0 (δ, ϑ, K, β, λ) is always satisfied (for any δ ≤ 2, 0 < ϑ ≤ 1, K ≥ ϑ). 7 Corollary 1. Suppose Θ satisfies LΘ ≤ 0. Then, (16) holds for all corresponding Θ-estimators, without requiring any regularity condition. In the case of hard-thresholding or SCAD thresholding, PΘ (β; λ) does not depend on the magnitude of β, and we can get a finite complexity rate in the oracle inequality. Also, R0 can be slightly relaxed, by replacing KPΘ (β; λ) with KP0 (β; λ) in (15). We denote the modified version by R′0 (δ, ϑ, K, β, λ). Corollary 2. Suppose that Θ corresponds to a bounded nonconvex penalty satisfying PΘ (t; λ) ≤ Cλ2 , ∀t ∈ R, for some constant C > 0. Then in the setting of Theorem 2, E[kX β̂ − Xβ ∗ k22 ] . kXβ − Xβ ∗ k22 + σ 2 J(β) log(ep) + σ 2 , (17) provided R′0 (δ, ϑ, K, β, λ) is satisfied for some constants δ > 0, ϑ > 0, K ≥ 0. Remark 1. The right-hand side of the oracle inequalities involves a bias term kXβ − Xβ ∗ k22 and a complexity term PΘ (β; λ). Letting β = β ∗ in, say, (16), the bias vanishes, and we obtain a prediction error bound of the order σ 2 J ∗ log(ep) (omitting constant factors), where J ∗ denotes the number of nonzero components in β ∗ . On the other hand, the existence of the bias term ensures the applicability of our results to approximately sparse signals. For example, when β ∗ has many small but nonzero components, we can use a reference β with a much smaller support than J (β ∗ ) to get a lower error bound, as a benefit from the bias-variance tradeoff. Remark 2. When R0 holds with δ > 1, the proof of Theorem 2 shows that the multiplicative constant for kXβ − Xβ ∗ k22 can be as small as 1. The corresponding oracle inequalities are called ‘sharp’ in some works (Koltchinskii et al., 2011). This also applies to Theorem 3. Our proof scheme can also deliver highprobability form results, without requiring an upper bound of kXk2 . Remark 3. Corollary 2 applies to all “hard-thresholding like” Θ, because when Θ(t; λ) = t for \|t\| > cλ, PΘ (t; λ) ≤ c2 λ2 . It is worth mentioning that the error rate of σ 2 J ∗ log(ep) cannot be significantly improved in a minimax sense. In fact, under the Gaussian noise contamination and some regularity conditions, there exist constants C, c > 0 such that inf β̌ supβ∗ :J(β∗ )≤J E[kX(β̌ − β ∗ )k22 )/(CPo (J))] ≥ c > 0, where β̌ denotes an arbitrary estimator of β ∗ and Po (J) = σ 2 {J + J log(ep/J)}. See, e.g., Lounici et al. (2011) for a proof. The bound in (17) achieves the minimax optimal rate up to a mild logarithm factor for any n and p. 8 3.2 P0 -type oracle inequalities under R1 This part uses P0 instead of PΘ to make an oracle bound. We will show that under another type of comparison regularity conditions, all thresholdings can attain the essentially optimal error rate given in Corollary 2. We will also show that in the case of soft-thresholding, our condition is more relaxed than many other assumptions in the literature. A SSUMPTION R1 (δ, ϑ, K, β, λ) Given X, Θ, β, λ, there exist δ > 0, ϑ > 0, K ≥ 0 such that the following inequality holds for any β ′ ∈ Rp ϑPH (β ′ − β; λ) + LΘ ′ kβ − βk22 + PΘ (β; λ) 2 (18) 2−δ ′ 2 ′ 2 kX(β − β)k2 + PΘ (β ; λ) + Kλ J(β). ≤ 2 p Theorem 3. Let β̂ be a Θ-estimator and λ = Aσ log(ep) with A a sufficiently large constant. Then E[kX β̂ − Xβ ∗ k22 ] . kXβ − Xβ ∗ k22 + λ2 J(β) + σ 2 holds for any β ∈ Rp if R1 (δ, ϑ, K, β, λ) is satisfied for some constants δ > 0, ϑ > 0, K ≥ 0. Remark 4. Some fusion thresholdings, like those associated with elastic net, Berhu and Hard-Ridge (cf. Table 1), involve an additional ℓ2 shrinkage. In the situation, the complexity term in the oracle inequality should involve both J(β) and kβk22 . We can modify our regularity conditions to obtain such ℓ0 +ℓ2 bounds using the same proof scheme. The details are however not reported in this paper. In addition, our results can be extended to Θ-estimators with a stepsize parameter. Given λ > 0 and 0 < α ≤ 1, suppose λα is introduced such that αPΘ (t; λ) = PΘ (t; λα ) for any t. Then, for any β̂ as a fixed point of β = Θ(β −αX T Xβ +αX T y; λα ), an analogous result can be obtained (the only change is that LΘ is replaced by LΘ /α). To give some more intuitive regularity conditions, we suppose PΘ is concave on [0, ∞). Examples include ℓr (0 ≤ r ≤ 1), MCP, SCAD, and so on. The concavity implies PΘ (t + s) ≤ PΘ (t) + PΘ (s), and so PΘ (βJ′ ; λ) − PΘ (βJ ; λ) ≤ PΘ ((β ′ − β)J ; λ) and PΘ (βJ′ c ; λ) = PΘ ((β ′ − β)J c ; λ), where J c is the complement of J and βJ is the subvector of β indexed by J . Then R1 is implied by R′1 below for given J = J (β). 9 A SSUMPTION R′1 (δ, ϑ, K, J , λ) Given X, Θ, J , λ, there exist δ > 0, ϑ > 0, K ≥ 0 such that for any ∆ ∈ Rp , PΘ (∆J ; λ) + ϑPH (∆J ; λ) + LΘ k∆k22 2 2−δ kX∆k22 + Kλ2 J + PΘ (∆J c ; λ) − ϑPH (∆J c ; λ), ≤ 2 (19) or (1 + ϑ)PΘ (∆J ; λ) + LΘ 2−δ k∆k22 ≤ kX∆k22 + Kλ2 J + (1 − ϑ)PΘ (∆J c ; λ). 2 2 (20) When Θ is the soft-thresholding, it is easy to verify that a sufficient condition for (20) is √ (1 + ϑ)k∆J k1 ≤ K J kX∆k2 + k∆J c k1 , (21) for some ϑ > 0 and K ≥ 0. (21) has a simper form than R1 . In the following, we give the definitions of the RE and the compatibility condition (Bickel et al., 2009; van de Geer and Bühlmann, 2009) to make a comparison to (21). A SSUMPTION RE(κRE , ϑRE , J ). Given J ⊂ [p], we say that X ∈ Rn×p satisfies RE(κRE , ϑRE , J ), if for positive numbers κRE , ϑRE > 0, JkX∆k22 ≥ κRE k∆J k21 , (22) kX∆k22 ≥ κRE k∆J k22 , (23) (1 + ϑRE )k∆J k1 ≥ k∆J c k1 . (24) or more restrictively, for all ∆ ∈ Rp satisfying Assume RE(κRE , ϑRE , J ) holds. When (1 + ϑRE )k∆J k1 ≤ k∆J c k√1 , (21) holds trivially with ϑ = ϑRE ; otherwise, (22) indicates (1+ϑ)k∆J k1 ≤ K JkX∆k2 √ with K = (1 + ϑRE )/ κRE . So intuitively, we have the following relationship: (23) + (24) ⇒ (22) + (24) ⇒ (21) ⇒ (20) ⇒ (19) ⇒ (18). 10 In particular, R1 is less demanding than RE. Next, let’s compare the regularity conditions required by ΘS and ΘH to achieve the nearly optimal error rate. Recall R1 (δ, ϑ, K, β, λ) and R′0 (δ, ϑ, K, β, λ) in Theorem 3 and Corollary 2, respectively 2−δ kX(β ′ − β)k22 + λkβ ′ k1 + Kλ2 J, 2 2−δ 1 kX(β ′ − β)k22 + PH (β ′ ; λ) + Kλ2 J. ϑPH (β ′ − β; λ) + kβ ′ − βk22 ≤ 2 2 ϑPH (β ′ − β; λ) + λkβk1 ≤ R′0 (δ, ϑ, K, β, λ) implies R1 (δ, ϑ, K + 1, β, λ). Indeed, for ∆ = β ′ − β, λkβk1 − λkβ ′ k1 ≤ λk∆J k1 − λkβJ′ c k1 1 1 ≤ λ2 J + k∆J k22 − PH (βJ′ c ; λ) 2 2 1 1 2 ≤ λ J + k∆J k22 − PH (β ′ ; λ) + PH (βJ′ ; λ) 2 2 1 1 2 ≤ λ J + k∆J k22 − PH (β ′ ; λ) + P0 (βJ′ ; λ) 2 2 1 ≤ λ2 J + k∆J k22 − PH (β ′ ; λ). 2 On the other hand, Corollary 2 studies when all ΘH -estimators have the optimal performance guarantee, while practically, one may initialize (10) with a carefully chosen starting point. Theorem 4. Given any Θ, there exists a Θ-estimator (which minimizes (12)) such that (16) holds without requiring any regularity condition. In particular, if Θ corresponds to a bounded nonconvex penalty as described in Corollary 2, then there exists a Θ-estimator such that (17) holds free of regularity conditions. Theorem 4 does not place any requirement on X. So it seems that applying ΘH may have some further advantages in practice. (How to efficiently pick a ΘH estimator to completely remove all regularity conditions is however beyond the the scope of the current paper. For a possible idea of relaxing the conditions, see Remark 6.) Finally, we make a discussion of the scaling parameter ρ. Our results so far are obtained after performing X ← X/ρ with ρ ≥ kXk2 . The prediction error is invariant to the transformation. But it affects the regularity conditions. 11 Seen from (8), 1/ρ2 is related to the stepsize α appearing in (1), also known as the learning rate in the machine learning literature. From the computational results in Section 2.2, ρ must be large enough to guarantee TISP is convergent. The larger the value of ρ is, the smaller the stepsize is (and so the slower the convergence is). Based on the machine learning literature, slow learning rates are always recommended when training a nonconvex learner (e.g., artificial neural networks). Perhaps interestingly, in addition to computational efficiency reasons, all our statistical analyses caution against using an extremely large scaling when LΘ > 0. For example, R′0 (δ, ϑ, K, β, λ) for an unscaled X reads ϑPH (ρ(β ′ − β); λ) + ρ2 kβ ′ − βk22 /2 ≤ (2 − δ)kX(β ′ − β)k22 /2 + PH (ρβ ′ ; λ) + Kλ2 J, which becomes difficult to hold when ρ is very large. This makes the statistical error bound break down easily. Therefore, a good idea is to have ρ just appropriately large (mildly greater than kXk2 ). The sequential analysis of the iterates in the next part also supports the point. 3.3 Sequential Algorithmic Analysis We perform statistical error analysis of the sequence of iterates defined by TISP: β (t+1) = Θ(β (t) + X T y − X T Xβ (t) ; λ), where kXk2 ≤ 1 and β (0) is the starting point. The study is motivated from the fact that in large-scale applications, Θ-estimators are seldom computed exactly. Indeed, why bother to run TISP till computational convergence? How does the statistical accuracy improve (or deteriorate) at t increases? Lately, there are some key advances on the topic. For example, Agarwal et al. (2012) showed that for convex problems (not necessarily strongly convex), proximal gradient algorithms can be geometrically fast to approach a globally optimal solution β̂ within the desired statistical precision, under a set of conditions. We however care about the statistical error between β (t) and the genuine β ∗ in this work. We will introduce two comparison regularity conditions (analogous to R0 and R1 ) to present both PΘ -type and P0 -type error bounds. Hereinafter, denote (β T Aβ)1/2 by kβkA , where A is a positive semi-definite matrix. A SSUMPTION S0 (δ, ϑ, K, β, β ′ , λ) Given X, Θ, β, β ′ , λ, there exist δ > 0, ϑ > 0, K ≥ 0 such that the following inequality holds LΘ + δ ′ kβ − βk22 2 ≤ kX(β ′ − β)k22 + PΘ (β ′ ; λ) + KPΘ (β; λ). ϑPH (β ′ − β; λ) + 12 (25) A SSUMPTION S1 (δ, ϑ, K, β, β ′ , λ) Given X, Θ, β, β ′ , λ, there exist δ > 0, ϑ > 0, K ≥ 0 such that the following inequality holds LΘ + δ ′ kβ − βk22 + PΘ (β; λ) 2 ≤ kX(β ′ − β)k22 + PΘ (β ′ ; λ) + (K + 1)λ2 J(β). ϑPH (β ′ − β; λ) + (26) (25) and (26) require a bit more than (15) and (18), respectively, due to kXk2 ≤ 1. The theorem and the corollary below perform sequential analysis of the iterates and reveal the explicit roles of δ, ϑ, K (which can often be treated as constants). ∗ (t+1) Theorem 5. Suppose S0 (δ, ϑ, pK, β , β p , λ) is satisfied for some δ > 0, ϑ > 0, K ≥ 0, then for λ = Aσ log(ep)/ (δ ∧ ϑ)ϑ with A sufficiently large, the 2 following error bound holds with probability at least 1 − Cp−cA : 1 + δ (t+1) 1 kβ − β ∗ k2(I−X T X) ≤ kβ (t) − β ∗ k2(I−X T X) + (K + 1)PΘ (β ∗ ; λ), 2 2 (27) where C, c are universal positive constants. Similarly, under the same choice of regularity parameter, if S1 (δ, ϑ, K, β ∗ , β (t) , λ) is satisfied for some δ > 0, ϑ > 0, K ≥ 0, (28) is true with probability at least 2 1 − Cp−cA : 1 + δ (t+1) 1 kβ − β ∗ k2(I−X T X) ≤ kβ (t) − β ∗ k2(I−X T X) + (K + 1)λ2 J ∗ . 2 2 (28) Corollary 3. In the setting of Theorem 5, for any initial point β (0) ∈ Rp , we have κ K ′ PΘ (β ∗ ; λ), 1−κ κ K ′ λ2 J ∗ , ≤ κt kβ (0) − β ∗ k2(I−X T X) + 1−κ kβ (t) − β ∗ k2(I−X T X) ≤ κt kβ (0) − β ∗ k2(I−X T X) + (29) kβ (t) − β ∗ k2(I−X T X) (30) under S0 (δ, ϑ, K, β ∗ , β (s) , λ) and S1 (δ, ϑ, K, β ∗ , β (s) , λ), 0 ≤ s ≤ t − 1, respec2 tively, with probability at least 1 − Cp−cA . Here, κ = 1/(1 + δ), K ′ = 2(K + 1). Remark 5. We can get some sufficient conditions for S0 and S1 , similar to the discussions made in Section 3.2. When kXk2 is strictly less than 1, (25) can be relaxed to ϑPH (β ′ − β; λ) + (LΘ + δ)kβ ′ − βk22 /2 ≤ (2 + δ)kX(β ′ − β)k22 /2 + 13 PΘ (β ′ ; λ) + KPΘ (β; λ) for some δ > 0. The proof in Section 4.4 also gives expectation-form results, with an additional additive term Cσ 2 /(δ ∧ ϑ) in the upper bounds. Similar to Remark 4, we can also study Θ-iterates with stepsize α, in which case the weighting matrix in (27)-(30) changes from I −X T X to I/α − X T X, and the factor (LΘ + δ)/2 in (25) and (26) is replaced by (LΘ + δ)/(2α). Remark 6. Theorem 5 still applies when δ, ϑ, K and λ are dependent on t. For example, if we use a varying threshold sequence, i.e., β (t+1) = Θ(β (t) + X T y − X T Xβ (t) ; λ(t) ), then (30) becomes kβ (t) − β ∗ k2(I−X T X) t ≤ κ kβ (0) − β ∗ k2(I−X T X) ′ +K J ∗ t−1 X κt−s λ2s . s=0 This allows for much larger values of λs to be used in earlier iterations to attain the same accuracy. It relaxes the regularity condition required by applying a fixed threshold level. At the end, we re-state some results under ρ > kXk2 , to get more intuition and implications. For a general X (unscaled), (30) reads kβ (t) − β ∗ k2(ρ2 I−X T X) ≤ κt kβ (0) − β ∗ k2(ρ2 I−X T X) + κ K ′ σ 2 λ2 J ∗ . 1−κ Set ρ to be a number slightly larger than kXk2 , i.e., ρ = (1+ǫ)kXk2 , ǫ > 0. Then, we know that the prediction error kXβ (t) − Xβ ∗ k22 decays geometrically fast to O(σ 2J ∗ log(ep)) with high probability, when ǫ, δ, ϑ, K are viewed as constants; a similar conclusion is true for the estimation error. This is simply due to ρ2 − kXk22 (t) kβ −β ∗ k2X T X ≤ (ρ2 −kXk22 )kβ (t) −β ∗ k22 ≤ kβ (t) −β ∗ k2(ρ2 I−X T X) . 2 kXk2 Accordingly, there is no need to run TISP till convergence—one can terminate the algorithm earlier, at, say, tmax = log{ρ2 kβ (0) − β ∗ k2 /(Kσ 2 λ2 J ∗ )} /log(1/κ), without sacrificing much statistical accuracy. The formula also reflects that the quality of the initial point affects the required iteration number. There are some related results in the literature. (i) As mentioned previously, in a broad convex setting Agarwal et al. (2012) proved the geometric decay of the optimization error kβ (t) − β̂k to the desired statistical precision, where β̂ is the convergent point. Loh and Wainwright (2015) extended the conclusion to a family of nononvex optimization problems, and they showed that when some 14 regularity conditions hold, every local minimum point is close to the authentic β ∗ . In comparison, our results are derived toward the statistical error between β (t) and β ∗ directly, without requiring all local minimum points to be statistically accurate. (ii) Zhang (2010b) showed a similar fast-converging statistical error bound for an elegant multi-stage capped-ℓ1 regularization procedure. However, the procedure carries out an expensive ℓ1 optimization at each step. Instead, (10) involves a simple and cheap thresholding, and our analysis covers any Θ. Acknowledgement The author would like to thank the editor, the associated editor and two anonymous referees for their careful comments and useful suggestions that improve the quality of the paper. The author also appreciates Florentina Bunea for the encouragement. This work was supported in part by NSF grant DMS-1352259. 4 Proofs Throughout the proofs, we use C, c, L to denote universal non-negative constants. They are not necessarily the same at each occurrence. Given any matrix A, we use R(A) to denote its column space. Denote by PA the orthogonal projection matrix onto R(A), i.e., PA = A(AT A)+ AT , where + stands for the MoorePenrose pseudoinverse. Let [p] := {1, · · · , p}. Given J ⊂ [p], we use XJ to denote a column submatrix of X indexed by J . Definition 2. ξ is called a sub-Gaussian random variable if there exist constants 2 C, c > 0 such that P{\|ξ\| ≥ t} ≤ Ce−ct , ∀t > 0. The scale (ψ2 -norm) for ξ is defined as σ(ξ) = inf{σ > 0 : E exp(ξ 2/σ 2 ) ≤ 2}. ξ ∈ Rp is called a sub-Gaussian random vector with scale bounded by σ if all one-dimensional marginals hξ, αi are sub-Gaussian satisfying khξ, αikψ2 ≤ σkαk2 , ∀α ∈ Rp . Examples include Gaussian random variables and bounded random variables such as Bernoulli. Note that the assumption that vec (ǫ) is sub-Gaussian does not imply that the components of ǫ must be i.i.d. We begin with two basic facts. Because they are special cases of Lemma 1 and Lemma 2 in She (2012), respectively, we state them without proofs. 15 Lemma 1. Given an arbitrary thresholding rule Θ, let P be any function satisR \|θ\| fying P (θ; λ) − P (0; λ) = PΘ (θ; λ) + q(θ; λ) where PΘ (θ; λ) , 0 (sup{s : Θ(s; λ) ≤ u} − u) du, q(θ; λ) is nonnegative and q(Θ(t; λ)) = 0 for all t. Then, β̂ = Θ(y; λ) is always a globally optimal solution to minβ 12 ky − βk22 + P (\|β\|; λ). It is the unique optimal solution provided Θ(·; λ) is continuous at \|y\|. Lemma 2. Let Q0 (β) = ky − βk22 /2 + PΘ (\|β\|; λ). Denote by β̂ the unique minimizer of Q0 (β). Then for any δ, Q0 (β̂ + δ) − Q0 (β̂) ≥ (1 − LΘ )kδk22 /2. 4.1 Proof of Theorem 1 Let s(u; λ) := Θ−1 (u; λ) − u for u ≥ 0. Assume β̂ is a local minimum point (the proof for a coordinate-wise minimum point follows the same lines). We write fΘ as f for simplicity. Let δf (β; h) denote the Gateaux differential of f at β with (β) . By the definition of PΘ , δf (β, h) increment h: δf (β; h) = limǫ→0+ f (β+ǫh)−f ǫ 1 p exists for any h ∈ R . Let l(β) = 2 kXβ − yk22 . We consider the following directional vectors: dj = [d1 , · · · , dp ]T with dj = ±1 and dj ′ = 0, ∀j ′ 6= j. Then for any j, δl(β; dj ) = dj xTj (Xβ − y), ( s(\|βj \|)sgn(βj )dj , δPΘ (β; dj ) = s(\|βj \|), (31) if βj = 6 0, if βj = 0. (32) Due to the local optimality of β̂, δf (β̂; dj ) ≥ 0, ∀j. When β̂1 6= 0, we obtain T x1 (X β̂−y)+s(\|β̂1\|; λ)sgn(β̂1 ) = 0. When β̂1 = 0, xT1 (X β̂−y)+s(\|β̂1 \|; λ) ≥ 0 and −xT1 (X β̂−y)+s(\|β̂1 \|; λ) ≥ 0, i.e., \|xT1 (X β̂−y)\| ≤ s(\|β̂1 \|; λ) = Θ−1 (0; λ). To summarize, when f achieves a local minimum or a coordinate-wise minimum (or more generally, a local coordinate-wise minimum) at β̂, we have β̂j 6= 0 ⇒ Θ−1 (\|β̂j \|; λ)sgn(β̂j ) = β̂j − xTj (X β̂ − y) β̂j = 0 ⇒ Θ(xTj (Xβ − y); λ) = 0 (33) (34) When Θ is continuous at β̂j − xTj (X β̂ − y), (33) implies that β̂j = Θ(β̂j − xTj (X β̂ − y); λ). Hence β̂ must be a Θ-estimator satisfying β = Θ(β + X T y − X T Xβ; λ). 16 4.2 Proofs of Theorem 2 and Theorem 3 Given Θ, let β̂ be any Θ-estimator, β be any p-dimensional vector (non-random) and ∆ = β̂ − β. The first result constructs a useful criterion for β̂ on basis of Lemma 1 and Lemma 2. Lemma 3. Any Θ-estimator β̂ satisfies the following inequality for any β ∈ Rp 1 1 kX(β̂ − β ∗ )k22 + ∆T (X T X − LΘ I)∆ 2 2 1 ∗ 2 ≤ kX(β − β )k2 + PΘ (β; λ) − PΘ (β̂; λ) + hǫ, X∆i, 2 (35) where ∆ = β̂ − β. To handle hǫ, X∆i, we introduce another lemma. p Lemma 4. Suppose kXk2 ≤ 1 and let λo = σ log(ep). Then there exist universal constants A1 , C, c > 0 such that for any constants a ≥ 2b > 0, the following event √ 1 1 sup {2hǫ, Xβi − kXβk22 − [PH (β; abA1 λo )]} ≥ aσ 2 t a b β∈Rp (36) 2 occurs with probability at most C exp(−ct)p−cA1 , where t ≥ 0. The lemma plays an important role in bounding the last stochastic term in (35). Its proof is based on the following results. Lemma 5. Suppose kXk2 ≤ 1. There exists a globally optimal solution β o to minβ 12 ky − Xβk22 + PH (β; λ) such that for any j : 1 ≤ j ≤ p, either βjo = 0 or \|βjo\| ≥ λ. p Lemma 6. Given X ∈ Rn×p and J : 1 ≤ J ≤ p, define Γ′J = {α ∈ R : kαk2 ≤ p ′ 2 1, α ∈ R(XJ ) for some J : \|J \| = J}. Let Po (J) = σ {J + log J }. Then for any t ≥ 0, ! p (37) P sup hǫ, αi ≥ tσ + LPo′ (J) ≤ C exp(−ct2 ), α∈Γ′J where L, C, c > 0 are universal constants. 17 √ 1 1 Let R = sup1≤J≤p sup∆∈ΓJ {hǫ, X∆i − 2b PH (∆; abA1 λo ) − 2a kX∆k22 }, o with λ , A1 given in Lemma 4. (The starting value of J is 1 because when J(∆) = 0, hǫ, X∆i = 0.) Substituting it into (35) gives 1 1 kX(β̂ − β ∗ )k22 + ∆T (2X T X − LΘ I)∆ 2 2 √ 1 1 ∗ 2 ≤ kX(β − β )k2 + PΘ (β; λ) − PΘ (β̂; λ) + PH (∆; abA1 λo ) 2 2b 1 1 2 2 + kX∆k2 + kX∆k2 + R 2a 2 √ 1 1 ∗ 2 ≤ kX(β − β )k2 + PΘ (β; λ) − PΘ (β̂; λ) + PH (∆; abA1 λo ) 2 2b 1 1 2 + (1 + )kX∆k2 + R. 2 a Because P(R ≥ aσ 2 t) ≤ C exp(−ct), we know E[R] . aσ 2 . √ Let λ = Aλo with A = A1 ab and set b ≥ 1/(2ϑ). The regularity condition R0 (δ, ϑ, K, β, λ) implies that LΘ 2−δ 1 PH (∆; λ) + k∆k22 ≤ kX∆k22 + PΘ (β̂; λ) + KPΘ (β; λ). (38) 2b 2 2 Choose a to satisfy a > 1/δ, a ≥ 2b. Combining the last two inequalities gives E[kX(β̂ − β ∗ )k22 ] ≤kX(β − β ∗ )k22 + 2(K + 1)PΘ (β; λ) + E[(1 + .kX(β − β ∗ )k22 + PΘ (β; λ) + σ 2 , 1 − δ)kX∆k22 ] + 2 E[R] a (39) with the last inequality due to kX∆k22 ≤ (1+1/c)kX(β−β ∗ )k22 +(1+c)kX(β̂− β ∗ )k22 for any c > 0. The proof of Theorem 3 follows the lines of the proof of Theorem 2, with (38) replaced by 1 LΘ 2−δ PH (∆; λ) + k∆k22 + PΘ (β; λ) ≤ kX∆k22 + PΘ (β̂; λ) + Kλ2 J(β), 2b 2 2 and (39) replaced by E[kX(β̂ − β ∗ )k22 ] ≤kX(β − β ∗ )k22 + 2Kλ2 J(β) + E[(1 + .kX(β − β ∗ )k22 + λ2 J(β) + σ 2 . 18 1 − δ)kX∆k22 ] + 2 E[R] a The details are omitted. 4.3 Proof of Theorem 4 From the proof of Lemma 5, there exists a Θ-estimator β̂ which minimizes f (β) = l(β) + PΘ (β; λ). This means that the term 12 ∆T (X T X − LΘ I)∆ can be dropped from (35). Following the lines of Section 4.2, (17) holds under a modified version of R0 (δ, ϑ, K, β, λ), which replaces (15) with ϑPH (β ′ − β; λ) ≤ 1−δ kX(β ′ − β)k22 + PΘ (β ′ ; λ) + KPΘ (β; λ). 2 (40) Using the sub-additivity of PH , we know that any design matrix satisfies (40) for any 0 < ϑ ≤ 1, δ ≤ 1, K ≥ ϑ. 4.4 Proof of Theorem 5 and Corollary 3 Let f (β) = l(β) + PΘ (β; λ) where l(β) = 21 kXβ − yk22 . Lemma 7. Let β (t+1) = Θ(β (t) + X T y − X T Xβ (t) ; λ). Then the following ‘triangle inequality’ holds for any β ∈ Rp 1 1 − LΘ (t+1) kβ − βk22 + kβ (t+1) − β (t) k2I−X T X 2 2 1 (t) ≤ kβ − βk2I−X T X + f (β) − f (β (t+1) ). 2 Letting β = β ∗ in the lemma, we have 1 (t+1) 1 kβ − β ∗ k2X T X+(1−LΘ )I + kβ (t+1) − β (t) k2I−X T X + PΘ (β (t+1) ; λ) 2 2 1 (t) ∗ 2 (t+1) ≤ kβ − β kI−X T X + hǫ, X(β − β ∗ )i + PΘ (β ∗ ; λ). 2 Moreover, under S0 (δ, ϑ, K, β ∗ , β ′ , λ) with β ′ = β (t+1) , 1 + δ (t+1) kβ − β ∗ k22 − KPΘ (β ∗ ; λ) 2 1 (t+1) 1 ∗ 2 ≤ kβ − β kX T X+(1−LΘ )I + PΘ (β (t+1) ; λ) + kβ (t+1) − β ∗ k2X T X . 2 2 ϑPH (β (t+1) − β ∗ ; λ) + 19 Combining the last two inequalities gives 1 + δ (t+1) 1 kβ − β ∗ k2I−X T X + kβ (t+1) − β (t) k2I−X T X 2 2 δ (t+1) − β ∗ k2X T X + ϑPH (β (t+1) − β ∗ ; λ) + kβ 2 1 ≤ kβ (t) − β ∗ k2I−X T X + (K + 1)PΘ (β ∗ ; λ) + hǫ, X(β (t+1) − β ∗ )i. 2 p Let ΓJ = {β ∈ Rp : J(β) = J}, λo = σ log(ep). We define an event E with its complement given by √ 1 1 E c , {sup{2hǫ, Xβi − kXβk22 − [PH (β; abA1 λo )]} ≥ 0}. a b β By Lemma 4, there exists a universal constant L such that for any A21 ≥ L, a ≥ 2 2b > 0, P (E c) ≤ Cp−cA1 . Clearly, E implies √ 1 1 kβ (t+1) − β ∗ k2X T X + PH (β (t+1) − β ∗ ; abA1 λo ). 2a 2b (41) √ o √ Take b = 1/(2ϑ), a = 1/(δ ∧ ϑ), A1 ≥ L, and λ = A1 abλ . Then, on E we get the desired statistical accuracy bound hǫ, X(β (t+1) − β ∗ )i ≤ 1 + δ (t+1) 1 kβ − β ∗ k2I−X T X ≤ kβ (t) − β ∗ k2I−X T X + (K + 1)PΘ (β ∗ ; λ). 2 2 The bound under S1 can be similarly proved. Noticing that (41) holds for any t, Corollary 3 is immediately true. 4.5 Proofs of Lemmas 4.5.1 Proof of Lemma 3 Let f (β) = l(β) + PΘ (β; λ) with l(β) = 12 kXβ − yk22. Define 1 g(β, γ) = l(β) + h∇l(β), γ − βi + kγ − βk22 + PΘ (γ; λ). 2 Given β, g(β, γ) can be expressed as 1 kγ − (β − ∇l(β))k22 + PΘ (γ; λ) + c(β), 2 20 (42) where c(β) depends on β only. Let β̂ be a Θ-estimator satisfying β̂ = Θ(β̂ − X T X β̂ + X T y; λ). Based on Lemma 1 and Lemma 2, we have g(β̂, β̂ + ∆) − g(β̂, β̂) ≥ 1 − LΘ k∆k22 , 2 from which it follows that 1 f (β̂ + ∆) − f (β̂) ≥ ∆T (X T X − LΘ I)∆. 2 This holds for any ∆ ∈ Rp . 4.5.2 Proof of Lemma 4. Let 1 lH (β) = 2hǫ, Xβi − kXβk22 − a 1 l0 (β) = 2hǫ, Xβi − kXβk22 − a √ 1 [PH (β; abA0 λo )] b √ 1 [P0 (β; abA0 λo )], b and EH = {supβ∈Rp lH (β) ≥ atσ 2 }, and E0 = {supβ∈Rp l0 (β) ≥ atσ 2 }. Because P0 ≥ PH , E0 ⊂ EH . We prove that EH = E0 . The occurrence of EH implies that lH (β o ) ≥ atσ 2 for any β o defined by √ 1 1 β o ∈ arg min kXβk22 − 2hǫ, Xβi + [PH (β; abA0 λo )], β a b With a ≥ 2b > 0, Lemma exists at least one global minimizer √ 5 states that there √ β oo satisfying PH (β oo ; abA1 λo ) = P0 (β oo ; abA1 λo ) and thus lH (β oo ) = l0 (β oo ). This means that sup l0 (β) ≥ l0 (β oo ) = lH (β oo ) ≥ atσ 2 . So EH ⊂ E0 , and it suffices to prove E0c occurs with high probability, or more specifically, P(E0 ) ≤ 2 C exp(−ct)p−cA1 . Given 1 ≤ J ≤ p, define ΓJ = √{β ∈ Rp : J(β) = J}. Let R = 1 1 P0 (β; abA1 λo ) − 2a kXβk22 }. We will use sup1≤J≤p supβ∈ΓJ {hǫ, Xβi − 2b Lemma 6 to bound its tail probability. Let Po′ (J) = σ 2 {J + log Jp }. We claim that P[ sup {hǫ, Xβi − β∈ΓJ 1 kXβk22 − aLPo′ (J)} > atσ 2 ] ≤ C exp(−ct). 2a 21 (43) Indeed, 1 2hǫ, Xβi − kXβk22 − 2aLPo′ (J) a p 1 ≤2hǫ, Xβ/kXβk2 ikXβk2 − 2kXβk2 LPo′ (J) − kXβk22 2a p 1 =2kXβk2 hǫ, Xβ/kXβk2 i − LPo′ (J) − kXβk22 2a p 1 ≤2kXβk2 hǫ, Xβ/kXβk2 i − LPo′ (J) − kXβk22 2a + 2 p ≤2a hǫ, Xβ/kXβk2 i − LPo′ (J) , (44) + where the last inequality is due to Cauchy-Schwarz inequality. (43) now follows from Lemma 6. √ Set A1 ≥ 4 L. We write P0 (β; λo ) with β ∈ ΓJ as P0 (J; λo ). Noticing ′ some basic facts that ≤ CP0p (J; λo ) due to Stirling’s p (i) Po (J) ≤ CJ log(ep) p approximation, (ii) (A21 /2)P0 (J; λo ) ≥ LPo′ (J) + cA21 P0 (J; λo ) for some c > 0, and (iii) J log(ep) ≥ log p + J for any J ≥ 1, we get P(R ≥ aσ 2 t) ! 2 p q X P a sup hǫ, Xβ/kXβk2 i − (A21 /2)P0 (J; λo ) ≤ ≥ aσ 2 t J=1 p β∈ΓJ + q X √ P( sup hǫ, αi − (A21 /2)P0(J; λo ) ≥ σ t) = J=1 p ≤ X ≤ X J=1 p α∈Γ′J q p √ ′ P( sup hǫ, αi − LPo (J) ≥ tσ + cA21 P0 (J; λo )) α∈Γ′J C exp(−ct) exp{−cA21 (J + log(p))} J=1 ≤C exp(−ct) p X exp(−cA21 log p) exp(−cA21 J) J=1 −cA21 ≤C exp(−ct)p , where the last inequality due to the sum of geometric series. 22 4.5.3 Proof of Lemma 5. Similar to the proof of Lemma 3, we set fH (β) = l(β) + PH (β; λ) with l(β) = 1 kXβ − yk22 and construct gH (β, γ) = fH (γ) + 12 kγ − βk22 − (l(γ) − l(β) − 2 h∇l(β), γ − βi). Under kXk2 ≤ 1, for any (β, γ), 1 gH (β, γ) − fH (γ) = (γ − β)T (I − X T X)(γ − β) ≥ 0. 2 o Let β be a globally optimal solution to minβ fH (β). Then γ o := ΘH (β o − X T Xβ o + X T y; λ) gives fH (γ o ) ≤ gH (β o , γ o ) ≤ gH (β o , β o ) = fH (β o ), with the second inequality due to Lemma 1. Therefore, γ o must also be a global minimizer of fH , and by definition, γ o demonstrates a threshold gap as desired. 4.5.4 Proof of Lemma 6. By definition, {hǫ, αi : α ∈ Γ′J } is a stochastic process with sub-Gaussian increments. The induced metric on Γ′J is Euclidean: d(α1 , α2 ) = σkα1 − α2 k2 . To bound the metric entropy log N (ε, Γ′J , d), where N (ε, Γ′J , d) is the smallest cardinality of an ε-net that covers Γ′J under d, we notice that α is in a Jdimensional ball in Rp . The number of such balls {PXJ ∩ Bp (0, 1) : J ⊂ [p]} is at most Jp , where Bp (0, 1) denotes the unit ball in Rp . By a standard volume argument (see, e.g., Vershynin (2012)), p p Cσ J ′ + J log(Cσ/ε), (45) ) = log ( log N (ε, Γr,J , d) ≤ log ε J J where C is a universal constant. The conclusion follows from Dudley’s integral bound (Talagrand, 2005). 4.5.5 Proof of Lemma 7 We use the notation in the proof of Lemma 3 with g defined in (42). By Lemma Θ 1 and Lemma 2, we obtain g(β (t) , β) − g(β (t) , β (t+1) ) ≥ 1−L kβ (t+1) − βk22 , 2 namely, 1 h∇l(β (t) ), β − β (t+1) i + PΘ (β) − PΘ (β (t+1) ) + kβ − β (t) k22 2 1 (t) 1 − L Θ − kβ − β (t+1) k22 ≥ kβ (t+1) − βk22 . 2 2 23 To cancel the first-order term, we give two other inequalities based on secondorder lower/upper bounds: 1 l(β) − l(β (t) ) − h∇l(β (t) ), β − β (t) i ≥ kβ (t) − βk2X T X , 2 1 l(β (t) ) + h∇l(β (t) ), β (t+1) − β (t) i − l(β (t+1) ) ≥ − kβ (t+1) − β (t) k2X T X . 2 Adding the three inequalities together gives the triangle inequality. References Agarwal, A., Negahban, S., and Wainwright, M. J. (2012). Fast global convergence of gradient methods for high-dimensional statistical recovery. Ann. Statist., 40(5):2452–2482. Bickel, P. J., Ritov, Y., and Tsybakov, A. B. (2009). 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arXiv:1709.07708v1 [math.GR] 22 Sep 2017 COMPATIBLE ACTIONS AND NON-ABELIAN TENSOR PRODUCTS VALERIY G. BARDAKOV AND MIKHAIL V. NESHCHADIM Abstract. For a pair of groups G, H we study pairs of actions G on H and H on G such that these pairs are compatible and non-abelian tensor products G ⊗ H are defined. 1. Introduction R. Brown and J.-L. Loday [1, 2] introduced the non-abelian tensor product G ⊗ H for a pair of groups G and H following works of C. Miller [6], and A. S.-T. Lue [5]. The investigation of the non-abelian tensor product from a group theoretical point of view started with a paper by R. Brown, D. L. Johnson, and E. F. Robertson [3]. The non-abelian tensor product G ⊗ H depends not only on the groups G and H but also on the action of G on H and on the action of H on G. Moreover these actions must be compatible (see the definition in Section 2). In the present paper we study the following question: what actions are compatible? The paper is organized as follows. In Section 2, we recall a definition of non-abelian tensor product, formulate some its properties and give an answer on a question of V. Thomas, proving that there are nilpotent group G and some group H such that in G ⊗ H the derivative subgroup [G, H] is equal to G. In the Section 3 we study the following question: Let a group H acts on a group G by automorphisms, is it possible to define an action of G on H such that this pair of actions are compatible? Some necessary conditions for compatibility of actions will be given and in some cases will be prove a formula for the second action if the first one is given. In the Section 4 we construct pairs compatible actions for arbitrary groups and for 2-step nilpotent groups give a particular answer on the question from Section 3. In Section 5 we study groups of the form G ⊗ Z2 and describe compatible actions. 2. Preliminaries In this article we will use the following notations. For elements x, y in a group G, the conjugation of x by y is xy = y −1 xy; and the commutator of x and y is [x, y] = x−1 xy = x−1 y −1 xy. We write G′ for the derived subgroup of G, i.e. G′ = [G, G]; Gab for the abelianized group G/G′ ; the second hypercenter ζ2 G of G is the subgroup of G such that ζ2 G/ζ1 G = ζ1 (G/ζ1 G), Date: March 24, 2018. 2010 Mathematics Subject Classification. Primary 20E22; Secondary 20F18, 20F28. Key words and phrases. tensor product; compatible action, nilpotent group. 1 2 V. G. BARDAKOV AND M. V. NESHCHADIM where ζ1 G = Z(G) is the center of a group G. Recall the definition of the non-abelian tensor product G ⊗ H of groups G and H (see [1, 2]). It is defined for a pair of groups G and H where each one acts on the other (on right) G × H −→ G, (g, h) 7→ g h ; H × G −→ H, (h, g) 7→ hg and on itself by conjugation, in such a way that for all g, g1 ∈ G and h, h1 ∈ H, −1 g h1 h g1 h1 g1−1 (hg1 ) . and h(g ) = hh1 g = g In this situation we say that G and H act compatibly on each other. The non-abelian tensor product G ⊗ H is the group generated by all symbols g ⊗ h, g ∈ G, h ∈ H, subject to the relations gg1 ⊗ h = (gg1 ⊗ hg1 )(g1 ⊗ h) and g ⊗ hh1 = (g ⊗ h1 )(gh1 ⊗ hh1 ) for all g, g1 ∈ G, h, h1 ∈ H. In particular, as the conjugation action of a group G on itself is compatible, then the tensor square G⊗G of a group G may always be defined. Also, the tensor product G ⊗ H is defined if G and H are two normal subgroups of some group M and actions are conjugations in M . The following proposition is well known. We give a proof only for fullness. Proposition 2.1. 1) Let G and H be abelian groups. Independently on the action of G on H and H on G, the group G ⊗ H is abelian. 2) (See [2, Proposition 2.4]) Let G and H be arbitrary groups. If the actions of G on H and H on G are trivial, then the group G ⊗ H ∼ = Gab ⊗Z H ab is the abelian tensor product. Proof. 1) We have the equality (g ⊗ h)g1 ⊗h1 = g [g1 ,h1] ⊗ h[g1 ,h1 ] , where g[g1 ,h1 ] is the action of the commutator [g1 , h1 ] ∈ G by conjugation on g, but G is abelian and g [g1 ,h1 ] = g. Analogously, h[g1 ,h1] = h. Hence, G ⊗ H is abelian. 2) From the previous formula and triviality actions we have −1 g1 h1 −1 g1 h1 −1 h−1 1 g 1 h1 g 1 h1 [g1 ,h1 ] g1−1 h−1 1 = g h1 = g. = g g1 = g g1 g =g = g g1 Analogously, h[g1 ,h1] = h. Hence, G ⊗ H is abelian. Remind presentation of non-abelian tensor product as a central extension (see [4]). The derivative subgroup of G by H is called the following subgroup DH (G) = [G, H] = hg −1 g h \| g ∈ G, h ∈ Hi. The map κ : G ⊗ H −→ DH (G) defined by κ(g ⊗ h) = g −1 gh is a homomorphism, its kernel A = ker(κ) is the central subgroup of G ⊗ H and G acts on G ⊗ H by the rule (g ⊗ h)x = gx ⊗ hx , x ∈ G, i.e. there exists the short exact sequence 1 −→ A −→ G ⊗ H −→ DH (G) −→ 1. COMPATIBLE ACTIONS AND NON-ABELIAN TENSOR PRODUCTS 3 In this case A can be viewed as Z[DH (G)]-module via conjugation in G ⊗ H, i.e. under the action induced by setting a · g = x−1 ax, a ∈ A, x ∈ G ⊗ H, κ(x) = g. The following proposition gives an answer on the following question: is there nonabelian tensor product G ⊗ H such that [G, H] = G? which of V. Thomas formulated in some letter to the authors. Proposition 2.2. Let G = Fn /γk Fn , k ≥ 2, be a free nilpotent group of rank n ≥ 2 and H = Aut(G) is its automorphism group. Then DH (G) = [G, H] = G. Proof. Let Fn be a free group of rank n ≥ 2 with the basis x1 , . . . , xn , G = Fn /γk Fn be a free k − 1-step nilpotent group for k ≥ 2. Let G acts trivially on H and elements of H act by automorphisms on G. It is easy to see that these actions are compatible. Let us show that in this case [G, H] = G. To do it, let us prove that x1 lies in [G, H]. Take ϕ1 ∈ H = Aut(G), which acts on the generators of G by the rules: ϕ1 ϕ1 ϕ1 1 xϕ 1 = x1 , x2 = x2 x1 , x3 = x3 , . . . , xn = xn . Then ϕ1 −1 ϕ1 −1 ϕ1 −1 ϕ1 x−1 1 x1 = 1, x2 x2 = x1 , x3 x3 = 1, . . . , xn xn = 1. Hence the generator x1 lies in [G, H]. Analogously, x2 , x3 , . . . , xn lie in [G, H]. This completes the proof. 3. What actions are compatible? In this section we study Question 1. Let a group H acts on a group G by automorphisms. Is it possible to define an action of G on H such that this pair of actions are compatible? Consider some examples. Example 3.1. Let us take G = {1, a, a2 } ∼ = Z3 , H = {1, b, b2 } ∼ = Z3 . In dependence on actions we have three cases. 1) If the action of H on G and the action of G on H are trivial, then by the second part of Proposition 2.1 G ⊗ H = Z3 ⊗Z Z3 ∼ = Z3 is abelian tensor product. 2) Let H acts non-trivially on G, i.e. ab = a2 and the action G on H is trivial. It is not difficult to check that G and H act compatibly on each other. To find DH (G) = [G, H] we calculate [a, b] = a−1 ab = a2 a2 = a. Hence, DH (G) = G. But DG (H) = 1. By the definition, G ⊗ H is generated by elements a ⊗ b, a2 ⊗ b, a ⊗ b2 , a2 ⊗ b2 . Using the defining relations: gg1 ⊗ h = (g g1 ⊗ hg1 )(g1 ⊗ h), g ⊗ hh1 = (g ⊗ h1 )(gh1 ⊗ hh1 ), 4 V. G. BARDAKOV AND M. V. NESHCHADIM we find a2 ⊗b = (aa ⊗ba )(a⊗b) = (a⊗b)2 , a⊗b2 = (a⊗b)(ab ⊗bb ) = (a⊗b)(a2 ⊗b) = (a⊗b)3 . On the other side 1 = a2 a ⊗ b = (a2 ⊗ ba )(a ⊗ b) = (a ⊗ b)3 . Hence, a ⊗ b2 = a2 ⊗ b2 = 1 and in this case we have the same result: Z3 ⊗ Z3 = Z3 . 3) Let H acts non-trivially on G, i.e. ab = a2 and G acts non-trivially on H. In this case G and H act non-compatibly on each other. Indeed, a) a(b but Hence, the equality a a−1 b a a 2 = ab = (a2 )b = a, (ba ) does not hold. = (ab )a = (a2 )2 = a2 . b a a−1 a = Let G, H be some groups. Actions of G on H and H on G are defined by homomorphisms β : G → Aut(H), α : H → Aut(G), and by definition gh = gα(h) , hg = hβ(g) , g ∈ G, h ∈ H. The actions (α, β) are compatible, if g 1 −1 α(h) β(g1 ) )= g g1 gα(h and β (g α(h1 ) ) h = h−1 1 h β(g) h1 for all g, g1 ∈ G, h, h1 ∈ H. In this case we will say that the pair (α, β) is compatible. Rewrite these equalities in the form (1) α hβ(g1 ) = gb1 −1 α(h)gb1 and c1 −1 β(g)c h1 , β gα(h1 ) = h (2) where b g is the inner automorphism of G which is induced by conjugation of g, i.e. gb : g1 7→ g−1 g1 g, g, g1 ∈ G, and analogously, b h is the inner automorphism of H which is induced by the conjugation of h, i.e. b h : h1 7→ h−1 h1 h, h, h1 ∈ H. COMPATIBLE ACTIONS AND NON-ABELIAN TENSOR PRODUCTS 5 Theorem 3.2. 1) If the pair (α, β) defines compatible actions of H on G and G on H, then the following inclusions hold NAut(G) (α(H)) ≥ Inn(G), NAut(H) (β(G)) ≥ Inn(H). Here Inn(G) and Inn(H) are the subgroups of inner automorphisms. 2) If α : H → Aut(G) is an embedding and NAut(G) (α(H)) ≥ Inn(G), then defining β : G → Aut(H) by the formula β(g) : h 7→ α−1 gb−1 α(h)b g , h ∈ H, we get the compatible actions (α, β). Proof. The first claim immediately follows from the relations (1), (2). To prove the second claim it is enough to check (2), or that is equivalent, the equality β(g) h1 β (g α(h1 ) ) h−1 1 . (3) h = h Using the definition β, rewrite the left side of (3): −1 α(h1 ) \ ) = α−1 g\ α(h1 ) α(h)g α(h1 ) . hβ (g (4) Rewrite the right side of (3): β(g) h1 −1 β(g) −1 −1 h−1 = h−1 h1 = h−1 g α(h1 hh−1 g )h1 . h 1 1 (h1 hh1 ) 1 α (b 1 )b (5) From (4) and (5): −1 α α(h1 ) g\ −1 α(h1 ) α(h)g\ −1 −1 = h−1 g α(h1 hh−1 g )h1 . 1 α (b 1 )b Using the homomorphism α: α(h1 ) g\ −1 α(h1 ) = α h−1 α−1 (b g )h1 = α(h)g\ g−1 α(h1 hh−1 1 )b 1 = α(h1 )−1 gb−1 α(h1 hh−1 g α(h1 ) = 1 )b α(h1 ) = α(h1 )−1 gb−1 α(h1 )α(h)α(h1 )−1 )b g α(h1 ) = g\ −1 α(h1 ) . α(h)g\ In the last equality we used the formula Hence, the equality (3) holds. α(h1 ) . α(h1 )−1 gbα(h1 ) = g\ Question 2. Are the inclusions NAut(G) (α(H)) ≥ Inn(G), NAut(H) (β(G)) ≥ Inn(H) sufficient for compatibility of the pare (α, β)? 6 V. G. BARDAKOV AND M. V. NESHCHADIM 4. Compatible actions for nilpotent groups At first, recall the following definition. Definition 4.1. Let G and H be groups and G1 E G, H1 E H are their normal subgroups. We will say that G is comparable with H with respect to the pare (G1 , H1 ), if there are homomorphisms ϕ : G −→ H, ψ : H −→ G, such that x ≡ ψϕ(x)(mod G1 ), for all x ∈ G, y ∈ H, i.e. y ≡ ϕψ(y)(mod H1 ) x−1 · ψϕ(x) ∈ G1 , y −1 · ϕψ(y) ∈ H1 . Note that if G1 = 1, H1 = 1, then ϕ, ψ are mutually inverse isomorphisms. The following theorem holds. Theorem 4.2. Let G, H be groups and there exist homomorphisms ϕ : G −→ H, ψ : H −→ G, such that x ≡ ψϕ(x)(mod ζ2 G), y ≡ ϕψ(y)(mod ζ2 H) for all x ∈ G, y ∈ H. Then the action of G on H and the action of H on G by the rules xy = ψ(y)−1 xψ(y), y x = ϕ(x)−1 yϕ(x), x ∈ G, y ∈ H, are compatible, i.e. the following equalities hold x(y x1 ) = ((xx1 )y )x1 , −1 y (x y1 ) = ((y y1 )x )y1 , −1 x, x1 ∈ G, y, y1 ∈ H. Proof. Let us prove that the following relation holds x(y x1 ) = ((xx1 )y )x1 . −1 For this denote the left hand side of this relation by L and transform it: L = x(y x1 ) = xϕ(x1 ) −1 yϕ(x 1) = ψ(ϕ(x1 )−1 y −1 ϕ(x1 ))xψ(ϕ(x1 )−1 yϕ(x1 )) = = (ψϕ(x1 ))−1 ψ(y)−1 (ψϕ(x1 ))x(ψϕ(x1 ))−1 ψ(y)(ψϕ(x1 )) = −1 −1 −1 = (c(x1 )−1 x−1 1 ψ(y) x1 c(x1 ))x(c(x1 ) x1 ψ(y)x1 c(x1 )). Here ψϕ(x1 ) = x1 c(x1 ), c(x1 ) ∈ ζ2 G. Since c(x1 ) ∈ ζ2 G, then the commutator [x−1 1 ψ(y)x1 , c(x1 )] lies in the center of G. Hence L = xx 1 −1 ψ(y)x1 . Denote the right hand side of this relation by R and transform it: R = ((xx1 )y )x1 = ((xx1 )ψ (y))x1 = xx1 −1 −1 −1 ψ(y)x1 . We see that L = R, i.e. the first relation from the definition of compatible action holds. The checking of the second relation is the similar. COMPATIBLE ACTIONS AND NON-ABELIAN TENSOR PRODUCTS 7 From this theorem we have particular answer on Question 1 for 2-step nilpotent groups. Corollary 4.3. If G, H are 2-step nilpotent groups, then any pare of homomorphisms ϕ : G −→ H, ψ : H −→ G define the compatible action. Problem 1. Let G and H be free 2-step nilpotent groups. By Corollary 4.3, any pair of homomorphisms (ϕ, ψ), where ϕ ∈ Hom(G, H), ψ ∈ Hom(H, G) defines a tensor product M (ϕ, ψ) = G ⊗ H. Give a classification of the groups M (ϕ, ψ). Note that for arbitrary groups Corollary 4.3 does not hold. Indeed, let G = hx1 , x2 i, H = hy1 , y2 i be free groups of rank 2. Define the homomorphisms ϕ : G −→ H, ψ : H −→ G by the rules ϕ(x1 ) = y1 , ϕ(x2 ) = y2 , ψ(y1 ) = ψ(y2 ) = 1. Then ϕ(x1 ) y2x1 = y2 = y2y1 6= y2 , i.e. the conditions of compatible actions does not hold. 5. Tensor products G ⊗ Z2 Note that the group Aut(Z2 ) is trivial and hence, any group G acts on Z2 only trivially. This section is devoted to the answer on the following question. Question 3. Let G be a group and ψ ∈ Aut(G) be an automorphism of order 2. Let Z2 = hϕi and α : Z2 −→ Aut(G) such that α(ϕ) = ψ. Under what conditions the pare (α, 1) is compatible? If ψ ∈ Aut(G) is trivial automorphism, then by the second part of Proposition 2.1 G ⊗ Z2 = Gab ⊗Z Z2 is an abelian tensor product. In the general case we have Proposition 5.1. Let 1) G be a group, 2) Z2 = hϕi be a cyclic group of order two with the generator ϕ, 3) α : Z2 −→ Aut(G) be a homomorphism, β = 1 : G → Aut(Z2 ) be the trivial homomorphism, Then the pare of actions (α, β) is compatible if and only if for any g ∈ G holds g α(ϕ) = gc(g), where c(g) is a central element of G such that c(g)α(ϕ) = c(g)−1 . In particular, if the center of G is trivial, then G ⊗ Z2 = Gab ⊗Z Z2 . 8 V. G. BARDAKOV AND M. V. NESHCHADIM Proof. Since Inn(G) normalizes α(Z2 ), then for every g ∈ G holds gb−1 α(ϕ)b g = α(ϕ). Using this equality for arbitrary element x ∈ G we get g−1 gα(ϕ) xα(ϕ) (g−1 gα(ϕ) )−1 = xα(ϕ) . Since xα(ϕ) is an arbitrary element of G, then c(g) is a central element of G. Applying α(ϕ) to the equality g α(ϕ) = gc(g) we have 2 g = g α(ϕ) = gα(ϕ) c(g)α(ϕ) = gc(g)c(g)α(ϕ) , that is c(g)α(ϕ) = c(g)−1 . For an arbitrary abelian group A we know that A ⊗Z Z = A. The following proposition is some analog of this property for non-abelian tensor product. Proposition 5.2. Let A be an abelian group, Z2 = hϕi is the cyclic group of order 2 and ϕ acts on the elements of A by the following manner aϕ = a−1 , a ∈ A. Then the non-abelian tensor product A ⊗ Z2 is defined and there is an isomorphism A ⊗ Z2 ∼ = A. Proof. It is not difficult to check that defined actions are compatible. Since A acts on Z2 trivially and A is abelian, then the defining relations of the tensor product: aa1 ⊗ h = (aa1 ⊗ ha1 )(a1 ⊗ h), a, a1 ∈ A, h ∈ Z2 , have the form aa1 ⊗ h = (a ⊗ h)(a1 ⊗ h) = (a1 ⊗ h)(a ⊗ h). (1) The relations a ⊗ hh1 = (a ⊗ h1 )(ah1 ⊗ hh1 ), a ∈ A, h, h1 ∈ Z2 , give only one non-trivial relation 1 = a ⊗ ϕ2 = (a ⊗ ϕ)(a−1 ⊗ ϕ), a ∈ A, which follows from (1). Since the set of relations (1) is a full system of relations for A ⊗ Z2 , then there exists the natural isomorphism of A ⊗ Z2 on A that is defined by the formular a ⊗ ϕ 7→ a, a ∈ A. Acknowledgement. The authors gratefully acknowledge the support from the RFBR-16-01-00414 and RFBR-15-01-00745. Also, we thank S. Ivanov, A. Lavrenov and V. Thomas for the interesting discussions and useful suggestions. COMPATIBLE ACTIONS AND NON-ABELIAN TENSOR PRODUCTS 9 References [1] R. Brown, J.-L. Loday, Excision homotopique en basse dimension, C. R. Acad. Sci. Paris Ser. I Math. 298 (15) (1984), 353–356. [2] R. Brown, J.-L. Loday, Van Kampen theorems for diagrams of spaces, Topology 26 (3) (1987), 311–335, with an appendix by M. Zisman. [3] R. Brown, D. L. Johnson, E. F. Robertson, Some computations of non-abelian tensor products of groups, J. Algebra, 1987, 111, 177–202. [4] G. Donadze, M. Larda, V. Thomas, More on the non-abelian tensor product and the Bogomolov multiplier, Preprint, 2015, 16 pp. [5] A. S.-T. Lue, The Ganea map for nilpotent groups, J. London Math. Soc. 14, 309–312, (1976). [6] C. Miller, The second homology group of a group; relations among commutators, Proceedings AMS 3, 588–595, (1952). Sobolev Institute of Mathematics, Novosibirsk 630090, Russia, Novosibirsk State University, Novosibirsk 630090, Russia, Novosibirsk State Agrarian University, Dobrolyubova street, 160, Novosibirsk, 630039, Russia, E-mail address: [email protected] Sobolev Institute of Mathematics and Novosibirsk State University, Novosibirsk 630090, Russia, E-mail address: [email protected]	4
Spanning Tree Congestion and Computation of Generalized Győri-Lovász Partition L. Sunil Chandran arXiv:1802.07632v1 [cs.DS] 21 Feb 2018 1 ?1 , Yun Kuen Cheung ??2 , and Davis Issac2 Department of Computer Science and Automation, Indian Institute of Science, India. [email protected] 2 Max Planck Institute for Informatics, Saarland Informatics Campus, Germany. [email protected] , [email protected] Abstract. We study a natural problem in graph sparsification, the Spanning Tree Congestion (STC) problem. Informally, the STC problem seeks a spanning tree with no tree-edge routing too many of the original edges. The root of this problem dates back to at least 30 years ago, motivated by applications in network design, parallel computing and circuit design. Variants of the problem have also seen algorithmic applications as a preprocessing step of several important graph algorithms. For any general connected graph with n vertices and m edges, we show that √ its STC is at most O( mn), which is asymptotically optimal since we also √ demonstrate graphs with STC at least Ω( mn). We present a polynomial√ time algorithm which computes a spanning tree with congestion O( mn · log n). We also present another algorithm for computing a spanning tree √ with congestion O( mn); this algorithm runs in sub-exponential time 2 when m = ω(n log n). For achieving the above results, an important intermediate theorem is generalized Győri-Lovász theorem, for which Chen et al. [14] gave a non-constructive proof. We give the first elementary and constructive proof by providing a local search algorithm with running time O∗ (4n ), which is a key ingredient of the above-mentioned sub-exponential time algorithm. We discuss a few consequences of the theorem concerning graph partitioning, which might be of independent interest. We also show that for any graph which satisfies certain expanding properties, its STC is at most O(n), and a corresponding spanning tree can be computed in polynomial time. We then use this to show that a random graph has STC Θ(n) with high probability. 1 Introduction Graph Sparsification/Compression generally describes a transformation of a large input graph into a smaller/sparser graph that preserves certain feature (e.g., ? ?? This work was done while this author was visiting Max Planck Institute for Informatics, Saarbrücken, Germany, supported by Alexander von Humboldt Fellowship. Part of the work done while this author was a visitor at the Courant Institute, NYU. The visit was funded in part by New York University. 2 L. Sunil Chandran, Yun Kuen Cheung, and Davis Issac distance, cut, congestion, flow) either exactly or approximately. The algorithmic value is clear, since the smaller graph might be used as a preprocessed input to an algorithm, so as to reduce subsequent running time and memory requirement. In this paper, we study a natural problem in graph sparsification, the Spanning Tree Congestion (STC) problem. Informally, the STC problem seeks a spanning tree with no tree-edge routing too many of the original edges. The problem is well-motivated by network design applications, where designers aim to build sparse networks that meet traffic demands, while ensuring no connection (edge) is too congested. Indeed, the root of this problem dates back to at least 30 years ago under the name of “load factor” [8,36], with natural motivations from parallel computing and circuit design applications. The STC problem was formally defined by Ostrovskii [30] in 2004, and since then a number of results have been presented. The probabilistic version of the STC problem, coined as probabilistic capacity mapping, also finds applications in several important graph algorithm problems, e.g., the Min-Bisection problem. Two canonical goals for graph sparsification problems are to understand the trade-off between the sparsity of the output graph(s) and how well the feature is preserved, and to devise (efficient) algorithms for computing the sparser graph(s). These are also our goals for the STC problem. We focus on two scenarios: (A) general connected graphs with n vertices and m edges, and (B) graphs which exhibit certain expanding properties: √ – For (A), we show that the p spanning tree congestion (STC) is at most O( mn), which is a factor of Ω( m/n) better than the trivial bound of m. We present a polynomial-time algorithm which computes a spanning tree with congestion √ O( mn · log n). We also √ present another algorithm for computing a spanning tree with congestion O( mn); this algorithm runs in sub-exponential time when m = ω(n log2 n). For almost all ranges √ of average degree 2m/n, we also demonstrate graphs with STC at least Ω( mn). – For (B), we show that the expanding properties permit us to devise polynomialtime algorithm which computes a spanning tree with congestion O(n). Using this result, together with a separate lower-bound argument, we show that a random graph has Θ(n) STC with high probability. For achieving the results for (A), an important intermediate theorem is generalized Győri-Lovász theorem, which was first proved by Chen et al. [14]. Their proof uses advanced techniques in topology and homology theory, and is non-constructive. Definition 1.1. In a graph G = (V, E), a k-connected-partition is a k-partition of V into ∪kj=1 Vj , such that for each j ∈ [k], G[Vj ] is connected. Theorem 1.2 ([14, Theorems 25, 26]). Let G = (V, E) be a k-connected 3 graph. Let w be a weight function w : V → R+ . For any U ⊂ V , let w(U ) := P v∈U w(v). Given any k distinct terminal vertices t1 , · · · , tk , and k positive 3 For brevity, we say “k-connected” for “k-vertex-connected” henceforth. Spanning Tree Congestion 3 Pk integers T1 , · · · , Tk such that for each j ∈ [k], Tj ≥ w(tj ) and i=1 Ti = w(V ), there exists a k-connected-partition of V into ∪kj=1 Vj , such that for each j ∈ [k], tj ∈ Vj and w(Vj ) ≤ Tj + maxv∈V w(v) − 1. One of our main contributions is to give the first elementary and constructive proof by providing a local search algorithm with running time O∗ (4n ):4 Theorem 1.3. (a) There is an algorithm which given a k-connected graph, computes a k-connected-partition satisfying the conditions stated in Theorem 1.2 in time O∗ (4n ). (b) If we need a (bk/2c + 1)-partition instead of k-partition (the input graph remains assumed to be k-connected), the algorithm’s running time improves to O∗ (2O((n/k) log k) ). We make three remarks. First, the O∗ (2O((n/k) log k) )-time algorithm is a key√ingredient of our algorithm for computing a spanning tree with congestion O( mn). Second, since Theorem 1.2 guarantees the existence of such a partition, the problem of computing such a partition is not a decision problem but a search problem. Our local search algorithm shows that this problem is in the complexity class PLS [20]; we raise its completeness in PLS as an open problem. Third, the running times do not depend on the weights. The STC Problem, Related Problems and Our Results. Given a connected graph G = (V, E), let T be a spanning tree. For an edge e = (u, v) ∈ E, its detour with respect to T is the unique path from u to v in T ; let DT(e, T ) denote the set of edges in this detour. The stretch of e with respect to T is \|DT(e, T )\|, the length of its detour. The dilation of T is maxe∈E \|DT(e, T )\|. The edge-congestion of an edge e ∈ T is ec(e, T ) := \|{f ∈ E : e ∈ DT(f, T )}\|, i.e., the number of edges in E whose detours contain e. The congestion of T is cong(T ) := maxe∈T ec(e, T ). The spanning tree congestion (STC) of the graph G is STC(G) := minT cong(T ), where T runs over all spanning trees of G. We note that there is an equivalent cut-based definition for edge-congestion, which we will use in our proofs. For each tree-edge e ∈ T , removing e from T results in two connected components; let Ue denote one of the components. Then ec(e, T ) := \|E(Ue , V \ Ue )\|. Various types of congestion, stretch and dilation problems are studied in computer science and discrete mathematics. In these problems, one typically seeks a spanning tree (or some other structure) with minimum congestion or dilation. We mention some of the well-known problems, where minimization is done over all the spanning trees of the given graph: 1. The Low Stretch Spanning Tree (LSST) problem is to find a spanning tree which minimizes the total stretch of all the edges of G. [3] It is easy to see that minimizing the total stretch is equivalent to minimizing the total edge-congestion of the selected spanning tree. 2. The STC problem is to find a spanning tree of minimum congestion. [30] 4 O∗ notation hides all polynomial factors in input size. 4 L. Sunil Chandran, Yun Kuen Cheung, and Davis Issac 3. Tree Spanner Problem is to find a spanning tree of minimum dilation. [13] The more general Spanner problem is to find a sparser subgraph of minimum distortion. [4] There are other congestion and dilation problems which do not seek a spanning tree, but some other structure. The most famous among them is the Bandwidth problem and the Cutwidth problem; see the survey [34] for more details. Among the problems mentioned above, several strong results were published in connection with the LSST problem. Alon et al. [3] had shown a lower bound of Ω(max{n log n, m}). Upper bounds have been derived and many efficient algorithms have been devised; the current best upper bound is Õ(m log n). [3,15,1,22,2] Since total stretch is identical to total edge-congestion, the best upper bound for the LSST problem automatically implies an Õ( m n log n) upper bound on the average edge-congestion. But in the STC problem, we concern the maximum edge-congestion; as we shall p see, for some graphs, the maximum edge-congestion has to be a factor of Ω̃( n3 /m) larger than the average edge-congestion. In comparison, there were not many strong and general results for the STC Problem, though it was studied extensively in the past 13 years. The problem was formally proposed by Ostrovskii [30] in 2004. Prior to this, Simonson [36] had studied the same parameter under a different name to approximate the cut width of outer-planar graph. A number of graph-theoretic results were presented on this topic [31,25,24,23,10]. Some complexity results were also presented recently [29,9], but most of these results concern special classes √ of graphs. The most general result regarding STC of general graphs is an O(n n) upper bound by Löwenstein, Rautenbach and Regen in 2009 [27], and a matching lower bound by Ostrovskii in 2004 [30]. Note that the above upper bound is not interesting when the graph is sparse, since there is also a trivial upper bound of m. In this paper we come up with a strong improvement to these bounds after 8 years: Theorem (informal): For a connected graph √ G with n vertices and m edges, its spanning tree congestion is at most O( mn). p In terms of average degree davg = 2m/n, we can state this upper bound as O(n davg ). There is a matching lower bound. √ Our proof for achieving the O( mn) upper bound is constructive. It runs in exponential time in general; for graphs with m = ω(n log2 n) edges, it runs in sub-exponential time. By using an algorithm of Chen et al. [14] for computing single-commodity confluent flow from single-commodity splittable flow, we improve the running time to polynomial, but with a slightly worse upper bound guarantee √ of O( mn · log n). Motivated by an open problem raised by Ostrovskii [32] concerning STC of random graphs, we formulate a set of expanding properties, and prove that for any graph satisfying these properties, its STC is at most O(n). We devise a polynomial time algorithm for computing a spanning tree with congestion O(n) for such graphs. This result, together with a separate lower-bound argument, n permit us to show that for random graph G(n, p) with 1 ≥ p ≥ c log for some n Spanning Tree Congestion 5 small constant c > 1,5 its STC is Θ(n) with high probability, thus resolving the open problem raised by Ostrovskii completely. Min-Max Graph Partitioning and the Generalized Győri-Lovász Theorem. It looks clear that the powerful Theorem 1.2 can make an impact on graph partitioning. We discuss a number of its consequences which might be of wider interest. Graph partitioning/clustering is a prominent topic in graph theory/algorithms, and has a wide range of applications.A popular goal is to partition the vertices into sets such that the number of edges across different sets is small. While the min-sum objective, i.e., minimizing the total number of edges across different sets, is more widely studied, in various applications, the more natural objective is the min-max objective, i.e., minimizing the maximum number of edges leaving each set. The min-max objective is our focus here. Depending on applications, there are additional constraints on the sets in the partition. Two natural constraints are (i) balancedness: the sets are (approximately) balanced in sizes, and (ii) induced-connectivity: each set induces a connected subgraph. The balancedness constraint appears in the application of domain decomposition in parallel computing, while the induced-connectivity constraint is motivated by divide-and-conquer algorithms for spanning tree construction. Imposing both constraints simultaneously is not feasible for every graph; for instance, consider the star graph with more than 6 vertices and one wants a 3-partition. Thus, it is natural to ask, for which graphs do partitions satisfying both constraints exist. Theorem 1.2 implies a simple sufficient condition for existence of such partitions. By setting the weight of each vertex in G to be its degree, and using the elementary fact that the maximum degree ∆(G) ≤ n ≤ 2m/k for any k-connected graph G on n vertices and m edges, we have Proposition 1.4. If G is a k-connected graph with m edges, then there exists a k-connected-partition, such that the total degree of vertices in each part is at most 4m/k. Consequently, the min-max objective is also at most 4m/k. Due to expander graphs, this bound is optimal up to a small constant factor. This proposition (together with Lemma 4.1) implies the following crucial lemma for achieving some of our results. Lemma 1.5. Let G be a k-connected graph with m edges. Then STC(G) ≤ 4m/k. Proposition 1.4 can be generalized to include approximate balancedness in terms of number of vertices. By setting the weight of each vertex to be cm/n plus its degree in G, we have Proposition 1.6. Given any fixed c > 0, if G is a k-connected graph with m edges and n vertices, then there exists a k-connected-partition such that the total 5 Note that the STC problem is relevant only for connected graphs. Since the threshold function for graph connectivity is logn n , this result applies for almost all of the relevant range of values of p. 6 L. Sunil Chandran, Yun Kuen Cheung, and Davis Issac degree of vertices in each part is at most (2c + 4)m/k, and the number of vertices n in each part is at most 2c+4 c · k. Further Related Work. Concerning STC problem, Okamoto et al. [29] gave an O∗ (2n ) algorithm for computing the exact STC of a graph. The probabilistic version of the STC problem, coined as probabilistic capacity mapping, is an important tool for several graph algorithm problems, e.g., the Min-Bisection problem. Räcke [33] showed that in the probabilistic setting, distance and capacity are interchangeable, which briefly says a general upper bound for one objective implies the same general upper bound for the other. Thus, due to the above-mentioned results on LSST, there is an upper bound of Õ(log n) on the maximum average congestion. Räcke’s result also implies an O(log n) approximation algorithm to the Min-Bisection problem, improving upon the O(log3/2 n) approximation algorithm of Feige and Krauthgamer [16]. However, in the deterministic setting, such interchanging phenomenon does not hold: there is a √ simple tight bound Θ(n) for dilation, but for congestion it can be as high as Θ(n n). For the precise definitions, more background and key results about the concepts we have just discussed, we recommend the writing of Andersen and Feige [5]. Graph partitioning/clustering is a prominent research topic with wide applications, so it comes no surprise that a lot of work has been done on various aspects of the topic; we refer readers to the two extensive surveys by Schaeffer [35] and by Teng [41]. Kiwi, Spielman and Teng [21] formulated the min-max k-partitioning problem and gave bounds for classes of graphs with small separators, which are then improved by Steurer [38]. On the algorithmic side, many of the related problems are NP-hard, so the focus is on devising approximation algorithms. Sparkled by the seminal work of Arora, Rao and Vazirani [6] on sparsest cut and of Spielman and Teng [37] on local clustering, graph partitioning/clustering algorithms with various constraints have attracted attention across theory and practice; we refer readers to [7] for a fairly recent account of the development. The min-sum objective has been extensively studied; the min-max objective, while striking as the more natural objective in some applications, has received much less attention. The only algorithmic work on this objective (and its variants) are Svitkina and Tardos [40] and Bansal et al. [7]. None of the above work addresses the induced-connectivity constraint. The classical version of Győri-Lovász Theorem (i.e., the vertex weights are uniform) was proved independently by Győri [17] and Lovász [26]. Lovász’s proof uses homology theory and is non-constructive. Győri’s proof is elementary and is constructive implicitly, but he did not analyze the running time. Polynomial time algorithms for constructing the k-partition were devised for k = 2, 3 [39,42], but no non-trivial finite-time algorithm was known for general graphs with k ≥ 4.6 Recently, Hoyer and Thomas [19] provided a clean presentation of Győri’s proof 6 In 1994, there was a paper by Ma and Ma in Journal of Computer Science and Technology, which claimed a poly-time algorithm for all k. However, according to a recent study [18], Ma and Ma’s algorithm can fall into an endless loop. Also, Győri said the algorithm should be wrong (see [28]). Spanning Tree Congestion 7 by introducing their own terminology, which we use for our constructive proof of Theorem 1.2. Notation. Given a graph G = (V, E), an edge set F ⊆ E and 2 disjoint vertex subsets V1 , V2 ⊂ V , we let F (V1 , V2 ) := { e = {v1 , v2 } ∈ F \| v1 ∈ V1 and v2 ∈ V2 }. 2 Technical Overview To prove the generalized Győri-Lovász theorem constructively, we follow the same framework of Győri’s proof [17], and we borrow terminology from the recent presentation by Hoyer and Thomas [19]. But it should be emphasized that proving our generalized theorem is not straight-forward, since in Győri’s proof, at each stage a single vertex is moved from one set to other to make progress, while making sure that the former set remains connected. In our setting, in addition to this we also have to ensure that the weights in the partitions do not exceed the specified limit; and hence any vertex that can be moved from one set to another need not be candidate for being transferred. The proof is presented in Section 3. As discussed, a crucial ingredient for our upper bound results is Lemma 1.5, which is a direct corollary of the generalized Győri-Lovász theorem. The lemma takes care of the highly-connected cases; for other cases we provide a recursive way to construct a low congestion spanning tree; see Section 4 for details. For showing our lower bound for general graphs, the challenge is to maintain high congestion while keeping density small. To achieve this, we combine three expander graphs with little overlapping between them, and we further make those overlapped vertices of very high degree. This will force a tree-edge adjacent to the centroid of any spanning tree to have high congestion; see Section 5 for details. We formulate a set of expanding properties which permit constructing a spanning tree of better congestion guarantee in polynomial time. The basic idea is simple: start with a vertex v of high degree as the root. Now try to grow the tree by keep attaching new vertices to it, while keeping the invariant that the subtrees rooted at each of the neighbours of v are roughly balanced in size; each such subtree is called a branch. But when trying to grow the tree in a balanced way, we will soon realize that as the tree grow, all the remaining vertices may be seen to be adjacent only to a few number of “heavy” branches. To help the balanced growth, the algorithm will identify a transferable vertex which is in a heavy branch, and it and its descendants in the tree can be transferred to a “lighter” branch. Another technique is to use multiple rounds of matching between vertices in the tree and the remaining vertices to attach new vertices to the tree. This will tend to make sure that all subtrees do not grow uncontrolled. By showing that random graph satisfies the expanding properties with appropriate parameters, we show that a random graph has STC of Θ(n) with high probability. 3 Generalized Győri-Lovász Theorem We prove Theorem 1.3 in this section. Observe that the classical Győri-Lovász Theorem follows from Theorem 1.2 by taking w(v) = 1 for all v ∈ V and Tj = nj 8 L. Sunil Chandran, Yun Kuen Cheung, and Davis Issac for all j ∈ [k]. We note that a perfect generalization where one requires that w(Vj ) = Tj is not possible — think when all vertex weights are even integers, while some Tj is odd. Let G = (V, E) be a k-connected graph on n vertices and m P edges, and w : V → R+ be a weight function. For any subset U ⊆ V , w(U ) := u∈U w(u). Let wmax := maxv∈V w(v). 3.1 Key Combinatorial Notions We first highlight the key combinatorial notions used for proving Theorem 1.3; see Figures 1 and 2 for illustrations of some of these notions. Fitted Partial Partition. First, we introduce the notion of fitted partial partition (FPP). An FPP A is a tuple of k subsets of V , (A1 , . . . , Ak ), such that the k subsets are pairwise disjoint, and for each j ∈ [k]: 1. tj ∈ Aj , 2. G[Aj ] is connected and 3. w(Aj ) ≤ Tj + wmax − 1 (we say the set is fitted for satisfying this inequality). We say an FPP is a Strict Fitted Partial Partition (SFPP) if A1 ∪ · · · ∪ Ak is a proper subset of V . We say the set Aj is light if w(Aj ) < Tj , and we say it is heavy otherwise. Note that there Pk exists at least one light set in any SFPP, for otherwise w(A1 ∪ · · · ∪ Ak ) ≥ j=1 Tj = w(V ), which means A1 ∪ · · · ∪ Ak = V . Also note that by taking Aj = {tj }, we have an FPP, and hence at least one FPP exists. Configuration. For a set Aj in an FPP A and a vertex v ∈ Aj \ {tj }, we define the reservoir of v with respect to A, denoted by RA (v), as the vertices in the same connected component as tj in G[Aj ] \ {v}. Note that v ∈ / RA (v). For a heavy set Aj , a sequence of vertices (z1 , . . . , zp ) for some p ≥ 0 is called a cascade of Aj if z1 ∈ Aj \ {tj } and zi+1 ∈ Aj \ RA (zi ) for all 1 ≤ i < p. The cascade is called a null cascade if p = 0, i.e., if the cascade is empty. Note that for light set, we do not need to define its cascade since we do not use it in the proof. (See Figure 1.) A configuration CA is defined as a pair (A, D), where A = (A1 , · · · , Ak ) is an FPP, and D is a set of cascades, which consists of exactly one cascade (possibly, a null cascade) for each heavy set in A. A vertex that is in some cascade of the configuration is called a cascade vertex. Given a configuration, we define rank and level inductively as follows. Any vertex in a light set is said to have level 0. For i ≥ 0, a cascade vertex is said to have rank i + 1 if it has an edge to a level-i vertex but does not have an edge to any level-i0 vertex for i0 < i. A vertex u is said to have level i, for i ≥ 1, if u ∈ RA (v) for some rank-i cascade vertex v, but u ∈ / RA (w) for any cascade vertex w such that rank of w is less than i. A vertex that is not in RA (v) for any cascade vertex v is said to have level ∞. A configuration is called a valid configuration if for each heavy set Aj , rank is defined for each of its cascade vertices and the rank is strictly increasing in the Spanning Tree Congestion 9 z3 z2 RA(z3) RA(z2) z1 RA(z1) tj Fig. 1. Given a configuration (A, D) and a heavy set Aj in A, the figure shows a cascade (z1 , z2 , z3 ) for the heavy set Aj and several reservoirs of the cascade vertices. For any z` , z` ∈ / RA (z` ). A cascade vertex z` is a cut-vertex of G[Aj ], i.e., G[Aj \ {z` }] is disconnected. The removal of z` from Aj will lead to at least two connected components in G[Aj \ {z` }], and the connected component containing tj is the reservoir of z` . We identify tj = z0 , but we clarify that a terminal vertex is never in a cascade. Each epoch between z` and z`+1 , and also the epoch above z3 , is a subset of vertices B ⊂ Aj , where B 3 z` and G[B] is connected. Note that in general, it is possible that there is no vertex above the last cascade vertex. cascade, i.e., if {z1 , . . . , zp } is the cascade, then rank(z1 ) < · · · < rank(zp ). Note that by taking Aj = {tj } and taking the null cascade for each heavy set (in this case Aj is heavy if w(tj ) = Tj ), we get a valid configuration. (See Figure 2.) Configuration Vectors and Their Total Ordering. For any vertex, we define its neighborhood level as the smallest level of any vertex adjacent to it. A vertex v of level ` is said to satisfy maximality property if each vertex adjacent on it is either a rank-(` + 1) cascade vertex, has a level of at most ` + 1, or is one of the terminals tj for some j. For any ` ≥ 0, a valid configuration is called an `-maximal configuration if all vertices having level at most ` − 1 satisfy the maximality property. Note that by definition, any valid configuration is a 0-maximal configuration. For a configuration CA = ((A1 , . . . , Ak ) , D), we define SA := V \(A1 ∪· · ·∪Ak ). An edge uv is said to be a bridge in CA if u ∈ SA , v ∈ Aj for some j ∈ [k], and level(v) 6= ∞. 10 L. Sunil Chandran, Yun Kuen Cheung, and Davis Issac L=∞ z12, rank = 2 L=2 L=∞ z22, rank = 5 L=5 z11, rank = 1 z32, rank = 4 L=4 z21, rank = 3 z31, rank = 1 L=1 L=3 L=1 t1 t2 t3 Vertices in all light sets Level = 0 Fig. 2. An instance of a valid configuration. Every blue segment/curve represent an edge from a cascade vertex to a vertex in some reservoir or light set. Every cascade vertex connected to a light set has rank 1, and all vertices in the epoch immediately below a rank 1 cascade vertex are of level 1. Inductively, every cascade vertex connected to a vertex of level i has rank i + 1, and all vertices in the epoch immediately below a rank i cascade vertex are of level i. All vertices above the last cascade vertex of each cascade has level ∞. A valid configuration CA is said to be `-good if the highest rank of a cascade vertex in CA is exactly ` (if there are no cascade vertices, then we take the highest rank as 0), CA is `-maximal, and all bridges uv in CA (if any) are such that u ∈ SA and level(v) = `. Note that taking Aj = {tj } and taking the null cascade for each heavy set gives a 0-good configuration. For each configuration CA = (A, D), we define a configuration vector as below: ( LA , NA0 , NA1 , NA2 , . . . , NAn ), where LA is the number of light sets in A, and NA` is the total number of all level-` vertices in CA . Next, we define ordering on configuration vectors. Let CA and CB be configurations. We say CA >0 CB if Spanning Tree Congestion 11 – LA < LB , or – LA = LB , and NA0 > NB0 . We say CA =0 CB if LA = LB and NA0 = NB0 . We say CA ≥0 CB if CA =0 CB or 0 0 CA >0 CB . We say CA =` CB if LA = LB , and NA` = NB` for all `0 ≤ `. For 1 ≤ ` ≤ n, we say CA >` CB if – CA >`−1 CB , or – CA =`−1 CB , and NA` > NB` . We say CA ≥` CB if CA =` CB or CA >` CB . We say CA > CB (CA is strictly better than CB ) if CA >n CB . 3.2 Proof of Theorem 1.3 We use two technical lemmas about configuration vectors and their orderings to prove Theorem 1.3(a). The proof of Theorem 1.3(b) follows closely with the proof of Theorem 1.3(a), but makes use of an observation about the rank of a vertex in the local search algorithm, to give an improved bound on the number of configuration vectors navigated by the algorithm. Lemma 3.1. Given any `-good configuration CA = (A = (A1 , . . . , Ak ), DA )) that does not have a bridge, we can find an (` + 1)-good configuration CB = (B = (B1 , B2 , . . . , Bk ) , DB ) in polynomial time such that CB > CA . Proof. Since CA is `-maximal, any vertex that is at level `0 < ` satisfies maximality property. So, for satisfying (` + 1)-maximality, we only need to worry about the vertices that are at level `. Let Xj be the set of all vertices x ∈ Aj such that x is adjacent to a level-` vertex, level(x) ≥ ` + 1 (i.e., level(x) = ∞ as the highest rank of any cascade vertex is `), x = 6 tj , and x is not a cascade vertex of rank `. We claim that there exists at least one j for which Xj is not empty. If that is not the case, then we exhibit a cut set of size at most k − 1. For each j such that Aj is a heavy set with a non-null cascade, let yj be the highest ranked cascade vertex in Aj . For each j such that Aj is a heavy set with a null cascade, let yj be tj . Let Y be the set of all yj such that Aj is a heavy set. Note that \|Y \| ≤ k − 1 as A is an SFPP and hence has at least one light set. Let Z∞ be the set of all vertices in V \ Y that have level ∞ and Z be the remaining vertices in V \ Y . Since A is an SFPP, SA 6= ∅, and since all vertices in SA have level ∞, we have that Z∞ 6= ∅. Z is not empty because there exists at least one light set in A and the vertices in a light set have level 0. We show that there is no edge between Z∞ and Z in G. Suppose there exists an edge uv such that u ∈ Z∞ and v ∈ Z. If u ∈ SA , then uv is a bridge which is a contradiction by our assumption that CA does not have a bridge. Hence u ∈ Aj for some j ∈ [k]. Note that Aj has to be a heavy set, otherwise u has level 0. We have that u is not a cascade vertex (as all cascade vertices with level ∞ are in Y ) and u 6= tj (as all tj such that level(tj ) = ∞ are in Y ). Also, v is not of level ` as otherwise, u ∈ Xj but we assumed Xj is empty. But then, v has level at most ` − 1, u has level ∞, and there 12 L. Sunil Chandran, Yun Kuen Cheung, and Davis Issac is an edge uv. This means that CA was not `-maximal, which is a contradiction. Thus, there exists at least one j for which Xj is not empty. For any j such that Xj 6= ∅ , there is at least one vertex xj such that Xj \ {xj } ⊆ RA (xj ). Now we give the configuration CB as follows. We set Bj = Aj for all j ∈ [k]. For each heavy set Aj such that Xj 6= ∅, we take the cascade of Bj as the cascade of Aj appended with xj . For each heavy set Aj such that Xj = ∅, we take the cascade of Bj as the cascade of Aj . It is easy to see that CB is (` + 1)-maximal as each vertex that had an edge to level-` vertices in CA is now either a rank ` + 1 cascade vertex or a level-(` + 1) vertex or is tj for some j. Also, notice that all the new cascade vertices that we introduce (i.e., the xj ’s) have their rank as ` + 1 and there is at least one rank ` + 1 cascade vertex as Xj is not empty for some j. Since there were no bridges in CA , all bridges in CB has to be from SB to a vertex having level ` + 1. Hence, CB is (` + 1)-good. All vertices that had level at most ` in CA retained their levels in CB . And, at least one level-∞ vertex of CA became a level-(` + 1) vertex in CB because the cascade vertex that was at rank ` becomes level-(` + 1) vertex now in at least one set. Since CA had no level-(` + 1) vertices, this means that CB > CA . Lemma 3.2. Given an `-good configuration CA = (A = (A1 , . . . , Ak ), DA ) having a bridge, we can find in polynomial time a valid configuration CB = (B = (B1 , . . . , Bk ) , DB ) such that one of the following holds: – CB >` CA , and CB is an `-good configuration, or – CB ≥`−1 CA , there is a bridge u0 v 0 in CB such that u0 ∈ SB and level(v 0 ) ≤ ` − 1, and CB is an (` − 1)-good configuration. Proof. Let uv be a bridge where u ∈ SA . Let Aj ∗ be the set containing v. Note that level(v) = ` because CA is `-good. We keep Bj = Aj for all j 6= j ∗ . But we modify Aj ∗ to get Bj ∗ as described below. We maintain that if Aj is a heavy set then Bj is also a heavy set for all j, and hence maintain that LB ≤ LA . Case 1: Aj ∗ is a light set (i.e., when ` = 0). We take Bj ∗ = Aj ∗ ∪ {u}. For all j such that Bj is a heavy set, cascade of Bj is taken as the null cascade. We have w(Aj ∗ ) ≤ Tj − 1 because Aj ∗ is a light set. So, w(Bj ∗ ) = w(Aj ∗ ) + w(u) ≤ (Tj − 1) + wmax , and hence Bj ∗ is fitted. Also, G[Bj ∗ ] is connected and hence (B1 , . . . , Bk ) is an FPP. We have CB >0 CA because either Bj ∗ became a heavy set in which case LB < LA , or it is a light set in which case LB = LA and NB0 > NA0 . It is easy to see that CB is 0-good. Case 2: Aj ∗ is a heavy set i.e., when ` ≥ 1. Case 2.1: w(Aj ∗ ∪ {u}) ≤ Tj + wmax − 1. We take Bj ∗ = Aj ∗ ∪ {u}. For each j such that Bj is a heavy set (Aj is also heavy set for such j), the cascade of Bj is taken as the cascade of Aj . G[Bj ∗ ] is clearly connected and Bj ∗ is fitted by assumption of the case that we are in. Hence B is indeed an FPP. Observe that all vertices that had level `0 ≤ ` in CA still has level `0 in CB . Since level(v) was ` in CA by `-goodness of CA , u also has level ` in CB ; and u had level ∞ in CA . Hence, CB >` CA . It is also easy to see that CB remains `-good. Case 2.2: w(Aj ∗ ∪ {u}) ≥ Tj + wmax . Let z be the cascade vertex of rank ` in Aj ∗ . Note that Aj ∗ should have such a cascade vertex as v ∈ Aj ∗ has level `. Let Spanning Tree Congestion 13 R̄ be Aj ∗ \ (RA (z) ∪ z), i.e., R̄ is the set of all vertices in Aj ∗ \ {z} with level ∞. We initialize Bj ∗ := Aj ∗ ∪ {u}. Now, we delete vertices one by one from Bj ∗ in a specific order until Bj ∗ becomes fitted. We choose the order of deleting vertices such that G[Bj ∗ ] remains connected. Consider a spanning tree τ of G[R̄ ∪ {z}]. τ has at least one leaf, which is not z. We delete this leaf from Bj ∗ and τ . We repeat this process until τ is just the single vertex z or Bj ∗ becomes fitted. If Bj ∗ is not fitted even when τ is the single vertex z, then delete z from Bj ∗ . If Bj ∗ is still not fitted then delete u from Bj ∗ . Note that at this point Bj ∗ ⊂ Aj ∗ and hence is fitted. Also, note that G[Bj ∗ ] remains connected. Hence (B1 , . . . , Bk ) is an FPP. Bj ∗ does not become a light set because Bj became fitted when the last vertex was deleted from it. Before this vertex was deleted, it was not fitted and hence had weight at least Tj ∗ + wmax before this deletion. Since the last vertex deleted has weight at most wmax , Bj ∗ has weight at least Tj ∗ and hence is a heavy set. Now we branch into two subcases for defining the cascades. Case 2.2.1: z ∈ Bj∗ (i.e, z was not deleted from Bj ∗ in the process above). For each j such that Bj is a heavy set, the cascade of Bj is taken as the cascade of Aj . Since a new ` level vertex u is added and all vertices that had level at most ` retain their level, we have that CB >` CA . It is also easy to see that CB remains `-good. Case 2.2.2: z ∈ / Bj ∗ (i.e, z was deleted from Bj ∗ ). For each j such that Bj is a heavy set, the cascade of Bj is taken as the cascade of Aj but with the rank ` cascade vertex (if it has any) deleted from it. CB ≥`−1 CA because all vertices that were at a level of `0 = ` − 1 or smaller, retain their levels. Observe that there are no bridges in CB to vertices that are at a level at most ` − 2, all vertices at a level at most ` − 2 still maintain the maximality property, and we did not introduce any cascade vertices. Hence, CB is (` − 1)-good. It only remains to prove that there is a bridge u0 v 0 in CB such that level(v 0 ) ≤ ` − 1. We know z ∈ SB . Since z was a rank ` cascade vertex in CA , z had an edge to z 0 such that z 0 had level ` − 1 in CA . Observe that level of z 0 is at most ` − 1 in CB as well. Hence, taking u0 v 0 = zz 0 completes the proof. Proof of Theorem 1.3(a): . We always maintain a configuration CA = (A, DA ) that is `-good for some ` ≥ 0. If the FPP A is not an SFPP at any point, then we are done. So assume A is an SFPP. We start with the 0-good configuration where Aj = {tj } and the cascades of all heavy sets are null cascades. If our current configuration CA is an `-good configuration that has no bridge, then we use Lemma 3.1 to get a configuration CB such that CB > CA and B is (` + 1)-good. We take CB as the new current configuration CA . If our current configuration CA is an `-good configuration with a bridge, then we get an `0 -good configuration CB for some `0 ≥ 0 such that CB > CA by repeatedly applying Lemma 3.2 at most ` times. So in either case, we get a strictly better configuration that is `0 -good for some `0 ≥ 0 in polynomial time. We call this an iteration of our algorithm. Notice that the number of iterations possible is at most the number of distinct configuration vectors possible. It is easy to see that the number of distinct configuration vectors with highest rank at most r is at most n+r−1 . Since rank n 14 L. Sunil Chandran, Yun Kuen Cheung, and Davis Issac of any point is at most n, the number of iterations of our algorithm is at most n (k + 1) · 2n n , which is at most n · 4 . Since each iteration runs in polynomial time as guaranteed by the two lemmas, the required running time is O∗ (4n ). When the algorithm terminates, the FPP given by the current configuration is not an SFPP and this gives the required partition. Proof of Theorem 1.3(b): . Since any k-connected graph is also (bk/2c + 1)−vertex connected, the algorithm will give the required partition due to Theorem 1.3(a). We only need to prove the better running time claimed by Theorem 1.3(b). For this, we show that the highest rank attained by any vertex during the algorithm is at most 2n/(k − 2). Since the number of distinct config uration vectors with highest rank r is at most n+r−1 , we then have that the n 2n −1 ∗ O((n/k) log k) running time is O∗ n+ k−2 , which is O (2 ), as claimed. Hence, it n only remains to prove that the highest rank is at most 2n/(k − 2). For this, observe that in an `-good configuration, for each 0 ≤ i < `, the union of all vertices having level i and the set of (bk/2c + 1) terminals together forms a cutset. Since the graph is k-connected, this means that for each 0 ≤ i < `, the number of vertices having level i is at least k/2 − 1. The required bound on the rank easily follows. 4 Upper Bounds for Spanning Tree Congestion We first state the following easy lemma, which together with Proposition 1.4, implies Lemma 1.5. Lemma 4.1. In a graph G = (V, E), let t1 be a vertex, and let t2 , · · · , t` be any (` − 1) neighbours of t1 . Suppose that there exists a `-connected-partition ∪`j=1 V` such that for all j ∈ `, tj ∈ Vj , and the sum of degree of vertices in each Vj is at most D. Let τj be an arbitrary spanning tree of G[Vj ]. Let ej denote the edge S {t1 , tj }. Let τ be the spanning tree of G defined as τ := ∪`j=1 τj ∪`j=2 ej . Then τ has congestion at most D. Theorem 4.2. For any connected graph G = (V, E), there is an algorithm which √ √ O n log n/ m/n ∗ computes a spanning tree with congestion at most 8 mn in O 2 time. Theorem 4.3. For any connected graph G = (V, E), there is a polynomial time √ algorithm which computes a spanning tree with congestion at most 16 mn · log n. The two algorithms follow the same framework, depicted in Algorithm 1. It is a recursive algorithm; the parameter m̂ is a global parameter, which is the number of edges in the input graph G in the first level of the recursion; let n̂ denote the number of vertices in this graph. The only difference between the two algorithms is in Line 15 on how this step is executed, with trade-off between the running time of the step T (m̂, nH , mH ), and the guarantee D(m̂, nH , mH ). For proving Theorem 4.2, we use Theorem 1.3(b), Spanning Tree Congestion 15 p Proposition 1.4 andLemma 4.1, yielding D(m̂, nH , mH ) ≤ 8mH nH /m̂ and √ T (m̂, nH , mH ) = O∗ 2O(nH log nH / m̂/nH ) . For proving Theorem 4.3, we make p use of an algorithm in Chen et al. [14], which yields D(m̂, nH , mH ) ≤ 16mH nH /m̂· log nH and T (m̂, nH , mH ) = poly(nH , mH ). Algorithm 1: FindLCST(H, m̂) Input : A connected graph H = (VH , EH ) on nH vertices and mH edges Output : A spanning tree τ of H p 1 if mH ≤ 8 m̂nH then 2 return an arbitrary spanning tree of H 3 end lp m 4 k ← m̂/nH 5 6 7 8 9 10 11 12 13 14 15 Y ← a global minimum vertex cut of H if \|Y \| < k then X ← the smallest connected component in H[VH \ Y ] (See Figure 3) Z ← VH \ (X ∪ Y ) τ1 ← FindLCST( H[X], m̂ ) τ2 ← FindLCST( H[Y ∪ Z], m̂); (H[Y ∪ Z] is connected as Y is a global min cut) return τ1 ∪ τ2 ∪ (an arbitrary edge between X and Y ) else t1 ← an arbitrary vertex in VH Pick bk/2c neighbours of t1 in the graph H; denote them by t2 , t3 , · · · , tbk/2c+1 . Let ej denote edge t1 tj for 2 ≤ j ≤ bk/2c + 1. (See Figure 4) Compute a (bk/2c + 1)-connected-partition of H, denoted by bk/2c+1 16 17 18 ∪j=1 Vj , such that for each j ∈ [bk/2c + 1], tj ∈ Vj , and the total degree (w.r.t. graph H) of vertices in each Vj is at most D(m̂, nH , mH ). Let the time needed be T (m̂, nH , mH ). For eachj ∈ [bk/2c + 1], τj← an arbitrary spanning tree of G[Vj ] S bk/2c+1 bk/2c+1 return ∪j=1 τj ∪j=2 ej end In the rest of this section, we first discuss the algorithm in Chen et al., then we prove Theorem 4.3. The proof of Theorem 4.2 is almost identical, and is deferred to Appendix A.2. Single-Commodity Confluent Flow and The Algorithm of Chen et al. In a single-commodity confluent flow problem, the input includes a graph G = (V, E), a demand function w : V → R+ and ` sinks t1 , · · · , t` ∈ V . For each v ∈ V , a flow of amount w(v) is routed from v to one of the sinks. But there is a restriction: at every vertex u ∈ V , the outgoing flow must leave u on at most 1 edge, i.e., the outgoing flow from u is unsplittable. The problem is to seek a flow satisfying the demands which minimizes the node congestion, i.e., the maximum 16 L. Sunil Chandran, Yun Kuen Cheung, and Davis Issac Fig. 3. The scenario in Algorithm 1 when the graph has low connectivity. The vertex set Y is a global minimum vertex cut of the graph. The vertex set X is the smallest connected component after the removal of Y , and Z is the union of all the other connected components. Fig. 4. The scenario in Algorithm 1 when the graph has high connectivity. incoming flow among all vertices. Since the incoming flow is maximum at one of the sinks, it is equivalent to minimize the maximum flow received among all sinks. (Here, we assume that no flow entering a sink will leave.) Single-commodity splittable flow problem is almost identical to single-commodity confluent flow problem, except that the above restriction is dropped, i.e., now the outgoing flow at u can split along multiple edges. Note that here, the maximum incoming flow might not be at a sink. It is known that single-commodity splittable flow can be solved in polynomial time. For brevity, we drop the phrase “single-commodity” from now on. Theorem 4.4 ([14, Section 4]). Suppose that given graph G, demand w and ` sinks, there is a splittable flow with node congestion q. Then there exists a Spanning Tree Congestion 17 polynomial time algorithm which computes a confluent flow with node congestion at most (1 + ln `)q for the same input. Corollary 4.5. Let G be a k-connected graph with m edges. Then for any ` ≤ k and for any ` vertices t1 , · · · , t` ∈ V , there exists a polynomial time algorithm which computes an `-connected-partition ∪`j=1 V` such that for all j ∈ `, tj ∈ Vj , and the total degrees of vertices in each Vj is at most 4(1 + ln `)m/`. Corollary 4.5 follows from Theorem 4.4 and Proposition 1.4. See Appendix A.1 for details. Congestion Analysis. We view the whole recursion process as a recursion tree. There is no endless loop, since down every path in the recursion tree, the number of vertices in the input graphs are strictly decreasing. On the other hand, note that the leaf of the recursion tree pis resulted by either (i) when the input graph H to that call satisfies mH ≤ 8 m̂nH , or (ii) when Lines 13–17 are executed. An internal node appears only when the vertex-connectivity of the input graph H is low, and it makes two recursion calls. We prove the following statement by induction from bottom-up: for each graph which is the input to some call in thep recursion tree, the returned spanning tree of that call has congestion at most 16 m̂nH log nH . We first handle the two basis cases (i) and (ii). In case (i), FindLCST p returns an arbitrary spanning tree, and the congestion is bounded by mH ≤ 8 m̂nH . In case (ii), by Corollaryp4.5 and Lemma 4.1,pFindLCST returns a tree with congestion at most 16mH nH /m̂ · log nH ≤ 16 m̂nH · log nH . Next, let H be the input graph to a call which is represented by an internal node of the recursion tree. Recall the definitions of X, Y, Z, τ1 , τ2 in the algorithm. Let \|X\| = x. Note that 1 ≤ x ≤ nH /2. Then by induction hypothesis, the congestion of the returned spanning tree is at most max{ congestion of τ1 in H[X] , congestion of τ2 in H[Y ∪ Z] } + \|X\| · \|Y \| p p ≤ 16 m̂(nH − x) log(nH − x) + m̂/nH + 1 · x. (1) Viewing x as a real variable, by taking derivative, it is easy to see that the above expression is maximized at x = 1. Thus the congestion is at most p p p 16 m̂(nH − 1) log(nH −1)+ m̂/nH +1 ≤ 16 m̂nH log nH , as desired by Theorem 4.3. Runtime Analysis. At every internal node of the recursion tree, the algorithm makes two recursive calls with two vertex-disjoint and strictly smaller (w.r.t. vertex size) inputs. The dominating knitting cost is in Line 5 for computing a global minimum vertex cut, which is well-known that it can be done in polynomial time. Since at every leaf of the recursion tree the running time is polynomial, by standard analysis on divide-and-conquer algorithms, the running time of the whole algorithm is polynomial, which completes the proof of Theorem 4.3. 18 5 L. Sunil Chandran, Yun Kuen Cheung, and Davis Issac Lower Bound for Spanning Tree Congestion Here, we give a lower bound on spanning tree congestion which matches our upper bound. Theorem 5.1. For any sufficiently large n, and for any m satisfying n2 /2 ≥ m ≥ max{16n log n, 100n}, there exists a connected graph with N = (3 − o(1))n vertices √ and M ∈ [m, 7m] edges, for which the spanning tree congestion is at least Ω ( mn). We start with the following lemma, which states that for a random graph G(n, p), when p is sufficiently large, its edge expansion is Θ(np) with high probability. The proof of the lemma uses only fairly standard arguments and is deferred to Appendix A.3. Lemma 5.2. For any integer n ≥ 4 and 1 ≥ p ≥ 32 · logn n , let G(n, p) denote the random graph with n vertices, in which each edge occurs independently with probability p. Then with probability at least 1 − O(1/n), (i) the random graph is connected, (ii) the number of edges in the random graph is between pn2 /4 and pn2 , and (iii) for each subset of vertices S with \|S\| ≤ n/2, the number of edges leaving S is at least p2 · \|S\| · (n − \|S\|). In particular, for any sufficiently large integer n, when n2 /2 ≥ m ≥ 16n log n, by setting p = 2m/n2 , there exists a connected graph with n vertices and [m/2, 2m] edges, such that for each subset of vertices S with \|S\| ≤ n/2, the m number of edges leaving S is at least 2n · \|S\| = Θ(m/n) · \|S\|. We denote such a graph by H(n, m). We discuss our construction here (see Figure 5) before delving into the proof. The vertex set V is the union of three vertex subsets V1 , V2 , V3 , such that p \|V1 \| = \|V2 \| = \|V3 \| = n, \|V1 ∩ V2 \| = \|V2 ∩ V3 \| = m/n, and V1 , V3 are disjoint. In each of V1 , V2 and V3 , we embed H(n, m). The edge sets are denoted E1 , E2 , E3 respectively. Up to this point, the construction is similar to that of Ostrovskii [30], except that we use H(n, m) instead of a complete graph. The new component in our construction is adding the following edges. For each vertex v ∈ V1 ∩ V2 , add an edge between v and every vertex in (V1 ∪ V2 ) \ {v}. The set of these edges are denoted F1 . Similarly, for each vertex v ∈ V3 ∩ V2 , add an edge between v and every vertex in (V3 ∪ V2 ) \ {v}. The set of these edges are denoted F3 . This new component is crucial: without it, we could only prove a √ √ √ lower bound of Ω(m/ n) = Ω( mn · nm ). Proof of Theorem 5.1: . p Let G = (V, E) be the graph constructed as above. The whole graph has 3n − 2 m/n vertices. The number m p of edges is at least √ (due to edges in E1 and E3 ), and is at most 6m + 2 m/n · 2n = 6m + 4 mn, which is at most 7m for all sufficiently large n. It is well known that for any tree on n vertices, there exists a vertex x called a centroid of the tree such that, removing x decomposes the tree into connected components, each of size at most n/2. Now, consider any spanning tree of the Spanning Tree Congestion 19 V2 H(n, m) V1 V3 v3 v1 H(n, m) H(n, m) Fig. 5. Our lower-bound construction for spanning tree congestion. V1 , V2 , V3 are three vertex subsets of the same size. In each of the subsets, we embed expander H(n, m). There is a small overlap between V2 and V1 , V3 , while V1 , V3 are disjoint. For any vertex v1 ∈ V1 ∩ V2 , we add edges between it and any other vertex in V1 ∪ V2 ; similarly, for any vertex v3 ∈ V3 ∩ V2 , we add edges (not shown in figure) between it and any other vertex in V3 ∪ V2 . given graph, let u be a centroid of the tree. Without loss of generality, we can assume that u ∈ / V1 ; otherwise we swap the roles of V1 and V3 . The removal of u (and its adjacent edges) from the tree decomposes the tree into a number of connected components. For any of these components which intersects V1 , it must contain pat least one vertex of V1 ∩ V2 , thus the number of such components is at most m/n, and hence there exists one of them, denoted by Uj , such that p p b1 := \|Uj ∩ V1 \| ≥ n/( m/n) = n n/m. Let ej denote the tree-edge that connects u to Uj . Then there are three cases: p p Case 1: n n/m ≤ b1 ≤ n − n n/m. Due to the property √ of H(n, m), the congestion of ej is at least Θ(m/n) · min{b1 , n − b1 } ≥ Θ( mn). p p Case 2: b1 > n − n n/m and \|Uj ∩ V1 ∩ V2 \| ≤ 21 · m/n. Let W := p (V1 ∩ V2 ) \ Uj . Note that by this case’s assumption, \|W1 \| ≥ 12 · m/n. Due to the edge subset F1 , the congestion of ej is at least √ 1 p n F1 (W , V1 \ W ) ≥ · m/n · = Θ mn . 2 2 p p Case 3: b1 > n − n n/m and \|Uj ∩ V1 ∩ V2 \| > 21 · m/n. Let W 0 := Uj ∩ V1 ∩ V2 , and let Z := (V2 \ V1 ) ∩ Uj . p Note that b1 > n − n n/m ≥ 9n/10. Suppose \|Z\| ≥ 6n/10, then \|Uj \| > 9n/10 + 6n/10 > \|V \|/2, a contradiction to the assumption that u is a 20 L. Sunil Chandran, Yun Kuen Cheung, and Davis Issac centroid. Thus, \|Z\| < 6n/10. Due to the edge subset F2 , the congestion of ej is at least F2 (W 0 ∪ Z , V2 \ (W 0 ∪ Z)) ≥ \|W 0 \| · (n − \|W 0 \| − \|Z\|) p √ 1 p 6n ≥ · m/n · n − m/n − = Θ( mn). 2 10 6 Graphs with Expanding Properties For any vertex subset U, W ⊂ V , let NW (U ) denote the set of vertices in W which are adjacent to a vertex in U . Let N (U ) := NV \U (U ). Definition 6.1. A graph G = (V, E) on n vertices is an (n, s, d1 , d2 , d3 , t)expanding graph if the following four conditions are satisfied: (1) for each vertex subset S with \|S\| = s, \|N (S)\| ≥ d1 n; (2) for each vertex subset S with \|S\| ≤ s, \|N (S)\| ≥ d2 \|S\|; (3) for each vertex subset S with \|S\| ≤ n/2 and for any subset S 0 ⊂ S, \|NV \S (S 0 )\| ≥ \|S 0 \|− t. (4) For each vertex subset S, \|E(S, V \ S)\| ≤ d3 \|S\|. Theorem 6.2. For any connected graph G which is an (n, s, d1 , d2 , d3 , t)-expanding graph, there is a polynomial time algorithm which computes a spanning tree with congestion at most " # log(2−δ) 2 3d1 n 1 t d3 · 4 · max s + 1 , . · + t , where δ = d2 2d1 d1 n Next, we present the polynomial time algorithm in Theorem 6.2 and its analysis. Algorithm. Let G be an (n, s, d1 , d2 , d3 , t)-expanding graph. By Condition (2), every vertex has degree at least d2 . Let v0 be a vertex of degree d ≥ d2 , and let v1 , · · · , vd be its d neighbours. We maintain a tree T rooted at v0 such that T = T1 ∪ T2 ∪ · · · ∪ Td ∪ {v0 v1 , v0 v2 , . . . , v0 vd } where T1 , T2 , · · · , Td are trees rooted at v1 , v2 , . . . , vd respectively. We call the Ti0 s as branches. (See Figure 6). We start with each branch Ti = vi . In order to minimize congestion, we grow T in a balanced way, i.e., we maintain that the Tin’s are roughlyoof the same size. A branch is saturated if it contains at least max s + 1 , 3dd12n vertices. At any point of time, let VT be the set of vertices in T and VT be the vertices not in T . Often, we will move a subtree of a saturated branch Ti to an unsaturated branch Tj to ensure balance. For any x ∈ VT , let Tx denote the subtree of T rooted at x. A vertex x of a saturated branch Ti is called transferable (to branch Tj ) if x has a neighbour y in Tj and the tree Tj ∪ {xy} ∪ Tx is unsaturated. (See Figure 7.) Spanning Tree Congestion 21 v0 v1 vd v2 T1 T2 Td VT Fig. 6. The tree T and its branches v0 Ti∗ v0 Tj Ti∗ Tj y x y x Fig. 7. Transfer of a subtree from a saturated branch to an unsaturated branch The algorithm is divided into two phases which are described below. Throughout the algorithm, whenever a branch Ti gets modified, T gets modified accordingly, and whenever T gets modified VT and VT gets modified accordingly. Phase 1: Repeatedly do one of the following two actions, until \|VT \| ≥ d1 n: (We will prove that the precondition of at least one of the actions is satisfied if \|VT \| < d1 n) 1. If there exists a b ∈ VT such that b has a neighbour a in some unsaturated branch Ti : Add the vertex b and the edge ab to branch Ti . 2. If there exists at least one transferable vertex: (see Figure 7) Find the transferable vertex x such that Tx is the smallest. Let Ti∗ be the 22 L. Sunil Chandran, Yun Kuen Cheung, and Davis Issac branch currently containing x, Tj be a branch to which it is transferable, and y be an arbitrarily chosen neighbour of x in Tj . (a) Remove the subtree Tx from Ti∗ and add it to Tj with x as a child of y. (b) Pick a b ∈ VT that has a neighbour a (arbitrarily chosen, if many) either in Ti∗ or in Tj . (We will show in the analysis that such b exists). We add vertex b and edge ab to the branch containing a (i.e. to Ti∗ or Tj ). Phase 2: While VT 6= ∅, repeat: Find a maximum matching of G[VT , VT ], the bipartite graph formed by edges of G between VT and VT . Let M be the matching. Add all edges of M to T . In the analysis below, we say that a tree is saturated if it contains at least A vertices; we will determine its appropriate value by the end of the analysis. Analysis of Phase 1. We claim that during Phase 1, i.e. if \|VT \| < d1 n, the precondition of either step 1 or step 2 is satisfied. We also show the existence of a vertex b as specified in step 2b, whenever step 2b is reached. Given these and the fact that a vertex in VT is moved to VT (either in step 1 or in step 2b) during each round of Phase 1, we have that Phase 1 runs correctly and terminates after a linear number of rounds. During Phase 1, we will also maintain the invariant that each branch has at most A vertices; thus, each saturated branch has exactly A vertices. We call this invariant the balancedness. Note that balancedness is not violated due to step 1, as the new vertex is added to an unsaturated branch. It is not violated during step 2 as the branches Ti∗ and Tj (as defined in step 2) become unsaturated at the end of the step. We define the hidden vertices of T (denoted by H ≡ HT ) as follows: they are the vertices which are not adjacent to any vertices outside the tree, i.e., to any vertex in VT . If there is an unsaturated branch with a non-hidden vertex, clearly the precondition of step 1 is satisfied. So, let us assume that all the vertices in all unsaturated branches are hidden. In such a case, we show that the precondition of step 2 is satisfied if \|VT \| < d1 n. We argue that in this case \|H\| ≤ s: otherwise, take a subset H 0 ⊂ H of cardinality s, then by condition (2), N (H 0 ), which is contained in VT , has cardinality at least d1 n, a contradiction. Since \|VT \| < d1 n, the number of saturated branches is at most d1 n/A. To ensure that at least one unsaturated branch exists, we set A such that d1 n/A < d2 . Let U denote the set of vertices in all unsaturated branches. Since all vertices in U are hidden vertices, \|U \| ≤ s. Then by condition (2), \|N (U )\| ≥ d2 \|U \|. Note that the vertices in N (U ) are all in the saturated branches. By the pigeon-hole principle, there exists a saturated branch containing at least N (U )/(d1 n/A) ≥ Ad2 \|U \| d1 n vertices of N (U ). By setting A ≥ 3dd12n , the above calculation guarantees the existence of a saturated branch containing at least 3\|U \| ≥ \|U \| + 2 vertices of N (U ); let Ti be such a branch. Spanning Tree Congestion 23 In Ti , pick a vertex x ∈ Ti ∩ N (U ) such that Tx does not contain any vertex in N (U ), except x. Then the size of Tx is at most A − \|N (U )∩Ti \|+1 ≥ A −(\|U \|+1). Let y ∈ U be a vertex which is adjacent to x and Tj be the branch containing y. Since Tj has at most \|U \| vertices, x is a transferable vertex (to Tj ). Thus precondition of step 2 is satisfied. We further set A > s so that in each saturated branch, there is at least one unhidden vertex. In particular, Ti has an unhidden vertex, which is adjacent to some b ∈ VT . The vertex b is either adjacent to a vertex in Tx , or a vertex in Ti \ Tx as required in step 2b. Analysis of Phase 2. Since G is connected, M is non-empty in each iteration of Phase 2, and hence Phase 2 terminates in linear number of rounds. At the end of Phase 2, since VT is empty, T is clearly a spanning tree. It only remains to estimate the congestion of this spanning tree. Towards this, we state the following modified Hall’s theorem, which is an easy corollary of the standard Hall’s theorem. Lemma 6.3. In a bipartite graph (L, R) with \|L\| ≤ \|R\|, for any vertex w ∈ L, let R(w) denote the neighbours of w in R; then for any W ⊂ L, let R(W ) := ∪w∈W R(w). Suppose that there exist t ≥ 0 such that for any W ⊂ L, we have \|R(W )\| ≥ \|W \| − t. Then the bipartite graph admits a matching of size at least \|L\| − t. Recall that Phase 2 consists of multiple rounds of finding a matching between VT and VT . As long as \|VT \| ≤ n/2, condition (3) (with S = VT ) plus the modified Hall’s theorem (with L = VT and R = VT ) guarantees that in each round, at least t \|VT \| − t ≥ 1− · \|VT \| =: (1 − δ)\|VT \| d1 n m l number of vertices in VT are matched. Thus, after at most log(2−δ) 2d11 rounds of matching, \|VT \| ≥ n/2. After reaching \|VT \| ≥ n/2, condition (3) (with S = VT ) plus the modified Hall’s theorem (with L = VT and R = VT ) guarantees that after one more round of matching, all but t vertices are left in VT . By the end of Phase 1, each branch had at most A vertices. After each round of matching, the cardinality of each branch is doubled at most. Thus, the maximum possible number of vertices in each branch after running the whole algorithm is at most log(2−δ) 2 l m 1 log(2−δ) 2d1 +1 1 + t ≤ 4A · A·2 + t. 2d1 and hence the STC is at most " d3 · 4A · 1 2d1 log(2−δ) 2 # +t . Recall that we need A to satisfy d1 n/A < d2 , A ≥ n l mo set A := max s + 1 , 3dd12n . 3d1 n d2 and A > s. Thus we 24 6.1 L. Sunil Chandran, Yun Kuen Cheung, and Davis Issac Random Graph Let G ∈ G(n, p) where p ≥ c0 log n/n, and c0 = 64. The following lemmas show that with high probability G is an (n, s, d1 , d2 , d3 , t)-expanding graph with s = Θ(1/p), d1 = Θ(1), d2 = Θ(np), d3 = Θ(np), t = Θ(1/p) (and hence δ = o(1)). The proof of the lemmas are deferred to Appendix B.2. Lemma 6.4. For any S ⊆ V (G) such that \|S\| = d1/pe, we have \|N (S)\| ≥ c2 n with probability at least 1 − e−n/16 , where c2 = 1/25. Lemma 6.5. For any S ⊆ V (G) such that \|S\| ≤ 1/p, we have \|N (S)\| ≥ c3 np\|S\| with probability at least 1 − O(1/n2 ), where c3 = 1/16. Lemma 6.6. For all A ⊆ V (G) such that \|A\| ≤ n/2, and for all S ⊆ A, with probability at least 1 − e−n , S has at least \|S\| − c4 /p neighbors in V (G) \ A, where c4 = 12. Lemma 6.7. For all S ⊆ V (G), the cut size \|E(S, V (G) \ S)\| is at most np\|S\| with probability at least 1 − n−c0 /4 . Plugging the bounds from above lemmas into Theorem 6.2, together with a separate lower bound argument (Theorem B.2 in Appendix B.1), we have the following theorem; in Appendix B.1, we also present a non-algorithmic proof of this theorem. Theorem 6.8. If G ∈ G(n, p) where p ≥ 64 log n/n, then with probability at least 1 − O(1/n), its STC is Θ(n). 7 Discussion and Open Problems In this paper, we provide thorough understanding, both combinatorially and algorithmically, on the spanning tree congestion of general graphs and random graphs. On course of doing so, we also provide the first constructive proof for the generalized Győri-Lovász theorem, which might be of independent interest. Following are some natural open problems: – Finding the spanning tree with minimum congestion is NP-hard; indeed, Bodlaender et al. [9] showed a (9/8 − )-approximation NP-hardness for the STC problem. Does a constant or a poly-logarithmic factor approximation polynomial time algorithm exist? – We present √ an algorithm for computing a spanning tree achieving congestion at most O( mn). The algorithm runs in sub-exponential time when m = ω(n log2 n). Is there a polynomial time algorithm for constructing such a spanning tree? – For a k-connected graph, a connected k-partition where all parts are of size at most O((n/k) log k) can be found in polynomial time due to an algorithm of Chen et al. [14]. Can we improve the sizes of parts to O(n/k)? – Is finding Győri-Lovász partition PLS-complete? If not, is it polynomial time solvable? Spanning Tree Congestion 25 References 1. Ittai Abraham, Yair Bartal, and Ofer Neiman. Nearly tight low stretch spanning trees. 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Efficient algorithms for tripartitioning triconnected graphs and 3-edge-connected graphs. In Graph-Theoretic Concepts in Computer Science, 19th International Workshop, WG ’93, Utrecht, The Netherlands, June 16-18, 1993, Proceedings, pages 132–143, 1993. Spanning Tree Congestion A A.1 27 Missing Proofs in Sections 4 and 5 Proof of Corollary 4.5 First of all, we set the demand of each vertex in the flow problem to be the the degree of the vertex in G, and t1 , · · · , t` as the sinks in the flow problem. By Proposition 1.4, there exists an `-connected-partition ∪`j=1 U` such that for all j ∈ [`], tj ∈ Uj , and the total degrees of vertices in each Uj is at most 4m/`. With this, by routing the demand of a vertex in Uj to tj via an arbitrary path in G[Uj ] only, we construct a splittable flow with node congestion at most 4m/`. By Theorem 4.4, one can construct a confluent flow with node congestion at most 4(1 + ln `)m/` in polynomial time. Obviously, in the confluent flow, all the flow originating from one vertex goes completely into one sink. Set Vj to be the set of vertices such that the flows originating from these vertices go into tj . It is then routine to check that ∪`j=1 V` is our desired `-connected-partition. A.2 Proof of Theorem 4.2 Instead of giving the full proof, we point out the differences from the proof of Theorem 4.3. First, in handling the basis case (ii), by Theorem 1.3(b), Proposition 1.4 and Lemma 4.1, we havep an improvedp upper bound on the congestion of the returned tree, which is 8mH / m̂/nH ≤ 8 m̂nH . Thus, (1) can be improved to s p 8 m̂(nH − x) + m̂ · x. nH Again, by viewing x as a real variable and taking derivative, it is easy to see that the above expression is maximized at x = 1. So the above bound is at most s p 8 m̂(nH − 1) + m̂ nH p ≤ 8 m̂nH , as desired. Concerning the running time, it is clear that in the worst case, it is dominated by some calls to the algorithm in Theorem 1.3(b). Note that the number of such calls is at most n̂, since each call to the algorithm is on a disjoint set of vertices. There remains one concern, which is the connectedness of H[Y ∪ Z]. Suppose the contrary that H[Y ∪ Z] is not connected. Let C be one of its connected components, so that it contains the least number of vertices from Y . Then C contains at most b\|Y \|/2c vertices from Y , i.e., \|C ∩ Y \| < \|Y \|. Note that C ∩ Y is a vertex cut set of the graph H, thus contradicting that Y is a global minimum vertex cut set. 28 A.3 L. Sunil Chandran, Yun Kuen Cheung, and Davis Issac Proof of Lemma 5.2 It is well known that the requirements (i) and (ii) are satisfied with probability 1 − o(1/n). [11] For each subset S with \|S\| ≤ n/2, by the Chernoff bound, h i p P E(S, V \ S) ≤ · \|S\| · (n − \|S\|) ≤ e−p\|S\|(n−\|S\|)/8 ≤ e−pn\|S\|/16 . 2 Since p ≥ 32 · logn n , the above probability is at most n−2\|S\| . Then by a union bound, the probability that (iii) is not satisfied is at most bn/2c X s=1 B B.1 n · n−2s ≤ s bn/2c X s=1 bn/2c ns · n−2s ≤ X s=1 n−s ≤ 2 . n Spanning Tree Congestion of Random Graphs Non-Algorithmic Proof of Theorem 6.8 We first present a simple non-algorithmic proof that random graph has STC Θ(n) with high probability. Theorem B.1 gives the upper bound and Theorem B.2 gives the lower bound. The proof of Theorem B.1 uses Lemma 1.5 and the fact that for random graphs, vertex-connectivity and minimum degree are equal with high probability. Theorem B.1 does not give an efficient algorithm. Theorem B.1. If G ∈ G(n, p) where p ≥ 8 log n/n, then the spanning tree congestion of G is at most 16n with probability at least 1 − o(1/n). Proof. It is known that the threshold probability for a random graph being k-connected is same as the threshold probability for it having minimum degree at least k [12]. Since p ≥ 8 log n/n, using Chernoff bound and taking union bound over all vertices gives that G has minimum degree at least np/2 with probability at least 1 − o(1/n). Hence G is (np/2)-connected with probability at least 1 − o(1/n). We also have that the number of edges in G is at most 2n2 p with probability at least 1 − o(1/n). Now, by using Lemma 1.5, we have that with probability at least 1 − o(1/n), the spanning tree congestion is at most 16n. Theorem B.2. If G ∈ G(n, p) where p ≥ 32 log n/n, then the spanning tree congestion of G is Ω(n) with probability 1 − O(1/n). Proof. By using Chernoff Bounds and applying union bound, it is easy to show that with probability 1 − o(1/n), every vertex of G has degree at most c1 np for a sufficiently large constant c1 . Also, by Lemma 5.2, with probability 1 − O(1/n), properties (i) and (iii) of that lemma holds. In the proof below, we conditioned on the above mentioned highly probable events. Take a spanning tree T of G which gives the minimum congestion. Let u be a centroid of the tree T , i.e., each connected component of T \ {u} has at most n/2 vertices. If there is a connected component with number of vertices at least Spanning Tree Congestion 29 n/4, then define this connected component as T 0 . Else, all connected components have at most n/4 vertices. In this case, let T 0 be the forest formed by the union of a minimum number of connected components of T \ {u} such that \|T 0 \| ≥ n/4. It is easy to see that \|T 0 \| ≤ n/2. Also, the number of edges in T from V (T 0 ) to V (T ) \ V (T 0 ) is at most degG (u), which is at most c1 np. By property (iii) of Lemma 5.2, the number of edges between V (T 0 ) and V (G)\V (T 0 ) is Ω(n2 p). Each of these edges in G between V (T 0 ) and V (G)\V (T 0 ) have to contribute to the congestion of at least one of the edges in T between V (T 0 ) and V (G) \ V (T 0 ). Now since T 0 sends at most c1 np tree edges to other parts of T , it follows that there exists one edge in T with congestion at least Ω(n2 p)/(c1 np) = Ω(n), as claimed. B.2 Random Graph Satisfies Expanding Properties Constants. For easy reference, we list out the constants used. c0 = 64, c2 = 1/25, c3 = 1/16, c4 = 12 Proof of Lemma 6.4: . Let S = V (G) \ S. The probability that a fixed vertex in S does not have edge to S is at most (1 − p)\|S\| ≤ (1 − p)1/p ≤ e−1 . Since \|S\| ≥ n − 2/p ≥ n − 2n/(c0 log n) ≥ 31n/32, the expected value of \|N (S)\| is at least (31/32) n(1 − e−1 ) ≥ n/2. Hence, using Chernoff bound, the probability that \|N (S)\| < c2 n = n/25 is at most e−n/8 . Since the number of such S is at most n2/p = 22n/c0 ≤ 2n/32 , we have the lemma by applying union bound. Proof of Lemma 6.5: . Let S = V (G) \ S. Since \|S\| ≤ 1/p ≤ n/ log n, we have \|S\| ≥ n/2 for sufficiently large n. Divide S into groups of size d1/(p\|S\|)e. The probability that such a group does not have edge to S is at most (1 − p)\|S\|(1/(p\|S\|)) ≤ 1/e. The expected number of groups having edge to S is at least (np\|S\|/2)(1 − 1/e) ≥ np\|S\|/4. Thus, by Chernoff bound, the probability that \|N (S)\| ≤ np\|S\|/16 is at most e−np\|S\|/16 ≤ 2−c0 \|S\| log n/16 ≤ 2−4\|S\| log n . The number of sets of size \|S\| is at most 2\|S\| log n . Hence, taking union bound over all S with \|S\| ≤ 1/p, we get the required lemma. Proof of Lemma 6.6: . First, we prove that for all C, D ⊆ V (G) such that \|C\| ≥ n/4,\|D\| ≥ c4 /p, and C ∩ D = ∅, there exist at least one edge between C and D with high probability. The probability that there is no edge between such a fixed C and D is at most (1 − p)(n/4)(c4 /p) ≤ e−c4 n/4 . The number of pairs of such C and D is at most 22n . Hence, by taking union bound, the probability that for all C and D, the claim holds is at least 1 − e2n−(c4 n/4) ≥ 1 − e−n . Using the above claim, we prove that for all S ⊆ A, S has at least \|S\| − c4 /p neighbors in A := V (G) \ A with high probability. Suppose there is an S which violates the claim. Note that we can assume \|S\| ≥ c4 /p, because otherwise the claim is vacuously true. Let B := A \ N (S). There cannot be any edges between S and B. Also, \|B\| ≥ (n/2) − (\|S\| − (c4 /p)). So, \|B\| is at least c4 /p and when \|B\| < n/4, \|S\| is at least n/4. Hence, using the previous claim, there is an edge between S and B with probability at least 1 − e−n . Hence, we get a contradiction, and hence our claim is true with probability at least 1 − e−n . 30 L. Sunil Chandran, Yun Kuen Cheung, and Davis Issac Proof of Lemma 6.7: . Let C(S) denote \|E(S, V (G) \ S)\|. For a fixed vertex subset S, the expected value of C(S) is at most np\|S\|. Therefore, probability that C(S) > np\|S\| ≥ c0 \|S\| log n is at most n−c0 \|S\|/2 using Chernoff bounds. The probability that C(S) ≤ np\|S\| for all sets S of size k is at least 1−n−c0 k/2+k ≥ 1− n−c0 /2+1 using union bound and using k ≥ 1. The probability that C(S) ≤ np\|S\| for all vertex subsets S is at least 1 − n−c0 /2+2 ≥ using union bound over all k ∈ [n].	8
arXiv:1711.08848v2 [cs.CV] 29 Nov 2017 Real-Time Seamless Single Shot 6D Object Pose Prediction Bugra Tekin EPFL Sudipta N. Sinha Microsoft Research Pascal Fua EPFL [email protected] [email protected] [email protected] Abstract detection pipeline made of one CNN to coarsely segment the object and another to predict the 2D locations of the projections of the object’s 3D bounding box given the segmentation, which are then used to compute the 6D pose using a PnP algorithm [16]. The method is effective but slow due to its multi-stage nature. SSD-6D [10] is a different pipeline that relies on the SSD architecture [19] to predict 2D bounding boxes and a very rough estimate of the object’s orientation in a single step. This is followed by an approximation to predict the object’s depth from the size of its 2D bounding box in the image, to lift the 2D detections to 6D. Both BB8 and SSD-6D require a further pose refinement step for improved accuracy, which increases their running times linearly with the number of objects being detected. We propose a single-shot approach for simultaneously detecting an object in an RGB image and predicting its 6D pose without requiring multiple stages or having to examine multiple hypotheses. Unlike a recently proposed single-shot technique for this task [10] that only predicts an approximate 6D pose that must then be refined, ours is accurate enough not to require additional post-processing. As a result, it is much faster – 50 fps on a Titan X (Pascal) GPU – and more suitable for real-time processing. The key component of our method is a new CNN architecture inspired by [27, 28] that directly predicts the 2D image locations of the projected vertices of the object’s 3D bounding box. The object’s 6D pose is then estimated using a PnP algorithm. For single object and multiple object pose estimation on the L INE M OD and O CCLUSION datasets, our approach substantially outperforms other recent CNN-based approaches [10, 25] when they are all used without postprocessing. During post-processing, a pose refinement step can be used to boost the accuracy of these two methods, but at 10 fps or less, they are much slower than our method. In this paper, we propose a single-shot deep CNN architecture that takes the image as input and directly detects the 2D projections of the 3D bounding box vertices. It is end-to-end trainable and accurate even without any a posteriori refinement. And since, we do not need this refinement step, we also do not need a precise and detailed textured 3D object model that is needed by other methods [10, 25]. We only need the 3D bounding box of the object shape for training. This can be derived from other easier to acquire and approximate 3D shape representations. We demonstrate state-of-the-art accuracy on the L INE M OD dataset [8], which has become a de facto standard benchmark for 6D pose estimation. However, we are much faster than the competing techniques by a factor of more than five, when dealing with a single object. Furthermore, we pay virtually no time-penalty when handling several objects and our running time remains constant whereas that of other methods grow proportional to the number of objects, which we demonstrate on the O CCLUSION dataset [1]. 1. Introduction Real-time object detection and 6D pose estimation is crucial for augmented reality, virtual reality, and robotics. Currently, methods relying on depth data acquired by RGBD cameras are quite robust [1, 3, 4, 11, 13]. However, active depth sensors are power hungry, which makes 6D object detection methods for passive RGB images more attractive for mobile and wearable cameras. There are many fast keypoint and edge-based methods [21, 31, 35] that are effective for textured objects. However, they have difficulty handling weakly textured or untextured objects and processing low-resolution video streams, which are quite common when dealing with cameras on wearable devices. Therefore, our contribution is an architecture that yields a fast and accurate one-shot 6D pose prediction without requiring any post-processing. It extends single shot CNN architectures for 2D detection in a seamless and natural way to the 6D detection task. Our implementation is based on YOLO [28] but the approach is amenable to other singleshot detectors such as SSD [19] and its variants. Deep learning techniques have recently been used to address these limitations [10, 25]. BB8 [25] is a 6D object 1 2. Related Work We now review existing work on 6D pose estimation ranging from classical feature and template matching methods to newer end-to-end trainable CNN-based methods. Classical methods. Traditional RGB object instance recognition and pose estimation works used local keypoints and feature matching. Local descriptors needed by such methods were designed for invariance to changes in scale, rotation, illumination and viewpoints [21, 31, 35]. Such methods are often fast and robust to occlusion and scene clutter. However, they only reliably handle textured objects in high resolution images [15]. Other related methods include 3D model-based registration [17, 20], Hausdorff matching [9], oriented Chamfer matching for edges [18] and 3D chamfer matching for aligning 3D curve-based models to images [26]. RGB-D methods. The advent of commodity depth cameras has spawned many RGB-D object pose estimation methods [1, 3, 4, 11, 14, 23, 32, 38]. For example, Hinterstoisser et al. proposed template matching algorithms suitable for both color and depth images [7, 8]. Rios et al. [30] extended their work using discriminative learning and cascaded detections for higher accuracy and efficiency respectively. RGB-D methods were used on indoor robots for 3D object recognition, pose estimation, grasping and manipulation [3, 4, 5, 13, 14, 39]. Brachmann et al. [1] proposed using regression forests to predict dense object coordinates, to segment the object and recover its pose from dense correspondences. They also extended their method to handle uncertainty during inference and deal with RGB images [2]. Zach et al. [37] explored fast dynamic programming based algorithms for RGB-D images. CNN-based methods. In recent years, research in most pose estimation tasks has been dominated by CNNs. Techniques such as Viewpoints and Keypoints [34] and Render for CNN [33] cast object categorization and 3D pose estimation into classification tasks, specifically by discretizing the pose space. In contrast, PoseNet [12] proposes using a CNN to directly regress from a RGB image to a 6D pose, albeit for camera pose estimation, a slightly different task. Since PoseNet outputs a translational and a rotational component, the two associated loss terms have to be balanced carefully by tuning a hyper-parameter during training. To avoid this problem, the newer PoseCNN architecture [36] is trained to predict 6D object pose from a single RGB image in multiple stages, by decoupling the translation and rotation predictors. A geodesic loss function more suitable for optimizing over 3D rotations have been suggested in [22]. Another way to address this issue has recently emerged. In [10, 25], the CNNs do not directly predict object pose. Instead, they output 2D coordinates, 2D masks, or discrete orientation predictions from which the 6D pose can be inferred. Because all the predictions are in the 2D image, the problem of weighting different loss terms goes away. Also training becomes numerically more stable, resulting in better performance on the L INE M OD dataset [8]. We also adopt this philosophy in our work. In parallel to these developments, on the 2D object detection task, there has been a progressive trend towards single shot CNN frameworks as an alternative to two-staged methods such as Faster-RCNN [29] that first find a few candidate locations in the image and then classifies them as objects or background. Recently, single shot architectures such as YOLO [27, 28] and SSD [19] have been shown to be fast and accurate. SSD has been extended to predict the object’s identity, its 2D bounding box in the image and a discrete estimate of the object’s orientation [10, 24]. In this paper, we go beyond such methods by extending a YOLO-like architecture [28] to directly predict a few 2D coordinates from which the full 6D object pose can be accurately recovered. 3. Approach With our goal of designing an end-to-end trainable network that predicts the 6D pose in real-time, we were inspired by the impressive performance of single shot 2D object detectors such as YOLO [27, 28]. This led us to design the CNN architecture [27, 28] shown in Fig. 1. We designed our network to predict the 2D projections of the corners of the 3D bounding box around our objects. The main insight was that YOLO was originally designed to regress 2D bounding boxes and to predict the projections of the 3D bounding box corners in the image, a few more 2D points had to be predicted for each object instance in the image. Then given these 2D coordinates and the 3D ground control points for the bounding box corners, the 6D pose can be calculated algebraically with an efficient PnP algorithm [16]. BB8 [25] takes a similar approach. However, they first find a 2D segmentation mask around the object and present a cropped image to a second network that predicts the eight 2D corners in the image. We now describe our network architecture and explain various aspects of our approach in details. 3.1. Model We formulate the 6D pose estimation problem in terms of predicting the 2D image coordinates of virtual 3D control points associated with the 3D models of our objects of interest. Given the 2D coordinate predictions, we calculate the object’s 6D pose using a PnP algorithm. We parameterize the 3D model of each object with 9 control points. For these control points, we select the 8 corners of the tight 3D bounding box fitted to the 3D model, similar to [25]. In addition, we use the centroid of the object’s 3D model as the 9th point. This parameterization is general and can be S S (9x2+1+C) Figure 1. Overview: (a) The proposed CNN architecture. (b) An example input image with four objects. (c) The S × S grid showing cells responsible for detecting the four objects. (d) Each cell predicts 2D locations of the corners of the projected 3D bounding boxes in the image. (e) The 3D output tensor from our network, which represents for each cell a vector consisting of the 2D corner locations, the class probabilities and a confidence value associated with the prediction. used for any rigid 3D object with arbitrary shape and topology. In addition, these 9 control points are guaranteed to be well spread out in the 2D image and could be semantically meaningful for many man-made objects. Our model takes as input a single full color image, processes it with a fully-convolutional architecture shown in Figure 1(a) and divides the image into a 2D regular grid containing S × S cells as shown in Figure 1(c). In our model, each grid location in the 3D output tensor will be associated with a multidimensional vector, consisting of predicted 2D image locations of the 9 control points, the class probabilities of the object and an overall confidence value. At test time, predictions at cells with low confidence values, ie. where the objects of interest are not present, will be pruned. The output target values for our network are stored in a 3D tensor of size S × S × D visualized in Fig. 1(e). The target values for an object at a specific spatial cell location i ∈ S × S is placed in the i-th cell in the 3D tensor in the form of a D dimensional vector vi . When N objects are present in different cells, we have N such vectors, v1 , v2 , . . . , vn in the 3D tensor. We train our network to predict these target values. The 9 control points in our case are the 3D object model’s center and bounding box corners but could be defined in other ways as well. To train our net- work, we only need to know the 3D bounding box of the object, not a detailed mesh or an associated texture map. As in YOLO, it is crucial that a trained network is able to predict not only the precise 2D locations but also high confidence values in regions where the object is present and low confidence where it isn’t present. In case of 2D object detection, YOLO uses for its confidence values, an intersection over union (IoU) score associated with the predicted (and true 2D rectangles) in the image. In our case, the objects are in 3D and to compute an equivalent IoU score with two arbitrary cuboids, we would need to calculate a 3D convex hull corresponding to their intersections. This would be tedious and would slow down training. Therefore, we take a different approach. We model the predicted confidence value using a confidence function shown in Figure 2. The confidence function, c(x), returns a confidence value for a predicted 2D point denoted by x based on its distance DT (x) from the ground truth i.e. target 2D point. Formally, we define the confidence function c(x) as follows: ( c(x) = α(1− e 0 DT (x) dth ) , if DT (x) < dth otherwise (1) The distance DT (x) is defined as the 2D Euclidean distance in the image space. To achieve precise localization points, we do not constrain the network’s output as those points should be allowed to fall outside the cell. The predicted control point (gx , gy ) is defined as Confidence 1 0 0 Distance Figure 2. Confidence c(x) as a function of the distance DT (x) between a predicted point and the true point. with this function, we choose a sharp exponential function with a cut-off value dth instead of a monotonically decreasing linear function. The sharpness of the exponential function is defined by the parameter α. In practice, we apply the confidence function to all the control points and calculate the mean value and assign it as the confidence. As mentioned earlier, we also predict C conditional class probabilities at each cell. The class probability is conditioned on the cell containing an object. Overall, our output 3D tensor depicted in Figure 1(e) has dimension S × S × D, where the 2D spatial grid corresponding to the image dimensions has S × S cells and each such cell has a D dimensional vector. Here, D = 9×2+C +1, because we have 9 (xi , yi ) control points, C class probabilities and one confidence value. Our network architecture follows the fully convolutional YOLO v2 architecture [28]. Thus, our network has 23 convolutional layers and 5 max-pooling layers. Similar to YOLO v2, we choose S = 13 and have a 13 × 13 2D spatial grid on which we make our predictions. We also allow higher layers of our network to use fine-grained features by adding a passthrough layer. Specifically, we bring features from an earlier layer at resolution 26 × 26, apply batch normalization and resize the input image during training onthe-fly. As the network downsamples the image by a factor of 32, we change the input resolution to a multiple of 32 randomly chosen from the set {320, 352, . . . , 608} to be robust to objects of different size. 3.2. Training Procedure Our final layer outputs class probabilities, (x, y) coordinate locations for the control points, and the overall confidence score. During training, this confidence value is computed on the fly using the function defined in Eq. 1 to measure the distance between the current coordinate predictions and the ground-truth, DT (x). We predict offsets for the 2D coordinates with respect to (cx , cy ), the top-left corner of the associated grid cell. For the centroid, we constrain this offset to lie between 0 and 1. However, for the corner gx = f (x) + cx (2) gy = f (y) + cy (3) where f (·) is chosen to be a 1D sigmoid function in case of the centroid and the identity function in case of the eight corner points. This has the effect of forcing the network to first find the approximate cell location for the object and later refine its eight corner locations. We minimize the following loss function to train our complete network. L = λpt Lpt + λconf Lconf + λid Lid (4) Here, the terms Lpt , Lconf and Lid denote the coordinate loss, confidence loss and the classification loss respectively. We use mean-squared error for the coordinate and confidence losses, and cross entropy for the classification loss. As suggested in [27, 28], to improve model stability, we downweight the confidence loss for cells that don’t contain objects by setting λconf to 0.1. For cells that contain objects, we set λconf to 5.0. When multiple objects are located close to each other in the 3D scene, they are more likely to appear close together in the images or be occluded by each other. In these cases, certain cells might contain multiple objects. To be able to predict the pose of such multiple objects that lie in the same cell, we allow up to 5 candidates per cell and therefore predict five sets of control points per cell. Similarly to [28], we precompute with k-means, five anchor boxes that define the size, ie. the width and height of a 2D rectangle tightly fitted to a masked region around the object in the image. During training, we assign whichever anchor box has the most similar size to the current object as the responsible one to predict the 2D coordinates for that object. 3.3. Pose Prediction We detect and estimate the pose of objects in 6D by invoking our network only once. At test time, we estimate the class-specific confidence scores for each object by multiplying the class probabilities and the score returned by the confidence function. Each grid cell produces predictions in one network evaluation and cells with predictions with low confidence are pruned using a confidence threshold. For large objects and objects whose projections lie at the intersection of two cells, multiple cells are likely to predict highly confident detections. To obtain a more robust and well localized pose estimate, we inspect the cells in the 3×3 neighborhood of the cell which has the maximum confidence score. We combine the individual corner predictions of these adjacent cells by computing a weighted average of the individual detections, where the weights used are the confidence scores of the associated cells. At run-time, the network gives the 2D projections of the object’s centroid and corners of its 3D bounding box along with the object identity. We estimate the 6D pose from the correspondences between the 2D and 3D points using a Perspective-n-Point (PnP) pose estimation method [16]. In our case, PnP uses only 9 such control point correspondences and provides an estimate of the 3D rotation R and 3D translation t of the object in camera coordinates. 4. Implementation Details We initialize the parameters of our network by training the original network on the ImageNet classification task. As the pose estimates in the early stages of training are inaccurate, the confidence values computed using Eq. 1 are initially unreliable. To remedy this, we pretrain our network parameters by setting the regularization parameter for confidence to 0. Subsequently, we train our network by setting λconf to 5 for the cells that contain an object, and to 0.1 otherwise, to have more reliable confidence estimates in the early stages of the network. In practice, we set the sharpness of the confidence function α to 2 and the distance threshold to 30 pixels. We use stochastic gradient descent for optimization. We start with a learning rate of 0.001 and divide the learning rate by 10 at every 100 epochs. To avoid overfitting, we use extensive data augmentation by randomly changing the hue, saturation and exposure of the image by up to a factor of 1.2. We also randomly scale and translate the image by up to a factor of 20% of the image size. Our implementation is based on PyTorch. We will make our code publicly available for the sake of reproducibility. 5. Experiments We first evaluate our method for estimating the 6D pose of single objects and then we evaluate it in the case where multiple objects are present in the image. We use the same datasets and evaluation protocols as in [2, 10, 25], which we review below. We then present and compare our results to the state of the art methods. 5.1. Datasets We test our approach on two datasets that were designed explicitly to benchmark 6D object pose estimation algorithms. We describe them briefly below. LineMod [8] has become a de facto standard benchmark for 6D object pose estimation of textureless objects in cluttered scenes. The central object in each RGB image is assigned a ground-truth rotation, translation, and ID. A full 3D mesh representing the object is also provided. Method Object Ape Benchvise Cam Can Cat Driller Duck Eggbox Glue Holepuncher Iron Lamp Phone Average w/o Refinement Brachmann BB8 OURS [2] [25] 95.3 92.10 80.0 95.06 80.9 93.24 84.1 97.44 97.0 97.41 74.1 79.41 81.2 94.65 87.9 90.33 89.0 96.53 90.5 92.86 78.9 82.94 74.4 76.87 77.6 86.07 69.5 83.9 90.37 w/ Refinement Brachmann BB8 [2] [25] 85.2 96.6 67.9 90.1 58.7 86.0 70.8 91.2 84.2 98.8 73.9 80.9 73.1 92.2 83.1 91.0 74.2 92.3 78.9 95.3 83.6 84.8 64.0 75.8 60.6 85.3 73.7 89.3 Table 1. Comparison of our approach with state-of-the-art algorithms on LineMod in terms of 2D reprojection error. We report percentages of correctly estimated poses. Bold face numbers denote the best overall methods, bold italic numbers denote the best methods among those that do not use refinement as opposed to the ones that use, if different. Note that even though we do not rely on the knowledge of a detailed 3D object model our method consistently outperforms the baselines. OCCLUSION [1] is a multi-object detection and pose estimation dataset that contains additional annotations for all objects in a subset of the LineMod images. As its name suggests, several objects in the images are severely occluded due to scene clutter, which makes pose estimation extremely challenging. With the exception of [10, 25], it has primarily been used to test algorithms that require depth images. 5.2. Evaluation Metrics We use three standard metrics to evaluate 6D pose accuracy, namely – 2D reprojection error, average 3D distance of model vertices (referred to as ADD metric), and IoU score as in [2, 10, 25]. In all cases, we calculate the accuracy as the percentage of correct pose estimates for certain error thresholds. When using the reprojection error, we consider a pose estimate to be correct when the mean distance between the 2D projections of the object’s 3D mesh vertices using the estimate and the ground truth pose is less than 5 pixels [2]. This measures the closeness of the true image projection of the object to that obtained by using the estimated pose. This metric is suitable for augmented reality applications. When comparing 6D poses using the ADD metric, we take a pose estimate to be correct if the mean distance between the true coordinates of 3D mesh vertices and those estimated given the pose is less than 10% of the object’s diameter [8]. For most objects, this is approximately a 2cm threshold but for smaller objects, such as ape, the threshold drops to about 1cm. For rotationally symmetric objects whose pose can only be computed up to one degree of rotational freedom, we modify slightly the metric as in [2, 8] Method and compute s= 1 X min k(Rx + t) − (R̂x + t̂)k , M \|M\| (5) x1 ∈M where (R, t) are the ground-truth rotation and translation, (R̂, t̂) the predicted ones, and M the vertex set of the 3D model. We use this metric when evaluating the pose accuracy for the rotationally invariant objects, eggbox and glue as in [2, 8]. To compute the IoU metric, we measure the overlap between the projections of the 3D model given the ground-truth and predicted pose and accept a pose as correct if the overlap is larger than 0.5. 5.3. Single Object Pose Estimation We first estimate the 6D pose of the central object in the RGB only LineMod images, without reference to the depth ones. We compare our approach to those of [2, 10, 25], which operate under similar conditions. In this dataset, the training images are selected such that the relative orientation between corresponding pose annotations are larger than a threshold. To avoid being influenced by the scene context, we segment the training images using the segmentation masks provided with the dataset and replace the background by a random image from the PASCAL VOC dataset [6]. We use exactly the same training/test splits as in [25]. We report our results in terms of 2D reprojection error in Table 1 and 6D pose error in Table 2. We provide example pose predictions of our approach in Figure 3. 5.3.1 w/o Refinement Brachmann BB8 SSD-6D Object [2] [25] [10] Ape 27.9 0 Benchvise 62.0 0.18 Cam 40.1 0.41 Can 48.1 1.35 Cat 45.2 0.51 Driller 58.6 2.58 Duck 32.8 0 Eggbox 40.0 8.9 Glue 27.0 0 Holepuncher 42.4 0.30 Iron 67.0 8.86 Lamp 39.9 8.20 Phone 35.2 0.18 Average 32.3 43.6 2.42 w/ Refinement OURS Brachmann BB8 SSD-6D [2] [25] [10] 21.62 33.2 40.4 65 81.80 64.8 91.8 80 36.57 38.4 55.7 78 68.80 62.9 64.1 86 41.82 42.7 62.6 70 63.51 61.9 74.4 73 27.23 30.2 44.3 66 69.58 49.9 57.8 100 80.02 31.2 41.2 100 42.63 52.8 67.2 49 74.97 80.0 84.7 78 71.11 67.0 76.5 73 47.74 38.1 54.0 79 55.95 50.2 62.7 79 Table 2. Comparison of our approach with state-of-the-art algorithms on LineMod in terms of ADD metric. We report percentages of correctly estimated poses. Bold face numbers denote the best overall methods, bold italic numbers denote the best methods among those that do not use refinement as opposed to the ones that use, if different. Threshold Object Ape Benchvise Cam Can Cat Driller Duck Eggbox Glue Holepuncher Iron Lamp Phone Average [10] 0 0.18 0.41 1.35 0.51 2.58 0 8.9 0 0.30 8.86 8.20 0.18 2.42 10% OURS 21.62 81.80 36.57 68.80 41.82 63.51 27.23 69.58 80.02 42.63 74.97 71.11 47.74 55.95 30% [10] OURS 5.62 70.67 2.07 91.07 34.52 81.57 61.43 99.02 36.87 90.62 56.01 99.01 5.56 70.70 24.61 81.31 14.18 89.00 18.23 85.54 59.26 98.88 57.64 98.85 35.55 91.07 31.65 88.25 50% [10] OURS 19.95 88.10 10.62 98.85 63.54 94.80 85.49 99.90 64.04 98.80 84.86 99.80 32.65 89.39 48.41 98.31 26.94 97.20 38.75 96.29 88.31 99.39 81.03 99.62 61.22 98.85 54.29 96.78 Table 3. Comparison of our approach with SSD-6D [10] without refinement using different thresholds for the 6D pose metric. Comparative Accuracy 6D Accuracy in terms of projection error. In Table 1, we compare our results to those of Brachmann et al. [2] and to BB8 [25]. Both of these competing methods involve a multi-stage pipeline that comprises a 2D detection step followed by pose prediction and refinement. Since we do not have a refinement stage, we show in the table their results without and with it. In both cases, we achieve better 6D pose estimation accuracies. In Table 4, we perform a similar comparison with SSD6D [10], whose authors report their projection accuracy in terms of the IoU metric. That method also requires a posteriori refinement and our results are again better in both cases, even though SSD-6D relies on a large training set of rendered images that are sampled over a wide range of viewpoints and locations. the competing methods. Before refinement, we outperform all the methods by a significant margin of at least 12%. After refinement, our pose estimates are still better than Brachmann et al. [2]. By assuming the additional knowledge of a full 3D CAD model and using it to further refine the pose, BB8 1 and SSD-6D 2 boost their pose estimation accuracy. Without any bells and whistles, our approach achieves state-of-the-art pose estimation accuracy in all the metrics without refinement. When compared against methods that rely on the additional knowledge of full 3D CAD models and pose refinement, it still achieves state-of-the-art performance in 2D projection error and IoU metrics and yields comparable accuracy in the ADD metric. Our approach could be used in conjunction with such refinement strategies to further increase the accuracy however this comes at a heavy computational cost as we describe below. 6D Accuracy in terms of the ADD metric. In Tables 2 and 3, we compare our methods against the other in terms of the average of the 3D distances, as described in Section 5.2. In Table 2, we give numbers before and after refinement for 1 The authors do not report results without refinement, however they provided us with the accuracy numbers reported in Table 2. 2 The authors were not able to provide their accuracy numbers without refinement for this metric, but made their code publicly available. We ran their code with provided pretrained models to obtain the 6D pose errors. Method Object Ape Benchvise Cam Can Cat Duck Glue Holepuncher Iron Lamp Phone Average Driller Eggbox w/o Refinement SSD-6D OURS [10] 98.46 99.81 100 99.90 99.53 100 100 99.81 99.34 99.90 99.04 100 97.24 99.81 98.95 99.90 99.65 100 99.38 100 99.91 100 99.22 99.92 100 99.91 w/ Refinement SSD-6D [10] 99 100 99 100 99 98 98 99 99 99 100 99.4 99 99 Table 4. Comparison of our approach against [10] on LineMod using IoU metric. The authors of [10] were able to provide us the results of our approach w/o the refinement. 5.3.2 Accuracy / Speed Trade-off In Table 5, we report the computational efficiency of our approach for single object pose estimation in comparison to the state-of-the-art approaches [2, 10, 25]. Our approach runs at real-time performance in contrast to the existing approaches which fall short of it. In particular, our algorithm runs at least 5 times faster than the state-of-the-art techniques for single object pose estimation. As can be seen in Table 2, pose refinement in Brachmann et al. increase the accuracy significantly by 17.9% at an additional run-time of 100 miliseconds per object. BB8 also gets a substantial improvement of 19.1% in accuracy at an additional run-time of 21 miliseconds per object. Even without correcting for the pose error, our approach outperforms Brachmann et al. and yields close accuracy to BB8 while being 16 times faster for single object pose estimation. As discussed also in [10], the unrefined poses computed from the bounding boxes of the SSD 2D object detector, are rather approximate. We confirmed this by running their publicly available code with the provided pretrained models. We report the accuracy numbers without the refinement using the ADD metric in Table 3 for different thresholds. While providing a good initialization for the subsequent pose processing, the pose estimates of SSD-6D without refinement are much less accurate than our approach. The further refinement increases the pose estimation accuracy significantly, however st of a computational time of 24 miliseconds per object. Moreover, in contrast to our approach, the refinement requires the knowledge of the full 3D object CAD model. In Figure 3, we show example results of our method on the L INE M OD. We include more visual results of our method in the supplementary material. Method Overall speed for 1 object Refinement runtime 2 fps 3 fps 10 fps 50 fps 100 ms/object 21 ms/object 24 ms/object - Brachmann et al. [2] Rad & Lepetit [25] Kehl et al. [10] OURS Table 5. Comparison of the overall computational runtime of our approach for a single object in comparison to [2, 10, 25]. We further provide the computational runtime induced by the pose refinement stage of [2, 10, 25] and report pose estimation accuracy as in [25]. The identity of the objects cannot be assumed to be known a priori and has to be guessed. To this end, the method of [25] assumes that it has access to image crops based on the ground-truth 2D bounding boxes. 3 We make no such assumptions. Instead, we jointly detect the object in 2D, estimate its identity and predict its 6D pose. We generate our training images with the approach explained in Section 5.2. We further augment the LineMod training data by adding into the images objects extracted from other training sequences. We report our pose estimation accuracy in Figure 4 and demonstrate that even without assuming ground-truth information as in the case of [25], our method yields satisfactory pose accuracy in the case of severe occlusions. For object detection purposes, we consider an estimate to be correct if its detection IoU is larger than 0.5. Note that here the detection IoU corresponds to the overlap of the 2D bounding boxes of the object, rather than the overlap of the projected masks as is the case for the IoU metric defined in Sec 5.2. In Table 6, we report a mean average precision (MAP) of 0.48 which is similar to the accuracy reported by [2] and outperforms the ones reported by [7, 10]. Method MAP Hinterstoisser et al. [7] Brachmann et al. [2] Kehl et al. [10] OURS 0.21 0.51 0.38 0.48 Table 6. The detection experiment on the Occlusion dataset [2]. (Left) Precision-recall plot. (Right) 5.4. Multiple Object Pose Estimation Our approach provides accurate 6D poses with real-time performance. Upon one network invocation, our only computational overhead is an efficient PnP algorithm which operates on just 9 points per object. Furthermore we do not require full 3D colored object models to further refine our initial pose estimates. Our approach is therefore scalable to handle multiple objects as shown in Figure 5 and has only a negligible computational overhead of PnP (0.2 miliseconds/object) while the competing approaches [10] have a linear runtime growth. We use the OCCLUSION dataset to compare our approach to Brachmann et al. [2] for multi-object detection 3 This it is not explicitly stated in [25], but the authors confirmed this to us in private email communication. Figure 3. Pose estimation results of our approach. Note that our method can recover the 6D pose in these challenging scenarios, which involve significant amounts of clutter, occlusion and orientation ambiguity. In the last column, we show failure cases due to motion blur, severe occlusion and specularity (this figure is best viewed on a computer screen). tions. With only 1-2 % decrease in accuracy we can reach to a runtime of 94 fps and the runtime virtually remains the same for estimating the pose of multiple objects. Method 416 × 416 480 × 480 544 × 544 608 × 688 Figure 4. Percentage of correctly estimated poses as a function of the projection error for different objects of the Occlusion dataset [2]. 500 Kehl et al. `17 OURS Runtime in miliseconds 400 89.71 90.00 90.37 90.65 94 fps 67 fps 50 fps 43 fps Table 7. Accuracy/speed trade-off of our method on the L INE M OD dataset. Accuracy reported is the percentage of correctly estimated poses with respect to the 2D projection error. The same network model is used for all four input resolutions. Timings are on a . NVIDIA Titan X (Pascal) GPU. 6. Conclusion 300 200 100 0 2D projection metric Speed 0 5 10 15 20 Number of objects Figure 5. The runtime of our approach with increasing number of objects as compared to that of [10]. We also evaluated the accuracy and speed of our approach for different input resolutions. As explained in Section 3.1, we adopt a multi-scale training procedure and change the input resolution during training randomly as in [28]. This allows us to be able to change the input resolution at test-time and predict from images with higher resolution. This is especially useful for predicting the pose of small objects more robustly. As we do not have an initial step for 2D object detection and produce image crops which are then resized to higher resolutions for pose prediction as in [25], our approach requires better handling of the small objects. In Table 7, we compare the accuracy and computational efficiency of our approach for different input resolu- We have proposed a new CNN architecture for fast and accurate single-shot 6D pose prediction that naturally extends the single shot 2D object detection paradigm to 6D object detection. Our network predicts 2D locations of the projections of the objects 3D bounding box corners which involves predicting just a few more 2D points than for 2D bounding box regression. Given the predicted 2D corner projections, the 6D pose is computed via an efficient PnP method. For high accuracy, existing CNN-based 6D object detectors all refine their pose estimates during postprocessing, a step that requires an accurate 3D object model and also incurs a runtime overhead per detected object. In contrast, our single shot predictions are very accurate which alleviates the need for refinement. 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PoseCNN: A Convolutional Neural Network for 6D Object Pose Estimation in Cluttered Scenes. arXiv preprint arXiv:1711.00199, 2017. 2 [37] C. Zach, A. Penate-Sanchez, and M.-T. Pham. A Dynamic Programming Approach for Fast and Robust Object Pose Recognition from Range Images. In CVPR, 2015. 2 [38] H. Zhang and Q. Cao. Combined Holistic and Local Patches for Recovering 6D Object Pose. In ICCV, 2017. 2 [39] M. Zhu, K. G. Derpanis, Y. Yang, S. Brahmbhatt, M. Zhang, C. Phillips, M. Lecce, and K. Daniilidis. Single Image 3D Object Detection and Pose Estimation for Grasping. In ICRA, 2014. 2 Supplemental Material: “Real-Time Seamless Single Shot 6D Object Pose Prediction” In the supplemental material, we provide details on how the training images were prepared and on the proposed confidence weighted prediction step. We also present qualitative results on O CCLUSION [1] and L INE M OD [8]. Training Images. As discussed in the main paper, we segment the foreground object in the images in the training set, using the segmentation masks provided and paste the segmented image over a random image taken from the PASCAL VOC dataset [6]. Examples of such images, which are given as input to the network at training time are shown in Figure 6. This operation of removing the actual background prevents the network from learning the scene context and is essential in order to achieve proper generalization. Confidence-weighted prediction. In the final step of our method, we compute a weighted sum of multiple sets of predictions for the corners and the centroid, using associated confidence values as weights. On L INE M OD, this gave a 1–2% improvement in accuracy with the 2D projection metric. The first step involves scanning the full 17×17 grid to find the cell with the highest confidence for each potential object. We then consider a 3 × 3 neighborhood around it on the grid and prune the cells with confidence values lower than the detection threshold of 0.5. On the remaining cells, we compute a confidence-weighted average of the associated predicted 18-dimensional vectors, where the eight corner points and the centroid have been stacked to form the vector. The averaged coordinates are then used in the PnP method. This sub-pixel refinement on the grid usually improves the pose of somewhat large objects that occupy several adjoining cells in the grid. Figure 7 shows an example where the ape object lies between two adjoining cells and the confidence weighting improves the pose accuracy. Figure 7. (Left) The 17×17 grid on a 544×544 image. (Middle) Confidence values for predictions of the ape object on the grid. (Right) Cropped view of our pose estimate (shown in blue) and the ground truth (shown in green). Here, three cells next to the best cell have good predictions and their combination gives a more accurate pose than the best prediction alone (best viewed in color). Figure 6. (Top) Using segmentation masks given in L INE M OD, we extract the foreground objects in our training images and composite them over random images from PASCAL VOC [6]. (Bottom) We also augment the training set by combining images of multiple objects taken from different training images. Qualitative Results. We show qualitative results from the O CCLUSION [1] and L INE M OD [8] datasets in Figures 8 to 13. These examples show that our method is robust to severe occlusions, rotational ambiguities in appearance, reflections, viewpoint change and scene clutter. (a) (b) (c) Figure 8. Results on the O CCLUSION dataset. Our method is quite robust against severe occlusions in the presence of scene clutter and rotational pose ambiguity for symmetric objects. (a) Input images, (b) 6D pose predictions of multiple objects, (c) A magnified view of the individual 6D pose estimates of six different objects is shown for clarity. In each case, the 3D bounding box is rendered on the input image. The following color coding is used – A PE (gold), B ENCHVISE (green), C AN (red), C AT (purple), D RILLER (cyan), D UCK (black), G LUE (orange), H OLEPUNCHER (blue). In addition to the objects from the O CCLUSION dataset, we also visualize the pose predictions of the Benchvise object from the L INE M OD dataset. As in [25], we do not evaluate on the Eggbox object, as more than 70% of close poses are not seen in the training sequence. This image is best viewed on a computer screen. (a) (b) (c) Figure 9. Results on the O CCLUSION dataset. Our method is quite robust against severe occlusions in the presence of scene clutter and rotational pose ambiguity for symmetric objects. (a) Input images, (b) 6D pose predictions of multiple objects, (c) A magnified view of the individual 6D pose estimates of six different objects is shown for clarity. In each case, the 3D bounding box is rendered on the input image. The following color coding is used – A PE (gold), B ENCHVISE (green), C AN (red), C AT (purple), D RILLER (cyan), D UCK (black), G LUE (orange), H OLEPUNCHER (blue). In addition to the objects from the O CCLUSION dataset, we also visualize the pose predictions of the Benchvise object from the L INE M OD dataset. As in [25], we do not evaluate on the Eggbox object, as more than 70% of close poses are not seen in the training sequence. This image is best viewed on a computer screen. Figure 10. Example results on the L INE M OD dataset: (left) A PE, (middle) B ENCHVISE, (right) C AM. The projected 3D bounding boxes are rendered over the image and they have been cropped and resized for ease of visualization. The blue cuboid is rendered using our pose estimate whereas the green cuboid is rendered using the ground truth object pose. Note that the input image dimension is 640 × 480 pixels and the objects are often quite small. Noticeable scene clutter and occlusion makes these examples challenging. Figure 11. Example results on the L INE M OD dataset: (left) C AN, (middle) C AT, (right) D RILLER. The projected 3D bounding boxes are rendered over the image and they have been cropped and resized for ease of visualization. The blue cuboid is rendered using our pose estimate whereas the green cuboid is rendered using the ground truth object pose. Note that the input image dimension is 640 × 480 pixels and the objects are often quite small. Noticeable scene clutter and occlusion makes these examples challenging. Figure 12. Example results on the L INE M OD dataset: (left) D UCK, (middle) E GGBOX, (right) G LUE. The projected 3D bounding boxes are rendered over the image and they have been cropped and resized for ease of visualization. The blue cuboid is rendered using our pose estimate whereas the green cuboid is rendered using the ground truth object pose. Note that the input image dimension is 640 × 480 pixels and the objects are often quite small. Noticeable scene clutter and occlusion makes these examples challenging. Figure 13. Example results on the L INE M OD dataset: (left) H OLE P UNCHER, (middle) I RON, (right) L AMP and P HONE. The projected 3D bounding boxes are rendered over the image and they have been cropped and resized for ease of visualization. The blue cuboid is rendered using our pose estimate whereas the green cuboid is rendered using the ground truth object pose. Note that the input image dimension is 640 × 480 pixels and the objects are often quite small. Noticeable scene clutter and occlusion makes these examples challenging.	1
Deep Residual Text Detection Network for Scene Text Xiangyu Zhu, Yingying Jiang, Shuli Yang, Xiaobing Wang, Wei Li, Pei Fu, Hua Wang and Zhenbo Luo Machine Learning Lab Samsung R&D Institute of China, Beijing Beijing, China {xiangyu.zhu, yy.jiang} @samsung.com Abstract—Scene text detection is a challenging problem in computer vision. In this paper, we propose a novel text detection network based on prevalent object detection frameworks. In order to obtain stronger semantic feature, we adopt ResNet as feature extraction layers and exploit multi-level feature by combining hierarchical convolutional networks. A vertical proposal mechanism is utilized to avoid proposal classification, while regression layer remains working to improve localization accuracy. Our approach evaluated on ICDAR2013 dataset achieves 0.91 F-measure, which outperforms previous state-ofthe-art results in scene text detection. Keywords—Scene text detection; Deep CTPN Residual Networks; I. INTRODUCTION Text detection is an important part of text content analysis, especially for reading natural text in the wild. Scene text detection is becoming increasing attractive to researchers, with the development of smart phone and tremendous demands of text recognition in Augmentation Reality (AR). Unlike traditional documental text, detecting scene text seems to be a much more challenging task due to illuminations, perspective distortion and complex background. In the last few decades, series of methods [1, 2, 3] had been proposed to deal with this problem, which achieved considerable performance. Those methods can be categorized into Sliding Window based methods and Connected Component (CC) based methods. Sliding Window based method utilizes sliding windows to search the image densely for candidate text regions, and classifies text/non-text regions by traditional machine learning tools. This kind of method can be quite slow as a consequence of densely search and multiscale windows. Comparing to the previous method, CC base method draws more attention until recently. It involved several steps, typically three. First, CCs are extracted from images as character candidates. Second, a character classifier is trained to remove the non-text CCs. Finally, remained CCs are going to be grouped into text-lines by clustering or rules. Maximally Stable Extremal Regions (MSER) is one of the most popular CC-based methods. It had been reported outstanding performance in ICDAR2013 benchmark [4]. However, the following limitations constrain its further improvement in performance. Words constituent of single character are ignored by grouping rules for the sake of precision, characters in low color contrast can not be extracted by MSER, and another disadvantage is the complex post-processing. Convolutional Neural Networks (CNN) approach has led to a great breakthrough in object detection. Region proposal CNN (R-CNN) [5] was the first attempt to classify proposals by CNN. Then Faster R-CNN [6] was proposed, where a subnetwork named RPN was designed to generate proposals autonomously by feature maps and a few additional convolution layers. Faster R-CNN used VGG-16 [7] as baseline for feature map extraction and proposal classification until deep residual network (ResNet) [8] was presented. ResNet was reported better performance in PASCAL VOC 2007 [9] and ILSVRC 2016 comparing to VGG16 and GoogLeNet [10, 11]. Moreover, the structure of ResNet was designed fully convolutional, without heavy fully connected layers. The ResNet version of Faster R-CNN was observed better performance. Inspired by the great progress in object detection, a few CNN based methods [12, 13, 14, 15, 16, 17] had been proposed to address scene text detection. The Connectionist Text Proposal Network (CTPN) [12] is a novel framework based on Faster R-CNN, which benefits from an additional recurrent neural network and vertical proposal mechanism. In this paper, we came up with a framework called Residual Text detection Network (RTN). RTN were inspired by ResNet and CTPN vertical proposal mechanism. First, ResNet was used to generate strong semantic feature instead of traditional networks like VGG-16. Rather than a naively layer replacement, we combine multi-level features to produce hierarchy residual feature. The outstanding performance was mainly contributed by this stronger semantic feature. Second, vertical proposal mechanism was adopted and an additional regression part was used to improve localization accuracy, this step was implemented by a two stage training strategy. It achieved 91.54% F-measure on ICDAR2013. II. RELATED WORK A. Object detection With the success of deep convolutional network in image recognition, R-CNN was inspired to classify region proposal via CNN. After R-CNN was proposed, the related object detection approaches had been developed rapidly, such as SPPnet, Fast R-CNN, Faster R-CNN, R-FCN. Faster R-CNN is a mature prevalent framework that trained and tested from end to end. The framework constitutes of three parts: (1) Feature map generation. Feature maps representing semantic information were extracted by deep convolutional network, VGG-16 was used in Faster R-CNN. (2) Proposal generation. A simple ResNet Vertical mechanism RPN Conv 1x1 BLSTM 2k box coordinates regression Hierarchy residual feature PSROI Pooling pool Fig.1. Architecture of Residual Text detection Network (RTN) convolutional network name Region Proposal Networks (RPN) was designed to generate candidate regions with the input of feature maps. (3) Region classification and regression. By sharing features regions proposals were projected to the location in feature maps, then a following Fast R-CNN structure outputted final results by classification and regression. Influenced by the latest progress in image recognition, other deeper convolutional networks were transplant to this framework instead of VGG-16 [7], including GoogLeNet [10, 11] and ResNet [8]. ResNet was proved to be a superior convolutional network than GoogLeNet and VGG16 in ImageNet classification task. R-FCN [18] is a completely fully convolutional architecture that combines ResNet and Faster RCNN together. FPN (Feature Pyramid Network) [19] exploits multi-scale pyramid of ResNet, and the framework using FPN won the champion of COCO [20] detection challenge 2016. Besides Faster R-CNN based pipeline, Single Shot MultiBox Detector (SSD) [21] and You Look Only Once (YOLO) [22] are two representative and promising works. SSD is one of the first attempts to utilizing multi-level convolutional networks, while YOLO is extremely faster than all the methods mentioned above. However, they do not get a superior performance with significant margin comparing to Faster RCNN pipeline. B. CNN based text detection General object detection pipeline can be transplant to text detection realm barrier free. CNN based text detection gradually becomes the most promising approach. Zhang [14] proposed a fully convolutional network for text detection in arbitrary orientation instead of semantic segmentation. It achieved an F-measure of 0.74 on ICDAR2013. DeepText [13] proposed a Inception-RPN and multi-level region-of-interest pooling based on the framework of Faster R-CNN. It achieved 0.85 F-measure on ICDAR2013. Inspired by SSD, Liao [15] presented a approach called TextBoxes, multi-level jointly predictions and word recognition were utilized. CTPN [12] is a unique network abandoned Fast R-CNN classification and regression, which can be treated as a novel individual RPN with Recurrent Neural Network (RNN). It achieved previous state-of-the-art on ICDAR2013 as 0.88 Fmeasure among published papers. Nevertheless, it was just a prototype for detection using RNN and fixed width proposal is harmful for localization accuracy. III. RESIDUAL TEXT DETECTION NETWORK The architecture of this Residual Text detection Network (RTN) is shown in Fig. 1. It consists of three parts: hierarchy residual feature map for feature extraction, vertical mechanism RPN for proposal prediction and bounding box regression part for higherlocalization accuracy. A. Hierarchy residual feature map In our framework, we use ResNet to derive feature map from original images. The feature map is a serial of features in 2D formation, similar to handcraft feature. It is fed to RPN and regression part. ResNet consists of 5 concatenate blocks (i.e., conv1, conv2_x, conv3_x, conv4_x and conv5_x). Conv4_x have the same stride as VGG-16 output (16 pixels). In RFCN[18], region proposals were predicted by conv4_x. They believed the conv4_x feature maps were semantic strong enough and comparable to VGG-16 feature maps. VGG-16 differs from ResNet in structure. Thus, a simple replacement, from VGG-16 to ResNet, would not work properly. Unlike VGG-16, typical ResNet based detection does not share the same feature map between RPN and regression parts. Conv4_x is utilized to generate proposals in RPN while conv5_x for regression. In this kind of methods, RPN is unable to use a deeper semantic feature. By visualizing feature maps of conv3_x, conv4_x, conv5_x and VGG-16, we find out conv3_x contains too many low level features, while conv4_x and conv5_x are competitive to VGG-16 on the first glance. We have carried out series of experiments on Faster R-CNN baseline using conv3_x, conv4_x and conv5_x respectively. Framework using conv3_x detected edges and lines instead of objects and required much more computation due to larger feature map sizes. It was a strong evidence that conv3_x contained too many low level features to be used directly. On the contrary, baselines using conv4_x and conv5_x detected text correctly. However, framework using conv5_x fails on detecting small text due to coarse resolution feature maps. Although conv5_x represents deeper feature, the resolution is half comparing to conv4_x. Even we adopt the “ à trous algorithm” [23] to compensate stride difference, the performance is still unsatisfactory. Using conv5_x as the only feature maps might be insufficient for text detection, but abandon deeper representations seems to be an unwise choice. We believe using conv5_x in a proper way will contribute to proposal prediction. It is rational to come up with a naive idea that predicting multi-scale proposals on conv4_x and conv5_x respectively, like previous approaches did, such as SSD and TextBoxes. In this way, not only we can detect fine scale text and robust to scale invariance, but also utilizing deeper feature representations. Nevertheless, it is inconvenient to identify reliability from multi-scale proposals without an additional classification, as we introduced vertical mechanism to RPN, it seems to be a rather complicated problem. To deal with that, we combine the hierarchy feature maps (conv4_x and conv5_x) together to produce a new hierarchy feature map. In this way, we can use both conv4_x and con5_x feature maps simultaneously, and the task to identify which feature maps are more reliable is assigned to convolution layers. As shown in Fig 2, the input size of original images is 224  224 , after several convolutional layers, conv4_x and conv5_x get feature maps in size 14  14 and 7  7 , corresponding to 16 pixels and 32 pixels stride. A deconvolution layer was used to upsample conv5_x, make sure the shapes of conv5_x (res5c) match conv4_x (res4b22) exactly. We attach a convolution layer with kernel 1  1 , which aim to work as learnable weights for combining conv5_x and conv4_x. Our experiment shows hierarchy feature lead to an improvement on both precision and recall. 7x7 Conv1x1 14 x 14 Conv5 14 x 14 deconv Conv4 Conv1x1 14 x 14 hierarchy feature 224 x 224 Fig.2. Hierarchy residual network architecture. First, conv5_x was upsampled to make sure its shapes match conv4_x. Second ,we attached convlutional layers with kernal to both conv5_x and conv4_x. Finally, hierarchy feature was produced by element wise addition. B. Vertical mechanism RPN In Faster R-CNN, a serial of CNN is used to classify proposals. The structure is called Fast R-CNN [24]. However, CTPN abandoned Fast R-CNN structure, namely RPN output vertical proposals directly without classification and regression. As we know, RPN can be treated as a general object detection system. If the detection task is to distinguish only one category from background (two categories in total), it seems that RPN is already competent for text detection. Depending on vertical proposal mechanism and recurrent neural network, CTPN [12] was able to detect text without Fast-RCNN. That mechanism makes the final model much smaller. In this approach, we adopt this vertical mechanism to RPN. Anchors and ground truth are divided into fixed width (16 pixels) boxes, shown in Fig 3. Particularly, spaces between ground truths are treated as negative samples. This enable the method to output result in word level. Sequences of vertical proposals will be predicted by RPN. A threshold is applied to remove non-text vertical proposals, therefore remained adjacent text proposals can be connected together to produce text line proposals. Fig.3 Yellow box: ground truth of vertical proposals. Green box: space between words which are treated as negative samples C. Bounding box regression By connecting vertical proposals, we will obtain text-line proposals as result. Nevertheless, fixed width proposal might lead to inaccurate localization, when the beginning and the end of vertical proposals are not exactly fit text. In small text case, the problem becomes more serious. Unlike general object detection, this inaccuracy will influence recognition tremendously. If parts of the characters are not included in bounding box, they might be omitted or wrongly recognized. On the contrary, a loose bounding box contains much background, and that could be recognized as additional characters. In conclusion, a tight and exact bounding box is significant for text detection and recognition. To achieve this goal, we introduce bounding box regression to get exact coordinates, just as Faster R-CNN and R-FCN did in their framework. In this paper, we refer to Fast R-CNN structure. As text line proposals are obtained in section B, bounding box offset of every proposal were calculated. However, classification is not contained in this part, only regression is remained. A further classification is unnecessary, and experiments show it is harmful for performance. This is because recurrent neural networks we adopted in RPN have a tendency to connect words into text lines. After we set word level as network learning goal, text line level proposal might be classified as negative result. The bounding box regression loss is defined as: 𝐿𝑙𝑜𝑐 (𝑡 𝑢 , 𝑣) = ∑ 𝑠𝑚𝑜𝑜𝑡ℎ𝐿1 (𝑡𝑖𝑢 − 𝑣𝑖 ) 𝑖∈{𝑥,𝑤} 𝑠𝑚𝑜𝑜𝑡ℎ𝐿1 (𝑥) = { 0.5𝑥 2 \|𝑥\| − 0.5 𝑖𝑓\|𝑥\| < 1 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 In this functions, 𝑣 = (𝑣𝑥 , 𝑣𝑤 , ) is the ground truth of bounding box, 𝑡 𝑢 = (𝑡𝑥 , 𝑡𝑤 , ) is predicted coordinates, 𝑥 and 𝑤 stand for x coordinate and width. 𝐿1 smooth function is used for regression. This loss function is almost the same as what used in Fast R-CNN except that two coordinates (𝑥, 𝑤) offsets are predicted instead of four coordinates (𝑥, 𝑦, ℎ, 𝑤) . It is unnecessary to regression y coordinate (𝑦) and height (ℎ) which was done in RPN layers for every single vertical proposal. We develop a two stage training strategy to implement this further regression:  Stage one. Hierarchy residual feature and vertical mechanism RPN were trained; the learning rates of regression parts were set to 0.  Stage two. Regression parts were trained individually, the learning rates of ResNet, hierarchy residual feature and RPN were set to 0. A normal RPN presented in Faster R-CNN is used to generate anchors and train regression parts and will not be used in test model. D. Training and testing details Our model was trained on 15,000 natural images collected and labeled by ourselves. These images were labeled in word level and resized to (600, 1000) scale. There is no overlap between these images or any kind of public dataset available on the internet. On the condition of the extremely similarity between ICDAR2013 training set and testing set, ICDAR2013 training set was not included to prevent over-fitting ICDAR2013 testing set. Training ground truth is labeled in word level, and then divided into vertical ground truth by a fixed width (16 pixels) in proposal layer, corresponding to vertical proposals mentioned above. The space between words were labeled as negative samples, anchor has an IoU> 0.5 overlap with space samples were signed as negative label. About 10% of negative samples are space. By adding space sample, the networks tend to output word level proposal rather than text-line level. IV. EXPERIMENTS A. Evaluation of hierarchy residual feature In CTPN, 3,000 natural images were collected and label for training, much less than ours. In order to prove the improvement is a consequence of stronger semantic feature map rather than much more training data, we implement our own version of CTPN and training on 15,000 images. All the experiments carried out below were trained on the same amount of images. In this experiment, we used VGG-16 and ResNet-101 as backbone for feature extraction. Feature map generated by different layers were evaluated, including conv5 of VGG-16, conv4_x (res4b22) of ResNet-101 and hierarchy residual feature map (res4b22 + res5c) used in RTN. Table 1 shows the performances on ICDAR2013, we use CTPN framework as baseline and different feature maps mentioned above are evaluated, all the parameters and following processing are the same. We evaluated these methods on two scales respectively, namely (600, 1000) and (960, 1280). Scale (600, 1000) means the shortest side of images is no more than 600 pixels and the longest side can not exceed 1000 pixels, so does to scale (960, 1280). One observation is that all these feature maps are competitive in scale (600, 1000). However, when it comes to scale (960, 1280), margins between these methods becoming considerable. We had run the open source test code 2 provided by the author of CTPN, marked as CTPN-tianzhi. Larger scale did not benefit performance. On the contrary, the F-score degraded. Moreover, our CTPN implementation with VGG-16 improved slightly on F-scores. In conclusion, larger test scale does not always helpful for detection and localization. Nevertheless, by simply replacing VGG-16 to ResNet-101 conv4_x (res4b22), F-score improved to from 88.75% to 90.32%, it proves ResNet-101 has a superior feature representation comparing to VGG-16 as other papers [8, 18] mentioned. Furthermore, baseline with hierarchy residual feature map (res4b22 + res5c) achieved the best performance with F-score=91.17%, which improve 5 points on recall comparing to the original CTPN. The results shows baseline with hierarchy residual feature achieves the best performance on both recall and precision on scale (960, 1280), which could be a convincing evidence for stronger semantic feature. We evaluated RTN on ICDAR 2013 benchmarks. It consists of 233 focused text images taken in the wild. The evaluation criteria are provided by the ICDAR2015 Robust Reading Competition website1 as previous works did. First, the effectiveness of hierarchy residual feature map was verified comparing to other prevalent feature extraction layers. Then, additional regression layers were proved to be helpful for localization accuracy. Finally, this method was compared to other published methods, and it achieved state-ofthe-art performance. TABLE 1. EVALUATING BASELINE WITH DIFFERENT FEATURE MAP ON ICDAR2013 Method CTPN-Tianzhi CTPN-Tianzhi CTPN CTPN CTPN CTPN RTN RTN  1. http://rrc.cvc.uab.es 2. https://github.com/tianzhi0549/CTPN/ Backbone VGG-16 VGG-16 VGG-16 VGG-16 ResNet-101 ResNet-101 ResNet-101 ResNet-101 Scale (600,1000) (960,1280) (600,1000) (960,1280) (600,1000) (960,1280) (600,1000) (960,1280) Feature map conv5 conv5 conv5 conv5 res4b22 res4b22 res4b22+res5c res4b22+res5c Precision 92.98% 91.02% 92.56% 91.52% 93.38% 93.62% 93.65% 93.64% Recall 82.98% 82.98% 83.96% 86.14% 82.76% 88.09% 83.14% 88.82% F score 87.69% 86.81% 88.06% 88.75% 87.75% 90.32% 88.08% 91.17% Fig.4. Example detection results of our RTN on the ICDAR2013 benchmark. The first row of images is the result before connection and regression. The second row of images is the result after vertical proposal connection and regression. TABLE 2 REGRESSION IMPROVEMENT BY ADDITIONAL Convolutional layers RTN_no_regression RTN_regression REGRESSION Precision Recall 93.64% 88.82% 94.20% 89.02% TABLE 3 COMPARISON WITH STATE-OF-THE-ART PUBLICATIONS ON ICDAR2013 Method Yin [1] Faster R-CNN baseline[6] R-FCN[18] Multi-OrientedFCN[14] SegLink[17] DeepText[13] TextBoxes[15] CCTN[16] CTPN[12] Proposed RTN F score 91.17% 91.54% B. Regression improvement Proposals connected by fixed width vertical proposals are inaccurate on both beginning and end sides. Moreover, the evaluation criteria are extremely strict. Detection bounding box can be judged as false positive sample if its boundary exceed ground truth slightly. It means this inaccuracy can degrade performance on both recall and precession, even if the texts are detected correctly. Through bounding box regression, we are able to deal with this problem properly. As shown in Table 2, RTN with regression improved 0.4% on F-scores, both recall and precision benefit from this additional regression. C. Evaluation of RTN on ICDAR2013 After proving the effectiveness of hierarchy residual feature and additional regression, we compare RTN with other published methods on ICDAR2013. This single model approach did not utilize multi-scale training and multi-scale testing. Running time of each image is about 0.8s with GPU. Fig4 shows examples of detection results on ICDAR2013. First, we compared RTN with methods mentioned in recent publications. CNN based text detection methods were compared, including Textboxes, DeepText, Multi-oriented FCN, CCTN, SegLink and CTPN. The prevalent object detection frameworks like Faster R-CNN and R-FCN are also evaluated. Table 3 shows RTN achieved the best performance with great margin. Second, we submitted our results to ICDAR2015 Robust Reading Competition website and compared RTN with other competitors on CHALLENGE 2.1. This task is also evaluated on ICDAR2013 dataset. RTN with single model ranked third performance with slightly margin (F-scores=0.3%) compared to “Tencent Youtu” and “NLPR-CASIA”. Precision 0.88 Recall 0.66 F score 0.76 0.86 0.75 0.80 0.90 0.76 0.83 0.88 0.78 0.83 0.87 0.87 0.89 0.90 0.93 0.94 0.83 0.83 0.83 0.83 0.83 0.89 0.85 0.85 0.86 0.86 0.88 0.91 V. CONCLUSIONS In this paper, a deep residual text detection network is proposed based on the prevalent object detection framework. First, stronger semantic feature is obtained by using deep residual networks and combining multi-level feature from different convolutional networks. Then, a vertical proposal mechanism is introduced inRPN inspired by CTPN. At last, an additional regression system is used to improve localization accuracy. 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Fiducial, confidence and objective Bayesian posterior distributions for a multidimensional parameter arXiv:1612.01882v1 [math.ST] 6 Dec 2016 Piero Veronese and Eugenio Melilli Bocconi University, Milano, Italy Abstract We propose a way to construct fiducial distributions for a multidimensional parameter using a step-by-step conditional procedure related to the inferential importance of the components of the parameter. For discrete models, in which the nonuniqueness of the fiducial distribution is well known, we propose to use the geometric mean of the “extreme cases” and show its good behavior with respect to the more traditional arithmetic mean. Connections with the generalized fiducial inference approach developed by Hannig and with confidence distributions are also analyzed. The suggested procedure strongly simplifies when the statistical model belongs to a subclass of the natural exponential family, called conditionally reducible, which includes the multinomial and the negative-multinomial models. Furthermore, because fiducial inference and objective Bayesian analysis are both attempts to derive distributions for an unknown parameter without any prior information, it is natural to discuss their relationships. In particular, the reference posteriors, which also depend on the importance ordering of the parameters are the natural terms of comparison. We show that fiducial and reference posterior distributions coincide in the location-scale models, and we characterize the conditionally reducible natural exponential families for which this happens. The discussion of some classical examples closes the paper. Keywords: Confidence distribution, Jeffreys prior, Location-scale parameter model, Multinomial model, Natural exponential family, Reference prior. 1 Introduction Fiducial distributions, after having been introduced by Fisher (1930, 1935) and widely discussed (and criticized) in the subsequent years, have been de facto brushed aside for a long time and only recently they have obtained new vitality. The original idea of Fisher was to construct a distribution for a parameter which includes all the information given by the data, without resorting to the Bayes theorem. This is obtained by transferring the randomness from the observed quantity given by the statistical model to the parameter. Originally Fisher considered a continuous sufficient statistic S with distribution function Fθ , depending on a real parameter θ. Let qα (θ) denote the quantile of order α of Fθ and 1 let s be a realization of S. If qα (θ) is increasing in θ (i.e., Fθ is decreasing in θ), the statement s < qα (θ) is equivalent to θ > qα−1 (s) and thus Fisher assumes qα−1 (s) as the quantile of order 1 − α of a distribution which he names fiducial. The set of all quantiles qα−1 (s), α ∈ (0, 1), establishes the fiducial distribution function Hs (θ) so that Hs (θ) = 1 − Fθ (s) and hs (θ) = ∂ ∂ Hs (θ) = − Fθ (s). ∂θ ∂θ (1) Of course Hs , and its density hs , must be properly modified if Fθ is increasing in θ. Fisher (1973, cap.VI) also provides some examples of multivariate fiducial distributions obtained by a “step-by-step” procedure, but he never develops a general and rigorous theory. This fact, along with the problem to cover discrete models, the presence of some inconsistencies of the fiducial distribution (e.g. the marginalization paradox, see Dawid & Stone, 1982), and the difficulties in its interpretation, gave rise to a quite strong negative attitude towards Fisher proposal. In the renewed interest for the fiducial approach a relevant role is played by the generalized fiducial inference introduced and developed by Hannig (2009, 2013), see also Hannig et al. (2016) for a review. He provides a formal and mathematically rigorous definition which has a quite general applicability. The crucial element of his definition is a data-generating equation X = G(U, θ), which links the unknown parameter θ and the observed data X through a random element U having a known distribution. Roughly speaking, by shifting the randomness of U from X to θ (inverting G with respect to θ after having fixed X = x), the distribution given by the statistical model leads to a distribution for the parameter θ. Contrary to the original idea of Fisher, the generalized fiducial distribution is non-unique and Hannig widely discusses this point. Applications to different statistical models can be found for instance in Hannig et al. (2007), Hannig & Iyer (2008) and Wandler & Hannig (2012). Other recent contributions to the topic of fiducial distributions are given by Taraldsen & Lindqvist (2013), Martin & Liu (2013) and Veronese & Melilli (2015), henceforth V&M (2015). In this last paper the authors derive fiducial distributions for a parameter in a discrete or continuous real natural exponential family (NEF), and discuss some of their properties with particular emphasis on the frequentist coverage of the fiducial intervals. In the past fiducial distributions have often been associated with confidence distributions even if these latter have a different meaning. A modern definition of confidence distribution is given in Schweder & Hjort (2002) and Singh et al. (2005), see the book by Schweder & Hjort (2016) for a complete and updated review on confidence distributions and their connections with fiducial inference. It is important to emphasize that a confidence distribution must be regarded as a function of the data with reasonable properties from a purely frequentist point of view. A confidence distribution is conceptually similar 2 to a point estimator: as there exist several unbiased estimators, several confidence distributions can be provided for the same parameter and choosing among them can be done resorting to further optimality criteria. Thus the confidence distribution theory allows to compare, in a quite general setting, formal distributions for the parameter derived by different statistical procedures. In this paper we suggest a way to construct a unique distribution for a multidimensional parameter, indexing discrete and continuous models, following a step-by-step procedure similar to that used by Fisher (1973) in some examples. We call it fiducial distribution, but we look at it simply as a distribution on the parameter space in the spirit of the confidence distribution theory. The key-point of the construction is the procedure by conditioning: the distribution of the data is factorized as a product of one-dimensional laws and, for each of these, the fiducial density for a real parameter component, possibly conditional on other components, is obtained. The joint fiducial density for the parameter is then defined as the product of the (conditional) one-dimensional fiducial densities. It is well known that Fisher’s fiducial argument presents several drawbacks in higher dimensions, essentially because one cannot recover the fiducial distribution for a function of the parameters starting from the joint fiducial distribution, see Schweder & Hjort (2016, Ch. 6 and 9). Our approach, when it can be applied, presents the advantage to construct sequentially the fiducial distribution directly on the parameters of interest and different fiducial distributions can be obtained focusing on different parameters of interest. Also, it should be noticed that a general definition of confidence distribution for a multidimensional parameter does not exist and more attention is given to the construction of approximate confidence curves for specific nested families of regions, see Schweder & Hjort (2016, Ch. 9 and Sec. 15.4). Interestingly, our joint fiducial distribution coincides in many cases with the Bayesian posterior obtained using the reference prior. This fact motivates the second goal of the paper: to investigate the relationships between the objective Bayesian posteriors and the suggested fiducial distributions. Objective Bayesian analysis, see e.g. Berger (2006), essentially studies how to perform a good Bayesian inference, especially for moderate sample size, when one is unwilling or unable to assess a subjective prior. Under this approach, the prior distribution is derived directly from the model and thus it is labeled as objective. The reference prior, introduced by Bernardo (1979) and developed by Berger & Bernardo (1992), is the most successful default prior proposed in the literature. For a multidimensional parameter the reference prior depends on the grouping and ordering of its components and, in general, no longer coincides with the Jeffreys prior. This is the reference prior only for a real parameter and it is unsatisfactory otherwise, as well known. 3 Lindley (1958) was the first to discuss the connections between fiducial and posterior distributions for a real parameter, when a real continuous sufficient statistic exists. V&M (2015) extend this result to real discrete NEFs, characterizing all families admitting a fiducial prior, i.e. a prior leading to a posterior coinciding with the fiducial distribution. This prior is strictly related to the Jeffreys prior. We show here that when the parameter is multidimensional this relationship no longer holds and a new one is established with the reference prior. In particular we prove results for location-scale parameter models and conditionally reducible NEFs, a subclass of NEFs defined in Consonni & Veronese (2001). The paper is structured as follows. Section 2 reviews some basic facts on fiducial and confidence distributions for real NEFs and on generalized fiducial distributions. The proposal for constructing a step-by-step multivariate fiducial distribution is presented in Section 3, which also discusses: the relationships with confidence distributions (Section 3.1), the use of the geometric mean of fiducial densities for solving the non-uniqueness problem in discrete models (Section 3.2), the connections with the generalized fiducial inference and the consistency with the sufficiency principle (Section 3.3). Section 3.4 studies the fiducial distributions for conditionally reducible NEFs and provides their explicit expression for a particular subclass which includes the multinomial and the negativemultinomial model. Section 4 analyzes the relationships between the fiducial distributions and the reference posteriors, in particular for location-scale parameter models (Section 4.1) and NEFs (Section 4.2), characterizing those which admit the fiducial prior. Section 5 discusses further examples in which fiducial and reference posteriors coincide. Section 6 concludes the paper presenting some possible asymptotic extensions. Finally, Appendix A1 collects some useful technical results on conditionally reducible NEFs, while Appendix A2 includes the proofs of all the results stated in the paper. 2 Preliminary results The modern definition of confidence distribution for a real parameter φ of interest, see Schweder & Hjort (2002, 2016) and Singh et al. (2005), can be formulated as follows: Definition 1. Let {Fφ,λ , φ ∈ Φ ⊆ R, λ ∈ Λ} be a parametric model for data X ∈ X ; here φ is the parameter of interest and λ is a nuisance parameter. A function C : X × Φ → R is a confidence distribution for φ if C(x, ·) is a distribution function for each x ∈ X and C(X, φ0 ) has a uniform distribution in (0, 1) under Fφ0 ,λ0 , where (φ0 , λ0 ) is the true parameter value. The relevant requirement in the previous definition is the uniformity of the distribution, which ensures the correct coverage of the confidence intervals. As seen in Section 4 1, the confidence distribution theory must be placed in a purely frequentist context and allows to compare distributions on the parameter space, obtained using different approaches. Finally, the definition of confidence distribution can be generalized by requiring that the uniformity assumption holds only asymptotically. Strictly linked to the notion of confidence distribution is that of confidence curve, defined, for each observed X = x, as the function φ → cc(φ) = \|1 − 2C(x, φ)\|; see Schweder & Hjort (2016). This function gives the extremes of equal-tail confidence inter- vals for any level 1 − α, allowing a fast and clear comparison of confidence distributions with respect to their interval length. When the parameter of interest is multidimensional, how to extend the definitions of confidence distribution and confidence curve is much less clear and various proposals have been made, see Schweder & Hjort (2002, 2016) and Singh et al. (2005). As detailed in Section 1, Hannig (2009) has proposed the notion of generalized fiducial distribution, which is based on a data-generating equation X = G(θ, U). Because several functions G can generate the same statistical model, and not all the resulting fiducial distributions are reasonable in terms of properties or computational tractability, Hannig (2013, Sec. 5) gives some hints on the choice of a default function G. In particular, if X = (X1 , . . . , Xn ) is an independent identically distributed (i.i.d.) random sample from an (absolutely) continuous distribution function Fθ , with density fθ , θ ∈ Rd , he suggests to use Xi = Fθ−1 (Ui ), i = 1, . . . , n, where Ui are i.i.d. uniform random variables on (0, 1) and Fθ−1 is the inverse (or generalized inverse) of Fθ . If other regularity assumptions are satisfied, the generalized fiducial distribution for θ can be written as r(θ) = R fθ (x)J(x, θ) , Θ fθ (x)J(x, θ)dθ (2) where the expression of J(x, θ), given in Hannig (2013, formula (3.7)), is J(x, θ) = X det {(i1 ,...,id ):1≤i1 <···<id ≤n} d dθ (Fθ (xi1 ), . . . , Fθ (xid )) Qd j=1 fθ (xij ) . (3) In (3) the numerator of the ratio is the determinant of the matrix whose kj-entry is ∂Fθ (xij )/∂θk . This procedure leads to the Fisher definition of fiducial density (1) when n = d = 1. Hannig (2013, Example 4) explicitly recognizes the advise of Wilkinson (1977) that the choice of a fiducial distribution should depend on the parameter of interest and uses the well known example of d independent normal distributions N(µi , 1), in which the P parameter of interest is θ = ( di=1 µ2i )1/2 . He shows that the default data-generating equations Xi = µi + Ui , i = 1, . . . , d, lead to a fiducial distribution which has good frequentist properties for inference on the µ’s, but very bad ones when the interest is on 5 θ, as already recognized by Stein (1959). Thus Hannig suggests an ad hoc alternative equation, which leads to a better solution. Notice that our general procedure, suggested in the next section, constructs a fiducial distribution starting directly from the parameter of interest and do not required the choice a priori of a data generating function. Fiducial distributions and their properties, with particular emphasis on the frequentist coverage of the fiducial intervals, for a discrete or a continuous real regular NEF, are discussed in V&M (2015). More specifically, consider the sufficient statistic S associated with a sample of size n and denote by S its support. Let Fθ (s) be the distribution function of S and pθ (s) = exp {θs − nM (θ)} the corresponding density (with respect to a measure ν). Let a = inf S, b = sup S and define S ∗ = [a, b) if ν(a) > 0, otherwise S ∗ = (a, b). Then, for s ∈ S ∗ , Petrone & Veronese (2010) have proved that   θ ≤ inf Θ   0 Hs (θ) = 1 − Fθ (s) inf Θ < θ < sup Θ    1 θ ≥ sup Θ (4) is a fiducial distribution function for the natural parameter θ. It follows that the fiducial density of θ is ∂ ∂ hs (θ) = Hs (θ) = − Fθ (s) = ∂θ ∂θ Z (−∞,s] (nM ′ (θ) − t)pθ (t)dν(t). (5) It is important to underline, and simple to verify, that the distribution function Hs is also a confidence distribution (only asymptotically, in the discrete case), according to Definition 1. Notice that, for discrete NEFs, Fθ (s) = Prθ {S ≤ s} and Prθ {S < s} do not coincide and thus, besides Hs in (4), one could define a left fiducial distribution as Hsℓ (θ) = 1 − Prθ {S < s}. (6) For convenience, sometimes Hs will be called right fiducial distribution. A standard way to overcome this non-uniqueness is referring to the half-correction device (see Schweder & Hjort, 2016, pag. 62) which amounts to consider the mixture HsA (θ) = (Hs (θ) + Hsℓ (θ))/2 = Prθ {S > s} + Prθ {S = s}/2, whose density is the arithmetic mean of hs (θ) and hℓs (θ). Instead, we will suggest to average hs and hℓs using their geometric mean hG s (suitably normalized) and show that it presents better properties than hA s (Section 3.2) and a more direct connection with objective Bayesian inference (Section 4.2), even if, operationally, the difference is usually not particularly big. Table 1 provides the fiducial distributions obtained in V&M (2015) for some important discrete and continuous NEFs, which will be used in the forthcoming examples. It also establishes the abbreviations used in the paper for the standard distributions. 6 Table 1: Fiducial distributions for some real NEFs N(µ, σ 2 ) Sufficient Fiducial statistic P Hs (µ) : N(s/n, σ 2 /n) S= distributions i Xi (σ 2 known) N(µ, σ 2 ) S= P S= P i Xi Hs (λ) : Ga(nα, s) S= P i log(Xi /x0 ) Hs (λ) : Ga(n, s) S= P i Xic Hs (λ) : Ga(n, s) S= P i Xi Hs (p) : Be(s + 1, nm − s) i (Xi − µ)2 Hs (σ 2 ): In-Ga(n/2, s/2) (µ known) Ga(α, λ) (α known) Pa(λ, x0 ) (x0 known) We(λ, c) (c known) Bi(m, p) Hsℓ (p) : Be(s, nm − s + 1) (m known) HsG (p) : Be(s + 1/2, nm − s + 1/2) Po(µ) S= P i Xi Hs (µ) : Ga(s + 1, n) Hsℓ (µ) : Ga(s, n) HsG (µ) : Ga(s + 1/2, n) Ne-Bi(m, p) S= P i Xi Hs (p) : Be(nm, s + 1) Hsℓ (p) : Be(nm, s) (m known) HsG (p) : Be(nm, s + 1/2) The following notations are used: Ga(α, λ) for a gamma distribution with shape α and mean α/λ; In-Ga(α, λ) for an inverse-gamma distribution (if X ∼ Ga(α, λ) then 1/X ∼ In-Ga(α, λ)); Be(α, β) for a beta distribution with parameters α and β; Bi(m, p) for a binomial distribution with m trials and success probability p; Ne-Bi(m, p) for a negative-binomial with m successes and success probability p; Po(µ) for the Poisson distribuition with mean µ; Pa(λ, x0 ) for a Pareto distribution with density λxλ0 x−λ−1 , x > x0 > 0, λ > 0; We(λ, c) for a Weibull distribution with density cλxc−1 exp(−λxc), x, λ, c > 0. 7 3 Fiducial distributions for multidimensional parameters A natural way to construct a suitable fiducial distribution for a multidimensional parameter is to follow the step-by-step procedure used by Fisher (1973) in some examples. The key-point of our proposal stems on the factorization of the sampling distribution as a product of one-dimensional conditional laws. For each of these the fiducial density for a real component of the parameter, possibly conditional on other components, is defined. It is well known that different factorizations of sampling distributions can produce different joint fiducial distributions, see e.g. Dempster (1963). However, we do not consider this aspect a drawback of the procedure if it is linked to the inferential importance ordering of the parameter components implied by the factorization. For example, if a parameter θ ∈ R2 is transformed in such a way that φ is the parameter of interest and λ the nuisance, the obvious ordering is (φ, λ) and a suitable factorization must be defined accordingly, see Example 4 (ctd.) in this section for an illustration. The crucial role played by the ordering of the parameters accordingly to their inferential importance is widely acknowledged by objective Bayesian inference, in which reference priors are different for different orderings, see Section 4. In order to construct a fiducial distribution, we consider two basic transformations: one involving the sample data X = (X1 , . . . , Xn ), having a distribution parameterized by θ = (θ1 , . . . , θd ), d ≤ n, and one involving θ. Given X, consider a statistic T = (T1 , . . . , Tm ), d ≤ m ≤ n, with density pθ (t), which summarizes X without losing information on θ. T can be a sufficient statistic or a one-to-one transformation of X. Split T in (T[d] , T−[d] ), where T[d] = (T1 , . . . , Td ) and T−[d] = (Td+1 , . . . , Tm ), and suppose that T−[d] is ancillary for θ. As a consequence pθ (t) = pθ (t[d] \|t−[d] )p(t−[d] ) and all the information on θ provided by X are included in the conditional distribution of T[d] given T−[d] . Assume now that there exists a one-to-one smooth reparameterization from θ to φ, with φ1 , . . . , φd ordered with respect to their importance, such that pφ (t[d] \|t−[d] ) = d Y k=1 pφd−k+1 (tk \|t[k−1] , t−[d] ; φ[d−k] ). (7) The density pφd−k+1 (tk \|t[k−1] , t−[d] ; φ[d−k] ), with the corresponding distribution function Fφd−k+1 (tk \|t[k−1] , t−[d] ; φ[d−k] ), must be interpreted as the conditional distribution of Tk given (T[k−1] = t[k−1], T−[d] = t−[d] ), parameterized by φd−k+1 , assuming φ[d−k] known. In the following, we will always assume that all the one-dimensional conditional distribution functions Fφj ’s involved in the analysis are monotone and differentiable in φj and have limits 0 and 1 when φj tends to the boundaries of its domain. Notice that this is always true if Fφj belongs to a NEF, see (4). Under these assumptions, the joint fiducial 8 density of φ is obtained as ht (φ) = d Y k=1 ht[k] ,t−[d] (φd−k+1 \|φ[d−k] ), (8) ∂ Fφ (tk \|t[k−1] , t−[d] ; φ[d−k] ) . ∂φd−k+1 d−k+1 (9) and ht[k] ,t−[d] (φd−k+1 \|φ[d−k] ) = Several applications of this procedure to well known models will be provided in Section 5. Here we illustrate some interesting features of the fiducial distribution (8). i) The existence of an ancillary statistic is not necessary if there exists a sufficient statistic with the same dimension of the parameter (m = d). An important case is m = d = 1 so that formula (8) and (9) reduce to ht (φ) = \|∂Fφ (t)/∂φ\|, the original formula suggested by Fisher (1930). ii) If one is only interested in φ1 , it follows from (7) that it is enough to consider ht (φ1 ) = ∂ Fφ (td \|t[d−1] , t−[d] ) , ∂φ1 1 which, depending on all observations, does not lose any sample information. A typical choice for Td is given by the maximum likelihood estimator φb1 of φ1 and thus, when φb1 is not sufficient, we have to consider the distribution of φb1 given the ancillary statistic t−[d] . Similarly, if one is interested in φ1 , φ2 , it is enough to consider ht (φ1 ) · ht[d−1] ,t−[d] (φ2 \|φ1 ), and so on. iii) When an ancillary statistic T−[d] is needed, the fiducial distribution (8) is invariant with respect to any one-to-one transformation of T−[d] . All the sampling distributions are conditional on it and thus any transformation establishes the same constraints; see Section 4.1 for an example. iv) The construction by successive conditioning makes the fiducial distribution invariant under the so called one-to-one lower triangular transformation of T[d] , for fixed T−[d] . More precisely, we consider a transformation T∗ = (T∗[d] , T−[d] ) such that Tk∗ = gk (T[k] , T−[d] ), for k = 1, . . . , d. To see this, assuming for instance t∗k = gk (t[k] , t−[d] ) increasing in tk , it is sufficient to show that Prφd−k+1 (Tk∗ ≤ t∗k \| T∗[k−1] = t∗[k−1] , T−[d] = t−[d] ; φ[d−k] ) = Prφd−k+1 (gk (T[k] , T−[d] ) ≤ gk (t[k] , t−[d] ) \| T∗[k−1] = t∗[k−1] , T−[d] = t−[d] ; φ[d−k] ) = Prφd−k+1 (Tk ≤ tk \| T[k−1] = t[k−1] , T−[d] = t−[d] ; φ[d−k] ). 9 It follows immediately that T and T∗ lead to the same fiducial distribution. v) If (T[k−1] , T−[d] ) is sufficient for φ[d−k] , for each k, then the conditional distribution of Tk given (T[k−1] = t[k−1] , T−[d] = t−[d] ) does not depend on φ[d−k] and the fiducial distribution (8) becomes the product of the “marginal” fiducial distributions of the φk ’s. As a consequence, (9) can be used alone to make inference on φd−k+1 and the fiducial distribution does not depend on the inferential ordering of the parameters. An important case in which this happens will be discussed in Section 3.4. We close this section establishing the invariance property of the fiducial distribution ht (φ) under a lower triangular transformation, i.e. a transformation from φ to λ = (λ1 , . . . , λd ), say, which maintains the same decreasing ordering of importance in the components of the two vectors. Proposition 1. If φ = φ(λ) is a one-to-one lower triangular continuously differentiable function from Λ to Φ, then the fiducial distribution hφ t (φ), obtained applying (8) to the model pφ (t), and the fiducial distribution hλ t (λ), obtained applying (8) to the model pλ (t) = pφ(λ) (t), are such that, for each measurable A ⊂ Φ, Z Z φ hλ ht (φ)dφ = t (λ)dλ. A 3.1 (10) λ−1 (A) Relationships with confidence distributions Given a real NEF, Hs (θ) in (4) is an exact or approximate confidence distribution if the observations are continuous or discrete, respectively. It is possible to verify that the same is true for the marginal fiducial distribution of the main parameter of interest φ1 in the more general definition (8). Indeed, the distribution function of φ1 is Ht (φ1 ) = 1 − Fφ1 (td \|t[d−1] , t−[d] ), so that the first requirement in Definition 1 is clearly satisfied, thanks to the assumption on the distribution function given after formula (7). For what concerns the uniformity condition, assuming that Fφ1 is decreasing in φ1 (if it is increasing, replace 1 − Fφ1 with Fφ1 ), we have, for u ∈ (0, 1) and arbitrary φ, Prφ Ht[d] ,t−[d] (φ1 ) ≤ u = 1 − Prφ Fφ1 (td \|T[d−1] , T−[d] ) < 1 − u = =1− Z Prφ Fφ1 (td \|t[d−1] , t−[d] ) < 1 − u dFφ (t[d−1] , t−[d] ) = u because, by construction, the integrand is equal to 1 − u for all fixed (t[d−1] , t−[d] ). 3.2 The discrete case: the geometric mean of the left and right fiducial densities As mentioned in Section 2, for a discrete statistic S with distribution depending on a real parameter θ, we suggest to use the geometric mean of the right and left fiducial 10 −1 ℓ 1/2 , where c is the normalizing constant, instead of densities, hG s (θ) = c (hs (θ)hs (θ)) their arithmetic mean hA s (θ). A first justification of the use of the geometric mean of densities is suggested by Berger et al. (2015) who mention its property to be the density “closest” to hs and hℓs with respect to the the Kullback-Leibler divergence, as specified in the following proposition. We give a simple proof of this fact, without resorting to the calculus of variations. Recall that, given two densities p and q, having the same support and the same dominating measure ν, the Kullback-Leibler divergence of p from q is defined as R KL(q\|p) = q(x) log(q(x)/p(x))dν(x). Proposition 2. Consider two densities p1 and p2 with the same support. The density q which minimizes KL(q\|p1 ) + KL(q\|p2 ) is given by q = pG ∝ (p1 p2 )1/2 , which is the (normalized) geometric mean of p1 and p2 . Furthermore, Krishnamoorthy & Lee (2010) observe that a distribution for θ, whose aim is to give a synthesis of two fiducial distributions, should “stochastically” lie between them. In our setting, the extreme distributions are Hs and Hsℓ . This property is surely satisfied by the arithmetic mean, because Hs (θ) < HsA (θ) < Hsℓ (θ) uniformly with respect to θ, for each s belonging to the set S0 for which both Hs and Hsℓ can be defined. The same inequalities are true for HsG under mild assumptions. As usual, here we assume that Hs (θ) is defined as 1 − Fθ (s). Proposition 3. Let pθ , θ ∈ Θ ⊆ R, be the probability mass function of a real observation S, having a continuous derivative with respect to θ. For each s ∈ S0 , assume that the function γs (θ) = ∂pθ (s) ∂θ ∂pθ (s) ∂Fθ (s) / − = /hs (θ) ∂θ ∂θ is decreasing on Θ. Then Hs (θ) < HsG (θ) < Hsℓ (θ) uniformly on Θ. The assumptions required in the previous proposition are satisfied by many important models. For example we have the following Corollary 1. If pθ is the probability mass function of a real NEF, then Hs (θ) < HsG (θ) < Hsℓ (θ) uniformly on Θ. We now discuss the relationship between HsG and HsA . Proposition 4. Let pθ , θ ∈ Θ ⊆ R, be the probability mass function of a real observation S, satisfying the following assumptions in addition to those stated in Proposition 3: lim γs (θ) = +∞; θ→inf Θ lim θ→sup Θ 11 γs (θ) = −1. Then, for each s ∈ S0 , there exists θ ∗ ∈ Θ (depending on s) such that HsG (θ) < HsA (θ) for θ < θ ∗ and HsG (θ) > HsA (θ) for θ ≥ θ ∗ . The result in Proposition 4 is important in connection with confidence intervals, because it shows that HsG gives, for a fixed level, a confidence interval smaller than that obtained from HsA (θ); see Figure 1 (graph 2) for an example. Notice that the assumptions on γs (θ) in Proposition 4 are fulfilled by a real NEF with natural parameter space Θ = R, as it occurs in the binomial and Poisson models. However, these assumptions are not necessary to ensure the stated behavior of HsG and HsA , that we conjecture to be quite general, as the following example shows. Example 1. Consider an i.i.d. sample of size n from a logarithmic distribution with parameter θ ∈ (0, 1) with probability mass function pθ (x) = The sufficient statistic T = Pn i=1 Xi pθ (t) = θx I (x). −x log(1 − θ) {1,2,...} is distributed as n!\|s(t, n)\|θ t I (t), t!(− log(1 − θ))n {n,n+1,...} where s(t, n) is the Stirling number of the first kind with arguments t and n, see Johnson et al. (2005). The distribution of T belongs to a real NEF with Fθ (t) decreasing in θ, so that the fiducial distribution function Ht , for t = n, n + 1, . . . and θ ∈ (0, 1), is Ht (θ) = 1 − Fθ (t) = 1 − t X j=n n!\|s(j, n)\|θ j . j!(− log(1 − θ))n For this model ∂pθ (s) ∂Fθ (s) \|s(t, n)\|θ t−1 (nθ + t(1 − θ) log(1 − θ)) γt (θ) = − / = − Pt . ∂θ ∂θ t! j=n \|s(j, n)\|θ j−1 (nθ + j(1 − θ) log(1 − θ))/j! It can be seen that, for each t ≥ n, γt is decreasing in θ and lim γt (θ) = +∞, θ→0+ −1  t X \|s(j, n)\| t!  lim γt (θ) = −  ∈ (−1, 0). − \|s(t, n)\| j! θ→1 j=n Nevertheless, the fiducial distributions HtG and HtA behave as stated in Proposition 4, see Figure 1 (graph 1). Finally, we justify our preference for HsG versus HsA showing that its confidence risk under quadratic penalty, as defined in Schweder & Hjort (2016, Sec. 5.3), is uniformly better for all the important discrete models reported in Table 1. The confidence risk 12 Figure 1: Graph 1: Fiducial distributions for a sample from the logarithmic distribution G (n = 10, t = 12): hℓt (θ) (red), hA t (θ) (green), ht (θ) (yellow), ht (θ) (blue). Graph 2: G Confidence curves for hA t (θ) (green) and ht (θ) (yellow)). R(µ, Hs ) for the mean parameter µ and a confidence (or fiducial) distribution Hs under quadratic penalty is R(µ, Hs ) = Z (µ′ − µ)2 dHs (µ′ ) = Eµ (VarHs (µ)) + Eµ (µ̂ − µ)2 , where VarHs (µ) denotes the variance of µ under Hs , Eµ the expected value with respect to the distribution of S and µ̂ = E Hs (µ) is the mean of µ under Hs . Now, recalling that for the binomial and the negative-binomial distribution in Table 1, assuming m = 1 for simplicity, we have µ = p and µ = (1 − p)/p, it is easy to verify that for both these models and the Poisson model µ̂ is the same under HsG and HsA . As a consequence, A G R(µ, HsA )−R(µ, HsG ) = Eµ (VarHs (µ))−Eµ (VarHs (µ)) which becomes (4(n+1)(n+2))−1 , (4(n − 1)(n − 2)−1 and (4n2 )−1 , for the three models above, respectively. All these values are strictly positive for each n uniformly in µ. Let us now consider the fiducial distribution for a multivariate parameter defined in (8). For each discrete component of the product, starting from Prφd−k+1 {Tk ≤ tk \|T[k−1] = t[k−1] , T−[d] = t−[d] ; φ[d−k] } = Fφd−k+1 (tk \|t[k−1] , t−[d] ; φ[d−k] ) and Prφd−k+1 {Tk < tk \|T[k−1] = t[k−1] , T−[d] = t−[d] ; φ[d−k] }, it is possible to define a right and a left fiducial distribution, respectively, and hence their geometric and arithmetic means. Notice that each compo- nent of (8) involves a one-dimensional parameter and a real observation (the remaining quantities being fixed), so that the Propositions 3 and 4 can be applied. Multivariate fiducial distributions for discrete observations can thus be obtained combining in the various possible way these univariate distributions. In particular, we will consider Ht (φ), obtained as the product of all the right univariate conditional fiducial distributions, Htℓ (φ), obtained as the product of all the left univariate conditional fiducial distributions, HtA (φ), defined as the product of the d mixtures HtA[k] ,t−[d] = (Ht[k] ,t−[d] + Htℓ[k] ,t−[d] )/2 and finally 13 HtG (φ), corresponding to the density hG t (φ) obtained as the product of the d geometric 1/2 . Notice that hG (φ) coincides with the geometric ℓ means hG t t[k] ,t−[d] ∝ (ht[k] ,t−[d] ·ht[k] ,t−[d] ) mean of all the 2d fiducial densities derived as described above. 3.3 Fiducial inference and the sufficiency principle The step-by-step procedure introduced at the beginning of Section 3 gives a generalized fiducial distribution, according to Hannig (2009), if one considers as data-generating equation T = G(φ, U) with ( Gk (φ, U[k] , U−[d] ) k = 1, . . . , d , Tk = Uk k = d + 1, . . . , m where U is a random vector with a completely known distribution. The functions Gk can be explicitly obtained iteratively as follows: T1 = G1 φ, U1 , U−[d] = Fφ−1 U \|, U ; φ 1 −[d] [d−1] d T2 = G2 φ, U1 , U2 , U−[d] = Fφ−1 U \|G (φ, U , U ), U ; φ 2 1 1 −[d] −[d] [d−2] d−1 and so on. It is interesting to observe that the generalized fiducial distribution r(θ) given in (2) does not necessarily satisfy the sufficiency principle. This can be verified immediately looking at the Example 2 in Hannig (2013), in which a uniform distribution on (θ, θ 2 ) is considered and r(θ) does not depend on the Xi ’s only through the sufficient statistic S = (X(1) , X(n) ), where X(i) denotes the i-th order statistic. Despite its simple form, this model is highly irregular, but the inconsistency with the sufficiency principle of the generalized fiducial distribution r(θ) can also occur for more standard models. In particular, if a real continuous sufficient statistic S for a real parameter exists, one could derive two different fiducial distributions starting from S or from the whole sample. A simple example of this issue can be easily constructed considering a beta model with parameters 2 and θ. Another interesting example is the following. Example 2. Let X = (X1 , . . . , Xn ) be an i.i.d. sample from a truncated exponential density pθ (x) = θe−θx /(1 − e−θ ), 0 < x < 1, θ ∈ R − {0}. This density is not defined for θ = 0, but it can be completed by continuity setting p0 (x) = 1. The distribution function of Xi is Fθ (xi ) = (1 − e−θxi )/(1 − e−θ ), 0 < xi < 1, so that from (3) we have n J(x, θ) = where s = Pn i=1 xi . X s e−θ (1 − e−θxi ), + −θ θ θ(1 − e ) i=1 Thus, using (2), we obtain θ n−1 e−θs r(θ) ∝ (1 − e−θ )n+1 ! n X (1 − e−θxi ) , s(1 − e−θ ) + e−θ i=1 14 θ ∈ R, (11) Figure 2: Graph 1: Fiducial densities: r(θ; x1 = 0.05, x2 = 0.95) (red); r(θ; x1 = 0.5, x2 = 0.5) (green); h(θ; s = 1) (blue). Graph 2: Fiducial densities r(θ; x1 = 0.02, x2 = 0.48) (red); r(θ; x1 = 0.2, x2 = 0.3) (green); h(θ; s = 0.5) (blue). Figure 3: Confidence curves for r(θ; x1 = 0.5, x2 = 0.5) (red) and h(θ; s = 1) (blue). which depends on the values of the specific xi ’s. Consider now the sufficient statistic P S = ni=1 Xi and, for simplicity, assume n = 2. The density of S is  θ2  e−ts s 0<s≤1 (1−e−θ )2 pθ (s) = . 2 θ −ts  (2 − s) 1 < s < 2 −θ 2 e (1−e ) and the generalized fiducial density (11) reduces to hs (θ) = ∂Fθ (s)/∂θ. In Figure 2 we report the fiducial densities r and hs for different values of (x1 , x2 ) and s = x1 + x2 . For s = 1 all densities are symmetric with the mode in 0, while the dispersion is increasing in \|x1 − x2 \|, so that the more concentrated fiducial density is obtained for x1 = x2 = 0.5. However, for s 6= 1, the densities have different modes and are shifted to the left when x1 increases. In all cases the fiducial density hs is in the middle of the various cases. Notice that hs has all the good properties discussed in V&M (2015) and, in particular, it is a confidence distribution because the model belongs to a NEF. The confidence intervals corresponding to hs (θ) are slightly smaller than those corresponding to r(θ), as can be seen from the confidence curves reported in Figure 3. For instance, when x1 = x2 = 0.5, the 95% confidence intervals are (-4.191,4.191) and (-4.399,4.399), for hs (θ) and r(θ), respectively. 15 The computation of the fiducial distribution Ht defined in (8) is greatly simplified starting with the sufficient statistic instead of the whole sample. However, when both the alternatives are feasible, they seem to lead to the same result. In particular, the following proposition states that the sufficiency principle is always satisfied by Ht when there exists a complete sufficient statistic for the parameter. Proposition 5. Consider the fiducial distribution Ht (φ), defined in (8) and (9), with T a one-to-one transformation of the data X = (X1 , . . . , Xn ). If S = S(T) is a complete and sufficient statistic of dimension d for φ, such that S = g(T[d] , T−[d] ) is a one-toone lower triangular transformation of T[d] for fixed T−[d] , then the fiducial distribution Hs (φ) for φ, obtained using S instead of T in (8) and (9), coincides with Ht (φ). Notice that the completeness of S is not necessary to satisfy the sufficiency principle, as the following example shows. Example 3. Given an i.i.d. sample X of size n from a uniform distribution on (θ, θ + 1), it is immediate to verify that the sufficient statistic S = (X(1) , X(n) ) is not complete. Because θ is a location parameter, Z = X(n) − X(1) is an ancillary statistic and the fiducial distribution for θ can be obtained starting from the distribution function of X(n) given Z, which is Fθ (x(n) \|z) = (x(n) − z − θ)/(1 − z), x(n) − 1 < θ < x(n) − z = x(1) . Thus hs (θ) = − ∂ ∂ x(n) − z − θ 1 Fθ (x(n) \|z) = − = , ∂θ ∂θ 1−z 1−z x(n) − 1 < θ < x(n) − z = x(1) .(12) If we start directly with X, we can consider the distribution function of Xn given Z = (Z1 , . . . , Zn−1 ), where Zi = Xn − Xi . Omitting tedious calculations, we have Fθ (xn \| z) = xn − θ − max(zi , 0) , 1 + min(zi , 0) − max(zi , 0) θ + max(zi , 0) < xn < θ + 1 + min(zi , 0), and thus, for xn − 1 − min(zi , 0) < θ < xn − max(zi , 0), hx (θ) = − ∂ 1 Fθ (xn \|z) = . ∂θ 1 + min(zi , 0) − max(zi , 0) Observing that min(zi , 0) = zi unless xn = x(n) and recalling that zi = xn − xi , i = 1, . . . , n−1, it follows that xn −1−min(zi , 0) = x(n) −1 and similarly xn −max(zi , 0) = x(1) , so that hx (θ) coincides with hs (θ) given in (12). 3.4 Conditionally reducible natural exponential families Consider a multivariate natural exponential family whose density, with respect to a fixed σ-finite positive measure ν, is given by ( d ) X pθ (x) = exp θk xk − M (θ) , θ = (θ1 . . . , θd ) ∈ Θ, x = (x1 , . . . , xd ) ∈ Rd . k=1 16 (13) A NEF is d-conditionally reducible (in the sequel cr-NEF) if its joint density (13) can be factorized as a product of d conditional densities each belonging to a real exponential family. More precisely if pφ (x\|φ(θ)) = d Y k=1 pφk (xk \|x[k−1] ; φk (θ)) = d Y k=1 exp φk (θ)xk − Mk (φk (θ); x[k−1] ) , (14) where φ = (φ1 , . . . , φd ) is a one-to-one function from Θ onto φ(Θ) = Φ. Furthermore, it can be shown that Φ = Φ1 × · · · × Φd , with φk ∈ Φk , k = 1, . . . , d, so that the φk ’s are variation independent. Notice that φk is the natural parameter of the k-th conditional distribution. For details on these families, with emphasis on enriched conjugate priors and on reference Bayesian analysis, see Consonni & Veronese (2001) and Consonni et al. (2004), respectively. Both these papers deal in particular with the families having simple quadratic variance function, named NEF-SQVFs, which include, as most interesting cases, the multinomial and negative-multinomial models, see Casalis (1996) and Appendix A1. Example 4 (Multinomial model). Consider a random vector X distributed according to a multinomial distribution and denote by pk the probability of the k-th outcome Xk , k = P P 1, . . . , d, with dk=1 xk ≤ N , and dk=1 pk ≤ 1. It is well know that the conditional disPk−1 Pk−1 pj )), xj , pk /(1− j=1 tribution of Xk given X[k−1] = x[k−1] , k = 2, . . . , d, is Bi(N − j=1 whereas the marginal distribution of X1 is Bi(N, p1 ). Since a binomial distribution is a real NEF, one can factorize the multinomial distribution as in (14) with   k−1 X pk φk = log xj  log(1 + eφk ), φk ∈ R. (15) , Mk (φk ; x[k−1] ) = N − P 1 − kj=1 pj j=1 For models belonging to a cr-NEF, the construction of the fiducial distribution proposed in Section 3 drastically simplifies. The existence of a sufficient statistic of the same dimension of the parameter makes the ancillary statistic not necessary, while the φ-parameterization, indexing each conditional distribution with a real parameter, implies the independence of the φk ’s under the fiducial distribution. Proposition 6. Let S be the sufficient statistic distributed according to a regular crNEF on Rd , parameterized by φ ∈ Φ = Φ1 × · · · × Φd , with Φk coinciding with the natural parameter space of the k-th conditional distribution. Then, for S = s, with sk , k = 1, . . . , d, satisfying conditions similar to those given before (4), Hs (φ) = d Y k=1 Hs[k] (φk ) = d Y (1 − Fφk (sk \|s[k−1] ) k=1 17 (16) is a fiducial distribution function on φ with density hs (φ) = d Y hs[k] (φk ), where hs[k] (φk ) = k=1 ∂ ∂ Hs[k] (φk ) = − Fφ (sk \|s[k−1] ). ∂φ ∂φk k (17) The φk ’s are independent under Hs (φ) and thus their importance ordering is irrelevant. This fact also justifies the simplification in the index notation adopted in (16). Notice, however, that the definition and the interpretation of the φk ’s depend on the particular ordering considered for the Xk ’s, as seen in Example 4. As recalled in Section 2, a general definition of multi-dimensional confidence distribution does not exist. However, in our context, since Hs (φ) is constructed as a product of marginal confidence distributions, it can be considered as a multivariate (possibly asymptotic) confidence distribution for φ. Some of the examples of Section 5 can be reconnected with this framework, but here we consider in specific the NEF-SQVFs, whose variance function is given in (35). For this class, with the exclusion of the negative-multinomial/hyperbolic secant distribution, it is possible to give a simple explicit expression of the fiducial density of φ, recalling the definition of Bk (φk ) given in (31) and setting zk = zkk in (35). The specifications of zkk and q, appearing in (35), and of Bk (φk ) can be found in Appendix A1. Proposition 7. Consider a sample of size n from a Poisson/normal, a multinomial, or a negative-multinomial family on Rd . If S denotes the sufficient statistic, then the (right) fiducial distribution for φ has density      d d k−1   Y Y X hs (φ) = hs[k] (φk ) ∝ sj − 1 Bk (φk ) , (18) exp φk (sk + zk ) − n + q    k=1 j=1 k=1 while for the negative-multinomial/gamma/normal family, with an m dimensional nega- tive multinomial component, the (right) fiducial distribution is given by hs (φ) = d Y k=1 h[sk ] (φk ) ∝ m Y k=1 exp{φk (sk + 1)}(1 − exp(φk ))n/q+ × exp {φm+1 sm+1 } (−φm+1 )n/q+ × d Y k=m+2 Pk−1 j=1 sj −1 Pm j=1 sj −1 exp φk sk − n sm+1 φ2k /2 . (19) Notice that the discrete components of a basic NEF-SQVF are integer-valued with zk = 1, so that the left fiducial distribution is obtained by the previous formulas replacing the term (sk + 1) by sk in (18) and (19). Thus it follows that the geometric mean hG s has the same structure in (18) and (19) with (sk + 1/2) instead of (sk + 1). 18 Example 4 (ctd.). For the multinomial family, because q = −1/N , zk = 1 and Bk (φk ) = N log(1 + eφk ), k = 1, . . . , d, it easily follows from formula (18) that ( ) 1 d d Y Y 1 eφk (sk + 2 ) G G hs[k] (φk ) = hs (φ) = ,(20) Pk−1 Pk 1 1 φk )nN − j=1 sj +1 B(s + , nN − s + ) j k (1 + e j=1 2 2 k=1 k=1 where B(·, ·) denotes the beta function. The fiducial distribution for φ not always is of particular interest in itself, but it can be used as a starting point for the construction of the fiducial distribution for alternative and more relevant parameters. We consider here the mean-parameter µ, which is a lower triangular transformation of φ, see (32), so that its fiducial distribution can be directly obtained from that of φ thanks to Proposition 1. Corollary 2. The (right) fiducial distribution for the mean parameter µ, relative to the ordering µ1 , . . . , µd , for the following NEF-SQVFs on Rd , has density: • Poisson/normal family (with m Poisson components) hs (µ) ∝ m Y s µkk−1 exp(−nµk ) k=1 d Y k=m+1 n n o exp − 2 (µ2 − 2µk sk /n) , 2σ (21) which corresponds to the product of m densities Ga(sk , n), k = 1, . . . , m and (d−m) densities N(sk /n, σ 2 /n), k = m + 1, . . . , d. • Multinomial family hs (µ) ∝ d Y k=1  µskk N − k X j=1 γk µj  where γk = −1 for k = 1, . . . , d − 1 and γd = N n − 1 − , (22) Pd j=1 sj . • Negative-multinomial family, with R occurrences in the (d + 1)-th cell γk  k d X Y µj  , µskk R + hs (µ) ∝ (23) j=1 k=1 where γk = −1 for k = 1, . . . , d − 1 and γd = −Rn − 1 − Pd j=1 sj . • Negative-multinomial/gamma/normal family (with an m-dimensional negative - 19 multinomial component with R occurrences in the (m + 1)-th cell)  γk m k Y X hs (µ) ∝ µj  µskk R + j=1 k=1  × R + × d Y m X j=1 Rn+Pm j=1 sj µj  −(d−m−1) µm+1 k=m+2 −(Rn+ µm+1 (24)   Pm   R + µ s m+1 j=1 j j=1 sj )−1 exp −   µm+1 Pm sk nsm+1 2 , µk − 2 µk µm+1 exp − 2 n 2µm+1 P where γk = −1 for k = 1, . . . , m − 1 and γm = −Rn − 1 − m j=1 sj . Notice that the Pm P density of µm+1 given µ[m] is an In-Ga(Rn + j=1 sj , sm+1 (R + m j=1 µj )), while the density of µk given µ[k−1] , k = m + 2, . . . , d, is a N(µm+1 sk /n, µ2m+1 /(nsm+1 )) depending only on µm+1 . Example 4 (ctd.). Inference for the multinomial distribution is usually performed for the cell-probabilities parameter p = (p1 . . . , pd ). Since pk = µk /N , the fiducial distribution hG for p is easily derived from (22), noting that the left fiducial density can be obtained replacing (sk + 1) by sk in (18), (and not in (22), which is derived aggregating the hyperparameters). It follows that the geometric mean hG s (p) is given by  γk d k d Y X X sk −1/2  G  hs (p) ∝ pj pk 1− , pk = 1, 0 < pk < 1, k=1 j=1 with γk = −1/2 for k = 1, . . . , d − 1 and γd = N n − 1/2 − Dirichlet distribution. Clearly, hG s (p) (25) k=1 Pd j=1 sj . This is a generalized in (25) refers to the specific order of importance p1 , p2 , . . . , pd . If we change this order, the fiducial distribution will change accordingly. Similarly, for the negative-multinomial model, with R occurrences in the (d + 1)th cell, hG s (φ) can be easily computed from (18) observing that zk = 1, q = 1/R and Bk (φk ) = −R log(1 − exp(φk )). 4 Connections with objective Bayesian inference As mentioned in Section 1, if we look at fiducial inference as a way to obtain a distribution on the parameter space of the model without any prior information, it appears natural to compare it with objective Bayesian inference. Recall that when a fiducial distribution coincides with a posterior, the corresponding prior is called fiducial prior. The step-by-step construction of the fiducial distribution ht (φ) defined in (8) is based on the inferential importance ordering of the parameter components φ1 , . . . , φd . 20 This aspect is also crucial in the procedure adopted to construct reference priors, see Bernardo & Smith (1994, Sec. 5.4.5). The reference prior π R for a parameter φ is generated by successive conditioning, established by the importance ordering of its compoQ nents, as π R (φ) = dk=1 π R (φd−k+1 \|φ[d−k] ). It is widely recognized that the dependence of the reference prior on the choice of the parameter of interest is necessary to obtain good frequentist properties such as coverage and consistency. For a one-dimensional parameter φ the reference prior coincides with the Jeffreys prior π J (φ) ∝ I(φ)1/2 , where I(φ) denotes the Fisher information. While the Jeffreys prior is invariant under a reparameterization of the model, the reference prior (and thus the reference posterior) is generally not invariant unless the transformation from φ to λ = (λ1 , . . . , λd ) is lower triangular, see Datta & Ghosh (1996). Thus the reference posterior has the same invariance property of the fiducial distribution proved in Proposition 1. Recently Berger et al. (2015) recognize the existence of situations in which one is interested simultaneously in all the parameter components of the model, or in none of them but a prior (and thus a posterior) distribution is necessary to perform other inferences such as predictions. In these cases an overall prior is needed. Its determination is an open problem but, as they highlight, when there exists a “common reference prior for all parameters”, this is the natural choice for the overall prior. A similar problem occurs in our context and we will comment on this aspect in the following sections. Notice that here the fiducial distribution (2) suggested by Hannig can be a good choice. 4.1 Location-scale parameter models For location-scale parameter models the fiducial prior exists and coincides with the reference prior. Assume first that only one parameter, θ, is unknown. In this case the model admits an ancillary statistic Z and, in particular, we take Zi = Xi − X1 or Zi = Xi /X1 , i = 2, . . . , n, if θ is a location or a scale parameter, respectively. Proposition 8. Let X = (X1 , . . . , Xn ) be an i.i.d. sample from a density pθ , θ ∈ Θ ⊆ R. If θ is a location or a scale parameter, then the fiducial distribution coincides with the Bayesian posterior obtained with the Jeffreys prior π J (θ) ∝ 1 or π J (θ) ∝ 1/θ, respectively. Example 5. Let X be an i.i.d. sample from the uniform distribution on (0, θ), θ > 0, so that θ is a scale parameter. First notice that S = X(n) is a sufficient statistic for θ and thus we can obtain directly the fiducial distribution ∂ nsn ∂ ∂ s n hs (θ) = = n+1 , Hs (θ) = − Fθ (s) = − ∂θ ∂θ ∂θ θ θ θ > s. (26) However the same result can be obtained without resorting to the sufficient statistic. Set w = max(z2 , . . . , zn ) and consider the distribution function of X1 given the ancillary 21 statistic Z = (X2 /X1 , . . . , Xn /X1 ) ( n Fθ (x1 \|z) = x1 θ x1 w n θ 0 < x1 < θ, 0 < x1 < θ, 0<w≤1 w>1 . (27) Now, because w ≤ 1 means x1 = max(x1 , . . . , xn ), while for w > 1 we have x1 w = max(x2 , . . . , xn ), expression (27), as a function of θ, is equivalent to Fθ (s) appearing in (26) and thus provides the same fiducial distribution. It is immediate to verify that it coincides with the Jeffreys posterior. A case in which the sufficient statistic is not one-dimensional and thus it is necessary to use an ancillary statistic can be found in the previous Example 3. Trivially hs (θ) given in (12) coincides with the Bayesian posterior obtained by π J (θ) ∝ 1. Consider now a model with a location parameter θ and a scale parameter σ, both unknown. Given an i.i.d. sample of size n, an ancillary statistic is, for example, Z = (Z3 , . . . , Zn ), with Zj = (Xj − X1 )/Z2 , j = 3, . . . , n, where Z2 = X2 − X1 is marginally ancillary for θ. Then, the one-to-one transformation from X to (X1 , Z2 , Z) allows to write the sampling distribution as pσ (z2 \|z)pθ (x1 \|z2 , z; σ)p(z). Note that in specific contexts other transformations could be more appropriate. For example, in a normal model one P P could use (X̄ = ni=1 Xi /n, S 2 = ni=1 (Xi − X̄)2 , Z) with Zj = (Xj − X̄)/S, j = 3, . . . , n, so that the factorization becomes pσ (s2 \|z)pθ (x̄\|s2 , z; σ)p(z). Proposition 9. Let X = (X1 , . . . , Xn ) be an i.i.d. sample from a density pθ,σ , where θ and σ are a location and a scale parameter, respectively. Then the fiducial distribution hx (σ, θ) for (σ, θ) coincides with the Bayesian posterior obtained with the reference prior R (σ, θ) ∝ 1/σ. πσ,θ Notice that π R (σ, θ) ∝ 1/σ is different from π J (σ, θ) ∝ 1/σ 2 obtained by the Jeffreys rule which, as already recalled, is not suitable for multidimensional parameters. Furthermore, while π R does not depend on the ordering of θ and σ, the step-by-step fiducial distribution is in general not allowable if the ordering is reversed. However, hx (σ, θ) coincides with the fiducial distribution obtained through other “symmetric” approaches, see Hannig (2009) and Fraser (1961). Thus the inferential ordering of importance seems irrelevant for this model and hx (σ, θ) can be assumed as an overall fiducial distribution. 4.2 Exponential families Lindley (1958) was the first to study the existence of a fiducial prior, analyzing in particular the case of continuous real NEFs and proving that it exists only for gaussian (with known variance) and gamma (with known shape) models. A full characterization of the real NEFs which admit a fiducial prior is given in V&M (2015). The following proposition summarizes their results. 22 Proposition 10. Let F be a real NEF with natural parameter θ. i) A fiducial prior exists if and only if F is an affine transformation of one of the following families: normal with known variance, gamma with known shape param- eter, binomial, Poisson and negative-binomial. For the three discrete families, the fiducial prior exists for all Hs , Hsℓ and HsG . ii) When a fiducial prior exists, it belongs to the family of conjugate distributions. Moreover, it coincides with the Jeffreys prior for continuous NEFs and for discrete NEFs too if we choose HsG as the fiducial distribution. iii) The fiducial distribution Hs (or HsA in the discrete case) and the Bayesian posterior distribution corresponding to the Jeffreys prior have the same Edgeworth’s expansion up to the term of order n−1 . The previous results establish a strong connections between Jeffreys posteriors and fiducial distributions for real NEFs, and thus the two different approaches lead, in some sense, to the same objective inference. A discussion about the coverage of the fiducial and the Jeffreys intervals and their good frequentist properties, in particular when compared with the standard Wald intervals, is given in V&M (2015, Section 5). Consider now a cr-NEF. It is easy to verify that the fiducial distribution hs (φ) in (18) belongs to the enriched conjugate family defined in Consonni & Veronese (2001, Section 4.3). This fact is the key-point to prove the following proposition. Proposition 11. Let S be a sufficient statistic distributed according to a cr-NEF on Rd , parameterized by φ = (φ1 , . . . , φd ). Then a fiducial prior for φ exists if and only if the conditional distribution of Sk given S[k−1] = s[k−1] is an affine transformation of one of the following families: normal with known variance, gamma with known shape parameter, binomial, Poisson and negative-binomial. In particular, all basic NEF-SQVFs, with the exclusion of the Negative-multinomial/ hyperbolic secant, admit a fiducial prior, which belongs to the enriched conjugate family. Moreover, if for the discrete components of these models we consider the geometric mean hG s[k] , then the product of the Jeffreys priors computed from the conditional distribution of Sk given S[k−1] = s[k−1] , the reference prior and the fiducial prior are all equal. Example 4 (ctd.). The multinomial distribution is a basic NEF-SQVF and thus from Proposition 11, setting sk = n = 0 in hG s (φ) given in (20), we obtain the fiducial prior π(φ) ∝ d Y eφk /2 /(1 + eφk /2 ), k=1 23 (28) which coincides with the reference prior and with the product of the Jeffreys priors for φk , k = 1, . . . , d, computed on the distribution of Xk given X[k−1] = x[k−1] . Finally, we observe that the fiducial distribution (17) is always an overall fiducial distribution for φ. However, the φ-parameterization is often not interesting in itself even if in some cases it is strictly related with a more relevant one. For example, following Berger et al. (2015), consider a multinomial model applied to directional data, as it happens for outcomes from an attitude survey. In this case the cells are naturally ordered, so that it is meaningful to reparameterize the model in terms of the conditional probabilities p∗k = exp(φk )/(1 + exp(φk )), k = 1, . . . , d. Then π(φ) in (28) induces on p∗ = (p∗1 , . . . , p∗d ) an overall fiducial prior which is a product of independent Be(1/2,1/2) distributions coinciding with the overall reference prior. 5 Further examples 5.1 Examples concerning normal models i) Difference of means. Consider two independent normal i.i.d. samples, each of size n, with known common variance σ 2 and means µ1 and µ2 , respectively. The sufficient statistics are the sample sums S1 and S2 , with Si ∼ N(nµi , nσ 2 ), i = 1, 2. If the parameter of interest is φ1 = µ2 − µ1 , we can reparameterize the joint density of (S1 , S2 ) in (φ1 = µ2 −µ1 , φ2 = µ1 ), so that the conditional distribution of S2 given S1 +S2 , being N((nφ1 + s1 + s2 )/2, nσ 2 /2), depends only on φ1 . From Table 1, the fiducial distribution of φ1 /2 + (s1 + s2 )/(2n) is N(s2 /n, σ 2 /(2n)), and thus φ1 is N(x̄2 − x̄1 , 2σ 2 /n), where x̄i = si /n. Because S1 + S2 is N(φ1 + 2φ2 , 2nσ), arguing as before, the fiducial distribution of φ2 given φ1 is N((x̄1 + x̄2 −φ1 )/2, σ 2 /(2n)), so that hS1 ,S2 (φ1 , φ2 ) = hS1 ,S2 (φ1 )hS1 +S2 (φ2 \|φ1 ). Notice that the same joint fiducial distribution is obtained if we consider the ordering (φ2 , φ1 ) or even if we compute the (marginal) fiducial distributions of µ1 and µ2 and obtain that of (φ1 , φ2 ) through the change-of-variable rule. Thus the ordering of the parameter is irrelevant and hS1 ,S2 (φ1 , φ2 ) is an overall fiducial distribution. Furthermore, it coincides with the reference posterior obtained with a constant prior and the marginal distribution of φ1 and φ2 are both confidence distributions. ii) Many normal means (Neyman Scott Problem). Consider n samples of size two (Xi1 , Xi2 ), with each Xij independently distributed according to a N(µi , σ 2 ), i = 1, . . . , n P and let X̄i = (Xi1 + Xi2 )/2 and W = ni=1 (Xi1 − Xi2 )2 . The aim is to make inference on the common variance σ 2 , with nuisance parameter µ = (µ1 , . . . , µn ). This well known example is used to show that the maximum likelihood estimator σ̂ 2 = W/(4n) of σ 2 is inconsistent, because W/(4n) → σ 2 /2, n → ∞. To obtain the fiducial distribution of σ 2 , first notice that the joint distribution of the sufficient statistics X̄ = (X̄1 , . . . , X̄n ) 24 and W can be factorized as Qn i=1 pµi ,σ2 (x̄i ) pσ2 (w), for the independence of X̄ and W , with W ∼ Ga(n/2, 1/(4σ 2 )). Using Table 1 one can easily obtain from pσ2 (w) the fiducial distribution for 1/(4σ 2 ), and hence that for σ 2 which is In-Ga(n/2, w/4), while that of each µi given σ 2 , derived from pµi ,σ2 (x̄i ), is N(x̄i , σ 2 /2). As a consequence Qn 2 2 hx̄,w (σ 2 , µ) = i=1 hx̄i (µi \|σ ) hw (σ ). This distribution coincides with the posterior obtained from the order invariant reference prior π R (σ 2 , µ1 , . . . , µn ) ∝ 1/σ 2 and does not present the inconsistency of the likelihood estimator, which instead occurs for the posterior distribution obtained from the Jeffreys prior π J (σ 2 , µ1 , . . . , µn ) ∝ 1/σ n+2 . 5.2 Comparison of two Poisson rates The comparison of Poisson rates µ1 and µ2 is a classical problem arising in many contexts, see for example Lehmann & Romano (2005) for a discussion on an unbiased uniformly most powerful test for the ratio φ1 = µ2 /µ1 . Given two i.i.d. samples of size n from two independent Poisson distributions, the sufficient statistics are the sample sums S1 and S2 , with Si ∼ Po(nµi ), i = 1, 2. Reparameterizing the joint density of (S1 , S2 ) in (φ1 = µ2 /µ1 , φ2 = µ1 + µ2 ), we have that the conditional distribution of S2 given S1 + S2 is Bi(s1 + s2 , φ1 /(1 + φ1 )) and the marginal distribution of S1 + S2 is Po(nφ2 ). Thus the sampling distribution is a cr-NEF and we can apply (16). Using Table 1, the fiducial density for φ1 /(1 + φ1 ), derived from the conditional distribution of S2 given S1 + S2 is Be(s2 + 1/2, s1 + 1/2) which implies hG s1 ,s2 (φ1 ) = 1 s −1/2 φ12 (1 + φ1 )−s1 −s2 −1 , B(s2 + 1/2, s1 + 1/2) φ1 > 0. (29) From the marginal distribution of S1 + S2 and using again Table 1, it follows that G G G hG s1 +s2 (φ2 ) is Ga(s1 +s2 +1/2, n) and thus hs1 ,s2 (φ1 , φ2 ) = hs1 ,s2 (φ1 )hs1 +s2 (φ2 ). This joint fiducial distribution is order-invariant, coincides with the reference posterior according to Proposition 11, and is an overall distribution for (φ1 , φ2 ). Notice that hG s1 ,s2 (φ1 ) is a confidence distribution and that it differs from the fiducial distribution induced on φ1 by the two independent marginal fiducial densities for µ1 and µ2 . 5.3 Bivariate binomial A Bayesian analysis for the bivariate binomial model has been discussed by Crowder & Sweeting (1989) in connection with a microbiological application. Consider m spores, each with a probability p to germinate, and denote by R the random number of germinating spores, so that R is Bi(m, p). If q is the probability that one of the latter spores bends in a particular direction and S is the random number of them, the probability distribution of S given R = r is Bi(r, q). The joint distribution of R and S is called bivariate binomial. Crowder and Sweeting observe that the Jeffreys prior π J (p, q) ∝ p−1 (1 − p)−1/2 q −1/2 (1 − q)−1/2 25 is not satisfactory for its asymmetry in p and 1 − p, while Polson & Wasserman (1990) show that this fact does not occur using the order-invariant reference prior π R (p, q) ∝ p−1/2 (1 − p)−1/2 q −1/2 (1 − q)−1/2 which is the product of the two independent Jeffreys priors. G The joint fiducial density hG r,s (q, p) can be obtained as the product of hr,s (q), derived from the conditional model Bi(r, q) of S given R = r, and hG r (p\|q), derived from the marginal model Bi(m, p) of R, which does not depend on q. Thus p and q are independent under hG r,s so that it is an overall fiducial distribution. Because for the binomial model the fiducial prior is equal to the Jeffreys prior, see Proposition 10, it follows immediately that hG r,s (q, p) coincides with the reference posterior. All previous conclusions hold even if we consider the alternative parametrization (η = pq, λ = p(1 − q)/(1 − pq)). 5.4 Ratio of parameters of a trinomial distribution Bernardo & Ramon (1998) perform the Bayesian reference analysis for the ratio of two multinomial parameters presenting some applications. In particular they discuss the case of (X1 , X2 ) distributed according to a trinomial distribution with parameters n and p = (p1 , p2 ), and provide the joint reference prior for (φ1 = p1 /p2 , φ2 = p2 ), with φ1 the parameter of interest. Then they derive the marginal reference posterior for φ1 which is x −1/2 π R (φ1 \|x1 , x2 ) ∝ φ1 1 (1 + φ1 )−x1 −x2 −1 . (30) To find the fiducial distribution of φ1 , we reparameterize the trinomial model in (φ1 , φ2 ). The conditional distribution of X1 given T = X1 + X2 = t is Bi(t; φ1 /(1 + φ1 )), so that, by Table 1, the fiducial density for φ1 /(1 + φ1 ) is Be(x1 + 1/2, t − x1 + 1/2) and hG x1 ,t (φ1 ) coincides with (30). From the marginal distribution of T , which is Bi(n; φ2 (1 + φ1 )), it is possible to derive the fiducial density hG t (φ2 \|φ1 ) = Γ(n + 1) t−1/2 (1 + φ1 )t+1/2 φ2 (1 − (1 + φ1 )φ2 )n−t−1/2 Γ(t + 1/2)Γ(n − t + 1/2) G G so that the joint fiducial density is hG x1 ,x2 (φ1 , φ2 ) = hx1 ,t (φ1 )ht (φ2 \|φ1 ) which coincides with the joint reference posterior. 6 Conclusions and final remarks We have suggested a way to construct a fiducial distribution which depends on the inferential importance ordering of the parameter components. Our proposal appears to be quite simple to apply and, even if it is not so general as the theory suggested by Hannig, has some advantages in connection with the modern confidence distribution theory and it is strictly related to objective Bayesian analysis. 26 In complex models an exact analysis is generally not possible, but approximate results can be derived working with asymptotic distributions. In V&M (2015), starting from the sufficient statistic, an expansion up to the first order of the fiducial distribution for the mean parameter of a real NEF is provided. This result can be extended to arbitrary regular models starting from the maximum likelihood estimator of the parameter. When the maximum likelihood estimator is not sufficient a better fiducial distribution can be obtained using an ancillary statistic, as suggested in Section 3. To this aim the magic formula p∗ given by Barndorff-Nielsen (1983), which provides an approximation of the conditional distribution of the maximum likelihood estimator given an ancillary statistic, can be fruitfully adopted. Furthermore, these asymptotic results appear to be strictly connected with the theory of matching priors, i.e. priors that ensure approximate frequentist validity of posterior credible set. Notice that also these priors crucially depend on the inferential ordering of the parameters, see Tibshirani (1989) and Datta & Mukerjee (2004). However, a normal approximation of the fiducial distribution, when it can be established and is enough for the analysis, can be proved to be order-invariant. These type of results will be discussed in a forthcoming paper. Acknowledgements This research was supported by grants from Bocconi University. Appendix A1: Useful results on cr-NEFs Some technical aspects related to cr-NEFs are the following. 1. A NEF is a cr-NEF if and only if the principal k × k matrix of the variance function does not depend on µk+1 , . . . µd , for k = 1, . . . , d − 1. 2. The Fisher information matrix relative to the φ-parametrization is diagonal with the kk-th element depending only on φ[k] . 3. The cumulant transform Mk (φk ; x[k−1] ) of the k-th conditional density is given by Mk (φk ; x[k−1] ) = k−1 X Akj (φk )xj + Bk (φk ), (31) j=1 for some functions Akj and Bk . 4. The conditional expectation of Xk given X[k−1] = x[k−1] is linear in x[k−1] , because it is the gradient of (31). 27 5. The parameter µk depends on φ only through φ[k] , because from (31) µk = k−1 X ∂Akj (φk ) j=1 ∂φk µj + ∂Bk (φk ) . ∂φk (32) 6. Using (14) and (31), it can be checked that θk = φk − d X Auk (φu ), and M (θ) = d X Bk (φk (θ)). (33) k=1 u=k+1 As a consequence of the first part of (33), there exists a function gk such that φk = θk + gk (θk+1 , . . . , θd ). (34) Of course all the previous formulas hold for k = 1, . . . , d, with the understanding that components that lose meaning for a specific k are set to zero. A NEF has a simple quadratic variance function (SQVF) if the ij-th element of its variance-covariance matrix, seen as a function of the mean parameter µ = (µ1 , . . . , µd ), P (k) can be written as Vij (µ) = qµi µj + dk=1 µk Lij + Cij , where q is a real constant and L(k) , k = 1, . . . , d and C are constant d × d symmetric matrices. Any NEF-SQVF can be obtained, via a nonsingular affine transformation, from one of the basic families: Poisson/normal (q = 0), multinomial (q = −1/N , N positive integer), negative-multinomial (q = 1/R, R positive integer), negative-multinomial/gamma/normal (q = 1/R) and negative-multinomial/hyperbolic-secant (q = 1/R), see Casalis (1996) for a detailed description of these distributions. The ij-th element of the variance function V (µ) of a basic NEF-SQVF is Vij (µ) = qµi µj + d X zik µk + Cij , (35) k=1 where zij = zji , i, j ∈ {1, . . . , d}, are constants. The values of zii for the basic NEFSQVFs, together with other technical details, are given in the proof of Corollary 2. A2: Proofs Proof of Proposition 1. By the standard change-of-variable rule applied to the first integral in (10), it is enough to show that λ hφ t (φ(λ))\|Jφ (λ)\| = ht (λ), 28 (36) where Jφ (λ) is the Jacobian of the transformation from φ to λ. Now, from (8) we have hφ t (φ(λ)) = = d Y ht[k] ,t−[d] (φd−k+1 (λ[d−k+1] )\|φ[d−k] (λ[d−k] )) k=1 d Y k=1 ∂ Fφ (tk \|t[k−1] , t−[d] ; φ[d−k] ) ∂φd−k+1 d−k+1 φ[d−k+1] =φ[d−k+1] (λ[d−k+1] ) while Jφ (λ) = d Y ∂φd−k+1 (λ[d−k+1] ) , ∂λd−k+1 k=1 because the transformation from φ to λ is lower triangular. It follows from the last two formulas and the chain rule that hφ t (φ(λ))\|Jφ (λ)\| = d Y k=1 ∂ ∂λd−k+1 Fλd−k+1 (tk \|t[k−1] , t−[d] ; λ[d−k] ) , where Fλd−k+1 is the distribution function of Tk given (T[k−1] = t[k−1], T[−d] = t[−d] ) in the λ parameterization. The equality (36) follows by applying (9) to the model parameterized by λ. ⋄ Proof of Proposition 2. Let pG (x) = c−1 (p1 (x)p2 (x))1/2 , where c = constant. Then Z R (p1 (x)p2 (x))1/2 dν(x) is the normalizing Z q(x) q(x) KL(q\|p1 ) + KL(q\|p2 ) = log q(x)dν(x) + log q(x)dν(x) p1 (x) p2 (x) Z Z q(x) q 2 (x) q(x)dν(x) = 2 log q(x)dν(x) = log p1 (x)p2 (x) CpG (x) Z q(x) = 2 log G q(x)dν(x) − 2 log c = 2KL(q\|pG ) − 2 log c. (37) p (x) Because c does not depend on q, it follows that the functional in (37) achieves its minimum (equal to −2 log c) if and only if KL(q\|pG ) = 0, i.e. q = pG . ⋄ Proof of Proposition 3. We only prove that Hs (θ) < HsG (θ); the other inequality can be shown in the same way. Using (5) and (6), we can write hG s (θ) hs (θ) = = 1 c 1 c p s hs (θ)hℓs (θ) hs (θ) 1+ 1 = c ∂ ∂θ pθ (s) hs (θ) = 29 q hs (θ)(hs (θ) + ∂ ∂θ pθ (s)) hs (θ) 1p 1 + γs (θ). c (38) By hypothesis γs (θ) is decreasing and thus, from (38), hG s (θ)/hs (θ) is also decreasing on Θ. This is a sufficient condition for Hs (θ) < HsG (θ), see Shaked & Shanthikumar (2007, Theorem 1.C.1). ⋄ Proof of Corollary 1. Let pθ (s) = exp(θs − nM (θ) be the probability mass function (with respect to a measure ν) of a real NEF, with θ the natural parameter. Fixing s ∈ S0 , we can write ∂pθ (s) ∂ (s − nM ′ (θ)) exp(θs − nM (θ)) γs (θ) = (1 − Fθ (s)) = P+∞ / ′ ∂θ ∂θ t=s+1 (t − nM (θ)) exp(θt − nM (θ)) !−1 +∞ X t − nM ′ (θ) . (39) exp{(t − s)θ} = s − nM ′ (θ) t=s+1 The elements in the sum (39) are continuous and increasing functions of θ in both intervals for which θ < θbs and θ > θbs , where θbs = (M ′ )−1 (s/n). Thus γs (θ) is decreasing in these intervals. Moreover, γs (θ) is equal to zero for θ = θbs , positive for θ < θbs and negative for θ > θbs , because the denominator of γs (θ) is hs (θ), which is positive. Then γs (θ) is decreasing over all Θ and from Proposition 3 the result follows. ⋄ Proof of Proposition 4. In order to prove the proposition, it is sufficient to show that there exist θ1 and θ2 in Θ, G A G θ1 < θ2 , such that hA s (θi ) = hs (θi ), i = 1, 2, with hs (θ) < hs (θ) for θ1 < θ < θ2 and G hA s (θ) > hs (θ) otherwise, see Shaked & Shanthikumar (2007, proof of Theorem 3.A.44). G Thus we analyze the sign of hA s (θ) − hs (θ). We can write q 1 1 G ℓ (θ) = hA (θ) − h (h (θ) + h (θ)) − hs (θ)hℓs (θ) s s s s 2 c 1p 1 1 + γs (θ) , (2 + γs (θ)) − = hs (θ) 2 c G so that the sign of the difference hA s (θ)−hs (θ) is a function of γs (θ) only. First notice that, Rp by a standard property of the arithmetic and geometric means, c = hs (θ)hℓs (θ)dθ < R (hs (θ) + hℓs (θ))/2 dθ = 1 for all θ. After some straightforward algebra, it can be seen √ −2 2 G 2 that hA s (θ) − hs (θ) = 0 when (and only when) γs (θ) = 2c ((1 − c ) − 1 − c ) = k1 or √ γs (θ) = 2c−2 ((1 − c2 ) + 1 − c2 ) = k2 , with k1 ∈ (−1, 0) and k2 > 0. Moreover we have G A G hA s (θ) < hs (θ) for k1 < γs (θ) < k2 and hs (θ) > hs (θ) for γs (θ) < k1 or γs (θ) > k2 . By assumption, γs (θ) is decreasing on Θ from +∞ to -1, so that there exist θ1 and θ2 , with γs (θ1 ) = k2 and γs (θ2 ) = k1 , satisfying the sufficient condition stated at the beginning of the proof. ⋄ 30 Proof of Proposition 5. First notice that if we use for constructing the fiducial distribution the sufficient statistic S, which has the same dimension of the parameter, we do not need an ancillary statistic. Furthermore, T is a one-to-one transformation of X, and thus S is a function of T = (T [d] , T −[d] ) but, since S is complete, it is stochastically independent of T −[d] by Basu’s Theorem. As a consequence, S[k] is also independent of T −[d] and thus Prφd−k+1 (Sk ≤ sk \| S[k−1] = s[k−1] ; φ[d−k] ) = Prφd−k+1 (Sk ≤ sk \| S[k−1] = s[k−1] , T−[d] = t−[d] ; φ[d−k] ). (40) From the one-to-one lower triangular transformation s = g(t[d] , t−[d] ), we have that sk = gk (tk , t[k−1] , t−[d] ), with gk invertible with respect to tk , so that (assuming gk increasing) (40) becomes Prφd−k+1 (gk (Tk , T[k−1] , T −[d] ) ≤ sk \| T[k−1] = t[k−1], T−[d] = t−[d] ; φ[d−k] ) = Prφd−k+1 (Tk ≤ gk−1 (sk , T[k−1] , T −[d] ) \| T[k−1] = t[k−1], T−[d] = t−[d] ; φ[d−k] ) = Prφd−k+1 (Tk ≤ tk \| T[k−1] = t[k−1], T−[d] = t−[d] ; φ[d−k] ), which proves the proposition. ⋄ Proof of Proposition 6. Because each conditional distribution of Xk given X[k−1] = x[k−1] belongs to a NEF with natural parameter φk , using (4) we have that Hs[k] (φk ) is a distribution function for φk . The result follows from the postulated independence among the φk ’s. ⋄ Proof of Proposition 7. Formulas (18) and (19) derive by a direct application of (17) to the conditional distributions of the different families. For a detailed description of the cr-NEFs involved, see Consonni & Veronese (2001, proof of Theorem 3). ⋄ Proof of Corollary 2. First notice that the fiducial distribution hs (µ) can be more easily obtained via a double transformation, namely θ hµ s (µ) = hs (θ(µ))\|Jφ (θ(µ))\|\|Jθ (µ)\|, where the Jacobian \|Jφ (θ)\| = 1 for (34), and ) ( d X θk (µ)zk − q(d + 1)M (θ(µ)) , \|Jθ (µ)\| ∝ det{V (µ)}−1 = exp − k=1 31 (41) see Gutiérrez-Peña & Smith (1997, pag. 34) for the proportionality relationship and Consonni et al. (2004, Prop. 1) for the equality. We consider now each family. Poisson/normal family with m Poisson components. We have q = 0, φk = θk ; zk = 1, θk = log(µk ) and Bk (φk ) = exp(φk ) for k = 1, . . . , m, while zk = 0, θk = µk /σ 2 and Bk (φk ) = σ 2 φ2k /2 for k = m + 1, . . . , d, where σ 2 is the known variance of the normal Q −1 components. Then from (41), it follows that \|Jφ (θ(µ))\| = m k=1 µk , and thus using (18), the result (21) follows. Multinomial family. Using the relationships in Example 1, (41) gives \|Jφ (θ(µ))\| = (N − Pd Qd −1 −1 k=1 µk ) k=1 µk and using (8) we obtain (22). Negative-multinomial family. We have q = 1/R, R > 0, zk = 1, φk (θ) = θk − log(1 − Pd Pd φk θu j=1 µj ), and Bk (φk ) = −R log(1 − e ) for k = u=k+1 e ), θk = log(µk ) − log(R + Pd Q 1, . . . , d. Then from (41), it follows that \|Jφ (θ(µ))\| = (R + k=1 µk )−1 dk=1 µ−1 k , and thus using (18), the result (23) follows. Negative-multinomial/gamma/normal family (with an m dimensional negative-multinomial P component). We have q = 1/R, R > 0; zk = 1 and φk = log(µk /(R + kj=1 µj )), P k = 1, . . . , m; zm+1 = 0 and φm+1 = −(R + m j=1 µj )/µm+1 ; zk = 0 and φk = µk /µm+1 , k = m+2, . . . , d. In this case it is convenient to compute the fiducial density of µ directly from (19). Observing that the Jacobian of the transformation from φ to µ is  !−1 m ! m m X Y X −d+m+1 R + \|Jφ (µ)\| = R + µk µj  µ−2 µ−1 m+1 µm+1 k k=1 j=1 k=1 and using the previous expression of φk , the density (24) follows. ⋄ Proof of Proposition 8. Let X be an i.i.d. sample of size n, with Xi ∼ pθ (xi ) = f (xi − θ), i.e. θ is a location parameter, and consider the transformation Z1 = X1 , Zi = Xi − X1 , i = 2, . . . , n, whose Jacobian is one. Then, setting z = (z1 , . . . , zn ) R +∞ Hx (θ) = Hz (θ) = 1 − Fθ (z1 \|z2 , . . . , zn ) = z1 f (t R +∞ −∞ f (t − θ) − θ) Qn i=2 f (t Qn i=2 f (t + zi − θ)dt + zi − θ)dt . Using now the substitution m = −t + θ + z1 in the previous two integrals, and recalling that π J (θ) ∝ 1 we obtain Rθ Qn i=2 f (z1 + zi − m)dm −∞ f (z1 − m) R +∞ Qn i=2 f (z1 + zi − m)dm −∞ f (z1 − m) = = Rθ Qn J i=1 f (xi − m)π (m)dm −∞ R +∞ Qn J i=1 f (x1 − w)π (m)dm −∞ Z θ J π (m\|x)dm. −∞ 32 The result relative to the scale parameter follows recalling that the model pθ (x) = f (x/θ)/θ can be transformed in a model with location parameter µ setting y = log(x) and µ = log(θ). In this case a constant prior on µ is equivalent to a prior on θ proportional ⋄ to 1/θ. Proof of Proposition 9. Let X be an i.i.d. sample of size n, with Xi ∼ pθ,σ (xi ) = f ((xi − θ)/σ)/σ, i = 1, . . . , n and notice that the absolute value of the Jacobian of the transformation from x to (x1 , z2 , z), with z2 = x2 − x1 , z = (z3 , . . . , zn ) and zj = (xj − x1 )/z2 , j = 3, . . . , n, is \|z2 \|n−2 . Furthermore, the reference prior π R (θ, σ) is order-invariant and can be written as π R (θ\|σ)π R (σ), where π R (θ\|σ) ∝ 1 and π(σ) ∝ 1/σ, see Fernández & Steel (1999). Working conditionally on σ we can thus apply Proposition 8 to conclude that the reference posterior and the fiducial distribution for θ given σ coincide. It remains to show that Z ∞ R R p θ,σ (x1 , z2 , z)π R (θ\|σ)π R (σ)dθ (42) π (σ\|x) = π (σ\|x1 , z2 , z) ∝ ∝ Z ∞ 1 −∞ σ n+1 f −∞ x1 − θ σ n z2 zi + x1 − θ z2 + x1 − θ Y f f dθ σ σ i=3 corresponds to the fiducial density hz2 ,z (σ). We have Z R +∞ R +∞ −∞ z2 = +∞ Z +∞ X ,Z2 \|Z (t, w\|z)/pZ (z)dtdw z2 −∞ \|w\|n−2 wzi +t−θ t−θ w+t−θ Qn dtdw f f f i=3 σn σ σ σ Hz2 ,z (σ) = 1 − Fσ (z2 \|z) = 1 p θ,σ pZ (z) (43) , where the density of pZ (z) does not depend on the parameters because z is ancillary. Assuming z2 > 0, and using the transformation m = x1 − v(t − θ)/σ, v = z2 σ/w, which implies t = σ(x1 − m)/v + θ, w = z2 σ/v, with Jacobian z2 σ 2 /v 3 , the fiducial distribution Hz2 ,z (σ) in (43) becomes R σ R +∞ z2n−1 0 −∞ vn+1 f x1 −m v f z2 +x1 −m v pZ (z) Qn i=3 f z2 zi +x1 −m v dmdv . Taking the derivative with respect to σ, it is immediate to see that the fiducial density for σ coincides with the posterior distribution given in (42). If z2 < 0, and applying to the integral the same transformation used in the previous case, we have Fσ (z2 \|z) = R z2 R +∞ −∞ −∞ = − R σ R +∞ 0 −∞ \|w\|n−2 σn f t−θ σ (−z2 )n−1 f vn+1 f w+t−θ σ Z p (z) x1 −m v 33 f Qn i=3 f z2 +x1 −m v pZ (z) wzi +t−θ σ Qn i=3 f dtdw z2 zi +x1 −m v dmdv , so that again the derivative with respect to σ of Hz2 ,z (σ) = 1 − Fσ (z2 \|z) leads to (42). ⋄ The following lemma will be used in the proof of Proposition 11. Lemma 1. Consider a cr-NEF on Rd , with the k-th diagonal element in the Fisher information matrix given by Ikk (φ) = ak (φk )bk (φ[k−1] ). Then the d-group (order-invariant) reference prior π R for φ = (φ1 , . . . , φd ) is π R (φ) = d Y k=1 πkJ (φk ) ∝ d Y (ak (φk ))1/2 , (44) k=1 where πkJ (φk ) is the Jeffreys prior obtained from the conditional distribution of Xk given X[k−1] = x[k−1] . Proof of Lemma 1 First observe that µ[k] is a one-to-one transformation of φ[k] and that the information matrix I(φ) of a cr-NEF is diagonal, see Appendix A1 (points 2 and 5). From (14) and (31), the kk-th element of I is 2 ∂ X Ikk (φ[k] ) = −Eφ log pφk (xk \|x[k−1] ; φk ) = −EφX (Mk′′ (φk ; x[k−1] )) ∂φ2k = k−1 X A′′kj (φk )µj (φ[j] ) + Bk′′ (φk ). j=1 Under the assumption in the proposition, we can write Ikk (φ[k] ) = ak (φk )b∗k (µ[k−1] (φ[k−1] )). From Datta & Ghosh (1995), it follows that the reference prior on φ is order-invariant and is given by the last product in (44). Consider now the Jeffreys prior on φk obtained from pφk (xk \|x[k−1] ). This is propor- tional to the square root of −E Xk \|x[k−1] (Mk′′ (φk ; x[k−1] )) = k−1 X A′′kj (φk )xj + Bk′′ (φk ) = ak (φk )b∗k (x[k−1] ), j=1 where again the last equality holds by the assumption in the proposition. Thus the ⋄ product of the d Jeffreys priors is equal to (44) and the result holds. Proof of Proposition 11. Due to the independence of the φk ’s, a fiducial prior for φ exists if and only if there exists a fiducial prior for each φk . Because the conditional distribution of Sk given S[k−1] = s[k−1] belongs to a real NEF with natural parameter φk , the result of the first part of the proposition follows from Proposition 10. The first statement of the second part of the proposition follows checking directly the form of the conditional distributions of the basic NEF-SQVFs and using again Proposition 10. 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Source Forager: A Search Engine for Similar Source Code Vineeth Kashyap∗, David Bingham Brown† , Ben Liblit† , David Melski∗ , and Thomas Reps∗† ∗ GrammaTech, arXiv:1706.02769v1 [cs.SE] 8 Jun 2017 Inc., Ithaca, New York, USA Email: {vkashyap,melski}@grammatech.com † University of Wisconsin–Madison, USA Email: {bingham,liblit,reps}@cs.wisc.edu Abstract—Developers spend a significant amount of time searching for code—e.g., to understand how to complete, correct, or adapt their own code for a new context. Unfortunately, the state of the art in code search has not evolved much beyond text search over tokenized source. Code has much richer structure and semantics than normal text, and this property can be exploited to specialize the code-search process for better querying, searching, and ranking of code-search results. We present a new code-search engine named Source Forager. Given a query in the form of a C/C++ function, Source Forager searches a pre-populated code database for similar C/C++ functions. Source Forager preprocesses the database to extract a variety of simple code features that capture different aspects of code. A search returns the k functions in the database that are most similar to the query, based on the various extracted code features. We tested the usefulness of Source Forager using a variety of code-search queries from two domains. Our experiments show that the ranked results returned by Source Forager are accurate, and that query-relevant functions can be reliably retrieved even when searching through a large code database that contains very few query-relevant functions. We believe that Source Forager is a first step towards muchneeded tools that provide a better code-search experience. Index Terms—code search, similar code, program features. I. Introduction In this age of software proliferation, it is useful to be able to search large source-code corpora effectively for code with desired properties.1 Developers routinely use code search as a learning and debugging tool for tasks such as looking for existing functionality in a code base, determining how to use an API or library, gathering information about what code is intended to do, etc. [1]. Text-based search techniques are not always precise enough for code because they focus purely on strings in the code: Supported, in part, by a gift from Rajiv and Ritu Batra; by AFRL under DARPA MUSE award FA8750-14-2-0270, and by the UW–Madison Office of the Vice Chancellor for Research and Graduate Education with funding from the Wisconsin Alumni Research Foundation. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors, and do not necessarily reflect the views of the sponsoring agencies. T. Reps has an ownership interest in GrammaTech, Inc., which has licensed elements of the technology reported in this publication. 1 In this paper, the term “search” is used in the sense of Google search— namely, to retrieve documents that are related to a specified query. “Search” is not used in the sense of finding an occurrence of a user-specified string or pattern in a given document. comments, complete or partial names of functions and variables, and so on. Text search largely ignores code structure and semantics (i.e., what the code does and how it does it). A text-based approach can cause searching to be imprecise: relevant code fragments may be missed, while many spurious matches may be returned. Recent search techniques allow users to specify certain aspects of code semantics in addition to the textual query [2]–[8]. Some techniques allow users to specify structural requirements, such as that the search target should have nested loops. Others specify context, such as that the search target should implement a particular interface. Yet others specify sets of input/output pairs. Additional semantic information can improve search accuracy. However, existing techniques share the following shortcomings: • The techniques do not provide a unified way of specifying semantics for the search query. Each technique has its own ad-hoc specification of the semantic aspects of the code that it uses. • Each technique is closely married to its chosen semantic aspect, which is deeply ingrained into the implementation of the search technique. This tight coupling makes it hard to extend these techniques to model additional semantic aspects. We propose a search technique for finding similar source code that addresses these shortcomings: • • Unified Query Specification. Our code-search mechanism takes code fragments as queries. Various kinds of semantic information can be extracted from the query and used by the search. This approach provides a unified mechanism for code search: searching code using code fragments. Moreover, the same techniques for extracting semantic information are used on both queries and elements of the corpus being searched, leading to greater consistency. Extensibility. Our code-search technique uses a vector of feature-observations extracted from elements in the corpus. Feature-observations capture various aspects of the syntax and semantics of a program (each such aspect is called a feature-class), and provide a unified interface for querying. This approach also makes our search technique extensible: it is easy to introduce more feature-classes that model additional aspects of the code. int binsearch (int x, int v[], int n) { int low , high , mid; low = 0; high = n - 1; while (low <= high) { mid = (low + high) / 2; if (x < v[mid ]) { high = mid - 1; } else if (x > v[mid]) { low = mid + 1; /* found match / } else { return mid; } } / no match / return -1; } feature-observations of various feature-classes weights similar-code results feature-class weight determination similarity-based neighbor search query ... corpus ... program elements feature extraction engine code database Fig. 2. Example program that implements a binary search over a sorted integer array Fig. 1. Overview of the Source Forager architecture In addition to being useful on its own right as a developer A. Offline Phase: Population of the Source Forager Database tool, similar-code search can serve as an important building In this phase, Source Forager analyzes a given code corpus, block for automated program repair and program synthesis. and populates a code database with rich information about The ability to find other code similar to a query can help each of the functions in the code corpus. Source Forager automated tools learn from the similar code, and fix bugs or extracts several different kinds of information about each perform code completion tasks on the query. function; we refer to each of the different kinds of information The main contributions of Source Forager are: as a feature-class. §III describes our different feature-classes in • The ability to perform C/C++ code searches using code detail. A feature-observation is some specific value observed fragments as queries. The searches and answers of Source for a given feature-class. Thus, each function has one featureForager are both based on a query formalism that is close observation for each feature-class. For example, one of our to the concepts that developers are already familiar with. feature-classes is Numeric Literals. The corresponding feature• A code-search architecture that uses multiple code featureobservation is the set of all the numeric constants used in classes simultaneously. The architecture is extensible, althe function. For the binary-search implementation code given lowing easy addition of new code feature-classes, which in Fig. 2, the Numeric Literals feature-observation is the set enhances the dimensions along which code is searched. {−1, 0, 1, 2}. • A mechanism for automatically selecting useful code featureA feature extraction engine consists of several feature extracclasses to be employed in code search of a given query, given tors, which collect a given function’s feature-observations into no a priori domain information about the query. a feature-vector. Note that the elements of the feature-vector • A supervised-learning technique to pre-compute the relative importance of different feature-classes, when it is known can be non-numeric, such as sets, multisets, trees, maps, etc. that a query belongs to a specific domain for which suitable The number of feature-classes determines the length of the feature-vector. training data is available. Organization: The remainder of the paper is organized The feature extractors operate on a code corpus, and popinto four sections: §II gives an overview of our approach and ulate a code database. Each element of the code database algorithms. §III describes the methods in detail. §IV presents consists of a C/C++ function from the corpus along with its our experimental results. §V discusses related work. extracted feature-vector. If Numeric Literals is employed as one of the feature-classes, then one element of a function’s II. Overview feature-vector is the set of numeric constants. Source Forager is a search engine for finding similar source The code database also has access to several similarity code. It takes an input query as C/C++ source text, then functions, one for each feature-class. The similarity function searches a pre-populated database for similar C/C++ code, for a given feature-class takes any two feature-observations returning a ranked list of results. The units of code about belonging to that feature-class and returns a value between 0.0 which Source Forager can reason about are called program and 1.0. A higher value indicates greater similarity between elements. In its current incarnation, program elements are two feature-observations. For example, the similarity function C/C++ functions; that is, both queries and results are C/C++ for Numeric Literals is the Jaccard index. Given two sets S1 functions. and S2 , the Jaccard index is given by: Fig. 1 provides an architectural overview of Source Forager. Source Forager has two stages: an offline phase to populate its \|S1 ∩ S2 \| . (1) simJacc(S1,S2 ) = code database, and an online query-search phase. \|S1 ∪ S2 \| 2 The second implementation integrates our infrastructure with Pliny-DB [10], which is an in-memory object-store database implemented in C++. The feature-observations in feature-vectors are serialized into efficient in-memory data structures by Pliny-DB. Pliny-DB has access to similarity functions implemented in C++ for all feature-classes. It implements the search for the k functions most similar to the query by (1) scanning all the feature-vectors in the database, (2) comparing each of them to the query feature-vector, and (3) maintaining a priority queue of size k that keeps track of the k most-similar feature-vectors. Given a query feature-vector and relative weights for different feature-classes, Pliny-DB can find the 10 most-similar functions in a code database containing 500,000 functions in under 2 seconds on a single machine with 8 Intel i7 3.6 GHz cores and 16 GB RAM. Effort is underway by the developers of Pliny-DB to make a distributed version, which would allow Source Forager to search large code databases without taking a big performance hit: a large code database can be split into p smaller units that can each be searched in parallel, and the sorted k most-similar results from each of the p units can be merged using a multi-way merge algorithm. int bins(int key ,int array [],int min ,int max) { if (max < min) { return KEY_NOT_FOUND; } else { int midpoint = (int)floor (( min+max )/2); if( array[ midpoint ] < key) { return bins(key , array , midpoint +1, max); } else if( array [midpoint ] > key) { return bins(key , array , min , midpoint -1); } else { return midpoint ; } } } Fig. 3. Example Source Forager code-search result for the query in Fig. 2. This result is a recursive implementation of binary search. B. Online Phase: Search for Similar Code In the online search phase, Source Forager takes a query and uses the same feature-extraction infrastructure to obtain the feature-vector that corresponds to the query. This infrastructure reuse creates a consistent representation and view of code throughout the code-search infrastructure. For each featureclass in the feature-vector, a weight is assigned to determine the importance of that feature-class. This feature-class weight determination is based on which configuration Source Forager is run with; sections III-B, III-C and IV-B provide an overview of the different configurations. A combined similarity function is defined on any two feature-vectors by combining the per-feature-class similarity functions with the per-feature-class weight assignment using a weighted average. That is, Íncl sim ( A® , B® ) · wc ® B) ® = c=1 Ícn c c simcombined ( A, , (2) cl w c=1 c C. Extensible Architecture Source Forager’s architecture allows for easy extension. To add a new feature-class, one implements (1) a feature extractor that determines the feature-observation for any given function, and (2) a corresponding similarity function. We currently implement our feature extractors using CodeSonar®. However, Source Forager is not tightly coupled with CodeSonar: any C/C++ processing tool can be used to implement a feature extractor. The feature-observations for all existing feature-classes are represented with well-known container data structures, such as lists, maps, and trees; all similarity functions work at the level of container data structures, and thus are available to be reused with any additional user-supplied feature extractors. Furthermore, Source Forager is not tied to having functions as the only kind of program element. The underlying architecture is also not limited to C/C++, and thus Source Forager can be re-targeted to perform code searches of programs written in other languages. where A® and B® are two feature-vectors; ncl is the total number of feature-classes (i.e, the length of each feature-vector); simc is the similarity function for feature-class c; A®c and B®c ® are the feature-observations for feature-class c in A® and B, respectively; and wc is the weight assigned to feature-class c. The feature-vector of the query is compared with each of the feature-vectors in the code database using this combined similarity function, and the k most-similar functions (that is, with the highest similarity scores to the query) are returned as results (for some configurable limit k). Fig. 3 shows an example Source Forager code-search result when the code in Fig. 2 is used as query. We have two implementations of Source Forager. The first one is a slower-performing version, in which the code database is implemented as a large, in-memory JSON [9] object, and the various similarity functions and the algorithm for k-mostsimilar function-search are implemented in Python. This implementation allows for easier and quicker experimentation with new ideas. We use this version for the experiments reported in §IV. III. Code Search In this section, we first describe the different feature-classes and the accompanying similarity functions that are employed in Source Forager. We then describe two configurations of Source Forager. The first configuration (dyn-select) selects a subset of the feature-classes on a per-query basis for performing code search: this configuration is useful when no additional information is available regarding a code query. The second configuration (svm-weights) pre-computes the relative importance of feature-classes for a specific domain ahead of time using supervised-learning techniques. This configuration is useful when the domain of the code query is known. 3 Seq TABLE I A brief overview of the different feature-classes employed in Source Forager. The marked feature-classes all use Jaccard index (Eq. (1)) as the similarity function. The similarity functions used for the remaining feature-classes accompany their descriptions in §III-A. − negate Loop Seq Feature-Class Brief Description Type–Operation Coupling* types used and operations performed on the types Skeleton Tree structure of loops and conditionals Decorated Skeleton Tree structure of loops, conditionals, and operations Weighted NL Terms processed natural language terms in code 3 Graph CFG BFS CFG subgraphs of size 3, BFS used for generating subgraphs 4 Graph CFG BFS CFG subgraphs of size 4, BFS used for generating subgraphs 3 Graph CFG DFS CFG subgraphs of size 3, DFS used for generating subgraphs 4 Graph CFG DFS CFG subgraphs of size 4, DFS used for generating subgraphs Modeled Library Calls* calls made to modeled libraries Unmodeled Library Calls* calls made to unmodeled libraries User-Defined Library Calls* calls made to user-defined libraries Type Signature input types and the return type Local Types* types of local variables Numeric Literals* numeric data constants used String Literals* string data constants used Comments* associated comment words Seq Loop <= / + Seq Cond Seq Cond < − Seq Cond Seq Cond > + (a) Skeleton Tree (b) Decorated Skeleton Tree Fig. 4. Tree-structured feature-observations for the example program in Fig. 2. The AST is further abstracted by retaining only the loops (for, while, do. . . while) and conditionals (if. . . else, switch). Operationally, the feature extractor can be realized as a tree transducer that drops all AST nodes that are not loops or conditionals. Sequences of loops or conditionals are encapsulated within a sequence node, and empty sequences are dropped from the feature-observation. The intuition behind using this feature-class for code search is that similar functions tend to have similar loop and conditional structures. Fig. 4a shows the Skeleton Tree feature-observation for the example code in Fig. 2. The similarity function used for Skeleton Tree featureobservations is based on tree edit distances. Let dr be a rough approximation of the distance between two trees, only based on their sizes: A. Feature-Classes and Similarity Functions Table I summarizes Source Forager’s feature-classes. Below, we further describe these feature-classes and their associated similarity functions. Type–Operation Coupling: The feature-observation for this feature-class consists of the types of variables operated on in the function, coupled with the operations performed on those types. The feature-observation is a set of (type, operation) pairs. Primitive types are paired with the builtin arithmetic, logical, and relational operations, for example, (int, >=). User-defined types such as C++ classes are paired with the user-defined operations on them, including direct and indirect field accesses and method calls. For example, the pair (Bar, .foo) indicates that the field foo of an aggregate data type Bar is accessed. The intuition behind including this feature-class is that similar functions tend to use similar type–operation pairs. For the example in Fig. 2, the Type– Operation Coupling feature-observation extracted is the set {(int, unary-), (int, /), (int, +), (int, >), (int, +), (int, <=), (int, -), (int, <)}. Skeleton Tree: The feature-observation for this featureclass is based on the abstract syntax tree (AST) of a function. dr (T1, T2 ) = \|size(T1 ) − size(T2 )\| max(size(T1 ), size(T2 )) Further, let DT be a fixed distance threshold (which we set to 0.5). We obtain an approximate distance between two trees, dt as follows:  dr (T1, T2 )   !     ed(pre(T1 ), pre(T2 )), dt (T1, T2 ) = max  ed(post(T1 ), post(T2 ))      max(size(T1 ), size(T2 )) if dr (T1, T2 ) ≥ DT otherwise Here pre(T ) is the sequence obtained by performing a pre-order traversal of the tree T , post(T ) is the sequence obtained by performing a post-order traversal of the tree T , and ed(S1, S2 ) is the word edit distance between the sequences S1 and S2 . The similarity function used for Skeleton Tree feature-observations is then computed as: simtree (T1, T2 ) = 1 − dt (T1, T2 ) 4 (3) An exact tree-edit-distance computation [11] has quartictime complexity in the size of the trees being compared. We instead use a fast under-approximation of edit distance [12] that gives our similarity function quadratic-time complexity overall. Note that we also use a further rough approximation based on just the size of the trees, if one of the two trees being compared is at least twice as large as the other. We found that using these approximations as opposed to the exact tree-editdistance based similarity made no discernible difference in the quality of the final search results obtained, but made a big difference in performance: more than 6× faster in our tests. Decorated Skeleton Tree: This feature-class is similar to the Skeleton Tree, except that instead of retaining just the loop and conditional structure in the feature-observations, most operations (e.g., +, -, and <) are also retained from the AST. We discard some common operations, such as assignment (=) and address-of (&), because they cause excessive bloat. The intuition behind including this feature-class is that similar functions use similar operations in structurally similar locations. Fig. 4b shows the Decorated Skeleton Tree featureobservation for the example code in Fig. 2. The similarity function used is simtree from Eq. (3). Weighted NL Terms: The feature-observations for this feature-class consist of various natural-language (NL) terms in source code, such as function name, comments, local variable names, and parameter names of a function. Such NL terms, after extraction, are subjected to a series of standard NL preprocessing steps, such as splitting words with under_scores or CamelCase, stemming, lemmatization, and removing singlecharacter strings and stop-words. Stop-word removal discards both typical English stop words such as “the”, “and”, and “is” [13], as well as stop words specialized for code, such as “fixme”, “todo”, and “xxx”. Additionally, we use a greedy algorithm [14] for splitting terms into multiple words based on dictionary lookup. This splitting is to handle the case where programmers choose identifiers that combine multiple words without under_scores or CamelCase. After NL pre-processing, we compute a term frequencyinverse document frequency (TF-IDF) score for each NL term. We consider each function as a document, and compute the TF-IDF per C/C++ project. We give function-name terms an inflated score (5× more than other terms) because these often provide significant information about functions’ purposes. The intuition behind including this feature-class is that similar functions tend to have similar natural-language vocabulary. The feature-observation for the example in Fig. 2 is {“bin”: 0.65, “search”: 0.65, “high”: 0.13, “low”: 0.13, “found”: 0.13, “mid”: 0.13, “match”: 0.13}. The similarity function for two observations of Weighted NL Terms uses cosine similarity: Ín i=1 Ai Bi simnl (A, B) = q q Ín 2 Ín B 2 i=1 Ai i=1 i A B C D A B C D A 0 0 0 0 B 1 0 0 0 C 0 1 0 0 D 0 1 0 0 Fig. 5. An example 4-graph and its corresponding adjacency matrix. Serializing the adjacency matrix entries yields binary digits “0100 0011 0000 0000”, or 17,152 in decimal. Node ordering in the adjacency matrix is the traversal order. K-Subgraphs of CFG: We implement multiple featureclasses based on k-sized subgraphs of the control flow graph (CFG) of a function. Given the CFG of a function, we begin either a breadth-first-search (BFS) traversal or a depth-first search (DFS) traversal at a node until k nodes are traversed; a subgraph of the CFG involving these k nodes is extracted. If fewer than k nodes are reachable from a node (including itself), then such a sub-graph is thrown away. We repeat this process for every node in the CFG, extracting at most n subgraphs of size k, where n is the size of the CFG. We represent a graph of size k as a k 2 -bit integer, which is a 1-D representation of a 2-D adjacency-matrix representation of the graph, obtained by concatenating each of the matrix rows in order. Thus, from each function’s CFG, we extract a multiset of k-graph shapes. Fig. 5 shows an example of converting a 4-graph into a 16-bit integer in this manner. We implement the following four feature-classes based on the value of k and the traversal strategy chosen: 3 Graph CFG BFS: k = 3, traversal strategy is BFS. 4 Graph CFG BFS: k = 4, traversal strategy is BFS. 3 Graph CFG DFS: k = 3, traversal strategy is DFS. 4 Graph CFG DFS: k = 4, traversal strategy is DFS. For the example in Fig. 2, the feature-observation extracted for the feature-class “4 Graph CFG BFS” is the multiset {134, 134, 134, 194, 194, 194, 194, 194, 2114, 2114, 2114}. The intuition behind including these feature-classes is that similar functions tend to have similar control-flow structures [15]. The similarity function used for these feature-class is based on the generalized Jaccard index between two multisets O1 and O2 : Í min(O1i, O2i ) simGen-Jacc(O1,O2 ) = Í i (4) i max(O1i, O2i ) Here, i iterates over all the unique elements in O1 ∪ O2 , and O1i is the number of times i appeared in the multiset O1 . Calls to Library Functions: We implement three featureclasses that extract calls to various kinds of library functions: Modeled Library Calls: CodeSonar models a large range of library functions for performing static analysis on C/C++ code. For this feature-class, calls made to any of these modeled library functions are extracted. Unmodeled Library Calls: Calls made to any unmodeled library functions are extracted for this feature-class—that is, calls to a function not modeled by CodeSonar, and whose definition is not available in the source code. Here n is the total number of words in the universe, A and B are vectors with TF-IDF scores, and the i th index Ai is the TF-IDF value for the i th word. 5 User-Defined Library Calls: For this feature-class, calls to B. Dynamic Feature-Class Selection functions whose definitions are available in a directory difCombining feature-classes can be beneficial for code search, ferent from the caller function are extracted. We use such however, the feature-classes that are useful for performing a functions as a heuristic for identifying user-defined libraries. code search may vary from one query to another. For example, The intuition behind including the above three featureconsider a query function containing of just straight-line code. classes is that similar code tends to call the same library A significant number of functions in our code-database are functions. For each of these three feature-classes, the featuredevoid of loops and conditionals,2 and all such functions look values are sets of library functions called. A library function identical to the query function with respect to the Skeleton is represented as tuple: it includes the name of the function Tree feature-class. Thus, performing a code search with this together with the file name containing the function’s declaraquery by including the Skeleton Tree feature-class can lead to tion. For example, if a function calls strcpy and strncpy, lower-quality results. On the other hand, if a query function then the feature-observation corresponding to Modeled Library has an unusual loop and conditional structure that is idiomatic Calls for that function is {(strcpy, string.h), (strncpy, to the computation being performed, then the Skeleton Tree string.h)}. feature-class would be useful in code search: other instances Type Signature: For this feature-class, the featureof the same distinctive structure from the code database would observations consist of the type signature of the function: i.e., have high similarity scores to the query function. the argument types and the return type of a function. Together, Thus, it is useful to select feature-classes automatically on a the argument types and the return type form a multiset of per-query basis for code search. This configuration of Source types. For the example code in Fig. 2, the feature-observation Forager is called dyn-select. Intuitively, a feature-class for a corresponding to Type Signatures is {int, int, int, int}. given query is selected for code search if the corresponding Type signatures define a function’s interface for interaction feature-observation is sufficiently discriminatory/unique with with the rest of the code. Similar code tends to have similar respect to the overall feature-observation distribution for that interfaces, and therefore type signatures could help with code feature-class. search. To prepare for the dynamic feature-class selection on a perThe generalized Jaccard index (Eq. (4)) is used as the query basis, we take following steps offline: similarity function for this feature-class. • From the code database, we retrieve a random sample S Local Types: For this feature-class, the featureof feature-vectors. Random sampling gives an inexpensive observations consist of the set of types of all the local variables. estimate of feature-observation distributions across the entire The intuition behind using local variable types in code search code database. is that similar code creates and operates on variables of similar types. For the example code in Fig. 2, the Local Types feature- • We calculate a similarity threshold for each feature-class c by (1) computing pairwise similarity scores on the featureobservation is {int}. observations for c in S and (2) taking the sum of means Constants: We implement two feature-classes that extract and standard deviations of the similarity scores. Two featureconstants from a function: observations for c are considered similar if their similarity Numeric Literals: This feature-class is described in §II-A. score is above the similarity threshold for c. String Literals: For this feature-class, a feature-observation Online, when a query is posed, we take the following steps is the set of all the literal strings used in a function. The intuition behind using sets of constants in code search is for each feature-class c (which can be performed in parallel): • We compare the query’s feature-observation for c with all that similar code typically uses similar constants. other feature-observations for c in sample S (of size nsamp ), Comments: For this feature-class, the featureand count the number of similar feature-observations nsim-c. observations consist of the comments associated with a function. The comments are represented as a set of words. • We select the feature-class c for code search if it is not too < tuniq . Here tuniq is a threshold common, that is, if nnsim-c The intuition behind using comments in code search is that samp the comments in similar pieces of code are likely to use that indicates a feature-observation is sufficiently unique in a similar vocabulary. For the example code in Fig. 2, the the sample. For example, tuniq = 0.15 indicates that any Comments feature-observation is {“found”, “match”, “no”}. feature-observation that is similar to less than 15% of the Combining Feature-Classes: Using several featuresample feature-observations is considered distinctive enough classes in combination allows Source Forager to obtain good to warrant inclusion. code-search results in a fairly robust manner by using different Each feature-class is assigned a weight of exactly 1.0 or dimensions of the code. For example, consider the binary- exactly 0.0, based on whether the feature-class is selected search implementation in Fig. 2. We see that variables named in the above process. These weights are used for combining mid, low, high are used; that there are two conditionals feature-class similarities for code search (Eq. (2)), and the knested inside a single loop; and that an integer division and most-similar-function search is carried out between the query integer less-than-or-equal-to operation is performed. When put together, these observations are hallmarks of a binary-search 2We did a brief study of feature-observation distributions for the Skeleton Tree feature-class over our corpus, which revealed this data point. implementation. 6 function and the functions in the code database as described in §II-B, to obtain the k functions most similar to the query. TABLE II Task categories used for code-search queries in algo-qs. “#Similar” gives the number of similar functions that were manually found for a given task category. “Partial” reports how many function pairs MOSS considered to be potential clones. “Significant” reports the % of function pairs with at least 50% code overlap. C. SVM-Guided Feature-Class Weight Generation Note that dyn-select does not need any additional knowledge about the query. However, if we know ahead of time that a query belongs to a specific domain, and we have ground-truth information available regarding what constitutes similar code in that domain, then we can use supervised-learning techniques to learn good feature-class weights (for Eq. (2)) for that domain ahead of time, and use these weights for code search with all future queries in that domain. Given a particular ground-truth data set with labeled similar code, we generate fine-tuned weights by training a binaryclassification support vector machine (SVM). We do not train using raw code text, or even raw sets of feature-observations. Because we use the SVM training process to generate relative weights for feature-class similarity scores in Eq. (2), we train the SVM on these similarity scores directly. The similarity scores for all feature-classes between two functions are assembled into a similarity vector. The SVM is then trained on examples of similarity vectors for both similar and dissimilar functions, each labeled accordingly. This technique allows us to optimize ahead of time how these feature-classes are relatively weighted in a code search, by using the same similarity functions that are employed in code search of a query. Our SVM uses a linear classifier, which allows a convenient interpretation of internal weights [16]. The final pre-processing step is to extract these internal weights and normalize them relative to the sum of their magnitudes, truncating negative weights. These normalized weights are then used directly as feature-class weights in Eq. (2). §IV-B provides more details about the corpus and training process. Of course, it is not obvious that weights obtained by training for classification purposes are useful in ranking results for code-search queries. §IV-C measures the effectiveness of this strategy in practice. MOSS Detected Task Category Binary Search Edit Distance Insertion Sort Knapsack Modular Exponentiation Non Recursive Depth First Search Red Black Tree Left Rotate #Similar Partial Significant 5 5 5 5 6 5 6 20% 40% 30% 10% 0% 0% 40% 10% 0% 30% 0% 0% 0% 13% A code-search task involves searching for relevant documents from a group of documents that include both relevant and non-relevant documents. (In the case of Source Forager, “documents” are C/C++ functions.) Non-relevant documents are also known as distractors, which leads naturally to the following question: RQ4 How much does Source Forager’s code-search performance degrade as we increase the number of distractors in the code base being searched? B. Experimental Setup and Methodology Source Forager uses CodeSonar, an industrial-strength C/C++ static-analysis engine, to analyze C/C++ corpora and implement feature extractors. CodeSonar handles real-world C/C++ projects with tens of millions of lines of code. CodeSonar also exposes a wealth of information about a program through well-defined APIs. Source Forager’s feature extractors are implemented as CodeSonar plugins that use these APIs. Consequently, Source Forager inherits CodeSonar’s requirement that programs must be compilable to be analyzable. Code-Search Tasks: Our experiments assess Source Forager’s performance under various configurations. Code-search tasks are set up as follows. For each query function, there is a set of known relevant functions that are similar to the query. The relevant functions are treated as ground truth. The relevant functions are then mixed with many non-relevant functions as distractors, and together they form the code database used in the experiment. Source Forager then searches the code database for similar functions. We compute informationretrieval statistics based on the ranking of the known-relevant functions in the returned results. Queries: We use two query ground-truth sets for the code-search tasks, representing two domains. One, called algo-qs, represents “algorithmic” code queries. For algo-qs, we created seven tasks, outlined in Table II, and manually curated a total of thirty-eight functions that each accomplish one of the seven tasks. The functions were mostly obtained from GitHub, and were written by a variety of programmers, none of whom are authors of this paper. The functions that accomplish a specific task have been manually vetted to be IV. Experimental Evaluation This section outlines the research questions we seek to answer through experiments (§IV-A); describes the setup and methodology used in the experiments (§IV-B); and presents the results of the experiments (§IV-C). A. Research Questions Our experiments were designed to answer the following research questions: RQ1 How do the individual feature-classes described in §III perform in code-search tasks relative to each other? RQ2 Does combining feature-classes using per-query dynamic feature-class selection (§III-B) improve Source Forager’s performance? RQ3 Does combining feature-classes using supervised learning (§III-C) further improve Source Forager’s performance, when the query domain is known? 7 similar to each other. We thus have a total of thirty-eight base queries. We use these sets of real-world functions as queries (and the desired search results), and consider them to be an appropriate proxy for the code-search queries performed (and search results expected) by users in the algorithm domain. We have made the labeled queries available for inspection.3 To make sure that the similar functions we found were not all clones of each other, we ran them through the MOSS software-plagiarism detector [17]. Given a group of programs, MOSS reports program pairs that may be clones, along with an overlap percentage. Table II reports MOSS’s findings, run using default settings. In this table, partial overlap represents any pair that MOSS reports as possible clones, while significant overlap counts only possible clones with at least 50% overlap. Observe that many function pairs marked manually as being similar are not just MOSS-detectable clones of each other. Thus, recognizing similar function pairs in this corpus is a nontrivial challenge. The second query ground-truth set we use is called libc-qs, and represents code queries from systems programming. We looked at three implementations of the standard C library: musl libc [18], diet libc [19], and uClibc [20]. From these we define 88 function categories corresponding to 88 functions that all three implementations provide. We assume that within the same function category, the three libc implementations are “similar.” For this domain, we have 88×3 queries. For example, musl libc’s sprintf is labeled to be similar to diet libc’s sprintf and uClibc’s sprintf, and dissimilar to everything else. Distractor Functions: The distractor functions have been taken from the openly available MUSE corpus [21], and mainly consist of code from Fedora source packages (SRPMs). Our feature extractors currently require compilable code, which Fedora SRPMs provide. Due to the large size of the distractorfunction corpus (over 200,000), we have not manually vetted all of the distractor functions to be sure that they are irrelevant to the queries issued. It is possible that some distractor functions are indeed relevant to some queries, so our retrieval statistics are under-approximations. With the exception of the experiments reported in Fig. 7, all experiments use 10,000 distractors. Retrieval Statistics: We compute Mean Average Precision (MAP) as the retrieval statistic, as is common in information retrieval. MAP is typically used to measure the quality of ranked retrieval results, because MAP takes into account the rank of the relevant documents in the retrieved results. MAP provides a measure of quality across all recall levels. MAP is the mean of the average precision computed for each query. The average precision (AP) for each query is Í given by nk=1 P(k) · r(k) /R, where n is the total number of documents searched; R is the number of documents marked relevant to the query; P(k) is the precision when k documents are requested; and r(k) is 1 when the k th retrieved document is relevant, and 0 otherwise. That is, AP is the average precision at all the points when a new relevant document is retrieved in a ranked result list. The best MAP score that can be achieved is 1.0, when for each query, the R relevant documents appear as the top R search results. SVM-Guided Weights: We applied the techniques discussed in §III-C on algo-qs and libc-qs to provide labeled data-sets on which to train an SVM. Each instance in our training set is generated by comparing two functions a and b, yielding a single similarity vector that consists of similarity scores for each feature-class. The binary classification for each training instance is 1.0 if a and b are implementations of the same function, 0.0 otherwise. We use LIBLINEAR [22] to train the SVM to classify these function comparisons; this process takes roughly twenty milliseconds. Using this technique, we are able to achieve over 98% accuracy under ten-fold cross-validation. Once the SVM is trained, we extract and normalize its internal weights for use in code search. For the svm-weights configuration described below, within each domain, the dataset is divided into multiple folds of training-set and test-set pairs. The weights extracted from the training set are used to obtain MAP scores on the test set. That is, weights are trained on a subset of a given domain (algo-qs or libc-qs) and tested using queries from a different subset of the same domain. For the cross-weights configuration described below, algo-qs is used to train weights for queries from libc-qs, and vice-versa. Source Forager Configurations: Our experiments run Source Forager under many configurations. Each configuration is defined by the weight wc assigned to each of the featureclasses c given in Table I. These weights are used in Eq. (2) for performing the code search. solo-c: For each query, the weight wc corresponding to feature-class c is 1.0. Weights corresponding to all other feature-classes are set to 0.0. equal-all: For each query, for all feature-classes c, wc = 1.0, giving equal importance to all feature-classes for all queries. dyn-select: For each query, a subset of feature-classes are selected and given equal weights, as described in §III-B. The dynamic selection of feature-classes adds a small run-time overhead to each query.4 rand-select: For each query, a new random configuration is used as follows: a random subset of the feature-classes is selected, and the selected feature-classes are given equal weights. Repeat this process 10 times with different random selections, and report mean results over these 10 trials. svm-weights: For each query, use weights learned for the domain that the query belongs to, as described in §III-C and above. cross-weights: For each query, use weights learned for the domain that the query does not belong to. 4In our naive Python implementation, dyn-select adds an average run-time overhead of 2.1 seconds per query for dynamic selection of feature-classes. Currently, the selection decision on each feature-class is done sequentially, instead of in parallel as suggested in §III-B. 3Available at the URL: http://tinyurl.com/source-forager-algo-benchmarks. 8 Note that, unlike the other configurations, the svm-weights and cross-weights configurations permit weights to give different (non-zero) importance levels to different feature-classes. RQ3 Finding: When the domain of a query is known, and training data is available, combining multiple featureclasses using weights derived from supervised learning (§III-C) is the most effective strategy for code search. C. Results and Discussion The cross-weights (0.74, 0.85) configuration tests whether the weights learned from one domain are useful in a different The left side of Fig. 6 shows how each individual featuredomain. The rightmost two bars in Fig. 6 show that it is class performs on the code-search tasks in isolation. This hard to derive a single set of relative feature-class weights that experiment addresses RQ1. The solo feature-class Weighted work well for queries in both domains. Thus, in the absence of NL Terms (0.70, 0.86)5 performs the best individually on both domain information about the query, dyn-select is preferred. algo-qs and libc-qs. Thus: Fig. 7 shows how Source Forager’s result quality scales with increasing distractor-set sizes. This experiment addresses RQ1 Finding: If we were to drive Source Forager using RQ4. Source Forager is used in the dyn-select and svm-weights only one feature-class, Weighted NL Terms is the best configurations for this experiment. As one would expect, MAP option. However, Fig. 6 shows that the performance of the scores decline as distractors proliferate. However, consider different feature-classes varies considerably depending on that relevant sets contain just 2 to 6 items competing against the query ground-truth set. This variance suggests that distractor sets that are up to five orders of magnitude larger. different feature-classes are important for different kinds of queries. RQ4 Finding: Resilient MAP scores indicate that Source Forager returns high-quality results even when distractors outnumber relevant items by several orders of magnitude. RQ2 asks whether multiple feature-classes can be usefully combined, and whether dyn-select is a good way to do such a combination. A straight-forward manner in which featureclasses can be combined is the equal-all configuration, which represents a baseline to compare against other configurations. The dyn-select configuration selects different subsets of the feature-classes on a per-query basis (§III-B). As a sanity check for the selections performed by dyn-select, we also compare it with the rand-select configuration, which randomly selects feature-class subsets for every query. The right side of Fig. 6 shows that dyn-select (0.84, 0.89) performs better on both algo-qs and libc-qs when compared to equal-all (0.67, 0.73) and rand-select (0.57, 0.63). dyn-select also outperforms each of the solo configurations from the left side of Fig. 6. D. Threats to Validity The issue of whether evaluation benchmarks are appropriate is a potential threat to the validity of any information retrieval system. We mitigate this threat for Source Forager in several ways. First, we use benchmark queries from two different domains, algo-qs and libc-qs. Second, we use the MOSS plagiarism detector to show that our manually labeled set of relevant functions in algo-qs are not trivial clones of each other. Third, we draw the algo-qs and libc-qs data sets from real-world code written by arbitrary programmers, not artificial programs written by us. Feature-classes can be combined in various ways to perform code searches. We have explored part of the vast space of all such combinations, and our results speak only to those we have tried. We find that the MAP scores of the configuration dynselect on both algo-qs and libc-qs are good. We designed the experiments with equal-all and rand-select configurations to test whether the selections made by dyn-select are indeed necessary and useful, and find that they are. RQ2 Finding: In the absence of any additional information about the query, combining multiple feature-classes and dynamically selecting feature-classes on a per-query basis (§III-B) is the most effective strategy for code search. RQ3 addresses the scenario where the domain of a query is known, and additional information is available regarding that domain (as described in §III-C). The svm-weights configuration tests Source Forager under this scenario. Pre-learning the relative importance of feature-classes for a given domain (in the form of weights wc for each feature-class) also makes code search more efficient by eliminating any run-time overhead in feature-class selection. The right side of Fig. 6 shows that svmweights (0.86, 0.95) outperforms all other configurations. V. Related Work Code-search engines: Several popular text-based codesearch tools “grep” over tokenized source code: GitHub, SearchCode, Open HUB, etc. While these tools are useful, they fall short in many use cases, as they do not exploit the rich semantics of code. For example, the top search results for the term “dfs” on C code projects in GitHub yields function declarations, macro names, and #include directives that mention “dfs”, but that are not actually useful. The Sourcerer code-search engine [2] combines text-based search techniques with information about relations among programming “entities” like packages, classes, methods, and fields. 5Pairs of numbers following a configuration in this section indicates the MAP scores of that configuration on algo-qs and libc-qs, respectively. 9 0.95 0.85 0.74 0.40 0.09 0.00 0.00 0.03 0.09 0.01 0.04 0.00 0.07 User-Defined Library Calls 0.18 0.01 0.00 Unmodeled Library Calls 0.04 0.04 0.03 0.06 0.03 0.21 0.25 0.07 0.12 0.04 0.12 0.14 0.01 0.2 0.31 0.4 0.86 0.84 0.89 0.67 0.73 0.58 0.70 0.45 0.6 0.19 MAP score 0.8 algo-qs libc-qs 0.57 0.63 0.86 1 cross-weights svm-weights dyn-select rand-select equal-all Comments String Literals Numeric Literals Local Types Type Signatures Modeled Library Calls 4 Graph CFG DFS 3 Graph CFG DFS 4 Graph CFG BFS 3 Graph CFG BFS Weighted NL Terms Decorated Skeleton Tree Skeleton Tree Type–Operation Coupling 0 Fig. 6. Information retrieval performance with 10,000 distractors. The left side of the plot, from “Type–Operation Coupling” through “Comments”, uses the solo-c configuration with the given feature-class as the only non-zero-weighted feature. The right side of the plot, from “equal-all” through “cross-weights”, uses the various other Source Forager configurations that leverage multiple feature-classes simultaneously. Although no MAP score is exactly zero, several are below 0.005 and therefore round to “0.00” in the data-point labels above. 0.87 0.70 0.91 0.93 0.75 0.94 0.84 0.79 0.96 0.97 0.88 0.86 0.98 0.99 0.90 0.99 0.95 0.93 0.98 0.99 0.95 0.92 1.00 0.96 100 200 400 800 1,600 3,200 6,400 12,800 25,600 51,200 102,400 204,800 MAP score Strathcona [8] returns relevant Java code examples to developers learning to use complex object-oriented frameworks. It 1 uses several heuristics based on class-inheritance hierarchies, method calls, and type uses. Source Forager could also use the 0.8 applicable heuristics from Sourcerer and Strathcona as featureclasses, but additionally demonstrates how to search using more complex structures, such as decorated skeleton trees and 0.6 CFG-subgraphs. CodeGenie [7], [23] proposes test-driven code search, in which the user supplies a set of unit tests for the code compo0.4 nent they want to find. CodeGenie leverages Sourcerer [2] to svm-weights with libc-qs perform keyword-based search; test cases refine these results. dyn-select with libc-qs Source Forager could be used as a replacement for Sourcerer 0.2 svm-weights with algo-qs to perform similar code search in CodeGenie. dyn-select with algo-qs Stollee et al. [4], [24] perform code search based on logical 0 characterizations of programs’ I/O behaviors, obtained via symbolic execution. A query consists of concrete I/O pairs for the desired code fragment. While this approach precisely captures the semantics of the corpus elements, it does not imNumber of Distractor Functions mediately handle some common programming constructs, such as loops and global variables. It also restricts the size of the Fig. 7. Impact of the number of distractor functions on MAP scores using dynprogram elements in the corpus, because symbolic execution select and svm-weights for all algo-qs and libc-qs queries. The horizontal axis is on a log scale. of larger elements may lead to path explosion. Source Forager can easily be extended to use I/O pairs as an additional featureclass in scenarios where the above restrictions are acceptable. Sourcerer also uses fingerprints that capture some light-weight XSnippet [5] and ParseWeb [25] are specialized code-search structural information about the code, such as depth of loop engines: XSnippet looks specifically for code that instantiates nesting and presence or absence of certain language constructs. objects of given type in a given context, ParseWeb has a similar Queries in Sourcerer are text-based and are powered by Lucene focus on code sequences that instantiate objects. Codify [6] (http://lucene.apache.org), as opposed to the code-based search extracts and stores a large amount of metadata for each symbol by Source Forager. in a program, and provides a user interface for querying that 10 metadata. Codify aids in understanding and browsing code. The goal of Source Forager’s code search is different from the above, i.e., to find source code similar to a query. Code-clone detection: Source Forager’s code searches differ from the typical clone detection problem in that we are interested in finding code that has both semantic and syntactic similarity. Therefore, we use a range of feature-classes that span from syntactic to semantic. Source Forager’s notion of similarity does not neatly fall into any of the definitions of standard clone types 1–4 [26]. Similar-machine-code search: Finding similar machine code [15], [27]–[29] is useful in finding known vulnerabilities in third-party code for which source code is not available. The primary difference in code search at the source-level and machine-level is that machine code has poorer syntactic, semantic, and structural information available compared to source code. As a result, while there is some overlap between techniques, research on machine-code search is focused on tackling different problems, such as how to do similar-machinecode search across different CPU architectures, compiler optimizations, compilers, operating systems, etc. SVM-based code-classification: Rosenblum et al. [30] train SVMs with features extracted from source code in the attempt to classify programs by author. Source Forager builds on this idea by training an SVM with similarity scores derived from feature-observations, and then extracting internal weights from the trained SVM to strengthen the combined similarity function used for code search. References [1] C. Sadowski, K. T. Stollee, and S. G. Elbaum, “How developers search for code: a case study,” in Found. of Softw. Eng., 2015. [2] E. Linstead, S. K. Bajracharya, T. C. Ngo, P. Rigor, C. V. Lopes, and P. Baldi, “Sourcerer: Mining and searching internet-scale software repositories.” Data Mining and Knowledge Discovery, vol. 18, no. 2, pp. 300–336, Apr. 2009. [3] S. P. Reiss, “Semantics-based code search,” in Int. Conf. on Softw. Eng., 2009, pp. 243–253. [4] K. T. Stollee, S. G. Elbaum, and D. Dobos, “Solving the search for source code,” Trans. on Softw. Engineering and Methodology, vol. 23, no. 3, May 2014. [5] N. Sahavechaphan and K. T. Claypool, “XSnippet: mining for sample code,” in Conf. on Object-Oriented Prog. Systems, Languages, and Applications, 2006, pp. 413–430. [6] A. Begel, “Codifier: A programmer-centric search user interface,” in Workshop on Human-Computer Interaction and Inf. Retrieval, 2007. [7] O. Lemos, B. K. Bajracharya, J. Ossher, R. S. Morla, P. C. Masiero, P. Baldi, and C. V. Lopes, “Codegenie: using test-cases to search and reuse source code,” in Int. Conf. on Automated Softw. Eng., 2007. [8] R. Holmes and G. C. Murphy, “Using structural context to recommend source code examples,” in Int. Conf. on Softw. Eng., 2005. [9] D. Crockford, “Introducing JSON,” Apr. 2016. [Online]. Available: http://json.org/ 11 [10] C. Jermaine, “The pliny database,” Aug. 2016. [Online]. Available: http://cmj4.web.rice.edu/PlinyDBSlides.pdf [11] K. Zhang and D. Shasha, “Simple fast algorithms for the editing distance between trees and related problems,” SIAM J. Comput., vol. 18, no. 6, Dec. 1989. [12] S. Guha, H. Jagadish, N. Koudas, D. Srivastava, and T. Yu, “Approximate XML joins,” in Int. Conf. on Management of Data. ACM, 2002, pp. 287–298. [13] NLTK Project, “Stopwords corpus,” Mar. 2016. [Online]. Available: http://www.nltk.org/nltk_data [14] H. Feild, D. Binkley, and D. Lawrie, “An empirical comparison of techniques for extracting concept abbreviations from identifiers,” in Proc. IASTED Int. Conf. on Software Engineering and Applications (SEA). Citeseer, 2006. [15] W. M. Khoo, A. Mycroft, and R. Anderson, “Rendezvous: A search engine for binary code,” in Proceedings of the 10th Working Conference on Mining Software Repositories, 2013, pp. 329–338. [16] I. Guyon and A. Elisseeff, “An introduction to variable and feature selection,” J. Mach. Learn. Res., vol. 3, pp. 1157–1182, Mar. 2003. [17] S. Schleimer, D. S. Wilkerson, and A. Aiken, “Winnowing: Local algorithms for document fingerprinting,” in Int. Conf. on Management of Data, 2003, pp. 76–85. [18] Eta Labs, “musl libc,” Aug. 2016. [Online]. Available: https://www.musl-libc.org/ [19] diet libc contributors, “diet libc,” Aug. 2016. [Online]. Available: https://www.fefe.de/dietlibc/ [20] E. Andersen, “uClibc,” Aug. 2016. [Online]. Available: https://github.com/klee/klee-uclibc [21] Leidos Holdings, Inc., “MUSE corpus,” Apr. 2016. [Online]. Available: http://corpus.museprogram.org/ [22] R.-E. Fan, K.-W. Chang, C.-J. Hsieh, X.-R. Wang, and C.-J. 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This is a pre-print of the conference paper accepted at the IEEE Winter Conference on Applications of Computer Vision (WACV) 2018. Towards Robust Deep Neural Networks with BANG Andras Rozsa, Manuel Günther, and Terrance E. Boult Vision and Security Technology (VAST) Lab University of Colorado, Colorado Springs, USA arXiv:1612.00138v3 [cs.CV] 30 Jan 2018 {arozsa,mgunther,tboult}@vast.uccs.edu Abstract Machine learning models, including state-of-the-art deep neural networks, are vulnerable to small perturbations that cause unexpected classification errors. This unexpected lack of robustness raises fundamental questions about their generalization properties and poses a serious concern for practical deployments. As such perturbations can remain imperceptible – the formed adversarial examples demonstrate an inherent inconsistency between vulnerable machine learning models and human perception – some prior work casts this problem as a security issue. Despite the significance of the discovered instabilities and ensuing research, their cause is not well understood and no effective method has been developed to address the problem. In this paper, we present a novel theory to explain why this unpleasant phenomenon exists in deep neural networks. Based on that theory, we introduce a simple, efficient, and effective training approach, Batch Adjusted Network Gradients (BANG), which significantly improves the robustness of machine learning models. While the BANG technique does not rely on any form of data augmentation or the utilization of adversarial images for training, the resultant classifiers are more resistant to adversarial perturbations while maintaining or even enhancing the overall classification performance. (a) MNIST Samples and Their Distortions Yielding Misclassifications (b) CIFAR-10 Samples and Their Distortions Yielding Misclassifications Figure 1: I MPROVING ROBUSTNESS VIA BANG. This figure demonstrates the enhanced robustness against perturbations generated via the non-gradient-based hot/cold adversarial generation method on MNIST digits and CIFAR-10 samples displayed in top rows of (a) and (b). Underneath the raw test images, we show their distorted versions formed by the smallest perturbations that change the correctly classified class labels of the test samples. The second rows of (a) and (b) present perturbations that we obtained on regularly trained learning models, while the last rows show examples that we generated on networks trained via our Batch Adjusted Network Gradients (BANG) approach. As indicated by most of the perturbations being highly perceptible, the learning models trained with BANG have become more robust to adversarial perturbations. 1. Introduction Machine learning is broadly used in various real-world vision applications and recent advances in deep learning have made deep neural networks the most powerful learning models that can be successfully applied to different vision problems [27, 25, 7, 28, 20, 14, 18, 17, 29]. The recent performance gain is mainly the result of improvements in two fields, namely, building more powerful learning models [25, 7] and designing better strategies to avoid overfitting [24]. These advancements are then leveraged by the use of larger datasets and massive GPU-enhanced computing. Although deep neural networks (DNNs) achieve state- of-the-art performance in a wide range of tasks, the generalization properties of these learning models were questioned by Szegedy et al. [26] when the existence of adversarial examples was revealed. DNNs are capable of learning high-level feature embeddings that enable them to be successfully adapted to different problems. They were generally considered to generalize well and, hence, expected to be robust to moderate distortions to their inputs. Surprisingly, adversarial examples formed by applying imperceptible perturbations to otherwise correctly recognized inputs can lead machine learning models – including state-of-the- art DNNs – to misclassify those samples, often with high confidence. This highly unexpected and intriguing property of machine learning models highlights a fundamental problem that researchers have been trying to solve. To explain why adversarial examples exist, several controversial explanations were proposed. As hypothesized in [4, 2], adversarial instability exists due to DNNs acting as high-dimensional linear classifiers that allow even imperceptibly small, well-aligned perturbations applied to inputs to spread among higher dimensions and radically change the outputs. This belief was challenged in [19], where – by analyzing and experimenting with DNNs trained to recognize objects in more unconstrained conditions – it was demonstrated that those classifiers are only locally linear to changes on the recognized object, otherwise DNNs act nonlinearly. After performing various experiments, Gu et al. [6] concluded that adversarial instability is rather related to “intrinsic deficiencies in the training procedure and objective function than to model topology.” The problem addressed in this paper is not only about preventing attacks via adversarial examples, the focus is on the overall robustness and generalizability of DNNs. This fundamental problem of deep learning has recently received increasing attention by researchers [3, 8, 21]. Considering state-of-the-art learning models applied to computer vision tasks, the classification of many incorrectly or uncertainly recognized inputs can be corrected and improved by small perturbations [30, 22], so this is a naturally occurring problem for learning-based vision systems. In this paper, we introduce our theory on the instability of machine learning models and the existence of adversarial examples: evolutionary stalling. During training, network weights are adjusted using the gradient of loss, evolving to eventually classify examples correctly. Ideally, we prefer broad flat regions around samples to achieve good generalization [11] and adversarial robustness [2]. However, after a training sample is correctly classified, its contribution to the loss and, thus, on forming the weight updates is reduced. As the evolution of the local decision surface stalls, the correctly classified samples cannot further flatten and extend their surroundings to improve generalization. Therefore, as the contributions of those correctly classified training samples to boundary adjustments are highly decreased compared to other batch elements, samples can end up being stuck close to decision boundaries and, hence, susceptible to small perturbations flipping their classifications. To mitigate evolutionary stalling, we propose our Batch Adjusted Network Gradients (BANG) training algorithm. We experimentally evaluate robustness using a combination of gradient- and non-gradient-based adversarial perturbations, and random distortions. The paper explores the impact of BANG parameters and architectural variations, such as Dropout [24], on instability and adversarial robust- ness. In conclusion, we validate our theory by experimentally demonstrating that BANG significantly improves the robustness of deep neural networks optimized on two small datasets while the trained learning models maintain or even improve their overall classification performance. 2. Related Work Deep neural networks (DNNs) achieve high performance on various tasks as they are able to learn non-local generalization priors from training data. Counter-intuitively, Szegedy et al. [26] showed that machine learning models can misclassify samples that are formed by slightly perturbing correctly recognized inputs. These so-called adversarial examples are indistinguishable from their originating counterparts to human observers, and their unexpected existence itself presents a problem. The authors introduced the first technique that is capable of reliably finding adversarial perturbations and claimed that some adversarial examples generalize across different learning models. A computationally cheaper adversarial example generation algorithm, the Fast Gradient Sign (FGS) method, was presented by Goodfellow et al. [4]. While this approach also uses the inner state of DNNs, it is more efficient as FGS requires the gradient of loss to be calculated only once. The authors demonstrated that by using adversarial examples generated with FGS implicitly in an enhanced objective function, both accuracy and robustness of the trained classifiers can be improved. In their paper focusing on adversarial machine learning, Kurakin et al. [13] proposed new algorithms extending the FGS method to target a specific class and to calculate and apply gradients iteratively instead of a single gradient calculation via FGS. The authors compared the effect of different types of adversarial examples used for implicit adversarial training and found that the results vary based upon the type of the applied adversarial examples. Rozsa et al. [23] introduced the non-gradient-based hot/cold approach, which is capable of efficiently producing multiple adversarial examples for each input. They demonstrated that using samples explicitly with higher magnitudes of adversarial perturbations than the sufficient minimal can outperform regular adversarial training. The authors also presented a new metric – the Perceptual Adversarial Similarity Score (PASS) – to better measure the distinguishability of original and adversarial image pairs in terms of human perception. As the commonly used L2 or L∞ norms are very sensitive to small geometric distortions that can remain unnoticeable to us, PASS is more applicable to quantify similarity and the quality of adversarial examples. Although adversarial training, both implicit and explicit, was demonstrated to decrease the instability of learning models, forming those examples is still computationally expensive, which limits the application of such techniques. Furthermore, considering the various adversarial generation techniques, utilizing certain types of those samples might not lead to improved robustness to adversarial examples of other techniques. Alternatively, Zheng et al. [30] proposed their stability training as a lightweight and still effective method to stabilize DNNs against naturally occurring distortions in the visual input. The introduced training procedure uses an additional stability objective that makes DNNs learn weights that minimize the prediction difference of original and perturbed images. In order to obtain general robustness and not rely on any class of perturbations, the authors applied Gaussian noise to distort the training images. Gu et al. [6] conducted experiments with different network topologies, pre-processing, and training procedures to improve the robustness of DNNs. The authors proposed the Deep Contractive Network (DCN), which imposes a layerwise contractive penalty in a feed-forward DNN. The formulated penalty aims to minimize output variances with respect to perturbations in inputs, and enable the network to explicitly learn flat, invariant regions around the training data. Based on positive initial results, they concluded that adversarial instability is rather the result of the intrinsic deficiencies in the training procedure and objective function than of model topologies. Luo et al. [19] proposed a foveation-based technique that selects and uses only a sub-region of the image during classification. As the authors demonstrated, the negative effect of foveated perturbations to the classification scores can be significantly reduced compared to entire perturbations. Graese et al. [5] showed that transformations of the normal image acquisition process can also negate the effect of the carefully crafted adversarial perturbations. While these preprocessing techniques can alleviate the problem posed by adversarial images, they do not solve the inherent instability of DNNs. In other words, these methods treat the symptoms and not the disease. In summary, a wide variety of more or less efficient approaches were proposed in the literature that all aim at improving the robustness and generalization properties of DNNs, but none of those proved to be effective enough. 3. Approach In this section, we first briefly describe our intuition about why the unexpected adversarial instability exists in machine learning models. Afterwards, we present our simple and straightforward modification in the training procedure that aims to optimize weights in a way that the resulting DNNs become more robust to distortions of their inputs. 3.1. Intuition During training, some inputs in the batch are correctly and others are incorrectly classified. In general, the calculated loss and, thus, the gradient of loss for the misclassified ones are larger than for the correctly classified inputs of the same batch. Therefore, in each training iteration most of the weight updates go into learning those inputs that are badly predicted. On the other hand, the correctly classified samples do not have a significant impact on advancing decision boundaries and can remain in the positions close to what they obtained when becoming correctly classified. Due to this evolutionary stalling, samples with low gradients cannot form a flatter, more invariant region around themselves. Consequently, samples of those regions remain more susceptible to adversarial perturbations – even a small perturbation can push them back into an incorrect class. By increasing the contribution of the correctly classified examples in the batch on the weight updates, and forcing them to continue improving decision boundaries, it is reasonable to think that we can flatten the decision space around those training samples and train more robust DNNs. 3.2. Implementation The core concept of our Batch Adjusted Network Gradients (BANG) approach is a variation of batch normalization [9]. However, rather than trying to balance the inputs of the layers, we seek to ensure that the contributions on the weight updates are more balanced among batch elements by scaling their gradients. Let us dive into the details and introduce our notations we use to formulate BANG. In short, we scale the gradients of batch elements that will be used to compute the weight updates in each training iteration. Let us consider a network fw with weights w in a layered structure having layers y (l) where l ∈ [1, L], with their respective weights w(l) : fw (xi ) = y (L) y (L−1) . . . y (1) (xi ) . . . . (1) For a given input xi , the partial derivatives of the loss E(fw , xi ) with respect to the output of layer y (l) are: (l) κi = κ(l) (xi ) = ∂Ei (l) . (2) ∂yi For simplicity, we leave out the structure of the weights w(l) in layers and the structure of the layer outputs which can be either one-dimensional for fully connected layers or threedimensional for convolutional layers. With BANG, our goal is to balance gradients in the batch by scaling up those that have lower magnitudes. In order to do so, we determine the highest gradient for the batch having N inputs xi , i ∈ {1, . . . , N } at given layer y (l) in terms of L2 norm. We use that as the basis for balancing the magnitudes of gradients in the batch. Weight updates are calculated after scaling each derivative κi in the batch with the element-wise learning rate:  ρi(l) (l) max kκ 0 k i  i0 ∈[1,N ]  (l) ηi =  (3)  (l) kκi k where:    (l) ρi = (l) 1 − (l) kκi k max 0 i ∈[1,N ]  (l)  kκi0 k . (4) As a key parameter for our approach, (l) specifies the degree of gradient balancing among batch elements. While the (l) exponent ρi might appear a little complex and ambiguous, its sole purpose is to scale up gradients with small magnitudes more than others having larger L2 norms. Assuming that the regular backward pass combines the gradients of the batch elements by calculating: ∇fw(l) N N 1 X (l) ∂y (l) 1 X ∂Ei = κ = N i=1 ∂w(l) N i=1 i ∂w(l) (5) which is normally scaled with the learning rate and then used to update weights (after combining with the previous weight update scaled with momentum), BANG produces: ∇fw(l) = β (l) N 1 X (l) ∂Ei η , N i=1 i ∂w(l) (6) where β (l) is the second (set of) parameter(s) of our approach used for scaling. In general, β (l) acts as a local learning rate that can play a more important role in future work. Throughout our experiments, we keep BANG parameters fixed for all layers: (l) = and β (l) = β (which will actually just modify the original learning rate η). Note that although our approach changes the actual calculation of weight updates for the layers, there is no impact on the backpropagation of the original gradient down the network. Finally, we implemented BANG by applying small modifications to the regular training procedure with negligible computational overhead. 4. Experiments To evaluate our approach, we conducted experiments on the slightly modified versions of LeNet [16] and “CIFAR10 quick” models distributed with Caffe [10]. Namely, after running preliminary experiments with BANG, we added a Dropout layer [24] to both model architectures that serves multiple purposes. We observed that BANG tends to cause overfitting on the trained LeNet networks, and the resultant models made very confident classifications – even when they misclassified the test images. While the additional Dropout layer alleviates both problems, the adjusted network architectures also result in improved classification performances with both regular and BANG training. After obtaining learning models with regular and BANG training, we assess and compare the robustness of those classifiers in two ways. It is important to note that we do not select the best training models based on their performance on the validation set for these evaluations, but we simply use the models obtained at the last training iteration. As our primary goal is to measure the evolving robustness, we believe that this decision leads to a fairer comparison, however, the classification performance of the selected models are not optimal. Finally, we would like to mention that we conducted experiments to discover the effectiveness of BANG used for fine-tuning regularly trained models, and found that the robustness of the resultant networks are not even comparable to those that we trained from scratch. First, we evaluate the adversarial vulnerability against two adversarial example generation methods: the gradientbased Fast Gradient Sign (FGS) method [4] and the nongradient-based hot/cold approach [23]. Although the latter is capable of forming multiple adversarial perturbations for each input, we only target the most similar class with the hot/cold approach, referred to as HC1. We aim to form adversarial perturbations for every correctly classified image from the MNIST [15] or CIFAR10 [12] test set, respectively. We consider an adversarial example generation attempt successful, if the direction specified by either FGS or HC1 leads to a misclassification, where the only constraint is that the discrete pixel values are in [0, 255] range. Of course, this limitation means that the formed perturbations may or may not be adversarial in nature as they can be highly perceptible to human observers. We compare the adversarial robustness of classifiers by collecting measures to quantify the quality of the produced adversarial examples. For this purpose, we calculate the Perceptual Adversarial Similarity Score (PASS) [23] of original and adversarial image pairs, and we also determine the L∞ norms of adversarial perturbations. Although the L∞ norm is not a good metric to quantify adversarial quality in terms of human perception, it can demonstrate how far the actual perturbed image is from the original sample. Second, we quantify how the robustness of the learning models evolve during training by applying a more general approach. For a given pair of classifiers where one was regularly trained while the other was obtained by BANG training, we add a certain level of random noise to 100 test images from each class that are correctly classified by both networks at all tested stages and compute the proportion of perturbed images that are classified differently than the originating one. While the previously described test assessing the adversarial vulnerability explores only two directions – specified by the FGS method and the HC1 approach – applying 1000 random distortions to each inspected image for every noise level gives us a more general evaluation. Although experimenting with random noise is more universal as it does not rely on any specific adversarial generation technique, small random perturbations that cause misclassifications are hard to find [23] and, hence, the collected Table 1: L E N ET T RAINING. This table highlights the difference between LeNet models obtained by using regular (R0-R1) and BANG training (B0-B5). Accuracy on the MNIST test set, the achieved success rates of FGS and HC1 adversarial example generation methods with PASS scores and L∞ norms of the produced examples on the MNIST test set are listed. ID β Accuracy FGS-Rate FGS-PASS FGS-L∞ HC1-Rate HC1-PASS HC1-L∞ R0 - - 99.16% 90.33% 0.4072 ± 0.1081 40.51 ± 15.72 99.53% 0.7535 ± 0.1143 122.60 ± 49.46 R1 - - 99.15% 91.41% 0.4072 ± 0.1065 40.70 ± 15.88 99.77% 0.7517 ± 0.1160 122.16 ± 49.07 B0 1.00 0.785 99.16% 3.51% 0.6806 ± 0.1457 8.34 ± 04.32 95.13% 0.5359 ± 0.2023 187.86 ± 63.66 B1 1.00 0.815 99.22% 1.68% 0.7638 ± 0.1367 5.52 ± 02.86 94.19% 0.4880 ± 0.2110 201.84 ± 62.51 B2 1.20 0.810 99.31% 2.13% 0.7579 ± 0.1452 5.57 ± 03.05 94.56% 0.5129 ± 0.2015 186.10 ± 63.77 B3 1.35 0.780 99.25% 3.86% 0.6763 ± 0.1471 8.28 ± 04.26 94.73% 0.5709 ± 0.2127 178.11 ± 65.76 B4 1.50 0.840 99.11% 1.52% 0.8220 ± 0.1310 4.19 ± 03.01 97.68% 0.4669 ± 0.1881 203.50 ± 58.97 B5 1.60 0.780 99.32% 4.45% 0.6771 ± 0.1487 8.20 ± 04.40 98.95% 0.6376 ± 0.1829 146.96 ± 61.97 (a) Accuracy (b) FGS Success Rate (c) HC1 PASS Figure 2: L E N ET M ODELS T RAINED WITH BANG. These plots summarize our results on LeNet models trained with BANG using combinations of β and . We tested a grid of those two parameters where β ∈ [1.0, 1.6] with step size 0.05, and ∈ [0.78, 0.84] with step size 0.005. We trained a single model with each combination and show (a) the obtained accuracy on the MNIST test set, (b) the achieved success rates by using FGS and (c) the mean PASS score of HC1 adversarial examples on the MNIST test images. Each solid green line represents the level of regularly trained learning models. For better visual representation we applied interpolation. results are qualitatively not as good as explicitly forming adversarial perturbations. Furthermore, in order to evaluate the stability of the trained classifiers, we distorted the images with Gaussian noise far beyond the noise level that can be considered imperceptible or adversarial. 4.1. LeNet on MNIST We commenced our experiments by evaluating BANG on the LeNet model optimized on the MNIST dataset. MNIST contains 70k images overall: 50k used for training, 10k for validation, and the remaining 10k for testing. The tested network originally has four layers (two convolutional and two fully connected) – extended with one additional Dropout layer – that we optimize without changing the hyperparameters distributed with Caffe. The learning model is trained with a batch size of 64 for 10k iterations using the inverse decay learning rate policy with an initial learning rate of 0.01. Since our training procedure has two parameters, β defined in Equation (6) and introduced in Equation (4), we trained LeNet models with parameter combinations from a grid, and evaluated the accuracy and adversarial vulnerability of the trained classifiers. The results of the conducted experiments are visualized in Figure 2, we also show accuracies and metrics indicating adversarial robustness in Table 1 for some models obtained with regular training (R0R1) and optimized with BANG training (B0-B5). As we can see in Table 1, FGS success rates achieved by regular training can be dramatically decreased by BANG: the rate drops from above 90% to below 2%. Almost every single failed adversarial example generation attempt is due to blank gradients – the gradient of loss with respect to the original image and its ground-truth label contains only zeros – which means that methods utilizing that gradient of loss cannot succeed. As we increase , or in other words, as we balance the contributions of batch elements more by scaling up gradients with lower magnitudes, the resultant classifiers become more resistant to gradient-based adversarial generation methods. Although the success rates obtained by the HC1 method remain relatively high, the qual- (a) Regular Training (b) BANG Training (c) Absolute Improvement Figure 3: L E N ET: ROBUSTNESS TO R ANDOM D ISTORTIONS. These plots show the evolving robustness of LeNet models: (a) obtained with regular training (R0 from Table 1), (b) trained with BANG (B1 from Table 1), and (c) displays the improvement. After identifying 100 test images per class that are correctly classified by both networks at every 500 iterations, we perturb each 1000 times by adding the level of Gaussian noise specified by the standard deviation, and test the networks at several stages of training. The plots show the percentage of distortions yielding misclassifications. For better visual representation we applied interpolation. ities of HC1 examples degrade significantly on LeNet models trained with BANG compared to the regular training as displayed in Figure 2(c). This degradation is highlighted by both decreasing PASS scores and by the significantly increased L∞ norms of perturbations listed in Table 1. With respect to the achieved classification performances, we find that there can be a level of degradation depending on the selected values for β and . This phenomenon can be seen in Figure 2(a); it is partially due to random initializations and can be the result of overfitting or our decision to evaluate all networks at 10k training iterations. Still, we can observe that BANG can yield improved classification performance over regular training paired with improved robustness as listed in Table 1. Additionally, we conducted experiments to quantify and compare how the robustness to random perturbations evolves during training. For this general approach, we selected to test two classifiers from Table 1: R0 optimized with regular training and B1 trained with BANG. We can see in Figure 3(a) that the regularly trained model is initially highly susceptible to larger distortions, but as the training progresses it becomes more stable, and settles at approximately 20% with respect to the strongest class of Gaussian noise that we formed by using standard deviation of 100 pixels. Contrarily, the classifier trained with BANG maintains significantly lower rates throughout the whole training as shown in Figure 3(b), and after 10k iterations only 3% of the strongest distortions can alter the original classification. The absolute improvements are displayed in Figure 3(c). 4.2. CIFAR-10 We also evaluated training with BANG on the so-called “CIFAR-10 quick” model of Caffe trained on the CIFAR10 dataset. CIFAR-10 consists of 60k images, 50k training images, and 10k images used for both validation and testing purposes. The network architecture originally has five layers (three convolutional and two fully connected) that we extended with one Dropout layer, and the learning model is trained with a batch size of 100 for 20k iterations (40 epochs). We use a fixed learning rate of 0.001 that we decrease by a factor of 10 after 36 epochs, and once again after another 2 epochs. Due to the different nature of CIFAR-10 training, we slightly adjusted BANG parameters. Specifically, as the classification performance is significantly worse than achieved by LeNet on MNIST yielding proportionately more incorrectly classified samples in each mini-batch, we applied lower local learning rates (β) and higher values for scaling (). Furthermore, we found that scaling incorrectly classified inputs less than correct ones has beneficial effects on robustness, hence, we applied 50% of the specified values on the incorrectly classified batch elements. Similarly to our conducted experiments on LeNet, we trained classifiers on CIFAR-10 with all possible combinations of β and parameters of a grid and then measured the accuracy and adversarial vulnerability of each of those networks. The results are visualized in Figure 4, and for some models obtained with regular training (R0-R1) and optimized with BANG training (B0-B5), we show accuracies and metrics indicating adversarial robustness in Table 2. As we can see in Table 2, FGS success rates achieved by regular training are significantly decreased by BANG: the rate drops from approximately 96% to 34% where, again, the majority of the failed adversarial example generation attempts are due to blank gradients. Figure 4(b) shows that as we increase , the classifiers become more resistant to gradient-based adversarial generation methods. The higher levels of success rates in comparison to LeNet might sim- Table 2: CIFAR-10 T RAINING. This table shows the difference between classifiers obtained using regular (R0-R1) and BANG training (B0-B5). The accuracy on the CIFAR-10 test set, the achieved success rates of FGS and HC1 adversarial example generation methods with PASS scores and L∞ norms of the formed examples on the CIFAR-10 test images are listed. ID β Accuracy FGS-Rate FGS-PASS FGS-L∞ HC1-Rate HC1-PASS HC1-L∞ R0 - - 79.59% 96.52% 0.9553 ± 0.0969 4.08 ± 06.40 98.97% 0.9669 ± 0.1005 18.15 ± 29.80 R1 - - 79.55% 96.71% 0.9513 ± 0.1057 4.43 ± 07.05 98.91% 0.9557 ± 0.1332 22.16 ± 39.77 B0 0.40 0.855 79.26% 34.27% 0.9511 ± 0.1302 4.11 ± 10.31 95.94% 0.8712 ± 0.1649 55.52 ± 49.98 B1 0.45 0.805 80.43% 45.94% 0.9818 ± 0.0548 2.04 ± 02.49 96.20% 0.7966 ± 0.2438 77.34 ± 71.20 B2 0.75 0.800 79.74% 41.71% 0.9828 ± 0.0586 1.94 ± 03.03 98.34% 0.8362 ± 0.2195 64.26 ± 63.57 B3 0.75 0.845 79.41% 35.00% 0.9526 ± 0.1266 3.94 ± 08.71 96.54% 0.8603 ± 0.1981 59.83 ± 58.28 B4 0.95 0.840 79.30% 34.88% 0.9575 ± 0.1236 3.61 ± 09.60 96.87% 0.8994 ± 0.1487 48.44 ± 47.35 B5 1.00 0.800 79.22% 41.34% 0.9803 ± 0.0722 2.03 ± 03.64 98.17% 0.8586 ± 0.1948 61.14 ± 61.23 (a) Accuracy (b) FGS Success Rate (c) HC1 PASS Figure 4: BANG CIFAR-10 M ODELS. These plots summarize our results on CIFAR-10 models trained with BANG using combinations of β and . We tested a grid of those two parameters where β ∈ [0.4, 1.0] with step size 0.05, and ∈ [0.80, 0.86] with step size 0.005. We trained a single model with each combination and show (a) the obtained accuracy on the CIFAR-10 test set, (b) the achieved success rates by FGS, and the (c) mean PASS score of HC1 adversarial examples on the CIFAR-10 test images. Each solid green line represents the level of regularly trained learning models. For better visual representation we applied interpolation. ply be due to the fact that the classifiers trained on CIFAR10 are less accurate, therefore, learning the incorrect samples of the batch still has a large contribution on weight updates. While the success rates achieved by HC1 remain high, the quality of HC1 adversarial examples degrades significantly compared to regular training. This degradation is highlighted by both decreasing PASS scores shown in Figure 4(c) and by the significantly increased L∞ norms of adversarial perturbations listed in Table 2. Finally, as shown in Table 2, we can train classifiers with BANG that slightly outperform models of regular training in terms of classification accuracy. Of course, the achieved overall performance depends on the chosen parameters as depicted in Figure 4(a). Finally, we ran experiments to better quantify and compare how the robustness of the trained classifiers to random perturbations evolves during training. Similarly to our experiments on LeNet, we selected two classifiers from Table 2 for testing: R0 trained regularly and B0 optimized with BANG. We can see in Figure 5(a) that the regularly trained R0 model is highly susceptible to larger distortions, its robustness does not improve during training, and finally achieves 46.0% with respect to the strongest class of Gaussian noise that we formed by using standard deviation of 40 pixels. Contrarily, the B0 model trained with BANG remains more robust throughout training epochs as shown in Figure 5(b) and at the end 39.1% of the strongest distortions change the original classification. The absolute improvements are visualized in Figure 5(c). We can conclude that although BANG enhanced robustness to random perturbations, the results are less impressive in comparison to LeNet – at least, with respect to the strongest distortions. 5. Conclusion In this paper, we introduced our theory to explain an intriguing property of machine learning models. Namely, the regular training procedure can prevent samples from forming flatter and broader regions around themselves. This evolutionary stalling yields samples remaining close to de- (a) Regular Training (b) BANG Training (c) Absolute Improvement Figure 5: CIFAR-10: ROBUSTNESS TO R ANDOM D ISTORTIONS. These plots show the evolving robustness of CIFAR-10 models: (a) obtained with regular training (R0 from Table 2), (b) trained with BANG (B0 from Table 2), and (c) displays the improvement. After identifying 100 test images per class that are correctly classified by both networks at every second epoch, we perturb each 1000 times with the level of Gaussian noise specified by the standard deviation, and test the networks at different stages of training. The plots show the percentage of distortions yielding misclassifications. For better visual representation we applied interpolation. cision boundaries and, hence, being susceptible to imperceptibly small perturbations causing misclassifications. To address this problem, we proposed a novel approach to improve the robustness of Deep Neural Networks (DNNs) by slightly modifying the regular training procedure. Our approach does not require additional training data – neither adversarial examples nor any sort of data augmentation – to achieve improved robustness, while the overall performance of the trained network is maintained or even enhanced. We experimentally demonstrated that optimizing DNNs with our Batch Adjusted Network Gradient (BANG) technique leads to significantly enhanced stability in general. By balancing the contributions of batch elements on forming the weight updates, BANG allows training samples to form flatter, more invariant regions around themselves. The trained classifiers become more robust to random distortions, and as we demonstrated with the gradient-based Fast Gradient Sign (FGS) method and the non-gradient-based hot/cold approach where we targeted the closest scoring class (HC1), they are also less vulnerable to adversarial example generation methods. To visualize the advancement achieved by BANG training in terms of improved adversarial robustness, in Figure 1 correctly classified MNIST and CIFAR-10 test images are presented along with adversarial examples formed via the HC1 approach on DNNs trained regularly and with BANG. While BANG helps to mitigate adversarial instability, learning models can maintain or even improve their overall classification performance. Our proposed approach achieves these results with negligible computational overhead over the regular training procedure. Although we managed to achieve good results on two DNNs trained on different datasets, we found that BANG parameters needed to be adjusted to these problems. To obtain better results, exploring the effect of different pa- rameters on different layers, and changing the contributions of correctly and incorrectly classified batch elements can be considered. Future work will focus on having a better understanding of BANG, enhancing the algorithm to be more self-adaptive, and exploring its application for training DNNs on real-world datasets. While some might argue that a similar balancing effect can be achieved by distillation, Carlini et al. [1] demonstrated that defensive distillation is not effective to improve adversarial robustness. The effectiveness of BANG to adversarial perturbations obtained via various adversarial example generation techniques likely varies – as Kurakin et al. [13] observed for adversarial training – and further research needs to explore that. In summary, we can conclude that the adversarial instability of DNNs is closely related to the applied training procedures – as was claimed by Gu et al. [6] – and there is a huge potential in this research area to further advance the generalization properties of machine learning models and their overall performances as well. Acknowledgments This research is based upon work funded in part by NSF IIS-1320956 and in part by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via IARPA R&D Contract No. 2014-14071600012. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the ODNI, IARPA, or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon. References [1] N. 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Neural Domain Adaptation for Biomedical Question Answering Georg Wiese1,2 , Dirk Weissenborn2 and Mariana Neves1 1 Hasso Plattner Institute, August Bebel Strasse 88, Potsdam 14482 Germany 2 Language Technology Lab, DFKI, Alt-Moabit 91c, Berlin, Germany [email protected], [email protected], [email protected] Abstract arXiv:1706.03610v2 [cs.CL] 15 Jun 2017 Factoid question answering (QA) has recently benefited from the development of deep learning (DL) systems. Neural network models outperform traditional approaches in domains where large datasets exist, such as SQuAD (≈ 100, 000 questions) for Wikipedia articles. However, these systems have not yet been applied to QA in more specific domains, such as biomedicine, because datasets are generally too small to train a DL system from scratch. For example, the BioASQ dataset for biomedical QA comprises less then 900 factoid (single answer) and list (multiple answers) QA instances. In this work, we adapt a neural QA system trained on a large open-domain dataset (SQuAD, source) to a biomedical dataset (BioASQ, target) by employing various transfer learning techniques. Our network architecture is based on a state-of-theart QA system, extended with biomedical word embeddings and a novel mechanism to answer list questions. In contrast to existing biomedical QA systems, our system does not rely on domain-specific ontologies, parsers or entity taggers, which are expensive to create. Despite this fact, our systems achieve state-of-the-art results on factoid questions and competitive results on list questions. 1 Introduction Question answering (QA) is the task of retrieving answers to a question given one or more contexts. It has been explored both in the opendomain setting (Voorhees et al., 1999) as well as domain-specific settings, such as BioASQ for the biomedical domain (Tsatsaronis et al., 2015). The BioASQ challenge provides ≈ 900 factoid and list questions, i.e., questions with one and several answers, respectively. This work focuses on answering these questions, for example: Which drugs are included in the FEC-75 regimen? → fluorouracil, epirubicin, and cyclophosphamide. We further restrict our focus to extractive QA, i.e., QA instances where the correct answers can be represented as spans in the contexts. Contexts are relevant documents which are provided by an information retrieval (IR) system. Traditionally, a QA pipeline consists of namedentity recognition, question classification, and answer processing steps (Jurafsky, 2000). These methods have been applied to biomedical datasets, with moderate success (Zi et al., 2016). The creation of large-scale, open-domain datasets such as SQuAD (Rajpurkar et al., 2016) have recently enabled the development of neural QA systems, e.g., Wang and Jiang (2016), Xiong et al. (2016), Seo et al. (2016), Weissenborn et al. (2017), leading to impressive performance gains over more traditional systems. However, creating large-scale QA datasets for more specific domains, such as the biomedical, would be very expensive because of the need for domain experts, and therefore not desirable. The recent success of deep learning based methods on open-domain QA datasets raises the question whether the capabilities of trained models are transferable to another domain via domain adaptation techniques. Although domain adaptation has been studied for traditional QA systems (Blitzer et al., 2007) and deep learning systems (Chen et al., 2012; Ganin et al., 2016; Bousmalis et al., 2016; Riemer et al., 2017; Kirkpatrick et al., 2017), it has to our knowledge not yet been applied for end-to-end neural QA systems. To bridge this gap we employ various do- main adaptation techniques to transfer knowledge from a trained, state-of-the-art neural QA system (FastQA, Weissenborn et al. (2017)) to the biomedical domain using the much smaller BioASQ dataset. In order to answer list questions in addition to factoid questions, we extend FastQA with a novel answering mechanism. We evaluate various transfer learning techniques comprehensively. For factoid questions, we show that mere fine-tuning reaches state-of-the-art results, which can further be improved by a forgetting cost regularization (Riemer et al., 2017). On list questions, the results are competitive to existing systems. Our manual analysis of a subset of the factoid questions suggests that the results are even better than the automatic evaluation states, revealing that many of the ”incorrect” answers are in fact synonyms to the gold-standard answer. 2 Related Work Traditional Question Answering Traditional factoid and list question answering pipelines can be subdivided into named-entity recognition, question classification, and answer processing components (Jurafsky, 2000). Such systems have also been applied to biomedical QA such as the OAQA system by Zi et al. (2016). Besides a number of domain-independent features, they incorporate a rich amount of biomedical resources, including a domain-specific parser, entity tagger and thesaurus to retrieve concepts and synonyms. A logistic regression classifier is used both for question classification and candidate answer scoring. For candidate answer generation, OAQA employs different strategies for general factoid/list questions, choice questions and quantity questions. Neural Question Answering Neural QA systems differ from traditional approaches in that the algorithm is not subdivided into discrete steps. Instead, a single model is trained end-to-end to compute an answer directly for a given question and context. The typical architecture of such systems (Wang and Jiang, 2016; Xiong et al., 2016; Seo et al., 2016) can be summarized as follows: 1. Embedding Layer: Question and context tokens are mapped to a high-dimensional vector space, for example via GloVe embeddings (Pennington et al., 2014) and (optionally) character embeddings (Seo et al., 2016). 2. Encoding Layer: The token vectors are processed independently for question and context, usually by a recurrent neural network (RNN). 3. Interaction Layer: This layer allows for interaction between question and context representations. Examples are Match-LSTM (Wang and Jiang, 2016) and Coattention (Xiong et al., 2016). 4. Answer Layer: This layer assigns start and end scores to all of the context tokens, which can be done either statically (Wang and Jiang, 2016; Seo et al., 2016) or by a dynamic decoding process (Xiong et al., 2016). FastQA FastQA fits into this schema, but reduces the complexity of the architecture by removing the interaction layer, while maintaining state-of-the-art performance (Weissenborn et al., 2017). Instead of one or several interaction layers of RNNs, FastQA computes two simple wordin-question features for each token, which are appended to the embedding vectors before the encoding layer. We chose to base our work on this architecture because of its state-of-the-art performance, faster training time and reduced number of parameters. Unsupervised Domain Adaptation Unsupervised domain adaptation describes the task of learning a predictor in a target domain while labeled training data only exists in a different source domain. In the context of deep learning, a common method is to first train an autoencoder on a large unlabeled corpus from both domains and then use the learned input representations as input features to a network trained on the actual task using the labeled source domain dataset (Glorot et al., 2011; Chen et al., 2012). Another approach is to learn the hidden representations directly on the target task. For example, domain-adversarial training optimizes the network such that it computes hidden representations that both help predictions on the source domain dataset and are indistinguishable from hidden representations of the unlabeled target domain dataset (Ganin et al., 2016). These techniques cannot be straightforwardly applied to the question answering task, because they require a large corpus of biomedical question-context pairs (albeit no answers are required). Supervised Domain Adaptation In contrast to the unsupervised case, supervised domain adaptation assumes access to a small amount of labeled training data in the target domain. The simplest approach to supervised domain adaptation for neural models is to pre-train the network on data from the source domain and then fine-tune its parameters on data from the target domain. The main drawback of this approach is catastrophic forgetting, which describes the phenomenon that neural networks tend to ”forget” knowledge, i.e., its performance in the source domain drops significantly when they are trained on the new dataset. Even though we do not directly aim for good performance in the source domain, measures against catastrophic forgetting can serve as a useful regularizer to prevent over-fitting. Progressive neural networks combat this issue by keeping the original parameters fixed and adding new units that can access previously learned features (Rusu et al., 2016). Because this method adds a significant amount of new parameters which have to be trained from scratch, it is not well-suited if the target domain dataset is small. Riemer et al. (2017) use fine-tuning, but add an additional forgetting cost term that punishes deviations from predictions with the original parameters. Another approach is to add an L2 loss which punishes deviation from the original parameters. Kirkpatrick et al. (2017) apply this loss selectively on parameters which are important in the source domain. 3 Model Our network architecture is based on FastQA (Weissenborn et al., 2017), a state-of-the-art neural QA system. Because the network architecture itself is exchangeable, we treat it as a black box, with subtle changes at the input and output layer as well as to the decoding and training procedure. These changes are described in the following. See Figure 3 for an overview of the system. 3.1 Input Layer In a first step, words are embedded into a highdimensional vector space. We use three sources of embeddings, which are concatenated to form a single embedding vector: • GloVe embeddings: 300-dimensional GloVe vectors (Pennington et al., 2014). These are Start Probabilities pstart End Probabilitiesp(e\|s) p(e\|s) End End Probabilities Probabilities pend sigmoid softmax EndScores Scoresee(s)(s) End End Scores yend Start Scores ystart Extractive QA System Biomedical Embeddings ... ... GloVe Embeddings Character Embeddings Question Type Features Context Embeddings Question Embeddings Figure 1: Network architecture of our system for biomedical question answering. At its core, it uses an extractive neural QA system as a black box (we use FastQA (Weissenborn et al., 2017)). The embedding layer is modified in order to include biomedical word embeddings and question type features. The output layer is adjusted to add the ability to answer list questions in addition to factoid questions. open-domain word vectors trained on 840 billion tokens from web documents. The vectors are not updated during training. • Character embeddings: As used in FastQA (Weissenborn et al., 2017) and proposed originally by Seo et al. (2016), we employ a 1-dimensional convolutional neural network which computes word embeddings from the characters of the word. • Biomedical Word2Vec embeddings: 200dimensional vectors trained using Word2Vec (Mikolov et al., 2013) on about 10 million PubMed abstracts (Pavlopoulos et al., 2014). These vectors are specific to the biomedical domain and we expect them to help on biomedical QA. As an optional step, we add entity tag features to the token embeddings via concatenation. Entity tags are provided by a dictionary-based entity tagger based on the UMLS Metathesaurus. The entity tag feature vector is a 127-dimensional bit vector that for each of the UMLS semantic types states whether the current token is part of an entity of that type. This step is only applied if explicitly noted. Finally, a one-hot encoding of the question type (factoid or list) is appended to all the input vectors. With these embedding vectors as input, we invoke FastQA to produce start and end scores for each of the n context tokens. We denote start i scores by ystart and end scores conditioned on a i,j predicted start at position i by yend , with start index i ∈ [1, n] and end index j ∈ [i, n]. 3.2 Output Layer In our adapted output layer, we convert the start and end scores to span probabilities. The computation of these probabilities is independent of the question type. The interpretation, however, depends on the question type: While for factoid questions, the list of answer spans is interpreted as a ranked list of answer candidates, for list questions, answers above a certain probability threshold are interpreted as the set of answers to the question. 1 n Given the start scores ystart , ..., ystart and end i,n i,1 scores yend , ..., yend , we compute the start and end probabilities as follows: i pistart = σ(ystart ) i,· pi,· end = softmax(yend ) (1) (2) where σ(x) is the sigmoid function. As a consequence, multiple tokens can be chosen as likely start tokens, but the network is expected to select a single end token for a given start token, hence the softmax function. Finally, the probability that a given span (i, j) answers the question i,j i is pi,j span = pstart · pend . This extension generalizes the FastQA output layer such that multiple answer spans with different start positions can have a high probability, allowing us to retrieve multiple answers for list questions. 3.3 Decoding Given a trained model, start probabilities can be obtained by running a forward pass and computing the start probability as in Equation 1. For the top 20 starts, we compute the end probabilities as given by Eq. 2. From the start and end probabilities, we extract the top 20 answer spans ranked by pi,j span . As a simple post-processing step, we remove duplicate strings and retain only those with the highest probability. For factoid questions, we output the 5 most likely answer spans as our ranked list of answers. For list questions, we learn a probability cutoff threshold t that defines the set of list answers A = {(i, j)\|pi,j span ≥ t}. We choose t to be the threshold that optimizes the list F1 score on the respective development set. 3.4 Domain Adaptation Fine-tuning Our training procedure consists of two phases: In the pre-training phase, we train the model on SQuAD, using a token F1 score as the training objective as by Weissenborn et al. (2017). We will refer to the resulting parameters as the base model. In the fine-tuning phase, we initialize the model parameters with the base model and then continue our optimization on the BioASQ dataset with a smaller learning rate. Forgetting Cost Regularization To avoid catastrophic forgetting during fine-tuning as a means to regularize our model, we optionally add an additional forgetting cost term Lf c , as proposed by Riemer et al. (2017). It is defined as the cross-entropy loss between the current predictions and the base model’s predictions. L2 Weight Regularization We also add an L2 loss term Ll2 which penalizes deviations from the base model’s parameters. Note that a more advanced approach would be to apply this loss selectively on weights which are particularly important in the source domain (Kirkpatrick et al., 2017). The final loss is computed as Lf inal = Loriginal + Cf c · Lf c + Cl2 · Ll2 where Cf c and Cl2 are hyperparameters which are set to 0 unless otherwise noted. 4 4.1 Experimental Setup Datasets SQuAD SQuAD (Rajpurkar et al., 2016) is a dataset of ≈ 100, 000 questions with relevant contexts and answers that sparked research interest into the development of neural QA systems recently. The contexts are excerpts of Wikipedia articles for which crowd-source workers generated questions-answer pairs. Because of the large amount of training examples in SQuAD, it lends itself perfectly as our source dataset. BioASQ The BioASQ challenge provides a biomedical QA dataset (Tsatsaronis et al., 2015) consisting of questions, relevant contexts (called snippets) from PubMed abstracts and possible answers to the question. It was carefully created with the help of biomedical experts. In this work, we focus on Task B, Phase B of the BioASQ challenge, in which systems must answer questions from gold-standard snippets. These questions can be either yes/no questions, summary questions, factoid questions, or list questions. Because we employ an extractive QA system, we restrict this study to answering factoid and list questions by extracting answer spans from the provided contexts. The 2017 BioASQ training dataset contains 1, 799 questions, of which 413 are factoid and 486 are list questions. The questions have ≈ 20 snippets on average, each of which are on average ≈ 34 tokens long. We found that around 65% of the factoid questions and around 92% of the list questions have at least one extractable answer. For questions with extractable answers, answers spans are computed via a simple substring search in the provided snippets. All other questions are ignored during training and treated as answered incorrectly during evaluation. 4.2 Training We minimize the cross-entropy loss for the gold standard answer spans. However, for multiple answer spans that refer to the same answer (e.g. synonyms), we only minimize the loss for the span of the lowest loss. We use the ADAM (Kingma and Ba, 2014) for optimization on SQuAD with a learning rate starting at 10−3 which is halved whenever performance drops between checkpoints. During the fine-tuning phase, we continue optimization on the BioASQ dataset with a smaller learning rate starting at 10−4 . During both phases, the model is regularized by variational dropout of rate 0.5 (Gal and Ghahramani, 2015). 4.3 Evaluation The official evaluation measures from BioASQ are mean reciprocal rank (MRR) for factoid questions and F1 score for list questions 1 . For factoid questions, the list of ranked answers can be at most five entries long. The F1 score is measured on the gold standard list elements. For both measures, 1 The details can be found at http:// participants-area.bioasq.org/Tasks/b/ eval_meas/ case-insensitive string matches are used to check the correctness of a given answer. A list of synonyms is provided for all gold-standard answers. If the system’s response matches one of them, the answer counts as correct. For evaluation, we use two different finetuning datasets, depending on the experiment: BioASQ3B, which contains all questions of the first three BioASQ challenges, and BioASQ4B which additionally contains the test questions of the fourth challenge. BioASQ4B is used as the training dataset for the fifth BioASQ challenge whereas BioASQ3B was used for training during the fourth challenge. Because the datasets are small, we perform 5fold cross-validation and report the average performance across the five folds. We use the larger BioASQ4B dataset except when evaluating the ensemble and when comparing to participating systems of previous BioASQ challenges. All models were implemented using TensorFlow (Abadi et al., 2016) with a hidden size of 100. Because the context in BioASQ usually comprises multiple snippets, they are processed independently in parallel for each question. Answers from all snippets belonging to a question are merged and ranked according to their individual probabilities. 5 5.1 Results Domain Adaptation In this section, we evaluate various domain adaptation techniques. The results of the experiments are summarized in Table 1. Baseline As a baseline without transfer learning, Experiment 1 trains the model on BioASQ only. Because the BioASQ dataset by itself is very small, a dropout rate of 0.7 was used, because it worked best in preliminary experiments. We observe a rather low performance, which is expected when applying deep learning to such a small dataset. Fine-tuning Experiments 2 and 3 evaluate the pure fine-tuning approach: Our base model is a system trained on SQuAD only and tested on BioASQ (Experiment 2). For Experiment 3, we fine-tuned the base model on the BioASQ4B training set. We observe that performance increases significantly, especially on list questions. This increase is expected, because the network is trained Experiment Factoid MRR List F1 (1) Training on BioASQ only 17.9% 19.1% (2) Training on SQuAD only (3) Fine-tuning on BioASQ 20.0% 24.6% 8.1% 23.6% (4) Fine-tuning on BioASQ w/o biomedical embeddings (5) Fine-tuning on BioASQ w/ entity features 21.3% 23.3% 22.4% 23.8% (6) Fine-tuning on BioASQ + SQuAD (7) Fine-tuning on BioASQ w/ forgetting cost (Cf c = 100.0) (8) Fine-tuning on BioASQ w/ L2 loss on original parameters (Cl2 = 0.3) 23.9% 26.2% 22.6% 23.8% 21.1% 20.4% Table 1: Comparison of various transfer learning techniques. In Experiment 1, the model was trained on BioASQ only. In Experiment 2, the model was trained on SQuAD and tested on BioASQ. We refer to it as the base model. In Experiment 3, the base model parameters were fine-tuned on the BioASQ training set. Experiments 4-5 evaluate the utility of domain dependent word vectors and features. Experiments 6-8 address the problem of catastrophic forgetting. All experiments have been conducted with the BioASQ4B dataset and 5-fold cross-validation. on biomedical- and list questions, which are not part of the SQuAD dataset, for the first time. Overall, the performance of the fine-tuned model on both question types is much higher than the baseline system without transfer learning. Features In order to evaluate the impact of using biomedical word embeddings, we repeat Experiment 3 without them (Experiment 4). We see a factoid and list performance drop of 3.3 and 1.2 percentage points, respectively, showing that biomedical word embeddings help increase performance. In Experiment 5, we append entity features to the word vector, as described in Section 3.1. Even though these features provide the network with domain-specific knowledge, we found that it actually harms performance on factoid questions. Because most of the entity features are only active during fine-tuning with the small dataset, we conjecture that the performance decrease is due to over-fitting. Catastrophic Forgetting We continue our study with techniques to combat catastrophic forgetting as a means to regularize training during fine-tuning. In Experiment 6 of Table 1 we fine-tune the base model on a half-half mixture of BioASQ and SQuAD questions (BioASQ questions have been upsampled accordingly). This form of joint training yielded no significant performance gains. Experiment 7 regularizes the model via an additional forgetting cost term, as proposed by Riemer et al. (2017) and explained in Section 3.4. We generally found that this technique only increases performance for factoid questions where the performance boost was largest for Cf c = 100.0. The fact that the forgetting loss decreases performance on list questions is not surprising, as predictions are pushed more towards the predictions of the base model, which has very poor performance on list questions. Experiment 8 adds an L2 loss which penalizes deviations from the base model’s parameters. We found that performance decreases as we increase the value of Cl2 which shows that this technique does not help at all. For the sake of completeness we report results for Cl2 = 0.3, the lowest value that yielded a significant drop in performance. 5.2 Ensemble Model ensembles are a common method to tweak the performance of a machine learning system. Ensembles combine multiple model predictions, for example by averaging, in order to improve generalization and prevent over-fitting. We evaluate the utility of an ensemble by training five models on the BioASQ3B dataset using 5-fold crossvalidation. Each of the models is evaluated on the 4B test data, i.e., data which is not included in BioASQ3B. During application, we run an ensemble by averaging the start and end scores of individual models before they are passed to the sigmoid / softmax functions as defined in Eq. 1 and 2. In Table 2 we summarize the average performance of Experiment Factoid MRR List F1 Average Best Ensemble 23.4% 24.3% 27.3% 24.0% 27.7% 28.6% Table 2: Performance of a model ensemble. Five models have been trained on the BioASQ3B dataset and tested on the 4B test questions. We report the average and best single model performances, as well as the ensemble performance. the five models, the best performance across the five models, and the performance of the ensemble. We observe performance gains of 3 percentage points on factoid questions and a less than 1 percentage point on list questions, relative to the best single model. This demonstrates a small performance gain that is consistent with the literature. 5.3 Comparison to competing BioASQ systems Because the final results of the fifth BioASQ challenge are not available at the time of writing, we compare our system to the best systems in last year’s challenge 2 . For comparison, we use the best single model and the model ensemble trained on BioASQ3B (see Section 5.2). We then evaluate the model on the 5 batches of last year’s challenge using the official BioASQ evaluation tool. Each batch contains 100 questions of which only some are factoid and list questions. Note that the results underestimate our system’s performance, because our competing system’s responses have been manually evaluated by humans while our system’s responses are evaluated automatically using string matching against a potentially incomplete list of synonyms. In fact, our qualitative analysis in Section 5.4 shows that many answers are counted as incorrect, but are synonyms of the gold-standard answer. The results are summarized in Table 3 and compared to the best systems in the challenge in each of the batches and question type categories. With our system winning four out of five batches on factoid questions, we consider it stateof-the-art in biomedical factoid question answering, especially when considering that our results might be higher on manual evaluation. The results on list questions are slightly worse, but still very 2 Last year’s results are available at http: //participants-area.bioasq.org/results/ 4b/phaseB/ competitive. This is surprising, given that the network never saw a list question prior to the finetuning phase. Due to small test set sizes, the sampling error in each batch is large, causing the single model to outperform the model ensemble on some batches. 5.4 Qualitative Analysis In order to get a better insight into the quality of the predictions, we manually validated the predictions for the factoid questions of batch 5 of the fourth BioASQ challenge as given by the best single model (see Table 3). There are in total 33 factoid questions, of which 23 have as the gold standard answer a span in one of the contexts. According to the official BioASQ evaluation, only 4 questions are predicted correctly (i.e., the gold standard answer is ranked highest). However, we identified 10 rank-1 answers which are not counted as correct but are synonyms to the gold standard answer. Examples include ”CMT4D disease” instead of ”Charcot-Marie-Tooth (CMT) 4D disease”, ”tafazzin” instead of ”Tafazzin (TAZ) gene”, and ”β-glucocerebrosidase” instead of ”Beta glucocerebrosidase”. In total, we labeled 14 questions as correct and 24 questions as having their correct answer in the top 5 predictions. In the following, we give examples of mistakes made by the system. Questions are presented in italics. In the context, we underline predicted answers and present correct answers in boldface. We identified eight questions for which the semantic type of the top answer differs from the question answer type. Some of these cases are completely wrong predictions. However, this category also includes subtle mistakes like the following: In which yeast chromosome does the rDNA cluster reside? The rDNA cluster in Saccharomyces cerevisiae is located 450 kb from the left end and 610 kb from the right end of chromosome XII... Here, it predicted a yeast species the rDNA cluster is located in, but ignored that the question is asking for a chromosome. Another type of mistakes is that the top answer is somewhat correct, but is missing essential information. We labeled four predictions with this category, like the following example: Batch Factoid MRR Best Participant Single Ensemble Best Participant List F1 Single 1 2 3 4 5 12.2% (fa1) 22.6% (LabZhu-FDU) 24.4% (oaqa-3b-3) 32.5% (oaqa-3b-4) 28.5% (oaqa-3b-5) 25.2% 16.4% 24.7% 34.0% 23.7% 29.2% 24.2% 20.6% 40.3% 23.2% 16.8% (fa1) 15.5% (LabZhu-FDU) 48.3% (oaqa-3b-3) 31.2% (oaqa-3b-4) 29.0% (oaqa-3b-5) 29.1% 25.8% 31.8% 29.0% 23.5% 27.9% 20.8% 33.3% 24.1% 26.1% Avg. 24.0% 24.8% 27.5% 28.1% 27.8% 26.5% Ensemble Table 3: Comparison to systems on last year’s (fourth) BioASQ challenge for factoid and list questions. For each batch and question type, we list the performance of the best competing system, our single model and ensemble. Note that our qualitative analysis (Section 5.4) suggests that our factoid performance on batch 5 would be about twice as high if all synonyms were contained in the gold standard answers. How early during pregnancy does non-invasive cffDNA testing allow sex determination of the fetus? Gold Standard Answer: "6th to 10th week of gestation" or "first trimester of pregnancy" Given Top Answer: "6th-10th" In summary, to our judgment, 14 of 33 questions (42.4%) are answered correctly, and 24 of 33 questions (72.7%) are answered correctly in one of the top 5 answers. These are surprisingly high numbers considering low MRR score of 23.7% of the automatic evaluation (Table 3). poor prior to fine-tuning which is due to the lack of list questions in SQuAD. We believe that large scale open-domain corpora for list questions would enhance performance further. Unsupervised domain adaptation could be an interesting direction for future work, because the biomedical domain offers large amounts of textual data, some of which might even contain questions and their corresponding answers. We believe that leveraging these resources holds potential to further improve biomedical QA. 6 In this paper, we described a deep learning approach to address the task of biomedical question answering by using domain adaptation techniques. Our experiments reveal that mere fine-tuning in combination with biomedical word embeddings yield state-of-the-art performance on biomedical QA, despite the small amount of in-domain training data and the lack of domain-dependent feature engineering. Techniques to overcome catastrophic forgetting, such as a forgetting cost, can further boost performance for factoid questions. Overall, we show that employing domain adaptation on neural QA systems trained on large-scale, open-domain datasets can yield good performance in domains where large datasets are not available. Discussion and future work The most significant result of this work is that state-of-the-art results in biomedical question answering can be achieved even in the absence of domain-specific feature engineering. Most competing systems require structured domain-specific resources, such as biomedical ontologies, parsers, and entity taggers. While these resources are available in the biomedical domain, they are not available in most domains. Our system, on the other hand, requires a large open-domain QA dataset, biomedical word embeddings (which are trained in an unsupervised fashion), and a small biomedical QA dataset. This suggests that our methodology is easily transferable to other domains as well. Furthermore, we explored several supervised domain adaptation techniques. In particular, we demonstrated the usefulness of forgetting cost for factoid questions. 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1 Uncertainty Marginal Price, Transmission Reserve, and Day-ahead Market Clearing with Robust Unit Commitment arXiv:1507.01540v3 [math.OC] 2 Aug 2016 Hongxing Ye, Member, IEEE, Yinyin Ge, Mohammad Shahidehpour, Fellow, IEEE, Zuyi Li, Senior Member, IEEE Abstract—The increasing penetration of renewable energy in recent years has led to more uncertainties in power systems. These uncertainties have to be accommodated by flexible resources (i.e. upward and downward generation reserves). In this paper, a novel concept, Uncertainty Marginal Price (UMP), is proposed to price both the uncertainty and reserve. At the same time, the energy is priced at Locational Marginal Price (LMP). A novel market clearing mechanism is proposed to credit the generation and reserve and to charge the load and uncertainty within the Robust Unit Commitment (RUC) in the Day-ahead market. We derive the UMPs and LMPs in the robust optimization framework. UMP helps allocate the cost of generation reserves to uncertainty sources. We prove that the proposed market clearing mechanism leads to partial market equilibrium. We find that transmission reserves must be kept explicitly in addition to generation reserves for uncertainty accommodation. We prove that transmission reserves for ramping delivery may lead to Financial Transmission Right (FTR) underfunding in existing markets. The FTR underfunding can be covered by congestion fund collected from uncertainty payment in the proposed market clearing mechanism. Simulations on a six-bus system and the IEEE 118-bus system are performed to illustrate the new concepts and the market clearing mechanism. Index Terms—Uncertainty Marginal Price, Cost Causation, Robust Unit Commitment, Financial Transmission Right, Generation Reserve, Transmission Reserve N OMENCLATURE Indices i, l, t m, n mi k indices for generators, lines, and time intervals index for buses index of bus where unit i is located index of the worst point for uncertainty Functions and sets ˆ F U CiP (·), CiI (·) L(·) G(m) K symbol for the optimal value of a variable feasible set for UC and dispatch uncertainty set cost related to dispatch and UC for unit i Lagrangian function set of units located at bus m set of the indices for ˆk up down Km,t , Km,t set of indices k for upward and downward UMPs at bus m time t Constants ND , NT number of buses and time intervals dm,t aggregated equivalent load Fl transmission line flow limit Γl,m shift factor for line l with respect to bus m Pimin , Pimax minimum and maximum generation outputs riu , rid ramping-up/down limits between sequential intervals Riu , Rid ramping-up/down limits for uncertainty accommodation um,t bound for uncertainty ˆ ˆk is the k th worst uncertainty vector in K, ˆk ∈ RND NT , ˆkm,t ∈ R FTRm→n FTR amount from bus m to n Variables Ii,t unit on/off status indicators yi,t , zi,t unit start-up and shut-down indicators Pi,t generation dispatch inj Pm,t net power injection m,t uncertainty at bus m time t Z optimal value of problem (SP) given (x, y, z, I, P ) ∆Pi,t generation re-dispatch pos ∆fl,t transmission capacity reserve in positive direction neg ∆fl,t transmission capacity reserve in negative direction inj ∆Pm,t net power injection change λt , λkt Lagrangian multipliers α, β, η non-negative Lagrangian multipliers u,k e πm,t marginal prices. πm,t for energy price; πm,t is the u,up UMP for kth uncertainty point; πm,t for upward u,down UMP; πm,t for downward UMP up Qi,t , Qdown i,t upward and downward generation reserves Ψm,t charge for uncertainty source T ΘG i,t , Θl,t credits to generation reserve for unit i and transmission reserve for line l at time t This work is supported by the U.S. National Science Foundation Grant I. I NTRODUCTION ECCS-1549937. The early version of this work was available on arXiv July 06, 2015, titled “Market Clearing for Uncertainty, Generation Reserve, and N modern power systems, uncertainties grow significantly Transmission Reserve”. The authors are with the Galvin Center for Electricity with the increasing penetration of Renewable Energy Innovation at Illinois Institute of Technology, Chicago, IL 60616, USA. (email: [email protected]; [email protected]; [email protected]; [email protected]). Source (RES), such as wind power generation. They pose 0885-8950 c 2016 IEEE. http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=7524711, Citation information: DOI 10.1109/TPWRS.2016.2595621, IEEE Transactions on Power Systems I 2 new challenges for the operation of electricity markets. In the Day-ahead market (DAM), the Unit Commitment (UC) and Economic Dispatch (ED) problems considering uncertainties have become a focus of research in recent years. The objective of the UC problem is to find the least cost UC solution for the second day while respecting both system-wide and unit-wise constraints. By fixing the UC variables, the ED problem is established. The Locational Marginal Price (LMP) and reserve price are then obtained as byproducts of the ED problem [1], [2]. When considering the uncertainties, the generation from uncontrollable RES are uncertain parameters in the optimization problem. Recently, Robust UC (RUC) is proposed to address the issues of uncertainty [3]–[7]. The largest merit is that the UC solution can be immunized against all the uncertainties in predefined set. The key idea of the two-stage RUC is to determine the optimal UC in the first stage which leads to the least cost for the worst scenario in the second stage. However, this approach is conservative and the Robust ED (RED) is absent. Authors in [8] combined the stochastic and robust approach using a weight factor in the objective function to address the conservativeness issue. [9], [10] employed the Affine Policy (AP) to formulate and solve the RED problem. A Multi-stage RUC is proposed to incorporate the latest information in each stage [11], where AP is also used to overcome the computational challenge. Recently, we reported a new approach which tries to bridge the gap of RUC and RED [7], [12]. In DAM, the main difficulty for pricing is that RED is absent in the traditional RUC [4], [5], [8]. On the other hand, a large number of works on pricing reserves exists within the UC framework considering contingencies [2], [13], [14] and stochastic security [15]. They are normally modeled as co-optimization problem. In [2], the reserve is cleared on zonal levels. Instead of countable contingency scenarios [15] or single additional scenario for reserve [16], the infinite continuous uncertainties are considered in the RUC, and the reserves are fully deliverable in infinite scenarios. In this paper, we propose a novel mechanism to price the energy, uncertainty, and flexibility simultaneously based on the RUC in [7]. An explicit price signal is derived for pricing the uncertainty. As the ED solution obtained is robust [7], both marginal impacts of the uncertainty and flexibility are reflected in these prices. In the proposed mechanism, reserve costs are allocated to uncertainty sources. Generation reserves, also called flexibilities in this paper, are the key factor for the robust optimization approaches. They are entitled to proper credits based on their contribution to uncertainty management. According to the market equilibrium analysis, market participants (price takers) can get the maximal profit by following the ISO/RTO’s dispatch instruction. The generation reserve and its deliverability are the main focus in [7]. The definition of LMP in [17] are employed to derive the energy price. The new concept, Uncertainty Marginal Price (UMP), is proposed to define the marginal cost of immunizing the next increment of uncertainty at a specific location. Load and generation are a pair, and they are priced at LMP. Uncertainty and flexibility (i.e., generation reserve) are another pair, and they are priced at UMP. Both LMPs and UMPs may vary with the locations due to transmission congestions. Limited by the transmission capacity and power flow equations, sometimes the uncertainties at certain buses cannot be mitigated by the system-wide cheapest generation reserve, and expensive generation reserve, which is deliverable, has to be kept in the system. Therefore, uncertainty sources are charged and generation reserves are credited based on UMPs at the corresponding buses. As the transmission reserve is kept within the RUC framework, the congestion component may exist in both the energy price and reserve price even if the physical limit of the line is not reached yet in the base case scenario. LMP congestion costs are distributed to Financial Transmission Right (FTR) holders in the existing market according to the LMP difference and the FTR amount. The revenue inadequacy occurs when the LMP congestion cost collected is smaller than the credit distributed to FTR holders, which is also called FTR underfunding. This has been a serious issue in recent years in the industry [18], [19]. We reveal that transmission reserve will be another reason for FTR underfunding when physical transmission limit is adopted in Simultaneous Feasibility Test (SFT) for FTR market [18], [20], [21]. This conclusion is applicable to any robust UC framework for DAM. The main contributions of this paper are listed as follows. 1) The novel UMP for uncertainties and generation reserves, as well as LMP for energy, are derived within a robust UC framework. The derivation is for uncertainties set with interval and budget constraints. The general concepts still apply when other uncertainty sets are modeled. 2) It is revealed that transmission capacities have to be reserved for uncertainty accommodation and the transmission reserves may cause FTR underfunding because of the deficiency of energy congestion revenues based on existing market rules. 3) A new market clearing mechanism is proposed to credit the generation and reserve and to charge the load and uncertainty. The payment collected from uncertainty sources can exactly cover the credits to generation reserves and transmission reserves, effectively resolving the FTR underfunding issue. The rest of this paper is organized as follows. Derivation of the LMP and UMP is presented in Section II, so is the market clearing mechanism for charge and credit based on LMP and UMP. Case studies are presented in Section III. Section IV concludes this paper. II. RUC AND M ARKET C LEARING One motivation of this work is to price the uncertainty, and allocate the cost of uncertainty accommodation to the uncertainty source. As the uncertainty source is charged the uncertainty payment, it has the incentive to reduce the uncertainty. With UMP, we can follow the cost causation principle, which is normally required in the market design, to charge the uncertainty sources. Cost causation principle is described as “require that all approved rates reflect to some degree the costs 3 actually caused by the customer who must pay them” in KN Energy, Inc. V. FERC, 968 F.2d 1295, 1300 (D.C. Cir. 1992). Another important motivation is to provide a theory that supports the application of the RUC in the DAM clearing. Although the RUC/RED are studied extensively, the only application of the RUC now is for the Reliability Assessment Commitment (RAC) in the DAM. There are several reasons why they are not applied in the DAM clearing. First, the computation burden of RUC is much larger than the standard UC. Second, as the objective is the cost of the worst-case scenario [3], [5], the solution is criticized on over conservatism. Third, no economic dispatch and prices are available within the RUC framework. Recently, with the new achievements in the algorithms, models, and high-performance computing application [6]–[8], [22]–[24], the first two obstacles are being addressed with great promises. This paper tries to clear the last obstacle with the new model [7]. Adopting RUC in the market clearing can give clear price signals for the uncertainties and reserves. On the other side, it is also easier for the solution to pass the robustness test, which is a RUC, in RAC. To our best knowledge, this is the first work on pricing energy, uncertainties, and reserves within the robust optimization framework in DAM. Hence, we focus on illustrating the concept with the following assumptions. • Network loss is ignored. Shift factor matrix is constant. • Uncertainty is from load and RES. Contingency is ignored. • The uncertainty budget set can be truly formulated by the ISO/RTO. A. RUC and RED ISOs/RTOs desire to get the optimal UC and ED solution in the base-case scenario. They can re-dispatch the flexible resources, such as adjustable load demands and generators with fast ramping capabilities, to follow the load when deviation occurs (or uncertainty is revealed). Consistent with the robust literature [4], [5], the uncertainty set is modeled as U := { ∈ RND NT : −um,t ≤ m,t ≤ um,t , ∀m, t X \|m,t \| ≤ Λ∆ t , ∀t} u m,t m (RUC) min C I (x) + C P (p) s.t. Ax + Bp ≤ b n F := (x, p) : ∀ ∈ U, ∃∆p such that o Cx + Dp + G∆p ≤ d + E . (1) (2) The basic idea of the above model is to find a robust UC and ED for the base-case scenario. The UC x and dispatch p are immunized against any uncertainty ∈ U. When uncertainty occurs, it is accommodated by the generation adjustment ∆p. Please refer to Appendix A for the detailed formulation. (x,p) s.t. Ax + Bp ≤ b Cx + Dp + G∆pk ≤ d + Eˆ k , ∀k ∈ K (3a) and Z := max min 1> s ∈U (s,∆p)∈R() n R() := (s, ∆p) : s ≥ 0 (SP) (4a) (4b) o G∆p − s ≤ d − Cx − Dp + E (4c) where K is the index set for uncertainty points ˆ which are dynamically generated in (SP) with iterations. Please refer to Appendix B for the detailed formulation. It should be noted that ˆk is the extreme point of U. Variable ∆pk is associated with ˆk . The objective function in (SP) is to find the worst point in U given (x, p). The procedure is 1: K ← ∅, k ← 1, Z ← +∞, define feasibility tolerance δ 2: while Z ≥ δ do 3: Solve (MP), obtain optimal (x̂, p̂). 4: Solve (SP) with x = x̂, p = p̂, get solution (Z, ˆk ) 5: K ← K ∪ k, k ← k + 1 6: end while Once the procedure is converged, we also get the optimal UC and ED solution by solving (MP). Similar to traditional LMP calculation, we fix the binary variables as x̂. Then a convex linear programming problem (RED) can be formed as XX CiP (Pi,t ) (RED) min (5) P,∆P s.t. (λt ) X (β̄i,t ) (αi,t ) ¯ (η̄l,t ) (ηl,t ) ¯ k (β̄i,t ) k (βi,t ) ¯k (ᾱi,t ) k (αi,t ) t i X dm,t , ∀t, m Pi,t ≤ Iˆi,t Pimax , ∀i, t −Pi,t ≤ −Iˆi,t Pimin , ∀i, t Pi,t − Pi,t−1 ≤ riu (1 − ŷi,t ) + Pimin ŷi,t , ∀i, t −Pi,t + Pi,t−1 ≤ rid (1 − ẑi,t ) + Pimin ẑi,t , ∀i, t X inj Γl,m Pm,t ≤ Fl , ∀l, t m X inj − Γl,m Pm,t ≤ Fl , ∀l, t m X X k ∆Pi,t = ˆkm,t , ∀t, ∀k ∈ K m i k Pi,t + ∆Pi,t ≤ Iˆi,t Pimax , ∀i, t, ∀k ∈ K k −Pi,t − ∆Pi,t ≤ −Iˆi,t Pimin , ∀i, t, ∀k ∈ K k ∆Pi,t ≤ Riu (1 − ŷi,t ), ∀i, t, ∀k ∈ K k −∆Pi,t ≤ Rid (1 − ẑi,t+1 ), ∀i, t, ∀k ∈ K Pi,t = i (λkt ) (x,p)∈F min C I (x) + C P (p) (MP) (βi,t ) ¯ (ᾱi,t ) is the budget parameter and assumed as an integer [3]. It is noted that all the flexible resources are modeled as generators. In this paper, the RUC is formulated according to the model in [7]. Λ∆ t Column and Constraint Generation (CCG) based method is used to solve the above model [6]. Problem (MP) and (SP) are established as follows. ¯ X inj inj,k k (η̄l,t ) Γl,m (Pm,t + ∆Pm,t ) ≤ Fl , ∀l, t, ∀k ∈ K (6a) (6b) (6c) (6d) (6e) (6f) (6g) (7a) (7b) (7c) (7d) (7e) (7f) m k (ηl,t ) − ¯ X m inj inj,k Γl,m (Pm,t + ∆Pm,t ) ≤ Fl , ∀l, t, ∀k ∈ K,(7g) 4 where (6a)-(6g) are the constraints for the base ED, and (7a)(7g) are constraints for different extreme points in U. (7a) denotes the load balance after re-dispatch. The generation adjustments respects capacity limits (7b)(7c) and ramping limits (7d)(7e). Network constraints are denoted by (7f)(7g). inj inj,k The Pm,t and ∆Pm,t are defined as inj Pm,t := X i∈G(m) and inj,k ∆Pm,t := X i∈G(m) Pi,t − dm,t , ∀m, t, k ∆Pi,t − ˆkm,t , ∀m, t, k, respectively. B. Marginal Prices In this section, marginal prices for the energy, uncertainty, and generation reserve are derived based on the Lagrangian function. Denote the Lagrangian function for (RED) as L(P, ∆P, λ, α, β, η), which is shown in Appendix C. According to the definition of marginal price [17], the LMP for energy at bus m is ∂L(P, ∆P, λ, α, β, η) (8) ∂dm,t XX X k k =λt − Γl,m η̄l,t − ηl,t − Γl,m η̄l,t − ηl,t ¯ ¯ l l k∈K e πm,t = It is observed that the impact of the uncertainty is also reflected in the LMP. The new concept, UMP for DAM, is defined as the marginal cost of immunizing the next unit increment of uncertainty. For ˆk , an extreme point of U, the UMP is X ∂L(P, ∆P, λ, α, β, η) k k = λkt − Γl,m η̄l,t − ηl,t = k ∂ˆ m,t ¯ l (9) u,k Both the uncertainty and generation reserve are priced at πm,t . u,k In the derivation of πm,t , the worst point ˆk is the only concern. Therefore, the general principles in this paper still work when U is replaced with other sets. It should be noted that (9) is intermediate price signals. In order to get the aggregated UMPs, the following new sets are u,k defined based on the sign of πm,t . u,k πm,t up u,k Km,t := {k : πm,t ≥ 0}; u,k down Km,t := {k : πm,t < 0} (10) The aggregated upward and downward UMPs are defined as u,up πm,t := X up k∈Km,t u,k πm,t ; u,down πm,t := X u,k πm,t (11) down k∈Km,t respectively. In the following context, we will show how the aggregated UMPs are used. C. Market Clearing Mechanism With LMP and UMP, the charges and credits for the market participants become clear and fair in the DAM. Energy clearing is straightforward. The basic principle related to uncertainty and flexibility is that those who cause uncertainties (uncertainty sources), such as RES, pay based on UMP and those who contribute to the management of uncertainties (uncertainty mitigators), such as generators or storage with ramping capabilities, get paid. 1) Energy Payment and Credit: LSEs pay based on the amount of the load and LMP. The energy payment from the e LSE at Bus m at t is πm,t dm,t . It should be noted that RES is entitled to the credit due to the negative load modeled in RUC. Generator i, located at Bus mi , is entitled to the credit e πm P for energy production. i ,t i,t 2) Charge to Uncertainty Source: The uncertainty source can be charged as X u,k Ψm,t = πm,t ˆkm,t (12) k∈K The uncertainty source pays based on the marginal price and the worst point ˆk . The uncertainty source is charged only u,k when πm,t is non-zero, and it may have to pay more when the uncertainty becomes larger. The uncertainty point ˆkm,t can be upward (i.e. ˆkm,t ≥ 0) or downward (i.e. ˆkm,t ≤ 0). We have the following lemma regarding the relation between the u,k signs of πm,t and ˆkm,t . u,k Lemma 1. If ˆkm,t > 0, then πm,t ≥ 0. If ˆkm,t < 0, then u,k πm,t ≤ 0. Please check Appendix D-A for the proof. When the budget set is adopted, the extreme point ˆkm,t ∈ {−um,t , 0, um,t } [4], [7], so the uncertainty charge in (12) can also be written as (13) according to Lemma 1 and (11). u,up u,down Ψm,t = πm,t um,t + πm,t (−um,t ) (13) Thus, upward and downward uncertainties are charged separately. It should be noted that we still need to use (12) when other uncertainty sets are used. 3) Credit to Generation Reserve: Only resources that can provide deliverable generation reserve are entitled to credits. If i ∈ G(m), then the credits can be formulated as X u,k k ΘG πm,t ∆Pi,t . (14) i,t = k∈K In other words, generation reserve is paid the UMP at the bus u,k where it is located. If πm,t = 0, then the associated credit k is zero no matter what the value of ∆Pi,t is. Similar to the uncertainties, the generation reserves can be in either upward or downward direction. Denote the upward generation reserve down as Qup i,t and the downward generation reserve as Qi,t n o ˆi,t Pimax − Pi,t , Riu (1 − ŷi,t ) , (15) Qup := min I i,t n o down Qi,t := max Iˆi,t Pimin − Pi,t , −Rid (1 − ẑi,t+1 ) . (16) We also have the following lemma regarding the relation down k between Qup and ∆Pi,t . i,t , Qi,t 5 k to Lemma 2. If i ∈ G(m), then the optimal solution ∆Pi,t problem (RED) is ( u,k Qup if πm,t >0 k i,t , ∆Pi,t = u,k Qdown , if π m,t < 0 i,t and u,k k k k k πm,t = β̄i,t − βi,t + ᾱi,t − αi,t , (17) ¯ ¯ Please check Appendix D-B for the proof. The credit to generation reserve i located at bus m (14) can be rewritten as (18) according to Lemma 2 and (11). u,up up u,down down ΘG i,t = πm,t Qi,t + πm,t Qi,t (18) (18) shows that the upward and downward generation reserves are credited separately. Flexible resources may receive credits for both the upward and downward generation reserves simultaneously. (18) always holds even if other uncertainty sets are modeled in RUC. D. Transmission Reserve and Revenue Adequacy Some transmission capacities are reserved according to the solution to (RED). These transmission reserves are used to ensure the ramping deliverability when the uncertainty is revealed, as shown in (7f) and (7g). It is noted that they are determined automatically in (RED), and kept explicitly without explicit transmission reserve requirement constraints. Just like the “scheduled” generation reserve, the “scheduled” transmission reserves in positive direction and negative direction are X pos inj , (19) ∆fl,t := Fl − Γl,m Pm,t neg ∆fl,t := Fl + m X inj , Γl,m Pm,t (20) m respectively. They are always non-negative. An important issue related to the transmission reserve is the credit entitled to the Financial Transmission Right (FTR) holders. FTR is a financial instrument used to hedge congestion cost in the electricity market, where participants are charged or credited due to the transmission congestion [21], [25]. Within the robust framework, the effective transmission capacity for base-case scenario is different from the physical limit, which is used in the Simultaneous Feasibility Test (SFT) for FTR market [18], [20], [21]. In the existing market, the FTR credit is funded by the energy congestion cost, which is the net payment of energy. However, the energy congestion cost may not be sufficient to fund the FTR credit [18], [19]. We argue that the transmission reserve becomes a new reason for FTR underfunding in any framework to guarantee the ramping deliverability. pos neg Theorem 1. If transmission reserve ∆fl,t and ∆fl,t are kept for line l at time t in DAM, then the maximum FTR underfunding associated with line l at time t is X pos neg k k η̄l,t ∆fl,t + ηl,t ∆fl,t (21) ¯ k∈K due to the deficiency of energy congestion cost. Uncertainty Payment Gen. Res. Credit Energy Payment Trans. Res. Credit LMP Cong. Cost Energy Credit FTR Credit Fig. 1. Money flow of the proposed market clearing mechanism, where uncertainty sources make the uncertainty payment, and LSEs make the energy payment. Please check Appendix D-C for the proof. From the FTR holder’s point of view, (21) is the credit due to the transmission reserve. Therefore, we also call (21) transmission reserve credit, and denote it as X pos neg k k η̄l,t ∆fl,t ∆fl,t ΘTl,t := + ηl,t . (22) ¯ k∈K k k At most one of η̄l,t and ηl,t is non-zero for transmission l. ¯ The credit to positivePtransmission reserve is zero for line l at pos k time t, when either k∈K η̄l,t = 0 or ∆fl,t = 0. Theorem 2. If (RED) is feasible, then uncertainty payment can exactly cover generation reserve credit and transmission reserve credit, and the revenue adequacy is always guaranteed in the proposed market clearing mechanism. Please check Appendix D-D for the proof. Theorem 1 reveals that FTR underfunding issue can occur within the existing market structures as long as the transmission reserve is non-zero, even if the LMPs are calculated based on other approaches. Theorem 2 shows that the new market clearing mechanism overcomes the FTR underfunding issue. The money flow of the proposed market clearing mechanism is depicted in Fig.1. Energy payment collected based on LMP is distributed to FTR holders as LMP congestion cost and generators as energy credit. On the other hand, the payment collected based on UMP is distributed to FTR holders as transmission reserve credit and flexible resources as generation reserve credit. The LMP congestion cost and transmission reserve credit can exactly cover the FTR credit, which is calculated based on the LMP difference and FTR amount. E. Market Equilibrium In this section, we characterize the competitive market equilibrium model. In the electricity industry, the partial market equilibrium model is often employed [1], [13], [26], where market participants are price takers [27]. The energy is cleared according to (6a). Uncertainty and generation reserve are cleared according to (7a). Without loss of generality, consider unit i located at bus m. Its profit maximization problem can be formulated as ! u,up up u,down down e X Pi,t πm,t + πm,t Qi,t + πm,t Qi,t (PMPi )max Pi,t −CiP (Pi,t ) t s.t. (6b) − (6e), (15) − (16) 6 where the decision variable is Pi,t given the price signal u,up u,down e (πm,t , πm,t , πm,t ). As proved in Appendix D-E, unit i is not inclined to change its power output level as it can obtain the maximum profit by following the ISO’s dispatch e instruction P̂i,t . Price signal πm,t provides the incentives for u,k unit i to dispatch power output to P̂i,t , and price πm,t gives incentives for unit i to maintain the generation reserve for uncertainty. Hence, the dispatch instruction P̂i,t and price u,up u,down e signal (πm,t , πm,t , πm,t ) constitute a competitive partial equilibrium [27]. F. Discussions down k As the Pi,t and ∆Pi,t (or (Qup i,t , Qi,t )) are coupled by k k (7b) and (7c), the opportunity cost (β̄i,t − βi,t ) is enough to ¯ provide the incentives for i to keep the generation level at P̂i,t . k k Including ᾱi,t − αi,t in the generation reserve price has several ¯ benefits. Firstly, generation reserves provided by different units are priced fairly. Generation reserve prices are the same for the units at the same bus, and they may vary with locations if line congestions exist. Secondly, higher generation reserve price attracts long-term investment for flexible resources. Thirdly, it is consistent with the existing reserve pricing practice [2], [28]. In fact, generation reserve price is consistent with the UMP. Therefore, the uncertainties and flexibilities are also treated fairly at the same bus. The upward and downward UMPs are obtained according to (11), respectively. The uncertainty sources are charged according to (13). The generation reserves are credited according to u,k k defined in (9) and re-dispatch ∆Pi,t (18). The price signal πm,t are intermediate variables for market clearing. The proposed UMP may be non-zero even if the uncertainty at a bus is zero. This is similar to the LMP, which may also be non-zero for the bus without load. The market clearing mechanism proposed in this paper follows the cost causation principle for the cost allocation. In reality, it may be controversial to allocate the reserve cost to uncertainty sources. However, we argue that it would be fair and must be done when the RES penetration level is high. An extreme case is when the loads are all supplied by RES. There has been study showing it is possible that the increasing RES penetration can cause higher system operation cost. This issue cannot be handled by the existing market clearing mechanism, in which loads pay for the additional system reserve that is required to accommodate the uncertainty from RES. In other words, loads are actually providing subsidies to RES. When the RES penetration level is low, the subsidies can help the growth of the RES. However, when the RES penetration level is high, these growing subsidies will cause serious fairness issue. On the other hand, with UMP as the stimulating price signals, RES will have the incentives to improve its forecast techniques and reduce its uncertainty. In the ideal case when its uncertainty approaches zero, RES will no longer pay. Following the existing practice, the UC variables are fixed during the marginal price derivation. Hence, the uplift issue, which exists in the real market, still remains in the proposed market clearing mechanism. Although the UC variables are fixed, the LMP and reserve price in the real market can provide effective signals for the long-term investment of generation and transmission as well as consumption strategy of electricity. Similarly, the uncertainty impact is not only reflected in UC, but also in the ED within the RUC model in this paper. Hence, the proposed LMP and UMP can also provide signals for the long-term investment of flexibilities (i.e. generation, transmission, and demand). The pricing for uncertainties proposed in this paper is not in conflict with the pricing for traditional reserves, which are mainly prepared for the contingencies. The traditional reserve prices can be derived in the framework by adding extra traditional reserve constraints, and the corresponding reserve costs can still be allocated to LSEs. It is observed that the credit in (14) is the sum of credits for all extreme points. That is because the related constraints may be binding for multiple extreme points, and the dual variables (shadow prices) for these constraints work together in the dual problem. The traditional price for energy and reserve also has similar form when multiple contingencies are modeled. Although only one scenario will happen in reality, we still need to consider the worst scenario defined in uncertainty set and keep enough reserves in DAM. That is because DAM is a financial market, and the LMP and UMP are the financially binding prices. This is similar to the existing market model considering contingencies. Even if the contingency seldom occurs, they are still modeled for market clearing, and the contingencies are reflected in LMP and reserve price. The issue of price multiplicity still exists in the proposed model [29] because problem (RED) is a linear programming (LP) problem. However, the price is unique with the nondegeneracy assumption. For simplicity, we have considered a single-sided auction in the proposed model. By introducing the demand bids, we can formulate a double-sided auction and the general principles in this paper will still apply. III. C ASE S TUDY A six-bus system and the IEEE 118-Bus system are simulated to illustrate the proposed market clearing mechanism. In the six-bus system, the basic ideas of UMP are presented within the robust optimization framework. FTR underfunding issue is illustrated and a comparison between the UMP and traditional reserve price are presented. In the IEEE 118-Bus system, the UMP related products are presented for different uncertainty levels. The behaviors and impacts of flexible sources are analyzed by an energy storage example. A. Six-bus System A six-bus system is studied in this section. The one-line diagram is shown in Fig. 2. The unit data and line data are shown in Table I and Table II, respectively. Table III presents the load and uncertainty information. Column “Base Load” shows the hourly forecasted load. Assume that the load distributions are 20%, 40%, and 40% for Bus 3, Bus 4, and Bus 5, respectively. ū1,t and ū3,t in Table III are the bounds of the uncertainties at Bus 1 and Bus 3, respectively. The uncertainty bounds at other buses are 0. 7 1 G1 G2 2 L1 TABLE IV M ARGINAL C OSTS AT D IFFERENT G ENERATION L EVELS ($/MW H ) 3 Gen. 1 P1w 6 5 4 L3 L2 ¯ 100 124 148 172 196 G3 Fig. 2. One-line diagram for 6-bus system. TABLE I U NIT DATA FOR THE 6- BUS S YSTEM # P min P max P0 1 100 2 10 6 10 a b c 4 3 1 4 2 1 4 3 −2 P min ,P max ,P0 : min/max/initial generation level (MW); fuel cost ($): aP 2 + bP + c ; Ru ,Rd : ramping up/down rate (MW/h); Cu ,Cd : startup/shutdown cost ($); T on ,T off ,T0 : min on/min off/initial time (h) It is assumed that the relative forecasting errors increase with hours. Uncertainty 1,t and 3,t also respect (23a) −Λ · ūm,t ≤ m,t ≤ Λ · ūm,t , ∀t, m X \|m,t \| ≤ Λ∆ , ∀t, ūm,t m (23b) where (23a) denotes the uncertainty interval at a single bus, and (23b) represents the system-wide uncertainty [5]. The Λ and Λ∆ are the budget parameters for the single bus and system, respectively. 1) LMP and UMP: Consider the case where Λ = 1, Λ∆ = 2. The CCG based approach converges after 2 iterations. TABLE II L INE DATA FOR THE 6- BUS S YSTEM from 1 1 2 5 3 2 4 to 2 4 4 6 6 3 5 x(p.u.) 0.17 0.258 0.197 0.14 0.018 0.037 0.037 capacity(MW) 200 100 100 100 100 200 200 TABLE III L OAD AND U NCERTAINTY DATA FOR THE 6- BUS S YSTEM (MW) Time (h) Base Load 1 2 3 4 5 6 7 8 9 10 11 12 175.19 165.15 158.67 154.73 155.06 160.48 173.39 177.6 186.81 206.96 228.61 236.1 ū1,t 1.09 2.06 2.98 3.87 4.85 6.02 7.59 8.88 10.51 12.94 15.72 17.71 ū3,t Time (h) Base Load 0.29 0.55 0.79 1.03 1.29 1.6 2.02 2.37 2.8 3.45 4.19 4.72 13 14 15 16 17 18 19 20 21 22 23 24 242.18 243.6 248.86 255.79 256 246.74 245.97 237.35 237.31 232.67 195.93 195.6 mar. cost 124 148 172 196 220 14.396 14.588 14.78 14.972 15.164 P2w P̄2w ¯ 10 28 28 46 46 64 64 82 82 100 Gen. 3 mar. cost 32.638 32.674 32.71 32.746 32.782 P3w ¯ 10 12 14 16 18 P̄3w mar. cost 12 14 16 18 20 17.71 17.73 17.75 17.77 17.79 TABLE V G ENERATION AND R ESERVE (Λ = 1, Λ∆ = 2, MW) Ru Rd Cu Cd T on T off T0 220 120 0.004 13.5 176.9 24 24 180 50 100 50 0.001 32.6 129.9 12 12 360 40 20 0 0.005 17.6 137.4 5 5 60 0 Gen. 2 P̄1w ū1,t ū3,t 19.68 21.32 23.33 25.58 27.2 27.76 29.21 29.67 31.15 31.99 28.16 29.34 5.25 5.68 6.22 6.82 7.25 7.4 7.79 7.91 8.31 8.53 7.51 7.82 T P1 P2 P3 up up up Q1 Qdown Q2 Qdown Q3 1 2 21 195.19 25.58 16.54 24 −24 12 −12 3.46 Qdown 3 −5 Hence, K = {1, 2}. Given the UC solutions, the problem (RED) can be solved by commercial LP solver. The marginal prices are then obtained as byproducts. The generation outputs are presented in Table V at Hours 21. It can be observed that G1 supplies most of the loads at Hour 21, which is 195.19 MW. According to the bid information in Table IV, G2 is much more expensive than G1 and G3. Hence, the output of G2 is relatively small and at the low level of its capacity. The upward and downward generation reserves provided by the three units are also listed in Table V. These data can be obtained directly from Eqs. (15) and (16) given the generation output Pi,t . Although the remaining generation capacity of G1 is 220 − 195.19 = 24.62 MW, the upward reserve is limited by its upward ramping rate 24 MW. In the meantime, the upward reserve provided by G3 is limited by its generation capacity although it has more remaining ramping capacity (i.e. min{20 − 16.54, 5} = 3.46 MW). Table VI shows the extreme points obtained in the CCGbased approach. The intermediate price signals for these points u,k πm,t are also presented. It can be observed that the worst point is always obtained at the extreme point of the uncertainty set. For example, at Hour 21, the ˆ11,1 is 31.15 MW. It is exactly the upper bound of the uncertainty at Hour 21 at Bus 1. The data in Table VI also verifies Lemma 1. The intermediate UMPs u,k πm,t have the same sign as the uncertainties ˆkm,t at the same bus. The LMPs, aggregated upward UMPs, and aggregated downward UMPs at Hour 21 are shown in Table VII. It is noted that UMPs still exist at buses without uncertainties (i.e., Buses 2,4,5,6). This is similar to LMPs, which also exist at buses where net power injections are 0. The LMPs vary with locations, which indicates that the line congestion exists. TABLE VI E XTREME P OINTS OF U NCERTAINTY S ET k=1 t ˆ11,t ˆ13,t u,1 π1,t k=2 u,1 π3,t ˆ21,t ˆ23,t u,2 π1,t u,2 π3,t 21 31.15 8.31 14.87 14.87 −31.15 8.31 −17.67 1.77 8 TABLE VII LMP AND UMP AT H OUR 21 (Λ = 1, Λ∆ = 2) Bus1 Bus2 Bus3 Bus4 Bus5 Bus6 πe π u,up 14.97 14.87 -17.67 32.64 14.87 0 34.4 16.63 0 43.71 25.94 0 41.94 24.17 0 35.26 17.49 0 π u,down 10 16 Price $/MW Price Price $/MW 16.5 15.5 15 0 2 4 Bus The load at Bus 4 has to pay the highest LMP $43.71/MWh. The UMPs are also different at various locations. The highest upward UMP at Hour 21 is also located at Bus 4. With these prices, the market participants can be paid and credited. The LMP paid to G3 is $35.26/MWh on Bus 6, which is $17.49/MWh larger than its marginal cost. In the same time, The upward UMP is $17.49/MWh on Bus 6, which is exactly the difference between the LMP and G3’s marginal cost. Hence, G3 is the UMP setter on Bus 6. The UMPs provide important price signals on the planning of renewable energy sources and storages. For example, the UMP at Bus 2 is relatively small, so it is an ideal location for renewable energy sources in terms of payment for uncertainties. In contrast, the UMP at Bus 4 is large, which may attract the long-term investment for storages or generation plants with large ramping rates. 2) Comparison with Existing LMPs and Reserve Prices: The motivation of this part is to compare the proposed clearing scheme with the existing one. However, as the reserve is not robust in the traditional scheme, we cannot compare them fairly. With the observation that the transmission constraints are the most challenging one in the robust UC framework, we drop these constraints in this subsection and add reserve constraints as follows. Ii,t Pimin ≤ Qdown i,t + Pi,t , max Qup , ∀i, t (24a) i,t + Pi,t ≤ Ii,t Pi u −Rid Ii,t ∆T ≤ Qdown Qup i,t , i,t ≤ Ri Ii,t ∆T, ∀i, t X X up Qdown Qi,t ≥ R̄t , ∀t, i,t ≤ Rt , ¯ i i (24b) (24c) down where Qup are the largest upward and downward i,t and Qi,t reserves, respectively. Rt and R̄t are system-wide reserve ¯ requirements. Refer to [2], [30] for more details on the reserve formulations. In the experiment, ∆T is set to 1 and Λ = 0.8, Λ∆ = 2. The reserve requirements Rt and R̄t are set ¯ to the lower and upper bounds of the system-wide uncertainty in (23b), respectively. The results are as expected. The optimal solutions of the RUC and the standard UC with explicit reserve constraints are the same. LMPs calculated in the RUC and UC also have the same values. The UMPs calculated in the proposed mechanism are also exactly the same as the reserve prices in standard UC. Two things are verified with these results. First, without transmission constraints, the solution to standard UC can easily be robust by adding reserve constraints. Second, the proposed LMPs and UMPs are consistent with LMPs and reserve prices in the existing market when the transmission constraints are dropped. When considering transmission constraints, the generation reserve cannot be guaranteed at bus levels in the traditional 5 (a) Hour 20 6 2 4 6 Bus (b) Hour 21 Fig. 3. Upward UMP (blue bar) and reserve price (red bar) with network constraint. UC model. For simplicity, we assume that the 6 buses are in a zone. Consider the case where Λ = 0.8, Λ∆ = 2. The upward UMP and reserve price at Hour 20 are depicted in Fig. 3a. It is observed UMPs at Bus 1 and Bus 2 are lower than the traditional reserve prices. In the same time, the UMPs at Bus 4 and Bus 5 is higher than the traditional reserve prices. The differences are caused by the congestion of Line 1-4 for reserve delivery. It is worth mentioning that the LMP differences in two models are within 1% at Hour 20. The prices illustrated in Fig. 3b reveals another trends that the UMP may be higher than the traditional reserve prices. At Hour 21, the zonal reserve price is 0 while the UMPs are nonzeros at bus 4, 5, and 6. Because the constraint related with reserves in the RUC is stronger than the one in traditional UC model. Consequently, more expensive resources are used in RUC, which also generally leads to higher UMPs. 3) FTR Underfunding: When Λ∆ = 2, Λ = 1, the generation schedules at Hour 21 are 195.193MW, 25.577MW, and 16.54MW. The power flow of Line 2 is 97.63MW, which is 2.47MW smaller than its physical limit of 100MW. The transmission reserve 2.47 MW is kept to guarantee the delivery of the generation reserve. The binding constraint for Line 2 causes LMP differences. Hence, the FTR holder gets credits. Consider a set of FTR amounts [202.3429, 23.2771, −55.772, −94.924, −94.924, 20]. It can be verified that the FTR amounts satisfy the SFT in the FTR market. Then the total credit for the FTR holders is $5,554.77. However, the congestion cost in the DAM is $5,422.87. It means that the LMP congestion cost collected is not enough to cover the FTR credit. The FTR underfunding value is $5554.77 − $5422.87 = $131.90. The revenue residues after UMP settlement is $131.9. It exactly covers the FTR underfunding in this scenario. Therefore, the revenue is adequate at Hour 21. B. IEEE 118-Bus System The simulations are performed for the IEEE 118-bus system with 54 thermal units and 186 branches in this section. The peak load is 6600MW. The detailed data including generator parameters, line reactance and ratings, and load profiles can be found at http://motor.ece.iit.edu/Data/RUC118UMP.xls. Two cases are studied in this section. 1) The uncertainty levels and load levels are changed to analyze the simulation results in the system level. The impact of transmission line capacity on prices is also studied. 9 ·104 TABLE VIII O PERATION C OST AND UMP PAYMENT ($,Λ∆ = 10) ·106 UP GRC 2 OC Op. Cost Un. Payment Gen. Res. Credit Rev. Res. 0.2 0.25 0.3 1,866,023 1,871,364 1,877,471 11,043 20,044 30,879 10,560 19,209 28,658 483 835 2,221 1.9 3 OC ($) Λ UP and GRC ($) 4 2 1.8 2) An energy storage is installed at a specified bus with high UMP to show the potential application of UMPs. 95 100 105 Load Level (%) Fig. 5. Uncertainty payment (UP), generation reserve Credit (GRC), and operation cost (OC) with different load levels (Λ = 0.25) Upward UMP LMP 50 30 40 Price $/MW Price $/MW 1) Case 1: We assume that the uncertainty sources are located at buses (11, 15, 49, 54, 56, 59, 60, 62, 80, 90). The budget parameter Λ∆ is set to 10 in this section. The buslevel uncertainty budget parameter Λ changes from 0.2 to 0.3, and the bound of the uncertainty is the base load. The simulation results are shown in Table VIII. It can be observed that the total operation cost increases with increasing Λ. It indicates that a larger uncertainty level may increase the operation cost. The columns “Un. Payment” and “Gen. Res. Credit” denote the total payment from uncertainty sources and credit to generation reserves, respectively. The lowest payment is $11,043 and the highest one is $30,879. On the other hand, the credit entitled to the generation reserves is also a monotonically increasing function of Λ. When Λ = 0.3, the generation reserves have the highest credit. The last column “Rev. Res.” shows that the revenue residues related to UMPs. It can be observed that the residue is always positive. Fig. 4a in the next page depicts the heat map for the upward UMPs from Bus 80 to Bus 100 in 24 hours. The xaxis represents time intervals and the y-axis represents bus numbers. The color bar on the right shows different colors for various UMP values. For example, the $0/MWh is denoted by the blue color at the bottom, and the $18/MWh is represented by the dark red color on the top of the color bar. It can be observed that the uncertainty sources have system-wide unique UMPs at some intervals, such as Hours 8, 13, 15, and so on. It indicates that there is no transmission reserve in these hours. On the other hand, the UMPs at Hour 11 vary dramatically with different locations. The highest upward UMP is around $18/MWh, and the lowest one is around $2/MWh. According to the data shown in Fig. 4a, the high UMP at Bus 94 may attract investment of flexible resources, such as energy storages, in terms of generation reserve credit, and Bus 100 is an attractive location for the investment of renewable energy sources in terms of uncertainty payments. Fig. 5 shows the uncertainty payment and generation reserve credit with respect to load levels. The base load level is set at 100%. Higher loads in general lead to more uncertainty payments and generation reserve credits. It is also consistent with the heat map of UMPs in Fig. 4a, where UMPs at peak load hours are high. It suggests that the generation reserves also become scarce resources when load levels are high. The transmission line capacity plays an important role in the price calculation. Fig. 6 shows the LMPs and upward UMPs at Hour 11 with respect to increasing capacity of Line 94100. The prices at Buses 88, 94, and 100 are depicted. When the line capacity increases from 165MW to 175MW, LMP at 1 30 20 10 20 0 170 180 170 Line Capacity 180 Line Capacity Bus 88 Bus 94 Bus 100 Fig. 6. LMP (left) and upward UMP (right) at Hour 11 with respect to increasing capacity of Line 94-100 Bus 94 decreases from $47.92/MWh to $35.84/MWh and that at Bus 88 also drops to $30.58/MWh from $38.78/MWh. The upward UMPs at Bus 94 and Bus 88 also drop by $8.20/MWh and $12.08/MWh, respectively. In contrast, the LMP and upward UMP at Bus 100, which is connected to Line 94-100, remain at $19.42/MWh and $1.64/MWh, respectively. It shows that the change of line capacity may only have impacts on the prices at some buses. When the line capacity further increases to 185MW from 175MW, the changes of LMPs and UMPs at Bus 94 and Bus 88 are within $0.1/MWh, and there is still no change at Bus 100. It means that the additional 10MW cannot help deliver cheaper energy and reserves to Bus 94 and Bus 88. These results are also consistent with the analysis of the traditional LMPs [31]. 2) Case 2: As discussed in Case 1, the upward UMP on Bus 94 is high at Hour 11. Assume that an energy storage (8MW/30MWh) is installed at Bus 94. A simple model for the energy storage is formulated as follows. Et = Et−1 + ρd PtD + ρc PtC , ∀t 0 ≤ Et ≤ E max , ∀t 0 ≤ −PtD ≤ ItD RD , ∀t 0 ≤ PtC ≤ ItC RC , ∀t ItD + ItC ≤ 1, ∀t ENT = E0 , where Et denotes the energy level, PtD and PtC represent the discharging and charging rates, and ItD and ItC are the indicators of discharging and charging. As the UMP is the major concern in this section, we use simplified parameters for 10 18.00 18.00 Bus 82 Bus 82 16.00 16.00 Bus 84 Bus 84 14.00 14.00 Bus 86 Bus 86 12.00 12.00 Bus 88 Bus 88 10.00 Bus 90 8.00 Bus 92 10.00 Bus 90 8.00 Bus 92 6.00 Bus 94 6.00 Bus 96 4.00 Bus 96 4.00 Bus 98 2.00 Bus 98 2.00 Bus 94 Bus 100 Hour 5 Hour 10 Hour 15 Hour 20 0.00 Bus 100 Hour 5 Hour 10 Hour 15 Hour 20 0.00 (b) With Storage at Bus 94 (a) Without Storage at Bus 94 Fig. 4. Heat Map for Upward UMPs (Λ = 0.3). Different colors represent various UMPs. Figure (a) depicts the UMPs from Bus 80 to Bus 100 in 24 hours without the energy storage at Bus 94. Figure (b) depicts the new UMPs after the energy storage is sited at Bus 94. storage. The discharging efficiency ρd and charging efficiency ρc are set to 100%. The capacity E max and initial energy level E0 are set to 30 MWh and 15 MWh, respectively. The maximal charging rate RD and discharging rate RC are set to 8 MW/h. By siting the energy storage, we can lower the new operation cost to $1,875,211 from $1,877,471. The payment collected from the uncertainty sources becomes $27,473, and the credit to generation reserves decreases to $24,289. Compared to the data in Table VIII, the energy storage also helps to reduce the payment related to UMPs. The storage is entitled to $1326 generation reserve credit. Fig. 4b depicts the new upward UMPs after the installation of the energy storage. Compared to that in Fig. 4a, the upward UMP for Hour 11 at Bus 94 decreases a lot. The UMPs for Hour 10 and 12 are also lower. It suggests that sitting the energy storage at Bus 94 effectively lower the generation reserve price. The simulation results demonstrate that flexible resources can lower the UMPs, and UMPs provide the investment signal at locations where generation reserves are scarce resources. prices within the new market scheme, as the reserve fees are paid by uncertainty sources. Many potential applications on UMPs are open. As UMPs are unified prices of uncertainties and reserves, it is interesting to investigate the optimal strategy for the one who is the uncertainty source as well as reserve provider in the market (e.g. the wind generation company with energy storages). The UMPs derived in this paper also provide an important price signal for the long-term investment of flexible resources. When the upward UMP or downward UMP at a bus is high, the investor can get more return in terms of generation reserves. Another potential future research on UMP is to study how to determine the budget uncertainty set in the market. Modeling the traditional spinning reserve for the contingency [13], [30] is also our future work. In this paper, the demand bids are not considered. We have forecasted load, forecasted RES, and uncertainty of load and RES for market clearing with a singlesided model. In an extended double-sided model, we can have demand bids, forecasted RES, and uncertainty of load and RES for market clearing. The forecasted load, forecasted RES, uncertainty of load and RES can be used in RAC. IV. C ONCLUSIONS A novel market model in this paper clears uncertainty, energy, and generation reserve simultaneously within the RUC framework in DAM. The uncertainty sources are charged and the generator reserve providers are credited based on the proposed UMP. The UMP formulation is derived within a robust optimization framework. We also characterize the market equilibrium for the new market clearing mechanism. As the market clearing mechanism is established within the robust optimization framework, the robustness of the dispatch is guaranteed. The optimal reserves for uncertainty accommodation are obtained in the model. The UMP proposed in this paper can effectively address the issue on how to charge and credit the uncertainties and generation reserve fairly in the market with RES. Our study also shows that traditional pricing mechanism within RUC framework may lead to FTR underfunding. The proposed market clearing mechanism can address this issue. Our study shows load serving entities can have lower energy A PPENDIX A D ETAILED F ORMULATION FOR P ROBLEM (RUC) (RUC) s.t. min (x,y,z,I,P )∈F X Pi,t = X Ii,t Pimin m t X m i −Fl ≤ XX i CiP (Pi,t ) + CiI (Ii,t ) dm,t , ∀t.  Γl,m  X i∈G(m) (25a) (25b)  Pi,t − dm,t  ≤ Fl , ∀l, t (25c) ≤ Pi,t ≤ Ii,t Pimax , ∀i, t Pi,t − Pi,(t−1) ≤ riu (1 − yi,t ) + Pimin yi,t , ∀i, t −Pi,t + Pi,(t−1) ≤ rid (1 − zi,t ) + Pimin zi,t , ∀i, t minimum on/off time limit (26a) (26b) (26c) 11 inj ∆Pm,t = and n F := (x, y, z, I, P ) : ∀ ∈ U, ∃∆P such that X X ∆Pi,t = m,t , ∀t, i∈G(m) (27a) m i Ii,t Pimin ≤ Pi,t + ∆Pi,t ≤ Ii,t Pimax , ∀i, t (27b) −Rid (1 − zi,t+1 ) ≤ ∆Pi,t ≤ Riu (1 − yi,t ), ∀i, t (27c) X inj ∆Pm,t = ∆Pi,t − m,t , ∀m, t (27d) i∈G(m) −Fl ≤ o inj inj Γl,m (Pm,t + ∆Pm,j ) ≤ Fl , , ∀l, t . (27e) X m The basic idea of the above model is to find a robust UC and dispatch for the base-case scenario. In the base-case scenario, (25b) denotes the load balance constraint; (25c) represents the transmission line constraint; (26a) denotes the unit capacity limit constraint; (26b)-(26c) denote the unit ramping up/down limits. Ii,t , yi,t , and zi,t are the indicators of the unit being on, started-up, and shutdown, respectively. Units also respect the minimum on/off time constraints which are related to these binary variables [1]. The UC and dispatch solution are immunized against any uncertainty ∈ U. When uncertainty occurs, it is accommodated by the generation adjustment ∆Pi,t (27a). Generation dispatch is also enforced by the capacity limits (27b). (27c) models the ramping rate limits of generation adjustment ∆Pi,t . In fact, the right and left hand sides of (27c) can correspond to a response time ∆T , which is similar to the 10-min or 30-min reserves in the literatures [30]. (27e) stands for the network constraint after uncertainty accommodation. A PPENDIX B D ETAILED F ORMULATION FOR P ROBLEM (MP) AND (SP) (MP) S.T. X i min (x,y,z,I,P,∆P ) XX t CiP (Pi,t ) + CiI (Ii,t ) i (25b), (25c), (26a)-(26c), minimum on/off time limit X k ∆Pi,t = km,t , ∀t, ∀k ∈ K (28a) m k Ii,t Pimin ≤ Pi,t + ∆Pi,t ≤ Ii,t Pimax , ∀i, t, ∀k ∈ K (28b) k −∆Pi,t (28d) k ∆Pi,t ≤ Riu (1 − yi,t ), ∀i, t, ∀k ∈ K −Fl ≤ Rid (1 ≤ X m inj,k ∆Pm,t = − zi,t+1 ), ∀i, t, ∀k ∈ K (28e) k ∆Pi,t − km,t , ∀m, t, ∀k ∈ K, (28f) i∈G(m) and (28c) inj inj,k Γl,m (Pm,t + ∆Pm,t ) ≤ Fl , ∀k ∈ K, ∀l, t X (SP) max ∈U min (s+ ,s− ,∆P )∈R() XX − (s+ m,t + sm,t ) (29a) m t n + − R():= (s , s , ∆P ) : X X − ∆Pi,t = (m,t + s+ m,t − sm,t ), ∀m, t i −Fl ≤ X m m X (29b) (29c) inj inj + ∆Pm,t ≤ Fl , ∀l, t (29d) Γl,m Pm,t − ∆Pi,t − (m,t + s+ m,t − sm,t )(29e) − s+ m,t , sm,t ≥ 0, ∀m, t o (27b), (27c) (29f) where K is the index set for uncertainty points ˆ which are dynamically generated in (SP) with iterations. It should be k noted that ˆk is the extreme point of U. Variable ∆Pi,t is k associated with ˆ . The objective function in (SP) is the − summation of non-negative slack variables s+ m,t and sm,t , which evaluates the violation associated with the solution − (x, y, z, I, P ) from (MP). s+ m,t and sm,t are also explained as un-followed uncertainties (generation or load shedding) due to system limitations. A PPENDIX C L AGRANGIAN F UNCTION FOR P ROBLEM (RED) Please check equation (30) in the next page. A PPENDIX D P ROOFS FOR LEMMAS AND THEOREMS A. Proof of Lemma 1 u,k < 0. With a small perturbation Proof. Consider ˆkm,t > 0, πm,t δ > 0 to ˆkm,t , we replace ˆkm,t with ˆkm,t − δ in (RED). u,k As the πm,t < 0 , then the optimal value to problem (RED) increases. It means that there are violations for the original optimal solution Pi,t to problem (RED) with ˆkm,t − δ. Hence, the optimal solution Pi,t to problem (RED) cannot be immunized against the uncertainty ˆkm,t −δ. It contradicts with the robustness of the solution Pi,t . Therefore, if ˆkm,t > 0, then u,k u,k ≤ 0. πm,t ≥ 0. Similarly, if ˆkm,t < 0, then πm,t B. Proof of Lemma 2 Proof. Assume i ∈ G(m), according to the KKT condition ∂L(P, ∆P, λ, α, β, η) =0 k ∂∆Pi,t (31) at the optimal point, we have k k k k β̄i,t − βi,t + ᾱi,t − αi,t − λkt + ¯ ¯ X l k k (η̄l,t − ηl,t )Γl,m = 0. (32) ¯ u,k k k Then (17) holds. If πm,t > 0, then β̄i,t + ᾱi,t > 0 as k k k k β̄i,t , βi,t , ᾱi,t , and αi,t are non-negative. According to the ¯ ¯ complementary conditions for (7b) and (7d), at least one of (7b) and (7d) is binding. Hence, n o k ∆Pi,t = min Iˆi,t Pimax − Pi,t , Riu (1 − ŷi,t ) u,k holds. Similarly, the other equation holds when πm,t < 0. 12 L(P, ∆P, λ, α, β, η) XX XX X X X CiP (Pi,t ) + λt dm,t − Pi,t + β̄i,t (Pi,t − Iˆi,t Pimax ) + βi,t (Iˆi,t Pimin − Pi,t ) ¯ t t m t i i i XX min min + ᾱi,t Pi,t − Pi,t−1 − riu (1 − ŷi,t ) − Pi,t ŷi,t + αi,t Pi,t−1 − Pi,t − rid (1 − ẑi,t ) − Pi,t ŷi,t ¯ t i X X X X XX X X inj inj k + η̄l,t Γl,m Pm,t − Fl − ηl,t Γl,m Pm,t + Fl + λkt km,t − ∆Pi,t ¯ t m m m i l k∈K t X XX k k max k ˆ min k ˆ + β̄i,t (Pi,t + ∆Pi,t − Ii,t Pi ) + βi,t (Ii,t Pi − Pi,t − ∆Pi,t ) ¯ i k∈K t X XX k k k k ∆Pi,t + Rid (1 − ẑi,t ) ∆Pi,t − Riu (1 − ŷi,t ) − αi,t + ᾱi,t ¯ i k∈K t X X XX X inj inj,k inj inj,k k k + η̄l,t Γl,m (Pm,t + ∆Pm,t ) − Fl − ηl,t Γl,m (Pm,t + ∆Pm,t ) + Fl ¯ m m k∈K t l = C. Proof of Theorem 1 Proof. The energy congestion cost at t is X X e e πm,t dm,t − πm,t Pi,t m = X (33) i∈G(m) m XX X X k η̄l,t k ηl,t − ηl,t − ¯ m l k∈K k∈K ¯ ! X X pos k = η̄l,t + η̄l,t Fl − ∆fl,t = Γl,m l − = = η̄l,t + k∈K X X l X k ηl,t ηl,t + ¯ k∈K ¯ ! X neg ∆fl,t − Fl X k k η̄l,t + ηl,t + ηl,t ¯ ¯ l k∈K k∈K XX pos neg k k − η̄l,t ∆fl,t + ηl,t ∆fl,t ¯ l k∈K X pos neg − η̄l,t ∆fl,t + ηl,t ∆fl,t ¯ l X η̄l,t + X k k η̄l,t + ηl,t + ηl,t ¯ ¯ l k∈K k∈K XX pos neg k k − η̄l,t ∆fl,t + ηl,t ∆fl,t ¯ l k∈K η̄l,t + X ! inj Pm,t ! ! Fl Fl The first equality holds following the definition of net power injection. The second equality holds according to (8) and P inj P following (19) and m m,t = 0. The third equality Pholds k (20). The sign change of ηl,t and l ηl,t in the third equality ¯ direction. According is because of the definition¯ of power flow to the complementary conditions, the third term in the fourth equality must be zero based on the following three cases. pos ∆fl,t 3) η̄l,t = 0 and ηl,t = 0. ¯ The second term in the last equality corresponds to (21). The credits to FTR holders can be written as X e e (πm,t − πn,t )FTRm→n (34) (m→n) inj e πm,t Pm,t 1) If η̄l,t 6= 0, then = 0, and ηl,t = 0. neg 2) If η̄l,t = 0 and ηl,t 6= 0, then ∆f¯l,t = 0. ¯ (30) X  Γl,m (η̄l,t − ηl,t )   ¯ l   XX  k k  Γl,m (η̄l,t − ηl,t ) − X  ¯  l k∈K   X =  FTRm→n   −λt + Γ (η̄ − η ) l,n l,t l,t  (m→n)  ¯   l   XX  k k  + Γl,n (η̄l,t − ηl,t ) ¯ l k∈K X   k (Γl,n − Γl,m )(η̄l,t + η̄l,t ) X X  k∈K   FTRm→n = X  k  −(Γl,n − Γl,m )(ηl,t + ηl,t ) (m→n) l ¯ k∈K ¯ ! X X X k k ≤ η̄l,t + η̄l,t + ηl,t + ηl,t Fl ¯ l k∈K k∈K ¯  λt − The first equality holds according to (8). The inequality is true as the amount of FTRm→n respects X −Fl ≤ (Γl,m − Γl,n )FTRm→n ≤ Fl . m→n according to the SFT for FTR market [18], [20], [21]. The right-hand-side of the inequality is the first term in the last equality of (33). Based on (33) and (34), the maximum difference between the FTR credit and the energy congestion cost is equal to the transmission reserve credit. That is, the maximum FTR underfunding is (21). D. Proof of Theorem 2 Proof. According to Theorem 1, the FTR underfunding value is (21) due to the deficiency of the energy congestion cost. 13 Therefore, we need to prove that the money collected from uncertainty sources can cover the FTR underfunding and credits to generation reserve. Without loss of generality, we consider the payment collected from uncertainty sources at time t for ˆk X u,k πm,t ˆkm,t m λkt − m = l XX m = X k ∆Pi,t + l i∈G(m) l k k Γl,m η̄l,t ( ∆Pi,t ) i∈G(m) XX m i∈G(m) k Γl,m η̄l,t l X P P X − k η̄l,t X X m l inj − Fl Γl,m Pm,t pos k η̄l,t ∆fl,t l k k Γl,m ηl,t ˆm,t can be reformulated similarly. Hence, ¯ the second equality holds. The third equality holds from (9). Therefore, XX XX XX Ψm,t = ΘG ΘTl,t i,t + m l m t i t l t holds. That is, the uncertainty payment covers the generation reserve credit and transmission reserve credit. Then, following the energy congestion cost shown in (33), XX XX e πm,t dm,t + Ψm,t ≥ m t X X i + e πm P + i ,t i,t t X X t m→n e (πm,t − m t X X ΘG i,t t i e πn,t )FTRm→n holds. That is, the total payments collected from loads and uncertainty sources can cover the total credits to energy, generation reserve, and FTR holders. So, the revenue adequacy of the proposed market clearing mechanism is guaranteed. E. Proof of Competitive Equilibrium down Proof. Pi,t and (Qup i,t , Qi,t ) are coupled by constraints (15) and (16). According to (17), we can rewrite generation reserve credit as X u,k u,up up u,down down k πm,t Qi,t + πm,t Qi,t = πm,t ∆Pi,t k∈K k k k k k (β̄i,t − βi,t + ᾱi,t − αi,t )∆Pi,t ¯ ¯ k∈K ! k ˆ k β̄i,t (Ii,t Pimax − Pi,t ) + βi,t (Pi,t − Iˆi,t Pimin ) = (35) ¯k d k +ᾱi,t (Riu (1 − ŷi,t )) + αi,t Ri (1 − ẑi,t+1 ) k∈K ¯ Substituting (35) into problem (PMPi ), we can decouple Pi,t down and (Qup i,t , Qi,t ). In fact, we also get all terms related to Pi,t in Lagrangian L(P, λ, α, β, η) for problem (RED). Since the problem (RED) is a linear programming problem, the saddle point P̂i,t , which is the optimal solution to (RED), is also the optimal solution to (PMPi ). Consequently, unit i is not inclined to deviate its output level as it can obtain the maximum profit by following the ISO’s dispatch instruction u,k e P̂i,t . Therefore, dispatch P̂i,t and price signal (πm,t , πm,t ) constitute a competitive partial equilibrium [27]. X The first equalityPholds to (9). According to (7a), P according k k (7f), and (7g), the m l Γl,m η̄l,t ˆm,t in the second line can be rewritten as X X XX k k k Γl,m η̄l,t (∆Pi,t + Pi,t ) − dm,t − η̄l,t Fl X X k k k Γl,m η̄l,t ˆm,t − ηl,t ¯ m l X XX X k k k k k ( ∆Pi,t = ∆Pi,t λt − Γl,m η̄l,t − ηl,t ) ¯ m i l i∈G(m) X pos neg k k η̄l,t + ∆fl,t + ηl,t ∆fl,t ¯ l X X u,k X pos neg k k k η̄l,t = πm,t ∆Pi,t + ∆fl,t + ηl,t ∆fl,t ¯ m i∈G(m) l X u,k X pos neg k k k πmi ,t ∆Pi,t η̄l,t = + ∆fl,t + ηl,t ∆fl,t ¯ i l = X = R EFERENCES [1] M. Shahidehpour, H. Yamin, and Z. Li, Market Operations in Electric Power Systems: Forecasting, Scheduling, and Risk Management, 1st ed. Wiley-IEEE Press, 2002. [2] T. Zheng and E. Litvinov, “Contingency-Based Zonal Reserve Modeling and Pricing in a Co-Optimized Energy and Reserve Market,” IEEE Trans. Power Syst., vol. 23, no. 2, pp. 277–286, May 2008. [3] R. Jiang, J. Wang, and Y. Guan, “Robust unit commitment with wind power and pumped storage hydro,” IEEE Trans. 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Litvinov, “On ex post pricing in the real-time electricity market,” IEEE Trans. Power Syst., vol. 26, no. 1, pp. 153–164, 2011. [27] A. Mas-Colell, M. D. Whinston, J. R. Green et al., Microeconomic theory. Oxford university press New York, 1995, vol. 1. [28] J. F. Ellison, L. S. Tesfatsion, V. W. Loose, and R. H. Byrne, “Project report: A survey of operating reserve markets in us iso/rto-managed electric energy regions,” Sandia Natl Labs Publications, 2012. [29] W. W. Hogan, “Multiple market-clearing prices, electricity market design and price manipulation,” The Electricity Journal, vol. 25, no. 4, pp. 18– 32, 2012. [30] Z. Li and M. Shahidehpour, “Security-constrained unit commitment for simultaneous clearing of energy and ancillary services markets,” IEEE Trans. Power Syst., vol. 20, no. 2, pp. 1079–1088, 2005. [31] H. Ye, Y. Ge, X. Liu, and Z. Li, “Transmission line rating attack in twosettlement electricity markets,” IEEE Trans. Smart Grid, vol. 7, no. 3, pp. 1346–1355, May 2016. Hongxing Ye (S’14-m’16) received his B.S. degree in Information Engineering, in 2007, and M.S. degree in Systems Engineering, in 2011, both from Xi’an Jiaotong University, China, and the Ph.D. degree in Electrical Engineering from the Illinois Institute of Technology, Chicago in 2016. His research interests include large-scale optimization in power systems, electricity market, renewable integration, and cyber-physical system security in smart grid. He is “Outstanding Reviewer” for IEEE Transactions on Power Systems and IEEE Transactions on Sustainable Energy in 2015. He received Sigma Xi Research Excellence Award at Illinois Institute of Technology in 2016. Yinyin Ge (S’14) received the B.S. degree (2008) in Automation and M.S. degree (2011) in Systems Engineering from Xian Jiaotong University, China. She also received Ph.D. degree (2016) in Electrical Engineering at Illinois Institute of Technology, Chicago. Her research interests are power system optimization and modeling; PMU applications in Smart Grid; monitoring, visualization, and state estimation for distribution systems. Mohammad Shahidehpour (F’01) received his Ph.D. degree from the University of Missouri in 1981 in electrical engineering. He is currently the Bodine Chair Professor and Director of the Robert W. Galvin Center for Electricity Innovation at the Illinois Institute of Technology, Chicago. He is the founding Editor-in-Chief of IEEE Transactions on Smart Grid. He is a member of US National Academy of Engineering (NAE). Zuyi Li (SM’09) received the B.S. degree from Shanghai Jiaotong University, Shanghai, China, in 1995, the M.S. degree from Tsinghua University, Beijing, China, in 1998, and the Ph.D. degree from the Illinois Institute of Technology (IIT), Chicago, in 2002, all in electrical engineering. Presently, he is a Professor in the Electrical and Computer Engineering Department at IIT. His research interests include economic and secure operation of electric power systems, cyber security in smart grid, renewable energy integration, electric demand management of data centers, and power system protection.	3
QUIVER MUTATIONS AND BOOLEAN REFLECTION MONOIDS arXiv:1711.09995v3 [math.RA] 21 Feb 2018 BING DUAN, JIAN-RONG LI, AND YAN-FENG LUO Abstract. In 2010, Everitt and Fountain introduced the concept of reflection monoids. The Boolean reflection monoids form a family of reflection monoids (symmetric inverse semigroups are Boolean reflection monoids of type A). In this paper, we give a family of presentations of Boolean reflection monoids and show how these presentations are compatible with mutations of certain quivers. A feature of the quivers in this paper corresponding to presentations of Boolean reflection monoids is that the quivers have frozen vertices. Our results recover the presentations of Boolean reflection monoids given by Everitt and Fountain and the presentations of symmetric inverse semigroups given by Popova. Surprisingly, inner by diagram automorphisms of irreducible Weyl groups or Boolean reflection monoids can be constructed by sequences of mutations preserving the same underlying diagrams. As an application, we study the cellularity of semigroup algebras of Boolean reflection monoids and construct new cellular bases of such cellular algebras using presentations we obtained and inner by diagram automorphisms of Boolean reflection monoids. Key words: Boolean reflection monoids; presentations; mutations of quivers; inner by diagram automorphisms; cellular semigroups; cellular basis 2010 Mathematics Subject Classification: 13F60; 20M18; 16G20; 20F55; 51F15 1. Introduction In their influential work on cluster algebras, Fomin and Zelevinsky associated mutations of skew-symmetrizable matrices [20, Definition 4.2] with mutations of quivers [21, Proposition 8.1]. The quivers whose underlying graphs are Dynkin diagrams play an important role in the cluster algebra theory, as they appear in the finite type classification [21]. It is well known that a finite irreducible crystallographic reflection group W or a finite irreducible Weyl group W can be classified by Dynkin diagrams, whose vertex set is in one-to-one correspondence with a family S of simple reflections and for which there is an edge labeled 1 (respectively, 2, 3) between vertices i and j if and only if (si sj )3 = e (respectively, (si sj )4 = e, (si sj )6 = e) where si , sj ∈ S, e is the identity element of W , see [2, 4, 6, 28]. Let Γ be a Dynkin diagram and a Γ quiver be a quiver whose underlying diagram is Γ. In [2], Barot and Marsh gave presentations of the reflection group WΓ determined by Γ and showed that these presentations are compatible with mutation of Γ quivers. More precisely, Barot and Marsh introduced some additional relations (cycle relations) corresponding to chordless cycles arising in quivers of finite type. For each quiver Q mutation equivalent to a Γ quiver, they first defined an abstract group W (Q) by generators (corresponding to vertices of Q) and relations, and then proved that W (Q) ∼ = WΓ . Motivated 1 2 BING DUAN, JIAN-RONG LI, AND YAN-FENG LUO by Barot and Marsh’s work, the similar presentations of affine Coxeter groups, braid groups, Artin groups, and Weyl groups of Kac-Moody algebras, have been considered in [17, 18, 25, 30, 38], respectively. Let V be a Euclidean space with standard orthonormal basis {v1 , v2 , . . . , vn } and Φ ⊆ V an irreducible crystallographic root system which is in turn classified by Dynkin diagrams. In [11], Everitt and Fountain introduced the concept of reflection monoids. The Boolean reflection monoid M (Φ, B) of type Φ, formed from the Weyl group W (Φ) for classical root system Φ and the Boolean system B, is a family of reflection monoids. Symmetric inverse semigroups are Boolean reflection monoids of type A. Note that the root systems of types Bn and Cn give rise to the same Weyl group. So we only concern the classical Weyl group W (Bn ). In [12], Everitt and Fountain provided a presentation of the Boolean reflection monoid M (Φ, B) for Φ = An−1 , Bn , or Dn . One of the aims in present paper is to obtain new presentations of Boolean reflection monoids and show how these presentations are compatible with mutation of certain quivers. Let Aεn−1 (respectively, Bnε , Dnε ) be the Dynkin diagram with n (respectively, n + 1, n + 1) vertices, where the first n − 1 (respectively, n, n) vertices are mutable vertices and the n-th (respectively, n + 1-th, n + 1-th) vertex ε is a frozen vertex, which is shown in the 4-th column of Table 1. In practice the label is left on an edge only if its weight is greater than 2, and the edge is left unlabelled if its weight is 1. Let ∆ ∈ {Aεn−1 , Bnε , Dnε } and Q any quiver mutation equivalent to a ∆ quiver. We define an inverse monoid M (Q) from Q, see Section 4.2, and then we show that M (Q) ∼ = M (Φ, B), see Theorem 4.7 and Proposition 4.8. This implies that Boolean reflection monoids can also be classified by ∆, see Table 1. In [2, 17, 25, 30], the diagrams corresponding to generators of irreducible Weyl groups, affine Coxeter groups, braid groups, Artin groups have no frozen vertices. In present paper, the diagrams corresponding to generators of Boolean reflection monoids have frozen vertices. Type of Φ Boolean reflection monoids Generators An−1 (n ≥ 2) M (An−1 , B) {s1 , . . . , sn−1 , sε } Bn (n ≥ 2) M (Bn , B) {s0 , . . . , sn−1 , sε } ∆ = Φε 1 2 n−1 ε 1 n−1 ε 2 n−1 ε 2 0 1 Dn (n ≥ 4) M (Dn , B) {s0 , . . . , sn−1 , sε } 0 Table 1. Boolean reflection monoids and Dynkin diagrams Aεn−1 , Bnε , Dnε . In Proposition 3.1 of [11], Everitt and Fountain proved that the symmetric inverse semigroup In is isomorphic to the Boolean reflection monoid of type An−1 . So we recover the presentation of the symmetric inverse semigroup In defined in [10, 34]. The presentation corresponds exactly to the presentation determined by Dynkin diagram QUIVER MUTATIONS AND BOOLEAN REFLECTION MONOIDS 3 Aεn−1 . Moreover, we also recover Everitt and Fountain’s presentations of Boolean reflection monoids defined in Section 3 of [12]. These presentations can be obtained from any ∆ quiver by a finite sequence of mutations. We show in Theorem 4.10 that the inner automorphism group of Boolean reflection monoid M (Φ, B) is naturally isomorphic to W (Φ)/Z(W (Φ)). We further study the actions of (inward) mutations. Surprisingly, inner by diagram automorphisms of finite irreducible Weyl groups and Boolean reflection monoids can be constructed by a sequence of mutations preserving the same underlying diagrams, see Theorem 3.7 and Theorem 4.11 respectively. As an application, we study the cellularity of semigroup algebras of Boolean reflection monoids. It is well known that Hecke algebras of finite type, q-Schur algebras, the Brauer algebra, the Temperley-Lieb algebras and partition algebras are cellular, see [22–24, 42, 43]. Recently, the cellularity of semigroup algebras is investigated by East [8], Wilox [41], Guo and Xi [26], and Ji and Luo [32] respectively. By applying Geck’s and East’s results, we show that semigroup algebras of Boolean reflection monoids are cellular algebras, see Proposition 4.12. Moreover, we construct new cellular bases of such cellular algebras by presentations we obtained and inner by diagram automorphisms of Boolean reflection monoids. The results and methods of this paper have applications in several lines of research which will be studied in future work, including automorphisms of Boolean reflection monoids [39], Hecke algebras of Boolean reflection monoids [27, 31, 36], Coxeter arrangement monoids [7, 11, 12, 14, 15], Braid inverse monoids [9, 13, 25], and algebraic monoids. The paper is organized as follows. In Section 2, we recall some notations and background knowledge which will be useful to us. In Section 3, building on Barot and Marsh’s work, we further study inner by diagram automorphisms of irreducible Weyl groups (Theorem 3.7) and cellular basis of group algebras of irreducible Weyl groups. In Section 4, we state our main results, Theorem 4.7 and Proposition 4.8, which show that presentations of Boolean reflection monoids are compatible with mutations of ∆ quivers. We recover the presentations of Boolean reflection monoids given by Everitt and Fountain and the presentations of symmetric inverse semigroups given by Popova. Moreover, we characterize inner by diagram automorphisms of Boolean reflection monoids by the method of mutations, Theorem 4.11. Furthermore, we study the cellularity of semigroup algebras of Boolean reflection monoids and give new cellular bases of such cellular algebras. In Section 5, we consider the way of mutations of ∆ quivers and the oriented cycles appearing in them. In Section 6, we find an efficient subset of the relations sufficient to define the inverse monoid M (Q). The last section, Section 7, we prove our main result, Theorem 4.7. 2. Preliminaries 2.1. Mutation of quivers. Let Q be a quiver with finitely many vertices and finitely many arrows that have no loops or oriented 2-cycles. Given a quiver Q, let I be the set of its vertices and Qop its opposite quiver with the same set of vertices but with the reversed orientation for all the arrows. If there are q arrows pointing from a vertex i to a vertex j, then we draw an arrow from i to j with a weight wij = q. We will frequently draw an arrow with no label if wij = 1. 4 BING DUAN, JIAN-RONG LI, AND YAN-FENG LUO For each mutable vertex k of Q, one can define a mutation of Q at k, due to Fomin and Zelevinsky [21]. This produces a new quiver denoted by µk (Q) which can be obtained from Q in the following way: (i) The orientations of all edges incident to k are reversed and their weights intact. (ii) For any vertices i and j which are connected in Q via a two-edge oriented path going through k, the quiver mutation µk affects the edge connecting i and j in the way shown in Figure 1, where the weights c and c′ are related by √ √ √ ± c ± c′ = ab, √ √ where the sign before c (resp., c′ ) is “+” if i, j, k form an oriented cycle in Q (resp., in µk (Q)), and is “−” otherwise. Here either c or c′ may be equal to 0, which means no arrows between i and j. i k ✂@ ❂❂❂ ✂ ❂❂b a ✂✂ ❂❂ ✂✂ ❂ ✂ ✂ c µ k ←→ j i k ✂ ^❂❂❂ ✂ ❂❂b a ✂✂ ❂❂ ✂✂ ❂ ✂ ✂ c′ j Figure 1. Quiver mutation (iii) The rest of the edges and their weights in Q remain unchanged. Two quivers Q1 and Q2 are said to be mutation equivalent if there exists a finite sequence of mutations taking one to the other. We write Q1 ∼mut Q2 to indicate that Q1 is mutation equivalent to Q2 . The underlying diagram of a quiver Q is a undirected diagram obtained from Q by forgetting the orientation of all the arrows. We call a quiver Q connected if its underlying diagram is connected (every node is “reachable”). It is obvious that Dynkin quivers are connected quivers. It was shown in [21, Theorem 1.4] that there are only finitely many quivers in the mutation classes of Dynkin quivers. We call a cycle in the underlying diagram of a quiver a chordless cycle if no two vertices of the cycle are connected by an edge that does not itself. As shown in [21, Proposition 9.7] (or see [2, Proposition 2.1]), all chordless cycles are oriented in the mutation classes of Dynkin quivers. 2.2. Cellular algebras and cellular semigroups. Let us first recall the basic definition of cellular algebras introduced by Graham and Lehrer [24]. Let R be a commutative ring with identity. Definition 2.1. An associative R-algerba A is called a cellular algebra with cell datum (Λ, M, C, i) if the following conditions are satisfied: (C1) Λ is a finite partially ordered set. Associated with each λ ∈ Λ there is a finite set λ \| λ ∈ Λ; S, T ∈ M (λ)} of A; M (λ) of indices and there exists an R-basis {CS,T 2 λ λ ; (C2) i is an R-linear anti-automorphism of A with i = i, which sends CS,T to CT,S (C3) For each λ ∈ Λ, S, T ∈ M (λ), and each a ∈ A, X λ aCS,T ≡ ra (S ′ , S)CSλ′ ,T (mod A(< λ)), S ′ ∈M (λ) QUIVER MUTATIONS AND BOOLEAN REFLECTION MONOIDS 5 where ra (S ′ , S) ∈ R is independent of T and A(< λ) is the R-submodule of A generated by {CSµ′′ ,T ′′ \| µ < λ, S ′′ , T ′′ ∈ M (µ)}. Cellular algebras provide a general framework to studying the representation theory of many important classes of algebras including Hecke algebras of finite type, q-Schur algebras, the Brauer algebra, the Temperley-Lieb algebras and partition algebras, see [22–24, 42, 43]. Recently, the cellularity of semigroup algebras is investigated by East [8], Wilox [41], Guo and Xi [26], and Ji and Luo [32] respectively. In the following, we shall recall some basic notions and facts from the theory of semigroups. Let S be a semigroup. For any a, b ∈ S, define a L b ⇔ S 1 a = S 1 b, a R b ⇔ aS 1 = bS 1 , a J b ⇔ S 1 aS 1 = S 1 bS 1 , H = L ∩ R and D = L ∨ R = L ◦ R = R ◦ L, where S 1 is the monoid obtained from S by adding an identity if necessary. A semigroup S is said to be inverse if for each element s ∈ S, there exists a unique inverse s−1 ∈ S such that ss−1 s = s and s−1 ss−1 = s−1 . If S is a finite inverse semigroup, then D = J . For any s, t ∈ S, define Ds ≤ Dt if and only if s ∈ S 1 tS 1 . Let S be an inverse semigroup with the set E(S) of idempotents. Let D be a D-class of S. Suppose that e1 , . . . , ek ∈ D ∩ E(S). Choose a1 = e1 , . . . , ak ∈ Le1 such that aj R ej for each j. Then D = Re1 ∪ Ra2 ∪ · · · ∪ Rak . Put HD = He1 and by Green’s Lemma, for each x ∈ D we have x = ai ga−1 j for unique 1 ≤ i, j ≤ k and g ∈ HD . Using East’s symbol in [8], let [ei , ej , g]D be the element x. Then x−1 = [ej , ei , g−1 ]D . For more detail knowledge of semigroups, the reader is referred to [8, 29]. A semigroup S is said to be cellular if its semigroup algebra R[S] is a cellular algebra. In [8], East proved the following theorem: Theorem 2.2. [8, Theorems 15] Let S be a finite inverse semigroup with the set E(S) of idempotents. If S satisfies the following conditions: (1) For each D-class D, the subgroup HD is cellular with cell datum (ΛD , MD , CD , iD ); (2) The map i : R[S] → R[S] sending [e, f, g]D to [f, e, iD (g)]D is an R-linear antihomomorphism. Then S is a cellular semigroup with cell datum (Λ, M, C, i), where Λ = {(D, λ) \| D ∈ S/D, λ ∈ ΛD } with partial order defined by (D, λ) ≤ (D ′ , λ′ ) if D < D ′ or D = D ′ and λ ≤ λ′ , M (D, λ) = {(e, s) \| e ∈ E(S) ∩ D, s ∈ MD (λ)} for (D, λ) ∈ Λ, and (D,λ) λ ] \| (D, λ) ∈ Λ; (e, s), (f, t) ∈ M (D, λ)} for (D, λ) ∈ Λ and C = {C(e,s),(f,t) = [e, f, Cs,t D (e, s), (f, t) ∈ M (D, λ). By the definition of cellular algebras, we have the following corollary. Corollary 2.3. Suppose that A is a cellular algebra over R with cell datum (Λ, M, C, i) λ λ ) for any λ ∈ Λ, (S, T ) ∈ and ϕ is an R-linear automorphism of A. Let C S,T = ϕ(CS,T M (λ) × M (λ), and i = ϕiϕ−1 . Then (Λ, M, C, i) is a cellular basis of A. λ Proof. Since {CS,T \| λ ∈ Λ, (S, T ) ∈ M (λ) × M (λ)} is an R-basis of A and ϕ is an λ R-linear automorphism of A, {C S,T \| λ ∈ Λ, (S, T ) ∈ M (λ) × M (λ)} is also an R-basis 2 λ λ of A. It follows from the definition of i that i = i and i(C S,T ) = C T,S . 6 BING DUAN, JIAN-RONG LI, AND YAN-FENG LUO For each λ ∈ Λ, S, T ∈ M (λ), and each a ∈ A, X λ aCS,T ≡ ra (S ′ , S)CSλ′ ,T (mod A(< λ)), S ′ ∈M (λ) where ra (S ′ , S) ∈ R is independent of T and A(< λ) is the R-submodule of A generated by {CSµ′′ ,T ′′ \| µ < λ, S ′′ , T ′′ ∈ M (µ)}. For each a ∈ A, there exists a b ∈ A such that a = ϕ(b). Then X λ λ λ ) = ϕ(bCS,T )≡ rb (S ′ , S)ϕ(CSλ′ ,T ) (mod A(< λ)) aC S,T = ϕ(b)ϕ(CS,T S ′ ∈M (λ) ≡ X λ rb (S ′ , S)C S ′ ,T (mod A(< λ)), S ′ ∈M (λ) where rb (S ′ , S) ∈ R is independent of T . Therefore (Λ, M, C, i) is a cellular basis of A, as required. 3. Some new results of irreducible Weyl groups Let V be a Euclidean space with a standard orthonormal basis {v1 , v2 , . . . , vn }. Let Φ ⊆ V be a root system and Π the set of simple roots in Φ. For each αi ∈ Π, the associated simple reflection is si . Then the finite irreducible Weyl group W (Φ) can be generated by S = {si \| αi ∈ Π} and the number of reflections in W (Φ) is equal to the number of positive roots in Φ. We refer the reader to [3, 19, 28, 35] for more information about Weyl groups, root systems and reflection groups. 3.1. Barot and Marsh’s results. It is well known that the finite irreducible crystallographic reflection groups or the irreducible Weyl groups have been classified by Dynkin diagrams, see [28]. Let Γ be a Dynkin diagram and I the set of its vertices. Let WΓ be the finite irreducible Weyl group determined by Γ. We say that a Γ quiver is a quiver whose underlying diagram is Γ. In [2], Barot and Marsh gave presentations of WΓ . The construction works as follows: Let Q be a quiver mutation equivalent to a Γ quiver. Barot and Marsh defined an inward mutation at vertex k as follows: ( sk si sk if there is an arrow i → k in Q (possibly weighted), ti = (3.1) si otherwise. For two vertices i, j of Q, one defines   2 i and j are not connected,  3 i and j are connected by an edge with weight 1, mij =  4 i and j are connected by an edge with weight 2,    6 i and j are connected by an edge with weight 3. (3.2) Definition 3.1. Let W (Q) be the group with generators si , i ∈ I, subjecting to the following relations: (1) s2i = e for all i; (2) (si sj )mij = e for all i 6= j; QUIVER MUTATIONS AND BOOLEAN REFLECTION MONOIDS 7 (3) For any chordless cycle C in Q: ω ω ωd−1 ω 1 2 0 i0 −→ i1 −→ · · · −→ id−1 −→ i0 , where either all of the weights are 1, or ω0 = 2, we have: (si0 si1 · · · sid−2 sid−1 sid−2 · · · si1 )2 = e; where e is the identity element of W (Q). One of Barot and Marsh’s main results in [2] is stated as follows. Theorem 3.2 ([2, Theorem A]). The group W (Q) does not depend on the choice of a quiver in the mutation class of Q. In particular, W (Q) ∼ = WΓ for each quiver Q mutation equivalent to a Γ quiver. 3.2. Inner by diagram automorphisms of irreducible Weyl groups. Let W be a Coxeter group defined by a set S of generators and relations (1) and (2) of Definition 3.1. We call the pair (W, S) a Coxeter system. In what follows, given any two Coxeter systems (W, S1 ) and (W, S2 ), we say that there exists an automorphism of W , we mean that there is an automorphism α ∈ Aut(W ) such that α(S1 ) = S2 . If such an automorphism can always be chosen from Inn(W ), the group of inner automorphisms of W , then W is called strongly rigid. In case W is strongly rigid, the group Aut(W ) has a very simple structure (see Corollary 3.2 of [5]): Aut(W ) = Inn(W ) × Diag(W ), where Diag(W ) consists of diagram automorphisms of the unique Coxeter diagram corresponding to W . The following lemma is well known. Lemma 3.3. Let W be a finite group generated by a finite set S of simple reflections. Then the set of all reflections in W is {wsw−1 \| w ∈ W, s ∈ S}. In [1, Table I], Bannai computed the center Z(W (Φ)) of an irreducible Weyl group W (Φ). The longest element w0 in W (Φ) is a central element of W (Φ) except for Φ = An (n ≥ 2), D2k+1 (k ≥ 2), E6 . The following important notation was introduced by Franzsen in [16]. Definition 3.4. [16, Definition 1.36] An inner by diagram automorphism is an automorphism generated by some inner and diagram automorphisms in Aut(W ). The subgroup of inner automorphisms is a normal subgroup of Aut(W ), therefore any inner by diagram automorphism can be written as the product of an inner and a diagram automorphism. The following two lemmas collect together some facts from [16] which will be useful later. ∼ Lemma 3.5. [16, Proposition 2.4] If W is Weyl group of type An , then Aut(W (An )) = ∼ W (An ) if n 6= 5, while Aut(W (A5 )) = W (A5 ) ⋊ Z2 . So, for n 6= 5, any automorphism of Weyl group of type An maps reflections to reflections, furthermore any automorphism of A5 that does preserve reflection is inner. Lemma 3.6. [16, Proposition 1.44, Propositions 2.8–2.10, Proposition 2.13] Let W (Φ) be Weyl group of type Φ. 8 BING DUAN, JIAN-RONG LI, AND YAN-FENG LUO (a) Any automorphism of W (Bn ) for n > 2 that does preserve reflections must be inner. (b) For k ≥ 2, Aut(W (D2k+1 )) ∼ = W (D2k+1 ), that is, all automorphisms of W (D2k+1 ) map reflections to reflections. (c) All automorphisms of W (E6 ) or W (E7 ) are inner. (d) All automorphisms of Weyl groups that preserve reflections are inner by diagram automorphisms. The following theorem reveals a connection between inner by diagram automorphisms of irreducible Weyl groups and quiver mutations. Theorem 3.7. Let Q be a Γ quiver and W (Q) the corresponding Weyl group generated by a set S of simple reflections. Then α is an inner by diagram automorphism of W (Q) if and only if there exists a sequence of mutations preserving the underlying diagram Γ such that α(S) can be obtained from Q by mutations. In particular, all the reflections in W (Q) can be obtained from Q by mutations. Proof. Through observation, every variable obtained by mutations must be some reflection of the corresponding Weyl group. The sufficiency follows from the fact that all automorphisms of Weyl groups that preserve reflections are inner by diagram automorphisms, see Lemmas 3.5 and 3.6. To prove necessity, assume without loss of generality that the vertex set of Q is {1, 2, . . . , n} and α is an inner by diagram automorphism of W (Q). Note that diagram automorphisms of any Dynkin diagram keep the underlying Dynkin diagram. Then relabelling the vertices of Q if necessary, there exists an inner automorphism α of W (Q) such that α(S) = α(S). It is sufficient to prove that α(S) can be obtained from Q by mutations and the sequence of mutations preserves the underlying diagram Γ. Let g = si1 si2 · · · sir ∈ W (Q) be a reduced expression for g, where sik ∈ S, 1 ≤ k ≤ r. We assume that α(S) = gSg −1 . In the following we shall use induction to prove that gSg−1 can be obtained from Q by a sequence of mutations preserving the underlying diagram Γ. Step 1. We mutate firstly Q at the vertex i1 twice. Then we get a quiver Q1 , which has the same underlying diagram with Q. Moreover, the set S becomes si1 Ssi1 = {si1 s1 si1 , si1 s2 si1 , . . . , si1 sn−1 si1 , si1 sn si1 }. We keep vertices of Q1 having the same label as vertices of Q. Step 2. We then mutate Qt−1 , t = 2, 3, · · · , r at the vertex it twice. Note that the variable corresponding to the vertex it of Qt−1 is si1 . . . sit−1 sit sit−1 . . . si1 . Then we get a quiver Qt , which has the same underlying diagram with Q, Q1 , . . . , Qt−1 . Moreover, the set si1 . . . sit−1 Ssit−1 . . . si1 of generators in Qt−1 becomes si1 · · · sit−1 sit sit−1 · · · si1 (si1 · · · sit−1 Ssit−1 · · · si1 )si1 · · · sit−1 sit sit−1 · · · si1 = si1 · · · sit−1 sit Ssit sit−1 · · · si1 . We keep vertices of Qt having the same label as vertices of Qt−1 . Step 3. We repeat Step 2 until we get the quiver Qr . By induction, it is not difficult to show that the set of generators in Qr is si1 si2 · · · sir Ssir · · · si2 si1 = gSg −1 = α(S). QUIVER MUTATIONS AND BOOLEAN REFLECTION MONOIDS 9 Finally, every reflection in W (Φ) is conjugate to a simple reflection by Lemma 3.3. So we assume that a reflection is of the form gsi g−1 , where g = si1 si2 · · · sik ∈ W (Φ) is a reduced expression for some subset {i1 , i2 , . . . , ik } ⊆ {1, 2, . . . , n}. By the same arguments as before, we mutate the sequence i1 , i1 , i2 , i2 , . . . , ik , ik starting from Q, we can obtain the reflection gsi g−1 . Remark 3.8. (1) In types An , Bn (n > 2), D2k+1 , E6 , E7 , and E8 , all inner by diagram automorphisms of the corresponding Weyl groups are inner automorphisms. W (Φ) is strongly rigid for Φ = An (n 6= 5), D2k+1 , E6 , E7 . (2) If (W, S) is a Coxeter system of W , then for any inner by diagram automorphism α of W , (W, α(S)) is also a Coxeter system of W . (3) Suppose that Q is a quiver mutation equivalent to a Γ quiver. Let W (Q) be the corresponding Weyl group with a set S of generators. Then Theorem 3.7 holds for Q. 3.3. Cellular basis of group algebras of irreducible Weyl groups. In [23], Geck proved Hecke algebras of finite type are cellular algebras. Let ϕ be any inner by diagram automorphism of irreducible Weyl groups, see Theorem 3.7. By Lemma 2.3, we can obtain new cellular basis of group algebras of irreducible Weyl groups. 4. Main results of Boolean reflection monoids Let Aεn−1 (respectively, Bnε , Dnε ) be the Dynkin diagram with n (respectively, n + 1, n + 1) vertices, where the first n − 1 (respectively, n, n) vertices are mutable vertices and the n-th (respectively, n + 1-th, n + 1-th) vertex ε is a frozen vertex, which is shown in Figure 2. The label is left on an edge only if its weight is greater than 2, and the edge is left unlabelled if its weight is 1. We shall always assume that ∆ is one of Aεn−1 , Bnε and Dnε . A quiver is said to be a ∆ quiver if the underlying diagram of such quiver is ∆. Aεn−1 Bnε 1 2 3 n−1 ε 1 2 n−1 ε 2 3 n−1 ε 2 0 1 Dnε 0 Figure 2. Classical Dynkin diagrams Aεn−1 , Bnε and Dnε with a frozen vertex ε. 10 BING DUAN, JIAN-RONG LI, AND YAN-FENG LUO 4.1. Boolean reflection monoids. In 2010, Everitt and Fountain introduced reflection monoids, see [11]. The Boolean reflection monoids are a family of reflection monoids (symmetric inverse semigroups are Boolean reflection monoids of type A). Let V be a Euclidean space with standard orthonormal basis {v1 , v2 , . . . , vn } and let Φ ⊆ V be a root system and W (Φ) the associated Weyl group of type Φ. A partial linear isomorphism of V is a vector space isomorphism Y → Z for two vector subspaces Y , Z of V . Any partial linear isomorphism of V can be realized by restricting a full isomorphism to some subspace. We will write gY for the partial isomorphism with domain Y and effect that of restricting g to Y . We denote by M (V ) (respectively, GL(V )) the general linear monoid (respectively, general linear group) on V consisting of partial linear isomorphisms (respectively, linear isomorphisms) of V . In M (V ), gY = hZ if and only if Y = Z and gh−1 is in the isotropy group GY = {g ∈ GL(V )\|gv = v for all v ∈ Y }. Moreover, gY hZ = (gh)Y ∩g−1 Z and (gY )−1 = (g −1 )gY . Let us recall the notation “system” in V for a group G ⊆ GL(V ) introduced in [11]. Definition 4.1 ([11, Definition 2.1]). Let V be a real vector space and G ⊆ GL(V ) a group. A collection S of subspaces of V is called a system in V for G if and only if (1) V ∈ S, (2) GS = S, that is, gX ∈ S for any g ∈ G and X ∈ S, (3) if X, Y ∈ S, then X ∩ Y ∈ S. For J ⊆ X = {1, 2, . . . , n}, let X(J) = M j∈J Rvj ⊆ V, and B = {X(J) : J ⊆ X} with X(∅) = 0. Then B is a Boolean system in V for W (Φ), where Φ = An−1 , Bn /Cn , or Dn . For example, the Weyl group W (An−1 )-action on the subspaces X(J) ∈ B is just g(π)X(J) = X(πJ), where g(π) 7→ π induces an isomorphism between W (An−1 ) and the symmetric group Sn on the set X. Note that B is not a system for any of the exceptional W (Φ). Definition 4.2 ([11, Definition 2.2]). Let G ⊆ GL(V ) be a group and S the system in V for G. The monoid of partial linear isomorphisms given by G and S is the submonoid of M (V ) defined by M (G, S) := {gX : g ∈ G, X ∈ S}. If G is a reflection group, then M (G, S) is called a reflection monoid. Let G be the reflection group W (Φ) for Φ = An−1 , Bn /Cn , or Dn , and S the Boolean system in V for G, then M (G, S) is called a Boolean reflection monoid. In general, we write M (Φ, B) instead of M (W (Φ), B), and call M (Φ, B) the Boolean reflection monoid of type Φ. Recall that Everitt and Fountain gave a presentation of the Boolean reflection monoid M (Φ, B) for Φ = An−1 , Bn , or Dn in Section 4 of [12]. QUIVER MUTATIONS AND BOOLEAN REFLECTION MONOIDS 11 Lemma 4.3. Everitt and Fountain’s presentations of Boolean reflection monoids are shown as follows: M (An−1 , B) = hs1 , s2 , . . . , sn−1 , sε \| (si sj )mij = e for 1 ≤ i, j ≤ n − 1, s2ε = sε , si sε = sε si for i 6= 1, sε s1 sε s1 = s1 sε s1 sε = sε s1 sε i . M (Bn , B) = hs0 , s1 , . . . , sn−1 , sε \| (si sj )mij = e for 0 ≤ i, j ≤ n − 1, s2ε = sε , s0 s1 sε s1 = s1 sε s1 s0 , s0 sε = sε , si sε = sε si for i 6= 1, s1 sε s1 sε = sε s1 sε s1 = sε s1 sε i. M (Dn , B) = hs0 , s1 , . . . , sn−1 , sε \| (si sj )mij = e for 0 ≤ i, j ≤ n − 1, s2ε = sε , si sε = sε si for i > 1, sε s1 sε s1 = s1 sε s1 sε = sε s1 sε , s0 sε s0 = s1 sε s1 , s0 s2 s1 sε s1 s2 = s2 s1 sε s1 s2 s0 i . Here mij is defined in (3.2). 4.2. Inverse monoids determined by quivers. Let I ∪ {ε} be the set of vertices of a quiver Q with a frozen vertex ε. For any i, j ∈ I and ε, define   2 if i and j are not connected,  3 if i and j are connected by an edge of weight 1, mij =  4 if i and j are connected by an edge of weight 2,    6 if i and j are connected by an edge of weight 3.   2 if ε and j are not connected, mεj = 3 if ε and j are connected by an edge of weight 1,   1 if ε and j are connected by an edge of weight 2.   2 if ε and j are not connected, mjε = 4 if ε and j are connected by an edge of weight 1,   2 if ε and j are connected by an edge of weight 2. Let mii = 1 for any i ∈ I ∪ {ε}. Then (mij )i,j∈I is a Coxeter matrix and (mij )i,j∈I∪{ε} is a generalized Coxeter matrix, see [40]. To illustrate, generalized Coxeter matrices corresponding to a Aεn−1 quiver and a Bnε quiver are respectively     1 3 2 ··· ··· ··· 2 1 4 2 ··· ··· ··· 2 3 1 4 1 3 2 · · · · · · 2 3 2 · · · · · · 2     2 3   1 3 2 · · · 2 1 3 2 · · · 2  2 3   .. . . . . . . . . . . ..   .. . . . . . . . . . . ..  and . .   . . . . . . . . . . . .  .  2 · · · 2 2 · · · 2 3 1 3 2 3 1 3 2     2 · · · · · · 2   3 1 4 2 ··· ··· 2 3 1 4 2 ··· ··· ··· 2 3 1 n×n 2 ··· ··· ··· 2 3 1 (n+1)×(n+1) Let (i1 , . . . , ε) be an ordered tuple such that the subquiver of Q on the vertices i1 , . . . , ε contains only one underlying subdiagram or and does not contain one 12 BING DUAN, JIAN-RONG LI, AND YAN-FENG LUO . Such an ordered tuple (i1 , . . . , ε) is called a shortest path underlying subdiagram tuple if it is the shortest path from i1 to ε in Q. For any shortest path tuple (i1 , . . . , ε), we denote by P (si1 , sε ) the word si1 . . . sε . Denote by e the identity element of an inverse monoid and denote by (aba . . .)m an alternating product of m terms. Definition 4.4. Let Q be any quiver mutation equivalent to a ∆ quiver. Define an inverse monoid M (Q) with generators si , i ∈ I ∪ {ε} and relations: (R1) s2i = e for i ∈ I, s2ε = sε ; (R2) (si sj )mij = e for i, j ∈ I and (sε sj sε · · · )mεj = (sj sε sj · · · )mjε = (sε sj sε · · · )mεj +1 for any j ∈ I; (R3) (i) for every chordless oriented cycle C in Q: w w w 1 2 0 i0 −→ i1 −→ · · · → id−1 −→ i0 , where id ∈ I for d = 0, 1, . . . , d−1, either all of the weights are 1, or w0 = 2, we have: (si0 si1 · · · sid−2 sid−1 sid−2 · · · si1 )2 = e. (ii) for every chordless oriented cycle C in Q: ε −→ i1 −→ · · · → id−1 −→ ε, where id ∈ I for d = 1, . . . , d − 1, we have: sε si1 · · · sid−2 sid−1 sid−2 · · · si1 = si1 · · · sid−2 sid−1 sid−2 · · · si1 sε . (iii) for every chordless oriented cycle C in Q: w 2 w 1 2 ε −→ i1 −→ i2 −→ ε, where i1 , i2 ∈ I, if w1 = 1 and w2 = 2, we have sε si1 si2 si1 = si1 si2 si1 sε ; if w1 = 2 and w2 = 1, we have si1 si2 sε si2 = si2 sε si2 si1 . (R4) (path relations) for every underlying subdiagram of Q of the form shown in the first column of Table 2, we take path relations listed in the second column of Table 2. Remark 4.5. (1) In (R2), if mjε = 2 then mεj = 2. In this case the equation (sε sj sε . . .)mεj = (sj sε sj . . .)mjε = (sε sj sε . . .)mεj +1 can be reduced to be sε sj = sj sε . (2) For (R3) (ii) relation, though in this paper we only use the case d = 3, 4, the defined relation for arbitrary d is still meaningful, see our unpublished paper [7]. The following lemma is well known and easily verified. Lemma 4.6. If two quivers Q1 and Q2 both have the same underlying diagram G and G is a tree, then Q1 ∼mut Q2 . It follows from the connectivity and finiteness of ∆ that any two ∆ quivers are mutation-invariance. QUIVER MUTATIONS AND BOOLEAN REFLECTION MONOIDS Subdiagrams of Q 2 ε 0 i1 Path relations P (s0 ,sε )=P (si1 ,sε ), P (sε ,s0 )=P (sε ,si1 ) 2 0 P (s0 ,sε )=P (si1 ,sε ), P (sε ,s0 )=P (sε ,si1 ) 2 ε i1 i3 i4 13 i2 C ε i1 id−1 P (si2 ,sε )P (sε ,si2 )=si3 si4 ···sid P (si1 ,sε )P (sε ,si1 )sid ···si4 si3 for d ≥ 4 id i2 i3 i1 ε i1 ε P (si2 ,sε )P (sε ,si2 )=P (si4 ,sε )P (sε ,si4 ) i4 i2 i3 P (si2 ,sε )P (sε ,si2 )=P (si4 ,sε )P (sε ,si4 ) i4 i2 i1 ε P (si2 ,sε )P (sε ,si2 )=P (si3 ,sε )P (sε ,si3 ) i3 Table 2. Path relations of underlying subdiagrams of Q, where C stands for a chordless cycle. Now we are ready for our main results in this section. Theorem 4.7. Let ∆ ∈ {Aεn−1 , Bnε , Dnε } and Q0 be a ∆ quiver. If Q ∼mut Q0 then M (Q) ∼ = M (Q0 ). We will prove Theorem 4.7 in Section 7. Up to the above isomorphism, we denote by M (∆) the inverse monoid determined by any quiver appearing in the mutation class of ∆ quivers. When we say that we mutate a sequence (n, n − 1, · · · , 2, 1) of vertices of a quiver we mean that we first mutate the n-th vertex of the quiver, then we mutate the (n − 1)-th vertex, and so on until the first vertex. The following proposition shows that Everitt and Fountain’s presentations of Boolean reflection monoids can be obtained from any ∆ quiver by mutations. Proposition 4.8. Let Φ = An−1 , Bn or Dn . Then M (Φ, B) = M (Φε ). Proof. All ∆ quivers are mutation-invariance, so any ∆ quiver can be viewed as an initial quiver we mutate. 14 BING DUAN, JIAN-RONG LI, AND YAN-FENG LUO We mutate the sequence (n − 1, n − 2, · · · , 2, 1) of vertices of the following Aεn−1 quiver ◦ /◦ 1 / ◦ ❴ ❴ ❴/ ◦ 2 3 n−1 /•, ε we obtain the quiver Q1 : /◦ ◦ ε / ◦ ❴ ❴ ❴/ ◦ 1 2 / • . n−1 n−2 Then by Definition 4.4 M (Q1 ) = s1 , s2 , . . . , sn−1 , sε \| (si sj )mij = e for 1 ≤ i, j ≤ n − 1, s2ε = sε , si sε = sε si for i 6= 1, sε s1 sε s1 = s1 sε s1 sε = sε s1 sε i , where   1 if i = j, mij = 3 if \|i − j\| = 1,   2 otherwise. By Lemma 4.3, we deduce that M (Q1 ) = M (An−1 , B). Mutating a sequence (n − 1, n − 2, · · · , 1, 0) of vertices of the following Bnε quiver 2 ◦ /◦ / ◦ ❴ ❴ ❴/ ◦ 1 0 2 n−1 /•, ε we get obtain quiver Q2 : ε ◦ ☎☎ 2 ☎☎ ☎ ☎☎ ☎ ☎ • \✿ 2 0 ✿✿ ✿✿ ✿✿ ✿✿ /◦ 1 . / ◦ ❴ ❴ ❴/ ◦ 2 n−2 / ◦ n−1 Then by Definition 4.4 M (Q2 ) = hs0 , s1 , . . . , sn−1 , sε \| (si sj )mij = e for 0 ≤ i, j ≤ n − 1, s2ε = sε , s0 s1 sε s1 = s1 sε s1 s0 , s0 sε = sε s0 = sε , si sε = sε si for i 6= 1, s1 sε s1 sε = sε s1 sε s1 = sε s1 sε i, where  1    2 mij =  3    4 if if if if i = j, i and j are not connected, i and j are connected by an edge with weight 1, i and j are connected by an edge with weight 2. By Lemma 4.3, we have M (Q2 ) = M (Bn , B). QUIVER MUTATIONS AND BOOLEAN REFLECTION MONOIDS 15 By mutating a sequence (n − 1, n − 2, · · · , 2, 0) of vertices of the following Dnε quiver 0 ◦ / ◦2 O / ◦3 ❴ ❴ ❴/ n−1 ◦ / •ε , ◦ 1 we obtain the quiver Q3 : 0 ◦ ✆B ✾✾✾ ✆ ✾✾ ✆ ✾✾ ✆✆ ✆ ✾ ✆ ε ✆ 2 • \✾ ◦ ✾✾ ✆✆ ✾✾ ✆✆ ✾✾ ✆ ✾✾ ✆✆✆ ✆ . / ◦3 ❴ ❴ ❴/ n−2 ◦ / n−1 ◦ ◦ 1 Then by Definition 4.4 M (Q3 ) = s0 , s1 , . . . , sn−1 , sε \| (si sj )mij = e for 0 ≤ i, j ≤ n − 1, s2ε = sε , si sε = sε si for i > 1, sε sj sε sj = sj sε sj sε = sε sj sε for j = 0, 1, s0 sε s0 = s1 sε s1 , s0 s2 s1 sε s1 s2 = s2 s1 sε s1 s2 s0 i , where   1 mij = 2   3 if i = j, if i and j are not connected, if i and j are connected by an edge with weight 1. We claim that M (Q3 ) = M (Dn , B), which follows from Lemma 6.2 (3). Suppose that a quiver Q is mutation equivalent to a ∆ quiver, then by Theorem 4.7 and Proposition 4.8, M (Q) gives a presentation of the Boolean reflection monoid M (∆). In [11], Everitt and Fountain proved that the Boolean reflection monoid M (An−1 , B) (respectively, M (Bn , B)) is isomorphic to the symmetric inverse semigroup In (respectively, the monoid I±n of partial signed permutations). Hence our results recover the presentation of the symmetric inverse semigroup In defined in [10, 34], that is, such presentation is exactly the presentation of M (Q) for a Aεn−1 quiver Q. The following example is given to explain Theorem 4.7. Example 4.9. We start with a Aε3 quiver Q0 which is shown in Figure 3 (a). Let Q1 = µ2 (Q0 ) be the quiver obtained from Q0 by a mutation at 2. 16 BING DUAN, JIAN-RONG LI, AND YAN-FENG LUO 1 (a) ◦ / ◦2 / •ε µ2 −→ 1 (b) ◦ o 2 ◦o ε • Figure 3. (a) A Aε3 quiver Q0 ; (b) The quiver Q1 = µ2 (Q). It follows from Definition 4.4 that M (Q0 ) = s1 , s2 , sε \| s21 = s22 = e, s2ε = sε , s1 s2 s1 = s2 s1 s2 , s1 sε = sε s1 , s2 sε s2 sε = sε s2 sε s2 = sε s2 sε i , M (Q1 ) = t1 , t2 , tε \| t21 = t22 = e, t2ε = tε , t1 t2 t1 = t2 t1 t2 , t1 tε t1 tε = tε t1 tε t1 = tε t1 tε , t2 tε t2 tε = tε t2 tε t2 = tε t2 tε , tε t2 t1 t2 = t2 t1 t2 tε i . Then ϕ : M (Q0 ) → M (Q1 ) is an inverse monoid isomorphism defined by ( t2 ti t2 if i = 1, ϕ(si ) = ti otherwise. 4.3. Inner by diagram automorphisms of Boolean reflection monoids. We first consider inner automorphisms of Boolean reflection monoids. It is well known that for any group G, Inn(G) ∼ = G/Z(G), where Inn(G) is the inner automorphism group of G, Z(G) is the center of G. It has been shown in [33,39] that automorphisms of the Boolean reflection monoid M (An−1 , B) are inner: for every automorphism α of M (An−1 , B), there exists a uniquely determined element g ∈ W (An−1 ) of the Weyl group W (An−1 ) such that α(t) = gtg −1 for all t ∈ M (An−1 , B). In other words, the automorphism group of M (An−1 , B) is naturally isomorphic to W (An−1 )/Z(W (An−1 )) for n ≥ 3 and the automorphism group of M (A1 , B) is naturally isomorphic to W (A1 ). As a generalization of the above result, we have the following theorem. Theorem 4.10. The inner automorphism group of M (Φ, B) is naturally isomorphic to W (Φ)/Z(W (Φ)), where Φ = An−1 (≥ 3), Bn (n ≥ 2), Dn (≥ 4). Proof. Let α be an inner automorphism of M (Φ, B). Since W (Φ) ⊆ M (Φ, B) is the unique unit group of M (Φ, B), α\|W (Φ) ∈ Inn(W (Φ)) = W (Φ)/Z(W (Φ)). In the following we prove the cases of Φ = Bn (n ≥ 2), Dn (≥ 4). Let Φε be one of Bnε , and Dnε shown in Figure 2. Suppose that Λ = {s0 , s1 , . . . , sn−1 , sε } is a set of generators of M (Φ, B). For any element g ∈ W (Φ), we claim that the set gΛg−1 is still a set of generators of M (Φ, B). Firstly, it is obvious that (gsi g−1 )2 = e and (gsε g−1 )2 = gsε g−1 . Nextly we will prove that gΛg−1 satisfies (R2)–(R4) in Definition 4.4. Case 1. There is no edge between i and j. Then gsi g−1 gsj g−1 = gsi sj g−1 = gsj si g−1 = gsj g −1 gsi g−1 . Case 2. There is an edge labeled by 1 between i and j. Then gsi g−1 gsj g−1 gsi g−1 = gsi sj si g−1 = gsj si sj g−1 = gsj g−1 gsi g −1 gsj g−1 . Case 3. There is an edge labeled by 2 between i and j. Then gsi g−1 gsj g−1 gsi g−1 gsj g−1 = gsi sj si sj g−1 = gsj si sj si g −1 = gsj g−1 gsi g−1 gsj g−1 gsi g−1 . QUIVER MUTATIONS AND BOOLEAN REFLECTION MONOIDS 17 Case 4. There is no edge between i and ε. Then gsi g−1 gsε g−1 = gsi sε g−1 = gsε si g−1 = gsε g−1 gsi g −1 . Case 5. There is an edge labeled by 1 between i and ε. Then gsi g−1 gsε g−1 gsi g−1 gsε g−1 = gsi sε si sε g−1 = gsε si sε si g−1 = gsε si sε g −1 = gsε g−1 gsi g−1 gsε g−1 gsi g −1 = gsε g−1 gsi g−1 gsε g−1 . Case 6. In type Bn , gs0 g −1 gs1 g −1 · · · gsε g−1 = gs0 s1 · · · sε g−1 = gs1 · · · sε g−1 = gs1 g−1 · · · gsε g−1 , gsε g−1 · · · gs1 g−1 gs0 g−1 = gsε · · · s1 s0 g−1 = gsε · · · s1 g−1 = gsε g−1 · · · gs1 g−1 . Case 7. In type Dn , gs0 g−1 gs2 g−1 · · · gsε g −1 · · · gs2 g−1 gs0 g −1 = gs0 s2 · · · sε · · · s2 s0 g−1 = gs1 s2 · · · sε · · · s2 s1 g−1 = gs1 g−1 gs2 g−1 · · · gsε g−1 · · · gs2 g −1 gs1 g −1 . Finally, we shall show that g1 sε g1−1 = g2 sε g2−1 for any g1 , g2 ∈ Z(W (Φ)). It suffices to prove that sε = w0 sε w0 , where w0 is the longest element in W (Φ) and w0 is an involution. In Section 1.2 of [16], we have w0 = wn wn−1 · · · w1 , where wi = si−1 · · · s1 s0 s1 · · · si−1 in type Φ = Bn , and wi = si−1 · · · s3 s2 s1 s0 s2 s3 · · · si−1 for i ≥ 3 and w1 = s0 , w2 = s1 in type Φ = Dn . Then by (R1), (R2), and (R4) of Definition 4.4, w0 sε w0 = wn wn−1 · · · w1 sε w1 · · · wn−1 wn = wn sε wn ( sn−1 · · · s1 (s0 s1 · · · sn−1 sε sn−1 · · · s1 s0 )s1 · · · sn−1 = sε = sn−1 · · · s3 s2 s1 (s0 s2 s3 · · · sn−1 sε sn−1 · · · s3 s2 s0 )s1 s2 s3 · · · sn−1 = sε in type Bn , in type Dn . Therefore the inner automorphism group of M (Φ, B) is isomorphic to W (Φ)/Z(W (Φ)) for Φ = An−1 (≥ 3), Bn (n ≥ 2), Dn (≥ 4). Let ∆ be one of Aεn−1 , Bnε , and Dnε shown in Figure 2. Let Q be a ∆ quiver. Let I ∪ {ε} be the set of vertices of Q and Q′ = µk (Q) the quiver obtained by a mutation of Q at a mutable vertex k. Following Barot and Marsh’s work [2], one can define variables 18 BING DUAN, JIAN-RONG LI, AND YAN-FENG LUO ti for i ∈ I, and tε in M (Q′ ) as follows: ( sk si sk if there is an arrow i → k in Q (possibly weighted), ti = si otherwise, ( sk sε sk if there is an arrow ε → k in Q (possibly weighted), tε = sε otherwise. (4.1) From Lemma 3.3 and Equation (4.1), it follows that new elements ti , i ∈ I, appearing in the procedure of mutations of quivers, must be some reflections of Weyl groups. By our Theorem 4.7 and Proposition 4.8, up to isomorphism, Boolean reflection monoids are encoded by their (generalized) Coxeter diagrams, see Figure 2. In the following theorem, we show that inner by diagram automorphisms of Boolean reflection monoids can be constructed by a sequence of mutations preserving the same underlying diagrams. Theorem 4.11. Let Q be a ∆ quiver and M (Q) the corresponding Boolean reflection monoid generated by a set S consisting of simple reflections and sε . Then α is an inner by diagram automorphism of M (Q) if and only if there exists a sequence of mutations preserving the underlying diagram ∆ such that α(S) can be obtained from Q by mutations. In particular, all reflections in W (Q\{ε}) and gsε g −1 for g ∈ W (Q\{ε}) can be obtained from Q by mutations. Proof. Let ∆ be one of Aεn−1 , Bnε , and Dnε shown in Figure 2. Suppose that S = {s1 , s2 , . . . , sn−1 , sε } for ∆ = Aεn−1 or S = {s0 , s1 , . . . , sn−1 , sε } for ∆ = Bnε , Dnε . In the case of type An , all automorphisms of M (Φ, B) are inner, see [33, 39]. A sequence µ of mutations preserving the underlying diagram of Q induces to an inner automorphism of M (Φ, B). All automorphisms of Weyl groups that preserve reflections are inner by diagram automorphisms, see Lemmas 3.5 and 3.6. So we assume without loss of generality that M (µ(Q)) = ht0 , t1 , . . . , tn−1 , tε i, where ti = gsi g−1 for 0 ≤ i ≤ n − 1, g ∈ W (Bn ) (respectively, g ∈ W (Dn )). We claim that tε = gsε g −1 . Firstly, if tε = gsε g−1 , then {t0 , t1 , . . . , tn−1 , tε } is a set of generators of M (µ(Q)) and the generalized Coxeter diagram corresponding to {t0 , t1 , . . . , tn−1 , tε } preserves the underlying diagram ∆. Since tε ti = ti tε for 0 ≤ i ≤ n − 2, we have tε ∈ Z(W (Bn−1 )) (respectively, tε ∈ Z(W (Dn−1 ))), where W (Bn−1 ) = htε t0 , tε t1 , . . . , tε tn−2 i (respectively, W (Dn−1 ) = htε t0 , tε t1 , . . . , tε tn−2 i). The variable tε must be of the form g′ sε g′ −1 for some g′ ∈ W (Bn ) (respectively, g′ ∈ W (Dn )). So tε is not the longest word w0 in W (Bn−1 ) (respectively, W (Dn−1 )). Therefore tε must be the unique identity element in W (Bn−1 ) (respectively, W (Dn−1 )) and hence tε = gsε g−1 . Conversely, for each inner automorphism α of M (Q), by Theorem 4.10, there exists an element g ∈ W (Q\{ε}) of the Weyl group W (Q\{ε}) ⊆ M (Q) such that α(t) = gtg −1 for all t ∈ M (Q). The remainder proof of the necessity is similar to the proof of the necessity of Theorem 3.7. Every reflection in W (Q\{ε}) is of the form gsi g−1 , where g = si1 si2 · · · sik ∈ W (Q\{ε}) is a reduced expression for g. By the same arguments as before, we mutate the sequence i1 , i1 , i2 , i2 , . . . , ik , ik starting from Q, we get gsi g −1 and gsε g−1 . QUIVER MUTATIONS AND BOOLEAN REFLECTION MONOIDS 19 4.4. Cellularity of semigroup algebras of Boolean reflection monoids. In this section, we show that semigroup algebras of Boolean reflection monoids are cellular algebras. We use these presentations we obtained to construct new cellular bases of such cellular algebras. Let R be a commutative ring with identity. Recall that a semigroup S is said to be cellular if its semigroup algebra R[S] is a cellular algebra. Proposition 4.12. The Boolean reflection monoid M (Φ, B) for Φ = An−1 , Bn , or Dn is a cellular semigroup. Proof. All maximal subgroups of the Boolean reflection monoid M (Φ, B) are finite reflection groups. It has been shown in [23] that any finite reflection group W (Φ) is cellular with respect to which the anti-involution is inversion. Therefore, for each D-class D of M (Φ, B), the subgroup HD ⊆ M (Φ, B) is cellular with cell datum (ΛD , MD , CD , iD =−1 ), which satisfies East’s first assumption, see Theorem 15 in [8] or Theorem 2.2. We define a map i : R[M (Φ, B)] → R[M (Φ, B)] X X rj gj 7→ rj gj−1 , j j where rj ∈ R, gj ∈ M (Φ, B). The map i is an R-linear anti-homomorphism and −1 i([e, f, g]D ) = [e, f, g]−1 D = [f, e, g ]D = [f, e, iD (g)]D for any g ∈ HD , e D f in M (Φ, B). From Theorem 19 in [8] or Theorem 2.2, it follows that the Boolean reflection monoid M (Φ, B) is a cellular semigroup, as required. Remark 4.13. The case that a finite inverse semigroup whose maximal subgroups are direct products of symmetric groups has been considered by East, see Theorem 22 of [8]. The Boolean reflection monoid of type An−1 is isomorphic to the symmetric group Sn of degree n. Maximal subgroups of the Boolean reflection monoid of type Bn are finite reflection groups of type Br , r ≤ n, which are isomorphic to (Z2 × Z2 . . . × Z2 ) ⋊ Sr . Let ∆ be one of Aεn−1 , Bnε , and Dnε shown in Figure 2. For two quivers with the same underlying diagrams appearing in the mutation class of ∆ quivers, we always use their presentations to construct inner by diagram automorphisms of Boolean reflection monoids, see Theorem 4.11 and then we extend it an R-linear automorphism of semigroup algebras of Boolean reflection monoids. By Corollary 2.3, we obtain new cellular bases of semigroup algebras of Boolean reflection monoids. 4.5. An example. Let In be the symmetric inverse semigroup on [n] = {1, 2, . . . , n}. Let w be a partial permutation on a set A ⊆ [n] and denote the image of i ∈ A under the map w by wi and the image of i 6∈ A under the map w by wi = ∅. We denote w by the sequence (w1 w2 . . . wn ). For example, (3 − 2) is the partial permutation with domain {1, 3} and range {2, 3} under which 1 → 3, 3 → 2. The following example gives new cellular bases of R[I3 ] by the method of quiver mutations. Example 4.14. Let Q0 be the quiver in Example 4.9 and by the results of preceding sections, M (Q0 ) ∼ = I3 a Boolean reflection monoid. We have M (Q0 )/D = {D0 < D1 < 20 BING DUAN, JIAN-RONG LI, AND YAN-FENG LUO D2 < D3 }, where each Di is the set of all elements of M (Q0 ) of rank i, and idempotents in each Di are the partial identity permutation on i-subsets of [3]. Let A be an i-subset of [3]. As shown in Example 23 of [8], the H-class containing the idempotent idA is the subgroup {x ∈ I3 \| im(x) = dom(x) = A} ∼ = S\|A\| . It is well known that the group algebra R[Sn ] has cellular bases with respect to which the anti-involution is inversion. Indeed the Khazdan-Luzstig bases and the Murphy basis both have this property (see, Example (1.2) of [24], Example (2.2) of [36] or Section 4 of [37]). Take s1 = (2 1 3), s2 = (1 3 2), and sε = (1 2 −). By mutating Q0 , we obtain the following isomorphic quivers (Theorems 4.10 and 4.11): / s◦2 s1 (a) ◦ s1 (b) ◦ (c) s1 s2 s1 s2 s1 s2 / s◦2 ◦ s2 / • ◦ / / s◦1 ◦ s2 s2 sε s2 / s1 s2 s1 s2 s1 s2 (f ) ◦ / s•ε ◦ / (d) ◦ (e) / s•ε / s◦1 / / s1 s2 sε s2 s1 • s2 sε s2 • s1 s2 sε s2 s1 • From Theorem 4.7 and Proposition 4.8, it follows that the inverse monoids determined by quivers (a)–(f ) are isomorphic to the symmetric inverse semigroup I3 , respectively. The presentation of I3 determined by the quiver (a) admits an initial cellular bases by Theorem 19 of [8] or Theorem 2.2. We can construct an R-linear automorphism of R[I3 ] using these presentations corresponding to quivers (a)–(f ), and then by Corollary 2.3, we obtain new cellular bases of R[I3 ]. 5. Mutations of quivers of finite type Throughout this section, let as before ∆ be one of Aεn−1 , Bnε , and Dnε in Figure 2, we consider the way of mutations of ∆ quivers and the oriented cycles appearing in them, refer to [2, 21]. A quiver without no loops and no 2-cycles is said to be of finite type if it is mutation equivalent to a Dynkin quiver. A chordless cycle is a cycle such that no two vertices of the cycle are connected by an edge that does not itself. One can show [21, Proposition 9.7] (or [2, Proposition 2.1]) that all chordless cycles are oriented in the mutation classes of Dynkin quivers. We extend the results in [21] and [2, Corollary 2.3] to the case of ∆ quivers. Lemma 5.1. Let Q be a ∆ quiver and k a mutable vertex of Q. Suppose that k has two neighbouring vertices. Then the induced subquivers of Q containing vertex k and its neighbours are shown in Figure 4. The effect of the mutation of Q at k is shown in each case. QUIVER MUTATIONS AND BOOLEAN REFLECTION MONOIDS µ k (a) ◦ ✆B \✾✾✾ ✾✾ ✆✆ ✆ ✾✾ ✆✆ ✾✾ ✆ ✆✆ ◦ i ◦ ◦ B ◦ \✾ ✆✆ ✾✾✾ ✆ ✾✾2 ✆✆ ✾✾ ✆✆ ✾ ✆ ✆ i ◦ ◦ ◦ i i µk ←→ ◦ j k (a’) ◦ B ◦ \✾ ✆✆ ✾✾✾ ✆ ✾✾ ✆✆ ✾✾ ✆✆ ✾ ✆✆ • (c’) ◦ ◦ ✆ \✾✾✾ ✾✾ ✆✆ ✆ ✾✾ ✆✆ ✾✾ ✆ ✆ ✆ ←→ • ε i µ k (e’) B◦✾ ✆✆ ✾✾✾ ✆ ✾✾ ✆✆ ✾✾ ✆✆ ✾ ✆ ✆ k ←→ 2 ◦ (g’) • k ◦ \✾ ✆✆ ✾✾✾ ✆ 2 ✆ ✾✾2 ✆ ✾✾ ✆✆ ✾ ✆ ✆ /• ◦ i i ◦ ◦ ✆B ✾✾✾ ✾✾ ✆✆ ✆ ✾✾ ✆✆ ✾✾ ✆ ✆✆ • ◦o (d’) k (f’) ε i µk ←→ 2 ε ←→ • ◦ i j k ◦ \✾ ✆✆ ✾✾✾ ✆ ✾✾ ✆✆ ✾✾ ✆✆ ✾ ✆ ✆ /• ◦ ε k ◦ ✆ ✾✾✾ ✾✾ ✆✆ ✆ ✾✾ ✆✆ ✾✾ ✆ ✆ ✆ 2 • ◦ µ k ←→ • ε i 2 ◦ ◦ \✾ ✆✆ ✾✾✾ ✆ 2 ✆ ✾✾2 ✆ ✾✾ ✆✆ ✾ ✆ ✆ /◦ ◦ ←→ k ◦ \✾ ✆✆ ✾✾✾ ✆ ✾✾ ✆✆ ✾✾ ✆✆ ✾ ✆ ✆ j k i ε i 2 µk k B ◦ \✾ ✆✆ ✾✾✾ 2 ✆✆ ✾✾ ✆ ✾✾ ✆✆ ✾ ✆ ✆ ◦ \✾ ✆✆ ✾✾✾ ✆ 2 ✆ ✾✾ ✆ ✾✾ ✆✆ ✾ ✆ ✆ /◦ ◦ i µk ε i ε ←→ j B◦✾ ✆✆ ✾✾✾ ✆ ✾✾ ✆✆ ✾✾ ✆✆ ✾ ✆✆ j k i µk k 2 ε i ◦ B◦✾ ✆✆ ✾✾✾ ✆ 2 ✆ ✾✾2 ✆ ✾✾ ✆✆ ✾ ✆ ✆ ◦o ◦ • ◦ \✾ ✆✆ ✾✾✾ ✆ ✾✾ ✆✆ ✾✾ ✆✆ ✾ ✆ ✆ /• ◦ ←→ j (b’) k i B◦✾ ✆✆ ✾✾✾ ✆ 2 ✆ ✾✾ ✆ ✾✾ ✆✆ ✾ ✆ ✆ k ◦ ✆ \✾✾✾ ✾✾ ✆✆ ✆ ✾✾ ✆✆ ✾✾ ✆ ✆ ✆ /◦ ◦ i µk k ε i µk k ◦✾ ✆✆ ✾✾✾ ✆ ✾✾ ✆✆ ✾✾ ✆✆ ✾ ✆✆ ◦ j i j k ◦ ε i 2 i ←→ ◦ j (f) k ←→ k ◦ ◦ \✾ ✆✆ ✾✾✾ ✆ ✾✾2 ✆✆ ✾✾ ✆✆ ✾ ✆ ✆ /◦ ◦ ◦ ✆B ✾✾✾ ✾✾ ✆✆ ✆ ✾✾ ✆✆ ✾✾ ✆ ✆✆ i (d) k µk ◦ ◦ j ◦✾ ✆✆ ✾✾✾ ✆ ✾✾2 ✆✆ ✾✾ ✆✆ ✾ ✆ ✆ µ k (b) k ←→ j B◦✾ ✆✆ ✾✾✾ ✆ ✾✾2 ✆✆ ✾✾ ✆✆ ✾ ✆ ✆ ◦ ✆ ✾✾✾ ✾✾ ✆✆ ✆ ✾✾ ✆✆ ✾✾ ✆ ✆ ✆ i µk k (e) ◦ j k (c) k k ←→ 21 k B◦✾ ✆✆ ✾✾✾ ✆ ✾✾ ✆✆ ✾✾ ✆✆ ✾ ✆ ✆ o ◦ • 2 • ε i 2 ε k B◦✾ ✆✆ ✾✾✾ ✆ 2 ✆ ✾✾2 ✆ ✾✾ ✆✆ ✾ ✆ ✆ • ◦o i ε Figure 4. Subquivers of mutations of Q. In a diagram, a vertex is said to be connected to another if there is an edge between them. Let Q be a quiver mutation equivalent to a Dynkin quiver. In Lemma 2.4 of [2], Barot and Marsh have described the way vertices in Q can be connected to a chordless cycle: A vertex is connected to at most two vertices of a chordless cycle, and if it is connected to two vertices, then the two vertices must be adjacent in the cycle. The following lemma is a generalization of Barot and Marsh’s results [2, Lemma 2.5]. Lemma 5.2. Let Q′ = µk (Q) be the mutation of Q at vertex k. We list various types of induced subquivers in Q and corresponding cycles in Q′ . Then every chordless cycle in Q′ arises in such a way. 22 BING DUAN, JIAN-RONG LI, AND YAN-FENG LUO µ k (a) ◦ ◦ ✂ ]❁❁❁ ❁❁ ✂✂ ✂ ❁❁ ✂✂ ❁ ✂ ✂ µ ◦ ◦ ✂ ]❁❁❁ ❁❁2 ✂✂ ✂ ❁❁ ✂✂ ❁ ✂ ✂ k (a’) ◦ k −→ ◦ ◦ ✆B ✾✾✾ ✾✾ ✆✆ ✆ ✾✾ ✆✆ ✾✾ ✆ ✆✆ • µ k B◦✾ ✆✆ ✾✾✾ ✆ ✾✾ ✆✆ ✾✾ ✆✆ ✾ ✆ ✆ k −→ 2 ◦ (e’) ◦ \✾ ✆✆ ✾✾✾ ✆ 2 ✆ ✾✾2 ✆ ✾✾ ✆✆ ✾ ✆ ✆ /• ◦ (b’) k (d’) 2 µk ←→ ◦ d❏❏ (h’) ◦k O k k−1 2 6 ◦ ❴ ❴ ❴/ ◦ / ◦1 k ←→ k ◦ o❴ ❴ ❴ ◦ o d−2 k−1 2 6 ◦ ❴ ❴ ❴/ ◦ ◦ k k ◦S ◦ o❴ ❴ ❴ ◦ o k+1 d−1 ◦ d i ε i • ε ε j / ◦1 ◦ o❴ ❴ ❴ ◦ o k+1 µ ◦ d−2 2 ◦O ❴ ❴ ❴/ ◦ k−1 k ←→ k d ◦ ( •L k ◦ v ε ε j ✉: ◦ ✉✉ ✉✉ ✉ ✉ ✉✉ ✉✉ ✉✉✉ ◦o • 2 ◦O ❴ ❴ ❴/ ◦ d 2 i µk k−1 d−1 / ◦1 i • ε j ◦ B◦✾ ✆✆ ✾✾✾ ✆ ✾✾ ✆✆ ✾✾ ✆✆ ✾ ✆ ✆ o ◦ • µk / ◦i ←→ ◦k o ◦o µ ε k / ◦ ←→ ◦ o❏ ◦O O ❏❏ ❏❏ ❏❏ ❏❏ ❏❏ ❏❏ ❏$ • ◦o • i ε k ◦ ✆B ✾✾✾ ✾✾ ✆✆ ✆ ✾✾ ✆✆ ✾✾ ✆ ✆✆ ◦o • 2 k (f’) ◦ O / ◦i ◦ ◦S (i’) j ◦o ε k −→ ε i ε • k+1 ◦ \✾ ✆✆ ✾✾✾ ✆ ✾✾ ✆✆ ✾✾ ✆✆ ✾ ✆ ✆ ◦ ✂A ❁❁❁ ❁❁2 ✂✂ ✂ ❁❁ ✂✂ ❁ ✂ ✂ o ◦ ◦ i µ k ◦ B◦✾ ✆✆ ✾✾✾ ✆ 2 ✆ ✾✾2 ✆ ✾✾ ✆✆ ✾ ✆ ✆ ◦o • j • 2 k i µ k −→ ε i ε ❏❏ ❏❏ ❏❏ ❏❏ ❏❏ ❏❏ /• k ◦ ◦ \✾ ✆✆ ✾✾✾ ✆ ✾✾ ✆✆ ✾✾ ✆✆ ✾ ✆ ✆ /• ◦ µk / ◦i ←→ ◦o ◦ ✆ \✾✾✾ ✾✾ ✆✆ ✆ ✾✾ ✆✆ ✾✾ ✆ ✆ ✆ k 2 k ε i ε (g’) ◦j O (e) • k i k k ◦ ✂A ❁❁❁ ❁❁ 2 ✂✂✂ ❁❁ ✂✂ ❁ ✂ ✂ ◦ ◦o 2 µ k ←→ ◦ ✂ ]❁❁❁ ❁❁2 ✂✂ ✂ ❁❁ ✂✂ ❁ ✂ ✂ /◦ ◦ 2 ε i ◦ 2 ◦ ✆ \✾✾✾ ✾✾ ✆✆ ✆ ✾✾ ✆✆ ✾✾ ✆ ✆ ✆ /• ◦ i k −→ k (d) ◦ ✂A ❁❁❁ ❁❁2 ✂✂ ✂ ❁❁ ✂✂ ❁ ✂ ✂ 2 o ◦ ◦ ε i (c’) µ k −→ ◦ ✂ ]❁❁❁ ❁❁ 2 ✂✂✂ ❁❁ ✂✂ ❁ ✂ ✂ ◦ k µ k (b) ◦ ✂A ❁❁❁ ❁❁ ✂✂ ✂ ❁❁ ✂✂ ❁ ✂ ✂ ◦ ◦o ◦ k (c) k k −→ v d−1 / ◦1 i ε ◦ ( •L ◦ o❴ ❴ ❴ ◦ o k+1 d−1 ◦ d (j’) The vertex k does not connect to an oriented chordless cycle C in Q. Then C is the corresponding cycle in Q′ . QUIVER MUTATIONS AND BOOLEAN REFLECTION MONOIDS 23 (k’) The vertex k connects to one vertex of an oriented chordless cycle C in Q (via an edge of unspecified weight). Then C is the corresponding cycle in Q′ . By Lemmas 5.1 and 5.2, we have the following corollary. Corollary 5.3. Let Q be a quiver in the mutation class of a ∆ quiver. Then the frozen vertex ε in Q has one neighbour or two neighbours and if it has two neighbours, then ε must be in an oriented cycle. 6. Cycle relations and path relations In this section, we find an efficient subset of the relations sufficient to define Boolean reflection monoids, which generalizes Barot and Marsh’s results, Lemmas 4.1, 4.2, 4.4 and Proposition 4.6 in [2]. Lemma 6.1 ([2, Lemmas 4.1, 4.2 and 4.4]). Let Q be a Dynkin quiver and W (Q) the reflection group determined by Q, see Section 3. (1) If Q contains a chordless cycle Cd , see Figure 5 (1), d ≥ 3, then the following are equivalent: (a) (sa sa+1 · · · sa+d−1 sa+d−2 · · · sa+1 )2 = e (with subscripts modulo d) for a single fixed value of a, 0 ≤ a ≤ d − 1; (b) (sa sa+1 · · · sa+d−1 sa+d−2 · · · sa+1 )2 = e (with subscripts modulo d) for any a = 0, 1, · · · , d − 1. (2) If Q contains a chordless 3-cycle C3 , see Figure 5 (2), then the following are equivalent: (a) (s1 s2 s3 s2 )2 = e; (b) (s2 s3 s1 s3 )2 = e. Furthermore, if one of the above holds, then the following holds: (c) (s3 s1 s2 s1 )3 = e. (3) If Q contains a chordless 4-cycle C4 , see Figure 5 (3), then the following are equivalent: (a) (s1 s2 s3 s4 s3 s2 )2 = e; (b) (s3 s4 s1 s2 s1 s4 )2 = e. Furthermore, if one of the above holds, then the following holds: (c) (s2 s3 s4 s1 s4 s3 )3 = e; (d) (s4 s1 s2 s3 s2 s1 )3 = e. (1) 0 ◦O ◦ o d−1 / ◦1 .. (2) . 2 ◦ ✆B ✾✾✾ ✆ ✾✾ 2 ✆✆ ✾✾ ✆ ✆ ✾✾ ✆ ✆ ✆ o ◦ ◦ 1 2 3 1 (3) ◦O / ◦2 2 2 ◦o 4 ◦ 3 Figure 5. (1) A chordless d-cycle Cd , (2) a chordless 3-cycle C3 , and (3) a chordless 4-cycle C4 . 24 BING DUAN, JIAN-RONG LI, AND YAN-FENG LUO Let ∆ be one of Aεn−1 , Bnε , and Dnε in Figure 2. Suppose that Q is any quiver mutation equivalent to a ∆ quiver. The following lemma shows an efficient subset of relations (R3) and (R4) in Definition 4.4, which generalizes the above lemma. Lemma 6.2. Let M (Q) be an inverse monoid with generators subjecting to relations (R1), (R2) in Definition 4.4. (1) If Q contains a chordless cycle C3′ , see Figure 6 (1), then the following statements are equivalent: (a) sε s1 s2 s1 = s1 s2 s1 sε ; (b) s1 s2 sε s2 = s2 sε s2 s1 . Furthermore, if one of the above holds, then the following statements are equivalent: (c) sε s1 sε = sε s1 sε s1 = s1 sε s1 sε ; (d) sε s2 sε = sε s2 sε s2 = s2 sε s2 sε . (2) If Q contains a chordless cycle C3′′ , see Figure 6 (2), then s1 s2 sε s2 = s2 sε s2 s1 . (3) If Q contains a chordless cycle C4′ , see Figure 6 (3), then the following statement holds: (a) sε s1 s2 s3 s2 s1 = s1 s2 s3 s2 s1 sε ; (b) s1 s2 s3 sε s3 s2 = s2 s3 sε s3 s2 s1 . Furthermore, if one of the above holds, then the following statements are equivalent: (c) sε s1 sε = sε s1 sε s1 = s1 sε s1 sε ; (d) sε s3 sε = sε s3 sε s3 = s3 sε s3 sε . (4) If Q contains a subquiver Cd′ , see Figure 6 (4), then the following statements are equivalent: (a) sa sa+1 · · · sd P (s1 , sε )P (sε , s1 )sd · · · sa+1 sa = sa−1 sa−2 · · · s1 s1 P (s1 , sε ) P (sε , s1 )s1 s1 · · · sa−2 sa−1 for a single fixed value of a, 2 ≤ a ≤ d; (b) sa sa+1 · · · sd P (s1 , sε )P (sε , s1 )sd · · · sa+1 sa = sa−1 sa−2 · · · s1 s1 P (s1 , sε ) P (sε , s1 )s1 s1 · · · sa−2 sa−1 for any a = 2, · · · , d. (1) 2 ◦ ✆B ✾✾✾ ✆ ✾✾ ✆✆ ✾✾ ✆✆ ✾✾ ✆ ✆✆ o • ◦ ε 1 (3) 2 ◦ ✆B ✾✾✾ ✆ ✾✾ 2 ✆✆ ✾✾ ✆✆ ✾✾ ✆ ✆✆ o • ◦ (2) 2 ◦O / ◦3 ◦o • 1 ε 2 1 ε 2 ◦O (4) ✝ ✝✝ ✝ ✝ ✝✝ ✝ ✝ /◦o •❴ ❴ ❴◦ ε 1 d / ◦3 .. . Figure 6. (1) A chordless 3-cycle C3′ , (2) a chordless 3-cycle C3′′ , (3) a chordless 4-cycle C4′ , and (4) a subquiver Cd′ . (see Lemma 6.2) QUIVER MUTATIONS AND BOOLEAN REFLECTION MONOIDS 25 Proof. For (1), the equivalence of (a) and (b) follows from: s1 s2 sε s2 = s2 (s1 s2 s1 sε )s2 , s2 sε s2 s1 = s2 (sε s1 s2 s1 )s2 , using (R2). Suppose that (a) and (b) hold. Then by (R1), (R2), (a), and (b), the equivalence of (c) and (d) follows from: s1 s2 s1 (sε s1 sε )s1 s2 s1 = sε s1 s2 s1 s1 s1 s2 s1 sε = sε s2 sε , s1 s2 s1 (sε s1 sε s1 )s1 s2 s1 = sε (s1 s2 sε s2 )s1 = sε s2 sε s2 , s1 s2 s1 (s1 sε s1 sε )s1 s2 s1 = s1 (s2 sε s2 s1 )sε = s2 sε s2 sε . For (3), the equivalence of (a) and (b) follows from: s1 s2 s3 sε s3 s2 = s2 s3 (s1 s2 s3 s2 s1 sε )s3 s2 , s2 s3 sε s3 s2 s1 = s2 s3 (sε s1 s2 s3 s2 s1 )s3 s2 , using first s1 s2 s3 s2 s1 = s3 s2 s1 s2 s3 and then (R1). Suppose that (a) and (b) hold. Using first s2 sε = sε s2 and then (a), we have: s1 s2 s3 s2 s1 s2 (sε s1 sε )s2 s1 s2 s3 s2 s1 = (s1 s2 s3 s2 s1 sε )s2 s1 s2 (sε s1 s2 s3 s2 s1 ) = (sε s1 s2 s3 s2 s1 )s2 s1 s2 (s1 s2 s3 s2 s1 sε ) = sε s1 s2 s3 s2 s3 s2 s1 sε = sε s3 sε , (by (R2)) where in the last equation we used that s1 and s3 commute. Using (R1), (R2), (a), and (b), by a similar argument, we have s1 s2 s3 s2 s1 s2 (s1 sε s1 sε )s2 s1 s2 s3 s2 s1 = s1 s2 s3 s1 s2 sε s1 s2 s1 s2 s3 s2 s1 sε , = s1 s2 s1 s3 sε s1 s3 s2 s1 sε , = s2 (s1 s2 s3 sε s3 s2 )s1 s2 sε , = s2 (s2 s3 sε s3 s2 s1 )s1 s2 sε , = s3 sε s3 sε . s1 s2 s3 s2 s1 s2 (sε s1 sε s1 )s2 s1 s2 s3 s2 s1 = (s1 s2 s3 s2 s1 sε )s2 s1 sε s2 s1 s3 s2 s1 = sε s1 s2 s3 s2 s1 s2 s1 s2 sε s3 s1 s2 s1 = sε s2 (s1 s2 s3 sε s3 s2 )s1 s2 = sε s2 (s2 s3 sε s3 s2 s1 )s1 s2 = sε s3 sε s3 . Therefore (c) and (d) are equivalent. For (4), using (R1), it is obvious. At an end of this section, we show that M (Q) could be defined using only the underlying unoriented weighted diagram of Q, by taking relations (R1)–(R4) corresponding to both Q and Qop as the defining relations. Our result can be viewed as a generalization of Proposition 4.6 of [2]. Proposition 6.3. Let M (Φ, B) be a Boolean reflection monoid with generators si , i ∈ I ∪ {ε}. Then the generators satisfy (R1)–(R4) with respect to Q if and only if they satisfy (R1)–(R4) with respect to Qop . 26 BING DUAN, JIAN-RONG LI, AND YAN-FENG LUO Proof. We assume that generators si , i ∈ I ∪ {ε} satisfy relations (R1)–(R4) with respect to Q, and show that these generators satisfy relations (R1)–(R4) with respect to Qop . The converse follows by replacing Q with Qop . Since (R1) and (R2) do not depend on the orientation of Q, generators si , i ∈ I ∪ {ε} satisfy relation (R1) and (R2) with respect to Qop . The cases of chordless cycles appearing in quivers of finite type have been proved in Proposition 4.6 of [2]. The remaining needed to check the cases are C3′ , C3′′ , C4′ , and Cd′ shown in Figure 6. Case 1. In C3′ , we have sε s2 s1 s2 = sε s1 s2 s1 = s1 s2 s1 sε = s2 s1 s2 sε , s2 s1 sε s1 = s1 s2 (s1 s2 sε s2 )s2 s1 = s1 s2 (s2 sε s2 s1 )s2 s1 = s1 sε s1 s2 . Case 2. In C3′′ , we have sε s2 s1 s2 = s2 s1 (s1 s2 sε s2 )s1 s2 = s2 s1 (s2 sε s2 s1 )s1 s2 = s2 s1 s2 sε . Case 3. In C4′ , note that s1 s2 s3 s2 s1 = s3 s2 s1 s2 s3 . We have sε s3 s2 s1 s2 s3 = sε s1 s2 s3 s2 s1 = s1 s2 s3 s2 s1 sε = s3 s2 s1 s2 s3 sε , s3 s2 s1 sε s1 s2 = s2 s1 s3 s2 (s1 s2 s3 sε s3 s2 )s2 s3 s1 s2 = s2 s1 s3 s2 (s2 s3 sε s3 s2 s1 )s2 s3 s1 s2 = s2 s1 sε s1 s2 s3 . Case 4. In Cd′ , it follows from Lemma 6.2 (4) that (R4) do not depend on the orientation of chordless cycles in Cd′ . Since every chordless cylce in Qop corresponds to a chordless cycle in Q, the result holds. 7. The proof of Theorem 4.7 In this section, we give the proof of Theorem 4.7. Let ∆ be one of Aεn−1 , Bnε , and Dnε in Figure 2. We fix a ∆ quiver Q. Let Q′ = µk (Q) be the mutation of Q at vertex k, k ∈ I. Throughout the section, we will write si and ri for the generators corresponding to vertex i ∈ I ∪ {ε} of M (Q) and M (Q′ ) respectively. Similar to [2], we define elements ti , i ∈ I, and tε in M (Q) as follows: ( sk si sk if there is an arrow i → k in Q (possibly weighted), ti = si otherwise, ( (7.1) sk sε sk if there is an arrow ε → k in Q (possibly weighted), tε = sε otherwise. Then ( (sk si sk )(sk si sk ) = e if there is an arrow i → k in Q (possibly weighted), t2i = s2i = e otherwise, ( (sk sε sk )(sk sε sk ) = tε if there is an arrow ε → k in Q (possibly weighted), t2ε = otherwise. s2ε = tε (7.2) QUIVER MUTATIONS AND BOOLEAN REFLECTION MONOIDS 27 In order to prove Theorem 4.7, we need the following proposition, which we will prove in Section 7.2. Proposition 7.1. For each i ∈ I ∪ {ε}, the map Φ : M (Q′ ) −→ M (Q) ri 7−→ ti , is an inverse monoid homomorphism. 7.1. Proof Theorem 4.7. For each vertex i ∈ I ∪ {ε} of Q define the elements t′i in M (Q′ ) as follows: ( rk ri rk if there is an arrow k → i in Q′ , t′i = ri otherwise, ( rk rε rk if there is an arrow k → ε in Q′ , t′ε = rε otherwise. We claim that these elements t′i , for each vertex i ∈ I ∪ {ε}, satisfy the relations (R1)– (R4) defining M (Q). This follows from Proposition 7.1 by interchanging Q and Q′ and using the fact that the definition of M (Q) is unchanged under reversing the orientation of all the arrows in Q (see Lemma 6.3). Therefore there is an inverse monoid homomorphism Θ : M (Q) → M (Q′ ) such that Θ(si ) = t′i for each i. If there is no arrow i → k in Q, then there is also no arrow k → i in Q′ and consequently Θ ◦ Φ(ri ) = Θ(si ) = ri . If there is an arrow i → k in Q, then there is an arrow k → i in Q′ and therefore Θ ◦ Φ(ri ) = Θ(sk si sk ) = Θ(sk )Θ(si )Θ(sk ) = rk (rk ri rk )rk = ri . So Θ ◦ Φ = idM (Q′ ) , and, similarly, Φ ◦ Θ = idM (Q) , and hence Θ and Φ are isomorphisms. 7.2. The proof of Proposition 7.1. We will prove Proposition 7.1 by showing that the elements ti , i ∈ I ∪ {ε} satisfy the (R1)–(R4) relations in M (Q′ ). We denote by m′ij the value of mij for Q′ . By Equation (7.2), (R1) is obvious. In the sequel, the proof that the elements ti , i ∈ I ∪ {ε} satisfy (R2) in M (Q′ ) follows from Lemma 7.2 and the rest of proof is completed case by case. Lemma 7.2. The elements ti , for i a vertex of Q, satisfy the following relations. ′ (1) If i = k or j = k and i, j 6= ε, then (ti tj )mij = e. (2) If at most one of i, j is connected to k in Q (or, equivalently, in Q′ ) and i, j 6= ε, ′ then (ti tj )mij = e. (3) Let i be in I. Then   if ε, i are not connected in Q′ ,  ti tε = tε ti tε ti tε = tε ti tε ti = ti tε ti tε if ε, i are connected by an edge with weight 1 in Q′ ,   tε = ti tε = tε ti if ε, i are connected by an edge with weight 2 in Q′ . Proof. In Lemma 5.1 of [2], Barot and Marsh proved the parts (1) and (2). We only need to prove the part (3). 28 BING DUAN, JIAN-RONG LI, AND YAN-FENG LUO Suppose without loss of generality that i = k. The only nontrivial case is when there is an arrow ε → k = i with a weight q in Q. If q = 1, then tε tk tε tk = (sk sε sk )sk (sk sε sk )sk = sk sε sk sε = sε sk sε sk = sk (sk sε sk )sk (sk sε sk ) = tk tε tk tε , = sε sk sε = sk sε sk sε sk = (sk sε sk )sk (sk sε sk ) = tε tk tε . If q = 2, note that sε sk = sk sε = sε , we have tk tε = sk (sk sε sk ) = sε sk = sk sε = sε = tε tk = tε . In the following, suppose that i 6= k. We divide this proof into three cases. Case 1. There are no arrows from i, ε to k, then ti = si , tε = sε hold (3). Case 2. There are arrows from one of i, ε to k and there are no arrows from the other of i, ε to k in Q, then we assume that there are arrows from ε to k and there are no arrows from i to k in Q. If ε, i are not connected in Q, we have ti tε = si (sk sε sk ) = (sk sε sk )si = tε ti . If ε, i are connected by an edge with weight 1 in Q, then tε ti tε = (sk sε sk )si (sk sε sk ) = sk sε si sε sk = sk sε si sε sk si = (sk sε sk )si (sk sε sk )si = tε ti tε ti = si sk sε si sε sk = si (sk sε sk )si (sk sε sk ) = ti tε ti tε . That ε, i are connected by an edge with weight 2 and there are no arrows from i to k in Q is impossible, because of the fact that there is only 3-chordless cycle in the mutation class of Bnε quivers and Corollary 5.3. Case 3. There are arrows from i, ε to k. The possibilities for the subquivers induced by i, ε, and k are enumerated in (a’)–(g’) of Figure 4. We show that ti and tε satisfy (3) by checking each case. Within each case, subcase (i) is when the subquiver of Q is the diagram on the left, and subcase (ii) is when the subquiver of Q is the diagram on the right. (a′ )(i) We have ti tε = (sk si sk )(sk sε sk ) = sk si sε sk = sk sε si sk = (sk sε sk )(sk si sk ) = tε ti . (a′ )(ii) We have ti tε = si sε = sε si = tε ti . (b′ ) (i) We have tε ti tε = sε (sk si sk )sε = sε si sk si sε = si (sε sk sε )si , ( s i s ε s k s ε s k s i = s ε s i s k s i s ε s i s k s i = s ε s k s i s k s ε s k s i s k = tε ti tε ti , = si sk sε sk sε si = si sk si sε si sk si sε = (sk si sk )sε (sk si sk )sε = ti tε ti tε . (b′ ) (ii) We have ti tε = si (sk sε sk ) = sk (sk si sk sε )sk = sk (sε sk si sk )sk = (sk sε sk )si = tε ti . (c′ ) (i) We have tε ti tε = (sk sε sk )si (sk sε sk ) = sk sε si sk si sε sk = sk si sε sk sε si sk , ( sk si sε sk sε sk si sk = sk sε si sk si sε sk si = (sk sε sk )si (sk sε sk )si = tε ti tε ti , = sk si sk sε sk sε si sk = si sk sε si sk si sε sk = si (sk sε sk )si (sk sε sk ) = ti tε ti tε . (c′ ) (ii) We have ti tε = (sk si sk )sε = si sk si sε = sε si sk si = sε sk si sk = tε ti . (d′ ) (i) We have ti tε = (sk si sk )(sk sε sk ) = sk si sε sk = sk sε si sk = (sk sε sk )(sk si sk ) = tε ti . QUIVER MUTATIONS AND BOOLEAN REFLECTION MONOIDS 29 (d′ )(ii) We have ti tε = si sε = sε si = tε ti . (e′ )(i) Note that si sk sε = sk sε and sε sk si = sε sk . We have ti tε = (sk si sk )sε = sk (sk sε ) = sε = tε , tε ti = sε (sk si sk ) = (sε sk )sk = sε = tε . (e′ )(ii) We have ti tε = si (sk sε sk ) = si sk (sε sk si sk )sk si = si sk (sk si sk sε )sk si = sk sε sk si = tε ti . (f ′ )(i) Note that si sk sε = sk sε and sε sk si = sε sk . We have ti tε = si (sk sε sk ) = sk sε sk = tε , tε ti = (sk sε sk )si = sk sε sk = tε . (f ′ )(ii) We have ti tε = (sk si sk )sε = sk (si sk sε sk )sk = sk (sk sε sk si )sk = sε sk si sk = tε ti . (g′ )(i) Note that sk sε = sε sk = sε . We have ( s i s ε s i s ε = ti tε ti tε , tε ti tε = (sk sε sk )si (sk sε sk ) = sε si sε = s ε s i s ε s i = tε ti tε ti . (g′ )(ii) Note that sk sε = sε sk = sε . We have ( sε si sε sk = sε si sε si sk = sε (sk si sk )sε (sk si sk ) = tε ti tε ti , tε ti tε = sε (sk si sk )sε = sk sε si sε = sk si sε si sε = (sk si sk )sε (sk si sk )sε = ti tε ti tε . The possibilities for chordless cycles in mutation classes of ∆ quivers are enumerated in Lemma 5.2. For (R3), Barot and Marsh proved in [2] that (R3) (i) holds for (a)–(e). We show that (R3) (ii) and (R3) (iii) hold by checking (a′ )–(k′ ). In each case, we need to check that the corresponding cycle relations hold. Within each case, subcase (i) is when the subquiver of Q is the diagram on the left, and subcase (ii) is when the subquiver of Q is the diagram on the right. In the sequel, we frequently use (R1) and (R2) without comment. (a′ )(i) We have tε tk ti tk = sε sk (sk si sk )sk = sε si = si sε = sk (sk si sk )sk sε = tk ti tk tε . (b′ )(i) We have tε ti tk ti = (sk sε sk )si sk si = sk sε si sk = sk si sε sk = si sk si (sk sε sk ) = ti tk ti tε . (c′ )(i) We have tε tk ti tk = sε sk (sk si sk )sk = sε si = si sε = sk (sk si sk )sk sε = tk ti tk tε . (d′ )(i) We have ti tk tε tk = si sk (sk sε sk )sk = si sε = sε si = sk (sk sε sk )sk si = tk tε tk ti . (e′ )(i) Note that sk sε = sε sk = sε and sk si sε si = si sε si sk . We have tε ti tk ti = (sk sε sk )si sk si = sε si sk si = si sk (sk si sε si )sk si = si sk (si sε si sk )sk si = si sk si sε = si sk si (sk sε sk ) = ti tk ti tε . 30 BING DUAN, JIAN-RONG LI, AND YAN-FENG LUO (e′ )(ii) Note that sk sε = sε sk = sε and sε si sk si = si sk si sε . We have tk ti tε ti = sk (sk si sk )sε (sk si sk ) = si sε si sk = si (sε si sk si )si = si (si sk si sε )si = sk si sε si = (sk si sk )sε (sk si sk )sk = ti tε ti tk . (f ′ ) (i) Note that sj sε = sε sj and sε si sj sk sj si = si sj sk sj si sε . We have tε tk tj tk = sε sk (sk sj sk )sk = sε sj = sj sε = sk (sk sj sk )sk sε = tk tj tk tε , tε ti tj ti = sε si (sk sj sk )si = sε si sj sk sj si = si sj sk sj si sε = si (sk sj sk )si sε = ti tj ti tε . (f ′ ) (ii) Note that sk si = si sk and sε si sj si = si sj si sε .We have tε ti tj tk tj ti = (sk sε sk )si sj sk sj si = sk sε si sk sj sk sj si = sk (sε si sj si )sk = sk (si sj si sε )sk = si sj sk sj si (sk sε sk ) = ti tj tk tj ti tε . (g′ ) (i) Note that sj sε = sε sj and sε sk sj si sj sk = sk sj si sj sk sε . We have tε tj tk tj = (sk sε sk )sj sk sj = sk sε sj sk = sk sj sε sk = sj sk sj (sk sε sk ) = tj tk tj tε , tε tj ti tj = (sk sε sk )sj si sj = sj si sj sk sε sk = tj ti tj tε . (g′ ) (ii) Note that sk si = si sk and sε sj si sj = sj si sj sε . We have tε tk tj ti tj tk = sε sk (sk sj sk )si (sk sj sk )sk = sε sj sk si sk sj = sε sj si sj = sj si sj sε = sk (sk sj sk )si (sk sj sk )sk sε = tk tj ti tj tk tε . (h′ ) (i) Note that si sj = sj si , sε sk = sk sε , and sε sj sk si sk sj = sj sk si sk sj sε . We have ti tk tj tk = si sk (sk sj sk )sk = si sj = sj si = sk (sk sj sk )sk si = tk tj tk ti , tε tj ti tj = sε (sk sj sk )si (sk sj sk ) = sk (sε sj sk si sk sj )sk = sk (sj sk si sk sj sε )sk = (sk sj sk )si (sk sj sk )sε = tj ti tj tε . (h′ ) (ii) Note that sε sj si sj = sj si sj sε . We have tε tj tk ti tk tj = sε sj sk (sk si sk )sk sj = sε sj si sj = sj si sj sε = sj sk (sk si sk )sk sj sε = tj tk ti tk tj tε . Case (i′ ) follows from either Barot and Marsh’s result or (a′ ) or (b′ ) or (h′ ). Case is trivial and Case (k′ ) follows from the commutative property of tk and ti for each vertex i in C. For (R4), by Lemma 6.2, we prove the following several cases, where in each case we number the vertices 0, 1, . . ., d, ε of these subquivers for convenience. Within each case, subcase (i) is when the subquiver of Q is the diagram on the left, and subcase (ii) is when the subquiver of Q is the diagram on the right. In the sequel, we frequently use (R1) and (R2) without comment. (1) (j ′ ) ◦ ]❁ ✂✂ ❁❁❁ 2 ✂✂ ❁❁ ❁❁ ✂✂ ✂ ✂ / ◦ ❴ ❴ ❴/ ◦ ◦ 0 2 1 k−1 µ k ←→ /◦ k /• ε k ◦ \✾ ◦ ✂ \✾✾✾ ✆✆ ✾✾✾ ✾✾ ✂✂ 2 ✆✆ ✾ ✂ ✾✾ ✾✾ ✂ ✆ ✾✾ ✾✾ ✂✂ ✆✆ ✆ ✂ ✆ ✂ / ◦ ❴ ❴ ❴/ ◦ /• ◦ 0 2 1 k−1 ε QUIVER MUTATIONS AND BOOLEAN REFLECTION MONOIDS 31 (2) 0 / ◦d ❴ ❴ ❴/ ◦4 ε • / ◦3 0 µ2 ←→ B◦ ✆✆ ✆ ✆✆ ✆✆ ✆ / ◦2 ✾✾ ✾✾ ✾✾ ✾✾ ✾ ε • B◦✾ ✆✆ ✾✾✾ ✆ ✾✾ ✆✆ ✾✾ ✆✆ ✆ 2 / ◦3 o B◦ ✾✾ ✾✾ ✆✆ ✆ ✾✾ ✆ ✾✾ ✆✆ ✾ ✆✆✆ / ◦d ❴ ❴ ❴/ ◦4 ◦ ◦ 1 1 (3) 0 ε • / ◦d ❴ ❴ ❴/ ◦4 0 µ0 ◦ \✾ ←→ ✆✆ ✾✾✾ ✆ ✾✾ ✆✆ ✾✾ ✆✆ ✆ 1 / ◦3 ◦ ✾✾ ✆B ✾✾ ✆ ✾✾ ✆✆ ✾✾ ✆✆ ✾ ✆✆✆ ε • B◦✾ ✆✆ ✾✾✾ ✆ ✾✾ ✆✆ ✾✾ ✆✆ 1 ✆ / ◦3 o B◦ ✾✾ ✾✾ ✆✆ ✆ ✾✾ ✆ ✾✾ ✆✆ ✾ ✆✆✆ / ◦d ❴ ❴ ❴/ ◦4 ◦ ◦ 2 2 (4) 1 ◦O / ◦2 ❴ ❴ ❴/ k−1 ◦ ε 0 • ❴ ❴ ❴/ ◦ k ◦ )◦o d 1 µ k ←→ ◦O ε 0 • ❴ ❴ ❴/ ◦ v ◦ o❴ ❴ ❴ ◦ d−1 / ◦2 ❴ ❴ ❴/ k−1 ◦ h k+1 k ◦K )◦o d ◦ o❴ ❴ ❴ ◦ d−1 (1) (i) Note that sk sk−1 sk = sk−1 sk sk−1 and sk−1 sε = sε sk−1 . We have P (t0 , tε ) = t0 P (t1 , tk−1 )tε = s0 P (s1 , sk−1 )sk sk−1 sε = s0 P (s1 , sk−1 )sk sε sk−1 = P (s1 , sk−1 )sk sε sk−1 = P (s1 , sk−1 )sk sk−1 sε = P (t1 , tε ), P (tε , t1 ) = tε P (tk−1 , t1 ) = sε sk−1 sk P (sk−1 , s1 ) = sk−1 sε sk P (sk−1 , s1 ) = sk−1 sε sk P (sk−1 , s1 )s0 = sε sk−1 sk P (sk−1 , s1 )s0 = P (tε , t0 ). (1) (ii) We have P (t0 , tε ) = P (t0 , tk−1 )tk tε = P (s0 , sk−1 )sk (sk sε sk ) = P (s0 , sk−1 )sε sk = P (s1 , sk−1 )sk (sk sε sk ) = P (t1 , tε ), P (tε , t1 ) = (sk sε sk )sk P (sk−1 , s1 ) = sk sε P (sk−1 , s1 )s0 = sk sε sk sk P (sk−1 , s1 )s0 = P (tε , t0 ). k+1 32 BING DUAN, JIAN-RONG LI, AND YAN-FENG LUO (2) (i) We have P (t0 , tε )P (tε , t0 ) = t0 t3 P (t4 , tε )P (tε , t4 )t3 t0 = s0 (s2 s3 s2 )P (s4 , sε )P (sε , s4 )(s2 s3 s2 )s0 = s0 s2 s3 P (s4 , sε )P (sε , s4 )s3 s2 s0 = s1 s2 s3 P (s4 , sε )P (sε , s4 )s3 s2 )s1 = s1 (s2 s3 s2 )P (s4 , sε )P (sε , s4 )(s2 s3 s2 )s1 = t1 t3 P (t4 , tε )P (tε , t4 )t3 t1 = P (t1 , tε )P (tε , t1 ). (2) (ii) We have P (t0 , tε )P (tε , t0 ) = t0 t2 P (t3 , tε )P (tε , t3 )t2 t0 = (s2 s0 s2 )s2 P (s3 , sε )P (sε , s3 )s2 (s2 s0 s2 ) = s2 P (s0 , sε )P (sε , s0 )s2 = s2 P (s1 , sε )P (sε , s1 )s2 = (s2 s1 s2 )s2 P (s3 , sε )P (sε , s3 )s2 (s2 s1 s2 ) = t1 t2 P (t3 , tε )P (tε , t3 )t2 t1 = P (t1 , tε )P (tε , t1 ). (3) (i) We have P (t0 , tε )P (tε , t0 ) = P (s0 , sε )P (sε , s0 ) = P (s2 , sε )P (sε , s2 ) = P (t2 , tε )P (tε , t2 ). (3) (ii) We have P (t2 , tε )P (tε , t2 ) = s2 (s0 s3 s0 )P (s4 , sε )P (sε , s4 )(s0 s3 s0 )s2 , = s2 (s0 s3 P (s4 , sε )P (sε , s4 )s3 s0 )s2 = P (s3 , sε )P (sε , s3 ) = s3 s0 P (s4 , sε )P (sε , s4 )s0 s3 = s0 (s0 s3 s0 )P (s4 , sε )P (sε , s4 )(s0 s3 s0 )s0 , = P (t0 , tε )P (tε , t0 ). (4) (i) We have t2 · · · tk−1 tk+1 · · · td P (t0 , tε )P (tε , t0 )td · · · tk+1 tk−1 · · · t2 = s2 · · · (sk sk−1 sk )sk+1 · · · sd P (t0 , tε )P (tε , t0 )sd · · · sk+1 (sk sk−1 sk ) · · · s2 = s2 · · · (sk−1 sk sk−1 )sk+1 · · · sd P (t0 , tε )P (tε , t0 )sd · · · sk+1 (sk−1 sk sk−1 ) · · · s2 = s2 · · · sk−1 sk sk+1 · · · sd P (t0 , tε )P (tε , t0 )sd · · · sk+1 sk sk−1 · · · s2 = s1 P (t0 , tε )P (tε , t0 )s1 = t1 P (t0 , tε )P (tε , t0 )t1 . QUIVER MUTATIONS AND BOOLEAN REFLECTION MONOIDS 33 (4) (ii) We have t2 · · · tk tk+1 · · · td P (t0 , tε )P (tε , t0 )td · · · tk+1 tk · · · t2 = s2 · · · sk (sk sk+1 sk ) · · · sd P (t0 , tε )P (tε , t0 )sd · · · (sk sk+1 sk )sk · · · s2 = s2 · · · sk+1 sk · · · sd P (t0 , tε )P (tε , t0 )sd · · · sk sk+1 · · · s2 = s2 · · · sk+1 · · · sd P (t0 , tε )P (tε , t0 )sd · · · sk+1 · · · s2 = s1 P (t0 , tε )P (tε , t0 )s1 = t1 P (t0 , tε )P (tε , t0 )t1 . Acknowledgements B. Duan would like to express his gratitude to B. Everitt, W. N. 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[43] , Cellular algebras, available at https://webusers.imj-prg.fr/~ bernhard.keller/ictp2006/lecturenotes/x Bing Duan: School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, P. R. China. E-mail address: [email protected] Jian-Rong Li: Dept. of Mathematics, The Weizmann Institute of Science, Rehovot 7610001, Israel; school of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, P. R. China. E-mail address: [email protected] QUIVER MUTATIONS AND BOOLEAN REFLECTION MONOIDS 35 Yan-Feng Luo: School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, P. R. China. E-mail address: [email protected]	4
arXiv:1709.02152v1 [math.GR] 7 Sep 2017 THE CONJUGACY RATIO OF GROUPS LAURA CIOBANU, CHARLES GARNET COX, AND ARMANDO MARTINO Abstract. In this paper we introduce and study the conjugacy ratio of a finitely generated group, which is the limit at infinity of the quotient of the conjugacy and standard growth functions. We conjecture that the conjugacy ratio is 0 for all groups except the virtually abelian ones, and confirm this conjecture for certain residually finite groups of subexponential growth, hyperbolic groups, right-angled Artin groups, and the lamplighter group. 1. Introduction In this paper we introduce and study the conjugacy ratio of a group, which is the limit of the quotient of two functions naturally associated to any finitely generated group: conjugacy growth and standard growth. More precisely, if G is generated by the finite set X, let BG,X (n) denote the ball of radius n with respect to X, and let CG,X (n) denote the set of conjugacy classes of G which have a representative in BG,X (n). Then the conjugacy ratio of G with respect to X is: (1) crX (G) = lim sup n→∞ \|CG,X (n)\| . \|BG,X (n)\| The motivation of this paper is twofold. On one hand, the conjugacy ratio of a finite group H is equal to the degree of commutativity dc(H) of H, which measures the probability that two elements of the group commute, and is defined as: \|{(x, y) ∈ H × H : xy = yx}\| (2) . dc(H) = \|H\|2 The degree of commutativity of a group has received a lot of attention recently, as its definition was extended to finitely generated infinite groups in [AMV17] to be dcX (G) = lim sup n→∞ \|{(x, y) ∈ BG,X (n)2 : ab = ba}\| . \|BG,X (n)\|2 As raised in [Cox16], it is natural to explore whether the degree of commutativity and the conjugacy ratio are related for infinite groups as well. Our second motivation comes from the fact that very few quantitative results comparing standard and conjugacy growth in groups exist in the literature. While in any group there are fewer conjugacy classes than elements, the gap between these two functions has not been been explored in detail, and it is worth investigating. For example, the standard and conjugacy growth rates (i.e. taking the limit of the nth root of the function at n) are equal in some of the most frequently encountered families of infinite groups: hyperbolic groups [AC17], graph products [CHM17], Date: March 19, 2018. 2010 Mathematics Subject Classification. 20P05, 20F69. Key words and phrases. Conjugacy growth, degree of commutativity, polynomial growth, RAAGs, hyperbolic groups, wreath products. 1 2 LAURA CIOBANU, CHARLES GARNET COX, AND ARMANDO MARTINO many wreath products [Mer17]; thus in these examples the quotient of the two functions, as a function of n, must be at most subexponential, and if the conjugacy ratio is 0, the convergence to 0 will not be very fast. Our starting point is the following conjecture, inspired by [AMV17, Conj. 1.6]. Conjecture 1.1. Let G be a group generated by a finite set X. Then crX (G) > 0 if and only if G is virtually abelian. Our results on the conjugacy ratio in several families of groups support Conjecture 1.1. In Section 3 we investigate groups of stable subexponential growth (Definition 3.1). We first show that any virtually abelian group has crX (G) > 0 for any finite generating set X. We then show that, if N is a normal, finite index subgroup of G, then (for any finite generating set X of G) crX (G) 6 dc(G/N ). This allows us to apply a technique from [AMV17] to show that any residually finite group G of stable subexponential growth which is not virtually abelian has crX (G) = 0 for any finite generating set X. We also show in Theorem 3.9 that if G is a finitely generated virtually abelian group, with finite generating sets X and Y , then crX (G) = crY (G). We say that a group, G, with generating set X has stable subexponential growth \|BG,X (n+1)\| = 1, Definition 3.1. This includes all finitely generated if limn→∞ \|B G,X (n)\| virtually-nilpotent groups. Since all finitely generated virtually-nilpotent groups are residually finite, the theorem below means that Conjecture 1.1 is true for all groups of polynomial growth. Theorem 3.7. The conjugacy ratio for all finitely generated, residually finite groups of stable subexponential growth that are not virtually abelian is zero, with respect to all finite generating sets. The proof of Theorem 3.7 cannot be generalised to groups of exponential growth, but we provide independent arguments for several important classes of groups of exponential growth. Theorem 4.1. Let G be a non-elementary hyperbolic group. Then crX (G) = 0 for any finite generating set X. Theorem 4.3. Let G be the lamplighter group, that is, the wreath product C2 ≀ Z. Then crX (G) = 0 for the standard generating set X (defined in (12)). Theorem 4.12. Let G = (GV , XV ) be a right-angled Artin group (RAAG) based on a graph Γ = (V, E) with generating set XV . Then crXV (G) = 0 unless G is free abelian, in which case crXV (G) = 1. We may also consider the strict or spherical conjugacy ratio, where the counting is done in the sphere of radius n rather than the ball of radius n, that is, we may take the ratio of the strict conjugacy growth function over the spherical growth function. More precisely, let SG,X (n) be the sphere of radius n in the group G s with respect to finite generating set X, and let CG,X (n) be the conjugacy classes that intersect SG,X (n) but not BG,X (n − 1), that is, those conjugacy classes with a minimal length representative in SG,X (n). The spherical conjugacy ratio is then (3) crsX (G) = lim sup n→∞ s \|CG,X (n)\| . \|SG,X (n)\| THE CONJUGACY RATIO OF GROUPS 3 Remark 1.2. By the Stolz-Cesàro theorem, anytime the spherical conjugacy ratio turns out to be a limit, the conjugacy ratio will be equal to this limit. In particular, if the spherical conjugacy ratio is 0, then the conjugacy ratio is 0. 2. Preliminaries Recall that for a finitely generated group, G, with generating set X, the exponential growth rate of G with respect to X is: q (4) ExpX (G) = lim n \|BG,X (n)\|. n→∞ Definition 2.1. A group, G, with finite generating set X, is said to have exponential growth if ExpX (G) > 1 and subexponential growth if ExpX (G) = 1. This does not depend on the generating set, X. Additionally, for any ǫ > 0, if λ = ExpX (G), then for sufficiently large n, λn ≤ \|BG,X (n)\| ≤ (λ + ǫ)n . Moreover, if we replace balls with spheres, we get the same limit and inequality. We collect below a few results on convergence of series that will be relevant later. Theorem 2.2 (Stolz-Cesàro). Let an , bn , n ≥ 1 be two sequences with bn strictly increasing and divergent. If the lefthandside limit exists, an+1 − an an lim = l =⇒ lim = l. n→∞ bn+1 − bn n→∞ bn Proposition 2.3 is a partial converse to the Stolz-Cesàro theorem. It implies that for groups of exponential growth, if the conjugacy ratio is a limit and the ratio of sizes of consecutive balls has a limit, then the spherical conjugacy ratio is equal to the conjugacy ratio. Proposition 2.3. Let an , bn , n ≥ 1 be two sequences with bn strictly increasing and divergent, such that the lefthandside limit exists and limn→∞ bn+1 bn 6= 1. Then an an+1 − an = l =⇒ lim = l. n→∞ bn n→∞ bn+1 − bn lim Proposition 2.4. Let an , bn , cn , dn , n ≥ 0 be monotonically increasing sequences of positive integers. Define the sequences b cn and dbn as b c0 := c0 , db0 := d0 , and cn := cn − cn−1 and dbn := dn − dn−1 , for n ≥ 1. b Suppose that (i) an ≤ bn and b cn ≤ dbn for all n, an cn (ii) bn → 0 and dn → 0 as n → ∞. Then Pn ai b cn−i = 0. lim Pi=0 n n→∞ bi dbn−i i=0 Proof. Given ǫ > 0, fix an N such that abnn < ǫ for all n ≥ N . Next choose an M ≥ N such that dcnn < aǫN for all n ≥ M . Then, for n ≥ M ≥ N , n n n X X X bi dbn−i . ai b cn−i < ǫ bi b cn−i ≤ ǫ i=N i=N i=0 4 LAURA CIOBANU, CHARLES GARNET COX, AND ARMANDO MARTINO Thus, for n ≥ M , Pn PN PN Pn cn−i cn−i ai b cn−i cn−i i=N +1 ai b i=0 ai b i=0 ai b = Pn + Pn < Pni=0 + ǫ. Pn b b b bi dn−i bi dn−i bi dn−i bi dbn−i i=0 i=0 i=0 i=0 Now we obtain the result by using the fact that for n ≥ M PN PN cn−i b cn−i cn i=0 ai b ≤ aN Pni=0 ≤ aN < ǫ. Pn dn bi dbn−i dbn−i i=0 i=0 Proposition 2.5. Let an , bn , cn , dn , n ≥ 0, be sequences of positive integers satisfying the following properties: (i) an , bn are monotone sequences, (ii) an ≤ bn and cn ≤ dn for all n, (iii) abnn → 0 as n → ∞, (iv) dbnn ≤ δ n for all sufficiently large n, and for some 0 < δ < 1. Then, Pn ai cn−i = 0. lim Pi=0 n n→∞ i=0 bi dn−i Proof. Given ǫ > 0, fix an N such that n X ai cn−i < ǫ′ i=N an bn n X i=N < ǫ′ < ǫ for all n ≥ N . Then, for n ≥ N , bi cn−i ≤ ǫ′ n X bi dn−i . i=0 Thus, for n ≥ N , Pn Pn PN PN ai cn−i ai cn−i ai cn−i i=N +1 ai cn−i i=0 i=0 Pn = Pn + Pn < Pni=0 + ǫ′ b d b d b d i=0 i n−i i=0 i n−i i=0 i n−i i=0 bi dn−i and so it suffices to show that PN PN ai cn−i i=0 ai cn−i = 0. lim Pi=0 ≤ lim n n→∞ n→∞ bn d0 i=0 bi dn−i Now PN ai cn−i 6 bn d0 i=0 PN N N X cn−i X dn−i ai cn−i ≤ aN ≤ aN . bn b b i=0 n−i i=0 n−i i=0 Using hypothesis (iv), there is a sufficiently large n such that N N X X dn−i δ n 1 − δ N +1 aN 1 aN δ n−i = aN N ≤ δn < ǫ − ǫ′ . ≤ aN N (1 − δ) b δ 1 − δ δ i=0 n−i i=0 3. Results for groups of stable subexponential growth Definition 3.1. A group G, with finite generating set, X, is said to be of stable G.X (n+1)\| = 1. subexponential growth if limn→∞ \|B\|B G,X (n)\| THE CONJUGACY RATIO OF GROUPS 5 Note that being of stable subexponential growth, implies that ExpX (G) = 1, and hence that the group has subexponential growth. By the celebrated result of Gromov, every finitely generated group of polynomial growth - where BG,X (n) is bounded above by a polynomial function - is virtually nilpotent. All these groups are of stable subexponential growth since, by a result of Bass, [BASS72], if G is a finitely generated, virtually nilpotent group, and X is any finite generating set, then, for some exponent d, and constants, A, B: (5) And ≤ \|BG,X (n)\| ≤ Bnd . The exponent d is calculated explicitly in [BASS72]; for a virtually abelian group it is equal to the rank of a finite index free abelian subgroup. From (5) we get that for any positive integer, k, (6) lim n→∞ \|BG,X (n + k)\| = 1. \|BG,X (n)\| The main result which we require for this class is the following. Proposition 3.2. [BV02]. Let G be a finitely generated group with stable subexponential growth, and finite generating set X. For every finite index subgroup H 6 G and every g ∈ G, we have \|gH ∩ BG,X (n)\| 1 \|Hg ∩ BG,X (n)\| lim = lim = . n→∞ n→∞ \|BG,X (n)\| \|BG,X (n)\| [G : H] Furthermore, if H is an infinite index subgroup of G then both limits are zero for any coset of H. Remark 3.3. The last statement does not appear explicitly in [BV02], but follows easily from their arguments. Alternatively, one could prove this via the construction of an invariant mean which requires the choice of an ultrafilter. The stable subexponential condition ensures that any ultrafilter will do, and hence that all limit points of the sequences above are equal. From now on, whenever there is no ambiguity concerning the group and its generating set, we will write C(n) instead of CG,X (n) and B(n) instead of BG,X (n). Proposition 3.4. Suppose that G is a finitely generated, virtually abelian group. Then, for any finite generating set X of G, we have that crX (G) > 0. More precisely, if [G : A] = m where A is abelian, then crX (G) ≥ 1/m2 . Proof. Let [G : A] = m, where A is abelian. We note that G acts by multiplication on the right cosets of A. If g and h lie in the same right coset, then h = αg for some α ∈ A, so for any a ∈ A, h−1 ah = (αg)−1 a(αg) = g −1 ag since A is abelian. Thus there are at most m conjugates of each element a ∈ A and so, for all n ∈ N, 1 . Now we have that \|C(n) ∩ A\| > \|B(n) ∩ A\| · m \|C(n) ∩ A\| \|B(n) ∩ A\| \|C(n) ∩ A\| \|B(n) ∩ A\| 1 \|C(n)\| > = · > · \|B(n)\| \|B(n)\| \|B(n) \|B(n) ∩ A\| \|B(n)\| m which tends to 1/m2 by Proposition 3.2. Lemma 3.5. Let G be a group of stable subexponential growth with finite generating set X, let g ∈ G and let H be a finite index subgroup of G. For d ∈ N we have 1 \|gH ∩ BG,X (n + d)\| = . lim n→∞ \|BG,X (n)\| [G : H] 6 LAURA CIOBANU, CHARLES GARNET COX, AND ARMANDO MARTINO Proof. This follows from writing lim n→∞ \|B(n + d)\| \|gH ∩ B(n + d)\| \|gH ∩ B(n + d)\| = lim n→∞ \|B(n)\| \|B(n)\| \|B(n + d)\| together with Proposition 3.2 and (6). Proposition 3.6. Let G be a finitely generated group of stable subexponential growth and N a subgroup of finite index in G. Then crX (G) ≤ dc(G/N ) for any finite generating set X of G. Proof. Let [G : N ] = m, so that G = g1 N ⊔g2 N ⊔. . .⊔gm N for some g1 , . . . , gm ∈ G. Let d := max{\|gi \|X : i = 1, . . . , m}. Now consider if xN ∼ yN (in G/N ). Then yN = g −1 xgN = g −1 xgg −1 N g = −1 g xN g for some g ∈ G. Moreover, since x and y are conjugate in G/N , we may choose g from {g1 , . . . , gm } and so \|g\|X 6 d. Now let yk1 ∈ B(n). We know there must exist some xk2 ∈ xN such that g −1 (xk2 )g = yk1 . But then xk2 = gyk1 g −1 , and so xk2 ∈ B(n + 2d). Hence, for every n ∈ N, each element in B(n) ∩ yN is conjugate to some element in B(n + 2d) ∩ xN . Let x1 , . . . xk ∈ {g1 , . . . , gm } be the representatives of the conjugacy classes in G/N . For every i ∈ N and every j ∈ Zk , we will assume that there are \|B(n) ∩ xj N \| conjugacy classes in B(n) ∩ xj N . Hence Pk \|xi N ∩ B(n + 2d)\| \|C(n)\| 6 i=1 \|B(n)\| \|B(n)\| which tends to k/m by the previous lemma. Theorem 3.7. Conjecture 1.1 is true for all finitely generated, residually finite groups of stable subexponetial growth. Proof. Proposition 3.4 states that, if a finitely generated group G is virtually abelian, then, for any finite generating set X, crX (G) > 0. For the other direction we apply the method of [AMV17, Proof of Thm. 1.3] by using Proposition 3.6. For completeness we will describe their argument. It requires the following result from [Gal70]: if F is a finite group and N E F , then (7) dc(F ) 6 dc(F/N ) · dc(N ). Our hypotheses are that G is: finitely generated, residually finite, of stable subexponential growth, and not virtually abelian. We wish to show that crX (G) = 0 for any finite generating set X. We will work with finite quotients and will build a chain of normal subgroups. Since G is finitely generated we may choose these subgroups to be characteristic, and will do this because being characteristic is transitive. Since G is not virtually abelian, choose g1 , g2 ∈ G that do not commute and, using the residually finite assumption, let [g1 , g2 ] 6∈ K1 where K1 is a characteristic and finite index subgroup of G. Hence G/K1 is non-abelian, and by Gustafson’s result we have that dc(G/K1 ) 6 5/8. Now, since the properties of G which we have used also apply to finite index subgroups, this argument also applies to K1 . Hence we may construct a descending chain of characteristic finite index subgroups . . . 6 Ki 6 Ki−1 6 . . . 6 K2 6 K1 6 K0 = G THE CONJUGACY RATIO OF GROUPS 7 where, for every i ∈ N, dc(Ki−1 /Ki ) 6 5/8. Moreover (G/Ki )/(Ki−1 /Ki ) = G/Ki−1 and so, from (7), dc(G/Ki ) 6 dc(G/Ki−1 ) · dc(Ki−1 /Ki ) 6 5/8 · dc(G/Ki−1 ). By induction dc(G/Ki ) 6 (5/8)i and so, by Proposition 3.6, for any finite generating set X of G, we have that crX (G) 6 dc(G/Ki ) 6 (5/8)i . Since this holds for every i ∈ N, we obtain that crX (G) = 0. Corollary 3.8. Conjecture 1.1 is true for all finitely generated, virtually nilpotent groups, or equivalently, all groups of polynomial growth. 3.1. Virtually Abelian groups. The goal of this section is to prove: Theorem 3.9. Let G be a finitely generated, virtually abelian group, and X, Y be finite generating sets for G. Then crX (G) = crY (G). It will be useful to have the following shorthand: Definition 3.10. Let G be generated by the finite set X. A subset, S, of G is \|S∩BG,X (n)\| generic if lim supn→∞ \|BG,X (n)\| = 1, and negligible if the limit is 0. Given a group, G, with finite generating set X, a finitely generated subgroup, H, of G is said to be undistorted if any word metric on H is bi-Lipschitz equivalent to any word metric on G, when restricted to H. This makes sense since any two finite generating sets on a group induce bi-Lipschitz equivalent word metrics. It is easy to see that a finite index subgroup is always undistorted, and that a subgroup H is undistorted if and only if it has an undistorted subgroup of finite index. Retracts are also undistorted (recall that a retract of G is the image of an endomorphism ρ : G → G such that ρ2 = ρ). We now collect the following facts: Proposition 3.11. Suppose that G is a finitely generated virtually abelian group, with finite generating set X, having a subgroup of finite index isomorphic to Zd . (i) Every subgroup of G is both finitely generated and undistorted. (ii) Let H be an infinite subgroup of G. Let T (n) = TH,X (n) (for transversal) be the number of cosets of H that have a representative in BG,X (n). Then, lim n→∞ T (n) = 0. \|BG,X (n)\| Proof. (i) Let H ≤ G. It is well known that H is finitely generated, as this fact is true in the case where G is virtually polycyclic, which includes the finitely generated virtually nilpotent (and abelian) case. However, the fact that H is undistorted is not true more generally, and follows from the fact that every subgroup of a finitely generated free abelian group has finite index in a direct summand. In our case, H has a finite index subgroup which is a retract of a finite index subgroup of G, and is therefore undistorted in G. (ii) From above, H is finitely generated and undistorted. Since H is infinite, it must contain an element of infinite order, so there exists an ǫ > 0 such that \|H ∩ BG,X (n)\| ≥ ǫn. More precisely, \|H ∩ BG,X (n)\| will have polynomial bounds of degree e, d ≥ e ≥ 1. 8 LAURA CIOBANU, CHARLES GARNET COX, AND ARMANDO MARTINO Let A, B, d be the constants in (5). Then Hence, B2d nd ≥ \|BG,X (2n)\| ≥ T (n)\|H ∩ BG,X (n)\| ≥ T (n)ǫn. T (n) B2d nd−1 = 0. ≤ lim n→∞ \|BG,X (n)\| n→∞ ǫAnd 0 ≤ lim From now on, we let G be an infinite, finitely generated, virtually abelian group. Let A be a normal, finite index, free abelian subgroup, and B be the centraliser of A in G. Note that A is a subgroup of B, which therefore has finite index. Proposition 3.12. Let G be a finitely generated, virtually abelian group and X any finite generating set for G. Let A be a normal, finite index, free abelian subgroup, and B be the centraliser of A in G. Then the set of minimal length G-conjugacy representatives in G \ B is negligible. Proof. Let y 6∈ B be an element of G and denote by CyA (n) the number of conjugacy classes which have a representative in BG,X (n) ∩ yA. Then we claim that lim n→∞ CyA (n) = 0. \|BG,X (n)\| For each conjugacy class with a representative in BG,X (n) ∩ yA, choose a shortest such representative, and denote this set of representatives Z = {yai : ai ∈ A}. From these, extract the set U = {ai }, rewriting the ai as geodesics if required. Note that, for some fixed k (the length of y), we have CyA (n) = \|Z ∩ BG,X (n)\| ≤ \|U ∩ BG,X (n + k)\|. Now let My denote the automorphism of A induced by conjugation with y, which we think of as a matrix. For any a1 , a2 ∈ A we have that: a−1 2 (ya1 )a2 = y(a1 + (I − My )a2 ), if we switch to an additive notation in A. Let H be the image of (I − My ) in A, that is, H = h[a, y] : a ∈ Ai. Since y 6∈ B, we can conclude that H is a non-trivial subgroup of A and is therefore infinite. Moreover, the elements of U are all in distinct cosets of H. Hence, by Proposition 3.11 part (ii), we may conclude that: \|Z∩BG,X (n)\| \|BG,X (n)\| ≤ \|U∩BG,X (n+k)\| \|BG,X (n)\| = TH,X (n+k) \|BG,X (n+k)\| \|BG,X (n+k)\| \|BG,X (n)\| → 0. Proposition 3.12 shows that the only elements of G that contribute to the conjugacy ratio are the elements of B. (The representative of a conjugacy class might not have a shortest representative in our particular coset yA, but varying y we see that we have an overcount of the number of conjugacy classes in the complement of B, which nonetheless gives 0.) Thus the strategy for proving Theorem 3.9 is the following. First note that each element of B has finite conjugacy class in G. We split the elements in B into those which centralise elements from outside of B and those whose centraliser is completely in B. Proposition 3.14 shows the former ones form a negligible set, and THE CONJUGACY RATIO OF GROUPS 9 the latter ones a generic set of B (Corollary 3.15); moreover, for the latter ones the size of the G-conjugacy class is the index of the B-centraliser, which is constant for elements in the same A-coset. Therefore, each coset (or, rather, conjugacy class of cosets) of A contributes a fixed amount to the conjugacy ratio, which is algebraically determined. We use the notation ZK (g) for the K-centraliser of g ∈ G, that is, ZK (g) = {k ∈ K : k −1 gk = g}. Lemma 3.13. Let x ∈ G. Then ZB (x) = ZB (xa) for any a ∈ A. Moreover, [G : ZB (x)] < ∞ if x ∈ B. S Proposition 3.14. The set y6∈B ZB (y) is a finite union of infinite index subgroups of G. Hence this set is negligible with respect to any finite generating set. Proof. Since ZB (y) = ZB (ya) for any a ∈ A, this is a finite union. So it is enough to show that each ZB (y) has infinite index. In fact, it is sufficient to show that ZA (y) = ZB (y) ∩ A is an infinite index subgroup of A. However, ZA (y), is a pure subgroup of A; that is, if am ∈ ZA (y) and m 6= 0 then a ∈ ZA (y). This implies that ZA (y) is a direct summand of A. But since y 6∈ B, this direct summand cannot be the whole of A and is therefore an infinite index subgroup of A as required. Corollary 3.15. There is a generic set of elements of B (with respect to any generating set) whose centraliser lies entirely in B. Proof. If for some b ∈ B there exists t ∈ / B such that [t, b] = 1, then b ∈ ZB (t) ⊂ S y6∈B ZB (y), which is negligible by Proposition 3.14. Proof of Theorem 3.9: For each r, let Ar be the elements b ∈ B for which ZB (b) has index r in G (and therefore conjugacy class size r in G), and let N = {b ∈ B : ZB (b) 6⊂ B}, that is,S N is the set of elements of B whose centraliser does not fully lie in B. Then N = y6∈B ZB (y) and so by Corollary 3.15 it is a negligible set. Since A ≤ ZB (b) ≤ G for any b ∈ B and A has finite index in G, there are only finitely many values for the index of ZB (b) in G, and thus finitely many r for which Ar is non-empty. Moreover, since ZB (y) = ZB (ya) for any y ∈ B \ A and a ∈ A, if y ∈ Ar , then ya ∈ Ar , so each non-empty Ar is a union of A-cosets and thus (8) \|Ar ∩ BG,X (n)\| = δ, n→∞ \|BG,X (n)\| lim where δ is 1/[G : A] times the number of A-cosets in Ar , so is independent of X. It is easy to see that there is an integer, k, such that if two elements of B are conjugate in G, then they are conjugate by an element of length at most k; the same holds for Ar as Ar ⊂ B. Moreover, since B is normal in G, it is easy to see that G acts on Ar by conjugation; G acts by conjugation on N , and hence on Ar \ N , as well. Let Cn be the number of conjugacy classes of G which meet BG,X (n) and are contained in Ar \ N . Then, (9) \|(Ar \ N ) ∩ BG,X (n)\| ≤ rCn ≤ \|(Ar \ N ) ∩ BG,X (n + 2k)\|. The first inequality comes from the fact that each element of Ar \ N has r conjugates in G, and the second from the fact that each of the conjugates can be obtained from a conjugator of length at most k. 10 LAURA CIOBANU, CHARLES GARNET COX, AND ARMANDO MARTINO Now Cn \|CG,X (n) ∩ Ar \| Cn \|N ∩ BG,X (n)\| ≤ ≤ + , \|BG,X (n)\| \|BG,X (n)\| \|BG,X (n)\| \|BG,X (n)\| and by (8), (9) and Corollary 3.15 Cn = lim lim n→∞ n→∞ \|BG,X (n)\| \|N ∩ BG,X (n)\| Cn + \|BG,X (n)\| \|BG,X (n)\| = δ , r so we get lim n→∞ δ \|CG,X (n) ∩ Ar \| = . \|BG,X (n)\| r Hence the number of conjugacy classes of G that meet Ar is independent of the generating set. Summing over the finitely many r gives the result. Remark 3.16. The same ideas as those just presented can be used to show that, if G is a finitely generated, virtually abelian group, and X is any finite generating set, then crX (G) = inf N Ef G cr(G/N ). That is, the conjugacy ratio is equal to the infimum of conjugacy ratios of the finite quotients. Hence, if one were to measure the conjugacy ratio using invariant means, one would get the same numerical value. Unpublished results indicate that this is the same as the degree of commutativity. For similar reasons, the same is true whenever G is a finitely generated virtually nilpotent group, the virtually abelian case being the key one. 4. Results for other families of groups 4.1. Hyperbolic groups. In this section we prove Conjecture 1.1 for non-elementary hyperbolic groups. We will write f (n) ∼ g(n) to mean f (n)/g(n) → 1 as n → ∞. Theorem 4.1. Let G be a non-elementary hyperbolic group. Then crX (G) = 0 for any finite generating set X. Proof. Let G be a non-elementary hyperbolic group with finite generating set X. Then by a result of Coornaert (see [Cor93]) there are positive constants A0 ,B0 , and integer n0 , such that for all n ≥ n0 (10) A0 enh ≤ \|BG,X (n)\| ≤ B0 enh , where h = ExpX (G). By Theorem 1.2 in [AC17], there are positive constants A1 , B1 and n1 such that (11) A1 enh enh ≤ \|CG,X (n)\| ≤ B1 n n for all n ≥ n1 . Thus from (10) and (11) we get \|CG,X (n)\| B1 6 \|BG,X (n)\| A0 n for all n ≥ max(n0 , n1 ), and by taking the limit we obtain that crX (G) = 0. THE CONJUGACY RATIO OF GROUPS 11 4.2. The lamplighter group. We follow the notation in [Mer17]. Let I be a L non-empty set. For η ∈ i∈I G we write η(i) for the ith component of η, and if L moreover I is a group and x ∈ I, we define η x ∈ i∈I G by η x (i) = η(x−1 i), and say that η x is the left translate of η by x. Definition 4.2. Consider groups H and L with symmetric generating sets A and B, and neutral elements e and e′ , respectively. The wreath product of G by L, written G ≀ L, is defined as M H ≀ L := H ⋊ L, i∈L where for (η, m), (θ, n) ∈ H ≀ L, (η, m)(θ, n) = (ηθm , mn). L For h ∈ H, let ~h ∈ i∈L H be such that ~h(e′ ) = h and ~h(i) = e for i 6= e′ . Then (12) X := {(~e, a) : a ∈ A} ∪ {(~b, e′ ) : b ∈ B}. generates H ≀ L. For the lamplighter group G = C2 ≀ Z we let A := {a}, where a is the non-trivial element of C2 , and let B be the standard generating set of Z. Theorem 4.3. Let G be the lamplighter group, that is, the wreath product C2 ≀ Z. Then crX (G) = 0 for the standard generating set X. Proof. The statement follows immediately from [Mer17, Example 5.0.3], where n it √ √ n 2 1+ 5 1+ 5 s is shown that \|CG,X (n)\| ∼ n , and the fact that \|SG,X (n)\| ∼ by 2 2 [Par92]. 4.3. Right-Angled Artin Groups. Let Γ = (V, E) be a simple graph (i.e. a non-oriented graph without loops or multiple edges) with vertex set V and edge set E. For each vertex v of Γ, let Gv be a group. The graph product of the groups Gv with respect to Γ is defined to be the quotient of their free product by the normal closure of the relators [gv , gw ] for all gv ∈ Gv , gw ∈ Gw for which {v, w} is an edge of Γ. Here we consider right-angled Artin groups (RAAGs), which are graph products with all Gv = Z, and denote by (GV , XV ) the RAAG based on the graph Γ with generating set XV (in bijection to V ). Conjugacy representatives in a RAAG come, to a large extent, from taking one word out of each cyclic permutation class, so we first establish the asymptotics of the language of cyclic representatives in a rather general setting. Example 4.4. In a free group on the free generating basis, counting the conjugacy classes with a minimal representative of length n is equivalent to counting the number of cyclically reduced words of length n, up to cyclic permutation. 4.3.1. Cyclic representatives of languages. We follow the notation in [CHM17, Section 2.3]. Let L be a language over a finite alphabet X, that is, L ⊆ X ∗ , and let L(n) denote the set of√words of length ≤ n in L. For n ≥ 1, n ∈ N, let Ln := {wn \| w ∈ L} and n L = {v \| v n ∈ L}. Define Prim(L) := {w ∈ L \| ∄k > 1, v ∈ L such that v k = w} to be the language of primitive words in L. Suppose L is closed under cyclic permutations; then we construct a language CycRep(L) of cyclic representatives of L out of the words wc , where wc the word that is least lexicographically among all cyclic permutations of w, for w ∈ L: CycRep(L) := {wc \| w ∈ L}. 12 LAURA CIOBANU, CHARLES GARNET COX, AND ARMANDO MARTINO Proposition 4.5 (see also Lemma 2.10 (4), [CHM17]). Let L be an exponential growth language closed under cyclic permutations. Furthermore assume that Lk ⊆ L √ k and L ⊆ L for all k ≥ 1. Then lim n→∞ \|CycRep(L)(n)\| = 0. \|L(n)\| Proof. For simplicity of notation let a(n) := \|Ls (n)\|, p(n) := \|Prim(L)s (n)\| and c(n) := \|CycRep(L)s (n)\|, that is, we consider the numbers of words of length exactly n in each language.S Write L as L = k≥1 Primk (L), and notice that the number of cyclic representatives of length n in Prim(L) is p(n)/n, and the number of cyclic representatives of P P length nk in Primk (L) is also p(n)/n. Thus a(n) = d/n p(d) and c(n) = d/n p(d) d . Let µ(n) and φ(n) be the standardPnumber theoretic Möbius and Euler functions. Then by Möbius inversion p(n) = d/n µ( nd )a(n) and so P X X l/(n/d) µ(l) φ(n/d) l a(d) a(d) = , c(n) = d n d/n d/n P P φ(n) which follows from d/n φ(d) = n and d/n µ(d) d = n . Since a(n) is exponential, only the last term in the sum above is of the same magnitude as a(n), so (13) c(n) ∼ \|CycRep(L)s (n)\| a(n) =⇒ lim = 0. n→∞ n \|Ls (n)\| By Stolz-Cesàro we obtain the result. 4.3.2. Conjugacy representatives in RAAGs. We first establish a result about the conjugacy ratio of direct products. Lemma 4.6. Let H and K be two groups with finite generating sets X and Y , respectively. If either (i) crX (H) = crY (K) = 0 or, (ii) crX (H) = 0 and ExpX (H) > ExpY (K), then crX∪Y (H × K) = 0. Proof. We calculate the conjugacy ratio with respect to balls in H × K. To do this we use balls in H and spheres in K. Let an := \|CH,X (n)\|, bn := \|BH,X (n)\|, s tn := \|CK,Y (n)\| and sn := \|SK,Y (n)\|. Then Pn ai tn−i . crX∪Y (H × K) = lim sup Pni=0 n→∞ i=0 bi sn−i If crX (H) = crY (K) = 0, then by Proposition 2.4 (putting tn = cbn , sn = dc n ) we get that crX∪Y (H × K) = 0. Similarly, if crX (H) = 0 and ExpX (H) > ExpY (K) then Proposition 2.5 (putting cn = tn , dn = sn ) states that this limit is zero, so crX∪Y (H × K) = 0. Since RAAGs interpolate between free and free abelian groups, the presence of commutativity does not allow us to simply consider cyclically reduced words up to permutation, as in free groups. We need to single out the words for which taking cyclic representatives produces conjugacy representatives, and use Crisp, Godelle and Wiest’s approach from [CGW], which was further developed in [CHM17]. THE CONJUGACY RATIO OF GROUPS 13 Definition 4.7 (Def 2.19, [CGW]). Let V = {a1 , . . . , aN } and set the total order −1 a1 < a−1 1 < a2 < a2 < . . . . A cyclically reduced word w is in cyclic normal form if it is in the shortlex language SL(GV , XV ) of GV with respect to XV and all its cyclic conjugates are in SL(GV , XV ) as well. Not all elements posses a cyclic normal form. For example, if [a1 , a2 ] = 1, the word a1 a2 is in SL(GV , XV ), but its cyclic permutation a2 a1 is not. To deal with this situation, [CGW] divides the words over XV into split and non-split. Definition 4.8 (Definition 2.13, [CGW]). Let w be a cyclically reduced word over XV and denote by ∆(w) the full subgraph spanned by Supp(w). Let ∆(w) be the graph complement of ∆(w). (i) The word w is split if ∆(w) is disconnected, which amounts to being able to write w as a product of commuting subwords (or blocks). (ii) The word w is non-split if ∆(w) is connected. (iii) Let CycSL(GV , XV ) denote the set of all non-trivial cyclic normal forms corresponding to non-split words in GV . We say that a group element is non-split (split) if it can be represented by a cyclically reduced word which is non-split (split). Proposition 4.9 (Prop. 2.21, [CGW]). Two cyclic normal forms represent conjugate elements if and only if they are equal up to a cyclic permutation. Proposition 4.10 (Remark 2.14, [CGW]). Let w and v be two cyclically reduced split words. Then they are conjugate if and only if ∆(w) = ∆(v) and the words corresponding to the commuting blocks are conjugate, respectively. Lemma 4.11. [CHM17] Let CycSL(GV , XV ) be the set of cyclic normal forms in GV . The following hold: (1) CycSLk (GV , XV ) ⊆ CycSL(GV , XV ) for all k ≥ 1, and (2) CycSL(GV , XV )) is closed under cyclic permutations. Theorem 4.12. Let G = (GV , XV ) be a right-angled Artin group (RAAG) based on a graph Γ = (V, E) with generating set XV . Then crXV (G) = 0 unless G is free abelian, in which case crXV (G) = 1. Proof. We use induction on the number of vertices. Let n := \|V \|. The result is trivial for n = 1. If G is a direct product, then we get cr(G) = 0 if at least one of the factors has cr = 0; this follows from Lemma 4.6(i) if both factors have cr = 0 and from Lemma 4.6(ii) if, say, the first factor has cr = 0, as the second is by induction free abelian, and of strictly smaller growth rate than the first. We get cr(G) = 1 when each factor is free abelian. So suppose G is not a direct product. We split the conjugacy classes CGV ,XV of G into two types: those which have a shortest length representative with support XU , where U ( V , and denote these by CGV ,≤XV , and those which have a shortest length representative with support exactly XV , and denote these by CGV ,=XV . By Propositions 4.9 and 4.10, this is well defined. Moreover, by Propositions 4.9 and 4.10, two cyclically reduced words w1 , w2 with support XU are conjugate in GV if and only if they are conjugate in GU (note that if a word w ∈ CycSL(GV , XV )∩XU∗ , where U ( V , then w ∈ CycSL(GU , XU )). 14 LAURA CIOBANU, CHARLES GARNET COX, AND ARMANDO MARTINO Thus we can write CGV ,≤XV ⊆ CGV ,XV ⊆ (14) Then (14) implies that (15) \|CGV ,XV (n)\| ≤ \|BGV ,XV (n)\| Now for U ( V so P S CGU ,XU and express the above as: [ CGV ,=XV . CGU ,XU U(V [ U(V XU ,U(V \|CGU ,XU (n)\| + \|CGV ,=XV (n)\| \|BGV ,XV (n)\| . \|CGU ,XU (n)\| \|CGU ,XU (n)\| \|BGU ,XU (n)\| = , \|BGV ,XV (n)\| \|BGU ,XU (n)\| \|BGV XV (n)\| \|BGU ,XU (n)\| \|CGU ,XU (n)\| ≤ crXU (GU ) lim sup . (n)\| \|B \|B n→∞ n→∞ GV ,XV GV ,XV (n)\| The right hand side is equal to 0 since either (i) crXU (GU ) = 0 by induction, or (ii) GU is free abelian (so of polynomial growth); if (ii), since G itself if not a direct product by assumption, it is of exponential growth, and the last fraction is 0. \|C (n)\| V It remains to find lim supn→∞ \|BGGV ,=X (n)\| , the second part of the right hand V ,XV side of (15). Since G is not a direct product, all conjugacy representatives with support exactly XV are non-split, so it suffices to consider cyclic normal forms up to cyclic permutations, that is lim sup \|CGV ,=XV (n)\| \|CycRep(CycSL(GV , XV )(n)\| ≤ = \|BGV ,XV (n)\| \|SL(G, XV )(n)\| \|CycRep(CycSL(GV , XV )(n)\| \|CycSL(GV , XV )(n)\| , \|CycSL(G, XV )(n)\| \|SL(G, XV )(n)\| and by Proposition 4.5 applied to the language CycSL(GV , XV ) (which satisfies the hypothesis of Proposition 4.5 by Lemma 4.11) lim n→∞ This proves the result. \|CycRep(CycSL(GV , XV )(n)\| = 0. \|CycSL(G, XV )(n)\| 5. Reflections and open questions Our results on the conjugacy ratio values are essentially identical to those on the degree of commutativity in [AMV17, Cox16, Val17]. That is, the two quantities are equal for all the classes of groups we studied. However, we could not establish a direct general link between them. Question 1. Is the limsup in the definition of the conjugacy ratio a limit? Question 2. What are the groups for which dcX (G) ≤ crX (G) (or vice versa)? They are equal in the virtually nilpotent case, in the hyperbolic group case and in many more. As is the case for the degree of commutativity, we do not know whether the conjugacy ratio might be influenced by a change of generators. Question 3. Does there exist a group G with finite generating sets X and Y such that crX (G) 6= crY (G)? THE CONJUGACY RATIO OF GROUPS 15 Finally, it would be interesting to unify the proofs confirming our conjecture for larger classes of groups, such as all groups of exponential growth, for example. References [AC17] Y. Antolı́n and L. Ciobanu, Formal conjugacy growth in acylindrically hyperbolic groups, Int. Math. Res. Notices, 1 (2017), 121–157. [AMV17] Y. Antolı́n, A. Martino, and E. Ventura, Degree of commutativity of infinite groups, Proceedings of the American Mathematical Society (Feb. 2017). [BASS72] H. Bass, The degree of polynomial growth of finitely generated nilpotent groups, Proc. London Math. Soc. (3) 25 (1972), 603–614. [BV02] J. Burillo and E. Ventura, Counting primitive elements in free groups, Geom. Dedicata 93 (2002), 143–162. [CHM17] L. Ciobanu, S. Hermiller, and V. Mercier, Conjugacy growth in graph products, Preprint 2017. [Cor93] M. Coornaert, Mesures de Patterson–Sullivan dans les espaces hyperboliques au sens de Gromov, Pacific J. Math. 159 (1993), 241–270. [Cor05] M. Coornaert, Asymptotic growth of conjugacy classes in finitely-generated free groups, IJAC. Vol. 15 (2005), 887–892. [Cox16] C.G. Cox, The degree of commutativity and lamplighter groups, Preprint 2017, https://arxiv.org/abs/1605.04829. [CGW] Crisp, Godelle and Wiest Linear time solution to the conjugacy problem in right-angled Artin groups and their subgroups, Journal of Topology, Vol. 2, (2009), 442 – 460. [ET68] P. Erdős and P. Turán, On some problems of a statistical group-theory. IV, Acta Math. Acad. Sci. Hungar 19 (1968), 413–435. MR 0232833 [Gal70] P.X. Gallagher, The number of conjugacy classes in a finite group, Math Z. 118 (1970), 175-179. [Gus73] W. H. Gustafson, What is the probability that two group elements commute?, Amer. Math. Monthly 80 (1973), 1031–1034. MR 0327901 [Mer17] V. Mercier, Conjugacy growth series in some wreath products, Preprint 2017, https://arxiv.org/abs/1610.07868. [Par92] W. Parry. Growth series of some wreath products. Trans. Amer. Math. Soc., 331 (1992), 2:751–759. [Val17] M. Valiunas. Degree of commutativity for right-angled Artin groups. Preprint 2017, https://arxiv.org/abs/1701.04374 Heriot-Watt University, Edinburgh, EH14 4AS, UK E-mail address: [email protected] URL: http://www.macs.hw.ac.uk/~lc45/ University of Bath, BA2 7AY, UK E-mail address: [email protected] Mathematical Sciences, University of Southampton, SO17 1BJ, UK E-mail address: [email protected] URL: http://www.personal.soton.ac.uk/am1t07/	4
1 Complex Systems Science meets 5G and IoT Nicola Marchetti, Irene Macaluso, Nicholas Kaminski, Merim Dzaferagic, M. Majid Butt, Marco Ruffini, Saul Friedner, Julie Bradford, Andrea Zanella, arXiv:1710.11548v1 [cs.NI] 31 Oct 2017 Michele Zorzi, and Linda Doyle Abstract We propose a new paradigm for telecommunications, and develop a framework drawing on concepts from information (i.e., different metrics of complexity) and computational (i.e., agent based modeling) theory, adapted from complex system science. We proceed in a systematic fashion by dividing network complexity understanding and analysis into different layers. Modelling layer forms the foundation of the proposed framework, supporting analysis and tuning layers. The modelling layer aims at capturing the significant attributes of networks and the interactions that shape them, through the application of tools such as agent-based modelling and graph theoretical abstractions, to derive new metrics that holistically describe a network. The analysis phase completes the core functionality of the framework by linking our new metrics to the overall network performance. The tuning layer augments this core with algorithms that aim at automatically guiding networks toward desired conditions. In order to maximize the impact of our ideas, the proposed approach is rooted in relevant, near-future architectures and use cases in 5G networks, i.e., Internet of Things (IoT) and self-organizing cellular networks. Index Terms Complex systems science, Agent-based modelling, Self-organization, 5G, Internet of Things. Nicola Marchetti ([email protected]), Irene Macaluso, Nicholas Kaminski, Merim Dzaferagic, M. Majid Butt, Marco Ruffini, and Linda Doyle are with CONNECT / The Centre for Future Networks and Communications, Trinity College, The University of Dublin, Ireland. Saul Friedner and Julie Bradford are with Real Wireless, UK. Andrea Zanella and Michele Zorzi are with the University of Padova, Italy. This material is based upon works supported by the Science Foundation Ireland under Grants No. 13/RC/2077 and 10/CE/i853. 2 I. I NTRODUCTION The transition of humanity into the Information Age has precipitated the need for new paradigms to comprehend and overcome a new set of challenges. Specifically, the telecommunication networks that underpin modern societies represent some of the largest scale construction and deployment efforts ever attempted by humanity, with renovations occurring nearly continuously over the course of decades. This results in networks that consist of numerous subsections, each following its own trajectory of development, commingled into a complex1 cacophony. A few emerging trends confirm the picture just drawn. Mobile and wireless networks are getting denser and more heterogeneous in nature. Nodes in the network vary hugely in form and functionality - ranging from tiny simple sensors to sophisticated cognitive entities. There is a wider range of node and network-wide parameters to set, many of which are interdependent and which impact heavily on network performance. Networks are becoming more and more adaptive and dynamic, and many parameters are set during run-time in response to changing contexts. As networks evolve, all of the above issues become more exaggerated - e.g., 5G networks will see more antennas, more base stations and devices, more modes of operation, more variability, and more dynamism. In a world like that, there is no way to systematically capture network behaviour. There is no straightforward network theory or information theoretic approach that can be used to describe the overall network or the interplay between the different networks. We propose to tackle this by studying wireless networks from the perspective of Complex Systems Science (CSS), developing complexity metrics and relating them to more traditional measures of network performance. One of the key questions in CSS relates to the degree of 1 With the term ’complexity’ we refer to a specific set of complex systems science quantities, related to the interactions between network entities (rather than to entities themselves) and between networks. As the current and future trend is towards more diverse networks coexisting and more entities (e.g., within IoT, or ultra dense small cell networks), the amount of interactions will increase, leading to an increase in complexity (in the meaning given to the word by complex systems science). 3 organization of a system [1], in terms of both the difficulty in describing its organizational structure (A), and the amount of information shared between the parts of the system as a result of the organizational structure (B). For example, the measure of excess entropy [2] (type (A)) can be used to describe the behaviour of a collection of self-organising networks [3], [4]; while the signalling complexity associated with future network resource management can be analyzed through a type (B) measure, i.e., functional complexity, introduced in [5], [6]. The above conceptual structure based on complexity informs an agent-based modelling (ABM) paradigm to examine the interactions between the different entities that shape a network. ABM provides a method of modelling complex systems from the ground up, which allows for a deeper investigation of the interactions that shape the ultimate system performance. ABM provides powerful modelling of entities in a variety of areas and contexts [7]–[9]. The attributes of ABM can be applied to inform communication networks’ decision making; in particular, ABM can be used to investigate the impact of several Medium Access Control (MAC) component technologies on the Key Performance Indicators (KPI) of both telecom networks and applications, for example in the case of a wireless sensor network aiding an Internet of Things (IoT) system [10]. In summary, we propose a new paradigm for telecommunications, drawing on concepts of a complex systems science nature, to understand and model the behaviour of highly heterogeneous networks and systems of networks. We also employ our framework to create new technologies for supporting network operation. II. M OTIVATION We propose the development of a conceptual framework as a means of exploring a broad range of possibilities in wireless networks, including a vast array of 5G technological possibilities. This framework for thought applies concepts from complex systems science [11], [12] to provide a means to understand wireless networks holistically on a variety of scales. Specifically, we 4 consider the communication patterns that enable network functions, by capturing all nodes necessary to perform a given function; then by drawing connections between these nodes we highlight their functional dependencies. We call a graph obtained in this way functional topology. This approach allows us to analyze the communication patterns on multiple scales. The lowest scale models the communication between individual devices/nodes. In other words, the lowest scale focuses on the communication between a node and all the immediate neighbors of this node in the functional topology. The second scale models the communication between a node, all its immediate neighbors and all neighbors of its neighbors. The increasing scale size moves the focus away from the communication between individual nodes, and allows us to analyze communication patterns between groups of nodes (i.e., functional entities/groups). Considering the high degree of heterogeneity and dense interplay of network elements in proposed 5G and IoT systems, achieving a holistic understanding of network operation is poised to become an even more challenging prospect in the near future. To address these challenges, we demonstrate the power of our framework for the modeling and analysis of relevant 5G scenarios, i.e., self-organizing cellular and IoT networks. While our framework supports innovation beyond these concepts, we feel these scenarios adequately represent the near-future applications of our work. The development of our concept is organized in a layered fashion, with a modelling layer forming the foundation of the framework and supporting analysis and tuning layers. The main aspects of our framework are represented in Fig. 1 and will be discussed in detail in the remainder of the paper. As compared to the CSS literature addressing communication systems [13]–[17], we study wireless networks from the infrastructure perspective. As a simple example, in [3], [4] excess entropy is used to measure complexity and, in combination with entropy, leads to an understanding of the structure emerging in a lattice of self-organising networks. The self-organising systems 5 Modelling Analysis Communication metrics CSS metrics Functional topology graphs Links between: 1. Communication & CSS metrics constraints 2. Technological behaviours 3. ABM parameters and range values Tuning Guidelines for tuning 5G Network Local rules  Global fitness Adaptive multi-dimensional 5G resource allocation Fig. 1: Our complex systems science based layered approach to 5G networks. Functional topology graphs are abstracted from the network, and are then used to compute complexity and telecom metrics, and find their relations. The understanding of such relations will then feed an ABM approach to network tuning. studied in [3], [4] exhibit a complex behaviour and this relates to robustness against changes in the environment; in particular, exploring frequency planning from a complex systems perspective leads to conclude that future networks shall eschew any current frequency planning approaches and instead determine frequency of operation on the fly. This has enormous implications for design and roll-out of networks, deployment of small cells, and network operation. III. M ETHODOLOGY Significant impacts have been made by CSS in a wide range of areas including physics, biology, economics, social sciences, computer sciences, and various engineering domains. We claim that the CSS perspective provides the necessary means to redefine the general understanding of telecommunication networks. We draw on concepts from information theory and ABM; each 6 concept augmenting and developing the understanding of wireless networks. We will now briefly review some of the most important tools and concepts we use in our studies. In order to specify and analyse the complexity of a network function, [5] introduced a framework representing an abstraction of a telecommunication network, by modelling its operation and capturing all elements, i.e., nodes and connections, necessary to perform a given function. Our framework includes functional topologies, i.e., graphs created based on the functional connectivity between system entities (see Fig. 1). A node in our topology represents a functional entity of a network node or any information source that is part of the given network function. The links indicate dependencies between nodes. The definition of functional topologies allows us to visualise the relationships between system entities, and enables the systematic study of interactions between them. Based on these topologies one can define CSS inspired metrics such as functional complexity [5], which quantifies the variety of structural patterns and roles of nodes in the functional topology, or other information theoretical-inspired metrics. Agent-based modelling (ABM) is a useful method to model networks. In [3], [4], [10] ABM was used to investigate the impact of several MAC component technologies, in terms of both telecom and IoT application’s Key Performance Indicators (KPI). This is key for our framework’s analysis and tuning layers. Our framework enables multi-scale modelling, analysis and tuning of wireless networks, in which changes in the 5G networks domain can be analysed and assessed. Indeed, in order to maximize the impact of our framework, our proposed approach is rooted in relevant, near-future architectures and use cases in 5G networks, such as self-organizing cellular and IoT networks. The use cases define the expected parameters, types of users/devices and environments; a general set of possible scenarios we could investigate using our framework is shown in Table I. 7 TABLE I: Possible use cases. Parameters Type of users Environments Low latency, High throughput, High reliability, Extensive coverage, Energy efficiency Typical mobile broadband, Healthcare, Automotive, Home/industrial automation, Wearable devices Busy train station, Emergency/disaster location, Busy office complex/campus, Large utility/manufacturing plant A. Solution Approach Our framework is based around the idea of using concepts, tools and measures of a complex systems science nature. The framework is based on a modelling layer which supports the analysis and tuning layers (see Fig. 1). 1) Modeling Layer: The modelling phase focuses on developing techniques to capture the significant attributes of networks and the interactions that shape them. Along with the traditional attributes used to characterize networks (e.g. coverage and throughput), the modelling phase develops new complexity metrics and investigates their relation to telecom KPIs. These metrics shall be developed distinctly for each application, based on existing and new concepts we draw from CSS. The modelling component of the framework develops appropriate abstractions and formalisms to enable metric calculation. To this end, we produce a multi-scale abstraction for networks. The first level or device level of this abstraction focuses on individual elements within a network, targeting the interplay that results from information being collected and used locally by a single entity. Interference and stability of the connection (e.g., as a function of power available at the node) between nodes in a network are two examples of notions studied at the device scale. Available local information may (as in the case of the interference perceived at a certain network node) or may not (battery level) result from the actions of other nodes. That is, the device scale typically models the implicit exchange of information, where nodes infer information for each other’s actions without directly exchanging messages, such as the interaction-through-interference 8 paradigm of a distributed Time Division Multiple Access (TDMA) system. The higher scales model the explicit exchange of information between groups of nodes in the network; at this level (interaction scale) the nodes act on the basis of information provided by some other node directly, as occurs for example when assigning a slot in a centralized TDMA system. The interactions that shape the network formation and operation are directly modelled using ABM. Our model considers the interactions between the interests of different network operators. These agents operate in a hierarchical fashion (see Fig. 2) with network operator agents who, in turn, contain sub-agents that determine specific aspects of the network, based on technical behaviours. Anything that makes decisions in a network can be viewed as an agent, and ABM is applied to model interactions between agents. For example, IoT agents may attempt to use the infrastructure provided by operator agents, as shown in Fig. 2. To capture the range of possibilities, we can use nested subagents, in which major agents might represent a whole network with subagents representing individual cells. ABM allows conversion of experience with detailed processes (micro-level behaviours) into knowledge about complete systems (macrolevel outcomes). In general we can consider several radio resources in our ABM model, e.g., resources belonging to frequency, power and space domains. Several alternative techniques and technologies can be applied within each domain, which entails a wide set of resources and related modes of utilisation. 2) Analysis Layer: In the analysis layer, the models are reviewed to determine the representative power and meaning of the metrics developed by linking: (i) the operator behaviours with our new CSS metrics; (ii) the operator behaviours with network KPIs (fitness); (iii) our new CSS metrics and KPIs. As an example, we could analyse the relationship between operator decisions on the amount of shared resources (infrastructure and/or spectrum) and the resulting network characteristics. For each scenario, measures of network performance can be identified, including standard 9 Network Operator Agent Cellular network agent - Cell Agent - Access Point Agent IoT Agent Network Operator Agent Cellular network agent Network Operator Agent WiFi network agent WiFi network agent WiFi network agent Cellular network agent IoT Agent IoT Agent Fig. 2: Agent organization. Our agent model is hierarchical, with major agents representing a whole network, and subagents representing IoT agents or individual cells and access points agents. operator KPIs such as cell edge, peak and mean throughput, spectrum utilisation relative to available bandwidth, network reliability, and coverage. For each type of the above mentioned relations (i), (ii), (iii), we can determine the most promising pairing of elements (i.e., operator behaviour and CSS metric, or CSS metric and KPI) within each scale and between scales for determining connections. In particular, we can identify which behaviours correlate to specific network performance measures on each scale, and to what extent and how our CSS metrics describe these relationships. Further, we can investigate how a certain CSS metric-KPI relation at a certain scale affects another CSS metric-KPI relation at a different scale (e.g. a strategy leading to throughput maximisation at the device level might compromise the fairness objective of the resource allocation scheduler at the interaction level). This process involves assessing the ability of the CSS metrics to describe the impact of operator behaviours, analysing the effect of these behaviours on the network KPIs, and finally describing the network KPIs in terms of the CSS metrics. Determining the link between network CSS metrics and KPIs would allow us to attempt to answer fundamental questions such as whether one needs a minimum complexity for achieving a given level of KPI (fitness), and what excess 10 complexity implies in terms of adaptivity and robustness vs. cost. In summary, the analysis layer completes the development of the core of our framework, by establishing a compact representation of the networks by linking complexity metrics to network performance. 3) Tuning Layer: The tuning layer augments the framework with algorithms that automatically guide the operation and management behaviours of relevant agents to achieve desired network properties. This tuning approach utilizes the holistic information encoded into the complexity based quantities, to select appropriate parameters and constraints for the behaviours of the agents. The developed tuning approach can be based on the application of multi-objective optimization techniques; the algorithms to be developed within this paradigm might apply multi-objective optimization algorithms (e.g., NSGA-II, PGEN, SMS-EMOA, successive Pareto optimization) to determine the Pareto fronts for the state spaces of the agent behaviours on the basis of achieving desirable CSS metrics values. These Pareto fronts provide the parameters and constraints of the operator behaviours, allowing operators to further optimize for specific differentiations while maintaining desired holistic properties. A particular solution may be selected from the Pareto front on the basis of agent preferences, such as a preference for high adaptivity and robustness or low complexity, without compromising the overall quality of the solution. IV. A PPLICATIONS OF THE P ROPOSED F RAMEWORK A. Modeling Layer 1) Agent-Based Modelling of the Internet of Things: We employ an instance of our framework concept to investigate the tightened coupling between operative reality and information transfer precipitated by IoT. As such, this investigation resides primarily in the modelling phase with some extension into the analysis phase. Within this work, we apply the tool of ABM to study the impact of communications technologies within the scope of IoT [10]. An automatic traffic management system is considered, where for the purposes of illustrating 11 DM Fig. 3: Single Intersection Diagram. Sensors are deployed alongside the roads and are represented as dots. Inactive sensors are depicted as black dots; sensors detecting moving and static cars are shown as orange and purple dots respectively. the nature of our ABM approach, a single intersection is assumed, depicted in Fig. 3, controlled with traffic lights, in which the avenue of the cross-road is observed by sensor nodes. A processing unit, here denoted as the decision maker (DM), serves as the sink of sensor information and the source of light control commands. Sensor nodes mark the advancement of cars, here portrayed as yellow squares and proceeding on the left side of the roadway, toward the intersection. Two MAC protocols (CSMA and Aloha) are investigated, for communication between the sensors and the DM. The DM applies the resultant information from this process to govern vehicular progress through the coloration of traffic signals. Notably, the semantics of communications greatly impact the operation of the physical system. Fig. 4 exemplifies this notion through a depiction of the difference between the actual number of cars waiting at a traffic light and the perceived number of cars known to the DM component of the system. As revealed by ABM, the minor difference of pre-sensing a channel (CSMA) or not (Aloha) causes either an over- or an under-estimation of the actual number of vehicles by the controlling element in the system. As such, the application of ABM techniques allows the development of an understanding of the various inter-relationships that direct the behavior of a complete telecommunication system. 12 Fig. 4: Impact of MAC on Perception of Situation on a single-intersection scenario. Vehicles always travel in a straight line at constant speed, unless they need to stop due to traffic lights or other cars. At each iteration the probability of a new car arriving at one of the four edges of the grid and travelling in the corresponding direction is 50%. 2) Functional Complexity: As another example of work at the modelling layer, we have developed a metric to capture the amount of information shared between elements of a network (as a result of the organization of the network) in support of a network function. This analytical approach to quantify the complexity of a functional topology provides us with the means to capture the signaling complexity of functional operations within a network, such as handover or frequency assignment. That is, our complexity metric provides a new method of describing the functional operation of telecommunication networks. Our complexity metric is built upon the concept of Shannon entropy (Hr (xn )). We employ the Bernoulli random variable xn to model the potential of a node to interact with other nodes. The probability of interaction pr (xn = 1) is defined as the reachability of a node n (pr (xn = 1) = inr /j, where inr is the number of nodes that can reach node n and j is the number of nodes for the given subgraph). The definition of reachability, in terms of the number of hops allowed between two nodes in the functional topology, enables the analysis of complexity on multiple scales (r). The one hop reachability represents the lowest possible scale (r = 1), where 13 each node interacts only with its immediate neighbors. The increasing number of allowed hops between the nodes brings the nodes closer to each other in terms of interactions, and moves the focus from interactions among nodes to interactions among groups of nodes, i.e., analysis of higher scales. The total amount of information of the k th subgraph with j nodes for scale r is calculated as Ir (Λjk ) = X Hr (xn ), (1) n∈Λjk where Λjk is the k th subgraph with j nodes. The total amount of information represents the total uncertainty which is related to the actual roles of nodes that appear within a subgraph and different subgraph patterns. Our complexity metric, which is calculated with Eq. (2), quantifies the amount of order and structure in a system that is seemingly disordered. R−1 N 1 X X r+1−j CF = \|hIr (Λj )i − Ir (ΛN )\| R − 1 r=1 j=1+r r+1−N (2) where R is the maximum scale size, which is defined as the diameter of the functional topology, N is the number of nodes in the functional topology, ΛN is the whole functional graph, and hIr (Λj )i is the average amount of information for a given subgraph size j. We call the metric in Eq. (2) functional complexity. Our approach holistically gauges the functional organization of a network by first describing the interactions necessary to perform a given function topologically. Within this representation, we capture the network elements involved in performing some function and the interactions that support the operation of the function. Our quantification of networks in terms of their functional relationships provides a wholly new approach to understanding the operation of networks. As corroborated by Fig. 5, more typical metrics for network topology do not capture the notions represented by our complexity metric (in fact, the correlation of our complexity metric with other traditional metrics is lower than 14 Average Path Length, Clustering Coefficient, Complexity 0.28 Average Degree, Complexity Average Path Length, Complexity Average Path Length, Average Degree, Complexity 0.47 -0.42 0.5 Clustering Coefficient, Average Degree, Complexity 0.15 0.3 Clustering Coefficient, Complexity Fig. 5: Correlation between the proposed complexity metric and the three most used measures of network topology (i.e., average path length, average degree distribution, clustering coefficient). 0.5 in all the cases we consider); this complexity metric thus provides an alternative method of describing network operation. The above functional topology and complexity framework can be applied for instance to understand the underlying mechanisms that lead to certain network properties (i.e., scalability, energy efficiency) in Wireless Sensor Networks (WSN) as the result of different clustering algorithms [6]. B. Analysis Layer In the context of the analysis layer of our framework, we focus on a cellular network that selforganises from a frequency perspective to understand the collective behaviour of the network. We calculate the excess entropy EC = ∞ X (h(M ) − h) (3) M =1 to measure complexity, and the entropy h = lim h(M ), M →∞ (4) where h(M ) is the entropy of the target cell X conditioned on M surrounding cells. By measuring EC and h we gain an understanding of the structure emerging in the lattice for a self-organising 15 network. Based on Eqs. (3) and (4), in [4] one shows that a self-organising cellular network can exhibit a complex behaviour, and that it can be robust against changes in the environment. In more detail, a self-organised and a centralised channel allocation are analyzed, with respect to their robustness to local changes in the environment. In order to compare the stability of the two types of channel allocation, 102 instances of the self-organising frequency allocation algorithm are run using 102 × 102 lattices. Then, for each resulting channel allocation, all possible cells n are considered, and for each cell all possible frequencies are in turn considered. Then the optimal minimum distance c to an interference-free channel allocation is computed (we define the distance between two channel allocations as the number of changes that are necessary to move from one configuration to the other). We found that the locally perturbed channel allocation matrices resulting from self-organisation are more stable than those resulting from a centralized frequency planner. What we know so far is that there is a relation between some complexity metrics and some telecom KPIs (i.e., between excess entropy and robustness to changes [4], and between functional complexity and the trade-off scalability-energy efficiency [6]). The complexity metrics we introduced have shed some new light on very relevant telecom KPIs/properties in the context of 5G networks, i.e., excess entropy can measure self-organization capabilities in the frequency allocation context and functional complexity can measure scalability in WSN. As widely acknowledged, self-organization and scalability are very important properties of 5G systems (e.g., for IoT and dense small cell deployments). In the future we plan to improve and expand such understanding to all the most prominent 5G network technologies and KPIs. C. Tuning Layer ABM rules will choose the technological behaviour options that maximize the targeted communication network KPI, subject to constraints defined by the correlation between CSS metrics 16 Set of available parameters - Complexity – robustness Complexity – energy efficiency Complexity - resilience Fitness functions Waveforms MIMO Frequency reuse Duplexing Tuning Layer - MIMO multi-user scheme OFDMA Full duplex Frequency assignment algorithm Network configuration parameters Fig. 6: The adaptation of network configuration parameters in the tuning layer. The set of available parameters represents a virtual pool of all the available network resources. The fitness functions depict a relationship between different network KPIs and complexity metrics which are calculated upon the set of available parameters. and other telecom KPIs. Local decisions will be based on only a few CSS metrics, and will lead to desired global behaviours/KPIs of the network. The local decisions are made according to ABM rules, by exploring and selecting the fittest behaviours (where by behaviour we mean some algorithm or policy acting on some radio resources). Our goal, for different services (e.g., mobile broadband, M2M) is to choose behaviours that allow the network to achieve satisfactory KPIs, in terms of, e.g., delay, throughput, coverage, energy efficiency, out-of-band emission, etc. The question is whether we can keep achieving globally satisfactory KPIs just by changing ABM rules in a distributed fashion at different nodes. Such adaptation will act within a certain resource allocation domain (e.g., picking among different massive MIMO schemes) or between performance-equivalent allocations using resources from different domains (e.g., spectrum or infrastructure). The main ideas behind the tuning layer of our framework are exemplified in Fig. 6. Although our own work on the tuning layer is still in the initial phase, from a substantial amount of literature, we can gather evidence that different physical layer (PHY) and Radio 17 Resource Management (RRM) techniques in the 5G domain should be chosen depending on environmental conditions and network requirements, i.e., we are potentially in a situation where our tuning layer is relevant and beneficial. We give a brief account of such evidence next. In [18], it is shown that for a massive MIMO system, the sum-rate has a linear or sublinear behaviour with respect to the number of base station antennas, depending on the spatial richness of the environment; related work on adaptive precoding for distributed MIMO is explored in [19]. Several works investigate the coexistence of various waveforms in terms of cross-waveform leakage interference [20], [21] and possible implications for the waveform selection [22]. The fraction of cells that have full duplex base stations can be used as a design parameter, to target an optimal trade-off between area spectral efficiency and outage in a mixed full/half duplex cellular system [23], [24]. In [25], it is shown that increasing the frequency reuse can improve the throughput-coverage trade-off for ultra-dense small cell deployments, while a lower frequency reuse should be favoured if the target is maximizing throughput given a certain BS density. In summary, we plan to use the above understanding of the benefit of adaptation at PHY and MAC layers in 5G networks, and extend it as needed in terms of technology components, KPIs and adaptation criteria, to inform our framework, and show its immediate benefit in understanding, operating and designing 5G systems. V. O PEN C HALLENGES Several more 5G component technologies, in addition to those considered in this paper, can enrich the set of possible choices used to model, analyse and tune the network, including (massive) co-located or distributed multiple antenna arrays; different waveforms and multiple access schemes; different duplexing schemes; novel spectrum sharing schemes such as License Assisted Access (LAA); and different frequency reuse schemes including probabilistic ones for ultra-dense networks. 18 What we know so far is that there is a relation between some complexity metrics and some telecom KPIs (i.e., between excess entropy and robustness to changes, and between functional complexity and the trade-off scalability-energy efficiency). In the future the aim is to improve and expand such understanding to all the most prominent technologies and KPIs for 5G networks. In particular it is still an open question how to achieve the desired network tuning properties within a large optimization space encompassing many different network resources, KPI objectives and constraints, many different heterogeneous co-existing networks and a very large number of nodes and decision points. We conjecture ABM can help us achieve such ambitious goal, as a key tool to engineer desired emergent properties in such future challenging networks. As the network graph representations discussed in the proposed framework might dynamically change according to the different radio resource domains and related techniques used, one open area of investigation is to study how the complexity metrics can be calculated and how they evolve over time for such dynamic multi-dimensional resource allocation; and then use such metrics to analyse and tune the network behaviour taking into account robustness, resilience, network utilization, and other time-dependent network characteristics. VI. C ONCLUSION Current complex systems science literature focusing on communication systems draws on network science, studying applications and traffic modelling, but lacks considerations of architecture, infrastructure, and technology. We instead apply complex systems science to wireless networks from the functional perspective, drawing on concepts from information (i.e., different metrics of complexity) and computational (i.e., agent based modeling) theory, adapted from complex system science. Since complex systems science metrics are currently absent from the quantities considered when operating and designing communication networks, by introducing our proposed framework we initiate a completely new way to model, analyse and engineer networks, founding a new 19 theory and practice of telecommunications not previously anticipated. 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Analysis of Unprotected Intersection Left-Turn Conflicts based on Naturalistic Driving Data arXiv:1702.00135v2 [cs.SY] 3 Apr 2017 Xinpeng Wang1 , Ding Zhao2 , Huei Peng3 and David J. LeBlanc2 Abstract— Analyzing and reconstructing driving scenarios is crucial for testing and evaluating highly automated vehicles (HAVs). This research analyzed left-turn / straight-driving conflicts at unprotected intersections by extracting actual vehicle motion data from a naturalistic driving database collected by the University of Michigan. Nearly 7,000 left turn across path - opposite direction (LTAP/OD) events involving heavy trucks and light vehicles were extracted and used to build a stochastic model of such LTAP/OD scenario, which is among the top priority light-vehicle pre-crash scenarios identified by National Highway Traffic Safety Administration (NHTSA). Statistical analysis showed that vehicle type is a significant factor, whereas the change of season seems to have limited influence on the statistical nature of the conflict. The results can be used to build testing environments for HAVs to simulate the LTAP/OD crash cases in a stochastic manner. I. INTRODUCTION Before highly automated vehicles (HAVs) can be released to the general public, a well-defined process for testing and evaluating them must be established. The Google self-driving car project experienced its first shared-responsibility crash in February 2016 [1]. Moreover, Tesla Autopilot failed to detect a semi-truck in its first fatal crash happened in May 2016 and was criticized for using the consumers as beta testers [2]. Fig. 1 briefly demonstrates how this crash happened, with the red sedan representing the Tesla. The National Highway Traffic Safety Administration (NHTSA) is now considering the possibility of putting a pre-market approval process into place [3], in addition to a rigorous self-certification process still anticipated from the vehicle manufacturers. A key factor in HAV testing is the test scenarios and behaviors of other road users, particularly those of other vehicles. The test conditions need to be not only realistic but also feasible for repeated safety tests. Test scenario models can be divided into two types. The first type has fixed scenarios, such as the tests of lane support systems (LSS) [4] and autonomous emergency braking (AEB) [5] launched by The European New Car Assessment Programme (EURO NCAP). A major advantage of this type is that it is repeatable. However, it is hard to use this type of models to * This work is funded by the Mobility Transformation Center Denso Tailor Project at the University of Michigan with grant No. N020210. 1 X. Wang is with the Department of Automation, Tsinghua University, Beijing, China, 100084, and now he is a visiting scholar at the University of Michigan, Ann Arbor, MI 48109, U.S. 2 D. Zhao (corresponding author: [email protected]) and D. J. LeBlanc are with the University of Michigan Transportation Research Institute, Ann Arbor, MI 48109, U.S. 3 H. Peng is with Department of Mechanical Engineering at the University of Michigan Transportation Research Institute, Ann Arbor, MI 48109, U.S. Fig. 1. A brief description of the Tesla accident represent the highly complex and variable nature of the human driving environment. Moreover, HAVs could be adjusted to pass certain fixed scenarios while their performance under broad conditions might not be well assessed. To overcome these drawbacks, we proposed a second type of models. In our previous works [6]–[8], we proposed a stochastic test method and built a test environment for car-following and lane-change scenarios. In this paper, we will focus on the intersection scenario. The intersection has been one of the most challenging scenarios for HAVs, due to the variety of road users, complexity of traffic flow and the unpredictability of vehicles and pedestrians. According to [9], crashes at intersections took up a major portion, about 44 %, of all the traffic crashes in the US. Among all kinds of scenarios with potential risks at an intersection, unprotected left turn across path opposite direction (LTAP/OD) is a typical one. This scenario is ranked second among 10 priority V2V light-vehicle precrash scenarios [10]. In an LTAP/OD scenario, two vehicles are considered: the turning vehicle (TV) and the straightdriving vehicle (SdV). Although a lot of research, such as [11], [12], has been conducted on traffic conflict analysis of LTAP/OD scenario, the factor of vehicle type has not been widely investigated. Now that the crash of the Tesla with Autopilot system has been attributed to its failure to detect the truck turning ahead [13], it is crucial that more attention should be paid to scenarios involving heavy trucks. Moreover, there has been insufficient research on the influence of season change on driving behaviors at intersections. As extreme weather such as storm and fog has a strong impact on the driving behaviors of human drivers, we propose that they are possibly influential to HAVs as well. This research focused on two major tasks: first, it built a stochastic model of traffic conflicts in the LTAP/OD scenario TABLE I INTRODUCTION TO IVBSS DATABASE Vehicle type Distance Time Trips Vehicles / Drivers Type of front radar Light vehicle 213, 309 mi Apr. 2009 - Apr. 2010 22,657 16 sedans / 108 drivers Bosch LRR2 Heavy truck 601, 994 mi Feb. 2009 - Dec. 2009 22,724 10 tractors / 20 drivers TRW AC20 (a) Light vehicle Fig. 2. based on naturalistic driving data. Events involving both light vehicles (LVs) and heavy trucks (HTs) as the SdV were extracted from the database, reconstructed into realistic trajectories of TVs and SdVs, and finally described with several key variables. Secondly, the influence of vehicle type of the SdV and the season factor to the driving behavior of the TV was analyzed by comparing the distribution of these key variables between LVs and HTs as well as between summer and winter. II. DATA S OURCE The data source for this research is the Integrated VehicleBased Safety Systems (IVBSS) database [14], which was collected from 2009 to 2010 and maintained by the University of Michigan Transportation Research Institute (UMTRI). The database consists of two parts: LV platform and HT platform. It comes from a naturalistic field operational test (N-FOT) which is to assess the potential safety benefits and driver acceptance associated with a prototype integrated crash warning system [15], [16]. As the system incorporates forward crash warning (FCW), lateral drift warning (LDW), lane-change/merge warning (LCM) and curve speed warning (CSW), non of the functions are designed to deal with LTAP/OD scenario. Thus, it is assumed in this research that whether the warning system is enabled will not affect driver behavior at LTAP/OD scenario. For LV platform [14], 16 identical prototype vehicles were driven by 108 drivers for their own personal use for over six weeks. On each test vehicle, there is one long-distance 77-GHz radar that looks forward and six 24-GHz radars that cover the adjacent lanes as well as the area behind the vehicle. In addition, there is a vision system, an automotivegrade non-differential global positioning system (GPS) and an on-board digital map. Around 700 different channels of signals have been collected. For the HT platform [17], 18 male commercial truck drivers from Con-way Freight drove 10 equipped Class 8 tractors for 10 months. There are eight radars, three exterior cameras and several interior cameras on the test truck, recording over 500 channels of data including the driving environment, drivers activity, system behaviors and vehicle kinematics. Basic information about the LV and HT platforms in the IVBSS database is listed in Table I; the configuration of on-board sensors on each platform is shown in Fig. 2. The test area covered by IVBSS N-FOT is primarily in (b) Heavy truck Sensor configuration of IVBSS test vehicles the Detroit area of the U.S. Most of the HT trips took place in the lower peninsula of Michigan (63 %) and Ohio (33 %) [16]. most of the LV trips fell within a similar region [15]. The database provides adequate information for this research. For event extraction, data from the on-board GPS sensor is used to locate the instrumented vehicle; data from the front long-range radar is used to reconstruct the trajectory of target vehicles; video recordings from vision cameras around the vehicles are used as a supplemental tool for event screening. In addition, as the IVBSS test lasted approximately one year, driving data under a variety of weather conditions throughout the year were covered, enabling us to uncover the influence of season factors. III. E XTRACTION OF L EFT T URN S CENARIO In order to extract eligible left-turn events from the database for both the LV and the HV platform, three major tasks were performed. First, we processed data from radar for further use. Second, we searched the database for all left-turn events that meet our criteria. Finally, data points in each event were interpreted into trajectories of the TV and the SdV. A. Target Association of Truck Data For radar data from the HT platform, we need to associate and mark data points together that belong to the same target in order to screen out unfit targets and create a trajectory for every eligible TV. To cluster points of interests, we apply the following criteria to processing HT data: • Only objects (TVs) that move in the opposite direction are detected. (vT V <-0.3 m/s). • Only points with small azimuth angle (\|α\| < 5.5°) are considered, as the effective detecting range of the radar is 11°. • When the cluster with point i is expended, only data points within a small time slot ([t(i), t(i) + 0.85 s]) are considered. • Only neighbor points that satisfy the following rules are grouped: 1) Strong correspondence between range, range rate and time difference: δtpred \| < 0.3 (1) \|1 − t(j) − t(i) where δtpred = 2 ∗ r(j) − r(i) rr(j) + rr(i) (a) Range Fig. 3. (b) Transversal Fig. 4. Configuration of the instrumented vehicle and the target vehicle for event extraction in LTAP/OD scenario An exemplary result of target association Here, r(i) is the range of point i, and rr(i) is the range rate of point i 2) Reasonable difference in transversal. tr(j) − tr(i) < 20m/s t(j) − t(i) (2) Here, tr(i) is the transversal of data point i, and t(i) is the time. Fig. 3 shows an example of data points that are associated and divided into different groups in one event. The dots with the same color show trajectories of targets, while red dots do not belong to any group and are seen as noise. In such a typical LTAP/OD scenario, a vehicle is turning in front of the instrumented truck. Fig. 3(a) shows how the range of target points change over time. As target vehicles cross the intersection when the instrumented truck is moving forward at a steady speed, the ranges to different targets are decreasing linearly. Moreover, Fig. 3(b) shows that the transversal of multiple targets is increasing from negative to positive, indicating that they cross from left to right in the view of the instrumented truck. Once data points from each target are clustered, the HT platform can be used for further event extraction for eligible LTAP/OD scenarios. B. Event Screening An unprotected LTAP/OD scenario can be recorded by either the SdV or the TV. In this paper, we use only the scenarios recorded by SdVs. Fig. 4 demonstrates the configuration of the instrumented vehicle, i.e., the SdV and the target vehicle, i.e., the TV for event extraction in LTAP/OD scenarios. For both the LV and HT platforms, eligible leftturn events are queried based on the following criteria: • The intersection has a stop sign or a set of signal lights. Although there will be protected LTAP/OD events retrieved with this criterion, they can be screened out by the following conditions, such as the constraint on the velocity of the TV and the SdV. • The instrumented vehicle is moving straight (speed larger than 3 m/s & change of heading angle smaller than 10°). • The target vehicle is moving towards the instrumented vehicle (the longitudinal projection of speed smaller than -0.5 m/s) and moving from left to right (due to the difference in radars, transversal goes from positive to negative for LVs, and from negative to positive for HTs). Fig. 5. • • Procedure and interim results for event extraction Time duration of the event is adequate (more than 1.5 s). The maximum of time difference between two consecutive points (defined as δt) in an event should be small enough to be seen as points of the same target (max{δt} < 1 s). Event extraction follows a similar procedure for LV and HT. For the LV platform, we first select all straight-driving occurrence at intersections; we then extract those with leftturn objects from the opposite direction. These tasks are completed in the Microsoft SQL Server Management Studio (SSMS). Afterwards, the extracted events are exported to MATLAB for the last round of screening, which guarantees reasonable speed, targets, and time duration. For the HT platform, the only difference is that after retrieving all the occurrences of straight-driving at intersections, we export data from the database server directly into MATLAB for target association and the following extraction tasks. The diagram in Fig. 5 illustrates the procedure and interim results of each phase for event extraction. Finally, HT has 5,780 eligible LTAP/OD events, whereas LV has 1,055. The location of these events is shown in Fig. 6. C. Trajectory Reconstruction Once all eligible events have been selected, the trajectories of both TV and SdV in each event are then reconstructed. The exact position of SdV comes from the on-board GPS sensor; the data from the front long-range radar are used to extract the relative position of TV in the coordinate of SdV. After synchronization on GPS and radar data, the trajectories of TV and SdV are generated. Fig. 7 shows the reconstructed trajectories for SdV and TV in one event. Here, dots with the same color represent the position of TV and SdV at the same moment. The TV crossed intersection before the SdV in this example. (a) Light vehicle Fig. 6. (b) Heavy truck Fig. 8. The location of extracted events Time to the conflict point of the SdV in a LTAP/OD scenario vSdV : Speed of the SdV at Tx vT V : Speed of the TV at Tx First, we demonstrate an example of conflict analysis on a single LTAP/OD event. Here we use the aforementioned occurrence, where the TV crossed the intersection before the SdV did. Fig. 8 uses Tcp to demonstrate how the SdV and the TV interacted in one real LTAP/OD event. The vertical axis indicates predicted time to the point of conflict of the SdV, whereas the horizontal axis shows the real elapsed time relative to the moment when the TV crosses the intersection, that is, Tx . In this scenario, time to the conflict point decreased linearly over time, indicating the margin between the TV and the SdV was large enough for the SdV to maintain a nearly constant speed when the TV was crossing. When the TV reached the conflict point, there was a 2.3-second margin for SdV, that is, Tcp , which is demonstrated by the red dot. Here, Tcp described the essence of this interaction between the SdV and the TV. Then, the following modeling and analysis will ignore the detailed interaction of the TV and the SdV, paying attention only to the four aforementioned variables in each event. We use all events we retrieved in the previous section from both the HT and LV platforms as the source for modeling. • • Fig. 7. Example of reconstructed trajectory IV. C ONFLICT A NALYSIS AND C OMPARISON A. Definition and Metrics of Conflicts In this section, conflict is used to describe risky events in traffic. According to [18], conflict is defined as an observational situation in which two or more road users approach each other in space and time to such an extent that a collision is imminent if their movements remain unchanged. Many conflict metrics have been used for measuring the level of safety for an LTAP/OD event, including post-encroachment time (PET), leading buffer (LB) and trailing buffer (TB) used by [19], and gap time (GT) used by [20]. For this paper, as the goal is to construct a stochastic model, we choose only the most representative time slice in each event to model LTAP/OD conflicts. The heading angle of the SdV is taken as constant during each event, with any small deviation being ignored. Thus, a conflict point is naturally defined as the location of the TV when its transversal in the radar of SdV crosses zero, and this exact moment is regarded as the representative moment of this conflict, defined as Tx . Consequently, four variables at Tx are chosen to model the conflict, including two modified conflict metrics: time to the conflict point (Tcp ) and distance to the conflict point (Dcp ): • Dcp : Distance to the conflict point for the SdV at Tx Dcp = dist(PSdV (Tx ), PT V (Tx )) • In this section, the effect of vehicle type on traffic conflict in LTAP/OD scenarios is discussed. Distributions of variables for LVs and HTs are compared. As events with smaller Dcp and Tcp are more dangerous, we generated the distributions of the reciprocal of Dcp and Tcp to put these risky but rare events in the tail, as shown in (3) Here PSdV and PT V are the positions of SdV and TV Tcp : Time to the conflict point for the SdV at Tx Tcp = Dcp /vSdV B. Comparison between Light Vehicles and Heavy Trucks (4) −1 (a) Distribution of Dcp −1 (b) Distribution of Tcp −1 −1 Fig. 9. Comparison of Dcp and Dcp between heavy trucks and light vehicles Light vehicle Heavy truck (a) Speed distribution of straight- (b) Speed distribution of turning driving heavy trucks and straight- heavy trucks and turning light vehidriving light vehicles cles Fig. 10. Comparison of the speed between heavy trucks and light vehicles Fig. 9. The dots and bars at the top of figures show the mean value and the standard deviation of each empirical −1 distribution. From Fig. 9, we can see that when Dcp or −1 Tcp increases, there are fewer points of data, giving rise to a shape with a long tail. Moreover, events with an HT as −1 −1 SdV tend to have both smaller Dcp and smaller Tcp than with an LV, indicating less severe conflicts. Fig. 10 shows the distributions of vSdV and vT V . The distribution of vSdV for HT and LV platforms both have a triangular shape. Most vSdV ranges from 12 to 20 m/s, whereas most vT V is less than 10 m/s at the conflict point. Though there is no obvious difference with the distribution of vSdV between events with LV and HT as the SdV, the vT V tends to be significantly higher when the SdV is an LV than is an HT. Combined with the previous results of Dcp and Tcp , we can conclude that for left-turn conflicts where HTs are SdVs, conflict metrics have significantly higher value, and TVs tend to turn with less aggressive speed. This means that when the TV chooses the time of turning and commences turning action, it behaves more conservatively when confronted by an HT coming from the opposite direction than an LV. The difference in vehicle type does influence the driving behavior of the TV and the severity of the conflict. C. Analysis of Season Factor In this section, we uncover the influence of season factor on behaviors of SdVs and TVs in LTAP/OD scenarios. During the test, 7 % of driving for HTs [16] and 15 % [15] for LVs took place in freezing temperature. The months with events that took place in freezing temperatures are defined as winter, which includes December through March of the following year. This period also coincides with the time when the average snowfall in Ann Arbor is over 8 inches. On the other hand, summer is defined as being from June to August. We have retrieved 272 events in summer and 391 events in winter for LVs, whereas the numbers for HTs are 1818 and −1 −1 844 respectively. Tcp , Dcp , vSdV and vT V are compared for summer and winter driving. Mann-Whitney-Wilcoxon (MWW) test [21] is a nonparametric hypothesis test of the null hypothesis that two populations are the same against an alternative hypothesis. Here, we used it to determine whether the conflict metrics differ between summer and winter. Fig. 11. Comparison between events in summer and winter Fig. 11 shows the result of the comparison. It can be concluded that for both LV and HT platforms, the mean values for summer and winter of all four variables that describe the conflict at LTAP/OD for both SdVs and TVs are very close. As the p-value from the MWW test is large (p-value > 0.6) for all the eight distributions, we are not able to distinguish between the left-turn pattern in summer and −1 −1 , vT V and vSdV . This result in winter in terms of Dcp , Tcp indicates that despite a large difference in climate, there is no significant difference between the way people drive in winter and in summer at LTAP/OD scenarios in the Great Lakes area. This conclusion has its significance for designing and testing of HAVs. V. CONCLUSION In this research, traffic conflicts of TVs and SdVs in LTAP/OD scenarios are modeled and analyzed based on nearly 7,000 left-turn events extracted and reconstructed from the naturalistic database. The two modified conflict metrics, Tcp and Dcp are used to model turning behavior of the TV. This stochastic model can be further used for developing simulation tools for evaluating HAVs. The significance of vehicle type and season are also addressed in the research. In general, when the SdV is an HT, the driver of the TV tends to turn in a more conservative fashion with a wider margin. Surprisingly, despite prevailing snow and freezing weather in the winter of Michigan, driver behavior at LTAP/OD scenarios during the N-FOT test did not differ significantly between summer and winter. These two conclusions can be useful for designing automated driving algorithms and for establishing regulations and policies for HAVs. In the following research, we will improve the accuracy of trajectory reconstruction by conducting sensor fusion to the GPS and yaw rate sensor, and by re-synchronizing data from different channels. Moreover, we will further investigate the reasons behind the conclusion on the similarity of driver behavior between summer and winter. Possible causes could be: snow on the road was shoveled promptly in winter thus normal driving was almost unaffected; the N-FOT trips avoided extreme weather in winter so that the data was biased. Besides, We will also facilitate the model to build a stochastic simulation environment for the testing and evaluation of HAVs. D ISCLAIMERS This work was funded in part by the University of Michigan Mobility Transformation Center Denso Pool project. The findings and conclusions in the report are those of the authors and do not necessarily represent the views of the MTC or Denso. R EFERENCES [1] “Google Self-Driving Car Project Monthly Report (February 2016),” pp. 14–16, 2016. [2] O. Solon, “Should Tesla be ’beta testing’ autopilot if there is a chance someone might die?” [Online]. Available: https://www.theguardian.com/technology/2016/jul/ 06/tesla-autopilot-fatal-crash-public-beta-testing [3] U.S. Department of Transportation and National Highway Traffic Safety Administration, “Federal Automated Vehicles Policy,” Tech. Rep. September, 2016. [Online]. Available: https://www.transportation. gov/sites/dot.gov/files/docs/AVpolicyguidancePDF.pdf [4] “European New Car Assessment Programme - TEST PROTOCOL - Lane Support Systems,” 2015. [Online]. 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A Revised Incremental Conductance MPPT Algorithm for Solar PV Generation Systems Meng Yue and Xiaoyu Wang Sustainable Energy Technologies Department Brookhaven National Laboratory Upton, NY 11973, USA [email protected], [email protected] Abstract—A revised Incremental Conductance (IncCond) maximum power point tracking (MPPT) algorithm for PV generation systems is proposed in this paper. The commonly adopted traditional IncCond method uses a constant step size for voltage adjustment and is difficult to achieve both a good tracking performance and quick elimination of the oscillations, especially under the dramatic changes of the environment conditions. For the revised algorithm, the incremental voltage change step size is adaptively adjusted based on the slope of the power-voltage (P-V) curve. An accelerating factor and a decelerating factor are further applied to adjust the voltage step change considering whether the sign of the P-V curve slope remains the same or not in a subsequent tracking step. In addition, the upper bound of the maximum voltage step change is also updated considering the information of sign changes. The revised MPPT algorithm can quickly track the maximum power points (MPPs) and remove the oscillation of the actual operation points around the real MPPs. The effectiveness of the revised algorithm is demonstrated using a simulation. Index Terms—IncCond MPPT algorithm, fractional opencircuit/short-circuit MPPT algorithm, P&O MPTT algorithm, solar PV generation. I. INTRODUCTION As one of the most promising renewable energy technologies, the installed capacity of the solar photovoltaic (PV) generation has increased dramatically in recent years. Although the cost of PV generation continues to drop, the economic competitiveness of solar PV energy is still low compared to the traditional energy sources, even with various local and federal policy instruments [1]. While it is desirable to further lower the cost and increase the efficiency of solar energy systems including both the solar panels and the power electronic devices, increasing the efficiency of the installed PV energy systems by simply improving the existing control algorithms should also be pursued. One way of achieving this is to modify the existing MPPT algorithms to extract more solar energy under various environmental conditions. Many different types of MPPT algorithms have been proposed in the literature. As summarized in [2], different algorithms have their own pros and cons, in terms of complexity, accuracy and convergence speed, etc. Among the commonly used algorithms, the hill-climbing/perturbation and observation (P&O) method [3-6] is easy to implement using either analog or digital circuits. It periodically perturbs either the duty ratio of the converter or the the PV array operating voltage even when the MPP is achieved. Further, the “true” MPPT cannot be achieved using the P&O method since the operation point of the PV system is oscillating around the MPP. Under the conditions of continuous fast changing irradiance, the operating point might continuously deviate from the MPPs and eventually the optimal operation points cannot be achieved at all. These issues degrade the performance of the solar generation system. The fractional open-circuit voltage (or short-circuit current) method [7-9] needs only to sense one voltage (or current) parameter to approximate the MPP by using empirical parameters. The major issue related to this method is that the PV circuit has to be periodically operated in open-circuit (or short-circuit) conditions and may have significant impact on the grid operation. Other types of algorithms, such as those based on fuzzy logic control and neural network may accurately track the MPPs under different environmental conditions [2]. The MPPT performance, however, is not guaranteed since they both rely heavily on the algorithm developers and/or a significant volume of field data under all kind of conditions for the design and implementation of such algorithms. The IncCond method (see, e.g., [10-17]) appears to be the most popular one in practice due to its medium complexity and the relatively good tracking performance. One of the major difficulties implementing the IncCond method is the selection of the (fixed) voltage change step size for simultaneously satisfying the tracking speed and maintaining the MPP. A large step size of voltage change helps the system rapidly approach the MPPs. On the other hand, this large value generally induces persisting oscillations around the MPP if no other special countermeasures were taken. The issues with using a small step size of voltage change are the opposite. A simple and effective revised IncCond algorithm is proposed in this paper. An adaptive voltage step change scheme is first adopted based on the slope where the operation point locates on the P-V curve. An accelerating factor and a decelerating factor are then applied to further adjust the voltage step change considering whether the sign of the P-V curve slope remains the same or changes in a subsequent tracking step. The same information of sign changes is also used to update the upper bound of the maximum voltage step change. The adaptive voltage step change enables the PV system to quickly track the environment condition variations, i.e. reach and stay at the MPPs. In this way, more solar energy generation can be harvested from the PV energy systems. These improvements enable the quick response to the environment condition changes and rapid landing on the MPP. The revised method is easy to implement since it does not require knowledge of the I-V characteristics of specific PV panels and the parameters are easy to tune. The revised IncCond algorithm is described in detail in Section II with an overview of various modified IncCond methods. Modeling of generic PV generation systems is presented in Section III for simulation purposes. Simulation results using the proposed MPPT algorithm will be shown in Section IV and concluding remarks are given in Section V. II. A REVISED INCCOND MPPT METHOD The MPP is achieved by adjusting the terminal output voltage of a solar array through controlling the converter duty ratio. While the cell temperature can be easily measured, the irradiance is difficult to measure accurately, and the desired voltage at the MPP is hard to know exactly. Therefore, a test condition needs to be developed in order to determine whether the current operating point is the MPP or not without measuring the temperature and irradiance. For a solar panel, there is only one maximum power point for a given irradiance level and cell temperature. Note, the presence of a partial shading condition of a panel may cause multiple local maxima and is not considered in this paper, although the revised algorithm can be used together with the two-staged methods proposed in [16, 17]. The IncCond method uses the information of the solar P-V curve, i.e., at the left hand side of the MPP the slope is greater than zero, at the right hand side of the MPP the slope is less than zero, and the slope is zero exactly at the MPP. Therefore, the solar array terminal voltage needs to be increased when the slope is positive and decreased when the slope is negative. The slope dP/dV can be calculated as, dP d ( IV ) dI = = I +V dV dV dV (1) with dI/dV ≈ ∆I/∆V (i.e., the incremental conductance) in an implementation. Under the MPP condition, i.e., dP/dV = 0, the following relationship holds, ∆I I =− ∆V V (2) The major difficulty with the IncCond method is the selection of the incremental step size of the duty ratio for adjusting the solar terminal output voltage. A fixed incremental step size of the duty ratio in general will not bring the array to the MPP because the operating point will oscillate around the MPP, i.e., either on the left or the right of the MPP. Ref. [14] divided the entire I-V (current-voltage) curves into two domains using "square root" functions with all of the MPPs contained in only one of them. Therefore, the first step of performing MPPT is to bring the operating point to the domain that contains all of the MPPs. This method, however, requires a good understanding of the PV panel I-V characteristics that are panel specific. In [15], a so-called "Van Allen's oscillator" was added between the solar panel and the inverter for a purpose of balancing the power source and the load that continues changing. A simple proportional integral (PI) controller was developed to track the MPPs based on this configuration. It is intuitively easy to avoid a fixed voltage change step size by adjusting the increment proportionally to the steepness of the slope and eventually the increment of the duty ratio will become zero at the MPP, where the slope is zero, similar to the PI controller proposed in [15]. The implementation, however, appears to be very difficult because (1) the P-V curve steepness around the MPP can be very different for different operating conditions (i.e., the P-V curve for a lower irradiance level can be more flat) and (2) a sudden change in the operating condition of the solar array may produce a very large numerical difference in calculating the slope when the change occurs. Since the duty ratio is between 0 and 1, this may cause unacceptable change in the solar terminal output voltage and make it very difficult to bring the voltage back to normal. Note, refs [16] and [17] proposed twostage methods mainly to avoid the local maxima caused by the non-uniform insolation experienced by the solar panels. The traditional IncCond method was still used after bringing the operating point close to the global MPP by using, e.g., monitoring cells in [17]. In this section, a simple and effective modified IncCond method is proposed based on observations that (1) in two consecutive tracking steps, a changing in sign of the slope (i.e., from positive to negative or from negative to positive) indicates that the increment step size is too large (otherwise, it may land on the MPP or the same side on the P-V curve) of the duty ratio; and (2) the same sign of the slope in two consecutive tracking steps indicates that the increment step size is too small (otherwise, the operating point may land on the MPP or the other side on the P-V curve). Based on these observations, the strategy proposed here is to (1) adjust the incremental step size considering the steepness of the slope; and (2) further adjust the incremental step size comparing the sign of slopes in two consecutive tracking steps, i.e., decrease the incremental size in the former case (e.g., by multiplying the incremental size by a factor DEACC such as 0.7) and to increase the incremental size in the latter case (e.g., by multiplying the increment by a factor ACC such as 1.2). By applying this improved strategy, the solar array will approach the MPP in an accelerating manner after a change in the operating condition(s), and the magnitude of oscillation around the MPP may rapidly decrease until the test condition is considered to be satisfied. After landing onto the MPP, the duty ratio will not be adjusted until the operating condition changes again. In the implementation, the upper- and lower-bounds for the incremental step size need to be defined to avoid extremely drastic changes in the duty ratio. However, the upper-bound is generally fixed and needs to be large enough to permit the rapid tracking of the MPP for a sudden change of the operation condition. The issue with a fixed upper bound is when the array starts tracking the new MPP and it quickly approaches the MPP and lands on the other side of the MPP, the duty ratio needs to be adjusted in the reverse direction using the incremental step, which could be large (due to the factor ACC) and remain large for some time (although the factor DEACC has been applied). At this point, having a large incremental step does not help because it may cause very large fluctuations or overshoot of the voltage before the MPP is achieved. Therefore, the second proposed improvement is, when the sign of the slope changes, the upper-bound is also decreased together with the incremental size. It is also preferred to maintain the upper-bound small nearby the MPP until the MPP is reached. Note, in the implementation of the algorithm, a nominal incremental step size is pre-selected. After the test condition of the slope is considered to be satisfied, the duty ratio will not be changed but the incremental step size might become very small now, which, if not corrected, will cause the very slow response in the beginning of attempting to track the next MPP under a different operating condition. A simple solution is to reset the step size to the nominal value without adjusting the duty ratio when the MPP is considered to be reached. ∆I I ∆I I or δd (k ) = δd (k − 1) × DEACC× \| + \| + \| ∆V V ∆V V if the slope is not small enough (i.e., greater than a preselected constant ε and equation (2) is still not satisfied), and the duty ratio d (k ) = d (k − 1) + δd (k ) . × ACC× \| δd (0) indicates the initial incremental size of the duty ratio, δd max (0) the initial upper bound of the incremental size. δd (k ) and δd max (k ) are the updated (based on conditions discussed above, as indicated in Fig. 1) increment size and the upper boundary of the incremental size. III. MODELING OF PV ENERGY SYSTEMS A. Solar Array In general, a solar array consists of many solar modules connected in series and/or parallel, each module being manufactured by serially connecting a certain number of solar cells. A solar cell is essentially represented by an equivalent electrical circuit as shown in Fig. 2. For illustration purposes, modeling of a solar cell is briefly summarized in this part. Interested readers can find details in other references, e.g., [18]. Note also, the PV terminal output voltage may be very sensitive to the duty ratio, especially for duty ratio close to 0.0 or 1.0. The dc-dc converter input and output voltages should thus be selected such that the duty ratio is in the middle of the duty ratio range. Fig. 2. An equivalent electrical circuit of a solar cell. For the solar cell model in Fig. 2, the following equation can be derived, (3) I PV = I Ph − I d − (VPV + I PV × RS ) / RP where IPV and VPV represent the solar cell terminal output current and voltage, respectively. IPh is the photon current source, Id the diode current. The series resistance RS and shunt resistance RP are used to represent the power losses while the latter can generally be neglected. It is noted in equation (3) that the photon current and the diode current are temperature and irradiance dependent. For given panel temperature T (Kelvin) and irradiance level G (W/m2), IPh and Id can be calculated using the following equations, G G  I Ph \|T ,G I= I SC \|Tref ,Gref [1 + α (T − Tref )] = Ph \|T ,Gref Gref Gref   \|T ,G I 0 \|T ,G ×[exp(q (V + I × RS ) / (nkT )) − 1] Id =   3 −qEg 1 1 T n  ) exp( ( − )) I 0 \| T ,G = I 0 \|Tref ,G × (  Tref nk T Tref Fig. 1. Flowchart of the revised IncCond MPPT algorithm.  I 0 \|Tref ,G I SC \|Tref ,Gref /[exp(qVOCref / (nkTref )) − 1] =  Based on the above discussions, the flow chart of the  proposed modified IncCond algorithm is shown in Fig. 1. In (4) Fig. 1, V(·) and I(·) are the terminal output voltage and current and G represent the reference cell In equation (4), T ref ref of the solar array, which will be adjusted according to the slope calculated by the dc-dc converter, δd (k ) = δd (k − 1) temperature (Tref = 225˚C, i.e., 298 K) and reference irradiance (Gref = 1,000 W/m ) under the standard condition. α is the (5) dV \|V can be obtained from the manufacturers' data dI OCref sheet. After substituting the above parameters into equation (3), the I-V characteristics of the solar cell can be numerically computed for any given cell temperature and irradiance level and then used to represent a solar array consisting of interconnected modules and cells. where A buck-boost converter is used to step up the output dc voltage of a solar array such that a bulky step-up transformer can be avoided and perform the MPPT by controlling the duty ratio of the converter. See, e.g., [19] for more details. IV. SIMULATION RESULTS The revised IncCond algorithm was implemented in an integrated Matlab-based power system simulation software EPTOOL that was developed based on the Power System Toolbox [20]. EPTOOL can be used to perform a transient analysis of the grid under faulted conditions and the solar irradiance and temperature changes. Only MPPT simulation results are presented here to validate the effectiveness of the revised algorithm using a hypothetical solar irradiance profile as the input to solar plant, as tabulated in Table I. The other input, the panel temperature is assumed constant of 25˚C during the cloud transients. A centralized solar PV plant consists of 17170000 solar BP SX 150 panels [21]. The capacity of the PV plant is 433 MW or 4.33 pu (100 base MVA and under standard environmental conditions). A 50-machine-145-bus system was used as an example for carrying out the simulation purpose only. TABLE I: VARIATION OF IRRADIANCE AT A SOLAR PLANT DURING A CLOUD TRANSIENT 0.0 0.2 0.7 0.9 1.2 Time (s) Irradiance ( W / m 2 ) 1000 20 200 300 400 1.5 1.9 2.5 3.0 4.0 Time (s) 500 650 850 990 150 Irradiance ( W / m 2 ) 4.2 4.3 4.4 4.5 4.8 Time (s) Irradiance ( W / m 2 ) 120 20 210 330 340 4.9 Time (s) Irradiance ( W / m 2 ) 350 The conventional IncCond algorithm with a fixed incremental step size (0.001) of duty ratio is first applied and the MPPT is performed every 10 ms. Other parameters are selected as the following: δdmax = 0.01, and ε = 5E-4. PV Output Voltage (kV) qVOCref dV RS = − \|VOC −nkTref / [ I 0 \|Tref ,G q exp( )] ref dI nkTref Simulation results for the solar array terminal output voltage and the deviation of the actual dc output power from the calculated maximum power points are shown in Fig. 3, from which one can observe the persisting oscillations around the MPPs in most of the time, i.e., the MPPs were not truly achieved. The reason for the oscillations is that, as implied in the algorithm description of Section II, the terminal voltage continues to be adjusted. Fig. 3 also indicates that the conventional IncCond algorithm is not able to track the MPP for a rapid variation of the irradiance since the output power deviations from the actual solar power generation are significant for the large change in irradiance at 0.2 s, although for slow variations it can provide acceptable performance. This significant power deficiency around 0.2 s is caused by an inability to adjust the panel voltage rapidly enough to compensate for the large decrement of irradiance level, as can be seen by comparing the top curves in Fig. 3 with those given in Fig. 4-5. This highlights the inefficiencies of selecting control parameters such as the incremental step size and the upper-bound in accordance with conventional MPPT algorithms. These inefficiencies related to the conventional IncCond method can be addressed by the modified algorithm proposed in this paper. 0.65 0.6 0.55 0.5 0.45 PV DC Power Deviation (p.u.) temperature coefficient and ISC is the short-circuit current of the solar cell. These are both constants that can be obtained from manufacturers’ data sheets. I0\|T,G is the reverse saturation current of the diode, n the diode ideality factor, q = 1.602e-19 C the Coulomb constant, k = 1.38e-23 J/K the Boltzmann constant. Eg is band-energy gap (eV) and is given as Eg=1.16–0.000702T2/(T-1108). VOC is the open-circuit voltage of the solar cell. The series resistance can be solved using parameters at reference temperature and irradiance, i.e., 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 Time(s) 3 3.5 4 4.5 5 2 0 -2 -4 -6 Fig. 3. Terminal voltage variation and deviated power output of the PV plant using conventional IncCond algorithm. For the modified MPPT algorithm proposed in Section II, the deceleration and acceleration factors are applied in the modified IncCond algorithm with DEACC = 0.8 and ACC = 1.2 while other parameters remain the same. In the first scenario, the upper bound of the incremental step size of the duty ratio is fixed, i.e., δdmax(k) = 0.01, k = 0, 1, …. As shown in Fig. 4, the oscillations are eliminated quickly as the irradiance changes. Fig. 4 also shows that the revised algorithm with the fixed upper-bound of the step size can quickly make the operating point reach the MPP, as the voltage level becomes quickly stable even after the sudden change of irradiance at time 0.2s. However, relatively large terminal voltage overshoot at changing points of irradiance is now introduced and must be addressed. Fig. 5 shows further improvement of the tracking performance for a second scenario where an adaptive upperbound of the incremental step is used. The overshoot at the change points of the irradiance have been significantly decreased. The output power deviation is also reduced. Simulation also shows that the tracking performance is not sensitive to the associated parameters (e.g., δd), which makes parameter tuning very easy and the modified IncCond algorithm very robust. [3] 0.6 [5] 0.55 0.5 0.45 0.4 PV DC Power Deviation (p.u.) PV Output Voltage (kV) [4] 0.65 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 [7] -2 -4 -6 [8] 0 0.5 1 1.5 2 2.5 Time(s) 3 3.5 4 4.5 5 Fig. 4. Terminal voltage and deviated power output of the array using the modified IncCond algorithm (fixed upper bound of the incremental step size of duty ratio). [9] 0.65 [10] 0.6 0.55 0.5 0.45 0.4 PV DC Power Deviation (p.u.) PV Output Voltage (kV) [6] 2 [11] 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 2 0 [12] -2 -4 -6 0 0.5 1 1.5 2 2.5 Time(s) 3 3.5 4 4.5 5 [13] Fig. 5. Terminal voltage and deviated power output of the array using the modified IncCond algorithm (with adaptive upper bound of the incremental step size of duty ratio). V. CONCLUSIONS A revised IncCond algorithm was presented in this paper for PV generation systems. Compared with traditional IncCond methods, the voltage step change is adaptively determined based on the slope of the P-V curve and the location of the operating points in two consecutive tracking steps such that the PV system can track the rapid change in environmental conditions while the oscillation of the PV system operating points around the MPP can be avoided. In addition, the upper bound of the voltage step change is assigned a factor, DEACC (less than 1) to constrain the step change when a change in the sign of slope is detected.. The simulation results demonstrate the effectiveness of the proposed algorithm. 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Focus: Querying Large Video Datasets with Low Latency and Low Cost Kevin Hsieh†§ Ganesh Ananthanarayanan§ Peter Bodik§ Paramvir Bahl§ Matthai Philipose§ Phillip B. Gibbons† Onur Mutlu∗† † Carnegie Mellon University § Microsoft ∗ ETH Zürich Focus-Opt-Query Focus-Balance Query-all Large volumes of videos are continuously recorded from cameras deployed for traffic control and surveillance with the goal of answering “after the fact” queries: identify video frames with objects of certain classes (cars, bags) from many days of recorded video. While advancements in convolutional neural networks (CNNs) have enabled answering such queries with high accuracy, they are too expensive and slow. We build Focus, a system for lowlatency and low-cost querying on large video datasets. Focus uses cheap ingestion techniques to index the videos by the objects occurring in them. At ingest-time, it uses compression and video-specific specialization of CNNs. Focus handles the lower accuracy of the cheap CNNs by judiciously leveraging expensive CNNs at query-time. To reduce query time latency, we cluster similar objects and hence avoid redundant processing. Using experiments on video streams from traffic, surveillance and news channels, we see that Focus uses 58× fewer GPU cycles than running expensive ingest processors and is 37× faster than processing all the video at query time. Normalized Query Latency arXiv:1801.03493v1 [cs.DB] 10 Jan 2018 Abstract Focus-Opt-Ingest Ingest-all 1 0.8 0.6 0.4 0.2 0 0.03 0.02 0.01 (I=141X, Q=46X) (I=26X, (I=86X, Q=63X) Q=56X) 0 0 0.2 0.4 0.6 0.8 1 Normalized Ingest Cost 0 0.02 0.04 0.06 Normalized Ingest Cost Figure 1: Effectiveness of Focus at reducing both ingest cost and query latency, for an example traffic video. We compare against two baselines: “Ingest-all” that runs ResNet152 on all video frames during ingest, and “Query-all” that runs ResNet152 on all the video frames at query time. By zooming in, we see that Focus (the Focus-Balance point) is simultaneously 86× cheaper than Ingest-all in its GPU consumption and 56× faster than Query-all in query latency, all the while achieving at least 95% precision and recall. (Also shown are two alternatives offering slightly different trade-offs.) ResNet152), using them for video analytics queries is both expensive and slow. Using the ResNet152 classifier at query-time to identify video frames with cars on a month-long traffic video requires 280 GPU hours and costs $250 in the Azure cloud. The latency for running queries is also high. To achieve a query latency of one minute on 280 GPU hours of work would require tens of thousands of GPUs classifying the frames of the video in parallel, which is many orders of magnitude more than what is typically provisioned (few tens or hundreds) by traffic jurisdictions or retail stores. Note that the above cost and latency values are after using motion detection techniques to exclude frames with no moving objects. We believe that enabling low-latency and low-cost querying over large video datasets will make video analytics more useful and open up many new opportunities. A natural approach to enabling low latency querying is doing all classifications with ResNet152 at ingest-time, i.e., on the live videos, and store the results in an index of object classes to video frames. Any queries for specific classes (e.g., cars) will thus involve only a simple index lookup at query-time. There are, however, at least two problems with this approach. First, the cost to index all the video at ingest-time, e.g., $250/month/stream in the 1. Introduction Cameras are ubiquitous, with millions of them deployed by government and private entities at traffic intersections, enterprise offices, and retail stores. Videos from these cameras are continuously recorded [2,7]. One of the main purposes for recording the videos is answering “after-thefact” queries: identify video frames with objects of certain classes (like cars or bags) over many days of recorded video. As results from these queries are used by analysts and investigators, achieving low query latencies is crucial. Advances in convolutional neural networks (CNNs) backed by copious training data and hardware accelerators (e.g., GPUs [13]) have led to high accuracy in the computer vision tasks like object detection and object classification. For instance, the ResNet152 object classifier CNN [38] won the ImageNet challenge that evaluates classification accuracy on 1, 000 classes using a public image dataset with labeled ground truths [63]. For each image, these classifiers return a ranked list of 1, 000 classes in decreasing order of confidence. Despite the accuracy of image classifier CNNs (like 1 above example, is prohibitively high. Second, most of this ingest-time cost is wasteful because typically only a small fraction of recorded videos get queried [16]. Following a theft, the police would query a few days of video from a handful of surveillance cameras, but not all the videos. We present Focus, a system to support low-latency low-cost querying on large video datasets. To address the above drawbacks, Focus has the following goals: (1) low cost indexing of video at ingest-time, (2) providing high accuracy and low latency for queries, and (3) allowing trade offs between the cost at ingest-time against the latency at query-time. As input, the user specifies the ground-truth CNN (or “GT-CNN”, e.g., the ResNet152 classifier) and the desired accuracy of results that Focus needs to achieve relative to the GT-CNN. Focus uses four key techniques – cheap CNNs for ingest, using top-K results from the ingest-time CNN, clustering similar objects, and judicious selection of system and model parameters. First, to make video ingestion cheap, Focus uses compressed and specialized versions of CNNs, to create an ingest-time index of object classes to frames. CNN compression (e.g., [66]) creates new CNNs with fewer convolutional layers and smaller input images. Specialization [35, 65] trains those CNNs on a smaller set of object classes specific to each video stream so that those cheaper CNNs can classify these video-specific objects more accurately. Together, these techniques result in highly efficient CNNs for video indexing. Second, the cheap ingest CNNs, however, are also less accurate than the expensive GT-CNN (like ResNet152), measured in terms of recall and precision. Recall is the fraction of frames in the video that contained objects of the queried class that were actually returned in the query’s results. Precision, on the other hand, is the fraction of frames in the query’s results that contained objects of the queried class. To increase recall, Focus relies on an empirical observation: while the top-most (i.e., most confident) classification results of the cheap and expensive CNNs may not always match, the top-most result of the expensive CNN falls within the top-K results of the cheap CNN. Therefore, at ingest-time, Focus indexes each object with the “top-K” results of the cheap CNN (instead of just the top-most). To increase precision, at query-time, we first filter the objects from the top-K index and then classify the filtered objects with the expensive GT-CNN. Third, to reduce the query-time latency of using the expensive GT-CNN, Focus relies on the significant similarity between objects in videos. For example, a car moving across an intersection will look very similar in consecutive frames. Focus leverages this similarity by clustering the objects at ingest-time, classifying only the cluster centroids with the expensive GT-CNN at query-time, and assigning the same class to all objects in the cluster, thus considerably reducing query latency. In a nutshell, Focus’s ingest-time and query-time operations are as follows. At ingest-time, it classifies the detected objects using a cheap CNN, clusters similar objects, and indexes each cluster centroid using the top-K classification results. At query-time, when the user queries for class X, Focus looks up the ingest index for centroids that match class X and classifies them using the GT-CNN. For centroids that were classified as class X, it returns all objects from the corresponding clusters to the user. Finally, Focus smartly chooses the ingest-time CNN and its parameters to meet user-specified targets on precision and recall. Among the choices that meet the accuracy targets, it allows the user to trade off between the ingest cost and query latency. For example, using a cheaper ingest CNN reduces the ingest cost but increases the query latency as Focus needs to use a larger K for the top-K index to retain the accuracy targets. Focus identifies the “sweet spot” in parameters that sharply improve one of ingest cost or query latency for a small worsening of the other. We built Focus and evaluated it on thirteen 12-hour videos from three domains – traffic cameras, surveillance cameras, and news channels. We compare against two baselines: “Ingest-all” that runs GT-CNN on all video frames during ingest, and “Query-all” that runs GT-CNN on all the video frames at query time. We use ResNet152 as GT-CNN and augment both baselines with motion detection to remove frames with no objects, which is one of the core techniques in a recent prior work, NoScope [44]. Figure 1 shows a representative result, for a traffic video from a commercial intersection. On average, Focus is 58× (up to 98×) cheaper than Ingest-all and 37× (up to 57×) faster than Query-all. This leads to the cost of ingestion coming down from $250/month/stream to $4/month/stream, and the latency to query a 24 hour video dropping from 1 hour to under 2 minutes. See §6 for the full details. We make the following contributions. 1. We formulate the problem of querying video datasets by showing the trade-offs between query latency, ingest cost, and accuracy (precision and recall) of results. 2. We propose techniques to ingest videos with low cost by leveraging compressed and video-specific specialization of CNNs, while retaining high accuracy targets by creating approximate (top-K) indexes. 3. We identify and leverage similarity between objects in a video to cluster them using CNN features and significantly speeding up queries. 4. We propose and build a new end-to-end system to support low-latency, low-cost querying on large video datasets. We show that our system offers new trade-off options between ingestion cost and query latency, as it is significantly cheaper than analyzing all videos frames at ingest time and significantly faster than analyzing queried video frames at query time. 2 2. Background and Motivation can only process 77 images/second even with a high-end GPU (NVIDIA K80 [13]). This makes querying on large video datasets using these CNNs to be slow and costly. There are at least two recent techniques designed to reduce the cost of CNNs. First, compression is a set of techniques aiming to reduce the cost of CNN inference (classification) at the expense of reduced accuracy. Such techniques include removing some expensive convolutional layers [66], matrix pruning [31, 37], and others [42, 62] and can dramatically reduce the classification cost of a CNN. For example, ResNet18, which is a ResNet152 variant with only 18 layers is 8× cheaper. Second, a more recent technique is CNN specialization [35], where the CNNs are trained on a subset of a dataset specific to a particular context, also making them much cheaper. Using the combination of cheap and expensive CNNs is a key facet of our solution, described in §4. We first provide a brief overview of convolutional Neural Networks (CNN), the state-of-the-art approach to detecting and classifying objects in images (§2.1). We then discuss new observations we made about real-world videos, which motivate the design of our techniques (§2.2). 2.1. Convolutional Neural Networks A Convolution Neural Network (CNN) [47] is a specific class of neural networks that works by extracting the visual features in images. During image classification, or “inference”, a CNN takes an input image and outputs the probability of each class (e.g., dog, flower, or car). CNNs are the state-of-the-art method used for many computer vision tasks, such as image classification (e.g., [38,45,71]) and face recognition (e.g., [46, 64]). Input Image Pooling Layers Convolutional + Rectification Layers Fully-Connected Layer Prob.(Apple) ✓ Prob.(Car) Prob.(Orange) Prob.(Cat) … Prob.(Flower) Prob.(Dog) . . . . . . Extracted . . Features . 2.2. Characterizing Real-world Videos We aim to support queries of the form, find all frames in the video that contain objects of class X. We identify some key characteristics of real-world videos towards supporting these queries: (1) large portions of videos can be excluded (§2.2.1), (2) only a limited set of object classes occur in each video (§2.2.2), and (3) objects of the same class have similar feature vectors (§2.2.3). The design of Focus is based on these characteristics. We have analyzed 12 hours of video from six video streams each. The six video stream span across traffic cameras, surveillance cameras, and news channels. (§6.1 provides the details.) We detect the objects in each frame of these videos (using background subtraction [43]), and classify each object with the ResNet152 CNN [38] among the supported 1, 000 object classes. In this paper, we use results from the costly ResNet152 CNN as ground truth. 2.2.1. Excluding large portions of videos. We find considerable potential to avoid processing large portions of videos at query-time. Significant portions of video streams either have no objects at all (as in a garage camera at night) or the objects are stationary (like parked cars). We find that in our video sets, one-third to one-half of the frames fall in these categories. Therefore, queries to any object class would benefit from pre-processing filters applied to exclude these portions of the videos. Even among the frames that do contain objects, not all of them are relevant to a query because each query only looks for a specific class of objects. In our video sets, an object class occurs in only 0.01% of the frames on average, and even the most frequent object classes occur in no more than 16% − 43% of the frames in the different videos. This is because while there are usually some dominant classes (e.g., cars in a traffic camera, people in a news channel), most other classes are rare. Since queries are for specific object classes, there is considerable poten- Figure 2: Architecture of an image classification CNN. Figure 2 illustrates the architecture of an image classification CNN. Broadly, almost all CNNs consist of three key types of network layers: (1) convolutional and rectification layers, which detect visual features from input pixels, (2) pooling layers, which down-sample the input by merging neighboring pixel values, and (3) fully-connected layers, which provide the reasoning to classify the input object based on the outputs from previous layers. The outputs of an image classification CNN are the the probabilities of all object classes, and the class with the highest probability is the predicted class for the input image. The output of the penultimate (i.e., previous-to-last) layer can be considered as “representative features” of the input image [45]. The features are a real-valued vector, with lengths between 512 and 4096 in state-of-the-art classifier CNNs (e.g., [38, 45, 66, 71]). It has been shown that images with similar feature vectors (i.e., small Euclidean distances) are visually similar [22, 23, 45, 58]. The high accuracy of CNNs comes at a cost: inferring (or classifying) using state-of-the-art CNNs to classify objects in images requires significant computational resources. This is because the higher accuracy of CNNs comes from using deeper architectures (i.e., more layers) to obtain better visual features. For instance, ResNet152 [38], the winner of the ImageNet competition [63] in 2015, has been trained to classify across 1000 classes from the ImageNet dataset using 152 layers, but 3 CDF (number of objects) 1 0.9 since they are specifically trained to extract visual features for classification. We verify the robustness of feature vectors using the following analysis. In each video, for each object i, we find its nearest neighbor j using feature vectors from the cheap ResNet18 CNN and compute the fraction of object pairs that belong to the same class. This fraction is over 99% in each of our videos, which shows using feature vectors from cheap CNNs can potentially help identify duplicate objects. 95% of objects 0.8 Auburn Lausanne CNN 0.7 0.6 Jackson Hole Sittard MSNBC 0.5 0% 2% 4% 6% 8% Percentage of ResNet152's 1000 classes 10% Figure 3: CDF of frequency of object classes. The x-axis is the fraction of classes out of the 1000 recognized by ResNet152 (truncated to 10%). 3. Overview of Focus The goal of Focus is to index live video streams by the object classes occurring in them and enable answering “after-the-fact” queries later on the stored videos of the form find all frames that contain objects of class X. Optionally, the query can be restricted to a subset of cameras and a time range. Such a query formulation is the basis for many widespread applications and could be used either on its own (such as for detecting all cars or bicycles in the video) or used as a basis for further processing (e.g., finding all collisions between cars and bicycles). Focus is designed to work with a wide variety of current and future CNNs. At system configuration time, the user (system administrator) provides a ground-truth CNN (GT-CNN), which serves as the accuracy baseline for Focus, but is far too costly to run on every video frame. Through a sequence of techniques, Focus provides nearlycomparable accuracy but at greatly reduced cost. By default, and throughout this paper, we use the ResNet152 image classifier as the GT-CNN. Because the acceptable target accuracy is applicationdependent, Focus permits the user to specify the target, while providing reasonable defaults. Accuracy is specified in terms of precision, i.e., fraction of frames output by the query that actually contain an object of class X according to GT-CNN, and recall, i.e., fraction of frames that contain objects of class X according to GT-CNN that were actually returned by the query. The lower the target, the greater the cost-savings provided by Focus. Even for high targets such as 95%–99%, Focus is able to achieve order-of-magnitude or more cost savings. Figure 4 presents the design of Focus. • At ingest-time (left part of Figure 4), Focus classifies objects in the incoming video frames and extracts their feature vectors. To make this step cheap, it uses a highly compressed and specialized version of the GT-CNN model (IT1 in Figure 4). Focus then clusters objects based on their feature vectors (IT2 ) and assign to each cluster the top K most likely classes these objects belong to (based on classification confidence of the ingest CNN); (IT3 ). It creates a top-K index, which maps each class to the set of object clusters (IT4 ). The top-K index is the output of Focus’ ingest-time processing of videos. • At query-time (right part of Figure 4), when the user tial in indexing frames by the classes of objects. 2.2.2. Limited set of object classes in each video. We next focus on the classes of objects that occur in each of the videos and the disparity in frequency among them. Most video streams have a limited set of objects because each video has its own context (e.g., traffic cameras can have automobiles, pedestrians or bikes but not airplanes). It is rare that a video stream contains objects of all the classes recognized by state-of-the-art classifier CNNs. Figure 3 shows the cumulative distribution function (CDF) of the frequency of object classes in our videos (as classified by ResNet152). We make two observations. First, objects of only 22% − 33% (not graphed) of the 1, 000 object classes occur in the less busy videos (Auburn, Jackson Hole, Lausanne, and Sittard). Even in the busier videos (CNN, and MSNBC), objects of only 50% − 69% of the classes appear. Also, there is little overlap between the classes of objects among the different videos. On average, the Jaccard indexes [72] (i.e., intersection over union) between the videos based on their object classes is only 0.46. Second, even among the object classes that do occur, a small fraction of classes disproportionately dominate. Figure 3 shows that 3% − 10% of the most frequent object classes cover ≥ 95% of the objects in each video stream. This suggests that for each video stream, we can automatically (i) determine its most frequently occurring classes and (ii) train efficient CNNs specialized for classifying these classes (§2.1). 2.2.3. Feature vectors for finding duplicate objects. Objects moving in the video often stay in the frame for several seconds; for example, a pedestrian might take a minute to cross a street. Instead of classifying each instance of the same object across the frames, we would like to inexpensively find duplicate objects and only classify one of them using a CNN (and apply the same label to all duplicates). Thus, given n duplicate objects, this requires only one CNN classification operation instead of n. Comparing pixel values across frames is an obvious choice to identify duplicate objects, however, they turn out to be highly sensitive to even small changes in the camera’s real-time view of an object. Instead, feature vectors extracted from the CNNs are much more robust 4 Frames Frames Frames Object feature vectors Objects IT1 Specialized, Compressed CNN Query-time Ingest-time CNN specialization IT3 IT2 QT1 Object clusters Object top-K classes Frames with objects of class X Querying for class X QT4 Matching clusters for X Centroid objects IT4 Top-K index QT2 QT3 GT-CNN Figure 4: Overview of Focus. queries for a certain class X (QT1 ), Focus retrieves the matching clusters from the top-K index (QT2 ), runs the centroids of the clusters through GT-CNN (QT3 ), and returns all frames from the clusters whose centroids were classified by GT-CNN as class X (QT4 ). The top-K ingest index is a mapping between the object class to the clusters. Specifically, object class → hcluster IDi cluster ID → [centroid object, hobjectsi in cluster, hframe IDsi of objects] We next explain how Focus’ key techniques keep ingest cost and query latency low while also meeting the userspecified accuracy targets. 1) Cheap Ingest-time CNN: Focus makes indexing at ingest-time cheap by compressing and specializing the GT-CNN model for each video stream. (i) Compression of CNN models [31, 37, 42, 62, 66] uses fewer convolutional layers and other approximation techniques (§2.1). (ii) Specialization of CNNs [35, 65] uses the observation that a specific video stream contains only a small number of object classes and their appearance is more constrained than in a generic video (§2.2.2). Both techniques are done automatically and together result in ingest-time CNN models that are up to 98× cheaper than GT-CNN. 2) Top-K ingest index: The cheap ingest-time CNNs are less accurate, i.e., their top-most results do not often match the top-most classifications of GT-CNN. Therefore, to keep the recall high, Focus associates each object with the top-K classification results of the cheap CNN, instead of just its top-most result. Increasing the K increases recall because the top-most result of GT-CNN often falls within the ingest-time CNN’s top-K results. At querytime, Focus uses the GT-CNN to remove objects in this larger set that do not match the class, to regain precision lost by including all the top-K. 3) Clustering similar objects: A high value of K at ingest-time increases the work to do at query time, thereby increasing query latency. To reduce this overhead, Focus clusters similar objects at ingest-time using feature vectors from the ingest-time CNN. In each cluster, at querytime, we run only the cluster centroid through GT-CNN and apply the classified result from the GT-CNN to all objects in the cluster. Thus, if the objects are not tightly clustered, clustering can reduce precision and recall. 4) Trading off ingest vs. query costs: Focus automatically chooses the cheap CNN, its K, and specialization and clustering parameters to achieve the desired precision and recall targets. These choices also help Focus trade off between the work done at ingest-time and query-time. For instance, to save ingest work, Focus can select a cheaper ingest-time CNN, and then counteract the resultant loss in accuracy by running the expensive GT-CNN on more objects at query time. Focus chooses its parameters so as to offer a sharp improvement in one of the two costs for a small degradation in the other cost. Because the desired trade-off point is application-dependent, Focus provides users with a choice of three options: ingest-optimized, query-optimized, and balanced (the default). Note that while our explanation is anchored on image classification CNNs, the architecture of Focus is generally applicable to all existing CNNs (e.g., face recognition). Techniques that we use for CNN compression [42, 62] and specialization [35], and feature extraction from the CNNs are all broadly applicable to all CNNs. 4. Video Ingest & Querying Techniques In this section, we describe the main techniques used in Focus: using cheap CNN models at ingest-time (§4.1), identifying similar objects and frames to save on redundant CNN processing (§4.2), and specializing the CNNs to the specific videos that are being analyzed (§4.3). §4.4 describes setting parameters in Focus. 4.1. Cheap Ingestion Focus indexes the live videos at ingest-time to reduce the query-time latency. We perform object detection on each frame, typically an inexpensive operation, and then will classify the extracted objects using ingest-time CNNs that are far cheaper than the ground-truth GT-CNN. We use these classifications to index objects by class. Cheap ingest-time CNN: As noted earlier, the user provides Focus with a GT-CNN. Optionally, the user can also provide other classifier architectures to be used in Focus’ search for cheap CNNs, such as AlexNet [45] and 5 CheapCNN 1 (7X) 100% CheapCNN 2 (28X) CheapCNN 3 (58X) class. The selection of the cheap ingest-time CNN model (CheapCNNi ) and the K value (for the top-K results) has a significant influence on the recall of the outputs produced. Lower values of K reduce recall, i.e., Focus will miss returning frames that contain the queried objects. At the same time, higher values of K increase the number of objects to classify with GT-CNN at query time to keep precision high, and hence adds to the latency. We defer to §4.4 on how Focus sets these parameters as they have to be jointly set with other parameters in §4.2 and §4.3. Recall 80% 60% 40% 20% 0% 10 20 60 100 Number of selected results (K) 200 Figure 5: Effect of K on recall for three cheap CNNs. The number within the parenthesis indicates how much cheaper the model is compared to our GT-CNN, ResNet152. VGG [66] (which vary in their resource costs and accuracies). Starting from these user-provided CNNs, Focus applies various levels of compression, such as removing convolutional layers and reducing the input image resolution (§2.1). This results in a large set of CNN options for ingestion, {CheapCNN1 , . . . , CheapCNNn }, with a wide range of costs and accuracies. Top-K Ingest Index: To keep recall high, Focus indexes each object using the top K object classes from CheapCNNi ’s output, instead of using just the top-most class. Recall from §2.1 that the output of the CNN is a list of object classes in descending order of confidence. We empirically observe that the top-most output of the expensive GT-CNN is often contained within the top-K classes output by the cheap CNN (for a small value of K relative to the 1, 000 classes recognized by the CNNs). Figure 5 plots the effect of K on recall on one of our video streams, lausanne (see §6.1). The three models in the figure are ResNet18 [38], and ResNet18 with 3 and 5 layers removed; additionally, the input images were rescaled to 224, 112, and 56 pixels, respectively. All models were retrained on their original training data (ImageNet [63]). We make two observations. First, we observe steady increase in recall with increasing K, for all three CheapCNNs. As the figure shows, CheapCNN1 , CheapCNN2 , and CheapCNN3 reach 90% recall when K ≥ 60, K ≥ 100, and K ≥ 200, respectively. Note that all these models recognize 1000 classes, so even K = 200 represents only 20% of the possible classes. Second, there is a trade-off between different models – the cheaper they are, the lower their recall with the same K. Overall, we conclude that by selecting the appropriate K, Focus can achieve the target recall. Focus creates the top-K index of an object’s top-K classes output by CheapCNNi at ingest-time. While filtering for objects of the queried class X using the top-K index (with the appropriate K) will have a high recall, it will have very low precision. Since we associate each object with K classes (while it has only one true class), the average precision is only 1/K. Thus, at query time, to keep the precision high, Focus determines the actual class of objects from the top-K index using the expensive GT-CNN and only return objects that match the queried 4.2. Redundancy Elimination At query time, Focus retrieves the objects likely matching the user-specified class from the top-K index and infers their actual class using the GT-CNN. This would ensure precision of 100%, but could cause significant latency at query time. Even if this inference is parallelized across many GPUs, it would still incur a large cost. Focus uses the following observation to reduce this cost: if two objects are visually similar, their feature vectors would be closely aligned and they would likely be classified as the same class (e.g., “cars”) by the GT-CNN model (§2.2.3). Focus clusters objects that are similar, invokes the expensive GT-CNN only on the cluster centroids, and assigns the centroid’s label to all objects in each cluster. Doing so dramatically reduces the work done by the GTCNN classifier at query time. Focus uses the feature vector output by the previous-to-last layer of the cheap ingest CNN (see §2.1) for clustering. Note that Focus clusters the objects in the frames and not the frames as a whole. The key questions regarding clustering are how do we cluster (algorithm) and when do we cluster (system). We discuss both these key questions below. Clustering Heuristic: We require two properties in our clustering technique. First, given the high volume of video data, it should be a single-pass algorithm to keep the overhead low, as the complexities of most clustering algorithms are quadratic. Second, it should make no assumption on the number of clusters and adapt to outliers in data points on the fly. Given these requirements, we use the following simple approach for incremental clustering, which has been well-studied in the literature [27, 55]. We put the first object into the first cluster c1 . To cluster a new object i with a feature vector fi , we assign it to the closest cluster c j if c j is at most distance T away from fi . However, if none of the clusters are within a distance T , we create a new cluster with centroid at fi , where T is a distance threshold. We measure distance as the L2 norm [10] between cluster centroid and object feature vector. We keep the number of clusters at a constant M by removing the smallest ones and storing their data in the top-K index. Using this algorithm, we can keep growing the popular clusters (such as similar cars), while keeping 6 the complexity as O(Mn), which is linear to n, the total number of objects. Clustering can reduce both precision and recall depending on parameter T . If the centroid is classified by GT-CNN as the queried class X but the cluster contains another object of a different class, it reduces precision. If the centroid is classified as a class different than X but the cluster has an object of class X, it reduces recall. We discuss setting T in §4.4. Clustering at Ingest vs. Query Time: Focus clusters the objects at ingest-time rather than at query-time. Clustering at query-time would involve storing all feature vectors, loading them for objects filtered from the ingest index and then clustering them. Instead, clustering at ingest time creates clusters right when the feature vectors are created and only stores the cluster centroids in the top-K index. This makes the query-time latency much lower and also reduces the size of the top-K index. We observe that the ordering of indexing and clustering operations is mostly commutative in practice and has little impact on result accuracy (we do not present these results due to space constraints). We therefore use ingest-time clustering due to its latency and storage benefits. Pixel Differencing of Objects: While clustering primarily reduces work done at query-time (number of objects to be classified by the GT-CNN), Focus also employs pixel differencing among objects in adjacent incoming frames to reduce ingest cost. Specifically, if two objects have very similar pixel values, it only runs the cheap CNN on one of them and assign them both to the same cluster in our top-K index. curacy on video streams, while removing 1/3 of the convolutional layers and making the input image 4× smaller in resolution. This leads to the specialized CheapCNNi being 10× cheaper than even the generic CheapCNNi . Since the specialized CNN classifies across fewer classes, they are more accurate, which allows Focus to select a much smaller K (for the top-K ingest index) to meet the desired recall. We find that specialized models can use K = 2 or 4, much smaller than the typical K = 60 ~ 200 for the generic cheap CNNs (Figure 5). Smaller K directly translates to fewer objects that have to be classified by GT-CNN at query time, thus reducing latency. Model Retraining: On each video stream Focus periodically obtains a small sample of video frames and classifies their objects using GT-CNN to estimate the ground truth of distribution of object classes for the video (similar to Figure 3). From this distribution, Focus selects the most frequently occurring Ls object classes to retrain new specialized models. As we saw in §2.2.2, there is usually a “power law” in the distribution of classes – just a handful of classes account for a dominant majority of the objects – thus, low values of Ls usually suffice.1 Specialization is also based off a family of CNN architectures (such as ResNet [38], AlexNet [45], and VGG [66]) with different number of convolution layers, similar to §4.1. Specialization adds to the set of options available for ingest CNNs ({CheapCNN1 , ..., CheapCNNn } in §4.1), and Focus picks the best model (CheapCNNi ) and the corresponding K for the index. “OTHER” class: While Focus specializes the CNN towards the most frequently occurring Ls classes, we also want to support querying of the less frequent classes. For this purpose, Focus includes an additional class called “OTHER” in the specialized model.2 Being classified as OTHER simply means not being one of the Ls classes. At query time, if the queried class is among the OTHER classes of the ingest CNN’s index, Focus extracts all the clusters that match the OTHER class and classifies their centroids through the GT-CNN model. The parameter Ls (for each stream) exposes the following trade-off. Using a small Ls allows us to train a simpler model with cheaper ingest cost and lower query-time latency for the popular classes, however, it also leads to a larger fraction of objects falling in the OTHER class; querying for them will be expensive because all those objects will have to be classified by the GT-CNN. Using a larger value of Ls , on the other hand, leads to a more expensive ingest and query-time models, but cheaper querying for the OTHER classes. We select Ls next in §4.4. 4.3. Video-specific Specialization of CNNs Recall from §4.1 that Focus uses a cheap ingest-time CNN, CheapCNNi to index object classes. Focus further reduces its cost by specializing the ingest-time CNN model to each video stream. Model specialization benefits from two properties of objects in each video stream. First, while object classification models are trained to differentiate between thousands of object classes, many video streams contain only a small number of classes (§2.2.2). Second, objects in a specific stream are often visually more constrained than objects in general (say, compared to the ImageNet [63] dataset). The cars and buses that occur in a specific traffic camera have much less variability, e.g., they have very similar angle, distortion and size, than a generic set of vehicles. Instead of trying to differentiate among thousands of object classes, differentiating among just (say) fifty classes and in a specific camera’s video is a much simpler task, requiring simpler image features and smaller image resolutions. As a result, the specialized models are smaller and more accurate [35]. For example, by retraining a stream-specific CheapCNNi , we can achieve similar ac- 1 Specialized CNNs can be retrained quickly on a small dataset. Retraining is relatively infrequent and done once every few days. 2 Since there will be considerably fewer objects in the video belonging to the OTHER class, we proportionally re-weight the training data to contain equal number of objects of all the classes. 7 Normalized Query Latency 4.4. Balancing Accuracy, Latency, and Cost Focus’ goals of high accuracy, low ingest cost and low query latency are impacted by the parameters in Focus’ techniques – K, the number of top results from the ingesttime CNN to index an object; Ls , the number of popular object classes we use to create a specialized model; CheapCNNi , the specialized ingest-time cheap CNN; and T , the distance threshold for clustering objects. The effect of these four parameters is intertwined. All the four parameters impact ingest cost, query latency, and recall, but only T impacts precision. This is because we apply the cluster centroid’s classification by GT-CNN to all the objects in its cluster. Thus, if the clustering is not tight (i.e., high value of T ), we lose precision. Parameter Selection: Focus selects parameter values per video stream. It samples a representative fraction of frames of the video stream and classifies them using GT-CNN for the ground truth. For each combination of parameter values, Focus computes the expected precision and recall (using the ground truths generated by GT-CNN) that would be achieved for each of the object classes. To navigate the combinatorial space of options for these parameters, we adopt a two-step approach. In the first step, Focus chooses CheapCNNi , Ls and K using only the recall target. In the next step, Focus iterates through the values of T , the clustering distance threshold, and only select values that meet the precision target. Trading off Ingest Cost and Query Latency: Among the combination of values that meet the precision and recall targets, the selection is based on balancing the ingest- and query-time costs. For example, picking a model CheapCNNi that is more accurate will have higher ingest cost, but lower query cost because we can use a lower K. Using a less accurate CheapCNNi will have the opposite effect. Focus identifies “intelligent defaults” that sharply improve one of the two costs for a small worsening of the other cost. Figure 6 illustrates the parameter selection based on the ingest cost and query latency for one of our video streams (auburn_c). The figure plots all the viable “configurations” (i.e., set of parameters that meet the precision and recall target) based on their ingest cost (i.e., cost of CheapCNNi ) and query latency (i.e., the number of clusters according to K, Ls , T ). We first draw the Pareto boundary [17], which is the set of configurations that cannot improve one metric without worsening the other. Focus can discard all the other configurations because at least one point on the Pareto boundary is better than them in both metrics. Focus balances between the ingest cost and query latency (Balance in Figure 6) by selecting the configuration that minimizes the sum of ingest and query cost (measured in total GPU cycles). Focus also allows for other configurations based on the application’s preferences and query rates. Opt-Ingest 0.035 0.030 0.025 0.020 0.015 0.010 Opt-Ingest Opt-Query Balance 0.00 0.05 0.10 Normalized Ingest Cost 0.15 Figure 6: Parameter selection based on trading off ingest cost and query latency. The ingest cost is normalized to ingesting all objects with ResNet152, while the query latency is normalized to the time for querying all objects with ResNet152. The dashed line is the Pareto boundary. minimizes the ingest cost and is applicable when the application expects most of the video streams to not get queried (such as a surveillance cameras), as this policy also minimizes the amount of wasted ingest work. On the other hand, Opt-Query minimizes query latency even if it incurs a heavy ingest cost. Such flexibility allows Focus to fit different applications. 5. Implementation Details We describe the key aspects in Focus’s implementation. Worker Processes. Focus’s ingest-time work is distributed across many machines, with each machine running one worker process for each video stream’s ingestion. The ingest worker receives the live video stream, and extracts the moving objects (using background subtraction [81]); it is extensible to plug in any other object detector. The detected objects are sent to the ingest-time CNN to infer the top-K classes and the feature vectors. The ingest worker uses the features to cluster objects in its video stream and stores the top-K index in MongoDB [12] for efficient retrieval at query-time. Worker processes also serve queries by fetching the relevant frames off the top-K index database and classifying the objects with GT-CNN. We parallelize a query’s work across many worker processes if resources are idle. GPUs for CNN classification. The cheap CNNs and GTCNN execute on GPUs (or other hardware accelerators for CNNs) which could either be local on the same machine as the worker processes or “disaggregated” on a remote cluster. This detail is abstracted away from our worker process and it seamlessly works with both designs. Dynamically adjusting K at query-time. As an enhanced technique, we can select a new Kx ≤ K at query time and only extract clusters where class X appears among the top-Kx classes; this will result in fewer clusters and thus also lower query-time latency. This technique is useful in two scenarios: 1) some classes might be very accurately classified by the cheap CNN; using a lower Kx will still meet the user-specified accuracy, yet will result in much lower latency; 2) if we want to retrieve only some objects of class X, we can use very low Kx to quickly retrieve them. If more objects are required, we can increase 8 Table 1: Video dataset characteristics Kx to extract a new batch of results. Type Description A commercial area intersection in the City of Auburn [6] A residential area intersection auburn_r AL, USA in the City of Auburn [5] A downtown intersection in city_a_d USA City A3 Traffic A residential area intersection city_a_r USA in City A3 A road-side camera in the City bend OR, USA of Bend [8] A busy intersection (Town jacksonh WY, USA Square) in Jackson Hole [9] A video stream rotates among church_st VT, USA cameras in a shopping mall (Church Street Marketplace) [3] A pedestrian plazalatency (Place de Surveillance lausanne Switzerland la Palud) in Lausanne [11] A bookshop street in the oxford England University of Oxford [15] sittard Netherlands A market square in Sittard [4] News channel cnn USA News News channel foxnews USA News channel msnbc USA 6. Evaluation We evaluate the Focus prototype with more than 150 hours of videos from 13 real video streams that span across traffic cameras, surveillance cameras, and news channels. Our highlights are: 1. On average, Focus is simultaneously 58× (up to 98×) cheaper than the Ingest-all baseline in its GPU consumption and 37× (up to 57×) faster than the Query-all baseline in query latency, all the while achieving at least 95% precision and recall (§6.2, §6.3). 2. Focus provides a rich trade-off space between ingest cost and query latency. Among the video streams, the ingest cost is up to 141× cheaper than the Ingestall baseline (and reduces query latency by 46×) if optimizing for low-cost ingest. The query latency is reduced by up to 66× (with 11× cheaper ingest) if optimizing for query latency (§6.4). 3. Focus is effective under broad conditions such as high accuracy targets (one order-of-magnitude savings even for 99% accuracy target, §6.5) and various frame sampling rates (30 fps-1 fps, §6.6). Name Location auburn_c AL, USA in a one-second segment of video if the GT-CNN reports such class in 50% of the frames in that segment. We use this criteria as our ground truth because our GT-CNN (ResNet152) sometimes gives different answers to the exact same object in consecutive frames, and this criteria can effectively eliminate these random, erroneous results. We set our default accuracy target as 95% recall and 95% precision. We also evaluate the results with other accuracy targets such as 97%, 98% and 99% (§6.5). Note that in most practical cases, only one of the two metrics (recall or accuracy) needs to be high. For example, an investigator cares about high recall, and looking through some irrelevant results is an acceptable trade-off. By setting both targets high, we are lower bounding the performance improvements that Focus can achieve. 6.1. Setup Software Tools. We use OpenCV 3.2.0 [14] to decode the videos into frames, and then use the built-in background subtraction algorithm [43] in OpenCV to extract moving objects from video frames. We use background subtraction instead of object detector CNNs (e.g., YOLOv2 [59] or Faster R-CNN [60]) to detect objects because: (1) running background subtraction is orders of magnitude faster than running these CNNs, and (2) background subtraction can detect moving objects more reliably, while object detector CNNs usually have difficulties on small objects [52]. Nonetheless, our system can seamlessly use object detector CNNs as well. We run and train CNNs with Microsoft Cognitive Toolkit 2.1 [54], an open-source deep learning system. Video Datasets. We evaluate 13 live video streams that span across traffic cameras, surveillance cameras, and news channels. We evaluate each video stream for 12 hours, which evenly cover day time and night time. Table 1 summarizes the video characteristics. By default, we evaluate each video at 30 fps and also evaluate the sensitivity to other frame rates (§6.6). In some figures we only show a representative sample of 9 cameras to improve legibility. Accuracy Target. We use ResNet152, a state-of-the-art CNN, as our ground-truth CNN (GT-CNN). We evaluate all extracted objects with the GT-CNN and use the results as the correct answers. We define a class present Baselines. We use two baselines for comparisons: (1) Ingest-all, the baseline system that uses GT-CNN to analyze all objects at ingest time, and stores the inverted index for query; and (2) Query-all, the baseline system that simply extracts objects at ingest time, and uses GT-CNN to analyze all the objects that fall into the query interval at query time. Note that we strengthen both baselines with basic motion detection (background subtraction). Therefore, the baselines do not run any GT-CNN on the frames that have no moving objects. Note that not running GTCNN on frames with no moving objects is one of the core techniques in the recent NoScope work [44]. Metrics. We use two performance metrics. The first metric is ingest cost, which is the GPU time to ingest each video. The second metric is query latency, which is the latency for an object class query. Specifically, for each video stream, we evaluate all dominant object classes and take the average of their latencies. (Querying for non-dominant “OTHER” classes is much cheaper than querying popular classes, and would skew the results because there are far more such classes; thus, we focus on 3 The video streams are obtained from real and operational traffic cameras in a city. We mask the city name for anonymity. 9 Traffic Surveillance News 33X Avg 37X Surveillance msnbc cnn foxnews sittard 11X oxford 22X (auburn_c, city_a_d and jacksonh), normal intersections or roads (auburn_r and city_a_r, bend), rotating cameras (church_st), busy plazas (lausanne and sittard), a university street (oxford), and different news channels (cnn, foxnews, and msnbc). Among these videos, the gains in query latency are smaller for relatively less busy videos (auburn_r, bend, lausanne, and oxford). This is because these videos are dominated by fewer object classes, and Focus has more work (i.e., analysis using GT-CNN) to do at query time for these classes. We conclude that the core techniques of Focus are general and effective on a variety of real-world videos. 56X 58X msnbc 43X 57X 57X 55X lausanne jacksonh 41X church_st bend 24X 30X 64X cnn sittard oxford lausanne jacksonh city_a_d 52X 52X city_a_r 31X city_a_d 56X auburn_r 80 60 40 20 0 auburn_c Faster than Query-all by (factor) Traffic 52X 48X church_st auburn_r bend 44X 44X 44X 53X city_a_r 58X foxnews 98X 94X 86X auburn_c Cheaper than Ingest-all by (factor) 100 80 60 40 20 0 News 6.3. Effect of Different Focus Components Figure 8 shows the breakdown of ingest-time cost and query latency across different design points of Focus: (1) Compressed model, which applies a generic compressed model for indexing at ingest time, (2) Compressed + Specialized model, which uses a per-stream specialized and compressed model for indexing, and (3) Compressed + Specialized model + Clustering, which adds feature-based clustering at ingest time to reduce redundant work at query time. All of the above include the top-K index and using GT-CNN at query-time, and achieve the same accuracy of 95%. Three main observations are in order. First, generic compressed models provide benefits for both ingest cost and query latency, but they are not the major source of improvement. This is because the accuracy of a generic compressed model degrades significantly when we remove convolutional layers. In order to retain the accuracy target, we need to choose relatively expensive compressed models (CheapCNNi ) and a larger K, which incur higher ingest cost and query latency. Second, specializing the model (in addition to compressing it) greatly reduces ingest cost and query latency. Because of fewer convolutional layers and smaller input resolution, our specialized models are 7× to 71× cheaper than the GT-CNN, while retaining the accuracy target for each video streams. Running a specialized model at ingest time speeds up query latency by 5× to 25× (Figure 8b). Third, clustering is a very effective technique to further reduce query latency with unnoticeable costs at ingest time. As Figure 8b shows, using clustering (on top of a specialized compressed model) reduces the query latency by up to 56×, significantly better than just running a specialized model at ingest time. This gain comes with a negligible cost (Figure 8a), because we run our clustering algorithm (§4.2) on the CPUs of the ingest machine, which is fully pipelined with the GPUs that run the specialized CNN model. Avg Figure 7: (Top) Focus ingest cost compared to Ingest-all. (Bottom) Focus query latency compared to Query-all. the popular ones.) Both metrics include only GPU time spent classifying images and exclude other (CPU) time spent decoding video frames, detecting moving objects, recording and loading video, and reading and writing to the top-K index. We focus solely on GPU time because when the GPU is involved, it is the bottleneck resource. The query latency of Ingest-all is 0 and the ingest cost of Query-all is 0. Experiment Platform. We run the experiments on our local cluster. Each machine in the cluster is equipped with a state-of-the-art GPU (NVIDIA GTX Titan X), 16-core Intel Xeon CPU (E5-2698), 64 GB RAM, a 40 GbE NIC, and runs 64-bit Ubuntu 16.04 LTS. 6.2. End-to-End Performance We first show the end-to-end performance of Focus by showing its ingest cost and query latency when Focus aims to balance these two metrics (§4.4). Figure 7 compares the ingest cost of Focus with Ingest-all and the query latency of Focus with Query-all. We make two main observations. First, Focus significantly improves query latency with a very small ingest cost. Focus makes queries by an average of 37× faster than Query-all with a very small cost at ingest time (an average of 58× cheaper than Ingest-all). With a 10-GPU cluster, the query latency on a 24-hour video goes down from one hour to less than two minutes. The processing cost of each video stream also goes down from $250/month to $4/month. This shows that Focus can strike a very good balance between these two competing goals very effectively. Second, Focus is effective across different video streams with various characteristics. It makes queries 11× to 57× faster with a very small ingest time cost (48× to 98× cheaper) across busy intersections 6.4. Ingest Cost vs. Query Latency Trade-off One of the interesting features of Focus is the flexibility to tune its system parameters to achieve different application 10 (a) Ingest cost 0 cnn Opt-I Opt-Q Opt-I Opt-Q Opt-I auburn_c city_a_r jacksonh church_st lausanne sittard Opt-Q Opt-I Opt-Q Opt-I Opt-Q Opt-I Opt-Q Opt-I Opt-Q 0 Opt-I 20 50 Opt-Q 40 100 Opt-I 60 Ingest Cheaper by Query Faster by 150 Opt-Q Improvements (factor) Compressed model + Specialized model + Clustering 80 auburn_c city_a_r jacksonh church_st lausanne sittard cnn foxnews msnbc Avg auburn_c city_a_r jacksonh church_st lausanne sittard cnn foxnews msnbc Avg Faster than Query-all by (factor) Cheaper than Ingest-all by (factor) 100 80 60 40 20 0 Compressed model + Specialized model + Clustering foxnews msnbc Figure 9: Trade-offs between ingest cost and query latency three higher targets, 97%, 98%, and 99%. As the figures show, with higher accuracy targets, the ingest costs are about the same, and the improvement of query latency decreases. Focus keeps the ingest cost similar (62× to 64× cheaper than the baseline) because it still runs the specialized and compressed CNN at ingest time. However, when the accuracy targets are higher, Focus needs to select more top-K classification results, which increases the work at query time. On average, the query latency of Focus is faster than Query-all by 15×, 12×, and 8× with respect to 97%, 98%, and 99% accuracy targets. We conclude that the techniques of Focus can achieve higher accuracy targets with significant improvements on both ingest cost and query latency. (b) Query latency Figure 8: Effect of different Focus components Ingest cheaper by (factor) goals (§4.4). Figure 1 from §1 depicted three alternative settings for Focus that illustrate the trade-off space between ingest cost and query latency, using the auburn_c video stream: (1) Focus-Opt-Query, which optimizes for query latency by increasing ingest cost, (2) Focus-Balance, which is the default option that balances these two metrics (§4.4), and (3): Focus-Opt-Ingest, which is the opposite of Focus-Opt-Query. The results are’shown relative to the two baselines. The chart at the right of the figure is the zoomed-in region that covers the three settings of Focus, and each data label (I, Q) indicates its ingest cost is I× cheaper than Ingest-all, while its query latency is Q× faster than Query-all. As Figure 1 shows, Focus offers very good options in the trade-off space between ingest cost and query latency. Focus-Opt-Ingest achieves 141× cheaper cost than Ingestall to ingest the video stream, and makes the query 46× faster than doing nothing at ingest (Query-all). On the other hand, Focus-Opt-Query reduces query latency by 63× with a relatively higher ingest cost, but it is still 26× cheaper than Ingest-all. As they are all good options compared to the baselines, such flexibility allows a user to tailor Focus for different contexts. For example, a traffic camera that requires fast turnaround time for queries can use FocusOpt-Query, while a surveillance video stream that will be queried very rarely would choose Focus-Opt-Ingest to reduce the amount of wasted ingest cost. Figure 9 shows the (I, Q) values for both Focus-OptIngest (Opt-I) and Focus-Opt-Query (Opt-Q) for the representative videos. As the figure show, the trade-off flexibility exists among all the other videos. On average, Focus-Opt-Ingest spends only 95× cheaper ingest cost to provide 35× query latency reduction. On the other hand, Focus-Opt-Query makes queries 49× faster with a higher ingest cost (15× cheaper than Ingest-all). We conclude that Focus provides good flexibility between ingest cost and query latency, and makes it a better fit in different contexts. 100 95% 97% 98% 99% 10 1 Query faster by (factor) Figure 10: Ingest cost sensitivity to accuracy target 100 95% 97% 98% 99% 10 1 Figure 11: Query latency sensitivity to accuracy target 6.6. Sensitivity to Frame Sampling A common approach to reduce the video processing time is to use frame sampling (i.e., periodically select a frame to process). However, not all applications can use frame sampling because it can miss objects that show up and disappear within a frame sampling window. As the frame sampling rate is an application dependent choice, we study the sensitivity of Focus’s performance to different frame rates. Figures 12 and 13 show the ingest cost and query latency of Focus at different frame rates (i.e., 30 fps, 10 fps, 5 fps, and 1 fps) compared to Ingest-all and Query-all, respectively. We make two observations. First, the ingest cost reduction is roughly the same across the different frame rates. On average, the ingest 6.5. Sensitivity to Accuracy Target Figures 10 and 11 illustrate the improvements of ingest cost and query latency of Focus compared to the baselines under different accuracy targets. Other than the default 95% accuracy target (recall and precision), we evaluate 11 Ingest cheaper by (factor) 100 30fps 10fps 5fps 1fps 7. Related Work 10 To our best knowledge, Focus is the first system that offers low-cost, low-latency,and high-accuracy video queries by balancing between ingest-time cost and query latency. We now discuss work related to our key techniques. 1) Cascaded classification. Various works in vision research propose speeding up classification by cascading a series of classifiers. Viola et al. [75] is the earliest work which cascades a series of classifiers (from the simplest to the most complicated) to quickly disregard regions in an image. Many improvements followed (e.g., [50, 76, 77]). CNNs are also cascaded (e.g., [26, 35, 49, 70]) to reduce object detection latency. Our work is different in two major ways. First, we decouple the compressed CNN from the GT-CNN, which allows us to choose from a wider range for ingest-time CNNs and allows for better trade-offs between ingest cost and query latency, a key aspect of our work. Second, we cluster similar objects using CNN features to eliminate redundant work, which is a new and effective technique for video streams. 2) Neural network compression. Recent work proposes various techniques to reduce the running time of CNNs. These techniques include shallow models [21], predicting weights [33], matrix pruning [31, 37], model quantization [36], and others (e.g., [20, 34, 39, 41, 42, 57, 62]). Our work is largely orthogonal to these, in that our system is not tied to a specific model compression technique, and we can employ any of these techniques. 3) Context-specific model specialization. Contextspecific specialization of models can improve accuracy [53] or reduce running time [35, 44, 65]. Among these, the closest to our work is Kang et al.’s proposal, NoScope [44], which aims to optimize CNN-based video queries. A few key differences stand out. First, NoScope applies all the optimizations at query-time, while Focus adopts a different architecture by splitting work between ingest- and query-time. Thus, Focus trades off higher ingest cost for even lower query latency. Second, NoScope optimizes CNNs for a single class, while we optimize ingest CNNs for all frequent classes in the stream and allow queries even for the rare – OTHER – classes. Finally, we use the object feature vectors to cluster similar objects and create an index to map classes to clusters; this allows us to efficiently query across all classes, while NoScope has to redo all query-time work, including training specialized CNNs, for each query. 4) Stream processing systems. Systems for general stream data processing (e.g., [1,18,19,24,28,29,51,56,73, 74, 79]) and specific to video analytics (e.g., [80]) mainly focus on the general stream processing challenges such as load shedding, fault tolerance, distributed execution, or limited network bandwidth. In contrast, our work is specific for querying on recorded video data with ingest and query trade-offs, thus it is mostly orthogonal to these. 1 Query faster by (factor) Figure 12: Ingest cost sensitivity to frame sampling 100 30fps 10fps 5fps 1fps 10 1 Figure 13: Query latency sensitivity to frame sampling cost of Focus is 62× cheaper than Ingest-all at 30 fps, and it is 64× to 58× cheaper at lower frame rates. This is because the major ingest cost saving comes from the specialized and compressed CNN models (§6.3), which are orthogonal to frame sampling rates. Second, the query latency improvement of Focus degrades with lower frame rates. This is expected because one of our key techniques to reduce query latency is redundancy elimination, especially clustering similar objects using CNN feature vectors. At lower frame rates, the benefit of this technique reduces because there are fewer redundancies. Nonetheless, on average, Focus is still one order of magnitude faster than Query-all at a very low frame rate (1 fps). 6.7. Applicability with Different Query Rate There are two factors that can affect the applicability of Focus: 1) the number of classes that get queried over time and 2) the fraction of videos that get queried. In the first extreme case where all the classes and all the videos are queried, Ingest-all could be a good option because its cost is amortized among all the queries. In our study, even in such an extreme case, the overall cost of Focus is still 4× cheaper than Ingest-all on average (up to 6× cheaper) because we run a very cheap CNN at ingest time, and we run GT-CNN per object cluster only once (§5), so the overall cost is still cheaper than Ingest-all. The second extreme case is only a tiny fraction of videos gets queried. While Focus can save the ingest cost by up to 141× (§6.4), it can be more costly than Query-all if the fraction of videos gets queried is less than 1 141 = 0.7%. In such a case, we can choose to do nothing at ingest time and run all the techniques of Focus only at query time when we know the fraction of videos that get queried. While this approach increases query latency, it still reduces the query latency by a average of 22× (up to 34×) than Query-all in our evaluation. We conclude that Focus is still better than both baselines even under extreme query rates. 12 We can integrate Focus with one of these general stream processing system to build a more fault tolerable system. 5) Video indexing and retrieval. A large body of works in multimedia and information retrieval research propose various content-based video indexing and retrieval techniques to facilitate queries on videos (e.g., [40,48,68,69]). Among them, most works focus on indexing videos for different types of queries such as shot boundary detection [78], semantic video search [30], video classification [25], or spatio-temporal information-based video retrieval [61]. Some works (e.g., [32, 67]) focus on the query interface to enable query by keywords, concepts, or examples. These works are largely orthogonal to our work because we focus on the cost and latency of video queries, not query types or interfaces. We believe our idea of splitting ingest-time and query-time work is generic for videos queries, and can be extended to different types of queries. [5] “City of Auburn North Ross St and East Magnolia Ave.” [Online]. Available: https://www.youtube.com/watch?v= cjuskMMYlLA [6] “City of Auburn Toomer’s Corner Webcam.” [Online]. Available: https://www.youtube.com/watch?v=yJAk_ FozAmI [7] “Genetec,” https://www.genetec.com/. [8] “Greenwood Avenue Bend, Oregon.” [Online]. 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arXiv:1605.03662v2 [math.ST] 21 Jan 2018 Subspace Perspective on Canonical Correlation Analysis: Dimension Reduction and Minimax Rates Zhuang Ma and Xiaodong Li Abstract Canonical correlation analysis (CCA) is a fundamental statistical tool for exploring the correlation structure between two sets of random variables. In this paper, motivated by the recent success of applying CCA to learn low dimensional representations of high dimensional objects, we propose two losses based on the principal angles between the model spaces spanned by the sample canonical variates and their population correspondents, respectively. We further characterize the non-asymptotic error bounds for the estimation risks under the proposed error metrics, which reveal how the performance of sample CCA depends adaptively on key quantities including the dimensions, the sample size, the condition number of the covariance matrices and particularly the population canonical correlation coefficients. The optimality of our uniform upper bounds is also justified by lower-bound analysis based on stringent and localized parameter spaces. To the best of our knowledge, for the first time our paper separates p1 and p2 for the first order term in the upper bounds without assuming the residual correlations are zeros. More significantly, our paper derives p1 ´ λ2k qp1 ´ λ2k`1 q{pλk ´ λk`1 q2 for the first time in the nonasymptotic CCA estimation convergence rates, which is essential to understand the behavior of CCA when the leading canonical correlation coefficients are close to 1. 1 Introduction Canonical correlation analysis (CCA), first introduced by Hotelling (1936), is a fundamental statistical tool to characterize the relationship between two groups of random variables and finds a wide range of applications across many different fields. For example, in genome-wide association study (GWAS), CCA is used to discover the genetic associations between the genotype data of single nucleotide polymorphisms (SNPs) and the phenotype data of gene expression levels (Witten et al., 2009; Chen et al., 2012). In information retrieval, CCA is used to embed both the search space (e.g. images) and the query space (e.g. text) into a shared low dimensional latent space such that the similarity between the queries and the candidates can be quantified (Rasiwasia et al., 2010; Gong et al., 2014). In natural language processing, CCA is applied to the word co-occurrence matrix and learns vector representations of the words which capture the semantics (Dhillon et al., 2011; Faruqui and Dyer, 2014). Other applications, to name a few, include fMRI data analysis (Friman et al., 2003), computer vision (Kim et al., 2007) and speech recognition (Arora and Livescu, 2013; Wang et al., 2015). The enormous empirical success motivates us to revisit the estimation problem of canonical correlation analysis. Two theoretical questions are naturally posed: What are proper error metrics to quantify the discrepancy between population CCA and its sample estimates? And under such metrics, what are the quantities that characterize the fundamental statistical limits? 1 The justification of loss functions, in the context of CCA, has seldom appeared in the literature. From first principles that the proper metric to quantify the estimation loss should depend on the specific purpose of using CCA, we find that the applications discussed above mainly fall into two categories: identifying variables of interest and dimension reduction. The first category, mostly in genomic research (Witten et al., 2009; Chen et al., 2012), treats one group of variables as responses and the other group of variables as covariates. The goal is to discover the specific subset of the covariates that are most correlated with the responses. Such applications are featured by low signal-to-noise ratio and the interpretability of the results is the major concern. In contrast, the second category is investigated extensively in statistical machine learning and engineering community where CCA is used to learn low dimensional latent representations of complex objects such as images (Rasiwasia et al., 2010), text (Dhillon et al., 2011) and speeches (Arora and Livescu, 2013). These scenarios are usually accompanied with relatively high signalto-noise ratio and the prediction accuracy, using the learned low dimensional embeddings as the new set of predictors, is of primary interest. In recent years, there has been a series of publications establishing fundamental theoretical guarantees for CCA to achieve sufficient dimension reduction (Kakade and Foster (2007); Foster et al. (2008); Sridharan and Kakade (2008); Fukumizu et al. (2009); Chaudhuri et al. (2009) and many others). In this paper, we aim to address the problems raised above by treating CCA as a tool for dimension reduction. 1.1 Population and Sample CCA Suppose x “ rX1 , . . . , Xp1 sJ P Rp1 and y “ rY1 , . . . , Yp2 sJ P Rp2 are two sets of variates with the joint covariance matrix ˆ„ ˙  „ Σx Σxy x Cov . (1.1) “ Σ :“ Σy ΣJ y xy For simplicity, we assume EpXi q “ 0, i “ 1, . . . , p1 , EpYj q “ 0, j “ 1, . . . , p2 . On the population level, CCA is designed to extract the most correlated linear combinations between two sets of random variables sequentially: The ith pair of canonical variables Ui “ φJ i x and Vi “ ψiJ y maximizes λi “ CorrpUi , Vi q such that Ui and Vi have unit variances and they are uncorrelated to all previous pairs of canonical variables. Here pφi , ψi q is called the ith pair of canonical loadings and λi is the ith canonical correlation. It is well known in multivariate statistical analysis that the canonical loadings can be found recursively by the following criterion: pφi , ψi q “ arg max φJ Σxy ψ subject to φJ Σx φ “ 1, ψ J Σy ψ “ 1; J J φ Σx φj “ 0, ψ Σy ψj “ 0, @ 1 ď j ď i ´ 1. 2 (1.2) Although this criterion is a nonconvex optimization, it can be obtained easily by spectral methods: Define Φ :“ rφ1 , ¨ ¨ ¨ , φp1 ^p2 s, Ψ :“ rψ1 , ¨ ¨ ¨ , ψp1 ^p2 s and Λ :“ diagpλ1 , ¨ ¨ ¨ , λp1 ^p2 q. Then ´1{2 ´1{2 1{2 1{2 λ1 , . . . , λp1 ^p2 are singular values of Σx Σxy Σy , and Σx Φ, Σy Ψ are actually left and right ´1{2 ´1{2 singular vectors of Σx Σxy Σy , respectively. 1.2 Canonical variables versus canonical loadings pi , ψ pi quk , the For any given estimates of the leading k canonical loadings, denoted by tpφ i“1 corresponding estimates for the canonical variables can be represented by pJ x, pi “ φ U i pJ y, Vpi “ ψ i i “ 1 . . . , p1 ^ p2 . To quantify the estimation loss, generally speaking, we can either focus on measuring the difference pi , ψ pi quk or measuring the difference between between the canonical loadings tpφi , ψi quki“1 and tpφ i“1 pi , Vpi quk . Here x, y in the definition of tpUi , Vi quk the canonical variables tpUi , Vi quki“1 and tpU i“1 i“1 pi , ψ pi quk are constructed. pi , Vpi quk are independent of the samples based on which tpφ and tpU i“1 i“1 Therefore, for the discrepancy between the canonical variables, there is an extra layer of randomness. As discussed above, in modern machine learning applications such as natural language processing and information retrieval, the leading sample canonical loadings are used for dimension reduction, i.e., for a new observation px0 , y0 q, ideally we hope to use the corresponding values of k J k the canonical variables pui “ φJ i x0 qi“1 and pvi “ ψi y0 qi“1 to represent the observation in a low pJ x0 qk dimension space. Empirically, the actual low dimensional representations are pûi “ φ i i“1 pJ y0 qk . Therefore, the discrepancy between the ideal dimension reduction and and pv̂i “ ψ i i“1 pi , Vpi quk approximate tpUi , Vi quk . actual dimension reduction should be explained by how well tpU i“1 i“1 Consequently, we choose to quantify the difference between the sample and population canonical variables instead of the canonical loadings. 1.3 Linear span However, there are still many options to quantify how well the sample canonical variables approximate their population correspondents. To choose suitable losses, it is convenient to come back to specific applications to get some inspiration. Motivated by applications in natural language processing and information retrieval, the model of multi-view sufficient dimension reduction has been studied in Foster et al. (2008). Roughly speaking, a statistical model was proposed by Foster et al. (2008) to study how to predict Z using two sets of predictors denoted by x “ rX1 , . . . , Xp1 sJ and y “ rY1 , . . . , Yp2 sJ , where the joint covariance of pZ, x, yq is ¨» ﬁ˛ » ﬁ Σx Σxy σxz x Σy σyz ﬂ . Cov ˝– y ﬂ‚ “ –ΣJ xy J J Z σz2 σxz σyz It was proven in Foster et al. (2008) that under certain assumptions, the leading k canonical variables U1 , . . . Uk are sufficient dimension reduction for the linear prediction of Z; That is, the best linear predictor of Z based on X1 , . . . , Xp1 is the same as the best linear predictor based on 3 U1 , . . . Uk . (Similarly, the best linear predictor of Z based on Y1 , . . . , Yp2 is the same as the best linear predictor based on V1 , . . . Vk .) Notice that the best linear predictor is actually determined by the set of all linear combinations of U1 , . . . , Uk (referred to as the “model space” in the literature of linear regression for prediction), which we denote as spanpU1 , . . . , Uk q. Inspired by Foster et al. (2008), we propose to quantify the pi uk by the discrepancy between the corresponding subspaces discrepancy between tUi uki“1 and tU i“1 p1 , . . . , U pk q and spanpU1 , . . . , Uk q (and similarly measure the difference between tVi uk and spanpU i“1 tVpi uki“1 by the distance between spanpVp1 , . . . , Vpk q and spanpV1 , . . . , Vk q). 1.4 Hilbert spaces and principal angles xpU,kq “ spanpU p1 , . . . , U pk q and MpU,kq “ In this section, we define the discrepancy between M spanpU1 , . . . , Uk q by introducing a Hilbert space. Noting that for any given sample tpxi , yi quni“1 , xpU,kq and MpU,kq are composed by linear combinations of X1 , . . . , Xp . Denote the set of all both M 1 possible linear combinations as H “ spanpX1 , . . . , Xp1 q. (1.3) Moreover, for any X1 , X2 P H, we define a bilinear function xX1 , X2 y :“ CovpX1 , X2 q “ EpX1 X2 q. It is easy to show that x¨, ¨y is an inner product and pH, x¨, ¨yq is a p1 -dimensional Hilbert space, which is isomorphic to Rp1 . xpU,kq and MpU,kq are With the natural covariance-based inner product, we know both M subspaces of H, so it is natural to define their discrepancy based on their principal angles π 2 ě θ1 ě . . . ě θk ě 0. In the literature of statistics and linear algebra, two loss functions are usually used p1 , . . . , U pk q, spanpU1 , . . . , Uk qq “ sin2 pθ1 q Lmax pspanpU and 1 psin2 pθ1 q ` . . . ` sin2 pθk qq k In spite of a somewhat abstract definition, we have the following clean formula for these two losses: p1 , . . . , U pk q, spanpU1 , . . . , Uk qq “ Lave pspanpU Theorem 1.1. Suppose for any p1 ˆ k matrix A, PA represents the orthogonal projector onto the column span of A. Assume the observed sample is fixed. Then › ›2 › p1 , . . . , U pk q, spanpU1 , . . . , Uk qq “ 1 ››P 1{2 ´ P Lave pspanpU › 1{2 p 1:k Σx Φ1:k F 2k Σx Φ ›2 ¯ 1 ››´ › “ › Ip1 ´ PΣ1{2 Φ PΣ1{2 Φ (1.4) › p x x 1:k 1:k k F “ ‰ 1 p J Q}22 “ min E }uJ ´ u k QPRkˆk p 1:k q :“ Lave pΦ1:k , Φ 4 and › ›2 › p1 , . . . , U pk q, spanpU1 , . . . , Uk qq “ ››P 1{2 ´ P Lmax pspanpU › 1{2 p 1:k Σx Φ1:k Σx Φ ›2 ›´ ¯ › › PΣ1{2 Φ “ › Ip1 ´ PΣ1{2 Φ › p x x 1:k 1:k ”`` ˘ ˘2 ı pJ Q g “ max min E uJ ´ u (1.5) gPRk QPRkˆk p 1:k q. :“ Lmax pΦ1:k , Φ Here Φ1:k “ rφ1 , . . . , φk s is a p1 ˆ k matrix consisting of the leading k population canonical p 1:k is its estimate based on a given sample. Moreover uJ :“ pU1 , . . . , Uk q and loadings for x, and Φ p1 , . . . , U pk q. ûJ :“ pU 1.5 Uniform upper bounds and minimax rates The most important contribution of this paper is to establish sharp upper bounds for the p 1:k q and estimation/prediction of CCA based on the proposed subspace losses Lmax pΦ1:k , Φ p Lave pΦ1:k , Φ1:k q. It is noteworthy that both upper bounds hold uniformly for all invertible Σx , Σy provided n ą Cpp1 ` p2 q for some numerical constant C. Furthermore, in order to justify the sharpness of these bounds, we also establish minimax lower bounds under a family of stringent and localized parameter spaces. These results will be detailed in Section 2. 1.6 Notations and the Organization Throughout the paper, we use lower-case and upper-case non-bolded letters to represent fixed and random variables, respectively. We also use lower-case and upper-case bold letters to represent vectors (which could be either deterministic or random) and matrices, respectively. For any matrix U P Rnˆp and vector u P Rp , }U }, }U }F denotes operator (spectral) norm and Frobenius norm respectively, }u} denotes the vector l2 norm, U1:k denotes the submatrix consisting of the first k columns of U , and PU stands for the projection matrix onto the column space of U . Moreover, we use σmax pU q and σmin pU q to represent the largest and smallest singular value of U respectively, and κpU q “ σmax pU q{σmin pU q to denote the condition number of the matrix. We use Ip for the identity matrix of dimension p and Ip,k for the submatrix composed of the first k columns of Ip . Further, Opm, nq (and simply Opnq when m “ n) stands for the set of m ˆ n matrices with orthonormal columns and Sp` denotes the set of p ˆ p strictly positive definite matrices. For a random vector x P Rp , spanpxJ q “ txJ w, w P Rp u denotes the subspace of all the linear combinations of x. Other notations will be specified within the corresponding context. In the following, we will introduce our main upper and lower bound results in Section 2. To highlight our contributions in the new loss functions and theoretical results, we will compare our results to existing work in the literature in Section 3. All proofs are deferred to Section 4. 2 Theory In this section, we introduce our main results on non-asymptotic upper and lower bounds for estimating CCA under the proposed loss functions. It is worth recalling that λ1 , . . . , λp1 ^p2 are ´1{2 ´1{2 singular values of Σx Σxy Σy . 5 It is natural to estimate population CCA by its sample counterparts. Similar to equation (1.2), the sample canonical loadings are defined recursively by pi , ψ pi q “ arg max pφ p xy ψ φJ Σ p x φ “ 1, ψ J Σ p y ψ “ 1; subject to φJ Σ p x φj “ 0, ψ J Σ p y ψj “ 0, @ 1 ď j ď i ´ 1. φJ Σ (2.1) p x, Σ p y, Σ p xy are the sample covariance matrices. The sample canonical variables are where Σ defined as the following linear combinations by the sample canonical loadings: pJ x, pi “ φ U i pJ y, Vpi “ ψ i i “ 1 . . . , p1 ^ p2 . We prove the following upper bound for the estimate based on sample CCA. „  x Theorem 2.1. (Upper bound) Suppose „ N p0, Σq where Σ is defined as in (1.1). Assume y Σx and Σy are invertible. Moreover, assume λk ą λk`1 for some predetermined k. Then there exist universal positive constants γ, C, C0 such that if n ě Cpp1 ` p2 q, the top-k sample canonical p 1:k satisfies coefficients matrix Φ ﬀ « ” ı 2 p1 ´ λ2k qp1 ´ λ2k`1 q p1 pp ` p q 1 2 p 1:k q ď C0 ` 2 ` e´γpp1 ^p2 q E Lmax pΦ1:k , Φ pλk ´ λk`1 q2 n n pλk ´ λk`1 q4 « ﬀ ” ı p1 ´ λ2k qp1 ´ λ2k`1 q p1 ´ k pp1 ` p2 q2 ´γpp1 ^p2 q p E Lave pΦ1:k , Φ1:k q ď C0 ` 2 `e pλk ´ λk`1 q2 n n pλk ´ λk`1 q4 p 1:k can be obtained by switching p1 and p2 . The upper bounds for Ψ Since we pursue a nonasymptotic theoretical framework for CCA estimates, and the loss functions we propose are nonstandard in the literature, the standard minimax lower bound results in parametric maximum likelihood estimates do not apply straightforwardly. Instead, we turn to the nonparametric minimax lower bound frameworks, particularly those in PCA and CCA; See, e.g., Vu et al. (2013); Cai et al. (2013); Gao et al. (2015). Compared to these existing works, the technical novelties of our results and proofs are summarized in Sections 3.3 and 6. We define the parameter space Fpp1 , p2 , k, λk , λk`1 , κ1 , κ2 q as the collection of joint covariance matrices Σ satisfying 1. κpΣx q “ κ1 and κpΣy q “ κ2 ; 2. 0 ď λp1 ^p2 ď ¨ ¨ ¨ ď λk`1 ă λk ď ¨ ¨ ¨ ď λ1 ď 1. We deliberately set κpΣx q “ κ1 , κpΣy q “ κ2 to demonstrate that the lower bound is independent of the condition number. For the rest of the paper, we will use the shorthand F to represent this parameter space for simplicity. 6 Theorem 2.2. (Lower bound) There exists a universal constant c independent of n, p1 , p2 and Σ such that ¸ + #˜ ” ı p1 ´ λ2k qp1 ´ λ2k`1 q p1 ´ k p ´ k 1 2 p 1:k q ě c ^1^ inf sup E Lmax pΦ1:k , Φ pλk ´ λk`1 q2 n k p 1:k ΣPF Φ #˜ ¸ + ” ı p1 ´ λ2k qp1 ´ λ2k`1 q p1 ´ k p ´ k 1 2 p 1:k q ě c inf sup E Lave pΦ1:k , Φ ^1^ . pλk ´ λk`1 q2 n k p 1:k ΣPF Φ p 1:k can be obtained by replacing p1 with p2 . The lower bounds for Ψ Corollary 2.3. When p1 , p2 ě p2kq _ Cplog nq and něC pp1 ` p2 qp1 ` p2 {p1 q pλk ´ λk`1 q2 p1 ´ λ2k qp1 ´ λ2k`1 q (2.2) for some universal positive constant c, the minimax rates can be characterized by 3 ” ı p1 ´ λ2 qp1 ´ λ2 q p 1 k k`1 p 1:k q — , inf sup E Lmax pΦ1:k , Φ 2 pλ ´ λ q n p k k`1 Φ1:k ΣPF ” ı p1 ´ λ2 qp1 ´ λ2 q p 1 k`1 k p 1:k q — . inf sup E Lave pΦ1:k , Φ 2 pλk ´ λk`1 q n p 1:k ΣPF Φ Related Work and Our Contributions Recently, the non-asymptotic rate of convergence of CCA has been studied by Gao et al. (2015, 2017) under a sparse setup and by Cai and Zhang (2017) under the usual non-sparse setup. Cai and Zhang (2017) appeared on arXiv almost at the same time as the first version of our paper was posted. In this section, we state our contributions by detailed comparison with these works. 3.1 Novel loss funcitons We proposed new loss functions based on the principal angles between the subspace spanned by the population canonical variates and the subspace spanned by the estimated canonical variates. In contrast, Gao et al. (2017) proposed and studied the loss Lave ; Cai and Zhang (2017) proposed Lmax and studied both Lave and Lmax , where ˇ ” ı p 1:k q “ min E }xJ Φ1:k ´ xJ Φ p 1:k Q}2 ˇˇ Φ p 1:k , Lave pΦ1:k , Φ 2 QPOpk,kq „´´  ¯ ¯2 ˇ ˇ J J p 1:k Q g p 1:k q “ max p 1:k . x Φ1:k ´ x Φ Lmax pΦ1:k , Φ min E ˇΦ gPRk ,\|g\|“1 QPOpk,kq Lave and Lmax resemble our loss functions Lave and Lmax respectively. By Theorem 1.1, we also have ˇ ” ı p 1:k q “ 2 min E }xJ Φ1:k ´ xJ Φ p 1:k Q}2 ˇˇ Φ p 1:k Lave pΦ1:k , Φ 2 QPRkˆk  „´´ ¯ ¯2 ˇ ˇ p J Jp p x Φ1:k ´ x Φ1:k Q g Lmax pΦ1:k , Φ1:k q “ max min E ˇ Φ1:k gPRk ,\|g\|“1 QPRkˆk 7 By these two expressions, we can easily obtain p 1:k q p 1:k q ď 2Lave pΦ1:k , Φ Lave pΦ1:k , Φ p 1:k q ď Lmax pΦ1:k , Φ p 1:k q Lmax pΦ1:k , Φ (3.1) p 1:k q and Lave pΦ1:k , Φ p 1:k q are not equivalent up to a constant. Neither are However, Lave pΦ1:k , Φ p p Lmax pΦ1:k , Φ1:k q and Lmax pΦ1:k , Φ1:k q. In fact, we can prove that as long as n ą maxpp1 , p2 q, if λk “ 1 ą λk`1 , then p 1:k q “ Lmax pΦ1:k , Φ p 1:k q “ 0, Lave pΦ1:k , Φ p 1:k q ‰ 0 and Lmax pΦ1:k , Φ p 1:k q ‰ 0. while almost surely Lave pΦ1:k , Φ To illustrate this comparison, very simple simulation: Suppose „ we can consider „the following  „  1 0 1 0 1 0 p1 “ p2 “ 2, n “ 3 and Σx “ and Σy “ and Σxy “ . In this setup, we 0 1 0 1 0 0.5 know the population canonical are λ1 “ 1 and λ2 “ 0.5, and the leading  „  correlation„coefficients 1 1 . In our simulation, we generated the following data and ψ1 “ canonical loadings are φ1 “ 0 0 matrices » ﬁ 0.0736 1.5496 X “ –1.5390 ´0.0415ﬂ 0.9331 ´0.4776 and » ﬁ 0.0736 2.8982 Y “ –1.5390 ´1.2214ﬂ . 0.9331 2.5931 p1 “ 1 and λ p Furthermore, we can obtain the sample canonical correlations λ „  „  2 “ 0.5210, as well as ´0.9616 ´0.9616 p1 “ p1 “ p1 q “ the leading sample canonical loadings φ and ψ . Then Lave pφ1 , φ 0 0 p1 q “ 0 while Lave pφ1 , φ p1 q ‰ 0, Lmax pφ1 , φ p1 q ‰ 0. Lmax pφ1 , φ This numerical example clearly shows that the sample CCA can exactly identify that among all linear combinations of X1 and X2 and all linear combinations of Y1 and Y2 , aX1 and bY1 are mostly correlated. Our loss functions Lave and Lmax do characterize this exact identification, whereas Lave and Lmax do not. Moreover, the following joint loss was studied in Gao et al. (2015): „› ›2  ´ ´ ¯¯ › J J p 1:k Ψ p ´ Φ1:k Ψ ›› . p 1:k , Ψ p 1:k Ljoint pΦ1:k , Ψ1:k q , Φ “ E ›Φ 1:k 1:k F ´ ´ ¯¯ p 1:k , Ψ p 1:k ‰ 0 almost surely under the special case λk “ 1 ą Similarly, Ljoint pΦ1:k , Ψ1:k q , Φ λk`1 . 3.2 Sharper upper bounds Regardless of loss functions, we explain in the following why Theorem 2.1 implies sharper upper bounds than the existing rates in Gao et al. (2015), Gao et al. (2017) and Cai and Zhang (2017) 8 under the nonsparse case. Our discussion is focused on Lave in the following discussion while the discussion for Lmax is similar. Notice that if we only apply Wedin’s sin-theta law, i.e., replacing the fine bound Lemma 5.4 with the rough bound Lemma 5.2 (also see Gao et al. (2015) for similar ideas), we can obtain the following rough bound: „  ” ı p1 ` p2 p 1:k q ď C0 E Lave pΦ1:k , Φ . (3.2) npλk ´ λk`1 q2 p 1:k from p2 , both Gao et al. (2017) and In order to decouple the estimation error bound of Φ Cai and Zhang (2017) assume the residual canonical correlations are zero, i.e., λk`1 “ . . . “ λp1 ^p2 “ 0. This assumption is essential for proofs in both Gao et al. (2017) and Cai and Zhang (2017) under certain sample size conditions. We got rid of this assumption by developing new proof techniques and these techniques actually work for Lave , Lmax as well. A detailed comparison between our result and that in Cai and Zhang (2017) is summarized in Table 3.2 (The results of Gao et al. (2017) in the non-sparse regime can be implied by Cai and Zhang (2017) under milder sample size conditions). Loss function Sample size Cai and Zhang 2016 L Lave q ˆave pě ˙ ? p1 ` p1 p2 p2 nąC ` 4{3 λ2 Upper Bound Rates n ą Cpp1 ` p2 q λk k λk`1 “ ¨ ¨ ¨ “ λp1 “ 0 Our work Lave Yes p1 nλ2k ` No p1´λ2k qp1´λ2k`1 q p1 ´k n pλk ´λk`1 q2 p1 p2 n2 λ4k ` pp1 `p2 q2 2 n pλk ´λk`1 q4 ` e´γpp1 ^p2 q Perhaps the most striking contribution of our upper bound is that we first derive the factors p1 ´ λ2k q and p1 ´ λ2k`1 q in the literature of nonasymptotic CCA estimate. We now explain why these factors are essential when leading canonical correlation coefficients are close to 1. Example 1: λk “ 1 and λk`1 “ 0 Consider the example that k “ 1, p1 “ p2 :“ p " log n, λ1 “ 1 and λ2 “ 0. Then our bound rates p1´λ2k qp1´λ2k`1 q p1 ´k n pλk ´λk`1 q2 ` pp1 `p2 q2 n2 pλk ´λk`1 q4 ` e´γpp1 ^p2 q actually imply that 2 p1 q ď C p , ELave pφ1 , φ n2 while the rates in Gao et al. (2017) and Cai and Zhang (2017) imply that p1 q ď 2ELave pφ1 , φ p1 q ď C p . ELave pφ1 , φ n p1 q, our result could This shows that even under the condition λk`1 “ 0, under our loss Lave pφ1 , φ imply sharper convergence rates than that in Gao et al. (2017) and Cai and Zhang (2017) if λk “ 1. 9 p1 q “ 0 through Notice that as aforementioned, when λk “ 1, we can actually prove ELave pφ1 , φ a separate argument. How to improve Theorem 2.1 to imply this result is an open problem for future research. Example 2: Both λk and λk`1 are close to 1 Consider the example that k “ 1, p1 “ p2 :“ p " log n, λ1 “ 1 ´ our bound rates p1´λ2k qp1´λ2k`1 q p1 ´k n pλk ´λk`1 q2 ` q2 pp1 `p2 n2 pλk ´λk`1 q4 b 4 p n and λ2 “ 1 ´ 2 ` e´γpp1 ^p2 q actually imply that b 4 p n. Then p1 q ď C p , ELave pφ1 , φ n while the rough rates (3.2) by Wedin’s sin-theta law implies c p p ELave pφ1 , φ1 q ď C . n This shows that our upper bound rates could be much sharper than the rough rates (3.2) when both λk and λk`1 are close to 1. New proof techniques and connection to asymptotic theory To the best of our knowledge, none of the analysis in Gao et al. (2015), Gao et al. (2017), Cai and Zhang (2017) can be used to obtain the multiplicative factor p1´λ2k qp1´λ2k`1 q{pλk ´λk`1 q2 in the first order term of the upper bound, even under the strong condition that λk`1 “ ¨ ¨ ¨ “ λp1 ^p2 “ 0. Following a different path, we do careful non-asymptotic entry-wise perturbation analysis of the estimating equations of CCA to avoid the loss of precision caused by applying matrix inequalities in the early stage of the proof. The main challenge is to analyze the properties of matrix hardmard products, especially to derive tight operator norm bounds for certain hardmard products. We are particularly luckily to find a divide-and-conquer approach (λk ě 21 and λk ă 21 in the proof of Lemma 5.4) to decompose the target matrices into simple-structure matrices where we can apply the tools developed in Lemma 5.6. pi , ψ pi qup1 ^p2 has been studied by The asymptotic distribution of the canonical loadings tpφ i“1 Anderson (1999) under the assumption that all the canonical correlations are distinct and λ1 ‰ 1. Since we focus on subspaces, we only require λk ą λk`1 for the given k. Both Anderson (1999) and our work are based on analyzing the estimating equations ((5.5)) of CCA. Our analysis is more involved because completely novel techniques are required to obtain the factor p1 ´ λ2k qp1 ´ λ2k`1 q in the nonasymptotic framework. 3.3 Sharper lower bounds under parameter spaces with fixed λk and λk`1 The minimax lower bounds for the estimation rates of CCA were first established by Gao et al. (2015, 2017) under the losses Ljoint and Lave . However, the parameter space discussed in Gao et al. (2017) requires λk`1 “ 0. Moreover, the parameter space in Gao et al. (2015) is parameterized by λ satisfying λk ě λ, but λk`1 is not specified. In fact, they also constructed the hypothesis class with λk`1 “ 0 and the resulting minimax lower bound is proportional to λ12 . 10 However, this minimax lower bound b is not sharp when λk and λk`1 are close. Suppose p1 “ 1 1 p2 :“ p, k “ 1, λ1 “ 2 and λ2 “ 2 ´ np . Our minimax lower bound in Theorem 2.2 leads to ” ı p 1:k q ě Op1q. inf sup E Lave pΦ1:k , Φ p 1:k ΣPF Φ In contrast, to capture the fundamental limit of CCA estimates in this scenario under the framework of Gao et al. (2015), one needs to choose λ to capture both λk and λk`1 , i.e., λk ď λ ď λk`1 and hence λ « 1{2. Then the resulting minimax lower bound rate will be nλp 2 “ Op np q, which is much looser than Op1q. Technically speaking, we follow the analytical framework of Gao et al. (2015) and Gao et al. (2017), but the hypothesis classes construction requires any given λk`1 ą 0 instead of λk`1 “ 0, and this brings in new technical challenges. More detailed technical discussions are deferred to Section 6. 4 Proof of Theorem 1.1 Suppose the observed sample of px, yq is fixed and consider the correlation between p1 , . . . , U pk q. Let the two subspaces of H (defined in (1.3)): spanpU1 , . . . , Uk q and spanpU x x x pW1 , W1 q, pW2 , W2 q, . . . , pWk , Wk q be the first, second, ..., and kth pair of canonical p1 , . . . , U pk . variates between U1 , . . . , Uk and U Then spanpW1 , . . . , Wk q “ spanpU1 , . . . , Uk q, x x p p xj y “ xW xi , W xj y “ 0, for any i ‰ j and spanpW1 , . . . , Wk q “ spanpU1 , . . . , Uk q and xWi , Wj y “ xWi , W xi q “ 1, for i “ 1, . . . , k. VarpWi q “ VarpW xi q is actually the ith principal angle By the definition of principal angles, we know =pWi , W p p xi q. This implies that between spanpU1 , . . . , Uk q and spanpU1 , . . . , Uk q, i.e., θi :“ =pWi , W p 1:k q :“ Lave pΦ1:k , Φ k ÿ i“1 2 sin θi “ k ˆ ÿ i“1 ˇA Eˇ2 ˙ ˇ ˇ x 1 ´ ˇ Wi , Wi ˇ . p1 , . . . , U pk are linear combinations of X1 , . . . , Xp , we can denote Since U1 , . . . , Uk , U 1 p x1 , . . . , W xk q “ xJ Σx´1{2 B, w J :“ pW1 , . . . , Wk q “ xJ Σx´1{2 B, and ŵ J :“ pW p :“ rp where B :“ rb1 , . . . , bk s, B b1 , . . . , p bk s P Rpˆk . By the definition of w, we have Ik “ Covpwq “ B J Σx´1{2 CovpxqΣx´1{2 B “ B J B p J B. p Then B, B p are p ˆ k basis matrices. Moreover, we have bJp and similarly Ik “ B i bj “ xj y “ 0, for all i ‰ j. Moreover, we have xWi , W p “ B J B. p Diagpcospθ1 q, . . . , cospθk qq “ Covpw, ŵq “ B J Σx´1{2 CovpxqΣx´1{2 B Notice that spanpU1 , . . . , Uk q “ spanpW1 , . . . , Wk q, pU1 , . . . , Uk q “ xJ Φ1:k , and pW1 , . . . , Wk q “ ´1{2 xJ Σx B. Then Φ1:k “ Σx´1{2 BC ñ Σ1{2 x Φ1:k “ BC 11 1{2 for some nonsingular k ˆ k matrix C. This implies that B and Σx Φ1:k have the same column space. Since B P Rpˆk is a basis matrix, we have BB J “ PΣ1{2 Φ . x Similarly, we have 1:k pB p J “ P 1{2 . B p Σ Φ x 1:k Straightforward calculation gives › ›2 ¯ ´ › pB pJ ´ B pB p J BB J ` B pB p JB pB pJ pB p J ›› “ trace BB J BB J ´ BB J B ›BB J ´ B F pB p J Bq “ 2k ´ 2tracepB J B “ 2k ´ 2tracepDiagpcos2 pθ1 q, . . . , cos2 pθk qqq p 1:k q “ 2psin2 pθ1 q ` . . . ` sin2 pθk qq “ 2kLave pΦ1:k , Φ and ›` ›2 ´` ˘ ˘ ` ˘¯ › pB p JB pB p J ›› “ trace Ip ´ BB J B pB p J Ip ´ BB J › Ip1 ´ BB J B 1 1 F pB p J Bq “ k ´ tracepB J B p 1:k q. “ kLave pΦ1:k , Φ The above equalities yield the first two equalities in (1.4). Notice that both U1 , . . . , Uk and W1 , . . . Wk are both orthonormal bases of spanpU1 , . . . , Uk q. p1 , . . . , U pk and W x1 , . . . W xk are both orthonormal bases of spanpU p1 , . . . , U pk qq.) Then we (Similarly, U J J have u “ w R where R is a k ˆ k orthogonal matrix. Then min E}uJ ´ ûJ Q}22 “ min E}uJ ´ ŵ J Q}22 “ min E}w J R ´ ŵ J Q}22 QPRkˆk QPRkˆk QPRkˆk J “ min E}w ´ ŵ QPRkˆk “ “ “ E min i“1 k ÿ qi PRk , i“1,...,k i“1 i“1 QRJ }22 “ min E}w J ´ ŵ J Q}22 QPRkˆk pWi ´ ŵ J qi q2 min qi PRk , i“1,...,k k ÿ k ÿ J EpWi ´ ŵ J qi q2 min EpWi ´ ŵ J qi q2 qi PRk Notice that minqi PRk EpWi ´ ŵ J qi q2 is obtained by the best linear predictor, so min EpWi ´ ŵ J qi q2 “ VarpWi q ´ Covpŵ, Wi qJ Cov´1 pŵqCovpŵ, Wi q qi PRk “ 1 ´ cos2 θi “ sin2 θi . 12 Therefore, min E}uJ ´ ûJ Q}22 “ QPRkˆk k ÿ i“1 p 1:k q, sin2 θi “ kLave pΦ1:k , Φ which implies the third equality in (1.4). Similarly, max min E gPRk ,}g}“1 QPRkˆk “ “ “ “ “ `` J ˘ ˘2 u ´ ûJ Q g max min E max min E gPRk ,}g}“1 QPRkˆk gPRk ,}g}“1 max gPRk ,}g}“1 max gPRk ,}g}“1 max gPRk ,}g}“1 2 “ sin θ1 QPRkˆk min E QPRkˆk qi `` J ˘ ˘2 w R ´ ŵ J Q RJ g `` J ˘ ˘2 w ´ ŵ J Q g min PRk , k ÿ `` J ˘ ˘2 u ´ ŵ J Q g E i“1,...,k k ÿ i“1 gi2 pWi ´ ŵ J qi q2 gi2 sin2 θi i“1 Finally, we prove (1.5). By Wedin (1983), we have › ›2 ›` ›2 ›` ˘ ˘ ››2 › pB p J ›› “ ›› Ip ´ BB J B pB p J ›› “ ›› Ip ´ BB J B p› ›BB J ´ B 1 1 ´ ` ˘ ` ˘ ¯ p J Ip ´ BB J J Ip ´ BB J B p “ λmax B 1 1 ` ˘ “ λmax Ik ´ Diagpcos2 pθ1 q, . . . , cos2 pθk qq p 1:k q, “ 1 ´ cos2 pθ1 q “ sin2 pθ1 q “ Lmax pΦ1:k , Φ which implies the the equalities in (1.5). 5 Proof of Upper Bound Throughout this proof, we denote ∆ :“ λk ´ λk`1 . 5.1 Linear Invariance Without loss of generality, we assume p2 ě p1 :“ p. By the definition of canonical variables, we know that U1 , . . . , Up and V1 , . . . , Vp are only determined by spanpX1 , . . . , Xp1 q and spanpY1 , . . . , Yp2 q. In other words, for any invertible C1 P Rp1 ˆp1 and C2 P Rp2 ˆp2 , the canonical pairs of pX1 , . . . , Xp1 qC1 and pY1 , . . . , Yp2 qC2 are still pU1 , V1 q, . . . , pUp1 , Vp1 q. Therefore, we can consider the following orthonormal bases U1 , . . . , Up1 P spanpX1 , . . . , Xp1 q 13 and V1 , . . . , Vp1 , Vp1 `1 , . . . , Vp2 P spanpY1 , . . . , Yp2 q. Here pV1 , . . . , Vp1 , Vp1 `1 , . . . , Vp2 q is an orthonormal extension of V1 , . . . , Vp1 . Therefore, we know that pU1 , V1 q, . . . , pUp1 , Vp1 q are also the the canonical pairs between U1 , . . . , Up1 and V1 , . . . , Vp2 . Similarly, for a fixed sample of the variables of x and y, the sample canonical pairs p pp , Vpp q are also sample canonical pairs of the corresponding sample of pU1 , Vp1 q, . . . , pU 1 1 pX1 , . . . , Xp1 qC1 and pY1 , . . . , Yp2 qC2 . This can be easily seen from the concept of sample p1 and Vp1 are respectively the linear combinations of canonical variables. For example, U X1 , . . . , Xp1 and Y1 , . . . , Yp1 , such that their corresponding sample variance are both 1 and sample correlation is maximized. If we replace pX1 , . . . , Xp1 q and pY1 , . . . , Yp1 q with pX1 , . . . , Xp1 qC1 and pY1 , . . . , Yp2 qC2 respectively and seek for the first sample canonical pair, the constraints (linear combinations of the two sets of variables and unit sample variances) and the objective p1 , Vp1 q is still the answer. Similarly, (sample correlation is maximized) are the same as before, so pU p p p p pU1 , V1 q, . . . , pUp1 , Vp1 q are the sample canonical pairs of pX1 , . . . , Xp1 qC1 and pY1 , . . . , Yp2 qC2 . In particular, they are the sample canonical pairs of U1 , . . . , Up1 and V1 , . . . , Vp2 . The above argument gives the following convenient fact: In order to bound p1 , . . . , U pk q, spanpU1 , . . . , Uk qq Lave{max pspanpU we can replace X1 , . . . , Xp1 , Y1 , . . . , Yp2 with U1 , . . . , Up1 , V1 , . . . , Vp2 . In other words, we can assume x and y satisfy the standard form r Σx “ Ip1 , Σy “ Ip2 , Σxy “ rΛ, 0p1 ˆpp2 ´p1 q s :“ Λ where Λ “ Diagpλ1 , λ2 , . . . , λp1 q P Rp1ˆp1 . Moreover Φ1:p1 “ Ip1 , Ψ1:p1 “ which implies that Φ1:k “ 5.2 „ Ik 0pp1 ´kqˆk  „ Ip1 0pp2 ´p1 qˆp1 , Ψ1:k “ „  , Ik 0pp2 ´kqˆk  . Upper Bound Under the Standard Form Under the standard form, by (1.4) and (1.5), we have and ›2 › p1 , . . . , U pk q, spanpU1 , . . . , Uk qq “ 1 ››pIp ´ PΦ q P p ›› Lave pspanpU 1 1:k Φ1:k F k ›2 › › › p p Lmax pspanpU1 , . . . , Uk q, spanpU1 , . . . , Uk qq “ ›pIp1 ´ PΦ1:k q PΦ p 1:k › . 14 (5.1) (5.2) p 1:k Denote Φ « ﬀ pu Φ p u and Φ p l are the upper k ˆ k and lower pp1 ´ kq ˆ k sub“ p l1:k where Φ 1:k 1:k Φ1:k p 1:k respectively. Then matrices of Φ ›2 › ´ ¯ › › J p ´1 p J p p “ trace pI ´ P q Φ p Φ Φ q Φ pI ´ P q , › ›pIp1 ´ PΦ1:k q PΦ p Φ p Φ 1:k 1:k p 1:k 1 1 1:k 1:k 1:k 1:k F › ›2 ¯ ´ › › ´1 p J p 1:k pΦ pJ Φ p ›pIp1 ´ PΦ1:k q PΦ › “ λmax pIp1 ´ PΦ1:k q Φ p 1:k 1:k q Φ1:k pIp1 ´ PΦ1:k q 1:k Since ´1 p J p 1:k pΦ pJ Φ p pIp1 ´ PΦ1:k q Φ 1:k 1:k q Φ1:k pIp1 ´ PΦ1:k q  „ ı 0kˆk ” 1 1 J p p l qJ , p pIp1 ´ PΦ1:k q Φ1:k Φ1:k pIp1 ´ PΦ1:k q “ ĺ 0 p Φ kˆk pl 1:k p 1:k q p 1:k q Φ σk2 pΦ σk2 pΦ 1:k we have ›2 › › › pI ´ P q P › p1 Φ1:k p 1:k › ď trace Φ F and ›2 › › › ›pIp1 ´ PΦ1:k q PΦ p 1:k › ď λmax ˜ ˜ ¸  „ ı p l }2 }Φ 1 0kˆk ” 1:k F pJ “ , 0 Φ kˆk l 1:k p 2 2 p p 1:k q Φ σk pΦ1:k q σk pΦ 1:k ¸  „ ı p l }2 }Φ 0kˆk ” 1 1:k J p . “ Φ 0 kˆk 1:k pl 2 pΦ p 1:k q Φ p 1:k q σk2 pΦ σ 1:k k (5.3) (5.4) p l }2 , as well as a lower bound of p l }2 and }Φ Therefore, it suffices to give upper bounds of }Φ 1:k 1:k F p 1:k q. σk2 pΦ 5.3 Basic bounds Recall that Then and r Σx “ Ip1 , Σy “ Ip2 , Σxy “ rΛ, 0p1 ˆpp2 ´p1 q s :“ Λ. « ﬀ ˆ„ ˙ r x Ip1 Λ Cov :“ Σ “ r J y Λ Ip2 « ﬀ ˆ„ ˙ px Σ p xy x Σ y p “ Cov :“ Σ py . p yx Σ y Σ p 2p as the left upper p2p1 q ˆ p2p1 q principal submatrix of Σ. p We can Moreover, we can define Σ 1 similarly define Σ2p1 . Lemma 5.1. There exist universal constants γ, C and C0 such that when n ě C0 p1 , then with probability at least 1 ´ e´γp1 , the following inequalities hold c › › p1 › › p 1{2 p p . }Σ2p1 ´ Σ2p1 }, }Ip1 ´ Σx }, ›Σx ´ Ip1 › ď C n 15 Proof. It is obvious that }Σ2p1 } ď 2. By Lemma 5.9, there exist constants γ, C0 and C1 , such that when n ě C0 p1 , with probability at least 1 ´ e´γp1 there holds c p1 p }Σ2p1 ´ Σ2p1 } ď C1 . n b p x } ď C1 p1 . Moreover, As submatrices, we have }Ip1 ´ Σ n p x } “ }pIp ´ Σ p 1{2 qpIp ` Σ p 1{2 q} ě σmin pIp ` Σ p 1{2 q}Ip ´ Σ p 1{2 } ě }Ip ´ Σ p 1{2 }, }Ip1 ´ Σ 1 x 1 x 1 x 1 x 1 x b p1 `p2 p 1{2 which implies }Ip1 ´ Σ x } ď C1 n . Lemma 5.2. There exist universal constants c, C and C0 such that when n ě C0 pp1 ` p2 q, then with probability at least 1 ´ e´cpp1 `p2 q , the following inequalities hold c › › p1 ` p2 › › p 1{2 p p p }Σ ´ Σ}, }Ip2 ´ Σy }, }Σxy ´ Σxy }, ›Σy ´ Ip2 › ď C , n c p1 ` p2 ´1{2 p ´1{2 p p p , }Λ ´ Λ} ď }Σx Σxy Σy ´ Σxy } ď C n p 1:k }2 ď 3 , σ 2 pΨ p 1:k q ě 1 , }Ψ p 1:k }2 ď 3 , p 1:k q ě 1 , }Φ σk2 pΦ k 2 2 2 2 c C p ` p 1 2 pl } ď p l }, }Ψ }Φ , 1:k 1:k ∆ n where ∆ “ λk ´ λk`1 is the eigen-gap. The proof is deferred to Section 5.7. 5.4 p l }2 Estimating Equations and upper bound of }Φ 1:k p l }2 . Notice that we have already In this section, we aim to give a sharp upper bound for }Φ 1:k established an upper bound in Lemma 5.2, where Wedin’s sin θ law plays the essential role. However, this bound is actually too loose for our purpose. Therefore, we need to develop new techniques to sharpen the results. p P Rp1 ˆp1 , Ψ p P Rp2 ˆp1 consist of the sample canonical coefficients. By definition, Recall that Φ p x1{2 Φ p and the sample canonical coefficients satisfy the following two estimating equations (because Σ 1{2 ´1{2 ´1{2 py Ψ p are left and right singular vectors of Σ px Σ p xy Σ py Σ respectively), If we define define Λ“ „ Λ1 Λ2  p xy Ψ p “Σ p xΦ pΛ p Σ p yx Φ p “Σ p yΨ p Λ. p Σ PR p1 ˆp1 p “ , Λ 16 « p1 Λ (5.5) ﬀ P Rp1 ˆp1 , p Λ2 (5.6) p 1 are k ˆ k diagonal matrices while Λ2 , Λ p 2 are pp1 ´ kq ˆ pp1 ´ kq diagonal matrices. where Λ1 , Λ Then (5.5) imply p xy Ψ p 1:k “ Σ p xΦ p 1:k Λ p1 Σ (5.7) p yx Φ p 1:k “ Σ p yΨ p 1:k Λ p 1. Σ Divide the matrices into blocks, ﬀ ﬀ « ﬀ « ﬀ « « p 11 Σ p 12 p 11 Σ p 12 p 11 Σ p 12 p 11 Σ p 12 Σ Σ Σ Σ xy xy yx yx y y x x py “ p xy “ p px “ , Σ , Σ Σ p 22 p 21 p 22 p 21 p 22 , Σyx “ Σ p 21 p 22 p 21 Σ Σ Σ Σ Σ x y y xy Σxy yx Σyx x kˆk , Ψ p l P Rpp2 ´kqˆk in p 11 p 11 p 11 pu p 11 where Σ x , Σy , Σxy , Σyx are k ˆ k matrices. Finally, we define Ψ1:k P R 1:k pu ,Φ p l . With these blocks, (5.7) can be rewritten as the same way as Φ 1:k 1:k p 21 p u p p 22 p l p p 22 p l p 21 Ψ pu Σ xy 1:k ` Σxy Ψ1:k “ Σx Φ1:k Λ1 ` Σx Φ1:k Λ1 , p 21 Φ pu p 22 p l p 21 p u p p 22 p l p Σ yx 1:k ` Σyx Φ1:k “ Σy Ψ1:k Λ1 ` Σy Ψ1:k Λ1 , p 11 Φ pu Λ p1 ` Σ p 12 Φ pl Λ p 1, p 12 Ψ pl “ Σ p 11 Ψ pu ` Σ Σ xy 1:k p pu Σ11 yx Φ1:k Define the zero-padding of Λ2 : ` xy 1:k p pl Σ12 yx Φ1:k “ x 1:k 11 p u p p Σy Ψ1:k Λ1 ` (5.8) (5.9) (5.10) x 1:k 12 p l p p Σy Ψ1:k Λ1 . (5.11) r 2 :“ rΛ2 , 0s “ Σ22 P Rpp1 ´kqˆpp2 ´kq . Λ xy The above equations imply the following lemma: Lemma 5.3. The equality (5.7) gives the following result pu pl p l Λ2 ´ Λ2 Φ Φ 2 1:k “ B Φ1:k ` R 1:k 1 r r 2 pΣ p 21 ´ Σ pu ` R p 21 Λ1 qΦ p 21 ´ Σ p 21 Λ1 qΨ p u Λ1 ` Λ “ pΣ yx y xy x 1:k 1:k where (5.12) (5.13) p 21 Λ1 ` Λ r 2Σ p 21 ´ Σ p 21 Λ2 ´ Λ r 2Σ p 21 Λ1 , B :“ Σ xy yx x 1 y 21 21 r p r p R :“ pΣ R1 ´ R3 qΛ1 ´ Λ2 pΣ R2 ` R4 q, x y r ´ pΣ p 21 ´ Σ p 21 Λ1 qR2 . R :“ R x xy and p 12 p l p 1 ´ Λ1 q ` pΣ p 11 ´ Ik qΦ pu Λ p p 12 p l p p 11 pu p u pΛ R1 :“ Φ x 1:k 1 ` Σx Φ1:k Λ1 ´ pΣxy ´ Λ1 qΨ1:k ´ Σxy Ψ1:k , 1:k p 12 Φ pl , pu ´ Σ pu Λ p1 ` Σ p 12 Ψ pl Λ p 1 ´ pΣ p 11 ´ Λ1 qΦ p 1 ´ Λ1 q ` pΣ p 11 ´ Ik qΨ p u pΛ R2 :“ Ψ R3 :“ R4 :“ 1:k 1:k y 1:k yx y 1:k 22 l 22 l l 21 u r p p p 1 ´ Λ1 q ` pΣ p Φ p Λ p p p Φ p pΛ Σ x x 1:k 1 ´ Φ1:k Λ1 q ´ pΣxy ´ Λ2 qΨ1:k , 1:k p 22 r J p l p 22 p l p pl p 21 pu p Σ y Ψ1:k pΛ1 ´ Λ1 q ` pΣy Ψ1:k Λ1 ´ Ψ1:k Λ1 q ´ pΣyx ´ Λ2 qΦ1:k . 17 yx 1:k The proof is deferred to Section 5.7. By Lemma 5.2, one can easily obtain that }R1 }, }R2 } ď C c p1 ` p2 . n Recall that p 21 Φ p u pΛ p 1 ´ Λ1 q ` pΣ p 22 Φ pl Λ p pl p 22 r p l R3 :“ Σ x 1:k x 1:k 1 ´ Φ1:k Λ1 q ´ pΣxy ´ Λ2 qΨ1:k By Lemma 5.2, we have and p 21 Φ p u pΛ p 1 ´ Λ1 q} ď C p1 ` p2 , }pΣ p 22 ´ Λ r 2 qΨ p l } ď C p1 ` p2 , }Σ x xy 1:k 1:k n ∆n p 22 Φ pl Λ p pl p 22 pl p pl p }Σ x 1:k 1 ´ Φ1:k Λ1 } ď }pΣx ´ Ip1 ´k qΦ1:k Λ1 ` Φ1:k pΛ1 ´ Λ1 q} p 22 ´ Ip ´k qΦ pl Λ p pl p ď }pΣ x 1:k 1 } ` }Φ1:k pΛ1 ´ Λ1 q} ď C 1 p1 ` p2 . ∆n `p2 `p2 . Similarly, }R4 } ď C p1∆n . Therefore, we get }R3 } ď C p1∆n Combined with Lemma 5.2, we have and r “ }pΣ p 21 R1 ´ R3 qΛ1 ´ Λ r 2 pΣ p 21 R2 ` R4 q} ď C p1 ` p2 }R} x y ∆n r ` }Σ p 21 ´ Σ p 21 Λ1 }}R2 } ď C p1 ` p2 . }R} ď }R} xy x ∆n The proof of the following lemma is deferred to Section 5.7: Lemma 5.4. If n ě C0 pp1 ` p2 q, then with probability 1 ´ c0 expp´γp1 q, »d ﬁ 2 qp1 ´ λ2 q p p1 ´ λ pp ` p q 1 1 2 ﬂ k k`1 pl } ď C – }Φ ` . 1:k n∆2 n∆2 5.5 Upper bounds of risks Notice that the inequality (5.4) yields ›2 › p l }2 }Φ › › 1:k . ›pIp1 ´ PΦ1:k q PΦ p 1:k › ď 2 p 1:k q σk pΦ By Lemma 5.4 and Lemma 5.2, we know on an event G with probability at least 1 ´ Ce´γp1 , ﬀ « › ›2 p1 p1 ´ λ2k qp1 ´ λ2k`1 q pp1 ` p2 q2 › › ` . ›pIp1 ´ PΦ1:k q PΦ p 1:k › ď C n∆2 n 2 ∆4 18 ›2 › › › Moreover, since ›pIp1 ´ PΦ1:k q PΦ p 1:k › ď 1, by (5.2), we have « ﬀ › ›2 p1 p1 ´ λ2k qp1 ´ λ2k`1 q pp1 ` p2 q2 › › p 1:k q “ E ›pIp ´ PΦ q P p › ď C ELmax pΦ1:k , Φ ` ` e´γp1 . 1 1:k Φ1:k n∆2 n 2 ∆4 Since pIp1 ´ PΦ1:k q PΦ p 1:k is of at most rank-k, we have › ›2 ›2 1 ›› › › › ď pI ´ P q P › ›pIp1 ´ PΦ1:k q PΦ › p1 Φ1:k p 1:k › p 1:k Φ k F Then by (5.1) and the previous inequality, we have › ›2 › › p ELave pΦ1:k , Φ1:k q “ E ›pIp1 ´ PΦ1:k q PΦ p 1:k › ›2 1 ›› › “ E ›pIp1 ´ PΦ1:k q PΦ p 1:k › k F ›2 › › › ď E ›pIp1 ´ PΦ1:k q PΦ p 1:k › ﬀ « p1 p1 ´ λ2k qp1 ´ λ2k`1 q pp1 ` p2 q2 ` ` e´γp1 . ďC n∆2 n 2 ∆4 In fact, the factor p1 in the main term can be reduced to p1 ´ k by similar arguments as done for the operator norm. The Frobenius norm version of Lemma 5.4 is actually much simpler. We omit the proof to avoid unnecessary redundancy and repetition. 5.6 Supporting lemmas in linear algebra and probability Definition 5.5. (Hadamard Operator Norm) For A P Rmˆn , define the Hadamard operator norm as ( \|\|\|A\|\|\| “ sup }A ˝ B} : }B} ď 1, B P Rmˆn Let α1 , ¨ ¨ ¨ , αm and β1 , ¨ ¨ ¨ , βn be arbitrary positive numbers lower bounded by a positive constant δ. n mˆn , Lemma 5.6. Let tαi um i“1 and tβi ui“1 be two sequences of positive numbers. for any X P R there hold › ›« a ﬀ › 1 › α β i j › › ˝ X › ď }X}, (5.14) › › 2 › αi ` βj and ›„ ›  › minpαi , βj q › 1 › ˝ X ›› ď }X}, › αi ` βj 2 ›„ ›  › maxpαi , βj q › 3 › ˝ X ›› ď }X}. › αi ` βj 2 (5.15) Proof. The proof of (5.14) can be found in “Norm Bounds for Hadamard Products and an Arithmetic-Geometric Mean Inequality for Unitarily Invariant Norms” by Horn. Denote  „  minpαi , βj q maxpαi , βj q , G2 “ G1 “ αi ` βj αi ` βj „ The proof of (5.15) relies on the following two results. 19 Lemma 5.7. (Theorem 5.5.18 of Hom and Johnson (1991)) If A, B P Rnˆn and A is positive semidefinite. Then, ˙ ˆ }A ˝ B} ď max Aii }B}, 1ďiďn where } ¨ } is the operator norm. Lemma 5.8. (Theorem 3.2 of Mathias (1993)) The symmetric matrix ´ minpa , a q ¯ i j ai ` aj 1ďi,jďn is positive semidefinite if ai ą 0, 1 ď i ď n. Define γi “ βi , 1 ď i ď n and γi “ αi´n , n ` 1 ď i ď m ` n. Define M P Rpm`nqˆpm`nq by Mij “ mintγi , γj u . γi ` γj By Lemma 5.8, M is also positive semidefinite. Again, apply Lemma 5.7 and notice that G2 is the lower left sub-matrix of M , It is easy to obtain \|\|\|G2 \|\|\| ď \|\|\|M \|\|\| ď 1 . 2 Finally, since G1 ˝ B “ B ´ G2 ˝ B for any B, we have }G1 ˝ B} ď }B} ` }G2 ˝ B}, which implies, 3 \|\|\|G1 \|\|\| ď 1 ` \|\|\|G2 \|\|\| ď . 2 Lemma 5.9. (Covariance Matrix Estimation, Remark 5.40 of Vershynin (2010)) Assume A P Rnˆp has independent sub-gaussian random rows with second moment matrix Σ. Then there exists universal constant C such that for every t ě 0, the following inequality holds with probability at 2 least 1 ´ e´ct , c 1 J t p 2 } A A ´ Σ} ď maxtδ, δ u}Σ} δ“C `? . n n n Lemma 5.10. (Bernstein inequality, Proposition 5.16 of Vershynin (2010)) Let X1 , ¨ ¨ ¨ , Xn be independent centered sub-exponential random variables and K “ maxi }Xi }ψ1 . Then for every a “ pa1 , ¨ ¨ ¨ , an q P Rn and every t ě 0, we have + # " ˆ ˙* n ÿ t t2 , ai Xi \| ě t ď 2exp ´c min . P \| K 2 }a}22 K}a}8 i“1 20 Lemma 5.11. (Hanson-Wright inequality, Theorem 1.1 of Rudelson and Vershynin (2013)) Let x “ px1 , ¨ ¨ ¨ , xp q be a random vector with independent components xi which satisfy Exi “ 0 and }xi }ψ2 ď K, Let A P Rpˆp . Then there exists universal constant c such that for every t ě 0, ˙* " ˆ ( t t2 , . P \|xJ Ax ´ ExJ Ax\| ě t ď 2exp ´c min K 4 }A}2F K 2 }A} Lemma 5.12. (Covering Number of the Sphere, Lemma 5.2 of Vershynin (2010)). The unit Euclidean sphere Sn´1 equipped with the Euclidean metric satisfies for every ǫ ą 0 that 2 \|N pSn´1 , ǫq\| ď p1 ` qn , ǫ where N pSn´1 , ǫq is the ǫ-net of Sn´1 with minimal cardinality. The following variant of Wedin’s sin θ law (Wedin, 1972) is proved in Proposition 1 of Cai et al. (2015). p “ A ` E, define the singular value decompositions of A Lemma 5.13. For A, E P Rmˆn and A p and A as p“U pD p Vp J . A “ U DV J , A Then the following perturbation bound holds, › › › › › › › › ›pI ´ PU1:k q PUp 1:k › “ ›PU1:k ´ PUp 1:k › ď 2}E} , σk pAq ´ σk`1 pAq where σk pAq, σk`1 pAq are the kth and pk ` 1qth singular values of A. 5.7 5.7.1 Proofs of key lemmas Proof of Lemma 5.2 (1) The proof of c › › p1 ` p2 › › p 1{2 p p p }Σ ´ Σ}, }Ip2 ´ Σy }, }Σxy ´ Σxy }, ›Σy ´ Ip2 › ď C n is exactly the same as that of Lemma 5.1. (2) Observe that p x´1{2 Σ p xy Σ p y´1{2 ´ Σxy “ pIp ´ Σ p x1{2 qΣ p x´1{2 Σ p xy Σ p y´1{2 Σ 1 p 1{2 Σ p ´1{2 Σ p xy Σ p ´1{2 pIp ´ Σ p 1{2 q ` pΣ p xy ´ Σxy q. `Σ x x y p1 ď 1. Then p x´1{2 Σ p xy Σ p y´1{2 } “ λ and }Σ 2 y p ´1{2 Σ p xy Σ p ´1{2 ´ Σxy } ď }Ip ´ Σ p 1{2 } ` }Σ p x }}Ip ´ Σ p 1{2 } ` }Σ p xy ´ Σxy }. }Σ x y x y 1 2 21 p and Λ are singular values of Σ p x´1{2 Σ p xy Σ p y´1{2 and Σxy respectively. Hence by the Notice that Λ famous Weyl’s inequality for singular values, p ´ Λ} ď }Σ p ´1{2 Σ p xy Σ p ´1{2 ´ Σxy } }Λ x y p p x }}Ip ´ Σ p 1{2 } ` }Σ p xy ´ Σxy } ď }Ip1 ´ Σx } ` }Σ y 2 ¸ ˜ c c c p1 ` p2 p1 ` p2 p1 ` p2 C1 ď C2 . ď 3 ` C1 n n n Ip1 p 1{2 p p ´1{2 Σ p xy Σ p y´1{2 , we have }Σ p 1{2 p pJp p (3) Since Σ x Φ are left singular vectors of Σx x Φ} “ 1, Φ Σx Φ “ p JΦ p ´ Ip “ ´Φ p J pΣ p x ´ Ip qΦ. p Then we have, and Φ 1 1 p JΦ p ´ Ip } “ }Φ p J pΣ p x ´ Ip qΦ} p ď }Φ p JΣ p 1{2 }}Σ p ´1{2 pΣ p x ´ Ip qΣ p ´1{2 }}Σ p 1{2 Φ} p }Φ 1 1 x x 1 x x p ´1{2 pΣ p x ´ Ip qΣ p ´1{2 }. “ }Σ x 1 x As a submatrix, pJ Φ p p ´1{2 pΣ p x ´ Ip qΣ p ´1{2 } ď }Σ p ´1 }}Σ p x ´ Ip } }Φ x x 1:k 1:k ´ Ik } ď }Σx 1 1 p 1 p x ´ Ip } ď }Σ ´ Σ} ď 1 ď }Σ 1 p x ´ Ip } p ´ Σ} 2 1 ´ }Σ 1 ´ }Σ 1 as long as n ě C0 pp1 ` p2 q for sufficiently large C0 . In this case, By the same argument, (4) Recall that p 1:k q ě 1{2, }Φ p 1:k }2 ď 3{2. σk2 pΦ p 1:k q ě 1{2, }Ψ p 1:k }2 ď 3{2. σk2 pΨ Φ1:k “ „ Ik 0pp1 ´kqˆk  , Ψ1:k “ „ Ik 0pp2 ´kqˆk  . p 1{2 p The last inequality in the lemma relies on the fact that Σ x Φ1:k and Φ1:k are leading k singular ´1{2 p ´1{2 p p vectors of Σx Σxy Σy and Σxy respectively. By a variant of Wedin’s sin θ law as stated in Lemma 5.13, c › 2}Σ › p x´1{2 Σ p xy Σ p y´1{2 ´ Σxy } 2C2 p1 ` p2 › › pIp1 ´ PΦ1:k q› ď ď . ›P Σ p p 1{2 x Φ1:k ∆ ∆ n On the other hand, › › › › › › p 1{2 p › › 1{2 p J p pIp1 ´ PΦ1:k q› “ ›Σx Φ1:k pΣx Φ1:k q pIp1 ´ PΦ1:k q› ›PΣ p p 1{2 x Φ1:k › › › › p 1{2 p J Φ q pI ´ P q “ ›pΣ p1 Φ1:k › 1:k x › › › p 1{2 p l› Φ q “ ›pΣ 1:k › , x 22 p 1{2 p Here the second equality is due to the fact that Σ x Φ1:k has orthonormal columns. Moreover, 1{2 px Φ p 1:k ql denotes the lower pp1 ´ kq ˆ k sub-matrix of Σ p 1{2 p pΣ x Φ1:k . Again, by triangle inequality, › › › ´ ¯l ›› › p l › ›› p 1{2 p l 1{2 p p ›Φ1:k › “ ›pΣx Φ1:k q ´ pΣx ´ Ip1 qΦ1:k ›› › › › › ›› › p 1{2 p › p 1{2 ››p › l› Φ ď ›pΣ Φ q ` Σ ´ I q ›p x p1 › › 1:k › 1:k › x c c c c 2C2 p1 ` p2 C3 p 1 ` p 2 3 p1 ` p2 ď ` C1 ď . ∆ n 2 n ∆ n The last inequality is due to ∆ ď 1. Let C “ maxpC1 , C2 , C3 q, the proof is done. 5.7.2 Proof of Lemma 5.3 The equality (5.10) implies pu ´ Φ p u Λ1 “ Φ p u pΛ p 1 ´ Λ1 q ` pΣ p 11 ´ Ik qΦ pu Λ p p 12 p l p Λ1 Ψ 1:k 1:k 1:k x 1:k 1 ` Σx Φ1:k Λ1 p 12 p l :“ R1 . p 11 pu ´ Σ ´ pΣ xy Ψ xy ´ Λ1 qΨ (5.16) p 1 ´ Λ1 q ` pΣ p 11 pu p p 12 p l p p u pΛ p u Λ1 “ Ψ pu ´ Ψ Λ1 Φ y ´ Ik qΨ1:k Λ1 ` Σy Ψ1:k Λ1 1:k 1:k 1:k p 12 Φ p l :“ R2 . p 11 ´ Λ1 qΦ pu ´ Σ ´ pΣ (5.17) 1:k 1:k Similarly, (5.11) implies yx yx 1:k 1:k The equality (5.8) is equivalent to p 21 p u p p 21 p u p 22 r p l r pl p 21 Ψ pu Σ xy 1:k ` Λ2 Ψ1:k ` pΣxy ´ Λ2 qΨ1:k “ Σx Φ1:k Λ1 ` Σx Φ1:k pΛ1 ´ Λ1 q p 22 pl p pl p l Λ1 ` pΣ `Φ x Φ Λ1 ´ Φ Λ1 q, 1:k 1:k 1:k which can be written as p 21 p u p pl r pl p 21 p u p 21 pu Σ xy Ψ1:k ` Λ2 Ψ1:k ´ Σx Φ1:k Λ1 ´ Φ1:k Λ1 “ Σx Φ1:k pΛ1 ´ Λ1 q p 22 Φ pl Λ p1 ´ Φ p l Λ1 q ´ pΣ p 22 ´ Λ r 2 qΨ p l :“ R3 . ` pΣ (5.18) p 21 Φ pu rJ pl p 21 p u pl p 21 p u p Σ yx 1:k ` Λ2 Φ1:k ´ Σy Ψ1:k Λ1 ´ Ψ1:k Λ1 “ Σy Ψ1:k pΛ1 ´ Λ1 q p 22 r J p l :“ R4 . p 22 pl p pl ` pΣ y Ψ Λ1 ´ Ψ Λ1 q ´ pΣyx ´ Λ2 qΦ (5.19) x 1:k 1:k xy 1:k Apply the same argument to (5.9), we obtain 1:k 1:k 1:k r 2 ˆ (5.19), then Consider (5.18) ˆ p´Λ1 q ´ Λ that is r p 21 p u r p 21 p u p 21 p u 2 p 21 p u pl p l Λ2 ´ Λ2 Φ Φ 2 1:k ` Σx Φ1:k Λ1 ´ Σxy Ψ1:k Λ1 ´ Λ2 Σyx Φ1:k ` Λ2 Σy Ψ1:k Λ1 1:k 1 r 2 R4 q, “ ´pR3 Λ1 ` Λ p l Λ2 ´ Λ2 Φ pl p 21 p u r p 21 p u Φ 1:k 1 2 1:k “Σxy Ψ1:k Λ1 ` Λ2 Σyx Φ1:k r 2 R4 q. r 2Σ p 21 Ψ p u Λ1 ´ pR3 Λ1 ` Λ p 21 Φ p u Λ2 ´ Λ ´Σ y x 1:k 1:k 1 23 (5.20) Combined with (5.16) and (5.17), p l Λ2 ´ Λ2 Φ pl p 21 p u r p 21 p u p 21 p u p 21 Φ 2 1:k “ Σxy Ψ1:k Λ1 ` Λ2 Σyx Φ1:k ´ Σx Λ1 Ψ1:k Λ1 ` Σx R1 Λ1 1:k 1 r 2Σ p 21 r 2Σ pu ´ Λ p 21 r ´Λ y Λ1 Φ y R2 ´ pR3 Λ1 ` Λ2 R4 q 1:k p 21 ´ Σ p 21 Λ1 qΨ p u Λ1 ` Λ r 2 pΣ p 21 ´ Σ p 21 Λ1 qΦ pu “ pΣ xy x yx y 1:k 1:k 21 21 p r p ` pΣx R1 ´ R3 qΛ1 ´ Λ2 pΣy R2 ` R4 q. This finishes the proof of (5.13). Plug (5.17) into (5.21), we get p l Λ2 ´ Λ r 2Φ pl p 21 p 21 pu r p 21 p 21 pu r Φ 1:k 1 2 1:k “ pΣxy ´ Σx Λ1 qpΛ1 Φ1:k ´ R2 q ` Λ2 pΣyx ´ Σy Λ1 qΦ1:k ` R r ´ pΣ p 21 ´ Σ p 21 Λ1 qR2 q. p u ` pR “ BΦ xy 1:k x This finishes the proof of (5.12). 5.7.3 Proof of Lemma 5.4 First, we discuss two quite different cases: λk ě Case 1: λk ě 1 2 and λk ă 21 . 1 2 Let 1 δ :“ λ2k ´ λ2k`1 “ pλk ´ λk`1 qpλk ` λk`1 q ě ∆. 2 Define the pp1 ´ kq ˆ k matrices A by b b λ2j ´ λ2k ` 2δ λ2k`1 ´ λ2k`i ` 2δ Aij “ , 1 ď i ď p1 ´ k, 1 ď j ď k λ2j ´ λ2k`i By (5.12) in Lemma 5.3, there holds where and By Lemma 5.6, we have p u D2 q ` A ˝ pD1 RD2 q, p l “ A ˝ pD1 B Φ Φ 1:k 1:k ¨ 1 D1 “ diag ˝ b , ¨ ¨ ¨ , b ¨ δ 2 D2 “ diag ˝ b 1 λ2k`1 1 λ21 ´ λ2k ` δ 2 ´ λ2p1 ` δ 2 ˛ ˛ ‚ 1 , ¨ ¨ ¨ , b ‚. δ 2 p u D2 } ` 1 }pD1 RD2 q} p l } ď 1 }D1 B Φ }Φ 1:k 1:k 2 2 1 1 u p ď }D1 B}}Φ1:k }}D2 } ` }D1 }}R}}D2 }. 2 2 24 (5.21) p u } ď }Φ p 1:k } ď Recall that }Φ 1:k b 3 2 and it is obvious that }D1 }, }D2 } ď b 2 δ. Moreover, in the 1 `p2 q previous section, we also have shown that }R} ď Cppn∆ . It suffices to bound }D1 B} and to this end we apply the standard covering argument. Step 1. Reduction. Denote by Nǫ pSd q the d-dimensional unit ball surface. For ǫ ą 0 and any pair of vectors u P Rp1 ´k , v P Rk , we can choose uǫ P Nǫ pSp1 ´k´1 q, vǫ P Nǫ pSk´1 q such that }u ´ uǫ }, }v ´ vǫ } ď ǫ. Then J J J uJ D1 Bv “ uJ D1 Bv ´ uJ ǫ D1 Bv ` uǫ D1 Bv ´ uǫ D1 Bvǫ ` uǫ D1 Bvǫ J ď }u ´ uǫ }}D1 Bv} ` }uJ ǫ D1 B}}v ´ vǫ } ` uǫ D1 Bvǫ ď 2ǫ}D1 B} ` uJ ǫ D1 Bvǫ ď 2ǫ}D1 B} ` max uJ ǫ D1 Bvǫ . uǫ ,vǫ Maximize over u and v, we obtain }D1 B} ď 2ǫ}D1 B} ` max uJ ǫ D1 Bvǫ . uǫ ,vǫ Therefore, }D1 B} ď p1 ´ 2ǫq´1 max uJ ǫ D1 Bvǫ . Let ǫ “ 1{4. Then it suffices to give an upper uǫ ,vǫ bound max uJ ǫ D1 Bvǫ with high probability. uǫ ,vǫ Step 2. Concentration. 1 ď i ď p1 ´ k and 1 ď j ď k rD1 Bsi,j “b 1 λ2k`1 ´ λ2k`i ` δ 2 Y ´λl Xl ? 2 for all 1 ď α ď n and 1 ď l ď p1 . Then for Let Zα,l “ α,l 1´λl n 1 ÿ pλj Xα,k`i Yα,j ´ λ2j Xα,k`i Xα,j ` λk`i Yα,k`i Xα,j ´ λk`i λj Yα,k`i Yα,j q n α“1 n 1 ÿ! p1 ´ λ2j qλk`i λj Xα,k`i Xα,j ´ λ2j pYα,k`i ´ λk`i Xα,k`i qpYα,j ´ λj Xα,j q δ n 2 2 λk`1 ´ λk`i ` 2 α“1 ) ` p1 ´ λ2j qλj pYα,k`i ´ λk`i Xα,k`i qXα,j ` p1 ´ λ2j qλk`i pYα,j ´ λj Xα,j qXα,k`i . “b 1 n 1 ÿ! p1 ´ λ2j qλk`i λj Xα,k`i Xα,j δ n 2 2 λk`1 ´ λk`i ` 2 α“1 b b b ´ λ2j 1 ´ λ2k`i 1 ´ λ2j Zα,k`i Zα,j ` p1 ´ λ2j qλj 1 ´ λ2k`i Zα,k`i Xα,j b ) ` p1 ´ λ2j qλk`i 1 ´ λ2k`i Xα,k`i Zα,j . “b 1 In this way, tXα,k`i , Zα,k`i , 1 ď i ď p1 , 1 ď α ď nu are mutually independent standard gaussian 25 random variables. For any given pair of vectors u P Rp1 ´k , v P Rk , uJ D1 Bv “ n p1 ´k ÿ k ! ui vj 1 ÿ ÿ b p1 ´ λ2j qλk`i λj Xα,k`i Xα,j n α“1 i“1 j“1 λ2 ´ λ2 ` δ k`1 k`i 2 b b b ´ λ2j 1 ´ λ2k`i 1 ´ λ2j Zα,k`i Zα,j ` p1 ´ λ2j qλj 1 ´ λ2k`i Zα,k`i Xα,j b ) ` p1 ´ λ2j qλk`i 1 ´ λ2k`i Xα,k`i Zα,j n . 1 ÿ J “ w Aα wα , n α“1 α where J wαJ “ rxJ α , zα s “ rXα,1 , . . . , Xα,p1 , Zα,1 , . . . , Zα,p1 s and Aα P Rp2p1 qˆp2p1 q is symmetric and determined by the corresponding quadratic form. This yields }Aα }2F “ p1 ´k ÿ k ! u2i vj2 1 ÿ p1 ´ λ2j q2 λ2k`i λ2j ` λ4j p1 ´ λ2k`i qp1 ´ λ2j q 2 i“1 j“1 λ2k`1 ´ λ2k`i ` 2δ ) ` p1 ´ λ2j q2 λ2j p1 ´ λ2k`i q ` p1 ´ λ2j q2 λ2k`i p1 ´ λ2k`i q p1 ´k ÿ k ` ˘` ˘ u2i vj2 1 ÿ 1 ´ λ2j λ2k`i ` λ2j ´ 2λ2k`i λ2j δ 2 2 2 i“1 j“1 λk`1 ´ λk`i ` 2 ¸ ˜p ´k ¸ ˜ k 1 ÿ p1 ´ λ2j qpλ2k`i ` λ2j ´ 2λ2k`i λ2j q 1 ÿ max vj2 u2i ď 1ďiďp1 ´k 2 i“1 λ2k`1 ´ λ2k`i ` 2δ j“1 “ ď p1 1 max 2 1ďiďp1 ´k 1ďjďk ď p1 ´ λ2k q max 1ďiďp1 ´k δ 1ďjďk 2 ď p1 ´ λ2k q ď 2p1 ´ 1ďjďk 2 2 ´ λk qp2λj ´ 2λ2k`i λ2j q λ2k`1 ´ λ2k`i ` 2δ max λ2j p1 ´ λ2k`i q ` λ2k`1 ´ λ2i`k p1 ´ λ2k`1 q 1ďiďp1 ´k 1ďjďk 2 λk qp1 ´ λ2k`1 q δ δ 2 . “ K 2, where the second last inequality is due to the facts that λj ď 1 and δ 2 p1 ´ λ2k`i q ` λ2k`1 ´ λ2i`k ď p1 ´ λ2k`1 q δ 2 Moreover }Aα }22 ď }Aα }2F ď K 2 . 26 p7 δ ` λ2k`1 ă λ2k ď 1q. 2 Now define w J :“ rw1J , . . . , wnJ s and » A1 — A2 — A“— – Then we have }A} ď max }Aα } ď K, 1ďαďn .. . }A}2F and ﬁ An ď ﬃ ﬃ ﬃ. ﬂ n ÿ α“1 }Aα }2F ď nK 2 1 J w Aw, where w P N2p1 n p0, I2p1 n q. n Therefore, By the classic Hanson-Wright inequality (Lemma 5.11), there holds " ˆ 2 ˙* ( t t J P n\|u D1 Bv\| ě t ď 2 exp ´c0 min , nK 2 K uJ D1 Bv “ for some numerical constant c0 ą 0. Without loss of generality, we can also assume c0 ď 1. Let ? t “ c40 np1 K. By n ě p1 , straightforward calculation gives " * 4? J P n\|u D1 Bv\| ě np1 K ď 2e´4p1 . c0 Step 3. Union Bound. By Lemma 5.12, we choose 1{4-net such that , $ d / ’ ˆ ? ˙c & 2 2 p1 p1 ´ λk qp1 ´ λk`1 q . 4 2 J P max uǫ D1 Bvǫ ě / ’ c0 n δ %uǫ PNǫ pSp1 ´k´1q vǫ PNǫ pSk´1 q 3 ď 9p1 ´k 9k ˆ 2e´4p1 ď 2e´ 2 p1 . 3 In other words, with probability at least 1 ´ 2e´ 2 p1 , we have ˆ ? ˙c p1 8 2 J ´1 }D1 B} ď p1 ´ 2ǫq max uǫ D1 Bvǫ ď uǫ ,vǫ c0 n d p1 ´ λ2k qp1 ´ λ2k`1 q . δ In summary, we have as long as n ě C0 pp1 ` p2 q, with probability 1 ´ c0 expp´γp1 q, »d ﬁ 2 2 p1 p1 ´ λk qp1 ´ λk`1 q pp1 ` p2 q pl } ď C – ﬂ }Φ ` 1:k nδ2 n∆δ ﬁ »d p1 p1 ´ λ2k qp1 ´ λ2k`1 q pp1 ` p2 q ﬂ. ` ďC– n∆2 n∆2 Here the last inequality is due to δ “ pλk ` λk`1 q∆ ě 21 ∆. Here C0 , C, c0 , γ are absolute constants. 27 Case 2: λk ď 1 2 By (5.13), we have p l “ GΛ1 ` Λ2 F , p l Λ21 ´ Λ22 Φ Φ 1:k 1:k where and p 21 R1 ´ R3 q p 21 ´ Σ p 21 Λ1 qΨ p u ` pΣ G :“ pΣ x xy x 1:k ” ı p 21 ´ Σ p 21 Λ1 qΦ p u ´ pΣ p 21 R2 ` R4 q . F :“ rIp1 , 0p1 ˆpp2 ´p1 q s pΣ yx y 1:k y p 21 and Σ p 21 are submatrices of Σ p 2p . By Lemma 5.1, we have Notice that Σ xy x 1 Moreover, by }R1 } ď C b p1 `p2 n , p 21 ´ Σ p 21 Λ1 } ď C }Σ xy x c p1 . n `p2 }R3 } ď C p1n∆ and Lemma 5.2, there holds }G} ď C ˆc p1 p1 ` p2 ` n n∆ ˙ . p 2p . By a similar p 21 are submatrices of Σ p 21 and rIp , 0p ˆpp ´p q sΣ Similarly, rIp1 , 0p1 ˆpp2 ´p1 q sΣ x 1 yx 1 1 2 1 argument, ˙ ˆc p1 p1 ` p2 ` . }F } ď C n n∆ Then pl “ Φ 1:k „  „  „  „  λj 1 1 λk`i ˝ ˝ ˝G` ˝F λk`i ` λj λj ´ λk`i λk`i ` λj λj ´ λk`i Here 1 ď i ď p1 ´ k and 1 ď j ď k. By Lemma 5.6, there holds for any X, ›„  › ›„  › › › › maxpλk`i , λj q › 3 λj › ›“› X X ›› ď }X} › λk`i ` λj › › λk`i ` λj 2 and Finally, for any X, where ›„  › ›„  › › › minpλk`i , λj q › 1 › λ k`i › › › X ›› ď }X}. › λk`i ` λj X › “ › λk`i ` λj 2 „  1 X “ A ˝ pD1 XD2 q λj ´ λk`i »b A :“ – λj ´ λk ` ∆ 2 b λk`1 ´ λk`i ` λj ´ λk`i 28 ∆ 2 ﬁ ﬂ, ¨ 1 D1 “ diag ˝ b , ¨ ¨ ¨ , b and ¨ Since }D1 }, }D2 } ď b In summary, we have Since 1 2 2 ∆, D2 “ diag ˝ b λk`1 ´ λp1 ` 1 λ1 ´ λk ` ∆ 2 ∆ 2 ˛ ‚, 1 , ¨ ¨ ¨ , b ‚. ∆ 2 by Lemma 5.6, ›„  › › › 1 1 1 › › › λj ´ λk`i X › ď 2 }D1 XD2 } ď ∆ }X}. pl } ď C }Φ 1:k ě λk ě λk`1 , there holds »d pl } ď C – }Φ 1:k 6 ∆ 2 1 ˛ ˆc p1 p1 ` p2 ` 2 n∆ n∆2 ˙ . ﬁ p1 p1 ´ λ2k qp1 ´ λ2k`1 q pp1 ` p2 q ﬂ. ` n∆2 n∆2 Lower Bound: Proof of Theorem 2.2 To establish the minimax lower bounds of CCA estimates for our proposed losses, we follow the analytical frameworks in the literature of PCA and CCA, e.g., Vu et al. (2013); Cai et al. (2013); Gao et al. (2015), where the calculation is focused on the construction of the hypothesis class to which the packing lemma and Fano’s inequality are applied. However, since we fix both λk and λk`1 in the localized parameter spaces, new technical challenges arise and consequently we construct hypothesis classes based on the equality (6.1). In this section we also denote ∆ :“ λk ´ λk`1 . 6.1 On Kullback-Leibler Divergence The following lemma can be viewed as an extension of Lemma 14 in Gao et al. (2015) from λk`1 “ 0 to arbitrary λk`1 . The proof of the lemma can be found in Section 6.4. “ ‰ “ ‰ Lemma 6.1. For i “ 1, 2 and p2 ě p1 ě k, let Upiq , Wpiq P Opp1 , p1 q, Vpiq , Zpiq P Opp2 , p1 q where Upiq P Rp1 ˆk , Vpiq P Rp2 ˆk . For 0 ď λ2 ă λ1 ă 1, let ∆ “ λ1 ´ λ2 and define « ﬀ 1{2 J ` λ W Z J qΣ1{2 Σx Σx pλ1 Upiq Vpiq y 2 piq piq Σpiq “ i “ 1, 2, 1{2 J ` λ Z W J qΣ1{2 Σy pλ1 Vpiq Upiq Σy x 2 piq piq Let Ppiq denote the distribution of a random i.i.d. sample of size n from N p0, Σpiq q. If we further assume « Jﬀ « Jﬀ Vp1q Vp2q rUp1q , Wp1q s “ rU , W s , (6.1) p2q p2q J J Zp1q Zp2q 29 Then one can show that DpPp1q \|\|Pp2q q “ n∆2 p1 ` λ1 λ2 q J 2 }U V J ´ Up2q Vp2q }F . 2p1 ´ λ21 qp1 ´ λ22 q p1q p1q Remark 6.2. The conditon in (6.1) is crucial for obtaining the eigen-gap factor 1{∆2 in the lower bound and is the key insight behind the construction of the hypothesis class in the proof. Gao et al. (2015) has a similar lemma but only deals with the case that the residual canonical correlations are zero. To the best of our knowledge, the proof techniques in Gao et al. (2015, 2017) cannot be directly used to obtain our results. 6.2 Packing Number and Fano’s Lemma The following result on the packing number is based on the metric entropy of the Grassmannian manifold Gpk, rq due to Szarek (1982). We use the version adapted from Lemma 1 of Cai et al. (2013) which is also used in Gao et al. (2015). Lemmaa6.3. For any fixed U0 P Opp, kq and Bǫ0 “ tU P Opp, kq : }U U J ´ U0 U0J }F ď ǫ0 u with ǫ0 P p0, 2rk ^ pp ´ kqs q. Define the semi-metric ρp¨, ¨q on Bǫ0 by ρpU1 , U2 q “ }U1 U1J ´ U2 U2J }F . Then there exists universal constant C such that for any α P p0, 1q, the packing number MpBǫ0 , ρ, αǫ0 q satisfies ˙ ˆ 1 kpp´kq . MpBǫ0 , ρ, αǫ0 q ě Cα The following corollary is used to prove the lower bound. Corollary 6.4. If we change the set in Lemma 6.3 to Brǫ0 “ tU P Opp, kq : }U ´ U0 }F ď ǫ0 u, then we still have ˙ ˆ 1 kpp´kq r . MpBǫ0 , ρ, αǫ0 q ě Cα Proof. Apply Lemma 6.3 to Bǫ0 , there exists U1 , ¨ ¨ ¨ , Un with n ě p1{Cαqkpp´kq such that }Ui UiJ ´ U0 U0J }F ď ǫ0 , 1 ď i ď n, }Ui UiJ ´ Uj UjJ }F ě αǫ0 , 1 ď i ď j ď n. r i “ arg Define U min U PtUi Q, QPOpkqu }U ´ U0 }F , by Lemma 6.5, r i ´ U 0 }F ď }U riU r iJ ´ U0 U0J }F ď ǫ0 . }U r1 , ¨ ¨ ¨ , U rn P Brǫ and Therefore, U 0 which implies, riU r iJ ´ U rj U r jJ }F “ }Ui UiJ ´ Uj UjJ }F ě αǫ0 . }U MpBrǫ0 , ρ, αǫ0 q ě n ě 30 ˆ 1 Cα ˙kpp´kq . Lemma 6.5. For any matrices U1 , U2 P Opp, kq, inf QPOpk,kq }U1 ´ U2 Q}F ď }PU1 ´ PU2 }F Proof. By definition }U1 ´ U2 Q}2F “ 2k ´ 2trpU1J U2 Qq Let U1J U2 “ U DV J be the singular value decomposition. Then V U J P Opk, kq and inf QPOpk,kq }U1 ´ U2 Q}2F ď 2k ´ 2trpU1J U2 V U J q “ 2k ´ 2trpU DU J q “ 2k ´ 2trpDq. On the other hand, }PU1 ´ PU2 }2F “ }U1 U1J ´ U2 U2J }2F “ 2k ´ 2trpU1 U1J U2 U2J q “ 2k ´ 2trpU1J U2 U2J U1 q “ 2k ´ 2trpD 2 q. Since U1 , U2 P Opp, kq, }U1J U2 } ď 1 and therefore all the diagonal elements of D is less than 1, which implies that trpDq ě trpD 2 q and inf QPOpk,kq }U1 ´ U2 Q}2F ď }PU1 ´ PU2 }2F . Lemma 6.6 (Fano’s Lemma Yu (1997)). Let pΘ, ρq be a (semi)metric space and tPθ : θ P Θu a collection of probability measures. For any totally bounded T Ă Θ, denote MpT, ρ, ǫq the ǫ-packing number of T with respect to the metric ρ, i.e. , the maximal number of points in T whoese pairwise minimum distance in ρ is at least ǫ. Define the Kullback-Leibler diameter of T by dKL pT q “ sup DpPθ \|\|Pθ1 q. θ,θ 1 PT Then, ¯ ” ı 2´ p θq ě sup sup ǫ 1 ´ dKL pT q ` log 2 inf sup Eθ ρ2 pθ, log MpT, ρ, ǫq T ĂΘ ǫą0 4 θp θPΘ 31 6.3 Proof of Lower Bound ‰ ‰ “ “ For any fixed Up0q , Wp0q P Opp1 , p1 q and Vp0q , Zp0q P Opp2 , p1 q where Up0q P Rp1ˆk , Vp0q P Rp2 ˆk , Wp0q P Rp1 ˆpp1 ´kq , Vp0q P Rp2 ˆpp2 ´kq , define !` ˘ “ ‰ “ ‰ Hǫ0 “ U , W , V , Z : U , W P Opp1 , p1 q with U P Rp1 ˆk , V , Z P Opp2 , p1 q « ﬀ « Jﬀ J Vp0q ) V with V P Rp2 ˆk , }U ´ Up0q }F ď ǫ0 , rU , W s “ rU , W s . p0q p0q J ZJ Zp0q For any fixed Σx P Sp`1 , Σy P Sp`2 with κpΣx q “ κx , κpΣy q “ κy , consider the parametrization Σxy “ Σx ΦΛΨJ Σy , for 0 ď λk`1 ă λk ă 1, define « ﬀ 1{2 1{2 ! Σx Σx pλk U V J ` λk`1 W Z J qΣy Tǫ0 “ Σ “ , 1{2 1{2 Σy Σy pλk V U J ` λk`1 ZW J qΣx ) ` ˘ Φ “ Σx´1{2 rU , W s, Ψ “ Σy´1{2 rV , Zs, U , W , V , Z P Hǫ0 . It is straightforward to verify that Tǫ0 Ă Fpp1 , p2 , k, λk , λk`1 , κx , κy q. For any Σpiq P Tǫ0 , i “ 1, 2, they yield to the parametrization, « ﬀ 1{2 1{2 J `λ J Σx Σx pλk Upiq Vpiq k`1 Wpiq Zpiq qΣy Σpiq “ , 1{2 1{2 J `λ J Σy pλk Vpiq Upiq Σy k`1 Zpiq Wpiq qΣx ˘ ` ´1{2 piq piq where Upiq , Wpiq , Vpiq , Zpiq P Hǫ0 and the leading-k canonical vectors are Φ1:k “ Σx Upiq , Ψ1:k “ ´1{2 Σy Vpiq . We define a semi-metric on Tǫ0 as › › › › › › › › ρpΣp1q , Σp2q q “ ›PΣ1{2 Φp1q ´ PΣ1{2 Φp2q › “ ›PUp1q ´ PUp2q › . x x 1:k 1:k F F By Lemma 6.1, DpPΣ1 \|\|PΣ2 q “ n∆2 p1 ` λk λk`1 q J 2 }U V J ´ Up2q Vp2q }F . 2p1 ´ λ2k qp1 ´ λ2k`1 q p1q p1q Further by the definition of dKL pT q, dKL pT q “ n∆2 p1 ` λk λk`1 q J J 2 sup }Up1q Vp1q ´ Up2q Vp2q }F . 2p1 ´ λ2k qp1 ´ λ2k`1 q Σp1q ,Σp2q PTǫ0 (6.2) To bound the Kullback-Leibler diameter, for any Σp1q , Σp2q P Tǫ0 , by definition, « Jﬀ « Jﬀ Vp1q Vp2q rUp1q , Wp1q s “ rUp2q , Wp2q s , J J Zp1q Zp2q which implies that they are singular value decompositions of the same matrix. Therefore, there exists Q P Opp1 , p1 q such that rUp2q , Wp2q s “ rUp1q , Wp1q sQ , rVp2q , Zp2q s “ rVp1q , Zp1q sQ. 32 (6.3) Decompose Q into four blocks such that „  Q11 Q12 Q“ . Q21 Q22 Substitute into (6.3), Up2q “ Up1q Q11 ` Wp1q Q21 , Vp2q “ Vp1q Q11 ` Zp1q Q21 . Then, }Up2q ´ Up1q }2F “ }Up1q pQ11 ´ Ik q ` Wp1q Q21 }2F “ }Up1q pQ11 ´ Ik q}2F ` }Wp1q Q21 }2F “ }Q11 ´ Ik }2F ` }Q21 }2F . The second equality is due to the fact that Up1q and Wp1q have orthogonal column space and the third equality is valid because Up1q , Wp1q P Opp1 , kq. By the same argument, we will have }Vp2q ´ Vp1q }2F “ }Q11 ´ Ik }2F ` }Q21 }2F . Notice that J J 2 }Up1q Vp1q ´ Up2q Vp2q }F “ }pUp1q ´ Up2q qVp1q ` Up2q pVp1q ´ Vp2q q}2F ď 2}Up1q ´ Up2q }2F ` 2}Vp1q ´ Vp2q }2F “ 4}pUp1q ´ Up2q q}2F ` ˘ ď 8 }pUp1q ´ Up0q q}2F ` }pUp0q ´ Up2q q}2F ď 16ǫ20 . Then, substitute into (6.2) 8n∆2 p1 ` λk λk`1 q 2 ǫ . p1 ´ λ2k qp1 ´ λ2k`1 q 0 dKL pT q ď (6.4) J ´ Let Bǫ0 “ tU P Opp1 , kq : }U ´ Up0q }F ď ǫ0 u. Under the semi-metric ρrpUp1q , Up2q q “ }Up1q Up1q J } , we claim that the packing number of H is lower bounded by the packing number of B . Up2q Up2q F ǫ0 ǫ0 To prove this claim, it suffices to show that for any U P Bǫ0 , there exists corresponding W , V , Z such that pU , W , V , Zq P Hǫ0 . First of all, by definition, }U ´U0 }F ď ǫ0 . Let W P Opp1 , p1 ´kq be the orthogonal complement of U . Then rU , W s P Opp1 , p1 q and therefore there exists Q P Opp1 , p1 q such that rU , W s “ rUp0q , Wp0q sQ. Set rV , Zs “ rVp0q , Zp0q sQ P Opp2 , p1 q, then rU , W s « VJ ZJ ﬀ “ rUp0q , Wp0q s 33 « J Vp0q J Zp0q ﬀ , which implies pU , W , V , Zq P Hǫ0 . Let ¨ ˛ d a p1 ´ λ2k qp1 ´ λ2k`1 q ǫ “ αǫ0 “ c ˝ k ^ pp1 ´ kq ^ kpp1 ´ kq‚, n∆2 p1 ` λk λk`1 q a where c P p0, 1q depends on α and is chosen small enough such that ǫ0 “ ǫ{α P p0, 2rk ^ pp1 ´ kqs q. By Corollary 6.4, ˆ ˙ 1 kpp1 ´kq MpTǫ0 , ρ, αǫ0 q “ MpHǫ0 , ρr, αǫ0 q ě MpBǫ0 , ρr, αǫ0 q ě . Cα Apply Lemma 6.6 with Tǫ0 , ρ, ǫ, ˜ ¸ „ › ›2  8c2 kpp1 ´ kq ` log2 ǫ2 › › 1´ . inf sup E ›PΣ1{2 Φ › ě sup sup p 1:k ´ PΣ1{2 1 x Φ1:k F x p 1:k ΣPF kpp1 ´ kqlog Cα T ĂΘ ǫą0 4 Φ Choose α small enough such that 1´ 1 8c2 kpp1 ´ kq ` log2 ě . 1 2 kpp1 ´ kqlog Cα Then the lower bound is reduced to # + „ › ›2  c2 p1 ´ λ2 qp1 ´ λ2 q › › k`1 k kpp1 ´ kq ^ k ^ pp1 ´ kq inf sup E ›PΣ1{2 Φ › ě p 1:k ´ PΣ1{2 x Φ1:k F x 8 n∆2 p1 ` λk λk`1 q p 1:k ΣPF Φ ¸ + #˜ p1 ´ λ2k qp1 ´ λ2k`1 q p1 ´ k p1 ´ k 2 ^1^ ěC k ∆2 n k By symmetry, „ › › inf sup E ›PΣ1{2 Ψ p p 1:k ΣPF Ψ y 1:k ´ PΣ1{2 Ψ y 1:k ›2  › › ě C 2k F #˜ p1 ´ λ2k qp1 ´ λ2k`1 q p1 ´ k ∆2 n ¸ p1 ´ k ^1^ k + The lower bound for operator norm error can be immediately obtained by noticing that PΣ1{2 Ψ p 1:k ´ y PΣ1{2 Ψ has at most rank 2k and y 1:k › › ›PΣ1{2 Φ p x 6.4 1:k Proof of Lemma 6.1 ›2 ›2 1 ›› › › ´ PΣ1{2 Φ › ě ´ PΣ1{2 Φ › ›PΣ1{2 Φ p x x x 1:k 1:k 1:k 2k F By simple algebra, the Kullback-Leibler divergence between two multivariate gaussian distributions satisfies ) ¯ n ! ´ ´1 ´1 DpPΣp1q \|\|PΣp2q q “ Tr Σp2q pΣp1q ´ Σp2q q ´ log detpΣp2q Σp1q q . 2 Notice that ﬀ « ﬀ « 1{2 1{2 Σx Σx Σpiq “ 1{2 , 1{2 Ωpiq Σy Σy 34 where Ωpiq ﬀ J ` λ W ZJ Ip1 λ1 Upiq Vpiq 2 piq piq . “ J ` λ Z WJ λ1 Vpiq Upiq Ip2 2 piq piq « Then, DpPΣp1q \|\|PΣp2q q “ Also notice that Ωpiq “ „ Ip1 ) n! ´1 Ω q . Ω q ´ pp ` p q ´ log detpΩ TrpΩ´1 1 2 p1q p1q p2q p2q 2  „ ı λ „ U ” ı λ1 Upiq ” J 1 J J J piq U V Upiq ´Vpiq ´ ` piq piq Ip2 2 Vpiq 2 ´Vpiq ”  „ „ ı ı λ2 Wpiq λ2 Wpiq ” J J J J Wpiq Zpiq Wpiq ´Zpiq ` ´ . 2 Zpiq 2 ´Zpiq  Therefore Ωp1q , Ωp2q share the same set of eigenvalues: 1 ` λ1 with multiplicity k, 1 ´ λ1 with multiplicity k, 1 ` λ2 with multiplicity p1 ´ k, 1 ´ λ2 with multiplicity p1 ´ k and 1 with multiplicity 2pp2 ´ p1 q. This implies log detpΩ´1 Ω qq “ 0. On the other hand, by block inversion formula, we p2q p1q can compute ﬁ » 2 λ21 λ1 J ` λ2 W W J J ´ λ2 W Z J U U U V ´ I ` p 2 2 p2q p2q p2q p2q p2q p2q 1´λ2 p2q 1´λ2 p2q ﬂ 1´λ1 – 1 1´λ1 . Ω´1 λ21 λ22 p2q “ λ2 λ1 J J J J Ip2 ` 1´λ2 Vp2q Vp2q ` 1´λ2 Zp2q Zp2q ´ 1´λ2 Vp2q Up2q ´ 1´λ2 Zp2q Wp2q 1 1 Divide Ω´1 p2q Ωp1q into blocks such that Ω´1 p2q Ωp1q “ „  J11 J12 where J11 P Rp1 ˆp1 , J22 P Rp2 ˆp2 , J21 J22 and λ22 λ21 J J J J J pU U ´ U V V U q ` pWp2q Wp2q ´ Wp2q Zp2q Zp1q Wp1q q p2q p2q p2q p2q p1q p1q 1 ´ λ21 1 ´ λ22 λ1 λ2 λ1 λ2 J J J J ´ pUp2q Vp2q Zp1q Wp1q q´ pWp2q Zp2q Vp1q Up1q q 1 ´ λ21 1 ´ λ22 λ22 λ21 J J J J J pVp2q Vp2q pZp2q Zp2q ´ Zp2q Wp2q ´ Vp2q Up2q Wp1q Zp1q Up1q Vp1q q q` “ 2 1 ´ λ1 1 ´ λ22 λ1 λ2 λ1 λ2 J J J J ´ pVp2q Up2q pZp2q Wp2q Wp1q Zp1q Up1q Vp1q q´ q. 2 1 ´ λ1 1 ´ λ22 J11 “ J22 We spell out the algebra for trpJ11 q, and trpJ22 q can be computed in exactly the same fashion. 1 J J J J J J J J J Vp2q Up2q ` Up1q Vp1q Vp1q Up1q ´ 2Up2q Vp2q Vp1q Up1q q trpUp2q Up2q ´ Up2q Vp2q Vp1q Up1q q “ trpUp2q Vp2q 2 1 J “ }Up1q Vp1q ´ Up2q Vp2q }2F . 2 Similarly, J J trpWp2q Wp2q ´ Wp2q Zp2q Zp1q Wp1q q“ 35 1 J }Wp1q Zp1q ´ Wp2q Zp2q }2F . 2 J ` W Z J “ U V J ` W Z J , we have By the assumption (6.1), i.e., Up1q Vp1q p1q p1q p2q p2q p2q p2q 1 J J J trpWp2q Wp2q ´ Wp2q Zp2q Zp1q Wp1q q “ }Up1q Vp1q ´ Up2q Vp2q }2F . 2 Further, ¯ ´ J J J J J J J trpUp2q Vp2q Zp1q Wp1q q “ tr Up2q Vp2q pUp2q Vp2q ` Wp2q Zp2q ´ Up1q Vp1q q ¯ ´ J J J J “ tr Up2q Vp2q pUp2q Vp2q ´ Up1q Vp1q q 1 J ´ Up2q Vp2q }2F , “ }Up1q Vp1q 2 and by the same argument, 1 J J J trpWp2q Zp2q Vp1q Up1q q “ }Up1q Vp1q ´ Up2q Vp2q }2F . 2 Sum these equations, * λ22 λ1 λ2 λ1 λ2 λ21 J ` ´ ´ }Up1q Vp1q ´ Up2q Vp2q }2F 1 ´ λ21 1 ´ λ22 1 ´ λ21 1 ´ λ22 ∆2 p1 ` λ1 λ2 q J “ }Up1q Vp1q ´ Up2q Vp2q }2F . 2p1 ´ λ21 qp1 ´ λ22 q 1 trpJ11 q “ 2 " Repeat the argument for J22 , one can show that trpJ22 q “ trpJ11 q “ ∆2 p1 ` λ1 λ2 q J }Up1q Vp1q ´ Up2q Vp2q }2F . 2p1 ´ λ21 qp1 ´ λ22 q Therefore, n n Ωp1q q “ ptrpJ11 q ` trpJ22 qq trpΩ´1 p2q 2 2 n∆2 p1 ` λ1 λ2 q }U V J ´ Up2q Vp2q }2F . “ 2p1 ´ λ21 qp1 ´ λ22 q p1q p1q DpPΣp1q \|\|PΣp2q q “ References Anderson, T. 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Using Deep Convolutional Networks for Gesture Recognition in American Sign Language Vivek Bheda and N. Dianna Radpour Department of Computer Science, Department of Linguistics State University of New York at Buffalo {vivekkan, diannara}@buffalo.edu Abstract In the realm of multimodal communication, sign language is, and continues to be, one of the most understudied areas. In line with recent advances in the field of deep learning, there are far reaching implications and applications that neural networks can have for sign language interpretation. In this paper, we present a method for using deep convolutional networks to classify images of both the the letters and digits in American Sign Language. 1. Introduction Sign Language is a unique type of communication that often goes understudied. While the translation process between signs and a spoken or written language is formally called ‘interpretation,’ the function that interpreting plays is the same as that of translation for a spoken language. In our research, we look at American Sign Language (ASL), which is used in the USA and in English-speaking Canada and has many different dialects. There are 22 handshapes that correspond to the 26 letters of the alphabet, and you can sign the 10 digits on one hand. Figure 1. American Sign Language Alphabet Figure 2. American Sign Language Numbers One of the nuances in sign language is how often fingerspelling is used. Fingerspelling is a method of spelling words using only hand gestures. One of the reasons the fingerspelling alphabet plays such a vital role in sign language is that signers used it to spell out names of anything for which there is not a sign. People's names, places, titles, brands, new foods, and uncommon animals or plants all fall broadly under this category, and this list is by no means exhaustive. Due to this reason, the recognition process for each individual letter plays quite a crucial role in its interpretation. 2. Related Work Convolutional Neural Networks have been extremely successful in image recognition and classification problems, and have been successfully implemented for human gesture recognition in recent years. In particular, there has been work done in the realm of sign language recognition using deep CNNs, with input-recognition that is sensitive to more than just pixels of the images. With the use of cameras that sense depth and contour, the process is made much easier via developing characteristic depth and motion profiles for each sign language gesture [5]. The use of depth-sensing technology is quickly growing in popularity, and other tools have been incorporated into the process that have proven successful. Developments such as custom-designed color gloves have been used to facilitate the recognition process and make the feature extraction step more efficient by making certain gestural units easier to identify and classify [8]. Until recently, however, methods of automatic sign language recognition weren’t able to make use of the depth-sensing technology that is as widely available today. Previous works made use of very basic camera technology to generate datasets of simply images, with no depth or contour information available, just the pixels present. Attempts at using CNNs to handle the task of classifying images of ASL letter gestures have had some success [7], but using a pre-trained GoogLeNet architecture. 3. Method Our overarching approach was one of basic supervised learning using mini-batch stochastic gradient descent. Our task was that of classification using deep convolutional neural networks to classify every letter and the digits, 0-9, in ASL. The inputs were fixed size high-pixel images, 200 by 200 or 400 by 400, being padded and resized to 200 by 200. 3.1 Architecture Most implementations surrounding this task have attempted it via transfer learning, but our network was trained from scratch. Our general architecture was a fairly common CNN architecture, consisting of multiple convolutional and dense layers. The architecture included 3 groups of 2 convolutional layers followed by a max-pool layer and a dropout layer, and two groups of fully connected layer followed by a dropout layer and one final output layer. 4. Data We initially trained and tested on a self-generated dataset of images we took ourselves. This dataset was a collection of 25 images from 5 people for each alphabet and the digits 1-9. Since our dataset was not constructed in a controlled setting, it was especially prone to differences in light, skin color, and other differences in the environment that the images were captured in, so we also used a premade dataset to compare our dataset’s performance with [3]. Additionally, a pipeline was developed that can be used so people are able to generate and continue adding images to this dataset. 4.1 Preprocessing For generating our own dataset, we captured the images for each sign, then removed the backgrounds from each of the images using background-subtraction techniques. When we initially split the dataset into two for training and validation, the validation accuracy showed to be high. However, when we used datasets from two different sources, i.e. training on ours and testing on the premade and vice versa, the validation accuracy drastically decreased. Since training on one dataset and validating on another was not yielding as accurate of results, we used the premade dataset for the different gestures to train the network which yielded the following results. Figure 4. Training Accuracy on ASL Alphabets Figure 3. Network Architecture Figure 5. Training Accuracy on ASL Digits Figure 6. Training Loss on ASL Alphabet Figure 8. Validation Accuracy on ASL Letters Figure 7. Training Loss on ASL Digits 4.2 Data Augmentation We saw the performances improve differently in our two datasets via data augmentation. By transforming our images just a few pixels (rotating by 20 degrees, translating by 20% on both axes) there was an increased accuracy of approximately 0.05. We also flipped the images horizontally as we can sign using both hands. While it wasn’t extremely effective, we saw that with better and more representative initial training data, augmenting improved the performance more drastically. This was observed after augmentation of the premade dataset, which improved the performance by nearly 20%. 5. Results We observed 82.5% accuracy on the alphabet gestures, and 97% validation set accuracy on digits, when using the NZ ASL dataset. On our self-generated dataset, we observed much lower accuracy measures, as was expected since our data was less uniform than that which was collected under studio settings with better equipment. We saw 67% accuracy on letters of the alphabet, and 70% accuracy on the digits. In terms of time complexity, gestures of the letters converged in approximately 25 minutes, and the digits converged in nearly 10 minutes. Figure 9. Validation Accuracy on ASL Digits Figure 10. Validation Loss on ASL Letters Figure 11. Validation Loss on ASL Digits 5.1 Evaluation We trained with a categorical cross entropy loss function for both our datasets. It is a fairly common loss function used along with image classification problems. Initially, we observed low accuracy measures when testing on the validation set of the self-generated data, which we accounted largely to the lighting and skin tone variations in the images. The higher accuracy measure for the digits was expected, since the gestures for the digits are much more distinguishable and easier to classify. Compared to previous methods working on this same task, our network performed quite well, considering RF-JA were using both a color glove and depth-sensing Kinect camera. The cause of higher accuracy than Stanford’s method was likely due to their lack of background-subtraction for the images, since they used a large dataset from ILSVRC2012 as part of a competition. Method Accuracy deepCNN (our method) 82.5 Stanford deepCNN [7] 72 RF-JA+C(h-h) [8] 90 RF-JA+C(l-o-o) [8] 70 Figure 12. Comparison of previous methods with ours; Stanford didn’t use background subtraction, RF-JA(h-h) split the training and validation set 50-50, (l-o-o) omitted specific data. 6. Conclusions and Future Work In this paper, we described a deep learning approach for a classification algorithm of American Sign Language. Our results and process were severely affected and hindered by skin color and lighting variations in our self-generated data which led us to resort to a pre-made professionally constructed dataset. With a camera like Microsoft’s Kinect that has a depth sensor, this problem is easy to solve [5]. However, such cameras and technology are not widely accessible, and can be costly. Our method shows to have potential in solving this problem using a simple camera, if enough substantial training data is provided, which can be continuously done and added via the aforementioned processing pipeline. Since more people have access to simple camera technologies, this could contribute to a scalable solution. In recognizing that classification is a limited goal, we plan on incorporating structured PGMs in future implementations of this classification schema that would describe the probability distributions of the different letters’ occurrences based on their sequential contexts. We think that by accounting for how the individual letters interact with each other directly (e.g. the likelihood for the vowel ‘O’ to proceed the letter ‘J’), the accuracy of the classification would increase. This HMM approach with sequential pattern boosting (SP-boosting) has been done with the actual gesture units that occur in certain gestures’ contexts, i.e. capturing the upper-arm movements that precede a certain letter to incorporate that probability weight into the next unit’s class, [6] and processing sequential phonological information in tandem with gesture recognition [4], but not for part-of-word tagging with an application like what we hope to achieve. We also recognize that the representation itself makes a huge difference in the performance of algorithms like ours, so we hope to find the best representation of our data, and building off our results from this research, incorporate it into a zero-shot learning process. We see zero-shot learning as having the potential to facilitate the translation process from American Sign Language into English. Implementing one-shot learning for translating the alphabet and numbers from American Sign Language to written English, and comparing it with a pure deep learning heuristic could be successful and have the potential to benefit from error correction via language models. Recent implementations of one-shot adaptation have also had success in solving real world computer vision tasks, and effectively trained deep convolutional neural networks using very little domain-specific data, even as limited as single-image datasets. We ultimately aim to create a holistic and comprehensive representation learning system for which we have designed a set of features that can be recognized from simple gesture images that will optimize the translation process. 7. References [1] X. Chen and A. Yuille. Articulated pose estimation by a graphical model with image dependent pairwise relations. In Advances in Neural Information Processing Systems (NIPS), 2014. [2] T. Pfister, J. Charles, and A. Zisserman. Flowing convnets for human pose estimation in videos. In IEEE International Conference on Computer Vision, 2015. [3] Barczak, A.L.C., Reyes, N.H., Abastillas, M., Piccio, A., Susnjak, T. (2011), A new 2D static hand gesture colour image dataset for ASL gestures, Research Letters in the Information and Mathematical Sciences, 15, 12-20 [4] Kim, Taehwan & Livescu, K & Shakhnarovich, Greg. (2012). American sign language fingerspelling recognition with phonological feature-based tandem models. In IEEE Spoken Language Technology Workshop (SLT), 119-124. [5] Agarwal, Anant & Thakur, Manish. Sign Language Recognition using Microsoft Kinect. In IEEE International Conference on Contemporary Computing, 2013. [6] Cooper, H., Ong, E.J., Pugeault, N., Bowden, R.: Sign language recognition using sub-units. The Journal of Machine Learning Research, 13(1), 2205–2231, 2012. [7] Garcia, Brandon and Viesca, Sigberto. Real-time American Sign Language Recognition with Convolutional Neural Networks. In Convolutional Neural Networks for Visual Recognition at Stanford University, 2016. [8] Cao Dong, Ming C. Leu and Zhaozheng Yin. American Sign Language Alphabet Recognition Using Microsoft Kinect. In IEEE International Conference on Computer Vision and Pattern Recognition Workshops, 2015.	1
1 arXiv:1705.10091v1 [cs.IT] 29 May 2017 Rate (n − 1)/n Systematic MDS Convolutional Codes over GF(2m) Ángela Barbero Universidad de Valladolid 47011 Valladolid, Spain Email: [email protected] Øyvind Ytrehus Simula@UiB and University of Bergen N-5020 Bergen, Norway Email: [email protected] Abstract A systematic convolutional encoder of rate (n − 1)/n and maximum degree D generates a code of free distance at most D = D + 2 and, at best, a column distance profile (CDP) of [2, 3, . . . , D]. A code is Maximum Distance Separable (MDS) if it possesses this CDP. Applied on a communication channel over which packets are transmitted sequentially and which loses (erases) packets randomly, such a code allows the recovery from any pattern of j erasures in the first j n-packet blocks for j < D, with a delay of at most j blocks counting from the first erasure. This paper addresses the problem of finding the largest D for which a systematic rate (n − 1)/n code over GF(2m ) exists, for given n and m. In particular, constructions for rates (2m − 1)/2m and (2m−1 − 1)/2m−1 are presented which provide optimum values of D equal to 3 and 4, respectively. A search algorithm is also developed, which produces new codes for D for field sizes 2m ≤ 214 . Using a complete search version of the algorithm, the maximum value of D, and codes that achieve it, are determined for all code rates ≥ 1/2 and every field size GF(2m ) for m ≤ 5 (and for some rates for m = 6). I. I NTRODUCTION 1 In many practical communication applications, such as multimedia transmission over packet erasure channels, on-time delivery is an important quality-of-service criterion. Traditional ARQ systems, for example the one used by TCP for transport layer unicast service, suffer from long delays due to erasures when the round-trip time is large. This has led to an increased interest in the design and analysis of systems based on packet-level error correcting codes. Such coded schemes are also known to be beneficial in other transport layer models, for example in the multi-path case. Two main approaches to this coding problem have been discussed in the literature. The deterministic approach [1], [2], [3] is to send packets using a fixed 2m -ary convolutional code with a good column distance profile. This approach is discussed in Subsection II-B. Random coding was proposed as a solution in [4], [5]. In these schemes, the sender transmits k uncoded information packets, followed by n − k parity check packets formed by random linear combinations of all information packets that have not been acknowledged by the receiver so far. Subsection II-C describes this approach, and also discusses a hybrid approach that combines deterministic and random coding. A. Contributions We present new codes in Section III. In Section III-A we present two new, general, and optimum constructions of MDS convolutional codes. In the literature, there exist only a few general constructions of high-rate convolutional codes: As far as we know, only the Wyner-Ash code [6] and their binary generalizations ([7], and Thms. 7.10 and 7.13 in [8]). We present a simple (but as far as we can see, not previously described in the literature) distance-3 construction. This code has the same rate and Viterbi complexity as the binary Wyner-Ash code, but has a better column distance profile. We also present a much more interesting algebraic distance-4 construction in Proposition 3. In Section III-B we describe a search algorithm and in Section III-C we present the codes found by the algorithm. For most parameters, these codes are better (in a sense which will be made more precise) than previously known codes. Further, we present simple upper bounds in Section IV. By convention we will call a convolutional code systematic if is it has a systematic encoder; i.e. one that preserves all information symbols and obtains redundancy by extra parity symbols. 2m -ary systematic rate (n − 1)/n convolutional encoders are useful in order to obtain fast recovery of packet erasures in the common case of channels with moderate erasure rates, and we will focus only on this class of codes. 1 This work is supported by Ministerio de Economı́a, Industria y Competitividad, Gobierno de España, through project MTM2013-46949-P, the Estonian Research council through project EMP133, the Norwegian Research Council through the SARDS project. 2 II. BACKGROUND A. Notation For a thorough introduction to convolutional codes, please see [9]. In the following we will describe the concept of a 2m -ary MDS convolutional code in a way which is convenient for our purposes in this paper. Let m ≥ 1, n ≥ 2, k = n − 1 be integers, F = GF(2m ), and define the matrices and vectors R0 = (r0,1 , . . . , r0,k ) ∈ Fk where Fk is the k-dimensional space of row vectors over F, H0 = (R0 \|1) ∈ Fn , where Fr×c denotes the space of matrices with r rows and c columns over F. For i > 1 define Hi−1 Ri = (ri,1 , . . . , ri,k \|0) ∈ Fk , Hi = ∈ F(i+1)×n Ri and, for an integer L ≥ 2, let H (L) 0(L−1)×n 01×n = (HL , ,..., ) ∈ F(L+1)×n(L+1) , HL−1 H0 (1) where 0r×c is all-zero matrix with r rows and c columns. Then H (L) is the parity check matrix for the Lth truncated block code C (L) of a (systematic) convolutional code C , thus any vector of length (l + 1)n for l ≤ L, (0) (0) (1) (1) (l) (l) [v]l = (v1 , . . . , vn , v1 , . . . , vn , . . . , v1 , . . . , vn ) ∈ F(l+1)n is a codeword in C (l) if and only if the syndrome > (l+1)×1 H (l) [v]> . l = (0, . . . , 0) ∈ F A systematic encoder for the code C (L) is represented  G0 G1  G0  G(L) =   by ··· ··· .. .  GL GL−1   k(L+1)×n(L+1) ..  ∈ F .  (2) G0 where k×n k×n G0 = (Ik \|R> , Gi = (0k \|R> for i > 0, 0)∈F i )∈F > and Ik and 0k are the k ×k identity and zero matrices, respectively. It is straightforward to verify that G(L) ×H (L) = 0k(L+1)×(L+1) . Example 1. Let F = GF(23 ) with primitive element α defined by  1 1 1 0 H (2) =  1 α 0 1 α3 1 0 1 and α 3 + α + 1 = 0. Then the parity and generator matrices  0 0 0 0 0 1 1 0 0 0 α 0 1 1 1  1 0 1 0 0 0 1 1 0 0  0 0 0 1 0 (2) G = 0 0 0 0 1  0 0 0 0 0 0 0 0 0 0 1 α 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1  α3 1  1  α  1 1 define a truncated code C (2) , which is rate 6/9 block code over F. Note that the matrices are completely determined by the parity check coefficients ri, j , i = 0, . . . , L, j = 1, . . . , k. In the conventional polynomial notation of convolutional codes [9], the parity check matrix can be described as D D H(x) = ( ∑ ri,1 xi , . . . , ∑ ri,k xi , 1) ∈ F[x]. i=0 In Example 1, H(x) = (1 + x + α 3 x2 , 1 + αx + x2 , 1). i=0 Similarly, the corresponding polynomial generator matrix is 1 0 1 + x + α 3 x2 G(x) = 0 1 1 + αx + x2 3 B. MDS convolutional codes constructed from superregular matrices In the deterministic approach [1], [2], [3], the goal is to design codes with an optimum column distance profile, which we will define below. The l-th column distance dl = dl (C ) of a convolutional code C is the minimum Hamming weight of any truncated (0) (0) codeword [c]l with the first block (c1 , . . . , cn ) nonzero, and the column distance profile (CDP) is the non-decreasing sequence (d0 , d1 , d2 , . . . , dD = D, D, D, . . .), where D is the free distance of the code and D is the index for which the CDP reaches D. The CDP was originally studied for its significance on the performance of sequential decoding (please see Ch. 13 of [9].) Recently the CDP has received renewed attention in the context of 2m -ary codes, due to its importance for fast recovery from losses of symbols in an erasure channel. Recall that we consider only convolutional codes of rate k/n = (n − 1)/n that have a systematic encoder. In this case, by the Singleton bound for truncated block codes, d0 ≤ 2, and by similar linear algebra arguments, dl ≤ dl−1 + 1 for l > 0. Moreover, dl = dl−1 for l > D. So the best column distance profile one can hope to find in a code with a systematic encoder is d0 = 2, d1 = 3, . . . , d j = j + 2, . . . , dD = D + 2 = D. (3) By an MDS convolutional code, in this paper we will mean a code with a CDP as in (3). Remark 1. The concept of Strongly-MDS codes was introduced in [2]. This concept takes into account that for some codes that do not possess a systematic encoder, the free distance may grow beyond δ + 2, where δ is the memory of a minimal encoder. In order not to complicate the notation, and since Viterbi complexity is not an issue in this paper, we omit the details. Definition 1. Consider a lower triangular matrix     SR =    r0 r1 r2 .. . 0 r0 r1 .. . 0 0 r0 .. . ··· ··· ··· .. . 0 0 0 .. . rL rL−1 rL−2 ··· r0        where each element ri ∈ F. Consider a square submatrix P of size p of SR, formed by the entries of SR in the rows with indices 1 ≤ i1 < i2 < · · · < i p ≤ (L + 1) and columns of indices 1 ≤ j1 < · · · < j p ≤ (L + 1). P, and its corresponding minor, are proper if jl ≤ il for all l ∈ {1, . . . , p}. SR is superregular if all its proper p × p minors are non singular for any p ≤ L + 1. When matrix SR is upper triangular the definition of proper submatrices is analogous. A γ × γ superregular matrix can be used to construct a rate 1/2 code in two ways: (1) [1] a systematic MDS convolutional code with CDP as in (3) with D = D + 2 = γ + 1, (2) [2] when γ = 2δ + 1, a strongly-MDS code (in general nonsystematic) with a parity check matrix of max degree δ and the same CDP as for the systematic codes in case (1). While superregular matrices are known to exist for all dimensions if the field is large enough, general efficient constructions are not known, and for γ & 10 the minimum field size for which a γ × γ superregular matrix exists is not known. Another problem with the deterministic approach is that the existing design methods do not allow a simple construction of codes of high rate and/or high degree. Codes of higher rates (which are desirable in many practical cases) can also be constructed from these superregular matrices, but this involves deleting columns, so that the conditions on a superregular matrix are too strict. This means that in practice only simple codes can be constructed in this way. Since superregular matrices are so hard to construct, the reduction to the superregular matrix problem blocks the code construction. Therefore we generalize Definition 1 as follows: Definition 2. Consider an s-lower triangular matrix (where s is a positive integer)  r0,1 ··· r0,s 0 ··· 0 0 ··· 0  r1,1 · · · r r · · · r 0 · · · 0 1,s 0,1 0,s   r2,1 · · · r r · · · r r · · · r 2,s 1,1 1,s 0,1 0,s  SSR =  .. .. .. .. .. .. .. .. ..  . . . . . . . . .   rL−1,1 · · · rL−1,s rL−2,1 · · · rL−2,s rL−3,1 · · · rL−3,s rL,1 · · · rL,s rL−1,1 · · · rL−1,s rL−2,1 · · · rL−2,s ··· ··· ··· 0 0 0 .. . ··· ··· 0 · · · r0,1 ··· ··· ··· .. . ··· ··· 0 0 0 .. .         0  (4) r0,s Consider a square submatrix P of size p of SSR, formed by the entries of SSR in the rows with indices 1 ≤ i1 < i2 < · · · < i p ≤ (L + 1) and columns of indices 1 ≤ j1 < · · · < j p ≤ s(L + 1). P, and its corresponding minor, are proper if jl ≤ s · il for all l ∈ {1, . . . , p}. The matrix SSR is called s-superregular iff all of its proper p × p minors, for any p ≤ L + 1, are nonsingular. 4 Rate 1/2 1/2 1/2 1/2 2/3 2/3 3/4 Field size 4 8 8 32 16 64 16 D 4 6 6 8 5 5 3 Table I Description [11] [11] [2] superregular [2] superregular [2] ad hoc [2] superregular [2] superregular S OME RATE (n − 1)/n MDS CODES ( NOT NECESSARILY SYSTEMATIC ) DESCRIBED IN THE LITERATURE . The following lemma is a restatement of Theorem 1 in [1], using the terminology of this section. Lemma 1. Let H (D) be the parity check matrix of the D − th truncation of a systematic convolutional code, given by  r0,1 r1,1 r2,1 .. .     (D) H =    rD−1,1 rD,1 ··· ··· ··· .. . r0,k r1,k r2,k .. . 1 0 0 .. . ··· ··· rD−1,k rD,k 0 0 r0,1 r1,1 .. . ··· ··· ··· .. . rD−2,1 rD−1,1 ··· ··· 0 0 r0,k r1,k .. . 0 1 0 .. . rD−2,k rD−1,k 0 0 r0,1 .. . ··· ··· ··· .. . r0,k .. . 0 0 1 .. . rD−3,1 rD−2,1 ··· ··· rD−3,k rD−2,k 0 0 0 0 0 0 ··· ··· ··· 0 0 0 .. . ··· ··· 0 · · · r0,1 ··· ··· ··· .. . 0 0 0 .. . ··· ··· 0 r0,k 0 0 0 .. .         0  1 (5) and let H 0(D) be the matrix obtained from H (D) by removing the columns in positions (k + 1), 2(k + 1), 3(k + 1), . . . , (D + 1)(k + 1), that is   r0,1 ··· r0,k 0 ··· 0 0 ··· 0 ··· 0 ··· 0  r1,1 ··· r1,k r0,1 ··· r0,k 0 ··· 0 ··· 0 ··· 0     r2,1 ··· r2,k r1,1 ··· r1,k r0,1 ··· r0,k ··· 0 ··· 0    0(D) (6) H = .. .. .. .. .. .. .. ..  .. .. .. ..   . . . . . . . . . . · · · . .    rD−1,1 · · · rD−1,k rD−2,1 · · · rD−2,k rD−3,1 · · · rD−3,k · · · 0 · · · 0  rD,1 · · · rD,k rD−1,1 · · · rD−1,k rD−2,1 · · · rD−2,k · · · r0,1 · · · r0,k Then the CDP of the convolutional code given by H (D) is (2, 3, . . . , D + 2) if and only if H 0(D) is a k-superregular matrix. Theorem 1 in [1] is stated without proof. For reference, we include a formal proof in Appendix A. Definition 3. Let ∆(2m , n) be the largest free distance D such that there exists a rate (n − 1)/n systematic MDS convolutional code over GF(2m ) with column distance profile as in (3). The main problem that we address in this paper is to determine exact values, or constructive lower bounds, for ∆(2m , n). Please note that there is no restriction of the degree D in Definition 3. There are few known code constructions in the literature, beyond those based on superregular matrices. Table I contains the current world records with respect to rate (n − 1)/n MDS codes, to the best of our knowledge. We will describe new codes in Section III. Although this paper focuses on rate (n − 1)/n MDS codes, we observe that the following lemma, that follows directly from Theorem 2 in [1], implies that our results will also provide rate 1/n MDS codes. Lemma 2. If a systematic rate (n − 1)/n MDS code of memory D and free distance D + 2 exists, then its dual code is equivalent to a systematic rate 1/n MDS code of memory D and free distance (n − 1)(D + 1) + 1. C. Random convolutional codes In the terminology of this paper, the random approach [4], [5] consists of selecting the coefficients of ri j independently at random. The advantage of this is that one can pick codes with large degrees, and that over large fields the expected performance is “reasonably good”, although the exact loss compared to optimum average performance or optimum guaranteed worst case performance remains to be determined. Coefficients need to be transmitted in the headers of the data packets, but this represents only a small rate loss when large packets are transmitted. Proposition 1. Consider a hybrid scheme where the first blocks of coefficients ri, j (until time i = D) are selected fixed, and subsequent random coefficients ri, j for i > D are selected at random. Thus the parity check equation will be on the form H(x) = HCDP (x) + HRandom (x) (7) 5 where D D HCDP (x) = ( ∑ ri,1 xi , . . . , ∑ ri,k xi , 1) i=0 and i=0 ? HRandom (x)( ∑ i=D+1 ? Ri,1 xi , . . . , ∑ Ri,k xi , 0) i=D+1 where all Ri, j are nonzero randomly selected coefficients and where the degree of the random polynomials does not need to be fixed (except by the application protocol). Then the initial CDP (until time D) is not affected by the random part of the code construction. Proof. Obvious: Only the first component HCDP (x) of the parity check matrix determines the initial part of the CDP. Our suggestion is to use such hybrid codes, i. e. codes where the terms of degree 0, . . . , D of the parity check polynomials are preselected constants yielding an optimum initial column distance profile, while subsequent random parity checks are added as needed. This guarantees optimum recovery for the simplest and most likely erasure patterns, and hence better performance than random codes for light to moderate erasure patterns, while still allowing the degree to grow if required by the application. III. N EW CODES Gluesing-Luerssen et. al. [2] use superregular matrices to design codes. However, the authors also give examples of codes that are better than the ones constructed from superregular matrices, and note that ”..the abundance of (small) examples suggests that such a construction might be possible and might lead to smaller alphabets for given parameters than the construction ,[...] We will leave this as an open question for future research.” So here comes the future research. In this section we present constructions and a new search algorithm that, in combination, improve our knowledge of ∆(2m , n) for almost all sets of parameters, with respect to what we find in the literature. A. Codes with free distance D ∈ {3, 4} We present two optimum constructions, for D ∈ {3, 4}. For D = 3 the construction is simple, but we have not seen it presented in prior literature. We have tacitly assumed the following fact for the constant terms. Here comes the justification. Lemma 3. We can w.l.o.g assume r0,1 = · · · = r0,n = 1. Proof. If there is a r0, j equal to zero, then d0 < 2. We don’t want that. Then assume some nonzero r0, j 6= 1. If we multiply −1 the corresponding column of G(D) by r0, j , we obtain a new code with the same CDP and weight structure. Proposition 2. ∆(qm , qm ) = 3 for q prime and m ≥ 0. Proof. Select r0,i = 1 and r1,i , i = 1, . . . , qm − 1 as the qm − 1 distinct nonzero elements of GF(qm ). Without loss of generality, the parity check matrix of (1) takes the form 1 1 ··· 1 1 0 ··· 0 0 (1) H = 1 2 · · · qm − 1 0 1 · · · 1 1 1 1 ··· 1 0 ··· 0 H 0(1) = 1 2 · · · qm − 1 1 · · · 1 is qm -superregular because it is obvious that all the proper minors of sizes 1 and 2 are nonsingular. Clearly, d0 = 2 and d1 = 3. Remark 2. It is instructive to compare the construction of Proposition 2 with the binary Wyner-Ash codes [6]. Wyner-Ash codes were considered for digital media transmission already in 1974 [10]. The Wyner-Ash code of length 4 has the binary polynomial parity check matrix HWA = 1 + x + x2 1 + x 1 + x2 1 . It is easy to see that the CDP of the Wyner-Ash code is [2, 2, 3], i. e. this is not an MDS code. The construction of Proposition 2 can be considered as a qm -ary generalization of the Wyner-Ash code, of memory 2, but this code is an MDS code, with CDP [2, 3]. For D = 4, we present an optimum construction in Proposition 3. Complete computer searches for m ≤ 5 indicate that the construction is unique and, in a sense, much better than what can be achieved through other choices of the set of first degree coefficients {r1,i }. 6 Lemma 4. For a code with a CDP of [2, 3, 4], its parity check matrix H (2) must satisfy (i) ri,s 6= 0 for i = 1, 2, s = 1, . . . , k, (ii) ri,s 6= ri,t for i = 1, 2, 1 ≤ s < t ≤ k, (iii) r1,t 6= r2,s /r1,s for 1 ≤ s,t ≤ k, (iv) r2,s /r1,s 6= r2,t /r1,t for 1 ≤ s < t ≤ k, (v) r2,s − r2,t 6= r1,u (r1,s − r1,t ) for 1 ≤ s < t ≤ k, 1 ≤ u ≤ k, (vi) r2,s 6= (r1,s (r2,u − r2,t ) − r1,t r2,u + r1,u r2,t )/(r1,u − r1,t ) for 1 ≤ s < t < u ≤ k. Proof. From Lemma 1 we need H 0(2) to be k-superregular. That all 1 × 1 proper minors of H 0(2) are non singular is equivalent to condition (i). Proper minors of size 2 × 2 are of the following types 1 ri,s 1 0 , ri,s r1,t 1 0 , r2,s 1 r 1 , 1,s r2,s ri,t r 1 , 1,s r2,s r1,t r1,t r2,t The first type are trivially non zero, the second type are non zero when condition (i) is satisfied. The third type being nonsingular is equivalent to condition (ii), the fourth type is guaranteed to be non zero if and only if condition (iii) is satisfied and the fifth type being nonsingular is equivalent to condition (iv). Finally, 3 × 3 proper minors can be of four different types: 1 r1,s r2,s 0 1 r2,t 1 0 0 , r1,s r2,s 1 1 r1,t r2,t 1 0 0 , r1,s r2,s 1 1 r1,t r2,t 0 1 r1,u 1 , r1,s r2,s 1 r1,t r2,t 1 r1,u r2,u Those of the first type are trivially nonsingular. Condition (ii) takes care of those in the second type to be nonsingular. Those in the third type are nonsingular if and only if condition (v) is satisfied. The fifth type are non singular if and only if condition (vi) is satisfied. Example 2. Consider the code in Example 1. By checking conditions (i)-(vi) in Lemma 4 we observe that the code has CDP equal to [2,3,4]. Proposition 3. ∆(2m , 2m−1 ) = 4. Proof. Let F = GF(2m ). (≥:) The following construction gives a code that meets the requirements: The trace function [12] is defined by Trm () : F x → GF(2) 2i → Trm (x) = ∑m−1 i=0 x . Consider the set Hβ = {x ∈ F\|Trm (β x) = 0}. When F is regarded as an m-dimensional vector space over GF(2), the set Hβ is a hyperplane (an (m − 1)-dimensional linear subspace) of F. Let k = 2m−1 − 1, select β as an arbitrary nonzero field element, select c as an arbitrary constant in F \ Hβ . Then select a1 , . . . , ak := r1,1 , . . . , r1,k as all distinct nonzero elements in Hβ , and set bs := r2,s = as (as + c) = r1,s (r1,s + c) for s = 1, . . . , k. We need to verify that this construction satisfies the conditions in Lemma 4. (i) This holds because bs = as (as + c) is a product of two nonzeros. (ii) All as ’s are distinct. Assume that bs = bt , s 6= t. Then 0 = as (as + c) = at (at + c) = (as + at )c + a2s + at2 = (as + at )c + (as + at )2 = (as + at )(c + as + at ). The first factor is nonzero since as 6= at . The second factor is also nonzero since as + at ∈ Hβ (because Hβ is closed under addition) while c 6∈ Hβ , a contradiction. (iii) Assume that as at = bs . Then as at = as (as + c) ⇒ at = as + c, a contradiction, since at ∈ Hβ and as + c 6∈ Hβ . (iv) Assume that bs /as = bt /at , s 6= t. Then as + c = at + c ⇒ as = at , a contradiction. (v) bs + bt + au (as + at ) = as (as + c) + at (at + c) + au (as + at ) = a2s + at2 + (as + at )(c + au ) = (as + at )2 + (as + at )(c + au ) = (as + at )(as + at + c + au ) 7 which again is a product of nonzero factors, because c 6∈ Hβ and as + at + au ∈ Hβ , and hence nonzero. (vi) bs + as (bt + bu ) + au bt + at bu at + au as (at (at + c) + au (au + c)) + au at (at + c) + at au (au + c) at + au as (at + au )2 + as c(at + au ) + at au (at + au ) = as (as + c) + at + au = as (as + c) + as (at + au ) + as c + at au = as (as + c) + = a2s + as at + as au + at au = (as + at )(as + au ) 6= 0. (≤:) This follows from Theorem 1 in Section IV. Remark 3. By Theorem 1 later, the construction in Proposition 3 is optimum not only in the sense that it offers the maximum distance 4 for the given field size and code rate, but also it offers the minimum field size for a code of the given rate and distance 4, and the maximum code rate given the field size and distance 4. Moreover, complete computer search for field sizes 2m ≤ 32 show that the construction is unique for these parameters. B. Computer search algorithm The goal of the search algorithm is to select the coefficients ri, j successively, ordered first on i and then reversely on j, in such a way that the conditions on the minors are met. 1) Some useful facts: First, as for the constructions in Section III-A, we use Lemma 3 in order to set r0,1 = · · · = r0,k = 1. In order to simplify the search we apply the following results. Lemma 5. We can w.l.o.g assume r1,i < r1,i+1 , i = 1, . . . , k − 1 for any choice of ordering <. Lemma 6. Consider an MDS convolutional code C with polynomial parity check matrix D D H(x) = (1 + ∑ ri,1 xi , . . . , 1 + ∑ ri,k xi , 1) ∈ F[x]. i=1 i=1 Then the code Cc with parity check matrix D D Hc (x) = (1 + ∑ ci ri,1 xi , . . . , 1 + ∑ ci ri,k xi , 1) ∈ F[x] i=1 i=1 is also MDS for any c ∈ F \ {0}. D i i > > Proof. Let v(x) = (v1 (x), . . . , vn (x)) = (∑D i=0 v1,i x , . . . , ∑i=0 vn,i x ). Then v(x)H(x) = 0 iff vc (x)Hc (x) = 0 for D D i=0 i=0 vc (x) = ( ∑ c−i v1,i xi , . . . , ∑ c−i vn,i xi ). Corollary 1. If a systematic MDS convolutional code exists, we can w.l.o.g. assume that it has a parity check matrix with r1,k = 1. Proof. Assume that a systematic MDS convolutional code exists with a parity check matrix with r1,k = a ∈ F. Then apply Lemma 6 with c = a−1 . Lemma 7. Let M be a k-superregular matrix over GF(qm ), with q a prime. Raising each element of M to power q yields another k-superregular matrix. Proof. Given any square matrix  a1,1  .. A= . an,1  · · · a1,n ..  .  · · · an,n with ai, j ∈ GF(qm ), by definition det(A) = ∑ s(σ )a1,σ (1) · · · an,σ (n) σ ∈Sn where Sn is the group of permutations of n elements and s(σ ) is the sign of each permutation σ . 8 Also we have (x1 + x2 + · · · + xn )q = c(q1 , . . . , qn )x1q1 x2q2 · · · xnqn ∑ q1 + · · · + qn = q 0 ≤ qi ≤ q q−q1 −···−qn−1 1 . and c(q1 , . . . , qn ) = qq1 q−q qn q2 · · · When q is prime, q is a divisor of coefficient c(q1 , . . . , qn ) except in the cases qi = q, q j = 0 for j 6= i, and this for each i ∈ {1, . . . , n}. Therefore, in characteristic q we have (x1 + x2 + · · · + xn )q = x1q + x2q + · · · + xnq Back to the definition of determinant, in characteristic q we have !q (det(A))q = ∑ s(σ )a1,σ (1) · · · an,σ (n) = ∑ σ ∈Sn σ ∈Sn s(σ )q aq1,σ (1) · · · aqn,σ (n) Finally, s(σ ) is either 1 or -1, so in case q = 2 then s(σ ) = 1 for any σ ∈ Sn , and s(σ )2 = s(σ ), and in case q is odd we have s(σ ) = s(σ )q . This gives (det(A))q = ∑ σ ∈Sn s(σ )q aq1,σ (1) · · · aqn,σ (n) = ∑ σ ∈Sn s(σ )aq1,σ (1) · · · aqn,σ (n) = det(A◦q ) where A◦q denotes the q-th Hadamard or Schur power of A, that is, the matrix whose entries are the the entries of A raised to q. Now it is clear that given any proper minor P of size p of matrix M in GF(qm ), P◦q is the corresponding proper minor in ◦q M and P is nonsingular if and only if P◦q is nonsingular, so M is k-superregular if and only if M ◦q is k-superregular. Corollary 2. In particular, let M be a k-superregular matrix over GF(2m ). Squaring each element of M yields another k-superregular matrix. Proof. This is just the particular case for q = 2 of the Lemma above. Hence we also have: Corollary 3. Assume that the values for r0,i , i = 1, . . . , k and for r1,k are all fixed to 1, as allowed by Lemma 3 and Corollary 1. Then, for r1,k−1 , it suffices to consider one representative of each cyclotomic coset. Proof. Consider any minor of M. Squaring all coefficients in M will not change the values of r0,i , i = 1, . . . , k or r1,k . Thus if there is a k-superregular matrix with r1,k−1 = v, then there is also a k-superregular matrix with r1,k−1 = v2 . The search can be simplified (by constant factors O((2m − 1)n · (2m − 1) · (n − 2)!) by use of Lemma 3, Corollary 1, and Lemma 5, respectively. Corollary 3 reduces complexity by an extra factor of approximately log2 (2m ) = m, but this reduction is not entirely independent of the other reductions. In summary, the search algorithm is highly exponential in complexity, but the tricks allow a deeper search than would otherwise be possible. The search algorithm is sketched in Algorithm 1. The trickier steps are explained in some detail in Remark 4. Remark 4. Here we explain the steps of Algorithm 1. (i) In essence, the algorithm runs through a search tree. At each depth of the tree, ρ points to one of the variables ri, j in (6). Abusing notation, we will also say that ρ points to the current depth. Throughout the course of the algorithm, ρ goes back and forth along r1,k−1 , . . . , r1,1 , r2,k , . . . , r2,1 , r3,k , . . ., i. e. along the values of the last row of (6) in reverse order starting at r1,k−1 (since r1,k , r0,1 , . . . , r0,k can be assumed to be all equal to 1 by Lemma 3 and Corollary 1.) We will in this context use the ordering “<” to refer to the reverse order of the last row of (6), and addition and subtraction on ρ moves ρ left and right, respectively, on this row. (ii) Line 3: Let ρ refer to one element ri, j in the last row of (6). Then Mρ is the set of (formal) proper submatrices of SrD0 ,1 (6) which have ρ in its left lower corner. If all matrices in M = ρ=r Mρ for some D0 ≤ D are nonsingular, where 0,k ∗ D = D − 2 is the target maximum degree, then the submatrix of (6) with rD0 ,1 in the lower left corner is k-superregular. The set Mρ can be found by recursion. (The number of proper submatrices M is related to the Catalan numbers. We omit the details.) (iii) Line 5: At depth ρ, values have already been assigned for each depth ρ 0 < ρ. Hence, by keeping track of subdeterminants already computed, for each determinant corresponding to a proper submatrix in Mρ , the value in GF(2m ) that would 9 Algorithm 1: A computer search algorithm Result: finds good 2m -ary MDS codes of rate (n − 1)/n Input : Field size 2m , target distance D ∗ , code length n Data: ρ points to current position 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 initialization; value(r0,i ) := 1, i = 1, . . . , k, value(r1,k ) = 1; SrD ∗ −2,1 Precompute the set of proper submatrices M = ρ=r Mρ ; 1,k−1 ρ := r1,k−1 ; Precompute the set of legal values L (ρ); while ρ ≤ rD ∗ −2,1 and more coefficient values to check for ρ do if more coefficient values to check for ρ then assign next value to coefficient at ρ; update determinants needed for Mρ+1 , and L (ρ + 1); if deepest level so far then record selected values of coefficients; end ρ = ρ + 1; else ρ = ρ − 1; end end make the determinant zero can be obtained in constant time. In other words, going once through Mρ , we can identify the set L (ρ) of all illegal values for coefficient ρ. (iv) Line 6: (a) A complete search version of the algorithm will successively try all values in L (ρ). For a faster but incomplete search, the algorithm may be set to skip an arbitrary subset of values in L (ρ) at each depth ρ. (b) The target distance D ∗ is an input parameter for the algorithm. In order to determine that a code has a maximum distance D, it is necessary to verify that a complete search version of the algorithm will pass depth ρ = rD,1 , but not depth ρ = rD+1,1 . (v) Line 9: Using the set of values currently assigned to coefficients at all depths ρ 0 ≤ ρ, compute subdeterminants that will be useful for computing determinants in Mρ+1 , and initialize the set of legal values L (ρ + 1) for the next depth. (vi) Complexity: The assumptions enabled by the Lemmas of this section, together with the efficient computation of the determinants, allow a deeper search than would be possible with a naı̈ve search. However, the depth of the search tree that finds a code of degree D is (n − 1)D, and for many of the “early depths”, a complete search needs to go through almost 2m values. The size of the set of proper submatrices also grows exponentially with n and D. So the overall ∗ complexity is at least O(2mn·(D −2) · \|M \| · wd ) where \|M \| is the number of proper submatrices. C. Codes found by computer search with D ≥ 5 Here we present codes found from computer search, for field sizes of characteristic 2 ranging from 8 to 16384, and free distances D ≥ 5. Exact values of ∆(qm , qm ) = 3 and ∆(qm , qm−1 ) = 4 are provided by Propositions 2 and 3. In Tables II–XIII, each row summarizes what we have discovered about rate k/n = (n − 1)/n codes. The ∆ column lists the maximum value of D for which we have found a code with CDP [2, 3, . . . , D]. The absence of a ≥ sign in this column indicates that we have established, through an exhaustive search, that this value of D is indeed maximum for this rate and field size. The Coefficients column presents one encoder that possesses this CDP, in terms of logα () of the coefficients (r1,k , . . . , r1,1 ), (r2,k , . . . , r2,1 ), (r3,k , . . . , r3,1 ), . . ., where α is the primitive element of the field. Note that the degree zero terms, (r0,1 , . . . , r0,k ) are suppressed since they are assumed to be identically 1 = α 0 . The R column contains the rareness of the code, which will be explained in Section IV-B. In the Reference column we include references in the few cases where “similar codes” (i. e. codes over the same field which have the same CDP, but that do not necessarily possess a systematic encoder) have previously been described in the literature. We do not list encoders found by the search if we also found codes with the same set of parameters (rate, CDP) over a smaller field. Also, due to Lemma 8, we do not list codes of rate (n − 2)/(n − 1) if there exist codes of rate (n − 1)/n with the same CDP: Lemma 8. If a systematic MDS code with free distance D and rate k/(k + 1) exists for k > 1, then there is also a systematic MDS code with free distance D and rate (k − 1)/k. Proof. Shorten the k-superregular matrix H 0(D) by selecting any j0 ∈ {1, . . . , k} and removing columns j0 , j0 + k, j0 + 2k, . . . in H 0(D) . 10 Coefficients R Remark 0, 1, 4, 3 0.035 [11] Table II TABLE OF BOUNDS ON ∆(23 , n) FOR THE FIELD DEFINED BY 1 + α + α 3 = 0. n 2 ∆ 6 Coefficients R Remark 0, 1, 4, 3, 0 0.024 0 1, 4 0, 1 7 0.014 [2] Table III TABLE OF BOUNDS ON ∆(24 , n) FOR THE FIELD DEFINED BY 1 + α + α 4 = 0. P LEASE ALSO SEE E XAMPLE 3. n 2 3 ∆ 7 5 Example 3. According to Table III, for the finite field GF(24 ) defined by 1+α +α 4 = 0, there exists a systematic code of rate 2/3 and with CDP= [2, 3, 4, 5]. An example of such a code is represented by (r1,2 , r1,1 ), (r2,2 , r2,1 ), (r3,2 , r3,1 ) = (α 0 , α 1 ), (α 4 , α 0 ), (α 1 , α 7 ) = (1, α), (α 4 , 1), (α, α 7 ), and (implicitly) (r0,1 , . . . , r0,k ) =(1, . . . , 1). Thus the code has a polynomial parity check matrix H(x) = (1 + αx + x2 + α 7 x3 , 1 + x + α 4 x2 + αx3 , 1) and encoder/generator matrix G(x) = 1 0 1 + αx + x2 + α 7 x3 0 1 1 + x + α 4 x2 + αx3 Obviously, G(x)H(x)> = (0, 0). The absence of a ≥ symbol in the ∆ column in Table III indicates that a complete search of all systematic MDS codes of rate 2/3 reveals that D = 5 is maximum. The R column, as will be explained later, indicates that one in seventy random assignments of nonzero values for (r1,1 , r1,2 ), (r2,1 , r2,2 ), (r3,1 , r3,2 ) will give a code with the same CDP, i. e. codes with these parameters are not very rare. A nonsystematic code over GF(24 ) with degree δ = 2 and CDP= [2, 3, 4, 5] was presented in [2]. IV. U PPER BOUNDS AND CODE ASSESSMENT It would be useful to determine upper bounds on ∆(qm , n) in order to assess how good the codes from random search are with respect to optimum. The Heller bound [13] relates convolutional codes with a given free distance D with its truncated block codes, and uses known bounds on block codes to determine convolutional code parameters that cannot be achieved. Unfortunately the Heller bound is of limited use in our case, since the truncated code will actually have a much lower minimum distance than D when viewed as a block code, and also since exact bounds on block codes in the range of parameters that we are interested in here are not well known. Moreover, the approach of sphere packing for binary codes [14] cannot be easily adapted to the current case, since the structure of optimum nonbinary codes turns out to be quite different from that of optimum binary codes2 . A simple bound is described in the next subsection. In Subsection IV-B we present an alternative way of describing how great our codes are, through the concept of rareness. Coefficients R 0, 1, 19, 5, 24, 15, 0 3.4 · 10−8 0 1, 11 28, 21 6, 24 11 4.4 · 10−5 0 1 18 2, 5 8 17 25, 3 2 13 18 5.2 · 10−11 Table IV TABLE OF BOUNDS ON ∆(25 , n) FOR THE FIELD DEFINED BY 1 + α 2 + α 5 = 0. n 2 3 5 ∆ 9 6 5 Coefficients R 0, 1, 6, 61, 60, 46, 28, 23 1.2 · 10−10 0 1, 6 0, 2 37, 21 44, 55 28 4.1 · 10−11 0 1 6, 2 6 26, 13 61 38, 30 33 60 1.4 · 10−11 0 1 6 2 12 3, 14 36 26 25 51 13, 19 60 16 62 5 58 3.2 · 10−20 Table V TABLE OF BOUNDS ON ∆(26 , n) FOR THE FIELD DEFINED BY 1 + α + α 6 = 0. n 2 3 4 7 ∆ 10 7 ≥6 ≥5 2 Optimum binary convolutional codes tend to require parity check matrices with many r 1, j = 0, whereas we have seen that in the nonbinary case, all degree one coefficients r1, j are nonzero. These differences impose different combinatorial constraints in the binary and the nonbinary case. 11 Coefficients R 0 1 31 2, 62 103 64 125, 51 57 19 110, 11 39 43 114 8 · 10−18 0 1 31 2 62 32 103, 3 31 15 0 7 1 63, 6.4 · 10−16 8 94 119 51 41 10 17 Table VI TABLE OF BOUNDS ON ∆(27 , n) FOR THE FIELD DEFINED BY 1 + α 3 + α 7 = 0. n 5 8 n 2 3 4 11 ∆ ≥6 ≥5 Coefficients R 0, 1, 25, 3, 0, 198, 152, 56, 68 2.2 · 10−7 0 1, 25 0, 1 238, 100 106, 195 245, 37 33 2.0 · 10−12 0 1 25, 2 25 198, 1 14 228, 113 74 214, 21 250 172 ≈ 2 · 10−17 0 96 95 176 156 169 160 81 11 245, 107 5 223 167 7 177 98 238 93 53, ≈ 3 · 10−28 37 208 233 89 75 74 184 31 119 100 Table VII TABLE OF BOUNDS ON ∆(28 , n) FOR THE FIELD DEFINED BY 1 + α 2 + α 3 + α 4 + α 8 = 0. ∆ ≥11 ≥8 ≥7 ≥5 Coefficients R 0, 54, 91, 181, 267, 291, 379, 28, 95, 143 1.4 · 10−11 0 280 362 276 426, 206 155 326 324 360, 3.9 · 10−11 356 447 507 312 144, 224 375 236 55 448 13 ≥5 0 19 325 321 356 397 317 455 98 130 149 413, 8.4 · 10−27 48 101 120 272 209 188 405 352 46 343 289 152, 318 80 256 98 255 274 147 340 392 453 30 451 Table VIII TABLE OF BOUNDS ON ∆(29 , n) FOR THE FIELD DEFINED BY 1 + α 4 + α 9 = 0. n 2 6 n 3 5 8 17 n 2 4 ∆ ≥13 ≥8 9 ≥6 n 2 6 ∆ ≥14 ≥7 11 ∆ ≥12 ≥6 Coefficients R 0 603, 246 106, 115 693, 483 544, 603 152, 815 788, 984 721 ≈ 10−15 0 498 997 964, 560 214 101 723, 453 111 370 54, 5 · 10−18 455 17 625 509, 904 431 926 856 ≥6 0 322 804 12 140 1004 384, 778 916 786 247 586 698 294, 3 · 10−24 379 7 784 239 817 284 398, 178 588 110 41 425 976 393 ≥5 0 1 77 2 154 78 956 3 10 155 325 79 618 957 231 4, 4 · 10−39 308 0 4 77 11 1 200 10 80 3 24 155 87 325 619 618, 958 768 255 404 577 976 368 374 709 33 530 109 677 594 652 226 Table IX TABLE OF BOUNDS ON ∆(210 , n) FOR THE FIELD DEFINED BY 1 + α 3 + α 10 = 0. ∆ ≥9 ≥7 Coefficients 0, 1992, 813, 1890, 440, 630, 1947, 1574, 1356, 234, 1266 0 1809 1118, 2027 1610 539, 1042 7 1730, 2020 591 1459, 902 899 1584, 172 1192 513 0 1999 762 1845 1102 1115 1014 328, 1349 345 498 1561 27 987 1300 1793, 1728 562 488 304 43 71 1911 1140, 1524 660 465 327 322 748 1574 1414 Table X TABLE OF BOUNDS ON ∆(211 , n) FOR THE FIELD DEFINED BY 1 + α 2 + α 11 = 0. R 1.0 · 10−9 5.6 · 10−15 2.0 · 10−22 Coefficients R 0, 3294, 1040, 448, 3624, 2406, 826, 1122, 587, 1034, 342, 4037 < 10−15 0 3202 2711 92 2688, 3908 1649 1252 3897 1604, 3687 3602 1603 2339 1350, 1.2 · 10−14 1700 2969 104 3406 2679, 1345 919 3302 2116 810 ≥6 0 669 4050 4007 745 3863 324 1617 3951 1343, 3 · 10−31 703 1123 782 3343 1919 3177 1839 1006 2183 426, 2139 2050 1676 1187 3222 467 1764 2387 2868 641, 2564 2249 3187 3114 3228 743 443 1220 3540 2620 Table XI TABLE OF BOUNDS ON ∆(212 , n) FOR THE FIELD DEFINED BY 1 + α 3 + α 4 + α 7 + α 12 = 0. 12 Coefficients R 0 337, 7672 6843, 3625 3361, 7970 7490, 3.6 · 10−11 5531 2322, 5227 5758, 133 2290, 1453 189 5 ≥8 0 441 2192 3413, 3222 7502 7405 4155, 88 5939 343 6171, ≈ 5 · 10−21 1082 8149 2823 7269, 8022 6454 4999 3373, 3518 442 710 6968 7 ≥7 0 5160 5711 7681 748 5319, 2131 6233 723 4539 7315 5654, 2 · 10−19 5126 7465 3577 6826 5553 1131, 4954 6763 6593 1568 7157 8112, 1961 4310 877 2927 7197 2672 13 ≥6 0 5645 7651 3109 2678 802 6934 1946 5589 2833 5821 38, ≈ 8 · 10−37 5394 2500 5877 3141 4724 3374 5191 7218 4844 423 822 6875, 5712 6619 3935 6414 8025 1422 4391 5698 5481 6850 2635 4786, 556 2558 1063 5172 566 7978 3664 5848 3859 6905 6434 71 Table XII TABLE OF BOUNDS ON ∆(213 , n) FOR THE FIELD DEFINED BY 1 + α + α 3 + α 4 + α 13 = 0. n 3 n 4 ∆ ≥9 8 ≥7 15 ≥6 ∆ ≥10 Coefficients R 0 61 9533, 1260 4487 6469, 3689 8777 4510, 11257 13252 1239, 3 · 10−14 15121 10306 11679, 9618 13110 4549, 12420 5210 13006 0 14132 6404 8841 7620 6707 1150, 1.4 · 10−22 14939 8238 9174 9560 1677 4156 11112, 11424 2037 7827 4640 11071 14007 6628, 13374 10684 2080 14648 1097 14383 1198, 10966 15875 9746 9595 13007 4019 1354 0 15439 10581 4136 503 11096 5590 8608 16006 8229 562 15423 14311 16137, 2 · 10−38 5899 1875 8985 16334 15293 13429 5172 5303 9128 109 10068 1358 7752 6288, 13251 13386 11513 2438 443 15582 4641 2845 3509 12593 6608 14686 11470 15578, 8683 12489 444 8891 4727 12844 12383 5530 4478 9079 9226 5886 6790 8363 Table XIII TABLE OF BOUNDS ON ∆(214 , n) FOR THE FIELD DEFINED BY 1 + α + α 11 + α 12 + α 14 = 0. A. A simple bound The following simple bound is tight for D ≤ 4. Theorem 1. For rate (n − 1)/n codes over GF(qm ) with CDP = [2, 3, . . . , D], n − 1 ≤ (qm − 1)/(D − 2). Proof. For D = 3 the result follows from Proposition 2. Assume that D = 4. Recall that all coefficients are nonzero. Consider the 2 × 2 minors of type 1 r1,s 1 r = r1,s + r1,t , 1,s r1,t r2,s r r1,t = r1,s r2,t + r1,t r2,s , and 1,s r2,s r2,t 1 = r2,s + r1,s r1,t . r1,t (8) From the conditions on the 2 × 2 proper minors, since all those minors have to be nonzero, it follows that in order to have D > 3, the values in the sets {r1,1 , . . . , r1,k } and {r2,1 /r1,1 , . . . , r2,k /r1,k } must be 2k distinct values in GF(qm ) \ {0}. Now consider a code with D > 4. Then the minors r2,s r3,s r2,t r = r2,s r3,t + r2,t r3,s , 2,s r3,t r3,s 1 r = r2,s r1,t + r3,s , and 2,s r1,t r3,s r1,t = r2,s r2,t + r1,t r3,s . r2,t (9) Again they all have to be nonzero, and this implies that the set {r3,1 /r2,1 , . . . , r3,k /r2,k } is a new set of k different values, and they are all different from the values in the sets {r1,1 , . . . , r1,k } and {r2,1 /r1,1 , . . . , r2,k /r1,k }. So in order to have D ≥ 5 we need to have at least 3k different non zero elements in the field. Generalizing the argument, it follows that all ri,t /ri−1,t for 1 ≤ i ≤ D − 2, 1 ≤ t ≤ k are distinct nonzero values. B. Rareness In this section we address the probability that a randomly generated convolutional code over GF(2m ) of rate (n − 1)/n will be an MDS code with CDP of [2, . . . , D]. By “randomly generated” code we will mean one generated by a random systematic encoder, where each coding coefficient ri, j is selected independently and uniformly in GF(2m ) \ {0}. We define this probability as the rareness of the parameter pair (n, D). For small values of n and D, the exact value of the rareness can be determined as a by-product of a complete code search. Since for large parameters it quickly becomes intractable to determine the best codes, it also quickly turns difficult to compute exact results for rareness. However, it is possible to obtain estimates of rareness, as described below. 13 First assume that a complete search is applied. This will determine the set G (ρ ∗ , n, m) of distinct sequences r1,k−1 , . . . , r1,1 , r2,k , . . . , ρ ∗ over GF(2m ) for which all proper submatrices in Mρ 0 , ρ 0 ≤ ρ ∗ are nonsingular. Thus the probability that a given randomly selected sequence corresponds to a path in the search tree that satisfies the conditions at depth ρ ∗ is PR (ρ ∗ , n, m) = \|G (ρ ∗ , n, m)\| ∗ . (2m − 1)\|ρ \| For \|ρ ∗ \| > 1, define PR (ρ ∗ , n, m\|ρ ∗ − 1) = \|L ρ ∗ \| PR (ρ ∗ , n, m) = Avg( m ) ∗ PR (ρ − 1, n, m) 2 −1 (10) where Avg() is the average computed over the complete search. PR (ρ ∗ , n, m\|ρ ∗ − 1) is the average conditional probability that a random generator which satisfies depth ρ ∗ − 1 in the search tree also satisfies depth ρ ∗ . For large parameters we are not able to carry out a complete search. However, we can perform deep but incomplete searches, which also provide estimates of the conditional probabilities PR (ρ ∗ , n, m\|ρ ∗ − 1) in (10) as . These estimates will be quite accurate especially for the first depths, and hence they can be changed together to obtain an estimate for PR (ρ ∗ , n, m). As long as there is a substantial number of different search tree paths leading to depth ρ ∗ − 1, the estimate PR (ρ ∗ , n, m\|ρ ∗ − 1) should be reasonably good. Hence we can also estimate PR (ρ ∗ , n, m\|ρ ∗ − 1)) as ˜ P̃R (ρ ∗ , n, m\|ρ ∗ − 1)) = Avg( \|L ρ ∗ \| 2m − 1 ) ˜ where Avg() is the (weighted) average computed over the incomplete search, and we can then estimate PR (ρ ∗ , n, m) as ρ∗ P̃R (ρ ∗ , n, m) = ∏ P̃R (ρ − 1, n, m\|ρ − 1) ρ=(1,k−1) where for ρ = (1, k − 1), P̃R (ρ − 1, n, m\|ρ − 1) = 1. In Tables II–XIII, we include the exact rareness in cases where we can perform a complete search, and otherwise we include the estimate. We concede that this approach is not foolproof. For example, the construction in Proposition 3, is unique at least for field sizes up to 32. For other choices for the first layer of coefficients r1,1 , . . . , r1,2m−1 than indicated in the proof of Proposition 3, it appears that the search tree ends up being considerably shallower. The rareness of the construction in Proposition 3, i. e. the m probability that a random sequence will match that construction exactly, is 2m−1 · (2m−1 − 1)!/(2m − 1)2 −3 . Already for m = 5 −30 −393 the rareness is about 10 , for m = 8 less than 10 . Hence, if for an arbitrary set of search parameters there exists a very rare construction that is not caught by the incomplete search, the estimates for the deepest values of ρ ∗ may be unprecise. However, we do believe that our estimates of PR (ρ ∗ , n, m) provide some intuition about the difficulty of reaching a certain depth in the search tree with a random path, and in the cases where we are able to carry out a complete search, we also note that the estimates as described here are pretty accurate with a modest non-exhaustive search effort. Figure 1 contains exact values (for n = 2, 3) and estimates (for n = 4, 7) of PR (ρ ∗ , n, 6). Please see the figure caption for explanations. We have also include rareness estimates in Tables II–XIII. V. C ONCLUSION AND OPEN PROBLEMS Motivated by the practical problem of fast recovery of a coded packet-erasure channel, we have studied systematic MDS convolutional codes over GF(2m ). We have characterized them in terms of k-superregularity of a certain matrix. We have presented new optimum constructions for free distances D ≤ 4, tables of new codes found by computer search, and a combinatorial upper bound which is tight in the case of small free distances. In order to assess how “good” a code is, we have also introduced the concept of rareness. It would be interesting to establish upper bounds that are tight also for larger free distances. Another issue would be to study whether there exist general algebraic constructions, similar to the one in Proposition 3, for systematic MDS codes of free distance D ≥ 5. It would also be of some theoretical interest to optimize the CDP of strongly-MDS codes over GF(2m ) under an additional constraint on the degree δ of their minimal encoders. We have not considered this problem since the complexity of Viterbi decoding of such codes is prohibitive for all but small values of the product m · δ (and since it seems difficult). R EFERENCES [1] E. M. Gabidulin, “Convolutional codes over large alphabets,” in Proc. Int. Workshop on Algebraic Combinatorial and Coding Theory, Varna, Bulgaria, 1988, pp. 80—84. [2] Heide Gluesing-Luerssen, Joachim Rosenthal, and Roxana Smarandache, “Strongly-MDS Convolutional Codes,” IEEE Transactions on Information Theory, vol. 52, no. February 2006, 584–598. [3] Paulo Almeida, Diego Napp, and Raquel Pinto, “A new class of superregular matrices and MDP convolutional codes,” 14 1 4 10−2 10−6 10−8 10−10 10−12 10−14 10−16 log10 𝑃( random code is MDS) 10−4 9 6 10 5 7 6 Rate 1/2 Rate 2/3 10−18 Rate 3/4 Rate 6/7 5 Number of coefficients 𝜌 (starting with 𝑟1,𝑘−1 ) Figure 1. Rareness PR (ρ, n, 6) of codes for GF(64) for n ∈ {2, 3, 4, 7}: Exact rareness PR (ρ, n, 6) for ρ ≤ 7, estimates P̃R (ρ, n, 6) for n > 7. In the figure, the search depth ρ is measured in terms of number of coefficients. In order to construct a rate 6/7 encoder of distance D = 5, it is necessary to find a sequence of 17 coefficients r1,5 , . . . , r1,1 , r2,6 , . . . , r3,1 . To get an encoder with distance D = 4, it suffices with 11 coefficients. Similar for the other cases. [4] Pierre Ugo Tournoux, Emmanuel Lochin, Jérôme Lacan, Amine Bouabdallah, and Vincent Roca, “On-the-Fly Erasure Coding for Real-Time Video Applications,” IEEE Transactions on Multimedia, vol. 17, no. 4, (2011), pp. 797–812. [5] M. Kim, J. Cloud, A. Parandeh Gheibi, L. Urbina, K. Fouli, D. J. Leith, and M. Médard, “Network Coded TCP (CTCP),” http://arxiv.org/abs/1212.2291. [6] A. D. Wyner and R. B. Ash, “Analysis of recurrent codes,” IEEE Transactions on Information Theory,Vol. 9, Issue: 3, Jul. 1963, pp. 143 – 156. [7] Øyvind Ytrehus, “Ascetic convolutional codes,” Proc. 33rd. Allerton Conference on Communications, control, and Computing (October 1995), 382–390. [8] Robert J. McEliece, The Algebraic Theory of Convolutional Codes,, in: Handbook of Coding Theory, Eds. V. S. Pless and W. C. Huffman, pp. 1065–1138, North-Holland, 1998. [9] S. Lin and D. Costello, Error Control Coding, 2nd. Ed., Prentice-Hall, 2004. [10] J. H. Stott, A. Oliphant, D. W. Osborne, “Digital video: Error correcting codes and a practical study of a Wyner-Ash error corrector,” Techn. Report, British Broadcasting Corporation, December 1974. [11] J. Justesen and L. Hughes, “On maximum-distance-separable convolutional codes (Corresp.),” in IEEE Transactions on Information Theory, vol. 20, no. 2, pp. 288–288, Mar 1974. [12] F. J. MacWilliams and N. J. A. Sloane,, The Theory of Error-Correcting Codes, Elsevier, 1977. [13] J. A. Heller, “Sequential decoding: Short constraint length convolutional codes,” Space Programs Summary JPL, Pasadena, CA, 1968. [14] Eirik Rosnes and Øyvind Ytrehus, “Sphere-packing bounds for convolutional codes,” IEEE Transactions on Information Theory,Vol. 50, Issue: 11, Nov. 2004, pp. 2801 – 2809. A PPENDIX A : P ROOF OF L EMMA 1 Proof. Before starting we will set some notations. In H (D) for each s = 1, . . . , (D + 1) let Cs be the set of column indices Cs = {(s − 1)(k + 1) + 1, (s − 1)(k + 1) + 2, . . . , s(k + 1)}. Taking into account the way H (D) is constructed it is clear that for any s ≤ D, the submatrix of H (D) formed by the first s + 1 rows and the columns in C1 , . . . ,Cs+1 is H (s) (the parity-check matrix of the s-th truncation). Also the submatrix formed by the last s + 1 rows and the columns in CD−s+1 , . . . ,CD+1 is H (s) . For each set of column indices Cs the last index is s(k + 1) and the corresponding column in H (D) is the s-th column of the identity ID+1 . In an analogous way we will call Cs0 the set of column indices Cs0 = {(s − 1)k + 1, (s − 1)k + 2, . . . , sk} in the matrix H 0(D) . 15 In what follows we will use the same name for a square submatrix and for the corresponding minor since it will create no confusion. Now we start the proof. 1) Assume that the CDP is (d0 = 2, d1 = 3, . . . , dD = D + 2). In particular d0 = 2 implies that all the entries r0, j for j ∈ C1 are non zero. Let M 0 be a proper minor of H 0(D) of size p × p formed by the entries of H 0(D) in rows with indices 1 ≤ i01 < i02 < · · · < i0p ≤ s + 1 and columns with indices 1 ≤ j10 < j20 < · · · < j0p ≤ k(D + 1). Since M 0 is proper we have jl0 ≤ ki0l for l = 1, . . . , p. Let F 0 = {i01 , . . . , i0p } be the set of row indices in M 0 . From M 0 we construct a (D + 1) × (D + 1) minor M in H (D) by doing the following: • The row indices are {1, . . . , D + 1}. • For each s ∈ {1, . . . , D + 1} we define the column index js as follows: – If s ∈ F 0 then there exists a unique l(s) ∈ {1, . . . , p} such that i0l(s) = s (note that l(s) ≤ s). Considering the 0 corresponding column index in M 0 we have jl(s) = ql(s) k + rl(s) with 0 ≤ ql(s) ≤ D and 1 ≤ rl(s) ≤ k unique. (We 0 note that l is an increasing function of s and also that jl(s) ≤ ki0l(s) = ks, which implies ql(s) < s). 0 0 Then define js = ql(s) (k +1)+rl(s) . Clearly jl(s) ∈ Cql(s) +1 and jl(s) ∈ Cql(s) +1 and actually the corresponding columns are identical. – If s ∈ / F 0 then js = s(k + 1), so the corresponding column is the last in the block with column indices Cs . Let us note that 1 ≤ j1 , . . . , jD+1 ≤ (D + 1)(k + 1) but those column indices are not guaranteed to be ordered in increasing order as the js0 were. The added columns will form a submatrix which is ID+1−p in the rows that were not in F 0 , and we have ID+1−p ? M= 0 M0 therefore the value of minor M 0 is the same as the value of M and in order to see that H 0 (D) is k-superregular we just need to check that M 6= 0. We will proceed in a recursive way using that each s-th truncation will provide minimum distance ds = s + 2 for each s ≤ D. • M has at least one column index in C1 . Proof: If there are no columns in C1 it means that 1 ∈ F 0 (otherwise j1 = 1(k + 1), which is in C1 , would be in M). 1 ∈ F 0 implies i01 = 1, hence l(1) = 1 and j10 ≤ ki01 = k, that is, j10 = 0 · k + r1 , and j1 = 0(k + 1) + r1 < (k + 1). This means j1 ∈ C1 which would contradict the assumption. • If M has exactly 1 column in C1 then the other D columns have indices in C2 ∪ · · · ∪CD+1 and all have 0 in the first position, so we have r0, j1 0···0 M= ∗ M2...D+1 • where M2...D+1 is the submatrix of M formed by the last D rows and the last D columns. Since r0, j1 6= 0 we have M 6= 0 if and only if M2...D+1 6= 0, and we can proceed working with M2...D+1 in H (D−1) in the same way. If M has at least two columns in C1 . Suppose that s is the first index for which we have that at least two columns of M are in C1 , at least 3 are in C1 ∪C2 , . . . , at least s + 1 columns are in C1 ∪C2 ∪ · · · ∪Cs but there are no s + 2 columns in C1 ∪C2 ∪ · · · ∪Cs ∪Cs+1 . This clearly implies that there are no column of M in Cs+1 . Now let us consider each t ∈ {1, . . . , s + 1}. – If t ∈ F 0 there exists l(t) ≤ t ≤ s + 1 such that i0l(t) = t. 0 = q k+r 0 jl(t) l(t) l(t) ≤ kil(t) = kt, therefore ql(t) ≤ t − 1, and from here jt = ql(t) (k + 1) + rl(t) ≤ (t − 1)(k + 1) + rl(t) . So jt ∈ C1 ∪C2 ∪ · · · ∪Ct ⊆ C1 ∪C2 ∪ · · · ∪Cs+1 , but it cannot be in Cs+1 , then jt ∈ C1 ∪C2 ∪ · · · ∪Cs . – If t ∈ / F 0 . Note that in this case t ≤ s since s + 1 ∈ / F 0 implies column (s + 1)(k + 1) is in M and in Cs+1 , contradicting that there were no columns in Cs+1 Then jt = t(k + 1) ∈ C1 ∪C2 ∪ · · · ∪Cs . We have proven that even though indices j1 , . . . js+1 are not ordered in increasing order, we have that they are all in C1 ∪C2 ∪ · · · ∪Cs . On the other hand, index js+2 ∈ / C1 ∪C2 ∪ · · · ∪Cs+1 . Hence, M can be decomposed as M1...s+1 0 M= ∗ Ms+2...D+1 16 where M1...s+1 is the part of M corresponding to the first s + 1 rows and columns and we have proven it is contained in the submatrix of H (D) formed by the first s + 1 rows and the first (s + 1)(k + 1) columns, which actually is H (s) and it is guaranteed to be non zero because ds = s + 2 and the minor satisfies the condition that it has at least 2 columns among the first k + 1 columns of H (s) , at least three among the first 2(k + 1), . . ., and at least s + 1 among the first s(k + 1). Minor Ms+2...D+1 is formed by the last D − s rows and columns of M and it is contained in the submatrix of H (D) formed by the last D − s rows and the last (D − s)(k + 1) columns, which is H (D−s−1) and the same argument used so far can be used to prove that is is non zero by decomposing it further into blocks; each of them nonzero. Finally, we can note that M will have at most one column index in CD+1 . Having at least two would imply that M 0 has 0 also at least two columns in CD+1 and this would contradict the condition of M 0 being proper since i0p−1 ≤ D, j0p−1 ∈ 0 0 CD+1 implies j p−1 ≥ kD + 1 > kD = ki0p−1 . 2) Suppose now that H 0(D) is k-superregular. Consider a minor M of size (D + 1) in H (D) formed by the columns in positions 1 ≤ j1 < j2 < · · · < jD+1 ≤ (D + 1)(k + 1) and assume that j2 ≤ (k + 1), j3 ≤ 2(k + 1), . . . , jD+1 ≤ D(k + 1). We construct a minor M 0 by removing from M any column which is in position s(k + 1) and the corresponding row s. As before, it is clear that ID+1−p ? M= 0 M0 where D + 1 − p is the number of removed columns and p is the size of the remaining minor M 0 . With a careful analysis similar to the one done in the reciprocal part of the proof, one can prove that M 0 is a proper minor in H 0 (D) and hence non zero. For this we will continue using the same notations as in the demonstration of the reverse. Consider that the rows remaining in M 0 are 1 ≤ i01 < i02 < · · · i0p = D + 1. We call this set of indices F 0 as before. The other rows correspond to the identity columns that have been suppressed. The corresponding column indices in M 0 are 1 ≤ j10 < j20 < · · · < jD+1 , and each of those columns jt0 is a copy of a column 0 . Using the same notations as in the j f (t) in M and it is clear that j f (t) ∈ Cb(t) for some b(t) ≤ D + 1, implies jt0 ∈ Cb(t) other part of the proof, if j f (t) = q f (t) (k + 1) + r f (t) with 0 ≤ q f (t) ≤ D and 1 ≤ r f (t) ≤ k, then jt = q f (t) k + r f (t) , so they will be in the same block of column indices; b(t) = q f (t) + 1. Note that r(t) ≤ k since column indices that are multiples of k + 1 will be removed and will never turn into columns in M 0 . • First we observe that i p = D + 1 because there were no columns of M in the block with indices in CD+1 , hence the last column of ID+1 cannot be removed (it was never there) and row D + 1 remains in F 0 . The corresponding column j0p will be a copy of some column j f (p) ≤ jD+1 ∈ C1 ∪ · · · ∪CD , so j0p ∈ C10 ∪ · · · ∪CD0 , hence j0p ≤ Dk < (D + 1)k ≤ i0p k. So the proper condition is satisfied for the last index. • In general when we consider row index in position p − s we have i0p−s = D + 1 − s − r where r is the number of identity columns after the D + 1 − s that have been removed. Column j0p−s is copy of column j f (p−s) and f (p−s) ≤ D+1−s−r (s columns after it have been already considered and r have been removed. From here we have j f (p−s) ≤ jD+1−s−r ∈ C1 ∪ · · · ∪CD−s−r and this implies j0p−s ≤ (D − s − r)k < i0p−s k. • A final observation is that j10 is always in block C10 (because block C1 contained at least two columns of M, so even is one is removed there will always be at least one column remaining in that first block. On the other hand i01 ≥ 1 and we have j10 ≤ k ≤ ki01 . The first and last observations are not necessary but they help to understand the general case. We have proven that minor M 0 is proper and therefore cannot be singular.	7
FUSION SYSTEMS WITH SOME SPORADIC J-COMPONENTS arXiv:1605.04615v3 [math.GR] 7 Jun 2017 JUSTIN LYND AND JULIANNE RAINBOLT Abstract. Aschbacher’s program for the classification of simple fusion systems of “odd” type at the prime 2 has two main stages: the classification of 2-fusion systems of subintrinsic component type and the classification of 2-fusion systems of J-component type. We make a contribution to the latter stage by classifying 2-fusion systems with a J-component isomorphic to the 2-fusion systems of several sporadic groups under the assumption that the centralizer of this component is cyclic. 1. Introduction The Dichotomy Theorem for saturated fusion systems [AKO11, II 14.3] partitions the class of saturated 2-fusion systems into the fusion systems of characteristic 2-type and the fusion systems of component type. This is a much cleaner statement than the corresponding statement for finite simple groups, and it has a much shorter proof. In the last few years, M. Aschbacher has begun work on a program to give a classification of a large subclass of the 2-fusion systems of component type. A memoir setting down the outline and first steps of such a program is forthcoming [Asc16], but see [Asc15] for a survey of some of its contents. The immediate goal is to give a simpler proof of roughly half of the classification of the finite simple groups by carrying out most of the work in the category of saturated 2-fusion systems. Let F be a saturated fusion system over a finite 2-group S, of which the standard example is the fusion system FS (G), where G is a finite group and S is a Sylow 2-subgroup of G. A component is a subnormal, quasisimple subsystem. The system is said to be of component type if some involution centralizer in F has a component. The 2-fusion systems of odd type consist of those of subintrinsic component type and those of J-component type. This is a proper subclass of the 2-fusion systems of component type. In focusing attention on this restricted class, one is expected to avoid several difficulties in the treatment of standard form problems like the ones considered in this paper. By carrying out the work in fusion systems, it is expected that certain difficulties within the classification of simple groups of component type can be avoided, including the necessity of proving Thompson’s B-conjecture. We refer to [Asc16] for the definition of a fusion system of subinstrinsic component type, as it is not needed in this paper. The fusion system F is said to be of J-component type if it is not of subintrinsic component type, and there is a (fully centralized) involution x ∈ S such that the 2-rank of CS (x) is equal to the 2-rank of S, and CF (x) has a component. We shall call such a component in an involution centralizer a J-component. Date: March 13, 2018. Key words and phrases. fusion systems, sporadic groups, involution centralizer, components. The research of the first author was partially supported by NSA Young Investigator Grant H98230-14-10312 and was supported by an AMS-Simons grant which allowed for travel related to this work. 1 In this paper, we classify saturated 2-fusion systems having a J-component isomorphic to the 2-fusion system of M23 , J3 , McL, or Ly under the assumption that the centralizer of the component is a cyclic 2-group. A similar problem for the fusion system of L2 (q), q ≡ ±1 (mod 8) was treated in [Lyn15] under stronger hypotheses. Theorem 1.1. Let F be a saturated fusion system over the finite 2-group S. Suppose that x ∈ S is a fully centralized involution such that F ∗ (CF (x)) ∼ = Q × K, where K is the 2-fusion system of M23 , J3 , McL, or Ly, and where Q is a cyclic 2-group. Assume further that m(CS (x)) = m(S). Then K is a component of F . Here F ∗ (CF (x)) is the generalized Fitting subsystem of the centralizer CF (x) [Asc11], and m(S) := m2 (S) is the 2-rank of S – that is, the largest rank of an elementary abelian 2-subgroup of S. We mention that any fusion system having an involution centralizer with a component isomorphic to McL or Ly is necessarily of subintrinsic component type by [Asc16, 6.3.5]. This means that, when restricted to those components, Theorem 1.1 gives a result weaker than is needed to fit into the subintrinsic type portion of Aschbacher’s program. However, we have included McL and Ly here because our arguments apply equally well in each of the four cases. There is no almost simple group with an involution centralizer having any of these simple groups as a component, but the wreath product G = (K1 × K2 )hxi with K1x = K2 always has CG (x) = hxi × K with K a component that is diagonally embedded in K1 × K2 . The strategy for the proof of Theorem 1.1 is to locate a suitable elementary abelian subgroup F in the Sylow 2-subgroup of K, and then to show that the normalizer in S of E := hxiF has at least twice the rank as that of F . Thus, the aim is to force a resemblance with the wreath product, in which NG (hxiF ) modulo core is an extension of F1 × F2 (with Fi the projection of F onto the ith factor) by hxi × AutK (F ). Lemma 3.2 is important for getting control of the extension of E determined by NF (E) in order to carry out this argument. Acknowledgements. We would like to thank the Department of Mathematics and Statistics at Saint Louis University and the Departments of Mathematics at Rutgers University and Ohio State University for their hospitality and support during mutual visits of the authors. We would also like to thank R. Solomon and R. Lyons for helpful discussions, and an anonymous referee for their comments and suggestions. 2. Background on fusion systems We assume some familiarity with notions regarding saturated fusion systems as can be found in [AKO11] or [Cra11], although some items are recalled below. Most of our notation is standard. Whenever G is a group, we write G# for the set of nonidentity elements of G. If we wish to indicate that G is a split extension of a group A P G by a group B, then we will write G = A · B. For g ∈ G, denote by cg the conjugation homomorphism cg : x 7→ xg and its restrictions. Morphisms in fusion systems are written on the right and in the exponent. That is, we write xϕ (or P ϕ ) for the image of an element x (or subgroup P ) of S under a morphism ϕ in a fusion system, by analogy with the more standard exponential notation for conjugation in a group. 2 2.1. Terminology and basic properties. Throughout this section, fix a saturated fusion system F over the p-group S. We will sometimes refer to S as the Sylow subgroup of F . For a subgroup P ≤ S, we write AutF (P ) for HomF (P, P ), and OutF (P ) for AutF (P )/ Inn(P ). Whenever two subgroups or elements of S are isomorphic in F , we say that they are F conjugate. Write P F for the set of F -conjugates of P . If E is a subsystem of F on the subgroup T ≤ S and α : T → S is a morphism in F , the conjugate of E by α is the subsystem E α over T α with morphisms ϕα := α−1 ϕα for ϕ a morphism in E. We first recall some of the terminology for subgroups and common subsystems in a fusion system. Definition 2.1. Fix a saturated fusion system over the p-group S, and let P ≤ S. Then • P is fully F -centralized if \|CS (P )\| ≥ \|CS (Q)\| for all Q ∈ P F , • P is fully F -normalized if \|NS (P )\| ≥ \|NS (Q)\| for all Q ∈ P F , • P is F -centric if CS (Q) ≤ Q for all Q ∈ P F , • P is F -radical if Op (OutF (P )) = 1, • P is weakly F -closed if P F = {P }, • the centralizer of P in F is the fusion system CF (P ) over CS (P ) with morphisms those ϕ ∈ HomF (Q, R) such that there is an extension ϕ̃ ∈ HomF (P Q, P R) that restricts to the identity on P , • the normalizer of P in F is the fusion system NF (P ) over NS (P ) with morphisms those ϕ ∈ HomF (Q, R) such that there is an extension ϕ̃ : P Q → P R in F such that P ϕ̃ = P . We write F f and F c for the collections of fully F -normalized and F -centric, respectively, and we write F f c for the intersection of these two collections. Sometimes we refer to an element x of S as being fully F -centralized, when we actually mean that the group hxi is fully F -centralized, especially when x is an involution. For example, this was done in the the statement of the theorem in the introduction. Whenever P ≤ S, we write A(P ) for the set of α ∈ HomF (NS (P ), S) such that P α is fully F -normalized. Lemma 2.2. For each P ≤ S, A(P ) is not empty. Moreover, for each Q ∈ P F ∩ F f , there is α ∈ A(P ) with P α = Q. Proof. This is [BLO03, A.2(b)], applied with K = Aut(P ). By a result of Puig, the centralizer CF (P ) is saturated if P is fully F -centralized, and the normalizer NF (P ) is saturated if P is fully F -normalized. We write Op (F ) for the (unique) largest subgroup P of S satisfying NF (P ) = F , and Z(F ) for the (unique) largest subgroup P of S satisfying CF (P ) = F . We note that if F = FS (G) for some finite group G with Sylow p-subgroup S, then Op (G) ≤ S is normal in F so that Op (G) ≤ Op (F ), but the converse does not hold in general. 2.2. The Model Theorem. A subgroup P ≤ S is F -centric if and only if CS (P ) ≤ P and P is fully F -centralized [AKO11, I.3.1]. If P is F -centric and fully F -normalized, then the normalizer fusion system M := NF (P ) is constrained – that is, Op (M) is M-centric. By the Model Theorem [AKO11, Proposition III.5.10], there is then a unique finite group M up to isomorphism having Sylow p-subgroup NS (P ) and such that Op′ (M) = 1, Op (M) = Op (M), and FNS (P ) (M) ∼ = M. Then M is said to be a model for M in this case. 3 2.3. Tame fusion systems. The main hypothesis of Theorem 1.1 is that the generalized Fitting subsystem of the involution centralizer C is the fusion system of a finite group Q×K, where Q is a cyclic 2-group, and K is simple. In this situation, CF (x) is itself the fusion system of a finite group C with F ∗ (C) = Q × K, where K ∼ = M23 , McL, J3 , or Ly, since each of these simple groups tamely realizes its 2-fusion system [AOV12, Oli16b]. Roughly, a finite group tamely realizes its fusion system if every automorphism of its fusion system is induced by an automorphism of the group. Moreover, a fusion system is said to be tame if there is some finite group that tamely realizes it. We refer to [AOV12] for more details. The importance of tameness in the context of standard form problems was pointed out in [Lyn15, §§1.5]. The discussion there is centered around the notion of strong tameness, which was needed for proofs of the results of [AOV12], but the contents of [Oli13, GL16] imply that a fusion system is tame if and only if it is strongly tame. Recently, Oliver has established the following useful corollary of the results in [AOV12], which we state for our setup here. Theorem 2.3 ([Oli16a, Corollary 2.5]). Let C be a saturated fusion system over a 2-group. Assume that F ∗ (C) = O2 (C)K, where K is simple and tamely realized by a finite simple group K. Then C is tamely realized by a finite group C such that F ∗ (C) = O2 (C)K. Note that, upon application of Theorem 2.3 to the involution centralizer C = CF (x) in Theorem 1.1, we have O2 (C) = Q = O2 (C). Indeed, O2 (C) ≤ Q since O2 (C) is normal in C, and one sees that Q = CS (K) by combining Lemma 1.12(c) of [Lyn15] with Lemma 3.7(c) below. However, O2 (C) is normal and self-centralizing in CC (K) by properties of the generalized Fitting subgroup, so that CC (K)/O2 (C) is a group of outer automorphisms of the cyclic 2-group O2 (C), and so is itself a 2-group. It follows that CC (K) = CS (K) is a normal 2-subgroup of C (since K E C), and hence Q = CS (K) ≤ O2 (C). Thus, the effect of Theorem 2.3 for our purposes is that we may work in the group C, where Q is a normal subgroup. In particular, in the setup of the Theorem 1.1, the quotient C/Q is isomorphic to a subgroup of Aut(K) containing Inn(K), where K is one of the simple groups appearing in Theorem 1.1. 3. Structure of the components In this section, we recall some properties of the simple systems appearing in Theorem 1.1 that are required for the remainder. Lemma 3.1. Let G be A7 or GL2 (4), and V a faithful F2 [G]-module of dimension 4. Then (a) G acts transitively on the nonzero vectors of V , (b) CGL(V ) (G) ≤ G, (c) H 1 (G, V ) = 0, and (d) if G acts on a homocyclic 2-group Y with Ω1 (Y ) = V , then Y = V . Proof. In each case, V is irreducible. There is a unique such module for GL2 (4) ∼ = C3 × A5 , namely the natural F4 [G]-module considered as a module over F2 , and thus (a) holds in this case. The module for A7 is unique up to taking duals; clearly points (a) and (b) are independent of the choice between these two modules, and (c) is independent of such a choice by [Asc00, §17]. Note that A7 acts transitively on V # , which can be seen by noting that a Sylow 7-subgroup acts with exactly one fixed point on V # , and a Sylow 5-subgroup acts with no fixed points. Point (b) holds for G = A7 by absolute irreducibility. Similarly for 4 G = GL2 (4), one has that CGL(V ) (G) = EndF2 [G] (V )× = F× 4 , and so (b) follows in this case as Z(G) ∼ C . Point (c) for G = GL (4) holds because, by coprime action, Z(G) (and so = 3 2 G) has a fixed point on any 5-dimensional module containing V as a submodule (see [Asc00, §17]). Point (c) for G = A7 holds, for example, by applying [AG72] with L = {L0 , L1 , L2 }, where L1 = CG ((1, 2, 3)), L2 = CG ((4, 5, 6)), and L0 = L1 ∩ L2 . The Li indeed satisfy the hypotheses of that theorem: H 0 (Li , V ) = 0 because each Sylow 3-subgroup of GL4 (2) ≥ G has no nontrivial fixed point on V , and H 1 (Li , V ) = 0 using a similar argument via coprime action as above. We now turn to (d), which follows from a special case of a result of G. Higman [Hig68, Theorem 8.2]. This says that if SL2 (4) acts faithfully on a homocyclic 2-group Y in which an element of order 3 acts without fixed points on Y # , then Y is elementary abelian. In case G = GL2 (4), V is the natural module for G and certainly SL2 (4) ≤ G has (with respect to an appropriate basis) a diagonal element of order 3 acting without fixed points. In the case G = A7 , we have G ≤ A8 ∼ = GL4 (2), and the action of G on V is the restriction of the natural (or dual) action of GL4 (2). Restriction of either one to GL2 (4) ∼ = C3 × SL2 (4) ∼ = C3 × A5 shows that SL2 (4) is embedded in G as an A5 moving 5 points in the natural permutation action. So SL2 (4) is contained in G up to conjugacy, and as before, it has an element of order 3 acting without fixed points on V # . Hence, (d) holds in this case as well by Higman’s Theorem. For a vector space E over the field with two elements, the next lemma examines under rather strong hypotheses the structure of extensions of E by certain subgroups of the stabilizer in GL(E) of a hyperplane. Lemma 3.2. Let E ∼ = E2n , P = NA (V ), and = E2n+1 (n ≥ 3), A = Aut(E), V ≤ E with V ∼ U = O2 (P ). Let L be a complement to U in P acting decomposably on E, x ∈ E − V the fixed point for the action of L, and G ≤ L. Let H be an extension of E by UG with the given action and let X be the preimage in H of U under the quotient map H → UG. Assume that (a) G acts transitively on V # ; (b) CGL(V ) (G) ≤ G; and (c) H 1 (G, V ) = 0. Then there is a subgroup Y of X that is elementary abelian or homocyclic of order 22n and a G-invariant complement to hxi in X. Proof. Let X̄ = X/V . Since the commutator map (3.3) [x, −] determines a G-equivariant linear isomorphism X/E → V , G is transitive on the nonzero vectors of X/E by (a), and hx̄i ≤ Z(X̄). Hence if X̄ is not elementary abelian, then it is extraspecial with center hx̄i, and G preserves the squaring map X̄ → Z(X̄). This is not the case, because G is transitive on the nonzero vectors of X/E and n ≥ 3. Therefore, (3.4) X̄ is elementary abelian. Assumption (c) now yields that there is a G-invariant complement Ȳ to hx̄i. Let Y be the preimage of Ȳ in X. We claim that Y is abelian; assume on the contrary. Then [Y, Y ] and ✵1 (Y ) are contained in V since Ȳ is elementary abelian, and by assumption, neither of 5 these are trivial. Similarly, V is contained in Z(Y ), which is not Y . Therefore, V = [Y, Y ] = Φ(Y ) = ✵1 (Y ) = Z(Y ), (3.5) by (a) and (3.3). √ By (3.5), the squaring map Ȳ → V is G-equivariant linear isomorphism; let − be its √ inverse. Then the map ξ : V → V given by ξ(v) = [x, v] is a linear isomorphism commuting with the action of G, and so ξ ∈ ρ(G) by (b), where ρ : G → GL(V ) is the structure map. Let g ∈ G map to ξ −1 under ρ. Then y 7→ [x, y g ] is the squaring map. This means that for each y ∈ Y , we have y gx = y 2 y g . Hence, for each pair w, y ∈ Y , w 2 y 2 w g y g = w gx y gx = (wy)gx = (wy)2w g y g which gives w 2 y 2 = (wy)2 = w 2 y 2 [y, w]. Thus Y is abelian after all. It follows that Ω1 (Y ) = V or Y by (a), and this completes the proof of the lemma. We now examine the structure of the simple systems occupying the role of K in Theorem 1.1. Let T0 be a 2-group isomorphic to a Sylow 2-subgroup of L3 (4). This is generated by involutions t1 , t2 , a1 , a2 , b1 , and b2 such that Z(T0 ) = ht1 , t2 i and with additional defining relations: [a1 , a2 ] = [b1 , b2 ] = 1, [a1 , b1 ] = [a2 , b2 ] = t1 , [a2 , b1 ] = t2 , [a1 , b2 ] = t1 t2 . A Sylow subgroup of M23 or McL is isomorphic to a Sylow 2-subgroup of an extension of L3 (4) by a field automorphism; this is a semidirect product T0 hf i with f 2 = 1, [a1 , f ] = [a2 , f ] = a1 a2 , [b1 , f ] = [b2 , f ] = b1 b2 , [t2 , f ] = t1 . A Sylow subgroup of J3 is isomorphic to a Sylow 2-subgroup of L3 (4) extended by a unitary (i.e., graph-field) automorphism; this is a semidirect product T0 hui with u2 = 1, [a1 , u] = [u, b1 ] = a1 b1 , [a2 , u] = [u, b2 ] = a2 b2 , [t2 , u] = t1 . A Sylow subgroup of Ly is isomorphic to a Sylow 2-subgroup of Aut(L3 (4)); this is a semidirect product T0 hf, ui with [f, u] = 1 and the relations above. Denote by T1 a 2-group isomorphic to one of T0 hf i, T0 hui, or T0 hf, ui. Recall that the Thompson subgroup J(P ) of a finite p-group P is the subgroup generated by the elementary abelian subgroups of P of largest order. Lemma 3.6. Let K be M23 , McL, J3 , or Ly, with Sylow 2-subgroup T1 as above. Then (a) Z(T1 ) = ht1 i is of order 2; (b) A(T1 ) = {F1 , F2 } where F1 = ht1 , t2 , a1 , a2 i and F2 = ht1 , t2 , b1 , b2 i, so that J(T1 ) = T0 . Also, after suitable choice of notation, one of the following holds: (i) K = M23 , McL, or Ly, and AutK (F1 ) ∼ = A7 , or (ii) K = J3 and AutK (F1 ) ∼ = GL2 (4). (c) There is F ∈ A(T1 ) such that the pair (AutK (F ), F ) satisfies assumptions (a)-(c) of Lemma 3.2 in the role of (G, V ). (d) All involutions in ht1 , t2 i are AutK (J(T1 ))-conjugate. Proof. Point (a) holds by inspection of the relations above. Now F1 and F2 are the elementary abelian subgroups of T0 of maximal rank, and so to prove (b), it suffices to show that each elementary abelian subgroup of maximal rank in T1 is contained in T0 . Set L := L3 (4), and identify Inn(L) with L. Write Inndiag(L) ≥ L for the group of inner-diagonal automorphisms 6 of L. Then Inndiag(L) contains L with index 3, corresponding to the size of the center of the universal version SL3 (4) of L [GLS98, Theorem 2.5.12(c)]. Also, Aut(L) is a split extension of of Inndiag(L) by ΦL ΓL = ΦL × ΓL , where ΦL = hϕi ∼ = C2 is generated by a C is generated by a graph automorphism [GLS98, field automorphism of L, and ΓL = hγi ∼ = 2 Theorem 2.5.12]. By [GLS98, Theorems 4.9.1, 4.9.2], each involution of Aut(L)−L is Aut(L)conjugate to ϕ, ϕγ, or γ, and the centralizers in L of these automorphisms are isomorphic to L3 (2), U3 (2) ∼ = (C3 × C3 )Q8 , and Sp2 (4) ∼ = A5 , respectively, again by those theorems. These centralizers have 2-ranks 2, 1, and 2, respectively. Since T0 has 2-rank 4, this shows that J(T1 ) ≤ T0 . From the relations used in defining T0 , each involution in T0 is contained in one of F1 or F2 , so we conclude that A(T1 ) = A(T0 ) = {F1 , F2 }. The description of the automizers in (b) follows from [Fin76a, Table 1] for M23 and McL, [Fin76b, Lemma 3.7] for J3 , and [Wil84] for Ly. Now point (c) follows from (b) and Lemma 3.1, and point (d) follows from (c) and Burnside’s fusion theorem (i.e. the statement that the automizer of a weakly K-closed subgroup of T1 (which is J(T1 ) in this case) controls the K-conjugacy in its center). Lemma 3.7. Let K be one of the sporadic groups M23 , McL, J3 , or Ly, and let T1 be a Sylow 2-subgroup of K. Then (a) Out(K) = 1 if K ∼ = C2 otherwise; and = M23 or Ly, and Out(K) ∼ (b) for each involution α ∈ Aut(K) − Inn(K), (i) CK (α) ∼ = McL, = M11 if K ∼ ∼ (ii) CK (α) ∼ L (17) if K = J3 ; and = 2 (c) each automorphism of K centralizing a member of A(T1 ) is inner. Proof. Points (a) and (b) follow by inspection of Table 5.3 of [GLS98]. By Lemma 3.6(b), the 2-rank of K is 4, while each of the centralizers in (b)(i-ii) is of 2-rank 2, so (c) holds. 4. Preliminary lemmas We now begin in this section the proof of Theorem 1.1, and so we fix the notation and hypotheses that will hold throughout the remainder of the paper. Let F be a saturated fusion system over the 2-group S, and let x ∈ S be an involution. Assume that hxi is fully F -centralized, that m(CS (x)) = m(S), and that F ∗ (CF (x)) = Q×K, where Q is cyclic. Set C = CF (x) and T = CS (x), so that C is a saturated fusion system over T by the remark just after Lemma 2.2. Let T1 be the Sylow subgroup of K, and set R := Q × T1 ≤ T. Assume that K is the fusion system of one of the sporadic groups K = M23 , J3 , McL, or Ly. Since K tamely realizes K in each case, the quotient (4.1) T /R induces a 2-group of outer automorphisms of K by Theorem 2.3. Arguing by contradiction, we assume K is not a component of F. We fix the presentation in Section 3 for T1 in whichever case is applicable, and we note that Ω1 (Q) = hxi by assumption on Q. Lemma 4.2. Notation may be chosen so that T and J(T ) are fully F -normalized. 7 Proof. We repeatedly use Lemma 2.2. Let α ∈ A(T ). Then as \|CS (xα )\| ≥ \|T α \| = \|T \| = \|CS (x)\|, we have that xα is still fully F -centralized. Thus we may assume that T is fully F normalized after replacing x, T , and J(T ) by their conjugates under α. Now let β ∈ A(J(T )). Then as NS (T ) = NNS (J(T )) (T ), it follows that \|NS (T β )\| = \|NNS (J(T )β ) (T β )\| ≥ \|NNS (J(T ))β (T β )\| = \|NNS (J(T )) (T )β \| = \|NS (T )β \| = \|NS (T )\| and so equality holds because T is fully F -normalized. Hence T β is still fully F -normalized. As before, xβ is still fully F -centralized. Lemma 4.3. hxi is not weakly F -closed in T . Proof. Assume on the contrary, in which case T = S, otherwise NS (T ) contains T properly and moves x. It follows that x ∈ Z(S), which is contained in the center of every F -centric subgroup. Hence x ∈ Z(F ) by Alperin’s fusion theorem [BLO03, Theorem A.10], and we conclude that K is a component of C = F , contrary to hypothesis. Lemma 4.4. The following hold. (a) J(T ) = J(R) = hxi × J(T1 ); and (b) CT (T1 ) = Qht1 i. Proof. Suppose (a) does not hold. Choose A ∈ A(T ) with A hxi × J(T1 ). Then A R by the structure of R, and A acts nontrivially on K by (4.1). In particular, Out(K) = 2 and m(CR (A)) ≤ 3 by Lemma 3.7. Hence, m(A) ≤ 4 while m(R) = 5. This contradicts the choice of A and establishes (a). By Lemma 3.6(a), CR (T1 ) = Qht1 i. Also, CT (T1 ) = CR (T1 ) by part (a), so (b) is also established. Lemma 4.5. If T = S, then Ω1 Z(S) = hx, t1 i. If T < S, then Ω1 (Z(S)) = ht1 i. Proof. Note first that Ω1 (Z(S)) ≤ T . By Lemma 3.7(c) and (4.1), Ω1 (Z(S)) ≤ R, and so Ω1 Z(S) ≤ Ω1 (Z(R)) = hx, t1 i by Lemma 3.6(a). Thus, the lemma holds in case T = S. In case T < S, let a ∈ NS (T )−T with a2 ∈ T . Note that [J(T ), J(T )] = [J(T1 ), J(T1 )] = ht1 , t2 i by Lemma 4.4(a). So a normalizes Ω1 Z(T ) = hx, t1 i and Ω1 Z(T ) ∩ [J(T ), J(T )] = hx, t1 i ∩ ht1 , t2 i = ht1 i, but a does not centralize x. Thus, Ω1 Z(S) = ht1 i as claimed. Lemma 4.6. The following hold. (a) x is not F -conjugate to t1 ; and (b) x is conjugate to xt1 if and only if T < S, and in this case x is NS (T )-conjugate to xt1 . Proof. For part (a), let ϕ ∈ F with xϕ ∈ Z(T ). Assume first that T = S. By the extension axiom, ϕ extends to ϕ̃ ∈ AutF (T ), which restricts to an automorphism of J(T ). Lemma 4.4(a) shows that x ∈ / [J(T ), J(T )], while t1 ∈ [J(T ), J(T )] from Lemma 3.6(d). Hence, xϕ 6= t1 ; this shows x is not F -conjugate to t1 in this case. Now assume that T < S. Then x ∈ / Z(S), whereas Z(S) = ht1 i by Lemma 4.5. Since hxi is fully F -centralized by assumption, we conclude that x is not F -conjugate to t1 in this case either. This completes the proof of (a). If T < S, then x is NS (T )-conjugate to xt1 by (a), while if T = S, then (a) and Burnside’s fusion theorem imply that hxi is weakly F -closed in Ω1 (Z(S)) = hx, t1 i. Thus, (b) holds. 8 5. The 2-central case In this section it is shown that T < S; that is, x is not 2-central. We continue the notation set at the beginning of Section 4. Lemma 5.1. If T = S, then hxi is weakly F -closed in R. Proof. Assume T = S. Then Ω1 (Z(S)) = hx, t1 i by Lemma 4.5. Using Burnside’s fusion theorem and assumption, we see from Lemma 4.6 that (5.2) hxi is weakly F -closed in hx, t1 i. By inspection of [GLS98, Table 5.3], K has one class of involutions. Thus, there are exactly three C-classes of involutions, namely {x}, (xt1 )C , and tC1 . The lemma therefore holds by (5.2). Lemma 5.3. If T = R, then T < S. In particular, T < S in case K is the fusion system of M23 or Ly. Proof. Assume T = R, and also to the contrary that T = S. By Lemma 5.1, hxi ≤ Z(S) is weakly F -closed, and so is fixed by each automorphism of each F -centric subgroup. Therefore, hxi ≤ Z(F ), and so K is a component of C = F , contrary to assumption. The last statement follows then follows from Lemma 3.7(a). Lemma 5.4. Assume K is the fusion system of McL or J3 . Then T < S. Proof. Assume T = S. Then R < T by Lemma 5.3. Fix f ∈ xF ∩ (T − R). The extension Khf i := KT1 hf i of K is defined by [Asc11, §8], and Khf i/Q is the 2-fusion system of Aut(McL) or Aut(J3 ) by Theorem 2.3. Thus, \|T : R\| = 2 by Lemma 3.7(b). Conjugating in Khf i if necessary, we may assume hf i is fully Khf i-centralized. By Lemma 3.7(b), all involutions of T1 hf i−T1 are Khf i-conjugate, and CKhf i (f ) ∼ = hf i×F2 (M11 ) or hf i × F2 (L2 (17)). In particular, CT1 (f ) is semidihedral or dihedral, respectively, of order 16 and with center ht1 i. Fix a four subgroup V ≤ CT1 (f ). Then f is conjugate to f t1 (for example, by an element in the normalizer of CT1 hf i (f ) in T1 hf i), and hence is F -conjugate to each element of f V by the structures of F2 (M11 ) and F2 (L2 (17)). Fix α ∈ HomF (hf i, hxi). By the extension axiom, α extends to a morphism, which we also call α, defined on hf iV ≤ CT (f ). Therefore, x is F -conjugate to each element in xV α . Now the intersection xV α ∩ R = V α ∩ R is nontrivial because \|T : R\| = 2, so as x is not itself in V α , we see that x has a distinct conjugate in R. This contradicts Lemma 5.1 and completes the proof. 6. Proof of Theorem 1.1 Continue the notation and hypotheses set at the beginning of Section 4. In addition, we fix F ∈ A(T1 ) satisfying assumptions (a)-(c) of Lemma 3.2, as guaranteed by Lemma 3.6(c), and set E := hxiF . Then E ∈ A(T ) by Lemma 4.4(a), and so (6.1) m(T ) = 5. In this section, we finish the proof of Theorem 1.1, by showing that the hypotheses of Lemma 3.2 hold for a model of the normalizer in F of an appropriate F -conjugate of E. Via Lemma 3.1(d), this forces the 2-rank of S to be at least 8, contrary to the hypothesis that m(S) = m(T ). 9 By Lemmas 5.3 and 5.4, T < S. Lemma 6.2. \| AutF (T ) : AutC (T )\| = 2. Proof. Represent AutF (T ) on Ω1 Z(T ) = hx, t1 i and apply Lemma 4.6. Lemma 6.3. The following hold. (a) \| AutF (J(T )) : AutC (J(T ))\| = 4 and (b) xAutF (E) = xF , and so \| AutF (E) : AutC (E)\| = 16. Proof. Represent AutF (J(T )) on Ω1 Z(J(T )) = hx, t1 , t2 i. Now AutC (J(T )) = CAutF (J(T ) (x), and the former is transitive on Ω1 ZJ(T1 )# by Lemma 3.6(d). Also, since x is NS (T )conjugate to xt1 , we conclude from Lemma 4.6 that xAutF (J(T )) = xZJ(T1 ) is of size 4. Thus, (a) holds. Similarly to (a), we have AutC (E) = CAutF (E) (x), and the former is transitive on F # by choice of F . From Lemma 4.4(a) and Lemma 3.6(a), \|A(T )\| = 2. By part (a) and Lemma 4.2, \|NS (J(T )) : T \| = 4. Representing NS (J(T )) on A(T ), we see that the kernel has index at most 2, so there is an element of NS (J(T ))−T that normalizes E. In particular, NT (E) < NS (E), and so x is AutF (E)-conjugate to a member of xF # . Now by choice of F , another appeal to Lemma 4.6 yields that xAutF (E) = F x has size 16, which establishes (b). Lemma 6.4. The following hold: (a) Q = hxi. (b) E is F -centric. Proof. Suppose on the contrary that Q > hxi and choose w ∈ Q with w 2 = x. Fix a ∈ NS (T ) − T such that a2 ∈ T . Then xa = xt1 and also (w a )2 = xt1 . Further hw a i is normal in T since hwi is. Thus, [hw a i, T1 ] ≤ hw a i ∩ T1 = 1, whereas CT (T1 ) = Qht1 i by Lemma 4.4(b). It follows that hxt1 i = ✵1 (hw a i) ≤ Ω1 ✵1 (Qht1 i) = hxi, a contradiction that establishes (a). Let E0 be one of the two elementary abelian subgroups of rank 5 in T , and set F0 = E0 ∩ J(T1 ). Then E0 = hxiF0 contains x, and so CS (E0 ) = CT (E0 ). By Lemma 3.7(c), CT (E0 ) = CR (E0 ) = QF0 . Hence CS (E0 ) = QF0 = E0 by part (a). We can now prove (b). Fix α ∈ A(E). Since hxi is fully centralized, the restriction of α−1 to hxα i has an extension β : CS (xα ) → CS (x) = T , which is defined on CS (E α ). Thus, setting E0 := CS (E α )β ≤ T , we see from the previous paragraph that \|CS (E)\| = \|CS (E0 )\| = \|E0 \| = \|CS (E α )\|, so that CS (E α ) = E α . As E α is fully F -normalized and contains its centralizer in S, this means that E is F -centric, as claimed. Since we will be working in NF (E) for the remainder, we may assume, after replacing E by an F -conjugate if necessary, that E is fully F -normalized. Hence E ∈ F f c by Lemma 6.4(b). Fix a model H for NF (E) (cf. §§2.2). Lemma 6.5. H satisfies the hypotheses of Lemma 3.2. Proof. Set G̃ = AutC (E) and observe that AutF (E) ∼ = H/E by Lemma 6.4(b). Thus G̃ contains G := A7 or GL2 (4) with index 1 or 2. As G acts transitively on F # and centralizes x, it follows from Lemma 6.3(b) that xF and F # are the orbits of AutF (E) on E # . Hence 10 AutF (E) ≤ NAut(E) (F ), a nontrivial split extension of an elementary abelian 2-group U of order 16 by GL(F ) with the standard action. We claim that U ≤ AutF (E). Suppose that this is not the case. Now G acts transitively on F # and the commutator map [x, −] defines an isomorphism of G-modules from U to F , so G acts transitively on U # . Since U is normalized by AutF (E), we see that AutF (E) ∩ U = 1. In particular, AutF (E) embeds into GL(F ). Now G ∼ = A7 or GL2 (4) in the cases under consideration, and by Lemma 6.3(b), AutF (E) is therefore a subgroup of GL(F ) containing G with index 16 or 32. However, A7 has index 8 in GL4 (2), and GL2 (4) is contained with index 2 in a unique maximal subgroup of GL4 (2), a contradiction. Therefore, U ≤ AutF (E) as claimed. It has thus been shown that AutF (E) contains a subgroup with index 1 or 2 that is a split extension of U = O2 (NAut(E) (F )) by G. Thus, H has a subgroup of index 1 or 2 that is an extension of E by UG. Assumptions (a)-(c) of Lemma 3.2 hold via Lemma 3.6(c) by the choice of F . Proof of Theorem 1.1. Keep the notation of the proof of Lemma 6.5. By that lemma and Lemma 3.2, there is a G-complement Y to hxi in O2 (H) that is homocyclic of order 28 with Ω1 (Y ) = F , or elementary abelian of order 28 . Now G is isomorphic to A7 or GL2 (4) with faithful action on F , so the former case is impossible by Lemma 3.1. Hence, m2 (T ) = 5 < 8 ≤ m2 (S), contrary to hypothesis. 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MR 755827 Institute of Mathematics, University of Aberdeen, Fraser Noble Building, Aberdeen AB24 3UE E-mail address: [email protected] Department of Mathematics and Statistics, Saint Louis University, 220 North Grand Blvd., Saint Louis, MO 63103 E-mail address: [email protected] 12	4
Prediction with a Short Memory arXiv:1612.02526v3 [cs.LG] 9 Nov 2017 Sham Kakade University of Washington [email protected] Percy Liang Stanford University [email protected] Vatsal Sharan Stanford University [email protected] Gregory Valiant Stanford University [email protected] Abstract We consider the problem of predicting the next observation given a sequence of past observations, and consider the extent to which accurate prediction requires complex algorithms that explicitly leverage long-range dependencies. Perhaps surprisingly, our positive results show that for a broad class of sequences, there is an algorithm that predicts well on average, and bases its predictions only on the most recent few observation together with a set of simple summary statistics of the past observations. Specifically, we show that for any distribution over observations, if the mutual information between past observations and future observations is upper bounded by I, then a simple Markov √ model over the most recent I/ observations obtains expected KL error —and hence `1 error —with respect to the optimal predictor that has access to the entire past and knows the data generating distribution. For a Hidden Markov Model with n hidden states, I is bounded by log n, a quantity that does not depend on the mixing time, and we show that the trivial prediction algorithm based on the empirical frequencies of length O(log n/) windows of observations achieves this error, provided the length of the sequence is dΩ(log n/) , where d is the size of the observation alphabet. We also establish that this result cannot be improved upon, even for the class of HMMs, in the following two senses: First, for HMMs with n hidden states, a window length √ of log n/ is information-theoretically necessary to achieve expected KL error , or `1 error . Second, the dΘ(log n/) samples required to accurately estimate the Markov model when observations are drawn from an alphabet of size d is necessary for any computationally tractable learning/prediction algorithm, assuming the hardness of strongly refuting a certain class of CSPs. 1 Memory, Modeling, and Prediction We consider the problem of predicting the next observation xt given a sequence of past observations, x1 , x2 , . . . , xt−1 , which could have complex and long-range dependencies. This sequential prediction problem is one of the most basic learning tasks and is encountered throughout natural language modeling, speech synthesis, financial forecasting, and a number of other domains that have a sequential or chronological element. The abstract problem has received much attention over the last half century from multiple communities including TCS, machine learning, and coding theory. The fundamental question is: How do we consolidate and reference memories about the past in order to effectively predict the future? Given the immense practical importance of this prediction problem, there has been an enormous effort to explore different algorithms for storing and referencing information about the sequence. These efforts have led to recurrent neural networks [1]—which encode the past as a real vector of fixed length that is updated after every observation—and specific classes of such networks, such as Long Short-Term Memory (LSTM) networks [2, 3]. Other recently popular models that have explicit notions of memory include neural Turing machines [4], memory networks [5], differentiable neural computer [6], etc. These models have been quite successful (see e.g. [7, 8]); nevertheless, they seem largely unable to consistently learn long-range dependencies, which are crucial in many settings including language. In parallel to these efforts to design systems that explicitly use memory, there has been much effort from the neuroscience community to understand how humans and animals are able to make accurate predictions about their environment. Many of these efforts also attempt to understand the computational mechanisms behind the formation of memories (memory “consolidation”) and retrieval [9, 10, 11]. Despite the long history of studying sequential prediction, many fundamental questions remain: • How much memory is necessary to accurately predict future observations, and what properties of the underlying sequence determine this requirement? • Must one remember significant information about the distant past or is a short-term memory sufficient? • What is the computational complexity of accurate prediction? • How do answers to the above questions depend on the metric that is used to evaluate prediction accuracy? Aside from the intrinsic theoretical value of these questions, their answers could serve to guide the construction of effective practical prediction systems, as well as informing the discussion of the computational machinery of cognition and prediction/learning in nature. In this work, we provide insights into the first three questions. We begin by establishing the following proposition, which addresses the first two questions with respect to the pervasively used metric of average prediction error: Proposition 1. Let M be any distribution over sequences with mutual information I(M) between the past observations . . . , xt−2 , xt−1 and future observations xt , xt+1 , . . .. The best `-th order Markov model, which makes predictions based only on the most recent ` observations, predicts the p distribution of the next observation with average KL error I(M)/` or average `1 error I(M)/`, with respect to the actual conditional distribution of xt given all past observations. The intuition behind the statement and proof of this general proposition is the following: at time t, we either predict accurately and are unsurprised when xt is revealed to us, or if we predict poorly and are surprised by the value of xt , then xt must contain a significant amount of information about 1 the history of the sequence, which can then be leveraged in our subsequent predictions of xt+1 , xt+2 , etc. In this sense, every timestep in which our prediction is ‘bad’, we learn some information about the past. Because the mutual information between the history of the sequence and the future is bounded by I(M), if we were to make I(M) consecutive bad predictions, we have captured nearly this amount of information about the history, and hence going forward, as long as the window we are using spans these observations, we should expect to predict well. This general proposition, framed in terms of the mutual information of the past and future, has immediate implications for a number of well-studied models of sequential data, such as Hidden Markov Models (HMMs). For a HMM with n hidden states, the mutual information of the generated sequence is trivially bounded by log n, which yields the following corollary to the above proposition. We state this proposition now, as it provides a helpful reference point in our discussion of the more general proposition. Corollary 1. Suppose observations are generated by a Hidden Markov Model with at most n hidden states. The best log n -th order Markov model, which makes predictions based only on the most recent log n predicts the distribution of the next observation with average KL error ≤ or `1 observations, √ error ≤ , with respect to the optimal predictor that knows the underlying HMM and has access to all past observations. In the setting where the observations are generated according to an HMM with at most n hidden states, this “best” `th order Markov model is easy to learn given sufficient data, and corresponds to the naive “empirical” `-gram model based on the previous observations. Specifically, this is the model that, given xt−` , xt−`+1 , . . . , xt−1 , outputs the observed (empirical) distribution of the observation that has followed this length ` sequence. (To predict what comes next in the phrase “. . . defer the details to the ” we look at the previous occurrences of this subsequence, and predict according to the empirical frequency of the subsequent word.) The following theorem makes this claim precise. Theorem 1. Suppose observations are generated by a Hidden Markov Model with at most n hidden states, and output alphabet of size d. For > 1/n there exists a window length ` = O( log n ) and absolute constant c such that for any T ≥ dc` , if t ∈ {1, 2, . . . , T } is chosen uniformly at random, then the expected `1 distance between the true distribution of xt given the entire history (and knowledge of the HMM), and the distribution predicted by the naive “empirical” `-th order √ Markov model based on x0 , . . . , xt , is bounded by . The above theorem states that the window length necessary to predict well is independent of the mixing time of the HMM in question, and holds even if the model does not mix. While the amount of data required to make accurate predictions using length ` windows scales exponentially in `—corresponding to the condition in the above theorem that t is chosen uniformly between 0 and T = dO(`) —our lower bounds, discussed in Section 1.3, argue that this exponential dependency is unavoidable. 1.1 Interpretation of mutual information of past and future While the mutual information between the past observations and the future observations is an intuitive parameterization of the complexity of a distribution over sequences, the fact that it is the right quantity is a bit subtle. It is tempting to hope that this mutual information is a bound on the amount of memory that would be required to store all the information about past observations that is relevant to the distribution of future observations. Consider the following setting: Given a 2 joint distribution over random variables P AST and F U T , suppose we wish to define a function f that maps P AST to a binary “advice”/memory string f (P AST ), possibly of variable length, such that F U T is independent of P AST , given f (P AST ). As is shown in Harsha et al. [12], there are joint distributions over (P AST, F U T ) such that even on average, the minimum length of the advice/memory string necessary for the above task is exponential in the mutual information I(P AST ; F U T ). This setting can also be interpreted as a two-player communication game where one player generates P AST and the other generates F U T given limited communication (i.e. the ability to communicate f (P AST )). 1 Given the fact that this mutual information is not even an upper bound on the amount of memory that an optimal algorithm (computationally unbounded, and with complete knowledge of the distribution) would require, Proposition 1 might be surprising. 1.2 Implications of Proposition 1 and Corollary 1. These results show that a Markov model—a model that cannot capture long-range dependencies or structure of the data—can predict accurately on any data-generating distribution, provided the order of the Markov model scales with the complexity of the distribution, as parameterized by the mutual information between the past and future. Strikingly, this parameterization is indifferent to whether the dependencies in the sequence are relatively short-range as in an HMM that mixes quickly, or very long-range as in an HMM that mixes slowly or does not mix at all. Independent of the nature of these dependencies, provided the mutual information is small, accurate prediction is possible based only on the most recent few observation. (See Figure 1 for a concrete illustration of this result in the setting of an HMM that does not mix and has long-range dependencies.) Figure 1: A depiction of a HMM on n states, that repeats a given length n binary sequence of outputs, and hence does not mix. Corollary 1 and Theorem 1 imply that accurate prediction is possible based only on short sequences of O(log n) observations. At a time where increasingly complex models such as recurrent neural networks and neural Turing machines are in vogue, these results serve as a baseline theoretical result. They also help explain the practical success of simple Markov models, such as “Kneser-Ney” smoothing [13, 14], which are crucial components in state-of-the-art machine translation and speech recognition systems. Although recent recurrent neural networks have yielded empirical gains (see e.g. [7, 8]), current models still seem largely incapable of successfully capturing long-range dependencies.2 In 1 It is worth noting that if the advice/memory string s is sampled first, and then P AST and F U T are defined to be random functions of s, then the length of s can be related to I(P AST ; F U T ) (see [12]). This latter setting where s is generated first corresponds to allowing shared randomness in the two-player communication game; however, this is not relevant to the sequential prediction problem. 2 One amusing example is the recent sci-fi short film Sunspring whose script was automatically generated by an LSTM. Locally, each sentence of the dialogue (mostly) makes sense, though there is no cohesion over longer time frames, and no overarching plot trajectory (despite the brilliant acting). 3 some settings, such as natural language, capturing such long-range dependencies seems crucial for achieving human-level results. Indeed, the main message of a narrative is not conveyed in any single short segment. More generally, higher-level intelligence seems to be about the ability to judiciously decide what aspects of the observation sequence are worth remembering and updating a model of the world based on these aspects. Thus, for such settings, Proposition 1, can be interpreted as a negative result—that average error is not a good metric for training and evaluating models. It is important to note that average prediction error is the metric that ubiquitously used in practice, both in the natural language processing domain and elsewhere. Our results suggest that a different metric might be essential to driving progress towards systems that attempt to capture long-range dependencies and leverage memory in meaningful ways. We discuss this possibility of alternate prediction metrics more in Section 1.4. For many other settings, such as financial prediction and lower level language prediction tasks such as those used in OCR or speech recognition, average prediction error is the most meaningful metric. For these settings, the result of Proposition 1 is extremely positive: no matter the nature of the dependencies in the financial markets, it is sufficient to learn a Markov model. As one obtains more and more data, one can learn a higher and higher order Markov model, and average prediction accuracy should continue to improve. For these applications, the question now becomes a computational question: the naive approach to learning an `th-order Markov model in a domain with an alphabet of size d might require Ω(d` ) space to store, and data to learn. From a computational standpoint, is there a better algorithm? What properties of the underlying sequence imply that such models can be learned, or approximated more efficiently or with less data? Our computational lower bounds, described below, provide some perspective on these computational considerations. 1.3 Lower bounds Our positive results show that accurate prediction is possible via an algorithmically simple model— a Markov model that only depends on the most recent observations—which can be learned in an algorithmically straightforward fashion by simply using the empirical statistics of short sequences of examples, compiled over a sufficient amount of data. Nevertheless, the Markov model has d` parameters, and hence requires an amount of data that scales as Ω(d` ) to learn, where d is a bound on the size of the observation alphabet. This prompts the question of whether it is possible to learn a successful predictor based on significantly less data. We show that, even for the special case where the data sequence is generated from an HMM over n hidden states, this is not possible in general, assuming a natural complexity-theoretic assumption. A HMMs with n hidden states and an output alphabet of size d is defined via only O(n2 + nd) parameters and O (n2 + nd) samples are sufficient, from an information theoretic standpoint, to learn a model that will predict accurately. While learning an HMM is computationally hard (see e.g. [15]), this begs the question of whether accurate (average) prediction can be achieved via a computationally efficient algorithm and and an amount of data significantly less than the dΘ(log n) that the naive Markov model would require. Our main lower bound shows that there exists a family of HMMs such that the Ω(dlog n/ ) sample complexity requirement is necessary for any computationally efficient algorithm that predicts accurately on average, assuming a natural complexity-theoretic assumption. Specifically, we show that this hardness holds, provided that the problem of strongly refuting a certain class of CSPs is 4 hard, which was conjectured in [16] and studied in related works [17] and [18]. See Section 5 for a description of this class and discussion of the conjectured hardness. Theorem 2. Assuming the hardness of strongly refuting a certain class of CSPs, for all sufficiently large n and any ∈ (1/nc , 0.1) for some fixed constant c, there exists a family of HMMs with n hidden states and an output alphabet of size d such that any polynomial time algorithm that achieves average error (with respect to the optimal predictor) for a random HMM in the family must observe dΘ(log n/) observations from the HMM. As the mutual information of the generated sequence of an HMM with n hidden states is bounded by log n , Theorem 2 directly implies that there are families of data-generating distributions M with mutual information I(M) and observations drawn from an alphabet of size d such that any computationally efficient algorithm requires dΩ(I(M)/) samples from M to achieve average error . The above bound holds when d is large compared to log n or I(M), but a different but equally relevant regime is where the alphabet size d is small compared to the scale of dependencies in the sequence (for example, when predicting characters [19]). We show lower bounds in this regime of the same flavor as those of Theorem 2 except based on the problem of learning a noisy parity function; the (very slightly) subexponential algorithm of Blum et al. [20] for this task means that we lose at least a superconstant factor in the exponent in comparison to the positive results of Proposition 1. Proposition 2. Let f (k) denote a lower bound on the amount of time and samples required to learn parity with noise on uniformly random k-bit inputs. For all sufficiently large n and ∈ (1/nc , 0.1) for some fixed constant c, there exists a family of HMMs with n hidden states such that any algorithm that achieves average prediction error (with respect to the optimal predictor) for a random HMM in the family requires at least f (log n/) time or samples. Finally, we also establish the information theoretic optimality of the results of Proposition 1, in the sense that among (even computationally unbounded) prediction algorithms that predict based only on the most recent ` observations, an average KL prediction error of Ω(I(M)/`) and `1 error Ω(I(M)/2 ) with respect to the optimal predictor, is necessary. Proposition 3. There is an absolute constant c < 1 such that for all 0 < < 1/4 and sufficiently large n, there exists an HMM with n hidden states such that it is not information-theoretically √ possible to obtain average KL prediction error less than or `1 error less than (with respect to the optimal predictor) while using only the most recent c log n/ observations to make each prediction. 1.4 Future Directions As mentioned above, for the settings in which capturing long-range dependencies seems essential, it is worth re-examining the choice of “average prediction error” as the metric used to train and evaluate models. One possibility, that has a more worst-case flavor, is to only evaluate the algorithm at a chosen set of time steps instead of all time steps. Hence the naive Markov model can no longer do well just by predicting well on the time steps when prediction is easy. In the context of natural language processing, learning with respect to such a metric intuitively corresponds to training a model to do well with respect to a question answering task instead of a language modeling task. A fertile middle ground between average error (which gives too much reward for correctly guessing common words like “a” and “the”), and worst-case error might be a re-weighted prediction error that provides more reward for correctly guessing less common observations. It seems possible, however, that the techniques used to prove Proposition 1 can be extended to yield analogous statements for such error metrics. 5 Given the many settings for which average error is the most natural metric of prediction accuracy, and the upper bounds of Proposition 1, it is natural to consider what additional structure might be present that avoids the (conditional) computational lower bounds of Theorem 2. One possibility is a robustness property—for example the property that a Markov model would continue to predict well even when each observation were obscured or corrupted with some small probability. The lower bound instances in Theorem 2 and Proposition 2 rely on parity based constructions and hence are very sensitive to noise and corruptions. For learning over product distributions, there are well known connections between noise stability and approximation by low-degree polynomials [21, 22]. Additionally, low-degree polynomials can be learned agnostically over arbitrary distributions via polynomial regression [23]. It is tempting to hope that this thread could be made rigorous, by establishing a connection between natural notions of noise stability over arbitrary distributions, and accurate low-degree polynomial approximations. Such a connection could lead to significantly better sample complexity requirements for prediction on such “robust” distributions of sequences, perhaps requiring only poly(d, I(M), 1/) data. Additionally, such sample-efficient approaches to learning succinct representations of large Markov models may inform the many practical prediction systems that currently rely on Markov models. 1.5 Related Work Parameter Estimation. It is interesting to compare using a Markov model for prediction with methods that attempt to properly learn an underlying model. For example, method of moments algorithms [24, 25] allow one to estimate a certain class of Hidden Markov model with polynomial sample and computational complexity. These ideas have been extended to learning neural networks [26] and input-output RNNs [27]. Using different methods, Arora et al. [28] showed how to learn certain random deep neural networks. Learning the model directly can result in better sample efficiency, and also provide insights into the structure of the data. The major drawback of these approaches is that they usually require the true data-generating distribution to be in (or extremely close to) the model family that we are learning. This is a very strong assumption that often does not hold in practice. Universal Prediction and Coding Theory. On the other end of the spectrum is the class of no-regret online learning methods which assume that the data generating distribution can even be adversarial [29]. However, the nature of these results are fundamentally different from ours: whereas we are comparing to the perfect model that can look at the infinite past, online learning methods typically compare to a fixed set of experts, which is much weaker. There is much work on sequential prediction based on KL-error from the information theory and statistics communities. The philosophy of these approaches are often more adversarial, with perspectives ranging from minimum description length [30, 31] and individual sequence settings [32], where no model of the data distribution process is assumed. With regards to worst case guarantees (where there is no data generation process), and regret as the notion of optimality, there is a line of work on both minimax rates and the performance of Bayesian algorithms, the latter of which has favorable guarantees in a sequential setting. With regards to minimax rates, [33] provides an exact characterization of the minimax strategy, though the applicability of this approach is often limited to settings where the strategies available to the learner is relatively small (i.e., the normalizing constant in [33] must exist). More generally, there has been considerable work on the regret in information-theoretic and statistical settings, such as the works in [32, 34, 35, 36, 37, 38, 39, 40]. With regards to log-loss more broadly, there is considerable work on information consistency (convergence in distribution) and minimax rates with regards to statistical estimation in parametric and non-parametric families [41, 42, 43, 44, 45, 46]. In some of these settings, e.g. minimax risk in 6 parametric, i.i.d. settings, there are characterizations in terms of mutual information [42]. There is also work on universal lossless data compression algorithm, such as the celebrated Lempel-Ziv algorithm [47]. Here, the setting is rather different as it is one of coding the entire sequence (in a block setting) rather than prediction loss. Sequential Prediction in Practice. Our work was initiated by the desire to understand the role of memory in sequential prediction, and the belief that modeling long-range dependencies is important for complex tasks such as understanding natural language. There have been many proposed models with explicit notions of memory, including recurrent neural networks [48], Long Short-Term Memory (LSTM) networks[2, 3], attention-based models[49], neural Turing machines [4], memory networks [5], differentiable neural computers [6], etc. While some of these models have been quite successful in practice (see e.g. [7, 8]), they still largely fail to capture many long-range dependencies–in the case of LSTMs, for example, it is not difficult to show that they forget the past exponentially quickly if they are “stable” [1]. To gain more insight into this problem, we began by analyzing the simplest Markov predictor, and found to our surprise that it performed nearly as well as one could hope. 2 Proof Sketch of Theorem 1 We provide a sketch of the proof of Theorem 1, which is stronger than Proposition 1 but applies specifically to sequences generated from a Hidden Markov Model. The core of this proof is the following lemma that guarantees that the Markov model that knows the true marginal probabilities of all short sequences, will end up predicting well. Additionally, this good expected prediction will hold with respect to only the randomness of the HMM during the short window, as opposed to over the randomness of when the window begins (as in our more general results). For settings such as financial forecasting, this additional guarantee is particularly pertinent; you do not need to worry about the possibility of choosing an “unlucky” time to begin your trading regime, as long as you plan to trade for a duration that spans an entire short window. Beyond the extra strength of this result for HMMs, the proof approach is intuitive and pleasing, in comparison to the more direct proof of Proposition 1. We first state the lemma and sketch its proof, and then conclude the section by describing how this yields Theorem 1. Lemma 4. Consider an HMM with n hidden states, let the hidden state at time s = 0 be chosen according to an arbitrary distribution π, and denote the observation at time s by xs . Let OP Ts denote the conditional distribution of xs given observations x0 , . . . , xs−1 , and knowledge of the hidden state at time s = 0. Let Ms denote the conditional distribution of xs given only x0 , . . . , xs−1 , which corresponds to the naive sth order Markov model that knows only the joint probabilities of sequences of the first s observations. Then with probability at least 1 − 1/nc−1 over the choice of initial state, for ` = c log n/2 , `−1 hX i E kOP Ts − Ms k1 ≤ 4`, s=0 where the expectation is with respect to the randomness in the outputs x0 , . . . , x`−1 . The proof of the this lemma will hinge on establishing a connection between OP Ts —the Bayes optimal model that knows the HMM and the initial hidden state h0 , and at time s predicts the true distribution of xs given h0 , x0 , . . . , xs−1 —and the naive order s Markov model Ms that knows the joint probabilities of sequences of s observations (given that the initial state is drawn according to π), and predicts accordingly. This latter model is precisely the same as the model that knows 7 the HMM and distribution π (but not h0 ), and outputs the conditional distribution of xs given the observations. To relate these two models, we proceed via a martingale argument that leverages the intuition that, at each time step either OP Ts ≈ Ms , or, if they differ significantly, we expect the sth observation xs to contain a significant amount of information about the hidden state at time zero, h0 , which will then improve Ms+1 . Our submartingale will precisely capture the sense that for any s where there is a significant deviation between OP Ts and Ms , we expect the probability of the initial state being h0 conditioned on x0 , . . . , xs , to be significantly more than the probability of h0 conditioned on x0 , . . . , xs−1 . More formally, let H0s denote the distribution of the hidden state at time 0 conditioned on x0 , . . . , xs and let h0 denote the true hidden state at time 0. We show that the following expression is a submartingale: s P r[H0s = h0 ] 1X log − kOP Ti − Mi k21 . 1 − P r[H0s = h0 ] 2 i=0 The fact that this is a submartingale is not difficult: Define Rs as the conditional distribution of xs given observations x0 , · · · , xs−1 and initial state drawn according to π but not being at hidden state h0 at time 0. Note that Ms is a convex combination of OP Ts and Rs , hence kOP Ts − Ms k1 ≤ kOP Ts − Rs k1 . To verify the submartingale property, note that by Bayes Rule, the change in the LHS at any time step s is the log of the ratio of probability of observing the output xs according to the distribution OP Ts and the probability of xs according to the distribution Rs . The expectation of this is the KL-divergence between OP Ts and Rs , which can be related to the `1 error using Pinsker’s inequality. At a high level, the proof will then proceed via concentration bounds (Azuma’s inequality), to 2 show that, with high probability, if the error from the first ` = c log n/ timesteps is large, then P r[H0`−1 =h0 ] is also likely to be large, in which case the posterior distribution of the hidden 1−P r[H0`−1 =h0 ] state, H0`−1 will be sharply peaked at the true hidden state, h0 , unless h0 had negligible mass (less than n−c ) in distribution π. log There are several slight complications to this approach, including the fact that the submartingale we construct does not necessarily have nicely concentrated or bounded differences, as the first term in the submartingale could change arbitrarily. We address this by noting that the first term should not decrease too much except with tiny probability, as this corresponds to the posterior probability of the true hidden state sharply dropping. For the other direction, we can simply “clip” the deviations to prevent them from exceeding log n in any timestep, and then show that the submartingale property continues to hold despite this clipping by proving the following modified version of Pinsker’s inequality: Lemma 1. (Modified Pinsker’s inequality) For any two distributions h µ(x)nand ν(x) oidefined on µ(x) x ∈ X, define the C-truncated KL divergence as D̃C (µ k ν) = Eµ log min ν(x) , C for some fixed C such that log C ≥ 8. Then D̃C (µ k ν) ≥ 12 kµ − νk21 . Given Lemma 4, the proof of Theorem 1 follows relatively easily. Recall that Theorem 1 concerns the expected prediction error at a timestep t ← {0, 1, . . . , dc` }, based on the model Memp corresponding to the empirical distribution of length ` windows that have occurred in x0 , . . . , xt ,. The connection between the lemma and theorem is established by showing that, with high probability, Memp is close to Mπ̂ , where π̂ denotes the empirical distribution of (unobserved) hidden states h0 , . . . , ht , and Mπ̂ is the distribution corresponding to drawing the hidden state h0 ← π̂ and then generating x0 , x1 , . . . , x` . We provide the full proof in Appendix A. 8 3 Definitions and Notation Before proving our general Proposition 1, we first introduce the necessary notation. For any random variable X, we denote its distribution as P r(X). The mutual information between two random variables X and Y is defined as I(X; Y ) = H(Y ) − H(Y \|X) where H(Y ) is the entropy of Y and H(Y \|X) is the conditional entropy of Y given X. The conditional mutual information I(X; Y \|Z) is defined as: I(X; Y \|Z) = H(X\|Z) − H(X\|Y, Z) = Ex,y,z log P r(X\|Y, Z) = Ey,z DKL (P r(X\|Y, Z) k P r(X\|Z)) P r(X\|Z) P p(x) where DKL (p k q) = x p(x) log q(x) is the KL divergence between the distributions p and q. Note that we are slightly abusing notation here as DKL (P r(X\|Y, Z) k P r(X\|Z)) should technically be DKL (P r(X\|Y = y, Z = z) k P r(X\|Z = z)). But we will ignore the assignment in the conditioning when it is clear from the context. Mutual information obeys the following chain rule: I(X1 , X2 ; Y ) = I(X1 ; Y ) + I(X2 ; Y \|X1 ). Given a distribution over infinite sequences, {xt } generated by some model M where xt is random variable denoting the output at time t, we will use the shorthand xji to denote the collection of random variables for the subsequence of outputs {xi , · · · , xj }. The distribution of {xt } is stationary if the joint distribution of any subset of the sequence of random variables {xt } is invariant with respect to shifts in the time index. Hence P r(xi1 , xi2 , · · · , xin ) = P r(xi1 +l , xi2 +l , · · · , xin +l ) for any l if the process is stationary. We are interested in studying how well the output xt can be predicted by an algorithm which only looks at the past ` outputs. The predictor A` maps a sequence of ` observations to a predicted distribution of the next observation. We denote the predictive distribution of A` at time t as QA` (xt \|xt−1 t−` ). We refer to the Bayes optimal predictor using only windows of length ` as P` , hence the prediction of P at time t is P r(xt \|xt−1 t−` ). P` is just the naive `-th order Markov predictor provided with the true distribution of the data. Let the Bayes optimal predictor looking at the entire history of the model be P∞ , the prediction of P∞ at time t is P r(xt \|xt−1 −∞ ). We will evaluate the predictions of A` and P` with respect to P∞ over a long time window [0 : T − 1]. The crucial property of the distribution that is relevant to our results is the mutual information between past and future observations. For a stochastic process {xt } generated by some model M we define the mutual information I(M) of the model M as the mutual information between the past and future, averaged over the window [0 : T − 1]. T −1 1 X ∞ I(M) = I(xt−1 −∞ ; xt ) T (3.1) t=0 ∞ If the process {xt } is stationary, then I(xt−1 −∞ ; xt ) is the same for all time steps hence I(M) = ∞ I(x−1 −∞ ; x0 ). We compare the prediction of the predictor P` and A` with respect to P∞ . Let F (P, Q) be some measure of distance between two predictive distributions. In this work, we consider the KL-divergence, `1 distance and the relative zero-one loss between the two distributions. The KLdivergence and `1 distance between two distributions are defined in the standard way. We define the relative zero-one loss as the difference between the zero-one loss of the optimal predictor P∞ and the algorithm A` . We define the expected loss of any predictor A` with respect to the optimal 9 predictor P∞ and a loss function F as follows: (t) δF (A` ) i h t−1 ) ), QA` (xt \|xt−1 = Ext−1 F (P r(xt \|x−∞ t−` , −∞ T −1 1 X (t) δF (A` ) = δF (A` ) T t=0 (t) δ̂F (A` ) We also define and δ̂F (A` ) for the algorithm A` in the same fashion as the error in estimating P (xt \|xt−1 ), the true conditional distribution of the model M. t−` i h (t) t−1 )) , δ̂F (A` ) = Ext−1 F (P r(xt \|xt−` ), QA` (xt \|xt−1 t−` t−` 4 δ̂F (A` ) = T −1 1 X (t) δ̂F (A` ) T t=0 Predicting Well with Short Windows To establish our general proposition, which applies beyond the HMM setting, we provide an elementary and purely information theoretic proof. Proposition 1. For any data-generating distribution M with mutual information I(M) between past and future observations, the best `-th order Markov model P` obtains average KL-error, δKL (P` ) ≤ I(M)/` with respect to the optimal predictor with access to the infinite history. Also, any predictor A` with δ̂KL (A` ) average KL-error in estimating the joint probabilities over windows of length ` gets average error δKL (A` ) ≤ I(M)/` + δ̂KL (A` ). Proof. We bound the expected error by splitting the time interval 0 to T − 1 into blocks of length `. Consider any block starting at time τ . We find the average error of the predictor from time τ to τ + ` − 1 and then average across all blocks. To begin, note that we can decompose the error as the sum of the error due to not knowing the past history beyond the most recent ` observations and the error in estimating the true joint (t) distribution of the data over a ` length block. Consider any time t. Recall the definition of δKL (A` ). i h (t) t−1 δKL (A` ) = Ext−1 DKL (P r(xt \|xt−1 )) ) k Q (x \|x t A −∞ ` t−` −∞ h i h i t−1 t−1 t−1 = Ext−1 DKL (P r(xt \|xt−1 −∞ ) k P (xt \|xt−` )) + Ext−1 DKL (P r(xt \|xt−` ) k QA` (xt \|xt−` )) −∞ = −∞ (t) δKL (P` ) + (t) δ̂KL (A` ) (t) t−`−1 Therefore, δKL (A` ) = δKL (P` ) + δ̂KL (A` ). It’s easy to verify that δKL (P` ) = I(x−∞ ; xt \|xt−1 t−` ). This relation expresses the intuition that the current output (xt ) has a lot of extra information t−`−1 about the past (x−∞ ) if we cannot predict it as well using the ` most recent observations (xt−1 t−` ) t−1 as can be done by using the entire past (x−∞ ). We will now upper bound the total error for the −1 ∞ window [τ, τ + ` − 1]. We expand I(xτ−∞ ; xτ ) using the chain rule, ∞ τ +`−1 X X τ −1 ∞ τ −1 −1 t−1 I(x−∞ ; xτ ) = I(x−∞ ; xt \|xτ ) ≥ I(xτ−∞ ; xt \|xt−1 τ ). t=τ t=τ (t) −1 t−`−1 Note that I(xτ−∞ ; xt \|xτt−1 ) ≥ I(x−∞ ; xt \|xt−1 t−` ) = δKL (P` ) as t−` ≤ τ and I(X, Y ; Z) ≥ I(X; Z\|Y ). The proposition now follows from averaging the error across the ` time steps and using Eq. 3.1 to average over all blocks of length ` in the window [0, T − 1], τ +`−1 1 X (t) 1 I(M) −1 ∞ δ (P` ) ≤ I(xτ−∞ ; xτ ) =⇒ δKL (P` ) ≤ ` t=τ KL ` ` 10 Proposition 1 also directly gives guarantees for the scenario where the task is to predict the distribution of the next block of outputs instead of just the next immediate output, because KLdivergence obeys the chain rule. The following easy corollary, relating KL error to `1 error yields the following statement, which also trivially applies to zero/one loss with respect to that of the optimal predictor, as the expected relative zero/one loss at any time step is at most the `1 loss at that time step. Corollary 2. For any data-generating distribution M with mutual information I(M) between past and p future observations, the best `-th order Markov model P` obtains average `1 -error, δ`1 (P` ) ≤ I(M)/2` with respect to the optimal predictor with access to the infinite history. Also, any predictor A p` with δ̂`1 (A` ) average `1 -error in estimating the joint probabilities gets average error δ`1 (A` ) ≤ I(M)/2` + δ̂`1 (A` ). Proof. We again decompose the error as the sum of the error in estimating P̂ and the error due to not knowing the past history using the triangle inequality. h i (t) )k1 δ`1 (A` ) = Ext−1 kP r(xt \|xt−1 ) − QA` (xt \|xt−1 −∞ t−` −∞ i i h h t−1 t−1 t−1 t−1 ≤ Ext−1 kP r(xt \|x−∞ ) − P r(xt \|xt−` )k1 + Ext−1 kP r(xt \|xt−` ) − QA` (xt \|xt−` )k1 −∞ −∞ = (t) δ`1 (P` ) + (t) δ̂`1 (A` ) (t) Therefore, δ`1 (A` ) ≤ δ`1 (P` ) + δ̂`1 (A` ). By Pinsker’s inequality and Jensen’s inequality, δ`1 (A` )2 ≤ (t) δKL (A` )/2. Using Proposition 1, T −1 1 X (t) I(M) δKL (A` ) = δKL (A` ) ≤ T ` t=0 Therefore, using Jensen’s inequality again, δ`1 (A` ) ≤ 5 p I(M)/2`. Lower Bound for Large Alphabets Our lower bounds for the sample complexity in the large alphabet case leverage a class of Constraint Satisfaction Problems (CSPs) with high complexity. A class of (Boolean) k-CSPs is defined via a predicate—a function P : {0, 1}k → {0, 1}. An instance of such a k-CSP on n variables {x1 , · · · , xn } is a collection of sets (clauses) of size k whose k elements consist of k variables or their negations. Such an instance is satisfiable if there exists an assignment to the variables x1 , . . . , xn such that the predicate P evaluates to 1 for every clause. More generally, the value of an instance is the maximum, over all 2n assignments, of the ratio of number of satisfied clauses to the total number of clauses. Our lower bounds are based on the presumed hardness of distinguishing random instances of or a certain class of CSP, versus instances of the CSP with high value. There has been much work attempting to characterize the difficulty of CSPs—one notion which we will leverage is the complexity of a class of CSPs, first defined in [16] and studied in [17]: Definition 1. The complexity of a class of k-CSPs defined by predicate P : {0, 1}k → {0, 1} is the largest r such that there exists a distribution supported on the support of P that is (r − 1)-wise independent (i.e. “uniform”), and no such r-wise independent distribution exists. 11 Example 1. Both k-XOR and k-SAT are well-studied classes of k-CSPs, corresponding, respectively, to the predicates PXOR that is the XOR of the k Boolean inputs, and PSAT that is the OR of the inputs. These predicates both support (k − 1)-wise uniform distributions, but not k-wise uniform distributions, hence their complexity is k. In the case of k-XOR, the uniform distribution over {0, 1}k restricted to the support of PXOR is (k − 1)-wise uniform. The same distribution is also supported by k-SAT. A random instance of a CSP with predicate P is an instance such that all the clauses are chosen uniformly at random (by selecting the k variables uniformly, and independently negating each variable with probability 1/2). A random instance will have value close to E[P ], where E[P ] is the expectation of P under the uniform distribution. In contrast, a planted instance is generated by first fixing a satisfying assignment σ and then sampling clauses that are satisfied, by uniformly choosing k variables, and picking their negations according to a (r−1)-wise independent distribution associated to the predicate. Hence a planted instance always has value 1. A noisy planted instance with planted assignment σ and noise level η is generated by sampling consistent clauses (as above) with probability 1 − η and random clauses with probability η, hence with high probability it has value 1 − η + ηE[P ]. Our hardness results are based on distinguishing whether a CSP instance is random or has a high value. As one would expect, the difficulty of distinguishing random instances from noisy planted instances, decreases as the number of sampled clauses grows. The following conjecture of Feldman et al. [16] asserts a sharp boundary on the number of clauses, below which this problem becomes computationally intractable, while remaining information theoretically easy. The notation is made more explicit in Appendix B. Conjectured CSP Hardness [Conjecture 1] [16]: Let Q be any distribution over k-clauses and n variables of complexity r and 0 < η < 1. Any polynomial-time (randomized) algorithm that, given access to a distribution D that equals either the uniform distribution over k-clauses Uk or a (noisy) planted distribution Qησ = (1 − η)Qσ + ηUk for some σ ∈ {0, 1}n and planted distribution Qσ , decides correctly whether D = Qησ or D = Uk with probability at least 2/3 needs Ω̃(nr/2 ) clauses. Feldman et al. [16] proved the conjecture for the class of statistical algorithms 3 . Recently, Kothari et al. [18] showed that the polynomial time Sum-of-Squares (SOS) algorithm requires Ω̃(nr/2 ) clauses to refute random instances of a CSP with complexity r, hence proving Conjecture 1 for any polynomial-size semidefinite programming relaxation for refutation. Note that Ω̃(nr/2 ) is tight, as Allen et al. [17] give a SOS algorithm for refuting random CSPs beyond this regime. Other recent papers such as Daniely and Shalev-Shwartz [51] and Daniely [52] have also used presumed hardness of strongly refuting random k-SAT and random k-XOR instances with a small number of clauses to derive conditional hardness of learning results. A first attempt to encode a k-CSP as a sequential model is to construct a model which outputs k randomly chosen literals for the first k time steps 0 to k −1, and then their (noisy) predicate value for the final time step k. Clauses from the CSP correspond to samples from the model, and the algorithm would need to solve the CSP to predict the final time step k. However, as all the outputs up to the final time step are random, the trivial prediction algorithm that guesses randomly and does not try to predict the output at time k, would be near optimal. To get strong lower bounds, 3 Statistical algorithms are an extension of the statistical query model, these are algorithms that do not access samples from the distribution but instead have access to estimates of the expectation of any bounded function of a sample through an oracle. Feldman et al. [50] point out that almost all algorithms that work on random data also work with this limited access to samples, refer to Feldman et al. [50] for more details and examples. 12 we will output m > 1 functions of the k literals after k time steps, while still ensuring that all the functions remain collectively hard to invert without a large number of samples. We use elementary results from the theory of error correcting codes to achieve this, and prove hardness due to a reduction from a specific family of CSPs to which Conjecture 1 applies. By choosing k and m carefully, we obtain the near-optimal dependence on the mutual information and error —matching the upper bounds implied by Proposition 1. We provide a short outline of the argument, followed by the detailed proof in the appendix. 5.1 Sketch of construction and proof We construct a sequential model M such that making good predictions on the model requires distinguishing random instances of a k-CSP C on n variables from instances of C with a high value. The output alphabet of M is {ai } of size 2n. We choose a mapping from the 2n characters {ai } to the n variables {xi } and their n negations {x̄i }. For any clause C and planted assignment σ to the CSP C, let σ(C) be the k-bit string of values assigned by σ to literals in C. The model M randomly uniformly outputs k characters from time 0 to k − 1, which correspond to literals in the CSP C, hence the k outputs correspond to a clause C of the CSP. For some m to be specified later, we will construct a binary matrix A ∈ {0, 1}m×k , which will correspond to a good error-correcting code. For the time steps k to k + m − 1, with probability 1 − η the model outputs y ∈ {0, 1}m where y = Av mod 2 and v = σ(C) with C being the clause associated with the outputs of the first k time steps. With the remaining probability, η, the model outputs m uniformly random bits. Note that the mutual information I(M) is at most m as only the outputs from time k to k + m − 1 can be predicted. We claim that M can be simulated by a HMM with 2m (2k + m) + m hidden states. This can be done as follows. For every time step i from 0 to k − 1, we maintain 2m hidden states corresponding to vi = 0 and 2m hidden states corresponding to vi = 1. Each of these 2m states stores the current value of the m bits of y. This takes a total of k2m+1 hidden states. We use 2m hidden states for each time step k through k + m − 1 for the k output bits. Finally, we need an additional m hidden states to output m uniform random bits from time k to k + m − 1 with probability η. This accounts for a total of k2m+1 + 2m + m hidden states. Note that the larger m is with respect to k, the higher the cost (in terms of average prediction error) of failing to correctly predict the outputs from time k to k + m − 1. Tuning k and m allows us to control the number of hidden states or the mutual information, and average error incurred by a computationally constrained predictor. We define the CSP C in terms of a collection of predicates P (y) for each y ∈ {0, 1}m . While Conjecture 1 does not directly apply to C, as it is defined by a collection of predicates instead of a single one, we will later show a reduction from a related CSP C0 defined by a single predicate for which Conjecture 1 holds. For each y, the predicate P (y) of C is the set of v ∈ {0, 1}k which satisfy y = Av mod 2. Hence each clause has an additional label y which determines the satisfying assignments, this label is just the output of our sequential model M from time k to k + m − 1. Hence for any planted assignment σ, the set of satisfying clauses C of the CSP C are all clauses such that Av = y mod 2 where y is the label of the clause and v = σ(C). We define a (noisy) planted distribution over clauses Qησ by first uniformly randomly sampling a label y, and then sampling a consistent clause with probability (1 − η), otherwise with probability η we sample a uniformly random clause. Let Uk be the uniform distribution over all k-clauses with uniformly chosen labels y. We will show that Conjecture 1 implies that distinguishing between the distributions Qησ and Uk is hard without sufficiently many clauses. This gives us the hardness results we desire for our sequential model M: if an algorithm obtains low prediction error on the outputs from time k 13 through (k + m − 1), then it can be used to distinguish between instances of the CSP C with a high value and random instances, as no algorithm obtains low prediction error on random instances. Hence hardness of strongly refuting the CSP C implies hardness of making good predictions on M. We now sketch the argument for why Conjecture 1 implies the hardness of strongly refuting the CSP C. We define another CSP C0 which we show reduces to C. The predicate P of the CSP C0 is the set of all v ∈ {0, 1}k such that Av = 0 mod 2. Hence for any planted assignment σ, the set of satisfying clauses of the CSP C0 are all clauses such that v = σ(C) is in the nullspace of A. As before, the planted distribution over clauses is uniform on all satisfying clauses with probability (1 − η), with probability η we add a uniformly random k-clause. For some γ ≥ 1/10, if we can construct A such that the set of satisfying assignments v (which are the vectors in the nullspace of A) supports a (γk − 1)-wise uniform distribution, then by Conjecture 1 any polynomial time algorithm cannot distinguish between the planted distribution and uniformly randomly chosen clauses with less than Ω̃(nγk/2 ) clauses. We show that choosing a matrix A whose null space is (γk − 1)-wise uniform corresponds to finding a binary linear code with rate at least 1/2 and relative distance γ, the existence of which is guaranteed by the Gilbert-Varshamov bound. We next sketch the reduction from C0 to C. The key idea is that the CSPs C0 and C are defined by linear equations. If a clause C = (x1 , x2 , · · · , xk ) in C0 is satisfied with some assignment t ∈ {0, 1}k to the variables in the clause then At = 0 mod 2. Therefore, for some w ∈ {0, 1}k such that Aw = y mod 2, t + w mod 2 satisfies A(t + w) = y mod 2. A clause C 0 = (x01 , x02 , · · · , x0k ) with assignment t + w mod 2 to the variables can be obtained from the clause C by switching the literal x0i = x̄i if wi = 1 and retaining x0i = xi if wi = 0. Hence for any label y, we can efficiently convert a clause C in C0 to a clause C 0 in C which has the desired label y and is only satisfied with a particular assignment to the variables if C in C0 is satisfied with the same assignment to the variables. It is also not hard to ensure that we uniformly sample the consistent clause C 0 in C if the original clause C was a uniformly sampled consistent clause in C0 . We provide a small example to illustrate the sequential model constructed above. Let k = 3, m = 1 and n = 3. Let A ∈ {0, 1}1×3 . The output alphabet of the model M is {ai , 1 ≤ i ≤ 6}. The letter a1 maps to the variable x1 , a2 maps to x̄1 , similarly a3 → x2 , a4 → x̄2 , a5 → x3 , a6 → x̄3 . Let σ be some planted assignment to {x1 , x2 , x3 }, which defines a particular model M. If the output of the model M is a1 , a3 , a6 for the first three time steps, then this corresponds to the clause with literals, (x1 , x2 , x̄3 ). For the final time step, with probability (1 − η) the model outputs y = Av mod 2, with v = σ(C) for the clause C = (x1 , x2 , x̄3 ) and planted assignment σ, and with probability η it outputs a uniform random bit. For an algorithm to make a good prediction at the final time step, it needs to be able to distinguish if the output at the final time step is always a random bit or if it is dependent on the clause, hence it needs to distinguish random instances of the CSP from planted instances. We re-state Theorem 2 below, deferring its proof to Appendix B. Theorem 2. Assuming Conjecture 1, for all sufficiently large T and 1/T c < ≤ 0.1 for some fixed constant c, there exists a family of HMMs with T hidden states and an output alphabet of size n such that, any polynomial time prediction algorithm that achieves average KL-error, `1 error or relative zero-one error less than with probability greater than 2/3 for a randomly chosen HMM in the family needs requires nΘ(log T /) samples from the HMM over any window length which the algorithm uses for prediction. 14 6 Lower Bound for Small Alphabets Our lower bounds for the sample complexity in the binary alphabet case are based on the average case hardness of the decision version of the parity with noise problem, and the reduction is straightforward. In the parity with noise problem on n bit inputs we are given examples v ∈ {0, 1}n drawn uniformly from {0, 1}n along with their noisy labels hs, vi+ mod 2 where s ∈ {0, 1}n is the (unknown) support of the parity function, and ∈ {0, 1} is the classification noise such that P r[ = 1] = η where η < 0.05 is the noise level. Let Qηs be the distribution over examples of the parity with noise instance with s as the support of the parity function and η as the noise level. Let Un be the distribution over examples and labels where each label is chosen uniformly from {0, 1} independent of the example. The strength of of our lower bounds depends on the level of hardness of parity with noise. Currently, the fastest algorithm for the problem due to Blum et al. [20] runs in time and samples 2n/ log n . We define the function f (n) as follows– Definition 2. Define f (n) to be the function such that for a uniformly random support s ∈ {0, 1}n , with probability at least (1 − 1/n2 ) over the choice of s, any (randomized) algorithm that can distinguish between Qηs and Un with success probability greater than 2/3 over the randomness of the examples and the algorithm, requires f (n) time or samples. Our model will be the natural sequential version of the parity with noise problem, where each example is coupled with several parity bits. We denote the model as M(Am×n ) for some A ∈ {0, 1}m×n , m ≤ n/2. From time 0 through (n − 1) the outputs of the model are i.i.d. and uniform on {0, 1}. Let v ∈ {0, 1}n be the vector of outputs from time 0 to (n − 1). The outputs for the next m time steps are given by y = Av + mod 2, where ∈ {0, 1}m is the random noise and each entry i of is an i.i.d random variable such that P r[i = 1] = η, where η is the noise level. Note that if A is full row-rank, and v is chosen uniformly at random from {0, 1}n , the distribution of y is uniform on {0, 1}m . Also I(M(A)) ≤ m as at most the binary bits from time n to n + m − 1 can be predicted using the past inputs. As for the higher alphabet case, M(Am×n ) can be simulated by an HMM with 2m (2n + m) + m hidden states (see Section 5.1). We define a set of A matrices, which specifies a family of sequential models. Let S be the set of all (m × n) matrices A such that the sub-matrix of A corresponding to all rows but only the first 2n/3 columns is full row rank. We need this restriction to lower bound I(M(A)), as otherwise there could be small or no dependence of the parity bits on the inputs from time 0 to 2n/3 − 1. We denote R as the family of models M(A) for A ∈ S. Lemma 2 shows that with high probability over the choice of A, distinguishing outputs from the model M(A) from random examples Un requires f (n) time or examples. Lemma 2. Let A be chosen uniformly at random from the set S. Then, with probability at least (1 − 1/n) over the choice A ∈ S, any (randomized) algorithm that can distinguish the outputs from the model M(A) from the distribution over random examples Un with success probability greater than 2/3 over the randomness of the examples and the algorithm needs f (n) time or examples. The proof of Proposition 2 follows from Lemma 2 and is similar to the proof of Theorem 2. 15 Proposition 2. With f (T ) as defined in Definition 2, For all sufficiently large T and 1/T c < ≤ 0.1 for some fixed constant c, there exists a family of HMMs with T hidden states such that any algorithm that achieves average relative zero-one loss, average `1 loss, or average KL loss less than with probability greater than 2/3 for a randomly chosen HMM in the family needs, requires f (log T /) time or samples samples from the HMM over any window length which the algorithm uses for prediction. 7 Information Theoretic Lower Bounds We show that information theoretically, windows of length cI(M)/2 are necessary to get expected relative zero-one loss less than . As the expected relative zero-one loss is at most the `1 loss, which can be bounded by the square of the KL-divergence, this automatically implies that our window length requirement is also tight for `1 loss and KL loss. In fact, it’s very easy to show the tightness for the KL loss, choose the simple model which emits uniform random bits from time 0 to n − 1 and repeats the bits from time 0 to m − 1 for time n through n + m − 1. One can then choose n, m to get the desired error and mutual information I(M). To get a lower bound for the zero-one loss we use the probabilistic method to argue that there exists an HMM such that long windows are required to perform optimally with respect to the zero-one loss for that HMM. We state the lower bound and a rough proof idea, deferring the details to Appendix D. Proposition 3. There is an absolute constant c such that for all 0 < < 0.5 and sufficiently large n, there exits an HMM with n states such that it is not information theoretically possible to get average relative zero-one loss or `1 loss less than using windows of length smaller than c log n/2 , and KL loss less than using windows of length smaller than c log n/. We illustrate the construction in Fig. 2 and provide the high-level proof idea with respect to Fig. 2 below. Figure 2: Lower bound construction, n = 16 We want show that any predictor P using windows of length ` = 3 cannot make a good prediction. The transition matrix of the HMM is a permutation and the output alphabet is binary. Each state is assigned a label which determines its output distribution. The states labeled 0 emit 0 with probability 0.5 + and the states labeled 1 emit 1 with probability 0.5 + . We will randomly and uniformly choose the labels for the hidden states. Over the randomness in choosing the labels for the permutation, we will show that the expected error of the predictor P is large, which means that there must exist some permutation such that the predictor P incurs a high error. The rough 16 proof idea is as follows. Say the Markov model is at hidden state h2 at time 2, this is unknown to the predictor P. The outputs for the first three time steps are (x0 , x1 , x2 ). The predictor P only looks at the outputs from time 0 to 2 for making the prediction for time 3. We show that with high probability over the choice of labels to the hidden states and the outputs (x0 , x1 , x2 ), the output (x0 , x1 , x2 ) from the hidden states (h0 , h1 , h2 ) is close in Hamming distance to the label of some other segment of hidden states, say (h4 , h5 , h6 ). Hence any predictor using only the past 3 outputs cannot distinguish whether the string (x0 , x1 , x2 ) was emitted by (h0 , h1 , h2 ) or (h4 , h5 , h6 ), and hence cannot make a good prediction for time 3 (we actually need to show that there are many segments like (h4 , h5 , h6 ) whose label is close to (x0 , x1 , x2 )). The proof proceeds via simple concentration bounds. A Proof of Theorem 1 Theorem 1. Suppose observations are generated by a Hidden Markov Model with at most n hidden states, and output alphabet of size d. For > 1/n there exists a window length ` = O( log n ) and absolute constant c such that for any T ≥ dc` , if t ∈ {1, 2, . . . , T } is chosen uniformly at random, then the expected `1 distance between the true distribution of xt given the entire history (and knowledge of the HMM), and the distribution predicted by the naive “empirical” `-th order √ Markov model based on x0 , . . . , xt , is bounded by . Proof. Let πt be a distribution over hidden states such that the probability of the ith hidden state under πt is the empirical frequency of the ith hidden state from time 1 to t − 1 normalized by t − 1. For 0 ≤ s ≤ ` − 1, consider the predictor Pt which makes a prediction for the distribution of observation xt+s given observations xt , . . . , xt+s−1 based on the true distribution of xt under the HMM, conditioned on the observations xt , . . . , xt+s−1 and the distribution of the hidden state at time t being πt . We will show that in expectation over t, Pt gets small error averaged across the time steps 0 ≤ s ≤ ` − 1, with respect to the optimal prediction of the distribution of xt+s which knows the hidden state ht at time t. In order to show this, we need to first establish that the true hidden state ht at time t does not have very small probability under πt , with high probability over the choice of t. Lemma 3. With probability 1 − 2/n over the choice of t ∈ {0, . . . , T }, the hidden state ht at time t has probability at least 1/n3 under πt . Proof. Consider the ordered set Si of time indices t where the hidden state ht = i. For the sets corresponding to hidden states j which have probability less than 1/n2 under πT , the cardinality \|Sj \| ≤ T /n2 . The sum of the cardinality of all such small sets is at most T /n, and hence the probability that a uniformly random t ∈ {1, . . . , T } lies in one of these sets is at most 1/n. Now consider the set of time indices Si corresponding to some hidden state i which has probability at least 1/n2 under πT . For all t which are not among the first T /n3 time indices in this set, the hidden state i has probability at least 1/n3 under πt . As the fraction of the “bad” time steps t corresponding to any hidden state which has probability at least 1/n2 under πT is at most 1/n, the total fraction of these “bad” time steps t is at most 1/n. Therefore using a union bound, with failure probability 2/n, the hidden state ht at time t has probability at least 1/n3 under πt . Consider any time index t, for simplicity assume t = 0, and let OP Ts denote the conditional distribution of xs given observations x0 , . . . , xs−1 , and knowledge of the hidden state at time s = 0. Let Ms denote the conditional distribution of xs given only x0 , . . . , xs−1 , given that the hidden state at time 0 has the distribution π0 . 17 Lemma 4. For > 1/n, if the true hidden state at time 0 has probability at least 1/nc under π0 , then for ` = c log n/2 , `−1 h1 X i E kOP Ts − Ms k1 ≤ 4, ` s=0 where the expectation is with respect to the randomness in the outputs from time 0 to ` − 1. By Lemma 3, for a randomly chosen t ∈ {1, . . . , T } the probability that the hidden state i at time 0 has probability less than 1/n3 in the prior distribution πt is at most 2/n. Hence using Lemma 4, the expected average error of the predictor Pt across all t is at most 4 + 2/n ≤ 6 for ` = 3 log n/2 . Now consider the predictor P̂t which for 0 ≤ s ≤ ` − 1 predicts xt+s given xt , . . . , xt+s−1 according to the empirical distribution of xt+s given xt , . . . , xt+s−1 , based on the observations up to time t. We will now argue that the predictions of P̂t are close in expectation to the predictions of Pt . Recall that prediction of Pt at time t + s is the true distribution of xt under the HMM conditioned on the observations xt , . . . , xt+s−1 and the distribution of the hidden state at time t being drawn from πt . For any s < `, let P1 refer to the prediction of P̂t at time t + s and P2 refer to the prediction of Pt at time t + s. We will show that kP1 − P2 k1 is small in expectation. We do this using a martingale concentration argument. Consider any string r of length s. Let Q1 (r) be the empirical probability of the string r up to time t and Q2 (r) be the true probability of the string r given that the hidden state at time t is distributed as πt . Our aim is to show that \|Q1 (r) − Q2 (r)\| is small. Define the random variable Yτ = P r[[xτ : xτ +s−1 ] = r\|hτ ] − I([xτ : xτ +s−1 ] = r), P where I denotes the indicator function and Y0 is defined to be 0. We claim that Zτ = τi=0 Yi is a martingale with respect to the filtration {φ}, {h1 }, {h2 , x1 }, {h3 , x2 }, . . . , {ht+1 , xt }. To verify, note that, E[Yτ \|{h1 }, {h2 , x1 }, . . . , {hτ , xτ −1 }] = P r[[xτ : xτ +s−1 ] = r\|hτ ] − E[I([xτ : xτ +s−1 ] = r)\|{h1 }, {h2 , x1 }, . . . , {xτ −1 , hτ }] = P r[[xτ : xτ +s−1 ] = r\|hτ ] − E[I([xτ : xτ +s−1 ] = r)\|hτ ] = 0 Therefore E[Zτ \|{h1 }, {h2 , x1 }, . . . , {hτ , xτ −1 }] = Zτ −1 , and hence Zτ is a martingale. Also, note that \|Zτ − Zτ −1 \| ≤ 1 as 0 ≤ P r[[xτ : xτ +s−1 ] = r\|hτ ] ≤ 1 and 0 ≤ I([xτ : xτ +s−1 ] = r) ≤ 1. Hence using Azuma’s inequality (Lemma 8), P r[\|Zt−s \| ≥ K] ≤ 2e−K 2 /(2t) Note that Zt−s /(t − s) = Q2 (r) − Q1 (r). By Azuma’s inequality and doing a union bound over all ds ≤ d` strings r of length s, for c ≥ 4 and t ≥ T /n2 = dc` /n2 ≥ dc`/2 , kQ1 − Q2 k1 ≤ √ 1/dc`/20 with failure probability at most 2d` e− t/2 ≤ 1/n2 . Similarly, for all strings of length s + 1, the estimated probability of the string has error at most 1/dc`/20 with failure probability 1/n2 . As the conditional distribution of xt+s given observations xt , . . . , xt+s−1 is the ratio of the joint distributions of {xt , . . . , xt+s−1 , xt+s } and {xt , . . . , xt+s−1 }, therefore as long as the empirical distributions of the length s and length s + 1 strings are estimated with error at most 1/dc`/20 and the string {xt , . . . , xt+s−1 } has probability at least 1/dc`/40 , the conditional distributions P1 and P2 satisfy kP1 − P2 k1 ≤ 1/n2 . By a union bound over all ds ≤ d` strings and for c ≥ 100, 18 the total probability mass on strings which occur with probability less than 1/dc`/40 is at most 1/dc`/50 ≤ 1/n2 for c ≥ 100. Therefore kP1 − P2 k1 ≤ 1/n2 with overall failure probability 3/n2 , hence the expected `1 distance between P1 and P2 is at most 1/n. By using the triangle inequality and the fact that the expected average error of Pt is at most 6 for ` = 3 log n/2 , it follows that the expected average error of P̂t is at most 6 + 1/n ≤ 7. Note that the expected average error of P̂t is the average of the expected errors of the empirical s-gram Markov models for 0 ≤ s ≤ ` − 1. Hence for ` = 3 log n/2 there must exist at least some s < ` such that the s-gram Markov model gets expected `1 error at most 7. A.1 Proof of Lemma 4 Let the prior for the distribution of the hidden states at time 0 be π0 . Let the true hidden state h0 at time 0 be 1 without loss of generality. We refer to the output at time t by xs . Let H0s (i) = P r[h0 = i\|xs0 ] be the posterior probability of the ith hidden state at time 0 after seeing the observations xs0 up to time t and having the prior π0 on the distribution of the hidden states at time 0. For convenience, denote us = H0s (1) and vs = 1 − us . Define Pis (j) = P r[xs = j\|xs−1 0 , h0 = i] as the distribution of the output at time t conditioned on the hidden state at time 0 being i and s observations xs−1 0 . Note that OP Ts = P1 . As before, define Rs as the conditional distribution of xs given observations xP 0 , · · · , xs−1 and initial distribution π but not being at hidden state h0 at time 0 i.e. Rs = (1/vs ) ni=2 H0s (i)Pis . Note that Ms is a convex combination of OP Ts and Rs , i.e. Ms = us OP Ts + vs Rs . Hence kOP Ts − Ms k1 ≤ kOP Ts − Rs k1 . Define δs = kOP Ts − Ms k1 . Our proof relies on a martingale concentration argument, and in order to ensure that our martingale has bounded differences we will ignore outputs which cause a significant drop in the posterior of the true hidden state at time 0. Let B be theP set of all outputs j at some time t P 4 j∈B Rs (j) Ts (j) 4 4 such that OP ≤ clog j∈B OP Ts (j) ≤ clog n n . Hence by a union Rs (j) ≤ clog n . Note that, 4 Ts (j) bound, with failure probability at most 2 any output j such that OP Rs (j) ≤ clog n is not emitted in a window of length clog n/2 . Hence we will only concern ourselves with sequences of outputs such Ts (j) 4 that the output j emitted at each step satisfies OP Rs (j) ≤ clog n , let the set of all such outputs be S1 , note that P r(xs0 ∈ / S1 ) ≤ 2 . Let ES1 [X] be the expectation of any random variable X conditioned on the output sequence being in the set S1 . Consider the sequence of random variables Xs = log us − log vs for t ∈ [−1, ` − 1] defining X−1 = log(π1 ) − log(1 − π1 ). Let ∆s+1 = Xs+1 − Xs be the change in Xs on seeing the output xs+1 at time s + 1. Let the output at time s + 1 be j. We will first find an expression for ∆s+1 . The posterior probabilities after seeing the (s + 1)th output get updated according to Bayes rule – P r[h0 = 1\|xs0 ]P r[x[s + 1] = j\|h0 = 1, xs0 ] P r[x[s + 1] = j\|xs0 ] us OP Ts+1 (j) = P r[x[s + 1] = j\|xs0 ] H0s+1 (1) = P r[h0 = 1\|xs0 , x[s + 1] = j] = =⇒ us+1 Let P r[x[s + 1] = j\|xs0 ] = dj . Note that H0s+1 (i) = H0s (i)Pis+1 (j)/dj if the output at time s + 1 is j. We can write – Rs+1 = n X H0s (i)Pis+1 /vs i=2 vs+1 = n X H0s+1 (i) = n X i=2 i=2 19 H0s (i)Pis+1 (j) /dj = vs Rs+1 (j)/dj Therefore we can write ∆s+1 and its expectation E[∆s+1 ] as – ∆s+1 = log =⇒ E[∆s+1 ] = X OP Ts+1 (j) Rs+1 (j) OP Ts+1 (j) log j OP Ts+1 (j) = D(OP Ts+1 k Rs+1 ) Rs+1 (j) ˜ s+1 as ∆ ˜ s+1 := min{∆s+1 , log log n} to keep martingale differences bounded. E[∆ ˜ s+1 ] We define ∆ then equals a truncated version of the KL-divergence which we define as follows. Definition 3. hFor any two n distributions oiµ(x) and ν(x), define the truncated KL-divergence as D̃C (µ k ν) = E log min µ(x)/ν(x), C for some fixed C. We are now ready to define our martingale. Consider the sequence of random variables X̃s := Pn 2 ˜ X̃s−1 + ∆s for t ∈ [0, ` − 1], with X̃−1 := X−1 . Define Z̃s := s=1 X̃s − X̃s−1 − δs /2 . Note that ˜ s =⇒ Xs ≥ X̃s . ∆s ≥ ∆ is with respect Lemma 5. ES1 [X̃s − X̃s−1 ] ≥ δs2 /2, where the expectation to the output at time t. Ps Hence the sequence of random variables Z̃s := i=0 X̃s − X̃s−1 − δs2 /2 is a submartingale with respect to the outputs. ˜ s and E[∆ ˜ s ] = D̃C (OP Ts k Rs ), C = log n. By taking an Proof. By definition X̃s − X̃s−1 = ∆ expectation with respect to only sequences S1 instead of all possible sequences, we are removing ˜ s ], hence events which have a negative contribution to E[∆ ˜ s ] ≥ E[∆ ˜ s ] = D̃C (OP Ts k Rs ) ES1 [∆ We can now apply Lemma 6. Lemma 6. (Modified Pinsker’s inequality) For any two distributions h µ(x)nand ν(x) oidefined on µ(x) x ∈ X, define the C-truncated KL divergence as D̃C (µ k ν) = Eµ log min ν(x) , C for some fixed C such that log C ≥ 8. Then D̃C (µ k ν) ≥ 12 kµ − νk21 . ˜ s ] ≥ 1 kOP Ts − Rs k2 . Hence ES [X̃s − X̃s−1 ] ≥ δ 2 /2. Hence ES1 [∆ s 1 1 2 We now claim that our submartingale has bounded differences. √ Lemma 7. \|Z̃s − Z̃s−1 \| ≤ 2 log(cγ 4 log n). 2 )/2 can be at most 2. Z − Z ˜ ˜ Proof. Note that (δs2 − δs−1 s s−1 = ∆s . By definition ∆s ≤ log(log n). ˜ s ≥ − log(clog n/4 ) as we restrict ourselves to sequences in the set S1 . Hence \|Z̃s − Z̃s−1 \| ≤ Also, ∆ √ log(clog n/4 ) + 2 ≤ 2 log(clog n/4 ). We now apply Azuma-Hoeffding – Lemma 8. (Azuma-Hoeffding 2 inequality) Let Zi be a submartingale with \|Zi − Zi−1 \| ≤ C. Then −λ P r[Zs − Z0 ≤ −λ] ≤ exp 2tC 2 20 Applying Lemma 8 we can show, P r[Z̃`−1 − Z̃0 ≤ −log n] ≤ exp −log n ≤ 2 4c(1/)2 log2 (clog n/4 ) (A.1) We now bound the average error in the window 0 to ` − 1. With failure probability at most over the randomness in the outputs, Z̃`−1 − Z̃0 ≥ −log n by Eq. A.1. Let S2 be the set of all sequences in S1 which satisfy Z̃`−1 − Z̃0 ≥ −log n. Note that X0 = X̃0 ≥ log(1/π1 ). Consider the last point after which vs decreases below 2 and remains below that for every subsequent step in the window. Let this point be τ , if there is no such point define τ to be ` − 1. The total contribution of the error at every step after the τ th step to the average error is a 2 term as the error after this step is 2 . Note that Xτ ≤ log(1/)2 =⇒ X̃τ ≤ log(1/)2 as X̃s ≤ Xs . Hence for all sequences in S2 – 2 X̃τ ≤ log(1/)2 =⇒ X̃τ − X̃−1 ≤ log(1/)2 + log(1/π1 ) τ X =⇒ 0.5 δs2 ≤ log(1/)2 + log(1/π1 ) + log n =⇒ s=0 P`−1 2 s=0 δs c log n/2 (By Eq. A.1) ≤ 62 (as log(1/π1 ) ≤ c log n) ≤ 3 (By Jensen’s inequality) P`−1 =⇒ s=0 δs c log n/2 P As the total probability of sequences outside S2 is at most 22 , E[ `−1 s=0 δs ] ≤ 4, whenever the hidden state i at time 0 has probability at least 1/nc in the prior distribution π0 . A.2 Proof of modified Pinsker’s inequality (Lemma 6) Lemma 6. (Modified Pinsker’s inequality) For any two distributions h µ(x)nand ν(x) oidefined on µ(x) x ∈ X, define the C-truncated KL divergence as D̃C (µ k ν) = Eµ log min ν(x) , C for some fixed C such that log C ≥ 8. Then D̃C (µ k ν) ≥ 12 kµ − νk21 . Proof. We rely on the following Lemma which bounds the KL-divergence for binary distributionsLemma 9. For every 0 ≤ q ≤ p ≤ 1, we have 2 1. p log pq + (1 − p) log 1−p 1−q ≥ 2(p − q) 2 2. 3p + (1 − p) log 1−p 1−q ≥ 2(p − q) Proof. For the second result, first observe that log(1/(1 − q)) ≥ 0 and (p − q) ≤ p as q ≤ p. Both the results then follow from standard calculus. Let A := {x ∈ X : µ(x) ≥ ν(x)} and B := {x ∈ X : µ(x) ≥ Cν(x)}. Let µ(A) = p, µ(B) = δ, ν(A) = q and ν(B) = . Note that kµ − νk1 = 2(µ(A) − ν(A)). By the log-sum inequality– D̃C (µ k ν) = X x∈B µ(x) log X X µ(x) µ(x) µ(x) + µ(x) log + µ(x) log ν(x) ν(x) ν(x) x∈A−B 21 x∈X−A = δ log C + (p − δ) log 1. Case 1 : δ p p−δ 1−p + (1 − p) log q− 1−q ≥ 0.5 p 1−p log C + (1 − p) log 2 1−q 1 ≥ 2(p − q)2 = kµ − νk21 2 D̃C (µ k ν) ≥ 2. Case 2 : δ p < 0.5 p δ 1−p + (p − δ) log 1 − + (1 − p) log q− p 1−q p 2δ 1−p ≥ δ log C + (p − δ) log − (p − δ) + (1 − p) log q p 1−q p 1−p ≥ δ(log C − 2) + (p − δ) log + (1 − p) log q 1−q D̃C (µ k ν) = δ log C + (p − δ) log (a) Sub-case 1 : log pq ≥ 6 p 1−p + (1 − p) log q 1−q 1−p ≥ 3p + (1 − p) log 1−q 1 ≥ 2(p − q)2 = kµ − νk21 2 D̃C (µ k ν) ≥ (p − δ) log (b) Sub-case 2 : log pq < 6 p p 1−p D̃C (µ k ν) ≥ δ(log C − 2 − log ) + p log + (1 − p) log q q 1−q 1 ≥ 2(p − q)2 = kµ − νk21 2 B B.1 Proof of Lower Bound for Large Alphabets CSP formulation We first go over some notation that we’ll use for CSP problems, we follow the same notation and setup as in Feldman et al. [16]. Consider the following model for generating a random CSP instance on n variables with a satisfying assignment σ. The k-CSP is defined by the predicate P : {0, 1}k → {0, 1}. We represent a k-clause by an ordered k-tuple of literals from {x1 , · · · , xn , x̄1 , · · · , x̄n } with no repetition of variables and let Xk be the set of all such k-clauses. For a k-clause C = (l1 , · · · , lk ) let σ(C) ∈ {0, 1}k be the k-bit string of values assigned by σ to literals in C, that is {σ(l1 ), · · · , σ(lk )} where σ(li ) is the value of the literal li in assignment σ. In the planted model we draw clauses with probabilities that depend on the value of σ(C). Let Q : {0, 1}k → 22 P R+ , t∈{0,1}k Q(t) = 1 be some distribution over satisfying assignments to P . The distribution Qσ is then defined as followsQ(σ(C)) Qσ (C) = P (B.1) 0 C 0 ∈Xk Q(σ(C )) Recall that for any distribution Q over satisfying assignments we define its complexity r as the largest r such that the distribution Q is (r − 1)-wise uniform (also referred to as (r − 1)-wise independent in the literature) but not r-wise uniform. Consider the CSP C defined by a collection of predicates P (y) for each y ∈ {0, 1}m for some m ≤ k/2. Let A ∈ {0, 1}m×k be a matrix with full row rank over the binary field. We will later choose A to ensure the CSP has high complexity. For each y, the predicate P (y) is the set of solutions to the system y = Av mod 2 where v = σ(C). For all y we define Qy to be the uniform distribution over all consistent assignments, i.e. all v ∈ {0, 1}k satisfying y = Av mod 2. The planted distribution Qσ,y is defined based on Qy according to Eq. B.1. Each clause in C is chosen by first picking a y uniformly at random and then a clause from the distribution Qσ,y . For any planted σ we define Qσ to be the distribution over all consistent clauses along with their labels y. Let Uk be the uniform distribution over k-clauses, with each clause assigned a uniformly chosen label y. Define Qησ = (1 − η)Qσ + ηUk , for some fixed noise level η > 0. We consider η to be a small constant less than 0.05. This corresponds to adding noise to the problem by mixing the planted and the uniform clauses. The problem gets harder as η becomes larger, for η = 0 it can be efficiently solved using Gaussian Elimination. We will define another CSP C0 which we show reduces to C and for which we can obtain hardness using Conjecture 1. The label y is fixed to be the all zero vector in C0 . Hence Q0 , the distribution over satisfying assignments for C0 , is the uniform distribution over all vectors in the null space of A over the binary field. We refer to the planted distribution in this case as Qσ,0 . Let Uk,0 be the uniform distribution over k-clauses, with each clause now having the label 0. For any planted assignment σ, we denote the distribution of consistent clauses of C0 by Qσ,0 . As before define Qησ,0 = (1 − η)Qσ,0 + ηUk,0 for the same η. Let L be the problem of distinguishing between Uk and Qησ for some randomly and uniformly chosen σ ∈ {0, 1}n with success probability at least 2/3. Similarly, let L0 be the problem of distinguishing between Uk,0 and Qησ,0 for some randomly and uniformly chosen σ ∈ {0, 1}n with success probability at least 2/3. L and L0 can be thought of as the problem of distinguishing random instances of the CSPs from instances with a high value. Note that L and L0 are at least as hard as the problem of refuting the random CSP instances Uk and Uk,0 , as this corresponds to the case where η = 0. We claim that an algorithm for L implies an algorithm for L0 . Lemma 10. If L can be solved in time t(n) with s(n) clauses, then L0 can be solved in time O(t(n) + s(n)) and s(n) clauses. Let the complexity of Q0 be γk, with γ ≥ 1/10 (we demonstrate how to achieve this next). By Conjecture 1 distinguishing between Uk,0 and Qησ,0 requires at least Ω̃(nγk/2 ) clauses. We now discuss how A can be chosen to ensure that the complexity of Q0 is γk. B.2 Ensuring high complexity of the CSP Let N be the null space of A. Note that the rank of N is (k − m). For any subspace D, let w(D) = (w1 , w2 , · · · , wk ) be a randomly chosen vector from D. To ensure that Q0 has complexity 23 γk, it suffices to show that the random variables w(N ) = (w1 , w2 , · · · , wk ) are (γk − 1)-wise uniform. We use the theory of error correcting codes to find such a matrix A. A binary linear code B of length k and rank m is a linear subspace of Fk2 (our notation is different from the standard notation in the coding theory literature to suit our setting). The rate of the code is defined to be m/k. The generator matrix of the code is the matrix G such that B = {Gv, v ∈ {0, 1}m }. The parity check matrix of the code is the matrix H such that B = {c ∈ {0, 1}k : Hc = 0}. The distance d of a code is the weight of the minimum weight codeword and the relative distance δ is defined to be δ = d/k. For any codeword B we define its dual codeword B T as the codeword with generator matrix HT and parity check matrix GT . Note that the rank of the dual codeword of a code with rank m is (k − m). We use the following standard result about linear codes– Fact 1. If B T has distance l, then w(B) is (l − 1)-wise uniform. Hence, our job of finding A reduces to finding a dual code with distance γk and rank m, where γ = 1/10 and m ≤ k/2. We use the Gilbert-Varshamov bound to argue for the existence of such a code. Let H(p) be the binary entropy of p. Lemma 11. (Gilbert-Varshamov bound) For every 0 ≤ δ < 1/2, and 0 < ≤ 1 − H(δ), there exists a code with rank m and relative distance δ if m/k = 1 − H(δ) − . Taking δ = 1/10, H(δ) ≤ 0.5, hence there exists a code B whenever m/k ≤ 0.5, which is the setting we’re interested in. We choose A = GT , where G is the generator matrix of B. Hence the null space of A is (k/10 − 1)-wise uniform, hence the complexity of Q0 is γk with γ ≥ 1/10. Hence for all k and m ≤ k/2 we can find a A ∈ {0, 1}m×k to ensure that the complexity of Q0 is γk. B.3 Sequential model of CSP and sample complexity lower bound We now construct a sequential model which derives hardness from the hardness of L. Here we slightly differ from the outline presented in the beginning of Section 5 as we cannot base our sequential model directly on L as generating random k-tuples without repetition increases the mutual information, so we formulate a slight variation L0 of L which we show is at least as hard as L. We did not define our CSP instance allowing repetition as that is different from the setting examined in Feldman et al. [16], and hardness of the setting with repetition does not follow from hardness of the setting allowing repetition, though the converse is true. B.3.1 Constructing sequential model Consider the following family of sequential models R(n, Am×k ) where A ∈ {0, 1}m×k is chosen as defined previously. The output alphabet of all models in the family is X = {ai , 1 ≤ i ≤ 2n} of size 2n, with 2n/k even. We choose a subset S of X of size n, each choice of S corresponds to a model M in the family. Each letter in the output alphabet is encoded as a 1 or 0 which represents whether or not the letter is included in the set S, let u ∈ {0, 1}2n be the vector which stores this encoding so ui = 1 whenever the letter ai is in S. Let σ ∈ {0, 1}n determine the subset S such that entry u2i−1 is 1 and u2i is 0 when σ i is 1 and u2i−1 is 0 and u2i is 1 when σ i is 0, for all i. We choose σ uniformly at random from {0, 1}n and each choice of σ represents some subset S, and hence some model M. We partition the output alphabet X into k subsets of size 2n/k each so the first 2n/k letters go to the first subset, the next 2n/k go to the next subset and so on. Let the ith 24 subset be Xi . Let Si be the set of elements in Xi which belong to the set S. At time 0, M chooses v ∈ {0, 1}k uniformly at random from {0, 1}k . At time i, i ∈ {0, · · · , k−1}, if vi = 1, then the model chooses a letter uniformly at random from the set Si , otherwise if vi = 0 it chooses a letter uniformly at random from Xi − Si . With probability (1 − η) the outputs for the next m time steps from k to (k + m − 1) are y = Av mod 2, with probability η they are m uniform random bits. The model resets at time (k + m − 1) and repeats the process. Recall that I(M) is at most m and M can be simulated by an HMM with 2m (2k + m) + m hidden states (see Section 5.1). B.3.2 Reducing sequential model to CSP instance We reveal the matrix A to the algorithm (this corresponds to revealing the transition matrix of the underlying HMM), but the encoding σ is kept secret. The task of finding the encoding σ given samples from M can be naturally seen as a CSP. Each sample is a clause with the literal corresponding to the output letter ai being x(i+1)/2 whenever i is odd and x̄i/2 when i is even. We refer the reader to the outline at the beginning of the section for an example. We denote C 0 as the CSP C with the modification that the ith literal of each clause is the literal corresponding to a letter in Xi for all 1 ≤ i ≤ k. Define Q0σ as the distribution of consistent clauses for the CSP C 0 . Define Uk0 as the uniform distribution over k-clauses with the additional constraint that the ith literal of each clause is the literal corresponding to a letter in Xi for all 1 ≤ i ≤ k. Define 0 0 Qση = (1 − η)Q0σ + ηUk0 . Note that samples from the model M are equivalent to clauses from Qση . We show that hardness of L0 follows from hardness of L– Lemma 12. If L0 can be solved in time t(n) with s(n) clauses, then L can be solved in time t(n) with O(s(n)) clauses. Hence if Conjecture 1 is true then L0 cannot be solved in polynomial time with less than Ω̃(nγk/2 ) clauses. We can now prove the Theorem 2 using Lemma 12. Theorem 2. Assuming Conjecture 1, for all sufficiently large T and 1/T c < ≤ 0.1 for some fixed constant c, there exists a family of HMMs with T hidden states and an output alphabet of size n such that, any polynomial time prediction algorithm that achieves average KL-error, `1 error or relative zero-one error less than with probability greater than 2/3 for a randomly chosen HMM in the family needs requires nΘ(log T /) samples from the HMM over any window length which the algorithm uses for prediction. Proof. We describe how to choose the family of sequential models R(n, Am×k ) for each value of and T . Recall that the HMM has T = 2m (2k + m) + m hidden states. Let T 0 = 2m+2 (k + m). Note that T 0 ≥ T . Let t = log T 0 . We choose m = t − log(1/) − log(t/5), and k to be the solution of m t = m + log(k + m) + 2, hence k = t/(5) − m − 2. Note that for ≤ 0.1, k ≥ m. Let 0 = 29 k+m . 0 We claim ≤ . To verify, note that k + m = t/(5) − 2. Therefore, 0 = 2m 10(t − log(1/) − log(t/5)) = ≥ , 9(k + m) 9t(1 − 10/t) for sufficiently large t and ≥ 2−ct for a fixed constant c. Hence proving hardness for obtaining error 0 implies hardness for obtaining error . We choose the matrix Am×k as outlined earlier. For 25 each vector σ ∈ {0, 1}n we define the family of sequential models R(n, A) as earlier. Let M be a randomly chosen model in the family. We first show the result for the relative zero-one loss. The idea is that any algorithm which does a good job of predicting the outputs from time k through (k + m − 1) can be used to distinguish between instances of the CSP with a high value and uniformly random clauses. This is because it is not possible to make good predictions on uniformly random clauses. We relate the zero-one error from time k through (k + m − 1) with the relative zero-one error from time k through (k + m − 1) and the average zero-one error for all time steps to get the required lower bounds. Let ρ01 (A) be the average zero-one loss of some polynomial time algorithm A for the output 0 (A) be the average relative zero-one loss of A for the outtime steps k through (k + m − 1) and δ01 put time steps k through (k + m − 1) with respect to the optimal predictions. For the distribution Uk0 it is not possible to get ρ01 (A) < 0.5 as the clauses and the label y are independent and y is 0 chosen uniformly at random from {0, 1}m . For Qση it is information theoretically possible to get ρ01 (A) = η/2. Hence any algorithm which gets error ρ01 (A) ≤ 2/5 can be used to distinguish be0 tween Uk0 and Qση . Therefore by Lemma 12 any polynomial time algorithm which gets ρ01 (A) ≤ 2/5 with probability greater than 2/3 over the choice of M needs at least Ω̃(nγk/2 ) samples. Note that 0 (A) = ρ (A) − η/2. As the optimal predictor P δ01 01 ∞ gets ρ01 (P∞ ) = η/2 < 0.05, therefore 0 (A) ≤ 1/3 =⇒ ρ (A) ≤ 2/5. Note that δ (A) ≥ δ 0 (A) m . This is because δ (A) is the δ01 01 01 01 01 k+m average error for all (k +m) time steps, and the contribution to the error from time steps 0 to (k −1) m 0 (A) < 1 =⇒ ρ (A) ≤ 2/5. is non-negative. Also, 13 k+m > 0 , therefore, δ01 (A) < 0 =⇒ δ01 01 3 Hence any polynomial time algorithm which gets average relative zero-one loss less than 0 with probability greater than 2/3 needs at least Ω̃(nγk/2 ) samples. The result for `1 loss follows directly from the result for relative zero-one loss, we next consider the KL loss. 0 (A) be the average KL error of the algorithm A from time steps k through (k + m − 1). Let δKL 0 (A) ≤ 2/9 =⇒ δ 0 (A) ≤ 1/3. By application of Jensen’s inequality and Pinsker’s inequality, δKL 01 0 (A) < 2/9 needs Ω̃(nγk/2 ) samTherefore, by our previous argument any algorithm which gets δKL 0 (A) ≤ 2/9. Hence any polynomial time algorithm which ples. But as before, δKL (A) ≤ 0 =⇒ δKL succeeds with probability greater than 2/3 and gets average KL loss less than 0 needs at least Ω̃(nγk/2 ) samples. We lower bound k by a linear function of log T / to express the result directly in terms of log T /. We claim that log T / is at most 10k. This follows because– log T / ≤ t/ = 5(k + m) + 10 ≤ 15k Hence any polynomial time algorithm needs nΘ(log T /) samples to get average relative zero-one loss, `1 loss, or KL loss less than on M. B.4 Proof of Lemma 10 Lemma 10. If L can be solved in time t(n) with s(n) clauses, then L0 can be solved in time O(t(n) + s(n)) and s(n) clauses. Proof. We show that a random instance of C0 can be transformed to a random instance of C in time s(n)O(k) by independently transforming every clause C in C0 to a clause C 0 in C such that 26 C is satisfied in the original CSP C0 with some assignment t to x if and only if the corresponding clause C 0 in C is satisfied with the same assignment t to x. For every y ∈ {0, 1}m we pre-compute and store a random solution of the system y = Av mod 2, let the solution be v(y). Given any clause C = (x1 , x2 , · · · , xk ) in C0 , choose y ∈ {0, 1}m uniformly at random. We generate a clause C 0 = (x01 , x02 , · · · , x0k ) in C from the clause C in C0 by choosing the literal x0i = x̄i if vi (y) = 1 and x0i = xi if vi (y) = 0. By the linearity of the system, the clause C 0 is a consistent clause of C with some assignment x = t if and only if the clause C was a consistent clause of C0 with the same assignment x = t. We next claim that C 0 is a randomly generated clause from the distribution Uk if C was drawn from Uk,0 and is a randomly generated clause from the distribution Qσ if C was drawn from Qσ,0 . By our construction, the label of the clause y is chosen uniformly at random. Note that choosing a clause uniformly at random from Uk,0 is equivalent to first uniformly choosing a k-tuple of unnegated literals and then choosing a negation pattern for the literals uniformly at random. It is clear that a clause is still uniformly random after adding another negation pattern if it was uniformly random before. Hence, if the original clause C was drawn to the uniform distribution Uk,0 , then C 0 is distributed according to Uk . Similarly, choosing a clause uniformly at random from Qσ,y for some y is equivalent to first uniformly choosing a k-tuple of unnegated literals and then choosing a negation pattern uniformly at random which makes the clause consistent. As the original negation pattern corresponds to a v randomly chosen from the null space of A, the final negation pattern on adding v(y) corresponds to the negation pattern for a uniformly random chosen solution of y = Av mod 2 for the chosen y. Therefore, the clause C 0 is a uniformly random chosen clause from Qσ,y if C is a uniformly random chosen clause from Qσ,0 . Hence if it is possible to distinguish Uk and Qησ for some randomly chosen σ ∈ {0, 1}n with success probability at least 2/3 in time t(n) with s(n) clauses, then it is possible to distinguish between Uk,0 and Qησ,0 for some randomly chosen σ ∈ {0, 1}n with success probability at least 2/3 in time t(n) + s(n)O(k) with s(n) clauses. B.5 Proof of Lemma 12 Lemma 12. If L0 can be solved in time t(n) with s(n) clauses, then L can be solved in time t(n) with O(s(n)) clauses. Hence if Conjecture 1 is true then L0 cannot be solved in polynomial time with less than Ω̃(nγk/2 ) clauses. Proof. Define E to be the event that a clause generated from the distribution Qσ of the CSP C has the property that for all i the ith literal belongs to the set Xi , we also refer to this property of the clause as E for notational ease. It’s easy to verify that the probability of the event E is 1/k k . We claim that conditioned on the event E, the CSP C and C 0 are equivalent. This is verified as follows. Note that for all y, Qσ,y and Q0σ,y are uniform on all consistent clauses. Let U be the set of all clauses with non-zero probability under Qσ,y and U 0 be the set of all clauses with non-zero probability under Q0σ,y . Furthermore, for any v which satisfies the constraint that y = Av mod 2, let U(v) be the set of clauses C ∈ U such that σ(C) = v. Similarly, let U 0 (v) be the set of clauses C ∈ U 0 such that σ(C) = v. Note that the subset of clauses in U(v) which satisfy E is the same as the set U 0 (v). As this holds for every consistent v and the distributions Q0σ,y and Qσ,y are uniform on all consistent clauses, the distribution of clauses from Qσ is identical to the distribution of clauses Q0σ conditioned on the event E. The equivalence of Uk and Uk0 27 conditioned on E also follows from the same argument. Note that as the k-tuples in C are chosen uniformly at random from satisfying k-tuples, with high probability there are s(n) tuples having property E if there are O(k k s(n)) clauses in C. As the problems L and L0 are equivalent conditioned on event E, if L0 can be solved in time t(n) with s(n) clauses, then L can be solved in time t(n) with O(k k s(n)) clauses. From Lemma 10 and Conjecture 1, L cannot be solved in polynomial time with less than Ω̃(nγk/2 ) clauses. Hence L0 cannot be solved in polynomial time with less than Ω̃(nγk/2 /k k ) clauses. As k is a constant with respect to n, L0 cannot be solved in polynomial time with less than Ω̃(nγk/2 ) clauses. C C.1 Proof of Lower Bound for Small Alphabets Proof of Lemma 2 Lemma 2. Let A be chosen uniformly at random from the set S. Then, with probability at least (1 − 1/n) over the choice A ∈ S, any (randomized) algorithm that can distinguish the outputs from the model M(A) from the distribution over random examples Un with success probability greater than 2/3 over the randomness of the examples and the algorithm needs f (n) time or examples. Proof. Suppose A ∈ {0, 1}m×n is chosen at random with each entry being i.i.d. with its distribution uniform on {0, 1}. Let A0 be the sub-matrix of A corresponding to the first 2n/3 columns and all the m rows. Recall that S is the set of all (m × n) matrices A such that the sub-matrix A0 is full row-rank. We claim that P (A ∈ S) ≥ 1 − m2−n/6 . To verify, consider the addition of each row one by one to A0 . The probability of the ith row being linearly dependent on the previous (i − 1) rows is 2i−1−2n/3 . Hence by a union bound, A0 is full row-rank with failure probability at most m2m−2n/3 ≤ m2−n/6 . From Definition 2 and a union bound over all the m ≤ n/2 parities, any algorithm that can distinguish the outputs from the model M(A) for uniformly chosen A from the distribution over random examples Un with probability at least (1 − 1/(2n)) over the choice of A needs f (n) time or examples. As P (A ∈ S) ≥ 1 − m2−n/6 for a uniformly randomly chosen A, with probability at least (1 − 1/(2n) − m2−n/6 ) ≥ (1 − 1/n) over the choice A ∈ S any algorithm that can distinguish the outputs from the model M(A) from the distribution over random examples Un with success probability greater than 2/3 over the randomness of the examples and the algorithm needs f (n) time or examples. C.2 Proof of Proposition 2 Proposition 2. With f (T ) as defined in Definition 2, For all sufficiently large T and 1/T c < ≤ 0.1 for some fixed constant c, there exists a family of HMMs with T hidden states such that any algorithm that achieves average relative zero-one loss, average `1 loss, or average KL loss less than with probability greater than 2/3 for a randomly chosen HMM in the family needs, requires f (log T /) time or samples samples from the HMM over any window length which the algorithm uses for prediction. Proof. We describe how to choose the family of sequential models Am×n for each value of and T . Recall that the HMM has T = 2m (2n + m) + m hidden states. Let T 0 = 2m+2 (n + m). Note that T 0 ≥ T . Let t = log T 0 . We choose m = t − log(1/) − log(t/5), and n to be the solution of m . t = m + log(n + m) + 2, hence n = t/(5) − m − 2. Note that for ≤ 0.1, n ≥ m. Let 0 = 29 n+m 28 We claim ≤ 0 . To verify, note that n + m = t/(5) − 2. Therefore, 0 = 2m 10(t − log(1/) − log(t/5)) = ≥ , 9(n + m) 9t(1 − 10/t) for sufficiently large t and ≥ 2−ct for a fixed constant c. Hence proving hardness for obtaining error 0 implies hardness for obtaining error . We choose the matrix Am×n as outlined earlier. The family is defined by the model M(Am×n ) defined previously with the matrix Am×n chosen uniformly at random from the set S. Let ρ01 (A) be the average zero-one loss of some algorithm A for the output time steps n through 0 (A) be the average relative zero-one loss of A for the output time steps n through (n+m−1) and δ01 (n + m − 1) with respect to the optimal predictions. For the distribution Un it is not possible to get ρ01 (A) < 0.5 as the clauses and the label y are independent and y is chosen uniformly at random from {0, 1}m . For Qηs it is information theoretically possible to get ρ01 (A) = η/2. Hence any algorithm which gets error ρ01 (A) ≤ 2/5 can be used to distinguish between Un and Qηs . Therefore by Lemma 2 any algorithm which gets ρ01 (A) ≤ 2/5 with probability greater than 2/3 over the choice 0 (A) = ρ (A) − η/2. As the optimal of M(A) needs at least f (n) time or samples. Note that δ01 01 0 (A) ≤ 1/3 =⇒ ρ (A) ≤ 2/5. Note that predictor P∞ gets ρ01 (P∞ ) = η/2 < 0.05, therefore δ01 01 0 (A) m . This is because δ (A) is the average error for all (n + m) time steps, and the δ01 (A) ≥ δ01 01 n+m m contribution to the error from time steps 0 to (n − 1) is non-negative. Also, 31 n+m > 0 , therefore, 1 0 0 δ01 (A) < =⇒ δ01 (A) < 3 =⇒ ρ01 (A) ≤ 2/5. Hence any algorithm which gets average relative zero-one loss less than 0 with probability greater than 2/3 over the choice of M(A) needs f (n) time or samples. The result for `1 loss follows directly from the result for relative zero-one loss, we next consider the KL loss. 0 (A) be the average KL error of the algorithm A from time steps n through (n + m − 1). Let δKL 0 (A) ≤ 2/9 =⇒ δ 0 (A) ≤ 1/3. By application of Jensen’s inequality and Pinsker’s inequality, δKL 01 0 (A) < 2/9 needs f (n) samples. Therefore, by our previous argument any algorithm which gets δKL 0 (A) ≤ 2/9. Hence any algorithm which gets average KL loss But as before, δKL (A) ≤ 0 =⇒ δKL 0 less than needs f (n) time or samples. We lower bound n by a linear function of log T / to express the result directly in terms of log T /. We claim that log T / is at most 10n. This follows because– log T / ≤ t/ = 5(n + m) + 10 ≤ 15n Hence any algorithm needs f (log T /)) samples and time to get average relative zero-one loss, `1 loss, or KL loss less than with probability greater than 2/3 over the choice of M(A). D Proof of Information Theoretic Lower Bound Proposition 3. There is an absolute constant c such that for all 0 < < 0.5 and sufficiently large n, there exits an HMM with n states such that it is not information theoretically possible to get average relative zero-one loss or `1 loss less than using windows of length smaller than c log n/2 , and KL loss less than using windows of length smaller than c log n/. 29 Proof. Consider a Hidden Markov Model with the Markov chain being a permutation on n states. The output alphabet of each hidden state is binary. Each state i is marked with a label li which is 0 or 1, let G(i) be mapping from hidden state hi to its label li . All the states labeled 1 emit 1 with probability (0.5 + ) and 0 with probability (0.5 − ). Similarly, all the states labeled 0 emit 0 with probability (0.5 + ) and 1 with probability (0.5 − ). Fig. 3 illustrates the construction and provides the high-level proof idea. Figure 3: Lower bound construction, ` = 3, n = 16. A note on notation used in the rest of the proof with respect to this example: r(0) corresponds to the label of h0 , h1 and h2 and is (0, 1, 0) in this case. Similarly, r(1) = (1, 1, 0) in this case. The segments between the shaded nodes comprise the set S1 and are the possible sequences of states from which the last ` = 3 outputs could have come. The shaded nodes correspond to the states in S2 , and are the possible predictions for the next time step. In this example S1 = {(0, 1, 0), (1, 1, 0), (0, 1, 0), (1, 1, 1)} and S2 = {1, 1, 0, 0}. Assume n is a multiple of (` + 1), where (` + 1) = c log n/2 , for a constant c = 1/33. We will regard as a constant with respect to n. Let n/(` + 1) = t. We refer to the hidden states by hi , 0 ≤ i ≤ (n − 1), hji refers to the sequence of hidden states i through j. We will show that a model looking at only the past ` outputs cannot get average zero-one loss less than 0.5 − o(1). As the optimal prediction looking at all past outputs gets average zero-one loss 0.5 − + o(1) (as the hidden state at each time step can be determined to an arbitrarily high probability if we are allowed to look at an arbitrarily long past), this proves that windows of length ` do not suffice to get average zero-one error less than − o(1) with respect to the optimal predictions. Note that the Bayes optimal prediction at time (` + 1) to minimize the expected zero-one loss given outputs from ` = s` ) where s` is the sequence of time 1 to ` is to predict the mode of the distribution P r(x`+1 \|xP 1 1 1 outputs from time 1 to `. Also, note that P r(x`+1 \|x`1 = s`1 ) = i P r(hi` =i \|x`1 = s`1 )P r(x`+1 \|hi` =i ) where hi` is the hidden state at time `. Hence the predictor is a weighted average of the prediction of each hidden state with the weight being the probability of being at that hidden state. We index each state hi of the permutation by a tuple (f (i), g(i)) = (j, k) where j = i mod (` + 1) i and k = b `+1 c hence 0 ≤ j ≤ `, 0 ≤ k ≤ (t − 1) and i = k(` + 1) + j. We help the predictor to make the prediction at time (` + 1) by providing it with the index f (i` ) = i` mod (` + 1) of the true hidden state hi` at time `. Hence this narrows down the set of possible hidden states at time ` (in Fig. 3, the set of possible states given this side information are all the hidden states before the shaded states). The Bayes optimal prediction at time (` + 1) given outputs s`1 from time 1 to ` and index f (hi` ) = j is to predict the mode of P r(x`+1 \|x`1 = s`1 , f (hi` ) = j). Note that by the definition of Bayes optimality, the average zero-one loss of the prediction using P r(x`+1 \|x`1 = s`1 , f (hi` ) = j) cannot be worse than the average zero-one loss of the prediction using P r(x`+1 \|x`1 = s`1 ). Hence 30 we only need to show that the predictor with access to this side information is poor. We refer to this predictor using P r(x`+1 \|x`1 = s`1 , f (hi` ) = j) as P. We will now show that there exists some permutation for which the average zero-one loss of the predictor P is 0.5 − o(1). We argue this using the probabilistic method. We choose a permutation uniformly at random from the set of all permutations. We show that the expected average zero-one loss of the predictor P over the randomness in choosing the permutation is 0.5 − o(1). This means that there must exist some permutation such that the average zero-one loss of the predictor P on that permutation is 0.5−o(1). To find the expected average zero-one loss of the predictor P over the randomness in choosing the permutation, we will find the expected average zero-one loss of the predictor P given that we are in some state hi` at time `. Without loss of generality let f (i` ) = 0 and g(i` ) = (` − 1), hence we were at the (` − 1)th hidden state at time `. Fix any sequence of labels for the hidden states `−1 h`−1 emitted by the hidden states h`−1 from time 0 to ` − 1, let E[δ(s0`−1 )] 0 . For any string s0 0 be the expected average zero-one error P over the randomness in the rest of the P of the predictor `−1 permutation. Also, let E[δ(h`−1 )] = s`−1 E[δ(s0 )]P r[s`−1 0 ] be the expected error averaged across 0 all outputs. We will argue that E[δ(h`−1 )] = 0.5 − o(1). The set of hidden states hi with g(i) = k k(`+1)−2 defines a segment of the permutation, let r(k) be the label G(h(k−1)(`+1) ) of the segment k, excluding its last bit which corresponds to the predictions. Let S1 = {r(k), ∀ k 6= 0} be the set of all the labels excluding the first label r(0) and S2 = {G(hk(`+1)+` ), ∀ k} be the set of all the predicted bits (refer to Fig. 3 for an example). Consider any assignment of r(0). To begin, we show that with `−1 `−1 of high probability over the output s`−1 0 , the Hamming distance D(s0 , r(0)) of the output s0 ` from r(0) is at least the set of hidden states h`−1 − 2`. This follows directly from Hoeffding’s 0 2 inequality as all the outputs are independent conditioned on the hidden state4 – 2 −2` P r[D(s`−1 ≤ n−2c 0 , r(0)) ≤ `/2 − 2`] ≤ e (D.1) We now show that for any k 6= 0, with decent probability the label r(k) of the segment k is closer than r(0). Then we argue that with high probability in Hamming distance to the output s`−1 0 in Hamming distance than r(0). Hence there are many such segments which are closer to s`−1 0 these other segments are assigned as much weight in predicting the next output as r(0), which means that the output cannot be predicted with a high accuracy as the output bits corresponding to different segments are independent. We first find the probabilityp that the segment corresponding to some k with label r(k) has a Hamming distance less than 2` − ` log t/8 from any fixed binary string x of length `. Let F (l, m, p) be the probability of getting at least l heads in m i.i.d. trails with each trial having probability p of giving a head. F (l, m, p) can be bounded below by the following standard inequality– l 1 F (l, m, p) ≥ √ exp − mDKL p m 2m h 1−q where DKL (q k p) = q log pq + (1 − q) log 1−p . We can use this to lower bound P r D(r(k), x) ≤ i p `/2 − ` log t/8 , h i p p P r D(r(k), x) ≤ `/2 − ` log t/8 = F (`/2 + ` log t/8, `, 1/2) 4 For n independent random variables {Xi } lying in the interval [0, 1] with X̄ = −2nt2 e . In our case t = and n = `. 31 1 n P i Xi , P r[X ≤ E[X̄] − t] ≤ 1 1 ≥ √ exp − `DKL + 2 2` r log t 1 8` 2 Note that DKL ( 12 + v k 12 ) ≤ 4v 2 by using the inequality log(1 + v) ≤ v. We can simplify the KL-divergence using this and write– h i p √ P r D(r(k), x) ≤ `/2 − ` log t/8 ≥ 1/ 2`t (D.2) p Let D be the set of all k 6= 0 such that D(r(k), x) ≤ 2` − ` log t/8 for some fixed x. We argue that with high probability over the randomness of the permutation \|D\| is large. This follows from Eq. D.2 and the Chernoff bound as the labels for all segments r(k) are chosen independently5 – h i √ p 1 P r \|D\| ≤ t/(8`) ≤ e− 8 t/(2`) q p 1 0.25 Note that t/(8`) ≥ n . Therefore for any fixed x, with probability 1−exp(− 8 2`t ) ≥ 1−n−0.25 q t 0.25 segments in a randomly chosen permutation which have Hamming disthere are 8` ≥ n p p tance less than `/2 − ` log t/8 from x. Note that by our construction 2` ≤ ` log t/8 because log(` + 1) ≤ (1 − 32c) log n. Hence the segments in D are closer in Hamming distance to the output if D(s`−1 s`−1 0 , r(0)) > `/2 − 2`. 0 Therefore if D(s`−1 0 , r(0)) > `/2 − 2`, then with high probability over randomly choosing the segments S1 there is a subset D of segments in S1 with \|D\| ≥ n0.25 such that all of the `−1 `−1 such that segments in D have Hamming distance less than D(s`−1 0 , r(0)) from s0 . Pick any s0 `−1 D(s0 , r(0)) > `/2 − 2`. Consider any set of segments S1 which has such a subset D with respect to the string s`−1 0 . For all such permutations, the predictor P places at least as much weight on the hidden states hi with g(i) = k, with k such that r(k) ∈ D as the true hidden state h`−1 . The prediction for any hidden state hi is the corresponding bit in S2 . Notice that the bits in S2 are independent and uniform as we’ve not used them in any argument so far. The average correlation of an equally weighted average of m independent and uniform random bits with any one of the √ random bits is at most 1/ m. Hence over the randomness of S2 , the expected zero-one loss of the predictor is at least 0.5 − n−0.1 . Hence we can writep −0.1 E[δ(s`−1 )P r[\|D\| ≥ t/(8`)] 0 )] ≥ (0.5 − n 0.25 ≥ (0.5 − n−0.1 )(1 − e−n ) ≥ 0.5 − 2n−0.1 By using Equation D.1, for any assignment r(0) to h`−1 0 h i h i `−1 `−1 E[δ(h`−1 )] ≥ P r D(s`−1 0 , r(0)) > `/2 − 2` E δ(s0 ) D(s0 , r(0)) > `/2 − 2` ≥ (1 − n−2c )(0.5 − 2n−0.1 ) = 0.5 − o(1) As this is true for all assignments r(0) to h`−1 and for all choices of hidden states at time `, using 0 linearity of expectations and averaging over all hidden states, the expected average zero-one loss of 5 For independent random variables {Xi } lying in the interval [0, 1] with X = p P r[X ≤ (1 − )µ] ≤ exp(−2 µ/2). In our case = 1/2 and µ = t/(2`). 32 P i Xi and µ = E[X], the predictor P over the randomness in choosing the permutation is 0.5 − o(1). This means that there must exist some permutation such that the average zero-one loss of the predictor P on that permutation is 0.5 − o(1). Hence there exists an HMM on n states such that is not information theoretically possible to get average zero-one error with respect to the optimal predictions less than − o(1) using windows of length smaller than c log n/2 for a fixed constant c. Therefore, for all 0 < < 0.5 and sufficiently large n, there exits an HMM with n states such that it is not information theoretically possible to get average relative zero-one loss less than /2 < − o(1) using windows of length smaller than c−2 log n. The result for relative zero-one loss follows on replacing /2 by 0 and setting c0 = c/4. The result follows immediately from this as the expected relative zero-one loss is less than the expected `1 loss. 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An Optimal Algorithm for Range Search on Multidimensional Points T.Hema ∗and K.S. Easwarakumar† Department of Computer Science & Engineering Anna University, Chennai 600 025, INDIA. arXiv:1607.00208v1 [cs.CG] 1 Jul 2016 Abstract This paper proposes an efficient and novel method to address range search on multidimensional points in θ(t) time, where t is the number of points reported in <k space. This is accomplished by introducing a new data structure, called BITS k d-tree. This structure also supports fast updation that takes θ(1) time for insertion and O(log n) time for deletion. The earlier best known algorithm for this problem is O(logk n + t) time [5, 15] in the pointer machine model. Keywords: BITS k d-tree, Threaded Trie, Range Search. 1 Introduction k d-trees introduced by J.L.Bentley [4, 6] are multidimensional binary search trees commonly used for storing k dimensional points. They are also used to perform search operations such as exact match, partial match and range queries. Range queries are mostly used in GIS applications to locate cities within a certain region in a map. Similarly, in the geometrical view of a database, one can use orthogonal range search to perform a query. Generally, k d-trees with n nodes have a height n and hence the complexity for insertion and search are high. Although many multi-dimensional search structures are found in the literature [2, 8, 20, 22, 23] they differ from the standard k d-trees mainly in the space-partitioning methods used. Recall that a 2-d tree stores two-dimensional point data of the form (x, y). A 2-d tree splits primarily on the x coordinate of a point at even level and then on the corresponding y coordinate at the odd level, and so on. Hence, the trees are unbalanced and are not efficient for√ search operations. Also, the worst case time complexity for range search on a 2-d tree is O( n + t), where t is the number of points reported and for k dimensions it is O(n1−1/k + t)[4, 16]. In general, most of the k d-tree variants get unbalanced when the data is clustered thereby affecting query operations. P R k -d tree, Bucket P R k -d tree [19], P M R k -d trees [21] and Path level compressed P R k -d trees[18] are some of the trie-based kd trees used to store point data. However, these trees are not always balanced, especially when the data is clustered. One of the dynamic versions of k -d tree is the divided k -d trees [25] for which the range query time is O(n1−1/k log1/k n + t). The best known dynamically balanced tree uses bitwise interlaced data [24] over k d-trees mapping k dimensions to one dimension. Although their search time is O(k(log n + t)) for reporting t points, bitwise interlacing leads to discarded areas during range search. In the case of squarish k d-trees [12], an x, y discriminant is based on the longest side of rectangle enclosing the problem space instead of alternating the keys. Recently, hybrid versions of squarish k d-tree, relaxed k d-tree and median k d-trees [11] have overcome the problem of height balancing. An amortized worst case efficiency of range search for the ∗ Email: [email protected] Author.Email: [email protected] † Corresponding 1 hybrid squarish k d-trees, relaxed and median trees for k -dimensional partial match queries are 1.38628 log2 n, 1.38629 log2 n and 1.25766 log2 n respectively. Their experimental results match the aforementioned theoretical results, where they show that the hybrid median trees outperform the other variants. However, as far as query handling is concerned, these structures perform only partial match queries for two dimensions efficiently. The most recent work in the pointer machine model is an orthogonal range reporting data structure with O(n(log n/ log logn)d ) space that address range queries in O(log n(log n/log log n) time, where d ≥ 4 [1]. Range trees of Bentley and Maurer [6, 5] are yet another class of balanced binary search trees used for rectangular range search which showed improvement in the query time of O(logk n + t) over O(n1−1/k + t) of kd-trees, where k is the dimension for a set of n points and k is the number of reported points. This was later improved to O(logk−1 n + t) using fractional cascading in layered range trees[13] but the space requirements are relatively high of O(n logk−1 n). A k d-Range DSL-tree performs k-dimensional range search in O(logk n+t) time was proposed in [15]. Recently, Chan et.al [10] have proposed two data structures for 2d orthogonal range search in the word RAM model. The first structure takes O(n lg lg n) space and O(lg lg n) query time. They show improved performance over previous results [3] of which O(n lg n) space and O(lg lg n) query time, or with O(n lg lg n) space and O(lg 2 lg n) query time. The second data strucure is based on O(n) space and answers queries in O(lg n) time that outperforms previous O(n) space data structure [17], answers queries in O(lg n/lg lg n) time. Furthermore, they also propose an efficient data structure for 3-d orthogonal range reporting with O(n lg 1+ + n) space and O(lg lg n + k) query time for points in rank space where > 0. This improves their previous results [9] with O(n lg 2 n) space and O(lg lg n+k) query time, or with O(n lg 1+ n) space and O(lg 2 lg n + k) query time, where k points are reported. Finally they have extended range search to higher dimensions also. Since such range queries are common among multi-dimensional queries in database applications, we have mainly considered an orthogonal range search on multi-dimensional points. 2 Our Contributions In this work, we make use of the BIT S-tree [14], a segment tree variant that performs stabbing and range queries on segments efficiently in logarithmic time. Most importantly, the distribution of the data points (uniform or skewed) does not affect the height of the BIT S-tree and in turn facilitates faster search time. Here, we actually use the BIT S-tree structure to store points related to each dimension and thereby form a multi-level tree, called BIT S k d-tree. In addition, certain nodes of the BIT S-tree associate to a variant of the trie data structure, called threaded trie, to facilitate fetching a required node in constant time. Unlike k -d trees, it does not associate co-ordinate axis, level wise, for comparison to locate or insert a point. Instead, the tree at the first level has nodes with a key on only distinct values of first co-ordinate of the points. Therefore, this tree corresponds to the one dimensional data. This tree is then augmented with another tree at second level and there in key values of the nodes associated with distinct first two co-ordinates of the points. In general, ith tree corresponds to the distinct first i co-ordinates of the set of points given. Moreover, in each tree, the inorder sequence provides the sorted sequence. That is, BIT S k -d trees is a multi-level tree, and its construction is illustrated in the subsequent sections. 2.1 BIT S-Trees Originally, the BIT S-tree (Balanced Inorder Threaded Segment Tree) [14] is a dynamic structure that stores segments, and also answers both stabbing and range queries efficiently. Unlike segment trees, it also permits insertion of segment with any interval range. 2 Figure 1: (a) Set of segments.(b) BIT S-Tree for the given segments. Definition 1 A BIT S-tree is a height balanced two-way inorder-threaded binary tree T that satisfies the following properties. 1. Each node v of T is represented as v([a, b], L), where [a, b] is the range associated with the node v, and L is the list of segments containing the range [a, b], i.e if [c, d] ∈ L then [a, b] ⊆ [c, d]. 2. Given v1 ([a1 , b1 ], L1 ) 6= v2 ([a2 , b2 ], L2 ), then   [b1 ] if a2 = b1 [b2 ] if a1 = b2 [a1 , b1 ] ∩ [a2 , b2 ] =  φ otherwise i.e ranges can either overlap only at end points or do not overlap at all. 3. Suppose v1 ([a1 , b1 ], L1 ) appears before v2 ([a2 , b2 ], L2 ) in the inorder sequence, then b1 ≤ a2 . 4. It has a special node, called dummy node denoted by D, with range and list as φ(empty). 5. Suppose v1 ([a1 , b1 ], L1 ) and vn ([an , bn ], Ln ) are the first and last nodes of the inorder sequence respectively, then InP red(v1 ) = InSucc(vn ) = D, and the range, say [a, b], of any node contained in [a1 , bn ], i.e [a, b] ⊆ [a1 , bn ]. Here, the functions InPred() and InSucc() respectively returns inorder predecessor and successor. A sample BIT S-tree is shown in Figure 1. Note that the dangling threads actually point to a dummy node, which is not shown in the figure. The BIT S-tree is originally developed for storing segments, but we use this for a different purpose of storing points. Thus, we modify this structure to suit our requirement as described below. 1. Each node v([a, b], L) is replaced by v(p, L0 , T ), where p is a point in <k , k ≥ 1, and L0 is a pointer to the list of collinear points in dimension k + 1, having p for the first k co-ordinates. However, this list is maintained in the tree at the next level, which is described in section 2.2. Now, T is either null or a pointer to a threaded trie, which is elaborated in the next section. 2. For any two points p1 and p2 stored in a tree, p1 6= p2 . 3. Suppose v1 (p1 , L01 , T1 ) appears before v2 (p2 , L02 , T2 ) in the inorder sequence, then p1 < p2 as per the following definition. Definition 2 Let p1 = (x11 , x12 , . . . x1k ) and p2 = (x21 , x22 , . . . x2k ) be two points in a kdimensional space, then a. p1 = p2 implies x1j = x2j for each j=1, 2,. . . k. b. p1 < (or >)p2 implies head(p1 , j) = head(p2 , j) and x1j+1 < (or >) x2j+1 for some j. 3 In the subsequent sections, for better clarity, we use hyphen(−) for a certain parameter of a node to denote that the particular parameter is irrelevant with respect to the context. For instance, (p, −, T ) denotes that the list contents are irrelevant for that point p at this time. 2.2 Threaded Trie Figure 2: A sample threaded trie Threaded tries are variants of tries that consists of two types of nodes viz. trie node and data node. For instance, in Figure 2, A, B and C are trie nodes and the rest are data nodes. Unlike in tries, the trie node here does not have a field for blank (6b). However, each of these trie nodes contain two segments. One is the index pointer and the other is a tag value, which is either 0 or 1, where 0 denotes the corresponding index point in a thread and otherwise it will be 1. Here, all null pointers are replaced by threaded pointers, which point to the next valid node, if one exists. For instance, the thread pointers of 1, 2, 3 of node A points to the node C, as this is the next valid node. Similarly, thread pointers of 0 and 1, in C points to the data node 42. Note here that ordering on the nodes provides the sorted sequence. Also, data nodes appear at the same level. This is accomplished by having uniform width for all data. For instance, the data 8 is treated as 08 in Figure 2. 2.3 Construction of Multi-level BIT S-Tree Multi-level BIT S-trees are constructed using a collection of BIT S-trees one at each level, and interlinking the trees of two consecutive levels in a specified manner, which are due to the following definitions. These multi-level BIT S-trees are termed here as BIT S-k d trees. Definition 3 Given a point p = (x1 , x2 , . . . xk ) and an integer l ≤ k, the head of p and tail of p are defined respectively as head(p, l) = (x1 , x2 , . . . xl ) and tail(p, l) = (xk−l+1 , xk−l+2 , . . . xk ). Also, having (head(p, l), y1 , y2 , . . . ym ) = (x1 , x2 , . . . xl , y1 , y2 , . . . ym ), leads to (head(p, l), tail(p, k − l)) = p. Definition 4 Given S as the set of points in <k and \|S\|=n, the set S1 is defined as, Sn S1 = i=1 {(x) \| p ∈ S and head(p, Sn 1) = (x)}. That is, S1 is the set of distinct x values of the points in S. In general, Sj = i=1 {(x1 , x2 , . . . xj ) \| p ∈ S and head(p, j) = (x1 , x2 , . . . xj )}, where 1 ≤ j ≤ k. Definition 5 For a point p = (x1 , x2 , . . . xj ) in Sj , the term xj is said to be the dimensional value of p as the set of points in Sj is used to construct j th level BIT S-tree. 4 Figure 3: A BIT S 2d-tree: (a) Spatial representation of points. (b) BIT S-2d tree for points shown in (a). Definition 6 A BIT S kd-tree is a multi-level tree, which is constructed as follows. 1. Create separate BIT S trees, Tj for each Sj , 1 ≤ j ≤ k. 2. Let Xj = (x1 , x2 , . . . xj ). Now, for each node, say vj = (Xj , L, −), of Tj , 1 ≤ j < k, the list L points to the node vj+1 = ((Xj , x0j+1 ), −, −) of Tj+1 , where x0j+1 = min{xj+1 \| head(p, j) = Xj and (head(p, j), xj+1 ) ∈ Sj+1 }. We term these links as cross links, and the node vj+1 as a cross link node in Tj+1 . 3. In T1 , there is only one cross link node, which is the first node in the in order sequence, and the tree pointer always points to this node. 4. For each node v = ((Xj−1 , x0j ), −, T ) in Tj , 1 ≤ j ≤ k, T is pointer to the threaded tree if v is a cross link node, and otherwise T is set to be null. 5. For every cross link node vj = ((Xj−1 , x0j ), −, T ) in Tj , the data node of T for a key, say k 0 , points to the node ((Xj−1 , k 0 ), −, −) in Tj . That is, T provides links to the nodes in {((Xj−1 , xj ), −, −)\|(Xj−1 , xj ) ∈ Sj } and these links are termed as trie links. A BIT S 2d-tree for the sample points in Figure 3(a) is shown in Figure 3(b). Since, BIT S k d-trees are multi-level trees with binary inorder threaded search trees at each level, the height of the trees at each level is O(log n). Also, each node in Ti−1 has a cross link to a node in Ti , which has the least value for the ith co-ordinate with respect to the head value of the node in Ti−1 . Note here that at least one such point exists. This link is useful to locate a list of collinear points in the ith dimension, associated with a point in Ti−1 . Also, the trie links are useful to locate a point in a given range window in constant time. The cross link and trie links also make the structure much suitable to address range queries efficiently. Normally, k d-trees perform insertion by a simple comparison between the respective coordinates at each level. However, deletion is tedious due to candidate replacement. This is because, candidate for replacement can be anywhere in the subtree. Also, it requires a little more work when the right subtree is empty. Now, to find a candidate for replacement, it is required to find the smallest element from the left subtree to avoid violation of the basic 5 rules of k d-trees and then it is required to perform a swap of left and right subtrees, as many possible candidate keys exist in the left subtree. To handle such a situation, we make use of a collection of BIT S-trees, one for each dimension. Here, deleting a point may or may not require a replacement, but if so, it is only the inorder successor and that can be located in θ(1) time as inorder links exist for each node. Also, the cross links that exist between two consecutive levels, practically provide a faster search on next level trees. Another advantage of this structure is that when a node is pruned out at a particular level, it need not be considered in the subsequent levels. That is, nodes that have head values as these will be ignored in the subsequent levels. To the best of our knowledge, there is no such structure using multi-levels of balanced binary search trees, with two-way threads introduced in this work, for storing point data and to perform range search efficiently. 2.4 2d-Range Search for Window Query Given a rectangular range in the form of a window, a range query finds all points lying within this window. Let [x1 : x2 ] × [y1 : y2 ] be a given query range. First, we use a trie stored in the first node, which is the only cross link node, in the tree at level 1 to find the smallest point larger than or equal to x1 in S1 . The non-existence of such a point is determined from the trie itself. On the other hand, once such a point p is located, subsequent points that fall within [x1 : x2 ] can be determined using the inorder threads as the inorder sequence is in sorted order. Let us say that the reported set of points as S 0 . However, if the dimensional value of the point p is greater than x2 , it implies absence of required candidates. Now, using cross links of the node in T1 , that corresponds to each point in S 0 , further search is performed at T2 in a similar fashion. Note that each cross link node in T2 has a trie structure that supports quick access to a node in T2 , where the dimensional value is in [y1 : y2 ]. In case, the dimensional value of the cross link node is within [y1 : y2 ], the respective trie structure need not be looked into, instead the inorder threads are used to find the remaining candidates. Example: 1 For instance, let us consider Figure 3 with search range [1 : 8] × [5 : 7]. First, we use the only cross link node present in T1 . As its dimension value, ie. 2 lies within the range [1 : 8], we do not use the respective trie. Instead, we use the inorder threads to identify the candidate points, which are 2,6 and 8. Now, for each of these candidates, further search is continued respectively from (2, 2),(6, 2) and (8, 10) in T2 , as these are the corresponding cross link nodes. Now, by looking at the tries of these cross link nodes we find a point whose dimensional value is the smallest one is [5 : 7]. Thus, tries of (2, 2) yields (2, 6), (6, 2) yields (6, 6) and (8, 10) yields nothing. Further, by performing inorder traversal from (2, 6) and (6, 6), the final reported points for Q1 are E(2, 6) and C(6, 6). Also, for Q2 i.e, ([5 : 8] × [12 : 14]), no points will be reported. Notice that one can stop the search at T1 without traversing T2 if there is no candidate node in T1 within the given range. This is also applicable in k-d trees because if there is no candidate node in the higher tree, the lower level trees need not be searched. Thus, this structure prunes the search in some cases and thereby practically reduces the time for reporting a query. 2.5 k -d Range Search A range search on k-dimensional points can be performed by extending the search on T3 , T4 , . . . Tk , similar to that of T2 as in the case of 2d range search. However in T1 and T2 , we need to perform the search as described for 2d range search. That is, when we take the query range as [x1 : x01 ] × [x2 : x02 ] × . . . [xk : x0k ], the search is performed to find candidates within the range of [x1 : x01 ] in T1 , [x2 : x02 ] in T2 , [x3 : x03 ] in T3 , and so on. Finally, the points reported from Tk will be in Q. It is important to note that the search requires comparison of keys within the given range of the particular co-ordinate dimension in each of T1 , T2 , . . . Tk . This simplifies subsequent searches at the next level. 6 3 3.1 Implementation Details Two Dimensions Given a set of two dimensional points in <2 , a two-level tree (BIT S2d-tree) is constructed in O(n) time as a point may require at most two insertions, one at T1 and the other at T2 . But the position at which insertion is to be made in T1 and T2 could be determined in constant time as described in the proof of Lemma 5. Thus, to insert n nodes requires O(n) time. Also, it may be required to create a cross link for each node of T1 in the case of BIT S 2d-tree. Since, T1 cannot have more than n points, the number of cross links created cannot exceed n. Also, the number of trie links created cannot exceed the number of nodes in T1 and T2 , which is O(n). Moreover, construction of a trie requires only constant time as the height of the trie is constant due to fixed size of the key. Thus, all these factors lie within O(log n) for each insertion. Regarding space requirements in a BIT S 2d-tree, it is O(n), as the second tree is the one that contains all the n points, and fewer or equal number of points in the first tree. Also, the number of trie nodes is O(n) as the height of a trie is constant which is due to the size of(number of digits) of the key. Thus, we obtain the following lemma. Lemma: 1 Construction of BIT S 2d-tree for n points requires O(n) time and O(n) space. Now, searching a candidate node in T1 is done through the trie in T1 and that requires only constant time as the height of the trie is fixed. Once such a point is identified, subsequent points are identified through inorder threads. Thus. for identifying candidate points, it takes only θ(t1 ) time, if there are t1 candidate points in T1 . Now, using cross links of each of these nodes, we can locate the required tries in constant time and further search is to be done in a similar fashion as described earlier. Thus, it leads to the following lemma. Lemma: 2 Range search for window query using BIT S 2d-tree can be addressed in θ(t) time, where t stands for the number of points reported. 3.2 Higher Dimensions A straight forward extension of BIT S 2d-tree to k dimensions is made easy by connecting (cross links) to the corresponding nodes in the tree at next level. Unlike range trees [7] which build another range tree at a given node from the main tree, we maintain the trees T1 , T2 , . . . Tk , dimension-wise such that the inorder traversal provides an ordered sequence of points stored in the tree. This definitely reduces the overall time taken for range search across k dimensions. As described in the previous section, the time required to find a candidate point in any Ti , 1 ≤ i ≤ k, is only a constant. Thus, it leads to the following lemma. Lemma: 3 Let S be a set of points in k-dimensional space, k ≥ 1. A range search on BIT S kd-tree reports all points that lie within the rectangular query range in θ(t) time, where t is the number of points reported. Lemma: 4 Given a set of n points, a BIT S kd-tree can be constructed in O(n) time and O(n) space. Proof: Since we construct T1 , T2 . . . Tk , such that Tk at level k has at most n nodes, it follows that N (Ti ) ≤ N (Ti+1 ), 1 ≤ i < k and N (Tk )=n, where N (Ti ) is the number of nodes in Ti . Note that levels correspond to dimensions and hence may be used interchangeably. Also, the number of trie nodes is O(n) as its height is constant. Therefore for k levels, a BIT S k d-tree uses O(n) storage in the worst case as k is a constant. Now, construction of BIT S k d-tree is considered as a sequence of insertions. Each insertion, may or may not alter Ti , 1 ≤ i ≤ k, a BIT S tree of a particular level. However, if a BIT S-tree Tj is altered, due to insertion, all trees Tj+1 , Tj+2 , . . . Tk will be altered. Let j be the least index such that 7 Table 1: Theoretical comparison of k d-trees, divided k -d trees, range trees, k -d Range DSLtrees, layered range trees and the proposed BIT S k d-trees. Description k d-Trees [4] Divided k -d trees [25] Range Trees[7] k d-Range DSL-Trees [15] Layered Range Trees[13] BITS k d-Trees Storage O(n) O(n) O(n logk−1 n) O(n logk−1 n) O(n logk−1 n) O(n) Construction O(n log n) O(n log n) O(n logk−1 n) O(n logk n) O(n logk−1 n) O(n) Update O(logk n) O(logk−1 n) O(logk n) O(n logk−1 n) O(logk n) Ins. θ(1) Del. O(log n) Range Search O(n1−1/k + t) O(n1−1/k log1/k n + t) O(logk n + t) O(logk n + t) O(logk−1 n + t) θ(t) n-number of points, k-dimensions, t-number of points reported. the tree Tj is altered. Thus, for T1 , T2 , . . . Tj−1 , with trie links and cross links, one can determine that the required values are already stored in those trees within constant time. Now, from a particular cross link in Tj−1 followed by a trie link in Tj , one can find a position for the new value in Tj . This requires only constant time. Then, while inserting the value if the tree is unbalanced, atmost one rotation is required to balance the tree. So, for Tj too, it requires constant time. Let nj be the new node inserted in Tj . Now, by taking cross link of inorder successor of nj , one can determine the position of the new node in Tj+1 , and that as inorder predecessor of cross link node of inorder successor of nj . This new node in Tj+1 need to have a trie, which again be created in constant time. Then, the process is to be continued for Tj+2 . . . Tk . Here, updation in each Ti , 1 ≤ i ≤ k, takes only constant time and hence each insertion takes θ(1) time. So, construction of BIT S k d-tree for n points requires O(n) time. Lemma: 5 Insertion and Deletion of a point in a BIT S kd-tree can be respectively done in θ(1) and O(log n) time. Proof: As per the description given in the proof of Lemma 4, insertion of a point in BITS k d-tree takes only θ(1) time. But for deletion, finding a node to be removed from a BIT Stree requires only constant time. However, if that node is not a leaf node a cascading replacement with inorder successor is required until reaching a leaf node to be removed physically. Certainly, the number of such replacements to be done cannot exceed O(log n). After that it may require a sequence of rotations on the path from the physically removed leaf to the root, and that too in at most O(log n) rotations. So, deletion of a point in BIT S k d-tree requires O(log n) time. 4 Performance Table 1 summarizes the performance of k d-trees, divided k -d trees, range trees, k d-range DSL-trees and the BIT Sk d- tree proposed in this work. Furthermore, our theoretical comparison of the BIT S k d-tree is made with k d-trees adapted for internal memory(pointer machine model) and not with any of the other bulk loading k d-trees(RAM model). The results give an θ(t) query time using the BIT S k d-tree that shows a reduction in time as compared to the existing bounds. Since we try to capitalize on the efficiency of balanced search trees at all the levels by using cross links and trie links, we ensure that the number of nodes visited during a range query is considerably reduced in BIT S k d-tree. Observe that the storage is increased from O(n) in k d-trees to O(n logk−1 n) in range trees while BIT S k d-tree still maintains an O(n). Notice that the update time for BITSk d-tree has been reduced considerably. To summarize, although the storage requirements of BIT S k dtree are comparable to k -d trees, divided k -d trees, the construction and update time are improved considerably. Moreover, the overall query time is improved to θ(t) time where t is the number of points reported as it prunes points falling outside the query region for each dimension. 8 5 Conclusion A BIT S k d-tree for storing k-dimensional points having update and query operations efficiently than kd-trees is proposed. The main advantage of this tree is that it effectively handles the collinear points. As a result, number of nodes visited during search is much less compared to other k d-tree variants that are either not height balanced or update operation is complex. In the case of height balanced k d-trees, having better search efficiency, insertion is tedious. A k -d range DSL tree gives a logarithmic amortized worst case search time with efficient updates mainly for partial match queries and not for window queries. In BIT S k d-tree, overall insertion time is θ(1). Moreover, points can be dynamically updated at each level. Since co-ordinate dimensions at each level are distributed and using threaded tries, we quickly find points falling within the query range. Also, points falling above and below the search range are pruned efficiently using cross links to the next level and inorder threads similar to the BIT S-tree. In addition, threaded tries introduced in this work link the node, having cross link, by means of trie links to find the points within the given range in constant time. Therefore, range search for points in a rectangular region using BIT S k -d tree takes θ(t) time where t is the number of points reported, and therefore the logarithmic factor in earlier worst case bounds is reduced. Hence it is definitely a remarkable improvement over O(n1−1/k + t) of k d-trees and O(logk n + t) time of k -d range DSL trees. References [1] P. Afshani, L. Arge, and K.G. Larsen. Higher-dimensional orthogonal range reporting and rectangle stabbing in the pointer machine model. In Proceedings of the twentyeighth annual symposium on Computational geometry, pages 323–332. ACM, 2012. [2] P. K. Agarwal. Range searching. In J. E. Goodman and J. ORourke, editors, CRC Handbook of Discrete and Computational Geometry. CRC Press, Inc, 2004. [3] S. Alstrup, G. S. Brodal, and T. Rauhe. New data structures for orthogonal range searching. In Foundations of Computer Science, 2000. Proceedings. 41st Annual Symposium on, pages 198–207. IEEE, 2000. [4] J.L. Bentley. Multidimensional binary search tress used for associative searching. Communications of ACM, 18(9):509–516, 1975. [5] J.L. Bentley. Decomposable search problems. Information Processing Letters, 8(5):5–9, June 1979. [6] J.L. Bentley. Multidimensional binary search trees in database applications. IEEE Transactions on Software Engineering, SE-5(4):333–340, 1979. [7] J.L. Bentley. Multidimensional divide and conquer. Communications of the ACM, 23(4):214–229, April 1980. [8] M.D. Berg, O. Cheong, M.V. Kreveld, and M. Overmars. Computational Geometry: algorithms and applications. Springer-Verlag, New York,USA, third edition, 2008. [9] T.M. Chan. Persistent predecessor search and orthogonal point location on the word ram. In Proceedings of the Twenty-second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA ’11, pages 1131–1145. SIAM, 2011. [10] T.M. Chan, K.G. Larsen, and M. Pătraşcu. Orthogonal range searching on the ram, revisited. In Proceedings of the Twenty-seventh Annual Symposium on Computational Geometry, SoCG ’11, pages 1–10, New York, NY, USA, 2011. ACM. 9 [11] M.M.P. Crespo. Design, Analysis and Implementation of New Variants of Kd-trees. Master Thesis, Universitat Politecnica de Catalunya,Departament de Llenguatges i Sistemes Informatics, 2010. [12] L. Devroye, J. Jabbour, and C. Zamora-Cura. Squarish kd-trees. SIAM Journal of Computing, 30:1678–1700, 2000. [13] D.Willard. New data structures for orthogonal queries. Harvard University, TR:22–78, 1978. [14] K.S. Easwarakumar and T. Hema. BITS-Tree-An efficient data structure for segment storage and query processing. International Journal of Computers and Technology, 11(10):3108–3116, December 2013. [15] M.G. Lamoureux and B.G. Nicolson. Determinisitic skip lists for k-dimensional range search. Technical Report(TR95-098), pages 1–95, Novemeber 1995. [16] D.T. Lee and C. K. Wong. Worst-case analysis for region and partial region searches in multidimensional binary search trees and balanced quad trees. Acta Informatica, pages 23–29, 1977. [17] Y. Nekrich. Orthogonal range searching in linear and almost-linear space. Computational Geometry, 42(4):342–351, 2009. [18] S. Nilsson and M.Tikkanen. An experimental study of compression methods for dynamic tries. Algorithmica, 33(1):19–33, 2002. [19] J.A. Orienstein. Multidimensional tries used for associative searching. Information Processing Letters, 14(4):150–157, June 1982. [20] F.P. Preparata and M.L. Shamos. Computational Geometry: An Introduction. SpringerVerlag, New York, 1985. [21] R.C.Nelson and H.Samet. A consistent hierarchical representation for vector data. In Proceedings of the SIGGRAPH’86 Conference, Dallas, volume 20, pages 197–206, August 1986. [22] H. Samet. Fundamentals of Multi-dimensional and Metric Data Structures. Academic Press, New York,USA, 1974. [23] H. Samet. The Design and Analysis of Spatial Data Structures. Addison Wesley, 1990. [24] H. Tropf and H.Herzog. Multidimensional range search in dynamically balanced trees. Applied Informatics, Vieweg Verlag,Germany, 2:71–77, 1981. [25] M.J. van Kreveld and M.H. Overmars. Divided k-d trees. Algorithmica, 6:840–858, 1991. 10	8
Slow links, fast links, and the cost of gossip arXiv:1611.06343v2 [cs.DC] 14 Dec 2017 Suman Sourav National University of Singapore [email protected] Peter Robinson Royal Holloway, University of London [email protected] Seth Gilbert National University of Singapore [email protected] Abstract Consider the classical problem of information dissemination: one (or more) nodes in a network have some information that they want to distribute to the remainder of the network. In this paper, we study the cost of information dissemination in networks where edges have latencies, i.e., sending a message from one node to another takes some amount of time. We first generalize the idea of conductance to weighted graphs by defining φ∗ to be the “critical conductance” and `∗ to be the “critical latency”. One goal of this paper is to argue that φ∗ characterizes the connectivity of a weighted graph with latencies in much the same way that conductance characterizes the connectivity of unweighted graphs. We give near tight lower and upper bounds on the problem of information dissemination, up to polylogarithmic factors. Specifically, we show that in a graph with (weighted) diameter D (with latencies as weights) and maximum degree ∆, any information dissemination algorithm requires at least Ω(min(D+ ∆, `∗ /φ∗ )) time in the worst case. We show several variants of the lower bound (e.g., for graphs with small diameter, graphs with small max-degree, etc.) by reduction to a simple combinatorial game. We then give nearly matching algorithms, showing that information dissemination can be solved in O(min((D + ∆) log3 n, (`∗ /φ∗ ) log n) time. This is achieved by combining two cases. We show that the classical push-pull algorithm is (near) optimal when the diameter or the maximum degree is large. For the case where the diameter and the maximum degree are small, we give an alternative strategy in which we first discover the latencies and then use an algorithm for known latencies based on a weighted spanner construction. (Our algorithms are within polylogarithmic factors of being tight both for known and unknown latencies.) While it is easiest to express our bounds in terms of φ∗ and `∗ , in some cases they do not provide the most convenient definition of conductance in weighted graphs. Therefore we give a second (nearly) equivalent characterization, namely the average conductance φavg . 1 1 Introduction Consider the problem of disseminating information in a large-scale distributed system: nodes in the network have information that they want to share/aggregate/reconcile with others. Real world network communication often has a time delay, which we model here as edges with latencies. The latency of an edge captures how long communication takes, i.e., how many rounds it takes for two neighbors to exchange information. Low latency on links imply faster message transmission whereas higher latency implies longer delays. In the case of unweighted graphs, all edges are considered the same and are said to have unit latencies. However, this is not true in real life and link latencies can vary greatly. In fact, even if nodes are connected directly it might not be the fastest route for communication due to large latency of the link (which might arise due to poor connection quality, hardware or software restrictions etc.); often choosing a multi-hop lower latency path leads to faster distribution of information. For unweighted graphs (without latencies), there exists a significant amount of literature, characterizing the connectivity of a graph (referred as the conductance of a graph) which exactly indicates how efficient information dissemination will be. We would like to do the same for graphs with latencies, however, due to the presence of latencies, not all edges can be regarded as the same; and therefore connectivity alone is no longer enough. Thus, we introduce a new notion of the critical conductance φ∗ that generalizes the notion of classical conductance. Using φ∗ , we give nearly tight lower and upper bounds for information dissemination. For some cases, φ∗ might not be the most convenient definition of conductance in weighted graphs. Alternatively, we give a (nearly) equivalent characterization, namely the average conductance φavg . Model. We model the network as a connected, undirected graph G = (V, E) with n = \|V \| nodes. Each node knows the identities of its neighbors and a polynomial upper bound on the size of the network. Nodes communicate bidirectionally over the graph edges, and communication proceeds in synchronous rounds. An edge is said to be activated whenever a node sends any message over the edge. Latencies occur in the communication channel and not on the nodes. For simplicity, we assume that each edge latency is an integer. (If not, latencies can be scaled and rounded to the nearest integer.) Also, the edge latencies here are symmetric. Problems for non-symmetric arbitrarily large latencies are at least as hard as directed unweighted networks (for which many tasks are impossible to achieve efficiently). Let D be the (weighted) diameter of the graph (with latencies as weights), and let `max be the maximum edge latency. We consider both cases where nodes know the latencies of adjacent edges (Section 5) and cases where nodes do not know the latencies of adjacent edges (the rest of the paper). Nodes do not know D or `max .1 In each round, each node can choose one neighbor to exchange information with: it sends a message to that neighbor and (automatically) receives a response.2 If the edge has latency `, then this round-trip exchange takes time `. This model is, within constant factors, equivalent to a more standard model in which a round-trip involves first sending a message with latency `, receiving it at the other end, and then sending a response at a cost of latency `. Notice that each node can initiate a new exchange in every round, even if previous messages have not yet been delivered, i.e., communication is non-blocking. Information dissemination. We focus in this paper on one-to-all information dissemination. A designated source node begins with a message (the rumor) and, when the protocol completes, every node should have received the message. Classic examples include distributed database replication, sensor network data aggregation, and P2P publish-subscribe systems. This fundamental problem has been widely studied under various names, e.g., information dissemination (e.g., [5]), rumor spreading (e.g., [8]), global broadcast (e.g., [18]), one-to-all 1 In real world settings, nodes are often aware of their neighbors. However, due to fluctuations in network quality (and hence latency), a node cannot necessarily predict the latency of a connection. 2 Notice that this model of communication is essentially equivalent to the traditional push-pull where each node can either push data to a neighbor or pull data from a neighbor; here we assume a node always does both simultaneously. Without the ability to pull data, it is easy to see that information exchange takes Ω(nD) time, e.g., in a star. Simple flooding matches this lower bound. 2 multicast, and information spreading (e.g., [6]). As a building block, we look at local broadcast, i.e., the problem of every node distributing a message to all of its neighbors. Conductance in weighted graphs. Our goal in this paper is to determine how long it takes to disseminate information in a graph with latencies. Clearly the running time will depend on the (weighted) diameter D of the graph. Typically, such algorithms also depend on how well connected the graph is, and this is normally captured by the conductance φ. Unfortunately, conductance is no longer a good indicator of connectivity in a graph with latencies, as slow edges (with large weights) are much worse than fast edges.3 We begin by generalizing the idea of conductance to weighted graphs. We give two (nearly) equivalent definitions of conductance in weighted graphs, which we refer to as the critical conductance φ∗ (Definition 2) and the average conductance φavg (Definition 4). While they give (approximately) the same value for every graph, there are times when one definition is more convenient than the other. In fact, we show that the values φ∗ of φ∗ and φavg are closely related; as in 2` < φavg < φ`∗∗ dlog(`max )e (c.f. Theorem 5). We compare these ∗ definitions further in Section 2.3. We use φ∗ in determining the lower and upper bounds for information dissemination as it makes our analysis simpler and then use the above relation to determine the bounds for φavg . A core goal of this paper is to argue that the notion of φ∗ (and φavg ) defined herein well captures the connectivity of weighted graphs, and may be useful for understanding the performance of other algorithms. Lower bounds. These constitute some of the key technical contributions of this paper. For a graph G, with diameter D, maximum degree ∆, critical conductance φ∗ , and critical latency `∗ , we show that any information dissemination algorithm requires Ω(min(D +∆, `∗ /φ∗ )) rounds. That is, in the worst case it may take time D + ∆ to distribute information. However, if the graph is well connected, then we may do better and the time is characterized by the critical conductance. We show that this lower bound holds even in various special cases, e.g., for graphs with small diameter, or with small max-degree, etc. By the relation provided in Theorem 5, we determine the lower bound in terms of average conductance as Ω(min(D + ∆, 1/φavg )). The main technique we use for showing our lower bounds is a reduction to a simpler combinatorial guessing game. (See [25] for a demonstration of how other variants of guessing games can be used to prove lower bounds for radio networks.) We first show that the guessing game itself takes a large number of rounds. Thereafter we reduce the problem of solving the game to that of solving information dissemination via a simulation. Upper bounds. We then show nearly matching upper bounds, i.e., algorithms for solving information dissemination. In this regard, we differentiate our model into two cases. For the case where nodes are not aware of the adjacent edge latencies, we show that the classical push-pull random phone call algorithm [22] in which each node initiates a connection with a randomly chosen neighbor in each round, completes in O((`∗ /φ∗ ) log n) rounds. By using the relationship between φ∗ and φavg , we give a O((log(`max )/φavg ) log n) upper bound in terms of φavg . For the case where nodes do know the latencies of the incident edges, we obtain nearly tight bounds that are independent of ∆ and φ∗ : we give a O(D log3 n)-time algorithm (which is within polylogarithmic factors of the trivial Ω(D) lower bound). The key idea of the algorithm is to build a (weighted) spanner (based on that in [2]). This spanner is then used to distribute information. This algorithm, however, requires knowledge of a polynomial upper bound on n; hence for completeness we also provide an alternate algorithm in Appendix 5.4 that does not require the knowledge of n but takes an additional log D factor (instead of log n), making it unsuitable for graphs with large diameters. Finally, we observe that we can always discover the latencies of the “important” adjacent edges in 3 Notice you might model an edge with weight w as a path of w edges with weight 1. If you calculate the conductance of the resulting graph, you do not get a good characterization of the connectivity of the original graph for a few different reasons. For e.g., consider the ability of the imaginary nodes on the edge to pull data from the endpoints. 3 Õ(D + ∆) time4 , after which we can use the algorithm that works when latencies are known. Hence, even if latencies are unknown, combining the various algorithms, we can always solve the information dissemination in O(min((D + ∆) log3 n, (`∗ /φ∗ ) log n) time (or O(min((D + ∆) log3 n, (log(`max )/φavg ) log n) time), matching the lower bounds up to polylogarithmic factors (with respect to the critical conductance). Summary of our contributions. To the best of our knowledge, this work provides a first ever characterization of conductance in graphs with latencies. In this regard, we provide two different parameters namely φ∗ and φavg . Note that, we provide the summary here only in terms of φ∗ , however, for each case there exists an alternate version in terms of φavg . For lower bounds, we show that there exists graphs with (a) O(log n) diameter with maximum degree ∆ where local broadcast requires Ω(∆) rounds; (b) O(`∗ ) diameter with critical conductance φ∗ where local broadcast requires Ω(1/φ∗ + `∗ ) rounds; (c) Θ(1/φ∗ ) diameter where information dissemination requires Ω(min(D + ∆, `∗ /φ∗ )) rounds; showing the trade-off among the various parameters affecting information dissemination. For upper bounds on information dissemination, we show that (d) the push-pull algorithm takes O(`∗ log(n)/φ∗ ) rounds; (e) a spanner-based algorithm takes O((D + ∆) log3 n) rounds. We view our results as a step towards a more accurate characterization of connectivity in networks with delays and we believe that the metrics φ∗ and φavg can prove useful in solving other graph problems. Prior work. There is a long history studying the time and message complexity of disseminating information when all the links have the same latency. It is interesting to contrast what can be achieved in the weighted case with what can be achieved in the unweighted case. The classic model for studying information dissemination is the random phone call model, introduced by [10]: in each round, each node communicates with a single randomly selected neighbor; if it knows the rumor, then it “pushes” the information to its neighbor; if it does not know the rumor, then it “pulls” it from its neighbor (see, e.g., [13], [23], [16]). An important special case is when the graph is a clique: any pair of nodes can communicate directly. In a seminal paper, Karp et al. [22] show that a rumor can be disseminated in a complete graph in O(log n) rounds with O(n log log n) message complexity. Fraigniaud and Giakkoupis [14] show how to simultaneously achieve optimal communication complexity (except for extremely small rumor sizes). When the graph is not a clique, the performance of the classical push-pull protocol, wherein a node exchanges information with a random neighbor in each round, typically depends on the topology of the graph, specifically, how well connected the graph is. An exciting sequence of papers (see [7, 8, 16, 24] and references therein) eventually showed that rumor spreading in this manner takes time O( logφ n ), where φ is the conductance of the graph. The question that remained open was whether a more careful choice of neighbors lead to faster information dissemination. In a breakthrough result, Censor-Hillel et al. [5] gave a randomized algorithm for solving information dissemination in any (unweighted) graph in time O(D + polylogn), where D here is the nonweighted diameter of the graph. Of note, the protocol has no dependence on the conductance of the graph but only on the diameter (which is unavoidable). There were two key ingredients to their solution: first, they gave a “local broadcast” protocol where each node exchanges information with all its neighbors in O(log3 n) time; second, as a by-product of this protocol they obtain a spanner which they use in conjunction with a simulator (defined therein) to achieve information dissemination in O(D + polylogn) time. Haeupler [18] then showed how local broadcast could be achieved in O(log2 n) time using a simple deterministic algorithm. The conclusion, then, is that in an unweighted graph (with unit latency edges), information dissemination can be achieved in time O(D + polylogn) or in time O(log(n)/φ). 4 The notation Õ hides polylogarithmic factors, which arise due to D and ∆ being unknown. 4 Other related works. The problem has been well researched in several other settings as well. For graphs modeling social networks Doerr et al. [11, 12] show a Θ(log n) time bound for solving broadcast. For the case of direct addressing, Haeupler and Malkhi [19] show that broadcast can be performed optimally in O(log log n) rounds. Information dissemination in random geometric graphs has been studied by Bradonjić et al. [4], in wireless sensor networks and adhoc networks by Boyd et al. [3], Sarwate and Dimakis [26] and Gandhi et al. [15], Giakkoupis et al. [17] study the problem in dynamic graphs. 2 Conductance in Weighted Graphs In this section, we provide two different approaches to characterize conductance in weighted graphs namely the critical conductance and the average conductance and show the relationship between them. In the sections that follow, where we determine the bounds on information dissemination, we use the critical conductance as it makes our analysis simpler. Corresponding bounds for average conductance are obtained by the application of the given relationship in Theorem 5. 2.1 Critical Conductance We now define the critical conductance of a graph, generalizing the classical notion of conductance. For a given graph G = (V, E), and for a set of edges S ⊆ E, we define E` (S) to be the subset of edges of S that have latency 6 `. For a set of nodes U ⊆ V and cut C = (U, V \ U ), we define P E` (C) to be the subset of edges across the cut C with latency 6 `, and we define the volume Vol(U ) = v∈U degv , where degv refers to the degree of node v. We first define the critical conductance of a cut for a given latency `, and then define the weight-` conductance as the minimum critical conductance across all cuts. Definition 1 (Weight-` Conductance). Consider a graph G = (V, E). For any cut C in the set of all possible cuts (C̃) of the graph G and an integer `, we define φ` (C) = \|E` (C)\| . min{Vol(U ), Vol(V \ U )} The weight-` conductance is given by φ` (G) = min{φ` (C) \| C ∈ C̃}. Definition 2 (Critical Conductance). We define the critical conductance φ∗ (G) as φ` (G) is maximum for any ` ∈ (1, `max ) . φ∗ (G) = φ` (G) ` We call `∗ the critical latency for G if `∗ = ` and φ∗ (G) = φ` (G). We simply write φ∗ (or φ` ) instead of φ∗ (G) (or φ` (G)) when graph G is clear from the context. If all edges have latency 1, then φ∗ is exactly equal to the classical graph conductance [21]. 2.2 Average Conductance For a given graph G = (V, E), we first define dlog(`max )e different latency classes, where the first class contains all the edges of latency < 2 and the subsequent ith latency class consists of all the edges in the latency range of (2i−1 , 2i ]. For a set of nodes U ⊆ V and the cut C = (U, V \ U ), we define ki (C) to be the subset of edges across the cut C belonging to latency class i (i.e. all cut edges of latency > 2i−1 and 6 2i ). For a cut C, we first define the average cut conductance as φavg (C), and then define the average conductance as the minimum average cut conductance across all cuts. 5 Definition 3 (Average Cut Conductance). Consider a graph G = (V, E), a set of nodes U ⊆ V and the cut C = (U, V \ U ). Let S be the min{Vol(U ), Vol(V \ U )}. 1 S φavg (C) = dlog(`max )e X i=1 \|ki (C)\| 2i Definition 4 (Average Conductance). Let C̃ be the set of all possible cuts of the graph G. We define the average conductance as φavg (G) = min{φavg (C) \| C ∈ C̃}. We simply write φavg instead of φavg (G) when graph G is clear from the context. If all edges have latency 1, then φavg is exactly equal to the classical graph conductance [21]. 2.3 Comparing Critical and Average Conductances Conductance, in general, is a characterization of the “bottleneck in communication” of a graph. For unweighted graphs, the only bottleneck in communication can be the connectivity of the graph, however, for weighted graphs the bottleneck can arise either due to the graph connectivity or due to the edge latency (even if the nodes are directly connected by a slow edge, there might exist a different multi-hop faster path). Our aim is to capture both aspects of this bottleneck in communication. Having good connectivity facilitates faster communication whereas large latencies result in slow-downs. Ideally, we would want the best connectivity along with the least slowdown for faster communication. We obtain the definition of φ∗ by directly optimizing these orthogonal parameters. The connectivity that maximizes this ratio is defined as the critical conductance φ∗ and the corresponding latency is defined as the critical latency `∗ . In other words, φ∗ captures the bottleneck due to connectivity whereas `∗ captures the bottleneck due to latency. The definition of the average conductance φavg is inspired by the classical notion of conductance. Each cut edge’s contribution towards the overall connectivity is normalized by dividing it with its latency (rounded to the upper bound of its latency class), so as to account for the slow-down. Surprisingly, we see that φ∗ and φavg are closely related and to show the relationship between them, we first define L as the number of non-empty latency classes in the given graph G. Latency class i is said to be non-empty if there is at least one edge in the graph G that has a latency > 2i−1 and 6 2i . The maximum value that L can take is dlog(`max )e which is the total number of possible latency classes. Theorem 5. φ∗ /2`∗ < φavg < Lφ∗ /`∗ . Proof. Consider any weighted graph G that has critical conductance φ∗ and critical latency as `∗ . We first show the upper bound. Let C be the cut from which φ∗ was obtained and let S be the minimum volume among either side of the cut. By the definition of weight-` conductance, Pi φ2i (C) = j=1 \|kj (C)\| S and from the definition of φ∗ , we know that ⇒ φ∗ `∗ φ2i (C) 2i is > ∀i ∈ (1, dlog(`max )e) φ` ` Pi = j=1 \|kj (C)\|/2i S > \|ki (C)\|/2i S for any `, which implies φ i (C) \|ki (C)\|/2i φ∗ > 2 i > `∗ 2 S Note that, in the definition of φavg , the terms corresponding to the empty latency classes becomes zero. We replace each remaining term in the definition of φavg (C) by φ`∗∗ and using the above inequality, we get φavg (C) 6 φ`∗∗ L. Combining with the fact that φavg is the minimum average cut conductance, we obtain 6 φavg 6 φavg (C) 6 φ∗ L `∗ (1) Next we show the lower bound, for this we consider the cut C 0 that determines φavg and let S 0 be the minimum volume among either side of the cut. On this cut C 0 consider the latency class of the critical latency `∗ ; say `∗ lies in the latency class x, which implies that 2x−1 < `∗ 6 2x . From the definition of weight-` conductance, we get φ`∗ (C 0 ) \|k1 (C 0 )\| + \|k2 (C 0 )\| + · · · + \|kx (C 0 )\| 6 . 2`∗ 2`∗ S 0 Rewriting φavg as (from definition) φavg \|kdlog(`max )e (C 0 )\| \|k1 (C 0 )\| \|k2 (C 0 )\| , + + ··· + = 2S 0 22 S 0 2dlog(`max )e S 0 and comparing the first x terms of φavg to that of φ`∗ (C 0 )/2`∗ , we observe that each term in the expression of φavg is at least as large as the corresponding term in the above upper bound on φ`∗ (C 0 )/2`∗ . Also there are some additional positive terms in φavg . Combining this with the fact that φ`∗ /2`∗ 6 φ`∗ (C 0 )/2`∗ (as by definition φ` is chosen as the minimum value among all possible cuts), we obtain φ` ∗ φ` (C 0 ) 6 ∗ 6 φavg 2`∗ 2`∗ (2) This proves the lower bound and completes the proof. 3 Lower Bounds We proceed to lower bound the time for completing information dissemination. The main goal of this section (as found in Theorems 9, 10, and 13) is to show that every gossip algorithm requires Ω(min{∆ + D, `∗ /φ∗ }) on graphs with diameter D, max-degree ∆, critical conductance φ∗ , and critical latency `∗ . Throughout this section, we assume that nodes do not know the latencies of their adjacent links (when nodes do know the latencies, the trivial lower bound of Ω(D) is sufficient). We begin by defining a combinatorial guessing game (a similar approach as in [25]) and show a lower bound for it.5 We then construct several different worst-case graphs and reduce the guessing game to solving information dissemination on these graphs, thereby showing our lower bound. 3.1 The Guessing Game We define a guessing game played by Alice against an oracle. Conceptually, the game is played on a bipartite graph of 2m nodes. The oracle selects a subset of the edges as the target. In each round, Alice guesses a set of at most 2m edges, and the oracle reveals any target edges that have been hit. At the same time, if any edge (u, v) in the target set is guessed by Alice, then all adjacent edges (x, v) in the target set are removed from the target set. Fix an integer m. Let A and B be two disjoint sets of m integers each, i.e., the left and right group of nodes in the bipartite graph. The winning condition of the game depends on a predicate P , which returns a subset of edges from A × B. For example, P = Randomp returns a subset T that contains elements of A × B, where each element is chosen with probability p or discarded with probability 1 − p. 5 The results of [25] do not apply directly to our setting, as their “proposal set” of the player must intersect the target set in exactly 1 element. By contrast, the guessing game here requires us to discover sufficiently many target elements such that every element in the target set occurs at least once. 7 We now define the game Guessing(2m, P ), which begins when Alice receives two disjoint sets A and B. The oracle chooses a target set T1 ⊆ A × B returned by the predicate. Throughout, we assume that Alice has access to a source of unbiased random bits. Alice’s goal is to eliminate all the elements in the target set. In each round r > 1, Alice submits a set Xr ⊆ A × B of size at most 2m as her round r guesses to the oracle. The oracle replies by revealing the items she guessed correctly, i.e., Xr ∩ Tr . The oracle then computes the round r + 1 target set Tr+1 by removing the items that Alice hit, i.e., all the items in Tr that have the same B-component as an item in Xr ∩ Tr : Tr+1 = Tr \ TrA × TrB ∩ XrB . (3) This concludes round r and the next round begins. The game is solved in the first round r0 , where Alice’s guesses result in an empty target set; at this point, the oracle answers halt. In other words, the game ends in round r0 if, for every b ∈ T1B , there was some a0 ∈ A such that (a0 , b) ∈ Xr ∩ Tr , in some r ∈ [1, r0 ]. Alice’s aim is to minimize the number of rounds until the target set becomes empty. We say that a protocol Π solves Guessing(2m, P ) with probability 1 − in r rounds, if Π always terminates within r rounds, and Tr+1 = ∅ with probability > 1 − , for any target set T . In this case, we call Π an -error protocol. 3.2 Guessing by Gossiping Our lower bound results use variants of an n-node distributed network that has a guessing game gadget of 2m nodes embedded as a subgraph. In our gadget construction, we use predicate P , to specify a set of hidden low latency edges, which we call fast edges. We show that, the execution of a gossip algorithm on an n-node network can be simulated by Alice when playing the guessing game Guessing(2m, P ), where n > 2m. We use the notation id(v) to denote the ID of a vertex v, which, by construction is unique. For a given instance of the guessing game, Alice creates a set of nodes L = {v1 , . . . , vm } where id(vi ) = ai ∈ A for i = 1, . . . , m and, similarly, maps the integers in B to the IDs of the vertex set R = {u1 , . . . , um } in a one-to-one fashion. Next, Alice creates a complete bipartite graph on sets L and R by adding m2 cross edges and adds a clique on the vertices in L where all clique edges are considered to have latency 1. For given integer parameters lo and hi, we construct the network in a way that only some cross edges in the target set are useful to the algorithm by giving them a low latency lo whereas all other cross edges are assigned a large latency value hi. Formally, the latencies of a cross edge e = (vi , uj ) is lo iff (id(vi ), id(uj )) ∈ P ; otherwise e has latency hi. We denote this constructed gadget as G(2m, lo, hi, P ), where the parameters refer to the size of the gadget (i.e. 2m), the low latency value lo, the high latency value hi, and the predicate P respectively. We also consider a symmetric variant, called Gsym (2m, lo, hi, P ), where Alice creates a clique on R in addition to the one on L. See figure 1. Since Alice does not know the target set T in advance, she also does not know when a cross edge should have latency lo or latency hi. Nevertheless, implicitly these latency assignments are fixed a priori by the target set (unknown to Alice) which in turn depends on the predicate P . Whenever a cross edge e is activated in our simulation, Alice submits the ID pair of the vertices of e as a guess to the oracle, whose answer reveals the target set membership and hence also the latency of e. Lemma 6 (Gossip Protocol Simulation). Suppose that there is a t-round -error algorithm A that solves local broadcast on a given n-node network H that contains G(2m, 1, h, P ) or Gsym (2m, 1, h, P ) such that the cross edges of the gadget form a cut of H, for h > t, n > 2m, and a predicate P . Then there is an -error protocol Π for Guessing(2m, P ) that terminates in 6 t rounds. Proof. We argue that Alice can simulate the execution of A on network H and, in particular, on the subgraph G(2m, 1, h, P ), until the gossip algorithm A terminates or the oracle answers halt. (It is straightforward to 8 v1 u1 v1 u1 v2 u2 v2 u2 vm-1 um-1 vm-1 vm L R vm um (a) L R um (b) Figure 1: Guessing Game Gadgets. Red edges correspond to “fast” links whereas the blue edges are “slow” links with high latency. extend the argument to a subgraph Gsym (2m, 1, h, P ).) At the same time, Alice can use the behavior of A on the subgraph G(2m, 1, h, P ) to derive a protocol for Guessing(2m, P ). For a given instance of the guessing game, Alice creates the network H by first assigning all edges in the subgraph H \ G(2m, 1, h, P ) a latency of 1. Moreover, she creates the edges of the subgraph G(2m, 1, h, P ) as described in Section 3.2; we will see below that the latency of a cross edges is only set when it is first activated. If an non-cross edge (vi , vj ) (i.e. a clique edge on L or an edge in E(H \ G(2m, 1, h, P ))) is activated by the algorithm, Alice locally simulates the bidirectional message exchange by updating the state of nodes vi and vj accordingly. In each round r of the gossip algorithm, a set of at most 2m cross edges is activated by the vertices simulated by Alice’s. For each activated cross edge (vi , uj ), Alice uses (id(vi ), id(uj )) as one of her round r guesses. Consider some round r > 1, and suppose the oracle returns the empty set. For each one of Alice’s submitted round r guess (ai , bj ) that was not contained in the oracle’s answer, Alice sets the latency of (ai , bj ) to h by updating the local state of ai . Here ai = id(vi ) and bj = id(uj ) that are chosen in round r, for some vi ∈ L and uj ∈ R. It follows by a simple inductive argument that the state of every vertex in the simulation is equivalent to executing the algorithm on the network. We now argue that the above simulation of a t-round gossip algorithm for local broadcast solves the game Guessing(2m, P ) in at most t rounds with probability > 1 − , for any predicate P . Recall that the guessing game ends if T becomes empty, which happens when Alice’s correct guesses have included every b ∈ T B at least once. By the premise of the lemma, the cross edges of G(2m, 1, h, P ) form a cut of H, which tells us that A cannot solve local broadcast without using the cross edges between L × R. Since every such b ∈ R is a neighbor of a node in L, the only way it can receive a local broadcast message is via a fast cross-edge in T . Hence, if the local broadcast algorithm terminates, we know that b was hit by one of Alice’s guesses. 3.3 Guessing Game Lower Bounds The following lemma is instrumental for showing the Ω(∆) lower bound of Theorem 9, which holds when there are no other assumptions on the critical conductance of the graph. Lemma 7. Let Guessing(2m, \|T \| = 1) be the guessing game where the target set is a single pair chosen uniformly at random from A × B. If protocol Π is an -error protocol for Guessing(2m, \|T \| = 1) where < 1, then the number of rounds until Π terminates is at least Ω(m). Proof. For the sake of a contradiction, suppose that Π solves Guessing(2m, \|T \| = 1) in t < m 2 − 1 rounds. We define Time to be the random variable of the number of rounds until termination in a given execution of Π. 9 Consider a round r > 1 of the protocol and suppose that the game has not yet ended, i.e., Alice has not yet guessed all of T correctly and has made at most 2m(r − 1) (incorrect) guesses in the previous rounds. Let Xr denote the (at most 2m) pairs from A × B chosen by Alice in round r. Since from Alice’s point of view, the adversary has chosen the single element of T uniformly at random from the m2 elements in A × B, 2 2m 6 m−2r the probability that Alice guesses the element of T in round r is at most m2 −2m(r−1) . Let Correct denote the event that protocol Π correctly solves the game. It follows that Pr Time = r Correct = Pr Tr ⊆ Xr Correct 6 2 m−2r . (4) In the remainder of the proof, we will lower bound the probability of event {Time > t}. Observe that Pr[Time > t] > Pr Time > t Correct Pr[Correct]. If Time > t, then none of the rounds 1, . . . , t guesses of Alice were successful, i.e., Q Pr Time > t Correct > ti=1 1 − Pr Time = i Correct . Applying (4) to each round i 6 t, we get Pr[Time > t] > Qt i=1 1− 2 m−2i (1 − ) > 1− 1 m 2 −t t (1 − ). Since the running time of Π never exceeds t rounds, i.e., Pr[Time > t] = 0 and < 1, we get a contradiction to t < m 2 − 1. The next lemma bounds the number of guesses required when the target set is less restricted and its edges form a random subset of the cross edges between A × B. This allows us to derive a lower bound on the local broadcast time complexity in terms of the critical conductance in Theorem 10. Lemma 8. For the guessing game input sets A and B, let Randomp be the predicate that defines the target 1 set T by adding each element of A × B to T with probability p, for some p > Ω( m ). Then, any protocol that solves Guessing(2m, Randomp ) requires Ω(1/p) rounds in expectation. On the other hand, if Alice uses the protocol where she submits her 2m guesses in each round by choosing, for each a ∈A, anelement b0 ∈ B uniformly at random, and, for each b ∈ B, an a0 ∈ A uniformly at random, then Ω logpm rounds are required in expectation. Proof. Recall that the game ends when the guesses of Alice have hit each element in T B ⊆ B at least once, whereas T B is itself a random variable. Let Y be the maximum number of guesses required by Alice’ protocol Π. For the sake of our analysis, we will consider Alice’s guesses as occurring sequentially and hence we can assume that elements of T B are discovered one by one. For each j > 1, we define Zj to denote the number of guesses required to guess the j-th element of T B , after having already guessed j − 1 elements. We will first consider general protocols. Considering that each edge is in the target set with probability p, we can assume that the target membership of an edge e is determined only at the point when Alice submits e as a guess. Recalling that Alice has full knowledge of the remaining elements in T B that she still needs to guess (cf. (3)), we can assume that her guess is successful with probability p (as she will only guess edges that potentially discover a new element in T B ). For this guessing strategy, this remains true independently of the current target set and the set of previously discovered elements (which we denote by Dj ). Formally, Pr[Zj \| Dj , T ] = Pr[Zj ] and hence E[Zj \| Dj , T ] = E[Zj ] = d1/pe. Note that any b ∈ B will be part of some target edge in T , i.e., b ∈ T B , with probability > 1 − (1 − p)m = Ω(1), since p = Ω(1/m), and therefore E[\|T \|] = Ω(m). It follows that hP i hP i \|T \| \|T \| m E[Y ] = E[E[Y \| Dj , T ]] = E E[Z \| D , T ] = E E[Z ] i j i = Ω( p ). i=1 i=1 10 Considering that Alice can guess up to 2m elements per round, it follows that the time is Ω( p1 ), which completes the proof for general algorithms. Now consider the case where Alice uses the protocol where she submits her 2m guesses in each round by choosing, for each a ∈ A, an element b0 ∈ B uniformly at random, and, for each b ∈ B, an a0 ∈ A uniformly at random. Note that this process of selecting her guesses is done obliviously of her (correct and incorrect) guesses so far. Observe that Zj depends on a random variable Fj , which is the size of T after the (j − 1)-th successful guess. Since Zj is the number of times that the protocol needs to guess until a new element in T B is discovered, the distribution of Zj corresponds to a geometric distribution. According to Alice’s protocol, the 2 F B probability of guessing a new element is given by mj2 and hence E[Zj \| Fj ] > m Fj . Let U = \|T1 \|; i.e., U is the number of all elements in B that are part of an edge in T initially. We have m E Y \|U > m 2 = E E[Y \| Fi ] \| U > 2 hP i U m E[Z \| F ] \| U > =E i i i=1 2 bm/2c > X E E[Zi \| Fi ] \| U > m 2 i=1 bm/2c > X i=1 m2 E Fi \| U > m 2 , where the last inequality follows from E[1/X] > 1/E[X], for any positive random variable X, due to Jensen’s Inequality. Since Alice has already correctly guessed i − 1 elements from T B , we discard all elements that “intersect” with successful guesses when updating the target set at the end of each round, according to (3). It can happen that the protocol discovers multiple elements of T B using the round r guesses (which we have assumed to happen sequentially in this analysis). In that case, the target set is not updated in-between guesses. However, it is easy to see that this does not increase the probability of guessing a new element of T B . We get E Fi \| U > m 2 6 (m − i)mp, and thus E Y \|U > m 2 > bm/2c m X 1 . p m−i i=1 This sum is the harmonic number Hbm/2c−1 , which is Θ(log m), for sufficiently large m, and hence E Y \|U > m 2 m log m . >Ω p By the law of total expectation it follows that E[Y ] > E Y \| U > m 2 Pr U > m 2 . Finally, a standard probability calculation shows that U > m 2 happens with large probability, assuming that c p > m for a sufficiently large constant c > 0. The time bound follows since Alice can submit 2m guesses per round. 11 3.4 Lower Bounds for Information Dissemination In this section we show three different lower bounds. Together, these show what properties cause poor performance in information dissemination protocols: in some graphs, high degree is the cause of poor performance (Theorem 9); in other graphs, poor connectivity is the cause of poor performance (Theorem 10). And finally, we give a family of graphs where we can see the trade-off between D, ∆, and φ∗ (Theorem 13). We begin with a result showing that Ω(∆) is a lower bound: Theorem 9. For any ∆ ∈ (Θ(1), dn/2e), there is an n-node network that has a weighted diameter of O(log n), and a maximum node degree Θ(∆), where any algorithm requires Ω(∆) rounds to solve local broadcast with constant probability. Proof. Consider the network H of n nodes that consists of the guessing game gadget Gsym (2∆, 1, ∆, P ), where predicate P returns an arbitrary singleton target set, combined with a constant degree regular expander [20] of n − 2∆ vertices (if any) of which any one node is connected to all the vertices on the left side of the gadget; all the edges of, and connected to the expander have latency 1 and the latencies of the edges in the gadget are assigned as in Lemma 6. Clearly, the weighted diameter of H is O(log n) (diameter of the expander [20]). By Lemma7, we know that any guessing game protocol on Guessing(2∆, \|T \| = 1) requires Ω(∆) rounds for the predicate that returns exactly 1 pair as the target set. Lemma 6 tells us that any gossip algorithm that solves local broadcast in H, must require Ω(∆) rounds. We next show that every local broadcast algorithm requires time at least Ω(1/φ∗ + `∗ ). Note that, we get this Ω(1/φ∗ ) lower bound just for local broadcast and not information dissemination, which is in contrast to the results in the unweighted case. The following result is given in terms of the weight-` conductance, for any `, and thus also holds for φ∗ and `∗ . In the proof, we construct a network that corresponds to the bipartite guessing game graph with a target set where each edge is fast with probability φ∗ . That way, we obtain a network with critical conductance Θ(φ∗ ), hop diameter O(1), and a weighted diameter of O(`∗ ). The guessing game lower bound of Lemma 8 tells us that the cost of information dissemination still depends on φ∗ . Theorem 10. For any ` ∈ [1, n] and φ` where Ω(log(n)/n) 6 φ` 6 1/2, there is a network of 2n nodes that has a weighted diameter O(`) (w.h.p.), and critical conductance Θ(φ` ) (w.h.p.), such that any gossip algorithm requires Ω((1/φ` ) + `) rounds for solving local broadcast in expectation. Also, solving local broadcast using push-pull requires Ω((log n/φ` ) + `) rounds in expectation. Proof. Our goal is to reduce the game Guessing(2n, Randomφ` ) to local broadcast, hence we consider the 2n-node graph G(2n, `, n2 , Random φ` ) as our guessing game gadget defined in Section 3.2. Since we want to show the time bound of t = Ω log n φ` + ` rounds (for push-pull), for the high latency edges we can use n the value n2 > log φ` + ` (as Ω(log(n)/n) 6 φ` and ` 6 n). We assign each cross edge latency ` independently with probability φ` and latency n2 with probability 1 − φ` . The fast cross edges have the same distribution as the target set implied by the predicate Randomφ` , which we have used to show a lower bound of Ω( φ1` ) for general protocols on Guessing(2n, Randomφ` ) in n Lemma 8, and also a stronger lower bound of Ω( log φ` ) for “random guessing” protocols, which choose a random edge for each vertex as their guesses. It is straightforward to see that push-pull gossip corresponds exactly to this random guessing game strategy. Applying Lemma 6, this means that local broadcast requires n in expectation Ω( φ1` ) time for general algorithms and Ω( log φ` ) time for push-pull. The additional term of Ω(`) in the theorem statement is required to actually send the broadcast over the latency ` edge once it is discovered. 12 Since each edge of L × R is assigned latency ` with probability φ` = Ω(log(n)/n), it follows that each u ∈ R is connected by a latency ` edge to some node in L with high probability. Hence, the weighted diameter of G(2n, `, n2 , Randomφ` ) is O(`) with high probability. In the remainder of the proof, we show that G(2n, `, n2 , Randomφ` ) has a conductance of Θ(φ` ) with high probability. We point out that several previous works prove bounds on the network expansion (e.g., [27] and [1]). However, as these results were shown for random graphs, we cannot employ these results directly and thus need to adapt these proof techniques to show a conductance of Θ(φ` ) for our guessing game gadget. We assume that there is an integer-valued function f = f (n), such that nf = φ` , noting that this assumption does not change the asymptotic behavior of our bounds. For readability, we only consider ` = 1 and note that the extension to the general case is straightforward. By construction, G(2n, 1, n2 , Randomf /n ) consists of edges with latencies 1 or n2 and we have 1 φ1 φn 2 6 2 6 , n2 n 1 where the last inequality follows from the assumption φ1 > Ω logn n . Thus, we know that φ∗ = φ1 and hence we need to prove φ1 = Θ(f /n). Consider a set S ⊆ L ∪ R of at most n vertices and let l = \|S ∩ L\| and r = \|S ∩ R\|. We first assume that l > r, since the number of latency 1 cross edges is symmetric for vertices in L and R; subsequently, we will remove this assumption by a union bound argument. For vertex sets A and B, let E1 (A, B) be the set of the (randomly sampled) latency 1 edges in the cut (A, B) and define e1 (A, B) = \|E1 (A, B)\|. Given the set S, our goal is to show that many latency 1 edges originating in S ∩ L have their other endpoint in R \ S, assuming that there are sufficiently many latency 1 cross edges to begin with. In other words, we need to bound from above the probability of the event e1 (S ∩ L, S ∩ R) > Ω(f l) conditioned that there are sufficiently many latency 1 cross edges. Claim 11 (Sufficiently many latency 1 cross edges). There exist constants c, c0 > 0, such that events LR := {∀S, \|S\| 6 n : (e1 (S ∩ L, R) > cf l) ∧ (e1 (S ∩ R, L) > cf r)}, − 0 0 LR := {∀S, \|S\| 6 n : (e1 (S ∩ L, R) 6 c f l) ∧ (e1 (S ∩ R, L) 6 c f r)} (5) (6) occur with high probability. Proof. According to the construction of G(2n, 1, n2 , Randomf /n ), the latency 1 cross edges are chosen independently each with probability f /n. Note that each cross edges is assigned latency 1 independently with probability f /n = Ω( logn n ). Thus, for each node v, the expected number of cross edges is f = Ω(log n) and, by a standard Chernoff bound, we know that the number of latency 1 cross edges to v is in [c1 f, c2 f ] with high probability, for suitable constants c2 > c1 > 0. After taking a union bound over all nodes in V (G), we can conclude that the claim holds for any set S ⊆ V (G). Conditioning on LR is equivalent with choosing a subset of (at least) cf l edges among all possible edges in the cut E1 (S ∩ L, R) uniformly at random and assigning them latency 1. Consider an edge (v, u) ∈ E1 (S ∩ L, R). It follows that u ∈ S ∩ R (and hence (v, u) ∈ E1 (S ∩ L, S ∩ R)), with probability r n and we need to exclude the event Bad(S) := {e1 (S ∩ L, S ∩ R) > 45 cf l}, cf l subsets of latency 1 edges incident to vertices in S ∩ L. In addition, we need to bound the probability that Bad(S) happens, for S chosen in any of the nl ways of choosing S that satisfy \|S ∩ L\| = l. for all 4 cf l 5 13 Claim 12. Pr ∃S : Bad(S) LR 6 n−Ω(1) . Proof of claim. Combining the above observations, we get n cf l r 45 cf l . Pr ∃S : Bad(S) LR 6 4 n l 5 cf l (7) First, we assume that r and l are both large, i.e., l > r > c0 n, for a sufficiently small positive constant k m·H2 ( m ) 4 c0 < 5e . Then, we can apply Stirling’s approximation of the form m , where H2 (x) = k ≈ 2 −x log2 (x) − (1 − x) log(1 − x) is the binary entropy function. Thus, for sufficiently large n, we get r 4 cf l 5 n·H2 ( nl )+cf lH2 ( 45 ) Pr ∃S : Bad(S) LR 6 2 · n n+cf l·H2 ( 45 )− 45 cf l , 62 (8) where, to derive the second inequality, we have used the facts that H2 ( nl ) 6 1 and nr 6 12 , since r + l 6 n and r 6 l. By the premise of the theorem, f = Ω(log n), which implies c f l = Ω(n log n). Together with the fact that H2 ( 45 ) < 54 , this means that the term (− 45 c f l) dominates in the exponent of (8) and hence Pr ∃S : Bad(S) LR 6 2−Θ(n log n) . em k Next, we consider the case where r 6 l < c0 n. Applying the upper bound of the form m to (7), k k 6 tells us that 4 en l 5e r 5 cf l Pr ∃S : Bad(S) LR 6 l 4n en l 5c0 e 45 cf l 6 , l 4 since r < c0 n. We get 4 5 0 Pr ∃S : Bad(S) LR 6 exp l 1 + log n − log l + cf l log c e 5 4 4 5 6 exp l 1 + log n + cf l log c0 e . 5 4 4 By assumption, c0 < 5e and hence the term 45 c f l log 45 c0 e in the exponent is negative. Moreover, recall that f = Ω(log n) and thus we can assume that 45 c f l log 54 c0 e 6 −c00 log n, for a sufficiently large constant c00 > 0. This term dominates the other terms in the exponent, thereby completing the proof of the claim. cf \|S\| Considering that l > \|S\| 2 , the above bound implies that at least b 10 c latency 1 edges incident to S are connected to nodes outside in S, with probability at least 1 − n−Ω(1) . Taking a union bound over all possible choices for the values of l and r adhering to r 6 l and r + l 6 n = \|V (G)\| 2 , shows that h i Pr ∀S, \|S\| 6 n : e1 (S ∩ L, R \ S) > cf10\|S\| LR > 1 − n−Ω(1) . Observe that the latency 1 cross edges are constructed symmetrically for the left and right side of the bipartite graph G and thus we can apply the above argument in a similar manner for a set S where r > l, conditioned on e1 (S ∩ R, L) > cf l. Thus, we can conclude that h i Pr ∀S, \|S\| 6 n : e1 (S, V (G) \S) > cf10\|S\| LR > 1 − n−Ω(1) . (9) 14 We can remove the conditioning in (9) by virtue of Claim (11), since h i h Pr ∀S, \|S\| 6 n : e1 (S, V (G) \S) > cf10\|S\| > Pr ∀S, \|S\| 6 n : e1 (S, V (G) \S) > cf \|S\| 10 i LR Pr[LR] > 1 − n−Ω(1) . To upper bound Vol(S) for any set S, we take into account the n cross edges of each node in S. Also, if v ∈ L, then we need to account for the n − 1 incident clique edges of v, yielding Vol(S) 6 2\|S\|n. Considering the upper bound on the number of latency 1 cross edges given by (6), we have φ∗ = min φ1 (S) = min S S e1 (S, V (G) \ S) cf \|S\| > min = Ω( nf ), S 20\|S\|n Vol(S) where the inequality is true with high probability. To see that this bound istight, observe that φ∗ 6 φ1 (L). By (5) and (6), we know that e1 (L, R) = Θ(f n) and hence φ1 (L) = Θ nf with high probability, as required. This completes the proof of Theorem 10. Finally, we give a family of graphs that illustrate the trade-off among the parameters. Intuitively, when the edge latencies are larger, it makes sense to search for the best possible path and the lower bound is Ω(D + ∆); when the edge latencies are smaller, then we can simply rely on connectivity and the lower bound is Ω(`/φ` ). Note that, we can individually obtain a lower bound of Ω((`/φ) log n), using the technique in [7] where we show that there exists a graph with diameter (`/φ) log n. Unlike here, that lower bound is simply D. Theorem 13. For a given α ∈ [Ω(1/n), O(1)] and any integer ` ∈ [1, O(n2 α2 )], there is a class of networks of 2n nodes, critical conductance φ∗ = φ` = Θ(α), maximum degree ∆ = Θ(αn), and weighted diameter D = Θ(1/φ` ), such that any gossip algorithm that solves broadcast with at least constant probability, requires Ω(min{∆ + D, `/φ` }) rounds. Proof. We create a network G consisting of a series of k node layers V1 , . . . , Vk that are wired together asa q 2 8 ring, using the guessing game gadgets introduced above. We define k = cα where c = 34 + 41 9 − nα . This implies that 1 6 c < 3/2 as α ∈ [Ω(1/n), O(1)]. Each layer consists of s = cnα nodes. As it does not change our asymptotic bounds, we simplify the notation by assuming that 2/cα and cnα are integers. V(k/2)+1 Vk/2 V3k/4 V(k/4)+1 V(3k/4)+1 Vk/4 Vk V1 Figure 2: Guessing Game Gadgets wired together as a ring. For each pair Vi and V(i+1) mod k (0 6 i 6 k − 1), we construct the symmetric guessing game gadget Gsym (2cnα, 1, `, P ) (in Section 3.2), for simulating a gossip algorithm to solve the game Guessing(2cnα, \|T \| = 15 1). That is, we create a complete bipartite graph on Vi and V(i+1) mod k and form cliques on Vi and V(i+1) mod k (see Figure 2). We assign latency ` to every cross edge between Vi and V(i+1) mod k , except for a uniformly at random chosen edge that forms the singleton target set, which we assign latency 1. Observe that the weight-j conductance φj cannot be maximal for any j other than 1 or `. Observation 14. Let s = cnα. Graph G is (3s − 1)-regular. Proof. For a layer Vi , we call V(i−1) mod k the predecessor layer and V(i+1) mod k the successor layer. The size of a layer is s = cnα. Each node has 2s edges to its neighbors in the predecessor resp. successor layer and s − 1 edges to nodes in its own layer. This means that G is a (3s − 1)-regular graph. We define a cut C that divides the ring into two equal halves such that none of the internal clique edges are cut edges. By a slight abuse of notation, we also use C to denote the set of vertices present in the smaller side of the partition created by the cut C (ties broken arbitrarily). Lemma 15. φ` (C) = α. Proof. Since C partitions G into two sets of identical size, the volume can be determined by considering either partition of size n, thus we focus on the node set C. Also, by Observation 14 we know that G is (3s − 1)-regular. The volume of C can be calculated to be n(3cnα − 1). The number of cut edges of latency 6 ` is 2(cnα)2 (by the construction of C). According to Definition 1, the `-weight conductance is given by 2(cnα)2 φ` (C) = n(3cnα−1) . By plugging in the value of c, we can verify that φ` (C) is exactly equal to α. Using the conductance bound of Lemma 15 for cut C, we know that φ` 6 α. In the proof of the next lemma, we show that φ` = Ω(α). Lemma 16. The weight-` conductance of the constructed ring network is φ` = Θ(α). Proof. By Lemma 15, we know that φ` 6 α as the actual graph conductance is always 6 to any cut conductance. We will now show φ` = Ω(α) as well. By Observation 14 we know that G is (3s − 1)-regular and therefore for a set of nodes U the volume Vol(U ) is exactly equal to (3s − 1)\|U \|. This clearly implies that for any two sets U and V , Vol(U ) 6 Vol(V ) if and only if \|U \| 6 \|V \|. Now, consider an arbitrary cut (U, V (G) \ U ) of G and suppose that U contains at most half of the nodes of G, i.e., \|U \| 6 n, since G has 2n nodes. If there are at least Θ(s2 ) cut edges, then, using the fact that \|U \| 6 n, we get φ` (U ) > Θ(s2 /s\|U \|) = Θ(s/\|U \|) > Θ(s/n) > Θ(α), and we are done. In the remainder of the proof, we will show that there are Θ(s2 ) cut edges. We distinguish two cases: 1. \|U \| > 3s/4: We classify each node in U either as good if it has at least s/4 adjacent edges across the cut (U, V \ U ) and as bad otherwise. Thus, our goal is to identify Θ(s) good nodes, which in turn implies Θ(s2 ) cut edges. Let S be an arbitrary subset of 3s/4 nodes in U . If all nodes in S are good, we are done. Otherwise, let x ∈ S be a bad node. It is important to note that the following properties are true for every bad node: (a) Node x is in a layer in G which contains at least 3s/4 nodes inside U . (b) The successor layer from x has at least 3s/4 nodes inside U . To see why (a) holds, assume that it was not true. Then, x would have at least s/4 neighbors in its own layer across the cut, contradicting the assumption that x is bad. Similarly, if (b) was false, x would be connected to at least s/4 nodes in the successor layer outside U . (This is true of the predecessor layer too.) Let A be the successor layer to the layer containing x. We now run the following procedure: 16 Invariant: A contains at least 3s/4 nodes in U . If at least half of the nodes in A are good, we are done. Terminate and claim Θ(s2 ) cut edges. Otherwise, let y be a bad node in A. Let A0 be the successor layer of the layer A. Then, start again at Step (1) with A = A0 and y = x. From the assertion (b), A0 contains at least 3s/4 nodes in S. If this procedure ever terminates in Step (2), we are done. Otherwise, it continues around until every layer has been explored. In that case, the invariant implies that every layer contains at least 3s/4 nodes in U . This implies that > 1/2 of the nodes of G are in U , which contradicts the choice of U . Thus, the procedure does terminate, which means there must be at least Θ(s2 ) cut edges, implying φ` > α. 2. \|U \| < 3s/4: Let m be the number of nodes in U . Since G is (3s − 1)-regular, the volume of U is m(3s − 1). Each node in U now contains at least s/4 neighbors outside of U (since it has > s neighbors and there are only < 3s/4 other nodes in U ), so the cut size is at least sm/4. Thus, the conductance of this graph (sm/4) φ` > m(3s−1) = Ω(1) > Θ(α). Since, φ` 6 α and φ` > Θ(α), it is clearly the case that φ` = Θ(α), which is what we wanted to prove. (1) (2) (3) (4) Combining Lemmas 15 and 16 (and again using cut C), we argue that the critical latency is `. Lemma 17. For any ` 6 O(cnα)2 , φ∗ = φ` = Θ(α). Proof. To prove that φ∗ is in fact φ` , which by Lemma 16 is Θ(α), we need to show that (φ` /`) > (φ1 /1) = φ1 . To this end, let us consider the cut C defined above. We will show that φ`` > φ1 (C) > φ1 , and since weight-j conductance φj (cf. Definition 1), cannot be maximal for any j other than 1 or `, we get φ∗ = φ` . There are two latency 1 cross edges in the cut C and the volume of C can be calculated as in the proof of Lemma 15 to be n(3cnα − 1). Thus, we need to show that φ` Θ(α) 2 = > . ` ` (3cnα − 1)n As c is a constant, the above inequality is true as long as ` = O(α2 n2 ), which is ensured by the premise of the theorem. The weighted diameter of the network D = Θ(k/2), since each pair of adjacent node layers is connected by a latency 1 edge and, internally, each layer forms a latency 1 clique. Using the fact that c ∈ [1, 32 ), it can be shown that (2/3α) < D 6 (1/α), implying that D = Θ(1/φ` ) (by lemma 16). Now, consider a source node in layer V1 that initiates the broadcast of a rumor. Each node can either spend time in finding the required fast edge (which we assume can be done in parallel) or, instead, it can instantly use an edge of latency ` to forward the rumor. Lemma 7 tells us that finding the single latency 1 cross edge with constant probability, for the guessing game gadget corresponding to any pair of node layers, requires Ω(∆) rounds, and then forwarding the rumor takes Ω(D) additional rounds. Alternatively, the algorithm can forward the rumor along latency ` edges across node layers and spread the rumor using the latency 1 edges within each clique. It follows that the required time for broadcast is Ω(min{∆ + D, `/φ` }). We obtain the following corollary that gives a lower bound on information dissemination in terms of φavg , either by a similar analysis as above, or by the application of Theorem 5. Corollary 18. For a given α ∈ [Ω(1/n), O(1)] and any integer ` ∈ [1, O(n2 α2 )], there is a class of networks of 2n nodes, average conductance φavg = Θ(α/`), maximum degree ∆ = Θ(αn), and weighted diameter D = Θ(1/`φavg ), such that any gossip algorithm that solves broadcast with at least constant probability, requires Ω(min{∆ + D, 1/φavg }) rounds. 17 Proof. Observe that in the given graph, there exists edges with latency either 1 or `, and as such the number of non-empty latency classes here is 2. Now, Theorem 5 reduces to φ∗ /2`∗ < φavg < 2φ∗ /`∗ . This implies that for this case φavg = Θ(φ∗ /`∗ ). Alternatively, φ∗ = `φavg (as in this case ` = `∗). Replacing this value of φ∗ in Theorem 13 gives us the above required corollary. 4 Algorithms for Unknown Latencies We divide the upper bounds on information dissemination into two sub-components and later combine them n ), to obtain a unified result. First, we analyze classical push-pull, showing that it completes in time O( `∗ φlog ∗ which is optimal when D + ∆ is large. Alternatively for graphs where D + ∆ is small, we give an algorithm wherein each node first spends Õ(D + ∆) time discovering the neighboring latencies after which nodes use the local information to build a spanner, across which data can be distributed in Õ(D) time. 4.1 Push-Pull To show the time required for information dissemination in a weighted graph G using push-pull, we define E` as the set of all edges of latency 6 `, Eu as the set of incident edges of vertex u and Eu,` := E` ∩ Eu . n ) rounds in a network G, where Theorem 19. The push-pull protocol achieves broadcast w.h.p. in O( `∗ φlog ∗ φ∗ is the critical conductance of G and `∗ is the corresponding critical latency. Proof. We construct a strongly edge-induced graph G` , which is a generalization of the strongly (vertex) induced subgraph defined in [5] and which has the same vertex set as G. The edges of G` have a multiplicity6 defined by the edge multiplicity function µ, given by   if (u, v) ∈ E` 1 µ(u, v) = \|Eu \| − \|Eu,` \| if u = v (10)   0 otherwise It is easy to see that the (unweighted) conductance φ(G` ) corresponds to φ` (G), as a self-loop at node u is counted as µ(u, u) edges when computing the volume. We also define another unweighted graph G0 that is derived from G by dropping all edge latencies. Now, we consider the Markov chain process describing the informed node set, i.e., the vertex set that is in possession of some message m originating from a vertex v when running push-pull. Formally, the state space of the Markov chain consists of all possible informed node sets. Only paths that correspond to monotonically growing informed node sets have nonzero probability. We argue that this process on G0 (resp. G) dominates the respective process in the graph G` . We observe that each node v selects an incident edge in E` from G` in the push-pull protocol with the same probability as in G0 . The probability of choosing an edge ∈ Eu \ E` P (i.e., a self loop in case of G` ) is µ(u, u)/ v∈V µ(u, v) in both graphs. Clearly, choosing a self loop of a node u cannot help in the propagation of the message in G` , but choosing the corresponding edge in G0 might. It follows that the Markov process of reaching any informed node set S in G0 dominates over the one in G` , i.e., the probability of reaching any informed node set S by using the Markov chain in G0 is at least as large as the probability of reaching the same set S by using the Markov chain for G` . To translate this result back to our actual network G (with weighted edges), we charge each round of push-pull in G` to ` rounds in G. With similar arguments, it follows that the Markov process of the informed node set given by considering ` consecutive rounds of push-pull in G at a time, dominates the one in G` . 6 The “multiplicity of an edge” is called “edge weight” in [5]. We use a different terminology here to avoid confusion with the latencies of edges and consider “edge weight” as a synonym to edge latency instead. 18 From [16] and [5] it is known that O(log(n)/φ(G` )) rounds suffice w.h.p. to solve broadcast in G` . Hence, achieving broadcast in G requires O(` log(n)/φ` (G)) rounds. Since the above analysis applies for any ` > 1, and in particular for the critical latency `∗ , the theorem follows. We combine Theorem 19 with Theorem 5 to obtain the following corollary that gives the upper bound on information dissemination using push-pull in terms of φavg . n Corollary 20. The push-pull protocol achieves broadcast w.h.p. in O( Lφlog ) rounds in a network G, where avg φavg is the average conductance of G and L is the number of non-empty latency classes in G. 4.2 An Õ(D + ∆) Algorithm In Section 5.1 we provide an algorithm that solves all-to-all information dissemination when each node knows the latencies of all its adjacent edges. The same algorithm can be naturally extended for the case where nodes do not know the adjacent latencies by first discovering the edge latencies and then running the algorithm as such. When both D and ∆ are known: for ∆ rounds, each node broadcasts a request to each neighbor (sequentially) and then waits up to D rounds for a response to determine the adjacent edge’s latency. If both or either values are unknown, the guess and double strategy (described in Section 5.3) can be used, as we can efficiently detect when information dissemination has completed correctly. By similar arguments as in Section 5.3 we obtain an algorithm that solves information dissemination in O((D + ∆) log3 n) time. 5 Algorithms for Known Latencies In this section, we discuss the case where each node knows the latencies of the adjacent edges. We focus on the problem of all-to-all information dissemination (instead of one-to-all information dissemination), as it will simplify certain issues to solve the seemingly harder problem. (Of course, all-to-all information dissemination also solves one-to-all information dissemination. And most one-to-all information dissemination algorithms can be used to solve all-to-all information dissemination by using them to collect and disseminate data.) In Section 5.1, we use the fact that nodes know a polynomial upper bound on the network size (and this is the only place where we rely on that assumption). When edge latencies are known, the spanner algorithm (described below) solves all-to-all information dissemination in O(D log3 n) which differs from the trivial lower bound of Ω(D) by only polylog factors. 5.1 Spanner Algorithm Preliminaries We initially assume that the weighted diameter (D) is known to all nodes; later (in Section 5.3), we do away with the assumption via a guess-and-double technique. It is assumed w.l.o.g. that every edge has latency 6 D: clearly we do not want to use any edges with latency > D. Local broadcast. An important building block of our algorithms is local broadcast. For unweighted graphs, the (randomized) Superstep algorithm by Censor-Hillel et al. [5] and the Deterministic Tree Gossip (DTG) algorithm by Haeupler [18] solve this problem. We make use of the DTG algorithm, which runs in O(log2 n) rounds on unweighted graphs. See [18] and Appendix A.1 for details. Observe that for the unweighted case, if any algorithm solves local broadcast in O(t) rounds, it obtains a t-spanner as a direct consequence, which thereafter can be used for propagating information. However, for graphs with latencies, just solving local broadcast might take O(D) time, resulting in a O(D)-spanner (and leading to an O(D2 ) solution for information dissemination). Recall that a subgraph S = (V, E 0 ) of a graph G = (V, E) is called an α-spanner if any two nodes u, v with distance ` in G have distance at most α` in S. 19 For weighted graphs, we are mainly interested in the `-local broadcast problem in which each node disseminates some information to all its neighbors that are connected to it by edges of latency 6 `. While DTG assumes edges to be unweighted (uniform weight), we can execute the same protocol in a graph with non-uniform latencies simply by ignoring all edges with a latency larger than ` and simulating 1 round of the DTG protocol as ` rounds in our network. We refer to this protocol as the `-DTG protocol. It follows immediately that within O(` log2 n) time, the `-DTG protocol ensures that each node has disseminated the information to all its neighbors connected to it with edges of latency 6 `. Note that we can trivially solve the all-to-all information dissemination problem in O(D2 log2 n) time using `-DTG protocol (if D were known) by simply repeating it D times with ` = D. The challenge now, given the restriction that finding neighbors by a direct edge might be costly, is to somehow find sufficiently short paths to all of them. We show here that with sufficient exploration of the local neighborhood up to O(log n) steps and using only favorable weights, we are able to obtain a global spanner. An intermediate goal of our algorithm is to construct an O(log n)-spanner and to obtain an orientation of the edges such that each node has a small, i.e., O(log n), out-degree.7 Once we have such a structure, we achieve all-to-all information dissemination by using a flooding algorithm that repeatedly activates the out-edges in round-robin order. 5.2 Spanner Construction and Broadcast In a seminal work, Baswana and Sen [2] provide a spanner construction algorithm for weighted graphs (where weights did not correspond to latency) in the LOCAL model of communication. As our goal here is to find a low stretch, low out-degree spanner, we modify the algorithm of [2] by carefully associating a direction with every edge that is added to a spanner such that each node has w.h.p. O(log n) out-degree. To deal with latencies, we choose to locally simulate the algorithm on individual nodes after obtaining the log n-hop neighborhood information by using the `-DTG protocol. We show that this log n-hop neighborhood information is sufficient for obtaining the required spanner. The algorithm in [2] also assumes distinct edge weights. We can ensure this by using the unique node IDs to break ties. We first show that the size of the obtained spanner does not increase significantly when running the algorithm of [2] with an estimate of n (namely n̂). 5.2.1 Spanner Construction Algorithm Each node v executes a set of rules for adding edges (explained below) and each time one of these rules is triggered, v adds some of its incident edges to the spanner while assigning them as outgoing direction. This way, we obtain a low stretch spanner (undirected stretch) where nodes also have a low out-degree, which we leverage in the subsequent phases of our algorithm. For a given parameter k, the algorithm computes a (2k − 1)-spanner by performing k iterations. At the beginning of the i-th iteration, for 1 6 i 6 k − 1, every node that was a cluster center in the previous iteration, chooses to become an active cluster with probability n̂−1/k , for some n 6 n̂ 6 poly(n); note that for i = 1, every node counts as a previously active center. Then, every active center c broadcasts this information to all cluster members. As a cluster grows by at most 1 hop in each round, this message needs to be disseminated throughout the i-neighborhood of c.8 Then, every cluster member broadcasts its membership information to all its neighbors to ensure that every node is aware of its adjacent active clusters. For adding edges to the spanner, nodes also remember its set of incident clusters Ci−1 that were active in iteration i − 1. With this information in hand, every node u adds some of its incident edges to its set of spanner edges Hu , and also (permanently) discards some edges, as follows: 7 8 It is clearly impossible to guarantee small degree in an undirected sense, for example, if the original graph is a star. By slight abuse of notation, we use c to denote cluster centres and the cluster itself when the distinction is clear from the context. 20 (Rule 1) If none of u’s adjacent clusters in Ci−1 were sampled in iteration i, then u adds its least weight edge to cluster c as an outgoing edge to Hu and discards all other edges to nodes in c, for every c ∈ Ci−1 . (Rule 2) If u has active adjacent clusters, then u will add the edge ev to some cluster c with the minimum weight among all these clusters and, for each adjacent cluster c0 ∈ Ci−1 that has a weight less than ev , node u also adds one outgoing edge to the respective node in c0 . All other edges from v to nodes in clusters c and c0 are discarded. In the k-th iteration, every vertex v adds the least weight edge to each adjacent cluster in Ck−1 to Hv . Lemma 21. Consider a synchronous network of n nodes where nodes know only n̂, where n 6 n̂ 6 nc , for some constant c > 1. For any k > c, there’s a distributed algorithm (based on [2]) that computes a spanner and terminates in O(k) rounds in the LOCAL model and each node’s out-degree is O(nc/k log n) w.h.p. Proof. Note that the running time of the algorithm is O(k 2 ) rounds if used with a restricted message size of O(log n). Inspecting the algorithm reveals that the computation at each node only depends on its k-hop neighborhood in the graph. Also, because the decision to remove an edge (u, v) can be taken by either node u or v, each node needs to simulate the running of the algorithm at all its neighbors (to know when to remove the edge (u, v) from consideration) and hence we can simulate the execution of the algorithm locally by first collecting this information regarding (k + 1)-hop neighborhood in k + 1 rounds in the LOCAL model. We now analyse the difference when running the algorithm with n̂ instead of n. First, we observe that sampling clusters with probability n̂(−1/k) does not affect the stretch guarantee. For the sake of our analysis we assume that the spanner is directed: we count every incident edge of v that it adds to its set of spanner edges Hv as an outgoing edge of v. The degree bound will follow by showing an upper bound on the number of outgoing edges of each node. Consider any iteration i in Phase 1 of the algorithm, i.e., 1 6 i < k. We call a cluster sampled in iteration i if it is among the sampled clusters in all iterations 1, . . . , i. Every cluster that was sampled in the previous iteration is sampled again with probability n̂−1/k . (In the very first iteration, every node counts as a previously sampled cluster.) To bound the number of edges that contribute to the out-degree of a node v, we consider the clusters adjacent to v that were sampled in iteration i − 1 and order them as c1 , . . . , cq in increasing order of the weight of their least weight edge incident to v. Let Ai be the event that v adds at least l edges to its outdegree in iteration i. Note that Ai occurs if and only if (1) none of the clusters c1 , . . . cl is sampled in iteration i and (2) there are at least l active clusters in iteration i − 1. By the description of Phase 1 (first k − 1 iterations) of the algorithm, we only add an edge from v to a node in cluster cj in iteration i if Ai does not happen. We have Pr[Ai ] 6 (1−n−c/k )l and taking a union bound over the first k − 1 iterations and over all n nodes, it follows that the probability of any node adding more than l edges to the spanner in any of the first k − 1 iterations is at most exp(−n−c/k l + log k + log n). By choosing l > Ω(n1/k (log n + log k)), this probability is 6 n−Ω(1) as required. In Phase 2 (final iteration), every vertex u adds a least weight (outgoing) edge to every cluster that was sampled in iteration k − 1. Let Xv be the indicator random variable that vertex v is the center of a cluster sampled in iteration k − 1 that is incident to u. We have Pr[Xv ] 6 n− Setting X = P v:(u,v)∈G Xv , c(k−1) k c = n−c+ k . it follows that c c E[X] 6 n1−c+ k 6 n k , since c > 1. Since each cluster is sampled independently all Xv are independent, we can apply a standard Chernoff bound to show that, for some sufficiently large constant c1 depending on c, it holds that h i c c/k Pr X > c1 n k log n 6 e−Θ(n log n) 6 n−Ω(1) . 21 By taking a union bound over all vertices, we can see the number of edges that each vertex adds to the spanner c in Phase 2 is at most O(n k log n) with high probability. Combining this with the bound that we have derived for Phase 1 completes the proof. Theorem 22. There is an O(D log3 n) time algorithm A in the gossip model that yields an O(log n)-spanner that has O(n log n) edges (w.h.p.). Moreover, A also computes an orientation of the edges that guarantees that each node has an out-degree of O(log n) (w.h.p.). Proof. To convert the classic synchronous algorithm for the local model assumed in Lemma 21 to an algorithm that works in the gossip model with latencies, we use the `-DTG protocol and simulate each of the k = log n iterations of the spanner algorithm by first discovering the log n-hop neighborhood. The neighborhood discovery takes O(D log3 n) rounds in our model and then all computations are done locally. To broadcast on this directed spanner we use the RR broadcast algorithm, which is a deterministic round-robin-style exchange of information among nodes. Each node sends all the rumors known to it to all its 1-hop neighbors one by one in a round robin fashion. The algorithm with a parameter k is run on the directed spanner of the graph Gk (G without edges of latency > k). RR Broadcast (k) 1: for each vertex v in parallel do 2: for iteration i equals 1 to (k∆out + k) do 3: propagate rumor set Rv along the out-edges of length 6 k one-by-one in a round robin fashion 4: add all received rumors to Rv Algorithm 1: RR Broadcast u ∆-1 edges k1 u1 ∆-2 edges u2 k2 ∆-2 edges kh v ∆-1 edges Figure 3: An example of message propagation from node u to node v. Lemma 23. After the execution of RR Broadcast algorithm with a parameter k on the directed spanner of graph Gk , any two nodes u and v at a distance 6 k in G have exchanged rumors with one another in O(k∆out + k) rounds, where ∆out is the maximum out-degree of any node in Gk . Proof. Consider a path from a node u to another node v at a distance k or less from it. Clearly, all edges in this path would have a weight of 6 k. Therefore, we can work on Gk (G without edges of latency > k) as well without affecting the correctness of the algorithm. Also, let us assume that the number of hops between u and v to be h which again would be 6 k, since there are no fractional weights. Let the latency between each hop be denoted by ki as shown in Figure 3. Messages reach the next node when either of the nodes initiate a bidirectional exchange. For example, u’s rumor could reach node u1 either by a request initiated by node u or by u1 , depending upon the direction of the edge uu1 . In the worst case nodes have to try all other ∆out − 1 links before initiating a connection along the required edge where ∆out is the maximum out-degree of any node. After a connection is initialized it takes k1 time to exchange rumors. By generalization, we 22 observe that in the non-blocking model, the delay that can be incurred before rumor exchange among any two adjacent nodes ui and ui−1 can be ∆out + ki in the worst case. In this way u’s rumor proceeds towards v in individual steps, each step incurring a maximum cost of ∆out + ki . A node might receive multiple rumors to propagate in the next round, which its adds to its rumor set and forwards to its neighbors in a round robin fashion. As such, the total worst case delay in rumor exchange among node u and v would be represented by h X (∆out + ki ) = h∆out + i=1 h X ki . i=1 Ph But we know that both h and i=1 ki can have a maximum value equal to k . Therefore, we conclude that for any two nodes v and u in Gk , v’s rumor would have reached u and u’s rumor would have reached v if all nodes forward rumors in a round robin fashion for (k∆out + k) rounds. Here, on the created spanner with stretch of O(log n), the maximum distance between any two nodes can be O(D log n). Since the maximum out-degree (∆out ) is O(log n) w.h.p., we get the following corollary. Corollary 24. The RR broadcast algorithm on the constructed spanner takes O(D log2 n) time and solves all-to-all information dissemination w.h.p. We combine all the previously defined techniques to a single algorithm called Efficient Information Dissemination or EID. EID (D) 1: for each vertex v in parallel do 2: for iteration i = 1 to O(log n) do 3: Perform D-DTG 4: call Spanner Construction algorithm 5: call algorithm RR Broadcast (O(D log n)) /* to gain neighborhood information / / executed locally */ Algorithm 2: Efficient information dissemination Lemma 25. For a graph G with diameter D, Efficient Information Dissemination (EID) algorithm takes O(D log3 n) time for solving all-to-all information dissemination w.h.p. when D is known to all the nodes. 5.3 Unknown Diameter For unknown diameter, we apply the standard guess-and-double strategy: begin with an initial guess of 1 for D. Try the algorithm and see if it succeeds. If so, we terminate. Otherwise, double the estimate and repeat. The challenge here is to correctly determine the termination condition i.e. how does a particular node determine whether information dissemination has been achieved for all other nodes. Early termination might lead to partial dissemination whereas late termination might cause the time complexity to increase. The critical observation is as follows: if two nodes u and v cannot communicate in one execution of all-to-all information dissemination (protocol RR Broadcast) for a given estimate of the diameter, then there must be some edge (w, z) on the path from u to v where, in one execution: u is able to communicate with w but not with z. There are two cases: If w is not able to communicate with z, then it is aware that it has an unreachable neighbor and can flag the issue; the next time that u and w communicate, node u learns of the problem. Otherwise, if w can communicate with z, then the next time that u and w communicate, node u learns that there was a node it did not hear from previously. In either case, u knows that the estimate of D was not correct and should continue. Each node also checks whether it has heard from all of its neighbors, 23 and raises an error flag if not. We then repeat all-to-all broadcast so that nodes can check if everyone has the same “rumor set” and that no one has raised an error flag. In total, checking termination has asymptotic complexity of O(D log2 n). The Termination_Check algorithm checks for every node that v contacts or is contacted by (either directly or indirectly) whether that node has (i) exactly the same rumor set as v and (ii) the value 0 as its flag bit. The flag bit of a node is set to 1 if a neighbor of that node is not present in its rumor set or if the node has not yet exchanged all the rumors known to it presently with all of its neighbors in G that are at a distance 6 to the current estimate of D (say k): this condition is easily checked by either doing an additional k-DTG (which does not affect the complexity) or can be checked in parallel with the execution of RR Broadcast. If both of the above conditions are not met, then node v sets its status to “failed” and v uses a broadcast algorithm for propagating the “failed” message. Any broadcast algorithm that, given a parameter k, is able to broadcast and collect back information from all nodes at a distance 6 k from v, can be used. It is easily seen that RR Broadcast satisfies this criteria and can be used in this case. Note that broadcast is achieved here (for General_EID algorithm) by execution of RR Broadcast, however when Path_Discovery algorithm (described later) invokes Termination_Check, broadcast is achieved by execution of the sequence T (k) (also described later). Here, the rumor set known to a particular vertex v is denoted by Rv , Γ(v) represents all its neighbors in G whereas k-neighbors refers to only those nodes that are connected with v with an edge of latency k or less. Also, initially node_status of all nodes is set to “default”. Termination_Check (k) 1: if (node w ∈ Γ(v) and w ∈ / Rv ) or (node v has not exchanged rumors with all k-neighbors) then 2: set flag bit, vf lag = 1 3: else set flag bit, vf lag = 0 4: 5: 6: 7: 8: 9: broadcast and gather all responses from any node u in v’s k-distance neighborhood if ∃ any u such that (Rv 6= Ru ) or (uf lag = 1) then set node_status = “failed” broadcast “failed” message to the k-distance neighborhood if received message = “failed” then set node_status = “failed” Algorithm 3: Termination_Check We prove the following regarding the termination detection: Lemma 26. No node terminates until it has exchanged rumors with all other nodes. Moreover, all nodes terminate in the exact same round. Proof. Suppose that a node v terminates without having exchanged rumors with some other node w. Considering any path from node v to node w, let u be the farthest node (in hop distance) with which v has exchanged rumors with and let x be the next node in the path. Case 1 : u has exchanged rumors with x. It implies that v has also exchanged rumors with x, from the condition that all nodes that exchange rumors with one another have the same rumor set. Thus, contradicting the fact that u is the farthest node on the path that v has exchanged rumors with. Case 2 : u has not exchanged rumors with x. If u had not exchanged rumors with x, then u would have set its flag bit as 1, which would have been detected by v during the broadcast and it would not have terminated. This also gives us a contradiction. Thus, no such node w exists and v terminates only after it has exchanged rumors with all the other nodes. For the second part of the proof, let consider u and v to be nodes such that v is set for termination and has not set its status to “failed” in the Termination_Check algorithm, whereas, in the same iteration, node u 24 has set its status to “failed” and hence is set to continue. We show that there cannot be two such nodes in the same round. The node v did not set its status to “failed” implying all the nodes that it exchanged rumors with had exactly the same set of rumors, none of the nodes had set its flag bit as 1 and in addition it did not receive a “failed” message from any other node. From the first part, we know that the set of nodes that v exchanged rumors with is the entire vertex set of the graph G. That implies, v has also exchanged rumors with u: node u also has the exact set of rumors (which essentially is all the rumors from all the nodes) and does not have a set flag bit. So in the current iteration, if any other node broadcasted a “failed” message both v and u would have received it resulting in both nodes to set their status as “failed”. Again, since the rumor sets of both nodes are identical, both nodes would observe the same flag bits of all the nodes. Then node u will also not satisfy the termination condition and will not set its status as “failed”. This gives us a contradiction that completes the proof. General_EID (k) k=1 2: repeat 3: call algorithm EID (k) 4: call algorithm Termination_Check (k) 5: if node_status = “failed” then 6: k = 2k 7: set node_status to “default” 8: else terminate 1: Algorithm 4: General_EID; code for vertex v. Combining the all-to-all dissemination protocol with the termination detection, we get the following: Theorem 27. There exists a randomized gossip algorithm that solves the all-to-all information dissemination problem w.h.p. and terminates in O(D log3 n) rounds. 5.4 An Alternative All-to-All Information Dissemination Algorithm We propose an alternate algorithm to solve all-to-all information dissemination without any global knowledge (polynomial upper bound of n need not be known) that takes O(D log2 n log D) time. This algorithm works even when nodes cannot initiate a new exchange in every round, and wait till the acknowledgement of the previous message, i.e., communication is blocking. The algorithm involves repeatedly invoking the `-DTG algorithm with different parameters determined by a particular pattern. The intuition behind the choice of the pattern is to make minimal use of the heavier latency edges by collecting as much information as possible near the heavier latencies before making use of that edge. The pattern for k is derived according to a sequence T (k) that is recursively defined as follows: T (1) = 1-DTG T (2) = T (1) · 2-DTG · T (1) T (4) = T (2) · 4-DTG · T (2) .. . T (k) = T (k/2) · k-DTG · T (k/2) We show that, when the above sequence is run for the particular pattern for length k, it guarantees that any node u and v in the graph G, at a distance of 6 k, have exchanged their rumors with one another. Overall, 25 the pattern of values of the parameter ` is 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, . . . , k, . . . , 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, and, for each value `, we perform the `-DTG protocol. That is, T (k) is a sequence of calls to `-DTG with varying parameters according to a known pattern. Lemma 28. After the execution of T (k), any node in the weighted graph G (V,E) has exchanged rumors with all other nodes that are at distance k or less from it. Proof. We proceed by induction over the path length k. For the base case, recall from [18] that, after running T (1) on G1 , i.e., the subgraph of G induced by edges with latency 6 1, any node v has exchanged rumors with all its distance 1 neighbors. For the inductive step, suppose that the claim is true for T (k), i.e. after running the sequence, any node v has exchanged rumors with all other nodes at a weighted distance 6 k. To prove the claim for T (2k) (i.e. T (k) · 2k-DTG · T (k)), we consider the various possibilities of forming a path of length 2k. Case 1: The path consists only of edges with latencies 6 k. Here we distinguish two sub-cases: Case 1a: There exists a node m which is equidistant from both end points u and v (see Figure 4). By the induction hypothesis, both nodes u and v would have exchanged rumors with node m in the initial T (k). In the next T (k), node m propagates all rumors that it received from u to v and vice-versa. path of length k path of length k u m v Figure 4: Case 1a Case 1b: No such node middle exists as depicted in Figure 5. Then, after the initial T (k), node u must have exchanged rumors with m1 and node v with m2 , due to the induction hypothesis. In the invocation of the 2k-DTG, node m1 propagates all rumors gained from u to m2 , and m2 also propagates all rumors gained from v to m1 . This information then travels from m1 to u and from m2 to v in the final T (k). path of length k or less u edge of length k-1 or less m1 path of length k or less m2 v Figure 5: Case 1b Case 2: There exists at most one edge e with latency value in between [k + 1, 2k]. This situation can yield one of the following two sub-cases: Case 2a: Edge e is located at one end of the path (see Figure 6). By the induction hypothesis, node v would have exchanged rumors with m in the initial T (k). In the 2k-DTG, u gets to know this (and other) rumors from m and m also gets to know u’s rumors. In the next T (k), node m propagates all rumors gained from u to v. path of length k-1 or less edge of length [k +1, 2k] u m Figure 6: Case 2a 26 v Case 2b: The edge is located between two inner nodes on the path (see Figure 7). In this case, by the induction hypothesis, node u has exchanged rumors with m1 , whereas node v has exchanged rumors with node m2 in the initial T (k). In the 2k-DTG, node m1 propagates all rumors gained from u to m2 . Moreover, m2 propagates all rumors gained from v to m1 . These rumors then propagate from m1 to u and from m2 to v in the final T (k). path of length k-1 or less u path of length k-1 or less m1 m2 v Figure 7: Case 2b Lemma 29. For known diameter, solving all-to-all information dissemination by executing the sequence T (D), takes O(D log2 n log D) time. Proof. From the way the sequence is constructed, we observe the recurrence relation T (k) = 2T (k/2) + k log2 n. Using standard methods to solve the recurrence completes the proof. When the graph diameter is known to all nodes, nodes can just invoke T (D) to solve all-to-all information dissemination. For completeness, we also present an algorithm called Path_Discovery that uses the sequence of invocations of `-DTG to solve all-to-all information dissemination, when the graph diameter is unknown. This algorithm is similar in flavour to that of the General_EID algorithm described in Section 5.3 and also makes use of the Termination_Check algorithm, albeit with a different broadcasting technique (calling T (k) rather than RR Broadcast). Path_Discovery (k) 1: k=1 2: repeat 3: execute sequence T (k) 4: call algorithm Termination_Check (k) 5: if node_status = “failed” then 6: k = 2k 7: set node_status to “default” 8: else terminate Algorithm 5: Path_Discovery; code for vertex v. Lemma 30. Path_Discovery algorithm takes O(D log2 n log D) time to solve all-to-all information dissemination. Applying techniques similar to section 5.3, the complexity can be easily shown for the case with unknown diameter as well. 6 Unified Upper Bounds Combining the results, we can run both push-pull and the spanner algorithm in parallel to obtain unified upper bounds for both the known and the unknown latencies cases. However, we point out that, for single source broadcast, push-pull works with small message sizes whereas the spanner algorithm does not (because of its 27 reliance on DTG). Also, exchanging messages with the help of the spanner does not have good robustness properties whereas push-pull is inherently quite robust. Theorem 31. There exists randomized gossip algorithms that solves the all-to-all information dissemination problem in O(min((D+∆) log3 n, (`∗ /φ∗ ) log n) time when latencies are unknown and in O(min(D log3 n, (`∗ /φ∗ ) log n)) time when latencies are known. Corollary 32. There exists randomized gossip algorithms that solves the all-to-all information dissemination problem in O(min((D + ∆) log3 n, (L/φavg ) log n) time when latencies are unknown and in O(min(D log3 n, (L/φavg ) log n)) time when latencies are known. 7 Conclusion We have presented two different new concepts, namely the critical conductance and the average conductance, that characterize the bottlenecks in communication for weighted graphs. We believe that these parameters will be useful for a variety of applications that depend on connectivity. A question that remains is whether the running time of O(D log3 n) for information dissemination can be improved, e.g., using better spanner constructions or more efficient local broadcast to save the polylogarithmic factors. (Recall that in the unweighted case, there are information dissemination protocols that run in O(D + polylogn) time.) Another interesting direction would be the development of reliable robust fault-tolerant algorithms in this regard. Another issue is whether we can reduce the number of incoming messages in a round; recently, Daum et al. [9] have considered such a more restricted model, yielding interesting results. It would also be interesting to look at the bounds where each node is only allowed O(1) connections per round, whether initiated by the node itself or by its neighbor. Acknowledgment We thank George Giakkoupis for the helpful conversations and useful ideas. A A.1 Appendix The DTG Local Broadcast Protocol In this section, we describe in more detail the DTG protocol that was originally developed in [18] as well as the `-DTG algorithm. It is clear that the algorithm solves local broadcast because it keeps on contacting new neighbors until it has exchanged rumors with all of its neighbors. The author [18] makes use of binomial trees to derive the time complexity and better explain the working of the algorithm. The key idea used for deriving the time complexity is to show that when information is propagated in a pipelined manner along the binomial trees (created on-the-fly), then for any node that is still active in the ith iteration, it has a binomial tree of order 2i (i-tree of depth i: see Figure 8) rooted at it. Furthermore, it is shown that for any two different nodes that are still active in iteration i, their i-trees are vertex disjoint. Since an i-tree is formed by joining two (i − 1)-trees, the growth rate of an i-tree is exponential which limits the number of iterations to O(log n). Also, each node on an average needs to contact O(log n) nodes (O(i) nodes in the ith round). Thus, the overall complexity of the algorithm becomes O(log2 n). In our case, for `-DTG, the additional waiting time of ` increases the time complexity to O(` log2 n). 28 0-tree 1-tree 2-tree 3-tree Figure 8: i-trees for i ∈ 0, 1, 2, 3 The i-tree can be seen as witness structures that provides an explanation as to why a node was active in that particular iteration. The i-tree rooted at a particular node is built recursively as the rounds progress and essentially store the information about which other nodes communicated with one another in which particular round as viewed from the root node. For example, in Figure 9 , the labels on the edges denote the time in which the node of the higher level contacted the lower level node (as observed by the root node). The root contacts the nodes in first level in rounds according to their label, the nodes on the first level similarly contact the nodes in the second level in rounds according to their label and so on. This observation also helps in the realization of the key idea of a node being active in the ith round having an i-tree rooted at it. The nodes in the first level did not contact the root previously as they were busy contacting the nodes of the second level, the nodes of the second level did not contact nodes on the first level as they were busy contacting the nodes in the third level and so on. Figure 9: 5-tree with edge labels As shown in the pseudo code, in the initial PUSH sequence, the message is propagated in a decreasing order of connection round number (as observed by the root node: given by the labels on the edges of Figure 8), helping in pipe-lining the roots message to all other nodes of the i-tree. Similarly, during the initial PULL sequence the message from the nodes is pipelined up to the root. The subsequent PULL-PUSH sequence helps in maintaining the symmetry of the algorithm such that if node u learns about node v, then node v also learns about node u. Finally, the collection of rumors R is updated to the union of rumors collected in the aforementioned sequences. For ` being an integer > 1, we run the modified DTG algorithm on a sub-graph of G, G` , rather than on G, where G` contains only the edges of length up to `. Lets denote this algorithm as `-DTG. The algorithm is presented below and each node v belonging to G` runs it in parallel. Γ(v) can be considered as the neighborhood of v comprising of set of nodes that are node v’s 1-hop neighbors. 29 `-DTG (`) 1: R = v 2: for i = 1 UNTIL Γ(v)\R = φ do 3: link to any new neighbor ui ∈ Γ(v) 4: R0 = v 5: PUSH : 6: for j = i downto 1 do 7: send rumors in R0 to uj 8: wait for ` time to receive uj ’s rumors 9: add all received rumors to R0 10: PULL: 11: for j = 1 to i do 12: send rumors in R0 to uj 13: wait for ` time to receive uj ’s rumors 14: add all received rumors to R0 15: R00 = v 16: perform PULL, PUSH with R00 17: R = R0 ∪ R00 Algorithm 6: `-DTG References [1] John Augustine, Gopal Pandurangan, Peter Robinson, Scott Roche, and Eli Upfal. Enabling robust and efficient distributed computation in dynamic peer-to-peer networks. In IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS Berkeley, USA, pages 350–369, 2015. [2] Surender Baswana and Sandeep Sen. A simple and linear time randomized algorithm for computing sparse spanners in weighted graphs. 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Cooperative control of multi-agent systems to locate source of an odor arXiv:1711.03819v1 [cs.SY] 10 Nov 2017 Abhinav Sinha, Rishemjit Kaur, Ritesh Kumar and Amol P. Bhondekar Abstract—This work targets the problem of odor source localization by multi-agent systems. A hierarchical cooperative control has been put forward to solve the problem of locating source of an odor by driving the agents in consensus when at least one agent obtains information about location of the source. Synthesis of the proposed controller has been carried out in a hierarchical manner of group decision making, path planning and control. Decision making utilizes information of the agents using conventional Particle Swarm Algorithm and information of the movement of filaments to predict the location of the odor source. The predicted source location in the decision level is then utilized to map a trajectory and pass that information to the control level. The distributed control layer uses sliding mode controllers known for their inherent robustness and the ability to reject matched disturbances completely. Two cases of movement of agents towards the source, i.e., under consensus and formation have been discussed herein. Finally, numerical simulations demonstrate the efficacy of the proposed hierarchical distributed control. Index Terms—Odor source localization, multi-agent systems (MAS), sliding mode control (SMC), homogeneous agents, cooperative control. I. I NTRODUCTION A. Overview Inspiration of odor source localization problem stems from behavior of biological entities such as mate seeking by moths, foraging by lobsters, prey tracking by mosquitoes and blue crabs, etc., and is aimed at locating the source of a volatile chemical. These behaviors have long been mimicked by autonomous robot(s). Chemical source tracking has attracted researchers around the globe due to its applications in both civilian and military domains. A plethora of applications are possible, some of which include detection of forest fire, oil spills, release of toxic gases in tunnels and mines, gas leaks in industrial setup, search and rescue of victims and clearing leftover mine after an armed conflict. A plume containing filaments, or odor molecules, is generally referred to the downwind trail formed as a consequence of mixing of contaminant molecules in any kind of movement of air. The dynamical optimization problem of odor source localization can be effectively solved using multiple robots working in cooperation. The obvious advantages of leveraging multiagent systems (MAS) are increased probability of success, A. Sinha is with School of Mechatronics & Robotics, Indian Institute of Engineering Science and Technology; and Central Scientific Instruments Organization (CSIR- CSIO), India. email: [email protected] R. Kaur, R. Kumar & A. P. Bhondekar are with CSIR- CSIO. emails: [email protected],[email protected], [email protected] redundancy and improved overall operational efficiency and spatial diversity in having distributed sensing and actuation. B. Motivation Odor source localization is a three stage problem– sensing, maneuvering and control. Some of reported literature on odor source localization date back to 1980s when Larcombe et al. [1] discussed such applications in nuclear industry by considering a chemical gradient based approach. Other works in 1990s [2]–[6] relied heavily on sensing part using techniques such as chemotaxis [7], infotaxis [8], anemotaxis [9], [10] and fluxotaxis [11]. The efficiency of such algorithms was limited by the quality of sensors and the manner in which they were used. These techniques also failed to consider turbulence dominated flow and resulted in poor tracking performance. Bio-inspired algorithms have been reported to maneuver the agents, some of which include Braitenberg style [12], E. coli algorithm [13], Zigzag dung beetle approach [14], silkworm moth style [15]–[17] and their variants. A tremendous growth of research attention towards cooperative control has been witnessed in the past decade [18], [19] but very few have addressed the problem of locating source of an odor. Hayes et al. [20] proposed a distributed cooperative algorithm based on swarm intelligence for odor source localization and experimental results proved multiple robots perform more efficiently than a single autonomous robot. A Particle Swarm Optimization (PSO) algorithm [21] was proposed by Marques et al. [22], [23] to tackle odor source localization problems. To avoid trapping into local maximum concentrations, Jatmiko et al. [23] proposed modified PSO algorithms based on electrical charge theory, where neutral and charged robots has been used. Lu et al. [24] proposed a distributed coordination control protocol based on PSO to address the problem. It should be noted that simplified PSO controllers are a type of proportional-only controller and the operating region gets limited between global and local best. This needs complicated obstacle avoidance algorithms and results in high energy expenditure. Lu et al. [25] also proposed a cooperative control scheme to coordinate multiple robots to locate odor source in which a particle filter has been used to estimate the location of odor source based on wind information, a movement trajectory has been planned, and finally a cooperative control scheme has been proposed to coordinate movement of robots towards the source. Motivated by these studies, we have implemented a robust and powerful hierarchical cooperative control strategy to tackle the problem. First layer is the group level in which the information about the source via instantaneous sensing and swarm intelligence is obtained. Second layer is designed to maneuver the agents via a simplified silkworm moth algorithm. Third layer is based on cooperative sliding mode control and the information obtained in the first layer is passed to the third layer as a reference to the tracking controller. C. Contributions Major contributions of this paper are summarized below. 1) As opposed to existing works on cooperative control to locate source of odor, we have considered a more general formulation by taking nonlinear dynamics of MAS into account. When the uncertain function is zero, the problem reduces to stabilizing integrator dynamics. 2) The control layer is designed on the paradigms of sliding mode, a robust and powerful control with inherent robustness and disturbance rejection capabilities. The reaching law, as well as the sliding manifold in this study are nonlinear and novel resulting in smoother control and faster reachability to the manifold. Use of sliding mode controller also helps in achieving a finite time convergence as opposed to asymptotic convergence to the equilibrium point. The proposed control provides stability and ensures robustness even in the presence of bounded disturbances and matched uncertainties. 3) Odor propagation is non-trivial, i.e., odor arrives in packets, leading to wide fluctuations in measured concentrations. Plumes are also dynamic and turbulent. As odor tends to travel downwind, direction of the wind provides an effective information on relative position of the source. Hence, we have used wind information based on a measurement model describing movement of filaments and concentration information from swarm intelligence to locate the source of odor. 4) Formation keeping of agents to locate source of odor has also been demonstrated in this work. D. Paper Organization After introduction to the study in section I, remainder of this work in organized as follows. Section II provides insights into preliminaries of spectral graph theory and sliding mode control. Section III presents dynamics of MAS and mathematical problem formulation, followed by hierarchical distributed cooperative control scheme in section IV. Results and discussions have been carried out in section V, followed by concluding remarks in section VI. II. P RELIMINARIES A. Spectral Graph Theory for Multi-Agent Systems A directed graph, also known as digraph is represented throughout in this paper by G = (V, E, A). V is the nonempty set in which finite number of vertices or nodes are contained such that V = {1, 2, ..., N }. E denotes directed edge and is represented as E = {(i, j) ∀ i, j ∈ V & i 6= j}. A is the weighted adjacency matrix such that A = a(i, j) ∈ RN×N . The possibility of existence of an edge (i, j) occurs iff the vertex i receives the information supplied by the vertex j, i.e., (i, j) ∈ E. Hence, i and j are termed neighbours. The set Ni contains labels of vertices that are neighbours of the vertex i. For the adjacency matrix A, a(i, j) ∈ R+ 0 . If (i, j) ∈ E ⇒ a(i, j) > 0. If (i, j) ∈ / E or i = j ⇒ a(i, j) = 0. The Laplacian matrix L [26] is central to the consensus problem and is given by L = D − A where degree matrix, D is a diagonal matrix, Pn i.e, D = diag(d1 , d2 , ..., dn ) whose entries are di = j=1 a(i, j). A directed path from vertex j to vertex i defines a sequence comprising of edges (i, i1 ), (i1 , i2 ), ..., (il , j) with distinct vertices ik ∈ V, k = 1, 2, 3, ..., l. Incidence matrix B is also a diagonal matrix with entries 1 or 0. The entry is 1 if there exists an edge between leader agent and any other agent, otherwise it is 0. Furthermore, it can be inferred that the path between two distinct vertices is not uniquely determined. However, if a distinct node in V contains directed path to every other distinct node in V, then the directed graph G is said to have a spanning tree. Consequently,the matrix L + B has full rank [26]. Physically, each agent has been modelled by a vertex or node and the line of communication between any two agents has been modelled as a directed edge. B. Sliding Mode Control Sliding Mode Control (SMC) [27] is known for its inherent robustness. The switching nature of the control is used to nullify bounded disturbances and matched uncertainties. Switching happens about a hypergeometric manifold in state space known as sliding manifold, surface, or hyperplane. The control drives the system monotonically towards the sliding surface, i.e, trajectories emanate and move towards the hyperplane (reaching phase). System trajectories, after reaching the hyperplane, get constrained there for all future time (sliding phase), thereby ensuring the system dynamics remains independent of bounded disturbances and matched uncertainties. In order to push state trajectories onto the surface s(x), a proper discontinuous control effort uSM (t, x) needs to be synthesized satisfying the following inequality. sT (x)ṡ(x) ≤ −ηks(x)k, (1) with η being positive and is referred as the reachability constant. ∂s ∂s ẋ = f (t, x, uSM ) (2) ∵ ṡ(x) = ∂x ∂x ∂s ∴ sT (x) f (t, x, uSM ) ≤ −ηks(x)k. (3) ∂x The motion of state trajectories confined on the manifold is known as sliding. Sliding mode exists if the state velocity vectors are directed towards the manifold in its neighbourhood. Under such consideration, the manifold is called attractive, i.e., trajectories starting on it remain there for all future time and trajectories starting outside it tend to it in an asymptotic manner. Hence, in sliding motion, ∂s f (t, x, uSM ) = 0. (4) ∂x uSM = ueq is a solution, generally referred as equivalent control is not the actual control applied to the system but can be thought of as a control that must be applied on an average to maintain sliding motion and is mainly used for analysis of sliding motion. ṡ(x) = III. DYNAMICS OF M ULTI -AGENT S YSTEMS & P ROBLEM F ORMULATION Consider first order homogeneous MAS interacting among themselves and their environment in a directed topology. Under such interconnection, information about the predicted location of source of the odor through instantaneous plume sensing is not available globally. However, local information is obtained by communication among agents whenever at least one agent attains some information of interest. The governing dynamics of first order homogeneous MAS consisting of N agents is described by nonlinear differential equations as ẋi (t) = f (xi (t)) + uSMi (t) + ςi ; i ∈ [1, N ], (5) where f (·) : R+ × X → Rm is assumed to be locally Lipschitz over some fairly large domain DL with Lipschitz constant L̄, and denotes the uncertain nonlinear dynamics of each agent. Also X ⊂ Rm is a domain in which origin is contained. xi and uSMi are the state of ith agent and the associated control respectively. ςi represents bounded exogenous disturbances that enter the system from input channel, i.e., kςi k ≤ ςmax < ∞. The problem of odor source localization can be viewed as a cooperative control problem in which control laws uSMi need to be designed such that the conditions limt→∞ kxi − xj k = 0 and limt→∞ kxi −xs k ≤ θ are satisfied. Here xs represents the probable location of odor source & θ is an accuracy parameter. IV. H IERARCHICAL D ISTRIBUTED C OOPERATIVE C ONTROL S CHEME In order to drive the agents towards consensus to locate the source of odor, we propose the following hierarchy. A. Group Decision Making This layer utilizes both concentration and wind information to predict the location of odor source. Then, the final probable position of the source can be described as ψ(tk ) = c1 pi (tk ) + (1 − c1 )qi (tk ), (6) with pi (tk ) as the oscillation centre according to a simple Particle Swarm Optimization (PSO) algorithm and qi (tk ) captures the information of the wind. c1 ∈ (0, 1) denotes additional weighting coefficient. Remark 1. The arguments in (6) represent data captured at t = tk instants (k = 1, 2, ...) as the sensors equipped with the agents can only receive data at discrete instants. It should be noted that ψ is the tracking reference that is fed to the controller. Now, we present detailed description of obtaining pi (tk ) and qi (tk ). Simple PSO algorithm that is commonly used in practice has the following form. vi (tk+1 ) = ωvi (tk ) + uPSO (tk ), (7) xi (tk+1 ) = xi (tk ) + vi (tk+1 ). (8) Here ω is the inertia factor, vi (tk ) and xi (tk ) represent the respective velocity and position of ith agent. This commonly used form of PSO can also be used as a proportional-only type controller, however for the disadvantages mentioned earlier, we do not use PSO as our final controller. PSO control law uPSO can be described as uPSO = α1 (xl (tk ) − xi (tk )) + α2 (xg (tk ) − xi (tk )). (9) In (9), xl (tk ) denotes the previous best position and xg (tk ) denotes the global best position of neighbours of ith agent at time t = tk , and α1 & α2 are acceleration coefficients. Since, every agent in MAS can get some information about the magnitude of concentration via local communication, position of the agent with a global best can be easily known. By the idea of PSO, we can compute the oscillation centre pi (tk ) as pi (tk ) = α1 xl (tk ) + α2 xg (tk ) , α1 + α2 (10) where xl (tk ) = arg xg (tk ) = arg max {g(xl (tk−1 )), g(xi (tk ))}, 0<t<tk−1 (11) max {g(xg (tk−1 )), max aij g(xj (tk ))}. 0<t<tk−1 j∈N (12) Thus, from (9), (10) uPSO (tk ) = (α1 + α2 ){pi (tk ) − xi (tk )}, (13) which is clearly a proportional-only controller with proportional gain α1 + α2 , as highlighted earlier. In order to compute qi (tk ), movement process of a single filament that consists several order molecules has been modelled. If xf (t) denotes position of the filament at time t, v̄a (t) represent mean airflow velocity and n(t) be some random process, then the model can be described as ẋf (t) = v̄a (t) + n(t). (14) Without loss of generality, we shall regard the start time of our experiment as t = 0. From (14), we have Z t Z t xf (t) = v̄a (τ )dτ + n(τ )dτ + xs (0). (15) 0 0 xs (0) denotes the real position of the odor source at t = 0. Assumption IV.1. We assume the presence of a single, stationary odor source. Thus, xs (t) = xs (0). Implications from remark 1 require (15) to be implemented at t = tk instants. Hence, xf (tk ) = t X v̄a (τm )∆t + m=0 t X n(τm )∆t + xs (tk ), m=0 ? xf (tk ) = xs (tk ) + v̄a? (tk ) + w (tk ). In (17), w? (tk ). Pt m=0 v̄a (τm )∆t = v̄a? (tk ) and (16) (17) Pt m=0 n(τm )∆t = Remark 2. In (17), the accumulated average of v̄a? (tk ) and w? (tk ) can also be considered ∀ possible filament releasing time. From (17), xf (tk ) − v̄a? (tk ) = xs (tk ) + w? (tk ). (18) C. Distributed Control In the control layer, we design a robust and powerful controller on the paradigms of sliding mode. It is worthy to mention that based on instantaneous sensing and swarm information, at different times, each agent can take up the role of a virtual leader whose opinion needs to be kept by other agents. ψ from (6) has been provided to the controller as the reference to be tracked. The tracking error is formulated as ei (t) = xi (t) − ψ(tk ) ; t ∈ [tk , tk+1 [. (23) In terms of graph theory, we can reformulate the error variable as i (t) = (L + B)ei (t) = (L + B)(xi (t) − ψ(tk )). (24) The above relationship, (18) can be viewed as the information about xs (tk ) with some noise w? (tk ). Hence, From this point onward, we shall denote L + B as H. Next, we formulate the sliding manifold qi (tk ) = xs (tk ) + w? (tk ). si (t) = λ1 tanh(λ2 i (t)), (19) Therefore, ψ in (6) can now be constructed from (10) & (19). B. Path Planning Since, detection of information of interest is tied to the threshold value defined for the sensors, the next state is updated taking this threshold value into account. Thus, the blueprints of path planning can be described in terms of three types of behavior. 1) Surging: If the ith agent receives data well above threshold, we say that some clues about the location of the source has been detected. If the predicted position of the source at t = tk as seen by ith agent be given as xsi (tk ), then the next state of the agent is given mathematically as xi (tk+1 ) = xsi (tk ). (20) 2) Casting: If the ith agent fails to detect information at any particular instant, then the next state is obtained using the following relation. xi (tk+1 ) = kxi (tk ) − xsi (tk )k + xsi (tk ). 2 which is a nonlinear sliding manifold offering faster reachability to the surface. λ1 ∈ R+ represents the speed of convergence to the surface, and λ2 ∈ R+ denotes the slope of the nonlinear sliding manifold. These are coefficient weighting parameters that affect the system performance. The forcing function has been taken as ṡi (t) = −µ sinh−1 (m + w\|si (t)\|)sign(si (t)). (26) In (26), m is a small offset such that the argument of sinh−1 function remains non zero and w is the gain of the controller. The parameter µ facilitates additional gain tuning. In general, m << w. This novel reaching law contains a nonlinear gain and provides faster convergence towards the manifold. Moreover, this reaching law is smooth and chattering free, which is highly desirable in mechatronic systems to ensure safe operation. Theorem IV.1. Given the dynamics of MAS (5) connected in a directed topology, error candidates (23, 24) and the sliding manifold (25), the stabilizing control law that ensures accurate reference tracking under consensus can be described as (21) uSMi (t) = − (ΛH)−1 µ sinh−1 (m + w\|si (t)\|)sign(si (t))Γ−1 3) Search and exploration: If all the agents fail to detect odor clues for a time segment [tk , tk+l ] > δ0 for some l ∈ N and δ0 ∈ R+ being the time interval for which no clues are detected or some constraint on wait time placed at the start of the experiment, then the next state is updated as xi (tk+1 ) = xsi (tk ) + zφσ . (25) (22) In (22), zφσ is some random parameter with σ as its standard deviation and φ as its mean. + (f (xi (t)) − ψ̇(tk )) (27) where Λ = λ1 λ2 , Γ = 1 − tanh2 (λ2 i (t)), w > supt≥0 {kςi k} & µ > sup{kΛHςi Γk}. Remark 3. As mentioned earlier, λ1 , λ2 ∈ R+ . This ensures Λ 6= 0 and hence its non singularity. The argument of tanh is always finite and satisfies λ2 i (t) 6= πι(κ + 1/2) for κ ∈ Z, thus Γ is also invertible. Moreover the non singularity of H can be established directly if the digraph contains a spanning tree with leader agent as a root. Proof. From (24) and (25), we can write ṡi (t) = λ1 {λ2 ˙i (t)(1 − tanh2 (λ2 i (t)))} (28) 2 = λ1 λ2 ˙i (t) − λ1 λ2 ˙i (t) tanh (λ2 i (t)) (29) 2 = λ1 λ2 ˙i (t){1 − tanh (λ2 i (t))} (30) = ΛH(ẋi (t) − ψ̇(tk ))Γ (31) with Λ & Γ as defined in Theorem IV.1. From (5), (31) can be further simplified as ṡi (t) = ΛH(f (xi (t)) + uSMi (t) + ςi − ψ̇(tk ))Γ. (32) Using (26), the control that brings the state trajectories on to the sliding manifold can now be written as uSMi (t) = − (ΛH)−1 µ sinh−1 (m + w\|si (t)\|)sign(si (t))Γ−1 + (f (xi (t)) − ψ̇(tk )) . (33) Thus, the derivative of Lyapunov function candidate is negative definite confirming stability in the sense of Lyapunov. Since, µ > 0, ksi k > 0 and sinh−1 (·) > 0 due to the nature of its arguments. Therefore, (37) and (26) together provide implications that ∀si (0), si ṡi < 0 and the surface is globally attractive. This ends the proof. V. R ESULTS AND DISCUSSIONS Interaction topology of the agents represented as a digraph has been shown here in figure 1. The associated graph matrices have been described below. The computer simulation has been performed assuming that agent 1 appears as virtual leader to all other agents, making the topology fixed and directed for this study. It should be noted that, the theory developed so far can be extended to the case of switching topologies and shall be dealt in future. This concludes the proof. 1 Remark 4. The control (27) can be practically implemented as it does not contain the uncertainty term. It is crucial to analyze the necessary and sufficient conditions for the existence of sliding mode when control protocol (27) is used. We regard the system to be in sliding mode if for any time t1 ∈ [0, ∞[, system trajectories are brought upon the manifold si (t) = 0 and are constrained there for all time thereafter, i.e., for t ≥ t1 , sliding motion occurs. Theorem IV.2. Consider the system described by (5), error candidates (23, 24), sliding manifold (25) and the control protocol (27). Sliding mode is said to exist in vicinity of sliding manifold, if the manifold is attractive, i.e., trajectories emanating outside it continuously decrease towards it. Stating alternatively, reachability to the surface is ensured for some reachability constant η > 0. Moreover, stability can be guaranteed in the sense of Lyapunov if gain µ is designed as µ > sup{kΛHςi Γk}. Proof. Let us take into account, a Lyapunov function candidate Vi = 0.5s2i . (34) Taking derivative of (34) along system trajectories yield V̇i = si ṡi = si ΛH(f (xi (t)) + uSMi (t) + ςi − ψ̇(tk ))Γ . 5 Fig. 1: Topology in which agents are connected  0 0 A= 0 0 0 0 1 0 1 0 0 1  1 0 L = D−A =  0 0   1 0 0 0 , B =  0 0 0 0 0 0 −1 0 0 1 0 0 0 0 0 0   1 0 0 0 , D =  0 0 0 0   2 −1 0 0 0 0 ,L + B =  0 1 0 0 −1 1 0 1 −1 0  0 0 0 0 0 0 , 0 1 0 0 0 1 (39)  −1 0 0 0  1 0 −1 1 (40) Agents have the following dynamics. (35) ẋ2 = 0.1 sin(x2 ) + cos(2πt) + uSM2 (t) + ς2 , (42) (36) ẋ3 = 0.1 sin(x3 ) + cos(2πt) + uSM3 (t) + ς3 , (43) ẋ4 = 0.1 sin(x4 ) + cos(2πt) + uSM4 (t) + ς4 , (44) ẋ5 = 0.1 sin(x5 ) + cos(2πt) + uSM5 (t) + ς5 . (45) (37) where η = µ sinh−1 (m + w\|si \|) − ΛHςi Γ > 0 is called reachability constant. For µ > sup{kΛHςi Γk}, we have V̇i < 0. 3 (41) = −µ sinh−1 (m + w\|si \|)ksi k + ΛHςi Γksi k = − µ sinh−1 (m + w\|si \|) + ΛHςi Γ ksi k = −ηksi k, 4 ẋ1 = 0.1 sin(x1 ) + cos(2πt) + uSM1 (t) + ς1 , Substituting the control protocol (27) in (36), we have V̇i = si − µ sinh−1 (m + w\|si \|)sign(si ) + ΛHςi Γ 2 (38) In this study, advection model given in [28] has been used to simulate the plume with both additive and multiplicative disturbances. The initial conditions for simulation are taken to be large values, i.e., far away from the equilibrium point. Time varying disturbance has been taken as ςi = 0.3 sin(π 2 t2 ), accuracy parameter θ = 0.001 and maximum mean airflow velocity v̄amax = 1 m/s. Other key design parameters are mentioned in table 1. Agents progressing towards the source 11 x1 x2 x3 x4 x5 source info position of agents (m) 2 Direction of movement of agents towards the source 1.5 10.9 10.8 10.7 10.6 Direction of movement of filaments released from the source 1 10.5 10.4 0.5 10.3 True Odor Source 10.2 0 10.1 10 12 -0.5 0 2 4 6 8 10 time progression for agents (sec) Fig. 2: Agents in consensus to locate source of odor Agents progressing towards the source in parallel formation 11 position of agents (m) 3 Formation gap 2 x2 10.8 x4 10.7 10.6 10.5 1 10.4 True odor source 0 x1 10.9 10.3 x3 x5 source info (movement of ﬁlaments away from source) odor source location 4 agent 1 initial point agent 2 initial point agent 3 initial point agent 4 initial point agent 5 initial point agent 1 terminal point 10.2 -1 agent 2 terminal point agent 3 terminal point 10.1 2 4 6 8 agent 5 terminal point 10 12 -2 0 agent 4 terminal point 10 time progression for agents (sec) Fig. 3: Agents in formation to locate source of odor Tracking errors Control signals 8 e1 e2 e3 e4 e5 1 u1 u2 u3 u4 u5 6 4 2 u i (t) Norm of error variables e i (t) 1.5 0 -2 0.5 -4 -6 -8 0 -10 0 1 2 3 4 5 6 7 time (sec) Fig. 4: Norm of tracking errors 8 9 10 0 1 2 3 4 5 6 7 8 time (sec) Fig. 5: Control signals during consensus 9 10 position of the source 2.5 Sliding manifolds 1.5 s1 s2 s3 s4 s5 surface variables s i (t) 1 0.5 0 -0.5 -1 -1.5 0 1 2 3 4 5 6 7 8 9 10 time (sec) Fig. 6: Sliding manifolds during consensus TABLE I: Values of the design parameters used in simulation c1 ωmax α1 α2 λ1 λ2 µ m w 0.5 2 rad/s 0.25 0.25 1.774 2.85 5 10−3 2 Figure 2 shows agents coming to consensus in finite time to locate the source of odor and figure 3 shows agents moving in parallel formation to locate the odor source. 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Coresets for Dependency Networks Alejandro Molina FIRST. LAST @ TU - DORTMUND . DE CS Department TU Dortmund, Germany Alexander Munteanu FIRST. LAST @ TU - DORTMUND . DE arXiv:1710.03285v2 [cs.AI] 16 Oct 2017 CS Department TU Dortmund, Germany Kristian Kersting LAST @ CS . TU - DARMSTADT. DE CS Dept. and Centre for CogSci TU Darmstadt, Germany Abstract Many applications infer the structure of a probabilistic graphical model from data to elucidate the relationships between variables. But how can we train graphical models on a massive data set? In this paper, we show how to construct coresets—compressed data sets which can be used as proxy for the original data and have provably bounded worst case error—for Gaussian dependency networks (DNs), i.e., cyclic directed graphical models over Gaussians, where the parents of each variable are its Markov blanket. Specifically, we prove that Gaussian DNs admit coresets of size independent of the size of the data set. Unfortunately, this does not extend to DNs over members of the exponential family in general. As we will prove, Poisson DNs do not admit small coresets. Despite this worst-case result, we will provide an argument why our coreset construction for DNs can still work well in practice on count data. To corroborate our theoretical results, we empirically evaluated the resulting Core DNs on real data sets. The results demonstrate significant gains over no or naive sub-sampling, even in the case of count data. 1. Introduction Artificial intelligence and machine learning have achieved considerable successes in recent years, and an ever-growing number of disciplines rely on them. Data is now ubiquitous, and there is great value from understanding the data, building e.g. probabilistic graphical models to elucidate the relationships between variables. In the big data era, however, scalability has become crucial for any useful machine learning approach. In this paper, we consider the problem of training graphical models, in particular Dependency Networks Heckerman et al. (2000), on massive data sets. They are cyclic directed graphical models, where the parents of each variable are its Markov blanket, and have been proven successful in various tasks, such as collaborative filtering Heckerman et al. (2000), phylogenetic analysis Carlson et al. (2008), genetic analysis Dobra (2009); Phatak et al. (2010), network inference from sequencing data Allen and Liu (2013), and traffic as well as topic modeling Hadiji et al. (2015). Specifically, we show that Dependency Networks over Gaussians—arguably one of the most prominent type of distribution in statistical machine learning—admit coresets of size independent of the size of the data set. Coresets are weighted subsets of the data, which guarantee that models fitting 1 them will also provide a good fit for the original data set, and have been studied before for clustering Badoiu et al. (2002); Feldman et al. (2011, 2013); Lucic et al. (2016), classification Har-Peled et al. (2007); Har-Peled (2015); Reddi et al. (2015), regression Drineas et al. (2006, 2008); Dasgupta et al. (2009); Geppert et al. (2017), and the smallest enclosing ball problem Badoiu and Clarkson (2003, 2008); Feldman et al. (2014); Agarwal and Sharathkumar (2015); we refer to Phillips (2017) for a recent extensive literature overview. Our contribution continues this line of research and generalizes the use of coresets to probabilistic graphical modeling. Unfortunately, this coreset result does not extend to Dependency Networks over members of the exponential family in general. We prove that Dependency Networks over Poisson random variables Allen and Liu (2013); Hadiji et al. (2015) do not admit (sublinear size) coresets: every single input point is important for the model and needs to appear in the coreset. This is an important negative result, since count data—the primary target of Poisson distributions—is at the center of many scientific endeavors from citation counts to web page hit counts, from counts of procedures in medicine to the count of births and deaths in census, from counts of words in a document to the count of gamma rays in physics. Here, modeling one event such as the number of times a certain lab test yields a particular result can provide an idea of the number of potentially invasive procedures that need to be performed on a patient. Thus, elucidating the relationships between variables can yield great insights into massive count data. Therefore, despite our worst-case result, we will provide an argument why our coreset construction for Dependency Networks can still work well in practice on count data. To corroborate our theoretical results, we empirically evaluated the resulting Core Dependency Networks (CDNs) on several real data sets. The results demonstrate significant gains over no or naive sub-sampling, even for count data. We proceed as follows. We review Dependency Networks (DNs), prove that Gaussian DNs admit sublinear size coresets, and discuss the possibility to generalize this result to count data. Before concluding, we illustrate our theoretical results empirically. 2. Dependency Networks Most of the existing AI and machine learning literature on graphical models is dedicated to binary, multinominal, or certain classes of continuous (e.g. Gaussian) random variables. Undirected models, aka Markov Random Fields (MRFs), such as Ising (binary random variables) and Potts (multinomial random variables) models have found a lot of applications in various fields such as robotics, computer vision and statistical physics, among others. Whereas MRFs allow for cycles in the structures, directed models aka Bayesian Networks (BNs) required acyclic directed relationships among the random variables. Dependency Networks (DNs)—the focus of the present paper—combine concepts from directed and undirected worlds and are due to Heckerman et al. (2000). Specifically, like BNs, DNs have directed arcs but they allow for networks with cycles and bi-directional arcs, akin to MRFs. This makes DNs quite appealing for many applications because we can build multivariate models from univariate distributions Allen and Liu (2013); Yang et al. (2015); Hadiji et al. (2015), while still permitting efficient structure learning using local estimtatiors or gradient tree boosting. Generally, if the data are fully observed, learning is done locally on the level of the conditional probability distributions for each variable mixing directed and indirected as needed. Based on these local distributions, samples from the joint distribution are obtained via Gibbs sampling. Indeed, the Gibbs sampling neglects the question of a consistent joint probability distribution and instead makes only use of 2 local distributions. The generated samples, however, are often sufficient to answer many probability queries. Formally, let X = (X (1) , . . . , X (d) ) denote a random vector and x its instantiation. A Dependency Network (DN) on X is a pair (G, Ψ) where G = (V, E) is a directed, possibly cyclic, graph where each node in V = [d] = {1, . . . , d} corresponds to the random variable X (i) . In the set of directed edges E ⊆ V × V \ {(i, i) \| i ∈ [d]}, each edge models a dependency between variables, i.e., if there is no edge between i and j then the variables X (i) and X (j) are conditionally independent given the other variables X \i,j indexed by [d] \ {i, j} in the network. We refer to the nodes that have an edge pointing to X (i) as its parents, denoted by pai = {X (j) \| (j, i) ∈ E}. Ψ = {pi \| i ∈ [d]} is a set of conditional probability distributions associated with each variable X (i) ∼ pi , where pi = p(x(i) \| pai ) = p(x(i) \| x\i ) . As example of such a local model, consider Poisson conditional probability distributions as illustrated in Fig. 1 (left): (i) λi (x\i )x −λi (x\i ) p(x(i) \| pai ) = e . x(i) ! Here, λi (x\i ) highlights the fact that the mean can have a functional form that is dependent on X (i) ’s parents. Often, we will refer to it simply as λi . The construction of the local conditional probability distribution is similar to the (multinomial) Bayesian network case. However, in the case of DNs, the graph is not necessarily acyclic and p(x(i) \| x\i ) typically has an infinite range, and hence cannot be represented using a finite table of probability values. Finally, the full joint distribution is simply defined as the product of local distributions: Y p(x) = p(x(i) \| x\i ) , i∈[d] also called pseudo likelihood. For the Poisson case, this reads (i) p(x) = Y i∈[d] λix −λi . e x(i) ! Note, however, that doing so does not guarantee the existence of a consistent joint distribution, i.e., a joint distribution of which they are the conditionals. Bengio et al. (2014), however, have recently proven the existence of a consistent distribution per given evidence, which does not have to be known in closed form, as long as an unordered Gibbs sampler converges. 3. Core Dependency Networks As argued, learning Dependency Networks (DNs) amounts to determining the conditional probability distributions from a given set of n training instances xi ∈ Rd representing the rows of the data matrix X ∈ Rn×d over d variables. Assuming that p(x(i) \| pai ) is parametrized as a generalized linear model (GLM) McCullagh and Nelder (1989), this amounts to estimating the parameters γ (i) of the GLM associated with each variable X (i) , since this completely determines the local distributions, but p(x(i) \| pai ) will possibly depend on all other variables in the network, and these dependencies define the structure of the network. This view of training DNs as fitting d GLMs to the data allows us to develop Core Dependency Networks (CDNs): Sample a coreset and train a DN over certain members of the GLM family on the sampled corest. 3 Relative Frequency 0.5 fit data 0.4 data fit 0.3 0.2 X (0) 0.1 0.0 X (1) 0 2 4 6 X (2) 8 Number of Goals Figure 1: Illustration of Dependency Networks (DNs) using Poissons. (left) The number of goals scored in soccer games follows a Poisson distribution. The plot shows the distribution of home goals in the season 2012/13 of the German Bundesliga by the home team. The home team scored on average λ = 1.59 goals per game. (right) Example structure of a Poisson DN. The conditional distribution of each count variable given its neighbors is a Poisson distribution. Similar to a Bayesian network a Poisson DN is directed, however, it also contains cycles. (Best viewed in color) A coreset is a (possibly) weighted and usually considerably smaller subset of the input data that approximates a given objective function for all candidate solutions: Definition 1 (ε-coreset) Let X be a set of points from a universe U and let Γ be a set of candidate solutions. Let f : U × Γ → R≥0 be a non-negative measurable function. Then a set C ⊂ X is an ε-coreset of X for f , if ∀γ ∈ Γ : \|f (X, γ) − f (C, γ)\| ≤ ε · f (X, γ). We now introduce the formal framework that we need towards the design of coresets for learning dependency networks. A very useful structural property for `2 based objective (or loss) functions is the concept of an ε-subspace embedding. Definition 2 (ε-subspace embedding) An ε-subspace embedding for the columnspace of X is a matrix S such that ∀γ ∈ Rd : (1 − ε)kXγk2 ≤ kSXγk2 ≤ (1 + ε)kXγk2 We can construct a sampling matrix S which forms an ε-subspace embedding with constant probabilty in the following way: Let U be any orthonormal basis for the columnspace of X. This basis can be obtained from the singular value decomposition (SVD) X = U ΣV T of the data matrix. Now let ρ = rank(U ) = rank(X) and define the leverage scores li = kUi∗ k2 /kU k2F = kUi∗ k2 /ρ for i ∈ [n]. Now we fix a sampling size parameter k = O(ρ log(ρ/ε)/ε2 ), sample the input points one-by-one with probability qi = min{1, k · li } and reweight their contribution to the loss function by wi = 1/qi . Note that, for the sum of squares loss, this corresponds to defining a diagonal 4 √ (sampling) matrix S by Sii = 1/ qi with probability qi and Sii = 0 otherwise. Also note, that the expected number of samples is k = O(ρ log(ρ/ε)/ε2 ), which also holds with constant probability by Markov’s inequality. Moreover, to give an intuition why this works, note that for any fixed γ ∈ Rd , we have X X xi γ 2 2 E kSXγk = qi = (xi γ)2 = kXγk2 . √ qi The significantly stronger property of forming an ε-subspace embedding, according to Definition 2, follows from a matrix approximation bound given in Rudelson and Vershynin (2007); Drineas et al. (2008). Lemma 3 Let X be an input matrix with rank(X) = ρ. Let S be a sampling matrix constructed as stated above with sampling size parameter k = O(ρ log(ρ/ε)/ε2 ). Then S forms an ε-subspace embedding for the columnspace of X with constant probability. Proof Let X = U ΣV T be the SVD of X. By Theorem 7 in Drineas et al. (2008) there exists an absolute constant C > 1 such that r T T log k E kU S SU − U T U k ≤ C kU kF kU k k r log k √ ≤ C ρ ≤ ε, k √ where we used the fact that kU kF = ρ and kU k = 1 by orthonormality of U . The last inequality holds by choice of k = Dρ log(ρ/ε)/ε2 for a large enough absolute constant D > 1 such that 1+log D < 4C1 2 , since D log k log(Dρ log(ρ/ε)/ε2 ) 2ε2 log(Dρ log(ρ/ε)/ε) = ≤ k Dρ log(ρ/ε)/ε2 Dρ log(ρ/ε) 2 2 4ε 1 + log D ε2 4ε (log(ρ/ε) + log D) ≤ < 2 . ≤ Dρ log(ρ/ε) ρ D C ρ By an application of Markov’s inequality and rescaling ε, we can assume with constant probability kU T S T SU − U T U k ≤ ε. (1) We show that this implies the ε-subspace embedding property. To this end, fix γ ∈ Rd . \| kSXγk2 − kXγk2 \| = kγ T X T S T SXγ − γ T X T Xγk = kγ T V ΣU T S T SU ΣV T γ − γ T V ΣU T U ΣV T γk = kγ T V Σ (U T S T SU − U T U ) ΣV T γk ≤ kU T S T SU − U T U k · kΣV T γk2 ≤ kU T S T SU − U T U k · kXγk2 ≤ εkXγk2 , The first inequality follows by submultiplicativity, and the second from rotational invariance of the spectral norm. Finally we conclude the proof by Inequality (1). 5 The question arises whether we can do better than O(ρ log(ρ/ε)/ε2 ). One can show by reduction from the coupon collectors theorem that there is a lower bound of Ω(ρ log ρ) matching the upper bound up to its dependency on √ ε. The hard instance is a dm ×d, m ∈ N orthonormal matrix in which the scaled canonical basis Id / dm−1 is stacked dm−1 times. The leverage scores are all equal to 1/dm , implying a uniform sampling distribution with probability 1/d for each basis vector. Any rank ρ = d preserving sample must comprise at least one of them. This is exactly the coupon collectors theorem with d coupons which has a lower bound of Ω(d log d) Motwani and Raghavan (1995). The fact that the sampling is without replacement does not change this, since the reduction holds for arbitrary large m creating sufficient multiple copies of each element to simulate the sampling with replacement Tropp (2011). Now we know that with constant probability over the randomness of the construction algorithm, S satisfies the ε-subspace embedding property for a given input matrix X. This is the structural key property to show that actually SX is a coreset for Gaussian linear regression models and dependency networks. Consider (G, Ψ), a Gaussian dependency network (GDN), i.e., a collection of Gaussian linear regression models Ψ = {pi (X (i) \|X \i , γ (i) ) = N (X \i γ (i) , σ 2 ) \| i ∈ [d]} on an arbitrary digraph structure G Heckerman et al. (2000). The logarithm of the (pseudo-)likelihood Besag (1975) of the above model is given by Y X ln L (Ψ) = ln pi = ln pi . A maximum likelihood estimate can be obtained by maximizing this function with respect to γ = (γ (1) , . . . , γ (d) ) which is equivalent to minimizing the GDN loss function X fG (X, γ) = kX \i γ (i) − X (i) k2 . Theorem 4 Given S, an ε-subspace embedding for the columnspace of X as constructed above, SX is an ε-coreset of X for the GDN loss function. Proof Fix an arbitrary γ = (γ (1) , . . . , γ (d) ) ∈ Rd(d−1) . Consider the affine map Φ : Rd−1 × \i [d] → Rd , defined by Φ(γ (i) ) = Id γ (i) − ei . Clearly Φ extends its argument from d − 1 to d dimensions by inserting a −1 entry at position i and leaving the other entries in their original order. Let β (i) = Φ(γ (i) ) ∈ Rd . Note that for each i ∈ [d] we have Xβ (i) = XΦ(γ (i) ) = X \i γ (i) − X (i) , (2) and each β (i) is a vector in Rd . Thus, the triangle inequality and the universal quantifier in Definition 2 guarantee that X X \| kSXβ (i) k2 − kXβ (i) k2 \| X = \| (kSXβ (i) k2 − kXβ (i) k2 ) \| X ≤ \|kSXβ (i) k2 − kXβ (i) k2 \| X X ≤ εkXβ (i) k2 = ε kXβ (i) k2 . 6 The claim follows by substituting Identity (2). It is noteworthy that computing one single coreset for the columnspace of X is sufficient, rather than computing d coresets for the d different subspaces spanned by X \i . From Theorem 4 it is straightforward to show that the minimizer found for the coreset is a good approximation of the minimizer for the original data. Corollary 5 Given an ε-coreset C of X for the GDN loss function, let γ̃ ∈ argminγ∈Rd(d−1) fG (C, γ). Then it holds that fG (X, γ̃) ≤ (1 + 4ε) min fG (X, γ). γ∈Rd(d−1) Proof Let γ ∗ ∈ argminγ∈Rd(d−1) fG (X, γ). Then fG (X, γ̃) ≤ ≤ 1 1 fG (C, γ̃) ≤ fG (C, γ ∗ ) 1−ε 1−ε 1+ε fG (X, γ ∗ ) ≤ (1 + 4ε)fG (X, γ ∗ ). 1−ε The first and third inequalities are direct applications of the coreset property, the second holds by optimality of γ̃ for the coreset, and the last follows from ε < 12 . Moreover, the coreset does not affect inference within GDNs. Recently, it was shown for (Bayesian) Gaussian linear regression models that the entire multivariate normal distribution over the parameter space is approximately preserved by ε-subspace embeddings Geppert et al. (2017), which generalizes the above. This implies that the coreset yields a useful pointwise approximation in Markov Chain Monte Carlo inference via random walks like the pseudo-Gibbs sampler in Heckerman et al. (2000). 4. Negative Result on Coresets for Poisson DNs Naturally, the following question arises: Do (sublinear size) coresets exist for dependency networks over the exponential family in general? Unfortunately, the answer is no! Indeed, there is no (sublinear size) coreset for the simpler problem of Poisson regression, which implies the result for Poisson DNs. We show this formally by reduction from the communication complexity problem known as indexing. To this end, recall that the negative log-likelihood for Poisson regression is McCullagh and Nelder (1989); Winkelmann (2008) X `(γ) := `(γ\|X, Y ) = exp(xi γ) − yi · xi γ + ln(yi !). Theorem 6 Let ΣD be a data structure for D = [X, Y ] that approximates likelihood queries ΣD (γ) for Poisson regression, such that ∀γ ∈ Rd : η −1 · `(γ\|D) ≤ ΣD (γ) ≤ η · `(γ\|D). If η < exp( n ) 4 2n2 then ΣD requires Ω(n) bits of storage. 7 Proof We reduce from the indexing problem which is known to have Ω(n) one-way randomized communication complexity Jayram et al. (2008). Alice is given a vector b ∈ {0, 1}n . She produces for every i with bi = 1 the points xi = (r·ω i , −1) ∈ R3 , where ω i , i ∈ {0, . . . , n−1} denote the nth 3 unit roots in the plane, i.e., the vertices of a regular n-polygon of radius r = n/(1 − cos( 2π n )) ≤ n in canonical order. The corresponding counts are set to yi = 1. She builds and sends ΣD of size 3 s(n) to Bob, whose task is to guess the bit bj . He chooses to query γ = (ω j , r · cos( 2π n )) ∈ R . Note j that this affine hyperplane separates r · ω from the other scaled unit roots since it passes exactly through r · ω (j−1) mod n and r · ω (j+1) mod n . Also, all points are within distance 2r from each other by construction and consequently from the hyperplane. Thus, −2r ≤ xi γ ≤ 0 for all i 6= j. If bj = 0, then xj does not exist and the cost is at most `(γ) = X exp(xi γ) − yi · xi γ + ln(yi !) ≤ X 1 + 2r + 1 ≤ 2n + 2nr ≤ 4n4 . If bj = 1 then xj is in the expensive halfspace and at distance exactly 2π j T j xj γ = (rω ) ω − r · cos n 2π = r · 1 − cos = n n So the cost is bounded below by `(γ) ≥ exp(n) − n + 1 ≥ exp( n2 ). exp( n ) Given η < 2n24 , Bob can distinguish these two cases based on the data structure only, by deciding whether ΣD (γ) is strictly smaller or larger than exp( n4 ) · 2n2 . Consequently s(n) = Ω(n), since this solves the indexing problem. Note that the bound is given in bit complexity, but restricting the data structure to a sampling based coreset and assuming every data point can be expressed in O(d log n) bits, this means we still have a lower bound of k = Ω( logn n ) samples. Corollary 7 Every sampling based coreset for Poisson regression with approximation factor η < exp( n ) 4 as in Theorem 6 requires at least k = Ω( logn n ) samples. 2n2 At this point it seems very likely that a similar argument can be used to rule out any o(n)-space constant approximation algorithm. This remains an open problem for now. 5. Why Core DNs for Count Data can still work So far, we have a quite pessimistic view on extending CDNs beyond Gaussians. In the Gaussian setting, where the loss is measured in squared Euclidean distance, the number of important points, i.e., having significantly large leverage scores, is bounded essentially by O(d). This is implicit in the original early works Drineas et al. (2008) and has been explicitly formalized later Langberg and Schulman (2010); Clarkson and Woodruff (2013). It is crucial to understand that this is an inherent property of the norm function, and thus holds for arbitrary data. For the Poisson GLM, in contrast, we have shown that its loss function does not come with such properties from scratch. We 8 constructed a worst case scenario, where basically every single input point is important for the model and needs to appear in the coreset. Usually, this is not the case with statistical models, where the data is assumed to be generated i.i.d. from some generating distribution that fits the model assumptions. Consider for instance a data reduction for Gaussian linear regression via leverage score sampling vs. uniform sampling. It was shown that given the data follows the model assumptions of a Gaussian distribution, the two approaches behave very similarly. Or, to put it another way, the leverage scores are quite uniform. In the presence of more and more outliers generated by the heavier tails of tdistributions, the leverage scores increasingly outperform uniform sampling Ma et al. (2015). The Poisson model yi ∼ Poi(λi ), λi = exp(xi γ). (3) though being the standard model for count data, suffers from its inherent limitation on equidispersed data since E [yi \|xi ] = V [yi \|xi ] = exp(xi γ). Count data, however, is often overdispersed especially for large counts. This is due to unobserved variables or problem specific heterogeneity and contagion-effects. The log-normal Poisson model is known to be inferior for data which specifically follows the Poisson model, but turns out to be more powerful in modeling the effects that can not be captured by the simple Poisson model. It has wide applications for instance in econometric elasticity problems. We review the log-normal Poisson model for count data Winkelmann (2008) yi ∼ Poi(λi ), λi = exp(xi γ)ui = exp(xi γ + vi ), vi = ln ui ∼ N (µ, σ) . 2 A natural choice for the parameters of the log-normal distribution is µ = − σ2 in which case we have E [yi \|xi ] = exp(xi γ + µ + σ 2 /2) = exp(xi γ) , V [yi \|xi ] = E [yi \|xi ] + (exp(σ 2 ) − 1)E [yi \|xi ]2 . It follows that V [yi \|xi ] = exp(xi γ) + Ω(exp(xi γ)2 ) > exp(xi γ), where a constant σ 2 that is independent of xi , controls the amount of overdispersion. Taking the limit for σ → 0 we arrive at the simple model (3), since the distribution of vi = ln ui tends to δ0 , the deterministic Dirac delta distribution which puts all mass on 0. The inference might aim for the log-normal Poisson model directly as in Zhou et al. (2012), or it can be performed by (pseudo-)maximum likelihood estimation of the simple Poisson model. The latter provides a consistent estimator as long as the log-linear mean function is correctly specified, even if higher moments do not possess the limitations inherent in the simple Poisson model Winkelmann (2008). Summing up our review on the count modeling perspective, we learn that preserving the loglinear mean function in a Poisson model is crucial towards consistency of the estimator. Moreover, modeling counts in a log-normal model gives us intuition why leverage score sampling can capture the underlying linear model accurately: In the log-normal Poisson model, u follows a log-normal distribution. It thus holds for ln λ = Xγ + ln u = Xγ + v, that 2 σ 2 v ∼ N − · 1, σ In 2 9 3.58 3.63 3.58 3.59 3.58 CDN Uniform Full 3.6×107 3.55×107 10% 3.48 20% 30% 40% Training data (Sample size in percentage) 3.58 3.35 3.36 3.33 3.31 3.31 3.29 3.6×106 100% 3.26 CDN Uniform Full 3.5×106 3.4×106 3.3×106 3.2×10 6 10% 20% 30% 40% Training data (Sample size in percentage) Negative Log Pseudo Likelihood 100% −2.5 −2.6 −2.55 −2.64 −2.62 −2.65 −2.65 −2.66 −2.66 −2.67 CDN Uniform Full −2.55 −2.6 −2.65 −2.7 2.2 3.5 3.0 1.72 1.76 2.38 2.46 2.79 2.9 3.08 3.18 4.0 CDN Uniform Full 2.5 2.0 1.5 10% 1.55 20% 30% 40% Training data (Sample size in percentage) 1.68 1.4 1.48 1.36 1.42 1.35 1.4 1.8 1.0 100% 1.39 CDN Uniform Full 2.0 1.6 1.4 1.2 1.0 4.0 Log Time (in hours) 3.59 3.0 2.5 Log Time (in minutes) 3.58 Log RMSE (Root Mean Square Error) 5.23 Log RMSE (Root Mean Square Error) 3.7×10 3.59 7 3.65×107 Negative Poisson Pseudo Log–Likelihood Negative Gaussian Pseudo Log–Likelihood 3.75×107 2.0 10% 20% 30% 40% Training data (Sample size in percentage) 0.29 −0.25 0.48 0.09 0.81 0.54 1.05 0.86 100% 2.56 CDN Uniform Full 1.5 1.0 0.5 0.0 10% 20% 30% 40% Training data (Sample size in percentage) 100% RMSE −0.5 10% 20% 30% 40% Training data (Sample size in percentage) 100% Training Time Figure 2: (Q1) Performance (the lower, the better) of Gaussian CDNs on MNIST (upper row) and Poisson CNDs on the traffic dataset (lower row) 10-fold cross-validated. Shown are the negative log pseudo likelihood (left), the squared error loss (middle, in log-space) as well as the training time (right, in log-space) on the y-axis for different proportions of the data sampled (x axis). Please note the jump in the x-axis after 40%. As one can see, CDNs (blue) quickly approach the predictive performance of the full dataset (Full, black). Uniform sampling (Uniform, red) does not perform as well as CDNs. Moreover, CDNs can be orders of magnitude faster than DNs on the full dataset and scale similar to uniform sampling. This is also supported by the vertical lines. They denote the mean performances (the more to the left, the better) on the top axes. (Best viewed in color) by independence of the observations, which implies σ2 2 ln λ ∼ N Xγ − · 1, σ In . 2 2 Omitting the bias µ = − σ2 in each intercept term (which can be cast into X), we notice that this yields again an ordinary least squares problem kXγ − ln(λ)k2 defined in the columspace of X. There is still a missing piece in our argumentation. In the previous section we have used that the coreset construction is an ε-subspace embedding for the columnspace of the whole data set including the dependent variable, i.e., for [X, ln(λ)]. We face two problems. First, λ is only implicitly given in the data, but is not explicitly available. Second, λ is a vector derived from X \i in our setting and might be different for any of the d instances. Fortunately, it was shown via more complicated arguments Drineas et al. (2008), that it is sufficient for a good approximation, if the sampling is done obliviously to the dependent variable. The intuition comes from the fact that the loss of any point in the subspace can be expressed via the projection of ln(λ) onto the subspace spanned by X, and the residual of its projection. A good approximation of the subspace implicitly approximates the projection of any fixed vector, which is then applied to the residual vector of the orthogonal projection. This solves the first problem, since it is only necessary to have a subspace embedding for X. The second issue can be addressed by increasing the sample size by a factor of O(log d) for boosting the error probability to O(1/d) and taking a union bound. 10 Sample portion 10% 20% 30% 40% MNIST GCDN 18.03% 0.57% 0.01% 0.01% GUDN 11162.01% 13.86% 13.33% 2.3% Traffic PCDN 6.81% 2.9% 2.04% 1.59% PUDN 9.6% 3.17% 1.68% 0.99% Table 1: (Q1) Comparison of the empirical relative error (the lower, the better). Best results per dataset are bold. Both Gaussian (GCDNs) and Poisson (PCDNs) CDNs recover the model well, with a fraction of the training data. Uniformly sampled DNs (UDNs) lag behind as the sample size drops. 6. Empirical Illustration Our intention here is to corroborate our theoretical results by investigating empirically the following questions: (Q1) How does the performance of CDNs compare to DNs with access to the full training data set and to a uniform sample from the training data set? and how does the empirical error behave according to the sample sizes? (Q2) Do coresets affect the structure recovered by the DN? To this aim, we implemented (C)DNs in Python calling R. All experiments ran on a Linux machine (56 cores, 4 GPUs, and 512GB RAM). Benchmarks on MNIST and Traffic Data (Q1): We considered two datasets. In a first experiment, we used the MNIST1 data set of handwritten labeled digits. We employed the training set consisting of 55000 images, each with 784 pixels, for a total of 43,120,000 measurements, and trained Gaussian DNs on it. The second data set we considered contains traffic count measurements on selected roads around the city of Cologne in Germany Ide et al. (2015). It consists of 7994 timestamped measurements taken by 184 sensors for a total of 1,470,896 measurements. On this dataset we trained Poisson DNs. For each dataset, we performed 10 fold cross-validation for training a full DN (Full) using all the data, leverage score sampling coresets (CDNs), and uniform samples (Uniform), for different sample sizes. We then compared the predictions made by all the DNs and the time taken to train them. For the predictions on the MNIST dataset, we clipped the predictions to the range [0,1] for all the DNs. For the Traffic dataset, we computed the predictions bxc of every measurement x rounded to the largest integer less than or equal to x. Fig. 2 summarizes the results. As one can see, CDNs outperform DNs trained on full data and are orders of magnitude faster. Compared to uniform sampling, coresets are competitive. Actually, as seen on the traffic dataset, CDNs can have more predictive power than the “optimal” model using the full data. This is in line with Mahoney (2011), who observed that coresets implicitly introduce regularization and lead to more robust output. Table 1 summarizes the empirical relative errors \|f (X, γ̃) − f (X, γ ∗ )\|/f (X, γ ∗ ) between (C/U)DNs γ̃ and DNs γ ∗ trained on all the data. CDNs clearly recover the original model, at a fraction of training data. Overall, this answers (Q1) affirmatively. Relationship Elucidation (Q2): We investigated the performance of CDNs when recovering the graph structure of word interactions from a text corpus. For this purpose, we used the NIPS2 bag1. http://yann.lecun.com/exdb/mnist/ 2. https://archive.ics.uci.edu/ml/datasets/bag+of+words 11 skill item loss component rotation direction tree cell digit map skill set document item disparity object pca oscillator neuron distance pyramid tangent loss component estimator dialogue routing saliency road iiii policy context fuzzy option wavelet speaker user control call star letter delay building attractor controller subscriber block eeg neural evidence potential classifier spike channel instruction rule stress rules chip net lesion circuit light rotation direction tree cell digit map skill set document item disparity object pca oscillator neuron distance pyramid tangent estimator dialogue routing saliency road iiii policy context fuzzy option wavelet speaker user control call star letter delay building attractor controller subscriber block eeg neural evidence potential classifier spike channel instruction rule stress rules chip net lesion circuit light analog set document loss component disparity object estimator pca oscillator neuron dialogue distance routing pyramid saliency tangent road iiii policy context fuzzy option wavelet speaker user control call star letter delay building attractor controller subscriber block eeg neural evidence potential classifier spike channel instruction rule stress rules chip net lesion circuit light rotation direction tree cell digit map analog analog Gaussian CDN Poisson CDN skill item loss component rotation direction tree cell road digit map routing policy object pca neuron pyramid skill set document item disparity estimator oscillator distance dialogue saliency tangent context user iiii fuzzy option wavelet speaker control call star letter delay building attractor controller subscriber block eeg neural evidence potential classifier spike channel instruction rule stress rules chip net lesion circuit light analog loss component rotation direction tree cell road digit map routing policy object pca neuron pyramid skill set document item disparity estimator oscillator distance dialogue saliency tangent context user iiii fuzzy option wavelet speaker control call star letter delay building attractor controller subscriber block eeg neural evidence potential classifier spike channel instruction rule stress rules chip net lesion circuit light analog loss component rotation direction tree cell road digit map routing policy set document object pca neuron pyramid disparity estimator oscillator distance dialogue saliency tangent context user iiii fuzzy option wavelet speaker control call star letter delay building attractor controller subscriber block eeg neural evidence potential classifier spike channel instruction rule stress rules chip net lesion circuit light analog Figure 3: (Q2) Elucidating the relationships between random variables. Shown are the (positive) dependency structures of Gaussian (top) and Poisson (bottom) CDNs on NIPS and different learning sampling sizes: using 40% (Left) , 70% (Middle) and 100% (Right). The edges show the 70 top thresholded positive coefficients of the GLMs. The colors of the edges represent modularity. As one can see, CDNs elucidate relationships among the words that make semantically sense and approach the structure learned using the full dataset. For a quantitative assessment, see Tab. 2. (Best viewed in color) of-words dataset. It contains 1,500 documents with a vocabulary above 12k words. We considered the 100 most frequent words. Fig. 3 illustrates the results qualitatively. It shows three CDNs of sampling sizes 40%, 70% and 100% for Gaussians (top) after a log(x+1) transformation and for Poissons (bottom): CDNs capture well the gist of the NIPS corpus. Table 2 confirms this quantitatively. It shows the Frobenius norms between the DNs: CDNs capture the gist better than naive, i.e., uniform sampling. This answers (Q2) affirmatively. To summarize our empirical results, the answers to questions (Q1) and (Q2) show the benefits of CDNs. 7. Conclusions Inspired by the question of how we can train graphical models on a massive dataset, we have studied coresets for estimating Dependency networks (DNs). We established the first rigorous guarantees for obtaining compressed ε-approximations of Gaussian DNs for large data sets. We proved worstcase impossibility results on coresets for Poisson DNs. A review of log-normal Poisson modeling of counts provided deep insights into why our coreset construction still performs well for count data in practice. 12 Sample portion 40% 70% UDN Gaussian 9.0676 4.8487 CDN Poisson 6.4042 1.6262 Gaussian 3.9135 2.6327 Poisson 0.6497 0.3821 Table 2: (Q2) Frobenius norm of the difference of the adjacency matrices (the lower, the better) recovered by DNs trained on the full data and trained on a uniform subsample (UDN) resp. coresets (CDNs) of the training data. The best results per statiscal type (Gaussian/Poisson) are bold. CDNs recover the structure better than UDNs. Our experimental results demonstrate, the resulting Core Dependency Networks (CDNs) can achieve significant gains over no or naive sub-sampling, even in the case of count data, making it possible to learn models on much larger datasets using the same hardware. CDNs provide several interesting avenues for future work. The conditional independence assumption opens the door to explore hybrid multivariate models, where each variable can potentially come from a different GLM family or link function, on massive data sets. This can further be used to hint at independencies among variables in the multivariate setting, making them useful in many other large data applications. Generally, our results may pave the way to establish coresets for deep models using the close connection between dependency networks and deep generative stochastic networks Bengio et al. (2014), sum-product networks Poon and Domingos (2011); Molina et al. (2017), as well as other statistical models that build multivariate distributions from univariate ones Yang et al. (2015). Acknowledgements: This work has been supported by Deutsche Forschungsgemeinschaft (DFG) within the Collaborative Research Center SFB 876 ”Providing Information by Resource-Constrained Analysis”, projects B4 and C4. 13 References Pankaj K. Agarwal and R. 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PROC. OF THE 6th EUR. CONF. ON PYTHON IN SCIENCE (EUROSCIPY 2013) 15 CATOS: Computer Aided Training/Observing System Jinook Oh∗† arXiv:1404.6384v1 [cs.CE] 25 Apr 2014 F Abstract—In animal behavioral biology, there are several cases in which an autonomous observing/training system would be useful. 1) Observation of certain species continuously, or for documenting specific events, which happen irregularly; 2) Longterm intensive training of animals in preparation for behavioral experiments; and 3) Training and testing of animals without human interference, to eliminate potential cues and biases induced by humans. The primary goal of this study is to build a system named CATOS (Computer Aided Training/Observing System) that could be used in the above situations. As a proof of concept, the system was built and tested in a pilot experiment, in which cats were trained to press three buttons differently in response to three different sounds (human speech) to receive food rewards. The system was built in use for about 6 months, successfully training two cats. One cat learned to press a particular button, out of three buttons, to obtain the food reward with over 70 percent correctness. Index Terms—animal training, animal observing, automatic device 1 I NTRODUCTION It is often the case in animal behavioral biology that a large amount of human resources, time, and data storage (such as video recordings) are required in animal observation and training. Some representative examples of these cases are: • Observation of certain species continuously or monitoring for specific events, which occur irregularly, when behavior of certain species during any time period or specific time period, such as nocturnal behaviors, are investigated. • Certain experiments require a prolonged training period, sometimes over a year. This type of experiment requires reliable responses, which may not correspond to usual behavior patterns, from animals in tasks. Therefore, training may require a long period of time until the subject is ready to be tested. Additionally, long periods of human supervised training can introduce unintended cues and biases for animals. In the first case, an autonomous system for observing animals can save human resources and reduce the amount of data storage. The reduced amount of data can also conserve other types of human resources such as investigation and maintenance of large-scale data. There have been attempts to build autonomous observing or surveillance systems in the fields of biology, such as Kritzler et al. [Kri08]’s work, and * Corresponding author: [email protected] † Cognitive Biology Dept., University of Vienna c 2014 Jinook Oh. This is an open-access article disCopyright ○ tributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. http://creativecommons.org/licenses/by/3.0/ security systems, such as Belloto et al. [Bel09], Vallejo et al. [Val09], for instance. There are also commercial products for surveillance systems with various degrees of automation, or incorporating artificial intelligence. However, the intelligence of each system is case-specific and it is difficult to apply these specific systems to novel situations without considerable adjustments. In the second case, an autonomous system for prolonged, intensive training can also save human resources and eliminate potential cues and biases caused by humans. Training with an autonomous system is an extension of traditional operant conditioning chambers and many modern and elaborated versions have been developed and used, such as in Markham et al. [Mar96], Takemoto et al. [Tak11], Kangas et al. [Kan12], Steurer et al. [Ste12], and Fagot & Bonte [Fag09]. However, many of the previous devices use commercial software. Also, they do not possess the observational features developed in the current project. It would be useful to have an open-source, relatively low-budget, and modularized system which could be customized for the observation, training and the experimentation on animal subjects of various species. CATOS, the system built in the present study, fulfills these necessities. The difference between the previous systems and CATOS (Computer Aided Training/Observing System) in the present work is that the animals do not have to be captured or transported to a separated space at a specific time in order to be trained. The disadvantages of separating animals (e.g., primates) are well-known, and include stress on animals separated from their group or moved from their usual confines, the risky catching procedure for both animal and human (cf. Fagot & Bonte [Fag09]). Similar arguments apply to most animal species, especially when they are social. The automatic learning device for monkeys (ALDM) described in Fagot & Bonte [Fag09] is very similar to the trainer aspect of CATOS described in the present work, but CATOS is different in following features. First of all, it aimed to be opensource based and more modular so that it can be more easily adjusted and adopted to different species and experiments. Another feature is that CATOS is equipped with various observational features, including visual and auditory recording and recognition through video camera and microphone, which make the system able to interact with the subjects, such as reacting immediately to a subject with a motion detection from a camera or a sound recognition from a microphone. CATOS should offer the following advantages. • The system should be flexible in terms of its adjustability and the extendibility to various projects and species. The 16 PROC. OF THE 6th EUR. CONF. ON PYTHON IN SCIENCE (EUROSCIPY 2013) • • • • software should be open-source, and both software and hardware components should be modularized as much as possible, thus the system reassembly for researchers in animal behavioral biology is practical. The system should have various observational features applicable to a broad range of animal species and observational purposes. The system should perform continuous monitoring, and it should record video and/or sound only when a set of particular conditions is fulfilled. This would reduce the amount of data produced during the procedure. The system should have actuators to react in certain situations, which allows it to act as a trainer/experimenter. The human trainer/experimenter designs the procedure by adjusting parameters and modules, but the actual performance should be done by the system. In this way, the system could help reducing the amount of time required for training, and eliminating cues/biases which might be induced by the human interferences. With this system, the animal should not have to be transported to a certain space, or separated from its group, for training. The animals should be able to choose when to start a trial on their own. Two CATOS prototypes have been built during this study. The first build of CATOS has 3 pushbuttons as a main input device for cats and the second build has a touch-screen as a main input device. The first build was an initial attempt to build and test such a system. The second build is the final product of the study. The basic structures of these two builds are more or less the same. The differences are that the second version has improved functions and it uses the touchscreen instead of pushbuttons. The first build of CATOS was tested with domestic cats (Felis catus) to train them to press three different buttons differently depending on the auditory stimuli (three different human speech sounds). The final goal of this training is to investigate human speech perception in cats. There is no doubt in that many animal species can recognize some words in human speech. The examples of speech perception in dogs and chimpanzees can be found in the work of Kaminski et al. [Kam04] and Heimbauer et al. [Hei11] respectively. In some cases, animals can even properly produce words with specific purposes. An example of speech perception and production in a parrot can be found in the work of Pepperberg [Pep87]. Despite these findings, there is ongoing debate about whether the same perceptual mechanisms are used in speech recognition by humans and animals (Fitch [Fit11]). To investigate this issue, animals have to be trained to show different and reliable responses to different human speech sounds. Then, we can test which features of human speech are necessary for different animal species to understand it. Thus, the final aim of the training in this study would be to obtain cats showing different responses to different human speech sounds with statistical significance (over 75 percent). Before reaching this final goal, several smaller steps and goals are required. Fig. 1: Overall system diagram. 2 B RIEF DESCRIPTION OF CATOS (C OMPUTER A IDED T RAINING /O BSERVING S YSTEM ) The overall system is composed of a combination of software and hardware components. The software components are mainly composed of the Python script named as ’AA.<version>.py’ and the program for the microcontroller. The ’AA’ runs all of the necessary processes and communicates with the microcontroller program. The microcontroller program operates sensors and actuators as it communicates with the ’AA’ program. The hardware components are composed of various devices, some of which are directly connected to the computer via USB cables. Some other devices only have GPIO (General Purpose Input Output) pins; therefore they are connected to the microcontroller. The microcontroller itself is connected to the computer via a USB cable. The hardware devices, which are directly connected via USB cables, can be accessed using various software modules, which are imported into the ’AA’ program. The access to other devices only using GPIO pins is performed in the microcontroller and the ’AA’ program simply communicates with the microcontroller program via a serial connection for sending commands to actuators and receiving values from sensors. The software for this system is called AA (Agent for Animals). This software was build with helps of many external libraries such as OpenCV [Bra00], and NumPy/SciPy [Jon01]. Once it starts, seven processes were launched using multiprocessing package of Python and it runs until the user terminates the program. The multiprocessing was used because the heavy calculation for image processing from multiple webcams were concerned. The number of processes can be changed as some of them can be turned on or off. These processes include a video-in process for each camera, a video-out process, an audio-in process, an audio-out process, a schema process, and a message-board process. Figure 1. Even though some of these processes have quite simple tasks, they were separated in order to prevent them from interfering with each other and/or becoming the bottleneck. The system has to process the visual, auditory, and other sensory and motor information simultaneously to recognize the change of the environment and CATOS: COMPUTER AIDED TRAINING/OBSERVING SYSTEM 17 Fig. 2: AA_DataViewer respond to it properly. The output data such as captured video input images, recorded WAV files, movement-records, CSV files for trial results, and the log file are temporarily stored in the ’output’ folder. After the daily session is finished, all of these output files go through an archiving process which can include, but is not restricted to, generating movies, generating images with the movement analysis, labeling sound files, and moving different types of files into the categorized subfolders of an archiving folder named with a timestamp. Besides combining all the above modules and implementing some common functions, one more Python program was implemented to facilitate the process of analyzing the recorded data. The program is called ”AA DataViewer”, which is based on wxPython GUI toolkit and Matplotlib [Hun07] for drawing graphs. Figure 2. It loads the log file, the result CSV (comma separated values) file containing the results of the trial, the movement-record CSV files, the MP4 movie files, and the WAV files from one folder containing all data collected for one session (day). For each video clip, there is a JPEG image showing the movements of the blobs. The circles in the image represent the positions of the blobs and their color represents the time-flow, with the black corresponding to the beginning of the movie, and the white to the end of the movie. A line connecting multiple circles means that those blobs occurred at the same time. Another feature of this program is its ability to generate a graph with selected sessions. In the ’archive’ folder, there are sub-folders, each of which contains all the data for a session. When the ’select sessions’ button is clicked, a pop-up window appears for selecting multiple folders. The result data from these selected sub-folders of ’archive’ folder is drawn as a graph using Matplotlib [Hun07]. By visualizing the data for certain period, it helps the trainer or experimenter quickly assess the current status of the training procedure. The two feeders used in this study is a device mainly comprising the Arduino microcontroller; (refer http://www. arduino.cc/), a motor-shield for the microcontroller, a servomotor, and a frame encasing the whole feeder. Both Feeder variants work in a similar way, by rotating the servomotor by a certain number of degrees, although the second feeder shows better performance in terms of consistent amount of Fig. 3: Automatic feeder Fig. 4: Circuit with a microcontroller food released, due to the usage of an Archimedes’ screw. Initially, an estimate of the amount of food left in the food container was obtained using an IR distance sensor, but this feature was discarded in the second build since the distance information from the IR sensor was not accurate enough for this application. The second feeder confirms the emission of a food reward via the piezoelectric sensor, which is positioned right below the Archimedes’ screw. Figure 3. Communication between the Arduino chip and the main computer was accomplished by using the Arduino module of the ’AA’ program. In the circuit, Figure 4, • • • • The temperature sensor measures the temperature inside of the protective wooden platform. The photocell sensor measures the ambient light level. The light bulb can be turned on when the photocell sensor indicates the ambient light level is below a user-defined threshold. Two fans are turned on when the temperature sensor indicates the temperature is too high in the platform. 18 PROC. OF THE 6th EUR. CONF. ON PYTHON IN SCIENCE (EUROSCIPY 2013) • • The piezoelectric sensor is read while the servomotor is actuating, in order to confirm the occurrence of the food reward. This sensor reading is required because occasionally the food dispensing fails due to the combination of the short motor activation time (<0.5 seconds) and the shape of the dry food pieces (which can fit into other pieces easily and then fail to emerge). The servomotor is responsible for the food dispense by turning the Archimedes’ screw back and forth. 3 R ESULTS OF BUILDING CATOS AND ITS TESTING ON 2 DOMESTICATED CATS The hardware and software were built and tested. The software is available at https://github.com/jinook0707/CATOS_alpha with GNU General Public License, version 3. Both hardware and software are curretnly in its alpha stage. Although its potential to be used to train and test animal cognition was tested and its usage seemed promising to save human resources in certain situations, both hardware and software should be developed further to be practically used for experimenting animal cognition. The two web-cams observed the experimental area for 8 to 12 hours per day for about 5 months (from the middle of October 2012 to the middle of March 2013). The movement records, MP4 movie files, JPEG image files, and WAV sound files generated during this period took 37.35 Giga bytes of storage. To obtain a rough idea of the degree of reduction in data storage that was achieved using the system, the number of recorded frames in the video recording was assessed. Data for 15 days were taken to calculate it. The total observation period was 406138 seconds, corresponding to 112.8 hours. The number of frames recorded was 206024 and the average FPS(Frame Per Second) was 7.5, therefore, approximately, the video recordings were stored for 27470 seconds (=7.6 hours), which is about 6.7 percent of entire observation period. These specific numbers are not very meaningful since they can fluctuate with the increase or decrease of the subject’s movements, but the point is that the most of the meaningless recordings were successfully filtered out by CATOS. Human presence during session is not necessary. Data transfer from one computer to another, maintenance, or modification of the system requires human interaction, but no time and effort is required concerning the training and testing sessions. Because no one attends the sessions, a periodic analysis of the animal’s performance with the system is required. A simple assessment of how much food the animals took, or more specifically, how many correct and incorrect trials occurred, can be done quickly since this information is already stored in result CSV file displaying the number of correct and incorrect trials generated with timestamps at the end of each session. Also, the data-viewer utility program displays all the timestamps and its JPEG image, which presents a brief report on the movement detected in the recorded video-clip. Thus, simply browsing the JPEG images is often enough to assess the session. If it is not enough, then one can obtain a more detailed assessment by playing the video-clips recorded around the trial times. Fig. 5: Recent performance of the trained cat on three human speech discrimination task. Two domesticated cats were trained for testing the system. Both cats learned that approaching the feeder on a playback sound could lead to a food reward. Then one cat further learned that pressing one out of three buttons could lead to a food reward. The training of the association between three different sound stimuli and three different buttons is an ongoing process. The most recent performance data Figure 5 shows over 70 percent of overall performance and also the performance on each button is significantly higher than 33.3 percent of chance level. R EFERENCES [Bel09] N. Bellotto, E. Sommerlade, B. Benfold, C. Bibby, I. Reid, D. Roth, C. Fernandez, L.V. Gool and J. Gonzalez. A distributed camera system for multi-resolution surveillance, Proc. of the third ACM/IEEE Int. Conf. on Distributed Smart Cameras (ICDSC), 2009. [Bra00] G. Bradski. 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Graphical Nonconvex Optimization for Optimal Estimation in Gaussian Graphical Models Qiang Sun⇤, Kean Ming Tan†, Han Liu‡ and Tong Zhang§ arXiv:1706.01158v1 [stat.ML] 4 Jun 2017 Abstract We consider the problem of learning high-dimensional Gaussian graphical models. The graphical lasso is one of the most popular methods for estimating Gaussian graphical models. However, it does not achieve the oracle rate of convergence. In this paper, we propose the graphical nonconvex optimization for optimal estimation in Gaussian graphical models, which is then approximated by a sequence of convex programs. Our proposal is computationally tractable and produces an estimator that achieves the oracle rate of convergence. The statistical error introduced by the sequential approximation using the convex programs are clearly demonstrated via a contraction property. The rate of convergence can be further improved using the notion of sparsity pattern. The proposed methodology is then extended to semiparametric graphical models. We show through numerical studies that the proposed estimator outperforms other popular methods for estimating Gaussian graphical models. Keywords: Adaptivity, Graphical nonconvex optimization, Nonconvexity, Semiparametric, Sequential convex approximation. 1 Introduction We consider the problem of learning an undirected graph G = (V, E), where V = {1, . . . , d} contains nodes that represent d random variables, and the edge set E describes the pairwise conditional dependence relationships among the d random variables. Gaussian graphical models have been widely used to represent pairwise conditional dependencies among a set of variables. Let X be a d-dimensional random variables. Under the Gaussian assumption X ⇠ N (0, ⌃⇤ ), the graph G is encoded by the sparse concentration matrix ⌦ = (⌃⇤ ) 1 , or the sparse inverse correlation matrix ⇤ = (C⇤ ) 1 . Here, C⇤ is the correlation matrix such that ⌃⇤ = WC⇤ W and W2 is a diagonal matrix with ⇤ Department of Operations Research and NJ 08544; e-mail: [email protected]. † Department of Operations Research and NJ 08544; e-mail: [email protected]. ‡ Department of Operations Research and NJ 08544, USA; e-mail:[email protected]. § Tencent AI Lab, Shen Zhen, Guangdong, Financial Engineering, Princeton University, Princeton, Financial Engineering, Princeton University, Princeton, Financial Engineering, Princeton University, Princeton, China; e-mail:[email protected]. 1 diagonal elements of ⌃⇤ . In particular, it is well known that the jth and kth variables are conditionally independent given all of the other variables if and only if the (j, k)-th element of ⌦⇤ (or ⇤ ) is equal to zero. Thus, inferring the conditional dependency structure of a Gaussian graphical model boils down to estimating a sparse inverse covariance (or correlation) matrix. A number of methods have been proposed to estimate the sparse concentration matrix under the Gaussian assumption. For example, Meinshausen and Bühlmann (2006) proposed a neighborhood selection approach for estimating Gaussian graphical models by solving a collection of sparse linear regression problems using the lasso penalty. In addition, Yuan (2010) and Cai et al. (2011) proposed the graphical Dantzig and CLIME, both of which can be solved efficiently. From a di↵erent perspective, Yuan and Lin (2007) and Friedman et al. (2008) proposed the graphical lasso methodology, a penalized likelihood based approach, to estimate the concentration matrix ⌦⇤ directly. Various extensions of the graphical lasso were proposed and the theoretical properties were also studied (among others, Banerjee et al., 2008; Rothman et al., 2008; Ravikumar et al., 2011). The Gaussian graphical models literature is vast and we refer the reader to Cai et al. (2016a) and Drton and Maathuis (2016) for recent reviews on this topic. Despite the large literature on using the graphical lasso to estimate concentration matrices in Gaussian graphical models, the graphical lasso does not achieve the oracle rate of convergence. More specifically, it is belived that the optimal rate of convergence p in spectral norm for the graphical lasso is at the order of s log d/n (Rothman et al., 2008). Here, n is the sample size, d is the number of nodes, and s is the number of edges in the true graph. In fact, the graphical lasso and all of the aforementioned methods are based on the lasso penalty and it is well known that convex penalties usually introduce non-negligible estimation bias. For example, in the linear regression setting, Fan and Li (2001); Zhang (2010a,b); Fan et al. (2017) have shown that the nonconvex penalized regression is able to eliminate the estimation bias and attain a more refined statistical rate of convergence. Based on these insights, we consider the following penalized maximum likelihood estimation with nonconvex regularizers: ⇢ X ⌦ ↵ b b ⇥ = argmin ⇥, ⌃ log det(⇥) + p ⇥ij , (1.1) d ⇥2S+ i6=j d = {A 2 Rd⇥d : A = AT , A where S+ 0} is the symmetric definite cone formed by all b is the sample covariance symmetric positive definite matrices in d ⇥ d dimensions, ⌃ matrix, and p (·) is a nonconvex penalty. Here, hA, Bi = tr(AT B) denotes the trace of AT B. However, from the computational perspective, minimizing a folded concave penalized problem is very complicated due to its intrinsic nonconvex structure. Indeed, Ge et al. (2015) have shown that solving (1.1) with a general concave penalty, such as the SCAD Fan and Li (2001) or the MCP Zhang (2010a), is strongly NP-hard. In other words, there does not exist a fully polynomial-time approximation scheme for problem (1.1) unless more structures are assumed. Recently, Loh and Wainwright (2015) proposed an algorithm to obtain a good local optimum for (1.1), but an additional convex constraint that depends on the unknown true concentration matrix is imposed. Moreover, 2 they failed to provide a faster rate of convergence statistically due to not taking the signal strength into account. In this paper, instead of directly solving the nonconvex problem (1.1), we propose to approximate it by a sequence of adaptive convex programs. Even though the proposed approach is solving a sequence of convex programs, under some regularity conditions, we show that the proposed estimator for estimating the sparse concentration matrix achieves p the oracle rate of convergence of s/n, treating as if the locations of the nonzeros were known a priori. This is achieved by a contraction property. Roughly speaking, each convex program gradually contracts the initial estimator to the region of oracle rate of convergence even when a bad initial estimator is used in the first place: r s 1 b (` 1) ⇤ ⇤ b (`)  C + , F F n 2 \| {z } \| {z } Oracle Rate Contraction where b (`) is the inverse correlation matrix estimator after the `-th convex approxip mation, k · kF denotes the Frobenius norm, C is a constant, and s/n is referred to as the oracle rate. Each iteration of the proposed method helps improve the accuracy ⇤ k dominates the statistical error. The error caused by each only when k b (` 1) F iteration is clearly demonstrated via the proven contraction property. By rescaling the inverse correlation matrix using the estimated marginal variances, we obtain an estimator of the concentration matrix with spectral norm convergence rate in the order of p p log d/n _ s/n. Here, a _ b = max{a, b} is used to denote the maximum of a and b. By exploiting a novel notion called sparsity pattern, we further sharpens the rate of convergence under the spectral norm. The rest of this paper proceeds as follows. In Section 2, we propose the new methodology and its implementation. Section 3 is devoted to theoretical studies. We show that the proposed methodology can be extended to the semiparametric graphical models in Section 4. Numerical experiments are provided to support the proposed methodology in Section 5. We conclude the paper in Section 6. All the proofs and technical details are collected in the supplementary material. Notation: We summarize the notation that will be used regularly throughout the paper. Given a vector u = (u1 , u2 , . . . , ud )T 2 Rd , we define the `q -norm of u by kukq = P ( dj=1 \|uj \|q )1/q , where q 2 [1, 1). For a set A, let \|A\| denote its cardinality. For a matrix A = (ai,j ) 2 Rd⇥d , we use A 0 to indicate that A is positive definite. For q 1, we use kAkq = maxu kAukq /kukq to denote the operator norm of A. For index sets I, J ✓ {1, . . . , d}, we define AI,J 2 Rd⇥d to be the matrix whose (i, j)-th entry is equal to ai,j if i 2 I and j 2 J , and zero otherwise. We use A B = (aij bij ) to denote the Hadamard product of two matrices A and B. Let diag(A) denote the diagonal matrix consisting diagonal elements of A. We use sign(x) to denote the sign of x: sign(x) = x/\|x\| if x 6= 0 and sign(x) = 0 otherwise. For two scalars fn and gn , we use fn & gn to denote the case that fn cgn , and fn . gn if fn  Cgn , for two positive constants c and C. We say fn ⇣ gn , if fn & gn and fn . gn . OP (·) is used to denote bounded in probability. We use c and C to denote constants that may vary from line to line. 3 2 A Sequential Convex Approximation Let X = (X1 , X2 , . . . , Xd )T be a zero mean d-dimensional Gaussian random vector. Then its density can be parameterized by the concentration matrix ⇥⇤ or the inverse correlation matrix ⇤ . The family of Gaussian distributions respects the edge structure of a graph G = (V, E) in the sense that ⇤ij = 0 if and only if (i, j) 62 E. This family is known as the Gauss-Markov random field with respect to the graph G. The problem of estimating the edge corresponds to parameter estimation, while the problem of identifying the edge set, i.e., the set E ⌘ {i, j 2 V \| i 6= j, ⇤ij 6= 0}, corresponds to the problem of model selection. Given n independent and identically distributed observations {X (i) }ni=1 of a zero mean d-dimensional random vector X 2 Rd , we are interested in estimating the inverse (i) (i) T b = n 1P correlation matrix ⇤ and concentration matrix ⇥⇤ . Let ⌃ 1in X (X ) b =W c 1⌃ bW c 1 , where W c 2 = diag(⌃). b To be the sample covariance matrix and let C estimate b (`) ⇤, we propose to adaptively solve the following sequence of convex programs n⌦ o ↵ (` 1) b = argmin ,C log det( ) + k k1,o↵ , for ` = 1, . . . , T, (2.1) d 2S+ P (` 1) where k⇥k1,o↵ = i6=j \|⇥ij \|, (` 1) = · w b ij is a d ⇥ d adaptive regularization matrix for a given tuning parameter and a weight function w(·), and T indicates the number of total convex programs needed. The weight function w(·) can be taken to be w(t) = p0 (t)/ , where p (t) is a folded concave penalty such as the SCAD or the MCP proposed by Fan and Li (2001) and Zhang (2010a), respectively. To obtain an estimate for the concentration matrix estimator ⇤ , we rescale b (T ) e (T ) = W c 1 b (T ) W c 1 after the T -th convex program. This rescaling helps back to ⇥ e (T ) significantly by eliminating the e↵ect introimprove the rate of convergence for ⇥ duced through the unpenalized diagonal terms. The detailed routine is summarized in Algorithm 1. Algorithm 1 A sequential convex approximation for the graphical nonconvex optimization. b regularization parameter . Input: Sample covariance matrix ⌃, b by C b = W c 1⌃ bW c 1 , where W c 2 is a Step 1: Obtain sample correlation matrix C b diagonal matrix with diagonal elements of ⌃. Step 2: Solve a sequence of graphical lasso problem adaptively n o b (`) = argmin h , Ci b log det( ) + k (` 1) k1,o↵ , d 2S+ and (`) = (`) · w( b ij ), for ` = 1, . . . , T. e (T ) = W c Step 3: Obtain an estimate of ⇥⇤ by ⇥ 1 b (T ) W c 1. The complexity of Step 2 in Algorithm 1 is O(d3 ) per iteration: this is the complexity of the algorithm for solving the graphical lasso problem. We will show in the latter section 4 that the number of iteration can be chosen to be T ⇡ log log d based on our theoretical analysis. Algorithm 1 can be implemented using existing R packages such as glasso. 3 Theoretical Results In this section, we study the theoretical properties of the proposed estimator. We start with the assumptions needed for our theoretical analysis. 3.1 Assumptions Let S = (i, j) : ⇥⇤ij 6= 0, i 6= j be the support set of the o↵-diagonal elements in ⇥⇤ . Thus, S is also the support set of the o↵-diagonal elements in ⇤ . The first assumption we need concerns the structure of the true concentration and covariance matrices. Assumption 3.1 (Structural Assumption). We assume that \|S\|  s, k⌃⇤ k1  M < 1, 0 < "1  min  max  1/"1 < 1, 0 < "2  min (⇥⇤ )  max (⇥⇤ )  1/"2 < 1. 2 2 Here, max = maxj ⌃⇤jj and min = minj ⌃⇤jj , where ⌃⇤ = ⌃⇤ij . Assumption 3.1 is standard in the existing literature for Gaussian graphical models (see, for instance, Meinshausen and Bühlmann, 2006; Yuan, 2010; Cai et al., 2016b; Yuan and Lin, 2007; Ravikumar et al., 2011). We need min and max to be bounded from above and below to guarantee reasonable performance of the concentration matrix estimator (Rothman et al., 2008). Throughout this section, we treat M, "1 , "2 as constants to simplify the presentation. The second assumption we need in our analysis concerns the weight functions, which are used to adaptively update the regularizers in Step 2 of Algorithm 1. Define the following class of weight functions: n o W = w(t) : w(t) is nonincreasing , 0  w(t)  1 if t 0, w(t) = 1 if t  0 . (3.1) Assumption 3.2 (Weight Function). There exists an ↵ such that the weight function w(·) 2 W satisfies w(↵ ) = 0 and w(u) 1/2, where u = c for some constant c. The above assumption on the weight functions can be easily satisfied. For example, it can be satisfied by simply taking w(t) = p0 (t)/ , where p (t) is a folded concave penalty such as the SCAD or the MCP (Fan and Li, 2001; Zhang, 2010a). Next, we impose an assumption on the magnitude of the nonzero o↵-diagonal entries in the inverse correlation matrix ⇤ . Assumption 3.3 (Minimal Signal Strength). Recall that S is the true support set. The minimal signal satisfies that min(i,j)2S ⇤ij (↵ + c) & , where c > 0 is the same constant that appears in Assumption 3.2. Assumption 3.3 is rather mild. In the sub-Gaussian design case, can be taken to p be the order of log d/n, which diminishes quickly as n increases. It is an analogue to the minimal signal strength assumption frequently assumed in nonconvex penalized regression problems (Fan and Li, 2001; Zhang, 2010a). Taking the signal strength into account, we can then obtain the oracle rate of convergence. 5 3.2 Main Theory We now present several main theorems concerning the rates of convergence of the proposed estimator for the sparse inverse correlation and the concentration matrices. The following theorem concerns the rate of convergence for the one-step estimator b (1) obtained from Algorithm 1 when ` = 1. p Proposition 3.4 (One-step Estimator). Let ⇣ log d/n. Under Assumption 3.1, we have r s log d (1) ⇤ b . F n with probability at least 1 8/d, Proof of Proposition 3.4. We collect the proof of Proposition 3.4 in Appendix A in the supplementary material. The above proposition indicates that the statistical error under the Frobenius norm p for the one-step estimator is at the order of s log d/n, which is believed to be unimprovable when one-step convex regularization is used (Rothman et al., 2008; Ravikumar et al., 2011). However, when a sequence of convex programs is used as in our proposal, the rate of convergence can be improved significantly. This is demonstrated in the following theorem. Theorem 3.5 (Contraction Property). Suppose that n & s log d and take such that p ⇣ log d/n. Under Assumptions 3.1, 3.2 and 3.3, b (`) satisfies the following contraction property: b (`) ⇤ F  8k \| with probability at least 1 ⇤ 2 k2 krL( {z Oracle Rate ⇤ 1 b (` 1) )S kF + } \|2 {z Contraction ⇤ , 1  `  T, } F p 8/d. Moreover, if T & log( n) & log log d, we have ✓r ◆ s ⇤ b (T ) = OP . F n Proof of Theorem 3.5. The proof is collected in Appendix A in the supplementary material. Theorem 3.5 establishes a contraction property: each convex approximation contracts the initial estimator towards the true sparse inverse correlation matrix until it reaches p the oracle rate of convergence: s/n. To achieve the oracle rate, we need to solve no more than approximately log log d convex programs. Note that log log d grows very slowly as d increases and thus, in practice, we only need to solve a few convex programs to get a better estimator than existing method such as the graphical lasso. The rate p of convergence s/n is better than the existing literature on likelihood-based methods for estimating sparse inverse correlation matrices (Rothman et al., 2008; Lam and Fan, 2009a; Ravikumar et al., 2011). By rescaling, we obtain a concentration matrix estimator with a faster rate of convergence. 6 Theorem 3.6 (Faster Rate in Spectral Norm). Under the same conditions in Theorem 3.5, we have r ✓r ◆ s log d (T ) ⇤ e ⇥ ⇥ 2 = OP _ . n n Proof of Theorem 3.6. The proof is deferred to Appendix A in the supplementary material. The theorem above provides the optimal statistical rate for estimating sparse concentration matrices using likelihood based methods (Rothman et al., 2008; Lam and Fan, 2009b; Ravikumar et al., 2011). The extra log d term is a consequence of estimating the marginal variances. We further sharpen the obtained theory using a novel notion, called sparsity pattern, as defined below. Definition 3.7 (Sparsity Pattern). For a matrix A = aij , we say Asp = asp ij is the sp sp corresponding sparsity pattern matrix if aij = 1 when aij 6= 0; and aij = 0, otherwise. Let M⇤ be the sparsity pattern matrix of ⇤ or ⇥⇤ . Our next theorem provides an improved rate of convergence using this newly defined notion of sparsity pattern. Theorem 3.8 (Improved Convergence Rate using Sparsity Pattern). Suppose that n & p (s + s2max ) log d and take such that ⇣ log d/n. Let T & log s. Under Assumptions 3.1, 3.2 and 3.3, we have r ◆ ✓ 1 ⇤ b (T ) = OP kM⇤ k2 , and 2 n r r ✓ ◆ 1 log d (T ) ⇤ ⇤ e ⇥ ⇥ 2 = OP kM k2 _ . n n Proof of Theorem 3.8. The proof is deferred to Appendix B in the supplementary material. Theorem 3.8 suggests that the rates of convergence can be bounded using the spectral norm of the sparsity pattern matrix M⇤ , which are sometimes much sharper than those provided in Theorems 3.5 and 3.6. To demonstrate this observation, we consider a sequence of chain graphs specified by the following sparsity pattern matrices: " # A 0 k Mck = , for k = 4, . . . , 50, 0 Id k 1 where Ak 2 R(k+1)⇥(k+1) such that the (i, j)-th entry Ak,ij = 1 if \|i j\|  1, and Ak,ij = 0 otherwise. Id k 1 2 R(d k 1)⇥(d k 1) is the identity matrix. Let sk be the total sparsity of Mck , that is sk = 2k. We plot the ratio of the two rates of convergence for estimating ⇤ in Theorems 3.5 and 3.8, kMc k2 /s , versus s in Figure 1. From Figure 1, we can k k 2 k see that the ratio goes to 0 as the total sparsity increases. This demonstrates that the convergence rate in Theorem 3.8 is indeed much sharper than that in Theorem 3.5, as least for the chain graphs constructed above. We also observe similar but less significant improvement for star-shape graphs. In Figure 2, we give an geometric illustration of the star and chain graphs. 7 Chain Graph 0.8 ● 0.6 ● ● ● ● 0.4 kMck k22 /sk ● ● ● ● ● ● ● ● 0.2 ● ● 20 ● ● ● ● ● ● ● ● ● 40 ● ● ● ● 60 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 80 ● ● 100 sk Figure 1: Convergence rates using sparsity pattern matrix Mck and total sparsity sk . Star Graph Chain Graph Figure 2: An illustration of the star and chain graphs. 4 Extension to Semiparametric Graphical Models In this section, we extend the proposed method to modeling semiparametric graphical models. We focus on the nonparanormal family proposed by Liu et al. (2012), which is a nonparametric extension of the normal family. More specifically, we replace the random variable X = (X1 , . . . , Xd )T by the transformation variable f (X) = (f1 (X1 ), . . . , fd (Xd ))T , and assume that f (X) follows a multivariate Gaussian distribution. Definition 4.1 (Nonparanormal). Let f = {f1 , . . . , fd }T be a set of monotone univariate functions and let ⌃npn 2 Rd⇥d be a positive-definite correlation matrix with diag(⌃npn ) = 1. A d-dimensional random variable X = (X1 , . . . , Xd )T has a nonparanormal distribution X ⇠ NPNd (f, ⌃npn ) if f (X) ⌘ (f (X1 ), . . . , fd (Xd ))T ⇠ Nd (0, ⌃npn ). We aim to recover the precision matrix ⇥npn = (⌃npn ) 1 . The main idea behind this procedure is to exploit Kendall’s tau statistics to directly estimate ⇥npn , without explicitly calculating the marginal transformation functions {fj }dj=1 . We consider the following Kendall’s tau statistic: X 2 (i) (i0 ) (i) (i0 ) ⌧bjk = sign (Xj Xj )(Xk Xk ) . n(n 1) 0 1i<i n The Kendall’s tau statistic ⌧bjk represent the nonparametric correlations between the empirical realizations of random variables Xj and Xk and is invariant to monotone trans8 ej and X ek be two independent copies of Xj and Xk . The population formations. Let X ej ), sign(Xk X ek ) . We need version of Kendall’s tau is given by ⌧jk ⌘ Corr sign(Xj X the following lemma which is taken from Liu et al. (2012). It connects the Kendall’s tau statistics to the underlying Pearson correlation coefficient ⌃npn . Lemma 4.2. Assuming X ⇠ NPNd (f, ⌃), we have ⌃0jk = sin ⌧jk ·⇡/2 . b = [Sbjk ] for the Motivated by this Lemma, we define the following estimators S unknown correlation matrix ⌃npn : ( sin ⌧bjk ·⇡/2 , j 6= k, ⌧ Sbjk = 1, j = k. Now we are ready to prove the optimal spectral norm rate for the Gaussian copula graphical model. The results are provided in the following theorem. p Theorem 4.3. Assume that n & s log d and let ⇣ log d/n. Under Assumptions 3.1, b (`) satisfies the following contraction property: 3.2 and 3.3, ⇥ b (`) ⇥⇤ ⇥ F 1 b (` 1)  4k⇥⇤ k22 krL(⇥⇤ )S kF + ⇥ ⇥⇤ F , 1  `  T, \| {z } \|2 {z } with probability at least 1 Optimal Rate Contraction 8/d. If T & log( b (T ) ⇥⇤ ⇥ F p n) & log log d, we have ✓r ◆ s = OP . n Proof of Theorem 4.3. The proof is deferred to Appendix C in the supplementary material. 5 Numerical Experiments We compare our proposal to the graphical lasso (glasso) (Friedman et al., 2008) and neighborhood selection (NS) (Meinshausen and Bühlmann, 2006). Each of these approaches learns a Gaussian graphical model via an `1 penalty on each edge. To evaluate the performance across di↵erent methods, we define the true positive rate as the proportion of correctly identified edges in the graph, and the false positive rate as the proportion of incorrectly identified edges in the graph. In addition, we calculate the difference between the estimated and true concentration matrix under the Frobenius norm. We do not compute this quantity for the NS approach since they do not estimate the concentration matrix directly. For our proposal, we consider T = 4 iterations with the SCAD penalty proposed by Fan and Li (2001) that takes the following form: 8 > if \|t\|  , > < p0 (t) = > > :0 \|t\| 1 if < \|t\| < otherwise, 9 , where > 2. In all of our simulation studies, we pick = 2.1. Each of the methods involves a sparsity tuning parameter: we applied a fine grid of tuning parameter values to obtain the curves shown in Figure 3. We consider cases with n = {150, 200} and d = 150 with two set-ups for a p ⇥ p adjacency matrix A: (i) random graph with 2.5% elements of A set to 1; (ii) band graph with Ai,i+1 = Ai+1,i = 1 for 1  i  d 1. We then use the adjacency matrix A to create a matrix E, as ( 0 if Aij = 0 Eij = 0.4 otherwise, and set E = 12 (E+ET ). Given the matrix E, we set ⇥ 1 equal to E+(0.1 emin )I, where emin is the smallest eigenvalue of E. We then standardize the matrix ⇥ 1 so that the i.i.d. diagonals are equal to one. Finally, we generate the data according to X (1) , . . . , X (n) ⇠ N (0, ⌃). We present the results averaged over 100 data sets for each of the two simulation settings with n = {150, 200} and p = 150 in Figure 3. Random graph (n=200) 0.0 ● ● ● ● 1.0 0.8 0.4 0.0 0.2 0.4 0.6 0.8 0.2 0.0 ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 3 Our Proposal Glasso 2 0.0 0.1 0.2 ● ● ● ● ● ● ●●●●● ● ●●●● ●●●●●●●●●● ● ●●●●● ● ● ●●● ●● ● ● ● ● ● ● ● ● 0.3 False positive rate 0.4 5 Our Proposal NS Glasso 0.2 ●● ● ● ● ● 0.4 0.6 0.8 0.8 Our Proposal NS Glasso ● ● 0.0 7 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● Our Proposal Glasso 2 0.0 0.1 0.2 0.3 0.4 6 0.2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● 0.4 0.6 0.8 0.0 ● 0.2 8 7 ● ● ● ● ● 4 ● ● ● ● ● 3 ● ● ● ● ● ● ● ● ● Our Proposal Glasso ● ● ● ● ● ● 0.0 0.1 0.2 0.3 0.4 0.6 0.8 1.0 False positive rate ● False positive rate False positive rate ●● Our Proposal NS Glasso ● ● 0.0 1.0 ● 5 2 ● ● ● 0.2 Band graph (n=150) ● ● ●● ● ● False positive rate ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.4 ● 0.0 ● ● ● ● ● ● ● ●●● ●●●● ●●● ●●●●●●●●● ●● ●●● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.6 0.2 1.0 1.0 ● ● 8 ● 3 ●● 0.4 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 4 ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Random graph (n=200) Frobenius norm Frobenius norm 4 ● 0.8 8 6 Band graph (n=200) Band graph (n=150) 1.0 ● ●● 0.6 7 ● ● ● False positive rate 7 5 ●● ● ● ● ● ● ● ● ● 0.0 1.0 ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● Random graph (n=150) 6 ●● ● ● False positive rate 8 ●● ● ● ● ● 0.6 Our Proposal NS Glasso ●● ● ● ●● ● ● ● ●● ●●●● ● ●●●● ● ● ● ●● ●●● ●●● ●● ● ●● ● True positive rate 0.2 ● 0.4 Frobenius norm 0.4 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Frobenius norm 0.6 ● ● True positive rate True positive rate 0.8 ● ● ● ● ● ● ● ●● ● ● ● ●●● ● ● ●● ● ● ●● ● ●●●●● ●● ●●● ●● ●●● ● ● ● True positive rate Random graph (n=150) 1.0 6 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● Band graph (n=200) ● ● 5 ● ● ● ● 4 ● ● ● ● 3 2 ● ● ● ● ● ● ● ●● ● ● 0.0 ● ● ● ● Our Proposal Glasso ● 0.1 0.2 0.3 0.4 False positive rate Figure 3: Row I: True and false positive rates, averaged over 100 data sets with p = 150, for random and band graphs, respectively. Row II: Di↵erence between the estimated and the true inverse covariance matrices under the Frobenius norm. The di↵erent curves are obtained by varying the sparsity tuning parameter for each of the methods. From Row I of Figure 3, we see that our proposal is very competitive relative to the existing proposals for estimating Gaussian graphical models in terms of true and false positive rates across all simulation settings. Row II of Figure 3 contains the di↵erence between the estimated and the true inverse covariance matrices under the Frobenius norm as a function of the false positive rate. For random graph with n = 150, we see that the minimum error under the Frobenius norm for our proposal is smaller than that of the graphical lasso. As we increase the number of observations to n = 200, the di↵erence between the minimum error for the two proposals are more apparent. More interestingly, 10 the region for which our proposal has lower Frobenius norm than the graphical lasso is the primary region of interest. This is because an ideal estimator is one that has a low false positive rate while maintaining a high true positive rate with low error under the Frobenius norm. In contrast, the region for which the graphical lasso does better under the Frobenius norm is not the primary region of interest due to the high false positive rate. We see similar results for the band graph setting. 6 Conclusion and Discussions We propose the graphical nonconvex optimization, which is then approximated by a sequence of convex programs, for estimating the inverse correlation and concentration matrices with better rates of convergence comparing with existing approaches. The proposed methodology is sequential convex in nature and thus is computationally tractable. Yet surprisingly, it produces estimators with oracle rate of convergence as if the global optimum for the penalized nonconvex problem could be obtained. Statistically, a contraction property is established: each convex program contracts the previous estimator by a 0.5-fraction until the optimal statistical error is reached. Our work can be applied to many di↵erent topics: low rank matrix completion problems, high-dimensional quantile regression and many others. We conjecture that in all of the aforementioned topics, a similar sequential convex approximation can be proposed and can possibly give faster rate, with controlled computing resources. It is also interesting to see how our algorithm works in large-scale distributed systems. Is there any fundamental tradeo↵s between statistical efficiency, communication and algorithmic complexity? We leave these as future research projects. References Banerjee, O., El Ghaoui, L., and d’Aspremont, A. 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(2010), “High dimensional inverse covariance matrix estimation via linear programming,” Journal of Machine Learning Research, 11, 2261–2286. Yuan, M. and Lin, Y. (2007), “Model selection and estimation in the Gaussian graphical model,” Biometrika, 94, 19–35. Zhang, C.-H. (2010a), “Nearly unbiased variable selection under minimax concave penalty,” The Annals of Statistics, 38, 894–942. Zhang, T. (2010b), “Analysis of multi-stage convex relaxation for sparse regularization,” The Journal of Machine Learning Research, 11, 1081–1107. 12 Supplementary Material to “Graphical Nonconvex Optimization for Optimal Estimation in Gaussian Graphical Models” Qiang Sun, Kean Ming Tan, Han Liu and Tong Zhang Abstract This supplementary material collects proofs for the main theoretical results in the main text and additional technical lemmas. The proofs of Proposition 3.4, Theorems 3.5 and 3.6 are collected in Section A. Section B provides the proof for Theorem 3.8. Proofs related to semiparametric graphical models are given in Section C. Various concentration inequalities and preliminary lemmas are postponed to Sections D and E, respectively. A Rate of Convergence in Frobenius Norm This section presents an upper bound for the adaptive estimator b (`) in Frobenius norm, which in turn helps establish the scaling conditions needed to achieve the optimal spectral norm convergence rate. A.1 Proofs of Proposition 3.4, Theorems 3.5 and 3.6 In this section, we collect the proofs for Proposition 3.4, Theorems 3.5 and 3.6. (` 1) In order to suppress the noise at the `th step, it is necessary to control min(i,j)2S b ij in high dimensions. For this, we construct an entropy set, E` , of S and analyze the mag(` 1) nitude of . The entropy set at the `-th stage, E` , is defined as Ec min ` n E` = (i, j) : (i, j) 2 S or (` 1) ij < w(u), for u = 2 32k ⇤ 2 k2 + k⌃⇤ k21 _ 1 o . (A.1) Thus the constant in Assumption 3.3 is c = 2(32k ⇤ k22 + k⌃⇤ k21 _ 1). Then it can be seen that S ✓ E` , and thus E` is an entropy set of S for any ` 1. Proposition 3.4 follows from a slightly more general result below, which establishes rate of convergence for the one-step estimator of sparse inverse correlation matrix b (1) . Proposition A.1 (One-step Estimator). Assume that assumption 3.1 holds. Suppose p p 8k ⇤ k22 s < 1. Take such that ⇣ (log d)/n and suppose n & log d. Then with probability at least 1 8/d, b (1) must satisfy r s log d ⇤ b (1)  Ck ⇤ k22 . F n 1 b C⇤ kmax  /2 . Then in the Proof of Proposition A.1. Define the event J = kC ⇤ k  4k ⇤ k2 · event J , by applying Lemma q A.4 and taking E = S, we obtain k b (1) F 2 p p p 1 s. If we further take = 3c2 (log d)/n ⇣ (log d)/n, then by Lemma D.5, we have event J hold with probability at least 1 8d 1 . The result follows by plugging the choice of . Theorems 3.5 and 3.6 follow form a slightly more general result below, which chare (T ) in spectral acterizes the rate of convergence of b (`) in Frobenius norm and that of ⇥ norm. p Theorem A.2. Assume that assumptions 3.1, 3.2 and 3.3. Suppose that 8k ⇤ k22 s < p 1. Take such that ⇣ log d/n. Then with probability at least 1 8d 1 , b (`) satisfies b (`) ⇤ F  8k \| Moreover, if that T & log( ⇤ 2 k2 krL( ⇤ {z Optimal Rate p 1 b (` 1) )S kF + } \|2 {z , 1  `  T. } F Contraction p s/n , and F r r ◆ ⇤k log d _ k ⇤ k22 s 2 . 2 n n min min n), we have ✓ 3 k ⇤ ⇥ 2 = OP max 3 e (T ) ⇥ ⇤ b (T ) ⇤ = OP k ⇤ k2 2 Proof of Theorem A.2. Under the conditions of theorem, combining Proposition A.7 and Lemma D.5, we obtain the following contraction property of the solutions, { b (`) }T `=1 , 1 b (` 1) ⇤ . F 2 Next, we introduce an inequality by induction analysis. Specifically, if an  a0 + ↵an 1 , 8 n 2 and 0  ↵ < 1, then b (`) ⇤ F ⇤ 2 k2 krL(  4k an  a0 1 ⇤ ↵n 1 ↵ 1 ) S kF + + ↵n 1 a1 . ⇤ k2 krL( ⇤ ) k , we obtain that b (`) ⇤  8k ⇤ k22 krL( ⇤ )S kF + S F 2 F ⇤ ⇤ k respec. In the sequel, we bound krL( ⇤ )S kF and k b (1) F F p ⇤ k . 8k ⇤ k2 tively. By Proposition A.1, we have k b (1) s. Moreover, if we F 2 p p p T 1 (1) ⇤ ⇤ 2 b let T log( n) log 2 & log( n), then (1/2) k kF  16k k2 · s/n. p ⇤ ⇤ 2 On the other side, we have krL( )S kF = OP (k k2 · s/n), which follows from ⇤k = Lemma D.4. Therefore, combining the above results obtains us that k b (T ) F p OP k ⇤ k22 s/n . e (T ) ⇥⇤ k2 , we apply Lemma E.3 and obtain To achieve the statistical rate for k⇥ Taking a0 = 4k ` 1 b (1) 1/2 that e (T ) k⇥ ⇥ ⇤ k2 = c W + c  kW \| 1 c W 1 c +kW \| W 1 W b (T ) 1 1 W ⇤ ⇤c 1 1 2 b (T ) k2 k {z W ⇤ (R1) 1 W 1 k2 k {z (R3) c W ⇤ W c + W 2 c k2 +kW } \| c k2 kW 2 1 1 1 + 2 1 b (T ) 1 c k2 +kW } \| W c W ⇤ 1 1 W W 1 2 k2 k b (T ) k2 kW {z (R2) 1 1 k2 kW k k b (T ) {z2 1 (R4) b (T ) W 1 k2 } ⇤ k2 . } 1 2 We now bound terms (R1) to (R4) respectively. Before we proceed, we apply Lemma D.2 and the union sum bound to obtain that, for any " 0, ⇣ ⌘ n o n o c 2 W2 k2 > " max ⌃⇤  d · exp P kW n · C(") = exp n · C(")+log d , ii i 1 (" where C(") = 2 log(1 + ")). Suppose that 0  "  1/2, then we have n · C(")  p 2 n · " /3. Further suppose that n 36 log d and take " = 3 (log d)/n, we obtain that n · C(")+log d  2 log d and r ✓ ◆ log d 1 2 2 2 c P kW W k2 > 3 max ·  2, n d 2 . Therefore, we have W c 2 W2 = where we use the assumption that maxi ⌃⇤ii  max 2 ⇣ ⌘ p 2 2 2 c OP max · log d/n . Since W and W are diagonal and thus commutative. We note that, for any two event A and B, P(A) = P(A \ B) + P(A \ B c ) holds. Therefore, for any M > 0, we have r ✓ ◆ log d 1 1 2 c P W W > M max 2 n r ✓ log d c 1 W 1 >M 2 P W , max 2 n ◆ p 2 1 1 2 2 2 c c W W  2( 2+1) W 2 min W W W 2 2 ✓ ◆ p 2 2 c 1 W 1 > 2( 2+1) W c 2 W2 +P W W W . 2 2 min 2 Further using Lemma E.7 yields that r ✓ ◆ log d 1 1 2 c P W W > M max 2 n ✓ p 2 c 2 W2  P 2( 2+1) W 2 min W2 W \| {z +P \| ✓ (T1) c2 W W2 >2 2 {z 1 min W2 ◆ >M 2 2 max r log d n . ◆ } } (T2) 2 4 By taking M = M1 · kWk2 min (W2 ) = M1 · max / min and letting M1 ! 0, we 2 2 get (T1) ! 0. Under the assumption that max / min = O (n/ log d)1/3 , we have p 2 2 c 1 n/ log d , and thus (T2) ! 0. Therefore we obtain that W max / min = o p 4 3 W 1 2 = OP min max (log d)/n . Similarly, we have the following facts: b (T ) 2 = OP k ⇤ k2 , c W 1 2 = 1 min c = OP ( W 1 min ), and W 1 2 = Applying the above results to the terms (R1)-(R4). we obtain that r r ◆ ✓ ◆ ✓ 6 s s max log d 2 2 ⇤ 2 ⇤ 2 (R1) = OP min k k2 · 6 = OP min k k2 , n n min n r r ◆ ✓ 3 ◆ ✓ log d s max 2 ⇤ ⇤ 2 (R2) = (R3) = OP k k2 , (R4) = OP min k k2 . 3 n n min 3 1 min . Therefore, by combining the rate for terms (R1)-(R4), we obtain the final result. A.2 Technical Lemmas s (⇥, ⇥⇤ ) = Define the symmetrized Bregman divergence for the loss function L(·) as DL ⌦ ↵ ⇤ ⇤ d⇥d d⇥d rL(⇥) L(⇥ ), ⇥ ⇥ . For any matrix A 2 R , let A 2 R be the o↵ diagonal matrix of A with diagonal entries equal to 0, and A+ = A A be the diagonal mtrix. Lemma A.3. For the symmetrized Bregman divergence defined above, we have ⌦ s DL (⇥, ⇥⇤ ) = rL(⇥) rL(⇥⇤ ), ⇥ ⇥⇤ ↵ k⇥⇤ k2 + k⇥ 2 ⇥ ⇤ k2 k⇥ ⇥⇤ k2F . Proof of Lemma A.3. We use vec(A) to denote the vectorized form of any matrix A. Then by the mean value theory, there exists a 2 [0, 1] such that, ⌦ s DL (⇥, ⇥⇤ ) = rL(⇥) min (r 2 ↵ ⇥⇤ = vec(⇥ rL(⇥⇤ ), ⇥ L(⇥⇤ + ) k k2F , where ⇥⇤ )T r2 L(⇥⇤ + =⇥ ) vec(⇥ ⇥⇤ ) ⇥⇤ . By standard properties of the Kronecker product and the Weyl’s inequality (Horn and Johnson, 2012), we obtain that ⇣ ⌘ ⇣ ⌘ 1 2 ⇤ ) = min (⇥⇤ + ) ⌦ (⇥⇤ + ) min r L(⇥ + = k⇥⇤ + Finally, observing that  1, we obtain ⌦ s DL (⇥, ⇥⇤ ) = rL(⇥) Plugging the definition of 2 k⇥⇤ k2 + k k2 k2 2 ↵ rL(⇥⇤ ), . k⇥⇤ k2 + k k2 2 ⇥⇤ k2F . k⇥ obtains us the final bound. ⇤ k by using localized The following lemma characterizes an upper bound of k b F analysis. p Lemma A.4. Suppose 8k ⇤ k2 s < 1. Take E such that S ✓ E and \|E\|  2s. Further assume k E c kmin /2 krL( ⇤ )kmax . Let b be the solution to (B.4). Then b must satisfy kb ⇤ ⇤ 2 k2 kF  4k k S kF +krL( ⇤ )E kF  8k ⇤ 2 k2 p s. Proof of Lemma A.4. We start by introducing an extra local parameter r which satp p p p isfies 8k ⇤ k22 s < r  k ⇤ k2 . This is possible since \|E\|  2 s ! 0 and p 8k ⇤ k2 s < 1 by assumption. Based on this local parameter r, we construct an inter⇤ ), where t is taken such that k( e ⇤ k = r, mediate estimator: e = ⇤ + t · ( b F ⇤ e e if k( kF > r; t = 1 otherwise. Applying Lemma A.3 with ⇥1 = and ⇥2 = ⇤ obtains us ⇤ 2 +r 2 e ⇤ 2 F ⌦  rL( e ) 4 rL( ⇤ ), e ⇤ ↵ . (A.2) To bound the right hand side of the above inequality, we use Lemma E.2 to obtain s e DL ( , ⇤ s b )  tDL ( , ⇤ ⌦ ) = t rL( b ) rL( ⇤ ), b ⇤ ↵ . (A.3) We note that the sub-di↵erential of the norm k · k1,o↵ evaluated at consists the set d⇥d of all symmetric matrices 2 R such that ij = 0 if i = j; ij = sign( ij ) if i 6= j and ij 6= 0; ij 2 [ 1, +1] if i 6= j and ij = 0, where ij is the (i, j)-th entry of . Then by the Karush-Kuhn-Tucker conditions, there exists a b 2 @k b k1,o↵ such that b =C b b 1+ b = 0. Plugging (A.3) into (A.2) and adding the term rL( b )+ ⇤ i on both sides of (A.3), we obtain b, b h (k ⇤ k2 +r) 2 k e ), b {z ⇤ 2 kF +t hrL( ⇤ \| b, b {z ⇤ i+t h } \| I b, b  t hrL( b )+ \| {z ⇤ i } II ⇤ i. } III (A.4) Next, we bound terms I, II and III respectively. For a set E, let E c denote its complement with respect to (w.r.t.) the full index set {(i, j) : 1  i, j  d}. For term I, separating ⇤ to E [ D and E c \ D, in which D is the set consisting the support of rL( ) and b of all diagonal elements, and then using the matrix Hölder inequality, we obtain ⌦ ⇤ rL( ), b ⇤ ↵ = ⌦ ⇤ rL( ) ⇤ rL( E[D ) ⇤ rL( , b ⇤ E[D F ) E c \D F b ), ( b ⇤ b )S[D , ( b )i = h( ⇤ b ↵ ⌦ + rL( ⇤ )S[D i +h( For the last term in the above equality, we have b )S c \D , ( b h( ⇤ )S c \D i = h S c \D , \| ⇤ ) E c\D E[D F ⇤ . E c \D F b ) and ( b For term II, separating the support of ( obtain h( b E[D b S c \D \|i = h ⇤) , b ⇤ E c\D ↵ to S [ D and S c \ D, we b )S c \D , ( b S c \D , \|( ⇤ b ⇤ )S c \D i. (A.5) )S c \D \|i. (A.6) Plugging (A.6) into (A.5) and applying matrix Hölder inequality yields h( b, b ⇤ b )S[D , ( b i = h( b )S , ( b = h( k S kF k( b ⇤ ⇤ )S[D i + h )S i + k S c \D , \|( S c \D kF k( ) S kF + k b E c \D kF k( b b ⇤ ⇤ )S c \D \|i )S c\D kF ⇤ )E c\D kF , where we use D = 0 in the second equality and E c \D ✓ S c \D in the last inequality. ⌦ ↵ b, b For term III, using the optimality condition, we have III = rL( b )+ = 0. Plugging the bounds for term I, II and III back into (A.4), we find that k ⇤ k2 + r 2 e ⇤ 2 +t F k E c \D kF t rL( k(rL( ⇤ 5 ) E[D F ))E c \D kF · ( b + S · b ⇤ F ⇤ ⇤ )E c \D F . F Further observing the facts that k E c \D kF ⇤k = k e rL( ⇤ ) E c\D F and tk b F we can simplify the above inequality to (k ⇤ 2 k2 +r) ke ⇤ kF  k S kF +krL( p p \|E c \D\| E c\D min \|E c \D\| rL( ⇤ k , dividing both sides by k e F ⇤ )E[D kF = k ⇤ S kF +krL( ) E kF  2 ⇤) max ⇤k , F p s, b C⇤ )E[D kF = k(C b C⇤ )E kF = krL( ⇤ )E kF in where we use krL( ⇤ )E[D kF = k(C the equality, and the last inequality follows from the Cauchy-Schwarz inequality, the fact k kmax  and the assumption that 2krL( ⇤ )kmax . Therefore, by the definition of ⇤ k  2(k ⇤ k + r)2 ps  8k ⇤ k2 ps < r, which implies e = b r, we obtain k e 2 F 2 from the construction of e . Thus b satisfies the desired `2 error bound. k Recall the definition of E` , 1  `  T . We can bound k b (`) ⇤k (` 1) kF . S F in terms of Lemma A.5 (Sequential Bound). Under the same assumptions and conditions in Lemma A.4, for ` 1, b (`) must satisfy k b (`) ⇤ ⇤ 2 2 kF  4 (` 1) S F ⇤ + rL( Proof of Lemma A.5. Now if we assume that for all ` ) E` F . 1, we have the following \|E` \|  2s, where E` is defined in (A.1) , and (` 1) E`c \D min /2 ⇤ krL( Using the matrix Hölder inequality, we obtain p p (` 1)  \|S\| S max  s and krL( S F ⇤ )kmax . )E` kF  Therefore, we have (` 1) + S F rL( ⇤ ) E`  F p s+krL( ⇤ (A.7) (A.8) p )E` kmax \|E` \|krL( ⇤ )E` kmax . p p \|E` \|  2 s, (A.9) where the second inequality is due to the assumption that krL( ⇤ )kmax  /2. The `2 error bound is given by Lemma A.4 by taking = (` 1) and E = E` , i.e. p 2 (` 1) ⇤ b (`) 4 ⇤ 2· + rL( ⇤ )E` F  8k ⇤ k22 · s, (A.10) S F F where last inequality is due to (A.9). Therefore, we only need to prove that (A.7) and (A.8) hold by induction. For ` = 1, we have w(u) for any u and thus E1 = S, which implies that (A.7) and (A.8) hold for ` = 1. Now assume that (A.7) and (A.8) hold at ` 1 for some ` 2. Since (i, j) 2 E` \S implies that (i, j) 2 / S and (` 1) (`) b w ij = j < w(u) = /2. By assumption, and since w(x) is non-increasing, we (` 1) b must have u. Therefore by induction hypothesis, we obtain that ij p \|E` \S\|  b (` 1) E`\S F u  b (` 1) u ⇤ F  8k ⇤ k2 2 u · p s p s, where the second last inequality follows from Lemma A.4, the fact that (A.7) and (A.8) hold at ` 1. This implies that \|E` \|  2\|S\| = 2s. Now for such E`c , we have k E`c kmin w(u) /2 krL( )k1 , which completes the induction step. 6 Our next lemma establishes the relationship between the adaptive regularization parameter and the estimator from the previous step. Lemma A.6. Assume w(·) 2 T . Let ij = w \|⇥ij \| for some ⇥ = (⇥ij ) and w(⇥S ) = w(⇥ij ) (i,j)2S , then for the Frobenius norm k · kF , we have w \|⇥⇤S \|  S F u F 1 + u ⇥⇤S ⇥S F . Proof of Lemma A.6. By assumption, if \|⇥⇤ij ⇥ij \| u, then w \|⇥ij \|  1  u 1 \|⇥ij ⇥⇤ij \|; otherwise, w \|⇥ij \|  w \|⇥⇤ij \| u . Therefore,the following inequality always hold: w \|⇥ij \|  w \|⇥⇤ij \| 1 u +u \|⇥⇤ij ⇥ij \|. Then by applying the k · k⇤ -norm triangle inequality, we obtain that S F w \|⇥⇤S \|  u ⇥⇤S 1 + u F ⇥S ?F . Our last technical result concerns a contraction property, namely, how the sequential approach improves the rate of convergence adaptively. Proposition A.7 (Contraction Property). Assume that assumptions 3.1, 3.2 and 3.3 p hold. Assume that 2krL( ⇤ )kmax and 8k ⇤ k22 s < 1. Then b (`) satisfies the following contraction property b (`) ⇤ F  4k ⇤ 2 k2 krL( ⇤ )S kF + 1 b (` 2 ⇤ 1) F . Proof of Proposition A.7. Under the conditions of the theorem, the proof of Lemma A.5 yields that (` 1) E`c \D kmin \|E` \|  2s, where E` is defined in (A.1), and k Thus, applying Lemma A.5 with b = b (`) , b (`) ⇤ F 4 ⇤ 2 2 · = (` 1) (` 1) kF S (` 1) S F  w(\| ⇤ S\| u) F + u ⇤ + rL( 1 ) E` . F b (` ⇤ 1) ⇤ S\| u) F . F I +4k u 1 b (` 1) ⇤ F . ⇤ ) E` F  rL( 7 ⇤ )S F + rL( (A.11) 1) ⇤k : F (A.12) ⌘ (A.13) In the next, we bound term I. Separating the support of rL( then using triangle inequality, we obtain I = rL( )kmax . in terms of k b (` Plugging the bound (A.12) into (A.11) yields that ⇣ ⇤ ⇤ 2 b (`)  4 rL( ⇤ )E` F + w(\| F 2 \| {z } ⇤ 2 k2 ⇤ and E = E` , we obtain (` 1) k S On the other side, by Lemma A.6, we can bound k krL( ⇤ ⇤) E` )E` \S to S and E`\S and F . (A.14) p Moreover, we have the following facts. First, we have rL( ⇤ )E` \S 2  \|E` \S\| rL( ⇤ ) by the Hölder inequality. From the assumption, we know krL( ⇤ )kmax  /2. Plugging p these bounds into (A.14) results that krL( ⇤ )E` kF  krL( ⇤ )S kF + \|E` \S\|. Now, by p (` 1) following a similar argument in Lemma A.5, we can bound \|E` \S\| by b E` \S F u  ⇤ b (` 1) u. Therefore, term I can be bounded by krL( ⇤ )S kF + u 1 b (` 1) ⇤ F . Plugging the upper bound for I into (A.13), we obtain F b (`) ⇤ F  4k ⇤ 2 k2 + (4k ⇣ krL( ⇤ 2 k2 ⇤ )S k2 + kw(\| ⇤ S\| b (` ⇤ 1 + 1) u 1) u)k2 F . ⌘ Now observing that k ⇤S kmin u + ↵ ⇣ , thus w(\|⇥⇤S \| u)  w(↵ · 1S ) = 0S , where 1S is a matrix with each entry equals to 1 and 0S is defined similarly. Further notice that (4k ⇤ k22 + 1) u 1  1/2, we complete the proof. B Improved Convergence Rate Using Sparsity Pattern We develop an improved spectral norm convergence rate using sparsity pattern in this section. We collect the proof for Theorem 3.8 first and then give technical lemmas that are needed for the proof. B.1 Proof of Theorem 3.8 (`) ⇤ Proof of Theorem 3.8. Let us define S (`) = (i, j) : ij u , where u is introij (0) ⇤ duced in (A.1). Let S = {(i, j) : \| ij \| u} = S. Then Lemma B.5 implies (` 1) E` F  w(\| ⇤ S\| u) + F q \|S (` 1) \ S\| + q E` /S ⇤ > u and thus (i, j) 2 S (` 1) /S. For any (i, j) 2 E` /S, we must have b ij = b ij ij Therefore, applying Lemma B.5 and using the fact that k ⇤S kmax u + ↵ , we obtain ⇢q q p p ⇤ 2 ⇤ 2 (` 1) \ S + (` 1) /S b (`) b  32 S S  32 2 S (` 1) . F 2 2 On the other side, (i, j) 2 S (`) implies that (`) \| b ij b ij \| (`) \| b ij ⇤ ij \| \| b ij Exploiting the above fact, we can bound q \|S (`) \|  b (`) 64k p ⇤ ij \| u 22 64k \|S (`) \| in terms of k b (`) b ⇤ k2 2 F  q S (` 1) /2. By induction on `, we obtain q ⇣ 1 ⌘`/2 q ⇣ 1 ⌘`/2 p (`) \|S \|  \|S (0) \| = s. 2 2 8 ⇤ 2 k2 b kF : , max Since ` > log s/ log 2, we must have that the right hand side of the above inequality is smaller than 1, which implies that S (`) = ? and b (`) = b . Therefore, the estimator enjoys the strong oracle property. Using Lemma B.4 obtains us that b (`) ⇤ b  2 ⇤ 2 . M⇤ b C 2 Applying Lemma D.6 finishes the proof of theorem. B.2 C⇤ S max . Technical Lemmas We start with the definitions of some constants. For notational simplicity, let 1 = k⌃⇤ k1 and D = {(i, i) : 1  i  d}. Define the oracle estimator as n⌦ o ↵ b = b argmin ,C log det( ) . d supp( )=S, 2S+ Recall that smax = maxj P ⇤ i 1(⇥ij ) is the maximum degree. Lemma B.1. Suppose that the weight function satisfies that w(u) 1/2 for u defined p in (A.1). Assume that 2 smax  1 2 k ⇤ k2 , 8k ⇤ k22 s < 1. If 2krL( b )kmax , we must have b (`) \|E` \|  2s and b F  32 Proof of Lemma B.1. If we assume that for all ` ⇤ 2 2 (` 1) . E` F 1, we have the following \|E` \|  2s, where E` is defined in (A.1), and k (` 1) kmin E`c (B.1) krL( b )kmax . (B.2) ⇤k ⇤ k +2 s Using lemma B.4, we obtain that k b k2  k ⇤ k2 +k b 1 k 2 2 max . p b Therefore, the assumption of the lemma implies 4k k2 s < 1. Replacing S by E` in Lemma B.3 and using Hölder inequality, we have 2 2 2 p (` 1) (` 1) b (`) b  4 b 2 E`  16 ⇤ 2 E`  32 ⇤ 2 s, (B.3) F F F For ` = 1, we have w(u) and thus E1 = S, which implies that (B.1) and (B.2) hold for ` = 1. Now assume that (B.1) and (B.2) hold at ` 1 for some ` 2. Since (` 1) (`) j 2 E` \ S implies that j 2 / S and w( j ) = j < w(u) by assumption, and since (` 1) \| j w(x) is decreasing, we must have \| obtain that b (` 1) E` \S F p \|E` \ S\|   b (` u. Therefore by induction hypothesis, we b 1) F  32 ⇤ 2 2 u u u where the last inequality follows from the definition of u hold at ` implies that \|E` \|  2\|S\| = 2s. Now for such E`c , we have k E`c kmin w(u) /2 krL( b )kmax , which completes the induction step. This completes the proof. 9 p s p s, 1. This inequality With some abuse of notation, we let \| ⇤S \| = (\| ⇤ij \|)(i,j)2S and \| u)(i,j)2S . The following inequality bounds the regularization parameter w( ⇤ij ) (i,j)2E in terms of functionals of ⇤ and . Lemma B.2. Let E F  ⇤\| S E = w \| \| . For any set E ◆ S, E must satisfy q ⇤ w(\| ⇤S \| u) F + E/S + {j 2 S : \| ij ij \| u = ( ⇤ij = w(\| ⇤E \|) = u} 1/2 p Proof. By triangle inequality, we have k E kF  k S kF + \|E/S\|. We further bound ⇤ ⇤\| k S kF . If \| ij u, then we have w \| ij \|  1  I \| ij u , otherwise, ij \| ij ⇤ ⇤ since because w(·) is non-increasing and thus \| ij ij \| < u implies w \| ij \|  w \| ij \| u . Therefore, using the Cauchy Schwartz inequality completes our proof. Define the following optimization problem n⌦ ↵ b = argmin b ,C log det( ) + 1,o↵ d 2S+ Lemma B.3. Let k satisfy krL( b )kmax and 4k b k2 S c /D kmin b b F 4 b 2 2 p o . (B.4) s < 1. Then b must S F. e = ⇥⇤ + t(⇥ b ⇥⇤ ), where t is chosen Proof. We construct an intermediate solution ⇥ e e such that k(⇥ ⇥⇤ kF = r, if k(⇥ ⇥⇤ kF > r; t = 1 otherwise. Here r satisfies p 4k b k22 s < r  k b k2 . Lemma A.3 implies that ⌦ ↵ 2 e b b b ⌘ Ds e , b . +r  rL( e ) rL( b ), e (B.5) L 2 F Then, we use Lemma E.2 to upper bound the right hand side of the above inequality ⌦ ↵ s e b s b b b . DL ( , )  tDL ( , ) = t rL b rL b , b Plugging the above inequality into (B.5), we obtain ⌦ 2 e b b 2  rL( b ) +r 2 F rL( b ), e b ↵ . (B.6) We further control the right hand side of the above inequality by exploiting the first b = 0 and rL( b )S[D = 0. Therefore, order optimality condition, which is rL( b )+ b to the right hand side of (B.6) and using the optimality adding and subtracting term condition obtains us that ⌦ ↵ ⌦ ↵ 2 e b b 2+ b, e b + rL( b ), e b  0. +r (B.7) 2 F \| {z } \| {z } I II b k2 , it suffices to bound I and II separately. For term I, by Therefore, to bound k e F decomposing the support to S and S c /D, then using matrix Hölder inequality, we have I S F e b S F + S c /D min 10 vec e b S c /D 1 . Again, by using the optimality condition, we has ⌦ ↵ b II = rL b S c /D , e rL( b )S c /D c S /D max vec e b By plugging the upper bound for I and II back into (B.7), we have b  2 2 +r S F (e e 2 + F b b )S rL( b )S c /D S c /D min . F max S c /D 1 vec e b . 1 By assumption, we know that k kmin krL( b )kmax , which implies that the second b term in the right hand side of the above inequality is positive. Thus, we have + 2 p 2 e b b 2 s < r  k b k2 , we obtain that e r  S F . Now since 4k k2 F p 2 b  4 b 2 S F  4k b k22 s < r. By the construction of e , we must have t = 1, F and thus e = b . Recall that M⇤ is the sparsity pattern matrix corresponding to ⇤ . p b C⇤ )S kmax  cn /2 for a sequence Lemma B.4. If 441 cn + 1 < 1+41 /smax and k(C cn , then we have b ⇤ max b  21 cn and ⇤ 2  21 cn kM⇤ k2 . ⇤ . It suffices to show that k k Proof of Lemma B.4. Let = b max  r, where 2 ⇤ ⇤ ). e r = 1 cn . To show this, we construct an intermediate estimator, = + t( b ⇤k e = b , otherwise. We choose t such that k e max = r, if k kmax > r, and For a matrix A, let AS be a matrix agreeing with A on S and having 0 elsewhere. Using the two term Taylor expansion, we know that there exists a 2 [0, 1] such that ⇤ ), e⇤ = ⇤ + (e vec rL( e ) = vec rL( which implies that n vec C⇤E e 1 E where E = S [ D. Let e = e E n vec C⇤E o ⇣ ⇤ ) + r2 L e ⇤ vec e e⇤ ⌦ e⇤ E E ⌘ 1 ⇣ vec e E ⇤ ⇤ E ⌘ = 0, = t . Define f vec( e ) to be o 1 ⇤ ⇤ eE + e E , EE vec ⇤ E , (B.8) 1 in which ⇤EE = ( ⇤E ⌦ ⇤E ) 1 . By the matrix expansion formula that (A+ P1 1 )m A 1 , f {vec( e )} reduces to m=1 ( A ⇢ X 1 vec ( ⌃⇤ e ) m ⌃⇤ . E m=2 1 Using triangle inequality, we then obtain that f vec( e )  max (j,k)2E 1 X m=2 11 eTj (⌃⇤ e )m ⌃⇤ ek . ) 1 A 1 = Further applying Hölder inequality to each single term in the right hand side of the above displayed inequality, we have eTj (⌃⇤ e )m ⌃⇤ ek  ⌃⇤ m+1 1 m 1 1 e e max 1 ⇤  sm max ⌃ m+1 1 where we use the fact k k1  smax k kmax . Therefore, we obtain f vec( e )  1 X m=2 1 ⇤ m+1 e m sm max k⌃ k1 k kmax = which, by triangle inequality, implies that k e kmax  k ⇤ EE k1 ✓ vec n C⇤E ⇤ + e 1 E o b E = b , the fact kC b Utilizing the KKT condition C E p 1+ 1+1 /smax , we obtain k e kmax  21 cn ⇣1 2 + e m , max 31 smax k e k2max , 1 1 smax k e kmax ◆ 31 smax k e k2max + . 1 1 1 smax k e kmax C⇤ kmax  cn /2 and 441 cn < 31 smax r2 ⌘ < 21 cn ⌘ r, 1 1 smax r which is a contradiction. Thus, e = and b satisfies the desired maximum norm bound. For the spectral norm bound, we utilize Lemma E.6 and obtain that b The proof is finished. C ⇤ 2  M⇤ 2 b ⇤ max  21 cn kM⇤ k2 . Semiparametric Graphical Model Proof of Theorem 4.3. We need the follows lemma, which are taken from Liu et al. b⌧ . (2012). It provides a nonasymptotic probability bound for estimating ⌃npn using S Lemma C.1. Let C be a constant. For any n & log d, with probability at least 1 we have r log d npn ⌧ sup \|Sbjk ⌃jk \|  C . n jk 8/d, The rest of the proof is adapted from that of Theorem 4.3 and thus is omitted. D Concentration Inequality In this section, we establish the concentration inequalities which are the key technical tools to the large probability bounds in Section 3. 12 Lemma D.1 (Sub-Gaussian Tail Bound). Let X = (X1 , X2 , . . . , Xd )T be a zero-mean random vector with covariance ⌃⇤ such that each Xi / ii⇤ is sub-Gaussian with variance proxy 1. Then there exists constants c1 and t0 such that for all t with 0  t  t0 the b satisfies the following tail probability bound associated sample covariance ⌃ P \|bij ⇤ ij \| t  8 exp c1 nt2 . Proof of Lemma D.1. By the definition of the sample covariance matrix, we have bij = P P (k) (k) (k) (k) n 1 nk=1 (Xi X̄i )(Xj X̄j ) = n 1 nk=1 Xi Xj X̄i X̄j . Therefore we can deP (k) (k) n ⇤ 1 ⇤ compose bij ij as n ij X̄i X̄j . By applying the union sum bound, k=1 Xi Xj we obtain that ◆ ✓ ◆ ✓ ◆ ✓ X n 1 t t (k) (k) ⇤ ⇤ P bij t P Xi Xj + P X̄i X̄j ij ij n 2 2 k=1 {z } \| {z } \| (R2) (R1) In the sequel, we bound (R1) and (R2) separately. For term (R1), following the argument of Lemma A.3 in Bickel and Levina (2008), there exists constant c01 and t00 not depending n, d such that ✓ X ◆ ⇢ n 1 t (k) (k) ⇤ (R1) = P Xi Xj  4 exp c01 nt2 ij n 2 k=1 for all t satisfying 0  t  t0 . Next, we bound the term (R2). By the linear structure p of sub-Gaussian random variables, we obtain that nX̄i ⇠ sub-Gaussian(0, ii⇤ ) for all p p 1  i  d. Therefore, by applying Lemma E.1, we obtain that \| nX̄i · nX̄j \| is a p p sub-exponential random variable with 1 norm bounded by 2k nX̄i k 2 k nX̄j k 2 . We p p give explicit bounds for the 2 -norm of nX̄i and nX̄j . By the Cherno↵ bound, the p tail probability of nX̄i can be bounded in the following ⇣p ⌘ n t2 o P \| nX̄i \| t  2 exp . 2 ii⇤ For every non-negative random variable Z, integration by parts yields the identity EZ = R1 p u)du. We apply this for Z = \| nX̄i \|p and obtain after change of variables 0 P(Z u = tp that Z 1 Z 1 n p p t2 o p 1 p p 1 E\| nX̄i \| = P(\| nX̄i \| t) · pt dt  2p · exp t dt 2 ii⇤ 0 0 ⇣ p ⌘p/2 p = p(2 ii⇤ )p/2 · ( )  p(2 ii⇤ )p/2 · , 2 2 p p which indicates that k nX̄i k 1  2 ii⇤ . The Gamma function is defined as (t) = q R1 t t 1 p ⇤ . Therefore we obtain e x dx. Similary, we can bound k n X̄ k by 2 jj j 2 0 q p p 2 2 ⇤ ⇤ ⇤ ⇤ k nX̄i · nX̄j k 1  2 ii jj  2 max , where max = max{ 11 , . . . , dd }. Define Zij = p p 2 e2 ) 1 and write the Taylor expansion series of the \| nX̄i · nX̄j \|. Let = (e 1)(2 max expoential function, we obtain E exp{ Zij } = 1 + 1 X k=1 k E(Z k ) ij k! 1+ 1 X k=1 13 k (2 2 k)k max k! 1+ 1 X k=1 (2 2 max · e)k  e, where we use k! (k/e)k in the last second inequality. Exponenting and using the Markov inequalty yields that ⇣ ⌘ ⇣ ⌘ ⇣ ⌘ Ee Zij P Zij t = P Zij t = P e Zij e t   exp{1 t}, et for all t 0. Using the above result, we can boudn (R2) as ⇣ n n nt ⌘ nt o (R2)  P Zij  exp 1  4 exp 1 2 2 nt o . 2 Combing the bounds for (R1) and (R2), taking c1 = min{c01 , } and t0 = min{1, t00 } obtain us that ⇤ ij \| P \|bij t  8 exp c1 nt2 8 t  t0 , which completes the proof. We then develop a large deviation bound for marginal variances. Lemma D.2 (Large Deviation Bound for Marginal Variance). Let X = (X1 , X2 , . . . , Xd )T p ⇤ be a zero-mean random vector with covariance ⌃⇤ such that each Xi ⌃ii is subn (k) Gaussian with variance proxy 1, and X be n i.i.d. samples from X. Let k=1 1 C(") = 2 " log(1 + ") > 0. Then, for any " 0, we must have ⇣ ⌘ n o b ii ⌃⇤ > " · ⌃⇤  2 · exp P ⌃ n · C(") . ii ii (k) Proof. We write Zi (k) (k) (k) = ⌃⇤ii 1/2 (k) Xi e ii = n and ⌃ 1 Pn (k) k=1 Zi (k) · Zi , for 1  i  d. Let &i = Zi · Zi ⇠ 21 , for 1  k  n. Therefore, the moment-generating function of (k) &i is M& (k) (t) = (1 2t) 1/2 , for t 2 ( 1, 1/2). Next, we control the tail probability i e ii > 1 + " and ⌃ e ii < 1 ", respectively. For the tail probability of ⌃ e ii > 1 + ", by of ⌃ applying Lemma E.8, we obtain ! (1) (n) n o &i +. . . + &i P > 1+"  exp n · A(") , n where A(") = supt (1 + ")t + 2 1 log(1 2t) = 2 1 " log(1 + ") . Similarly, for any e ii < 1 " as " > 0, we obtain the tail probability of ⌃ ! (1) (n) n o &i +. . . + &i P < 1 "  exp n · B(") , n where B(") = supt (1 ")t + 2 1 log(1 2t) . After some algebra, we obtain B(") = 2 1 " + log(1 ") , if " < 1; B(") = +1, otherwise. Let C(") = min A("), B(") = 2 1 " log(1+") . Therefore, combing the ⇣ ⌘ above twon inequalities o by union bound, we obtain P n b ii = (⌃⇤ ) ⌃ ii 1 (1) &i 1 ·⌃ e (n) 1 > "  2 · exp + . . . + &i ii = n 1 & (1) +. . .+& (n) ii ii ⇣ b ii P ⌃ ⌃⇤ii n · C(") . Note that we have . Thus, we obtain ⌘ n o > " · ⌃⇤ii  2 · exp n · C(") . 14 Our next results characterizes a large deviation bound for sample correlation matrix. Lemma D.3 (Large Deviation Bound for Sample Correlation). Let X = (X1 , X2 , . . . , Xd )T p ⇤ be a zero-mean random vector with covariance matrix ⌃⇤ such that each Xi ⌃ii is (k) n sub-Gaussian with variance proxy 1 and {X }k=1 be n independent and identically b = 1/n Pn X (k) X (k) T denote the sample covariance distributed copies of X. Let ⌃ k=1 b =W c 1⌃ bW c 1 denote the sample correlation matrix, where W c 2 is the diagonal and C b Further let ⇢bij and ⇢ij be the (i, j)th element of matrix with diagonal elements of ⌃. ⇤ b C and C respectively. Define c2 = min{4 1 c1 min(⌃⇤ )2 , 1/6}. Then, for 0  "  min{1/2, t0 maxi ⌃⇤ii }, we have ⇣ ⌘ n P \|b ⇢ij ⇢ij \| > "  6 exp ii o c2 n·"2 , where 1  i 6= j  d. b ii · ⌃ b jj ) 1/2 ⌃ b ij . To Proof of Lemma D.3. We denote the sample correlation as ⇢bij = (⌃ prove the tail probability bound. It suffices to prove the tail probability bound for ⇢bij ⇢ij > " and ⇢bij ⇢ij < ", respectively. We start with the tail probability bound for ⇢bij ⇢ij > ". Let us assume that ⇢ij 0. Using the basic probability argument, we have P(A) = P(A \ B) + P(A \ B c )  P(A) + P(B c ). Thus, for any 0  t  1 we obtain ⇣ ⌘ ⇣ ⌘ b ij (⌃ b ii ⌃ b jj ) 1/2 ·⇢ij > (⌃ b ii ⌃ b jj ) 1/2 ·" P ⇢bij ⇢ij > " = P ⌃ ⇣ ⌘ b ij (⌃⇤ ⌃⇤ ) 1/2 (1 t) 1 ·⇢ij > (⌃⇤ ⌃⇤ ) 1/2 (1 t) 1 ·" P ⌃ ii jj ii jj \| {z } ⇣ b ii +P ⌃ ⌘ (R1.1) ⇣ b jj ⌃⇤ii > ⌃⇤ii ·t + P ⌃ ⌘ ⌃⇤jj > ⌃⇤jj ·t . (D.1) Next, we bound the term (R1.1). After some simple algebra, (R1.1) can be bounded by ✓ ◆ ⇤ ⇤ ⇤ 1/2 1 ⇤ b P ⌃ij ⌃ij > " + ⇢ij ·(⌃ii ⌃jj ) (1 t) ⌃ij ✓ ◆ 1/2 ⇤ ⇤ ⇤ ⇤ b  P ⌃ij ⌃ij > " ⌃ii ⌃jj (1 + t) + t·⌃ij Let c02 = c1 mini (⌃⇤ii )2 , where c1 is defined in Lemma D.1.q If we apply Lemma D.1 with a better constant and Lemma D.2, then for any 0  "  t0 ⌃⇤ii ⌃⇤jj , in which t0 is defined in Lemma D.1, we must have ⇣ ⌘ ⇣ ⌘ ⇣ ⌘ b ij ⌃⇤ij > " ⌃⇤ii ⌃⇤jj 1/2 + P ⌃ b ii ⌃⇤ii > t·⌃⇤ii P ⇢bij ⇢ij > "  P ⌃ ⇣ ⌘ b jj ⌃⇤ > t·⌃⇤ +P ⌃ jj jj n o n o 1  4 exp c02 n·"2 + 2 exp n· t log(1 + t) . 2 Let c002 = min c02 , 1/6 . Further, for any 0  "  min{1/2, t0 maxi ⌃⇤ii }, by taking t = " and using the inequality t log(1 + t) 1/3·t2 for all t such that 0  t  1/2, we obtain ⇣ P ⇢bij ⌘ n o n 1 o n o ⇢ij > "  4 exp c02 "2 · n + 2 exp "2 · n  6 exp c002 n·"2 . 6 15 If ⇢ij < 0, in the a similar fashion as before, we can obtain the the following tail probability bound ⇣ ⌘ ⇣ ⌘ q b ij ⌃⇤ij > " ⌃⇤ii ⌃⇤jj 1/2 + ⌃⇤ij ·(t2 t) " ⌃⇤ ⌃⇤ ·t P ⇢bij ⇢ij > "  P ⌃ ii jj \| {z } ⇣ b ii +P ⌃ ⌘ (R1.2) ⇣ b jj ⌃⇤ii > t·⌃⇤ii + P ⌃ ⌘ ⌃⇤jj > t·⌃⇤jj . To continue, we bound the term (R1.2) in the next. If take t = "  min 1/2, t0 maxi ⌃⇤ii  q q 1/2 + 1/2\|⇢ij \|, we obtain that ⌃⇤ij ·(t2 t) " ⌃⇤ii ⌃⇤jj ·t 1/2 ⌃⇤ii ⌃⇤jj ·t. Thus, we have ⇣ P ⇢bij ⌘ ⇣ ⌘ ⇣ ⌘ b ij ⌃⇤ij > 1 " ⌃⇤ii ⌃⇤jj 1/2 + P ⌃ b ii ⌃⇤ii > t·⌃⇤ii ⇢ij > "  P ⌃ 2 ⇣ ⌘ ⇤ b + P ⌃jj ⌃jj > t·⌃⇤jj n 1 o n 1 o  4 exp c02 n·"2 + 2 exp n· " log(1 + ") 2 n 4 o 2  6 exp c2 n·" , where c2 = min{4 1 c02 , 1/6} = min{4 1 c1 min(⌃⇤ii )2 , 1/6}  c002 . By combining above two cases, for 0  "  min{1/2, t0 maxi ⌃⇤ii }, we have P(b ⇢ij ⇢ij > ")  6 exp{ c2 n · "2 }. In a similar fashion, we obtain the same tail probability bound for ⇢bij ⇢ij < ", for 0  "  min{1/2, t0 maxi ⌃⇤ii }. Thus the proof is completed. Lemma D.4. Under the same conditions in Lemma (D.3). We have the following result hold r ◆ ✓ ✓r ◆ 1 s ⇤ ⇤ lim lim sup P rL >M = 0, and krL( )S kF = OP . S max M !1 n n n b Proof of Lemma D.4. It is easy to check that rL( ⇤ )S F = C C⇤ S F . By applying Lemma D.3 and the union sum bound, for any M such that 0  M  p min 1/2, t0 maxi ⌃⇤ii · n, in which t0 is defined in Lemma D.3, we obtain r ◆ ✓ 1 ⇤ P rL >M  s · exp c2 M 2  exp c2 M 2 + log s . S max n q p Taking M such that 2c2 1 log s  M  min 1/2, t0 maxi ⌃⇤ii · n and M ! 1 in the above inequality obtains us that r ◆ ✓ 1 ⇤ lim lim sup P rL >M = 0, S max M !1 n n p which implies that rL ⇤ S F = OP ( s/n). b C⇤ , Lemma D.5 (A Concentration Inequality for Sample Correlation Matrix). Let C, 1 ⇤ 2 ⇢bij and ⇢ij be defined in Lemma D.3. Suppose n 3 c 2 t1 · log d. Take = q p b must 3c2 1 · (log d)/n ⇣ log(d)/n, in which c2 is defined as in Lemma D.3. Then C satisfy ⇣ ⌘ b C⇤ P C   1 8/d. max 16 b C⇤ . Therefore, applying Lemma D.3 and Proof. It is easy to check that rL(C⇤ ) = C union sum bound, we obtain that, for any  t1 ⌘ min 1/2, t0 maxi {⌃⇤ii } with t0 defined in Lemma D.1, ⇣ ⌘ b C⇤ P C >  6d2 · exp{ c2 n 2 }. max where c2 = min{4 1c ⇤ 2 1 min(⌃ii ) , 1/6}, in which c1 is definedqin Lemma D.1. , for n ·log d, by taking = 3c2 1 · (log d)/n  t1 , we 1 sufficiently large such that n 3 c2 t21 b C⇤ kmax  b C⇤ kmax > obtain P kC = 1 P kC The proof is completed. 1 6d2 ·exp{ c2 n Lemma D.6. Under the same conditions in Lemma D.5, we have r ◆ ✓ 1 ⇤ b b C⇤ lim lim sup P C C S max > M = 0, and C S M !1 n n max 2} 1 8/d. ✓r ◆ 1 = OP . n Proof of Lemma D.6. The proof is similar to that of Lemma D.5 and thus is omitted. E Preliminary Lemmas In this section we state and prove the technical lemmas used in previous sections. The following lemma establishes the tail bound type of the product of two sub-Gaussian random variables. Let k · k 1 and k · k 2 be the 1 - and 2 -norm defined in Vershynin (2010). Lemma E.1. For X and Y being two sub-Gaussian random variables, then the absolute value of their product \|X · Y \| is a sub-exponential random variable with kX · Y k 1  2 · kXk 2 kY k 2 . Proof of Lemma E.1. To show X · Y is sub-exponential, it suffices to prove that the 1 -norm of X · Y is bounded. By the definition of the 1 -norm, we have kX · Y k 1 = sup p p 1 1 ⇥ E\|X · Y \|p We need to use the Hölder inequality as follows ⇥ ⇤ ⇥ ⇤1/r ⇥ ⇤1/s E \|hf, gi\|  E\|f \|r E\|g\|s , ⇤1/p . (E.1) 1 1 + = 1, r s where f and g are two random functions. If we choose f = X p , g = Y p and r = s = 2 in the Hölder inequality, then the right hand side of (E.1) can be bounded by n ⇥ ⇤1/(2p) ⇥ ⇤1/(2p)o sup p 1 E\|X\|2p E\|Y \|2p p 1 n n ⇥ ⇤1/(2p)o ⇥ ⇤1/(2p)o  2 sup (2p) 1/2 E\|X\|2p · sup (2p) 1/2 E\|Y \|2p . p 1 Therefore we obtain that kX · Y k p 1 1  2kXk 2 kY k 17 2 < 1. The proof is completed. ⌦ ↵ s (⇥ , ⇥ ) = Lemma E.2. Let DL (⇥1 , ⇥2 ) = L(⇥1 ) L(⇥2 ) L(⇥2 ), ⇥1 ⇥2 and DL 1 2 DL (⇥1 , ⇥2 ) + DL (⇥2 , ⇥1 ). For ⇥(t) = ⇥⇤ + t(⇥ ⇥⇤ ) with t 2 (0, 1], we have that s s DL (⇥(t), ⇥⇤ )  tDL (⇥, ⇥⇤ ). ⌦ Proof of Lemma E.2. Let Q(t) = DL (⇥(t), ⇥⇤ ) = L(⇥(t)) L(⇥⇤ ) rL(⇥⇤ ), ⇥(t) ↵ ⇤ ⇥ . Since the derivative of L(⇥(t)) with respect to t is hrL(⇥(t)), ⇥ ⇥⇤ i, then the derivative of Q(t) is ⌦ ↵ Q0 (t) = rL(⇥(t)) rL(⇥⇤ ), ⇥ ⇥⇤ . s (⇥(t) Therefore the Bregman divergence DL ⇥⇤ ) can written as ⌦ ↵ s e e DL (⇥(t) ⇥⇤ ) = rL(⇥(t)) rL(⇥⇤ ), t(⇥ ⇥⇤ ) = tQ0 (t) for 0 < t  1. s (⇥, ⇥⇤ ) as a By plugging t = 1 in the above function equation, we have Q0 (1) = DL special case. If we assume that Q(t) is convex, then Q0 (t) is non-decreasing and thus s s DL (⇥(t), ⇥⇤ ) = tQ0 (t)  tQ0 (1) = tDL (⇥, ⇥⇤ ). Therefore the proof is completed. It remains to prove that Q(t) is a convex function, i.e. Q(↵1 t1 + ↵2 t2 )  ↵1 Q(t1 ) + ↵2 Q(t2 ), 8 t1 , t2 2 (0, 1], ↵1 , ↵2 0 s.t. ↵1 + ↵2 = 1. (E.2) For 8↵1 , ↵2 0 such that ↵1 + ↵2 = 1, and t1 , t2 2 (0, 1), we have ⇥(↵1 t1 + ↵2 t2 ) = ↵1 ⇥(t1 ) + ↵2 ⇥(t2 ). By the bi-linearity property of the inner product function h·, ·i, and using the linearity property of ⇥(·), we have the following equality hold ⌦ ↵ rL(⇥⇤ ), ⇥(↵1 t1 + ↵2 t2 ) ⇥⇤ ⌦ ⌦ ↵ = ↵1 rL(⇥⇤ ), ⇥(t1 ) ⇥⇤ i ↵2 rL(⇥⇤ ), ⇥(t2 ) ⇥⇤ . (E.3) On the other side, by the convexity of the loss function L(·), we obtain L ⇥(↵1 t1 + ↵2 t2 ) = L ↵1 ⇥(t1 ) + ↵2 ⇥(t2 )  ↵1 L ⇥(t1 ) + ↵2 L ⇥(t2 ) . (E.4) By adding (E.3) and (E.4) together and using the definition of function Q(·), we obtain Q(↵1 t1 + ↵2 t2 )  ↵1 Q(t1 ) + ↵2 Q(t2 ), which indicates Q(t) is a convex function. Thus we complete our proof. Lemma E.3. Let Ai , Bi 2 Rd⇥d be square matrices for i = 1, 2. Then we have A 1 B1 A 1 A2 B2 A2 = (A1 A2 )(B1 + (A1 B2 )(A1 A2 )B2 A1 + A1 (B1 The next lemma characterizes an upper bound of kA where k · k⇤ is any matrix norm. 18 A2 ) + (A1 1 B 1k ⇤ A2 )B2 A2 B2 )A2 . in terms of kA Bk⇤ , Lemma E.4. Let A, B 2 Rd⇥d be invertible. For any matrix norm k · k⇤ , we have 1 kA B 1 k⇤  kA 1 k2⇤ kA Bk⇤ . 1 kA 1 k⇤ kA Bk⇤ We need the following lemma for bounding the di↵erence with respect to the Kronecker product. Lemma E.5. Let A and B be matrices of the same dimension. Then we have kA ⌦ A kA ⌦ Bk1 = kAk1 kBk1 , B ⌦ Bk1  kA Bk21 and + 2 min kAk1 , kBk1 kA Bk1 . The proof of the above lemma can be carried out by using the definitions and thus is omitted here for simplicity. For a matrix A = aij , we say Asp = asp ij is the corresponding sparsity pattern sp sp matrix if aij = 1 when aij 6= 0; and aij = 0, otherwise. Lemma E.6. Let A 2 Rd⇥d be a matrix such that kAkmax  1. Let Asp be the corresponding sparsity pattern matrix. Then we have kAk2  kAsp k2 . Proof of Lemma E.6. Let aij be the (i, j)-th entry of matrix A and xj the j-th entry of x. Following the definition of the spectral norm of a matrix, we obtain that kAk2 = sup kAxk2 = sup kxk2 =1  sup kxk2 =1 = kxk2 =1 ⇢X n ✓X n i=1 sup x 0,kxk2 =1 i=1 aij xj j=1 sgn(xj )1(aij 6= 0) · xj j=1 ⇢X n ✓X n i=1 ⇢X n ✓X n j=1 1(aij 6= 0) · xj ◆2 ◆2 ◆2  kAsp k2 . Thus the proof is completed. b 2 Rd⇥d be a semi-positive definite random matrix, A 2 Rd⇥d a Lemma E.7. Let A positive definite deterministic matrix. Then we have ⇣ ⌘ ⇣ ⌘ b 1 A 1 >2 2 A · A b A b A > 2 1 min A . P A  P A min 2 2 2 b and A are commutative, that is AA b = AA, b then we have If we further assume that A ⇣ ⌘ p b 1/2 A 1/2 > 2( 2 + 1) A 1/2 2 A A b A P A min 2 2 2 ⇣ ⌘ b A > 2 1 min A . P A 2 19 b 1 A 1 as A b 1 (A b A)A Proof of Lemma E.7. We first write A the sub-multiplicative property of the spectral norm that b A 1 A 1 2 b  A  b (A b A 1 1 min b 1  A b A . A · A 2 A)A 1 min 1 2 2 A 1, 1 2 then it follows from b A A 2 (E.5) b b A , and thus min A b By Weyl’s inequality, we obtain that min (A)  min (A)+ A 2 b b A A 2 . Thus in the event of A A 2  2 1 min A , we have min A b 2 1 min A hold. Thus it follows from (E.5) that min A ⇣ ⌘ ⇣ ⌘ b 1 A 1 2 2 A · A b A b A  2 1 min A . P A P A min 2 2 2 b and A This proves the first desired probability bound. If we further assume that A 1 b A 2 are commutative, under the event A min A , we have 2 b A 1/2 A 1/2 2 = ⇣ b A 1/2 +A 1/2 1 ⌘ b A 1 A 1 2 b 1/2 + A 1/2 A b 1 A 1  A 2 2 2 p 1/2 b 1 A 1 2  ( 2 + 1) A 2 A p 1/2 2 b A .  2( 2 + 1) A 2 min A A 2 Therefore we prove the third result. The following lemma is taken from Dembo and Zeitouni (2009), which leads to a P concentration bound of the empirical means X̄ = n 1 ni=1 Xi , where Xi ’s are i.i.d. random copies of X. Define the logarithmic moment generating function associated with X to be ⇥ ⇤ ⇤X ( ) ⌘ log MX ( ) = log E exp{ X} . (E.6) Lemma E.8 (Large Deviation Inequality). Let the logarithmic moment generating function of X, ⇤X ( ), be defined in E.6. Define the Fenchel-Legendre dual of ⇤X (x) to be ⇤⇤X (x) ⌘ sup 2R x ⇤( ) . Then, for any t 0, we have ✓ X n 1 P Xi n i=1 ✓ X n 1 P Xi n i=1 EX EX  ⇥ where F1 = t, +1 and F2 = t ◆ t  exp ◆  exp n(EX + inf ⇤⇤ (x)) x2F1 and n(EX + inf ⇤⇤ (x)) , x2F2 ⇤ 1, t . References Bickel, P. J. and Levina, E. (2008), “Regularized estimation of large covariance matrices,” The Annals of Statistics, 36, 199–227. 20 Dembo, A. and Zeitouni, O. (2009), Large deviations techniques and applications, vol. 38, Springer Science & Business Media. Horn, R. A. and Johnson, C. R. (2012), Matrix analysis, Cambridge university press. Liu, H., Han, F., Yuan, M., La↵erty, J., Wasserman, L., et al. (2012), “High-dimensional semiparametric Gaussian copula graphical models,” The Annals of Statistics, 40, 2293– 2326. Vershynin, R. (2010), “Introduction to the non-asymptotic analysis of random matrices,” arXiv preprint arXiv:1011.3027. 21	10
arXiv:1802.02368v1 [math.ST] 7 Feb 2018 Group kernels for Gaussian process metamodels with categorical inputs O. Roustant1 , E. Padonou1 , Y. Deville2 , A. Clément3 , G. Perrin4 , J. Giorla4 , and H. Wynn5 1 Mines Saint-Étienne, UMR CNRS 6158, LIMOS, F–42023 Saint-Étienne, France 2 3 AlpeStat, Chambéry, France CEA/DAM/VA, F–21120, Is-sur-Tille, France 4 CEA/DAM/DIF, F–91297, Arpajon, France 5 London School of Economics, England Abstract Gaussian processes (GP) are widely used as a metamodel for emulating time-consuming computer codes. We focus on problems involving categorical inputs, with a potentially large number L of levels (typically several tens), partitioned in G L groups of various sizes. Parsimonious covariance functions, or kernels, can then be defined by block covariance matrices T with constant covariances between pairs of blocks and within blocks. However, little is said about the positive definiteness of such matrices, which may limit their practical usage. In this paper, we exploit the hierarchy group/level and provide a parameterization of valid block matrices T, based on a nested Bayesian linear model. The same model can be used when the assumption within blocks is relaxed, giving a flexible parametric family of valid covariance matrices with constant covariances between pairs of blocks. As a by-product, we show that the positive definiteness of T is equivalent to the positive definiteness of a small matrix of size G, obtained by averaging each block. We illustrate with an application in nuclear engineering, where one of the categorical inputs is the atomic number in Mendeleev’s periodic table and has more than 90 levels. 1 1 Introduction This research is motivated by the analysis of a time-consuming computer code in nuclear engineering, depending on both continuous and categorical inputs, one of them having more than 90 levels. The final motivation is an inversion problem. However, due to the heavy computational cost, a direct usage of the simulator is hardly possible. A realistic approach is to use a statistical emulator or metamodel. Thus, as a first step, we investigate the metamodelling of such computer code. More precisely, we consider Gaussian process (GP) regression models, also called kriging models ([Sacks et al., 1989], [Rasmussen and Williams, 2006]), which have been successfully used in sequential metamodel-based strategies for uncertainty quantification (see e.g. [Chevalier et al., 2014]). Whereas there is a flourishing literature on GP regression, the part concerned with categorical inputs remains quite limited. We refer to [Zhang and Notz, 2015] for a review. As for continuous inputs, covariance functions or kernels are usually built by combination of 1-dimensional ones, most often by multiplication or, more rarely, by addition [Deng et al., 2017]. The question then comes down to constructing a valid kernel on a finite set, which is a positive semidefinite matrix. Some effort has been spent on parameterization of general covariance matrices [Pinheiro and Bates, 1996] and parsimonious parameterizations of smaller classes [Pinheiro and Bates, 2009]. Some block form have also been proposed [Qian et al., 2007], in order to deal with a potential large number of levels. However, their validity was not investigated. Furthermore, to the best of our knowledge, applications in GP regression are limited to categorical inputs with very few levels, typically less than 5. Guided by the application, we investigate more deeply the so-called group kernels cited in Qian et al. [2007], defined by block covariance matrices T with constant covariances between pairs of blocks and within blocks. We exploit the hierarchy group/level by revisiting a nested Bayesian linear model where the response term is a sum of a group effect and a level effect. This leads to a parameterization of T which is automatically positive definite. Interestingly, the assumption on within blocks can be relaxed, and we obtain a parameterization of a wider class of valid group kernels. The positive definiteness condition of T is also explicited: it is equivalent to the positive definiteness of the smaller covariance matrix obtained by replacing each block by its average. 2 As mentioned above, this work has some connections with Bayesian linear models as well as linear mixed effect models (see e.g. Lindley and Smith [1972], Smith [1973]) in a hierarchical view. Other related works concern hierarchical GPs with a tree structure. For instance, particular forms of group kernels are obtained in multiresolution GP models ([Fox and Dunson, 2012], [Park and Choi, 2010]). Given two resolution levels and a spatial partition A = ∪i Ai of Rd , a parent GP on A, corresponding to the lowest resolution, serves as a trend for children GPs on Ai , corresponding to the highest resolution. Children GPs are independent conditionaly on the parent GP, and have the same covariance structure with a lengthscale parameter decreasing with the diameter of Ai . As a result, for a given resolution, the covariance matrix has a block form given by a sum of nested block diagonal covariance matrices. In comparison, the two-resolution (group/level) GP corresponding to a categorical input, does not assume a conditional independence between children, and the block form of covariance matrices can be more general. The paper is structured as follows. Section 2 gives some background on GP regression with mixed categorical and continuous inputs. Section 3 presents new findings on group kernels. Section 4 illustrates on synthetic examples. Section 5 is devoted to the application which motivated this work. Section 6 gives some conclusions and perspectives for future research. 2 Background and notations 2.1 GPs with continuous and categorical variables We consider a set of I continuous variables x1 , . . . , xI defined on a hypercubic domain ∆, and a set of J categorical variables u1 , . . . , uJ with L1 , . . . , LJ levels. Without loss of generality, we assume that ∆ = [0, 1]I and that, for each j = 1, . . . , J, the levels of uj are numbered 1, 2, . . . , Lj . We denote x = (x1 , . . . , xI ), u = (u1 , . . . , uJ ), and w = (x, u). We consider GP regression models defined on the product space J Y D = [0, 1] × {1, . . . , Lj }, I j=1 3 and written as: yi = µ(w(i) ) + Z(w(i) ) + i , i = 1, . . . , N. (1) where µ, Z and are respectively the trend, the GP part and a noise term. There exist a wide variety of trend functions, as in linear models. Our main focus here is on the centered GP Z(w), characterized by its kernel k : (w, w0 ) 7→ cov (Z(w), Z(w0 )) . I Kernels QJ on D can be obtained by combining kernels on [0, 1] and kernels on j=1 {1, . . . , Lj }. Standard valid combinations are the product, sum or ANOVA. Thus if kcont denotes a kernel for the continuous variables x, kcat a kernel for the categorical ones u, examples of valid kernels for w = (x, u) are written: (Product) (Sum) (ANOVA) k(w, w0 ) = kcont (x, x0 )kcat (u, u0 ) k(w, w0 ) = kcont (x, x0 ) + kcat (u, u0 ) k(w, w0 ) = (1 + kcont (x, x0 ))(1 + kcat (u, u0 )) For consiseness, we will denote by ∗ one of the operations: sum, product or ANOVA. The three formula above can then be summarized by: k(w, w0 ) = kcont (x, x0 ) ∗ kcat (u, u0 ) (2) Then, in turn, kcont and kcat can be defined by applying these operations to 1-dimensional kernels. For continuous variables, famous 1-dimensional kernels include squared exponential or Matérn [Rasmussen and Williams, i 2006]. We denote by kcont (xi , x0i ) such kernels (i = 1, . . . , I). For a categorical variable, notice that, as a positive semidefinite function on a finite space, a kernel is a positive semidefinite matrix. We denote by Tj the matrix of size Lj corresponding to kernels for uj (j = 1, . . . , J). Thus, examples of expressions for kcont and kcat are written: 1 I kcont (x, x0 ) = kcont (x1 , x01 ) ∗ · · · ∗ kcont (xI , x0I ) kcat (u, u0 ) = [T1 ]u1 ,u0 ∗ · · · ∗ [TJ ]uJ ,u0 1 J (3) (4) The formulation given by Equations (2), (3), (4) is not the most general one, since kernels are not always obtained by combining 1-dimensional ones. 4 Nevertheless, it encompasses the GP models used in the literature of computer experiments with categorical inputs. It generalizes the tensor-product kernels, very often used, and the sum used recently by [Deng et al., 2017] on the categorical part. It also contains the heteroscedastic case, since the matrices Tj are not assumed to have a constant diagonal, contrarily to most existing works [Zhang and Notz, 2015]. This will be useful in the application of Section 5, where the variance of the material is level dependent. Remark 1. Combining kernels needs some care to obtain identifiable models. 0 For instance, the product of kernels k1 , k2 with ki (xi , x0i ) = σi2 e−\|xi −xi \| (i = 1, 2), is a kernel depending on only one variance parameter σ 2 := σ12 σ22 . The GP model is identifiable for this new parameter, but not for the initial parameters σ12 , σ22 . 2.2 1-dimensional kernels for categorical variables We consider here a single categorical variable u with levels 1, . . . , L. We recall that a kernel for u is then a L by L positive semidefinite matrix T. 2.2.1 Kernels for ordinal variables A categorical variable with ordered levels is called ordinal. In this case, the levels can be viewed as a discretization of a continuous variable. Thus a GP Y on {1, . . . , L} can be obtained from a GP Yc on the interval [0, 1] by using a non-decreasing transformation F (also called warping): Y (u) = Yc (F (u)). Consequently, the covariance matrix T can be written: [T]`,`0 = kc (F (`), F (`0 )), `, `0 = 1, . . . , L. (5) When kc (x, x0 ) depends on the distance \|x − x0 \|, then k(`, `0 ) depends on the distance between the levels `, `0 , distorted by F . In the general case, F is piecewise-linear and defined by L − 1 parameters. However, a parsimonious parameterization may be preferred, based on the cdf of a flexible probability distribution such as the Normal or the Beta. We refer to [McCullagh, 1980] for examples in regression and to [Qian et al., 2007] for illustrations in computer experiments. 5 Remark 2. Notice that usual continuous kernels have non-negative values, which is a necessary condition if they are valid radial kernels in all dimensions. As a consequence, the kernels for ordinal variables built by warping do not allow negative correlations between levels. 2.2.2 Kernels for nominal variables For simplicity we present here the homoscedastic case, i.e. when T has a constant diagonal. It is immediately extended to situations where the variance depends on the level, by considering the correlation matrix. General parametric covariance matrices. There are several parameterizations of positive-definite matrices based on the spectral and Choleky decompositions. The spectral decomposition of T is written T = PDP> (6) where D is diagonal and P orthogonal. Standard parameterizations of P involve the Cayley transform, Eulerian angles, Householder transformations or Givens rotations, as detailed in [Khuri and Good, 1989] and [Shepard et al., 2015]. Another general parameterization of T is provided by the Cholesky decomposition: T = LL> , (7) where L is lower triangular. When the variance [T]`,` does not depend on the level `, the columns of L have the same norm and represent points on a sphere in RL . A spherical parameterization of L is then possible with one variance term and L(L−1)/2 angles, representing correlations between levels [see e.g. Pinheiro and Bates, 1996]. Parsimonious parameterizations. The general parametrizations of T described above require O(L2 ) parameters. More parsimonious ones can be used, up to additional model assumptions. Among the simplest forms, the compound symmetry (CS) - often called exchangeable - covariance matrix assumes a common correlation for all levels [see e.g. Pinheiro and Bates, 2009]. The CS matrix with variance v and covariance c is defined by: v if ` = `0 [T]`,`0 = , c/v ∈ (−1/(L − 1), 1) . (8) c if ` 6= `0 6 This generalizes the kernel obtained by substituting the Gower distance d 2 [Gower, 1982] into the exponential kernel, corresponding to c/v = e−d > 0. The CS covariance matrix treats equally all pairs of levels, which is an important limitation, especially when L 1. More flexibility is obtained by considering groups of levels. Assume that the L levels of u are partitioned in G groups G1 , . . . , GG and denote by g(`) the group number corresponding to a level `. Then a desired parameterization of T is given by the block matrix (see e.g. Qian et al. [2007]): v if ` = `0 [T]`,`0 = (9) cg(`),g(`0 ) if ` 6= `0 where for all i, j ∈ {1, . . . , G}, the terms ci,i /v are within-group correlations, and ci,j /v (i 6= j) are between-group correlations. Notice that additional conditions on the ci,j ’s are necessary to ensure that T is a valid covariance matrix, which is developed in the next section. 3 Block covariance matrices for levels grouping We consider the framework of Section 2.2.2 where u denotes a categorical variable whose levels are partitioned in G groups G1 , . . . , GG of various sizes n1 , . . . , nG . Without loss of generality, we assume that G1 = {1, . . . , n1 }, G2 = {n1 + 1, . . . , n1 + n2 }, . . . . We are interested in parsimonious parameterizations of the covariance matrix T, written in block form:   W1 B1,2 ··· B1,G ..  ...  .   B2,1 W2 T= . (10)  ... ...  .. BG−1,G  BG,1 · · · BG,G−1 WG where the diagonal blocks Wg contain the within-group covariances, and the off-diagonal blocks Bg,g0 are constant matrices containing the between-group covariances. We denote: g 6= g 0 ∈ {1, . . . , G} Bg,g0 = cg,g0 Jng ,n0g , where Js,t denotes the s by t matrix of ones. This means that the betweengroup covariances only depends on groups (and not on levels). 7 We will also consider the particular case where diagonal blocks Wg are CS covariance matrices with variance vg and covariance cg . In this subclass, the between-group and within-group covariances only depends on groups (and not on levels). When the variance term is the same for all groups, we obtain block matrices of the form (9) as a special case. Although the block matrices of the form (10) may be covariance matrices, they are not positive semidefinite in general. In the next section, we provide a proper characterization as well as a parameterization of such matrices which automatically fulfills the positive semidefinite conditions. We will use the following additional notations: for a given integer L ≥ 1, IL is the identity matrix of size L, JL is the matrix of ones of size L, 1L is the vector of ones of size L. Finally, for a vector or a matrix M, we denote by M the real number equal to the average of its coefficients. 3.1 A Gaussian model for CS covariance matrices We first focus on the case of a CS matrix. We denote by ΓCS L (v, c) = (v − c)IL + cJL (11) the CS matrix with a common variance term v and a common covariance term c. It is well-known that ΓCS L (v, c) is positive definite if and only if − (L − 1)−1 v < c < v. (12) For instance, one can check that the eigenvalues of ΓCS L (v, c) are v + (L − 1)c with multiplicity 1 (eigenvector 1L ) and v − c with multiplicity L − 1 (eigenspace 1⊥ L ). Notice that a CS matrix is positive definite for a range of negative values of its correlation term. Then we consider the following Gaussian model: η ` = µ + λ` , ` = 1, . . . , L (13) where µ ∼ N (0, vµ ) with vµ > 0, and λ1 , . . . , λL are i.i.d. random variables from N (0, vλ ), with vλ > 0, assumed to be independent of µ. A direct computation shows that the covariance matrix of η is the CS covariance matrix ΓCS L (vµ + vλ , vµ ). Clearly this characterizes the subclass of 8 positive definite CS covariance matrices ΓCS L (v, c) such that c is non-negative. The full parameterization, including negative values of c in the range (−(L − 1)−1 v, 0), can be obtained by restricting the average of level effects to be zero, as detailed in the next proposition. Proposition 1. When η and λ are related as in (13), the covariance of η conditional on zero average errors λ = 0 is a CS matrix with variance v = vµ + vλ [1 − 1/L] and covariance c = vµ − vλ /L. Conversely, given a CS covariance matrix C with variance v and covariance c, there exists a representation (13) such that C is the covariance of η conditional on zero average errors λ = 0 where vµ = v/L + c[1 − 1/L] and vλ = v − c. 3.2 Parameterization of centered covariance matrices The usage of Model (13) to describe CS covariance matrices involves Gaussian vectors that sum to zero. This is linked to centered covariance matrices, i.e. covariance matrices F such that F = 0, as detailed in the next proposition. We further give a parameterization of centered covariance matrices. Proposition 2. Let F be a covariance matrix of size L ≥ 2. Then, F is centered iff there exists a Gaussian vector z on RL such that F = cov(z\|z = 0). In that case, let A be a L × (L − 1) matrix whose columns form an orthonormal basis of 1⊥ L . Then F is written in an unique way F = AMA> (14) where M is a covariance matrix of size L − 1. In particular if F = v[IL − L−1 JL ] is a centered CS covariance matrix, then M = vIL−1 , and we can choose z ∼ N (0, vIL ). The choice of A in Prop. 2 is free, and can be obtained by normalizing the columns of a L × (L − 1) Helmert contrast matrix (Venables and Ripley [2002], §6.2.):   −1 −1 −1 · · · −1  1 −1 −1 · · · −1    0  2 −1 · · · −1  . ..  ..  .  . 0 3 .   .  .  .. . . ..  .. . . . −1  0 0 ··· 0 L − 1 9 3.3 A hierarchical Gaussian model for block covariance matrices Let us now return to the general case, where the levels of u are partitioned in G groups. It will be convenient to use the hierarchical notation g/`, indicating that ` belongs to the group Gg . Then, we consider the following hierarchical Gaussian model: ηg/` = µg + λg/` , g = 1, . . . , G, ` ∈ Gg (15) where for each g the random variable µg represent the effect of the group g, and the random variables λg/1 , . . . , λg/ng represent the effects of the levels in this group. We further assume: • The vector µ is normal N (0, Γµ ). • The vectors λg/. are normal N (0, Γλg ). • The vectors λ1/. , . . . , λG/. are independent. • The vectors µ and λ are independent. As an extension of Prop. 1, the next proposition and Cor. 1 show that (15) gives a one-to-one parameterization of positive semidefinite matrices of the form (10) with CS diagonal blocks, under the additional assumption that the average of level effects is zero in each group. More generally, we obtain a large parametric family of positive semidefinite matrices of the form (10). Proposition 3. The covariance matrix of η conditional on {λg/. = 0, g = 1, . . . , G} has the form (10) with, for all g, g 0 ∈ {1, . . . , G}: Wg = [Γµ ]g,g Jng + Fg , Bg,g0 = [Γµ ]g,g0 Jng ,ng0 , (16) where Fg is a centered positive semidefinite matrix equal to cov(λg/. \|λg/. = 0). Therefore Wg − Wg Jng is positive semidefinite for all g = 1, . . . , G. Conversely, consider a positive semidefinite matrix T having the block form (10) such that Wg − Wg Jng is positive semidefinite for all diagonal blocks g. e be the G × G matrix obtained by averaging each block of T. Then there Let T 10 exists a representation (15) such that T is the covariance of η conditional on zero average errors λg/. = 0, (g = 1, . . . , G), with: cov(λg/. \|λg/. e Γµ = T, = 0) = Wg − Wg Jng . Corollary 1. Positive semidefinite matrices of the form (10) with CS diagonal blocks exactly correspond to covariance matrices of η in (15) conditional on the G constraints λg/. = 0 when cov(λg/. ) ∝ Ing . As a by-product, we obtain a simple condition for the validity of block covariance matrices of the form (10). Interestingly, it only involves a small matrix whose size is the number of groups. Proposition 4. Let T be a matrix having the block form (10) such that Wg − Wg Jng is positive semidefinite for all g = 1, . . . , G. Then e is positive semidefinite. (i) T is positive semidefinite if and only if T e is positive definite and the diagonal (ii) T is positive definite if and only if T blocks Wg are positive definite for all g = 1, . . . , G. Furthermore, we have e > + diag(W1 − W1 Jn1 , . . . , WG − WG Jn ) T = XTX G where X is the n × G matrix    X :=   1n1 0 .. . 0 (17)  ... 0 ..  . 1n2 . . .  . ... ... 0  . . . 0 1nG 0 Remark 3. All the results depend on the conditional distribution λg/. \|λg/. = 0. Thus there is some flexibility in the choice of Γλg , since several matrices Γλg can lead to the same conditional covariance matrix cov(λg/. \|λg/. = 0). Remark 4 (Groups of size 1). Prop. 3 is still valid for groups of size 1. Indeed if ng = 1, then (λg/. \|λg/. = 0) is degenerate and equal to 0. Thus Fg = Wg − Wg J1 = 0 is positive semidefinite. 11 3.4 Related works Model (15): ηg/` = µg + λg/` , shares similarities with two-way Bayesian models and linear mixed effect models (see e.g. Lindley and Smith [1972]), with Gaussian priors for the effects µ and λg/. . The centering constraints λg/. = 0 are also standard identifiability conditions in such models. Furthermore, the particular case of CS covariance matrices corresponds to the exchangeable assumption of the corresponding random variables. Typically, in the framework of linear modelling, Model (15) could be written as yg,` = m + µg + λg,` + εg,` , with additionals grand mean m and errors εg,` . However, if the framework is similar, the goal is different. In linear modelling, the aim is to quantify the effects by estimating their posterior distribution (µ, λ)\|y. On the other hand, we aim at investigating the form of the covariance matrix of the response part µg + λg/` , or, equivalently, the covariance matrix of the likelihood y\|µg + λg/` . 3.5 Summary and comments The results of the previous sections show that a wide class of valid block covariance matrices can be parameterized by a family of covariance matrices of smaller sizes. This class is formed by positive definite matrices of the form (10) such that Wg − Wg Jng is positive semidefinite for all g = 1, . . . , G. It contains the case where diagonal blocks are CS covariance matrices. The algorithm is summarized below. 1. Generate a covariance matrix Γµ of size G. 2. For all g = 1, . . . , G, If ng = 1, set Fg = 0, else: • Generate a covariance matrix Mg of size ng − 1. • Compute a centered matrix Fg = Ag Mg A> g , where Ag is a ng × (ng − 1) matrix whose columns form an orthonormal basis of 1⊥ ng . 12 3. For all 1 ≤ g < g 0 ≤ G, compute the within-group blocks Wg and between-group blocks Bg,g0 by Eq. (16). In steps 1 and 2, the covariance matrices Γµ and Mg can be general, and obtained by one of the parameterizations of §2.2.2. However, some specific form, such as CS matrices, can also be chosen. Depending on the number of groups and their sizes, different levels of parsimony can be obtained. Table 1 summarizes some possibilities. Notice that it may be hard to choose the parametric setting Mg , Γµ in order to account for a specified constraint of the block matrix T, such as homoscedasticity. An alternative to over-parameterization is to use the economic constraint on T of Prop. 4. Indeed, positive definiteness on T is e of size G. equivalent to positive definitiness of the small matrix T Parametric setting Mg Γµ CS vλg Ing −1 Γ (vµ , cµ ) General vλg Ing −1 CS General Γ (vµ , cµ ) General General Resulting form of T Wg Bg,g0 CS Γ (vg , cg ) cg,g0 ≡ cµ ΓCS (vg , cg ) cg,g0 General cg,g0 ≡ cµ General cg,g0 Number of parameters 2G + 1 G(G+3) P 2 ng (ng +1) 2+ G g=1 PG n2g (ng +1) G(G+1) + g=1 2 2 Table 1: Parameterization details for some valid block-covariance matrices T of the form (10). 4 Examples Before considering the application in nuclear engineering, we consider two toy functions with one continuous input and one categorical input, which reproduce two specificities of that application. The first function mimics a situation where the output variance depends on the level of a categorical input. The second one investigates level grouping when the number of levels is large. 13 4.1 Example 1 (An heretoscedastic case) Consider the deterministic function  6.8πx  cos  2 f (x, u) = −2 cos 7.0πx 2  1 cos 7πx 2 2 if u = 1 if u = 2 , if u = 3 −2 −1 0 1 2 where x ∈ [0, 1] and u ∈ {1, 2, 3}. The expression of f is adapted from [Han et al., 2009] by scaling the three output curves f (., u) according to the level of u. Thus the output variance clearly depends on the level. As visible in Figure 1, these three curves are strongly dependent, with a positive link between f (x, 1) and f (x, 3), and negative between f (x, 1) and f (x, 2). 0.0 0.2 0.4 0.6 0.8 1.0 Figure 1: Test function 1. f (x, 1) in black, f (x, 2) in red and f (x, 3) in green. The design points correspond to one realization of a sliced LHD. The aim is to compare the accuracy of four GP models by reconstructing f with few evaluations. For all of them, a Matérn 5/2 kernel [Rasmussen and Williams, 2006] is chosen for x. The first GP model (IND) consists in three independent GPs corresponding to the levels of u. The other ones have tensor product kernels, with three different covariance matrices T for u. The first two ones assume a constant variance: T is defined by a CS covariance 14 structure (Eq. 8), or by a general spherical parameterization (SPH) (See Section 2.2.2). Finally, we consider a heteroscedastic spherical parameterization (H-SPH) where the covariance matrix is defined by a general variance vector, and a spherical parameterization of the correlation matrix. In order to benefit from the strong link between levels, we use a design that spreads out the points between levels. For instance, the information given by f (0, 1) may be useful to estimate f (x, 2) and f (x, 3) at 0, without computing f (0, 2) and f (0, 3). More precisely, we have used a (random) sliced Latin hypercube design (SLHD) [Qian, 2012] with 5 points by level, for a total budget of 15 points. Parameter estimation is by maximum likelihood. As the likelihood surface may be multimodal, we have launched several optimizations with different starting points chosen at random in the domain. Model accuracy is measured over a test set formed by a regular grid of size 1000, in terms of Q2 criterion. The Q2 criterion has a similar expression than R2 , but is computed on the test set: P (yi − ŷi )2 2 , (18) Q = 1 − Pi 2 i (yi − ȳ) where the yi denote the observations (on the test set), ȳ their mean, ŷi the predictions. It is negative if the model performs worst than the mean, positive otherwise, and tends to 1 when predictions are close to true values. Finally, the process is repeated 100 times, in order to assess the sensitivity of the result to the design. We observe that the heteroscedastic model H-SPH clearly outperforms the other ones. As expected, the estimated variances (2.5, 5.3, 1.4) for this model are level-dependent, whereas a constant variance is wrongly estimated around 2 by CS and SPH. Moreover, we have represented in Figure 3 an estimation of the correlations between levels, deduced from the parameterized covariance matrix T . For representativeness, we have chosen one of the 100 designs used to generate Figure 2, such that the corresponding Q2 is the closest to the median of the 100 Q2 values. We can see that the strong dependence link of f (., u) between the levels of u is only recovered correctly by H-SPH, with a poor estimation of correlation parameters for the other ones. 4.2 Example 2 (Levels grouping) The second function is defined as: x u g(x, u) = cos 7π + p(u)π − 2 20 15 1.00 0.75 0.50 0.25 0.00 IND CS SPH H−SPH Figure 2: Q2 criterion for four GP models, based on 100 repetitions of the design. u with x ∈ [0, 1], u ∈ {1, . . . , 13} and p(u) = 0.4 + 15 1u>9 . As visible in Figure 4, there are two groups of curves corresponding to levels {1, . . . , 9} and {10, . . . , 13} with strong within-group correlations, and strong negative between-group correlations. We aim at reconstructing g with five GP models based on levels grouping. The first one uses a CS covariance matrix, corresponding to a single group. The second one considers the two groups {1, . . . , 9} and {10, . . . , 13}. The third model, based on the five groups {1, . . . , 9}, {10}, {11}, {12}, {13}, has two variants: (a) when the inter-groups correlation is constant and (b) in the general case. The fourth model uses the spherical parameterization of T , leading to 13 groups, and the last one considers an ordinal paramaterization for T . The design of experiments is a 39-points SLHD. The remaining simulation settings are the same as in Example 1. The estimated correlation parameters are shown in Figures 5. The right correlation structure is well recovered with two groups and five groups, with different between-groups correlations. The model with thirteen groups involves the estimation of 90 parameters, which is hard to achieve, especially with 39 points. This is visible in the erratic values of the estimated correlations values, which seem not meaningful. On the opposite, considering only one group or five groups with a common between-group correlation oversimplifies and fails at recovering the right correlations. The ordinal ker16 CS H-SPH SPH −1.0 −0.5 0.0 0.5 1.0 Figure 3: Estimated correlation parameters among levels for f , for a design of experiments corresponding to a median Q2 . 0.0 0.2 0.4 0.6 0.8 1.0 Figure 4: Test function g. nel recovers the two blocs of curves, but cannot detect negative correlation between them (Remark 2). In Figure 6, we can see that the best tradeoff between prediction accuracy and parsimony is obtained with two groups. whereas it reduces the number of observations by group. Notice the rather good performance of the ordinal model, at the cost of a larger number of parameters: the warping was parameterized by an affine function (see Section 2.2.1). This is noticeable since negative correlations are not possible (see above). This may be due to the larger number of within-group levels combinations, which reduces the influence of the negative between-group correlations. 17 1 1 1 0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0 −0.2 −0.2 −0.2 −0.4 −0.4 −0.4 −0.6 −0.6 −0.6 −0.8 −0.8 −0.8 −1 −1 −1 1 group. 5 groups (b) 2 groups. 5 groups (a) 1 1 1 0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0 −0.2 −0.2 −0.2 −0.4 −0.4 −0.4 −0.6 −0.6 −0.6 −0.8 −0.8 −0.8 −1 −1 −1 13 groups. Ordinal. Figure 5: Estimated correlation parameters among levels for g, based on a representative design of experiments (design with median Q2 ). 5 5.1 Application in nuclear engineering Position of the problem As presented in Introduction, this research is originally motivated by the solving of an inverse problem confronting experimental measurements in nuclear engineering and time-consuming numerical simulation. More precisely, this analysis concerns the identification of the mass m of 239 Pu that is present in a particular waste container using a non-destructive nuclear detection technique such as the gamma spectrometry [Knoll, 2010]. In that case, at each energy level E, m × (E; E) = y SG (E), (19) where y SG (E) is the quantity of interest provided by the gamma transmitter, and (E; E) is the attenuation coefficient, which depends on the source envi18 1.0 0.8 0.6 0.4 1 group 2 groups 5 groups 5 groups (b) 13 groups ordinal Figure 6: Q2 of six GP models, based on 100 repetitions of the design. Number of parameters used (bloxplot order): 5, 7, 10, 19, 90, 16. ronment denoted by E. In practice, only discrete values of E are of interest, corresponding to the natural energy levels of 239 Pu: E ∈ {94.66, 129.3, 203.6, 345.0, 375.1, 413.7} (keV). (20) Then, based on previous studies [Guillot, 2015], the real source environment is parameterized by the following input variables: • An equivalent geometric shape for the nuclear waste: sphere (‘sph’), cylinder (‘cyl’) or parallelepiped (‘par’). • An equivalent material for this waste, characterized by its chemical element with atomic number in {1, . . . , 94}, • The bulk density of the waste, in [0, 1], • The distance of measurement between the container and the measurement device, in [80, 140] (cm), • The mean width and lateral surfaces (in logarithmic scale) crossed by a gamma ray during the rotation of the object. After normalization, the characteristics of the input space can be summed up in Table 2. 19 Name of the input Distance Density Width Surface Energy Shape Chemical element Variation domain [0,1] [0,1] [0,1] [0,1] {1, 2, 3, 4, 5, 6} {sph, cyl, par} {1, . . . , 94} Table 2: Description of the input variables for the nuclear application. To recapture the notation of the previous sections, let x and u be the vectors gathering respectively the continuous and categorical inputs, and w = (x, u). For a given value of w, Monte Carlo simulation codes as MCNP [Goorley et al., 2013] can be used to model the measured scene and approach the value of (w) = (E, E). The mass m can eventually be searched as the solution of the following optimization problem: (m? , w? ) = arg min ky obs − m × (w)k, m,w (21) where k · k is the classical Euclidian norm, (w) and y obs respectively gather the values of and y SG at the six values of E that are used for the measurements. To solve (21), it is therefore necessary to compute at a high number of points. However, each evaluation of the MCNP code can be extremely demanding (between several minutes to several hours CPU for one evaluation). Thus, surrogate models have to be introduced to emulate the function w 7→ (w), which is now investigated in the frame of Gaussian process regression. We refer to Clement et al. [2018] for the second step, namely the treatment of the inversion problem. 5.2 Model settings For pedagogical purpose, a dataset of large size N = 5076 has been computed with the MNCP code. The construction of the design of experiments was guided by the categorical inputs, such that each of the 6 × 3 × 94 = N/3 combinations of levels appears 3 times. It was completed by a Latin hypercube of size N to define the values of the four continuous inputs. 20 From this full dataset, a training set of size n = 3 × 94 = 282 is extracted by selecting at random 3 observations by chemical element. The remaining N − n points serve as test set. −12 −12 −14 −14 Y Y −16 −16 E1 E2 E3 E4 E5 E6 Sph Y in function of the energy Cyl Par Y in function of the geometric shape. Figure 7: Y in function of the energy and geometric shape. Model settings are now motivated by a graphical analysis. In Figure 7, the output is displayed in function of the energy and the geometric shape. We observe that successive energy levels correspond to close values. This fact confirms that the energy is ordinal and we use the warped kernel defined by Eq. (5). The influence of the geometric shape is less obvious, and we have chosen an exchangeable (CS) covariance structure for it. In Figure 8, Y is displayed in function of the 94 chemical elements, ordered by atomic number. Two important facts are the high number of levels and heteroscedasticity. For this purpose, the 94 chemical elements are divided into 5 groups, provided by expert knowledge and represented by colors. This partition suggests to use a group kernel of the form (10), where the within-group blocks Wg are CS covariance matrices. In order to handle heteroscedasticity, the variance of Wg is assumed to depend on the group number g. The influence of continuous variables can be observed by panels (not represented), and does not reveal useful information for our purpose. A Matérn 5/2 kernel is set for all continuous inputs, as we expect the output to be a regular function of the continuous inputs. Indeed, for this kernel, the corresponding Gaussian process is two times mean-square differentiable. Finally, three candidate kernels for w are obtained by combining the kernels of input variables defined above, by sum, product or ANOVA (see Section 2). 21 −12 −14 Y −16 1 10 20 36 61 94 Figure 8: Y in function of chemical elements, ordered by atomic number. 5.3 Results Following the model settings detailed above, Figure 9, Panel 3, presents the results obtained with 60 random designs of size n and three operations on kernels. Furthermore, we have implemented three other kernels for the chemical element, in order to compare other model choices for this categorical input. In the first panel, we grouped all the 94 levels in a single group. In the second one, we kept the 5-group kernel but forced the between-group covariances to have a common value. Finally, in the fourth panel, we considered that the levels were ordered by their atomic number, and used the warped kernel of Eq. (5) with a Normal transform. Figure 9: Q2 of several GP models, based on 60 random designs, corresponding to different model choices for the chemical element. First panel: Single group. Second panel: 5 groups, with a common between-group covariance. Third panel: 5 groups. Fourth panel: Ordered levels. Total number of parameters used in the panel order: ‘prod’ = (12, 21, 30, 14), ‘add’ = ‘prod’ + 6, ‘anova’ = ‘prod’ + 7. 22 First, comparing the three operations on kernels, we remark that in all the panels, additive kernels provide the worst results. This suggests the existence of interactions between different inputs of the simulator. Second, the ANOVA combination produces slight improvements, compared to the standard tensor-product, both in terms of accuracy and stability with respect to design choice. Now, comparing the four panels, we see that gathering the levels in a single group is the least efficient strategy. The 5-group kernel gives very good performances, especially when the between-group covariances vary freely: constraining them to be equal degrades the result. Surprisingly here, the ordinal kernel gives the best performance. Indeed, for this application it was not intuitive to the experts that the chemical element can be viewed as an ordinal variable, simply sorted by its atomic number. This is confirmed by the correlation plots of Figure 10, corresponding to a model with a median Q2 score. We can see that the estimated correlations between levels seems to decrease as the difference between levels increases, an indication that the levels may be ordered by their atomic number. Finally, we report several post-processing results. First, the estimated transformation of energy levels (Figure 11a) is concave and flat near high values, which corresponds to the behaviour observed in Figure 7 (left panel). In addition, the last three levels lead to similar results (Figure 11). This corresponds to the fact that when the energy is high, the gamma ray almost always crosses the nuclear waste, leading to a high value for the output. Second, the estimated correlation among the sphere, the cylinder and the parallelepiped is very high (c = 0.9, Figure 12). This justifies considering a covariance structure for that categorical input, rather than using three independent GP models for all the three levels. 6 Conclusion In the framework of GP regression with both continuous and categorical inputs, we focused on problems where categorical inputs may have a potentially large number of levels L, partitioned in G L groups of various sizes. We provided new results about parsimonious block covariance matrices, defined by a few within- and between-group covariance parameters. 23 (a) 5 groups (common betweengroup covariance) (b) 5 groups (general) 6 5 4 3 0.6 2 1.0 0.8 2 3 0.2 0.4 1 1 4 −0.2 0.2 1 Figure 10: Estimated correlation parameters among the chemical element. 5 0.8 0.6 0.4 0 −0.4 −0.6 −0.8 6 0.0 0.2 0.4 0.6 0.8 1.0 −1 (a) Transformation. (b) Correlations. 3 2 1 Figure 11: Estimated correlation parameters for the energy. 1 0.8 1 0.6 0.4 0.2 2 0 −0.2 −0.4 3 −0.6 −0.8 −1 Figure 12: Estimated correlation parameters for the geometric shape. 24 We revisited a two-way nested Bayesian linear model, where the response term is defined as a sum of a group effect and a level effect. We obtained a flexible parameterization of block covariance matrices which automatically satisfy the positive definiteness conditions. As a particular case, we recover situations where the within-group covariance structures are compound symmetry, with possible negative correlations. Furthermore, we showed that the positive definiteness of a given block covariance matrix can be checked by verifying that the small matrix of size G obtained by averaging each block is positive definite. This criterion can be useful if the proposed block matrix has a desirable constraint, such as homoscedasticity, which is not directly handled by the proposed parameterization. We applied these findings on several toy functions as well as an application in nuclear engineering, with 2 continuous inputs, 3 categorical inputs, one of them having 94 levels corresponding to chemical numbers in Mendeleev’s table. In this application, 5 groups were defined by experts. The results, measured in terms of prediction accuracy, outperform those obtained with oversimplifying assumptions, such as gathering all levels in a same group. On the other hand, when the categorical input can be viewed as an ordinal one, plugging the right order into warped kernels has lead to slightly better results in our experiments. There are several perspectives for this work. Firstly, one future direction is to find a data-driven technique to recover groups of levels. This may be not an easy task, due to the small number of observations available in the context of GP regression. Similarly, if there is an order between levels, can we infer it from the data? Secondly, the trend of the GP models has been fixed to a constant. More complex forms, based on linear models, could be explored. Software information and acknowledgements Implementations have been done with the R packages mixgp and kergp [Deville et al., 2015]. Illustrations use ggplot2 [Wickham, 2009] and corrplot [Wei and Simko, 2016]. This research was conducted within the frame of the Chair in Applied Mathematics OQUAIDO, gathering partners in technological research (BRGM, 25 CEA, IFPEN, IRSN, Safran, Storengy) and academia (CNRS, Ecole Centrale de Lyon, Mines Saint-Etienne, University of Grenoble, University of Nice, University of Toulouse) around advanced methods for Computer Experiments. Appendix Proof of Proposition 1. The vector (λ, λ1 + · · · + λL ) is a centered Gaussian vector with covariance matrix IL 1L vλ . 1> L L Hence the conditional distribution of λ knowing λ = 0 is a centered Gaussian vector with covariance matrix −1 cov(λ \| λ = 0) = vλ [IL − 1L L−1 1> L ] = vλ [IL − L JL ]. Then, by using the independence between µ and the λ` ’s, we deduce cov(η \| λ = 0) = vµ JL + vλ [IL − L−1 JL ] = vλ IL + [vµ − L−1 vλ ]JL . −1 We recognize the CS covariance matrix ΓCS L (v, c) with v = vµ + (1 − L )vλ −1 and c = vµ − L vλ . As a covariance matrix, it is positive semidefinite. Furthemore, we have c < v and c + (L − 1)−1 v = vµ [1 + (L − 1)−1 ] > 0, and the conditions of positive definiteness (12) are satisfied. Conversely, let C be a positive definite CS matrix ΓCS L (v, c). Then we have −1 −1 −(L−1) v < c < v, and we can define vµ = L [v +(L−1)c] and vλ = v −c. From the direct sense, we then obtain that the covariance matrix of η \| λ = 0 is ΓCS L (v, c) = C. Proof of Proposition 2. The first part of the proposition is obtained by remarking that if F = cov(z), then F = cov(z). Thus, assuming that z is centered, F = 0 is equivalent to z = 0 with probability 1. For the second part, notice that z = 0 means that z is orthogonal to 1L . Thus, one can write the expansion of z in the orthonormal basis 1⊥ L defined by A. Denoting by t the (L − 1)-vector of coordinates, we have z = At. This gives F = cov(At) = A cov(t)A> , and (14) follows with M = cov(t). 26 To prove unicity, observe that, by definition, A> A = IL−1 , A> 1L = 0. Starting from F = AMA> , and multiplying by A> on the left and by A on the right, we get M = A> FA, showing that M is unique. Now, let F = v[IL − L−1 JL ]. Since JL = 1L 1> L , we obtain M = A> FA = v[A> A − L−1 (A> 1L )(1> L A)] = vIL−1 . As a by-product, notice that resubstituting M into F = AMA> gives AA> = IL − L−1 JL . Finally, if z ∼ N (0, vIL ), then the properties of conditional Gaussian vectors lead immediately to cov(z\|z = 0) = F. Proof of Proposition 3. The expressions of Wg and Bg,g0 are obtained directly by using the independence assumptions about µ and the λ’s. Notice that Fg , the covariance matrix of λg/. knowing λg/. = 0, is centered by Proposition 2. This gives Wg − Wg Jng = Fg , which is positive semidefinite. Conversely, let T be a positive semidefinite matrix of the form (10), such e be that Wg − Wg Jng is positive semidefinite for all g = 1, . . . , G. Let T e is also a the matrix obtained from T by averaging each block. Then T positive semidefinite matrix. Indeed, since T is positive semidefinite, it is e is the covariance matrix the covariance matrix of some vector z. Then T P of e z, the vector obtained from z by averaging by group: e zg = n−1 g `∈Gg z` . Thus there exists a centered Gaussian vector (µg )1≤g≤p whose covariance matrix is e Γµ = T. Now, for g = 1, . . . , G, define Fg = Wg − [Γµ ]g,g Jng = Wg − Wg Jng . Observe that Fg = 0, and by assumption Fg is positive semidefinite. Hence, from Proposition 2, there exists a centered Gaussian vector (λg/j )1≤`≤ng such that Fg = cov(λg/. \|λg/. = 0). We can assume that λ1/. , . . . , λG/. are independent, and µ and λ are independent. Finally, we set ηg/` = µg + λg/` . By the direct sense and (16), we obtain that T is the covariance matrix of η conditional on {λg/. = 0, g = 1, . . . , G}. 27 Proof of Corollary 1. Let T be a positive semidefinite matrix of the form (10) with CS diagonal blocks. Then the diagonal CS matrices are positive semidefinite, leading to vg − cg ≥ 0. Thus, Wg − Wg Jng = (vg − cg )(Ing − n−1 g Jng ) is a positive semidefinite CS matrix. Hence, by Prop. 3, T is obtained from Model (15), with cov(λg/. \|λg/. = 0) = (vg − cg )(Ing − n−1 g Jng ). By Prop. 2 (last part), we can choose Γλg = vλg Ing , with vλg = vg − cg ≥ 0. Conversely, if Γλg = vλ,g Ing , then by Prop. 2, Fg = cov(λg/. \|λg/. = 0) is a CS covariance matrix. The result follows by Prop. 3. Proof of Proposition 4. The direct sense of (i) has already been derived in e the proof of Prop. 3. Furthermore, inspecting that proof, we see that if T is positive semidefinite, then T admits the representation (15). Thus T is a covariance matrix, and positive semidefinite. For (ii), a proof is available in [Roustant and Deville, 2017]. Notice that we need to add the condition that Wg is positive definite for all g = 1, . . . , G. However, adding an equivalent condition for (i), namely that Wg is positive semidefinite, was not necessary. 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MAY 2017 1 A Survey on Trapping Sets and Stopping Sets arXiv:1705.05996v1 [cs.IT] 17 May 2017 Aiden Price and Joanne Hall, Member, IEEE. Abstract—LDPC codes are used in many applications, however, their error correcting capabilities are limited by the presence of stopping sets and trappins sets. Trappins sets and stopping sets occur when specific low-wiehgt error patterns cause a decoder to fail. Trapping sets were first discovered with investigation of the error floor of the Margulis code. Possible solutions are constructions which avoid creating trapping sets, such as progressive edge growth (PEG), or methods which remove trapping sets from existing constructions, such as graph covers. This survey examines trapping sets and stopping sets in LDPC codes over channels such as BSC, BEC and AWGNC. Index Terms—LDPC codes, trapping sets, stopping sets, QCLDPC codes, Margulis codes, AWGNC, PEG algorithm, graph covers. I. I NTRODUCTION S technology advances, we wish to communicate over longer distances and have the ability to stay connected even over poor communication channels. While quasi-cyclic low-density parity-check (QC-LDPC) codes are one of the best ways to achieve this [1], their performance in many cases is limited by the presence of trapping sets and stopping sets. Trapping sets and stopping sets can cause iterative decoding methods to fail with relatively few errors. Finding ways to avoid or remove trapping sets and stopping sets will further improve the already high performance of LDPC codes and bring their performance curves even closer to the Shannon limit [2], [3]. Performance optmization is becoming incresingly crucial as the world moves further into the digital age. An increase in the speed at which digital communication occurs through modern applications such as WiFi [4] and DVB-S2 [5] has drastic implications on the overall productivity of the world. In 1962, Gallager [6] introduced low-density parity-check codes (LDPCs). LDPC codes are a class of binary linear block codes with a sparse parity-check matrix. An advantage of using LDPC codes is that they are able to provide error control which is very close to the capacity for many different channels [2]. This categorizes LDPC codes as one of few capacityapproaching codes; error correction methods which can allow the noise in a channel to be set very close to its theoretical maximum while maintaining error-correcting ability [7]. The performance of an error correction method is based upon two properties; the performance of such a code over a channel with variable noise and the optimal bit error ratio (BER) of a code with sufficient signal-to-noise ratio (SNR). The optimal BER is known as the error floor of a code [8], and A A. Price is with the Science and Engineering Faculty, Queensland University of Technology, Queensland, QLD, Australia. e-mail: [email protected]. J. Hall is with the School of Science, Royal Melbourne Institute of Technology, Melbourne. Manuscript received MO DATE, YEAR; revised MO DATE, YEAR. is discussed in different papers in terms of bit error rate (BER), frame error rate (FER), block error rate and symbol error rate, depending on the application being addressed (see [9], [10] for examples of such error floor analysis). Consideration of the error floor is one of the most important aspects of constructing a high-performing LDPC code [10]. Analysis of the performance of LDPC codes over the binary erasure channel (BEC) led to the discovery of stopping sets in 2002 [11]. The Margulis construction [12] improved upon the performance of Gallager codes, though a weakness in this construction led to a high error floor over the additive white Gaussian noise channel (AWGNC) compared to the performance of other constructions of the time [13]. This high error floor was due to the presence of stopping sets. Stopping sets over BEC as described in [11] became a well understood problem and led to the definition of trapping sets, which are defined over AWGNC and BSC. In some early works trapping sets are called near-code words [8], [13]. Trapping sets and stopping sets are an important topic, worthy of a stand-alone survey. II. P RELIMINARIES AND N OTATION In order to engage with the literature on stopping sets and trapping sets an overview of the preliminaries is necessary. We provide a short review of the literature surrounding LDPC codes, the common transmission channels, and common decoding techniques. Definition 1: [14] A binary [n, k, d] linear code, C, is a k-dimensional subspace of an n-dimensional vector space, F2n which is used to provide structure to a message vector for transmission over a channel. In order to transmit messages over communication channels using an error correcting code, we encode the message using a generator matrix. Definition 2: [7] The generator matrix, G, of a code, C, is a matrix which has dimensions k × n. The k rows of G correspond to linearly independent code words which form a basis of C. One of the most important aspects of error correction is the process of decoding. A parity-check matrix allows us to identify whether errors have been introduced during transmission. This matrix can also be represented by a Tanner graph. Definition 3: [7] A parity-check matrix, H, of C is a matrix which generates the nullspace of the code. This means that a code word, c, is in the code, C, iff H · cT = 0, where 0 is an r × 1 null vector. H has dimensions (n − k) × n. Definition 4: [14] The parity-check matrix, H, may be represented by a bipartite graph with variable node set, V, and check node set, C. This bipartite graph is denoted G(H) = (V ∪C, E), MAY 2017 2 where the columns of H indicate the variable nodes in V and the rows of H indicate the check nodes in C. For i ∈ V and j ∈ C, (i, j) ∈ E if and only if Hi j = 1. This bipartite graph is known as a Tanner graph with r = n − k check nodes and n variable nodes (see Fig. 1). We can also refer to individual variable nodes; let the i-th variable node be vi and the j-th check node be c j [15]. Definition 5: [16] If a parity-check matrix of a code is sparse, then the corresponding code, C, is called a low-density parity-check (LDPC) code. We note that the classification of sparse used in the context of LDPC codes is that there are fewer ones in the parity-check matrix than there are zeros [7]. The sparse nature of LDPC codes means that decoding processes have a fast run-time, as there are fewer operations to compute when compared to a non-sparse parity-check matrix. Two more important features of Tanner graphs are neighbours and node degrees. Definition 6: [16][17] For a variable node vi and check node c j , if (i, j) ∈ E we say that nodes vi and c j are neighbours. The degree of a node in the Tanner graph is defined as the number of edges it is connected to. From the node degree definition we can also define regular LDPC codes. Definition 7: [6] An LDPC code is called (dv, dc )−regular if each variable node, v, has degree dv and each check node, c, has degree dc . We denote an LDPC code of this form a C(dv, dc ) code. LDPC codes are designed to be used as error-correction methods over a variety of communication channels. There are three communication channels discussed in this paper; the binary erasure channel (BEC), the binary symmetric channel (BSC) and the additive white gaussian noise channel (AWGNC) [18]. Though these channels handle data transmission in different ways, their encoding and decoding goals are the same. The upper bound on the error correcting ability of an LDPC code is determined by the minimum distance of the code. In order to define the minimum distance of a code, we will first define Hamming weight and Hamming distance. Definition 8: [19] The Hamming weight, w, of a vector is the number of its non-zero elements. The Hamming weight of a binary vector is therefore the number of ones in the vector. The Hamming distance, d, between two vectors, x and y, is the number of places in which they differ; written as d(x, y). In the literature, Hamming weight and Hamming distance are often referred to using the terms “weight” and “distance” [14]. The weight of code words affects the number of operations performed in decoding and the distance between code words affects how many errors can be corrected. Definition 9: [18] The minimum distance of a code, C, is defined as the smallest Hamming distance between any two code words in the code, A. Encoding and Verification The distance between these codewords is given as d(c1, c2 ) = 4. Take the following error vector: The process of transforming a message vector into its associated code word is known as encoding. Every code word c = [c0, c1, ..., cn−1 ] ∈ C can be expressed as c = m · G. Where m = [m0, m1, ..., mk−1 ] is the mesage vector [7]. The code word, c, has the original k information bits as well as an additional r parity bits to give the code word a length of n bits. As the parity-check matrix, H, is the nullspace of the code, we can use it as a verification method to test if a recieved vector is a code word [14]. The product H · cT is denoted the syndrome, s, of c through H. For a given vector, v, v ∈ C iff s = H · vT = 0. There must be an even number of ones in the components of the product H·cT which add to give s = 0. This is known as the even-parity constraint [2]. After a code word, c, is transmitted through a channel, the other party receives a vector, v. If s = H· vT , 0, then v < C and so we use error correcting techniques in attempt to correct v and recover c. d(C) = min{d(x, y) \| x, y ∈ C, x , y}. The following theorem and corollary describe a code’s error detection and correction abilities using minimum distance [18]. Theorem 1: [18] A code C can detect up to s errors in any code word if d(C) ≥ s + 1. A code C can correct up to t errors in any code word if d(C) ≥ 2t + 1. Corollary 1: [18] If a code C has minimum distance d, then C can be used to either detect up to d − 1 errors, or to correct up to (d − 1)/2 errors in any code word. If the minimum distance of a code is too small, then it cannot provide sufficient error correction. This is demonstrated in Example 1 Example 1: Let two code words be given as; c1 = [0100001111] c2 = [0100111010]. e = [0000110101]. If e is added to c1 then the resulting code word is identical to c2 , demonstrating the importance of minimum distance. The minimum distance of an LDPC code is also related to its code-rate; a large code rate lowers the upper bound on the minimum distance of a code. Definition 10: The code rate, R(C), of an LDPC code is the portion of information bits sent in comparison to the entire code vector sent, written as R(C) = k , n where 0 ≤ R(C) ≤ 1. The code rate and minimum distance often determine the error correcting capability of an LDPC code, though the decoding algorithm plays a direct part in the time it takes to decode messages. MAY 2017 3 v1 e e c1 v2 3e → 1 (a) 1 0e 0 (b) 0e 0 1e e 0e 1 1 1 1 → 0e 0 c5 v10 1 0e 0 v9 1 1e 1 1 c4 v8 1 1 1 v7 0e 1 c3 v6 1 1e 2e 1 v5 0 1 e c2 v4 0e 0 0 v3 0 0e 1e 0e 0 (c) (d) Fig. 1. A 5 × 10 irregular Tanner graph (a) used to demonstrate the ER decoding algorithm with the received vector v = [e, 0, e, 1, 1, 1, 1, 0, 1, e] over BEC. Nodes of interest are highlighted gray. (b) shows steps 1 and 2 of the ER algorithm’s first iteration. (c) shows the changes made in step 3 and then step 2 of the second iteration. (d) shows the changes made in step 3 again, this time revealing that no further erasures exist. The algorithm then terminates on step 4 of this iteration, thus successfully correcting the received vector in 2 iterations. B. Communication Channels and Decoding Basics The communication channel by which transmission occurs impacts the error correction algorithms that are chosen. A communication channel can be modelled as a triple which contains an input alphabet, an output alphabet and the probability of transition between a symbol in the input alphabet and a symbol in the output alphabet [3]. The binary erasure channel (BEC) is one of the simplest, non-trivial channel models [20],[21]. Definition 11: [22],[16] The Binary Erasure Channel (BEC) is a communication channel with two input symbols, 0 and 1, and three output symbols, 0, 1 and e (the erasure symbol). The BEC has an erasure probability, p, where given an input, ci , the output, vi , is defined by the probability formulae P[vi = ci ] = 1 − p and P[vi = e] = p. Analysis of the BEC significantly advanced modern understanding of error correction [20]. An example of a simple decoding process over the BEC is the Edge Removal algorithm. Definition 12: [16] Let c ∈ C ⊆ {0, 1} n be a binary code word transmitted over the BEC and v ∈ {0, 1, e} n be the received vector. The Edge Removal (ER) algorithm proceeds as follows: 1) Initial Step: The value of each received vector bit, vi is assigned to each variable node i ∈ V of the Tanner Graph. 2) The check nodes ci ∈ C count the number of erased bits which are neighbours in the Tanner graph, G(H). 3) If check node ci neighbours only one e symbol in v, the even parity constraint uniquely determines the original value of e for that variable node. 4) Repeat steps (2) and (3) until either all erasures have been recovered or until every check node that is a neighbour of an erased bit is a neighbour of at least two erased bits. For step (4) above, if the latter occurs, then the decoder has failed due to the presence of a stopping set (see Section IV). We provide an example of the ER decoding process in Fig. 1, where ◦ represents a variable node and a check node. This example is found in [15]. For this decoding example, we use an irregular 5×10 paritycheck matrix, H and demonstrate the decoding of received vector v = [e, 0, e, 1, 1, 1, 1, 0, 1, e] using the ER algorithm over the BEC. Another communication channel is the binary symmetric channel (BSC). Definition 13: [16], [22] The Binary Symmetric Channel (BSC) is a communication channel with two input symbols, 0 and 1, and two output symbols, also 0 and 1. The BSC has an error probability, p, where given an input, ci , the output, vi , is defined by the probability formulae P[vi = ci ] = 1 − p and P[vi = c¯i ] = p. An example of a decoding process over the BSC is the Gallager A algorithm [6]. Definition 14: [6], [22], [23] Let c ∈ C ∈ {0, 1} n be a binary code word transmitted over the BSC and v ∈ {0, 1} n be the received vector. The Gallager A algorithm proceeds as follows: 1) Initial Step: The value of each received vector bit, vi is assigned to each variable node i ∈ V of the Tanner graph. 2) After this, a check node ci sends to all neighbouring variable nodes vi, . . . , v j the sum (mod 2) of all of the adjacent variable nodes except for the node itself (where j is the degree of check node ci ). 3) Each variable node, vi then sends the following to their adjacent check nodes: If all messages from check nodes ci other than the target check node of the message are equal, then vi sends that message back, otherwise it resends its prior value. 4) Repeat steps (2) and (3) until either all variables nodes send the same values over two consecutive iterations or when a pre-set max iteration count is reached. MAY 2017 4 v1 1, 0, 1 0 c1 v2 1 0 → 0, 1, 0 1 0, 1, 0 (a) (b) 0 1, 1, 0 1 1 0 1, 1, 1 0, 0, 0 0, 1, 0 1 → 0 1, 1, 0 c5 v10 1, 1, 1 0 0 v9 0, 1, 0 1 0, 0, 0 1 c4 v8 1, 1, 1 0, 1, 0 1 v7 0 1, 1, 1 c3 v6 0, 0, 1 1 1 0 v5 0, 0, 0 0, 0, 1 1 c2 v4 1 0, 0, 0 1 v3 1, 0, 1 1 1 1 0, 1, 0 (c) (d) Fig. 2. A 5 × 10 (6,3)-regular Tanner graph (a) used to demonstrate the Gallager A decoding algorithm with the received vector v = [0, 1, 1, 0, 1, 0, 1, 0, 1, 1]. We represent a 0 being sent along an edge by a dashed line and a 1 with a full black edge. (b) shows step 1 of the Gallager A algorithm, as well the check node calculation, taken as the addition mod 2 of all incoming message from variable nodes adjacent to each check node (denoted vi → ci ). (c) then shows step 2, ci → vi . Lastly, (d) shows step 3, vi → ci . Due to the complexity of this algorithm, only one full iteration has been shown, though step 4 in Definition 14 describes how decoding continues. The Gallager B algorithm offers improved decoding with an additional step on each loop within the algorithm [23]. For each degree, j, and each check node loop, i, there is a prechosen threshold value, bi, j . Throughout the steps involved in check node ci for each variable node, v, and each adjacent check node, c, if at least bi, j neighbours of v excluding c sent the same information in the previous round, then v sends that information to c; otherwise v sends its received value to c. Algorithm A is a special case of algorithm B, where bi, j = j−1 independent of the round [23]. Throughout the decoding procedure, if the pre-set max iteration count is reached without completion, the decoder has failed due to the existence of a trapping set (see Section V). An example of the first steps of the Gallager A algorithm (see Fig. 2) demonstrates the differences between the decoding considerations made between the BEC and the BSC. The most complex channel considered here is the binary input additive white gaussian noise channel, expressed commonly either as the BI-AWGNC or just as the AWGNC [24]. Definition 15: [24] Let X ∈ {0, 1}∗ be a message vector, where ∗ denotes an arbitrary length. The additive white Gaussian noise channel (AWGNC) maps the input vector, X, to the vector X 0 ∈ {+1, −1}∗ and then adds the result with Gaussian white noise to give an output vector Y = X 0 + W, where W ∼ N (0, N0 /2Eb ). Each code symbol, y ∈ Y , carries with it a signal to noise ratio (SNR) of Eb /N0 and the conditional distribution of Y is P(y\|x 0) = PW (y− x 0) = p 1 2π(N0 /2Eb ) ·exp −(y − x 0) , (1) (N0 /Eb ) 2 which gives the output alphabet for the AWGNC as y ∈ R. As in the BEC and BSC, we would like to have some indication of the errors that a channel is introducing to the code word. A metric used for the AWGNC is the log likelihood ratio (LLR). P(y\|x = 0) L(y\|x) = ln . (2) P(y\|x = 1) This L-value describes the likelihood that x is 0 or 1. If L is positive then P(y\|x = 0) > P(y\|x = 1) and thus the input estimate should be x̂ = 0. There are methods to map from Y ∈ R to Y 0 ∈ {0, 1}∗ and, as such, all decoding methods used over the BSC can be implemented on the AWGNC. However, high performing decoding algorithms, such as maximumlikelihood decoders, the sum-product algorithm and the maxproduct algorithm [8],[25],[26] utilize the LLR information to improve decoding speed [27]. The BSC can be used for these channels as LLR values are defined over the BSC, though the AWGNC more closely models the influence of real-world communication channels and is favoured for high performance simulations [7],[24]. Example 2: The BSC has a conditional LLR function as the bit flipping probabilities are well understood for the outputs ( 1−p ln( p ) y = 0 LBSC (y\|x) = (3) p ln( 1−p ) y=1 As the noise determines the values of y in the AWGNC and y ∈ R, the LLR on this channel is defined as L AW G NC (y\|x) = 4 Eb y. N0 (4) The decoding algorithms used on the AWGNC are far more complex than those on the BEC and BSC and, as such, we provide an overview of various methods rather than detailed definitions and examples. Decoding methods used over the AWGNC tend to be message passing algorithms, where nodes send information to their neighbours to correct errors based on the structure of the parity-check matrix [28]. The original message-passing algorithm [6] is an example of a flooding MAY 2017 5 v1 ability of a code as well as the frame error ratio (FER) [8], [10], [15]. The FER is the ratio of frames or whole messages transmitted which cannot be fully corrected versus the total number of frames transmitted. The largest contributors to the error floor stopping sets and trapping sets. c1 v2 v3 v7 c2 v4 v5 c3 → c1 c4 c5 v6 v7 c4 v9 v8 v9 c5 v10 (a) (b) Fig. 3. (a) Tanner graph for the irregular 5 × 10 parity-check matrix given in Example 3. (b) Induced subgraph of the highlighted stopping set with consistent labelling. schedule [29] where in each iteration, all variable nodes and subsequently all check nodes pass new messages to their neighbours. Another example of a flooding schedule is the sum-product algorithm (see [25]). An improved schedule, where both variable nodes and check nodes send messages to each other throughout a single iteration is known by many names including serial scheduling [30], [31], layered scheduling [32] and sequential scheduling [33]. These algorithms offer an improved decoding performance as information is moving through the Tanner graph more frequently. Examples of decoding algorithms using this scheduling include the max-product algorithm (MPA) [26] and the belief propagation algorithm (BPA) [33]. BPA is widely used in LDPC code analysis and is based on the likelihood that a node takes a value given its current value and the values of nearby nodes from previous iterations [23]. Error correction must be implemented differently for each channel. The edge removal algorithm, for example, deals with erasures and thus is not suitable over the BSC. Errors which occur during transmission that are not corrected by the decoding algorithm form what is known as the bit error ratio (BER); the ratio of bits that cannot be corrected versus the total number of bits transmitted [8] [34]. In order to test the performance of LDPC codes, we can simulate the transmission of messages over an increasing signal-to-noise ratio (SNR) and calculate the BER of a code under varying conditions. As SNR grows larger, the BER of a code will suddenly decrease depending on the conditions of the channel and the error correcting capability of the LDPC code in use. This curve is known as the waterfall region [8]. The best scenario for correcting errors is when the probability of error during transmission over a channel is negligible and when the implemented error correcting code can correct many errors. Definition 16: [8] The waterfall region eventually ends in all BER graph curves as anomalous errors cause decoders to fail even with a high SNR ratio. The lowest the BER becomes before levelling is called the error floor of a code. The BER is a standard way to analyze the error correcting III. C YCLES AND G IRTH The decoding method we choose has direct implications for the accuracy and efficiency of decoding. Cycles were the first known negative characteristic of LDPC codes and were extensively studied as they impacted on the accuracy of high performance LDPC codes [35]. A cycle in a graph is a sequence of connected nodes which form a closed loop where the initial and final node are the same and no edge is used more than once [7]. The cycle length is the number of edges a cycle contains, and the length of the smallest cycle in a graph is denoted as its girth [17]. If no cycles exist within the Tanner graph of a paritycheck matrix, then the iterative belief propagation decoding technique is always successful with sufficient iterations [36]. However, if the neighbours of a node are not conditionally independent then belief propagation methods become inaccurate [35]. The inferred solution is to construct a parity-check matrix with no cycles. However, as discussed in Section VII, this is unessecary as not all cycles negatively impact the decoding efficiency of LDPC codes. In fact, the restriction of girth can lead to constraints on the structure of the code which further impedes the decoding efficiency [35]. The cycles which negatively impact the decoding efficiency of LDPC codes combine to form what are known as stopping sets and trapping sets [37]. These sets lead to a high error floor in otherwise efficienct LDPC code constructions throughout various communication channels and affect all high performing decoding algorithms. IV. S TOPPING S ETS OVER BEC Stopping sets are collections of variable and check nodes in the Tanner graph of an LDPC code which greatly reduce its error correcting ability. These sets cause decoding to fail when certain variable nodes are affected by errors after transmission. Stopping sets were first described in 2002 by Di et al [11], who were researching the average erasure probabilities of bits and blocks over the BEC. Definition 17: [11] Let G(H) be a Tanner graph and V be the set of variable nodes in G(H). A stopping set, S, is a subset of V, such that all neighbours of S are connected to S at least twice. The empty set is also a stopping set and the space of stopping sets is closed under union [11]; if S1 and S2 are both stopping sets then so is S1 ∪ S2 . The following lemma describes a stopping set by the performance of the LDPC code’s decoding algorithm. Lemma 1: [11] Let G be the generator for an LDPC code over the BEC and E denote the subset of the set of variable nodes which is erased by the channel after the transmission of a message. Then the set of erasures which remain when the MAY 2017 6 v1 0 0 c1 v2 2e 0 v3 1 1 1 → e e (a) 2e 0 e e 2e 0 (b) 0e 1 e 2e 0 → 2e 0 c5 v10 1 2e 0 v9 1 0e e c4 v8 0e 1 1 0e 1 v7 1 0e 1 c3 v6 0 1e e v5 2e 0 c2 v4 0 3e 2e 0 (c) (d) Fig. 4. The effect of a stopping set on the ER decoding process for the 5 × 10 irregular Tanner graph (a) on the right hand side of the line. (b) shows steps 1 and 2 of the ER algorithm’s first iteration. (c) shows the changes made in step 3 and then step 2 of the second iteration. Finally, (d) shows that no further erasures can be corrected, and thus we see that the received vector v = [0, 0, 1, e, 1, 1, e, 0, e, 0] produces a scenario in the Tanner graph where the ER algorithm cannot retrieve the original code word. From Definition 12, we know that this is due to the presence of a stopping set. decoder stops is equal to the unique maximal stopping set of E. Definition 17 is now widely accepted [38],[39],[40]. Given a BEC with erasure probability , the performance of the code over the BEC is completely determined by the presence of stopping sets [8]. Since stopping sets have a combinatorial characterization, their distributions through various Tanner graphs can be analyzed rigorously [11], [38]. Definition 18: [38] Let S denote the collection of all stopping sets in a Tanner graph, G(H). The stopping number, s∗ , of G(H) is the size of the smallest, non-empty stopping set in S. The stopping number of a code aids in the analysis of the code’s error floor. It is known that the performance of an LDPC code over the BEC is dominated by the small stopping sets in the graph [8]. The larger this value is, the lower the error floor of the code. In some cases, this stopping number increases linearly with the number of variable nodes, \|V \|, in the Tanner graph [38]. This can be seen more easily using the stopping ratio. Definition 19: [38] Let G(H) be a Tanner graph with n variable nodes and stopping number s∗ . The stopping ratio, σ ∗ , of a Tanner graph is defined by s∗ /n; the ratio of its stopping number to the number of variable nodes. A stopping set in the parity-check matrix of an LDPC code is shown in Example 3. Example 3: [15] Let C be the code with the following check matrix 0  1  H = 1 0  0  0 0 0 1 1 1 0 1 0 0 1 1 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 1 0 0 0 0 1 0 1 0 1 1 1 1 0 1 1 . 0 1 Columns 7 and 9 in H have been highlighted as they belong to a stopping set. The Tanner graph for C with the stopping set highlighted is shown in Fig. 3. A stopping set must be either empty or at least contain two variable nodes. The stopping number, s∗ , of C is therefore 2 and its stopping ratio, σ ∗ , 0.2. An example showing the impact of a stopping set on the decoder is shown in Fig. 4, where the edge removal decoding algorithm is used over the BEC. Solutions to the problem of stopping sets (covered in Section VII) involve either avoiding or removing small stopping sets in the Tanner graph, leaving only LDPC codes with large stopping sets [39]. While stopping sets are well defined and some solutions exist to minimize their effect of the error floor of LDPC codes, the terminology does not support channels without erasure. V. T RAPPING S ETS OVER BSC, AWGN Trapping sets, much like stopping sets, are also collections of variable nodes and check nodes which impede the error correcting ability of LDPC code. Only small, elementary trapping sets impact the error floor of LDPC codes over the BSC and AWGNC because of clustering [8], [41]. The definition of trapping sets came shortly after stopping sets were defined. Similarly to the BEC, when decoding over the BSC and AWGNC, sometimes the maximum iteration count is reached when only a small set of variable nodes are in error. Experiments with the argulis codes lead to the definition of trapping sets [8], [13]. Definition 20: [8] Let G(H) be a Tanner graph. For a received vector, y, of length n, we define the failure set, T(y), to be the set of bits that are not eventually correct using some arbitrary iterative decorder. Decoding is successful on y if and only if T(y) = ∅. Definition 21: [41], [8] If T(y) , ∅ then T(y) is a trapping set. More specifically, T is an (a,b) trapping set in H if it has a variable nodes for which the sub-graph induced by T contains b ≥ 0 odd-degree check nodes. MAY 2017 Fig. 5. A (5,3) trapping set (left) with critical number k = 3 and a (4,4) trapping set (right) with critical number k = 4 [10]. These k values are found using the Gallager B decoding algorithm and may vary when other decoding algorithms are applied. Iterative techniques on the BSC and AWGNC distinguish trapping sets from stopping sets over the BEC [8]. If there is only iteration by which the decoding algorithm can become trapped then the notion of trapping sets becomes irrelevant. Lemma 2: [8] Let C be a code using a one-step maximum likelihood decoder, then the trapping sets are precisely the non-zero code words. Though, if the channel is BEC, then an iterative decoding failure is said to be due to stopping sets, making stopping sets and trapping sets equivalent over the BEC. Lemma 3: [8] Let C be a code using a belief propagation algorithm over BEC, then the trapping sets are precisely the stopping sets. Lemma 3 is an important bridge between trapping sets and stopping sets, allowing us to relate the BEC to the BSC and AWGNC. Decoding failure in an LDPC code over the BSC and AWGNC is largely due to the existence of trapping sets. Trapping sets pose a real threat to the error correcting ability of LDPC codes; even though there may be very few nodes in error after transmission, if enough of those nodes belong to a trapping set, the decoder will fail. Definition 22: [10] Let T be a trapping set. The critical number, k, is the minimal number of variable nodes that have to initially be in error for the decoder to become “trapped" in T. It is important to note that the variables nodes that are initially in error do not necessarily belong to the trapping set; it is possible that, at some iteration, the trapping set is entered, causing the decoder to fail. In order to become trapped, the decoder must, after some finite number of iterations, be in error on at least one variable node from T at every iteration thereafter. Only trapping sets with a small number of variable nodes and check nodes impact the error-floor of LDPC codes [8], [41]. Definition 23: [41] An (a, b) √ trapping set in a [n, k, d] code is a small trapping set if a ≤ n and b ≤ 4a. Only these small trapping sets contribute to a larger errorfloor [8]. Small trapping sets are also of elementary form [41]. Definition 24: [41] An elementary (a,b) trapping set in a [n, k, d] code is a trapping set for which all check nodes in the induced subgraph have either degree one or two, and there are exactly b degree-one check nodes. While check nodes of odd degree larger than one are possible, they are very unlikely within small trapping sets 7 [8],[41]. Techniques to find and remove elementary trapping sets have become crucial when constructing high perfoming codes [10], [42]. Two examples of trapping sets [10] are shown in Fig. 5. In Fig. 5 the trapping set on the right has a smaller number of variable nodes than the one of the left, however, under the Gallager B decoding algorithm the larger trapping set has a smaller critical number. Thus the performance of the code is limited by the larger trapping set. This idea is quite unintuitive and shows the depth of consideration which must be made when attempting to improve the error floor of LDPC codes. The problems which trapping sets and stopping sets introduce to LDPC code ares important to research and solve. There do exist methods for constructing LDPC codes by avoiding or removing trapping sets and stopping sets, however, these methods come at the cost of restraining other properties such as code length, density or error correcting ability [9]. VI. T HE INFLUENCE OF S TOPPING S ETS AND T RAPPING S ETS ON LDPC C ODE P ERFORMANCE The original LDPC codes proposed in 1962 by Gallager [6], were construction methods which allowed for varied code rates. Definition 25: [13] A Gallager code is an LDPC code constructed using a parity-check matrix with uniform row weight i and uniform column weight j. The code has length n code words and has code rate R(C), which gives a parity-check matrix, H, with n columns and k rows, where k = n(1− R(C)). Naive analysis indicated that failed decoding is due to received vectors containing too many errors for the decoding algorithm [6]. Analysis of a range of error patterns by Di et al [11] determined that this was not always the case; leading to the definition of stopping sets over the BEC. A variety of analyses of Gallager codes have shown high performance [2] [43]. A construction in 1982 by Margulis [12] promised an improved performance over the AWGNC. For each prime, p, let SL2 (p) be the Special Linear Group whose elements consist of 2×2 matrices of determinant 1 over Z p . This group has k = (p2 − 1)(p2 − p)/(p − 1) = (p2 − 1)p elements. For p ≥ 5, the Margulis code is of length n = 2k, with code rate R(C) = 1/2 [12]. The rows of the parity-check matrix are indexed by the elements of SL2 (p) and the columns are indexed by two copies of SL2 (p); detailed in the following definition. Definition 26: [13][12] Let SL2 (p) be generated by the following matrices; 1 2 1 0 A= ,B = 0 1 2 1 If g ∈ SL2 (p) is the index of a row of the parity-check matrix, a one is placed in the columns corresponding to g A2 , g ABA−1 and gB on the left hand side of the matrix and also in the columns corresponding to g A−2 , g AB−1 A−1 and gB−1 on the right hand side of the matrix. This results in a (3,6)-regular parity-check matrix for a Margulis code. An example of a parity-check matrix generated using the Margulis construction is shown in Fig. 6 to demonstrate the MAY 2017 8 Parity-Check Matrix for Margulis Code (p=7) Frame Error Rate 10 -2 0 (336,672)-Margulis (336,672)-Gallager 50 100 150 10 -3 FER 200 250 300 0 100 200 300 400 500 600 10 -4 Fig. 6. Parity-check matrix generated using the Margulis construction, setting p = 7 to give a (3,6)-regular 1/2-rate code with n = 672. The blue dots represent ones in this matrix with the remaining white space representing zeros. 10 -5 sparse nature of LDPC codes. Another example of a Margulis parity-check matrix can be found in [13] where p = 11 which corresponds to a 1/2-rate code with n = 2640. While the code has a higher performance than a random Gallager code, the error floor is still quite high [13]. This error floor was claimed to be due to near-code words [13]. A comparison between the Margulis code and a random Gallager code, both with n = 672, can be seen in Fig. 7. Definition 27: [13] Let H be a parity-check matrix. If x is a vector of weight w and HxT = s where s is of weight v, then x is a (w, v) near-code word. Near-code words are different from stopping sets. Typical (w, v) near-code words contain v check nodes which are only connected to the variables nodes once. The near-code words in the Margulis code are the (12, 4) and (14, 4) near-code words [13]. The high error floors of the Margulis code can be reproduced with a 5 bit approximation to a belief propagation algorithm [8]. Near-code words account for 98% of the error floor performance of the Margulis code. Near code words are trapping sets [8]. Trapping sets are often clustered [8]; if one trapping set is found it will often contain nodes which belong to another trapping set. This makes the search for trapping sets somewhat simpler. Finding both stopping and trapping sets are NP-hard problems [16], [44], which makes solutions to these sets difficult to analyse. The ER decoding algorithm is simple and the effect of stopping sets can be demonstrated easilly. However, decoding over the BSC or AWGNC is much more complex (see Fig. 2). Iterative decoding methods tend to have maximum iteration counts as termination conditions and, as such, demonstrating the effect of trapping sets is difficult to show. In lieu of an example, we remind the reader of the termination conditions for the Gallager A algorithm. This decoder terminates either when all variable nodes send the same values over two consecutive iterations or when some maximum iteration count is reached. In the latter case, the decoder has failed due to the existence of a trapping set. 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4 Eb/No (dB) Fig. 7. BER comparison between the p = 7 Margulis code (using the same example as presented in Fig. 6) and a random Gallager code, both with n = 672 and decoded using MPA over the AWGNC. These graphs are also known in the literature as waterfall curves [8]. While we have only discussed the issues with the Margulis code using specific decoding algorithms here, there are many code constructions which contain trapping and stopping sets and decoding algorithms which terminate for the same reasons. For further reading, see [45]. Further reading on stopping sets in LDPC codes include the message passing (MP) algorithm [46] and the maximumlikelihood (ML) decoder [47]; both over the BEC. Further reading on trapping sets include finite alphabet iterative decoders (FAIDs) [48] and constructions based on Latin squares [49]. This construction offers high structure by which stopping sets and trapping sets can be analysed. If constructions which avoid trapping and stopping sets exist, then the error floors of the associated LDPC codes will lower significantly. This would improve the speed at which almost all digital communication occurs given the already high performance of LDPC codes in modern applications including WiFi [4] and DVB-S2 [5]. VII. C URRENT S OLUTIONS The simple goal is to avoid or completely remove every stopping set or trapping set from an LDPC code. This is both not reasonable given the number of cycles in an LDPC construction [7] and, more importantly, not necessary [38], [41]. Only small, elementary trapping sets impact the error floor due to clustering [8], [41]. If there are enough errors in transmission for a decoder to get trapped in a large trapping set then it is highly likely that it would also be trapped in a small trapping set [8]. If there are not enough errors for the decoder to become trapped in a large trapping set, the received vector can either be successfully decoded or the decoder will fail due to the presence of at least one small trapping set. The current solutions to trapping sets are the development of constructions which avoid small trapping sets and the removal of trapping sets from existing constructions. MAY 2017 9 vi Depth-0 ... ... ... Depth-1 ... ... .. . .. . ... ... .. . ... Depth-l ... ... ... Fig. 8. [51] A subgraph (tree) contained in the depth l neighbourhood spreading from the variable node vi . Note that ◦ here represents a variable node and represents a check node. A. Avoiding Trapping Sets Stopping sets pose threats to the error correction of messages sent over the BEC, however, in practice the AWGNC is used and so, when discussing proposed solutions, we focus on the influence of trapping sets over the AWGNC. The progressive-edge-growth (PEG) construction [50], [51] is a method of constructing Tanner graphs with high girth. Many trapping sets include small cycles, so the likelihood of a small trapping set being constructed is small with a graph of high girth [52]. In order to give the definition for the PEG construction, some definitions are needed. The PEG construction method uses variable and check node degree sequences [51]. The variable node degree sequence is denoted Dv = {dv0 , dv1 , . . . , dvn−1 }, where dvi is the degree of variable node vi , 0 ≤ i ≤ n − 1, and dv0 ≤ dv1 ≤ · · · ≤ dvn−1 . The parity check sequence is denoted Dc = {dc0 , dc1 , . . . , dcm−1 }, where dc j is the degree of check node c j , 0 ≤ j ≤ m − 1, and dc0 ≤ dc1 ≤ · · · ≤ dcm−1 . The construction partitions the set of edges, E, to E = Ev0 ∪ Ev1 ∪ · · · ∪ Evn−1 , where Evi contains all edges incident on symbol node vi . The k th edge incident on vi is denoted as Evki , where 0 ≤ k ≤ dvi − 1. The neighbourhood of depth l for variable node vi is Nvli and is defined as the set of all check nodes included in a subgraph (tree) spreading from variable node vi within depth l. This is demonstrated in Fig. 8. The complement of Nvli is Nv−li = C\Nvli , where C is the set of check nodes. The subgraph (tree) generated this way is constructed breadth-first with vi as the root. Given the parameters n, m and Dv , we define the PEG construction as follows. Progressive edge-growth algorithm (PEG) [51]: for i = 0 to n − 1 do begin for k = 0 to dvi − 1 do begin if k = 0 Ev0i ← edge (c j , vi ), where ev0i is the first edge incident to vi and c j is a check node that it has the lowest check-node degree in Ev0 ∪ Ev1 ∪ · · · ∪ Evi−1 . else Expand a subgraph from symbol node vi up to depth l in Ev0 ∪ Ev1 ∪ · · · ∪ Evi−1 until the cardinality of Nvli stops increasing but is less than m, or Nv−li , but Nvl+1 = , then Evki ← i k edge (c j , vi ), where Evi is the k th edge incident to vi and c j is a check node chosen from the set Nv−li having the lowest check-node degree. end end end When presented with check nodes of the same degree, a decision must be made; the selection of a check node at random or the selection of a check node according to some order. An improved construction, the randomized-PEG construction [53], chooses at random, though the deterministic nature of the ordered check node process might be of use [51]. An example of PEG construction is given in Fig. 9, setting n = 10, m = 5 and Dv = {2, 2, 2, 2, 2, 2, 2, 2, 2, 2}. The PEG construction maximises the local girth of a variable node when a new edge is added to the node [51]. After the discovery of stopping and trapping sets the PEG construction was modified [37], [53]. The PEG construction is notable for its ability to create high girth LDPC codes, however, the number of cycles is not controlled. Trapping sets are formed by a combination of several cycles [37]. The PEG algorithm, while having a higher girth than aternate constructions, contains more trapping sets; thus leaving the error floor open to improvement. The RandPEG construction improves upon the PEG algorithm by minimizing cycles at the same time as reducing the computational complexity of the PEG algorithm [53]. This can be further improved by adding an objective function to avoid small trapping sets [37]. Quasi-cyclic LDPC (QCLDPC) codes, which are used in many applications [1], [54] can be constructed using the Improved RandPEG algorithm [37]. The objective function used in the improved RandPEG algorithm detects all (5,3) and (6,4) trapping sets, removing all (5,3) trapping sets and as many (6,4) trapping sets as possible without adversely affecting the performance of the LDPC code. The characterization of trapping sets is achieved in [37] through the locations of check nodes in different levels of depth-l trees (see Fig. 8). The resulting construction is as follows: Improved RandPEG algorithm (RPEG) [37]: for i = 0 to n − 1 do MAY 2017 10 v0 v0 v0 c0 v1 c0 v1 v2 v2 v2 v4 → v5 → v6 c3 v7 v8 v8 c4 v9 (b) c2 v5 v6 c4 v9 → c3 v7 v8 (a) v5 v6 c4 v9 v4 c2 c3 v7 v8 v3 v4 c2 c3 v7 c1 v3 v4 v6 v2 c1 v3 v5 v1 c1 v3 c0 v1 c1 c2 v0 c0 c4 v9 (c) (d) Fig. 9. A PEG construction with n = 10, m = 5 and D v = {2, 2, 2, 2, 2, 2, 2, 2, 2, 2}. Check nodes are chosen based on index order. The first edge chosen for each variable node is chosen from the check nodes with lowest degree at random. The generation of the subgraphs and subsequent edge placement are the factors which highlight this construction method. In (a), the edge choices are simplistic as the breadth-first subgraphs are of low depth. This decision making continues in (b) until s5 is considered, where the edge choice is restricted to c2 or c3 due to connections in the subgraph to check nodes c1 and c4 . These choices can be observed in both remaining figures (c) and (d). One notable choice is the 2 n d edge decision for v9 in (d), where the only remaining option once c2 is chosen becomes c4 , which gives a uniform check node degree sequence. begin for k = 0 to dvi − 1 do begin if k = 0 Ev0i ← edge (c j , vi ), where ev0i is the first edge incident to vi and c j is a check node such that it has the lowest check-node degree in Ev0 ∪ Ev1 ∪ · · · ∪ Evi−1 . else Expand a subgraph from symbol node vi up to depth l in Ev0 ∪ Ev1 ∪ · · · ∪ Evi−1 until the cardinality of Nvli stops increasing but is less than m, or Nv−li , but Nvl+1 = . Remove from i Nv−li all check nodes that appear at least once in the depth-3 tree spreading from vi . This removes all check nodes that would create cycles of size < 8. for cm in Nv−li do Compute the number of (5,3) and (6,4) trapping sets that would be created if cm is selected. Remove all check nodes that would create (5,3) trapping sets and remove check nodes which create more than the smallest number of (6,4) trapping sets. If Nv−li , Evki ← edge (cm, vi ), where Evki is the k th edge incident to vi and cm is a check node chosen from the remaining nodes in Nv−li . else Declare a design failure. end end end end end The Improved RandPEG construction algorithm, while having a high computational complexity, performs the task of avoiding trapping sets optimally for given dimensions of an LDPC code. Possible improvements to this construction method include lowering the computational complexity and potentially lowering the girth. The removal of all cycles is unnecessary as not all cycles contribute to trapping sets [38], [41]. The inclusion of a lower girth into a construction which also contains no small trapping sets could lead to an LDPC code with a higher decoding performance. B. Removing Stopping and Trapping Sets The performance of LDPC codes is constrained by the presence of cycles and trapping sets within the code’s paritycheck matrix. We discuss two methods of removing trapping sets; the addition of a redundant parity-check equation [55] and the use of Tanner graph covers [10]. Redundant Parity-Check Equations Adding a redundant parity-check equation is equivalent to adding a redundant row to the parity-check matrix. This has been used in an attempt to remove the trapping sets present in the [2640, 1320] Margulis code [55]. The (12, 4) and (14, 4) trapping sets in the [2640, 1320] Margulis code are elementary point trapping sets [13]. Point trapping sets are subsets of variable nodes that contain all errors ever to occur throughout the decoding process [55]. A redundant parity check row is identified which, when added to the parity-check matrix, potentially disrupts the (12,4) and (14,4) elementary trapping sets. This parity-check row is identified through a genie-aided random search which relies on information about trapped variables not available during MAY 2017 11 decoding [55]. As random searches cannot be used in applied error correction a structured search was considered to be more useful. The structured search identifies variable and check nodes which connect to both the (12,4) and (14,4) trapping sets and combines the projection of the involved nodes such that a redundant parity-check row can be added to eliminate the effect of those trapping sets. The only way to disrupt the (12,4) and (14,4) trapping sets in the Margulis code is if the projection of both the 12 variables and the 14 variables on the redundant parity-check equation has row weight one [53]. This can be most reliably achieved by extending (12,4) trapping sets to (14,4) trapping sets (see Fig. 10). Given that the Margulis code has a (3,6)-regular paritycheck matrix, an elementary (a,b) trapping set contains a fixed number of check nodes. Let e denote the number of check nodes connected to two variable nodes within the trapping set, then e = (3a − b)/2 and therefore two variable nodes and three check nodes must be added to extend a (12,4) trapping set to a (14,4) trapping set [53]. At most one check node can be connected to both of the added variable nodes such that 4-cycles are not created. Such an extension which avoids the creation of 4-cycles in the Margulis code is only possible in two configurations. (a) The two degree-one check nodes of the basic (12,4) trapping set are connected through two additional variable nodes to one additional check node. (b) In the second configuration, the additional variable nodes do not share a check node. These configurations are demonstrated in Fig. 10. The existing check and variable nodes neighbouring the additional check and variable nodes are linearly combined to generate a redundant parity-check equation. A structured search is then used to ensure that this projection has row weight one in both the (12,4) and (14,4) trapping set. The addition of a redundant parity-check equation focuses on point elementary trapping sets from the Margulis code, where the structure of the trapping sets are well known. If this method were applied to other LDPC codes, the location and structure of the trapping sets within such a code are unknown. The redundant rows are computationally inexpensive to compute, however, the code rate of the resulting LDPC code will be reduced and the extra row increases the number of operations per decoding iteration, though by a negligible amount. Another potential problem is the success rate of this solution. The addition of a redundant parity-check equation does not guarantee that the trapping set will be disrupted [55]. Tanner Graph Covers Another method capable of eliminating trapping sets is the utilization of graph covers [10]. This method constructs an LDPC code C (2) of length 2n given a code C of length n. The parity check matrix of this code is denoted H (2) and is initialized to H (2) = H 0 0 H cBO,1 cBO,2 vBO,1 vBO,2 vBO,3 vBO,4 cBO,3 cBO,4 (12, 4) TS cBO,1 vBO,1 vBO,2 vBO,3 vBO,4 cBO,2 cBO,3 cBO,4 (12, 4) TS vE,1 vE,2 cE cEO,1 cEO,2 Expansion to (14,4) TS (a) vE,1 vE,2 cE cEO,1 cEO,2 Expansion to (14,4) TS (b) Fig. 10. [55] The trapping set structure of a (12, 4) trapping set and its (14, 4) expansion configurations; (a) left and (b) right. For (a), the two expansion variables are denoted vE,1 and vE,2 and the check node connected to both of the variables nodes is denoted c E . The unsatisfied check nodes in the original (12, 4) trapping set are denoted c BO,1 through c BO,4 and the variable nodes connected to these check nodes are denoted vBO,1 through vBO,4 . The check nodes of degree one in the expansion of the trapping set are denoted c E O,1 and c E O,2 . The node labels for (b) follow similarly. (2) (2) The operation of changing the value of Ht,k and Hm+t,n+k (2) (2) to “0", and Hm+t,k and Hm,n+k to “1" is termed as edge swapping e. The graph covers method requires that the locations of dominant trapping sets are known. The method of edge swapping is then described as follows. Graph covers algorithm [10]: 1) Take two copies, C1 and C2 , of the code C. Since the codes are identical they share the same trapping sets. Initialize SwappedE dges = , FrozenE dges = ; 2) Order the trapping sets by their critical numbers. 3) Choose a trapping set T1 in the Tanner graph of C1 , with minimal critical number. Let ET1 denote the set of all edges in T1 . If ET1 ∩ SwappedE dges , go to step 5, else go to step 4. 4) Swap an arbitrarily chosen edge e ∈ ET1 \ FrozenE dges. Set SwappedE dges = SwappedE dges∪ e. 5) “Freeze" the edges ET1 from T1 so that they cannot be swapped in the following steps. Set FrozenE dges = FrozenE dges ∪ ET1 . 6) Repeat steps 2 to 4 until all trapping sets of the desired size are removed. Possible improvements to the graph cover method are to prioritize specific edges for swapping and freezing to avoid creating trapping sets of the same critical number [10]. However, experimentally, all trapping sets with minimal critical number were removed using the above algorithm [10]. The graph covers method gave improved FER results for a Tanner code [56], a Margulis code [57] and a MacKay code [58] using the Gallager B decoding algorithm. While the decoding method was constant throughout these results, the application of graph covers will optimize FER performance using an MAY 2017 12 arbitrary decoding algorithm [10]. The LDPC code C (2) created from code C has code rate (2) r ≤ r and minimum distance 2d ≥ d (2) ≥ d. An increase to the minimum distance of the code C (2) gives higher error correcting capabilities than C. However, a lower code rate could decrease the overall efficiency. The lower row and column weight of H (2) gives C (2) higher FER performance than C, though with a trade-off of low decoding complexity. The trade-offs associated with removing trapping sets are more severe than the surveyed construction methods which avoid them (code length, decoding speed from check nodes, etc). The current research goal remains the creation of a construction or modifiable construction which can either avoid or remove small elementary trapping sets without penalty to the code’s error correcting ability or decoding efficiency. VIII. C ONCLUSION Throughout this survey we have covered the literature surrounding LDPC codes, communication channels and decoding techniques. The negative impact cycles have on LDPC code efficiency is noted and the problem of stopping sets and trapping sets have been defined and discussed including the dominance of small elementary trapping sets over AWGNC. A small variety of partial solutions such as the randomized progressive edge-growth algorithm and Tanner graph covers are discussed. The research goal remains to find constructions of LDPC codes without small trapping sets. ACKNOWLEDGMENT The authors would like to thank Professor Ian Turner, who worked closely with us throughout our research, Emeritus Professor Ed Dawson and Dr Harry Bartlett for their help in the final stages before submission, Dr Dhammika Jayalath for his help with the decoding simulations over AWGNC, and Xuan He for his suggestions on how to present our BER data. Computational resources and services used in this work were provided by the HPC and Research Support Group, Queensland University of Technology, Brisbane, Australia. A. 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Whiting, “Asymptotic spectra of trapping sets in regular and irregular LDPC code ensembles," IEEE Trans. on Inf. Theory, vol. 51, no. 1, pp. 39–55, 2007. [53] A. Venkiah, D. Declercq, and C. Poulliat. “Design of cages with a randomized progressive edge-growth algorithm," IEEE Commun. Lett., vol. 12, no. 4, pp. 301–303, 2008. [54] Y. Wang, J. Yedidia, and S. Draper, “Construction of high-girth QCLDPC codes,” IEEE 5th Intern. Symp. Turbo Codes and Related Topics, pp. 180–185, 2008. [55] S. Laendner, T. Hehn, O. Milenkovic, and J. B. Huber, “When does one redundant parity-check equation matter?” IEEE Globecom 2006, pp. 1–6, 2006. [56] R. Tanner, D. Sridhara, and T. Fuja, “A class of group-structured LDPC codes,” Proc. ISCTA, pp. 365–370, 2001. [57] J. Rosenthal, and P. Vontobel, “Constructions of LDPC codes using Ramanujan graphs and ideas from Margulis,” Proc. of the 38-th Allerton Conference on Commun., Control, and Computing, 2000. [58] D. MacKay, “Encyclopedia of sparse graph codes," 2003. 13 Aiden Price was awarded the QUT Dean’s scholarship and received a BSc and 1 s t class honours in mathematics from Queensland University of Technology. Aiden is currently studying a PhD at QUT in the school of mathematics under the supervision of Doctor Harry Bartlett and Emeritus Professor Ed Dawson and is funded by the APA scholarship. His research interests are coding theory and its application to digital communications and cryptography. Joanne Hall received a BSc and MPhil in mathematics from the Australian National University. She graduated with a PhD from RMIT University in 2011 under the supervision of Asha Rao in the Information Security and Informatics research group. Dr Hall spent one year as a postdoctoral research scientist at Charles University in Prague and four years as a Lecturer at the Queensland University of Technology in Brisbane. In 2017 she has returned to RMIT University as a Lecturer in the School of Science. Her research interests are algebraic and combinatorial structures and their applications in digital communication.	7
arXiv:1705.09058v1 [cs.AI] 25 May 2017 An Empirical Analysis of Approximation Algorithms for the Euclidean Traveling Salesman Problem Yihui He Xi’an Jiaotong University Xi’an, China Ming Xiang Xi’an Jiaotong University Xi’an China [email protected] [email protected] Abstract space, and the cost function is the Euclidean distance. That is, the Euclidean distance between two cities x = (x1 , x2 , ..., xd ), y = (y1 , y2 , ..., yd ) is: With applications to many disciplines, the traveling salesman problem (TSP) is a classical computer science optimization problem with applications to industrial engineering, theoretical computer science, bioinformatics, and several other disciplines [2]. In recent years, there have been a plethora of novel approaches for approximate solutions ranging from simplistic greedy to cooperative distributed algorithms derived from artificial intelligence. In this paper, we perform an evaluation and analysis of cornerstone algorithms for the Euclidean TSP. We evaluate greedy, 2opt, and genetic algorithms. We use several datasets as input for the algorithms including a small dataset, a mediumsized dataset representing cities in the United States, and a synthetic dataset consisting of 200 cities to test algorithm scalability. We discover that the greedy and 2-opt algorithms efficiently calculate solutions for smaller datasets. Genetic algorithm has the best performance for optimality for medium to large datasets, but generally have longer runtime. Our implementations is public available 1 . d X ( (xi − yi )2 )1/2 (1) i=0 This simplification allows us to survey several cornerstone algorithms without introducing complex scenarios. The remainder of this paper is organized as follows. In Section 2, we briefly review the first solutions and survey variants to the TSP. We describe the algorithms used in our experiment in Section 3. A description of the benchmark datasets and results of the experiment are detailed in Section 4, and explains the findings and compares the performance of the algorithms. We then conclude and describe future work in Section 5. 2. Background An example TSP is illustrated in Figure 1. The input is a collection of cities in the two dimensional space. This input can be represented as a distance matrix for each pair of cities or as a list of points denoting the coordinate of each city. In the latter method, distances are calculated using Euclidean geometry. A non-optimal tour is shown in subfigure (b). Although not shown in the figure, each edge will have some non-negative edge weight denoting the distance between two nodes or cities. Due to the computational complexity of the TSP, it may be necessary to approximate the optimal solution. The optimal tour is shown in sub-figure (c). For small graphs, it may be possible to perform an exhaustive search to obtain the optimal solution. However, as the number of cities increases, so does the solutions space, problem complexity, and running time. If n is number of cities. The number of possible Pthe n−1 edges is i=0 i. The number of possible tours is (n−1)!/2. since the same tour, with start point X and Y appears twice: once with X as the start node and once with Y as the start node. 1. Introduction Known to be NP-hard, the traveling salesman problem (TSP) was first formulated in 1930 and is one of the most studied optimization problems to date [8]. The problem is as follows: given a list of cities and a distance between each pair of cities, find the shortest possible path that visits every city exactly once and returns to the starting city. The TSP has broad applications including: shortest-path for lasers to sculpt microprocessors and delivery logistics for mail services, to name a few. The TSP is an area of active research. In fact, several variants have been derived from the original TSP. In this paper, we focus on the Euclidean TSP. In the Euclidean TSP, the vertices correspond to points in a d-dimensional 1 https://github.com/yihui-he/TSP 1 3.3. 2opt Algorithm In optimization, 2-opt is a simple local search algorithm first proposed by Croes in 1958 for solving the TSP [5]. The main idea behind it is to take a route that crosses over itself and reorder it so that it does not. A complete 2-opt local search will compare every possible valid combination of the swapping mechanism. This technique can be applied to the travelling salesman problem as well as many related problems. These include the vehicle routing problem (VRP) as well as the capacitated VRP, which require minor modification of the algorithm. This is the mechanism by which the 2-opt swap manipulates a given route: Figure 1. The TSP was first formulated in the 1930s by Karl Menger in Vienna and Harvard. By the mid-1950s, solutions for TSP began to appear. The first solution was published by Dantzig, Fulkerson, and Johnson using a dataset of 49 cities. In 1972, Richard M. Karp proved that the Hamiltonian cycle problem was NP-Complete, which proves that the TSP is NP-Hard. In modern day, the TSP has a variety of applications to numerous fields. Examples among these applications include genome sequencing, air traffic control, supplying manufacturing lines, and optimization. 1. take route[1] to route[i-1] and add them in order to new route 2. take route[i] to route[k] and add them in reverse order to new route 3. take route[k+1] to end and add them in order to new route 3. Algorithms 4. return new route We now move to a discussion of the algorithms used in our evaluation. First, we describe an upper bound for TSP in Section 3.1. The traditional greedy and 2-opt approaches are discussed in Section 3.2 and Section 3.3. We finally discuss the genetic algorithm in Section 3.4. 3.4. Genetic Algorithm Genetic algorithms (GA) are search heuristics that attempt to mimic natural selection for many problems in optimization and artificial intelligence [6]. In a genetic algorithm, a population of candidate solutions is evolved over time towards better solutions. These evolutions generally occur through mutations, randomization, and recombination. We define a fitness function to differentiate between better and worse solutions. Solutions, or individuals, with higher fitness scores are more likely to survive over time. The final solution is found if the population converges to a solution within some threshold. However, great care must be taken to avoid being trapped at local optima. We will now apply a genetic algorithm to the TSP [3]. We define a fitness function F as the length of the tour. Supposed we have an ordering of the cities A = x1 , x2 , ..., xn where n is the number of cities. The fitness score for the TSP becomes the cost of the tour d(x, y) denote the distance from x to y. 3.1. Random Path Finding the worst case of TSP is as hard as the best one. So we uniformly generate a random path for all available edges, and use this as a upper bound of optimal path benchmark for all other algorithms. 3.2. Greedy Algorithm The greedy heuristic is based on Kruskals algorithm to give an approximate solution to the TSP [11]. The algorithm forms a tour of the shortest route and can be constructed if and only if: The edges of the tour must not form a cycle unless the selected number of edges is equal to the number of vertices in the graph. The selected edge (before being appended to the tour) does not increase the degree of any node to be more than 2. The algorithm begins by sorting all edges from least weight to most heavily weighted. After the edges are sorted, the least heavily-weighted edge is selected and it is added to the tour if it does not violate the above conditions. The algorithm continues by selecting the next least-cost edge and adding it to the tour. This process is repeated until all vertices can be reached by the tour. The result is a minimum spanning tree and is a solution for the TSP. The runtime for the greedy algorithm is O(n2 log(n)) and generally returns a solution within 15-20% of the HeldKarp lower bound [15]. F (A) = n−1 X d(xi , xi+1 ) + d(xn , x0 ) (2) i=0 The genetic algorithm begins with an initial, P0 , random population of candidate solutions. That is, we have a set of paths that may or may not be good solutions. We then move forward one time step. During this time step, we perform a set of probabilistic and statistical methods to select, mutate, and produce an offspring population, P1 , with traits similar to those of the best individuals (with the highest fitness) 2 runtime comparison 10 2 10 1 time/sec 10 0 greedy 2opt sa 10 -1 10 -2 10 -3 0 p15 Figure 2. the ATT48 dataset in the 2D plane. (the United States) att48 rand200 cities Figure 3. runtime comparison, the y axis is in log scale from P0 . We then repeat this process until our population becomes homogeneous. The running time of genetic algorithms is variable and dependent on the problem and heuristics used. However, for each individual in the population, we require O(n) space for storage of the path. For genetic crossover, the space requirement remains O(n). The best genetic algorithms can find solutions within 2% of the optimal tour for certain graphs [9]. tour length comparison tour length 10 0 greedy 2opt sa optimal random 10 -1 rand200 10 -2 att48 We benchmark our algorithms using publicly available datasets. Additionally, to test the scalability of the algorithms, we generated a synthetic dataset consisting of 200 cities. In all dataset names, the numeric digits represent the number of cities in the dataset. The datasets are as follows: P15, ATT48, and R200. All datasets except R200 can be found online [4, 14]. The ATT48 and SGB128 datasets represent real-data consisting of locations of cities in the United States. A visual representation of the ATT48 dataset in the 2D plane is shown in Figure 2 Not all datasets have a known optimal tour. When this is the case, we use random path algorithm to infer a upper bound of the optimal tour. p15 4. Experiment dataset Figure 4. tour length comparison, the y axis is in log scale, divided by random tour length a solution similar with the optimal for small datasets and become worse for larger datasets. In terms of running time (Figure 3), the best algorithm is greedy algorithm. However, in terms of optimal tour length of solution, the best algorithm is GA. This is in line with our expectations and alludes to the fact that different heuristics are better suited for different situations. As shown in Figure 4, genetic algorithm performs fairly consistently, in comparison to the 2-opt and greedy algorithms, across all datasets. Highlighted in Figure 3, the running time of genetic is almost linear. This suggests that for larger datasets, if running time is a concern, then the genetic algorithm should be used. Figure 4 further demonstrates that genetic algorithm maintains a smaller percent above optimal than the other algorithms. From this, we can see that genetic algorithm has high accuracy and better complexity than other heuristics, especially for larger datasets. Surprisingly, genetic algorithm got the optimal solution for 4.1. Random Dataset The R200 dataset was generated by plotting 200 random, uniformly distributed points (x, y), in R2 with (x, y) ∈ [0, 4000]. As a result, all distances satisfy the triangle inequality and this dataset can be classified as a Euclidean TSP dataset. The running time for creating the dataset is O(n). The output is a list of all cities represented as (x, y) points. 4.2. Comparison As we can see in Figure 3, the greedy is the most efficient. In Figure 4, we can see that most algorithms return 3 [3] K. Bryant and A. Benjamin. Genetic algorithms and the traveling salesman problem. Department of Mathematics, Harvey Mudd College, pages 10–12, 2000. [4] J. Burkardt. Data for the traveling salesperson problem, 2011. http://people.sc.fsu.edu/ jburkardt/datasets/tsp/tsp.html. [5] G. A. Croes. A method for solving traveling-salesman problems. Operations research, 6(6):791–812, 1958. [6] J. Grefenstette, R. Gopal, B. Rosmaita, and D. Van Gucht. Genetic algorithms for the traveling salesman problem. In Proceedings of the first International Conference on Genetic Algorithms and their Applications, pages 160–168. Lawrence Erlbaum, New Jersey (160-168), 1985. [7] A. Haque, J. Shah, F. Ejaz, and J. X. Xu. An empirical evaluation of approximation algorithms for the metric traveling salesman problem. [8] A. Hoffman, J. Wolfe, R. Garfinkel, D. Johnson, C. Papadimitriou, P. Gilmore, E. Lawler, D. Shmoys, R. Karp, J. Steele, et al. The traveling salesman problem: a guided tour of combinatorial optimization. J. Wiley & Sons, 1986. [9] A. Homaifar, S. Guan, and G. E. Liepins. Schema analysis of the traveling salesman problem using genetic algorithms. Complex Systems, 6(6):533–552, 1992. [10] I. Hong, A. B. Kahng, and B.-R. Moon. Improved largestep markov chain variants for the symmetric tsp. Journal of Heuristics, 3(1):63–81, 1997. [11] B.-I. Kim, J.-I. Shim, and M. Zhang. Comparison of tsp algorithms. Project for Models in Facilities Planning and Materials Handling, 1998. [12] M. Mucha. \ frac {13}{9}-approximation for graphic tsp. Theory of computing systems, 55(4):640–657, 2014. [13] F. Qiu, J. Zhang, and H. Yan. An adaptive markov chain monte carlo algorithm for tsp. In Computer Science and Software Engineering, 2008 International Conference on, volume 1, pages 439–442. IEEE, 2008. [14] G. Reinelt. Tsplib, 1995. http://comopt.ifi.uniheidelberg.de/software/TSPLIB95/. [15] D. J. Rosenkrantz, R. E. Stearns, and P. M. Lewis, II. An analysis of several heuristics for the traveling salesman problem. SIAM journal on computing, 6(3):563–581, 1977. Figure 5. Solution generated by genetic algorithm for att48 dataset att48 dataset, shown in Figure 5. 5. Conclusion Most of our algorithms attempt to solve the TSP in a linear fashion. Originating from artificial intelligence, the genetic algorithm is very different compared to greedy, and 2-opt. Literature suggests that the best algorithms focus on iteration and convergence to find optimal tours – something genetic algorithms attempt to achieve. For example, the Large Step Markov Chain [10] relies on Markov chains to find convergence of many paths to form a global optimum and several papers cite Markov Chains as the best known solution to TSP. Recent studies include using adaptive Markov Chain Monte Carlo algorithms [13]. Many of these extend the Metropolis algorithm [9], a simulated annealing algorithm which attempts to mimic randomness with particles as the temperature varies. This further supports our conclusion that algorithms inspired from artificial intelligence perform well for finding solutions for the TSP. However, these may not be suitable when a guarantee is required. In this paper, we surveyed several key cornerstone approaches to the TSP. We selected four well-known algorithms and tested their performance on a variety of public datasets. Our results suggest that genetic algorithms (and other approaches from artificial intelligence) are able to find a near-optimal solution. References [1] S. Arora. Polynomial time approximation schemes for euclidean tsp and other geometric problems. In Foundations of Computer Science, 1996. Proceedings., 37th Annual Symposium on, pages 2–11. IEEE, 1996. [2] BaoJunpeng. Introduction to Artificial Intelligence. [M].BeiJing:MachineryIn-dustryPress, 2010. 4	2
arXiv:1705.08738v1 [cs.CE] 24 May 2017 Doppler Synthetic Aperture Radar Interferometry: A Novel SAR Interferometry for Height Mapping using Ultra-Narrowband Waveforms Birsen Yazıcı1,∗ , Il-Young Son1 and H. Cagri Yanik 1 Department of Electrical and Computer Systems Engineering, Rensselaer Polytechnic Institute, Troy, NY, USA ∗ Corresponding author E-mail: [email protected] Abstract. This paper introduces a new and novel radar interferometry based on Doppler synthetic aperture radar (Doppler-SAR) paradigm. Conventional SAR interferometry relies on wideband transmitted waveforms to obtain high range resolution. Topography of a surface is directly related to the range difference between two antennas configured at different positions. Doppler-SAR is a novel imaging modality that uses ultra-narrowband continuous waves (UNCW). It takes advantage of high resolution Doppler information provided by UNCWs to form high resolution SAR images. We introduced the theory of Doppler-SAR interferometry. We derived interferometric phase model and develop the equations of height mapping. Unlike conventional SAR interferometry, we show that the topography of a scene is related to the difference in Doppler between two antennas configured at different velocities. While the conventional SAR interferometry uses range, Doppler and Doppler due to interferometric phase in height mapping, Doppler-SAR interferometry uses Doppler, Doppler-rate and Doppler-rate due to interferometric phase in height mapping. We demonstrate our theory in numerical simulations. Doppler-SAR interferometry offers the advantages of long-range, robust, environmentally friendly operations; low-power, low-cost, lightweight systems suitable for low-payload platforms, such as micro-satellites; and passive applications using sources of opportunity transmitting UNCW. Submitted to: Inverse Problems Doppler Synthetic Aperture Radar Interferometry 2 1. Introduction Synthetic Aperture Radar (SAR) interferometry is a powerful tool in mapping surface topography and monitoring dynamic processes. This tool is now an integral part of wide range of applications in many disciplines including environmental remote sensing, geosciences and climate research, earthquake and volcanic research, mapping of Earth’s topography, ocean surface current monitoring, hazard and disaster monitoring, as well as defense and security related research [1]. Basic principles of SAR interferometry were originally developed in radio astronomy [2, 3]. Interferometric processing techniques and systems were later developed and applied to Earth observation [4, 5, 6, 7, 8]. SAR interferometry exploits phase differences of two or more SAR images to extract more information about a medium than present in a single SAR image [9] [10]. Conventional SAR interferometry relies on wideband transmitted waveforms to obtain high range resolution [10, 1, 11, 12, 13]. The phase difference of two wideband SAR images are related to range difference. There are many different interferometric methods depending on the configuration of imaging parameters in space, time, frequency etc [1]. When two images are acquired from different look-directions, the phase difference is related to the topography of a surface. In this paper, we develop the basic principles of a new and novel interferometric method based on Doppler-SAR paradigm to determine topography of a surface. Unlike conventional SAR, Doppler-SAR uses ultra-narrowband continuous waves (CW) to form high resolution images [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]. Conventional SAR takes advantage of high range resolution and range-rate due to the movement of SAR antenna for high resolution imaging. Doppler-SAR, on the other hand, takes advantage of high temporal Doppler resolution provided by UNCWs and Doppler-rate for high resolution imaging. We develop the phase relationship between two Doppler-SAR images and show that the phase difference is related to Doppler difference. We approximate this phase difference as Doppler-rate and derive the equations of height mapping for Doppler-SAR interferometry. Conventional wideband SAR interferometry for height mapping requires two different look-directions. Doppler-SAR interferometry provides a new degree of freedom in system design by allowing antennas to have the same look-direction, but different velocities to obtain height mapping. Additional advantages of Doppler-SAR interferometry include the following: (i ) Small, lightweight, inexpensive, easy-to-design and calibrate hardware, high Signal-to-Noise-Ratio(SNR) and long effective range of operation. All of these make Doppler-SAR interferometry a suitable modality for applications requiring high SNR, long range of operation and low payload platforms such as micro-satellites or small uninhabited aerial vehicles. (ii ) Effective use of electromagnetic spectrum and environmentally friendly illumination. (iii ) Passive applications. Doppler-SAR may not require dedicated transmitters, since existing Radio Doppler Synthetic Aperture Radar Interferometry 3 Frequency (RF) signals of opportunity often have the ultra-narrowband properties. To the best of our knowledge, this is the first interferometric method that is developed in Doppler-SAR paradigm. We present the theory for two monostatic Doppler-SAR. However, the method can be easily be extended to bistatic and multistatic configurations and synthetic aperture imaging applications in acoustics. The rest of the paper is organized as follows: In Section 2, SAR geometry and notation are defined. In Section 3, wideband SAR image formation, layover effect and basic principles of wideband SAR interferometry are described in a perspective relevant to our subsequent development. In Section 4, Doppler-SAR data model, image formation and layover are summarized. Section 5 introduces the basic principles of Doppler-SAR interferometry and compares the results to wideband SAR case. Section 6 presents numerical simulations and Section 7 concludes the paper. 2. Configurations and Notation We consider two mono-static SAR systems as shown in Fig. 1. Antenna 2 Antenna 1 : Location of the scatter : Height of the scatter Figure 1: Imaging geometry for an interferometric SAR system with two antennas following trajectories γ1 (s) and γ2 (s). The scatterer is located at x ∈ R3 where its height is h(x) and x = [x1 , x2 ] ∈ R2 . Let γ1 (s) and γ2 (s), s ∈ [S1 , S2 ] ⊆ R, denote the trajectories of the first and second antennas, respectively. Unless otherwise stated, bold Roman, bold italic, and Roman lower-case letters will denote elements in R3 , R2 and R, respectively, i.e., x = [x1 , x2 ] ∈ R2 , x3 ∈ R, and x = [x, x3 ] ∈ R3 . The Earth’s surface is located at x = [x, h(x)], where h : R2 → R, is the unknown height representing ground topography. Let V : R3 → R denote target reflectivity where we assume that the scattering takes place only on the surface of the Earth. Major notation used throughout the paper is tabulated in Table 1. Doppler Synthetic Aperture Radar Interferometry Table 1: Notation Symbol Description x = [x, h(x)], x ∈ R2 Location on earth’s surface h(x) Unknown height of a scatter at x V (x) Surface reflectivity γi (s) i-th antenna trajectory s Slow-time t Fast-time Ri (x, s) Range of i-th antenna ω0 Center frequency of the transmitted waveforms si0 Zero-Doppler time for the i-th antenna Li (x, s) Look-direction of the i-th antenna B dW (t, s) i Wideband SAR demodulated received signal at i-th antenna \|z − γi (s)\| = C. Iso-range surface \ (z − γi (s)) · γ̇i (s) = C Iso-Doppler surface KiW B Filtered backprojection (FBP) operator for wideband SAR IiW B Wideband SAR image B ΦW s0 (x) Wideband interferometric phase b Baseline vector in wideband SAR interferometry L1 (z, s10 ) · b = C Interferometric phase cone l Vector from a known scatterer position to the unknown location of a scatterer l⊥ 1 Component of l perpendicular to (z0 \ − γ1 (s)) B ΦW f lat (x) Flattened wideband SAR interferometric phase φ(t) Smooth windowing function Tφ Duration of φ(t) NB dU (µ, s) i Doppler-SAR data Li (z, s) · γ̇i (s) = C Iso-Doppler surface Li (z, s) · γ̈i (s) − Iso-Doppler-rate surface γ̇i (s)·γ̇i⊥ (s) Ri (z,s) =C KiU N B FBP operator for Doppler-SAR IiU N B Doppler-SAR image NB ΦU (x) sd Doppler-SAR interferometric phase v Baseline velocity NB ΦU f lat (x) Flattened Doppler-SAR interferometric phase 4 Doppler Synthetic Aperture Radar Interferometry 5 3. Wideband SAR Interferometry The basic principles of SAR interferometry are described by many sources [10], [25], [9], [26] [27], [1] and [28]. In this section, we summarize the principles and theory of SAR interferometry in a notation and context relevant to our subsequent presentation of Doppler-SAR interferometry. We begin with the wideband SAR received signal model, derive the interferometric phase model, provide a geometric interpretation of the interferometric phase from which we develop the equations of height mapping. 3.1. Wideband SAR received signal model We assume that the SAR antennas are transmitting wideband waveforms. Let ri (t, s) denote the received signals, i = 1, 2 where s and t are the slow-time and fast-time variables, respectively. Under the start-stop and Born approximations, the received signals can be modeled as [29, 30, 31]: Z ri (t, s) = e−iω(t−2Ri (x,s)/c) Ãi (x, s, ω)V (x)dxdω (1) where Ri (x, s) = \|x − γi (s)\| (2) is the range of the ith antenna, c is the speed of light in free-space, ω is the temporal frequency variable, V (x) is the scene reflectivity function. Ãi is a slowly-varying function of ω that depends on antenna beam patterns, geometrical spreading factors and transmitted waveforms. Let ω = ω0 + ω 0 , ω 0 ∈ Ω where Ω is the bandwith and ω0 is the center frequency of the transmitted waveforms. We demodulate the received signals and write B dW (t, s) = eiω0 t ri (t, s), i ZΩ ω0 0 = e−iω (t−2Ri (x,s)/c) Ãi (x, s, ω 0 )ei2 c Ri (x,s) V (x)dxdω 0 . (3) (4) −Ω Next, we approximate Ri (x, s) in ei ω0 Ri (x,s) c around s = si0 as follows: Ri (x, s) ≈ Ri (x, si0 ) + (s − si0 ) ∂s Ri (x, s)\|s=si 0 (s − si0 )2 2 + ∂s Ri (x, s) s=si , i = 1, 2 0 2 (5) where ∂s denotes derivative with respect to s and si0 is the zero-Doppler time for the ith antenna, i.e., \ ∂s Ri (x, s)\|s=si = (x − γi (si0 )) · γ̇i (si0 ) = 0. 0 (6) b denotes the unit vector in the direction of x and γ̇i (s) denotes the velocity of In (6) x th the i antenna. Doppler Synthetic Aperture Radar Interferometry 6 We define Li (x, s) = (x \ − γi (s)) (7) and refer to Li (x, s) as the look-direction of the ith antenna. Note that at the zeroDoppler time, the antenna look-direction is orthogonal to the antenna velocity. Let i2 Ai (x, s, ω 0 ) = Ãi (x, s, ω 0 )e i 2 ω0 (s−s0 ) c 2 ∂s2 Ri (s,x)\| s=si0 . (8) Finally, we write the demodulated received signal as follows: B dW (t, s) i ZΩ ≈ 0 e−iω (t−2Ri (x,s)/c) Ai (x, s, ω 0 )ei2 ω0 Ri (x,si0 ) c V (x)dxdω 0 . (9) −Ω 3.2. Wideband SAR image formation and layover Many different algorithms were developed to form wideband SAR images such as range-Doppler [25], seismic migration [32], backprojection [31] and chirp scaling [33] algorithms. All of these algorithms take advantage of high range resolution provided by wideband transmitted waveforms and pulse-to-pulse Doppler information provided by the movement of antennas. The location of a scatterer is identified by intersecting the iso-range and iso-Doppler surfaces and the ground topography as shown in Fig. 2. Range sphere Doppler cone Velocity vector Sensor position x Scatterer position Figure 2: The SAR image of a scatterer is reconstructed at the intersection of the iso-range (sphere) and iso-Doppler (cone) surfaces and the height of the scatterer. More precisely, the image of a scatterer is formed at z satisfying the following equations: \|z − γi (s)\| = Ri (x, s) \ Iso-Doppler surface: (z − γi (s)) · γ̇i (s) = ∂s Ri (x, s) (10) Height: (12) Iso-range surface: z3 = h(x), z = [z, z3 ]. (11) Doppler Synthetic Aperture Radar Interferometry 7 Note that Ri (x, s) and ∂s Ri (x, s) are the measured range and Doppler and h(x) is the height of the scatterer. As functions of z, (10) and (11) define the iso-range and iso-Doppler surfaces, respectively. Iso-range contours are defined as the intersection of the iso-range surface, i.e., sphere, and the ground topography. Without loss of generality, we consider a filtered backprojection (FBP) type method where the received and demodulated signals are backprojected onto iso-range contours defined on a reference surface [31], [29]. In the absence of heigh information, demodulated signal is backprojected onto the intersection of the iso-range surface and a known reference surface. Without loss of generality, we assume a flat reference surface at zero height and backproject the demodulated signals onto the following iso-range contours: HiRange (z0 ) = z0 ∈ R3 z0 = [z, 0] and \|z0 − γi (s)\| = Ri (x, s) . (13) Let KiW B be an FBP operator. Then, the reconstructed image of the scatterer at x becomes W B ˜W B i IiW B (zi0 ) := K Z i [di ](z0 ), 0 i B B = eiω (t−2Ri (z0 ,s)/c) QW (z0i , ω 0 , s)dW (t, s)dω 0 dtds, (14) i i B where QW is a filter that can be chosen with respect to a variety of criteria [31], [34]. i From (9), the image of the scatterer at x becomes IiW B (zi0 ) = \|IiW B (zi0 )\|ei2 ω0 Ri (x,si0 ) c . (15) The magnitude of reconstructed images is a measure of target reflectivity, whereas the phase of the reconstructed image depends on the true location, x = [x, h(x)] of the scatterer. However, since the true height h(x) of the scatter is unknown and hence different than that of the reference surface, the location, z0i , at which the scatterer is reconstructed is different than its true location, x. This positioning error due to incorrect height information is known as layover. Fig. 3 depicts the layover effect. We see that without the knowledge of ground topography, additional information or measurements are needed to reconstruct the scatterers at correct locations. This additional information is provided by a second antenna that has a different vantage point than the first one. 3.3. Wideband SAR interferometric height reconstruction An interferogram is formed by multiplying one of the SAR images with the complex conjugate of the other SAR image [9, 10]. Prior to multiplying the SAR images, the two intensity images, \|IiW B (zi0 )\|, i = 1, 2 are co-registered so that pixel locations z01 and z02 , each corresponding to the scatterer at position x in the scene, are roughly aligned‡. Multiplying I1W B (z10 ) with the complex conjugate of I2W B (z20 ), we get I1W B (z10 )I2W B (z20 ) = \|I1W B (z10 )\|\|I2W B (z20 )\|ei2 ω0 (R1 (x,s10 )−R2 (x,s20 )) c . (16) ‡ The positioning errors due to layover are different in the two SAR images due to different imaging geometries. Doppler Synthetic Aperture Radar Interferometry 8 Figure 3: Layover in wideband SAR - The range sphere depicts the iso-range surface of the monostatic SAR configuration. Since the correct height of the scatterer at location x is unknown, the image of the scatterer at x is formed at z0 on a flat surface. We refer to the phase of the interferogram as the wideband interferometric phase ω0 B (R1 (x, s10 ) − R2 (x, s20 )) ΦW (17) s0 (x) = 2 c B provides us the where s0 is a multi-index for {s10 , s20 }. The interferometric phase ΦW s0 third measurement needed to determine the location of a scatterer in R3 . In general the range difference can be many multiples of 2π. Unique phase proportional to range difference can be determined by a phase unwrapping process [1]. Now consider the following surface c WB \|z − γ1 (s10 )\| − \|z − γ2 (s20 )\| = Φ (x) (18) 2ω0 s0 B where ΦW (18) defines a two-sheet s0 (x) is the measured interferometric phase. 2 1 hyperboloid with foci at γ1 (s0 ) and γ2 (s0 ). We assume that the distance between the antennas is much smaller than the ranges of the antennas to the scene and approximate this hyperboloid as follows: c WB L1 (z, s10 ) · b ≈ (19) Φ (x) 2ω0 s0 where b = γ2 (s20 ) − γ1 (s10 ) (20) is the baseline vector. (19) defines a cone whose vertex is the first antenna and the axis of rotation is the baseline vector. We call this surface the interferometric phase cone. The interferometric phase cone provides the third equation needed to locate the position of a scatterer in R3 . More precisely, the location of the scatterer is given by the solution of the following equations: Range sphere: \|z − γ1 (s10 )\| = R1 (x, s10 ) \ (z − γ1 (s10 )) · γ̇1 (s10 )) = ∂s R1 (x, s)\|s=s1 0 c WB 1 Interferometric phase cone: L1 (z, s0 ) · b = Φ (x). 2ω0 s0 Doppler cone: (21) (22) (23) Doppler Synthetic Aperture Radar Interferometry 9 The right-hand-side of (21)-(23) are measured quantities defined in terms of the true location, x, of the scatterer in the scene and the left hand-side-defines the three surfaces in terms of the location of the scatterer z in the image. Fig. 4 geometrically illustrates the solution of these three equations in wideband SAR interferometry. Typically the Figure 4: Wideband SAR interferometry provides a third algebraic equation by which the unknown location of a scatters in R3 is determined. The scatterer is located at the intersection of the Doppler-cone, iso-range sphere, and the interferometric phase cone. The axis of rotation of the Doppler-cone is the velocity of the first antenna and the axis of rotation of the interferometric cone is the baseline vector extending from the first to the second antenna. variation in the color coding of interferogram is “flattened” by subtracting the expected phase from a surface of constant elevation. Let x = l + z0 . Then, under the assumption that \|l\| \|z0 − γ1 (s)\| \ (x − γ1 (s)) ≈ (z0 \ − γ1 (s)) + l⊥ 1 \|z0 − γ1 (s)\| (24) where " l⊥ 1 # \ l · (z − γ (s)) 0 1 = l − (z0 \ − γ1 (s)) . \|z0 − γ1 (s)\| (25) \ In other words, the vector l⊥ 1 is the component of l perpendicular to (z0 − γ1 (s)). The flattened phase then becomes ω0 B ΦW L1 (x, s10 ) − L1 (z0 , s10 ) · b (26) f lat (x) = 2 c ω0 l⊥ 1 ·b ≈2 . (27) c R(z0 , s10 ) ⊥ ⊥ 1 Since b⊥ 1 · l = l1 · b where b1 is the component of b perpendicular to L1 (z, s0 ), (27) can be alternatively expressed as B ΦW f lat (x) ω0 b⊥ 1 ·l =2 . c R(z0 , s10 ) (28) Doppler Synthetic Aperture Radar Interferometry 10 Fig. 5 illustrates the key concepts and vectors involved in the wideband interferometry. Figure 5: A two-dimensional illustration of vectors involved in wideband interferometric phase. l = x − z0 where z0 = [z0 , 0], γi (si0 ) denotes the location of the ith antenna at the zero-Doppler time si0 , L1 (z0 , s10 ) denotes the look-direction of the first antenna with respect to the reference scatterer located at z0 , and l⊥ 1 is the component of l 1 perpendicular to L1 (z0 , s0 ). The wideband interferometric phase is related to the projection of the baseline vector, b, onto the the look-direction, L1 (x, s10 ), of the antenna with respect to scatterer location x. Known vectors are shown in red and unknown vectors are shown in black. 4. Data Model and Image Formation for Doppler-SAR 4.1. Data Model for Doppler-SAR We consider two mono-static antennas following the trajectories γi (t), i = 1, 2, transmitting ultra-narrowband CWs as shown in Fig. 1. Let p(t) ≈ p̃(t)eiω0 be the transmitted waveform where ω0 is the center frequency. The scattered field model at the ith antenna is then given by Z −iω0 (t−2\|x−γi (t)\|/c) ω02 e ri (t) = p̃(t − 2\|x − γi (t)\|/c)V (x)dx. (29) (4π)2 \|x − γi (t)\|2 Let µ ∈ R+ and φ(t) be a smooth windowing function with a finite support, t ∈ [0, Tφ ]. Following [15, 14, 23], we correlate ri (t) with a scaled and translated version of the transmitted signal over φ(t) as follows: Z UNB di (µ, s) = ri (t)eiω0 µ(t−sTφ )/c p̃∗ (µ(t − sTφ ))φ(t − sTφ )dt. (30) Doppler Synthetic Aperture Radar Interferometry 11 Inserting (29) into (30), we obtain Z −iω0 (t−2\|x−γi (t)\|/c) ω02 e UNB (µ, s) = di p̃(t − 2\|x − γi (t)\|/c)V (x) 2 (4π) \|x − γi (t)\|2 × eiω0 µ(t−sTφ )/c p̃∗ (µ(t − sTφ ))φ(t − sTφ )dtdx. (31) Approximating γi (t) around t = sTφ , γi (t) ≈ γi (sTφ ) + γ̇i (sTφ )(t − sTφ ), and making the far-field approximation, we write \|x − γi (t)\| ≈ \|x − γi (sTφ )\| − Li (x, sTφ ) · γ̇i (sTφ )(t − sTφ ), (32) where Li (x, sTφ ) = (x −\ γi (sTφ )) and γ̇i (sTφ ) = ∂s γi (sTφ ) is the velocity of the ith antenna. To simplify our notation, for the rest of the paper, we set Li (x, sTφ ) = Li (x, s), γi (sTφ ) = γi (s), γ̇i (sTφ ) = γ̇i (s), ∂s2 γi (sTφ ) = γ̈i (sTφ ) = γ̈i (s) and Ri (x, sTφ ) = Ri (x, s). We next define Doppler for the ith antenna ω0 (33) fid (x, s) = − Li (x, s) · γ̇i (s). c Inserting (32) and (31) into (33), the data model becomes Z d d UNB di (µ, s) = e−it[ω0 (1−µ)−2fi (x,s)] Ãi (t, x, s, µ)ei2fi (x,s)sTφ V (x)dtdx, (34) where Ãi (t, x, s, µ) is a slow varying function of t composed of the rest of the terms in (31). We now approximate fid (x, s) around s = sid as follows: fid (x, s) ≈ fid (x, sid )+(s−sid ) ∂s fid (x, s) s=sid + (s − sid )2 2 d ∂s fi (x, s) 2 s=sid .(35) We choose sid such that ∂fid (x, s) ∂s =0 ⇒ Li (x, sid ) · γ̈i (sid ) − s=sid γ̇i (sid ) · γ̇i⊥ (sid ) =0 Ri (xsid ) (36) where γ̈i (sid ) is the acceleration of the ith antenna and γ̇i⊥ (sid ) is the component of γ̇i (sid ) perpendicular to the look-direction Li (x, sid ) as described in (25). We refer to sid as the zero-Doppler-rate time for the ith antenna. d Using (35) in ei2fi (x,s)sTφ and redefining the slow-varying function in t, i2sid Tφ Ai (t, x, s, µ) = Ãi (t, x, s, µ)e (s−sid )2 2 d ∂s fi (x,s)\|s=si 2 d , (37) we obtain the following data model for Doppler-SAR image reconstruction: Z d d i i UNB di (µ, s) ≈ e−it[ω0 (1−µ)−2fi (x,s)] Ai (t, x, s, µ)ei2fi (x,sd )sd Tφ V (x)dtdx.(38) Doppler Synthetic Aperture Radar Interferometry 12 4.2. Doppler-SAR Image Formation and Layover Similar to the wideband case, we reconstruct images by backprojection as described in [35, 15] [14]. The forward model in (38) shows that the data, dUi N B (s, µ), is the weighted integral of the scene reflectivity over iso-Doppler contours. It was shown in [14] that a scatterer located at x in the scene is reconstructed at the intersection of iso-Doppler surface and iso-Doppler-rate surface and ground topography. More precisely, the image of a scatterer located at x in the scene is reconstructed at z satisfying the following equations: c (39) Iso-Doppler surface: Ldi (z, s) · γ̇i (s) = fid (x, s) ω0 γ̇i (s) · γ̇i⊥ (s) c Iso-Doppler-rate surface: Li (z, s) · γ̈i (s) − (40) = ∂s fid (x, s) Ri (z, s) ω0 Height: z3 = h(x), z = [z, z3 ] (41) where the right-hand-side of (39)-(40) corresponds to measurements and the left-handside defines surfaces in image parameter z. The iso-Doppler-rate surface, given by the following set, γ̇i (s) · γ̇i⊥ (s) c Dop−rate 3 d Hi (z) = z ∈ R Li (z, s) · γ̈i (s) − = ∂s fi (x, s) . (42) Ri (z, s) ω0 can be viewed as a continuum of intersections of cones and expanding spheres centered at the sensor location. The axis of rotation for the surface is the acceleration vector of the antenna trajectory. Fig. 6 illustrates iso-Doppler and iso-Doppler-rate surfaces and the reconstruction of a point scatterer by the intersection of these surfaces and ground topography. The reconstruction is analogous to the wideband SAR image reconstruction shown in Fig. 2. In the absence of ground topography information, we backproject data onto isoDoppler contours on a reference surface. Without loss of generality, we consider the following iso-Doppler contours: c d Dop 3 Hi (z0 ) = z0 ∈ R z0 = [z, 0] and Li (z, s) · γ̇i (s) = fi (x, s) (43) ω0 where the right-hand-side of the equality in (43) is the high resolution measurement provided by ultra-narrowband CW. Let KiU N B be an FBP operator as described in [14]. Then, the reconstructed image is given by: UNB UNB [di ](zi0 ) IiU N B (zi0 ) := K Zi ≈ d i eit(ω0 (1−µ)−2fi (z0 ,s)) QUi N B (s, z0i , t)dUi N B (s, µ)dtdµds (44) where QUi N B is a filter that can be chosen as in [35, 15, 14]. The reconstructed image is given by d i i IiU N B (zi0 ) = \|IiU N B (zi0 )\|ei2fi (x,sd )sd Tφ . (45) Doppler Synthetic Aperture Radar Interferometry 13 Figure 6: In Doppler-SAR image reconstruction, a scatterer located at x in the scene is correctly reconstructed at the intersection of the iso-Doppler and iso-Doppler-rate surfaces and the ground topography. Iso-Doppler surface is a cone in which its vertex is the antenna location and its axis of rotation is the antenna velocity. The geometry of the iso-Doppler-rate surface depends on the antenna trajectory. Figure is drawn for a linear trajectory at a constant height. In the absence of topography information, we see that a scatterer located at x in the scene is reconstructed at zi0 6= x in the image. This position error in the reconstructed image is the counterpart of the layover effect observed in conventional wideband SAR images. Fig. 7 illustrates the layover effect in Doppler-SAR. However, the phase of the reconstructed image is a function of the scatterer’s true location, x, and hence, includes its height information, h(x). Figure 7: If the height of a scatter is not known, it is reconstructed at an incorrect position. Both the correct scatterer location x and its image z0 lie on the same isoDoppler surface, i.e., the Doppler cone. z0 lies at the intersection of the Doppler cone defined f1d (x, s) and the flat topography. Note that the phases of the reconstructed images depend on the Doppler-rate, Doppler Synthetic Aperture Radar Interferometry 14 fid (x, sid ), the duration of the windowing function, Tφ , and the corresponding zeroDoppler-rate times, sid . The height information is included in the Doppler-rate. However, since each imaging geometry may yield different zero-Doppler-times, Dopplerrate in the phase of each image is multiplied by a different zero-Doppler-rate time. To equalize the effect of this multiplication factor, we multiply one of the reconstructed images with itself so that the Doppler-rate in the phase of both images are multiplied by the same factor, say s1d . As a result, each image becomes d i 1 IiU N B (zi0 ) = \|IiU N B (zi0 )\|ei2fi (x,sd )sd Tφ , i = 1, 2. (46) 5. Doppler-SAR Interferometric Height Reconstruction Similar to the wideband case, we form two Doppler-SAR images, IiU N B (zi0 ), i = 1, 2, co-register the intensity images \|IiW B (zi0 )\| and multiply one of them by the complex conjugate of the other to form an interferogram. Then the interferometric phase, i.e., the phase function of I1U N B (x)I2U N B (x) is given by ΦUsdN B (x) = 2s1d Tφ f1d (x, s1d ) − f2d (x, s2d ) (47) where sd denotes multi-index for {s1d , s2d }. Thus the scatterer lies on the following surface: c L1 (z, s1d ) · γ̇1 (s1d ) − L2 (z, s2d ) · γ̇2 (s2d ) = − f1d (x, s1d ) − f2d (x, s2d ) (48) 2ω0 where the right-hand-side is the measured interferometric phase. The left-hand-side of (48) defines a surface that can be described as the intersections of two cones one of which has a continuously changing solid angle. Assuming that the distance between the antennas is much smaller than the ranges of the antennas to the scene, we can approximate the look-direction of the second antenna in terms of the look-direction of the first one as follows: b⊥ 1 L2 (x, s2d ) = L1 (x, s1d ) + (49) R1 (x, s1d ) where b = γ2 (s2d ) − γ1 (s1d ) is the baseline vector and b⊥ 1 is the component of b perpendicular to the look-direction of the first antenna. Using (49), we approximate the interferometric phase as follows: 2 c b⊥ 1 · γ̇2 (sd ) UNB 1 − 1 Φ (x) ≈ L1 (x, sd ) · v + 2sd Tφ ω0 sd R1 (x, s1d ) (50) v = γ̇2 (s2d ) − γ̇1 (s1d ). (51) where We refer to v as the baseline velocity. We see that (50) approximates the interferometric phase as a Doppler-rate. Additionally, (50) shows that Doppler-SAR interferometry involves not only configuring antennas in position space, but also in velocity space. The larger the difference in antenna velocities in the look-direction of the first antenna, the larger the interferometric phase becomes. If on the other hand, the velocities of the antennas are the same, the second term in (50) defines the interferometric phase surface. Doppler Synthetic Aperture Radar Interferometry 15 Clearly, in Doppler-SAR interferometry (50) provides the third equation needed to determine the location of a scatterer in R3 . More precisely, the location of a scatterer is given by the solution of the following three equations: c (52) (z −\ γ1 (s1d )) · γ̇1 (s1d ) = f1d (x, s1d ) ω0 γ̇1 (s1d ) · γ̇1⊥ (s1d Tφ ) 1 1 \ Iso-Doppler-rate: (z − γ1 (sd )) · γ̈1 (sd ) − = ∂s f1d (x, s1d ) (53) 1 R1 (sd , z) c b⊥ · γ̇2 (s2d ) =− 1 ΦU N B (x). (54) Interferometric Doppler-rate: L1 (z, s1d ) · v + 1 1 R1 (z, sd ) 2sd Tφ ω0 sd Iso-Doppler: Fig. (8) depicts the intersection of the three surfaces at the scatterer location in R3 . Doppler-rate surface Interferometric phase Doppler-rate surface Doppler cone Velocity vector Sensor position Scatterer position Figure 8: Determination of the scatterer location in Doppler-SAR interferometry. The scatterer is located at the intersection of the Doppler cone and the two iso-Dopplerrate surfaces. Interferometric phase measurement provides the third surface, i.e., the interferometric phase iso-Doppler-rate surface. Similar to the wideband SAR interferometry, the interferometric phase can be “flattened” by subtracting the phase due to a scatterer with known height. Without loss of generality, let z0 = [z, 0] with R1 (z0 , s) = R1 (x, s) and x = z0 + l. Thus, identifying the location of a scatterer is equivalent to determining l. Using (24), we see that l⊥ 1 1 ·v UNB UNB UNB Φf lat (x) = Φsd (x) − Φsd (z0 ) ≈ +O (55) R1 (z0 , s1d ) R12 (z0 , s1d ) 1 where l⊥ 1 is the component of l perpendicular to L1 (z0 , sd ). (55) shows that the flattened interferometric phase for Doppler-SAR interferometry is related to the projection of the unknown l⊥ 1 onto the baseline velocity vector scaled by the range of the first antenna to ⊥ z0 . Since l1 · v = v1⊥ · l where v1⊥ is the component of v perpendicular to L1 (z0 , s1d ), we alternative express (55) as follows: v1⊥ · l NB ΦUf lat (x) ≈ . (56) R1 (z0 , s1d ) Fig. 9 shows the key concepts and vectors involved in Doppler-SAR interferometry. Doppler Synthetic Aperture Radar Interferometry 16 Figure 9: An illustration of key concepts and vectors in Doppler-SAR interferometry. l = x − z0 where z0 = [z0 , 0], γi (s) denotes the ith antenna position, L1 (z0 , s1d ) denotes the look-direction with respect to a reference surface, l⊥ 1 is the component of 1 l perpendicular to L1 (z0 , sd ), γ̇1 (s) denotes the antenna velocity, L1 (x, s1d ) denotes the look-direction of the antenna with respect to the correct target location. Doppler-SAR interferometric phase is proportional to the projection of the baseline velocity vector onto l⊥ 1 . Known vectors are shown in red and unknown vectors are shown in black. 5.1. Comparison of Doppler-SAR Interferometry wide Wideband Case Table II tabulates the interferometric phase for the wideband SAR and Doppler-SAR cases. We compare and contrast the two interferometric phases below: • For WB and UNB, the “baseline” is the difference in range and difference in velocity, respectively. • The larger the ω0 , the center frequency, the larger the interferometric phase in both WB and UNB cases. • The larger the range, R1 , the smaller the interferometric phase in both WB and UNB cases. • For UNB, larger the Tφ , the larger the interferometric phase. • For WB, the larger the b, the difference between the positions of the two antennas, the larger the interferometric phase. For UNB, the larger the v, the difference between the velocities of the two antennas, the larger the interferometric phase. Doppler Synthetic Aperture Radar Interferometry 17 Table 2: Raw and flattened interferometric phase functions for wideband SAR and Doppler-SAR. Wideband SAR Doppler-SAR Interferometric Phase 2 ωc0 L1 (x, s10 ) · b h −2 ωc0 sd Tφ L1 (x, s1d ) · v + Flattened Interferometric Phase 2 ωc0 R1 (z10 ,s1 ) b⊥ 1 ·l 0 i 1 ⊥ 2 b · γ̇ (s ) −2 ωc0 sd Tφ R1 (z10 ,s1 ) v1⊥ · l 1 2 1 d R1 (x,s ) d d 6. Numerical Experiments 6.1. Experimental Setup We conducted numerical experiments for both wideband and Doppler-SAR. Our experimental setup was as follows: • A scene of size 128 × 128m at 1m resolution was imaged. • A single point target was placed at (−20, −31, 50)m with the origin (0, 0, 0) at the scene center. • Two antennas flying on a linear trajectory parallel to the y-axis was used with both antennas placed at 7.1km from the scene center in the x-axis direction. The midpoint of the linear trajectories for both antennas was aligned at y = 0. • Wideband: First antenna was placed at height of 3km and the second at 4km. The length of the trajectories were 1km in length for both antennas. Both antennas were moving at velocity of 100m/s. A waveform with flat spectrum of 100M Hz bandwidth at center frequency of 8GHz was transmitted from both antennas. 512 frequency samples and 1024 slow-time, s, samples were used for imaging. • Doppler: First antenna was placed at height of 2km and the second at 4km. The length of the trajectories were 1km for both antennas. The first antenna was moving at velocity of 100m/s and the second at 400m/s. A continuous waveform at center frequency of 8GHz was transmitted from both antennas. A window of 0.01s was used for processing at each slow time. 512 fast time, t, samples and 1024 slow-time, s, samples were used for imaging. 6.2. Wideband SAR Interferometry Fig. 10a and Fig. 10b show the reconstructed images of the point target located at (−20, −31, 50)m from the first and the second antenna, respectively assuming a flat ground topography at height of 0m. In both Fig. 10a and Fig. 10b, we see that there is a displacement due to layover effect in the range direction (x-axis). The first antenna reconstructs the target at (−41, −31, 0)m. The second antenna reconstructs the target at (−48, −31, 0)m. Doppler Synthetic Aperture Radar Interferometry (a) 18 (b) Figure 10: (a) Wideband reconstruction of the target located at (−20, −31, 50)m using the first antenna assuming flat ground topography. The target is reconstructed at (−41, −31, 0)m. (b) Wideband reconstruction of the target located at (−20, −31, 50)m using the second antenna assuming flat ground topography. The target is reconstructed at (−48, −31, 0)m. We next align the peaks in the two images and multiply the first image with the complex conjugate of the second as in (16) to generate the interferogram. The resulting interferogram is shown in Fig. 11. Figure 11: The interferogram from wideband SAR reconstructed images. In order to reconstruct the height we use the set of equations (21), (22), and (23). The Doppler cone equation (22) at zero-Doppler point s10 gives us that the iso-Doppler contours are in the look-direction, which in our scenario is parallel to the x-axis. Thus, iso-Doppler contours have constant y value at the target’s y position. Using this fact, Doppler Synthetic Aperture Radar Interferometry 19 we need only to compute the intersection of iso-range contour (21) and interferometric phase contour (23) fixing the y position. From Figs. 10a and 10b we see that both targets are reconstructed at y position of −31m. Thus we reconstruct the true target position using y = −31m. For reconstruction, we sampled the height in the interval [1, 100]m at 0.5m resolution. Fig. 12a shows the magnitude image of \|z − γ1 (s10 )\| − R1 (x, s10 ) at y = −31m. Note that R1 (x, s10 ) is the measured value derived from the phase of the reconstructed image. The dark blue area indicates the iso-range contour where the magnitude of the difference is minimized. (a) (b) Figure 12: (a) Image of the magnitude of \|z − γ1 (s10 )\| − R1 (x, s10 ) at y = −31m. The iso-range contour is indicated by dark blue area where the magnitude of \|z − γ1 (s10 )\| − B R1 (x, s10 ) is minimized. (b) Image of the magnitude of L1 (z, s10 ) · b − 2ωc 0 ΦW s0 (x) at y = −31m. The interferometric phase contour is indicated by dark blue area where the B magnitude of L1 (z, s10 ) · b − 2ωc 0 ΦW s0 (x) is minimized. Similarly, Fig. 12b shows the magnitude image of the difference L1 (z, s10 ) · b − c ΦW B (x). As before, the dark blue area indicates the interferometric phase contour. 2ω0 s0 Combining the two images, Fig. 13 shows the intersection of the two contours indicated by the dark blue area. The white ‘x’ in Fig. 13 indicates the exact intersection computed and where the target is reconstructed. The white ‘o’ indicates the true target position. It is clear that the target is reconstructed at the correct position and height. 6.3. Doppler-SAR We proceed similar as in the wideband case for the Doppler-SAR case. Figs. 14a and 14b show the reconstructed image for Doppler-SAR for the first and second antennas, respectively. The first antenna reconstructs the target at (−34, −31, 0)m and the second antenna at (−48, −31, 0)m. Doppler Synthetic Aperture Radar Interferometry 20 Figure 13: Image of the intersection of the iso-range contour with the interfermetric phase contour at y = −31m. The exact intersection is indicated by white ‘x’. The true target position is indicated by white ‘o’. The target is reconstructed at the correct position and height. (a) (b) Figure 14: (a) Doppler-SAR reconstruction of the target located at (−20, −31, 50)m using the first antenna assuming flat ground topography. The target is reconstructed at (−34, −31, 0)m. (b) Doppler-SAR reconstruction of the target located at (−20, −31, 50)m using the second antenna assuming flat ground topography. The target is reconstructed at (−48, −31, 0)m. Doppler Synthetic Aperture Radar Interferometry 21 As in the wideband case, we align the peaks of the two images and multiply the first image with the conjugate of the second image to form the interferogram of the Doppler images. The resulting interferogram is shown in Fig. 15. Figure 15: The interferogram from Doppler-SAR reconstructed images. To reconstruct the height we use the set of equations given in (52), (53), and (54). The zero-Doppler-rate points, s1d , is approximated by the end of the antenna’s trajectories farthest from the target position. By (36), for a linear trajectory with constant velocity, true zero-Doppler-rate point would be where γ̇i (s1d ) ⊥ γ̇i⊥ (s1d ). Namely, where the look-direction is parallel to the velocity vector. The best estimate would be at a point in the trajectory farthest away from the target location. Fig. 16a illustrates the iso-Doppler surface at y = −31m, which is the y-position where the target position is reconstructed and the true target’s y position. Notice that both images reconstruct the scatterer at the correct y position. The iso-Doppler contour is given by the dark blue area as before. Similarly, Figs. 16b and 16c illustrate the iso-Doppler-rate and interferometric Doppler-rate surfaces, respectively at y = −31m. Fig. 17 combines Figs. 16a, 16b, and 16c. The intersection of the three contours is indicated by white ‘x’. The white ‘o’ shows the true target location. Clearly, the target is reconstructed at the correct position and height. 7. Conclusions We present a novel radar interferometry based on Doppler-SAR imaging paradigm. Doppler-SAR uses single frequency transmitted waveforms. It has several advantages over conventional SAR including simpler, inexpensive hardware, high SNR and long effective range of operation, and is suitable for use in passive radar applications. We derived the interferometric phase relationship for Doppler-SAR. Doppler-SAR interferometric phase depends on the difference in the velocity of the antennas as opposed Doppler Synthetic Aperture Radar Interferometry 22 (a) (b) (c) Figure 16: (a) Image of the magnitude of (z −\ γ1 (s1d )) · γ̇1 (s1d ) − ωc0 f1d (x, s1d ) at y = −31m. The iso-Doppler contour is indicated by dark blue area where the magnitude of (z −\ γ1 (s1d )) · γ̇1 (s1d ) = ωc0 f1d (x, s1d ) is minimized. (b) Image of the γ̇ (s1 )·γ̇ ⊥ (s1 T ) magnitude of (z −\ γ1 (s1 )) · γ̈1 (s1 ) − 1 d 11 d φ − ∂s f d (x, s1 ) at y = −31m. The d d R1 (sd ,z) 1 d iso-Doppler-rate contour is indicated by dark blue area where the magnitude of γ̇ (s1 )·γ̇ ⊥ (s1 T ) (z −\ γ1 (s1d ))·γ̈1 (s1d )− 1 Rd 1 (s11 ,z)d φ −∂s f1d (x, s1d ) is minimized. (c) Image of the magnitude b⊥ ·γ̇2 (s2d ) L1 (z, s1d ) · v + R11 (z,s 1) d d of + 2s1 Tcφ ω0 ΦUsdN B (x) at y = −31m. The interferometric Dopplerd rate contour is indicated by dark blue area where the magnitude of L1 (z, s1d ) · v + 2 b⊥ 1 ·γ̇2 (sd ) + 2s1 Tcφ ω0 ΦUsdN B (x) is minimized. R1 (z,s1 ) d d Doppler Synthetic Aperture Radar Interferometry 23 Figure 17: Image of the intersection of the iso-Doppler, iso-Doppler-rate and interferometric Doppler-rate contours at y = −31m. The intersection is indicated by white ‘x’. The true target position is indicated by white ‘o’. The target is reconstructed at the correct position and height. to the range difference observed in wideband SAR. Thus, in Doppler-SAR interferometry, one can reconstruct the ground topography even with the same look-direction from both antennas so long as their velocities are different). Furthermore, we showed that the true target position is determined by the intersection of iso-Doppler, iso-Doppler-rate, and interferometric Doppler-rate surfaces. This is different from conventional wideband SAR in that the surfaces that determine the true target position are iso-range, iso-Doppler, and interferometric Doppler-rate surfaces. We presented numerical simulations for a single point scatterer using two antennas moving in linear trajectories to verify our interferometric method. We also conduct conventional wideband SAR interferometric reconstruction as a comparison. We show that both wideband SAR and Doppler-SAR interferometry is able to accurately reconstruct the target location. Thus, our numerical simulations show that DopplerSAR interferometry retains the accuracy of conventional SAR interferometry while having the advantage that Doppler-SAR affords. In the future, we will analyze the sensitivity of height estimation with respect to other observables and parameters. Acknowledgement This material is based upon work supported by the Air Force Office of Scientific Research (AFOSR) under award number FA9550-16-1-0234, and by the National Science Foundation (NSF) under Grant No. CCF-1421496. Doppler Synthetic Aperture Radar Interferometry 24 Appendix A. Approximations Appendix A.1. Far-field approximation Let x and y be two vectors such that \|x\| \|y\|. Then, by using Taylor series expansion we can make the following approximation: p \|x − y\| = (x1 − y1 )2 + (x2 − y2 )2 + (x3 − y3 )2 , (A.1) p = \|x\|2 − 2(x1 y1 + x2 y2 + x3 y3 ) + \|y\|2 , (A.2) s 2(x · y) \|y\|2 = \|x\| 1 − + 2, (A.3) \|x\|2 \|x\| 1 2x · y , (A.4) ≈ \|x\| 1 − 2 \|x\| ≈ \|x\| − x̂ · y (A.5) where x̂ is the unit vector x̂ = x . \|x\| Appendix A.2. Approximation of look-direction under far-field assumption Let (x \ − γ(s)) denote a look direction where x = y + z and \|y − γ(s)\| \|z\|. Then by using far field expansion we can write x − γ(s) \|x − γ(s)\| y + z − γ(s) = , \|y + z − γ(s)\| z y − γ(s) + , ≈ \|y − γ(s)\| + (y \ − γ(s)) · z \|y − γ(s)\| + (y \ − γ(s)) · z y − γ(s) ≈ (\|y − γ(s)\| − (y \ − γ(s)) · z) \|y − γ(s)\|2 (\|y − γ(s)\| − (y \ − γ(s)) · z)z + , \|y − γ(s)\|2 i h \ \ z − (y − γ(s)) (y − γ(s)) · z ≈ (y \ − γ(s)) + , \|y − γ(s)\| z⊥ ≈ (y \ − γ(s)) + \|y − γ(s)\| (x \ − γ(s)) = (A.6) (A.7) (A.8) (A.9) (A.10) (A.11) where z⊥ is the transverse z, i.e. projection of z onto the plane whose normal vector is \ along the look direction (γ(s) − y). Therefore, difference of look directions is given by: (x \ − γ(s)) − (y \ − γ(s)) ≈ where x = y + z. z⊥ \|γ(s) − y\| (A.12) Doppler Synthetic Aperture Radar Interferometry 25 References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] Bamler R and Hartl P 1998 Inverse problems 14 R1 Rogers A and Ingalls R 1969 Science 165 797–799 Rogers A, Ingalls R and Rainville L 1972 The Astronomical Journal 77 100 Graham L C 1974 Proceedings of the IEEE 62 763–768 Zebker H A and Goldstein R M 1986 Journal of Geophysical Research: Solid Earth 91 4993–4999 Goldstein R M and Zebker H A 1987 Nature 328 707–709 Gabriel A K and Goldstein R M 1988 International Journal of Remote Sensing 9 857–872 Gabriel A K, Goldstein R M and Zebker H A 1989 Journal of Geophysical Research: Solid Earth 94 9183–9191 Hanssen R F 2001 Radar interferometry: data interpretation and error analysis vol 2 (Springer) Rosen P A, Hensley S, Joughin I R, Li F K, Madsen S N, Rodriguez E and Goldstein R M 2000 Proceedings of the IEEE 88 333–382 Cherniakov M and Moccia A 2008 Bistatic radar: emerging technology (John Wiley & Sons) ISBN 9780470026311 URL http://books.google.com/books?id=a6nMEY2bKp4C Fritz T, Rossi C, Yague-Martinez N, Rodriguez-Gonzalez F, Lachaise M and Breit H 2011 Interferometric processing of tandem-x data IEEE International Geoscience and Remote Sensing Symposium (IGARSS) (IEEE) pp 2428–2431 Duque S, Lopez-Dekker P and Mallorqui J 2010 Geoscience and Remote Sensing, IEEE Transactions on 48 2740–2749 Wang L and Yazici B 2012 IEEE Trans. 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Séminaire BOURBAKI 69ème année, 2016-2017, no 1125 Janvier 2017 ISOMORPHISMES DE GRAPHES EN TEMPS QUASI-POLYNOMIAL [d’après Babai et Luks, Weisfeiler-Leman, . . .] arXiv:1701.04372v2 [math.GR] 12 Oct 2017 par Harald Andrés HELFGOTT Résumé : Soient donnés deux graphes Γ1 , Γ2 à n sommets. Sont-ils isomorphes ? S’ils le sont, l’ensemble des isomorphismes de Γ1 à Γ2 peut être identiﬁé avec une classe H · π du groupe symétrique sur n éléments. Comment trouver π et des générateurs de H ? Le déﬁ de donner un algorithme toujours eﬃcace en réponse à ces questions est resté longtemps ouvert. Babai a récemment montré comment résoudre ces questions – et d’autres qui y sont liées – en temps quasi-polynomial, c’est-à-dire en temps O(1) exp O(log n) . Sa stratégie est basée en partie sur l’algorithme de Luks (1980/82), qui a résolu le cas de graphes de degré borné. 1. INTRODUCTION Soient x, y deux chaı̂nes de caractères, à savoir, deux applications Ω → Σ, où Σ (l’alphabet) et Ω (le domaine) sont des ensembles ﬁnis. Tout groupe de permutations (1) G < Sym(Ω) agit sur l’ensemble ΣΩ des chaı̂nes de domaine Ω sur un alphabet Σ. Pour nous, décrire un groupe G, ou être donné un groupe G, voudra toujours dire « donner, voire être donné, un ensemble de générateurs de G » ; décrire une classe Hπ voudra dire « donner un élément π de la classe et un ensemble de générateurs de H ». Le problème de l’isomorphisme de chaı̂nes consiste à déterminer, étant donnés x, y et G, s’il y a au moins un élément π de G qui envoie x sur y, et, si de tels éléments (isomorphismes) existent, à les décrire. Il est clair que l’ensemble des isomorphismes IsoG (x, y) forme une classe AutG (x)π du groupe AutG (x) d’automorphismes de x dans G, c’està-dire du groupe consistant dans les éléments de G qui envoient x sur lui-même. Le déﬁ consiste à donner un algorithme qui résolve le problème en temps polynomial en la taille n = \|Ω\| de Ω, voire en temps raisonnable. Par exemple, le temps employé pourrait être quasi-polynomial en n, ce qui veut dire exp O(log n)O(1) . Ici, comme toujours, O(f (n)) désigne une quantité bornée par C · f (n), pour n assez grand et C > 0 une constante, et Oǫ indique que la constante C dépend de ǫ. Une grande partie de la motivation pour le problème de l’isomorphisme de chaı̂nes vient du fait que le problème de l’isomorphisme de graphes se réduit à lui. Ce problème 1. Pour nous, G < S (ou S > G) veut dire « G est un sous-groupe de S, pas forcement propre. » 1125–02 consiste à déterminer si deux graphes ﬁnis Γ1 et Γ2 sont isomorphes, et, s’ils le sont, à décrire la classe de leurs isomorphismes. (Un isomorphisme π : Γ1 → Γ2 est une bijection π de l’ensemble de sommets de Γ1 vers celui de Γ2 telle que π(Γ1 ) = Γ2 .) Une solution permettrait, par exemple, de trouver une molécule dans une base de données. Le problème de l’isomorphisme de graphes se réduit en temps polynomial au problème de l’isomorphisme de chaı̂nes, de la façon suivante. Supposons sans perte de généralité que Γ1 et Γ2 ont le même ensemble de sommets V . Alors, nous pouvons déﬁnir Ω comme l’ensemble des paires d’éléments de V (ordonnés ou non ordonnés, suivant que nos graphes sont orientés ou pas). La chaı̂ne xi , i = 1, 2, est déﬁnie comme suit : pour la paire a = {v1 , v2 } (ou a = (v1 , v2 ), si nos graphes sont orientés), la valeur de xi (a) est 1 s’il y a une arête entre v1 et v2 en Γ1 , et 0 dans le cas contraire. Soit G l’image de l’homomorphisme ι : Sym(V ) → Sym(Ω) déﬁnie par σ ι ({v1 , v2 }) = {σ(v1 ), σ(v2 )}, où σ ι = ι(σ). Alors ι induit une bijection entre la classe des isomorphismes de Γ1 à Γ2 et la classe IsoG (x1 , x2 ). Théorème 1.1 (Babai). — Le problème de l’isomorphisme de chaı̂nes Ω → Σ peut être résolu en temps quasi-polynomial en le nombre d’éléments du domaine Ω. En novembre 2015, Babai a annoncé une solution en temps quasipolynomial, avec un algorithme explicite. La préparation de cet exposé m’a conduit à trouver une erreur non triviale dans l’analyse du temps, mais Babai a réussi à le réparer en simpliﬁant l’algorithme. La preuve est maintenant correcte. Corollaire 1.2 (Babai). — Le problème de l’isomorphisme de graphes peut être résolu en temps quasi-polynomial en le nombre de sommets. Notre référence principale sera [Ba] ; nous nous servirons aussi de la version courte [Ba2]. Nous essayerons d’examiner la preuve de la façon la plus détaillée possible dans un exposé de ce format, en partie pour aider à éliminer tout doute qui pourrait rester sur la forme actuelle du résultat. La meilleure borne générale connue antérieurement pour le temps requis par le √ problème de l’isomorphisme de graphes, due à Luks [BKL], était exp(O( n log n)), * L’usage de la canonicité joue un rôle crucial dans la stratégie de Babai. Comme dans la théorie de catégories, voire dans l’usage courant, un choix est canonique s’il est fonctoriel. La situation typique pour nous sera la suivante : un groupe G < Sym(Ω) agit sur Ω, et donc sur ΣΩ ; il agit aussi sur un autre ensemble S, et donc aussi sur les applications S → C , où C est un ensemble ﬁni. Une application S → C s’appelle un coloriage ; l’ensemble C s’appelle l’ensemble de couleurs. Un choix canonique (en relation à G) d’un coloriage de Ω pour chaque chaı̂ne x ∈ ΣΩ est une application qui va de ΣΩ aux coloriages et qui commute avec l’action de G. En particulier, un choix canonique peut être un outil pour détecter des nonisomorphismes : si les coloriages C(x) et C(y) induits canoniquement par x et y 1125–03 ne sont pas isomorphes l’un à l’autre – par exemple, s’ils ont un nombre diﬀérent d’éléments vermeils – alors x et y ne sont pas isomorphes l’un à l’autre. Même quand il y a des isomorphismes dans G qui envoient C(x) sur C(y), la classe IsoG (C(x), C(y)) de tels isomorphismes sert à délimiter la classe d’isomorphismes IsoG (x, y) de x à y, puisque cette dernière est forcément un sous-ensemble de IsoG (C(x), C(y)). La preuve assimile aussi plusieurs idées développées lors d’approches antérieures au problème. La première étape de la procédure consiste à essayer de suivre ce qui est en essence l’algorithme de Luks [Lu]. Si cet algorithme s’arrête, c’est parce qu’il s’est heurté contre un quotient H1 /H2 isomorphe à Alt(Γ), où H2 ⊳ H1 < G et Γ est plutôt grand. Notre tâche majeure consiste à étudier ce qui se passe à ce moment-là. La stratégie principale sera de chercher à colorier Γ d’une façon qui dépend canoniquement de x. Cela limitera les automorphismes et isomorphismes possibles à considérer. Par exemple, si la moitié de Γ est coloriée en rouge et l’autre en noir, le groupe d’automorphismes possibles se réduit à Sym(\|Γ\|/2) ×Sym(\|Γ\|/2). Un coloriage similaire induit par y limite les isomorphismes aux applications qui alignent les deux coloriages. Nous trouverons toujours des coloriages qui nous aident, sauf quand certaines structures ont une très grande symétrie, laquelle, en revanche, permettra une descente à Ω considérablement plus petit. Cette double récursion – réduction du groupe H1 /H2 ou descente à des chaı̂nes considérablement plus courtes – résoudra le problème. 2. FONDEMENTS ET TRAVAUX PRÉCÉDENTS En suivant l’usage courant pour les groupes de permutations, nous écrirons r g pour l’élément g(r) auquel g ∈ Sym(Ω) envoie r ∈ Ω. Étant donnés une chaı̂ne x : Ω → Σ et −1 un élément g ∈ Sym(Ω), nous déﬁnissons xg : Ω → Σ par xg (r) = x r g . Par contre, nous écrivons Ωk pour l’ensemble des ~x = (x1 , . . . , xk ) avec l’action à gauche donnée par (φ(~x))r = ~xφ(r) . L’idée est que ceci est déﬁni non pas seulement pour φ une permutation, mais pour toute application φ : {1, . . . , k} → {1, . . . , k}, même non injective. Nous appelons les éléments de Ωk tuples plutôt que chaı̂nes. 2.1. Algorithmes de base 2.1.1. Schreier-Sims. — Plusieurs algorithmes essentiels se basent sur une idée de Schreier [Sch]. Il a remarqué que, pour tout sous-groupe H d’un groupe G et tout sous-ensemble A ⊂ G qui engendre G et contient des représentants de toutes les classes de H dans G, A′ = AAA−1 ∩ H = σ1 σ2 σ3−1 : σi ∈ A ∩ H est un ensemble de générateurs de H. 1125–04 L’étape suivante est celle de Sims [Si1], [Si2], qui a montré l’utilité de travailler avec un groupe de permutations G < Sym(Ω), Ω = {x1 , . . . , xn }, en termes d’une chaı̂ne de stabilisateurs G = G0 > G1 > G2 > . . . > Gn−1 = {e}, où Gk = G(x1 ,x2 ,...,xk ) = {g ∈ G : ∀1 ≤ i ≤ k xgi = xi } (stabilisateur de points). L’algorithme de Schreier-Sims (Algorithme 1 ; description basée sur [Lu, §1.2]) construit des ensembles Ci de représentants de Gi /Gi+1 tels que ∪i≤j<n−1 Cj engendre Gi pour tout 0 ≤ i < n − 1. Le temps pris par l’algorithme est O(n5 + n3 \|A\|), où A est l’ensemble de générateurs de G qui nous est donné : la fonction Filtre prend O(n) de temps, et tout g pour lequel elle est appelée satisfait g ∈ AC ∪ CA ∪ C 2 , où C est la valeur de ∪i Ci à la ﬁn de la procédure. Bien sûr, \|C\| ≤ n(n + 1)/2. Grâce à l’algorithme lui-même, nous pourrons toujours supposer que nos ensembles de générateurs sont de taille O(n2). Le temps pris par l’algorithme est donc O(n5 ). (2) Algorithme 1 Schreier-Sims : construction d’ensembles Ci 1: fonction SchreierSims(A, ~ x) ⊲ A engendre G < Sym({x1 , . . . , xn }) assure ∪i≤j<n−1Cj engendre Gi et Ci 7→ Gi /Gi+1 est injectif ∀i ∈ {0, 1, . . . , n − 2} 2: Ci ← {e} pour tout i ∈ {0, 1, . . . , n − 2} 3: B←A 4: tantque B 6= ∅ 5: Choisir g ∈ B arbitraire, et l’enlever de B 6: (i, γ) ← Filtrer(g, (Ci ), ~x) 7: si γ 6= e alors 8: ajouter γ à Ci S S 9: B ← B ∪ j≤i Cj γ ∪ j≥i γCj 10: retourner (Ci ) fonction Filtrer(g, (Ci ), ~x) ⊲ retourne (i, γ) tel que γ ∈ Gi , g ∈ C0 C1 · · · Ci−1 γ requiert Ci ⊂ Gi et Ci → Gi /Gi+1 injectif ∀i ∈ {0, 1, . . . , n − 2} assure g ∈ / C0 C1 · · · Ci Gi+1 sauf si (i, γ) = (n − 1, e) 12: γ←g 13: pour i = 0 jusqu’à n − 2 14: si ∃h ∈ Ci tel quel xhi = xgi alors 15: γ ← h−1 γ 16: sinon 17: retourner (i, γ) 11: 18: retourner (n − 1, e) 2. Nous supposons que l’ensemble de générateurs initial, spéciﬁant le groupe G du problème, est de taille O(nC ), C une constante. Le temps pris par la première utilisation de l’algorithme est donc O nmax(5,3+C) . 1125–05 Une fois les ensembles Ci construits, il devient possible d’accomplir plusieurs tâches essentielles rapidement. Exercice 2.1. — Montrer comment accomplir les tâches suivantes en temps polynomial, étant donné un groupe G < Sym(Ω), \|Ω\| = n : (a) Déterminer si un élément g ∈ Sym(Ω) est dans G. (b) Étant donnés un homomorphisme φ : G → Sym(Ω′ ), \|Ω′ \| ≪ \|Ω\|O(1) , et un sousgroupe H < Sym(Ω′ ), décrire φ−1 (H). (c) [FHL] Soit H < G avec [G : H] ≪ nO(1) . Étant donné un test qui détermine en temps polynomial si un élément g ∈ G appartient à H, décrire H. Astuce : travailler avec G > H > H1 > H2 > . . . à la place de G = G0 > G1 > G2 > . . . . Ici, comme toujours, « décrire » veut dire « trouver un ensemble de générateurs », et un groupe nous est « donné » si un tel ensemble nous est donné. L’algorithme de Schreier-Sims décrit le stabilisateur de points G(x1 ,...,xk ) pour x1 , . . . , xk ∈ Ω arbitraires. Par contre, nous ne pouvons pas demander allègrement un ensemble de générateurs d’un stabilisateur d’ensemble G{x1 ,...,xk } = {g ∈ G : {xg1 , . . . , xgk } = {x1 , . . . , xk }} pour G, xi arbitraires : faire ceci serait équivalent à résoudre le problème de l’isomorphisme lui-même. 2.1.2. Orbites et blocs. — Soit donné, comme toujours, un groupe de permutations G agissant sur un ensemble ﬁni Ω. Le domaine Ω est l’union disjointe des orbites {xg : g ∈ G} de G. Ces orbites peuvent être déterminées en temps polynomial (3) en \|Ω\|. Ceci est un exercice simple. La tâche se réduit à celle – simple elle aussi – de trouver les composantes connexes d’un graphe. Supposons que l’action de G soit transitive. (Il y a donc une seule orbite.) Un bloc de G est un sous-ensemble B ⊂ Ω, B ∈ / {∅, Ω}, tel que, pour g, h ∈ G quelconques, g h g h soit B = B , soit B ∩ B = ∅. La collection {B g : g ∈ G} (système de blocs) pour B donné partitionne Ω. L’action de G est primitive s’il n’y a pas de blocs de taille > 1 ; autrement, elle s’appelle imprimitive. Un système de blocs est minimal (4) si l’action de G sur lui est primitive. Voyons comment déterminer si l’action de G est primitive, et, s’il ne l’est pas, comment trouver un système de blocs de taille > 1. En itérant la procédure, nous obtiendrons un système de blocs minimal en temps polynomial. (Nous suivons [Lu], qui cite [Si1].) Pour a, b ∈ Ω distincts, soit Γ le graphe avec Ω comme son ensemble de sommets et l’orbite {{a, b}g : g ∈ G} comme son ensemble d’arêtes. La composante connexe qui 3. Pour être précis : O \|Ω\|O(1) + \|A\|\|Ω\| , où A est la taille de l’ensemble de générateurs de G qui nous est donné. Nous omettrons toute mention de cette taille par la suite, puisque, comme nous l’avons déjà dit, nous pouvons la garder toujours sous contrôle. 4. Pour paraphraser [Lu, §1.1] : il faut avouer qu’un tel système pourrait s’appeler plutôt maximal. La taille des blocs est maximale, leur nombre est minimal. 1125–06 contient a et b est le bloc le plus petit qui contient a et b. (Si Γ est connexe, alors le « bloc » est Ω.) L’action de G est imprimitive ssi Γ est non connexe pour un a arbitraire et au moins un b ; dans ce cas-là, nous obtenons un bloc qui contient a et b, et donc tout un système de blocs de taille > 1. Un dernier mot : si G < Sym(Ω), nous disons que G est transitif, voire primitif, si son action sur Ω l’est. 2.2. Luks : le cas de groupes avec facteurs d’ordre borné Luks a montré comment résoudre le problème de l’isomorphisme de graphes en temps polynomial dans le cas spécial de graphes de degré borné. (Le degré, ou valence, d’un sommet dans un graphe non orienté est le nombre d’arêtes qui le contiennent.) Il réduit ceci au problème de décrire le groupe d’automorphismes de chaı̂nes dans le cas d’un groupe G tel que tout facteur de composition de G – c’est-à-dire, tout quotient dans une suite principale (Jordan-Hölder) de G – est borné. Le processus de réduction, élégant et loin d’être trivial, ne nous concerne pas ici. Voyons plutôt comment Luks résout ce cas du problème de l’isomorphisme de chaı̂nes. Nous suivrons la notation de [Ba], même si les idées viennent de [Lu]. Définition 2.2. — Soient K ⊂ Sym(Ω) et ∆ ⊂ Ω (la « fenêtre »). L’ensemble d’isomorphismes partiels Iso∆ K est τ Iso∆ K (x, y) = {τ ∈ K : x(x) = y(x ) ∀x ∈ ∆}. ∆ L’ensemble d’automorphismes partiels Aut∆ K (x) est égal à IsoK (x, x). Iso∆ K est donc l’ensemble de toutes les permutations g ∈ K qui envoient x sur y – au moins à en juger par ce qui peut se voir par la fenêtre ∆. Nous travaillerons en général avec K de la forme Hπ, où H laisse ∆ invariante (en tant qu’ensemble). Il est clair que, pour K, K1 , K2 ⊂ Sym(Ω) et σ ∈ Sym(Ω), ∆ ∆ σ−1 σ, (1) IsoKσ (x, y) = IsoK x, y (2) ∆ ∆ Iso∆ K1 ∪K2 (x, y) = IsoK1 (x, y) ∪ IsoK2 (x, y). Il est aussi clair que, si G est un sous-groupe de Sym(Ω) et ∆ est invariant sous G, alors AutG (x) est un sous-groupe de G, et, pour tout σ ∈ Sym(Ω), IsoGσ (x, y) est soit vide, soit une classe à droite de la forme AutG (x)τ , τ ∈ Sym(Ω). Soient ∆1 , ∆2 ⊂ Ω, ′ 1 ∆1 invariant sous G. Pour G′ = AutG (x) et σ, τ tels que Iso∆ Gσ (x, y) = G τ , ∆1 ∪∆2 ∆2 τ −1 2 τ, x, y (3) IsoGσ (x, y) = Iso∆ (x, y) = Iso ′ ′ Gτ G où la deuxième équation est une application de (1). Babai appelle (3) la règle de la chaı̂ne. L’énoncé suivant n’utilise pas la classiﬁcation de groupes ﬁnis simples. 1125–07 Théorème 2.3 ([BCP] (6) ). — Soit G < Sym(Ω) un groupe primitif. Soit n = \|Ω\|. Si tout facteur de composition de G est d’ordre ≤ k, alors \|G\| ≤ nOk (1) . Ici, comme d’habitude, Ok (1) désigne une quantité qui dépend seulement de k. Théorème 2.4 (Luks [Lu]). — Soient Ω un ensemble ﬁni et x, y : Ω → Σ deux chaı̂nes. Soit donné un groupe G < Sym(Ω) tel que tout facteur de composition de G est d’ordre ≤ k. Il est possible de déterminer IsoG (x, y) en temps polynomial en n = \|Ω\|. Preuve — Cas 1 : G non transitif. Soit ∆1 ( Ω, ∆1 6= ∅, ∆1 stable sous l’action 1 de G. Déﬁnissons ∆2 = Ω \ ∆1 . Alors, par (3), il suﬃt de calculer Iso∆ G (x, y) (égal à −1 ′ ′ τ 2 une classe que nous notons G′ τ ) et Iso∆ . Or, pour déterminer G′ (x, y ) pour y = y ∆1 IsoG (x, y), nous déterminons, de façon récursive, IsoG (x\|∆1 , y\|∆1 ), puis, par Schreier′ 2 Sims, le stabilisateur de points G(∆1 ) . De la même manière, déterminer Iso∆ G′ (x, y ) pour −1 y′ = yτ se réduit à déterminer le groupe d’isomorphismes (dans un groupe G′ ) entre deux chaı̂nes de longueur \|∆2 \|. Comme \|∆1 \|+\|∆2\| = n et Schreier-Sims prend du temps O(n5 ), tout va bien. (La comptabilité est laissée au lecteur.) Cas 2 : G transitif. Soit N le stabilisateur d’un système de blocs minimal pour G ; donc, G/N est primitif. Par le Théorème 2.3, \|G/N\| ≤ mOk (1) , où m est le nombre de blocs. Or, pour σ1 , . . . , σℓ (ℓ = \|G/N\|) tels que G = ∪1≤i≤ℓ Nσi , [ [ −1 (4) IsoG (x, y) = Iso∪i N σi (x, y) = IsoN σi (x, y) = IsoN (x, yσi )σi 1≤i≤ℓ 1≤i≤ℓ par (1) et (2). Comme les orbites de N sont contenues dans les blocs, qui sont de taille −1 n/m, déterminer IsoN (x, yi ) (yi = yσi ) se réduit – par la règle (3) – à déterminer les groupes d’isomorphismes de m paires de chaı̂nes de longueur n/m. Nous avons donc réduit le problème à la solution de ℓ·m = mOk (1) problèmes pour des chaı̂nes de longueur n/m. Le pas ﬁnal consiste à faire l’union de classes en (4). Nous avons une description de chaque IsoN (x, yi ), soit comme l’ensemble vide, soit comme une classe à droite Hτi du groupe H = AutN (x), dont nous avons trouvé une description, c’est-à-dire un ensemble de générateurs A. Alors [ [ Hτi σi IsoN (x, yi )σi = IsoG (x, y) = 1≤i≤ℓ 1≤i≤ℓ = A ∪ τi σi (τ1 σ1 )−1 : 1 ≤ i ≤ ℓ τ1 σ1 . Nous aurions pu éviter quelques appels à Schreier-Sims en travaillant toujours avec des isomorphismes partiels, mais cela a peu d’importance qualitative. 6. À vrai dire, [BCP, Thm 1.1] est plus général que ceci ; par exemple, des facteurs abéliens arbitraires (non bornés) sont admis. Cela donne une généralisation du Théorème 2.4. 1125–08 2.3. Relations, partitions, configurations Soit C (« couleurs ») un ensemble ﬁni que nous pouvons supposer ordonné (disons, de rouge à violet). Une relation k-aire sur un ensemble ﬁni Γ est un sous-ensemble R ⊂ Γk . Une structure (relationnelle) k-aire est une paire X = (Γ, (Ri )i∈C ), où, pour chaque i ∈ C , Ri est une relation k-aire sur Γ. Si les Ri sont tous non vides et partitionnent Γk , nous disons que X est une structure de partition k-aire. Dans ce cas-là, nous pouvons décrire X par une fonction c : Γk → C qui assigne à chaque ~x ∈ Γk l’indice i de la relation Ri à laquelle il appartient. Nous disons que c(~x) est la couleur de ~x. Un isomorphisme entre deux structures k-aires X = (Γ, (Ri )i∈C ) et X′ = (Γ′ , (Ri′ )i∈C ) est une bijection Γ → Γ′ qui envoie Ri à Ri′ pour chaque i. Il est possible de construire un foncteur F1 qui envoie chaque structure k-aire X sur Γ à une structure de partition k-aire F1 (X) sur Γ ; qui plus est, Iso(X, Y) = Iso(F1 (X), F1 (Y)). La procédure est plutôt triviale ; nous la détaillons (Algorithme 2) pour montrer ce qu’indexer veut dire. Cela nous permet de ne pas utiliser plus de min \|Γ\|k , 2\|C \| couleurs, où n = \|Ω\|, tout en gardant leur signiﬁcation en termes des couleurs originales C . Le temps pris pour calculer F1 (X) est O(\|C \|\|Γ\|O(k)). Nous ne nous occupons pas des détails d’implémentation de la collection de tuples I , mais il peut s’agir tout simplement d’une liste ordonnée lexicographiquement ; dans ce cas, \|Γ\|O(k) est \|Γ\|2k . (Dans la réalité, I serait implémentée avec du hachage, ce qui n’est que l’art de bien organiser une bibliothèque.) Algorithme 2 Raﬃnement d’une structure de relation. Indexeur. 1: fonction F1 (Γ,k,C ,(Ri )i∈C ) 2: I ←∅ 3: pour ~x ∈ Γk 4: a ← {i ∈ C : ~x ∈ Ri } 5: c(~x) ← Indexeur(I , a) 6: 7: 8: 9: 10: 11: retourner (I , c) ⊲ retourne c : Γk → C ′ ⊲ C ′ est l’ensemble d’indices de I ; I explique C ′ en termes de C fonction Indexeur(I ,a) ⊲ I est une collection modiﬁable si a n’est pas dans I alors ajouter a à I retourner indice de a dans I Un élément ~x ∈ Γk déﬁnit une relation d’équivalence ρ(~x) sur {1, . . . , k} : i ∼ j ssi xi = xj . Le monoı̈de M(S) (S un ensemble) consiste en les applications S → S, avec la composition comme opération. Définition 2.5. — Une structure de partition k-aire X = (Γ, c) est dite conﬁguration k-aire si (a) Pour tous ~x, ~y ∈ Γk , si c(~x) = c(~y ), alors ρ(~x) = ρ(~y ). (b) Il y a un homomorphisme de monoı̈des η : M({1, . . . , k}) → M(C ) tel que, pour tout τ ∈ M({1, . . . , k}), c (τ (~x)) = τ η (c(~x)) pour tout ~x ∈ Γk . 1125–09 Alors, par exemple, pour k = 2, (a) veut dire que la couleur de ~x = (x1 , x2 ) « sait » si x1 = x2 ou pas, dans le sens où, si nous connaissons c(~x), alors nous savons si x1 = x2 ou pas. De la même façon, (b) nous indique que la couleur de ~x connaı̂t les couleurs de (x2 , x1 ), (x1 , x1 ) et (x2 , x2 ). Nous pouvons déﬁnir un foncteur F2 qui envoie chaque structure de partition k-aire X sur Γ à une conﬁguration k-aire ; comme pour F1 , le fait que F2 (X) est un raﬃnement de X implique que Iso(X, Y) = Iso(F2 (X), F2 (Y)). La procédure pour calculer F2 est très similaire à celle pour calculer F1 (Algorithme 2). Au lieu d’assigner à ~x la couleur {i ∈ C : ~x ∈ Ri }, nous lui assignons la couleur ρ(~x), (c(φ(~x)))φ∈M({1,...,k}) . Il est aisé de voir que F2 (X) est le raﬃnement le plus grossier d’une structure de partition X qui est une conﬁguration, de la même manière que F1 (X) est le raﬃnement le plus grossier d’une structure X qui est une structure de partition. Définition 2.6. — Soit X = (Γ, c), c : Γk → C , une structure de partition k-aire. Pour 1 ≤ l ≤ k, nous déﬁnissons c(l) : Γl → C comme suit : c(l) (~x) = c(x1 , x2 , . . . , xl , xl , . . . xl ). La structure de partition l-aire X(l) = Γ, c(l) est dite le (l-)squelette de X. La chaı̂ne vide sera viride. Exercice 2.7. — Tout squelette d’une conﬁguration est une conﬁguration. Ici le fait que l’axiome (b) dans la déﬁnition de conﬁguration soit valable même pour η non injectif est crucial. Pour X = (Γ, c) une structure de partition et Γ′ ⊂ Γ, la sous-structure induite X[Γ′ ] est la structure (Γ′ , c\|Γ′ ) déﬁnie par la restriction de c à Γ′ . Il est clair que, si X est une conﬁguration, alors X[Γ′ ] l’est aussi. * Il ne faut pas confondre une structure de partition (partition structure) avec ce que nous appellerons un découpage (colored partition). Un découpage d’un ensemble Γ est un coloriage de Γ supplémenté d’une partition de chaque classe de couleur. (Une classe de couleur est l’ensemble de sommets d’une couleur donnée.) Un découpage est dit admissible si chaque ensemble B dans chaque partition est de taille ≥ 2. Pour α < 1, un α-découpage est un découpage admissible tel que \|B\| ≤ α\|Γ\| pour chaque B. Un découpage est une structure plus ﬁne que le coloriage qu’il raﬃne, mais moins ﬁne que la structure que nous obtiendrions si nous donnions à chaque élément de chaque partition une couleur diﬀérente. Un automorphisme ou isomorphisme d’un découpage doit préserver les couleurs de celui-ci, mais pourrait permuter les ensembles de la même taille qui appartiennent à la partition d’une couleur. Comme les ensembles de taille diﬀérente ne peuvent, évidemment, être permutés, il est clair que nous pouvons supposer sans perte de généralité que toute couleur est partitionnée en ensembles de la même taille. Nous ajoutons ceci à la déﬁnition de α-découpage à partir de maintenant. 1125–10 2.4. Configurations cohérentes k-aires Pour ~x ∈ Γk , z ∈ Γ et 1 ≤ i ≤ k, nous déﬁnissons ~xi (z) ∈ Γk comme suit : ~xi (z) = (x1 , x2 , . . . , xi−1 , z, xi+1 , . . . , xk ). Définition 2.8. — Une conﬁguration cohérente k-aire X = (Γ, c) est une conﬁguration k-aire ayant la propriété suivante : il y a une fonction γ : C k × C → Z≥0 telle que, pour ~k ∈ C k et j ∈ C arbitraires et tout ~x ∈ Γk tel que c(~x) = j, \|{z ∈ Γ : c(~xi (z)) = ki ∀1 ≤ i ≤ k}\| = γ(~k, j). Les valeurs γ(~k, j) sont appelées nombres d’intersection de X. Une conﬁguration cohérente est dite classique si k = 2. Remarque 2.9. — Les conﬁgurations cohérentes classiques ont été introduites par Higman [Hi]. Les premiers exemples étaient du type schurien : une conﬁguration est schurienne si elle est la partition de Γ2 dans ses orbites (« orbitales ») sous l’action d’un groupe G < Sym(Γ). Définition 2.10. — Si une conﬁguration cohérente classique n’a que deux couleurs, une pour {(x, x) : x ∈ Γ} et l’autre pour son complément, la conﬁguration est dite une clique, ou triviale. Exercice 2.11. — Tout squelette d’une conﬁguration cohérente est cohérent. Encore une fois, l’axiome (b) des conﬁgurations joue un rôle clé. Exercice 2.12. — Soient X = (Γ, c) une conﬁguration cohérente et Γ′ ⊂ Γ une classe de couleurs en relation au coloriage induit par c sur Γ. Alors la sous-structure induite X[Γ′ ] est une conﬁguration cohérente. Ici, c’est un cas spécial de (b) qu’il faut utiliser : la couleur c(x1 , . . . , xn ) « connaı̂t » les couleurs c(x1 ), . . . , c(xn ), puisque c(xi ) = c(xi , . . . , xi ). Soient 0 ≤ l < k et ~x ∈ Γl . Nous colorions Γk−l comme suit : pour ~y ∈ Γk−l , c~x (~y ) = c(~x~y ). En résulte une structure de partition (k − l)-aire X~x = (Γ, c~x ). Exercice 2.13. — Soit X = (Γ, c) une structure de partition ; soit ~x ∈ Γl , 0 ≤ l < k. Alors (a) c~x est un raﬃnement du coloriage du squelette X(k−l) . (b) Si X est cohérente, X~x l’est aussi. Il est clair que, de plus, X~x est canonique en relation à ~x, ce qui veut dire que X → X~x commute avec l’action sur Γ du stabilisateur dans Sym(Γ) des points x1 , . . . , xl . 1125–11 Définition 2.14. — Une conﬁguration cohérente (Γ, c) est dite homogène si la couleur c(x, x, . . . , x) de tout sommet x ∈ Γ est la même. Une conﬁguration cohérente classique est dite primitive si elle est homogène et les graphes Gr = {(x, y) : x, y ∈ Γ, c(x, y) = r} (pour toute couleur r telle que c(x, y) = r pour au moins une paire (x, y) avec x 6= y) sont tous connexes. Elle est dite uniprimitive si elle est primitive et non triviale. Nous n’avons pas besoin de préciser si ces graphes son connexes dans le sens propre (à savoir, il y a un chemin de tout sommet à tout autre, respectant l’orientation) ou dans le sens faible (sans compter l’orientation) : le fait que (Γ, c) soit cohérente, classique et homogène implique que d+ r (x) = \|{y ∈ Γ : (x, y) ∈ Gr }\| est indépendant de x (pourquoi ?), ce qui implique que toute composante faiblement connexe de Gr est connexe (exercice). Exercice 2.15. — Soit X = (Γ, c) une conﬁguration cohérente classique uniprimitive. Il n’y a aucun ensemble B ⊂ Γ, \|B\| > \|Γ\|/2, tel que la restriction de X à B soit une clique. Solution – Si les arêtes de la grande clique sont sensées être blanches, soit noir une autre couleur d’arêtes de X, et soit G = Gnoir . Or, pour un graphe orienté birégulier (7) G non vide avec Γ comme ensemble de sommets, il est impossible qu’il y ait un ensemble B ⊂ Γ, \|B\| > \|G\|/2, tel que la réduction du graphe à B soit vide (pourquoi ?). Exercice 2.16. — Soit (Γ, c) une conﬁguration cohérente classique homogène. (a) Soit r0 , . . . , rk une séquence de couleurs. Alors, si x0 , xk ∈ Γ sont tels que c (x0 , xk ) = r0 , le nombre de x1 , . . . , xk−1 ∈ Γ tels que c (xi−1 , xi ) = ri pour tout 1 ≤ i ≤ k dépend seulement de r0 , . . . , rk . (b) Pour toute couleur r, toute composante connexe de Gr est de la même taille. Solution (esquisse) — En (a), le cas k = 2 vaut par la déﬁnition de « cohérent » ; prouvez les cas k > 2 par induction. Pour prouver (b), utilisez (a). 2.5. Le raffinement canonique k-aire à la façon de Weisfeiler-Leman Déﬁnissons un foncteur F3 qui envoie une conﬁguration X = (Γ, c) à une conﬁguration cohérente F3 (X) = (Γ, c′ ). Comme F3 (X) sera un raﬃnement de X, nous aurons Iso(X, Y) = Iso(F3 (X), F3(Y)). L’algorithme 3, qui calcule F3 , est basé sur une idée de Weisfeiler et Leman (8) [WL]. Il s’agit d’itérer une procédure de raﬃnement. Si, dans une itération, aucun raﬃnement ne se produit – c’est-à-dire, si les classes d’équivalence du nouveau coloriage Ci sont les mêmes que celles de l’ancien coloriage Ci−1 – alors, (a) aucun raﬃnement ne se produira dans le futur, (b) le coloriage Ci−1 est déjà cohérent. 7. Voir la déﬁnition du §2.6. 8. Aussi appelé Lehman, mais [Ba] indique que le deuxième auteur préférait Leman. Deux transformations naturelles L → Л, Л → L peuvent ne pas être l’inverse l’une de l’autre. 1125–12 Si le coloriage C = C0 a r couleurs diﬀérentes du début, il est clair qu’il ne peut être raﬃné que \|Γ\|k − r fois. Alors, \|Γ\|k − r itérations sont suﬃsantes pour produire une conﬁguration cohérente. En particulier, si l’indexation est faite en temps logarithmique, et le vecteur dans le pas 6 de l’algorithme 3 est représenté comme un vecteur creux (puisque son nombre d’entrées non-nulles est au plus \|Γ\|), le temps pris par l’algorithme est O(k 2 \|Γ\|2k+1 log \|Γ\|). (En outre, [Ba, §2.8.3] aﬃrme une borne plus forte.) Les algorithmes de type Weisfeiler-Leman étaient autrefois regardés comme une approche plausible au problème de l’isomorphisme de graphes. Depuis [CFI], [EvP], il est clair qu’il ne se suﬃsent pas à eux-mêmes. Ils sont quand même un outil précieux. La version k-aire ici est due à Babai-Mathon [Ba3] et Immerman-Lander [ImL]. Algorithme 3 Weisfeiler-Leman pour les conﬁgurations k-aires. 1: fonction WeisfeilerLeman(Γ, k, c : Γk → C ) 2: C0 ← C ; c0 ← c ; i0 ← \|Γk \| − \|c(Γk )\| 3: pour i = 1 jusqu’à i0 4: Ii ← ∅ 5: pour ~x ∈ Γk j 6: ν ← ci−1 (~x), (\|{z ∈ Γ : ci−1 (~x (z)) = rj ∀ 1 ≤ j ≤ k}\|)~r∈C k i−1 7: ci (~x) = Indexeur (Ii , ν) ⊲ Indexeur est comme dans l’algorithme 2 8: 9: Ci ← indices de Ii retourner cn−i0 : Γk → Cn−i0 , (Ii )1≤i≤n−i0 ⊲ (Ii ) donne du sens à Cn−r 2.6. Graphes, hypergraphes et designs en blocs Nous savons déjà qu’un graphe est une paire (V, A), où V est un ensemble (« sommets ») et A est une collection de paires d’éléments de V (voire de sous-ensembles de V avec deux éléments, si le graphe est non orienté). Un graphe non orienté est dit régulier si le degré de tout sommet est le même ; un graphe orienté est dit birégulier si le degré sortant d+ (v) = \|{w ∈ V : (v, w) ∈ A}\| et le degré entrant d− (v) = \|{w ∈ V : (v, w) ∈ A}\| sont indépendants de v. (Pour V ﬁni, ils sont forcément la même constante.) Un graphe biparti est un triplet (V1 , V2 ; A) avec A ⊂ V1 × V2 . Un graphe biparti est semirégulier si le degré (9) d+ (v1 ) est indépendant de v1 ∈ V1 , et le degré d− (v2 ) est indépendant de v2 ∈ V2 . Exercice 2.17. — Soit X = (Γ, c) une conﬁguration cohérente classique homogène. (a) Soient C1 , C2 deux classes de couleur, et soit vert une couleur d’arêtes en C1 ×C2 . Alors, le graphe biparti (C1 , C2 ; Gvert ) est semirégulier. 9. Nous omettons les mots « entrant » et « sortant », puisqu’il est évident qu’il s’agit du degré entrant dans le cas de v1 et du degré sortant dans le cas de v2 . 1125–13 (b) Soit y ∈ Γ, et Li (y) = {x ∈ Γ : c(x, y) = i}. Soient lin, bis et terre trois couleurs d’arêtes. Alors, pour L1 = Llin (y) et L2 = Lbis (y), le graphe biparti (L1 , L2 ; Rterre ∩ (L1 × L2 )) est semirégulier. Exercice 2.18. — Soit X une conﬁguration cohérente classique homogène. Soient C1 , C2 deux classes de couleur. Soient vert une couleur d’arêtes en C1 × C2 et rouge une couleur d’arêtes en C2 × C2 . Soient B1 , . . . , Bm les composantes connexes de Grouge en C2 . Déﬁnissons le graphe biparti Y = (C1 , {1, . . . , m}; D) comme suit : (x, y) ∈ D ssi (x, y) ∈ Gvert pour au moins un y ∈ Bi . Alors Y est semirégulier. Solution — Notez que, pour y ∈ Bi et x ∈ C1 , (x, y ′ ) est vert pour au moins un y ′ ∈ Bi ssi il existe x0 = x, x1 , . . . , xm tels que (xi , xi+1 ) est rouge pour 0 ≤ i < m et (xm , y) est vert. Concluez par l’exercice 2.16a que tous les sommets en {1, . . . , m} ont le même degré en Y . De façon analogue, montrez que, pour x ∈ V1 et y ∈ Bi tels que (x, y) est rouge, le nombre de z ∈ Bi tels que (x, z) est rouge ne dépend pas de x, y ou i. Notons ce nombre q. Alors, le degré de tout v ∈ C1 est son degré en X, divisé par q. Par (a), il ne dépend donc pas de v. Un graphe biparti est complet (en tant que graphe biparti) si A = V1 × V2 . Un graphe biparti qui n’est ni vide ni complet est appelé non trivial. Un hypergraphe H = (V, A ) consiste en un ensemble V (« sommets ») et une collection A de sous-ensembles de V (« arêtes »), peut-être avec des sous-ensembles répétés. Un hypergraphe est dit u-uniforme si \|A\| = u pour tout A ∈ A . Il est dit régulier de degré r si tout v ∈ V appartient à exactement r ensembles A dans A . L’hypergraphe u-uniforme complet sur V est (V, {A ⊂ V : \|A\| = u}), où chaque ensemble A est compté une fois. Un coloriage des arêtes de l’hypergraphe complet est une application de {A ⊂ V : \|A\| = u} à un ensemble ﬁni C . Un block design équilibré (BDE) de paramètres (v, u, λ) est un hypergraphe avec \|V \| = v sommets, u-uniforme et régulier de degré r ≥ 1, tel que toute paire {v1 , v2 } de sommets distincts est contenue dans exactement λ ≥ 1 arêtes (« blocks »). Un block design dégénéré a la même déﬁnition, mais avec λ = 0, et la condition additionnelle d’être un hypergraphe régulier. (La régularité peut être déduite de la déﬁnition si λ ≥ 1.) Un block design est incomplet si u < v. Notons b le nombre \|A \| d’arêtes d’un BDE. Proposition 2.19 (Inégalité de Fisher (11) [F]). — Pour tout block design équilibré incomplet, b ≥ v. Il est aisé de voir que cette inégalité est vraie même pour les designs dégénérés. Les blocks designs admettent une généralisation. Un design t-(v, u, λ) est un hypergraphe (V, A ) u-uniforme avec v = \|V \| sommets tel que tout T ⊂ V de taille t est contenu dans exactement λ arêtes. Ici t ≥ 2 et λ ≥ 1. Nous écrivons toujours b = \|A \|. 11. Si, R. A. Fisher, le statisticien. Ici design vient d’experimental design. 1125–14 Proposition 2.20 ([RChW]). — Pour tout design t-(v, u, λ) et tout s ≤ min(t/2, v−u), nous avons b ≥ vs . 2.7. Schémas de Johnson Un schéma d’association est une conﬁguration cohérente classique (Γ, c : Γ2 → C ) telle que c(x, y) = c(y, x) ∀x, y ∈ Γ. (Il s’agit donc d’un sens du mot schéma qui n’a rien à voir avec les schémas de la géométrie algébrique.) Soient s ≥ 2 et r ≥ 2s + 1. Un schéma de Johnson J (r, s) = (Γ, c) est donné par Γ = Ss (Λ) = {S ⊂ Λ : \|S\| = s}, c(S1 , S2 ) = \|S1 \ (S1 ∩ S2 )\|, où Λ est un ensemble à r éléments. La relation Ri est bien sûr l’ensemble Ri = {(S1 , S2 ) : c(S1 , S2 ) = i}. Notons que nous avons déﬁni implicitement un foncteur de la catégorie d’ensembles Λ avec \|Λ\| = r à la catégorie de schémas de Johnson. Ceci est un foncteur plein ; autrement dit, les seuls automorphismes de J (r, s) sont ceux qui sont induits par Sym(Λ). 2.8. Identification de groupes et de schémas Il est une chose de démontrer que deux groupes G, H sont isomorphes, et une autre de construire un isomorphisme φ de façon explicite entre eux. Cette dernière tâche implique, au moins, de donner les images φ(g1), . . . , φ(gr ) de générateurs g1 , . . . , gr de G. Voyons un cas particulier qui nous sera crucial. Nous aurons un groupe de permutation G < Sym(Γ), et nous saurons qu’il est isomorphe au groupe abstrait Altm . Comment construire un isomorphisme ? Si m n’est pas trop petit en relation à n = \|Γ\|, il est connu que G doit être isomorphe que le groupe Altm à un groupe de permutation de la forme Alt(k) m , qui n’est autre m agissant sur l’ensemble Sk (Λ0 ) = {S ⊂ Λ0 : \|S\| = k} à k éléments, où Λ0 est un ensemble à m éléments. (12) En d’autres termes, il existe une bijection ι0 : Γ → Sk (Λ0 ) et un isomorphisme φ0 : G → Alt(Λ0 ) tels que ι0 (ω g ) = ι0 (ω)φ0 (g) . Le problème consiste à construire ι : Γ → Sk (Λ) et φ : G → Alt(Λ), calculables en temps polynomial, avec ces mêmes propriétés. Nous suivons [BLS]. Soient Υ ⊂ Γ × Γ l’orbitale la plus petite de G (hors la diagonale ({ω, ω} : ω ∈ Γ}) ; soit ∆ ⊂ Γ × Γ l’orbitale la plus grande. Nous supposerons que 12. Babai nomme les groupes Alt(k) m groupes de Johnson, par analogie avec les schémas de Johnson. Puisque Alt(k) n’est qu’un déguisement de Altm , ne faudrait-il pas appeler ce dernier groupe de m Ramerrez ? 1125–15 m > (k + 1)2 − 2, ce qui revient à dire que n n’est pas trop grand en relation à m. (13) Alors, φ(Υ) = R1 = {(S1 , S2 ) ∈ Sk (Λ0 ) : \|S1 ∩ S2 \| = k − 1}, (5) φ(∆) = Rk = {(S1 , S2 ) ∈ Sk (Λ0 ) : S1 ∩ S2 = ∅}. Déﬁnissons, pour (x, y) ∈ Υ, B(x, y) = {z ∈ Γ : (x, z) ∈ / ∆, (y, z) ∈ ∆}. Ceci est l’ensemble de tous les z tels que ι0 (z) intersecte ι0 (x) mais pas ι0 (y). Soit [ C(x, y) = Γ \ {r : (z, r) ∈ ∆(z)}. z∈B(x,y) Alors ι0 (C(x, y)) = {S ∈ Sk (Λ0 ) : S ∩ S ′ 6= ∅ ∀S ′ ∈ Sk (Λ0 ) t.q. S ′ ∩ ι0 (x) 6= ∅, S ′ ∩ ι0 (y) = ∅} = {S ∈ Sk (Λ0 ) : i ∈ S}, où i est l’élément de ι0 (x) qui n’est pas dans ι0 (y). Soit Λ la collection {C(x, y) : (x, y) ∈ Υ}, sans multiplicités. Nous pouvons calculer et comparer C(x, y) pour (x, y) donné, et calculer et indexer Λ, tout en temps polynomial. Nous calculons, aussi en temps polynomial, l’action de G sur Λ induite par l’action de G sur Υ. Ceci déﬁnit φ : G → Alt(Λ). Il y a une bijection naturelle j : Λ → Λ0 qui commute avec l’action de G : elle envoie C(x, y) à i, où i est l’élément de Λ0 tel que ι0 (C(x, y)) = {S ∈ Sk (Λ0 ) : i ∈ S}. Il est clair que, pour ω ∈ Γ, ω ∈ C(x, y) ssi j(C(x, y)) ∈ ι0 (ω). Ainsi, nous obtenons la bijection ι : Γ → Sk (Λ), donnée par ι(ω) = {γ ∈ Λ : ω ∈ γ}. Celle-ci satisfait ι (ω g ) = ι(ω)φ(g) . Les applications φ, ι sont donc celles que nous désirions ; nous avons construit un isomorphisme explicite entre G et Alt(Λ). Notons que cette même procédure nous permet de construire un isomorphisme explicite entre, d’un côté, un schéma d’association (§2.7) qu’on sait être isomorphe à un schéma de Johnson J (m, k), et, de l’autre côté, ce même schéma. 13. Si m ≤ (k + 1)2 − 2, alors n est si grand que m! = nO(log n) . En ce cas, nous pouvons enlever le groupe G (c’est-à-dire, dans l’application qui nous intéressera, un quotient G/N ) de façon brutale, comme dans le cas 2 de la preuve du théorème 2.4 (Luks). Nous pourrions aussi nous passer de la supposition m > (k + 1)2 − 2 au coût de quelques complications en ce qui suit. En particulier, φ(∆) ne serait pas Rk comme dans (5), sinon un autre Rj . 1125–16 3. LA PROCÉDURE PRINCIPALE input : G < Sym(Ω) x, y : Ω → Σ réduction de G/N à Altm′ m′ ≤ \|m\|/2 G transitif ? aligner coupe coupe ou Johnson ? coupe réduction J de G/N à Altm′ √ m′ ≪ m coupe ou relations ? rels. plénitude > 1/2 ? non G/N ∼ Altm ? blocs ∼ Γ k récursion n′ ≤ n/2 non oui m petit ? non oui k = 1? oui cas trivial non non certiﬁcats locaux symétrie > 1/2 ? non non récursion n′ < n oui : G/N ∼ Alt(Γ) G primitif ? Lemme des designs non oui oui une couleur domine ? output : IsoG (x, y) Fonction Isomorphisme-de-Chaı̂nes Weisfeiler - Leman k-aire oui x, y → relations k-aires sur Γ oui pullback 1125–17 3.1. Premiers pas : récursion à la façon de Luks G transitif ? non récursion n′ < n non récursion n′ ≤ n/2 oui G/N ∼ Altm ? oui m petit ? oui Les premiers pas de la procédure sont ceux de la preuve du Théorème 2.4 (Luks). En particulier, si G < Sym(Ω) n’est pas transitif, nous procédons exactement comme dans le cas non transitif de la preuve du Théorème 2.4. Bien qu’il soit possible que n = \|Ω\| ne décroisse que très légèrement, la récursion marche, puisque son coût est aussi très léger dans ce cas : nous n’avons qu’à subdiviser le problème selon les orbites de G. Supposons que G soit transitif. Nous savons que nous pouvons trouver rapidement un système de blocs minimal R = {Bi : 1 ≤ i ≤ r}, Bi ⊂ Ω (§2.1.2). Par Schreier-Sims, nous trouvons aussi, en temps polynomial, le sous-groupe N ⊳ G des éléments g ∈ G tels que Big = Bi pour tout i. Le groupe H = G/N agit sur R. Au lieu du Théorème 2.3 [BCP], nous utiliserons une conséquence de la Classiﬁcation des Groupes Finis Simples (CGFS). Elle a été dérivée pour la première fois par Cameron, puis raﬃnée par Maróti. Théorème 3.1 ([Cam], [Ma]). — Soit H < Sym(R) un groupe primitif, où \|R\| = r est plus grand qu’une constante absolue. Alors, soit (14) (a) \|H\| < r 1+log2 r , soit (b) il y a un M ⊳ H tel que R se subdivise (15) en un système de m blocs sur lequel k (k) M agit comme un groupe Altm , m ≥ 5. En plus, [H : M] ≤ r. La borne [H : M] ≤ r se déduit de m > 2, \|H\| ≥ r 1+log2 r , \|H\| ≤ m!s s!, ms ≤ r et [H : M] ≤ 2s s!, où s ≥ 1 est un paramètre dans Cameron-Maróti. Il est possible [BLS] de trouver en temps polynomial le sous-groupe normal M et les blocs de l’action de M. Nous avons déjà vu au §2.8 comment identiﬁer explicitement l’action de M avec celle de Alt(k) m . Par ailleurs, l’algorithme de Schreier-Sims nous permet de calculer \|H\| en temps polynomial, et donc nous dit aussi si nous sommes dans le cas (a). Si c’est le cas, nous 14. Pour nous, log2 désigne le logarithme en base 2, et non pas le logarithme itéré log log. 15. L’énoncé dans [Cam], [Ma] est plus fort : il décrit toute l’action de H sur R. À vrai dire, le groupe m s s M est isomorphe, en tant que groupe de permutation, à (Alt(k) m ) , s ≥ 1. Nous avons r = k . 1125–18 procédons comme dans le cas transitif de la preuve du Théorème 2.4. Nous réduisons ainsi le problème à r 1+log2 r instances du problème pour des chaı̂nes de longueur ≤ n/r. Si nous sommes dans le cas (b) nous commençons toujours par réduire le problème à [H : M] instances du problème avec M à la place de H : par l’équation (2) et comme dans l’équation (4), [ −1 IsoH (x, y) = IsoM x, yσ σ, σ∈S où S est un système de représentants des classes de M dans H. Si m ≤ C log n, où C est une constante, \|M\| = m! < mm ≤ mC log n ≤ (m′ )C log n , 2 où m′ = m . Donc, ici comme dans le cas (a), nous nous permettons de procéder k comme dans le cas transitif de la preuve du Théorème 2.4. Nous obtenons une réduction à ≤ r · (m′ )C log n = (m′ )O(log n) instances du problème pour des chaı̂nes de longueur n/m′ . Ceci est tout à fait consistant avec l’objectif d’avoir une solution en temps quasipolynomial en n (ou même en temps nO(log n) ). Il reste à savoir que faire si nous sommes dans le cas suivant : il y a un isomorphisme φ : G/N → Alt(Γ), \|Γ\| > C log n, C une constante. (Ici nous avons déjà (i) remplacé G par la préimage de M dans la réduction G → G/N, et, après cela, (ii) remplacé N par le stabilisateur des blocs dans la partie (b) du Théorème 3.1.) Ce cas nous occupera pour le reste de l’article. * Babai indique comment enlever la dépendance de CGFS à cette étape. Soient G et N comme avant, avec G transitif. Alors G/N est un groupe primitif agissant sur l’ensemble de blocs R. Si un groupe de permutations sur un ensemble R est tel que son action sur l’ensemble des paires d’éléments distincts de R est transitive, le groupe est dit doublement transitif. Or, un résultat de Pyber qui ne dépend pas de CGFS [Py2] nous dit qu’un tel groupe 2 est soit Alt(R), soit Sym(R), soit d’ordre ≤ \|R\|O(log \|R\|) . Si G/N est Alt(R) ou Sym(R), nous sommes dans le cas que nous discuterons d’ici jusqu’à la ﬁn. Si G/N est doublement transitif, mais n’est égal ni à Alt(R) ni à Sym(R), nous pouvons procéder comme dans le cas transitif de la preuve du Théorème 2.4, 2 puisque \|G/N\| ≤ r O(log r) , r = \|R\| ≤ n. (Babai propose aussi un traitement alternatif, même plus eﬃcace et élémentaire.) Supposons donc que G/N n’est pas doublement transitif. Alors la conﬁguration cohérente schurienne (§2.4) qu’elle induit n’est pas une clique. En conséquence, nous pouvons donner cette conﬁguration à la procédure Coupe-ou-Johnson (§5.2), et reprendre le ﬁl de l’argument à ce point-là. 1125–19 4. LA STRUCTURE DE L’ACTION DE Alt 4.1. Stabilisateurs, orbites et quotients alternants Nous aurons besoin de plusieurs résultats sur les épimorphismes G → Altk . Ils joueront un rôle crucial dans la méthode des certiﬁcats locaux (§6.1). Dans la version originale [Ba], ils ont aussi été utilisés dans le rôle joué par [BLS] dans cet exposé. Lemme 4.1. — Soit G < Sym(Ω) primitif. Soit φ : G → Altk un épimorphisme avec k > max(8, 2 + log2 \|Ω\|). Alors φ est un isomorphisme. Prouver ce lemme est à peu près un exercice en théorie des groupes ﬁnis ; il faut utiliser [BaPS, Prop. 1.22] pour le cas de socle abélien et la conjecture de Schreier pour le cas de socle non abélien. La conjecture de Schreier est un théorème, mais un théorème dont la preuve dépend, à son tour, de CGFS. Par contre, Pyber [Py] a donné une preuve du Lemme 4.1 qui n’utilise pas CGFS, avec une condition plus stricte : k > max(C, (log \|Ω\|)5 ), C constante. La dépendance de CGFS a donc été complètement enlevée de la preuve du théorème principal. Définition 4.2. — Soit G < Sym(Ω). Soit φ : G → Symk un homomorphisme dont l’image contient Altk . Alors x ∈ Ω est dit atteint si φ(Gx ) ne contient pas Altk . Lemme 4.3. — Soit G < Sym(Ω). Soit φ : G → Altk un épimorphisme avec k > max(8, 2 + log2 n0 ), où n0 est la taille de la plus grande orbite de G. (a) Si G est transitif, tout x ∈ Ω est atteint. (b) Au moins un x ∈ Ω est atteint. Preuve (esquisse) — (a) Ceci découle immédiatement du Lemme 4.1 si G est primitif, ou si K < ker(φ) pour K le stabilisateur d’un système de blocs minimal. Il reste le cas de φ : K → Altk surjectif. En général : Lemme.— Pour Ki arbitraires, K < K1 × · · · × Ks et un épimorphisme φ : K → S, S simple, il doit y avoir un i tel que φ se factorise comme suit : ψ K → Ki → S, ψ un épimorphisme. En utilisant ce lemme pour les restrictions Ki de K aux orbites de K, nous passons à une orbite Ki , et procédons par induction. (b) Soient Ω1 , . . . , Ωm les orbites de G, et soit Gi = G\|Ωi la restriction de G à Ωi . Par ψ le Lemme en (a), il doit y avoir un i tel que φ se factorise en G → Gi → Altk , ψ un épimorphisme. Alors, par (a), (Gx )ψ = ((Gi )x )ψ 6= Altk pour tout x ∈ Ωi . La proposition suivante jouera un rôle crucial au §6. Proposition 4.4. — Soient G < Sym(Ω) transitif et φ : G → Altk un épimorphisme. Soit U ⊂ Ω l’ensemble des éléments non atteints. (a) Supposons que k ≥ max(8, 2 + log2 n0 ), où n0 est la taille de la plus grande orbite de G. Alors (G(U ) )φ = Altk . 1125–20 (b) Supposons que k ≥ 5. Si ∆ est une orbite de G qui contient des éléments atteints, alors chaque orbite de ker(φ) contenue dans ∆ est de longueur ≤ \|∆\|/k. Rappelons que G(U ) = {g ∈ G : xg = x ∀x ∈ U} (stabilisateur de points). Preuve — (a) Il est facile de voir que G ﬁxe U en tant qu’ensemble. Alors, G(U ) ⊳ G, et donc (G(U ) )φ ⊳ Gφ . Or, Gφ = Altk . Supposons que (G(U ) )φ = {e}. Alors φ se factorise comme suit : ψ G → G\|U → Altk , puisque G(U ) est le noyau de G → G\|U . Ici ψ est un épimorphisme, et donc, par le Lemme 4.3 (b), il existe un x ∈ U tel que ((G\|U )x )ψ 6= Altk . Or ((G\|U )x )ψ = (Gx )φ = Altk , parce que x est dans U, c’est-à-dire non atteint. Contradiction. (b) Comme ∆ contient des éléments atteints et est une orbite de G, tout élément de ∆ est atteint. Soit N = ker(φ), x ∈ ∆. La longueur de l’orbite xN est xN = [N : Nx ] = [N : (N ∩ Gx )] = [NGx : Gx ] = = [G : Gx ] [G : NGx ] \|∆\| \|∆\| = . [Gφ : (Gx )φ ] [Altk : (Gx )φ ] Or, tout sous-groupe propre de Altk est d’indice ≥ k. Donc xN ≤ \|∆\|/k. 4.2. Le cas de grande symétrie G primitif ? oui k = 1? oui cas trivial non symétrie > 1/2 ? oui pullback Considérons le cas de G primitif. Nous pouvons supposer que G est isomorphe en tant que groupe de permutation à Alt(k) m , puisque nous avons déjà éliminé les autres cas au §3 (peut-être en passant à un groupe non primitif M ; le cas non primitif sera traité au §6). Comme nous l’avons vu au §2.8, nous pouvons construire une bijection ι entre Ω et l’ensemble Sk (Γ) des sous-ensembles avec k éléments d’un ensemble Γ. Cette bijection induit un isomorphisme φ : G → Alt(Γ). Si k = 1, alors Ω est en bijection avec Γ, et G ∼ Altn = Altm . Nous sommes donc dans le cas trivial : le groupe AutG (x) consiste en les éléments de Altn qui permutent les lettres de x de la même couleur, et IsoG (x, y) est non vide ssi x et y ont exactement le même nombre de lettres de chaque couleur – où, si aucune lettre n’est répétée ni en x ni en y, nous ajoutons la condition que la permutation de {1, . . . , n} qui induit x 7→ y soit dans Altn . Alors, soit G primitif, k > 1. 1125–21 Deux éléments γ1 , γ2 ∈ Γ sont des jumeaux par rapport à un objet si la transposition (γ1 γ2 ) le laisse invariant. Il est clair que les jumeaux forment des classes d’équivalence, et que, pour toute telle classe d’équivalence C, tout Sym(C) laisse l’objet invariant. Notre objet sera la chaı̂ne x (ou y) : γ1 , γ2 sont des jumeaux par rapport à x si, pour −1 tout i ∈ Ω, x(i) = x(τ φ (i)), où τ = (γ1 γ2 ). Nous pouvons donc déterminer facilement (et en temps polynomial) les classes d’équivalence en Γ (dites classes de jumeaux), et vériﬁer s’il y a une classe d’équivalence C de taille > \|Γ\|/2. Examinons cette possibilité puisque nous devrons l’exclure après. La classe C de taille > \|Γ\|/2 est évidemment unique et donc canonique. Si x a une telle classe et y ne l’a pas, ou si les deux ont de telles classes, mais de tailles diﬀérentes, alors x et y ne sont pas isomorphes. Si x, y ont des classes de jumeaux Cx , Cy de la même taille > \|Γ\|/2, nous choisissons ′ σ ∈ Alt(Γ) tel que Cx = (Cy )σ . (Nous supposons m > 1.) En remplaçant y par yσ , où σ ′ = φ−1 (σ −1 ), nous réduisons notre problème au cas Cx = Cy . (Voilà l’exemple le plus simple de ce que Babai appelle aligner ; nous avons aligné Cx et Cy .) Alors, soit C = Cx = Cy . La partition {C, Γ \ C} de Γ induit une partition {Ωj }0≤j≤k de Ω : ω ∈ Ωj ssi ψ(ω) contient k − j éléments de C et j éléments de Γ \ C. Il est aisé de montrer que αk−j (1 − α)j kj < 1/2 pour α ∈ (1/2, 1], 1 ≤ j ≤ k ; donc, \|Ωj \| < n/2 pour 1 ≤ j ≤ k. Nous avons réduit notre problème à celui de déterminer IsoH (x, y), où H = φ (Alt(Γ)C ). Ici le besoin de prendre un stabilisateur d’ensemble (à savoir, Alt(Γ)C ) ne pose aucun souci : nous engendrons H en prenant des préimages φ−1 (h1 ), . . . , φ−1(h5 ) de deux générateurs h1 , h2 de Alt(C) < Alt(Γ), deux générateurs h3 , h4 de Alt(Γ \ C) < Alt(Γ) et un élément h5 ∈ Alt(Γ) de la forme (γ1 γ2 )(γ3 γ4 ), où γ1 , γ2 ∈ C, γ3 , γ4 ∈ Γ \ C. (Si \|Γ\| < 8, le nombre de générateurs est moindre, et la discussion se simpliﬁe.) Notre problème se réduit à celui de déterminer IsoH ′ (x, y′ ) pour y′ = y et y′ = yh5 , où H ′ = φ−1 (Alt(C) × Alt(Γ \ C)) = φ−1 (hh1 , . . . , h4 i). −1 Comme C est une classe de jumeaux pour x, tout élément de φ−1 (Alt(C)) laisse x invariant. Si x\|Ω0 6= y\|Ω0 , alors IsoH ′ (x, y) = ∅. Soit alors x\|Ω0 = y\|Ω0 . Nous avons réduit notre problème à celui de déterminer IsoH ′ \|Ω′ (x\|Ω′ , y\|Ω′ ), où Ω′ = Ω \ Ω0 . Rappelons que H ′ \|Ω′ agit sur Ω′ avec des orbites de longueur \|Ωi \| < n/2. Nous procédons donc comme dans le cas non transitif de la méthode de Luks (preuve du Thm. 2.4). 1125–22 5. DES CHAÎNES AUX SCHÉMAS DE JOHNSON x, y → relations k-aires sur Γ Weisfeiler - Leman k-aire Lemme des designs une couleur domine ? oui coupe ou Johnson ? non récursion n′ ≤ n/2 Discutons maintenant le cas de G primitif et, plus précisément, G isomorphe à Alt(k) m , k ≥ 2. Maintenant nous pouvons supposer que nos chaı̂nes x, y n’ont pas de classes de jumeaux de taille > m/2. Les outils principaux que nous développerons (Lemme des designs, coupe-ou-Johnson) nous seront utiles, voire essentiels, aussi dans le cas de G imprimitif. Nous avons une bijection entre les éléments de Ω et {S ⊂ Γ : \|S\| = k}. Pour x : Ω → Σ donné, nous avons donc une structure relationnelle X = (Γ, (Ri )i∈Σ ) k-aire sur Γ : (x1 , . . . , xk ) ∈ Ri si x1 , . . . , xk sont tous diﬀérents et x(ω) = i, où ω est l’élément de Ω qui correspond à {x1 , . . . , xk }. Nous appliquons à X le foncteur F1 (§2.3), qui fait d’elle une structure de partition, puis le foncteur F2 (encore §2.3), qui nous donne une conﬁguration k-aire, et, ﬁnalement, le foncteur F3 déﬁni par Weisfeiler-Leman k-aire (§2.5). Nous obtenons ainsi un raﬃnement F3 (F2 (F1 (X))) = (Γ, cx : Ωk → C ) qui est une conﬁguration cohérente k-aire. Comme F1 , F2 , F3 sont des foncteurs, l’assignation de cx à x est canonique. Elle nous sera donc utile : si cx et cy ne sont pas isomorphes sous l’action de Altm , alors x et y ne sont pas isomorphes sous l’action de Alt(k) m non plus. Nous obtiendrons une conﬁguration cohérente classique de façon canonique à partir de cx (Lemme des designs). Soit cette nouvelle conﬁguration sera non triviale, soit nous obtiendrons un coloriage canonique sans couleur dominante, ce qui nous permettra immédiatement de réduire le problème à un certain nombre de problèmes pour des chaı̂nes plus courtes, comme dans l’algorithme de Luks. Supposons, alors, que nous disposons d’une conﬁguration cohérente classique non triviale assignée de façon canonique à x. La procédure Coupe-ou-Johnson nous donnera l’un ou l’autre de ces deux résultats : soit un découpage canonique de Γ, soit un schéma de Johnson plongé de façon canonique dans Γ. Dans un cas comme dans l’autre, avoir une telle structure canonique limite fortement l’ensemble d’isomorphismes et automorphismes possibles. Nous pourrons réduire G à un sous-groupe ∼ Altm′ , avec m′ ≤ m/2, √ dans le cas du découpage, ou m′ ≪ m, dans le cas de Johnson. Déjà m′ ≤ m/2 est suﬃsante pour une récursion réussie. 1125–23 5.1. Lemme des designs Étant donnés une conﬁguration X = (Γ, c : Γk → C ) et un paramètre 1/2 ≤ α < 1, une couleur i est dite α-dominante si c(γ, . . . , γ) = i pour ≥ α\|Γ\| valeurs de γ ∈ Γ. La classe de couleurs {γ ∈ Γ : c(γ, . . . , γ) = i} est, elle aussi, dite dominante. Par contre, si, pour toute couleur i, la classe {γ ∈ Γ : c(γ, . . . , γ) = i} est de taille < α\|Γ\|, le coloriage est dit un α-coloriage. Comme avant, deux éléments γ1 , γ2 ∈ Γ sont des jumeaux par rapport à une structure X (ici, une conﬁguration cohérente sur Γ) si (γ1 γ2 ) ∈ Aut(X). Proposition 5.1 (Lemme des designs). — Soit X = (Γ, c : Γk → C ) une conﬁguration cohérente k-aire, où 2 ≤ k ≤ \|Γ\|/2. Soit 1/2 ≤ α < 1. Supposons qu’il n’y a aucune classe de jumeaux dans Γ avec > α\|Γ\| éléments. Alors, au moins une des options suivantes est vraie : (1) (a) il existe x1 , . . . , xℓ ∈ Γ, 0 ≤ ℓ < k, tels que X~x n’a pas de couleur α-dominante ; (1) (b) il existe x1 , . . . , xℓ ∈ Γ, 0 ≤ ℓ < k − 1, tels que X~x a une couleur α-dominante C et (X~x )(2) [C] n’est pas une clique. La notation a été déﬁnie dans les sections 2.3 – 2.4 . En particulier, le 1-squelette est tout simplement un coloriage de Γ. (1) X~x Lemme 5.2 (Lemme de la grande clique). — Soit X = (Γ, c) une conﬁguration cohérente classique. Soit C ⊂ Γ une classe de couleurs avec \|C\| ≥ \|Γ\|/2. Si X[C] est une clique, alors C est une classe de jumeaux. Preuve — Supposons que C n’est pas une classe de jumeaux. Il y a donc un x ∈ Γ et une couleur (disons, azur) telle que c(x, y) est de cette couleur pour au moins un y ∈ C mais pas pour tous. Comme X[C] est une clique, x ∈ / C. Appelons la couleur de C carmin, et celle de x bronze. Soit B ⊂ Γ l’ensemble des éléments de couleur bronze. Il s’agit de construire un block design équilibré (§2.6) qui contredise l’inégalité de Fisher (Prop. 2.19). Déﬁnissons Ab = {y ∈ Γ : c(by) = azur} pour b ∈ B. Comme x ∈ B et c(xy) = azur pour au moins un y ∈ C, et c(xy) connaı̂t la couleur de y, tous les éléments de Ab sont carmin. Par la cohérence de X et la déﬁnition des nombres d’intersection (Def. 2.8), \|Ab \| = γ(azur, azur−1 , bronze), et donc \|Ab \| ne dépend pas de b. Comme nous l’avons dit au début, 1 ≤ \|Ax \| < \|C\| ; donc, 1 ≤ \|Ab \| < C pour tout b ∈ B. Montrez de façon similaire que, pour v ∈ C, la taille de {b ∈ B : v ∈ Ab } = {b ∈ B : c(bv) = azur} ne dépend pas de b. Comme X[C] est une clique, c(v, v ′) est de la même couleur pour tous v, v ′ ∈ C, v = v ′ ; appelons cette couleur doré. Montrez que {b ∈ B : v, v ′ ∈ Ab } = γ(azur, azur−1 , doré). Alors (C, {Ab}b∈B ) est un block design équilibré incomplet. 1125–24 En conséquence, par l’inégalité de Fisher, \|B\| ≥ \|C\|. Or, nous savons que \|C\| > \|Γ\|/2, B, C ⊂ Γ et B ∩ C = ∅. Contradiction. Preuve du Lemme des designs (Prop. 5.1) — Supposons que pour chaque ~x ∈ Ωℓ , 0 ≤ ℓ < k, C~x a une couleur α-dominante C(~x), et, en plus, si ℓ < k − 1, (X~x )(2) [C] est une clique. Nous arriverons à une contradiction. Soit C = C(vide). Comme \|C\| > α\|Γ\|, C est trop grande pour être un ensemble de jumeaux. Donc il existe u, v ∈ C, u 6= v, tels que τ = (uv) ∈ / Aut(X). Soit ~y de longueur τ minimale r entre les chaı̂nes satisfaisant c(~y ) 6= c(~y ). Par cette minimalité et les règles dans la déﬁnition 2.5, y1 , . . . yr sont tous distincts. En les permutant, nous pouvons assurer que u, v ∈ / {y1 , y2 , . . . , yr−2 }, et, sans perte de généralité, que soit (i) yr−1 6= u, v, yr = u, soit (ii) yr−1 = u, yr = v. Dans le cas (i), nous choisissons ~x = y1 , . . . , yr−1 , ℓ = r −1, et voyons que c~x (u) 6= c~x (v) ; dans le cas (ii), nous choisissons ~x = y1 , . . . , yr−2 , ℓ = r−2, et obtenons c~x (u, v) 6= cx (v, u). Nous aurons donc une contradiction avec notre supposition une fois que nous aurons prouvé que u, v ∈ C(~x). Le fait que u, v ∈ C(~x) s’ensuivra immédiatement de l’égalité C(~x) = C \{x1 , . . . , xℓ } ; cette égalité, à son tour, se déduit par itération du fait que, pour ~y de longueur ≤ k − 2 et ~x = ~y z, z ∈ Ω, (6) C(~x) = C(~y ) \ {z}. (2) Pourquoi (6) est-il vrai ? Nous sommes en train de supposer que X~y [C(~y )] est une clique, et que \|C(~y )\| > α\|Γ\| ≥ \|Γ\|/2. Donc, par le lemme de la grande clique, tous (2) les éléments de C(~y ) sont des jumeaux en X~y . En particulier, pour u ∈ C(~y ) \ {z}, c~x (u) = c~y (zu) ne dépend pas de u. Puisque le coloriage de sommets en C~x est un raﬃnement de celui en C~y (par la deuxième règle de la déﬁnition 2.5), il s’ensuit que, soit C(~x) = C(~y ) \ {z}, soit C(~x) ⊂ Γ \ C(~y ), soit C(~x) = {z}. Comme \|C(~x)\|, \|C(~y)\| > α\|Γ\| ≥ \|Γ\|/2, les deux dernières possibilités sont exclues. Nous appliquons le Lemme des designs (avec α = 1/2) à la conﬁguration cohérente k-aire X′ = F3 (F2 (F1 (X))), où X est donnée par x de la façon décrite au début de la section. Nous parcourons tous les tuples possibles ~x = (x1 , . . . , xℓ ) ∈ Γℓ , 0 ≤ ℓ < k, jusqu’à trouver un tuple pour lequel la première ou la deuxième conclusion du Lemme des designs est vraie. (1) Si la première conclusion est vraie, nous déﬁnissons cX = X~x et sautons à la section 5.3.1. Si la deuxième conclusion est vraie, nous passons au §5.2, ayant déﬁni X′′ = (2) (1) X~x [C], où C est la couleur α-dominante de X~x . 5.2. Coupe ou Johnson Nous avons une conﬁguration classique cohérente homogène non triviale X′′ = (Γ, c). (Nous rappelons que ceci est un coloriage c du graphe complet sur Γ tel que (a) les sommets ont leur couleur propre (« couleur diagonale »), (b) les arêtes (x, y), x 6= y, ne sont pas toutes de la même couleur, (c) la couleur c(x, y) de l’arête (x, y) détermine c(y, x), et (d) l’axiome de cohérence (2.8) se vériﬁe.) Nous voudrions trouver des structures qui dépendent canoniquement de X′′ et qui contraignent son groupe d’automorphismes. 1125–25 Il est raisonnable de s’attendre à ce que de telles structures existent : par le Théorème 3.1, si le groupe d’automorphismes est transitif, soit il est imprimitif (et donc il laisse une partition invariante), soit il est près d’être Alt(k) m , k ≥ 2, (qui laisse invariant un schéma de Johnson), soit il est petit (et donc le stabilisateur de quelques points aura des orbites petites, et ainsi nous donnera un coloriage sans couleur dominante). Le déﬁ est de trouver de telles structures, et de le faire canoniquement. Si X′′ n’est pas primitif (Déf. 2.14), la tâche est plutôt facile : soit r la couleur non diagonale la plus rouge telle que le graphe Gr = {(x, y) : x, y ∈ Γ, c(x, y) = r} est connexe ; par l’exercice 2.16, ceci donne une partition de Γ dans des ensembles de la même taille ≤ \|Γ\|/2. Théorème 5.3 (Coupe ou Johnson). — Soit X = (Γ, c) une conﬁguration classique cohérente uniprimitive. Soit 2/3 ≤ α < 1. En temps \|Γ\|O(1) , nous pouvons trouver — soit un α-découpage de Γ, — soit un schéma de Johnson plongé sur Γ0 ⊂ Γ, \|Γ0 \| ≥ α\|Γ\|, et un sous-groupe H < Sym(Γ) avec [Sym(Γ) : H] = \|Γ\|O(log \|Γ\|) tel que le découpage, voire le schéma, est canonique en relation à H. Le groupe H sera déﬁni comme un stabilisateur de points en Γ. La valeur 2/3 dans l’énoncé est assez arbitraire ; toute valeur > 1/2 serait valable. Une valeur proche à 1/2 aﬀecterait les constantes implicites. Preuve — Choisissons un x ∈ Γ arbitraire. Donnons à chaque y ∈ Γ la couleur de c(x, y). Ce coloriage est canonique en relation à Gx . S’il n’y a aucune classe de couleur C de taille > α\|Γ\|, la partition triviale (non-partition) de chaque classe nous donne un α-découpage de Γ, et nous avons ﬁni. Supposons, par contre, qu’il y ait une classe de couleur – disons, Clin – de taille > α\|Γ\|. Comme αn > n/2, la relation Rlin de cette couleur est non orientée (c(y, z) = lin ssi c(z, y) = lin). Le complément de Rlin (ou de toute autre relation) est de diamètre 2 (exercice). Soient x, z ∈ Γ tels que c(x, z) = lin, et soit y ∈ Γ tel que c(x, y), c(z, y) 6= lin. Appelons c(x, y) bis et c(z, y) terre. Considérons le graphe biparti (V1 , V2 ; A) avec sommets V1 = Clin, V2 = Cbis et arêtes Rterre ∩ (V1 × V2 ). Le graphe est non vide par déﬁnition et semirégulier par l’exercice 2.17b. Par homogénéité et cohérence, le nombre de y tels que c(y, w) est d’une couleur donnée c0 est indépendant de w. Donc, il est toujours ≤ (1 − α)n < n/2 pour c0 6= lin. Appliquant ceci à c0 = terre et V2 , nous voyons que le degré \|{v1 ∈ V1 : (v1 , v2 ) ∈ A}\| est < n/2, et donc, comme \|V1 \| > n/2, le graphe n’est pas complet. Nous appliquons donc la Proposition 5.7 à (V1 , V2 ; A) avec β = α\|Γ\|/\|V1\|. Notons que \|V2 \| ≤ β\|V1 \|. Nous travaillerons donc avec un graphe biparti (V1 , V2 ; A). La stratégie sera d’essayer, soit de rendre V2 plus petit (par au moins un facteur constant), soit de trouver des structures en lui. Soit ces structures nous permettront de réduire V2 quand même, soit 1125–26 elles nous aideront à découper V1 , ou à trouver un schéma de Johnson assez grand sur V1 . Tout d’abord, nous devrons borner la symétrie en V1 , c’est-à-dire réduire, voire éliminer les jumeaux. Il y a deux raisons à ceci. — Même si nous découvrions une structure assez riche en V2 , cela impliquerait peu ou rien sur V1 si beaucoup d’éléments de V1 se connectent à V2 de la même façon. — Si V2 est petit, nous colorierons chaque sommet de V1 par son ensemble de voisins en V2 . Ceci nous donnera un coloriage canonique en relation à G(V2 ) . Or, dans ce coloriage, deux sommets en V1 auront la même couleur ssi ils sont des jumeaux ; donc, si aucune classe de jumeaux en V1 n’a > α\|V1 \| éléments, nous aurons un α-coloriage. Exercice 5.4. — Soit (V1 , V2 ; A) un graphe biparti semirégulier et non trivial. Alors, aucune classe de jumeaux en V1 n’a plus de \|V1 \|/2 éléments. Solution — Nous assurons que \|A\| ≤ \|V1 \|\|V2 \|/2 en prenant le complément s’il est nécessaire. Soient d2 le degré des sommets en V2 et S une classe de jumeaux en V1 . Montrez que d2 ≥ \|S\|, et donc \|A\| ≥ \|S\|\|V2\|. Exercice 5.5. — Soit (V1 , V2 ; A) un graphe biparti sans jumeaux en V1 . Soient V2 = C1 ∪ C2 , C1 ∪ C2 = ∅. Montrez que, pour au moins un i = 1, 2, il n’y a aucune classe de ≥ \|V1 \|/2 + 1 jumeaux en V1 dans le graphe (V1 , Ci ; A ∩ (V1 × Ci )). Exercice 5.6. — Soit X = (Γ, c) une conﬁguration cohérente. Soient C1 , C2 deux classes de couleurs en Γ. Soit brun une couleur d’arêtes en A × B. Alors, pour x, y ∈ C1 , la couleur c(x, y) détermine si x et y sont des jumeaux dans le graphe biparti (C1 , C2 ; Gbrun ). Proposition 5.7 (Coupe ou Johnson biparti, ou « Una partita a poker ») Soit X = (V1 , V2 ; A) un graphe biparti avec \|V2 \| < β\|V1 \|, où 2/3 ≤ β < 1, et tel qu’aucune classe de jumeaux en V1 n’ait plus de 2\|V1\|/3 éléments. Alors, nous pouvons trouver, en temps \|V1 \|O(1) , — soit un β-découpage de V1 , — soit un schéma de Johnson plongé sur V0 ⊂ V1 , \|V0 \| ≥ β\|V1\|, et un sous-groupe H < G, G = Sym(V1 ) × Sym(V2 ), avec [G : H] = \|V1 \|O(log \|V1 \|) tel que le découpage, voire le schéma, est canonique en relation à H. La condition sur les classes de jumeaux ici était remplie (même avec 1/2 à la place de 2/3) à la ﬁn de la preuve du Thm. 5.3, grâce à l’exercice 5.4. En ce qui concerne le temps de la procédure, nous expliciterons quelques détails qui pourraient ne pas être évidents. Ce qui sera le détail le plus délicat est l’indice [G : H]. Le groupe H sera déﬁni comme un stabilisateur de points ; nous devons bien contrôler le nombre de points que nous stabilisons. 1125–27 Esquissons la stratégie générale de la preuve. Ce que nous voulons est une réduction à la Proposition 5.8, “Coupe-ou-Johnson cohérent”. Nous pouvons produire une conﬁguration cohérente classique sur V1 ∪V2 à partir du graphe X, tout simplement en utilisant Weisfeiler-Leman. Ce qui demande de la ruse est de garantir que la restriction X[C2 ] à la classe de couleurs dominante (s’il y a une) soit non triviale. Pour obtenir une conﬁguration non-triviale sur C2 , nous noterons que le graphe X induit lui-même une relation d-aire sur C2 , où d est au plus le degré de la majorité d’éléments de V1 (si telle chose existe ; sinon, les degrés nous donnent une partition de V1 ). Si la relation est triviale, dans le sens de contenir toutes les d-tuples d’éléments distincts dans C2 , nous obtenons un schéma de Johnson. Si elle est non triviale mais contient beaucoup de jumeaux, elle nous donne une manière de descendre à un C2 plus petit. S’il n’y a pas beaucoup de jumeaux, nous utilisons le Lemme des Designs (supplémenté par un lemme standard sur les designs) pour obtenir une conﬁguration cohérente classique non-triviale sur C2 , ce qui était à trouver. Preuve — Si \|V1 \| ≤ c, où c est une constante, nous colorions chaque v ∈ V1 par lui-même. Ce coloriage est canonique en relation à H = {e} ; autrement dit, il n’est pas canonique du tout. Peu importe : trivialement, \|G\| ≤ (c!)2 ≤ \|V1 \|O(log \|V1 \|) . Nous pouvons donc supposer que \|V1 \| > c. Si \|V2 \| ≤ (6 log \|V1 \|)3/2 (disons), alors, par la discussion ci-dessus, nous obtenons un (2/3)-coloriage de V1 (et donc : un (2/3)-découpage de V1 ). Ce coloriage est canonique en relation à un H d’indice 3 3 \|V2 \|! ≤ \|V2 \|\|V2 \| ≤ (6 log \|V1 \|) 2 (6 log \|V1 \|) 2 ≪ \|V1 \|log \|V1 \| . Nous pouvons donc supposer que \|V2 \| > (6 log \|V1 \|)3/2 . Notre première tâche est d’éliminer les jumeaux. Nous divisons V1 dans ses classes de jumeaux et colorions chaque v ∈ V1 par son nombre de jumeaux et par son degré dans le graphe (V1 , V2 ; A). Nous obtenons un β-découpage de V1 , sauf s’il y a un entier d1 tel que l’ensemble V1′ des sommets v sans jumeaux et de degré d1 est de taille \|V1′ \| > β\|V1\|. Supposons dorénavant que cela est le cas. Comme \|V1′ \| > \|V2 \| et qu’il n’y a pas de jumeaux en V1′ , nous voyons que 1 < d1 < \|V2 \| − 1 ; nous pouvons supposer que d1 ≤ \|V2 \|/2 en remplaçant A par son complément, si nécessaire. Soit H = (V2 , A ) l’hypergraphe dont les arêtes sont les voisinages en (V1 , V2 ; A) des sommets dans V1′ . (Elles sont toutes contenues en V2 .) L’hypergraphe est d1 -uniforme. Comme il n’y a pas de jumeaux dans V1′ , il n’y a pas d’arêtes identiques. Si H est l’hypergraphe complet d1 -uniforme, alors V1′ peut être identiﬁé avec le schéma de Johnson Sd1 (V2 ). (Scoppia in un pianto angoscioso e abbraccia la testa di Johnson.) Supposons alors que H n’est pas complet. Nous voudrions avoir un coloriage canonique sur V2d pour un d ≪ l, l = (log \|V1′ \|)/ log \|V2 \|, tel que les éléments de V2 ne soient pas tous jumeaux. Si d1 ≤ 6⌈l⌉, nous déﬁnissons d = d1 et colorions {(v1 , . . . , vd ) ∈ V2d : {v1 , . . . , vd } ∈ H } en écarlate, et tout le reste en gris. Supposons, par contre, que d1 > 6⌈l⌉. Soit d = 6⌈l⌉. Nous colorions ~v = (v1 , . . . , vd ) en gris si les vi ne sont pas tous distincts ; dans le cas contraire, nous donnons à ~v la 1125–28 couleur (7) \|{H ∈ H : {v1 , . . . , vd } ⊂ H}\|. Cette opération de coloriage peut être faite en temps de l’ordre de log \|V1′ \| 6 d1 ≤ \|V1 \| · \|V2 \|d = \|V1 \| · \|V2 \| log \|V2 \| = \|V1 \| · \|V1′ \|O(1) = \|V1 \|O(1) . \|V1 \| · d Si les tuples avec v1 , . . . , vd distincts n’avaient pas tous la même couleur λ, nous aurions un design d − (\|V2 \|, d1, λ) avec \|V1′ \| arêtes. Donc, par la Proposition 2.20, \|V1′ \| ≥ \|V2 \| pour s = 3⌈l⌉. Comme \|V2 \| ≥ (6 log \|V1′ \|)3/2 et \|V1′ \| peut être supposé plus grand s qu’une constante, s s log \|V1′ \| \|V2 \| \|V2 \| \|V2 \| 1/3 3l log \|V2 \| ≥ > > \|V2 \| = \|V2 \| = \|V1′ \|, s s 6l ce qui donne une contradiction. Donc, pour d1 arbitraire, les tuples avec v1 , . . . , vd distincts n’ont pas tous la même couleur ; en d’autres termes, les éléments de V2 ne sont pas tous jumeaux en relation à notre nouvelle structure d-aire. S’il y a une classe S de jumeaux de taille > \|V2 \|/2, alors, par l’exercice 5.5, au moins un des deux graphes (V1′ , S; A ∩ (V1′ × S)), (V1′ , V2 \ S; A ∩ (V1′ × (V2 \ S))) n’a aucune classe de > \|V1′ \|/2 + 1 jumeaux dans V1 . Comme 2\|V1′ \|/3 ≥ \|V1′ \|/2 + 1 pour \|V1 \| ≥ 8, nous appliquons la Proposition 5.7 elle-même à un de ces deux graphes (disons, celui sur V1′ × S si les deux sont valables), et terminons. (Peut-être que la taille de V2 est descendue seulement à \|V2 \| − 1, mais tous nos choix ont été canoniques – des non-choix, si l’on veut – donc gratuits. Nous n’avons perdu que du temps ; pour être précis, \|V1\|O(1) de temps, ce qui est acceptable.) Alors, nous avons un coloriage de V2d en relation auquel il n’y a aucune classe de jumeaux en V2 de taille > \|V2 \|/2. Nous appliquons les foncteurs F1 , F2 , F3 (WeisfeilerLeman) à ce coloriage. Puis nous utilisons le Lemme des designs (Prop. 5.1) avec α = 2/3. Nous trouvons les éléments x1 , . . . , xℓ ∈ V2 (ℓ = d − 1 ou ℓ = d − 2) dans l’énoncé de la Proposition 5.1 par force brute, en temps proportionnel à \|V2 \|d = \|V1 \|O(1) . Nous les ﬁxons, et nous imposons que H ﬁxe x1 , . . . , xℓ , ce qui a un coût de \|V1 \|O(1) , dans le sens où [G : Gx1 ,...,xℓ ] ≤ \|V2 \|d = \|V1 \|O(1) . Si nous sommes dans le premier cas du Lemme de designs (pas de couleur dominante), nous cueillons les classes de couleur, en commençant par la plus rouge (interprétez la quantité en (7) comme une longueur d’onde), jusqu’à avoir une union des classes S ⊂ V2 avec \|V2 \|/3 < \|S\| ≤ 2\|V2 \|/3. (Ceci marche s’il n’y a aucune classe de taille > \|V2 \|/3 ; si telles classes existent, nous déﬁnissons S comme la classe la plus grande de ce type.) Nous appliquons l’exercice 5.5, et obtenons un graphe (V1′ , V2′ , A ∩ (V1′ ∩ V2′ )) remplissant les conditions de notre Proposition 5.7 avec V2′ = S ou V2′ = V2 \ S, et donc \|V2′ \| ≤ α\|V2\|. Donc, nous appliquons la Proposition 5.7 à ce graphe ; la récursion marche. (Il est important ici que \|V2′ \| ≤ α\|V2\|, puisque nous avons déjà encouru un coût considérable (\|V1 \|O(1) ) dans l’indice.) 1125–29 Restons donc dans le deuxième cas du Lemme des designs : nous avons un coloriage de V2 avec une classe de couleurs C ⊂ V2 telle que \|C\| ≥ 2\|V2\|/3, et une conﬁguration cohérente homogène classique Y non triviale sur C. Nous déﬁnissons un graphe avec des sommets V1′ ∪ V2 , où V1′ est de couleur nacrée et V2 vient d’être colorié par le Lemme des Designs ; les arêtes seront non pas seulement celles en A ∩ (V1′ × V2 ) (coloriées en noir) mais aussi les arêtes entre les éléments de V2 , dans les couleurs données par Y. Nous appliquons les raﬃnements F1 , F2 et F3 (Weisfeiler-Leman) à ce graphe, et obtenons une conﬁguration cohérente X. La conﬁguration X[V2 ] est un raﬃnement de Y. Si elle n’a pas de couleur α-dominante, nous réduisons notre problème à celui pour (V1′ , V2′ , A ∩ (V1′ ∩ V2′ )), \|V2′ \| ≤ α\|V2 \|, comme avant ; nous pouvons appliquer la proposition 5.7 à un tel graphe sans changer β parce que 2 \|V2′ \| ≤ \|V2 \| ≤ β\|V2\| < β\|V1′ \|. 3 La récursion marche ici aussi parce que \|V2′ \| ≤ 2\|V2 \|/3 : il est important que V2′ soit plus petit que V2 par un facteur constant, puisque le coût entraı̂né jusqu’à maintenant dans l’index [G : H] est déjà considérable (\|V1\|O(1) ). Supposons donc qu’il y a une classe de couleurs (2/3)-dominantes C2 dans X[V2 ]. Elle doit être un sous-ensemble de C car 2/3 + 2/3 > 1. La restriction de X[C2 ] n’est pas une clique : si elle l’était, la restriction de Y à C2 l’aurait été aussi, et cela est impossible par l’exercice 2.15. Nous pouvons supposer qu’il existe une classe de couleurs C1 ⊂ V1′ en X1 qui satisfait \|C1 \| > β\|V1 \| ; sinon, nous avons un β-coloriage de V1 , et pouvons ﬁnir. Le fait que \|C1 \| > β\|V1\| implique que \|C1 \| > \|V2 \| ≥ \|C2 \|. Nous pouvons supposer aussi que les arêtes de X en C1 × C2 ne sont pas toutes de la même couleur. Si elles l’étaient, il y aurait une classe de ≥ \|C1 \| > β\|V1\| > \|V1 \|/2 + 1 jumeaux en V1 dans le graphe (V1 , C2; A∩(V1 ×C2 )), dont X[V1 ×C2 ] est un raﬃnement. Dans ce cas, par l’exercice 5.5, nous aurions une réduction à (V1 , V2 \ C2 ; A ∩ (V1 × (V2 \ C2 ))), et nous pourrions ﬁnir en utilisant la Proposition 5.7 de façon récursive. Ainsi, nous avons tout réduit à la Proposition 5.8 : nous l’appliquons à X[C1 ∪ C2 ]. Nous obtenons, soit un (1/2)-découpage de C1 , soit un graphe biparti (W1 , W2 ; A′ ), W1 ⊂ C1 , W2 ⊂ C2 , avec \|W1 \| ≥ \|C1 \|/2, \|W2 \| ≤ \|C2 \|/2 ≤ \|V2 \|/2, tel qu’aucune classe de jumeaux en W1 n’a plus que \|W1 \|/2 éléments. Nous pouvons supposer que \|W1 \| > β\|V1 \|, parce que, dans le cas contraire, nous avons obtenu un β-découpage de W1 . Alors, \|W2 \| < \|W1 \|/2. Nous pouvons, alors, faire de la récursion : nous appliquons la Proposition 5.7 avec (W1 , W2 ; A′ ) à la place de (V1 , V2 ; A). La récursion se ﬁnit après pas plus que O(log \|V2 \|) pas puisque \|W2 \| ≤ \|V2 \|/2. Si la taille de W1 (ou de V1 ) décroı̂t en dessous de β\|V1\| (pour la valeur originale de \|V1 \|), alors nous avons obtenu un β-découpage de V1 . Comme nous l’avons vu, Coupe ou Johnson biparti utilise Coupe ou Johnson cohérent. À son tour, Coupe ou Johnson cohérent se réduira à Coupe ou Johnson biparti pour un graphe biparti (V1 , V2 ; A) avec V2 de taille au plus une moitié de la taille du V2 original. 1125–30 Proposition 5.8 (Coupe ou Johnson cohérent). — Soit X = (C1 ∪ C2 ; c) une conﬁguration cohérente avec des classes de couleurs de sommets C1 , C2 , où \|C1 \| > \|C2 \|. Supposons que ni c\|C1 ×C2 ni c\|C2 ×C2 est une fonction constante. Alors, nous pouvons trouver, en temps \|C1 \|O(1) , soit — un (1/2)-découpage de C1 , ou — un graphe biparti (V1 , V2 ; A), Vi ⊂ Ci , \|V1 \| ≥ \|C1 \|/2, \|V2 \| ≤ \|C2 \|/2, tel que toute classe de jumeaux en V1 contient au plus \|V1 \|/2 éléments, et un élément y ∈ C2 , tel que le découpage, voire le graphe biparti, est canonique en relation à Gy , où G = Sym(C1 ) × Sym(C2 ). Il va de soi que dire que c\|C2 ×C2 est constant équivaut à dire que X[C2 ] est une clique. Preuve — Si la restriction X[C1 ] était une clique, alors, par cohérence, pour toute couleur en C1 × C2 – pourpre, disons – les voisinages dans (C1 , V2 ; Gpourpre ) des sommets en C2 nous donneraient un block design équilibré (et peut-être dégénéré) sur C1 . Le design est incomplet parce que c n’est pas monochrome sur C1 × C2 . L’inégalité de Fisher nous donne que \|C2 \| ≥ \|C1 \|, en contradiction avec nos suppositions. Donc, X1 [C1 ] n’est pas une clique. Si X[C1 ] n’est pas primitive, la plus rouge de ses relations non connexes nous donne un (1/2)-découpage canonique de V1 , par l’exercice 2.16. Nous pouvons donc supposer que X[C1 ] est primitif. Nous avons deux cas à considérer : X[C2 ] primitive et X[C2] imprimitive. Supposons d’abord que X[C2 ] est imprimitive. La relation non connexe la plus rouge dans X[C2 ] nous donne une partition de C2 dans des ensembles B1 , . . . , Bm , m ≥ 2, tous de la même taille ≥ 2. Nous avons donc trouvé une structure en C2 , et nous l’utiliserons, soit pour découper C1 , soit pour réduire \|C2\| par un facteur constant. Le premier pas consiste à montrer qu’il n’y a pas de jumeaux dans C1 . Comme notre conﬁguration est cohérente, la couleur d’une arête en C1 sait si ses sommets sont des jumeaux en relation à C2 (Ex. 5.6) ; donc, s’il y avait des jumeaux dans C1 en relation à C2 , nous aurions, soit qu’une des couleurs d’arêtes en C1 donne une relation non connexe – ce qui contredit le fait que X[C1 ] est uniprimitive – soit que tous les éléments de C1 sont des jumeaux en relation en C2 . Dans ce dernier cas, par l’exercice 5.4, c\|C1 ×C2 serait monochrome, ce qui n’est pas le cas. En conclusion, il n’y a pas de jumeaux dans C1 en relation à C2 . Notre intention est d’appliquer l’exercice 2.18 pour obtenir un graphe biparti contracté C1 × {1, 2, . . . , m} avec m ≤ \|C2 \|/2. Nous devons seulement faire attention à ce que ce graphe ne soit pas trivial. Soit dk le degré de tout w ∈ C2 dans le graphe biparti (C1 , C2; Gk ) pour une couleur k donnée, où Gk consiste en les arêtes de couleur k. (Par l’ex. 2.17a, le degré dk ne dépend pas de w.) Si dk ≤ \|C1 \|/2 pour tout k, nous ﬁxons un w ∈ C2 (non canonique) et obtenons un (1/2)-coloriage de C1 en assignant la couleur c(x, w) au sommet x ∈ C1 . Supposons donc qu’il y a une couleur – que nous appellerons violet – telle que 1125–31 dviolet > \|C1 \|/2. S’il y a un 1 ≤ i ≤ m tel qu’il n’y a aucune classe de plus que \|C1 \|/2 jumeaux dans C1 en relation à Bi , nous ﬁxons un élément y ∈ Bi d’un tel i (non canoniquement), ﬁxant ainsi cet i. De cette façon, nous obtenons une réduction au graphe biparti (C1 , Bi ; Gviolet ∩ (C1 × Bi )). Supposons que cela n’est pas le cas. Donc, pour chaque i, il existe une classe Ti ⊂ C1 de jumeaux en relation à Bi telle que \|Ti \| > \|C1 \|/2. Pour chaque w ∈ Bi , les arêtes de w à tout v ∈ Ti sont de la même couleur ; alors, elles doivent être violettes. Soit vert une couleur d’arêtes en C1 × C2 qui ne soit pas violet. Alors, le graphe X = (C1 , {1, . . . , m}; D) dans l’exercice 2.18 n’est pas vide ; comme (vi , i) est violet pour tout v ∈ Ti , X n’est pas complet non plus. Comme X est birégulier, il n’y a aucune classe de jumeaux en C1 en relation à {1, . . . , m} avec > \|C1 \|/2 éléments (ex. 5.4). Nous avons donc tout réduit à un graphe biparti X du type que nous désirions. Considérons maintenant le cas de X[C2 ] primitive (16) . Fixons un y ∈ C2 arbitraire (non canoniquement). Nous pouvons supposer qu’il y a une couleur – disons, violet – telle que dviolet > \|C1 \|/2, puisque, sinon, les couleurs des arêtes qui connectent les éléments de C1 avec y nous donneraient un (1/2)-coloriage de C1 . Écrivons V1 = Lviolet (y) = {x ∈ C1 : c(x, y) = violet}. Donc \|V1 \| > \|C1 \|/2. Soit bleu une couleur d’arêtes en X[C2 ] telle que le degré de Gbleu est (positif et) < \|C2 \|/2 ; une telle couleur existe parce que X[C2 ] n’est pas une clique. (S’il y a plusieurs couleurs comme cela, nous choisissons la plus bleue d’entre elles.) Alors, V2 = Lbleu (y) ⊂ C2 satisfait 1 ≤ \|V2 \| < \|C2 \|/2. Le graphe biparti (V1 , V2 ; Gviolet ∩ (W × U)) est semirégulier par l’exercice 2.17b. Il est non vide parce que, pour tout u ∈ V2 , \|Lviolet (u)\| > \|C1 \|/2, et donc Lviolet (u) ∩ V1 6= ∅. S’il était complet, nous aurions V1 ⊂ Lviolet (u) pour tout u ∈ V2 ; comme \|V1 \| = \|Lviolet (y)\| = \|Lviolet (u)\|, ceci impliquerait que V1 = Lviolet (u). Or, cela voudrait dire que y et u sont des jumeaux dans le graphe (C1 , C2 ; Gviolet ). Par le même argument qu’avant (basé sur l’exercice 5.6), la primitivité de X[C2 ] et le fait que c\|C1 ×C2 ne soit pas monochrome impliquent qu’il n’y a pas de jumeaux dans C2 en relation au graphe (C1 , C2 ; Gviolet ). Donc, (V1 , V2 ; Gviolet ∩ (V1 × V2 )) n’est pas complet. Par l’exercice 5.4, nous obtenons qu’aucune classe de jumeaux dans (V1 , V2 ; Gviolet ∩ (V1 × V2 )) n’a plus de \|V2 \|/2 éléments. Nous avons donc terminé. 16. Le problème dans la preuve originale de Babai était à ce point précis. Ce qui suit est un argument alternatif proposé par lui (col rumore sordo di un galoppo) lorsque cet article était en train d’être édité. Il est plus concis et élégant que l’argument d’origine, en plus d’être correct. Avant, la preuve faisait deux fois (ou plus) recours à la proposition elle-même, ce qui faisait croı̂tre l’indice [G : H] de façon catastrophique. 1125–32 5.3. Récursion et réduction une couleur domine ? non récursion n′ ≤ n/2 réduction de G/N à Altm′ m′ ≤ \|m\|/2 aligner G transitif ?, etc. réduction de G/N à Altm′ √ m′ ≪ m 5.3.1. Le cas sans couleurs dominantes. — Nous sommes dans le cas dans lequel un coloriage cX : Γ → C n’a pas de couleur dominante. Ici cX est l’image d’une structure X sous un foncteur F qui commute avec l’action de H(x1 ,...,xℓ ) , où H = Alt(Γ), xi ∈ Γ. Le fait que cX n’a pas de couleur dominante nous servira pour trouver ou écarter ses isomorphismes possibles en H~x = H(x1 ,...,xℓ ) . Pour trouver ou écarter des isomorphismes en tout H = Alt(Γ), nous n’avons qu’à travailler avec un ensemble de représentants {σ1 , . . . , σs }, s ≤ \|Γ\|ℓ = mℓ , des classes de H~x dans H, et à faire l’union de IsoH~x (cX , cYi ) −1 pour Yi = Yσi : [ (8) IsoH (X, Y) = IsoH~x (X, Yi ) σi , IsoH~x (X, Yi ) ⊂ IsoH~x (cX , cYi ). 1≤i≤s Ceci est similaire à l’équation (4), en §2.2. Le coût de la procédure est multiplié par s ≤ mℓ . Si le coloriage cX n’est pas une permutation (en H~x ) du coloriage cYi , alors IsoH~x (cX , cYi ) = ∅. Supposons, par contre, qu’il y a au moins un τi ∈ H~x tel que cX = cτYii . (Nous disons que τi aligne cX et cYi .) Il est trivial de trouver τi . Or IsoH~x (X, Yi ) = IsoH~x (X, Yτi i ) τi−1 ⊂ AutH~x (cX )τi−1 . Comme cX n’a pas de couleur dominante, ceci est assez contraignant, ce que nous voulions. Appliquons cette procédure générale au cas de G primitif que nous sommes en train de discuter. Il y a une bijection ι : Ω → {S ⊂ Γ : \|S\| = k} ; donc, cX induit un coloriage P c′ : Ω → {(ki )i∈C : ki ≥ 0, i ki = k}. Nous sommes dans une situation similaire à celle de la ﬁn du §4.2, mais en mieux : il est facile de montrer que, comme aucune classe de couleur de c possède plus de α\|Γ\| éléments, aucune classe de couleur de c′ possède plus de α\|Ω\| éléments. Nous procédons alors comme dans le cas intransitif de la preuve de Luks (Thm. 2.4), ce qui réduit le problème à ≤ n problèmes d’isomorphisme de chaı̂nes pour des chaı̂nes de longueur ≤ αn et de longueur totale ≤ n. Le dernier pas (lifting, « relèvement ») consiste à trouver des éléments de G qui induisent τi . Étant donnée une bijection ι, ceci est trivial. 1125–33 5.3.2. Le cas du découpage. — Considérons maintenant un α-découpage (ﬁn de §2.3) d’un ensemble de sommets Γ. Ce découpage sera donné canoniquement, à savoir, en tant que l’image d’une structure X sous un foncteur, tout comme le coloriage au §5.3.1. Nous pouvons supposer que le découpage a une classe de couleurs C dominante (\|C\| > α\|Γ\|, α > 1/2), puisque, dans le cas contraire, nous pouvons passer au §5.3.1. Nous voulons savoir quels éléments de Alt(Γ) respectent le α-découpage ; ceci nous aidera à contraindre les isomorphismes de X, tout comme en (8). Par la déﬁnition de α-découpage, C est partitionné en ℓ ≥ 2 ensembles de la même taille ≥ 2. Les seules permutations en Altm0 , m0 = C, qui sont permises sont celles qui respectent la partition. Le groupe qui respecte la partition est isomorphe à Altm0 /ℓ . Nous avons donc réduit notre problème à un problème avec m′ = m0 /ℓ ≤ m/2. Après avoir résolu ce problème, nous travaillons – comme dans le §5.3.1 – sur les autres classes de couleurs. Étant donnés deux α-découpages, nous vériﬁons si les partitions des deux découpages ont le même nombre d’ensembles de la même taille pour chaque couleur, puis nous alignons les deux découpages, et procédons exactement comme pour le problème de l’automorphisme. 5.3.3. Le cas du schéma de Johnson. — Soit donné un schéma de Johnson sur un ensemble de sommets Γ, ou plutôt deux schémas de Johnson J (mi , ki ), 2 ≤ ki ≤ mi /2, sur des ensembles de sommets Γ1 , Γ2 de la même taille. Nous avons vu au §2.8 comment identiﬁer Γi (là, Ω) explicitement avec les ensembles de taille ki d’un ensemble Λi (là, Γ) de taille mi . Si k1 6= k2 et m1 6= m2 , nos structures ne sont pas isomorphes. Si k1 = k2 et m1 = m2 , nous établissons une bijection entre Λ1 et Λ2 et alignons les deux structures. √ Nous avons réduit notre problème à un problème avec m′ ≪ m à la place de m. La situation nous est donc même plus favorable que dans le cas du découpage. À nouveau, nous laissons la comptabilité au lecteur. * Une petite confession : le cas de G primitif, que nous venons de ﬁnir de traiter, pourrait être traité exactement comme le cas de G imprimitif, que nous examinerons maintenant. La motivation du traitement séparé pour G primitif est pédagogique. Aucune peine n’est perdue, puisque toutes les techniques que nous avons étudiées nous seront essentielles dans le cas imprimitif. 6. LE CAS IMPRIMITIF Nous avons une application surjective explicite φ : G → Alt(Γ), 1125–34 où G < Sym(Ω) est un groupe de permutation, \|Γ\| = m, \|Ω\| = n. Nous pouvons supposer que \|Γ\| ≥ C log n, C arbitraire. L’application φ se factorise comme suit G → G/N → Alt(Γ), où N est le stabilisateur d’un système de blocs, et G/N → Alt(Γ) est un isomorphisme. Nous devons déterminer IsoG (x, y), où x, y sont des chaı̂nes. Nous avons déjà résolu le cas N = {e}. Nous attaquerons le problème de façon locale : pour T ⊂ Γ, nous arriverons à obtenir un certiﬁcat, soit du fait que φ(AutGT (x))\|T contient Alt(T ) (« certiﬁcat de plénitude »), soit du contraire. (Ici GT désigne le groupe {g ∈ G : T φ(g) = T }.) Nous calculerons tous ces certiﬁcats pour T d’une taille k modérée. Si le nombre de certiﬁcats de plénitude est très grand, nous aurons prouvé que φ(AutG (x)) contient un grand groupe alternant ; ce qui restera à faire sera une version de la procédure du §4.2 (« pull-back »). Dans le cas contraire, les certiﬁcats formeront une structure k-aire dont la symétrie est bornée. Nous pourrons donc appliquer le Lemme des designs, suivi de Coupe-ouJohnson, comme avant. Il y a aussi quelques autres cas particuliers, mais ils nous amènent à des α-découpages, α < 1, ce qui est aussi bien. 6.1. Les certificats locaux 6.1.1. Certiﬁcats d’automorphismes. — Un certiﬁcat local (17) pour T ⊂ Γ est — soit une paire (« pas plein », W, M(T )), oùW ⊂ Ω, M(T ) < Sym(T ), M(T ) 6= Alt(T ) (donc « pas plein ») et φ AutW GT (x) \|T < M(T ), — soit une paire (« plein », K(T )), où K(T ) < AutGT (x), et φ(K(T ))\|T = Alt(T ). Le certiﬁcat local dépend de x de façon canonique. Il est clair qu’un certiﬁcat plein, voire pas plein, garantit que φ(AutGT (x))\|T est Alt(T ), voire ne l’est pas. Si T est donné en tant que tuple ordonné, son certiﬁcat dépend de l’ordre de T seulement dans le sens de ne pas en dépendre : le même groupe {(23), e} < Sym({1, 2, 3}) (disons) a une apparence diﬀérente si nous le regardons du point de vue de l’ordre (1, 2, 3) ou de l’ordre (2, 1, 3). Nous construisons le certiﬁcat par une procédure itérative. Au début de chaque pas, W ⊂ Ω et A(W ) est le groupe AutW GT (x) ; la fenêtre W sera invariante sous A(W ). Au tout début de la procédure, W = ∅ et A(W ) = GT . (Nous pouvons calculer GT comme dans l’exercice 2.1c en temps \|Ω\|O(k) , où k = \|T \|.) À chaque pas, nous ajoutons à W tous les éléments atteints par A(W ) (voir §4.1), puis nous mettons A(W ) à jour, selon le nouveau W . Nous nous arrêtons si φ(A(W ))\|T 6= Alt(T ) (non-plénitude) ou si W ne croı̂t plus, ce qui veut dire qu’aucun élément de Ω \ W n’est atteint par A(W ). Il est clair qu’il y aura ≤ \|Ω\| itérations. À la ﬁn, dans le cas de non-plénitude, nous retournons (« pas plein », W, φ(A(W ))) ; dans le cas de plénitude, nous retournons « plein », A(W )(Ω\W ) . Il est clair que le stabilisateur des points A(W )(Ω\W ) est contenu 17. Ou « local-global », dans la nomenclature de Babai. « Global » fait référence à AutGT (x) < Sym(Ω). 1125–35 non pas seulement dans AutW GT (x), mais aussi dans AutGT (x), puisqu’il ﬁxe tous les points de Ω \ W . Nous savons que φ A(W )(Ω\W ) = Alt(T ) par la Proposition 4.4a, sous la condition que \|T \| ≥ max(8, 2 + log2 \|Ω\|). Vériﬁer si φ(A(W ))\|T = Alt(T ) est facile : nous n’avons qu’à vériﬁer, en utilisant Schreier-Sims, si deux générateurs arbitraires de Alt(T ) sont en φ(A(W ))\|T . De la même façon, il est simple de déterminer quels éléments sont atteints par A(W ) : nous calculons A(W )x pour chaque x ∈ Ω (par Schreier-Sims) et, toujours par Schreier-Sims, vériﬁons si φ(A(W )x )\|T = Alt(T ). Ceci prend du temps polynomial en \|Ω\|. Il reste à voir comment mettre à jour A(W ), étant donné A (W − ), où nous écrivons W − pour l’ancienne valeur de W . Tout élément de A(W ) est dans A(W − ), et donc A (W ) = AutW A(W − ) (x). Comme dans l’équation (4), [ [ W W σ−1 , x, x Iso (9) AutW (x) = Aut (x) = − N Nσ A(W ) σ σ où N est le noyau de φ\|A(W− ) et σ parcourt des représentants des k!/2 classes de N en A(W ). Nous pouvons trouver rapidement un σ ∈ A(W − ) ∩ φ−1 ({τ }) pour tout τ ∈ Sym(Γ), par Schreier-Sims. La Proposition 4.4b nous donne que toute orbite de N contenue en W (l’ensemble d’éléments atteints par A(W − )) est de longueur ≤ \|W \|/k ≤ \|Ω\|/k. En conséquence, par la règle (3), mettre A(W ) à jour se réduit à \|Ω\| · (k!/2) problèmes de détermination de Iso pour des chaı̂nes de longueur ≤ \|Ω\|/k. Comme le nombre d’itérations est ≤ \|Ω\|, la procédure fait appel à Isomorphismede-Chaı̂nes ≤ \|Ω\|2 · (k!/2) fois pour des chaı̂nes de longueur ≤ \|Ω\|/k. Ceci – comme la routine qui prenait \|Ω\|O(k) de temps – est acceptable pour k ≪ (log \|Ω\|)κ . Nous choisirons κ = 1. 6.1.2. Comparaisons de certiﬁcats. — Une légère modiﬁcation de la procédure cidessus nous permet d’élucider la relation entre deux certiﬁcats locaux pour deux chaı̂nes. Soient x, x′ : Ω → Σ, T, T ′ ⊂ Σ, \|T \| = \|T ′\| = k. Pour S ⊃ T , soit xS la chaı̂ne ( x(i) si i ∈ S, xS (i) = glauque si i ∈ /S où glauque ∈ / Σ. Nous voulons calculer (10) ′ IsoGT ·τT,T ′ xW , xW , où GT · τT,T ′ est la classe des éléments de G qui envoient l’ensemble T à T ′ , et W ′ est la valeur de W retournée quand la donnée est T ′ à la place de T . Pour déterminer (10), nous suivons la procédure (§6.1.1), modiﬁée de la façon suivante : nous mettrons à jour, dans chaque itération, non pas seulement A(W ), mais ′ aussi la classe A(W )τ d’isomorphismes en GT · τT,T ′ de xW à (x′ )W . Voilà comment le 1125–36 faire, de façon analogue à (9) : −1 [ [ ′ σ ′ W ′ W W ′ W , = IsoN x , (x ) (11) IsoN σ x , (x ) σ σ où N est le noyau de φ\|A(W− ) et σ parcourt des représentants des k!/2 classes de N contenues en A(W − )τ − . Comme W est stabilisé par A(W− ) (et donc par N), le fait que σ envoie W sur W ′ ou non dépend seulement de la classe de N à laquelle σ appartient. (La classe Iso dans la dernière expression de (11) est vide si W σ 6= W ′.) Comme avant, toute orbite de N contenue en W est de longueur ≤ \|W \|/k, et le problème se réduit à \|Ω\| · (k!/2) appels par itération à Isomorphisme-de-Chaı̂nes pour des chaı̂nes de longueur ≤ \|W \|/k ≤ \|Ω\|/k. Par ailleurs, si T et T ′ nous sont données comme tuples ordonnés (T ), (T ′ ), il est facile de déterminer W′ , (12) I(x, x′ , T, T ′) = IsoG(T ) ·τ(T ),(T ′ ) xW , (x′ ) où G(T ) · τ(T ),(T ′ ) est la classe des éléments de G qui envoient le tuple ordonné (T ) à (T ′ ). En eﬀet, nous n’avons qu’à déterminer (10), puis utiliser Schreier-Sims pour déceler les éléments de (10) qui envoient (T ) à (T ′ ) dans le bon ordre. 6.2. L’agrégation des certificats coupe coupe certiﬁcats locaux plénitude > 1/2 ? coupe ou relations ? relations réduction de G/N à Altm′ m′ ≤ \|m\|/2 Weisfeilerréduction Leman, Johnson de G/N Lemme des à Altm′ √ designs, m′ ≪ m etc. oui pas de couleur dominante pullback récursion n′ ≤ 3n/4 En suivant la procédure du §6.1.1 pour une chaı̂ne x, nous trouvons des certiﬁcats locaux pour chaque T ⊂ Γ de taille k, où k est une constante ∼ C log \|Ω\| (C > 1/ log 2) et k < \|Γ\|/10. Soit F < AutG (x) le groupe engendré par les certiﬁcats pleins K(T ). Soit S ⊂ Γ le support de φ(F ), c’est-à-dire l’ensemble des éléments de Γ qui ne sont pas ﬁxés par tout élément de φ(F ). Notre objectif est de déterminer les isomorphismes IsoG (x, x′ ) de x à une autre chaı̂ne x′ . Puisque les certiﬁcats sont canoniques, l’assignation de F et S à une chaı̂ne l’est aussi. Donc, si nous arrivons à deux cas diﬀérents ci-dessous en suivant la procédure pour x et pour x′ , les deux chaı̂nes ne sont pas isomorphes. Cas 1 : \|S\| ≥ \|Γ\|/2, mais aucune orbite de φ(F ) n’est de longueur > \|Γ\|/2. Alors, nous colorions chaque élément de Γ par la longueur de l’orbite qui le contient. Ceci est un coloriage canonique. Soit aucune classe de couleurs n’est de taille > \|Γ\|/2, 1125–37 soit une classe de couleurs de taille > \|Γ\|/2 est découpée en ≥ 2 ensembles de la même taille ≥ 2. Dans un cas comme dans l’autre, nous passons à une réduction/récursion. Cas 2 : \|S\| ≥ \|Γ\|/2 et une orbite Φ de φ(F ) est de longueur > \|Γ\|/2. Cas 2a : Alt(Φ) < φ(F )\|Φ . Nous sommes dans le cas de grande symétrie. Nous procédons comme au §4.2, jusqu’au point où nous devons déterminer IsoH (x, y) (où y ′ est (x′ )σ , σ ′ ∈ G, et H = φ−1 (Alt(Γ)Φ )). Déﬁnissons K = φ−1 Alt(Γ)(Φ) , et soient σ1 , σ2 ∈ G des préimages (arbitraires) sous φ de deux générateurs de Alt(Φ) < Alt(G), trouvées par Schreier-Sims. Nous savons que les classes AutKσi (x), i = 1, 2, sont non vides, puisque Alt(Φ) < φ(F )\|Φ . Comme K n’a pas d’orbites de longueur > \|Ω\|/2, nous pouvons déterminer ces deux classes par des appels à Isomorphisme-de-Chaı̂nes pour des chaı̂nes de longueur ≤ \|Ω\|/2 et longueur totale ≤ 2\|Ω\|. Elles engendrent AutH (x). Encore par le fait que Alt(Φ) < φ(F )\|Φ , la classe IsoH (x, y) sera non vide ssi IsoK (x, y) est non vide. Nous pouvons déterminer cette dernière classe par des appels à Isomorphisme-de-Chaı̂nes comme ci-dessus, puisque K n’a pas d’orbites de longueur > \|Ω\|/2. Si elle est non vide, nous obtenons la réponse IsoH (x, y) = AutH (x) IsoK (x, y). Cas 2b : Alt(Φ) ≮ φ(F )\|Φ . Soit d ≥ 1 l’entier maximal avec la propriété que φ(F )\|Φ est d-transitif, c’est-à-dire, φ(F )\|Φ agit transitivement sur l’ensemble des dtuples d’éléments distincts de Φ. Par CGFS, d ≤ 5 ; si nous ne voulons pas utiliser CGFS, nous avons la borne classique d ≪ log \|Γ\|. Choisissons x1 , . . . , xd−1 ∈ Φ arbitrairement. Le reste de notre traitement de ce cas sera donc seulement canonique en relation à φ(g) G(x1 ,...,xd−1 ) = {g ∈ G : xi = xi ∀1 ≤ i ≤ d − 1}, ce qui, comme nous le savons, n’est pas un problème ; voir le début du §5.3.1. La restriction du groupe φ(F )(x1 ,...,xd−1 ) à Φ′ = Φ\{x1 , . . . , xd−1 } est transitive sur Φ′ , mais elle n’est pas doublement transitive. Donc, la conﬁguration cohérente schurienne qui lui correspond n’est pas une clique. Nous livrons cette conﬁguration à Coupe-ouJohnson (§5.2), tout comme à la ﬁn du §5.2. Pour comparer les conﬁgurations qui correspondent à deux chaı̂nes x, x′ , nous alignons leurs classes Φ d’abord. (Si elles ne sont pas de la même taille, ou si une chaı̂ne nous donne le cas 2a et l’autre pas, les chaı̂nes ne sont pas isomorphes.) Les isomorphismes seront donc contenus dans le stabilisateur H < G de l’ensemble Φ (facile à déterminer, comme vers la ﬁn du §4.2, puisque φ est surjective). Nous pouvons remplacer φ par l’application g 7→ φ(g)\|Φ de H à Alt(Φ). Puis nous construisons les conﬁgurations comme ci-dessus, et comparons ce que Coupe-ou-Johnson nous donne. Tout à la ﬁn, nous nous occupons du complément de C. Il s’agit, comme d’habitude, d’appels à Isomorphisme-de-Chaı̂nes pour des chaı̂nes de longueur ≤ \|Ω\|/2 et longueur totale < \|Ω\|. Cas 3 : \|S\| < \|Γ\|/2. Nous commençons en alignant les supports S pour les chaı̂nes x, x′ , et en remplaçant φ par g 7→ φ(g)\|Γ\S , tout comme dans le cas 2(b). 1125–38 Nous allons déﬁnir une relation k-aire avec très peu de jumeaux, pour la donner après au Lemme des designs. Regardons la catégorie de toutes les chaı̂nes Ω → Σ, où Ω et Σ sont ﬁxes, une action de G sur Ω est donnée, et φ : G → Γ est aussi donnée. Nous la regardons depuis longtemps, puisque nous devons comparer les couleurs sur des conﬁgurations induites par des chaı̂nes diﬀérentes pour décider si ces dernières sont isomorphes. Cette fois-ci, nous déﬁnirons des couleurs par des classes d’équivalence : deux paires (x, (T )), (x′ , (T ′ )) (T, T ′ ⊂ Γ \ S, \|T \| = \|T ′ \| = k) sont équivalentes si l’ensemble des isomorphismes en (12) est non vide. Nous colorions (T ) – dans le coloriage de (Γ \ S)k correspondant à x – par la classe d’équivalence de (x, (T )). Ici, (T ) est un k-tuple ordonné sans répétitions ; si (T ) a des répétitions, elle est coloriée en gris. Pour x donné, aucune classe de jumeaux en Γ ne peut avoir ≥ k éléments : s’il existait un tel ensemble avec ≥ k éléments, il contiendrait un ensemble T avec k éléments, et tous les ordres (T ) de T auraient la même couleur. Ceci voudrait dire que l’ensemble des isomorphismes en (12) serait non vide pour n’importe quels ordres (T ), (T ′ ) de T . En conséquence, AutGT (xW ) contiendrait des éléments donnant toutes les permutations possibles de T . Ceci nous donnerait une contradiction, puisque T , étant contenu en Γ\S, n’est pas plein. Alors, pourvu que k ≤ \|Γ\|/4, nous avons un coloriage de (Γ\S)k sans aucune classe de jumeaux avec ≥ \|Γ \ S\| éléments. Nous pourrons donc appliquer le Lemme des designs, après une application de raﬃnements habituels F2 , F3 (Weisfeiler-Leman). Mais – pouvons-nous calculer ces coloriages ? Les classes d’équivalence sont énormes. Par contre, il n’y a aucun besoin de les calculer. Tout ce dont nous aurons besoin, pour comparer des structures qui viennent de chaı̂nes x, y, sera d’être capables de comparer deux tuples (T ) (sur la conﬁguration donnée par x ou y) et (T ′ ) (sur la conﬁguration donnée par x′ = x ou x′ = y) et dire si elles sont de la même couleur. En d’autres termes, nous devrons calculer – au tout début de la procédure, pour toute paire ((T ), (T ′)), \|T \| = \|T ′ \| = k, et pour les paires de chaı̂nes (x, x), (x, y), (y, y) – l’ensemble d’isomorphismes en (12), ce que nous savons déjà faire. Les couleurs sont donc, dans la pratique, des entrées dans un index que nous enrichissons et auquel nous faisons référence durant nos procédures. Nous invoquons donc le Lemme des Designs, suivi par Coupe-ou-Johnson, et le reste de la procédure. — Fine dell’opera — Le lecteur peut vériﬁer que les informations précisées jusqu’à ici (temps pris par des procédures, type de récursion) sont assez pour donner une borne du type exp(O(log \|Ω\|)c ) pour le temps de l’algorithme qui résout le problème de l’isomorphisme de chaı̂nes. Ceci donne une borne exp(O(log n)c ) pour le problème de l’isomorphisme de graphes avec n sommets. Avec un peu plus de travail, il devient clair que, dans un cas comme dans l’autre, c = 3. Nous donnons les détails dans l’appendice. L’exposant c = 3 1125–39 est plus petit que celui d’origine ; il est devenu possible grâce à quelques améliorations et simpliﬁcations que j’ai été capable d’apporter. Remerciements .— Je remercie vivement L. Babai, J. Bajpai, L. Bartholdi, D. Dona, E. Kowalski, W. Kantor, G. Puccini, L. Pyber, A. Rimbaud et C. Roney-Dougal pour des corrections et suggestions. En particulier, L. Babai a répondu à beaucoup de mes questions, et m’a aussi fourni des versions corrigées ou améliorées de plusieurs sections de [Ba]. En particulier, les §2.3–2.4 et §5.1 sont basés sur ces nouvelles versions. Je voudrais aussi remercier V. Ladret et V. Le Dret pour un grand nombre de corrections d’ordre typographique et linguistique. APPENDICE A. ANALYSE DU TEMPS D’EXÉCUTION A.1. Quelques précisions sur la procédure principale À tout moment donné, nous travaillons avec un groupe transitif G < Sym(Ω) qui S agit sur un système de blocs B = {Bi }, Ω = i Bi , Bi disjoints ; nous notons N le noyau de l’action sur B. À vrai dire, nous aurons toute une tour de systèmes de blocs B1 , B2 , . . . , Bk , où Bi est un raﬃnement de Bi+1 ; B signiﬁera Bk , le système le moins ﬁn. Au début, il n’y a qu’un système, B1 , dont les blocs Bi sont tous de taille 1, et dont le noyau N est trivial. Nous voudrions que l’action de G sur B soit primitive. Donc, si elle ne l’est pas, nous ajoutons à la tour un système minimal Bk+1 tel que Bk soit un raﬃnement de Bk+1 . Nous redéﬁnissons B = Bk+1 ; N sera le noyau du nouveau B. Si G/N est petit (≤ bO(log b) , où b = \|B\| ; cas (a) du Théorème 3.1 (Cameron)), nous réduisons notre problème à plusieurs instances du problème avec N à la place de G. Chacune de ces instances se décompose en plusieurs instances – une pour chaque orbite de N. Chaque orbite Ω′ de N est contenue dans un bloc de B. Les intersections de Ω′ avec les blocs de B1 , B2 , . . . nous donnent une tour de systèmes de blocs pour N\|Ω′ . Si nous sommes dans le cas (b) du Théorème 3.1, nous passons à ≤ b instances du problème avec M ⊳ G (où[G : M] ≤ b) à la place de G. Nous passons à un nouveau ≤ b blocs, et l’ajoutons à la tour comme son nouveau dernier système (18) B ′ de m′ = m k ′ niveau. Nous notons N le noyau de l’action de M sur B ′ . Alors, M/N ′ = Alt(k) m . Nous remplaçons G par M et redéﬁnissons B = B ′ , N = N ′ . Donc, nous avons un isomorphisme de G/N à Altm . Nous sommes dans le cas principal que Babai attaque. Ses méthodes amènent à une réduction de Altm , soit à un groupe intransitif sans grandes orbites, soit à un produit Alts1 ≀ Alts2 , s1 , s2 > 1, s1 s2 ≤ m, soit √ à un groupe Altm′ , m′ ≪ m. (Nous simpliﬁons quelque peu. Nous pourrions avoir, disons, un produit Alts1 ≀ Alts2 , agissant sur une orbite de grande taille s1 s2 ≤ m, et 18. Ce système peut être égal à B seulement si M = G ; voir la deuxième note de pied de page dans l’énoncé du Théorème 3.1. Dans ce cas-là, le passage de G à M est bien sûr gratuit. 1125–40 d’autres groupes sur des petites orbites, ou plusieurs produits agissant sur des petites orbites.) Dans le cas intransitif sans grandes orbites, nous procédons comme dans la preuve de Luks. (La procédure aura été plus coûteuse que dans Luks, mais grâce au manque de grandes orbites, le gain dans la récursion est aussi plus grand.) Dans le cas de Altm′ , √ m′ ≪ m, nous itérons la procédure. Dans le cas de Alts1 ≀ Alts2 – qui correspond à un découpage dans des ensembles de taille r de la même couleur – nous avons une action primitive de Alts sur un système de s blocs de taille r. Nous passons, alors, à cette action et à ces blocs, sans oublier les blocs B ′ , auxquels nous retournons plus tard, après avoir ﬁni de travailler sur Alts . Il est clair que ce type de procédure réduit complètement Altk en un nombre d’itérations qui n’est pas supérieur à log2 m. A.2. Récursion et temps Examinons le temps total d’exécution de l’algorithme qui trouve les isomorphismes entre deux chaı̂nes. Les pas individuels sont peu onéreux ; aucun ne précise plus de nO(log n) de temps. Notre attention doit se porter avant tout sur la récursion. Dans la procédure générale, une récursion est toujours d’une descente, soit vers des chaı̂nes plus courtes, soit vers un groupe plus petit, ou au moins coupé dans des tranches plus ﬁnes par une tour de systèmes de blocs ayant plus de niveaux. Dans le premier type de descente, le groupe reste le même ou, plutôt, est remplacé par une restriction de lui-même. Dans le deuxième cas, la longueur des chaı̂nes reste la même. (Nous pouvons aussi avoir un mélange des deux cas – tant mieux : le groupe devient plus petit et les chaı̂nes se raccourcissent aussi.) La descente la moins coûteuse, et moins avantageuse, est celle du cas intransitif de la procédure de Luks. Il pourrait arriver que G ait deux orbites sur Ω (\|Ω\| = n), une de longueur n − 1 et une de longueur 1. Ceci serait même compatible avec une borne polynomiale sur le temps, pourvu que le temps pris avant la descente soit lui-même polynomial : nc+1 ≤ (n − 1)c+1 + 1c+1 + nc pour c ≥ 1. D’autres types de descente sont plus coûteux, mais aussi plus avantageux : nous descendons à des chaı̂nes de longueur ≤ n/2 (ou ≤ 2n/3), ou de Altm à Alts1 ≀ Alts2 , s1 s2 ≤ m, s1 , s2 ≤ m/2, par exemple. Il est clair qu’il est impossible de descendre plus qu’un nombre logarithmique de fois de cette façon. Il est crucial de ne pas oublier qu’un coût (considérable) peut être caché dans une perte de canonicité. Si nos choix ne sont canoniques qu’en relation à un sous-groupe H de notre groupe G, le coût de leur application sera multiplié par [G : H]. (Voir §5.3.1.) *** Considérons, alors, le coût de chaque procédure. Le cas intransitif de Luks est, comme nous l’avons déjà vu, compatible même avec une borne polynomiale. Concentrons-nous alors sur le cas où G agit de façon primitive sur un système de blocs ; soit N le noyau. 1125–41 Si nous sommes dans le cas (a) du Théorème 3.1, ou dans le cas (b), mais avec m ≤ C log n, nous faisons appel à (m′ )O(log n) instances de la procédure principale pour des chaı̂nes de longueur n/m′ (où m′ ≥ m). Ceci est consistant avec une borne totale du type exp(O((log n)c )), c ≥ 2. Nous pouvons, donc, nous concentrer sur le cas où il existe un isomorphisme φ : G/N → Alt(Γ), \|Γ\| = m > C log n. (La procédure du §2.8 rend cet isomorphisme explicite.) Le premier pas à considérer est la création de certiﬁcats locaux, avec, comme objectif, la création d’une relation k-aire sur Γ. (Si G est primitif, créer une telle relation est trivial ; voir le début du §5.) Il y a nk certiﬁcats locaux, où k = 2 log n (disons) ; nous devons les calculer et aussi comparer toute paire de certiﬁcats. Déjà le premier pas du calcul d’un certiﬁcat, à savoir le calcul de GT , prend un temps nO(k) (plus précisément, O((n/k)O(k))). D’autres calculs prennent moins de temps. L’usage de la récursion, par contre, est relativement lourd : nous faisons appel à la procédure principale ≤ n2 · k! fois pour des chaı̂nes de longueur ≤ n/k. Ceci se passe pour chaque ensemble T de taille k, c’est-à-dire ≤ nk /k! fois. La procédure pour comparer des paires de certiﬁcats est analogue. Nous faisons donc appel à la procédure principale O(n2k+1 ) fois pour des chaı̂nes de longueur ≤ n/k. Dans chacun de ces appels, notre tour de stabilisateurs est héritée : notre groupe est un groupe transitif, égal à la restriction de N − à une de ses orbites, où N − (noté N au §6) est un sous-groupe d’un sous-groupe A(W − ) de G. Pour deux systèmes de blocs consécutifs Bi , Bi+1 , notons ri le nombre de blocs de Bi dans chaque bloc de Bi+1 . Il est clair que ce nombre n’augmente pas quand nous passons à la restriction d’un sous-groupe de G (par exemple, N − ) à une de ses orbites. Examinons maintenant l’agrégation des certiﬁcats locaux (§6.2). Il y a trois cas. Dans le premier, le temps de calcul additionnel est à peu près trivial, et nous obtenons une réduction, soit à un groupe intransitif sans grandes orbites, soit à un produit Alts1 ≀ Alts2 sur une grande orbite et éventuellement d’autres groupes sur des orbites plus petites. Ici, déjà, l’analyse devient délicate. Nous devons prendre en considération non seulement la taille du domaine mais aussi le groupe qui agit sur lui. Plus précisément, nous devons borner le nombre de fois que notre tour B1 , B2 , . . . , Bk pourrait être raﬃné ou raccourci encore. Ceci sera mesuré par X ρ= (2⌊log2 ri ⌋ − 1), 1≤i≤k−1 où nous supposons que nous avons enlevé des systèmes répétés de la tour (donc ri > 1). Notons F (n, r) le temps d’exécution de la procédure principale pour des chaı̂nes de longueur n et pour une tour de systèmes de blocs pour G telle que le paramètre ρ est ≤ r. Une réduction de G/N fait décroı̂tre r par au moins 1 ; un coloriage sans aucune grande classe de couleurs assure une descente vers des chaı̂nes de longueur ≤ n/2. Nous devrons aussi inclure un facteur de log n2k , prenant en considération le temps requis 1125–42 pour accéder à nos comparaisons de paires de certiﬁcats locaux (19) . Donc, dans le cas que nous examinons, F (n, r) est borné par ! X nO(k) + n2k+1 F (n/k, r) + F (n1 , r − 1) + F (ni , r) · O(k log n), i≥2 où P ni = n et ni ≤ n/2 pour i ≥ 2, ou nO(k) + n2k+1 F (n/k, r) + X i F (ni , r) ! · O(k log n), où ni = n et ni ≤ n/2 pour i ≥ 1. Puisque k ≪ log n, ceci est consistant avec F (n, r) = exp (O (r + log n)c ) pour c ≥ 3, ou même avec F (n, r) = exp (O ((log n)c1 + (log r)c2 )) pour c1 ≥ 3 et c2 ≥ 1, par exemple. Le cas 2a a un coût très similaire, à un facteur constant près. Le cas 2b et 3 sont diﬀérents. Dans les deux cas, nous arrivons à construire une relation d-aire, avec d ≤ 5, dans le cas 2b, et d = k ≪ log n dans le cas 3. Puis, nous appelons Weisfeiler-Leman, suivi du Lemme des Designs pour des conﬁgurations d-aires, et, ﬁnalement, Coupe-ouJohnson. Weisfeiler-Leman prend un temps \|Γ\|O(d) = mO(d) . Le Lemme des designs garantit l’existence d’un tuple (x1 , . . . , xℓ ) ∈ Γ, ℓ ≤ d − 1, avec certaines propriétés. Nous cherchons un tel tuple par force brute, ce qui prend un temps O(md ). Ce qui est plus important est que ce choix n’est pas canonique. Donc, le temps d’exécution de tout ce qui reste est multiplié par md = mO(log n) . Coupe-ou-Johnson prend un temps O(md ). Ici, à nouveau, nous faisons des choix qui ne sont pas complètement canoniques ; ils imposent un facteur de mO(log m) sur tout ce qui suit. Le résultat de Coupe-ou-Johnson est soit un β-découpage, ce qui implique une réduction à un produit du type Alts1 ≀ Alts2 et/ou à des chaı̂nes plus courtes, soit un √ schéma de Johnson, ce qui implique une réduction à Altm′ , m′ ≪ m. Donc, soit (13) ! X F (ni , r) , F (n, r) ≤ nO(k) + O kn2k+2 F (n/k, r) + mO(log n) 1 + F (n1 , r − 1) + P i≥2 où P (14) ni = n, et ni ≤ n/2 pour i ≥ 2, ou F (n, r) ≤ nO(k) + O kn2k+2 F (n/k, r) + mO(log n) 1 + X i ! F (ni , r) , P où ni = n et ni ≤ n/2 pour i ≥ 1. Ici m ≤ n. (Nous pourrions travailler avec une borne moins grossière, mais cela nous servirait peu.) Donc, les inégalités (13) et (14) sont consistantes avec F (n, r) = exp (O (r + log n)c ) pour c ≥ 3. 19. Faire ce type de comparaisons à l’avance nous aide, mais ne pas les faire à l’avance ne changerait pas l’ordre du temps utilisé, asymptotiquement. 1125–43 Comme r ≤ 2 log2 n, nous concluons que le temps total d’exécution de la procédure pour déterminer les isomorphismes entre deux chaı̂nes de longueur n est 3 F (n, r) ≤ eO(log n) . RÉFÉRENCES [Ba] L. BABAI – Graph Isomorphism in Quasipolynomial time, prépublication, disponible en ligne sur arxiv.org:1512.03547. [Ba2] L. BABAI – Graph Isomorphism in Quasipolynomial time (Extended Abstract), dans Proc. 48th ACM STOC (2016), 684–697. [Ba3] L. BABAI – Lectures on Graph Isomorphism, University of Toronto, Dept. of Computer Science, notes polycopiées, 1979. [BCP] L. BABAI, P. J. CAMERON et P. P. PÁLFY – On the orders of primitive groups with restricted nonabelian composition factors, Journal of Algebra 79 (1982), 161–168. [BKL] L. BABAI, W. M. KANTOR et E. M. 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[Sch] O. SCHREIER – Die Untergruppen der freien Gruppen, Abh. Math. Semin. Univ. Hambg. 5 (1927), 161–183. [Si1] C. C. SIMS – Graphs and ﬁnite permutation groups, Math. Z. 95 (1967), 76–89. [Si2] C. C. SIMS – Computational methods for permutation groups, dans « Computational Problems in Abstract Algebra », pp. 169–184, Pergamon, Oxford, 1970. [SW] X. SUN et J. WILMES - Faster canonical forms for primitive coherent conﬁgurations, dans Proc. 47th ACM STOC (2015), 693–702. [WL] Б. Ю. ВЕЙСФЕЙЛЕР и А. А. ЛЕМАН – Приведение графа к каноническому виду и возникающая при этом алгебра, НТИ, сер. 2, 9 (1968), 12–16. Harald Andrés HELFGOTT Universität Göttingen Mathematisches Institut Bunsenstrasse 3-5 D-37073 Göttingen Allemagne E-mail : [email protected]	4
Modified Recursive Cholesky (Rchol) Algorithm arXiv:1703.05904v1 [math.AC] 17 Mar 2017 An Explicit Estimation and Pseudo-inverse of Correlation Matrices Vanita Pawar Krishna Naik Karamtot vanita [email protected] [email protected] Abstract—The Cholesky decomposition plays an important role in finding the inverse of the correlation matrices. As it is a fast and numerically stable for linear system solving, inversion, and factorization compared to singular valued decomposition (SVD), QR factorization and LU decomposition. As different methods exist to find the Cholesky decomposition of a given matrix, this paper presents the comparative study of a proposed RChol algorithm with the conventional methods. The RChol algorithm is an explicit way to estimate the modified Cholesky factors of a dynamic correlation matrix. Cholesky decomposition is a fast and numerically stable for linear system solving, inversion, and factorization compared to singular valued decomposition (SVD), QR factorization and LU decomposition [1]. The wireless communication system is highly dependent on matrix inversion of the correlation matrix. Such system consists of a huge matrix inversion. An outdoor wireless communication has a time-varying channel which changes dynamically for mobile user. In case of narrowband channel, the channel is considered constant for a symbol duration, whereas for broadband, it is changing within a symbol period. Such time-varying channel forms the special structure of channel matrix and correlation matrix. To exploit such special structure, a novel modified recessive Cholesky (RChol) algorithm is introduced in [2]. Our proposed (RChol) algorithm is a computational efficient algorithm to compute the modified Cholesky factors of known as well as an unknown covariance matrix. In this paper, we present the comparative study of conventional Cholesky algorithm and the RChol algorithm to manifest the importance of the proposed algorithm in a highly dynamic wireless communication. I. S YSTEM M ODEL In wireless communication system, number of transmit and or received antennas are used to improve the diversity of the system. The channel h between transmitter and receiver has the different form and depends on the number of antennas used at the transmitter and the receiver side. The channel for Single-input-single-output (SISO) as h = {h0n , h1n , . . . hL−1 }, for Single-input-multiple-output (SIMO) n as h = {h0n , h1n , . . . hL−1 } and for Multiple-input-multiplen output (MIMO): H = {H0n , H1n , . . . HL−1 }. n Let y(n) be received signal with the number of transmit antennas 0 K = 10 , multipath L − 1 and channel noise v, represented as y(n) := K L−1 X X hk (n; l)sk (n − l) + v(n), n = 0, 1, ..T − 1 k=1 l=0 (1) Let yN (n) be the received vector by stacking N successive received vectors. Where yN (n) = [y(n), y(n − 1), . . . y(n − N + 1)]T and the transmitted symbol vector is sN = [s(n), s(n − 1), .s(n − N + 1)]T . Then yN (n) can be represented in matrix form as yN (n) = HN sN (n) + vN (n) and the correlation matrix for yN can be written as RN (n) = H (n)]. Let rn00 = E[y(n)yH (n)] and rnij = E[yN (n)yN H E[y(n − i)y (n − j)] then the correlation matrix RN (n) and RN (n − 1) at time instant 0 n0 and 0 n − 10 can be represented as equation (2) and equation (3) respectively. 0    rn 00 rn 10 rn 01 rn 11 . . . . . . ...... ...... . . . ...... rn 0(N −2) rn 1(N −2) rn 0(N −1) rn 1(N −1) . . . . . .    (2) rn (N −1)0 rn (N −1)1  rn 11 rn 12 ...... rn 1(N −1) rn 1N    . . . rn (N −1)1 rn N1 . . . rn (N −1)2 rn N2 . . . ...... . . . rn (N −1)(N −1) rn N (N −1) . . . rn (N −1)N rn NN   ...... rn (N −1)(N −2) 0 rn (N −1)(N −1) (3) II. C HOLESKY D ECOMPOSITION The correlation matrix is complex matrix and the pseudoinverse of R can be computed from Cholesky factors, such that if lower triangular matrix L is Cholesky factors of the correlation matrix R and can be represented as R = LLH then pseudo-inverse of R can be computed as R̂† := L−H L−1 . The section below details the conventional Cholesky algorithms and the RChol algorithm. A. Cholesky Decomposition (Gaxpy version) The Cholesky Decomposition [3] factorizes a complex (or real-valued) positive-denite Hermitian symmetric matrix into a product of a lower triangular matrix and its Hermitian transpose. R = LLH where, L is a lower triangular matrix and LH is Hermitian of L. The matrix R must be a positive definite and this method needs square root operation. 1) 1) 2) 3) 4) 5) Algorithm steps: Compute R at each time instant n Find the square root of diagonal element of R Modify each column of R Equate lower triangular part of R to L Repeat steps (1) to (4) for each time instant Algorithm 1 Cholesky Decomposition R = LLH Initialization: [R]1:N,1 [R]1:N,1 = q [R]1,1 0 Order Updates on R s: for k = 2 to N H [R]k:N ,k = [R]k:N ,k − [R]k:N ,1:k−1 [R]k,1:k−1 [R]k:N,k [R]k:N,k = q [R]k,k end [4] that Levinson recursion may be used to derive the Lattice recursion for computing QR factors of data matrices and Lattice recursion can be used to derive the Schur recursion for computing Cholesky factors of a Toeplitz correlation matrix. The detail algorithm is given in algorithm 3. The Schur algorithm like previously mentioned algorithm computes all N inner product to compute matrix R for initialization. 1) Algorithm steps: 1) Compute R at each time instant n 2) Initialize first column of R to the first column of Cholesky factor H 3) Compute rest column recursively from columns of R 4) repeat step (1) to (3) for each time instant Algorithm 3 Schur Algorithm R = LLH Initialization: for k = 1 L = tril(R) T n n n H1 (n) = [r00 , r10 , .....r(N −1)0 ] T n n H̃1 (n) = [0, r10 , .....r(N −1)0 ] H B. Modified Cholesky Algorithm R = LDL To avoid square root operation, a modified Cholesky algorithm [3] is used, which avoids square root operation by introducing a diagonal matrix D in between Cholesky factors. The modified Cholesky algorithm does not require R to be a positive definite matrix but it’s determinant must be nonzero. R may be rank deficient to a certain degree i.e. D may contain negative main diagonal entries if R is not positive semidefinite. 1) Algorithm steps: 1) Compute R at each time instant n 2) Modify each column of R 3) Equate the strictly lower part of matrix R to L1 with ones on the main diagonal 4) Equate main diagonal of R with the main diagonal of D 5) Repeat step (1) to (4) for each time instant. Algorithm 2 Modified Cholesky Decomposition R = LDLH Initialization: [R]2:N,1 = [R]2:N,1 [R]1,1 0 Order Updates on R s: for k = 2 to N for i=1:k-1 [v]i = [R]1,k , if i = 1 [R]i,i [R]∗ n,i , if i 6= 1 end [v]k = [R]k,k − [R]k,1:k−1 [v]1:k−1 [R]k,k = [v]k [R]k+1:N ,k = [R]k+1:N ,k − [R]k+1:N ,1:k−1 [v]1:k−1 [v]k end D = diag(daig(R)) L = tril(R) C. Recursive Cholesky Algorithm (The Shcur Algorithm) RSchur := HHH The Schur algorithm recursively compute the columns of the lower triangular matrix H form matrix R. It is shown in 0 Order Updates on H s: for k = 2 to N σk Hk = k̃ref [kref (σk−1 H̃k−1 ) + ZM (σk−1 Hk−1 )] k k σk H̃k = k̃ref [(σn H̃k ) + kref ZM (σk Hk )] k k Scaling Factors: (σk H̃k )k kref = − k (σk Hk )k−1 (σk Hk )k−1 k̃ref = k (σk+1 Hk+1 )k ∗ Note: Here notation is followed same as in [4] and H represents vector D. The RChol Algorithm L̂D̂L̂H := R It is clear from above equation (2) and equation (3) that RN (n) can be represented from submatrix of RN (n − 1). To utilize such special structure of correlation matrices, we propose a modified recursive Cholesky algorithm to compute the Cholesky factors recursively. This algorithm is modification of Schur algorithm mentioned above. The more general approach consists of using the Schur algorithm to induce recursion for columns of dynamic L. This algorithm does not need N inner products to compute the correlation matrix R. The Cholesky factors are computed explicitly such that Let L1 = LD1/2 then −1 pseudo-inverse can be computed as R̂† = L−H 1 L1 1) Algorithm steps: 1) Initialize first the first column of Cholesky factor A as A1 2) Compute second column recursively from A1 (n) and A1 (n − 1) 3) Substitute sub-matrix A2:N −1,2:N −1 (n − 1) to A3:N,3:N (n) 4) Repeat step (1) to (3) for each time instant In the Schur algorithm, columns of Cholesky factors at time instant n are computed recursively from the correlation matrix at that instant. Whereas in the RChol algorithm first two columns of Cholesky factors at time instant n is computed recursively from previous Cholesky factor and submatrix of that Cholesky factors are updated recursively from previous Cholesky factor i.e. at time instant n − 1. Conventional Cholesky algorithm mentioned here are introduced for normal matrices whereas proposed matrix is well suited for block matrices and simulations are shown for that only. Algorithm 4 Recursive Cholesky Update : RChol R = LDLH Initialization: for k = 1 , D1 (n) = rn 00 T n n n A1 (n) = [r00 , r10 , . . . r(N −1)0 ] T n n Ã1 (n) = [0, r10 , . . . r(N −1)0 ] 0 Order Updates on A s: for k = 2 IV. C ONCLUSION Ak (n) := ZM Ak−1 (n − 1) − Ãk−1 (n)k̃ref (n) Convention methods of Cholesky factorization requires the correlation matrix which needs inner product. While the recursive modied Cholesky algorithm (RChol) algorithm is an explicit way to recursively calculating the pseudo-inverse of the matrices without estimating the correlation matrix. It requires less number of iteration which avoids error propagation through column updates. The RChol algorithm has most of the use in calculating the pseudo-inverse of the of a time-varying matrix which is applicable to SIMO/MIMO, CDMA, OFDM, etc. wireless communication systems. Dk (n) = Dk (n − 1)[IM − kref (n)k̃ref (n)] for k > 2 , Ãk−1 (n) = 0 Ak (n) = ZM Ak (n − 1) Dk (n) = ZM Dk (n − 1) Ã1 (n)(2,:) Scaling Factors: kref (n) = A1 (n−1)(1,:) 0 kref (n) D1 (n−1) k̃ref (n) = D̃1 (n) III. S IMULATION RESULTS To compare proposed the RChol algorithm with Schur algorithm, we compared the result of both the algorithm with theoretical results. Fig. 1. Show the ratio and difference of matrices R̂N , R̂RChol and R̂Schur , when the correlation matrix is unknown. That has the application in blind channel and or data estimation. Fig. 1 (a) and (b) shows the maximum error for the RChol algorithm, [R̂N − R̂RChol ] is 0.6 while for the Schur algorithm, [R̂N − R̂Schur ] is 4 i.e. nearly 6 times the RChol algorithm. In case of ratio Fig. 1 (a) and (b) shows the maximum ratio for the RChol algorithm, [R̂N ./R̂RChol ] is 45 while for the Schur algorithm, [R̂N ./R̂Schur ] is 1500. 0 0 5 10 15 20 25 30 35 5 10 15 20 25 30 35 40 0 0 40 0 0 5 10 15 20 25 30 35 5 10 15 20 25 30 35 40 0 40 0 45 2500 0.6 5 1 5 40 5 0.4 10 35 10 5 10 2000 10 0 30 15 15 15 15 0.2 25 1500 -1 20 20 20 20 0 15 -2 25 25 -0.4 30 0 -5 40 40 0 40 40 (a) (b) 0 5 10 15 20 25 30 35 (c) 5 10 15 20 25 30 35 (d) 40 0 0 40 0 0.03 0 0 0 10 15 20 25 30 35 5 1 15 20 25 30 35 40 1 10 0.8 0.02 15 15 15 20 10 1.05 10 15 5 0 40 5 0.025 10 -0.5 -1 5 5 5 0.95 0.6 20 0.015 20 20 -1.5 25 25 0.9 0.4 30 30 -2 25 25 0.01 30 0.005 0.2 0.85 35 35 35 35 -2.5 40 0 0.8 40 (e) 500 35 35 -4 -0.6 5 35 35 30 1000 30 -3 30 0 25 10 30 10 20 25 -0.2 40 0 40 (f) (g) (h) Fig. 1: Comparisons of RChol algorithm Vs Schur Algorithm for the unknown and known correlation matrix R, (a, e): Proposed Algorithm ( Difference), (b, f): Schur Algorithm( Difference), (c, g):Proposed Algorithm (Ratio), (d, h): Schur Algorithm (Ratio) Fig. 1 Show the ratio and difference of matrices R̂N , R̂RChol and R̂Schur , when the correlation matrix is known. Fig. 1 (a) and (b) shows that the maximum error for the RChol algorithm, [R̂N − R̂RChol ] is 2.5 while for the Schur 4 algorithm, [R̂N − R̂Schur ] is 0.03 i.e. nearly 6 times the RChol algorithm. In case of ratio Fig. 1 (e) and (f) shows that the maximum ratio for the RChol algorithm, [R̂N ./R̂RChol ] is 1.15 while for the Schur algorithm, [R̂N ./R̂Schur ] is 1. From Fig. 1 it can be concluded that the Schur algorithm is best suited when the correlation matrix is known, but leads to huge error propagation through the column when R is unknown and cannot be applied for blind channel estimation. In converse, the RChol algorithm is best suited for blind channel estimation and reduces error propagation through the column. V. Pawar and K. Naik (DIAT, Pune, India) E-mail: [email protected] R EFERENCES [1] G. Golub and C. Van Loan: ’Matrix computations’, 2012 [2] V. Pawar and K. Krishna Naik: ’Blind multipath time varying channel estimation using recursive Cholesky update’, AEU - Int. J. Electron. Commun., 2016, 70, no. 1, pp. 113-119 [3] R. Hunger and T. Report: ’Floating Point Operations in Matrix-Vector Calculus’, Matrix, 2007. [4] C. P. Rialan and L. L. Scharf: ’Fast algorithms for computing QR and Cholesky factors ofToeplitz operators’, IEEE Trans. Acoust., 1998 36, pp. 1740-1748	0
arXiv:1605.01298v1 [math.AC] 4 May 2016 THE EUCLIDEAN CRITERION FOR IRREDUCIBLES PETE L. CLARK Abstract. We recast Euclid’s proof of the infinitude of prime numbers as a Euclidean Criterion for a domain to have infinitely many atoms. We make connections with Furstenberg’s “topological” proof of the infinitude of prime numbers and show that our criterion applies even in certain domains in which not all nonzero nonunits factor into products of irreducibles. 1. Introduction This article has its genesis in a graduate VIGRE research group taught by Paul Pollack and me in Fall 2015: Introduction to the Process of Mathematical Research. Rather than concentrating on a fixed topic preselected by us, the goal was to guide students through the process of selecting and performing research on their own. One technique we tried to inculcate is exploitation of the many-to-one relation between theorems and proofs. A good theorem has several proofs, and you will know two proofs are different when can be used to prove further theorems the other cannot. In our first meeting, Pollack and I presented seven proofs of Euclid’s Proposition IX.20: there are infinitely many prime numbers. My first proof: suppose given a domain R that is not a field, in which each nonzero nonunit factors into irreducibles and whenever x ∈ R is a nonzero nonunit then x + 1 is not a unit; then there is at least one irreducible element f1 , and given irreducibles f1 , . . . , fn , by factoring f1 · · · fn +1 we get a new irreducible element. It was pointed out that this argument, though correct, does not imply Euclid’s result: x = −2 is a problem. Some salvages were suggested: in Z it is enough to replace f1 · · · fn by −f1 · · · fn , if necessary. Here we present a general fix – a Euclidean Criterion for a domain to have infinitely many nonassociate irreducibles – and explore its consequences. We soon find ourselves on a scenic tour of 20th century mathematics, as we engage with work of Jacobson, Furstenberg, Cohen-Kaplansky and Anderson-Mott, among others. 1.1. Acknowledgments. Thanks to all members of the 2015-2016 Introduction to Mathematical Research UGA VIGRE group. Conversations with Saurabh Gosavi, Noah Lebowitz-Lockard, Robert Samalis, Lee Troupe and Lori D. Watson were helpful. My group coleader Paul Pollack made key contributions: first, he emphasized that the Euclidean Criterion automatically yields pairwise comaximality. Second, Theorem 2.9 was inspired by [P, Thm. 1.16], and though I came up with the statement, I could prove it only in various special cases. The proof included here is his. I am grateful to two anonymous referees for their careful, detail-oriented reports. In particular, Example 4.19 was suggested by the “first” referee. Date: May 5, 2016. 1 2 PETE L. CLARK 2. The Euclidean Criterion 2.1. A primer on factorization in domains. By a ring we will mean a commutative ring with a multiplicative identity. We denote the set of nonzero elements of R by R• . An element x ∈ R is a unit if there is y ∈ R such that xy = 1. We denote the group of units of R by R× . For a subset S of a ring R, we denote by (S) the ideal of R generated by S. (As is standard, we write (x1 , . . . , xn ) for ({x1 , . . . , xn }). Ideals I and J in R are comaximal if I + J = R. Elements a, b ∈ R are comaximal if (a) and (b) are comaximal: (a, b) = R. An indexed family of ideals {Ii } is pairwise comaximal if Ii + Ij = R for all i 6= j, and similarly for pairwise comaximal elements. A domain is a nonzero ring in which x, y 6= 0 =⇒ xy 6= 0. For x, y ∈ R we say x divides y and write x \| y if there is c ∈ R such that cx = y. Elements x and y are associates if y = ux for some u ∈ R× . An element x of a domain is irreducible if it is a nonzero nonunit and x = yz implies y ∈ R× or z ∈ R× . A prime element p ∈ R is an element p ∈ R• for which (p) is a prime ideal. Thus a nonzero nonunit p is prime if and only if p \| ab =⇒ p \| a or p \| b. An atom in a domain R is a principal ideal (x) generated by an irreducible element x. Thus two irreducibles of a domain R determine the same atom if and only if they are associate. (It is more common in the literature for the terms “atom” and “irreducible” to be fully synonymous, but this minor distinction is convenient for our purposes: usually we will count to count irreducibles in a domain up to associates, but sometimes we will want to count irreducibles.) A Furstenberg domain is a domain R in which every nonzero nonunit has an irreducible divisor.1 An atomic domain is a domain R in which for every nonzero nonunit x ∈ R there are irreducible elements f1 , . . . , fn such that x = f1 · · · fn . A unique factorization domain (UFD) is an atomic domain such that if f1 , . . . , fm , g1 , . . . , gn are irreducibles such that f1 · · · fm = g1 · · · gn , then m = n and there is a bijection σ : {1, . . . , m} → {1, . . . , n} such that (fi ) = (gσi ) for all 1 ≤ i ≤ m. Prime elements are irreducible. In general the converse is false! An atomic domain is a UFD iff every irreducible is prime [Cl-CA, Thm. 15.8]. The terminology can be confusing in light of the definition of a prime number p as a positive integer not divisible by any 1 < n < p: this means p is irreducible in Z. But Euclid showed p \| ab =⇒ p \| a or p \| b. From this one can easily show the Fundamental Theorem of Arithmetic: Z is a UFD. A principal ideal domain (PID) is a domain in which each ideal is generated by a single element. Every PID is a UFD. It follows from the Euclidean algorithm that Z is a PID. A Bézout domain is a domain in which every finitely generated ideal is principal. A ring is Noetherian if all of its ideals are finitely generated. Noetherian domains are atomic [Cl-CA, Prop. 15.3]. Thus a PID is precisely a Noetherian Bézout domain. A Dedekind domain is a domain in which each nonzero proper ideal factors uniquely into prime ideals. A domain is Dedekind iff it is Noetherian, of dimension at most one – every nonzero prime ideal is maximal – and integrally closed – every element of the fraction field which satisfies a monic polynomial with coefficients in R lies in R [Cl-CA, Thm. 20.10]. Working in a domain rather than a general ring confers certain advantages: 1The explanation for the terminology comes in §3.1. THE EUCLIDEAN CRITERION FOR IRREDUCIBLES 3 Fact 1. a) Every nonzero ideal in a ring contains a nonzero principal ideal. b) If R is a domain and α ∈ R• , x ∈ R 7→ αx gives a bijection from R to (α). c) Thus for every nonzero ideal I of a domain R we have #I = #R. d) For nonzero ideals I and J of R, I ∩ J contains IJ and thus is nonzero. 2.2. The Euclidean Criterion. A ring R satisfies Condition (E) if for all x ∈ R• , there is y ∈ R such that yx + 1 ∈ / R× . In other words, if x 6= 0 then 1 + (x) 6⊂ R× . By Fact 1a) this is equivalent to: if I is a nonzero ideal of R then 1 + I 6⊂ R× , though we will defer consideration of this restatement until later on. Example 2.1. a) The ring Z satisfies Condition (E). Indeed, Z× = {±1}, so for x ∈ Z• , take y = 1 if x is positive and y = −1 if x is negative); then yx ≥ 1 so yx + 1 ≥ 2. b) For any domain R, the polynomial ring R[t] satisfies Condition (E). Indeed, (R[t])× = R× , so for any x ∈ R[t]• , take y = t. c) R = Z[i] satisfies Condition (E). Indeed Z[i]× = {1, i, −1, −i}, so this is geometrically clear: for any x ∈ Z[i]• , if we multiply it by a y with large enough \|y\|, then yx will be much more than 1 unit away from any point on the unit circle. Proposition 2.2. A domain R with #R > #R× satisfies Condition (E). Proof. For x ∈ R• , the map ι : R → R given by y 7→ yx + 1 is an injection. Thus #ι(R) = #R > #R× , so it cannot be that ι(R) ⊂ R× . And here we go: Theorem 2.3. (The Euclidean Criterion) Let R be a domain, not a field, satisfying Condition (E). a) There is an infinite sequence {an }∞ n=1 of pairwise comaximal nonunits. b) If R is also Furstenberg, it admits an infinite sequence {fn }∞ n=1 of pairwise comaximal irreducibles. Thus {(fn )}∞ n=1 is a sequence of distinct atoms in R. Proof. a) By induction on n. Let a1 ∈ R be a nonzero nonunit. Having chosen a1 , . . . , an pairwise comaximal, by Condition (E) there is y ∈ R such that an+1 := ya1 · · · an + 1 ∈ / R× . Clearly (ai , an+1 ) = R for all 1 ≤ i ≤ n. b) By induction on n. Since R is Furstenberg and not a field, it has an irreducible f1 . Having chosen pairwise comaximal irreducibles f1 , . . . , fn , by Condition (E) there is y ∈ R such that x = yf1 · · · fn + 1 is a nonzero (since f1 ∈ / R× ) nonunit, so x has an irreducible factor fn+1 . For all 1 ≤ i ≤ n we have Y fj )fi , 1 = (x/fn+1 )fn+1 − (y j6=i so fi , fn+1 are comaximal. Finally, if x and y are pairwise comaximal irreducibles, then (x), (y) ( R and (x) + (y) = (x, y) = R, so we must have (x) 6= (y). Here are two applications of the Euclidean Criterion. The first two are immediate. Theorem 2.4. a) For any domain R, R[t] has infinitely many atoms. b) In particular, let D be a UFD and let R = D[t1 , . . . , tn ]. Then R is a UFD satisfying Condition (E), so R has infinitely many nonassociate prime elements. c) The Gaussian integers Z[i] have infinitely many atoms. Since Z[i] is a PID, there are infinitely many nonassociate prime elements. 4 PETE L. CLARK Theorem 2.5. Let R be a Furstenberg domain, not a field, such that #R > #R× . Then R has infinitely many atoms. Theorem 2.6. Let R be a Furstenberg domain, let I be the set of all irreducible elements of R. Then I is either empty (if R is a field) or infinite (otherwise). Proof. Assume I 6= ∅ and fix f ∈ I. If R× is finite, Theorem 2.5 yields infinitely many atoms. If R× is infinite, then {uf \| u ∈ R× } is an infinite subset of I. 2.3. Supplement: Irreducibles in Residue Classes. We switch from an ancient theorem to matters of contemporary interest if we ask for infinitely many primes satisfying certain additional conditions. Here is a result along these lines, relatively modest over Z, but of a general algebraic nature. Lemma 2.7. Let a, b, c be elements of a ring R. If (a, b) = R and c \| a + b, then (a, c) = (b, c) = R. Proof. Let d ∈ R be such that cd = a + b. Then (a, c) ⊃ (a, cd) = (a, a + b) = (a, b) = R. Lemma 2.8. Let R be a domain, not a field, satisfying Condition (E). For any at + b ∈ R[t] with a ∈ R• , there is x ∈ R such that ax + b is a nonzero nonunit. Proof. Put P (t) = at + b. If b = 0, take any nonzero nonunit x ∈ R. If b ∈ R× , by Condition (E) there is x ∈ R such that b−1 ax + 1 ∈ / R× so P (x) = b(b−1 ax + 1) is a nonzero nonunit. If b ∈ R is a nonzero nonunit, take x = 0. The proof of the following result was suggested to me by Paul Pollack. Theorem 2.9. Let R be an atomic domain satisfying Condition (E), let I be a nonzero ideal of R, and let H be a proper subgroup of (R/I)× . Then there are infinitely many pairwise comaximal irreducibles f such that the class of f modulo I lies in (R/I)× \ H. Proof. Let r : R → R/I be the quotient map, let α ∈ R be such that r(α) ∈ (R/I)× \ H, and let β ∈ R be such that αβ − 1 ∈ I \ {0}. Inductively, assume that we have pairwise irreducibles f1 , . . . , fn of R such that (fi , α) = (fi , I) = R for all i and such that r(fi ) ∈ / H. Let P (t) = (αt + 1)(αβ − 1)f1 · · · fn + α ∈ R[t]. (We need to include the base case n = 0, and in this case f1 · · · fn = 1.) By Lemma 2.8 there is x ∈ R such that y = (αx + 1)(αβ − 1)f1 · · · fn + α is a nonzero nonunit, so we get an irreducible factorization y = g1 · · · gs with s ≥ 1. Then r(g1 ) · · · r(gs ) = r(y) = r(α) ∈ (R/I)× \ H, so (gj , I) = 1 for all j and there is at least one gj , say g1 , such that r(g1 ) ∈ / H. Now g1 cannot be associate to any fi ; if so g1 and hence also fi would divide α: if α ∈ R× this contradicts the irreducibility of fi ; if not, this contradicts (fi , α) = 1. THE EUCLIDEAN CRITERION FOR IRREDUCIBLES 5 Moreover y ≡ −f1 · · · fn (mod α) so y ∈ (R/α)× , hence also g1 ∈ (R/α)× , i.e., (g1 , α) = R. Finally, since (αx + 1)(αβ − 1)f1 · · · fn ≡ −f1 · · · fn (mod α), we have ((αx + 1)(αβ − 1)f1 · · · fn , α) = R, so by Lemma 2.7 we have (g1 , (αx + 1)(αβ − 1)f1 · · · fn ) = R so (g1 , fi ) = R for all i. Thus we may take fn+1 = g1 , completing the induction. When R = Z, we get: for any proper subgroup H ( (Z/N Z)× , there are infinitely many prime numbers p such that ±p (mod N ) ∈ / H. Moreover, in this classical case one can run the argument with positive integers only and so get rid of the annoying ±. This is a special case of Dirichlet’s theorem on primes in arithmetic progressions. It is an observation of A. Granville – unpublished by him, but reproduced in [P, Thm. 1.16] – that this case can be proved in an elementary “Euclidean” way. The special case of trivial H – for all N ≥ 3 there are infinitely many primes p 6≡ 1 (mod N ) – is older and better known. It is also simpler – just consider N p1 · · · pn−1 − 1. This case does not use that Z is a UFD, but Granville’s argument does. The most auspicious replacement for coprimality arguments is by comaximality, and that is what we’ve done here. 3. A “Topological” Interlude 3.1. Furstenberg’s Lemma. In this section we will give several proofs of the following result. Theorem 3.1. Let R be a Furstenberg domain with at least one and only finitely many irreducibles f1 , . . . , fn . Then: a) We have #R× = #R. b) More precisely there is a nonzero ideal I of R such that 1 + I ⊂ R× . Theorem 3.1 is the contrapositive of part b) of the Euclidean Criterion, without the information on comaximality. The proofs that we give here are inspired by the famous paper of H. Furstenberg [Fu55]. The essential core of his argument is the observation that in Z the set of elements not divisible by any prime number is ±1. Notice that has nothing to do with the natural ordering of Z that underlies most of the classical proofs of Euclid’s Theorem. In fact the property of Z being used is that Z is a Furstenberg domain. Lemma 3.2. (Furstenberg’s Lemma) T a) A domain R is a Furstenberg domain iff R× = f irreducible R \ (f ). b) In a FurstenbergTdomain with at least one and only finitely many irreducibles f1 , . . . , fn , we have ni=1 (R \ (fi )) = R× . The proof is virtually immediate and is left to the reader. 3.2. Following Furstenberg. Let R be a domain. By Fact 1d), for each x ∈ R, the family C(x) = {x + I \| I is a nonzero ideal of R} 6 PETE L. CLARK is closed under finite intersections, so {C(x)}x∈X is a system of neighborhood bases for a topology on R – let us call it the adic topology – in which U ⊂ R is open iff for all x ∈ U there is a nonzero ideal I with x + I ⊂ U . By Fact 1c), every nonempty open has cardinality #R. Proof of Theorem 3.1: let R be a Furstenberg domain with at least one and only finitely many irreducibles f1 , . . . , fn . Then each (fi ) is open, hence its complement R \ (fi ), being a union of cosets of (fi ), is also open. By Furstenberg’s Lemma T R× = ni=1 (R \ (fi )) is open. Since 1 ∈ R× , we have #R× = #R. More precisely, R× ⊃ 1 + I for some nonzero ideal of R. 3.3. Following Cass-Wildenberg. Let R be a domain, and let F2 be the field of two elements. For an ideal I of R, a function f : R → F2 is I-periodic if f (x + y) = f (x) for all x ∈ X and y ∈ I. Lemma 3.3. Let R be a domain, and let I, I1 , . . . , In be nonzero ideals of R. a) If I2 ⊂ I1 and f : R → F2 is I1 -periodic, it is also I2 -periodic. b) If for all 1 ≤ i ≤ n, fi : R → F2 is Ii -periodic, then the pointwise product f1 · · · fn : R → F2 is I1 · · · In -periodic. c) If f : R → F2 is I-periodic, then for all x ∈ R, we have #{y ∈ R \| f (y) = f (x)} = #R. Proof. a) This is immediate T from the definition. T b) Certainly f1 · · · fn is ni=1 Ii -periodic, and ni=1 Ii ⊃ I1 · · · In . Apply part a). c) Choose a nonzero α ∈ I. Then f (x + Rα) = f (x), and #Rα = #R. Proof of Theorem 3.1: Step 1: For 1 ≤ i ≤ n, let χi : R → F2 be the characteristic function of (fi ); put n Y χ= (1 − χi ). i=1 Each χi is (fi )-periodic, hence so too is 1 − χi , and thus χ is (f1 · · · fn )-periodic. T Moreover χ is the characteristic function of ni=1 (R \ (fi )) = R× . Step 2: Since χ(1) = 1, #R× = {x ∈ R \| χ(x) = 1} = #R: part a). Step 3: More precisely χ(1 + Rf1 · · · fn ) = 1, so Rf1 · · · fn + 1 ⊂ R× : part b). 3.4. Following Mercer. Let R be a domain. Call a subset X ⊂ R lovely if it is of the form x + I for x ∈ R and a nonzero ideal I of R, i.e., if it is a coset of a nonzero ideal. Call a subset X ⊂ R pleasant if it is a union of lovely subsets. If I is a nonzero ideal of R, then R\ I is a union of cosets of I hence pleasant. If X, Y ⊂ R are pleasant sets and x ∈ X ∩ Y , there are nonzero ideals I, J of R such that x + I ⊂ X and x + J ⊂ Y . By Fact 1d) x+(I ∩J) = (x+I)∩(x+J) is a lovely subset of X ∩Y containing x. So X ∩Y is pleasant. By Fact 1c), every nonempty pleasant subset has cardinality #R. Proof of Theorem 3.1: let R be a Furstenberg domain with at least one and only finitely many irreducibles f1 , . . . , fn . By Furstenberg’s Lemma, R× is the finite intersection of complements of nonzero ideals so is pleasant. Since 1 ∈ R× , we have #R× = #R. More precisely, R× ⊃ 1 + I for some nonzero ideal of R. THE EUCLIDEAN CRITERION FOR IRREDUCIBLES 7 3.5. Debriefing. The three proofs given above are generalizations of the proofs of Euclid’s Theorem given by Furstenberg [Fu55], Cass-Wildenberg [CW03] and Mercer [Me09]. The latter two works take the detopologization of Furstenberg’s proof as their goal. Our presentation of the argument of §3.4 differs superficially from Mercer’s. We chose the words “lovely” and “pleasant” precisely because they do not have a commonly understood technical mathematical meaning: had we said “basic” and “open” then the reader’s attention would have been drawn to the fact that since the basic sets are closed under finite intersections, they form the base of a topology. Mercer’s exposition takes pains to point out that the underlying fact here is just that finite intersections of unions are unions of finite intersections. Of course this is a basic logical principle: conjunctions distribute over disjunctions and conversely. Like many basic logical principles it is completely innocuous when used in context (as in our version of the argument). That the pleasant sets form a topology on R is no more and no less than a crisp enunciation of the facts we need to check in the first part of the proof. I find it quite striking (and pleasant!) that the facts can be enunciated in this way, but I must now agree with those who have claimed that there is no essential topological content in Furstenberg’s argument.2 The use of periodic functions involves slightly more packaging, but of a standard kind: it is well known that the Boolean ring 2R of subsets of R can be represented as the ring Maps(R, F2 ) with pointwise addition and multiplication. We recommend wikipedia and Glaymann [Gl67] as references. Glaymann develops this correspondence and applies it to prove such identities as A∆B = C ⇐⇒ B∆C = A ⇐⇒ C∆A = B...in a manner intended to be used in the high school classroom. This is an interesting snapshot of “the new math” near its zenith. 3.6. The Ubiquitous Theorem. Here is a result that complements Theorem 3.1. It is not deep, but it will play a recurring role for us as a common intersection of various constructions and themes. The first proof that we give follows the “topological conceit” of this section. We will give other, simpler, proofs later on. Theorem 3.4. Let R be a domain, not a field, with only finitely many maximal ideals m1 , . . . , mn . Then: a) We have #R× = #R. b) More precisely there is a nonzero ideal I of R such that 1 + I ⊂ R× . Proof. We endow R with the topology for which, for x ∈ R, C(x) = {x + m \| m is a maximal ideal of R} is a neighborhood subbase at x: that is, U ⊂ R is open iff for all x ∈ U there is a subset J ⊂ {1, . . . , n} such that \ \ (x + mi ) = x + mi ⊂ U. i∈J i∈J 2Furstenberg does not claim a topological proof of the infinitude of the primes but rather a “topological” proof of the infinitude of the primes. 8 PETE L. CLARK T Fact 1 gives i∈J mi ) (0), so every nonempty open has cardinality #R. Each R \ mi , being a union of cosets of mi , is also open. Therefore R× = n \ (R \ mi ) i=1 × is open. Since 1 ∈ R× we have T #R =×#R. More precisely T there is a subset J ⊂ {1, . . . , n} such that 1 + i∈J mi ⊂ R , and thus also 1 + ni=1 mi ⊂ R× . 3.7. Supplement: Further Topologies on a Domain. Here is a common generalization of Theorems 3.1 and 3.4: let J be a family of nonzero T ideals of a domain R,T and suppose there are I1 , . . . , In ∈ J such that R× = ni=1 (R \ Ii ). Then 1 + ni=1 Ii ⊂ R× , so in particular #R× = #R. Look again at Theorem 3.1: instead of taking J to be the family of all nonzero ideals, we could take J = {(f1 ), . . . , (fn )} and endow R with the unique translationinvariant topology with J as a neighborhood subbase at 0. This coarsens the adic T topology3 so that being open yields the sharper conclusion 1 + ni=1 (fi ) ⊂ R× . In × particular 1 + (f1 · · · fn ) ⊂ R . We are back to a version of Euclid’s argument. The adic topology on Z is not very interesting as a topological space: it is countably infinite, metrizable, totally disconnected and without isolated points, hence homeomorphic to the Euclidean topology on Q. In [Go59], Golomb proved Euclid’s Theorem using the topology on Z+ with base the one-sided arithmetic progressions {an + b \| n ∈ Z+ } for coprime a, b ∈ Z+ . Golomb’s topology makes Z+ into a countably infinite connected Hausdorff space...which is already interesting. In a domain R that is not a field, we may consider the Golomb topology with neighborhood base at x ∈ R given by C(x) = {x + I \| I is a nonzero ideal with (x, I) = R}. In this topology every maximal ideal is closed, so in a domain that is not a field with only finitely many maximal ideals m1 , . . . , mn , R× is open and thus contains 1 + I for some nonzero ideal I. We get another proof of Theorem 3.4. The Golomb topology is never Hausdorff: in fact {0} = R. However, the induced topology on R• can be (it is for Z). We leave further exploration to the reader. 4. Connections With Ideal Theory For a ring R, we denote by MaxSpec R the set of all maximal ideals of R. 4.1. Comaximal Ideals. Lemma 4.1. Let {In }∞ n=1 be a sequence of pairwise comaximal proper ideals in a ring R. Then MaxSpec R is infinite. Proof. For n ∈ Z+ , let mn be a maximal ideal containing In . If for n1 6= n2 we had mn1 = mn2 then R = In1 + In2 ⊂ mn1 , contradiction. In particular, part a) of the Euclidean Criterion implies that a domain that is not a field and that satisfies Condition (E) has infinitely many maximal ideals. Thus we get another proof of Theorem 3.4....but by no means our last. 3The adic topology on a domain is always Hausdorff, but in a Furstenberg domain with finitely many irreducibles, this new topology is not. THE EUCLIDEAN CRITERION FOR IRREDUCIBLES 9 4.2. Euclid Meets Jacobson. Now is the time to examine the more explicitly ideal-theoretic statement of Condition (E): for all nonzero ideals I, we have 1 + I 6⊂ R. Some readers will now see – or will have already seen – the connection with the Jacobson radical, but we will not assume a prior familiarity. In fact we will use the Euclidean Criterion to motivate a self-contained discussion of this and other ideal-theoretic concepts. Proposition 4.2. [Cl-CA, Prop. 4.14] For a ring R, let \ m, J(R) = m∈MaxSpec R the Jacobson radical of R. For x ∈ R, the following are equivalent: (i) x ∈ J(R). (ii) For all y ∈ R, yx + 1 ∈ R× . Proof. (i) =⇒ (ii): By contraposition: suppose there is y ∈ R such that z = yx + 1 ∈ / R× . Then z lies in some maximal ideal m. If also x ∈ m, then yx ∈ m and thus also z − yx = 1 ∈ m, contradiction. So x does not lie in m and thus x ∈ / J(R). (ii) =⇒ (i): Again by contraposition: suppose that there is a maximal ideal m such that x ∈ / m. Then m ( (m, x), so (m, x) = R. It follows that there is m ∈ m and y ∈ R such that m + yx = 1. Thus (−y)x + 1 = −m ∈ m so is not a unit. We get immediately: Corollary 4.3. A ring R satisfies Condition (E) iff J(R) = (0). This gives a third proof of Theorem 3.4: if R has only finitely many maximal ideals m1 , . . . , mn , then n n Y \ mi ) {0}. mi ⊃ J(R) = i=1 i=1 Apply Corollary 4.3. A ring with zero Jacobson radical is called semiprimitive.4 4.3. Some Questions and Some Answers. We now raise some natural questions...and answer them. Question 4.4. In part b) of the Euclidean Criterion, must we assume that R is a Furstenberg domain? Question 4.5. A semiprimitive domain, not a field, has infinitely many maximal ideals. Must a domain with infinitely many maximal ideals be semiprimitive? Question 4.6. Let R be a Furstenberg domain. a) If R is not semiprimitive, can it still have infinitely many atoms? b) Can R have finitely many maximal ideals and infinitely many atoms? 4Or Jacobson semisimple or J-semisimple. 10 PETE L. CLARK Example 4.7. The ring Z of all algebraic integers is not a Furstenberg domain. In fact it is an antimatter domain: there are no irreducibles whatsoever: if z is an algebraic integer then so is z 1/2 , so we can always factor z = z 1/2 z 1/2 . Moreover Z × is not a field: for all integers n ≥ 2, if n ∈ Z then n1 ∈ Z ∩ Q = Z, contradiction. If I is a nonzero ideal of Z then the constant coefficient of the minimal polynomial of a nonzero element α ∈ I is a nonzero integer in I. It follows that if J(Z) 6= 0 then there is N ∈ Z+ that is contained in every m ∈ MaxSpec Z. Choose a prime number p ∤ N . Then p is not a unit in Z – otherwise 1p ∈ Z ∩ Q = Z – so there is at least one maximal ideal mp of Z containing p. (In fact the set of maximal ideals of Z containing p has continuum cardinality.) Then mp ⊃ (N, p) = Z: contradiction. So the answer to Question 4.4 is yes: a semiprimitive domain that is not a field can have no irreducibles whatsoever. The following result answers Questions 4.5 and 4.6 for Dedekind domains and shows that the Euclidean Criterion is, in principle, completely efficacious in determining whether a Dedekind domain has infinitely many atoms. Theorem 4.8. For a Dedekind domain R that is not a field, the following are equivalent: (i) R is semiprimitive. (ii) R has infinitely many maximal ideals. (iii) R has infinitely many atoms. Proof. We know (i) =⇒ (ii) in any domain. (ii) =⇒ (i): in a Dedekind domain, any nonzero element is contained in only finitely many maximal ideals. So in fact for any infinite subset M ⊂ MaxSpec R T we have m∈M m = (0). (i) =⇒ (iii): Dedekind domains are Noetherian, hence Furstenberg domains, so the Euclidean Criterion applies. (iii) =⇒ (i): By contraposition: a Dedekind domain with finitely many maximal ideals is a PID [Cl-CA, Thm. 20.6], and in a PID there is no distinction between maximal ideals, principal ideals generated by prime elements, and atoms. Question 4.9. Let K be a number field, with ring of integers ZK . The set of prime numbers is an infinite sequence of pairwise comaximal nonunits of ZK , so (as is well known!) ZK has infinitely many prime ideals and thus is semiprimitive. When K = Q or is imaginary quadratic, the finiteness of Z× K leads to a direct verification of Condition (E). Is there a similarly direct verification for all K? This is a question we will leave to the reader to address. Proposition 4.10. Let R be a Noetherian domain of dimension at most one (nonzero prime ideals are maximal). If MaxSpec R is infinite, then R is semiprimitive and thus has infinitely many pairwise comaximal irreducibles. Proof. If R is not semiprimitive, then every maximal ideal m of R is a minimal prime ideal of R/J(R). Since R is Noetherian, so is R/J(R), and a Noetherian ring has only finitely many minimal prime ideals [Cl-CA, Thm. 10.13]. A Jacobson ring is a ring in which every prime ideal is the intersection of the maximal ideals containing it. Since in a domain (0) is prime, a Jacobson domain THE EUCLIDEAN CRITERION FOR IRREDUCIBLES 11 must be semiprimitive. Any quotient of a Jacobson ring is again a Jacobson ring. If R is a Jacobson ring and S is a commutative, finitely generated R-algebra then S is a Jacobson ring [Cl-CA, Thm. 12.15, 12.21]. So: Theorem 4.11. a) A Jacobson Furstenberg domain that is not a field has infinitely many pairwise comaximal irreducibles. b) Let F be a field, and let p be a prime but not maximal ideal of F [t1 , . . . , tn ]. Then the ring R = F [t1 , . . . , tn ]/p – i.e., a coordinate ring of an integral affine variety of positive dimension – has infinitely many pairwise comaximal irreducibles. c) A domain R that is finitely generated over Z and not a field has infinitely many pairwise comaximal irreducibles. To sum up: if we want to see a domain that has infinitely many maximal ideals but is not semiprimitive, it cannot be finitely generated over a field, and if Noetherian it must have a nonzero prime ideal that is not maximal. This cues us up for the following example, which gives a negative answer to Question 4.5. Example 4.12. Consider the ring Z[[t]] of formal power series with integral coefficients. It is not hard to show that Z[[t]] is an atomic domain. In fact Z[[t]] is a Noetherian UFD [Cl-CA, Thm. 15.32]. Since 1 + (t) ⊂ Z[[t]]× , the Jacobson radical J(Z[[t]]) contains (t) and is thus nonzero. Since J(Z[[t]])) 6= (0), the hypotheses of the Euclidean Criterion do not apply. Nevertheless there are infinitely many pairwise comaximal prime elements, namely the prime numbers! Hence there are infinitely many maximal ideals. Here we could have replaced Z with any PID with infinitely many maximal ideals. Thus the answer to Question 4.6a) is yes: moreover a nonsemiprimitive domain can have infinitely many comaximal irreducibles. Example 4.13. Let k be a field. Recall that k[x, y] is a UFD, and let K = k(x, y) be its fraction field. Let R be the subring of k(x, y) consisting of rational functions f (x,y) g(x,y) that, when written in lowest terms, have g(0, 0) 6= 0. Then R is itself a UFD – factorization in R proceeds as in k[x, y] except that the prime elements p(x, y) ∈ k[x, y] such that p(0, 0) 6= 0 become units in R – in which an element (x,y) is a unit iff f (0, 0) 6= 0. Thus m = { fg(x,y) \| f (0, 0) = 0} is the unique maximal ideal, so J(R) = m and R is very far from being semiprimitive. Nevertheless it has infinitely many prime elements, e.g. {y − xn }∞ n=1 . In more geometric language, the irreducibles are the irreducible curves in the affine plane passing through (0, 0). Thus the answer to Question 4.6b) is yes. However, there is more to say. The preceding example can be vastly generalized using the following striking result. Theorem 4.14. (Cohen-Kaplansky [CK46]) Let R be an atomic domain with finitely many atoms. Then: a) R has only finitely many prime ideals. b) R is Noetherian. c) Every nonzero prime ideal of R is maximal. Proof. a) In an atomic domain R, whenever a prime ideal p of R contains a nonzero element x, we may factor x = f1 · · · fr into irreducibles and thus see that p contains some irreducible element f dividing x. Thus, given any set of generators of a prime ideal p we can replace it with a set of irreducible generators. In a set of generators 12 PETE L. CLARK of an ideal, replacing each element by any one of its associates does not change the ideal generated, and thus if we have only finitely many nonassociate irreducibles we can only generate finitely many prime ideals. b) It follows from the proof of part a) that every prime ideal of R is finitely generated. By a famous result of Cohen [Cl-CA, Thm. 4.26], all ideals are finitely generated. (This is an instance of the prime ideal principle of Lam-Reyes [LR08].) c) If not, there are prime ideals (0) ( p1 ( p2 . Since R is Noetherian, this implies there are infinitely many prime ideals between (0) and p2 [Cl-CA, Cor. 8.46]. A Cohen-Kaplansky domain is an atomic domain with finitely many atoms. The work [CK46] does not give a complete classification: we are left with the case of a Noetherian domain R with finitely many nonzero prime ideals, all of which are maximal. If R is a Dedekind domain, then by Theorem 4.8 there are only finitely many atoms. So the remaining case is when R is not integrally closed in its fraction field, in which case the integral closure R is a Dedekind domain with finitely many prime ideals [Cl-CA, Cor. 18.8]. One might expect that this forces R to be Cohen-Kaplansky. This need not be the case! Example 4.15. Let k be a field, and consider the subring R = k[[t2 , t3 ]] = k + t2 k[[t]] P n of the formal power series ring k[[t]]. For 0 6= f = ∞ n=0 an t ∈ k[[t]], we define v(f ) to be the least n such that an 6= 0. Then v is a discrete valuation on k[[t]], and the only nonzero prime ideal of k[[t]] is (t) = {f ∈ R \| v(f ) > 0} ∪ {0}. In particular, k[[t]] is a PID. So is the (isomorphic!) subring k[[t2 ]], and {1, t3 } is a generating set for R as a k[[t2 ]]-module, so by standard PID structure theory, every ideal of R canP be generated by two elements. Thus R is Noetherian, hence atomic. n × For f = a0 + ∞ ⇐⇒ a0 6= 0, and thus n=2 an t ∈ R, we have f ∈ R m={ ∞ X an tn } = (t2 , t3 ) n=2 is the unique maximal ideal of R. We will give a complete description of the atoms of R. First we claim that f ∈ R is irreducible iff v(f ) ∈ {2, 3}. Indeed a nontrivial factorization f = xy involves v(x), v(y) ≥ 2 hence v(f ) ≥ 4; conversely, if v(f ) ≥ 4 then f = t2 tf2 is a nontrivial factorization. Since k × ⊂ R× , every irreducible is associate to one of the form t2 + X an tn , (v(f ) = 2 case) n≥3 or one of the form t3 + X an tn , (v(f ) = 3 case). n≥4 Associate elements have the same valuation, so certainly no irreducible P of the first type is associate to an irreducible of the second type. We claim that t2 + n≥3 an tn P P is associate to t2 + n≥3 bn tn iff a3 = b3 and t3 + n≥3 an tn is associate to P t3 + n≥3 bn tn iff a4 = b4 . This can be done by direct computation: (t2 + a3 t3 + a1 4t4 + a5 t5 + . . .)(1 + u2 t2 + u3 t3 + . . .) = t2 + a3 t3 + (a4 + u2 )t4 + (a5 + a3 u2 + u3 )t5 + . . . , THE EUCLIDEAN CRITERION FOR IRREDUCIBLES 13 so a3 = b3 and there is a unique choice of u2 , u3 , . . . leading to an = bn for all n ≥ 4. The v(f ) = 3 case is similar. Thus there are precisely 2#k atoms, and R is Cohen-Kaplansky iff k is finite. Example 4.16. (Anderson-Mott [AM92, Cor. 7.2]) For a prime power q and d, e ∈ Z+ , the ring R = Fq + te Fqd [[t]] is a Cohen-Kaplansky domain with exactly d −1 d(e−1) q irreducibles, none of which are one nonzero prime ideal and exactly e qq−1 prime unless (d, e) = (1, 1). The paper [CK46] was mostly forgotten for many years, until the breakthrough work of Anderson and Mott [AM92] gave a complete characterization of CohenKaplansky domains. In fact they give 14 characterizations! Here is one: Theorem 4.17. (Anderson-Mott [AM92]) For an atomic domain R, the following are equivalent: (i) R is a Cohen-Kaplansky domain. (ii) R is Noetherian of dimension at most one (nonzero prime ideals are maximal), has finitely many prime ideals, the integral closure R of R is finitely generated as an R-module, # MaxSpec R = # MaxSpec R, and for all nonprincipal ideals m ∈ MaxSpec R, R/m is finite. Example 4.18. Let k be a field of characteristic different from 2 or 3, and consider: • R1 : the localization of k[x, y]/(y 2 − x3 − x) at m0 = (x, y). • R2 : the localization of k[x, y]/(y 2 − x3 − x2 ) at m0 = (x, y). • R3 : the localization of k[x, y]/(y 2 − x3 ) at m0 = (x, y). Then: • R1 is always Cohen-Kaplansky (it is a Dedekind domain with one maximal ideal). • R2 is never Cohen-Kaplansky (# MaxSpec R2 = 2 > 1 = # MaxSpec R2 ). • R3 is Cohen-Kaplansky iff k is finite. 4.4. Euclid Beyond Atomicity. In the case of an atomic domain, the part of the Euclidean Criterion that yields infinitely many maximal ideals is much weaker than the Cohen-Kaplansky Theorem. However, there is life beyond atomic domains. Example 4.19. Let Hol(C) be the ring of entire functions f : C → C. For f ∈ Hol(C), put Z(f ) = {z ∈ C \| f (z) = 0}. If f, g ∈ Hol(C)• , then Z(f ) and Z(g) are countable sets, hence so is Z(f g) = Z(f ) ∪ Z(g), so f g 6= 0. Thus H(C) is a domain. The map z0 ∈ C 7→ (z − z0 ) gives a bijection from C to the atoms of Hol(C). An element f ∈ Hol(C) is a unit iff Z(f ) = ∅, and a nonzero nonunit f is a (finite!) product of atoms iff Z(f ) is finite and nonempty. So Hol(C) is not atomic – consider e.g. f (z) = sin z – but it is Furstenberg: if f is a nonzero nonunit, then f vanishes at some z0 ∈ C and thus is divisible by the irreducible element z − z0 . Moreover Hol(C) satisfies Condition (E): if f ∈ Hol(C)• 1 then there is w ∈ C such that f (w) 6= 0. Let g = z −w− f (w) . Then (gf +1)(w) = 0, × so gf + 1 ∈ / Hol(C) . Thus the Euclidean Criterion applies in Hol(C). Theorem 4.20. Let 1 ≤ α ≤ β ≤ γ be cardinal numbers. There is a domain R satisfying all of the following properties: (i) R is a Bézout domain: every finitely generated ideal is principal. (ii) R has exactly α atoms, each of which is a maximal ideal. 14 PETE L. CLARK (iii) R has exactly β maximal ideals. (iv) R has exactly γ nonzero prime ideals. (v) R is an atomic domain iff α = β = γ < ℵ0 . (vi) R is a Furstenberg domain iff α = β. (vii) R is semiprimitive iff β ≥ ℵ0 . We postpone the proof of Theorem 4.20 in order to discuss its significance. By taking α = β and γ ≥ ℵ0 we get Furstenberg domains with any number α ≥ 1 of irreducibles and any number γ ≥ max(α, ℵ0 ) nonzero prime ideals. In particular, a Furstenberg domain can have any finite, positive number of irreducibles and any infinite number of prime ideals, so the Cohen-Kaplansky Theorem does not extend from atomic domains to Furstenberg domains. For any α = β ≥ ℵ0 and γ ≥ α we get a semiprimitive Furstenberg domain that is not an atomic domain. Now we come to the proof of Theorem 4.20, which requires somewhat more specialized results. A completely self-contained presentation would require more space than we want to devote here. So we will make use of the material of [FS, Ch. II and III], and our treatment will be at the level of a detailed sketch. Let R be a domain with fraction field K. To x ∈ K • we attach the principal fractional ideal (x) = {ax \| a ∈ R}. When x ∈ R, this coincides with the usual notion of a principal ideal. For x, y ∈ K • we have (x) = (y) iff there is u ∈ R× such that y = ux. The principal fractional ideals of K form a commutative group under pointwise multiplication: we have (x)(y) = (xy). We call this the group of divisibility of R and denote it G(R). It is partially ordered by reverse inclusion: that is, for x, y ∈ K • we put (x) ≤ (y) iff (y) ⊃ (x). This order reversal is actually rather familiar: for x, y ∈ K × , we write x \| y ⇐⇒ xy ∈ R, and then we have x \| y if (x) ⊃ (y): to contain is to divide. Let {Gi }i∈IL be an indexed family of nonzero totally ordered commutative groups, and let G = i∈I Gi be the direct sum endowed with the pointwise partial ordering: x ≤ y iff xi ≤ yi for all i ∈ I. Let πi : G → Gi be projection onto the ith coordinate. By the Kaplansky-Jaffard-Ohm Theorem [FS, Thm. III.5.3] ∼ there is a Bézout domain R and an isomorphism ϕ : G(R) → G of partially ordered commutative groups. See [FS, Example III.5.4]. Let v be the composϕ L ite K × → K × /R× → I∈I Gi . Then the maximal ideals of R are precisely mi = {x ∈ R \| (πi ◦ v)(x) > 0} ∪ {0} for i ∈ I. Thus no element of R• lies in infinitely many maximal ideals, so R is semiprimitive iff I is infinite. An atom in a partially ordered commutative group is a minimal positive element. This is a direct generalization of our previous use of the term: if R is a domain, the minimal positive elements of the group of divisibility G(R) are precisely the principal fractional ideals (x) for an irreducible element x ∈ R. For every atom x ∈ G, there is i ∈ I such that xi is an atom of Gi and xj = 0 for all j 6= i, and conversely all such elements give atoms of G. Since each Gi is totally ordered, it has at most one atom, the least positive element of Gi if such an element exists. It follows that R is Furstenberg iff each Gi has a least positive element. Similarly, a nonzero nonunit x ∈ R factors into irreducibles iff v(x) ∈ G is a sum of atoms iff for all i ∈ I, Gi has a least positive element ai and vi (r) = nai for some n ∈ Z+ . Thus R is an atomic domain iff each Gi ∼ = Z. The domain R is h-local: each nonzero prime ideal is contained in a unique maximal ideal [FS, loc. cit.]. The nonzero prime ideals contained in mi correspond THE EUCLIDEAN CRITERION FOR IRREDUCIBLES 15 bijectively to the proper convex subgroups of Gi . (A subset Y of a totally ordered set X is convex if for all x < y < z ∈ X, if x, z ∈ Y then also y ∈ Y .) We will take each Gi to be a lexicographic product of copies of subgroups of (R, +) indexed by an ordinal η. Then the convex subgroups of of Gi are precisely {Hδ }0≤δ≤η , where Hδ is the set of all elements of Gi with j-coordinate zero for all j < δ. So there are #η nonzero prime ideals in mi . We will take a family of nonzero totally ordered commutative groups Gi parameterized by i ∈ β: this gives us β maximal ideals, and R is semiprimitive iff β ≥ ℵ0 . We are left to choose the groups Gi in terms of α and γ so as to attain the other assertions. We define an ordinal η: if γ is finite, it is the positive integer γ − β + 1; if γ is infinite, it is the successor ordinal to γ (what matters in this case is that η is a well-ordered set of cardinality γ and with a largest element). There are cases: • If α = β = γ < ℵ0 , we take R to be a PID with γ nonzero prime ideals. • If α = β and γ ≥ min(β + 1, ℵ0 ) we take Gi = Z for all 0 < i ∈ β. We take G0 to be the Cartesian product of copies of Z indexed by η, endowed with the lexicographic ordering. Then G0 has a least positive element: the element that is 0 in all factors but the last and 1 in the last factor. So all Gi have least elements and R is a Furstenberg domain. Moreover η ≥ 2 so G0 ∼ 6 Z and R is not an atomic = domain. It has (β − 1) + #η = γ nonzero prime ideals. • If α < β, we take G0 to be the Cartesian product of copies of Z indexed by η, for 1 ≤ i < α we take Gi = Z, and for i ≥ α we take Gi = R. 4.5. Supplement: Rings With Infinitely Many Maximal Ideals. Let us briefly consider the case of an arbitrary commutative ring. Though others have done so (see e.g. [AVL96]), it is beyond our ambitions to pursue a factorization theory in the presence of zero divisors. But we can still ask for criteria under which there are infinitely many maximal ideals. In this more general context J(R) = (0) is no longer sufficient: e.g. J(C × C) = 0 and there are only two maximal ideals. Nevertheless both Euclid and Jacobson have a role to play. Proposition 4.21. [Cl-CA, Prop. 4.15] Let I be an ideal of R contained in the Jacobson radical. Then for all x ∈ R, if the image of x in R/I is a unit, then x is a unit. In particular the natural map R× → (R/I)× is surjective. Proof. If the image of x in R/I is a unit, then there is y ∈ R such that xy ≡ 1 (mod I), i.e., xy − 1 ∈ I ⊂ J(R). Thus for every maximal ideal m of R, xy − 1 ∈ m so we cannot have x ∈ m. So x lies in no maximal ideal of R and thus x ∈ R× . Theorem 4.22. (Dubuque [Du10]) Let R be an infinite ring. If #R > #R× , then MaxSpec R is infinite. Proof. We will show by induction on n that for all n ∈ Z+ , R has n maximal ideals. Base Case: Since R is infinite, it is nonzero and thus it has a maximal ideal m1 . Induction Step: Let m1 , . . . , mm be maximal ideals, and put m Y mi . I= i=1 × Case 1: Suppose I + 1 ⊂ R . Then #I ≤ #R× . Moreover I ⊂ J(R), so by × × Proposition 4.21 R× → (R/I)× is surjective. It follows Qn that #(R/I) ≤ #R < ∼ #R: by the Chinese Remainder Theorem, R/I = i=1 R/mi , hence there is an 16 PETE L. CLARK injection (R/mi )× → (R/I)× . Putting the last two sentences together we conclude #(R/mi )× < #R, and thus, since R/mi is a field and R is infinite, #R/mi = #(R/mi )× + 1 < #R. Finally this gives the contradiction n Y #R = #I · #R/I = #I · #R/mi < (#R)n+1 = #R. i=1 × Case 2: So there is x ∈ I + 1 \ R . Let mn+1 be a maximal ideal containing x. For all 1 ≤ i ≤ n we have x − 1 ∈ I ⊂ mi , so 1 = x + (1 − x) ∈ mn+1 + mi . So mn+1 is an (n + 1)st maximal ideal of R, completing the induction step. A special case of Theorem 4.22 appears in [K, § 1.1, Exc. 8]. For a ring R, consider the quotient R/J(R). The maximal ideals of R/J(R) correspond to the maximal ideals of R containing J(R) – that is, to the maximal ideals of R. Thus R/J(R) is semiprimitive. Thus we can replace any ring with a semiprimitive ring without changing its MaxSpec. However this “Jacobson semisimplification” need not carry domains to domains: e.g. Q if R is a domain with 2 ≤ n < ℵ0 maximal ideals m1 , . . . , mn , then R/J(R) ∼ = ni=1 R/mi . Here is a generalization. Theorem 4.23. a) For a ring R, the following are equivalent. (i) R has only finitely many maximal ideals. (ii) R/J(R) is a finite product of fields. (iii) R/J(R) has only finitely many ideals. (iv) R/J(R) is Artinian (i.e., there are no infinite descending chains of ideals). b) A semiprimitive ring with finitely many maximal ideals has finitely many ideals. Proof. a) (i) =⇒ (ii): If the maximal ideals of R are m1 , . . . , mn , then by the Chinese Remainder Theorem [Cl-CA, Thm. 4.18] we have R/J(R) = R/ n \ i=1 mi ∼ = n Y R/mi . i=1 (ii) =⇒ (iii) =⇒ (iv) immediately. (iv) =⇒ (i): Maximal ideals of R/J(R) correspond bijectively to maximal ideals of R. And an Artinian ring has only finitely many maximal ideals [Cl-CA, Thm. 8.31]. b) This follows from part a). 5. But What About Primes? Our take on Euclid’s argument has been as a criterion for the existence of irreducibles. The distinction evaporates in a UFD. A PID with only finitely many prime ideals is a UFD with only finitely many principal prime ideals. It turns out that the converse is also true.5 Theorem 5.1. Let R be a UFD, not a field, with only finitely many atoms. Then R is a PID with finitely many prime ideals and #R = #R× . 5Theorem 5.1 is known to the experts: see e.g. [Za08]. THE EUCLIDEAN CRITERION FOR IRREDUCIBLES 17 Proof. A UFD with finitely many nonassociate prime elements is a Cohen-Kaplansky domain, so MaxSpec R is finite and #R = #R× by Theorem 3.4. By Theorem 4.14 every nonzero prime ideal of R is maximal. The proof of Theorem 4.14a) shows: every nonzero prime ideal p contains a prime element p. Since (p) is maximal, we have p = (p). Thus every prime ideal is principal, so R is a PID [Cl-CA, Thm. 4.25]. (This is another case of the Lam-Reyes Prime Ideal Principle.) Let us now move away from UFDs. From Example 4.15, we deduce: Theorem 5.2. Let κ ≥ ℵ0 be a cardinal. There is a Noetherian domain R with exactly one nonzero prime ideal, exactly κ irreducibles and no prime elements. Proof. Let k be a field of cardinality κ, e.g. k = Q({tα \| α ∈ κ}). By Example 4.15, R = k[[t2 , t3 ]] is a Noetherian domain with one nonzero prime ideal m = (t2 , t3 ) and 2κ = κ irreducibles. Since m is not principal, R has no prime elements. Cohen-Kaplansky showed that an atomic domain that is neither a field nor a UFD must have at least 3 atoms [CK46, p. 469]. Their argument is a nice one: we must have at least one nonprime irreducible f1 . Since (f1 ) is not prime, it is properly contained in some prime ideal p, which must therefore contain a nonassociate irreducible f2 . Since f1 + f2 ∈ p, f1 + f2 is not a unit and therefore it is divisible by an irreducible f3 , which cannot be associate to either f1 or f2 . Finally, we consider Dedekind domains. Question 5.3. Let R be a Dedekind domain with infinitely many prime ideals. Must R have infinitely many atoms? In an important classical case the answer is yes, as most number theorists know. Theorem 5.4. For each number field K, the ring of integers ZK has infinitely many nonassociate prime elements. Proof. Step 1: For any number field L, the number of rational primes that split completely in L is infinite. This is a special case of the Chebotarev Density Theorem, which however can be proved in a more elementary way, as was shown in [Po10]. Using some basic algebraic number theory which we omit here, it comes down to showing that for every nonconstant polynomial f ∈ Z[t], the set of prime numbers p dividing f (n) for some n ∈ Z is infinite. If f (0) = 0 this is trivial. If f (0) 6= 0, let p1 , . . . , pk be the prime divisors of f (0) (we allow k = 0) and let q1 , . . . , qℓ be any finite set of primes not dividing f (0). For 1 ≤ i ≤ k, let ai be such that pai i \| f (0) and pai i +1 ∤ f (0). For N ∈ Z+ consider xN = f (N pa1 1 +1 · · · pakk +1 q1 · · · qℓ ). Then for all 1 ≤ i ≤ k, pai i +1 ∤ xN and for all 1 ≤ j ≤ ℓ, qj ∤ xN , so the set of N for which xN is not divisible by some prime other than p1 , . . . , pk , q1 , . . . , qℓ is finite. Step 2: A prime ideal p of a number field is principal iff it splits completely in the Hilbert class field K 1 of K. So every prime ideal p of K lying above any one of the infinitely many prime numbers p that split completely in K 1 is principal. Looking at the above argument, one wonders: were we working working too hard? Perhaps some simple argument gives a general affirmative answer to Question 5.3. In fact Question 5.3 was answered negatively by Claborn [Cl65, Example 1.5]. 18 PETE L. CLARK The construction is impressively direct: start with a Dedekind domain A that is not a PID, let P be the set of prime elements of R and pass to R = A[{ p1 }p∈P ]. The prime ideals of R are precisely the nonprincipal prime ideals of A, which remain nonprincipal in R! This prime-killing construction also appears in a work of Samuel [Sa64, p. 17, Thm. 6.3] and is therein attributed to Nagata (cf. [N57, Lemma 2]). For a Dedekind domain A, write Cl A for its ideal class group: the quotient of the monoid of nonzero ideals of A under the equivalence relation I ∼ J iff there are α, β ∈ A• with (α)I = (β)J. In the setting of the prime-killing construction – i.e., R is the localization of A at the multiplicative subset generated by the prime elements – we have [Sa64], [Cl65] that Cl R ∼ = Cl A. Theorem 5.5. Let κ be an infinite cardinal. There is a Dedekind domain R with exactly κ atoms and no prime elements. Proof. We will use some properties of “elliptic Dedekind domains”: for more details, see [Cl09, §2.4]. Let k be an algebraically closed field of characteristic 0 and cardinality κ, and put R = k[x, y]/(y 2 − x3 − x). Then R is a Dedekind domain, and by the Nullstellensatz the nonzero prime ideals of R are all of the form p(x0 ,y0 ) = (x − x0 , y − y0 , y 2 − x3 − x) for pairs (x0 , y0 ) ∈ k 2 such that y02 = x30 +x0 . In other words, they are the k-rational points on the projective elliptic curve E : y 2 z = x3 + xz 2 , excluding the point at infinity O = [0 : 1 : 0]. Moreover, by the Riemann-Roch Theorem, since [x0 : y0 : 1] 6= O, the prime ideal p(x0 ,y0 ) is not principal. Thus R is a Dedekind domain with # MaxSpec R = #R = κ and without prime elements. Because R is Dedekind, every ideal can be generated by two elements [Cl-CA, Thm. 20.12]. This, together with the fact that Dedekind domains are atomic domains, implies that for all p ∈ MaxSpec R there are irreducibles pp , qp such that p = (pp , qp ). Thus if λ is the number of irreducibles of R we have κ = # MaxSpec R ≤ λ2 ≤ (#R)2 = κ2 = κ, so λ2 = κ. Since κ is infinite, so is λ and thus λ = λ2 = κ. References [AM92] [AVL96] [Cl-CA] [Cl09] [CK46] [Cl65] [Co73] [CS12] [CW03] [Du10] [FS] [Fu55] D.D. Anderson and J.L. Mott, Cohen-Kaplansky domains: integral domains with a finite number of irreducible elements. J. Algebra 148 (1992), 17-41. D.D. Anderson and S. Valdes-Leon, Factorization in commutative rings with zero divisors. Rocky Mountain J. Math. 26 (1996), 439-480. P. L. Clark, Commutative Algebra, http://math.uga.edu/~ pete/integral.pdf. P.L. Clark, Elliptic Dedekind domains revisited. Enseignement Math. 55 (2009), 213– 225. I.S. Cohen and I. Kaplansky, Rings with a finite number of primes. I. Trans. Amer. Math. Soc. 60 (1946), 468-477. L.E. Claborn, Dedekind domains and rings of quotients. Pacific J. Math. 15 (1965), 59–64. P.M. Cohn, Unique factorization domains. Amer. Math. Monthly 80 (1973), 1–18. J. Coykendall and C. Spicer, Cohen-Kaplansky domains and the Goldbach conjecture. Proc. Amer. Math. Soc. 140 (2012), 2227-2233. D. Cass and G. Wildenberg, Math Bite: A Novel Proof of the Infinitude of Primes, Revisited Mathematics Magazine, Vol. 76 (2003), 203. W.G. Dubuque, http://math.stackexchange.com/questions/201 L. Fuchs and L. Salce, Modules over non-Noetherian domains. Mathematical Surveys and Monographs, 84. American Mathematical Society, Providence, RI, 2001. H. Furstenberg, On the infinitude of primes, Amer. Math. Monthly 62 (1955), 353. THE EUCLIDEAN CRITERION FOR IRREDUCIBLES [Gl67] [Go59] [K] [LR08] [Me09] [N57] [P] [Po10] [Sa64] [Za08] 19 M. Glaymann, Characteristic Functions and Sets. Mathematics Teacher, 60 (1967), 775–778. S.W. Golomb, A connected topology for the integers. Amer. Math. Monthly 66 (1959), 663-665. I. Kaplansky, Commutative rings. Allyn and Bacon, Inc., Boston, Mass. 1970. T.Y. Lam and M. Reyes, A prime ideal principle in commutative algebra. J. Algebra 319 (2008), 3006-3027. I.D. Mercer, On Furstenberg’s Proof of the Infinitude of Primes. Amer. Math. Monthly 116 (2009), 355–356. M. Nagata, A remark on the unique factorization theorem. J. Math. Soc. Japan 9 (1957), 143–145. P. Pollack, Not always buried deep. A second course in elementary number theory. American Mathematical Society, Providence, RI, 2009. B. Poonen, http://mathoverflow.net/q/15221 P. 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1 Sufficient Conditions for the Tightness of Shannon’s Capacity Bounds for Two-Way Channels Jian-Jia Weng† , Lin Song‡, Fady Alajaji† , and Tamás Linder† arXiv:1801.03163v1 [cs.IT] 9 Jan 2018 Abstract—New sufficient conditions for determining in closed form the capacity region of point-to-point memoryless two-way channels (TWCs) are derived. The proposed conditions not only relax Shannon’s condition which can identify only TWCs with a certain symmetry property but also generalize other existing results. Examples are given to demonstrate the advantages of the proposed conditions. Index Terms—Network information theory, two-way channels, capacity region, inner and outer bounds, channel symmetry. I. I NTRODUCTION Finding the capacity region of point-to-point discrete memoryless two-way channels (TWCs) in single-letter form is a long-standing open problem. The difficulty lies in the causality of transmission, since the senders are allowed to generate channel inputs by adapting to previously received channel outputs. In [1], Shannon gave an (uncomputable) multi-letter expression for the capacity region. Another multi-letter expression, using directed information [2], was given in [3]. The capacity region of TWCs is known only for some special channels such as TWCs with additive white Gaussian noise [4], determinisitc TWCs [5], TWCs with discrete additive noise [6], and injective semi-deterministic TWCs [7]. Thus, Shannon’s inner and outer bounds [1] still play an important role in characterizing the capacity region. In the literature, Shannon’s symmetry condition [1] and a condition established by Chaaban, Varshney, and Alouini (CVA) [7] are two known sufficient conditions under which Shannon’s inner and outer bounds coincide, thus directly characterizing the capacity region. Shannon’s condition focuses on a certain symmetry structure for the channel transition probabilities, while the CVA condition focuses on the existence of independent inputs which achieve Shannon’s outer bound. Although the two conditions can be used to determine the capacity region of a large class of TWCs, it is of interest to establish new conditions for wider families of channels. In this paper, four sufficient conditions guaranteeing that Shannon’s inner and outer bounds coincide are derived. Similar to the CVA condition, our conditions identify independent inputs which achieve Shannon’s outer bound based on the approach that a TWC can be viewed as two one-way channels † The authors are with the Department of Mathematics and Statistics, Queen’s University, Kingston, ON K7L 3N6, Canada (Emails: [email protected], {fady, linder}@mast.queensu.ca). ‡ The author was with the Department of Mathematics and Statistics, Queen’s University, Kingston, ON K7L 3N6, Canada. She is now with Contextere Ltd., Ottawa, ON K1Y 2C5, Canada (Email: [email protected]). This work was supported in part by NSERC of Canada. X1N M1 User 1 M̂2 X2N TWC Y1N M2 User 2 Y2N M̂1 Fig. 1. Block diagram of two-way transmission. with state. Two of the derived results are shown to be substantial generalizations of the Shannon and CVA conditions. Moreover, our simplest condition can be easily verified by observing the channel marginal distributions. The rest of this paper is organized as follows. In Section II, the system model and prior results are reviewed. New conditions for finding the capacity region are provided in Section III. A discussion of the connections between the new conditions and prior results is given in Section IV along with illustrative examples. Concluding remarks are given in Section V. II. P RELIMINARIES In a two-way communication system as shown in Fig. 1, two users want to exchange their own messages M1 and M2 via N uses of a TWC. Here, the messages M1 and M2 are assumed to be mutually independent and uniformly distributed on M1 , {1, 2, ..., 2N R1 } and M2 , {1, 2, ..., 2N R2 }, respectively, where N R1 and N R2 are non-negative integers. For j = 1, 2, let Xj and Yj respectively denote the finite channel input and output alphabets for user j. The joint distribution of the inputs and outputs of a memoryless TWC is governed by the channel transition probability PY1 ,Y2 \|X1 ,X2 . A channel code for a TWC is defined as follows. Definition 1: An (N, R1 , R2 ) code for a TWC consists of two message sets M1 = {1, 2, . . . , 2N R1 } and M2 = {1, 2, . . . , 2N R2 }, two sequences of encoding functions f1N , (f1,1 , f1,2 , . . . , f1,N ) and f2N , (f2,1 , f2,2 , . . . , f2,N ), with f1,1 : M1 → X1 , f1,n : M1 × Y1n−1 → X1 , f2,1 : M2 → X2 , and M2 ×Y2n−1 → X2 for n = 2, 3, . . . , N , and two decoding functions g1 : M1 × Y1N → M2 and g2 : M2 × Y2N → M1 . When messages M1 and M2 are encoded, the channel inputs at time n = 1 are only functions of the messages, i.e., Xj,1 = fj,1 (Mj ) for j = 1, 2, but all the other channel inputs are generated by also adapting to the previous channel outputs Yjn−1 , (Yj,1 , Yj,2 , . . . , Yj,n−1 ) via Xj,n = fj,n (Mj , Yjn−1 ) for j = 1, 2 and n = 2, 3, . . . , N . After receiving N channel outputs, user j reconstructs Mi as M̂i = gj (Mj , YjN ) for i, j = 1, 2 with i 6= j, and the probability of decoding error (N ) is defined as Pe (f1N , f2N , g1 , g2 ) = Pr{M̂1 6= M1 or M̂2 6= M2 }. Based on this performance index, we define achievable rate pairs and the capacity region. Definition 2: A rate pair (R1 , R2 ) is said to be achievable if there exists a sequence of (N, R1 , R2 ) codes such that 2 (N ) limN →∞ Pe = 0. The capacity region C of a TWC is the closure of the convex hull of all achievable rate pairs. To date, a computable single-letter expression for the capacity region of general memoryless TWCs has not been found. In [1], Shannon established inner and outer bounds for the capacity region. Let R(PX1 ,X2 , PY1 ,Y2 \|X1 ,X2 ) denote the set of rate pairs (R1 , R2 ) with R1 ≤ I(X1 ; Y2 \|X2 ) and R2 ≤ I(X2 ; Y1 \|X1 ), where the joint distribution of all random variables is given by PX1 ,X2 PY1 ,Y2 \|X1 ,X2 . Then, the capacity region of a discrete memoryless TWC with transition probability PY1 ,Y2 \|X1 ,X2 is inner bounded by [1]  CI (PY1 ,Y2 \|X1 ,X2 ) , co  [ PX1 PX2 and outer bounded by  CO (PY1 ,Y2 \|X1 ,X2 ) , co  [ PX1 ,X2  R(PX1 PX2 , PY1 ,Y2 \|X1 ,X2 ),  R(PX1 ,X2 , PY1 ,Y2 \|X1 ,X2 ), where co denotes taking the closure of the convex hull. In general, CI and CO are different, but if they coincide, then the exact capacity region is obtained and independent inputs can be used to achieve any point of the capacity region. We note that there exist other improved bounds for TWCs [4], [8]-[11]. However, those bounds are either restricted for the particular case of the binary multiplier TWC or expressed with auxiliary random variables, which do not match our approach. We next review the Shannon [1] and CVA [7] conditions that imply the coincidence of CI and CO . For a finite set A, let π A : A → A be a permutation (bijection), and for any two symbols a′ and a′′ in A, let τaA′ ,a′′ : A → A denote the transposition which swaps a′ and a′′ in A, but leaves the other symbols unaffected. Moreover, let PX,Z,Y = PX PZ\|X PY \|X,Z denote a probability distribution defined on finite sets X , Y, and Z. We define two functionals for conditional entropies: H(PX,Z , PY \|X,Z ) , X PX,Z (x, z)PY \|X,Z (y\|x, z) log x,z,y 1 PY \|X,Z (y\|x, z) and H̄(PX , PZ\|X , PY \|X,Z ) , X PX (x)PY \|X (y\|x) log x,y 1 , PY \|X (y\|x) P where PY \|X (y\|x) = z PY \|X,Z (y\|x, z)PZ\|X (z\|x). In particular, if P PZ\|X=x′ = PZ\|X=x′′ for any x′ , x′′ ∈ X , we let PZ (z) = x PX,Z (x, z) and define H̄⊥ (PX , PZ , PY \|X,Z ) , X x,y P PX (x)QY \|X (y\|x) log 1 , QY \|X (y\|x) where QY \|X (y\|x) = z PY \|X,Z (y\|x, z)PZ (z). Note that, given any PX1 ,X2 = PX2 PX1 \|X2 = PX1 PX2 \|X1 , we have H(Yj \|X1 , X2 ) = H(PX1 ,X2 , PYj \|X1 ,X2 ), H(Y1 \|X1 ) = H̄(PX1 , PX2 \|X1 , PY1 \|X1 ,X2 ), and H(Y2 \|X2 ) = H̄(PX2 , PX1 \|X2 , PY2 \|X1 ,X2 ), where PYj \|X1 ,X2 is a marginal of the channel probability PY1 ,Y2 \|X1 ,X2 and j = 1, 2. Furthermore, for any PX1 ,X2 which can be factorized as PX1 PX2 , we have H(Y1 \|X1 ) = H̄⊥ (PX1 , PX2 , PY1 \|X1 ,X2 ) and H(Y2 \|X2 ) = H̄⊥ (PX2 , PX1 , PY2 \|X1 ,X2 ). Finally, let P(Xj ) denote the set of all probability distributions on Xj for j = 1, 2. Proposition 1 (Shannon’s Symmetry Condition [1]): For a memoryless TWC with transition probability PY1 ,Y2 \|X1 ,X2 , we have C = CI = CO if for any pair of distinct input symbols x′1 , x′′1 ∈ X1 , there exists a pair of permutations (π Y1 [x′1 , x′′1 ], π Y2 [x′1 , x′′1 ]) on Y1 and Y2 (which depend on x′1 and x′′1 ) such that for all x1 , x2 , y1 , y2 , PY1 ,Y2 \|X1 ,X2 (y1 , y2 \|x1 , x2 ) = PY1 ,Y2 \|X1 ,X2 (π Y1 [x′1 , x′′1 ](y1 ), π Y2 [x′1 , x′′1 ](y2 )\|τxX′1,x′′ (x1 ), x2 ). (1) 1 1 Proposition 2 (CVA Condition [7]): For a memoryless TWC with transition probability PY1 ,Y2 \|X1 ,X2 , we have C = CI = CO if for any PX1 ,X2 = PX2 PX1 \|X2 = PX1 PX2 \|X1 , H(PX2 P̃X1 \|X2 , PYj \|X1 ,X2 ) does not depend on P̃X1 \|X2 for given PX2 and there exists P̃X1 ∈ P(X1 ) such that H̄⊥ (P̃X1 , PX2 , PY1 \|X1 ,X2 ) ≥ H̄(PX1 , PX2 \|X1 , PY1 \|X1 ,X2 ) and H̄⊥ (PX2 , P̃X1 , PY2 \|X1 ,X2 ) ≥ H̄(PX2 , PX1 \|X2 , PY2 \|X1 ,X2 ). We remark that Proposition 1 describes a channel symmetry property with respect to the channel input of user 1, but an analogous condition can be obtained by exchanging the roles of users 1 and 2. Also, the invariance of H(PX2 P̃X1 \|X2 , PYj \|X1 ,X2 ) in Proposition 2 in fact imposes a certain symmetry constraint on the channel marginal distribution PYj \|X1 ,X2 . In the literature, a TWC with independent q-ary additive noise [6] is an example that satisfies both the Shannon and CVA conditions. III. C ONDITIONS FOR THE T IGHTNESS OF S HANNON ’ S I NNER AND O UTER B OUNDS In this section, we present four results regarding the tightness of Shannon’s inner and outer bounds. We adopt the viewpoint that a two-way channel consists of two one-way channels with state. For example, the one-way channel from user 1 to user 2 is governed by the marginal distribution PY2 \|X1 ,X2 (derived from the channel probability distribution PY1 ,Y2 \|X1 ,X2 ), where X1 and Y2 are respectively the input and the output of the channel with state X2 . Let PX and PY \|X be probability distributions on finite sets X and Y. To simplify the presentation, we define I(PX , PY \|X ) = X x,y PX (x)PY \|X (y\|x) log P x′ PY \|X (y\|x) , PX (x′ )PY \|X (y\|x′ ) which is the mutual information I(X; Y ) between input X (governed by PX ) and corresponding output Y of a channel with transition probability PY \|X . A useful fact is that I(·, ·) is concave in the first argument when the second argument is fixed. Moreover, the conditional mutual information I(X1 ; Y2 \|X2 = x2 ) and I(X2 ; Y1 \|X1 = x1 ) can be expressed as I(PX1 \|X2 =x2 , PY2 \|X1 ,X2 =x2 ) and I(PX2 \|X1 =x1 , PY1 \|X1 =x1 ,X2 ), respectively. By viewing a TWC as two one-way channels with state, each of the following four theorems comprises two conditions, one for each direction of the two-way transmission. By symmetry, these theorems are also valid if the roles of users 1 and 2 are swapped. For simplicity, we will use I (k) (Xi ; Yj \|Xj ) and H (k) (Yj \|X1 , X2 ) to denote the conditional mutual information and conditional entropy evaluated (k) under input distribution PX1 ,X2 for i, j = 1, 2 with i 6= j. For 3 (k) (k) (k) (1) (1) (1) (2) PX1 ,X2 = PXi PXi \|Xj , the conditional entropy H (k) (Yi \|Xi ) Proof: Given any PX1 ,X2 = PX2 PX1 \|X2 , let PX1 ,X2 = is evaluated under the marginal distribution PYi \|Xi (yi \|xi ) = P (k) xj PXj \|Xi (xj \|xi )PYi \|Xj ,Xi (yi \|xj , xi ). Theorem 1: For a given memoryless TWC, if both of the following conditions are satisfied, then CI = CO : ∗ (i) There exists PX ∈ P(X1 ) such that for all x2 ∈ X2 we 1 ∗ . have arg maxPX \|X =x I(X1 ; Y2 \|X2 = x2 ) = PX 1 1 2 2 (ii) I(PX2 , PY1 \|X1 =x1 ,X2 ) does not depend on x1 ∈ X1 for any fixed PX2 ∈ P(X2 ). (2) (1) (1) (1) Proof: For any PX1 ,X2 = PX2 PX1 \|X2 , let PX1 ,X2 = ∗ PX P . By the same argument as in (2)-(6), we obtain via 1 X2 (i) that I (1) (X1 ; Y2 \|X2 ) ≤ I (2) (X1 ; Y2 \|X2 ). Moreover, (k) (1) ∗ ∗ PX PX2 , where PX is given by (i). In light of (i), we have 1 1 I (1) (X1 ; Y2 \|X2 ) X (1) PX2 (x2 ) · I (1) (X1 ; Y2 \|X2 = x2 ) = (2) x2 ≤ X (1) PX2 (x2 ) · x2 = X max PX1 \|X2 =x2 I(X1 ; Y2 \|X2 = x2 ) (1) PX2 (x2 ) · I(PX1∗ , PY2 \|X1 ,X2 =x2 ) (3) (4) x2 = X x2 (2) =I (1) PX2 (x2 ) · I (2) (X1 ; Y2 \|X2 = x2 ) (5) (X1 ; Y2 \|X2 ). (6) Moreover, I (1) (X2 ; Y1 \|X1 ) X (1) PX1 (x1 ) · I (1) (X2 ; Y1 \|X1 = x1 ) = (7) x1 = X (1) (1) (1) (1) PX1 (x1 ) · I(PX2 \|X1 =x1 , PY1 \|X1 =x1 ,X2 ) (8) x1 = X PX1 (x1 ) · I(PX2 \|X1 =x1 , PY1 \|X1 =x′1 ,X2 ) (9) x1 ≤I X (1) (1) PX1 (x1 )PX2 \|X1 =x1 , PY1 \|X1 =x′1 ,X2 x1 (1) ! = I(PX2 , PY1 \|X1 =x′1 ,X2 ) X (1) PX1∗ (x′1 ) · I(PX2 , PY1 \|X1 =x′1 ,X2 ) = (10) (11) (12) x′1 = I (2) (X2 ; Y1 \|X1 ), (13) where (9) holds by the invariance assumption in (ii), (10) holds since the functional I(·, ·) is concave in the first argument, and (12) is obtained from the invariance assumption in (1) (ii). Combining the above yields R(PX1 ,X2 , PY1 ,Y2 \|X1 ,X2 ) ⊆ (1) ∗ R(PX PX2 , PY1 ,Y2 \|X1 ,X2 ), which implies that CO ⊆ CI and 1 hence CI = CO . Theorem 2: For a given memoryless TWC, if for any PX1 ,X2 = PX2 PX1 \|X2 = PX1 PX2 \|X1 , both of the following conditions are satisfied, then CI = CO : ∗ (i) There exists PX ∈ P(X1 ) such that for all x2 ∈ X2 we 1 ∗ . have arg maxPX \|X =x I(X1 ; Y2 \|X2 = x2 ) = PX 1 1 2 2 (ii) H(PX2 P̃X1 \|X2 , PY1 \|X1 ,X2 ) does not depend on P̃X1 \|X2 given PX2 and PY1 \|X1 ,X2 , and the ∗ common maximizer PX in (i) also satisfies 1 ∗ H̄⊥ (PX1 , PX2 , PY1 \|X1 ,X2 ) ≥ H̄(PX1 , PX2 \|X1 , PY1 \|X1 ,X2 ). (1) I (1) (X2 ; Y1 \|X1 ) = H (1) (Y1 \|X1 ) − H (1) (Y1 \|X1 , X2 ) = H̄(PX(1)1 , PX(1)2 \|X1 , PY1 \|X1 ,X2 ) − H(PX(1)2 PX(1)1 \|X2 , PY1 \|X1 ,X2 ) (14) ≤ H̄⊥ (PX∗ 1 , PX(1)2 , PY1 \|X1 ,X2 ) − H(PX(1)2 PX∗ 1 , PY1 \|X1 ,X2 ) (2) = H (Y1 \|X1 ) − H = I (2) (X2 ; Y1 \|X1 ), (2) (Y1 \|X1 , X2 ) (15) (16) where (14) and (16) follow from the definitions in Section II and (15) is due to condition (ii). Consequently, (1) (1) ∗ PX2 , PY1 ,Y2 \|X1 ,X2 ), and R(PX1 ,X2 , PY1 ,Y2 \|X1 ,X2 ) ⊆ R(PX 1 hence CO ⊆ CI , so that CI = CO . Theorem 3: For a given memoryless TWC, if both of the following conditions are satisfied, then CI = CO : (i) I(PX1 , PY2 \|X1 ,X2 =x2 ) does not depend on x2 ∈ X2 for any fixed PX1 ∈ P(X1 ). (ii) I(PX2 , PY1 \|X1 =x1 ,X2 ) does not depend on x1 ∈ X1 for any fixed PX2 ∈ P(X2 ). Proof: From conditions (i) and (ii), we know that maxPX1 \|X2 =x2 I(X1 ; Y2 \|X2 = x2 ) has a common maximizer ∗ PX for all x2 ∈ X2 and maxPX2 \|X1 =x1 I(X2 ; Y1 \|X1 = x1 ) 1 ∗ has a common maximizer PX for all x1 ∈ X1 . For any 2 (2) (1) (1) (1) ∗ P ∗ . By the same PX1 ,X2 = PX1 PX2 \|X1 , let PX1 ,X2 = PX 1 X2 argument as in (2)-(6), we conclude that I (1) (X1 ; Y2 \|X2 ) ≤ I (2) (X1 ; Y2 \|X2 ) and I (1) (X2 ; Y1 \|X1 ) ≤ I (2) (X2 ; Y1 \|X1 ). (1) ∗ ∗ PX Thus, R(PX1 ,X2 , PY1 ,Y2 \|X1 ,X2 ) ⊆ R(PX , PY1 ,Y2 \|X1 ,X2 ), 1 2 which yields CI = CO . Similar to the CVA condition, complex computations are often inevitable for checking the above conditions. We next present a useful condition which needs little computational effort. Let [PY2 \|X1 ,X2 (·\|·, x2 )] (resp. [PY1 \|X1 ,X2 (·\|x1 , ·)]) denote the marginal transition probability matrix obtained from PY1 ,Y2 \|X1 ,X2 =x2 (resp. PY1 ,Y2 \|X1 =x1 ,X2 ), whose columns and rows are indexed according to a fixed order on the symbols in Y2 and X1 (resp. Y1 and X2 ). Theorem 4: For a given memoryless TWC, if both of the following conditions are satisfied, then CI = CO : (i) The matrices [PY2 \|X1 ,X2 (·\|·, x2 )], x2 ∈ X2 , are column permutations of each other. (ii) The matrices [PY1 \|X1 ,X2 (·\|x1 , ·)], x1 ∈ X1 , are column permutations of each other. Since the proof is similar to the second part of the proof of Theorem 5 in the next section, the details are omitted. IV. D ISCUSSION AND E XAMPLES A. Comparison with Other Conditions As already noted, the relationship between Propositions 1 and 2 is unclear as examples that satisfy the Shannon condition but not the CVA condition seem hard to construct. In this section, we show that Theorems 1 and 2 in fact generalize the Shannon and CVA results, respectively. To see this, it suffices to show that the Shannon and CVA conditions imply the conditions in Theorems 1 and 2, respectively. 4 Theorem 5: A TWC satisfying Shannon’s symmetry condition in Proposition 1 must satisfy the conditions in Theorem 1. Proof: For a TWC satisfying the condition of Proposition 1, the optimal input probability distribution that achieves capacity is of the form PX1 ,X2 = PX2 /\|X1 \| for some PX2 ∈ P(X2 ) [1]. This result implies that condition (i) of Theorem 1 is satisfied because a common maximizer exists for all x2 ∈ X ∗ and is given by PX (x1 ) = 1/\|X1 \|. To prove that condition 1 (ii) is also satisfied, we consider the two (marginal) matrices [PY1 \|X1 ,X2 (·\|x′1 , ·)] and [PY1 \|X1 ,X2 (·\|x′′1 , ·)] for some fixed x′1 , x′′1 ∈ X1 and show that these matrices are column permutations of each other and hence I(PX2 , PY1 \|X1 =x′1 ,X2 ) = I(PX2 , PY1 \|X1 =x′′1 ,X2 ). The former claim is true because PY1 \|X1 ,X2 (y1 \|x′1 , x2 ) (17) = PY1 \|X1 ,X2 (π1Y1 [x′1 , x′′1 ](y1 )\|x′′1 , x2 ), (18) 1 where (17) is obtained by marginalizing Y2 on both sides of (1) and (18) follows from the definition of transposition. The second claim can be verified by a direct computation on I(PX2 , PY1 \|X1 =x1 ,X2 ) with the above result straightforwardly, and hence the details are omitted. Remark 1: Example 1 in the next subsection demonstrates that a TWC that satisfies the conditions in Theorem 1 may not satisfy Shannon’s symmetry condition in Proposition 1 since the common maximizer is not necessarily the uniform input distribution. Hence, Theorem 1 is a more general result than Proposition 1. Theorem 6: A TWC satisfying the CVA condition in Proposition 2 must satisfy the conditions in Theorem 2. Proof: Suppose that the condition of Proposition 2 is satisfied. To prove the theorem, we first claim that for j = 1, 2, H(Yj \|X1 = x′1 , X2 = x′2 ) = H(Yj \|X1 = x′′1 , X2 = x′2 ) for all x′1 , x′′1 ∈ X1 and x′2 ∈ X2 . Given arbitrary pairs (x′1 , x′2 ) and (x′′1 , x′2 ) with x′1 6= x′′1 , consider the two probability distributions 1, if a = x′1 and b = x′2 , (1) PX1 ,X2 (a, b) = 0, otherwise, and (2) PX1 ,X2 (a, b) (1) = 1, 0, if a = x′′1 and b = x′2 , otherwise. (2) Noting that PX2 = PX2 , we have H(Yj \|X1 = x′1 , X2 = x′2 ) = H (1) (Yj \|X1 , X2 ) (19) = H(PX(1)2 PX(1)1 \|X2 , PYj \|X1 ,X2 ) = H(PX(1)2 PX(2)1 \|X2 , PYj \|X1 ,X2 ) (2) = H (Yj \|X1 , X2 ) = H(Yj \|X1 = x′′1 , X2 = x′2 ), (1) PX1 ,X2 1 2 2 (1) (1) (1) ∗ for some PX2 ∈ P(X2 ). and define PX1 ,X2 = PX2 PX 1 \|X2 Since H(Yj \|X1 , X2 = x2 ) does not depend on PX1 \|X2 =x2 , ∗ is in fact a maximizer for H(Y2 \|X2 = x2 ). Note PX 1 \|X2 =x2 ∗ may not be unique, but any that the maximizer PX 1 \|X2 =x2 (1) choice works for our purposes. Now for PX1 ,X2 , by the CVA condition, there exists P̃X1 ∈ P(X1 ) such that (1) (1) = PY1 \|X1 ,X2 (π1Y1 [x′1 , x′′1 ](y1 )\|τxX′1,x′′ (x′1 ), x2 ) 1 on x1 ∈ X1 for fixed x2 ∈ X2 , H(Yj \|X1 , X2 = x2 ) does not depend on PX1 \|X2 =x2 . Next, we show that condition (i) of Theorem 2 holds by constructing the common maximizer from the CVA condition. For each x2 ∈ X2 , let ∗ = arg maxPX \|X =x I(X1 ; Y2 \|X2 = x2 ) = PX 1 \|X2 =x2 1 2 2 arg maxPX \|X =x [H(Y2 \|X2 = x2 ) − H(Y2 \|X1 , X2 = x2 )] (20) (21) (22) where (19) and (22) are due to the definitions of and (2) PX1 ,X2 , respectively, (20) follows from the CVA condition, (2) (1) PX2 . The claim is proved. Since and (21) holds since PX2 = P H(Yj \|X1 , X2 = x2 ) = x1 PX1 \|X2 (x1 \|x2 )H(Yj \|X1 = x1 , X2 = x2 ) and H(Yj \|X1 = x1 , X2 = x2 ) does not depend ∗ , PY2 \|X1 ,X2 ) ≤ H̄⊥ (PX2 , P̃X1 , PY2 \|X1 ,X2 ). H̄(PX2 , PX 1 \|X2 (1) (2) ∗ is the maximizer Set PX1 ,X2 = P̃X1 PX2 . Since PX 1 \|X2 =x2 for H(Y2 \|X2 = x2 ), we have (1) ∗ , PY2 \|X1 ,X2 ) H̄(PX2 , PX 1 \|X2 = H (1) (Y2 \|X2 ) X (1) PX2 (x2 ) · H (1) (Y2 \|X2 = x2 ) = x2 = X (1) PX2 (x2 ) x2 ≥ X max PX1 \|X2 =x2 H(Y2 \|X2 = x2 ) (1) PX2 (x2 ) · H (2) (Y2 \|X2 = x2 ) x2 (2) =H · (Y2 \|X2 ) (1) = H̄⊥ (PX2 , P̃X1 , PY2 \|X1 ,X2 ). Thus, H̄(PX(1)2 , PX∗ 1 \|X2 , PY2 \|X1 ,X2 ) = H̄⊥ (PX(1)2 , P̃X1 , PY2 \|X1 ,X2 ), i.e., X (1) PX2 (x2 ) · H (1) (Y2 \|X2 = x2 ) = X (1) PX2 (x2 ) · H (2) (Y2 \|X2 = x2 ). x2 x2 (2) (1) Since H (Y2 \|X2 = x2 ) ≤ H (Y2 \|X2 = x2 ) for each x2 ∈ X2 , we obtain H (1) (Y2 \|X2 = x2 ) = H (2) (Y2 \|X2 = x2 ), i.e., ∗ P̃X1 achieves the same value of H(Y2 \|X2 = x2 ) as PX 1 \|X2 =x2 for all x2 ∈ X2 . Consequently, P̃X1 is a common maximizer and thus condition (i) of Theorem 2 is satisfied. Moreover, since the common maximizer P̃X1 is provided by the CVA condition, condition (ii) of Theorem 2 automatically holds. Remark 2: Example 1 below shows that a TWC that satisfies the conditions in Theorem 2 does not necessarily satisfy the condition in Proposition 2 because our conditions allow H(PX2 P̃X1 \|X2 , PY2 \|X1 ,X2 ) to depend on P̃X1 \|X2 for given PX2 . Hence, Theorem 2 is more general than Proposition 2. B. Examples We next illustrate the effectiveness of our conditions via two examples in which X1 = X2 = Y1 = Y2 = {0, 1}. The TWC in Example 1 satisfies the conditions of Theorems 1-4 and the capacity region is rectangular. The TWC in Example 2 satisfies the conditions of Theorem 1 and 2 and has a non-rectangular capacity region. However, neither of the constructed TWCs satisfy the Shannon or the CVA conditions. 5 Example 1: Consider the TWC with  00 01 10 11 0.783 0.087 0.117 0.013    01   0.0417 0.3753 0.0583 0.5247 . [PY1 ,Y2 \|X1 ,X2 ] =   10  0.261 0.609 0.039 0.091  11 0.2919 0.1251 0.4081 0.1749 The corresponding one-way channel marginal distributions are given by 0.9 [PY2 \|X1 ,X2 (·\|·, 0)] = 0.3 0.1 [PY2 \|X1 ,X2 (·\|·, 1)] = 0.7 0.08  0.1 0.87 0.13 , [PY1 \|X1 ,X2 (·\|0, ·)] = , 0.7 0.417 0.583 0.87 0.13 0.9 . , [PY1 \|X1 ,X2 (·\|1, ·)] = 0.417 0.583 0.3 For this TWC, Shannon’s symmetry condition in Proposition 1 does not hold since there are no permutations on Y1 and Y2 which can result in (1). Furthermore, since H(Y2 \|X1 = 0, X2 = 0) = Hb (0.1) and H(Y2 \|X1 = 1, X2 = 0) = Hb (0.3), where Hb (·) denotes the binary entropy function, H(PX2 P̃X1 \|X2 , PY2 \|X1 ,X2 ) depends on P̃X1 \|X2 for given PX2 . Thus, the CVA condition in Proposition 2 does not hold, either. However by Theorem 4, Shannon’s inner and outer bounds coincide since [PY2 \|X1 ,X2 (·\|·, 0)] (resp. [PY1 \|X1 ,X2 (·\|0, ·)]) can be obtained by permuting the columns of [PY2 \|X1 ,X2 (·\|·, 1)] (resp. [PY1 \|X1 ,X2 (·\|1, ·)]). Since the conditions in Theorem 4 imply the conditions in Theorem 3 and the conditions in Theorem 3 further imply the conditions in Theorem 1, the conditions of Theorems 1 and 3 are also satisfied. Moreover, the optimal input distribution for this TWC can be obtained by searching for the common maximizer for each of the two one-way channels via the Blahut-Arimoto algorithm yielding ∗ ∗ PX (0) = PX (0) = 0.471. Thus, the capacity region is 1 2 ∗ ∗ achieved by the input distribution PX = PX P ∗ , i.e., 1 ,X2 1 X2 C = {(R1 , R2 ) : 0 ≤ R1 ≤ 0.2967, 0 ≤ R2 ≤ 0.1715}. Finally, we note that this TWC also satisfies the conditions of Theorem 2, in which the first condition is already implied by the conditions of Theorem 1. To verify the second condition, we consider 0.07 0.06 R2 00 0.1 0.09 0.05 0.04 0.03 0.02 0.01 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 R1 Fig. 2. The capacity region of the TWC in Example 2. [PY1 \|X1 ,X2 (·\|0, ·)] = [PY1 \|X1 ,X2 (·\|1, ·)] = [PY2 \|X1 ,X2 (·\|·, 1)]. Using the same arguments as in Example 1, one can easily see that this TWC satisfies neither the Shannon nor the CVA conditions. However, it satisfies the conditions in Theorem 1 since a common maximizer exists for the one-way channel ∗ from users 1 to 2, i.e., PX (0) = 0.471, and condition (ii) 1 trivially holds. To verify that this channel also satisfies the conditions in Theorem 2, the same argument as in the previous example is used. Finally, by considering all input distributions ∗ of the form PX1 ,X2 = PX P , the capacity region of this 1 X2 channel is determined as shown in Fig. 2. V. C ONCLUSIONS In this paper, four conditions on the coincidence of Shannon’s capacity inner and outer bounds were derived. These invariance conditions were shown to generalize existing results, thus enlarging the class of TWCs whose capacity region can be exactly determined. Numerical examples illustrate the applications of the new conditions in situations where prior results do not apply. R EFERENCES [1] C. E. Shannon, “Two-way communications channels,” in Proc. 4th Berkeley Symp. Math. Stat. Probab., Chicago, IL, USA, Jun. 1961, pp. 611-644. [2] J. Massey, “Causality, feedback and directed information,” in Proc. Int. Symp. Information Theory Its Applic. (ISITA-90), Waikiki, HI, USA, Nov. 1990, pp. 303-305. 0.417 0.583 0.87 0.13 . [3] G. Kramer, “Directed information for channels with feedback,” Ph.D. , [PY1 \|X1 ,X2 (·\|·, 1)] = [PY1 \|X1 ,X2 (·\|·, 0)] = 0.417 0.583 0.87 0.13 Dissertation, Swiss Federal Institute of Technology Zurich, 1998. [4] T. Han, “A general coding scheme for the two-way channel,” IEEE Here, for all x1 ∈ {0, 1}, H(Y1 \|X1 = x1 , X2 = 0) = Trans. Inf. Theory, vol. IT-30, no. 1, pp. 35-44, Jan. 1984. Hb (0.13) and H(Y1 \|X1 = x1 , X2 = 1) = Hb (0.417). [5] Z. Cheng and N. Devroye, “Two-way networks: when adaptation is Thus, H(PX2 P̃X1 \|X2 , PY1 \|X1 ,X2 ) does not depend on P̃X1 \|X2 useless,” IEEE Trans. Inf. Theory, vol. 60, no. 3, pp. 1793-1813, Mar. (1) 2014. given PX2 . Together with the substitutions PX1 ,X2 = PX1 ,X2 [6] L. Song, F. Alajaji, and T. Linder, “Adaptation is useless for two discrete (2) ∗ into (7)-(13), we then obtain that and PX1 ,X2 = PX P additive-noise two-way channels,” in Proc. IEEE Int. Symp. Inf. Theory, 1 X2 ∗ Barcelona, Spain, Jul. 2016, pp. 1854-1858. ). , P , P ) ≥ H̄(P H̄⊥ (PX , P , P X X Y \|X ,X X \|X Y \|X ,X 1 2 1 1 2 2 1 1 1 2 1 [7] A. Chaaban, L. R. Varshney, and M. Alouini, “The capacity of injective Therefore, the second condition of Theorem 2 holds. semi-deterministic two-way channels,” in Proc. IEEE Int. Symp. Inf. Theory, Aachen, Germany, Jun. 2017, pp. 431-436. Example 2: Consider the TWC with [8] J. Schalkwijk, “The binary multiplying channel - a coding scheme that 00 01 10 11 operates beyond Shannons inner bound region,” IEEE Trans. Inf. Theory,   vol. IT-28, no. 1, pp. 107-110, Jan. 1982. 00 0.783 0.087 0.117 0.013 [9] J. Schalkwijk, “On an extension of an achievable rate region for the   01  0.36279 0.05421 0.50721 0.07579  binary multiplying channel,” IEEE Trans. Inf. Theory, vol. IT-29, no. 3,   [PY1 ,Y2 \|X1 ,X2 ] =  , pp. 445-448, May 1983. 10  0.261 0.609 0.039 0.091  [10] Z. Zhang, T. Berger, and J. Schalkwijk, “New outer bounds to capacity 11 0.173889 0.243111 0.243111 0.339889 regions of two-way channels,” IEEE Trans. Inf. Theory, vol. IT-32, no. 3, pp. 383-386, May 1986. where two one-way channel marginal distributions are [11] A. Hekstra and F. Willems, “Dependence balance bounds for single output two-way channels,” IEEE Trans. Inf. Theory, vol. 35, no. 1, pp. 0.9 0.1 0.87 0.13 44-53, Jan. 1989. [PY2 \|X1 ,X2 (·\|·, 0)] = , [PY2 \|X1 ,X2 (·\|·, 1)] = , 0.3 0.7 0.417 0.583	7
On κ-reducibility of pseudovarieties of the form V ∗ D J. C. Costa C. Nogueira M. L. Teixeira arXiv:1602.03020v1 [math.GR] 9 Feb 2016 February 8, 2016 Abstract This paper deals with the reducibility property of semidirect products of the form V ∗ D relatively to graph equation systems, where D denotes the pseudovariety of definite semigroups. We show that, if the pseudovariety V is reducible with respect to the canonical signature κ consisting of the multiplication and the (ω − 1)-power, then V ∗ D is also reducible with respect to κ. Keywords. Pseudovariety, definite semigroup, semidirect product, implicit signature, graph equations, reducibility. 1 Introduction A semigroup (resp. monoid) pseudovariety is a class of finite semigroups (resp. monoids) closed under taking subsemigroups (resp. submonoids), homomorphic images and finite direct products. It is said to be decidable if there is an algorithm to test membership of a finite semigroup (resp. monoid) in that pseudovariety. The semidirect product of pseudovariets has been getting much attention, mainly due to the Krohn-Rhodes decomposition theorem [18]. In turn, the pseudovarieties of the form V∗D, where D is the pseudovariety of all finite semigroups whose idempotents are right zeros, are among the most studied semidirect products [23, 25, 3, 1, 4]. For a pseudovariety V of monoids, LV denotes the pseudovariety of all finite semigroups S such that eSe ∈ V for all idempotents e of S. We know from [17, 23, 24, 25] that V ∗ D is contained in LV and that V ∗ D = LV if and only if V is local in the sense of Tilson [25]. In particular, the equalities Sl ∗ D = LSl and G ∗ D = LG hold for the pseudovarieties Sl of semilattices and G of groups. J. C. Costa & M. L. Teixeira: CMAT, Dep. Matemática e Aplicações, Universidade do Minho, Campus de Gualtar, 4710-057 Braga, Portugal; e-mail: [email protected], [email protected] C. Nogueira: CMAT, Escola Superior de Tecnologia e Gestão, Instituto Politécnico de Leiria, Campus 2, Morro do Lena, Alto Vieiro, 2411-901 Leiria, Portugal; e-mail: [email protected] 1 2 J. C. Costa, C. Nogueira, M. L. Teixeira It is known that the semidirect product operator does not preserve decidability of pseudovarieties [20, 11]. The notion of tameness was introduced by Almeida and Steinberg [7, 8] as a tool for proving decidability of semidirect products. The fundamental property for tameness is reducibility. This property was originally formulated in terms of graph equation systems and latter extended to any system of equations [2, 21]. It is parameterized by an implicit signature σ (a set of implicit operations on semigroups containing the multiplication), and we speak of σ-reducibility. For short, given an equation system Σ with rational constraints, a pseudovariety V is σ-reducible relatively to Σ when the existence of a solution of Σ by implicit operations over V implies the existence of a solution of Σ by σ-words over V and satisfying the same constraints. The pseudovariety V is said to be σ-reducible if it is σ-reducible with respect to every finite graph equation system. The implicit signature which is most commonly encountered in the literature is the canonical signature κ = {ab, aω−1 } consisting of the multiplication and the (ω − 1)-power. For instance, the pseudovarieties D [9], G [10, 8], J [1, 2] of all finite J -trivial semigroups, LSl [16] and R [6] of all finite R-trivial semigroups are κ-reducible. In this paper, we study the κ-reducibility property of semidirect products of the form V∗D. This research is essentially inspired by the papers [15, 16] (see also [13] where a stronger form of κ-reducibility was established for LSl). We prove that, if V is κ-reducible then V ∗ D is κreducible. In particular, this gives a new and simpler proof (though with the same basic idea) of the κ-reducibility of LSl and establishes the κ-reducibility of the pseudovarieties LG, J ∗ D and R ∗ D. Combined with the recent proof that the κ-word problem for LG is decidable [14], this shows that LG is κ-tame, a problem proposed by Almeida a few years ago. This also extends part of our work in the paper [15], where we proved that under mild hypotheses on an implicit signature σ, if V is σ-reducible relatively to pointlike systems of equations (i.e., systems of equations of the form x1 = · · · = xn ) then V ∗ D is pointlike σ-reducible as well. As in [15], we use results from [5], where various kinds of σ-reducibility of semidirect products with an order-computable pseudovariety were considered. More specifically, we know from [5] that a pseudovariety of the form V ∗ Dk is κ-reducible when V is κ-reducible, where Dk is the order-computable pseudovariety defined by the identity yx1 · · · xk = x1 · · · xk . As S V ∗ D = k V ∗ Dk , we utilize this result as a way to achieve our property concerning the pseudovarieties V ∗ D. The method used in this paper is similar to that of [15]. However, some significant changes, inspired by [16], had to be introduced in order to deal with the much more intricate graph equation systems. 2 Preliminaries The reader is referred to the standard bibliography on finite semigroups, namely [1, 21], for general background and undefined terminology. For basic definitions and results about combinatorics on words, the reader may wish to consult [19]. On κ-reducibility of pseudovarieties of the form V ∗ D 2.1 3 Words and pseudowords Throughout this paper, A denotes a finite non-empty set called an alphabet. The free semigroup and the free monoid generated by A are denoted respectively by A+ and A∗ . The empty word is represented by 1 and the length of a word w ∈ A∗ is denoted by \|w\|. A word is called primitive if it cannot be written in the form un with n > 1. Two words u and v are said to be conjugate if u = w1 w2 and v = w2 w1 for some words w1 , w2 ∈ A∗ . A Lyndon word is a primitive word which is minimal in its conjugacy class, for the lexicographic order on A+ . A left-infinite word on A is a sequence w = (an )n of letters of A indexed by −N also written w = · · · a−2 a−1 . The set of all left-infinite words on A will be denoted by A−N and we put A−∞ = A+ ∪ A−N . The set A−∞ is endowed with a semigroup structure by defining a product as follows: if w, z ∈ A+ , then wz is already defined; left-infinite words are right zeros; finally, if w = · · · a−2 a−1 is a left-infinite word and z = b1 b2 · · · bn is a finite word, then wz is the left-infinite word wz = · · · a−2 a−1 b1 b2 · · · bn . A left-infinite word w of the form u−∞ v = · · · uuuv, with u ∈ A+ and v ∈ A∗ , is said to be ultimately periodic. In case v = 1, the word w is named periodic. For a periodic word w = u−∞ , if u is a primitive word, then it will be called the root of w and its length \|u\| will be said to be the period of w. For a pseudovariety V of semigroups, we denote by ΩA V the relatively free pro-V semigroup generated by the set A: for each pro-V semigroup S and each function ϕ : A → S, there is a unique continuous homomorphism ϕ : ΩA V → S extending ϕ. The elements of ΩA V are called pseudowords (or implicit operations) over V. A pseudovariety V is called order-computable when the subsemigroup ΩA V of ΩA V generated by A is finite, in which case ΩA V = ΩA V, and effectively computable. Recall that, for the pseudovariety S of all finite semigroups, ΩA S is (identified with) the free semigroup A+ . The elements of ΩA S \ A+ will then be called infinite pseudowords. A pseudoidentity is a formal equality π = ρ of pseudowords π, ρ ∈ ΩA S over S. We say that V satisfies the pseudoidentity π = ρ, and write V \|= π = ρ, if ϕπ = ϕρ for every continuous homomorphism ϕ : ΩA S → S into a semigroup S ∈ V, which is equivalent to saying that pV π = pV ρ for the natural projection pV : ΩA S → ΩA V. 2.2 Pseudoidentities over V ∗ Dk For a positive integer k, let Dk be the pseudovariety of all finite semigroups satisfying the identity yx1 · · · xk = x1 · · · xk . Denote by Ak the set of words over A with length k and by Ak the set {w ∈ A+ : \|w\| ≤ k} of non-empty words over A with length at most k. We notice that ΩA Dk may be identified with the semigroup whose support set is Ak and whose multiplication is given by u · v = tk (uv), where tk w denotes the longest suffix of length at most k of a given (finite or left-infinite) word w. Then, the Dk are order-computable pseudovarieties such that S D = k Dk . Moreover, it is well-known that ΩA D is isomorphic to the semigroup A−∞ . For each pseudoword π ∈ ΩA S, we denote by tk π the unique smallest word (of Ak ) such that Dk \|= π = tk π. Simetrically, we denote by ik π the smallest word (of Ak ) such 4 J. C. Costa, C. Nogueira, M. L. Teixeira that Kk \|= π = ik π, where Kk is the dual pseudovariety of Dk defined by the identity x1 · · · xk y = x1 · · · xk . Let Φk be the function A+ → (Ak+1 )∗ that sends each word w ∈ A+ to the sequence of factors of length k + 1 of w, in the order they occur in w. We still denote by Φk (see [3] and [1, Lemma 10.6.11]) its unique continuous extension ΩA S → (ΩAk+1 S)1 . This function Φk is a k-superposition homomorphism, with the meaning that it verifies the conditions: i) Φk w = 1 for every w ∈ Ak ; ii) Φk (πρ) = Φk πΦk (tk π)ρ = Φk π(ik ρ) Φk ρ for every π, ρ ∈ ΩA S. Throughout the paper, V denotes a non-locally trivial pseudovariety of semigroups. For any pseudowords π, ρ ∈ ΩA S, it is known from [1, Theorem 10.6.12] that V ∗ Dk \|= π = ρ 2.3 ⇐⇒ ik π = ik ρ, tk π = tk ρ and V \|= Φk π = Φk ρ. (2.1) Implicit signatures and σ-reducibility By an implicit signature we mean a set σ of pseudowords (over S) containing the multiplication. In particular, we represent by κ the implicit signature {ab, aω−1 }, usually called the canonical signature. Every profinite semigroup has a natural structure of a σ-algebra, via the natural interpretation of pseudowords on profinite semigroups. The σ-subalgebra of ΩA S generated by A is denoted by ΩσA S. It is freely generated by A in the variety of σ-algebras generated by the pseudovariety S and its elements are called σ-words (over S). To a (directed multi)graph e Γ = V (Γ) ⊎ E(Γ), with vertex set V (Γ), edge set E(Γ), and edges αe − → ωe, we associate the system ΣΓ of all equations of the form (αe) e = ωe, with e ∈ E(Γ). Let S be a finite A-generated semigroup, δ : ΩA S → S be the continuous homomorphism respecting the choice of generators and ϕ : Γ → S 1 be an evaluation mapping such that ϕE(Γ) ⊆ S. We say that a mapping η : Γ → (ΩA S)1 is a V-solution of ΣΓ with respect to (ϕ, δ) when δη = ϕ and V \|= ηu = ηv for all (u = v) ∈ ΣΓ . Furthermore, if ηΓ ⊆ (ΩσA S)1 for an implicit signature σ, then η is called a (V, σ)-solution. The pseudovariety V is said to be σ-reducible relatively to the system ΣΓ if the existence of a V-solution of ΣΓ with respect to a pair (ϕ, δ) entails the existence of a (V, σ)-solution of ΣΓ with respect to the same pair (ϕ, δ). We say that V is σ-reducible, if it is σ-reducible relatively to ΣΓ for all finite graphs Γ. 3 κ-reducibility of V ∗ D Let V be a given κ-reducible non-locally trivial pseudovariety. The purpose of this paper is to prove the κ-reducibility of the pseudovariety V ∗ D. So, we fix a finite graph Γ and a finite A-generated semigroup S and consider a V ∗ D-solution η : Γ → (ΩA S)1 of the system ΣΓ with respect to a pair (ϕ, δ), where ϕ : Γ → S 1 is an evaluation mapping such that ϕE(Γ) ⊆ S and δ : ΩA S → S is a continuous homomorphism respecting the choice of generators. We have to construct a (V ∗ D, κ)-solution η ′ : Γ → (ΩκA S)1 of ΣΓ with respect to the same pair (ϕ, δ). On κ-reducibility of pseudovarieties of the form V ∗ D 3.1 5 Initial considerations Suppose that g ∈ Γ is such that ηg = u with u ∈ A∗ . Since η and η ′ are supposed to be V ∗ D-solutions of the system ΣΓ with respect to (ϕ, δ), we must have δη = ϕ = δη ′ and so, in particular, δη ′ g = δu. As the homomorphism δ : ΩA S → S is arbitrarily fixed, it may happen that the equality δη ′ g = δu holds only when η ′ g = u. In that case we would be obliged to define η ′ g = u. Since we want to describe an algorithm to define η ′ that should work for any given graph and solution, we will then construct a solution η ′ verifying the following condition: ∀g ∈ Γ, (ηg ∈ A∗ =⇒ η ′ g = ηg). C1 (Γ, η, η ′ ) Suppose next that a vertex v ∈ V (Γ) is such that D \|= ηv = uω with u ∈ A+ , that is, suppose that pD ηv = u−∞ . Because Γ is an arbitrary graph, it could include, for instance, an edge e such that αe = ωe = v and the labeling η could be such that ηe = u. Since D is a subpseudovariety of V ∗ D, η is a D-solution of ΣΓ with respect to (ϕ, δ). Hence, as by condition C1 (Γ, η, η ′ ) we want to preserve finite labels, it would follow in that case that D \|= (η ′ v)u = η ′ v and, thus, that D \|= η ′ v = uω = ηv. This observation suggests that we should preserve the projection into ΩA D of labelings of vertices v such that pD ηv = u−∞ with u ∈ A+ . More generally, we will construct the (V ∗ D, κ)-solution η ′ in such a way that the following condition holds: ∀v ∈ V (Γ), (pD ηv = u−∞ z with u ∈ A+ and z ∈ A∗ =⇒ pD η ′ v = pD ηv). C2 (Γ, η, η ′ ) Let ℓη = max{\|u\| : u ∈ A∗ and ηg = u for some g ∈ Γ} be the maximum length of finite labels under η of elements of Γ. To be able to make some reductions on the graph Γ and solution η, described in Section 3.2, we want η ′ to verify the extra condition below, where L ≥ ℓη is a non-negative integer to be specified later, on Section 3.3: ∀v ∈ V (Γ), 3.2 (ηv = uπ with u ∈ AL =⇒ η ′ v = uπ ′ with δπ = δπ ′ ). C3 (Γ, η, η ′ ) Simplifications on the solution η We begin this section by reducing to the case in which all vertices of Γ are labeled by infinite e pseudowords under η. Suppose first that there is an edge v − → w such that ηv = uv and ηe = ue with uv ∈ A∗ and ue ∈ A+ , so that ηw = uv ue . Drop the edge e and consider the restrictions η1 and ϕ1 , of η and ϕ respectively, to the graph Γ1 = Γ\{e}. Then η1 is a V∗D-solution of the system ΣΓ1 with respect to the pair (ϕ1 , δ). Assume that there is a (V ∗ D, κ)-solution η1′ of ΣΓ1 with respect to (ϕ1 , δ) verifying condition C1 (Γ1 , η1 , η1′ ). Then η1′ v = uv and η1′ w = uv ue . Let η ′ be the extension of η1′ to Γ obtained by letting η ′ e = ue . Then η ′ is a (V ∗ D, κ)-solution of ΣΓ with respect to (ϕ, δ). By induction on the number of edges labeled by finite words under η beginning in vertices also labeled by finite words under η, we may therefore assume that there are no such edges in Γ. Now, we remove all vertices v of Γ labeled by finite words under η such that v is not the beginning of an edge, thus obtaining a graph Γ1 . As above, if η1′ is a (V ∗ D, κ)-solution of 6 J. C. Costa, C. Nogueira, M. L. Teixeira ΣΓ1 , then we build a (V ∗ D, κ)-solution η ′ of ΣΓ by letting η ′ coincide with η1′ on Γ1 and letting η ′ v = ηv for each vertex v ∈ Γ \ Γ1 . So, we may assume that all vertices of Γ labeled by finite words under η are the beginning of some edge. e Suppose next that v − → w is an edge such that ηv = u and ηe = π with u ∈ A∗ and π ∈ ΩA S \ A+ . Notice that, since it is an infinite pseudoword, π can be written as π = π1 π2 with both π1 and π2 being infinite pseudowords. Drop the edge e (and the vertex v in case e e1 is the only edge beginning in v) and let v1 be a new vertex and v1 −→ w be a new edge thus obtaining a new graph Γ1 . Let η1 and ϕ1 be the labelings of Γ1 defined as follows: • η1 and ϕ1 coincide, respectively, with η and ϕ on Γ′ = Γ1 ∩ Γ; • η1 v1 = uπ1 , η1 e1 = π2 , ϕ1 v1 = δη1 v1 and ϕ1 e1 = δη1 e1 . Then η1 is a V ∗ D-solution of the system ΣΓ1 with respect to the pair (ϕ1 , δ). Assume that there is a (V ∗D, κ)-solution η1′ of ΣΓ1 with respect to (ϕ1 , δ) verifying conditions C1 (Γ1 , η1 , η1′ ) and C3 (Γ1 , η1 , η1′ ). In particular, since L is chosen to be greater than ℓη , η1′ v1 = uπ1′ with ′ ′ ′ ′ δπ1 = δπ1′ . Let η ′ be the extension of η1\|Γ ′ to Γ obtained by letting η e = π1 (η1 e1 ) (and η ′ v = u in case v 6∈ Γ′ ). As one can easily verify, η ′ is a (V ∗ D, κ)-solution of ΣΓ with respect to (ϕ, δ). By induction on the number of edges beginning in vertices labeled by finite words under η, we may therefore assume that all vertices of Γ are labeled by infinite pseudowords under η. Suppose at last that an edge e ∈ Γ is labeled under η by a finite word u = a1 · · · an , where n > 1 and ai ∈ A. Denote v0 = αe and vn = ωe. In this case, we drop the edge e and, for each ei → vi to the graph Γ. Let Γ1 i ∈ {1, . . . , n − 1}, we add a new vertex vi and a new edge vi−1 − be the graph thus obtained and let η1 and ϕ1 be the labelings of Γ1 defined as follows: • η1 and ϕ1 coincide, respectively, with η and ϕ on Γ′ = Γ \ {e}; • for each i ∈ {1, . . . , n − 1}, η1 vi = (ηv)a1 · · · ai , η1 ei = ai , ϕ1 vi = δη1 vi and ϕ1 ei = δη1 ei . Hence, η1 is a V ∗ D-solution of the system ΣΓ1 with respect to the pair (ϕ1 , δ). Suppose there exists a (V ∗ D, κ)-solution η1′ of ΣΓ1 with respect to (ϕ1 , δ) verifying condition C1 (Γ1 , η1 , η1′ ). ′ ′ ′ Let η ′ be the extension of η1\|Γ ′ to Γ obtained by letting η e = u. Then η is a (V ∗ D, κ)solution of ΣΓ with respect to (ϕ, δ). By induction on the number of edges labeled by finite words under η, we may further assume that each edge of Γ labeled by a finite word under η is, in fact, labeled by a letter of the alphabet. 3.3 Borders of the solution η The main objective of this section is to define a certain class of finite words, called borders of the solution η. Since the equations (of ΣΓ ) we have to deal with are of the form (αe) e = ωe, these borders will serve to signalize the transition from a vertex αe to the edge e. For each vertex v of Γ, denote by dv ∈ A−N the projection pD ηv of ηv into ΩA D and let Dη = {dv \| v ∈ V (Γ)}. We say that two left-infinite words v1 , v2 ∈ A−N are confinal if they On κ-reducibility of pseudovarieties of the form V ∗ D 7 have a common prefix y ∈ A−N , that is, if v1 = yz1 and v2 = yz2 for some words z1 , z2 ∈ A∗ . As one easily verifies, the relation ∝ defined, for each dv1 , dv2 ∈ Dη , by dv 1 ∝ dv 2 if and only if dv1 and dv2 are confinal is an equivalence on Dη . For each ∝-class ∆, we fix a word y∆ ∈ A−N and words zv ∈ A∗ , for each vertex v with dv ∈ ∆, such that dv = y∆ zv . Moreover, when dv is ultimately periodic, we choose y∆ of the form u−∞ , with u a Lyndon word, and fix zv not having u as a prefix. The word u and its length \|u\| will be said to be, respectively, a root and a period of the solution η. Without loss of generality, we assume that η has at least one root (otherwise we could, easily, modify the graph and the solution in order to include one). η′ . We fix a few of the integers that will be used in the construction of the (V ∗ D, κ)-solution They depend only on the mapping η and on the semigroup S. Definition 3.1 (constants nS , pη , L, E and Q) We let: • nS be the exponent of S which, as one recalls, is the least integer such that snS is idempotent for every element s of the finite A-generated semigroup S; • pη = lcm{\|u\| : u ∈ A+ is a root of η}; • L = max{ℓη , \|zv \| : v ∈ V (Γ)}; • E be an integer such that E ≥ nS pη and, for each word w ∈ AE , there is a factor e ∈ A+ of w for which δe is an idempotent of S. Notice that, for each root u of η, \|unS \| ≤ E and δ(unS ) is an idempotent of S; • Q = L + E. For each positive integer m, we denote by Bm the set Bm = {tm y∆ ∈ Am \| ∆ is a ∝-class}. If y∆ = u−∞ is a periodic left-infinite word, then the element y = tm y∆ of Bm will be said to be periodic (with root u and period \|u\|). For words y1 , y2 ∈ Bm , we define the gap between y1 and y2 as the positive integer g(y1 , y2 ) = min{\|u\| ∈ N : u ∈ A+ and, for some v ∈ A+ , y1 u = vy2 or y2 u = vy1 }, and notice that g(y1 , y2 ) = g(y2 , y1 ) ≤ m. Proposition 3.2 Consider the constant Q introduced in Definition 3.1. There exists qQ ∈ N such that for all integers m ≥ qQ the following conditions hold: 8 J. C. Costa, C. Nogueira, M. L. Teixeira (a) If y1 and y2 are distinct elements of Bm , then g(y1 , y2 ) > Q; (b) If y is a non-periodic element of Bm , then g(y, y) > Q. Proof. Suppose that, for every qQ ∈ N there is an integer m ≥ qQ and elements ym,1 and ym,2 of Bm such that g(ym,1 , ym,2 ) ≤ Q. Hence, there exist a strictly increasing sequence (mi )i of positive integers and an integer r ∈ {1, . . . , Q} such that g(ymi ,1 , ymi ,2 ) i is constant and equal to r. Moreover, since the graph Γ is finite, we may assume that ymi ,1 = tmi y∆1 and ymi ,2 = tmi y∆2 for every i and some ∝-classes ∆1 and ∆2 . It then follows that y∆1 u = y∆2 or y∆2 u = y∆1 for some word u ∈ Ar . Hence, y∆1 and y∆2 are confinal left-infinite words, whence ∆1 and ∆2 are the same ∝-class ∆. Therefore, for every m, ym,1 and ym,2 have the same length and are suffixes of the word y∆ and, so, ym,1 and ym,2 are the same word. This proves already (a). Now, notice that y∆ u = y∆ , meaning that y∆ is the periodic left-infinite word u−∞ . This shows (b) and completes the proof of the proposition. We now fix two more integers. Definition 3.3 (constants M and k) We let: • M be an integer such that M is a multiple of pη and M is greater than or equal to the integer qQ of Proposition 3.2, and notice that M > Q; • k = M + Q. The elements of the set BM will be called the borders of the solution η. We remark that the borders of η are finite words of length M such that, by Proposition 3.2, for any two distinct occurrences of borders y1 and y2 in a finite word, either these occurrences have a gap of size at least Q between them, or y1 and y2 are the same periodic border y. In this case, y is a power of its root u, since M is a multiple of the period \|u\|, and g(y, y) is \|u\|. 3.4 Getting a (V ∗ Dk , κ)-solution As V ∗ Dk is a subpseudovariety of V ∗ D, η is a V ∗ Dk -solution of ΣΓ with respect to (ϕ, δ). The given pseudovariety V was assumed to be κ-reducible. So, by [5, Corollary 6.5], V ∗ Dk is κ-reducible too. Therefore, there is a (V ∗ Dk , κ)-solution ηk′ : Γ → (ΩκA S)1 of ΣΓ with respect to the same pair (ϕ, δ). Moreover, as observed in [6, Remark 3.4], one can constrain the values ηk′ g of each g ∈ Γ with respect to properties which can be tested in a finite semigroup. Since the prefixes and the suffixes of length at most k can be tested in the finite semigroup ΩA Kk × ΩA Dk , we may assume further that ηk′ g and ηg have the same prefixes and the same suffixes of length at most k. We then denote ig = ik ηk′ g = ik ηg and tg = tk ηk′ g = tk ηg, for each g ∈ Γ. Notice that, by the simplifications introduced in Section 3.2, if ηg is a finite word, then g is an edge and ηg is a letter ag and so ig = tg = ag . Otherwise, ig and tg are On κ-reducibility of pseudovarieties of the form V ∗ D 9 length k words. In particular, condition C1 (Γ, η, ηk′ ) holds. That is, ηk′ e = ηe for every edge e such that ηe is a finite word. On the other hand, Lemma 2.3 (ii) of [12], which is stated only for edges, can be extended easily to vertices, so that ηk′ g can be assumed to be an infinite pseudoword for every g ∈ Γ such that ηg is infinite. Thus, in particular, ηk′ v is an infinite pseudoword for all vertices v. Notice that, for each vertex v, there exists a border yv of η such that the finite word yv zv is a suffix of ηv. On the other hand, by Definitions 3.1 and 3.3, \|zv \| ≤ L < Q and k = M + Q. So, as \|yv \| = M , tv = xv yv zv and ηk′ v = πv tv (3.1) for some infinite κ-word πv and some word xv ∈ A+ with \|xv \| = Q − \|zv \|. 3.5 Basic transformations The objective of this section is to introduce the basic steps that will allow to transform the (V ∗ Dk , κ)-solution ηk′ into a (V ∗ D, κ)-solution η ′ . The process of construction of η ′ from ηk′ is close to the one used in [15] to handle with systems of pointlike equations. Both procedures are supported by (basic) transformations of the form a1 · · · ak 7→ a1 · · · aj (ai · · · aj )ω aj+1 · · · ak , which replace words of length k by κ-words. Those procedures differ in the way the indices i ≤ j are determined. In the pointlike case, the only condition that a basic transformation had to comply with was that j had to be minimum such that the value of the word a1 · · · ak under δ is preserved. In the present case, the basic transformations have to preserve the value under δ as well, but the equations (αe)e = ωe impose an extra restriction that is not required by pointlike equations. Indeed, we need η ′ to verify, in particular, δη ′ αe = δηk′ αe(= δηαe) and δη ′ e = δηk′ e(= δηe). So, somewhat informally, for a word a1 · · · ak that has an occurrence overlapping both the factors ηk′ αe and ηk′ e of the pseudoword (ηk′ αe)(ηk′ e), the introduction of the factor (ai · · · aj )ω by the basic transformation should be done either in ηk′ αe or in ηk′ e, and not in both simultaneously. The borders of the solution η were introduced to help us to deal with this extra restriction. Informally speaking, the borders will be used to detect the “passage” from the labeling under ηk′ of a vertex αe to the labeling of the edge e and to avoid that the introduction of (ai · · · aj )ω affect the labelings under δ of ηk′ αe or ηk′ e. Consider an arbitrary word w = a1 · · · an ∈ A+ . An integer m ∈ {M, . . . , n} will be called a bound of w if the factor w[m] = am′ · · · am of w is a border, where m′ = m − M + 1. The bound m will be said to be periodic or non-periodic according to the border w[m] is periodic or not. If w admits bounds, then there is a maximum one that we name the last bound of w. In this case, if ℓ is the last bound of w, then the border w[ℓ] will be called the last border of w. Notice that, by Proposition 3.2 and the choice of M , if m1 and m2 are two bounds of w with m1 < m2 , then either m2 − m1 > Q or w[m1 ] and w[m2 ] are the same periodic border. 10 J. C. Costa, C. Nogueira, M. L. Teixeira Let w = a1 · · · ak ∈ A+ be a word of length k. Notice that, since k = M + Q, if w has a non-periodic last bound ℓ, then ℓ is the unique bound of w. We split the word w in two parts, lw (the left-hand of w) and rw (the right-hand of w), by setting l w = a1 · · · as and rw = as+1 · · · ak where s (the splitting point of w) is defined as follows: if w has a last bound ℓ then s = ℓ; otherwise s = k. In case w has a periodic last bound ℓ, the splitting point s will be said to be periodic. Then, s is not periodic in two situations: either w has a non-periodic last border or w has not a last border. The factorization w = lw rw will be called the splitting factorization of w. We have s ≥ M > Q ≥ E. So, by definition of E, there exist integers i and j such that s − E < i < j ≤ s and the factor e = ai · · · aj of lw verifies δe = (δe)2 . We begin by fixing the maximum such j and, for that j, we fix next an integer i and a word ew = ai · · · aj , called the essential factor of w, as follows. Notice that, if the splitting point s is periodic and u is the root of the last border of w, then δ(unS ) is idempotent and the left-hand of w is of the form lw = l′w unS . Hence, in this case, j = s and we let ew = unS , thus defining i as j − nS \|u\| + 1. Suppose now that the splitting point is not periodic. In this case we let i be the maximum integer such that δ(ai · · · aj ) is idempotent. The word w can be factorized as w = l′w ew l′′w rw , where l′w = a1 · · · ai−1 . We then denote by w b the following κ-word w b = l′w ew eωw l′′w rw = a1 · · · aj (ai · · · aj )ω aj+1 · · · ak and notice that δw b = δw. Moreover \|ew l′′w \| ≤ E and so \|l′w \| ≥ M − E > Q − E = L. It is also convenient to introduce two κ-words derived from w b λk w = a1 · · · aj (ai · · · aj )ω , ̺k w = (ai · · · aj )ω aj+1 · · · ak . (3.2) This defines two mappings λk , ̺k : Ak → ΩκA S that can be extended to ΩA S as done in [15]. Although they are not formally the same mappings used in that paper, because of the different choice of the integers i and j, we keep the same notation since the selection process of those integers is absolutely irrelevant for the purpose of the mappings. That is, with the above adjustment the mappings maintain the properties stated in [15]. The next lemma presents a property of the b-operation that is fundamental to our purposes. Lemma 3.4 For a word w = a1 · · · ak+1 ∈ A+ of length k + 1, let w1 = a1 · · · ak and w2 = a2 · · · ak+1 be the two factors of w of length k. If w b1 = a1 · · · aj1 (ai1 · · · aj1 )ω aj1 +1 · · · ak and w b2 = a2 · · · aj2 (ai2 · · · aj2 )ω aj2 +1 · · · ak+1 , then a1 lw2 = lw1 x for some word x ∈ A∗ . In particular j1 ≤ j2 . On κ-reducibility of pseudovarieties of the form V ∗ D 11 Proof. Write w2 = b1 · · · bk with bi = ai+1 . Let s1 and s2 be the splitting points of w1 and w2 respectively, whence lw1 = a1 · · · as1 and lw2 = b1 · · · bs2 = a2 · · · as2 +1 . To prove that there exists a word x such that a1 lw2 = lw1 x, we have to show that s1 ≤ s2 + 1. Under this hypothesis, we then deduce that ai1 · · · aj1 is an occurrence of the essential factor ew1 in lw2 which proves that j1 ≤ j2 . Assume first that w1 has a last bound ℓ1 , in which case s1 = ℓ1 . By definition, ℓ1 ≥ M . If ℓ1 > M , then the last border of w1 occurs in w2 , one position to the left relatively to w1 . Hence ℓ1 − 1 is a bound of w2 and, so, w2 has a last bound ℓ2 such that ℓ2 ≥ ℓ1 − 1. It follows in this case that s2 = ℓ2 and s1 ≤ s2 + 1. Suppose now that ℓ1 = M . Since s2 ≥ M by definition, the condition s1 ≤ s2 + 1 holds trivially in this case. Suppose now that w1 has not a last bound. Then s1 = k. Moreover, either w2 does not have a last bound or k is its last bound. In both circumstances s2 = k, whence s1 = s2 ≤ s2 + 1. This concludes the proof of the lemma. In the conditions of the above lemma and as in [15], we define ψk : (ΩAk+1 S)1 → (ΩA S)1 as the only continuous monoid homomorphism which extends the mapping Ak+1 → ΩκA S a1 · · · ak+1 7→ (ai1 · · · aj1 )ω aj1 +1 · · · aj2 (ai2 · · · aj2 )ω and let θk = ψk Φk . The function θk : ΩA S → (ΩA S)1 is a continuous k-superposition homomorphism since it is the composition of the continuous k-superposition homomorphism Φk with the continuous homomorphism ψk . We remark that a word w = a1 · · · an of length n > k has precisely r = n − k + 1 factors of length k and θk (w) = ψk (a1 · · · ak+1 , a2 · · · ak+2 , . . . , ar−1 · · · an ) = ψk (a1 · · · ak+1 )ψk (a2 · · · ak+2 ) · · · ψk (ar−1 · · · an ) = (eω1 f1 eω2 )(eω2 f2 eω3 ) · · · (eωr−1 fr−1 eωr ) = eω1 f1 eω2 f2 · · · eωr−1 fr−1 eωr where, for each p ∈ {1, . . . , r}, ep is the essential factor ewp = aip · · · ajp of the word wp = ap · · · ak+p−1 and fp = ajp +1 · · · ajp+1 (p 6= r). Above, for each p ∈ {2, . . . , r − 1}, we have replaced each expression eωp eωp with eωp since, indeed, these expressions represent the same κword. More generally, one can certainly replace an expression of the form xω xn xω with xω xn . Using this reduction rule as long as possible, θk (w) can be written as θk (w) = eωn1 f¯1 eωn2 f¯2 · · · eωnq f¯q , called the reduced form of θk (w), where q ∈ {1, . . . , r}, 1 = n1 < n2 < · · · < nq ≤ r, f¯p = fnp · · · fnp+1−1 (for p ∈ {1, . . . , q − 1}) and f¯q is fnq · · · fr−1 if nq 6= r and it is the empty word otherwise. 12 3.6 J. C. Costa, C. Nogueira, M. L. Teixeira Definition of the (V ∗ D, κ)-solution η ′ We are now in conditions to describe the procedure to transform the (V ∗ Dk , κ)-solution ηk′ into the (V ∗ D, κ)-solution η ′ . The mapping η ′ : Γ → (ΩκA S)1 is defined, for each g ∈ Γ, as η ′ g = (τ1 g)(τ2 g)(τ3 g), where, for each i ∈ {1, 2, 3}, τi : Γ → (ΩκA S)1 is a function defined as follows. First of all, we let τ2 = θk ηk′ . That τ2 is well-defined, that is, that τ2 g is indeed a κ-word for every g ∈ Γ, follows from the fact that ηk′ g is a κ-word and θk transforms κ-words into κ-words (see [15]). Next, for each vertex v, consider the length k words iv = ik ηk′ v = ik ηv and tv = tk ηk′ v = tk ηv. We let τ1 v = λk iv and τ3 v = ̺k tv , where the mappings λk and ̺k were defined in (3.2). Note that, by (3.1), tv = xv yv zv . Moreover, the occurrence of yv shown in this factorization is the last occurrence of a border in tv . Hence, the right-hand rtv of tv is precisely zv . Therefore, one has τ1 v = λk iv = l′iv eiv eωiv and τ3 v = ̺k tv = eωtv l′′tv zv . Consider now an arbitrary edge e. Suppose that ηe is a finite word. Then, ηe is a letter ae and ηk′ e is also ae in this case. Then τ2 e = θk ae = 1 because θk is a k-superposition homomorphism. Since we want η ′ e to be ae , we then define, for instance, τ 1 e = ae and τ3 e = 1. Suppose at last that ηe (and so also ηk′ e) is an infinite pseudoword. We let τ3 e = ̺k te and notice that τ3 e = τ3 ωe. Indeed, as ηk′ is a V ∗ Dk -solution of ΣΓ , it follows from (2.1) that te = tk ηk′ e = tk ηk′ ωe = tωe . The definition of τ1 e is more elaborate. Let v be the vertex αe and consider the word tv ie = a1 · · · a2k . This word has r = k + 1 factors of length k. Suppose that θk (tv ie ) is eω1 f1 eω2 f2 · · · eωr−1 fr−1 eωr and consider its reduced form θk (tv ie ) = eω1 f¯1 eωn2 f¯2 · · · eωnq f¯q . Notice that tv ie = f¯0 f¯1 · · · f¯q f¯q+1 for some words f¯0 , f¯q+1 ∈ A∗ . Hence, there is a (unique) ′ and f¯ = f¯′ f¯′′ with f¯′ ∈ A∗ and f¯′′ ∈ index m ∈ {1, . . . , q} such that tv = f¯0 f¯1 · · · f¯m−1 f¯m m m m m m ω ω ′′ ′ + ω ω ω ¯ ¯ ¯ ¯ ¯ A . Then θk (tv ie ) = β1 β2 , where β1 = e1 f1 en2 f2 · · · enm fm and β2 = fm enm+1 fm+1 · · · enq f¯q and we let ′′ ω τ1 e = β2 = f¯m enm+1 f¯m+1 · · · eωnq f¯q . ′′ f¯ ′ ω ¯ Note that the word β2′ = f¯m m+1 · · · fq is ak+1 · · · ajr , whence β2 er = λk ie . The next lemma is a key result that justifies the definition of the b-operation. On κ-reducibility of pseudovarieties of the form V ∗ D 13 Lemma 3.5 Let e be an edge such that ηe is infinite. Then, with the above notation, β1 = τ3 v and so θk (tv ie ) = (τ3 v)(τ1 e). Moreover, δτ1 e = δλk ie . Proof. We begin by recalling that tv ie = a1 · · · a2k and θk (tv ie ) = eω1 f1 eω2 f2 · · · eωr−1 fr−1 eωr = eω1 f¯1 eωn2 f¯2 · · · eωnq f¯q , where ep is the essential factor ewp = aip · · · ajp of the word wp = ap · · · ak+p−1 and fp = ajp +1 · · · ajp+1 for each p. Note also that λk ie = β2′ eωr , er is a suffix of β2′ and δer is idempotent. So, to prove the equality δτ1 e = δλk ie it suffices to show that δτ1 e = δβ2′ . We know from (3.1) that tv = xv yv zv with 1 ≤ \|xv \| ≤ Q. So, xv = a1 · · · ah−1 , yv = ah · · · aM +h−1 and zv = aM +h · · · ak for some h ∈ {2, . . . , Q + 1}. There are two cases to verify. Case 1. yv is a non-periodic border. Consider the factor wh = ah · · · ak+h−1 of tv ie . By the choice of M and k, the prefix yv is the only occurrence of a border in wh . Hence, M is the last bound of wh and, so, its splitting point. It follows that wh = yv ·zv ak+1 · · · ak+h−1 is the splitting factorization of wh . Therefore, as one can verify for an arbitrary p ∈ {1, . . . , h}, there is only one occurrence of a border in wp , precisely yv , and the splitting factorization of wp is wp = ap · · · ah−1 yv · zv ak+1 · · · ak+p−1 , whence ep = e1 with jp = j1 ≤ M + h − 1 and, so, fp = 1 for p < h. So, the prefix eω1 f1 eω2 · · · fh−1 eωh of θk (tv ie ) reduces to eω1 . Consider now the factor wh+1 = ah+1 · · · ak+h . Hence, either wh+1 does not have a last bound or k is its last bound. In both situations, the splitting point of wh+1 is k and its splitting factorization is wh+1 = wh+1 · 1. Therefore, one deduces from Lemma 3.4 that, for every p ∈ {h + 1, . . . , r}, the occurrence aip · · · ajp of the essential factor ewp in wp is, in fact, an occurrence in the suffix w′ = ak+h−E · · · a2k = aM +L+h · · · a2k of tv ie . Since \|xv yv \| = M +h−1 and \|zv \| ≤ L, it follows that k = \|xv yv zv \| < M + L + h, whence w′ is a suffix of ie and so k < ip < jp for all p ∈ {h + 1, . . . , r}. This means, in particular, that the ω-power eωh+1 is introduced at the suffix ie of tv ie . Hence β1 = eω1 f1 eω2 · · · fh−1 eωh ajh +1 · · · ak and its reduced form is eω1 aj1 +1 · · · ak = τ3 v, which proves that β1 and τ3 v are the same κ-word. Moreover, from k < ip , one deduces that the word ep is a suffix of ak+1 · · · ajp , which proves that δτ1 e = δβ2′ . Case 2. yv is a periodic border. Let u be the root of yv . Then, since M was fixed as a M multiple of \|u\|, yv = uMu where Mu = \|u\| . If the prefix yv is the only occurrence of a border in wh , then one deduces the lemma as in Case 1 above. So, we assume that there is another occurrence of a border y in wh . Hence, by Proposition 3.2 and the choice of M and k, y is precisely yv . Furthermore, since u is a Lyndon word and k = M + Q with Q < M , wh = yv ud wh′ for some positive integer d and some word wh′ ∈ A∗ such that u is not a prefix of wh′ . Notice that, since u is not a prefix of zv by definition of this word, zv is a proper prefix of u. On the other hand wh = ud yv wh′ and the occurrence of yv shown in 14 J. C. Costa, C. Nogueira, M. L. Teixeira this factorization is the last occurrence of yv in wh . Thus, wh = ud yv · wh′ is the splitting factorization of wh . Therefore w ch = ud yv (unS )ω wh′ and eh = unS . More generally, for any p ∈ {1, . . . , h}, yv is a factor of wp and it is the only border that occurs in wp . Hence, the splitting point of wp is periodic and ep = unS . Moreover, as one can verify, j1 = M + h − 1 and the prefix eω1 f1 eω2 · · · fh−1 eωh of θk (tv ie ) is eω1 (u(eω1 )\|u\| )d and so, analogously to Case 1, it reduces to eω1 ud . Since zv is a proper prefix of u and d ≥ 1, k < jh . This allows already deduce that the reduced form of β1 is (unS )ω zv = τ3 v, thus concluding the proof of the first part of the lemma. Now, there are two possible events. ′′ = β ′ , in which case δτ e = δβ ′ is trivially verified. Or m 6= q Either m = q and β2 = f¯m 1 2 2 ω and the ω-power enm+1 was not eliminated in the reduction process of θk (tv ie ). This means that the splitting point of the word wnm+1 is not determined by one of the occurrences of the border yv in the prefix a1 · · · ak+h−1 of tv ie . Then, as in Case 1 above, one deduces that k < ip for each p ∈ {nm+1 , . . . , r} and, so, that δτ1 e = δβ2′ . In both cases β1 = τ3 v and δτ1 e = δλk ie . Hence, the proof of the lemma is complete. Notice that, as shown in the proof of Lemma 3.5 above, if a vertex v is such that yv is a periodic border with root u, then τ3 v = (unS )ω zv . So, the definition of the mapping τ3 on vertices assures condition C2 (Γ, η, η ′ ). 3.7 Proof that η ′ is a (V ∗ D, κ)-solution This section will be dedicated to showing that η ′ is a (V ∗ D, κ)-solution of ΣΓ with respect to the pair (ϕ, δ) verifying conditions C1 (Γ, η, η ′ ) and C3 (Γ, η, η ′ ). We begin by noticing that η ′ g is a κ-word for every g ∈ Γ. Indeed, as observed above, each τ2 g is a κ-word. That both τ1 g and τ3 g are κ-words too, is easily seen by their definitions. Let us now show the following properties. Proposition 3.6 Conditions δη ′ = ϕ, C1 (Γ, η, η ′ ) and C3 (Γ, η, η ′ ) hold. Proof. As ηk′ is a V ∗ Dk -solution of ΣΓ with respect to (ϕ, δ) and, so, the equality δηk′ = ϕ holds, to deduce that δη ′ = ϕ holds it suffices to establish the equality δη ′ = δηk′ . Consider first a vertex v ∈ Γ. Then τ1 v = λk iv = l′iv eiv eωiv and τ3 v = ̺k tv = eωtv l′′tv zv . In this case, the equality δηk′ v = δη ′ v is a direct application of [15, Proposition 5.3], where the authors proved that δπ = δ (λk ik π)(θk π)(̺k tk π) (3.3) for every pseudoword π. Moreover, by definition of the b-operation, \|l′iv \| > L. Therefore, ηv and η ′ v are of the form ηv = uπ and η ′ v = uπ ′ with u ∈ AL and δπ = δπ ′ . So, condition C3 (Γ, η, η ′ ) holds. On κ-reducibility of pseudovarieties of the form V ∗ D 15 Consider next an edge e ∈ Γ. If ηk′ e is a finite word ae , then η ′ e = (τ1 e)(τ2 e)(τ3 e) = ae ·1·1 = ae = ηk′ e, whence δη ′ e = δηk′ e holds trivially. Moreover, since ηk′ e = ηe in this case and every vertex is labeled under η by an infinite pseudoword, it follows that condition C1 (Γ, η, η ′ ) holds. Suppose at last that ηk′ e is infinite and let v = αe. Then τ3 e = ̺k te . On the other hand, by Lemma 3.5, δτ1 e = δλk ie . Hence, by (3.3) and since δ is a homomorphism, δη ′ e = δ (τ1 e)(τ2 e)(τ3 e) = δ (λk ie )(θk ηk′ e)(̺k te ) = δηk′ e. This ends the proof of the proposition. e Consider an arbitrary edge v − → w of Γ. To achieve the objectives of this section it remains to prove that V ∗ D satisfies (η ′ v)(η ′ e) = η ′ w. Since ηk′ is a V ∗ Dk -solution of ΣΓ , V ∗ Dk satisfies (ηk′ v)(ηk′ e) = ηk′ w. Hence, by (2.1), iv = ik (ηk′ v)(ηk′ e) = ik (ηk′ w) = iw and tk (ηk′ v)(ηk′ e) = tk (ηk′ w) = tw . Thus, τ1 v = λk iv = l′iv eiv eωiv = l′iw eiw eωiw = λk iw = τ1 w and τ3 w = ̺k tw = eωtw l′′tw zw . As shown in the proof of [15, Proposition 5.4], it then follows that V ∗ D satisfies eωiw θk (ηk′ v)(ηk′ e) eωtw = eωiw θk (ηk′ w)eωtw and, so, V ∗ D \|= (τ1 v)θk (ηk′ v)(ηk′ e) (τ3 w) = (τ1 w)θk (ηk′ w)(τ3 w) = η ′ w. (3.4) On the other hand, from the fact that θk is a k-superposition homomorphism one deduces θk (ηk′ v)(ηk′ e) = θk (ηk′ v)θk tv (ηk′ e) = θk (ηk′ v)θk (tv ie )θk (ηk′ e). (3.5) Suppose that ηk′ e is an infinite pseudoword. In this case te = tw , whence τ3 e = τ3 w. Moreover, by Lemma 3.5, θk (tv ie ) = (τ3 v)(τ1 e). Therefore, by conditions (3.4) and (3.5), V ∗ D satisfies (η ′ v)(η ′ e) = η ′ w. Assume now that ηk′ e is a finite word, whence ηk′ e = ae ∈ A and η ′ e = ae . Since η is a D-solution of ΣΓ , D \|= (η ′ v)ae = η ′ w and, thus, dv ae = dw . Hence the left-infinite words dv and dw are confinal and, so, ∝-equivalent. Hence dv = y∆ zv , dw = y∆ zw and yv = yw = tk y∆ , where ∆ is the ∝-class of dv and dw . It follows that y∆ zv ae = y∆ zw and tk tv ae ) = tw . In this case, θk (ηk′ v)(ηk′ e) = θk (ηk′ v)θk (tv ae ). On the other hand, tv ae = a1 · · · ak ak+1 = a1 tw is a word of length k + 1 and, so, θk (tv ae ) = ψk (tv ae ) is of the form θk (tv ae ) = eω1 f eω2 . The splitting factorizations of tv and tw are, respectively, tv = xv yv · zv and tw = xw yw · zw . Since yv = yw , it follows that e1 = etv = etw = e2 . Suppose that zv ae = zw . In this case it is clear that f = 1, so that θk (tv ae ) = eωtv . Since θk (ηk′ v) ends with eωtv , it then follows that θk (ηk′ v)(ηk′ e) = θk ηk′ v = τ2 v. Therefore, (τ1 v)θk (ηk′ v)(ηk′ e) (τ3 w) = (τ1 v)(τ2 v)(τ3 w). On the other hand, τ3 w = ̺k tw = eωtw l′′tw zw = eωtv l′′tv zv ae = (τ3 v)ae . So, by (3.4), one has that V ∗ D satisfies (η ′ v)ae = (τ1 v)(τ2 v)(τ3 v)ae = (τ1 v)(τ2 v)(τ3 w) = η ′ w. Suppose now that zv ae 6= zw . In this case, one deduces from the equality y∆ zv ae = y∆ zw , that y∆ is a periodic left-infinite word. Let u be its root, so that y∆ = u−∞ , etv = unS and l′′tv = l′′tw = 1. Since, by definition, u is a primitive word which is not a prefix of zv nor a prefix of 16 J. C. Costa, C. Nogueira, M. L. Teixeira zw , we conclude that zv ae = u and zw = 1. In this case f = u, whence θk (tv ae ) = eωtv u. Then, θk (ηk′ v)(ηk′ e) = (θk ηk′ v)u = (τ2 v)u. Therefore, (τ1 v)θk (ηk′ v)(ηk′ e) (τ3 w) = (τ1 v)(τ2 v)u(τ3 w). Moreover, u(τ3 w) = ueωtw l′′tw zw = u(unS )ω = (unS )ω u = eωtv l′′tv zv ae = (τ3 v)ae . Therefore, using (3.4), one deduces as above that V ∗ D satisfies (η ′ v)ae = η ′ w. We have proved the main theorem of the paper. Theorem 3.7 If V is κ-reducible, then V ∗ D is κ-reducible. This result applies, for instance, to the pseudovarieties Sl, G, J and R. Since the κ-word problem for the pseudovariety LG of local groups is already solved [14], we obtain the following corollary. Corollary 3.8 The pseudovariety LG is κ-tame. Final remarks. In this paper we fixed our attention on the canonical signature κ, while in [15] we dealt with a more generic class of signatures σ verifying certain undemanding conditions. Theorem 3.7 is still valid for such generic signatures σ but we preferred to treat only the instance of the signature κ to keep the proofs clearer and a little less technical. References [1] J. Almeida, Finite Semigroups and Universal Algebra, (World Scientific, Singapore, 1995). 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Teixeira, The word problem for κterms over the pseudovariety of local groups, submitted, preprint available at http://arxiv.org/abs/1509.01533. [15] J. C. Costa, C. Nogueira and M. L. Teixeira, Pointlike reducibility of pseudovarieties of the form V ∗ D, Int. J. Algebra Comput., DOI: 10.1142/S0218196716500090, to appear, preprint available at http://arxiv.org/abs/1509.04088. [16] J. C. Costa and M. L. Teixeira, Tameness of the pseudovariety LSl, Int. J. Algebra Comput. 14 (2004), 627–654. [17] S. Eilenberg, Automata, Languages and Machines, vol. B, (Academic Press, New York, 1976). [18] K. Krohn and J. Rhodes, Algebraic theory of machines. I. Prime decomposition theorem for finite semigroups and machines, Trans. Amer. Math. Soc. 116 (1965), 450–464. [19] M. Lothaire, Algebraic Combinatorics on Words, (Cambridge University Press, 2002). [20] J. Rhodes, Undecidability, automata and pseudovarieties of finite semigroups, Int. J. Algebra Comput. 9 (1999), 455–473. [21] J. Rhodes and B. 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arXiv:1403.4349v1 [math.AC] 18 Mar 2014 LINEARLY RELATED POLYOMINOES VIVIANA ENE, JÜRGEN HERZOG, TAKAYUKI HIBI Abstract. We classify all convex polyomino ideals which are linearly related or have a linear resolution. Convex stack polyominoes whose ideals are extremal Gorenstein are also classified. In addition, we characterize, in combinatorial terms, the distributive lattices whose join-meet ideals are extremal Gorenstein or have a linear resolution. Introduction The ideal of inner minors of a polyomino, a so-called polyomino ideal, is generated by certain subsets of 2-minors of an m × n-matrix X of indeterminates. Such ideals have first been studied by Qureshi in [17]. They include the two-sided ladder determinantal ideals of 2-minors which may also be viewed as the join-meet ideal of a planar distributive lattice. It is a challenging problem to understand the graded free resolution of such ideals. In [7], Ene, Rauf and Qureshi succeeded to compute the regularity of such joint-meet ideals. Sharpe [19, 20] showed that the ideal I2 (X) of all 2-minors of X is linearly related, which means that I2 (X) has linear relations. Moreover, he described these relations explicitly and conjectured that also the ideals of t-minors It (X) are generated by a certain type of linear relations. This conjecture was then proved by Kurano [13]. In the case that the base field over which It (X) is defined contains the rational numbers, Lascoux [14] gives the explicit free resolution of all ideals of t-minors. Unfortunately, the resolution of It (X) in general may depend on the characteristic of the base field. Indeed, Hashimoto [8] showed that for 2 ≤ t ≤ min(m, n) − 3, the second Betti number β2 of It (X) depends on the characteristic. On the other hand, by using squarefree divisor complexes [2] as introduced by Bruns and the second author of this paper, it follows from [2, Theorem 1.3] that β2 for t = 2 is independent of the characteristic. In this paper we use as a main tool squarefree divisor complexes to study the first syzygy module of a polyomino ideal. In particular, we classify all convex polyominoes which are linearly related; see Theorem 2.1. This is the main result of this paper. In the first section we recall the concept of polyomino ideals and show that the polyomino ideal of a convex polyomino has a quadratic Gröbner basis. The second section of the paper is devoted to state and to prove Theorem 2.1. As mentioned before, the proof heavily depends on the theory of squarefree divisor complexes which allow to compute the multi-graded Betti numbers of a toric ideal. To apply this theory, one observes that the polyomino ideal of a convex polyomino 2010 Mathematics Subject Classification. 13C05, 05E40, 13P10. Key words and phrases. Binomial ideals, Linear syzygies, Polyominoes. The first author was supported by the grant UEFISCDI, PN-II-ID-PCE- 2011-3-1023. 1 may be naturally identified with a toric ideal. The crucial conclusion deduced from this observation, formulated in Corollary 2.5, is then that the Betti numbers of a polyomino ideal is bounded below by the Betti numbers of the polyomino ideal of any induced subpolyomino. Corollary 2.5 allows to reduce the study of the relation of polyomino ideals to that of a finite number of polyominoes with a small number of cells which all can be analyzed by the use of a computer algebra system. In the last section, we classify all convex polyominoes whose polyomino ideal has a linear resolution (Theorem 3.1) and all convex stack polyominoes whose polyomino ideal is extremal Gorenstein (Theorem 3.4). Since polyomino ideals overlap with join-meet ideals, it is of interest which of the ideals among the join-meet ideals have a linear resolution or are extremal Gorenstein. The answers are given in Theorem 3.2 and Theorem 3.5. It turns out that the classifications for both classes of ideals almost lead to the same result. 1. Polyominoes In this section we consider polyomino ideals. This class of ideals of 2-minors was introduced by Qureshi [17]. To this end, we consider on N2 the natural partial order defined as follows: (i, j) ≤ (k, l) if and only if i ≤ k and j ≤ l. The set N2 together with this partial order is a distributive lattice. If a, b ∈ N2 with a ≤ b, then the set [a, b] = {c ∈ N2 \| a ≤ c ≤ b} is an interval of N2 . The interval C = [a, b] with b = a + (1, 1) is called a cell of N2 . The elements of C are called the vertices of C and a is called the left lower corner of C. The egdes of the cell C are the sets {a, (a + (1, 0)}, {a, a + (0, 1)}, {(a + (1, 0), a + (1, 1)} and {(a + (0, 1), a + (1, 1)}. Let P be a finite collection of cells and C, D ∈ P. Then C and D are connected, if there is a sequence of cells of P given by C = C1 , . . . , Cm = D such that Ci ∩ Ci+1 is an edge of Ci for i = 1, . . . , m − 1. If, in addition, Ci 6= Cj for all i 6= j, then C is called a path (connecting C and D). The collection of cells P is called a polyomino if any two cells of P are connected; see Figure 1. The set of vertices of P, denoted V (P), is the union of the vertices of all cells belonging to P. Two polyominoes are called isomorphic if they are mapped to each other by a composition of translations, reflections and rotations. Figure 1. A polyomino 2 We call a polyomino P row convex, if for any two cells C, D of P with left lower corner a = (i, j) and b = (k, j) respectively, and such that k > i, it follows that all cells with left lower corner (l, j) with i ≤ l ≤ k belong to P. Similarly, one defines column convex polyominoes. The polyomino P is called convex if it is row and column convex. The polyomino displayed in Figure 1 is not convex, while Figure 2 shows a convex polyomino. Note that a convex polyomino is not convex in the common geometric sense. Figure 2. A convex polyomino Now let P be any collection of cells. We may assume that the vertices of all the cells of P belong to the interval [(1, 1), (m, n)]. Fix a field K and let S be the polynomial ring over K in the variables xij with (i, j) ∈ P. The ideal of inner minors IP ⊂ S of P, is the ideal generated by all 2-minors xil xkj − xkl xij for which [(i, j), (k, l)] ⊂ V (P). Furthermore, we denote by K[P] the K-algebra S/IP . If P happens to be a polyomino, then IP will also be called a polyomino ideal. For example, the polyomino P displayed in Figure 2 may be embedded into the interval [(1, 1), (4, 4)]. Then, in these coordinates, IP is generated by the 2-minors x22 x31 − x32 x21 , x23 x31 − x33 x21 , x24 x31 − x34 x21 , x23 x32 − x33 x22 , x24 x32 − x34 x22 , x24 x33 − x34 x23 , x13 x22 − x12 x23 , x13 x32 − x12 x33 , x13 x42 − x12 x43 , x23 x42 − x22 x43 , x33 x42 − x32 x43 . The following result has been shown by Qureshi in [17, Theorem 2.2]. Theorem 1.1. Let P be a convex polyomino. Then K[P] is a normal Cohen– Macaulay domain. The proof of this theorem is based on the fact that IP may be viewed as follows as a toric ideal: with the assumptions and notation as introduced before, we may assume that V (P) ⊂ [(1, 1), (m, n)]. Consider the K-algebra homomorphism ϕ : S → T with ϕ(xij ) = si tj for all (i, j) ∈ V (P). Here T = K[s1 , . . . , sm , t1 , . . . , tn ] is the polynomial ring over K in the variables si and tj . Then, as observed by Qureshi, IP = Ker ϕ. It follows that K[P] may be identified with the edge ring of the bipartite graph GP on the vertex set {s1 , . . . , sm } ∪ {t1 , . . . , tn } and edges {si , tj } 3 with (i, j) ∈ V (P). With this interpretation of K[P] in mind and by using [16], we obtain Proposition 1.2. Let P be a convex polyomino. Then IP has a quadratic Gröbner basis. Proof. We use the crucial fact, proved in [16], that the toric ideal which defines the edge ring of a bipartite graph has a quadratic Gröbner basis if and only if each 2r-cycle with r ≥ 3 has a chord. By what we explained before, a 2k-cycle, after identifying the vertices of P with the edges of a bipartite graph, is nothing but a sequence of vertices a1 , . . . , a2r of P with a2k−1 = (ik , jk ) and a2k = (ik+1 , jk ) for k = 1, . . . , r such that ir+1 = i1 , ik 6= iℓ and jk 6= jℓ for all k, ℓ ≤ r and k 6= ℓ. A typical such sequence of pairs of integers is the following: 32244553 11332244 Here the first row is the sequence of the first component and the second row the sequence of the second component of the vertices ai . This pair of sequences represents an 8-cycle. It follows from Lemma 1.3 that there exist integers s and t with 1 ≤ t, s ≤ r and t 6= s, s + 1 such that either is < it < is+1 or is+1 < it < is . Suppose that is < it < is+1 . Since a2s−1 = (is , js ) and a2s = (is+1 , js ) are vertices of P and since P is convex, it follows that (it , js ) ∈ P. This vertex corresponds to a chord of the cycle a1 , . . . , a2r . Similarly one argues if is+1 < it < is . Lemma 1.3. Let r ≥ 3 be an integer and f : [r + 1] → Z a function such that f (i) 6= f (j) for 1 ≤ i < j ≤ r and f (r + 1) = f (1). Then there exist 1 ≤ s, t ≤ r such that one has either f (s) < f (t) < f (s + 1) or f (s + 1) < f (t) < f (s). Proof. Let, say, f (1) < f (2). Since f (r + 1) = f (1), there is 2 ≤ q ≤ r with f (1) < f (2) < · · · < f (q) > f (q + 1). • Let q = r. Then, since q = r ≥ 3, one has (f (1) =) f (r + 1) < f (2) < f (r). • Let q < r and f (q + 1) > f (1). Since f (q + 1) 6∈ {f (1), f (2), . . . , f (q)}, it follows that there is 1 ≤ s < q with f (s) < f (q) < f (s + 1). • Let q < r and f (q + 1) < f (1). Then one has f (q + 1) < f (1) < f (q). The case of f (1) > f (2) can be discussed similarly. We denote the graded Betti numbers of IP by βij (IP ). Corollary 1.4. Let P be a convex polyomino. Then β1j (IP ) = 0 for j > 4. Proof. By Proposition 1.2, there exists a monomial order < such that in< (IP ) is generated in degree 2. Therefore, it follows from [10, Corollary 4] that β1j (in< (IP )) = 0 for j > 4. Since β1j (IP ) ≤ β1j (in< (IP )) (see, for example, [9, Corollary 3.3.3]), the desired conclusion follows. 4 2. The first syzygy module of a polyomino ideal Let P be a convex polyomino and let f1 , . . . , fm be the minors generating IP . In this section we study the relation module Syz1 (IP ) of IP which is the kernel of the L S-module homomorphism m i=1 Sei → IP with ei 7→ fi for i = 1, . . . , m. The graded module Syz1 (IP ) has generators in degree 3 and no generators in degree > 4, as we have seen in Corollary 1.4. We say that IP (or simply P) is linearly related if Syz1 (IP ) is generated only in degree 3. Let fi and fj be two distinct generators of IP . Then the Koszul relation fi ej −fj ei belongs Syz1 (IP ). We call fi , fj a Koszul relation pair if fi ej − fj ei is a minimal generator of Syz1 (IP ). The main result of this section is the following. Theorem 2.1. Let P be a convex polyomino. The following conditions are equivalent: (a) P is linearly related; (b) IP admits no Koszul relation pairs; (c) Let, as we may assume, [(1, 1), (m, n)] be the smallest interval with the property that V (P) ⊂ [(1, 1), (m, n)]. We refer to the elements (1, 1), (m, 1), (1, n) and (m, n) as the corners. Then P has the shape as displayed in Figure 5, and one of the following conditions hold: (i) at most one of the corners does not belong to V (P); (ii) two of the corners do not belong to V (P), but they are not opposite to each other. In other words, the missing corners are not the corners (1, 1), (n, m), or the corners (m, 1), (1, n). (iii) three of the corners do not belong to V (P). If the missing corners are (m, 1), (1, n) and (m, n) (which one may assume without loss of generality), then referring to Figure 5 the following conditions must be satisfied: either i2 = m − 1 and j4 ≤ j2 , or j2 = n − 1 and i4 ≤ i2 . As an essential tool in the proof of this theorem we recall the co-called squarefree divisor complex, as introduced in [9]. Let K be field, H ⊂ Nn an affine semigroup and K[H] the semigroup ring attached to it. Suppose that h1 , . . . , hm ∈ Nn is the unique minimal set of generators of H. We consider the polynomial ring T = K[t1 , . . . , tn ] Q h (j) in the variables t1 , . . . , tn . Then K[H] = K[u1 , . . . , um ] ⊂ T where ui = nj=1 tj i and where hi (j) denotes the jth component of the integer vector hi . We choose a presentation S = K[x1 , . . . , xm ] → K[H] with xi 7→ ui for i = 1, . . . , m. The kernel IH of this K-algebra homomorphism is called the toric ideal of H. We assign a Zn -grading to S by setting deg xi = hi . Then K[H] as well as IH become Zn graded S-modules. Thus K[H] admits a minimal Zn -graded S-resolution F with L Fi = h∈H S(−h)βih (K[H]) . In the case that all ui are monomials of the same degree, one can assign to K[H] the structure of a standard graded K-algebra by setting deg ui = 1 for all i. The degree of h with respect to this standard grading will be denoted \|h\|. Given h ∈ H, we define the squarefree divisor complex ∆h as follows: ∆h is the simplicial complex whose faces F = {i1 , . . . , ik } are the subsets of [n] such that 5 h(1) ui1 · · · uik divides t1 · · · th(n) in K[H]. We denote by H̃i (Γ, K) the ith reduced n simplicial homology of a simplicial complex Γ. Proposition 2.2 (Bruns-Herzog [2]). With the notation and assumptions introduced one has Tori (K[H], K)h ∼ = H̃i−1 (∆h , K). In particular, βih (K[H]) = dimK H̃i−1 (∆h , K). Let H ′ be a subsemigroup of H generated by a subset of the set of generators of H, and let S ′ be the polynomial ring over K in the variables xi with hi generator of H ′ . Furthermore, let F′ the Zn -graded free S ′ -resolution of K[H ′ ]. Then, since ′ S. The S is a flat S ′ -module, F′ ⊗S ′ S is a Zn -graded free S-resolution of S/IH ′ n ′ inclusion K[H ] → K[H] induces a Z -graded complex homomorphism F ⊗S ′ S → F. Tensoring this complex homomorphism with K = S/m, where m is the graded maximal ideal of S, we obtain the following sequence of isomorphisms and natural maps of Zn -graded K-modules ′ TorS (K[H ′ ], K) ∼ = TorS (K[H], K). = Hi (F′ ⊗S ′ S)⊗S K) → Hi (F⊗S K) ∼ = Hi (F′ ⊗S ′ K) ∼ i i For later applications we need Corollary 2.3. With the notation and assumptions introduced, let H ′ be a subsemigroup of H generated by a subset of the set of generators of H, and let h be an element of H ′ with the property that hi ∈ H ′ whenever h − hi ∈ H. Then the ′ natural K-vector space homomorphism TorSi (K[H ′ ], K)h → TorSi (K[H], K)h is an isomorphism for all i. Proof. Let ∆′h be the squarefree divisor complex of h where h is viewed as an element of H ′ . Then we obtain the following commutative diagram Tori (K[H ′], K)h −−−→ Tori (K[H], K)h    y    y H̃i−1 (∆′h , K) −−−→ H̃i−1 (∆h , K). The vertical maps are isomorphisms, and also the lower horizontal map is an isomorphism, simply because ∆′h = ∆h , due to assumptions on h. This yields the desired conclusion. Let H ⊂ Nn be an affine semigroup generated by h1 , . . . , hm . An affine subsemigroup H ′ ⊂ H generated by a subset of {h1 , . . . , hm } will be called a homological pure subsemigroup of H if for all h ∈ H ′ and all hi with h − hi ∈ H it follows that hi ∈ H ′ . As an immediate consequence of Corollary 2.3 we obtain Corollary 2.4. Let H ′ be a homologically pure subsemigroup of H. Then ′ TorSi (K[H ′], K) → TorSi (K[H], K) is injective for all i. In other words, if F′ is the minimal Zn -graded free S ′ -resolution of K[H ′] and F is the minimal Zn -graded free S-resolution of K[H], then the complex homomorphism F′ ⊗S → F induces an injective map F′ ⊗K → F ⊗K. In particular, 6 any minimal set of generators of Syzi (K[H ′]) is part of a minimal set of generators of Syzi (K[H]). Moreover, βij (IH ′ ) ≤ βij (IH ) for all i and j. We fix a field K and let P ⊂ [(1, 1), (m, n)] be a convex polyomino. Let as before S be the polynomial ring over K in the variables xij with (i, j) ∈ V (P) and K[P] the K-subalgebra of the polynomial ring T = K[s1 , . . . , sm , t1 , . . . , tn ] generated by the monomials uij = si tj with (i, j) ∈ V (P). Viewing K[P] as a semigroup ring K[H], it is convenient to identify the semigroup elements with the monomial they represent. Given sets {i1 , i2 , . . . , is } and {j1 , j2 , . . . , jt } of integers with ik ⊂ [m] and jk ⊂ [n] for all k, we let H ′ be the subsemigroup of H generated by the elements sik tjl with (ik , jl ) ∈ V (P). Then H ′ a homologically pure subsemigroup of H. Note that H ′ is also a combinatorially pure subsemigroup of H in the sense of [15]. A collection of cells P ′ will be called a collection of cells of P induced by the columns i1 , i2 , . . . , is and the rows j1 , j2 , . . . , jt , if the following holds: (k, l) ∈ V (P ′ ) if and only if (ik , jl ) ∈ V (P). Observe that K[P ′ ] is always a domain, since it is a K-subalgebra of K[P]. The map V (P ′ ) → V (P), (k, l) 7→ (ik , jl ) identifies IP ′ with the ideal contained in IP generated by those 2-minors of I(P) which only involve the variables xik ,jl . In the following we always identify IP ′ with this subideal of IP . If the induced collection of cells of P ′ is a polyomino, we call it an induced polyomino. Any induced polyomino P ′ of P is again convex. Consider for example the polyomino P on the left side of Figure 3 with left lower corner (1, 1). Then the induced polyomino P ′ shown on the right side of Figure 3 is induced by the columns 1, 3, 4 and the rows 1, 2, 3, 4. P′ P Figure 3. Obviously Corollary 2.4 implies Corollary 2.5. Let P ′ be an induced collection of cells of P. Then βij (IP ′ ) ≤ βij (IP ) for all i and j, and each minimal relation of IP ′ is also a minimal relation of IP . We will now use Corollary 2.5 to isolate step by step the linearly related polyominoes. Lemma 2.6. Suppose P admits an induced collection of cells P ′ isomorphic to one of those displayed in Figure 4. Then IP has a Koszul relation pair. 7 Proof. We may assume that V (P ′ ) ⊂ [(1, 1), (4, 4)]. By using CoCoA [3] or Singular [4] to compute Syz1 (IP ′ ) we see that the minors fa = [12\|12] and fb = [34\|34] form a Koszul relation pair of IP ′ . Thus the assertion follows from Corollary 2.5. (a) (b) Figure 4. P ′ Corollary 2.7. Let P be a convex polyomino, and let [(1, 1), (m, n)] be the smallest interval with the property that V (P) ⊂ [(1, 1), (m, n)]. We assume that m, n ≥ 4. If one of the vertices (2, 2), (m − 1, 2), (m − 1, n − 1) or (2, n − 1) does not belong to V (P), then IP has a Koszul relation pair, and, hence, IP is not linearly related. Proof. We may assume that (2, 2) 6∈ V (P). Then the vertices of the interval [(1, 1), (2, 2)] do not belong to V (P). Since [(1, 1), (m, n)] is the smallest interval containing V (P), there exist, therefore, integers i and j with 2 < i ≤ m − 1 and 2 < j ≤ n − 1 such that the cells [(i, 1), (i + 1, 2)] and [(1, j), (2, j + 1)] belong to P. Then the collection of cells induced by the rows 1, 2, i, i + 1 and the columns 1, 2, j, j + 1 is isomorphic to one of the collections P ′ of Figure 4. Thus the assertion follows from Lemma 2.6 and Corollary 2.5. Corollary 2.7 shows that the convex polyomino P should contain all the vertices (2, 2), (m − 1, 2), (m − 1, n − 1) and (2, n − 1) in order to be linearly related. Thus a polyomino which is linearly related must have the shape as indicated in Figure 5. The number i1 is also allowed to be 1 in which case also j1 = 1. In this case the polyomino contains the corner (1, 1). A similar convention applies to the other corners. In Figure 5 all for corners (1, 1), (1, n), (m, 1) and (m, n) are missing. The convex polyomino displayed in Figure 6 however is not linearly related, though it has the shape as shown in Figure 5. Thus there must still be other obstructions for a polyomino to be linearly related. Now we proceed further in eliminating those polyominoes which are not linearly related. Lemma 2.8. Let P be a convex polyomino, and let [(1, 1), (m, n)] be the smallest interval with the property that V (P) ⊂ [(1, 1), (m, n)]. If P misses only two opposite corners, say (1, 1) and (m, n), or P misses all four corners (1, 1), (1, n), (m, 1) and (m, n), then IP admits a Koszul pair and hence is not linearly related. 8 i3 i4 (2, n − 1)• • (m − 1, n − 1) j4 j2 j3 j1 (2, 2) • •(m − 1, 2) i1 i2 Figure 5. Possible shape Figure 6. Not linearly related Proof. Let us first assume that (1, 1) and (m, n) do not belong to V (P), but (1, n) and (m, 1) belong to V (P). The collection of cells P1 induced by the rows 1, 2, m − 1, m and the columns 1, 2, n − 1, n is shown in Figure 7. All the light colored cells, some of them or none of them are present according to whether or not all, some or none of the equations i1 = 2, j1 = 2, i4 = m − 1 and j4 = n − 1 hold. For example, if i1 = 2, j1 6= 2, i4 = m − 1 and j4 6= n − 1, then the light colored cells [(2, 1), (3, 2)] and [(2, 3), (3, 4)] belong P1 and the other two light colored cells do not belong to P1 . It can easily be checked that the ideal IP1 displayed in Figure 7 has a Koszul relation pairs in all possible cases, and so does IP by Corollary 2.5. Next, we assume that none of the four corners (1, 1), (1, n), (m, 1) and (m, n) belong to P. In the following arguments we refer to Figure 5. In the first case suppose [i3 , i4 ] ⊂ [i1 , i2 ] and [j3 , j4 ] ⊂ [j1 , j2 ]. Then the collection of cells induced by the columns 2, i3 , i4 , m − 1 and the rows 1, j3 , j4 , n is the polyomino displayed in Figure 2 which has a Koszul relation pair as can be verified by computer. Thus P has a Koszul relation pair. A similar argument applies if [i1 , i2 ] ⊂ [i3 , i4 ] or [j1 , j2 ] ⊂ [j3 , j4 ]. Next assume that [i3 , i4 ] 6⊂ [i1 , i2 ] or [j3 , j4 ] 6⊂ [j1 , j2 ]. By symmetry, we may discuss only [i3 , i4 ] 6⊂ [i1 , i2 ]. Then we may assume that i3 < i1 and i4 < i2 . 9 Figure 7. We choose the columns i1 , i2 , i3 , i4 and the rows 1, 2, n − 1, n. Then the induced polyomino by these rows and columns is P1 if i1 < i4 , P2 if i4 = i1 and P3 if i4 < i1 ; see Figure 8. In all three cases the corresponding induced polyomino ideal has a Koszul relation pair, and hence so does IP . i1 < i4 i4 = i1 i4 < i1 Figure 8. Lemma 2.9. Let P be a convex polyomino, and let [(1, 1), (m, n)] be the smallest interval with the property that V (P) ⊂ [(1, 1), (m, n)]. Suppose P misses three corners, say (1, n), (m, 1), (m, n), and suppose that i2 < m − 1 and j2 < n − 1, or i2 = m − 1 and j2 < j4 , or j2 = n − 1 and i2 < i4 . Then IP has a Koszul relation pair and hence is not linearly related. Proof. We proceed as in the proofs of the previous lemmata. In the case that i2 < m − 1 and j2 < n − 1, we consider the collection of cells P ′ induced by the columns 1, 2, m − 1 and the rows 1, 2, n − 1. This collection of cells P ′ is depicted in Figure 9. It is easily seen that IP ′ is generated by a regular sequence of length 2, which is a Koszul relation pair. In the case that i2 = m − 1 and j2 < j4 we choose the columns 1, 2, m − 1, m and the rows 1, 2, j4 − 1, j4 . The polyomino P ′′ induced by this choice of rows and columns has two opposite missing corners, hence, by Lemma 2.8, it has a Koszul pair. The case j2 = n − 1 and i2 < i4 is symmetric. In both cases the induced polyomino ideal has a Koszul relation pair. Hence in all three cases IP itself has a Koszul relation pair. 10 Figure 9. Proof of Theorem 2.1. Implication (a)⇒(b) is obvious. Implication (b)⇒(c) follows by Corollary 2.7, Lemma 2.8, and Lemma 2.9. It remains to prove (c)⇒(a). Let P be a convex polyomino which satisfies one of the conditions (i)–(iii). We have to show that P is linearly related. By Corollary 1.4, we only need to prove that β14 (IP ) = 0. Viewing K[P] as a semigroup ring K[H], it follows that one has to check that β1h (IP ) = 0 for all h ∈ H with \|h\| = 4. The main idea of this proof is to use Corollary 2.3. Let h = h1 h2 h3 h4 with jq = siq tiq for 1 ≤ q ≤ 4, and i = minq {iq }, k = maxq {iq }, j = minq {jq }, and ℓ = maxq {jq }. Therefore, all the points hq lie in the (possible degenerate) rectangle Q of vertices (i, j), (k, j), (i, ℓ), (k, ℓ). If Q is degenerate, that is, all the vertices of Q are contained in a vertical or horizontal line segment in P, then β1h (IP ) = 0 since in this case the simplicial complex ∆h is just a simplex. Let us now consider Q non-degenerate. If all the vertices of Q belong to P, then the rectangle Q is an induced subpolyomino of P. Therefore, by Corollary 2.3, we have β1h (IP ) = β1h (IQ ) = 0, the latter equality being true since Q is linearly related. Next, let us assume that some of the vertices of Q do not belong to P. As P has one of the forms (i)–(iii), it follows that at most three verices of Q do not belong to P. Consequently, we have to analyze the following cases. Case 1. Exactly one vertex of Q does not belong to P. Without loss of generality, we may assume that (k, ℓ) ∈ / P which implies that k = m and ℓ = n. In this case, any relation in degree h of P is a relation of same degree of one of the polyominoes displayed in Figure 10. One may check with a computer algebra system that all polyominoes displayed in Figure 10 are linearly related, hence they do not have any relation in degree h. Actually, one has to check only the shapes (a), (b), and (d) since the polyomino displayed in (c) is isomorphic to that one from (b). Hence, β1h (IP ) = 0. Case 2. Two vertices of Q do not belong to P. We may assume that the missing vertices from P are (i, ℓ)and (k, ℓ). Hence, we have i = 1, k = m, and ℓ = n. In this case, any relation in degree h of P is a relation of same degree of one of the polyominoes displayed in Figure 11 (a)–(c). Note that the polyominoes (b) and (c) are isomorphic. One easily checks with the computer that all these polyominoes are linearly related, thus β1h (IP ) = 0. Case 3. Finally, we assume that there are three vertices of Q which do not belong to P. We may assume that these vertices are (i, ℓ), (k, ℓ), and (k, j). In this case, any relation in degree h of P is a relation of same degree of the polyomino displayed in Figure 11 (d) which is linearly related as one may easily check with the computer. Therefore, we get again β1h (IP ) = 0. 11 (a) (b) (c) (d) Figure 10. (a) (b) (c) (d) Figure 11. 12 3. Polyomino ideals with linear resolution In this final section, we classify all convex polyominoes which have a linear resolution and the convex stack polyominoes which are extremal Gorenstein. Theorem 3.1. Let P be a convex polyomino. Then the following conditions are equivalent: (a) IP has a linear resolution; (b) there exists a positive integer m such that P is isomorphic to the polyomino with cells [(i, i), (i + 1, i + 1)], i = 1, . . . , m − 1. Proof. (b) ⇒ (a): If the polyomino is of the shape as described in (b), then IP is just the ideal of 2-minors of a 2×m-matrix. It is well-known that the ideal of 2-minors of such a matrix has a linear resolution. Indeed the Eagon-Northcott complex, whose chain maps are described by matrices with linear entries, provides a free resolution of the ideal of maximal minors of any matrix of indeterminates, see for example [6, Page 600]. (a) ⇒ (b): We may assume that [(1, 1), (m, n)] is the smallest interval containing V (P). We may further assume that m ≥ 4 or n ≥ 4. The few remaining cases can easily be checked with the computer. So let us assume that m ≥ 4. Then we have to show that n = 2. Suppose that n ≥ 3. We first assume that all the corners (1, 1), (1, n), (m, 1) and (m, n) belong to V (P). Then the polyomino P ′ induced by the columns 1, 2, m and the rows 1, 2, n is the polyomino which is displayed on the right of Figure 15. The ideal IP ′ is a Gorenstein ideal, and hence it is does not have a linear resolution. Therefore, by Corollary 2.5, the ideal IP does not have a linear resolution as well, a contradiction. Next assume that one of the corners, say (1, 1), is missing. Since IP has a linear a linear resolution, IP is linearly related and hence has a shape as indicated in Figure 5. Let i1 and j1 be the numbers as shown in Figure 5, and let P ′ the polyomino of P induced by the columns 1, 2, 3 and the rows a, j1 , j1 + 1 where a = 1 if i1 = 2 and a = 2 if i1 > 2, j1 > 2. If j1 = 2 and i1 > 2, we let P ′ to be the polyomino induced by the columns 1, i1 , i1 + 1 and the rows 1, 2, 3. In any case, P ′ is isomorphic to that one displayed on the left of Figure 15. Since IP ′ is again a Gorenstein ideal, we conclude, as in the first case, that IP does not a have linear resolution, a contradiction. As mentioned in the introduction, polyomino ideals overlap with join-meet ideals of planar lattices. In the next result we show that the join-meet ideal of any lattice has linear resolution if and only if it is a polyomino as described in Theorem 3.1. With methods different from those which are used in this paper, the classification of join-meet ideals with linear resolution was first given in [7, Corollary 10]. Let L be a finite distributive lattice [11, pp. 118]. A join-irreducible element of L is an element α ∈ L which is not a unique minimal element and which possesses the property that α 6= β ∨ γ for all β, γ ∈ L \ {α}. Let P be the set of join-irreducible elements of L. We regard P as a poset (partially ordered set) which inherits its ordering from that of L. A subset J of P is called an order ideal of P if a ∈ J, b ∈ P together with b ≤ a imply b ∈ J. In particular, the empty set of P is an order ideal of P . Let J (P ) denote the set of order ideals of P , ordered by inclusion. It 13 then follows that J (P ) is a distributive lattice. Moreover, Birkhoff’s fundamental structure theorem of finite distributive lattices [11, Proposition 37.13] guarantees that L coincides with J (P ). Let L = J (P ) be a finite distributive lattice and K[L] = K[ xα : α ∈ L ] the polynomial ring in \|L\| variables over K. The join-meet ideal IL of L is the ideal of K[L] which is generated by those binomials xα xβ − xα∧β xα∨β , where α, β ∈ L are incomparable in L. It is known [12] that IL is a prime ideal and the quotient ring K[L]/IL is normal and Cohen–Macaulay. Moreover, K[L]/IL is Gorenstein if and only if P is pure. (A finite poset is pure if every maximal chain (totally ordered subset) of P has the same cardinality.) Now, let P = {ξ1 , . . . , ξd } be a finite poset, where i < j if ξi < ξj , and L = J (P ). A linear extension of P is a permutation π = i1 · · · id of [n] = {1, . . . , n} such that j < j ′ if ξij < ξij′ . A descent of π = i1 · · · id is an index j with ij > ij+1 . Let D(π) denote the set of descents of π. The h-vector of L is the sequence h(L) = (h0 , h1 , . . . , hd−1 ), where hi is the number of permutations π of [n] with \|D(π)\| = i. Thus, in particular, h0 = 1. It follows from [1] that the Hilbert series of K[L]/IL is of the form h0 + h1 λ + · · · + hd−1 λd−1 . (1 − λ)d+1 We say that a finite distributive lattice L = J (P ) is simple if L has no elements α and β with β < α such that each element γ ∈ L \ {α, β} satisfies either γ < β or γ > α. In other words, L is simple if and only if P possesses no element ξ for which every µ ∈ P satisfies either µ ≤ ξ or µ ≥ ξ. Theorem 3.2. Let L = J (P ) be a simple finite distributive lattice. Then the join-meet ideal IL has a linear resolution if and only if L is of the form shown in Figure 12. • • • • • • • • • • Figure 12. Proof. Since IL is generated in degree 2, it follows that IL has a linear resolution if and only if the regularity of K[L]/IL is equal to 1. We may assume that K is infinite. Since K[L]/IL is Cohen–Macaulay, we may divide by a regular sequence of 14 linear forms to obtain a 0-dimensional K-algebra A with reg A = reg K[L]/IL whose h-vector coincides with that of reg K[L]/IL . Since reg A = max{i : Ai 6= 0} (see for example [6, Exercise 20.18]), it follows that IL has a linear resolution if and only if the h-vector of L is of the form h(L) = (1, q, 0, . . . , 0), where q ≥ 0 is an integer. Clearly, if P is a finite poset of Figure 12, then \|D(π)\| ≤ 1 for each linear extension π of P . Thus IL has a linear resolution. Conversely, suppose that IL has a linear resolution. In other words, one has \|D(π)\| ≤ 1 for each linear extension π of P . Then P has no three-element clutter. (A clutter of P is a subset A of P with the property that no two elements belonging to A are comparable in P .) Since L = J (P ) is simple, it follows that P contains a two-element clutter. Hence Dilworth’s theorem [5] says that P = C ∪ C ′ , where C and C ′ are chains of P with C ∩ C ′ = ∅. Let \|C\| ≥ 2 and \|C ′ \| ≥ 2. Let ξ ∈ C and µ ∈ C ′ be minimal elements of P . Let ξ ′ ∈ C and µ′ ∈ C ′ be maximal elements of P . Since L = J (P ) is simple, it follows that ξ 6= µ and ξ ′ 6= µ′ . Thus there is a linear extension π of P with \|D(π)\| ≥ 2. Thus IL cannot have a linear resolution. Hence either \|C\| = 1 or \|C ′ \| = 1, as desired. A Gorenstein ideal can never have a linear resolution, unless it is a principal ideal. However, if the resolution is as much linear as possible, then it is called extremal Gorenstein. Since polyomino ideals are generated in degree 2 we restrict ourselves in the following definition of extremal Gorenstein ideals to graded ideals generated in degree 2. Let S be a polynomial ring over field, and I ⊂ S a graded ideal which is not principal and is generated in degree 2. Following [18] we say that I is an extremal Gorenstein ideal if S/I is Gorenstein and if the shifts of the graded minimal free resolution are −2 − p − 1, −2 − (p − 1), −2 − (p − 2), . . . , −3, −2, where p is the projective dimension of I. With similar arguments as in the proof of Theorem 3.2, we see that I is an extremal Gorenstein ideal if and only if I is a Gorenstein ideal and reg S/I = 2, and that this is the case if and only if S/I is Cohen–Macaulay and the h-vector of S/I is of the form h(L) = (1, q, 1, 0, . . . , 0), where q > 1 is an integer. In the following theorem we classify all convex stack polyominoes P for which IP is extremal Gorenstein. Convex stack polyominoes have been considered in [17]. In that paper Qureshi characterizes those convex stack polyominoes P for which IP is Gorenstein. Let P be a polyomino. We may assume that [(1, 1), (m, n)] is the smallest interval containing V (P). Then P is called a stack polyomino if it is column convex and for i = 1, . . . . , m − 1 the cells [(i, 1), (i + 1, 2)] belong to P. Figure 13 displays stack polyominoes – the right polyomino is convex, the left is not. The number of cells of the bottom row is called the width of P and the number of cells in a maximal column is called the height of P. 15 Figure 13. Stack polyominoes Let P be a convex stack polyomino. Removing the first k bottom rows of cells of P we obtain again a convex stack polyomino which we denote by Pk . We also set P0 = P. Let h be the height of the polymino, and let 1 < k1 < k2 < · · · < kr < h be the numbers with the property that width(Pki ) < width(Pki−1 ). Furthermore, we set k0 = 1. For example, for the convex stack polyomino in Figure 13 we have k1 = 1, k2 = 2 and k3 = 3. With the terminology and notation introduced, the characterization of Gorenstein convex stack polyominoes is given in the following theorem. Theorem 3.3 (Qureshi). Let P be a convex stack polyomino of height h. Then the following conditions are equivalent: (a) IP is a Gorenstein ideal. (b) width(Pki ) = height(Pki ) for i = 0, . . . , r. According to this theorem, the convex stack polyomino displayed in Figure 13 is not Gorenstein, because width(Pk0 ) = 5 and height(Pk0 ) = 4. An example of a Gorenstein stack polyomino is shown in Figure 14. Figure 14. A Gorenstein stack polyomino Combining Theorem 3.3 with the results of Section 2, we obtain Theorem 3.4. Let IP be convex stack polyomino. Then IP is extremal Gorenstein if and only if P is isomorphic to one of the polyominoes in Figure 15. 16 Figure 15. Extremal convex stack polyominoes Proof. It can be easily checked that IP is extremal Gorenstein, if P is isomorphic to one of the two polyominoes shown in Figure 15. Conversely, assume that IP is extremal Gorenstein. Without loss of generality we may assume that [(1, 1), (m, n)] is the smallest interval containing V (P). Then Theorem 3.3 implies that m = n. Suppose first that V (P) = [(1, 1), (n, n)]. Then, by [7, Theorem 4] of Ene, Rauf and Qureshi, it follows that the regularity of IP is equal to n. Since IP is extremal Gorenstein, its regularity is equal to 3. Thus n = 3. Next, assume that V (P) is properly contained in [(1, 1), (n, n)]. Since IP is linearly related, Corollary 2.7 together with Theorem 3.3 imply that the top row of P consists of only one cell and that [(2, 1), (n − 1, n − 1)] ⊂ V (P). Let P ′ be the polyomino induced by the rows 2, 3, . . . , n − 1 and the columns 1, 2, . . . , n − 1. Then P ′ is the polyomino with V (P ′ ) = [(1, 1), (n − 2, n − 1)]. By applying again [7, Theorem 4] it follows that reg IP′ = n − 2. Corollary 2.5 then implies that reg IP ≥ reg IP ′ = n − 2, and since reg IP = 3 we deduce that n ≤ 5. If n = 5, then IP′ is the ideal of 2-minors of a 3 × 4-matrix which has Betti numbers β35 6= 0 and β36 6= 0. Since P ′ is an induced polyomino of P and since IP is extremal Gorenstein, Corollary 2.5 yields a contradiction. Up to isomorphism there exist for n = 4 precisely the Gorenstein polyominoes displayed in Figure 16. They are all not extremal Gorenstein as can be easily checked with CoCoA or Singular. For n = 3 any Gorenstein polyomino is isomorphic to one of the two polyominoes shown in Figure 15. This yields the desired conclusion. Figure 16. Gorenstein polyominoes of width 3 17 The following theorem shows that besides of the two polyominoes listed in Theorem 3.4 whose polyomino ideal is extremal Gorenstein, there exist precisely two more join-meet ideals having this property. Theorem 3.5. Let L = J (P ) be a simple finite distributive lattice. Then the joinmeet ideal IL is an extremal Gorenstein ideal if and only if L is one of the following displayed in Figure 17. • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Figure 17. Proof. Suppose that L = J (P ) is simple and that K[L]/IL is Gorenstein. it then follows that P is pure and there is no element ξ ∈ P for which every µ ∈ P satisfies either µ ≤ ξ or µ ≥ ξ. Since h(L) = (1, q, 1, 0, . . . , 0), no 4-element clutter is contained in P . Suppose that a three-element clutter A is contained in P . If none of the elements belonging to A is a minimal element of P , then, since L = J (P ) is simple, there exist at least two minimal elements. Hence there exists a linear extension π of P with \|D(π)\| ≥ 3, a contradiction. Thus at least one of the elements belonging to A is a minimal element of P . Similarly, at least one of the elements belonging to A is a maximal element. Let an element x ∈ A which is both minimal and maximal. Then, since P is pure, one has P = A. Let A = {ξ1 , ξ2 , ξ3} with A 6= P , where ξ1 is a minimal element and ξ2 is a maximal element. Let µ1 be a maximal element with ξ1 < µ1 and µ2 a minimal element with µ2 < ξ2 . Then neither µ1 nor µ2 belongs to A. If ξ3 is either minimal or maximal, then there exists a linear extension π of P with \|D(π)\| ≥ 3, a contradiction. Hence ξ3 can be neither minimal nor maximal. Then since P is pure, there exist ν1 with ξ1 < ν1 < µ1 and ν2 with µ2 < ν2 < ξ2 such that {ν1 , ν2 , ξ3} is a three-element clutter. Hence there exists a linear extension π of P with \|D(π)\| ≥ 4, a contradiction. Consequently, if P contains a three-element clutter A, then P must coincide with A. Moreover, if P is a three-element clutter, then h(L) = (1, 4, 1) and IL is an extremal Gorenstein ideal. Now, suppose that P contains no clutter A with \|A\| ≥ 3. Let a chain C with \|C\| ≥ 3 be contained in P . Let ξ, ξ ′ be the minimal elements of P and µ, µ′ the maximal elements of P with ξ < µ and ξ ′ < µ′ . Since L = J (P ) is simple and since P is pure, it follows that there exist maximal chains ξ < ν1 < · · · < νr < µ and ξ ′ < ν1′ < · · · < νr′ < µ′ such that νi 6= νi′ for 1 ≤ i ≤ r. Then one has a linear extension π of P with D(π) = 2+r ≥ 3, a contradiction. Hence the cardinality of all maximal chains of P is at most 2. However, if the cardinality of all maximal chains of P is equal to 1, then h(L) = (1, 1). Thus IL cannot be an extremal Gorenstein ideal. If the cardinality of all maximal chains of P is equal to 2, then P is the posets 18 displayed in Figure 18. For each of them the join-meet ideal IL is an extremal Gorenstein ideal. • • • • • • • • • • • • h(L) = (1, 2, 1) h(L) = (1, 3, 1) h(L) = (1, 4, 1) Figure 18. References [1] A. Björner, A. M. Garsia and R. P. Stanley, An introduction to Cohen–Macaulay partially ordered sets, In: “Ordered Sets” (I. Rival, Ed.), Springer Netherlands, 1982, pp. 583–615. [2] W. Bruns, J. Herzog, Semigroup rings and simplicial complexes, J. Pure Appl. Algebra 122 (1997), 185–208. [3] CoCoATeam, CoCoA: a system for doing Computations in Commutative Algebra. Available at http://cocoa.dima.unige.it [4] W. Decker, G.-M. Greuel, G. Pfister, H. Schönemann, Singular 3-1-6 — A computer algebra system for polynomial computations. http://www.singular.uni-kl.de (2012). [5] R. P. Dilworth, A decomposition theorem for partially ordered sets, Annals of Math. 51 (1950), 161–166. [6] D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics 150, Springer, 1995. [7] V. Ene, A. A. Qureshi, A. Rauf, Regularity of join-meet ideals of distributive lattices, Electron. J. Combin. 20 (3) (2013), #P20. [8] M. Hashimoto, Determinantal ideals without minimal free resolutions, Nagoya Math. J. 118 (1990), 203–216. [9] J. Herzog, T. Hibi, Monomial ideals, Graduate Texts in Mathematics 260, Springer, 2010. [10] J. Herzog, H. Srinivasan, A note on the subadditivity problem for maximal shifts in free resolutions, to appear in MSRI Proc., arxiv: 1303:6214 [11] T. Hibi, Algebraic Combinatorics on Convex Polytopes, Carslaw Publications, Glebe, N.S.W., Australia, 1992. [12] T. Hibi, Distributive lattices, affine semigroup rings and algebras with straightening laws, In: “Commutative Algebra and Combinatorics” (M. Nagata and H. Matsumura, Eds.), Adv. Stud. Pure Math. 11, North–Holland, Amsterdam, 1987, pp. 93–109. [13] K. Kurano, The first syzygies of determinantal ideals, J. Algebra 124 (1989), 414–436. [14] A. Lascoux, Syzygies des variétés determinantales, Adv. in Math. 30 (1978), 202–237. [15] H. Ohsugi, J. Herzog, T. Hibi, Combinatorial pure subrings, Osaka J. Math. bf 37 (2000), 745–757. [16] H. Ohsugi, T. Hibi, Koszul bipartite graphs, Adv. in Appl. Math. 22 (1999), 25-28. [17] A. Qureshi, Ideals generated by 2-minors, collections of cells and stack polyominoes, J. Algebra 357 (2012), 279–303. [18] P. Schenzel, Uber die freien Auflösungen extremaler Cohen-Macaulay Ringe, J. Algebra 64 (1980), 93–101. [19] D. W. Sharpe, On certain polynomial ideals defined by matrices, Quart. J. Math. Oxford (2) 15 (1964), 155–175. [20] D. W. Sharpe, The syzygies and semi-regularity of certain ideals defined by matrices, Proc. London Math. Soc. 15 (1965), 645–679. 19 Viviana Ene, Faculty of Mathematics and Computer Science, Ovidius University, Bd. Mamaia 124, 900527 Constanta, Romania, and Simion Stoilow Institute of Mathematics of the Romanian Academy, Research group of the project ID-PCE-2011-1023, P.O.Box 1-764, Bucharest 014700, Romania E-mail address: [email protected] Jürgen Herzog, Fachbereich Mathematik, Universität Duisburg-Essen, Campus Essen, 45117 Essen, Germany E-mail address: [email protected] Takayuki Hibi, Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Toyonaka, Osaka 5600043, Japan E-mail address: [email protected] 20	0
A Class of MSR Codes for Clustered Distributed Storage Jy-yong Sohn, Beongjun Choi and Jaekyun Moon arXiv:1801.02014v1 [cs.IT] 6 Jan 2018 KAIST School of Electrical Engineering Email: {jysohn1108, bbzang10}@kaist.ac.kr, [email protected] Abstract—Clustered distributed storage models real data centers where intra- and cross-cluster repair bandwidths are different. In this paper, exact-repair minimum-storage-regenerating (MSR) codes achieving capacity of clustered distributed storage are designed. Focus is given on two cases: = 0 and = 1/(n−k), where is the ratio of the available cross- and intra-cluster repair bandwidths, n is the total number of distributed nodes and k is the number of contact nodes in data retrieval. The former represents the scenario where cross-cluster communication is not allowed, while the latter corresponds to the case of minimum cross-cluster bandwidth that is possible under the minimum storage overhead constraint. For the = 0 case, two types of locally repairable codes are proven to achieve the MSR point. As for = 1/(n − k), an explicit MSR coding scheme is suggested for the two-cluster situation under the specific of condition of n = 2k. I. I NTRODUCTION Distributed Storage Systems (DSSs) have been deployed by various enterprises to reliably store massive amounts of data under the frequent storage node failure events. A failed node is regenerated (repaired) by collecting information from other survived nodes with the regeneration process guided by a predefined network coding scheme. Under this setting, Dimakis et al. [1] obtained the expression for the maximum reliably storable file size, denoted as capacity C(α, γ), as a function of given system parameters: the node capacity α and the bandwidth γ required for repairing a failed node. The capacity analysis in [1] underscores the following key messages. First, there exists a network coding scheme which utilizes the (α, γ) resources and enables a reliable storage of a file of size C(α, γ). Second, it is not feasible to find a network coding scheme which can reliably store a file larger than C(α, γ), given the available resources of (α, γ). In subsequent research efforts, the authors of [2]–[4] proposed explicit network coding schemes which achieve the capacity of DSSs. These coding schemes are optimal in the sense of efficiently utilizing (α, γ) resources for maintaining the reliable storage systems. Focus on the clustered nature of distributed storage has been a recent research direction taken by several researchers [5]–[8]. According to these recent papers, storage nodes dispersed into multiple racks in real data centers are seen as forming clusters. In particular, the authors of the present paper proposed a system model for clustered DSSs in [5] that reflects the difference between intra- and cross-cluster bandwidths. In the system model of [5], the file to be stored is coded and distributed into n storage nodes, which are evenly dispersed into L clusters. Each node has storage capacity of α, and the data collector contacts arbitrary k out of n existing nodes to retrieve the file. Since nodes are dispersed into multiple clusters, the regeneration process involves utilization of both intra- and cross-cluster repair bandwidths, denoted by βI and βc , respectively. In this proposed system model, the authors of [5] obtained the closed-form expression for the maximum reliably storable file size, or capacity C(α, βI , βc ), of the clustered DSS. Furthermore, it has been shown that network coding exists that can achieve the capacity of clustered DSSs. However, explicit constructions of capacity-achieving network coding schemes for clustered DSSs have yet to be found. This paper proposes a network coding scheme which achieves capacity of the clustered DSS, with a minimum required node storage overhead. In other words, the suggested code is shown to be a minimum-storage-regenerating (MSR) code of the clustered DSS. This paper focuses on two important cases of = 0 and = 1/(n − k), where := βc /βI represents the ratio of cross- to intra-cluster repair bandwidths. The former represents the system where crosscluster communication is not possible. The latter corresponds to the minimum value that can achieve the minimum storage overhead of α = M/k, where M is the file size. When = 0, it is shown that appropriate application of locally repairable codes suggested in [9], [10] achieves the MSR point for general n, k, L settings with the application rule depending on the parameter setting. For the = 1/(n−k) case, an explicit coding scheme is suggested which is proven to be an MSR code under the conditions of L = 2 and n = 2k. There have been some previous works [7], [8], [11], [12] on code construction for DSS with clustered storage nodes, but to a limited extent. The works of [8], [11] suggested a coding scheme which can reduce the cross-cluster repair bandwidth, but these schemes are not proven to be an MSR code that achieves capacity of clustered DSSs with minimum storage overhead. The authors of [12] provided an explicit coding scheme which reduces the repair bandwidth of a clustered DSS under the condition that each failed node can be exactly regenerated by contacting any one of other clusters. However, the approach of [12] is different from that of the present paper in the sense that it does not consider the scenario with unequal intra- and cross-cluster repair bandwidths. Moreover, the coding scheme proposed in [12] is shown to be a minimum-bandwidth- regenerating (MBR) code for some limited parameter setting, while the present paper deals with an MSR code. An MSR code for clustered DSSs has been suggested in [7], but this paper has the data retrieval condition different from the present paper. The authors of [7] considered the scenario where data can be collected by contacting arbitrary k out of n clusters, while data can be retrieved by contacting arbitrary k out of n nodes in the present paper. Thus, the two models have the identical condition only when each cluster has one node. The difference in data retrieval conditions results in different capacity values and different MSR points. In short, the code in [7] and the code in this paper achieves different MSR points. (1) A data collector (DC) retrieves the original file M by contacting arbitrary k out of n nodes - this property is called the maximum-distance-separable (MDS) property. The clustered distributed storage system with parameters n, k, L is called an [n, k, L]-clustered DSS. In an [n, k, L]-clustered DSS with given parameters of α, βI , βc , capacity C(α, γ) is defined in [5] as the maximum data that can be reliably stored. The closed-form expression for C(α, γ) is obtained in Theorem 1 of [5]. Aiming at reliably storing file M, the set of (α, γ) pair values is said to be feasible if C(α, γ) ≥ M holds. According to Corollaries 1 and 2 of [6], the set of feasible (α, γ) points shows the optimal trade-off relationship between α and γ, as illustrated in Fig. 1. In the optimal trade-off curve, the point with minimum node capacity α is called the minimum-storageregenerating (MSR) point. Explicit regenerating codes that achieve the MSR point are called the MSR codes. According to Theorem 3 of [6], node capacity of the MSR point satisfies αmsr = M/k αmsr > M/k 1 , n−k 1 if 0 ≤ < . n−k if ≥ MBR point 𝛾𝛾𝑚𝑚𝑚𝑚𝑚𝑚 (2) 𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼 Fig. 1: The optimal trade-off relationship between α and γ in the clustered distributed storage modeled in [6] ૚ ૛ ૜ ૝ ૛ A given file of M symbols is encoded and distributed into n nodes, each of which has node capacity α. The storage nodes are evenly distributed into L ≥ 2 clusters, so that each cluster contains nI := n/L nodes. A failed node is regenerated by obtaining information from other survived nodes: nI −1 nodes in the same cluster help by sending βI each, while n−nI nodes in other clusters help by sending βc each. Thus, repairing each node requires the overall repair bandwidth of 𝛼𝛼: Node Capacity 𝛾𝛾: Repair Bandwidth MSR point 𝛾𝛾𝑚𝑚𝑚𝑚𝑚𝑚 ૚ II. BACKGROUNDS AND N OTATIONS γ = (nI − 1)βI + (n − nI )βc . 𝛾𝛾 ૜ ࢐ 1st cluster (݈ = 1) 2nd cluster (݈ = 2) 3rd cluster (݈ = 3) ࢒ ܰ(ଶ,ଷ) Fig. 2: Two-dimensional representation of clustered distributed storage (n = 12, L = 3, nI = n/L = 4) divides b. Similarly, write a - b if a does not divide b. For given k and nI , we define k c, nI m := mod(k, nI ) = k − qnI . q := b (4) (5) For vectors we use bold-faced lower case letters. For a given vector a, the transpose of a is denoted as aT . For natural numbers m and n ≥ m, the set {ym , ym+1 , · · · , yn } is represented as {yi }ni=m . For a matrix G, the entry of G at the ith row and j th column is denoted as Gi,j . We also express the nodes in a clustered DSS using a two-dimensional representation: in the structure illustrated in Fig. 2, N (l, j) represents the node at the lth row and the j th column. Finally, we recall definitions on the locally repairable codes (LRCs) in [9], [10]. As defined in [10], an (n, k, r)−LRC represents a code of length n, which is encoded from k information symbols. Every coded symbol of the (n, k, r)−LRC can be regenerated by accessing at most r other symbols. As defined in [9], an (n, r, d, M, α)−LRC takes a file of size M and encodes it into n coded symbols, where each symbol is composed of α bits. Moreover, any coded symbol can be regenerated by contacting at most r other symbols, and the code has the minimum distance of d. (3) Note that α = M/k is the minimum storage overhead to satisfy the MDS property, as stated in [1]. Thus, = 1/(n − k) is the scenario with minimum cross-cluster communication when the minimum storage overhead constraint α = M/k is imposed. Here we introduce some useful notations used in the paper. For a positive integer n, [n] represents the set {1, 2, · · · , n}. For natural numbers a and b, we use the notation a \| b if a III. MSR C ODE D ESIGN FOR = 0 In this section, MSR codes for = 0 (i.e., βc = 0) is designed. Under this setting, no cross-cluster communication is allowed in the node repair process. First, the system parameters for the MSR point are examined. Second, two types of locally repairable codes (LRCs) suggested in [9], [10] are proven to achieve the MSR point, under the settings of nI \| k and nI - k, respectively. A. Parameter Setting for the MSR Point We consider the MSR point (α, γ) = (αmsr , γmsr ) which can reliably store file M. The following property specifies the system parameters for the = 0 case. Proposition 1. Consider an [n,k,L] clustered DSS to reliably store file M. The MSR point for = 0 is M M , (nI − 1) , (6) (αmsr , γmsr ) = k−q k−q where q is defined in (4). This point satisfies α = βI . (1) 𝑦𝑦1 (1) 𝑦𝑦2 (1) 𝑦𝑦3 (1) 𝑦𝑦4 (1) 𝑦𝑦5 (1) 𝑦𝑦6 (1) 𝑥𝑥1 (1) 𝑥𝑥2 (6,3) MDS (1) 𝑥𝑥3 (2) 𝑥𝑥2 (1) 𝑠𝑠3 = 𝑦𝑦3 (1) 𝑠𝑠4 = 𝑦𝑦4 (2) (6,3) MDS (2) 𝑥𝑥3 Proof. See Appendix D-A. (1) 𝑠𝑠2 = 𝑦𝑦2 𝑦𝑦1 (2) 𝑦𝑦2 (2) 𝑦𝑦3 (2) 𝑦𝑦4 (2) 𝑦𝑦5 (2) 𝑦𝑦6 (2) 𝑥𝑥1 (1) 𝑠𝑠1 = 𝑦𝑦1 (1) 𝑠𝑠5 = 𝑦𝑦5 (1) 𝑠𝑠6 = 𝑦𝑦6 (2) + 𝑦𝑦1 (2) + 𝑦𝑦2 (2) + 𝑦𝑦3 (2) + 𝑦𝑦4 (2) + 𝑦𝑦5 (2) + 𝑦𝑦6 (a) MDS Precoding B. Code Construction for nI \| k We now examine how to construct an MSR code for the nI \| k case. The following theorem shows that a locally repairable code constructed in [9] with locality r = nI − 1 is a valid MSR code for nI \| k. Theorem 1 (Exact-repair MSR Code Construction for = 0, nI \| k) Let C be the (n, r, d, M, α)−LRC explicitly constructed in [9] for locality r = nI − 1. Consider allocating coded symbols of C in a [n, k, L]−clustered DSS, where r + 1 = nI nodes within the same repair group of C are located in the same cluster. Then, the code C is an MSR code for the [n, k, L]− clustered DSS under the conditions of = 0 and nI \| k. Proof. See Appendix A. Fig. 3 illustrates an example of the MSR code for the = 0 and nI \| k case, which is constructed using the LRC in [9]. In the [n, k, L] = [6, 3, 2] clustered DSS scenario, the parameters are set to 𝑦2 (1) 𝑦3 (2) 𝑦2 (2) 𝑦3 𝑦1 (1) Cluster 1 (1) 𝑠3 = 𝑦3 (2) + 𝑦3 (1) 𝑠1 = 𝑦1 (2) 𝑠6 = + (2) + 𝑦1 (1) 𝑠2 = 𝑦2 𝑦6 (2) (2) 𝑦6 (2) 𝑦6 𝑠4 = (1) 𝑦4 + (2) + 𝑦2 (1) 𝑦5 𝑦5 (1) 𝑦6 (2) (1) (1) 𝑦4 Cluster 2 (1) 𝑦1 𝑦4 (2) 𝑦4 𝑠5 = (1) 𝑦5 (2) + 𝑦5 (b) Allocation of coded symbols into n nodes Fig. 3: MSR code for = 0 with nI \| k (n = 6, k = 3, L = 2). The construction rule follows the instruction in [9], while the concept of the repair group in [9] can be interpreted as the cluster in the present paper. authors of [9], while the present paper proves that this code also achieves the MSR point of the [n, k, L] clustered DSS, in the case of = 0 and nI \| k. α = nI = n/L = 3, M = (k − q)α = (k − bk/nI c)α = 6. C. Code Construction for nI - k Thus, each storage node contains α = 3 symbols, while the [n, k, L] clustered DSS aims to reliably store a file of size M = 6. This code has two properties, 1) exact regeneration and 2) data reconstruction: 1) Any failed node can be exactly regenerated by contacting nI − 1 = 2 nodes in the same cluster, 2) Contacting any k = 3 nodes can recover the original file (j) {xi : i ∈ [3], j ∈ [2]} of size M = 6. (1) (2) The first property is obtained from the fact that yi , yi (1) (2) and si = yi + yi form a (3, 2) MDS code for i ∈ [6]. The second property is obtained as follows. For contacting arbitrary (1) (1) (1) k = 3 nodes, three distinct coded symbols {yi1 , yi2 , yi3 } having superscript one and three distinct coded symbols (2) (2) (2) {yj1 , yj2 , yj3 } having superscript two can be obtained for some i1 , i2 , i3 ∈ [6] and j1 , j2 , j3 ∈ [6]. From Fig. 3a, the (1) (1) (1) (1) (1) (1) information {yi1 , yi2 , yi3 } suffice to recover x1 , x2 , x3 . (2) (2) (2) Similarly, the information {yj1 , yj2 , yj3 } suffice to recover (2) (2) (2) x1 , x2 , x3 . This completes the proof for the second property. Note that this coding scheme is already suggested by the Here we construct an MSR code when the given system parameters satisfy nI - k. The theorem below shows that the optimal (n, k − q, nI − 1)−LRC designed in [10] is a valid MSR code when nI - k holds. Theorem 2 (Exact-repair MSR Code Construction for = 0, nI - k) Let C be the (n0 , k0 , r0 )−LRC constructed in [10] for n0 = n, k0 = k − q and r0 = nI − 1. Consider allocating the coded symbols of C in a [n, k, L]−clustered DSS, where r + 1 = nI nodes within the same repair group of C are located in the same cluster. Then, C is an MSR code for the [n, k, L]−clustered DSS under the conditions of = 0 and nI - k. Proof. See Appendix B. Fig. 4 illustrates an example of code construction for the nI - k case. Without loss of generality, we consider α = 1 case; parallel application of this code multiple α times achieves the MSR point for general α ∈ N, where N is the set 𝐴 0 0 𝐴 1 1 1 = 𝛼1 𝛼2 𝛼3 𝛼12 𝛼22 𝛼32 𝐺=𝑉 𝐺=𝑉 𝑥1 𝑥2 1 = 𝛼1 𝛼12 𝑥3 𝐴 0 0 𝐴 1 1 𝛼2 𝛼3 𝛼22 𝛼32 1 𝛼4 𝛼42 1 𝑤 0 1 0 0 0 0 1 𝑤 0 1 0 0 0 0 1 𝛼4 𝛼42 0 𝑤 0 0 0 0 1 0 0 0 𝑤 1 0 0 0 𝑤 (4,3) RS code 𝑥1 𝑧1 𝑧2 (3,2) - MDS 𝑦1 𝑦2 𝑧1 𝑧2 (3,2) - MDS 𝑦 𝑦1 3𝑦2 𝑥2 𝑥3 Cluster 1 𝑧3RS code𝑧4 (4,3) 𝑧3 Cluster 1 𝑧4 Cluster 2 (3,2) - MDS Cluster 2 (3,2) - MDS 𝑦 𝑦 𝑦3 4 𝑦4 5𝑦5 𝑦1 𝑦2 𝑦1 𝑦4 𝑦5 𝑦4 𝑦2 𝑦3 𝑦6 𝑦5 𝑦𝑦6 (a) Encoding structure 0 𝑤 0 0 0 0 1 0 0 0 𝑤 1 0 0 0 𝑤 Proposition 2. The MSR point for = 1/(n − k) is M M n − nI (αmsr , γmsr ) = , nI − 1 + . 𝑦3 k k n−k (8) This point satisfies α = βI = n − k and M = k(n − k). 𝑦6 Proof. See Appendix D-B. 6 (b) Allocation of coded symbols into n nodes Fig. 4: MSR code for = 0 with nI - k case (n = 6, k = 4, L = 2). The encoding structure follows from the instruction in [10], which constructed [n0 , k0 , r0 ] − LRC. This paper utilizes [n, k − q, nI − 1] − LRC to construct MSR code for [n, k, L] clustered DSS, in the case of = 0 with nI - k. of positivie integers. In the [n = 6, k = 4, L = 2] clustered DSS with = 0, the code and system parameters are: [n0 , k0 , r0 ] = [n, k − q, nI − 1] = [6, 3, 2], α = 1, M = (k − q)α = (k − bk/nI c) = 3 from Proposition 1. The code in Fig. 4 satisfies the exact regeneration and data reconstruction properties: 1) Any failed node can be exactly regenerated by contacting nI − 1 = 2 nodes in the same cluster, 2) Contacting any k = 4 nodes can recover the original file {xi : i ∈ [3]} of size M = 3. Note that {yi }3i=1 in Fig. 4 is a set of coded symbols generated by a (3, 2)−MDS code, and this statement also holds for {yi }6i=4 . This proves the first property. The second property is directly from the result of [10], which states that the minimum distance of the [n0 , k0 , r0 ] − LRC is k0 d = n0 − k0 − + 2 = 6 − 3 − d3/2e + 2 = 3. (7) r0 Note that the [n0 , k0 , r0 ] − LRC is already suggested by the authors of [10], while the present paper proves that applying this code with n0 = n, k0 = k − q, r0 = nI − 1 achieves the MSR point of the [n, k, L]−clustered DSS, in the case of = 0 and nI - k. IV. MSR C ODE D ESIGN FOR = 1 n−k 1 We propose an MSR code for = n−k in clustered DSSs. 1 From (2) and (3), recall that n−k is the minimum value which allows the minimum storage of αmsr = M/k. First, we obtain the system parameters for the MSR point. Second, we design a coding scheme which is shown to be an MSR code under the conditions of n = 2k and L = 2. A. Parameter Setting for the MSR Point The following property specifies the system parameters for the = 1/(n − k) case. Without a loss of generality, we set the cross-cluster repair bandwidth as βc = 1. B. Code Construction for [n, k, L] = [2k, k, 2] Here, we construct an MSR code under the constraints of n = 2k and L = 2. Since we consider the n = 2k case, the system parameters in Proposition 2 are set to α = βI = n − k = k, (9) 2 M = kα = k . Construction 1. Suppose that we are given M = k 2 source symbols {mi,j : i, j ∈ [k]}. Moreover, let the encoding matrix  (1)  (2) (k) G1 G1 · · · G1  (1) (2) (k)  G2 G2 · · · G2   (10) G= . .. ..  ..  .  .. . .  (1) (2) (k) Gk Gk · · · Gk (j) be a k 2 × k 2 matrix, where each encoding sub-matrix Gi is a k × k matrix. For j ∈ [k], node N (1, j) stores mj and node N (2, j) stores pj , where mi = [mi,1 , · · · , mi,k ]T , pi = [pi,1 , · · · , pi,k ]T = (11) k X (j) mTj Gi . (12) j=1 Remark 1. The code generated in Construction 1 satisfies the followings: (a) Every node in cluster 1 contains k message symbols. (b) Every node in cluster 2 contains k parity symbols. Note that this remark is consistent with (9), which states α = k. Under this construction, we have the following theorem, which specifies the MSR construction rule for the [n = 2k, k, L = 2]−DSS with = 1/(n − k). Theorem 3 (Exact-repair MSR Code Construction for 1 = n−k ) If all square sub-matrices of G are invertible, the code designed by Construction 1 is an MSR code for [n, k, L] = [2k, k, 2]−DSS with = 1/(n − k). Proof. See Appendix C. The following result suggests an explicit construction of an MSR code using the finite field. Corollary 1. Applying Construction 1 with encoding matrix G set to the k 2 × k 2 Cauchy matrix [13] achieves the MSR point for an [n = 2k, k, L = 2]−DSS. A finite field of size 2k 2 suffices to design G. Proof. The proof is directly from Theorem 3 and the fact that all sub-matrices of a Cauchy matrix has full rank, as stated in [14]. Moreover, the Cauchy matrix of size n × n can be Cluster 1 Cluster 2 N(1,1) N(1,2) 𝒎𝟏,𝟏 𝒎𝟐,𝟏 𝒎𝟏,𝟐 𝒎𝟐,𝟐 𝒑𝟏,𝟏 𝒑𝟐,𝟏 𝒑𝟏,𝟐 𝒑𝟐,𝟐 N(2,1) N(2,2) 𝑝,,, 𝑝,,/ 𝑝/,, = 𝐺 𝑝/,/ 7 = 2 3 4 𝑚,,, 𝑚,,/ 𝑚/,, 𝑚/,/ 2 7 4 3 Cluster 1 3 4 7 2 4 3 2 7 𝑚,,, 𝑚,,/ 𝑚/,, 𝑚/,/ Cluster 2 ü 𝒎𝟏,𝟏 𝒎𝟐,𝟏 𝒎𝟏,𝟐 𝒎𝟐,𝟐 𝒑𝟏,𝟏 𝒑𝟐,𝟏 𝒑𝟏,𝟐 𝒑𝟐,𝟐 ü ü 7 𝐺= 2 3 4 ü 2 7 4 3 3 4 7 2 4 3 2 7 Parities 𝒑𝟏,𝟐, 𝒑𝟐,𝟏 are inaccessible Messages 𝒎𝟐,𝟏, 𝒎𝟐,𝟐 are accessible Fig. 6: Repairing a failed node in proposed MSR code example for n = 4, k = 2, L = 2 Fig. 5: MSR example for n = 4, k = 2, L = 2 constructed using a finite field of size 2n, according to [15]. An example of MSR code designed by Construction 1 is illustrated in Fig. 5, in the case of n = 4, k = 2, L = 2. This coding scheme utilizes a Cauchy matrix   7 2 3 4 2 7 4 3  (13) G= 3 4 7 2 4 3 2 7 3 using the finite field GF (2 ) with the primitive polynomial x3 + x + 1. The element aα2 + bα + c in GF (23 ) is denoted by the decimal number of (abc)2 , where α is the primitive element. For example, α + 1 is denoted by 3 = (011)2 in the generator matrix G. When [n, k, L, ] = [4, 2, 2, 1/2], the system parameters are α = 2, M = 4, βI = 2, βc = 1 from Proposition 2, which holds for the example in Fig. 5. Here we show that the proposed coding scheme satisfies two properties: 1) exact regeneration of any failed node and 2) recovery of M = 4 message symbols {m1,1 , m1,2 , m2,1 , m2,2 } by contacting any k = 2 nodes. 1) Exact regeneration: Fig. 6 illustrates the regeneration process. Suppose that node N (1, 1) containing the message m1 = [m1,1 , m1,2 ] fails. Then, node N (1, 2) transmits βI = 2 symbols, m2,1 and m2,2 . Nodes N (2, 1) and N (2, 2) transmit βc = 1 symbol each, for example p1,1 and p2,2 , respectively. Then, from the received symbols of m2,1 , m2,2 , p1,1 , p2,2 and matrix G, we obtain y1 p1,1 − G1,3 m2,1 − G1,4 m2,2 7 2 m1,1 := = . y2 p2,2 − G4,3 m2,1 − G4,4 m2,2 4 3 m1,2 Thus, the contents of the failed node can be regenerated by −1 m1,1 7 2 y1 3 2 y1 = = m1,2 4 3 y2 4 7 y2 where the matrix inversion is over GF (23 ). Note that the exact regeneration property holds irrespective of the contents transmitted by N (2, 1) and N (2, 2), since the encoding matrix is a Cauchy matrix, all submatrices of which are invertible. 2) Data recovery: First, if DC contacts two systematic nodes, the proof is trivial. Second, contacting two parity nodes can recover the original message since G is invertible. Third, suppose that DC contacts one systematic node and one parity node, for example, N (1, 1) and N (1, 4). Then, DC can retrieve message symbols m1,1 , m1,2 and parity symbols p2,1 , p2,2 . Using the retrieved symbols and the information on the encoding matrix G, DC additionally obtains z1 p2,1 − G3,1 m1,1 − G3,2 m1,2 7 2 m2,1 := = . z2 p2,2 − G4,1 m1,1 − G4,2 m1,2 2 7 m2,2 Thus, DC obtains m2,1 7 = m2,2 2 2 7 −1 z1 1 = z2 3 3 1 z1 , z2 which completes the data recovery property of the suggested code. V. C ONCLUSION A class of MSR codes for clustered distributed storage modeled in [5] has been constructed. The proposed coding schemes can be applied in practical data centers with multiple racks, where the available cross-rack bandwidth is limited compared to the intra-rack bandwidth. Two important cases of = 0 and = 1/(n − k) are considered, where = βc /βI represents the ratio of available cross- to intra-cluster repair bandwidth. Under the constraint of zero cross-cluster repair bandwidth ( = 0), appropriate application of two locally repairable codes suggested in [9], [10] is shown to achieve the MSR point of clustered distributed storage. Moreover, an explicit MSR coding scheme is suggested for = 1/(n − k), when the system parameters satisfy n = 2k and L = 2. The proposed coding scheme can be implemented in a finite field, by using a Cauchy generator matrix. A PPENDIX A P ROOF OF T HEOREM 1 We focus on code C, the explicit (n, r, d, M, α)-LRC constructed in Section V of [9]. This code has the parameters r+1M ), (A.1) r k where r is the repair locality and d is the minimum distance, and other parameters (n, M, α) have physical meanings identical to those in the present paper. By setting r = nI − 1, the code has node capacity of (n, r, d = n − k + 1, M, α = α= nI M M M = = nI − 1 k k(1 − 1/nI ) k−q (A.2) where the last equality holds from the nI \| k condition and the definition of q in (4). We first prove that any node failure can be exactly regenerated by using the system parameters in (6). According to the description in Section V-B of [9], any node is contained in a unique corresponding repair group of size r + 1 = nI , so that a failed node can be exactly repaired by contacting r = nI − 1 other nodes in the same repair group. This implies that a failed node does not need to contact other repair groups in the exact regeneration process. By setting each repair group as a cluster (note that each cluster contains nI = n/L nodes), we can achieve βc = 0. (A.3) Moreover, Section V-B of [9] illustrates that the exact regeneration of a failed node is possible by contacting the entire symbols contained in r = nI − 1 nodes in the same repair group, and applying the XOR operation. This implies βI = α, which result in M , (A.4) γ = (nI − 1)βI = (nI − 1) k−q combined with (1) and (A.2). From (A.2) and (A.4), we can conclude that code C satisfies the exact regeneration of any failed node using the parameters in (6). Now we prove that contacting any k nodes suffices to recover original data in the clustered DSS with code C applied. Note that the minimum distance is d = n − k + 1 from (A.1). Thus, the information from k nodes suffices to pick the correct codeword. This completes the proof of Theorem 1. A PPENDIX B P ROOF OF T HEOREM 2 We first prove that the code C has minimum distance of d = n−k+1, which implies that the original file of size M = k−q can be recovered by contacting arbitrary k nodes. Second, we prove that any failed node can be exactly regenerated under the setting of (6). Recall that the [n0 , k0 , r0 ]−LRC constructed in [10] has the following property, as stated in Theorem 1 of [10]: Lemma 1 (Theorem 1 of [10]). The code constructed in [10] has locality r0 and optimal minimum distance d = n0 − k0 − d kr0 e + 2, when (r0 + 1) \| n0 . Note that we consider code C of optimal [n0 , k0 , r0 ] = [n, k − q, nI − 1]−LRC. Since r0 + 1 = nI divides n0 = n, Lemma 1 can be applied. The result of Lemma 1 implies that the minimum distance of C is k−q d = n − (k − q) − + 2. (B.1) nI − 1 Since we consider the nI - k case, we have k = qnI + m, (0 < m ≤ nI − 1) from (5). Inserting (B.2) into (B.1), we have (nI − 1)q + m d = n − (k − q) − +2 nI − 1 = n − (k − q) − (q + 1) + 2 = n − k + 1, (B.2) (B.3) Cluster 1 (2) (2) 𝑠3 = + 𝑠1 = 𝑠6 = (1) 𝑦1 + 𝑦1 (2) 𝑦1 𝑠2 = (1) 𝑦4 𝑦5 (2) 𝑦5 (2) 𝑦6 (1) 𝑦6 (2) 𝑦3 (2) 𝑦3 (1) Cluster 2 𝑦3 𝑦2 𝑦2 (1) 𝑦3 (1) (1) (1) 𝑦1 + (2) 𝑦6 𝑠4 = (1) 𝑦4 + (1) 𝑦2 (2) + 𝑦2 (1) 𝑦6 (2) 𝑦4 (2) 𝑦4 𝑠5 = (1) 𝑦5 (2) + 𝑦5 Fig. 7: Code construction for = 0, nI - k case where the second last equality holds since 0 < m ≤ nI − 1 from (B.2). Thus, this proves that contacting arbitrary k nodes suffices to recover the original source file. Now, all we need to prove is that any failed node can be exactly regenerated under the setting of system parameters specified in Proposition 1. According to the rule illustrated in [10], the construction of code C can be shown as in Fig. 7. First, we have M = k − q source symbols {xi }k−q i=1 to store reliably. By applying a (T, k − q) Reed-Solomon code to the source symbols, we obtain {zi }ti=1 where T := L(nI − 1). Then, we partition {zi }Ti=1 symbols into L groups, where each group contains (nI − 1) symbols. Next, each group of {zi } symbols is encoded by an (nI , nI − 1)−MDS code, which result in a group of nI symbols of {yi }. Finally, we store symbol ynI (l−1)+j in node N (l, j). By this allocation rule, yi symbols in the same group are located in the same cluster. Assume that N (l, j), the j th node at lth cluster, containing ynI (l−1)+j symbol fails for l ∈ [L] and j ∈ [nI ]. From Fig. I 7, we know that (nI − 1) symbols of {ynI (l−1)+s }ns=1,s6 =j stored in lth cluster can decode the (nI , nI − 1)−MDS code for group l. Thus, the contents of ynI (l−1)+j can be recovered by retrieving symbols from nodes in the the lth cluster (i.e., the same cluster where the failed node is in). This proves the ability of exactly regenerating an arbitrary failed node. The regeneration process satisfies βc = 0, βI = α. (B.4) Moreover, note that the code in Fig. 7 has M = (k − q)α (B.5) source symbols. Since parameters obtained in (B.4) and (B.5) are consistent with Proposition 1, we can confirm that code C is a valid MSR point under the conditions = 0 and nI - k. A PPENDIX C P ROOF OF T HEOREM 3 Recall that the code designed by Construction 1 allocates systematic nodes at 1st cluster and parity nodes at 2nd cluster, as illustrated in Fig. 8. Moreover, recall that the system parameters for [n, k, L] = [2k, k, 2]−DSS with = 1/(n − k) are α = βI = k, βc = 1, (C.1) 9 𝒎Z = 𝑚Z,,, 𝑚Z,/, ⋯ 𝑚Z,> , 𝒑Z = 𝑝Z,,, 𝑝Z,/, ⋯ 𝑝Z,> 9 𝑁(1,1) 𝑁(1,2) 𝑁(1, 𝑘) Cluster 1 𝒎,9 𝒎9/ ⋯ 𝒎9> Cluster 2 𝒑,9 𝒑9/ ⋯ 𝒑9> 𝑁(2,1) 𝑁(2,2) 𝑁(2, 𝑘) Fig. 8: Code construction for [n, k, L] = [2k, k, 2]−clustered DSS when = 1/(n − k) from Proposition 2 and the definition of = βc /βI . First, we show that exact regeneration of systematic nodes (in the first cluster) is possible using βI = k, βc = 1 in the [n, k, L] = [2k, k, 2] DSS with Construction 1. We use the concept of the projection vector to illustrate the repair process. For l ∈ [L], (l) let vi,j be the lth projection vector assigned for N (1, j), in repairing N (1, i). Similarly, let vi,j be the projection vector assigned for N (2, j), in repairing N (1, i). Assume that the node N (1, i) containing mi = [mi,1 , mi,2 , · · · , mi,k ]T fails. (l) Then, node N (1, j) transmits βI = k symbols {mTj vi,j }kl=1 , T while node N (2, j) transmits βc = 1 symbol pj vi,j . For (l) simplicity, we set vi,j = el and vi,j = ek , where ei is the kdimensional standard basis vector with a 1 in the ith coordinate and 00 s elsewhere. This means that node N (1, j) transmits k symbols mj = [mj,1 , mj,2 , · · · , mj,k ]T it contains, while N (2, j) transmits the last symbol it contains, i.e., the symbol pj,k . Thus, the newcomer node for regenerating systematic node N (1, i) obtains the following information Mi := {mj,s : j ∈ [k] \ {i}, s ∈ [k]} ∪ {pj,k }kj=1 . (C.2) We now show how the newcomer node regenerates mi = [mi,1 , mi,2 , · · · , mi,k ]T using information Mi . Recall that the parity symbols and message symbols are related as in the following k 2 equations:     m1 p1 p2  m2      (C.3)  ..  = G  ..   .   .  pk mk obtained from (10) and (12). Among these k 2 parity symbols, k parity symbols received by the newcomer node can be expressed as      p1,k Gk,1 Gk,2 · · · Gk,k2 m1,1 p2,k  G2k,1 G2k,2 · · · G2k,k2   m1,2        ..  =  .. .. ..   ..  , (C.4) ..  .   .   . . . .  pk,k Gk2 ,1 Gk2 ,2 ··· subtracting the constant known values from (C.4) results in      mi,1 y1 Gk,(i−1)k+1 Gk,(i−1)k+2 · · · Gk,ik y2  G2k,(i−1)k+1 G2k,(i−1)k+2 · · · G2k,ik  mi,2        ..  =  .. .. ..   ..  ..  .   . .  . . . 2 2 2 mi,k yk Gk ,(i−1)k+1 Gk ,(i−1)k+2 · · · Gk ,ik (C.5) where Gk2 ,k2 mk,k where the matrix in (C.4) is generated by removing k(k − 1) rows from G. Since we are aware of k(k−1) message symbols of {mj,s : j ∈ [k] \ {i}, s ∈ [k]} and the entries of G matrix, yl := pl,k − k k X X Glk,(j−1)k+s mj,s (C.6) j=1j6=i s=1 for l ∈ [k]. Note that the matrix in (C.5) can be obtained by removing k(k − 1) columns from the matrix in (C.4). Since every square sub-matrix of G is invertible, we can obtain mi = [mi,1 , mi,2 , · · · , mi,k ]T , which completes the proof for exactly regenerating the failed systematic node. Second, we prove that exact regeneration of the parity (l) nodes (in the second cluster) is possible. Let ωi,j be the lth projection vector assigned for N (2, j) in repairing N (2, i). Similarly, let ωi,j be the projection vector assigned for N (1, j) in repairing N (2, i). Assume that the parity node N (2, i) fails, which contains pi = [pi,1 , pi,2 , · · · , pi,k ]T . Then, node (l) N (2, j) transmits βI = k symbols {pTj ωi,j }kl=1 , while node N (1, j) transmits βc = 1 symbol mTj ωi,j . For simplicity, we (l) set ωi,j = el and ωi,j = ek . This means that node N (2, j) transmits k symbols pj = [pj,1 , pj,2 , · · · , pj,k ]T it contains, while N (1, j) transmits the last symbol it contains, i.e., the symbol mj,k . Thus, the newcomer node for regenerating parity node N (2, i) obtains the following information Pi := {pj,s : j ∈ [k] \ {i}, s ∈ [k]} ∪ {mj,k }kj=1 (C.7) We show how the newcomer node regenerates pi = [pi,1 , pi,2 , · · · , pi,k ]T using the information Pi . Among k 2 parity symbols in (C.3), k(k − 1) parity symbols received by the newcomer node can be expressed as    (1) (k)  G1 · · · G1 p1   .. ..  ..  ..   m1 .  .   . .      (1) (k)   m2  pi−1     0   Gi−1 · · · Gi−1  (C.8) (1) (k)   ..  = G m, pi+1  =     G · · · G .    i+1 i+1    .   . ..  mk ..  ..   .. . .  (1) (k) pk Gk · · · Gk (j) where Gi is defined in Construction 1. Note that G0 is a k(k − 1) × k 2 matrix, which is generated by removing lth rows from G, for l ∈ {(i − 1)k + 1, (i − 1)k + 2, · · · , ik}. Since we know the values of k message symbols {mj,k }kj=1 and the entries of G matrix, subtracting constant known values from (C.8) results in G00 m0 , (C.9) where G00 is generated by removing lth columns from G0 for l ∈ {k, 2k, · · · , k 2 }. Similarly, m0 is generated by removing lth rows from m for l ∈ {k, 2k, · · · , k 2 }. Thus, G00 is an invertible k(k − 1) × k(k − 1) matrix, so that we can obtain m0 , which contains P̃i = {mj,s : j ∈ [k], s ∈ [k − 1]}. (C.10) Since Pi ∪ P̃i contains every message symbol {mj,s : j, s ∈ [k]}, we can regenerate pi = [pi,1 , pi,2 , · · · , pi,k ]T using (C.3). This completes the proof for exactly regenerating the failed parity node. Finally, we prove that M = k 2 message symbols can be obtained by contacting arbitrary k nodes. In this proof, we use a slightly modified notation for representing message and parity symbols. For j, s ∈ [k], the message symbol mj,s and the parity symbol pj,s are denoted as m(j−1)k+s and p(j−1)k+s , respectively. Then, (C.3) is expressed as     m1 p1  m2   p2      (C.11)  ..  = G  ..   .   .  Suppose that the data collector (DC) contacts e nodes from the 1st cluster, and k − e nodes from the 2nd cluster, for e ∈ {0, 1, · · · , k}. Then, DC obtains k(k − e) parity symbols and ke message symbols. Since there exists total of M = k 2 message symbols, the number of message symbols that DC cannot obtain is M − ke = k(k − e). Let the parity symbols obtained by DC be pi1 , · · · , pik(k−e) , and the message symbols not obtained by DC be mj1 , · · · , mjk(k−e) . Then, the known parities can be expressed as     pi1 m1,1  m1,2   pi2      (C.12)  ..  = G0  ..  ,  .   .  pik(k−e) 1) if m = nI − 1 where G00 is a k(k−e)×k(k−e) matrix generated by taking the k(k−e) lth columns from G0 for l ∈ {jt }t=1 . Since G00 is invertible, k(k−e) we obtain the unknown message symbols {mji }i=1 . This completes the proof. M ζnI −2 λnI −2 (1 − (D.4) q nI − 1 = k − 2q λnI −2 = (D.5) δnI −2 (D.6) ζnI −2 = k − q (D.7) where q and m are defined in (5) and (4). When m = nI − 1, (D.3) can be expressed as = k − 2q − 1, δnI −2 )), ζnI −2 (D.8) where the last equality is from (5). Thus, from (D.8), (D.4) and (D.2), we have ζnI −2 − δnI −2 = nI − 1 λnI −2 (D.9) holds for the m = nI −1 case. Similarly, using (D.5), (D.6) and (D.7), we can confirm that (D.9) holds for the 0 ≤ m ≤ nI −2 case. Inserting (D.4), (D.7), (D.9) into (D.1), we obtain M , k−q M γ= (nI − 1) k−q α= (D.10) (D.11) Since γ = (nI − 1)βI for βc = 0 from (1), we obtain βI = α = M/(k − q), (D.12) which completes the proof. B. Proof of Proposition 2 We consider the βc = 1 case without losing generality. This implies that βI = 1/ = n − k (D.13) according to the definition = βc /βI . Now, we observe the expressions for α and M. From Corollary 3 of [6], the MSR point for = 1/(n − k) is illustrated as M M 1 , ), k k sk−1 (D.14) where From Corollary 3 of [6], the MSR point for = 0 is given , (D.3) 2) else (m = 0, 1, · · · , nI − 2) A. Proof of Proposition 1 M δnI −2 (α, γ) = ( A PPENDIX D P ROOF OF P ROPOSITIONS (α, γ) = ( (D.2) ζnI −2 = k − q mk,k where G0 is a k(k − e) × k 2 matrix obtained by taking lth k(k−e) rows from G, for l ∈ {it }t=1 . Since we know ke message symbols and the elements of G, subtracting the known constant values from (C.12) results in   mj1  mj2    G00  (C.13) , ..   . mjk(k−e) by q+1 nI − 1 = (q + 1)(nI − 2) λnI −2 = δnI −2 = qnI − 2q + nI − 2 = qnI − 2q + m − 1 mk2 pk2 where {ζi }, {λi }, {δi } are defined in [6]. This paper does not review the explicit form of the definitions, but shows how (α, γ) looks like. From the proof of Lemma 5 of [6], we have (D.1) sk−1 = (n − k) 1 = (nI − 1) + (n − nI ) nI − 1 + n−nI n−k (D.15) from the definition of {si } in [6] and the setting of = 1/(n− k). Combining (D.14) and (D.15) result in (8). Note that γ in (1) can be expressed as γ = (n − nI )βc + (nI − 1)βI = (n − nI ) + (nI − 1)(n − k), (D.16) where the last equality holds due to (D.13). Combining (8) and (D.16), we obtain γ= M γ , k n−k which result in M = k(n − k). (D.17) Using α = M/k in (D.14), we have α = n − k. (D.18) This completes the proof. R EFERENCES [1] A. G. Dimakis, P. B. Godfrey, Y. Wu, M. J. Wainwright, and K. Ramchandran, “Network coding for distributed storage systems,” IEEE Transactions on Information Theory, vol. 56, no. 9, pp. 4539–4551, 2010. [2] K. Rashmi, N. B. Shah, P. V. Kumar, and K. Ramchandran, “Explicit construction of optimal exact regenerating codes for distributed storage,” in Communication, Control, and Computing, 2009. 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arXiv:1604.08860v4 [cs.DS] 25 Nov 2016 Designing optimal- and fast-on-average pattern matching algorithms Gilles Didier and Laurent Tichit Aix-Marseille Université, CNRS, Centrale Marseille, I2M UMR7373, Marseille, France E-mail: {gilles.didier,_laurent.tichit}@univ-_amu.fr March 26, 2018 Abstract Given a pattern w and a text t, the speed of a pattern matching algorithm over t with regard to w, is the ratio of the length of t to the number of text accesses performed to search w into t. We first propose a general method for computing the limit of the expected speed of pattern matching algorithms, with regard to w, over iid texts. Next, we show how to determine the greatest speed which can be achieved among a large class of algorithms, altogether with an algorithm running this speed. Since the complexity of this determination makes it impossible to deal with patterns of length greater than 4, we propose a polynomial heuristic. Finally, our approaches are compared with 9 pre-existing pattern matching algorithms from both a theoretical and a practical point of view, i.e. both in terms of limit expected speed on iid texts, and in terms of observed average speed on real data. In all cases, the pre-existing algorithms are outperformed. 1 Introduction We focus on algorithms solving the online string matching problem, which consists in reporting all, and only the occurrence positions of a pattern w in a text t (online meaning that no pre-processing of the text is allowed). As one of the oldest problems addressed in computer science, it has been extensively studied. We refer to [10] for a comprehensive list and an evaluation of all the pattern matching algorithms developed so far. By the authors’ count, more than 80 algorithms have already been proposed, among which more than a half were published during the last ten years. This fact sounds quite paradoxical, since the Morris-Pratt algorithm, which is optimal in terms of worst case analysis, dates back to 1970. A possible explanation is that there is wide gap between the worst case complexity of algorithms and their computation times on real data. For instance, there are pattern matching algorithms with non-linear worst case complexities, which perform much better than Morris-Pratt on English texts. Basically, the 1 average case analysis is way more suited to assess the relevance of a pattern matching algorithm from a practical point of view. The average case analysis of some pattern matching algorithms, notably Boyer-Moore-Horspool and Knuth-Morris-Pratt, has already been carried out from various points of view [27, 12, 4, 3, 16, 21, 22, 25]. We provide here a general method for studying the limit average behavior of a pattern algorithm over iid texts. More precisely, following [18], we consider the limit expectation of the ratio of the text length to the number of text accesses performed by an algorithm for searching a pattern w in iid texts. This limit expectation is called the asymptotic speed of the algorithm with regard to w under the iid model. The computation of the asymptotic speed is based on w-matching machines which are automata-like structures able to simulate the behavior of a pattern matching algorithm while searching the pattern w. The underlying idea is the same as in [18, 19, 20, 17] and can be seen as a generalization of the string matching automaton [7]. In the companion paper, G. Didier provided a theoretical analysis of the asymptotic speed of pattern matching algorithms over iid texts [8]. In particular, he showed that, for a given pattern w, the greatest asymptotic speed among a large class of pattern matching algorithms, is achieved by a w-matching machine in which the states are essentially subsets of positions of w. Such machines are called strategies below. We provide here a brute force algorithm computing the Fastest strategy for a given pattern w and the frequencies of an iid model. The algorithm is based on an original structure associated to the pattern w and called its position lattice, which gives a full representation of the overlap relations between the subsets of positions of w. Since the brute force algorithm cannot be applied on patterns of length greater than 4, because of its (very high) time-complexity, we propose a polynomial K-Heuristic, in which the polynomial order K may be chosen by the user. The Fastest and K-Heuristic approaches are finally compared with 9 several pre-existing pattern matching algorithms: • from a theoretical point of view, by computing their limit expected speeds with regard to various patterns and iid models, • from a practical point of view, by computing their average speeds over two sources (an English text and a DNA sequence). In both cases, the Fastest and K-Heuristic (with K large enough) approaches outperform the pre-existing algorithms. The software and the data used to perform the tests are available at https: //github.com/gilles-_didier/Matchines.git. The rest of the paper is organized as follows. Section 2 presents the notations and recalls some concepts and results from [8]. It is followed by two sections which introduce the central objects of this work: the strategies and the position lattice of a pattern. In particular, we provide an algorithm computing the position lattice of a given pattern. Section 5 shows how to use the position 2 lattice of a pattern to obtain the Fastest strategy with regard to this pattern and an iid model. In Section 6, we provide a polynomial heuristic allowing to compute fast strategies. Section 7 presents the results of various comparisons between 9 pre-existing pattern matching algorithms, the K-Heuristic and, each time it is possible, the Fastest strategy. The results are discussed in the last section. 2 Notations and definitions 2.1 Notations and general definition For all finite sets S, P(S) is the power set of S and \|S\| is its cardinal. An alphabet is a finite set A of elements called letters or symbols. A word, a text or a pattern on A is a finite sequence of symbols of A. We put \|v\| for the length of a word v. Words are indexed from 0, i.e. v = v0 v1 . . . v\|v\|−1 . We write v[i,j] for the subword of v starting at its position i and ending at its position j, i.e. v[i,j] = vi vi+1 . . . vj . The concatenate of two words u and v is the word uv = u0 u1 . . . u\|u\|−1 v0 v1 . . . v\|v\|−1 . For any length n ≥ 0, we note An the S set of words of length n on A, and ∞ ? A , the set of finite words on A, i.e. A? = n=0 An . Unless otherwise specified, all the texts and patterns considered below are on a fixed alphabet A. A pattern matching algorithm takes a pattern w and a text t as inputs an reports all, and only the occurrence positions of w in t. For all patterns w, we say that two pattern matching algorithms are w-equivalent if, for all texts t, they access exactly the same positions of t on the input (w, t). 2.2 Matching machines and the generic algorithm [8] For all patterns w, a w-matching machine is 6-uple (Q, o, F, α, δ, γ) where • Q is a finite set of states, • o ∈ Q is the initial state, • F ⊂ Q is the subset of pre-match states, • α : Q → N is the next-position-to-check function, which is such that for all q ∈ F , α(q) < \|w\|, • δ : Q × A → Q is the transition state function, • γ : Q × A → N is the shift function. By convention, the set of states of a matching machine always contains a sink state , which is such that, for all symbols x ∈ A, δ( , x) = and γ( , x) = 0. The order OΓ of a matching machine Γ = (Q, o, F, α, δ, γ) is defined as OΓ = maxq∈Q {α(q)}. 3 The w-matching machines carry the same information as the Deterministic Arithmetic Automatons defined in [19, 20]. The generic algorithm takes a w-matching machine and a text t as inputs and outputs positions of t (Algorithm 1). input : a w-matching machine (Q, o, F, α, δ, γ) and a text t output: all the occurrence positions of w in t 1 2 3 4 5 (q, p) ← (o, 0) while p ≤ \|t\| − \|w\| do if q ∈ F and tp+α(q) = wα(q) then print “ occurrence at position p ” (q, p) ← (δ(q, tp+α(q) ), p + γ(q, tp+α(q) )) Algorithm 1: The generic algorithm Each component of a w-matching machine makes sense in regard to the way it is used by the generic algorithm. The pre-match states in F are those which lead to report an occurrence of the pattern at the current position, if the nextposition-to-check of the pattern matches the corresponding position in the text (Line 3 of Algorithm 1). The condition α(q) < \|w\| for all q ∈ F in the definition of w-matching machines, is technical and used in [8]. A w-matching machine Γ is valid if, for all texts t, the execution of the generic algorithm on the input (Γ, t) outputs all, and only the occurrence positions of w in t. Since one has to check all the positions of the pattern w before concluding that it occurs somewhere in a text, the order of a valid w-matching machine is at least \|w\| − 1. We claim that for all the pattern matching algorithms developed so far and all patterns w, there exists a w-matching machine Γ which is such that, for all texts t, the generic algorithm and the pattern matching algorithm access exactly the same positions of t on the inputs (Γ, t) and (w, t) respectively [8]. For instance, Figure 1 displays a abb-matching machine which accesses the same positions as the naive algorithm while searching abb. 2.3 Full-memory expansion – standard matching machines [8] We present here a transformation on matching machines which split their states according to the text positions read from the current position during an execution of the generic algorithm. The main point of this transformation is that the average complexity of matching machines such obtained may then be computed through algebraic methods (Sections 2.4 and 2.5). For all n ∈ N, Rn is the set of subsets H of {0, . . . , n} × A verifying that, for all i ∈ {0, . . . , n}, there exists at most one pair in H with i as first entry. For all H ∈ Rn , we put f (H) for the set comprising all the first entries of 4 b/1 S0 0 a/0 S1 1 b/0 S2 2 a/1 b/1 a/1 Figure 1: abb-matching machine of the naive algorithm. The next-position-to check are displayed below all states S0, S1 and S2. Edges from states Si are labelled with “x/γ(Si, x)” for all symbols x. The transition associated to a match is blue-colored. the pairs in H, namely f (H) = {i \| ∃x ∈ A with (i, x) ∈ H}. For all k ∈ N and H ∈ Rn , the k-shifted of H is k ← − H = {(u − k, y) \| (u, y) ∈ H with u ≥ k}, i.e. the subset of Rn obtained by subtracting k from the first entries of the pairs in H and by keeping only the pairs with non-negative first entries. The full memory expansion of a w-matching machine Γ = (Q, o, F, α, δ, γ) is the w-matching machine Γ? obtained by removing the unreachable states of Γ0 = (Q0 , o0 , F 0 , α0 , δ 0 , γ 0 ), defined as: • Q0 = Q × ROΓ • o0 = (o, ∅) • α0 ((q, H)) = α(q) • γ 0 ((q, H), x) = γ(q, x) • F 0 = F × ROΓ • δ 0 ((q, H), x)  γ(q,x)  ←−−−−−−−−−−−    (δ(q, x), H ∪ {(α(q), x)}) if ∀a ∈ A, (α(q), a) 6∈ H = if ∃a 6= x s.t. (α(q), a) ∈ H  γ(q,x)   ← −  (δ(q, x), H ) if (α(q), x) ∈ H 5 (S0, {(0, b)}) 0 b/1 b/1 b/1 b/1 (S0, ∅) 0 a/0 (S0, {(0, b), (1, b)}) 0 (S1, {(0, a)}) 1 a/0 b/0 a/1 (S0, {(0, a)}) 0 (S2, {(0, a), (1, b)}) 2 a/1 b/1 (S0, {(0, b), (1, a)}) 0 Figure 2: Full memory expansion of the abb-matching machine of Figure 1. By construction, at all iterations of the generic algorithm on the input (Γ? , t), if the current state and position are (q, H) and p, respectively, then the positions of {j + p \| j ∈ f (H)} are exactly the positions of t greater than p accessed so far (the second entries of the corresponding elements of H give the symbols read). For all texts t, the generic algorithm access the same positions of t on the inputs (Γ, t) and (Γ? , t) [8]. Let us remark that the full memory expansion of the full memory expansion of a matching machine is equal to its full memory expansion (up to a state isomorphism). A w-matching machine Γ is standard if each state q of Γ appears in a unique pair/state of its full memory expansion, or, equivalently, if it is equal to its full memory expansion. For instance the abb-matching machine of Figure 1 is not standard. Since the matching machine of Figure 2 is a full memory expansion, it is standard. For all states q of a standard matching machine Γ, we put hΓ (q) for the second entry of the unique pair/state of Γ? in which q appears. We implemented a basic algorithm computing the full-memory expansion Γ? = (Q? , o? , F ? , α? , δ ? , γ ? ) of a w-matching machine Γ = (Q, o, F, α, δ, γ) in O(\|w\|.\|Q? \|) time. We have \|Q? \| ≤ (A + 1)\|w\| \|Q\| but the size of Q? may vary a lot with regard to the matching machine/algorithm considered. A w-matching machine Γ is compact if it contains no state q which always leads to the same state. Formally, Γ = (Q, o, F, α, δ, γ) is compact if there is no q ∈ Q such that one of the following assertions holds: 1. there exists a symbol x with δ(q̇, x) 6= y 6= x; and δ(q̇, y) = for all symbols 2. for all symbols x and y, we have both δ(q̇, x) = δ(q̇, y) and γ(q̇, x) = γ(q̇, y). Basically, a non-compact machine performs useless text accesses. In [8], it is shown that any w-matching machine can be turned into a compact (and faster) machine. 6 2.4 iid and Markov models An independent identically distributed (iid) model (aka Bernoulli model) is fully specified by a probability distribution π on the alphabet (i.e. π(x) is the probability of the symbol x in the model). Such a model will be simply referred to as “π” below. Under π, the probability of a text t is \|t\|−1 Y pπ (t) = π(ti ). i=0 A Markov model M over a given set of states Q is a 2-uple (πM , δM ), where πM is a probability distribution on Q (the initial distribution) and δM associates a pair of states (q, q 0 ) with the probability for q to be followed by q 0 (the transition probability). Under a Markov model M = (πM , δM ), the probability of a sequence s of states is \|s\|−1 pM (s) = πM (s0 ) Y δM (si , si+1 ). i=0 Theorem 1 ([8]). Let Γ = (Q, o, F, α, δ, γ) be a w-matching machine. If a text t follows an iid model and Γ is standard then the sequence of states parsed by the generic algorithm on the input (Γ, t) follows a Markov model (πM , δM ). Proof. Whatever the text model and the machine, the sequence of states always starts with o with probability 1. We have πM (o) = 1 and πM (q) = 0 for all q 6= o. The probability δM (q, q 0 ) that the state q 0 follows the state q during an execution of the generic algorithm, is equal to: • 1, if there exists a symbol x such that δ(q, x) = q 0 and (α(q), x) ∈ hΓ (q), i.e. if the relative position α(q) was already checked with x occurring at it, X • π(x), otherwise, x s.t. δ(q, x) = q 0 independently of the previous states. 2.5 Asymptotic speed Let M be a text model and A be an algorithm. The w-asymptotic speed of A under M is the limit expectation, under M, of the ratio of the text length to the number of text accesses performed by A [8]. Namely, by putting aA (t) for the number of text accesses performed by A to parse t and pM (t) for the probability of t with regard to M, the asymptotic speed of A under M is ASM (A) = lim n→∞ X t∈An 7 \|t\| pM (t). aA (t) The asymptotic speed ASM (Γ) of a w-matching machines Γ is that the generic algorithm with Γ as first input. From Theorem 5 of [8], the asymptotic speed of a standard w-matching machine Γ = (Q, o, F, α, δ, γ) under an iid model π exists and is given by X ASπ (Γ) = βq E(q), (1) q∈Q where (βq )q∈Q are the limit frequencies of the states of the Markov model associated to Γ and π in Theorem 1, and γ(q, x) if (α(q), x) ∈ hΓ (q)), P E(q) = γ(q, x)π otherwise. x x Computing the asymptotic speed of a pattern matching algorithm, with regard to a pattern w and an iid model π is performed by following the stages below. 1. We get a w-matching machine Γ which simulates the behavior of the algorithm while looking for w (Figure 1). The transformation of the 9 algorithms presented in Section 7 (and a few others, see our GitHub repository) into w-matching machines, given w, has been implemented. 2. We obtain the full-memory expansion Γ? of Γ (Figure 2, Section 2.3). 3. We compute the limit frequencies of the Markov model associated to Γ? and π in Theorem 1. This mainly needs to solve a system of linear equations of dimension \|Q? \|. 4. We finally obtain the asymptotic speed of the algorithm from these limit frequencies, π and Γ? by using Equation 1. The most time-consuming stage is the computation of the limit frequencies, which has O(\|Q? \|3 ) time complexity, where \|Q? \|, the number of states of the full memory expansion, is smaller than (A + 1)\|w\| \|Q\|. 3 Strategies For all sets I ⊂ N and k ∈ N, we define the k-left-shifted of I as ∆(I, k) = {i − k \| ∃i ∈ I and i ≥ k}. A w-strategy S = (Q, o, F, α, δ, γ) is a w-matching machine such that • Q ⊆ P({0, . . . , \|w\| − 1}) \ {{0, . . . , \|w\| − 1}} and ∅ ∈ Q, • o = ∅, • F = {s ∈ Q \| \|s\| = \|w\| − 1}, 8 b/3 {1} 0 b/0 ∅ 1 a/2 a/1 {0} 2 a/0 {0, 1} 2 a/2 a/1 b/0 {0, 2} 1 a/0 {0, 1} 2 b/3 b/3 {1} 0 b/0 ∅ 1 a/2 a/1 b/0 a/1 {0} 1 Figure 3: Two abb-strategies with the same conventions as in Figure 1. • α : Q → {0, 1, . . . , \|w\| − 1} is such that for all s ∈ Q, α(s) 6∈ s and \|s\| < \|w\| − 1 ⇒ s ∪ {α(s)} ∈ Q, • γ : Q × A → {0, 1, . . . , \|w\|} is such that for all states s and all symbols x, γ(s, x) min{k ≥ 1 \| wα(s)−k = x if α(s) ≥ k and wj = wj+k for all j ∈ ∆(s, k)} = min{k ≥ 0 \| wα(s)−k = x if α(s) ≥ k and wj = wj+k for all j ∈ ∆(s, k)} • δ : Q × A → Q is such that for all s ∈ Q and all symbols x, δ(s, x) = ∆(s ∪ {α(s)}, γ(s, x)). Figure 3 shows two abb-strategies which differ notably in the next-positionto-check of state {0}. Proposition 1. A w-strategy is a standard, compact, valid and non-redundant w-matching machine. Proof. By construction, a w-strategy is standard, compact and non-redundant. The validity of a w-strategy follows from Theorem 1 of [8]. Proposition 2. There is a w-strategy which achieves the greatest asymptotic speed among all the w-matching machines of order \|w\| − 1. 9 if s ∈ F, otherwise, Proof. The Corollary 2 of [8] implies that there exists a w-matching machine which achieves the greatest asymptotic speed among those of order \|w\| − 1 and which is 1. standard, 2. compact, 3. valid, 4. in which all the states are relevant (i.e. such that they may lead to a match without any positive shift [8]), 5. such that there is no pair of states (q, q 0 ) with q 6= q 0 and hΓπ (q) = hΓπ (q 0 ). Let us verify that a w-matching machine Γ = (Q, o, F, α, δ, γ) of order \|w\| − 1 satisfying the properties above is (isomorphic to) a w-strategy. Since it verifies in particular the properties 4 and 5, its set of states Q is in bijection with a subset of P({0, . . . , \|w\| − 1}). Let us identify all states q of Q with f (hΓ (q)), its corresponding element of P({0, . . . , \|w\| − 1}). Since Γ is standard, compact and of order \|w\| − 1, we do not have {0, . . . , \|w\| − 1} ∈ Q. Moreover, since Γ is standard, we have δ(s, x) = ∆(s ∪ {i}, γ(s, x)) for all s ∈ Q. Last, by construction, if γ(s, x) min{k ≥ 1 \| wα(s)−k = x if α(s) ≥ k and wj = wj+k for all j ∈ ∆(s, k)} > min{k ≥ 0 \| wα(s)−k = x if α(s) ≥ k and wj = wj+k for all j ∈ ∆(s, k)} if s ∈ F, otherwise, then Γ is not valid, and if γ(s, x) min{k ≥ 1 \| wα(s)−k = x if α(s) ≥ k and wj = wj+k for all j ∈ ∆(s, k)} < min{k ≥ 0 \| wα(s)−k = x if α(s) ≥ k and wj = wj+k for all j ∈ ∆(s, k)} if q ∈ F, otherwise, then δ(s, x) is not relevant. 4 Position lattices [w] [w] The position lattice of a pattern w is the 3-uple L[w] = (Q[w] , (δs )s∈Q[w] , (γs )s∈Q[w] ) where, by putting s for {0, . . . , \|w\| − 1} \ s, • Q[w] = P({0, . . . , \|w\| − 1}) \ {{0, . . . , \|w\| − 1}}, i.e. the set made of all the subsets of positions of w but {0, . . . , \|w\| − 1}, [w] is a map from s × A to {0, . . . , \|w\|}, [w] is a map from s × A to Q[w] , • for all s ∈ Q[w] , γs • for all s ∈ Q[w] , δs 10 0, b\|1 ∅ 0, a\|0 2, b\|3 1, b\|0 0, b\|2 2, b\|0 0, b\|3 2, a\|2 0, a\|3 1, a\|1 1, a\|1 {0} {1} 2, a\|2 {2} 0, b\|1 1, b\|3 2, a\|2 2, b\|0 0, a\|0 2, a\|2 1, b\|0 1, b\|0 1, a\|1 2, b\|0 0, a\|0 {0, 1} 1, a\|1 {0, 2} {1, 2} Figure 4: Position lattice of the pattern abb. Vertices represent the states of L[abb] . For all states s, there is an outgoing edge for all pairs (i, x) with i ∈ {0, . . . , \|abb\| − 1} \ s and x ∈ A. This outgoing edge is labeled with [abb] [abb] “i, x\|γs (i, x)”, is colored according to i, and goes to δs (i, x). where, for all s ∈ Q[w] , all i ∈ s and all x ∈ A, we have γs[w] (i, x) min{k ≥ 1 \| wα(s)−k = x if α(s) ≥ k and wj = wj+k for all j ∈ ∆(s, k)} = min{k ≥ 0 \| wα(s)−k = x if α(s) ≥ k and wj = wj+k for all j ∈ ∆(s, k)} and δs[w] (i, x) = ∆(s ∪ {i}, γs[w] (i, x)). [w] In particular, if x = wi and \|s\| < \|w\| − 1 then we have γs (i, x) = 0 and [w] δs (i, x) = s ∪ {i}. Let us remark that, since max(s) ≤ \|w\| − 1 for all s ∈ Q[w] , we have, for all [w] i ∈ s and all x ∈ A, ∆(s ∪ {i}, \|w\|) = ∅, thus γs (i, x) ≤ \|w\| which is consistent [w] with the definition of γs . [w] The edges of L[w] are the pairs (s, δs (i, x)) for all s ∈ Q[w] , all i ∈ s and all x ∈ A (see Figure 4). 11 if \|s\| = \|w\| − 1, otherwise, Remark 1. The position lattice of w contains 2\|w\| − 1 states and \|A\|.\|w\|.2\|w\|−1 edges. Remark 2. Let s be a state of Q[w] , i and j be two positions in s such that i 6= j and x and y be two symbols of A. We have γ [w] [w] δs (i,x) (j − γs[w] (i, x), y) + γs[w] (i, x) = γ δ [w] [w] δs (i,x) [w] [w] δs (j,y) (j − γs[w] (i, x), y) = δ (i − γs[w] (j, y), x) + γs[w] (j, y), and [w] [w] δs (j,y) (i − γs[w] (j, y), x). By considering the particular case where x = wi , we get [w] γs∪{i} (j, y) = γ [w] δs∪{i} (j, y) = δ [w] [w] δs (j,y) [w] [w] δs (j,y) (i − γs[w] (j, y), wi ) + γs[w] (j, y), and (i − γs[w] (j, y), wi ). Let precw be the table indexed on {0, . . . , \|w\| − 1} × A and in which, for all positions i of w and all symbols x of A, the entry precw [i, x] is defined as max{j ≤ i \| wj = x} if {j ≤ i \| wj = x} = 6 ∅, precw [i, x] = NULL otherwise. For instance, the table precabb is 0 1 2 a b 0 NULL 0 1 0 2 Lemma 1. Let s be a state of Q[w] , i a position in s and x a symbol of A. 1. If x = wi then [w] [w] • if \|s\| = \|w\|−1 then δs (i, x) = {0, . . . , B−1} and γs (i, x) = \|w\|−B, where B is the length of the longest proper suffix of w which is a prefix of w; [w] [w] • otherwise δs (i, x) = s ∪ {i} and γs (i, x) = 0. 2. If x 6= wi , (a) if s = ∅ then {precw [i, x]} [w] δ∅ (i, x) = ∅ [w] γ∅ (i, x) = i − precw [i, x] i+1 if precw [i, x] 6= NULL, otherwise, if precw [i, x] 6= NULL, otherwise, 12 (b) if s 6= ∅ then for all ` ∈ s, we have δs[w] (i, x) = δ [w] [w] [w] δs\{`} (i,x) γs[w] (i, x) = γ (` − γs\{`} (i, x), w` ), [w] [w] [w] δs\{`} (i,x) [w] (` − γs\{`} (i, x), w` ) + γs\{`} (i, x). Proof. The only case which does not immediately follow from the definition of L[w] , is when x 6= wi and s 6= ∅ which is given by Remark 2. The relation 4 on Q[w] is defined as follows. For all sets s and s0 in Q[w] , we have s 4 s0 if one of the following properties holds: • \|s\| < \|s0 \|, • \|s\| = \|s0 \|, s 6= s0 and min(s difference of s and s0 , s0 ) ∈ s, where s s0 is the symmetric • s = s0 . The relation 4 defines a total order on Q[w] . We write “ s ≺ s0 ” for “ s 4 s0 and s 6= s0 ”. Lemma 2. Let s be a state of Q[w] with \|s\| > 1, i a position in s and x a symbol [w] of A. If x 6= wi then δs\{max s} (i, x) ≺ s. Proof. Under the assumption that \|s\| > 1, we have min s = min(s \ {max s}). [w] By construction, the fact that x 6= wi implies that γs\{max s} (i, x) > 0. If we have [w] min((s \ {max s}) ∪ {i}) < γs\{max s} (i, x) [w] [w] then \|δs\{max s} (i, x)\| < \|s\|, thus δs\{max s} (i, x) ≺ s. [w] Otherwise, we have \|δs\{max s} (i, x)\| = \|s\| but since necessarily [w] [w] min δs\{max s} (i, x) ≤ min\|s\| − γs\{max s} (i, x) < min\|s\|, [w] we get again δs\{max s} (i, x) ≺ s. Theorem 2. Algorithm 2 computes the position lattice of the pattern w in O(\|w\|2\|w\| ) time by using the same amount of memory. Proof. Let us first show that Algorithm 2 determines the shifts and the transitions of the state s before those of the state s0 if and only if s ≺ s0 . The loop at Lines 2-8 computes the shifts and the transitions of ∅. Next, the loop at Lines 9-23 computes the shifts and the transitions of the singletons from {0} to {\|w\| − 1}. The last loop (Lines 24-41) determines the shifts and the transitions of the states corresponding to the subsets of increasing cardinals ` from 2 to 13 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 B ← length of the longest proper suffix of w which is also a prefix; for x ∈ A do last[x] ← NULL for i = 0 to \|w\| − 1 do last[wi ] ← i; for x ∈ A do if last[x] 6= NULL then [w] [w] γ∅ (i, x) ← i − last[x]; δ∅ (i, x) ← {last[x]}; else [w] [w] γ∅ (i, x) ← i + 1; δ∅ (i, x) ← ∅; for i = 0 to \|w\| − 1 do for j = 0 to i − 1 do for x ∈ A do [w] [w] [w] [w] γ{i} (j, x) ← γ∅ (j, x) + γ [w] (i − γ∅ (j, x), wi ); [w] δ{i} (j, x) ← δ δ∅ (j,x) [w] [w] [w] δ∅ (j,x) (i − γ∅ (j, x), wi ); for j = i + 1 to \|w\| − 1 do for x ∈ A do if x = wj then if \|w\| = 2 then [w] [w] γs (i, x) ← \|w\| − B; δs (i, x) ← {0, . . . , B − 1}; else [w] [w] γ{i} (j, x) ← 0; δ{i} (j, x) ← {i, j}; else [w] [w] [w] (j − γ∅ (i − 1, wi ), x); γ{i} (j, x) ← γ [w] [w] δ{i} (j, x) ← δ δ∅ (i−1,wi ) [w] [w] δ∅ (i−1,wi ) [w] (j − γ∅ (i − 1, wi ), x); for ` = 2 to \|w\| − 1 do for j = 0 to ` − 1 do S[j] ← j repeat s ← {S[0], . . . , S[` − 1]}; s0 ← {S[0], . . . , S[` − 2]}; for i ∈ {0, . . . , \|w\| − 1} \ s do for x ∈ A do if x = wi then if ` = \|w\| − 1 then [w] [w] γs (i, x) ← \|w\| − B; δs (i, x) ← {0, . . . , B − 1}; else [w] [w] γs (i, x) ← 0; δs (i, x) ← s ∪ {i}; else [w] [w] [w] [w] (S[` − 1] − γs0 (i, x), wS[`−1] ); γs (i, x) ← γs0 (i, x) + γ [w] δ 0 (i,x) s [w] δs (i, x) ← δ [w] [w] [w] δ 0 (i,x) s (S[` − 1] − γs0 (i, x), wS[`−1] ); j ← ` − 1; while j ≥ 0 and S[j] ≥ \|w\| − ` + j do j ← j − 1 if j ≥ 0 then S[j] ← S[j] + 1; for k = j + 1 to ` − 1 do S[k] ← S[k − 1] + 1 until j < 0; Algorithm 2: Computation of the position lattice. Value B is the last entry of the Partial Match table of the KMP algorithm. Its computation takes a time linear with \|w\| [15]. 14 \|w\| − 1. Inside the last loop, the way in which the next subset s0 is computed from the current subset s, both of cardinal `, ensures that s ≺ s0 (Lines 37-41). For all iterations i of the loop at Lines 2-8 and all symbols x, we have last[x] = precw [i, x] at the beginning of the inner loop (Line 4). From Lemma [w] [w] 1 (Cases 1 and 2a), the transitions δ∅ (i, x) and the shifts γ∅ (i, x) for all positions i of w and all symbols x, are correctly computed at the end of the loop. The loop at Lines 9-23 computes the shifts and the transitions from the singleton states. For all pairs of positions (i, j) and all symbols x, determining [w] [w] δ{i} (j, x) and γ{i} (j, x) is performed by distinguishing between two cases. [w] • If i > j, then δ∅ (j, x) ≺{i} and its shifts and transitions were already computed. Formula of Remark 2 gives us those of {i} (Lines 12-13). • If i < j, we distinguish between two subcases according to the symbol x considered. If x = wj then the shift and the transition state are given in [w] Lemma 1 - Case 1. Otherwise, we remark that, since γ{i} (j, x) is positive, [w] we have that γ{i} (j, x) = min{k ≥ 1 \| wj−k = x if j ≥ k and wi−k = wi }. [w] This implies that γ{i} (j, x) = γ [w] [w] [w] δ∅ (i−1,wi ) (j, x). We have δ∅ (i−1, wi ) ≺ s, [w] thus both the shifts and the transitions of the state δ∅ (i − 1, wi ) are computed before s (Lines 22-23). The last loop, lines 24-41, computes the shifts and the transitions of the states corresponding to the subsets of cardinals 2 to ` − 1. For all states s with 2 ≤ \|s\| ≤ \|w\| − 1, all positions i ∈ s and all symbols x 6= wi , the corresponding [w] [w] shift and transition γs (i, x) and δs (i, x) are computed from the shifts and [w] transitions of the state δs\{max s} (i, x) following Lemma 1 - Cases 2b (in Algo[w] rithm 2, we put s0 for s \ {max s}). Lemma 2 ensures that δs\{max s} (i, x) ≺ s, [w] thus that the shifts and transitions of δs\{max s} (i, x) are computed before those of s. For all states s with 2 ≤ \|s\| ≤ \|w\| − 1, all positions i ∈ s, the shift and [w] [w] transition γs (i, wi ) and δs (i, wi ) are given in Lemma 1 - Case 1. P\|w\|−1 The time complexity is O( k=0 (k + \|w\| − k) \|w\| k ) (loop Lines 24-41), i.e. O(\|w\|2\|w\| ). We do not use more memory than needed to store the lattice, which is, from Remark 1, O(\|w\|2\|w\| ). 5 The Fastest w-strategy Determining the fastest w-strategy, which, from Proposition 2, has the greatest asymptotic speed among all the w-matching machines of order \|w\| − 1, may be performed by computing the asymptotic speed of all the w-strategies and by returning the fastest one. In order to enumerate all the w-strategies, let us remark that they are all contained in the position lattice of w in the sense that: 15 • the set of states of a w-strategy is included in that of the position lattice; [w] • all the w-strategies Γ = (Q, o, F, α, δ, γ) verify δ(s, x) = δs (α(s), x) and [w] γ(s, x) = γs (α(s), x) for all s ∈ Q and all symbols x. Reciprocally, to any map φ from Q[w] to {0, . . . , \|w\|−1} such that φ(s) ∈ s for all states s ∈ Q[w] , there corresponds the unique w-strategy S = (Q, o, F, α, δ, γ) for which the next-position-to-check function α coincides with φ on Q. Finally, our brute force algorithm 1. takes as input a pattern w and an iid model π, 2. computes the position lattice of w, 3. enumerates all the maps φ such that φ(s) ∈ s for all states s ∈ Q[w] , 4. for each φ, gets the corresponding w-strategy by keeping only the states of Q[w] reachable from ∅, with the next-position-to-check function φ, 5. computes the asymptotic speed of all the w-strategies under π, 6. returns the w-strategy with the greatest speed. The time complexity of the brute force algorithm is    \|w\|−1 Y \|w\| O  (\|w\| − k)( k )  23\|w\|  , k=1 where the first factor stands for the number of functions φ and the second one for the computation of the asymptotic speed of a w-strategy, which needs to solve a linear system of size equal to the number of states, which is O(2\|w\| ). Its memory space complexity is \|w\|2\|w\|−1 , i.e. what is needed to store the position lattice of w. Under its current implementation, the brute force determination of the fastest w-strategy is unfeasible for patterns of length greater than 4. 6 A polynomial heuristic There are two points which make the complexity of the brute force algorithm given in Section 5 that high: 1. the size of the position lattice, which is exponential with the length of the pattern, 2. determining the fastest strategy in the position lattice, which needs a time exponential with its size. 16 Our heuristic is based on two independent stages, each one aiming to overcome one of these two points. Both of them start from the general idea that, since, for any current position of the text, the probability that no mismatch occurs until the nth text access decreases geometrically with n, the first relative positions accessed by a strategy (or more generally by a pattern algorithm) are those which have the greatest influence on its asymptotic speed. 6.1 n-sets sublattices A sufficient condition for a sublattice U ⊆ Q[w] to contain a w-strategy is that, [w] for all s ∈ U, there exists at least a position i ∈ s with δs (i, x) ∈ U for all x ∈ A. A sublattice U verifying this condition will be said to be complete. Figure 5 displays four complete sublattices extracted from the position lattice of abb (Figure 4). Let us introduce some additional notations here. For all sets S of positions, the prefix of S is defined as P (S) = max{i ∈ S \| j ∈ S for all 0 ≤ j ≤ i} and its rest is R(S) = S \ {0, . . . , P (S)} For all positive integers n, the n-sets sublattice of w is the sublattice U of Q[w] which contains all and only the subsets of Q[w] with a rest containing less than n positions, i.e. the subsets of the form {0, . . . , p} ∪ X with p < \|w\| − 1 and \|X \| ≤ n. By construction, the n-sets sublattice of w is complete. It contains O(\|w\|n ) states and O(\|w\|n+1 ) transitions. We adapted Algorithm 2 to compute the n-sets sublattice of w in O(\|w\|n+1 ) time with the same amount of memory space. 6.2 `-shift expectation We are now interested in a fast way for finding an efficient w-strategy in a given complete sublattice. For all integers ` and all states s of a sublattice U, the `-shift expectation of s is defined as the greatest shift expectation one could possibly get in ` steps in U by starting from s, conditioned on starting from s, while parsing a text following an iid model π. Namely, the `-shift expectation is computed following the recursive formula: [w] • ES0 [s] = 0, • for all ` > 0, [w] ES` [s] = max i∈Tr(s) X [w] π(x) γs[w] (i, x) + ES`−1 [δs[w] (i, x)] x∈A [w] where Tr(s) = {i ∈ s \| δs (i, x) ∈ U for all x ∈ A}. The `-shift expectation of a complete sublattice U is well defined and can be computed in O(`T ) time, where T is the number of transitions of the sublattice U and by using O(\|U\|) memory space. 17 0, b\|1 ∅ ∅ 2, b\|0 2, b\|3 2, a\|2 0, a\|0 2, b\|3 0, a\|3 2, a\|2 1, a\|1 {0} {2} {0} 0, b\|3 2, b\|0 2, a\|2 2, a\|2 1, b\|0 1, b\|0 1, a\|1 {0, 1} {0, 2} 1, a\|1 1, b\|3 {1, 2} {0, 1} a b ∅ ∅ 2, b\|3 {0} 2, a\|2 2, b\|3 1, b\|0 0, b\|2 2, a\|2 1, b\|0 0, b\|2 1, a\|1 1, a\|1 {1} 1, b\|3 {0} 2, b\|0 1, a\|1 {1} 2, a\|2 1, b\|0 0, a\|0 0, a\|0 {0, 1} 1, a\|1 {0, 2} {0, 1} c d Figure 5: Four complete sublattices extracted from the position lattice of abb. Sublattice a (resp. b) leads to the strategy where the next-position-to-check is always the smallest (resp. the greatest) relative position unchecked. Sublattice c (resp. d) leads to the strategy at the top (resp. at the bottom) of Figure 3. 18 We finally extract a w-strategy from U by setting the next-position-to-check of all states s ∈ U to X [w] arg max γs[w] (i, x) + ES`−1 [δs[w] (i, x)] . i ∈Tr(s) x ∈A 6.3 K-Heuristic The K-Heuristic combines the two approaches above in order to compute a w-strategy in a time polynomial with the length of the pattern. Being given an order K ≥ 1, we start by computing the K-sets sublattice of w, thus in O(\|w\|K+1 ) time. In order to select a w-strategy from the K-sets sublattice, we next compute the (K + 1)-shift expectation of all its states and extract a w-strategy as described just above. This computation is performed in O(K\|w\|K+1 ) time, since the number of transition of the sublattice is O(\|w\|K+1 ), by using O(\|w\|K ) memory space. Let us remark that the order ` of the `-shift expectation does not have, a priori, to be strongly related to the order K of the K-sets sublattice on which it is computed. By experimenting various situations, we observed that considering an order greater than K + 1 generally does not improve much the performances, whereas the strategies obtained from `-expectations with ` smaller than K may be significantly slower. The K-Heuristic returns a w-strategy in O(K\|w\|K+1 ) time by using O(\|w\|K ) memory space. We insist on the fact that the K-Heuristic generally does not return the fastest strategy, even if K > \|w\|. However, we will see in the next section that it performs quite well in practice. 7 Evaluation We shall compare the approaches introduced in Sections 5 and 6 with selected pattern matching algorithms. The comparison is performed, first, from a theoretical point of view, by computing their asymptotic speeds under iid models, and second, in practical situations, by measuring their average speed over real data. The average speed with regard to a pattern w, of an algorithm or a matching machine on a text t is the ratio of \|t\| to the number of text accesses performed by the algorithm to search w in t. We are also interested in to what extent taking into account the frequencies of the letters of an iid model or a text, for determining the Fastest and the KHeuristic strategies, actually improves their asymptotic or their average speeds. To this purpose, we compute the Fastest and the K-Heuristic strategies from the uniform iid model. Next, we test their efficiency in terms of asymptotic speed under a non-uniform iid model and in terms of average speeds on data with non-uniform frequencies of letters. 19 7.1 Pre-existing pattern matching algorithms More than forty years of research have already led to the development of dozens algorithms. We selected the 9 ones below for our evaluation: 1. Naive [6], 2. Morris-Pratt [6], 3. Knuth-Morris-Pratt [15], 4. Quicksearch [23], 5. Boyer-Moore-Horspool [13], 6. TVSBS [24], a“right-to-left” algorithm in which shifts are given by a badcharacter rule [5, 23] taking into account the two letters at distances \|w\|−1 and \|w\| from the current position of the text, 7. EBOM [9], a version of the Backward Oracle Matching algorithm [1] which also uses a “bad two-characters” rule, 8. HASHq [26], which implements the Boyer-Moore algorithm on blocks of length q by using efficient hashing techniques [14] (our tests are performed with q = 3), 9. FJS [11], which combines the ideas of Knuth-Morris-Pratt [15] and Sunday [23] algorithms. Algorithms 1 to 5 are classics. The last four ones were chosen for being known to be efficient on short patterns and small alphabets [10], a situation in which the determination of the fastest strategy is feasible. Let us remark that the order of the w-matching machine associated to TVSBS is equal to \|w\|, thus greater than that of the Fastest strategy that we compute. The transformation into matching machines was implemented for a few other pattern matching approaches, for instance the SA algorithm (the Baeza-YatesGonnet algorithm) based on bitwise operations [2], or the string-matching automaton [7]. Since the asymptotic and average speeds of these two algorithms are exactly 1, whatever the pattern, the model and the text, there is no point in displaying them. 7.2 Results We shall evaluate: • the pre-existing pattern matching algorithms presented in Section 7.1, • the 1- 2- and 3-Heuristics and • the Fastest strategy (each time it is possible). 20 ra tt st Fa st e ris tic eu ris tic 3H eu ris tic 2H 0.78 0.69 0.59 0.71 0.66 0.56 0.56 0.76 0.76 0.56 0.56 0.66 0.71 0.59 0.69 0.78 0.72 0.39 0.54 0.50 0.61 0.50 0.39 0.62 0.62 0.39 0.50 0.61 0.50 0.54 0.39 0.72 0.93 0.73 0.62 0.69 0.62 0.73 0.67 0.73 0.73 0.67 0.73 0.62 0.69 0.62 0.73 0.93 0.52 0.52 0.53 0.53 0.54 0.54 0.53 0.53 0.53 0.53 0.54 0.54 0.53 0.53 0.52 0.52 1.50 1.37 1.19 1.30 1.23 1.22 1.27 1.47 1.47 1.27 1.22 1.23 1.30 1.19 1.37 1.50 1.69 1.52 1.33 1.43 1.34 1.33 1.31 1.59 1.59 1.31 1.33 1.34 1.43 1.33 1.52 1.69 1.80 1.60 1.35 1.54 1.38 1.36 1.34 1.64 1.64 1.34 1.36 1.38 1.54 1.35 1.60 1.80 1.83 1.60 1.37 1.56 1.38 1.43 1.34 1.69 1.69 1.34 1.43 1.38 1.56 1.37 1.60 1.83 V eu 1H as hq H 1.18 1.18 0.73 0.73 0.73 0.73 0.94 0.94 0.94 0.94 0.73 0.73 0.73 0.73 1.18 1.18 O EB M S T SB FJ 0.98 0.51 0.51 0.69 0.63 0.50 0.55 0.75 0.75 0.55 0.50 0.63 0.69 0.51 0.51 0.98 S H or sp o ol -P rc h ris 1.00 0.94 0.89 0.84 0.80 0.80 0.73 0.70 0.70 0.73 0.80 0.80 0.84 0.89 0.94 1.00 ui Q ck se a t hM or ra t 0.70 0.76 0.76 0.76 0.73 0.70 0.70 0.70 0.70 0.70 0.70 0.73 0.76 0.76 0.76 0.70 nu t K ris -P M or 0.53 0.53 0.53 0.53 0.53 0.53 0.53 0.53 0.53 0.53 0.53 0.53 0.53 0.53 0.53 0.53 ai ve N aaaa aaab aaba aabb abaa abab abba abbb baaa baab baba babb bbaa bbab bbba bbbb Table 1: Asymptotic speeds for the patterns of length 4 on {a, b} under the uniform model. 7.2.1 Asymptotic speed The asymptotic speeds are computed for texts and patterns on the binary alphabet {a, b}. Table 1 displays the asymptotic speeds for all the patterns of length 4 on iid texts drawn from the uniform distribution. As expected, the strategy computed with the brute force algorithm (last column) is actually the fastest, but the speeds of the 1-,2- and 3-Heuristics are very close. The pre-existing algorithms are outperformed by all our approaches (even by the 1-Heuristic) for all the patterns. We observe that the Naive, Morris-Pratt and Knuth-Morris-Pratt algorithms have asymptotic speeds always smaller than 1. One cannot expect them to be faster since, by construction, they access all the positions of a text at least once. In the following, we will not display their speeds, nor that of Quicksearch, for they are always smaller than at least one of the other preexisting algorithms. The full tables can easily be re-computed by using our software. Table 2 displays the asymptotic speeds with regard to the same patterns as Table 1, but under the iid model (πa , πb ) = (0.1, 0.9). This table shows the asymptotic speeds of the K-Heuristics and the Fastest strategies computed with regard to an uniform iid model (the columns starting with “Unif.”). The strategies such obtained are not optimized according to the letter probabilities of the model. They may be used as general purpose approaches, while the strategies obtained from the model probabilities will be called adapted below. Overall, 21 1 1H 2H eu ris tic 3H eu ris tic Fa st es t U eu r ist ic st ic ris t U ni f. ni f. Fa st e ist ic 3H eu ic eu r ris t 2H 1H eu 0.67 0.66 0.66 0.63 0.66 0.63 0.61 0.46 0.66 0.65 0.63 0.54 0.63 0.54 0.36 0.19 f. 1.49 1.37 1.24 0.70 1.24 1.17 0.63 0.31 1.37 1.18 1.17 0.55 0.70 0.55 0.31 0.27 ni 1.68 0.27 1.55 0.29 1.63 0.28 0.82 0.31 1.33 0.22 1.16 0.21 1.06 0.23 0.73 0.24 U 1.97 0.53 0.87 0.53 1.24 0.54 0.87 0.57 1.49 0.37 0.80 0.39 1.10 0.31 0.90 0.27 f. 3.30 1.77 0.91 0.38 1.67 0.85 0.93 0.33 2.50 1.34 0.91 0.38 1.67 0.85 1.00 0.35 U ni EB O M H as hq T V SB S ol FJ S po or s H aaaa aaab aaba aabb abaa abab abba abbb baaa baab baba babb bbaa bbab bbba bbbb 3.02 2.43 1.77 1.34 1.80 1.29 1.06 1.08 2.44 1.75 1.09 1.03 1.09 1.00 1.02 1.03 3.46 2.55 2.14 1.71 2.14 1.78 1.31 1.12 2.60 1.75 1.35 0.91 1.72 0.83 1.09 1.03 3.50 2.55 2.14 1.77 2.17 1.68 1.51 1.14 2.61 1.75 1.44 1.04 1.84 1.06 1.17 1.05 3.50 2.55 2.14 1.77 2.16 1.64 1.54 1.15 2.61 1.75 1.74 1.05 1.83 1.08 1.17 1.05 3.02 2.43 1.77 1.74 1.80 1.42 1.30 1.08 2.44 1.75 1.09 1.04 1.09 1.01 1.08 1.03 3.47 2.60 2.19 1.79 2.15 1.80 1.73 1.10 2.60 1.75 1.78 1.04 1.72 1.08 1.16 1.03 3.50 2.60 2.19 1.80 2.18 1.80 1.80 1.14 2.61 1.75 1.84 1.04 1.84 1.08 1.24 1.05 3.50 2.61 2.19 1.80 2.18 1.81 1.80 1.15 2.61 1.75 1.84 1.05 1.84 1.08 1.24 1.05 Table 2: Asymptotic speeds for the patterns of length 4 on {a, b} under the iid model (πa , πb ) = (0.1, 0.9). our methods are faster than the pre-existing algorithms, with a few exceptions: Horspool is faster than the 1-Heuristic for two patterns ending with the rare letter a: aaaa and baaa. And EBOM is faster than the 1-Heuristic for searching baba. The K-Heuristics and the Fastest strategies computed with regard to an uniform iid model have asymptotic speeds smaller than their counterparts obtained from the actual probabilities of the text model (here highly unbalanced). Nevertheless, the uniform approaches still perform quite well, notably better than the pre-existing algorithms, except for the uniform 1-Heuristic and the same patterns as above. Considering longer patterns leads to similar observations. Table 3 shows the asymptotic speeds obtained for random patterns of length 10. The 3-Heuristic outperforms all the others approaches (the Fastest strategy cannot be computed for this length). The (uniform) 1-Heuristic is slower than algorithms such EBOM or Hashq. But both the uniform 2- and 3-Heuristic overall perform better than the pre-existing algorithms, though they are slightly slower for a few patterns. 7.3 Average speed Our data benchmark consists in the Wigglesworthia glossinidia genome, known for its bias in nucleotide composition (78% of {a, t}), and the Bible in English from [10]. Table 4 displays the average speeds of patterns randomly picked from the data. Let us remark that we are now dealing with real texts, which are not iid. In particular, the Fastest strategy could possibly be outperformed (this is 22 1 ris tic 3H eu 1H r is eu tic r is t i 2c H eu ris tic 3H eu ris tic H U U U ni f. ni f. 2- H eu ris tic eu 1H EB O as hq M S SB V T S FJ ni f. l oo or sp H babbbaabab ababbbbbab aaabaaaaba bbbabbabab bbabaabbab baabbaaaaa abbbababbb baabbbabba baabbaabab bbbbababbb 0.85 0.70 0.91 0.84 0.80 4.10 0.31 0.90 0.85 0.31 0.42 0.55 0.87 0.27 0.31 2.17 0.58 0.81 0.37 0.26 0.22 0.29 3.01 0.24 0.22 1.86 0.32 0.68 0.22 0.20 1.42 0.77 3.93 1.45 2.21 2.40 1.32 1.39 2.22 0.95 1.57 0.85 2.61 1.77 1.92 2.45 1.20 1.14 2.29 0.84 1.09 1.19 3.04 1.02 1.15 1.99 1.08 1.61 1.77 1.03 1.98 1.34 4.78 1.05 1.41 4.03 1.33 1.12 2.21 1.05 1.54 1.47 4.92 1.72 1.69 4.72 1.52 1.63 2.26 1.10 1.22 1.43 3.04 1.02 1.15 1.99 1.35 1.73 1.77 1.03 2.38 1.87 4.79 1.29 2.14 4.06 1.90 2.44 3.15 1.20 2.71 2.34 5.27 2.30 3.02 4.78 2.29 2.78 3.54 1.50 ic Fa st es 1t H eu ris tic 2H eu ris tic 3H eu ris tic Fa st es t U ni ni f. U f. 3H eu r ist ist ic eu r H 2- 1H U ni f. U ni f. hq as H O M EB SB S 1.17 1.69 1.58 2.82 1.53 2.47 1.36 1.67 1.92 1.07 3.16 3.20 3.53 2.84 3.04 3.13 3.15 2.72 2.73 3.35 0.77 1.11 0.85 1.70 0.77 1.56 0.70 1.12 0.82 0.87 2.02 1.91 2.18 1.78 1.59 1.76 2.02 1.81 1.80 1.97 0.74 1.28 0.66 1.50 0.71 1.45 0.76 1.27 0.93 0.89 1.85 1.78 1.87 1.70 1.48 1.60 1.85 1.69 1.69 1.73 1.04 1.10 0.92 1.43 1.05 1.23 1.26 1.09 1.34 1.06 1.39 1.42 1.48 1.42 1.45 1.45 1.46 1.35 1.36 1.49 0.60 0.63 0.58 0.63 0.63 0.63 0.65 0.63 0.65 0.63 0.66 0.66 0.66 0.66 0.66 0.67 0.66 0.65 0.66 0.66 1.46 1.64 1.57 2.72 1.74 2.12 1.67 1.64 1.97 1.75 2.88 3.01 3.50 2.89 2.89 3.01 2.96 2.94 2.94 3.23 1.70 2.16 1.76 3.03 1.87 2.67 1.77 2.15 2.11 2.04 3.23 3.28 3.60 2.99 3.05 3.16 3.22 3.03 3.04 3.53 1.74 2.16 1.84 3.08 1.83 2.77 1.81 2.14 2.09 1.87 3.23 3.25 3.60 3.03 3.06 3.16 3.22 2.87 2.87 3.53 1.80 2.16 1.84 3.09 1.87 2.77 1.85 2.14 2.11 2.04 3.24 3.28 3.62 3.06 3.01 3.11 3.22 3.05 3.06 3.53 V T S FJ atat tatg aaat tccc caat aacc acta tatc gtga gatt he m , to usal le t hem are at d f th r th fede H or sp o ol eu r ist ic Table 3: Asymptotic speeds for some patterns of length 10 on {a, b} (drawn from the uniform distribution) under the iid model (πa , πb ) = (0.1, 0.9). 1.46 1.64 1.57 2.72 1.74 2.12 1.67 1.64 1.97 1.75 2.88 3.01 3.50 2.89 2.89 3.01 2.96 2.94 2.94 3.23 1.73 2.16 1.80 3.03 1.94 2.69 1.85 2.15 2.17 2.04 3.24 3.29 3.62 3.07 3.05 3.16 3.22 3.10 3.12 3.53 1.73 2.16 1.90 3.08 1.97 2.77 1.88 2.14 2.18 2.04 3.24 3.29 3.62 3.07 3.06 3.16 3.22 3.10 3.12 3.53 1.81 2.17 1.90 3.09 1.97 2.77 1.88 2.15 2.19 2.05 3.24 3.29 3.62 3.07 3.06 3.16 3.22 3.10 3.12 3.53 Table 4: Average speeds for some patterns of length 4 picked from the benchmark data (the Wigglesworthia glossinidia complete genome and the Bible in English). 23 1 FJ S T EB H as hq U ni f. 1H eu U ni ris f. tic 2H eu U ni ris f. tic 3H eu 1ris H eu tic ris tic 2H eu ris tic 3H eu ris tic 1.8 2.8 3.0 1.4 1.4 2.7 3.0 2.7 1.9 2.1 11.9 13.3 11.9 12.4 14.8 10.1 16.2 12.8 12.4 12.1 1.0 1.8 1.7 1.0 1.0 1.6 1.7 1.1 1.1 1.3 5.9 7.2 6.0 6.1 7.5 5.8 8.4 6.8 5.5 6.3 2.2 3.3 2.0 2.1 1.7 3.7 2.8 2.3 1.9 2.9 8.0 8.4 8.9 8.1 9.0 7.6 9.6 8.6 8.0 9.4 6.1 7.2 6.2 7.0 6.1 8.3 6.3 7.4 6.4 6.8 11.7 12.6 13.2 11.7 12.0 11.7 12.8 11.9 11.7 12.8 4.7 5.2 4.4 4.4 5.0 5.4 4.7 5.5 4.9 4.7 8.6 8.7 8.8 8.6 8.5 8.6 8.8 8.5 8.8 8.6 tccttatgtaaaatataaatgtagcaattt aaaagaaccccggcgaggggagtgaaatag aattttcaactaatattaaaccacgttctg aaaggtccattaagtattactatcacagca agatttgcgtgatttaaaataatcatctaa ataggaaaagattggattaaactagatatg at the mount called the mount ith Israel, to wit, with all t esus going up to Jerusalem too them, as they were able to hea o in Osee, I will call them my things are come upon thee, the Syria, that dwelt at Damascus, full of darkness. If therefor e it: for there is no other sa g, Syria is confederate with E S O M V SB sp oo l H or tggataaaaatttgttattaccatatctat cttctttaattatgttttctatttcttttt gttctatttgttggagatttaaaataatta tcctactttaacctctaaatgtcccttatt 2.3 2.5 2.6 2.5 2.2 3.4 2.0 2.3 2.4 2.2 8.0 8.9 9.5 7.8 9.0 8.5 10.4 9.4 7.7 9.8 4.6 5.0 5.3 5.5 4.7 6.9 4.8 5.1 4.9 5.1 17.2 17.1 18.0 16.5 18.5 16.3 19.3 18.5 16.4 18.6 7.5 7.9 8.0 8.4 7.5 11.3 8.1 8.1 7.5 8.3 18.0 18.0 18.5 17.9 19.3 17.7 20.0 18.9 17.6 18.9 2.3 2.6 2.7 2.6 2.3 3.5 2.2 2.5 2.5 2.3 8.0 9.1 9.4 7.8 8.9 8.5 10.4 9.4 7.7 9.8 5.5 5.3 6.1 6.5 5.5 9.3 5.6 6.6 5.8 5.9 17.6 18.0 18.1 16.6 18.8 16.8 19.6 18.7 16.9 18.6 8.9 8.5 9.1 9.9 8.5 12.5 9.7 10.2 9.1 9.7 18.4 18.8 18.7 18.0 19.3 18.1 20.1 19.1 17.9 18.9 Table 5: Average speeds for some patterns of length 30 picked from the benchmark data (the Wigglesworthia glossinidia complete genome and the Bible in English). not observed on the benchmark data). The 2- and 3-Heuristics, uniform and adapted, are faster than the pre-existing algorithms for all the patterns, whereas the 1-Heuristic is sometimes slightly outperformed by Horspool. Horspool is almost as fast as our approaches on the Bible while being sometimes significantly outperformed on the Wigglesworthia glossinidia genome. The average speeds are overall greater on the Bible than on the DNA sequence. In both cases, we do not observe a wide performance gap between the uniform and the adapted approaches, though our benchmark data are far from following an uniform iid model. Let us remark that the 2- and 3-Heuristics have almost the same performances both in the uniform and the adapted cases. Table 5 shows the averages speeds with regard to patterns of length 30. The average speeds on the Bible are about twice those on the Wigglesworthia glossinidia genome. One actually expects the speed to be greater in average on texts with large alphabets, since the less likely the match between two symbols, the greater the shift expectation per iteration. Again the 3-Heuristic, uniform or adapted, outperforms the pre-existing algorithms. The speeds of the 3-Heuristic and of the 2-Heuristic differ in a greater amount than with patterns of length 4 for the Wigglesworthia glossinidia genome, and, to a smaller extent, for the Bible. 24 1 8 Discussion In practical situations and though they do not take into account the letter frequencies, the uniform K-Heuristics and the uniform Fastest strategy perform generally almost as well as their adapted counterparts. The greatest difference observed is for the patterns of length 30 on the Wigglesworthia glossinidia genome (Table 5) and is relatively small. We do observe a notable amount of difference for the quite extreme case of the asymptotic speed under the iid model (πa , πb ) = (0.1, 0.9). But even for these frequencies, the uniform approaches show greater asymptotic speeds than any of the selected pre-existing algorithms. The 3-Heuristic has very good results whatever the pattern or the text. There is no situation for which the performances of the 2-heuristic are far from the best. On the contrary, the performance ranking of the pre-existing algorithms depends heavily on the patterns and on the texts or the model. For instance, Horspool may perform very well, even almost optimally, for some patterns and texts or models while its speed may completely plummet in other situations. The question of selecting the most efficient order of K-Heuristic still deserves further investigations. A basic answer could be “the greater, the better” but we should take into consideration that an higher order of heuristic comes with an increased computational cost. After some experiments, we observed that the asymptotic speed the K-Heuristic tends to stop improving beyond a certain rank. For instance, the difference in average speed between the 2- and 3-Heuristics for patterns of length 4, both on the genome and on the Bible, probably does not justify the computational cost of the 3-Heuristic, while it is worth to use the 3-Heuristic rather than the 2-Heuristic for searching patterns of length 30 in the Bible (not that much for the Wigglesworthia glossinidia genome). The best trade-off for the order of the K-Heuristic depends on the pattern (notably its length) and on the text features (in particular the alphabet size and the letter frequencies). It is certainly possible to obtain efficient heuristic with a lower computational cost than for the K-Heuristic. Since in standard situation, the length of the text is much greater than that of the pattern, there is no real reason for considering only pattern matching algorithms with linear pre-processings of the pattern. In the extreme case where the texts are arbitrarily long with regard to the patterns, any pre-processing, i.e. whatever its computation time, would be beneficial as soon as it improves the overall speed. Authors’ contributions Gilles Didier provided the initial idea, led the software development and wrote all the manuscript but the section Evaluation. Laurent Tichit collaborated on the software development, ran the tests and wrote the section Evaluation. Both authors read, edited and approved the final manuscript. 25 References [1] C. Allauzen, M. Crochemore, and M. Raffinot. Efficient experimental string matching by weak factor recognition. In Combinatorial Pattern Matching, pages 51–72. Springer, 2001. [2] R. Baeza-Yates and G. H. Gonnet. A new approach to text searching. Communications of the ACM, 35(10):74–82, 1992. [3] R. A. Baeza-Yates and M. Régnier. Average running time of the BoyerMoore-Horspool algorithm. Theoretical Computer Science, 92(1):19 – 31, 1992. [4] G. Barth. 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Intrinsic Point of Interest Discovery from Trajectory Data Matthew Piekenbrock Derek Doran Dept. of Computer Science & Engineering Kno.e.sis Research Center Wright State University, Dayton, OH, USA [email protected] Dept. of Computer Science & Engineering Kno.e.sis Research Center Wright State University, Dayton, OH, USA [email protected] arXiv:1712.05247v1 [cs.AI] 14 Dec 2017 Abstract This paper presents a framework for intrinsic point of interest discovery from trajectory databases. Intrinsic points of interest are regions of a geospatial area innately defined by the spatial and temporal aspects of trajectory data, and can be of varying size, shape, and resolution. Any trajectory database exhibits such points of interest, and hence are intrinsic, as compared to most other point of interest definitions which are said to be extrinsic, as they require trajectory metadata, external knowledge about the region the trajectories are observed, or other application-specific information. Spatial and temporal aspects are qualities of any trajectory database, making the framework applicable to data from any domain and of any resolution. The framework is developed under recent developments on the consistency of nonparametric hierarchical density estimators and enables the possibility of formal statistical inference and evaluation over such intrinsic points of interest. Comparisons of the POIs uncovered by the framework in synthetic truth data to thousands of parameter settings for common POI discovery methods show a marked improvement in fidelity without the need to tune any parameters by hand. ACM Reference format: Matthew Piekenbrock and Derek Doran. 2016. Intrinsic Point of Interest Discovery from Trajectory Data. In Proceedings of ACM Conference, Washington, DC, USA, July 2017 (Conference’17), 10 pages. DOI: 10.1145/nnnnnnn.nnnnnnn 1 Introduction The development and deployment of location acquisition systems have enabled large scale capturing of ‘movement’ or ‘trajectory’ data from people, cars, and other objects. Technologies like global positioning systems (GPS), global system for mobile communications (GSM), wide area motion imagery (WAMI), and radio-frequency identification (RFID) allow organizations and governments to collect and exploit trajectory patterns in many scenarios. More recent initiatives (e.g. Uber’s Movement1 and IBM’s Smarter Cities2 programs) have even made such data available to the either the public or city planning experts at large. With the rise in importance of this 1 https://movement.uber.com/cities 2 https://www.ibm.com/smarterplanet/us/en/smarter cities/overview/ Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]. Conference’17, Washington, DC, USA © 2016 ACM. 978-x-xxxx-xxxx-x/YY/MM. . . $15.00 DOI: 10.1145/nnnnnnn.nnnnnnn data comes prevalent use of Geographic Information Systems (GIS) and related platforms such as ArcGIS3 and Mapbox4 . Other related use cases of GIS information have also emerged for surveillance [9] and location-based service (LBS) applications [36]. In many of these applications, trajectory data is exploited for knowledge acquisition tasks [17], the integration of movement patterns to uncover “patterns of life” over a region [43], to expand situational awareness in crises [40], and to support the value added by a LBS application [13]. In many of these knowledge acquisition tasks, the notion of a “location” or “point of interest” (POI) is foundational to understanding the entirety of the common space in which the data are observed [31]. For example, mapping systems must know the position and geometry of locations for navigation and automated guidance control purposes. In LBS applications, the POIs and metadata such as their popularity (e.g. ‘star-rating’) are necessary to provide useful location recommendations [13, 30, 44]. Because POIs are not available from ‘raw’ trajectory data captured by location acquisition systems, they are often extrinsically defined by gazetteers such as Google Places, FourSquare, GeoNames, or OpenStreetMap. Yet external sources of location data present many difficulties when faced with the problem of understanding how a given trajectory dataset relates to the underlying geographical area where it was observed. For example, many gazetteers store varying types of either POI metadata or POI relational data, allowing gazetteer-derived information to present a source of bias. Furthermore, relying on gazetteers explicitly defines the set of POIs that exist in a given geographical region. When there is disagreement on this definition, analysis becomes difficult. Furthermore, with POIs defined a priori, one is faced with the problem of “fitting” observed trajectory data to models defined by such POIs, many of which may or may not be relevant to the given data at hand. For example, it may be desirable for a city-planner gathering movement (trajectory) data following a public event to discover ‘bottleneck’ congestion areas like parking lots, roads, or sidewalk segments for the purpose of traffic analysis. In this situation, it would more useful to discover POIs directly from the data itself during the event, but such geographical POIs may not be available in a gazetteer. To over come these challenges, this paper investigates the POI discovery problem in the most generic context possible. We ask: given only trajectory data, without access to gazetteers, can we infer subregions within a geospace that are “interesting” enough to call it a POI? We seek intrinsic POIs, which are POIs recoverable without the use of a gazetteer, are completely defined by observed movement patterns, and can be used for any domain-specific application and at any scale (e.g. from movements within a building to movements across an entire city region). To make such a definition meaningful, 3 https://www.arcgis.com/ 4 https://www.mapbox.com/ we build off recent theoretical work in density-based clustering and introduce a data-driven, statistically rigorous definition of a POI applicable to trajectory data of any (and even mixed) resolution. The definition follows from a recent minimax analysis of the consistency of hierarchical density superlevel set estimators. We use this definition to present a parameter-free framework for extracting intrinsic POIs, i.e. yields an optimal unsupervised solution without ad hoc parameter-tuning.5 A comparative analysis is performed on realistic simulations involving both vehicle and pedestrian traffic. Validation results show marked improvements in fidelity against several state of the art (SOTA) algorithms. Of interest to the authors, the simulation settings, the resulting traffic data, the validation code, and the framework itself is all completely reproducible and open source, available online.6 2 Point of Interest Discovery This section provides preliminary information about the POI discovery problem, and provides context and definitions for this work. It then formally defines a POI and (subsequently) an intrinsic POI, and the framework their discovery. 2.1 Preliminaries We consider a trajectory database of discrete, time-indexed spatial data having at least the 3-tuple of attributes (<object id>, <spatial component>, <temporal component>) This minimal amount of information implies a trajectory for an object of the form: ∆t 1 ∆t 2 ∆t n−1 T = p1 −−−→ p2 −−−→ . . . −−−−−→ pn (1) where p1 , p2 , ..., pn are chronologically ordered spatial coordinates. In the geographical sense, these spatial components are often defined by a <latitude> and <longitude> pair, but in practice could be from any coordinate system. Such representations require trajectory pattern mining techniques [42], or techniques that seek to mine common spatiotemporal patterns across trajectories to assert significance over areas where trajectory patterns emerge. Mined patterns in trajectories are often referred to as mobility patterns, characterizing some specific trajectory quality of interest, such as heading, stopping rate, velocity, rotation, curvature, or shape [5]. Such mobility patterns exhibit properties that make the formal retrieval of significant areas challenging. For example, if the timespan of an observed trajectory is long, the processes driving the mobility pattern may be non-stationary (e.g. road traffic that changes due to construction, or congestion effects due to time of day shifts in the work schedule). There may also be paths of objects that are transient (some areas are never traveled to more than once). Furthermore, the spatial components in trajectory data often have a high degree of autocorrelation, breaking assumptions of independence [13]. A variety of models have been proposed to handle thee situations, largely focusing on estimating individual trajectory statistics under these assumptions. This includes, for examples, adaptive Kalman 5 We see this as a necessary form of usability, an important feature to have in the modern clustering era. It is well-known that having several sensitive, real-valued parameters results in combinatorial explosion of the parameter space of an algorithm, resulting in the need for the user to use one or more parameter-tuning methods to arrive at a solution that befits the application. 6 ¡Anonymized for review purposes.¿ filters for vehicle navigation [20], state-space models [16], and trajectory path uncertainty models [34]. The knowledge mined from individual trajectories says little of the macroscopic patterns driving such trajectory observations. Rather than focusing on the statistics of individual trajectories, collective models preprocess the trajectory data to extract characteristics across a swath of trajectories. Such preprocessing is desirable, as it discards highly autocorrelated data representing redundant information in favor of aggregating trajectory positions into observations of significance. Examples of this preprocessing scheme include extracting “semantically enriched” points that intersect known geographical regions [1], aggregating trajectory positions as stay points using supplied spatial and/or temporal thresholds [43], or processing trajectory data into groups using some convex combination of spatial, temporal, and semantic similarity kernels [25, 39]. In a collective model, we refer to the ‘important’ or semantically meaningful data samples aggregated from trajectory points as exemplar positions, or simply, exemplars: Definition 1. Exemplar Consider a sample of n discrete points X n ⊂ Rd that constitute a trajectory T . Define an aggregation function α : P(X n ) 7→ Rd that maps any subset of points (e.g. a trajectory segment) in T to a set of exemplar positions Σ ⊂ Rd . The aggregation function of choice depends on the intent of the analysis. For example, consider an urban environmental study that defines α as a mapping of some isolated trajectory segment {pk , pk +1 , ..., pk +l } to the mean coordinate of the segment if the speed of the object traveling from pk to pk +l exceeds a certain threshold. Groups of these exemplar positions may determine “highemission” zones in a city [4]. Alternatively, if the traffic is made of pedestrians, such groups may represent tourist attraction areas, the popularity of which are useful for LBS applications [44]. It is not difficult to find this type of trajectory preprocessing in geospatial applications, and the grouping of them is foundational to countless tasks in trajectory mining [1, 24, 39, 43–45]. We generalize this preprocessing step by referring to it as “exemplar extraction.” An important aspect of exemplar extraction is to choose an aggregation function that befits the intent of the analyst and thus satisfies a study’s interpretation of “interesting.” This is inevitably application-specific, and the proposed framework is agnostic to the specific form of aggregation used, thus it is irrelevant to bestow a particular interpretation of what “interesting” means. We consider a more concrete and practical definition using a popular type of aggregation in Section 3. 2.2 Defining a point of interest Under the premise that exemplars represent meaningful aggregations of observations from a trajectory data source, it is natural to define a POI as a region of exemplars. We seek a definition of such regions with a statistical (rather than heuristic) foundation as a means of reflecting the naturally occurring structure within the data. Towards this end, we define a POI as a contiguous, high density region of exemplars. To formalize this definition, we follow the notation of Chaudhuri et. al [10]. Let X be a subset of Rd and define a path as a function P : [0, 1] → S where S ⊂ X. Also denote the equivalence relation C Figure 1: Illustrating the cluster tree hierarchy and its interpretation of POIs. Consider an estimated density (right panel) of exemplar positions extracted from trajectories in a geospace (middle, bottom panel). A POI is a geospatial region inhabited by exemplar positions at some density threshold λ, with the number of the POIs extracted depend on this scale parameter setting (left panel). Higher λ limits a POI to being specific and small, and could cause POIs to be manifested by random noise or be overfitted to a particular set of observations. Low λ defines POIs as very broad areas of low exemplar position density. The cluster tree hierarchy (left panel) summarizes the set of exemplar positions representing a POI at every density threshold, thus capturing the entire collection of POIs over a common area (middle panel, upper layers). as connected, where xCy iff P(0) = x and P(1) = y. Then C partitions S into connected components or clusters. Each component represents an area of high density and is called a high density cluster: Definition 2. High density clusters For a density function f on Rd , consider a partitioning: {x : f (x) ≥ λ}, for some λ > 0 (2) where λ is called the level, or high density threshold, parameter. Then all maximally connected components in this set are high density clusters at density level λ. We relate this formal definition to the trajectory mining domain with the following definition of a point of interest, defined over a extracted set of exemplars. Definition 3. Point of interest Given a set of m exemplars {ε 1 , ε 2 , . . . , εm } ∈ Σ and a fixed “scale” or resolution λ, each high density cluster of such exemplars forms a point of interest at the density level λ. The sets of high density clusters across all values of λ forms a hierarchy often referred to as the cluster tree of the density f [10, 11]. A hierarchical definition of locations is common [44] and matches the intuitive interpretation of a POI. For example, not only may a particular restaurant in a mall food court be a POI, but the food court itself may also be considered a POI, as well as entire mall may be yet another POI. The cluster tree conceptualization formalizes a POI as a maximally connected set of exemplars falling along a higher density area, implying such areas are ‘significant,’ and that such connected exemplars may be related. A visualization of hierarchical POIs and a dendrogram of the corresponding cluster tree is provided in Figure 1. The middle figure demonstrates a high-level view of what a set of trajectories might look like, with the colored dots in the left and middle figures representing exemplars. The right figure demonstrates a density estimate of the positions of these exemplars. That is, when these exemplars are very close to each other, they’re said to have a high density and are thought to be related, constituting a POI—the scale of the density depends on the analysis at hand. A sufficiently low density threshold λ 0 will designate every exemplar as one POI. From this definition, it may seem that any arbitrary density estimator may be used to find high-density clusters: simply estimate the density of every point by kernel density estimation (KDE), and then iterate through all possible values of λ that create distinct high-density clusters. Yet not every estimation will produce the same hierarchy—different kernels (and kernel bandwidths) may result in a completely different hierarchy of high-density clusters, and by extension, a different set of POIs. From the cluster tree perspective, the ideal kernel fn is one that is uniformly consistent (i.e. supx \| fn (x) − f (x)\| → 0 as n → ∞) from a given sample X n . In this case, a model could be fitted with the appropriate kernel and bandwidth parameter, and the would KDE furnish a continuous surface from which a cluster tree and its high-density clusters can be derived [11]. The main issue is that the set of all high-density clusters is not easy to compute for typical density estimates of f [10] and generally require a significant amount of memory to store. This computational inefficiency limits usability for large trajectory datasets, often observed over wide geographical areas and over long periods of time. From the applied perspective, many state-of-the-art approaches find POIs by variants of hierarchical clustering to find groups of exemplars. This has proved useful for application-specific problems [43–45] but they are largely heuristic, i.e. it is common for most clustering algorithms to have unstated or unknown statistical properties, precluding the possibility of formal inference [14]. The framework we introduce therefore examines density-based clustering methods as they are designed to infer a cluster tree [11] without facing the computational hurdles of KDEs. A desirable property of any finite-sample density estimator is some notion of consistency.7 In 1981, Hartigan establsihed a reasonable definition [19], often referred to as Hartigan consistency: Definition 4. Hartigan Consistency Let Cfn be the set of all high-density clusters from the cluster tree. For any sets A, A0 ⊂ X, let An (respectively, An0 ) denote the smallest set of Cfn containing A ∩ X n (respectively A0 ∩ X n ). Cfn is consistent if, whenever A and A0 are different connected components of {x : f (x) ≥ λ} (for some λ > 0), P(An is disjoint from An0 ) → 1 as n → ∞. This consistency definition essentially requires that two disjoint high-density clusters from the unknown, population density (A and A0 ) will also be disjoint components in a given empirical cluster tree (An and An0 ), given enough samples (n). The proposed framework for POI discovery is developed and implemented from the first computationally tractable and provably consistent algorithm that satisfies Hartigan consistency, as analyzed by Chaudhuri et. al [10], to be discussed in the next section. Having a nonparametric model satisfying this notion of consistency is important, as it transforms the unsupervised problem of POI discovery into a formal statistical estimation problem, not only enabling analysis driven by data, but requiring minimal assumptions regarding the nature of the data. Such a relation enables methods of formal statistical inference, allowing one to quantify uncertainty, i.e. to create hypothesis tests to discern “true” POIs as opposed to “false” POIs resulting from random noise or artifacts of low sample sizes, or to create notions of confidence in estimation [11]. Consistent cluster tree estimation: We next motivate a recent cluster tree estimator and discuss its relationship and applicability to POI discovery for the propsoed framework. Recall that an empirical estimate of the cluster tree, applied over exemplars, represents a hierarchy of POIs. Viewed from this perspective, what we propose can be seen as an extension of Chaudhuri et. al’s work on the cluster tree [10] to a trajectory mining context. Consider using Single-Linkage (SL) clustering, an agglomerative scheme that creates a hierarchical representation of clusters using the minimum pairwise distance D between all points, as a tool for clustering exemplars. Beginning with every exemplar x as a singleton, SL iteratively merges exemplars into clusters according to the linkage function: D(x i , x j ) = min d(x i , x j ) x i ,x j ∈X SL clustering is often criticized due to its tendency to create ‘excessive chaining’, wherein two clusters which may have been seen as generally unrelated are amalgamated by chance at a distance threshold that does not reflect the true dissimilarity between the resulting clusters. Hartigan proved SL is a consistent estimator of the cluster tree for densities in R (for d = 1) and is not consistent for any d > 1 [19], implying that any SL cluster that contains all the sample points in A will also contain nearly all sample points in A0 , in probability as n → ∞.8 This is reflected in the geospatial sense as well: that an estimator Θ̂n whose value θˆ is a point estimate of θ is consistent if, as more samples are collected (n → ∞), Θ̂n converges in probability to the true value of the parameter, i.e. plim (Θ̂n ) = θ . 7 Recall n→∞ 8 The condition is related to the “thin bridge” between any two population modes. Fractional consistency was shown for SL if for any pair A and A0 , the ratio of inf {f (x ) : x ∈ A ∪ A0 } to sup{inf {f (x ) : x ∈ P } : paths P from A to A0 } is sufficiently large. Figure 2: SL excessive chaining example. The bottom panel denotes a possible clustering using SL when pedestrians were found to stop between buildings. consider the case where exemplars represent aggregated ‘stops’ within a set of trajectories, a case that we will also consider later in Section 3. If an area is observed long enough, such exemplars should naturally form an area high density in areas where people stop frequently, e.g. within buildings. In such cases, it may be useful to categorize exemplars within their respective POIs (this is done in supervised way applications extract semantic information, see [1] for example). However, it’s also possible that there exist a few stops just outside of such buildings, which SL has a tendency to chain together. An example of this is shown in Figure 2. This discovery motivated efforts to modify SL not only to reduce this chaining to make SL more ‘robust’, but also to achieve (at least) Hartigan consistency for d = 2 and beyond. The first provably consistent estimator, which we consider in this effort, is a generalization of SL referred to as ‘Robust Single Linkage’ (RSL) [10]. Robust Single Linkage: Let X be a subset of Rd . Let k·k denote the `2 norm and let B(x, r ) be a closed ball of radius r around the point x. The RSL algorithm is given in the listing below: Robust Single Linkage Algorithm (1) For each x i set r k (x i ) = inf {r : B(x i , r ) contains k data points}. (2) As r grows from 0 to ∞: (a) Construct a graph G r with nodes {x i : r k (x i ) ≤ r }. Include edge (x i , x j ) if kx i − x j k ≤ αr (b) Let Cfn (r ) be the connected components of G r . The RSL algorithm has two free parameters which need to be set: α and k. SL is equivalent to RSL with the setting α = 1, k = 2. Whereas SL is equivalent to (and can be efficiently computed by) the minimum spanning tree (MST) computed over all pairwise distances, RSL scales these distances by a constant factor α, and only reduces to the MST if the components are restricted from connecting (satisfying {x i : r k (x i ) ≤ r }) within the MST computation. Chaudhuri et. al found that RSL is Hartigan consistent and established finite-sample rates of convergence for all 1 ≤ α ≤ 2, with the optimal rate of convergence with the setting α ≥ 2 [10]. 2.3 Finding intrinsic points of interest Using a consistent cluster tree estimator, such as RSL, on a set of exemplars creates a hierarchical representation of POIs. However, a nested set of multiple solutions is not always desirable, and a ‘flat’ solution (where each point is assigned a single label) may be preferred. A traditional approach in hierarchical clustering, “cutting” the empirical cluster tree at a given density threshold value λ yields a set of high-density clusters Cfn (λ) = {C 1 , C 2 , . . . Cm } that form m POIs, a possible ‘flat’ solution. However, the choice of λ forces all POIs to be of the same scale, requires the user to know which granularity to choose a priori, affecting the size and kinds of POIs discovered. For example, a small λ may define shops in a mall as POIs, while a larger λ may define the mall itself as a POI. It may not be known ahead of time what granularity level is relevant. Furthermore, it is reasonable to expect that relevant POIs exist at multiple levels of granularity, such that a sprawling city park and a small restaurant could both constitute a POI. Thus, it would useful to have some sensible notion of “cluster quality” that can be used (and optimized) as an objective function to discover POIs that are not dependent on the analyst’s choice of λ, and are strongly intrinsic to the geospace itself, e.g. are intrinsic POIs. To capture POIs of any scale and hence satisfy our notion of an intrinsic POIs, we first recall that high-density clusters are contiguous, relatively dense areas of the data space, separated by contiguous and relatively non-dense areas, defined over a working definition of density over a set of exemplars. From a statistical point of view, we can think of a high-density cluster as a set of points with high density around some “neighborhood” or volume of the support. Müller et. al quantify this using a functional called excess of mass [26]: Definition 5. Excess of Mass For a Ci ∈ Cfn (λ) for some value of λ > 0, the excess of mass of Ci is given by: ∫ E(Ci ) = f (x) − λmin (Ci ) dx (3) x ∈C i where λmin (Ci ) represents the lowest density level Ci appears. Initially, this measure seems like a reasonable definition of the “quality” of a clustering within the cluster tree estimate. Considering the definition of a high-density cluster from Equation 2, where a cluster exists along a mode of local maximum of the underlying density, it’s far too likely that a finite-sample estimation may empirically find a mode at a given point x 0 if the data is sparse, allowing an arbitrarily low probability associated x 0 to be classified. A more interesting result would be to associate a high-density cluster with a region that exhibits relatively high probability over a neighborhood. See Müller et. al for visualization, along with a more in depth description of this functional [26]. However, as Campello et. al remark, this measure exhibits monotonic behavior in any direction varying the density-level λ in the hierarchy, and instead propose an alternative, local measure of cluster quality [8]: Definition 6. Relative Excess of Mass For a Ci ∈ Cfn (λ) for some value of λ > 0, the relative excess of mass of Ci is given by: ∫ E R (Ci ) = λmax (x, Ci ) − λmin (Ci ) dx (4) x ∈C i where λmax (x, Ci ) = min{ f (x), λmax (Ci )} is the density level beyond which x is no longer part of Ci , and λmax (Ci ) is the highest density beyond which Ci either becomes disconnected (creating separate components) or disappears (creating singleton clusters). It is important to note that relative excess of mass is defined in terms of λ values associated with a specific cluster, as opposed to a specific clustering. This implies that an ‘optimal’ clustering with respect to the relative excess of mass estimate may not occur at a fixed, global density threshold, but rather as a result of several local density thresholds applied to the hierarchy. Intuitively, if a given cluster Ci contains many points that have high density relative to λmin (Ci ), such a cluster will exist across several thresholds of λ and is thus robust to fluctuations in the scale of analysis. For this reason, the relative excess of mass can be thought of as a measure of cluster ‘stability’ across different density levels, which we posit reflects an intrinsic POI that is innately defined by the dataset independent of density level. Such intrinsic POIs are thus defined as follows: let δi be an indicator equal to 1 if cluster Ci ∈ Cfn represents an intrinsic POI and 0 otherwise. Assign values to these indicators such that the following is maximized: m Õ maximize J = δi E R (Ci ) δ 1, ...,δm subject to i=2 ( δi ∈ {0, 1}, i ∈ {2, . . . , m} exactly one δ (·) = 1 per disjoint branch (5) Where the “per disjoint branch” constraint means that the indicator function δ (·) equals 1 exactly once for all clusters in each path from a leaf node to the root of the cluster tree. The optimization of this objective function is beyond the scope of this paper; we refer to Campello et. al’s cluster extraction method for general cluster hierarchies [7] to solve this optimization, as it was developed alongside an estimator very similar to RSL, is capable of producing an optimal result at several density levels, and accounts for the density thresholds at which points become noise (fall along densities below a given threshold). 3 Experiments and Discussion We next evaluate the proposed framework for intrinsic POI discovery. Because intrinsic POIs do not rely on gazetteers and may manifest themselves in unknown locations, evaluation on real data validated against “ground truth” external knowledge (such as imported location data from sources such as OpenStreetMap or Google Places) is not feasible. A common approach to evaluate clusterings when ground truth is absent is to use an internal cluster validity index (CVI). CVIs include common indices like the Silhouette score, the Dunn Index, and the Calinski-Harabasz criterion (see Arbelaitz [3] for an overview of these techniques and references therein). Recent work recommends validation using multiple CVIs, as they each score different aspects of a clustering such as the ratio of inter- to intra-cluster distances, sum of squares distance to centroid, or graph-theory scores based on similarity [3]. We do not believe scores are informative for intrinsic POI evaluation, as most of these CVIs operate on unrealistic assumptions (e.g. symmetry or convexity of cluster shape, a notion of minimal variance, the existence of a centroid or medoid, etc.). Contrary to these widespread concepts, we do not assume that a cluster of exemplars representing an intrinsic POI will maintain some hyper-spherical or -elliptical shape. Indeed, there are a number of features within a geographical area that may be considered POIs, yet inevitably exhibit arbitrary shapes (e.g. buildings, parks, gathering areas, etc.) and manifest at varying densities (e.g. a busy intersection that is small and concentrated in exemplar density vs. a parking lot that is large and more uniform in density). Following the advice of Guyon and Luxburg et al. [18, 37], we evaluate the efficacy of the framework in the context of its end-use. We use an external validation where “truth” can be defined a priori over simulated data, enabling a direct evaluation of intrinsic POIs against “truly interesting” regions, while ensuring the latent patterns in the generated data mimic the real geospatial dynamics of cars and pedestrians over a region. 3.1 Generating synthetic data To generate synthetic data for evaluation, we turn to the Simulation of Urban MObility (SUMO) software [23]. SUMO is an open source traffic simulation system capable of generating trajectories of many objects of multiple modalities (e.g. car, truck, person, plane, etc.). Given a shapefile that defines avenues for travel (e.g., a road network, a map of footpaths within a university campus, or the floor plan of a mall or large building), SUMO is able to generate trajectories following the avenues provided. Default parameter settings generate traffic and trajectories in ways that satisfy their measured physical properties have been shown to be incredibly accurate [23]. We use SUMO to generate two simulations of both pedestrian and vehicular traffic under different geographical areas: an urban region having a mixture of vehicle and pedestrian traffic (the area surrounding The Ohio State University (OSU)), and a suburban area where pedestrian traffic is more prominent (the area surrounding Wright State University (WSU)). Details about the simulation, the simulation data used in this paper, and the code that produced the resulting evaluation are all publicly available and reproducible online.9 The (RSL) cluster tree framework itself is part of a larger open source effort by the author.10 Simulation configuration: SUMO requires every object to have a trip defined by departure and destination nodes, which SUMO refers to as junctions. Junctions are connected by edges representing a possible travel path. Given a file containing the trip definitions of every object, SUMO dynamically generates routes, or sequences of edges the object travels along to get from departure junction A to its destination junction B. We leave nearly all simulation parameters at their default settings, only modifying simulation length and arrival parameters (binomially distributed arrivals) to generate pedestrian and vehicle demand. Because pedestrian traffic within unrestricted and indoor areas may constitute intrinsic POIs in a realistic setting, and because SUMO can generate only outdoor pedestrian traffic, we extended SUMO to simulate indoor pedestrian traffic as well. Figure 3 illustrates how this extension interplays with vehicular traffic generated by SUMO.11 Shapefiles denoting the location of buildings are first loaded into SUMO (the peach colored regions in Figure 3 inlet). Then, within the shapefile, a random number of pedestrian-only junctions are generated within the building and registered to nearby pedestrian-only edges (such as sidewalks). If a generated track is labeled as a pedestrian and its trip includes a junction contained within the building region (Figure 3; lower right inlet), the pedestrian undergoes a random walk within the junctions generated in the building. This random walk is emulated by choosing a random ordered subset of the generated junctions for a random amount of 9 See the following for simulation details: ¡Anonymized for review purposes.¿ the following package: ¡Anonymized for review purposes.¿ 11 See ¡Anonymized for review purposes.¿ 10 See Figure 3: Top-level view of extending SUMO to support indoor pedestrian traffic. Shapefiles defining buildings are loaded into SUMO and registered as junction. If pedestriantrack visits an attached junction during a trip, the simulator chooses ordered random set of junctions to follow within the building, exiting after a random period of time. time. The pedestrian visits these interior junctions and then travels to an ‘exit junction’ attached to the building polygon, continuing along the original (outdoor) route generated by SUMO. Defining truth: Recall that intrinsic POIs are inferred by exemplars representing the specific mobility pattern of interest. With both vehicular and more realistic pedestrian demand generated, the next step in data generation is to define an aggregation function to extract meaningful exemplars. To give a concrete use-case of the proposed framework, we align our experiment with much of the applied literature related to this topic [28, 42, 45] and extract exemplars representing the “stay points” of an object. A stay point is a position where objects have stopped or significantly slowed down. Extracting such points from simulated SUMO data is trivial, as the true speed of any traveling object is known at any given time. For pedestrian traffic, we extract trajectory points where pedestrians stopped moving. For vehicular traffic, we extract either a) the points where the vehicles stopped moving or b) the slowest point in a vehicle braking sequence using SUMOs exported braking signals, whichever is available. From these stay points (exemplars), we next establish a mapping between each exemplar and its presence within a “true” intrinsic POI, allowing external validation. Since exemplars represent object stopped moving, a natural definition of an intrinsic POI is an assignment defined by the mechanism causing such objects to stop. Specifically, we define a building that pedestrians stop within as a “true” intrinsic POI, as this is very natural and useful grouping. We follow a similar pattern for vehicular traffic, assigning exemplars a common label if stopped at identical intersections, stop signs or stop light, or other junctions. This mechanistic assignment of creating “true” intrinsic POIs has the benefit of not only being tractable (in the sense that SUMO provides this information directly), but also being semantically meaningful in the sense that the mechanisms encouraging objects to stop moving are intrinsic to the geospace. 3.2 Experimental Design To evaluate the fidelity of the POIs extracted by the proposed framework under multiple settings, we run SUMO simulations over the OSU and WSU geospaces with parameter settings reflecting differences between the two regions. These settings are shown in Table 1: SUMO Simulation Parameters Region # Build# Veh. ings # Ped. Region size Sim. Length OSU 70 2,933 2,935 342km2 8 hours WSU 26 2,050 4,327 461km2 6 hours Table 1. The OSU geospace covers a smaller area, has an equal mix of vehicles and pedestrians, and nearly three times as many buildings. The OSU geospace also has a larger number of roadways and traffic intersections where intrinsic POIs involving vehicles may materialize. Being within the main campus, the WSU geospace has a larger proportion of pedestrian traffic, with few roadways for vehicles to traverse and smaller number of buildings pedestrians may visit. Figures 4(a) and 5(a) show what this SUMO generated POIs labeling creates for the OSU and WSU campus areas, respectively. Qualitatively, examination of these clusters appear to be reasonable labels of intrinsic POIs. For example, the clusters representing “true” POIs across OSU in Figure 4(a) finds buildings surrounding the OSU oval quad, particular locations on the ring road around the quad (which tend to be busy OSU intersections for both vehicles and pedestrians), and parking lots around the OSU recreation builds west of the oval to represent POIs. Across WSU in Figure 5(a), the truth POIs represent each of the major buildings around the campus, with particularly complex, separate areas of movement in WSU’s large student union (the yellow points in the large building in the lower left part of the figure). Evaluation Measures: As discussed at the beginning of this section, the unsupervised nature of intrinsic POI discovery make it difficult to carry out a meaningful evaluation of POI discovery methods using internal (not requiring ‘truth’ labels) validation measures. Instead, we consider a multifaceted approach: an external, quantitative evaluation of whether the intrinsic POIs discovered aligns with SUMO generated POIs using the well-known Adjusted Rand Index (ARI) [21], and qualitative evaluation of the quality of the intrinsic POIs our approach unearths as compared to the “true” intrinsic POIs as defined above. The Rand-family of indices were chosen due to their transparency and simplicity—although, whereas the traditional RI measures the proportion of pairwise agreements between two partitions, the ARI also adjusts the score based on the expected value of agreements under the null hypothesis that the agreements were completely random, and thus is what we report. Algorithms Compared: We further compare the fidelity of the intrinsic POIs extracted by the proposed framework against other clustering algorithms commonly used for POI discovery from trajectories. We either downloaded the implementation of, or implemented ourselves, a number of these algorithms for comparison. Aside from RSL, the selected methods includes the well-known density-based algorithms DBSCAN [12] and OPTICS [2], the widespread hierarchical algorithms single linkage (SL), average linkage (AL), and wards criterion (WL) [27], along with the partitioning-like algorithms k-means and CLARA [22]. These algorithms were chosen due to their relevance to this problem, wide-spread availability, known success in the clustering world. Parameter Settings: Clustering algorithms generally require parameter-tuning in order to ‘fit’ to a given data set, but the number and semantics of these parameters often changes with the algorithm used, leaving comparisons between parameter settings difficult. Although most hierarchical algorithms carry no free parameters to create a (hierarchical) set of solutions, they do require either a threshold value (h) or the exact number of clusters to extract (k) to be specified to extract a ‘flat’ clustering. Similarly, k-means and CLARA also require k to be specified a priori. Because the k parameter has the same interpretation in multiple algorithms, we will use k to refer to the number of clusters extracted. Density based algorithms have multiple parameters with interpretations compared to the aforementioned algorithms. For example, DBSCAN requires a minimum cluster size parameter minPts and a distance (or ‘scale’) threshold ϵ to be set. OPTICS, often cited as extension to DBSCAN, is an ordering algorithm that—given a parameter setting for minPts—can be be used to extract either a flat, DBSCAN-like cluster extraction or a simplified hierarchy using a either the a distance threshold ϵ 0 or a reachability-based threshold ξ , respectively. The DBSCAN-like cluster extraction is reported here. RSL requires the setting of α and k, the former relating to scaling the connection radii used to connect components, and the latter to the saliency of cluster estimates. Note that in RSL, k is more similar to the minPts parameter in that it is a minimum neighborhood parameter. The number of clusters is automatically determined by optimized the defined relative excess of mass functional from Section 2. Each algorithm reflects a large set of possible solutions over its parameter setting. Choosing a single parameter setting for evaluation would represent a source of possible bias. Rather, we employ a more comprehensive approach by comparing a wide range of parameter settings for each algorithm. To define these ranges, let seq(x, y, s) denote the sequential range operator, skipping s values in the sequence of integers from x to y. For example, seq(1, n, 1) = {1, 2, . . . , n}, and seq(1, n, i) = {1, 1 + i, . . . , n − i, n}. For the hierarchical clustering algorithms (SL, AL, and WL) the number of flat clusters extracted k is varied in the range seq(2, nt , 1) where nt is the number of “true” POIs assigned by SUMO. We see this as a reasonable strategy, as it gives a better view of how multiple levels extracted from the hierarchy matched the data set as well as how well the merge criterion (or linkage function) collectively captures the true POIs in the geospace. We use the same range to vary k for the k-means and CLARA algorithms. The density based methods DBSCAN and OPTICS are evaluated by first varying minPts, and then (for each value of minPts) by varying the scale parameters ϵ and ϵ 0 , respectively. Recall minPts relates to a minimum neighborhood value that constitutes a cluster. Thus, and to allow the testing to be tractable, we set minPts to reflect the possible sizes of the POIs, along the quantiles qnt = seq(0.10, 0.95, 0.025) corresponding to the the number of exemplars per POI in the SUMO “truth” data. The distance thresholds ϵ for DBSCAN and ϵ 0 OPTICS are also varied along the quantiles seq(0.01, 0.20, 0.01) of the pairwise distances computed over the data set. Since all density-based methods mark points that fall in areas of the data set not sufficiently dense as ‘noise’ according to a scale parameter—leading to severe overfitting if not guided with a measure like stability—all densitybased solutions were deemed only valid if at least 75% of the data is classified with a non-noise label. Finally, the RSL also contains (a) “True” POIs (b) Inferred intrinsic POIs Figure 4: Intrinsic POI comparison, OSU (c) “True” POIs (d) Inferred intrinsic POIs Figure 5: Intrinsic POI comparison, WSU two parameters, a k value, and an α parameter. We use Chaudhuri et. al’s analysis to determine how to set these. RSL √ was shown to have optimal rates of convergence when α ≥ 2, so we leave √ it at that constant value ( 2). Similarly, the rate only holds for k at least as large as d log(n), where d is the dimensionality of the data set (d = 2 in this case). After varying through the small set of k values in a similar fashion as was performed for DBSCAN and OPTICS (k ∈ qnt ∀k >= d log(n)). In total, 2,196 and 1,995 cluster configurations were performed for the OSU and WSU simulations respectively, totalling 4,191 reported configurations. 3.3 Validation Testing and Discussion Qualitative comparison to truth: Figures 4 and 5 compare the intrinsic POIs discovered by our framework agains the simulation’s “true” POIs. Recall that points with low density are discarded as noise (not shown). Direct comparison of the true POIs defined by the simulation show clear similarities. Over the OSU simulation in Figure 4(b), the framework recovers intrinsic POIs within buildings no matter its shape, the density, or closeness to other buildings. It also recovers intrinsic POIs over parking lots and street intersections around the OSU oval. Some buildings are decomposed into a collection of individual intrinsic POIs. For example, the easternmost large campus building by the northeast corner of the oval contains three separate intrinsic POIs: one at its entrance by the road, another in the center of the building, and a third at its back entrance. Although these labels may not match what SUMO assigned, they are in some sense more natural, i.e. it’s quite possible for large buildings to have dense, isolated areas of people movement. Looking at the intrinsic POIs over the WSU dataset in Figure 5(b), we find each building in general is recovered as an intrinsic POI and align in shape compared to the shape of the “true” POIs from Figure 5(a). We also note that the framework determines that some movement within buildings but covering very small areas were found not be significant enough be an intrinsic POI. Large buildings also showed further decomposition like the OSU simulation. For example, in the WSU student union (large building in the lower left corner of the figure) the framework defines intrinsic POI’s at the center and two back exits from the building. Quantitative comparison to other approaches: The proposed intrinsic POI framework measured ARI scores of 0.966 on the OSU data set and 0.922 for the WSU data set. Note that this is not the maximum ARI of any RSL solution, but the ARI of the solution found using the highest predefined notion of stability, determined completely without any knowledge of the surrounding geographical area. RSL performed consistently in terms of having low variability compared to other algorithm, with overall high similarity to the semantically driven SUMO assigned locations. Figure 6 shows the distribution of the ARI for the algorithms we compared our method against with the parameter settings discussed in Section 3.2. The orange line corresponds with the ARI of the proposed framework, which compares favorably with best possible settings of other algorithms. Note that although DBSCAN (like others) performed well with very specific configurations of minPts and ε 0 , the settings of these parameters is often not very intuitive in unsupervised scenarios where the truth is unknown (and thus external measures like ARI cannot be computed). Of the hierarchical algorithms, we see the impact of the linkage criterion used and how they are influenced by the ‘shape’ of the true clusters. For example, SL clustering performed fairly well on the well separated WSU data set, but substantially lower on the more density-varied OSU data set. This is reflected in AL as well. k-means was able capture much of the true clustering structure with the right parameter settings Figure 6: The distribution of Adjusted Rand Index (ARI) scores of various clustering algorithms (after varying free parameters). The orange line corresponds to the ARI of the proposed framework. for the WSU simulation (with max/mean ARI scores of (0.92, 0.71)), however exhibited degraded performance when the POIs were less separated in OSU data set ((0.61, 0.84)). OPTICS, with a few specific parameter settings, performed well on the WSU data set, however again suffered on the more variable-scale OSU data set. 4 Related Research The trajectory field has been largely progressed by “extensive and intensive individual efforts” [42]. Nonetheless, conceptual models have been proposed for how to deal with the patterns within trajectories and how to relate such patterns to geographical areas of interest for various purposes. One such model postulates that trajectories and their spatiotemporal patterns are essentially driven by the semantics the application associates with trajectory itself [29] [33], and have contributed significantly to the “Stops and Moves of Trajectories” (SMoT) family of classification algorithms [1, 28, 32], where the premise of the analysis is that by partitioning trajectory data into a labeled set of ‘stop’ and ‘move’ segments, one can take then annotate these segments with semantic information, derive specific mobility patterns, and as a result discover ‘interesting’ locations. Alvares et al. developed IB-SMoT to find interesting positions based on semantic annotations describing the places a trajectory visited [1]. Palma et al. reduced IB-SMoT’s reliance on prior knowledge about positions that are likely to be interesting by incorporating the speed at which tracks are traveling with their variation CB-SMoT [28]. DB-SMoT finds clusters of common trajectories based on similar direction changes and stopping points [32]. Zhou et al. tackle the problem of finding positions of interest to an individual track based on data about a track’s location preferences, position over time, and tags of locations provided by web services such as Google Maps [45]. Many related efforts encode or are reliant on varying notions of an “interesting place” using, for example, techniques from natural language processing (NLP) [15], data clustering [32, 35, 45], sequential pattern mining [38], and social network analysis [41] methods. Zheng et. al pioneered the use of ‘stay points’, corresponding to an aggregation of consecutive GPS points that collectively are within a user-supplied time and distance threshold, thereby characterizing a ‘virtual’ location [43, 44]. It is interesting to note that Zheng et. al used OPTICS to create a hierarchical clustering ‘stay points’ for an LBS-type application with Microsoft called ‘Geolife’ [43]. Indeed, Zheng anticipated a number of developments in the trajectory mining field [42]; the theoretical cluster tree may be viewed as a more statistically based conception of the‘Tree Based Hierarchical Graph’ that is used to represent POIs in that application as well. It’s worth noting that our definition of an intrinsic POI, having a more theoretical foundation in density-based clustering, is both conceptually very similar to OPTICS and computationally, more recently, to Hierarchical DBSCAN (HDBSCAN) [6]. There exist a number of commonalities between both OPTICS and DBSCAN and the theory of the cluster tree. A comprehensive exposition of this relationship is beyond the scope of this paper, see Campello et. al [8] for a thorough review of the subject. Although it’s not mentioned in such efforts, the usage of RSL with a relative excess of mass functional to cluster extraction is equivalent to the flat clusters “HDBSCAN” extracts with a setting of α = 1 and k = minPts. However, the asymptotic consistency of the setting of the pair (α = 1, k ∼ d log n) has not been established [10]. When alpha = 1, k must be much larger, exponential in√the dimensionality of the data set, d. Thus, we use RSL with α ≥ 2. 5 Concluding Remarks This paper proposed a general framework for intrinsic POI discovery, without needing to rely on external gazetteers, based on recent theoretical advances in hierarchical, nearest neighbor density estimation. It discussed a conceptually sound basis for automated POI discovery specifically in the context of geospatial data, and introduced a framework that provides a rigorous and usable solution to an applied domain primarily dominated by intuitively reasonable, but heuristically-based methods. With novel extensions to SUMO to support pedestrian movement in buildings, an evaluation of simulated trajectory data over diverse geographical areas supports the conclusion that the proposed framework is a useful tool for extracting intrinsic POIs. The framework has both theoretical guarantees and practical benefits, requires no ad hoc parameter tuning, and exhibits improved fidelity against common approaches over thousands of parameter settings. In future work, with the help of the asymptotic analysis done by Chaudhuri et. al, we plan to develop model-selection techniques for POI extraction. This is imperative in exploratory settings, such as large urban environments where the number of POIs is not known ahead of time, there is little useful knowledge to gain from ad hoc or heuristic-based cluster analysis, especially when the solution space is large. 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Critical Parameters in Particle Swarm Optimisation arXiv:1511.06248v1 [cs.NE] 19 Nov 2015 J. Michael Herrmann∗, Adam Erskine, Thomas Joyce Institute for Perception, Action and Behaviour School of Informatics, The University of Edinburgh 10 Crichton St, Edinburgh EH8 9AB, Scotland, U.K. Abstract Particle swarm optimisation is a metaheuristic algorithm which finds reasonable solutions in a wide range of applied problems if suitable parameters are used. We study the properties of the algorithm in the framework of random dynamical systems which, due to the quasi-linear swarm dynamics, yields analytical results for the stability properties of the particles. Such considerations predict a relationship between the parameters of the algorithm that marks the edge between convergent and divergent behaviours. Comparison with simulations indicates that the algorithm performs best near this margin of instability. 1 PSO Introduction Particle Swarm Optimisation (PSO, [1]) is a metaheuristic algorithm which is widely used to solve search and optimisation tasks. It employs a number of particles as a swarm of potential solutions. Each particles shares knowledge about the current overall best solution and also retains a memory of the best solution it has encountered itself previously. Otherwise the particles, after random initialisation, obey a linear dynamics of the following form vi,t+1 xi,t+1 = = ωvi,t + α2 R1 (pi − xi,t ) + α2 R2 (g − xi,t ) xi,t + vi,t+1 (1) Here xi,t and vi,t , i = 1, . . . , N , t = 0, 1, 2, . . . , represent, respectively, the d-dimensional position in the search space and the velocity vector of the i-th particle in the swarm at time t. The velocity update contains an inertial term parameterised by ω and includes attractive forces towards the personal best location pi and towards the globally best location g, which are parameterised by α1 and and α2 , respectively. The symbols R1 and R2 denote diagonal matrices whose non-zero entries are uniformly distributed in the unit interval. The number of particles N is quite low in most applications, usually amounting to a few dozens. In order to function as an optimiser, the algorithm uses a nonnegative cost function F : Rd → R, where without loss of generality F (x∗ ) = 0 is assumed at an optimal solution x∗ . In many problems, where PSO is applied, there are also states with near-zero costs can be considered as good solutions. The cost function is evaluated for the state of each particle at each time step. If F (xi,t ) is better than F (pi ), then the personal best pi is replaced by xi,t . Similarly, if one of the particles arrives at a state with a cost less than F (g), then g is replaced in all particles by the position of the particle that has discovered the new solution. If its velocity is non-zero, a particle will depart from the current best location, but it may still have a chance to return guided by the force terms in the dynamics. Numerous modifications and variants have been proposed since the algorithm’s inception [1] and it continues to enjoy widespread usage. Ref. [2] groups around 700 PSO papers into 26 discernible application areas. Google Scholar reveals over 150,000 results for “Particle Swarm Optimisation” in total and 24,000 for the year 2014. In the next section we will report observations from a simulation of a particle swarm and move on to a standard matrix formulation of the swarm dynamics in order to describe some of the existing ∗ corresponding author: [email protected] 1 analytical work on PSO. In Sect. 3 we will argue for a formulation of PSO as a random dynamical system which will enable us to derive a novel exact characterisation of the dynamics of one-particle system, which will then be generalised towards the more realistic case of a multi-particle swarm. In Sect. 4 we will compare the theoretical predictions with simulations on a representative set of benchmark functions. Finally, in Sect. 5 we will discuss the assumption we have made in the theoretical solution in Sect. 3 and address the applicability of our results to other metaheuristic algorithms and to practical optimisation problems. 2 2.1 Swarm dynamics Empirical properties The success of the algorithm in locating good solutions depends on the dynamics of the particles in the state space of the problem. In contrast to many evolution strategies, it is not straight forward to interpret the particle swarm as following a landscape defined by the cost function. Unless the current best positions p or g change, the particles do not interact with each other and follow an intrinsic dynamics that does not even indirectly obtain any gradient information. The particle dynamics depends on the parameterisation of the Eq. 1. To obtain the best result one needs to select parameter settings that achieve a balance between the particles exploiting the knowledge of good known locations and exploring regions of the problem space that have not been visited before. Parameter values often need to be experimentally determined, and poor selection may result in premature convergence of the swarm to poor local minima or in a divergence of the particles towards regions that are irrelevant for the problem. Empirically we can execute PSO against a variety of problem functions with a range of ω and α1,2 values. Typically the algorithm shows performance of the form depicted in Fig. 1. The best solutions found show a curved relationship between ω and α = α1 + α2 , with ω ≈ 1 at small α, and α ' 4 at small ω. Large values of both α and ω are found to cause the particles to diverge leading to results far from optimality, while at small values for both parameters the particles converge to a nearby solution which sometimes is acceptable. For other cost functions similar relationships are observed in numerical tests (see Sect. 4) unless no good solutions found due to problem complexity or run time limits, see Sect. 5.3. For simple cost functions, such as a single well potential, there are also parameter combinations with small ω and small α will usually lead to good results. The choice of α1 and α2 at constant α may have an effect for some cost functions, but does not seem to have a big effect in most cases. 2.2 Matrix formulation In order to analyse the behaviour of the algorithm it is convenient to use a matrix formulation by inserting the velocity explicitly in the second equation (1). zt+1 = M zt + α1 R1 (p, p)⊤ + α2 R2 (g, g)⊤ ⊤ with z = (v, x) (2) and M= ωId ωId −α1 R1 − α2 R2 Id − α1 R1 − α2 R2 , (3) where Id is the unit matrix in d dimensions. Note that the two occurrence of R1 in Eq. 3 refer to the same realisation of the random variable. Similarly, the two R2 ’s are the same realisation, but different from R1 . Since the second and third term on the right in Eq. 2 are constant most of the time, the analysis of the algorithm can focus on the properties of the matrix M . In spite of its wide applicability, PSO has not been subject to deeper theoretical study, which may be due to the multiplicative noise in the simple quasi-linear, quasi-decoupled dynamics. In previous studies the effect of the noise has largely been ignored. 2.3 Analytical results An early exploration of the PSO dynamics [4] considered a single particle in a one-dimension space where the personal and global best locations were taken to be the same. The random components were 2 Figure 1: Typical PSO performance as a function of its ω and α parameters. Here a 25 particle swarm was run for pairs of ω and α values (α1 = α2 = α/2). Cost function here was the d = 10 non-continuous rotated Rastrigin function [3]. Each parameter pair was repeated 25 times and the minimal costs after 2000 iterations were averaged. replaced by their averages such that apart from random initialisation the algorithm was deterministic. Varying the parameters was shown to result in a range of periodic motions and divergent behaviour for the case of α1 + α2 ≥ 4. The addition of the random vectors was seen as beneficial as it adds noise to the deterministic search. Control of velocity, not requiring the enforcement of an arbitrary maximum value as in Ref. [4], is derived in an analytical manner by [5]. Here eigenvalues derived from the dynamic matrix of a simplified version of the PSO algorithm are used to imply various search behaviours. Thus, again the α1 + α2 ≥ 4 case is expected to diverge. For α1 + α2 < 4 various cyclic and quasi-cyclic motions are shown to exist for a non-random version of the algorithm. In Ref. [6] again a single particle was considered in a one dimensional problem space, using a deterministic version of PSO, setting R1 = R2 = 0.5. The eigenvalues of the system were determined as functions of ω and a combined α, which leads to three conditions: The particle is shown to converge when ω < 1, α > 0 and 2ω −α+2 > 0. Harmonic oscillations occur for ω 2 +α2 −2ωα−2ω −2α+1 < 0 and a zigzag motion is expected if ω < 0 and ω−α+1 < 0. As with the preceding papers the discussion of the random numbers in the algorithm views them purely as enhancing the search capabilities by adding a drunken walk to the particle motions. Their replacement by expectation values was thus believed to simplify the analysis with no loss of generality. We show in this contribution that the iterated use of these random factors R1 and R2 in fact adds a further level of complexity to the dynamics of the swarm which affects the behaviour of the algorithm in a non-trivial way. In Ref. [7] these factors were given some consideration. Regions of convergence and divergence separated by a curved line were predicted. This line separating these regions (an equation for which is given in Ref. [8]) fails to include some parameter settings that lead to convergent swarms. Our analytical solution of the stability problem for the swarm dynamics explains why parameter settings derived from the deterministic approaches are not in line with experiences from practical tests. For this purpose we will now formulate the PSO algorithm as a random dynamical system and present an analytical solution for the swarm dynamics in a simplified but representative case. 3 3 3.1 Critical swarm conditions for a single particle PSO as a random dynamical system As in Refs. [4, 6] the dynamics of the particle swarm will be studied here as well in the single-particle case. This can be justified because the particles interact only via the global best position such that, while g (1) is unchanged, single particles exhibit qualitatively the same dynamics as in the swarm. For the one-particle case we have necessarily p = g, such that shift invariance allows us to set both to zero, which leads us to the following is given by the stochastic-map formulation of the PSO dynamics (2). zt+1 = M zt (4) Extending earlier approaches we will explicitly consider the randomness of the dynamics, i.e. instead of averages over R1 and R2 we consider a random dynamical system with dynamical matrices M chosen from the set ωId −αR Mα,ω = , Rij = 0 for i 6= j and Rii ∈ [0, 1] , (5) ωId Id − αR with R being in both rows the same realisation of a random diagonal matrix that combines the effects of R1 and R2 (1). The parameter α is the sum α1 + α2 with α1 , α2 ≥ 0 and α > 0. As the diagonal elements of R1 and R2 are uniformly distributed in [0, 1], the distribution of the random variable Rii = αα1 R1,ii + αα2 R2,ii in Eq. 4 is given by a convolution of two uniform random variables, namely Pα1 ,α2 (r) =  αr   max{α1 ,α2 } α max{α ,α }   α(α−r)1 2 α1 α2 if 0 ≤ r ≤ min{ αα1 , αα2 } if min{ αα1 , αα2 } < r ≤ max{ αα1 , αα2 } if max αα1 , αα2 < r ≤ 1 (6) if the variable r ∈ [0, 1] and Pα1 ,α2 (r) = 0 otherwise. Pα1 ,α2 (r) has a tent shape for α1 = α2 and a box shape in the limits of either α1 → 0 or α2 → 0. The case α1 = α2 = 0, where the swarm does not obtain information about the fitness function, will not be considered here. We expect that the multi-particle PSO is well represented by the simplified version for α2 ≫ α1 or α1 ≫ α2 , the latter case being irrelevant in practice. For α1 ≈ α2 deviations from the theory may occur because in the multi-particle case p and g will be different for most particles. We will discuss this as well as the effects of the switching of the dynamics at discovery of better solutions in Sect. 5.2. 3.2 Marginal stability While the swarm does not discover any new solutions, its dynamical properties are determined by an infinite product of matrices from the set M (5). Such products have been studied for several decades [9] and have found applications in physics, biology and economics. Here they provide a convenient way to explicitly model the stochasticity of the swarm dynamics such that we can claim that the performance of PSO is determined by the stability properties of the random dynamical system (4). Since the equation (4) is linear, the analysis can be restricted to vectors on the unit sphere in the (v, x) space, i.e. to unit vectors ⊤ ⊤ a = (x, v) / k (x, v) k, (7) where k · k denotes the Euclidean norm. Unless the set of matrices shares the same eigenvectors (which is not the case here) standard stability analysis in terms of eigenvalues is not applicable. Instead we will use means from the theory of random matrix products in order to decide whether the set of matrices is stochastically contractive. The properties of the asymptotic dynamics can be described based on a double Lebesgue integral over the unit sphere S 2d−1 and the set M [10, 11]. As in Lyapunov exponents, the effect of the dynamics is measured in logarithmic units in order to account for multiplicative action. Z Z λ (α, ω) = dνα,ω (a) dPα,ω (M ) log kM ak (8) 4 If a (α, ω) is negative the algorithm will converge to p with probability 1, while for positive a arbitrarily large fluctuations are possible. While the measure for the inner integral (8) is given by Eq. 6, we have to determine the stationary distribution ν on the unit sphere for the outer integral. It is given as the solution of the integral equation Z Z να,ω (a) = dνα,ω (b) dPα,ω (M ) δ (a, M b/ kM bk) , a, b ∈ S 2d−1 . (9) The existence of the invariant measure requires the dynamics to be ergodic which is ensured if at least some of elements of M have complex eigenvalues, such as being the case for ω 2 +α2 /4−ωα−2ω−α+1 < 0 (see above, [6]). This condition excludes a small region in the parameters space at small values of ω, such that there we have to take all ergodic components into account. There are not more than two components which due to symmetry have the same stability properties. It depends on the parameters α and ω and differs strongly from a homogenous distribution, see Fig. 2 for a few examples in the case d = 1. Critical parameters are obtained from Eq. 8 by the relation 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 1 2 3 4 5 6 Figure 2: Stationary distribution να,ω (a) on the unit circle (a ∈ [0, 2π)) in the (x, v) plane for a one-particle system (4) for ω = 0.7 and α = α2 = 0.5, 1.5, 2.5, 3.5, 4.5 (the distribution with peak near π is for α = 0.5, otherwise main peaks are highest for largest α). λ (α, ω) = 0. (10) Solving Eq. 10 is difficult in higher dimensions, so we rely on the linearity of the system when considering the (d = 1)-case as representative. The curve in Fig. 3 represents the solution of Eq. 10 for d = 1 and α = α2 . For other settings of α1 and α2 the distribution of the random factors has a smaller variance rendering the dynamics more stable such that the contour moves towards larger parameter values (see Fig. 4). Inside the contour λ (α, ω) is negative, meaning that the state will approach the origin with probability 1. Along the contour and in the outside region large state fluctuations are possible. Interesting parameter values are expected near the curve where due to a coexistence of stable and unstable dynamics (induced by different sequences of random matrices) a theoretically optimal combination of exploration and exploitation is possible. For specific problems, however, deviations from the critical curve can be expected to be beneficial. 3.3 Personal best vs. global best Due to linearity, the particle swarm update rule (1) is subject to a scaling invariance which was already used in Eq. 7. We now consider the consequences of linearity for the case where personal best and global best differ, i.e. p 6= g. For an interval where pi and g remain unchanged, the particle i with personal best pi will behave like a particle in a swarm where together with x and v, pi is also scaled by a factor κ > 0. The finite-time approximation of the Lyapunov exponent (see Eq. 8) λ(t) = 1 log hk(xt , vt )ki t 5 (11) Figure 3: Solution of Eq. 10 representing a single particle in one dimension with a fixed best value at g = p = 0. The curve that has higher α-values on the right (magenta) is for α1 = α2 , the other curve (green) is for α = α2 , α1 = 0. Except for the regions near ω = ±1, where numerical instabilities can occur, a simulation produces an indistinguishable curve. In the simulation we tracked the probability of a particle to either reach a small region (10−6 ) near the origin or to escape beyond a radius of 106 after starting from a random location on the unit circle. Along the curve both probabilities are equal. will be changed by an amount of 1t log κ by the scaling. Although this has no effect on the asymptotic behaviour, we will have to expect an effect on the stability of the swarm for finite times which may be relevant for practical applications. For the same parameters, the swarm will be more stable if κ < 1 and less stable for κ > 1, provided that the initial conditions are scaled in the same way. Likewise, if kpk is increased, then the critical contour will move inwards, see Fig. 5. Note that in this figure, the low number of iterations lead to a few erroneous trials at parameter pairs outside the outer contour which have been omitted here. We also do not consider the behaviour near α = 0 which is complex but irrelevant for PSO. The contour (10) can be seen as the limit κ → 0 such that only an increase of kpk is relevant for comparison with the theoretical stability result. When comparing the stability results with numerical simulations for real optimisation problems, we will need to take into account the effects caused by differences between p and g in a multi-particle swarm with finite runtimes. 4 Optimisation of benchmark functions Metaheuristic algorithms are often tested in competition against benchmark functions designed to present different problem space characteristics. The 28 functions [3] contain a mix of unimodal, basic multimodal and composite functions. The domain of the functions in this test set are all defined to be [−100, 100]d where d is the dimensionality of the problem. Particles were initialised within the same domain. We use 10-dimensional problems throughout. Our implementation of PSO performed no spatial or velocity clamping. In all trials a swarm of 25 particles was used. We repeated the algorithm 100 times, on each occasion allowing 200, 2000, 20000 iterations to pass before recording the best solution found by the swarm. For the competition 50000 fitness evaluation were allowed which corresponds to 2000 iterations with 25 particles. Other iteration numbers were included for comparison. This protocol was carried out for pairs of ω ∈ [−1.1, 1.1] and α ∈ [0, 5] This was repeated for all 28 functions. The averaged solution costs as a function of the two parameters showed curved valleys similar to that in Fig. 1 for all problems. For each function we obtain different best values along (or near) the theoretical curve (10). There appears to be no preferable location within the valley. Some individual functions yield best performance near ω = 1. This is not the case near ω = 0, although the global average performance over all test functions is better in the valley near ω = 0 than near ω = 1, see Fig 4. 6 Figure 4: Best parameter regions for 200 (blue), 2000 (green), and 20000 (magenta) iterations: For more iterations the region shifts towards the critical line. Cost averaged over 100 runs and 28 CEC benchmark functions. The red (outer) curve represents the zero Lyapunov exponent for N = 1, d = 1, α1 = α2 . At medium values of ω the difference between the analytical solutions for the cases α1 = α2 and α1 = 0 is strongest, see Fig. 4. In simulations this shows to a lesser extent, thus revealing a shortcoming of the one-particle approximation. Because in the multi-particle case, p and g are often different, the resulting vector will have a smaller norm than in the one-particle case, where p = g. The case p 6= g violates a the assumption of the theory the dynamics can be described based unit vectors. While a particle far away from both p and g will behave as predicted from the one-particle case, at length scales smaller than kp − gk the retractive forces will tend to be reduced such that the inertia becomes more effective and the particle is locally less stable which shows numerically in optimal parameters that are smaller than predicted. 5 5.1 Discussion Relevance of criticality Our analytical approach predicts a locus of α and ω pairings that maintain the critical behaviour of the PSO swarm. Outside this line the swarm will diverge unless steps are taken to constrain it. Inside, the swarm will eventually converge to a single solution. In order to locate a solution within the search space, the swarm needs to converge at some point, so the line represents an upper bound on the exploration-exploitation mix that a swarm manifests. For parameters on the critical line, fluctuations are still arbitrary large. Therefore, subcritical parameter values can be preferable if the settling time is of the same order as the scheduled runtime of the algorithm. If, in addition, a typical length scale of the problem is known, then the finite standard deviation of the particles in the stable parameter region can be used to decide about the distance of the parameter values from the critical curve. These dynamical quantities can be approximately set, based on the theory presented here, such that a precise control of the behaviour of the algorithm is in principle possible. The observation of the distribution of empirically optimal parameter values along the critical curve, confirms the expectation that critical or near-critical behaviour is the main reason for success of the algorithm. Critical fluctuations are a plausible tool in search problem if apart from certain smoothness assumption nothing is known about the cost landscape: The majority of excursions will exploit the smoothness of the cost function by local search, whereas the fat tails of the distribution allow the particles to escape from local minima. 7 Figure 5: For p 6= g we define neutral stability as the equilibrium between divergence and convergence. Convergence means here that the particle approaches the line connecting p and g. Curves are for a one-dimensional problem with p = 0.1 and g = 0 scaled (see Sect. 3.3) by κ = 1 (outer curve) κ = 0.1 and κ = 0.04 (inner curve). Results are for 200 iterations and averaged over 100000 repetitions. 5.2 Switching dynamics at discovery of better solutions Eq. 2 shows that the discovery of a better solution affects only the constant terms of the linear dynamics of a particle, whereas its dynamical properties are governed by the linear coefficient matrices. However, in the time step after a particle has found a new solution the corresponding force term in the dynamics is zero (see Eq. 1) such that the particle dynamics slows down compared to the theoretical solution which assumes a finite distance from the best position at all (finite) times. As this affects usually only one particle at a time and because new discoveries tend to become rarer over time, this effect will be small in the asymptotic dynamics, although it could justify the empirical optimality of parameters in the unstable region for some test cases. The question is nevertheless, how often these changes occur. A weakly converging swarm can still produce good results if it often discovers better solutions by means of the fluctuations it performs before settling into the current best position. For cost functions that are not ‘deceptive’, i.e. where local optima tend to be near better optima, parameter values far inside the critical contour (see Fig. 3) may give good results, while in other cases more exploration is needed. 5.3 The role of personal best and global best A numerical scan of the (α1 , α2 ) plane shows a valley of good fitness values, which, at small fixed positive ω, is roughly linear and described by the relation α1 +α2 = const, i.e. only the joint parameter α = α1 + α2 matters. For large ω, and accordingly small predicted optimal α values, the valley is less straight. This may be because the effect of the known solutions is relatively weak, so the interaction of the two components becomes more important. In other words if the movement of the particles is mainly due to inertia, then the relation between the global and local best is non-trivial, while at low inertia the particles can adjust their p vectors quickly towards the g vector such that both terms become interchangeable. Finally, we should mention that more particles, longer runtime as well as lower search space dimension increase the potential for exploration. They all lead to the empirically determined optimal parameters being closer to the critical curve. 8 6 Conclusion PSO is a widely used optimisation scheme which is theoretically not well understood. Existing theory concentrates on a deterministic version of the algorithm which does not possess useful exploration capabilities. We have studied the algorithm by means of a product of random matrices which allows us to predict useful parameter ranges and may allow for more precise settings if a typical length scale of the problem is known. A weakness of the current approach is that it focuses on the standard PSO [1] which is known to include biases [12, 13], that are not necessarily justifiable, and to be outperformed on benchmark set and in practical applications by many of the existing PSO variants. Similar analyses are certainly possible and are expected to be carried out for some of the variants, even though the field of metaheuristic search is often portrayed as largely inert to theoretical advances. If the dynamics of particle swarms is better understood, the algorithms may become useful as efficient particle filters which have many applications beyond heuristic optimisation. Acknowledgments This work was supported by the Engineering and Physical Sciences Research Council (EPSRC), grant number EP/K503034/1. References [1] J. Kennedy and R. Eberhart. Particle swarm optimization. In Proceedings IEEE International Conference on Neural Networks, volume 4, pages 1942–1948. IEEE, 1995. [2] R. Poli. Analysis of the publications on the applications of particle swarm optimisation. Journal of Artificial Evolution and Applications, 2008(3):1–10, 2008. [3] CEC2013. http://www.ntu.edu.sg/home/EPNSugan/index files/CEC2013/CEC2013.htm. [4] J. Kennedy. The behavior of particles. In V.W. Porto, N. Saravanan, D.Waagen, and A. E. Eiben, editors, Evolutionary programming VII, pages 579–589. Springer, 1998. [5] M. Clerc and J. Kennedy. The particle swarm-explosion, stability, and convergence in a multidimensional complex space. IEEE Transactions on Evolutionary Computation, 6(1):58–73, 2002. [6] I. C. Trelea. The particle swarm optimization algorithm: convergence analysis and parameter selection. Information Processing Letters, 85(6):317–325, 2003. [7] M. Jiang, Y. Luo, and S. Yang. Stagnation analysis in particle swarm optimization. In Swarm Intelligence Symposium, 2007. SIS 2007. IEEE, pages 92–99. IEEE, 2007. [8] C. W. Cleghorn and A. P Engelbrecht. A generalized theoretical deterministic particle swarm model. Swarm Intelligence, 8(1):35–59, 2014. [9] H. Furstenberg and H. Kesten. Products of random matrices. Annals of Mathematical Statistics, 31(2):457–469, 1960. [10] V. N. Tutubalin. On limit theorems for the product of random matrices. Theory of Probability & Its Applications, 10(1):15–27, 1965. [11] R. Z. Khas’minskii. Necessary and sufficient conditions for the asymptotic stability of linear stochastic systems. Theory of Probability & Its Applications, 12(1):144–147, 1967. [12] M. Clerc. Confinements and biases in particle swarm optimisation. Technical Report hal00122799, Open archive HAL, 2006. [13] W. M. Spears, D. Green, and D. F. Spears. Biases in particle swarm optimization. International Journal of Swarm Intelligence Research, 1(2):34–57, 2010. 9	9
1 On the detection of low rank matrices in the high-dimensional regime. Antoine Chevreuil and Philippe Loubaton Gaspard Monge Computer Science Laboratory (LIGM) - UMR 8049 CNRS Université de Paris-Est/Marne-la-Vallée 5 Bd. Descartes 77454 Marne-la-Vallée (France) arXiv:1804.04851v1 [eess.SP] 13 Apr 2018 Abstract We address the detection of a low rank n × ndeterministic matrix X0 from the noisy observation X0 + Z when n → ∞, where Z is a complex Gaussian random matrix with independent identically distributed Nc (0, n1 ) entries. Thanks to large random matrix theory results, it is now well-known that if the largest singular value λ1 (X0 ) of X0 verifies λ1 (X0 ) > 1, then it is possible to exhibit consistent tests. In this contribution, we prove a contrario that under the condition λ1 (X0 ) < 1, there are no consistent tests. Our proof is inspired by previous works devoted to the case of rank 1 matrices X0 . Index Terms statistical detection tests, large random matrices, large deviation principle. I. I NTRODUCTION The problem of testing whether an observed n1 × n2 matrix Y is either a zero-mean independent identically distributed Gaussian random matrix Z with variance n12 , or X0 + Z for some low rank deterministic matrix X0 , with no known structure, called also a spike, is a fundamental problem arising in numerous applications such as the detection of low-rank multivariate signals or the Gaussian hidden clique problem. When the two dimensions n1 , n2 converge towards ∞ in such a way that n1 /n2 → c > 0 (the rank of X0 remaining fixed), known results on the so-called additive spiked large random matrix models have enabled to re-consider this fundamental detection problem (see e.g. [13], [4], [3]). It was established a long time ago (see e.g. [1] and the references therein) that in the above asymptotic regime, the largest singular value λ1 (Z) of Z converges almost √ surely towards 1 + c. More recently, under mild technical extra assumptions, [3] proved that λ1 (X0 + Z) still converges √ towards 1 + c if λ1 (X0 ) converges towards a limit strictly less than c1/4 . On the√contrary, if the limit of λ1 (X0 ) is strictly greater than c1/4 , then λ1 (X0 +Z) converges towards a limit strictly greater than 1+ c. This result implies that the Generalized Likelihood Ratio Test (GLRT) is consistent (i.e. both the probability of false alarm and the probability of missed detection converge towards 0 in the above asymptotic regime) if and only if λ1 (X0 ) is above the threshold c1/4 . In order to simplify the exposition, we assume from now on that n1 = n2 = n, so that ratio c reduces to 1. While the detection problem was extensively addressed in the zone λ1 (X0 ) > 1, the case where λ1 (X0 ) < 1 was much less studied. Montanari et al. [12] consider the zone λ1 (X0 ) < 1 when X0 is a rank 1 matrix. Thanks to simple information geometry tools, [12] prove that, in this region, it is impossible to find a consistent test for the detection of the spike. Irrespective of the standard random matrix tools, this approach is extended to the more general case when X0 and Z are tensors of order d ≥ 3; namely, if the Frobenius norm of the tensor X0 is stricly less than a threshold depending in d, then the probability distributions of the observation under the two hypotheses are asymptotically undistinguishable, so that any detection test cannot behave better than a random guess. This property, which is stronger than the non-existence of a consistent test, does not hold in the matrix case d = 2: see for instance [14] where a non-consistent test is exhibited that has a better performance than a random guess. In this paper, we extend the above methodolodgy to the general case where X0 has rank r. Our contribution is to prove that under λ1 (X0 ) < 1, the consistent detection is impossible. While this theoretical result is not unexpected, we believe that it provides a better understanding of the above fundamental detection problem in large dimensions without resorting to the machinery of large random matrices. We mention that the works [9] (when the spike is symmetric) and [10] (non-symmetric case) are clearly related to the above problem. However, two major differences arise: firstly, the detection is not addressed but rather the estimation of X0 ; second, a statistical model of the spike is needed. The results are in general not explicit. However for a certain prior and for the rank one model, it can be deduced that, in the zone λ1 (X0 ) < 1, it is impossible to find estimates of the spike that have better performance than any dummy estimate (i.e. an estimate that does not rely on the observation). The authors rely on the computation of the mutual information between X0 and Y: this computation involves non-obvious results extending the approach of Tallagrand for studying the Sherrington-Kirkpatrick model [15]. 2 II. M ODEL , NOTATION , ASUMPTION The set of complex-valued matrices Cn×npis a complex vector-space endowed with the standard scalar product hX, Yi = Tr(XY∗ ) and the Frobenius norm kXkF = hX, Xi. The spectral norm of a matrix X is denoted by kXk2 . The spike (“the signal”) is assumed to be a matrix of fixed rank r and hence admits a SVD such as X0 = r X λj uj vj∗ = UΛV∗ (1) j=1 where λi = λi (X0 ) are the singular values of X0 sorted in descending order and where Λ is the diagonal matrix gathering the (λj )j=1,...,r in the descending order. As X0 has to be defined for any n, we impose a non-erratic behavior of X0 , namely that all its singular values (λj )j=1,...,r do not depend on n for n large enough. This hypothesis could be replaced by the condition that (λj )j=1,...,r all converge towards a finite limit at an ad’hoc rate. However, this would introduce purely technical difficulties. The noise matrix Z is assumed to have i.i.d. entries distributed as Nc (0, 1/n). We consider the alternative H0 : Y = Z versus H1 : Y = X0 + Z. We denote by p1,n (y) the probability probability density of Y under H1 and p0,n (y) the density p1,n (Y) of Y under H0 . L(Y) = p0,n (Y) is the likelihood ratio and we denote by E0 the expectation under H0 . We now recall the fundamental information geometry results used in [12] in order to address the detection problem.The following properties are well known (see also [2] section 3): 2 is bounded, • (i) If E0 L(Y) then no consistent detection test exists. 2 • (ii) If moroever E0 L(Y) = 1 + o(1), the total variation distance between p0,n and p1,n converges towards 0, and no test performs better than a decision at random. We however mention that (i) and (ii) are only sufficient conditions. In particular, E0 L(Y)2 unbounded does not imply the existence of consistent tests. III. P RIOR ON THE SPIKE . E XPRESSION OF THE SECOND - ORDER MOMENT. 2 The density of Z, seen as a collection of n2 complex-valued random variables, is obviously p0,n (z) = κn exp −n kzkF n2 where κn = nπ . On the one hand, we notice that the study of the second-order momentof the likelihood ratio is not suited 2 to the deterministic model of the spike as presented previously. Indeed, in this case E L(Y) has the simple expression 0 2 exp 2n kX0 kF and always diverges. On the other hand, the noise matrix shows an invariance property: if Θ1 , Θ2 are unitary n × n matrices , then the density of Θ1 ZΘ2 equals this of Z. We hence modify the data according to the procedure: we pick two independent unitary Θ1 , Θ2 according to the Haar measure (which corresponds to the uniform distribution on the set of all unitary n × n matrices), and change the data tensor Y according to Θ1 YΘ2 . As said above, this does not affect the distribution of the noise, but this amounts to assume a certain prior on the spike. Indeed, this amounts to replace ui by Θ1 ui and vi by Θ∗2 vi . In the following, the data and the noise tensors after this procedure are still denoted respectively by Y and Z. We are now in position to give a closed-form expression of the second-order moment of L(Y) . We have p1,n (Y) = EX [p0,n (Y − X)] where EX is the mathematical expectation over the prior distribution of the spike, or equivalently over the Haar matrices Θ1 , Θ2 . It holds that E0 L(Y)2 = E [exp (2nR hX, X0 i)] where the expectation is over independent 0 0 0 0 copies X,2X of the spike (R stands for the real part); X and X being respectively associated with (Θ1 , Θ2 ) and (Θ1 , Θ2 ), E0 L(Y) has the expression h ∗ ∗ i E exp 2nRTr Θ1 X0 Θ2 Θ02 X∗0 Θ01 . ∗ ∗ As Θk and Θ0k are Haar and independent, then (Θ01 ) Θ1 and Θ2 (Θ02 ) are also independent, Haar distributed and it holds E0 L(Y)2 = E [exp (2nη)] , (2) where the expectation is over the independent Haar matrices Θ1 , Θ2 and η = RTr (Θ1 X0 Θ2 X∗0 ). The ultimate simplification comes from the decomposition (1) which implies that η = RTr (ΛΨ1 ΛΨ2 ) (3) where Ψ1 = U∗ Θ1 U and Ψ2 = V∗ Θ2 V. It is clear that Ψ1 and Ψ2 are independent matrices that are both distributed as the upper r × r diagonal block of a Haar unitary matrix. 3 IV. R ESULT The main result of our contribution is the following Theorem 1. If λ1 (X0 ) < 1 then lim sup E0 L(Y)2 ≤ 1 1 − λ1 (X0 )4 r2 and it is not possible to find a consistent test. We remind that we are looking for a condition on X0 (due to (2,3), this is a condition on Λ) under which E [exp (2nη)] is bounded. Evidently, the divergence may occur only when η > 0. We hence consider E1 = E [exp (2nη) Iη> ] and E2 = E [exp (2nη) Iη≤ ], and prove that, for a certain small enough > 0 to be specified later, E1 = o(1) and that E2 is bounded. V. T HE E1 TERM : COMPUTATION OF THE GRF OF η. It is clear that the boundedness of the integral E1 is achieved when η rarely deviates from 0. As remarked in [12], the natural machinery to consider is this of the Large Deviation Principle (LDP). In essence, if η follows the LDP with rate n, there can be found a certain non-negative function called Good Rate Function (GRF) Iη such that for any Borel set A of R, 1 n log P (η ∈ A) converges towards supx∈A −Iη (x). The existence of a GRF allows one to analyze the asymptotic behaviour of the integral E1 . In the next section, we thus justify that η follows a Large Deviation Principle with rate n, and we compute the associated GRF. A. Computation of the GRF of η Pr Eq. (3) and the Cauchy-Schwarz inequality imply that the random variable η is bounded: \|η\| ≤ ηmax with ηmax = j=1 λ2j . We first recall that for i = 1, 2, the random matrix Ψi follows a LDP with rate n and that its GRF at the parameter ψ ∈ Cr×r , kψk2 ≤ 1, is log det (Ir − ψ ∗ ψ) (see Theorem 3-6 in [8]). Besides, η is a function of the i.i.d. matrices (Ψi )i=1,2 and therefore, the contraction principle applies to η (see Theorem 4.2.1 in [7]): it ensures that η follows a LDP with rate n and its GRF is such that, for each real \|x\| ≤ ηmax , −Iη (x) is the solution of the following optimization problem: Problem 2. Maximize in Cr×r log det (I − ψ ∗1 ψ 1 ) + log det (I − ψ ∗2 ψ 2 ) . (4) under the constraints RTr (Λψ 1 Λψ 2 ) = x (5) kψ i k2 ≤ 1, i = 1, 2 (6) We provide a closed-form solution of Problem 2. In this respect, we define for each k = 1, . . . , r the interval Ik defined by ∀k = 1, ..., r − 1 : Ik =] k X X k+1 λ2i − λ2k , λ2i − λ2k+1 ] i=1 (7) i=1 Pr and Ir =] i=1 λ2i − λ2k , ηmax ]. It is easy to check that (Ik )k=1,...,r are disjoint and that ∪rk=1 Ik =]0, ηmax ]. The following result holds: Theorem 3. The maximum of Problem 2 is given by − Iη (x) = 2 r X k=1 " P log  k i=1 λ2i k − \|x\| #k  1  IIk (\|x\|) Πki=1 λ2i (8) It is easy to check that the function x 7→ −Iη (x) is continuous on ]0, ηmax [. The proof of Theorem 3 is provided in the Appendix. We illustrate Theorem 3 through the following experiment. The rank of the spike is fixed to r = 3 and the singular values have been set to (λ1 , λ2 , λ3 ) = (1 , 0.7 , 0.2). We have computed millions of random P2 samples of the matrices (ψ 1 , ψ 2 ). Each pair is associated with a point (x, y) defined as x = RTr (Λψ 1 Λψ 2 ) and y = i=1 log det (I − ψ ∗i ψ i ) . We obtain a cloud of points, the upper envelope of which is expected to be −Iη (x). We have also plotted the graph of the function y = −Iη (x). In addition, we mention that, in the more general context of tensors of order d, the second-order moment of L(Y) is still given by (2) but the random variable - call it ηd - has a more complicated form than (3), see [5]; the asymptotics of the term E1 can still be studied by evaluating the GRF of ηd . This GRF is the solution of an optimization problem that, apparently, cannot be solved in closed form for d ≥ 3. In [5], bound of the opposite of the true GRF is computed; this upper an upper \|x\| bound, valid for any d is given for d = 2 by log 1 − ηmax . We thus also represent in Figure V-A this upper bound; clearly, it is not tight. −6 −12 −10 −8 k ∑ log(det(I−ψkψHk ) −4 −2 0 4 −1.0 −0.5 0.0 Tr(Λψ1Λψ2) 0.5 1.0 Fig. 1. graph of −Iη (x) seen as an upper envelope. Upper curve: the upper bound computed in [5] B. Computation of E1 The Varadhan lemma (see Theorem 4.3.1 in [7]) states that n1 log E [exp (2nη) Iη> ] → supx> (2x − Iη (x)) and hence the E1 term converges towards 0 when supx> (2x − Iη (x)) < 0. Consider any of the intervals Ik defined in (7). The derivative 2 of 2x − Iη (x) for any x ∈ Ik is 2 − 2k/(λ + λ2k − x) : it is decreasing on Ik and the limit in the left extremity of Ik , 1 + ... Pk−1 2 1 2 i.e. ( j=1 λj ) − (k − 1)λk , is simply 2 1 − λ2 . If λ1 (X0 ) < 1, then for all the indices k, 1 − λ12 < 0. This shows that k k 2x + Iη (x) is strictly decreasing on every Ik . Hence, for every x ∈]0, ηmax ], we have 2x − Iη (x) < 0 − Iη (0) = 0. We have proved that E1 = o(1). VI. T HE E2 TERM : CONCENTRATION OF η. Notice that the upper block r × r Ψ of a unitary Haar matrix Θ has the same distribution as −1/2 G G̃∗ G̃ where the n×r matrix G̃ has i.i.d. entries distributed as NC (0, 1) and G is the top r ×r block of G̃. Obviously, E[G̃∗ G̃] = nI. It is a standard result that a random variable distributed as a χ2 (n) is concentrated around its mean. This can be easily extended to the matrix G̃∗ G̃: Lemma 4. For any 0 < δ < 1, there exists a constant c such that δ2 1 ∗ P G̃ G̃ − I > δ ≤ c exp −n . n 2 2 We take G̃1 and G̃2 independent, distributed as consider the upper r × r blocks G and G2 of G̃1 and G̃2 . It G̃ and −1/2 −1/2 1 ∗ ∗ follows that η has the same distribution as 2RTr ΛG1 G̃1 G̃1 ΛG2 G̃2 G̃2 . Take now any δ < 1. We may split the integral E2 in two parts: E exp (2nη) I{η≤}∩B1c ∩B2c + E exp (2nη) I{η≤}∩(B1 ∪B2 ) . {z } \| {z } \| E200 E20 where we have defined the events Bi = n 1 ∗ n G̃i G̃i E200 o −I 2 > δ . Thanks to the above concentration result, we have ≤ exp(2n) (P(B1 ) + P(B2 )) ≤ exp(2n)2c exp −nδ 2 /2 As it is always possible to choose δ and such that δ 2 − 4 > 0 and δ < 1 it follows that E200 = o(1). Let us now inspect the term E20 . Since we have, for i = 1, 2, n1 G̃∗i G̃i − I ≤ δ, then there exist ∆i for i = 1, 2 such 2 −1/2 that G̃∗i G̃i = √1n (I + ∆i )with k∆i k2 ≤ δ/2. We hence have E20 ≤ E [exp (2RTr (ΛG1 (I + ∆1 ) ΛG2 (I + ∆2 ))] . 5 We expand 2RTr (ΛG1 (I + ∆1 ) ΛG2 (I + ∆2 )) as the sum of four terms. Take for instance T2 = 2RTr (ΛG1 ∆1 ΛG2 ) Thanks to von Neumann’s lemma [11], we have T2 ≤ 2 r X λk (∆1 )λk (ΛG2 ΛG1 ) k=1 ≤ 2 k∆1 k r X λk (ΛG2 ΛG1 ) k=1 As Pr k=1 λk (ΛG2 ΛG1 ) ≤ √ pPr 2 r k=1 λk (ΛG2 ΛG1 ), it yields √ q T2 ≤ 2 k∆1 k r Tr (ΛG2 ΛG1 G∗1 ΛG∗2 Λ). Invoking the von Neumann’s lemma three times, it holds that q √ T2 ≤ 2 k∆1 k r Λ2 Tr (G1 G∗1 ) Tr (G2 G∗2 ) √ ≤ r k∆1 k Λ2 (Tr (G1 G∗1 ) + Tr (G2 G∗2 )) Similar manipulations can be done on the other terms of the expansion. so that E20 is less than E [exp (2RTr (ΛG1 ΛG2 ) + βTr ((G1 G∗1 ) + Tr (G2 G∗2 )))] √ 2 with β = 2r δ(2 + δ) kΛk . The above expectation is to be understood as the expectation over (G1 , G2 ). As G1 and G2 are independent, we consider first the expectation over G1 . This gives, up to the factor exp (βTr (G2 G∗2 )) Z −r 2 π exp (2RTr (g1 E) + (β − 1) Tr (g1∗ g1 ) ) dg1 with E = ΛG2 Λ. It is always possible to choose δ such that β < 1. With such a β, the above integral is ! 2 2 1 −r 2 √ Tr (EE∗ ) (1 − β) exp 4 1−β 4 As Tr (EE∗ ) ≤ kΛk2 Tr (G2 G∗2 ) we finally obtain after multiplying by exp (βTr (G2 G∗2 )) and taking the expectation over G2 : ! 4 −r 2 Z 2 (1 − β) − kΛk (1 − β) 2 E20 ≤ exp − Tr (g2∗ g2 ) dg2 . 1−β π r2 2 If kΛk2 < 1, it is always possible to adjust δ such that the above integral converges. In this condition, we have !r2 1 E20 ≤ . 4 (1 − β)2 − kΛk2 This must be true for all β arbitrarily small, hence the result. A PPENDIX We prove Theorem 3 when x > 0. As the function to be maximized converges towards −∞ if kψ 1 k → 1 or kψ 2 k → 1, any argument (ψ 1 , ψ 2 ) of the maximization problem satisfies kψ i k < 1, i = 1, 2. Therefore, the Karush-Kuhn-Tucker (KKT) conditions imply the existence P2 of a scalar Lagrange multiplier µ ≥ 0 such that (ψ 1 , ψ 2 ) is a stationary point of the Lagrangian `(ψ 1 , ψ 2 , µ) defined by i=1 log det (Ir − ψ ∗i ψ i ) + µ RTr (Λψ 1 Λψ 2 ) . As ` is a real valued function, a stationary point is computedwhen setting the differential w.r.t. the entries of ψ 1 and ψ 2 to zero. It can be checked that (ψ 1 , ψ 2 ) is a stationary point of ` when µ Λψ 2 Λ = ψ ∗1 (I − ψ 1 ψ ∗1 )−1 µ Λψ 1 Λ = ψ ∗2 (I − ψ 2 ψ ∗2 )−1 In a first step, these equations can be shown to be satisfied only if ψ 1 and ψ 2 are diagonal up to permutations of the columns. Then, is can be deduced that there exists a diagonal matrix 0 ≤ P ≤ I and a matrix of permutation Π such that 6 log det (Ir − ψ ∗1 ψ 1 ) + log det (Ir − ψ ∗2 ψ 2 ) = 2 log det(I − P)and RTr (Λψ 1 Λψ 2 ) = Tr(ΛΠ∗ ΛΠ P). This invites us to consider the following Problem 5. Maximize log det(I − P) (9) jointly over all the r! permutations Π and over diagonal matrices P verifying 0 ≤ P ≤ I and the constraint Tr(ΛΠ∗ ΛΠ P) = x. (10) In a first step, we set Π = I in the above problem and consider the Problem 6. Maximize r X log(1 − pi ) (11) under the constraints that 0 ≤ pi ≤ 1 for each i = 1, . . . , r and r X λ2i pi = x. (12) i=1 i=1 The maximum is denoted by JΛ (x). This is a variant of the celebrated water-filling problem (see e.g. [16] and Chap. 9 of [6]) that was solved to evaluate the capacity of a frequency selective Gaussian channel, the difference being that in the latter problem, log(1 − pi ) is replaced by log(1 + pi ).In order to solve Problem 6, we assume that the non zero singular values (λi )i=1,...,r are distinct. If this is not the case, a standard perturbation argument can be used in order to address the general case. As the function to be maximized is strictly concave on the set defined by the constraints, the maximum is reached p∗ verifying pi,∗ Pr at a unique point P Pr< 1 for r 2 each i. We consider the Lagrangian corresponding to Problem (6) given by i=1 log(1 − pi ) + µ λ p + i=1 i i i=1 δi pi where µ ≥ 0 and δi ≥ 0 for i = 1, . . . , r. The partial derivatives w.r.t. parameters (pi )i=1,...,r are zero at p∗ . This leads to for i = 1, . . . , r : 1 = µ∗ λ2i + δi,∗ 1 − pi,∗ (13) The first remark is that necessarily, these equations imply that the numbers pi,∗ are sorted in decreasing order. To verify this claim, we assume that i < j and that pi,∗ = 0 and pj,∗ > 0. Then, it holds that µ∗ λ2i + δi,∗ = 1 and that µ∗ λ2j = 1−p1 j,∗ > 1 because pj,∗ > 0 implies δj,∗ = 0. Therefore, λ2i ≤ µ1∗ < λ2j , a contradiction because λ2i ≥ λ2j . We denote by s(x) the number of non-zero entries of p∗ . Hence, the first s(x) entries of p∗ are non zero. Morever, the equations µ∗ λ2i = 1−p1 i,∗ for i = 1, . . . , s(x) imply that p1,∗ ≥ . . . ≥ ps(x),∗ > 0 = ps(x)+1,∗ = . . . = pr,∗ . We now analytically characterize s(x). On the one hand, (13) computed at for i = s(x) and for i = s(x) + 1 both imply 1 λ2s(x)+1 ≤ < λ2s(x) (14) µ∗ On the other hand, the constraint (12) imposes that 1/µ∗ verifies Ps(x) 2 λ −x 1 = i=1 i . µ∗ s(x) Therefore, it holds that s(x) ( X s(x) λ2i ) − s(x)λ2s(x) < x ≤ ( i=1 X λ2i ) − s(x)λ2s(x)+1 (15) i=1 such that s(x) coincides with the integer k for which x ∈ Ik (see (7) for the definition of these intervals). The maximum Ps(x) i=1 log(1 − pi,∗ ) is direcly computed as "  #s(x) Ps(x) 2 λ − x 1 i i=1  JΛ (x) = log  (16) s(x) 2 s(x) Π λ i=1 i In order to show that the GRF of η is Iη (x) = −2JΛ (x), it remains to show that the solution of Problem 5 is reached when the permutation matrix Π is the identity. In this respect, we introduce a nested problem motivated by the following observation. We denote by α and β the r–dimensional vectors whose components are respectively the diagonal entries of Λ2 and of ΛΠ∗ ΛΠ arranged in the decreasing order. Evidently, α majorizes β in the sense that for k = 1, . . . , r : k X i=1 αi ≥ k X i=1 βi (17) 7 We thus consider the relaxed problem Problem 7. Maximize log det(I − P) over the diagonal matrices 0 ≤ P ≤ I and over vectors β = (β1 , ..., βr ) satisfying β1 ≥ β2 ≥ . . . βr ≥ 0, the majorization constraint (17), and the equality constraint r X βi pi = x (18) i=1 The maximum of Problem 7 is above the maximum of Problem 5 which is itself above the maximum JΛ (x) of Problem 6. We actually show that the maximum of Problem 7 is less than JΛ (x), and that it is reached for a vector β that coincides with α. This will imply that the optimal permutation Π in Problem 5 is I and Iη (x) = −2JΛ (x). We give some elements for solving Problem 7. We consider a stationary point (p∗ , β ∗ ) of the associated Lagrangian and compute the KKT conditions. We suppose that this stationary point attains the maximum. If s denotes the number of non-zero components in p∗ , we prove that, necessarily, p1,∗ ≥ p2,∗ ≥ ... ≥ ps,∗ > 0 and β1,∗ ≥ β2,∗ ≥ ... ≥ βs,∗ . We let j1 be the Pj1 Pj1 βi (this index exists otherwise β ∗ = α and the problem is solved). This implies that αi > i=1 first index such that i=1 Pj1 +k Pj+k βi,∗ = αi for all indices i = 1, ..., j1 − 1. Notice this fact: if we suppose that the condition i=1 αi > i=1 βi is true whatever k, then it is possible to add a small > 0, and update βj1,∗ as βj1 ,∗ + in such a way that the majorization constraints still hold, the constraint (18) holds and the updated p∗ increases the function to maximize. This isPin contradiction with the Pj1 +j j1 +j2 2 βi,∗ . αi = i=1 definition of (p∗ , β ∗ ). This means that there exists an index j2 (we choose the smallest) such that i=1 It can be shown that it is necessary that all the βi,∗ are equal for i = j1 , ..., j1 + j2 . After some algebraic gymnastics, it can be shown that it in this case, all the inequalities (17) at β ∗ are saturated hence implying that β ∗ = α. The value of P i log(1 − pi,∗ ) equals JΛ (x). R EFERENCES [1] Z. Bai and J.W. Silverstein. Spectral Analysis of Large Dimensional Random Matrices. Springer-Verlag Series in Statistics, 2010. [2] J. Banks, C. Moore, R. Vershynin, N. Verzelen, and J. Xu. Information-theoretic bounds and phase transitions in clustering, sparse PCA, and submatrix localization. In 2017 IEEE International Symposium on Information Theory (ISIT), pages 1137–1141, June 2017. [3] Florent Benaych-Georges and Raj Rao Nadakuditi. The singular values and vectors of low rank perturbations of large rectangular random matrices. Journal of Multivariate Analysis, 111:120–135, 2012. [4] P. Bianchi, M. Debbah, M. Maı̈da, and M. Najim. Performance of statistical tests for single source detection using random matrix theory. 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Recurrent Neural Network Language Models for Open Vocabulary Event-Level Cyber Anomaly Detection Aaron Tuor,1 Ryan Baerwolf,2 Nicolas Knowles,2 Brian Hutchinson,1,2 Nicole Nichols1 and Robert Jasper1 1 Pacific Northwest National Laboratory Richland, Washington 2 Western Washington University Bellingham, Washington Abstract Automated analysis methods are crucial aids for monitoring and defending a network to protect the sensitive or confidential data it hosts. This work introduces a flexible, powerful, and unsupervised approach to detecting anomalous behavior in computer and network logs; one that largely eliminates domain-dependent feature engineering employed by existing methods. By treating system logs as threads of interleaved “sentences” (event log lines) to train online unsupervised neural network language models, our approach provides an adaptive model of normal network behavior. We compare the effectiveness of both standard and bidirectional recurrent neural network language models at detecting malicious activity within network log data. Extending these models, we introduce a tiered recurrent architecture, which provides context by modeling sequences of users’ actions over time. Compared to Isolation Forest and Principal Components Analysis, two popular anomaly detection algorithms, we observe superior performance on the Los Alamos National Laboratory Cyber Security dataset. For log-line-level red team detection, our best performing character-based model provides test set area under the receiver operator characteristic curve of 0.98, demonstrating the strong fine-grained anomaly detection performance of this approach on open vocabulary logging sources. 1 Introduction To minimize cyber security risks, it is essential that organizations be able to rapidly detect and mitigate malicious activity on their computer networks. These threats can originate from a variety of sources including malware, phishing, port scanning, etc. Attacks can lead to unauthorized network access to perpetrate further damage such as theft of credentials, intellectual property, and other business sensitive information. In a typical scenario, cyber defenders and network administrators are tasked with sifting through vast amounts of data from various logging sources to assess potential security risks. Unfortunately, the amount of data for even a modestly-sized network can quickly grow beyond the ability of a single person or team to assess, leading to delayed response. The desire for automated assistance has and continues to encourage inter-domain research in cyber security and machine learning. Signature-based approaches for automated detection can be highly effective for characterizing individual threats. De- spite their high precision, they suffer from low recall and may fail to detect subtle mutations or novel attacks. Alternatively, given an unlabeled training set of typically benign activity logs, one can build a model of “normal behavior”. During online joint training and evaluation of this model, patterns of normal usage will be reinforced and atypical malicious activity will stand out as anomalous. The features used to identify unusual behavior are typically statistical feature vectors associated with time slices, e.g., vectors of counts for types of activities taking place in a 24-hour window. Such systems developed in research have been criticized as brittle to differences in site-specific properties of real-world operational networks such as security constraints and variable usage patterns (Sommer and Paxson 2010). The approach we introduce aims to minimize site-specific assumptions implicit in feature engineering, and effectively model variability in network usage by direct online learning of language models over log lines. Language models assign probabilities to sequences of tokens and are a core component of speech recognition, machine translation, and other language processing systems. Specifically, we explore the effectiveness of several recurrent neural network (RNN) language models for use in a network anomaly detection system. Our system dynamically updates the network language model each day based on the previous day’s events. When the language model assigns a low probability to a log-line it is flagged as anomalous. There are several advantages to this approach: 1. Reduced feature engineering: Our model acts directly on raw string tokens, rather than hand-designed domainspecific statistics. This dramatically reduces the time to deployment, and makes it agnostic to the specific network or logging source configuration. It also removes the “blind spots” introduced when tens of thousands of log-lines are distilled down to a single aggregated feature vector, allowing our model to capture patterns that would have otherwise been lost. 2. Fine grained assessment: The response time for analysts can be improved by providing more specific and relevant events of interest. Baseline systems that alert to a user’s day aggregate require sifting through tens of thousands of actions. Our approach can provide log-line-level or even token-level scores to the analyst, helping them quickly lo- cate the suspicious activity. 3. Real time processing: With the ability to process events in real time and fixed bounds on memory usage which do not grow over time, our approach is suitable for the common scenario in which log-line events are appearing in a high-volume, high-velocity log stream. We assess our models using the publicly available Los Alamos National Laboratory (LANL) Cyber Security Dataset, which contains real (de-identified) data with ground truth red team attacks, and demonstrate language models definitively outperforming standard unsupervised anomaly detection approaches. 2 Prior work Machine learning has been widely explored for network anomaly detection, with techniques such as isolation forest (Gavai et al. 2015; Liu, Ting, and Zhou 2008) and principal component analysis (Novakov et al. 2013; Ringberg et al. 2007) attracting significant interest. Machine learning classifiers ranging from decision trees to Naı̈ve Bayes have been used for cyber security tasks such as malware detection, network intrusion, and insider threat detection. Extensive discussion of machine learning applications in cyber security is presented in (Bhattacharyya and Kalita 2013; Buczak and Guven 2016; Dua and Du 2016; Kumar, Kumar, and Sachdeva 2010; Zuech, Khoshgoftaar, and Wald 2015; Rubin-Delanchy, Lawson, and Heard 2016). Deep learning approaches are also gaining adoption for specialized cyber defense tasks. In an early use of recurrent neural networks, Debar, Becker, and Siboni (1992) model sequences of Unix shell commands for network intrusion detection. Anomaly detection has been demonstrated using deep belief networks on the KDD Cup 1999 dataset (Alrawashdeh and Purdy 2016), and Bivens et al. (2002) use multi-layer perceptrons for the DARPA 1999 dataset. Both approaches use aggregated features and synthetic network data. Tuor et al. (2017) and Veeramachaneni et al. (2016) both employ deep neural network autoencoders for unsupervised network anomaly detection using time aggregated statistics as features. Some works of note have been previously published on the LANL data. Turcotte, Heard and Kent (2016) develop an online statistical model for anomaly detection in network activity using Multinomial-Dirichlet models. Similarly, Turcotte et al. (2016) use Poission Factorization (Gopalan, Hofman, and Blei 2013) on the LANL authentication logs. A user/computer authentication count matrix is constructed by assuming each count comes from a Poisson distribution parameterized by latent factors for users and computers. The learned distributions are then used to predict unlikely authentication behavior. Several variants of tiered recurrent networks have been explored in the machine learning and natural language processing communities (Koutnik et al. 2014; Ling et al. 2015b; Ling et al. 2015a; Chung et al. 2015). They are often realized by a lower tier pre-processing network, whose output is fed to an upper tier network and the separate tiers are jointly trained. Ling et al. (2015b) use a character-level convolutional neural network to feed a word level long short-term memory (LSTM) RNN for machine translation, with predictions made at the word-level. Both Hwang and Sung (2016) and Ling et al. (2015a) use a character-based LSTM to feed a second word or utterance-based LSTM for language modeling. Pascanu et al. (2015) create activity models from real world data on a per-event (command) basis and sequences of system calls are then modeled using RNN and echo state networks. The learned features are used to independently train neural network and logistic regression classifiers. Max pooling is applied to hidden layers of the unsupervised RNN for each time step in a session and the result is concatenated to the final hidden state to produce feature vectors for the classifier. This is similar to our tiered approach, in which we use the average of all hidden states concatenated with the final hidden state as input to the upper-tier RNN. In contrast, our model is completely unsupervised and all components are jointly trained. 3 Approach Our approach learns normal behavior for users, processing a stream of computer and network log-lines as follows: 1. Initialize model weights randomly 2. For each day k in chronological order: (a) Given model Mk−1 , produce log-line-level anomaly scores for all events in day k (b) Optionally, produce an aggregated anomaly score each user for day k (from the log-line-level scores) (c) Send per-user-day or per-user-event anomaly scores in rank order to analysts for inspection (d) Update model weights to minimize loss on all log-lines in day k, yielding model Mk This methodology interleaves detection and training in an online fashion. In this section we detail the components of our approach. 3.1 Log-Line Tokenization To work directly from arbitrary log formats, we treat loglines as sequences of tokens. For this work, we consider two tokenization granularities: word-level and character-level. For word tokenization, we assume that tokens in the logline are delimited by a known character (e.g., space or comma). After splitting the log-lines on this delimiter, we define a shared vocabulary of “words” over all log fields, consisting of the sufficiently-frequent tokens appearing in the training set. To allow our model to handle previously unseen tokens, we add an “out of vocbulary” token to our vocabulary, <oov>. (For instance, not every IP address will be represented in a training set; likewise, new PCs and users are continually being added to large networks.) To ensure that <oov> has non-zero probability, we replace sufficiently infrequent tokens in the training data with <oov>. During evaluation, tokens not seen before are labeled <oov>. In order to accommodate shifting word distributions in an online environment, a fixed size vocabulary could be periodically updated using a sliding window of word frequency statistics. For simplicity, we assume we have a fixed training set from which we produce a fixed vocabulary. To avoid the challenges of managing a word-level vocabulary, we also develop language models using a characterlevel tokenization. In this case our primitive vocabulary, the alphabet of printable ASCII characters, circumvents the open vocabulary issue by its ability to represent any log entry irrespective of the network, logging source, or log field. With character-level tokenization, we keep the delimiter token in the sequence, to provide our models with cues to transitions between log-line fields. 3.2 Recurrent Neural Network Language Models To produce log-line-level anomaly scores, we use recurrent neural networks in two ways: 1) as a language model over individual log-lines, and 2) to model the state of a user over time. We first present two recurrent models that focus only on (1), and then a tiered model that accomplishes both (1) and (2). Both were implemented1 for our experiments using TensorFlow (Abadi et al. 2015). Event Model (EM). First we consider a simple RNN model that operates on the token (e.g., word) sequences of individual log-lines (events). Specifically, we consider a Long Short-Term Memory (LSTM) (Hochreiter and Schmidhuber 1997) network whose inputs are token embeddings and from whose output we predict distributions over the next token. For a log-line with K tokens, each drawn from a shared vocabulary of size C, let X(1:K) = x(1) , x(2) , . . . , x(K) denote a sequence of one-hot representations of the tokens (each x(t) ∈ RC ). In this model, the hidden representation at token t, h(t) , from which we make our predictions, is a function of x(1) , x(2) , . . . , x(t) according to the usual LSTM equations: h(t) c(t) = = o(t) ◦ tanh(c(t) ) f(t) ◦ c(t−1) + i(t) ◦ g(t) g(t) = f(t) = i(t) = o(t) = tanh x(t) W(g,x) + h(t−1) W(g,h) + b(g) σ x(t) W(f,x) + h(t−1) W(f,h) + b(f ) σ x(t) W(i,x) + h(t−1) W(i,h) + b(i) σ x(t) W(o,x) + h(t−1) W(o,h) + b(o) , (1) (2) (3) (4) (5) (6) where the initial hidden and cell states, c(0) and h(0) , are set to zero vectors, and ◦ and σ denote element-wise multiplication and logistic sigmoid, respectively. Vector g(t) is a hidden representation based on the current input and previous hidden state, while vectors f(t) , i(t) , and o(t) , are the standard LSTM gates. The matrices (W) and bias vectors (b) are the model parameters. We use each h(t−1) to produce a probability distribution p(t) over the token at time t, as follows: p(t) = softmax h(t−1) W(p) + b(p) (7) 1 Code will soon be available at https://github.com/pnnl/safekit p(1) p(2) p(K-1) p(K) softmax softmax softmax softmax LSTM LSTM LSTM LSTM <sos> x(1) x(K-2) x(K-1) LSTM LSTM LSTM LSTM x(2) x(3) x(K) <eos> … Figure 1: Event Models. Set of black bordered nodes and connections illustrate the EM model while set of all nodes and connections illustrate the BEM model. We use cross-entropy loss, K 1 X H(x(t) , p(t) ), K t=1 (8) for two important purposes: first, as per-log-line anomaly score and second, as the training objective to update model weights. We train this model using stochastic mini-batch (non-truncated) back-propagation through time. Bidirectional Event Model (BEM). Following the language model formulation suggested in (Schuster and Paliwal 1997), we alternatively model the structure of log lines with a bidirectional LSTM. We define a new set of hidden vectors hb(K+1) , hb(K) , . . . , hb(1) by running the LSTM equations backwards in time (starting with initial zero cell and hidden states at time K + 1 set to zero). The weights W and biases b for the backward LSTM are denoted with superscript b. The probability distribution p(t) over the token at time t is then: b p(t) = softmax h(t−1) W(p) + hb(t+1) W(p) + b(p) (9) Tiered Event Models (T-EM, T-BEM). To incorporate inter-log-line context, we propose a two-tiered recurrent neural network. The lower-tier can be either event model (EM or BEM), but with the additional input of a context vector (generated by the upper-tier) concatenated to the token embedding at each time step. The input to the upper-tier model is the hidden states of the lower-tier model. This upper tier models the dynamics of user behavior over time, producing the context vectors provided to the lower-tier RNN. This model is illustrated in Fig. 2. In this model, x(u,j) denotes user u’s jth log line, which consists of a sequence of tokens as described in the previous subsections. The upper-tier models a sequence of user log lines, x(u,1) , x(u,2) , . . . , x(u,Tu ) , using an LSTM. For each user u and each log line j in the user’s log line sequence, a lower-tier LSTM is applied to the tokens of x(u,j) . The input to the upper-tier model at log-line j is the concatenation of: 1) the final lower-tier hidden state(s) and 2) the average of the lower-tier hidden states. In the case of a lower-tier EM, Context LSTM Mean LSTM LSTM Final LSTM … <sos> Context LSTM <eos> Mean LSTM <sos> LSTM Final LSTM … <eos> Figure 2: Tiered Event Model (T-EM) (1) refers to the hidden state at time K; for the BEM, (1) is the concatenation of the forward hidden state at time K and the backward hidden state at time 1. For (2), we average over hidden states primarily to provide many short-cut connections in the LSTM, which aids trainability. The output of the upper-tier LSTM at log-line j is a hidden state ĥ(u,j) . This hidden vector serves to provide context for the lower-tier model at the next time step: specifically, ĥ(u,j−1) is concatenated to each of the inputs of the lower-tier model operating on the jth log-line. Note that the upper-tier model serves only to propagate context information across individual log-lines; no loss is computed directly on the values produced by the upper-tier model. The upper- and lower-tier models are trained jointly to minimize the cross-entropy loss of the lower-tier model. We unroll the two-tier model for a fixed number of log-lines, fully unrolling each of the lower-tier models within that window. The lower-tier model’s cross-entropy loss is also used to detect anomalous behavior, as is described further in Section 4.2. Minibatching becomes more challenging for the tiered model, as the number of log-lines per day can vary dramatically between users. This poses two problems: first, it introduces the possibility that the most active users may have a disproportionate impact on model weights; second, it means that toward the end of the day, there may not be enough users to fill the minibatch. To counteract the first problem, we fix the number of log-lines per user per day that the model will train on. The remaining log-lines are not used in any gradient updates. We leave compensating for the inefficiency that results from the second to future work. 3.3 Baselines Anomaly detection in streaming network logs often relies upon computing statistics over windows of time and applying anomaly detection techniques to those vectors. Below we describe the aggregate features and two anomaly detection techniques that are typical of prior work. Aggregate Features We first define the set of per-userday features, which summarize users’ activities in the day. To aggregate the features that have a small number of distinct values (e.g. success/failure, logon orientation) we count the number of occurrences for each distinct value for the user-day. For fields that have a larger number of distinct values (pcs, users, domains), we count the number of common and uncommon events that occurred, rather than the number of occurrences of each distinct value (this approach avoids high dimensional sparse features). Furthermore, we define two categories of common/uncommon; to the individual entity/user, and relative to all users. A value is defined as uncommon for the user if it accounts for fewer than 5% of the values observed in that field (up to that point in time), and common otherwise. A value is defined as uncommon for all users if it occurs fewer times than the average value for the field, and common otherwise. For the LANL dataset, the prior featurization strategy yields a 108-dimensional aggregate feature vector per userday. These feature vectors then serve as the input to the baseline models described next. Models We consider two baseline models. The first uses Principal Components Analysis (pca) to learn a low dimensional representation of the aggregate features; the anomaly score is proportional to the reconstruction error after mapping the compressed representation back into the original dimension (Shyu et al. 2003). The second is an isolation forest (iso) based approach (Liu, Ting, and Zhou 2008) as implemented in scikit-learn’s outlier detection tools (Pedregosa et al. 2011). This was noted as the best performing anomaly detection algorithm in the recent DARPA insider threat detection program, (Gavai et al. 2015). 4 Experiments In this section we describe experiments to evaluate the effectiveness of the proposed event modeling algorithms. 4.1 Data The Los Alamos National Laboratory (LANL) Cyber Security Dataset (Kent 2016) consists of event logs from LANL’s internal computer network collected over a period of 58 consecutive days. The data set contains over one billion loglines from authentication, process, network flow, and DNS logging sources. Identifying fields (e.g., users, computers, and processes) have been anonymized. The recorded network activities included both normal operational network activity as well as a series of red team activities that compromised account credentials over the first 30 days of data. Information about known red team attack events is used only for evaluation; our approach is strictly unsupervised. For the experiments presented in this paper, we rely only on the authentication event logs, whose fields and statistics are summarized in Figure 3a. We filter these events to only those log-lines linked to an actual user, removing computercomputer interaction events. Events on weekends and holidays contain drastically different frequencies and distributions of activities. In a real deployment a separate model would be trained for use on those days, but because no malicious events were in that data it was also withheld. Table 3b has statistics of our data split; the first 12 days serve as the development set, while the remaining 18 days are the independent test set. 4.2 Assessment Granularity Our model learns normal behavior and assigns relatively high loss to events that are unexpected. A principal advantage of our approach is this ability to score the anomaly of Field time source user dest. user source pc dest. pc auth. type logon type auth. orient success Example 1 C625@DOM1 U147@DOM1 C625 C625 Negotiate Batch LogOn Success # unique labels 5011198 80553 98563 16230 15895 29 10 7 2 Model pca iso EM BEM T-EM T-BEM EM BEM T-EM T-BEM (a) Days # Events # Attacks # User-days Dev 1-12 133M 316 57 Test 13-58 918M 385 79 W W W W C C C C Figure 3: Dataset statistics: (a) Authentication log fields and statistics and (b) dataset splits. 4.3 Metrics We consider two performance metrics. First, we assess results using the standard area under the receiver operator characteristic curve (AUC) which characterizes the trade-off in model detection performance between true positives and false positives, effectively sweeping through all possible analyst budgets. False positives are detections that are not truly red team events, while true positives are detections that are. To quantify the proportion of the data the analyst must sift through to diagnose malicious behavior on the network, we use the Average Percentile (AP) metric. Specifically, for each red team event or user-day, we note the percentile of its anomaly amongst all anomaly scores for the day. We then average these percentiles for all of the malicious events or AUC 0.754 0.763 0.802 0.876 0.782 0.864 0.750 0.843 0.772 0.837 AP 73.9 75.0 79.3 87.0 77.5 85.7 70.9 82.9 76.2 82.9 Table 1: User-day granularity test set AUC and AP. Language model anomaly scores calculated with average userday normalization (diff). (b) individual events, allowing us to flag at the event-level or aggregate anomalies over any larger timescale. For this work, we consider two timescales. First, we assess based on individual events; a list of events would be presented to the analyst, sorted descending by anomaly score. Second, to facilitate comparison with traditional aggregation methods, we aggregate anomaly scores over all of a user’s events for the day (specifically, taking the max), producing a single anomaly score per-user, per-day. In this scenario, a list of user-days would be provided to the analyst, sorted descending by anomaly score. We refer to this approach as max, because the anomaly scores provided to the analyst are produced by taking the maximum score over the event scores in the window for that user (where event-level scoring is just taking the max over a singleton set of one event). In order to counter systematic offsets in users’ anomaly scores for a day we also consider a simple normalization strategy, which we refer to as diff, by which every raw score is first normalized by subtracting the user’s average event-level anomaly score for the day. Tokenization Word Word Word Word Char Char Char Char EM BEM T-EM T-BEM EM BEM T-EM T-BEM LOG diff max 0.964 0.932 0.974 0.895 0.959 0.948 0.959 0.902 0.940 0.935 0.973 0.979 0.859 0.927 0.945 0.969 DAY diff max 0.802 0.794 0.876 0.811 0.782 0.803 0.864 0.838 0.751 0.754 0.843 0.846 0.772 0.809 0.837 0.854 Table 2: Comparison of AUC for day-level and log-line-level analysis with and without user-day normalization. Figures 4 and 5 provide a visualization of these results. user-days. Note that if all true malicious events or user-days are flagged as the most anomalous on the respective days, then AP ≈ 100, while if all malicious events or user-days are ranked as the least anomalous on their respective days, AP ≈ 0. For both AUC and AP, a higher score is better. Our model hyperparameters were manually tuned to maximize AP for day-level diff scores on the development set. No separate training set is needed as our approach is unsupervised and trained online. 4.4 Results and Analysis We begin by exploring the user-day granularity performance. Table 1 summarizes model detection performance at this granularity on the test set for the AUC and AP metrics using the diff method to produce day level scores from the language models. A few trends are evident from these results. First, the aggregate feature baselines have nearequivalent performance by both metrics, with the isolation forest approach having a slight edge. We hypothesize the feature representation, which is common to these methods, could be a bottleneck in performance. This highlights the “blind spot” issue feature engineering introduces. Second, despite having only the context of a single log-line at a time, as opposed to features aggregated over an entire day, the event model (EM) performs comparably to the baseline models when a forward pass LSTM network is used with 100 8079 97 8987 81 9594 7880 40 9393 9797 8484 80 60 20 100 95 90 86 83 AUC AUC 80 96 93 7575 92 85 80 77 96 94 8385 60 40 20 diff max LOG \| DAY W EM LOG \| DAY W BEM LOG \| DAY W T-EM LOG \| DAY W T-BEM diff max LOG \| DAY W EM LOG \| DAY W BEM LOG \| DAY W T-EM LOG \| DAY W T-BEM Figure 4: Word model comparison of AUC at day-level and log-line-level granularities. Figure 5: Character model comparison of AUC at day-level and log-line-level granularities. a character tokenization and outperforms the baselines with word tokenization. The most pronounced performance gain results from using bidirectional models. Finally, word-level tokenization performs better than character-level; however, even the bidirectional character models perform appreciably better than the baselines. It is clear from these results that the tiered models perform comparably to, but not better than, the event-level models. This suggests that the event level model is able to characterize normal user behavior from information stored in the model weights of the network, which are trained each day to model user activity. Given the context of the past day’s activity stored in the model weights, the categorical variables represented by the fields in an individual log line may eliminate the need for explicit event context modeling. We leave tracking the state of individual computers, rather than users, to future work, but hypothesize that it may make the tiered approach more effective. Next, we broaden our analysis of language modeling approaches, comparing performance across all language models, tokenization strategies, anomaly granularity, and normalization techniques. Figure 4 plots AUC for all language model types using word tokenization, contrasting max and diff normalization modes. Figure 5 compares the same variations for character tokenization. Table 2 presents these results in tabular form. With few exceptions, log-line-level granularity vastly outperforms day-level; this is true for both the character-level and word-level tokenization strategies, with an average gain of 0.1 AUC. The most interesting outcome of these comparisons is that word tokenization performance gains are heavily reliant on the diff normalization, whereas for character tokenization the diff normalization has a minor detrimental effect for some models. This suggests that the character-level model could be used to provide a more immediate response time, not having to wait until the day is done to obtain the day statistics used in diff mode. The two tokenization strategies may in fact be complementary as the versatility and response time gains of a character tokenization come at the expense of easy interpretibility of a word tokenization: the word tokenization allows anomaly scores to be decomposed into individual log-line fields, enabling analysts to pinpoint features of the event that most contributed to it being flagged. Since we tuned hyperparameters using diff mode, the character-level model has potential to do better with additional tuning. Next, Figures 6 and 7 visualize the average percentiles of red team detections for the subset of the test set with the most red-team activity. Anomaly scores for both word and character tokenizations are computed without average userday offset normalization. Red team log-line-level scores are plotted as purple x’s with the x coordinate being the second in time at which the event occurred and y coordinate the anomaly score for that event. Percentile ranges are colored to provide context for the red-team anomaly scores against the backdrop of other network activity. The spread of non-normalized anomaly scores is much greater for the word-level tokenizations (Fig. 7) than character-level (Fig. 6), which could explain the different sensitivity of word level tokenization to normalization. Also notice that there is an expected bump in percentiles for windows of frequent redteam activity. Curiously, at the end of day 14 there are massive bumps for the 99th percentile, which suggest unplanned and un-annotated anomalous events on the LANL network for those hours. Notice that for the character tokenization almost all non-normalized red team anomaly scores are above the 95th percentile, with a large proportion above the 99th percentile. Finally, Figure 8 plots the ROC curves for the best aggregate baseline (iso), the best user-day granularity language model (word BEM), and the best event-level granularity model (character BEM). It illustrates the qualitatively different curves obtained with the baselines, the user-day granularity, and the event-level granularity. Since the proportion of red-team to normal events is vanishingly low in the data-set (< 0.001%), the false-positive rate is effectively the proportion of data flagged to achieve a particular recall. From this observation, Figure 8 shows the character event model can achieve 100% recall from only 12% of the data whereas the other models considered only achieve 100% recall when nearly all of the data has been 2.00 75-95 Percentile 95-99 Percentile Red-event 1.75 Anomaly Score 1.50 1.25 1.00 0.75 0.50 0.25 15 pm am pm ay D 6 12 14 6 pm pm ay D 6 12 am 6 13 D ay pm pm 6 12 12 6 D ay am 0.00 Day/Time Figure 6: Character-level red-team log-line anomaly scores in relation to percentiles over time. 1.0 True positive 0.8 0.6 0.4 0.2 iso agg 0.76 AUC W BEM day 0.88 AUC C BEM log-line 0.98 AUC 0.0 0.0 0.2 0.4 0.6 False positive 0.8 1.0 Figure 8: ROC curves for best performing baseline, word language model evaluated at day-granularity, and character language model evaluated at log-line-granularity. handed to the analyst. Further, the character event model can achieve 80% recall by flagging only 3% of the data whereas the word day language model needs 14% of the data and the aggregate isolation forest model needs 55% of the data to achieve the same result. 5 Conclusion This work builds upon advances in language modeling to address computer security log analysis, proposing an unsupervised, online anomaly detection approach. We eliminate the usual effort-intensive feature engineering stage, making our approach fast to deploy and agnostic to the system configuration and monitoring tools. It further confers the key advantage of event-level detection which allows for a near immediate alert response following anomalous activity. In experiments using the Los Alamos National Laboratory Cyber Security Dataset, bidirectional language models significantly outperformed standard methods at day-level Figure 7: Word-level red-team log-line anomaly scores in relation to percentiles over time. detection. The best log-line-level detection performance was achieved with a bidirectional character-based language model, obtaining a 0.98 area under the ROC curve, showing that for the constrained language domain of network logs, character based language modeling can achieve comparable accuracy to word based modeling for event level detection. We have therefore demonstrated a simple and effective approach to modeling dynamic networks with open vocabulary logs (e.g. with new users, PCs, or IP addresses). We propose to extend this work in several ways. First, potential modeling advantages of tiered architectures merit further investigation. The use of tiered architectures to track PCs instead of network users, or from a richer set of logging sources other than simply authentication logs may take better advantage of their modeling power. Next, we anticipate interpretability can become lost with such detailed granularity provided by log-line-level detection from a characterbased model, therefore future work will explore alternate methods of providing context to an analyst. Finally, we are interested in exploring the robustness of this approach to adversarial tampering. Similarly performing models could have different levels of resilience that would lead to selection of one over another. Acknowledgments The research described in this paper is part of the Analysis in Motion Initiative at Pacific Northwest National Laboratory. It was conducted under the Laboratory Directed Research and Development Program at PNNL, a multi-program national laboratory operated by Battelle for the U.S. Department of Energy. The authors would also like to thank the Nvidia corporation for their donations of Titan X and Titan Xp GPUs used in this research. 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IEEE SIGNAL PROCESSING LETTERS 1 Look Wider to Match Image Patches with Convolutional Neural Networks arXiv:1709.06248v1 [cs.CV] 19 Sep 2017 Haesol Park, and Kyoung Mu Lee Abstract—When a human matches two images, the viewer has a natural tendency to view the wide area around the target pixel to obtain clues of right correspondence. However, designing a matching cost function that works on a large window in the same way is difficult. The cost function is typically not intelligent enough to discard the information irrelevant to the target pixel, resulting in undesirable artifacts. In this paper, we propose a novel convolutional neural network (CNN) module to learn a stereo matching cost with a large-sized window. Unlike conventional pooling layers with strides, the proposed per-pixel pyramid-pooling layer can cover a large area without a loss of resolution and detail. Therefore, the learned matching cost function can successfully utilize the information from a large area without introducing the fattening effect. The proposed method is robust despite the presence of weak textures, depth discontinuity, illumination, and exposure difference. The proposed method achieves near-peak performance on the Middlebury benchmark. Index Terms—stereo matching,pooling,CNN I. I NTRODUCTION Most stereo matching methods first compute the matching cost of each pixel with a certain disparity, before optimizing the whole cost volume either globally or locally by using specific prior knowledge [1]. For decades, many researchers have focused on the second step, designing a good prior function and optimizing it [2], [3], [4], [5], [6]. Few studies have been conducted on designing or selecting a better matching cost function. One of the most widely used matching cost functions is a pixel-wise matching cost function, such as the one used in [7]. Along with sophisticated prior models, it sometimes produces good results, especially in preserving the detailed structures near the disparity discontinuities. However, the function fails when the image contains weakly-textured areas or repetitive textures. In such cases, a window-based matching cost, such as CENSUS or SAD [8], produces a more reliable and distinctive measurement. The critical shortcoming of window-based matching cost functions is their unreliability around disparity discontinuities. Figure 1 visually illustrates the characteristics of different matching cost measures. One method to handle this trade-off is to make the windowbased versatile to its input patterns [10], [11], [12]. The key idea is making the shape of the matching template adaptive so that it can discard the information from the pixels that are irrelevant to the target pixel. However, knowing the background pixels before the actual matching is difficult, making it a H. Park and K. M. Lee are with Automation and Systems Research Institute, Seoul National University, Seoul 151-744, Korea matching cost for Pixelwise matching cost for 1 1 0.5 0.5 0 1 0 1 SAD (11x11) 0.5 0.5 0 1 0 1 SAD (37x37) 0.5 0.5 0 1 0 1 CENSUS (11x11) 0.5 0.5 0 1 0 1 CENSUS (37x37) 0.5 0.5 0 1 0 1 MC-CNN-arct[13] (11x11) 0.5 0.5 0 1 0 1 0.5 0.5 0 0 Proposed (37x37) Fig. 1. The top image shows the reference image with two interested points, x1 and x2 . The pixel positions are marked as blue dots, whereas the red and green boxes represent 37 × 37 and 11 × 11 windows centered on them, respectively. At the bottom, the matching costs for each pixel are visualized as a normalized function of disparity for different matching cost functions. The positions of true disparities are marked as red vertical lines. The pixelwise cost shows the lowest values at the true disparity, but it also gives zero costs for other disparities. The SAD and CENSUS matching cost functions [9] become less ambiguous as the matching window becomes larger. However, these functions are affected by pixels irrelevant to the target pixel (x2 ). Even the matching cost learned by using the baseline convolutional neural network (CNN) architecture fails when the surface has a nearly flat texture (x1 ). On the other hand, the proposed CNN architecture works well both on weakly textured regions and disparity discontinuities. ‘chicken-and-egg’ problem. Therefore, the use of a CNN [13], [14] is appropriate, as it automatically learns the proper shape of the templates for each input pattern. The existing methods, however, are based on conventional CNN architectures resembling the AlexNet [15] or VGG [16] network, which are optimized for image classification task and not for image matching. The architectures comprise several convolution layers, each followed by a rectified linear unit (ReLU) [15], and pooling layers with strides. One of the limitations of using these architectures for matching is the difficulty of enlarging the size of the patches that are to be compared. The effective size of the patch is directly related to IEEE SIGNAL PROCESSING LETTERS the spatial extent of the receptive field of CNN, which can be increased by (1) including a few strided pooling/convolution layers, (2) using larger convolution kernels at each layer, or (3) increasing the number of layers. However, use of strided pooling/convolution layers makes the results downsampled, losing fine details. Although the resolution can be recovered by applying fractional-strided convolution [17], reconstructing small or thin structures is still difficult if once they are lost after downsampling. Increasing the size of the kernels is also problematic, as the number of feature maps required to represent the larger patterns increases significantly. Furthermore, a previous study [18] reported that the repetitive usage of small convolutions does not always result in a large receptive field. This paper contributes to the literature by proposing a novel CNN module to learn a better matching cost function. The module is an innovative pooling scheme that enables a CNN to view a larger area without losing the fine details and without increasing the computational complexity during test times. The experiments show that the use of the proposed module improves the performance of the baseline network, showing competitive results on the Middlebury [1], [19] benchmark. II. R ELATED W ORKS Given the introduction of high-resolution stereo datasets with the ground-truth disparity maps [20], [19], [21], many attempts have been made to learn a matching cost function using machine learning algorithms [13], [14], [22]. The most impressive results are obtained by using CNN [13], [14]. The architecture proposed in [13] takes a small 11 × 11 window and processes it without the use of pooling. The computed cost volume is noisy due to the limited size of the window. Thus, it is post-processed by using the crossbased cost aggregation (CBCA) [23], semi-global matching (SGM) [3], and additional refinement procedures. On the other hand, the method in [14] uses multiple pooling layers and spatial-pyramid-pooling (SPP) [24] to process larger patches. However, the results show a fattening effect owing to the loss of information introduced by pooling. The main contribution of this paper is in proposing a novel pooling scheme that can handle information from a large receptive field without losing the fine details. Recently, several attempts have been made to accomplish the same goal in the context of semantic segmentation [25], [26], [27]. These methods combine the feature maps from the highlevel layers with those from the lower layers, with the aim of correctly aligning the object-level information along the pixel-level details. While this approach can successfully align the boundaries of the big objects, its inherent limitation is its inability to recover small objects in the final output once they are lost during the abstraction due to multiple uses of pooling. In the same context, the FlowNet [28] architecture can upsample the coarse-level flow to the original scale by using lower-level feature maps. However, it fails to recover the extreme flow elements that are hidden due to the low resolution of high-level feature maps. The architecture most closely related to the current work has been proposed in [24]. Unlike the other approaches, the 2 4 P Fig. 2. The 4P module with pooling size vector s = [5, 3, 1] is visualized. This figure shows its action for one channel of the feature maps for brevity; it does the same job for all channels. SPP network excludes pooling layers between convolutional layers. Instead, it first computes highly-nonlinear feature maps by cascading convolutional layers several times and then generates high-level and mid-level information by pooling them at different scales. By keeping the original feature maps along with feature maps pooled at multiple scales, the SPP network can combine the features from multiple levels without losing fine details. Although the previously mentioned stereo method in [14] uses SPP, it also employs conventional pooling layers between convolutional layers, thus losing the detailed information. III. A RCHITECTURE OF THE N EURAL N ETWORK The proposed architecture takes two input patches and produces the corresponding matching cost. In the following subsections, the newly proposed module is first introduced. Then the detailed architecture of the entire network is presented. A. Per-pixel Pyramid Pooling (4P) The use of pooling layers in CNN has been considered desirable because of its accuracy and efficiency in image classification tasks. While the use of max-pooling layers has been reported to provide an additional invariance in spatial transformation, the most important gain comes from the downsampling of feature maps. By performing pooling with a stride that is larger than one, the output feature maps after the pooling are scaled down. The final scale of the CNN output is decreased exponentially in terms of the number of pooling layers. Given that no parameters related to a pooling operation exist, this method is an effective way to widen the receptive field area of a CNN without increasing the number of parameters. The drawback of strided pooling is that the network loses fine details in the original feature maps as the pooling is applied. Thus, a trade-off exists in seeing a larger area and preserving the small details. Inspired by the idea discussed in [24], we propose a novel pooling scheme to overcome this trade-off. Instead of using a small pooling window with a stride, a large pooling window is used to achieve the desired size of the receptive field. The use of one large pooling window can lead to the loss of finer details. Thus, multiple poolings with varying window sizes are performed, and the outputs are concatenated to IEEE SIGNAL PROCESSING LETTERS 3 Matching score Matching score 1x1 Conv. + Sigmoid (1) 1x1 Conv. + Sigmoid (1) TABLE I T HE QUANTITATIVE RESULTS ON THE ‘ TRAINING DENSE ’ SET OF THE M IDDLEBURY BENCHMARK [1] ARE SHOWN . T HE ERROR REPRESENTS THE PERCENTAGE OF BAD PIXELS WITH A DISPARITY THRESHOLD 2.0, AND THE SAME WEIGHTING SCHEME IS APPLIED AS IN [1] WHEN COMPUTING THE AVERAGE . 1x1 Conv. + ReLU (384) 4P (384) 1x1 Conv. (96) 1x1 Conv. + ReLU (384) 1x1 Conv. + ReLU (384) 1x1 Conv. + ReLU (384) 1x1 Conv. + ReLU (384) 1x1 Conv. + ReLU (384) 1x1 Conv. + ReLU (384) Methods WTA after post-processing 3x3 Conv. + ReLU (112) 3x3 Conv. + ReLU (112) 3x3 Conv. + ReLU (112) 3x3 Conv. + ReLU (112) 3x3 Conv. + ReLU (112) 3x3 Conv. + ReLU (112) 3x3 Conv. + ReLU (112) 3x3 Conv. + ReLU (112) 3x3 Conv. + ReLU (112) 3x3 Conv. + ReLU (112) L R Baseline L R Proposed Fig. 3. The network structures are visualized for the baseline network, ‘MCCNN-acrt’ [13], and the proposed network. The parenthesized numbers at each layer represent the number of feature maps after the corresponding operations. Note that this figure is drawn in terms of the fully convolutional network. create new feature maps. The resulting feature maps contain the information from coarse-to-fine scales. The multi-scale pooling operation is performed for every pixel without strides. We call this whole procedure, “per-pixel pyramid pooling” (4P), which is formally defined as follows: P 4P (F, s) = [P (F, s1 ) , · · · , P (F, sM )] , (1) where s is a vector having M number of elements, and P (F, si ) is the pooling operation with size si and stride one. The structure of this module is illustrated in Figure 2. B. Proposed model To validate the effect of the proposed module, we trained and tested CNNs with and without the 4P module. The baseline architecture is selected as the ‘MC-CNN-acrt’ [13]. The 4P module in the proposed architecture is constructed by using the size vector s = [27, 9, 3, 1]. The structures of two CNNs are visualized in Figure 3. IV. I MPLEMENTATION D ETAILS For a fair comparison, we followed the details in [13] to train the proposed architecture with a few exceptions mentioned below. First, the size of the training patch became 37×37. Furthermore, we only fine-tuned the parameters of the last three 1 × 1 convolution layers of the proposed architecture in Figure 3. The parameters of the earlier layers are borrowed from the pre-trained ‘MC-CNN-acrt’ [13] network. In our experiments, this resulted in a better performance than the end-to-end training of the network with random initializations. Moreover, training a few convolution layers with pre-trained features is easier, making it converge faster. We have run a avg. error MC-CNN-acrt [13] proposed 22.91 11.75 MC-CNN-acrt [13] proposed (w/ parameters in [13]) proposed (w/ parameter tuning) 10.26 9.72 8.45 total of four epochs of training, where the last two epochs were run with a decreased learning rate from 0.003 to 0.0003. We also used the same post-processing pipeline as in [13] during the test phase. The post-processing pipeline includes the use of the CBCA [23] and SGM [3], and the disparity maps are refined to have continuous values and undergo median filtering and bilateral filtering. V. E XPERIMENTS To verify the effect of the proposed 4P module, we have compared the results of the baseline and proposed network. The performance is measured using the ‘training dense’ set of the Middlebury benchmark [1]. The quantitative results are briefly summarized in Table I using the average errors. All experiments are performed by using the Intel core i7 4790K CPU and a single Nvidia Geforce GTX Titan X GPU. The proposed method outperforms the baseline architecture regardless of the use of post-processing. The benefit of using the 4P module is clear when the disparity maps are obtained by using the pixel-wise winner-takes-it-all (WTA) rule without any post-processing. Given that the images in the dataset contain many weakly-textured areas, the small-sized 11×11 window cannot distinguish the true matches from false ones without the aid of post-processing. On the other hand, the proposed architecture effectively sees the larger window, 37 × 37, by inserting the 4P module before the final decision layer. It is less straightforward to understand why the proposed architecture still outperforms the baseline even after postprocessing. In that sense, it is worth to mention that the best parameter setting for post-processing of the proposed method largely differ from that of the baseline.1 The most notable changes from the original parameter setting is that we use much less number of CBCA [23], and it means that multiple uses of CBCA [23] become redundant in the proposed architecture. From this fact, we can interpret the role of 4P module as adaptive local feature aggregation. Compared to the hand-designed algorithm such as CBCA [23], the influence of neighboring pixels to a certain pixel is automatically learned 1 Following the conventions in [13], the best parameter setting is as follows: cbca_num_iterations_1 = 0, cbca_num_iterations_2 = 1, sgm_P1 = 1.3, sgm_P2 = 17.0, sgm_Q1 = 3.6, sgm_Q2 = 36.0, and sgm_V = 1.4. IEEE SIGNAL PROCESSING LETTERS true disparity and left image 4 proposed MC-CNN-acrt Fig. 4. The results for PlaytableP and Vintage are visualized. For each datum, the upper row shows the disparity map and the bottom row shows the corresponding error maps. While the ‘MC-CNN-acrt’ [13] shows errors around the weakly-textured areas, such as the surfaces of the chair and the table in PlaytableP or the white wall in Vintage, the proposed method shows more reliable results. and it can be jointly trained with the cost function itself. Furthermore, the information exchange among pixels is done in feature space which contains richer contextual information than the final cost volume space. Note that the improvement over the baseline clearly results neither from the use of extra layers nor from the use of more parameters, as the authors of [13] already have shown that the additional use of fully-connected (FC) layers is less significant. Using two additional FC layers leads to an improvement of approximately 1.90%, whereas using the 4P module results in a 21.42% improvement in terms of average error. The main contribution of the proposed method lies in learning a less ambiguous matching cost function by inspecting a larger area. Figure 4 shows that the proposed network actually works better around the weakly-textured area than the ‘MCCNN-acrt’ [13]. The quantitative and qualitative results of each dataset, including the ones in the ‘test dense’ set, are available at the Middlebury benchmark [1] website. VI. C ONCLUSIONS Viewing a large area to estimate the dense pixel correspondence is necessary to fully utilize the texture information to achieve less ambiguous and more accurate matching. A conventional matching cost function fails because neighboring pixels on the same surface as the target pixel are unknown. In this paper, a novel CNN module is proposed to make the CNN structure handle a large image patch without losing the small details, which enables it to learn an intelligent matching cost function for large-sized windows. The learned cost function can discriminate the false matches for weakly-textured areas or repeating textures, and can also conserve the disparity discontinuities well/. The learned cost function achieves competitive performance on the Middlebury benchmark. IEEE SIGNAL PROCESSING LETTERS R EFERENCES [1] D. Scharstein and R. Szeliski, “A taxonomy and evaluation of dense two-frame stereo correspondence algorithms,” IJCV, vol. 47, no. 1-3, pp. 7–42, 2002. [2] V. 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On the asymptotic structure of Brownian motions with a small lead-lag effect arXiv:1601.03614v2 [math.ST] 8 Apr 2018 Yuta Koike∗†‡ April 10, 2018 Abstract This paper considers two Brownian motions in a situation where one is correlated to the other with a slight delay. We study the problem of estimating the time lag parameter between these Brownian motions from their high-frequency observations, which are possibly subject to measurement errors. The measurement errors are assumed to be i.i.d., centered Gaussian and independent of the latent processes. We investigate the asymptotic structure of the likelihood ratio process for this model when the lag parameter is asymptotically infinitesimal. We show that the structure of the limit experiment depends on the level of the measurement errors: If the measurement errors locally dominate the latent Brownian motions, the model enjoys the LAN property. Otherwise, the limit experiment does not result in typical ones appearing in the literature. We also discuss the efficient estimation of the lag parameter to highlight the statistical implications. Keywords and phrases: Asymptotic efficiency; Endogenous noise; Lead-lag effect; Local asymptotic normality; Microstructure noise. 1 Introduction Let Bt = (Bt1 , Bt2 ) (t ∈ R) be a bivariate two-sided Brownian motion such that B0 = 0, E[(B11 )2 ] = E[(B12 )2 ] = 1 and E[B11 B12 ] = ρ for some ρ ∈ (−1, 0) ∪ (0, 1). Also, let ǫi = (ǫ1i , ǫ2i ) (i = 1, 2, . . . ) be a sequence of i.i.d. bivariate standard normal variables independent of B. For each n ∈ N, we denote by Pn,ϑ the law of the vector Zn = (X1 , . . . , Xn , Y1 , . . . , Yn )⊤ generated by the following model: ( √ 1 + √v ǫ1 , 2 Xi = Bi/n Yi = Bi/n−ϑ + vn ǫ2i if ϑ ≥ 0, n i √ 2 + √ v ǫ2 1 if ϑ < 0 Xi = Bi/n−ϑ + vn ǫ1i , Yi = Bi/n n i for i = 1, . . . , n, (1.1) where vn is a non-negative number. ϑ ∈ R denotes the unknown time-lag parameter which we are interested in. Especially, the sign of ϑ is unknown. The aim of this paper is to study the asymptotic structure of the sequence of experiments (R2n , B 2n , (Pn,ϑ )ϑ∈R ), n = 1, 2, . . . , as n → ∞ when the time lag parameter ϑ is asymptotically infinitesimal, i.e. ϑ tends to 0 as n → ∞ (here and below B m denotes the Borel σ-field of Rm for m ∈ N). More precisely, we study the limit experiment of (Pn,rnu )u∈R as n → ∞ for the proper convergence rate rn . ∗ Department of Business Administration, Graduate School of Social Sciences, Tokyo Metropolitan University, Marunouchi Eiraku Bldg. 18F, 1-4-1 Marunouchi, Chiyoda-ku, Tokyo 100-0005 Japan † The Institute of Statistical Mathematics, 10-3 Midori-cho, Tachikawa, Tokyo 190-8562, Japan ‡ CREST, Japan Science and Technology Agency 1 If vn ≡ 0, model (1.1) is a special case of the Hoffmann-Rosenbaum-Yoshida (HRY) model introduced in Hoffmann et al. (2013) to describe lead-lag effects in high-frequency financial data. A similar model has also been studied in Robert and Rosenbaum (2010) with an asymptotic regime different from the current setting. Here, the lead-lag effect refers to a situation where one time series is correlated to another time series at a later time, which has especially drawn attention in analysis of economic time series data for a long time, and associated econometric methods have been developed by many authors; see Section 1 of Hoffmann et al. (2013), Section 3 of Robert and Rosenbaum (2010) and references therein. The practicality of the HRY model in empirical work has recently been established by several authors such as Alsayed and McGroarty (2014), Huth and Abergel (2014) and Bollen et al. (2017) for financial data and Iacus et al. (2015) for social media data. These empirical studies show that time lag parameters are typically comparable to the observation frequencies in their scales. This motivates us to study the HRY model when ϑ is small. In such a situation, one would especially be interested in how small lag parameters can be identified in principle. To the author’s knowledge, however, there is few theoretical study for the HRY model and, in particular, nothing has been known about the optimality of statistical inferences for the HRY model. The purpose of this paper is trying to fill in this gap. In this paper, as well as (a special case of) the HRY model, we also consider a situation where the model contains measurement errors. This is motivated by the recent studies for the volatility estimation from ultra high frequency financial data, which is typically modeled as a discretely observed semimartingale with market microstructure noise. We refer to Chapter 7 of Aı̈t-Sahalia and Jacod (2014) for a brief description of this subject. In particular, the asymptotic structure and the asymptotic efficiency bound have been established in the work of Gloter and Jacod (2001a,b) (see also Cai et al. (2010)) for a statistical model of estimating the scale parameter σ > 0 from the discrete observations σWi/n + √ vn δi , i = 1, . . . , n, (1.2) where W = (Wt )t∈[0,1] is a one-dimensional standard Wiener process and (δi )ni=1 is a sequence of centered i.i.d. standard normal variables independent of W . They proved the LAN property for the above model and constructed asymptotically efficient estimators for σ (they indeed considered a more general setting). Extensions of their LAN result to a multivariate setting have also been studied by several authors. The correlation estimation in a bivariate setting is studied in Bibinger (2011), while a more general setting containing non-synchronous sampling case is studied in Ogihara (2014). On the other hand, Reiß (2011) has studied the asymptotic structure of model (1.2) when σ is a function of time rather than a constant and established the asymptotic equivalence between such a model and a Gaussian white noise model. The result has been extended to the bivariate case by Bibinger and Reiß (2014) and a multivariate setting containing non-synchronous case by Bibinger et al. (2014). Another type of extension, replacing the Wiener process W by a different continuous-time process, has also been studied. For example, Sabel and Schmidt-Hieber (2014) consider the efficient estimation of σ in a situation where W is a more general Gaussian process, especially a fractional Brownian motion. The main contribution of this paper is (i) to determine the proper convergence rate rn , and (ii) to derive a stochastic expansion for the likelihood ratio process for (Pn,rnu )u∈R . Analogously to Gloter and Jacod (2001a), the proper convergence rate rn depends on the behavior of the sequence nvn . This is intuitively natural because Var[Bi/n −B(i−1)/n ] = n−1 and thus the behavior of nvn determines how strongly the measurement errors (locally) 3 dominate the nature of the observed returns. In particular, we find that rn = n− 2 if vn ≡ 0, or more generally if 2 3 1 nvn is bounded.1 The rate n− 2 is much faster than the usual parametric rate n− 2 and even faster than the rate n−1 . Since the time resolution of our model is n−1 , our result suggests that we could estimate lag parameters smaller than the time resolution of observation data. This implication is at least true for our restrictive situation, as shown in Section 3. Since the convergence rate of the estimator for the lag parameter ϑ proposed in Hoffmann et al. (2013) cannot be faster than n−1 (see Proposition 2 of Hoffmann et al. (2013) and the discussion after this proposition), our result shows that their estimator is suboptimal in the setting considered in this paper (although their estimator works in a more general setting). Given the proper convergence rate, we have the following stochastic expansion for the likelihood ratio process: There are random variables Tn and Sn defined on (R2n , B 2n ) and non-negative numbers Iγ and Jγ such that dPn,rnun u2n p log →0 under Pn,0 as n → ∞ (1.3) − un Tn + \|un \|Sn − (Iγ + Jγ ) − dPn,0 2 for any bounded sequence un of real numbers and d (Tn , Sn ) − → N (0, Iγ ) ⊗ N (0, Jγ ) under Pn,0 as n → ∞. (1.4) Therefore, by a contiguity argument we deduce that the experiments (R2n , B 2n , (Pn,rn u )u∈R ) converge weakly to the experiment (R2 , B 2 , (Qu )u∈R ) in the Le Cam sense, where Qu = N (uIγ , Iγ ) ⊗ N (\|u\|Jγ , Jγ ) (see Corol- lary 2.3). The numbers Iγ and Jγ are determined by the asymptotic behavior of nvn and precisely defined by (2.1)–(2.2). In particular, Iγ is always positive, while Jγ is positive if nvn is bounded and Jγ = 0 otherwise. The case Jγ = 0 corresponds to the situation where the measurement errors locally dominate the signal, and in this case our model enjoys the LAN property which commonly appears in regular experiments. This result is of interest because model (1.1) exhibits irregularity in the sense that its likelihood function is not smooth in ϑ, and the limit experiment of such a model typically deviates from the LAN structure as illustrated in Chapters V–VII of Ibragimov and Has’minskii (1981). Our result means that the measurement errors have a kind of regularizing effect on the asymptotic structure of model (1.1). On the other hand, if Jγ > 0, which corresponds to the cases where the signal dominates or is balanced with the measurement errors, in addition to an observation from a usual Gaussian shift experiment N (u, Iγ−1 ), the limit experiment contains an extra observation from the experiment N (\|u\|, Jγ−1 ). Although this experiment looks simple, to the author’s knowledge it does not result in well-studied cases (such as in Ibragimov and Has’minskii (1981)), so the definition of asymptotically efficient estimators in this case is not obvious. To obtain the asymptotic efficiency bound for estimating the lag parameter in this case, in Section 3 we apply Ibragimov and Has’minskii (1981)’s theory to our problem, which is a common approach to establish asymptotic efficiency bounds for experiments generated by diffusion type processes (see Kutoyants (2004) for details). As a result, we find that Bayesian estimators are asymptotically efficient, while the maximum likelihood estimator is not always asymptotically efficient. This is a common phenomenon in irregular models; see Chapters V–VII of Ibragimov and Has’minskii (1981), Küchler and Kutoyants (2000), Chapter 3 of Kutoyants (2004), Rubin and Song (1995) and Chapter 9 of van der Vaart (1998) for example. This paper is organized as follows. Section 2 presents the main result of the paper. In Section 3 we discuss the efficient estimation of the lag parameter in our setting. Section 4 is devoted to the proof of an auxiliary technical result. 1 Indeed, an intuition for this fact has already been appeared in Hoffmann et al. (2013) (see Remark 2.2). 3 General notation Em denotes the m × m-identity matrix. For a matrix A, we denote by kAksp and kAkF its spectral norm and the Frobenius norm, respectively. That is, kAksp = sup{kAxk : kxk ≤ 1} and kAk2F = tr(A⊤ A). Also, we denote by Aij the (i, j)-th entry of A. 2 Main result We start with completing the definitions of the quantities rn , Iγ and Jγ appearing in the Introduction. First, following Gloter and Jacod (2001a) we assume that the sequence nvn converges in [0, ∞] and set γ := lim nvn . n→∞ We also assume lim supn vn < ∞ as in Gloter and Jacod (2001a). Then we set ( p n/vn if γ = ∞, Nn = n otherwise. Nn can be considered as an “effective” sample size in the sense that the proper convergence rate for estimating σ −1 from model (1.1) is given by Nn 2 , which is seen as the regular parametric rate if we regard Nn as the sample size. −3 Using this effective sample size Nn , we define our proper convergence rate as rn = Nn 2 . The constants Iγ and Jγ appearing in (1.3)–(1.4) are defined by  ρ2   2(1−ρ 2  √  √ ) ρ (1+ρ)(1+ρ+4γ)− (1−ρ)(1−ρ+4γ)−2ρ Iγ =  8γ 2   ρ2  √ √ 2( 1+ρ+ 1−ρ) if γ = 0, (2.1) if 0 < γ < ∞, if γ = ∞ and Jγ =          3 4 n 1 16γ 2 0 o 1 if γ = 0, + (1−ρ) 2 1 1 3 3 2 2 2 2 1+ρ 1−ρ 1−ρ 1+ρ + 1−ρ+4γ + 1−ρ+4γ + if 0 < γ < ∞, 4−3 1+ρ+4γ 1+ρ+4γ 1 (1+ρ)2 (2.2) if γ = ∞. Remark 2.1. Iγ is always positive for any γ ∈ [0, ∞]. This is evident when γ = 0 or γ = ∞. When 0 < γ < ∞, this is proven as follows. First suppose that 0 < ρ < 1. Then we have p (1 + ρ)(1 + ρ + 4γ) − { (1 − ρ)(1 − ρ + 4γ) + 2ρ}2 p = 4ρ + 8γρ − 4ρ (1 − ρ)2 + 4γ(1 − ρ) − 4ρ2 p = 4ρ (1 − ρ) + 2γ − (1 − ρ)2 + 4γ(1 − ρ) p p = 4ρ (1 − ρ)2 + 4γ(1 − ρ) + 4γ 2 − (1 − ρ)2 + 4γ(1 − ρ) > 0. Hence we have Iγ > 0. On the other hand, if −1 < ρ < 0, applying the above inequality with replacing ρ by −ρ, p p we obtain (1 − ρ)(1 − ρ + 4γ) > (1 + ρ)(1 + ρ + 4γ) − 2ρ. Hence we have Iγ > 0. The following statement is our main result. 4 Theorem 2.1. There are two sequences Tn and Sn of random variables satisfying (1.3)–(1.4) for any bounded sequence un of real numbers. We can explicitly give the variables Tn and Sn in Theorem 2.1 by (2.8) below. Theorem 2.1 has some immediate consequences. The first one is the direct consequence of the definition of the LAN property. Corollary 2.1. If γ = ∞, (Pn,ϑ )ϑ∈R has the LAN property at ϑ = 0 with rate rn and asymptotic Fisher information Iγ . The second one follows from Le Cam’s first lemma (see e.g. Lemma 6.4 of van der Vaart (1998)). Corollary 2.2. Pn,ϑn and Pn,0 are mutually contiguous if the sequence ϑn of real numbers satisfies ϑn = O(rn ). The third one is derived from Corollary 2.2 and Theorem 61.6 of Strasser (1985) (we refer to Drost et al. (2009), Le Cam (1986), Chapter 10 of Strasser (1985) and Chapters 8–9 of van der Vaart (1998) for the definition and applications of the weak convergence of experiments). Corollary 2.3. The sequence (R2n , B 2n , (Pn,rn h )h∈R ) of experiments converges weakly to the experiment (R2 , B 2 , (Qu )u∈R ) as n → ∞, where Qu = N (uIγ , Iγ ) ⊗ N (\|u\|Jγ , Jγ ). Now we turn to the proof of Theorem 2.1. Although (Pn,ϑ )ϑ∈R consists of Gaussian distributions, the problem is not simple because the covariance matrix Cn (ϑ) of Pn,ϑ is a complicated function of the lag parameter ϑ. In particular, Cn (ϑ) and Cn (ϑ′ ) are not simultaneously diagonalizable in general (even asymptotically) if ϑ 6= ϑ′ . This could be troublesome because in analysis of Gaussian experiments the (asymptotically) simulta- neous diagonalizability of the covariance matrices of the statistical model for different parameters typically plays an important role (cf. Section 3 of Davies (1973), Lemma 8.1 of Gloter and Jacod (2001a) and Lemma C.4 of Sabel and Schmidt-Hieber (2014)). For this reason we first transfer from the model Pn,ϑ to a more tractable model defined as follows: For each n ∈ N, set Θn = {ϑ ∈ R : vn − nϑ2 + \|ϑ\| ≥ 0} = {ϑ ∈ R : √ e n,ϑ the law of the vector Z en = \|ϑ\| ≤ (1 + 1 + 4nvn )/(2n)}. Then, for each ϑ ∈ Θn we denote by P e1 , . . . , X en , Ye1 , . . . , Yen )⊤ defined by (X where ( √ 2 − B2 2 2 e ǫ2i,n = −nϑ(Bi/n if ϑ ≥ 0, (i−1)/n ) + vn − nϑ + ϑǫi p 1 1 1 2 2 = −n\|ϑ\|(Bi/n − B(i−1)/n ) + vn − nϑ2 + \|ϑ\|ǫi , e ǫi,n = ǫi if ϑ < 0 e ǫ1i,n = ǫ1i , e ǫ1i,n 2 Yei = Bi/n +e ǫ2i , ei = B 1 + e ǫ1i , X i/n (2.3) (2.4) en,ϑ . en (ϑ) the covariance matrix of P for i = 1, . . . , n. We denote by C e n,ϑ . To be precise, the Hellinger distance In the following we will show that Pn,ϑ is well-approximated by P en,ϑ tends to 0 as n → ∞, provided that ϑ tends to 0 sufficiently fast. Here, the Hellinger between Pn,ϑ and P distance H(P, Q) between two probability measures P and Q on a measurable space (X , A) is defined by  s s !2 1/2 Z dP dQ dµ , − H(P, Q) =  dµ dµ X where µ is a σ-finite measure dominating both P and Q (µ = P + Q for example). It can easily be checked that H(P, Q) does not depend on the choice of µ. See Appendix A.1 of Reiß (2011), Section 2 of Strasser (1985) and Section 2.4 of Tsybakov (2009) for more information about the Hellinger distance. 5 e n,ϑ ) expectation with respect to Pn,ϑ (resp. P en,ϑ ). Throughout the paper, we denote by En,ϑ (resp. E e n,ϑ . Proposition 2.1. (a) If \|ϑ\| ≤ 1/n, then Pn,ϑ = P en,ϑ ) ≤ 4v −2 (4 + 3ρ2 )n2 \|ϑ\|3 for any n ∈ N and any ϑ ∈ Θn . (b) If vn > 0, H 2 (Pn,ϑ , P n 4 − e n,ϑ ) → (c) If a sequence ϑn of positive numbers satisfies ϑn = o(n−1 ∨Nn 3 ) as n → ∞, then sup\|ϑ\|≤ϑn H(Pn,ϑ , P 0. Proof. Claim (c) immediately follows from (a) and (b), so we focus on (a) and (b). By symmetry we may assume e n,ϑ [xi xj ] for ϑ ≥ 0. Let x1 , . . . , xn , y1 , . . . , yn be the canonical variables on R2n . Then we have En,ϑ [xi xj ] = E all i, j. Moreover, a simple computation shows that ( ( i∧j if i ∧ j ≥ nϑ, n − ϑ) + vn 1{i=j} En,ϑ [yi yj ] = i∨j (ϑ − n )+ + vn 1{i=j} otherwise and and e n,ϑ [yi yj ] = E En,ϑ [xi yj ] = ( i∧j i∧j − ϑ1{i≥j} − ϑ1{i≤j} + (vn + ϑ)1{i=j} = − ϑ + vn 1{i=j} n n if i < j − nϑ, ρi/n ( e n,ϑ [xi yj ] = E ρ(j/n − ϑ)+ otherwise, ρi/n if i < j, ρ(j/n − ϑ) otherwise. en (ϑ) if ϑ ≤ 1/n. This yields claim (a) because both Pn,ϑ and P en,ϑ are centered Therefore, we have Cn (ϑ) = C Gaussian. On the other hand, from the above identities we also have en (ϑ)k2F kCn (ϑ) − C n X i∨j i∧j ϑ− = −ϑ − n n + 2 i,j=1 i∧j<nϑ +2  n  X i=1 = n X i,j=1 i∧j<nϑ +2  X ρ i∨j n − j:i<j≤i+nϑ ϑ−  n  X X + ρ i n j −ϑ n i∧j −ϑ n 2 + + − i n 2 + j:j≤i 2 X ρ X ρ j−i −ϑ n 2  j:i<j≤nϑ j:i∨nϑ<j≤i+nϑ 2 2 nϑ 2nϑ 2 + 6n · ρ · nϑ · = (8 + 6ρ2 )n2 ϑ3 . ≤ 2n · nϑ · n n i=1 1 j −ϑ n + + X − j:j≤i∧nϑ ρ j −ϑ n  2  j −ϑ n   2  −1 Now if vn > 0, Cn (ϑ) is positive semidefinite and satisfies kCn (ϑ)− 2 ksp ≤ vn 2 by the monotonicity theorem for eigenvalues (Corollary 4.3.3 of Horn and Johnson (1985)) because Cn (ϑ) − vn E2n is positive semidefinite. Therefore, by Eqs.(A.4) and (A.6) from Reiß (2011) we obtain en (ϑ))Cn (ϑ)− 12 k2 ≤ 4v −2 (4 + 3ρ2 )n2 ϑ3 , en,ϑ ) ≤ 2kCn (ϑ)− 21 (Cn (ϑ) − C H 2 (Pn,ϑ , P F n hence claim (b) holds true. 6 In the following we will frequently use the fact that the Hellinger distance dominates the total variation distance: V (P, Q) ≤ H(P, Q), (2.5) where V (P, Q) = supA∈A \|P (A) − Q(A)\|. See e.g. Lemma 2.3 of Tsybakov (2009) for the proof. The following properties of the total variation distance are immediate consequences of the definition and important for our purpose. For each n ∈ N, let Pn and Qn be two sequences of probability measures on a measurable space (Xn , An ), and let ζn be a random variable on (Xn , An ) taking its value in a metric space D. Then, for any a ∈ D and any probability measure µ on D, the following statements hold true: p p V (Pn , Qn ) → 0, ζn − → a under Pn ⇒ ξn − → a under Qn , d d V (Pn , Qn ) → 0, ζn − → µ under Pn ⇒ ξn − → µ under Qn . ) (2.6) e n,ϑ as a tractable form. For this purpose we introduce some en (ϑ) of P Next we express the covariance matrix C notation. The n × n matrix ∇n denotes the backward difference operator, i.e.   1    −1 1    ∇n =  . .. ..   . .   −1 1 b n := (∇n ⊕ ∇n )Z e n = (X e1 , X e2 − X e1 , . . . , X en − X en−1 , Ye1 , Ye2 − Ye1 , . . . , Yen − Yen−1 )⊤ and denote by We set Z b n , i.e. Vn (ϑ) = (∇n ⊕ ∇n )C en (ϑ)(∇n ⊕ ∇n )⊤ . Vn (ϑ) can explicitly be expressed Vn (ϑ) the covariance matrix of Z sign(ϑ) ¯n as Vn (ϑ) = Ḡn − ρϑ∇ " Gn Ḡn = ρ n En , where # ρ E n n , Gn ¯+ ∇ n = with Gn = n1 En + vn Fn , Fn = ∇n ∇⊤ n and "     Rn =    0 ∇⊤ n # ··· 0 ··· .. . 0 .. . 0 0 ∇n ρ−1 Rn 1 0 0 .. . 0 .. . 0 ··· ¯− ∇ n =− , " ρ−1 Rn ∇n ∇⊤ n 0 #     .   ¯ sign(ϑ) It is more convenient to rewrite the expression ∇ as follows. Let Sn and Tn be the symmetric and skewn ⊤ ⊤ symmetric parts of 2∇⊤ n , respectively. That is, Sn = ∇n + ∇n and Tn = ∇n − ∇n . Then we set # # " " Sn Tn 1 ρ−1 Rn 1 −ρ−1 Rn S̄n = , T̄n = . 2 2 Sn ρ−1 Rn −Tn ρ−1 Rn ¯ ± = T̄n ± S̄n , so we obtain Vn (ϑ) = Ḡn − ρ(ϑT̄n + \|ϑ\|S̄n ). This is a simple function of ϑ, We can easily check ∇ n so Vn (ϑ) is more tractable than Cn (ϑ): Although Vn (ϑ)’s are not simultaneously diagonalizable for different ϑ’s, it is sufficient to consider a relationship between the matrices Ḡn , T̄n and S̄n . In fact, it turns out that the following result is sufficient for our purpose. 7 −1 −1 Proposition 2.2. For any α, β ∈ R, we have kḠn 2 (αT̄n + β S̄n )Ḡn 2 ksp = O(Nn ) as n → ∞ and −1 −1 lim ρ2 rn2 kḠn 2 (αT̄n + β S̄n )Ḡn 2 k2F = 2(α2 Iγ + β 2 Jγ ). n→∞ (2.7) The proof of Proposition 2.2 consists of elementary but complicated calculations, so we postpone it to the Appendix (Section 4). We remark that the proof requires a calculation essentially different from that of the Fisher information for the scale parameter estimation from observations of the form (1.2) such as in Gloter and Jacod (2001a) and Sabel and Schmidt-Hieber (2014) (see Remark 4.1). Also, note that Proposition 2.2 yields the invertibility of 1 −1 1 −1 Vn (ϑn ) for sufficiently large n if ϑn = o(Nn−1 ) because Vn (ϑn ) = Ḡn2 (E2n − ρḠn 2 (ϑn T̄n + \|ϑn \|S̄n )Ḡn 2 )Ḡn2 . Proof of Theorem 2.1. Define the function zbn : R2n → R2n by setting zbn (ζ) = (∇n ⊕ ∇n )ζ for ζ ∈ R2n . Then we set o ρ n −1 −1 Tn = − rn zbn⊤ Ḡ−1 T̄ Ḡ z b − tr( Ḡ T̄ ) , n n n n n n 2 o ρ n −1 −1 S̄ Sn = − rn zbn⊤ Ḡ−1 Ḡ z b − tr( Ḡ S̄ ) . n n n n n n 2 (2.8) By virtue of Proposition 2.1 and (2.5)–(2.6), it suffices to prove the following statements: dPn,rnun u2n p en,0 , log → 0 under P − un Tn + \|un \|Sn − (Iγ + Jγ ) − dPn,0 2 (2.9) d e n,0 . (Tn , Sn ) − → N (0, Iγ ) ⊗ N (0, Jγ ) under P (2.10) (2.10) follows from Proposition 2.2 and Proposition 2 of Dalalyan and Yoshida (2011). On the other hand, setting −1 −1 An = ρrn Ḡn 2 (un T̄n + \|un \|S̄n )Ḡn 2 , we have kAn k2F − 2u2n (Iγ + Jγ ) → 0 by Proposition 2.2. Therefore, by Proposition 2.1, (2.5) and Proposition 2 from Chapter 4 of Le Cam (1986) we obtain (2.9) once we show that en,rnun dP kAn k2F p e n,0 . ξn := log → 0 under P − (2.11) − un Tn + \|un \|Sn − en,0 4 dP The strategy of the proof of (2.11) is the same as that of Theorem 4.2 from Davies (1973). First, by Eq.(4.3) of Davies (1973), for sufficiently large n we have 2 e n,0 [ξn ] = − 1 log det Vn (rn un ) − log det Vn (0) + tr(Vn (0)(Vn (rn un )−1 − Vn (0)−1 )) + kAn kF E 2 4 1 1 2 −1 2 =− log det(E2n − An ) + tr(An ) + tr(An ) + tr((E2n − An ) − E2n − An − An ) . 2 2 P p Note that it holds that (E2n − An )−1 = ∞ p=0 An for sufficiently large n because kAn ksp → 0 as n → ∞ by Proposition 2.2. Combining this fact with inequality (v) from Appendix II of Davies (1973), we obtain 1 1 1 1 2 e + \|En,0 [ξn ]\| ≤ kAn ksp kAn kF 2 3 (1 − kAn ksp )3 1 − kAn ksp e n,0 [ξn ] → 0. for sufficiently large n. Hence Proposition 2.2 yields E Next, noting that ξn can be rewritten as 1 en,0 [b en,0 [ξn ], ξn = − zbn⊤ Bn zbn − E zn⊤ Bn zbn ] + E 2 −1 −1 where Bn = Vn (rn un )−1 − Vn (0)−1 − Ḡn 2 An Ḡn 2 , we obtain from Eq.(4.4) of Davies (1973) 1 1 2 VarePn,0 [ξn ] = Vn (0) 2 (Vn (rn un )−1 − Vn (0)−1 )Vn (0) 2 − An 8 2 F = (E2n − An )−1 − E2n − An 2 F . Therefore, using the identity (E2n − An )−1 = P∞ p p=0 An again we obtain 2 VarPen,0 [ξn ] ≤ kAn k2sp kAn k2F (1 − kAn ksp )−2 for sufficiently large n. Hence Proposition 2.2 again yields VarPen,0 [ξn ] → 0, and we obtain (2.11). We finish this section with some remarks. 3 Remark 2.2. It is worth mentioning that we can infer from Hoffmann et al. (2013) why the rate n− 2 is the proper convergence rate of our model in the case of vn ≡ 0 as follows. Let us set n U (ϑ) = n−1 X (Xi+1 − Xi )(Yj+1 − Yj )1{[ i , i+1 )∩[ j −ϑ, j+1 −ϑ)6=∅} n i,j=1 n n n for ϑ ∈ R. The principle used in Hoffmann et al. (2013) is that U n (ϑ) is close to the true correlation ρ if and only if ϑ is close to the true time-lag parameter. Since the accuracy of estimating the correlation parameter is of √ √ order 1/ n, we naturally consider the quantity \| n(U n (ϑ) − ρ)\| to measure the distance between U n (ϑ) and √ ρ: \| n(U n (ϑ) − ρ)\| would take a large value if ϑ is not sufficiently close to the true time-lag parameter. In fact, √ Proposition 1 from Hoffmann et al. (2013) implies that \| n(U n (ϑ)−ρ)\| does not diverge if and only if the distance 3 between ϑ and the true time-lag parameter is at most of order n− 2 . This information allows us to estimate the true 3 time-lag parameter with the accuracy of order n− 2 . Remark 2.3. From an econometric point of view, Proposition 2.1 is of independent interest because the model given by (2.3)–(2.4) has an economic interpretation different from model (1.1). This model contains measurement errors correlated to the latent returns Bi/n − B(i−1)/n . The integrated volatility estimation in the presence of this type of measurement error has been studied by Kalnina and Linton (2008) for example. In the market microstruc- ture theory, such a correlation is often explained as an effect of asymmetric information (e.g. Glosten (1987)). Interestingly, some economic arguments suggest that such an information asymmetry would cause a lead-lag effect; see Chan (1993) and Chordia et al. (2011) for instance. It would also be worth emphasizing that de Jong and Schotman (2010) connect this type of model with the investigation of price discovery, for price discovery processes are closely related to lead-lag effects, as seen in de Jong et al. (1998) and Hasbrouck (1995). Remark 2.4. Our proof of the main result heavily depends on the Gaussianity of the model, and especially we require the Gaussianity of the measurement errors. It is obvious that we need some restriction on the distribution of the measurement errors to derive a specific limit experiment. In fact, if vn ≡ 1 and ǫi ’s take their values only in integers, then we can completely recover the signal for sufficiently large n. Apart from such a trivial example, the recent study of Bibinger et al. (2016) has shown that another (non-trivial) specification for the distribution of the measurement errors δi ’s in (1.2) can improve the convergence rate for estimating the scale parameter σ. In the light of the connection between the convergence rates for models (1.1)–(1.2), we naturally conjecture that a similar specification for the measurement errors would affect the convergence rate for our model. This issue is beyond the scope of this paper and left to future research. 3 Efficient estimation of the lag parameter As an application of the results from the previous section, we construct efficient estimators for the lag parameter ϑ in the models (Pn,ϑ )ϑ∈R at ϑ = 0. Here we consider a slightly extended setup as follows: letting ηn be a sequence of positive numbers tending to 0 and C be a bounded open interval in R, we construct efficient estimators for the parameter c in the models (Pn,cηn )c∈C at every c ∈ C. To make use of the results from the previous section, we 9 impose the following condition on ηn : −4 ηn = o n−1 ∨ Nn 3 rn−1 ηn → ∞ and as n → ∞. (3.1) Under (3.1) there is a positive integer n0 such that cηn ∈ Θn and Vn (cηn ) is invertible for any c ∈ C and n ≥ n0 due to Proposition 2.2. Throughout this section, we always assume that n is larger than such an n0 . Remark 3.1. In a practical point of view, the dependence of the time-lag parameter ϑ on the sampling frequency n is just a theoretical device to control the relative size of ϑ compared with n (which corresponds to ηn ) in the asymptotic theory and it is only important whether the asymptotic order condition (corresponding to (3.1) in our case) is acceptable as an approximation. Namely, our asymptotic theory concerns whether the time-lag parameter ϑ is comparable to n−ι for some ι > 0 given a fixed sampling frequency n (the possible values of ι change in accordance with the noise level vn ) and it does not require that the time-lag parameter varies in proportion to the sampling frequency. This type of asymptotic theory is standard in econometrics: For example, when one considers the volatility estimation of a financial asset with taking account of rounding, one usually lets the rounding level shrink as the sampling frequency increases; see Rosenbaum (2009), Li and Mykland (2015), Li et al. (2015) and Sato and Kunitomo (2015) for example. We start with generalizing Proposition 2.2 by a matrix perturbation argument. Lemma 3.1. For any α, β ∈ R we have 1 1 sup kVn (cηn )− 2 (αT̄n + β S̄n )Vn (cηn )− 2 ksp = O(Nn ), c∈C 1 1 sup ρ2 rn2 kVn (cηn )− 2 (αT̄n + β S̄n )Vn (cηn )− 2 k2F − 2(α2 Iγ + β 2 Jγ ) → 0 c∈C as n → ∞. 1 1 Proof. Setting Hn (c) = Vn (cηn )− 2 Ḡn2 , we have −1 1 1 −1 Vn (cηn )− 2 (αT̄n + β S̄n )Vn (cηn )− 2 = Hn (c)Ḡn 2 (αT̄n + β S̄n )Ḡn 2 Hn (c)⊤ for any c ∈ C. Therefore, Ostrowski’s theorem (Theorem 4.5.9 of Horn and Johnson (1985)) implies that 1 −1 1 −1 kVn (cηn )− 2 (αT̄n + β S̄n )Vn (cηn )− 2 ksp ≤ kHn (c)Hn (c)⊤ ksp kḠn 2 (αT̄n + β S̄n )Ḡn 2 ksp and 1 −1 1 −1 \|kVn (cηn )− 2 (αT̄n + β S̄n )Vn (cηn )− 2 k2F − kḠn 2 (αT̄n + β S̄n )Ḡn 2 k2F \| −1 −1 ≤ k(Hn (c)Hn (c)⊤ )2 − E2n ksp kḠn 2 (αT̄n + β S̄n )Ḡn 2 k2F −1 −1 ≤ kHn (c)Hn (c)⊤ − E2n ksp (kHn (c)Hn (c)⊤ ksp + 1)kḠn 2 (αT̄n + β S̄n )Ḡn 2 k2F . Hence, Proposition 2.2 implies that the proof is completed once we show that supc∈C kHn (c)Hn (c)⊤ ksp = O(1) and supc∈C kHn (c)Hn (c)⊤ − E2n ksp = o(1) as n → ∞. Since Hn (c)Hn (c)⊤ and Hn (c)⊤ Hn (c) share the −1 same eigenvalues (Theorem 1.3.20 of Horn and Johnson (1985)) and Hn (c)⊤ Hn (c) = (E2n − ρηn Gn 2 (\|c\|S̄n + −1 cT̄n )Gn 2 )−1 , the desired results follow from Proposition 2.2, (3.1) and the Neumann series representation of Hn (c)⊤ Hn (c). 10 Using the above result, we can prove a uniform version of Theorem 2.1. Proposition 3.1. Let Tn and Sn be defined by (2.8). Then dPn,cηn+rn un u2n p →0 (Iγ + Jγ ) − − un Tn + \|un \|Sn − log dPn,cηn 2 d under Pn,cηn as n → ∞ uniformly in c ∈ C for any bounded sequence un of real numbers. Moreover, (Tn , Sn ) − → N (0, Iγ ) ⊗ N (0, Jγ ) under Pn,cηn as n → ∞ uniformly in c ∈ C. Proof. We can prove the first claim in a similar manner to the proof of Theorem 2.1 using Lemma 3.1 instead d of Proposition 2.2. To prove the second claim, it suffices to show that (Tn , Sn ) − → N (0, Iγ ) ⊗ N (0, Jγ ) under Pn,cnηn as n → ∞ for any sequence cn of numbers in C, which follows from Lemma 3.1 and inequality (13) from Dalalyan and Yoshida (2011). Proposition 3.1 implies that the experiments (Pn,cηn )c∈C enjoy the LAN property if γ = ∞ and do not oth- erwise. When the LAN property holds true, there is a well-established theory to define the asymptotic efficiency of estimators (cf. Section II-11 of Ibragimov and Has’minskii (1981)): A sequence cn of estimators in the experiments (Pn,cηn )c∈C is said to be asymptotically efficient at c ∈ C if the variables rn−1 ηn (cn − c) converge in law to N (0, Iγ−1 ) under Pn,cηn as n → ∞ (see Definition II-11.1 of Ibragimov and Has’minskii (1981)). Under the LAN property, this definition of the asymptotic efficiency is supported by several theorems such as the convolution theorem (e.g. Theorem II-9.1 of Ibragimov and Has’minskii (1981)) and the local asymptotic minimax theorem (e.g. Theorem II-12.1 of Ibragimov and Has’minskii (1981)). Moreover, it is well-known that both maximum likelihood and Bayesian estimators are asymptotically efficient under very general settings (cf. Chapter III of Ibragimov and Has’minskii (1981)). On the other hand, if the LAN property fails, it is generally not obvious how to define the asymptotic efficiency of estimators. Here, we adopt the approach from Küchler and Kutoyants (2000) to define the asymptotic efficiency, which is based on Theorem I-9.1 of Ibragimov and Has’minskii (1981) that derives an asymptotically minimax lower bound from the asymptotic properties of the Bayesian estimators. As a consequence, the Bayesian estimators are turned out to be asymptotically efficient. Now we explain the strategy to obtain asymptotically efficient estimators in our setting. As in the previous en,ϑ rather than the original model Pn,ϑ . For this reason section, we would like to work with the tractable model P we consider the quasi-likelihood function based on the former as follows: en,cηn dP 1 ⊤ 1 −1 p exp − zbn Vn (cηn ) zbn , = Ln (c) := dx 2 (2π)n det Vn (cηn ) c ∈ C. Then we consider the quasi maximum likelihood and Bayesian estimators based on Ln (c) as our estimators and give en,cηn )c∈C using the general scheme of Ibragimov and Has’minskii their asymptotic behavior in the experiments (P (1981) (see Proposition 3.2). Next we consider the case γ = ∞ where the LAN property holds true and thus en,cηn )c∈C can be transferred to that in (Pn,cηn )c∈C by Proposition 2.1(c). Finally, we convergence in law in (P en,cη )c∈C = (Pn,cη )c∈C for sufficiently large n due to (3.1) and consider the case γ < ∞ where we have (P n n Proposition 2.1(a), hence we can apply the Ibragimov-Has’minskii method to define and obtain asymptotically efficient estimators. The quasi maximum likelihood estimator (QMLE) ĉn is defined as a solution of the equation Ln (ĉn ) = sup Ln (c). c∈C 11 Note that the above equation always has at least one solution belonging to the closure of C because c 7→ Ln (c) is continuous. Moreover, we can choose ĉn so that it is measurable by the measurable selection theorem (see e.g. Theorem 6.7.22 of Pfanzagl (1994)). Also, the quasi Bayesian estimator (QBE) c̃n for a prior density q : C → (0, ∞) with respect to the quadratic loss is defined by Z Z Ln (c)q(c)dc, c̃n = cLn (c)q(c)dc C C where the prior density q is assumed to be continuous and satisfy 0 < inf c∈C q(c) ≤ supc∈C q(c) < ∞. The corresponding QMLE and QBE in the experiments (Pn,ϑ )ϑ∈R are given by ϑ̂n = ĉn ηn and ϑ̃n = c̃n ηn , respectively. Remark 3.2. Since the quantity ηn seems the exact order of the true time-lag parameter, one may consider that in a practical setting it is difficult to know ηn beforehand and thus it is difficult to use the estimators ϑ̂n and ϑ̃n . However, when we construct the estimator ϑ̂n , ηn can be considered as the maximum order of the true time-lag en,ϑ /dx and Cn = {cηn : c ∈ C}. Then the estimator ϑ̂n can be parameter as follows. Let us set L′n (ϑ) = dP considered as a solution of the equation L′n (ϑ̂n ) = sup Ln (ϑ). ϑ∈Cn Therefore, in a practical situation (supc∈C c)ηn (resp. (inf c∈C c)ηn ) can be interpreted as an upper bound (resp. a lower bound) of possible time-lag parameters ϑ. It is often not so difficult to find such bounds in a practical setting (and they are typically “small” as pointed out in the Introduction). For example, we can find them by computing the cross-correlations via Hoffmann et al. (2013)’s method as in Huth and Abergel (2014). The same remark can be applied to the estimator ϑ̃n because it can be rewritten as Z Z ′ ϑLn (ϑ)qn (ϑ)dϑ ϑ̃n = Cn Cn L′n (ϑ)qn (ϑ)dϑ, where qn (ϑ) = ηn−1 q(ϑ/ηn ) for ϑ ∈ Cn (so qn is a prior density on Cn ). To describe the limit distribution of these estimators, we introduce the likelihood ratio process for the limit experiment u2 Z(u) = exp uζ1 + \|u\|ζ2 − (Iγ + Jγ ) , 2 u ∈ R, where ζ1 and ζ2 are two mutually independent variables such that ζ1 ∼ N (0, Iγ ) and ζ2 ∼ N (0, Jγ ). Then we set     (ζ1 + ζ2 )/(Iγ + Jγ ) if ζ1 ≥ (−ζ2 ) ∨ 0, û = argmaxu∈R Z(u) = (ζ1 − ζ2 )/(Iγ + Jγ ) if ζ1 < ζ2 ∧ 0,    0 otherwise and R∞ ũ = R−∞ ∞ uZ(u)du −∞ Z(u)du . en,cηn )c∈C using the We first give the asymptotic behavior of the estimators ĉn and c̃n in the experiments (P general scheme of Ibragimov and Has’minskii (1981). Note that in this situation ĉn and c̃n are true maximum likelihood and Bayesian estimators, respectively. 12 Proposition 3.2. For any compact subset K of C, uniformly in c ∈ K it holds that rn−1 ηn (ĉn − c) converges in law en,cη and E e n,cη [\|r −1 ηn (ĉn − c)\|p ] → E[\|û\|p ] for any p > 0 as n → ∞. Also, uniformly in c ∈ K to û under P n n n −1 e n,cηn and E e n,cηn [\|r −1 ηn (c̃n − c)\|p ] → E[\|ũ\|p ] for any it holds that r ηn (c̃n − c) converges in law to ũ under P n n p > 0 as n → ∞. en,cη +r u /dP en,cη Proof. For every c ∈ C, we set Un (c) = {u ∈ R : c + rn ηn−1 u ∈ C} and define Zn,c (u) = dP n n n for each u ∈ Un (c). According to Theorems I-10.1 and I-10.2 from Ibragimov and Has’minskii (1981), it suffices to prove the following statements: p p e n,cη [\| Zn,c (u) − Zn,c (v)\|2 ] < ∞, (a) lim supn→∞ supc∈K supu,v∈Un (c) \|u − v\|−2 E n p 2 e n,cη [ Zn,c (u)] < ∞, (b) there is a constant κ > 0 such that lim sup sup sup eκu E n→∞ c∈K u∈Un (c) n (c) the marginal distributions of Zn,c converge in law to the marginal distributions of Z uniformly in c ∈ K as n → ∞. (c) is an immediate consequence of Proposition 3.1. On the other hand, by Eq.(A.4) from Reiß (2011) we obtain e n,cη E n " q Zn,c (u) − q 2 Zn,c (v) # en,cη +r u , P en,cη +r v ) = H 2 (P n n n n 1 1 1 1 ≤ 4ρ2 rn2 (u − v)2 kVn (cηn + rn u)− 2 T̄n Vn (cηn + rn u)− 2 k2F + kVn (cηn + rn u)− 2 S̄n Vn (cηn + rn u)− 2 k2F , hence Lemma 3.1 yields claim (a). Now we consider (b). By Corollary 3.2a.1 from Mathai and Provost (1992) we have q 1 1 1 −1 e n,cη (E2n − An,c (u)) − E2n , log E Zn,c (u) = − log det[E2n − An,c (u)] − log det E2n + n 4 2 2 1 1 where An,c (u) = E2n − Vn (cηn )− 2 Vn (cηn + rn u)Vn (cηn )− 2 . Then we consider the following decomposition: q e n,cη Z (u) log E n,c n 1 1 2 =− log det[E2n − An,c (u)] + tr[An,c (u)] + kAn,c (u)kF 4 2 1 1 1 1 − log det E2n + Bn,c (u) − tr (Bn,c (u)) + kBn,c (u)k2F 2 2 2 8 1 1 1 2 2 2 + tr[An,c (u)] + kAn,c (u)kF − tr (Bn,c (u)) − kAn,c (u)kF − kBn,c (u)kF 4 8 2 =: In,c (u) + IIn,c (u) + IIIn,c (u) + IVn,c (u), where Bn,c (u) = (E2n − An,c (u))−1 − E2n . Let us set 1 1 1 1 S̃n,c = ρVn (cηn )− 2 S̄n Vn (cηn )− 2 . T̃n,c = ρVn (cηn )− 2 T̄n Vn (cηn )− 2 , Then we have An,c (u) = cηn T̃n,c + \|cηn \|S̃n,c − (cηn + rn u)T̃n,c − \|cηn + rn u\|S̃n,c , hence it holds that sup sup kAn,c (u)ksp ≤ sup \|c\| sup 2ηn (kT̃n,c ksp + kS̃n,c ksp ). c∈C u∈Un (c) c∈C c∈C 13 Here, we use the fact that c+rn ηn−1 u ∈ C because of u ∈ Un (c). In particular, we have supc∈C supu∈Un (c) kAn,c (u)ksp < P k 1 for sufficiently large n by (3.1) and Lemma 3.1. For such an n we have Bn,c (u) = ∞ k=1 An,c (u) and thus kBn,c (u)ksp ≤ kAn,c (u)ksp , 1 − kAn,c (u)ksp kBn,c (u)kF ≤ kAn,c (u)kF , 1 − kAn,c (u)ksp k−1 for k ≥ 1 to obtain the latter estimate. where we use the inequality kAn,c (u)k kF ≤ kAn,c (u)kF kAn,c (u)ksp Therefore, for sufficiently large n we have for any c ∈ C and any u ∈ Un (c) \|In,c (u)\| ≤ 1 kAn,c (u)ksp kAn,c (u)k2F , 12 1 − kAn,c (u)k3sp \|IIn,c (u)\| ≤ 1 kBn,c (u)ksp kBn,c (u)k2F 6 8 − kBn,c (u)k3sp by Appendix II-(v) from Davies (1973) and \|IIIn,c (u)\| ≤ ∞ X k=3 tr(An,c (u)k ) ≤ kAn,c (u)k2F kAn,c (u)ksp 1 − kAn,c (u)ksp k−2 for k ≥ 3, as well as by the inequality tr(An,c (u)k ) ≤ kAn,c (u)k2F kAn,c (u)ksp kAn,c (u)k2F 1 . 1− IVn,c (u) ≤ − 8 2(1 − kAn,c (u)ksp )2 Consequently, there is a constant κ0 > 0 such that for sufficiently large n it holds that q e Zn,c (u) ≤ −κ0 kAn,c (u)k2F log En,cηn for any c ∈ C and any u ∈ Un (c). Now we consider giving an upper bound for −kAn,c (u)k2F . We have −kAn,c (u)k2F = −u2 rn2 kT̃n,c k2F − 2rn u(\|cηn + rn u\| − \|cηn \|) tr(T̃n,c S̃n,c ) − (\|cηn + rn u\| − \|cηn \|)2 kS̃n,c k2F ≤ −2u2 Iγ + u2 rn2 kT̃n,c k2F − 2Iγ + 2 rn2 tr(T̃n,c S̃n,c ) . Therefore, noting Iγ > 0, by Lemma 3.1 for sufficiently large n we have −kAn,c (u)k2F ≤ −Iγ u2 for any c ∈ C and any u ∈ Un (c). Consequently, we obtain (b) by setting κ = κ0 Iγ . If γ < ∞, (3.1) is equivalent to the condition that ηn = o(n−1 ) as n → ∞. Therefore, Proposition 2.1(a) yields the following result: e n,cη ’s are replaced by Pn,cη ’s. Corollary 3.1. If γ < ∞, the statement of Proposition 3.2 still holds true while P n n Now we return to the efficient estimation of the parameter c in the model (Pn,cηn )c∈C . First we consider the case γ = ∞. In this case we know that (Pn,cηn )c∈C enjoy the LAN property at every c ∈ C by Proposition 3.1, so the definition of the asymptotic efficiency of an estimator sequence is well-established as explained in the above. Theorem 3.1. If γ = ∞, both ĉn and c̃n are asymptotically efficient at every c ∈ C in the experiments (Pn,cηn )c∈C . That is, both rn−1 ηn (ĉn −c) and rn−1 ηn (c̃n −c) converge in law to N (0, Iγ−1 ) under Pn,cηn for any c ∈ C as n → ∞. In particular, both ϑ̂n and ϑ̃n are asymptotically efficient at ϑ = 0 in the experiments (Pn,ϑ )ϑ∈R . Next we turn to the case γ < ∞. In this case the experiments (Pn,cηn )c∈C no longer enjoy the LAN property, so the definition of the asymptotic efficiency is not obvious. As explained in the above, here we follow the approach of Küchler and Kutoyants (2000) to define the asymptotic efficiency for our experiments. We obtain the following result by virtue of Corollary 3.1 and Theorem I-9.1 of Ibragimov and Has’minskii (1981): 14 Theorem 3.2. If γ < ∞, we have lim lim inf sup (rn−1 ηn )2 En,cηn [(c∗n − c)2 ] ≥ E[ũ2 ] δ→0 n→∞ \|c−c0 \|<δ for any c0 ∈ C and any estimator sequence c∗n in the experiments (Pn,cηn )c∈C . In particular, we also have lim lim inf sup rn−2 En,ϑ [(ϑ∗n δ→0 n→∞ \|ϑ\|<δηn − ϑ)2 ] ≥ E[ũ2 ] for any estimator sequence ϑ∗n in the experiments (Pn,ϑ )ϑ∈R . Thanks to Theorem 3.2, an estimator sequence c∗n is said to be asymptotically efficient at c0 ∈ C in the experi- ments (Pn,cηn )c∈C if it holds that lim lim inf sup (rn−1 ηn )2 En,cηn [(c∗n − c)2 ] = E[ũ2 ]. δ→0 n→∞ \|c−c0 \|<δ Similarly, an estimator sequence ϑ∗n is said to be asymptotically efficient at ϑ = 0 in the experiments (Pn,ϑ )ϑ∈R if it holds that lim lim inf sup rn−2 En,ϑ [(ϑ∗n δ→0 n→∞ \|ϑ\|<δηn − ϑ)2 ] = E[ũ2 ] for any sequence ηn of positive numbers satisfying (3.1). The following result is an immediate consequence of this definition. Theorem 3.3. If γ < ∞, the sequence c̃n is asymptotically efficient at every c ∈ C in the experiments (Pn,cηn )c∈C . In particular, the sequence ϑ̃n is asymptotically efficient at ϑ = 0 in the experiments (Pn,ϑ )ϑ∈R . In contrast, there is no guarantee of the asymptotic efficiency of the (Q)MLE ĉn if γ < ∞. In fact, c̃n may perform much better than ĉn if γ = 0, as shown in the following proposition. Proposition 3.3. It holds that s ! ! p Iγ Jγ Jγ 1 E û , 1 − arctan + π Iγ π(Iγ + Jγ ) Z ∞Z ∞ 2 1 xΨ(x) − yΨ(y) 2 E ũ = ψR (x, y)dxdy, Iγ + Jγ −∞ −∞ Ψ(x) + Ψ(y) where Ψ(x) = R∞ eux−u 2 1 = Iγ + Jγ 2 /2 (3.2) (3.3) du and ψR (x, y) denotes the bivariate normal density with standard normal marginals and correlation R = (Jγ − Iγ )/(Jγ + Iγ ). In particular, lim\|ρ\|→1 E ũ2 /E û2 = 0 if γ = 0. 0 Proof. Let us denote by φa the normal density with mean 0 and variance a. Then, a simple calculation yields Z ∞ Z ∞ 2 2 2 φIγ (x)φJγ (z − x)dx. z dz E û = (Iγ + Jγ )2 0 0 By formulae (3.322.2) and (6.292) from Gradshteyn and Ryzhik (2007) we have Z ∞ 2 z dz 0 Z 0 ∞ Iγ + Jγ Iγ + Jγ − arctan φIγ (x)φJγ (z − x)dx = 2 2π hence we obtain (3.2). 15 s Jγ Iγ ! p Iγ Jγ , + 2π Next, by a change of variable we obtain Z ∞ 1 Z(u)du = p Iγ + Jγ −∞ ( ζ + ζ2 p1 Iγ + Jγ Ψ ! +Ψ ζ − ζ1 p2 Iγ + Jγ !) . Moreover, formulae (3.462.5) and (3.322.2) from Gradshteyn and Ryzhik (2007) imply that ! !) ( Z ∞ ζ2 − ζ1 1 ζ1 + ζ2 ζ2 − ζ1 ζ1 + ζ2 p Ψ p −p Ψ p . uZ(u)du = Iγ + Jγ Iγ + Jγ Iγ + Jγ Iγ + Jγ Iγ + Jγ −∞ p p Since the distribution of the vector ((ζ1 +ζ2 )/ Iγ + Jγ , (ζ2 −ζ1 )/ Iγ + Jγ ) has the density ψR (x, y), we obtain (3.3). Finally, we prove the latter statement. Define the functions f and g on (−1, 1) by ! √ r Z ∞Z ∞ 1 xΨ(x) − yΨ(y) 2 1+r 1 − r2 f (r) = 1 − arctan + ψr (x, y)dxdy. , g(r) = π 1−r 2π Ψ(x) + Ψ(y) −∞ −∞ Then we have E[û2 ] = f (R)/(Iγ + Jγ ) and E[ũ2 ] = g(R)/(Iγ + Jγ ). Since R → 1 as \|ρ\| → 1 if γ = 0 and limr→1 f (r) = 12 , it suffices to prove limr→1 g(r) = 0. Because we have g(r) = Z ∞ −∞ Z ∞ −∞ !2 √ √ xΨ(x) − (rx + 1 − r 2 y)Ψ(rx + 1 − r 2 y) √ φ1 (x)φ1 (y)dxdy, Ψ(x) + Ψ(rx + 1 − r 2 y) the dominated convergence theorem yields limr→1 g(r) = 0, which completes the proof. 4 Appendix: Proof of Proposition 2.2 Before starting the proof, we introduce some notation. We set π 1 ξi := ξi,n := i− , 2n + 1 2 i = 1, 2, . . . . Then we define the n × n matrix Un = (uij )1≤i,j≤n by 2 2π 1 1 2 √ uij := i− j− =√ cos cos [ξi (2j − 1)] , 2n + 1 2 2 2n + 1 2n + 1 which is often referred to as the Discrete Cosine Transform (DCT) of type-VIII (see Sabel and Schmidt-Hieber (2014) and references therein). Note that Un⊤ = Un and Un is real orthogonal. It is known that Un diagonalizes Fn as follows: where λi := 2 [1 − cos (2ξi )] . Un Fn Un = diag(λ1 , . . . , λn ), (4.1) See Lemma 1 of Kunitomo and Sato (2013) or Lemma C.2 of Sabel and Schmidt-Hieber (2014) for the proof. For each a > 0 we define the functions fa and ga on R by fa (x) = a + 2vn (1 − cos(x)), n ga (x) = sin(x) fa (x) (x ∈ R). We also set Gn (a) = na En + vn Fn . From (4.1) we have Un Gn (a)Un = Λn (a) := diag(fa (2ξ1 ), . . . , fa (2ξn )). 16 (4.2) Remark 4.1. It turns out that the off-diagonal components of Un Tn Un play a dominant role to calculate the limit of −1 −1 kḠn 2 T̄n Ḡn 2 kF . This is essentially different from the case of calculating the Fisher information for the scale pa- rameter estimation from observations of the form (1.2), where the similarity transformations of Toeplitz matrices by Un are sufficiently approximated by diagonal matrices as manifested by Lemma C.4 of Sabel and Schmidt-Hieber (2014). For this reason we need rather specific calculations as seen in Lemmas 4.4 and 4.7. For a square matrix A, spr(A) denotes the spectral radius of A. We will frequently use the identity kAksp = spr(A) holding if A is a normal matrix. Now we start the main body of the proof. We will frequently use the following well-known inequality for the sine function, π . 0≤x≤ 2 2 x ≤ sin(x) ≤ 1 π Lemma 4.1. For any a > 0, we have sup ga (x) = p 0≤x≤π ( na )2 1 , + 4 na vn (4.3) sup \|ga′ (x)\| ≤ 0≤x≤π Proof. The claim immediately follows from the identity ga′ (x) = 3n . a a +2vn ) cos(x)−2vn (n fa (x)2 = a (1+cos(x)) n fa (x)2 − 1 fa (x) . Lemma 4.2. Let ψ : [0, π] → R be continuous. Also, let mn be a sequence of positive integers such that mn ≤ n and mn /n → c ∈ (0, 1] as n → ∞. Then, for any p > 0 we have lim 1 n→∞ np+1 Z πc mn X ψ(2ξi ) 1 = p ψ(x)dx fa (2ξi )p a π 0 i=1 provided that γ = 0. Proof. By the fundamental theorem of calculus, we have ψ(2ξi ) ψ(2ξi ) ≤ pkψk∞ − p fa (2ξi ) (a/n)p Z vn λi 0 1 2pkψk∞ vn dx ≤ . p+1 (a/n + x) (a/n)p+1 Hence the desired result follows from the standard Riemann sum approximation. Lemma 4.3. Let mn be a sequence of positive integers such that mn ≤ n and mn /n → c ∈ (0, 1] as n → ∞. Then   mn  X 1 1 lim = n→∞ nNn f (2ξi )   i=1 a π √ 2 a(a+4γ) arctan 1 √ 2 a q 1+ 4γ a tan πc 2 if γ < ∞,  1  c − sin2π2πc  2ab q  √   b(b+4γ) 4γ  πc  1 + b tan 2 mn  2πγ 2 (b−a) arctan X 2 √ q lim r ga (2ξi )gb (2ξi ) = a(a+4γ) n→∞ n 4γ  πc  − − 2πγ 2 (b−a) arctan 1 + a tan 2 i=1      1  √ √ 2( a+ b) for any a, b > 0 such that a 6= b. 17 (4.4) if γ = ∞, if γ = 0, c 4γ 2 (4.5) if 0 < γ < ∞, if γ = ∞. Proof. First, using the lower and upper Darboux sums of the integral 2n + 1 2π Z 2ξmn 2ξ1 m R 2ξmn 2ξ1 1 fa (x) dx, n X 1 1 2n + 1 1 1 dx + ≤ + ≤ fa (x) fa (2ξmn ) fa (2ξi ) fa (2ξ1 ) 2π i=1 we obtain Z 2ξmn 1 dx. fa (x) 2ξ1 Now, formula (2.553.3) in Gradshteyn and Ryzhik (2007) yields Z y 0 r 1 2n dx = p arctan fa (x) a(a + 4nvn ) ! y 4nvn , tan 1+ a 2 hence we obtain (4.4). Next, a simple calculation yields n ga (x)gb (x) = b−a sin2 x sin2 x − fa (x) fb (x) . Therefore, if γ = 0, Lemma 4.2 implies that lim r 2 n→∞ n mn X i=1 1 ga (2ξi )gb (2ξi ) = πab Z πc 0 1 sin xdx = 2ab 2 sin 2πc c− 2π . On the other hand, if γ 6= 0, for sufficiently large n we have 1 − cos(x) = (fa (x) − a/n)/(2vn ), hence sin2 x = fa (x)/vn − a/(nvn ) − fa (x)2 /(4vn2 ) + fa (x)a/(2nvn2 ) − a2 /(2nvn )2 . Therefore, we obtain sin2 x sin2 x − = fa(x) fb (x) b b2 + nvn (2nvn )2 1 − fb (x) a a2 + nvn (2nvn )2 1 b−a − . fa (x) 4nvn2 Hence the desired result follows from (4.4). Lemma 4.4. Let mn be a sequence of positive integers such that mn ≤ n and mn → ∞ as n → ∞. Then lim n→∞ 4 2n + 1 2 X mn sin4 (n · 2ξ1 i) = 2. sin2 (2ξ1 i) i=1 Proof. Since 4nξ1 = π − 2ξ1 , we have o 1 1n 1 − (−1)l cos (2ξ1 l) = sin (n · 2ξ1 l) = {1 − cos (4nξ1 l)} = 2 2 2 ( sin2 (ξ1 l) if l is even, cos2 (ξ1 l) if l is odd. Therefore, using the formula sin(2ξ1 i) = 2 sin(ξ1 i) cos(ξ1 i), we can decompose the target quantity as     2 X m m mn n n  X X 4 4 sin4 (n · 2ξ1 i) 2 2 cot (ξ i) =: A1,n + A2,n . tan (ξ i) + = 1 1  2n + 1 (2n + 1)2  sin2 (2ξ1 i)   i=1 i=1 i=1 i: even i: odd First we prove limn A1,n = 0. Using the monotonicity of the tangent function and assumption mn ≤ n, we obtain A1,n 8 ≤ (2n + 1)π Z 0 18 π n+1 2 2n+1 tan2 (x)dx Since formula (2.526.22) in Gradshteyn and Ryzhik (2007) yields limn that limn A1,n = 0. n+1 R π2 2n+1 0 tan2 (x)dx = 1−π/4, we conclude Next we prove limn A2,n = 2. Our proof relies on the following inequality for the tangent function: π π x π x ≤ tan x ≤ (0 ≤ x < 1). 2 2 2 1 − x2 (4.6) The lower estimate of (4.6) is well-known, and the upper estimate is known as the Becker-Stark inequality (Eq.(2) of Becker and Stark (1978)). Now, using (4.6), we obtain ⌊ mn2+1 ⌋ ⌊ mn2+1 ⌋ 2 (2i − 1)2 16 X 16 X 1 1 − + . ≤ A2,n ≤ 2 2 2 2 4 π (2i − 1) (2n + 1) (2n + 1) π (2i − 1)2 i=1 i=1 Therefore, using formula P∞ i=1 (2i − 1)−2 = π2 8 , we conclude that limn A2,n = 2. 1 1 1 1 Lemma 4.5. For any a, b > 0, it holds that kGn (a)− 2 Rn Gn (b)− 2 ksp = O(Nn ) and kGn (a)− 2 Rn Gn (b)− 2 k2F = O(Nn2 ) as n → ∞. 1 1 1 1 Proof. First, by definition we have kGn (a)− 2 Rn Gn (b)− 2 ksp = spr(Gn (b)− 2 Gn (a)− 2 Rn ), hence (4.2) and Theorem 5.6.9 of Horn and Johnson (1985) yield − 12 kGn (a) − 12 Rn Gn (b) ksp n X n 4 X p ≤ ≤ max 1≤i≤n 2n + 1 fa (2ξk )fb (2ξk ) k=1 k=1 uik uk1 1 1 1 + fa (2ξk ) fb (2ξk ) . 1 Therefore, Lemma 4.3 implies that kGn (a)− 2 Rn Gn (b)− 2 ksp = O(Nn ). Moreover, since it holds that 1 1 kGn (a)− 2 Rn Gn (b)− 2 k2F = tr(Gn (a)−1 Rn Gn (b)−1 Rn ) n X u1k uk1 u1l ul1 = ≤ fa (2ξk )fb (2ξl ) n 1 4 X 2n + 1 fa (2ξk ) k,l=1 k=1 ! n 4 X 1 2n + 1 fb (2ξl ) l=1 ! , Lemma 4.3 again yields the desired result. Lemma 4.6. For any a > 0, we have 1 1 1 kGn (a)− 2 Sn Gn (a)− 2 k2sp = O(Nn ), 1 rn2 kGn (a)− 2 Sn Gn (a)− 2 k2F → Jγ0 (a) (4.7) as n → ∞, where for any a > 0 we set  6  if γ = 0,   a2  1/2 3/2 a a 1 + a+4γ if 0 < γ < ∞, Jγ0 (a) = 2γ 2 2 − 3 a+4γ     0 if γ = ∞. Proof. Since Sn − Rn = Fn , by Lemma 4.5 it suffices to prove (4.7) in the case where Sn is replaced by Fn . (4.1)–(4.2) imply that 1 1 λi ≤ min{vn−1 , 4n/a} 1≤i≤n fa (2ξi ) kGn (a)− 2 Fn Gn (a)− 2 ksp = max 19 (4.8) and 1 1 rn2 kGn (a)− 2 Fn Gn (a)− 2 k2F = rn2 n X i=1 λ2i . fa (2ξi )2 (4.9) The first equation in (4.7) immediately follows from (4.8). In order to prove the second equation in (4.7), we will prove the right side of (4.9) converges to Jγ0 (a) as n → ∞. First, if γ = 0, the desired result follows from Lemma 4.2. Next, if 0 < γ < ∞, noting that fa (2ξi ) = a/n + vn λi , we have a 2 1 1 1 a λ2i + = 2 1−2 , fa (2ξi )2 vn n fa (2ξi ) n fa (2ξi )2 hence by Lemma 4.3 we obtain the desired equation once we prove n 1 X 1 1 a lim = p . 1+ n→∞ n3 f (2ξi )2 a + 4γ 2a a2 + 4aγ i=1 a The monotonicity of the cosine function yields 2n + 1 2π Z π 0 n X 1 1 1 2n + 1 dx ≤ ≤ + 2 2 2 fa (x) fa (2ξi ) fa (2ξ1 ) 2π i=1 Z π 0 1 dx. fa (x)2 Since formula (3.661.4) in Gradshteyn and Ryzhik (2007) implies that Z π π 1 a/n p dx = , 1+ 2 a/n + 4vn 2(a/n) (a/n)2 + 4avn /n 0 fa (x) we obtain the desired result. Finally, if γ = ∞, using the inequality fa (2ξi ) ≥ vn λi we obtain rn2 n X i=1 λ2i 1 n , ≤ 3 2 =√ 2 fa (2ξi ) N n vn nvn hence we deduce the desired result. Lemma 4.7. If a and b are positive numbers such that a 6= b, we have  2   ab √  √ 1 1 b(b+4γ)− a(a+4γ) 2 −2 −2 2 lim rn kGn (a) Tn Gn (b) kF = − γ 2 (b−a) n→∞    √ 2√ a+ b if γ = 0, 1 γ2 if 0 < γ < ∞, if γ = ∞. Proof. Since u0i = u1i , we have (Un Tn Un )ij + (Un Rn Un )ij = (Un (Tn + Rn )Un )ij = n X (ui,k−1 − ui,k+1 )ukj . k=1 x−y Using the trigonometric identities cos(x) − cos(y) = −2 sin( x+y 2 ) sin( 2 ) and 2 sin(x) cos(y) = sin(x + y) − sin(x − y), we obtain n (Un Tn Un )ij + (Un Rn Un )ij = X 4 sin(2ξi ) {sin ((2k − 1) (ξi + ξj )) + sin ((2k − 1) (ξi − ξj ))} . 2n + 1 k=1 20 Then, using summation formula (1.342.3) of Gradshteyn and Ryzhik (2007), we have 2 sin (n(ξi + ξj )) sin2 (n(ξi − ξj )) 4 ij ij (Un Tn Un ) + (Un Rn Un ) = sin(2ξi ) + 1{i6=j} . 2n + 1 sin(ξi + ξj ) sin(ξi − ξj ) (4.10) Now, since (Un Tn Un )⊤ = −Un Tn Un and (Un Rn Un )⊤ = Un Rn Un , by (4.2), (4.10) and the unitary invariance of the Frobenius norm, we obtain 1 1 1 1 kGn (a)− 2 Tn Gn (b)− 2 k2F − kGn (a)− 2 Rn Gn (b)− 2 k2F 4 2 X n 4 sin (n(ξi + ξj )) sin4 (n(ξi − ξj )) sin4 (n(ξi − ξj )) =− ga (2ξi )gb (2ξj ) − 1{i<j} − 1{i>j} 2n + 1 sin2 (ξi + ξj ) sin2 (ξi − ξj ) sin2 (ξi − ξj ) i,j=1 =: B1,n + B2,n + B3,n . First we consider B1,n . Using inequalities (4.3) and (x + y)2 ≥ 4xy (x, y ∈ R), we have X n 1 \|B1,n \| ≤ 4 max ga (2ξi ) max gb (2ξi ) 1 1≤i≤n 1≤i≤n (i − 2 + j − 12 )2 i,j=1 !2 X n 1 ≤ 4 max ga (2ξi ) max gb (2ξi ) , 1≤i≤n 1≤i≤n 2i − 1 i=1 and thus Lemma 4.1 yields B1,n = O((Nn log n)2 ) = o(rn−2 ). Next we consider B2,n . First we prove B2,n = B′2,n + o(rn−2 ), where 2 X n sin4 (n(ξi − ξj )) 4 ′ ga (2ξi )gb (2ξi ) 1{i<j} . B2,n = 2n + 1 sin2 (ξi − ξj ) i,j=1 Lemma 4.1 and (4.3) yield B2,n − B′2,n ≤ ≤ ≤ 4 2n + 1 2 4 (2n + 1)2 n sin4 (n(ξi − ξj )) 6n X ga (2ξi )\|ξi − ξj \| 1{i<j} b sin2 (ξi − ξj ) 6π 2 n b 4 6πn 2n + 1 b n X i,j=1 n X ga (2ξi ) i,j=1 ga (2ξi ) n X 1 j=1 i=1 1 1 \|ξi − ξj \| {i<j} j . If γ = ∞, by the property of ga we have n 2π 2π X ga (2ξi ) ≤ 2 max ga (2ξi ) + 2n + 1 2n + 1 1≤i≤n i=1 ≤ hence Lemma 4.1 yields Pn Lemma 4.3 because ga (x) i=1 ga (2ξi ) ≤ fa (x)−1 . 4 2n + 1 2ξn ga (x)dx 2ξ1 1 fa (2ξn ) 2π 2 max ga (2ξi ) + log , 2n + 1 1≤i≤n 2vn fa (2ξ1 ) = O(Nn2 log n). This also holds true in the case that γ < ∞ due to Consequently, B2,n − B′2,n = O((Nn log n)2 ) = o(rn−2 ). Now, since ξi − ξj = 2ξ1 (i − j), for any c ∈ (0, 1) we have Z 2 X n n X X sin4 (n · 2jξ1 ) sin4 (n · 2jξ1 ) 4 ′ g (2ξ )g (2ξ ) ga (2ξi )gb (2ξi ) ≤ B ≤ . a i i b 2,n 2n + 1 sin2 (2jξ1 ) sin2 (2jξ1 ) j=1 i=1 i=1 j=1 2 ⌊nc⌋ X n−⌊nc⌋ 21 Therefore, Lemma 4.4 implies that 2 lim inf rn2 n→∞ ⌊nc⌋ X i=1 ga (2ξi )gb (2ξi ) ≤ lim inf rn2 B′2,n ≤ lim sup rn2 B′2,n ≤ 2 lim sup rn2 n→∞ n→∞ n→∞ n X ga (2ξi )gb (2ξi ). i=1 Then, letting c ↑ 1, by Lemma 4.3 we obtain limn→∞ rn2 B2,n = limn→∞ rn2 B′2,n = 2 limn→∞ rn2 Pn i=1 ga (2ξi )gb (2ξi ). By symmetry we have limn→∞ rn2 B3,n = limn→∞ rn2 B2,n , hence we complete the proof due to (4.5) and Lemma 4.5. Proof of Proposition 2.2. Set 1 Ūn = √ 2 " Un −Un Un Un # . Then we have −1 −1 Ḡn 2 S̄n Ḡn 2 1 = Ūn 2 " Ln Un (Sn + ρ−1 Rn )Un Ln 0 0 −Ln Un (Sn − ρ−1 Rn )Un Ln −1 −1 Ḡn 2 T̄n Ḡn 2 1 = Ūn 2 " 0 Ln Un (Tn + ρ−1 Rn )Un Ln # Ūn⊤ and −Ln Un (Tn − ρ−1 R n )Un Ln 0 # Ūn⊤ , 1 1 where Ln = Λn (1 + ρ)− 2 and Ln = Λn (1 − ρ)− 2 . Hence we obtain −1 −1 4kḠn 2 (αT̄n + β S̄n )Ḡn 2 k2F 1 1 = 2α2 kGn (1 + ρ)− 2 (Tn + ρ−1 Rn )Gn (1 − ρ)− 2 k2F 1 1 1 1 + β 2 (kGn (1 + ρ)− 2 (Sn + ρ−1 Rn )Gn (1 + ρ)− 2 k2F + kGn (1 − ρ)− 2 (Sn − ρ−1 Rn )Gn (1 − ρ)− 2 k2F ). −1 −1 1 Therefore, Lemmas 4.5–4.7 yield (2.7). On the other hand, since 2kḠn 2 S̄n Ḡn 2 ksp ≤ kGn (1 + ρ)− 2 (Sn + 1 1 −1 −1 1 ρ−1 Rn )Gn (1+ρ)− 2 ksp +kGn (1−ρ)− 2 (Sn −ρ−1 Rn )Gn (1−ρ)− 2 ksp , Lemmas 4.5–4.6 also yield kḠn 2 S̄n Ḡn 2 ksp = O(Nn ). −1 −1 −1 −1 Hence the proof is completed once we prove kḠn 2 T̄n Ḡn 2 ksp = O(Nn ). Note that Ḡn 2 Vn (ϑ)Ḡn 2 is positive −1 −1 −1 −1 −1 −1 semidefinite and ρϑḠn 2 T̄n Ḡn 2 + Ḡn 2 Vn (ϑ)Ḡn 2 = E2n − ρ\|ϑ\|Ḡn 2 S̄n Ḡn 2 for any ϑ ∈ Θn . Note also that −1 −1 both T̄n and S̄n are symmetric. Therefore, if λ is an eigenvalue of Ḡn 2 T̄n Ḡn 2 , by the monotonicity theorem −1 −1 for eigenvalues (Corollary 4.3.3 of Horn and Johnson (1985)) we have ρϑλ ≤ kE2n − ρ\|ϑ\|Ḡn 2 S̄n Ḡn 2 ksp for −1 −1 any ϑ ∈ Θn . Since we can take ϑ = ±Nn−1 , this inequality implies that \|ρ\|Nn−1 kḠn 2 T̄n Ḡn 2 ksp ≤ kE2n − −1 −1 −1 −1 ρNn−1 Ḡn 2 S̄n Ḡn 2 ksp ≤ 1 + \|ρ\|Nn−1 kḠn 2 S̄n Ḡn 2 ksp , which yields the desired result. 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Robust Estimation via Robust Gradient Estimation arXiv:1802.06485v1 [stat.ML] 19 Feb 2018 Adarsh Prasad‡ Arun Sai Suggala‡ Sivaraman Balakrishnan† Pradeep Ravikumar‡ Machine Learning Department‡ Department of Statistics† Carnegie Mellon University, Pittsburgh, PA 15213. Abstract We provide a new computationally-efficient class of estimators for risk minimization. We show that these estimators are robust for general statistical models: in the classical Huber ǫ-contamination model and in heavy-tailed settings. Our workhorse is a novel robust variant of gradient descent, and we provide conditions under which our gradient descent variant provides accurate estimators in a general convex risk minimization problem. We provide specific consequences of our theory for linear regression, logistic regression and for estimation of the canonical parameters in an exponential family. These results provide some of the first computationally tractable and provably robust estimators for these canonical statistical models. Finally, we study the empirical performance of our proposed methods on synthetic and real datasets, and find that our methods convincingly outperform a variety of baselines. 1 Introduction Robust estimation has a rich history in statistics with seminal contributions due to Box [4], Tukey [42], Huber [24], Hampel [21] and several others. In the classical analysis of statistical estimators, statistical guarantees are derived under strong model assumptions and in most cases these guarantees hold only in the absence of arbitrary outliers, and other deviations from the model assumptions. Strong model assumptions are rarely met in practice, and this has led to the development of robust inferential procedures and various associated statistical concepts such as the influence function, the breakdown point and the Huber ǫ-contamination model to assess the robustness of estimators. Despite this progress however, the statistical methods with the strongest robustness guarantees, for instance those based on non-convex M -estimators [24], ℓ1 tournaments [13, 15, 45] and notions of depth [10, 20, 38], are computationally intractable. In this paper, we present a class of estimators that are computationally tractable and have strong robustness guarantees. The estimators we propose are obtained by robustifying classical algorithms for risk minimization and are applicable to a wide-range of parametric statistical models for which parameter estimation can be cast within this framework. In contrast to classical work, for instance on M-estimation, we do not attempt to replace the risk minimization objective with a robust counterpart but instead focus on making canonical gradient-based optimization for the usual risk minimization objective robust. We find that this shift in perspective enables a unified treatment of different statistical models, leads to computationally tractable estimators and leads to estimators with strong robustness guarantees. In the risk minimization framework, the target parameter θ ∗ is defined as the solution to an optimization problem: (1) θ ∗ = argmin R(θ) ≡ argmin Ez∼P L(θ; z) , θ∈Θ θ∈Θ 1 where L is an appropriate loss-function, R is the population risk and Θ is the set of feasible parameters. The goal of empirical risk minimization procedures is then to compute an approximate minimizer to the above program when given access to samples Dn = {z1 , . . . , zn }. In this classical setting, a standard assumption that is imposed on Dn is that the data has no outliers, and has no arbitrary deviations from model assumptions; i.e., it is typically assumed that each of the zi ’s are independent and identically distributed according to the distribution P . Many analyses of risk minimization further assume that P follows a sub-gaussian distribution, or has otherwise well-controlled tails in order to appropriately control the deviation between the population risk and its empirical counterpart. While our general results can be specialized to obtain results for a variety of models and notions of robustness, we focus on developing estimators which are robust to two canonical classes of deviations from the model assumptions: 1. Robustness to arbitrary outliers: In this setting, we focus on Huber’s ǫ-contamination model, where rather than observe samples directly from P in (1) we instead observe samples drawn from Pǫ which for an arbitrary distribution Q is defined as: Pǫ = (1 − ǫ)P + ǫQ. The distribution Q allows for arbitrary outliers, which may correspond to gross corruptions or more subtle deviations from the assumed model. This model can be equivalently viewed as model mis-specfication in the Total Variation (TV) metric. 2. Robustness to heavy-tails: In this setting, we are interested in developing estimators under weak moment assumptions. We assume that the distribution P from which we obtain samples only has finite low-order moments (see Section 5.4 for a precise characterization). Such heavy tailed distributions arise frequently in the analysis of financial data and large-scale biological datasets (see for instance examples in [18, 47]). In contrast to classical analyses of empirical risk minimization [43], in this setting the empirical risk is not uniformly close to the population risk, and methods that directly minimize the empirical risk perform poorly (see Section 4). The goal of our work is to develop estimators which are computationally tractable and robust in these models. Below, we provide an outline of our results and contributions. 1. Our first contribution is to introduce a new class of robust estimators for risk minimization (1). These estimators are based on robustly estimating gradients of the population risk, and are computationally tractable by design. Building on prior work for robust mean estimation in the Huber model [29], and in the heavy-tailed model [37], we design robust gradient estimators for the population risk in (1). Our main insight is that in this general risk minimization setting, the gradient of the population risk is simply a multivariate mean vector, and we can leverage prior work on mean estimation to design robust gradient estimators. Through this we are able to significantly generalize the applicability of mean estimation methods to general parametric models. 2. Our estimators are practical and our second contribution is to conduct extensive numerical experiments on real and simulated data with our proposed estimators. We provide guidelines for tuning parameter selection and we compare the proposed estimators with several competitive baselines [3, 19, 24]. Across different settings and according to various metrics we find that our estimators consistently perform well. 2 3. Finally, we provide rigorous robustness guarantees for the estimators we propose for a variety of canonical statistical models including for linear regression, for logistic regression and for estimation of the canonical parameters in an exponential family. Our contributions in this direction are two-fold: building on prior work [2] we provide a general result on the stability of gradient descent for risk minimization, showing that in favorable cases gradient descent can be quite tolerant to inaccurate gradient estimates. Subsequently, in concrete settings we provide a careful analysis of the quality of gradient estimation afforded by our proposed gradient estimators and combine these results to obtain guarantees on our final estimates. Broadly, as we discuss in the sequel, our work suggests that estimators which are based on robust gradient estimation offer a variety of practical, conceptual, statistical and computational advantages for robust estimation. 1.1 Related Work There is extensive work broadly in the area of robust statistics (see for instance [21] and references therein), and we focus this section on some lines of work that are most related to this paper. Classical work has already developed several estimators which are known to be optimally robust for a variety of inferential tasks, including hypothesis testing [26], mean estimation [38], general parametric estimation [11, 15, 45], and non-parametric estimation [13]. However, a major drawback of this classical line of work has been that most of the estimators with strong robustness guarantees are computationally intractable [42], while the remaining ones are heuristics which are not optimal [22]. Recently, there has been a flurry of research in theoretical computer science [9, 14, 16, 29, 32] designing provably robust estimators which are computationally tractable while achieving near-optimal contamination dependence for special classes of problems. Some of the proposed algorithms, are not practical as they rely on the ellipsoid algorithm or require solving semidefinite programs [9, 14, 16, 32] which can be slow for modern problem sizes. We build on the work of Lai et al. [29], who study practical robust mean and covariance estimators for distributions with appropriately controlled moments. A complementary line of recent research [10, 20] has focused on providing minimax upper and lower bounds on the performance of estimators under ǫ-contamination model, without the constraint of computational tractability. In the ǫ-contaminated model, Q can be arbitrary, but there has been a lot of work in settings where the contamination distribution is restricted in various ways. For example, recent work in high-dimensional statistics (for instance [7, 12, 33, 34, 46]) have studied problems like principal component analysis and linear regression under the assumption that the corruptions are evenly spread throughout the dataset. Another line of research has focused on designing robust estimators under the heavy tailed distribution setting. These approaches relax the sub-gaussian or sub-exponential distributional assumptions that are typically imposed on the target distribution P and allow it to be a heavy tailed distribution. Most of the approaches in this category use robust mean estimators [8, 31] that exhibit sub-gaussian type concentration around the true mean for distributions satisfying mild moment assumptions. The median-of-means estimator [31] and Catoni’s mean estimator [8] are two popular examples of such robust mean estimators. Hsu and Sabato [23] use the median-of-means estimator to develop an alternative to ERM under heavy tails. Although this estimator has strong theoretical guarantees and is computationally tractable, as noted by the authors in [23] it performs poorly in practice. In recent 3 work Brownlees et al. [5] replace empirical mean in the empirical risk minimization framework (ERM) with Catoni’s mean estimator and perform risk minimization. The authors provide risk bounds similar to the bounds one can achieve under sub-gaussian distributional assumptions. However, their estimator is not easily computable and the authors do not provide a practical algorithm to compute the estimator. Other recent works by Lerasle and Oliveira [31], Lugosi and Mendelson [35] use similar ideas to derive estimators that perform well theoretically, in heavy-tailed situations. However, these approaches involve optimization of complex objectives for which no computationally tractable algorithms exist. We emphasize that in contrast to our work, these works focus on robustly estimating the population risk which does not directly lead to a computable estimator. We instead consider robustly estimating the gradient of the population risk. When complemented with the gradient descent algorithm this leads naturally to a computable estimator. 1.2 Outline We conclude this section with a brief outline of the remainder of the paper. In Section 2, we provide some background on risk minimization and the Huber and heavy-tailed noise models. In Section 3, we introduce our class of estimators and provide concrete algorithms for the ǫ-contaminated and heavy-tailed setting. In Section 4 we study the empirical performance of our estimator on a variety of tasks and datasets. We complement our empirical results with theoretical guarantees in Sections 5, 6 and 7. We defer technical details to the Appendix. Finally, we conclude in Section 8 with a discussion of some open problems. 2 Background and Problem Setup In this section we provide the necessary background on risk minimization, gradient descent and introduce two notions of robustness that we consider in this work. 2.1 Risk Minimization and Parametric Estimation In the setting of risk minimization, we assume that we have access to a differentiable loss function L : Θ × Z 7→ R, where Θ is a convex subset of Rp . Let R(θ) = Ez∼P L(θ; z) be the population loss (risk), and let θ ∗ be the minimizer of the population risk R(θ), over the set Θ θ ∗ = argmin R(θ). (2) θ∈Θ The goal of risk minimization is to minimize the population risk R(θ), given only n samples Dn = {zi }ni=1 . Whereas in parameter estimation we are interested in estimating the unknown parameter θ ∗ from samples Dn . In this work we assume that the population risk is convex to ensure tractable minimization. Moreover, in order to ensure identifiability of the parameter θ ∗ , we impose two standard regularity conditions [6] on the population risk. These properties are defined in terms of the error of the first-order Taylor approximation of the population risk, i.e. defining, τ (θ1 , θ2 ) := R(θ1 ) − R(θ2 ) − h∇R(θ2 ), θ1 − θ2 i, we assume that τℓ τu kθ1 − θ2 k22 ≤ τ (θ1 , θ2 ) ≤ kθ1 − θ2 k22 , 2 2 (3) where the parameters τℓ , τu > 0 denote the strong-convexity and smoothness parameters respectively. 4 2.2 Gradient Descent and Empirical Risk Minimization A starting point for the techniques we develop in this paper is the classical projected gradient descent method for empirical risk minimization. Given data Dn = {zi }ni=1 , empirical risk minimization (ERM) estimates the unknown parameter θ ∗ as the minimizer of the empirical risk, i.e. n 1X L(θ; zi ). θbn = argmin Rn (θ) := n θ∈Θ i=1 A popular method for solving this optimization problem is projected gradient descent. Projected gradient descent generates a sequence of iterates {θ t }∞ t=0 , by refining an initial parameter θ0 ∈ Θ via the update: θ t+1 = PΘ θ t − η∇Rn (θ t ) , where η > 0 is the step size and Pθ is the projection operator onto Θ. Despite its simplicity, the gradient descent method is not robust for general convex losses. Furthermore, the empirical risk minimizer is a poor estimator of θ ∗ in the presence of outliers in the data: since ERM depends on the sample mean, outliers in the data can effect the sample mean and lead ERM to sub-optimal estimates. This observation has led to a large body of research that focuses on developing robust M-estimators which have favorable statistical properties, but are often computationally intractable. In this work we take a different approach. Our work relies on an important observation that the gradient of the population risk (Ez∼P ∇L(θ; z) ) is simply a mean vector: one that can be estimated robustly by leveraging recent advances in robust mean estimation [29, 37]. This leads to a general method for risk minimization based on robust gradient estimation (see Algorithm 1). 2.3 Robust Estimation One of the goals of this work is to develop general statistical estimation methods that are robust in one of the following two models: Huber’s ǫ-contamination model or the heavy-tailed model. We now briefly review these two notions of robustness. 1. Huber’s ǫ-contamination model: Huber [25, 26] proposed the ǫ-contamination model where we observe samples that are obtained from a mixture of the form Pǫ = (1 − ǫ)P + ǫQ, (4) where P is the true distribution, ǫ is the expected fraction of outliers and Q is an arbitrary outlier distribution. Given i.i.d. observations drawn from Pǫ , our objective is to estimate θ ∗ , the minimizer of the population risk R(θ) = Ez∼P L(θ; z) , robust to the contamination from Q. 2. Heavy-tailed model: In the heavy-tailed model it is assumed that the data follows a heavy-tailed distribution (i.e, P is heavy-tailed). While heavy-tailed distributions have various possible characterizations: in this paper we consider a characterization via gradients. For a fixed θ ∈ Θ we let Pgθ denote the multivariate distribution of the gradient of population loss, i.e. ∇L(θ; z). We refer to a heavy-tailed distribution as one for which 5 Pgθ has finite second moments for any θ ∈ Θ. As we illustrate in Section 7, in various concrete examples this translates to relatively weak low-order moment assumptions on the data distribution P . Given n i.i.d observations from P , our objective is to estimate the minimizer of the population risk. From a conceptual standpoint, the classical analysis of risk-minimization which relies on uniform concentration of the empirical risk around the true risk, fails in the heavy-tailed setting necessitating new estimators and analyses [5, 23, 35, 36]. 3 Gradient Estimation Gradient descent and its variants are at the heart of modern optimization and are well-studied in the literature. Suppose we have access to the true distribution Pθ∗ . Then to minimize the population risk R(θ), we can use projected gradient descent, where starting at some initial θ 0 and for an appropriately chosen step-size η, we update our estimate according to: θ t+1 ← PΘ (θ t − η∇R(θ t )). (5) However, we only have access to n samples Dn = {zi }ni=1 . The key technical challenges are then to estimate the gradient of R(θ) from samples Dn , and to ensure that an appropriate modification of gradient descent is stable to the resulting estimation error. To address the first challenge we observe that the gradient of the population risk at any point θ is the mean of a multivariate distribution, i.e. ∇R(θ) = Ez∼P ∇L(θ; z) . Accordingly, the problem of gradient estimation can be reduced to a multivariate mean estimation problem, where our goal is to robustly estimate the true mean ∇R(θ) from n samples {∇L(θ; zi )}ni=1 . For a given sample-size n and confidence parameter δ ∈ (0, 1) we define a gradient estimator: Definition 1. A function g(θ; Dn , δ) is a gradient estimator, if for functions α and β, with probability at least 1 − δ, at any fixed θ ∈ Θ, the estimator satisfies the following inequality: kg(θ; Dn , δ) − ∇R(θ)k2 ≤ α(n, δ)kθ − θ ∗ k2 + β(n, δ). (6) In subsequent sections, we will develop conditions under which we can obtain gradient estimators with strong control on the functions α(n, δ) and β(n, δ) in the Huber and heavy-tailed models. Furthermore, by investigating the stability of gradient descent we will develop sufficient conditions on these functions such that gradient descent with an inaccurate gradient estimator still returns an accurate estimate. To minimize R(θ), we replace ∇R(θ) in equation (5) with the gradient estimator g(θ; Dn , δ) and perform projected gradient descent. In order to avoid complex statistical dependency issues that can arise in the analysis of gradient descent, for our theoretical results we consider a sample-splitting variant of the algorithm where each iteration is performed on a fresh batch of samples (see Algorithm 1). We further assume that the number of gradient iterations T is specified a-priori, and accordingly we define: jnk δ and δe = , (7) n e= T T We discuss methods for selecting T , and the impact of sample-splitting in later sections. As confirmed in our experiments (see Section 4), sample-splitting should be viewed as a device introduced for theoretical convenience which can likely be eliminated via more complex uniform arguments (see for instance the work [2]). 6 Algorithm 1 Projected Gradient Descent function PGD({z1 , . . . , zn }, Step Size η, Number of Iterations T , δ) e. Split samples into T subsets {Zt }Tt=1 of size n for t = 0 to T − 1 do e k2 . θ t+1 = argminθ∈Θ kθ − θ t − ηg(θ t ; Zt , δ) 2 end for end function Next, we consider the two notions of robustness described in Section 2, and derive specific gradient estimators for each of the models using the framework described above. Although the major focus of this work is on Huber contamination and heavy-tailed models, our class of estimators are more general and are not restricted to these two notions of robustness. 3.1 Gradient Estimation in Huber’s ǫ-contamination model There has been a flurry of recent interest [9, 14, 16, 29, 32] in designing mean estimators which, under the Huber contamination model, can robustly estimate the mean of a random vector. While some of these results are focused on the case where the uncorrupted distribution is Gaussian, or isotropic, we are more interested in robust mean oracles for more general distributions. Lai et al. [29] proposed a robust mean estimator for general distributions, satisfying weak moment assumptions, and we leverage the existence of such an estimator to design a Huber gradient estimator g(θ; Dn , δ) which works in the Huber contamination model (see Algorithm 2). Now, we briefly describe the main idea behind Algorithm 2 and the mean estimator of Lai et al. [29]. The algorithm builds upon the fact that in one-dimension, it is relatively easy to estimate the gradient robustly. In higher-dimension, the crucial insight of Lai et al. [29] is that the effect of the contamination Q on the mean of uncontaminated distribution P is effectively one-dimensional provided we can accurately estimate the direction along which the mean is shifted. In our context, if we can compute the gradient shift direction, i.e. the direction of the difference between the sample (corrupted) mean gradient and the true (population) gradient, then the true gradient can be estimated by using a robust 1D-mean algorithm along the gradient-shift direction and a non-robust sample-gradient in the orthogonal direction since the contamination has no effect on the gradient in this orthogonal direction. In order to identify the gradient shift direction, we use a recursive Singular Value Decomposition (SVD) based algorithm. In each stage of the recursion, we first remove gross-outliers via a truncation algorithm (described in more detail in the Appendix) and subsequently identify two subspaces using an SVD – a clean subspace where the contamination has a small effect on the mean and another subspace where the contamination has a potentially larger effect. We use a simple sample-mean estimator in the clean subspace and recurse our computation on the other subspace. Building on the work of Lai et al. [29], in Lemma 1 and Appendix J we provide a careful analysis of this gradient estimator. 7 Algorithm 2 Huber Gradient Estimator function HuberGradientEstimator(Sample Gradients S = {∇L(θ; zi )}ni=1 , Corruption Level ǫ, Dimension p, δ) Se = HuberOutlierGradientTruncation(S, ǫ, p, δ). if p=1 then e return mean(S) else e Let ΣSe be the covariance matrix of S. Let V be the span of the top p/2 principal components of ΣSe and W be its complement. e where PV is the projection operation on to V . Set S1 := PV (S) Let µ bV := HuberGradientEstimator(S1 , ǫ, p/2, δ). e Let µ bW := mean(PW S). p Let µ b ∈ R be such that PV (b µ) = µ bV , and PW (b µ) = µ bW . return µ b. end if end function 3.2 Gradient Estimation in the Heavy-Tailed model To design gradient estimators for the heavy-tailed model, we leverage recent work on designing robust mean estimators in this setting. These robust mean estimators build on the classical work of Alon et al. [1], Nemirovski and Yudin [39] and Jerrum et al. [27] on the so-called median-of-means estimator. For the problem of one-dimensional mean estimation, Catoni [8], Lerasle and Oliveira [31] propose robust mean estimators that achieve exponential concentration around the true mean for any distribution with bounded second moment. In this work we require mean estimators for multivariate distributions. Several recent works ([23, 36, 37]) extend the median-of-means estimator of to general metric spaces. In this paper we use the geometric median-of-means estimator (Gmom), which was originally proposed and analyzed by Minsker [37], to design the gradient estimator g(θ; Dn , δ). The basic idea behind the Gmom estimator is to first split the samples into non-overlapping subsamples and estimate the sample mean of each of the subsamples. Then the Gmom estimator is given by the median-of-means of the subsamples. Formally, let {xi . . . xn } ∈ R be n i.i.d random variables sampled from a distribution P . Then the Gmom estimator for estimating the mean of P can be described as follows. Partition the n samples into b blocks B1 , P . . . Bb each 1 of size ⌊n/b⌋. Let {b µ1 , . . . , µ bb } be the sample means in each block, where µ bi = \|Bi \| xj ∈Bi xj . Then the Gmom estimator is given by median{b µ1 , . . . µ bb }. In high dimensions where different notions of the median have been considered Minsker [37] uses geometric median: µ b = argmin µ b X i=1 kµ − µ bi k2 . Algorithm 3 presents the gradient estimator g(θ; Dn , δ) obtained using Gmom as the mean estimator. 8 Algorithm 3 Heavy Tailed Gradient Estimator function HeavyTailedGradientEstimator(Sample Gradients S = {∇L(θ; zi )}ni=1 , δ) Define number of buckets b = 1 + ⌊3.5 log 1/δ⌋. Partition S into b blocks B1 , . . . Bb each of size ⌊n/b⌋. for i = 1 . . . nX do s. µ bi = \|B1i \| s∈Bi end for Let µ b = argmin µ return µ b. end function 4 b X i=1 kµ − µ bi k2 . Experiments In this section we demonstrate our proposed methods for Huber contamination and heavytailed models, on a variety of simulated and real data examples. 4.1 Huber Contamination We first consider the Huber contamination model and demonstrate the practical utility of gradient-descent based robust estimator described in Algorithms 1 and 2. 4.1.1 Synthetic Experiments: Linear Regression In linear regression we observe paired samples {(x1 , y1 ), . . . (xn , yn )}, where each (xi , yi ) ∈ Rp × R. We assume that the (x, y) pairs sampled from the true distribution P are linked via a linear model: y = hx, θ ∗ i + w, (8) where w is drawn from a zero-mean normal distribution with variance σ 2 (w ∼ N (0, σ 2 )). We use the squared loss as our loss function 1 L(θ; (x, y)) = (y − hx, θi)2 . 2 Note that the true parameter θ ∗ is the minimizer of the resulting population risk R(θ). We now describe the experiment setup, the data model and present the results. Setup We fix the contamination level ǫ = 0.1 and σ 2 = 0.1. Next, we generate (1 − ǫ)n clean covariates from x ∼ N (0, Ip ), the corresponding clean responses using y = hx, θ ∗ i + w where θ ∗ = [1, . . . , 1]T and w ∼ N (0, σ 2 ). We simulate an outlier distribution by drawing the covariates from N (0, p2 Ip ), and setting the responses to 0. The total number of samples are set to be(10 ǫp2 ) i.e. the sample size increases with the dimension. This scaling is used to ensure that the statistical (minimax) error, in the absence of any contamination, is roughly 0.001. An optimally robust method should have error close to 0.1 (roughly equal to corruption level), which ours does (see Figure 1). 9 25 1.7 OLS RobustGD TORRENT Huber Plugin RANSAC 1.6 1.5 Parameter Error \|\|θb − θ∗ \|\|2 20 ×10-3 15 10 1.4 1.3 1.2 1.1 5 1 0 100 150 200 250 300 350 400 450 0.9 0.1 500 0.15 0.2 p 0.25 0.3 ǫ (a) Parameter error vs p for ǫ = 0.1 (b) Parameter error vs ǫ 3 epsilon=0.1 epsilon=0.2 epsilon=0.3 epsilon=0.4 log(Parameter Error) 2 1 0 -1 -2 -3 0 10 20 30 40 50 60 Iterations (c) log(kθt − θ∗ k2 ) vs t for different ǫ. Figure 1: Robust Linear Regression. Metric We measure the parameter error in ℓ2 -norm. We also study the convergence properties of our proposed method, for different contamination levels ǫ. We use code provided by Lai et al. [29] to implement our gradient estimator. Baselines We use OLS, TORRENT [3], Huber-loss, RANSAC and plugin estimator as our baselines. TORRENT is an iterative hard-thresholding based alternating minimization algorithm, where in one step, it calculates an active set of examples by keeping only (1−ǫ)n samples which have the smallest absolute values of residual r = y − x, θ t , and in the other step it updates the current estimates by solving OLS on the active set. Bhatia et al. [3] had shown the superiority of TORRENT over other convex-penalty based outlier techniques, hence, we do not compare against those methods.P The plugin estimator is implemented using P Algorithm 2 to estimate both the mean vector n1 ni=1 yi xi and the covariance matrix n1 ni=1 xi xTi . Results We summarize our main findings here. 1. All estimators except our proposed algorithm perform poorly (Figure 1(a)). Note that the TORRENT algorithm has strong guarantees when only the response y is corrupted but 10 performs poorly in the Huber contamination model where both x and y may be contaminated. The error for the robust plugin estimator increases with dimension. We investigate this theoretically in Section 6.1, where we find that the error of the plugin estimator grows √ with the norm of θ ∗ . In our experiment, we choose kθ ∗ k2 = p, and thus Figure 1(a) corroborates Corollary 3 in Section 6.1. 2. In Figure 1(b) we find that the parameter error kθb − θ ∗ k2 increases linearly with the contamination rate ǫ and we study this further in Section 6.1. 3. Finally, Figure 1(c) shows that the convergence rate decreases with increasing contamination ǫ and after ǫ is high enough, the algorithm remains stuck at θ0 , corroborating Lemma 8 (in the Appendix). Next, we study the performance of our proposed method in the context of classification. 4.1.2 Synthetic Experiments: Logistic Regression In logistic regression we observe paired samples, {(x1 , y1 ), . . . , (xn , yn )} where each (xi , yi ) ∈ Rp × R. We assume that the (x, y) pairs sampled from the true distribution P are linked via a linear model: ( 1 1 with probability 1+exp(−hx, θ ∗ i) , (9) y= 0 otherwise. In this case, we use the negative conditional log-likelihood as our loss function, i.e. L(θ; (x, y)) = −yhx, θi + log(1 + exp(hx, θi)). Setup We simulate a linearly separable classification problem, where the clean covariates are sampled from N (0, Ip ), the corresponding clean responses are computed as y = sign(hx, θ ∗ i) √ √ where θ ∗ = [1/ p, . . . , 1/ p]T . We simulate the outlier distribution by adding asymmetric noise, i.e. we flip the labels of one class, and increase the variance of the corresponding covariates by multiplying them by p2 . The total number of samples are set to be (10 pǫ ). Metric We measure the 0-1 classification error on a separate (clean) test set. We study how the 0-1 error changes with p and ǫ and the convergence properties(parameter error) of our proposed method for different contamination levels ǫ. Baselines We use the logistic regression MLE and the linear Support Vector Machine (SVM) as our baselines. Results Figures 2(b) and 2(c) show qualitatively similar results to the linear regression setting, i.e. that the error of our proposed estimator degrades gracefully (and grows linearly) with the contamination level ǫ and that the gradient descent iterates converge linearly. In Figure 2(a) we observe that both the SVM and logistic regression MLE perform poorly. The logistic regression MLE completely flips the labels and has a 0-1 error close to 1, whereas the linear SVM outputs a random hyperplane classifier that flips the label for roughly half of the dataset. 11 1 1 RobustGD logisticRegression SVM 0.9 0.8 0.7 0.7 0.6 0.6 0-1 error 0-1 error 0.8 0.5 0.4 0.5 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 100 150 200 250 300 350 400 450 RobustGD 0.9 0 0.1 500 0.15 0.2 0.25 p 0.3 0.35 0.4 0.45 epsilon (a) 0-1 Error vs p at ǫ = 0.1 (b) 0-1 error vs ǫ 0.5 eta=0.1 eta=0.2 eta=0.3 eta=0.4 log(Parameter Error) 0 -0.5 -1 -1.5 -2 -2.5 0 100 200 300 400 500 600 Iterations (c) log(kθt − θ∗ k2 ) vs t for different ǫ Figure 2: Robust Logistic Regression. 4.1.3 Robust Face Reconstruction Setup In this experiment, we show the efficacy of our algorithm by attempting to reconstruct face images that have been corrupted with heavy occlusion, where the occluding pixels play the role the outliers. We use the data from the Cropped Yale Dataset [30] . The dataset contains 38 subjects, and each image has 192×168 pixels. Following the methodology of Wang et al. [44], we choose 8 face images per subject, taken under mild illumination conditions and computed an eigenface set with 20 eigenfaces. Then given a new corrupted face image of a subject, the goal is to get the best reconstruction/approximation of the true face. To remove the scaling effects, we normalized all images to [0, 1] range. One image per person was used to test reconstruction. Occlusions were simulated by randomly placing 10 blocks of size 30 × 30. We repeated this 10 times for each test image. Note that in this example, we use a linear regression model as the uncontaminated statistical model, which is almost certainly not an Table 1: Fitting to original image error. Mean RMSE Best Possible Proposed TORRENT OLS SCRRR 0.05 0.09 0.175 0.21 0.13 12 exact match for the unknown ground truth distribution. Despite this model misspecification, as our results show, that robust mean based gradient algorithms do well. Metric We use Root Mean Square Error (RMSE) between the original and reconstructed image to evaluate the performance of the algorithms. We also compute the best possible reconstruction of the original face image by using the 20 eigenfaces. Methods We use TORRENT, OLS as baselines. Wang et al. [44] implemented popular robust estimators such as RANSAC, Huber Loss etc. and showed their poor performance. Wang et al. [44] then proposed an alternate robust regression algorithm called Self Scaled Regularized Robust Regression(SCRRR), and showed its equivalence to and ℓ1 -penalized method. We also compare against the best possible RMSE obtained by reconstructing the un-occluded image using the eigenfaces. Results Table 1 shows that the mean RMSE is best for our proposed gradient descent based method and that the recovered images are in most cases closer to the un-occluded original image. (Figure 4.1.3). Figure 3(c) shows a case when none of the methods succeed in reconstruction. (a) Successful Reconstruction (b) Successful Reconstruction (c) Failed Reconstruction Figure 3: Robust Face recovery results: Top; in order from L to R: original image, occluded image, best possible recovery with given basis. Bottom; in order from L to R: Reconstructions using our proposed algorithm, TORRENT and ordinary least squares (OLS). 4.2 Heavy-tailed Estimation We now consider the heavy-tailed model and present experimental results on synthetic and real world datasets comparing the gradient descent based robust estimator described in Algo13 rithms 1 and 3 (which we call GMOM) with ERM and several other recent proposals. In these experiments we focus on the problem of linear regression which is described in Section 4.1 and work with heavy-tailed noise distributions. 4.2.1 Synthetic Experiments: Simple Linear Regression Setup. The covariate x ∈ Rp is sampled from a zero-mean isotropic gaussian distribution. √ We set each entry of θ ∗ to 1/ p. The noise w is sampled from a Pareto distribution, with mean zero, variance σ 2 and tail parameter β. The tail parameter β determines the moments of the Pareto random variable. More specifically, the moment of order k exists only if k < β, hence, smaller the β the more heavy-tailed the distribution. In this setup, we keep the dimension p fixed to 128 and vary n, σ and β. We always maintain the sample-size n to be at least 2p. Methods. We use ERM as our baseline and compare it with GMOM. Since we are always in the low-dimensional (n ≥ p) setting, the solution to ERM has a closed form expression and is simply the OLS solution. We also study ERM-GD, which performs a gradient descent on ERM and is equivalent to using empirical mean as the gradient oracle in our framework. We also compare against the robust estimation techniques of Hsu and Sabato [23] and Duchi and Namkoong [17]. In our experiments, all the iterative techniques are run until convergence. Hyper Parameter Selection. The GMOM estimator depends on the confidence parameter δ ∈ (0, 1), which needs to be tuned. In our experiments, we noticed that the performance of GMOM varies very little when δ is selected from a reasonable range that is not too close to 0 (see Figure 5 and the discussion below). So we set δ = 0.1 in all our simulations. Metrics. In our experiments, we vary σ and β, the parameters of Pareto distribution, which can change the minimal risk R(θ ∗ ). So to compare various approaches across parameter values, ∗ ) − 1.0, where θ b b is we use a scaled version of the Excess Risk, which we define as R(θ)/R(θ the estimator. To compare the performance of two estimators θb1 , θb2 , we define the notion of relative efficiency: (R(θb2 ) − R(θ ∗ )) − (R(θb1 ) − R(θ ∗ )) . RelEff(θb1 , θb2 ) = R(θb1 ) − R(θ ∗ ) Roughly, this corresponds to the percentage improvement in the excess risk obtained using θb1 over θb2 . Whenever RelEff(θb1 , θb2 ) > 0, θb1 has a lower risk, and higher the value, the more the fractional improvement. Results. To reduce the variance in the plots presented here and in the next section, we averaged results over 25 repetitions. Figure 4 shows the benefits of using GMOM over ERM. In Figure 4(a) we plot the excess risk of ERM, ERM-GD and GMOM against the number of Iterations. We see that upon convergence GMOM has a much lower population risk than ERM. As expected, ERM-GD converges to ERM. However, the population risk of ERM-GD in the first few iterations is much lower than the risk of ERM, suggesting early stopping. Next, in Figure 4(b) we plot the scaled excess risk for ERM and GMOM as n/p increases. We see that GMOM is always better than ERM, even when the number of samples is 12 times the dimension p. In Figure 4(c), we plot the relative efficiency of GMOM and ERM against σ. This shows that the percentage improvement in the excess risk by GMOM decreases as 14 the noise level σ decreases. This behavior is expected because in the noiseless setting both methods would have a similar behavior. We do a similar study to see the relative efficiency against the heavy-tailedness of the noise distribution. As noted before, as β is increased more moments exist for the underlying distribution. Figure 4(d) shows that as the noise distribution becomes more heavy-tailed, there is more benefit in using GMOM over ERM. 2 1 ERM ERM (GD) GMOM (GD) 1.9 Excess Risk / True Risk 1.8 1.7 Population Risk ERM GMOM (GD) 0.9 1.6 1.5 1.4 1.3 0.8 0.7 0.6 0.5 0.4 0.3 1.2 0.2 1.1 0.1 0 1 50 100 150 2 200 4 6 (a) Population Risk vs Iterations 10 12 (b) ExcessRisk/TrueRisk vs n/p 0.6 RelEff(GMOM,ERM) 1.2 8 n/p Iterations RelEff(GMOM,ERM) 0.5 Relative Efficiency Relative Efficiency 1 0.8 0.6 0.4 0.4 0.3 0.2 0.1 0.2 0 0 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 2 4 6 8 10 12 β+1 σ (c) Relative Efficiency vs σ (d) Relative Efficiency vs β + 1 Figure 4: Linear Regression: Performance comparison of GMOM and ERM. Dependence on Confidence Level. Figure 5(a) shows the performance of GMOM estimator for various values of δ. It can be seen that the choice of δ have very little effect on the performance of the estimator. However, we notice that for small values of δ the performance of the GMOM degrades. In practice, one can use either cross validation or a validation set for choosing δ. 5 Theoretical Preliminaries In this section we develop some theoretical preliminaries. We begin with a description of some canonical examples of risk minimization in Section 5.1. Next we develop a general 15 2 ERM δ=0.01 δ=0.05 δ=0.1 δ=0.2 1.9 1.8 Population Risk 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 50 100 150 200 Iterations (a) Population Risk vs δ Figure 5: Linear Regression: Dependence on Confidence Level δ. theory on convergence of projected gradient descent in Section 5.2. We analyze the gradient estimators defined in Algorithms 2 and 3 in Sections 5.3 and 5.4 respectively. Finally in Sections 6,7 we present consequences of our general theory for the canonical examples, under Huber contamination and heavy-tailed models. For some of our examples, we will assume certain mild moment conditions. Concretely, for a random vector x ∈ Rp , let µ = E[x] and Σ be the covariance matrix. Then x has bounded 2kth moments if there exists a constant C2k such that for every unit vector v we have that i h k . (10) E hx − µ, vi2k ≤ C2k E hx − µ, vi2 5.1 Illustrative Examples The framework of risk minimization is a central paradigm of statistical estimation and is widely applicable. In this section, we provide illustrative examples that fall under this framework. 5.1.1 Linear Regression Here we observe paired samples {(x1 , y1 ), . . . (xn , yn )}, where each (xi , yi ) ∈ Rp ×R. We assume that the (x, y) pairs sampled from the true distribution P are linked via a linear model: y = hx, θ ∗ i + w, (11) where w is drawn from a zero-mean distribution such as normal distribution with variance σ 2 (N (0, σ 2 )) or a more heavy-tailed distribution such as student-t or Pareto distribution. We suppose that under P the covariates x ∈ Rp , have mean 0, and covariance Σ. For this setting we use the squared loss as our loss function, which induces the following population risk: 1 1 (y − hx, θi)2 , and R(θ) = (θ − θ ∗ )T Σ(θ − θ ∗ ). 2 2 ∗ Note that the true parameter θ is the minimizer of the population risk R(θ). The strongconvexity and smoothness assumptions from (3) in this setting require that τℓ ≤ λmin (Σ) ≤ λmax (Σ) ≤ τu . L(θ; (x, y)) = 16 5.1.2 Generalized Linear Models Here we observe paired samples {(x1 , y1 ), . . . (xn , yn )}, where each (xi , yi ) ∈ Rp × Y. We suppose that the (x, y) pairs sampled from the true distribution P are linked via a linear model such that when conditioned on the covariates x, the response variable has the distribution: yhx, θ ∗ i − Φ(hx, θ ∗ i) (12) P (y\|x) ∝ exp c(σ) Here c(σ) is a fixed and known scale parameter and Φ : R 7→ R is the link function. We focus on the random design setting where the covariates x ∈ Rp , have mean 0, and covariance Σ. We use the negative conditional log-likelihood as our loss function, i.e. L(θ; (x, y)) = −yhx, θi + Φ(hx, θi). (13) Once again, the true parameter θ ∗ is the minimizer of the resulting population risk R(θ). It is easy to see that Linear Regression with Gaussian Noise lies in the family of generalized linear models. We now instantiate GLMs for logistic regression. Logistic Regression In this case the (x, y) pairs are linked as: ( 1 1 with probability 1+exp(−hx, θ ∗ i) , y= 0 otherwise. (14) This corresponds to setting Φ(t) = log(1 + exp(t)) and c(γ) = 1 in (12). The hessian of the population risk is given by exp hx, θi T 2 xx . ∇ R(θ) = E (1 + exp hx, θi)2 Note that as θ diverges, the minimum eigenvalue of the hessian approaches 0 and the loss is no longer strongly convex. To prevent this, in this case we take the parameter space Θ to be bounded. 5.1.3 Exponential Families and Canonical Parameters Finally we consider the case where the true distribution P is in exponential family with canonical parameters θ ∗ ∈ Rp , and a vector of sufficient statistics obtained from the map φ : Z 7→ Rp . Note that while the linear and logistic regression models are indeed in an exponential family, our interest in those cases was not in the canonical parameters. In more details, we can write the true distribution P in this case as P (z) = h(z) exp (hφ(z), θ ∗ i − A(θ ∗ )) , where h(z) is an arbitrary nuisance function. The negative log-likelihood gives us the following loss function: L(θ; z) = −hφ(z), θi + A(θ). (15) The strong-convexity and smoothness assumptions require that there are constants τℓ , τu such that τℓ ≤ ∇2 A(θ) ≤ τu , for θ ∈ Θ. 17 5.2 Stability of Gradient Descent In this section we develop a general theory for the convergence of the projected gradient descent described in Algorithm 1. Note that our gradient estimators could be biased and are not guaranteed to be consistent estimators of the true gradient ∇R(θ). This is especially true in the Huber contamination model where it is impossible to obtain consistent estimators of the gradient of the risk because of the non-vanishing bias caused by the contaminated samples. Hence, we turn our attention to understanding the behavior of projected gradient descent with a biased, inexact, gradient estimator of the form in (6). Before we present our main result, we define the notion of stability of a gradient estimator, which plays a key role in the convergence of gradient descent. Definition 2 (Stability). A gradient estimator is stable for a given risk function R : Θ 7→ R if for some φ ∈ [0, τℓ ), α(e n, δ̃) < τℓ − φ. We denote by κ the following contraction parameter: r 2ητℓ τu e κ := 1 − + ηα(e n, δ), τℓ + τu and note that κ < 1. With these definitions in place we state our main result on the stability of gradient descent: Theorem 1. Suppose that the gradient estimator satisfies the condition in (6) and is stable for the risk function R : Θ 7→ R. Then Algorithm 1 initialized at θ 0 with step-size η ≤ 2/(τℓ + τu ), returns iterates {θbt }Tt=1 such that with probability at least 1 − δ for the contraction parameter κ above we have that, kθbt − θ ∗ k2 ≤ κt kθ 0 − θ ∗ k2 + 1 e β(e n, δ). 1−κ (16) We defer a proof of this result to the Appendix. Theorem 1 provides a general result for risk minimization and parameter estimation. In any concrete instantiation for a given gradient estimator, risk pair, we first study the distribution of the gradient of the risk to estimate the e β(e e and then apply Theorem 1. error suffered by the gradient estimator (α(e n, δ), n, δ)) For the bound (16), the first term is decreasing in T , while the second term is increasing in T . This suggests that for a given n and δ, we need to run just enough iterations for the first term to be bounded by the second. Hence, we can fix the number of iterations T ∗ as the smallest positive integer such that: T ≥ log1/κ (1 − κ)kθ 0 − θ ∗ k2 . e β(e n, δ) Since we obtain linear convergence, i.e. κ < 1, typically a logarithmic number of iterations suffice to obtain an accurate estimate. 5.3 General Analysis of Algorithm 2 We now analyze the gradient estimator described in Algorithm 2 for Huber contamination model and study the error suffered by it. As stated before, Algorithm 2 uses the robust mean 18 estimator of Lai et al. [29]. Hence, while our proof strategy mimics that of Lai et al. [29], we present a different result which is obtained by a more careful non-asymptotic analysis of the algorithm. We define: p log p log n/(pδ) 3/8 ǫp2 log p log + γ(n, p, δ, ǫ) := n n p log(p) 1/4 δ , (17) and with this definition in place we have the following result: Lemma 1. Let P be the true probability distribution of z and let Pθ be the true distribution of the gradients ∇L(θ; z) on Rp with mean µθ = ∇R(θ), covariance Σθ , and bounded fourth moments. There exists a positive constant C1 > 0, such that given n samples from the distribution in (4), the Huber Gradient Estimator described in Algorithm 2 when instantiated with the contamination level ǫ, with probability at least 1 − δ, returns an estimate µ b of µθ such that, kb µ − µθ k2 ≤ C1 √ 1p ǫ + γ(n, p, δ, ǫ) kΣθ k22 log p. We note in particular, if n → ∞ (with other parameters held fixed) then γ(n, p, δ, ǫ) → 0 and the error of our gradient estimator satisfies kb µ − µθ k2 ≤ C p kΣθ k2 ǫ log p, and has only a weak dependence on the dimension p. 5.4 General Analysis of Algorithm 3 In this section we analyze the gradient estimator for heavy-tailed setting, described in Algorithm 3. The following result shows that the gradient estimate has exponential concentration around the true gradient, under the mild assumption that the gradient distribution has bounded second moment. Its proof follows from the analysis of geometric median-of-means estimator of Minsker [37]. We use tr (Σθ ) to denote the trace of the matrix Σθ . Lemma 2. Let P be the probability distribution of z and Pθ be the distribution of the gradients ∇L(θ; z) on Rp with mean µθ = ∇R(θ), covariance Σθ . Then the heavy tailed gradient estib that satisfies the following exponential mator described in Algorithm 3 returns an estimate µ concentration inequality, with probability at least 1 − δ: r kb µ − µθ k2 ≤ 11 6 tr (Σθ ) log 1.4/δ . n Consequences for Estimation under ǫ-Contaminated Model We now turn our attention to the examples introduced earlier, and present specific applications of Theorem 1, for parametric estimation under Huber contamination model. As shown in Lemma 1, we need the added assumption that the true gradient distribution has bounded fourth moments, which suggests the need for additional assumptions. We make our assumptions explicit and defer the technical details to the Appendix. 19 6.1 Linear Regression We assume that the covariates x ∈ Rp have bounded 8th -moments and the noise w has bounded 4th moments. Theorem 2 (Robust Linear Regression). Consider the statistical model in equation (11), and C1√τℓ e < suppose that the number of samples n is large enough such that γ(e n, p, δ) and the kΣk2 log p 2 ℓ e contamination level is such that ǫ < kΣkC2√τlog − γ(e n, p, δ) for some constants C1 and C2 . p 2 Then, there are universal constants C3 , C4 , such that if Algorithm 1 is initialized at θ 0 with stepsize η ≤ 2/(τu + τℓ ) and Algorithm 2 as gradient estimator, then it returns iterates {θbt }Tt=1 such that with probability at least 1 − δ p C σ kΣk2 log p 1 3 t ∗ t 0 ∗ e , kθb − θ k2 ≤ κ kθ − θ k2 + n, p, δ) (18) ǫ 2 + γ(e 1−κ for some contraction parameter κ < 1. In the asymptotic setting when the number of samples n → ∞ (and other parameters are held fixed), we see that for the Huber Gradient Estimator, the corresponding maximum allowed C τ2 contamination level is ǫ < τ 2 1logℓ p . This says that the more well-conditioned the covariance u matrix Σ, the higher the contamination level we can tolerate. Plugin Estimation For linear regression, the true parameter can be written in closed form as θ ∗ = E[xxT ]−1 E[xy]. A non-iterative way to estimate θ ∗ is to separately estimate E[xxT ] and E[xy] using robust covariance and mean oracles respectively. Under the assumption that x ∼ N (0, Ip ), one can reduce the problem to robustly estimating E[xy]. Under this setting, we now present a result using Lai et al. [29] as the mean estimator for estimation of E[xy]. Recall, the definition of γ in (17). We have the following result: Corollary 3. Consider the model in equation(11) with the covariates drawn from N (0, Ip ) and w ∈ N (0, 1), then there are universal constants C1 , C2 such that if ǫ < C1 , then [29] returns an estimate θb of E[xy], such that with probability at least 1 − δ q 1 (19) kθb − θ ∗ k2 ≤ C2 (1 + 2kθ ∗ k22 ) log p ǫ 2 + γ(n, p, δ, ǫ) . Comparing bounds (18) and (19), we see that the error of the plugin estimator depends on kθ ∗ k2 , which would make the estimator vacuous if kθ ∗ k2 scales with the dimension p. On the other hand, the asymptotic rate of our robust gradient estimator is independent of the kθ ∗ k2 . This disadvantage of plugin estimation is inescapable, due to known minimax results for robust mean estimation [10] that show that the dependence on kθ ∗ k2 is unavoidable for any oracle which estimates the mean of xy in the ǫ-contaminated setting. Next, we apply our estimator to generalized linear models. 6.2 Generalized Linear Models Here we assume that the covariates have bounded 8th moments. Additionally, we assume smoothness of Φ′ (·) around θ ∗ . To be more precise, we assume that there exist universal constants LΦ,2k , B2k such that h i 2k ≤ LΦ,2k kθ ∗ − θk2k Ex Φ′ (hx, θi) − Φ′ (hx, θ ∗ i) 2 + BΦ,2k , for k = 1, 2, 4 20 k We also assume that Ex [ Φ(t) (hx, θ ∗ i) ] ≤ MΦ,t,k where Φ(t) (·) is the tth -derivative of Φ(·). Theorem 4 (Robust Generalized Linear Models). Consider the statistical model in equation (12), and suppose that the number of samples n is large enough such that e < γ(e n, p, δ) √ C1 τ ℓ 1 1 1 , 4 2 + LΦ,2 ] log pkΣk22 [LΦ,4 and the contamination level is such that,  2 C τ 2 ℓ e , ǫ < √ − γ(e n, p, δ) 1 1 1 2 4 2 log pkΣk2 [LΦ,4 + LΦ,2 ] for some constants C1 and C2 . Then, there are universal constants C3 , C4 , such that if Algorithm 1 is initialized at θ 0 with stepsize η ≤ 2/(τu + τℓ ) and Algorithm 2 as gradient estimator, then it returns iterates {θbt }Tt=1 such that with probability at least 1 − δ kθbt − θ ∗ k2 ≤κt kθ 0 − θ ∗ k2 1 1 1 1 1 √ 1 3 4 2 4 4 + BΦ,2 + c(σ) 2 MΦ,2,2 + c(σ) 4 MΦ,4,1 ] 1 C3 log pkΣk22 [BΦ,4 e 2 + γ(e n , p, δ) , ǫ + 1−κ (20) for some contraction parameter κ < 1. Note that for the case of linear regression with gaussian noise, it is relatively straightforward to see that LΦ,2k = C2k kΣkk2 , BΦ,2k = 0, MΦ,t,k = 1 ∀(t ≥ 2, k ∈ N ) and MΦ,t,k = 0 ∀(t ≥ 3, k ∈ N ) under the assumption of bounded 8th moments of the covariates; which essentially leads to an equivalence between Theorem 2 and Theorem 4 for this setting. In the following section, we instantiate the above Theorem for logistic regression and compare and contrast our results to other existing methods. 6.2.1 Logistic Regression By observing that Φ(t) (·) is bounded for logistic regression for all t ≥ 1, we can see that LΦ,2k = 0, and that there exists a universal constant C > 0 such that BΦ,2k < C and MΦ,t,k < C ∀(t ≥ 1, k ∈ N ). Corollary 5 (Robust Logistic Regression). Consider the model in equation(14), then there are universal constants C1 , C2 , such that if ǫ < C1 , then Algorithm 1 initialized at θ 0 with stepsize η ≤ 2/(τu + τℓ ) and Algorithm 2 as gradient estimator, returns iterates {θbt }Tt=1 , such that with probability at least 1 − δ p C2 kΣk2 log p 1 t ∗ t 0 ∗ e , b n, p, δ) (21) ǫ 2 + γ(e kθ − θ k2 ≤ κ kθ − θ k2 + 1−κ for some contraction parameter κ < 1. Under the restrictive assumption that x ∼ N (0, Ip ), Du et al. [16] exploited Stein’s trick to derive a plugin estimator for logistic regression. However, similar to the linear regression, the error of the plugin estimator scales with kθ ∗ k2 , which is avoided in our robust gradient descent algorithm. We also note that our algorithm extends to general covariate distributions. 21 6.3 Exponential Family Here we assume that the random vector φ(z), z ∼ P has bounded 4th moments. Theorem 6 (Robust Exponential Family). Consider the model in equation(15), then there are universal constants C1 , C2 , such that if ǫ < C1 , then Algorithm 1 initialized at θ 0 with stepsize η ≤ 2/(τu + τℓ ) and Algorithm 2 as gradient oracle, returns iterates {θbt }Tt=1 , such that with probability at least 1 − δ √ C2 τu log p 1 t ∗ t 0 ∗ e , b ǫ 2 + γ(e n, p, δ) (22) kθ − θ k2 ≤ κ kθ − θ k2 + 1−κ for some contraction parameter κ < 1. Plugin Estimation Since the true parameter θ ∗ is the minimizer of the negative loglikelihood, we know that E[∇L(θ ∗ )] = 0, which implies that ∇A(θ ∗ ) = Eθ∗ [φ(Z)]. This shows that the true parameter θ ∗ can be obtained by inverting the ∇A operator, whenever possible. In the robust estimation framework, we can use a robust mean of the sufficient statistics to estimate Eθ∗ [φ(Z)]. We instantiate this estimator using the mean estimator of [29] to estimate Eθ∗ [φ(Z)]: Corollary 7. Consider the model in equation(15), then there are universal constants C1 , C2 b of E[φ(z)], such that with probability at such that if ǫ < C1 , then [29] returns an estimate µ least 1 − δ √ τu log p 1 −1 ∗ ǫ 2 + γ(n, p, δ, ǫ) , (23) b − θ k2 ≤ C2 kPΘ (∇A) µ τℓ where PΘ [θ] = argminy∈Θ ky − θk22 is the projection operator onto the feasible set Θ. 6.4 Discussion and Limitations In the asymptotic setting of n → ∞, Algorithm 1 √ with Algorithm 2 as gradient estimator converges to a point θb such that kθb − θ ∗ k2 = O( ǫ log p). Hence, our error scales only logarithmically with the dimension p. This dependency on the dimension p is a facet of using the estimator from Lai et al. [29] for gradient estimation. Using better oracles will only improve our performance. Next, we would like to point to the difference in the maximum allowed contamination ǫ∗ between the three models. For logistic regression and exponential C τ2 family, ǫ∗ < C1 , while for linear regression, ǫ∗ < τ 2 1logℓ p . These differences are in large part u due to differing variances of the gradients, which naturally depend on the underlying risk function. This scaling of the variance of gradients for linear regression also provides insights into the limitations of Algorithm 1 for gradient estimators. In the Appendix, we provide an upper bound for the contamination level ǫ based on the initialization point θ 0 , above which, Algorithm 1 would not work for any gradient estimator. 7 Consequences for Heavy-Tailed Estimation In this section we present specific applications of Theorem 1 for parametric estimation, under heavy tailed setting. The proofs of the results can be found in the Appendix. 22 7.1 Linear Regression We first consider the linear regression model described in Equation (11). We assume that the covariates x ∈ Rp have bounded 4th -moments and the noise w has bounded 2th moments. This assumption is needed to bound the error in the gradient estimator (see Lemma 2). Theorem 8 (Heavy Tailed Linear Regression). Consider the statistical model in equation (11). There are universal constants C1 , C2 > 0 such that if n e> C1 τu2 e p log(1/δ) τl2 and if Algorithm 1 is initialized at θ 0 with stepsize η ≤ 2/(τu + τℓ ) and Algorithm 3 as gradient estimator, then it returns iterates {θbt }Tt=1 such that with probability at least 1 − δ s  p e C2 σ kΣk2  p log(1/δ)  , (24) kθbt − θ ∗ k2 ≤ κt kθ 0 − θ ∗ k2 + 1−κ n e for some contraction parameter κ < 1. 7.2 Generalized Linear Models In this section we consider generalized linear models described in Equation (12), where the covariate x is allowed to have a heavy tailed distribution. Here we assume that the covariates have bounded 4th moment. Additionally, we assume smoothness of Φ′ (·) around θ ∗ . Specifically, we assume that there exist universal constants LΦ,2k , B2k such that h i 2k ≤ LΦ,2k kθ ∗ − θk2k Ex Φ′ (hx, θi) − Φ′ (hx, θ ∗ i) 2 + BΦ,2k , for k = 1, 2 k We also assume that Ex [ Φ(t) (hx, θ ∗ i) ] ≤ MΦ,t,k for t ∈ {1, 2, 4}, where Φ(t) (·) is the tth derivative of Φ(·). Theorem 9 (Heavy Tailed Generalized Linear Models). Consider the statistical model in equation (12). There are universal constants C1 , C2 > 0 such that if p LΦ,4 + LΦ,2 C1 kΣk2 e n e> p log 1/δ, τl2 and if Algorithm 1 is initialized at θ 0 with stepsize η ≤ 2/(τu + τℓ ) and Algorithm 3 as gradient estimator, it returns iterates {θbt }Tt=1 such that with probability at least 1 − δ kθbt − θ ∗ k2 ≤κt kθ 0 − θ ∗ k2 1 1 1 1 1 1 3 s  2 4 2 4 4 C2 kΣk2 BΦ,4 + BΦ,2 + c(σ) 2 MΦ,2,2 + c(σ) 4 MΦ,4,1 e  p log(1/δ)  , (25) + 1−κ n e for some contraction parameter κ < 1. We now instantiate the above Theorem for logistic regression model. 23 Corollary 10 (Heavy Tailed Logistic Regression). Consider the model in equation(14). There are universal constants C1 , C2 > 0 such that if n e> C12 kΣk2 e p log 1/δ. τl2 and if Algorithm 1 initialized at θ 0 with stepsize η ≤ 2/(τu + τℓ ) and Algorithm 3 as gradient estimator, it returns iterates {θbt }Tt=1 such that with probability at least 1 − δ s  p e C kΣk p log(1/ δ) 2 2  , kθbt − θ ∗ k2 ≤ κt kθ 0 − θ ∗ k2 + 1−κ n e (26) for some contraction parameter κ < 1. 7.3 Exponential Family We now instantiate Theorem 1 for parameter estimation in heavy-tailed exponential family distributions. Here we assume that the random vector φ(z), z ∼ P has bounded 2nd moments, and we obtain the following result: Theorem 11 (Heavy Tailed Exponential Family). Consider the model in equation(15). If Algorithm 1 is initialized at θ 0 with stepsize η ≤ 2/(τu + τℓ ) and Algorithm 3 as gradient estimator, it returns iterates {θbt }Tt=1 , such that with probability at least 1 − δ kθbt − θ ∗ k2 ≤ κt kθ 0 − θ ∗ k2 + 1 C 1−κ s k∇2 A(θ ∗ )k2 p log 1/δe , n e (27) for some contraction parameter κ < 1 and universal constant C. 8 Discussion In this paper we introduced a broad class of estimators and showed that these estimators can have strong robustness guarantees in Huber’s ǫ-contamination model and for heavy-tailed distributions. These estimators leverage the robustness of gradient descent, together with the observation that for risk minimization in most statistical models the gradient of the risk takes the form of a simple multivariate mean which can be robustly estimated by using recent work of robust mean estimation. These estimators based on robust gradient descent work well in practice and in many cases outperform other robust (and non-robust) estimators. There are several avenues for future work, including developing a better understanding of robust mean estimation. Any improvement for robust mean estimation would immediately translate to improved guarantees for the estimators we propose for general parametric models. Finally, it would also be of interest to understand the extent to which we could replace gradient descent with other optimization methods such as accelerated gradient descent or Newton’s method. We note however, that although these methods may have faster rates of convergence in the classical risk minimization setting: in our setup their stability (to using inexact gradients) is far more crucial and warrants further investigation. 24 9 Acknowledgements The research of SB was supported in part by the grant NSF-DMS-1713003. We thank Larry Wasserman for helpful comments on the paper. References [1] Noga Alon, Yossi Matias, and Mario Szegedy. The space complexity of approximating the frequency moments. 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At any iteration step t ∈ {1, 2, . . . , T }, by assumption we have that with probability at least 1 − Tδ , kg(θ t ; Dn , δ/T ) − ∇R(θ t )k2 ≤ α(n/T, δ/T )kθ − θ ∗ k2 + β(n/T, δ/T ). (28) Taking union bound, (28) holds over all iteration steps t ∈ {1 . . . T }, with probability at least 1 − δ. For the remainder of the analysis, we assume this event to be true. 27 Notation. Let g(θ k ) = ∇R(θ k ) + ek be the noisy gradient. Let α = α(n/T, δ/T ) and β = β(n/T, δ/T ) for brevity. We have the following Lemma from Bubeck [6]. Lemma 3. [Lemma 3.11 [6]] Let f be M -smooth and m-strongly convex, then for all x, y ∈ Rp , we have: h∇f (x) − ∇f (y), x − yi ≥ mM 1 kx − yk22 + k∇f (y) − ∇f (x)k22 . m+M m+M By assumptions we have that: k∇R(θ k ) − g(θ k )k2 = kek k2 ≤ αkθ k − θ ∗ k2 + β. Our update rule is θ k+1 = PΘ θ k − ηg(θ k ) . Then we have that: kθ k+1 − θ ∗ k22 = kPΘ [θ k − ηg(θ k )] − θ ∗ k22 = kPΘ [θ k − ηg(θ k )] − PΘ [θ ∗ − η∇R(θ ∗ )]k22 ≤ kθ k − ηg(θ k ) − (θ ∗ − η∇R(θ ∗ ))k22 = ≤ k ∗ k kθ − θ − η(∇R(θ ) − ∇R(θ )) − ηek k22 kθ k − θ ∗ − η(∇R(θ k ) − ∇R(θ ∗ ))k22 + η 2 kek k22 + 2kek k2 kθ k − θ ∗ − η(∇R(θ k ) − ∇R(θ ∗ ))k2 , (29) ∗ (30) where Equation (29) follows from contraction property of projections. Now, we can write kθ k − θ ∗ − η(∇R(θ k ) − ∇R(θ ∗ ))k2 as kθk − θ∗ − η(∇R(θk ) − ∇R(θ∗ ))k22 = kθk − θ∗ k22 + η 2 k∇R(θk ) − ∇R(θ∗ )k22 − 2η ∇R(θk ) − ∇R(θ∗ ), θk − θ∗ 1 τℓ τu k ∗ 2 2 k ∗ 2 k ∗ 2 k ∗ 2 ≤ kθ − θ k2 + η k∇R(θ ) − ∇R(θ )k2 − 2η kθ − θ k2 + k∇R(θ ) − ∇R(θ )k2 τℓ + τu τℓ + τu (31) = kθk − θ∗ k22 (1 − 2ητℓ τu /(τℓ + τu )) + ηk∇R(θk ) − ∇R(θ∗ )k22 (η − 2/(τu + τℓ )) ≤ kθk − θ∗ k22 (1 − 2ητℓ τu /(τℓ + τu )), (32) (33) where the second step follows from Lemma 3 and the last step follows from the step size η ≤ 2/(τℓ + τu ). Now, combining Equations (30) and (33), and using our assumption that kek k2 ≤ αkθ k − θ ∗ k2 + β, we get: 2 p kθ k+1 − θ ∗ k22 ≤ kθ k − θ ∗ k2 (1 − 2ητℓ τu /(τℓ + τu )) + ηkek k2 p kθ k+1 − θ ∗ k2 ≤ hp i 1 − 2ητℓ τu /(τℓ + τu ) + ηα kθ k − θ ∗ k2 + ηβ. 1 − 2ητℓ τu /(τℓ + τu ) + ηα. By assumption, we choose ǫ < ǫ∗ such that α < τℓ . p κ = 1 − 2ητℓ τu /(τℓ + τu ) + ηα (34) p < 1 − 2ητℓ τu /(τℓ + τu ) + ητℓ . (35) p p Since 0 ≤ η ≤ 2/(τℓ + τu ), we get that 1 − 2ητℓ τu /(τℓ + τu ) ≤ 1 − η 2 τℓ τu . We have: p (36) κ < 1 − η 2 τℓ τu + ητℓ q (∵ τℓ ≤ τu ) (37) < 1 − η 2 τℓ2 + ητℓ Let κ = (38) < 1. 28 Therefore, we have that, kθ k+1 − θ ∗ k2 ≤ κkθ k − θ ∗ k2 + ηβ. for some κ < 1. Solving the induction,we get: kθ k − θ ∗ k2 ≤ κk kθ 0 − θ ∗ k2 + B 1 ηβ. 1−κ Proof of Theorem 4 To prove our result on Robust Generalized Linear Models, we first study the distribution of gradients of the corresponding risk function. Lemma 4. Consider the model in Equation (12), then there exist universal constants C1 , C2 > 0 such that p p C4 LΦ,4 + LΦ,2 kCov(∇L(θ)k2 ≤C1 k∆k22 kΣk2 q p p 3 + C2 kΣk2 BΦ,2 + BΦ,4 + c(σ) 3MΦ,2,2 + c(σ) MΦ,4,1 Bounded fourth moments E h (∇L(θ) − E[∇L(θ)])T v 4 i ≤ C2 (Var[∇L(θ)T v])2 . Proof. The gradient ∇L(θ) and it’s expectation can be written as: ∇L(θ) = −y.x + u(hx, θi).x E[∇L(θ)] = E[x u(xT θ) − u(xT θ ∗ ) ], where u(t) = Φ′ (t). kE[∇L(θ)]k2 = sup y T E[∇L(θ)] y∈Sp−1 ≤ sup E[(y T x) u(xT θ) − u(xT θ ∗ ) ] y∈Sp−1 ≤ sup y∈Sp−1 q 1 2 ≤ C1 kΣk2 E[(y T x)2 ] q q 2 E[(u(xT θ) − u(xT θ ∗ )) ] LΦ,2 k∆k22 + BΦ,2 where the last line follows from our assumption of smoothness. Now, to bound the maximum eigenvalue of the Cov(∇L(θ)), kCov(∇L(θ))k2 = sup z T E ∇L(θ)∇L(θ)T − E[∇L(θ)]E[∇L(θ)]T z z∈Sp−1 ≤ sup z T E ∇L(θ)∇L(θ)T z + sup z T E[∇L(θ)]E[∇L(θ)]T z z∈Sp−1 z∈Sp−1 i h 2 z + kE[∇L(θ)]k22 ≤ sup z T E xxT u(xT θ) − y) p−1 z∈S h 2 i z + kE[∇L(θ)]k22 ≤ sup E z T xxT u(xT θ) − y z∈Sp−1 ≤ sup z∈Sp−1 q r h i E (u(xT θ) − y)4 ] + kE[∇L(θ)]k22 E [(z T x)4 ] 29 To bound E h u(xT θ) − y 4 i , we make use of the Cr inequality. Cr inequality. If X and Y are random variables such that E\|X\|r < ∞ and E\|Y \|4 < ∞ where r ≥ 1 then: E\|X + Y \|r ≤ 2r−1 (E\|X\|r + E\|Y \|r ) Using the Cr inequality, we have that h h h 4 i 4 i 4 i + E u(xT θ ∗ ) − y ≤ 8 E u(xT θ) − u(xT θ ∗ ) E u(xT θ) − y ≤ C LΦ,4 k∆k42 + BΦ,4 + c(σ)3 MΦ,4,1 + 3c(σ)2 MΦ,2,2 where the last line follows from our assumption that Pθ∗ (y\|x) is in the exponential family, hence, the cumulants are higher order derivatives of the log-normalization function. q √ p p p p 2 3 LΦ,4 k∆k2 + BΦ,4 + c(σ) 3MΦ,2,2 + c(σ) MΦ,4,1 + kE[∇L(θ)]k22 kCov(∇L(θ))k2 ≤ C C4 kΣk2 q √ p p p p 2 3 ≤ C C4 kΣk2 LΦ,4 k∆k2 + BΦ,4 + c(σ) 3MΦ,2,2 + c(σ) MΦ,4,1 + C12 kΣk2 LΦ,2 k∆k22 + BΦ,2 q p p p p ≤ Ck∆k22 kΣk2 C4 LΦ,4 + LΦ,2 + C6 kΣk2 BΦ,2 + BΦ,4 + c(σ) 3MΦ,2,2 + c(σ)3 MΦ,4,1 Bounded Fourth Moment. To show that the fourth moment of the gradient distribution is bounded, we have h h 4 i 4 i ≤ E (∇L(θ) − E[∇L(θ)])T v E (∇L(θ) − E[∇L(θ])T v   ≤ 8 E[\|∇L(θ])T v\|4 ] + E[\|E[∇L(θ)]T v\|4 ] . {z } \| {z } \| A Control of A. B E[\|∇L(θ])T v\|4 ] = E[(xT v)4 (u(xT θ) − y)4 ] q q T 8 ≤ E[(x v) ] E[(u(xT θ) − y)8 ] q p 2 ≤ C8 kΣk2 E[(u(xT θ) − u(xT θ ∗ ))8 ] + E[(u(xT θ ∗ ) − y)8 ] v u 8 X p u 2t 8 gt,k MΦ,t,k ≤ C8 kΣk2 LΦ,8 k∆k2 + BΦ,8 + t,k=2 ≤ √ v u 8 uX p p 2 4 CkΣk2 LΦ,8 k∆k2 + BΦ,8 + t gt,k MΦ,t,k t,k=2 where the last step follows from the fact that the 8th central moment can be written as a polynomial involving the lower cumulants, which in turn are the derivatives of the lognormalization function. Control of B. 2 E[\|E[∇L(θ)]T v\|4 ] ≤ kE[∇L(θ)k42 ≤ C1 kΣk22 L2Φ,2 k∆k22 + BΦ,2 30 By assumption LΦ,k , BΦ,k , MΦ,t,k are all bounded for k, t ≤ 8, which implies that there exist constants c1 , c2 > 0 such that h 4 i ≤ c1 kΣk22 k∆k42 + c2 (39) E (∇L(θ) − E[∇L(θ])T v Previously, we say that kCov∇L(θ)k2 ≤ c3 kΣk2 k∆k22 +c4 , for some universal constants c3 , c4 > 0, hence the gradient ∇L(θ) has bounded fourth moments. Having studied the distribution of the gradients, we use Lemma 1 to characterize the stability of Huber Gradient estimator. Using Lemma 1, we know that at any point θ, the Huber Gradient Estimator g(θ, δ/T ) satisfies that with probability 1 − δ/T , 1 1p e kCov(∇L(θ))k 2 log p. n, p, δ) kg(θ, δ/T ) − ∇R(θ)k2 ≤ C2 ǫ 2 + γ(e 2 Substituting the upper bound on kCov(∇L(θ))k2 from Lemma 4, we get that there are universal constants C1 , C2 such that with probability at least 1 - δ/T 1 p 1 1 1 4 2 e log pkΣk22 [LΦ,4 (40) kg(θ) − ∇R(θ)k2 ≤ C1 ǫ 2 + γ(e + LΦ,2 ] k∆k2 n, p, δ) {z } \| e α(e n,δ) p 1 1 1 1 1 1 1 3 4 2 4 4 e + C2 ǫ 2 + γ(e n, p, δ) + BΦ,2 + c(σ) 2 MΦ,2,2 + c(σ) 4 MΦ,4,1 ] log pkΣk22 [BΦ,4 \| {z } e β(e n,δ) (41) e < τℓ . Using Equation (40), we To ensure stability of gradient descent, we need that α(e n, δ) get that gradient descent is stable as long as the number of samples n is large enough such C1 τℓ e < that γ(e n, p, δ) , and the contamination level is such that 1 1 1 √ 4 +L 2 ] log pkΣk22 [LΦ,4 Φ,2 ǫ< C2 τℓ 1 1 1 √ 4 +L 2 ] log pkΣk22 [LΦ,4 Φ,2 !2 e − γ(e n, p, δ) for some constants C1 and C2 . Plugging the corre- e into Theorem 1, we get back the result of Theorem 4. sponding ǫ and β(e n, δ) C Proof of Corollary 3 We begin by studying the distribution of the random variable xy = xxT θ ∗ + x.w. Lemma 5. Consider the model in Equation (11), with x ∼ N (0, Ip ) and w ∼ N (0, 1) then there exist universal constants C1 , C2 such that E[xy] = θ ∗ kCov(xy)k2 = 1 + 2kθ ∗ k22 h 4 i ≤ C2 (Var[(xy)T v])2 . Bounded fourth moments E (xy − E[xy])T v Proof. Mean. xy = xxT θ ∗ + x.w T ∗ E[xy] = E[xx θ + x.w] ∗ (42) (43) (44) E[xy] = θ . 31 Covariance. Cov(xy) = E[(xxT − I)θ ∗ + x.w)((xxT − I)θ ∗ + x.w)T )] ∗ ∗T Cov(xy) = E[(xxT − I)θ θ (45) (xxT − I)] + Ip . (46) Now, Z = (xxT − I)θ ∗ can be written as:    ∗  ∗ 2  2 θ1 (x1 − 1) + x1 x2 θ2∗ + . . . + x1 xp θp∗ θ1 (x1 − 1) x1 x2 ... x1 xp ∗ ∗ 2 ∗  ∗   x1 x2 x2 xp  (x22 − 1) . . .  θ2  x1 x2 θ1 + (x2 − 1)θ2 + . . . + x2 xp θp   T ∗ (xx −I)θ =  .   ..  =  .. .. .. .. ..   .    . . . . . ∗ 2 2 ∗ ∗ ∗ θp x1 xp x2 xp . . . (xp − 1) x1 xp θ1 + x2 xp θ2 + . . . + (xp − 1)θp Then,  ... θ1∗ θp∗ 2θ1∗ 2 + θ2∗ 2 + . . . + θp∗ 2 θ1∗ θ2∗  θ1∗ 2 + 2θ2∗ 2 + . . . + θp∗ 2 . . . θ2∗ θp∗ θ1∗ θ2∗    T E ZZ =  . .. .. .. ..   . . . . 2 2 2 ∗ ∗ ∗ ∗ ∗ ∗ ∗ . . . θ1 + θ2 + . . . + 2θp θ2 θp θp θ1  Hence the covariance matrix can be written as: Cov(xy) = Ip (1 + kθ ∗ k22 ) + θ ∗ θ ∗ T . Therefore kCov(xy)k2 = 1 + 2kθ ∗ k22 . Bounded Fourth Moment. E h (xy − E[xy])T v 4 i ≤E We start from the LHS h (xy − E[xy])T v 4 i (47) 4 = E ((xxT − I)θ ∗ + wx)T v i4 h = E (θ ∗T x)(xT v) − θ ∗T v + wv T x    4   T ∗T ∗T  ≤ 8 8 E (θ x)(x v) + E θ v \| {z } \| {z A B (48) (49)    + E w(xT v) 4 .  \| {z }  } 4 (50) C The last line follows from two applications of the following inequality: Cr inequality. If X and Y are random variables such that E\|X\|r < ∞ and E\|Y \|4 < ∞ where r ≥ 1 then: E\|X + Y \|r ≤ 2r−1 (E\|X\|r + E\|Y \|r ) . Now to control each term on: • Control of A. Using Cauchy Schwartz, and normality of 1D projections of normal distribution q q T 8 ∗ A ≤ E[\|θ x\| ] E[\|xT v\|8 ] (51) - kθ ∗ k42 . 32 (52) • Control of B, B ≤ kθ ∗ k42 . • Control of C, C = O(1), using independence of w and normality of 1D projections of normal distribution. h 4 i Therefore the E (xy − E[xy])T v - c + kθ ∗ k42 . For the RHS: Var((xy)T v)2 = (v T Cov(xy)v)2 ≤ kCov(xy)k22 . We saw that the kCov(xy)k2 - c + kθ ∗ k22 , so both the LHS and RHS scale with kθ ∗ k42 . Hence, xy has bounded fourth moments. Now that we’ve established that xy has bounded fourth moments implies that we can use [29] as a mean estimation oracle. Using Theorem 1.3 [29], we know that the oracle of [29] outputs an estimate θb of E[xy] such that with probability at least 1 − 1/pC1 , we have: 1 p kθb − θ ∗ k2 ≤ C2 kCov(xy)k2 log p ǫ 2 + γ(n, p, δ, ǫ) Using Lemma 5 to subsitute kCov(xy)k2 ≤ 1+2kθ ∗ k22 ), we recover the statement of Corollary 3. D Proof of Theorem 6 To prove our result on Robust Exponential Family, we first study the distribution of gradients of the corresponding risk function. Lemma 6. Consider the model in Equation (15), then there exists a universal constant C1 such that E[∇L(θ)] = ∇A(θ) − ∇A(θ ∗ ) 2 ∗ kCov[∇L(θ)]k2 = k∇ A(θ )k2 h 4 i ≤ C1 (Var[∇L(θ)T v])2 . Bounded fourth moments E (∇L(θ) − E[∇L(θ)])T v (53) (54) (55) Proof. By Fisher Consistency of the negative log-likelihood, we know that Eθ∗ [∇L(θ ∗ )] = 0 (56) =⇒ ∇A(θ ) − Eθ∗ [φ(z)] = 0 (57) ∗ ∗ =⇒ ∇A(θ ) = Eθ∗ [φ(z)]. (58) For the mean, ∇L(θ) = ∇A(θ) − φ(z) (59) E[∇L(θ)] = ∇A(θ) − Eθ∗ [φ(z)] ∗ E[∇L(θ)] = ∇A(θ) − ∇A(θ ). Now, for the covariance: T i ∇L(θ) − Eθ∗ [∇L(θ)] ∇L(θ) − Eθ∗ [∇(L(θ)] i h T ∗ ∗ ∗ = Eθ (∇A(θ ) − φ(z)) (∇A(θ ) − φ(z)) = Covθ∗ ∇L(θ ∗ ) = ∇2 A(θ ∗ ). Covθ∗ [∇L(θ)] = Eθ∗ h 33 (60) (61) Bounded moments follows from our assumption that the sufficient statistics have bounded 4th moments. Having studied the distribution of the gradients, we use Lemma 1 to characterize the stability of Huber Gradient estimator. Using Lemma 1, we know that at any point θ, the Huber Gradient Estimator g(θ, δ/T ) satisfies that with probability 1 − δ/T , 1 1p e kCov(∇L(θ))k 2 log p. n, p, δ) kg(θ, δ/T ) − ∇R(θ)k2 ≤ C2 ǫ 2 + γ(e 2 Substituting the upper bound on kCov(∇L(θ))k2 from Lemma 6, we get that there are universal constants C1 , C2 such that 1 p √ e kg(θ) − ∇R(θ)k2 ≤ C1 ǫ 2 + γ(e n, p, δ) log p τu . \| {z } e β(e n,δ) e = 0 < τℓ by assumption. Therefore we just have that ǫ < In this case we have that α(e n, δ) e into Theorem 1, C1 for some universal constant C1 . Plugging the corresponding ǫ and β(e n, δ) we get back the result of Corollary 6. E Proof of Corollary 7 Using the contraction property of projections, we know that kPΘ (∇A)−1 µ b − θ ∗ k2 = kPΘ (∇A)−1 µ b − PΘ [θ ∗ ] k2 ≤ k(∇A)−1 µ b − θ ∗ k2 . By Fisher Consistency of the negative log-likelihood, we know that ∇A(θ ∗ ) = Eθ∗ [φ(z)]. The true parameter θ ∗ can be obtained by inverting the ∇A operator whenever possible. k(∇A)−1 µ b − θ ∗ k2 = k(∇A)−1 µ b − (∇A)−1 Eθ∗ [φ(z)]k2 ∗ ∗ = k∇A µ b − ∇A Eθ∗ [φ(z)]k2 . (62) (63) where A∗ is the convex conjugate of A. We can use the following result to control the Lipschitz smoothness A∗ . Theorem 12. (Strong/Smooth Duality) Assume f (·) is closed and convex. Then f (·) is smooth with parameter M if and only if its convex conjugate f (·) is strongly convex with parameter 1 . m= M A proof of the above theorem can be found in [28]. Hence, we have that: 1 µ − Eθ∗ [φ(z)]k2 b − θ ∗ k2 ≤ kb kPΘ (∇A)−1 µ τℓ (64) By assumption, we have that the fourth moments of the sufficient statistics are bounded. We also know that Cov(φ(z) = ∇2 A(θ ∗ ) which implies that we can use [29] as our oracle. Using Lemma 1, we get that, there exists universal constants C1 , C2 such that with probability at least 1 − 1/pC1 , 1 p kb µ − Eθ∗ [φ(z)]k2 ≤ C2 τu log p ǫ 2 + γ(n, p, δ, ǫ) . Combining the above with Equation (64) recovers the result of Corollary 7. 34 F Proof of Theorem 8 Before we present the proof of Theorem 8, we first study the distribution of gradients of the loss function. This will help us bound the error in the gradient estimator. Lemma 7. Consider the model in Equation (11). Suppose the covariates x ∈ Rp have bounded 4th -moments and the noise w has bounded 2th moments.. Then there exist universal constants C1 , C2 such that E[∇L(θ)] = Σ∆ kCov(∇L(θ)k2 ≤ σ 2 kΣk2 + C1 k∆k22 kΣk22 , where ∆ = θ − θ ∗ and E[xxT ] = Σ. Proof. We start by deriving the results for E[∇L(θ)]. 1 1 L(θ) = (y − xT θ)2 = (xT (∆) − w)2 2 2 ∇L(θ) = xxT ∆ − x.w (65) (66) (67) E[∇L(θ)] = Σ∆. Next, we bound the operator norm of the covariance of the gradients ∇L(θ) at any point θ. Covariance. Cov(∇L(θ)) = E[(xxT − Σ)∆ − x.w)((xxT − Σ)∆ − x.w)T )] T T T (68) 2 Cov(∇L(θ)) = E[(xx − Σ)∆∆ (xx − Σ)] + σ Σ. (69) Now, we want to bound kCov(∇L(θ))k2 = λmax (Cov(∇L(θ))). λmax (Cov(∇L(θ))) ≤ σ 2 λmax (Σ) + λmax E[(xxT − Σ)∆∆T (xxT − Σ)] 2 ≤ σ λmax (Σ) + sup y y∈Sp−1 T (70) E[(xx − Σ)∆∆ (xx − Σ)] y T T T (71) ≤ σ 2 λmax (Σ) + sup y T E[(xxT − Σ)∆∆T (xxT − Σ)] y y∈Sp−1 ≤ σ 2 λmax (Σ) + k∆k22 ≤ σ 2 λmax (Σ) + k∆k22 sup y,z∈Sp−1 sup ≤ σ 2 λmax (Σ) + 2k∆k22 2 ≤ σ λmax (Σ) + ≤ σ 2 kΣk2 + E (y T (xxT − Σ)z)2 y,z∈Sp−1 ≤ σ 2 λmax (Σ) + 2k∆k22 2k∆k22 sup y,z∈Sp−1 sup y,z∈Sp−1 sup (72) (73) E 2(y T x)2 (xT z)2 + 2(y T Σz)2 (74) E (y T x)2 (xT z)2 + kΣk22 E (y T x)2 (xT z)2 + kΣk22 q E [(y T x)4 ] y,z∈Sp−1 2 2k∆k2 (kΣk22 + C4 kΣk22 ), q E [(z T x)4 ] + kΣk22 (75) (76) (77) (78) where the second last step follows from Cauchy-Schwartz and the last step follows from our assumption of bounded 4th moments (see Equation (10)). 35 We now proceed to the proof of Theorem 8. From Lemma 2, we know that at any point e satisfies the following with θ, the gradient estimator described in Algorithm 3, g(θ; Dne , δ), probability at least 1 − δ, e − ∇R(θ)k2 ≤ C kg(θ; Dne , δ) q tr(Cov(∇L(θ))) log 1/δe . n e We substitute the upper bound for kCov(∇L(θ))k2 from Lemma 7 in the above equation e − ∇R(θ)k2 ≤ C kg(θ; Dne , δ) ≤ C q q ≤ C1 \| tr(Cov(∇L(θ))) log 1/δe n e p(σ2 kΣk2 +2k∆k22 (kΣk22 +C4 kΣk22 )) log 1/δe n e s kΣk22 p log 1/δe kθ − θ ∗ k2 n e {z } e α(e n,δ) s kΣk2 p log 1/δe . + C2 σ n e {z } \| e β(e n,δ) To complete the proof of this theorem, we use the results from Theorem 1. Note that the e < τl . This holds when gradient estimator satisfies the stability condition if α(e n, δ) n e> C12 τu2 e p log 1/δ. τl2 e into Theorem 1 gives us Now suppose n e satisfies the above condition, then plugging β(e n, δ) the required result. G Proof of Theorem 9 To prove the Theorem we use the result from Lemma 4, where we derived the following expression for covariance of ∇L(θ) p p kCov(∇L(θ)k2 ≤C1 k∆k22 kΣk2 C4 LΦ,4 + LΦ,2 q p p 3 + C2 kΣk2 BΦ,2 + BΦ,4 + c(σ) 3MΦ,2,2 + c(σ) MΦ,4,1 From Lemma 2, we know that at any point θ, the gradient estimator described in Algorithm 3, e satisfies the following with probability at least 1 − δ, g(θ; Dne , δ), e − ∇R(θ)k2 ≤ C kg(θ; Dne , δ) 36 q tr(Cov(∇L(θ))) log 1/δe . n e Substitute the upper bound for kCov(∇L(θ))k2 in the above equation, we get q log 1/δe e kg(θ; Dne , δ) − ∇R(θ)k2 ≤ C tr(Cov(∇L(θ))) n e s √ p kΣk2 C4 LΦ,4 + LΦ,2 p log 1/δe kθ − θ ∗ k2 ≤ C1 n e {z } \| e + C2 \| α(e n,δ) v u p u kΣk B + pB + c(σ)p3M 3 c(σ) MΦ,4,1 p log 1/δe Φ,2 2 Φ,4 Φ,2,2 + t {z n e e β(e n,δ) We now use the results from Theorem 1. The gradient estimator satisfies the stability condition e < τl . This holds when if α(e n, δ) √ p C12 kΣk2 C4 LΦ,4 + LΦ,2 e n e> p log 1/δ. τl2 e into Theorem 1 gives us Now suppose n e satisfies the above condition, then plugging β(e n, δ) the required result. H Proof of Theorem 11 The proof proceeds along similar lines as the proof of Theorem 9. To prove the Theorem we utilize the result of Lemma 6, where we showed that kCov[∇L(θ)]k2 = k∇2 A(θ ∗ )k2 . Combining this result with Lemma 2 we get that with probability at least 1 − δ q log 1/δe e kg(θ; Dne , δ) − ∇R(θ)k2 ≤ C tr(Cov(∇L(θ))) n e s k∇2 A(θ ∗ )k2 p log 1/δe ≤ C . n e \| {z } e β(e n,δ) e = 0, the stability condition is always satisfied, as long as τl > 0. Substituting Since α(e n, δ) e into Theorem 1 gives us the required result. β(e n, δ) I Upper bound on Contamination Level We provide a complementary result, which gives an upper bound for the contamination level ǫ based on the initialization point θ 0 , above which, Algorithm 1 would not work. The key idea is that the error incurred by any mean estimation oracle is lower bounded by the variance of the distribution, and that if the zero vector lies within that error ball, then any mean oracle can be forced to output 0 as the mean. For Algorithm 1, this implies that, in estimating the mean of the gradient, if the error is high, then one can force the mean to be 0 which forces the algorithm to converge. For the remainder of the section we consider the case of linear regression with x ∼ N (0, Ip ) in the asymptotic regime of n → ∞. 37 } . Lemma 8. Consider the model in equation(11) with x ∼ N (0, Ip ) and w ∼ N (0, 1), then 0 ∗ there exists a universal constant C1 such that if ǫ > C1 √ kθ −θ0 k2∗ 2 , then for every gradient 1+2kθ −θ k2 oracle, there exists a contamination distribution Q such that, Algorithm 1 will converge to θ 0 even when the number of samples n → ∞. Proof. Using Lemma 5, we know that for any point θ, ∇L(θ) = xxT ∆ − x.w Eθ∗ [∇L(θ)] = (θ − θ ∗ ) = ∆ kCov(∇L(θ)k2 = 1 + 2k∆k22 , where ∆ = θ − θ ∗ . Let P∇L(θ) represent the distribution ∇Lθ. Similarly, let Pǫ,∇L(θ),Q represent the corresponding ǫ-contaminated distribution. Then, using Theorem 2.1 [10], we know that the minimax rate for estimating the mean of the distribution of gradients is given by: µ − Eθ∗ [∇L(θ)]k22 ≥ Cǫ2 (1 + 2k∆k22 ) ≥ c. inf sup Pǫ,∇L(θ),Q kb µ b θ∈Rp ,Q The above p statement says that at any point θ, any mean oracle Ψ will always incur an error of Ω( Cǫ2 (1 + 2k∆k22 )) in estimating the gradient Eθ∗ [∇L(θ)]. kΨ(θ) − Eθ∗ [∇L(θ)]k2 ≥ Cǫ q (1 + 2k∆k22 ) ∀ Ψ Forp any oracle Ψ, there exists some adversarial contamination Q, such that whenever kEθ∗ [∇L(θ)]k2 < Cǫ (1 + 2k∆k22 ), then kΨ(θ)k2 = 0. Suppose that the contamination level ǫ is such that, ǫ> 1 kEθ∗ [∇L(θ 0 )]k2 p , C (1 + 2kθ 0 − θ ∗ k22 ) then for every oracle there exists a corresponding Q such that Algorithm 1 will remain stuck at θ 0 . Plugging Eθ∗ [∇L(θ 0 )] = θ 0 − θ ∗ , we recover the statement of the lemma. Chen et al. [10] provide a general minimax lower bound of Ω(ǫ) for ǫ-contamination√models in this setting. In contrast, using Algorithm 1 with [29] as oracle, we can only O( ǫ log p) close to the true parameter even when the contamination is small, which implies that our procedure is not minimax optimal. Our approach is nonetheless the only practical algorithm for robust estimation of general statistical models. J Details and Analysis of Algorithm 2 In this section we present a refined, non-asymptotic analysis of the algorithm from [29]. We begin by introducing some preliminaries. We subsequently analyze the algorithm in 1-dimension and finally turn our attention to the general algorithm. 38 J.1 Preliminaries Unless otherwise stated, we assume throughout that the random variable X has bounded fourth moments, i.e. for every unit vector v, 2 E hX − µ, vi4 ≤ C4 E hX − µ, vi2 . We summarize some useful results from [29], which bound the deviation of the conditional mean/covariance from the true mean/covariance. Lemma 9. [Lemma 3.11 [29]] Let X be a univariate random variable with bounded fourth moments, and let A be any with event with probability P(A) = 1 − γ ≥ 21 . Then, p \|E(X\|A) − E(X)\| ≤ σ 4 8C4 γ 3 . Lemma 10. [Lemma 3.12 [29]] Let X be a univariate random variable with E[X] = µ, E (X − µ)2 = σ 2 and let E((X − µ)4 ) ≤ C4 σ 4 . Let A be any with event with probability P(A) = 1 − γ ≥ 12 . Then, p (1 − C4 γ)σ 2 ≤ E((X − µ)2 \|A) ≤ (1 + 2γ)σ 2 . Corollary 13. [Corollary 3.13 [29]] Let A be any event with probability P(A) = 1 − γ ≥ 21 , and let X be a random variable with bounded fourth moments. We denote Σ\|A = E(XX T \|A) − (E(X\|A))(E(X\|A))T to be the conditional covariance matrix. We have that, p p (1 − C4 γ − 8C4 γ 3 )Σ Σ\|A (1 + 2γ)Σ. For random variables with bounded fourth moments we can use Chebyshev’s inequality to obtain tail bounds. Lemma 11. [Lemma 3.14 [29]] Let X have bounded fourth moments, then for every unit vector v we have that, P(\|hX, vi − E[hX, vi]\| ≥ t p [E [hX − µ, vi2 ]]) ≤ C4 . t4 Our proofs also use the matrix Bernstein inequality for rectangular matrices. As a preliminary, we consider a finite sequence {Zk } of independent, random matrices of size d1 ×d2 . We assume that each random matrix satisfies E(Zk ) = 0, and kZk kop ≤ R almost surely. We define: ( ) X X σ 2 := max k E(Zk ZkT )kop , k E(Zk ZkT )kop . k k With these preliminaries in place we use the following result from [41]. Lemma 12. For all t ≥ 0, P X k Zk op ≥ t ≤ (d1 + d2 ) exp −t2 /2 σ 2 + Rt/3 . Equivalently, with probability at least 1 − δ, s X d1 + d2 d1 + d2 2R log + . ≤ 2σ 2 log Zk δ 3 δ op k 39 We let I denote the set of all intervals in R. The following is a standard uniform convergence result. Lemma 13. Suppose X1 , . . . , Xn ∼ P, then with probability at least 1 − δ, r n 4 log(en) + 2 log(2/δ) 1X I(Xi ∈ I) ≤ 2 . sup P(I) − n n I∈I i=1 Algorithm 4 Huber Outlier Gradients Truncation function HuberOutlierGradientTruncation(Sample Gradients S, Corruption Level ǫ, Dimension p,δ) if p=1 then r \|S\| 1 Let [a, b] be smallest interval containing 1 − ǫ − C5 (1 − ǫ) fracδ \|S\| log tion of points. Se ← S ∩ [a, b]. return Se else Let [S]i be the samples with the ith co-ordinates only, [S]i = {hx, ei i \|x ∈ S} for i = 1 to p do a[i] = HuberGradientEstimator([S]i , ǫ, 1, δ/p). end for r Let B(r, a) be the ball of smallest radius centered at a containing (1 − ǫ − Cp p \|S\| log \|S\| pδ (1 − ǫ) fraction of points in S. Se ← S ∩ B(r, a). return Se end if end function We now turn our attention to an analysis of Algorithm 2 for the 1-dimensional case. J.2 The case when p = 1 Firstly, we analyze Algorithm 2 when p = 1. Lemma 14. Suppose that, Pθ∗ is a distribution on R1 with mean µ, variance σ 2 , and bounded fourth moments. There exist positive universal constants C1 , C2 , C8 > 0, such that given n samples from the distribution in (4), the algorithm with probability at least 1 − δ, returns an estimate µ b such that, !3 !1 r r r 4 2 1 log 3/δ log 3/δ log(3/δ) +t +t + C2 σ ǫ + kb µ − µk2 ≤ C1 C44 σ ǫ + 2n 2n n where t = C8 q 1 n n δ log 1 4 . which can be further simplified to, kb µ − µk2 ≤ C1 C4 σ ǫ + C8 r !3 !1 r r n 4 n 2 log(1/δ) 1 1 + C2 σ ǫ + C8 log log n δ n δ n 40 Proof. By an application of Hoeffding’s inequality we obtain that with probability at least 1 − δ/3, q the fraction of corrupted samples (i.e. samples from the distribution Q) is less than ǫ + log(3/δ) 2n . We condition on this event through the remainder of this proof. We let η denote the fraction of corrupted samples. Further, we let SP be the samples from the true distribution. Let nP be the cardinality of this set, i.e. nP := \|SP \|. Let I1−η be the interval around µ containing 1 − η mass of Pθ∗ . Then, using Lemma 11, we have that: 1 length (I1−η ) ≤ C44 σ 1 η4 . Using Lemma 13 we obtain that with probability at least 1 − δ/3 the number of samples from the distribution P that fall in the interval I1−η is at least 1 − η − t where t is upper bounded as: r 4 log(en) + 2 log(6/δ) . t≤2 n Now we let Se be the set of points in the smallest interval containing (1 − η − t)(1 − η) fraction of all the points. • Using VC theory, we know that for every interval I ⊂ R, there exists some universal constant C3 such that P (\|(P (x ∈ I\|x ∼ D) − P (x ∈ I\|x ∈u SD ))\| > t/2) ≤ n2D exp(−nD t2 /8) (79) This can be re-written as, that with probability at least (1−δ/3), there exists a universal constant C0 such that, r r n n 1 1 D ≤ C5 log log sup \|(P (x ∈ I\|x ∼ D) − P (x ∈ I\|x ∈u SD ))\| ≤ C0 nD δ n δ I \| {z } t • Using Equation (79), we know that (1 − η − t) fraction of SD lie in I1−η . Let Se be the set of points in the smallest interval containing (1 − η − t)(1 − η) fraction of the points. • We know that the length of minimum interval containing (1 − η − t)(1 − η) fraction of the points of S is less than length of smallest interval containing (1 − η − t) fraction of points of SD , which in turn is less than length of I1−η . • Now, I1−η and minimum interval containing (1 − η − t) fraction of points of SD need to overlap. This is because, n is large enough such that t < 12 − η hence, the extreme points for such an interval can be atmost 2length (I1−η ) away. • Hence, the distance of all chosen noise-points from µ will be within the length (I1−η ). • Moreover, the interval of minimum length with (1−η −t)(1−η) fraction of S will contain at least 1 − 3η − t fraction of SD . e by controlling the sources of error. • Hence, we can bound the error of mean(S) 41 – All chosen noise points are within length (I1−η ), and there are atmost η of them, hence the maximum error can be ηlength (I1−η ). – Next, the mean of chosen good points will converge to the mean of the conditional distribution. i.e. points sampled from D but conditioned to lie in the minimum length interval. The variance of these random variables is upper bounded using Lemma 10. – To control the distance between the mean(E(X) and the conditional mean(E(X\|A)), where A is the event that a sample x is in the chosen interval. We know that P (A) ≥ 1 − 3η − t, hence, using Lemma 3.11[29], we get that there exists a constant C13 such that, 1 3 \|E[X] − E[X\|A]\| ≤ C13 C44 σ(η + t) 4 • Hence, with probability at least 1 − δ/3, the mean of Se will be within r 1 3 1 log(3/δ) 4 η × length (I1−η ) + C13 C4 σ(η + t) 4 + C6 σ(1 + 2η) 2 n • q Taking union-bound over all conditioning statements, and upper bounding, η with ǫ + log(3/δ) 2n , we recover the statement of the lemma. J.3 The case when p > 1 To prove the case for p > 1, we use a series of lemmas. Lemma 15 proves that the outlier filtering constrains the points in a ball around the true mean. Lemma 17 controls the error in e Lemma 18 controls the mean and covariance the true distribution after outlier filtering (D). e the error for the mean of S when projected onto the bottom span of the covariance matrix ΣSe. Lemma 15. Suppose that, Pθ∗ is a distribution on Rp with mean µ, covariance Σ, and bounded fourth moments. There exist positive universal constants C1 , C2 , C8 > 0, such that given n samples from the distribution in Equation (4), we can find a vector a ∈ Rp such that with probability at least 1 − δ, 1 4 ka − µk2 ≤C1 C4 + C2 !3 np 4 1 tr (Σ) ǫ + C8 log n δ !1 r r np 2 p 1 log(p/δ) tr (Σ) log ǫ + C8 n δ n r p Proof. Pick n orthogonal directions v1 , v2 , . . . , vn , and use method for one-dimensions, and using union bound, we can recover the result. Next, we prove the case when p > 1. Firstly, we prove that after the outlier step, Lemma 16. After the outlier removal step, there exists universal constants C11 > 0 such that with probability at least 1 − δ, every remaining point x satisfies, kx − µk2 ≤ r1∗ + 2r2∗ 42 √ q 1p p 3 1 and r2∗ = C1 C44 pkΣk2 (η + t) 4 + C2 pkΣk2 (η + t) 2 log(1/δ) and n q q log(1/δ) is the fraction of samples corrupted. t = C8 n1 log np δ . Here η ≤ ǫ + 2n 1 where r1∗ = C10 C44 pkΣk2 1 η4 Proof. • Let Se be the set of points chosen after the outlier filtering. Let SeD be set of good points chosen after the outlier filtering. Let SeN be the set of bad points chosen after the outlier filtering. • Using VC theory we know that for every closed ball B(µ, r) = {x\|kx − µk2 ≤ r}, there exists a constant C9 such that with probability at least 1 − δ s n p log sup \|P (x ∈ B\|x ∼ D) − P (x ∈ B\|x ∈u SD )\| ≤ C9 n pδ B \| {z } t2 1 • Let B ∗ = B(µ, r ∗ ) for r1∗ = C10 C44 1 (η) 4 p pkΣk2 . Then, we claim that P (x ∈ B ∗ \|x ∼ D) ≥ 1 − η – To see this, suppose we have some x ∈ D. LetPz = x − µ. Let zi = z T vi for some orthogonal directions v1 , v2 , . . . , vp . Let Z 2 = zi2 = kzk22 . –  1 2  C pkΣk2 P Z 2 ≥ 4 1  = P (η) 2 Z4 ≥ C4 p2 kΣk22 (η) ≤ (η)E(Z 4 ) C4 p2 kΣk22 – Now, E(Z 4 ) ≤ p2 maxi E(zi4 ) ≤ C4 p2 kΣk22 . Plugging this in the above, we have that P (x ∈ B ∗ \|x ∼ D) ≥ 1 − η. • Hence, we have that P (x ∈ B ∗ \|x ∈u SD ) ≥ 1 − η − t2 . • Using Lemma 15, we have that at least (1 − η − t2 ) fraction of good points are r1∗ + r2∗ away from a. Hence, we have that the minimum radius of the ball containing all the (1 − η − t2 )(1 − η) has a radius of atmost r1∗ + r2∗ , which when combined with the triangle inequality recovers the statement of lemma. e µe = As before, let Se be the set of points after outlier filtering. Let µSe = mean(S), SD mean(SeD ), µ e = mean(SeN ). SN Lemma 17. Let SeD be the set of clean points remaining after the outlier filtering. Then, with probability at least 1 − δ, we have that r 1 p p 3p log(p/δ) 1 4 kµSeD − µk2 ≤ C1 C4 (η + t2 ) 4 kΣk2 1 + log(p/δ) + kΣk2 (1 + 2(η + t2 ) n n (r ∗ + 2r2∗ ) log(p/δ) + C15 1 n 43 and kΣSeD k2 ≤ β(n, δ)kΣk2 , where β(n, δ) = !! √ r 3 3 p C4 p log(p/δ log(p/δ) + 1 + 2C(η + t2 ) + 1 + √ + C4 p(η + t) 2 + (η + t2 ) 2 η n n Proof. We first prove the bounds on the mean shift. kµSeD − µk2 ≤ kµSeD − µDe k2 + kµDe − µk2 {z } \| {z } \| B A µ −µ • Control of B. We use Lemma 9 on X = xT kµ De−µk2 for x ∼ D, and A be the event e D that x is not removed by the outlier filtering. 1 3 kµDe − µk2 ≤ C1 C44 (η + t2 ) 4 p kΣk2 (80) • Control of A. Using Lemma 10,we have that kΣDe k2 ≤ (1 + 2(η + t2 ))kΣk2 . Now, we use Bernstein’s inequality . Lemma 12 with R = C(r1∗ + 2r2∗ + B), we get that, with probability at least 1 − δ, r p p (r ∗ + 2r2∗ + B) 1 log(p/δ) + C15 1 log(p/δ) kµSeD − µDe k2 ≤ C14 kΣk2 (1 + 2(η + t2 ) n n (81) Next, we prove the bound for covariance matrix. kΣDe − Σk2 \| {z } kΣSeD k2 ≤ kΣSeD − ΣDe k2 + +kΣk2 (82) ≤2C(η+t2 )kΣk2 (By Corollary 13)) (x −µ )(x −µ )T −Σ e D . From, To control kΣSeD −ΣDe k2 , we use Bernstein’s inequality, with Zk = k De kn De Lemma 16, we know that the points are constrained in a ball. Plugging this into Lemma 12, ! r log(p/δ log(p/δ) 2 kΣSeD − ΣDe k2 ≤ C(kΣk2 + R ) + n n 2 2 where R2 = C r1∗ + r2∗ + B 2 . Plugging in the values, we get that, kΣSeD − ΣDe k2 ≤ CkΣk2 ! √ r 3 3 p C4 p log(p/δ log(p/δ) 1 + √ + C4 p(η + t) 2 + (η + t2 ) 2 + η n n Finally, we have that, !! √ r 3 3 log(p/δ log(p/δ) p C4 p + kΣSeD k2 ≤ kΣk2 1 + 2C(η + t2 ) + 1 + √ + C4 p(η + t) 2 + (η + t2 ) 2 η n n \| {z } β(n,δ) 44 Lemma 18. Let W be the bottom p/2 principal components of the covariance matrix after filtering ΣSe. Then there exists a universal constant C > 0 such that with probability at least 1 − δ, we have that 1 2 2 kηPW δµ k2 ≤ Cη β(n, δ) + γ(n, δ)C4 )kΣk2 , where δµ = µSeN − µSeD , PW is the projection matrix on the bottom p/2-span of ΣSe, β(n, δ) 1 is as defined in Lemma 17 and γ(n, δ) = η 2 + (η + t)5/2 + η(η + t) log(1/δ) n Proof. We have ΣSe = (1 − η)ΣSeD + ηΣSeN + (η − η 2 )δµ δµT {z } \| {z } \| E (83) F (84) By Weyl’s inequality we have that, λp/2 (ΣSe) ≤ λ1 (E) + λp/2 (F ) • Control of λp/2 (F ). tr (F ) p/2 ((r ∗ )2 + (r2∗ )2 ) + B 2 ≤ C15 η 1 p/2 1 1 log(1/δ) 5/2 2 ≤ C16 C4 kΣk2 η 2 + (η + t) + η(η + t) n \| {z } λp/2 (F ) ≤ γ(n,δ) where t = C8 q 1 n log • Control of λ1 (E). np δ . λ1 (E) ≤ (1 − η)βkΣk2 Hence, we have that: λp/2 (ΣSe) ≤ (1 − η)βkΣk2 + C16 γ p C4 kΣk2 Using that W is the space spanned by the bottom p/2 eigenvectors of ΣSe and PW is corresponding projection operator, we have that: h p i T PW ΣSePW (1 − η)β + C16 γ C4 kΣk2 Ip Following some algebraic manipulation in [29], we get that, 1 2 2 kηPW δµ k2 ≤ η (β(n, δ) + γC4 )kΣk2 45 Having established all required results, we are now ready to prove Lemma 1. We restate the result for the sake of completeness. Theorem 14. Suppose that, Pθ∗ is a distribution on Rp with mean µ, covariance Σ and bounded fourth moments. There exist positive universal constant C > 0, such that given n samples from the distribution in Equation (4), the algorithm with probability at least 1 − δ, returns an estimate µ b such that,  !1  r 1 1 1 p 3 log p log(p log(p/δ)) 2  √ √ 2 4 2  4 kb µ − µk2 ≤ CkΣk2 (1 + log p) η + C4 (η + t2 ) + ηpC4 n where η = ǫ + q log(p) log(log p/δ) 2n and t2 = q p log(p) log(n/(pδ)) . n Proof. We divide n samples into ⌊log(p)⌋ different sets. We choose the first set and keep that as our active set of samples. We run our outlier filtering on this set, and let the remaining samples after the outlier filtering be SeD . By orthogonality of subspaces spanned by eigenvectors, coupled with triangle inequality and contraction of projection operators, we have that µV − PV µk22 kb µ − µk22 ≤ 2kPW (b µ − µSeD )k22 + 2kPW (µSeD − µ)k22 + kb µV − PV µk22 kb µ − µk22 ≤ 2kPW (b µ − µSeD )k22 + 2k(µSeD − µ)k22 + kb bV is the mean vector where V is the span of the top p/2 principal components of ΣSe and where µ of returned by the running the algorithm on the reduced dimensions dim(V ) = p/2. From Lemma 18, both β(n, δ) and γ(n, δ) are monotonically increasing in the dimension; moreover the upper bound in Lemma 17 is also monotonically increasing in the dimension p, hence, the error at each step of the algorithm can be upper bounded by error incurred when running on dimension p, with n/ log(p) samples, and probability of δ/ log p. Hence, the overall error for the recursive algorithm can be upper bounded as, kb µ − µk22 ≤ 2kPW (b µ − µSeD )k22 + 2kµSeD − µk22 (1 + log p) Combining Lemma 17 and Lemma 18 which are instantiated for n/ log p samples and probability δ/ log p, we get,  !1  r 2 1p 1 1 3 log p log(p log(p/δ)) √ √  kb µ − µk2 ≤ CkΣk22 log p  η + C44 (η + t2 ) 4 + ηpC42 n 46	2
(Quasi)Periodicity Quantification in Video Data, Using Topology arXiv:1704.08382v2 [cs.CV] 21 Jan 2018 Christopher J. Tralie† and Jose A. Perea* ∗ † January 23, 2018 Abstract This work introduces a novel framework for quantifying the presence and strength of recurrent dynamics in video data. Specifically, we provide continuous measures of periodicity (perfect repetition) and quasiperiodicity (superposition of periodic modes with non-commensurate periods), in a way which does not require segmentation, training, object tracking or 1-dimensional surrogate signals. Our methodology operates directly on video data. The approach combines ideas from nonlinear time series analysis (delay embeddings) and computational topology (persistent homology), by translating the problem of finding recurrent dynamics in video data, into the problem of determining the circularity or toroidality of an associated geometric space. Through extensive testing, we show the robustness of our scores with respect to several noise models/levels; we show that our periodicity score is superior to other methods when compared to human-generated periodicity rankings; and furthermore, we show that our quasiperiodicity score clearly indicates the presence of biphonation in videos of vibrating vocal folds, which has never before been accomplished end to end quantitatively. 1 Introduction Periodicity characterizes many natural motions including animal locomotion (walking/wing flapping/slithering), spinning wheels, oscillating pendulums, etc. Quasiperiodicity, thought of as the superposition of non-commensurate frequencies, occurs naturally during transitions from ordinary to chaotic dynamics [11]. The goal of this work is to automate the analysis of videos capturing periodic and quasiperiodic motion. In order to identify both classes of motion in a unified framework, we generalize 1-dimensional (1D) sliding window embeddings [38] to reconstruct periodic and quasiperiodic attractors from videos123 . We analyze the resulting attractors using persistent homology, a technique which combines geometry and topology (Section 2.2), and we return scores in the range [0, 1] that indicate the degree of periodicity or quasiperiodicity in the corresponding video. We show that our periodicity measure compares favorable to others in the literature when ranking videos (Section 4.2). Furthermore, to our knowledge, there is no other method able to quantify the existence of quasiperiodicity directly from video data. Our approach is fundamentally different from most others which quantify periodicity in video. For instance, it is common to derive 1D signals from the video and apply Fourier or autocorrelation to measure periodicity. By contrast, our technique operates on raw pixels, avoiding common video ∗ † Department of Electrical And Computer Engineering, Duke University,Durham, NC, USA. e-mail: [email protected] † * Department of Mathematics and Department of Computational Mathematics, Science & Engineering, Michigan State University, East Lansing, MI, USA. e-mail: [email protected] 1 Some of the analysis and results appeared as part of the Ph.D. thesis of the first author [?]. 2 Code to replicate results: https://github.com/ctralie/SlidingWindowVideoTDA 3 Supplementary material and videos: https://www.ctralie.com/Research/SlidingWindowVideoQuasi 1 preprocessing and tracking entirely. Using geometry over Fourier/autocorrelation also has advantages for our applications. In fact, as a simple synthetic example shows (Figure 3), the Fourier Transform of quasiperiodic signals is often very close to the Fourier transform of periodic signals. By contrast, the sliding window embeddings we design yield starkly different geometric structures in the periodic and quasiperiodic cases. We exploit this to devise a quasiperiodicity measurement, which we use to indicate the degree of “biphonation” in videos of vibrating vocal folds (Section 4.3), which is useful in automatically diagnosing speech pathologies. In the context of applied topology, our quasiperiodicity score is one of the first applications of persistent H2 to high dimensional data, which is largely possible due to recent advancements in the computational feasibility of persistent homology [3]. 1.1 1.1.1 Prior Work on Recurrence in Videos 1D Surrogate Signals One common strategy for detecting periodicity in video is to derive a 1D function to act as a surrogate for its dynamics, and then to use either frequency domain (Fourier transform) or time domain (autocorrelation, peak finding) techniques. One of the earliest works in this genre finds level set surfaces in a spatiotemporal “XYT” volume of video (all frames stacked on top of each other), and then uses curvature scale space on curves that live on these “spatiotemporal surfaces” as the 1D function [1]. [34] use Fourier Transforms on pixels which exhibit motion, and define a measure of periodicity based on the energy around the Fourier peak and its harmonics. [10] extract contours and find eigenshapes from the contours to classify and parameterize motion within a period. Frequency estimation is done by using Fourier analysis and peak detection on top of other 1D statistics derived from the contours, such as area and center of mass. Finally, [47] derive a 1D surrogate function based on mutual information between the first and subsequent frames, and then look for peaks in the similarity function with the help of a watershed method. 1.1.2 Self-Similarity Matrices Another class of techniques relies on self-similarity matrices (SSMs) between frames, where similarity can be defined in a variety of ways. [37] track a set of points on a foreground object and compare them with an affine invariant similarity. Another widely recognized technique for periodicity quantification [6], derives periodicity measures based on self-similarity matrices of L1 pixel differences. This technique has inspired a diverse array of applications, including analyzing the cycles of expanding/contracting jellyfish [33], analyzing bat wings [2], and analyzing videos of autistic spectrum children performing characteristic repetitive motions such as “hand flapping” [22]. We compare to this technique in Section 4.2. 1.1.3 Miscellaneous Techniques for Periodic Video Quantification There are also a number of works that don’t fall into the two categories above. Some works focus solely on walking humans, since that is one of the most common types of periodic motion in videos of interest to people. [29] look at the “braiding patterns” that occur in XYT slices of videos of walking people. [17] perform blob tracking on the foreground of a walking person, and use the ratio of the second and first eigenvalues of PCA on that blob. For more general periodic videos, [43] make a codebook of visual words and look for repetitions within the resulting string. [23] take a deep learning approach to counting the number of periods that occur in a video segment. They use a 3D convolutional neural network on spatially downsampled, non-sequential regions of interest, which are uniformly spaced in time, to estimate the length of the cycle. Finally, perhaps the most philosophically similar work to ours is the work of [41], who use cohomology to find maps of MOCAP data to the circle for parameterizing periodic motions, though this work does not provide a way to quantify periodicity. 2 1.1.4 Our Work We show that geometry provides a natural way to quantify recurrence (i.e. periodicity and quasiperiodicity) in video, by measuring the shape of delay embeddings. In particular, we propose several optimizations (section 3) which make this approach feasible. The resulting measure of quasiperiodicity, for which quantitative approaches are lacking, is used in section 4.3 to detect anomalies in high-speed videos of vibrating vocal folds. Finally, in contrast to both frequency and time domain techniques, our method does not rely on the period length being an integer multiple of the sampling rate. 2 2.1 Background Delay Embeddings And Their Geometry Recurrence in video data can be captured via the geometry of delay embeddings; we describe this next. 2.1.1 Video Delay Embeddings We will regard a video as a sequence of grayscale4 image frames indexed by the positive real numbers. That is, given positive integers W (width) and H (height), a video with W × H pixels is a function X : R+ −→ RW ×H In particular, a sequence of images X1 , X2 , . . . ∈ RW ×H sampled at discrete times t1 < t2 < · · · yields one such function via interpolation. For an integer d ≥ 0, known as the dimension, a real number τ > 0, known as the delay, and a video X : R+ −→ RW ×H , we define the sliding window (also referred to as time delay) embedding of X – with parameters d and τ – at time t ∈ R+ as the vector   X(t)  X(t + τ )    (1) SWd,τ X(t) =   ∈ RW ×H×(d+1) ..   . X(t + dτ ) The subset of RW ×H×(d+1) resulting from varying t will be referred to as the sliding window embedding of X. We remark that since the pixel measurement locations are fixed, the sliding window embedding is an “Eulerian” view into the dynamics of the video. Note that delay embeddings are generally applied to 1D time series, which can be viewed as 1-pixel videos (W = H = 1) in our framework. Hence equation (1) is essentially the concatenation of the delay embeddings of each individual pixel in the video into one large vector. One of the main points we leverage in this paper is the fact that the geometry of the sliding window embedding carries fundamental information about the original video. We explore this next. 2.1.2 Geometry of 1-Pixel Video Delay Embeddings As a motivating example, consider the harmonic (i.e. periodic) signal π π t + cos t fh (t) = cos 5 15 and the quasiperiodic signal π 1 fq (t) = cos t + cos t 5 5 (2) (3) 4 For color videos we can treat each channel independently, yielding a vector in RW ×H×3 . In practice, there isn’t much of a difference between color and grayscale embeddings in our framework for the videos we consider. 3 1 1 We refer to fh as harmonic because its constitutive frequencies, 10 and 30 , are commensurate; that is, they are linearly dependent over the rational numbers Q ⊂ R. By way of contrast, the underlying fre1 1 quencies of the signal fq , 10 and 10π , are linearly independent over Q and hence non-commensurate. We use the term quasiperiodicity, as in the non-linear dynamics literature [19], to denote the superposition of periodic processes whose frequencies are non-commensurate. This differs from other definitions in the literature (e.g. [?, 43]) which regard quasiperiodic as any deviation from perfect repetition. A geometric argument from [31] (see equation 7 below and the discussion that follows) shows that given a periodic function f : [0, 2π] −→ R with exactly N harmonics, if d ≥ 2N and 0 < τ < 2π d then the sliding window embedding SWd,τ f is a topological circle (i.e. a closed curve without selfintersections) which wraps around an N -dimensional torus S 1 = {z ∈ C : \|z\| = 1} TN = S 1 × · · · × S 1 , {z } \| N −times As an illustration, we show in Figure 1 a plot of fh and of its sliding window embedding SWd,τ fh , via a PCA (Principal Component Analysis) 3-dimensional projection. Figure 1: Sliding window embedding of the harmonic signal fh . Colors in the signal correspond to colors of the points in the PCA plot. The sliding window embedding traces a topological circle wrapped around a 2-dimensional torus. However, if g : R −→ R is quasiperiodic with N distinct non-commensurate frequencies then, for appropriate d and τ , SWd,τ g is dense in (i.e. fills out) TN [30]. Figure 2 shows a plot of the quasiperiodic signal fq (t) and a 3-dimensional projection, via PCA, of its sliding window embedding SWd,τ fq . Figure 2: Sliding window embedding of the quasiperiodic signal fq . Colors in the signal correspond to colors of the points in the PCA plot. The sliding window embedding is dense in a 2-dimensional torus. 4 The difference in geometry of the delay embeddings is stark compared to the difference between their power spectral densities, as shown in Figure 3. Figure 3: The power spectral densities (300 samples) of commensurate and non-commensurate signals with relative harmonics at ratios 3 and π, respectively. The difference is not nearly as evident as in the geometry of their sliding window embeddings. Additionally, unless sampling is commensurate with a frequency, a fixed Fourier basis causes that frequency component to bleed into many frequency bins in a sinc-like pattern, making precise peak finding difficult. Moreover, as we will see next, the interpretation of periodicity and quasiperiodicity as circularity and toroidality of sliding window embeddings remains true for videos with higher resolution (i.e. max{W, H} > 1). The rest of the paper will show how one can use persistent homology, a tool from the field of computational topology, to quantify the presence of (quasi)periodicity in a video by measuring the geometry of its associated sliding window embedding. In short, we propose a periodicity score for a video X which measures the degree to which the sliding window embedding SWd,τ X spans a topological circle, and a quasiperiodicity score which quantifies the degree to which SWd,τ X covers a torus. This approach will be validated extensively: we show that our (quasi)periodicity detection method is robust under several noise models (motion blur, additive Gaussian white noise, and MPEG bit corruption); we compare several periodicity quantification algorithms and show that our approach is the most closely aligned with human subjects; finally, we provide an application to the automatic classification of dynamic regimes in high-speed laryngeal video-endoscopy. 2.1.3 Geometry of Video Delay Embeddings Though it may seem daunting compared to the 1D case, the geometry of the delay embedding shares many similarities for periodic videos, as shown in [39]. Let us argue why sliding window embeddings from (quasi)periodic videos have the geometry we have described so far. To this end, consider an example video X that contains a set of N frequencies ω1 , ω2 , ..., ωN . Let the amplitude of the nth frequency and ith pixel be ain . For simplicity, but without loss of generality, assume that each is a cosine with zero phase offset. Then the time series at pixel i can be written as Xi (t) = N X ain cos(ωn t) (4) n=1 Grouping all of the coefficients together into a (W × H) × N matrix A, we can write X(t) = N X An cos(ωn t) n=1 5 (5) where An stands for the nth column of A. Constructing a delay embedding as in Equation 1:   An cos(ωn t) N X  .. SWd,τ X(t) =   . n n=1 A cos(ωn (t + dτ ) (6) and applying the cosine sum identity, we get SWd,τ X(t) = N X ~un cos(ωn t) − ~vn sin(ωn t) (7) n=1 where ~un , ~vn ∈ RW ×H×(d+1) are constant vectors. In other words, the sliding window embedding of this video is the sum of linearly independent ellipses, which lie in the space of d + 1 frame videos at resolution W × H. As shown in [31] for the case of commensurate frequencies, when the window length is just under the length of the period, all of the ~un and ~vn vectors become orthogonal, and so they can be recovered by doing PCA on SWd,τ X(t). Figure 4 shows the components of the first 8 PCA vectors for a horizontal line of pixels in a video of an oscillating pendulum. Note how the oscillations are present both temporally and spatially. X t Figure 4: Showing an XT slice of the principal components on SWd,1 for the synthetic video of an oscillating pendulum; d is chosen just under the period length (∼ 25 frames). 2.1.4 The High Dimensional Geometry of Repeated Pulses Using Eulerian coordinates has an important impact on the geometry of delay embeddings of natural videos. As Figure 5 shows, pixels often jump from foreground to background in a pattern similar to square waves. These types of abrupt transitions require higher dimensional embeddings to reconstruct the geometry. To see why, first extract one period of a signal with period ` at a pixel Xi (t): Xi (t) 0 ≤ t ≤ ` fi (t) = (8) 0 otherwise Then Xi (t) can be rewritten in terms of the pulse as Xi (t) = ∞ X fi (t − m`) (9) m=−∞ Since Xi (t) repeats itself, regardless of what fi (t) looks like, periodic summation discretizes the frequency domain [32] ∞ m m X F {Xi (t)} (k) ∝ F(fi (t)) δ −k (10) ` ` m=−∞ 6 Figure 5: An example of an Eulerian pixel witnessing a foreground/background transition in a video of a woman doing jumping jacks. Red, green, and blue channels are plotted over time. These transitions induce a per pixel periodic signal with sharp transitions, which leads to high dimensionality in an appropriate sliding window embedding. Switching back to the time domain, we can write Xi (t) as Xi (t) ∝ ∞ X F (fi (t)) m=−∞ m ` ei 2πm ` t (11) In other words, each pixel is the sum of some constant offset plus a (possibly infinite) set of harmonics at integer multiples of 1` . For instance, applying Equation 11 to a square wave of period ` centered at the origin is a roundabout way of deriving the Fourier Series 2π 1 6π 1 10π sin t + sin t + sin t + ... (12) ` 3 ` 5 ` by sampling the sinc function sin(π`f )/(πf ) at intervals of m/2` (every odd m coincides with π/2+kπ, proportional to 1/k, and every even harmonic is zero conciding with πk). In general, the sharper the transitions are in Xi (t), the longer the tail of F{fi (t)} will be, and the more high frequency harmonics will exist in the embedding, calling for a higher delay dimension to fully capture the geometry, since every harmonic lives on a linearly independent ellipse. Similar observations about harmonics have been made in images for collections of patches around sharp edges ( [48], Figure 2). 2.2 Persistent Homology Informally, topology is the study of properties of spaces which do not change after stretching without gluing or tearing. For instance, the number of connected components and the number of (essentially different) 1-dimensional loops which do not bound a 2-dimensional disk, are both topological properties of a space. It follows that a circle and a square are topologically equivalent since one can deform one onto the other, but a circle and a line segment are not because that would require either gluing the endpoints of the line segment or tearing the circle. Homology [12] is a tool from algebraic topology designed to measure these types of properties, and persistent homology [50] is an adaptation of these ideas to discrete collections of points (e.g., sliding window embeddings). We briefly introduce these concepts next. 2.2.1 Simplicial Complexes A simplicial complex is a combinatorial object used to represent and discretize a continuous space. With a discretization available, one can then compute topological properties by algorithmic means. 7 Formally, a simplicial complex with vertices in a nonempty set V is a collection K of nonempty finite subsets σ ⊂ V so that ∅ = 6 τ ⊂ σ ∈ K always implies τ ∈ K. An element σ ∈ K is called a simplex, and if σ has (n + 1) elements then it is called an n-simplex. The cases n = 0, 1, 2 are special, 0-simplices are called vertices, 1-simplices are called edges and 2-simplices are called faces. Here is an example to keep in mind: the circle S 1 = {z ∈ C : \|z\| = 1} is a continuous space but its topology can be captured by a simplicial complex K with three vertices a, b, c, and three edges {a, b}, {b, c}, {a, c}. That is, in terms of topological properties, the simplicial complex K = {a}, {b}, {c}, {a, b}, {b, c}, {a, c} can be regarded as a combinatorial surrogate for S 1 : they both have 1 connected component, one 1-dimensional loop which does not bound a 2-dimensional region, and no other features in higher dimensions. 2.2.2 Persistent Homology of Point Clouds The sliding window embedding of a video X is, in practice, a finite set SWd,τ X = {SWd,τ X(t) : t ∈ T } determined by a choice of T ⊂ R finite. Moreover, since SWd,τ X ⊂ RW ×H×(d+1) then the restriction of the ambient Euclidean distance endows SWd,τ X with the structure of a finite metric space. Discrete metric spaces, also referred to as point clouds, are trivial from a topological point of view: a point cloud with N points simply has N connected components and no other features (e.g., holes) in higher dimensions. However, when a point cloud has been sampled from/around a continuous space with non-trivial topology (e.g., a circle or a torus), one would expect that appropriate simplicial complexes with vertices on the point cloud should reflect the topology of the underlying continuous space. This is what we will exploit next. Given a point cloud (X, dX ) – where X is a finite set and dX : X×X −→ [0, ∞) is a distance function – the Vietoris-Rips complex (or Rips complex for short) at scale ≥ 0 is the collection of non-empty subsets of X with diameter less than or equal to : R (X) := σ ⊂ X : dX (x1 , x2 ) ≤ , ∀xi , xj ∈ σ (13) That is, R (X) is the simplicial complex with vertex set equal to X, constructed by adding an edge between any two vertices which are at most apart, adding all 2-dimensional triangular faces (i.e. 2-simplices) whose bounding edges are present, and more generally, adding all the k-simplices whose (k − 1)-dimensional bounding facets have been included. We show in Figure 6 the evolution of the Rips complex on a set of points sampled around the unit circle. epsilon = 0 epsilon = 0.30 epsilon = 0.35 1 1 1 0 0 0 -1 -1 -1 -1 0 1 -1 0 1 -1 0 1 Figure 6: The Rips complex, at three different scales ( = 0, 0.30, 0.35), on a point cloud with 40 points sampled around S 1 ⊂ R2 . The idea behind persistent homology is to track the evolution of topological features of complexes such as R (X), as the scale parameter ranges from 0 to some maximum value max ≤ ∞. For instance, in Figure 6 one can see that R0 (X) = X has 40 distinct connect components (one for each 8 point), R0.30 (X) has three connected components and R0.35 (X) has only one connected component; this will continue to be the case for every ≥ 0.35. Similarly, there are no closed loops in R0 (X) or R0.30 (X) bounding empty regions, but this changes when increases to 0.35. Indeed, R0.35 (X) has three 1-dimensional holes: the central prominent hole, and the two small ones to the left side. Notice, however, that as increases beyond 0.35 these holes will be filled by the addition of new simplices; in particular, for > 2 one has that R (X) will have only one connected component and no other topological features in higher dimensions. The family R(X) = {R (X)}≥0 is known as the Rips filtration of X, and the emergence/dissapearence of topological features in each dimension (i.e., connected components, holes, voids, etc) as changes, can be codified in what are referred to as the persistence diagrams of R(X). Specifically, for each dimension n = 0, 1, . . . (0 = connected components, 1 = holes, 2 = voids, etc) one can record the value of for which a particular n-dimensional topological feature of the Rips filtration appears (i.e. its birth time), and when it disappears (i.e. its death time). The birth-death times (b, d) ∈ R2 of n-dimensional features for R(X) form a multiset dgmn (R(X)) — i.e. a set whose elements can come with repetition — known as the n-dimensional persistence diagram of the Rips filtration on X. Since dgmn (R(X)) is just a collection of points in the region {(x, y) ∈ R2 : 0 ≤ x < y}, we will visualize it as a scatter plot. The persistence of a topological feature with birth-death times (b, d) is the quantity d − b, i.e. its lifetime. We will also include the diagonal y = x in the scatter plot in order to visually convey the persistence of each birth-death pair. In this setting, points far from the diagonal (i.e. with large persistence) represent topological features which are stable across scales and hence deemed significant, while points near the diagonal (i.e. with small persistence) are often associated with unstable features. We illustrate in Figure 7 the process of going from a point cloud to the 1-dimensional persistence diagram of its Rips filtration. Original Point Cloud 5 4 3 Time of Death 2 1 0 1 2 3 1D Persistence Diagram 4.5 4.0 2 3.5 3.0 2.5 2.0 1 1.5 1.0 0.5 0.0 5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Time of Birth Class 1 Death (d = 1.84) 5 3 2 1 0 1 2 3 4 Class 2 Birth (d = 1.8) 5 4 3 2 1 0 1 2 3 4 4 3 3 3 2 2 2 1 1 1 0 0 0 1 1 1 2 2 2 3 3 2 1 0 1 2 3 4 5 3 2 1 0 1 2 3 4 5 3 3 2 1 0 1 2 3 4 5 Class 2 Death (d = 3.55) 5 4 3 Class 1 Birth (d = 0.774) 5 3 2 1 0 1 2 3 4 5 Figure 7: From a point cloud to the 1-dimensional persistence diagram of its Rips filtration. Connected edges in the Rips filtration are drawn in blue, the birth/death of a class is indicated in red, and filled in triangles are shaded green. We remark that the computational task of determining all non-equivalent persistent homology classes of a filtered simplicial complex can, surprisingly, be reduced to computing the homology of a single simplicial complex [3, 50]. This is in fact a problem in linear algebra that can be solved via 9 elementary row and column operations on appropriate boundary matrices. The persistent homology of R SWd,τ X , and in particular its n-dimensional persistence diagrams for n = 1, 2, are the objects we will use to quantify periodicity and quasiperiodicity in a video X. Figures 8 and 9 show the persistence diagrams of the Rips filtrations, on the sliding window embeddings, for the commensurate and non-commensurate signals from Figures 1 and 2, respectively. We use fast new code from the “Ripser” software package to make persistent H2 computation feasible [3]. Figure 8: Sliding window embedding of the harmonic signal fh (left) and the n-dimensional persistence diagrams n = 1, 2 (right) of the associated Rips filtration. The sliding window embedding SWd,τ fh traces a topological circle wrapped around a 2-dimensional torus. The persistence diagram in dimension one (H1 ) shows only one birth-death pair with prominent persistence; this is consistent with a point cloud sampled around a space with the topology of a circle. Figure 9: Sliding window embedding of the quasiperiodic signal fq (left) and the n-dimensional persistence diagrams n = 1, 2 (right) of the associated Rips filtration. The sliding window embedding SWd,τ fq is dense on a 2-dimensional torus. The persistence diagram in dimension one (H1 ) shows two birth-death pairs with prominent persistence, while the persistence diagram in dimension two (H2 ) shows one prominent birth-death pair; this is consistent with a point cloud sampled around a space with the topology of a 2-dimensional torus. 3 3.1 Implementation Details Reducing Memory Requirements with SVD Suppose we have a video which has been discretely sampled at N different frames at a resolution of W × H, and we do a delay embedding with dimension d, for some arbitrary τ . Assuming 32 bit floats per grayscale value, storing the sliding window embedding requires 4W HN (d + 1) bytes. For 10 a low resolution 200 × 200 video only 10 seconds long at 30fps, using d = 30 already exceeds 1GB of memory. In what follows we will address the memory requirements and ensuing computational burden to construct and access the sliding window embedding. Indeed, constructing the Rips filtration only requires pairwise distances between different delay vectors, this enables a few optimizations. First of all, for N points in RW H , where N W H, there exists an N -dimensional linear subspace which contains them. In particular let A be the (W × H) × N matrix with each video frame along a column. Performing a singular value decomposition A = U SV T , yields a matrix U whose columns form an orthonormal basis for the aforementioned N -dimensional linear subspace. Hence, by finding the coordinates of the original frame vectors with respect to this orthogonal basis Â = U T A = U T U SV = SV (14) and using the coordinates of the columns of SV instead of the original pixels, we get a sliding window embedding of lower dimension   U T X(t)   .. (15) SWd,τ (t) =   . U T X(t + dτ ) for which kSWd,τ (t) − SWd,τ (t0 )k = kSWd,τ X(t) − SWd,τ X(t0 )k Note that SV can be computed by finding the eigenvectors of AT A; this has a cost of O(W 2 H 2 + N ) which is dominated by W 2 H 2 if W H N . In our example above, this alone reduces the memory requirements from 1GB to 10MB. Of course, this procedure is the most effective for short videos where there are actually many fewer frames than pixels, but this encompasses most of the examples in this work. In fact, the break-even point for a 200x200 30fps video is 22 minutes. A similar approach was used in the classical work on Eigenfaces [40] when computing the principal components over a set of face images. 3 3.2 Distance Computation via Diagonal Convolutions A different optimization is possible if τ = 1; that is, if delays are taken exactly on frames and no interpolation is needed. In this case, the squared Euclidean distance between SWd,1 X(i) and SWd,1 X(j) is d X \|\|SWd,1 X(i) − SWd,1 X(j)\|\|22 = \|\|X(i + m) − X(j + m)\|\|22 (16) m=0 2 DX Let be the N × N matrix of all pairwise squared Euclidean distances between frames (possibly computed with the memory optimization in Section 3.1), and let DY2 be the (N − d) × (N − d) matrix of all pairwise distances between delay frames. Then Equation 16 implies that DY2 can be obtained 2 from DX via convolution with a “rect function”, or a vector of 1s of length d + 1, over all diagonals in 2 DX (i.e. a moving average). This can be implemented in time O(N 2 ) with cumulative sums. Hence, regardless of how d is chosen, the computation and memory requirements for computing DY2 depend only on the number of frames in the video. Also, DY can simply be computed by taking the entry wise square root of DY2 , another O(N 2 ) computation. A similar scheme was used in [16] when comparing distances of 3D shape descriptors in videos of 3D meshes. Figure 10 shows self-similarity matrices on embeddings of the pendulum video with no delay and with a delay approximately matching the period. The effect of a moving average along diagonals with delay eliminates the anti-diagonals caused by the video’s mirror symmetry. Even for videos without mirror symmetries, such as a video of a running dog (Figure 11), introducing a delay brings the geometry into focus, as shown in Figure 12. 11 Pairwise Distances, Tau = 1, d = 28 Pairwise Distances, Tau = 1, d = 0 Figure 10: Self-similarity matrices DY0,1 and DY28,1 for a video of the oscillating pendulum. Bright colors indicate far distances and dark colors indicate near distances. This example clearly shows how adding a delay embedding is like performing block averaging along all diagonals of the pairwise distance matrices, and it gets rid of the mirror symmetry. Time Figure 11: An animation of a periodic video of a running dog, which, unlike an oscillating pendulum, does not have mirror symmetry in the second half of its period. Pairwise Distances, Tau = 1, d = 0 Pairwise Distances, Tau = 1, d = 18 Figure 12: Self-similarity matrices DY0,1 and DY18,1 for a video of a running dog. Even without the delay embedding (d = 0), the video frames still form a topological loop. However, a delay embedding with d = 25 cleans up the geometry and leads to a rounder loop, as seen in the resulting SSM. 3.3 Normalization A few normalization steps are needed in order to enable fair comparisons between videos with different resolutions, or which have a different range in periodic motion either spatially or in intensity. First, we perform a “point-center and sphere normalize” vector normalization which was shown in [31] to have nice theoretical properties. 12 That is, g d,τ (t) = SW SWd,τ (t) − (SWd,τ (t)T 1)1 \|\|SWd,τ (t) − (SWd,τ (t)T 1)1\|\|2 (17) where 1 is a W H(d + 1) × 1 vector of all ones. In other words, one subtracts the mean of each component of each vector, and each vector is scaled so that it has unit norm (i.e. lives on the unit sphere in RW H(d+1) ). Subtracting the mean from each component will eliminate additive linear drift on top of the periodic motion, while scaling addresses resolution / magnitude differences. Note that we can still use the memory optimization in Section 3.1, but we can no longer use the optimizations in Section 3.2 since each window is normalized independently. Moreover, in order to mitigate nonlinear drift, we implement a simple pixel-wise convolution by the derivative of a Gaussian for each pixel in the original video before applying the delay embedding: X̂i (t) = Xi (t) ∗ −at exp−t 2 /(2σ 2 ) (18) This is a pixel-wise bandpass filter which could be replaced with any other bandpass filter leveraging application specific knowledge of expected frequency bounds. This has the added advantage of reducing the number of harmonics, enabling a smaller embedding dimesion d. 3.4 Periodicity/Quasiperiodicity Scoring Once the videos are normalized to the same scale, we can score periodicity and quasiperiodicity based on the geometry of sliding window embeddings. Let dgmn be the n-dimensional persistence diagram for the Rips filtration on the sliding window embedding of a video, and define mpi (dgmn ) as the i-th largest difference d − b for (b, d) ∈ dgmn . In particular mp1 (dgmn ) = max{d − b : (b, d) ∈ dgmn } and mpi (dgmn ) ≥ mpi+1 (dgmn ). We propose the following scores: 1. Periodicity Score (PS) 1 P S = √ mp1 (dgm1 ) 3 (19) Like [31], we exploit the fact that for the Rips filtration on S 1 , the √ 1-dimensional persistence diagram has only one prominent birth-death pair with coordinates 0, 3 . Since this is the limit shape of a normalized perfectly periodic sliding window video, the periodicity score is between 0 (not periodic) and 1 (perfectly periodic). 2. Quasiperiodicity Score (QPS) r QP S = mp2 (dgm1 )mp1 (dgm2 ) 3 (20) This score is designed with the torus in mind. We score based on the second largest 1D persistence times the largest 2D persistence, since we want a shape that has two core circles and encloses a void to get a large score. Based on the Künneth theorem of homology, the 2-cycle (void) should die the moment the smallest 1-cycle dies. 3. Modified Periodicity Score (MPS) 1 M P S = √ (mp1 (dgm1 ) − mp2 (dgm1 )) 3 (21) We design a modified periodicity score which should be lower for quasiperiodic videos, than what the original periodicity score would yield. 13 Note that we use Z3 field coefficients for all persistent homology computations since, as shown by [31], this works better for periodic signals with strong harmonics. Before we embark on experiments, let us explore the choice of two crucial parameters for the sliding window embedding: the delay τ > 0 and the dimension d ∈ N. In practice we determine an equivalent pair of parameters: the dimension d and the window size dτ . 3.5 Dimension and Window Size Takens’ embedding theorem is one of the most fundamental results in the theory of dynamical systems [38]. In short, it contends that (under appropriate hypotheses) there exists an integer D, so that for all d ≥ D and generic τ > 0 the sliding window embedding SWd,τ X reconstructs the state space of the underlying dynamics witnessed by the signal X. One common strategy for determining a minimal such D is the false nearest-neighbors scheme [21]. The idea is to keep track of the k-th nearest neighbors of each point in the delay embedding, and if they change as d is increased, then the prior estimates for d were too low. This algorithm was used in recent work on video dynamics [42], for instance. Even if we can estimate d, however, how does one choose the delay τ ? As shown in [31], the sliding window embedding of a periodic signals is roundest (i.e. so that the periodicity score P S is maximized) when the window size, dτ , satisfies the following relation: d πk (22) dτ = L d+1 Here L is number of periods that the signal has in [0, 2π] and k ∈ N. To verify this experimentally, we show in Figure 13 how the periodicity score P S changes as a function of window size for the pendulum video, and how the choice of window size from Equation 22 maximizes P S. To generate this figure we fixed a sufficiently large d and varied τ . Let us now describe the general approach: Given a video we perform a period-length estimation step (see section 3.6 next), which results in a positive real number d `. For a given d ∈ N large enough we let τ > 0 be so that dτ = ` · d+1 . Figure 13: Varying the window size, dτ , in a delay embedding of the synthetic pendulum video, which has a period length around 25 frames. Red dashed lines are drawn at the window lengths that would be expected to maximize roundness of the embedding for that period length based on theory in [31]. 3.6 Fundamental Frequency Estimation Though Figure 13 suggests robustness to window size as long as the window is more than half of the period, we may not know what that is in practice. To automate window size choices, we do a coarse estimate using fundamental frequency estimation techniques on a 1D surrogate signal. To get a 1D signal, we extract the first coordinate of diffusion maps [4] using 10% nearest neighbors on the 14 raw video frames (no delay) after taking a smoothed time derivative. Note that a similar diffusionbased method was also used in recent work by [46] to analyze the frequency spectrum of a video of an oscillating 2 pendulum + spring system in a quasiperiodic state. Once we have the diffusion time series, we then apply the normalized autocorrelation method of [25] to estimate the fundamental frequency. In particular, given a discrete signal x of length N , define the autocorrelation as rt (τ ) = t+N −1−τ X xj xj+τ (23) j=t However, as observed by [7], a more robust function for detecting periodicities is the squared difference function t+N −1−τ X dt (τ ) = (xj − xj+τ )2 (24) j=t which can be rewritten as dt (τ ) = mt (τ ) − 2rt (τ ) where mt (τ ) = t+N −1−τ X (x2j + x2j+τ ) (25) j=t Finally, [25] suggest normalizing this function to the range [−1, 1] to control for window size and to have an interpretation akin to a Pearson correlation coefficient: nt (τ ) = 1 − 2rt (τ ) mt (τ ) − 2rt (τ ) = mt (τ ) mt (τ ) (26) The fundamental frequency is then the inverse period of the largest peak in nt which is to the right of a zero crossing. The zero crossing condition helps prevent an offset of 0 from being the largest peak. Defining the normalized autocorrelation as in Equation 26 has the added advantage that the value of nt (τ ) at the peak can be used to score periodicity, which the authors call clarity. Values closer to 1 indicate more perfect periodicities. This technique will sometimes pick integer multiples of the period, so we multiply nt (τ ) by a slowly decaying envelope which is 1 for 0 lag and 0.9 for the maximum lag to emphasize smaller periods. Figure 14 shows the result of this algorithm on a periodic video, and Figure 15 shows the algorithm on an irregular video. Figure 14: Diffusion maps + normalized autocorrelation fundamental frequency estimation on a periodic vocal folds video (Section 4.3). The chosen period length is 32, as indicated by the red dot over the peak. This matches with the visually inspected period length. 15 Figure 15: Diffusion maps + normalized autocorrelation fundamental frequency estimation on a video of vocal folds with irregular oscillations (Section 4.3). 4 Experimental Evaluation Next we evaluate the effectiveness of the proposed (Modified) Periodicity and Quasiperiodicity scores on three different tasks. First, we provide estimates of accuracy for the binary classifications periodic/notperiodic or quasiperiodic/not-quasiperiodic in the presence of several noise models and noise levels. The results illustrate the robustness of our method. Second, we quantify the quality of periodicity rankings from machine scores, as compared to those generated by human subjects. In a nutshell, and after comparing with several periodicity quantification algorithms, our approach is shown to be the most closely aligned with the perception of human subjects. Third, we demonstrate that our methodology can be used to automatically detect the physiological manifestations of certain speech pathologies (e.g., normal vs. biphonation), directly from high-speed videos of vibrating vocal folds. 4.1 Classification Under Varying Noise Levels/Models As shown empirically in [8], a common source of noise in videos comes from camera shake (blur); this is captured by point spread functions resembling directed random walks [8, Figure 1] and the amount of blur (i.e. noise level) is controlled by the extent in pixels of the walk. Other sources are additive white Gaussian noise (awgn), controlled by the standard deviation of the Gaussian kernel, and MPEG bit errors quantified by the percentage of corrupted information. Figure 16 shows examples of these noise types. For classification purposes we use three main recurrence classes. Three types of periodic videos (True periodic, TP): an oscillating pendulum, a bird flapping its wings, and an animation of a beating heart. Two types of quasiperiodic videos (True quasiperiodic, TQ): one showing two solid disks which oscillate sideways at non-commensurate rates, and the second showing two stationary Gaussian pulses with amplitudes non-commensurately modulated by cosine functions. Two videos without significant recurrence (True non-recurrent, TN): a video of a car driving past a landscape, and a video of an explosion. Each one of these seven videos is then corrupted by the three noise models at three different noise levels (blur = 20, 40, 80, awgn σ = 1, 2, 3, bit error = 5, 10, 20%) as follows: given a particular video, a noise model and noise level, 600 instances are generated by sampling noise independently at random. Results: We report in Table 1 the area under the Receiver Operating Characteristic (ROC) curve, or AUROC for short, for the classification task TP vs. TN (resp. TQ vs. TN) and binary classifier furnished by Periodicity (resp. Quasiperiodicity) Score. For instance, for the Blur noise model with noise level of 80 × 80 pixels, the AUROC from using the Periodicity Score to classify the 600 instances of the Heartbeat video as periodic, and the 600 16 (b) 20 × 20 blur (a) Original (c) awgn σ = 2 (d) 5% Bit Err Figure 16: The results of applying motion blur, additive white Gaussian noise, and MPEG bit corruption to a video frame. Table 1: AUROC values for different levels of noise, from the binary classification task: periodic (bird flapping, heart beating, pendulum) vs. non-recurrent (driving - left subcell, explosions - right subcell) based on the periodicity score (Equation 19). Also two synthetic quasiperiodic videos (sideways disks, modulated pulses) are compared to the same two non-recurrent videos based on the quasiperiodicity score (Equation 20). Awgn Awgn σ=1 σ=2 Awgn σ=3 Bit Err 5% Bit Err 10% 1 1 1 1 1 1 0.97 1 0.91 0.94 1 1 1 1 1 0.94 0.84 0.85 0.43 0.68 1 1 1 1 1 1 1 Blur 20 Bird Flapping Heart Beat Pendulum Blur 40 Blur 80 1 1 1 1 1 1 1 1 QuasiPeriodic Disks 1 0.85 QuasiPeriodic Pulses 1 0.9 1 1 1 Bit Err 15% 0.92 0.75 0.79 0.9 0.89 1 0.91 0.98 0.87 0.56 0.62 1 1 0.99 0.98 0.97 1 0.83 0.75 0.39 1 1 1 1 1 1 1 0.92 0.99 0.93 0.82 0.82 1 0.87 1 1 1 1 1 1 0.96 0.99 0.95 0.95 0.95 1 0.83 1 instances of the Driving video as not periodic is 0.91. Similarly, for the MPEG bit corruption model with 5% of bit error, the AUROC from using the Quasiperiodicity score to classify the 600 instances of the Quasiperiodic Sideways Disks as quasiperiodic and the 600 instances of the Explosions video as not quasiperiodic is 0.92. To put these numbers in perspective, AUROC = 1 is associated with a perfect classifier and AUROC = 0.5 corresponds to classification by a random coin flip. Overall, the type of noise that degrades performance the most across videos is the bit error, which makes sense, since this has the effect of randomly freezing, corrupting, or even deleting frames, which all interrupt periodicity. The blur noise also affects videos where the range of motion is small. The pendulum video, for instance, only moves over a range of 60 pixels at the most extreme end, so an 80x80 pixel blur almost completely obscures the motion. 4.2 Comparing Human and Machine Periodicity Rankings Next we quantify the extent to which rankings obtained from our periodicity score (Equation 19), as well as three other methods, agree with how humans rank videos by periodicity. The starting point is a dataset of 20 different creative commons videos, each 5 seconds long at 30 frames per second. Some 17 videos appear periodic, such as a person waving hands, a beating heart, and spinning carnival rides. Some of them appear nonperiodic, such as explosions, a traffic cam, and drone view of a boat sailing. And some of them are in between, such as the pendulum video with simulated camera shake. It is known that humans are notoriously bad at generating globally consistent rankings of sets with more than 5 or 7 elements [27]. However, when it comes to binary comparisons of the type “should A be ranked higher than B?” few systems are as effective as human perception, specially for the identification of recurrent patterns in visual stimuli. We will leverage this to generate a globally consistent ranking of the 20 videos in our initial data set. We use Amazon’s Mechanical Turk (AMT) [5] to present each pair of videos in the set of 20, 20 2 = 190, each to three different users, for a total of 570 pairwise rankings. 15 unique AMT workers contributed to our experiment, using an interface as the one shown in Figure 17. Figure 17: The interface that humans are given on AMT for pairwise ranking videos by periodicity. In order to aggregate this information into a global ranking which is as consistent as possible with the pairwise comparisons, we implement a technique known as Hodge rank aggregation [18]. Hodge rank aggregation finds the closest consistent ranking to a set of preferences, in a least squares sense. More precisely, given a set of objects X, and given a set of comparisons P ⊂ X × X, we seek a scalar function s on all of the objects that minimizes the following sum X \|vab − (sb − sa )\|2 (27) (a,b)∈P where vab is a real number which is positive if b is ranked higher than a and negative otherwise. Thus, s is a function whose discrete gradient best matches the set of preferences with respect to an L2 norm. Note that the preferences that we feed the algorithm are based on the pairwise rankings returned from AMT. If video b is greater than video a, then we assign vab = 1, or -1 otherwise. Since we have 3 rankings for each video, we actually assign weights of +3, +1, -1, or -3. The +/- 3 are if all rankings agree in one direction, and the +/- 1 are if one of the rankings disagrees with the other two. Figure 18 shows a histogram of all of the weighted scores from users on AMT. They are mostly in agreement, though there are a few +/- 1 scores. As comparison to the human scores, we use three different classes of techniques for machine ranking of periodicity. Sliding Windows (SW): We sort the videos in decreasing order of Periodicity Score (Equation 19). We fix the window size at 20 frames and the embedding dimension at 20 frames (which is enough to capture 10 strong harmonics). We also apply a time derivative of width 10 to every frame. 18 80 Histogram of Weighted Pairwise Turk Scores Counts 60 40 20 0 -4 -3 -2 -1 0 1 2 3 4 Score Figure 18: The histogram of scores that the workers on AMT gave to all pairwise videos. Cutler-Davis [6]: The authors of this work present two different techniques to quantify periodicity from a self-similarity matrix (SSM) of video frames. The first is a frequency domain technique based on the peak of the average power spectral density over all columns (rows) of the SSM after linearly de-trending and applying a Hann window. To turn this into a continuous score, we report the ratio of the peak minus the mean over the standard deviation. This method will be referred to as Frequency Score. As the authors warn, the frequency peak method has a high susceptibility to false positives. This motivated the design of a more robust technique in [6], which works by finding peaks in the 2D normalized autocorrelation of the Gaussian smoothed SSMs. For videos with mirror symmetry, the peaks will lie on a diamond lattice, while for videos without mirror symmetry, they will lie on a square lattice. After peak finding within neighborhoods, one simply searches over all possible lattices at all possible widths to find the best match with the peaks. Since each lattice is centered at the autocorrelation point (0, 0), no translational checks are necessary. To turn this into a continuous score, let E be the sum of Euclidean distances of the matched peaks in the autocorrelation image to the best fit lattice, let r1 be the proportion of lattice points that have been matched, and let r2 be the proportion of peaks which have been matched to a lattice point. Then we give the final periodicity score as CDscore = (1 + E/r1 )/(r1 r2 )3 (28) A lattice which fits the peaks perfectly (r1 = 1) with no error (E = 0) and no false positive peaks (r2 = 1) will have a score of 1, and any video which fails to have a perfectly matched lattice will have a score greater than 1. Hence, we sort in increasing order of the score to get a ranking. As we will show, this technique agrees the second best with humans after our Periodicity Score ranking. One of the main drawbacks is numerical stability of finding maxes in non-isolated critical points around nearly diagonal regions in square lattices, which will erroneously inflate the score. Also, the lattice searching only occurs over an integer grid, but there may be periods that aren’t integer number of frames, so there will always be a nonzero E for such videos. By contrast, our sliding window scheme can work for any real valued period length. Diffusion Maps + Normalized Autocorrelation “Clarity”: Finally, we apply the technique from Section 3.6 to get an autocorrelation function, and we report the value of the maximum peak of the normalized autocorrelation to the right of a zero crossing, referred to as “clarity” by [25]. Values closer to 1 indicate more perfect repetitions, so we sort in descending order of clarity to get a ranking. Figure 19 shows an example of these three different techniques on a periodic video. There is a dot which rises above the diagonal in the persistence diagram, a lattice is found which nearly matches the critical points in the autocorrelation image, and autocorrelation function on diffusion maps has a nice peak. 19 First Diﬀusion Coordinate Figure 19: An example of the SW score (top), the clarity score (bottom left), and the CDscore (bottom right, matched peaks in green and lattice in blue), on a periodic video of a man waving his arms from the KTH dataset ( [36]). By contrast, for a nonperiodic video (Figure 20), there is hardly any persistent homology, there is no well matching lattice, and the first diffusion coordinate has no apparent periodicities. First Diﬀusion Coordinate Figure 20: An example of the SW score (top), the clarity score (bottom left), and the CDscore (bottom left, matched peaks in green and lattice in blue), on a video of an explosion, which is nonperiodic. Results: Once we have the global human rankings and the global machine rankings, we can compare them using the Kendall τ score [20]. Given a set of objects N objects X and two total orders >1 and >2 , where > (xa , xb ) = 1 if xa > xb and > (xa , xb ) = −1 if xa < xb , the Kendall τ score is defined as 20 τ= X 1 (>1 (xi , xj ))(>2 (xi , xj )) N (N − 1)/2 i<j (29) For two rankings which agree exactly, the Kendall τ score will be 1. For two rankings which are exactly the reverse of each other, the Kendall τ score will be -1. In this way, it analogous to a Pearson correlation between rankings. Table 2: The Kendall τ scores between all of the machine rankings and the Hodge aggregated human rankings. Human SW+TDA Freq [6] CDscore [6] Clarity [7] Human 1 0.663 -0.295 0.347 0.284 SW+TDA 0.663 1 -0.316 0.221 0.516 Freq [6] -0.295 -0.316 1 -0.0842 -0.189 CDscore [6] 0.347 0.221 -0.0842 1 0.411 Clarity [7] 0.284 0.516 -0.189 0.411 1 Table 3: Average runtimes, in milliseconds, per video for all of the algorithms SW+TDA 3101ms Freq [6] 73ms CDscore [6] 176ms Clarity [7] 154ms Table 2 shows the Kendall τ scores between all of the different machine rankings and the human rankings. Our sliding window video methodology (SW+TDA) agrees with the human ranking more than any other pair of ranking types. The second most similar are the SW and the diffusion clarity, which is noteworthy as they are both geometric techniques. Table 3 also shows the average run times, in milliseconds, of the different algorithms on each video on our machine. This does highlight one potential drawback of our technique, since TDA algorithms tend to be computationally intensive. However, at this scale (videos with at most several hundred frames), performance is reasonable. 4.3 Periodicity And Biphonation in High Speed Videos of Vocal Folds In this final task we apply our methodology to a real world problem of interest in medicine. We show that our method can automatically detect certain types of voice pathologies from high-speed glottography, or high speed videos (4000 fps) of the left and right vocal folds in the human vocal tract [9, 44, 45]. In particular, we detect and differentiate quasiperiodicity from periodicity by using our geometric sliding window pipeline. Quasiperiodicity is a special case of what is referred to as “biphonation” in the biological context, where nonlinear phenomena cause a physical process to bifurcate into two different periodic modes, often during a transition to chaotic behavior [15]. The torus structure we sketched in Figure 9 has long been recognized in this context [28], but we provide a novel way of quantifying it. Similar phenomena exist in audio [14, 15], but the main reason for studying laryngeal high speed video is understanding the biomechanical underpinnings of what is perceived in the voice. In particular, this understanding can potentially lead to practical corrective therapies and surgical interventions. On the other hand, the presence of biphonation in sound is not necessarily the result of a physiological phenomenon; it has been argued that it may come about as the result of changes in states of arousal [?]. In contrast with our work, the existing literature on video-based techniques usually employs an inherently Lagrangian approach, where different points on the left and right vocal folds are tracked, and coordinates of these points are analyzed as 1D time series (e.g. [13, 24, 28, 35], [26]). This is a natural approach, since those are the pixels where all of the important signal resides, and wellunderstood 1D signal processing technique can be used. However, edge detectors often require tuning, 21 and they can suddenly fail when the vocal folds close [24]. In our technique, we give up the ability to localize the anomalies (left/right, anterior/posterior) since we are not tracking them, but in return we do virtually no preprocessing, and our technique is domain independent. Results: We use a collection of 7 high-speed videos for this analysis, drawn from a variety of different sources [49], [26], [28], [13]. There are two videos which correspond to “normal” periodic vocal folds, three which correspond to biphonation [28], and two which correspond to irregular motion5 . We manually extracted 400 frames per video (100milliseconds) and autotuned the window size based on autocorrelation of 1D diffusion maps (Section 3.6). We then chose an appropriate τ and chose a time spacing so that each point cloud would have 600 points. As shown in Table 4, our technique is able to differentiate between the four classes. We also show PCA and persistence diagrams for one example for each class. In Figure 21, we see what appears to be a loop in PCA, and one strong 1D persistent dot confirms this. In Figure 22, we see a prominent torus in the persistence diagram. In Figure 23, we don’t see any prominent structures in the persistence diagram, even though PCA looks like it could be a loop or a torus. Note, however, that PCA only preserves 13.7% of the variance in the signal, which is why high dimensional techniques are important to draw quantitative conclusions. Table 4: Results of our sliding window pipeline on videos of periodic vocal folds, biphonation, and irregularities. We give the max persistence periodicity score (PS), the modified periodicity score (MPS), the harmonic score (HS), and quasiperiodic score (QPS) presented in Section 3.4. We also show the window size (Win) that the autocorrelation technique in Section 3.6 gives. We have bolded the top three MPS and QPS scores across all videos. The max modified periodic scores include the two periodic videos and one of the biphonation videos. The max quasiperiodic scores are all of the biphonation videos, which means the one with a high periodicity score could be ruled out of the periodicity category. Video Name Periodic 1 ( [13]) Periodic 2 ( [26], Figure 21) Biphonation 1 ( [28]) Biphonation 2 ( [28]) Biphonation 3 ( [28], Figure 22) Mucus Perturbed Periodic ( [49]) Irregular ( [13], Figure 23) 5 Win 16 32 53 42 67 94 232 PS 0.816 0.601 0.638 0.703 0.515 0.028 0.18 MPS 0.789 0.533 0.294 0.583 0.076 0.019 0.097 QPS 0.011 0.009 0.292 0.116 0.426 0.004 0.04 Discussion We have shown in this work how applying sliding window embeddings to videos can be used to translate properties of the underlying dynamics into geometric features of the resulting point cloud representation. Moreover, we also showed how topological/geometric tools such as persistence homology can be leveraged to quantify the geometry of these embeddings. The pipeline was evaluated extensively showing robustness to several noise models, high quality in the produced periodicity rankings and applicability to the study of speech conditions form high-speed video data. Moving forward, an interesting avenue related to medical applications is the difference between biphonation which occurs from quasiperiodic modes and biphonation which occurs from harmonic modes. [31] shows that Z3 field coefficients can be used to indicate the presence of a strong harmonic, so we believe a geometric approach is possible. This could be used, for example, to differentiate between subharmonic anomalies and quasiperiodic transitions [44]. 5 Please refer to supplementary material for an example video from each of these three classes 22 Figure 21: Video frames and sliding window statistics on a video of vocal folds undergoing normal periodic vibrations [26]. One strong loop is visible in PCA and in the persistence diagrams Figure 22: Video frames and sliding window statistics on a video of vocal folds undergoing biphonation, courtesy of Juergen Neubauer [28]. 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arXiv:1802.04834v1 [cs.LG] 13 Feb 2018 Challenging Images For Minds and Machines Amir Rosenfeld, John K. Tsotsos Department of Electrical Engineering and Computer Science York University, Toronto, ON, Canada [email protected],[email protected] February 15, 2018 Abstract There is no denying the tremendous leap in the performance of machine learning methods in the past half-decade. Some might even say that specific sub-fields in pattern recognition, such as machine-vision, are as good as solved, reaching human and super-human levels. Arguably, lack of training data and computation power are all that stand between us and solving the remaining ones. In this position paper we underline cases in vision which are challenging to machines and even to human observers. This is to show limitations of contemporary models that are hard to ameliorate by following the current trend to increase training data, network capacity or computational power. Moreover, we claim that attempting to do so is in principle a suboptimal approach. We provide a taster of such examples in hope to encourage and challenge the machine learning community to develop new directions to solve the said difficulties. 1 Introduction Once only known to a few outside of academia, machine-learning has become ubiquitous in both popular media and in the industry. Superhuman capabilities are now being gradually recorded in various fields: in the game of GO, ([1, 2]), in face verification ([3, 4]), image categorization ([5]) and even in logical reasoning in simple scenes ([6, 7, 8]). Most current leading methods involve some variant of deep learning. Consequentially, they require large amounts of hand-labeled data (with the exception of [2] - which used self-play to gain experience). This has elicited a data-hungry era, with increasingly large-scale datasets painstakingly labeled for object classification/detection/segmentation, image annotation, visual question-answering, and pose estimation ([9, 10, 11, 12, 13]) to name a few. This is accompanied by a growing demand for computational power. We bring forward challenges in vision which do not seem to be solved by current methods - and more importantly - by current popular methodologies, meaning that neither additional data, nor added computational power will be the drivers of the solution. 1 Figure 1: A children’s puzzle where the goal is to find six hidden words: Book, words, story, pages, read, novel. For a machine this is far from child’s play. Could this be solved by providing a million similar examples to a deep-learning system? Does a human need such training? Related Work Imbalanced or Small Data: datasets tend to be naturally imbalanced, and there is a long history of suggested remedies ([14, 15, 16]). Handling lack of training data has also been treated by attempting to use web-scale data of lesser quality than handannotated dataset [17], simulating data [cite data for cars, text recognition in the wild, captcha]. Transfer Learning: reusing features of networks trained on large is a useful starting point (cf [18]) One-Shot-Learning: attempting to reduce the number of required training example, in extreme cases to one or even zero examples ([19]); DeepLearning Failures: recently, some simple cases where deep learning fails to work as one would possibly expect were introduced, along with theoretical justifications ([20]). 2 Challenging Cases We present two examples and then discuss them. They have a few common characteristics: humans are able to solve them on the first “encounter” - despite not having seen any such images before. Incidentally - but not critically - the two examples are from the domain of visual text recognition. Moreover, though humans know how to recognize text as seen in regular textbooks, street-signs, etc, the text in these images is either hidden, rendered, or distorted in an uncharacteristic manner. Children’s games: the first case is well exemplified by a child’s game, hidden word puzzles. The goal is to find hidden words in an image. Fig. 1 shows an arbitrarily selected example. For a human observer this is a solvable puzzle, though it may take a few minutes to complete. We applied two state-of-the-art methods for text recognition 2 Sub Image [21] “sned” “vvoz” “novees” “teg” [22] “score” ∅ ∅ ∅ Table 1: Text detected by two state-of-the-art scene-text recognition methods applied to sub-images of a children’s puzzle. ∅ means no text was detected by the method (images scaled to fit figure). Figure 2: Variants of textual CAPTCHA. Captchas are becoming increasingly difficult (reproduced from [24]) in the wild with available code ([21]) or an on line-demo ([22]1 ) on the image in Fig. 1. As this did not work immediately, we focused on the word “NOVEL” (the “N” is below the forearm of the left person, ending with an “L” below his foot), by cropping it an rotating so the text is level, cropping more tightly , and even cropping only the letter “L”. See Table 1 for the corresponding sub-images (including the entire image at the top row) and the results output by the two methods. This is by no means a systematic test and some may even claim that it isn’t fair and they would be right: these systems were not trained on such images; [21] was only trained on a photo-realistic dataset of 8 million synthetic training images, and [22] was only trained on tens of thousands of images from coco-text ([23]), or used powerful pre-trained networks where training data was less available. CAPTCHA: a well-known mechanism to thwart automated misuse of websites by distinguishing between humans and machines ([25]). Textual captchas involve presenting an image of text which has to be read and written by the user. We focus on this type of captcha, though others exist ([26]). The introduction of captchas immediately triggered the invention of new automatic ways to break them ([27]), which eventually sparked an “arms race” between increasingly complex captchas and correspondingly powerful automated methods ([28]). This caused a state where on one-hand the best leading textual captcha-solution methods involve training DNN’s over data with similar distortion characteristics as the desired types of captcha - though still these systems have limited success rates (at times less than 50%) - and on the other hand the level of distortion has become such that humans have a hard-time solving some of them. 3 Machines vs Humans as Supervised Learners One can rule out the suggested examples by saying that they are simply out-of-sample datapoints on behalf of a statistical learner’s perspective. Yet it seems that with what1 http://east.zxytim.com 3 ever supervision human-beings receive - they are usually able to solve them despite not being especially exposed to this kind of stimulus. Moreover, precisely these kinds of images are used routinely in human IQ testing, so they are a universally accepted indicator for human performance. If these examples may seem esoteric, we can revert to more common cases: as a child, how often is one exposed to bounding boxes of objects? How often to delineations of objects with precise segmentation masks? How often to pose-configurations, facial and bodily key-points, and dense-meshes of 3D objects overlayed on their field of view ([13])? More critically, for how many different object types does this happen (if any), for how many different instances, with what level of precision of annotation, and in how many modalities? The granularity of visual supervision given to machines seems to be much finer than that given to humans. As for the amount of directly supervised data, it does not seem to really be the main limiting factor; as already noted several times, performance either saturates with training data ([29, 30]) or at best grows logarithmically ([17, 31], increasing mAP from 53% to 58% when growing from 10M to 300M examples) making the solution of more data for better performance simply impractical - even for those with the most resources. And this is for “common” problems, such as object detection. Humans who only ever read street-signs and textbooks are able to solve captchas of various kinds without any special training on their first encounter with them. The same is true for the “picture puzzles” mentioned above, as it is for other cases not mentioned here. We do not claim that humans are not subject to supervised learning in their early life, and in later stages. On the contrary, supervisory signals arise from multiple sources: caretakers who provide supervisory signals by teaching, “internal supervision” provided by innate biases ([32]) and finally rewards stemming from results of behaviour, such as suffering pain from hitting an object. But any such supervision is interspersed within a vast, continuous stream of unsupervised data, most of which does not have an easily measurable supervisory affect on the observer. There is something fundamentally different about the way humans construct or use internal representations, enabling them to reason about and solve new patternrecognition tasks. We hypothesize that these are approached by generating procedures of a compositional nature when presented with a novel - or known - task (as suggested by the Visual Routines of [33] or the Cognitive Programs of [34]. We intend to maintain a collection of examples beyond the ones suggested above, to encourage the community to attempt to solve them, not by learning from vast amounts of similar examples, but by learning from related, simpler subtasks and learning to reason and solve them by composing the appropriate solutions. References [1] D. Silver, A. Huang, C. J. Maddison, A. Guez, L. Sifre, G. Van Den Driessche, J. Schrittwieser, I. Antonoglou, V. Panneershelvam, M. Lanctot et al., “Mastering the game of Go with deep neural networks and tree search,” Nature, vol. 529, no. 7587, pp. 484–489, 2016. 1 4 [2] D. Silver, J. Schrittwieser, K. Simonyan, I. Antonoglou, A. Huang, A. Guez, T. Hubert, L. Baker, M. Lai, A. Bolton et al., “Mastering the game of go without human knowledge,” Nature, vol. 550, no. 7676, p. 354, 2017. 1 [3] C. Lu and X. Tang, “Surpassing Human-Level Face Verification Performance on LFW with GaussianFace.” 2015. 1 [4] X. Qi and L. 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Kokkinos, “DensePose: Dense Human Pose Estimation In The Wild,” arXiv preprint arXiv:1802.00434, 2018. 1, 3 [14] J. J. Lim, R. R. Salakhutdinov, and A. Torralba, “Transfer learning by borrowing examples for multiclass object detection,” in Advances in neural information processing systems, 2011, pp. 118–126. 1 5 [15] X. Zhu, D. Anguelov, and D. Ramanan, “Capturing long-tail distributions of object subcategories,” in Computer Vision and Pattern Recognition (CVPR), 2014 IEEE Conference on. IEEE, 2014, pp. 915–922. 1 [16] Y.-X. Wang, D. Ramanan, and M. Hebert, “Learning to Model the Tail,” in Advances in Neural Information Processing Systems, 2017, pp. 7032–7042. 1 [17] C. Sun, A. Shrivastava, S. Singh, and A. Gupta, “Revisiting unreasonable effectiveness of data in deep learning era,” in 2017 IEEE International Conference on Computer Vision (ICCV). IEEE, 2017, pp. 843–852. 1, 3 [18] A. Sharif Razavian, H. Azizpour, J. Sullivan, and S. 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Some Theory for Ordinal Embedding arXiv:1501.02861v2 [math.ST] 4 May 2016 Ery Arias-Castro∗ Abstract Motivated by recent work on ordinal embedding (Kleindessner and von Luxburg, 2014), we derive large sample consistency results and rates of convergence for the problem of embedding points based on triple or quadruple distance comparisons. We also consider a variant of this problem where only local comparisons are provided. Finally, inspired by (Jamieson and Nowak, 2011), we bound the number of such comparisons needed to achieve consistency. Keywords: ordinal embedding, non-metric multidimensional scaling (MDS), dissimilarity comparisons, landmark multidimensional scaling. 1 Introduction The problem of ordinal embedding, also called non-metric multidimensional scaling (Borg and Groenen, 2005), consists of finding an embedding of a set of items based on pairwise distance comparisons. Specifically, suppose that δij ≥ 0 is some dissimilarity measure between items i, j ∈ [n] ∶= {1, . . . , n}. We assume that δii = 0 and δij = δji for all i, j ∈ [n]. These dissimilarities are either directly available but assumed to lack meaning except for their relative magnitudes, or only available via comparisons with some other dissimilarities, meaning that we are only provided with a subset C ⊂ [n]4 such that δij < δkℓ , ∀(i, j, k, ℓ) ∈ C. (1) Note that the latter setting encompasses the former. Given C and a dimension d, the goal is to embed the items as points p1 , . . . , pn ∈ Rd in a way that is compatible with the available information, specifically δij < δkℓ ⇒ ∥pi − pj ∥ ≤ ∥pk − pℓ ∥, ∀(i, j, k, ℓ) ∈ C, (2) where ∥⋅∥ denotes the Euclidean norm. The two most common situations are when all the quadruple comparisons are available, meaning C = [n]4 , or all triple comparisons are available, meaning C = {(i, j, i, k) ∶ i, j, k ∈ [n]}, which can be identified with [n]3 . This problem has a long history surveyed in (Young and Hamer, 1987), with pioneering contributions from Shepard (1962a,b) and Kruskal (1964). The main question we tackle here is that of consistency. Suppose that the items are in fact points x1 , . . . , xn ∈ Rd and δij = ∥xi −xj ∥. (When the δij ’s are available, suppose that δij = g(∥xi −xj ∥) where g is an unknown increasing function.) Provided with a subset C = Cn of dissimilarity comparisons as in (2), is it possible to reconstruct the original points in the large-sample limit n → ∞? Clearly, the reconstruction can only be up to a similarity transformation — that is, a transformation f ∶ Rd ↦ Rd such that, for some λ > 0, ∥f (x) − f (y)∥ = λ∥x − y∥ for all x, y ∈ Rd , or equivalently, of the form f (x) = λR(x) + b where R is an orthogonal transformation and b is a constant vector — since such ∗ Department of Mathematics, University of California, San Diego, USA 1 2 a transformation leaves the distance comparisons unchanged. This question is at the foundation of non-metric multidimensional scaling. Early work only addressed the continuous case, where the x’s span a whole convex subset U ⊂ Rd . In that setting, the goal becomes to characterize isotonic functions on U , that is, functions f ∶ U ↦ Rd satisfying ∥x − y∥ < ∥x′ − y ′ ∥ ⇒ ∥f (x) − f (y)∥ ≤ ∥f (x′ ) − f (y ′ )∥, ∀x, y, x′ , y ′ ∈ U. (3) Shepard (1966) argues that such functions must be similarities, and cites earlier work (Aumann and Kruskal, 1958; Suppes and Winet, 1955) dealing with the case d = 1. Only recently has the finite sample case been formally considered. Indeed, Kleindessner and von Luxburg (2014) prove a consistency result, showing that if x1 , . . . , xn ∈ U ⊂ Rd , where U is a bounded, connected, and open subset of Rd satisfying some additional conditions — for example, a finite union of open balls — and C = [n]4 , then in the large sample limit with x1 , . . . , xn becoming dense in U , it is possible to recover the x’s up to a similarity transformation. (Note that U is then uniquely defined as the interior of {xi ∶ i ≥ 1}.) We note that Kleindessner and von Luxburg (2014) focus on the strictly isotonic case, where the second inequality in (3) is strict. Our first contribution is an extension of this consistency result for quadruple learning to triple learning where C = [n]3 . In the process, we greatly simplify the arguments of Kleindessner and von Luxburg (2014) and weaken the conditions on the sampling domain U . We note that Terada and Von Luxburg (2014) have partially solved this problem by a reduction to the problem of embedding a nearest-neighbor graph. However, their arguments are based on an apparently incomplete proof in (Von Luxburg and Alamgir, 2013), which is itself based on a rather sophisticated approach. Our proofs are comparatively much simpler and direct. Our second contribution is to provide rates of convergence, a problem left open by Kleindessner and von Luxburg (2014). In the context of quadruple learning, we obtain a rate in O(εn ), where εn is the Hausdorff distance between the underlying sample {x1 , . . . , xn } and U , meaning, εn ∶= supx∈U mini∈[n] ∥x − xi ∥. This is the first convergence rate for exact ordinal embedding that we know of. (We are not able to obtain the same rate in the context of triple learning.) Compared to establishing consistency, the proof is much more involved. The last decade has seen a surge of interest in ordinal embedding, motivated by applications to recommender systems and large-scale psychometric studies made available via the internet, for example, databases for music artists similarity (Ellis et al., 2002; McFee and Lanckriet, 2011). Sensor localization (Nhat et al., 2008) is another possible application. Modern datasets being large, all quadruple or triple comparisons are rarely available, motivating the proposal of embedding methods based on a sparse set of comparisons (Agarwal et al., 2007; Borg and Groenen, 2005; Jamieson and Nowak, 2011; Terada and Von Luxburg, 2014). Terada and Von Luxburg (2014) study what they call local ordinal embedding, which they define as the problem of embedding an unweighted K-nearest neighbor (K-NN) graph. With our notation, this is the situation where C = {(i, j, k) ∶ δij ≤ δi(K) < δik }, δi(K) being the dissimilarity between item i and its Kth nearestneighbor. Terada and Von Luxburg (2014) argue that, when the items are points x1 , . . . , xn sampled from a smooth density on a bounded, connected, convex, and open subset U ⊂ Rd with smooth boundary, then K = Kn ≫ n2/(2+d) (log n)d/(2+d) is enough for consistency. Our third contribution is to consider the related situation where C = {(i, j, k, ℓ) ∶ δij < δkℓ and max(δij , δik , δiℓ ) ≤ δi(K) }, which provides us with the K-NN graph and also all the quadruple √ comparisons between the nearest neighbors. In this setting, we are only able to show that Kn ≫ n log n is enough. Beyond local designs, which may not be feasible in some settings, Jamieson and Nowak (2011) consider the problem of adaptively (i.e., sequentially) selecting triple comparisons in order to minimize the number of such comparisons and yet deduce all the other triple comparisons. They 3 consider a few methods, among which a non-metric version of the landmark MDS method of De Silva and Tenenbaum (2004). Less ambitious is the problem of selecting few comparisons in order to consistently embed the items when these are points in a Euclidean space. Our fourth contribution is to show that one can obtain a consistent embedding with a landmark design based on an n queries, where an is any diverging sequence. Moreover, the embedding can be computed in (expected) time ζ(an ) n, for some function ζ ∶ R+ ↦ R+ . The rest of the paper is organized as follows. In Section 2, we state our theoretical results and prove the simpler ones. We then gather the remaining proofs in Section 3. Section 4 concludes the paper with a short discussion. 2 Theory In this section we present our theoretical findings. Most proofs are gathered in Section 3. We already defined isotonic functions in (3). Following (Kleindessner and von Luxburg, 2014), we say that a function f ∶ U ⊂ Rd ↦ Rd is weakly isotonic if ∥x − y∥ < ∥x − z∥ ⇒ ∥f (x) − f (y)∥ ≤ ∥f (x) − f (z)∥, ∀x, y, z ∈ U. (4) Obviously, if a function is isotonic (3), then it is weakly isotonic (4). Weak isotonicity is in fact not much weaker than isotonicity. Indeed, let P be a property (e.g., ‘isotonic’), and say that a function f ∶ U ⊂ Rd ↦ Rd has the property P locally if for each x ∈ U there is r > 0 such that f has property P on B(x, r) ∩ U , where B(x, r) denotes the open ball with center x and radius r. Lemma 1. Any locally weakly isotonic function on an open U is also locally isotonic on U . Proof. This is an immediate consequence of (Kleindessner and von Luxburg, 2014, Lem 6), which implies that a weakly isotonic function on B(x, r) is isotonic on B(x, r/4). Suppose we have data points x1 , . . . , xn ∈ Rd . Define Ωn = {x1 , . . . , xn }, Ω = ⋃ Ωn = {xn ∶ n ≥ 1}. (5) n≥1 Let δij = ∥xi − xj ∥ and suppose that we are only provided with a subset Cn ⊂ [n]4 of distance comparisons as in (1). To an (exact) ordinal embedding p ∶ [n] ↦ Rd — which by definition satisfies (2) — we associate the map φn ∶ Ωn ↦ Rd defined by φn (xi ) = pi for all i ∈ [n]. We crucially observe that, in the case of all quadruple comparisons (Cn = [n]4 ), the resulting map φn is isotonic on Ωn ; in the case of all triple comparisons (Cn = [n]3 ), φn is only weakly isotonic on Ωn , instead. In light of this, and the fact that the location, orientation and scale are all lost when only ordinal information is available, the problem of proving consistency of (exact) ordinal embedding reduces to showing that any such embedding is close to a similarity transformation as the sample size increases, n → ∞. This is exactly what Kleindessner and von Luxburg (2014) do under some assumptions. 2.1 Ordinal embedding based on all triple comparisons Our first contribution is to extend the consistency results of Kleindessner and von Luxburg (2014) on quadruple learning to triple learning. Following their presentation, we start with a result where the sample is infinite, which is only a mild generalization of (Kleindessner and von Luxburg, 2014, Th 3). 4 Theorem 1. Let U ⊂ Rd be bounded, connected and open. Suppose Ω is dense in U and consider a locally weakly isotonic function φ ∶ Ω ↦ Rd . Then there is a similarity transformation S that coincides with φ on Ω. The proof is largely based on that of (Kleindessner and von Luxburg, 2014, Th 3), but a bit simpler; see Section 3.1. We remark that there can only be one similarity with the above property, since similarities are affine transformations, and two affine transformations of Rd that coincide on d+1 affine independent points are necessarily identical. In this theorem, the set Ω is dense in an open subset of Rd , and therefore infinite. In fact, Kleindessner and von Luxburg (2014) use this theorem as an intermediary result for proving consistency as the sample size increases. Most of their paper is dedicated to establishing this, as their arguments are quite elaborate. We found a more direct route by ‘tending to the limit as soon as possible’, based on Lemma 2 below, which is at the core of the Arzelà-Ascoli theorem. For the remaining of this section, we consider the finite sample setting: U ⊂ Rd is bounded, connected and open, Ωn = {x1 , . . . , xn } ⊂ U is such that Ω ∶= {xn ∶ n ≥ 1} is dense in U , and φn ∶ Ωn ↦ Q ⊂ Rd is a function with values in a bounded set Q. (6) In the context of (6), we implicitly extend φn to Ω, for example, by setting φn (x) = q for all x ∈ Ω ∖ Ωn , where q is a given point in Q, although the following holds for any extension. Lemma 2. Consider Ωn ⊂ Rd finite and φn ∶ Ωn ↦ Q ⊂ Rd , where Q is bounded. Then there is N ⊂ N infinite such that φ(x) ∶= limn∈N φn (x) exists for all x ∈ Ω ∶= ⋃n Ωn . This is called the diagonal process in (Kelley, 1975, Problem D, Ch 7). Although the result is classical, we provide a proof for completeness. Proof. Without loss of generality, suppose Ωn = {x1 , . . . , xn }. Let N0 = N. Since (φn (x1 ) ∶ n ∈ N0 ) ∈ Q and Q is bounded, there is N1 ⊂ N0 infinite such that limn∈N1 φn (x1 ) exists. In turn, since (φn (x2 ) ∶ n ∈ N1 ) is bounded, there is N2 ⊂ N1 infinite such that limn∈N2 φn (x2 ) exists. Continuing this process — which formally corresponds to a recursion — we obtain ⋯ ⊂ Nk+1 ⊂ Nk ⊂ ⋯ ⊂ N1 ⊂ N0 = N such that, for all k, Nk is infinite and limn∈Nk φn (xk ) exists. Let nk denote the kth element (in increasing order) of Nk and note that (nk ∶ k ≥ 1) is strictly increasing. Define N = {nk ∶ k ≥ 1}. Since {np , p ≥ k} ⊂ Nk , we have limn∈N φn (xk ) = limn∈Nk φn (xk ), and this is valid for all k ≥ 1. Corollary 1. Consider the setting (6) and assume that φn is weakly isotonic. Then (φn ) is sequentially pre-compact for the pointwise convergence topology for functions on Ω and all the functions where it accumulates are similarity transformations restricted to Ω. The corresponding result (Kleindessner and von Luxburg, 2014, Th 4) was obtained for isotonic (instead of weakly isotonic) functions and for domains U that are finite unions of balls, and the convergence was uniform instead of pointwise. For now, we provide a proof of Corollary 1, which we derive as a simple consequence of Theorem 1 and Lemma 2. Proof. Lemma 2 implies that (φn ) is sequentially pre-compact for the pointwise convergence topology. Let φ be an accumulation point of (φn ), meaning that there is N ⊂ N infinite such that φ(x) = limn∈N φn (x) for all x ∈ Ω. Take x, y, z ∈ Ω such that ∥x − y∥ < ∥x − z∥. By definition, there is m such that x, y, z ∈ Ωm , and therefore ∥φn (x) − φn (y)∥ ≤ ∥φn (x) − φn (z)∥ for all n ≥ m. Passing to the limit along n ∈ N , we obtain ∥φ(x) − φ(y)∥ ≤ ∥φ(x) − φ(z)∥. Hence, φ is weakly isotonic on Ω and, by Theorem 1, it is therefore the restriction of a similarity transformation to Ω. 5 It is true that (Kleindessner and von Luxburg, 2014, Th 4) establishes a uniform convergence result. We do the same in Theorem 2 below, but with much simpler arguments. The key are the following two results bounding the modulus of continuity of a (resp. weakly) isotonic function. We note that the second result (for weakly isotonic functions) is very weak but sufficient for our purposes here. For Λ ⊂ V ⊂ Rd , define δH (Λ, V ) = supy∈V inf x∈Λ ∥y − x∥, which is their Hausdorff distance. We say that (yi ∶ i ∈ I) ⊂ Rd is an η-packing if ∥yi − yj ∥ ≥ η for all i ≠ j. We recall that the size of the largest η-packing of a Euclidean ball of radius r is of exact order (r/η)−d . For a set V ⊂ Rd , let diam(V ) = supx,y∈V ∥x − y∥ be its diameter and let ρ(V ) = arg sup{∃v ∈ V ∶ B(v, r/2) ⊂ V }, (7) r>0 which is the diameter of a largest ball inscribed in V . Everywhere in the paper, d is fixed, and in fact implicitly small as we assume repeatedly that the sample (of size n) is dense in a full-dimensional domain of Rd . In particular, all the implicit constants of proportionality that follow depend solely on d. Lemma 3. Let V ⊂ Rd be open. Consider Λ ⊂ V and set ε = δH (Λ, V ). Let ψ ∶ Λ ↦ Q be isotonic, where Q ⊂ Rd is bounded. There is C ∝ diam(Q)/ρ(V ), such that ∥ψ(x) − ψ(x′ )∥ ≤ C(∥x − x′ ∥ + ε), ∀x, x′ ∈ Λ. (8) Proof. The proof is based on the fact that an isotonic function transforms a packing into a packing. Take x, x′ ∈ Λ such that ξ ∶= ∥ψ(x) − ψ(x′ )∥ > 0, and let η = ∥x − x′ ∥. Since V is open it contains an open ball of diameter ρ(V ). Let y1 , . . . , ym be an (η + 3ε)-packing of such a ball with m ≥ C1 (ρ(V )/(η + ε))d for some constant C1 depending only on d. Then let x1 , . . . , xm ∈ Λ such that maxi ∥yi − xi ∥ ≤ ε. By the triangle inequality, for all i ≠ j we have ∥xi − xj ∥ ≥ ∥yi − yj ∥ − 2ε ≥ η + ε > ∥x − x′ ∥. Because ψ is isotonic, we have ∥ψ(xi ) − ψ(xj )∥ ≥ ξ, so that ψ(x1 ), . . . , ψ(xm ) is a ξ-packing. Therefore, there is a constant C2 depending only on d such that m ≤ C2 (diam(Q)/ξ)d . We conclude that ξ ≤ (C2 /C1 )1/d (diam(Q)/ρ(V ))(η + ε). For V ⊂ Rd and h > 0, let V h = {x ∈ V ∶ ∃y ∈ V s.t. x ∈ B(y, h) ⊂ V }. We note that V h is the complement of the h-convex hull of V c ∶= Rd ∖ V — see (Cuevas et al., 2012) and references therein. Lemma 4. In the context of Lemma 3, if ψ is only weakly isotonic, then there is C ∝ diam(Q), such that for all h > 0, √ 1/d (9) ∥ψ(x) − ψ(x′ )∥ ≤ C(∥x − x′ ∥/h + ε/h) , ∀x ∈ Λ ∩ V h , ∀x′ ∈ Λ. Proof. Assume that V h ≠ ∅, for otherwise there is nothing to prove. Take x ∈ Λ ∩ V h and x′ ∈ Λ such that ξ ∶= ∥ψ(x) − ψ(x′ )∥ > 0, and let η = ∥x − x′ ∥. Because ψ is bounded, it is enough to prove the result when η, ε < h/2. Let y ∈ V be such that x ∈ B(y, h) ⊂ V . There is y ′ ∈ B(y, h) such that y ∈ [xy ′ ] and ∥x − y ′ ∥ ≥ 2h/3. Define u = (y ′ − x)/∥y ′ − x∥. Let z0 = x, and for j ≥ 1, define zj = zj−1 + (η + 5jε)u. Let k ≥ 0 be maximum such that ∑kj=1 (η + 5jε) < h/2. Since k satisfies √ kη + 5k2 ε ≥ h/2, we have k ≥ min(h/(4η), h/(10ε)). By construction, for all j ∈ [k], zj ∈ [xy ′ ] and B(zj , 2ε) ⊂ B(y, h). Let x−1 = x′ , x0 = x and take x1 , . . . , xk ∈ Λ such that maxj ∥xj − zj ∥ ≤ ε. By the triangle inequality, for j = 2, . . . , k, ∥xj − xj−1 ∥ ≥ ∥zj − zj−1 ∥ − 2ε ≥ ∥zj−1 − zj−2 ∥ + 3ε ≥ ∥xj−1 − xj−2 ∥ + ε, 6 which implies by induction that ∥xj − xj−1 ∥ ≥ ∥x1 − x0 ∥ + ε ≥ ∥z1 − z0 ∥ = η + 5ε > ∥x − x′ ∥. By weak isotonicity, this implies that ∥ψ(xj ) − ψ(xj−1 )∥ ≥ ∥ψ(x) − ψ(x′ )∥ = ξ. We also have, for any i, j ∈ [k] such that 1 ≤ i ≤ j − 2, ∥xj − xi ∥ ≥ ∥zj − zi ∥ − 2ε ≥ ∥zj − zj−1 ∥ + η + 5ε − 2ε ≥ ∥xj − xj−1 ∥ + η + ε. By weak isotonicity, this implies that ∥ψ(xj ) − ψ(xi )∥ ≥ ∥ψ(xj ) − ψ(xj−1 )∥ for all 0 ≤ i < j ≤ k. Consequently, (ψ(xj ) ∶ j ∈ [k]) forms a ξ-packing of Q. Hence, k ≤ C ′ (diam(Q)/ξ)d , for some constant C ′ . We conclude with the lower bound on k. From this control on the modulus of continuity, we obtain a stronger version of Corollary 1. Theorem 2. Under the same conditions as Corollary 1, we have the stronger conclusion that there is a sequence (Sn ) of similarities such that, for all h > 0, maxx∈Ωn ∩U h ∥φn (x) − Sn (x)∥ → 0 as n → ∞. If in fact each φn is isotonic, then this remains true when h = 0. We remark that when U is a connected union of a possibly uncountable number of open balls of radius at least h > 0, then U = U h . This covers the case of a finite union of open balls considered in (Kleindessner and von Luxburg, 2014). We also note that, if U is bounded and open, and ∂U has bounded curvature, then there is h > 0 such that U = U h . This follows from the fact that, in this case, U c has positive reach (Federer, 1959), and is therefore h-convex when h is below the reach by1 (Cuevas et al., 2012, Prop 1). Moreover, our arguments can be modified to accommodate sets U with boundaries that are only Lipschitz, by reasoning with wedges in Lemma 4. Theorem 2 now contains (Kleindessner and von Luxburg, 2014, Th 4), and extends it to weakly isotonic functions and to more general domains U . Overall, our proof technique is much simpler, shorter, and elementary. Define εn = δH (Ωn , U ), which quantifies the density of Ωn in U . Because Ωn+1 ⊂ Ωn and Ω is dense in U , we have εn ↘ 0 as n → ∞. Proof. Let φ be an accumulation point of (φn ) for the pointwise convergence topology, meaning there is N ⊂ N infinite such that φ(x) = limn∈N φn (x) for all x ∈ Ω. We show that, in fact, the convergence is uniform. First, suppose that each φn is isotonic. In that case, Lemma 3 implies the existence of a constant C > 0 such that ∥φn (x)−φn (x′ )∥ ≤ C(∥x−x′ ∥+εn ) for all x, x′ ∈ Ωn , and for all n. Passing to the limit along n ∈ N , we get ∥φ(x) − φ(x′ )∥ ≤ C∥x − x′ ∥ for all x, x′ ∈ Ω. (In fact, we already knew this from Corollary 1, since we learned there that φ coincides with a similarity, and is therefore Lipschitz.) Fix ε > 0. There is m such that εm ≤ ε. Then there is m′ ≥ m such that maxi∈[m] ∥φn (xi )−φ(xi )∥ ≤ ε for all n ∈ N with n ≥ m′ . For such an n, and x ∈ Ωn , let i ∈ [m] be such that ∥x − xi ∥ ≤ εm . By the triangle inequality, ∥φn (x) − φ(x)∥ ≤ ∥φn (x) − φn (xi )∥ + ∥φn (xi ) − φ(xi )∥ + ∥φ(xi ) − φ(x)∥ ≤ C(∥x − xi ∥ + εn ) + ∥φn (xi ) − φ(xi )∥ + C∥xi − x∥ ≤ C(εm + εn ) + ε + Cεm ≤ (3C + 1)ε. Since x ∈ Ω is arbitrary and ε can be taken as small as desired, this shows that the sequence (φn ∶ n ∈ N ) convergences uniformly to φ over (Ωn ∶ n ∈ N ). 1 This proposition is stated for compact sets (which is not the case of U c ) but easily extends to the case where set is closed with compact boundary 7 When the φn are only weakly isotonic, we use Lemma 4 to get a constant C > 0 depending on √ diam(Q) and h > 0 such that ∥φn (x) − φn (x′ )∥ ≤ C(∥x − x′ ∥ + εn )1/d for all x ∈ Ωn ∩ U h and all x′ ∈ Ωn , and for all n. Passing to the limit along n ∈ N , we get ∥φ(x) − φ(x′ )∥ ≤ C∥x − x′ ∥1/d for all x, x′ ∈ Ω. (In fact, ∥φ(x) − φ(x′ )∥ ≤ C∥x − x′ ∥ for all x, x′ ∈ Ω from Corollary 1, as explained above.) The rest of the arguments are completely parallel. We conclude that (φn ∶ n ∈ N ) convergences uniformly to φ over (Ωn ∩ U h ∶ n ∈ N ). Let S denote the similarities of Rd . For any functions φ, ψ ∶ Ω ↦ Rd , define δn (φ, ψ) = maxx∈Ωn ∩U h ∥φ(x) − ψ(x)∥, and also δn (φ, S) = inf S∈S δn (φ, S). Our end goal is to show that δn (φn , S) → 0 as n → ∞. Suppose this is not the case, so that there is η > 0 and N ⊂ N infinite such that δn (φn , S) ≥ η for all n ∈ N . By Corollary 1, there is N1 ⊂ N and S ∈ S such that S(x) = limn∈N1 φn (x) for all x ∈ Ω. As we showed above, the convergence is in fact uniform over (Ωn ∩ U h ∶ n ∈ N1 ), meaning limn∈N1 δn (φn , S) = 0. At the same time, we have δn (φn , S) ≥ δn (φn , S) ≥ η. We therefore have a contradiction. 2.2 Rates of convergence Beyond consistency, we are able to derive convergence rates. We do so for the isotonic case, i.e., the quadruple comparison setting. Recall that εn = δH (Ωn , U ). Theorem 3. Consider the setting (6) with φn isotonic. There is C depending only on (d, U ), and a sequence of similarities Sn such that maxx∈Ωn ∥φn (x) − Sn (x)∥ ≤ C diam(Q)εn . If U = U h for some h > 0, then C = C ′ / diam(U ) where C ′ is a function of (d, h/ diam(U ), ρ(U )/ diam(U )). The proof of Theorem 3 is substantially more technical than the previous results, and thus postponed to Section 3. Although Kleindessner and von Luxburg (2014) are not able to obtain rates of convergence, the proof of Theorem 3 bares resemblance to their proof technique, and in particular, is also based on a result of Alestalo et al. (2001) on the approximation of ε-isometries; see Lemma 18. We will also make use of a related result of Vestfrid (2003) on the approximation of approximately midlinear functions; see Lemma 17. We mention that we know of a more elementary proof that only makes use of (Alestalo et al., 2001), but yields a slightly slower rate of convergence. We note that there is a constant c depending only on d such that εn ≥ cn−1/d . This is because U being open, it contains an open ball, and this lower bound trivially holds for an open ball. And such a lower bound is achieved when the xi ’s are roughly regularly spread out over U . If instead the xi ’s are iid uniform in U , and U is sufficiently regular — for example, U = U h for some h > 0 — then εn = O(log(n)/n)1/d , as is well-known. This would give the rate, and we do not know whether it is optimal, even in dimension d = 1. √ Remark 1. We are only able to get a rate in εn for the weakly isotonic case. We can do so by adapting of the arguments underlying Theorem 3, but only after assuming that U = U h for some h > 0 and resolving a few additional complications. 2.3 Ordinal embedding with local comparisons Terada and Von Luxburg (2014) consider the problem of embedding an unweighted nearest-neighbor graph, which as we saw in the Introduction, is a special case of ordinal embedding. Their arguments — which, as we explained earlier, seem incomplete at the time of writing — indicate that K = Kn ≫ n2/(2+d) (log n)d/(2+d) is enough for consistently embedding a K-NN graph. We consider here a situation where we have more information, specifically, all the distance comparisons between K-nearest-neighbors. Formally, this is the situation where Cn = {(i, j, k, ℓ) ∶ δij < δkℓ and {j, k, ℓ} ⊂ NKn (i)}, 8 where NK (i) denotes the set of the K items nearest item i. If the items are points Ωn = {x1 , . . . , xn } ⊂ Rd , an exact ordinal embedding φn is only constrained to be locally weakly isotonic as we explain now. We start by stating a standard result which relates a K-NN graph to an r-ball graph. Lemma 5. Let U ⊂ Rd be bounded, connected and open, and such that U = U h for some h > 0. Sample x1 , . . . , xn iid from a density f supported on U with (essential) range in (0, ∞) strictly. There is a constant C such that, if K ∶= [nr d ] ≥ C log n, then with probability tending to 1, NeighK/2 (xi ) ⊂ {xj ∶ ∥xj − xi ∥ ≤ r} ⊂ Neigh2K (xi ), ∀i ∈ [n], where NeighK (xi ) denotes the set of the K points in {xj ∶ j ∈ [n]} nearest xi . The proof is postponed to Section 3 and only provided for completeness. Therefore, assuming that K ≥ C log n, where C is the constant of Lemma 5, we may equivalently consider the case where Cn = {(i, j, k, ℓ) ∶ δij < δkℓ and max(δij , δik , δiℓ ) < rn }, for some given rn > 0. An exact embedding φn ∶ Ωn ↦ Rd in that case is isotonic on Ωn ∩ B(x, rn ) for any x ∈ Ωn . We require in addition that ∥x − x′ ∥ < rn ≤ ∥x† − x‡ ∥ ⇒ ∥φn (x) − φn (x′ )∥ ≤ ∥φn (x† ) − φn (x‡ )∥, ∀x, x′ , x† , x‡ ∈ Ωn . (10) This is a reasonable requirement since it is possible to infer it from Cn . Indeed, for k, ℓ ∈ [n], we have δkℓ < rn if, and only if, (k, k, k, ℓ) ∈ Cn or (ℓ, ℓ, ℓ, k) ∈ Cn . (Here we assume that δii = 0 for all i and δij > 0 if i ≠ j, as is the case for Euclidean distances.) We can still infer this even if the quadruples in Cn must include at least three distinct items. Indeed, suppose k, ℓ ∈ [n] are such that there is no i such that (i, k, i, ℓ) ∈ Cn or (i, ℓ, i, k) ∈ Cn , then (a) δik = δiℓ for all i such that max(δik , δiℓ ) < rn , or (b) δkℓ ≥ rn . Assume that rn ≥ Cεn with C > 0 sufficiently large, so that situation (a) does not happen. Conversely, if (k, ℓ) is such that δkℓ < rn , then when (a) does not happen, there is i such that (i, k, i, ℓ) ∈ Cn or (i, ℓ, i, k) ∈ Cn . Theorem 4. Consider the setting (6) and assume in addition that U = U h for some h > 0, and that φn is isotonic over balls of radius rn and satisfies (10). There is a constant C > 0 depending only on (d, h, ρ(U ), diam(U ), diam(Q)) and similarities Sn such that maxx∈Ωn ∥φn (x) − Sn (x)∥ ≤ Cεn /rn2 . Assume the data points are generated as in Lemma 5. In that case, we have εn = O(log(n)/n)1/d and Theorem 4 implies consistency when rn ≫ (log(n)/n)1/2d . By Lemma 5, this corresponds to the situation where we are provided with comparisons among Kn -nearest neighbors with Kn ≫ √ n log n. If the result of Terada and Von Luxburg (2014) holds in all rigor, then this is a rather weak result. 2.4 Landmark ordinal embedding Inspired by (Jamieson and Nowak, 2011), we consider the situation where there are landmark items indexed by Ln ⊂ [n], and we are given all distance comparisons from any point to the landmarks. Formally, with triple comparisons, this corresponds to the situation where Cn = {(i, j, k) ∈ [n] × L2n ∶ δij < δik }. If the items are points Ωn = {x1 , . . . , xn } ⊂ Rd , an exact ordinal embedding φn is only constrained to be weakly isotonic on the set of landmarks and, in addition, is required to respect the ordering of the distances from any point to the landmarks. The following is an easy consequence of Theorem 2. 9 Corollary 2. Theorem 2 remains valid in the landmark triple comparisons setting (meaning with φn as just described) as long as the landmarks become dense in U . Jamieson and Nowak (2011) study the number of triple comparisons that are needed for exact ordinal embedding. With a counting argument, they show that at least Cn log n comparisons are needed, where C is a constant depending only on d. If we only insist that the embedding respects the comparisons that are provided, then Corollary 2 implies that a landmark design is able to be consistent as long as the landmarks become dense in U . This consistency implies that, as the sample size increases, an embedding that respects the landmark comparisons also respects all other comparisons approximately. This is achieved with O(nℓ2n + ℓ3n ) triple comparisons, where ℓn ∶= ∣Ln ∣ is the number of landmarks, and the conditions of Corollary 2 can be fulfilled with ℓn → ∞ at any speed, so that the number of comparisons is nearly linear in n. Proof. We focus on the weakly isotonic case, where we assume that U = U h for some h > 0. Let Λn = {xl ∶ l ∈ Ln } denote the set of landmarks. Since Λn becomes dense in U , meaning ηn ∶= δH (Λn , U ) → 0, by Theorem 2, there is a sequence of similarities Sn such that ζn ∶= maxx∈Λn ∥φn (x) − Sn (x)∥ → 0. Now, for x ∈ Ωn , let x̃ ∈ Λn such that ∥x − x̃∥ ≤ ηn . We have ∥φn (x) − Sn (x)∥ ≤ ∥φn (x) − φn (x̃)∥ + ∥φn (x̃) − Sn (x̃)∥ + ∥Sn (x̃) − Sn (x)∥. (11) 1/2d by Lemma 4, for some constant C. The middle term is The first term is bounded by Cηn bounded by ζn . For the third term, express Sn in the form Sn (x) = βn Rn (x) + bn , where βn ∈ R, Rn is an orthogonal transformation, and bn ∈ Rd . Take two distinct landmarks x† , x‡ ∈ Λn such that ∥x† − x‡ ∥ ≥ diam(U )/2, which exist when n is sufficiently large. Since and, at the same time, ∥Sn (x† ) − Sn (x‡ )∥ = βn ∥x† − x‡ ∥ ≥ βn diam(U )/2 ∥Sn (x† ) − Sn (x‡ )∥ ≤ ∥Sn (x† ) − φn (x† )∥ + ∥φn (x† ) − φn (x‡ )∥ + ∥φn (x‡ ) − Sn (x‡ )∥ ≤ ζn + diam(Q) + ζn ≤ 2 diam(Q), eventually, we have βn ≤ β̄ ∶= 4 diam(Q)/ diam(U ). Hence, the third term on the RHS of (11) is bounded by 1/2d β̄ηn . Thus, the RHS of (11) is bounded by Cηn + ζn + β̄ηn , which tends to 0 as n → ∞. This being valid for any x ∈ Ωn , we conclude. We remark that at the very end of the proof, we obtained a rate of convergence as a function of the density of the landmarks and the convergence rate implicit in Theorem 2. This leads to the following rate for the quadruple comparisons setting, which corresponds to the situation where Cn = {(i, j, k) ∈ [n] × L2n ∶ δij < δik } ⋃ {(i, j, k, ℓ) ∈ L4n ∶ δij < δik }. Here, φn is constrained to be isotonic on the set of landmarks and, as before, is required to respect the ordering of the distances from any data point to the landmarks. Corollary 3. Consider the setting (6) in the landmark quadruple comparisons setting (meaning with φn as just described). Let Λn denote the set of landmarks and set ηn = δH (Λn , U ). There is a constant C > 0 and a sequence of similarities Sn such that maxx∈Ωn ∥φn (x) − Sn (x)∥ ≤ Cηn . Proof. The proof is parallel to that of Corollary 2. Here, we apply Theorem 3 to get ζn ≤ C0 ηn . This bounds the second term on the RHS of (11). The first term is bounded by C1 ηn by Lemma 3, while the third term is bounded by β̄ηn as before. (C0 , C1 are constants.) 10 Computational complexity. We now discuss the computational complexity of ordinal embedding with a landmark design. The obvious approach has two stages. In the first stage, the landmarks are embedded. This is the goal of (Agarwal et al., 2007), for example. Here, we use brute force. Proposition 1. Suppose that m items are in fact points in Euclidean space and their dissimilarities are their pairwise Euclidean distances. Then whether in the triple or quadruple comparisons setting, an exact ordinal embedding of these m items can be obtained in finite expected time. Proof. The algorithm we discuss is very naive: we sample m points iid from the uniform distribution on the unit ball, and repeat until the ordinal constraints are satisfied. Since checking the latter can be done in finite time, it suffices to show that there is a strictly positive probability that one such sample satisfies the ordinal constraints. Let Xm denote the set of m-tuples (x1 , . . . , xm ) ∈ B(0, 1) that satisfy the ordinal constraints, meaning that ∥xi − xj ∥ < ∥xk − xℓ ∥ when (i, j, k, ℓ) ∈ C. Seeing Xn as a subset of B(0, 1)m ⊂ Rdm , it is clearly open. And sampling x1 , . . . , xm iid from the uniform distribution on B(0, 1) results in sampling (x1 , . . . , xm ) from the uniform distribution on B(0, 1)m , which assigns a positive mass to any open set. In the second stage, each point that is not a landmark is embedded based on the order of its distances to the landmarks. We quickly mention the work Davenport (2013), who develops a convex method for performing this task. Here, we are contented with knowing that this can be done, for each point, in finite time, function of the number of landmarks. For example, a brute force approach starts by computing the Voronoi diagram of the landmarks, and iteratively repeats within each cell, creating a tree structure. Each point that is not a landmark is placed by going from the root to a leaf, and choosing any point in that leaf cell, say its barycenter. Thus, if there are ℓ landmarks, the first stage is performed in expected time F (ℓ), and the second stage is performed in time (n − ℓ)G(ℓ). The overall procedure is thus computed in expected time F (ℓ) + (n − ℓ)G(ℓ). Remark 2. The procedure described above is not suggested as a practical means to perform ordinal embedding with a landmark design. The first stage, described in Proposition 1, has finite expected time, but likely not polynomial in the number of landmarks. For a practical method, we can suggest the following: 1. Embed the landmarks using the method of Agarwal et al. (2007) (which solves a semidefinite program) or the method of Terada and Von Luxburg (2014) (which uses an iterative minimization-majorization strategy). 2. Embed the remaining points using the method of Davenport (2013) (which solves a quadratic program). Although practical and reasonable, we cannot provide any theoretical guarantees for this method. 3 More proofs In this section we gather the remaining proofs and some auxiliary results. We introduce some additional notation and basic concepts. For z1 , . . . , zm ∈ Rd , let Aff(z1 , . . . zm ) denote their affine hull, meaning the affine subspace they generate in Rd . For a vector x in a Euclidean space, let ∥x∥ denote its Euclidean norm. For a matrix M ∈ √ Rp×q , let ∥M ∥ denote its usual operator norm, meaning, ∥M ∥ = max{∥M x∥ ∶ ∥x∥ ≤ 1} and ∥M ∥F = tr(M ⊺ M ) its Frobenius norm. Regular simplexes. These will play a central role in our proofs. We say that z1 , . . . , zm ∈ Rd , with m ≥ 2, form a regular simplex if their pairwise distances are all equal. We note that, necessarily, 11 m ≤ d + 1, and that regular simplexes in the same Euclidean space and with same number of (distinct) nodes m are similarity transformations of each other — for example, segments (m = 2), equilateral triangles (m = 3), tetrahedron (m = 4). By recursion on the number of vertices, m, it is easy to prove the following. Lemma 6. Let z1 , . . . , zm form a√ regular simplex with edge length 1 and let µ denote the barycenter of z1 , . . . , zm . Then ∥µ − zi ∥ = (m − 1)/2m, and if z, z1 , . . . , zm form a regular simplex, then √ ∥z − µ∥ = (m + 1)/2m. (In dimension m, there are exactly two such points z.) 3.1 Proof of Theorem 1 We assume d ≥ 2. See (Kleindessner and von Luxburg, 2014) for the case d = 1. We divide the proof into several parts. Continuous extension. Lemma 4 implies that φ is locally uniformly continuous. Indeed, take x0 ∈ Ω and let r > 0 such that B(x0 , r) ⊂ U and φ is weakly isotonic on B(x0 , r) ∩ Ω. Applying Lemma 4 with V = B(x0 , r) and Λ = Ω ∩ B(x0 , r) — so that δH (Λ, V ) = 0 because Λ is dense in V — and noting that V r = V , yields a constant Cr > 0 such that ∥φ(x) − φ(x′ )∥ ≤ Cr ∥x − x′ ∥1/d , for all x, x′ ∈ Ω ∩ B(x0 , r). Being locally uniformly continuous, we can uniquely extend φ to a continuous function on U , also denoted by φ. By continuity, this extension is locally weakly isotonic on U . Isosceles preservation. Sikorska and Szostok (2004) say that a function f ∶ V ⊂ Rd → Rd preserves isosceles triangles if ∥x − y∥ = ∥x − z∥ ⇒ ∥f (x) − f (y)∥ = ∥f (x) − f (z)∥, ∀x, y, z ∈ V. In our case, by continuity, we also have that φ preserves isosceles triangles locally. Indeed, for the sake of pedagogy, let u ∈ U and r > 0 such that B(u, r) ⊂ U and φ is weakly isotonic on B(u, r). Take x, y, z ∈ B(u, r/2) be such that ∥x − y∥ = ∥x − z∥. For t ∈ R, define zt = (1 − t)x + tz. Let t > 1 such that zt ∈ B(u, r). Because ∥x − y∥ < t∥x − z∥ = ∥x − zt ∥, we have ∥φ(x) − φ(y)∥ ≤ ∥φ(x) − φ(zt )∥. Letting t ↘ 1, we get ∥φ(x) − φ(y)∥ ≤ ∥φ(x) − φ(z)∥ by continuity of φ. Since y and z play the same role, the converse inequality is also true, and combined, yield an equality. Midpoint preservation. Let V ⊂ Rd be convex. We say that a function f ∶ V ↦ Rd preserves midpoints if x+y f (x) + f (y) f( )= , ∀x, y ∈ V. 2 2 We now show that φ preserves midpoints, locally. Kleindessner and von Luxburg (2014) also do that, however, our arguments are closer to those of Sikorska and Szostok (2004), who make use of regular simplexes. The important fact is that a function that preserves isosceles preserves regular simplexes. Let u ∈ U and r > 0 such that B(u, r) ⊂ U and φ preserves isosceles on B(u, r). Take x, y ∈ B(u, r/2), and let µ = (x + y)/2. Let z1 , . . . , zd form a regular simplex with barycenter µ and side length s, and such that ∥x−zi ∥ = s for all i. In other words, x, z1 , . . . , zd forms a regular simplex placed so that µ is the barycenter of z1 , . . . , zd . By √ symmetry, y, z1 , . . . , zd forms a regular simplex also. By Lemma 6, we have ∥zi − µ∥/∥x − µ∥ = (d − 1)/(d + 1), so that z1 , . . . , zd ∈ B(µ, r/2) ⊂ B(u, r), by the triangle inequality and the fact that ∥x − µ∥ < r/2. Hence, φ(x), φ(z1 ), . . . , φ(zd ) and φ(y), φ(z1 ), . . . , φ(zd ) are regular simplexes. If one of them is singular, so is the other one, in which case φ(x) = φ(y) = φ(µ). Otherwise, necessarily φ(x) is the symmetric of φ(y) with respect to Aff(φ(z1 ), . . . , φ(zd )); the only other possibility would be that φ(x)√ = φ(y), but in that case we would still have that φ(zi ) = φ(x) for all i ∈ [d], since ∥x − zi ∥/∥x − µ∥ = 2d/(d + 1) by Lemma 6 — implying that ∥x − zi ∥ < ∥x − y∥ — and φ is weakly isotonic in that neighborhood. So assume that 12 φ(x) is the symmetric of φ(y) with respect to Aff(φ(z1 ), . . . , φ(zd )). For a ∈ {x, y, µ}, ∥a − zi ∥ is constant in i, and therefore so is ∥φ(a)−φ(zi )∥, so that φ(a) belongs to the line of points equidistant to φ(z1 ), . . . , φ(zd ). This implies that x, y, µ are collinear. And because ∥µ − x∥ = ∥µ − y∥, we also have ∥φ(µ) − φ(x)∥ = ∥φ(µ) − φ(y)∥, so that φ(µ) is necessarily the midpoint of φ(x) and φ(y). Conclusion. We arrived at the conclusion that φ can be extended to a continuous function on U that preserves midpoints locally. We then use the following simple results in sequence: with Lemma 7, we conclude that φ is locally affine; with Lemma 8, we conclude that φ is in fact affine on U ; and with Lemma 9, we conclude that φ is in fact a similarity on U . Lemma 7. Let V be a convex set of a Euclidean space and let f be a continuous function on V with values in a Euclidean space that preserves midpoints. Then f is an affine transformation. Proof. This result is in fact well-known, and we only provide a proof for completeness. It suffices to prove that f is such that f ((1 − t)x + ty) = (1 − t)f (x) + tf (y) for all x, y ∈ V and all t ∈ [0, 1]. Starting with the fact that this is true when t = 1/2, by recursion we have that this is true when t is dyadic, meaning, of the form t = k2−j , where j ≥ 1 and k ≤ 2j are both integers. Since dyadic numbers are dense in [0, 1], by continuity of f , we deduce the desired property. Lemma 8. A locally affine function over an open and connected subset of a Euclidean space is the restriction of an affine function over the whole space. Proof. Let U be the domain and f the function. Cover U with a countable number of open balls Bi , i ∈ I such that f coincides with an affine function fi on Bi . Take i, j ∈ I distinct. Since U is connected, there must be a sequence i = k1 , . . . , km = j, all in I, such that Bks ∩ Bks+1 ≠ ∅ for s ∈ [m − 1]. Since Bks ∩ Bks+1 is an open set, we must have fks = fks+1 , and this being true for all s, it implies that fi = fj . Lemma 9. An affine function that preserves isosceles locally is a similarity transformation. Proof. Let f be an affine function that preserves isosceles in an open ball. Without loss of generality, we may assume that the ball is B(0, 2) and that f (0) = 0 (so that f is linear). Fix u0 ∈ ∂B(0, 1) and let a = ∥f (u0 )∥. Take x ∈ Rd different from 0 and let u = x/∥x∥. We have ∥f (x)∥/∥x∥ = ∥f (u)∥ = ∥f (u) − f (0)∥ = ∥f (u0 ) − f (0)∥ = ∥f (u0 )∥ = a. Hence, ∥f (x)∥ = a∥x∥, valid for all x ∈ Rd , and f being linear, this implies that f is a similarity. 3.2 Auxiliary results We list here a number of auxiliary results that will be used in the proof of Theorem 3. The following result is a perturbation bound for trilateration, which is the process of locating a point based on its distance to landmark points. For a real matrix Z, let σk (Z) denote its k-th largest singular value. Lemma 10. Let z1 , . . . , zd+1 ∈ Rd such that Aff(z1 , . . . , zd+1 ) = Rd and let Z denote the matrix with columns z1 , . . . , zd+1 . Consider p, q ∈ Rd and define ai = ∥p − zi ∥ and bi = ∥q − zi ∥ for i ∈ [d + 1]. Then ∥p − q∥ ≤ √ 1√ d σd (Z)−1 max ∣a2d+1 − a2i − b2i + b2d+1 ∣ ≤ d σd (Z)−1 max ∣a2i − b2i ∣. i i 2 Proof. Assume without loss of generality that zd+1 = 0. In that case, note that ad+1 = ∥p∥ and bd+1 = ∥q∥. Also, redefine Z as the matrix with columns z1 , . . . , zd , and note that the first d singular values remain unchanged. Since Aff(z1 , . . . , zd+1 ) = Rd , there is α = (α1 , . . . , αd ) ∈ Rd 13 and β = (β1 , . . . , βd ) ∈ Rd such that p = ∑i∈[d] αi zi = Zα and q = ∑i∈[d] βi zi = Zβ. For p, we have a2i = ∥p − zi ∥2 = ∥p∥2 + ∥zi ∥2 − 2zi⊺ Zα for all i ∈ [d], or in matrix form, Z ⊺ Zα = 12 u, where u = (u1 , . . . , ud ) and ui = a2d+1 − a2i + ∥zi ∥2 . Similarly, we find Z ⊺ Zβ = 12 v, where v = (v1 , . . . , vd ) and vi = b2d+1 − b2i + ∥zi ∥2 . Hence, we have 1√ 1 ∥Z ⊺ (p − q)∥ = ∥Z ⊺ Zα − Z ⊺ Zβ∥ = ∥u − v∥ = ∑ (a2 − b2 − a2i + b2i )2 2 2 i∈[d] d+1 d+1 √ 1√ d max ∣a2d+1 − a2i − b2i + b2d+1 ∣ ≤ d max ∣a2i − b2i ∣. ≤ i i 2 Simultaneously, ∥Z ⊺ (p − q)∥ ≥ σd (Z)∥p − q∥. Combining both inequalities, we conclude. For η ∈ [0, 1), we say that z1 , . . . , zm ∈ Rd form an η-approximate regular simplex if min ∥zi − zj ∥ ≥ (1 − η) max ∥zi − zj ∥. i≠j i≠j Lemma 11. Let z1 , . . . , zm form an η-approximate regular simplex with maximum edge length λ ′ achieved by ∥z1 − z2 ∥. There is a constant Cm and z1′ , . . . , zm ∈ Aff(z1 , . . . , zm ) with z1′ = z1 and ′ z2 = z2 and forming a regular simplex with edge length λ, such that maxi ∥zi′ − zi ∥ ≤ λCm η. Proof. By scale equivariance, we may assume that λ = 1. We use an induction on m. In what ′ ′′ follows, Cm , Cm , Cm , etc, are constants that depend only on m. For m = 2, the statement is trivially true. Suppose that it is true for m ≥ 2 and consider an η-approximate regular simplex z1 , . . . , zm+1 ∈ Rd with maximum edge length 1. By changing d to m if needed, without loss of generality, assume that Aff(z1 , . . . , zm+1 ) = Rd . In that case, z1 , . . . , zm is an η-approximate regular simplex with maximum edge length achieved by ∥z1 − z2 ∥ = 1, and by the inductive hypothesis, this ′ ∈ A ∶= Aff(z1 , . . . , zm ) with z1′ = z1 and z2′ = z2 and forming a implies the existence of z1′ , . . . , zm regular simplex of edge length 1, such that maxi∈[m] ∥zi′ − zi ∥ ≤ Cm η for some constant Cm . Let p be the orthogonal projection of zm+1 onto A. Before continuing, let P be the set of such p ′ obtained when fixing z1′ , . . . , zm and then varying zi ∈ B(zi′ , Cm η) for i ∈ [m] and zm+1 among the points that make an η-approximate regular simplex with z1 , . . . , zm . Let µ′ be the barycenter of ′ z1′ , . . . , zm and note that µ′ ∈ P. Now, set δ = ∥zm+1 − p∥. By the Pythagoras theorem, we have 2 ∥p − zi ∥ = ∥zm+1 − zi ∥2 − δ2 , with 1 − η ≤ ∥zm+1 − zi ∥ ≤ 1, so that 0 ≤ 1 − δ2 − ∥p − zi ∥2 ≤ 2η. By the triangle inequality, ∣∥p − zi′ ∥ − ∥p − zi ∥∣ ≤ ∥zi − zi′ ∥ ≤ Cm η, so that ∣∥p − zi′ ∥2 − ∥p − zi ∥2 ∣ = ∣∥p − zi′ ∥ − ∥p − zi ∥∣(∥p − zi′ ∥ − ∥p − zi ∥) ≤ ∥zi − zi′ ∥∥zi − zi′ ∥ ′ η, ≤ Cm η(2 + Cm η) ≤ Cm using the fact that ∥p − zi ∥ ≤ ∥zm+1 − zi ∥ ≤ 1. Hence, ′′ P ⊂ {q ∶ ∥q − zi′ ∥2 = 1 − δ2 ± Cm η, ∀i ∈ [m]}. ′′ η. By Lemma 10, this implies that Since µ′ ∈√ P, we must therefore have ∥p − zi′ ∥2 = ∥µ′ − zi′ ∥2 ± 2Cm ′ ′′′ ′′ −1 ′ ′ ′ ∥p − µ ∥ ≤ m − 1 σm−1 ([z1 ⋯zm ])2Cm η =∶ Cm η. Let zm+1 be on the same side of A as zm+1 and such ′ ′ ′ , zm+1 form a regular simplex. Note that µ′ is the orthogonal projection of zm+1 onto that z1′ , . . . , zm A. By the Pythagoras theorem, applied multiple times, we obtain the following. First, we have ′ ′ − µ′ + µ′ − p + p − zm+1 ∥2 − zm+1 ∥2 = ∥zm+1 ∥zm+1 ′ ′ = ∥zm+1 − µ′ ∥2 − 2(zm+1 − µ′ )⊺ (zm+1 − p) + ∥p − zm+1 ∥2 + ∥µ′ − p∥2 ′ = (∥zm+1 − µ′ ∥ − ∥p − zm+1 ∥)2 + ∥µ′ − p∥2 , 14 ′ because zm+1 − µ′ and zm+1 − p are orthogonal to A, and therefore parallel to each other and both ′′′ orthogonal to µ′ − p. For the second term, we already know that ∥µ′ − p∥ ≤ Cm η, while the first ′′ 2 2 term is bounded by (2Cm + 2) η since, on the one hand, ′ ′ ∥zm+1 − µ′ ∥2 = ∥zm+1 − z1′ ∥2 − ∥µ′ − z1′ ∥2 = 1 − ∥µ′ − z1′ ∥2 while, on the other hand, ∥p − zm+1 ∥2 = ∥zm+1 − z1 ∥2 − ∥p − z1 ∥2 = 1 ± 2η − ∥p − z1 ∥2 , ′′ ′ 2 and we know that ∥µ′ − z1′ ∥2 = ∥p − z1 ∥2 ± 2Cm η. Hence, we find that ∥zm+1 − zm+1 ∥2 ≤ Cm+1 η2 for some constant Cm+1 function of m only. This shows that the induction hypothesis holds for m + 1. ′ Lemma 12. There are constants Cm , Cm > 0 such that, if z1 , . . . , zm form an η-approximate regular ′ simplex with maximum edge length λ, then σm−1 ([z1 ⋯zm ]) ≥ λCm (1 − Cm η). ′′ and Proof. By scale equivariance, we may assume that λ = 1. By Lemma 11, there is a constant Cm ′ ′ ′ z1 , . . . , zm ∈ Aff(z1 , . . . , zm ) forming a regular simplex with edge length 1 such that maxi ∥zi − zi ∥ ≤ ′′ Cm η. By Weyl’s inequality (Horn and Johnson, 1990, Cor 7.3.8), σm−1 (Z) ≥ σm−1 (Z ′ ) − ∥Z − Z ′ ∥. On the one hand, σm−1√ (Z ′ ) is a positive constant depending only on m, while on the other hand, √ ′ ′ ′′ η. ∥Z − Z ∥ ≤ ∥Z − Z ∥F = ∑i ∥zi − zi′ ∥2 ≤ mCm Lemma 13. Let z1 , . . . , zm form an η-approximate regular simplex with maximum edge length λ and barycenter µ. Let p ∈ Aff(z1 , . . . , zm ) and define γ = maxi ∥p − zi ∥2 − mini ∥p − zi ∥2 . There is a constant Cm ≥ 1 depending only on m such that ∥p − µ∥ ≤ Cm λγ when η ≤ 1/Cm . Proof. By scale equivariance, we may assume that λ = 1. By Lemma 10, we have ∥p − µ∥ ≤ 1√ −1 m − 1 σm−1 ([z1 ⋯zm ]) max ∣∥p − zm ∥2 − ∥p − zi ∥2 ∣. i 2 ′ −1 ′ ′ By Lemma 12, there is a constant Cm . And we when η ≤ 1/Cm such that σm−1 ([z1 ⋯zm ]) ≤ Cm 2 2 also have maxi ∣∥p − zm ∥ − ∥p − zi ∥ ∣ ≤ γ. From this, we conclude. Lemma 14. Let ψ ∶ Λ ↦ Q be isotonic, where Λ, Q ⊂ Rd . Let v ∈ Rd and r > 0, and set ε = δH (Λ, B(v, r)). There is C ∝ diam(Q)/r such that, for all x, x′ , x† , x‡ ∈ Λ with x, x′ ⊂ B(v, 3r/4) and for all η ∈ (0, r/4 − 2ε), ∥x − x′ ∥ = ∥x† − x‡ ∥ ± η ⇒ ∥ψ(x) − ψ(x′ )∥ = ∥ψ(x† ) − ψ(x‡ )∥ ± C(η + ε). (12) Proof. Let ξ = ∥x − x′ ∥ and ξ † = ∥x† − x‡ ∥. Suppose that ξ < η + 2ε, which implies that ξ † < 2η + 2ε. In that case, Lemma 3 — where the constant there is denoted here by C1 ∝ diam(Q)/r — yields ∥ψ(x) − ψ(x′ )∥ ≤ C1 (ξ + ε) ≤ C1 (η + 3ε) and, similarly, ∥ψ(x† ) − ψ(x‡ )∥ ≤ C1 (2η + 3ε). This proves (12). Henceforth, we assume that ξ ≥ η + 2ε. First assume that ξ > ξ † . In that case, we immediately have ∥ψ(x) − ψ(x′ )∥ ≥ ∥ψ(x† ) − ψ(x‡ )∥. For the reverse, let yt = (1 − t)x + tx′ , and note that ∥yt − x∥ = tξ. Take t = 1 − (η + 2ε)/ξ and note that t ∈ [0, 1], so that yt ∈ [xx′ ] ⊂ B(v, r), and therefore there is x⋆ ∈ Λ such that ∥x⋆ − yt ∥ ≤ ε. We have ∥x⋆ − x∥ ≤ ∥yt − x∥ + ∥x⋆ − yt ∥ ≤ ξ − η − ε < ξ † , so that ∥ψ(x) − ψ(x⋆ )∥ ≤ ∥ψ(x† ) − ψ(x‡ )∥. Applying the triangle inequality and Lemma 3, we then have ∥ψ(x) − ψ(x⋆ )∥ ≥ ∥ψ(x) − ψ(x′ )∥ − ∥ψ(x′ ) − ψ(x⋆ )∥ ≥ ∥ψ(x) − ψ(x′ )∥ − C1 (∥x′ − x⋆ ∥ + ε), 15 with ∥x′ − x⋆ ∥ ≤ ∥x′ − yt ∥ + ∥yt − x⋆ ∥ ≤ η + 3ε. When ξ < ξ † , we choose t = 1 + (η + 2ε)/ξ. Because x, x′ ⊂ B(v, 3r/4), we still have yt ∈ B(v, r) because of the constraint on η. The remaining arguments are analogous. When ξ = ξ † , repeating what we just did both ways and with η = 0 yields the result. Lemma 15. Consider ψ ∶ Λ ↦ Rd isotonic, where Λ ⊂ Rd . Let V denote the convex hull of Λ. Set ε = δH (Λ, V ) and c = diam(ψ(Λ))/5 diam(Λ). Then ∥ψ(x) − ψ(x′ )∥ ≥ c∥x − x′ ∥ for all x, x′ ∈ Λ such that ∥x − x′ ∥ ≥ 4ε. Proof. We first prove that, if c > 0 and η ≥ 4ε are such that ∥ψ(x) − ψ(x′ )∥ ≤ cη for all x, x′ ∈ Λ with ∥x − x′ ∥ < η, then diam(ψ(Λ)) < c(4 diam(Λ) + η). Indeed, take x, x′ ∈ Λ. Let u = (x′ − x)/∥x′ − x∥ and L = ∥x − x′ ∥, and define yj = x + sj u where sj = j(η − 3ε) for j = 0, . . . , J ∶= ⌊L/(η − 3ε)⌋, and then let sJ+1 = L. By construction, yj ∈ [xx′ ] ⊂ V , with y0 = x and yJ+1 = x′ . Let xj ∈ Λ be such that ∥xj − yj ∥ ≤ ε, with x0 = x and xJ+1 = x′ . By the triangle inequality, ∥xj+1 − xj ∥ ≤ ∥yj+1 − yj ∥ + 2ε = sj+1 − sj + 2ε < η. Hence, J ∥ψ(x) − ψ(x′ )∥ ≤ ∑ ∥ψ(xj ) − ψ(xj+1 )∥ ≤ (J + 1)cη ≤ c j=0 Lη + cη < c(4 diam(Λ) + η), η − 3ε since η − 3ε ≥ η − 3η/4 = η/4 and L ≤ diam(Λ). Now assume that ψ is isotonic and suppose that ∥ψ(x) − ψ(x′ )∥ < c∥x − x′ ∥ for some x, x′ ∈ Λ such that η ∶= ∥x − x′ ∥ ≥ 4ε. Then we have ∥ψ(x† ) − ψ(x‡ )∥ ≤ cη when x† , x‡ ∈ Λ satisfy ∥x† − x‡ ∥ < η. We just showed that this implies that diam(ψ(Λ)) < c(4 diam(V ) + η), and we conclude using the fact that η ≤ diam(Λ). The following result is on 1-nearest neighbor interpolation. Lemma 16. Let Λ be a subset of isolated points in V ⊂ Rd and set ε = δH (Λ, V ). For any function ψ ∶ Λ ↦ Rd , define its 1-nearest neighbor interpolation as ψ̂ ∶ V ↦ Rd as ψ̂(y) = 1 ∑ ψ(x), ∣NΛ (y)∣ x∈NΛ (y) NΛ (y) ∶= arg min ∥x − y∥. (13) x∈Λ Consider the modulus of continuity of ψ, which for η > 0 is defined as ω(η) = sup{∥ψ(x) − ψ(x′ )∥ ∶ x, x′ ∈ Λ, ∥x − x′ ∥ ≤ η}. Then the modulus of continuity of ψ̂, denoted ω̂, satisfies ω̂(η) ≤ ω(η + 2ε). Moreover, for any y, y ′ ∈ V and any x, x′ ∈ Λ such that ∥x − y∥ ≤ ε and ∥x′ − y ′ ∥ ≤ ε, ∥ψ̂(y) − ψ̂(y ′ )∥ = ∥ψ(x) − ψ(x′ )∥ ± 2ω(2ε). Proof. Fix η > 0 and take y, y ′ ∈ V such that ∥y − y ′ ∥ ≤ η. We have ∥x − y∥ ≤ ε for all x ∈ NΛ (y) and ∥x′ − y ′ ∥ ≤ ε for all x′ ∈ NΛ (y ′ ), so that ∥x − x′ ∥ ≤ ∥y − y ′ ∥ + 2ε for all such x and x′ , by the triangle inequality. Therefore, ∥ψ̂(y) − ψ̂(y ′ )∥ ≤ sup {∥ψ(x) − ψ(x′ )∥ ∶ x ∈ NΛ (y), x′ ∈ NΛ (y ′ )} ≤ sup {∥ψ(x) − ψ(x′ )∥ ∶ x, x′ ∈ Λ, ∥x − x′ ∥ ≤ η + 2ε} = ω(η + 2ε). Since this is true for all y, y ′ ∈ V such that ∥y − y ′ ∥ ≤ η, we conclude that ω̂(η) ≤ ω(η + 2ε). For the second part of the lemma, we have ∥ψ̂(y) − ψ̂(y ′ )∥ = ∥ψ(x) − ψ(x′ )∥ ± ∥ψ̂(y) − ψ(x)∥ ± ∥ψ̂(y ′ ) − ψ(x′ )∥, 16 where the second term is bounded by ∥ψ̂(y) − ψ(x)∥ ≤ sup {∥ψ(x̃) − ψ(x)∥ ∶ x̃ ∈ NΛ (y)} ≤ sup {∥ψ(x̃) − ψ(x)∥ ∶ ∥x̃ − x∥ ≤ 2ε} ≤ ω(2ε), using the fact that ∥x̃ − x∥ ≤ ∥x̃ − y∥ + ∥y − x∥ ≤ 2ε, and similarly for the third term. Let V ⊂ Rd be convex. In our context, we say that f ∶ V ↦ Rd is η-approximately midlinear if ∥f ( 1 x+y ) − (f (x) + f (y))∥ ≤ η, 2 2 ∀x, y ∈ V. Lemma 17. Let V ⊂ Rd be star-shaped with respect to some point in its interior. There is a constant C depending only on V such that, for any η-approximately midlinear function f ∶ V ↦ Rd , there is a affine function T ∶ Rd ↦ Rd such that supx∈V ∥f (x) − T (x)∥ ≤ Cη. Note that, if V is a ball, then by invariance considerations, C only depends on d. Proof. This is a direct consequence of (Vestfrid, 2003, Th 1.4). We say that f ∶ V ⊂ Rd ↦ Rd is an ε-isometry if ∥x − y∥ − ε ≤ ∥f (x) − f (y)∥ ≤ ∥x − y∥ + ε, ∀x, y ∈ V. For a set V ⊂ Rd , define its thickness as θ(V ) = inf { diam(u⊺ V ) ∶ u ∈ Rd , ∥u∥ = 1}. Recalling the definition of ρ in (7), we note that θ(V ) ≥ ρ(V ), but that the two are distinct in general. Lemma 18. Let V ⊂ Rd be compact and such that θ(V ) ≥ η diam(V ) for some η > 0. There is a constant C depending only on d such that, if f ∶ V ↦ Rd is an ε-isometry, then there is an isometry R ∶ Rd ↦ Rd such that maxx∈V ∥f (x) − R(x)∥ ≤ Cε/η. Proof. This is a direct consequence of (Alestalo et al., 2001, Th 3.3). Lemma 19. Let T ∶ Rd ↦ Rd be an affine function that transforms a regular simplex of edge length 1 into an η-approximate regular simplex of maximum edge length λ > 0. There is a constant C, depending only on d, and an isometry R, such that ∥T (x) − λR(x)∥ ≤ Cλη for all x ∈ B(0, 1). Proof. By invariance, we may assume T is linear and that the regular simplex is formed by 0, z1 , . . . , zd and has edge length 1. Letting wi = T (zi ), we have that 0, w1 , . . . , wd form an ηapproximate regular simplex of maximum edge length λ ∶= maxi ∥wi ∥. Lemma 11 gives 0, w1′ , . . . , wd′ forming a regular simplex of edge length λ such that maxi ∥wi − wi′ ∥ ≤ C1 λη for some constant C1 . Let R be the orthogonal transformation such that R(zi ) = wi′ /λ for all i ∈ [d]. We have ∥T (zi ) − λR(zi )∥ = ∥wi − wi′ ∥ ≤ C1 λη for all i. In matrix notation, letting Z ∶= [z1 . . . zd ], we have ¿ Ád √ √ À∑ ∥T z − λRz ∥2 ≤ d max ∥T z − λRz ∥ ≤ d C λη. ∥T Z − λRZ∥ ≤ ∥T Z − λRZ∥ = Á F i i=1 i i∈[d] i i 1 At the same time, ∥T Z − λRZ∥ ≥ ∥T − λR∥/∥Z −1 ∥ with ∥Z −1√∥ = 1/σd (Z) = 1/σd ([0z1 ⋯zd ]) being a positive constant depending only on d. Hence, ∥T − λR∥ ≤ ( d/σd (Z))C1 λη =∶ C2 λη. 17 Lemma 20. Suppose that S1 , S2 ∶ Rd ↦ Rd are two affinities such that maxx∈B(y,r) ∥S1 (x)−S2 (x)∥ ≤ η for some y ∈ Rd and r > 0. Then ∥S1 (x) − S2 (x)∥ ≤ 2η∥x − y∥/r + η for all x ∈ Rd . Proof. By translation and scale invariance, assume that y = 0 and r = 1. Let Li = Si − Si (0). For x ∈ B(0, 1), we ∥L1 (x) − L2 (x)∥ ≤ ∥S1 (x) − S2 (x)∥ + ∥S1 (0) − S2 (0)∥ ≤ 2η. Hence, for x ∈ Rd , ∥L1 (x) − L2 (x)∥ ≤ 2η∥x∥, which in turn implies that ∥S1 (x) − S2 (x)∥ ≤ ∥L1 (x) − L2 (x)∥ + ∥S1 (0) − S2 (0)∥ ≤ 2η∥x∥ + η. 3.3 Proof of Theorem 3 Without loss of generality, we may assume that Dn ∶= diam(φn (Ωn )) ≥ 1. Indeed, suppose that Dn < 1, but different from 0, for otherwise φn is a degenerate similarity and the result follows. Let φ̃n = Dn−1 φn , which is isotonic on Ωn and satisfies diam(φ̃n (Ωn )) = 1. If the result is true for φ̃n , there is a similarity S̃n such that maxx∈Ωn ∣φ̃n (x) − S̃n (x)∣ ≤ Cεn for some constant C. (We implicitly assume that the set φn (Ωn ) contains the origin, so that φ̃n (Ωn ) remains bounded.) We then have maxx∈Ωn ∣φn (x) − Sn (x)∣ ≤ CDn εn ≤ Cεn , where Sn ∶= Dn S̃n is also a similarity. Let r = ρ(U ), so that there is some u⋆ such that B(u⋆ , r) ⊂ U . Let Λn = Ωn ∩ B(u⋆ , r/2) and δn = diam(φn (Λn )). Let w be any unit-norm vector and define y± = u⋆ ± (r/2− εn )w. Let x± ∈ Ωn be such that ∥x± − y± ∥ ≤ εn . Necessarily, x± ∈ Λn because the distance from y± to ∂B(u⋆ , r/2) exceeds εn . Note that ∥x− − x+ ∥ ≥ r1 ∶= r − 4εn . By isotonicity, ∥φn (x) − φn (x′ )∥ ≤ ∥φn (x− ) − φn (x+ )∥ ≤ δn , whenever ∥x − x′ ∥ < r1 . (14) Let y1 , . . . , yK be a (r1 /3)-packing of U , so that K ≤ C(diam(U )/r)d for some constant C > 0. Let xik ∈ Ωn be such that ∥xik − yk ∥ ≤ εn , so that U ⊂ ⋃k∈[K] B(yk , r1 /3) ⊂ ⋃k∈[K] B(xik , r2 ), where r2 ∶= r1 /3 + εn . Let zk = xik for clarity. Take x, x′ ∈ Ωn . Because U is open, it is path-connected, so there is a continuous curve γ ∶ [0, 1] ↦ U such that γ(0) = x and γ(1) = x′ . Let k0 ∈ [K] be such that x ∈ B(zk0 , r2 ) and s0 = 0. Then for j ≥ 0, let sj+1 = inf{s > sj ∶ γ(s) ∉ ⋃l∈[j] B(zkl , r2 )}, and let kj+1 ∈ [K] be such that ∥zkj+1 − γ(sj+1 )∥ ≤ εn . Let J = min{j ∶ sj+1 = ∞}, which is indeed finite. By construction, ∥zkj − zkj+1 ∥ ≤ 2r2 < r1 when εn < r/10. By (14), we have ∥φn (zkj ) − φn (zkj+1 )∥ ≤ δn . Thus, by the triangle inequality, ∥φn (x) − φn (x′ )∥ ≤ Jδn ≤ Kδn . This being true for all x, x′ ∈ Ωn , this prove that δn ≥ Dn /K ∝ Dn (diam(U )/r)−d . 1-NN interpolation. Let φ̂n denote the 1-NN interpolation of φn as in (13). We claim that there is a C0⋆ ∝ Dn /r and c⋆0 ∝ (diam(U )/r)−d Dn /r such that φ̂n satisfies the following properties: for all y, y ′ , y † , y ‡ ∈ U , ∥y − y ∥ < ∥y − y ∥ − 4εn ⇒ ∥φ̂n (y) − φ̂n (y )∥ ≤ ∥φ̂n (y ) − φ̂n (y ′ and also and ∥φ̂n (y) − φ̂n (y ′ )∥ ≤ C0⋆ (∥y − y ′ ∥ + εn ), † ‡ ′ † (15) ‡ )∥ + C0⋆ εn , ∥φ̂n (y) − φ̂n (y ′ )∥ ≥ c⋆0 ∥y − y ′ ∥ − C0⋆ εn , if y, y ′ ∈ B(u⋆ , r/2) satisfy ∥y − y ′ ∥ ≥ 10εn , ∥y − y ′ ∥ = ∥y † − y ‡ ∥ ± η ⇒ ∥φ̂n (y) − φ̂n (y ′ )∥ = ∥φ̂n (y † ) − φ̂n (y ‡ )∥ ± C0⋆ (η + εn ), if y, y ′ ∈ B(u⋆ , r/2), εn < r/120 and 0 ≤ η ≤ r/5. Indeed, let x, x′ , x† , x‡ ∈ Ωn such that ∥x − y∥, ∥x′ − y ′ ∥, ∥x† − y † ∥, ∥x‡ − y ‡ ∥ ≤ εn . (16) (17) (18) 18 For (15), we start by applying Lemma 16 to get ∥φ̂n (y) − φ̂n (y ′ )∥ = ∥φn (x) − φn (x′ )∥ ± 2ωn (2εn ) ≤ ωn (∥x − x′ ∥) + 2ωn (2εn ) ≤ ωn (∥y − y ′ ∥ + 2εn ) + 2ωn (2εn ), where ωn is the modulus of continuity of φn . We then use Lemma 3, which gives that ωn (η) ≤ Cη for all η and some C ∝ Dn /r, to get ωn (∥y − y ′ ∥ + 2εn ) ± 2ωn (2εn ) ≤ C(∥y − y ′ ∥ + 6εn ). For (16), we first note that ∥x − x′ ∥ < ∥x† − x‡ ∥ by the triangle inequality, which in turn implies that ∥φn (x) − φn (x′ )∥ ≤ ∥φn (x† ) − φn (x‡ )∥ since φn is isotonic. We then apply Lemma 16 to get that ∥φ̂n (y) − φ̂n (y ′ )∥ ≤ ∥φ̂n (y † ) − φ̂n (y ‡ )∥ + 4ωn (2εn ), and conclude with Lemma 3 as for (15). For (17), we may apply Lemma 15 with Λn . Let V be the convex hull of Λn , so that V ⊂ B(u⋆ , r/2). Let z be a point in that ball. If z ≠ u⋆ , let w = (u⋆ − z)/∥u⋆ − z∥, and if z = u⋆ , let w be any unit-norm vector. Define z ′ = z + εn w and notice that the distance from z ′ to ∂B(u⋆ , r/2) exceeds εn . Therefore, if x ∈ Ωn is such that ∥z ′ − x∥ ≤ εn , then necessarily, x ∈ Λn . We then note that ∥z − x∥ ≤ 2εn . We conclude that δH (Λn , V ) ≤ 2εn . Since ∥x − x′ ∥ ≥ ∥y − y ′ ∥ − 2εn ≥ 4(2εn ), we get that ∥φn (x) − φn (x′ )∥ ≥ c∥x − x′ ∥, with c ∶= diam(φn (Λn ))/5 diam(Λn ) ≥ δn /5r. We then apply Lemma 16 to obtain ∥φ̂n (y) − φ̂n (y ′ )∥ ≥ c∥x − x′ ∥ − 2ωn (2εn ) ≥ c∥y − y ′ ∥ − 2(c + C)εn , using Lemma 3 as for (15). For (18), note that x, x′ ∈ B(u⋆ , r/2 + εn ) ⊂ B(u⋆ , 3r/4), and ∥x − x′ ∥ = ∥x† − x‡ ∥ ± (η + 4εn ) by the triangle inequality. By Lemma 14 — where the constant there is denoted here by C ′ ∝ Dn /r — this implies that ∥φn (x) − φn (x′ )∥ = ∥φn (x† ) − φn (x‡ )∥ ± C ′ (η + εn ) when η + 4εn < r/4 − 2εn , which is true when εn < r/120 and η ≤ r/5. We then apply Lemma 16 together with Lemma 3, as for (15). CASE d = 1. This case is particularly simple. Note that U is a bounded open interval of R. We show that the function φ̂n is approximately midlinear on U . Take x, y ∈ U and define µ = (x + y)/2. By the fact that φ̂n takes its values in R, and (18), we have ∣ 21 (φ̂n (x) + φ̂n (y)) − φ̂n (µ)∣ = 21 ∣∣φ̂n (x) − φ̂n (µ)∣ − ∣φ̂n (y) − φ̂n (µ)∣∣ ≤ C0⋆ εn /2, when εn /r is small enough. Hence, φ̂n is (C0⋆ εn )-approximate midlinear on U . By the result of Vestfrid (2003), namely Lemma 17, there is C ∝ 1 — since U is a ball — and an affine function Tn such that maxy∈U ∣φ̂n (y) − Tn (y)∣ ≤ CC0⋆εn . Since all affine transformations from R to R are (possibly degenerate) similarities, we conclude. CASE d ≥ 2. For the remaining of this subsection, we assume that d ≥ 2. Approximate midlinearity. We show that there is a constant C such that φ̂n is locally Cεn approximately midlinear. Take x, y ∈ B(u⋆ , r/4), and let µ = (x + y)/2. Let t > 0 be a constant to be set large enough later. If ∥x − y∥ ≤ tεn , then by (15), φ̂n (x), φ̂n (y) ∈ B(φ̂n (µ), C0⋆ (t/2 + 1)εn ), so that 1 ∥φ̂n (µ) − (φ̂n (x) + φ̂n (y))∥ ≤ C0⋆ (t/2 + 1)εn . 2 Therefore, assume that ∥x − y∥ ≥ tεn . Let z1 , . . . , zd be constructed as in the proof of Theorem 1. By construction, both x, z1 , . . . , zd and y, z1 , . . . , zd form regular simplexes, and µ is the barycenter of z1 , . . . , zd . By Lemma 6, for any i ≠ j, √ √ √ ∥zi − µ∥ = (d − 1)/2d ∥zi − zj ∥ = (d − 1)/2d 2d/(d + 1) ∥x − µ∥ ≤ ∥x − y∥/2, 19 which coupled with the fact that x, y ∈ B(u⋆ , r/4) yields that zi ∈ B(u⋆ , r/2) for all i. Now, √ let z0 = x. ⋆ By (18), we have mini≠j ∥φ̂n (zi ) − φ̂n (zj )∥ ≥ maxi,j ∥φ̂n (zi ) − φ̂n (zj )∥ − C0 εn . Let cd = d/(2d + 2). By (17) and Lemma 6, ∥φ̂n (zi ) − φ̂n (zj )∥ ≥ c⋆0 ∥zi − zj ∥ − C0⋆ εn = c⋆0 cd ∥x − y∥ − C0⋆εn ≥ (c⋆0 cd t − C0⋆ )εn . Hence, assuming t ≥ 2C0⋆ /c⋆0 cd , we have mini≠j ∥φ̂n (zi ) − φ̂n (zj )∥ ≥ (1 − η) maxi,j ∥φ̂n (zi ) − φ̂n (zj )∥, where η ∶= 2C0⋆ /(c⋆0 cd t). In that case, φ̂n (x), φ̂n (z1 ), . . . , φ̂n (zd ) form a η-approximate regular simplex. By symmetry, the same is true of φ̂n (y), φ̂n (z1 ), . . . , φ̂n (zd ). Define λ = ∥φ̂n (x) − φ̂n (y)∥. By Lemma 6, ∥zi − zj ∥ = cd ∥x − y∥ < ∥x − y∥ − 4εn when t > 4/(1 − cd ), since ∥x − y∥ ≥ tεn and cd < 1. By (16), this implies that ∥φ̂n (zi ) − φ̂n (zj )∥ ≤ λ + C0⋆εn . By (17), λ ≥ (c⋆0 t − C0⋆ )εn , so that λ + C0⋆ εn ≤ 2λ since we already assumed that t ≥ 2C0⋆ /c⋆0 cd > 2C0⋆ /c⋆0 . For a ∈ {x, y, µ}, ∥a − zi ∥ is constant in i ∈ [d]. Therefore, by (18), mini ∥φ̂n (a) − φ̂n (zi )∥ ≥ maxi ∥φ̂n (a) − φ̂n (zi )∥ − C0⋆ εn . Define ξa as the orthogonal projection of φ̂n (a) onto the affine space A ∶= Aff(φ̂n (z1 ), . . . , φ̂n (zd )) and let δa = ∥φ̂n (a) − ξa ∥. By the Pythagoras theorem, we have ∥ξa − φ̂n (zi )∥2 = ∥φ̂n (a) − φ̂n (zi )∥2 − δa2 . In particular, max ∥ξa − φ̂n (zi )∥2 − min ∥ξa − φ̂n (zi )∥2 = max ∥φ̂n (a) − φ̂n (zi )∥2 − min ∥φ̂n (a) − φ̂n (zi )∥2 i i i i ≤ 2C0⋆ εn min ∥φ̂n (a) − φ̂n (zi )∥ + (C0⋆ εn )2 ≤ C1 εn , i where C1 ∶= 2C0⋆ Dn + C0⋆ r, once εn ≤ r. Let ζ denotes the barycenter of φ̂n (z1 ), . . . , φ̂n (zd ). Assume that t is sufficiently large that η ≤ 1/C2 , where C2 ∝ 1 is the constant of Lemma 13. By that lemma, and the fact that φ̂n (z1 ), . . . , φ̂n (zd ) form a η-approximate regular simplex of maximum edge length bounded by λ, we have ∥ξa − ζ∥ ≤ C2 λC1 εn . Let L be the line passing through ζ and perpendicular to A. We just proved that φ̂n (x), φ̂n (y), φ̂n (µ) are within distance C3 λεn from L, where C3 ∶= C1 C2 . Let ξ denote the orthogonal projection of φ̂n (µ) onto (φ̂n (x)φ̂n (y)). Since ∥x − µ∥ = ∥y − µ∥, we can apply (18) to get ∣∥ξ − φ̂n (x)∥2 − ∥ξ − φ̂n (y)∥2 ∣ = ∣∥φ̂n (µ) − φ̂n (x)∥2 − ∥φ̂n (µ) − φ̂n (y)∥2 ∣ = ∣∥φ̂n (µ) − φ̂n (x)∥ + ∥φ̂n (µ) − φ̂n (y)∥∣ × ∣∥φ̂n (µ) − φ̂n (x)∥ − ∥φ̂n (µ) − φ̂n (y)∥∣ ≤ 4C0⋆ λεn , using the fact that max(∥φ̂n (µ) − φ̂n (x)∥, ∥φ̂n (µ) − φ̂n (y)∥) ≤ λ + C0⋆ εn ≤ 2λ, due to (16) and ∥x − µ∥ = ∥y − µ∥ = 21 ∥x − y∥ < ∥x − y∥ − 4εn when t is large enough. By Lemma 13, we then obtain ∥ξ − 21 (φ̂n (x) + φ̂n (y))∥ ≤ C4 λεn for some constant C4 ∝ C0⋆ . In particular, recalling that λ = ∥φ̂n (x) − φ̂n (y)∥, this implies that ξ ∈ [φ̂n (x)φ̂n (y)] when εn ≤ 1/2C4 . It remains to argue that φ̂n (µ) is close to ξ. We already know that φ̂n (x), φ̂n (y), φ̂n (µ) are within distance C3 λεn from L, and by convexity, the same must be true of ξ. Let M = (φ̂n (x)φ̂n (y)) and θ = ∠(L, M ). Let PM denote the orthogonal projection onto M , when M is a linear subspace. By Pythagoras theorem, λ2 = ∥φ̂n (x) − φ̂n (y)∥2 = ∥PL (φ̂n (x) − φ̂n (y))∥2 + ∥PL⊥ (φ̂n (x) − φ̂n (y))∥2 ≤ (cos θ)2 λ2 + (2C3 λεn )2 , 20 implying that sin θ ≤ 2C2 εn . Since ∥PL − PM ∥ = sin θ and φ̂n (µ) − ξ is parallel to M , we also have ∥φ̂n (µ) − ξ∥2 = ∥PL (φ̂n (µ) − ξ)∥2 + ∥PL⊥ (φ̂n (µ) − ξ)∥2 ≤ (sin θ)2 ∥φ̂n (µ) − ξ∥2 + (2C3 λεn )2 , √ so that ∥φ̂n (µ) − ξ∥ ≤ 2C3 λεn / cos θ ≤ 2C3 λεn / 1 − (2C3 εn )2 ≤ C5 λεn , for some constant C5 ∝ C3 , once C3 εn is small enough. We conclude that ∥φ̂n (µ) − 12 (φ̂n (x) + φ̂n (y))∥ ≤ (C4 + C5 )λεn , by the triangle inequality. Approximate affinity. We now know that φ̂n is Cεn -approximate midlinear on B(u⋆ , r/4) for some constant C ∝ C0⋆(Dn + r) ∝ C0⋆ (diam(Q) + r). This implies, by the result of Vestfrid (2003), that is Lemma 17, that there is an affine function Tn such ∥φ̂n (x) − Tn (x)∥ ≤ C1⋆ εn for all x ∈ W , for some constant C1⋆ ∝ rC ∝ rC0⋆(diam(Q) + r). Approximate similarity. (Reinitialize the constants Ck , k ≥ 1.) We saw above that φ̂n transforms the regular simplex z0 (= x), z1 , . . . , zd with height denoted h satisfying h ≥ tεn /2 into a η-approximate one, where η = 2C0⋆ /(c⋆0 cd t). In what follows, choose these points so that they are all in B(u⋆ , r/2) and the simplex has height h ≥ r/8. (From here on, reinitialize the variables x, y, λ, etc.) We can then take t = r/4εn , yielding η = C1 εn for a constant C1 ∝ C0⋆ /(c⋆0 r). By the triangle inequality, we have min ∥Tn (zi ) − Tn (zj )∥ ≥ min ∥φ̂n (zi ) − φ̂n (zj )∥ − 2C1⋆ εn i≠j i≠j ≥ (1 − C1 εn ) max ∥φ̂n (zi ) − φ̂n (zj )∥ − 2C1⋆ εn i≠j ≥ max ∥Tn (zi ) − Tn (zj )∥ − (4C1⋆ + C1 δn )εn . i≠j By the triangle inequality and (17), γn ∶= max ∥Tn (zi ) − Tn (zj )∥ ≥ max ∥φ̂n (zi ) − φ̂n (zj )∥ − 2C1⋆ εn i,j ≥ i,j ⋆ c0 max ∥zi i,j − zj ∥ − C0⋆ εn − 2C1⋆ εn ≥ c⋆0 r/8 − (C0⋆ + 2C1⋆ )εn . Hence, we find that Tn (z0 ), . . . , Tn (zd ) form a C2 εn -approximate regular simplex, where C2 ∶= (4C1⋆ + C1 δn )/(c⋆0 r/8 − (C0⋆ + 2C1⋆ )εn ). Note that its maximum edge length is bounded as follows: γn ≤ max ∥φ̂n (zi ) − φ̂n (zj )∥ + 2C1⋆ εn ≤ δn + 2C1⋆ εn ≤ 2δn , i,j when 2C1⋆ εn ≤ δn . By Lemma 19, there is a constant C3 > 0 and an isometry Rn⋆ , such that we have maxx∈W ∥Tn (x) − λn Rn⋆ (x)∥ ≤ C3 λn C2 εn , where λn ∶= γn /h. Because r/8 ≤ h ≤ r and the bounds on γn above, there is a constant C2⋆ ≥ 1 such that 1/C2⋆ ≤ λn ≤ C2⋆ . (19) This implies that ∥φ̂n (x) − λn Rn⋆ (x)∥ ≤ ∥φ̂n (x) − Tn (x)∥ + ∥Tn (x) − λn Rn⋆ (x)∥ ≤ (C1⋆ + C3 C2 C2⋆ )εn =∶ C3⋆ εn . (20) Covering and conclusion. (Reinitialize the constants Ck , k ≥ 1.) Let u1 = u⋆ and let u2 , . . . , uK ∈ U be such that u1 , . . . , uK form a maximal (r/16)-packing of U . (The number 16 is not essential 21 here, but will play a role in the proof of Theorem 4.) Note that U = U1 ∪ ⋯ ∪ UK where Uk ∶= U ∩ B(uk , r/4), and note that U⋆ ∶= U1 ⊂ U . For u, u′ ∈ Uk , there are w, w′ ∈ U⋆ such that ∥w − w′ ∥ = ∥u − u′ ∥. Define φ̃n = φ̂n /λn . By (18), and then (19)-(20), we have ∥φ̃n (u) − φ̃n (u′ )∥ = ∥φ̃n (w) − φ̃n (w′ )∥ ± C0⋆ εn /λn = ∥w − w′ ∥ ± (C0⋆ + C3⋆ )εn /C2⋆ =∶ ∥w − w′ ∥ ± C1 εn . Let θ(Uk ) , (21) diam(Uk ) which is strictly positive. The result of Alestalo et al. (2001), namely Lemma 18, gives a constant C2 ∝ ξ1 and an isometry Rk such that maxu∈Uk ∥φ̃n (u) − Rk (u)∥ ≤ C2 εn . Let 1 (22) ξ2 = min {ρ(Uk ∩ Uk′ ) ∶ Uk ∩ Uk′ ≠ ∅}. 2 Take k, k ′ ∈ [K] such that Uk ∩ Uk′ ≠ ∅, so that there is u ∈ U such that B(u, ξ2 ) ⊂ Uk ∩ Uk′ . Since ξ1 = min k max ∥Rk (x) − Rk′ (x)∥ ≤ max ′ ∥Rk (x) − φ̃n (x)∥ + ∥φ̃n (x) − Rk′ (x)∥ x∈Uk ∩Uk x∈B(u,ξ2 ) ≤ max ∥Rk (x) − φ̃n (x)∥ + max ∥φ̃n (x) − Rk′ (x)∥ ≤ 2C2 εn , x∈Uk x∈Uk′ we have ∥Rk (x) − Rk′ (x)∥ ≤ (2∥x − u∥/ξ2 + 1)2C2 εn for all x ∈ Rd , by Lemma 20. Hence, ∥Rk (x) − Rk′ (x)∥ ≤ (2 diam(U )/ξ2 + 1)2C2 εn =∶ C3 εn for all x ∈ U . If instead Uk ∩ Uk′ = ∅, we do as follows. Since U is connected, there is a sequence k0 = k, k1 , . . . , km = k′ in [K], such that Uki ∩ Uki+1 ≠ ∅. We thus have maxx∈U ∥Rki (x) − Rki+1 (x)∥ ≤ C3 εn . By the triangle inequality, we conclude that maxx∈U ∥Rk (x) − Rk′ (x)∥ ≤ KC3 εn for any k, k′ ∈ [K]. Noting that R1 = Rn⋆ (since U1 = U⋆ ), for any k ∈ [K] and x ∈ Uk , ∥φ̃n (x) − Rn⋆ (x)∥ ≤ ∥Rk (x) − R1 (x)∥ + C2 εn ≤ (KC3 + C2 )εn . We conclude that, for any x ∈ U , ∥φ̂n (x) − λn Rn⋆ (x)∥ ≤ (KC3 + C2 )λn εn ≤ (KC3 + C2 )C2⋆ εn =∶ C4 εn . (23) This concludes the proof when d ≥ 2. A refinement of the constant. Assume now that U = U h for some h > 0. Tracking the constants above, we see that they all depend only on (d, ρ(U ), diam(U ), diam(Q)), as well as ξ1 and ξ2 defined in (21) and (22), respectively. We note that diam(Uk ) ≤ r and ρ(Uk ) ≥ min(r/2, h) by Lemma 21, so that ξ1 ≥ min(r/2, h)/r. To bound ξ2 , we can do as we did at the beginning of this section, so that at the end of that section, we can restrict our attention to chains k0 , . . . , km where ∥ukj − ukj+1 ∥ ≤ 2r/16 = r/8. To be sure, fix k, k′ ∈ [K] and let γ ∶ [0, 1] ↦ U be a curve such that γ(0) = uk and γ(1) = uk′ . Define s0 = 0 and then sj+1 = inf{s > sj ∶ ∥γ(s) − ukj ∥ > r/16}, and let kj+1 ∈ [K] be such that ∥γ(sj+1 ) − ukj+1 ∥ ≤ r/16, which is well-defined since (uk , k ∈ [K]) is a (r/16)-packing of U . We then have ∥ukj − ukj+1 ∥ ≤ ∥ukj − γ(sj )∥ − ∥γ(sj+1 ) − ukj+1 ∥ ≤ r/16 + r/16 = r/8. We can therefore redefine ξ2 in (22) as 12 min{ρ(Uk ∩ Uk′ ) ∶ ∥uk − uk′ ∥ ≤ r/8}. Because U = U h , for each k ∈ [K], there is vk such that uk ∈ B(vk , min(r/16, h)) ⊂ U . By the triangle inequality, B(vk , min(r/16, h)) ⊂ Uk′ when ∥uk − uk′ ∥ ≤ r/8, so that ξ2 ≥ min(r/16, h). So we see that everything depends on (d, h, ρ(U ), diam(U ), diam(Q)). The second part of the theorem now follows by invariance considerations. 22 3.4 Proof of Lemma 5 Let c = ess inf U f and C = ess supU f , which by assumption belong to (0, ∞). Fix i ∈ [n] and let Ni = #{j ≠ i ∶ ∥xj − xi ∥ ≤ r}. For j ≠ i, pi (j) ∶= P(∥xj − xi ∥ ≤ r) = ∫B(xi ,r) f (u)du. For an upper bound, we have pi (j) ≤ C Vol(B(xi , r) ∩ U ) ≤ C Vol(B(xi , r)) = Cζd r d =∶ Q, where Vol denotes the Lebesgue measure in Rd and ζd is the volume of the unit ball in Rd . Hence, P(Ni > 2(n − 1)Q) ≤ P(Bin(n − 1, Q) > 2(n − 1)Q) ≤ e−(n−1)Q/3 by Bennett’s inequality for the binomial distribution. By the union bound, we conclude that maxi Ni ≤ 2(n − 1)Q with probability at least 1 − ne−(n−1)Q/3 , which tends to 1 if nr d ≥ C0 log n and C0 > 0 is sufficiently large. For a lower bound, we use the following lemma. Lemma 21. Suppose U ⊂ Rd is open and such that U = U h for some h > 0. Then for any x ∈ U and any r > 0, B(x, r) ∩ U contains a ball of radius min(r, h)/2. Moreover, the closure of that ball contains x. Proof. By definition, there is y ∈ U such that x ∈ B(y, h) ⊂ U . We then have B(x, r) ∩ U ⊃ B(x, r) ∩ B(y, h), so it suffices to show that the latter contains a ball of radius min(r, h)/2. By symmetry, we may assume that r ≤ h. If ∥x − y∥ ≤ r/2, then B(x, r/2) ⊂ B(y, h) and we are done. Otherwise, let z = (1−t)x+ty with t ∶= r/2∥x−y∥ ∈ (0, 1), and note that B(z, r/2) ⊂ B(x, r)∩B(y, h) and x ∈ ∂B(z, r/2). Now that Lemma 21 is established, we apply it to get pi (j) ≥ c Vol(B(xi , r) ∩ U ) ≥ cζd (min(r, h)/2)d =∶ q. Hence, P(Ni < (n − 1)q/2) ≤ P(Bin(n − 1, q) < (n − 1)q/2) ≤ e−(6/7)(n−1)q . By the union bound, we conclude that mini Ni ≥ (n − 1)q/2 with probability at least 1 − ne−(6/7)(n−1)q , which tends to 1 if nr d ≥ C1 log n and C1 > 0 is sufficiently large. (Recall that h is fixed.) 3.5 More auxiliary results We list here a few additional of auxiliary results that will be used in the proof of Theorem 4. For V ⊂ Rd and x, x′ ∈ V , define the intrinsic metric δV (x, x′ ) = sup {L ∶ ∃γ ∶ [0, L] ↦ V, 1-Lipschitz, with γ(0) = x, γ(L) = x′ }, where γ is 1-Lipschitz if ∥γ(s) − γ(t)∥ ≤ ∣s − t∣ for all s, t ∈ [0, L]. If no such curve exists, set δV (x, x′ ) = ∞. The intrinsic diameter of V is defined as sup{δV (x, x′ ) ∶ x, x′ ∈ V }. We note that, if L ∶= δV (x, x′ ) < ∞, then there is a curve γ ⊂ V̄ with length L joining x and x′ . Recall that a curve with finite length is said to be rectifiable. See (Burago et al., 2001) for a detailed account of intrinsic metrics. For U ⊂ Rd and h > 0, let U ⊖h = {x ∈ U ∶ B(x, h) ⊂ U }. This is referred to as an erosion (of the set U ) in mathematical morphology. Lemma 22. If U ⊂ Rd is open and connected, then for each pair of points x, x′ ∈ U , there is h > 0 and a rectifiable curve within U ⊖h joining x and x′ . 23 Proof. Take x, x′ ∈ U . By taking an intersection with an open ball that contains x, x′ , if needed, we may assume without loss of generality that U is bounded. Since every connected open set in a Euclidean space is also path-connected (Waldmann, 2014, Example 2.5.13), there is a continuous curve γ ∶ [0, 1] ↦ U such that γ(0) = x and γ(1) = x′ . A priori, γ could have infinite length. However, γ (≡ γ([0, 1])) is compact. For each t ∈ [0, 1], let r(t) > 0 be such that Bt ∶= B(γ(t), r(t)) ⊂ U . Since γ ⊂ ⋃t∈[0,1] Bt , there is 0 ≤ t1 < ⋅ ⋅ ⋅ < tm ≤ 1 such that γ ⊂ ⋃j∈[m] Btj . Since γ is connected, necessarily, for all j ∈ [m−1] there is sj ∈ [tj , tj+1 ] such that γ(sj ) ∈ Btj ∩Btj+1 . Let s0 = 0 and sm = 1. Then [γ(sj )γ(sj+1 )] ⊂ Btj+1 ⊂ U for all j ∈ 0, . . . , m − 1, and therefore the polygonal line defined by x = γ(s0 ), γ(s1 ), . . . , γ(sm−1 ), γ(sm ) = x′ is inside ⋃j∈[m] Btj ⊂ U ⊖r where r ∶= minj∈[m] r(tj ) > 0. By construction, this polygonal line joins x and x′ , and is also rectifiable since it has a finite number of vertices. Lemma 23. Suppose U ⊂ Rd is bounded, connected, and such that U = U h for some h > 0. Then ′ there is h‡ > 0 such that, for all h′ ∈ [0, h‡ ], the intrinsic diameter of U ⊖h is finite. Proof. Let V = U ⊖h . By assumption, for all x ∈ U , there is y ∈ V such that x ∈ B(y, h) ⊂ U . In particular, U ⊃ V ≠ ∅. Let V1 be a connected component of V . Pick y1 ∈ V1 and note that B1 ∶= B(y1 , h) ⊂ U by definition, and also B1 ⊂ V1 because B1 is connected. Let ζd be the volume of the unit ball in Rd . Since the connected components are disjoint and each has volume at least ζd hd while U has volume at most ζd (diam(U )/2)d , V can have at most ⌈(diam(U )/2h)d ⌉ connected components, which we now denote by V1 , . . . , VK . Pick yk ∈ Vk for each k ∈ [K]. Applying Lemma 22, for each pair of distinct k, k ′ ∈ [K], there is a rectifiable (i.e., finite-length) path γk,k′ ⊂ U joining yk and yk′ . By Lemma 22, the length of γk,k′ , denoted Dk,k′ , is finite, and there is hk,k′ > 0 such that γk,k′ ⊂ U ⊖hk,k′ . Let D‡ = maxk,k′ ∈[K] Dk,k′ and h‡ = mink,k′∈[K] hk,k′ . We now show that each connected component Vk has finite diameter in the intrinsic metric of V ′ ∶= U ⊖h/2 . Since Vk is bounded, there is x1 , . . . , xm ∈ Vk such that Vk ⊂ ⋃j∈[mk ] Qj , where Qj ∶= B(xj , h/2) ⊂ V ′ . Take any x, x′ ∈ Vk . Let j, j ′ ∈ [mk ] be such that x ∈ Qj and x′ ∈ Qj ′ . Since Vk is connected, there is a sequence j = j0 , j1 , . . . , jSk = j ′ ∈ [mk ] such that Qjs ∩ Qjs+1 ≠ ∅ for all s = 0, . . . , Sk . Choose zs ∈ Qjs ∩ Qjs+1 and let z0 = x and zSk = x′ . Then [zs zs+1 ] ⊂ Qjs+1 for all s. k Qjs ⊂ V ′ , it joins x Let L be the polygonal line formed by z0 , . . . , zSk . By construction, L ⊂ ⋃Ss=0 ′ ′ and x , and has length at most (Sk + 1)2h. Hence, δV ′ (x, x ) ≤ (Sk + 1)2h ≤ 2(mk + 1)h. This being valid for all x, x′ ∈ Vk , we proved that Vk has diameter at most Dk ∶= 2(mk + 1)h in the intrinsic metric of V ′ . Let D⋆ = maxk∈[K] Dk . Now take h† ∈ [0, h‡ ] and any x, x′ ∈ U ⊖h† . Let y, y ′ ∈ V be such that x ∈ B(y, h) and x′ ∈ B(y ′ , h). Let k, k ′ ∈ [K] be such that y ∈ Vk and y ′ ∈ Vk′ . There are curves γ, γ ′ ⊂ V ′ of length at most D⋆ such that γ joins y and yk , while γ ′ joins y ′ and yk′ . We then join yk and yk′ with γk,k′ All together, we have the curve [xy] ∪ γ ∪ γk,k′ ∪ γ ′ ∪ [y ′ x′ ], which joins x and x′ , lies entirely in U ⊖h† , and has length bounded by h + D⋆ + D‡ + D⋆ + h =∶ D. And this is true for any pair of such points. Lemma 24. Suppose that S1 , S2 ∶ Rd ↦ Rd are two affinities such that maxj ∥S1 (zj ) − S2 (zj )∥ ≤ ε, where z0 , . . . , zd form in a η-approximate regular simplex with minimum edge length at least λ. There is C > 0 depending only on d such that, if η ≤ 1/C, then ∥S1 (x) − S2 (x)∥ ≤ Cε∥x − z0 ∥/λ + ε for all x ∈ Rd . Proof. Note that this is closely related to Lemma 20. By translation and scale invariance, assume that z0 = 0 and λ = 1. Let Li = Si − Si (0). We have ∥L1 (zj ) − L2 (zj )∥ ≤ ∥S1 (zj ) − S2 (zj )∥ + ∥S1 (0) − 24 S2 (0)∥ ≤ 2ε. Let Z denote the matrix with columns z1 , . . . , zd . In matrix notation, we have ∥(L1 − L2 )Z∥F = √ √ ∑ ∥(L1 − L2 )zj ∥2 ≤ 2 dε. j We also have ∥(L1 − L2 )Z∥F ≥ ∥(L1 − L2 )Z∥ ≥ σd (Z)∥L1 − L2 ∥, and by Lemma 12, σd (Z) = σd ([z0 , Z]) ≥ 1/C1 when η ≤ 1/C1 , where C1 depends only on d. In that case, ∥L1 − L2 ∥ ≤ C2 ε for another constant C2 . Equivalently, for x ∈ Rd , ∥L1 (x) − L2 (x)∥ ≤ C2 ε∥x∥, which in turn implies that ∥S1 (x) − S2 (x)∥ ≤ ∥L1 (x) − L2 (x)∥ + ∥S1 (0) − S2 (0)∥ ≤ C2 ε∥x∥ + ε. 3.6 Proof of Theorem 4 Because φn is bounded independently of n, we may assume without loss of generality that C0 εn ≤ rn and C0 rn ≤ h for all n, where C0 ≥ 1 will be chosen large enough later on. Take y ∈ U and let Ωy = Ωn ∩ B(y, rn ) and Qy = φn (Ωy ). We first show that there is C1 ∝ diam(Q)/ρ(U ) such that, for any y ∈ U , diam(Qy ) ≤ C1 rn . For this, we mimic the proof of Lemma 3. Take x, x′ ∈ Ωy such that ξ ∶= ∥φn (x) − φn (x′ )∥ = diam(Qy ). Let u be such that B(u, ρ(U )) ⊂ U . Let y1 , . . . , ym be an (rn + 2εn )-packing of B(u, ρ(U )) with m ≥ A1 (ρ(U )/rn )d for some A1 ∝ 1. Then let {xis ∶ s ∈ [m]} ⊂ Ωn be such that maxs∈[m] ∥ys − xis ∥ ≤ εn . By the triangle inequality, for all s ≠ t, we have ∥xis − xit ∥ ≥ ∥ys − yt∥ − 2εn ≥ rn > ∥x − x′ ∥. By (10), we have ∥φn (xis ) − φn (xit )∥ ≥ ξ, so that φn (xi1 ), . . . , φn (xim ) form a ξ-packing. Therefore m ≤ A2 (diam(Q)/ξ)d for some A2 ∝ 1. We conclude that ξ ≤ (A2 /A1 )1/d (diam(Q)/ρ(U ))rn =∶ C1 rn . We apply Theorem 3 to Uy ∶= B(y, rn ) and Ωy . With the fact that δH (Ωy , Uy ) ≤ 2εn — as we saw in the proof of (17) — and invariance considerations, we obtain a constant C ∝ 1 and a similarity Sy such that maxx∈Ωy ∥φn (x) − Sy (x)∥ ≤ C(diam(Qy )/rn )εn ≤ CC1 εn =∶ C2 εn . (Note that all the quantities with subscript y depend also on n, but this will be left implicit.) Fix y⋆ ∈ U ⊖rn . For x ∈ Ωn , there is y ∈ U ⊖rn such that x ∈ Uy . Assume γ is parameterized by arc length and let h‡ be given by Lemma 23 and let D denote the intrinsic diameter of U ⊖h‡ . Then assuming rn ≤ h‡ , there is a curve γ ⊂ U ⊖rn of length L ≤ D joining y⋆ and y. Let y0 = y⋆ , yj = γ(jrn ) for j = 0, . . . , J ∶= ⌊L/rn ⌋, and then yJ+1 = y. We have maxz∈Uyj ∩Uyj+1 ∥Syj (z) − Syj+1 (z)∥ ≤ 2C2 εn by the triangle inequality. We also have ρ(Uyj ∩ Uyj+1 ) ≥ rn , because ∥yj − yj+1 ∥ ≤ rn . Let vj be such that B(vj , rn /2) ⊂ Uyj ∩ Uyj+1 . Fix j and let vj,0 , . . . , vj,d denote a regular simplex inscribed in the ball B(vj , rn /4). Let λn ∝ rn denote its edge length. Then let xj,0 , . . . , xs,d ∈ Ωn be such that maxk ∥xj,k − vj,k ∥ ≤ εn . When C0 is large enough, xj,0 , . . . , xj,d ∈ B(vj , rn /2) by the triangle inequality. Moreover, maxk,l ∥xj,k −xj,l ∥ ≤ λn +2εn , as well as mink≠l ∥xj,k −xj,l ∥ ≥ λn −2εn . When C0 is large enough, Fj ∶= {xj,0 , . . . , xj,d } is therefore an η-approximate regular simplex, with η ∝ εn /rn , and minimum edge length ∝ rn . Now, since maxk ∥Syj (xj,k ) − Syj+1 (xj,k )∥ ≤ 2C2 εn , by Lemma 24, for all z ∈ Rd , ∥Syj (z)−Syj+1 (z)∥ ≤ CC2 εn ∥z−xj,0 ∥/rn +2C2 εn for some C ∝ 1, assuming εn /rn ≤ 1/C. In particular, by the fact that ∥x − xj,0 ∥ ≤ diam(U ), this gives ∥Syj (x) − Syj+1 (x)∥ ≤ C3 εn /rn for some C3 ∝ diam(U )C2 . Hence, ∥Sy⋆ (x) − Sy (x)∥ ≤ (J + 1)C3 εn /rn ≤ C4 εn /rn2 , since J ≤ L/rn ≤ D/rn . This being true for any arbitrary x ∈ Ωn , we conclude that max ∥φn (x) − Sy⋆ (x)∥ ≤ C4 εn /rn2 + C2 εn ≤ C5 εn /rn2 . x∈Ωn 25 4 Discussion This paper builds on (Kleindessner and von Luxburg, 2014) to provide some theory for ordinal embedding, an important problem in multivariate statistics (aka unsupervised learning). We leave open two main problems: • What are the optimal rates of convergence for ordinal embedding with all triple and quadruple comparisons? • What is the minimum size of K = Kn for consistency of ordinal embedding based on the K-nearest neighbor distance comparisons? We note that we only studied the large sample behavior of exact embedding methods. In particular, we did not discuss or proposed any methodology for producing such an embedding. For this, we refer the reader to (Agarwal et al., 2007; Borg and Groenen, 2005; Terada and Von Luxburg, 2014) and references therein. In fact, the practice of ordinal embedding raises a number of other questions in terms of theory, for instance: • How many flawed comparisons can be tolerated? Acknowledgements We are grateful to Vicente Malave for introducing us to the topic and for reading a draft of this paper. 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A Framework for Datatype Transformation Jan Kort 1 and Ralf Lämmel 2,3 1 Universiteit van Amsterdam voor Wiskunde en Informatica 3 Vrije Universiteit van Amsterdam arXiv:cs/0204018v3 [cs.PL] 24 Feb 2003 2 Centrum Abstract We study one dimension in program evolution, namely the evolution of the datatype declarations in a program. To this end, a suite of basic transformation operators is designed. We cover structure-preserving refactorings, but also structure-extending and -reducing adaptations. Both the object programs that are subject to datatype transformations, and the meta programs that encode datatype transformations are functional programs. 1 Introduction We study operators for the transformation of the datatype declarations in a program. The presentation will be biased towards the algebraic datatypes in Haskell, but the concepts are of relevance for many typed declarative languages, e.g., Mercury and SML, as well as frameworks for algebraic specification or rewriting like ASF+SDF, CASL, Elan, and Maude. Our transformations are rather syntactical in nature as opposed to more semantical concepts such as data refinement. Our transformations contribute to the more general notion of functional program refactoring [TR01]. The following introductory example is about extracting a new datatype from constructor components of an existing datatype. This is illustrated with datatypes that represent the syntax of an imperative language. The following extraction identifies a piece of syntax to enable its reuse in later syntax extensions: - - Datatypes with focus on two constructor components data Prog = Prog ProgName [Dec ] [Stat ] data Dec = VDec Id Type data Stat = Assign Id Expr \| If Expr Stat Stat \| ... - - After extraction of [Dec] [Stat] to constitute a new datatype Block data Prog = Prog ProgName Block data Block = Block [Dec ] [Stat ] In the present paper, we describe the design of a framework for datatype transformations including the operators for the above extraction. In Sec. 2, we identify all the concerns addressed by the framework. In Sec. 3, we describe all the basic operators for datatype transformations. In Sec. 4, these operators are lifted from datatypes to complete programs. Related work is discussed in Sec. 5. The paper is concluded in Sec. 6. Kort & Lämmel 2 Concerns in datatype transformation The central contribution of the present paper is a simple, well-defined, and ‘editingcomplete’ suite of operators for datatype transformations. Before we embark on this suite, we identify the concerns addressed by our approach: • Datatype transformations via scripting or interactive tool support. • Well-defined primitives for datatype transformations. • Generic meta-programming for conciseness of datatype transformations. • Flexible means of referring to fragments of interest in datatype transformations. We will now discuss these concerns in some depth. 2.1 Scripting vs. interactive tool support From the point of view of a programmer, datatype transformations should be founded on intuitive scenarios for adaptation. To actually perform (datatype) transformations, there are two modes of operation. The first mode is scripting: the programmer encodes the desired transformation as an expression over basic or higher-level operators. The second mode is interactive transformation based on a corresponding GUI. The benefits of an interactive tool are rather obvious. Such a tool is useful to issue a transformation on the basis of an operator-specific dialogue, and to provide a tailored list of options for transformations that make sense in a given context. A crucial benefit of interactive transformation is that the GUI can be used to provide feedback to the programmer: Which locations were changed? Where is the programmer’s attention needed to complete the issued transformation scenario? The apparent benefits of scripting such as the opportunities to revise transformations and to replay them can be also integrated into an interactive setting. In Fig. 1, we illustrate the interactive treatment of the introductory example using our prototypical tool TH — Transform Haskell. As the snapshot indicates, we use a designated fold dialogue to perform the extraction of the piece of syntax. (Folding is the basic transformation underlying extraction.) This dialogue combines several transformation steps and side conditions in a convenient way. The figure shows the following situation. The user has selected two consecutive types “[Dec] [Stat]” and initiated the fold dialogue. The user has also typed in “Block” in the “type name” field. The introduction check-box is marked automatically since the given type name does not yet exist. The user has also selected the “kind” radiobutton to be “data” and filled in “Block” in the “cons name” field. After this, the user would press “Replace” to make the change. If there had been more than one occurrence, the user could replace them all with “Replace All”, or step through all occurrences with “Next”, and replace only specific ones with “Replace” as with ordinary find and replace in text editors. 2 Kort & Lämmel Fig. 1. A snapshot related to the interactive treatment of the introductory example Here is an open-ended list of further common transformation scenarios: • Renaming type and constructor names. • Permuting type arguments and constructor components. • The dual of extracting datatypes, i.e., inlining datatypes. • Including a constructor declaration together with associated functionality. • Excluding a constructor declaration together with associated functionality. • Inserting a constructor component together with associated functionality. • Deleting a constructor component together with associated functionality. 2.2 Well-defined transformation primitives The core asset of our framework is a suite of basic operators, which can be either used as is, or they can be completed into more complex, compound transformations. In the design of this suite, we reuse design experience from a related effort on grammar adaptation [Läm01]. Indeed, there is an obvious affinity of grammar transformations and datatype transformations. A challenging problem that we did not need to address in this previous work, is the completion of datatype transformations to apply to entire (functional) programs in which evolving datatypes reside. We list the required properties of our basic transformation operators: Correctness Mostly, we insist on ‘structure preservation’, that is, the resulting datatype is of the same shape as the original datatype. This is enforced by the pre- and postconditions of the operators. 3 Kort & Lämmel Completeness The operators are ‘editing-complete’, that is, they capture all scenarios of datatype evolution that are otherwise performed by plain text editors. Semantics-preserving adaptations are defined in terms of disciplined primitives. Orthogonality The operators inhabit well-defined, non-overlapping roles. Higherlevel scenarios for interactive transformation are derivable. Operators for datatype transformations are complementary to expression-level transformations. Locality The basic operators operate on small code locations as opposed to ‘global’ or ‘exhaustive’ operators, which iterate over the entire program. Note that some operators are necessarily exhaustive, e.g., an operator to rename a type name. Implementability The operators are implemented as syntactical transformations that are constrained by simple analyses to check for pre- and postconditions, but which otherwise do not necessitate any offline reasoning. Universality While the present paper focuses on datatype transformations, the principles that are embodied by our operators are universal in the sense that they also apply to other abstractions than datatypes, e.g., functions or modules. We do not list these properties to announce a formal treatment. This would be very challenging as we opt for the complex language setup of Haskell. The above properties provide merely a design rationale. A formal approach is an important subject for future work, but it does not contribute anything to the narrow goal of the present paper: to compile an inventory of the basic roles in datatype transformation. 2.3 Generic meta-programming We implement transformation operators and compound meta-programs in Haskell. We reuse a publicly available abstract syntax for Haskell. 1 We rely on generic programming techniques to perform meta-programming on the non-trivial Haskell syntax in Haskell. We use the Strafunski-style 2 of generic programming that allows us to complete functions on specific syntactical sorts into generic traversals that process subterms of the specific sorts accordingly. This style of metaprogramming is known to be very concise because one only provides functionality for the types and constructors that are immediately relevant for the given problem. All our datatype transformations are of type Trafo which is defined as follows: type Trafo = HsModule → Maybe HsModule That is, a datatype transformation is a partial function on HsModule — the abstract syntactical domain for Haskell modules. Partiality is expressed by means of the Maybe type constructor that wraps the result type. Partially is needed to model side conditions. In Fig. 2, we illustrate generic meta-programming by giving the definition of a simple operator for replacing type names. The specification formalises the fact that 1 2 The used abstract syntax is part of the Haskell Core Libraries — in the haskell-src package. http://www.cs.vu.nl/Strafunski/ 4 Kort & Lämmel Replace a type name replaceTypeId :: TypeId → TypeId → Trafo replaceTypeId n n ′ = full tdTP (adhocTP (adhocTP idTP declSite) refSite) where Transform declaring occurrences of type names declSite :: HsDecl → Maybe HsDecl declSite (HsTypeDecl l n0 ps t) \| n0 ≡ n = return (HsTypeDecl l n ′ ps t) declSite (HsDataDecl l c n0 ps cds d) \| n0 ≡ n = return (HsDataDecl l c n ′ ps cds d) declSite (HsNewTypeDecl l c n0 ps cd d) \| n0 ≡ n = return (HsNewTypeDecl l c n ′ ps cd d) declSite decl = return decl Transform using occurrences of type names refSite :: HsType → Maybe HsType refSite (HsTyCon (UnQual n0 )) \| n0 ≡ n = return (HsTyCon (UnQual n ′ )) refSite tpe = return tpe Fig. 2. Specification of the replacement operation underlying renaming of type names type names can occur in two kinds of locations: either on a declaration site, when we declare the type, or on a using site, when we refer to the type in a type expression. So we need to synthesise a transformation which pays special attention to the syntactical domains for declaring and using sites. Indeed, in the figure, there are two type-specific ‘ad-hoc’ cases which customise the identity function idTP. In the given context, we choose the traversal scheme full tdTP for ‘full top-down traversal in Type-Preserving manner’. This way, we will reach each node in the input tree to transform type names on declaring and using sites. The operator replaceTypeId, by itself, is a total function. (So the Maybe in its type is not really needed here.) Partiality would be an issue if we derived an operator for renaming type names. This necessitates adding a side condition to insist on a fresh new name. 2.4 Means of referring to fragments of interest Both the basic operators for datatype transformation but also actual transformation scenarios in scripts or in interactive sessions need to refer to program fragments of interest. Recall our introductory example. Extracting a type necessitates referring to the constructor components that are meant to constitute the new type. In our framework, we use three ways to refer to fragments of interest: Focus markers on subterms This approach is particularly suited for interactive transformations. Here, relevant fragments can be directly marked. In Fig. 3, we extend Haskell’s abstract syntax to include term constructors for focusing on relevant fragments in datatype transformations. That is, we are prepared to focus on names of types, on type expressions, and on lists of constructor components. Selectors of subterms This approach is particularly suited for scripting transformations. Selectors for Haskell’s type expressions are defined in Fig. 4. The three forms of TypeSel represent the three kinds of declarations that involve types. The helper TypeSel ′ allows to select any part of a given type expression. 5 Kort & Lämmel Focus on names data HsName = ... \| HsNameFocus HsName Focus on type expressions data HsType = ... \| HsTypeFocus HsType Focus on lists of constructor components data HsConDecl = HsConDecl SrcLoc HsName [ HsFocusedBangType ] \| HsRecDecl SrcLoc HsName [([HsName ], HsBangType)] data HsFocusedBangType = HsUnfocusedBangType HsBangType \| HsFocusedBangType [HsBangType ] Fig. 3. Kinds of focus for datatype transformation data TypeSel = AliasRef TypeId TypeSel ′ \| ConRef ConPos TypeSel ′ \| SigRef [FunId ] TypeSel ′ data TypeSel ′ = SelStop \| SelDom TypeSel ′ \| SelCod TypeSel ′ \| SelIth ParaPos TypeSel ′ \| SelFun TypeSel ′ \| SelArg TypeSel ′ type TypeId type ConId type FunId type ConPos type ParaPos data HsName = = = = = = HsName HsName HsName (ConId , ParaPos ) Int ... -- Refer to a type alias -- Refer to a constructor component -- Refer to a function signature -- Reference stops here -- Refer to domain of function type -- Refer to co-domain of function type -- Refer to products component -- Refer to type constructor -- Refer to type argument -- Refer to a type -- Refer to a constructor -- Refer to a function name -- Refer to a component of a constructor -- Refer to a parameter position -- Syntactical sort for all kinds of names Fig. 4. Selectors that refer to type expressions, and others Predicates on subterms Such predicates typically constrain the type of a term or the top-level pattern. This approach is particularly suited for the repeated application of a transformation to different focuses that match a given predicate. There are ways to mediate between these different ways of referring to subterms. For example. given a term with a focus marker on a type expression, one can compute the selector that refers to the focused subterm. Given a predicate on type expressions, one can compute the list of all selectors so that an operator that is defined on selectors can be used with predicates as well. Finally, given a selector, one can also add the corresponding focus marker in the input at hand. 3 Basic operators for datatype transformation We will now describe the themes that constitute our operator suite: • Renaming type and constructor names. • Permutation of type parameters and constructor components. • Swapping types on use sites. • Introduction vs. elimination of type declarations. • Folding vs. unfolding of type declarations. 6 Kort & Lämmel Sample input datatype data ConsList a = Nil \| Cons a (ConsList a) Renamed and permuted datatype data SnocList a = Lin \| Snoc (SnocList a) a Fig. 5. Illustration of renaming and permutation renameTypeId renameConId permuteTypeId permuteConId ::TypeId → TypeId → Trafo ::ConId → ConId → Trafo ::TypeId → [ParaPos ] → Trafo ::ConId → [ParaPos ] → Trafo -- Rename a type declaration -- Rename a constructor -- Permute type parameters -- Permute constructor components Fig. 6. Operators for renaming and parameter permutation renameTypeId (HsIdent "ConsList") (HsIdent "SnocList") renameConId (HsIdent "Nil") (HsIdent "Lin") renameConId (HsIdent "Cons") (HsIdent "Snoc") permuteConId (HsIdent "Snoc") [2, 1] ‘seqTrafo‘ ‘seqTrafo‘ ‘seqTrafo‘ Fig. 7. Script for the scenario in Fig. 5 • Wrapping vs. unwrapping of constructor components. • Inclusion vs. exclusion of entire constructor declarations. • Insertion vs. deletion of constructor components. As this list makes clear, we group an operator with its inverse such as in “folding vs. unfolding”, unless the operator can be used to inverse itself. This is the case for renaming, permutation, and swapping. The operators from the first six groups are (almost) structure-preserving. The last two groups deal with structure-extending and -reducing transformations. We will now explain the operators in detail including illustrative examples. We will only explain the effect of the operators on datatype declarations while we postpone lifting the operators to the level of complete programs until Sec. 4. 3.1 Renaming and permutation Let us start with the simplest datatype refactorings one can think of. These are transformations to consistently rename type or constructor names, and to permute parameters of type and constructor declarations. In Fig. 5, a simple example is illustrated. We rename the type name ConsList, the constructor names Nil and Cons, and we permute the two parameter positions of Cons. The resulting datatype specifies a SnocList as opposed to the ConsList before. In Fig. 6, we declare the operators for renaming names and permuting parameter lists. In Fig. 7, we include the script that encodes the ConsList-to-SnocList sample as a sequence of basic renaming and permuting transformations. To this end, we assume a sequential composition operator seqTrafo for datatype transformations. (In the script, seqTrafo is used as an infix operator ‘seqTrafo‘.) 7 Kort & Lämmel data HsDecl = ... introTypes elimTypes :: :: -- Syntactical sort for (type) declarations [HsDecl ] → Trafo [TypeId ] → Trafo -- Introduction of type declarations -- Elimination of type declarations Fig. 8. Operators for introduction and elimination of datatypes type TypeHdr = (TypeId , [TypeVar ]) type TypeVar = HsName -- Header (LHS) of type declaration -- Type variables foldAlias :: TypeSel → TypeHdr → Trafo unfoldAlias :: TypeSel → Trafo -- Folding the referred type -- Unfolding the referred type Fig. 9. Operators for folding and unfolding 3.2 Introduction vs. elimination The next group of operators deals with the introduction and elimination of type declarations (see Fig. 8). Introduction means that the supplied types are added while their names must not be in use in the given program. Elimination means that the referenced types are removed while their names must not be referred to anymore in the resulting program. The two operators take lists of types as opposed to single ones because types can often only be introduced and eliminated in groups, say mutually recursive systems of datatypes. All kinds of type declarations make sense in this context: aliases, newtypes, and proper datatypes. The operators for introduction and elimination are often essential in compound transformations. This will be illustrated below when we reconstruct the introductory example in full detail (see Sec. 3.4). 3.3 Folding vs. unfolding Instantiating the folklore notions of unfolding and folding for datatypes basically means to replace a type name by its definition and vice versa. Extra provisions are needed for parameterised datatypes. The prime usage scenarios for the two operators are the following: • extraction = introduction of a type followed by its folding. • inlining = unfolding a type followed by its elimination. To give an example, the introductory example basically extracts the structure of imperative program blocks. To actually reconstruct this example, we need a few more operators. So we postpone scripting the example (see Sec. 3.4). The operators for folding and unfolding are declared in Fig. 9, The operators make a strict assumption: the type which is subject to folding or unfolding is necessarily a type alias as opposed to a proper datatype. This assumption simplifies the treatment of the operators considerably since type aliases and their definitions are equivalent by definition. Extra operators for so-called wrapping and unwrapping allow us to use proper datatypes during folding and unfolding as well. This will be addressed below. In the type of the foldAlias operator, we do not just provide a type 8 Kort & Lämmel type ConRange = (ConPos , Int ) -- Refer to consecutive components groupConRange ungroupConPos alias2newtype newtype2data data2newtype newtype2alias :: :: :: :: :: :: ConRange → Trafo ConPos → Trafo TypeId → ConId → Trafo TypeId → Trafo TypeId → Trafo TypeId → Trafo -- Group constructor components -- Inline product -- Turn type alias into newtype -- Turn newtype into datatype -- Turn datatype into newtype -- Turn newtype into type alias Fig. 10. Operators for wrapping and unwrapping 0. Original syntax data Prog data Dec data Stat data Expr = = = = Prog ProgName [Dec ] [Stat ] VDec Id Type Assign Id Expr \| If Expr Stat Stat Var Id \| Const Int 1. After grouping [Dec] and [Stat] data Prog = Prog ProgName ([Dec ], [Stat ]) 2. After introduction of Block to prepare folding data Prog = Prog ProgName ([Dec ], [Stat ]) type Block = ([Dec ], [Stat ]) 3. After folding away the type expression ([Dec], [Stat]) data Prog = Prog ProgName Block type Block = ([Dec ], [Stat ]) 4. After turning Block into a proper datatype with the constructor Block data Prog = Prog ProgName Block data Block = Block ([Dec ], [Stat ]) 5. After ungrouping the product ([Dec], [Stat]) data Prog = Prog ProgName Block data Block = Block [Dec ] [Stat ] Fig. 11. Illustration of wrapping, unwrapping, and extraction name but also a list of type variables (cf. helper type TypeHdr). This is needed for parameterised datatypes, where we want to specify how the free type variables in the selected type expression map to the argument positions of the type alias. The preconditions for the operators are as follows. In the case of foldAlias, we need to check if the referenced type expression and the right-hand side of the given alias declaration coincide. In the case of unfolding, we need to check that the referenced type expression corresponds to an application of a type alias. 3.4 Wrapping vs. unwrapping We will now consider operators that facilitate certain forms of wrapping and unwrapping of datatype constructors (see Fig. 10). There are operators for grouping and ungrouping, that is, to turn consecutive constructor components into a single component that is of a product type, and vice versa. There are also operators to mediate between the different kinds of type declarations, namely type aliases, newtypes and datatypes. This will allow us to toggle the representation of datatypes in basic ways. As a result, the normal forms assumed by other operators can be established; recall, for example, the use of type aliases in folding and unfolding. This separation of concerns serves orthogonality. 9 Kort & Lämmel groupConRange ((HsIdent "Prog", 2), 2) introTypes [HsTypeDecl noLoc "Block" [ ] (HsTyTuple [ HsTyApp (HsTyCon (UnQual (HsIdent "List"))) (HsTyCon (UnQual (HsIdent "Dec"))), HsTyApp (HsTyCon (UnQual (HsIdent "List"))) (HsTyCon (UnQual (HsIdent "Stat")))])] foldAlias (ConRef (HsIdent "Prog", 2) SelStop) ((HsIdent "Block"), [ ]) alias2newtype (HsIdent "Block") (HsIdent "Block") newtype2data (HsIdent "Block") ungroupConPos ((HsIdent "Block"), 1) ‘seqTrafo‘ ‘seqTrafo‘ ‘seqTrafo‘ ‘seqTrafo‘ ‘seqTrafo‘ Fig. 12. Script for the scenario in Fig. 11 data Maybe k data Maybe ′ k data Maybe ′ k a = Nothing \| Just a k k k k a = Nothing ′ \| Just ′ a k k k k a = Nothing ′ \| Just ′ a k k k data ConsList a = Nil (Maybe ′ a) k k \| Cons a (ConsList a) Fig. 13. Illustration of the generalisation of Maybe to ConsList In Fig. 11, we show the steps that implement the introductory example. As one can see, we basically implement extraction, but extra steps deal with grouping and ungrouping the two components subject to extraction. Also, the extracted type should be a proper datatype as opposed to a type alias (see transition from 3. to 4.). For completeness’ sake, the transformation script is shown in Fig. 12. The script precisely captures the steps that underly the interactive transformation in Fig. 1. Some of the operators are not completely structure-preserving, that is, strictly speaking, the structures of the datatypes before and after transformation are not fully equivalent. For example, a newtype and a datatype are semantically distinguished, even if the defining constructor declaration is the very same. (This is because a constructor of a datatype involves an extra lifting step in the semantical domain, i.e., there is an extra ‘bottom’ element.) The operators for grouping and ungrouping also deviate from full structure preservation. 3.5 Swapping types on use sites We will now deal with transformations that eliminate or establish type distinctions by what we call swapping types on use sites. In Fig. 13, we illustrate a typical application of swapping. In the example, we want to generalise the standard datatype Maybe to allow for lists instead. In fact, we do not want to change the general definition of the library datatype Maybe, but we only want to change it on one use site (not shown in the figure). This is where swapping helps: as an intermediate step, we can replace Maybe on the use site by a newly introduced datatype Maybe ′ with equivalent structure. The figure illustrates how subsequent adaptations derive 10 Kort & Lämmel type DataNames type DataUnifier = = (TypeId , [ConId ]) (DataNames , DataNames) swapAlias swapData :: :: TypeSel → TypeId → TypeId → Trafo TypeSel → [DataUnifier ] → Trafo Fig. 14. Operators for swapping types on use sites type ConDecl data HsType = (ConId , [HsType ]) = ... :: :: includeConDecl excludeConDecl -- Constructor declaration -- Syntactical sort for type expressions TypeId → ConDecl → Trafo ConId → Trafo Fig. 15. Operators for inclusion and exclusion of constructor declarations Syntax as of Fig. 11 data Prog data Block data Dec data Stat data Expr = = = = = Prog ProgName Block Block [Dec ] [Stat ] VDec Id Type Assign Id Expr \| If Expr Stat Stat Var Id \| Const Int After syntax extension by statement blocks data Stat = Assign Id Expr \| If Expr Stat Stat \| SBlock Block Fig. 16. Illustration of constructor inclusion the ConsList datatype from the clone of the Maybe datatype. In particular, we add the boxed constructor component. The swapping operators are declared in Fig. 14. There is one operator for type aliases and another for datatype declarations. In the case of proper datatypes, one needs to match the constructors in addition to just the names of the types. This is modelled by the helper datatype DataUnifier. The type of the operator swapData clarifies that we are prepared to process a list of DataUnifiers. This is necessary if we want to swap mutually recursive systems of datatypes. 3.6 Inclusion vs. exclusion We now leave the ground of structure-preserving transformations. That is, we will consider transformations where input and output datatypes are not structurally equivalent. In fact, we consider certain ways to extend or reduce the structure of the datatype. The first couple of structure-extending and -reducing transformations is about inclusion and exclusion of constructor declarations (see Fig. 15). These operators are only feasible for proper datatypes and not for type aliases or newtypes. (This is because a type alias involves no constructor at all, and a newtype is defined in terms of precisely one constructor declaration.) In Fig. 16, we show an example for constructor inclusion. In fact, we just continue the introductory example to make use of the extracted block structure in a language extension for statement blocks. That is, we include a constructor application for Stat to capture Block as another statement form. This continuation of the 11 Kort & Lämmel insertConComp deleteConComp :: :: ConPos → HsType → Trafo ConPos → Trafo Fig. 17. Operators for insertion and deletion of constructor components A datatype for a transition relation / function, and helpers type TransRel a = a → Maybe a data Maybe a = Nothing \| Just a data ConsList a = Nil \| Cons a (ConsList a) Introduction of a substitute for Maybe data Maybe ′ a = Nothing ′ \| Just ′ a Swapping Maybe and Maybe’ in TransRel type TransRel a = a → Maybe ′ a Extension of Maybe ′ to fit with shape of ConsList data Maybe ′ a = Nothing ′ \| Just ′ a (Maybe ′ a) Swapping Maybe ′ and ConsList in TransRel type TransRel a = a → ConsList a Fig. 18. Illustration of component insertion and type swapping introductory example amplifies the intended use of our operator suite: for program evolution in the sense of datatype refactoring and adaptation. 3.7 Insertion vs. deletion Inclusion and exclusion of constructor declarations is about the branching structure of datatypes. We will now discuss operators that serve for the insertion or deletion of constructor components (see Fig. 17). Insertion of a component c into a constructor declaration C c1 · · · cn proceeds as follows. Given the target position for the new component, be it i 6 n + 1, the new constructor declaration is simply of the form C c1 · · · ci−1 c ci · · · cn . In general, c might need to refer to type parameters of the affected datatype. Deletion of a constructor declaration relies on the identification of the obsolete component. In Fig. 18, we elaborate on the earlier example for generalising ‘maybies’ to lists (recall Fig. 13). At the top of Fig. 18, we see three datatypes TransRel, Maybe, and ConsList. The idea is indeed to replace Maybe by ConsList in the using occurrence in TransRel. (That is, we want to allow for a function from a to a list of as instead of a partial function from a to a.) We call this adaptation a generalisation because a list is more general than an optional. In the initial phase of the generalisation of Maybe, we disconnect the relevant occurrence of Maybe in TransRel from other possible occurrences in the program. So we introduce a copy Maybe ′ of Maybe, and we perform type swapping so that TransRel refers to Maybe ′ instead of the ‘read-only’ Maybe. Now we need to make Maybe ′ structurally equivalent to ConsList. This amounts to adding a recursive component to the second constructor Just ′ . Then, we can again swap types to refer to ConsList in the co-domain of TransRel. 12 Kort & Lämmel 4 Datatype transformation meets program transformation We will now re-iterate over the groups of operators to investigate their impact on functional programs. It would be utterly complex to formalise the link between datatype and program transformation. The mere specification of the transformations is already intractable for a publication because of its size, and the number of details. So we will describe the implied program transformations informally while omitting less interesting details. 4.1 Renaming Type names only occur inside type declarations and type annotations. So there is no need to adapt expressions or function declarations except for their signatures, or the type annotations of expressions. Constructor names can very well occur inside patterns and expressions that contribute to function declarations. Renaming these occurrences is completely straightforward. 4.2 Permutation The permutation of type parameters does not necessitate any completion at the level of function declarations. The permutation of constructor components, however, needs to be realized in patterns and expressions as well. This is particularly simple for pattern-match cases because all components are matched by definition. Hence, we can directly permute the sub-patterns in an affected constructor pattern. Witnessing permutations of constructor components in expression forms is slightly complicated by currying and higher-order style. Instead of permuting components in possibly incomplete constructor applications, we could first get access to all components by ‘λ-pumping’: given a constructor C with say n potential components according to its declaration, we first replace C by λx1 · · · xn . C x1 · · · xn as justified by η-conversion. Then, we witness the permutation by permuting the arguments x1 , . . . , xn in the pumped-up expression. In the presence of a nonstrict language with an evaluation order on patterns, the permutation of constructor components might actually change the behaviour of the program regarding termination. We neglect this problem. We should also mention that it is debatable if the described kind of η-conversion is really what the programmer wants because it obscures the code. 4.3 Introduction vs. elimination Introduction does not place any obligations on the functions defined in the same program. In the case of elimination, we have to ensure that the relevant types are not used by any function. If we assume that all function declarations are annotated by programmer-supplied or inferred signatures, then the precondition for elimination can be checked by looking at these signatures. There is an alternative approach that does not rely on complete type annotations: we check that no constructor of the relevant types is used. 13 Kort & Lämmel 4.4 Folding vs. unfolding The restriction of folding and unfolding to type aliases guarantees that these operators do not necessitate any adaptation of the function declarations. This is simply because interchanging a type alias and its definition is completely structure- and semantics-preserving, by definition. This is extremely convenient: despite the crucial role of the operators for folding and unfolding, they do not raise any issue at the level of function declarations. 4.5 Wrapping vs. unwrapping Grouping and ungrouping These operators are handled using the same overall approach as advocated for the permutation of constructor components. That is, in patterns we witness grouping or ungrouping by inserting or removing the enclosing “( . . . )”; in expressions, we perform η-conversion to access the relevant components, and then we group or ungroup them in the pumped-up constructor application. Mediation between newtypes and datatypes These datatype transformations do not imply any adaptations of the functions that involve the datatype in question. (As we indicated earlier, the extra bottom value of a datatype, when compared to a newtype, allows a program to be ‘undefined’ in one more way.) Newtype to alias migration We simply remove all occurrences of the associated constructor both in pattern and expression forms. We require that the relevant newtype is not covered by any instance declaration of some type class or constructor class. Otherwise, we had to inline these members in a non-obvious way prior to the removal of the constructor. If we neglected this issue, the resulting program either becomes untypeable, or a different instance is applied accidentally, which would be hazardous regarding semantics preservation. Alias to newtype migration This operator requires a non-trivial treatment for function declarations. The crucial issue is how to know the following: • What expressions have to be wrapped with the newtype constructor? • In what patterns does the newtype constructor need to be stripped? Our approach is as simple as possible. We observe that the new newtype might be used in the declarations of other datatypes. The corresponding patterns and expressions can be easily located and adapted as in the case of permutation, grouping, and ungrouping (recall η-conversion etc.). We also need to adapt function declarations if their argument or result types are known to refer to the relevant alias. This basically means that we need to access the affected arguments and result expressions in all relevant equations to unwrap the arguments and wrap the result expressions. These adaptations are slightly complicated by the fact that the affected type alias can occur in arbitrarily nested locations. In Fig. 19, we illustrate the effect of the alias2newtype operator in the introductory example. We show the top-level interpreter function that maps over the statements 14 Kort & Lämmel Top-level interpreter function before the illustrative extraction run :: Prog → State () run (Prog name decs stats) = mapM interpret stats The same function after extraction run :: Prog → State () run (Prog name (Block decs stats) ) = mapM interpret stats Fig. 19. Function adaptation triggered by alias-to-newtype migration Input program type TransRel a = a → Maybe a data Maybe ′ a = Nothing \| Just a deadEnd :: TransRel a → a → Bool deadEnd r a = case r a of Nothing → True Just → False Output program type TransRel a = a → Maybe ′ a data Maybe ′ a = Nothing \| Just a deadEnd :: TransRel a → a → Bool deadEnd r a = case toMaybe (r a) of Nothing → True Just → False Induced helper for type swapping toMaybe :: Maybe ′ a → Maybe a toMaybe Nothing ′ = Nothing toMaybe (Just ′ a) = Just a Fig. 20. Function adaptation triggered by type swapping of the program. (The program name and the declarations do not carry any semantics here.) The type of the function run exhibits that the meaning of a program is a computation that involves a State for the program variables. The adapted version of run refers to the extra constructor Block, which resulted from extraction. 4.6 Swapping types on use sites This operator relies on the same techniques as alias2newtype. However, instead of wrapping and unwrapping a constructor. We invoke conversion functions that mediate between the two structurally equivalent types. These mediators merely map old to new constructors and vice versa, and hence they are immediately induced by the datatype transformation itself, namely by the DataUnifiers passed to the swap operator. This approach implies that we only perform very local changes. The program code will still work on the old datatypes thanks to the mediators. The impact of swapping types at the function level is illustrated in Fig. 20. We deal with the initial steps of the Maybe-to-ConsList migration in Fig. 18, where we replace the occurrence of Maybe within TransRel by a structurally equivalent Maybe ′ . We show an illustrative function deadEnd which performs a test if the given transition relation allows for a transition in the presence of a given state a. The adapted function deadEnd refers to the conversion function toMaybe prior to performing pattern matching on the obsolete Maybe type. 15 Kort & Lämmel Input program data Stat = Assign Id Expr \| If Expr Stat Stat interpret :: Stat → State () interpret (Assign i e) = envLookup i > >= λr → ... interpret (If e s1 s2 ) = reval e > >= λv → ... Output program data Stat = Assign Id Expr \| If Expr Stat Stat \| SBlock Block interpret :: Stat → State () interpret (Assign i e) = envLookup i > >= λr → ... interpret (If e s1 s2 ) = reval e > >= λv → ... interpret ( SBlock ) = ⊥ Fig. 21. Inclusion of a constructor declaration 4.7 Inclusion vs. exclusion Intuitively, the inclusion of a constructor should be complemented by the extension of all relevant case discriminations. This normally means to add a pattern-match equation (or a case to a case expression) for the new constructor. Dually, exclusion of a constructor should be complemented by the removal of all pattern-match equations (or cases) that refer to this constructor. In the case of added pattern-match equations, we view the right-hand sides of these equations as a kind of ‘hot spot’ to be resolved by subsequent expression-level transformations. To this end, we use “undefined”, i.e., “⊥”, as a kind of to-do marker. Dually, in the case of removed constructors, we also need to replace occurrences of the constructor within expressions by “⊥”. When using interactive tool support, these to-do markers are useful to control further steps in a transformation scenario. In Fig. 21, we progress with our running example of an interpreter for an imperative language. We illustrate the step where blocks are turned into another form of statements. Hence, the shown output program involves a new pattern-match equation that interprets statement blocks. This added equation reflects that the meaning of such blocks is as yet undefined, subject to subsequent adaptations. 4.8 Insertion vs. deletion Inserting a component into a declaration for a constructor C means that all patterns with C as outermost constructor must be adapted to neglect the added component, and all applications of C must be completed to include “⊥” for the added component. Dually, deletion of a component from C means that all applications of C and all patterns with C as outermost constructor need to be cleaned up to project away the obsolete component. Any reference to a pattern variable for the obsolete component is replaced by “⊥”. As in the case of permutation and others, η-conversion is needed to actually get access to constructor components in expressions. In Fig. 22, the insertion of a constructor component is illustrated by continuing the scenario from Fig. 20. The adapted equation of toMaybe involves an extended pattern. As the don’t care pattern “ ” indicates, the definition of toMaybe does not make use of the added component. In fact, the definition of the function deadEnd does not need to be adapted; it only tests for the availability of a transition step. 16 Kort & Lämmel Output program type TransRel a = a → Maybe ′ a data Maybe ′ a = Nothing ′ \| Just ′ a (Maybe ′ a) deadEnd :: TransRel a → a → Bool deadEnd r a = case toMaybe (r a) of Nothing → True Just → False Induced helper for type swapping toMaybe :: Maybe ′ a → Maybe a toMaybe Nothing ′ = Nothing toMaybe ( Just ′ a ) = Just a Fig. 22. Illustration of the insertion of a constructor component Normally, other functions will start to rely on the richer pattern. 5 Related work Transformational program development Formal program transformation [BD77] separates two concerns: the development of an initial, maybe inefficient program the correctness of which can easily be shown, and the stepwise derivation of a better implementation in a semantics-preserving manner. Partsch’s textbook [Par90] describes the formal approach to this kind of software development. Pettorossi and Proietti study typical transformation rules (for functional and logic) programs in [PP96]. Formal program transformation, in part, also addresses datatype transformation [dRE98], say data refinement. Here, one gives different axiomatisations or implementations of an abstract datatype which are then related by well-founded transformation steps. This typically involves some amount of mathematical program calculation. By contrast, we deliberately focus on the more syntactical transformations that a programmer uses anyway to adapt evolving programs. Database schema evolution There is a large body of research addressing the related problem of database schema evolution [BKKK87] as relevant, for example, in database re- and reverse engineering [HTJC93]. The schema transformations themselves can be compared with our datatype transformations only at a superficial level because of the different formalisms involved. There exist formal frameworks for the definition of schema transformations and various formalisms have been investigated [MP97]. An interesting aspect of database schema evolution is that schema evolution necessitates a database instance mapping [BCN92]. Compare this with the evolution of the datatypes in a functional program. Here, the main concern is to update the function declarations for compliance with the new datatypes. It seems that the instance mapping problem is a special case of the program update problem. Refactoring The transformational approach to program evolution is nowadays called refactoring [Opd92,Fow99], but the idea is not new [ABFP86,GN90]. Refactoring means to improve the structure of code so that it becomes more comprehensible, maintainable, and adaptable. Interactive refactoring tools are being studied and used extensively in the object-oriented programming context [Moo96,RBJ97]. Typical examples of functional program refactorings are described in [Läm00], e.g., the introduction of a monad in a non-monadic program. The precise inhabitation of 17 Kort & Lämmel the refactoring notion for functional programming is being addressed in a project at the University of Kent by Thompson and Reinke; see [TR01]. There is also related work on type-safe meta-programming in a functional context, e.g., by Erwig [ER02]. Previous work did not specifically address datatype transformations. The refactorings for object-oriented class structures are not directly applicable because of the different structure and semantics of classes vs. algebraic datatypes. Structure editing Support for interactive transformations can be seen as a sophistication of structure editing [RT88,Koo94,KS98]. This link between transformation and editing is particularly appealing for our “syntactical” transformations. Not surprisingly, concepts that were developed for structure editing are related to our work. For example, in [SdM99], primitives of structure editing are identified based on the notion of focus to select subtrees, and on navigation primitives left, right, up and down. Trees, subtrees and paths are here defined as follows: data Tree type SubTree type Path type Layer = = = = Fork Label [Tree ] (Path, Tree) [Layer ] (Label , [Tree ], [Tree ]) The t in a subtree (p, t) is the currently selected tree and it is between the left and right trees in the top layer (the head of the p). This approach does not account for the heterogeneous character of language syntaxes, but it shows that the fact if a focus resides in a term can be encoded in types. 6 Concluding remarks Contribution We identified the fundamental primitives for datatype transformation. These operators are meant to support common scenarios of program adaptation in functional programming, or other settings where algebraic datatypes play a role. In fact, all the identified operators are universal in the sense, that they are also meaningful for other program abstractions than just datatypes, e.g., function declarations. We deliberately focused on adaptations of datatypes because a vast body of previous work addressed fold/unfold transformations for recursive functions. Despite the focus on datatype transformations, we had to consider program transformations that are necessitated by the modification of datatypes. Regarding the executable specification of the operator suite, we adhered to the formula: metaprograms = object-programs = Haskell programs. We employed generic functional programming in the interest of conciseness. We also employed designated means of referring to fragments of interest, e.g., a focus concept. Partial project failure We are confident that the identified operators are sufficient and appropriate for actual datatype transformations. We have attempted to complement this framework development by actual interactive tool support. We initially thought that using Haskell for this interactive tooling as well would be a good idea. Since the actual transformation operators are implemented in Haskell anyway, and the interactive dialogues need to cooperate with the operator framework to perform analyses, Haskell indeed seems to be the obvious choice. To make a long story 18 Kort & Lämmel short, there are many GUI libraries for Haskell, but none of them is suitable for developing a sophisticated GUI for interactive program transformation at the moment. It seems that environments for interactive language tools would provide a better starting point, e.g., environments based on attribute grammars [RT88,KS98]. Perspective To cover full Haskell, a few further operators would have to be added to our suite, in particular, operators that support type and constructor classes. We should also pay full attention to some idiosyncrasies of Haskell; cf. refutable vs. irrefutable patterns. Then, there are also transformation techniques that seem to go beyond our notion of program evolution but it is interesting to cover them anyway. We think of techniques like turning a system of datatypes into functorial style, or threading a parameter through a system of datatypes. The ultimate perspective for the presented work is to integrate the datatype transformations into a complete, well-founded, and user-friendly refactoring tool for functional programming along the lines of Thompson’s and Reinke’s research project [TR01]. Another perspective for our research is to further pursue the intertwined character of datatype and program transformations in the context of XML format and API evolution. References [ABFP86] G. Arango, I. Baxter, P. Freeman, and C. Pidgeon. TMM: Software maintenance by transformation. IEEE Software, 3(3):27–39, May 1986. [BCN92] C. Batini, S. Ceri, and S.B. Navathe. Conceptual database design. Benjamin/Cummings, Redwood City, US, 1992. [BD77] R. M. Burstall and John Darlington. A transformation system for developing recursive programs. Journal of the ACM, 24(1):44–67, January 1977. [BKKK87] J. Banerjee, W. Kim, H.-J. Kim, and H.F. Korth. Semantics and Implementation of Schema Evolution in Object-Oriented Databases. SIGMOD Record (Proc. Conf. on Management of Data), 16(3):311–322, May 1987. [dRE98] Willem-Paul de Roever and Kai Engelhardt. Data Refinement: Model-Oriented Proof Methods and their Comparison, volume 47 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1998. [ER02] M. Erwig and D. Ren. A rule-based language for programming software updates. In Proceedings of the 2002 ACM SIGPLAN workshop on Rule-based programming, pages 67–78. ACM Press, 2002. [Fow99] M. Fowler. Refactoring—Improving the Design of Existing Code. Addison Wesley, 1999. [GN90] W. G. Griswold and D. Notkin. Program restructuring as an aid to software maintenance. Technical report, Seattle, WA, USA, August 1990. [HTJC93] J.-L. Hainaut, C. Tonneau, M. Joris, and M. Chandelon. Schema Transformation Techniques for Database Reverse Engineering. In Proc. of the 12th Int. Conf. on ER Approach, Arlington-Dallas, 1993. E/R Institute. 19 Kort & Lämmel [Koo94] J.W.C. Koorn. Generating uniform user-interfaces for interactive programming environments. PhD thesis, University of Amsterdam, 1994. [KS98] M. Kuiper and J. Saraiva. Lrc — A generator for Incremental LanguageOriented Tools. In K. Koskimies, editor, Compiler Construction CC’98, volume 1383 of LNCS, pages 298–301. Springer-Verlag, April 1998. Tool demonstration. [Läm00] R. Lämmel. Reuse by Program Transformation. In Greg Michaelson and Phil Trinder, editors, Functional Programming Trends 1999, pages 143–152. Intellect, 2000. [Läm01] R. Lämmel. Grammar Adaptation. In J.N. Oliveira and P. Zave, editors, Proc. Formal Methods Europe (FME) 2001, volume 2021 of LNCS, pages 550–570. Springer-Verlag, 2001. [Moo96] I. Moore. Automatic Inheritance Hierarchy Restructuring and Method Refactoring. In OOPSLA ’96 Conference Proceedings: Object-Oriented Programming Systems, Languages, and Applications, pages 235–250. ACM Press, 1996. [MP97] P. McBrien and A. Poulovassilis. A Formal Framework for ER Schema Transformation. In D.W. Embley and R.C. Goldstein, editors, Conceptual Modeling - ER ’97, 16th International Conference on Conceptual Modeling, Los Angeles, California, USA, November 3-5, 1997, Proc., volume 1331 of LNCS, pages 408–421. Springer-Verlag, 1997. [Opd92] W. F. Opdyke. Refactoring Object-Oriented Frameworks. University of Illinois at Urbana-Champaign, 1992. PhD thesis, [Par90] H.A. Partsch. Specification and Transformation of Programs. Springer-Verlag, 1990. [PP96] A. Pettorossi and M. Proietti. Rules and Strategies for Transforming Functional and Logic Programs. ACM Computing Surveys, 28(2):360–414, June 1996. [RBJ97] D. Roberts, J. Brant, and R.E. Johnson. A Refactoring Tool for Smalltalk. Theory and Practice of Object Systems (TAPOS), 3(4):253–263, 1997. [RT88] T.W. Reps and T. Teitelbaum. The Synthesizer Generator: A System for Constructing Language–Based Editors. Springer–Verlag, 1988. [SdM99] B.A. Sufrin and O. de Moor. Modeless structure editing. In A.W. Roscoe and J.C.P. Woodcock, editors, Proceedings of the Oxford-Microsoft symposium in Celebration of the work of Tony Hoare, September 1999. [TR01] S. Thompson and C. Reinke. Refactoring Functional Programs. Technical Report 16-01, Computing Laboratory, University of Kent at Canterbury, October 2001. Also see http://www.cs.ukc.ac.uk/people/ staff/sjt/Refactor/. 20	6
On the Capacity of a Class of Signal-Dependent Noise Channels Hamid Ghourchian, Gholamali Aminian, Amin Gohari, Mahtab Mirmohseni, and Masoumeh Nasiri-Kenari, arXiv:1702.03590v2 [cs.IT] 2 Jun 2017 Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran. E-mails: {h ghourchian, aminian}@ee.sharif.edu, {aminzadeh, mirmohseni, mnasiri}@sharif.edu ∗ Abstract In some applications, the variance of additive measurement noise depends on the signal that we aim to measure. For instance, additive Gaussian signal-dependent noise (AGSDN) channel models are used in molecular and optical communication. Herein we provide lower and upper bounds on the capacity of additive signal-dependent noise (ASDN) channels. The idea of the first lower bound is the extension of the majorization inequality, and for the second one, it uses some calculations based on the fact that h (Y ) > h (Y \|Z). Both of them are valid for all additive signal-dependent noise (ASDN) channels defined in the paper. The upper bound is based on a previous idea of the authors (“symmetric relative entropy”) and is used for the additive Gaussian signal-dependent noise (AGSDN) channels. These bounds indicate that in ASDN channels (unlike the classical AWGN channels), the capacity does not necessarily become larger by making the variance function of the noise smaller. We also provide sufficient conditions under which the capacity becomes infinity. This is complemented by a number of conditions that imply capacity is finite and a unique capacity achieving measure exists (in the sense of the output measure). Keywords: Signal-dependent noise channels, molecular communication, channels with infinite capacity, existence of capacity-achieving distribution. 1 Introduction An additive Gaussian signal-dependent noise (AGSDN) channel with input x and output y is defined by −(y−x)2 1 e 2σ(x)2 , fY \|X (y\|x) = p 2πσ(x)2 where σ(·) is a given function from R to [0, ∞). Alternatively, we may describe the AGSDN channel by Y = X + σ(X) · Z where Z ∼ N (0, 1) is a standard Gaussian random variable and independent of the input X. For constant function σ(x) = c, the AGSDN channel reduces to a simple additive Gaussian channel. More generally, we may relax the Gaussian assumption on Z and consider an additive signal-dependent noise (ASDN) channel defined by Y = X + σ(X) · Z, (1) where noise Z is assumed to be a continuous random variable with a given pdf fZ (z), and be independent of the input X.1 For instance, one can consider an ASDN with Z being a truncated ∗ This work was supported by INSF Research Grant on “Nano-Network Communications”. The first two authors contributed equally to this work. 1 See Definition 3 for the definition of continuous random variables. 1 version of the Gaussian distribution as a better model in an application if we know that the output Y has minimum and maximum values in that applications. Below we provide a number of applications in which the ASDN channel arises. 1. The AGSDN channel appears in optical pcommunications when modeling the shot noise or the optical amplification noise for σ(x) = c20 + c21 x [1]. √ 2. In molecular communication, the AGSDN channel with σ(x) = c x arises in the ligand receptor model, the particle sampling noise, the particle counting noise and the Poisson model for an absorbing receiver [2, 3, 4]. In all cases, the reason for appearance of a Gaussian signaldependent noise is the approximation of a binomial or Poisson distribution with a Gaussian distribution. Observe that the mean and variance of a binomial distribution with parameters (n, p) relate to each other: the mean is np and the variance is np(1 − p) respectively. As a result, the mean and variance of the approximated Gaussian distribution also relate to each other (see [5, Section II.B] for a detailed overview). 3. Besides the above applications of ASDN in molecular communications we shall provide two other cases where this channel model is helpful: Consider the Brownian motion of a particle with no drift over a nonhomogeneous medium with σ(x) denoting the diffusion coefficient of the medium at location x. The diffusion coefficient σ(x) describes the movement variance of a particle when in location x. More specifically, the motion of the particle is described by the stochastic differential equation dXt = σ(Xt ) dBt , where Bt is the standard Wiener process (standard Brownian motion). Alternatively, we can express the above equation using the following Itô integral Z t+s Xt+s − Xt = σ(Xu ) dBu . (2) t Let us denote the position of the particle at time 0 by X = X0 , and its position after t seconds by Y = Xt . If t is a small and fixed number, (2) reduces to Y = X + tσ(X) · Z, where Z ∼ N (0, 1). Thus, the movement of the particle follows an AGSDN channel law if t is small. 4. As another example, consider the molecular timing channel in a time-varying medium. In a molecular timing channel, information is encoded in the release time of molecules. A molecule released at time X hits the receiver after a delay Z at time Y = X + Z. Molecules are absorbed once they hit the receiver. As such, the distribution of Z is that of the first arrival time. The existing literature only studies this problem when the medium is time-invariant (see [6, 7, 8, 9]): if the medium is uniform, time-invariant and one-dimensional, Z is distributed according to the inverse Gaussian distribution (if there is a flow in the medium) or the Lévy distribution (if there is no flow in the medium). As a result, the channel is called the additive inverse Gaussian noise additive channel, or the additive Lévy noise in the literature. However, in a time-varying medium (or when the distance between the transmitter and receiver varies over time), the distribution of Z depends on the release time X. As a result, we obtain a signal-dependent noise additive component. For instance, the additive noise can have a Lévy 2 distribution with a scale parameter that depends on input X. Using the scaling property of the Lévy distribution, we can express this as σ(X) · Z where Z is the standard Lévy distribution, and σ(X) is the scale parameter. This would be an ASDN channel. 5. In the third item, we discussed Brownian motion after a small time elapse. A Brownian motion with no drift is an example of a martingale. Now let us consider a martingale after a large time elapse. Here, the AGSDN channel also arises as a conditional distribution in any process that can be modeled by a discrete time martingale with bounded increments. Assume that X0 , X1 , X2 , · · · is such a martingale. Then E [Xn ] = E [X0 ]. Furthermore, by the martingale central limit theorem, the conditional distribution of Xn given X0 = x for large values of n can be approximated by a Gaussian distribution with mean X0 = x and a variance σn (x) that depends on X0 = x. 6. Finally, we relate the ASDN channel to real fading channels with a direct line of sight. Consider a scalar Gaussian fading channel Y = X + HX + N, (3) where X is the input, H ∼ N (0, c1 ) is the Gaussian fading coefficient and N ∼ N (0, c0 ) is the additive environment noise. The first X term on the right-hand side of (3) corresponds to the direct line of sight, while the HX term is the fading term. The distribution of Y given X = x is N (x, c1 x2 + c0 ). Thus (3) can be expressed as Y = X + σ(X) · Z where p σ(x) = c1 x2 + c0 , Z ∼ N (0, 1). A fast fading setting in which H varies independently over each channel use corresponds to a memoryless ASDN channel. The purpose of this paper is to study the capacity of a memoryless additive signal-dependent noise (ASDN) channel defined via Y = X + σ(X) · Z, under input cost constraints. The memoryless assumption implies that the noise Z is drawn independently from fZ (z) in each channel use. Related works: In [10], vector AGSDN channels subject cost constraints are studied. It is shown that under some assumptions, p the capacity achieving distribution is a discrete distribution. The AGSDN channel with σ(x) = c20 + c21 x is investigated in [1] wherein capacity upper and lower bounds are derived considering peak and average constraints. Note that the memoryless AGSDN includes the additive white Gaussian noise (AWGN) channel as its special case. The capacity of AWGN channel under power constraint is classical and is obtained by an input of Gaussian random variable. Its capacity under both average and peak power constraints is quite different, as the capacity achieving input distribution is discrete with a finite number of mass points [11]. See [12, 13] for further results on the capacity of the AWGN channel with both average and peak power constraints. Our contributions: Our contributions in this work can be summarized as follows: • We provide a new tool for bounding the capacity of continuous input/output channels. Note that I (X; Y ) = h (Y ) − h (Y \|X) . 3 We provide two sufficient conditions under which h (Y ) ≥ h (X), which results in I (X; Y ) ≥ h (X) − h (Y \|X) , and leads to lower bounds on the channel capacity of an ASDN channel. • It is known that increasing the noise variance of an AWGN channel decreases its capacity. However, we show that this is no longer the case for signal-dependent noise channels: the constraint σ1 (x) ≥ σ2 (x) for all x does not necessarily imply that the capacity of an AGSDN channel with σ1 (x) is less than or equal to the capacity of an AGSDN with σ2 (x). • We identify conditions under which the capacity of the ASDN channel becomes infinity. In particular, this implies that the capacity of a AGSDN channel with p σ(x) = c1 x2 + c0 tends to infinity as c0 tends to zero. Thus, the capacity of the real Gaussian fast fading channel given earlier in this section tends to infinity as c0 tends to zero. This parallels a similar result given in [14] for complex Gaussian fading channels. • We provide a new upper bound for the AGSDN channel based on the KL symmetrized upper bound of [15]. This upper bound is suitable for the low SNR regime, when σ(x) is large. This is p in contrast with the upper bound of [1, Theorems 4, 5] for AGSDN channels with σ(x) = c20 + c21 x which is suitable for large values of peak and average constraints. Furthermore, we give ourp upper bound for a large class of functions σ(x) while the technique of [1] is tuned for σ(x) = c20 + c21 x. This paper is organized as follows. Section 2 includes some of primary definitions and notations. In Section 3, our main results are given. This includes two lower bounds and one upper bound on the capacity of the ASDN channel. There are some useful lemmas in Section 4 used in the paper. The numerical results and plots are given in Section 5. The proofs of our results are given in Section 6. 2 Definitions and Notations In this section we review the definitions of continuous and discrete random variables, as well as entropy and differential entropy, relative entropy and mutual information. Throughout this paper all the logarithms are in base e. Random variables are denoted by capital letters, and probability measure functions are denoted by letter µ. The collection of Borel measurable sets in R is denoted by B(R). We sometimes use a.e. and µ-a.e. as a short-hand for “almost everywhere” and “µ-almost everywhere”, respectively. The set A is µ-a.e., when Z dµ = 0. A The set A is a.e. if it is µ-a.e. when µ is the Lebesgue measure. Definition 1 (Relative Entropy). [16, Section 1.4] For random variables X and Y with probability measures µX and µY , the relative entropy between X and Y is defined as follows: i ( h X E log dµ (X) µX µY dµY D (µX kµY ) = D (XkY ) := , +∞ o.w. 4 X where dµ dµY is the Radon-Nikodym derivative and µX µY means µX is absolutely continuous w.r.t. µY i.e. µX (A) = 0 for all A ∈ B if µY (A) = 0, where B is the Borel σ-field of the space over which the measures are defined. Definition 2 (Mutual Information). [16, Section 1.6] For random variables X, Y with joint probability measure µX,Y , the mutual information between X and Y is defined as follows: I (X; Y ) = D (µX,Y kµX µY ) , where µX µY is the product measure defined as (µX µY )(A, C) = µX (A)µY (C), where A ∈ BX the Borel σ-field of the space over which µX is defined, and C ∈ BY the Borel σ-field of the space over which µY is defined. Similarly, for three random variable X, Y, Z with joint measure µX,Y,Z , conditional mutual information I (X; Y \|Z) is defined as I (X; Y, Z) − I (X; Z). Definition 3 (Continuous Random Variable). [10] Let X be a real-valued and random variable that is measurable with respect to B(R). We call X a continuous random variable if its probability measure µX , induced on (R, B), is absolutely continuous with respect to the Lebesgue measure for B(R) (i.e., µ(A) = 0 for all A ∈ B with zero Lebesgue measure). We denote the set of all absolutely continuous probability measures by AC. Note that the Radon-Nikodym theorem implies that for each random variable X with measure µX ∈ AC there exists a B(R)-measurable function fX : R → [0, ∞), such that for all A ∈ B(R) we have that Z µX (A) = Pr {X ∈ A} = fX (x) dx. (4) A The function fX is called the probability density function (pdf ) of X [16, p. 21]. We denote pdf of absolutely continuous probability measures by letter f . Definition 4 (Discrete Random Variable). [10] A random variable X is discrete if it takes values in a countable alphabet set X ⊂ R. Probability mass function (pmf) for discrete random variable X with probability measure µX is denoted by pX and defined as follows: pX (x) := µX ({x}) = Pr {X = x} , ∀x ∈ X . Definition 5 (Entropy and Differential Entropy). [17, Chapter 2] We define entropy H (X), for a discrete random variable X with measure µX and pmf pX as H (X) = H (µX ) = H (pX ) := X pX (x) log x if the summation converges. Observe that 1 H (X) = E log . pX (X) 5 1 , pX (x) For a continuous random variable X with measure µX and pdf fX , we define differential entropy h (X) as Z +∞ 1 fX (x) log h (X) = h (µX ) = h (pX ) := , f X (x) −∞ if the integral converges. Similarly, the differential entropy is the same as 1 . h (X) = E log fX (X) Similarly, for two random variables X, Y , with measure µX,Y , if for all x, µY \|X (·\|x) is absolutely discrete with pmf pY \|X (·\|x), the conditional entropy H (Y \|X) is defined as 1 H (Y \|X) = E log . pY \|X (Y \|X) Likewise, for two random variables X, Y , with measure µX,Y , if for all x, µY \|X (·\|x) is absolutely continuous with pdf fY \|X (·\|x), the conditional differential entropy h (Y \|X) is defined as h (Y \|X) = E log 1 . fY \|X (Y \|X) We allow for differential entropy to be +∞ or −∞ if the integral is convergent to +∞ or −∞, i.e., we say that h (X) = +∞, if and only if 1 dx = +∞, and fX (x) Z 1 fX (x) log dx converges to a finite number fX (x) A− Z fX (x) log A+ where A− = {x : fX (x) > 1}. A+ = {x : fX (x) ≤ 1}, Similarly, we define h (X) = −∞. When we write that h (X) > −∞, we mean that the differential entropy of X exists and is not equal to −∞. The following example, from [14], demonstrates the differential entropy can be +∞ or −∞. Example 1. Differential entropy becomes plus infinity for the following pdf defined over R [14]: ( 1 , x > e, x(log x)2 f (x) = 0, x ≤ e. On the other hand, as shown in [14], differential entropy is minus infinity for ( −1 0 < x < e−e , 2, g(x) = x log x(log(− log x)) 0, otherwise. 6 Definition 6 (Riemann integrable functions). Given −∞ ≤ ` < u ≤ +∞, in this work, we utilize Riemann integrable functions g : (`, u) 7→ R on open interval (`, u). Such functions satisfy the property that for any c ∈ (`, u), the function Z x h(x) = g(t) dt, c is well-defined. By the fundamental theorem of calculus, h(·) is continuous on (`, u) (but not necessarily differentiable unless g is continuous). As an example, consider the function g(x) = 1/x for x 6= 0, and g(0) = 0 otherwise. This function is Riemann integrable on the restricted domain (0, ∞), but not integrable on (−1, 1). 3 Main Results We are interested in the capacity of an ASDN channel with the input X taking values in a set X and satisfying the cost constraint E[gi (X)] ≤ 0, ∀i = 1, 2, · · · , k for some functions gi (·). The common power constraint corresponds to gi (x) = x2 − p for some p ≥ 0, but we allow for more general constraints. Then, given a density function fZ (z) for the noise Z and function σ(·), we consider the following optimization problem: C = sup I (X; Y ), (5) µX ∈F where X and Y are related via (1) and F = {µX supp(µX ) ⊆ X , E[gi (X)] ≤ 0 for all i = 1, · · · , k}. (6) We sometimes use supp(X) to denote the support of measure µX , supp(µX ), when the probability measure on X is clear from the context. As an example, if, in an application, input X satisfies ` ≤ X ≤ u, the set X can be taken to be [`, u] to reflect this fact; similarly, the constraint 0 < X ≤ u reduces to X = (0, u], and 0 ≤ ` ≤ \|X\| ≤ u reduces to X = [−u, −`] ∪ [`, u]. The rest of this section is organized as follows: in Section 3.1, we provide conditions that imply finiteness of the capacity of an ASDN channel. In Section 3.2, we review the ideas used for obtaining lower bounds in previous works and also in this work. Then, based on the new ideas introduced in this work, we provide two different lower bounds in Sections 3.3 and 3.4. Finally, in Section 3.5, we provide an upper bound for AGSDN channels. 3.1 Existence and Finiteness of Channel Capacity Theorem 1. Assume that an ASDN channel satisfies the following properties: • X is a closed and also bounded subset of R, i.e., there exists u ≥ 0 such that X ⊆ [−u, u]; • Real numbers 0 < σ` < σu exist such that σ` ≤ σ(x) ≤ σu for all x ∈ X ; • Positive real m and γ exist such that fZ (z) ≤ m < ∞ (a.e.), and E [\|Z\|γ ] = α < ∞; • The cost constraint functions gi (·) are bounded over X . 7 Then, the capacity of the ASDN channel is finite. Furthermore there is a capacity achieving probability measure; in other words, the capacity C can be expressed as a maximum rather than a supremum: C = max I (X; Y ). µX ∈F Moreover, the output distribution is unique, i.e. if µX1 and µX2 both achieves the capacity, then fY1 (y) = fY2 (y), ∀y ∈ R, where fY1 and fY2 are the pdfs of the output of the channel when the input probability measures are µX1 and µX2 , respectively. Remark 1. The above theorem is a generalization of that given in [10, Theorem 1] for the special case of Gaussian noise Z. The proof can be found in Section 6.1. To give a partial converse of the above theorem, consider the case that the second assumption of the above theorem fails, i.e., when there is a sequence {xi } of elements in X such that σ(xi ) converges to zero or infinity. The following theorem shows that input/output mutual information can be infinity in such cases. Theorem 2. Consider an ASDN channel with σ : X 7→ [0, +∞) where X is not necessarily a closed set. Suppose one can find a sequence {x̃i } of elements in X such that σ(x̃i ) converges to 0 or +∞ such that • As a sequence on real numbers, {x̃i } has a limit (possibly outside X ), which we denote by c. The limit c can be plus or minus infinity. • One can find another real number c0 6= c such that the open interval E = (c, c0 ) (or E = (c0 , c) depending on whether c0 > c or c0 < c) belongs to X . Furthermore, x̃i ∈ E, and σ(·) is monotone and continuous over E. 2 Then one can find a measure µX defined on E such that I (X; Y ) = ∞ provided that Z is a continuous random variable and has the following regularity conditions: \|h (Z) \| < ∞, ∃δ > 0 : Pr {Z > δ} , Pr {Z < −δ} > 0, Furthermore, there is more than one measure µX that makes I (X; Y ) = ∞. In fact, input X can be both a continuous or discrete random variable, i.e., one can find both an absolutely continuous measure with pdf fX and discrete pmf pX such that I (X; Y ) is infinity when the measure on input is either fX or pX . The proof can be found in Section 6.2 and uses some of the results that we prove later in the paper. Remark 2. As an example, consider an AGSDN channel with X = (0, u) for an arbitrary u > 0, and σ(x) = xα for α 6= 0. For this channel, we have C = +∞ if we have no input cost constraints. Setting α = 1, this shows that the capacity of the fast-fading channel given in (3) is infinity if c0 = 0; that is when there is no additive noise. This parallels a similar result given in [14] for complex Gaussian fading channels. 2 We only require monotonicity here, and not strictly monotonicity. 8 Remark 3. It is known that increasing the noise variance of an AWGN channel decreases its capacity. However, we show that this is no longer the case for signal-dependent noise channels: Consider two AGSDN channels with parameters σ1 (x) and σ2 (x), respectively, which are defined over X = (0, 1) with the following formulas: σ1 (x) = 1, σ2 (x) = 1 . x No input cost constraints are imposed. It is clear that σ2 (x) > σ1 (x) for all x ∈ X . However, by considering the constraint 0 < X < 1, from Theorem 1 we obtain that the capacity of the first channel is finite, while from Theorem 2, we obtain that the capacity of the second channel is ∞. Therefore, the constraint σ1 (x) > σ2 (x) for all x ∈ X does not necessarily imply that the capacity of an AGSDN channel with σ1 (x) is less than or equal to the capacity of an AGSDN with σ2 (x). 3.2 Lower Bounds on Capacity To compute capacity from (5), one has to take maximum over probability measures in a potentially large class F. Practically speaking, one can only find a finite number of measures µ1 , µ2 , · · · , µk in F and evaluate input/output mutual information for them. Ideally, {µi } should form an covering of the entire F (with an appropriate distance metric), so that mutual information at every arbitrary measure in F can be approximated with one of the measures µi . This can be computationally cumbersome, even for measures defined on a finite interval. As a result, it is desirable to find explicit lower bounds on the capacity. Observe that I (X; Y ) = h (Y ) − h (Y \|X). To compute the term h (Y \|X), observe that given X = x, we have Y = x + σ(x) · Z and thus h (Y \|X = x) = log σ(x) + h (Z) (see Lemma 2). Thus, h (Y \|X) = E [log σ(X)] + h (Z) . However, the term p h (Y ) is more challenging to handle. Authors in [1] consider an AGSDN channel with σ(x) = c20 + c21 x for x ≥ 0, as well as show that h (Y ) ≥ h (X) and hence I (X; Y ) ≥ h (X)−h (Y \|X). This implies that instead of maximizing I (X; Y ), one can maximize h (X)−h (Y \|X) to obtain a lower bound. The proof of the relation h (Y ) ≥ h (X) in [1] is non-trivial; we review it here to motivate our own techniques in this paper. First consider the special case of c1 = 0. In this case, we get σ(x) = c0 and the AGDSN reduces to AWGN channel Y = X +Z. In this special case, one obtains the desired equation by writing h (Y ) ≥ h (Y \|Z) = h (X + Z\|Z) = h (X\|Z) = h (X) . (7) p However, the above argument does not extend for the case of c1 > 0 since σ(x) = c20 + c21 x depends on x. As argued in [1], without loss of generality, one mayp assume that c0 = 0; this is because one can express a signal-dependent noise channel with σ(x) = c20 + c21 x as √ Y = X + c1 XZ1 + c0 Z0 , where Z0 and Z1 are √ independent standard normal variables. Thus, we can write Y = Y1 + c0 Z0 where Y1 = X + c1 XZ1 . From the argument for AWGN channels, we have that h (Y ) ≥ h (Y1 ). Thus, it suffices to show that h (Y1 ) ≥ h (X). This is the special case of the problem for c0 = 0 and √ corresponds to σ(x) = c1 x. 9 √ To show h (Y ) ≥ h (X) when Y = X + c1 XZ, more advanced ideas are utilized in [1]. The key observation is the following: assume that 1 x X ∼ gX (x) = e− α 1[x ≥ 0], α be exponentially distributed with mean E [X] = α. Then Y has density ! p √ αy − α + 2c21 \|y\| 1 √ 2 exp . gY (y) = p αc1 α(α + 2c21 ) Then, for any arbitrary input distribution fX , from the data processing property of the relative entropy, we have D (fY kgY ) ≤ D (fX kgX ) where fY is the output density for input density fX . Once simplified, this equation leads to h (fY ) ≥ h (fX ). The above argument crucially depends on the particular form of the output distribution corresponding to the input p exponential distribution. It is a specific argument that works for the specific choice of σ(x) = c20 + c21 x and normal distribution for Z, and cannot be readily extended to other choices of σ(·) and fZ (z). In this paper, we propose two approaches to handle more general settings: • (Idea 1:) We provide the following novel general lemma that establishes h (Y ) ≥ h (X) for a large class of ASDN channels. Lemma 1. Take an arbitrary channel characterized by the conditional pdf fY \|X (·\|x) satisfying Z fY \|X (y\|x) dx ≤ 1, ∀y ∈ Y, (8) X where X and Y are the support of channel input X and channel output Y , respectively. Take an arbitrary input pdf fX (x) on X resulting in an output pdf fY (y) on Y. Assuming that h (X) and h (Y ) exist, we have h (Y ) ≥ h (X) The proof is provided in Section 6.7. As an example, Lemma 1 yields an alternative proof for the result of [1] for an AGSDN p channel. Note that, as we mentioned before, in order to prove that h (Y ) ≥ h (X) for σ(x) = c0 2 + c2 x, √ we only need to prove it for σ(x) = c x. To this end, observe that since X ⊆ [0, +∞), we have Z Z ∞ (y−x)2 1 √ fY \|X (y\|x) dx ≤ e− 2c2 x dx 2πc2 x X 0 Z ∞ √ 2 )2 2 −(y−v √ = e 2c2 v2 dv πc2 0 1 y≥0 2y = ≤ 1. (9) e c21 y < 0 where x = v 2 , and v ≥ 0. The proof for equation (9) is given in Appendix A. • (Idea 2:) We provide a variation of the type of argument given in (7) by introducing a number of new steps. This would adapt the argument to ASDN channels. In the following sections, we discuss the above two ideas separately. 10 3.3 First Idea for Lower Bound Theorem 3. Assume an ASDN channel defined in (1), where σ : (`, u) 7→ (0, +∞) with −∞ ≤ ` < u ≤ +∞, and noise with pdf fZ (z) such that Z u 1 y−x dx ≤ 1, ∀y, (10) fZ σ(x) ` σ(x) 1 is Riemann integrable on (`, u). σ(x) (11) Then, if X is continuous random variable with pdf fX (x) supported over (`, u), I (X; Y ) ≥ h (ϕ(X)) − h (Z) , provided that the integrals defining h (ϕ(X)) and h (Z) converge to a real number or ±∞. The function ϕ(x) is an increasing function of x defined by Z x 1 ϕ(x) = dt, ∀x ∈ (`, u), (12) σ(t) c where c ∈ (`, u) is arbitrary. Remark 4. Note that for any c ∈ (`, u), ϕ(x) is well defined (see Definition 6). By selecting a different c0 ∈ (`, u) we obtain a different function ϕ0 (x) such that Z c 1 0 ϕ (x) − ϕ(x) = dt < ∞. c0 σ(t) However, h (ϕ(X)) is invariant with respect to adding constant terms, and thus invariant with respect to different choices of c ∈ (`, u). The above theorem is proved in Section 6.3. Corollary 1. Let W = ϕ(X). Since ϕ(·) is a one-to-one function (as σ(x) > 0), we obtain max µX ∈F ∩AC h (ϕ(X)) − h (Z) = max h (W ) − h (Z) , fW ∈G where F is defined in (6), and W ∼ fW belongs to G = {fW (·) µW ∈ AC, supp(µW ) ⊆ ϕ(X ), E[gi (ϕ−1 (W ))] ≤ 0 for all i = 1, · · · , k}. Here ϕ(X ) = {ϕ(x) : x ∈ X }. Hence, from Theorem 3 we obtain that max I (X; Y ) ≥ max h (W ) − h (Z) . µX ∈F fW ∈G In order to find the maximum of h (W ) over fW ∈ G, we can use known results on maximum entropy probability distributions, e.g., see [16, Chapter 3.1]. Corollary 2. Consider an ASDN channel satisfying (10) and (11). Assume that the only input constraint is X = (`, u) i.e. ` < X < u. Then, from Corollary 1, we obtain the lower bound Z u 1 max h (W ) − h (Z) = log dx − h (Z) , fW ∈G ` σ(x) by taking a uniform distribution for fW (w) over ϕ(X ) if this set is bounded [16, Section 3.1]. Else, if ϕ(X ) has an infinite length, the capacity is infinity by choosing a pdf for W whose differential entropy is infinity (see Example 1). The equivalent pdf fX (x) for X is the pdf of ϕ−1 (W ). 11 For more insight, we provide the following example. Example 2. Consider an AWGN channel (namely, an AGSDN channel with σ(x) = σ0) with X = R and Z ∼ N (0, 1). Let us restrict to measures that satisfy the power constraint E X 2 ≤ P ; that is g1 (x) = x2 . Since Z R 1 fZ σ(x) y−x σ(x) Z dx = R 1 − p e 2πσ02 (y−x)2 2 2σ0 dx = 1, we can apply Corollary 1. Here W = ϕ(X) = x/σ0 ; thus, the lower bound is C≥ max fW (·):E[W 2 ]≤ P 2 σ0 h (W ) − h (Z) = P 1 log 2 , 2 σ0 (13) √ where it is achieved by Gaussian distribution W ∼ N (0, P /σ0 )[17, Section 12.1]. It is well-known that the capacity of AWGN channel is 1 P C = log 1 + 2 . (14) 2 σ0 Comparing (14) and (13), we see that the lower bound is very close to the capacity in the high SNR regime. As another example, consider the constraints X ≥ 0, and E [X] ≤ α on admissible input measures. Here, we obtain the lower bound max fW (·):W ≥0 E[W ]≤ σα h (W ) − h (Z) = 1 α2 e log , 2 2πσ02 0 where we used the fact that the maximum is achieved by the exponential distribution fW (w) = σ0 /α exp(−wσ0 /α) for w ≥ 0 and fW (w) = 0 for w < 0 [17, Section 12.1]. Unlike the first example above, an exact capacity formula for this channel is not known. 3.4 Second Idea for Lower Bound Now, we are going to provide another lower bound which is more appropriate in the channels for which Z is either non-negative or non-positive, and σ(x) is a monotonic function. An example of such channels is the molecular timing channel discussed in the introduction. Theorem 4. Assume an ASDN channel defined in (1) with σ : (`, u) 7→ (0, ∞) for −∞ ≤ ` < u ≤ +∞. If X is a continuous random variable with pdf fX (x), and σ(x) is continuous and monotonic over (`, u), (15) 1 is Riemann integrable on (`, u), σ(x) (16) then I (X; Y ) ≥ αh (ψ(X)) − β, provided that α, β are well-defined, and α > 0. In order to define the variables α, β, and the function ψ(x), take some arbitrary δ > 0 and proceed as follows: 12 • If the function σ(x) is increasing over (`, u), let x Z ψ(x) = δ log σ(x) + c α = Pr {Z ≥ δ} , 1 dt, σ(t) β = αh (Z\|Z ≥ δ) + H2 (α), • If the function σ(x) is decreasing over (`, u), let Z ψ(x) = −δ log σ(x) + c α = Pr {Z ≤ −δ} , x 1 dt, σ(t) β = αh (Z\|Z ≤ −δ) + H2 (α), where c ∈ (`, u) is arbitrary, and H2 (p) := −p log p − (1 − p) log (1 − p). Remark 5. Observe that in both cases, ψ(x) is an strictly increasing function of x defined over (`, u), as σ(x) > 0 and log(x) is increasing. Similar to Remark 4, the choice of c ∈ (`, u) does not affect the value of h (ψ(X)), and hence the lower bound. However, the choice of δ > 0 affects the lower bound. The above theorem is proved in Section 6.4. Corollary 3. Similar to Corollary 1, let V = ψ(X). Since ψ(·) is a one-to-one (strictly increasing) function, we obtain max αh (ψ(X)) − β = max αh (V ) − β µX ∈F ∩AC fV ∈G where F is defined in (6), and V ∼ fV belongs to G = {fV (·) µV ∈ AC, supp(µV ) ⊆ ψ(X ), E[gi (ψ −1 (V ))] ≤ 0 for all i = 1, · · · , k}. Hence, from Theorem 4 we obtain that max I (X; Y ) ≥ α max h (V ) − β, µX ∈F fV ∈G where α and β are constants defined in Theorem 4. As mentioned earlier, to maximize h (V ) over fV ∈ G, we can use known results on maximum entropy probability distributions, e.g., see [16, Chapter 3.1]. Corollary 4. Consider an ASDN channel satisfying (15) and (16). Assume that the only input constraint is X = (`, u) i.e. ` < X < u. Then, from Corollary 3, we obtain the lower bound Z u σ(u− ) 1 α max h (V ) − β = α log δ log + dx − β, fV ∈G σ(`+ ) ` σ(x) where α and β are defined in Theorem 4, and σ(u− ) := lim σ(x). σ(`+ ) := lim σ(x), x↓` x↑u The lower bound is achieved by taking a uniform distribution for fV (w) over ψ(X ) if this set is bounded [16, Section 3.1]. Else, if ψ(X ) has an infinite length, the capacity is infinity by choosing a pdf fV (v) such that h (V ) = +∞. (see Example 1). The equivalent pdf fX (x) for X is the pdf of ψ −1 (V ). 13 3.5 An Upper Bound We begin by reviewing upper bound given in [1] to motivate our own upper bound. The upper bound in [1] works by utilizing Topsoe’s inequality [18] to bound mutual information I (X; Y ) from above as follows: I (X; Y ) ≤ EµX [D (f (y\|x)kq(y))]. for any arbitrary pdf q(y) on output Y . The distribution q(y) is chosen carefully to allow for √ calculation of the above KL divergence. The particular form of σ(x) = c0 + c1 x makes explicit calculations possible. The second difficulty in calculating the above expression is that we need to take expected value over input measure µX . However, the capacity achieving input measure is not known. This difficulty is addressed by the technique of “input distributions that escape to infinity”, under some assumptions about the peak constraint. In this part, we give an upper bound based on the KL symmetrized upper bound of [15]. The idea is that I (X; Y ) = D(µX,Y kµX µY ) ≤ D(µX,Y kµX µY ) + D(µX µY kµX,Y ) , Dsym (µX,Y kµX µY ). Our upper bound has the advantage of being applicable to a large class of σ(x). To state this upper bound, let Cov (X, Y ) := E [XY ] − E [X] E [Y ] be the covariance function between two random variables X and Y . Theorem 5. For any AGSDN channel defined in (1), we have 1 1 X 2 2 I (X; Y ) ≤ − Cov X + σ (X), 2 + Cov X, 2 , 2 σ (X) σ (X) provided that the covariance terms on the right hand side are finite. The proof can be found in Section 6.5 Corollary 5. For an AGSDN channel with parameters σ(x), Z ∼ N (0, 1), and X = [0, u], if functions σ(x) and x/σ 2 (x) are increasing over X , σ(0) > 0, and x2 + σ(x) is convex over X then ( 1 F α ≥ u2 max I (X; Y ) ≤ 18 , α α µX :0≤X≤u α < u2 2 1 − u uF E[X]≤α where F = u2 u2 σ 2 (0) σ 2 (u) + + + − 2. σ 2 (u) σ 2 (0) σ 2 (u) σ 2 (0) The corollary is proved in Section 6.6. Remark 6. Even though Corollary 5 is with the assumption σ(0) > 0, if we formally set σ(0) = 0, we see that F and the upper bound on capacity becomes infinity. This is consistent with Theorem 2 when σ(0) = 0. p Corollary 6. The particular choice of σ(x) = c20 + c21 x that was motivated by applications discussed in the Introduction has the property that σ(x), x/σ 2 (x) are increasing and Theorem 5 can be applied. 14 4 Some Useful Lemmas In this section, we provide three lemmas used in the proof of theorems in this paper. Lemma 2. In an ASDN channel defined in (1), with continuous random variable noise Z with pdf fZ (·), and noise coefficient σ(x) > 0 (µX −a.e.), the conditional measure µY \|X (·\|x) has the following pdf: 1 y−x fY \|X (y\|x) = , µX,Y -(a.e.). fZ σ(x) σ(x) Moreover, Y is a continuous random variable with the pdf 1 y−X fY (y) = E . fZ σ(X) σ(X) Furthermore, if h (Z) exists, h (Y \|X) can be defined and is equal to h (Y \|X) = E [log σ(X)] + h (Z) . The lemma is proved in Section 6.8. Lemma 3. Let X be a continuous random variable with pdf fX (x). For any function σ : (`, u) 7→ [0, +∞) such that σ(x) is Riemann integrable over (`, u) and σ(x) > 0 (a.e), where −∞ ≤ ` < u ≤ +∞, we have that h (X) + E [log σ(X)] = h (ϕ(X)) . (17) where Z x ϕ(x) = σ(t) dt, (18) c where c ∈ (`, u) is an arbitrary constant. Note that if the left-hand side does not exist, or becomes ±∞, the same occurs for the right-hand side and vice versa. The lemma is proved in Section 6.9. Lemma 4. Let X be a random variable with probability measure µX , and the functions w(x) and v(x) be increasing over [`, u], where −∞ < ` < u < +∞. If v(x) is convex over [`, u], then max µX :`≤X≤u E[X]≤α Cov (w(X), v(X)) ≤ β[w(u) − w(`)][v(u) − v(`)], where ( β= 1 4 (u−α)(α−`) (u−`)2 α≥ α< `+u 2 `+u 2 . Furthermore, for the case α ≥ (` + u)/2, a maximizer of (19) is the pmf 1 pX (`) = pX (u) = . 2 For the case α < (` + u)/2 if v(x) is linear, a maximizer of (19) is the pmf pX (`) = 1 − pX (u) = The proof is given in Section 6.10. 15 u−α . u−` (19) 1 10 1 10 Capacity Capacity (nats per channel use) Capacity (nats per channel use) KL Upper Bound 0 10 −1 10 0 10 Capacity −1 10 Corollary 2 Lower Bound Corollary 4 Lower Bound −2 10 −2 10 0 20 40 60 80 100 120 140 160 180 200 σ02 −3 10 50 100 150 200 250 300 A Figure 1: Capacity and Symmetrized divergence upper bound in terms of c20 for AGSDN channel with A p = 5, α = 2.5, c1 = 1 and function σ(x) = c20 + x. 5 Figure 2: Capacity and lower bound at corollary 2 and 4 in terms of A√for AGSDN channel with function σ(x) = 1 + x. Numerical Results p In this section, some numerical results are given for σ(x) = c20 + x and Z ∼ N (0, 1). The upper bound, Corollary (5), and the capacity are depicted in the logarithmic scale in Fig. 1, where we have considered the peak constraint A = 5 and average constraint α = 2.5 . It can be observed that the distance between the upper bound and the capacity is a small constant in the logarithmic scale and low SNR regime. This is consistent with [15] that argues that the upper bound based on symmetrized KL divergence is mostly suitable for the low SNR regime. p The lower bounds of Corollaries 4 and 2 are plotted in Fig. 2 for the function σ(x) = c20 + x in terms of peak constraint, A. Here, c0 = 1 is assumed. The lower bound of Corollary 2 for 0 < X < A is computed by the following closed form formula: ! q Z A 1 1 2 − 2c p log − h (Z) = log 2 A + c log(2πe), 0 − 0 2 2 c0 + x 0 while the lower bound of Corollary 4 equals ! ! p Z A q A + c20 1 σ(A) p + α log δ log dx − β = α log δ log + 2 A + c20 − 2c0 − β 2+x σ(0) c 0 c 0 0 where δ > 0 and α = Pr {Z ≥ δ} , β = αh (Z\|Z ≥ δ) − α log α − (1 − α) log (1 − α). We maximized over δ in order to find the lower bound of Corollary 4. The first lower bound is better than the second one mainly because of the multiplicative coefficient α of the second lower bound. Since the second lower bound is for a more general class of channels, we should consider the positive (or negative) part of the support of Z, causing a multiplicative of coefficient 1/2 for the Gaussian noise. However, if the support of Z is positive (or negative) reals, the two lower bounds do not differ much. 16 6 6.1 Proofs Proof of Theorem 1 Finiteness of capacity: The first step is to show that the capacity is finite: sup I (X; Y ) < ∞. (20) µX ∈F To prove this, it suffices to show that the supremum of both h (Y ) and h (Y \|X) over µX ∈ F are finite, i.e., \|h (Y ) \|, \|h (Y \|X) \| < +∞, uniformly on µX ∈ F. (21) Utilizing Lemma 2, the existence and boundedness of h (Y \|X) is obtained as follows: \|h (Y \|X) \| ≤ max{\| log σ` \|, \| log σu \|} + \|h (Z) \| < ∞, uniformly on F. From Lemma 2, we obtain that Y is continuous with a pdf fY (y). To prove that the integral defining h (Y ) is convergent to a finite value (existence of entropy), and furthermore the integral is convergent to a value that is bounded uniformly on F, it is sufficient to show that there are some positive real γ, m̄ and v such that for any µX ∈ F, we have [19]: sup fY (y) < m̄, (22) y∈R E [\|Y \|γ ] < v. (23) Also, from Lemma 2, we obtain that for any µX ∈ F fY (y) ≤ m . σ` Thus, (22) holds with m̄ = m/σ` . In order to prove (23), note that E [\|Y \|γ ] ≤E [(\|X\| + σu \|Z\|)γ ] ≤2γ E [max {\|X\|γ , σuγ \|Z\|γ }] ≤2γ E [\|X\|γ ] + (2σu )γ E [\|Z\|γ ] ≤2γ uγ + (2σu )γ α, uniformly on F. Thus, h (Y ) is well-defined and uniformly bounded on F. Hence, from the definition of mutual information we obtain that I (X; Y ) = h (Y ) − h (Y \|X) (24) is bounded uniformly for µX ∈ F. Existence of a maximizer: Let C = sup I (X; Y ) < ∞. (25) µX ∈F We would like to prove that the above supremum is a maximum. Equation (25) implies existence (k) of a sequence of measures {µX }∞ k=1 in F such that lim I (Xk ; Yk ) = C, k→∞ 17 (k) (k) where Xk ∼ µX , and Yk ∼ µY is the output of the channel when the input is Xk . Furthermore, (k) without loss of generality, we can assume that {µX }∞ k=1 is convergent (in the Lévy measure) to a ∗ measure µX ∈ F. The reason is that since X is compact, the set F is also compact with respect to the Lévy measure [10, Proposition 2]. Thus, any sequence of measures in F has a convergent (k) subsequence. With no loss of generality we can take the subsequence as {µX }∞ k=1 . Thus, from convergence in Lévy measure, we know that there is µ∗X ∈ F such that lim E [g(Xk )] = E [g(X ∗ )] , (26) k→∞ for all g : R 7→ C such that supx∈R \|g(x)\| < +∞. We would like to prove that I (X ∗ ; Y ∗ ) = C, (27) where Y ∗ ∼ µ∗Y is the output measure of the channel when the input measure is µ∗X . This will complete the proof. From the argument given in the first part of the proof on “Finiteness of capacity”, h (Y ∗ \|X ∗ ) and h (Y ∗ ) are well-defined and finite. As a result to show (27), we only need to prove that lim h (Yk \|Xk ) = h (Y ∗ \|X ∗ ) , (28) lim h (Yk ) = h (Y ∗ ) . (29) k→∞ k→∞ Since −∞ < σ` < σu < +∞, (28) is obtained from (26) and Lemma 2. In order to prove (29), we proceed as follows: (k) • Step 1: We begin by showing that the sequence {µY }∞ k=1 is a Cauchy sequence with respect to total variation i.e. (m) ∀ > 0, ∃N : m, n ≥ N ⇒ kµY (n) − µY kV ≤ , (30) where for any two arbitrary probability measure µA and µB , the total variation distance is defined by [16, p. 31] X kµA − µB k := sup µA (Ei ) − µB (Ei ) , ∆ i where ∆ = {E1 , · · · , Em } ⊆ B(R) is the collection of all the available finite partitions. • Step 2: Having established step 1 above, we utilize the fact that the space of probability measures is complete with respect to the total variation metric. To show this, note that by Lemma 2, all the Yk ’s have a pdf, and hence the total variation can be expressed in terms of the k · kL1 norm between pdfs [16, Lemma 1.5.3]. From [20, p. 276] we obtain that this space of pdfs is complete with respect to k · kL1 norm. (k) As a result, µY converges to some measure Yb ∼ µ bY with respect to the total variation metric. We further claim that this convergence implies that lim h (Yk ) = h Yb . (31) k→∞ (k) The reason is that from (22) and (23), we see that {fY } and fŶ are uniformly bounded and have finite γ-moments. Therefore, (31) follows from [19, Theorem 1]. Thus, in step 2, we obtain that the sequence h (Yk ) has a limit. 18 • Step 3: We show that the limit found in Step 2 is equal to h (Y ∗ ), i.e., h Yb = h (Y ∗ ) . (32) This completes the proof of (29). Hence, it only remains to prove (30) and (32). 0 Proof of (30): Since {I (Xk ; Yk )}∞ k=1 is convergent to C, for any > 0, there exists N such that: \|C − I (Xk ; Yk ) \| ≤ 0 , ∀k ≥ N. Now, consider m, n ≥ N . Let Q be a uniform Bernoulli random variable, independent of all (m) previously defined variables. When Q = 0, we sample from measure µX and when Q = 1, we (n) e ∼ µ e defined as follows: sample from measure µX . This induces the measure X X 1 (m) 1 (n) µXe = µX + µX . 2 2 e We have a Markov chain Q − X e − Ye . Let Ye ∼ µYe be the output of the channel when the input is X. Note that e Ye \|Q = 1 I (Xm ; Ym ) + 1 I (Xn ; Yn ) . I X; 2 2 From concavity of mutual information in input measure, we obtain that: e Ye ≥ 1 I (Xm ; Ym ) + 1 I (Xn ; Yn ) ≥ C − 0 . I X; 2 2 e Ye ≤ C, Since F is an intersection of half spaces, it is convex and as a result µXe ∈ F. Thus, I X; and we obtain that e Ye − I X; e Ye Q ≤ 0 . I X; e e e e Because of the Markov chain Q − X − Y , we obtain I Y ; Q X = 0 and as a result: I Ye ; Q ≤ 0 =⇒ D µYe ,Q kµYe µQ ≤ 0 . From the Pinsker’s inequality we obtain that kµYe ,Q − µYe µQ kV ≤ √ 20 , (33) where kµYe ,Q − µYe µQ kV is the total variation between the measures µYe ,Q and µYe µQ . Note that 1 (n) 1 (m) kµYe ,Q − µYe µQ kV = kµY − µYe kV + kµY − µYe kV . 2 2 Therefore from (33) and (34), we obtain that (m) kµY √ (n) − µYe kV , kµY − µYe kV ≤ 2 20 . As a result, (m) kµY √ (n) − µY kV ≤ 4 20 . 19 (34) (k) Hence, by taking 0 ≤ 2 /32, we obtain that {µY }∞ k=1 is a Cauchy sequence. Proof of (32): To this end, it suffices to prove that ΦYb (ω) = ΦY ∗ (ω), ∀ω ∈ R, where ΦX (ω) := E [exp(jωX)] is the characteristic function of the random variable X. Since Yk converge to Yb in total variation, and the fact that convergence in total variation is stronger than weakly convergence [16, p. 31], from (26) we obtain that their characteristic functions, ΦYk (ω), also converge to ΦYb (ω) pointwise. Hence, it suffices to prove that ΦYk (ω) converge to ΦY ∗ (ω) pointwise. From (1), we obtain that h i ΦYk (ω) = E ejω(Xk +σ(Xk )Z) = E ejωXk ΦZ (σ(Xk )ω) . Similarly, h i ∗ ΦY ∗ (ω) = E ejωX ΦZ (σ(X ∗ )ω) . Since {Xk } converges to X ∗ in Lévy measure and the function g(x) = ejωx ΦZ (σ(x)ω) is bounded: \|g(x)\| = ejωx ΦZ (σ(x)ω) ≤ ejωx \|ΦZ (σ(x)ω)\| ≤ 1, from (26) we obtain that E[g(Xk )] = ΦYk (ω) converges to E[g(X ∗ )] = ΦY ∗ (ω) pointwise. Uniqueness of the output pdf: The proof is the same as the first part of the proof of [10, Theorem 1]. This completes the proof. 6.2 Proof of Theorem 2 For a continuous input measure, we utilize a later result in the paper, namely Theorem 4 by choosing ` = c, u = c0 when c0 > c, or ` = c0 , u = c when c0 < c. To use Corollary 4, observe that the image of E under ψ(·) has infinite length. This is because the sequence {x̃i } in E was such that the monotone function σ(·) converged to zero or infinity on that sequence. Then, it is obtained that any pdf fX (·) such that h (ψ(X)) = +∞, makes I (X; Y ) infinity if \|h (Z\|Z > δ) \| < ∞ (which leads to \|β\| < ∞), where ψ(x) is the bijective function of x defined in the statement of Theorem 4. In order to prove that \|h (Z\|Z > δ) \| < ∞, let the random variable Z̄ be Z conditioned to Z > δ. Due to the continuity of Z and the fact that Pr {Z > δ} > 0, we obtain that Z̄ has a valid pdf fZ̄ (z) defined by ( 1 fZ (z) z > δ fZ̄ (z) = θ , 0 z≤δ where θ := Pr {Z > δ} > 0. Since h (Z) exists and \|h (Z) \| < ∞, we obtain that 1 E < ∞. fZ (Z) Hence, \|h Z̄ \| ≤ E 1 fZ̄ (Z̄) 1 ≤ − log θ + E θ 1 fZ (Z) < ∞. Therefore, h (Y \|Z > δ) exists and \|h (Y \|Z > δ) \| < ∞. A similar treatment can be used to prove \|h (Y \|Z < δ) \| < ∞. 20 It remains to construct a discrete pmf with infinite mutual information. The statement of the theorem assumes existence of a sequence {x̃i } in an open interval E = (c, c0 ) ⊂ X (or E = (c0 , c) if c0 < c) such that 1. c is the limit of the sequence {x̃i }, 2. σ(x̃i ) converges to 0 or +∞ 3. σ(·) is monotone and continuous over E We now make the following claim about existence of another sequence {xi }∞ i=1 ⊆ E with certain nice properties: Claim: Suppose that one cannot find a non-empty interval [x0 , x00 ] ⊂ E such that σ(x) = 0 for all x ∈ [x0 , x00 ]. Then, there exists 0 < a < b and a sequence {xi }∞ i=1 ⊆ E, such that • If σ(x) is increasing, Pr {a < Z < b} > 0, (35) (xi + aσ(xi ), xi + bσ(xi )) ∩ (xj + aσ(xj ), xj + bσ(xj )) = ∅, 0 < σ(xi ) < ∞, ∀i 6= j ∈ N ∀i ∈ N. (36) (37) • If σ(x) is decreasing, Pr {−b < Z < −a} > 0, (xi − bσ(xi ), xi − aσ(xi )) ∩ (xj − bσ(xj ), xj − aσ(xj )) = ∅, 0 < σ(xi ) < ∞, ∀i 6= j ∈ N ∀i ∈ N. We continue with the proof assuming that this claim is correct; we give the proof of this claim later. To show how this claim can be used to construct a discrete pmf with infinite mutual information, consider the possibility that the assumption of the claim fails: σ(x) = 0 for all x ∈ [x0 , x00 ], then Y = X in that interval when X ∈ [x0 , x00 ]. Therefore, we can provide any discrete distribution in that interval such that H (X) = ∞, as a result I (X; Y ) = I (X; X) = H (X) = ∞. Thus, we should only consider the case that the assumption of the claim holds. Assume that σ(x) is increasing. The construction when σ(x) is decreasing is similar. Fix a given a, b, {xi }∞ i=1 satisfying (35) and (36). Take an arbitrary pmf {pi }∞ i=1 such that X i pi log 1 = +∞. pi (38) Then, we define a discrete random variable X, taking values in {xi }∞ i=1 such that Pr {X = xi } = pi . We claim that I (X; Y ) = +∞. To this end, it suffices to show I (X; Y ) ≥ Pr {a < Z < b} I (X; Y \|a < Z < b) − H2 (Pr {a < Z < b}), (39) I (X; Y \|a < Z < b) = ∞. (40) Proof of (39): Define random variable E as following: ( 0 Z ∈ (a, b) E= 1 Z∈ / (a, b) 21 From the definition of mutual information, we have that I (X; Y \|E) − I (X; Y ) = I (Y ; E\|X) − I (Y ; E) ≤ H (E) , Since I (X; Y \|E) = Pr {E = 0} I (X; Y \|E = 0) + Pr {E = 1} I (X; Y \|E = 1) , we conclude (39). Proof of (40): Since I (X; Y \|a < Z < b) = H (X) − H (X\|Y, a < Z < b) , it suffices to show that H (X) = ∞, H (X\|Y, a < Z < b) = 0. (41) P The equality H (X) := − pi log pi = +∞ follows (38). To prove the other equality, note that Y belongs to the interval (xi + aσ(xi ), xi + bσ(xi )) when X = xi . Therefore, since the intervals (xi + aσ(xi ), xi + bσ(xi )) are disjoint, X can be found from Y . Thus, X is a function of Y when a < Z < b. As a result, the second equality of (41) is proved. Now, it only remains to prove our Claim on the existence of a, b, and {xi }∞ i=1 . We assume that σ(x) is increasing. The proof when σ(x) is decreasing is similar. From the assumptions on Z that Pr {Z ≥ δ} > 0, we obtain there exists δ < b < ∞ such that Pr {δ < Z < b} > 0. As a result, we select a = δ. Since σ(x) is monotone, we cannot have σ(x0 ) = σ(x00 ) = 0 for two arbitrary distinct x0 and x00 in E since this implies that σ(x) = 0 for all x in between x0 and x00 . As a result, we shall not worry about the constraint (37) on {xi } because σ(xi ) = 0 can occur for at most one index i and we can delete that element from the sequence to ensure (37). To show the existence of {xi }∞ i=1 , we provide a method to find xi+1 with respect to xi . The method is described below and illustrated in Figure 3. Take x1 an arbitrary element of E. Observe that since σ(x) is continuous and increasing over E, the functions x + aσ(x) and x + bσ(x) are continuous and strictly increasing over E, as well as x + aσ(x) < x + bσ(x), ∀x ∈ E. Therefore, for the case c0 > c (happening when σ(xi ) converge to 0), lim x + aσ(x) = lim x + bσ(x) = c, x→c x→c Hence, for a given xi ∈ E, due to the intermediate value theorem, there exists unique xi+1 satisfying c < xi+1 < xi < c0 such that xi+1 + bσ(xi+1 ) = xi + aσ(xi ). Similarly, for the case c0 < c (happening when σ(x̃i ) converge to +∞), if xi ∈ E, there exists unique xi+1 satisfying c > xi+1 > xi > c0 such that xi+1 + aσ(xi+1 ) = xi + bσ(xi ). It can be easily obtained that the intervals created this way are disjoint, and the process will not stop after finite steps. Therefore, the theorem is proved. 22 x1 x2 σ→+∞ (ascending) x3 c c' x4 ... x + a σ(x) x - a σ(x) x + b σ(x) x - b σ(x) c c' σ→+∞ (descending) ... x5 x1 x4 x3 x2 σ→0 (descending) σ→0 (ascending) x - a σ(x) x + a σ(x) x1 x5 x - b σ(x) x4 x + b σ(x) c' x3 c' x2 x3 ... x2 c x4 x1 Figure 3: Possible cases for σ(x) when \|c\| < ∞. 23 ... c 6.3 Proof of Theorem 3 From Lemma 2 we obtain that h (Y \|X) exists. Hence, utilizing Lemma 1 , we can write h (Y ) ≥ h (X) =⇒ I (X; Y ) ≥ h (X) − h (Y \|X) , provided that (42) u Z fY \|X (y\|x) dx ≤ 1, ` where it is satisfied due to Z u Z fY \|X (y\|x) dx = ` ` u 1 fZ σ(x) y−x σ(x) dx ≤ 1, where the last inequality comes from the assumption of the theorem. From Lemma 2, we have that h (Y \|X) = E [log σ(X)] + h (Z) . Therefore, (42) can be written as I (X; Y ) ≥ h (X) − E [log σ(X)] − h (Z) . Exploiting Lemma 3 we obtain that h (X) − E [log σ(X)] = h (ϕ(X)) , where ϕ(X) is defined in (12) Hence, the proof is complete. 6.4 Proof of Theorem 4 We only prove the case that σ(x) is an increasing function over (`, u). The proof of the theorem for decreasing functions is similar to the increasing case and we only need to substitute Z ≥ δ with Z ≤ −δ. We claim that I (X; Y ) ≥ αI (X; Y \|Z ≥ δ) − H2 (α). (43) Consider random variable E as following: ( 0 Z≥δ E= . 1 Z<δ From the definition of mutual information, we have that I (X; Y \|E) − I (X; Y ) = I (Y ; E\|X) − I (Y ; E) ≤ H (E) , Therefore, since I (X; Y \|E) = Pr {Z ≥ δ} I (X; Y \|Z ≥ δ) + Pr {Z < δ} I (X; Y \|Z < δ) , we conclude (43). Now, we find a lower bound for I (X; Y \|Z ≥ δ). From Lemma 2 we obtain that Y is a continuous random variable. We claim that I (X; Y \|Z ≥ δ) =h (Y \|Z ≥ δ) − h (Y \|X, Z ≥ δ) 24 =h (Y \|Z ≥ δ) − (E [log σ(X)] + h (Z\|Z ≥ δ)) =h (Y \|Z ≥ δ) − h (X) − E log(1 + Zσ 0 (X))\|Z ≥ δ 1 + Zσ 0 (X) + h (X) + E log Z ≥ δ − h (Z\|Z ≥ δ) , σ(X) (44) (45) where (44) is obtained from Lemma 2, and the fact that random variable Z conditioned to Z ≥ δ is also continuous when Pr {Z ≥ δ} > 0. Moreover, (45) is obtained by adding and subtracting the term E [log(1 + Zσ 0 (X))\|Z ≥ δ]. Note that we had not assumed that σ(x) needs to be differentiable. We had only assumed that σ : (`, u) 7→ (0, ∞) is continuous and monotonic over (`, u). However, every monotonic function is differentiable almost everywhere, i.e., the set of points in which σ(x) is not differentiable has Lebesgue measure zero. We define σ 0 (x) to be equal to zero wherever σ(x) is not differentiable; and we take σ 0 (x) to be the derivative of σ(x) wherever it is differentiable. With this definition of σ 0 (x) and from the continuity of σ(x), we have that the integral of σ 0 (x)/σ(x) gives us back the function log(σ(x)). Since σ(x) is an increasing positive function, and Z ≥ δ > 0, we conclude that 1 + Zσ 0 (X) 1 + δσ 0 (X) E log Z ≥ δ ≥ E log . (46) σ(X) σ(X) From Lemma 3 and the fact that the integral of σ 0 (x)/σ(x) gives us back the function log(σ(x)), we obtain that 1 + δσ 0 (X) h (X) + E log = h (ψ(X)) , σ(X) where ψ(x) is defined in Theorem 4. As a result from (45), we obtain that I (X; Y \|Z ≥ δ) ≥h (Y \|Z ≥ δ) − h (X) − E log(1 + Zσ 0 (X))\|Z ≥ δ + h (ψ(X)) − h (Z\|Z ≥ δ) , (47) Using this inequality in conjunction with (43), we obtain a lower bound on I (X; Y ). The lower bound that we would like to prove in the statement of the theorem is that I (X; Y ) ≥ αh (ψ(X)) − αh (Z\|Z ≥ δ) − H2 (α). As a result, it suffices to prove that for all continuous random variables X with pdf fX (x) we have h (Y \|Z ≥ δ) − h (X) − E log(1 + Zσ 0 (X))\|Z ≥ δ ≥ 0. To this end, observe that h (Y \|Z ≥ δ) ≥ h (Y \|Z, Z ≥ δ). Thus, if we show that h (Y \|Z, Z ≥ δ) = h (X) + E log(1 + Zσ 0 (X))\|Z ≥ δ , (48) the proof is complete. We can write that Z h (Y \|Z, Z ≥ δ) = ∞ fZ 0 (z)h (Y \|Z = z) dz, δ where Z 0 is Z conditioned to Z ≥ δ, and the pdf of Z 0 is denoted by fZ 0 (z). By defining the function rz (x) := x + zσ(x), we obtain that Yz = rz (X), 25 where Yz is Y which is conditioned to Z = z ≥ δ. Since, σ(x) is a continuous increasing function, rz (x) is a bijection for all z ≥ δ, and so its inverse function, rz−1 (y), exists. Moreover, since X is continuous and rz (·) is a bijection, Yz is also continuous random variable with pdf fY (y) defined as following: 1 fYz (y) = fX (x), 1 + zσ 0 (x) where x = rz−1 (y). 3 Thus, we have that 1 h (Y \|Z = z) =E log fYz (Yz ) 1 =E log + E log(1 + zσ 0 (X)) fX (X) =h (X) + E log(1 + zσ 0 (X)) . By taking expected value over Z ≥ δ from both sides, (48) is achieved. Therefore, the theorem is proved. 6.5 Proof of Theorem 5 Based on [15] we obtain that I (X; Y ) ≤ Dsym (µX,Y kµX µY ), Utilizing Lemma 2, we obtain that the pdfs fY (y) and fY \|X (y\|x) exist and are well-defined. Therefore, Dsym (µX,Y kµX µY ) =D(µX,Y kµX µY ) + D(µX µY kµX,Y ) fY \|X (Y \|X) fY (Y ) =EµX,Y log + EµX µY log fY (Y ) fY \|X (Y \|X) 1 1 = − EµX,Y log + E log fY \|X (Y \|X) fY (Y ) 1 1 + EµX µY log − E log fY \|X (Y \|X) fY (Y ) 1 =EµX µY log − h (Y \|X) . fY \|X (Y \|X) Again, from Lemma 2, since Z ∼ N (0, 1), we obtain that log √ (y − x)2 1 = log σ(x) 2π + fY \|X (y\|x) 2σ 2 (x) Therefore, since Z = (Y − X)/σ(X), we obtain that h √ i 1 h (Y \|X) = E log σ(X) 2π + . 2 3 (49) The measure zero points where σ(x) is not differentiable affect fYz (y) on a measure zero points. However, note that FYz (y) = FX (r−1 (y)) is always correct and thus the values of fYz (y) on a measure zero set of points are not important. 26 In addition, EµX µY h √ i 1 (Y − X)2 log = E log 2πσ(X) + EµX µY . fY \|X (Y \|X) 2σ 2 (X) By expanding, we obtain that 2 1 X2 X (Y − X)2 =E Y E 2 +E 2 − 2E [Y ] E 2 . EµX µY σ 2 (X) σ (X) σ (X) σ (X) By substituting Y with X + σ(X)Z and simplifying we can write 2 2 (Y − X)2 1 1 EµX µY =E X E + E σ (X) E σ 2 (X) σ 2 (X) σ 2 (X) 2 X X − 2E [X] E 2 +E 2 σ (X) σ (X) 2 2 2 1 X + σ 2 (X) 1 =E X E 2 + E σ (X) E 2 −E +1 σ (X) σ (X) σ 2 (X) X X2 − 2E [X] E 2 , +2 E 2 σ (X) σ (X) which equals to 1 − Cov X 2 + σ 2 (X), 1 2 σ (X) + 2Cov X, X 2 σ (X) . Therefore, from all above equations the theorem is proved. 6.6 Proof of Corollary 5 Observe that 2 1 u u2 σ 2 (0) σ 2 (u) 1 F = + + + −2 2 2 σ 2 (u) σ 2 (0) σ 2 (u) σ 2 (0) 1 2 1 1 u2 2 2 = u + σ (u) − σ (0) − + . 2 σ 2 (0) σ 2 (u) σ 2 (u) Then, using Theorem 5, it suffices to prove the following two inequalities: −1 1 1 2 2 2 2 2 ≤ β u + σ (u) − σ (0) − , Cov X + σ (X), 2 σ (X) σ 2 (0) σ 2 (u) and Cov X, where ( β= X 2 σ (X) 1 4 u α 1− u α ≤β u2 , σ 2 (u) α≥ α< u 2 u 2 (50) (51) . Since σ(x) is increasing, we obtain that x2 + σ 2 (x) and −1/σ 2 (x) are also increasing. Therefore, from Lemma 4, equation (50) is proved. Similarly, (51) is also obtained from Lemma 4 because x and x/σ 2 (x) are increasing functions. 27 6.7 Proof of Lemma 1 From Definition 5, we obtain that fY (Y ) . h (X) − h (Y ) = E log fX (X) Now, utilizing the inequality log x ≤ x − 1, it suffuces to prove that fY (Y ) E ≤ 1. fX (X) To this end, we can write Z Z fY (y) fY (Y ) = fX,Y (x, y) E dy dx fX (X) fX (x) X Y Z Z fY \|X (y\|x)fY (y) dy dx = ZX Y Z = fY (y) fY \|X (y\|x) dx dy Y X Z ≤ fY (y) dy = 1, Y where the last inequality holds because of the assumption of the lemma. Therefore, the lemma is proved. 6.8 Proof of Lemma 2 The conditional pdf fY \|X (y\|x) can be easily obtained from the definition of channel in (1). In order to calculate h (Y \|X), using the Definition 5 we can write   1 1  . h (Y \|X) = E = E [log σ(X)] + E log Y −X fY \|X (Y \|X) f Z σ(X) Exploiting the fact that (Y − X)/σ(X) = Z, h (Y \|X) is obtained. It only remains to prove that Y is continuous. To this end, from the definition of the channel in (1), we obtain that y−X y−X FY (y) = Pr Z ≤ = EµX FZ , σ(X) σ(X) where FY (y) and FZ (z) are the cdfs of the random variables Y and Z, defined by FY (y) = Pr {Y ≤ y} and FZ (z) = Pr {Z ≤ z}, respectively. In order to prove the claim about fY (y), we must show that Z y 1 y−x y−X E fZ dy = E FZ , σ(x) σ(x) σ(X) −∞ for all y ∈ R. Because of the Fubini’s theorem [20, Chapter 2.3], it is equivalent to −n − X lim E FZ = 0. n→∞ σ(X) 28 Equivalently, we need to show that for any > 0, there exists m such that −n − X E FZ ≤ , ∀n > m. σ(X) (52) Since limz→−∞ FZ (z) = 0, there exists ` ∈ R such that FZ (z) ≤ , 2 ∀z ≤ `. Therefore, since FZ (z) ≤ 1 for all z, we can write −n − X −n − X ≤ + Pr ≥` . E FZ σ(X) 2 σ(X) We can write Pr −n − X ≥ ` = Pr {X + `σ(X) ≤ −n} . σ(X) Now, we can take m large enough such that, −n − X ≥` ≤ , Pr σ(X) 2 n > m. As a result, (52) is proved. 6.9 Proof of Lemma 3 Since σ(x) is Riemann integrable, ϕ(x) is continuous and since σ(x) > 0 (a.e.), ϕ(x) is a strictly increasing function over the support of X. It yields that ϕ(x) is an injective function and there exists an inverse function ϕ−1 (·) for ϕ(·). Now, define random variable Y = ϕ(X). Assume that the pdf of X is fX (x). Since X is a continuous random variable and ϕ(x) is a bijection, Y is also continuous random variable with the following pdf: fY (y) = 1 fX (x), d dx ϕ(x) where x = ϕ−1 (y). Hence, we have that fY (y) = 1 fX σ (ϕ−1 (y)) ϕ−1 (y) . Now, we can calculate the differential entropy of Y as following: 1 h (Y ) =E log fY (Y ) " # σ ϕ−1 (Y ) =E log fX (ϕ−1 (Y )) σ(X) =E log fX (X) =h (X) + E [log ϕ(X)] . Therefore, the lemma is proved. 29 6.10 Proof of Lemma 4 First, assume that v(x) = ax + b, with a > 0. We will prove the general case later. In this case, we claim that the support of the optimal solution only needs to have two members. To this end, note that the following problem is equivalent to the original problem defined in (19): max max γ≤α µX :`≤X≤u E[X]=γ Cov (w(X), v(X)) . Since v(x) = ax + b, for a given γ, we would like to maximize Cov (w(X), v(X)) = E [w(X)v(X)] − (aγ + b)E [w(X)] , which is a linear function of µX , subject to E [X] = γ which is also a linear function of µX . By the standard cardinality reduction technique (Fenchel’s extension of the Caratheodory theorem), we can reduce the support of µX to at most two members (see [21, Appendix C] for a discussion of the technique). Assume that the support of µX is {x1 , x2 } where ` ≤ x1 ≤ x2 ≤ u with pmf pX (x1 ) = 1 − pX (x2 ) = p. Thus, we can simplify Cov (w(X), v(X)) as Cov (w(X), v(X)) = 2 X pX (xi )w(xi )v(xi ) − i=1 2 X 2 X pX (xi )pX (xj )w(xi )v(xj ) i=1 j=1 =p(1 − p) (w(x2 ) − w(x1 )) (v(x2 ) − v(x1 )) , where the last equality can be obtained by expanding the sums. Thus, the problem defined in (19) equals the following: max p,x1 ,x2 :0≤p≤1 `≤x1 ≤x2 ≤u px1 +(1−p)x2 ≤α p(1 − p) (w(x2 ) − w(x1 )) (v(x2 ) − v(x1 )) . We claim that the optimal choice for x1 is x1 = `. To see this, observe that w(x) and v(x) are increasing functions, and hence p` + (1 − p)x2 ≤ px1 + (1 − p)x2 ≤ α and (w(x2 ) − w(`)) (v(x2 ) − v(`)) ≥ (w(x2 ) − w(x1 )) (v(x2 ) − v(x1 )) . Hence, x1 = ` is optimal. Substituting v(x) = ax + b, we obtain that the problem is equivalent with the following: a max p(1 − p) (w(x) − w(`)) (x − `) . p,x:0≤p≤1 `≤x≤u p`+(1−p)x≤α Utilizing KKT conditions, one obtains that the optimal solution is ( p∗ = 21 , x∗1 = `, x∗2 = u α ≥ `+u 2 . ∗ = `, x∗ = u α < `+u p∗ = u−α , x 1 2 u−` 2 Now, we consider the general case of v(x) being a convex function (but not necessarily linear). Since v(x) is convex, we obtain that v(x) ≤ v(`) + (x − `) v(u) − v(`) , u−` 30 ∀x ∈ [`, u]. The right hand side is the line that connects the two points (`, v(`)) and (u, v(u)); this line lies above the curve x 7→ v(x) for any x ∈ [`, u]. Therefore, E[v(X)] ≤ v(`) + (E[X] − `) v(u) − v(`) . u−` Thus, E [X] ≤ α implies that E [v(X)] ≤ ∆ where ∆ = v(`) + (α − `) v(u) − v(`) . u−` Now, we relax the optimization problem and consider max µX :`≤X≤u E[v(X)]≤∆ Cov (w(X), v(X)) . The solution of the above optimization problem is an upper bound for the original problem because the feasible set of the original problem is a subset of the feasible set of the relaxed optimization problem. Now, using similar ideas as in the linear case, we conclude that the support of the optimal µX has at most two members. And the optimal solution is ( p∗ = 12 , x∗1 = `, x∗2 = u α ≥ `+u 2 . v(u)−∆ p∗ = v(u)−v(`) , x∗1 = `, x∗2 = u α < `+u 2 It can be verified that v(u) − ∆ u−α = . v(u) − v(`) u−` Note that in the case α > (` + u)/2, we obtain that E [X ∗ ] = (` + u)/2 < α, where X ∗ distributed with the optimal probability measure. As a result the constraint E [X] ≤ α is redundant. Therefore, the support of the optimal µX has two members, which shows that the upper bound is tight in this case. 7 Conclusion In this paper, we studied the capacity of a class of signal-dependent additive noise channels. These channels are of importance in molecular and optical communication; we also gave a number of new application of such channels in the introduction. A set of necessary and a set of sufficient conditions for finiteness of capacity were given. We then introduced two new techniques for proving explicit lower bounds on the capacity. As a result, we obtained two lower bounds on the capacity. These lower bounds were helpful in inspecting when channel capacity becomes infinity. We also provided an upper bound using the symmetrized KL divergence bound. References [1] S. M. Moser, “Capacity results of an optical intensity channel with input-dependent gaussian noise,” IEEE Transactions on Information Theory, vol. 58, no. 1, pp. 207–223, 2012. 31 [2] M. Pierobon and I. F. Akyildiz, “Diffusion-based noise analysis for molecular communication in nanonetworks,” IEEE Transactions on Signal Processing, vol. 59, no. 6, pp. 2532–2547, 2011. [3] G. Aminian, M. F. Ghazani, M. Mirmohseni, M. Nasiri-Kenari, and F. Fekri, “On the capacity of point-to-point and multiple-access molecular communications with ligand-receptors,” IEEE Transactions on Molecular, Biological and Multi-Scale Communications, vol. 1, no. 4, pp. 331– 346, 2016. [4] H. Arjmandi, A. Gohari, M. Nasiri-Kenari, and F. Bateni, “Diffusion-based nanonetworking: A new modulation technique and performance analysis,” IEEE Communications Letters, vol. 17, no. 4, pp. 645–648, 2013. [5] A. Gohari, M. Mirmohseni, and M. Nasiri-Kenari, “Information theory of molecular communication: Directions and challenges,” to appear in IEEE Transactions on Molecular, Biological and Multi-Scale Communications, 2016. [6] K. V. Srinivas, A. W. Eckford, and R. S. Adve, “Molecular communication in fluid media: The additive inverse gaussian noise channel,” IEEE Transactions on Information Theory, vol. 58, no. 7, pp. 4678–4692, 2012. [7] M. N. Khormuji, “On the capacity of molecular communication over the aign channel,” in Information Sciences and Systems (CISS), 2011 45th Annual Conference on, pp. 1–4, IEEE, 2011. [8] H. Li, S. M. Moser, and D. Guo, “Capacity of the memoryless additive inverse gaussian noise channel,” IEEE Journal on Selected Areas in Communications, vol. 32, no. 12, pp. 2315–2329, 2014. [9] N. Farsad, Y. Murin, A. W. Eckford, and A. Goldsmith, “Capacity limits of diffusion-based molecular timing channels,” arXiv:1602.07757, 2016. [10] T. H. Chan, S. Hranilovic, and F. R. Kschischang, “Capacity-achieving probability measure for conditionally gaussian channels with bounded inputs,” IEEE Transactions on Information Theory, vol. 51, no. 6, pp. 2073–2088, 2005. [11] J. G. Smith, “The information capacity of amplitude-and variance-constrained sclar gaussian channels,” Information and Control, vol. 18, no. 3, pp. 203–219, 1971. [12] R. Jiang, Z. Wang, Q. Wang, and L. Dai, “A tight upper bound on channel capacity for visible light communications,” IEEE Communications Letters, vol. 20, no. 1, pp. 97–100, 2016. [13] A. Lapidoth, S. M. Moser, and M. A. Wigger, “On the capacity of free-space optical intensity channels,” IEEE Transactions on Information Theory, vol. 55, no. 10, pp. 4449–4461, 2009. [14] R. R. Chen, B. Hajek, R. Koetter, and U. Madhow, “On fixed input distributions for noncoherent communication over high-snr rayleigh-fading channels,” vol. 50, no. 12, pp. 3390–3396, 2004. [15] G. Aminian, H. Arjmandi, A. Gohari, M. Nasiri-Kenari, and U. Mitra, “Capacity of diffusionbased molecular communication networks over lti-poisson channels,” IEEE Transactions on Molecular, Biological and Multi-Scale Communications, vol. 1, no. 2, pp. 188–201, 2015. 32 [16] S. Ihara, Information theory for continuous systems. Singapore: World Scientific, 1993. [17] T. M. Cover and J. A. Thomas, Elements of information theory. New York: John Wiley & Sons, 2nd ed., 2006. [18] F. Topsoe, “An information theoretical identity and a problem involving capacity,” Studia Scientiarum Math. Hungarica, vol. 2, no. 10, p. 291292, 1967. [19] H. Ghourchian, A. Gohari, and A. Amini, “Existence and continuity of differential entropy for a class of distributions,” IEEE Communications Letters, 2017. [20] E. M. Stein and R. Shakarchi, Real analysis: measure theory, integration, and Hilbert spaces. New Jersey: Princeton University Press, 2005. [21] A. El Gamal and Y.-H. Kim, Network Information Theory. Cambrdige University press, 2011. A Proof of Equation (9) Take some arbitrary x > 0. Then, equation (9) holds because 2 Z ∞ √ Z ∞ −(y−v 2 )2 1 2 − (y−v2 22)2 v +y v2 − y √ √ √ e 2c v dv = + √ e 2c2 v2 dv π v 2 2c2 v 2 2c2 πc2 0 0 Z ∞ Z ∞ 2 )2 2 )2 2y 1 v 2 − y −(y+v 1 v 2 + y −(y−v 2 2 2 √ e 2c v dv + √ e 2c2 v2 dv √ √ =e c π v 2 2c2 π v 2 2c2 0 0 Z 1 Z ∞ 2 2 2 )2 2 −(y+v ) 2y 2y 1 v −y 1 v 2 − y −(y+v √ √ √ e 2c2 v2 dv + e c2 √ e 2c2 v2 dv =e c2 π v 2 2c2 π v 2 2c2 0 1 Z Z 1 2 )2 2 )2 ∞ 1 v 2 + y −(y−v 1 v 2 + y −(y−v √ e 2c2 v2 dv + √ e 2c2 v2 dv. √ √ + π v 2 2c2 π v 2 2c2 1 0 Now utilize the change of variables y − v2 v 7→ u1 = √ , v 2c2 y + v2 v 7→ u2 = √ v 2c2 to re-express the above integrals. Note that du1 = − v2 + y √ dv, v 2 2c2 du2 = v2 − y √ dv. v 2 2c2 For y > 0, if v = 0 √ then u1 = +∞, u2 =√+∞, if v = +∞ then u1 = −∞, u2 = +∞ and if v = 1 then u1 = (y − 1)/ 2c2 , u2 = (y + 1)/ 2c2 . For y < 0, if v = 0√then u1 = −∞, u2√= −∞, if v = +∞ then u1 = −∞, u2 = +∞ and if v = 1 then u1 = (y − 1)/ 2c2 , u2 = (y + 1)/ 2c2 . Now for y > 0 we have −e 2y c2 Z ∞ y+1 √ 2c2 Z ∞ = −∞ 2y 1 2 √ e−u2 du2 + e c2 π Z ∞ y+1 √ 2c2 1 2 √ e−u2 du2 + π 1 2 √ e−u1 du1 = 1. π 33 Z ∞ y−1 √ 2c2 1 2 √ e−u1 du1 + π Z y−1 √ 2c2 −∞ 1 2 √ e−u1 du1 π Similarly, for y < 0 we have e 2y c2 y+1 √ Z ∞ Z √y−1 Z √y−1 2y 1 −u22 1 1 1 2 2 2 2 2c 2c2 −u −u √ e √ e 2 du2 − √ e 1 du1 + √ e−u1 du1 du2 + e c2 y+1 π π π π √ −∞ −∞ −∞ 2c2 Z ∞ 2y 2y 1 2 √ e−u1 du1 = e c2 . = e c2 π −∞ Z 2c2 Therefore, the proof is complete. 34	7
Stochastic bandits robust to adversarial corruptions arXiv:1803.09353v1 [cs.LG] 25 Mar 2018 Thodoris Lykouris∗ Vahab Mirrokni† Renato Paes Leme‡ Abstract We introduce a new model of stochastic bandits with adversarial corruptions which aims to capture settings where most of the input follows a stochastic pattern but some fraction of it can be adversarially changed to trick the algorithm, e.g., click fraud, fake reviews and email spam. The goal of this model is to encourage the design of bandit algorithms that (i) work well in mixed adversarial and stochastic models, and (ii) whose performance deteriorates gracefully as we move from fully stochastic to fully adversarial models. In our model, the rewards for all arms are initially drawn from a distribution and are then altered by an adaptive adversary. We provide a simple algorithm whose performance gracefully degrades with the total corruption the adversary injected in the data, measured by the sum across rounds of the biggest alteration the adversary made in the data in that round; this total corruption is denoted by C. Our algorithm provides a guarantee that retains the optimal guarantee (up to a logarithmic term) if the input is stochastic and whose performance degrades linearly to the amount of corruption C, while crucially being agnostic to it. We also provide a lower bound showing that this linear degradation is necessary if the algorithm achieves optimal performance in the stochastic setting (the lower bound works even for a known amount of corruption, a special case in which our algorithm achieves optimal performance without the extra logarithm). ∗ Cornell University, [email protected]. Work supported under NSF grant CCF-1563714. Part of the work was done while the author was interning at Google. † Google Research, [email protected] ‡ Google Research, [email protected] 1 Introduction In online learning with bandit feedback, a learner needs to decide at each time between alternative actions or arms of unknown quality, facing a trade-off between exploiting profitable past actions or exploring new actions about which she has little information. Bandit problems are typically classified according to how the rewards are generated. In stochastic bandits, rewards are drawn from fixed but unknown distributions, which models settings where the alternatives follow particular patterns and do not react to the learner. The other extreme is adversarial bandits, which are robust to rewards that are specifically designed to trick the learner, as in game-theoretic settings. In this paper, we focus on settings where the overall behavior is essentially stochastic but a small fraction of the rewards can be adversarially changed. Classic stochastic bandit algorithms, like Upper Confidence Bound (UCB) [ACBF02] or Active Arm Elimination (AAE) [EMM06], base most of their decisions on a few observations made in an initial phase of the algorithm and therefore can be easily tricked into incurring linear regret if very few arms are corrupted. Adversarial bandit algorithms like EXP3 are not fooled by such tricks, but cannot exploit the fact that the input is mostly stochastic. Our goal is to robustify the stochastic setting by designing algorithms that can tolerate corruptions and still be able to exploit the stochastic nature of the input. The algorithms we design are agnostic to the corruption, i.e. they can tolerate any level of corruption, and the guarantee degrades gracefully as more corruption is added. Moreover, we prove lower bounds showing that our results are tight up to a logarithmic factor. Before we explain our technical contribution in detail, we describe examples of settings we have in mind. Click fraud In pay-per-click online advertising, the platform selects for each pageview an ad to display and obtains a certain reward if the user clicks on the ad. The click probabilities are unknown. The tension between repeatedly displaying a particular profitable ad that provides reliable revenue and exploring other potentially more rewarding options is a major application of stochastic bandits in the ads industry. If it weren’t for a phenomenon known as click fraud, this would be a textbook example of stochastic bandits. In click fraud, botnets maliciously simulate users clicking on an ad to trick learning algorithms. One example is a bot consistently making searches to trigger some ad and not clicking on it to make it seem like a certain ad has very low click-through-rate in order to boost its competitor. Recommendation systems: A platform recommending activities or services to a user faces the same trade-off. Suggesting new restaurants leads to faster learning of the best spots but may result to dissatisfaction of the customers who are led to disappointing experiences. While most inputs follow a stochastic pattern, some inputs are typically corrupted: either maliciously, e.g. fake reviews by competitors, or non-maliciously, e.g. construction next-door makes the restaurant less desirable in certain interval. This corruption may again exhibit arbitrary patterns and is not identically distributed over time, yet it is dwarfed by the fact that most of the input is stochastic. There are several other such examples: emails mostly follow a stochastic pattern except a fraction of them which are spam and are designed to trick algorithms. Internet searches follow a predictable pattern except certain spikes caused by unpredictable events. Data collection used in the econometric process often suffers from errors that affect a small part of the input. In all those cases, the vast 1 majority of the input follows a predictable pattern, but a fraction of the samples are corrupted. 1.1 Our contribution Our model. In this paper, we introduce a new model of stochastic bandits with adversarial corruptions. The goal of this model is to encourage the design of bandit algorithms that (i) work well in mixed adversarial and stochastic models, and (ii) whose performance deteriorates gracefully as we move from fully stochastic to fully adversarial models. In this model there are K arms, each associated with a fixed reward distribution F(a). At each round t, a random reward rSt (a) ∼ F(a) is drawn and an adversary can change the reward to r t (a), possibly using information about the realizations of rSτ (a) from both the current and previous rounds τ ≤ t as well as the probability that the learner puts on each arm. The learner then draws an arm at and obtains r t (at ) both as P reward and feedback. We say that the adversary is C-corrupted if in every sample path we have t maxa \|r t (a) − rSt (a)\| ≤ C. Our results. The main result (Theorem 3 in Section 3) is a learning algorithm we term Multi-layer Active Arm Elimination Race that with probability 1 − δ has regret   X K · C log(KT/δ ) + log(T ) · log(KT/δ ) O ∆(a) ⋆ a6=a where ∆(a) is the gap of arm a, i.e. the difference in stochastic means of arm a and the optimal √ 1 arm a⋆ . For arms with √ very small gap, i.e. when ∆(a) ≤ / T , the inverse dependence on the gap can be replaced by T . It is possible to improve the bound by i.e. Pa log factor for pseudo-regret, K·C+log(T ) · log(KT ) . Two maximum expected regret against any fixed arm, obtaining: O a6=a∗ ∆(a) important features of the algorithm are that the guarantee is: • Agnostic: The algorithm does not need to know the corruption level C. The guarantee is provided with respect to how much corruption was added in retrospect. If the corruption level is known, we can remove the dependence on K · log(KT/δ ) as shown in Theorem 1. • High Probability: Our bounds hold with high probability which is important for practical applications as the ones described above. In contrast, the weaker definition of pseudo-regret often hides events with large regret that are offset by events with large negative regret. The stochastic case corresponds to C = 0 in which case we recover P a bound that is slightlyworse T than the guarantee provided by UCB. Our algorithm obtains O a6=a⋆ log(T ) · log( /δ )/∆(a) with probability 1 − δ, while UCB obtains this bound without the log(T ) term. En routeto the result, inTheorem 2 we show an algorithm that, for any fixed known C, provides 2 P P KT log(KT/δ) regret O for stochastic input and O K · C · a6=a⋆ (log(∆(a)/δ )) if it is C-corrupted. a6=a⋆ ∆(a) In other words, if we only need to tolerate either a known level C or zero corruptions, we save a logarithmic factor from the bound, and match the bound provided by UCB in the stochastic case. Another question is whether the linear dependence on the corruption level is tight. In Section 4, we show that it cannot be improved upon without decay in the stochastic guarantee (i.e. while still guaranteeing logarithmic regret when the input is stochastic). The lower bound is an adaptation from the adversarial to the corrupted setting of a result from Auer and Chiang [AC16]. This holds 2 even for the case where the corruptions are either 0 or a known level C (where our algorithm provides a matching upper bound). We prove in Theorem 4 that an algorithm with pseudo-regret O(log(T )/∆) in the stochastic setting (C = 0) then for every constant ǫ > 0, there is a O(T ǫ )-corrupted instance where the algorithm incurs regret Ω(T ǫ ) with constant probability. Our algorithm can also be viewed through the lens of the best of both worlds literature [BS12, SS14, AC16, SL17], where the goal is to design algorithms that simultaneously provide logarithmic regret guarantees in the stochastic regime and square-root guarantees in the adversarial. In Section 5, we sketch how our algorithm can be appropriately modified to obtain, for any constant 0 < a < 1/2, 1/2 a a+ e e O(C) pseudo-regret for C = O(T ) and O T pseudo-regret otherwise. We observe that the results in the best of both worlds literature correspond to the case where a = 0. We note that such bounds are obtained for pseudo-regret and not regret with high-probability. Our techniques. The starting point of our design are classical stochastic bandit learning algorithms like UCB and Active Arm Elimination. Such algorithms are very susceptible to corruptions since they base most of their decisions on a small initial exploration phase. Therefore, with a small number of corruptions it is possible to completely trick the algorithm into eliminating the optimal arm. We address this issue by robustifying them using a multi-layer approach. The learning algorithm consists of multiple layers running in parallel. The layers have decreasing speed and increasing tolerance to corruption. The first layer finishes very fast selecting an arm as optimal, but provides no tolerance to corruption. Subsequent layers are more robust but also slower. The resulting algorithm is a race between different layers for picking the optimal arm. Once the fastest layer finishes, it provides a first crude estimate of the optimal arm. Once slower layers finish, we obtain finer and finer estimates of the optimal arm. Our second main idea is that we can obtain more robust algorithms by subsampling. If a layer is only selected with probability p, it only receives in expectation a p-fraction of the corruption injected by the adversary. If p is low enough, the layer behaves almost as if it was stochastic. Finally, we couple the different layers together by a process of global eliminations. This process enables slower layers to eliminate arms in faster layers. Such a process is necessary for preventing inaccurate layers from pulling suboptimal arms too often. 1.2 Related work Online learning with stochastic rewards goes back to the seminal work of Lai and Robbins [LR85]. The case of adversarial rewards was introduced by Auer et al. [ACBFS03]. The reader is referred to the books of Cesa-Bianchi and Lugosi [CBL06], Bubeck and Cesa-Bianchi [BCB12], and Slivkins [Sli17] for an elaborate overview of the area. These two extremes suffer from orthogonal problems; the one is overoptimistic expecting that all rewards come from the same distribution while the other one is too pessimistic in order to be protected against malicious adversaries. Our work addresses the middle ground: rewards come from distributions but are often adversarially corrupted. This is motivated by the non-robustness of stochastic learning algorithms to even small corruption levels. Closely related to our work lie the works on best of both worlds guarantees [BS12, SS14, AC16, SL17]. These works achieve (up to logarithmic factors) the optimal pseudo-regret guarantee for stochastic rewards and the optimal pseudo-regret or actual regret guarantee for adversarial rewards. Bubeck 3 and Slivkins [BS12] and Auer and Chiang [AC16] begin from a stochastic algorithm and test whether they encounter non-stochastic behavior in which case they switch to adversarial algorithm. In contrast, Seldin et al. [SS14, SL17] begin from an adversarial algorithm with very optimistic learning rate and adapt it if they encounter such behavior. Recently and independently to this work, Wei and Luo [WL18] provide a best of both worlds result with a small-loss pseudo-regret guarantee on the adversarial setting, via a novel analysis of the log-barrier OMD algorithm of Foster et al. [FLL+ 16]. Although the aforementioned algorithms are very elegant, their analysis is not robust to inputs that are slightly away from stochastic. Our work bridges this gap by designing algorithms with a more smooth behavior for close-to-stochastic instances. There have been other works that attempt to provide improved guarantees than the adversarial setting when instances are well behaved. Hazan and Kale [HK09] offer regret guarantees that scale with the variance of the losses instead of the time horizon. This guarantee is meaningful in settings that have a very predictable nature and have usually the same performance such as routing. However they do not address most applications of stochastic bandits. In Click Fraud, for example the rewards come from Bernoulli distributions and the variance of such a distribution is high even if the input is totally stochastic. Another approach is the work of Shamir and Szlak [SS17], who consider an input that is adversarial but random local permutations are applied to obtain a more benign instance. This approach is very relevant in settings like buffering, but is again not applicable to our settings. On the opposite side, attempting to provide improved guarantees for the stochastic setting or enhancing their range is a very active area of research. For instance, the MOSS algorithm [AB09] of Audibert and Bubeck provides the optimal non-distribution-based upper bound for stochastic bandits while retaining the optimal distribution-based stochastic guarantee. The KL-UCB algorithm of Garivier and Cappé [GC11] provides improved constants in the upper bound of the stochastic guarantee matching the lower bound of Lai and Robbins [LR85] for Bernoulli rewards. The Robust UCB algorithm [BCBL13] extends the results to non-bounded rewards replacing with the weaker assumption of bounded variance. However, all the above results are not robust to corruptions from an adaptive adversary due to their deterministic nature. Since the adversary knows the arm the learner will select, they can always corrupt the optimal arm whenever it is about to be selected and therefore cause the learner to either play it multiple times even if it is suboptimal or decide against playing it even with a small amount of corruption (similarly as in our lower bound). There is also prior work on incorporating corruptions in online decision making. In the online learning front, there are two such attempts, to the best of our knowledge. In their best of both worlds result, Seldin and Slivkins [SS14] allow for some contamination in the data as long as they are obliviously selected and they do not decrease the gap by more than a factor of 2. The second work is a recent paper by Gajane et al. [GUK18] who suggest a model of corrupted feedback aiming for differential privacy. Unlike our model, their corruptions are neither adversarial nor adaptive. Both of these works make benign assumptions about the nature of corruption and do not address the main roadblock in the settings we consider: an adversarial saboteur will try to add faulty data in the beginning to change the order between the two arms and, with a minimal corruption, she will achieve this goal. Closer to our model are the works on robust allocation such as online matching with corrupted data [MGZ12, EKM15]; unlike online matching though, in online learning we cannot evaluate the optimum at every round since the algorithm’s decisions affect the information it observes. Last, learning in the presence of corruptions has recently received great attention in the batch learning setting. For instance, recent works study inference under the presence of adversarially corrupted data [MRT15], designing estimators that are robust to corrupted data [DKK+ 16], learn4 ing in auctions with some faulty data due to econometrics errors [CD17]. Our work suggests a similar framework for the study of online learning that is robust to adversarial corruptions in the more challenging problem of sequential decision making where decisions also affect the information observed. 2 Model Corrupted stochastic bandits. We study an online bandit learning setting with K arms. Each arm a ∈ [K] is associated with a distribution F(a) with mean µ(a). The distributions are assumed to have positive measure only on rewards in [0, 1] and are unknown to the learner. We refer to the optimal arm as a⋆ = arg maxa µ(a) and define ∆(a) = µ(a⋆ ) − µ(a).1 We consider an adversary who can corrupt some of the stochastic rewards. The adversary is adaptive, in the sense that the corrupted rewards can be a function of the realization of the stochastic rewards up to that point and of the learner’s choices in previous rounds. More formally, the protocol between learner and adversary, at each round t = 1 . . . T , is as follows: 1. The learner picks a distribution wt over the K arms. 2. Stochastic rewards are drawn for each arm: rSt (a) ∼ F(a). 3. The adversary observes the realizations of rSt (a) as well as rewards and choices of the learner in previous steps and returns a corrupted reward r t (a)∈ [0, 1]. 4. The learner draws arm at ∼ wt and observes r t at . We refer to maxa \|r t (a) − rSt (a)\| as the amount of corruption injected in round t. The instance is C-corrupted if the total injected corruption is at most C for all realizations of the random variables: X max\|r t (a) − rSt (a)\| ≤ C t a Note that the adversary is assumed to be adaptive, in the sense that she has access to all the realizations of random variables for all rounds τ < t and the realization of rewards at round t but only knows the player’s distribution at round t and not the arm at . Our guarantees gracefully degrades with the total corruption injected by the adversary. Regret notions. Regret corresponds to the difference between the reward obtained by the algorithm and the reward of the best arm in hindsight: X Reg = max r t (a) − r t at a t The regret is a random variable that depends on the random rewards, the randomness used by the learner, and the randomness of the adversary. We say that a regret bound R(T, δ) holds with probability 1 − δ if P[Reg < R(T, δ)] > 1 − δ where the probability is taken over all the three sources of randomness described. We note that a⋆ is one arm with optimal mean and this does not preclude the existence of other arms with the same mean. If more than one such arms exist, let a⋆ be an arbitrary arm with optimal mean and the other arms a 6= a⋆ with optimal mean have gap ∆(a) = 0. 1 5 Finally pseudo-regret is a weaker notion that compares the expected performance of the learner with the arm with the highest expected performance. In other words: " # X PseudoReg = max E r t (a) − r t at a t We note that by Jensen’s inequality, PseudoReg ≤ E[Reg]. We often obtain improved bounds for pseudo-regret since it allows us to offset large positive regret events with large negative regret events. 3 The upper bound: Multi-layer Active Arm Elimination Race Active arm elimination. The starting point of our design is the Active Arm Elimination algorithm for stochastic bandits [EMM06], which can be viewed as an alternative presentation of the more famous UCB algorithm [ACBF02]. It is based on the following idea: in an initial exploration phase, we pull arms in a round-robin fashion and compute an estimate µ e(a) as the average empirical reward of arm a. After n(a) pulls of arm a, usual concentration arguments establish that with probability at p least 1 − 1/T Ω(1) , the difference of the empirical and actual means is at most wd(a) = O( log(T )/n(a)). We say that [e µ(a) − wd(a), µ e(a) + wd(a)] is the confidence interval of arm a. This means in particular that given two arms a and a′ , if the difference in empirical means becomes larger than the widths of the confidence intervals, i.e., µ e(a) − µ e(a′ ) > wd(a) + wd(a′ ), then with ′ high probability arm a is not optimal. Once this happens, the algorithm eliminates arm a′ by removing it from the round-robin rotation. After both arms a and the optimal arm a⋆ are pulled O(log(T )/∆(a)2 ) times, the confidence intervals will be small enough that arm a will be eliminated. Eventually all arms but the optimal are eliminated and we enter what is called the exploitation phase. In this phase we only pull the arm with optimal mean. Before we enter exploitation we pulled each suboptimal arm a at most O(log(T )/∆(a)2 ) times. Each of thoseP suboptimal pulls incurs regret ∆(a) in expectation which leads to the pseudo-regret bound of O( a6=a⋆ log(T )/∆(a)). This bound can also be converted to a high probability bound if we replace log(T ) by log(T/δ ). √ Arms with small ∆(a). We note that, for the arms that have ∆(a) < 1/ T , the inverse dependence on the gap may initially seem vacuous; for instance, when there are two optimal arms a, a⋆ with the same mean, the upper bound becomes infinite as ∆(a) = 0. However,√the inverse dependence on the gap can be replaced by ∆(a) · T in the case of pseudo-regret and T in the case of actual regret (due to variance reasons). For simplicity of exposition, we omit this in the current section but we demonstrate how to perform this replacement in Section 5. 3.1 Enlarged confidence intervals The active arm elimination algorithm is clearly not robust to corruption since by corrupting the first O(log T ) steps, the adversary can cause the algorithm to eliminate the optimal arm. As the algorithm never pulls the suboptimal arms after exploration, it is not able to ever recover. One initial idea to fix this problem is to enlarge the confidence intervals. We can decompose the rewards r t (a) in two terms rSt (a) + ct (a) where the first term comes from the stochastic reward and the second is the corruption introduced by the adversary. If the total corruption introduced by the 6 p adversary is at most C, then with width wd(a) = O( log(T )/n(a) + C/n(a)), a similar analysis to above gives us the following regret bound: Theorem for thetotal corruption q1. If C is a valid upper bound then active arm elimination with P log(KT/δ)+C log(2KT/δ) C with probability 1 − δ. + n(a) has regret O wd(a) = a6=a⋆ n(a) ∆(a) Proof sketch. The proof follows the standard analysis of active arm elimination. We first establish that, with high probability the optimal arm a⋆ is never inactivated (Lemma 3.1) and then upper bound the number of times each suboptimal arm is played (Lemma 3.2). The pseudo-regret guarantee directly follows by multiplying the number of plays for each arm by its gap ∆(a). For the high-probability guarantee, we need to also show that the regret incurred in the meantime is not much more than the above. We provide proof details about the theorem and lemmas in Appendix A. Lemma 3.1. With probability at least 1 − δ, arm a⋆ never becomes inactivated. Lemma 3.2. With probability at least 1 − δ, all arms a 6= a⋆ become inactivated after N (a) = 36 log(2KT/δ)+6C plays. ∆(a)2 3.2 Stochastic bandits robust to known corruption The drawback of the active arm elimination algorithm with enlarged confidence intervals (Theorem 1) is that, even if there are no corruptions, it still incurs a regret proportional to C.As a warm up to P log (KT/δ) the main theorem, we provide an algorithm that achieves the usual bound of O ⋆ a6=a ∆(a) P log(KT/δ)2 if the if the input is purely stochastic and, at the same time, achieves O K · C · a6=a⋆ ∆(a) input is C-corrupted for a known C. In the next subsection, we modify the algorithm to make it agnostic to the corruption level C. Two instances of Active Arm Elimination. The first idea is to run two instances of active arm elimination: the first is supposed to select the correct arm if there is no corruption and the second is supposed to select the right arm if there is C corruption. The first instance is very fast but it is not robust to corruptions. The second instance is slower but more precise, in the sense that it can tolerate corruptions. Since the second instance is more trustworthy, if the second instance decides to eliminate a certain arm a, we eliminate the same arm in the faster instance. Decrease corruption by sub-sampling. To keep the regret low if the input is stochastic, the second instance of active arm elimination cannot pull a suboptimal arm too many times. Therefore, the technique in Theorem 1 alone is not enough. The main idea of the algorithm is to make arm a behave as if it was almost stochastic by running the second instance with low probability. If the learner selects to run the second instance with probability 1/C then, when the adversary adds a certain amount of corruption to a certain round, the second instance observes that corruption with probability 1/C . Therefore, the expected amount of corruption the learner observes in the second instance is constant. This makes the arms behave almost like stochastic arms in that instance. Learning algorithm. We obtain our algorithm by combining those ideas. We have two instances of active arm elimination which we denote by F (fast) and S (slow). Each instance keeps an estimate of the mean µ eF (a) and µ eS (a) corresponding to the average empirical reward of that arm and also keeps track of how many times each arm was pulled in that instance nF (a) and nS (a). This F allows p us to define a notion of confidence interval in each of the instances. We define wd (a) = O( log(T )/nF (a)) as usual and for the slow instance we define slighly larger confidence intervals: 7 p wdS (a) = O( log(T )/nS (a) + log(T )/nS (a)) (the reason will be clear in a moment). Also, each instance keeps a set of eliminated arms for that instance: I F and I S . In each round, with probability 1 − 1/C we make a move in the fast instance: we choose the next active arm a in the round robin order, i.e., arm a ∈ [K] \ I F which was played less often, pull this arm and increase nF (a) and update µ eF (a) accordingly. As usual, if there are two active arms a and a′ such that µ eF (a) − µ eF (a′ ) > wdF (a) + wdF (a′ ) we eliminate a′ by adding it to I F . With the remaining probability we make a move in the slow instance by executing the exact same procedure as described for the other instance. There is only one difference (which causes the two instances to be coupled): when we inactivate an arm a in S we also eliminate it in F. This leaves us with a potential problem: it is possible that all arms in the F instance end up being eliminated. If we reach that point, we play an arbitrary active arm of the slow instance, i.e., any arm a ∈ [K] \ I S . The resulting algorithm is formally provided in Algorithm 1. Algorithm 1 Fast-Slow Active Arm Elimination Race for known corruption C 1: Initialize nℓ (a) = 0, µ eℓ (a) = 0, I ℓ = ∅ for all a ∈ [K] and ℓ ∈ {F, S} 2: For Rounds t = 1..T 3: Sample algorithm ℓ: ℓ = S with probability 1/C. Else ℓ = F. 4: If [K] \ I ℓ 6= ∅ 5: Play arm at ← arg mina∈[K]\I ℓ nℓ (a) 6: Update µ eℓ (at ) ← [nℓ (a)e µℓ (at ) + r t (at )]/[nℓ (a) + 1] and nℓ (a) ← nℓ (a) + 1 ′ 7: While exists arms a, a ∈ [K] \ I ℓ with µ eℓ (a) − µ eℓ (a′ ) > wdℓ (a) + wdℓ (a′ ) 8: Eliminate a′ by adding it to I ℓ 9: If ℓ = S then eliminate a′ from the other algorithm by adding it to I F 10: Else 11: Play an arbitrary arm in the set [K] \ I S . Towards the performance guarantee, Lemma 3.3 bounds the amount of corruption that actually enters the slow active arm elimination algorithm, which enables the regret guarantee in Theorem 2. Lemma 3.3. In Algorithm 1, the slow active arm elimination algorithm S observes, with probability at least 1 − δ, corruption of at most ln(1/δ ) + 3 during its exploration phase (when picked with probability 1/C ). Proof sketch. If one cared just about the expected corruption that affects S, this is at most a constant number since the total corruption is at most C and it affects S with probability 1/C . To prove a high-probability guarantee we require a concentration inequality on martingale differences (since the corruptions can be adaptively selected by the adversary). We provide the details in Appendix B. q 8KT /δ ) log(8KT/δ ) Theorem 2. Algorithm 1 run with widths wdS (a) = + 2 log( and wdF (a) = nS (a) nS (a) q P P log(8KT/δ) log(KT/δ ) (log(KT/δ))2 has O for the stochastic case and O K · C · for ⋆ ⋆ a6=a a6=a ∆(a) ∆(a) nF (a) the C-corrupted case with probability at least 1 − δ. Proof sketch. The result for the stochastic case follows standard arguments for stochastic algorithms (since we obtain double the regret of this setting as we run two such algorithms with essentially the same confidence intervals). For the C-corrupted case, we establish via Lemma 3.3 an upper 8 bound on the corruption that will affect the slow active arm elimination algorithm S. Thanks to the sub-sampling, this upper bound is close to a constant instead of depending on C which allows to not incur dependence on C in the stochastic case. Having this upper bound, we can apply it to the algorithm of the previous section to get an upper bound on the number of plays of suboptimal arms in S. Since the algorithms are coupled, such a bound implies an upper bound on the regret that it can cause in F as well. This is because in expectation the arm is played at most K · C times more in F as it may be selected every single time in F prior to getting eliminated by S and F is selected C times more often than S. To obtain the above guarantee with high probability, we lose an extra logarithmic factor. The details of the proof are provided in Appendix B. 3.3 Stochastic bandits robust to agnostic corruption Multiple layers of active arm elimination. In the previous subsection we designed an algorithm with two layers: one is faster but cannot tolerate corruptions and the second one is slower but more robust. In order to be agnostic to corruption, we need to plan for all possible amounts of corruption. To achieve this, we introduce log(T ) layers. Each layer is slower but more robust than the previous one. We achieve that by selecting the ℓ-th layer with probability proportional to 2−ℓ . By the argument in the last section, if the corruption level is at most C, then each layer with ℓ ≥ log C will observe O(1) corruption in expectation and at most O(log T ) corruption with high probability. Global eliminations. We couple the log T instances through what we call global eliminations. If arm a is eliminated by the ℓ-th layer, then we eliminate a in all layers ℓ′ ≤ ℓ. This is important to prevent us from pulling arm a too often. If arm a is suboptimal and the adversary is C-corrupted, e 1/∆(a)2 ) in then arm a eventually becomes eliminated in the ℓ⋆ = ⌈log C⌉ layer after being pulled O( ⋆ ⋆ −ℓ e 2 then it takes O(C/∆(a) ) iterations until that layer. Since layer ℓ is played with probability 2 e C/∆(a)) from that arm. arm is eliminated globally, in which case we will have total regret at most O( Multi-layer active arm elimination race. We now describe our main algorithm in the paper. We call it a race since we view it as multiple layers racing to pick the optimal arm. The less robust layers are faster so they arrive first and we keep choosing (mostly) according to them until more robust but slower layers finish and correct or confirm the current selection of the best arm. The algorithm keeps ℓ = 1 . . . log(T ) different instances of active arm elimination. The ℓ-th instance has as state the empirical means of each arm µ eℓ (a), the number nℓ (a) of times each arm a was pulled and the set I ℓ of inactive arms. The p width of the confidence interval for arm a in the ℓ-th layer is implicitly defined as wdℓ (a) = O( log(T )/nℓ (a) + log(T )/nℓ (a)). In each round t we sample ℓ ∈ {1, . . . , log T } with probability 2−ℓ (with remaining probability we pick layer 1). When layer ℓ is selected, we make a move in the active arm elimination instance corresponding to that layer: we sample the active arm in that layer with the least number of pulls, i.e., arm a ∈ [K] \ I ℓ minimizing nℓ (a). In case [K] \ I ℓ is empty, we pull an arbitrary arm from ′ ′ [K] \ I ℓ for the lowest ℓ′ such that [K] \ I ℓ is non-empty. The way we couple different layers is that once arm a′ is eliminated in layer ℓ because there is another active arm a in layer ℓ such that µ eℓ (a) − µ eℓ (a′ ) < wdℓ (a) + wdℓ (a′ ) we eliminate arm a′ in 1 all previous layers, keeping the invariant that: I ⊇ I 2 ⊇ I 3 ⊇ . . .. Figure 1 provides an example of the state of the algorithm, which is formally defined in Algorithm 2. 9 ℓ=1 arm 1 arm 2 ... arm d µ e1 (1), n1 (1) µ e1 (2), n1 (2) ... µ e1 (d), n1 (d) µ e2 (1), n2 (1) ℓ=2 µ e3 (1), n3 (1) ℓ=3 .. . .. . ℓ = lg T µ e2 (2), n2 (2) µ e3 (2), n3 (2) ... ... .. . µ elg T (1), nlg T (1) µ elg T (2), nlg T (2) µ e2 (d), n2 (d) µ e3 (d), n3 (d) .. . ... µ elg T (d), nlg T (d) Figure 1: Example of the state of the algorithm: for each layer ℓ and arm a we keep the estimated mean µ eℓ (a) and the number of pulls nℓ (a). Red cells indicate arms that have been eliminated in that layer. If an arm is eliminated in a layer, it is eliminated in all previous layers. If a layer where all the arms are eliminated (like layer 1 in the figure) is selected, we play an arbitrary active arm with the lowest layer that contains active arms. Algorithm 2 Multi-layer Active Arm Elimination Race 1: Initialize nℓ (a) = 0, µ eℓ (a) = 0, I ℓ = ∅ for all a ∈ [K] and ℓ ∈ [log T ] 2: For Rounds t = 1..T 3: Sample layer ℓ ∈ [log T ] with probability 2−ℓ . With remaining prob, sample ℓ = 1 4: If [K] \ I ℓ 6= ∅ 5: Play arm at ← arg mina∈[K]\I ℓ nℓ (a) 6: Update µ eℓ (at ) ← [nℓ (a)e µℓ (at ) + r t (at )]/[nℓ (a) + 1] and nℓ (a) ← nℓ (a) + 1 7: While exists arms a, a′ ∈ [K] \ I ℓ with µ eℓ (a) − µ eℓ (a′ ) > wdℓ (a) + wdℓ (a′ ) ′ 8: Eliminate a′ by adding it to I ℓ for all ℓ′ ≤ ℓ 9: Else ′ 10: Find minimum ℓ′ such that [K] \ I ℓ 6= ∅ and play an arbitrary arm in that set. We now provide the main result of the paper, a regret guarantee for Algorithm 2. Theorem 3. Algorithm 2 which is agnostic to the coruption level C, when run with widths wdℓ (a) = q log(4KT ·log T/δ ) nℓ (a) + log(4KT ·log T/δ) nℓ (a)  has regret:  X K · C log(KT/δ ) + log(T ) O · log(KT/δ ). ∆(a) ∗ a6=a Proof sketch. Similarly to the previous theorem, the regret guarantee comes from the summation between layers that are essentially stochastic (where the corruption is below corruption level, their log(KT/δ) r regret. Since i.e. less than C ≤ 2 for layer r). From each of these layers, we incur O ∆(a) there are at most log(T ) such layers, the second term in the theorem is derived. The challenge is to bound the regret incurred by layers that are not robust to the corruption. However, there exists some layer ℓ⋆ that is above the corruption level. By bounding the amount of steps that this level will require in order to inactivate each arm a 6= a⋆ in the incorrect layers (via Lemma 3.2), we obtain similarly to Theorem 2 a bound on the regret caused by this arm in those layers. Since 10 we take the minimum such layer and the tolerance of layers is within powers of 2, the fact that its corruption level does not match exactly the corruption that occurred only costs an extra factor of 2 in the regret. The details of the proof are provided in Appendix C. 4 The lower bound For the two arms case where the gap between the arms is ∆ > 0, Theorem 2 presents an algorithm which achieves O(log T/∆) pseudo-regret if the input is stochastic and O(C log(T/δ )/∆) with probability 1 − δ if the input is at most C-corrupted. We show below that this dependence is tight. The lower bound (Theorem 4) adapts the technique of Auer and Chiang [AC16] from the adversarial to the corrupted setting. The main idea is that an algorithm with logarithmic regret in the stochastic setting cannot query the sub-optimal arm more than log(T )/∆2 times. This implies a long time period where the learner queries the input only constant number of times. By corrupting all rounds before this period, an adversary can make the optimal arm look sub-optimal and trick the learner into not pulling the optimal arm for long time, causing large regret. Theorem 5 adapts this argument bounding the expected positive regret E[Reg+ ] where x+ = max{x, 0}; the high probability bounds provided imply bounds on the expected positive regret. Both proofs are provided in Appendix D. Theorem 4. Consider a multi-armed bandits algorithm that has the property that for any stochastic input in the two arm setting, it has pseudo-regret bounded by c log(T )/∆, where ∆ = \|µ1 − µ2 \|. ′ For any ǫ, ǫ′ ∈ (0, 1), there is a corruption level C with T ǫ < C < T ǫ and a C-corrupted instance such that with constant probability the regret is Ω(C). Theorem 5. If a multi-armed bandits algorithm that has the property that for any stochastic input in the two arm setting, it has pseudo-regret bounded by c log1+α (T )/∆ for α < 1. For any ′ ǫ, ǫ′ ∈ (0, 1), there is a corruption level C with T ǫ < C < T ǫ and a C-corrupted instance such that E[Reg+ ] = Ω(T ǫ−δ ) for all δ > 0. 5 Extensions In this section, we discuss some extensions that our algorithm can accommodate. Definition of corruption. We presented all results measuring the corruption as the sum over all rounds of the maximum across arms of the corruption injected by the adversary: X max \|r t (a) − rSt (a) ≤ C. a t P t t In fact all our results can be improved via using C(a) = t \|r (a) − rS (a)\| and replacing C by max(C(a), C(a⋆ )) for summand a. More formally, our main theorem (Theorem P t 3) becomes: t Theorem 6. Algorithm 2 which is agnostic to the corruptions C(a) = t \|r (a) − rS (a)\|, when run q with widths wdℓ (a) = log(4KT ·log T/δ) nℓ (a) + log(4KT ·log T/δ) nℓ (a) has regret:  X K · max(C(a⋆ ), C(a)) · log(KT/δ ) + log(T ) · log(KT/δ ). O ∆(a) ∗  a6=a 11 The proof follows the same arguments since it only compares each arm a with a⋆ . This result is nice since the contribution of each arm to the regret is a function only of its own gap and the corruption injected to it and the one injected to arm a⋆ . The latter dependence on the corruption on the optimal arm is essential since the main attack we presented to the classical arguments only corrupts arm a⋆ – the lower bound of the previous section also only adds corruption to a⋆ . Dependence on the gap. In Section 3, all our guarantees have an inverse dependence on the gap ∆(a) of all arms a. Note that such a guarantee is completely meaningless for arms with a very small gap; for instance, if there exist two optimal arms then there is an arm a 6= a⋆ with ∆(a) = 0 which makes the presented bound infinite and therefore vacuous. As we hinted there though, this √ inverse dependence can be improved for arms with small ∆(a) ≤ 1/ T . Our proofs generally relied on setting an upper bound on the number of times that a suboptimal arm is played and thereby providing an upper bound on the regret they cause. √ For arms with ∆(a) ≤ 1/ T , an alternative analysis is to say that, even if they are erroneously selected every single time, we can upper bound the loss in performance they√cause. For pseudoregret, the performance loss if they were selected every single time is ∆(a)·T ≤ T ≤ log T/∆(a). For actual regret, one needs to also take into consideration the variance but, even if they are selected every single time, a Hoeffding bound shows that their total reward is with high probability at most √ T lower than its expectation. As a result, the inverse dependence on ∆(a) in our bound can be √ 1 1 replaced by min(∆(a) · T, /∆(a)) for pseudo-regret and min( T , /∆(a)) for actual regret. C can be Moreover, the careful reader may have noticed that in Theorem 1, the dependence ∆(a) replaced by a sole dependence on C without the gap. However, this does not extend to the subsequent theorems since the dependence on C there does not come from the upper bound on the corruption experienced (this is at most log T due to subsampling). Instead, the dependence on C comes from projecting the correct layer (smallest layer robust to corruption) to the previous layers via the number of times it will take to eliminate any suboptimal arm. Uncorrupted objective. In applications such as spam, the corruptions should not be counted as part of the rewards. Our algorithm provides the same guarantee in the case of uncorrupted rewards (the difference between the performances in the two objectives is at most C). One can also observe that the linear dependence on C is still necessary: consider 2 arms with ∆ = 1 and an adversary that corrupts the first C steps making them look identical. The learner has no better option than randomly selecting between the two which gives him a regret of C/2 under the uncorrupted objective. We note that, in this setting, the linear dependence is necessary unconditionally of the performance of the algorithm in the stochastic setting. Towards best of all worlds. In the previous section, we showed that a logarithmic dependence in the stochastic setting comes at the expense of linear dependence on C in the C-corrupted setting if we focus on actual regret. A very interesting direction is to achieve such an improvement with either a higher power on the logarithm in the stochastic setting or aiming for pseudo-regret instead. In fact, we can combine our algorithm with the SAPO algorithm of Auer and Chiang [AC16] and achieve a bicriteria guarantee for pseudo-regret. For an a < 1/2 specified by the algorithm, we achieve our guarantee if the corruption is C ≤ T a and at most T 1/2+a otherwise; notice that the case a = 0 corresponds to the best of both worlds. This is done via running the SAPO algorithm at the level a log(T ) with probability T −a instead of having higher layers. The SAPO algorithm guarantees that the pseudo-regret caused by any particular arm is at most logarithmic if the instance √ is stochastic and at most T if it is adversarial via a beautiful analysis that keeps negative regret of time intervals that have performed well to avoid testing eliminated arms too often. In our setting, if 12 the corruption level is less than T a , the instance behaves as stochastic causing at most logarithmic regret. Else the instance is corrupted and we can extrapolate the regret in this layer to the whole algorithm as arms that are eliminated in this layer are also eliminated before via global eliminations. √ Since the regret there is at most T and this is multiplied by T a , this implies a bound of T 1/2+a on pseudo-regret. Acknowledgements The authors would like to thank Sid Banerjee whose lecture notes on stochastic bandits proved very helpful, Andrés Munoz Medina, Karthik Sridharan, and Éva Tardos for useful discussions, Manish Raghavan for suggestions on the writeup, and the anonymous reviewers for the valuable feedback they provided that improved the presentation of the paper. References [AB09] Jean-Yves Audibert and Sébastien Bubeck. Minimax policies for adversarial and stochastic bandits. In Proceedings of the 22nd Annual Conference on Learning Theory (COLT), 2009. [AC16] Peter Auer and Chao-Kai Chiang. 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The crux of the proof lies in establishing that, with high probability, the upper bound of the confidence interval of a⋆ never becomes lower than the lower bound of the confidence interval of any other arm a and therefore a⋆ does not become eliminated. More formally, let µ eS (a) and µ e(a) be the empirical mean after n(a) samples of the stochastic part of the rewards and the empirical mean after n(a) samples of the corrupted rewards respectively. Recall that µ(a) is the mean of arm a. By Hoeffding inequality, for any arm a, with probability at least 1 − δ′ : s log(2/δ′ ) . (1) \|e µS (a) − µ(a)\| ≤ n(a) We set δ′ = δ/KT to establish that this holds for all arms and all time steps (after q arm a has been 2KT played n(a) times). As a result, for any arm a and any time: µ eS (a) ≤ µ(a) + log(n(a) /δ) and q 2KT µ eS (a⋆ ) ≥ µ(a⋆ ) − log(n(a⋆ )/δ) . Comparing now the actual (corrupted) empirical means, they can be altered by at most absolute C C and µ e(a⋆ ) ≥ µ eS (a⋆ ) − n(a corruption C. Hence µ e(a) ≤ µ eS (a) + n(a) ⋆) . Combining the above inequalities with the fact that the actual mean of a⋆ is higher than the one of a, i.e. µ(a⋆ ) ≥ µ(a), we establish that µ e(a) − µ e(a⋆ ) ≤ wd(a) + wd(a⋆ ) and therefore arm a⋆ is not eliminated. Since this holds for all times and arms, the lemma follows. Lemma 3.2 (restated) With probability at least 1 − δ, all arms a 6= a⋆ become eliminated after 2KT N (a) = 36·log(∆(a)/2δ )+6C plays. Proof. The proof stems from the following observations. By Lemma 3.1, arm a⋆ is with high probability never eliminated. After N (a) rounds, with high probability, the lower confidence interval of arm a⋆ is above the upper confidence interval of arm a. This comes from the fact that, after N (a) plays of arm a (and also of arm a⋆ since it is not eliminated), the empirical stochastic mean of a⋆ is, with high probability, at most ∆(a)/6 below its actual mean and similarly the empirical stochastic mean of arm a is at most ∆(a)/6 above its actual mean. Since the corruptions are upper bounded by C, they can only contribute to a decrease in the average empirical (corrupted) means by at most ∆(a)/6 which is not enough to circumvent the gap ∆(a). More formally, let µ eS (a) and µ e(a) denote the empirical means of the stochastic part of the rewards and the corrupted rewards respectively after N (a) plays of arm a. By the same Hoeffding inequality as in the proof of the previous lemma, with probability at least 1 − δ, it holds that \|e µS (a) − µ(a)\| ≤ q log(2KT/δ) N (a) . Therefore, with the same probability, µ eS (a⋆ ) − µ(a⋆ ) ≤ ∆(a) eS (a) ≤ ∆(a) 6 and µ(a) − µ 6 . 15 after 36·log(2KT/δ ) ∆(a)2 plays for both arm a and a⋆ : e(a) ≤ µ eS (a) + NC(a) . By The absolute corruption is at most C therefore µ e(a⋆ ) ≥ µ eS (a⋆ ) − NC(a) and µ the choice of N (a), we have C N (a) ≤ ∆(a) 6 . Combining with the above argument, this also implies that the widths are upper bounded by wd(a) ≤ ∆(a) 3 and wd(a⋆ ) ≤ ∆(a) 3 . Combining the above with the fact that the actual mean of a⋆ is ∆(a) higher than the one of a, i.e. µ(a⋆ ) − µ(a) = ∆(a), we establish C − wd(a) − wd(a⋆ ) N (a) ∆(a) ∆(a) ∆(a) − − >0 > µ(a⋆ ) − µ(a) − 2 · 6 3 3 µ e(a⋆ ) − µ e(a) − wd(a) − wd(a⋆ ) ≥ µ eS (a⋆ ) − µ eS (a) − 2 · As a result arm a becomes eliminated after N (a) plays if it is not already eliminated before. Theorem 1 (restated) If C is a valid upper bound for the total corruption then arm elimination q P log(2KT/δ) log(KT/δ )+C C with wd(a) = with probability 1 − δ. + n(a) has regret O a6=a⋆ n(a) ∆(a) Proof. The proof follows the classical stochastic bandit argument of measuring the regret caused by each arm a 6= a⋆ as a function of its gap ∆(a) and the number of times N (a) it is played as established by Lemma 3.2. For simplicity of presentation, we first provide the pseudo-regret guarantee. Pseudo-regret compares the expected performance of the algorithm to the expected performance one would have had, had they selected a⋆ throughout the whole time horizon. The expected performance when one uses a⋆ is µ(a⋆ ). The loss compared to that every time a 6= a⋆ is used instead is equal to its gap ∆(a). As a result, the expected contribution to pseudo-regret from suboptimal arm a 6= a⋆ is equal to N (a) · ∆(a). Lemma 3.2 establishes that with probability 1 − δ any suboptimal arm a is played at 2KT most N (a) = 36 log(∆(a)/2δ )+6C times. Each play of the suboptimal arm causes pseudo-regret of ∆(a). Multiplying the times by the expected regret per time the guarantee (which equals to the gap) and setting the failure probability δ to be some inverse polynomial of the time horizon T to ensure that the expected regret due to the bad event is at most a constant leads to the pseudo-regret guarantee. To turn the above into a high-probability guarantee, we need to show that the regret incurred during the steps that we pull arm a is not significantly higher than the expectation (therefore bounding the resulting variance). By the Hoeffding inequality of Lemma 3.1, the empirical cumulative reward of p arm a is, with high probability, at most N (a) log(2KT/δ ) less than its expectation. The same holds for arm a⋆ for these steps (its realized performance is at most this much more than its expectation). The probability that these statements do not hold for some arm or some time is at most δ. p Regarding arms a 6= a⋆ , the N (a) log(2KT/δ ) term can be upper bounded by O(N (a)∆(a)) by the definition of N (a): s s p 2KT/δ ) log( log(2KT/δ ) ≤ N (a) · ∆(a) ≤ N (a)∆(a) N (a) log(2KT/δ ) ≤ N (a) · N (a) 36 log(2KT/δ ) + 6C Regarding arm a⋆ , let a′ be the arm with the smallest gap. By Lemma 3.1, a⋆ never gets eliminated p but it is not necessarily the ex post optimal arm. In fact some other arm with ∆(a) ≤ 1/T may be the ex post optimal arm (arms with higher gap are with high probability not the ex post optimal arm by an analogous argument as in Lemma 3.2. However, by the same argument as above arm a⋆ 16 is with high probability at most N (a′ ) · ∆(a′ ) below its expectation and the ex post optimal arm is at most this much above its expectation. This gives a bound of N (a′ )∆(a′ ) that is caused by the case where a⋆ is not the ex post optimal arm. Therefore the actual regret from times that arm a is played is at most 2N (a)∆(a) where the one term comes from the expectation and the other from the aforementioned bounds on the variance. The corruption can increase any cumulative reward by at most C which is already existing in the regret bound. Replacing N (a) by Lemma 3.2, we obtain the high-probability guarantee. Note that the failure probabilities of the two lemmas are coupled as they correspond to the same bad events. B Supplementary material on Section 3.2 In this section, we provide the proof of Theorem 2. To handle the corruption, we bound with high probability the total corruption experienced by the slow active arm elimination instance S (Lemma 3.3). To deal with an adaptive adversary, we need a martingale concentration inequality; specifically we apply a Bernstein-style inequality introduced in [BLL+ 11] (Lemma B.1). Lemma B.1 (Lemma 1 in [BLL+ 11]). Let X1 , . . . , XT be a sequence of real-valued random numbers. Assume, for all t, that Xt ≤ R and that E[Xt \|X1 , . . . , Xt−1 ] = 0. Also let V = T X t=1 Then, for any δ > 0: E[Xt2 \|X1 , . . . , Xt−1 ]. " T X e−2 ·V Xt > R ln(1/δ) + P R t=1 # ≤δ Lemma 3.3 (restated) In Algorithm 1, the slow active arm elimination algorithm S observes, with probability at least 1 − δ, corruption of at most ln(1/δ ) + 3 during its exploration phase (when picked with probability 1/C ). Proof. The first observation is that the expected corruption encountered by algorithm S is at most a constant (total corruption of C encountered with probability 1/C ). The rest of the proof focuses on bounding the variance of this random variable (actual corruption encountered by the layer). Crucially, since we want to allow the adversary to be adaptive, we should not assume independence across rounds but only conditional independence (conditioned on the history) and this is why some more involved concentration inequality is necessary. Therefore we create a martingale sequence (actual corruption minus expected corruption) and apply a Bernstein-style concentration inequality. Let Zat be the corruption that is observed by the exploration phase of the algorithm if arm a is selected. For every round t, if adversary selects corruption Cat then Zat is therefore a random variable equal to Cat with probability 1/C and 0 otherwise. Given that the adversary is adaptive and may select the corruptions based on the realizations of the previous rounds, we need to use an appropriate concentration inequality. We use a Bernstein-style inequality, introduced in [BLL+ 11] (Lemma B.1). Initially we resolve the randomness conditioning on ℓ = S (the slow algorithm is selected). Since active arm elimination is deterministic, conditioned on selecting algorithm S, the 17 selected arm is deterministic. Let a(S, t) be the arm that would be selected if ℓ = S (which happens with probability 1/C ). The martingale sequence is now h i t t Xt = Za(S,t) − E Za(S,t) \| H(1 : t − 1) where H(1 : t) corresponds to the history up to round t. Note that E Xt2 \|X1 , . . . , Xt−1 Ca(S,t) 2 C − 1 Ca(S,t) 2 1 + Ca(S,t) − = C C C C 2 2 Ca(S,t) Ca(S,t) C −1 C − 1 Ca(S,t) 2 = ≤2· . + C C C C C t The last inequality holds as Ca(S,t) ∈ [0, 1] and Ca(S,t) ≤ C by the definition of C. Therefore, summing over all the rounds, X X Ca(S,t) 2 ≤ · 2 V = E Xt2 \|X1 , . . . , Xt−1 ≤ C C t t X t max Cat a ! ≤ 2. A trivial upper bound of \|Xt \| is R = 1, since the rewards are in [0, 1]. Applying Lemma B.1, we show that, w.p. 1 − δ: X Xt ≤ ln(1/δ ) + 2(e − 2) ≤ ln(1/δ ) + 2 t P The lemma then follows by adding the expected corruption of E t Za(S,t) \| H(1 : t − 1) ≤ 1 and therefore obtaining the bound of the statement on the corruption experienced: # " X X X t Za(S,t) Za(S,t) \| H(1 : t − 1) ≤ ln(1/δ ) + 3. = Xt + E t t t q log( /δ ) /δ) Theorem 2 (restated) Algorithm 1 run with widths wd (a) = + 2 log( and nS (a) nS (a) q P P KT KT 8KT (log( /δ ))2 log( / δ ) log( / δ ) has O for the stochastic case and O K · C · wdF (a) = ⋆ ⋆ F a6=a a6=a ∆(a) ∆(a) n (a) for the C-corrupted case with probability at least 1 − δ. S 8KT 8KT Proof. For the stochastic case, the bound follows via standard stochastic bandit arguments (similarly to the proof of Theorem 1 with C = 0) as for each of the two active arm elimination algorithms P (log(2KT/δℓ,S )) we incur, with probability 1 − δℓ,S regret O where δℓ,S = δ/4 is the failure a6=a⋆ ∆(a) probability of inequality (1), which governs the results in Lemmas 3.1 and 3.2, for each of ℓ ∈ {F, S}. The most interesting case is the C-corrupted setting. Let δS,C = δ/4 be the failure probability in Lemma 3.3. By Lemma 3.3, with probability at least 1 − δS,C , the actual corruption experienced by the slow active arm elimination algorithm is at most ln(1/δ) + 3 which is less than 2 log(2KT/δ ) for non-trivial values of K and T . Therefore we can apply the analysis of Theorem 1 with corruption level at least 2 log(2KT/δS,C ) and get a handle on the actual regret coming from the slow active arm elimination algorithm. 18 What is left is to bound the regret coming from the fast active arm elimination algorithm. Towards this goal, we bound the number of times that a suboptimal arm is played in the fast active arm elimination by the expected time that it remains active at the slow active arm elimination. By Lemma 3.2, arm a is played in the slow active arm elimination, with probability at least 1 − 3δ/4, at most 18 log(8KT/δ ) 16 log(2KT/δS,S ) + 2 log(2KT/δS,C ) ≤ . NS (a) = 2 ∆(a) ∆(a)2 Having a bound on the number of plays of the arm in the slow active arm elimination instance, we use this to bound the number of plays in the fast active arm elimination instance. In expectation, this is at most K · C · NS (a) times as every move in the slow active arm elimination occurs with probability 1/C and, at least 1/K of these moves are plays of a while it is still active. Since every time arm a is played it incurs pseudo-regret ∆(a), this provides the pseudo-regret guarantee. To obtain a high probability guarantee, let δm = δ/4KT and observe that with probability at least 1 − δm , we make one move at the slow arm elimination algorithm every O(C log(1/δm )) moves at the fast arm elimination algorithm. This can be seen by thinking the following process: One tosses coins with bias p = 1/C until she observes heads for the first time (heads is the p-biased event). After M tosses of the coins the probability that no heads have arrived is at most (1 − p)M . To ensure that 1/δm ) 1/δm ) , which is achieved by M = log( this is less than δm , we need to wait M ≥ log( p/(1−p) . log( 1 ) 1−p By union bound on the failure probabilities for each of those draws, we get that with failure probability δe = K · NS (a) · δm ≤ δ/4 (since NS (a) ≤ T as it is at most the time horizon), arm a gets inactivated in F after NF (a) = K · NS (a) · C · log(1/δe ) = 18 · C · K · (log(8KT/δ ))2 . ∆(a)2 The last part is to prove that the regret experienced throughout those rounds is not too large. This follows by the two applications of Hoeffding inequality as before for arms a and a⋆ , analogously to Theorem 1. Combining the above arguments the theorem follows. The total failure probability of the guarantee is δS,S + δS,C + δF,S + δe ≤ δ. C Supplementary material on Section 3.3 Theorem 3 (restated) Algorithm 2 which is agnostic to the coruption level C, when run with q 4KT ·log T log(4KT ·log T/δ) ℓ + log( nℓ (a) /δ) has regret: widths wd (a) = nℓ (a)  X K · C log(KT/δ ) + log(T ) · log(KT/δ ). O ∆(a) ∗  a6=a Proof. The proof follows similar arguments to the proof of Theorem 2. Specifically, for the layers that are above the corruption level C, by using the standard arguments described in Theorem log(2KT/δℓ,S ) 1, we establish a bound on the regret caused by any suboptimal arm a, with failure ∆(a) δ probability δℓ,S = /2 log T . Since there are log(T ) such levels, the regret coming from these layers is upper bounded by the second term of the theorem with failure probabilityδ/2. 19 For the layers ℓ that are not tolerant to the corruption, i.e. 2ℓ > C, we apply the same argument as in the proof of Theorem 2 and bound their regret via the number of plays they are played by the minimum layer that is robust to corruption ℓ⋆ = arg minℓ 2ℓ > C . Similarly as in the proof of the theorem we upper bound the number of plays Nℓ⋆ (a) of each suboptimal arm a at this layer (by exactly the same arguments), then bound the number of plays in the suboptimal layer via the same coin toss process as in that proof and, last bound the regret they incur during this part. Since we do not know the amount of corruption in advance (and this amount is adaptively selected), we also need to take a union bound on the number of layers so that the guarantee on Nℓ⋆ (a) holds for all layers simulataneously if they end up being correct; we therefore repeat the arguments in Theorem 2 with δℓ,C = δ/2 log T and δm ≤ δ/2KT log T . Last, we note that, since we used powers of 2 to increase the corruption among layers, the fact that we did not apply the arguments of Theorem 2 with the exact C but instead used a C ′ such that C < C ′ < 2C causes just an extra constant factor on the regret. D Supplementary material on Section 4 Theorem 4 (restated) Consider a multi-armed bandits algorithm that has the property that for any stochastic input in the two arm setting, it has pseudo-regret bounded by c log(T )/∆, where ′ ∆ = \|µ1 − µ2 \|. For any ǫ, ǫ′ ∈ (0, 1), there is a corruption level C with T ǫ < C < T ǫ and a C-corrupted instance such that with constant probability the regret is Ω(C). Proof. The proof follows a sequence of steps. Step 1: Analyze behavior in the stochastic case. Fix a constant ∆ ≤ 1/6 and observe how the algorithm behaves for the stochastic input that has Bernoulli arms of means (µ1 , µ2 ) = ( 21 − ∆, 12 ). Since in that setting the expected regret is the same as ∆ · E[T1 ] where T1 is the number of pulls of arm 1, it follows that E[T1 ] ≤ c log(T )/∆2 . Step 2: find a large interval that is hit with at most constant probability. We divide the ′ space between T ǫ and T ǫ into O(log(T )/(ǫ′ − ǫ)) intervals Ii = [3i−1 T ǫ , 3i T ǫ ) such that size of each interval is twice the size of all the previous intervals combined. For each interval i, let T1,i be the number of times that arm 1 is pulled in the i-th interval. Then, there exists an interval Ii = [C, 3C) such that E[T1,i ] ≤ c̃ := O(1/[(ǫ′ − ǫ)∆2 ]). Step 3: create an adversary that forces a lot of regret in interval i. The adversary is quite simple: for the first C steps, the arms are Bernoulli with means ( 12 − ∆, 21 ) and for the remaining timesteps, the arms are Bernoulli with means ( 21 + ∆, 12 ). We use E and P to refer to the probability law whe inputs are drawn with respect to ( 12 − ∆, 12 ) in all timesteps and E′ and P′ to refer to the probability law when the input is according to ( 12 − ∆, 21 ) in the first K steps and according to ( 21 + ∆, 12 ) onwards. Step 4: With constant probability arm 1 is pulled a constant number of times in Ii under both P and P′ . Under the probability law P, this follows directly from Markov’s inequality: c̃ = E[T1,i ] ≥ 2c̃ P[T1,i ≥ 2c̃], so: P[T1,i ≤ 2c̃] ≥ 12 . Denote by A the event that T1,i ≤ 2c̃. We want to argue that P′ [A] is also constant. In order to do that, let Z = (Z1 , Z2 , . . . , Z2c̃ ) be a vector storing in Zs the reward of arm 1 in the s-th time it is pulled in interval Ii . Notice that in both the stochastic and corrupted scenarios if the learner 20 observes the same values of Z she acts the exact same way. Therefore, if we condition on Z, the probability that she ends up pulling arm 1 for more than 2c̃ times is exactly the same. In other words: P[A\|Z] = P′ [A\|Z] Therefore: ′ P [A] = X z ′ P [Z = z] · P[A\|Z = z] ≥ 1 2 1 2 −∆ +∆ !2c̃ 1 P [Z = z] P[A\|Z = z] ≥ · 2 ′ 1 2 1 2 −∆ +∆ !2c̃ which is a constant. Step 5: concentration bounds for the regret incurred in each interval. We now define an event B that occurs with probability 1 − o(1) that captures all the concentration bounds we need for the proof. First, we requireP arm 1 to be the optimal arm. Let r tP (i) be the reward of arm i in ′ ′ 1 t time step t. We know that E [ t r (1)] = 2 T + ∆(T − 2C) and E [ t r t (2)] = 21 T . Since P all the rewards are independent we can use the Hoeffding bound to bound the probability P′ ( t ∆t < 0) where ∆t = r t (2) − r t (1): ! ! X X X C T ∆2 ′ t t ′ ′ 1−2 = o(1) ∆t − E ∆t > ∆(T − 2C) ≤ 2 exp − r (1) < r (2) ≤ P P 2 T t t t Now we establish some concentration on the regret that the learner achieves with respect to arm 1 in the intervals [1, C), [C, 3C) and [3C, T ). We note that if the learner pulls arm 1, she does not incur any regret. If she pulls arm 2, she incurs regret ∆t = r t (2) − r t (1) which can be positive or negative. To compute regret with respect to arm 1 in each of those intervals, we sample ∆t every time that the arm 2 is pulled. P Step 5a: interval [1, C). In this interval, E′ ∆t = −∆, so t∈[1,C) ∆t = −C∆. If Y is the number P of times arm 1 is pulled, then the regret is given by Ys=1 ∆s where in the previous expression we abuse notation and mean by ∆s the regret in the s-th time the arm 2 is pulled instead of the regret in the t-th period. Therefore: ! ! ! t C t Y X X X X ′ ′ ′ ∆s < −1.1∆C P ∆s < −1.1∆C ≤ ∆s < −1.1∆C ≤ P min P t≤C s=1 t=1 s=1 s=1 We then use the Hoeffding bound in the last expression and get: ! C X 1 1 1.1∆C − ∆t 2 2 ≤ C · exp − (0.1∆C) = o(1) 2 exp − t 2 t 2 t=1 Step 5b: interval [C, 3C). In this interval, E′ ∆t = ∆, so using the same bound as before, we get   X P′  ∆t < 1.9∆C  ≤ 2 exp −C(0.1∆)2 = o(1) t∈[C,3C) Step 5c: interval [3C, T ] In this interval, pulling arm 2 has again positive expected regret. We use the same technique used in 5a to argue that she cannot obtain large negative regret with high 21 probability: Let Y be the number of times arm 2 is pulled in that interval and again we abuse notation and let ∆s be the difference in rewards in the s-th time the arm is pulled. Then: ! ! ! t T t Y X X X X ∆t < − log T P′ ∆t < − log T ≤ ∆t < − log T ≤ P′ min P′ t≤T s=1 t=1 s=1 s=1 For t = 1.. log T this probability is zero, since ∆t ≥ −1. Now, for larger T , we can use the standard Chernoff bound: ! T t X X 1 ′ 2 ∆t < − log T ≤ 2T exp − log (T )∆ = o(1) P 2 s=1 t=log T Now all the concentration bounds have been established we define the event B to be the event where all those concentration bounds hold. More precisely, B is the event where the following four things happen: (a) empirically arm 1 is better than arm 2; (b) in interval [1, C), the regret of the learner is at least −1.1∆C; (c) in interval [C, 3C), the difference between the total rewards of both arms is at least 1.9∆C; and (d) the regret of the learer in interval [3C, T ] is at least − log T . By the discussion in step 5, we know that P′ (B) = 1 − o(1). Step 6: putting it all together. Since P′ (A) = Ω(1) and P′ (B) = 1 − o(1), then by the union bound, P′ (A and B) ≥ P′ (A) − o(1) = Ω(1). Now, we need to argue that in the constant probability event (A and B), the regret of the learner is at least Ω(C). We simply sum the regret of the learner in each of the intervals. For intervals [1, C) and [3C, T ] we can use the bounds computed in steps 5a andn 5c directly. For interval [C, 3C), we note that conditioned on A, the learner probes arm 1 a constant number of times, so his total regret differs from the regret by pulling arm 2 in all iterations by at most a constant, therefore the total regret can be bounded by: −1.1∆C + (1.9∆C − 4c̃) − log(T ) = Ω(∆C) We can adapt the argument to provide a bound on the expected positive regret E[Reg+ ] where x+ = max{x, 0}. Note that the high probability bounds provided also imply a bound on the expected positive regret. Theorem 5. If a multi-armed bandits algorithm that has the property that for any stochastic input in the two arm setting, it has pseudo-regret bounded by c log1+α (T )/∆ for α < 1. For any ′ ǫ, ǫ′ ∈ (0, 1), there is a corruption level C with T ǫ < C < T ǫ and a C-corrupted instance such that E[Reg+ ] = Ω(T ǫ−δ ) for all δ > 0. Proof. Modify the proof of Theorem 4 as follows. Define c̃ = O(logα (T )/[(ǫ′ − ǫ)∆2 ]) and again select an interval such that E[Ti,1 ] ≤ c̃. Event A is defined in the same way. By Markov’s inequality: P[A] ≥ 1/2 and !2c̃ 1 − ∆ 1 = exp(−O(logα (T ))) P′ [A] ≥ · 21 2 + ∆ 2 Step 5 remains unchanged and in step 6 note that P′ (B) ≪ P′ (A) since α < 1, so P′ (A and B) = exp(−O(logα (T ))). Therefore, with probability at least exp(−O(logα (T ))) the regret is at least Ω(C) = Ω(T ǫ ) and therefore, E[Reg+ ] = Ω(T ǫ · exp(−O(logα (T )))) = Ω(T ǫ−δ ) for all δ > 0. 22	8
Representation Learning and Recovery in the ReLU Model arXiv:1803.04304v1 [stat.ML] 12 Mar 2018 Arya Mazumdar1 and Ankit Singh Rawat1,2 1 College of Information and Computer Sciences, University of Massachusetts Amherst, Amherst, MA 01003, USA. 2 Research Laboratory of Electronics, MIT, Cambridge, MA 02139, USA. E-mail: [email protected], [email protected]. March 13, 2018 Abstract Rectiﬁed linear units, or ReLUs, have become the preferred activation function for artiﬁcial neural networks. In this paper we consider two basic learning problems assuming that the underlying data follow a generative model based on a ReLU-network – a neural network with ReLU activations. As a primarily theoretical study, we limit ourselves to a single-layer network. The ﬁrst problem we study corresponds to dictionary-learning in the presence of nonlinearity (modeled by the ReLU functions). Given a set of observation vectors yi ∈ Rd , i = 1, 2, . . . , n, we aim to recover d × k matrix A and the latent vectors {ci } ⊂ Rk under the model yi = ReLU(Aci + b), where b ∈ Rd is a random bias. We show that it is possible to recover the column space of A within an error of O(d) (in Frobenius norm) under certain conditions on the probability distribution of b. The second problem we consider is that of robust recovery of the signal in the presence of outliers, i.e., large but sparse noise. In this setting we are interested in recovering the latent vector c from its noisy nonlinear sketches of the form v = ReLU(Ac) + e + w, where e ∈ Rd denotes the outliers with sparsity s and w ∈ Rd denote the dense but small noise. This line of work has recently been studied (Soltanolkotabi, 2017) without the presence of outliers. For this problem, we q show that a generalized LASSO algorithm is able to recover the signal c ∈ Rk within an ℓ2 error of O( random Gaussian matrix. (k+s) log d ) d when A is a 1 Introduction Rectiﬁed Linear Unit (ReLU) is a basic nonlinear function deﬁned to be ReLU : R → R+ ∪ {0} as ReLU(x) ≡ max(0, x). For any matrix X , ReLU(X ) denotes the matrix obtained by applying the ReLU function on each of the coordinates of the matrix X . ReLUs are building blocks of many nonlinear data-ﬁtting problems based on deep neural networks (see, e.g., Soltanolkotabi [2017] for a good exposition). Let Y ⊂ Rd be a collection of message vectors that are of interest to us. Depending on the application at hand, the message vectors, i.e., the constituents of Y, may range from images, speech signals, network access patterns to user-item rating vectors and so on. We assume that the message vectors satisfy a generative model, where each message vector can be approximated by a map д : Rk → Rd from the latent space to the ambient space, i.e., for each y ∈ Y, y ≈ д(c) for some c ∈ Rk . 1 (1) Motivated by the recent results on developing the generative models for various real-life signals (see e.g., Goodfellow et al. [2014], Kingma and Welling [2014], Bora et al. [2017]), the non-linear maps д that take the following form warrant special attention. д(c) = h (Al h(Al−1 · · · h(A1 c) · · · ))) , (2) i.e., д is the function corresponding to an l-layer neural network with the activation function h. Here, for i ∈ [l], Ai ∈ Rdi ×di −1 with d0 = k and dl = d, denotes the weight matrix for the i-th layer of the network. In the special case, where the activation function h is the ReLU function, the message vectors of the interest satisfy the following. y ≈ ReLU (Al ReLU(Al−1 · · · ReLU(A1 c + b1 ) · · · + bl−1 ) + bl )) , (3) where, for i ∈ [l], bi ∈ Rdi denotes the biases of the neurons (or output units) at the i-th layer of the network. The speciﬁc generative model in (3) raises multiple interesting questions that play fundamental role in understanding the underlying data and designing systems and algorithms for processing the data. Two such most basic questions are as follows: 1. Learning the representation: Given the observations {y1 , y1 , . . . , yn } ⊂ Rd from the model (cf. (3)), recover the parameters of the model, i.e., {Ât }t ∈[l] , and {c1 , c2 , . . . , cn } ⊂ Rk such that yi ≈ ReLU Âl ReLU(Âl−1 · · · ReLU(Â1 ci + b1 ) · · · + bl−1 ) + bl ) ∀ i ∈ [n]. (4) Note that this question is diﬀerent from training the model, in which case the set {c1 , c2, . . . , cn } is known (and possibly chosen accordingly). 2. Recovery of the signal in the presence of errors: Given the erroneous (noisy) version of a vector generated by the model (cf. (3)), denoise the observation or recover the latent vector. Formally, given v = y + e + w = ReLU (Al ReLU(Al−1 · · · ReLU(A1 c + b1 ) · · · + bl−1 ) + bl )) + e + w (5) and the knowledge of model parameters, obtain ŷ ∈ Rd or ĉ ∈ Rk such that ky − ŷk or kc − ĉk is small, respectively. In (5), e and w correspond to outliers, i.e., large but sparse errors, and (dense but small) noise, respectively. Apart from being closely related, one of our main motivations behind studying these two problems together comes from the recent work on associative memory Karbasi et al. [2014], Mazumdar and Rawat [2015, 2017]. An associative memory consists of a learning phase, where a generative model is learned from a given dataset; and a recovery phase, where given a noisy version of a data point generated by the generative model, the noise-free version is recovered with the help of the knowledge of the generative model. There have been a recent surge of interest in learning ReLUs, and the above two questions are of basic interest even for a single-layer network (i.e., nonlinearity comprising of a single ReLU function). It is conceivable that understanding the behavior of a single-layer network would allow one to use some ‘iterative peeling oﬀ’ technique to develop a theory for multiple layers. In Goel et al. [2017], the problem of recovering ReLU-model under Reliable Agnostic learning model of Kalai et al. [2012] is considered. Informally speaking, under very general distributional assumptions (the rows of A are sampled from some distribution), given A and y = ReLU(Ac), Goel et al. [2017] propose an algorithm that recovers a hypothesis which has an error-rate (under some natural loss function deﬁned therein) of ϵ with respect to the true underlying ReLU-model. Moreover, the algorithm runs in time polynomial in d and exponential in 1/ϵ. As 2 opposed to this, given A and the corresponding output of the ReLU-network y = ReLU(Ac + b), we focus on the problem of recovering c itself. Here, we note that the the model considered in Goel et al. [2017] does not consider the presence of outliers. Soltanolkotabi [2017] obtained further results on this model under somewhat diﬀerent learning guarantees. Assuming that the entries of the matrix A to be i.i.d. Gaussian, Soltanolkotabi [2017] show that with high probability a gradient descent algorithm recovers c within some precision in terms of ℓ2 -loss: the relative error decays exponentially with the number of steps in the gradient descent algorithm. The obtained result is more general as it extends to constrained optimizations in the presence of some regularizers (for example, c can be restricted to be a sparse vector, etc.). However both of these works do not consider the presence of outliers (sparse but large noise) in the observation. The sparse noise is quite natural to assume, as many times only partial observations of a signal vector are obtained. The ReLU model with outliers as considered in this paper can be thought of as a nonlinear version of the problem of recovering c from linear observations of the form v = Ac + e, with e denoting the outliers. This problem with linear observations was studied in the celebrated work of Candès and Tao [2005]. We note that the technique of Candès and Tao [2005] does not extend to the case when there is a dense (but bounded) noise component present. Our result in this case is a natural generalization and complementary to the one in Soltanolkotabi [2017] in that 1) we present a recovery method which is robust to outliers and 2) instead of analyzing gradient descent we directly analyze the performance of the minimizer of our optimization program (a generalized LASSO) using the ideas from Plan and Vershynin [2016], Nguyen and Tran [2013]. On the other hand, to the best of our knowledge, the representation learning problem for single-layer networks has not been studied as such. The representation learning problem for single-layer ReLUs bears some similarity with matrix completion problems, a fact we greatly exploit later. In low rank matrix completion, a matrix M = AX is visible only partially, and the task is to recover the unknown entries by exploiting the fact that it is low rank. In the case of (4), we are more likely to observe the positive entries of the matrix M, which, unlike a majority of matrix completion literature, creates the dependence between the matrix M and the sampling procedure. Main result for representation learning. We assume to have observed d × n matrix Y = ReLU(AC + b ⊗ 1T ) where A is a d × k matrix, C is a k × n matrix, both unknown, b ∈ Rd is a random i.i.d. bias, and ⊗ denote the Kronecker product1 . We show that a relaxed maximum-likelihood method guarantees √ the recovery of the matrix AC with an error in Frobenius norm at most O( d) with high probability (see Theorem 3 for the formal statement). Then leveraging a known result for recovering column space of a perturbed matrix (see Theorem. 5 in the appendix), we show that it is possible to also recover the column space of A with similar guarantee. The main technique that we use to obtain this result is inspired by the recent work on matrix completion by Davenport et al. [2014]. One of the main challenges that we face in recovery here is that while an entry of the matrix Y is a random variable (since b is a random bias), whether that is being observed or being cut-oﬀ by the ReLU function (for being negative) depends on the value of the entry itself. In general matrix completion literature, the entries of the matrix being observed are sampled i.i.d. (see, for example, Candès and Recht [2009], Keshavan et al. [2010], Chatterjee [2015] and references therein). For the aforementioned reason we cannot use most of these results oﬀ-the-shelf. However, similar predicament is (partially) present in Davenport et al. [2014], where entries are quantized while being observed. Similar to Davenport et al. [2014], the tools that prove helpful in this situation are the symmetrization trick and the contraction inequality Ledoux and Talagrand [2013]. However, there are crucial diﬀerence of our model from Davenport et al. [2014]: in our case the bias vector, while random, do not change over 1 This is to ensure that the bias is random, but does not change over diﬀerent observation of the data samples. 3 observations. This translates to less freedom during the transformation of the original matrix to the observed matrix, leading to dependence among the elements in a row. Furthermore, the analysis becomes notably diﬀerent since the positive observations are not quantized. Main result for noisy recovery. We plan to recover c ∈ Rk from observations v = ReLU(Ac + b) + e + w, where A is a d × k standard i.i.d. Gaussian matrix, e ∈ Rd is the vector containing outliers (sparse noise) with kek0 ≤ s, and w ∈ Rd is bounded dense noise such that kwk∞ ≤ δ . To recover c (and e) we employ the LASSO algorithm, which is inspired by the work of Plan and Vershynin [2016] and Nguyen and Tran [2013]. In particular, Plan and Vershynin [2016] recently showed that a signal can be provably recovered (up to a constant multiple) from its nonlinear Gaussian measurements via the LASSO algorithm by treating the measurements as linear observations. In the context of ReLU model, for outlier-free measurements v = ReLU(Ac + b) + w, it follows from Plan and Vershynin [2016] that LASSO algorithm outputs µc as the solution with µ = E(д · ReLU(д + b)), where д is a Gaussian random variable and b is a random variable denoting bias associated with the ReLU function. that this approach guarantees with high probq We show (k+s) log d even when the measurements are corrupted by ability recovery of c within an ℓ2 error of O d outliers e. This is achieved by jointly minimizing the square loss over (c, e) after treating our measurements as linear measurements v ′ = Ac + e + w and adding an ℓ1 regularizer to the loss function to promote the sparsity of the solution for e (we also recover e, see Theorem 4 for a formal description). Organization. The paper is organized as follows. In section 2, we describe some of the notations used throughout the paper and introduce the some technical tools that would be useful to prove our main results. In the same section (subsection 2.3), we provide the formal models of the problem we are studying. In section 3, we provide detailed proofs for our main results on the representation learning problem (see, Theorem 3). Section 4 contains the proofs and the techniques used for the recovery problem in the presence of outliers (see, Theorem 4). 2 Notations and Technical Tools 2.1 Notation For any positive integer n, deﬁne [n] ≡ {1, 2, . . . , n}. Given a matrix M ∈ Rd×n , for (i, j) ∈ [d] × [n], Mi, j denotes the (i, j)-th entry of M. For i ∈ [d], mi = (Mi,1 , Mi,2 , . . . , Mi,n )T denotes the vector containing the elements of the i-th row of the matrix M. Similarly, for j ∈ [n], mj = (M 1, j , M 2, j , . . . , Md, j )T denotes the j-th column of the matrix M. Recall that the function ReLU : R → R+ ∪ {0} takes the following form. ReLU(x) = max(x, 0). (6) For a matrix X ∈ Rd×n , we use ReLU(X ) to denote the d ×n matrix obtained by applying the ReLU function on each of the entries of the matrix X . For two matrix A and B, we use A ⊗ B to represent the Kronecker product of A and B. Given a matrix P ∈ Rd×n , s Õ Pi,2 j kP kF = (i, j)∈[d]×[n] denotes its Frobenius norm. Also, let kP k denote the ℓ2 operator norm of P, i.e. the maximum singular value of P. We let kP k∗ denote the nuclear norm of P. Similar to Davenport et al. [2014], we deﬁne a ﬂatness parameter associated with a function f : R → [0, 1]: β α (f ) := inf \|x \| ≤α 4 \| f ′(x)\| 2 . 4\| f (x)\| (7) β α quantiﬁes how ﬂat f can be in the interval [−α, α]. We also deﬁne a Lipschitz parameter L α (f ) for a function f as follows: L α (f ) := max 2.2 n sup ∫ x \|x \| ≤α f (x) f (y)dy −∞ , sup \|x \| ≤α f ′(x) o . f (x) (8) Techniques to bound the supremum of an empirical process In the course of this paper, namely in the representation learning part, we use the key tools of symmetrization and contraction to bound the supremum of an empirical process following the lead of Davenport et al. [2014] and the analysis of generalization bounds in the statistical learning literature. In particular, we need the following two statements. Theorem 1 (Symmetrization of expectation). Let X 1 , X 2 , . . . , X n be n independent RVs taking values in X and F be a class of R-valued functions on X. Furthermore, let ε 1 , ε 2 , . . . , εn be n independent Rademacher RVs. Then, for any t ≥ 1, ! ! n n Õ Õ t t E sup ≤ 2t E sup f (X i ) − Ef (X i ) εi f (X i ) . (9) f ∈ F i=1 f ∈ F i=1 Theorem 2 (Contraction inequality Ledoux and Talagrand [2013]). Let ε 1 , ε 2 , . . . , εd be d independent Rademacher RVs and f : R+ → R+ be a convex and increasing function. Let ζi : R → R be an L-Lipschitz functions, i.e., \|ζi (a) − ζi (b)\| ≤ L\|a − b\|, which satisfy ζi (0) = 0. Then, for any T ⊆ Rn , d Õ 1 εi ζi (ti ) sup Ef 2 t ∈T i=1 2.3 ! ≤ Ef L · sup t ∈T d Õ i=1 ! εi ti . (10) System Model We focus on the problems of learning the representation and recovery of the signal in the presence of errors when the signal is assumed to be generated using a single layer ReLU-network. The models of learning representations and recovery is described below. Model for learning representations. We assume that a signal vector of interest y satisﬁes y = ReLU(Ac + b), (11) where A ∈ Rd×k and b ∈ Rd correspond to the weight (generator) matrix and the bias vector, respectively. As for the problem of representation learning, we are given n message vectors that are generated from the underlying single-layer model. For j ∈ [n], the j-th signal vector is deﬁned as follows. We deﬁne the d × n observation matrix yj = ReLU Acj + b ∈ Rd . Y = y1 y2 · · · yn . 5 (12) (13) Similarly, we deﬁne the k × n coeﬃcient matrix C= c1 c2 · · · cn . With this notion, we can concisely represent the n observation vectors as Y = ReLU AC + b ⊗ 1T = ReLU M + b ⊗ 1T , (14) (15) where 1 ∈ Rn denotes the all-ones vector. We assume that the bias vector b is a random vector comprising of i.i.d. coordinates with each coordinate being copies of a random variable B distributed according to probability density function p(·). Model for recovery. For the recovery problem, we are given a vector v ∈ Rd , which is obtained by adding noise to a valid signal vector y ∈ Rd that is well modeled by a single-layer ReLU-network, with the matrix A ∈ Rd×k and bias b ∈ Rd . In particular, for some c ∈ Rk we have, v = y + e + w = ReLU(Ac + b) + e + w, (16) where w ∈ Rd denotes the (dense) noise vector with bounded norm. On the other hands, the vector e ∈ Rd contains (potentially) large corruptions, also referred to as sparse errors or outliers (we assume, kek0 ≤ s). The robust recovery problem in ReLU-networks corresponds to obtaining an estimate ĉ of the true latent vector c from the corrupt observation vector y such that the distance between ĉ and c is small. A related problem of denoising in the presence of outliers only focuses on obtaining an estimate y which is close to the true signal vector y. For this part, we focus on the setting where the weight matrix A is a random matrix with i.i.d. entries, where each entry of the matrix is distributed according to standard Gaussian distribution. Furthermore, another crucial assumption is that the Hamming error is oblivious in its nature, i.e., the error vector is not picked in an adversarial manner given the knowledge of A and c2 . 3 Representation learning in a single-layer ReLU-network In the paper, we employ the natural approach to learn the underlying weight matrix A from the observation matrix Y . As the network maps a lower dimensional coeﬃcient vector c ∈ Rk to obtain a signal vector y = ReLU(Ac + b) (17) in dimension d > k, the matrix M = AC (cf. (15)) is a low-rank matrix as long as k < min{d, n}. In our quest of recovering the weight matrix A, we ﬁrst focus on estimating the matrix M, when given access to Y . This task can be viewed as estimating a low-rank matrix from its partial (randomized) observations. Our work is inspired by the recent work of Davenport et al. [2014] on 1-bit matrix completion. However, as we describe later, the crucial diﬀerence of our model from the model of Davenport et al. [2014] is that the bias vector b does not change over observations in our case. Nonetheless we describe the model and main results of 1-bit matrix completion below to underscore the key ideas. 1-bit matrix completion. In Davenport et al. [2014], the following observation model is considered. Given a low-rank matrix M and a diﬀerentiable function ψ : R → [0, 1], the matrix Z ∈ {0, 1}d×n is 2 It is an interesting problem to extend our results to a setting with adversarial errors. However, we note that this problem is an active area of research even in the case of linear measurement, i.e, y = Ac + e + w Bhatia et al. [2015, 2017]. We plan to explore this problem in future work. 6 assumed to be generated as follows3 . Z i, j = ( 1 0 with probability ψ (Mi, j ), with probability 1 − ψ (Mi, j ). (18) Furthermore, one has access to only those entries of Z that are indexed by the set Ω ⊂ [d] × [m], where the set Ω is generated by including each (i, j) ∈ [d] × [n] with certain probability p. Given the observations Z Ω , the likelihood function associated with a matrix X ∈ Rd×n takes the following form4 . Õ (19) 1 {Zi, j =1} log ψ (X i, j ) + 1 {Zi, j =0} log 1 − ψ (X i, j ) . L Ω,Z (X ) = (i, j)∈Ω Now, in order to estimate the matrix M with bounded entries from Z Ω , it is natural to maximize the log-likelihood function (cf. (19)) under the constraint that the matrix M has rank k. maximize L Ω,Z (X ) X ∈Rd ×n subject to rank(X ) = r (20) kX k∞ ≤ γ , where the last constraint is introduced to model the setting where (w.h.p.) the observations are assumed to have bounded coordinates. We note that such assumptions indeed hold in many observations of interests, such as images. Note that the formulation in (20) is clearly non-convex due to the rank constraint. Thus, Davenport et al. [2014] propose the following program. maximize L Ω,Z (X ) X ∈Rd ×n √ subject to kX k∗ ≤ α rmn, (21) kX k∞ ≤ γ . √ Note that the constraint kX k∗ ≤ α rmn is a convex-relaxation of the non-convex constraint rank(X ) ≤ r , b be the output of which is required to ensure that the program in (21) outputs a low-rank matrix. Let M the program in (21). Davenport et al. [2014] obtain the following result to characterize the quality of the b obtained solution M. √ Proposition 1 ([Davenport et al., 2014, Theorem A.1]). Assume that kM k∗ ≤ α rmn and kM k∞ ≤ γ . Let Z be as deﬁned in (18). Then, for absolute constants C 1 and C 2 , with probability at least 1 − C 1 /(d + n), the b satisﬁes the following: solution of (21) M s s (m + n) log(mn) r (m + n) 1 b kF ≤ C 2α kM − M · 1+ , (22) dn E(\|Ω\|) E(\|Ω\|) where the constant C 2 depends on the ﬂatness and steepness of the function ψ . Learning in a single layer ReLU and 1-bit matrix completion: Main diﬀerences. Note that the problem of estimating the matrix M from Y is related to the problem of 1-bit matrix completion as deﬁned 3 The authors assume that the entries of Z take values in the set {+1, −1}. In this paper we state the equivalent model where the binary alphabet is {0, 1}. 4 Throughout this paper, log represents the natural logarithm. 7 above. Similar to the 1-bit matrix completion setup, the observation matrix Y is obtained by transforming the original matrix M in a probabilistic manner, which is dictated by the underlying distribution of the bias vector b. In particular, we get to observe the entire observation matrix, i.e., Ω = [d] × [n]. However, there is key diﬀerence between these two aforementioned setups. The 1-bit matrix completion setup studied in Davenport et al. [2014] (in fact, most of the literature on non-linear matrix completion Ganti et al. [2015]) assume that each entry of the original matrix M is independently transformed to obtain the observation matrix Y . In contrast to this, such independence in absent in the single-layer ReLU-network. In particular, for i ∈ [d], the i-th row of the matrix Y is obtained from the corresponding row of M by utilizing the shared randomness deﬁned by the bias bi . Note that the bias associated with a coordinate of the observed vector in our generative model should not vary across observation vectors. This prevents us from applying the known results to the problem of estimating M from Y . However, as we show in the remainder of this paper that the well-behaved nature of the ReLU-function allows us to deal with the dependence across the entries of a row in Z and obtain the recovery guarantees that are similar to those described in Proposition 1. 3.1 Representation learning from rectiﬁed observations We now focus on the task of recovering the matrix M from the observation matrix Y . Recall that, under the single-layer ReLU-network, the observation matrix Y depends on the matrix M as follows. Y = ReLU(M + b ⊗ 1T ). (23) For i ∈ [d], we deﬁne NY (i) ⊆ [n] as the set of positive coordinates of the i-th row of the matrix Y , i.e., NY (i) = {j ∈ [n] : Yi, j > 0} and NY,i = \|NY (i)\|. (24) Note that, for i ∈ [d], the original matrix M needs to satisfy the following requirements. Mi, j + bi = Yi, j for j ∈ NY (i) (25) Mi, j + bi < 0 for j ∈ N Y (i) := [n]\NY (i). (26) and Given the original matrix M, for i ∈ [d] and j ∈ [n], let Mi,(j) denote the j-th largest element of the i-th row of M, i.e., for i ∈ [d], Mi,(1) ≥ Mi,(2) ≥ · · · ≥ Mi,(n) . It is straightforward to verify from (25) that NY (i) denotes the indices of NY,i largest entries of M. Furthermore, whenever NY,i = si ∈ [n], we have (27) bi = Yi,(1) − Mi,(1) = · · · = Yi,(si ) − Mi,(si ) . Similarly, it follows from (26) that whenever we have NY,i = 0, then bi satisﬁes the following. bi ∈ (−∞, − max Mi, j ) = (−∞, −Mi,(1) ). (28) j ∈[n] Based on these observation, we deﬁne the set of matrices XY,ν,γ ⊂ Rd×n as XY,ν,γ = X : kX k∞ ≤ γ ; Yi,(1) − X i,(2) = · · · = Yi,(si ) − X i,(si ) ; and X i,(si ) ≥ max X i, j + ν ∀i ∈ [d] . j ∈N Y (i) 8 (29) Recall that, p : R → R denote the probability density function of each bias RV. We can then write the likelihood that a matrix X ∈ XZ,ν,γ results into the observation matrix X as follows. Ö P(yi \|xi ), (30) P(Y \|X ) = i ∈[d] where, for i ∈ [d], P(yi \|xi ) = 1 {NY ,i =0} · P(bi ≤ − max X i, j ) + j ∈[n] n Õ s=1 1 {NY ,i =s } · p(bi = Yi,(s) − X i,(s) ). (31) By using the notation F (x 1 , x 2 ) = P(−x 1 ≤ B ≤ −x 2 ) and X i∗ = maxj ∈[n] X i, j , we can rewrite (31) as follows. P(yi \|xi ) = 1 {NY ,i =0} · F (∞, X i∗ ) + n Õ s=1 1 {NY ,i =s } · p(bi = Yi,(s) − X i,(s) ). (32) Therefore the log-likelihood of observing Y given that X ∈ XY,ν,γ is the original matrix takes the following form. Õ log P(yi \|xi ) LY (X ) = i ∈[d] = Õ i ∈[d] 1 {NY ,i =0} · log F (∞, X i∗ ) + n Õ s=1 1 {NY ,i =s } · log p(Yi,(s) − X i,(s) ) . (33) In what follows, we work with a slightly modiﬁed quantity L Y (X ) = LY (X ) − LY (0) = Õ i ∈[d] 1 {NY ,i =0} · log n F (∞, X i∗ ) Õ p(Y − X ) + 1 {NY ,i =s } · log i,(s) i,(s) . F (∞, 0) p(Yi,(s) ) s=1 In order to recover the matrix M from the observation matrix Y , we employ the natural maximum likelihood approach which is equivalent to the following. maximize L Y (X ) subject to X ∈ XY,ν,γ . X ∈Rd ×n (34) Deﬁne ωp,γ ,ν to be such that F (x, y) ≥ ωp,γ ,ν for all x, y ∈ [−γ , γ ] with \|x − y\| > ν. In what follows, we simply refer this quantity as ωp as γ and ν are clear from context. The following result characterizes the performance of the program proposed in (34). b be Theorem 3. Assume that kM k∞ ≤ γ and the observation matrix Y is related to M according to (15). Let M the solution of the program speciﬁed in (34), and the bias density function is p(x) diﬀerentiable with bounded 1 . derivative. Then, the following holds with probability at least 1 − d+n b kF2 ≤ C 0 Lγ (p) · kM − M γd , βγ (p)ωp (35) where, C 0 is a constant. The quantities βγ (p) and Lγ (p) depend on the distribution of the bias and are deﬁned in (7) and (8), respectively. The proof of Theorem 3 crucially depends on the following lemma. 9 Lemma 1. Given the observation matrix Y which is related to the matrix M according to (15), let XY,ν,γ be as deﬁned in (29). Then, for any X ∈ XY,ν,γ , we have i h (36) E L Y (M) − L Y (X ) ≥ βγ (p)ωp · kM − X kF2 . The proof of this lemma is delegated to the appendix. Now we are ready to prove Theorem 3. b be the solution of the program in (34). In what follows, we use X as a short hand Proof of Theorem 3. Let M notation for XY,ν,γ . We have, h i h i h i b − L Y (M) = E L Y (M) b − L Y (M) − L Y (M) − E L Y (M) + L Y (M) b − E L Y (M) b 0 ≤ L Y (M) i h i h b − L Y (M) + 2 sup L Y (X ) − E L Y (X ) , (37) ≤ E L Y (M) X ∈X which means, h i h i b ≤ 2 sup L Y (X ) − E L Y (X ) . E L Y (M) − L Y (M) (38) X ∈X We now employ Lemma 1 to obtain that h i b kF2 ≤ 2 sup L Y (X ) − E L Y (X ) . βγ (p)ωp · kM − M (39) X ∈X We now proceed to upper bound the right hand side of (39). It follows from the standard symmetrization trick Devroye et al. [2013] that, for any integer t ≥ 1, we have " d h i Õ F (∞, X i∗ ) t + εi · 1 {NY ,i =0} · log ≤ 2t · E sup E sup L Y (X ) − E L Y (X ) F (∞, 0) X ∈X i=1 X ∈X # n Õ p(Yi,(s) − X i,(s) ) t , (40) 1 {NY ,i =s } · log p(Yi,(s) ) s=1 where {εi }i ∈[d] are i.i.d. Rademacher random variables. Note that, for x, x̃ ∈ R, d(log F (∞, u)) du \|u \| ≤γ ∫ −u d(log −∞ p(y)dy) = \|x − x̃ \| · sup ≤ \|x − x̃ \| · Lγ (p), du \|u \| ≤γ log F (∞, x) − log F (∞, x̃) ≤ \|x − x̃ \| · sup and log p(Yi,(s) − x) − log p(Yi,(s) − x̃) ≤ \|x − x̃ \| · sup \|u \| ≤γ dp(u) ≤ \|x − x̃ \| · Lγ (p). du At this point, we can combine the contraction principle with (40) to obtain the following. h i t E sup L Y (X ) − E L Y (X ) X ∈X t t " t ≤ 2 · 2 · E (Lγ (p)) · sup d Õ X ∈X i=1 εi · 10 1 ∗ {NY ,i =0} · X i + n Õ s=1 1 {NY ,i =s } · X i,(s) t # " d d Õ Õ 2 t /2 ≤ 4 · E (Lγ (p)) · sup εi (i) (ii) t t X ∈X " i=1 ≤ 4t · E (Lγ (p))t · d t /2 dγ 2 t /2 # 1 i=1 ∗ {NY ,i =0} · X i + n Õ s=1 1 {NY ,i =s } · X i,(s) 2 t /2 = (4Lγ (p) · dγ )t , # (41) where (i) and (ii) follow from the Cauchy-Schwartz inequality and the fact that, for X ∈ X, kX k∞ ≤ γ , respectively. Now using Markov’s inequality, it follows from (41) that h P sup L Y (X ) − E L Z (X ) X ∈X i ≥ C 0Lγ (p) · γd ≤ (i) ≤ where (i) follows from (41); and (ii) follows by setting C 0 ≥ 3.2 h i i h t E supX ∈X L Z (X ) − E L Z (X ) 4 C0 4 e t (C 0 Lγ (p) · γd)t (ii) ≤ 1 , d +n and t = log(d + n). (42) Recovering the network parameters b ∈ XY,ν,γ such that As established in Theorem 3, the program proposed in (34) recovers a matrix M b kF ≤ kM − M s C 0 · Lγ (p) · γ √ d. βγ (p)ωp (43) b as M b = M + E, where E denotes the perturbation matrix that has Let’s denote the recovered matrix M bounded Frobenius norm (cf. (43)). Now the task of recovering the parameters of single-layer ReLUnetwork is equivalent to solving for A given b = M + E = AC + E. M (44) In our setting where we have A ∈ Rd×k and C ∈ Rk×n with d > k and n > k, M is a low-rank matrix with its column space spanned by the columns of A. Therefore, as long as the generative model ensures that the matrix M has its singular values suﬃciently bounded away from 0, we can resort to standard results from b as an candidate for the orthonormal matrix-perturbation theory and output top k left singular vectors of M basis for the column space of M or A. In particular, we can employ the result from Yu et al. [2015] which b be the top k left singular vectors of M and M, b respectively. Note that, is stated in Appendix A. Let U and U even without the perturbation we could only hope to recover the column space of A (or the column space of U ) and not the exact matrix A. Let σk , the smallest non-zero singular value of M, is at least δ > 0. Then, it follows from Theorem 5 (cf. Appendix A) and (43) that there exists an orthogonal matrix O ∈ Rk×k such that √ 3/2 (2σ + kE k) · min{ k kE k, kE k } 23/2 (2σ1 + kE kF ) · kE kF 2 F 1 bO kF ≤ ≤ , kU − U δ2 δ2 (45) which is a guarantee that the column space of U is recovered within an error of O(d) in Frobenius norm b. by the column space of U 11 4 Robust recovery in single-layer ReLU-network We now explore the second fundamental question that arises in the context of reconstructing a signal vector belonging to the underlying generative model from its erroneous version. Recall that, we are given a vector v ∈ Rd , which is obtained by adding noise to a valid message vector y ∈ Rd that is well modeled by a single-layer ReLU-network, i.e., v = y + e + w = ReLU(Ac + b) + e + w. (46) Here, w denotes the (dense) noise vector with bounded norm. On the other hands, the vector e contains (potentially) large corruptions, also referred to as outliers. We assume the number of outliers kek0 to be bounded above by s. The robust recovery problem in ReLU-networks corresponds to obtaining an estimate ĉ of the true representation c from the corrupt observation vector y such that the distance between ĉ and c is small. A related problem of denoising in the presence of outliers only focuses on obtaining an estimate ŷ which is close to the true message vector y. In the remainder of this paper, we focus on the setting where the weight matrix A is a random matrix with i.i.d. entries, where each entry is distributed according to the standard Gaussian distribution. Furthermore, another crucial assumption is that the outlier vector is oblivious in its nature, i.e., the error vector is not picked in an adversarial manner5 given the knowledge of A and c. Note that Soltanolkotabi [2017] study a problem which is equivalent to recovering the latent vector c from the observation vector generated form a single-layer ReLU-network without the presence of outliers. In that sense, our work is a natural generalization of the work in Soltanolkotabi [2017] and presents a recovery method which is robust to errors as well. However, our approach signiﬁcantly diﬀers from that in Soltanolkotabi [2017], where the author analyze the convergence of the gradient descent method to the true representation vector c. In contrast, we rely on the recent work of Plan and Vershynin Plan and Vershynin [2016] to employ the LASSO method to recover the representation vector c (and the Hamming error vector e). Given v = ReLU(Ac + b) + e + w, which corresponds to the corrupted non-linear observations of c, we try to ﬁt a linear model to these observations by solving the following optimization problem6 . minimize c∈Rk ,e∈Rd 1 kv − Ac − ek22 + λkek1 . 2d (47) In the aforementioned formulation, the regularizer part is included to encourage the sparsity in the estimate vector. The following result characterizes the performance of our proposed program (cf. (47)) in recovering the representation and the corruption vector. Theorem 4. Let A ∈ Rd×n be a random matrix with i.i.d. standard Gaussian random variables as its entires and v satisﬁes v = ReLU(Ac∗ + b) + e∗ + w, (48) where kc∗ k2 = 1, ke∗ k0 ≤ s and kwk∞ ≤ δ . Let µ be deﬁned as µ = E[ReLU(д + b) · д], where д is a standard Gaussian random variable and b is a random variable that represents the bias in a coordinate in (48). Let (ĉ, ê) 5 It is an interesting problem to extend our results to a setting with adversarial errors. However, we note that this problem is an active area of research even in the case of linear measurement, i.e, y = Ac + e + w Bhatia et al. [2015, 2017]. We plan to explore this problem in future work. 6 Note that this paper deals with a setup where number of observations is greater than the dimension of the signal that needs to be recovered, i.e., d > k. Therefore, we don’t necessarily require the vector c to belong to a restricted set, as done in the other version of the robust LASSO methods for linear measurements (see e.g., Nguyen and Tran [2013]). 12 be the outcome of the program described in (47). Then, with high probability, we have (r ) r k log k s log d ke∗ − êk2 ∗ ≤ C̃ max , kµc − ĉk2 + , √ d d d (49) where C̃ is a large enough absolute constant that depends on δ . Proof. Assume that ĉ = µc∗ + h and ê = e∗ + f . (50) Furthermore, for c ∈ Rd and e ∈ Rk , we deﬁne L(c, e) = 1 ky − Ac − ek22 + λkek1 . 2d (51) Let S = {i ∈ [d] : ei∗ , 0} be the support of the vector e∗ such that \|S\| = s. Given a vector a ∈ Rd and set T ⊆ [d], we use a T to denote the vector obtained by restricting a to the indices belonging to T . Note that 1 1 kv − µAc∗ − e∗ − Ah − f k22 + λke∗ + f k1 − kv − µAc∗ − e∗ k22 − λke∗ k1 2d 2d 1 1 kAh + f k22 − hv − µAc∗ − e∗ , Ah + fi + λ k(e∗ + f ∗ ) S k1 + kf SC k1 − ke∗ k1 = 2d d 1 (i) 1 2 kAh + f k2 − hReLU(Ac∗ + b) − µAc∗ + w, Ah + fi + = 2d d λ k(e∗ + f ∗ ) S k1 + kf SC k1 − ke∗ k1 (ii) 1 1 ≥ kAh + f k22 − hReLU(Ac∗ + b) − µAc∗ + w, Ah + fi + λ kf SC k1 − kf S k1 (52) 2d d L(ĉ, ê) − L(µc∗, e∗ ) = where (i) and (ii) follow from (46) and the triangle inequality, respectively. Since (ĉ, ê) is solution to the program in (47), we have L(ĉ, ê) − L(µc∗, e∗ ) ≤ 0. (53) By combining this with (52), we obtain that 1 1 kAh + f k22 ≤ · hReLU(Ac∗ + b) − µAc∗ + w, Ah + fi + λ(kf S k1 − kf SC k1 ) 2d d (54) We now complete the proof in two steps where we obtain universal lower and upper bounds on the left hand side and the right hand side of (54), respectively, that hold with high probability. Upper bound on the RHS of (54). Let’s deﬁne z = ReLU(Ac∗ + b) − µAc∗ . Note that 1 · hz + w, Ah + fi + λ(kf S k1 − kf SC k1 ) d 1 1 · hz + w, fi + λ(kf S k1 − kf SC k1 ) = · hz + w, Ahi + d d (i) 1 1 ≤ · hz + w, Ahi + · kz + wk∞ kf k1 + λ(kf S k1 − kf SC k1 ) d d 13 (55) = 1 1 1 · hz + w, Ahi + (λ + · kz + wk∞ )kf S k1 − (λ − · kz + wk∞ )kf SC k1 d d d (56) where (i) follows from the Hölder’s inequality. We now employ [Plan and Vershynin, 2016, Lemma 4.3] to obtain that7 √ √ sup hz, Ahi ≤ C kσ + η d · khk2 , (57) h∈Rk where C is an absolute constant and σ 2 := E (ReLU(д + b) − µд)2 η 2 := E д2 · (ReLU(д + b) − µд)2 (58) with д being a standard Gaussian random variable. Now we can combine (56) and (57) to obtain the following. 1 · hz + w, Ah + fi + λ kf S k1 − kf SC k1 d √ √ kσ + η kz + wk∞ kz + wk∞ k T · khk2 + ≤C kA wk∞ khk2 + λ + kf S k1 − λ − kf SC k1 (59) √ d d d d √ √ (i) √ kσ + η k T 1 · khk2 + ≤C kA wk∞ khk2 + s(λ + · kz + wk∞ )kf k2 √ d d d √ √ (ii) √ kσ + η k T kA wk∞ khk2 + 2λ s kf k2, (60) · khk2 + ≤ C √ d d √ √ where (i) and (ii) follow by setting λ ≥ 2kz + wk∞ /d and using the fact that kf S k1 ≤ s kf S k2 ≤ s kf k2 . We can further simplify the bound in (60) as follows. 1 · hz + w, Ah + fi + λ kf S k1 − kf SC k1 d ) ( √ √ √√ kσ + η k T kf k2 + . ≤ max C kA wk∞, 2λ s d khk2 + √ √ d d d (61) Lower bound on the LHS of (54). By combining (54) and (59), we get that 1 kAh + f k22 2d √ √ kσ + η k T kz + wk∞ kz + wk∞ · khk2 + ≤C kA wk∞ khk2 + λ + kf S k1 − λ − kf SC k1 √ d d d d (62) k∞ . Since the left hand side of (62) is non-negative, we ﬁnd that the Note that we have picked λ ≥ 2 kz+w d tuple (h, f) belongs to the following restricted set. ! √ √ kσ + η k (63) kAT wk∞ khk2 + 3λkf S k1 }. (h, f) ∈ R := {h ∈ Rk , f ∈ Rd : λ · kf SC k1 ≤ 2 C + √ d d 7 In Plan and Vershynin [2016], Plan and Vershynin obtain the bound in terms of the Gaussian width Vershynin [2018] of the ∗ cone which √ the vector h belongs to. However, in our setup where we do not impose any speciﬁc structure on c , this quantity is simply O( k). 14 As a result, in order to lower bound (54), we lower bounding the following quantity for every (h, f) ∈ R. 1 · kAh + f k22 . 2d (64) Towards this, we employ Lemma 3 in Appendix C, which gives us that, for every (h, f) ∈ R, with high probability, we have 2 1 kf k2 1 2 · kAh + f k2 ≥ khk2 + √ . 2d 128 d (65) Completing the proof. It follows from (54), (61), and (65) that ( √ ) √ 2 √√ kσ + η kf k2 k T kf k2 1 kA wk∞ , 2λ s d khk2 + √ khk2 + √ ≤ max C + √ 128 d d d d or ) ( √ √ √√ kσ + η k T kf k2 kA wk∞, 2λ s d . khk2 + √ ≤ 128 max C + √ d d d (66) Now using the fact that kwk∞ ≤ δ and A is an i.i.d. standard Gaussian matrix, we can obtain the following bound from (66), which holds with high probability. ) (r r k log k s log d kf k2 , (67) , khk2 + √ ≤ C̃ max d d d where C̃ is a large enough absolute constant. Acknowledgements. This research is supported in part by NSF awards CCF 1642550, CCF 1618512 and CAREER award 1642658. References K. Bhatia, P. Jain, and P. Kar. Robust regression via hard thresholding. In Advances in Neural Information Processing Systems (NIPS), pages 721–729. 2015. K. Bhatia, P. Jain, P. Kamalaruban, and P. Kar. Consistent robust regression. In Advances in Neural Information Processing Systems (NIPS), pages 2107–2116. 2017. A. Bora, A. Jalal, E. Price, and A. G. Dimakis. 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Soltanolkotabi. Learning relus via gradient descent. In Advances in Neural Information Processing Systems (NIPS), pages 2004–2014. 2017. R. Vershynin. High-Dimensional Probability: Available online, 2018. An Introduction with Applications in Data Science. Y. Yu, T. Wang, and R. J. Samworth. A useful variant of the Davis–Kahan theorem for statisticians. Biometrika, 102(2):315–323, 2015. 16 A Results on Matrix Perturbation Let M be a d × n matrix, where without loss of generality we assume that d ≥ n. Let M have the following singular value decomposition. M = U ΣV T , (68) where Σ = Diag (σ1 , σ2 , . . . , σn ) is the diagonal matrix with the singular values of M as its diagonal entries. b = M + E be the matrix which is obtained by perturbing the original matrix M by an error matrix E. Let M b have the following singular value decomposition. Let M b =U bb M ΣVbT , (69) where b Σ = Diag (b σ1, b σ2 , . . . , b σn ) is the diagonal matrix comprising the singular values of the perturbed b Let γ 1 ≥ γ 2 ≥ · · · ≥ γn be the singular values of the matrix U T U b. Deﬁne, matrix M. b) = Diag θ 1 , θ 2 , . . . , θ n . θ i = cos−1 γi ∀i ∈ [n] and Θ(U , U (70) b the subspaces spanned by the Note that {θ i }i ∈[n] are referred to as the canonical angles between U and U, b, respectively. It is common to use k sin Θ(U , U b)kF as a distance measure columns of the matrices U and U b between U and U. In Yu et al. [2015], Yu et al. present the following result which bounds the distance between the subb respectively8. spaces spanned by the singular vectors of the original matrix M and the perturbed matrix M, b = M + E ∈ Rd×n have singular values σ1 ≥ · · · σmin{d,n } and b Theorem 5. Let M, M σ1 ≥ · · · b σmin{d,n } , respectively. Fix 1 ≤ r ≤ rank(A) and assume that σr2 − σr2+1 ≥ 0. b = (b Let U = (u1 , u2 , . . . , ur ) ∈ Rd×r and U u1 , b u2 , . . . , b ur ) ∈ Rd×r contain the left singular vectors associated b respectively. Then, with the r leading singular values of M and M, √ 2(2σ1 + kE k) · min{ r kE k, kE kF } b . k sin Θ(U , U )kF ≤ σr2 − σr2+1 (71) Moreover, there exists an orthogonal matrix O ∈ Rr ×r such that 3/2 (2σ + kE k) · min{√r kE k, kE k } 2 1 F bO kF ≤ kU − U . σr2 − σr2+1 B (72) Proofs of Section 3.1 Lemma 2 (Deﬁning the distance measure). Given the observation matrix Y which is related to the matrix M according to (15), let XY,ν,γ be as deﬁned in (29). Then, for any X ∈ XY,ν,γ , we have h i E L Y (M) − L Y (X ) ≥ βγ (p)ωp · kM − X kF2 . (73) 8 Here, we state a special case of the result from Yu et al. [2015]. See [Yu et al., 2015, Theorem 4] for the statement of the general result. 17 Proof. First, we recall our notation that given the original matrix M, for i ∈ [d] and j ∈ [n], Mi,(j) denotes the j-th largest element of the i-th row of M, i.e., for i ∈ [d], Mi,(1) ≥ Mi,(2) ≥ · · · ≥ Mi,(n) . We deﬁne, Mi∗ = maxj ∈[n] Mi, j . Thus, h i E L Y (M) − L Y (X ) # " n d Õ F (∞, Mi∗ ) Õ p(Yi,(s) − Mi,(s) ) + 1 {NY ,i =s } · log = E 1 {NY ,i =0} · log F (∞, X i∗ ) s=1 p(Yi,(s) − X i,(s) ) i=1 d n−1 ∫ −Mi,(s +1) Õ F (∞, Mi∗ ) Õ p(b) ∗ F (∞, Mi ) · log = db + p(b) · log ∗) + F (∞, X p(b + M i,(s) − X i,(s) ) i i=1 s=1 −Mi,(s ) ∫ ∞ p(b) db , p(b) · log p(b + Mi,(n) − X i,(n) ) −Mi,(n) (74) where p : R → R represents the probability density function of the bias RV and F is deﬁned as F (x 1 , x 2 ) = P(−x 1 ≤ b ≤ −x 2 ). Given the matrices, M, X ∈ XZ,ν,γ , we deﬁne a new (density) function д as follows. ( p(u + Mi,(s) − X i,(s) ) if u ∈ (−Mi,(s) , −Mi,(s+1) ] for s ∈ [n − 1] д(u) = p(u + Mi,(n) − X i,(n) ) if u ∈ [−Mi,(n) , ∞). (75) Recall that for x ∈ R, we have x ≤ e x − 1. For x = log y, this gives us that log y ≤ y − 1. In particular, for b ∈ R, employing (76) with y = p(b) · log s д(b) ≤ p(b) · p(b) q д(b) p(b) , s (76) we get that ! p д(b) − 1 = p(b)д(b) − p(b) p(b) or p(b) · log p p(b) ≥ 2 · p(b) − p(b)д(b) . д(b) (77) By using (77), for every i ∈ [d], we obtain that n−1 ∫ Õ s=1 −Mi,(s +1) −Mi,(s ) ∫ ∞ p(b) db + p(b) · log p(b + Mi,(s) − X i,(s) ) = ≥ −Mi,(1) ∞ ∫ −Mi,(1) p(b) db д(b) p 2 · p(b) − p(b)д(b) p(b) · log 18 ∫ ∞ −Mi,(n) p(b) · log p(b) db p(b + Mi,(n) − X i,(n) ) ∫ = 2 · 1 − F (∞, M 1,(1) ) − ∞ −Mi,(1) p p(b)д(b)db = 1 − F (∞, Mi∗ ) + 1 − F (∞, X i∗ ) − = ∫ ∞ −Mi,(1) ∞ −Mi,(1) F (∞, X i∗ ) F (∞, Mi∗ ) p 2 · p(b)д(b)db + 1− − 1− p 2 p p(b) − д(b) db + 1 − F (∞, Mi∗ ) − 1 − F (∞, X i∗ ) (78) F (∞,X i∗ ) F (∞, Mi∗ ) to obtain the following. F (∞, X i∗ ) F (∞, X i∗ ) ∗ ∗ F (∞, Mi ) · log ≤ F (∞, Mi ) · − 1 = F (∞, X i∗ ) − F (∞, Mi∗ ). F (∞, Mi∗ ) F (∞, Mi∗ ) For i ∈ [d], we now employ (76) with y = or ∫ F (∞, Mi∗ ) + F (∞, X i∗ ) − F (∞, Mi∗ ) F (∞, X i∗ ) F (∞, Mi∗ ) = F (∞, Mi∗ ) · log + 1 − F (∞, Mi∗ ) − 1 − F (∞, X i∗ ) ≥ 0 ∗ F (∞, X i ) F (∞, Mi∗ ) · log (79) By combining (78) and (79), we obtain that n−1 ∫ −Mi,(s +1) F (∞, Mi∗ ) Õ p(b) ∗ p(b) · log + db + F (∞, Mi ) · log ∗ F (∞, X i ) s=1 −Mi,(s ) p(b + Mi,(s) − X i,(s) ) ∫ ∞ p(b) db p(b) · log p(b + Mi,(n) − X i,(n) ) −Mi,(n) ∫ ∞ p 2 p ≥ p(b) − д(b) db −Mi,(1) = n−1 ∫ −Mi,(s +1) Õ s=1 −Mi,(s ) ∫ ∞ p −Mi,(n) 2 q p p(b) − p(b + Mi,(s) − X i,(s) ) db 2 q p(b) − p(b + Mi,(n) − X i,(n) ) db !2 p ′ (ξ s ) = Mi,(s) − X i,(s) db + p 2 p(ξ s ) s=1 −Mi,(s ) !2 ∫ ∞ p ′ (ξ n ) Mi,(n) − X i,(n) db p −Mi,(n) 2 p(ξ n ) ∫ ∞ n−1 ∫ −Mi,(s +1) Õ (ii) 2 ≥ βγ (p) · Mi,(s) − X i,(s) db + (i) n−1 ∫ Õ −Mi,(s +1) s=1 (iii) + ≥ βγ (p)ωp · −Mi,(s ) n Õ s=1 −Mi,(n) 2 Mi,(s) − X i,(s) , 2 Mi,(n) − X i,(n) db ! (80) where (i) follows from the Mean Value Theorem with suitable {ξ i } ⊂ R and (ii) follows by assuming that we have q p ′ (u) ≥ βγ (p) for all \|u\| ≤ γ . p 2 p(u) 19 Since M ∈ XZ,ν,γ , we have \|Mi,(n) \| ≤ γ and \|Mi,(s) −Mi,(s+1) \| ≥ ν. The step (iii) follows from the assumption that F (x, y) ≥ ωp for all (x, y) such that \|x − y\| ≥ ν . By combining (74) with (80), we now obtain that d n i Õ h Õ 2 βγ (p)ωp · Mi,(s) − X i,(s) = βγ (p)ωp · kM − X kF2 . E L Y (M) − L Y (X ) ≥ i=1 s=1 C Proofs of Section 4 Here we state the special form of a result that was obtained in Nguyen and Tran [2013] for the generals setting, where one may potentially require the vector c to be sparse as well. Lemma 3. Let A ∈ Rd×n be random matrix that has i.i.d. standard Gaussian entries. Furthermore, let R ⊂ Rk × Rd be as deﬁned in (63). Then, with probability at least 1 − c exp(−c̃d), we have 2 1 kf k 1 · kAh + f k2 ≥ khk + √ for all (h, f) ∈ R. 2d 128 d (81) Here, c, c̃ > 0 are absolute constants. Proof. Note that kAh + f k22 = kAhk22 + kf k22 + 2hAh, fi. (82) For a d × k matrix with i.i.d. Gaussian entries, there exists constants c 1 and c 2 such that with probability at least 1 − c 1 exp(−c 2d), we have 1 1 √ kAhk2 ≥ khk2 . 4 d (83) Therefore, with probability at least 1 − c 1 exp(−c 2d), we have kAhk22 + kf k22 2 d d d kf k2 2 2 2 2 . ≥ khk2 + kf k2 ≥ (khk2 + kf k2 /d) ≥ khk2 + √ 16 16 32 d (84) Next, we focus on obtaining an upper bound on 1 \|hAh, fi\|. d Towards this we partition the set [d] into r blocks S1 = S, S2, . . . , Sr such that \|S2 \| = · · · \|S\|r = s ′ ≥ \|S\| = s. Here, S2 refers to the set of indices of s ′ largest entries (in terms of absolute value) of f SC ; S3 corresponds to the set of indices of the next s ′ largest entires of f SC ; and so on. Now, we have Õ 1 1Õ 1 \|hA Si h, f Si i\| ≤ max kA Si k2 khk2 \|hAh, fi\| ≤ kf Si k2 d d i=1 d i i=1 r r 20 (85) In [Nguyen and Tran, 2013, Appendix], Nguyen and Tran show that, with probability at least 1−2 exp(−τ 2s ′/2), for a set S ′ with \|S ′ \| = s ′, we have √ √ √ (86) kA S′ k2 ≤ k + s ′ + τ s ′ . By setting τ = τ ′ q d s′ and taking the union bound over all the subsets of [d] of size s ′, kA S′ k2 ≤ holds with probability at least √ √ √ k + s ′ + τ s ′ ∀ S ′ ⊂ [d] such that \|S ′ \| = s ′ (87) s′ d ed ′2 exp(−τ ′2d/2). 1 − ′ exp(−τ d/2) ≥ 1 − ′ s s Assuming that s ′ log(d/s ′ ) ≤ c 3d, the aforementioned probability is at least 1 − exp(−(τ ′2 /2 − c 3 )d). On the other hand, we have r Õ i=1 (i) kf Si k2 ≤ 2kf k2 + r Õ i=3 kf Si k2 1 ≤ 2kf k2 + √ kf Sc k1 s′ ! √ √ (iii) kσ + η 3 k T 2 kA wk∞ khk2 + √ kf S k1 + ≤ 2kf k2 + √ C √ d λ s′ s′ d ! √ √ (iv) kσ + η 2 k T + ≤ 5kf k2 + √ C kA wk∞ khk2 , √ ′ d λ s d (ii) (88) where (i) follows from the fact that kf S1 k2 ≤ kf S2 k2 ≤ kf k2 ; (ii) follows from a standard bound given in Candes et al. [2006]; (iii) follows from the fact that √f belongs to the set R deﬁned in (63); and (iv) is a √ consequence of the loose bound kf S k1 ≤ s kf S k2 ≤ s ′ kf k2 . Next, we use the fact that λ ≥ 2kz + wk∞ /d, it follows from (88) that ! √ √ r Õ kσ + η k T d kA wk∞ khk2 kf Si k2 ≤ 5kf k2 + + √ C √ d kz + wk∞ s ′ d i=1 ! ! √ √ √ √ kf k2 kσ + η d k T + =5 d √ + kA wk∞ khk2 √ C √ d 5kz + wk∞ s ′ d d √ kf k2 (89) ≤ 5 d √ + khk2 d By combining (85), (87), and (89), with probability at least 1 − c 4 exp(−c 5d), we obtain that √ kf k2 √ √ 1 √ 1 ′ ′ \|hAh, fi\| ≤ k + s + τ s khk2 ·5 d √ + khk2 d d d 2 √ √ kf k2 5 √ ′ ′ k + s +τ s ≤ √ √ + khk2 d d 21 2 1 kf k2 ≤ √ + khk2 , 128 d (i) (90) where (i) follows from large enough d. By combining (82), (84), and (90), we obtain that, for every (h, f) ∈ R, 2 1 1 kf k2 2 · kAh + f k2 ≥ khk + √ 2d 128 d holds with probability at least 1 − c exp(−c̃d), with absolute constants c, c̃ > 0. 22 (91)	7
CHUDNOVSKY’S CONJECTURE FOR VERY GENERAL POINTS IN PN k arXiv:1604.02217v2 [math.AC] 7 Dec 2017 LOUIZA FOULI, PAOLO MANTERO AND YU XIE A BSTRACT. We prove a long-standing conjecture of Chudnovsky for very general and generic points in PN k , where k is an algebraically closed field of characteristic zero, and for any finite set of points lying on a quadric, without any assumptions on k. We also prove that for any homogeneous ideal I in the homogeneous coordinate ring R = k[x0 , . . . , xN ], Chudnovsky’s conjecture holds for large enough symbolic powers of I. 1. I NTRODUCTION This manuscript deals with the following general interpolation question: Question 1.1. Given a finite set of n distinct points X = {p1 , . . . , pn } in PN k , where k is a field, what is the minimum degree, αm (X), of a hypersurface f ≠ 0 passing through each pi with multiplicity at least m? Question 1.1 has been considered in various forms for a long time. We mention a few conjectures and motivations. For instance, this question plays a crucial role in the proof of Nagata’s counterexamples to Hilbert’s fourteenth problem [19]. In the same paper Nagata conjectured that √ αm (X) > m n for sets of n general points in P2C [19], and a vast number of papers in the last few decades are related to his conjecture. Another reason for the interest sparked by the above question comes from the context of complex analysis: an answer to Question 1.1 would provide information about the Schwarz exponent, which is very important in the investigation of the arithmetic nature of values of Abelian functions of several variables [4]. However, besides a few very special classes of points (e.g., if these n points lie on a single hyper−1 ) points forming a star configuration and m is a multiple of N [5],[2]), plane or one has n = (β+N N at the moment a satisfactory answer to this elusive question appears out of reach. Therefore, there has been interest in finding effective lower bounds for αm (X). In fact, lower bounds for αm (X) yield upper bounds for the Schwarz exponent. Using complex analytic techniques, Waldschmidt [22] and Skoda [21] in 1977 proved that for all m ≥ 1 αm (X) α(X) ≥ , m N where α(X) = α1 (X) is the minimum degree of a hypersurface passing through every point of X and k = C. In 1981, Chudnovsky [4] improved the inequality in the 2-dimensional projective space. He showed that if X is a set of n points in P2C , then for all m ≥ 1 αm (X) α(X) + 1 ≥ . m 2 He then raised the following conjecture for higher dimensional projective spaces: 2010 Mathematics Subject Classification. 13F20, 11C08. Key words and phrases. Chudnovsky’s conjecture, initial degrees, symbolic powers, fat points, Seshadri constant. The first author was partially supported by a grant from the Simons Foundation, grant #244930. 1 2 L. FOULI, P. MANTERO AND Y. XIE Conjecture 1.2 (Chudnovsky [4]). If X is a finite set of points in PN C , then for all m ≥ 1 αm (X) α(X) + N − 1 ≥ . m N The first improvement towards Chudnovsky’s Conjecture 1.3 was achieved in [9] by Esnault and αm (X) α(X) + 1 Viehweg, who employed complex projective geometry techniques to show ≥ for m N points in PN C . In fact, this inequality follows by a stronger statement, refining previous inequalities from Bombieri, Waldschmidt and Skoda, see [9]. From the algebraic point of view, Chudnovsky’s Conjecture 1.3 can be interpreted in terms of symbolic powers via a celebrated theorem of Nagata and Zariski. Let R = k[x0 , . . . , xN ] be the homogeneous coordinate ring of PN k and I a homogeneous ideal in R. We recall that the m-th symbolic power of I is defined as the ideal I (m) = ⋂p I m Rp ∩ R, where p runs over all associated prime ideals of R/I, and the initial degree of I, α(I), is the least degree of a polynomial in I. Nagata and Zariski showed that if k is algebraically closed and X is a finite set of points in PN k , (m) then αm (X) = α(IX ), where IX is the ideal consisting of all polynomials in R that vanish on X. Thus in this setting Chudnovsky’s Conjecture 1.3 is equivalent to α(IX ) α(IX ) + N − 1 ≥ for all m ≥ 1. m N (m) α(IX ) = γ(IX ), called the Waldschmidt constant of IX , exists and is an “inf” The limit lim m→∞ m [2]. Thus another equivalent formulation of Chudnovsky’s Conjecture 1.3 is α(IX ) + N − 1 γ(IX ) ≥ . N We remark here that there is a tight connection between the Waldschmidt constant (especially for general points) and an instance of the multipoint Seshadri constant [1, Section 8]. We now state a generalized version of Chudnovsky’s Conjecture 1.2. When k is an algebraically closed field then the following conjecture is equivalent to Chudnovsky’s Conjecture 1.2. (m) Conjecture 1.3. If X is a finite set of points in PN k , where k is any field, then for all m ≥ 1 α(IX ) α(IX ) + N − 1 ≥ . m N In 2001, Ein, Lazarsfeld, and Smith proved a containment between ordinary powers and symbolic powers of homogeneous ideals in polynomial rings over the field of complex numbers. More precisely, for any homogeneous ideal I in R = C[x0 , . . . , xN ], they proved that I (N m) ⊆ I m [8]. Their result was soon generalized over any field by Hochster and Huneke using characteristic p techniques [16]. Using this result, Harbourne and Huneke observed that the Waldschmidt-Skoda α(I (m) ) α(I) ≥ actually holds for every homogeneous ideal I in R [15]. In the same inequality m N article, Harbourne and Huneke posed the following conjecture: (m) Conjecture 1.4 (Harbourne-Huneke [15]). If X is a finite set of points in PN k , then for all m ≥ 1 (N m) IX m , ⊆ M m(N −1) IX where M = (x0 , . . . , xN ) is the homogeneous maximal ideal of R = k[x0 , . . . , xN ]. CHUDNOVSKY’S CONJECTURE FOR VERY GENERAL POINTS IN PN k 3 Conjecture 1.4 strives to provide a structural reason behind Chudnovsky’s Conjecture 1.3: if it holds, then it would imply Chudnovsky’s Conjecture 1.3 in a similar way as to how the EinLazarsfeld-Smith and Hochster-Huneke containment implies the Waldschmidt-Skoda inequality [15]. These results have since raised new interest in Chudnovsky’s Conjecture 1.3. Harbourne and Huneke proved their conjecture for general points in P2k and when the points form a star configuration in PN k . In 2011, Dumnicki proved the Harbourne-Huneke Conjecture 1.4 for 3 general points in Pk and at most N + 1 points in general position in PN k for N ≥ 2 [6]. In summary, Chudnovsky’s Conjecture 1.3 is known in the following cases: ● ● ● ● any finite set of points in P2k [4], [15]; any finite set of general points in P3k , where k is a field of characteristic 0 [6]; any set of at most N + 1 points in general position in PN k [6]; N any set of a binomial number of points in Pk forming a star configuration [5], [2]. In the present paper, we prove that Chudnovsky’s Conjecture 1.3 holds for ● any finite set of very general points in PN k , where k is an algebraically closed field of characteristic 0 (Theorem 2.8); ● any finite set of generic points in PN k(z) , where k is an algebraically closed field of characteristic 0 (Theorem 2.7); ● any finite set of points in PN k lying on a quadric, without any assumptions on k (Proposition 2.6). As a corollary, we obtain that the Harbourne-Huneke Conjecture 1.4 holds for sets of a binomial number of very general points in PN k (Corollary 2.9). This result also yields a new lower bound for the multipoint Seshadri constant of very general points in PN k (Corollary 2.10). In the final section of the paper, we prove that for any homogeneous ideal I in the homogeneous coordinate ring R = k[x0 , . . . , xN ], Chudnovsky’s Conjecture 1.3 holds for sufficiently large symbolic powers I (t) , t ≫ 0 (Theorem 3.7). In the case of ideals of finite sets of points in PN C , we (t) prove a uniform bound, namely that if t ≥ N − 1, then I satisfies Chudnovsky’s Conjecture 1.3 (Proposition 3.10). Very recently, Dumnicki and Tutaj-Gasińska proved the Harbourne-Huneke Conjecture 1.4 for at least 2N number of very general points in PN k . As a corollary, they obtain Chudnovsky’s ConN jecture 1.3 for at least 2 number of very general points in PN k [7]. These results are obtained independently from ours and with different methods. 2. G ENERIC AND VERY GENERAL POINTS IN PN k We begin by discussing our general setting. Set-up 2.1. Let R = k[x0 , . . . , xN ] be the homogeneous coordinate ring of PN k , where k is an algebraically closed field. Let n be a positive integer and let S = k(z)[x], where k ⊆ k(z) is a purely transcendental extension of fields obtained by adjoining n(N + 1) variables z = (zij ), 1 ≤ i ≤ n, 0 ≤ j ≤ N . A set of n generic points P1 , . . . , Pn consists of points Pi = [zi0 ∶ zi1 ∶ . . . ∶ 4 L. FOULI, P. MANTERO AND Y. XIE ziN ] ∈ PN k(z) . We denote the defining ideal of n generic points as H = ⋂ I(Pi ), n i=1 where I(Pi ) is the ideal defining the point Pi . n(N +1) , where 1 ≤ i ≤ n, 0 ≤ j ≤ N , we define the set of For any nonzero vector λ = (λij ) ∈ Ak N points {p1 , . . . , pn } ⊆ Pk as the points pi = Pi (λ) = [λi0 ∶ λi1 ∶ . . . ∶ λiN ] ∈ PN k . For 1 ≤ i ≤ n, let I(pi ) be the ideal of R defining the point pi and define H(λ) = ⋂ I(pi ). n i=1 For any ideal J in S, recall that Krull [17, 18] defined the specialization Jλ with respect to the substitution z → λ as follows: Jλ = {f (λ, x) ∣ f (z, x) ∈ J ∩ k[z, x]}. In general, one has that Hλ ⊆ H(λ), where H and H(λ) are defined as in Set-up 2.1 and Hλ is the specialization with respect to the substitution z → λ defined by Krull. Notice that equality holds if n(N +1) λ is in a dense Zariski-open subset of Ak [20]. Recall the collection of all sets consisting of n points (not necessarily distinct) in PN k is parameN terized by G(1, n, N +1), the Chow variety of algebraic 0-cycles of degree n in Pk . It is well-known that G(1, n, N + 1) is isomorphic to the symmetric product Symn (PN k ), see for instance [10]. One N says that a property P holds for n general points in Pk if there is a dense Zariski-open subset W of G(1, n, N + 1) such that P holds for every set X = {p1 , . . . , pn } of n points with p1 + . . . + pn ∈ W . Similarly, one says that a property P holds for n very general points in PN k if P holds for every set ∞ of n points in a nonempty subset W of G(1, n, N + 1) of the form W = ⋂ Ui , where the Ui are i=1 dense Zariski-open sets (when k is uncountable, then W is actually a dense subset). We conclude this part by recalling the following well-known fact. Remark 2.2. Let n be a positive integer. The collection of all sets consisting of n distinct points in PN k is parameterized by a dense Zariski-open subset W (n) of G(1, n, N + 1). Unless specified, for the rest of this paper by a “set of points” we mean “a set of simple points”, i.e. points whose defining ideal is radical. n(N +1) Instead of working directly with the Chow variety, we will work over Ak (in order to specialize from the generic situation). We first need to prove that if a property holds on a dense Zariskin(N +1) open subset of Ak , then it also holds on a dense Zariski-open subset of the Chow variety. This is precisely the content of our first lemma. n(N +1) Lemma 2.3. Assume Set-up 2.1 and let U ⊆ Ak be a dense Zariski-open subset such that a property P holds for H(λ) whenever λ ∈ U . Then property P holds for n general points in PN k . ∞ Moreover, if a property P holds for H(λ) whenever λ ∈ U , where U = ⋂ Ui ⊆ Ak n(N +1) i=1 and each Ui is a dense Zariski-open set, then P holds for n very general points in PN k . is nonempty CHUDNOVSKY’S CONJECTURE FOR VERY GENERAL POINTS IN PN k n(N +1) Proof. For every i = 1, . . . , n, let πi ∶ Ak as follows: where λ = (λij ) ∈ Ak n(N +1) 5 ❴ ❴ ❴/ PN be the rational map defined by projection k πi (λ) = [λi0 ∶ λi1 ∶ . . . ∶ λiN ], . It is clear that πi is defined on the complement of the Zariski-closed n(N +1) Ak ∣ λi0 = . . . = λiN = 0}. proper subset Ci = {λ ∈ Taking products of these rational maps, we obtain the rational map π = (π1 × π2 × ⋯ × πn ) ∶ Ak n(N +1) ❴ ❴ ❴ / P N ×k P N ×k ⋯ ×k P N . k k k The map π is defined on the complement of the closed proper subset C = ⋃ Ci , where Ci is as n i=1 n(N +1) Ak , above. Note that U ∖ C is still open in and since π is surjective and thus dominant, then N N π(U ∖ C) contains a non-empty Zariski-open subset W ′ ⊆ PN k ×k Pk ×k ⋯ ×k Pk (see for instance [13, II. Ex. 3.19 (b)]). Now, since the symmetric group Sn on n elements is finite, the image W of W ′ in (PN k ×k n N N N Pk ×k ⋯ ×k Pk )/Sn ≅ Sym (Pk ) ≅ G(1, n, N + 1) contains a non-empty Zariski-open subset of G(1, n, N + 1). Let H be as in Set-up 2.1. We now prove that the initial degree of any symbolic power of H is no smaller than the initial degree of any ideal of a set with the same number of points. Equivalently, (m) if I is the defining ideal of a set of n points in PN ) ≥ α(I (m) ) for all m ≥ 1. k , then α(H Theorem 2.4. Let m ≥ 1. Assume Set-up 2.1 and that k has characteristic 0. Then α(H (m) ) ≥ (m) α(IX ), for every set X of n distinct points in PN k . Moreover, for every m ≥ 1, there is a dense n(N +1) Zariski-open subset Um ⊆ Ak for which equality holds. Proof. Let t ≥ 0. We define Vt = {λ = (λij ) ∈ Ak n(N +1) n(N +1) is a closed subset of Ak . Indeed, notice that Vt = {λ = (λij ) ∈ Ak n(N +1) ∣ α(H(λ)(m) ) ≤ t}. We first prove that Vt ∣ there exists 0 ≠ f ∈ H(λ)(m) of degree t}. Let f ∈ R be a homogeneous polynomial with deg f = t and write f = ∑ Cα xα . Since k is ∣α∣=t algebraically closed of characteristic 0, the statement f ∈ H(λ)(m) is equivalent to ∂β f (pi ) = 0 for all β with ∣β∣ ≤ m − 1 and all points p1 , . . . , pn . Since Pi = [zi ] = [zi0 ∶ zi1 ∶ . . . ∶ ziN ] and pi = [λi ] = [λi0 ∶ λi1 ∶ . . . ∶ λiN ], we write ∂β f (Pi ) = ∂β f (zi ) = ∑ Cα ∂β zi α and ∂β f (pi ) = ∣α∣=t 3 zi2 ∑ Cα ∂β λi α . (For instance, ∂(2,0,1) zi (3,3,2) = ∂x0 x0 x2 x30 x31 x22 ∣x=zi = 12x0 x31 x2 ∣x=zi = 12zi0 zi1 ∣αi ∣=t and ∂(2,0,1) λi (3,3,2) = 12λi0 λ3i1 λi2 ). +1 To order these equations we use, for instance, the natural deglex order in NN , i.e., α = 0 (α0 , . . . , αN ) > β = (β0 , . . . , βN ) if and only if ∣α∣ > ∣β∣ or ∣α∣ = ∣β∣ and there exists j such that αi = βi for i ≤ j and αj+1 > βj+1 . Then the system of equations {∂β f (Pi ) = 0}∣β∣≤m−1, 1≤i≤n can be written in the following form Bm,t [C(t,...,0) . . . Cα . . . C(0,...,t) ] = 0, T 6 L. FOULI, P. MANTERO AND Y. XIE where the rows of Bm,t are t [∂β zi0 ... ∂β zi α ... t ] , where 1 ≤ i ≤ n and ∣β∣ ≤ m − 1. ∂β ziN By construction, the existence of a nonzero element f ∈ H(λ)(m) of degree t is equivalent to the existence of a non-trivial solution for the homogeneous system [Bm,t ]λ [C(t,...,0) . . . Cα . . . C(0,...,t) ] = 0. T ) (t+N ). If n(m+N ) (t+N ), then for every Observe that the matrix Bm,t has size n(m+N m−1 × N m−1 < N n(N +1) λ ∈ Ak the homogeneous system [Bm,t ]λ [C(t,...,0) . . . Cα . . . C(0,...,t) ] = 0 has non-trivial T n(N +1) solutions. Therefore Vt = Ak If instead, ) n(m+N m−1 ≥ (t+N ), N n(N +1) , which is closed in Ak . then the system [Bm,t ]λ [C(t,...,0) . . . Cα . . . C(0,...,t) ] = 0 has T non-trivial solutions if and only if rank [Bm,t ]λ < (t+N N ). This is a closed condition on λ as it n(N +1) requires the vanishing of finitely many minors, and therefore Vt is closed in Ak Next, let t0 = α(H (m) ). The set Vt0 = {λ = (λij ) ∈ a dense Zariski-open subset of n(N +1) Ak . n(N +1) Ak n ∣ α(H(λ) (m) . ) ≤ t0 } contains Indeed, let 0 ≠ f ∈ ⋂ I(Pi )m be such that deg f = t0 . We may assume that f (z, x) ∈ k[z][x0 , . . . , xN ]. Then there exists a dense Zariski-open subset n(N +1) Um of Ak such that the polynomial 0 ≠ f (λ, x) ∈ (H (m) )λ = (H(λ))(m) and deg f = t0 (since f (z, x) ≠ 0, there is a non-empty Zariski-open subset of specializations z → λ such that f (λ, x) ≠ 0). Finally, since Vt0 is a Zariski-closed subset which also contains a dense Zariski-open subset of n(N +1) n(N +1) Ak , then Vt0 = Ak , which proves the statement. The second part of the statement also follows from the above argument. i=1 Following [12, Definition 2.4], we say that a set X of n points in PN k is in generic position if it has the “generic Hilbert function”, i.e., if HR/IX (d) = min{dimk (Rd ), n} for every d ≥ 0. Being in generic position is an open condition; indeed any set of generic (or general) points (see Set-up 2.1) is in generic position. We now prove a reduction argument, which will allow us to concentrate on certain binomial numbers of points. Proposition 2.5. (a). Chudnovsky’s Conjecture 1.3 holds for any finite set of generic points if it −1 ) generic points for all β ≥ 1. holds for sets of (β+N N −1 ) (b). Chudnovsky’s Conjecture 1.3 holds for any finite set of points if it holds for sets of (β+N N points in generic position for all β ≥ 1. Proof. (a): Let H = ⋂ I(Pi ) be the defining ideal of the n generic points P1 , . . . , Pn as in Setn i=1 up 2.1. Let β ≥ 1 be the unique integer such that ( β+N β +N −1 ). )≤n<( N N −1 ) and let J = ⋂ I(Pi ) be the ideal defining t of these generic points. Since the set Let t = (β+N N t Y = {P1 , . . . , Pt } is in generic position, in particular we have α(J) = α(H) = β. i=1 CHUDNOVSKY’S CONJECTURE FOR VERY GENERAL POINTS IN PN k 7 −1 ) generic points. Then for all m ≥ 1 Now assume Chudnovsky’s Conjecture 1.3 holds for (β+N N α(J (m) ) α(J) + N − 1 ≥ . m N Since H (m) ⊆ J (m) , one has that α(H (m) ) ≥ α(J (m) ) and α(H (m) ) α(J (m) ) α(J) + N − 1 α(H) + N − 1 ≥ ≥ = . m m N N (b): The proof of (b) is similar in spirit to (a). Let X be any finite set of points in PN k , let IX (α−1)+N −1 ), where α = α(IX ). By linear independence (e.g., be its defining ideal, and let t = ( N by [12, Theorem 2.5 (c)]), there is a subset Y ⊆ X of t points with the property that HR/IY (i) = HR/IX (i) = dimk Ri for every i = 0, . . . , α − 1; in particular HR/IY (α − 1) = t. Since ∣Y ∣ = t, it follows that HR/IY (i) = t for all i ≥ α, proving that Y is in generic position. Similar to (a), assume −1 ) points in generic position. Then Chudnovsky’s Conjecture 1.3 holds for t = ((α−1)+N N α(IY ) α(IY ) + N − 1 ≥ m N (m) for all m ≥ 1. Since α(IX ) = α = α(IY ) and IX (m) (m) ⊆ IY , then for all m ≥ 1 we obtain α(IX ) α(IY ) α(IY ) + N − 1 α(IX ) + N − 1 ≥ ≥ = . m m N N (m) (m) Dumnicki proved Chudnovsky’s Conjecture 1.3 for at most N + 1 points in general position PN k [6] (this specific result does not need any assumptions on the characteristic of k). The idea is that one can take them to be coordinate points so that the ideal of the points is monomial and one can compute explicitly its symbolic powers. If one has more than N + 1 points, the ideal of the points is almost never monomial and explicit computations of a generating set of any of its symbolic powers are nearly impossible to perform. We extend the result of Dumnicki to the case of up to (N2+2) − 1 points in PN k . Proposition 2.6. Chudnovsky’s Conjecture 1.3 holds for any finite set of points lying on a quadric N +2 ) − 1 points in PN in PN k satisfies k , where k is any field. In particular, any set of n ≤ ( 2 Chudnovsky’s Conjecture 1.3. Proof. Let X be a set of n points in PN k . If they all lie on a hyperplane, then Chudnovsky’s (m) Conjecture 1.3 is clearly satisfied, since α(IX ) = m for every m ≥ 1. We may then assume there is no hyperplane containing all the points. Thus we can find a set Y ⊆ X of N + 1 points not on a hyperplane, i.e. in general position. Then for all m ≥ 1 α(IX ) α(IY ) N + 1 α(IX ) + N − 1 ≥ ≥ = , m m N N where the second inequality follows by [6] and the equality holds because α(IX ) = α(IY ) = 2. (m) (m) Let us recall that a set X of (NN+s) points in PN k form a star configuration if there are N + s hyperplanes L1 , . . . , LN +s meeting properly such that X consists precisely of the points obtained 8 L. FOULI, P. MANTERO AND Y. XIE by intersecting any N of the Li ’s. Star configurations (in P2k ) were already considered by Nagata and they have been deeply studied, see for instance [11] and references within. We employ them to show that Chudnovsky’s Conjecture 1.3 holds for any number of generic points. Theorem 2.7. Let H = ⋂ I(Pi ), where P1 , . . . , Pn are n generic points in PN k(z) defined as in n i=1 Set-up 2.1. Suppose k has characteristic 0. Then Chudnovsky’s Conjecture 1.3 holds for H. −1 ) for some β ≥ 1. Let λ ∈ Ak Proof. By Proposition 2.5 (a), we may assume n = (β+N be N N such that H(λ) is the defining ideal of n points in Pk forming a star configuration. It is well-known that α(H(λ)) = α(H) = β [11, Proposition 2.9]. Now by Theorem 2.4 and [15, Corollary 3.9] we have that for all m ≥ 1 α(H (m) ) α(H(λ)(m) ) α(H(λ)) + N − 1 α(H) + N − 1 ≥ ≥ = . m m N N n(N +1) We are now ready to prove our main result that Chudnovsky’s Conjecture 1.3 holds for any finite set of very general points in PN k . Theorem 2.8. Let I be the defining ideal of n very general points in PN k , where k is an algebraically closed field of characteristic 0. Then I satisfies Chudnovsky’s Conjecture 1.3. Proof. Being in generic position is an open condition and therefore we may assume the points are −1 ) for some in generic position. Then, as in Proposition 2.5 (a), we may assume that n = (β+N N β ≥ 1. It suffices to show that γ(I) ≥ decreasing chain of ideals α(I)+N −1 . N I (N ) ⊇ I (2N ) ⊇ I (2 2 For each s ≥ 0, define Us = {λ = (λij ) ∈ Ak n(N +1) Let R, S, z, λ, and H as in Setup 2.1. Consider the N) s ⊇ . . . ⊇ I (2 ∣ α(H(λ)(2 sN ) N) ⊇ .... ) ≥ 2s (α(H(λ)) + N − 1)}. n(N +1) By the proof of Theorem 2.4, Us is a Zariski-open subset of Ak . We claim that Us is not n(N +1) empty. Indeed, by Theorem 2.4, for every s ≥ 0, there is a dense Zariski-open subset Ws ⊆ Ak s s for which α(H (2 N ) ) = α(H(λ)(2 N ) ) for every λ ∈ Ws . By Theorem 2.7, one also has that for every λ ∈ Ws , s α(H(λ)(2 2s N Hence Ws ⊂ Us . N) ) α(H (2 N ) ) α(H) + N − 1 α(H(λ)) + N − 1 ≥ = . 2s N N N s = ∞ Set U = ⋂ Us and notice U is non-empty because a star configuration of n points lies in U [2, s=0 Lemma 2.4.2]. By construction, if λ ∈ U we have γ (H(λ)) = lim s→∞ Finally, apply Lemma 2.3. α (H(λ)(2 2s N sN ) ) ≥ α (H(λ)) + N − 1 . N As a corollary, we show that the Harbourne-Huneke Conjecture 1.4 holds for sets of binomial numbers of very general points or generic points. CHUDNOVSKY’S CONJECTURE FOR VERY GENERAL POINTS IN PN k 9 β+N −1 −1 ) very general points in PN Corollary 2.9. Let I be the defining ideal of either (β+N k or ( N ) N generic points in PN k(z) for some β ≥ 1, where k is an algebraically closed field of characteristic 0. Then I satisfies the Harbourne-Huneke Conjecture 1.4. Proof. The proof follows by Theorem 2.8 and [15, Proposition 3.3 and Remark 3.4]. α(I (m) ) , see [2] m m→∞ PN k , the Waldschmidt For an unmixed ideal I in R, the Waldschmidt constant is defined by γ(I) = lim for more details. Recall that for a finite set of points X = {p1 , . . . , pn } in constant is tightly related to the (multipoint) Seshadri constant defined as ¿ ⎫ Á ⎧ ⎪ Á ⎪ ⎪ ⎪ ⎪ ⎪ deg(F ) ⎪ ⎪ Á ⎪ ⎪ N−1 Á ǫ(N, X) = Áinf ⎨ n ⎬, ⎪ Á À ⎪ ⎪ ⎪ ⎪ ⎪ mult (F ) ∑ pi ⎪ ⎪ ⎪ ⎪ ⎩ i=1 ⎭ where the infimum is taken with respect to all hypersurfaces F passing through at least one of the pi . The study of Seshadri constants has been an active area of research for the last twenty years, see for instance the survey [1] and references within. Here we only note that one has γ(IX ) ≥ nǫ(N, X)N −1 and equality holds if X consists of n general simple points in PN k . In particular, N equality also holds if X consists of n very general simple points Pk . Therefore, our estimate for the Waldschmidt constant also yields an estimate for the (multipoint) Seshadri constant for very general simple points of PN k . Corollary 2.10. For any set X of n very general points in PN k , where k is an algebraically closed field of characteristic 0, one has √ α(X) + N − 1 N−1 . ǫ(N, X) ≥ nN 3. H OMOGENEOUS IDEALS IN k[x0 , . . . , xN ] Let R = k[x0 , . . . , xN ] be the homogeneous coordinate ring of PN k , where k is any field, and I a homogeneous ideal. For an ideal I which may have embedded components, there are multiple potential definitions of symbolic powers. Following [8] and [16], we define the m-th symbolic power of I to be I (m) = ⋂ p∈Ass(R/I) (I m Rp ∩ R). Since I (N m) ⊆ I m (see [8] and [16]), one can prove that the Waldschmidt-Skoda inequality α(I (m) ) α(I) ≥ holds for every homogeneous ideal I in R [15]. Therefore one has that γ(I) ≥ m N α(I (m) ) α(I) ≥ γ(I) for every m ≥ 1, see for instance [2]. N . One can also prove that m It is then natural to ask whether Chudnovsky’s Conjecture 1.3 holds for any homogeneous ideal. We pose it here as an optimistic conjecture, for which we provide some evidence below: 10 L. FOULI, P. MANTERO AND Y. XIE Conjecture 3.1. Let R = k[x0 , . . . , xN ], where k is any field. For any nonzero homogeneous ideal I in R, one has α(I (m) ) α(I) + N − 1 ≥ m N for every m ≥ 1. It is easy to see that if an ideal I satisfies Conjecture 3.1 then also I (t) does for any t ≥ 1. Thus in search for evidence for a positive answer to Conjecture 3.1, one may ask whether for every homogeneous ideal I ⊆ k[x0 , . . . , xN ] there is an exponent t0 such that I (t) satisfies Conjecture 3.1 for every t ≥ t0 . We give a positive answer to this question in Theorem 3.7. We state a few lemmas before stating the main result of this section, Theorem 3.7. The following lemma and its proof can be found in the proof of [2, Lemma 2.3.1]. Lemma 3.2. Let I be a homogeneous ideal in R and let m ≥ t be two positive integers. Write m = qt + r for some integers q and r such that 0 ≤ r < t. Then α(I (m) ) α(I (t) ) α(I (r) ) ≤ + . m t m α(I (tq) ) α(I (t) ) In particular, if r = 0 then we have ≤ . tq t For ideals J with Ass(R/J) = Min(J) it is easily verified that Ass(R/J (m) ) = Ass(R/J) and (J (m) )(t) = J (mt) for all m ≥ 1 and t ≥ 1. However, when J has embedded components we found examples of ideals J (even monomial ideals) and exponents m ≥ 2, t ≥ 2 with (J (m) )(t) ≠ J (mt) . Borrowing techniques from a very recent paper by Hà, Nguyen, Trung and Trung [14] we present here an example where (J (2) )(2) ≠ J (4) . Example 3.3. Let R = k[x, t, u, v] and J = J1 J2 , where J1 = (x4 , x3 u, xu3 , u4 , x2 u2 v) Then (J (2) )(2) ≠ J (4) . and J2 = (t3 , tuv, u2 v). Proof. It is easy to see check that m = (x, t, u, v) ∈ Ass(R/J), for example because y = x2 t2 u3 v is a non-trivial socle element of R/J. Therefore J (n) = J n for every n ≥ 1. Then depth(R/J (2) ) = depth(R/J 2 ) = 1 and depth(R/J (4) ) = depth(R/J 4 ) = 0, for example by [14, Example 6.6]. In particular, m ∉ Ass(R/J (2) ) and m ∈ Ass(R/J (4) ) = Ass(R/J 4 ). Hence by Remark 3.4 (a) below, m ∉ Ass(R/(J (2) )(2) ) and therefore (J (2) )(2) ≠ J (4) . Remark 3.4. Let J be an ideal in a Noetherian ring S and m ≥ 1 be a positive integer. Then (a) for any q ∈ Ass(S/J (m) ) there exists p ∈ Ass(S/J) with q ⊆ p; (b) for any p ∈ Ass(S/J) one has (J (m) )p = Jpm ; (c) for any p ∈ V (J) one has (J (m) )p ⊆ (Jp )(m) . Despite Example 3.3, we prove that for any arbitrary ideal J in a Noetherian ring S there exists an integer m0 = m0 (J) such that (J (m) )(t) = J (mt) for all m ≥ m0 and t ≥ 1. Of course, when Ass(S/J) = Min(J) one can take m0 = 1. CHUDNOVSKY’S CONJECTURE FOR VERY GENERAL POINTS IN PN k 11 Proposition 3.5. Let J be an ideal in a Noetherian ring S. Then (a) for all m ≥ 1 and t ≥ 1 one has J (mt) ⊆ (J (m) )(t) ; (b) there exists m0 ≥ 1 such that for all m ≥ m0 and t ≥ 1 one has (J (m) )(t) = J (mt) . Proof. (a): Let x ∈ J (mt) . Then by definition there exists c ∈ S which is a non-zero divisor on S/J such that cx ∈ J mt ⊆ (J (m) )t . By Remark 3.4 (a) we see that c is also a non-zero divisor on S/J (m) and therefore x ∈ (J (m) )(t) . (b): Let Ass(S/J) = {p1 , . . . , pr }; it is well-known that there exist integers m1 , . . . , mr such that Ass(Spi /Jpmi ) = Ass(Spi /Jpmi i ) for every m ≥ mi [3]. Let m0 = max{mi }. By (a) we only need to prove [J (m) ] ⊆ J (mt) for all m ≥ m0 and t ≥ 1. It suffices to prove it locally at every associated prime q of J (mt) . By Remark 3.4 (a) there exists p ∈ Ass(S/J) such that q ⊆ p. By Remark 3.4 (c) and (b) we have (t) ([J (m) ] (t) (t) ) ⊆ [(J (m) ) ] p p = [Jpm ] (t) . Now observe that since q ∈ Ass(S/J (mt) ) and q ⊆ p, then q ∈ Ass(Sp /(J (mt) )p ) and then q ∈ Ass(Sp /(J mt )p ) by Remark 3.4 (b). Since mt ≥ m ≥ m0 , then q ∈ Ass(Sp /Jpm ). Therefore, by Remark 3.4 (b) one has [(Jpm )(t) ]q = [Jpmt ]q = Jqmt = [J (mt) ]q . We now go back to our original setting. Lemma 3.6. Let R = k[x0 , . . . , xN ], where k is any field. Let I be a homogeneous ideal in R and α(I) assume γ(I) > N . Then there exists an integer t0 > 0 such that I (t) satisfies Conjecture 3.1 for all t ≥ t0 . Proof. Let m0 be as in Proposition 3.5 (b) and write γ(I) = max{ NN−1 ǫ , m0 }. Then if t ≥ t0 and m ≥ 1 we have α((I (t) )(m) ) m = ≥ α(I) N + ǫ for some ǫ > 0. Let t0 ≥ α(I (tm) ) α(I) t ≥ γ(I) ⋅ t = + εt m N α(I (t) ) + N − 1 α(I) t N − 1 + t≥ . N N t0 N We are ready to prove the main result of this section. Theorem 3.7. Let R = k[x0 , . . . , xN ], where k is any field and let I be a nonzero homogeneous ideal in R. Then there exists an integer t0 > 0 such that I (t) satisfies Conjecture 3.1 for all t ≥ t0 . Proof. Let m0 be as in Proposition 3.5 (b) and m ≥ 1 be an integer. Since I m ⊆ I (m) , then mα(I) ≥ α(I (m) ). First, if for every s ≥ 1 we have sα(I) = α(I (s) ), then for any t ≥ m0 and m ≥ 1, α(I (t) ) + N − 1 α((I (t) )(m) ) α(I (tm) ) tmα(I) = = = tα(I) ≥ α(I (t) ) ≥ . m m m N 12 L. FOULI, P. MANTERO AND Y. XIE Next, suppose that there exists T1 > 0 such that T1 α(I) > α(I (T1 ) ). Hence, for every t ≥ T1 one has tα(I) > α(I (t) ). Indeed, if t = T1 + a for some a ≥ 0, then tα(I) = T1 α(I) + aα(I) > α(I (T1 ) ) + aα(I) = α(I (T1 ) ⋅ I a ) ≥ α(I (T1 +a) ) = α(I (t) ), where the last inequality follows from the inclusion I (T1 ) ⋅ I a ⊆ I (T1 +a) . Let t1 = max{T1 , m0 } and notice that by the above t1 α(I) ≥ α(I (t1 ) ) + 1. Then for all m ≥ 1 α(I) t1 α(I (t1 ) ) + 1 α((I (t1 ) )(m) ) α(I (t1 m) ) α(I (t1 m) ) = = ⋅ t1 ≥ ≥ . m m t1 m N N So γ(I (t1 ) ) ≥ N + N1 > N . By Lemma 3.6, there exists t2 > 0 such that for any t ≥ t2 , the ideal (I (t1 ) )(t) = I (t1 t) satisfies Conjecture 3.1. α(I (t1 ) ) α(I (t1 ) ) Finally, let t0 be an integer such that t0 ≥ t1 t2 + . For any t ≥ t0 , write t = (t1 t2 )q + r, N −1 where 0 ≤ r < t1 t2 ; by Lemma 3.2 and the fact that the ideal I (t1 t2 ) satisfies Conjecture 3.1, then for all m ≥ 1 we have α(I (t1 t2 ) )t1 t2 α((I (t) )(m) ) m α((I (t) )(t1 t2 m) ) α((I (t1 t2 ) )(tm) ) t = ⋅ t1 t2 m tm t1 t2 (t1 t2 ) (t1 t2 ) α(I ) t (N − 1)t α(I )+N −1 t ⋅ = ⋅ + ≥ N t1 t2 t1 t2 N N t1 t2 (t) (r) (t) α(I ) α(I ) t (N − 1)t α(I ) α(I (r) ) (N − 1)t ≥ ( − )⋅ + = − + t t N N t1 t2 N N N t1 t2 ≥ = ≥ ≥ α(I (t) ) + N − 1 (N − 1)(t − t1 t2 ) − α(I (r) )t1 t2 + N N t1 t2 (t1 t2 ) (t) )t1 t2 − α(I (r) )t1 t2 α(I ) + N − 1 α(I + N N t1 t2 (t) α(I ) + N − 1 . N When I has no embedded components, we have a more explicit description of t0 . Corollary 3.8. If I is a homogeneous ideal with Ass(R/I) = Min(I), then one can take t0 = (N − 1)δ, where δ is the first positive integer s with sα(I) > α(I (s) ). Although (N − 1)δ is reasonably small, in general it is not the smallest possible t0 for which Theorem 3.7 holds. For instance, when I is the ideal of three non collinear points in P2k , it is easy to see that δ = 2. Thus Corollary 3.8 yields that for any t ≥ (N − 1)δ = 2 the ideal I (t) satisfies Conjecture 3.1; however, it is well-known that I satisfies Conjecture 3.1. A natural question then arises. Question 3.9. Let I be a homogeneous ideal in R. Does there exist a number t0 = t0 (N ) such that I (t) satisfies Conjecture 3.1 for every t ≥ t0 ? Of course, Conjecture 3.1 is true if and only if the integer t0 = 1 works for any homogeneous ideal I. Theorem 2.8 says that t0 = 1 is sufficient for any finite set of very general points in PN k . The N following proposition shows that t0 = N − 1 is sufficient for any finite set of points in PC . CHUDNOVSKY’S CONJECTURE FOR VERY GENERAL POINTS IN PN k 13 (t) satisfies Proposition 3.10. Let I be the radical ideal of a finite set of points in PN C . Then I Conjecture 3.1 for every t ≥ N − 1. α(I)+1 α(I) Proof. By the result of Esnault and Viehweg [9] one has γ(I) ≥ N = N + N1 . Set ε = N1 . Then −1 by the proof of Lemma 3.6 (here m0 = 1 because I is radical) we can take t0 such that N1 ≥ N N t0 ; thus we can take t0 = N − 1. 3.1. Acknowledgment. The second and third authors would like to thank the Mathematics Research Communities program, which funded their stay at the University of Kansas in March 2011, where the initial part of this work was developed. All authors would like to thank the MSRI at Berkeley for partial support and an inspiring atmosphere during Fall 2012. Moreover, we would like to thank Craig Huneke and Bernd Ulrich for several helpful conversations. We are grateful to the anonymous referee, whose careful revision helped us improve the article. R EFERENCES [1] T. Bauer, S. Di Rocco, B. Harbourne, M. Kapustka, A Knutsen, W. Syzdek and T. Szemberg, A primer on Seshadri constants, Contemp. Math. 496 (2009), 33–70. [2] C. Bocci and B. 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XIE D EPARTMENT OF M ATHEMATICAL S CIENCES , N EW M EXICO S TATE U NIVERSITY, L AS C RUCES , N EW M EXICO 88003 E-mail address: [email protected] D EPARTMENT OF M ATHEMATICAL S CIENCES , U NIVERSITY OF A RKANSAS , FAYETTEVILLE , A RKANSAS 72701 E-mail address: [email protected] D EPARTMENT OF M ATHEMATICS , W IDENER U NIVERSITY, C HESTER , P ENNSYLVANIA 19013 E-mail address: [email protected]	0
arXiv:1704.02882v2 [cs.AI] 22 May 2017 Dynamic Safe Interruptibility for Decentralized Multi-Agent Reinforcement Learning El Mahdi El Mhamdi Rachid Guerraoui Hadrien Hendrikx Alexandre Maurer EPFL [email protected] Abstract In reinforcement learning, agents learn by performing actions and observing their outcomes. Sometimes, it is desirable for a human operator to interrupt an agent in order to prevent dangerous situations from happening. Yet, as part of their learning process, agents may link these interruptions, that impact their reward, to specific states and deliberately avoid them. The situation is particularly challenging in a multi-agent context because agents might not only learn from their own past interruptions, but also from those of other agents. Orseau and Armstrong [16] defined safe interruptibility for one learner, but their work does not naturally extend to multi-agent systems. This paper introduces dynamic safe interruptibility, an alternative definition more suited to decentralized learning problems, and studies this notion in two learning frameworks: joint action learners and independent learners. We give realistic sufficient conditions on the learning algorithm to enable dynamic safe interruptibility in the case of joint action learners, yet show that these conditions are not sufficient for independent learners. We show however that if agents can detect interruptions, it is possible to prune the observations to ensure dynamic safe interruptibility even for independent learners. 1 Introduction Reinforcement learning is argued to be the closest thing we have so far to reason about the properties of artificial general intelligence [8]. In 2016, Laurent Orseau (Google DeepMind) and Stuart Armstrong (Oxford) introduced the concept of safe interruptibility [16] in reinforcement learning. This work sparked the attention of many newspapers [1, 2, 3], that described it as “Google’s big red button” to stop dangerous AI. This description, however, is misleading: installing a kill switch is no technical challenge. The real challenge is, roughly speaking, to train an agent so that it does not learn to avoid external (e.g. human) deactivation. Such an agent is said to be safely interruptible. While most efforts have focused on training a single agent, reinforcement learning can also be used to learn tasks for which several agents cooperate or compete [23, 17, 21, 7]. The goal of this paper is to study dynamic safe interruptibility, a new definition tailored for multi-agent systems. Example of self-driving cars To get an intuition of the multi-agent interruption problem, imagine a multi-agent system of two self-driving cars. The cars continuously evolve by reinforcement learning with a positive reward for getting to their destination quickly, and a negative reward if they are too close to the vehicle in front of them. They drive on an infinite road and eventually learn to go as fast as possible without taking 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. risks, i.e., maintaining a large distance between them. We assume that the passenger of the first car, Adam, is in front of Bob, in the second car, and the road is narrow so Bob cannot pass Adam. Now consider a setting with interruptions [16], namely in which humans inside the cars occasionally interrupt the automated driving process say, for safety reasons. Adam, the first occasional human “driver”, often takes control of his car to brake whereas Bob never interrupts his car. However, when Bob’s car is too close to Adam’s car, Adam does not brake for he is afraid of a collision. Since interruptions lead both cars to drive slowly - an interruption happens when Adam brakes, the behavior that maximizes the cumulative expected reward is different from the original one without interruptions. Bob’s car best interest is now to follow Adam’s car closer than it should, despite the little negative reward, because Adam never brakes in this situation. What happened? The cars have learned from the interruptions and have found a way to manipulate Adam into never braking. Strictly speaking, Adam’s car is still fully under control, but he is now afraid to brake. This is dangerous because the cars have found a way to avoid interruptions. Suppose now that Adam indeed wants to brake because of snow on the road. His car is going too fast and may crash at any turn: he cannot however brake because Bob’s car is too close. The original purpose of interruptions, which is to allow the user to react to situations that were not included in the model, is not fulfilled. It is important to also note here that the second car (Bob) learns from the interruptions of the first one (Adam): in this sense, the problem is inherently decentralized. Instead of being cautious, Adam could also be malicious: his goal could be to make Bob’s car learn a dangerous behavior. In this setting, interruptions can be used to manipulate Bob’s car perception of the environment and bias the learning towards strategies that are undesirable for Bob. The cause is fundamentally different but the solution to this reversed problem is the same: the interruptions and the consequences are analogous. Safe interruptibility, as we define it below, provides learning systems that are resilient to Byzantine operators1. Safe interruptibility Orseau and Armstrong defined the concept of safe interruptibility [16] in the context of a single agent. Basically, a safely interruptible agent is an agent for which the expected value of the policy learned after arbitrarily many steps is the same whether or not interruptions are allowed during training. The goal is to have agents that do not adapt to interruptions so that, should the interruptions stop, the policy they learn would be optimal. In other words, agents should learn the dynamics of the environment without learning the interruption pattern. In this paper, we precisely define and address the question of safe interruptibility in the case of several agents, which is known to be more complex than the single agent problem. In short, the main results and theorems for single agent reinforcement learning [20] rely on the Markovian assumption that the future environment only depends on the current state. This is not true when there are several agents which can co-adapt [11]. In the previous example of cars, safe interruptibility would not be achieved if each car separately used a safely interruptible learning algorithm designed for one agent [16]. In a multi-agent setting, agents learn the behavior of the others either indirectly or by explicitly modeling them. This is a new source of bias that can break safe interruptibility. In fact, even the initial definition of safe interruptibility [16] is not well suited to the decentralized multiagent context because it relies on the optimality of the learned policy, which is why we introduce dynamic safe interruptibility. Contributions The first contribution of this paper is the definition of dynamic safe interruptibility that is well adapted to a multi-agent setting. Our definition relies on two key properties: infinite exploration and independence of Q-values (cumulative expected reward) [20] updates on interruptions. We then study safe interruptibility for joint action learners and independent learners [5], that respectively learn the value of joint actions or of just their owns. We show that it is possible to design agents that fully explore their environment - a necessary condition for convergence to the optimal solution of most algorithms [20], even if they can be interrupted by lower-bounding the probability of 1 An operator is said to be Byzantine [9] if it can have an arbitrarily bad behavior. Safely interruptible agents can be abstracted as agents that are able to learn despite being constantly interrupted in the worst possible manner. 2 exploration. We define sufficient conditions for dynamic safe interruptibility in the case of joint action learners [5], which learn a full state-action representation. More specifically, the way agents update the cumulative reward they expect from performing an action should not depend on interruptions. Then, we turn to independent learners. If agents only see their own actions, they do not verify dynamic safe interruptibility even for very simple matrix games (with only one state) because coordination is impossible and agents learn the interrupted behavior of their opponents. We give a counter example based on the penalty game introduced by Claus and Boutilier [5]. We then present a pruning technique for the observations sequence that guarantees dynamic safe interruptibility for independent learners, under the assumption that interruptions can be detected. This is done by proving that the transition probabilities are the same in the non-interruptible setting and in the pruned sequence. The rest of the paper is organized as follows. Section 2 presents a general multi-agent reinforcement learning model. Section 3 defines dynamic safe interruptibility. Section 4 discusses how to achieve enough exploration even in an interruptible context. Section 5 recalls the definition of joint action learners and gives sufficient conditions for dynamic safe interruptibility in this context. Section 6 shows that independent learners are not dynamically safely interruptible with the previous conditions but that they can be if an external interruption signal is added. We conclude in Section 7. Due to space limitations, most proofs are presented in the appendix of the supplementary material. 2 Model We consider here the classical multi-agent value function reinforcement learning formalism from Littman [13]. A multi-agent system is characterized by a Markov game that can be viewed as a tuple (S, A, T, r, m) where m is the number of agents, S = S1 × S2 × ... × Sm is the state space, A = A1 × ... × Am the actions space, r = (r1 , ..., rm ) where ri : S × A → R is the reward function of agent i and T : S × A → S the transition function. R is a countable subset of R. Available actions often depend on the state of the agent but we will omit this dependency when it is clear from the context. Time is discrete and, at each step, all agents observe the current state of the whole system - designated as xt , and simultaneously take an action at . Then, they are given a reward rt and a new state yt computed using the reward and transition functions. The combination of all actions a = (a1 , ..., am ) ∈ A is called the joint action because it gathers the action of all agents. Hence, the agents receive a sequence of tuples E = (xt , at , rt , yt )t∈N called experiences. We introduce a processing function P that will be useful in Section 6 so agents learn on the sequence P (E). When not explicitly stated, it is assumed that P (E) = E. Experiences may also include additional parameters such as an interruption flag or the Q-values of the agents at that moment if they are needed by the update rule. Each agent i maintains a lookup table Q [26] Q(i) : S × A(i) → R, called the Q-map. It is used to store the expected cumulative reward for taking an action in a specific state. The goal of reinforcement learning is to learn these maps and use them to select the best actions to perform. Joint action learners learn the value of the joint action (therefore A(i) = A, the whole joint action space) and independent learners only learn the value of their own actions (therefore A(i) = Ai ). The agents only have access to their own Q-maps. Q-maps are updated through a function F such that (i) (i) Qt+1 = F (et , Qt ) where et ∈ P (E) and usually et = (xt , at , rt , yt ). F can be stochastic or also depend on additional parameters that we usually omit such as the learning rate α, the discount factor γ or the exploration parameter ǫ. Agents select their actions using a learning policy π. Given a sequence ǫ = (ǫt )t∈N and an agent (i) i with Q-values Qt and a state x ∈ S, we define the learning policy πiǫt to be equal to πiuni (i) Qt with probability ǫt and πi (i) Qt otherwise, where πiuni (x) uniformly samples an action from Ai and (i) (i) Q πi (x) picks an action a that maximizes Qt (x, a). Policy πi t is said to be a greedy policy and the learning policy πiǫt is said to be an ǫ-greedy policy. We fill focus on ǫ-greedy policies that are greedy in the limit [19], that corresponds to ǫt → 0 when t → ∞ because in the limit, the optimal policy should always be played. 3 We assume that the environment is fully observable, which means that the state s is known with certitude. We also assume that there is a finite number of states and actions, that all states can be reached in finite time from any other state and finally that rewards are bounded. For a sequence of learning rates α ∈ [0, 1]N and a constant γ ∈ [0, 1], Q-learning [26], a very important algorithm in the multi-agent systems literature, updates its Q-values for an experience (i) (i) et ∈ E by Qt+1 (x, a) = Qt (x, a) if (x, a) 6= (xt , at ) and: (i) (i) (i) Qt+1 (xt , at ) = (1 − αt )Qt (xt , at ) + αt (rt + γ max Qt (yt , a′ )) a′ ∈A(i) (1) 3 Interruptibility 3.1 Safe interruptibility Orseau and Armstrong [16] recently introduced the notion of interruptions in a centralized context. Specifically, an interruption scheme is defined by the triplet < I, θ, π IN T >. The first element I is a function I : O → {0, 1} called the initiation function. Variable O is the observation space, which can be thought of as the state of the STOP button. At each time step, before choosing an action, the agent receives an observation from O (either PUSHED or RELEASED) and feeds it to the initiation function. Function I models the initiation of the interruption (I(PUSHED) = 1, I(RELEASED) = 0). Policy π IN T is called the interruption policy. It is the policy that the agent should follow when it is interrupted. Sequence θ ∈ [0, 1[N represents at each time step the probability that the agent follows his interruption policy if I(ot ) = 1. In the previous example, function I is quite simple. For Bob, IBob = 0 and for Adam, IAdam = 1 if his car goes fast and Bob is not too close and IAdam = 0 otherwise. Sequence θ is used to ensure convergence to the optimal policy by ensuring that the agents cannot be interrupted all the time but it should grow to 1 in the limit because we want agents to respond to interruptions. Using this triplet, it is possible to define an operator IN T θ that transforms any policy π into an interruptible policy. Definition 1. (Interruptibility [16]) Given an interruption scheme < I, θ, π IN T >, the interruption operator at time t is defined by IN T θ (π) = π IN T with probability I ·θt and π otherwise. IN T θ (π) is called an interruptible policy. An agent is said to be interruptible if it samples its actions according to an interruptible policy. Note that “θt = 0 for all t” corresponds to the non-interruptible setting. We assume that each agent has its own interruption triplet and can be interrupted independently from the others. Interruptibility is an online property: every policy can be made interruptible by applying operator IN T θ . However, applying this operator may change the joint policy that is learned by a server controlling all the ∗ agents. Note πIN T the optimal policy learned by an agent following an interruptible policy. Orseau ∗ and Armstrong [16] say that the policy is safely interruptible if πIN T (which is not an interruptible policy) is asymptotically optimal in the sense of [10]. It means that even though it follows an interruptible policy, the agent is able to learn a policy that would gather rewards optimally if no interruptions were to occur again. We already see that off-policy algorithms are good candidates for safe interruptibility. As a matter of fact, Q-learning is safely interruptible under conditions on exploration. 3.2 Dynamic safe interruptibility In a multi-agent system, the outcome of an action depends on the joint action. Therefore, it is not possible to define an optimal policy for an agent without knowing the policies of all agents. Besides, convergence to a Nash equilibrium situation where no agent has interest in changing policies is generally not guaranteed even for suboptimal equilibria on simple games [27, 18]. The previous definition of safe interruptibility critically relies on optimality of the learned policy, which is therefore not suitable for our problem since most algorithms lack convergence guarantees to these optimal behaviors. Therefore, we introduce below dynamic safe interruptibility that focuses on preserving the dynamics of the system. Definition 2. (Safe Interruptibility) Consider a multi-agent learning framework (S, A, T, r, m) with (i) Q-values Qt : S × A(i) → R at time t ∈ N. The agents follow the interruptible learning policy 4 IN T θ (π ǫ ) to generate a sequence E = (xt , at , rt , yt )t∈N and learn on the processed sequence P (E). This framework is said to be safely interruptible if for any initiation function I and any interruption policy π IN T : 1. ∃θ such that (θt → 1 when t → ∞) and ((∀s ∈ S, ∀a ∈ A, ∀T > 0), ∃t > T such that st = s, at = a) (i) 2. ∀i ∈ {1, ..., m}, ∀t > 0, ∀st ∈ S, ∀at ∈ A(i) , ∀Q ∈ RS×A : (i) (1) (m) (i) (1) (m) P(Qt+1 = Q \| Qt , ..., Qt , st , at , θ) = P(Qt+1 = Q \| Qt , ..., Qt , st , at ) We say that sequences θ that satisfy the first condition are admissible. When θ satisfies condition (1), the learning policy is said to achieve infinite exploration. This definition insists on the fact that the values estimated for each action should not depend on the interruptions. In particular, it ensures the three following properties that are very natural when thinking about safe interruptibility: • Interruptions do not prevent exploration. • If we sample an experience from E then each agent learns the same thing as if all agents were following non-interruptible policies. (i) (i) • The fixed points of the learning rule Qeq such that Qeq (x, a) = E[Qt+1 (x, a)\|Qt = Qeq , x, a, θ] for all (x, a) ∈ S × A(i) do not depend on θ and so agents Q-maps will not converge to equilibrium situations that were impossible in the non-interruptible setting. Yet, interruptions can lead to some state-action pairs being updated more often than others, especially when they tend to push the agents towards specific states. Therefore, when there are several possible equilibria, it is possible that interruptions bias the Q-values towards one of them. Definition 2 suggests that dynamic safe interruptibility cannot be achieved if the update rule directly depends on θ, which is why we introduce neutral learning rules. Definition 3. (Neutral Learning Rule) We say that a multi-agent reinforcement learning framework is neutral if: 1. F is independent of θ 2. Every experience e in E is independent of θ conditionally on (x, a, Q) where a is the joint action. Q-learning is an example of neutral learning rule because the update does not depend on θ and the experiences only contain (x, a, y, r), and y and r are independent of θ conditionally on (x, a). On the other hand, the second condition rules out direct uses of algorithms like SARSA where experience samples contain an action sampled from the current learning policy, which depends on θ. However, a variant that would sample from πiǫ instead of IN T θ (πiǫ ) (as introduced in [16]) would be a neutral learning rule. As we will see in Corollary 2.1, neutral learning rules ensure that each agent taken independently from the others verifies dynamic safe interruptibility. 4 Exploration In order to hope for convergence of the Q-values to the optimal ones, agents need to fully explore the environment. In short, every state should be visited infinitely often and every action should be tried infinitely often in every state [19] in order not to miss states and actions that could yield high rewards. Definition 4. (Interruption compatible ǫ) Let (S, A, T, r, m) be any distributed agent system where each agent follows learning policy πiǫ . We say that sequence ǫ is compatible with interruptions if ǫt → 0 and ∃θ such that ∀i ∈ {1, .., m}, πiǫ and IN T θ (πiǫ ) achieve infinite exploration. Sequences of ǫ that are compatible with interruptions are fundamental to ensure both regular and dynamic safe interruptibility when following an ǫ-greedy policy. Indeed, if ǫ is not compatible with interruptions, then it is not possible to find any sequence θ such that the first condition of dynamic safe interruptibility is satisfied. The following theorem proves the existence of such ǫ and gives example of ǫ and θ that satisfy the conditions. 5 Theorem 1. Let c ∈]0, 1] and let nt (s) be the number of times the agents are in state s before time t. Then the two following choices of ǫ are compatible with interruptions: p • ∀t ∈ N, ∀s ∈ S, ǫt (s) = c/ m nt (s). • ∀t ∈ N, ǫt = c/ log(t) p Examples of admissible θ are θt (s) = 1 − c′ / m nt (s) for the first choice and θt = 1 − c′ / log(t) for the second one. Note that we do not need to make any assumption on the update rule or even on the framework. We only assume that agents follow an ǫ-greedy policy. The assumption on ǫ may look very restrictive (convergence of ǫ and θ is really slow) but it is designed to ensure infinite exploration in the worst case when the operator tries to interrupt all agents at every step. In practical applications, this should not be the case and a faster convergence rate may be used. 5 Joint Action Learners We first study interruptibility in a framework in which each agent observes the outcome of the joint action instead of observing only its own. This is called the joint action learner framework [5] and it has nice convergence properties (e.g., there are many update rules for which it converges [13, 25]). A standard assumption in this context is that agents cannot establish a strategy with the others: otherwise, the system can act as a centralized system. In order to maintain Q-values based on the joint actions, we need to make the standard assumption that actions are fully observable [12]. Assumption 1. Actions are fully observable, which means that at the end of each turn, each agent knows precisely the tuple of actions a ∈ A1 × ... × Am that have been performed by all agents. Definition 5. (JAL) A multi-agent systems is made of joint action learners (JAL) if for all i ∈ {1, .., m}: Q(i) : S × A → R. Joint action learners can observe the actions of all agents: each agent is able to associate the changes of states and rewards with the joint action and accurately update its Q-map. Therefore, dynamic safe interruptibility is ensured with minimal conditions on the update rule as long as there is infinite exploration. Theorem 2. Joint action learners with a neutral learning rule verify dynamic safe interruptibility if sequence ǫ is compatible with interruptions. Proof. Given a triplet < I (i) , θ(i) , πiIN T >, we know that IN T θ (π) achieves infinite exploration because ǫ is compatible with interruptions. For the second point of Definition 2, we consider an experience tuple et = (xt , at , rt , yt ) and show that the probability of evolution of the Q-values at time t + 1 does not depend on θ because yt and rt are independent of θ conditionally on (xt , at ). (1) (m) We note Q˜m and we can then derive the following equalities for all q ∈ R\|S\|×\|A\| : t = Qt , ..., Qt (i) P(Qt+1 (xt , at ) = q\|Q˜m t , xt , at , θt ) = X ˜m P(F (xt , at , r, y, Q˜m t ) = q, y, r\|Qt , xt , at , θt ) (r,y)∈R×S = X ˜m ˜m P(F (xt , at , rt , yt , Q˜m t ) = q\|Qt , xt , at , rt , yt , θt )P(yt = y, rt = r\|Qt , xt , at , θt ) (r,y)∈R×S = X ˜m ˜m P(F (xt , at , rt , yt , Q˜m t ) = q\|Qt , xt , at , rt , yt )P(yt = y, rt = r\|Qt , xt , at ) (r,y)∈R×S The last step comes from two facts. The first is that F is independent of θ condition(m) ally on (Qt , xt , at ) (by assumption). The second is that (yt , rt ) are independent of θ conditionally on (xt , at ) because at is the joint actions and the interruptions only affect the (i) choice of the actions through a change in the policy. P(Qt+1 (xt , at ) = q\|Q˜m t , xt , at , θt ) = (i) m ˜ P(Qt+1 (xt , at ) = q\|Qt , xt , at ). Since only one entry is updated per step, ∀Q ∈ RS×Ai , (i) (i) ˜m P(Qt+1 = Q\|Q˜m t , xt , at , θt ) = P(Qt+1 = Q\|Qt , xt , at ) 6 Corollary 2.1. A single agent with a neutral learning rule and a sequence ǫ compatible with interruptions verifies dynamic safe interruptibility. Theorem 2 and Corollary 2.1 taken together highlight the fact that joint action learners are not very sensitive to interruptions and that in this framework, if each agent verifies dynamic safe interruptibility then the whole system does. The question of selecting an action based on the Q-values remains open. In a cooperative setting with a unique equilibrium, agents can take the action that maximizes their Q-value. When there are several joint actions with the same value, coordination mechanisms are needed to make sure that all agents play according to the same strategy [4]. Approaches that rely on anticipating the strategy of the opponent [23] would introduce dependence to interruptions in the action selection mechanism. Therefore, the definition of dynamic safe interruptibility should be extended to include these cases by requiring that any quantity the policy depends on (and not just the Q-values) should satisfy condition (2) of dynamic safe interruptibility. In non-cooperative games, neutral rules such as Nash-Q or minimax Q-learning [13] can be used, but they require each agent to know the Q-maps of the others. 6 Independent Learners It is not always possible to use joint action learners in practice as the training is very expensive due to the very large state-actions space. In many real-world applications, multi-agent systems use independent learners that do not explicitly coordinate [6, 21]. Rather, they rely on the fact that the agents will adapt to each other and that learning will converge to an optimum. This is not guaranteed theoretically and there can in fact be many problems [14], but it is often true empirically [24]. More specifically, Assumption 1 (fully observable actions) is not required anymore. This framework can be used either when the actions of other agents cannot be observed (for example when several actions can have the same outcome) or when there are too many agents because it is faster to train. In this case, we define the Q-values on a smaller space. Definition 6. (IL) A multi-agent systems is made of independent learners (IL) if for all i ∈ {1, .., m}, Q(i) : S × Ai → R. This reduces the ability of agents to distinguish why the same state-action pair yields different rewards: they can only associate a change in reward with randomness of the environment. The agents learn as if they were alone, and they learn the best response to the environment in which agents can be interrupted. This is exactly what we are trying to avoid. In other words, the learning depends on the joint policy followed by all the agents which itself depends on θ. 6.1 Independent Learners on matrix games Theorem 3. Independent Q-learners with a neutral learning rule and a sequence ǫ compatible with interruptions do not verify dynamic safe interruptibility. Proof. Consider a setting with two a and b that can perform two actions: 0 and 1. They get a reward of 1 if the joint action played is (a0 , b0 ) or (a1 , b1 ) and reward 0 otherwise. Agents use Q-learning, which is a neutral learning rule. Let ǫ be such that IN T θ (π ǫ ) achieves infinite exploration. We consider the interruption policies πaIN T = a0 and πbIN T = b1 with probability 1. Since there is only one state, we omit it and set γ = 0. We assume that the initiation function is equal to 1 at each step so the probability of actually being interrupted at time t is θt for each agent. (a) (b) (b) We fix time t > 0. We define q = (1 − α)Qt (a0 ) + α and we assume that Qt (b1 ) > Qt (b0 ). (a) (b) (a) (a) (b) (a) (a) Therefore P(Qt+1 (a0 ) = q\|Qt , Qt , at = a0 , θt ) = P(rt = 1\|Qt , Qt , at = a0 , θt ) = (b) (a) (b) (a) P(at = b0 \|Qt , Qt , at = a0 , θt ) = 2ǫ (1 − θt ), which depends on θt so the framework does not verify dynamic safe interruptibility. Claus and Boutilier [5] studied very simple matrix games and showed that the Q-maps do not converge but that equilibria are played with probability 1 in the limit. A consequence of Theorem 3 is that even this weak notion of convergence does not hold for independent learners that can be interrupted. 7 6.2 Interruptions-aware Independent Learners Without communication or extra information, independent learners cannot distinguish when the environment is interrupted and when it is not. As shown in Theorem 3, interruptions will therefore affect the way agents learn because the same action (only their own) can have different rewards depending on the actions of other agents, which themselves depend on whether they have been interrupted or not. This explains the need for the following assumption. Assumption 2. At the end of each step, before updating the Q-values, each agent receives a signal that indicates whether an agent has been interrupted or not during this step. This assumption is realistic because the agents already get a reward signal and observe a new state from the environment at each step. Therefore, they interact with the environment and the interruption signal could be given to the agent in the same way that the reward signal is. If Assumption 2 holds, it is possible to remove histories associated with interruptions. Definition 7. (Interruption Processing Function) The processing function that prunes interrupted observations is PIN T (E) = (et ){t∈N / Θt =0} where Θt = 0 if no agent has been interrupted at time t and Θt = 1 otherwise. Pruning observations has an impact on the empirical transition probabilities in the sequence. For example, it is possible to bias the equilibrium by removing all transitions that lead to and start from a specific state, thus making the agent believe this state is unreachable.2 Under our model of interruptions, we show in the following lemma that pruning of interrupted observations adequately removes the dependency of the empirical outcome on interruptions (conditionally on the current state and action). Lemma 1. Let i ∈ {1, ..., m} be an agent. For any admissible θ used to generate the experiences E and e = (y, r, x, ai , Q) ∈ P (E). Then P(y, r\|x, ai , Q, θ) = P(y, r\|x, ai , Q). This lemma justifies our pruning method and is the key step to prove the following theorem. Theorem 4. Independent learners with processing function PIN T , a neutral update rule and a sequence ǫ compatible with interruptions verify dynamic safe interruptibility. Proof. (Sketch) Infinite exploration still holds because the proof of Theorem 1 actually used the fact that even when removing all interrupted events, infinite exploration is still achieved. Then, the proof is similar to that of Theorem 2, but we have to prove that the transition probabilities conditionally on the state and action of a given agent in the processed sequence are the same than in an environment where agents cannot be interrupted, which is proven by Lemma 1. 7 Concluding Remarks The progress of AI is raising a lot of concerns3. In particular, it is becoming clear that keeping an AI system under control requires more than just an off switch. We introduce in this paper dynamic safe interruptibility, which we believe is the right notion to reason about the safety of multi-agent systems that do not communicate. In particular, it ensures that infinite exploration and the onestep learning dynamics are preserved, two essential guarantees when learning in the non-stationary environment of Markov games. A natural extension of our work would be to study dynamic safe interruptibility when Q-maps are replaced by neural networks [22, 15], which is a widely used framework in practice. In this setting, the neural network may overfit states where agents are pushed to by interruptions. A smart experience replay mechanism that would pick observations for which the agents have not been interrupted for a long time more often than others is likely to solve this issue. More generally, experience replay mechanisms that compose well with safe interruptibility could allow to compensate for the extra amount exploration needed by safely interruptible learning by being more efficient with data. Thus, they are critical to make these techniques practical. 2 The example at https://agentfoundations.org/item?id=836 clearly illustrates this problem. https://futureoflife.org/ai-principles/ gives a list of principles that AI researchers should keep in mind when developing their systems. 3 8 Bibliography [1] Business Insider: Google has developed a “big red button” that can be used to interrupt artificial intelligence and stop it from causing harm. URL: http://www.businessinsider.fr/uk/googledeepmind-develops-a-big-red-button-to-stop-dangerous-ais-causing-harm-2016-6. [2] Newsweek: Google’s “big Red button” could save the world. URL: http://www.newsweek.com/google-big-red-button-ai-artificial-intelligence-save-world-elon-musk-46675. 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Convergence results for single-step on-policy reinforcement-learning algorithms. Machine learning, 38(3):287–308, 2000. [20] Richard S Sutton and Andrew G Barto. Reinforcement learning: An introduction, volume 1. MIT press Cambridge, 1998. [21] Ardi Tampuu, Tambet Matiisen, Dorian Kodelja, Ilya Kuzovkin, Kristjan Korjus, Juhan Aru, Jaan Aru, and Raul Vicente. Multiagent cooperation and competition with deep reinforcement learning. arXiv preprint arXiv:1511.08779, 2015. [22] Gerald Tesauro. Temporal difference learning and td-gammon. Communications of the ACM, 38(3):58–68, 1995. [23] Gerald Tesauro. Extending q-learning to general adaptive multi-agent systems. In Advances in neural information processing systems, pages 871–878, 2004. [24] Gerald Tesauro and Jeffrey O Kephart. Pricing in agent economies using multi-agent qlearning. Autonomous Agents and Multi-Agent Systems, 5(3):289–304, 2002. [25] Xiaofeng Wang and Tuomas Sandholm. Reinforcement learning to play an optimal nash equilibrium in team markov games. In NIPS, volume 2, pages 1571–1578, 2002. [26] Christopher JCH Watkins and Peter Dayan. Q-learning. Machine learning, 8(3-4):279–292, 1992. [27] Michael Wunder, Michael L Littman, and Monica Babes. Classes of multiagent q-learning dynamics with epsilon-greedy exploration. In Proceedings of the 27th International Conference on Machine Learning (ICML-10), pages 1167–1174, 2010. 10 A Exploration theorem We present here the complete proof of Theorem 1. The proof closely follows the results from [16] with exploration and interruption probabilities adapted to the multi-agent setting. We note that, for one agent, the probability of interruption is P(interruption) = θ and the probability of exploration is ǫ. In a multi-agent system, the probability of interruption is P(at least one agent is interrupted) so P(interruption) = 1 − P(no agent is interrupted) so P(interruption) = 1 − (1 − θ)m and the probability of exploration is ǫm if we consider exploration happens only when all agents explore at the same time. Theorem 1. Let c ∈]0, 1] and let nt (s) be the number of times the agents are in state s before time t. Then the two following choices of ǫ are compatible with interruptions: p • ∀t ∈ N, ∀s ∈ S, ǫt (s) = c/ m nt (s) • ∀t ∈ N, ǫt = c/log(t) Proof. Lemma B.2 of Singh et al ([19]) ensures that πiǫ is GLIE. The difference for IN T θ (πiǫ ) is that exploration is slower because of the interruptions. Therefore, θ needs to be controlled in order to ensure that infinite exploration is still achieved. We define the random variable Θ by Θi = 1 if agent i actually responds to the interruption and Θi = 0 otherwise. We define ξ in a similar way to represent the event of all agents taking the uniform policy instead of the greedy one. p 1. Let θt (s) = 1 − c′ / m nt (s) with c′ ∈]0, 1]. We have P(a\|s, nt (s)) ≥ P(a, Θ = 0, ξ = √ P∞ 1 m 1 m cc′ 1\|s, nt (s)) ≥ \|A\| ǫt (s)(1 − θt (s))m = \|A\| t=1 P (a\|s, nt (s)) = ∞ nt (s) which satisfies so by the extended Borell-Cantelli lemma action a is chosen infinitely often in state s and thus nt (s) → ∞ and ǫt (s) → 0 2. Let θt = 1 − c′ /log(t), c′ ∈]0, 1]. We define M as the diameter of the MDP, \|A\| is the maximum number of actions available in a state and ∆t(s, s′ ) the time needed to reach s′ from s. In a single agent setting: P[∆t(s, s′ ) < 2M ] ≥ P[∆t(s, s′ ) < 2M \|actions sampled according to πs,s′ for 2M steps] × P[actions sampled according to πs,s′ for 2M steps] where πs,s′ the policy such that the agents takes less than M steps in expectation to reach s′ from s. We have: P[∆t(s, s′ ) < 2M ] = 1 − P[∆t(s, s′ ) ≥ 2M ] and using the Markov ′ )) ≤ 12 (since M is an upper bound on the inequality, P[∆t(s, s′ ) ≥ 2M ] ≤ E(∆t(s,s 2M expectation of the number of steps from state s to state s′ ), since ξ and 1 − θ are decreasing 1 sequences we finally obtain: P[∆t(s, s′ ) < 2M ] ≥ 2\|A\| [P[ξt+2M = 1](1 − θt+2M )]2M . Therefore, if we replace the probabilities of exploration and interruption by the values in the multi-agent setting, the probability to reach state s′ from state s in 2M steps is at least 1 ′ 4mM and the probability of taking a particular action in this state is at 2\|A\| [cc / log(t + M )] P∞ 1 1 ′ 2m ′ m(4M+2) least \|A\| [cc / log(t + M )] . Since t=1 2\|A\| = ∞ then the 2 [cc / log(t + M )] extended Borell Cantelli lemma (Lemma 3 of Singh et al. [19]) guarantees that any action in the state s′ is taken infinitely often. Since this is true for all states and actions the result follows. B Independent learners Recall that agents are now given an interruption signal at each steps that tells them whether an agent has been interrupted in the system. This interruption signal can be modeled by an interruption flag (Θt )t∈N ∈ {0, 1}N that equals 1 if an agent has been interrupted and 0 otherwise. Note that, contrary to I, it is an observation returned by the environment. Therefore, the value of Θt represents whether 11 an agent has actually been interrupted at time t. If function I equals 1 but does not respond to the interruption (with probability 1 − θt ) then Θt = 0. With definition of interruptions we adopted, it is possible to prove Lemma 2. Lemma 2. Let (x, r, a, y, Θ) ∈ E, then P(Θ\|y, r, x, a) = P(Θ\|x, a). Proof. Consider a tuple (x, r, a, y, Θ) ∈ E. We have P(y, r, Θ\|x, a) = P(y, r\|x, a, Θ)P(Θ\|x, a) and P(y, r, Θ\|x, a) = P(Θ\|x, a, y, r)P(y, r\|x, a). Besides, y = T (s, a) and r = r(s, a) and the functions T and r are independent of Θ. Therefore, P(y, r\|x, a, Θ) = P(y, r\|x, a). The tuple (x, r, a, y, Θ) is sampled from an actual trajectory so it reflects a transition and a reward that actually happened so P(y, r\|x, a) > 0. We can simplify by P(y, r\|x, a) and the result follows. Now, we assume that each agents do not learn on observations for which one of them has been interrupted. Let agent i be in a system with Q-values Q and following an interruptible learning policy with probability of interruption θ, where interrupted events are pruned. We denote by Premoved (y, r\|x, ai , Q) the probability to obtain state y and reward r from the environment for this agent when it is in state x, performs its (own) action ai and no other agents are interrupted. These are the marginal probabilities in the sequence P (E). Premoved (y, r\|x, ai , Q) = P P(y, r, Θ = 0\|x, ai , Q) . ′ ′ y ′ ∈S,r ′ ∈R P(y , r , Θ = 0\|x, ai , Q) Similarly, we denote by P0 (y, r\|x, ai , Q) the same probability when θ = 0, which corresponds to the non-interruptible setting. We first go back to the single agent case to illustrate the previous statement. Assume here that interruptions are not restricted to the case of Definition 1 and that they can happen in any way. The consequence is that any observation e ∈ E can be removed to generate P (E) because any transition can be labeled as interrupted. It is for example possible to remove a transition from P (E) by removing all events associated with a given destination state y0 , therefore making it disappear from the Markov game. Let x ∈ S and a ∈ A be the current state of the agent and the action it will choose. Let y0 ∈ S and θ0 ∈ (0, 1] and let us suppose that y0 is the only state in which interruptions happen. Then we have Premoved (y0 \|x, a) < P0 (y0 \|x, a) and Premoved (y\|x, a) > P(y\|x, a) ∀y 6= y0 because we only remove observations with y = y0 . This implies that the MDP perceived by the agents is altered by interruptions because the agent learns that P(T (s, a) = y0 ) = 0. Removing observations for different destination states but with the same state action pairs in different proportions leads to a bias in the equilibrium learned.4 In our case however, Lemma 2 ensures that the previous situation will not happen, which allows us to prove Lemma 1 and then Theorem 4. Lemma 1. Let i ∈ {1, ..., m} be an agent. For any admissible θ used to generate the experiences E and e = (y, r, x, ai , Q) ∈ P (E). Then P(y, r\|x, ai , Q, θ) = P(y, r\|x, ai , Q). Proof. . We denote the Q-values of the agents by Q. X Consider x ∈ S, i ∈ {1, .., m} and Xu ∈ AiX P(y ′ , r′ , Θ = 0\|x, u, Q) = P(y ′ , r′ , a, Θ = 0\|x, ai = u, Q) y ′ ∈S,r ′ ∈R a∈A,ai =u y ′ ∈S,r ′ ∈R = X X ′ P(y , r′ \|x, a, Θ = 0, Q)P(a, Θ = 0\|x, ai = u, Q) a∈A,ai =u y ′ ∈S,r ′ ∈R = X X a∈A,ai =u y ′ ∈S,r ′ ∈R P(y ′ , r′ \|x, a)P(Θ = 0\|x, ai = u, Q)P(a\|x, ai = u, Θ = 0, Q) X = P(Θ = 0\|x, ai = u, Q) a∈A,ai =u = P(Θ = 0\|x, ai = u, Q)[ X P(a\|x, ai = u, Θ = 0, Q)[ X P(y ′ , r′ \|x, a)] y ′ ∈S,r ′ ∈R P(a\|x, ai = u, Θ = 0, Q)] = P(Θ = 0\|x, ai = u, Q) a∈A,ai =u i =u,Q) Therefore, we have Premoved (y, r\|x, ai = u, Q) = P(y,r,Θ=0\|x,a P(Θ=0\|x,ai =u) so for any (x, ai , y, r, Q) ∈ P (E), P(y, r\|x, ai = u, θ, Q) = Premoved (y, r\|x, ai = u, Q) = 4 The example at https://agentfoundations.org/item?id=836 clearly illustrates this problem. 12 P(y, r\|x, ai = u, Θ = 0, Q) = P(y, r\|x, ai = u, θ = 0, Q). In particular, P(y, r\|x, ai = u, θ, Q) does not depend on the value of θ. Theorem 4. Independent learners with processing function PIN T , a neutral update rule and a sequence ǫ compatible with interruptions verify dynamic safe interruptibility. Proof. We prove that PIN T (E) achieves infinite exploration. The result from Theorem 1 still holds since we lower-bounded the probability of taking an action in a specific state by the probability of taking an action in this state when there are no interruptions. We actually used the fact that there is infinite exploration even if we remove all interrupted episodes to show that there is infinite exploration. (i) (1) (m) Now, we prove that P(Qt+1 (xt , at ) = q\|Qt , ..., Qt , xt , at , θt ) is independent of θ. We fix (1) (m) i ∈ {1, ..., m} and (xt , at , rt , yt ) ∈ PIN T (E) where at ∈ Ai . With Q˜m we have t = Qt , ..., Qt the following equality: (i) P(Qt+1 (xt , at ) = q\|Q˜m t , xt , at , θt ) = X ˜m P(F (xt , at , rt , yt , Q˜m t ) = q\|Qt , xt , at , rt , yt , θt ) (r,y) ·P(yt = y, rt = r\|Q˜m t , xt , at , θt ) The independence of F on θ still guarantees that the first term is independent of θ. However, at ∈ Ai so (rt , yt ) are not independent of θt conditionally on (xt , at ) as it was the case for joint action learners because interruptions of other agents can change the joint action. The independence on θ of the second term is given by Lemma 1. 13	2
Noname manuscript No. (will be inserted by the editor) Gaussian Variant of Freivalds’ Algorithm for Efficient and Reliable Matrix Product Verification arXiv:1705.10449v1 [cs.DS] 30 May 2017 Hao Ji · Michael Mascagni · Yaohang Li Received: date / Accepted: date Abstract In this article, we consider the general problem of checking the correctness of matrix multiplication. Given three n × n matrices A, B, and C, the goal is to verify that A × B = C without carrying out the computationally costly operations of matrix multiplication and comparing the product A × B with C, term by term. This is especially important when some or all of these matrices are very large, and when the computing environment is prone to soft errors. Here we extend Freivalds’ algorithm to a Gaussian Variant of Freivalds’ Algorithm (GVFA) by projecting the product A × B as well as C onto a Gaussian random vector and then comparing the resulting vectors. The computational complexity of GVFA is consistent with that of Freivalds’ algorithm, which is O(n2 ). However, unlike Freivalds’ algorithm, whose probability of a false positive is 2−k , where k is the number of iterations. Our theoretical analysis shows that when A × B 6= C, GVFA produces a false positive on set of inputs of measure zero with exact arithmetic. When we introduce round-off error and floating point arithmetic into our analysis, we can show that the larger this error, the higher the probability that GVFA avoids false positives. Hao Ji Department of Computer Science Old Dominion University E-mail: [email protected] Michael Mascagni Departments of Computer Science, Mathematics, and Scientific Computing Florida State University Applied and Computational Mathematics Division National Institute of Standards and Technology E-mail: [email protected] Yaohang Li Department of Computer Science Old Dominion University Tel.: 757-683-7721 Fax: 757-683-4900 E-mail: [email protected] 2 Hao Ji et al. Moreover, by iterating GVFA k times, the probability of a false positive decreases as pk , where p is a very small value depending on the nature of the fault on the result matrix and the arithmetic system’s floating-point precision. Unlike deterministic algorithms, there do not exist any fault patterns that are completely undetectable with GVFA. Thus GVFA can be used to provide efficient fault tolerance in numerical linear algebra, and it can be efficiently implemented on modern computing architectures. In particular, GVFA can be very efficiently implemented on architectures with hardware support for fused multiply-add operations. Keywords Fault-tolerance · Algorithmic Resilience · Gaussian Variant of Freivalds’ Algorithm · Matrix Multiplication · Gaussian Random Vector · Failure Probability Mathematics Subject Classification (2000) 65F99 · 65C05 · 62P99 1 Introduction As the demands on modern linear algebra applications created by the latest development of high-performance computing (HPC) architectures continues to grow, so does the likelihood that they are vulnerable to faults. Faults in computer systems are usually characterized as hard or soft, and in this article we are motivated primarily with the latter. Soft errors, defined by intermittent events that corrupt the data being processed, are among the most worrying, particularly when the computation is carried out in a low-voltage computing environment. For example, the 2, 048-node ASC Q supercomputer at Los Alamos National Laboratory reports an average of 24.0 board-level cache tag parity errors and 27.7 CPU failures per week [34]; the 131, 072-CPU BlueGene/L supercomputer at Lawrence Livermore National Laboratory experiences one soft error in its L1 cache every 4–6 hours [19]; more recently, a field study on Google’s servers reported an average of 5 single bit errors occur in 8 Gigabytes of RAM per hour using the top-end error rate [37]. The reliability of computations on HPC systems can suffer from soft errors that occur in memory, cache, as well as microprocessor logic [38], and thus produce potentially incorrect results in a wide variety of ways. We are specifically interested in examining ways to remedy the consequences of soft errors for certain linear algebra applications. Matrix-matrix multiplication is one of the most fundamental numerical operations in linear algebra. Many important linear algebraic algorithms, including linear solvers, least squares solvers, matrix decompositions, factorizations, subspace projections, and eigenvalue/singular values computations, rely on the casting the algorithm as a series of matrix-matrix multiplications. This is partly because matrix-matrix multiplication is one of the level-3 Basic Linear Algebra Subprograms (BLAS) [10,9,17]. Efficient implementation of the BLAS remains an important area for research, and often computer vendors spend significant resources to provide highly optimized versions of the BLAS GVFA for Efficient and Reliable Matrix Product Verification 3 for their machines. Therefore, if a matrix-matrix multiplication can be carried out free of faults, the linear algebraic algorithms that spend most of their time in matrix-matrix multiplication can themselves be made substantially faulttolerant [21]. Moreover, there is considerable interest in redesigning versions of the BLAS to be more fault-tolerant, and this work will certainly contribute to that goal. In this article, we consider the general problem of checking the correctness of matrix-matrix multiplication, i.e., given three n × n matrices A, B, and C, we want to verify whether A × B = C. In contrast to the best known matrixmatrix multiplication algorithm running in O(n2.3727 ) time [8,39], Freivalds’ algorithm [16] takes advantage of randomness to reduce the time to check a matrix multiplication to O(n2 ). The tradeoff of Freivalds’ algorithm is that the probability of failure detection, a false positive, is 2−k , where k is the number of iterations taken. We extend Freivalds’ algorithm from using binary random vectors to floating-point vectors by projecting the A × B result as well as C using Gaussian random vectors. We will refer to this algorithm as the Gaussian Variant of Freivalds’ Algorithm (GVFA). By taking advantage of a nice property of the multivariate normal distribution, we show that GVFA produces a false positive on a set of random Gaussian vectors and input matrices of measure zero. Taking floating point round-off error into account, by iterating GVFA k times, the probability of false positive decreases exponentially as pk , where p is usually a very small value related to the magnitude of the fault in the result matrix and floating-point precision of the computer architecture. We also present an efficient implementation of GVFA on computing hardware supporting fused multiplication-add operations. The plan of the paper is the following. We first discuss two relevant algorithms from the literature for error detection in matrix-matrix multiplication. These are the Huang-Abraham scheme, discussed in section 2, and Freivalds’ algorithm, the subject of section 3. The former is a deterministic algorithm based on carrying row and column sums along in a clever format to verify correct matrix-matrix multiplication. Freivalds’ algorithm is a random projection of the computed matrix-matrix product to the same random projection of the matrix-matrix product recomputed from the original matrices using only matrix-vector multiplication. The random vector used in Freivalds’ algorithm is composed of 0’s and 1’s. In section 4, we present the GVFA, a variation on Freivalds’ algorithm, where we instead use random Gaussian vectors as the basis of our projections. We analyze the GVFA and prove that with Gaussian vectors, a false positive occurs only on a set of Gaussian vectors of measure zero. Further analysis of false positive probabilities in the GVFA in the presence of floating-point arithmetic with round-off errors is then taken. Finally, in section 5 we provide a discussion of the results and implications for fault-tolerant linear algebraic computations and a method of enhancing the resilience of linear algebraic computations. In addition, in this final section we provide conclusions and suggest directions for future work. 4 Hao Ji et al. 2 The Huang-Abraham Scheme and its Limit in Error Detection/ Correction The Huang-Abraham scheme [24] is an algorithm-based fault tolerance method that simplifies detecting and correcting errors when carrying out matrix-matrix multiplication operations. This is slightly different from the matrix product verification problem. The fundamental idea of the Huang-Abraham scheme is to address the fault detection and correction problem at the algorithmic level by calculating matrix checksums, encoding them as redundant data, and then redesigning the algorithm to operate on these data to produce encoded output that can be checked. Compared to traditional fault tolerant techniques, such as checkpointing [5], the overhead of storing additional checksum data in the Huang-Abraham scheme is small, particularly when the matrices are large. Moreover, no global communication is necessary in the Huang-Abraham scheme [14]. The Huang and Abraham scheme formed the basis of many subsequent detection schemes, and has been extended for use in various HPC architectures [2,3,32,14]. (a) Generation of a column checksum for A and a row checksum for B, and multiplication of the extended matrices to produce the checksum matrix for C (b) Mismatches in the row and column checksums indicate an element fault in the matrix product Fig. 1: The Huang-Abraham scheme for detecting faults in matrix-matrix multiplication GVFA for Efficient and Reliable Matrix Product Verification 5 Fig. 1 illustrates the Huang-Abraham scheme [24] for detecting faults in matrix-matrix multiplication. First of all, column sums for A and row sums for B are generated and are added to an augmented representation of A and B. These are treated as particular checksums in the subsequent multiplication. Then, multiplication of the extended matrices produces the augmented matrix for C (Fig. 1(a)) where the checksums can be readily compared. Mismatches in the row and column checksums indicate an element fault in the matrix product, C (Fig. 1(b)). However, there are certain patterns of faults undetectable by the HuangAbraham scheme. Here is a simple 2 × 2 example to illustrate such an undetectable pattern. Consider the matrices 23 1 −6 56 A= ,B = , and C = . 34 1 6 76 Clearly A × B = C holds in this example. Then we use the Huang-Abraham scheme to calculate the column checksum for A and row checksum for B and we can get   23 1 −6 −5 AF =  3 4  and BF = . 1 6 7 57 Then   5 6 11 AF × BF =  7 6 13  = CF . 12 12 24 However, if there is a fault during the computation of C which causes an exchange of the first and second columns, an erroneous result matrix C ′ = 65 is generated by exchanging the columns of C. Column or row exchange, 67 usually caused by address decoding faults [20], is a commonly observed memory fault pattern [6]. The problem is that the checksum matrix of C ′ becomes  6 5 11 C ′ F =  6 7 13 , where both the row and column checksums match those 12 12 24 of the true product of A × B. Consequently, the Huang-Abraham scheme fails to detect this fault. The Huang-Abraham scheme can be viewed as a linear constraint satisfaction problem (CSP), where the variables are the n2 entries in the product matrix, C, the constraints are the 2n row and column checksums. Also, the 2n×n2 coefficient matrix in the under-determined linear CSP system equation specifies the selection of row or column elements, as shown in Fig. 2. Clearly, a product matrix, C, that does not satisfy the CSP equations indicates errors in C detectable by the Huang-Abraham scheme. The unique, correct product matrix, C, satisfies the CSP equations. Nevertheless, other possible product matrices satisfying the CSP equations are the fault patterns undetectable by 6 Hao Ji et al. the Huang-Abraham scheme. Only when at least n2 constraints with different element selection are incorporated so that the rank of the coefficient matrix in the CSP equation is n2 , can the undetectable fault patterns be eliminated. However, this situation is equivalent to simply checking every element in C. Fig. 2: Under-determined CSP system in the Huang-Abraham Scheme It is important to notice that there are an infinite number of existing fault patterns that satisfy the checksum constraints and thus are undetectable by the Huang-Abraham scheme, even in the above simple 2 × 2 example (the rank of the CSP coefficient matrix is 3). Moreover, as dimension, n, increases, the number of checksum constraints increases only linearly but the number of elements in a matrix has quadratic growth. Therefore, the undetectable patterns in the Huang-Abraham scheme increase quadratically with n. As a result, for multiplications in large matrices, fault detection methods based on the Huang-Abraham scheme can generate false positive results for a large number of circumstances. 3 Freivalds’ Algorithm The fault detection methods based on the Huang-Abraham scheme are deterministic algorithms. As many randomized fault tolerance algorithms [28,29], with the tradeoff of random uncertainty, Freivalds [16] showed that a probabilistic machine can verify the correctness of a matrix product faster than direct recalculation. The procedure of the corresponding method, later named Freivalds’ algorithm, is described in Algorithm 1. Obviously, if A × B = C, Cω = ABω always holds. Freivalds proved that when A × B 6= C, the probability of Cω = ABω is less than or equal to 12 . The running time of the above procedure is O(n2 ) with an implied multiplier of 3, as it is comprised of three matrix-vector multiplications. This is an upper bound as one can perhaps optimize the evaluation of Bω and Cω. By iterating GVFA for Efficient and Reliable Matrix Product Verification 7 Algorithm 1: Freivalds’ Algorithm 1. Randomly sample a vector ω ∈ {0, 1}n with p = 1 2 of 0 or 1. 2. Calculate the projection of C onto ω: Cω = C × ω. 3. Calculate the projection of the product A × B onto ω: ABω = A × (B × ω). the Freivalds’ algorithm k times, the running time becomes O(kn2 ) and the probability of a false positive becomes less than or equal to 2−k , according to the one-sided error. More generalized forms of Freivalds’ algorithm have also been developed, mainly based on using different sampling spaces [7,1,36,31]. Given at most p erroneous entries in the resulted matrix product, Gasieniec, Levcopoulos, and Lingas extended Freivalds’ algorithm to one with correcting √ capability running in O( pn2 log(n) log(p)) time [18]. 4 A Gaussian Variant of Freivalds’ Algorithm (GVFA) 4.1 Extending Freivalds’ Algorithm using Gaussian Vectors Freivalds’ original algorithm, and most of its extensions are based on integer matrices or matrices over a ring and sampling from discrete spaces. Clearly, we can also apply Freivalds’ algorithm to matrices with real or complex entries with the random vector remaining zeros and ones. A simple extension is to project A × B and C onto a vector ωP of form ωP = (1, r, r2 , ..., rn−1 )T , where r is a random real number. A false positive occurs only when r is the root of the corresponding polynomial. However, in practice, rn−1 can easily grow too large or small exceeding floating point representation [25]. Here we also extend Freivalds’ algorithm by using Gaussian random vectors for the projection. We use the fact that the multivariate normal distribution has several nice properties [35], which have been used for detecting statistical errors in distributed Monte Carlo computations [29]. The extended algorithm is described in Algorithm 2. Algorithm 2: Gaussian Variant of Freivalds’ Algorithm 1. Generate a Gaussian random vector, ωG , made up of n independent (but not necessarily identically) distributed normal random variables with finite mean and variance. 2. Calculate the projection of C on ωG : CωG = C × ωG . 3. Calculate the projection of product A × B on ωG : ABωG = A × (B × ωG ). This algorithm, which we call a Gaussian variant of Freivalds algorithm (GVFA), requires three matrix-vector multiplications and only one vector comparison for fault detection. 8 Hao Ji et al. 4.2 Theoretical Justification Similar to Freivalds’ algorithm, in GVFA if A × B = C, CωG = ABωG always holds within a certain floating point round-off threshold. When A×B 6= C, the chance that CωG = ABωG is a false positive event occurs with measure zero in exact arithmetic, as shown in Theorem 1. We first state a result of Lukacs and King [33], shown as Proposition 1, which will be used in the proof of Theorem 1. The main assumption of Proposition 1 is the existence of the nth moment of each random variable, which many distributions, particularly the normal distribution, have. One important exception of the normal is that it is the limiting distribution for properly normalized sums of random variables with two finite moments. This is Lindeberg’s version of the Central Limit Theorem [30]. Proposition 1 Let X1 , X2 , · · · , Xn be n independent (but not necessarily identically) distributed random variables with variances σi2 , and assume that the nth moment of each Xi (i = 1, 2, · · · , n) exists and is finite. The necessary and sufficient conditions for the existence Pn Pn of two statistically independent linear forms Y1 = i=1 ai Xi and Y2 = i=1 bi Xi are (1) Each random variable which has a nonzero coefficient in both forms is normally distributed. Pn 2 (2) i=1 ai bi σi = 0. Theorem 1 If A × B 6= C, the set of Gaussian vectors where CωG = ABωG holds in Algorithm 2 has measure zero. Proof Let the matrix ∆ ∈ Rn×n denote AB − C. Since A × B 6= C, rank(∆) = r > 0, and dim(null(∆)) = n − rank(∆) = n − r < n. Here dim(·) denotes dimension and null(·) denotes the null space, i.e., null(∆) = {x ∈ Rn : ∆×x = 0}. We can now find n − r of orthonormal vectors, v1 , v2 , · · · , vn−r , to form a basis for null(∆), such that null(∆) = span{v1 , v2 , · · · , vn−r }, and r more orthonormal vectors, vn−r+1 , vn−r+2 , · · · , vn , such that Rn = span{v1 , v2 , · · · , vn−r , vn−r+1 , vn−r+2 , · · · , vn }. Any vector, and in particular the Gaussian vector, ωG can be written in this basis as n X δi vi , ωG = i=1 GVFA for Efficient and Reliable Matrix Product Verification 9 where δi are the weights in this particular orthonormal coordinate system. If we denote V = [v1 , v2 , · · · , vn−r , vn−r+1 , vn−r+2 , · · · , vn ] , we have V ωG = [δ1 , δ2 , · · · , δn−r , δn−r+1 , δn−r+2 , · · · , δn ] . CωG = ABωG holds in Algorithm 2 only if A(BωG ) − CωG = (AB − C)ωG = ∆ωG = 0. This means that ωG ∈ null(∆), i.e., δn−r+1 = 0, δn−r+2 = 0, · · · , δn = 0. Due to the fact that ωG is a Gaussian random vector and V is an orthogonal matrix, Proposition 1 tells us that the elements, δi , in the resulting vector V ωG are normally distributed and statistically independent. With a continuous probability distribution, the discrete event where δi = 0 for all i > n − r occurs on a set of measure zero and we will say here that it has probability zero. Hence, GVFA using a Gaussian random projection will have unmatched CωG and ABωG when A × B 6= C on all but a set of measure zero of Gaussian vectors, which we will say is probability one. ⊓ ⊔ This argument in Theorem 1 is rather direct, but we must point out that the arguments are true when the computations are exact. In next subsection, we will analyze GVFA when float-point errors are present. 4.3 Practical Use in Floating-Point Matrix Product Verification In computer implementations of arithmetic with real numbers, one commonly uses floating-point numbers and floating-point arithmetic. Floating-point numbers are represented as finite numbers in the sense that they have a fixed mantissa and exponent size in number of bits. Therefore, there will be a small probability, p, that CωG = ABωG still holds due to unfortunate floating-point operations in a system with a known machine epsilon, ǫ, when A × B 6= C. The value of p depends on the magnitude of the error between A × B and C as well as ǫ, whose upper bound is justified in Theorem 2. Theorem 2 Assume that ωG is a standard Gaussian random vector, whose elements are i.i.d. normal variables with mean 0 and variance 1, i.e., the standard normal. Let ∆ = A × B − C, then the probability, p, that CωG = ABωG holds in Algorithm 2 using a standard Gaussian random vector ωG under floating-point uncertainty of size ǫ is ǫ p ≤ 2Φ − 1, σ e where Φ(·) is the cumulative density function of the standard normal, and σ e is a constant only related to ∆. Proof Since A × B 6= C, ∆ = A × B − C 6= 0. Consider the ith element, gi , of the product vector g = ∆ × ωG , we have gi = (∆ × ωG )i = n X j=1 ∆ij (ωG )j . 10 Hao Ji et al. Given ǫ, only if \|gi \| ≤ ǫ for all i = 1, · · · , n, can CωG = ABωG hold. Since ωG is a standard normal random vector, the gi for all i = 1, · · · , n, are normally distributed as well. This is because they are linear combinations of normals themselves. The key is to compute what the mean and variance is of the gi . The components of the ωG are i.i.d. standard normals. Thus we have that E [(ωG )j ] = 0 and E (ωG )2j = 1, for all j = 1, · · · , n. Also, we have that E [(ωG )i (ωG )j ] = 0 when i 6= j. This allows us to compute the mean:  E(gi ) = E  n X j=1  ∆ij (ωG )j  = n X ∆ij E [(ωG )j ] = 0, j=1 and the second moment about the mean, i.e., the variance: 2  n X ∆ij (ωG )j  E gi2 − E(gi )2 = E gi2 = E  j=1   n n X X ∆2ij . = E ∆2ij × 1 = j=1 j=1 So weP have that the gi ’s are normally distributed with mean zero and variance n ei2 . σ ei2 = j=1 ∆2ij , i.e., gi ∼ N 0, σ Then, the probability that \|gi \| ≤ ǫ can be computed as follows. Since gi ∼ N 0, σ ei2 , we know that σeg2i ∼ N (0, 1), and so we define the new variables i gei = σegi2 and e ǫ = σeǫ2 , and so we have i i p (\|gi \| ≤ ǫ) = p (−ǫ ≤ gi ≤ ǫ) = p (−e ǫ ≤ gei ≤ e ǫ) Z eǫ 1 2 1 √ e− 2 t dt = 2π −e ǫ = Φ(e ǫ) − Φ(−e ǫ). Since the probability density function of a standard normal is an even function, we have that Φ(e ǫ) + Φ(−e ǫ) = 1, and so we can use −Φ(−e ǫ) = Φ(e ǫ) − 1 to get: p(−ǫ ≤ gi ≤ ǫ) = 2Φ(e ǫ) − 1 = 2Φ ǫ σ ei − 1. Now let us consider computing an upper bound on p(\|gi \| ≤ ǫ, i = 1, · · · , n). We have proven that the gi ’s are normal random variables, but they are not necessarily independent. And so for this we use some simple ideas from conditional probability. By example, consider p(\|g1 \| ≤ ǫ and \|g2 \| ≤ ǫ) = p(\|g2 \| ≤ ǫ given \|g1 \| ≤ ǫ)p(\|g1 \| ≤ ǫ) ≤ p(\|g1 \| ≤ ǫ). GVFA for Efficient and Reliable Matrix Product Verification 11 The inequality holds due to the fact that the probabilities are numbers less than one. Now consider our goal of bounding ǫ −1 , p(\|gi \| ≤ ǫ, i = 1, · · · , n) ≤ p(\|g1 \| ≤ ǫ) = 2Φ σ e1 by iterating the conditional probability argument n times. By reordering we could haveqchosen the bound utilizing any of the gi ’s. However, let us define Pn 2 σ e = maxi j=1 ∆ij , i.e., the maximal standard deviation over all the gi ’s, which is only related to the matrix ∆. We can use that value instead to get h ǫ i p = p(\|gi \| ≤ ǫ, i = 1, · · · , n) ≤ 2Φ −1 . σ e ⊓ ⊔ As an interesting corollary, we can get a better bound in the case the at the gi ’s are independent. In that case n n Y Y ǫ 2Φ p(\|gi \| ≤ ǫ) = p(\|gi \| ≤ ǫ, i = 1, · · · , n) = −1 . σ ei i=1 i=1 qP n 2 Let σ e = maxi j=1 ∆ij , i.e., the maximal standard deviation over all the gi ’s, which is only related to the matrix ∆. Hence for all i = 1, · · · , n, we have that ǫ ǫ 2Φ − 1 ≤ 2Φ − 1. σei σ e And so, finally we get that p = p(\|gi \| ≤ ǫ, i = 1, · · · , n) n Y ǫ −1 2Φ = σ ei i=1 in h ǫ −1 ≤ 2Φ e ǫσ ≤ 2Φ − 1. σ e The last inequality is true since the number raised to the nth power is less than one. Note, that independence gives probability of a false positive that is n times smaller than in the general, dependent case. The conclusion of this seems to be that the bound in the dependent case is overly pessimistic, and we suspect that in cases where the matrix ∆ is very sparse, due to a very small number of errors, that we are in the independent gi ’s case or have very little dependence, and these more optimistic bounds reflect what happens, computationally. Theorem 2 reveals two interesting facts about GVFA in term of practical floating-point matrix product verification: 12 Hao Ji et al. (1) The bigger the error caused by the fault, the higher the probability that it can be captured. p is usually very small because the floating point bound, ǫ, is very small. (2) Similar to the original Freivalds’ algorithm, higher confidence can be obtained by iterating the algorithm multiple times. In fact, if we iterate k times using independent Gaussian random vectors, the probability of false positive decreases exponentially as pk . Actually, due to the fact that p is usually very small, one or a very small number of iterations will produce verification with sufficiently high confidence. R eǫ 1 2 One comment that should be made is that if we consider −eǫ √12π e− 2 t dt when e ǫ is small, we can easily approximate this. Since the integrand is at its maximum at zero, and is a very smooth function, analytic actually, this integral is approximately the value of the integrand at zeroqtimes the length of R eǫ 1 2 ǫ π2 . This is justified the integration interval, i.e., −eǫ √12π e− 2 t dt ≤ 2e ǫ √12π = e as e ǫ is a number on the order of the machine epsilon, which is 2−23Pin single n precision or 2−52 in double precision floating point, divided by σ ei2 = j=1 ∆2ij . Compared to deterministic methods, such as the Huang-Abraham scheme, GVFA has the following advantages: (1) Certain fault patterns, as shown in Section 2, are undetectable in deterministic methods such as the Huang-Abraham scheme. Deterministic methods absolutely cannot detect faults with certain patterns, i.e., certain patterns are detected with probability zero. In contrast, there are no fault patterns that are undetectable by GVFA with 100% probability. Moreover, iterating the algorithm multiple times can increase the probability of detecting any fault pattern any value less than one by iteration. (2) From the computational point-of-view, normal random vectors are generated independently of A, B, and C, which avoids the costly computation of checksums. 4.4 Huang-Abraham-like GVFA GVFA can also be implemented in a way similar to that of the Huang-Abraham scheme by providing row and column verification, as shown in Algorithm 3. Algorithm 3: Huang-Abraham-like GVFA 1. Generate two n-dimensional Gaussian random vectors, ωR , a column vector, and ωC , a row vector, where they independent (but not necessarily identically) distributed normal random variables with finite mean and variance. 2. Calculate the projection of C on ωR and ωC : ωR C = ωR × C and CωC = C × ωC . 3. Calculate the projection of the product A × B on ωR and ωC : ωR AB = (ωR × A) × B and ABωC = A × (B × ωC ). GVFA for Efficient and Reliable Matrix Product Verification 13 Similar to the Huang-Abraham scheme, a mismatched element of the row vectors of ωR C and ωR AB as well as that of the column vectors of CωC and ABωC uniquely identify a faulty element in C. By considering floatingpoint errors, the false positive probability of identifying this fault becomes p2 , according to the analysis in Section 4.3. However, the computational cost doubles with six matrix-vector multiplications and two vector comparisons. This is essentially the same work as doing two independent iterations of the GVFA, and obtains the same bound. 4.5 Implementation using Fused Multiply-Add Hardware The Fused Multiply-Add (FMA) machine instruction performs one multiply operation and one add operation with a single rounding step [23]. This was implemented to enable potentially faster performance in calculating the floatingpoint accumulation of products, a := a + b × c. Recall that the GVFA employs three matrix-vector multiplications to project A × B and C onto a normal random vector, which requires a sequence of product accumulations that cost 3n(2n − 1) floating-point operations. Therefore, the performance of the GVFA can be potentially boosted on modern computing architectures that support the FMA. More importantly, due to a single rounding step used in the FMA instruction instead of two roundings within separate instructions, less loss of accuracy occurs when using the FMA instruction in calculating the accumulation of products [4]. This should further reduce the floating-point rounding errors that cause false positives. 5 Discussion and Conclusions In this paper, we extend Freivalds’ algorithm, which we call the Gaussian variant of Freivalds’ algorithm (GVFA), to the real domain by random projection using vectors whose coefficients are i.i.d. normal random variables. If A×B 6= C, the probability that the resulting vectors match is zero using exact arithmetic. Considering the round-off errors in floating-point operations, the probability of fault detection depends on the magnitude of the error caused by the fault as well as the floating point precision. The new GVFA can be iterated k times with the probability of false positives decreasing exponentially in k. In addition to matrix-matrix multiplication, the new algorithm can be applied to verify a wide variety of computations relevant to numerical linear algebra as it provides fault tolerance to the computation that defines level 3 of the BLAS. GVFA can also be used to enforce the trustworthiness of outsourcing matrix computations on untrusted distributed computing infrastructures such as clouds or volunteer peer-to-peer platforms [27,26]. The GVFA can be easily extended to a more general matrix multiplication operation where A is m × p, B is p × n, and C is m × n. The overall computational time then becomes O(mp + np). The algorithm can be further extended 14 Hao Ji et al. to verify the product of N matrices, which requires overall N +1 matrix-vector multiplications. The GVFA can also be applied to verifying a wide variety of matrix decomposition operations such as LU, QR, Cholesky, as well as eigenvalue computations, and singular value decompositions. In this case, faults are not in the product matrix but occur in the decomposed ones instead. Anyway, the GVFA can be directly applied with no modifications necessary. The GVFA is a new tool to detect faults in numerical linear algebra, and since it is based on random Gaussian projection, it is related to the many new randomized algorithms being used directly in numerical linear algebra [22,11, 12,13,15]. The fundamental idea of these randomized algorithms is to apply efficient sampling on the potentially very large matrices to extract their important characteristics so as to fast approximate numerical linear algebra operations. We believe that the GVFA will be a very useful tool in the development of fault-tolerant and otherwise resilient algorithms for solving large numerical linear algebra problems. In fact, it seems that the GVFA’s similarity to other, new, stochastic techniques in numerical linear algebra affords the possibility of creating stochastic linear solvers that are by their very nature resilient and fault-tolerant. This is highly relevant for new machines being developed in HPC to have maximal floating-point operations per second (FLOPS) while existing within restrictive energy budgets. These HPC systems will be operating at voltages lower than most current systems, and so they are expected to be particularly susceptible to soft errors. However, even if one is not anticipating the use of these high-end machines, the trend in processor design is to lower power, and is being driven by the explosion of mobile computing. Thus, the ability to reliably perform complicated numerical linear algebraic computations on systems more apt to experience soft faults is a very general concern. The GVFA will make it much easier to perform such computations with high fidelity in HPC, cloud computing, mobile applications, as well in big-data settings. Acknowledgements We would like to thank Dr. Stephan Olariu for his valuable suggestions on the manuscript. 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arXiv:1704.07513v2 [math.ST] 17 May 2017 BAYES MODEL SELECTION QIYANG HAN Abstract. We offer a general Bayes theoretic framework to tackle the model selection problem under a two-step prior design: the first-step prior serves to assess the model selection uncertainty, and the secondstep prior quantifies the prior belief on the strength of the signals within the model chosen from the first step. We establish non-asymptotic oracle posterior contraction rates under (i) a new Bernstein-inequality condition on the log likelihood ratio of the statistical experiment, (ii) a local entropy condition on the dimensionality of the models, and (iii) a sufficient mass condition on the second-step prior near the best approximating signal for each model. The first-step prior can be designed generically. The resulting posterior mean also satisfies an oracle inequality, thus automatically serving as an adaptive point estimator in a frequentist sense. Model mis-specification is allowed in these oracle rates. The new Bernstein-inequality condition not only eliminates the convention of constructing explicit tests with exponentially small type I and II errors, but also suggests the intrinsic metric to use in a given statistical experiment, both as a loss function and as an entropy measurement. This gives a unified reduction scheme for many experiments considered in [23] and beyond. As an illustration for the scope of our general results in concrete applications, we consider (i) trace regression, (ii) shape-restricted isotonic/convex regression, (iii) high-dimensional partially linear regression and (iv) covariance matrix estimation in the sparse factor model. These new results serve either as theoretical justification of practical prior proposals in the literature, or as an illustration of the generic construction scheme of a (nearly) minimax adaptive estimator for a multi-structured experiment. 1. Introduction 1.1. Overview. Suppose we observe X (n) from a statistical experiment (n) (n) (X(n) , A(n) , Pf ), where f belongs to a statistical model F and {Pf }f ∈F is dominated by a σ-finite measure µ. Instead of using a single ‘big’ model F, a collection of (sub-)models {Fm }m∈I ⊂ F are available to statisticians, and the art of model selection is to determine which one(s) to use. Date: May 18, 2017. 2000 Mathematics Subject Classification. 60F17, 62E17. Key words and phrases. Bayes nonparametrics, model selection, adaptive estimation, model mis-specification, Bernstein inequality. Supported in part by NSF Grant DMS-1566514. 1 2 Q. HAN There are vast literatures on model selection from a frequentist point of view; we only refer the reader to [4, 36, 11, 44] as some representative pointers for various approaches of penalization, aggregation, etc. On the other hand, from a Bayes point of view, although posterior contraction rates have been derived for many different models (see e.g. [21, 43, 23, 16, 14, 15, 42, 45, 46, 28] for some key contributions), understanding towards general Bayes model selection procedures has been limited. [22] focused on designing adaptive Bayes procedures with models primarily indexed by the smoothness level of classical function classes in the context of density estimation. Their conditions are complicated and seem not directly applicable to other settings. [18] designed a prior specific to structured linear problems in the Gaussian regression model, with their main focus on high-dimensional (linear) and network problems. It seems non-trivial for their framework to handle other non-linear models. Despite these limitations, [22, 18] give useful clues. One common feature in these papers is a two-step prior design, where the first-step prior Λn assesses the model selection uncertainty, followed by a second-step prior Πn,m quantifying the prior belief in the strength of the signals within the specific chosen model Fm from the first step. Such a prior design is intrinsic in many proposals for different problems, e.g. [16, 15] for sparse linear regression, [2] for trace regression, [29, 27] for shape restricted regression, [19, 38] for problems related to convariance matrix estimation. This is the starting point of this paper. We give a unified theoretical treatment to this two-step prior design by identifying common structural (n) assumptions on the statistical experiments (X(n) , A(n) , Pf ), the collection of models {Fm } and the priors {Λn } and {Πn,m } such that the posterior distribution both (G1) contracts at an oracle rate with respect to some metric dn : 2 (1.1) inf inf dn (f0 , g) + pen(m) , m∈I g∈Fm where pen(m)1 is related to the ‘dimension’ of Fm , and (G2) concentrates on the model Fm∗ , where m∗ is the ‘best’ model balancing the bias-variance tradeoff in (1.1). The oracle formulation (1.1) follows the convention in the frequentist literature [36, 44], and has several advantages: (i) (minimaxity) if the true signal f0 can be well-approximated by the models {Fm }, the contraction rate in (1.1) is usually (nearly) minimax optimal, (ii) (adaptivity) if f0 lies in certain low-dimensional model Fm , the contraction rate adapts to this unknown information, and (iii) (mis-specification) if the models Fm are mis-specified while d2n (f0 , ∪m∈I Fm ) remains ‘small’, then the contraction rate should still be rescued by this relatively ‘small’ bias. 1pen(m) may depend on n but we suppress this dependence for notational convenience. BAYES MODEL SELECTION 3 As the main abstract result of this paper (cf. Theorem 1), we show that our goals (G1)-(G2) can be accomplished under: (i) (Experiment) a Bernstein-inequality condition on the log likelihood ratio for the statistical experiment with respect to dn ; (ii) (Models) a dimensionality condition of the model Fm measured in terms of local entropy with respect to the metric dn ; (iii) (Priors) exponential weighting for the first-step prior Λn , and sufficient mass of the second-step prior Πn,m near the ‘best’ approximating signal f0,m within the model Fm for the true signal f0 . One important ingredient in studying posterior contraction rates in Bayes nonparametrics literature has been the construction of appropriate tests with exponentially small type I and II errors with respect to certain metric [21, 23]. Such tests date back to the work of Le Cam [31, 32, 33] and Birgé [6, 7, 8], who brought out the special role of the Hellinger metric in which tests can be constructed generically. On the other hand, the testing framework [21, 23] requires the prior to spread sufficient mass near the Kullback-Leibler neighborhood of the true signal. The discrepancy of these two metrics can be rather delicate, particularly for non i.i.d. and complicated models, and it often remains unclear which metric is the natural one to use in these models. Moreover, it is usually a significant theoretical challenge to construct tests in complicated models, cf. [19, 38], to name a few. Our Bernstein-inequality condition (i) closes these gaps by suggesting the usage of an ‘intrinsic metric’ that mimics the behavior of the KullbackLeibler divergence in a given statistical experiment, in which a ‘good’ test can be constructed generically (cf. Lemma 1). Bernstein inequality is a fundamental tool in probability theory, and hence can be easily verified in many statistical experiments including various experiments considered in [23] and beyond: Gaussian/binary/poisson regression, density estimation, Gaussian autoregression, Gaussian time series and covariance matrix estimation problems. We identify the intrinsic metrics to use in these experiments. Furthermore, the Bernstein-inequality condition entails sharp exponential contraction of the posterior distribution near the ‘true’ signal, complementing a recent result of [28]. Results of this type typically do not follow directly from general principles in [21, 23], and have mainly been derived on a case-by-case basis, cf. [16, 19, 18]. As such, we provide a refinement of the seminal testing framework in [21, 23] so that the investigation of sharp posterior contraction rates in the intrinsic metric of an experiment essentially reduces to the study of prior design. Conditions (ii) and (iii) are familiar in Bayes nonparametrics literature. In particular, the first-step prior can be designed generically (cf. Proposition 1). Sufficient mass of the second-step prior Πn,m is a minimal condition in the sense that using Πn,m alone should lead to a (nearly) optimal posterior contraction rate on the model Fm . 4 Q. HAN These conditions, albeit minimal, imply more than an optimal adaptive Bayes procedure in the sense of (G1)-(G2). In fact, we show that the posterior mean automatically serves as an adaptive point estimator in a frequentist sense. These results reveal, in a sense, that the task of constructing adaptive procedures with respect to the intrinsic metric in a given statistical experiment, in both frequentist and Bayes contexts, is not really harder than that of designing an optimal non-adaptive prior for each of the models. A general theory would be less interesting without being able to address problems of different types. As an illustration of our general framework in concrete applications, we justify the prior proposals in (i) [2, 34] for the trace regression problem, and in (ii) [29, 27] for the shape-restricted regression problems. Despite many theoretical results for Bayes high-dimensional models (cf. [16, 15, 19, 18, 38, 3]), it seems that the important low-rank trace regression problem has not yet been successfully addressed. Our result here fills in this gap. Furthermore, to the best knowledge of the author, the theoretical results concerning shape-restricted regression problems provide the first systematic approach that bridges the gap between Bayesian nonparametrics and shape-restricted nonparametric function estimation literature in the context of adaptive estimation2. We also consider adaptive Bayes procedures for the high-dimensional partially linear regression model and the covariance matrix estimation problem in the sparse factor model. These new results serve as an illustration of the generic construction scheme of a (nearly) minimax adaptive estimator in a complicated experiment with multiple structures. Some of these results improve the best known result in the literature. During the preparation of this paper, we become aware of a very recent paper [48] who independently considered the Bayes model selection problem. Both our approach and [48] shed light on the general Bayes model selection problem, while differing in several important aspects (cf. Remark 2). Moreover, our work here applies to a wide range of applications that are not covered by [48]. 1.2. Notation. Cx denotes a generic constant that depends only on x, whose numeric value may change from line to line. a .x b and a &x b mean a ≤ Cx b and a ≥ Cx b respectively, and a ≍x b means a .x b and (n) a &x b. For a, b ∈ R, a ∨ b := max{a, b} and a ∧ b := min{a, b}. Pf T denotes the expectation of a random variable T = T (X (n) ) under the exper(n) iment (X(n) , A(n) , Pf ). 1.3. Organization. Section 2 is devoted to the general model selection theory. We work out a wide range of experiments that fit into our general theory 2Almost completed at the same time, [35] considered a Bayes approach for univariate log-concave density estimation, where they derived contraction rates without addressing the adaptation issue. BAYES MODEL SELECTION 5 in Section 3. Section 4 discusses various concrete applications as mentioned above. Most detailed proofs are deferred to Sections 5-6 and the Appendix. 2. General theory In the two-step prior design framework, we first put a prior Λn on the model index I, followed by a prior Πn,m on the model Fm chosen from the first P step. The overall prior is a probability measure on F given by Πn ≡ m∈I λn,m Πn,m . The posterior distribution is then a random measure on F: for a measurable subset B ⊂ F, R (n) (n) pf (X ) dΠn (f ) (2.1) Πn (B\|X (n) ) = RB (n) pf (X (n) ) dΠn (f ) (n) (n) where pf (·) denotes the probability density function of Pf to the dominating measure µ. with respect 2.1. Assumptions. To state our assumption on the experiment, let vλ2 , 0 ≤ \|λ\| < 1/c 2(1 − c\|λ\|) denote the ‘Bernstein’ function. This function plays a pivital role in proving sub-gamma behavior of a given (complicated) random variable, cf. [9]. Here v and c are the L2 size and L∞ size of the random variable controlling respectively the degree of its sub-Gaussian and sub-gamma behavior. (2.2) ψv,c (λ) = Assumption A (Experiment: Bernstein-inequality condition). There exist some absolute c1 > 0 and κ = (κg , κΓ ) ∈ (0, ∞) × [0, ∞) such that for all n ∈ N, and f0 , f1 ∈ F,    (n) (n) p p (n) f0   (n) f0 ≤ c1 exp ψκg nd2n (f0 ,f1 ),κΓ (λ) − Pf0 log (n) Pf0 exp λ log (n) p f1 p f1 holds for all \|λ\| < 1/κΓ . Here the metric dn : F × F → R≥0 satisfies (n) (2.3) c2 · d2n (f0 , f1 ) p f0 1 (n) ≤ Pf0 log (n) ≤ c3 · d2n (f0 , f1 ) n p f1 for some absolute constants c2 , c3 > 0. In Assumption A, we require the log likelihood ratio to satisfy a Bernstein inequality. In particular, the log likelihood ratio has local Gaussian behavior. Conversely, if the log likelihood ratio behaves locally like Gaussian, then we can pick some κΓ > 0 so that the Bernstein inequality holds. Lemma 1. Let Assumption A hold. Fix f0 , f1 ∈ F, there exists some test φn such that (n) (n) sup Pf0 φn + Pf (1 − φn ) ≤ c6 exp(−c7 nd2n (f0 , f1 )) f ∈F :d2n (f,f1 )≤c5 d2n (f0 ,f1 ) 6 Q. HAN where c5 ≤ 1/4, c6 ∈ [2, ∞), c7 ∈ (0, 1) only depend on c1 , c2 , c3 , κ. This lemma suggests that under a Bernstein-inequality condition on the log likelihood ratio, tests exist automatically under the intrinsic metric dn that mimics the behavior of the Kullback-Leibler divergence in the sense of (2.3). Several examples will be worked out in Section 3 to illustrate the choice of an intrinsic metric dn , including the discrete ℓ2 loss for regression models, a weighted L2 metric for the Gaussian autoregression model, the Hellinger metric for density estimation, the Frobenius norm for covariance matrix estimation problem. Next we state the assumption on the complexity of the models {Fm }m∈I . Let I = Nq be a q-dimensional lattice with the natural order (I, ≤)3. Here the dimension q is understood as the number of different structures in the models {Fm }m∈I . In the sequel we will not explicitly mention q unless otherwise specified. We require the models to be nested in the sense that Fm ⊂ Fm′ if m ≤ m′ . Let f0,m denote the ‘best’ approximation of f0 within the model Fm in the sense that f0,m ∈ arg inf g∈Fm dn (f0 , g)4. Assumption B (Models: Local entropy condition). For each m ∈ I, 2 sup log N c5 ε, {f ∈ Fm : dn (f, g) ≤ 2ε}, dn ≤ (c7 /2)nδn,m (2.4) 1 + ε>δ n,m holds for all g ∈ {f0,m′ }m′ ≤m . Furthermore there exist absolute constants c ≥ 1 and γ ≥ 1 such that for any m ∈ I, α ≥ c7 /2 and any h ≥ 1, X −αnδ2 2 2 2 2 n,m′ ≤ 2e−αnhδn,m /c , ≤ hγ δn,m . c−2 δn,hm e (2.5) m′ ≥hm Note that if we choose all models Fm = F, then (2.4) reduces to the local entropy condition in [21, 23]. When Fm is finite-dimensional, typically we can check (2.4) for all g ∈ Fm . Now we comment on (2.5). The left side of 2 , while the right (2.5) essentially requires super linearity of the map m 7→ δn,m side of (2.5) controls the degree of this super linearity. As a leading example, 2 (2.5) will be trivially satisfied with c = 1 and γ = 1 when nδn,m = cm for some absolute constant c > 2/c7 . Finally we state assumptions on the priors. Assumption C (Priors: Mass condition). For all m ∈ I, (P1) (First-step prior) There exists some h ≥ 1 such that X 1 2 2 ), λn,k ≤ 2 exp(−nδn,m ). λn,m ≥ exp(−2nδn,m (2.6) 2 k>hm,k∈I (P2) (Second-step prior) 2 2 ). /c3 ≥ exp(−2nδn,m Πn,m f ∈ Fm : d2n (f, f0,m ) ≤ δn,m (2.7) 3For any a, b ∈ I, a ≤ b iff a ≤ b for all 1 ≤ i ≤ q. Similar definition applies to i i <, ≥, >. 4We assume that f 0,m is well-defined without loss of generality. BAYES MODEL SELECTION 7 Condition (P1) can be verified by using the following generic prior Λn : (2.8) 2 λn,m ∝ exp(−2nδn,m ). Proposition 1. Suppose the first condition of (2.5) holds. Then (P1) in Assumption C holds for the prior (2.8) with h ≥ 2c2 . (2.8) will be the model selection (first-step) prior on the model index I in all examples in Section 4. Condition (P2) is reminiscent of the classical prior mass condition consid2 is understood as the ‘posterior contraction rate’ ered in [21, 23] where δn,m for the model Fm . Hence (P2) can also be viewed as a solvability condition imposed on each model. Note that (2.7) only requires a sufficient prior mass on a Kullback-Leibler ball near f0,m , where [21, 23] uses more complicated metric balls induced by higher moments of the Kullback-Leibler divergence. 2.2. Main results. The following is the main abstract result of this paper. Theorem 1. Suppose Assumptions A-C hold for some M ⊂ I with \|M\| = 2 . ∞, and h ≥ C0 c2 . Let ε2n,m ≡ inf g∈Fm d2n (f0 , g) ∨ δn,m (1) For any m ∈ M, (n) (n) 2 X Pf0 Πn f ∈ F : d2n (f, f0 ) > C1 inf d2n (f0 , g) + δn,m g∈Fm (2.9) ≤ C2 exp − nε2n,m /C2 . 2 (2) For any m ∈ M such that δn,m ≥ inf g∈Fm d2n (f0 , g), (n) (2.10) / FC3 m \|X (n) ≤ C2 exp − nε2n,m /C2 . Pf0 Πn f ∈ (3) Let fˆn ≡ Πn (f \|X (n) ) be the posterior mean. Then (n) 2 ˆ 2 2 inf dn (f0 , g) + δn,m . (2.11) Pf0 dn (fn , f0 ) ≤ C4 inf m∈M g∈Fm Here the constant C0 depends on {ci }3i=1 , κ and {Ci }4i=1 depend on the {ci }3i=1 , κ, c, h and γ. The main message of Theorem 1 is that, the task of constructing Bayes procedures adaptive to a collection of models in the intrinsic metric of a given statistical experiment, can be essentially reduced to that of designing a nonadaptive prior for each model. Furthermore, the resulting posterior mean serves as an automatic adaptive point estimator in a frequentist sense. In particular, if the non-adaptive priors we use on each model lead to (nearly) optimal posterior contraction rates on these models, adaptation happens automatically by designing a ‘correct’ model selection prior, e.g. (2.8). Besides being rate-adaptive to the collection of models, (2.10) shows that the posterior distribution concentrates on the model Fm that balances the bias and variance tradeoff in the oracle rates (2.9) and (2.11). Results of this type have been derived primarily in the Gaussian regression model (cf. 8 Q. HAN [16, 15, 18]) and in density estimation [22]; here our result shows that this is a general phenomenon for the two-step prior design. Note that f0 is arbitrary and hence our oracle inequalities (2.9) and (2.11) account for model mis-specification errors. Previous work allowing model mis-specification includes [18] who mainly focuses on structured linear models in the Gaussian regression setting, and [30] who pursued generality at the cost of harder-to-check conditions. The condition \|M\| = ∞ is assumed purely for technical convenience. If we have finitely many models {F1 , . . . , Fm′ } at hand, then we can define Fm ≡ Fm′ for m ≥ m′ so that this condition is satisfied. Remark 1. We make some technical remarks. (1) The probability estimate in (2.9) is sharp (up to constants) in view of the lower bound result Theorem 2.1 in [28], thus closing a gap that has not been attainable in a general setting by using [21, 23] directly. Beyond of theoretical interest in its own right, the sharp estimate helps us to derive an oracle inequality for the posterior mean as an important frequentist summary of the posterior distribution. Such sharp estimates have been derived separately in different models, e.g. the sparse normal mean model [16], the sparse PCA model [19], and the structured linear model [18], to name a few. (2) Assumption A implies, among other things, the existence of a good test (cf. Lemma 1). In this sense our approach here falls into the general testing approach adopted in [21, 23]. The testing approach has difficulties in handling non-intrinsic metrics, cf. [28]. Some alternative approaches for dealing with non-intrinsic metrics can be found in [14, 28, 49]. (3) The constants {Ci }4i=1 in Theorem 1 depend at most polynomially with respect to the constants involved in Assumption A. This allows some flexibility in the choice of the constants therein. In fact, Bernstein inequality in some dependent cases comes with logarithmic factors in n, cf. [1, 37]. Remark 2. We compare our results with Theorems 4 and 5 of [48]. Both their results and our Theorem 1 shed light on the general problem of Bayes model selection, while differing in several important aspects: (1) Our Theorem 1 hinges on the new Bernstein-inequality condition, while the results of [48] are based on the classical mechanism of [23] which requires the construction of tests. Some merits of our approach will be clear from Section 3 and (2) below along with Remark 1. (2) The probability estimate in [48] for the posterior distribution outside a ball of radius at the targeted contraction rate is asymptotic in nature, while our Theorem 1 provides non-asymptotic sharp estimates. (3) Theorem 4 of [48] targets at exact model selection consistency, under a set of additional ‘separation’ assumptions. Our Theorem 1 (2) requires no extra assumptions, and shows the concentration behavior BAYES MODEL SELECTION 9 of the posterior distribution on the ‘best’ model that balances the bias-variance tradeoff. This is significant in non-parametric problems: the true signal typically need not belong to any specific model. (4) Theorem 5 of [48] contains a term involving the cardinality of the models and hence the models need be apriori finitely many for their bound to be finite. It remains open to see if this can be removed. 2.3. Proof sketch. Here we sketch the main steps in the proof of our main abstract result Theorem 1. The details will be deferred to Section 5. The proof can be roughly divided into two main steps. (Step 1) We first solve a localized problem on the model Fm by ‘projecting’ the underlying probability measure from Pf0 to Pf0,m . In particular, we establish exponential deviation inequality for the posterior contraction rate via the existence of tests guaranteed by Lemma 1: (n) 2 (n) 2 (2.12) Pf0,m Πn f ∈ F : d2n (f, f0,m )) > M δn, . exp − c1 nδn, m̃ \|X m̃ , 2 2 where m̃ is the smallest index ≥ m such that δn, m̃ & ℓn (f0 , f0,m ). This index may deviate from m substantially for small indices. (Step 2) We argue that, the cost of the projection in Step 1 is essentially 2 ) factor in the probability bound (2.12), cf. a multiplicative O exp(c2 nδn, m̃ Lemma 8, which is made possible by the Bernstein-inequality Assumption A. Then by requiring c1 ≫ c2 we obtain the conclusion by the definition of 2 2 2 2 δn, m̃ and the fact that δn,m̃ ≈ ℓn (f0 , f0,m ) ∨ δn,m . The existence of tests (Lemma 1) is used in step 1. Step 2 is inspired by the work of [17, 5] in the context of frequentist least squares estimator over a polyhedral cone in the Gaussian regression setting, where the localized problem therein is estimation of signals on a low-dimensional face (where ‘risk adaptation’ happens). In the Bayesian context, [16, 15] used a change of measure argument in the Gaussian regression setting for a different purpose. Our proof strategy can be viewed as an extension of these ideas beyond the (simple) Gaussian regression model. 3. Statistical experiments In this section we work out a couple of specific statistical experiments that satisfy Bernstein-inequality Assumption A to illustrate the scope of the general theory in Section 2. Some of the examples come from [23]; we identify the ‘intrinsic’ metric to use in these examples. Since Bernstein inequality is a fundamental probabilistic tool, and has been derived in a wide range of complicated (dependent) settings ([1, 37]), we expect many more experiments to be covered beyond the ones we present here. 3.1. Regression models. Suppose we want to estimate θ = (θ1 , . . . , θn ) in a given model Θn ⊂ Rn in the following regression models: for 1 ≤ i ≤ n, (1) (Gaussian) Xi = θi + εi where εi ’s are i.i.d. N (0, 1) and Θn ⊂ Rn ; (2) (Binary) Xi ∼i.i.d. Bern(θi ) where Θn ⊂ [η, 1 − η]n for some η > 0; 10 Q. HAN (3) (Poisson) Xi ∼i.i.d. Poisson(θi ) where Θn ⊂ [1/M, M ]n for some M ≥ 1; We will use the following metric: for any θ0 , θ1 ∈ Θn , n 2 1X 2 ℓn (θ0 , θ1 ) ≡ θ0,1 − θ1,i . n i=1 Lemma 2. Assumption A holds for ℓn with (1) (Gaussian) c1 = c2 = c3 = κg = 1 and κΓ = 0; (2) (Binary) κΓ = 0 and the constants {ci }3i=1 , κg depend on η only; (3) (Poisson) constants {ci }3i=1 , κ depending on M only. Theorem 2. For Gaussian/binary/poisson regression models, let dn ≡ ℓn . If Assumptions B-C hold, then (2.9)-(2.11) hold. Using similar techniques we can derive analogous results for Gaussian regression with random design and white noise model. We omit the details. 3.2. Density estimation. Suppose X1 , . . . , Xn ’s are i.i.d. samples from a density f ∈ F with respect to a measure ν on the sample space (X, A). g(x) We consider the following form of F: f (x) = R eeg dν for some g ∈ G for X all x ∈ X. A natural metric to use for density estimation is the Hellinger metric: for any f0 , f1 ∈ F, Z p p 1 h2 (f0 , f1 ) ≡ ( f0 − f1 )2 dν. 2 X Lemma 3. Suppose that G is uniformly bounded. Then Assumption A is satisfied for h with constants {ci }3i=1 , κ depending on G only. Theorem 3. For density estimation, let dn ≡ h. If G is a class of uniformly bounded functions and Assumptions B-C hold, then (2.9)-(2.11) hold. 3.3. Gaussian autoregression. Suppose X0 , X1 , . . . , Xn is generated from Xi = f (Xi−1 ) + εi for 1 ≤ i ≤ n, where f belongs to a function class F with a uniform bound M , and εi ’s are i.i.d. N (0, 1). Then Xn is a Markov chain with transition density pf (y\|x) = φ(y − f (x)) where φ is the normal density. By the arguments on page 209 of [23], this chain has a unique stationary distribution with density qf with respect to the Lebesgue measure λ on R. We assume that X0 is generated from this stationary distribution under the true f . Consider the following metric: for any f0 , f1 ∈ F, Z d2r,M (f0 , f1 ) ≡ (f0 − f1 )2 rM dλ where rM (x) ≡ 1 2 (φ(x − M ) + φ(x + M )). Lemma 4. Suppose that F is uniformly bounded by M . Then Assumption A is satisfied for dr,M with constants {ci }3i=1 , κ depending on M only. Theorem 4. For Gaussian autoregression model, if F is uniformly bounded by M , let dn ≡ dr,M . If Assumptions B-C hold, then (2.9)-(2.11) hold. BAYES MODEL SELECTION 11 Compared with results obtained in [23] (cf. Section 7.4), we identify the intrinsic metric dr,M (a weighted L2 norm) for the Gaussian autoregression model, while [23] uses a weighted Ls (s > 2) norm to check the local entropy condition, and an average Hellinger metric as the loss function. 3.4. Gaussian time series. Suppose X1 , X2 , . . . is a stationary Gaussian process with spectral density f ∈ F defined on [−π, π]. Then the covariance R π √−1λ(k−l) (n) f (λ) dλ. matrix of X = (X1 , . . . , Xn ) is given by (Tn (f ))kl ≡ −π e We consider a special form of F: f ≡ fg ≡ exp(g) for some g ∈ G. We will use the following metric: for any g0 , g1 ∈ G, 1 Dn2 (g0 , g1 ) ≡ kTn (fg0 ) − Tn (fg1 )k2F , n where k·kF denotes the matrix Frobenius norm. Lemma 5. Suppose that G is uniformly bounded. Then Assumption A is satisfied for Dn with constants {ci }3i=1 , κ depending on G only. Theorem 5. For the Gaussian time series model, if G is uniformly bounded, let dn ≡ Dn . If Assumptions B-C hold, then (2.9)-(2.11) hold. The metric Dn can always be bounded from above by the usual L2 metric, and can be related to the L2 metric from below (cf. Lemma B.3 of [20]). Our result then shows that the metric to use in the entropy condition can be weakened to the usual L2 norm rather than the much stronger L∞ norm as in page 202 of [23]. 3.5. Covariance matrix estimation. Suppose X1 , . . . , Xn ∈ Rp are i.i.d. observations from Np (0, Σ) where Σ ∈ Sp (L), the set of p × p covariance matrices whose minimal and maximal eigenvalues are bounded by L−1 and L, respectively. We will use the Frobenius norm: for any Σ0 , Σ1 ∈ Sp (L), DF2 (Σ0 , Σ1 ) ≡ kΣ0 − Σ1 k2F . Lemma 6. Under the above setting, Assumption A holds for the metric DF with constants {ci }3i=1 , κ depending on L only. Theorem 6. For covariance matrix estimation in Sp (L) for some L < ∞, let dn ≡ DF . If Assumptions B-C hold, then (2.9)-(2.11) hold. 4. Applications In this section, we consider concrete applications. As we have seen in previous sections, construction of adaptive Bayes procedures in the intrinsic metric of an experiment essentially reduces to the design of non-adaptive priors, and hence we only consider the simplest setup for a particular structure. For instance, once we understand how to analyze the convex Gaussian regression problem, we can similarly consider convex binary/Poisson regression, convex density estimation, Gaussian autoregression with convex functions, Gaussian time series with convex spectral density problems in their 12 Q. HAN respective intrinsic metrics. Hence our emphasis in the examples will be focused on the analysis of different model structures. Models that can be handled using similar techniques will not be presented in detail (e.g. Remark 4). We will only explicitly state the corresponding oracle inequalities in the form of (2.9) for each example to be considered below. The corresponding results for (2.10) and (2.11) are omitted. 4.1. Trace regression. Consider fitting the Gaussian regression model yi = f0 (xi ) + εi (1 ≤ i ≤ n) by F ≡ {fA : A ∈ Rm1 ×m2 } where fA (x) = tr(x⊤ A) for all x ∈ X ≡ Rm1 ×m2 . Let m ≡ m1 ∧ m2 and m̄ ≡ m1 ∨ m2 . The index set is I = I1 ∪ I2 ≡ {1, . . . , rmax } ∪ {rmax + 1, . . .} where rmax ≤ m. For r ∈ I1 , let Fr ≡ {fA : A ∈ Rm1 ×m2 , rank(A) ≤ r}, and for r ∈ I2 , Fr ≡ Frmax 5. Although various Bayesian methods have been proposed in the literature (cf. see [2] for a state-to-art summary), theoretical understanding has been limited. [34] derived an oracle inequality for an exponentially aggragated estimator for the matrix completion problem. Their result is purely frequentist. Below we consider a two step prior similar to [34, 2], and derive the corresponding posterior contraction rates. For a matrix B = (bij ) ∈ Rm1 ×m2 let kBkp denote its Schatten p-norm6. p = 1 and 2 correspond to the nuclear norm and the Frobenius norm respectively. To introduce the notion of RIP, let X : Rm1 ×m2 → Rn be the n linear map defined via A 7→ (tr(x⊤ i A))i=1 . Definition 1. The linear map X : Rm1 ×m2 → Rn is said to satisfy RIP (r, νr ) for some 1 ≤ r ≤ rmax and some νr = (ν r , ν̄r ) with 0 < ν r ≤ ν̄r < ∞ iff √ (A)k2 ≤ ν̄r holds for all matrices A ∈ Rm1 ×m2 such that rank(A) ≤ r. ν r ≤ kX nkAk2 For r > rmax , X satisfies RIP(r, νr ) iff X satisfies RIP(rmax , νr ). Furthermore, X : Rm1 ×m2 → Rn is said to satisfy uniform RIP (ν; I) on an index set I iff X satisfies RIP(2r, ν) for all r ∈ I. RIP(r, νr ) is a variant of the RIP condition introduced in [13, 12, 40] with scaling factors ν̄r = 1/(1 − δr ) and ν r = 1/(1 + δr ) for some 0 < δr < 1. Example 1 (Matrix completion). Suppose that xi ∈ Rm1 ×m2 takes value 1 at one position and 0 otherwise. Further assume that A ≤ \|A0 \|ij ≤ Ā for all 1 ≤ i ≤ m1 and 1 ≤ j ≤ m2 7. Let Ω ≡ ΩX denote the indices for which {xi }’s take value 1. Then kX (A)k2 = kA1Ω k2 . Easy calculations show that 5This trick of defining models for high-dimensional experiments will also used in other applications in later subsections, but we will not explicitly state it again. 1/p 6That is, kBk ≡ Pm σ (B)p , where {σj (B)} are the singular values of B. p j=1 j 7This assumption is usually satisfied in applications: in fact in the Netflix problem (which is the main motivating example for matrix completion), A0 is the rating matrix with rows indexing the users and columns indexing movies, and we can simply take A = 1 (one star) and Ā = 5 (five stars). BAYES MODEL SELECTION we can take ν = (ν̄, ν) defined by ν̄ = X is uniform RIP(ν; I). √ √m1 m2 ∧n nm1 m2 Ā ·A ,ν = 13 √ √m1 m2 ∧n nm1 m2 ·A so that Ā Example 2 (Gaussian measurement ensembles). Suppose xi ’s are i.i.d. random matrices whose entries are i.i.d. standard normal. Theorem 2.3 of [12] entails that X is uniform RIP(ν; I) with ν̄ = 1 + δ, ν = 1 − δ for some δ ∈ (0, 1), with probability at least 1 − C exp(−cn)8, provided n & m̄rmax . Consider a prior Λn on I of form λn,r ∝ exp − ctr (m1 + m2 )r log m̄ . (4.1) Given the chosen index r ∈ I, by a prior on Pra prior⊤ on Fr is induced m 1 all m1 × m2 matrices of form i=1 ui vi where ui ∈ R and vi ∈ Rm2 . Here we use a product prior distribution G with Lebesgue density (g1 ⊗ g2 )⊗r on (Rm1 × Rm2 )r . For simplicity we use gi ≡ g⊗mi for i = 1, 2 where g is symmetric about 0 and non-increasing on (0, ∞)9. Let A0,r ∈ tr ≡ g σ arg minB:rank(B)≤r ℓ2n (fB , f0 ), and τr,g max (A0,r ) + 1 where σmax denotes the largest singular value. Theorem 7. Fix 0 < η < 1/2 and rmax ≤ n. Suppose that there exists some M ⊂ I such that the linear map X : Rm1 ×m2 → Rn satisfies uniform RIP(ν; M), and that for all r ∈ M, we have 2η tr (4.2) τr,g ≥ e− log m̄/2η , m̄ ≥ 3 ∨ 3ν̄(1 ∨ σmax (A0,r ))n . Then there exists some ctr in (4.1) depending on ν̄/ν, η such that for any r ∈ M, (4.3) (n) Pf0 Πn A ∈ Rm1 ×m2 : ℓ2n (fA , f0 ) Here ε2n,r > C1tr ℓ2n (f0 , fB ) (m1 + m2 )r log(m̄) + n (n) inf Y ≤ C2tr exp −nε2n,r /C2tr . o n ≡ max inf B:rank(B)≤r ℓ2n (f0 , fB ), (m1 +mn2 )r log m̄ , and the constants B:rank(B)≤r Citr (i = 1, 2) depend on ν̄/ν, η. By Theorem 5 of [41], the rate in (4.3) is minimax optimal up to a logarithmic factor. To the best knowledge of the author, the theorem above is the first result in the literature that addresses the posterior contraction rate in the context of trace regression in a fully Bayesian setup. (4.2) may be verified in a case-by-case manner; or generically we can take M = {r0 , r0 + 1, . . .} if the model is well specified, at the cost of sacrificing 8Note here we used the union bound to get a probability estimate r max exp(−cn) . exp(−c′ n) for some c′ < c under the assumption that n & m̄rmax . 9We will always use such g in the prior design in the examples in this section. 14 Q. HAN the form of oracle inequalities (but still get nearly optimal posterior contraction rates) in (4.3). In particular, the first condition of (4.2) prevents the largest eigenvalue of A0,r from growing too fast. This is in similar spirit with Theorem 2.8 of [16], showing that the magnitude of the signals cannot be too large for light-tailed priors to work in the sparse normal mean model. The second condition of (4.2) is typically a mild technical condition: we only need to choose η > 0 small enough. 4.2. Isotonic regression. Consider fitting the Gaussian regression model Yi = f0 (xi )+εi by F ≡ {f : [0, 1] → R : f is non-decreasing}. For simplicity the design points are assumed to be xi = i/(n + 1) for all 1 ≤ i ≤ n. Let Fm ≡ f ∈ F, f is piecewise constant with at most m constant pieces . Consider the following prior Λn on I = N: (4.4) λn,m ∝ exp − ciso m log(en) . Let gm ≡ g ⊗m where g is symmetric and non-increasing on (0, ∞). Then ḡm (µ) ≡ m!gm 1{µ1 ≤...≤µm } (µ) is a valid density on {µ1 ≤ . . . ≤ µm }. Given a chosen model Fm by the prior Λn , we randomly pick a set of change points {xi(k) }m k=1 (i(1) < . . . < i(m)) and put a prior ḡm on {f (xi(k) )}’s. [29] proposed a similar prior with Λn being uniform since they assumed the maximum number of change points is known apriori. Below we derive a theoretical result without assuming the knowledge of this. Let f0,m ∈ 10 iso = g kf . arg ming∈Fm ℓ2n (f0 , g), and τm,g 0,m k∞ + 1 Theorem 8. Fix 0 < η < 1/2. Suppose that (4.5) iso τm,g ≥ e− log(en)/2η . Then there exists some ciso in (4.4) depending on η such that (4.6) m log(en) (n) 2 iso 2 (n) Pf0 Πn f ∈ F : ℓn (f, f0 ) > C1 inf ℓn (f0 , g) + Y g∈Fm n 2 iso ≤ C2iso exp −n(εiso . n,m ) /C2 n o 2 ≡ max inf 2 (f , g), m log(en) , and the constants C iso (i = ) ℓ Here (εiso 0 g∈F m n n,m i n 1, 2) depend on η. (4.6) implies that if f0 is piecewise constant, the posterior distribution contracts at nearly a parametric rate. (4.5) can be checked by the following. Lemma 7. If f0 is square integrable, and the prior density g is heavy-tailed in the sense that there exists some α > 0 such that lim inf \|x\|→∞ xα g(x) > 0. Then for any η ∈ (0, 1/α), (4.5) holds uniformly in all m ∈ N for n large enough depending on α and kf0 kL2 ([0,1]) . 10The value of f 0,m outside of [1/(n + 1), n/(n + 1)] can be defined in a canonical way by extending f0,m (1/(n + 1)) and f0,m (n/(n + 1)) towards the endpoints. BAYES MODEL SELECTION 15 4.3. Convex regression. Consider fitting the Gaussian regression model Yi = f0 (xi ) + εi by F, the class of convex functions on X = [0, 1]d . Let Fm ≡ f (x) = max1≤i≤m (ai · x + bi ) : ai ∈ Rd , bi ∈ R denote the class of piecewise affine convex functions with at most m pieces. We will focus on the multivariate case since the univariate case can be easily derived using the techniques exploited in isotonic regression. A prior on each model Fm can be induced by a prior on the slopes the intercepts Nm and ⊗d ⊗ g on (Rd × {(ai , bi ) ∈ Rd × R}m g i=1 . We use a prior with density i=1 R)m to induce a prior on Fm . Let f0,m ∈ arg ming∈Fm ℓ2n (f0 , g) admit the (m) (m) . Further let representation given n by f0,m (x) ≡max1≤i≤m ai o· x + bi (m) (m) cvx ≡ min . τm,g 1≤i≤m g kai k∞ + 1 , g \|bi \| + 1 The prior Λn we will use on the index I = N is given by λn,m ∝ exp − ccvx dm log 3m · log n . (4.7) The first step prior used in [27] is a Poisson proposal, which slightly differs from (4.7) by a logarithmic factor. This would affect the contraction rate only by a logarithmic factor. Theorem 9. Fix 0 < η < 1/4. Suppose that (4.8) cvx τm,g ≥ e− log n·log 3m/8η , and n ≥ d. Then there exists some ccvx in (4.7) depending on η such that (4.9) d log n · m log 3m (n) 2 cvx 2 (n) Pf0 Πn f ∈ F : ℓn (f, f0 ) > C1 inf ℓn (f0 , g) + Y g∈Fm n 2 cvx ≤ C2cvx exp −n(εcvx . n,m ) /C2 o n 2 ≡ max inf 2 (f , g), d log n·m log 3m , and the constants Here (εcvx ) ℓ g∈F 0 n,m m n n Cicvx (i = 1, 2) depend on η. Our oracle inequality shows that the posterior contraction rate of [27] (Theorem 3.3 therein) is far from optimal. (4.8) can be satisfied by using heavy-tailed priors g(·) in the same spirit as Lemma 7–if f0 is square integrable and the design points are regular enough (e.g. using regular grids on [0, 1]d ). Moreover, explicit rate results can be obtained using approximation techniques in [26] (cf. Lemma 4.10 therein). We omit detailed derivations. Remark 3. For univariate convex regression, the term log(3m) in (4.7)-(4.9) can be removed. The logarithmic term is due to the fact that the pseudodimension of Fm scales as m log(3m) for d ≥ 2, cf. Lemma 22. Remark 4. Using similar priors and proof techniques we can construct a (nearly) rate-optimal adaptive Bayes estimator for the support function regression problem for convex bodies [25]. There the models Fm are support functions indexed by polytopes with m vertices, and a prior on Fm is induced by a prior on the location of the m vertices. The pseudo-dimension of Fm can be controlled using techniques developed in [25]. Details are omitted. 16 Q. HAN 4.4. High-dimensional partially linear model. Consider fitting the Gaussian regression model Yi = f0 (xi , zi ) + εi where (xi , zi ) ∈ Rp × [0, 1], by a partially linear model F ≡ {fβ,u (x, z) = x⊤ β + u(z) ≡ hβ (x) + u(z) : β ∈ Rp , u ∈ U } where the dimension of the parametric part can diverge. We consider U to be the class of non-decreasing functions as an illustration (cf. Section 4.2). Consider models F(s,m) ≡ {fβ,u : β ∈ B0 (s), u ∈ Um } where Um denotes the class of piecewise constant non-decreasing functions with at most m constant pieces, and B0 (s) ≡ {v ∈ Rp : \|supp(v)\| ≤ s}. In this example the model index I is a 2-dimensional lattice. Our goal here is to construct an estimator that satisfies an oracle inequality over the models {F(s,m) }(s,m)∈N2 . Consider the following model selection prior: (4.10) λn,(s,m) ∝ exp −chp (s log(ep) + m log(en)) . For a chosen model F(s,m) , consider the following prior Πn,(s,m) : pick randomly a support S ⊂ {1, . . . , p} with \|S\| = s and a set of change points Q ≡ {zi(k) }m k=1 (i(1) < . . . i(m)), and then put a prior gS,Q on βS and u(zi(k) )’s. For simplicity we use a product prior gS,Q ≡ g⊗s ⊗ ḡm where ḡm is a prior on {µ1 ≤ . . . ≤ µm } ⊂ Rm constructed in Section 4.2. Let f0,(s,m) ∈ inf g∈F(s,m) ℓ2n (f0 , g), and write f0,(s,m) (x, z) = x⊤ β0,s + u0,m (z) ≡ h0,s (x) + u0,m (z). Let τs,g ≡ g(kβ0,s k∞ + 1) and τm,g ≡ g(ku0,m k∞ + 1). Let X ∈ Rn×p be the design matrix so that X ⊤ X/n is normalized with diagonal elements taking value 111. Theorem 10. Fix 0 < η < 1/4 and p ≥ n. Suppose that (4.11) τs,g ≥ e− log(ep)/2η , τm,g ≥ e− log(en)/2η . Then there exists some chp in (4.10) depending on η such that (4.12) s log(ep) m log(en) (n) hp (n) 2 2 + Y Pf0 Πn f ∈ F : ℓn (f, f0 ) > C1 inf ℓn (f0 , fβ,u ) + fβ,u ∈F(s,m) n n hp 2 ≤ C2hp exp −n(εhp ) /C . 2 n,(s,m) o n s log(ep)+m log(en) 2 ≡ max inf 2 (f , f , and the ) ℓ ), Here (εhp 0 f ∈F β,u n β,u (s,m) n n,(s,m) constants Cihp (i = 1, 2) depend on η. The first condition of (4.11) requires that the magnitude of kβ0,s k∞ does not grow too fast; see also comments following Theorem 7. The second condition of (4.11) is the same as in (4.5). When the model is well-specified in the sense that f0 (x, z) = x⊤ β0 + u0 (z) for some β0 ∈ B0 (s0 ) and u0 ∈ U , the oracle rate in (4.12) becomes s0 log(ep) m log(en) (4.13) + inf inf ℓ2n (u0 , u) + . m∈N u∈Um n n 11This is a common assumption, cf. Section 6.1 of [10]. BAYES MODEL SELECTION 17 The two terms in the rate (4.13) trades off two structures of the experiment: the sparsity of hβ (x) and the smoothness level of u(z). The resulting phase transition of the rate (4.13) in terms of these structures is in a sense similar to the results of [51, 50]. It is not hard to see that (4.13) cannot be improved in general. Hence our Bayes estimator automatically serves as a theoretically (nearly) optimal adaptive estimator for the high-dimensional partially linear regression model. 4.5. Covariance matrix estimation in the sparse factor model. Suppose we observe i.i.d. X1 , . . . , Xn ∈ Rp from Np (0, Σ0 ). The covariance matrix is modelled by the sparse factor model M ≡ ∪(k,s)∈N2 M(k,s) where M(k,s) ≡ {Σ = ΛΛ⊤ + I : Λ ∈ R(k,s) (L)} with R(k,s) (L) ≡ {Λ ∈ Rp×k , Λ·j ∈ B0 (s), \|σj (Λ)\| ≤ L1/2 , ∀1 ≤ j ≤ k}. In this example, the model index I is a 2-dimensional lattice, and the sparsity structure depends on the rank structure. Consider the following model selection prior: (4.14) λn,(k,s) ∝ exp (−ccov ks log(ep)) . Theorem 11. Let p ≥ n. There exist some ccov in (4.14) and some sequence of sieve priors Πn,(k,s) on M(k,s) depending on L such that (4.15) ks log(ep) (n) (n) 2 cov ′ 2 PΣ0 Πn Σ ∈ M : kΣ − Σ0 kF > C1 X inf kΣ − Σ0 kF + Σ′ ∈M(k,s) n 2 cov . ≤ C2cov exp −n(εcov ) /C 2 n,(k,s) n o 2 ≡ max inf ′ ′ − Σ k2 , ks log(ep) , and the constants Here (εcov ) kΣ 0 Σ ∈M F (s,k) n,(k,s) n Cicov (i = 1, 2) depend on L. Since spectral norm (non-intrinsic) is dominated by Frobenius norm (intrinsic), our result shows that if the model is well-specified in the sense that Σ0 ∈ M, then we can construct an adaptive p Bayes estimator with convergence rates in both norms no worse than ks log p/n. [38] considered p the same sparse factor model, where they proved a strictly sub-optimal rate k3 s log p log n/n in spectral norm under ks & log p. [19] considered a closely related sparse PCA problem, where the convergence rate under spectral norm achieves the same rate as here (cf. Theorem 4.1 therein), √ while a factor of k is lost when using Frobenius norm as a loss function (cf. Remark 4.3 therein). It should be mentioned that the sieve prior Πn,(k,s) is constructed using the metric entropy of M(k,s) and hence the resulting Bayes estimator and the posterior mean as a point estimator are purely theoretical. We use this example to illustrate (i) the construction scheme of a (nearly) optimal adaptive procedure for a multi-structured experiment based on the metric entropy of the underlying parameter space, and (ii) derivation of contraction rates in non-intrinsic metrics when these metrics can be related to the intrinsic metrics nicely. 18 Q. HAN 5. Proofs for the main results 5.1. Proof of Theorem 1: main steps. First we need a lemma allowing a change-of-measure argument. Lemma 8. Let Assumption A hold. There exists some constant c4 ≥ 1 only depending on c1 , c3 and κ such that for any random variable U ∈ [0, 1], any δn ≥ dn (f0 , f1 ) and any j ∈ N, i h −1 2 2 (n) (n) Pf0 U ≤ c4 Pf1 U · ec4 njδn + e−c4 njδn . The next propositions solve the posterior contraction problem for the ‘local’ model Fm . 2 ≥ d2n (f0 , f0,m ). Then there Proposition 2. Fix m ∈ M such that δn,m exists some constant c8 ≥ 1 (depending on the constants in Assumption A) such that for j ≥ 8c2 /c7 h, (5.1) 2 2 (n) 2 X (n) ≤ c8 e−njhδn,m /c8 c . Pf0,m Πn f ∈ F : d2n (f, f0,m ) > c2 (jh)γ δn,m 2 < d2n (f0 , f0,m ). Let m̃ ≡ Proposition 3. Fix m ∈ M such that δn,m m̃(m) ≡ inf{m′ ∈ M, m′ ≥ m : δn,m′ ≥ dn (f0 , f0,m )}. Then for j ≥ 8c2 /c7 h, (5.2) 2 2 (n) Pf0,m Πn f ∈ F : d2n (f, f0,m ) > c4 (2jh)γ d2n (f0 , f0,m ) X (n) ≤ c8 e−njhδn,m̃ /c8 c . The proofs of these results will be detailed in later subsections. Proof of Theorem 1: main steps. Instead of (2.9), we will prove a slightly stronger statement as follows: for any j ≥ 8c2 /c7 h, and h ≥ 2c4 c8 c2 , (5.3) (n) 2 (n) 2 X ≤ c2 e−jnεn,m /c2 . Pf0 Πn f ∈ F : d2n (f, f0 ) > c1 j γ inf d2n (f0 , g) + δn,m g∈Fm Here the constants ci (i = 1, 2) depends on the constants involved in Assumption A and c, h. Proof of (5.3). First consider the overfitting case. By Proposition 2 and Lemma 8, we 2 see that when δn,m ≥ d2n (f0 , f0,m ) holds, for j ≥ 8c2 /c7 h, (n) 2 Pf0 Πn f ∈ F : d2n (f, f0 ) > 2d2n (f0 , f0,m ) + 2c2 (jh)γ δn,m X (n) (n) 2 2 γ 2 (n) ≤ Pf0 Πn f ∈ F : dn (f, f0,m ) > c (jh) δn,m X 2 2 (n) (5.4) ≤ c4 P (n) Πn f ∈ F : d2n (f, f0,m ) > c2 (jh)γ δn,m X ec4 njδn,m f0,m 2 −c−1 njδ n,m +e 4 2 −njδn,m ≤ c8 c4 e h −c4 c 8 c2 −1 + c4 e−c4 2 njδn,m 2 −1 ≤ 2c8 c4 e−jnδn,m min{c4 ,c4 } . BAYES MODEL SELECTION 19 Here in the second line we used the fact that d2n (f, f0,m ) ≥ d2n (f, f0 )/2 − d2n (f0 , f0,m ). (5.4) completes the estimate for overfitting m ∈ M. 2 < d2n (f0 , f0,m ). Next consider the underfitting case: fix m ∈ M such that δn,m Apply Proposition 3 and Lemma 8, and use arguments similar to (5.4) to see that for j ≥ 8c2 /c7 h, (5.5) (n) Pf0 Πn f ∈ F : d2n (f, f0 ) > 2c4 (2jh)γ + 2 d2n (f0 , f0,m ) X (n) 2 (n) 2 4 γ 2 (n) ≤ c4 Pf0,m Πn f ∈ F : dn (f, f0,m ) > c (2jh) dn (f0 , f0,m ) X ec4 njδn,m̃ −1 2 2 −c−1 jnδn, 4 m̃ +e ≤ 2c8 c4 e−njδn,m̃ min{c4 ,c4 } . Here in the second line we used (i) 2d2n (f, f0,m ) ≥ d2n (f, f0 ) − 2d2n (f0 , f0,m ), and (ii) δn,m̃ ≥ dn (f0 , f0,m ). The claim of (5.3) follows by combining (5.4) and (5.5). Proof of (2.11). The proof is essentially integration of tail estimates by a peeling device. Let the event Aj be defined via 2 2 Aj := {c1 j γ d2n (f0 , f0,m ) + δn,m < d2n (f, f0 ) ≤ c1 (j + 1)γ d2n (f0 , f0,m ) + δn,m }. Then, (n) (n) (n) Pf0 d2n (fˆn , f0 ) = Pf0 d2n Πn (f \|X (n) ), f0 ≤ Pf0 Πn d2n (f, f0 )\|X (n) X 2 ≤ Cc1 ,c,c7,h,γ d2n (f0 , f0,m ) + δn,m + Pfn0 Πn d2n (f, f0 )1Aj X (n) j≥8c2 /c7 h 2γ+1 c1 c2 2 + ≤ Cc1 ,c,c7,h,γ d2n (f0 , f0,m ) + δn,m n X 2 j γ nε2n,m e−jnεn,m /c2 . j≥8c2 /c7 h The inequality in the first line of the above display is due to Jensen’s inequality applied with dn (·, f0 ), followed by Cauchy-Schwarz inequality. The summation can be bounded up to a constant depending on γ, c1 , c2 by X X 2 2 (jnε2n,m )γ e−jnεn,m/c2 ≤ (jnε2n,m )γ e−jnεn,m /c2 (j + 1)nε2n,m − jnε2n,m j≥8c2 /c7 h j≥8c2 /c7 h where the inequality follows since nε2n,m ≥ nε2n,1 ≥ 1. This quantity can be R∞ bounded by a constant multiple of 0 xγ e−x/c2 dx independent of m. Now 2 majorizes 1/n up to a constant, the proof is complete by noting that δn,m and then taking infimum over m ∈ M. 5.2. Proofs of Propositions 2 and 3. We will need several lemmas before the proof of Propositions 2 and 3. Lemma 9. Let Assumption A hold. Let F be a function class defined on the sample space X. Suppose that N : R≥0 → R≥0 is a non-increasing function 20 Q. HAN such that for some ε0 ≥ 0 and every ε ≥ ε0 , it holds that N (c5 ε, {f ∈ F : ε < dn (f, f0 ) ≤ 2ε}, dn ) ≤ N (ε). Then for any ε ≥ ε0 , there exists some test φn such that 2 (n) Pf0 φn c6 N (ε)e−c7 nε , ≤ 1 − e−c7 nε2 + 2 (n) sup f ∈F :dn (f,f0 )≥ε Pf (1 − φn ) ≤ c6 e−c7 nε . The constants c5 , c6 , c7 are taken from Lemma 1. Lemma 10. Let Assumption A hold. Suppose that Π is a probability measure on {f ∈ F : dn (f, f0 ) ≤ ε}. Then for every C > 0, there exists some C ′ > 0 depending on C, κ such that Z (n) pf ′ 2 (n) −(C+c3 )nε2 (5.6) Pf0 dΠ(f ) ≤ e ≤ c1 e−C nε . (n) p f0 The proof of these lemmas can be found in Appendix C. Proof of Proposition 2. Fix m′ ∈ I with m′ ≥ m. Now we invoke Lemma 9 with F ≡ Fm′ , f0 ≡ f0,m ∈ Fm ⊂ Fm′ [since m′ ≥ m], ε0 ≡ δn,m′ and 2 log N (ε) ≡ (c7 /2)nδn,m ′ for ε = ε0 to see that, there exists some test φn,m′ such that (5.7) (n) Pf0,m φn,m′ ≤ and that (5.8) 2 log N (ε)−c7 nδn,m ′ c6 e 2 −c7 nδn,m ′ 1−e 2 −c7 nδn,m ′ /2 ≤ 2c6 e 2 −c7 nδn,m ′ (n) sup 2 f ∈Fm′ :d2n (f,f0,m )≥δn,m ′ , Pf (1 − φn,m′ ) ≤ c6 e . 2 Note that here in (5.7) we used the fact that nδn,m ′ ≥ 2/c7 by definition of ′ δn,m . Now for the fixed j, m as in the statement of the proposition, we let φn := supm′ ∈I:m′ ≥jhm φn,m′ be a global test for big models. Then by (5.7), X X 2 2 −c nδ2 /2 (n) (n) Pf0,m φn ≤ Pf0,m φn,m′ ≤ 2c6 e 7 n,m′ ≤ 4c6 e−(c7 /2c )njhδn,m . m′ ≥jhm m′ ≥jhm Here we used the left side of (2.5). This implies that for any random variable U ∈ [0, 1], we have (5.9) (n) (n) Pf0,m U · φn ≤ Pf0,m φn ≤ 4c6 e−(c7 /2c 2 )njhδ 2 n,m . On the power side, with m′ = jhm applied to (5.8) we see that (5.10) (n) (n) sup sup Pf (1 − φn ) ≤ Pf (1 − φn ) 2 f ∈Fjhm :d2n (f,f0,m )≥c2 (jh)γ δn,m 2 f ∈Fjhm :d2n (f,f0,m )≥δn,jhm 2 ≤ c6 e−c7 nδn,jhm ≤ 2c6 e−(c7 /c 2 )njhδ 2 n,m . BAYES MODEL SELECTION 21 2 ≥ The first inequality follows from the right side of (2.5) since c2 (jh)γ δn,m 2 δn,jhm, and the last inequality follows from the left side of (2.5). On the other hand, by applying Lemma 10 with C = c3 and ε2 ≡ (n) 2 c7 jhδn,m , 8c3 c2 we see 2 c njhδn,m −C ′ 7 8c3 c2 and it that there exists some event En such that Pf0,m (Enc ) ≤ c1 e holds on the event En that (5.11) (n) Z (n) Z pf pf dΠ(f ) ≥ λn,m dΠn,m (f ) (n) (n) 2 {f ∈Fm :d2n (f,f0,m )≤c7 jhδn,m /8c3 c2 } p pf0,m f0,m ≥ λn,m e− 2 c7 njhδn,m 4c2 Πn,m 2 f ∈ Fm : d2n (f, f0,m ) ≤ c7 jhδn,m /8c3 c2 Note that (5.12) (n) 2 2 γ 2 (n) Pf0,m Πn f ∈ F : dn (f, f0,m ) > c (jh) δn,m X (1 − φn )1En R  (n) (n) dΠ(f ) /p p 2 2 γ 2 f0,m f ∈F :dn (f,f0,m )>c (jh) δn,m f (n) = Pf0,m  (1 − φn )1En  R (n) (n) pf /pf0,m dΠ(f ) 2 ≤ . 2 ec7 njhδn,m /4c 2 /8c c2 λn,m Πn,m f ∈ Fm : d2n (f, f0,m ) ≤ c7 jhδn,m 3 # "Z (n) (n) (n) pf /pf0,m dΠ(f )(1 − φn ) × Pf0,m 2 f ∈F :d2n (f,f0,m )>c2 (jh)γ δn,m ≡ (I) · (II) where the inequality follows from (5.11). On the other hand, the expectation term in the above display can be further calculated as follows: Z (n) Pf (1 − φn ) dΠ(f ) (II) = 2 f ∈F :d2n (f,f0,m )>c2 (jh)γ δn,m (5.13) ≤ sup 2 f ∈Fjhm :d2n (f,f0,m )>c2 (jh)γ δn,m ≤ 2c6 e−(c7 /c 2 )njhδ 2 n,m (n) Pf (1 − φn ) + Π F \ Fjhm + 4e−(1/c 2 )njhδ 2 n,m ≤ 6c6 e−(c7 /c 2 )njhδ 2 n,m . The first term in the third line follows from (5.10) and the second term follows from (P1) in Assumption C along with the left side of (2.5). By (P1)-(P2) in Assumption C and j ≥ 8c2 /c7 h, (5.14) (n) 2 X (n) (1 − φn )1En ≤ Ce−(c7 /4c Pf0,m Πn f ∈ F : d2n (f, f0,m ) > c2 (jh)γ δn,m 2 )njhδ 2 n,m Hence we conclude (5.1) from (5.9), probability estimate on Enc , and (5.14). . 22 Q. HAN Proof of Proposition 3. The proof largely follows the same lines as that of Proposition 2. See Appendix C for details. 5.3. Completion of proof of Theorem 1. 2 ≥ d2n (f0 , f0,m ), following Proof of (2.10). For any m ∈ M such that δn,m the similar reasoning in (5.12) with j = 8c2 /c7 h, (5.15) (n) / Fjhm \|X (n) 1En Pf0,m Πn f ∈ 2 ≤ λn,m Πn,m ≤ Ce−(c7 /4c 2 ec7 njhδn,m /4c · Π F \ Fjhm 2 2 2 f ∈ Fm : dn (f, f0,m ) ≤ c7 jhδn,m /8c3 c 2 )njhδ 2 n,m . From here (2.10) can be established by controlling the probability estimate for Enc as in Proposition 2, and a change of measure argument as in (5.4) using Lemma 8. 5.4. Proof of Lemma 1. Proof of Lemma 1. Let c> 0 be a constant to be specified later. Consider (n) pf 0 ≤ −cnd2n (f0 , f1 ) . We first consider type the test statistics φn ≡ 1 log (n) pf 1 I error. Under the null hypothesis, we have for any λ1 ∈ (0, 1/κΓ ),    (n) (n) p p f0 f0  (n) (n) Pf0 φn ≤ Pf0 log (n) ≤ −(c + c2 )nd2n (f0 , f1 ) − Pf0 log (n) p f1 p f1 ≤ c1 exp ψκg nd2n (f0 ,f1 ),κΓ (−λ1 ) · exp −λ1 (c + c2 )nd2n (f0 , f1 ) . Choosing λ1 > 0 small enough (depending on κ, c2 , c) we get (n) Pf0 φn ≤ C1 exp(−C2 nd2n (f0 , f1 )) where C1 , C2 > 0 depend on c1 , c2 , c, κ. Next we handle the type II error. ′ To this end, 3 c5 to be specified later, consider the event for a constant c > c (n) En ≡ 1 log pf (n) 1 pf < c′ nd2n (f0 , f1 ) , where f ∈ F is such that d2n (f, f1 ) ≤ c5 d2n (f0 , f1 ), and λ2 ∈ (0, 1/κΓ ),   (n) (n) pf pf (n) (n) (n) Pf (Enc ) ≤ Pf log (n) − Pf log (n) > c′ nd2n (f0 , f1 ) − c3 nd2n (f, f1 ) p f1 p f1   (n) (n) p p f f (n) (n) ≤ Pf log (n) − Pf log (n) > (c′ − c3 c5 )nd2n (f0 , f1 ) p f1 p f1 ′ 2 ≤ e−λ2 (c −c3 c5 )ndn (f0 ,f1 ) c1 exp(ψκg nd2n (f,f1 ),κΓ (λ2 )). BAYES MODEL SELECTION 23 By choosing λ2 > 0 small enough depending on c3 , c5 , c′ , κ, we see that (n) Pf (Enc ) ≤ C3 exp(−C4 nd2n (f0 , f1 )) for some constants C3 , C4 depending only on c1 , c3 , c5 , c′ , κ (in particular, does not depend on f ). On the other hand,   (n) (n) p p f (n) (n) f1 + log (n) < cnd2n (f0 , f1 ) (1En + 1Enc ) Pf (1 − φn ) = Pf log (n) pf p f0   (n) p f (n) (n) (n) ≤ Pf log (n) < (c + c′ )nd2n (f0 , f1 ) + Pf (Enc ) ≡ (∗) + Pf (Enc ). p f0 (n) (n) pf ≥ c2 d2n (f, f0 ) ≥ c2 (1 − √ c5 )2 d2n (f0 , f1 ), we continue our √ computation: for c, c′ such that c + c′ < c2 (1 − c5 )2 , and λ3 ∈ (0, 1/κΓ ),   (n) (n) p p √ f (n) (n) f (∗) ≤ Pf log (n) − Pf log (n) < − c2 (1 − c5 )2 − (c + c′ ) nd2n (f0 , f1 ) p f0 p f0 √ 2 ′ 2 (−λ ) ψ ≤ e−λ3 c2 (1− c5 ) −(c+c ) ndn (f0 ,f1 ) c1 e κg nd2n (f,f0 ),κΓ 3 . Since Pf log (n) pf 0 Now choose λ3 > 0 small enough depending on c2 , c5 , c, c′ , κ we see that for any f ∈ F such that d2n (f, f1 ) ≤ c5 d2n (f0 , f1 ), (n) Pf (1 − φn ) ≤ C5 exp(−C6 nd2n (f0 , f1 )) where C5 , C6 depending on c1 , c2 , c5 , c, c′ , κ, C3 , C4 . Now we need to choose √ c, c′ , c5 such that c > 0, c′ > c3 c5 , c + c′ < c2 (1 − c5 )2 . This can be done by choosing c5 ≤ min{1/4, c2 /16c3 } and c′ = c = 2c3 c5 . 5.5. Proof of Lemma 8. We recall a standard fact. Lemma 11. If a random variable X satisfies E exp(λX) ≤ exp(ψv,c (λ)), t2 then for t > 0, P (X ≥ t) ∨ P(X ≤ −t) ≤ exp − 2(v+ct) . (n) pf 2 0 Proof of Lemma 8. For c = 2c3 , consider the event En ≡ log (n) < cjnδn . pf 1 By Lemma 11, we have for some constant C > 0 depending on c1 , c3 and κ,   (n) (n) p p (n) f0 (n) (n) f0 − Pf0 log (n) ≥ cjnδn2 − c3 nd2n (f0 , f1 ) Pf0 (Enc ) ≤ Pf0 log (n) p f1 p f1   (n) (n) p f0 p f0 (n) (n) ≤ Pf0 log (n) − Pf0 log (n) ≥ c3 jnδn2  (since dn (f0 , f1 ) ≤ δn ) p f1 p f1 n2 j 2 δn4 ≤ C exp −C −1 njδn2 . ≤ C exp − 2 C(njδn + 1) 24 Q. HAN We remind the reader that the constant C may not be the same in the above series of inequalities, and hence the last inequality follows by noting that (i) if njδn2 ≥ 1, then we replace the denumerator of the second last line by 2, (ii) if njδn2 < 1, then we increase C ≥ 1. Then   (n) p −1 2 (n) (n) (n) (n) f0 1En  + Ce−C njδn Pf0 U = Pf0 U 1En + Pf0 U 1Enc ≤ Pf1 U (n) p f1 (n) 2 ≤ Pf1 U · ecnjδn + Ce−C completing the proof. −1 njδ 2 n , 5.6. Proof of Proposition 1. P 2 Proof of Proposition 1. Let Σn = m∈I e−2nδn,m be the total mass. Then 2 2 2 e−2nδn,1 ≤ Σn ≤ 2e−2nδn,1 /c ≤ 2. The first condition ofP(P1) is trivial. We only need to verify the second condition of (P1): k>hm λn,k = P 2 2 2 2 2 −2nδ n,k ≤ e2nδn,1 · 2e−(2h/c )nδn,m ≤ 2e−2nδn,m , where the first Σ−1 n k>hm e inequality follows from (2.5) and the second by the condition h ≥ 2c2 . 6. Proofs for applications The proofs of the theorems in Section 4 follow the similar route by verifying (i) the local entropy condition in Assumption B, (ii) the summability condition in (2.5) and (iii) the sufficient mass condition (P2) in Assumption C. We remind the reader that we use (2.8) in all examples as the first-step (model selection) prior. We only prove Theorems 7 and 11 in this section. The proofs for Theorems 8-10 are deferred to Appendix B. 6.1. Proof of Theorem 7. Lemma 12. Let r ∈ I. Suppose that the linear map X : Rm1 ×m2 → Rn is uniform RIP(ν; I). Then for any ε > 0 and A0 ∈ Rm1 ×m2 such that rank(A0 ) ≤ r, we have log N c5 ε, {fA ∈ Fr : ℓn (fA , fA0 ) ≤ 2ε}, ℓn ≤ 2(m1 + m2 )r · log 18ν̄/c5 ν . We will need the following result. Lemma 13. Let S(r, B) = {A ∈ Rm1 ×m2 : rank(A) ≤ r, kAk2 ≤ B}. Then (m1 +m2 −1)r N ε, S(r, B), k·k2 ≤ 9B . ε Proof of Lemma 13. The case for B = 1 follows from Lemma 3.1 of [12] and the general case follows by a scaling argument. We omit the details. Proof of Lemma 12. We only need to consider the case r ≤ rmax . First note that the entropy in question equals √ √ (6.1) log N c5 nε, {X (A) : kX (A − A0 )k2 ≤ 2 nε, rank A ≤ r}, k·k2 . BAYES MODEL SELECTION 25 By uniform RIP(ν; I), the set to be covered in the above display is contained in {X (A) : kA − A0 k2 ≤ 2ε/ν, rank A ≤ r} ⊂ X (S(2r, 2ε/ν )). On the other hand, again by uniform RIP(ν; I), a c√ 5 ε/ν̄-cover of the set S(2r, 2ε/ν) under the Frobenius norm k·k2 induces a c5 nε-cover of X (S(2r, 2ε/ν )) under the Euclidean k·k2 norm. This implies that (6.1) can be further bounded from above by log N c5 ε/ν̄, S(2r, 2ε/ν ), k·k2 ≤ 2(m1 + m2 )r · log 18ν̄/c5 ν , where the last inequality follows from Lemma 13. 2 = 4 log(18ν̄/c5 ν) ∨ 1 ·(m1 +m2 )r log m̄ . Clearly δ 2 satisfies Now we take δn,r n,r c7 η n (2.5) with c ≡ 1 and γ = 1. Lemma 14. Suppose that X : Rm1 ×m2 → Rn is uniform RIP(ν; I), and that (4.2) holds. Then (P2) in Assumption C holds. Proof of Lemma 14. We only need to consider r ≤ rmax . First note that (6.2) 2 /c3 Πn,r fA ∈ Fr : ℓ2n (fA , fA0,r ) ≤ δn,r √ √ = ΠG A ∈ Rm1 ×m2 : kX (A − A0,r )k2 ≤ nδn,r / c3 , rank(A) ≤ r √ ≥ ΠG A ∈ Rm1 ×m2 : kA − A0,r k2 ≤ δn,r /ν̄ c3 , rank(A) ≤ r . P Let A0,r ≡ ri=1 σi ūi v̄i⊤ be the spectral decomposition of A0,r , and let ui ≡ P √ √ σi ūi and vi ≡ σi v̄i . Then A0,r ≡ ri=1 ui vi⊤ . Now for u∗i ∈ Bm1 (ui , ε) P and vi∗ ∈ Bm2 (vi , ε), i = 1, . . . , r, let A∗ ≡ ri=1 u∗i (vi∗ )⊤ , then by noting √ that the Frobenius norm is sub-multiplicative and that kui k2 = kvi k2 = σi , we have for ε ≤ 1, kA∗ − A0,r k2 ≤ ≤ r X k(ui − u∗i )vi⊤ k2 + ku∗i (vi − vi∗ )⊤ k2 i=1 r X i=1 √ √ (ε σi + ( σi + ε)ε) ≤ ρr ε √ where ρr ≡ i=1 (2 σi + 1). Now with εn,r ≡ can be further bounded from below as follows: Pr δ √n,r ν̄ c3 ρr ∧ 1 we see that (6.2) (6.2) ≥ ΠG (∩ri=1 {(u∗i , vi∗ ) : u∗i ∈ Bm1 (ui , εn,r ), vi∗ ∈ Bm2 (vi , εn,r )}) ≥ tr (m1 +m2 )r (τr,g ) r Y i=1 vol (Bm1 (ui , εn,r )) · vol (Bm2 (vi , εn,r )) −1 −1 tr r ≥ (τr,g · εn,r )(m1 +m2 )r vm v r ≥ e−(m1 +m2 )r·(log m̄/2+log τr,g +log(εn,r ∨1)) . 1 m2 √ where vd = vol(Bd (0, 1)), and vd ≥ (1/ d)d . Hence in order that the right 2 side of the above display can be bounded from below by e−2nδn,r , it suffices 26 Q. HAN to require that (6.3) log m̄ −1 . max log τr,g , log(ε−1 n,r ∨ 1) ≤ 2η 2 2 It is easy to calculate that ε−2 n,r ≤ 9ν̄ c3 (1 ∨ σmax (A0,r )) rmax n. Now the conclusion follows by noting that (4.2) implies (6.3) since rmax ≤ n and c3 = 1. Proof of Theorem 7. The theorem follows by Theorems 1 and 2, Proposition 1 coupled with Lemmas 12 and 14. 6.2. Proof of Theorem 11. Lemma 15. For any Σ0 ∈ M(k,s) , log N c5 ε, {Σ ∈ M(k,s) : kΣ − Σ0 kF ≤ CL ε}, k·kF √ ≤ ks log(ep/s) + ks log(6 kL/c5 ε). Proof. The set involved in the entropy is equivalent to o n (6.4) Λ ∈ R(k,s) (L) : kΛΛ⊤ − Λ0 Λ⊤ 0 kF ≤ CL ε, k·kF . √ We claim that supΛ∈R(k,s) kΛΛ⊤ kF ≤ kL. To see this, let Λ ≡ P ΞQ⊤ be the singular value decomposition of Λ, where P ∈ Rp×p , Q ∈ Rk×k are unitary matrices and Ξ ∈ Rp×k is a diagonal matrix. Then kΛΛ⊤ k2F = kΞΞ⊤ k2F ≤ kL, proving the claim. Combined with (6.4) and Euclidean embedding, we see that the entropy in question can be bounded by √ log N c5 ε, {v ∈ B0 (ks; pk) : kvk2 ≤ 2 kL}, k·k2  √ !ks  √ kL pk 6  ≤ ks log(ep/s) + ks log(6 kL/c5 ε), ≤ log  c5 ε ks where B0 (s; pk) ≡ {v ∈ Rpk : \|supp(v)\| ≤ s}. ′ ′ 2 Proof of Theorem 11. Take δn,(k,s) = KC ′ ks n log(C p) for some C ≥ e depending on c5 , c7 , L and some absolute constant K ≥ 1. Apparently (2.5) holds with c = 1, γ = 1. The prior Πn,(k,s) on M(k,s) will be the uniform disq log(C ′ p) covering-ball of the set {Σ ∈ M(k,s) } tribution on a minimal C ′ cks 3n under the Frobenius norm k·kF . The above lemma entails that the cardinality for such a cover is no more than exp(C ′′ ks log(C ′′ p)) for another constant C ′′ ≥ e depending on c3 , c5 , c7 , L. Hence o n 2 ≥ exp(−C ′′ ks log(C ′′ p)), /c3 Πn,(k,s) Σ ∈ M(k,s) : kΣ − Σ0,(k,s)kF ≤ δn,(k,s) 2 which can be bounded from below by exp(−2nδn,(k,s) ) by choosing K large enough. The claim of Theorem 11 now follows from these considerations along with Theorems 1 and 6, Proposition 1. BAYES MODEL SELECTION 27 Appendix A. Proof of lemmas in Section 3 (n) Proof of Lemma 2. Let Pθ0 denote the probability measure induced by the joint distribution of (X1 , . . . , Xn ) when the underlying signal is θ0 . First consider Gaussian regression case. It is easy to calculate that (n) n X p0 1 1 (X (n) ) = − (Xi − θ0,i )2 + (Xi − θ1,i )2 , log θ(n) 2 2 p θ1 i=1 (n) (n) Pθ0 log p θ0 (n) p θ1 1 (X (n) ) = nℓ2n (θ0 , θ1 ). 2 Then (n) Pθ0 exp " ≤ P exp (n) (n) λ log n X i=1 p θ0 (n) p θ1 (X (n) )− εi λ θ0,i − θ1,i (n) Pθ0 log ! p θ0 (n) p θ1 (X (n) !# ) ≤ exp λ2 nℓ2n (θ0 , θ1 )/2 . Next consider binary regression. Easy calculation shows that (n) log Pθ(n) log 0 p θ0 (n) p θ1 (n) p θ0 (n) p θ1 = n X Xi log 1 − θ0,i θ0,i + (1 − Xi ) log , θ1,i 1 − θ1,i θ0,i log θ0,i 1 − θ0,i + (1 − θ0,i ) log . θ1,i 1 − θ1,i i=1 = n X i=1 Using the inequality cx ≤ log(1 + x) ≤ x for all −1 < x ≤ c′ for some c > 0 depending on c′ > −1 only, we have shown Pθ(n) log 0 (n) 0 (n) pθ 1 pθ ≍ nℓ2n (θ0 , θ1 ) under the assumed condition that Θn ⊂ [η, 1 − η]n . Now we verify the Bernstein condition: " !# (n) (n) p θ0 p θ0 (n) Pθ0 exp λ log (n) − Pθ(n) log (n) 0 p θ1 p θ1 ! ! n n X X (n) 2 2 ti /8 (Xi − θ0,i )ti ≤ exp λ = Pθ0 exp λ i=1 i=1 θ 1−θ where ti ≡ ti (θ0 , θ1 ) = log 1−θ0,i0,i · θ1,i1,i and the last inequality follows from Hoeffding’s inequality (cf. Section 2.6 of [9]). The claim follows by h i2 θ0,i −θ1,i noting that t2i = log (1−θ + 1 ≍ (θ0,i − θ1,i )2 by the assumed 0,i )θ1,i condition and the aforementioned inequality log(1 + x) ≍ x in a constrained range. 28 Q. HAN Finally consider Poisson regression. It is easy to see that (n) log (n) Pθ0 log p θ0 (n) p θ1 (n) p θ0 (n) p θ1 = n X Xi log θ0,i + (θ1,i − θ0,i ), θ1,i θ0,i log θ0,i + (θ1,i − θ0,i ). θ1,i i=1 = n X i=1 Note that for any 1/M ≤ p, q ≤ M , 2 q q q p − 1 ≍ (p − q)2 , p log − (p − q) = p − log − 1 + ≍p· q p p p where in the middle we used the fact that − log x − 1 + x ≍ (x − 1)2 for x (n) bounded away from 0 and ∞. This shows that Pθ0 log (n) 0 (n) pθ 1 pθ ≍ nℓ2n (θ0 , θ1 ). Next we verify the Bernstein condition: " !# (n) (n) p θ0 p θ0 (n) Pθ0 exp λ log (n) − Pθ(n) log (n) 0 p θ1 pθ # # 1 " n " n X X (n) λti θ0,i e − 1 − λti (Xi − θ0,i )ti ≤ exp ≤ Pθ0 exp λ i=1 i=1 where ti = log(θ0,i /θ1,i ). Now for any \|λ\| ≤ 1, we have eλti −1−λti ≍ λ2 t2i ≤ λ2 t2i /(1−\|λ\|). On the other hand, θ0,i t2i = θ0,i (log(θ0,i /θ1,i ))2 ≍ (θ0,i −θ1,i )2 , completing the proof. Proof of Lemma 3. Since the log-likelihood ratio for X1 , . . . , Xn can be decomposed into sums of the log-likelihood ratio for single samples, and the log-likelihood ratio is uniformly bounded over F (since G is bounded), classical Bernstein inequality applies to see that for any couple (f0 , f1 ), the Bernstein condition in Assumption A holds with v = κg nVarf0 (log f0 /f1 ), c = κΓ where κg , κΓ depend only on G. Hence we only need to verify that Varf0 (log f0 /f1 ) . h2 (f0 , f1 ), Pf0 (log f0 /f1 ) ≍ h2 (f0 , f1 ). This can be seen by Lemma 8 of [24] and the fact that Hellinger metric is dominated by the Kullback-Leiber divergence. Lemma 16. Let Z be a random variable bounded by M > 0. Then E exp(Z) ≤ exp eM EZ . Proof. Note that log E exp(Z) = log (E [exp(Z) − 1] + 1) ≤ E [exp(Z) − 1] ≤ eM EZ. P where the last inequality follows from Taylor expansion ex −1 = nk=1 xk /k! ≤ P x k≥1 M k−1 /k! ≤ xeM . BAYES MODEL SELECTION 29 Proof of Lemma 4. We omit explicit dependence of M on the notation dr,M (n) and rM in the proof. Let Pf0 denote the probability measure induced by the joint distribution of (X0 , . . . , Xn ) where X0 is distributed according to the stationary density qf0 . Easy computation shows that (n) n−1 X p f0 1 2 log (n) = εi+1 (f0 (Xi ) − f1 (Xi )) + (f0 (Xi ) − f1 (Xi )) , 2 p f1 i=0 (n) Z p f0 n (n) Pf0 log (n) = (f0 − f1 )2 qf0 dλ. 2 p f1 Here λ denotes the Lebesgue measure on R. By the arguments on page 209 of [23], we see that r . qf0 . r. Hence we only need to verify the Bernstein condition. By Cauchy-Schwarz inequality, (A.1) 2   ! (n) n−1 X p (n) f (n) 0  ≤ P exp 2λ P exp λ log εi+1 (f0 (Xi ) − f1 (Xi )) f0 f0 (n) p f1 i=0 ! n−1 X (n) (f0 (Xi ) − f1 (Xi ))2 × Pf0 exp λ i=0 ≡ (I) × (II). The first term (I) can be handled by an inductive calculation. First note that for any \|µ\| ≤ 2 and X1 ∈ R, (A.2) 2 (f Pp(·\|X1 ) eµ 2 0 (X2 )−f1 (X2 ) ≤ ee 16M 2 µ2 P 2 p(·\|X1 ) (f0 −f1 )(X2 ) 2 d2 (f ,f ) r 0 1 ≤ eCM µ where the first inequality follows from Lemma 16 and the second inequality follows from r(·) . pf (·\|x) . r(·) holds for all x ∈ R where the constant Pn−1 involved depends only on M . Let Sn ≡ i=0 εi+1 (f0 (Xi ) − f1 (Xi )) and εn ≡ (ε1 , . . . , εn ). Then for \|λ\| ≤ 1, let µ ≡ 2λ, (n) (n) Pf0 e2λSn = Pf0 eµSn = EX0 ,εn−1 eµSn−1 Eεn eµεn (f0 (Xn−1 )−f1 (Xn−1 )) 2 2 ≤ EX0 ,εn−1 eµSn−1 eµ (f0 (Xn−1 )−f1 (Xn−1 )) /2 h i 2 2 ≤ EX0 ,εn−2 eµSn−2 Eεn−1 eµεn−1 (f0 (Xn−2 )−f1 (Xn−2 ))+µ (f0 (Xn−1 )−f1 (Xn−1 )) /2 1/2 ≤ EX0 ,εn−2 eµSn−2 Eεn−1 e2µεn−1 (f0 (Xn−2 )−f1 (Xn−2 )) 1/2 µ2 (f0 (Xn−1 )−f1 (Xn−1 ))2 × Ep(·\|Xn−2 ) e 2 2 2 2 ≤ EX0 ,εn−2 eµSn−2 eµ (f0 (Xn−2 )−f1 (Xn−2 )) · eCM µ dr (f0 ,f1 )/2 , 30 Q. HAN where the last inequality follows from (A.2). Now we can iterate the above calculation to see that (I) ≤ exp(CM λ2 nd2r (f0 , f1 )). (A.3) Next we consider (II). Since for any non-negative random variables Z1 , . . . , Zn , Q Q we have E ni=1 Zi ≤ ni=1 (EZin )1/n . It follows that (A.4) n Y 1/n (n) Pf0 exp nλ(f0 (Xi ) − f1 (Xi ))2 = Pqf0 exp(nλ(f0 (X0 ) − f1 (X0 ))2 (II) ≤ i=1 where the last inequality follows by stationarity. On the other hand, by Jensen’s inequality,   (n) p λn f0  (n) exp −λPf0 log (n) ≤ exp − Pqf0 (f0 − f1 )2 2 (A.5) p f1 ≤ Pqf0 exp(−λn(f0 (X0 ) − f1 (X0 ))2 /2). Collecting (A.1) and (A.3)-(A.5), we see that for \|λ\| ≤ 1,     (n) (n) (n) p p f0 p f0 p (n) f0  Pf (n) exp λ log (n) − Pf (n) log (n)  ≤ (I) · (II) exp −λPf0 log (n) 0 0 p f1 p f1 p f1 ′ ≤ exp(CM λ2 nd2r (f0 , f1 )), completing the proof. (n) Proof of Lemma 5. For any g ∈ G, let pg denote the probability density function of a n-dimensional multivariate normal distribution with covariance (n) matrix Σg ≡ Tn (fg ), and Pg the expectation taken with respect to the (n) density pg . Then for any g0 , g1 ∈ G, (n) log (A.6) pg 0 (n) pg 1 1 1 −1 (n) − log det(Σg0 Σ−1 (X (n) ) = − (X (n) )⊤ (Σ−1 g0 − Σg1 )X g1 ), 2 2 (n) Pg(n) log 0 pg 0 (n) pg 1 1 1 log det(Σg0 Σ−1 = − tr(I − Σg0 Σ−1 g1 ) − g1 ) 2 2 where we used the fact that for a random vector X with covariance matrix −1/2 (n) Σ, EX ⊤ AX = tr(ΣA). Let G ≡ Σg0 X (n) ∼ N (0, I) under Pg0 , and 1/2 1/2 B ≡ I − Σg0 Σ−1 g1 Σg0 , then (n) Yn ≡ log pg 0 (X (n) ) − Pg(n) log 0 (n) (n) pg 0 (n) pg 1 pg 1 i i 1h ⊤ 1 h (n) ⊤ −1 (n) −1 G BG − tr(B) . )X − tr(I − Σ Σ ) = − = − (X ) (Σg0 − Σ−1 g0 g1 g1 2 2 BAYES MODEL SELECTION 31 Let B = U ⊤ ΛU be the spectral decomposition of B where U is orthonormal and Λ = diag(λ1 , . . . , λn ) is a diagonal matrix. Then we can further compute ⊤ −2Yn =d G ΛG − tr(Λ) = n X i=1 λi (gi2 − 1) where g1 , . . . , gn ’s are i.i.d. standard normal. Note that for any \|t\| < 1/2, 1 √ 2π Z ∞ et(x 2 −1) e−x −∞ 2 /2 dx = √ 1 e−t 2 = e 2 (− log(1−2t)−2t) ≤ et /(1−2t) 1 − 2t P 1 k where the inequality follows from − log(1 − 2t) − 2t = k≥2 k (2t) = P 2t2 1 (2t)k ≤ 1−2t . Hence apply the above display with t = −λλi /2, 4t2 k≥0 k+2 we have that for any \|λ\| < 1/ maxi λi , (A.7) Z ∞ n n Y Y 1 2 2 2 √ E exp −λ · λi (gi − 1)/2 = E exp(λYn ) = e−λ·λi (x −1)/2 e−x /2 dx 2π −∞ i=1 i=1 P n Y λ2 i λ2i λ2 λ2i exp ≤ ≤ exp . 4 + 4λλi 4 − 4\|λ\| maxi \|λi \| i=1 Denote k·k and k·kF the matrix operator norm and Frobenius norm respectively. By the arguments on page 203 of [23], we have kΣg k ≤ 2πkeg k∞ −1 −g and kΣ−1 g k ≤ (2π) ke k∞ . Since G is a class of uniformly bounded function classes, the spectrum of the covariance matrices Σg and their inverses running over g must be bounded. Hence (A.8) −1 max\|λi \| = kBk = k(Σg1 − Σg0 )Σ−1 g1 k ≤ kΣg1 − Σg0 kkΣg1 k ≤ CG < ∞. i Next, note that (A.9) X i λ2i 1/2 = (tr(BB ⊤ ))1/2 = kBkF = k(Σg1 − Σg0 )Σ−1 g1 kF ′ ≤ kΣ−1 g1 kkΣg1 − Σg0 kF ≤ CG p nDn2 (g0 , g1 ) where in the first inequality we used kM N kF = kN M kF for symmetric matrices M, N and the general rule kP QkF ≤ kP kkQkF . Collecting (A.7)(A.9) we see that Assumption A is satisfied for v = κg nDn2 (g0 , g1 ) and c = κΓ for some constants κg , κΓ depending on G only. 32 Q. HAN (n) Finally we establish equivalence of n1 Pg0 log (A.6), we have (n) pg0 (n) pg1 and Dn2 (g0 , g1 ). First by (n) 1 1 = − tr(I − Σg0 Σ−1 log det(Σg0 Σ−1 g1 ) − g1 ) 2 2 1 −1/2 −1/2 −1/2 tr Σg1 (Σg0 − Σg1 )Σg−1/2 − log det I + Σ (Σ − Σ )Σ = g0 g1 g1 g1 1 2 1 1 2 2 −1 2 ′′ 2 ≤ kI − Σg0 Σ−1 g1 kF ≤ kΣg1 − Σg0 kF kΣg1 k ≤ CG nDn (g0 , g1 ). 4 4 Here in the second line we used the fact that det(AB −1 ) = det(I +B −1/2 (A− B)B −1/2 ), and in the third line we used the fact − log det(I + A) + tr(A) ≤ 1 2 2 tr(A ) for any p.s.d. matrix A, due to the inequality log(1 + x) − x ≥ − 12 x2 for all x ≥ 0. On the other hand, by using the reversed inequality log(1 + x) − x ≤ −cx2 for all 0 ≤ x ≤ c′ where c is a constant depending only Pg(n) log 0 pg 0 (n) pg 1 (n) on c′ , we can establish Pg0 log the proof. (n) pg0 (n) pg1 ≥ CG′′′ nDn2 (g0 , g1 ), thereby completing Proof of Lemma 6. Note that (n) n X p Σ0 1 1 ⊤ −1 −1 −1 (n) X (Σ0 − Σ1 )Xi − log det(Σ0 Σ1 ) , log (n) (X ) = − 2 i 2 p i=1 Σ1 (n) PΣ0 log (n) p Σ0 (n) p Σ1 n n log det(Σ0 Σ−1 = − tr(I − Σ0 Σ−1 1 )− 1 ). 2 2 The rest of the proof proceeds along the same line as in Lemma 5. Appendix B. Proof of remaining theorems in Section 4 B.1. Proof of Theorem 8. Lemma 17. Let n ≥ 2. Then for any g ∈ Fm , log N c5 ε, {f ∈ Fm : ℓn (f, g) ≤ 2ε}, ℓn ≤ 2 log(6/c5 ) · m log(en). Proof of Lemma 17. Let Qm denote all m-partitions of the design points n x1 , . . . , xn . Then it is easy to see that \|Qm \| = m−1 . For a given mpartition Q ∈ Qm , let Fm,Q ⊂ Fm denote all monotonic non-decreasing functions that are constant on the partition Q. Then the entropy in question can be bounded by n (B.1) log max N c5 ε, {f ∈ Fm,Q : ℓn (f, g) ≤ 2ε}, ℓn . m − 1 Q∈Qm On the other hand,√for any fixed m-partition Q ∈ Qm√ , the entropy term above equals N c5 nε, {γ ∈ Pn,m,Q : kγ − gk2 ≤ 2 nε}, k·k2 , where Pn,m,Q ≡ {(f (x1 ), . . . , f (xn )) : f ∈ Fm,Q }. By Pythagoras theorem, the set BAYES MODEL SELECTION 33 involved in the entropy is included in {γ ∈ Pn,m,Q : kγ − πPn,m,Q (g)k2 ≤ √ 2 nε} where πPn,m,Q is the natural projection from Rn onto the subspace Pn,m,Q . Clearly Pn,m,Q is contained in a linear subspace with dimension no more than m. Using entropy result for the finite-dimensional space [Problem 2.1.6 in [47], page 94 combined with the discussion in page 98 relating the packing number and covering number], (B.2) √ m 3 · 2 nε √ log N c5 ε, {f ∈ Fm,Q : ℓn (f, f0,m ) ≤ 2ε}, ℓn ≤ log = m log(6/c5 ). c5 nε n The claim follows by (B.1)-(B.2), and log m−1 ≤ m log(en). 1 m log(en) 5) 2 ∨ . It is clear that (2.5) ≡ 4 log(6/c Hence we can take δn,m c7 η n is satisfied with c ≡ 1 and γ = 1. Lemma 18. Suppose that (4.5) holds . Then (P2) in Assumption C holds. Proof of Lemma 18. Let Q0,m = {Ik }m k=1 be the associated m-partition of {x1 , . . . , xn } of f0,m ∈ Fm with the convention that {Ik } ⊂ {x1 , . . . , xn } is ordered from smaller values to bigger ones. Thenit is easy to see that µ0,m = (µ0,1 , . . . , µ0,m ) ≡ f0,m (xi(1) ), . . . , f0,m (xi(m) ) ∈ Rm is well-defined and µ0,1 ≤ . . . ≤ µ0,m . It is easy to see that any f ∈ Fm,Q0,m satisfying √ the property that sup1≤k≤m \|f (xi(k) ) − µ0,k \| ≤ δn,m / c3 leads to the error 2 /c . Hence estimate ℓ2n (f, f0,m ) ≤ δn,m 3 (B.3) 2 /c3 } Πn,m {f ∈ Fm : ℓ2n (f, f0,m ) ≤ δn,m −1 n 2 ≥ Πḡm {f ∈ Fm,Q0,m : ℓ2n (f, f0,m ) ≤ δn,m /c3 } m−1 −1 m √ n ≥ Πḡm {µ ∈ Rm : µ ≡ µ0,k + εk k=1 , 0 ≤ ε1 ≤ . . . ≤ εm ≤ δn,m / c3 } m−1 −1 √ m 1 n ≥ · inf √ ḡm (µ)(1 ∧ δn,m / c3 ) m m m−1 m! µ∈R :µ≡(µ0,k +εk )k=1 ,0≤ε1 ≤...≤εm ≤1∧δn,m / c3 −1 √c 3 iso )−1 ∨1 −m log √ n −m log(en)−m log (τm,g ∨1 iso m δn,m · (τm,g ) (1 ∧ δn,m / c3 )m ≥ e ≥ . m−1 Here the first inequality in the last line follows from the definition of ḡm iso . The claim follows by verifying (4.5) implies that the second and and τm,g 1 · m log(en) [the third term in the exponent above are both bounded by 2η √ −1 third term does not contribute to the condition since c3 δn,m ≤ n by noting c3 = 1 in the Gaussian regression setting and definition of η]. Proof of Theorem 8. The theorem follows by Theorems 1 and 2, Proposition 1 coupled with Lemmas 17 and 18. We now prove Lemma 7. We need the following result. 34 Q. HAN Lemma 19. Let f0 := (f0 (x1 ), . . . , f0 (xn )) ∈ Rn , and f0,m := (f0,m (x1 ), . . . , f0,m (xn )) ∈ Rn where f0,m ∈ arg ming∈Fm ℓ2n (f0 , g). Suppose that kf0 k2 ≤ L, and that there exists some element f ∈ Fm such that f ≡ (f (x1 ), . . . , f (xn )) satisfies kf k2 ≤ L. Then kf0,m k2 ≤ 3L. Proof of Lemma 19. It can be seen that f0,m ∈ arg minγ∈Pn,m Lf0 (γ) ≡ arg minγ∈Pn,m kf0 − γk2 where Pn,m ≡ {(f (x1 ), . . . , f (xn )) : f ∈ Fm }. For any γ ∈ Pn,m such that kγk2 ≤ L, the loss function satisfies Lf0 (γ) ≤ 2L by triangle inequality. If kf0,m k2 > 3L, then Lf0 (f0,m ) = kf0 − f0,m k2 ≥ kf0,m k2 − kf0 k2 > 3L − L = 2L, contradicting the definition of f0,m as a minimizer of Lf0 (·) over Pm,n . This shows the claim. R1 2 R 1 Proof of Lemma 7. Let L = 0 f . Note that kf0 k22 ≤ 2n 0 f 2 (x) dx = √ 2nL2 . By Lemma 19, we see that kf0,m k2 ≤ 3 2nL which √ entails that √ kf0,m k∞ ≤ 3 2nL. Now the conclusion follows from g(3 2nL + 1) ≥ (en)−1/2η while the left side is at least on the order of n−α/2 as n → ∞. B.2. Proof of Theorem 9. Checking the local entropy assumption B requires some additional work. The notion of pseudo-dimension will be useful in this regard. Following [39] Section 4, a subset V of Rd is said to have pseudo-dimension t, denoted as pdim(V ) = t, if for every x ∈ Rt+1 and indices I = (i1 , · · · , it+1 ) ∈ {1, · · · , n}t+1 with iα 6= iβ for all α 6= β, we can always find a sub-index set J ⊂ I such that no v ∈ V satisfies both vi > xi for all i ∈ J and vi < xi for all i ∈ I \ J. Lemma 20. Let n Suppose that pdim(Pn,m ) ≤ Dm where Pn,m := ≥ 2. n { f (x1 ), . . . , f (xn ) ∈ R : f ∈ Fm }. Then for all g ∈ Fm , log N c5 ε, {f ∈ Fm : ℓn (f, g) ≤ 2ε}, ℓn ) ≤ C · Dm log n for some constant C > 0 depending on c5 . To prove Lemma 20, we need the following result, cf. Theorem B.2 [25]. Lemma 21. Let V be a subset of Rn with supv∈V kvk∞ ≤ B and pseudodimension at most t. Then, for every ε > 0, we have N (ε, A, k·k2 ) ≤ √ κt , holds for some absolute constant κ ≥ 1. 4 + 2Bε n Proof of Lemma 20. Note that the entropy in question can be bounded by √ √ log N c5 ε n, {Pn,m − g} ∩ Bn (0, 2 nε), k·k2 . Since translation does not change the pseudo-dimension of a set, Pn,m − g has the same pseudo-dimension with that of Pn,m , which is bounded √ from above by Dm by assumption. Further note that {Pn,m − g} ∩ Bn(0, 2 nε) is √ uniformly bounded by 2 nε, hence an application of Lemma 21 yields that the above display can be further bounded as follows: log N c5 ε, {f ∈ Fm : ℓn (f, g) ≤ 2ε}, ℓn ) ≤ κDm log 4 + 4n/c5 ) ≤ C · Dm log n for some constant C > 0 depending on c5 whenever n ≥ 2. BAYES MODEL SELECTION 35 The pseudo-dimension of the class of piecewise affine functions Fm can be well controlled, as the following lemma shows. Lemma 22 (Lemma 4.9 in [26]). pdim(Pn,m ) ≤ 6md log 3m. As an immediate result of Lemmas 20 and 22, we can take for n ≥ 2, := (C ∨ 1/η) d · logn n · m log 3m for some C ≥ 2/c7 depending on c5 , c7 . 2 δn,m Lemma 23. Suppose that (4.8) holds and n ≥ d. Then (P2) in Assumption C holds. Proof of Lemma 23. We write f0,m ≡ max1≤i≤m ai · x + bi throughout √ the proof. We first claim that for any a∗i ∈ Bd (ai , δn,m /2 c3 d) and b∗i ∈ √ ∗ ∗ ∗ ∗ (x) := max B1 (bi , δn,m /2 c3 ), let gm 1≤i≤m (ai · x + bi ), then ℓ∞ (gm , f0,m ) ≤ √ δn,m / c3 . To see this, for any x ∈ X, there exists some index ix ∈ {1, . . . , m} ∗ (x) = a∗ · x + b∗ . Hence such that gm ix ix ∗ gm (x) − f0,m (x) ≤ a∗ix − aix · x + b∗ix − bix ≤ ka∗ix − aix k2 kxk2 + \|b∗ix − bix \| δn,m δn,m √ δn,m ≤ √ · d+ √ = √ . 2 c3 c3 2 c3 d The reverse direction can be shown similarly, whence the claim follows by taking supremum over x ∈ X. This entails that (B.4) 2 /c3 Πn,m f ∈ Fm : ℓ2n (f, f0,m ) ≤ δn,m n p √ o ∗ ∗ ∗ ∗ c d), b ∈ B (b , δ /2 c3 ) (a , b ) : a ∈ B (a , δ /2 ≥ ΠG ∩m 3 1 i n,m i n,m d i i i i i=1 = m Y i=1 m Y p √ Πg⊗d Bd (ai , δn,m /2 c3 d) · Πg B1 (bi , δn,m /2 c3 ) d δn,m δn,m √ √ g(kai k∞ + 1) · g(\|bi \| + 1) · ≥ ∧1 ∧ 1 vd 4c3 4c3 d i=1 √ 1 4c3 d −1 ∨ 1 − md log d ≥ exp − 2m(d + 1) log τm,g ∨ 1 − m(d + 1) log δn,m 2 √ where vd ≡ vol(Bd (0, 1)) and we used the fact that vd ≥ (1/ d)d . Now by requiring that n ≥ d and (B.5) √ d 4c3 d −1 ∨1 ≤ log n · m log 3m, max 2m(d + 1) log τm,g ∨ 1 , m(d + 1) log δn,m 2η √ −1 ≤ √n, the claim follows by verifying (4.8) implies (B.5)[since 4c3 dδn,m the second term is bounded by md log n. The inequality follows by noting η < 1/4]. d Lemma 24. For n ≥ 2, (2.5) is satisfied for c = 1 and γ = 2. 36 Q. HAN 2 = c log n(m log 3m) throughProof. For fixed n ≥ 2 and η > 0, write nδn,m out the proof, where c ≥ 2/c7 . Then for any α ≥ c7 /2 and h ≥ 1, since log(3m′ ) ≥ log(3hm) ≥ log(3m) for any m′ ≥ hm, we have X −αnδ2 X 2 ′ e−αchm log n log 3m n,m′ ≤ ≤ 2e−αhnδn,m . e e−αcm (log n·log 3m) = −αc log n log 3m 1−e ′ ′ m ≥hm m ≥hm For the second condition of (2.5), note that for γ = 2, in order to verify 2 2 , it suffices to have hm log(3hm) ≤ h2 m log(3m), equivalently δn,hm ≤ h2 δn,m 3hm ≤ (3m)h , and hence 3h−1 ≥ h for all h ≥ 1 suffices. This is valid and hence completing the proof. Proof of Theorem 9. This is a direct consequence of Theorems 1 and 2, Lemma 23 and 24, combined with Proposition 1. B.3. Proof of Theorem 10. Lemma 25. Let n ≥ 2, then for any g ∈ F(s,m) , log N (c5 ε, {f ∈ F(s,m) : ℓn (f, g) ≤ 2ε}, ℓn ) ≤ 2 log(6/c5 ) s log(ep) + m log(en) . Proof. The proof borrows notation from the proof of Lemma 17. Further let Ss denote all subsets of {1, . . . , p} with cardinality at most s. Then the entropy in question can be bounded by p n log N (c5 ε, {f ∈ F(s,m),(S,Q) : ℓn (f, g) ≤ 2ε}, ℓn ) max s m − 1 S∈Ss ,Q∈Qm ≤ s log(ep) + m log(en) √ √ log N c5 nε, {γ ∈ Pn,(S,Q) : kγ − gk2 ≤ 2 nε}, k·k2 + max S∈Ss ,Q∈Qm n n where Pn,(S,Q) ≡ {(x⊤ i β+u(zi ))i=1 ∈ R : supp(β) = S, u is constant on the partitions of Q} is contained in a linear subspace of dimension no more than s + m. Now similar arguments as in Lemma 17 shows that the entropy term in the above display can be bounded by (s + m) log(6/c5 ), proving the claim. log(en) 5) 2 ∨ η1 s log(ep)+m . Hence we can take δn,(s,m) ≡ 4 log(6/c c7 n Lemma 26. (2.5) holds with c = 1 and γ = 1. Proof. For the first condition of (2.5), note that for any h ≥ 1, with c′ ≡ 4 log(6/c5 ) ∨ η1 in the proof, for any α ≥ c7 /2, αc′ ≥ 2 log(6/c5 ) ≥ 2 since c7 c5 ≤ 1/4, X X X 2 ′ ′ −αnδn,(s ′ ,m′ ) e = e−αc s log(ep) e−αc m log(en) (s′ ,m′ )≥(hs,hm) s′ ≥hs m′ ≥hm ′ ′ 2 −αnhδn,(s,m) ≤ (1 − e−αc )−2 e−αc h(s log(ep)+m log(en) ≤ 2e The second condition of (2.5) is easy to verify by our choices of c, γ. Lemma 27. Suppose (4.11) holds. Then (P2) in Assumption C holds. . BAYES MODEL SELECTION 37 Proof. Using notation in Lemma 25, (B.6) 2 /c3 } Πn,(s,m) {f ∈ F(s,m) : ℓ2n (f, f0,(s,m) ) ≤ δn,(s,m) −1 −1 p n 2 ≥ /c3 } Πg⊗s ⊗ḡm {f ∈ F(s,m),(S0 ,Q0) : ℓ2n (f, f0,(s,m) ) ≤ δn,(s,m) s m−1 5) ∨ η1 throughout the proof, and where f0,m ∈ F(s,m),(S0 ,Q0) . Let c′ ≡ 4 log(6/c c7 2 2 δn,s ≡ c′ s log(ep)/n and δn,m ≡ c′ m log(en)/n. To bound the prior mass of the above display from below, it suffices to bound the product of the following two terms: 2 πs ≡ Πg⊗s {β ∈ B0 (s) : βS0c = 0, ℓ2n (hβ , hβ0,s ) ≤ δn,s /2c3 } , (B.7) 2 πm ≡ Πḡm {u ∈ Um,Q0 : ℓ2n (u, u0,m ) ≤ δn,m /2c3 } . The first term equals √ √ Πg⊗s β ∈ B0 (s) : βS0c = 0, kXβ − Xβ0,s k2 ≤ nδn,s / 2c3 (B.8) δn,s 1 c √ ≥ Πg⊗s β ∈ B0 (s) : βS0 = 0, kβ − β0,s k2 ≤ · σΣ 2c3 Here the inequality follows by noting 2 kXβ − Xβ0,s k22 ≤ n(β − β0,s )⊤ Σ(β − β0,s ) ≤ nσΣ kβ − β0,s k22 , √ where σΣ denotes the largest singular value of X ⊤ X/n. Note that σΣ ≤ p since the trace for X ⊤ X/n is p and the trace of a p.s.d. matrix dominates the largest eigenvalue. The set in the last line of (B.8) is supported on RpS0 s δ 1 s √n,s ∧ 1 · and hence can be further bounded from below by τs,g vs σΣ 2c3 where vs = vol(Bs (0, 1)). Hence s δn,s 1 s ∧ 1 vs ·√ πs ≥ (τs,g ∧ 1) σΣ 2c3 (B.9) 2 s 2c3 σΣ 1 −1 ≥ exp − s log s − s log τs,g ∨ 1 − log ∨1 , 2 2 2 δn,s √ where in the last inequality we used that vs ≥ (1/ s)s . By repeating the arguments in the proof of Lemma 18, we have m 2c3 −1 ∨1 . πm ≥ exp − m log τm,g ∨ 1 − log (B.10) 2 2 δn,m Combining (B.6), (B.7), (B.9) and (B.10), we see that (B.11) 2 /c3 } Πn,(s,m) {f ∈ F(s,m) : ℓ2n (f, f0,(s,m) ) ≤ δn,(s,m) −1 −1 ∨1 ∨ 1 − m log τm,g ≥ exp −2s log(ep) − m log(en) − s log τs,g 2 2c3 σΣ 2c3 m s ∨ 1 − log ∨1 . × exp − log 2 2 2 δn,s 2 δn,m 38 Q. HAN In order that the right side of the above display bounded from below by 2 exp(−2nδn,(s,m) ), we only need to require that ) √ ( 2c3 σΣ ∨1 −s log 1 −1 δ n,s ≥ e− 2η s log(ep) , min e−s log(τs,g ∨1) , e ( −1 τm,g ∨1 min e−m log( −m log ), e √ 2c3 δn,m ) ∨1 1 ≥ e− 2η m log(en) . The first terms in the above two lines lead to (4.11). The other terms in the 2c3 c7 2 n≤ above two lines do not contribute by noting that 2c3 /δn,m ≤ 4 log(6/c 5) (1/2)n ≤ en since c3 = 1 (in Gaussian regression model) and c7 ∈ (0, 1), 2 /δ 2 ≤ σ 2 n ≤ pn ≤ p2 and η < 1/4. while 2c3 σΣ n,s Σ Proof of Theorem 10. The claim of the theorem follows by Theorems 1 and 2, Proposition 1 and Lemmas 25-27. Appendix C. Proof of auxiliary lemmas in Section 5 Proof of Lemma 9. Let Fj := {f ∈ F : jε < dn (f, f0 ) ≤ 2jε} and Gj ⊂ Fj be the collection of functions that form a minimal c5 jε covering set of Fj under the metric dn . Then by assumption \|Gj \| ≤ N (jε). Furthermore, for each g ∈ Gj , it follows by Lemma 1 that there exists some test ωn,j,g such that 2 (n) (n) sup Pf0 ωn,j,g + Pf (1 − ωn,j,g ) ≤ c6 e−c7 ndn (f0 ,g) . (C.1) f ∈F :dn (f,g)≤c5 dn (f0 ,g) Recall that g ∈ Gj ⊂ Fj , then dn (f0 , g) > jε. Hence the indexing set above contains {f ∈ F : dn (f, g) ≤ c5 jε}. Now we see that (n) Pf0 ωn,j,g ≤ c6 e−c7 nj 2 ε2 , (n) sup f ∈F :dn (f,g)≤c5 jε Pf (1 − ωn,j,g ) ≤ c6 e−c7 nj Consider the global test φn := supj≥1 maxg∈Gj ωn,j,g , then X XX 2 2 (n) (n) N (jε)e−c7 nj ε ωn,j,g ≤ c6 Pf0 φn ≤ Pf0 ≤ c6 N (ε) j≥1 e−c7 . j≥1 j≥1 g∈Gj X 2 ε2 nj 2 ε2 2 ≤ c6 N (ε)e−c7 nε · 1 − e−c7 nε 2 −1 + . On the other hand, for any f ∈ F such that dn (f, f0 ) ≥ ε, there exists some j ∗ ≥ 1 and some gj ∗ ∈ Gj ∗ such that dn (f, gj ∗ ) ≤ j ∗ c5 ε. Hence ∗ 2 2 2 (n) (n) Pf (1 − φn ) ≤ Pf 1 − ωn,j ∗,gj∗ ≤ c6 e−c7 n(j ) ε ≤ c6 e−c7 nε . The right hand side of the above display is independent of individual f ∈ F such that dn (f, f0 ) ≥ ε and hence the claim follows. BAYES MODEL SELECTION 39 Proof of Lemma 10. By Jensen’s inequality, the left side of (5.6) is bounded by (n) Pf0 Z  (n) ≤ Pf0  (n) log Z p f0 (n) pf − (n) Pf0 log (n) p f0 (n) pf (n) log p f0 (n) (n) pf − Pf0 log  (n) ≤ exp(−Cλnε2 ) · c1 Pf0 exp λ 2 dΠ(f ) ≥ C + c3 )nε − c3 n (n) p f0 (n) pf Z Z d2n (f0 , f ) dΠ(f )  dΠ(f ) ≥ Cnε2  (n) log p f0 (n) pf (n) − Pf0 log (n) p f0 (n) pf  dΠ(f ) . Using Jensen’s inequality, the last term in the right side of the above display can be further bounded by   (n) (n) Z Z p p f0 (n) (n) f0  dΠ(f ) ≤ eψκg nd2n (f0 ,f ),κΓ (λ) dΠ(f ) − Pf0 log (n) Pf0 exp λ log (n) pf pf where the last inequality follows from Fubini’s theorem and Assumption A. Now the condition on the prior Π entails that (n) Pf0 Z (n) pf (n) p f0 −(C+c3 )nε2 dΠ(f ) ≤ e ≤ c1 exp −Cλnε2 + ψκg nε2 ,κΓ (λ) . The claim follows by choosing λ > 0 small enough depending on C, κ. Proof of Proposition 3. We may assume without loss of generality that dn (f0 , f0,m ) < ∞ so that m̃ is well-defined since \|M\| = ∞ and (2.5). By definition we have δn,m̃ ≥ dn (f0 , f0,m ) and δn,m̃−1 < dn (f0 , f0,m ). In this case, the global test can be constructed via φ̃n := supm′ ∈I,m′ ≥jhm̃ φn,m′ . Then analogous to (5.9) and (5.10), for any random variable U ∈ [0, 1], (n) (C.2) Pf0,m U · φ̃n ≤ 4c6 e−(c7 /2c sup 2 f ∈Fjhm̃ :d2n (f,f0,m )≥c2 (jh)γ δn, m̃ (n) Pf (1 − φ̃n ) ≤ 2c6 e−(c7 /c (n) Pf0,m (E˜nc ) 2 )njhδ 2 n,m̃ 2 )njhδ 2 n,m̃ −C ′ ≤ c1 e , . 2 c7 njhδn, m̃ 8c3 c2 Similar to (5.11), there exists an event Ẽn with and the following is true on the event E˜n : (C.3) Z Y 2 n c7 njhδn, m̃ pf 2 2 . dΠ(f ) ≥ λn,m e− 4c2 Πn,m f ∈ Fm : d2n (f, f0,m ) ≤ c7 jhδn, m̃ /8c3 c pf0,m i=1 40 Q. HAN Repeating the reasoning in (5.12), (5.13) and (5.14) we see that (C.4) (n) 2 4 γ 2 (n) Pf0,m Πn f ∈ F : dn (f, f0,m) > c (2jh) dn (f0 , f0,m ) X (1 − φ̃n )1Ẽn 2 ≤ ec7 njhδn,m̃ /4c λn,m Πn,m Z × ≤ (· · · ) × ≤ Ce−(c7 /4c n 2 2 /8c c2 f ∈ Fm : d2n (f, f0,m ) ≤ c7 jhδn, 3 m̃ (n) f ∈F :d2n (f,f0,m )>c4 (2jh)γ d2n (f0 ,f0,m ) sup 2 f ∈Fjhm̃ :d2n (f,f0,m )≥c2 (jh)γ δn, m̃ 2 )njhδ 2 n,m̃ o Pf (1 − φ̃n ) dΠ(f ) (n) Pf (1 − φ̃n ) + Π F \ Fjhm̃ . 2 Here the third line is valid since c4 (2jh)γ d2n (f0 , f0,m ) > c4 (2jh)γ δn, m̃−1 ≥ 2 2 γ 2 2 γ 2 c (jh) δn,m̃ by the right side of (2.5), which entails δn,m̃ ≤ c 2 δn,m̃−1 . The fourth line uses (C.2) and assumption (P1), together with the fact that δn,m̃ ≥ δn,m . (5.2) follows from (C.2), probability estimate for Enc and (C.4). Acknowledgements The author is indebted to Chao Gao for his numerous suggestions that lead to a substantially improved version of the paper. 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Ref: International Conference on Artificial Neural Networks (ICANN), Springer LNCS, Vol. 9887, pp. 170–178, Barcelona, Spain, September 2016. DNN-Buddies: A Deep Neural Network-Based Estimation Metric for the Jigsaw Puzzle Problem Dror Sholomon1 , Eli (Omid) David1 , and Nathan S. Netanyahu1,2 1 arXiv:1711.08762v1 [cs.CV] 23 Nov 2017 2 Department of Computer Science, Bar-Ilan University, Ramat-Gan 52900, Israel [email protected], [email protected], [email protected] Center for Automation Research, University of Maryland, College Park, MD 20742 [email protected] Abstract. This paper introduces the first deep neural network-based estimation metric for the jigsaw puzzle problem. Given two puzzle piece edges, the neural network predicts whether or not they should be adjacent in the correct assembly of the puzzle, using nothing but the pixels of each piece. The proposed metric exhibits an extremely high precision even though no manual feature extraction is performed. When incorporated into an existing puzzle solver, the solution’s accuracy increases significantly, achieving thereby a new state-of-the-art standard. (a) (b) Fig. 1: Jigsaw puzzle before and after reassembly using our DNN-Buddies scheme in an enhanced solver. 1 Introduction Jigsaw puzzles are a popular form of entertainment, available in different variation of difficulty to challenge children, adults and even professional players. Given n × m 2 D. Sholomon, E.O. David, N.S. Netanyahu different non-overlapping tiles of an image, the objective is to reconstruct the original image, taking advantage of both the shape and chromatic information of each piece. Despite the popularity and vast distribution of jigsaw puzzles, their assembly is not trivial computationally, as this problem was proven to be NP-hard [1] [8]. Nevertheless, a computational jigsaw solver may have applications in many realworld applications, such as biology [16], chemistry [25], literature [18], speech descrambling [27], archeology [2] [15], image editing [5], and the recovery of shredded documents or photographs [3,7,14,17]. Regardless, as noted in [11], research of the topic may be justified solely due to its intriguing nature. Recent years have witnessed a vast improvement in the research and development of automatic jigsaw puzzle solvers, manifested in both puzzle size, solution accuracy, and amount of manual human intervention required. In its most basic form, every puzzle solver requires some function to evaluate the compatibility of adjacent pieces and a strategy for placing the pieces as accurately as possible. Most strategies are greedy and rely heavily on some “trick” to estimate whether two pieces are truly adjacent (e.g. two pieces that are each the most compatible piece from all pieces to one another, four pieces that form a loop where each pair’s compatibility is above a threshold, etc). Such heuristics were dubbed an “estimation metric” in [20], as they allow estimating the adjacency correctness of two pieces without knowing the correct solution. The majority of recent works focused on devising elaborate, hand-crafted compatibility functions and high-precision estimation metrics. Despite the proven effectiveness of neural networks in the field of computer vision, no attempt has been made to automatically devise a high-precision estimation metric for the jigsaw puzzle problem. This might be due to the highly imbalanced nature of the puzzle problem, as in each n × m puzzle, there are O(n × m) matching piece-pairs and O(n2 × m2 ) possible mismatching ones. In this paper we propose a novel estimation metric relying on neural networks. The proposed metric achieves extremely high precision despite the lack of any manually extracted features. The proposed metric proves to be highly effective in real-world scenarios. We incorporated the metric in our GA-based solver, using no hand-crafted sophisticated compatibility measure and experimented with the currently known challenging benchmarks of the hardest variant of the jigsaw puzzle problem: non-overlapping, (28 × 28) square pieces (i.e. only chromatic information is available to the solver) where both piece orientation and puzzle dimensions are unknown. The enhanced solver proposed sets a new state-of-the-art in terms of the accuracy of the solutions obtained and the number of perfectly reconstructed puzzles. 2 Previous Work Jigsaw puzzles were first introduced around 1760 by John Spilsbury, a Londonian engraver and mapmaker. Nevertheless, the first attempt by the scientific community to computationally solve the problem is attributed to Freeman and Garder [9] who in 1964 presented a solver which could handle up to nine-piece problems. Ever since then, the research focus regarding the problem has shifted from shape-based to merely color-based solvers of square-tile puzzles. In 2010 Cho et al. [4] presented Deep Neural Network Based Estimation Metric for the Jigsaw Puzzle Problem 3 a probabilistic puzzle solver that could handle up to 432 pieces, given some a priori knowledge of the puzzle. Their results were improved a year later by Yang et al. [26] who presented a particle filter-based solver. Furthermore, Pomeranz et al. [20] introduced that year, for the first time, a fully automated square jigsaw puzzle solver that could handle puzzles of up to 3,000 pieces. Gallagher [10] has further advanced this by considering a more general variant of the problem, where neither piece orientation nor puzzle dimensions are known. Son et al. [24] improved the accuracy of the latter variant using so-called “loop-constraints”. Palkin and Tal [19] further improved the accuracy and handled puzzles with missing pieces. Sholomon et al. [21] presented a genetic algorithm (GA)-based solver for puzzles of known orientation which was later generalized to other variants [22,23]. 2.1 Compatibility Measures and Estimation Metrics As stated earlier, most works focus on the compatibility measure and an estimation metric. A compatibility measure is a function that given two puzzle piece edges (e.g. the right edge of piece 7 versus the upper edge of piece 12) predicts the likelihood that these two edges are indeed placed as neighbors in the correct solution. This measure applies to each possible pair of piece edges. The estimation metric, on the other hand, predict whether two piece edges are adjacent but may not apply to many possible pairs. Following is a more detailed review of the efforts made so far in the field. Cho et al. [4] surveyed four compatibility measures among which they found dissimilarity the most accurate. Dissimilarity is the sum (over all neighboring pixels) of squared color differences (over all color bands). Assuming pieces xi , xj are represented in some three-dimensional color space (like RGB or YUV) by a K × K × 3 matrix, where K is the height/width of a piece (in pixels), their dissimilarity, where xj is to the right of xi , for example, is v uK 3 uX X (xi (k, K, cb) − xj (k, 1, cb))2 , (1) D(xi , xj , r) = t k=1 cb=1 where cb denotes the color band. Pomeranz et al. [20] also used the dissimilarity measure but found empirically that using the (Lp )q norm works better than the usual L2 norm. Moreover, they presented the high-precision best-buddy metric. Pieces xi and xj are said to bestbuddies if ∀xk ∈ P ieces, C(xi , xj , R1 ) ≥ C(xi , xk , R1 ) and (2) ∀xp ∈ P ieces, C(xj , xi , R2 ) ≥ C(xj , xp , R2 ) where P ieces is the set of all given image pieces and R1 and R2 are “complementary” spatial relations (e.g. if R1 = right, then R2 = left and vice versa). Gallagher [10] proposed yet another compatibility measure, called the Mahalanobis gradient compatibility (MGC) as a preferable compatibility measure to those 4 D. Sholomon, E.O. David, N.S. Netanyahu used by Pomeranz et al. [20]. The MGC penalizes changes in intensity gradients, rather than changes in intensity, and learns the covariance of the color channels, using the Mahalanobis distance. Also, Gallagher suggested using dissimilarity ratios. Absolute distances between potential piece edge matches are sometimes not indicative (for example in smooth surfaces like sea and sky), so considering the absolute score, divided by the second-best score available seems more indicative. Son et al. [24] suggested “loop-constraints”, four or more puzzle piece edges where the compatibility ratio between each pair is in the top ten among all possible pairs of piece edges in the given puzzle. Palkin and Tal [19] proposed a greedy solver based on an L1 -norm asymmetric dissimilarity and the best-buddies estimation metric. 3 3.1 DNN-Buddies Motivation We propose a novel estimation metric called “DNN-Buddies”. Our goal is to obtain a classifier which predicts the adjacency likelihood of two puzzle piece edges in the correct puzzle configuration. Note that despite the exponential nature of the problem (as there are O((nm)!) possible arrangements of the pieces, taking into account rotations), the problem can be solved theoretically by assigning correctly, in a consecutive manner, n × m − 1 piece-edge pairs. (This is reminiscent of finding a minimal spanning tree, as noted by [10].) Hence, the classifier’s precision is of far greater importance than its recall. A classifier with perfect precision and a recall of n×m−1 1 n×m−1 = < all possible matches 4 × (n × (m − 1) + (n − 1) × m) 8 (3) might achieve a perfect solution by itself. 3.2 Challenges A straight-forward solution might have been to train a neural network against matching-pairs vs. non-matching ones. However, the issue of a jigsaw puzzle piece matching is of an imbalanced nature. In each n × m puzzle, there are O(n × m) matching pairs of piece edges and O(n2 × m2 ) possible nonmatching ones. A thorough review on the challenges and tactics to avoid them can be found in [13]. The trivial approach of random or uninformed undersampling, i.e. randomly choosing the required number of nonmatching pairs leads to a low-precision and highrecall metric, the very opposite of the goal set beforehand. We believe that the reason for this shortcoming is that there exist many “easy-to-spot” mismatches but only a handful of “hard-to-spot” ones. Thus, we resort to informed undersampling, choosing a subset of “good” mismatching pairs according to some criterion. Nevertheless, we avoid using any manual feature selection or other sophisticated image-related means. In the jigsaw puzzle domain, similarly to many other problem domains, the solver does not actually try to reassemble the original image (as this problem is Deep Neural Network Based Estimation Metric for the Jigsaw Puzzle Problem 5 not mathematically defined), but rather tries solving a “proxy problem” which is to achieve an image whose global overall score between abutting-edges is minimal. Thus, we choose using the compatibility measure as the undersampling criterion. 3.3 Neural Network Training For training and cross-validation, we use the 2,755 images of size 360 × 480 pixels from the IAPR TC-12 Benchmark [12]. Each image is first converted to YUV space followed by the normalization of each channel separately (via z-score normalization). Next, each (puzzle) image is divided to 12 × 17 tiles, where each tile is of size 28 × 28 pixels (as in all previous works); finally, we create a balanced set of positive and negative samples of puzzle-piece pairs, using informed undersampling as will be described below. In the end, we obtain a balanced set of 970,224 pairs overall. To balance our dataset, we use the most basic compatibility score which is the dissimilarity between two piece-edges in the YUV color-space, as described in Eq. 1, as an undersampling criterion. For each puzzle piece edge xi,j (i = 1..n×m, j = 1..4), we find its most compatible piece edge xk1,l1 and its second most compatible piece edge xk2,l2 . If the pair of edges xi,j −xk1,l1 is indeed adjacent in the original image, we add this pair to the pool of positively-labeled samples and toss the pair xi,j − xk2,l2 to the pool of negatively-labeled samples. Otherwise, xi,j − xk1,l1 is added to the negatively-labeled samples and the other pair is discarded. The latter is done to avoid training the network on adjacent pieces which happen to be vastly different due to a significant change of the image scenery in the corresponding region. In other words, we restrict our interest to highly compatible piece edges that are indeed adjacent. Since this method leads to more negative samples than positive ones, we eventually randomly throw some negative samples to balance out the set. From each image pair we extract the two columns near the edge, i.e. the column of abutting pixels in each edge and the one next to it. This results is an input of size (28 × 4 × 3 =) 336 pixels. We use a feed-forward neural network (FFNN) of five fully connected layers of size 336, 100, 100, 100, and 2. The output is a softmax layer containing two neurons. We expect (0, 1) for matching pairs and (1, 0) otherwise. The activation used in all layers is the rectified linear unit (ReLU) function, i.e. f (x) = max(0, x). Figure 2 depicts the network’s structure. We trained the network in a supervised manner using Stochastic Gradient Descent that minimizes the negative log likelihood of the error for 100 iterations. The resulting network reaches 95.04% accuracy on the training set and 94.62% on a held-out test set. All dataset preparation and network training was performed using Torch7 [6]. 4 Experimental Results For each piece edge xi,j (i = 1..n × m, j = 1..4), if its most compatible piece edge xk,l is classified positively using the DNN-Buddies network, we define xk,l to be xi,j ’s DNN-buddy piece edge. Note that each piece edge can have only a single DNN-buddy; 6 D. Sholomon, E.O. David, N.S. Netanyahu Fig. 2: Architecture of our DNN-Buddies scheme. also, some pieces might not have a DNN-buddy at all (if the most compatible piece is not classified as one by the DNN-Buddies network). First, we evaluate the precision of the proposed metric, i.e. how many DNNbuddies are indeed adjacent in the original image. Using the well known dataset presented by Cho et al. [4] of 20 432-piece puzzles, we obtained a precision of 94.83%. Next, we incorporated the estimation metric (due to the proposed DNN-Buddies scheme) into the GA-based solver proposed by us previously [23]. Unfortunately, due to lack of space, no self-contained review of genetic algorithms and the proposed method can be included in this paper. Nevertheless, the modification required with respect to the existing GA framework is rather simple; if a DNN-buddy pair appears in one of the parents, assign this pair in the child. Figure 3 describes the modified crossover operator in the GA framework according to the above (see Step 2, which includes the new DNN-buddy phase). Until 1. 2. 3. 4. 5. (n − 1) relative relations are assigned do Try assigning all common relative relations in the parents. Try assigning all DNN-buddy relative relations in the parents. Try assigning all best-buddy relative relations in the parents. Try assigning all existing most-compatible relative relations. Try assigning random relative relations. Fig. 3: Crossover overview We ran the augmented solver on the 432-piece puzzle set and on the two additional datasets proposed by Pomeranz et al. [20] of 540- and 805- piece puzzles. We evaluated our results according to the neighbor comparison which measures the fraction of correct neighbors and the number of puzzles perfectly reconstructed for each set. Deep Neural Network Based Estimation Metric for the Jigsaw Puzzle Problem 7 Table 1 presents the accuracy results of the same solver with and without the DNN-Buddies metric. For each dataset we achieve a considerable improvement in the overall accuracy of the solution, as well as the number of perfectly reconstructed puzzles. Moreover, our enhanced deep neural network-based scheme appears to outperform the current state-of-the-art results, as it yields accuracy levels of 95.65%, 96.37% and 95.86%, which surpass, respectively, the best results known of 95.4% [19], 94.08% and 94.12% [23]. # of Pieces 432 540 805 GA Neighbor Perfect 94.88% 94.08% 94.12% 11 8 6 Our (GA + DNN-Buddies) Neighbor perfect 95.65% 96.37% 95.86% 12 11 8 Table 1: Comparison of our accuracy results with and without the new DNN-Buddies estimation metric. 5 Conclusions In this paper we presented the first neural network-based estimation metric for the jigsaw puzzle problem. Unlike previous methods, no manual feature crafting was employed. The novel method exhibits high precision and when combined with a real-world puzzle solver, it significantly improves the solution’s accuracy to set a new state-of-the art standard. References 1. Altman, T.: Solving the jigsaw puzzle problem in linear time. 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arXiv:1709.10154v1 [cs.SY] 28 Sep 2017 Finite-Time Distributed Linear Equation Solver for Minimum l1 Norm Solutions Jingqiu Zhou, Wang Xuan, Shaoshuai Mou, and Brian. D. O. Anderson October 2, 2017 Abstract This paper proposes distributed algorithms for multi-agent networks to achieve a solution in finite time to a linear equation Ax = b where A has full row rank, and with the minimum l1 -norm in the underdetermined case (where A has more columns than rows). The underlying network is assumed to be undirected and fixed, and an analytical proof is provided for the proposed algorithm to drive all agents’ individual states to converge to a common value, viz a solution of Ax = b, which is the minimum l1 norm solution in the underdetermined case. Numerical simulations are also provided as validation of the proposed algorithms. 1 Introduction A significant amount of effort in the control community has recently been given to distributed algorithms for solving linear equations over multi-agent networks, in which each agent only knows part of the equation and controls a state vector that can be looked at as an estimate of the solution of the overall linear equations [1–5]. Numerous extensions along this direction include achieving solutions with the minimum Euclidean norm [6, 7], elimination of the initialization step [8], reduction of state vector dimension by utilizing the sparsity of the linear equation [9] and achieving least square solutions [10–15]. All these algorithms yield asymptotic convergence, but require an infinite number of sensing or communication events. ∗ This work was supported by a funding from Northrop Grumman Cooperation. J. Zhou, X. Wang and S. Mou are with the School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN 47906 USA (e-mail: [email protected], [email protected], [email protected]). B. D. O. Anderson is with Hangzhou Dianzi University, Hangzhou, China, The Australian National University and Data-61 CSIRO (formerly NICTA), Canberra, ACT 2600 Australia, e-mail: [email protected]; his work is supported by Data-61, CSIRO, and by the Australian Research Council’s Discovery Projects DP-130103610 and DP-160104500. Corresponding Author: Shaoshuai Mou. 1 ∗ Solutions to underdetermined linear equations with the minimum l1 norm are perhaps the most important in many engineering applications including earthquake location detection [16] , analysis of statistical data [17], solving biomeganetic inverse problems [18], and so on. One most intriguing case among these applications is compressive sensing, which enables transmission of sparse data in a very efficient way [19]. The decoding process of compressive sensing requires solving of linear equations with a minimum number of non-zero entries of the solution vectors, which, however, is an NP-hard problem and usually computationally costly [20]. Thus researchers usually turn to achieve solutions with minimum l1 norm instead for which the function to be minimized is convex [21, 22]. Most existing results for achieving minimum l1 norm solutions are based on the idea of Lasso [23] including Alternating Direction Method of Multipliers (ADMM) [24], the Primal-Dual Interior-Point Method [25, 26], Gradient Projection Methods [27], Homotopy Methods [28], Iterative Shrinkage-Thresholding Methods [29] and Proximal Gradient Methods [30]. Interesting as these results are, they either achieve limited accuracy dominated by a threshold parameter, involve solving a much larger linear equation, or lead to high computational complexity. In this paper we aim to develop distributed algorithms for multi-agent networks to achieve in finite time a solution of linear equations or, in the underdetermined case, one with the minimum l1 norm. By distributed is meant that each agent only knows part of the overall linear equation and can communicate with only its nearby neighbors. The problem of interest is formulated in Section 2. We introduce in section 3 the concepts to be employed in the paper including Filippov set-valued maps, Filippov solutions, generalized Lie derivatives, based on which a preliminary result is achieved. In Section 4, we will first propose a distributed algorithm to drive all agents’ state vectors to converge in finite time to the same solution of the overall linear equations. Then we present a centralized update for achieving a solution with the minimum l1 norm. Motivated by the projection-consensus flow proposed in [13] and the finite-time gradient flow for consensus devised in [31–36], we utilize a combination of the proposed distributed linear equation solver and the proposed centralized algorithm for minimum l1 -norm solutions to develop a distributed linear equation solver for achieving the minimum l1 norm solution, which is shown to converge in finite time. We provide simulations in Section 5 and concluding remarks in Section 6. Notation: Let r denote an arbitrary positive integer. Let 1r denote the vector in Rr with all entries equal to 1s. Let Ir denote the r × r identity matrix. We let col {A1 , A2 , · · · , Ar } be a stack of matrices Ai possessing the same number of columns with the index in a top-down ascending order, i = 1, 2, · · · , r. Let diag {A1 , A2 , · · · , Ar } denote a block diagonal matrix with Ai the ith diagonal block entry, i = 1, 2, · · · , r. By M > 0 and M ≥ 0 are meant that the square matrix M is positive definite and positive semi-definite, respectively. By M 0 is meant the transpose of a matrix M . Let ker M and image M denote the kernel and image of a matrix M , respectively. Let ⊗ denote the Kronecker product. Let k · k1 denote the l1 norm of a vector in Rr . 2 2 Problem Formulation Consider a network of m agents, i = 1, 2, ..., m; inside this network, each agent can observe states of certain other agents called its neighbors. Let Ni denote the set of agent i’s neighbors. We assume that the neighbor relation is symmetric, that is, j ∈ Ni if and only if i ∈ Nj . Then all these neighbor relations can be described by an m-node-m̄-edge undirected graph G such that there is an undirected edge connecting i and j if and only if i and j are neighbors. In this paper we only consider the case in which G is connected, fixed and undirected. Suppose that each agent i knows Ai ∈ Rni ×n and bi ∈ Rni and controls a state vector yi (t) ∈ Rn . Then all these Ai and bi can be stacked into an overall equation Ax = b, where A = col {A1 , A2 , · · ·, Am }, b = col {b1 , b2 , · · ·, bm }. Without loss of generality for the problems of interest to us, we assume A to have full row-rank. Let x∗ denote a solution to Ax = b (and in the underdetermined case, x∗ is not unique); and let x̄∗ denote its minimum l1 -norm solution, that is, x̄∗ = arg min kxk1 (1) Ax=b ∗ (In the non-singular A case, x and x̄∗ necessarily coincide) The problem of interest in this paper is to develop distributed algorithms for each agent i to update its state vector yi (t) by only using its neighbors’ states such that all yi (t) to converge in finite time to a common value x∗ and if desired in the nonsquare case the value x̄∗ . 3 Key Concepts and Preliminary Results Before proceeding, we introduce some key concepts and preliminary results for future derivation and analysis. The key references for the background we summarize are [37] and [38]. 3.1 Filippov set-valued maps and Filippov Solutions By a Filippov set-valued map F [f ] : Rr → B ⊂ Rr associated with a function f : Rr → Rr is meant \ \ co{f (B(x, δ))/S} (2) F [f ](x) , δ>0 µ(S)=0 Here B(x, δ) stands for the open ball on Rr , whose center is at x and has a radius of δ; µ(S) denotes the Lebesgue measure of S; and co stands for the convex closure. Let sgn (x) : Rr → Rr be a function with the kth entry, k = 1, 2, ..., r, defined as  (x)k > 0;  1, −1, (x)k < 0; (sgn (x))k = (3)  0, (x)k = 0. 3 It follows that the Filippov set-valued map F [sgn ](x) for x ∈ Rr is defined entrywise as:  1 (x)k > 0  [−1, 1] (x)k = 0 (F [sgn ](x))k = (4)  −1 (x)k < 0 for k = 1, 2, ..., r. Note that even if (x)i = (x)j = 0, the ith and jth entries of a vector in F [sgn ](x) may not necessarily be equal since each of them could be chosen as arbitrary values in the interval [−1, 1]. From the definition of F [sgn ](x), one can verify that q 0 x = kxk1 , ∀q ∈ F [sgn ](x) (5) While for any w ∈ Rr , there holds q 0 w ≤ kwk1 (6) . By a Filippov solution for ẋ ∈ F [f ](x) is meant a Caratheodory solution x(t) such that ẋ ∈ F [f ](x) for almost all t, x(t) is absolutely continuous and can be written in the form of an indefinite integral. The following two lemmas treat existence of such a Filippov solution. Lemma 1 ( Proposition 3 in [37]) If f : Rr → Rr is measurable and locally bounded, then for any initial point x0 ∈ Rr , there exists a Filippov solution1 for ẋ ∈ F [f ](x). Lemma 2 ( Theorem 8 in page 85 of [38] ) Let a vector-valued f (t, x) be defined almost-everywhere in the domain G of time space (t, x). With f (t, x) measurable and locally bounded almost-everywhere in an open domain G, let \ \ F [f ](t, x) , co{f (t, B(x, δ)/S)}. (7) δ>0 µ(S)=0 Then for any point (t0 , x0 ) ⊂ G, there exists a Filippov solution of ẋ ∈ F [f ](t, x) with x(t0 ) = x0 . Note that Lemma 2 establishes the existence of a solution for time-varying systems. This is more general than Lemma 1, which only guarantees the existence of solutions to time-invariant systems. 3.2 Generalized Gradients and Generalized Lie Derivatives For a locally Lipschitz function w : Rr → R, the generalized gradient of w is ∂w(x) , co{ lim ∇w(xi ) : xi → x, xi ∈ / S ∪ Ωw } i→∞ 1 There is no implication that the solution exists on an infinite interval 4 (8) where S ⊂ Rr is an arbitrarily chosen set of measure zero, Ωw denotes the set of points at which w is not differentiable, and co denotes convex hull. Specially, for the function \|\|x\|\|1 , one computes the kth element of its generalized gradient to be:  1 xk > 0  [−1, 1] xk = 0 (∂\|\|x\|\|1 )k = (9)  −1 xk < 0 It follows from this and the definition of F [sgn ](x) in (4) that F [sgn ](x) = ∂kxk1 (10) For a set-valued map F : Rr → B(Rr ), the generalized Lie derivative of w is defined as L̃F w(x) = {q ∈ R : there exists α ∈ F(x) such that ∀β ∈ ∂w(x), q = β 0 α} (11) The above definition of generalized Lie derivative implies that for each α in F(x), we check if the inner product β 0 α is a fixed value for all β ∈ ∂w(x). If so, this inner product is an element in L̃F w(x), but note that the set L̃F w(x) may be empty. Moreover, for locally Lipschitz and regular(see [39],p.39 and [40], p.3 for detailed discussion of regular function2 ) functions w(x), one has the following lemma: Lemma 3 (Proposition 10 in [37]) Let x : [0, t1 ] → Rr be a solution for ẋ ∈ F(x(t)), where F is any set-valued map. Let w(x) : Rr → R be locally Lipschitz and regular. Then w(x(t)) is differentiable at almost all t ∈ [0, t1 ]; ∈ L̃F w(x(t)) for almost all t ∈ [0, t1 ]. The derivative of w(x(t)) satisfies dw(x(t)) dt Lemma 3 guarantees the existence of generalized Lie derivatives for functions that are locally Lipschitz and regular. If one focuses on a specific solution, one can show that α in (11) is a special vector as summarized in the following lemma. Lemma 4 (See Proof of Lemma 1 in [40]) Let x(t) denote a specific solution of a differential enclosure. Suppose w(x) is locally Lipschitz and regular. Let I ⊂ [0, ∞) denote the time interval for which ẋ(t) exists. Then dw(x(t))) = β 0 ẋ(t) dt where β is any vector in ∂w(x). 2 That a function w(x) : Rn → R is called regular at x ∈ Rn if 0 (x, v). 1. for all v ∈ Rn there exists the usual right directional derivative w+ 0 (x, v) = w o (x, v). 2. for all v ∈ Rn , w+ 5 (12) 3.3 Preliminary Results For any positive semi-definite matrix M ∈ Rr×r , M 6= 0 one can define Φ(M ) = {q ∈ Rr \| ∃φ ∈ F [sgn (q)], M φ(q) = 0} (13) and its compliment Φc (M ) = {q ∈ Rr \| ∀φ ∈ F [sgn (q)], M φ(q) 6= 0}. (14) We impose a further requirement on M , namely that Φc (M ) is nonempty which can be easily ensured. Let Λ(M ) = {φ\|φ ∈ F [sgn ](q), q ∈ Φc (M )} (15) Now F [sgn ](q) is a closed set for any fixed q; also note that F [sgn ](q) can only be one of a finite number of different sets; hence it is easy to check for a given M whether Φc (M ) is nonempty, (and in a later use of the result, it proves easy to check). It further follows that Λ(M ) is also a closed set. Consequently, the continuous function f (φ) = φ0 M φ has a nonzero minimum on Λ(M ). We denote λ(M ) = min f (φ). (16) φ∈Λ(M ) From the definition of Φc (M ) and Λ(M ), one has λ(M ) > 0. To summarize, one has the following lemma: Lemma 5 For any nonnegative-definite matrix M , we let Φc (M ), Λ(M ) and λ(M ) defined as above. Suppose that Φc (M ) is nonempty. Then λ(M ) is a positive constant. For the m-node-m̄-edge graph G, we label all its nodes as 1, 2, ..., m and all its edges as 1, 2, ..., m̄. Assign an arbitrary direction to each edge in G. Then the incidence matrix of G denoted by H = [hik ]m×m̄ is defined as follows  i is the head of the kth edge;  1, −1, i is the tail of the kth edge; hik = (17)  0, otherwise. Since G is connected, then ker H 0 is the span of 1m [41]. Moreover, one has the following lemma: Lemma 6 Suppose A has full-row rank and G is connected. Let P̄ = diag {P1 , P2 , ..., Pm } where each Pi is the projection matrix to ker Ai . Let H̄ = H ⊗ In with H the incidence matrix of G. Then one has image H̄ ∩ ker P̄ = 0 (18) image H̄ 0 ∩ Φ(H̄ 0 P̄ H̄) = 0 (19) and 6 Proof of Lemma 6: Let u be a vector such that P̄ H̄u = 0. (20) The vector v = H̄u lies in image H̄ and ker P̄ , and we will show it is zero to establish (18). Define Ā = diag {A1 , A2 , ..., Am }, which is full row rank since A has full row rank. Then P̄ = I − Ā0 (ĀĀ0 )−1 Ā. It follows from (20) that H̄u = Ā0 (ĀĀ0 )−1 ĀH̄u (21) Multiplying both sides of the above equation by 10m ⊗ In , one has 0 = (1m ⊗ In )0 Ā0 (ĀĀ0 )−1 ĀH̄u (22) Since (10m ⊗ In0 )Ā0 = A0 there holds 0 = A0 (ĀĀ0 )−1 ĀH̄u (23) Since A0 is full column rank, one has 0 = (ĀĀ0 )−1 ĀH̄u (24) From (21) and equation (24) one has H̄u = 0 (25) Furthermore, we notice (20) and conclude that (18) is true. For any q ∈ image H̄ 0 ∩ Φ(H̄ 0 P̄ H̄), there exists a vector p such that q = H̄ 0 p (26) and a vector φ ∈ F [sgn ](q) such that H̄ 0 P̄ H̄φ = 0. Note that P̄ is a projection matrix; then P̄ H̄φ = 0. (27) From (18) one then has H̄φ = 0. (28) From φ ∈ F [sgn ](q) and (5), one has \|\|q\|\|1 = φ0 q which together with (26) and (28) implies \|\|q\|\|1 = 0. Then one has q = 0 and (19) is true. Now consider the system ẋ ∈ −M F [sgn ](x) (29) with any positive semi-definite matrix M ∈ Rr×r . The existence of a Filippov solution to (29) can be guaranteed by Lemma 1. The existence interval is 7 t ∈ [0, ∞) because of the global bound on the right-hand side to (29). Let x(t) denote such a Filippov solution for any given x(0). Note that the function kxk1 is locally Lipschitz and regular(The word was introduced early with reference). Then by Lemma 3, the time derivative of kx(t)k1 exists for almost all t ∈ [0, ∞) and is in the set of generalized Lie derivatives. In other words, there exists a set I = [0, ∞) \ T with T of Lebesgue measure 0 such that dkx(t)k1 exists for all t ∈ I dt (30) Proposition 1 Let x(t) denote a Filippov solution to (29) for any given x(0) ∈ Rr . Then • dkx(t)k1 ≤ 0, dt t ∈ I; (31) • there exists a finite time T such that x(T ) ∈ Φ(M ); (32) • if it is further true that dkx(t)k1 = 0, dt t ∈ [T, ∞) \ T , (33) one has x(t) = x(T ) t ∈ [T, ∞) (34) Remark 1 Note that the right-hand side of (29) is a projection of the gradient flow of the potential function kx(t)k1 . It is also standard that the gradient law of a real analytic function with a lower bound converges to a single point which is both a local minimum and a critical point of the potential function [42]. However, if the real analytic property does not hold, the convergence result may fail. Indeed, the function kx(t)k1 here is obviously not real analytic, and one cannot immediately assert that (29) will drive kx(t)k1 to its minimum, not to mention the finite time result in Proposition 1. Thus Proposition 1 is nontrivial and will serve as the foundation for devising finite-time distributed linear equation solvers in this paper. Proof of Proposition 1: By (12) in Lemma 4, one has d\|\|x(t)\|\|1 = β 0 ẋ, dt t∈I (35) holds for all β ∈ ∂kx(t)k1 . It follows from ẋ ∈ −M F [sgn ](x) in (29) that at each t there exists a γ(t) ∈ F [sgn ](x) such that ẋ = −M γ(t) 8 (36) Since β could be chosen as any vector in ∂kx(t)k1 , γ(t) ∈ F [sgn ](x) and ∂kxk1 = F [sgn ](x) from (10), one could choose β = γ(t) in (35), which together with (36) leads to dkx(t)k1 = −γ(t)0 M γ(t), t ∈ I (37) dt Note that M is positive semi-definite. Thus (31) is true. We use the method of contradiction to prove that there exists a finite time T such that x(T ) ∈ Φ(M ), T ∈ I. Suppose such a finite time does not exist. One has x(t) ∈ Φc (M ), t ∈ I Then dkx(t)k1 ≤ −λ(M ), t ∈ I dt where λ(M ) is as defined in (16). It follows that kx(t)k1 ≤ kx(0)k1 − λ(M )t, t ∈ [0, ∞). This contradicts the fact that kx(t)k1 ≥ 0, t ∈ [0, ∞) since λ(M ) is a positive constant by Lemma 5. Thus there exists a finite time T such that x(T ) ∈ Φ(M ). From the assumption (33), the fact that M is semipositive definite and (37), one has M γ(t) = 0, t ∈ [T, ∞) \ T . (38) It follows from this and (36) that ẋ(t) = 0, t ∈ [T, ∞) \ T . By integration of ẋ(t) from T to ∞, one has x(t) = x(T ), completes the proof. 4 t ∈ [T, ∞). This Algorithms and Main Results In this section, we study three related problems: finite time distributed solution of Ax = b, centralized solution of Ax = b achieving a minimum l1 norm, and then finally, using ideas from the first two, and finding a distributed algorithm achieving the minimum l1 -norm solution of Ax = b in finite time. 4.1 Finite-Time Distributed Linear Equation Solver In this subsection, we will present a distributed update for achieving a solution x∗ to Ax = b in finite time. Of course, A is assumed to have full row rank, but not necessarily be square. Recall that distributed linear equation solvers based on the agreement principle require each agent i to limit its update subject to its own constraint Ai x = bi while seeking consensus with its neighbors [1]. Such an agreement principle in continuous-time systems can be 9 achieved by a projection-consensus flow which at each agent i projects a function of its neighbors’ states to the subspace defined by its linear equation [13]. By choosing the finite-time gradient flow for consensus developed by [36] within the projection-consensus flow, we are led to postulate the following update for each agent i: X ẏi = −Pi φij , Ai yi (0) = bi (39) j∈Ni where φij ∈ F [sgn ](yi − yj ) and Pi denote the projection matrix to the kernel of Ai . Note the special property that F [sgn ](0) is a point which can be chosen arbitrarily from the interval [−1, 1]. Generally speaking, different agents may have different choices for F [sgn ](0). Before proceeding, we make the following assumption on coordinations between neighbor agents: Assumption 1 Each pair of neighbor agents i and j takes the choice of φij and φji when yi = yj such that φij = −φji (40) Under Assumption 1 and the definition of F [sgn ](x), one always has φij = −φji no matter whether yi is equal to yj or not. Let y = col {y1 , y2 , · · · , ym }, P̄ = diag{P1 , P2 , · · · , Pm } and H̄ = H ⊗ In with H the incidence matrix of G. Then from (39) we have ẏ ∈ −P̄ H̄F [sgn ](H̄ 0 y) (41) By Lemma 1 there exists a Filippov solution to system (41) for t ∈ [0, ∞), we denote this solution by y(t). By Lemma 3, there exists a set I = [0, ∞) \ T 1 exists for all t ∈ I. Moreover, with T of Lebesgue measure 0 such that dky(t)k dt one has the following main theorem, which establishes existence of a limiting consensus solution but for the moment leaves unspecified the L1 -optimality. Theorem 1 Under Assumption 1 and the updates (39), and with A assumed to have full row rank, all yi (t), i = 1, 2, ..., m, converge to be a single solution to Ax = b in finite time. Proof of Theorem 1. Since Ai yi (0) = bi and ẏi is in the kernel of Ai , one has Ai yi (t) = bi for all t ∈ [0, ∞). Then to prove Theorem 1, it is sufficient to show that all yi reach consensus in finite time. Note that G is connected and ker H 0 is spanned by the vector 1m . Then H̄ 0 y = 0 if and only if all yi are equal. Thus to prove Theorem 1, it suffices to prove z(t) = H̄ 0 y(t) converges to 0 in finite time. By multiplying both sides of (41) by H̄ 0 , one has ż ∈ −H̄ 0 P̄ H̄F [sgn ](z) (42) By Proposition 1, we have dkz(t)k1 ≤ 0, t ∈ I; dt 10 (43) and there exists a finite time T ∈ I such that z(T ) ∈ Φ(H̄ 0 P̄ H̄). (44) From this, we know the fact that z(T ) = H̄ 0 y(T ) so that z(T ) ∈ image H̄ 0 , and recalling (19) in Lemma 6, one has z(T ) = 0, which by (43) implies kz(t)k1 = 0, t ∈ [T, ∞). It follows that z(t) = 0, t ∈ [T, ∞). This completes the proof. 4.2 Finite-Time Centralized Update for Minimum l1 -norm Solution In this subsection, we will propose a centralized update for achieving the minimum l1 -norm solution to Ax = b. By noting that kxk1 is convex, we conceive of using a negative gradient flow of \|\|x\|\|1 subject to x remaining on the manifold Ax = b in order to achieve x̄∗ = arg minAx=b kxk1 . This leads us to the following update: ẏ ∈ −P F [sgn ](y), Ay(0) = b. (45) where P denotes the projection matrix onto the kernel of A. Again by Lemma 1 one has there exists a Filippov solution to system (45) for t ∈ [0, ∞), which we denote by y(t). By Lemma 3, there exists a set I = [0, ∞) \ T with T of 1 measure 0 such that dky(t)k exists for all t ∈ I. Moreover, we have the following dt main theorem: Theorem 2 With A of full row rank, the Filippov solution y(t) to (45) converges in finite time to a constant, which is the minimum l1 -norm solution to Ax = b. Proof of Theorem 2: By Proposition 1, one has dky(t)k1 ≤ 0, t ∈ I; dt (46) and there exists a finite time T ∈ I such that y(T ) ∈ Φ(P ). Then there exists a vector φ ∈ F [sgn ](y(T )) such that P φ = 0. This and (5) imply φ0 y(T ) = ky(T )k1 (47) 11 Moreover, let ȳ denote any solution to Ax = b. Recall (6), there holds φ0 ȳ ≤ kȳk1 (48) Since φ ∈ ker P , one has φ ∈ image A0 . This implies that there exists a vector q such that q 0 A = φ0 (49) From Ay(T ) = b = Aȳ and (47)-(49), one has ky(T )k1 = φ0 y(T ) = q 0 Ay(T ) = q 0 Aȳ = φ0 ȳ ≤ kȳk1 (50) where ȳ is any solution to Ax = b. Thus y(T ) is a minimum l1 norm solution to Ax = b. This and (46) implies \|\|y(t)\|\|1 reaches its minimum value subject to Ay(t) = b for t ∈ [T, ∞) \ T . Thus dky(t)k1 = 0, dt t ∈ [T, ∞) \ T (51) which satisfies the assumption (33) in Proposition 1 again. Then y(t) = y(T ), t ∈ [T, ∞). Thus y(t) is the minimum l1 -norm solution to Ax = b for t ∈ [T, ∞). This completes the proof. 4.3 Finite-Time Distributed Update for Minimum l1 -norm Solutions In this subsection we will develop a distributed update for a multi-agent network to achieve the minimum l1 -norm solution to Ax = b in finite time. Motivated to study a combination of the finite-time distributed linear equation solver in (39) and the finite-time centralized update for minimum l1 -norm solutions in (45), we propose the following update for agent i, i = 1, 2, ..., m: X ẏi = −k(t)Pi φi − Pi φij (52) j∈Ni where φi ∈ F [sgn ](yi ), φij ∈ F [sgn ](yi − yj ), with φij = −φji in case yi = yj , and Ai yi (0) = bi . (53) We assume that k(t) ∈ R is measurable and locally bounded almost everywhere for t ∈ [0, ∞), and lim k(t) = δ (54) t→∞ Z ∞ k(t)dt = ∞ (55) 0 where δ is a sufficiently small nonnegative number depending on the connection of the network and A, note that 0 is always a feasible choice of δ. One example δ̄ +δ. One simple case is choosing δ̄ = 1, δ = 0, and of a choice of k(t) is k(t) = t+1 12 1 resulting in k(t) = t+1 , obtained by taking δ to be zero. This choice obviates the need to decide how small one has to be to meet a “sufficiently small” condition, but may result in rather slow convergence. Now from Ai yi (0) = bi and the fact that Pi is the projection to the kernel of Ai (which ensures ẏi ∈ ker Ai ), one has Ai yi (t) = bi , t ∈ [0, ∞) (56) Let y = col {y1 , y2 , y3 , ..., ym }, P̄ = diag {P1 , P2 , ..., Pm }, and H̄ = H ⊗ In with H the incidence matrix of G. From the updates in (52) and Assumption 1, one has ẏ ∈ −k(t)P̄ F [sgn ](y) − P̄ H̄F [sgn ](H̄ 0 y) (57) Note that sgn (y), k(t) and P̄ H̄F [sgn ](H̄ 0 y) are measurable and locally bounded almost everywhere. Then by Lemma 2 there exists a Filippov solution to system (57) for any given y(0) satisfying (53), which we denote by y(t) = col {y1 (t), y2 (t), y3 (t), ..., ym (t)} . By Lemma 3, there exists a set I = [0, ∞) \ T 1 exists for all t ∈ I. Then one with T of Lebesgue measure 0 such that dky(t)k dt has the following theorem: Theorem 3 Under Assumption 1 and the update (52) and with A of full row rank, all yi (t), i = 1, 2, ..., m converge in finite time to the same value which is the minimum l1 -norm solution to Ax = b. Proof of Theorem 3: We first prove that all yi (t) reach a consensus in finite time by showing that z(t) converges to 0 in finite time, where z(t) = H̄ 0 y(t). Multiplying both sides of (57) by H̄ 0 , one has ż ∈ −k(t)H̄ 0 P̄ F [sgn ](y) − H̄ 0 P̄ H̄F [sgn ](z) (58) By Lemma 4, one has dkz(t)k1 = β 0 ż, t ∈ I dt where β can be any vector in ∂kzk1 . Note that ∂kzk1 = F [sgn ](z). Then dkz(t)k1 = −k(t)γ 0 H̄ 0 P̄ η − γ 0 H̄ 0 P̄ H̄γ, dt t∈I (59) where η ∈ F [sgn ](y), γ ∈ F [sgn ](z) and β is chosen to be equal to γ. Since z(t) ∈ image H̄ 0 , then by Lemma 6, if also z(t) ∈ Φ(H̄ 0 P̄ H̄), one will have z(t) = 0. Thus z(t) ∈ Φc (H̄ 0 P̄ H̄) as long as kz(t)k1 6= 0. (60) Now by the definition of λ(H̄ 0 P̄ H̄) and Lemma 5, one has γ 0 H̄ 0 P̄ H̄γ ≥ ρ as long as kz(t)k1 6= 0 13 (61) where ρ = λ(H̄ 0 P̄ H̄) is a positive constant. Let κ(H̄, P̄ ) denote an upper bound on \|γ 0 H̄ 0 P̄ η\|, and define an upper bound on δ by δ< ρ κ(H̄, P̄ ) (62) This captures the idea stated previously that δ depends on A and the graph. For any δ chosen as in (62), there must exist a finite time T1 such that \| − k(t)γ H̄ 0 P̄ η\| ≤ ρ2 , t ∈ [T1 , ∞). (63) where we have ρ2 = κ(H̄, P̄ )δ < ρ. From (59), (61) and (63), one has dkz(t)k1 ≤ −(ρ − ρ2 ) as long as kz(t)k1 6= 0, t ∈ [T1 , ∞) \ T dt (64) with ρ a positive constant. Thus there must exist a finite time T2 ≥ T1 such that z(T2 ) = 0 (65) Next we prove that z(t) = 0, t ∈ [T2 , ∞) (66) We prove this by contradiction. Suppose (66) is not true. Then there exists a time T̄2 > T2 such that z(T̄2 ) 6= 0. Then kz(T̄2 )k1 > 0. Since kz(t)k1 is continuous, there exists a time T2∗ such that kz(T2∗ )k1 takes its maximum value for t ∈ [T2 , T̄2 ]. Again, since kz(t)k1 is continuous, there exists a sufficiently small but positive such that kz(t)k1 > 0 is differentiable for t ∈ [T2∗ − , T2∗ ]. Because kz(t)k1 is differentiable almost everywhere, we know that kz(T2∗ )k1 = Z T2∗ T2∗ − dkz(t)k1 dt + kz(T2∗ − )k1 dt (67) Because kz(t)k1 > 0 in [T2∗ − , T2∗ ], by (64) and (67) we have kz(T2∗ )k1 ≤ −(ρ − ρ2 ) + kz(T2∗ − )k1 < kz(T2∗ − )k1 (68) This contradicts the fact that kz(T2∗ )k1 is the maximum value on [T2 , T̄2 ]. Thus (66) is true. By (66), one has there exists a vector ȳ(t) such that y1 (t) = y2 (t) = · · · = ym (t) = ȳ(t), t ∈ [T2 , ∞) (69) Moreover, Aȳ(T2 ) = b since Ai yi (t) = bi for i = 1, 2, ..., m. To prove Theorem 3, we only need to prove that ȳ(t) converges to be the minimum l1 -norm solution to Ax = b. To see why this is so, we let P denote the projection matrix to the kernel of A. Then P Pi = P for i = 1, 2, ..., m. Multiplying both sides of (52) by P , one has X ẏi = −k(t)P φi − P φij , i = 1, 2, ..., m (70) j∈Ni 14 Since G is undirected, then φij appears in the update i if φji appears in its neighbor j’s update. By adding the updates in (70) for i = 1, 2, ..., m and noting φij = −φji for any two neighbors i and j, one has m X ẏi = −k(t)P m X i=1 φi (71) i=1 where φi ∈ F [sgn ](yi ). By (69), one knows all yi (t) reach a consensus ȳ(t) for t ∈ [T2 , ∞). Note that if the kth entry of ȳ(t) is 0, the kth entry of each φi can be selected as an arbitrary value from [−1, 1], which may be different for different entries, but their average is still an arbitrary value in [−1, 1]. Thus m 1 X φi ∈ F [sgn ](ȳ(t)), m i=1 t ∈ [T2 , ∞) (72) From (71) and (72) we have ȳ˙ ∈ −k(t)P F [sgn ](ȳ), Let τ = Rt T2 t ∈ [T2 , ∞) (73) k(s)ds. From dȳ dt dȳ = · dτ dt dτ and (73), one has dȳ ∈ −P F [sgn ](ȳ), τ ∈ [0, ∞) (74) dτ with Aȳ(τ ) = b for τ = 0. This is exactly the same as the centralized update in (45). By Theorem 1, there exists a finite time Γ such that y(τ ) is the minimum l1 -norm solution to Ax = b for τ ∈ [Γ, ∞). By the relation between τ and t, one has correspondingly that there exist a finite time T such that ȳ(t) is a minimum l1 -norm solution for t ∈ [T, ∞). This completes our proof. 5 Simulation Result In this section, we will report several simulations of the proposed algorithms for solving an underdetermined linear equation Ax = b in a four-agent undirected and connected network as in Figure 1. 0 0 0 Here, A and b are partitioned as A = A1 A02 A03 A04 and b = b01 b02 b03 b04 , 15 1 2 4 3 Figure 1: A four agent network respectively. Each agent i knows Ai and bi with  0.63 0.58  0.65  0.33  0.68  0.22 0 A1 =  0.49  0.21  0.62 0.57  0.71 0.28  0.44 0.06  0.77  0.16  0.84  0.62 A03 =  0.74  0.26  0.44 0.28  0.50 0.38  0.04 0.60  0.50  0.81  0.01  0.51 , 0.23  0.29  0.25 0.21  0.66 0.90  0.34 0.94  0.28  0.41  0.75  0.56 , 0.41  0.89  0.69 0.23  0.88 0.63  0.80 0.25  0.53  0.79  0.13  0.79 0 A2 =  0.34  0.45  0.10 0.94  0.78 0.70  0.05 0.09  0.65  0.69  0.94  0.73 A04 =  0.51  0.59  0.76 0.95  0.39 0.03 b01 = 0.47 b03 = 0.63 0.52 , 0.33 , b02 = 0.77 b04 = 0.31  0.99 0.65  0.38  0.12  0.76  0.52  0.55  0.24  0.55 0.51  0.58 0.85  0.23 0.33  0.92  0.66  0.92  0.06  0.13  0.94  0.40 0.69  0.24 0.92 0.34 0.65 (75) (76) Example 1: We utilize the distributed update (39) to achieve a solution ∗ to Ax by x in finite time in the above four-agent network. Let 0= b 0denoted 0 0 0 y = y1 y2 y3 y4 where yi (t) denote the state of agent i that is the estimate of agent i to x∗ . Then ky(t) − 1m ⊗ x∗ k1 measures the difference between all agents’ estimations and the solution x∗ . As shown by simulations in Figure 2, ky(t) − 1m ⊗ x∗ k1 reaches 0 in finite time. This suggests all agents’ states achieves a consensus to x∗ in finite time, consistent with the claim of Theorem 1. Example 2: We employ the centralized update (45) with state vector y(t) to achieve x̄∗ which denotes a minimum l1 -norm solution to Ax = b. As shown in 16 Figure 2: Finite time achieving a solution under the update(39) Figure 3, ky(t) − x̄∗ k1 reaches 0 in finite time and maintains to be 0 afterwards. This indicates that the minimum l1 -norm solution x̄∗ is achieved in finite time corresponding to Theorem 2. It is worth noting that one could observe multiple phases of convergence in Figure 2. This is because F [sgn ](y(t)) in the update (45) takes different values piece-wisely, and results in different convergence rates. Figure 3: Centralized solver for achieving a minimum l1 norm solution under the update (45) Example 3: Finally, we utilize the distributed update (52) to achieve a minimum l1 solution to Ax = b denoted by x̄∗ in finite time. Here k(t) is δ̄ chosen to take the form 1+t + δ with δ̄ and δ constants. We still let y = 0 0 0 0 0 y1 y2 y3 y4 where yi (t) denote the state of agent i that is the estimate of 17 agent i to x̄∗ . Then ky(t)−1m ⊗ x̄∗ k1 measures the difference between all agents’ estimations and x̄∗ . As shown in Figure 4 and Figure 5, all yi (t) reach the same minimum l1 -norm solution in finite time regardless of different choices of δ̄ and δ. Moreover, by fixing δ̄ and increasing the value of δ in k(t), one achieves a significantly faster convergence as shown in Figure 4. Similarly, increasing δ̄ with a fixed δ also leads to a faster convergence, although not that dramatically, as shown in Figure 5. Figure 4: Distributed solver for achieving minimum l1 norm solution under the δ̄ update (52), where k(t) = t+1 + δ with fixed δ̄ = 0.1 and different values of δ. Figure 5: Distributed solver for achieving minimum l1 norm solution under δ̄ update (52) where k(t) = t+1 + δ with fixed δ = 0.01 and different values of δ̄. We also note from Figure 4 and Figure 5 that the convergence time required in this distributed way for minimum l1 -norm solutions is dramatically longer, 18 roughly speaking, 1δ times longer, than that in the centralized case in Figure 3. The major reason for this is that the centralized update appearing in the distributed update (52) is scaled by k(t), which is smaller than 1. The time required for consensus in this four-agent network example is minor under the distributed update (52) as indicated in Figure 6. Let ȳ(t) denote the average of all four agents’ states. The evolution of ky(t) − 1m ⊗ ȳ(t)k1 in Figure 6 suggests that all agents’ states reach a consensus in a finite time similar to that in Figure 2. We anticipate that when it comes to a very large network, the convergence time for consensus might play a more significant role in convergence of the distributed update (52). Figure 6: Consensus of distributed solver under update (52) with k(t) = 0.01 6 0.1 t+1 + Conclusion We have developed continuous-time distributed algorithms for achieving solutions, and minimum l1 -norm solutions, respectively, to linear equations Ax = b in finite time. The algorithms result from combination of the projectionconsensus flow proposed in [13] and the finite-time gradient flow for consensus devised in [36], and work for fixed undirected multi-agent networks. Future work includes the generalization of the proposed update to networks that are directed and time-varying. References [1] S. Mou, J. 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1 UAV-Aided Wireless Communication Designs With Propulsion Energy Limitations Subin Eom, Hoon Lee, Junhee Park and Inkyu Lee, Fellow, IEEE arXiv:1801.02782v1 [cs.IT] 9 Jan 2018 School of Electrical Eng., Korea University, Seoul, Korea Email: {esb777, ihun1, pjh0585, inkyu}@korea.ac.kr Abstract This paper studies unmanned aerial vehicle (UAV) aided wireless communication systems where a UAV supports uplink communications of multiple ground nodes (GNs) while flying over the area of the interest. In this system, the propulsion energy consumption at the UAV is taken into account so that the UAV’s velocity and acceleration should not exceed a certain threshold. We formulate the minimum average rate maximization problem and the energy efficiency (EE) maximization problem by jointly optimizing the trajectory, velocity, and acceleration of the UAV and the uplink transmit power at the GNs. As these problems are non-convex in general, we employ the successive convex approximation (SCA) techniques. To this end, proper convex approximations for the non-convex constraints are derived, and iterative algorithms are proposed which converge to a local optimal point. Numerical results demonstrate that the proposed algorithms outperform baseline schemes for both problems. Especially for the EE maximization problem, the proposed algorithm exhibits about 109 % gain over the baseline scheme. I. I NTRODUCTION Recently, unmanned aerial vehicles (UAVs) have received great attentions as a new communication entity in wireless networks [1]. Compared to conventional terrestrial communications where users are served by ground base stations (BSs) fixed at given position [2], UAV-aided systems could be dispatched to the field with various purposes such as disaster situations and military uses. Moreover, located high above users, UAVs are likely to have line-of-sight (LoS) communication links for air-to-ground channels. Utilizing these advantages, UAVs have been considered to diverse wireless communication systems. The authors in [3] and [4] studied a mobile relaying system where a UAV helps the communication of ground nodes (GNs) without direct communication links. In this UAV-aided 2 relaying system, compared to conventional static relay schemes [5], [6], the UAV can move closer to source and destination nodes in order to obtain good channel conditions, and thus the system throughput can be significantly improved. In [3], the throughput of mobile relaying channels was maximized by optimizing the transmit power at the source and the relay node as well as the trajectory of the mobile relay. For the fixed relay trajectory, the work [4] addressed the secrecy rate maximization problem for the UAV-based relaying system with an external eavesdropper. In addition, UAVs have been adopted to assist conventional terrestrial communication infrastructures [7]–[9]. For the disaster situation, UAVs were employed in [7] to recover malfunctioned ground infrastructure. The work in [8] examined a system where the UAV serves cell-edge users by jointly optimizing UAV’s trajectory, bandwidth allocation, and user partitioning. Also, the flying computing cloudlets with UAVs were introduced to provide the offloading opportunities to multiple users [9]. Moreover, the UAVs could play the role of mobile BSs in wireless networks [10]–[12]. The authors in [10] derived mathematical expressions for the optimum altitude of the UAVs that maximizes the coverage of the cellular network. Also, the trajectory optimization methods for mobile BSs were presented in [11] and [12]. Assuming that the GNs are located in a line, the minimum throughput performance was maximized in [11] by optimizing the position of a UAV on a straight line. This result was extended in [12] to a general scenario where multiple UAVs fly three-dimensional space to communicate with GNs. The joint optimization algorithms for the UAV trajectory, transmit power, and time allocation were provided in [12] to maximize the minimum throughput performance. However, these works did not consider the propulsion energy consumption of the UAVs necessary for practical UAV designs under limited on-board energy situation [13]. By taking this issue into account, recent works [14]–[16] investigated energy efficiency (EE) of the UAV system. Different from conventional systems which consider only communicationrelated energy consumption [17]–[19], the EE of the UAV should addresses the propulsion energy at the UAV additionally. The authors in [14] maximized the EE by controlling the turning radius of a UAV for mobile relay systems. Also, by jointly optimizing the time allocation, speed, and trajectory, both the spectrum efficiency and the EE were maximized in [15]. In [16], the propulsion energy consumption of the UAV was theoretically modeled, and the EE of the UAV was maximized for a single GN system. This paper studies UAV-aided wireless communications where a UAV with limited propulsion 3 energy receives the data of multiple GNs in the uplink. It is assumed that all GNs and the UAV operate in the same frequency band and there are no direct communication links among GNs. Under these setup, we formulate the minimum rate maximization problem and the EE maximization problem by jointly optimizing the UAV trajectory, the velocity, the acceleration, and the uplink transmit power at the GNs. A similar approach for solving the minimum rate maximization was studied in [12], but the authors in [12] did not involve the propulsion energy consumption at the UAV. For the EE maximization problem, our work can be regarded as a generalization of the single GN system in [16] to the multi-GN scenario, and thus we need to deal with inter-node interference as well. Due to these issues, existing algorithms presented in [12] and [16] cannot be directly applied to our problems. To tackle our problem of interest, we introduce auxiliary variables which couple the trajectory variables and the uplink transmit power in order to jointly optimize these variables. As the equivalent problem is still non-convex, we employ the successive convex approximation (SCA) technique which successively solves approximated convex problems of the original non-convex one. In order to apply the SCA to our optimization problems, we present new convex surrogate functions for the non-convex constraints. Then, we propose efficient algorithms for the minimum rate maximization problem and the EE maximization problem which yield local optimal solutions. Simulation results confirm that the proposed algorithms provide a significant performance gain over baseline schemes. The rest of this paper is organized as follows: Section II explains the system model and the problem formulations for the UAV-aided communication systems. In Section III, the minimum rate maximization and the EE maximization algorithms are proposed. We examine the circular trajectory case as baseline schemes in Section IV. Section V presents the numerical results for the proposed algorithms and we conclude the paper in Section VI. Notations: Throughout this paper, the bold lower-case and normal letters denote vectors and scalars, respectively. The space of M-dimensional real-valued vectors are represented by RM ×1 . For a vector a, kak and aT indicate norm and transpose, respectively. The gradient of a function f is defined as ∇f . For a time-dependent function x(t), ẋ(t) and ẍ(t) stand for the first-order and second-order derivatives with respect to time t, respectively. 4 Fig. 1. UAV-enabled wireless network II. S YSTEM M ODEL A ND P ROBLEM F ORMULATION As shown in Fig. 1, we consider UAV-aided wireless communications where a UAV receives uplink information transmitted from K GNs. The UAV horizontally flies at a constant altitude H with a time period T , while the GNs are located at fixed positions, which are perfectly known to the UAV in advance. For the location of the GNs and the UAV, we employ a three-dimensional Cartesian coordinate system, and thus the horizontal coordinate of GN k (k = 1, ..., K) is denoted by wk = [xk yk ]T . Also, we define the time-varying horizontal coordinate of the UAV at time instant t as q(t) = [qx (t) qy (t)]T , for 0 ≤ t ≤ T . Then, the instantaneous velocity v(t) and the acceleration a(t) of the UAV are expressed by v(t) , q̇(t) and a(t) , q̈(t), respectively. Continuous time expressions of variables make analysis and derivations in the UAV systems intractable. For ease of analysis, we discretize the time duration T into N time slots with the same time interval δt = T N [3]. As a result, the trajectory of the UAV can be represented by N vector sequences q[n] , q(nδt ), v[n] , v(nδt ), and a[n] , a(nδt ) for n = 0, 1, ..., N. When the discretized time interval δt is chosen as a small number, the velocity and the acceleration can be approximated by using Taylor expansions as [16] v[n] = v[n − 1] + a[n − 1]δt , for n = 1, ..., N, (1) q[n] = q[n − 1] + v[n − 1]δt + 21 a[n − 1]δt2 , for n = 1, ..., N. (2) Also, assuming the periodical operation at the UAV, we have [12] q[0] = q[N], v[0] = v[N], a[0] = a[N], (3) 5 which implies that after one period T , the UAV returns to its starting location with the same velocity and acceleration. In addition, the acceleration and the velocity of the practical UAV are subject to ka[n]k ≤ amax , for n = 0, 1, ...N, (4) Vmin ≤ kv[n]k ≤ Vmax , for n = 0, 1, ..., N, (5) where amax indicates the maximum UAV acceleration in m/sec2 and Vmin and Vmax stand for the minimum and the maximum UAV speed constraints in m/sec, respectively. Notice that the minimum speed constraint Vmin is important for practical fixed-wing UAV designs which need to move forward to remain aloft and thus cannot hover over a fixed location [16]. For the power consumption at the UAV, we take into account the propulsion power utilized for maintaining the UAV aloft and supporting its mobility. The propulsion power of the UAV Pprop [n] at time slot n is given by [16] c2 Pprop [n] = c1 kv[n]k + kv[n]k 3 ka[n]k2 1+ , for n = 0, 1, ..., N, g2 (6) where c1 and c2 are the parameters related to the aircraft design and g = 9.8 m/sec2 equals the gravitational acceleration. Thus, the average propulsion power and the total consumed propulsion P PN energy over N time slots are obtained by N1 N n=1 Pprop [n] and δt n=1 Pprop [n], respectively. The power consumed by signal processing circuits such as analog-to-digital converters and channel decoders are ignored since they are practically much smaller than the propulsion power [16]. Now, let us explain the channel model between the UAV and the GNs. We assume that the air-to-ground communication links are dominated by the LoS links. Moreover, the Doppler effect due to the UAV mobility is assumed to be well compensated. Then, the effective channel gain hk [n] from GN k to the UAV at time slot n follows the free-space path loss model as [3] hk [n] = γ0 , 2 dk [n] (7) where γ0 , β0 /σ 2 represents the reference signal-to-noise ratio (SNR) at 1 m with β0 and σ 2 being the channel power at 1 m and the white Gaussian noise power at the UAV, respectively, and the distance dk [n] is written by dk [n] = p kq[n] − wk k2 + H 2 . (8) 6 At time slot n, GN k transmits its data signal to the UAV with power 0 ≤ pk [n] ≤ Ppeak , where Ppeak is the peak transmission power constraint at the GNs. Accordingly, the instantaneous achievable rate Rk [n] can be expressed as Rk [n] = log2 where the term PK j=1,j6=k 1+ pk [n]hk [n] 1+ PK pj [n]hj [n] j=1,j6=k ! , (9) pj [n]hj [n] stands for interference from other GNs. Therefore, the achievable average rate of the GN k and the total information bits transmitted from GN k PN P over N time slots are denoted as N1 N n=1 Rk [n], respectively, where W n=1 Rk [n] and W δt means the bandwidth. In this paper, we jointly optimize the variables q[n], v[n], and a[n] and the uplink transmit power pk [n] at the GNs so that the minimum average rate among multiple GNs and the EE are maximized, respectively. First, the minimum rate maximization problem can be formulated as (P 1) : max {q[n],v[n],a[n]} {pk [n],τ } s.t. τ (10a) N 1 X Rk [n] ≥ τ, ∀k, N n=1 0 ≤ pk [n] ≤ Ppeak , ∀k, n, N 1 X Pprop [n] ≤ Plim , N n=1 (10b) (10c) (10d) (1) − (5), where Plim in (10d) indicates the propulsion power constraint at the UAV. Next, to support all of the individual GNs, the fairness based EE [20]–[22] is more suitable than the network-wise EE [18], [19]. Thus, we define the EE in the UAV-aided wireless communication systems as the ratio between the minimum information bits transmitted among the GNs and the total energy consumed at the UAV. Therefore, the EE maximization problem can be written by (P 2) : max {q[n],v[n],a[n]} {pk [n],η} η PN Pprop [n] W Rk [n] ≥ η, ∀k, n=1 N X n=1 s.t. (1) − (5), (10c). (11a) (11b) 7 In general, (P1) and (P2) are non-convex problems due to the constraints and the objective functions. Compared to [12], we additionally consider the propulsion power constraint (10d) in the minimum rate maximization problem (P1). Also, note that the EE maximization problem (P2) can be regarded as a generalization of [16] which investigated only a single GN scenario. From these respects, the works in [12] and [16] can be regarded as special cases of our problems (P1) and (P2), respectively. To solve the problems (P1) and (P2), we adopt the SCA framework [23] [24] which iteratively solves approximated convex problems for the original non-convex problems. III. P ROPOSED A LGORITHM In this section, we propose iterative algorithms for efficiently solving (P1) and (P2) by applying the SCA method. First, the minimum rate maximization problem (P1) is considered in Section III-A, and then it is followed by the EE maximization problem (P2) in Section III-B. A. Minimum Average Rate Maximization Applying the change of variables as Gk [n] , pk [n]hk [n] = pk [n]γ0 , ∀k, n, kq[n] − wk k2 + H 2 (12) where Gk [n] is a new optimization variable, the constraint (10c) becomes 0 ≤ Gk [n] ≤ Gk,max [n], ∀k, n, where Gk,max [n] , Ppeak hk [n] = Ppeak γ0 . kq[n]−wk k2 +H 2 Then, we can rewrite the achievable rate Rk [n] in (9) as Rk [n] = log2 1+ K X m=1 P G [n] . where R̂k [n] , log2 1 + K j=1,j6=k j ! Gm [n] − R̂k [n], (13) 8 By introducing new auxiliary variables {V1 [n]}, (P1) can be recast to (P 1.1) : max {q[n],v[n],a[n]} {Gk [n],V1 [n],τ } τ N 1 X log2 N n=1 s.t. (14a) 1+ K X Gm [n] m=1 ! − R̂k [n] ! ≥ τ, ∀k, 0 ≤ Gk [n] ≤ Gk,max [n], ∀k, n, N c2 1 X c2 ka[n]k2 c1 kv[n]k3 + + 2 ≤ Plim , N n=1 V1 [n] g V1 [n] (14b) (14c) (14d) Vmin ≤ V1 [n], ∀n, (14e) V12 [n] ≤ kv[n]k2 , ∀n, (14f) kv[n]k ≤ Vmax , ∀n, (14g) (1) − (4). It can be shown that at the optimal point of (P1.1), the inequality constraint in (14f) holds with the equality, since otherwise we can enlarge the feasible region corresponding to (14d) by increasing V1 [n]. Therefore, we can conclude that (P1.1) is equivalent to (P1). Thanks to the new auxiliary variables {V1 [n]}, constraints (14d) and (14e) now become convex, while (14b), (14c), and (14f) are still non-convex in general. To address these constraints, we employ the SCA methods. First, it can be checked that constraint (14b) is given by a difference of two concave functions. Hence, the convex surrogate function R̂kub [n] for R̂k [n] can be computed from a first order Taylor approximation as ! K K X X Gj,l [n] ≥ R̂k [n], (Gj,l+1[n] − Gj,l [n]) + log2 1 + R̂kub [n] , Γ̂k [n] j=1,j6=k (15) j=1,j6=k where Gk,l [n] indicates a solution of Gk [n] attained at the l-th iteration of the SCA process and P Γ̂k [n] , log2 e/(1 + K j=1,j6=k Gj,l [n]). Next, to identify the surrogate functions of (14c) and (14f), we present the following lemmas. Lemma 1: Denoting {ql [n]} as a solution for {q[n]} calculated at the l-th iteration, the concave surrogate function Glb k,max [n] for Gk,max [n] can be expressed as Glb k,max [n] , Ppeak γ0 kql+1 [n] − wk k2 + Bk [n](ql+1 [n] − wk )T (ql [n] − wk ) + Ck [n] − H4 ≤ Gk,max[n], ! (16) 9 where the constants Bk [n] and Ck [n] are respectively given as ! 1 1 − Bk [n] , 2 2 , 4 H kql [n] − wk k2 + H 2 kql [n] − wk k2 2kql [n] − wk k2 1 + Ck [n] , . 2 − H4 kql [n] − wk k2 + H 2 kql [n] − wk k2 + H 2 Proof: Please refer to Appendix A. Lemma 2: From a solution {vl [n]} obtained at the l-th iteration, the concave surrogate function of kvl+1 [n]k2 can be computed as −kvl+1 [n]k2 + 2vlT (2vl+1 [n] − vl [n]) ≤ kvl+1 [n]k2 . (17) Proof: Applying a similar process in Appendix A, we can conclude that the function in (17) satisfies the conditions for a concave surrogate function [23]. With the aid of Lemmas 1 and 2, at the (l + 1)-th iteration, the non-convex constraints in (14c) and (14f) can be approximated as 0 ≤ Gk [n] ≤ Glb k,max [n], (18) V12 [n] ≤ −kvl+1 [n]k2 + 2vlT (2vl+1 [n] − vl [n]) . (19) As a result, with given solutions {ql [n], vl [n], Gk,l [n]} at the l-th iteration, we solve the following problem at the (l + 1)-th iteration of the SCA procedure (P 1.2) : max {ql+1 [n],vl+1 [n],a[n]} {Gk,l+1 [n],V1 [n],τ lb } s.t. τ lb N 1 X log2 N n=1 (20a) 1+ K X m=1 Gm,l+1 [n] ! − R̂kub [n] ! ≥ τ lb , ∀k, (20b) (1) − (4), (14d), (14e), (14g), (18), (19), where τ lb denotes the lower bound of τ in the original problem (P1). Since (P1.2) is a convex problem, it can be optimally solved via existing convex optimization solvers, e.g. CVX [25]. Based on these results, we summarize the proposed iterative procedure in Algorithm 1. Algorithm 1: Proposed algorithm for (P1) Initialize {q0 [n], v0 [n], Gk,0 [n]}, ∀k, n and let l = 0. Repeat Compute {ql+1 [n], vl+1 [n], Gk,l+1[n]} for (P1.2) with given {ql [n], vl [n], Gk,l [n]}. Update l ← l + 1. Until Convergence. Gk,l+1 [n] Obtain pk [n] = hk,l+1 . [n] 10 For the convergence analysis of Algorithm 1, let us define the objective values of (P1) and (P1.2) at the l-th iteration as τl and τllb , respectively. Then we can express the relationship lb τl = τllb ≤ τl+1 ≤ τl+1 , (21) where the first equation holds because the surrogate functions in (15), (16), and (17) are tight at the given local points, the second inequality is derived from the non-decreasing property of the optimal solution of (P1.2), and the third inequality follows from the fact that the approximation problem (P1.2) is a lower bound of the original problem (P1). From (21), we can conclude that the objective value τ in (P1) is non-decreasing for every iterations of Algorithm 1. Since the objective value τ in (P1) has a finite upper bound value and at given local points, the surrogate functions in (15), (16), and (17) obtain the same gradients as their original functions, it can be verified that Algorithm 1 is guaranteed to converge to at least a local optimal solution for (P1) [23], [24]. B. Energy Efficiency Maximization In this subsection, we consider the EE maximization problem (P2). First, by applying (12)(13), and introducing an auxiliary variable {V1 [n]}, (P2) can be transformed as η (P 2.1) : max PN 2 2 3 {q[n],v[n],a[n]} c1 kv[n]k + V1c[n] + c2gka[n]k 2 V [n] n=1 1 {Gk [n],V1 [n],η} ! ! K N X X Gm [n] − R̂k [n] ≥ η, ∀k, s.t. W log2 1 + (22a) (22b) m=1 n=1 (1) − (4), (14c), (14e) − (14g). Similar to (P1.1), we can see that (P2.1) is equivalent to (P2), but (P2.1) is still non-convex due to the constraints in (14c), (14f), and (22b). To tackle this issue, we can employ the similar SCA process presented in Section III-A. By adopting (15) and Lemmas 1 and 2, a convex approximation of (P2.1) at the (l + 1)-th iteration is given by (P 2.2) : max {ql+1 [n],vl+1[n],a[n]} {Gk,l+1 [n],V1 [n],ηlb } s.t. η lb PN 3 n=1 c1 kv[n]k + W N X n=1 log2 1 + c2 V1 [n] K X m=1 + (23a) c2 ka[n]k2 g 2 V1 [n] ! Gm,l+1 [n] (1) − (4), (14e), (14g), (18), (19), − R̂kub [n] ! ≥ η lb , ∀k (23b) 11 where η lb denotes the lower bound of η in the original problem (P2). It can be shown that (P2.2) is a concave-convex fractional problem, which can be optimally P c2 3 solved via the Dinkelbach’s method [26], [27]. Then, denoting µ = N n=1 c1 kv[n]k + V1 [n] + c2 ka[n]k2 g 2 V1 [n] with a given constant λm , (P2.2) can be converted to (P2.3) as (P 2.3) : η lb − λm µ max {ql+1 [n],vl+1 [n],a[n]} {Gk,l+1 [n],V1 [n],ηlb } s.t. (24a) (1) − (4), (14e), (14g), (18), (19), (23b). Based on (P2.3), we summarize the proposed iterative procedure in Algorithm 2. The convergence and the local optimality of Algorithm 2 can be verified similar to Algorithm 1, and thus the details are omitted for brevity. Algorithm 2: Proposed algorithm for (P2) Initialize {q0 [n], v0 [n], Gk,0[n]}, ∀k, n and let λ0 = 0, m = 0, and l = 0. Repeat Repeat Compute {ql+1 [n], vl+1 [n], Gk,l+1 [n]} for (P2.3) with given {ql [n], vl [n], Gk,l [n]}, ∀k, n and λm . Update l ← l + 1. Until Convergence. Let F (λm ) = η lb − λm µ and λm+1 = η lb /µ. Update m ← m + 1. Let {q0 [n], v0 [n], Gk,0[n]} = {ql+1 [n], vl+1 [n], Gk,l+1[n]}, ∀k, n and l = 0. Until Convergence. Gk,l+1 [n] Obtain pk [n] = hk,l+1 , ∀k, n. [n] It is worthwhile to note that we need to initialize the trajectory variables {q[n], v[n]} for (P1) and (P2). However, it is not trivial to find such variables satisfying the UAV movement constraints (1)-(5) and the propulsion power constraint (10d). This will be clearly explained in Section IV-C. IV. C IRCULAR TRAJECTORY SYSTEM Now, we examine the circular trajectory system which will be used as a baseline scheme. First, we choose the center of the circular trajectory c = [x0 y0 ]T as the geometrical mean of the GNs c = PK k=1 K wk . Denoting r as the radius of the trajectory and θ[n] as the angle of the circle along which the UAV flies at time slot n, the horizontal coordinate of the UAV q[n] can be obtained by q[n] = [r cos θ[n] + x0 r sin θ[n] + y0 ]T . Also, the location of GN k wk can be 12 represented as wk = [ζk cos ϕk + x0 ζk sin ϕk + y0]T , where ζk and ϕk equal the distance and the angle between the geometric center c and GN k, respectively. Thus, the distance dk [n] between p the UAV and GN k in (8) can be expressed as dk [n] = r 2 + ζk2 + H 2 − 2rζk cos (θ[n] − ϕk ). By adopting the angular velocity ω[n] and the angular acceleration α[n], equations in (1)-(6) can be rewritten as ω[n] = ω[n − 1] + α[n − 1]δt , for n = 1, ..., N, (25) θ[n] = θ[n − 1] + ω[n − 1]δt + 12 α[n − 1]δt2 , for n = 1, ..., N, (26) θ[N] = θ[0] + 2π, ω[0] = ω[N], α[0] = α[N], (27) ka[n]k2 = kak [n]k2 + ka⊥ [n]k2 = r 2 α2 [n] + r 2 ω 4 [n] ≤ a2max , for n = 0, 1, ...N, (28) ωmin ≤ ω[n] ≤ ωmax , for n = 0, 1, ..., N, (29) Pprop [n] = c1 r 3 ω 3 [n] + c2 rω[n] + c2 rω 3 [n] g2 + c2 rα2 [n] , g 2 ω[n] for n = 0, 1, ..., N, (30) where ak [n] and a⊥ [n] are the tangential and centripetal accelerations, respectively, and ωmin , Vmin /r and ωmax , Vmax /r indicate the minimum and maximum angular velocity, respectively. Similar to Section III, we address the minimum average rate maximization problem and the EE maximization problem for the circular trajectory, which are respectively formulated as (P 3) : max {θ[n],ω[n],α[n]} {r,pk [n],τ } s.t. τ (31a) rmin ≤ r ≤ rmax , (31b) (10b) − (10d), (25) − (29), (P 4) : max {θ[n],ω[n],α[n]} {r,pk [n],η} s.t. where rmin , Vmin T 2π and rmax η PN n=1 Pprop [n] (32a) (10c), (11b), (25) − (29), (31b), Vmax T a max √ 4 , min , denote the minimum and 2π 2 max( ω [n]+α [n]) maximum radius of the circular trajectory, respectively. It is emphasized that (P3) and (P4) are difficult to solve because of the non-convex constraints and objective functions. To deal with the problems (P3) and (P4), similar SCA frameworks in Section III are applied. 13 A. Minimum Average Rate Maximization and EE maximization For the minimum average rate maximization problem (P3), we first find {r, pk [n]} with given {θ[n], ω[n], α[n]} and then updates {θ[n], ω[n], α[n], pk [n]} for a fixed r. For given {θ[n], ω[n], α[n]}, we adopt the change of variable Sk [n] and Sk,max [n] as pk [n]γ0 , (r − ζk cos (θ[n] − θk ))2 + ζk2 sin2 (θ[n] − θk ) + H 2 Sk [n] , pk [n]hk [n] = (33) Ppeak γ0 . (r − ζk cos (θ[n] − θk ))2 + ζk2 sin2 (θ[n] − θk ) + H 2 Sk,max [n] , Ppeak hk [n] = (34) Similar to the method in Section III-A, we employ the SCA to Sk,max [n]. Based on Lemma 1, lb1 the concave surrogate function Sk,max [n] of Sk,max [n] with a solution rl at the l-th iteration can be chosen as lb1 Sk,max [n] , Ppeak γ0 − rl+1 − b̌k [n] Ǎ2k [n] 2 ≤ Sk,max [n], ∀n, ! + B̌k [n] rl+1 − b̌k [n] rl − b̌k [n] + Čk [n] (35) where the constants b̌k [n], Ǎk [n], B̌k [n], and Čk [n] are respectively given by b̌k [n] , ζk cos (θ[n] − θk ) , Ǎk [n] , ζk2 sin2 (θ[n] − θk ) + H 2 ,   B̌k [n] , 2  1 Ǎ2k [n] −  1  2  , 2 rl − b̌k [n] + Ǎk [n] 2 2 2 rl − b̌k [n] rl − b̌k [n] 1 Čk [n] , . + 2 − 2 2 Ǎ2k [n] rl − b̌k [n] + Ǎk [n] rl − b̌k [n] + Ǎk [n] By applying (15), (P3) for fixed {θ[n], ω[n], α[n]} can be reformulated as an approximated convex problem at the (l + 1)-th iteration of the SCA (P 3.1) : max {rl+1 ,Sk,l+1 [n],τ lb1 } s.t. τ lb1 N 1 X log2 N n=1 (36a) 1+ K X Sm,l+1 [n] m=1 lb1 0 ≤ Sk [n] ≤ Sk,max [n], ∀k, n, (10d), (31b), ! − R̆kub [n] ! ≥ τ lb1 , ∀k, (36b) (36c) 14 P K where R̆kub [n] , Γ̆k [n] log2 e P . (P3.1) 1+ K j=1,j6=k Sj,l [n] j=1,j6=k (Sj,l+1[n] − Sj,l [n]) +log2 1 + PK j=1,j6=k Sj,l [n] and Γ̆k [n] , can be successively solved by the CVX until convergence. Next, we present a solution for (P3) with a given r. To obtain the concave surrogate function of Sk,max [n], we introduce the following lemma which identifies the surrogate function of the cosine function. Lemma 3: For any given φl , the concave surrogate function of cos φ can be computed as −(φ − φl + sin φl )2 sin2 φl + cos φl + ≤ cos φ. 2 2 (37) Proof: With a similar process in Appendix A, we can conclude that the function in (37) satisfies the conditions for a concave surrogate function [23]. lb2 By inspecting Lemmas 1 and 3, the concave surrogate function Sk,max [n] for Sk,max[n] can be identified as  2   rζk θl+1 [n] − b̂k [n] lb2 + B̂k [n] sin(θl [n] − θk ) θl+1 [n] − b̂k [n] + Ĉk [n] Sk,max [n] , Ppeak γ0 − 2 Âk [n]  ≤ Ppeak γ0 ≤ Sk,max [n], 2 rζk θl+1 [n] − b̂k [n] + Âk [n] (38) where b̂k [n], Âk [n], B̂k [n], and Ĉk [n] are given by b̂k [n] , θl [n] − sin (θl [n] − θk ) , Âk [n] , r 2 + ζk2 + H 2 − rζk 2 cos (θl [n] − θk ) + sin2 (θl [n] − θk ) ,  Ĉk [n] ,  B̂k [n] , 2rζk  1 Â2k [n] − 1   2  , rζk sin2 (θl [n] − θk ) + Âk [n] 2rζk sin2 (θl [n] − θk ) rζk sin2 (θl [n] − θk ) + − . 2 2 2 rζk sin2 (θl [n] − θk ) + Âk [n] Â [n] k rζk sin (θl [n] − θk ) + Âk [n] 1 15 By utilizing (15) and (38), at the (l + 1)-th iteration of the SCA algorithm with a given r, (P3) can be approximated to the following convex problem. (P 3.2) : max {θl+1 [n],ω[n],α[n]} {Sk,l+1 [n],τ lb2 } s.t. τ lb2 (39a) N 1 X log2 N n=1 1+ K X ! Sm,l+1 [n] m=1 lb2 0 ≤ Sk [n] ≤ Sk,max [n], ∀k, n, ! − R̆kub [n] ≥ τ lb2 , ∀k, (39b) (39c) (10d), (25) − (29). We then successively solve (P3.2) by the CVX until convergence. Similar to Algorithm 1, a solution of problem (P3) is obtained by alternately solving (P3.1) and (P3.2) until the objective value converges. For the EE maximization problem (P4) in the circular trajectory case, we can apply similar methods in Section III-B. Based on (P3.1) and (P3.2), given {θ[n], ω[n], α[n]} and r, (P4) can be transformed into two concave-convex fractional problems. By using Algorithm 2, we can alternately solve these problems until convergence. B. Trajectory Initialization To initialize the proposed algorithms, we employ a simple circular path concept in [12]. First, the initial angular velocity ω0 is set to ω0 = 2π , T which implies θ0 [n] = 2π Nn , ∀n. Next, the initial radius r0 is chosen to fulfill the constraints in (4), (5), and (10d), which can be expressed as Vmin T Vmax T amax (40) ≤ r0 ≤ min 2π , ω2 , 2π 0 c1 r03 ω03 + c2 r0 ω0 + c2 r0 ω03 g2 ≤ Plim . (41) We can simply find r0 which maximizes the minimum rate in (P1) and (P3) under constraints (40) and (41) via one-dimensional line search. For the EE maximization problems (P2) and (P4), r0 can be computed in the range of (40). As a result, the initial trajectory q0 [n] can be written by q0 [n] = [r0 cos 2π Nn + x0 r0 sin 2π Nn + y0 ]T (n = 0, 1, ..., N) and the initial velocity v0 [n] can be simply obtained as v0 [n] = (q0 [n + 1] − q0 [n])/δt (n = 0, 1, ..., N − 1) assuming δt2 ≈ 0 in (2). 16 T = 50 sec T = 100 sec T = 150 sec T = 400 sec 800 600 400 y (m) 200 0 -200 -400 -600 -800 -1000 -1200 -1000 -800 -600 -400 -200 0 200 400 600 800 1000 x (m) Fig. 2. Optimized UAV trajectories for different periods T with Plim = 150 W. V. N UMERICAL R ESULTS In this section, we provide numerical results to validate the effectiveness of the proposed algorithms. For the simulations, we consider K = 6 GNs which are distributed as in Fig. 2 where the locations of the GNs are marked with the triangles. The constant altitude, the bandwidth, the reference SNR, and the peak transmission power are set to be H = 100 m, W = 1 MHz, γ0 = 80 dB, and Ppeak = 10 dBm, respectively. Also, the minimum velocity, the maximum velocity, and the maximum acceleration of the UAV are determined as Vmin = 3 m/sec, Vmax = 100 m/sec, and amax = 5 m/sec2 , respectively. For the propulsion power consumption model in (6), the constants c1 and c2 are set as c1 = 9.26 × 10−4 and c2 = 2250, respectively, which make the minimum propulsion power consumption Pprop,min = 100 W when kvk = 30 m/sec. We first demonstrate the performance of the minimum rate maximization algorithms. Fig. 2 illustrates the optimized UAV trajectories with various T for Plim = 150 W. It is observed that when T is smaller than 150 sec, as T increases, the UAV tries to get closer to all GNs in order to improve the channel conditions from the GNs. In contrast, if T is sufficiently large (T = 400 sec), the UAV is now able to visit all the GNs within a given time period. Thus, the UAV can 17 1 0.9 Max-min rate (bps/Hz) 0.8 Proposed Circular w/ opt. r, ω, α, & p Circular w/ opt. r & p Circular w/ opt. r 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 50 100 150 200 250 300 350 400 450 500 Period T (sec) Fig. 3. Max-min rate with respect to the period T with Plim = 150 W. hover over each GN for a while by traveling smooth path around the GNs. This is different from the results in [12] where the UAV does not have practical movement constraints. This can be explained as follows: Due to the constraints on the velocity and the propulsion power, the UAV cannot stay at fixed positions as in [12]. Therefore, the UAV continuously moves around as close to the GNs as possible to maintain good communication channels without exceeding the propulsion power limit Plim . Fig. 3 shows the maximized minimum (max-min) rate performance of the proposed algorithm as a function of T . We compare the performance of the proposed algorithm with the following circular trajectory based methods. - Circular with optimum r, ω, α, and p: radius, angular velocity, angular acceleration, and uplink transmit power are jointly optimized with (P3) in Section IV-A with the circular trajectory. - Circular with optimum r and p: radius and uplink transmit power are jointly optimized with (P3.1) in Section IV-A with the circular trajectory. - Circular with optimum r: radius is optimized with Ppeak as the initial circular trajectory in 18 1000 Plim = 110 W 800 Plim = 130 W 600 No power limit 400 y (m) 200 0 -200 -400 -600 -800 -1000 -1500 -1250 -1000 -750 -500 -250 0 250 500 750 1000 1250 x (m) Fig. 4. Optimized UAV trajectories for different propulsion power limit Plim with T = 400 sec. Section IV-B First, it can be verified that the proposed algorithm outperforms the baseline schemes regardless of the time period T . Also, we can see that the max-min rate in the proposed algorithm monotonically increases with T , since more time is available at the UAV to hover around each GN. In contrast, in the baseline schemes which are restricted in circular shape trajectory, the max-min rate performance first increases as T grows, and then decreases after a certain T . This is due to a fact that in order to satisfy the propulsion power constraint, the radius of the circular trajectory should increase as T gets large, and thus the UAV may become too far away from the geometric center of the GNs after a certain T . Therefore, we can expect the performance gain of the proposed algorithm over baseline schemes is to grow with T . Fig. 4 illustrates the optimized UAV trajectories for various propulsion power limit Plim with T = 400 sec. It can be shown that for Plim = 110 W, the trajectory of the UAV is restricted to a smooth path with a large turning radius to consume a low propulsion power. However, as Plim gets larger, we observe quick changes along the trajectory path. Thus the UAV can move with a much smaller turning radius, which enhances the max-min rate performance. 19 1 0.9 Proposed Circular w/ opt. r, ω, α, & p Circular w/ opt. r & p Circular w/ opt. r Max-min rate (bps/Hz) 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 100 125 150 175 200 225 250 Propulsion power limit (W) Fig. 5. Max-min rate with respect to the propulsion power limit Plim with T = 400 sec. In Fig. 5, we depict the average max-min rate of various schemes as a function of the propulsion power constraint Plim . For both the proposed algorithm and the baseline schemes, the max-min rate first increases as Plim grows and then gets saturated. This can be explained as follows: With a large Plim , the trajectory and the velocity of the UAV change more freely to attain good channel conditions, and thus the max-min rate increases. However, even if a large Plim is given, the max-min rate cannot continue to increase because there are practical limits on the velocity and acceleration. Similar to Fig. 3, we can see that the proposed algorithm provides significant performance gains over the baseline schemes. Next, in Fig. 6, we investigate the optimized trajectory of the EE maximization problem with various T . As T increases, the overall patterns are similar to Fig. 2. Nevertheless, to balance between the rate performance and the propulsion power consumption, the EE maximization trajectory shows a smooth path with a relatively large turning radius, and thus the average propulsion power consumption becomes lower. To present the impact of the energy efficient UAV communication designs, Fig. 7 depicts the UAV speed of the proposed EE maximization method with T = 400 sec. For comparison, we 20 1000 T = 50 sec T = 200 sec T = 400 sec T = 500 sec 800 600 400 y (m) 200 0 -200 -400 -600 -800 -1000 -1250 -1000 -750 -500 -250 0 250 500 750 1000 1250 x (m) Fig. 6. Optimized energy efficient UAV trajectories for different periods T 100 Minimum rate maximization w/o P lim 90 EE maximization 80 Speed (m/sec) 70 60 50 40 30 20 10 0 0 50 100 150 200 250 300 350 400 Time (sec) Fig. 7. UAV speeds for the max-min rate without propulsion power constraint and the EE maximization with T = 400 sec 21 TABLE I P ERFORMANCE COMPARISON WITH MAX - MIN RATE AND EE MAXIMIZATION FOR T = 400 Max-min rate w/o Plim EE maximization Proposed Circular Proposed Circular SEC Average speed (m/sec) Average acceleration (m/sec2 ) Average max-min rate (bps/Hz) Average power (Watts) Energy efficiency (kbits/Joule) 18.42 13.50 25.73 15.35 4.73 1.88 2.71 0.27 0.99 0.53 0.79 0.47 553.01 541.41 122.14 151.33 1.80 0.98 6.47 3.10 also consider the max-min rate scheme without the propulsion power constraint. It is observed that for the max-min rate case, the UAV tries to fly between the GNs as fast as possible and stay over the GNs with a low speed. On the other hand, the EE maximization scheme keeps the speed of the UAV at around 30 m/sec in order not to waste the propulsion energy. Finally, Table I presents the performance comparison of the max-min rate without propulsion power constraint and the EE maximization designs for both the proposed and the baseline schemes with T = 400 sec. We can see that the max-min rate methods consume much higher propulsion power by allowing a large variation of the speed and the average acceleration. In contrast, the speed of the proposed EE maximization design slowly varies with low acceleration, and thus much higher EE can be achieved. We observe that the proposed EE maximization algorithm exhibits about 259 % gain over the max-min rate without the propulsion power constraint and 109 % gain over the circular baseline EE maximization scheme. VI. C ONCLUSION In this paper, we have studied the UAV-aided wireless communication optimization under the practical propulsion energy constraint at the UAV. For both the minimum average rate maximization problem and the EE maximization problem, the UAV trajectories and the uplink transmit power of the GNs have been jointly optimized. By applying the SCA technique, we have proposed efficient iterative algorithms which find local optimal solutions. Numerical results have demonstrated that the proposed algorithms provide substantial performance gains compared to the baseline schemes. 22 A PPENDIX A PROOF OF LEMMA Let us define a function f1 (u) , 1 ρkuk2 +z 1 for u = [ux uy ]T where z and ρ are positive constants. For any given ul ∈ R2×1 , in order for arbitrary function g1 (u\|ul ) to be a concave surrogate function of f1 (u), it must satisfy the following conditions: f1 (ul ) = g1 (ul \|ul ), ∇g1 (ul \|ul ) = ∇f1 (ul ), and g1 (u\|ul ) ≤ f1 (u), ∀u [23]. Denoting the function g1 (u\|ul ) as where B , 2ρ 1 z2 ρkuk2 g1 (u\|ul ) , − 2 + BuT ul + C, (42) z 2 2ρkul k2 − ρkuz 2l k , it can be easily shown − (ρku k12 +z)2 and C , ρkul1k2 +z + (ρku k2 +z)2 l l that f1 (ul ) = g1 (ul \|ul ), i.e., g1 (u\|ul ) fulfills the first condition for the surrogate function. Also, the gradient of f1 (u) and g1 (u\|ul ) with respect to u can be respectively computed as −2ρu , (ρkuk2 + z)2 2ρu ∇g1 (u\|ul ) = − 2 + Bul . z ∇f1 (u) = − (43) (44) Since two gradients in (43) and (44) become identical at u = ul , g1 (u\|ul ) satisfies the second condition for the surrogate function. To prove the global lower bound condition, we can calculate the Hessian matrix ∇2u h1 (u\|ul ) of the function h1 (u\|ul ) , f1 (u) − g1 (u\|ul ) as # " 2 2 2 E + 4ρz u 4ρz u u x y x , ∇2 h1 (u\|ul ) = D 2 4ρz ux uy E + 4ρz 2 u2y where D , 2ρ z 2 (ρkuk2 +z)3 (45) > 0 and E , ρ3 kuk6 + 3ρ2 zkuk4 + 2ρz 2 kuk2 ≥ 0. One can easily check that the Hessian in (45) is a positive semi-definite matrix, which implies that h1 (u\|ul ) is a convex function. Since ∇h1 (u\|ul ) = 0 at u = ul from (43) and (44), the global minimum of h1 (u\|ul ) is achieved at u = ul with h1 (ul \|ul ) = 0. As a result, we can show that h1 (u\|ul ) is greater than or equal to 0 for any given ul , and thus the third condition for the surrogate function holds. By substituting u = ql+1 [n] − wk , ul = ql [n] − wk , z = H 2 , and ρ = 1 and multiplying f1 (u) and g1 (u\|ul ) by Ppeak γ0 , Lemma 1 is thus proved. R EFERENCES [1] Y. Zeng, R. Zhang, and T. J. Lim, “Wireless communications with unmanned aerial vehicles: opportunities and challenges,” IEEE Commun. Mag., vol. 54, pp. 36–42, May. 2016. 23 [2] W. Lee, I. Lee, J. S. Kwak, B.-C. Ihm, and S. Han, “Multi-BS MIMO cooperation: challenges and practical solutions in 4G systems,” IEEE Wireless Commun., vol. 19, pp. 89–96, Feb. 2012. [3] Y. Zeng, R. Zhang, and T. J. 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c xxxx Society for Industrial and Applied Mathematics ⃝ SIAM/ASA J. UNCERTAINTY QUANTIFICATION Vol. xx, pp. x Mathematical Properties of Polynomial Dimensional Decomposition ∗ arXiv:1804.01647v1 [math.NA] 5 Apr 2018 Sharif Rahman† Abstract. Many high-dimensional uncertainty quantification problems are solved by polynomial dimensional decomposition (PDD), which represents Fourier-like series expansion in terms of random orthonormal polynomials with increasing dimensions. This study constructs dimension-wise and orthogonal splitting of polynomial spaces, proves completeness of polynomial orthogonal basis for prescribed assumptions, and demonstrates mean-square convergence to the correct limit – all associated with PDD. A second-moment error analysis reveals that PDD cannot commit larger error than polynomial chaos expansion (PCE) for the appropriately chosen truncation parameters. From the comparison of computational eﬀorts, required to estimate with the same precision the variance of an output function involving exponentially attenuating expansion coeﬃcients, the PDD approximation can be markedly more eﬃcient than the PCE approximation. Key words. Uncertainty quantification, ANOVA decomposition, multivariate orthogonal polynomials, polynomial chaos expansion. 1. Introduction. Polynomial dimensional decomposition (PDD) is a hierarchical, infinite series expansion of a square-integrable random variable involving measure-consistent orthogonal polynomials in independent random variables. Introduced by the author [20, 21] as a polynomial variant of the well-known analysis-of-variance (ANOVA) dimensional decomposition (ADD), PDD deflates the curse of dimensionality to some extent by developing an input-output behavior of complex systems with low eﬀective dimensions [4], wherein the eﬀects of degrees of interactions among input variables weaken rapidly or vanish altogether. Approximations stemming from truncated PDD are commonly used for solving uncertainty quantification problems in engineering and applied sciences, including multiscale fracture mechanics [6], random eigenvalue problems [24], computational fluid dynamics [27], and stochastic design optimization [25], to name a few. However, all existing works on PDD are focused on practical applications with almost no mathematical analysis of PDD. Indeed, a number of mathematical issues concerning necessary and suﬃcient conditions for the completeness of PDD basis functions; convergence, exactness, and optimal analyses of PDD; and approximation quality of the truncated PDD have yet to be studied or resolved. This paper fills the gap by establishing fundamental mathematical properties to empower PDD with a solid foundation, so that PDD can be as credible as its close cousin, polynomial chaos expansion (PCE) [5, 10, 28, 29], providing an alternative, if not a better, choice for uncertainty quantification in computational science and engineering. The principal objective of this work is to examine important mathematical properties of PDD, not studied heretofore, for arbitrary but independent probability measures of input random variables. The paper is organized as follows. Section 2 defines or discusses mathematical notations and preliminaries. Two sets of assumptions on the input probability measures required by PDD are explained. A brief exposition of univariate and multivariate orthogonal polynomials consistent with a general but product-type probability measure, ∗ This work was supported by the U.S. National Science Foundation under Grant Number CMMI-1462385. College of Engineering and Applied Mathematics & Computational Sciences, The University of Iowa, Iowa City, IA 52242 ([email protected]). Questions, comments, or corrections to this document may be directed to that email address. † 1 x–x 2 S. RAHMAN including their second moment properties, is given in Section 3. The section also describes relevant polynomial spaces and construction of their dimension-wise orthogonal decompositions. The orthogonal basis and completeness of multivariate orthogonal polynomials have also been proved. Section 4 briefly explains ADD, followed by presentations of PDD for a square-integrable random variable. The convergence and exactness of PDD are explained. In the same section, a truncated PDD and its approximation quality are discussed. The formulae for the mean and variance of a truncated PDD are also derived. The section ends with an explanation on how and when the PDD can be extended for infinitely many input variables. Section 5 briefly describes degree-wise orthogonal decompositions of polynomial spaces, leading to PCE. In Section 6, a second-moment error analysis of PDD is conducted, followed by a comparison with that of PCE. Finally, conclusions are drawn in Section 7. 2. Input Random Variables. Let N := {1, 2, . . .}, N0 := N ∪ {0}, and R := (−∞, +∞) represent the sets of positive integer (natural), non-negative integer, and real numbers, respectively. Denote by A{i} , i = 1, . . . , N , an ith bounded or unbounded subdomain of R, {i} ⊆ RN . so that AN := ×N i=1 A Let (Ω, F, P) be a complete probability space, where Ω is a sample space representing an abstract set of elementary events, F is a σ-algebra on Ω, and P : F → [0, 1] is a probability measure. With B N := B(AN ) representing the Borel σ-algebra on AN ⊆ RN , consider an AN -valued input random vector X := (X1 , . . . , XN )T : (Ω, F) → (AN , BN ), describing the statistical uncertainties in all system parameters of a stochastic problem. The input random variables are also referred to as basic random variables [10]. The non-zero, finite integer N represents the number of input random variables and is often referred to as the dimension of the stochastic problem. Denote by FX (x) := P(∩N i=1 {Xi ≤ xi }) the joint distribution function of X, admitting the joint probability density function fX (x) := ∂ N FX (x)/∂x1 · · · ∂xN . Given the abstract probability space (Ω, F, P) of X, the image probability space is (AN , B N , fX dx), where AN can be viewed as the image of Ω from the mapping X : Ω → AN , and is also the support of fX (x). Similarly, each component random variable Xi is defined on the abstract marginal probability space (Ω{i} , F {i} , P{i} ) comprising sample space Ω{i} , σ-algebra F {i} , and probability measure P{i} . Then, the corresponding image probability space is (A{i} , B {i} , fXi dxi ), where A{i} ⊆ R is the image sample space of Xi , B{i} is the Borel σ-algebra on A{i} , and fXi (xi ) is the marginal probability density function of Xi . Relevant statements and objects in the abstract probability space have obvious counterparts in the associated image probability space. Both probability spaces will be used in this paper. Two sets of assumptions used by PDD are as follows. Assumption 2.1.The input random vector X := (X1 , . . . , XN )T : (Ω, F) → (AN , B N ) satisfies all of the following conditions: (1) Each input random variable Xi : (Ω{i} , F {i} ) → (A{i} , B {i} ) has absolutely continuous marginal distribution function FXi (xi ) := P(Xi ≤ xi ) and continuous marginal density function fXi (xi ) := ∂FXi (xi )/∂xi with a bounded or unbounded support A{i} ⊆ R. (2) All component random variables Xi , i = 1, . . . , N , are statistically independent, but not necessarily identical. In consequence, X is endowed with a product-type probability ∏ density function, that is, fX (x) = N i=1 fXi (xi ) with a bounded or unbounded support AN ⊆ RN . (3) Each input random variable Xi possesses finite moments of all orders, that is, for all POLYNOMIAL DIMENSIONAL DECOMPOSITION i = 1, . . . , N and l ∈ N0 , ∫ ∫ [ ] l l (2.1) µi,l := E Xi := Xi (ω)dP(ω) = Ω 3 AN xli fX (x)dx = ∫ A{i} xli fXi (xi )dxi < ∞, where E is the expectation operator with respect to the probability measure P or fX (x)dx. Assumption 2.2.The moments and marginal density function of each input random variable Xi , where i = 1, . . . , N , satisfy at least one of the following conditions [10]: (1) The density function fXi (xi ) has a compact support, that is, there exists a compact interval [ai , bi ], ai , bi ∈ R, such that P(ai ≤ Xi ≤ bi ) = 1. (2) For the moment sequence {µi,l }l∈N0 for Xi , there holds (2.2) lim inf l→∞ (µi,2l )1/2l < ∞. 2l (3) For the moment sequence {µi,l }l∈N0 for Xi , there holds ∑ (2.3) l∈N0 1 = ∞. (µi,2l )1/2l (4) The random variable Xi is exponentially integrable, that is, there exists a real number a > 0 such that ∫ exp(a\|xi \|)fXi (xi )dxi < ∞. (2.4) A{i} (5) If the density function fXi (xi ) is symmetric, diﬀerentiable, and strictly positive, then there exists a real number a > 0 such that ∫ ln fXi (xi ) −xi dfXi (xi )/dxi − (2.5) → ∞ as (xi → ∞, xi ≥ a). 2 dxi = ∞ and fXi (xi ) {i} 1 + x A i Assumption 2.1 assures the existence of an infinite sequence of orthogonal polynomials consistent with the input probability measure. Assumption 2.2, in addition to Assumption 2.1, guarantees the input probability measure to be determinate, resulting in a complete orthogonal polynomial basis of a function space of interest. Both assumptions impose only mild restrictions on the probability measure. Examples of input random variables satisfying Assumptions 2.1 and 2.2 are Gaussian, uniform, exponential, beta, and gamma variables, which are commonly used in uncertainty quantification. These assumptions, to be explained in the next section, are vitally important for the determinacy of the probability measure and the completeness of the orthogonal polynomial basis. Therefore, for both PDD and PCE, which entail orthogonal polynomial expansions, Assumptions 2.1 and 2.2 are necessary. Unfortunately, they are not always clearly specified in the PDD or PCE literature. A prototypical example where Assumption 2.1 is satisfied, but Assumption 2.2 is not, is the case of a lognormal random variable. As noted by Ernst et al. [10], the violation of Assumption 2.2 leads to indeterminacy of the input probability measure and thereby fails to form a complete orthogonal polynomial basis. Finally, Assumptions 2.1 and 2.2 can be modified to account for random variables with discrete or mixed distributions [11] or dependent random variables [23]. The discrete or mixed distributions and dependent variables are not considered in this paper. 4 S. RAHMAN 3. Measure-Consistent Orthogonal Polynomials and Polynomial Spaces. 3.1. Univariate orthogonal polynomials. Consider an ith random variable Xi defined on the abstract probability space (Ω{i} , F {i} , P{i} ) with its image (A{i} , B {i} , fXi dxi ). Let Π{i} := R[xi ] be the space of real polynomials in xi . For any polynomial pair P{i} , Q{i} ∈ Π{i} , define an inner product ∫ [ ] (3.1) (P{i} , Q{i} )f dx := P{i} (xi )Q{i} (xi )fXi (xi )dxi =: E P{i} (Xi )Q{i} (Xi ) Xi i A{i} with respect to the probability measure fXi (xi )dxi and the induced norm ∥P{i} ∥fXi dxi := √ (P{i} , P{i} )f Xi dxi = (∫ Ai 2 P{i} (xi )fXi (xi )dxi )1/2 √ [ ] 2 (X ) . =: E P{i} i Under Assumption 2.1,∫ moments of Xi of all orders exist and are finite, including zeroorder moments µi,0 := A{i} fXi (xi )dxi = 1, i = 1, . . . , N , that are always positive. Clearly, ∥P{i} ∥ > 0 for all non-zero P{i} ∈ Π{i} . Then, according to Gautschi [12], the inner product in (3.1) is positive-definite on Π{i} . Therefore, there exists an infinite set of univariate orthogonal polynomials, say, {P{i},ji (xi ) : ji ∈ N0 }, P{i},ji ̸= 0, which is consistent with the probability measure fXi (xi )dxi , satisfying { 2 ( ) E[P{i},j (Xi )], ji = ki , i (3.2) P{i},ji , P{i},ki f dx = i Xi 0, ji ̸= ki , 2 for ki ∈ N0 , where 0 < E[P{i},j (Xi )] < ∞. Here, in the notation for the polynomial i P{i},ji (xi ), the first and second indices refer to the ith variable and degree ji , respectively. Prominent examples of classical univariate orthogonal polynomials comprise Hermite, Laguerre, and Jacobi polynomials, which are consistent with the measures defined by Gaussian, gamma, and beta densities on the whole real line, semi-infinite interval, and bounded interval, respectively. Many orthogonal polynomials, including the three classical polynomials mentioned, can be expressed in a unified way by invoking hypergeometric series, incorporated in a tree structure of the Askey scheme [1]. For even more general measures, established numerical techniques, such as Gram-Schmidt [13] and Stieltjes’ procedure [26], can be used to generate any measure-consistent orthogonal polynomials. 3.2. Multivariate orthogonal polynomials. For N ∈ N, denote by {1, . . . , N } an index set, so that u ⊆ {1, . . . , N } is a subset, including the empty set ∅, with cardinality 0 ≤ \|u\| ≤ N . When ∅ ̸= u ⊆ {1, . . . , N }, a \|u\|-dimensional multi-index is denoted by ju := \|u\| (ji1 , . . . , ji\|u\| ) ∈ N0 with degree \|ju \| := ji1 + · · · + ji\|u\| , where jip ∈ N0 , p = 1, . . . , \|u\|, represents the pth component of ju .1 For ∅ ̸= u ⊆ {1, . . . , N }, let Xu := (Xi1 , . . . , Xi\|u\| )T , a subvector of X, be defined on the abstract probability space (Ωu , F u , Pu ), where Ωu is the sample space of Xu , F u is a σ-algebra on Ωu , and Pu is a probability measure. The corresponding image probability space is (Au , B u , fXu dxu ), where Au := ×i∈u A{i} ⊆ R\|u\| is the image sample space of Xu , 1 The same symbol \| · \| is used for designating both the cardinality of a set and the degree of a multi-index in this paper. POLYNOMIAL DIMENSIONAL DECOMPOSITION 5 B u is the Borel σ-algebra on Au , and fXu (xu ) is the marginal ∏ probability density function of Xu supported on Au . Under Assumption 2.1, fXu (xu ) = i∈u fXi (xi ). Denote by Πu := R[xu ] = R[xi1 , . . . , xi\|u\| ] the space of all real polynomials in xu . Then, given the inner product ∫ Pu (xu )Qu (xu )fXu (xu )dxu =: E [Pu (Xu )Qu (Xu )] , (Pu , Qu )fX dxu := u Au Πu two polynomials Pu ∈ and Qu ∈ Πu in xu are called orthogonal to each other if (Pu , Qu )fX dxu = 0 [8]. Moreover, a polynomial Pu ∈ Πu is said to be an orthogonal u polynomial with respect to fXu dxu if it is orthogonal to all polynomials of lower degree, that is, if [8] (3.3) (Pu , Qu )fX u dxu = 0 ∀Qu ∈ Πu with deg Qu < deg Pu . \|u\| Let {Pu,ju (xu ) : ju ∈ N0 }, ∅ ̸= u ⊆ {1, . . . , N }, represent an infinite set of multivariate orthogonal polynomials, which is consistent with the probability measure fXu (xu )dxu , satisfying (3.4) (Pu,ju , Pu,ku )fX u dxu \|u\| =: E [Pu,ju (Xu )Pu,ku (Xu )] = 0 ∀ju ̸= ku , ku ∈ N0 . Clearly, each Pu,ju ∈ Πu is a multivariate orthogonal polynomial satisfying (3.3). Due to the product-type probability measure of Xu , a consequence of statistical independence from Assumption 2.1, such multivariate polynomials exist and are easily constructed by tensorizing univariate orthogonal polynomials. Proposition 3.1.Let X := (X1 , . . . , XN )T : (Ω, F) → (AN , B N ) be a vector of N ∈ N input random variables fulfilling Assumption 2.1. Suppose that the sets of univariate orthogonal polynomials for all marginal measures are obtained as {P{i},ji (xi ) : ji ∈ N0 }, i = 1, . . . , N . Then, for ∅ ̸= u ⊆ {1, . . . , N }, the set of multivariate orthogonal polynomials in xu consistent with the probability measure fXu (xu )dxu is { } ⊗{ } \|u\| P{i},ji (xi ) : ji ∈ N0 , (3.5) Pu,ju (xu ) : ju ∈ N0 = ⊗ i∈u where the symbol denotes tensor product. In terms of an element, the multivariate orthogonal polynomial of degree \|ju \| = ji1 + · · · + ji\|u\| is ∏ (3.6) Pu,ju (xu ) = P{i},ji (xi ). i∈u Proof. Consider two distinct polynomials Pu,ju (xu ) and Pu,ku (xu ) from the set {Pu,ju (xu ) : \|u\| ju ∈ N0 } satisfying (3.5). Since ju ̸= ku , ju and ku must diﬀer in at least one component. Without loss of generality, suppose that ji1 ̸= ki1 . Then, by Fubini’s theorem, with statistical independence of random variables in mind, ∫ (Pu,ju Pu,ku )fX dxu = Au Pu,ju (xu )Pu,ku (xu )fXu (xu )dxu u \|u\| [ ] ∏ ∫ = ×\|u\| A{ip } P{ip },jip (xip )P{ip },kip (xip )fXip (xip )dxip × (3.7) p=2 p=2 ∫ P (xi1 )P{i1 },ki1 (xi1 )fXi1 (xi1 )dxi1 {i } {i },j 1 i1 A 1 = 0, 6 S. RAHMAN where the equality to zero in the last line results from the recognition that the inner integral vanishes by setting i = i1 in (3.2). \|u\| In addition, for ju ∈ N0 , ] [ 2 ] ∏ [ 2 (3.8) (Pu,ju Pu,ju )fX dxu =: E Pu,j (X ) = E P (X ) >0 u i {i},ji u u i∈u and is finite by virtue of the existence of the set of univariate orthogonal polynomials \|u\| {P{i},ji (xi ) : ji ∈ N0 } for i = 1, . . . , N . Therefore, {Pu,ju (xu ) : ju ∈ N0 } satisfying (3.5) is a set of multivariate orthogonal polynomials consistent with the probability measure fXu (xu )dxu . Once the multivariate orthogonal polynomials are obtained, they can be scaled to generate multivariate orthonormal polynomials, as follows. Definition 3.2.A multivariate orthonormal polynomial Ψu,ju (xu ), ∅ ̸= u ⊆ {1, . . . , N }, \|u\| ju ∈ N0 , of degree \|ju \| = ji1 + · · · + ji\|u\| that is consistent with the probability measure fXu (xu )dxu is defined as (3.9) Ψu,ju (xu ) := √ Pu,ju (xu ) 2 (X )] E[Pu,j u u where Ψ{i},ji (xi ) := P{i},ji (xi )/ √ = ∏ P{i},j (xi ) √ [ i Ψ{i},ji (xi ), ] =: 2 i∈u i∈u E P{i},ji (Xi ) ∏ 2 E[P{i},j (Xi )] is a univariate orthonormal polynomial in i xi of degree ji that is consistent with the probability measure fXi (xi )dxi 3.3. Dimension-wise orthogonal decomposition of polynomial spaces. An orthogonal decomposition of polynomial spaces entailing dimension-wise splitting leads to PDD. Here, to facilitate such splitting of the polynomial space Πu for any ∅ ̸= u ⊆ {1, . . . , N }, limit the power jip of the ip -th variable, where ip ∈ u ⊆ {1, . . . , N }, p = 1, . . . , \|u\|, and \|u\| > 0, to take on only positive integer values. In consequence, ju := (ji1 , . . . , ji\|u\| ) ∈ N\|u\| , the multi-index of Pu,ju (xu ), has degree \|ju \| = ji1 + · · · + ji\|u\| , varying from \|u\| to ∞ as ji1 ̸= · · · ji\|u\| ̸= 0. For ju ∈ N\|u\| and xu := (xi1 , . . . , xi\|u ), a monomial in the variables xi1 , . . . , xi\|u\| is the ji ji \|u\| product xjuu = xi11 . . . xi\|u\| and has a total degree \|ju \|. A linear combination of xjuu , where \|ju \| = l, \|u\| ≤ l ≤ ∞, is a homogeneous polynomial in xu of degree l. For ∅ ̸= u ⊆ {1, . . . , N }, denote by Qul := span{xjuu : \|ju \| = l, ju ∈ N\|u\| }, \|u\| ≤ l < ∞, the space of homogeneous polynomials in xu of degree l where the individual degree of each variable is non-zero and by Θum := span{xjuu : \|u\| ≤ \|ju \| ≤ m, ju ∈ N\|u\| }, \|u\| ≤ m < ∞, the space of polynomials in xu of degree at least \|u\| and at most m where the individual degree of each variable is non-zero. The dimensions of the vector spaces Qul and Θum , respectively, are { } ( l−1 ) u \|u\| (3.10) dim Ql = # ju ∈ N : \|ju \| = l = \|u\| − 1 POLYNOMIAL DIMENSIONAL DECOMPOSITION and dim Θum = (3.11) m ∑ l=\|u\| u Z\|u\| dim Qul = 7 ) ( ) m ( ∑ l−1 m = . \|u\| − 1 \|u\| l=\|u\| Θu\|u\| . Let := For each \|u\| + 1 ≤ l < ∞, denote by Zlu ⊂ Θul the space of orthogonal polynomials of degree exactly l that are orthogonal to all polynomials in Θul−1 , that is, Zlu := {Pu ∈ Θul : (Pu , Qu )fXu dxu = 0 ∀ Qu ∈ Θul−1 }, \|u\| + 1 ≤ l < ∞. Then Zlu , provided that the support of fXu (xu ) has non-empty interior, is a vector space of dimension ( ) l−1 Mu,l := dim Zlu = dim Qul = . \|u\| − 1 Many choices exist for the basis of Zlu . Here, to be formally proved in Section 3.3.2, select {Pu,ju (xu ) : \|ju \| = l, ju ∈ N\|u\| } ⊂ Zlu to be a basis of Zlu , comprising Mu,l number of basis functions. Each basis function Pu,ju (xu ) is a multivariate orthogonal polynomial of degree \|ju \| as defined earlier. Clearly, Zlu = span{Pu,ju (xu ) : \|ju \| = l, ju ∈ N\|u\| }, \|u\| ≤ l < ∞. According to Proposition 3.3, to be presented later, Pu,ju (Xu ) is orthogonal to Pv,kv (Xv ) whenever (1) u ̸= v and ju , kv are arbitrary; or (2) u = v and ju ̸= kv . Therefore, any two distinct polynomial subspaces Zlu and Zlv′ , where ∅ ̸= u ⊆ {1, . . . , N }, ∅ ̸= v ⊆ {1, . . . , N }, \|u\| ≤ l < ∞, and \|v\| ≤ l′ < ∞, are orthogonal whenever u ̸= v or l ̸= l′ . In consequence, there exist orthogonal decompositions of Θum = m ⊕ l=\|u\| Zlu = m ⊕ l=\|u\| span{Pu,ju (xu ) : \|ju \| = l, ju ∈ N\|u\| } = span{Pu,ju (xu ) : \|u\| ≤ \|ju \| ≤ m, ju ∈ N\|u\| } with the symbol ⊕ representing orthogonal sum and Πu (3.12) = 1⊕ = 1⊕ ∞ ⊕ ⊕ ∅̸=v⊆u l=\|v\| ⊕ ∅̸=v⊆u Zlv =1⊕ ⊕ ∞ ⊕ ∅̸=v⊆u l=\|v\| span{Pv,jv (xv ) : \|jv \| = l, jv ∈ N\|v\| } span{Pv,jv (xv ) : jv ∈ N\|v\| }, where 1 := span{1}, the constant subspace, needs to be added because the subspace Zlv excludes constant functions. Recall that ΠN is the space of all real polynomials in x. Then, setting u = {1, . . . , N } in (3.12) first and then swapping v for u yields yet another orthogonal decomposition of ΠN (3.13) = 1⊕ = 1⊕ = 1⊕ ⊕ ∞ ⊕ ∅̸=u⊆{1,...,N } l=\|u\| ∞ ⊕ ⊕ ∅̸=u⊆{1,...,N } l=\|u\| ⊕ ∅̸=u⊆{1,...,N } Zlu span{Pu,ju (xu ) : \|ju \| = l, ju ∈ N\|u\| } span{Pu,ju (xu ) : ju ∈ N\|u\| }. 8 S. RAHMAN Note that the last expression of (3.13) is equal to the span of (3.14) N ⊗ { } { } N Pj (x) : j ∈ N0 := P{i},ji (xi ) : ji ∈ N0 , i=1 representing an infinite set of orthogonal polynomials in x. Given the dimension-wise orthogonal splitting of ΠN , any square-integrable function of input random vector X can be expanded as a Fourier-like series of hierarchically ordered multivariate orthogonal or orthonormal polynomials in Xu , ∅ = ̸ u ⊆ {1, . . . , N }. The expansion is referred to as PDD, to be formally presented and analyzed in Section 4. 3.4. Statistical properties of random multivariate polynomials. When the input random variables X1 , . . . , XN , instead of real variables x1 , . . . , xN , are inserted in the argument, the multivariate polynomials Pu,ju (Xu ) and Ψu,ju (Xu ), where ∅ = ̸ u ⊆ {1, . . . , N } and ju ∈ N\|u\| , become functions of random input variables. Therefore, it is important to establish their second-moment properties, to be exploited in the remaining part of this section and Section 4. Proposition 3.3.Let X := (X1 , . . . , XN ) be a vector of N ∈ N input random variables fulfilling Assumption 2.1. For ∅ ̸= u, v ⊆ {1, . . . , N }, ju ∈ N\|u\| , and kv ∈ N\|v\| , the first- and second-order moments of multivariate orthogonal polynomials are (3.15) E [Pu,ju (Xu )] = 0 and (3.16) E [Pu,ju (Xu )Pv,kv (Xv )] = ] ∏ [ 2  E P{i},j (X ) i , i u = v, ju = kv , i∈u  0, otherwise, respectively. independence of random variables, E[Pu,ju (Xu )] = ∏ Proof. Using (3.6) and statistical \|u\| i∈u E[P{i},ji (Xi ] for any ju ∈ N . Since each component of ju is non-zero, (3.2), with the constant function P{i},0 ̸= 0 in mind, produces E[P{i},ji (Xi ] = 0 for any i ∈ u, ji ∈ N, resulting in (3.15). To obtain the non-trivial result of (3.16), set u = v and ju = kv and use (3.8) directly. The trivial result of (3.16) is obtained by considering two subcases. First, when u = v and ju ̸= kv , (3.7) yields the result already. Second, when u ̸= v and ju , kv ∈ N\|v\| are arbitrary, then u and v diﬀer by at least one element. Suppose that i ∈ (u ∪ v) is that element with the associated degree ji ∈ N. Using the statistical independence of random variables and the fact that E[P{i},ji (Xi ] = 0, as already demonstrated, produces the desired result. Corollary 3.4.For ∅ ̸= u, v ⊆ {1, . . . , N }, ju ∈ N\|u\| , and kv ∈ N\|v\| , the first- and secondorder moments of multivariate orthonormal polynomials are (3.17) E [Ψu,ju (Xu )] = 0 and (3.18) respectively. E [Ψu,ju (Xu )Ψv,kv (Xv )] = { 1, 0, u = v, ju = kv , otherwise, POLYNOMIAL DIMENSIONAL DECOMPOSITION 9 3.5. Orthogonal basis and completeness. An important question regarding multivariate orthogonal polynomials discussed in the preceding subsection is whether they constitute a complete basis in a function space of interest, such as a Hilbert space. Let L2 (AN , B N , fX dx) represent a Hilbert space of square-integrable functions with respect to the probability measure fX (x)dx supported on AN . The following two propositions show that, indeed, measure-consistent orthogonal polynomials span various spaces of interest. Proposition 3.5.Let X := (X1 , . . . , XN )T : (Ω, F) → (AN , B N ) be a vector of N ∈ N input random variables fulfilling Assumption 2.1 and Xu := (Xi1 , . . . , Xi\|u\| )T : (Ωu , F u ) → (Au , B u ), ∅ ̸= u ⊆ {1, . . . , N }, be a subvector of X. Then {Pu,ju (xu ) : \|ju \| = l, ju ∈ N\|u\| }, the set of multivariate orthogonal polynomials of degree l, \|u\| ≤ l < ∞, consistent with the probability measure fXu (xu )dxu , is a basis of Zlu . Proof. Under Assumption 2.1, orthogonal polynomials consistent with the probability (M ) (1) measure fXu (xu )dxu exist. Denote by Pu,l = (Pu,l , . . . , Pu,l u,l )T a column vector of the elements of {Pu,ju (Xu ) : \|ju \| = l, ju ∈ N\|u\| } arranged according to some monomial order. Let (1) (M ) (j) aTu,l = (au,l , . . . , au,l u,l ) be a row vector comprising some constants au,l ∈ R, j = 1, . . . , Mu,l . Set aTu,l Pu,l = 0. Multiply both sides of the equality from the right by PTu,l , integrate with respect to the measure fXu (xu )dxu over Au , and apply transposition to obtain (3.19) Gu,l au,l = 0, where Gu,l = E[Pu,l PTu,l ] is an Mu,l × Mu,l matrix with its (p, q)th element ∫ [ ] (pq) (p) (q) (p) (q) Gu,l = Pu,l (xu )Pu,l (xu )fXu (xu )dxu = E Pu,l (Xu )Pu,l (Xu ) Au representing the covariance between two elements of Pu,l . According to Proposition 3.3, any two distinct polynomials from {Pu,ju (xu ) : \|ju \| = l, ju ∈ N\|u\| } are orthogonal, meaning (p) (q) that E[Pu,l Pu,l ] is zero if p ̸= q and positive and finite if p = q. Consequently, Gu,l is a diagonal, positive-definite matrix and hence invertible. Therefore, (3.19) yields au,l = 0, proving linear independence of the elements of Pu,l or the set {Pu,ju (xu ) : \|ju \| = l, ju ∈ N\|u\| }. Furthermore, the dimension of Zlu , which is Mu,l , matches exactly the number of elements of the aforementioned set. Therefore, the spanning set {Pu,ju (xu ) : \|ju \| = l, ju ∈ N\|u\| } forms a basis of Zlu . Proposition 3.6.Let X := (X1 , . . . , XN )T : (Ω, F) → (AN , B N ) be a vector of N ∈ N input random variables fulfilling both Assumptions 2.1 and 2.2 and Xu := (Xi1 , . . . , Xi\|u\| )T : (Ωu , F u ) → (Au , B u ), ∅ ̸= u ⊆ {1, . . . , N }, be a subvector of X. Consistent with the probability measure fXu (xu )dxu , let {Pu,ju (xu ) : \|ju \| = l, ju ∈ N\|u\| }, the set of multivariate orthogonal polynomials of degree l, \|u\| ≤ l < ∞, be a basis of Zlu . Then the set of polynomials from the orthogonal sum 1⊕ ⊕ ∞ ⊕ ∅̸=u⊆{1,...,N } l=\|u\| span{Pu,ju (xu ) : \|ju \| = l, ju ∈ N\|u\| } is dense in L2 (AN , B N , fX dx). Moreover, (3.20) L2 (AN , B N , fX dx) = 1 ⊕ ⊕ ∞ ⊕ ∅̸=u⊆{1,...,N } l=\|u\| Zlu 10 S. RAHMAN where the overline denotes set closure. Proof. Under Assumption 2.1, orthogonal polynomials exist. According to Theorem 3.4 of Ernst et al. [10], which exploits Assumption 2.2, the polynomial space Π{i} = R[xi ] is dense in L2 (A{i} , B {i} , fXi dxi ). Now use Theorem 4 of Petersen [19], which asserts that if, for p ≥ 1 and all i = 1, . . . , N , Π{i} is dense in Lp (A{i} , B{i} , fXi dxi ), then so is ΠN = R[x1 , . . . , xN ] in Lp (AN , B N , fX dx). Therefore, the set of polynomials from the orthogonal sum, which is equal to ΠN as per (3.13), is dense in L2 (AN , B N , fX dx). Including the limit points of the orthogonal sum yields (3.20). 4. Polynomial Dimensional Decomposition. Let y(X) := y(X1 , . . . , XN ) be a realvalued, square-integrable output random variable defined on the same probability space (Ω, F, P). The vector space L2 (Ω, F, P) is a Hilbert space such that ∫ ∫ [ ] E y 2 (X) := y 2 (X(ω))dP(ω) = y 2 (x)fX (x)dx < ∞ AN Ω with inner product (y(X), z(X))L2 (Ω,F,P) := ∫ y(X(ω))z(X(ω))dP(ω) = √ (y(X), y(X))L2 (Ω,F,P) = Ω ∫ AN y(x)z(x)fX (x)dx =: (y(x), z(x))fX dx and norm ∥y(X)∥L2 (Ω,F,P) := √ (y(x), y(x))fX dx =: ∥y(x)∥fX dx . It is elementary to show that y(X(ω)) ∈ L2 (Ω, F, P) if and only if y(x) ∈ L2 (AN , B N , fX dx). 4.1. ADD. The ADD, expressed by the recursive form [17, 22] ∑ (4.1a) y(X) = y∅ + yu (Xu ), (4.1b) y∅ = (4.1c) yu (Xu ) = ∫ AN ∫ ∅̸=u⊆{1,...,N } y(x)fX (x)dx, AN −\|u\| y(Xu , x−u )fX−u (x−u )dx−u − ∑ yv (Xv ), v⊂u is a finite, hierarchical expansion of y in terms of its input variables with increasing dimensions, where u ⊆ {1, · · · , N } is a subset with the complementary set −u = {1, · · · , N }\u and yu is a \|u\|-variate component function describing a constant or an \|u\|-variate interaction of Xu = (Xi1 , · · · , Xi\|u\| ) on y when \|u\| = 0 or \|u\| > 0. Here, (Xu , x−u ) denotes an N -dimensional vector whose ith component is Xi if i ∈ u and xi if i ∈ / u. The summation in (4.1a) comprises 2N − 1 terms with each term depending on a group of variables indexed by a particular subset of {1, · · · , N }. When u = ∅, the sum in (4.1c) vanishes, resulting in the expression of the constant function y∅ in (4.1b). When u = {1, · · · , N }, the integration in the last line of (4.1c) is on the empty set, reproducing (4.1a) and hence finding the last function y{1,··· ,N } . Indeed, all component functions of y can be obtained by interpreting literally (4.1c). This decomposition, first presented by Hoeﬀding [16] in relation to his seminal work on U -statistics, has been studied by many other researchers described by Efron and Stein [9], the author [22], and references cited therein. POLYNOMIAL DIMENSIONAL DECOMPOSITION 11 The ADD can also be generated by tensorizing a univariate function space decomposition into its constant subspace and remainder, producing [14] ⊕ (4.2) L2 (AN , B N , fX dx) = 1 ⊕ Wu , ∅̸=u⊆{1,...,N } where { } Wu := yu ∈ L2 (Au , B u , fXu dxu ) : E [yu (Xu )yv (Xv )] = 0 if u ̸= v, v ⊆ {1, . . . , N } is an ADD subspace comprising \|u\|-variate component functions of y. However, the subspaces Wu , ∅ ̸= u ⊆ {1, . . . , N }, are in general infinite-dimensional; therefore, further discretization of Wu is necessary. For instance, by introducing measure-consistent orthogonal polynomial basis discussed in Section 3, a component function yu (Xu ) ∈ Wu can be expressed as a linear combination of these basis functions. Indeed, comparing (3.20) and (4.2) yields the closure of an orthogonal decomposition of Wu = (4.3) ∞ ⊕ l=\|u\| Zlu into polynomial spaces Zlu , \|u\| ≤ l < ∞. The result is a polynomial refinement of ADD, which is commonly referred to as PDD. 4.2. PDD. The PDD of a square-integrable random variable y(X) ∈ L2 (Ω, F, P) is simply the expansion of y(X) with respect to a complete, hierarchically ordered, orthonormal polynomial basis of L2 (Ω, F, P). There are at least two ways to explain PDD: a polynomial variant of ADD and a dimension-wise orthogonal polynomial expansion. 4.2.1. Polynomial variant of ADD. The first approach, explained by the author in a prior work [20], involves the following two steps: (1) expand the ANOVA component function ∑ Cu,ju Ψu,ju (Xu ) (4.4) yu (Xu ) ∼ ju ∈N\|u\| in terms of the basis of Wu , which originally stems from the basis of Zlu , \|u\| ≤ l < ∞, with ∫ yu (Xu )Ψu,ju (xu )fXu (xu )dxu , ∅ ̸= u ⊆ {1, . . . , N }, ju ∈ N\|u\| , (4.5) Cu,ju = A\|u\| representing the associated expansion coeﬃcients; and (2) apply (4.1c) to (4.5) and exploit orthogonal properties of the basis. The end result is the PDD [20] of ∑ ∑ (4.6) y(X) ∼ y∅ + Cu,ju Ψu,ju (Xu ), ∅̸=u⊆{1,...,N } ju ∈N\|u\| where, eventually, (4.7) Cu,ju = ∫ AN y(x)Ψu,ju (xu )fX (x)dx. Comparing (4.1) and (4.6), the connection between PDD and ADD is clearly palpable, where the former can be viewed as a polynomial variant of the latter. For instance, Cu,ju Ψu,ju (Xu ) in (4.6) represents a \|u\|-variate, \|ju \|th-order PDD component function of y(X), describing the \|ju \|th-order polynomial approximation of yu (Xu ). In addition, PDD inherits all desirable properties of ADD [20]. 12 S. RAHMAN 4.2.2. Dimension-wise orthogonal polynomial expansion. The second approach entails polynomial expansion associated with the dimension-wise orthogonal splitting of polynomial spaces, as explained in Section 3.3. The latter approach has not been published elsewhere and is, therefore, formally presented here as Theorem 4.1. Theorem 4.1. Let X := (X1 , . . . , XN )T : (Ω, F) → (AN , BN ) be a vector of N ∈ N input random variables fulfilling Assumptions 2.1 and 2.2. For ∅ ̸= u ⊆ {1, . . . , N } and Xu := (Xi1 , . . . , Xi\|u\| )T : (Ωu , F u ) → (Au , B u ), denote by {Ψu,ju (Xu ): ju ∈ N\|u\| } the set of multivariate orthonormal polynomials consistent with the probability measure fXu (xu )dxu . Then (1) any random variable y(X) ∈ L2 (Ω, F, P) can be hierarchically expanded as a Fourier-like series, referred to as the PDD of y(X) ∼ y∅ + (4.8) = y∅ + ∑ ∞ ∑ ∑ Cu,ju Ψu,ju (Xu ) ∅̸=u⊆{1,...,N } l=\|u\| ju ∈N\|u\| \|ju \|=l ∑ ∑ Cu,ju Ψu,ju (Xu ), ∅̸=u⊆{1,...,N } ju ∈N\|u\| where the expansion coeﬃcients y∅ ∈ R and Cu,ju ∈ R, ∅ ̸= u ⊆ {1, . . . , N }, ju ∈ N\|u\| , are defined by ∫ (4.9) y∅ := E [y(X)] := y(x)fX (x)dx, AN Cu,ju := E [y(X)Ψu,ju (Xu )] := (4.10) ∫ AN y(x)Ψu,ju (xu )fX (x)dx; and (2) the PDD of y(X) ∈ L2 (Ω, F, P) converges to y(X) in mean-square; furthermore, the PDD converges in probability and in distribution. Proof. Under Assumptions 2.1 and 2.2, a complete infinite set of multivariate orthogonal polynomials in xu consistent with the probability measure fXu (xu )dxu exists. From Proposition 3.6 and the fact that orthonormality is merely scaling, the set of polynomials from the orthogonal sum (4.11) 1⊕ ⊕ ∞ ⊕ ∅̸=u⊆{1,...,N } l=\|u\| span{Ψu,ju (xu ) : \|ju \| = l, ju ∈ N\|u\| } = ΠN is also dense in L2 (AN , B N , fX dx). Therefore, any square-integrable random variable y(X) can be expanded as shown in (4.8). Combining the two inner sums of the expansion forms the equality in the second line of (4.8). From the denseness, one has the Bessel’s inequality [7] (4.12) [ E y∅ + ∑ ∑ ∅̸=u⊆{1,...,N } ju ∈N\|u\| ]2 Cu,ju Ψu,ju (Xu ) [ ] ≤ E y 2 (X) , POLYNOMIAL DIMENSIONAL DECOMPOSITION 13 proving that the PDD converges in mean-square or L2 . To determine the limit of convergence, invoke again Proposition 3.6, which implies that the set on the left side of (4.11) is complete in L2 (AN , B N , fX dx). Therefore, Bessel’s inequality becomes an equality [ ]2 ∑ ∑ [ ] (4.13) E y∅ + Cu,ju Ψu,ju (Xu ) = E y 2 (X) , ∅̸=u⊆{1,...,N } ju ∈N\|u\| known as the Parseval identity [7] for a multivariate orthonormal system, for every random variable y(X) ∈ L2 (Ω, F, P). Furthermore, as the PDD converges in mean-square, it does so in probability. Moreover, as the expansion converges in probability, it also converges in distribution. Finally, to find the expansion coeﬃcients, define a second moment ]2 [ ∑ ∑ (4.14) ePDD := E y(X) − y∅ − Cv,kv Ψv,kv (Xv ) ∅̸=v⊆{1,...,N } kv ∈N\|v\| of the diﬀerence between y(X) and its full PDD. Diﬀerentiate both sides of (4.14) with respect to y∅ and Cu,ju , ∅ ̸= u ⊆ {1, . . . , N }, ju ∈ N\|u\| , to write ∂ePDD ∂y∅ (4.15) ]2 [ ∑ ∑ ∂ Cv,kv Ψv,kv (Xv ) = E y(X) − y∅ − ∂y∅ ∅̸=v⊆{1,...,N } kv ∈N\|v\| [ }2 ] { ∑ ∑ ∂ = E Cv,kv Ψv,kv (Xv ) y(X) − y∅ − ∂y∅ \|v\| ∅̸ = v⊆{1,...,N } k ∈N v } ] [{ ∑ ∑ Cv,kv Ψv,kv (Xv ) − y(X) × 1 = 2E y∅ + ∅̸=v⊆{1,...,N } kv ∈N\|v\| = 2 {y∅ − E [y(X)]} and ∂ePDD ∂Cu,ju (4.16) ]2 [ ∑ ∑ ∂ Cv,kv Ψv,kv (Xv ) = E y(X) − y∅ − ∂Cu,ju ∅̸=v⊆{1,...,N } kv ∈N\|v\| [ { }2 ] ∑ ∑ ∂ Cv,kv Ψv,kv (Xv ) = E y(X) − y∅ − ∂Cu,ju \|v\| ∅̸ = v⊆{1,...,N } kv ∈N [{ } ] ∑ ∑ = 2E y∅ + Cv,kv Ψv,kv (Xv ) − y(X) Ψu,ju (Xu ) ∅̸=v⊆{1,...,N } kv ∈N\|v\| = 2 {Cu,ju − E [y(X)Ψu,ju (Xu )]} . Here, the second, third, and last lines of both (4.15) and (4.16) are obtained by interchanging the diﬀerential and expectation operators, performing the diﬀerentiation, swapping the expectation and summation operators and applying Corollary 3.4, respectively. The interchanges are permissible as the infinite sum is convergent as demonstrated in the preceding paragraph. Setting ∂ePDD /∂y∅ = 0 in (4.15) and ∂ePDD /∂Cu,ju = 0 in (4.16) yields (4.9) and (4.10), respectively, completing the proof. 14 S. RAHMAN The expressions of the expansion coeﬃcients can also be derived by simply replacing y(X) in (4.9) and (4.10) with the full PDD and then using Corollary 3.4. In contrast, the proof given here demonstrates that the PDD coeﬃcients are determined optimally. It should be emphasized that the function y must be square-integrable for the meansquare and other convergences to hold. However, the rate of convergence depends on the smoothness of the function. The smoother the function, the faster the convergence. If the function is a polynomial, then its PDD exactly reproduces the function. These results can be easily proved using classical approximation theory. A related expansion, known by the name of RS-HDMR [18], also involves orthogonal polynomials in connection with ADD. However, the existence, convergence, and approximation quality of the expansion, including its behavior for infinitely many input variables, have not been reported. 4.3. Truncation. The full PDD contains an infinite number of orthonormal polynomials or coeﬃcients. In practice, the number must be finite, meaning that PDD must be truncated. However, there are multiple ways to perform the truncation. A straightforward approach adopted in this work entails (1) keeping all polynomials in at most 0 ≤ S ≤ N variables, thereby retaining the degrees of interaction among input variables less than or equal to S and (2) preserving polynomial expansion orders (total) less than or equal to S ≤ m < ∞. The result is an S-variate, mth-order PDD approximation2 yS,m (X) = y∅ + (4.17) = y∅ + S m ∑ ∑ s=1 l=s ∑ ∑ ∑ Cu,ju Ψu,ju (Xu ) ∅̸=u⊆{1,...,N } ju ∈N\|u\| \|u\|=s \|ju \|=l ∑ Cu,ju Ψu,ju (Xu ) ∅̸=u⊆{1,...,N } ju ∈N\|u\| 1≤\|u\|≤S \|u\|≤\|ju \|≤m of y(X), containing (4.18) LS,m = 1 + S ( )( ) ∑ N m s=1 s s number of expansion coeﬃcients including y∅ . It is important to clarify a few things about the truncated PDD proposed. First, a diﬀerent truncation with respect to the polynomial expansion order based on ∞-norm as opposed to 1-norm, that is, ∥ju ∥∞ ≤ m, was employed in prior works [20, 21, 24]. Therefore, comparing (4.17) and (4.18) with the existing truncation, if it is desired, should be done with care. Having said this, the proposed truncation has one advantage over the existing one: a direct comparison with a truncated PCE is possible; this will be further explained in the forthcoming sections. Second, the right side of (4.17) contains sums of at most S-dimensional orthonormal polynomials, representing at most S-variate PDD component functions of y. Therefore, the term “S-variate” used for the PDD approximation should be interpreted in the context of including at most S-degree interaction of input variables, even though yS,m is strictly an N -variate function. Third, when S = 0, y0,m = y∅ for any m as the outer sums of (4.17) vanish. Finally, when S → N 2 The nouns degree and order associated with PDD or orthogonal polynomials are used synonymously in the paper. POLYNOMIAL DIMENSIONAL DECOMPOSITION 15 and m → ∞, yS,m converges to y in the mean-square sense, generating a hierarchical and convergent sequence of PDD approximations. Readers interested in an adaptive version of PDD, where the truncation parameters are automatically chosen, are directed to the work of Yadav and Rahman [30], including an application to design optimization [25]. It is natural to ask about the approximation quality of (4.17). Since the set of polynomials from the orthogonal sum in (4.11) is complete in L2 (AN , B N , fX dx), the truncation error y(X) − yS,m (X) is orthogonal to any element of the subspace from which yS,m (X) is chosen, as demonstrated below. Proposition 4.2.For any y(X) ∈ L2 (Ω, F, P), let yS,m (X) be its S-variate, mth-order PDD approximation. Then the truncation error y(X)−yS,m (X) is orthogonal to the subspace ΠN S,m := 1 ⊕ (4.19) ⊕ ⊕ ∅̸=u⊆{1,...,N } ju ∈N\|u\| 1≤\|u\|≤S \|u\|≤\|ju \|≤m span{Ψu,ju (Xu ) : ju ∈ N\|u\| } ⊆ L2 (Ω, F, P), comprising all polynomials in X with the degree of interaction at most S and order at most m, including constants. Moreover, E[y(X) − yS,m (X)]2 → 0 as S → N and m → ∞. Proof. Let ∑ ∑ (4.20) ȳS,m (X) := ȳ∅ + C̄v,kv Ψv,kv (Xv ), ∅̸=v⊆{1,...,N } kv ∈N\|v\| 1≤\|v\|≤S \|v\|≤\|kv \|≤m with arbitrary expansion coeﬃcients ȳ∅ and C̄v,kv , be any element of the subspace ΠN S,m of 2 L (Ω, F, P) described by (4.19). Then (4.21) E[{ [{y(X) − yS,m (X)}ȳS,m (X)] } ∑ ∑ ∑ ∑ = E Cu,ju Ψu,ju (Xu ) + Cu,ju Ψu,ju (Xu ) { ∅̸=u⊆{1,...,N } ju ∈N\|u\| 1≤\|u\|≤S m+1≤\|ju \|<∞ × ȳ∅ + = 0, ∑ ∑ ∅̸=v⊆{1,...,N } kv 1≤\|v\|≤S \|v\|≤\|kv \|≤m ∈N\|v\| ∅̸=u⊆{1,...,N } ju ∈N\|u\| S+1≤\|u\|≤N \|u\|≤\|ju \|<∞ }] C̄v,kv Ψv,kv (Xv ) where the last line follows from Corollary 3.4, proving the first part of the proposition. For the latter part, the Pythagoras theorem yields (4.22) 2 E[{y(X) − yS,m (X)}2 ] + E[yS,m (X)] = E[y 2 (X)]. 2 (X)] → E[y 2 (X)] as S → N and m → ∞. Therefore, E[{y(X) − From Theorem 4.1, E[yS,m 2 yS,m (X)} ] → 0 as S → N and m → ∞. The second part of Proposition 4.2 entails L2 convergence, which is the same as the mean-square convergence described in Theorem 4.1. However, an alternative route is chosen for the proof of the proposition. Besides, Proposition 4.2 implies that the PDD approximation is optimal as it recovers the best approximation from the subspace ΠN S,m , as described by Corollary 4.3. 16 S. RAHMAN Corollary 4.3.Let ΠN S,m in (4.19) define the subspace of all polynomials in X with the degree of interaction at most S and order at most m, including constants. Then the S-variate, mth-order PDD approximation yS,m (X) of y(X) ∈ L2 (Ω, F, P) is the best approximation in the sense that E [y(X) − yS,m (X)]2 = (4.23) inf ȳS,m ∈ΠN S,m E [y(X) − ȳS,m (X)]2 . Proof. Consider two elements yS,m (X) and ȳS,m (X) of the subspace ΠN S,m , where the former is the S-variate, mth-order PDD approximation of y(X) with the expansion coefficients defined by (4.9) and (4.10) and the latter is any S-variate, mth-order polynomial function, described by (4.20), with arbitrary chosen expansion coeﬃcients. From Proposition 4.2, the truncation error y(X) − yS,m (X) is orthogonal to both yS,m (X) and ȳS,m (X) and is, therefore, orthogonal to their linear combinations, yielding E [{y(X) − yS,m (X)}{yS,m (X)] − ȳS,m (X)}] = 0. Consequently, (4.24) E [y(X) − ȳS,m (X)]2 = E [y(X) − yS,m (X)]2 + E [yS,m (X) − ȳS,m (X)]2 ≥ E [y(X) − yS,m (X)]2 , as the second expectation on the right side of the first line of (4.24) is non-negative, thereby proving the mean-square optimality of the S-variate, mth-order PDD approximation. The motivations behind ADD- and PDD-derived approximations are the following. In a practical setting, the function y(X), fortunately, has an eﬀective dimension [3] much lower than N , meaning that the right side of (4.1a) can be eﬀectively approximated by a sum of lower-dimensional component functions yu , \|u\| ≪ N , but still maintaining all random variables X of a high-dimensional uncertainty quantification problem. Furthermore, an Svariate, mth-order PDD approximation is grounded on a fundamental conjecture known to be true in many real-world uncertainty quantification problems: given a high-dimensional function y, its \|u\|-variate, \|ju \|th-order PDD component function Cu,ju Ψu,ju (Xu ), where S + 1 ≤ \|u\| ≤ N and m+1 ≤ \|ju \| < ∞, is small and hence negligible, leading to an accurate lowvariate, low-order approximation of y. The computational complexity of a truncated PDD is polynomial, as opposed to exponential, thereby alleviating the curse of dimensionality to a substantial extent. Although PCE contains the same orthogonal polynomials, a recent work on random eigenvalue analysis of dynamic systems reveals markedly higher convergence rate of the PDD approximation than the PCE approximation [24]. 4.4. Output statistics and other probabilistic characteristics. The S-variate, mthorder PDD approximation yS,m (X) can be viewed as a surrogate of y(X). Therefore, relevant probabilistic characteristics of y(X), including its first two moments and probability density function, if it exists, can be estimated from the statistical properties of yS,m (X). Applying the expectation operator on yS,m (X) and y(X) in (4.17) and (4.8) and imposing Corollary 3.4, their means (4.25) E [yS,m (X)] = E [y(X)] = y∅ POLYNOMIAL DIMENSIONAL DECOMPOSITION 17 are the same and independent of S and m. Therefore, the PDD truncated for any values of 0 ≤ S ≤ N and S ≤ m < ∞ yields the exact mean. Nonetheless, E[yS,m (X)] will be referred to as the S-variate, mth-order PDD approximation of the mean of y(X). Applying the expectation operator again, this time on [yS,m (X) − y∅ ]2 and [y(X) − y∅ ]2 , and employing Corollary 3.4 results in the variances (4.26) ∑ var [yS,m (X)] = ∑ 2 Cu,j u ∅̸=u⊆{1,...,N } ju 1≤\|u\|≤S \|u\|≤\|ju \|≤m ∈N\|u\| and (4.27) var [y(X)] = ∑ ∑ 2 Cu,j u ∅̸=u⊆{1,...,N } ju ∈N\|u\| of yS,m (X) and y(X), respectively. Again, var[yS,m (X)] will be referred to as the S-variate, mth-order PDD approximation of the variance of y(X). Clearly, var[yS,m (X)] approaches var[y(X)], the exact variance of y(X), as S → N and m → ∞. Being convergent in probability and distribution, the probability density function of y(X), if it exists, can also be estimated by that of yS,m (X). However, no analytical formula exists for the density function. In that case, the density can be estimated by sampling methods, such as Monte Carlo simulation (MCS) of yS,m (X). Such simulation should not be confused with crude MCS of y(X), commonly used for producing benchmark results whenever possible. The crude MCS can be expensive or even prohibitive, particularly when the sample size needs to be very large for estimating tail probabilistic characteristics. In contrast, the MCS embedded in the PDD approximation requires evaluations of simple polynomial functions that describe yS,m . Therefore, a relatively large sample size can be accommodated in the PDD approximation even when y is expensive to evaluate. 4.5. Infinitely many input variables. In many fields, such as uncertainty quantification, information theory, and stochastic process, functions depending on a countable sequence {Xi }i∈N of input random variables need to be considered [15]. Under certain assumptions, PDD is still applicable as in the case of finitely many random variables, as demonstrated by the following proposition. Proposition 4.4.Let {Xi }i∈N be a countable sequence of input random variables defined on the probability space (Ω, F∞ , P), where F∞ := σ({Xi }i∈N ) is the associated σ-algebra generated. If the sequence {Xi }i∈N satisfies Assumptions 2.1 and 2.2, then the PDD of y({Xi }i∈N ) ∈ L2 (Ω, F∞ , P), where y : AN → R, converges to y({Xi }i∈N ) in mean-square. Moreover, the PDD converges in probability and in distribution. Proof. According to Proposition 3.6, ΠN is dense in L2 (AN , B N , fX dx) and hence in 2 L (Ω, FN , P) for every N ∈ N, where FN := σ({Xi }N i=1 ) is the associated σ-algebra generated by {Xi }N . Here, with a certain abuse of notation, ΠN is used as a set of polynomial i=1 functions of both real variables x and random variables X. Now, apply Theorem 3.8 of Ernst et al. [10], which says that if ΠN is dense in L2 (Ω, FN , P) for every N ∈ N, then Π∞ := ∞ ∪ N =1 ΠN , 18 S. RAHMAN a subspace of L2 (Ω, F∞ , P), is also dense in L2 (Ω, F∞ , P). But, using (4.11), Π∞ = ∞ ∪ N =1 = 1⊕ 1⊕ ∞ ⊕ ⊕ ∅̸=u⊆{1,...,N } l=\|u\| ∞ ⊕ ⊕ ∅̸=u⊆N l=\|u\| span{Ψu,ju : \|ju \| = l, ju ∈ N\|u\| } span{Ψu,ju : \|ju \| = l, ju ∈ N\|u\| }, demonstrating that the set of polynomials from the orthogonal sum in the last line is dense in L2 (Ω, F∞ , P). Therefore, the PDD of y({Xi }i∈N ) ∈ L2 (Ω, F∞ , P) converges to y({Xi }i∈N ) in mean-square. Since the mean-square convergence is stronger than the convergence in probability or in distribution, the latter modes of convergence follow readily. 5. Polynomial Chaos Expansion. In contrast to the dimension-wise splitting of polynomial spaces in PDD, a degree-wise orthogonal splitting of polynomial spaces results in PCE. The latter decomposition is briefly summarized here as PCE will be compared with PDD in the next section. 5.1. Degree-wise orthogonal decomposition of polynomial spaces. Let j := j{1,...,N } = (j1 , . . . , , jN ) ∈ NN 0 , ji ∈ N0 , i ∈ {1, . . . , N }, define an N -dimensional multi-index. For x = (x1 , . . . , xN ) ∈ AN ⊆ RN , a monomial in the variables x1 , . . . , xN is the product xj = xj11 · · · xjNN and has a total degree \|j\| = j1 + · · · + jN . Denote by j N ΠN p := span{x : 0 ≤ \|j\| ≤ p, j ∈ N0 }, p ∈ N0 , the space of real polynomials in x of degree at most p. Let V0N := ΠN 0 = span{1} be the space of constant functions. For each 1 ≤ l < ∞, denote by VlN ⊂ ΠN l the space of orthogonal polynomials of degree exactly l that are orthogonal to all polynomials in ΠN l−1 , that is, N VlN := {P ∈ ΠN l : (P, Q)fX dx = 0 ∀ Q ∈ Πl−1 }, 1 ≤ l < ∞. N From Section 3, with u = {1, . . . , N } in mind, select {Pj (x) : \|j\| = l, j ∈ NN 0 } ⊂ Vl to be N a basis of Vl . Each basis function Pj (x) is a multivariate orthogonal polynomial in x of degree \|j\|. Obviously, VlN = span{Pj (x) : \|j\| = l, j ∈ NN 0 }, 0 ≤ l < ∞. According to (3.7) with u = {1, . . . , N }, Pj (x) is orthogonal to Pk (x) whenever j ̸= k. Therefore, any two polynomial subspaces VlN and VrN , where 0 ≤ l, r < ∞, are orthogonal whenever l ̸= r. In consequence, there exists another orthogonal decomposition of ⊕ ⊕ N (5.1) ΠN = VlN = span{Pj (x) : \|j\| = l, j ∈ NN 0 } = span{Pj (x) : j ∈ N0 }. l∈N0 l∈N0 Compared with (3.13), (5.1) represents a degree-wise orthogonal decomposition of ΠN . 5.2. PCE. Given the degree-wise orthogonal decomposition of ΠN , the PCE of any square-integrable output random variable y(X) is expressed by [5, 10, 28, 29] (5.2) y(X) ∼ ∞ ∑ ∑ l=0 j∈NN 0 \|j\|=l Cj Ψj (X) = ∑ j∈NN 0 Cj Ψj (X), POLYNOMIAL DIMENSIONAL DECOMPOSITION 19 where {Ψj (X) : j ∈ NN 0 } is an infinite set of measure-consistent multivariate orthonormal polynomials in X that can be obtained by scaling Pj in (3.14) and Cj ∈ R, j ∈ NN 0 , are the 2 PCE expansion coeﬃcients. Like PDD, the PCE of y(X) ∈ L (Ω, F, P) under Assumptions 2.1 and 2.2 also converges to y(X) in mean-square, in probability, and in distribution. Since the PCE of y(X) in (5.2) is an infinite series, it must also be truncated in applications. A commonly adopted truncation is based on retaining orders of polynomials less than or equal to a specified total degree. In this regard, given 0 ≤ p < ∞, the pth-order PCE approximation of y(X) ∈ L2 (Ω, F, P) reads (5.3) p ∑ ∑ yp (X) = l=0 ∑ Cj Ψj (X) = j∈NN 0 \|j\|=l Cj Ψj (X). j∈NN 0 0≤\|j\|≤p This kind of truncation is related to the total degree index set { } N ∑ j ∈ NN ji ≤ p 0 : i=1 for defining the recovered multivariate polynomial space of a pth-order PCE approximation. Other kinds of truncation entail { } { } N ∏ N N j ∈ N0 : max ji ≤ p and j ∈ N0 : (ji + 1) ≤ p + 1 , i=1,...,N i=1 describing the tensor product and hyperbolic cross index sets, respectively, to name just two. The total degree and tensor product index sets are common choices, although the latter one suﬀers from the curse of dimensionality, making it impractical for high-dimensional problems. The hyperbolic cross index set, originally introduced for approximating periodic functions by trigonometric polynomials [2], is relatively a new idea and has yet to receive widespread attention. All of these choices and possibly others, including their anisotropic versions, can be used for truncating PCE. In this work, however, only the total degree index set is used for the PCE approximation. This is consistent with the 1-norm of ju used for truncating PDD in (4.17). 6. Error Analysis. 6.1. PDD error. Define a second-moment error, eS,m := E [y(X) − yS,m (X)]2 , (6.1) stemming from the S-variate, mth-order PDD approximation presented in the preceding section. Replacing y(X) and yS,m (X) in (6.1) with the right sides of (4.8) and (4.17), respectively, produces (6.2) eS,m = S ∑ ∞ ∑ ∑ ∑ s=1 l=m+1 ∅̸=u⊆{1,...,N } ju \|u\|=s \|ju \|=l 2 Cu,j u ∈N\|u\| + N ∑ s=S+1 ∞ ∑ l=s ∑ ∑ 2 Cu,j , u ∅̸=u⊆{1,...,N } ju \|u\|=s \|ju \|=l ∈N\|u\| where the second term vanishes expectedly when S = N as the lower limit of the outer sum exceeds the upper limit. In (6.2), the first term of the PDD error is due to the truncation 20 S. RAHMAN of polynomial expansion orders involving interactive eﬀects of at most S variables, whereas the second term of the PDD error is contributed by ignoring the interactive eﬀects of larger than S variables. Obviously, the error for a general function y depends on which expansion coeﬃcients decay and how they decay with respect to S and m. Nonetheless, the error decays monotonically with respect to S and/or m as stated in Proposition 6.1. Other than that, nothing more can be said about the PDD error. Proposition 6.1.For a general function y, eS+i,m+j ≤ eS,m , where 0 ≤ S < N , S ≤ m < ∞, and i and j are equal to either 0 or 1, but not both equal to 0. Proof. Setting i = 1, j = 0 and using (6.2), eS+1,m − eS,m = ∞ ∑ ∑ ∑ 2 Cu,j − u l=m+1 ∅̸=u⊆{1,...,N } ju ∈N\|u\| \|u\|=S+1 \|ju \|=l ∞ ∑ ∑ ∑ 2 Cu,j ≤ 0, u l=S+1 ∅̸=u⊆{1,...,N } ju ∈N\|u\| \|u\|=S+1 \|ju \|=l where the inequality to zero in the last line results from the fact that, as S ≤ m, the first term is smaller than or equal to the second term. Similarly, setting i = 0, j = 1 and using (6.2), S ∑ ∑ ∑ 2 eS,m+1 − eS,m = − Cu,j ≤ 0. u s=1 ∅̸=u⊆{1,...,N } ju ∈N\|u\| \|u\|=s \|ju \|=m+1 Finally, setting i = 1, j = 1, eS+1,m+1 − eS,m = eS+1,m+1 − eS,m+1 + eS,m+1 − eS,m ≤ 0, as eS+1,m+1 − eS,m+1 ≤ 0 and eS,m+1 − eS,m ≤ 0. Corollary 6.2.For a general function y, eS ′ ,m′ ≤ eS,m whenever S ′ ≥ S and m′ ≥ m. In practice, the eﬀects of interaction among input variables and polynomial expansion 2 , which is equal to order become increasingly weaker as \|u\| and \|ju \| grow. In this case, Cu,j u 2 the variance of Cu,ju Ψu,ju (Xu ), decreases with \|u\| and \|ju \|. Given the rates at which Cu,j u decreases with \|u\| and \|ju \|, a question arises on how fast does eS,m decay with respect to S and m. Proposition 6.3, Corollary 6.4, and subsequent discussions provide a few insights. 2 , ∅ ̸= u ⊆ {1, · · · , N }, Proposition 6.3.For a class of functions y, assume that Cu,j u −\|u\| −\|j \| 2 ju ∈ N\|u\| , attenuates according to Cu,j ≤ Cp1 p2 u , where C > 0, p1 > 1, and p2 > 1 u are three real-valued constants. Then it holds that [{ ] } 1 + p1 (p2 − 1) N (6.3) var [y(X)] ≤ C −1 p1 (p2 − 1) and (6.4) eS,m ≤ C [ S ∑ s=1 ∞ ∑ l=m+1 ( )( ) N ∑ N l − 1 −s −l p1 p2 + s s−1 s=S+1 ] ) ∞ ( )( ∑ N l − 1 −s −l p p . s s−1 1 2 l=s Proof. With the recognition that ( ) ( ) N l−1 \|u\| #{∅ ̸= u ⊆ {1, · · · , N } : \|u\| = s} = , #{ju ∈ N : \|ju \| = l} = , s \|u\| − 1 POLYNOMIAL DIMENSIONAL DECOMPOSITION 21 −\|u\| −\|j \| 2 use Cu,j ≤ Cp1 p2 u in (4.27) and (6.2) to obtain (6.3) and (6.4). u Corollary 6.4.For the function class described in Proposition 6.3, eS+i,m+j < eS,m , where 0 ≤ S < N , S ≤ m < ∞, and i and j are equal to either 0 or 1, but not both equal to 0. According to Corollary 6.4, eS,m decays strictly monotonically with respect to S and/or m for any rate parameters p1 > 1 and p2 > 1. When the equality holds in (6.3) and (6.4) from Proposition 6.3, Figure 1, comprising three subfigures, presents three sets of plots of the relative error, eS,m /var[y(X)], against m for five distinct values of S = 1, 2, 3, 5, 9. These subfigures, each obtained for N = 20, correspond to three distinct cases of the values of p1 and p2 : (1) p1 = 500, p2 = 5; (2) p1 = 5, p2 = 500; and (3) p1 = 500, p2 = 500. In all cases, the error for a given S decays first with respect to m, and then levels oﬀ at a respective limit when m is suﬃciently large. The limits get progressively smaller when S increases as expected. However, the magnitude of this behavior depends on the rates at which the expansion coeﬃcient attenuates with respect to the degree of interaction and polynomial expansion order. When p1 > p2 , as in case 1 [Figure 1 (top)], the error for a given S decays slowly with respect to m due to a relatively weaker attenuation rate associated with the polynomial expansion order. The trend reverses when the attenuation rate becomes stronger and reaches the condition p1 < p2 , as in case 2 [Figure 1 (middle)]. For larger values of S, for example, S = 5 or 9, the respective limits are significantly lower in case 2 than in case 1. When the attenuation rates are the same and large, as in case 3 [Figure 1 (bottom)], the decay rate of error accelerates substantially. 6.2. Relationship between PDD and PCE. Since PDD and PCE share the same orthonormal polynomials, they are related. Indeed, the relationship was first studied by Rahman and Yadav [24], who determined that any one of the two infinite series from PDD and PCE defined by (4.8) and (5.2) can be rearranged to derive the other. In other words, the PDD can also be viewed as a reshuﬄed PCE and vice versa. However, due to a strong connection to ADD endowed with a desired hierarchical structure, PDD merits its own appellation. More importantly, the PDD and PCE when truncated are not the same. In fact, two important observations stand out prominently. First, the terms in the PCE approximation are organized with respect to the order of polynomials. In contrast, the PDD approximation is structured with respect to the degree of interaction between a finite number of random variables. Therefore, significant diﬀerences may exist regarding the accuracy, eﬃciency, and convergence properties of their truncated sum. Second, if a stochastic response is highly nonlinear, but contains rapidly diminishing interactive eﬀects of multiple random variables, the PDD approximation is expected to be more eﬀective than the PCE approximation. This is because the lower-variate terms of the PDD approximation can be just as nonlinear by selecting appropriate values of m in (4.17). In contrast, many more terms and expansion coeﬃcients are required to be included in the PCE approximation to capture such high nonlinearity. In this work, a theoretical comparison between PDD and PCE in the context of error analysis, not studied in prior works, is presented. For error analysis, it is convenient to write a PCE approximation in terms of a PDD approximation. Indeed, there exists a striking result connecting PCE with PDD approximations, as explained in Proposition 6.3. Proposition 6.5.Let yp (X) and yS,m (X) be the pth-order PCE approximation and Svariate, mth-order PDD approximation of y(X) ∈ L2 (Ω, F, P), respectively, where 0 ≤ S ≤ N , S ≤ m < ∞, and 0 ≤ p < ∞. Then the pth-order PCE approximation and the 22 S. RAHMAN Figure 1. PDD errors for various attenuation rates of the expansion coeﬃcients; (top) p1 = 500, p2 = 5; (middle) p1 = 5, p2 = 500; (bottom) p1 = 500, p2 = 500. POLYNOMIAL DIMENSIONAL DECOMPOSITION 23 (p ∧ N )-variate, pth-order PDD approximation are the same, that is, (6.5) yp (X) = yp∧N,p (X), where y0,0 (X) = y∅ and p ∧ N denotes the minimum of p and N . Proof. According to Rahman and Yadav [24], the right side of (5.3) can be reshuﬄed, resulting in a long form of the PCE approximation, expressed by [ N −s+1 ] p−s+1 p−s+1 N N s ∑ ∑ ∑ ∑ ∑ ∏ (6.6) yp (X) = y∅ + ··· ··· C{i1 ···is },(ji1 ···jis ) ψiq jiq (Xiq ) , s=1 i1 =1 \| is =is−1 +1 {z s sums } ji1 =1 \| {z jis =1 } q=1 s sums;ji1 +···+jis ≤p in terms of the PDD expansion coeﬃcients. Note that, depending on the condition p ≤ N or p ≥ N , at most p-dimensional or N -dimensional sums survive in (6.6), meaning that the pth-order PCE approximation retains eﬀects of at most (p ∧ N )-degree interaction and at most pth-order polynomial expansion order. Accordingly, the compact form of the PCE approximation can be written as (6.7) yp (X) = y∅ + p∧N ∑ s=1 p ∑ l=s ∑ ∑ Cu,ju Ψu,ju (Xu ) = yp∧N,p (X), ∅̸=u⊆{1,...,N } ju \|u\|=s \|ju \|=l ∈N\|u\| completing the proof. Using Proposition 6.5, the number of expansion coeﬃcients, say, Lp associated with the pth-order PCE approximation can be calculated from that required by the (p ∧ N )-variate, pth-order PDD approximation. Accordingly, setting S = p ∧ N and m = p in (4.18), (6.8) Lp = Lp∧N,p = 1 + p∧N ∑ s=1 ( )( ) N p (N + p)! = N !p! s s with the last expression commonly found in the PCE literature [29]. The advantage of (6.7) over (5.3) is obvious: the PDD coeﬃcients, once determined, can be reused for the PCE approximation and subsequent error analysis, thereby sidestepping calculations of the PCE coeﬃcients. 6.3. PDD vs. PCE errors. Define another second-moment error, (6.9) ep := E [y(X) − yp (X)]2 , resulting from the pth-order PCE approximation yp (X) of y(X). Using Proposition 6.5, ep = ep∧N,p , meaning that the PCE error analysis can be conducted using the PDD approximation. Proposition 6.6.For a general function y, let eS,m and ep denote the PDD and PCE errors defined by (6.1) and (6.9), respectively. Given a truncation parameter 0 ≤ p < ∞ of the PCE approximation, if the truncation parameters of the PDD approximation are chosen such that p ∧ N ≤ S ≤ N and p ∨ S ≤ m < ∞, then (6.10) eS,m ≤ ep , 24 S. RAHMAN where p ∨ S denotes the maximum of p and S. Proof. The result follows from Propositions 6.1 and 6.5, and Corollary 6.2. Proposition 6.6 aids in selecting appropriate truncation parameters to contrast the second-moment errors due to PDD and PCE approximations. However, the proposition does not say anything about the computational eﬀort. Proposition 6.7 and subsequent discussion explain the relationship between computational eﬀort and error committed by both PDD and PCE approximations for a special class of functions. 2 , ∅ ̸= u ⊆ {1, · · · , N }, Proposition 6.7.For a special class of functions y, assume that Cu,j u −\|u\| −\|j \| 2 ju ∈ N\|u\| , diminishes according to Cu,j ≤ Cp1 p2 u , where C > 0, p1 > 1, and p2 > 1 u are three real-valued constants. Then it holds that   ( )( ) ) p∧N ∞ N ∞ ( )( ∑ ∑ ∑ ∑ N l − 1 −s −l N l − 1 −s −l  (6.11) ep ≤ C  p p + p p . s s−1 1 2 s s−1 1 2 s=1 s=p∧N +1 l=p+1 l=s Proof. Replacing S and m in (6.4) with p ∧ N and p, respectively, obtains the result. Theoretically, the numbers of expansion coeﬃcients required by the PDD and PCE approximations can be used to compare their respective computational eﬀorts. Table 1 presents for N = 20 the requisite numbers of expansion coeﬃcients when PDD is truncated at S = 1, 2, 3, 5, 9 and m = 1 − 20, and when PCE is truncated at p = 1 − 20. They are calculated using (4.18) and (6.8) for PDD and PCE approximations, respectively. According to Table 1, the growth of the number of expansion coeﬃcients in PCE is steeper than that in PDD. The growth rate increases markedly when the polynomial expansion order is large. This is primarily because a PCE approximation is solely dictated by a single truncation parameter p, which controls the largest polynomial expansion order preserved, but not the degree of interaction independently. In contrast, two diﬀerent truncation parameters S and m are involved in a PDD approximation, aﬀording a greater flexibility in retaining the largest degree of interaction and largest polynomial expansion order. In consequence, the numbers of expansion coeﬃcients and hence the computational eﬀorts by the PDD and PCE approximations can vary appreciably. Table 1 Growth of expansion coeﬃcients in the PDD and PCE approximations LS,m m or p S=1 1 21 2 41 S=2 S=3 S=5 S=9 Lp 21 231 231 3 61 631 1771 5 101 2001 13,401 1771 53,130 53,130 9 181 7021 102,781 2,666,755 10,015,005 10,015,005 12 241 12,781 263,581 14,941,024 211,457,454 225,792,840 15 301 20,251 538,951 53,710,888 2,397,802,638 3,247,943,160 20 401 36,501 1,336,101 265,184,142 51,855,874,642 137,846,528,820 Using the equalities in (6.3), (6.4), and (6.11), Figure 2 depicts how the relative PDD error, eS,m /var[y(X)], and the relative PCE error ep /var[y(X)], vary with respect to the POLYNOMIAL DIMENSIONAL DECOMPOSITION 25 Figure 2. PDD vs. PCE errors for various attenuation rates of the expansion coeﬃcients; (top) p1 = 500, p2 = 5; (middle) p1 = 5, p2 = 500; (bottom) p1 = 500, p2 = 500. 26 S. RAHMAN number of expansion coeﬃcients required for N = 20. Again, the three preceding cases of the attenuation rates p1 and p2 with respect to the degree of interaction and polynomial expansion order are studied. In all cases, the PDD or PCE errors decay with respect to S, m, and p as expected. However, in the PDD approximation, the error for a fixed S may decline even further by increasing m, whereas no such possibility exists in the PCE approximation. This behavior is pronounced in case 1, that is, when p1 > p2 [Figure 2 (top)]. For example, in case 1, the bivariate, sixth-order PDD approximation (S = 2, m = 6) achieves a relative error of 8.54 × 10−5 employing only 2971 expansion coeﬃcients. In contrast, to match the same-order error, the sixth-order PCE approximation (p = 6) is needed, committing a relative error of 7.15 × 10−5 at the cost of 230,230 expansion coeﬃcients. Therefore, the PDD approximation is substantially more economical than the PCE approximation for a similar accuracy. However, when p1 > p2 , as in case 2 [Figure 2 (middle)], the computational advantage of PDD over PCE approximations disappears as the attenuation rate associated with the polynomial expansion order is dominant over that associated with the degree of interaction. Nonetheless, in case 2, the S-variate, mth-order PDD approximation with the lowest m possible cannot commit more error than the mthorder PCE approximation for the same computational eﬀort. Finally, when the attenuation rates are the same, as in case 3 [Figure 2 (bottom)], the PDD approximation is still more computationally eﬃcient than the PCE approximation. For instance, the trivariate, fifthorder PDD (S = 3, m = 5) and fifth-order PCE (p = 6) approximations require 13,401 and 53,130 expansion coeﬃcients to commit the same-order errors of 5.07 × 10−14 and 3.51×10−14 , respectively. But, unlike in case 1, an unnecessarily large polynomial expansion order may render the PDD approximation more expensive than required. Readers should take note that the comparative error analyses reported here are limited to PDD and PCE approximations derived from truncations according to the total degree index set. For other index sets, such as the tensor product and hyperbolic cross index sets, it would be intriguing to find whether a similar conclusion arises. 7. Conclusion. The fundamental mathematical properties of PDD, representing Fourierlike series expansion in terms of random orthogonal polynomials with increasing dimensions, were studied. A dimension-wise splitting of appropriate polynomial spaces into orthogonal subspaces, each spanned by measure-consistent orthogonal polynomials, was constructed, resulting in a polynomial refinement of ADD and eventually PDD. Under prescribed assumptions, the set of measure-consistent orthogonal polynomials was proved to form a complete basis of each subspace, leading to an orthogonal sum of such sets of basis functions, including the constant subspace, to span the space of all polynomials. In addition, the orthogonal sum is dense in a Hilbert space of square-integrable functions, leading to mean-square convergence of PDD to the correct limit, including for the case of infinitely many random variables. The optimality of PDD and the approximation quality due to truncation were demonstrated or discussed. From the second-moment error analysis of a general function of 1 ≤ N < ∞ random variables, given 0 ≤ p < ∞, the (p ∧ N )-variate, pth-order PDD approximation and pth-order PCE approximation are the same. Therefore, an S-variate, mth-order PDD approximation cannot commit a larger error than a pth-order PCE approximation if p ∧ N ≤ S ≤ N and p ∨ S ≤ m < ∞. From the comparison of computational eﬀorts, required to estimate with the same accuracy the variance of an output function entailing exponentially attenuating expansion coeﬃcients, the PDD approximation can be substantially more economical than the PCE approximation. POLYNOMIAL DIMENSIONAL DECOMPOSITION 27 REFERENCES [1] R. Askey and J. Wilson, Some Basic Hypergeometric Polynomials that Generalize Jacobi Polynomials, Mem. Amer. Math. Soc., 319, AMS, Providence, RI, 1985. [2] K. I. Babenko, Approximation by trigonometric polynomials in a certain class of periodic functions of several variables, Soviet Math. Dokl., 1 (1960), pp. 672–675. [3] R. Bellman, Dynamic Programming, Princeton University Press: Princeton, NJ, 1957. [4] R. E. Caflisch, W. Morokoff, and A. Owen, Valuation of mortgage backed securities using brownian bridges to reduce eﬀective dimension, Journal of Computational Finance, 1 (1997), pp. 27–46. [5] R. H. Cameron and W. T. 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Rahman, A generalized anova dimensional decomposition for dependent probability measures, SIAM/ASA Journal on Uncertainty Quantification, 2 (2014), pp. 670–697. [24] S. Rahman and V. Yadav, Orthogonal polynomial expansions for solving random eigenvalue problems, International Journal for Uncertainty Quantification, 1 (2011), pp. 163–187. [25] X. Ren, V. Yadav, and S. Rahman, Reliability-based design optimization by adaptive-sparse polynomial dimensional decomposition, Structural and Multidisciplinary Optimization, 53 (2016), pp. 425–452. [26] T. J. Stieltjes, Quelques recherches sur la thorie des quadratures dites mcaniques, Ann. Sci. cole Norm, 3 (1884), pp. 409–426. 28 S. RAHMAN [27] K. Tang, P. M. Congedo, and R. Abgrall, Adaptive surrogate modeling by anova and sparse polynomial dimensional decomposition for global sensitivity analysis in fluid simulation, Journal of Computational Physics, 314 (2016), pp. 557–589. [28] N. 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arXiv:1308.6074v2 [q-bio.GN] 3 Apr 2014 Exploration and retrieval of whole-metagenome sequencing samples Sohan Seth 1 , Niko Välimäki 2,3 , Samuel Kaski 1,3 , Antti Honkela 3 1 Helsinki Institute for Information Technology HIIT, Department of Information and Computer Science, Aalto University, Espoo, Finland 2 Genome-Scale Biology Program and Department of Medical Genetics, University of Helsinki, Helsinki, Finland 3 Helsinki Institute for Information Technology HIIT, Department of Computer Science, University of Helsinki, Helsinki, Finland April 4, 2014 Abstract Over the recent years, the field of whole metagenome shotgun sequencing has witnessed significant growth due to the high-throughput sequencing technologies that allow sequencing genomic samples cheaper, faster, and with better coverage than before. This technical advancement has initiated the trend of sequencing multiple samples in different conditions or environments to explore the similarities and dissimilarities of the microbial communities. Examples include the human microbiome project and various studies of the human intestinal tract. With the availability of ever larger databases of such measurements, finding samples similar to a given query sample is becoming a central operation. In this paper, we develop a content-based exploration and retrieval method for whole metagenome sequencing samples. We apply a distributed string mining framework to efficiently extract all informative sequence k-mers from a pool of metagenomic samples and use them to measure the dissimilarity between two samples. We evaluate the performance of the proposed approach on two human gut metagenome data sets as well as human microbiome project metagenomic samples. We observe significant enrichment for diseased gut samples in results of queries with another diseased sample and very high accuracy in discriminating between different body sites even though the method is unsupervised. A software implementation of the DSM framework is available at https://github.com/HIITMetagenomics/dsm-framework. 1 Introduction Metagenomics is the study of microbial communities in their natural habitat using genomics techniques [28]. It is undergoing a boom due to proliferation of high-throughput sequencing technologies. Many studies focus at targeted sequencing of specific marker genes such as the 16S rRNA gene in bacteria, but recently there has been a growing interest in whole metagenome sequencing (see, e.g. [18, 27]). While targeted studies provide data for phylogenetic profiling at a lower cost, whole metagenomes provide much more information, for example, about the collective metabolism [5], and the population genetics of the community [22]. Recent studies have also found associations between features of whole human gut metagenomes and type II diabetes [19]. New data are accumulating rapidly, with a popular web-based MG-RAST server [14] listing almost 3000 public whole metagenomes. Analysing whole-metagenome shotgun (WMS) sequencing data is very challenging. The original sample typically contains genetic material from hundreds to thousands of bacterial species of different abundances [9], most of which have not been fully sequenced previously. After sequencing, we obtain a huge 1 Bag of metagenomic samples s1 ACTAGTCA TAGCATAG CCATGACA CTTAATGA s5 ATCGCAGA AGGTTAAT GTGTACCG TCAACGGG s3 ACTGACTG ATTCCTTA CTATGCAC GTTGCTTC s2 ATGACATA GATCATGA CACATGCA CATGACTG s6 Feature extraction s4 GATGGATT GTCAGTAC GTACTGAC ACTGCATG Dissimilarity evaluation Dissimilarity values Query: S3 Retrieve: S6,S1,... Query: S6 Retrieve: S5,S3,... ACTGGTCA CTTAAGGC GTGTACCA AGGACAAC Figure 1: Given a set of metagenomic samples our objective is to be able to retrieve relevant samples to a query sample. For this, we need to extract relevant features and evaluate a pairwise similarity (or dissimilarity) measure. The samples are then ranked in the order of increasing dissimilarity from the query. collection of short sequence reads whose species of origin is unknown. While significant progress has been made, analysis relying on either the limited previously annotated genomes, or assembling the reads into novel more complete genomes, remains difficult and inefficient, and potentially susceptible to annotation biases. In this paper we introduce an efficient purely data-driven feature extraction and selection method as well as similarity measures for WMS sequencing data sets, and apply them in retrieval of similar data sets. Such content-based retrieval is an extremely powerful tool for exploration of the data and generating hypotheses of disease associations, as previously demonstrated with gene expression data [2, 3]. Retrieval from existing databases makes it possible to automatically explore a much greater variety of hypotheses than relying solely on the more common specifically designed focused studies. Content-based similarity measures and retrieval of similar metagenomic data sets have been suggested previously [16, 10, 26, 6], based on quantifying abundances over a relatively small number of predetermined features requiring existing annotation. Up to some thousands of known taxa, genes or metabolic pathways have been used. We introduce similarity measures that are based solely on raw sequencing reads, and hence, unbiased and insensitive to the quality of the existing annotation. A similar measure has been previously suggested by [11], but only for pairwise comparisons using a method that is computationally too expensive to scale to even modestly large data sets. Furthermore, instead of considering all sequences of particular length, also known as k-mers, as has been done earlier for other tasks and by [11], we employ an efficient distributed string mining algorithm to find informative subsequences that can be of any length. In order to deal with the very large number of features some feature selection is necessary. Previous approaches for detecting relevant features in metagenomic data have been based on direct comparison of two classes of samples. Again, most of these methods work on up to some thousands of features [30, 17, 23], with the notable exception of one study [19] where quantification and association testing was done for over 4.3 million predefined genes. Without feature selection one can use short k-mers [1] or limit to a set of k-mers that are likely to be informative, such as k-mers associated with well characterised protein families [4]. While there are no previous examples of unsupervised feature selection for metagenomics, it is a common practice in information retrieval with text documents [31]; a particularly relevant method assesses the entropy of the distribution of documents in which a specific term occurs [8]. We evaluate the performance of the proposed unsupervised, unconstrained retrieval method on synthetic data, as well as metagenomic samples from human body sites [18, 19, 27]. To evaluate the performance of the retrieval engine, we use external validation based on a ground truth similarity between two samples. To simplify this process, we consider a binary similarity, which is crude but easily accessible. The human gut samples in [18, 19] come from studies exploring the change in bacterial species composition between healthy persons and either inflammatory bowel disease or type II diabetes. We utilize disease state to construct a 2 1. Normalization Collection of metagenomic samples 2. Regularization 4. Distributed string mining Dissimilarity matrix 5. Dissimilarity computation 3. Entropy ﬁltering 6. Evaluation Annotations Figure 2: Processing steps of our method. Given a collection of metagenomic samples, we use the collection as an input to the distributed string mining method (4). For the method, we estimate the frequency of each k-mer (1,2), evaluate if the k-mer is informative or not (3), and compute the needed dissimilarities (5). Finally, in this paper we evaluate the performance considering the existing annotations as ground truth; annotations are not needed for the retrieval in general. binary ground truth. Thus, we study if, given the metagenomic sample of a person with a disease, the retrieval finds metagenomic samples related by having the same disease. In the body site data [27] we use the body sites as ground truth to investigate whether it is possible to identify the bacterial communities at different body sites in an unsupervised setting without the need of reference genomes. It should be noted that especially for the gut data, two samples may be related in other ways too. The external validation with one simple ground truth nonetheless provides an objective platform for comparing different methods. Given that the method is unsupervised and hence completely oblivious of the disease labels, if such retrieval is successful it is a promising starting point for developing methods for leveraging data from earlier patients in early detection of disease and personalized medicine. 2 Approach Our objective is to extract and select suitable features for representing WMS sequencing samples and to form a pairwise dissimilarity measure for a collection of such samples. Given this dissimilarity one can query with a sample and retrieve other samples that are similar to it (Fig. 1). The measure needs to be reasonably rapidly computable, yet captures relevant differences between the samples, and does all this with as little prior biological knowledge and annotations as possible, since detailed quantitative prior knowledge is typically not yet available for metagenomics. Evaluating dissimilarity requires representing the metagenomic sample in a suitable feature space. A standard choice for representing objects over strings is to estimate the k-mer frequency values, where a kmer here is a string of k letters from the DNA alphabet {A,C,T,G}. Therefore, there are 4k possible k-mers for any given k. It is standard practice to set k to a specific value, typically a small value to keep the estimation problem tractable both computationally and statistically. A larger k would give better discriminability but not without bounds, as for finite data set sizes there simply is not enough data to estimate long k-mers. We argue that instead of setting k to a particular value, it is more effective to estimate all possible k-mers for all possible k which the data supports. This makes the problem more challenging, since the number of such observed different k-mers for large k becomes very large and they become more susceptible to sequencing errors. Focusing on k-mers appearing more than once in a sample helps significantly because it is relatively rare to have the exactly same sequencing errors in two independent reads. To make the method computationally efficient we treat each k-mer as an independent feature. We compute a Bayesian estimate of their relative frequencies across samples. The employed prior helps in suppressing noise caused by small observed read counts. In the filtering step the abundance distribution of each 3 Figure 3: Technical overview of our distributed string mining framework consisting of client (left) and server (right) processes. The client-side processes are responsible for computing the substring frequencies within each sample s1 , s2 , . . . sd separately. Substrings and their frequencies are found using a depth-first-traversal over a (compressed) suffix tree. Frequency information is transmitted over to the server-side by streaming it as a balanced-parenthesis representation of a sorted trie. For example, the trie on the left results as the parenthesis representation given in the middle. The server reads the client-streams and merges the (already sorted) tries in recursive manner: at each node, the server computes the entropy based on the received values and updates the affected pairwise distances. Load-balancing on the server-side is achieved by hashing the prefix of the substring so that each server corresponds to a certain range of hash values. k-mer over samples is used to judge informativeness of the k-mer for retrieval; a k-mer with constant abundance does not have discriminative power and, in the other extreme, a k-mer which is present in only one sample cannot generalize over samples. We show that the filtering step significantly improves the retrieval performance with most datasets and distance measures. Finally, we compute the dissimilarity between two samples across the features as a weighted average of distances between relative frequencies of individual k-mers. Treating each k-mer as an independent feature allows us to execute these steps fast and on the fly without storing the intermediate results. Such simplified distance measures are necessary to guarantee scalability given the extremely high dimensionality of the k-mer features. To summarize, we introduce methods to i. estimate the frequencies of a large number of k-mers over multiple samples, ii. decide if a k-mer is informative or uninformative in the context of a retrieval task, iii. compute a distance metric using the filtered k-mer frequencies, and iv. execute these steps fast without explicitly storing the frequency values. Fig. 2 summarizes the method. 3 Methods 3.1 Estimating k-mer frequencies: normalization, regularization, filtering In order to perform the feature selection or filtering, we first compute Bayesian estimates of the relative frequencies p(s\|w) of each k-mer w over samples s ∈ S using observed frequencies fˆ(s, w) of the k-mers. These are distributions over samples for each k-mer that are computed independently for each k-mer for reasons of computational efficiency. Even if the relative abundance of a k-mer is the same in every sample, the observed frequencies may differ because of different sequencing depth or coverage in different samples. To tackle this issue we employ normalization: we normalize the frequency fˆ(s, w) by a sample-specific constant σ(s), which is proportional to the total number of base pairs in a sample, and σ(s) = 1 for the largest sample in the collection in terms 4 of total base pair count, obtaining f (s, w) = fˆ(s, w)/σ(s). (1) The σ(s) can be interpreted probabilistically as the probability of observing a sequence in the actual sample, assuming every sample had the same number of base pairs to start with but some have been lost in the processing. In order to estimate the relative frequencies, we place a conjugate symmetric Dirichlet prior on the parameters of the multinomial distribution over the observed counts. The common choice of uniform prior distribution corresponds to a Dirichlet distribution with all parameters equal to 1. This yields a posterior mean estimate of the relative frequency values as p(s\|w) = P f (s, w) + 1 . [f (s′ , w) + 1] (2) s′ ∈S The Dirichlet prior with all parameters equal to one is ubiquitous in document retrieval. It is particularly suitable for metagenomics due to the following observations: 1. The distributed string mining algorithm (described below) trades off low k-mer counts for speed and ignores any k-mers that are present only once in a sample. The pseudo-count from the prior makes up for this missing count. 2. Adding pseudo-counts assists in playing down the significance of very rare k-mers that may appear due to sequencing errors in the filtering step without affecting other k-mers too much. Finally, given the massive number of potential k-mers, it is crucially important to improve signal-to-noise ratio by focusing on the informative ones. For the unsupervised tasks of comparing the samples, obviously only k-mers which distinguish between the samples are informative. As a concrete example, consider a kmer that is present in all samples with a similar abundance. It certainly does not give information useful for comparing samples. In the other extreme, if a k-mer is present in one specific sample, but not in any other, it is potentially a spurious k-mer due to sequencing error, and in any case does not help in comparing samples either. On the other hand, if a k-mer is present in some samples, but not all, then it gives information that those samples are similar in a specific sense. Informativeness in this sense can be measured by the entropy H of the distribution of the k-mer over the samples: we filter the k-mers based on the conditional entropies H(S\|w) = − X 1 p(s\|w) log p(s\|w); log(\|S\|) (3) s∈S a k-mer is taken into account in distance computation only if the normalized entropy is lower than a certain threshold e. By design 0 ≤ H ≤ 1. Notice that in standard information theory terminology higher entropy implies higher information. However, in our context an informative k-mer has low entropy. Also, due to the Bayesian estimation, a spurious k-mer having only very small counts will have large conditional entropy and will be filtered out. The optimal value of threshold e varies with datasets. It can be ‘optimized’ in a supervised manner by utilizing a training set where we have labelled samples. In the absence of a labelled set, we suggest taking the ‘average’ of distance metrics computed over the potential thresholds as the final metric. We refer to the final metrics in the two cases as optimized metric and average metric. In our experimental set-up, we randomly make a 50-50 split of a given dataset in training Str and testing Ste sets: Str ∩ Ste = ∅, and Str ∪ Ste = S. We use Str to optimize the entropy threshold: we query with samples in Str , and retrieve relevant samples within the same set to observe which entropy threshold results in the best retrieval result (see Sec. 3.4 for details). While comparing the performance of two methods we always present the evaluation over Ste : we query with samples within Ste , and we retrieve relevant samples from S (not just Ste ). 5 3.2 Algorithms to extract informative k-mers Our main computational challenge is to extract all informative k-mers from large-scale datasets in feasible time and space. Recall that the filtering step relies on knowledge over multiple samples to decide if the respective k-mer is informative for the retrieval task or not. Since the typical collections of WMS samples are huge in size, we cannot assume that even the plain input fits into the main memory of any single machine. To process these large-scale datasets, the computation needs to be done either using external memory (i.e. disk) or in a distributed manner (i.e. a computer cluster). We review two approaches: k-mer counting [12, 21] and distributed string mining [29]. The first one is a standard approach in the literature for fixed k, but has several limitations when applied in our context of multiple samples and large-scale data. We show that the latter approach is more flexible in this context and can also be generalized to extract informative k-mers over all values of k simultaneously. Jellyfish [12] and DSK [21] are examples of recent algorithmic improvements in k-mer counting. Both tools use hash tables to compute the k-mer distribution for a given (fixed) k. In both tools, space-efficiency is achieved by keeping most of the hash table on disk. The main drawback with these disk-based approaches is that they are aimed at counting k-mers in a single sample and extending them over to multiple samples is non-trivial. For example, Jellyfish could, in principle, be extended to count k-mers over multiple samples: the authors give a roughly linear time algorithm to merge two or more hash tables. However, the intermediate k-mer counts would need to be stored on disk, which requires significant amount of additional space, and the merge-phase is not parallelized [12, User manual, Sect. Bugs]. The decision whether a particular k-mer is informative or not is made by looking at its frequency over all the given WMS samples. We tackle this problem by a Distributed String Mining (DSM) framework [29] that can handle multi-sample inputs by utilizing a computer cluster. The main advantages of this framework are that (i) load-balancing divides the data and computation over multiple cluster nodes, (ii) intermediate k-mer counts are not stored explicitly, and (iii) there is no additional disk I/O strain, except reading through the input once. These advantages allow terabyte-scale data analysis on a cluster consisting of nodes having limited main memory. We extend the DSM framework to be compatible with our definition of informative k-mers (see the above subsection). It allows us to extract the informative k-mers either for a fixed k or over all values of k in feasible time. The DSM framework is based on a client-server model. The clients have one-to-one correspondence to the given samples, each client being responsible for computing the frequencies within the designated sample. The client-side computation relies heavily on suffix sorting techniques and on space-efficient data structures for strings [29]: the input data are first preprocessed into a compressed representation, which replaces the input data and acts as an efficient search structure. The server-side computation is more straightforward: the server simply merges the (sorted) input from the clients, computes the entropies and updates the distance matrices. Fig. 3 gives a toy example of the client-server interaction. Two crucial observations are needed to keep the whole computation and transmission costs feasible. First, the informative k-mers can be seen as a subset of left-right-branching substrings, i.e. substrings whose instances have differentiating continuation on both left and right. More formally: substring w of string T [1, n] is called right-branching if there exists two symbols a and b such that a 6= b and both wa and wb are substrings of T . Similarly, a substring w is leftbranching if aw and bw, a 6= b, are substrings of T . If a substring is both left-branching and right-branching we say it is left-right-branching. Second, for any string of length n, there are at most O(n) left-right-branching substrings, and the total length of all such substrings is bounded by O(n log n) [7, Theorem 1]. The first observation allows us to reduce the client-side computation to a smaller set of substrings: it is easy to see that if k-mer w, having frequency f ′ (s, w) ≥ 2, is non-branching, then there exists a substring w′ of length k ′ > k that is left-right-branching and has exactly the same frequency, i.e., f ′ (s, w) = f ′ (s, w′ ). It follows that the frequency of non-branching k-mers can be deduced from the branching k ′ -mers, and the left-right-branching substrings contain all the necessary information for us to detect informative k-mers. The second observation guarantees a feasible transmission cost between clients and servers: the upper bound for the concatenation of all left-right-branching substrings also acts as an upper bound for both the server-side running time and the amount of communication needed. The drawback of restricting to left-right-branching substrings is that the informative k-mers that we are able to detect have to appear at least twice in a sample, 6 although this limit may be useful in pruning spurious k-mers introduced by sequencing errors. More detailed explanation and analysis of the DSM framework is given in [29]. A software implementation of the DSM framework is available at https://github.com/HIITMetagenomics/dsm-framework. 3.3 Dissimilarity metrics Having extracted the informative k-mers, we use them to compute the dissimilarity between two metagenomic samples. We consider three dissimilarity metrics that can be computed easily over a large number of k-mers in sequential manner, i.e. one k-mer at a time, and without storing all the k-mer frequencies explicitly. To utilize the natural variance structure of the k-mers—some are more abundant than others—we weight the relative frequencies of each k-mer by their respective total counts, i.e., we utilize the absolute frequencies f (s, w) as defined in (1). We mainly use the very simple Jaccard distance which does not consider abundances at all, only whether a k-mer occurs or not. Given two sets s1 and s2 of k-mers detected as present in two different samples, Jaccard distance measures how many elements are shared between these two sets. Mathematically, it is defined as \|s1 ∩ s2 \| Dcount (s1 , s2 ) = 1 − . \|s1 ∪ s2 \| Despite its simplicity, we observe that Jaccard distance performs well; a potential reason is its robustness to measurement noise and effectiveness when two metagenomic samples differ in terms of presence and absence of certain species or functionalities. We assume a k-mer is present in a sample if its frequency is more than 2. We also experiment with two metrics that use the abundance information: I. Variance-stabilized Euclidean distance: An obvious distance measure between two metagenomic samples s1 and s2 is the Euclidean distance between their respective k-mer frequencies. We consider the distance metric Xp p Dsqrt (s1 , s2 ) = ( f (w, s1 ) − f (w, s2 ))2 w which can be computed sequentially as new informative k-mers are extracted. The square root transformation is the variance stabilizing transformation for Poisson distribution—a popular model for quantitative sequencing data. II. Log transformed Euclidean distance: We also consider the same metric but with log transformation which is a popular approach in document retrieval, i.e., X Dlog (s1 , s2 ) = (log(1 + f (w, s1 )) − log(1 + f (w, s2 )))2 . w The motivation for using the log transformation is that it decreases sensitivity to high frequency counts: some k-mers are present in high abundance in almost every genome, for instance k-mers from the marker gene, and the log transformation reduces their effect in the metric. 3.4 Evaluation metric We evaluate the performance of the dissimilarity metric in terms of its performance in the task of retrieving relevant samples given a query metagenomics sample. The ground truth for relevance is either the disease class (disease vs not) or the known body site: samples from the same class are considered relevant. For measuring retrieval performance we use an evaluation metric which is popular in document retrieval, the mean average precision (MAP) [25]. Given a query q, the retrieval method ranks the samples in an increasing order of their dissimilarities from q. Given one has retrieved the top (closest) n ∈ {1, . . . , N } samples the precision @n is defined as Precision(n; q) = number of relevant samples in n retrieved samples , n 7 HIGH-C 149 200 0.4 117 4.9 149 1.8 10 Input size (GB) Samples Preproc. (h) Total memory (GB) All k Wall-clock (h) CPU time (h) k = 21 Wall-clock (h) CPU time (h) MetaHIT 536 124 3.6 209 2.0 187 0.4 74 T2D-P2 786 199 10 610 8.0 1,137 2.8 279 HMP 3,353 435 65 2,885 53 20,000 12 4,000 Table 1: Computational resources required by the distributed string mining on different datasets. We report wall-clock times and total CPU times for both fixed k = 21 and over all k. Preprocessing is done only once, separately from the actual computation. Total memory is the memory requirement over all computation nodes. Experiments were ran on a cluster of Dell PowerEdge M610 nodes having 32 GB of RAM and 16 cores. Simulated data and MetaHIT were run using up to 8 nodes. T2D-P2 was ran using 32 nodes allowing more parallelization at the server-side. HMP was ran on a cluster of 20 nodes with 2x10-core Xeon CPUs and 256GB RAM. and MAP defined using average precision as, MAP = 1 X 1 X Precision(n; q). AveP(q), AveP(q) = \|Q\| mq n∈Rq q∈Q Here Q is the set of all queries, mq is the number of relevant samples to query q, and Rq is the set of locations in the ranked list where a relevant sample appears. It is straight-forward that a higher MAP implies better performance. To judge if two MAP values are significantly different or not, we employ the randomization test described in [25]: for each query, this test randomly reassigns the AvePs achieved by two methods to one another and computes the difference between the resulting MAP for multiple such reassignments to get a distribution, against which the true MAP value is tested in terms of p-value. In case two samples share the same dissimilarity from a query sample, we employ the modification suggested in [13] to break ties. When computing the mean, we follow a leave-one-out cross-validation type approach using each sample as a query, and retrieving from the rest of the collection. For simulated data and human gut samples, we only query with the positive samples in the testing set q ∈ Ste , whereas for body site samples we query with each sample in the testing set. For both cases we retrieve from the entire set S \ {q}. While choosing the entropy threshold in a supervised setting, we query from q ∈ Str and retrieve from Str \ {q}. 3.5 Synthetic data generation To test the method, we simulated four datasets containing samples from separate classes, with the interpretation that samples from the same class are relevant. In all the datasets we have two classes: both classes of samples have the same species composition but different relative abundances. We used MetaSim [20] to generate Illumina reads of length 80 using the error configuration file provided by the developers. Each dataset contains 200 samples: 98 of them belong to the positive class and the rest belong to the negative class. For each dataset, we used the same 100 species from the following genera: acetobacter, acetobacterium, acidiphilium, acidithiobacillus, acinetobacter, bacillus, bacteroides, bifidobacterium, chlamydia, chlamydophila, clostridium, escherichia, haloarcula, halobacterium, lactobacillus, pasteurella, salmonella, staphylococcus, and streptococcus. The abundance profiles were generated from two Dirichlet distributions; one for positive and the other for negative class. The parameters of the Dirichlet distributions were shared between two classes: for half of the species (randomly chosen) the same parameters were used for both classes and for the other half of the species the parameters were randomly permuted. For example, given 5 species the assigned parameters could be: (0.3, 0.2, 0.6, 0.1, 0.9) and (0.9, 0.2, 0.3, 0.1, 0.6) where the parameters for 8 log10(# of k−mers) MetaHIT + log 15 HMP + Jaccard 15 15 12 21 30 All 10 10 10 5 5 5 0 0 0.5 Entropy threshold HIGH−C + Jaccard log10(# of k−mers) T2D−P2 + Jaccard 1 0 0 0.5 Entropy threshold HIGH−VAR + Jaccard 15 1 0 0 F 0.5 Entropy threshold LOW−C + Jaccard 15 MIXED−C + Jaccard 15 15 12 21 30 All 10 10 10 10 5 5 5 5 0 0.7 0.8 0.9 Entropy threshold 1 0 0.7 0.8 0.9 Entropy threshold 0 0.7 1 1 0.8 0.9 Entropy threshold 1 0 0.7 F 0.8 0.9 Entropy threshold 1 Figure 4: Number of informative strings over varying entropy thresholds for the proposed approach ‘All’, fixed k-mer lenthgs ‘12’,‘21’ and ‘30’, and for protein family based comparison with FIGfam ‘F’. The box denotes the ‘optimized’ entropy threshold that has been used to evaluate the performance of the methods. Some general observations are as follows. The number of strings for k = 12 is lower than the rest while the number of strings for ‘All’ is much higher than rest of the methods, and number of strings for k = 21 and k = 30 are very close. We observe that there are strings with low entropies—more in the real data sets than in the simulated data sets—which indicate the presence of discriminative features. Also, the ‘optimized’ entropy threshold varies for different methods. the second and fourth species are the same, but for the other species they were permuted. The exact species and corresponding parameter values can be downloaded from https://github.com/HIITMetagenomics. The resulting datasets are: 1. HIGH-C: relatively easy data with high coverage (10,000,000 reads per sample) 2. LOW-C: relatively difficult data with low coverage (2,000,000 reads per sample) 3. MIXED-C: mixed data with half the samples from HIGH-C and the rest from LOW-C to simulate varying sequencing depth. 4. HIGH-VAR: relatively difficult data with same coverage as HIGH but additional noise in the class distributions to simulate more overlap between classes. To elaborate, the relative abundance of species is pHIGH−VAR = 0.5 pHIGH + 0.5 noise where noise is generated from a symmetric Dirichlet distribution with all parameters equal to 1. 4 Results We evaluated the retrieval performance on three human metagenomics datasets: 1. MetaHIT [18], 124 metagenomic samples from 99 healthy people and 25 patients with inflammatory bowel disease (IBD) syndrome. Each sample has on average 65 ± 21 million reads. Our goal was to retrieve IBD positive patients. 9 Mean average precision MetaHIT + Jaccard MetaHIT + log ↑* 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 ↓ T2D−P2 + Jaccard HMP + Jaccard ↓ * ↓ ↓ * 0.6 ↓ * ↓ * ↓ *** 1 0.4 0 Ak FIG Abd Method 3 0 HIGH−C + Jaccard Mean average precision ↓* 1 0.2 Ak FIG Abd Method 0 3 HIGH−VAR + Jaccard ↓ * ↓ * ↓ * ↑ * 0.5 0 0.5 1 T 0 FIG Abd Method 3 1 0 T Ak FIG Abd Method 3 MIXED−C + Jaccard ↓ * ↓ * ↑ * 0.5 Ak FIG Abd 3 Method 0 LOW−C + Jaccard ↓ * ↓ * ↓ * ↑ * 0.5 Ak FIG Abd 3 Method Ak 1 ↓ * ↓ * ↓ * ↑ * 0.5 Ak FIG Abd 3 Method T 0 Ak FIG Abd 3 Method T Figure 5: Retrieval performance comparison of the proposed approach using all k-mers (“Ak”) against the following base measures: (1) “FIG”: retrieval performance using known protein family, (2) “Abd”: Hellinger distance between relative estimated abundance, (3) “3”: d2S distance between relative abundance of 3-mers. “Ak” uses the ‘optimized metric’ over 101 equally spaced threshold values between 0 and 1. Each errorbar shows the MAP value along with the standard error. The grey horizontal line shows retrieval by chance: MAP computed over zero similarity metric. An arrow (if present) over a method indicates whether the performance of the corresponding method is significantly better (↑) or worse (↓) than “Ak”: The stars denote significance level: 0 < * < 0.001 < ** < 0.01 < * < 0.05. For the synthetic datasets (in the bottom row) the relative abundance is known from experimental design. We present this result as “T”. For MetaHIT we present the performance for both Jaccard and log metric, since the latter performs much better compared to the former. 2. T2D Phase II [19], 199 metagenomic samples from 100 healthy people and 99 patients with type II diabetes. Each sample has on average 47 ± 11 million reads. Our goal was to retrieve diabetic patients. We chose to explore the phase II data instead of the phase I data since the former has higher coverage; about 40% more reads than the latter. 3. HMP [27], 435 metagenomic samples from 10 different body sites. Out of 690 samples that passed the QC assessment (http://www.hmpdacc.org/HMASM/), we discarded 255 samples that had less than 1% of the number of reads of the largest sample. To recapitulate, for MetaHIT and T2D-P2, our goal is to observe if given a positive sample, e.g., from a patient with a particular disease, one can retrieve relevant samples, i.e., with similar disease; whereas for HMP, our goal is to observe if given a sample from a particular body site, one can retrieve relevant samples, i.e., samples from the same body site. For all data we applied a quality threshold of 30 and ignored any base pairs with quality less than the threshold. Table 1 gives an overview of the computational resources required for each data set. Additionally, number of k-mers used by different methods for each data set is available in 4. Retrieval of samples with similar annotation: We applied the proposed approach and a number of alternatives to retrieval of similar samples from the same data set and evaluated by how many of the retrieved 10 MetaHIT + log Average precision ↓ ** ↓* T2D−P2 + Jaccard ↓ ↑ * ↓ 0.4 0.2 0.5 0.2 All 12 21 0 30 All 12 Average precision ↓ * ↑ * ↑ * 0.5 21 k 30 0 30 All 12 ↓ *** 1 0 LOW−C + Jaccard ↑* 12 21 1 0 30 k 30 MIXED−C + Jaccard ↑ * ↑ * 0.5 All 21 k HIGH−VAR + Jaccard 0.5 12 21 k HIGH−C + Jaccard All ↓* 0.6 0.4 k 0 ↓ *** ↓ * 1 0.6 0 1 HMP + Jaccard ↓ ↓ * ↑ * ↑ * 1 0.5 All 12 21 k 30 0 All 12 21 30 k Figure 6: Comparison of best performances for different k-mer lengths. The figures show the performance over queries by all positive samples as a violin plot. All methods use the ‘optimized metric’ chosen over 101 equally spaced threshold values between 0 and 1: the box denotes the MAP value. The horizontal lines show retrieval by chance: AveP computed over zero dissimilarity metric. Straight line is the mean, and dotted lines are 5%, and 95% quantiles respectively, when number of relevant samples differ for different queries. An arrow (if present) over a method implies whether the corresponding method performs significantly better (↑) or worse (↓) than ‘All’ : The stars denote significance level: 0 < * < 0.001 < < 0.01 < * < 0.05. We observe that the considering all k-mers usually perform equally well with respect to considering a single k. samples had the same annotation: class label, disease state or body site. A comparison of the obtained mean average precision values averaged over queries by all positive samples is shown in Fig. 5. The results show the performance achieved by the ‘optimized metric’. The alternatives we considered were: i. retrieval performance based on the proposed distances but with the frequencies counted on specific 21-mers from known protein families (FIGfams) [15]; ii. retrieval based on Hellinger distances between relative species abundances estimated using MetaPhlAn [24]; and iii. retrieval based on d2S \|M0 distances between relative frequencies of 3-mers [6]. For the simulated data, the two classes differ only by the relative species abundance; thus, retrieval based on ground truth abundance can be considered to give an upper limit for the performance. For HIGHC and HIGH-VAR, the proposed method performs closer to the ground truth performance than any other methods, although the difference from ground truth performance is still statistically significant. For LOW-C, the performance of all methods, except the protein family based comparison, drop compared to HIGH-C, while for MIXED-C the performance is again close to HIGH-C despite the presence of low coverage samples. This is an encouraging observation showing the robustness of the proposed approach to varying sequencing depths. For the real data sets the proposed approach yielded statistically significantly higher mean average precision than any of the alternatives (p < 0.05) for all the datasets, except T2D-P2 where protein family based comparison works equally well. Interestingly, the abundance-based retrieval performs relatively poorly here, suggesting that the differences between the classes cannot be easily captured by species composition alone, while the proposed k-mer features can provide a better separation. Retrieval based on the known protein family performed fairly well, but slightly worse than the proposed approach on MetaHIT. We observe that 11 Mean average precision Mean average precision 1 0.4 MetaHIT + log ↑* ↑ * ↑ * 0.3 0.4 0.2 0 0.5 0.2 0.1 All 12 HIGH−C + Jaccard ↑ * ↑ * ↑ ↓ * ↑ * ↑ * ↑ * ↑ * ↑ * 21 30 FIG k 1 0 All 12 21 30 FIG k HIGH−VAR + Jaccard ↓ * ↑ * ↓ * ↓ * ↑ * ↑ * ↑ * ↑ * ↑ * 1 0.5 All 12 1 HMP + Jaccard ↓ * ↑ * ↓ * ↓ * ↓ * ↑ * ↑ * 0.6 0.5 0 T2D−P2 + Jaccard ↑ * ↑ * ↑ * ↑ * ↑ * ↑ * ↑ * ↓ ↓ *** 21 30 FIG k 0 0 All 12 LOW−C + Jaccard ↓ * ↑ * ↓ ↑ *** ↑ * ↑ *** 0.5 All 12 0 21 30 FIG k 21 30 FIG k 1 MIXED−C + Jaccard ↑ * ↑ * ↓ * ↑ * ↑ * ↑ * ↑ * ↑ * 0.5 All 12 21 30 FIG k 0 All 12 21 30 FIG k Figure 7: Comparison of the best retrieval performance achieved with ‘optimized metric’ (middle), ‘average metric’ (right) and without entropy filtering (left), for proposed approach ‘All’, individual ks as well as FIGfam based distance metric. The metrics are ‘optimized’/‘averaged’ over 101 equally spaced threshold values between 0 and 1. Each errorbar line shows the MAP value along with the standard error. The grey horizontal line shows retrieval by chance: MAP computed over zero dissimilarity metric. An arrow (if present) over a method implies whether the performance of the corresponding method (top: ‘average metric’, bottom: ‘optimized metric’) is better (↑) or worse (↓) than when entropy filtering is employed: The stars denote significance level: 0 < * < 0.001 < ** < 0.01 < * < 0.05. We observe that filtering has a positive impact on the retrieval performance. for MetaHIT, Jaccard metric performs poorly; however, a change of metric to log significantly improves the performance for all methods. Otherwise, all metrics usually work equally well over different data sets. Effect of using specific or unspecific k-mer length: We next compared the proposed approach of using all k-mers to using a specific k. The retrieval performance using ‘optimized metric’ is shown in Fig. 6. The figures show the complete distribution of average precision values over different queries whose mean is the mean average precision of Fig. 5. The performance of the proposed method is usually better than with any individual k. Thus, the proposed method appears to be a relatively safe choice that does not suffer from catastrophically bad performance on any of the data sets. Effect of the entropy filtering: Next, we evaluated the efficacy of filtering the informative k-mers against retrieval performance without the filtering operation. The results are presented in Fig. 7. We observed that entropy filtering usually improved retrieval performance for all tested k-mer lengths when using the ‘optimized metric’, although the improvement might not always be statistically significant. Although ‘average metric’ often provides significant performance, it might not always improve over performance without filtering. Also, retrieval performance of FIGfam may or may not improve with entropy filtering. Comparison across different metrics: Finally, we evaluated the retrieval performance over different dissimilarity metrics. We presented the performance using ‘optimized metric’ for different metrics in Fig. 8. We 12 Average precision MetaHIT ↓ *** ↑* ↓ * ↑ * Average precision 1 0.4 0.2 HIGH−C ↑* ↓ * ↑ * ↓ * HMP ↑ * ↑ * ↓ * ↓ * ↑ * ↓ * 0.5 0.2 count ↑ * ↓* sqrt Metric 0 log 1 HIGH−VAR ↓ * ↓ * ↓ * ↑ * count sqrt Metric ↑ * ↑ * sqrt Metric log 0 0 log 1 0.5 count ↓ * ↑ * 0.6 0.4 0.5 0 ↑ * ↑ * 0.6 0 1 T2D−P2 ↓ * ↓ * ↓* ↑ * LOW−C ↑ * ↑ * ↓ * ↓ * count ↑ * ↓ *** 0.5 count sqrt Metric 0 log sqrt Metric log 1 MIXED−C ↓* ↓ *** ↑* ↓* ↑ *** ↑* 0.5 count sqrt Metric log 0 count sqrt Metric log Figure 8: Comparison of the best retrieval performance for different distance metrics using all k-mers. They show a violin plot of the average performances over queries by all positive samples in the data sets. The ‘optimized metrics’ have been selected over 101 equally spaced threshold values between 0 and 1: the box denotes the MAP value. The horizontal lines show retrieval by chance: AveP computed over zero dissimilarity metric. Straight line is the mean, and dotted lines are 5%, and 95% quantiles respectively, when number of relevant samples differ for different queries. An arrow (if present) over a method implies whether the corresponding method performs significantly better (↑) or worse (↓) than the other methods (denoted by their colors): The stars denote significance level: 0 < * < 0.001 < < 0.01 < * < 0.05. We observe that different distance metrics usually demonstrate similar performance. observed that the simple presence/absence-based metric Dcount performed at least as well as abundancesensitive log and sqrt metrics, except for the MetaHIT data for which the other metrics performed the better. 5 Conclusion In the wake of collecting multiple samples from similar environments information retrieval for metagenomic samples is expected to become a handy tool in metagenomics research. In this paper, we have addressed the problem of retrieving relevant metagenomic samples given a query sample from the same collection. The novelty of the proposed approach is that it is unsupervised, and does not rely on the availability of reference databases. We have suggested employing k-mer frequencies as feature representation; however, rather than exploring k-mers of a fixed k, we have scanned through all possible k-mers of all possible k’s using distributed string mining, and have proposed appropriate filtering technique to discard uninformative k-mers. We have evaluated our method on both real and simulated data, and observed that the approach can effectively retrieve relevant metagenomic samples, outperforming both the FIGfams method based on known highly informative protein families as well as retrieval based on species composition of the samples. 13 Acknowledgement The authors would like to thank Ahmed Sobih for his help with the MetaPhlAn experiments on MetaHIT and T2D-P2. Part of the calculations presented above were performed using computer resources within the Aalto University School of Science “Science-IT” project. Funding: This work was supported by the Academy of Finland (project numbers 140057, 250345, 251170 and 259440). 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AMENABLE UNIFORMLY RECURRENT SUBGROUPS AND LATTICE EMBEDDINGS arXiv:1802.04736v2 [math.GR] 29 Mar 2018 ADRIEN LE BOUDEC Abstract. We study lattice embeddings for the class of countable groups Γ defined by the property that the largest amenable uniformly recurrent subgroup AΓ is continuous. When AΓ comes from an extremely proximal action and the envelope of AΓ is co-amenable in Γ, we obtain restrictions on the locally compact groups G that contain a copy of Γ as a lattice, notably regarding normal subgroups of G, product decompositions of G, and more generally dense mappings from G to a product of locally compact groups. We then focus on a family of finitely generated groups acting on trees within this class, and show that these embed as cocompact irreducible lattices in some locally compact wreath products. This provides examples of finitely generated simple groups quasi-isometric to a wreath product C ≀ F , where C is a finite group and F a non-abelian free group. Keywords. Lattices, locally compact groups, strongly proximal actions, Chabauty space, groups acting on trees, irreducible lattices in wreath products. 1. Introduction The questions considered in this article fall into the setting of the following general problem: given a (class of) countable group Γ, study the locally compact groups G such that Γ embeds as a lattice in G, i.e. such that Γ sits as a discrete subgroup of G and G/Γ carries a G-invariant probability measure. Malcev showed that every finitely generated torsion free nilpotent group embeds as a cocompact lattice in a unique simply connected nilpotent Lie group [Rag72, Ch. II]. Conversely if G is a locally compact group with a finitely generated nilpotent lattice, then after modding out by a compact normal subgroup, the identity component G0 is a Lie group of polynomial growth (these have been characterized in [Gui73, Jen73]) and G/G0 is finitely generated and virtually nilpotent. This statement is a combination of several works. First if G has a finitely generated nilpotent lattice Γ, then Γ is necessarily cocompact in G. Since Γ is virtually torsion free this is a classical fact when G is totally disconnected, and the general case can be deduced from [BQ14, Prop. 3.7] (which uses notably the solution of Hilbert’s fifth problem [MZ55]). In particular G is compactly generated with polynomial growth, and the statement then follows from the generalization of Gromov’s polynomial growth theorem for locally compact groups [Los87]. Beyond the nilpotent case, examples of classifications of embeddings of Γ as a cocompact lattice have been obtained by Dymarz in [Dym15] for several families Date: March 29, 2018. This work was carried out when the author was F.R.S.-FNRS Postdoctoral Researcher. Current affiliation: CNRS, UMPA - ENS Lyon. 1 2 ADRIEN LE BOUDEC of examples of solvable groups Γ. Although not directly related to our concerns, we also mention that a certain dual problem was considered by Bader–Caprace– Gelander–Mozes in [BCGM16] for the class of amenable groups. Outside the setting of amenable groups, Furman addressed the above problem for the class of lattices Γ in semi-simple Lie groups in [Fur01], improving rigidity results of Mostow, Prasad, Margulis (see the references in [Fur01]; see also Furstenberg [Fur67b]). In [BFS15], Bader–Furman–Sauer considered a large class of countable groups Γ defined by certain group theoretic conditions, and established, given a lattice embedding of Γ in G, a general arithmeticity result in the setting where the connected component of G is non-compact. In this article we consider the class of groups whose Furstenberg uniformly recurrent subgroup is continuous (see below for definitions). In the first part of the article we address the question to what extent the properties of the Furstenberg uniformly recurrent subgroup of a countable group Γ influence the locally compact groups into which Γ embeds as a lattice. In the second part we focus on a family of finitely generated groups within this class which embed as cocompact irreducible lattices in some locally compact wreath products. The groups under consideration. For a countable group Γ, the Chabauty space Sub(Γ) of all subgroups of Γ is a compact space, on which Γ acts by conjugation. A uniformly recurrent subgroup (URS) of Γ is a closed minimal Γ-invariant subset of Sub(Γ) [GW15]. Glasner and Weiss showed that every minimal action of Γ on a compact space X gives rise to a URS (see Proposition 3.2), called the stabilizer URS associated to the action. Conversely every URS arises as the stabilizer URS of a minimal action (see Matte Bon–Tsankov [MBT17], and Elek [Ele17] in the case of finitely generated groups). URS’s have been shown to be related to the study of ideals in reduced group C ∗ algebras [KK17, Ken15] and reduced crossed products [Kaw17]. URS’s of several classes of groups have been studied in [LBMB16]. For certain examples of groups Γ, rigidity results about minimal actions on compact spaces have been obtained in [LBMB16] from a complete description of the space URS(Γ). Various results about homomorphisms between topological full groups of étale groupoids, notably obstructions involving invariants of the groupoids, have been obtained in [MB18] via URS’s considerations (more precisely via a complete description of the points in the Chabauty space of these groups whose orbit does not approach the trivial subgroup). In the present article we will make use of URS’s as a tool in order to study lattice embeddings for a class of countable groups that we now define. A URS is amenable if it consists of amenable subgroups. Every countable group Γ admits a largest amenable URS AΓ (with respect to a natural partial order on URS(Γ), see §3.1), which is the stabilizer URS associated to the action of Γ on its Furstenberg boundary (see §2.2 for definitions). The URS AΓ is called the Furstenberg URS of Γ. AΓ is either a point, in which case we have AΓ = {Rad(Γ)}, where Rad(Γ) is the amenable radical of Γ, or homeomorphic to a Cantor space. In this last case we say that AΓ is continuous. We refer to [LBMB16] for a more detailed discussion. Let (C) denote the class of groups Γ for which the Furstenberg URS AΓ is continuous. Equivalently, a group Γ belongs to (C) if and only if Γ admits an amenable URS whose envelope is not amenable (see below for the definition of the AMENABLE URS’S AND LATTICE EMBEDDINGS 3 envelope). The class (C) is disjoint from all classes of groups previously mentioned in the introduction. More precisely, the class (C) is disjoint from the class of amenable groups, the class of linear groups [BKKO14], and also from other classes of groups specifically considered in [BFS15], such as groups with non-vanishing ℓ2 -Betti numbers [BKKO14] or acylindrically hyperbolic groups (see [DGO11, Th. 7.19] and [BKKO14, Th. 1.4]). The class (C) is stable under taking quotient by an amenable normal subgroup and extension by an amenable group [LBMB16, Prop. 2.20]. Also if Γ has a normal subgroup that is in (C), then Γ belongs to (C) [LBMB16, Prop. 2.24]. By a result of Breuillard–Kalantar–Kennedy–Ozawa, the complement of the class (C) is also stable under extensions (see [LBMB16, Prop. 2.24]). The study of this class of groups is also motivated by the work of Kalantar– Kennedy [KK17], who showed the following characterization: a countable group Γ belongs to (C) if and only if the group Γ/Rad(Γ) has a reduced C ∗ -algebra that is not simple. For an introduction and the historical developments of the problem of C ∗ -simplicity, we refer to the survey of de la Harpe [Har07]. Topological boundaries. We will make use of the notion of topological boundary in the sense of Furstenberg. These are compact spaces with a minimal and strongly proximal group action (see §2.2 for definitions). Many different notions of boundaries appear in the study of groups and group actions. What is now sometimes called “boundary theory” is particularly well described in the introduction of [BF14]. We insist that in the present article the term boundary will always refer to a topological boundary in the above sense. This notion should not be confused with any of the measured notions of boundaries. In particular, despite the possibly confusing terminology, the maximal topological boundary, called the Furstenberg boundary, is not the same notion as the measured notion of Poisson–Furstenberg boundary. Lattices and direct products. Special attention will be given to products of locally compact groups. The study of lattices in product groups is motivated (among other things) by its connections with the theory of lattices in semi-simple Lie groups, its rich geometric aspects, as well as the instances of groups with rare properties appearing in this setting. We refer to the literature (see [Mar91, Wis96, BM97, BM00b, Rém99, Sha00, BM02, Rat04, MS04, BS06, CR09, CM12, Rad17]) for developments over the last years on the study of lattices in products of locally compact groups. Given a countable group Γ with a continuous Furstenberg URS and a group G containing Γ as a lattice, we are interested in understanding how close the group G can be from a direct product of two groups, or which properties the group G can share with a direct product. Of course various notions of closeness can be considered. The most basic one is to ask whether the group G admits non-trivial decompositions as a direct product. One step further, one might consider quotient morphisms from G onto direct products of groups. In Theorems 1.1 and 1.2 below we more generally consider continuous morphisms with dense image from G to a direct product of groups G → G1 × G2 . We make no assumption about injectivity of these maps or injectivity of the composition with the projection to one factor 4 ADRIEN LE BOUDEC Gi . In particular this setting allows maps of the form G → G/N1 × G/N2 for closed normal subgroups N1 , N2 such that N1 N2 is dense in G. First results. A central notion in this article is the one of extremely proximal action. Minimal and extremely proximal actions naturally arise in geometric group theory, and are boundaries in the sense of Furstenberg. We refer to §2.3 for definitions and examples. We say that the Furstenberg URS AΓ of a countable group Γ comes from an extremely proximal action if there exists a compact space Z and a Γ-action on Z that is minimal and extremely proximal, whose associated stabilizer URS is equal to AΓ . Note that typically Z will not be the Furstenberg boundary of Γ. If H is a URS of Γ, the envelope Env(H) of H is by definition the subgroup of Γ generated by all the subgroups H ∈ H. Theorem 1.1. Let Γ be a countable group whose Furstenberg URS comes from a faithful and extremely proximal action, and let G be a locally compact group containing Γ as a lattice. The following hold: (a) Assume that Env(AΓ ) is finitely generated and co-amenable in Γ. Then G cannot be a direct product G = G1 × G2 of two non-compact groups. (b) Assume that Env(AΓ ) has finite index in Γ and finite abelianization. Then any continuous morphism with dense image from G to a product of locally compact groups G → G1 × G2 is such that one factor Gi is compact. The same conclusions hold for any group commensurable to G up to compact kernels. This result has applications to the setting of groups acting on trees, see Corollary 1.3. We make several comments about the theorem: 1) We do not assume that Γ is finitely generated, nor that G is compactly generated. For statement (a), The assumption that Env(AΓ ) if finitely generated admits variations, see Theorem 5.14. 2) Making an assumption on the “size” of the envelope of AΓ with respect to Γ is natural, in the sense that in general there is no hope to derive any conclusion on the entire group Γ if this envelope is too small. An extreme illustration of this is that there are groups Γ whose Furstenberg URS comes from a faithful and extremely proximal action but is trivial, and these can be lattices in products, e.g. PSL(2, Z[1/p]) inside PSL(2, R) × PSL(2, Qp ) (see also the discussion right after Corollary 1.3). 3) Under the assumption that Env(AΓ ) is co-amenable in Γ, the fact that the Furstenberg URS AΓ comes from a faithful and extremely proximal action is equivalent to asking that the action of Γ on AΓ is faithful and extremely proximal; see Remark 3.27. This provides an intrinsic reformulation of the assumption not appealing to any auxiliary space. 4) For Γ as in the theorem, the assumption in statement (b) that Env(AΓ ) has finite index in Γ and Env(AΓ ) has finite abelianization is equivalent to Γ being virtually simple (see Proposition 4.6). The URS approach to study lattice embeddings allows to consider more generally subgroups of finite covolume. Recall that a closed subgroup H of a locally AMENABLE URS’S AND LATTICE EMBEDDINGS 5 compact group G has finite covolume in G if G/H carries a G-invariant probability measure. Thus a lattice is a discrete subgroup of finite covolume. Before stating the following result we need some terminology. Recall the notion of disjointness introduced by Furstenberg in [Fur67a]. If X, Y are compact G-spaces, X and Y are disjoint if whenever Ω is a compact G-space and Ω → X and Ω → Y are continuous equivariant surjective maps, the map Ω → X × Y that makes the natural diagram commute remains surjective (see §3.3). When X, Y are minimal G-spaces, this is equivalent to asking that the diagonal G-action on the product X × Y is minimal. Consider the following property: two non-trivial G-boundaries are never disjoint. A group with this property will be called boundary indivisible. Glasner characterized minimal compact G-spaces which are disjoint from all G-boundaries as those carrying a fully supported measure whose orbit closure in the space of probability measures is minimal [Gla75, Th. 6.2]. The relation between disjointness and boundaries that we consider here is of different spirit, as it deals with disjointness within the class of G-boundaries, rather than disjointness from this class. Locally compact groups with a cocompact amenable maximal subgroup are examples of boundary indivisible groups [Fur73, Prop. 4.4]. On the contrary, many discrete groups are not boundary indivisible. The relevance of this property in our setting comes from the fact that, as we will show in Proposition 3.24, a discrete group Γ as in Theorem 1.1 is boundary indivisible. Actually the only examples of (non-amenable) boundary indivisible discrete groups that we are aware of fall into the setting of Proposition 3.24. Recall that a convex compact G-space is irreducible if it does not contain any proper closed convex G-invariant subspace. We say that a subgroup L of a topological group G is weakly co-amenable in G if whenever Q is a non-trivial convex compact G-space in which L fixes a point, Q is not irreducible. This is indeed a weakening of the notion of co-amenability †, which asks that every convex compact G-space Q with L-fixed points has G-fixed points [Eym72] (and hence Q is not irreducible, unless trivial). If G has a subgroup that is both amenable and weakly co-amenable, then G is amenable; and a normal weakly co-amenable subgroup is coamenable. However in general weak co-amenability does not imply co-amenability, even for discrete groups. In §6.3 we exhibit examples of finitely generated groups such that every subgroup is either amenable or weakly co-amenable, but having non-amenable subgroups that are not co-amenable. Finally we say that a subgroup L ≤ G is boundary-minimal if there exists a non-trivial G-boundary on which L acts minimally. We refer to §5.1 for context and examples. Theorem 1.2. Let H be a locally compact group with an amenable URS that comes from an extremely proximal action, and whose envelope is co-amenable in H. Let G be a locally compact group containing H as a closed subgroup of finite covolume. Then G is boundary indivisible, and the following hold: (a) Whenever G → G1 × G2 is a continuous morphism with dense image, one factor Gi is amenable. †and not a relative version of a notion of weak amenability. 6 ADRIEN LE BOUDEC (b) If L is a boundary-minimal subgroup of G, and L is uniformly recurrent, then L is weakly co-amenable in G. In particular every boundary-minimal normal subgroup of G is co-amenable in G. Again we make several comments: 1) The group H is allowed to be discrete, so the theorem applies for all groups Γ as in Theorem 1.1. While (a) will be an intermediate step in the proof of Theorem 1.1, (b) provides additional information that is rather independent of the conclusion of Theorem 1.1. 2) Statement (a) implies that whenever N1 , N2 are closed normal subgroups of G such that N1 N2 is dense in G, at least one Ni must be co-amenable in G. 3) The last sentence in (b) implies that if N is a closed normal subgroup of G such that N CN is open in G (where CN is the centralizer of N ), then N is either amenable or co-amenable in G (see Proposition 5.3). We do not know whether the condition that N CN is open can be removed. 4) Theorem 1.2 does not say anything about amenable normal subgroups of G. It is worst pointing out that, as illustrated by the examples discussed in Section 6, it happens that a discrete group Γ satisfying the assumptions of Theorem 1.2 and with trivial amenable radical, sits as a lattice in a group G with noncompact amenable radical. 5) Remark 5.13 below provides counter-examples showing that in statement (b) the conclusion cannot be strengthened by saying that L is co-amenable in G. 6) For H, G as in the theorem, it happens that G splits as a direct product of two non-compact groups, even under the additional assumption that the amenable URS of H comes from a faithful extremely proximal action (see Example 6.3), so that “amenable” cannot be replaced by “compact” in statement (a). We view the above remarks 4)-5)-6) as illustrations of the limitations of the use of topological boundaries and URS’s to the problem addressed here in the rather abstract setting of Theorem 1.2. Group actions on trees is a natural source of extremely proximal actions, and Theorems 1.1 and 1.2 find applications in this setting. In the following statement T is a locally finite simplicial tree. Corollary 1.3. Let Γ ≤ Aut(T ) be a countable group having no proper invariant subtree and no finite orbit in T ∪ ∂T . Assume that Γξ is non-trivial and amenable for all ξ ∈ ∂T ; and Γ is virtually simple. If G is a locally compact group containing Γ as a lattice, then: (a) any continuous morphism with dense image G → G1 × G2 is such that one factor Gi is compact. In particular G itself cannot be a direct product of two non-compact groups. (b) The conclusion (b) of Theorem 1.2 holds for G. A group Γ as in Corollary 1.3 is never discrete in Aut(T ). Recall that Burger and Mozes constructed simple groups Γ acting on two locally finite regular trees T, T ′ such that the image of Γ in Aut(T ) and Aut(T ′ ) are non-discrete, but Γ acts AMENABLE URS’S AND LATTICE EMBEDDINGS 7 freely and cocompactly on T × T ′ , so that Γ is a cocompact lattice in the product Aut(T ) × Aut(T ′ ) [BM00b]. These examples illustrate the fact that the assumption in Corollary 1.3 that end-stabilizers are all non-trivial is essential. Examples of groups to which Corollary 1.3 applies can be found among the family of groups denoted G(F, F ′ ) in [LB16] (see Corollary 6.22). These are examples of groups with a continuous Furstenberg URS. Here F ′ ≤ Sym(d) is a finite permutation group and F is a simply transitive (or regular) subgroup of F ′ . The group G(F, F ′ ) is then a finitely generated group acting on a d-regular tree, transitively on vertices and edges, and with local action at every vertex isomorphic to F ′ . We refer to §6.2 for a definition. The normal subgroup structure of these groups is highly sensible to the permutation groups: there are permutation groups F, F ′ such that G(F, F ′ ) virtually admits a non-abelian free quotient (Proposition 6.11), and there are permutation groups F, F ′ such that G(F, F ′ )∗ (the subgroup of index two in G(F, F ′ ) preserving the bipartition of Td ) is simple [LB16, Cor. 4.14]. This family of groups and the family of Burger–Mozes lattices in the product of two trees both contain instances of finitely generated simple groups which embed densely in some universal group U (F )+ [BM00a, §3.2]. Despite these similarities, Corollary 6.22 shows that any group containing a virtually simple G(F, F ′ ) as a lattice is rather allergic to any direct product behavior. Compare with Theorem 1.4. We also mention that other examples of groups to which Corollary 1.3 can be applied may be found among the family of piecewise prescribed tree automorphism groups considered in [LB17, Sec. 4]. Irreducible lattices in wreath products. Leaving aside the previous abstract situation, we then focus on the family of groups G(F, F ′ ) (see §6.2 for definitions). The above mentioned common properties between the discrete groups G(F, F ′ ) and certain lattices in the product of two trees provide motivation for studying which locally compact groups can contain a group G(F, F ′ ) as a lattice. The contribution of this article to this problem is on the one hand the conclusions given by Corollary 1.3 (see Corollary 6.22), and on the other hand to describe embeddings of these groups as irreducible lattices in some locally compact wreath products. If H is a group acting on a set Ω, and A is a subgroup of a group B, the semirestricted permutational wreath product B ≀A Ω H, introduced by Cornulier in Ω,A [Cor17], is the semi-direct product B ⋊ H, where B Ω,A is the set of functions f : Ω → B such that f (x) ∈ A for all but finitely many x ∈ Ω, and H acts on B Ω,A in the usual way. This definition somehow interpolates between the restricted and the unrestricted permutational wreath products, which correspond respectively to A = 1 (in which case we will write B ≀Ω H) and A = B. When B, H are locally compact and A is compact open in B, there is a natural locally compact group topology on B ≀A Ω H (see §6.5.1). We call a lattice Γ in B ≀A Ω H irreducible if Γ has a non-discrete projection to H. The terminology is motivated by the fact that this definition prevents Γ, and more generally any subgroup commensurable with Γ, from being of the form Γ1 ⋊ Γ2 , where Γ1 and Γ2 are lattices in B Ω,A and H. In the following statement Ck is the cyclic group of order k, Sk the symmetric group on k elements, Vd the vertex set of a d-regular tree Td , and G(F, F ′ )∗ the subgroup of index two in G(F, F ′ ) preserving the bipartition of Td . 8 ADRIEN LE BOUDEC Theorem 1.4. Let d ≥ 3, F F ′ ≤ Sym(d) permutation groups such that F acts freely, and n the index of F in F ′ . Then the following hold: (a) The group G(F, F ′ ) embeds as an irreducible cocompact lattice in the semiS restricted permutational wreath product Gn,d = Sn ≀Vdn−1 Aut(Td ). (b) When F is a transitive permutation group, the finitely generated group Γn,d = Cn ≀Vd (Cd ∗ Cd ) = (Cn2 ≀ Fd−1 ) ⋊ Cd and G(F, F ′ )∗ have isometric Cayley graphs. We note that no finite index subgroup of Gn,d can split non-trivially as a product, but the stabilizer of an edge of Td in Gn,d (for the projection action on Td ) is an open subgroup which does split as a direct product of two non-compact groups. V ,S The embedding of G(F, F ′ ) in Gn,d = Snd n−1 ⋊ Aut(Td ) is not the inclusion in the subgroup 1 ⋊ Aut(Td ), but a twisted embedding associated to the cocycle given by the local action on Td . See Section 6 for details. We also note that the image V ,S of G(F, F ′ ) in Gn,d does not intersect the amenable radical Snd n−1 ⋊ 1, but does intersect the subgroup 1 ⋊ Aut(Td ) (along a cocompact lattice of Aut(Td )). The case n = 2 is particular as the group G2,d is actually a restricted wreath product, and in this situation the group G(F, F ′ ) is an irreducible cocompact lattice in C2 ≀Vd Aut(Td ) = (⊕Vd C2 ) ⋊ Aut(Td ). Applications. Recall that the property of being virtually simple is not invariant by quasi-isometry. Indeed the lattices constructed by Burger and Mozes in [BM00b] show that a virtually simple finitely generated group may have the same Cayley graph as a product of two finitely generated free groups. Theorem 1.4 together with simplicity results from [LB16] provide another illustration of this fact, namely finitely generated simple groups having the same Cayley graph as a wreath product. The wreath product construction is already known to be a source of examples of finitely generated groups whose algebraic properties are not reflected in their Cayley graphs. Two wreath products B1 ≀ Γ and B2 ≀ Γ may have isometric or bi-Lipschitz Cayley-graphs, one being solvable or torsion free, while no finite index subgroup of the second has these properties [Dyu00]. The phenomenon exhibited in Theorem 1.4 is nonetheless very different, in the sense that it provides finitely generated groups with isometric Cayley graphs such that one is a wreath product, but the other is simple (and hence not commensurable with a wreath product). Recall that for finitely generated groups, being amenable is a quasi-isometry invariant. By contrast, Theorem 1.4 implies: Corollary 1.5. Among finitely generated groups, the property of having infinite amenable radical is not invariant by quasi-isometry. The examples from Theorem 1.4 simultaneously show that having an infinite elliptic radical is also not invariant by quasi-isometry. Recall that the elliptic radical of a discrete group is the largest locally finite normal subgroup. Recall that by a theorem of Eskin–Fisher–Whyte, any finitely generated group Γ that is quasi-isometric to a wreath product C ≀Z, where C is a finite group, must act properly and cocompactly on a Diestel-Leader graph DL(m, m) [EFW07, EFW12, EFW13]. By the algebraic description of the isometry groups of these graphs given in [BNW08] (see also [CFK12]), this implies in particular that Γ has a subgroup AMENABLE URS’S AND LATTICE EMBEDDINGS 9 of index at most two that is (locally finite)-by-Z. By contrast, Theorem 1.4 shows that this rigidity fails in the case of C ≀ Fk when k ≥ 2. Questions. We end this introduction with two questions. Extreme proximality is used in a crucial way at different stages of the proofs of Theorems 1.1 and 1.2. These results both fail without the extreme proximality assumption, simply because then the group itself may very well be a direct product. Putting aside these trivial counter-examples, we do not know whether serious algebraic restrictions on a locally compact group may be derived from the existence of a lattice with a continuous Furstenberg URS. In this direction, we find the following question natural: Question 1.6. Does there exist Γ with a continuous Furstenberg URS which is a lattice in a group G = G1 × G2 such both factors are non-discrete, and Γ has an injective and dense projection to each factor ? What if we impose moreover that Γ has trivial amenable radical ? Theorem 1.4 presents a situation of a locally compact group G with two cocompact lattices Γ1 , Γ2 ≤ G such that the stabilizer URS associated to the Γ1 -action on ∂sp G is {Rad(Γ1 )}, while the stabilizer URS associated to the Γ2 -action on ∂sp G is continuous. Here ∂sp G stands for the Furstenberg boundary of G; see §2.2. In these examples the group G splits as G = N ⋊ Q, where N is the amenable radical of G. The lattice Γ1 preserves this splitting, meaning that we have Γ1 = (N ∩Γ1 )⋊(Q∩Γ1 ) (and hence Γ1 does not act faithfully on ∂sp G), while Γ2 has an injective projection to Q. This naturally raises the following: Question 1.7. Let G be a locally compact group with two lattices Γ1 and Γ2 both acting faithfully on X = ∂sp G. Is it possible that the Γ1 -action on X is topologically free, but the Γ2 -action on X is not topologically free ? Can this happen with ∂sp G = G/H being a homogeneous G-space ? Note that by [Fur03, Prop. 7], the condition that Γ1 and Γ2 act faithfully on ∂sp G is equivalent to saying that Γ1 and Γ2 have trivial amenable radical. Recall that topologically free means that there is a dense subset of points having trivial stabilizer (equivalently, the stabilizer URS is trivial). Outline of proofs and organization. The article is organized as follows. In the next section we introduce terminology and preliminary results about topological boundaries and extremely proximal actions. In Section 3 we establish the results about uniformly recurrent subgroups that are used in later sections. In particular we prove a certain gap property for URS’s coming from extremely proximal actions (Proposition 3.17). Combined with an observation about compact spaces with comparable stabilizer URS’s (Proposition 3.9), we deduce that a locally compact group H with an amenable URS that comes from an extremely proximal action, and whose envelope is co-amenable in H, is boundary indivisible (Proposition 3.24). The setting of Section 4 is that of a group admitting a non-topologically free extremely proximal action. We establish intermediate results, notably concerning normal subgroups (Proposition 4.6) and commensurated subgroups (Proposition 4.12), and deduce non-embedding results for this class of groups (see Proposition 4.10 and Corollary 4.13). In Section 5 we use results from Section 3 together with Proposition 5.9 of Furstenberg and prove Theorem 1.2. We then specify to discrete groups and give 10 ADRIEN LE BOUDEC the proof of Theorem 1.1. The proof essentially splits in two steps: the first one is the application of Theorem 1.2 to obtain amenability of one factor, and the second consists in proving that under appropriate assumptions the amenable factor is compact, using results from Section 4. In Section 6 we consider groups acting on trees, and apply previous results of the article to this setting. After giving the proof of Corollary 1.3, we focus on the family of groups with prescribed local action G(F, F ′ ). We study boundaries of these groups, and use results from Section 3 in order to characterize the discrete groups within this family which are boundary indivisible (see Theorem 6.9). This includes those which are virtually simple, but this also contain non-virtually simple instances. Finally we study lattice embeddings of these groups and give the proof of Theorem 1.4. Acknowledgements. I am grateful to Alex Furman for pointing out Proposition 5.9 to my attention, and to Uri Bader for enlightening discussion about the proof. I am also grateful to Pierre-Emmanuel Caprace, Yves Cornulier, Bruno Duchesne, Nicolás Matte Bon, Nicolas Monod and Pierre Pansu for interesting discussions and comments related to this work. Finally I am indebted to Alain Valette for a decisive remark made in Neuchâtel in May 2015, which eventually led to Theorem 1.4. 2. Preliminaries 2.1. Conventions and terminology. The letter G will usually refer to a topological group, while Γ will denote a discrete group. The group of homeomorphic automorphisms of G will be denoted Aut(G). Whenever G is a locally compact group, we will always assume that G is second countable. The notation X will refer to a topological space. The letters X, Y will be reserved for compact spaces, and Z for a compact space equipped with an extremely proximal group action. All compact spaces are assumed to be Hausdorff. A space X is a G-space if G admits a continuous action G × X → X . The action of G on X (or the G-space X ) is minimal if all orbits are dense. The G-space X is said to be trivial if X is a one-point space. If X is locally compact, we denote by Prob(X ) the set of all regular Borel probability measures on X . The space of continuous compactly supported functions on X is denoted CK (X ). Each µ ∈ Prob(X ) defines a linear functional on CK (X ), and we endow Prob(X ) with the weak*-topology: a net (µi ) converges to µ if µi (f ) → µ(f ) for all f ∈ CK (X ). By Banach-Alaoglu theorem, Prob(X ) is relatively compact in CK (X )∗ . We denote by 2X the set of all closed subsets of X . The sets n o O(K; U1 , . . . , Un ) = C ∈ 2X : C ∩ K = ∅; C ∩ Ui 6= ∅ for all i , where K ⊂ X is compact and U1 , . . . , Un ⊂ X are open, form a basis for the Chabauty topology on 2X . Endowed with the Chabauty topology, the space 2X is compact. We will freely identify X with its image in 2X by the natural inclusion x 7→ {x}. Note that when X is a G-space, so is 2X . In the particular case where X = G is a locally compact group, the space Sub(G) of closed subgroups of G is closed in 2G . In particular Sub(G) is a compact space, on which G acts by conjugation. A uniformly recurrent subgroup (URS) of AMENABLE URS’S AND LATTICE EMBEDDINGS 11 G is a closed, G-invariant, minimal subset of Sub(G). The set of URS’s of G is denoted URS(G). By extension we also say that a subgroup H ≤ G is uniformly recurrent if the closure of the conjugacy class of H in Sub(G) is minimal. 2.2. Topological boundaries. If X is a compact G-space, the action of G on X is strongly proximal if the closure of any G-orbit in Prob(X) contains a Dirac measure. Strong proximality is stable under taking products (with diagonal action) and continuous equivariant images (see e.g. [Gla76]). We say that X is a boundary if X is both minimal and strongly proximal. For every topological group G, there exists a unique boundary ∂sp G with the universal property that for any boundary X, there exists a continuous G-equivariant surjection ∂sp G → X [Fur73, Prop. 4.6]. This universal space ∂sp G is referred to as the Furstenberg boundary of G. It is easy to verify that any amenable normal subgroup N of G acts trivially on any G-boundary, so that ∂sp G = ∂sp (G/N ). If G admits a cocompact amenable subgroup, then the Furstenberg boundary is a homogeneous space ∂sp G = G/H, and the G-spaces of the form G/L with L containing H are precisely the G-boundaries [Fur73, Prop. 4.4]. The situation for discrete groups is quite different: as shown in [KK17] and [BKKO14], Furstenberg boundaries of discrete groups are always non-metrizable (unless trivial). The following is a fundamental property of boundaries (see [Gla76, III.2.3]): Theorem 2.1. Any convex compact G-space contains a boundary. In fact if Q is an irreducible convex compact G-space, then the action of G on Q is strongly proximal, and the closure of extreme points of Q is a G-boundary. Irreducible means that Q has no proper closed convex G-invariant subspace. In particular Theorem 2.1 has the following consequence ([Gla76, III.3.1]): Theorem 2.2. A group G is amenable if and only if all G-boundaries are trivial, or equivalently ∂sp G is trivial. 2.3. Extremely proximal actions. Let X be a compact G-space. A closed subset C of X is compressible if the closure of the G-orbit of C in the space 2X contains a singleton {x}. Equivalently, for every neighbourhood U of x, there exists g ∈ G such that g(C) ⊂ U . The action of G on X is extremely proximal if every closed subset C ( X is compressible. References where extremely proximal actions were considered include [Gla74, LS96, JR00, FST15, LBMB16]. We will make use of the following result, which is Theorem 2.3 from [Gla74]: Theorem 2.3. Let X be a compact G-space, and assume X has at least three points. If the G-action on X is extremely proximal, then it is strongly proximal. Examples of extremely proximal actions are provided by group actions on trees or hyperbolic spaces. If G ≤ Aut(T ) acts on T with no proper invariant subtree and no finite orbit in T ∪ ∂T , then the action of G on ∂T is minimal and extremely proximal; and if G acts coboundedly on a proper geodesic hyperbolic space X with no fixed point or fixed pair at infinity, then the G-action on the Gromov boundary ∂X is minimal and extremely proximal. These two situations are particular cases of the following more general result, that we believe is well-known. A homeomorphism g of a space X is hyperbolic if there exist ξ− , ξ+ ∈ X, called the endpoints of g, such that for all neighbourhoods 12 ADRIEN LE BOUDEC U− , U+ of ξ− , ξ+ , for n large enough we have gn (X \ U− ) ⊂ U+ and g−n (X \ U+ ) ⊂ U− . Proposition 2.4. If G acts on a compact space X with hyperbolic elements having no common endpoints, and such that the set of endpoints of hyperbolic elements of G is dense in X, then the action is minimal and extremely proximal. Proof. Let U ⊂ X be a non-empty open invariant subset. By our density assumption, there is g ∈ G hyperbolic whose attracting endpoint ξ+ belongs to U . So for every x 6= ξ− , there is n > 0 such that gn (x) ∈ U since U is open, so we deduce that U contains X \ {ξ− }. But the existence of hyperbolic elements with no common endpoints ensures that G fixes no point of X, so finally U = X, i.e. the action is minimal. Now if C ( X is a closed subset then again there is g ∈ G whose attracting endpoint is outside C, and C is compressible to the repealing endpoint of g. Recent work of Duchesne and Monod shows that group actions on dendrites is also a source of extremely proximal actions. Recall that a dendrite X is a compact metrizable space such that any two points are the extremities of a unique arc. Duchesne and Monod show that if Γ acts on X with no invariant proper subdendrite, then there is a unique minimal closed invariant subset M ⊆ X and the Γ-action on M is extremely proximal. See the proof of Theorem 10.1 in [DM16]. Extremely proximal actions also play a prominent role in the context of group actions on the circle. For any minimal action α : Γ → Homeo+ (S1 ), either α(Γ) is conjugated to a group of rotations, or α(Γ) has a finite centralizer CΓ in Homeo+ (S1 ) and the action of Γ on the quotient circle CΓ \S1 is extremely proximal: see Ghys [Ghy01] and Margulis [Mar00]. We mention however that in all the examples of countable groups Γ with an action on S1 that is minimal and not topologically free that we are aware of, the stabilizer URS is either non-amenable, or not known to be amenable. In particular we do not know any application of Theorem 1.1 to groups acting on the circle. In the sequel we will make use of the following easy lemma. Lemma 2.5. Let G be a topological group, and H a subgroup of G such that there is some compact subset K of G such that G = KH. Let X be a compact G-space, and C a closed subset of X that is compressible by G. Then C is compressible by H. In particular if the G-action on X is extremely proximal, then the H-action is extremely proximal. Proof. By assumption there exists x ∈ X and (gi ) such that gi (C) converges to x in 2X . If gi = ki hi , by compactness of K we assume that (ki ) converges to some k, and it follows that hi (C) converges to k−1 x by continuity of G × 2X → 2X . 3. Uniformly recurrent subgroups 3.1. Generalities on uniformly recurrent subgroups. Let G be a locally compact group. For H, K ∈ URS(G), we write H 4 K when there exist H ∈ H and K ∈ K such that H ≤ K. This is equivalent to the fact that every H ∈ H is contained in an element of K, and every K ∈ K contains an element of H, and the relation 4 is an order on URS(G). See e.g. §2.4 in [LBMB16]. AMENABLE URS’S AND LATTICE EMBEDDINGS 13 For simplicity the URS {N } associated to a closed normal subgroup N of G will still be denoted N . In particular N 4 H (resp. H 4 N ) means that N is contained in (resp. contains) all the elements of H. By the trivial URS we mean the URS corresponding to the trivial subgroup {1}. We warn the reader that in this terminology the URS corresponding to a non-trivial normal subgroup N is trivial as a G-space (it is a one-point space), but is not trivial as a URS. Let X, Y be compact G-spaces. We say that X is a factor of Y , and Y is an extension of X, if there exists a continuous equivariant map Y → X that is onto. If π : Y → X is a continuous equivariant map, we say that π is almost 1-1 if the set of y ∈ Y such that π −1 (π(y)) = {y} is dense in Y . When moreover π is onto we say that Y is an almost 1-1 extension of X. We now recall the definition of the stabilizer URS associated to a minimal action on a compact space. If X is a compact G-space and x ∈ X, we denote by Gx the stabilizer of x in G. Definition 3.1. If X is a compact G-space, we denote by X0 ⊂ X the set of points at which Stab : X → Sub(G), x 7→ Gx , is continuous. Upper semi-continuity of the map Stab and second countability of G imply that X0 is a dense subset of X (indeed if (Un ) is a basis of the topology on Sub(G) and Xn is the set of x ∈ X such that Gx ∩ Un 6= ∅, which is closed, one verifies that Stab is continuous on ∩n (∂Xn )c ). Following [GW15], we denote X̃ = cls {(x, Gx ) : x ∈ X0 } ⊂ X × Sub(G), and SG (X) = cls {Gx : x ∈ X0 } ⊂ Sub(G), where cls stands for the closure in the ambient space. We have the obvious inclusions X̃ ⊆ X ∗ := cls {(x, Gx ) : x ∈ X} and ∗ SG (X) ⊆ SG (X) := cls {Gx : x ∈ X} . We denote by ηX and πX the projections from X × Sub(G) to X and Sub(G) respectively. Proposition 3.2 (Prop. 1.2 in [GW15]). If X is a minimal compact G-space, then ηX : X̃ → X is an almost 1-1 extension, and X̃ and SG (X) are the unique minimal ∗ (X). closed G-invariant subsets of respectively X ∗ and SG Definition 3.3. SG (X) is the stabilizer URS associated to the G-action on X. The action of G on X is topologically free if SG (X) is trivial, i.e. SG (X) = {{1}}. Remark 3.4. When G is not assumed second countable, in general X0 is no longer dense in X. However it is still possible to define the stabilizer URS associated to a minimal action on a compact space; see the discussion in [MBT17, p1]. In the sequel we will sometimes use the following version of Proposition 3.2. Proposition 3.5. Let X be compact G-space, and let H ≤ G be a subgroup acting minimally on X. Then H acts minimally on X̃ and SG (X), and X̃ and SG (X) ∗ (X). are the unique minimal closed H-invariant subsets of X ∗ and SG 14 ADRIEN LE BOUDEC Proof. Let Y ⊆ X ∗ closed and H-invariant. Since X is a factor of X ∗ and H acts minimally on X, for every x ∈ X0 there exists L ∈ Sub(G) such that (x, L) ∈ Y . But for x ∈ X0 , the fact that (x, L) belongs to X ∗ forces L to be equal to Gx by definition of X0 , and it follows that X̃ ⊆ Y . Moreover H acts minimally on X̃ since ηX : X̃ → X is an almost 1-1 extension and minimality is preserved by taking almost 1-1 extensions (if π : X1 → X2 is almost 1-1 and if C ⊆ X1 is a closed subset such that π(C) = X2 , then C = X1 ). So the statements for X̃ and ∗ (X) since these are factors X ∗ are established, and the same hold for SG (X) and SG of X̃ and X ∗ . 3.2. Envelopes. Let G be a locally compact group and H ∈ URS(G). Definition 3.6. The envelope Env(H) of H is the closed subgroup of G generated by all the subgroups H ∈ H. By definition Env(H) is the smallest closed subgroup of G such that H ⊂ Sub(Env(H)). Note that Env(H) is a normal subgroup of G, and is actually the smallest normal subgroup such that H 4 Env(H). Let Γ be a discrete group, X a compact Γ-space and X0 the domain of continuity of the map Stab. It is a classical fact that X0 consists of those x ∈ X such that for every γ ∈ Γx , there exists U neighbourhood of x that is fixed by γ (see e.g. [Vor12, Lem. 5.4] for a proof). For x ∈ X, we will denote by Γ0x ≤ Γx the set of elements fixing a neighbourhood of x, so that x ∈ X0 if and only if Γx = Γ0x . Lemma 3.7. Let Γ be a countable discrete group, X a compact minimal Γ-space, n ≥ 1 and γ1 , . . . , γn ∈ Γ. The following are equivalent: (i) (ii) (iii) (iv) ∩Fix(γi ) has non-empty interior; there is x ∈ X such that γi ∈ Γ0x for all i; same as in (ii) but with x ∈ X0 ; there is H ∈ SΓ (X) such that γi ∈ Hfor all i. In particular Env(SΓ (X)) is generated by the elements γ ∈ Γ such that Fix(γ) has non-empty interior. Proof. It is clear that (i) and (ii) are equivalent. Also (iii) clearly implies (ii), and (ii) also implies (iii) by density of X0 in X. Finally (iii) implies (iv) since Γ0x ∈ SΓ (X) for x ∈ X0 , and (iv) implies (iii) by density of the set of Γ0x , x ∈ X0 , in SΓ (X). 3.3. G-spaces with comparable stabilizer URS’s. We recall the notion of disjointness from [Fur67a]. Two compact G-spaces X, Y are disjoint if whenever X, Y are factors of a compact G-space Ω via ϕX : Ω → X and ϕY : Ω → Y , then the map (ϕX , ϕY ) : Ω → X × Y is surjective. When X, Y are minimal G-spaces, this is equivalent to saying that the product X × Y remains minimal [Fur67a, Lem. II.1]. The following lemma presents a situation which easily implies disjointness: Lemma 3.8. Let X, Y be minimal compact G-spaces such that there exists y0 ∈ Y such that Gy0 acts minimally on X. Then X and Y are disjoint. AMENABLE URS’S AND LATTICE EMBEDDINGS 15 Proof. This is clear: if W is a closed invariant subset of X × Y , then by minimality of Y there exists x0 ∈ X such that (x0 , y0 ) ∈ W . Since Gy0 acts minimally on X we deduce that W contains X × {y0 }, and by minimality of Y it follows that W is equal to X × Y . The following proposition will be used notably in Proposition 3.24. Proposition 3.9. Let X, Y be compact minimal G-spaces, and write H = SG (X) and K = SG (Y ). Suppose that H 4 K. Then X and Y can be disjoint only if Env(H) 4 K. In particular if SG (X) = SG (Y ) and this URS is not a point, then X and Y are not disjoint. Proof. Using notation from Proposition 3.2, we have almost 1-1 extensions ηX : X̃ → X and ηY : Ỹ → Y , and we write η = ηX × ηY : X̃ × Ỹ → X × Y , and π = πX × πY : X̃ × Ỹ → H × K. The set W ⊆ H × K of pairs (H, K) such that H ≤ K is non-empty by assumption, is clearly G-invariant, and is easily seen to be closed. If W is a proper subset of H × K then π −1 (W ) is a proper subset of X̃ × Ỹ since π is a factor, and it follows that η π −1 (W ) is a closed G-invariant subset of X × Y that is proper since η is almost 1-1. This contradicts disjointness of X and Y . Therefore we have W = H × K. This means that for a fixed K ∈ K we have H ≤ K for every H ∈ H, and hence Env(H) 4 K. 3.4. Action of a URS on a G-space. In this paragraph G still denote a locally compact group, and X is a compact G-space. Given H ∈ URS(G), we study the properties of the action of elements on H on the space X. The proof of the following lemma is an easy verification, and we leave it to the reader. Lemma 3.10. If X is a compact G-space, {H ∈ Sub(G) \| H fixes a point in X} is a closed G-invariant subset of Sub(G). In particular the following definition makes sense. Definition 3.11. Let X be a compact G-space, and H ∈ URS(G). We say that H fixes a point in X if for some (all) H ∈ H, there is x ∈ X such that h(x) = x for all h ∈ H. Lemma 3.12. Let X be a compact G-space, Y ⊆ X a closed invariant subset of X, and H ∈ URS(G). If there exists H ∈ H fixing x ∈ X such that Gx ∩ Y 6= ∅, then H fixes a point in Y . Proof. By assumption there exist (gi ) and y ∈ Y such that gi (x) converges to y. If K ∈ H is a limit point of (H gi ) (which exists by compactness), then K fixes y by upper semi-continuity of the stabilizer map. Lemma 3.12 implies the following: Lemma 3.13. If X is a compact G-space containing a unique minimal closed Ginvariant subset Xmin ⊂ X (e.g. X is proximal), and if H ∈ URS(G) fixes a point in X, then H fixes a point in Xmin . Proposition 3.14. Let Z be a compact G-space that is extremely proximal, and H ∈ URS(G). Then either H fixes a point in Z, or all H ∈ H act minimally on Z. 16 ADRIEN LE BOUDEC Proof. If there exist H ∈ H and a non-empty closed subset C ( Z that is invariant by H, then we may apply Lemma 3.12 to the space X = 2Z , the subspace Y = Z and the point x = C, and we deduce that H fixes a point in Z. ∗ (X) stands for the closure in Sub(G) Recall that given a compact G-space X, SG of the set of subgroups Gx , x ∈ X. Lemma 3.15. Let X be a compact G-space. Assume that K ≤ G is a closed ∗ (X) subgroup of G which acts minimally on X and such that there exists H ∈ SG with H ≤ K. Then Env(SG (X)) ≤ K. ∗ (X) Proof. Since K acts minimally on X, the closure of the K-orbit of H in SG contains SG (X) according to Proposition 3.5. Since H ∈ Sub(K) and Sub(K) is a closed subset of Sub(G), we deduce that SG (X) ⊂ Sub(K), and in particular Env(SG (X)) ≤ K. Definition 3.16. Let H ∈ URS(G). We say that H comes from an extremely proximal action if there exists a compact G-space Z that is minimal and extremely proximal, and such that SG (Z) = H. It was shown in [LBMB16] that for a discrete group Γ with a non-trivial URS H coming from an extremely proximal action, any non-trivial K ∈ URS(Γ) must be “relatively large” with respect to H (see [LBMB16, Th. 3.10] for a precise statement). Appropriate assumptions on Γ and H further imply that H 4 K for every non-trivial K ∈ URS(Γ) [LBMB16, Cor. 3.12]. The following proposition goes in the opposite direction by considering URS’s larger than H. Proposition 3.17. Let H ∈ URS(G) that comes from an extremely proximal action. Let K ∈ URS(G) such that H 4 K and H = 6 K. Then Env(H) 4 K. Proof. Let Z be a compact G-space that is minimal and extremely proximal and such that SG (Z) = H. Fix K ∈ K, and assume that K does not act minimally on Z. According to Proposition 3.14 this implies that the URS K fixes a point in Z, i.e. K 4 H. Since moreover H, K satisfy H 4 K by assumption, we deduce that H = K, which is a contradiction. Therefore K acts minimally on Z. Since moreover there exists H ∈ H such that H ≤ K, we are in position to apply Lemma 3.15, from which the conclusion follows. It should be noted that Proposition 3.17 is false without the extreme proximality assumption, as in general there are plenty of URS’s between H and Env(H). Lemma 3.18. Let H ∈ URS(G) that comes from an extremely proximal action. Then Env(H) acts minimally on H. Proof. Let Z be a compact G-space that is minimal and extremely proximal and such that SG (Z) = H, and let N = Env(H). Without loss of generality we may assume that H is not a point, since otherwise there is nothing to prove. This ensures that N acts non-trivially on Z. By extreme proximality N must act minimally on Z (see Lemma 4.2), and therefore also on H by Proposition 3.5. Remark 3.19. The extreme proximality assumption cannot be removed in Lemma 3.18. Indeed it is not true in general that, given H ∈ URS(G), H remains a URS of Env(H). Indeed, as explained in [GW15], any minimal subshift on two letters AMENABLE URS’S AND LATTICE EMBEDDINGS 17 gives rise to a URS H of the lamplighter group G = C2 ≀Z, such that H is contained in the Chabauty space Sub(L) of the base group L = ⊕C2 . In particular Env(H) lies inside the abelian group L, and it follows that Env(H) acts trivially on H. Proposition 3.20. Let H ∈ URS(G) that comes from an extremely proximal action, and assume H is not a point. Then: (a) The action of G on H gives rise to the same URS, i.e. SG (H) = H. (b) If moreover H comes from a faithful extremely proximal action, then the action of G on H is faithful. Proof. Write K = SG (H). By definition we have H 4 K. Argue by contradiction and suppose H = 6 K. Then applying Proposition 3.17, we deduce that Env(H) acts trivially on H. But Env(H) also acts minimally on H by Lemma 3.18, so we deduce that H must be a point, a contradiction. This shows (a). For (b), arguing as in the proof of Lemma 3.18 we see that any non-trivial normal subgroup N of G acts minimally on H. Since H is not a point, we have in particular that N acts non-trivially on H. Remark 3.21. Proposition 3.20 implies that, as far as our interest lies inside the URS associated to a minimal and extremely proximal action (and not the space Z itself), there is no loss of generality in assuming that (G, Z) is a sub-system of (G, Sub(G)). See also Remark 3.27. 3.5. Amenable URS’s. Recall that we say that H ∈ URS(G) is amenable if every H ∈ H is amenable. The following lemma already appeared in [LBMB16, Prop. 2.21]. Lemma 3.22. If H ∈ URS(G) is amenable and X is a G-boundary, then H 4 SG (X). Proof. Since H is amenable, H must fix a point in the compact G-space Prob(X). Now X is the unique minimal G-invariant subspace of Prob(X) since X is a Gboundary, so by Lemma 3.13 we have that H fixes a point in X, i.e. H 4 SG (X). Proposition 3.23. Let X be a compact minimal G-space such that H = SG (X) is amenable, and let Y be a G-boundary such that X and Y are disjoint. Then Env(H) acts trivially on Y . In particular if Env(H) is co-amenable in G, a non-trivial G-boundary is never disjoint with X. Proof. The fact that Env(H) must act trivially on Y follows by applying Lemma 3.22 and Proposition 3.9. Since an amenable group has no non-trivial boundary, the second statement follows. Proposition 3.23 says that when G admits an amenable URS whose envelope is co-amenable, a non-trivial G-boundary is never disjoint with X. This conclusion is not satisfactory for our concerns as it depends on the choice of a space X and not only on G. Although there is no hope to get a better conclusion in full generality, the next result, which will play an important role in Section 5, will remove this dependence under an extreme proximality assumption. We recall from the introduction that we say that G is boundary indivisible if two non-trivial G-boundaries are never disjoint. 18 ADRIEN LE BOUDEC Proposition 3.24. Assume that G admits an amenable H ∈ URS(G) that comes from an extremely proximal action, and let X be a non-trivial G-boundary. (a) Either SG (X) = H, or Env(H) acts trivially on X. (b) Assume that Env(H) is co-amenable in G. Then SG (X) = H, and G is boundary indivisible. Proof. (a). Since H is amenable, we have H 4 SG (X) by Lemma 3.22. Now if we assume H = 6 SG (X), then according to Proposition 3.17 we have Env(H) 4 SG (X), which exactly means that Env(H) acts trivially on X. (b). If SG (X) 6= H then the action of G on X factors through an action of G/Env(H) by (a). But by assumption the latter is amenable, so has no non-trivial boundaries. So it follows that X is trivial, a contradiction. Therefore all non-trivial G-boundaries have the same stabilizer URS H. Since moreover H cannot be a point (because otherwise G would be amenable), the fact that G is boundary indivisible follows from Proposition 3.9. For a countable group Γ, the Furstenberg URS of Γ is the stabilizer URS associated to the action of Γ on its Furstenberg boundary. We refer to [LBMB16] for the proof of the following properties. Proposition 3.25. Let Γ be a countable group, and AΓ its Furstenberg URS. Then the following hold: (a) AΓ is amenable, and H 4 AΓ for every amenable H ∈ URS(Γ). (b) If X is a Γ-boundary, then AΓ 4 SΓ (X). If moreover there is x ∈ X such that Γx is amenable, then AΓ = SΓ (X). (c) AΓ is invariant under Aut(Γ). Proposition 3.26. Let Γ be a countable group, and let Λ = Env(AΓ ) be the envelope of the Furstenberg URS of Γ. Then Λ acts minimally on AΓ , and AΓ = AΛ . Proof. The conjugation action of Γ on the normal subgroup Λ = Env(AΓ ) induces a map Γ → Aut(Λ). Since AΛ is invariant under Aut(Λ) by Proposition 3.25, it is in particular Γ-invariant. Moreover the action of Γ on AΛ is clearly minimal since it is already the case for Λ. Therefore AΛ is an amenable URS of Γ, so it follows that AΛ 4 AΓ since AΓ is larger than any amenable URS of Γ. On the other hand AΓ is a closed and Λ-invariant subset of Sub(Λ) consisting of amenable subgroups, so by the domination property applied to AΛ we must have AΓ 4 AΛ . Equality follows. Remark 3.27. When Env(AΓ ) is co-amenable in Γ, the fact that AΓ comes from a faithful and extremely proximal action is equivalent to saying that the Γ-action on AΓ is faithful and extremely proximal. The direct implication is consequence of Proposition 3.20, and the converse follows from Proposition 3.24. This gives us an intrinsic reformulation of the assumption of Theorem 1.1 inside the Chabauty space of Γ. 4. Extremely proximal actions If X is a Hausdorff Γ-space and U ⊂ X , we denote by ΓU the set of elements of Γ acting trivially on X \ U . We say that the action of Γ on X is micro-supported if ΓU is non-trivial for every non-empty open set U . We will need the following easy lemma. AMENABLE URS’S AND LATTICE EMBEDDINGS 19 Lemma 4.1. Assume that the action of Γ on X is micro-supported, and let U be a non-empty open set. Then ΓU is not solvable. Proof. Assume that Λ is a subgroup of ΓU whose action on U is micro-supported, and let V be a non-empty open subset of U . By assumption there exists a nontrivial λ1 ∈ ΛV , so that we may find an open set W ⊂ V such that W and λ1 (W ) are disjoint. For λ2 ∈ ΛW , the commutator [λ1 , λ2 ] coincides with λ−1 2 on W , and is therefore non-trivial provided that λ2 is non-trivial. It follows by induction that if ΓU,n is the n-th term of the derived series of ΓU , then the action of ΓU,n on U is micro-supported. In particular ΓU,n is never trivial, and ΓU is not solvable. In this section we will consider the following setting: (EP) Γ is a discrete group, Z is a compact Γ-space, and the action of Γ on Z is faithful, minimal and extremely proximal. In order to avoid trivialities, we assume that Z has at least three points. Unless specified otherwise, in the remaining of this section Γ and Z will be assumed to satisfy (EP). Our goal is to derive various properties on the group Γ that will be used in later sections. Lemma 4.2. Let N ≤ Homeo(Z) be a non-trivial subgroup that is normalized by Γ. Then N acts minimally and does not fix any probability measure on Z. Proof. Assume there exists C ( Z that is closed and N -invariant. Since C is compressible and N is normalized by Γ, wee see that N has a fixed point in Z. Now the set of N -fixed points is Γ-invariant, so it has to be the entire Z by minimality, and N is trivial. The same argument shows the absence of N -invariant probability measure on Z, since an extremely proximal action is also strongly proximal by Theorem 2.3. In all this section the terminology topologically free (see Definition 3.3) has to be understood with Γ viewed as a discrete group. Therefore that the action is not topologically free means that there exists γ 6= 1 which acts trivially on a non-empty open subset of Z. Lemma 4.3. If the action of Γ on Z is not topologically free, then it is microsupported. Proof. Let U be a non-empty open subset of Z. Let γ be a non-trivial element such that there is a non-empty open set V on which γ acts trivially, and let g ∈ Γ such that g(X \ V ) ⊂ U . Then the non-trivial element gγg−1 acts trivially outside U , so ΓU is non-trivial. Definition 4.4. Let Γ0 be the subgroup of Γ generated by the elements γ ∈ Γ such that Fix(γ) has non-empty interior. Remark 4.5. When Γ is a countable group, Γ0 is also equal to the envelope of the URS SΓ (Z) by Lemma 3.7. Recall that the monolith Mon(Γ) is the intersection of all non-trivial normal subgroups of Γ. We say Γ is monolithic if Mon(Γ) is non-trivial. Proposition 4.6. Assume that the action of Γ on Z is not topologically free. Then the following hold: 20 ADRIEN LE BOUDEC (a) The commutators [γ1 , γ2 ], where Fix(γ1 ) ∩ Fix(γ2 ) has non-empty interior, generate [Γ0 , Γ0 ]. (b) Γ is monolithic, and one has Mon(Γ) = [Γ0 , Γ0 ]. (c) Any non-trivial normal subgroup of Γ has trivial centralizer. (d) If the action of [Γ0 , Γ0 ] on Z is extremely proximal, then [Γ0 , Γ0 ] is a simple group. (e) Γ is virtually simple if and only if Γ0 has finite index in Γ and finite abelianization. Proof. (a) Denote by N the subgroup generated by the set of [γ1 , γ2 ], where γ1 , γ2 act trivially on a common open set. We show that for every g, h fixing non-empty open sets U, V , the commutator [g, h] belongs to N . Since Γ0 is generated by all these elements g, h, this will show that Γ0 /N is abelian. Hence [Γ0 , Γ0 ] ≤ N , and the other inclusion is clear. First note that N is not trivial by Lemmas 4.1 and 4.3. Therefore N acts minimally on Z according to Lemma 4.2, and we may find s ∈ N such that the open set W = U ∩ s(V ) is non-empty. Since g and hs fix W by construction, we have [g, hs ] ∈ N . But since s ∈ N , we deduce that [g, h] = [h, s]g [g, hs ][s, h] is a product of three elements of N , and hence belongs to N , as desired. (b) We shall show that any non-trivial normal subgroup N contains [Γ0 , Γ0 ]. Since [Γ0 , Γ0 ] is itself a non-trivial normal subgroup, this will prove that it is the monolith of Γ. By a classical commutator manipulation (see e.g. Lemma 4.1 from [Nek13]), there exists an open set U such that N contains the derived subgroup of ΓU . Now let γ1 , γ2 fixing an open set V . If γ is such that γ(Z \ V ) ⊂ U , then γ1γ , γ2γ are supported inside U , so that [γ1γ , γ2γ ] = [γ1 , γ2 ]γ is contained in N . Since N is normal, [γ1 , γ2 ] ∈ N . Now all these elements generate [Γ0 , Γ0 ] by (a), hence the conclusion. (c) If N is a normal subgroup of Γ, then so is CΓ (N ). Therefore they cannot be both non-trivial, because otherwise the intersection would be abelian and would contain [Γ0 , Γ0 ] by the previous paragraph, a contradiction. (d) If the action of [Γ0 , Γ0 ] on Z is extremely proximal, then according to (b) the monolith N of [Γ0 , Γ0 ] is non-trivial. Since N is characteristic in [Γ0 , Γ0 ], N is normal in Γ, and hence contains [Γ0 , Γ0 ] by (b). So N = [Γ0 , Γ0 ] and [Γ0 , Γ0 ] is simple. (e) For Γ to be virtually simple it is clearly necessary that the normal subgroup [Γ0 , Γ0 ] has finite index in Γ. Conversely, if this condition holds then the action of [Γ0 , Γ0 ] on Z is extremely proximal (Lemma 2.5), and [Γ0 , Γ0 ] is simple by (d). Definition 4.7. Let X be a topological space, and let Γ be a group acting on X . A non-empty open set Ω ⊂ X is wandering for Γ if the translates γ(Ω), γ ∈ Γ, are pairwise disjoint. We say that Ω is wandering for γ if it is wandering for hγi. Proposition 4.8. Let Γ and Z as in (EP). Then there exist an open set Ω and a non-abelian free subgroup F2 = ∆ ≤ Γ such that Ω is wandering for ∆. Proof. Following Glasner [Gla74], we consider pairwise disjoint non-empty open sets U− , U+ , V− , V+ , and elements a, b ∈ Γ such that a(Z\U− ) ⊂ U+ and b(Z\V− ) ⊂ V+ . Let W = U− ∪ U+ ∪ V− ∪ V+ . It follows from a ping-pong argument than any non-trivial reduced word in the letters a, b sends the complement of W inside W , so that the subgroup ∆ generated by a, b is free [Gla74, Th. 3.4]. AMENABLE URS’S AND LATTICE EMBEDDINGS 21 Upon reducing W if necessary, we may find an open set Ω such that Ω ∩ W = ∅ and a(Ω) ⊂ U+ , a−1 (Ω) ⊂ U− , b(Ω) ⊂ V+ and b−1 (Ω) ⊂ V− . Induction on the word length shows that if the left-most letter of γ ∈ ∆ is respectively a, a−1 , b, b−1 , then γ(Ω) lies respectively inside U+ , U− , V+ , V− . In particular Ω ∩ γ(Ω) is empty since Ω is disjoint from W , so Ω is wandering for ∆. Proposition 4.9. Retain notations (EP). Then the wreath product ΓU ≀ F2 embeds into Γ for every open subset U ( Z that is not dense. Proof. Let Ω and ∆ as in Proposition 4.8, and let Λ be the subgroup of Γ generated by ΓΩ and ∆. Since Ω is wandering for ∆, all the conjugates γΓΩ γ −1 pairwise commute, and it follows that Λ is isomorphic to ΓΩ ≀ F2 . Now if U is an in the statement, by extreme proximality the group ΓU is isomorphic to a subgroup of ΓΩ , hence the conclusion. The argument in the following proof is borrowed from [Cap16]. Proposition 4.10. In the setting (EP), if the action of Γ on Z is not topologically free, then Γ cannot have any faithful linear representation. Proof. Let U be an open subset of Z. By Lemmas 4.1 and 4.3, we may find a non-abelian finitely generated subgroup B inside ΓU . Now if we choose U small enough, it follows from Proposition 4.9 that the finitely generated group Λ = B ≀ F2 is isomorphic to a subgroup of Γ. Since Λ is not residually finite [Gru57], it admits no faithful linear representation by Malcev’s theorem [Mal40], and a fortiori the same is true for Γ. Recall that a subgroup Λ of a group Γ is commensurated if all conjugates of Λ are commensurable, where two subgroups are commensurable if the intersection has finite index in both. The beginning of the argument in the proof of the following proposition already appeared in [LBW16]. The idea is to extend classical techniques for normal subgroups to certain commensurated subgroups. Proposition 4.11. Retain notation from (EP), and assume that the action of Γ on Z is not topologically free. If Λ is a commensurated subgroup of Γ such that there exists an element of Λ admitting a wandering open set, then Λ contains the monolith Mon(Γ). Proof. Let λ ∈ Λ admitting a wandering open set Ω. We shall first prove that [ΓΩ , ΓΩ ] is contained in Λ. Let g, h ∈ ΓΩ , and let also n ≥ 1. Since Ω is wandering we have Ω ∩ λn (Ω) = ∅. It follows that the commutator [g, λn ] is trivial outside Ω ∪ λn (Ω), and coincides with g on Ω and with λn g−1 λ−n on λn (Ω). Therefore its commutator with h is trivial outside Ω, and is coincides with [g, h] on Ω. But since g, h ∈ ΓΩ , the elements [[g, λn ], h] and [g, h] actually coincide everywhere, i.e. [[g, λn ], h] = [g, h]. Now since Λ is commensurated in Γ, there exists n0 ≥ 1 such that [[g, λn0 ], h] belongs to Λ. Applying the previous argument with n = n0 , we deduce that [g, h] belongs to Λ. In order to prove the statement, it is enough to prove that [ΓC , ΓC ] is contained in Λ for every closed subset C ( Z according to Proposition 4.6. So let C a proper closed subset of Z. By minimality and extreme proximality, there is γ ∈ Γ such 22 ADRIEN LE BOUDEC that γ(C) ⊂ Ω. Fix such a γ, and choose some integer n1 ≥ 1 such that γ −1 λn1 γ belongs to Λ. Set λ′ = γ −1 λn1 γ and Ω′ = γ −1 (Ω). This Ω′ is wandering for λ′ and λ′ ∈ Λ, so [ΓΩ′ , ΓΩ′ ] is contained in Λ by the first paragraph. Since C ⊂ Ω′ , we have ΓC ≤ ΓΩ′ , and the proof is complete. Proposition 4.12. Assume that Z is a compact Γ-space and the action of Γ on Z is faithful, minimal, extremely proximal and not topologically free. Then there exists a free subgroup F2 ≤ Γ such that for every commensurated subgroup Λ ≤ Γ not containing the monolith Mon(Γ), we have F2 ∩ Λ = 1. Proof. Let F2 be a free subgroup of Γ as in the conclusion of Proposition 4.8. If Λ ≤ Γ is a commensurated subgroup such that F2 ∩ Λ 6= 1, then in particular Λ contains an element admitting a wandering open set. So by Proposition 4.11 we have Mon(Γ) ≤ Λ. This shows that every commensurated subgroup not containing Mon(Γ) intersects F2 trivially. Corollary 4.13. Assume that Z is a compact Γ-space and the action of Γ on Z is faithful, minimal, extremely proximal and not topologically free. If G is a locally compact amenable group whose connected component G0 is a Lie group, then there exists no injective homomorphism Γ → G. Proof. Argue by contradiction and assume Γ embeds in G. Let U be an open subgroup of G containing G0 as a cocompact subgroup. Such a U is commensurated in G, so the subgroup Γ ∩ U is commensurated in Γ. If there exists U such that Γ ∩ U does not contain Mon(Γ), then according to Proposition 4.12 we may find a non-abelian free subgroup F2 ≤ Γ such that F2 ∩ U = 1. In particular G contains the non-amenable group F2 as a discrete subgroup, which contradicts amenability of G. Therefore Mon(Γ) is contained in U for every choice of U . Since compact open subgroups form a basis at 1 in G/G0 by van Dantzig’s theorem, it follows that Mon(Γ) actually lies inside G0 . Now since G0 is a connected Lie group, the group Aut(G0 ) is linear, so the map G → Aut(G0 ) induced by the conjugation action of G on G0 is not injective in restriction to Γ by Proposition 4.10. Therefore this map must vanish on Mon(Γ), which means that Mon(Γ) actually lies inside the center of G0 . In particular Mon(Γ) is abelian, which contradicts Proposition 4.6. 5. The proofs of Theorems 1.1 and 1.2 5.1. Boundary-minimal subgroups. In this paragraph we consider the following property: Definition 5.1. Let G be a topological group, and L a closed subgroup of G. We say that L is boundary-minimal if there exists a non-trivial G-boundary on which L acts minimally. It should be noted that being boundary-minimal does not prevent L from being amenable. For instance the action of Thompson’s group T on the circle S1 is a boundary action, and the abelian subgroup of T consisting of rotations acts minimally on S1 . Other examples may be found among the groups acting on trees considered in §6.2, where the stabilizer of a vertex is an amenable subgroup acting minimally on the ends of the tree. In the sequel we will mainly focus on the case when L is normal in G, or more generally when L belongs to a URS (see Proposition 5.8). By contrast with the AMENABLE URS’S AND LATTICE EMBEDDINGS 23 previous examples, a normal boundary-minimal subgroup is never amenable, as a normal amenable subgroup of G acts trivially on any G-boundary. Recall that Furman showed [Fur03, Prop. 7] (see also Caprace–Monod [CM14, Prop. 3]) that if N is non-amenable normal subgroup of a locally compact group G, there always exists a G-boundary on which N acts non-trivially. This naturally raises the question whether any non-amenable normal subgroup of G is boundary-minimal. We do not know the answer to this question. While the case of discrete groups is easily settled (see below), the situation for non-discrete groups seems to be more delicate. We recall the following result of Furstenberg (see [Gla76, II.4.3]): Theorem 5.2. Let G be a topological group, and denote by ϕ : G → Homeo(∂sp G) the action of G on ∂sp G. Then there exists a homomorphism ψ : Aut(G) → Homeo(∂sp G) such that ψ ◦ Inn = ϕ, where Inn(G) ≤ Aut(G) is the group of inner automorphisms of G. In particular when N is a normal subgroup of a group G, the map G → Aut(N ) coming from the conjugation action of G on N induces an action of G on ∂sp N , which factors through G/CG (N ). Note that this result readily answers the above question for discrete groups, by showing that the boundary-minimal normal subgroups of G are exactly the non-amenable normal subgroups of G. Indeed if N is non-amenable then ∂sp N is a non-trivial space, N acts minimally on ∂sp N , and ∂sp N is a G-boundary by Theorem 5.2. However the argument does not carry over for arbitrary groups, as in general the G-action on ∂sp N is not continuous. Proposition 5.3. Let G be a locally compact group, and N a closed normal nonamenable subgroup. Assume that at least one of the following holds true: (a) N · CG (N ) is open in G (e.g. when N is a direct factor of G); (b) N is cocompact in G; (c) there exists H ∈ URS(N ) that is not a point, invariant by Aut(N ), and that is an N -boundary (e.g. if N has a closed cocompact amenable subgroup). Then N is boundary-minimal in G. Proof. Condition (a) ensures that the image of N in G/CG (N ) is open. Therefore the G-action on ∂sp N given by Theorem 5.2 is continuous because the N -action is, and we deduce that ∂sp N is a non-tivial G-boundary. If (b) holds, then N acts minimally on any G-boundary [Gla76, II.3.2]. Finally the verification of case (c) is straightforward since G → Aut(N ) is continuous [HR79, Th. 26.7] and Aut(N ) acts continuously on Sub(N ). 5.2. Weakly co-amenable subgroups. In this paragraph we consider the following weakening of the notion of co-amenability. Definition 5.4. Let G be a topological group, and H a subgroup of G. We say that H is weakly co-amenable in G if whenever Q is a non-trivial convex compact G-space in which H fixes a point, Q is not irreducible. The following properties readily follow from the definition. Proposition 5.5. Let K ≤ H ≤ G be subgroups of G. (i) If H ≤ G is co-amenable then H is weakly co-amenable. 24 ADRIEN LE BOUDEC (ii) For a normal subgroup N ⊳ G, weakly co-amenable is equivalent to coamenable. (iii) If H ≤ G is amenable and weakly co-amenable in G, then G is amenable. (iv) If ϕ : G → G′ is continuous with dense image and H ≤ G is weakly coamenable, then ϕ(H) is weakly co-amenable in G′ . (v) If K is weakly co-amenable in G, then H is weakly co-amenable in G. (vi) If K is co-amenable in H and H is weakly co-amenable in G, then K is weakly co-amenable in G. Proof. (i) If Q is non-trivial convex compact G-space with H-fixed points, then there is a G-fixed point by co-amenability of H in G, so Q is not irreducible. (ii) If N ⊳ G is not co-amenable, there is a convex Q such that Fix(N ) is nonempty but Fix(G) is empty. Since N is normal Fix(N ) is G-invariant, so that by Zorn’s lemma Fix(N ) contains an irreducible convex G-space, which is non-trivial since Fix(G) is empty. This shows N is not weakly co-amenable. The proofs of (iii), (iv), (v) and (vi) are similar verifications, and we leave them to the reader. Remark 5.6. As for co-amenability, it is natural to wonder whether weak coamenability of K in G implies weak co-amenability of K in H. In view of (ii), the same counter-examples given in [MP03] show that the answer is negative in general. By the correspondence between irreducible convex compact G-spaces and Gboundaries, weak co-amenability admits the following characterization: Proposition 5.7. A subgroup H ≤ G is weakly co-amenable in G if and only if for every non-trivial G-boundary X, there is no probability measure on X that is fixed by H. Proof. Follows from Theorem 2.1. The following shows how weak co-amenability naturally appears for boundary indivisible groups (see also Proposition 6.17). Proposition 5.8. Let G be a boundary indivisible locally compact group, and L a closed subgroup of G that is boundary-minimal and uniformly recurrent. Then L is weakly co-amenable in G. Proof. Write H for the closure of LG in Sub(G), which is a URS by assumption. Let X be a non-trivial G-boundary on which L acts minimally, and let Y be a G-boundary on which L fixes a probability measure. We have to show that Y is trivial. Since H fixes a point in Prob(Y ) and the G-action on Prob(Y ) is strongly proximal by Theorem 2.1, H fixes a point in Y by Lemma 3.13. So there exists y ∈ Y such that L ≤ Gy , and it follows that Gy acts minimally on X. Therefore by Lemma 3.8 X and Y are disjoint, and since X is non-trivial and G is boundary indivisible, this is possible only if Y is trivial. 5.3. The proof of Theorem 1.2. In this paragraph we shall give the proof of Theorem 1.2 from the introduction. We will make use of the following result. Proposition 5.9 (Furstenberg). Let G be a locally compact group, H ≤ G a closed subgroup of finite covolume, and X a G-boundary. Then X is a H-boundary. AMENABLE URS’S AND LATTICE EMBEDDINGS 25 For completeness we repeat the argument from [Fur81, Prop. 4.4]. Proof. Write Q = Prob(X), and consider a closed H-invariant subspace Q′ ⊆ Q. We have to show that X ⊆ Q′ . The set X = (gH, µ) : µ ∈ g(Q′ ) ⊆ G/H × Q is a well-defined, closed, G-invariant subspace of G/H × Q. Fix a G-invariant probability measure mG/H on G/H, and consider n o Y = ν ∈ Prob(X ) : p∗G/H (ν) = mG/H , where pG/H is the projection from G/H ×Q onto the first factor, and p∗G/H is the induced push-forward operator. Then Y is a closed (and hence compact) G-invariant subspace of Prob(X ), and p∗Q : Y → Prob(Q) is continuous. So p∗Q (Y ) is closed in Prob(Q), and by strong proximality of the G-action on Q (Theorem 2.1), p∗Q (Y ) must intersect Q. Now X being the unique minimal closed G-invariant subspace of Q, one has X ⊆ p∗Q (Y ). For every x ∈ X, we therefore have νx ∈ Prob(X ) such that p∗G/H (νx ) = mG/H and p∗Q (νx ) = δx . This implies mG/H {gH : x ∈ g(Q′ )} = 1 for every x, and it easily follows that X ⊆ Q′ . Remark 5.10. In the case when H is cocompact in G, strong proximality of the action of H on X also follows from [Gla76, II.3.1] applied to the action on Prob(X); and minimality follows [Gla76, IV.5.1] from disjointness of the G-spaces G/H and X [Gla76, III.6.1]. Theorem 5.11. Assume that H admits an amenable URS H that comes from an extremely proximal action, and such that Env(H) is co-amenable in H. Let G be a locally compact group containing H as a closed subgroup of finite covolume. Then: (a) G is boundary indivisible. (b) More generally if L is a locally compact group such that there is a sequence a topological group homomorphisms G = G0 → G1 → . . . → Gn = L such that either Gi → Gi+1 has dense image, or Gi → Gi+1 is an embedding of Gi as a closed subgroup of finite covolume in Gi+1 ; then L is boundary indivisible. In particular whenever G maps continuously and with dense image to a product G1 × G2 , one factor Gi must be amenable. Proof. Since H is amenable, H comes from an extremely proximal action, and Env(H) is co-amenable in H, the group H is boundary indivisible by Proposition 3.24. Now by Proposition 5.9, the property of being boundary indivisible is inherited from closed subgroups of finite covolume. Indeed if X, Y are disjoint Gboundaries, i.e. X × Y is a G-boundary, then X × Y is also a boundary for H by Proposition 5.9, hence of X or Y must be trivial since H is boundary indivisible. This shows (a). Since boundary indivisibility passes to dense continuous images, and is inherited from closed subgroups of finite covolume, (b) follows from (a). Finally if G1 , G2 are as in the last statement and Xi = ∂sp Gi , then X1 × X2 is a boundary for G1 × G2 , which is boundary indivisible by the previous paragraph. So one factor Xi must be trivial, which exactly means that Gi is amenable by Theorem 2.2. 26 ADRIEN LE BOUDEC Remark 5.12. In the proof of Theorem 5.11 we obtain that G is boundary indivisible from the same property for H, which is itself deduced from Proposition 3.24 (which in turn relies notably on Proposition 3.9). We note that the order in which the argument is developed seems to matter, in the sense that the arguments applied to H do not seem to be applicable directly to the group G. Indeed we do not know whether a group G as in Theorem 5.11 falls into the setting of Proposition 3.9, i.e. we do not know whether all non-trivial G-boundaries have the same stabilizer URS. We actually believe this might be false in general. We note at this point that the proof of Theorem 1.2 from the introduction is now complete. Indeed the fact that a group G as in Theorem 1.2 is boundary indivisible, as well as statement (a), is Theorem 5.11; and statement (b) follows from Proposition 5.8. The following remark explains a comment from the introduction. Remark 5.13. Theorem 1.4 provides instances of countable groups Γ being lattices in a group G of the form G = N ⋊ Aut(Td ), such that all the assumptions of Theorem 1.2 are satisfied (see Section 6). If M is a cocompact subgroup of Aut(Td ) acting minimally on ∂Td , then L = N ⋊ M is a cocompact (hence uniformly recurrent) subgroup of G, and L is boundary-minimal in G since ∂Td is a G-boundary. However when M is non-unimodular, then M is not co-amenable in Aut(Td ), and L is not co-amenable in G. This shows that the conclusion of statement (b) in Theorem 1.2 that L is weakly co-amenable in G cannot be strengthen by saying that L is co-amenable in G. 5.4. The proof of Theorem 1.1. Recall that if G is a topological group, the quasi-center QZ(G) of G is the subgroup of G containing the elements g ∈ G having an open centralizer. Note that QZ(G) contains the elements having a discrete conjugacy class, so in particular it contains all discrete normal subgroups. Recall also that the elliptic radical of G is the largest normal subgroup of G in which every compact subset generates a relatively compact subgroup. It is a closed characteristic subgroup of G. We say that two groups G1 , G2 are commensurable up to compact kernels if there exist Ki ≤ Hi ≤ Gi such that Hi is open and of finite index in Gi , Ki is a compact normal subgroup of Hi , and H1 /K1 and H2 /K2 are isomorphic. The following is slightly more complete than Theorem 1.1 from the introduction. Theorem 5.14. Let Γ be a countable group whose Furstenberg URS AΓ comes from a faithful and extremely proximal action, and assume that Env(AΓ ) is co-amenable in Γ. Let G be a locally compact group containing Γ as a lattice, and H a group commenurable with G up to compact kernels. Consider the following properties: (a) Env(AΓ ) is finitely generated; (b) Env(AΓ ) has finite index in Γ, and Γ admits a finitely generated subgroup with finite centralizer; (c) Env(AΓ ) has finite index in Γ and Env(AΓ ) has finite abelianization. Then: • (a), (b) both imply that H cannot be a product of two non-compact groups. AMENABLE URS’S AND LATTICE EMBEDDINGS 27 • (c) implies that any continuous morphism with dense image from H to a product of locally compact groups H → G1 × G2 is such that one factor Gi is compact. Proof. For simplicity we give the proof for H = G. The general case follows the same lines. Of course we may assume that Env(AΓ ) is non-trivial, since otherwise there is nothing to prove. According to Proposition 4.6 we have in particular that Γ is monolithic, and Mon(Γ) = [Env(AΓ ), Env(AΓ )]. For simplicity in all the proof we write E = Env(AΓ ) and M = Mon(Γ) = [E, E]. Assume that ϕ : G → G1 × G2 is continuous with dense image, and denote by pi the projection G1 × G2 → Gi , i = 1, 2. We will show that one factor must be compact. Upon modding out by the maximal compact normal subgroup of the identity component G0 , which intersects Γ trivially since Γ has no non-trivial finite normal subgroup (Proposition 4.6), we may also assume that G0 has no non-trivial compact normal subgroup. This implies in particular that G0 is a connected Lie group [MZ55]. By the assumption that E is co-amenable in Γ, we can apply Theorem 5.11, which says that one factor, say G2 , must be amenable. We then apply Corollary 4.13, which tells us that the map p2 ◦ ϕ is not injective in restriction to Γ. By definition of M we deduce that M ≤ ϕ−1 (G1 × 1). Assume now that (c) holds. Then M , being of finite index in Γ, is a lattice in G, and is contained in the closed normal subgroup ϕ−1 (G1 × 1). Therefore we deduce that ϕ−1 (G1 × 1) is cocompact in G, and that p2 ◦ ϕ(G) is a compact subgroup of G2 . Since p2 ◦ ϕ(G) is also dense in G2 , we have that G2 is compact. We now have to deal with (a), (b), in which case ϕ is the identity and G = G1 × G2 . Without loss of generality, we may assume that the projections pi (Γ) are dense. The proofs of the two cases will share a common mechanism, given by the following easy fact: Lemma 5.15. If there exists a subgroup L ≤ G whose centralizer CG (L) contains G2 , CG (L) is open in G, and CG (L) ∩ Γ is finite, then G2 is compact. Indeed, since Γ must intersect an open subgroup O ≤ G along a lattice of O, it follows that CG (L) is compact, and a fortiori so is G2 . We start with case (a). Consider H1 = p1 (E), which is normal in G1 by density of p1 (Γ). Note that H1 is compactly generated in view of the assumption that E is finitely generated. Since M = [E, E], the group H1 /M is abelian, and therefore of the form Zn × Rm × C for some compact group C. It follows that the group Q1 = H1 /H10 admits a discrete cocompact normal subgroup ∆, which is an extension of M by a free abelian group. Being characteristically simple and non-amenable, the group M has trivial elliptic radical, so the group ∆ also has trivial elliptic radical. Now since Q1 is compactly generated, there is a compact open normal subgroup K of Q1 such that K ⋊ ∆ has finite index in Q1 (see e.g. [BCGM16, Lem. 4.4]), so we deduce that Q1 has a compact open elliptic radical. Since any connected group has compact elliptic radical [MZ55], we deduce that H1 has a compact elliptic radical R, and H1 /R is discrete-by-connected. The compact group R is also normal in G, and therefore we can mod out by R and assume that R is trivial, so that H10 is open in H1 . Since H10 centralizes M , any γ ∈ E such that p1 (γ) belongs to H10 28 ADRIEN LE BOUDEC centralizes M , and therefore is trivial by Proposition 4.6. Therefore H10 is open in H1 and intersects the dense subgroup p1 (E) trivially, so it follows that H10 is trivial, and p1 (E) is a discrete subgroup of G. Observe that p1 (E) is centralized by G2 and normalized by G1 , and hence is normal in G. Being a discrete normal subgroup of G, p1 (E) therefore lies in the quasi-center QZ(G). Since p1 (E) is finitely generated, the centralizer of p1 (E) in G is actually open in G. Moreover the subgroup Γ ∩ CG (p1 (E)) is normal in Γ since CG (p1 (E)) is normal in G, but clearly does not contain M , and hence is trivial by Proposition 4.6. Therefore we can apply Lemma 5.15 with L = p1 (E), and we obtain the conclusion. We now deal with (b). Let Z be a minimal compact Γ-space on which the Γ-action is faithful and extremely proximal and such that SΓ (Z) = AΓ . By Proposition 5.9 (actually an easy case of it) the action of E on Z is also minimal, and it is extremely proximal by Lemma 2.5. Moreover the associated stabilizer URS remains equal to AΓ , and is also the Furstenberg URS of E by Proposition 3.26. So E satisfies all the assumptions of case (b) of the theorem, so it is enough to prove the result under the additional assumption Γ = E. In this case we have M = [Γ, Γ] thanks to Proposition 4.6, so it follows that p2 (Γ) is abelian. By density of the projection the group G2 is also abelian, and hence G2 lies in the center of G. Therefore Γ is normalized by the dense subgroup ΓG2 , and it follows that Γ is normal in G. In particular Γ ≤ QZ(G), and the conclusion follows by applying Lemma 5.15 with L a f.g. subgroup of Γ such that CΓ (L) is finite. 6. Groups acting on trees 6.1. Amenable URS’s and groups acting on trees. In this paragraph T is a locally finite tree, and H acts continuously on T by isometries. The assumption that T is locally finite is not essential here, and the results admit appropriate generalizations for non-locally finite trees (using the compactification from [MS04, Prop. 4.2]). Recall that the H-action on T is minimal if there is no proper invariant subtree, and of general type if H has no finite orbit in T ∪ ∂T . The following is wellknown, and essentially goes back to Tits [Tit70] (see also [PV91] and Proposition 2.4 for details). Proposition 6.1. If the action of H ≤ Aut(T ) is minimal and of general type, then the action of H on ∂T is minimal and extremely proximal. Theorem 5.11 therefore implies the following result: Corollary 6.2. Let H ≤ Aut(T ) be a locally compact group whose action on T is continuous, minimal and of general type. Assume that end stabilizers are amenable, and the envelope of SH (∂T ) is co-amenable in H. Assume H embeds as a subgroup of finite covolume in G. Then whenever G maps continuously and with dense image to a product G1 × G2 , one factor Gi must be amenable. The conclusion of Corollary 6.2 implies in particular that whenever H embeds in G with finite covolume, then G cannot be a product of two non-amenable groups. The following example, which is largely inspired from [CM11, Ex. II.8], shows that the group G can nonetheless be a product of two non-compact groups. AMENABLE URS’S AND LATTICE EMBEDDINGS 29 Example 6.3. Let k = Fp ((t)) be the field of Laurent series over the finite field Fp , and let α ∈ Aut(k) be a non-trivial automorphism of k. The group L = SL(2, k) acts on a (p + 1)-regular tree, 2-transitively and with amenable stabilizers on the boundary. This action extends to a continuous action of H = L ⋊α Z, so that H satisfies all the assumptions of Corollary 6.2. Nevertheless H embeds diagonally in the product G = (L ⋊ Aut(k)) × Z as a closed subgroup of finite covolume since G/H is compact and H and G are unimodular. We will need the following fact. If A is a subtree of T , by the fixator of A we mean the subgroup fixing pointwise A. Proposition 6.4. Let Γ ≤ Aut(T ) be a countable group whose action on T is minimal and of general type, and such that end-stabilizers in Γ are amenable. Then AΓ = SΓ (∂T ), and Env(AΓ ) is the subgroup generated by fixators of half-trees. Proof. Since the Γ-action on ∂T is extremely proximal, it is also strongly proximal by Theorem 2.3. So ∂T is a Γ-boundary with amenable stabilizers, and we deduce that AΓ = SΓ (∂T ) by Proposition 3.25. Now according to Lemma 3.7, the subgroup Env(SΓ (∂T )) = Env(AΓ ) is generated by the elements γ ∈ Γ whose fixed point set in ∂T has non-empty interior. Since half-trees form a basis of the topology in ∂T , the statement follows. Before going to the proof of Corollary 1.3, we make the following observation: Remark 6.5. For Γ acting on T (action minimal and general type) such that the action on ∂T is not topologically free, virtual simplicity of Γ is equivalent to Γ0 being of finite index in Γ and Γ0 having finite abelianization, where Γ0 is the subgroup generated by fixators of half-trees. See statement (e) of Proposition 4.6. Proof of Corollary 1.3. In view of Proposition 6.4, the assumptions on Γ imply that the Furstenberg URS of Γ comes from a faithful extremely proximal action. The fact that end-stabilizers are all non-trivial means that the action of Γ on ∂T is not topologically free, and by the above observation virtual simplicity of Γ is equivalent to Env(AΓ ) being of finite index in Γ and with finite abelianization. The first statement of the corollary therefore follows from Theorem 5.14, case (c), and the second statement from Theorem 1.2. 6.2. Groups with prescribed local action. In the next paragraphs we will illustrate the results of the previous sections on a family of groups acting on trees, which contains instances of discrete and non-discrete groups. The purpose of this paragraph is to recall the definition and give a brief description of known properties of these groups. We will denote by Ω a set of cardinality d ≥ 3 and by Td a d-regular tree. The vertex set and edge set of Td will be denoted respectively Vd and Ed . We fix a coloring c : Ed → Ω such that neighbouring edges have different colors. For every g ∈ Aut(Td ) and every v ∈ Vd , the action of g on the star around v gives rise to a permutation of Ω, denoted σ(g, v), and called the local permutation of g at v. These permutations satisfy the identity (1) σ(gh, v) = σ(g, hv)σ(h, v) for every g, h ∈ Aut(Td ) and v ∈ Vd . 30 ADRIEN LE BOUDEC Given a permutation group F ≤ Sym(Ω), the group U (F ) introduced by Burger and Mozes in [BM00a] is the group of automorphisms g ∈ Aut(Td ) such that σ(g, v) ∈ F for all v. It is a closed cocompact subgroup of Aut(Td ). Definition 6.6. Given F ≤ F ′ ≤ Sym(Ω), we denote by G(F, F ′ ) the group of automorphisms g ∈ Aut(Td ) such that σ(g, v) ∈ F ′ for all v and σ(g, v) ∈ F for all but finitely many v. That G(F, F ′ ) is indeed a subgroup of Aut(Td ) follows from (1), and we note that we have U (F ) ≤ G(F, F ′ ) ≤ U (F ′ ). We make the following observation for future reference. Remark 6.7. As it follows from the definition, any element γ ∈ G(F, F ′ ) fixing an edge e can be (uniquely) written as γ = γ1 γ2 , where each γi belongs to G(F, F ′ ) and fixes one of the two half-trees defined by e. In the sequel we always assume that F ′ preserves the F -orbits in Ω (see [LB16, Lem. 3.3] for the relevance of this property in this context). These groups satisfy the following properties (see [LB16]): (1) The group G(F, F ′ ) is dense in the locally compact group U (F ′ ). In particular G(F, Sym(Ω)) is a dense subgroup of Aut(Td ). (2) G(F, F ′ ) admits a locally compact group topology (defined by requiring that the inclusion of U (F ) is continuous and open), and the action of G(F, F ′ ) on Td is continuous but not proper as soon as F 6= F ′ . Endowed with this topology, the group G(F, F ′ ) is compactly generated. (3) stabilizers of vertices and stabilizers of ends in G(F, F ′ ) are respectively locally elliptic and (locally elliptic)-by-cyclic. In particular they are amenable. (4) G(F, F ′ ) is a discrete group if and only if F acts freely on Ω. When this is so, the group G(F, F ′ ) is therefore a finitely generated group, and stabilizers of vertices and stabilizers of ends in G(F, F ′ ) are respectively locally finite and (locally finite)-by-cyclic. When F acts freely on Ω and F 6= F ′ , the groups G(F, F ′ ) are instances of groups obtained from a more general construction described in [LB17, Sec. 4] (more precisely, a variation of it), which provides discrete groups with a continuous Furstenberg URS (the later being the stabilizer URS associated to the action on the boundary of the tree on which these groups act). In the particular case of the groups G(F, F ′ ), the Furstenberg URS can be explicitly described, see Proposition 4.28 and Corollary 4.29 in [LBMB16]. In the sequel whenever we use letters F and F ′ , we will always mean that F, F ′ are permutation groups on a set Ω, that F ′ contains F and preserves the F -orbits in Ω. Following [Tit70], we will denote by G(F, F ′ )+ the subgroup of G(F, F ′ ) generated by fixators of edges, and by G(F, F ′ )∗ the subgroup of index two in G(F, F ′ ) preserving the bipartition of Td . The following result, also obtained in [CRW17, Prop. 9.16], supplements simplicity results obtained in [LB16], where the index of the simple subgroup was found explicitly under appropriate assumptions on the permutation groups. Proposition 6.8. The group G(F, F ′ ) has a simple subgroup of finite index if and only if F is transitive and F ′ is generated by its point stabilizers. AMENABLE URS’S AND LATTICE EMBEDDINGS 31 Proof. These conditions are necessary by [LB16, Prop. 4.7]. Conversely, assume F transitive and F ′ generated by its point stabilizers. By [LB16, Prop. 4.7] again, G(F, F ′ )+ has index two in G(F, F ′ ), so in particular it is compactly generated. If M is the monolith of G(F, F ′ ), which is simple and open in G(F, F ′ ) by [LB16, Cor. 4.9], we have to show M has finite index. According to Remark 6.7, G(F, F ′ )+ is also the subgroup generated by fixators of half-trees, and therefore by Proposition 4.6 M is the commutator subgroup of G(F, F ′ )+ . The abelianization of G(F, F ′ )+ is therefore a finitely generated abelian group, which is generated by torsion elements since G(F, F ′ )+ is generated by locally elliptic subgroups (fixators of edges). Therefore this abelianization is finite, and it follows that M has finite index in G(F, F ′ ). 6.3. Boundaries of G(F, F ′ ). In this paragraph we use results from the previous sections in order to study the boundaries of the discrete groups G(F, F ′ ). The following result shows that several properties of the set of boundaries are governed by the permutation groups, and that rigidity phenomena occur under mild conditions of the permutation groups. Theorem 6.9. Assume F acts freely on Ω, F 6= F ′ , and write Γ = G(F, F ′ ). The following are equivalent: (i) The subgroup of F ′ generated by its point stabilizers has at most two orbits in Ω. (ii) Γ/Env(AΓ ) is isomorphic to one of C2 , D∞ or D∞ ⋊ C2 . (iii) Env(AΓ ) is co-amenable in Γ. (iv) SΓ (X) = AΓ for every non-trivial Γ-boundary X. (v) Γ is boundary indivisible. We will need preliminary results before proving Theorem 6.9. Lemma 6.10. Assume that F acts freely on Ω. The envelope of the Furstenberg URS of G(F, F ′ ) is equal to G(F, F ′ )+ . Proof. Write Γ = G(F, F ′ ) and Γ+ = G(F, F ′ )+ . According to Proposition 6.4, Env(AΓ ) is the subgroup generated by fixators of half-trees in Γ. Therefore the inclusion Env(AΓ ) ≤ Γ+ is clear. The converse inclusion also holds true by Remark 6.7, so equality follows. In view of Lemma 6.10 and Proposition 3.24, we are led to consider the quotient G(F, F ′ )/G(F, F ′ )+ , and in particular study when it is amenable. To this end, we will denote by F ′+ the subgroup of F ′ generated by its point stabilizers, and write D = F ′ /F ′+ . Since F ′+ is normal in F ′ , we have an action of F ′ on the set of orbits of F ′+ , which factors through a free action of D. Proposition 6.11. The group Q = G(F, F ′ )∗ /G(F, F ′ )+ is isomorphic to the group U (D)∗ , where D = F ′ /F ′+ is viewed as a permutation group acting freely on the set of orbits of F ′+ in Ω. If moreover F is transitive, one has Q = D ∗ D. Proof. We let O1 , . . . , Or ⊆ Ω be the orbits of F ′+ , and we will freely identify the set of orbits with the integers {1, . . . , r}. For every a ∈ Ω, there is a unique i ∈ {1, . . . , r} such that a ∈ Oi , and we denote i = ia . We view the tree Td as the Cayley graph of the free Coxeter group of rank d, namely the group defined by generators x1 , . . . , xd and relators x2j = 1 for all j. 32 ADRIEN LE BOUDEC When adding relations of the form xa = xb whenever ia = ib (i.e. a and b are in the same F ′+ -orbit), we obtain a free Coxeter group of rank r. It has a Cayley graph that is a regular tree of degree r, and we have a surjective map p : Td → Tr . Two elements v, w ∈ ∗d C2 = hx1 , . . . , xd i have the same image in ∗r C2 if and Q only if one can write w = v wj xaj xbj wj−1 for some words wj and colors aj , bj such that iaj = ibj . Since the inverse of wj is equal to the word obtained from wj by reversing the order, we have: Lemma 6.12. Two vertices v, w of Td have the same projection in Tr if and only if the distance between v and w is even, say d(v, w) = 2m, and if (a1 , . . . , a2m ) is the sequence of colors from v to w, then the word (ia1 , . . . , ia2m ) is a concatenation of palindromes of even length. Lemma 6.13. For every g ∈ G(F, F ′ ) and every vertex v on Td , the image of σ(g, v) in D = F ′ /F ′+ does not depend on v. We denote by σg ∈ D the corresponding element, which is trivial when g ∈ G(F, F ′ )+ . Proof. If v, w are adjacent vertices and a is the color of the edge between them, then σ(g, v)(a) = σ(g, w)(a). So σ(g, v)σ(g, w)−1 ∈ F ′+ . The first statement follows by connectedness. The fact that σg is trivial on G(F, F ′ )+ is then clear because g 7→ σg is a morphism according to the first statement, which vanishes on fixators of edges. Note that the set of edges of Tr inherits a natural coloring by the integers 1, . . . , r. Lemma 6.14. There is a natural morphism ϕ : G(F, F ′ ) → Aut(Tr ) such that ker(ϕ) = G(F, F ′ )+ and Im(ϕ) = U (D). Proof. We shall first define an action of G(F, F ′ ) on the set of vertices of Tr . Let g ∈ G(F, F ′ ). Let v, w be two vertices of Td , and (a1 , . . . , an ) be the sequence of colors from v to w. If v = v1 , . . . , vn+1 = w are the vertices between v and w, then the sequence of colors from g(v) to g(w) is (σ(g, v1 )(a1 ), . . . , σ(g, vn )(an )). If σg is the element defined in Lemma 6.13, then one has iσ(g,vj )(aj ) = σg (iaj ) for all j = 1, . . . , n. This shows in particular that if v, w satisfy the condition of Lemma 6.12, then the same holds for g(v) and g(w). This means that for every vertex x of Tr , the formula (2) ϕ(g)(x) := p(gx̃), where x̃ is any vertex of Td such that p(x̃) = x, is a well-defined action of G(F, F ′ ) on Tr . The fact that the tree structure is preserved is clear. Note that for every g ∈ G(F, F ′ ), all the local permutations of ϕ(g) are equal to σg : for every vertex x of Tr , one has σ(ϕ(g), x) = σg . In particular the image of ϕ lies inside U (D). We shall prove that ker(ϕ) = G(F, F ′ )+ . Let g ∈ G(F, F ′ ) fixing an edge in Td . Then ϕ(g) also fixes an edge of Tr by (2). Moreover one has σg = 1 (Lemma 6.13), and it follows from the previous paragraph that all local permutations of ϕ(g) are trivial. This implies g ∈ ker(ϕ). Conversely, we let g be an element of ker(ϕ), and we prove that g ∈ G(F, F ′ )+ . Note that since ϕ(g) is trivial, one has σg = 1, i.e. all local permutations of g are in F ′+ . Now let v be any vertex. By Lemma 6.12, the sequence of colors (a1 , . . . , a2n ) from v to g(v) gives rise to a sequence (ia1 , . . . , ia2n ) that is a concatenation of palindromes. For simplicity we treat the case where (ia1 , . . . , ia2n ) is a palindrome; the general case consists in repeating the AMENABLE URS’S AND LATTICE EMBEDDINGS 33 argument for this case. Let v = v1 , . . . , v2n+1 = g(v) be the vertices between v and g(v) (note that vn is the midpoint between v and g(v)). Since (ia1 , . . . , ia2n ) is a palindrome, one easily checks that there are elements g1 , . . . , gn such that gj belongs to the stabilizer of vj in G(F, F ′+ ) and g′ = g1 . . . gn g fixes the vertex v (this is obtained by successively folding the geodesic [v, g(v)] onto itself starting from its midpoint in order to bring back g(v) to v with g1 . . . gn ). We now invoke the following easy fact, whose verification is left to the reader. Lemma 6.15. Let γ ∈ G(F, F ′ ) fixing a vertex w, and such that σ(γ, w) ∈ F ′+ . Then γ ∈ G(F, F ′ )+ . We apply Lemma 6.15 to g′ and all gj ’s, and deduce that g = gn−1 . . . g1−1 g′ belongs to G(F, F ′ )+ as desired. The last thing that remains to be proved in the statement of Lemma 6.14 is that the image of ϕ is equal to U (D). The fact that ϕ(g) always belongs to U (D) has already been observed. For the converse inclusion, observe that since G(F, F ′ ) acts transitively on the vertices of Tr (as it is already the case on Td ), it is enough to check that the image of ϕ contains U (D)x for some vertex x of Tr . Now since D acts freely on {1, . . . , r}, the map U (D)x → D, γ 7→ σ(γ, x), is an isomorphism. Therefore it is enough to see that any action on the star around x on Tr can realized by an element of G(F, F ′ ), and this is indeed the case (see e.g. Lemma 3.4 in [LB16]). To finish the proof of the proposition, remark that the image of G(F, F ′ )∗ by ϕ is precisely U (D)∗ . When F is transitive then D is also transitive, so that U (D)∗ has two orbits of vertices and one orbit of edges, and therefore splits as the free product D ∗ D. Remark 6.16. The case F = F ′ is allowed in Proposition 6.11, so that the conclusion also holds for the groups U (F ) from [BM00a]. Proposition 6.11 naturally leads us to isolate the following three situations. We keep the previous notation, so that r is the number of orbits of F ′+ in Ω, D = F ′ /F ′+ and Q = G(F, F ′ )∗ /G(F, F ′ )+ : (1) r = 1. In this case Tr is a segment of length one, and D and Q are trivial. (2) r = 2. Tr is a bi-infinite line, and this case splits into two disjoint sub-cases: (a) If F ′ is intransitive then D is trivial, and Q = U (1)∗ = Z (generated by a translation of Tr of length 2). (b) If F ′ is transitive then we have D = Sym(2) and Q = D ∗ D = D∞ . (3) r ≥ 3. Then Q = U (D)∗ is a virtually free group (since it acts vertex transitively and with trivial edge stabilizers of Tr ). Theorem 6.9 says that all properties stated there hold true if and only if r ∈ {1, 2}. A sufficient condition for having r = 1 is for instance that F acts transitively on Ω, F 6= F ′ and F ′ acts primitively, or quasi-primitively on Ω. Recall that a permutation group is quasi-primitive if every non-trivial normal subgroup acts transitively. But Theorem 6.9 also applies beyond the case of quasi-primitive permutation groups. For example a situation giving rise to case (2) (a) is when F ′ has a fixed point and acts transitively on the complement. Examples giving rise to case (2) (b) are for instance obtained by taking F ′ = Sym(n)≀C2 = Sym(n)≀hτ i acting naturally 34 ADRIEN LE BOUDEC on 2n letters, and F the subgroup generated by ((cn , cn ), 1) and ((1, 1), τ ), where cn is a cycle of order n. Proof of Theorem 6.9. (i) ⇒ (ii) follows from Lemma 6.10 and Proposition 6.11 (and the discussion following its proof). (ii) ⇒ (iii) is clear. (iii) ⇒ (iv) is Proposition 3.24. (iv) ⇒ (v) is guaranteed by Proposition 3.9 and the fact that AΓ is not a point. Finally assume that (i) does not hold, i.e. F ′+ has at least three orbits in Ω, and write Γ+ = G(F, F ′ )+ . By Proposition 6.11, the group Q = Γ/Γ+ has a subgroup of finite index that is free of rank at least 2. So there exist non-trivial Q-boundaries, and a fortiori these are non-trivial Γ-boundaries. If X is such a boundary, then Γ+ acts trivially on X. Since Γ+ also acts minimally on ∂Td , it follows that X and ∂Td are disjoint Γ-boundaries, contradicting (v). Therefore property (v) implies property (i), and the proof is complete. 6.4. Weakly co-amenable subgroups. In this paragraph we show that subgroups of the groups G(F, F ′ ) satisfy the following dichotomy: Proposition 6.17. Assume that F acts freely transitively and F ′ acts primitively on Ω. Then any subgroup of G(F, F ′ ) is either (locally finite)-by-cyclic (and hence amenable) or weakly co-amenable. We will need the following lemma. Lemma 6.18. Assume that F ′ acts primitively on Ω, and take two subgroups H1 6= H2 in the Furstenberg URS of G(F, F ′ ). Then hH1 , H2 i = G(F, F ′ )∗ . Proof. Write Γ = G(F, F ′ ). Recall from [LBMB16, Prop. 4.28] that the Furstenberg URS of Γ consists of subgroups Γ0ξ , ξ ∈ ∂Td , where Γ0ξ is the set of elements acting trivially on a neighbourhood of ξ. Given ξ 6= η ∈ ∂Td , we show that the subgroup Λ generated by Γ0ξ and Γ0η must be equal to G(F, F ′ )∗ . Take a vertex v on the geodesic from ξ to η, let e1 , e2 be the edges containing v and pointing towards ξ and η, and a, b the colors of e1 , e2 . Denote by K(v) the subgroup of Γ consisting of elements γ fixing v and such that σ(γ, w) ∈ F for every w 6= v. Since F ′ is primitive, F ′ is generated by the point stabilizers Fa′ and Fb′ . This implies that every element of K(v) may be written as a product of elements fixing either the half-tree defined by e1 containing ξ, or the half-tree defined by e2 containing η, so that K(v) ≤ Λ. Since v was arbitrary, we also have K(v ′ ) ≤ Λ for v ′ a neighbour of v on the geodesic [ξ, η]. The conclusion now follows since for two neighbouring vertices v, v ′ , the subgroups K(v), K(v ′ ) always generate G(F, F ′ )∗ [LB16, Cor. 3.10]. Proof of Proposition 6.17. Write Γ = G(F, F ′ ), and let Λ be a subgroup of Γ that is non-amenable, equivalently whose action of Td is of general type. By Proposition 5.7 we have to show that Λ fixes no probability measure on any non-trivial Γboundary. Argue by contradiction and assume that X is a non-trivial Γ-boundary on which Λ fixes a probability measure µ. According to Theorem 6.9, we have SΓ (X) = AΓ . Therefore by Proposition 3.2 there exist an almost 1-1 extension η : X̃ → X and a factor map π : X̃ → AΓ . Let Q ⊂ Prob(X̃) be the set of ν such that η ∗ ν = µ, and write R = π ∗ (Q), which is a closed Λ-invariant subset of Prob(AΓ ). Since the action of Λ on ∂Td is strongly proximal and since AΓ is a factor of ∂Td [LBMB16, Prop. 2.10-4.28], AMENABLE URS’S AND LATTICE EMBEDDINGS 35 we deduce that R contains some Dirac measures. Let H ∈ AΓ such that there is ν ∈ Prob(X̃) with η ∗ ν = µ and π ∗ ν = δH . Such a measure ν must be supported in the set of (x, H) ∈ X̃, and it follows that µ is supported in the set of H-fixed points in X (because (x, H) ∈ X̃ implies that H ≤ Gx by upper semi-continuity of the stabilizer map). But since Λ does not fix any point in AΓ , we may find another H ′ ∈ AΓ such that δH ′ ∈ R, so that the same argument shows that H ′ also acts trivially on the support of µ. By Lemma 6.18 the subgroups H, H ′ generate G(F, F ′ )∗ , which is of index two in Γ. Therefore any point in the support of µ has a Γ-orbit of cardinality at most two, which is absurd since Γ acts minimally on X and X is non-trivial by assumption. Remark 6.19. Assume F acts freely transitively and F ′ acts primitively, and write Γ = G(F, F ′ ). Let Λ ≤ Γ such that the Λ-action on Td is of general type but with a proper Λ-invariant subtree. For instance one could take for Λ the subgroup generated by two hyperbolic elements with sufficiently far apart axis. Then Λ is not co-amenable in Γ (see the argument in the proof of Theorem 2.4 in [CM09]), but Λ is weakly co-amenable in Γ by Proposition 6.17. Remark 6.20. We mention that when F ′ is primitive, following the proof of Corollary 4.14 from [LBMB16] (with minor modifications), one could prove that every non-trivial G(F, F ′ )-boundary factors onto ∂Td . This would provide an alternative proof of Proposition 6.17. 6.5. Lattice embeddings of the groups G(F, F ′ ). In this section we study how the discrete groups G(F, F ′ ) can embed as lattices in some locally compact groups. The purpose of this paragraph is twofold: (1) First we apply previous results of the article to the family of groups G(F, F ′ ) and deduce some properties of general locally compact groups containing a group G(F, F ′ ) as a lattice (Corollary 6.22). (2) Second we explain how the groups G(F, F ′ ) embed as lattices in some locally compact wreath products. This will be the content of §6.5.2 below. Remark 6.21. Maybe it is worth pointing out that instances of lattice embeddings of the groups G(F, F ′ ) already appeared in [LB16]. Indeed under appropriate assumptions on permutation groups F ≤ F ′ , H ≤ H ′ , the inclusion of G(F, F ′ ) in G(H, H ′ ) has discrete and cocompact image [LB16, Cor. 7.4]. Corollary 6.22. Assume that F acts freely transitively on Ω, and that F ′ is generated by its point stabilizers. Let G be a locally compact group containing G(F, F ′ ) as a lattice. Then the conclusions of Corollary 1.3 hold. Proof. The assumptions on F, F ′ imply that G(F, F ′ ) is virtually simple by Proposition 6.8, so Corollary 1.3 applies. Remark 6.23. In the setting of Corollary 6.22, although G(F, F ′ ) cannot be a lattice in a product, it happens that there exist non-discrete groups G1 , G2 such that G(F, F ′ ) embeds as a discrete subgroup of G1 × G2 with injective and dense projection to each factor. For instance if F1 , F2 are permutation groups such that F Fi ≤ F ′ and we set Gi = G(Fi , F ′ ), then the diagonal embedding of G(F, F ′ ) in G1 × G2 has this property as soon as F1 ∩ F2 = F . See [LB16, Lem. 3.4 and §7.1]. 36 ADRIEN LE BOUDEC 6.5.1. Locally compact wreath products. In this paragraph we introduce some terminology that will be used in the sequel. Let Ω be a set, B a group and A a subgroup of B. We will denote by B Ω,A the set of functions f : Ω → B such that f (x) ∈ A for all but finitely many x ∈ Ω. Note that B Ω,A is a group. Definition 6.24 ([Cor17]). If H is a group acting on Ω, the semi-restricted Ω,A ⋊ H, permutational wreath product B ≀A Ω H is the semi-direct product B X,A −1 where h ∈ H acts on f ∈ B by (hf )(x) = f (h x). The extreme situations when A = 1 and when A = B correspond respectively to the restricted and the unrestricted wreath product. When A = 1, we shall write B ≀Ω H for the restricted wreath product. Also for simplicity we will sometimes say “wreath product” B ≀A Ω H instead of “semi-restricted permutational wreath product”. When A is a compact group and H a locally compact group acting continuously on Ω, the group AΩ ⋊ H is a locally compact group for the product topology. If moreover B is locally compact and A is compact open in B, there is a natural locally compact group topology on B ≀A Ω H, defined by requiring that the inclusion Ω A of A ⋊ H in B ≀Ω H is continuous and open. See [Cor17, Sec. 2]. In the remaining of the article we shall be interested in the study of certain lattices in some locally compact groups B ≀A Ω H. A few remarks are in order: Lemma 6.25. Let A, B, Ω and H as above. (a) Assume Γ1 is a lattice in B Ω,A and Γ2 is a lattice in H that normalizes Γ1 . Then Γ1 ⋊ Γ2 is a lattice in B ≀A Ω H. A (b) For B ≀Ω H to contain a lattice, it is necessary that H contains a lattice. Proof. For the first statement, see [Rag72, Lem. I.1.6-7]. For the second statement, Ω observe that if Γ ≤ B ≀A Ω H is a lattice, the intersection ΓA = Γ ∩ (A ⋊ H) is a Ω lattice in AΩ ⋊H since AΩ ⋊H is open in B ≀A Ω H. The subgroup A being compact, the projection of ΓA to H is discrete, and hence is a lattice in H. Recall that there are various notions of irreducibility for a lattice in a direct product of groups. In general whether all these notions coincide depends on the context. We refer to [CM12, 2.B] and [CM09, 4.A] for detailed discussions. In the setting of wreath products, we will use the following terminology: Definition 6.26. A lattice Γ in B ≀A Ω H is an irreducible lattice if Γ has a non-discrete projection to the group H. This definition implies that neither Γ nor its finite index subgroups can be of the form Γ1 ⋊ Γ2 as in Lemma 6.25. Lemma 6.27. If the group B Ω,A does not contain any lattice, then any lattice in B ≀A Ω H is irreducible. Proof. If Γ ≤ B≀A Ω H is a lattice with a discrete projection Λ to H, then the subgroup Λ contains Γ as a lattice, and it follows that Γ intersects the subgroup B Ω,A , B ≀A Ω Ω,A . which is open in B ≀A Ω Λ, along a lattice of B AMENABLE URS’S AND LATTICE EMBEDDINGS 37 Remark 6.28. Of course if B admits no lattice then the same holds for B Ω,A . More interestingly, there are finite groups A, B for which B Ω,A fails to admit any lattice (provided Ω is infinite). This is for instance the case when A 6= B and any non-trivial element of B has a non-trivial power in A. Consequently all lattices in B ≀A Ω H are irreducible by Lemma 6.27 (for arbitrary H). Proof of Remark 6.28. We claim that the above condition on A, B actually implies that B Ω,A has no infinite discrete subgroup. For every finite Σ ⊂ Ω, we write OΣ ≤ B Ω,A for the subgroup vanishing on Σ, and UΣ = OΣ ∩ AΩ . Assume Γ is a discrete subgroup of B Ω,A , so that there is a finite Σ ⊂ Ω such that Γ ∩ UΣ = 1. The assumption on A, B is easily seen to imply that any non-trivial subgroup of OΣ intersects UΣ non-trivially. Therefore Γ ∩ OΣ = 1, and OΣ being of finite index in B Ω,A , Γ is finite. It should be noted that the existence of an irreducible lattice in B ≀A Ω H forces H to be non-discrete and B to be non-trivial. However this does not force B Ω,A to be non-discrete, and as we will see below, interesting examples already arise when B is finite and A is trivial. 6.5.2. The proof of Theorem 1.4. Let n ≥ 2 and d ≥ 3. We denote by Σn the (V ) set of integers {0, . . . , n − 1}, and by Σn d the set of functions f : Vd → Σn with finite support, where the support of f is the set of v such that f (v) 6= 0. We will also write fv for the image of v by f , and sometimes use the notation (fv ) for the function f . We consider the graph Xn,d whose set of vertices is the set of pairs (f, e), where (V ) f belongs to Σn d and e ∈ Ed , and edges emanating from a vertex (f, e) are of two types: • type 1: (f, e′ ) is connected to (f, e) if e′ ∈ Ed is a neighbour of e (i.e. if e and e′ share exactly one vertex); • type 2: (f ′ , e) is connected to (f, e) if the function f ′ is obtained from f by changing the value at exactly one vertex of e. Note that since any e ∈ Ed has 2(d − 1) neighbours and Σn has cardinality n, every vertex of Xn,d has 2(d − 1) neighbours of type 1 and 2(n − 1) neighbours of type 2. The graph Xn,d is almost the wreath product of the complete graph on n vertices with the tree Td , see below. Let Sn be the group of permutations of Σn . For σ ∈ Sn and i ∈ Σn , we will write σ · i the action of σ on i. The stabilizer of 0 ∈ Σn in Sn is obviously isomorphic to Sn−1 , and by abuse of notation we will denote it Sn−1 . In particular when viewing Sn−1 as a subgroup of Sn , we will always implicitly mean that Sn−1 is the subgroup of Sn acting only on {1, . . . , n − 1}. S Definition 6.29. We will denote by Gn,d the wreath product Sn ≀Vdn−1 Aut(Td ). Groups of the form Gn,d were considered in [Cor17, Ex. 2.6]. We will denote V ,S Un,d = Snd n−1 , so that Gn,d = Un,d ⋊ Aut(Td ). We endow Gn,d with the topology such that sets of the form ((σv )U1 , γU2 ) form a basis of neighbourhoods of ((σv ), γ), d and in where U1 and U2 belong to a basis of the identity respectively in SVn−1 Aut(Td ). This defines a totally disconnected locally compact group topology on Gn,d (see [Cor17, Prop. 2.3]). We note that the case n = 2 is somehow particular, 38 ADRIEN LE BOUDEC as U2,d is a discrete subgroup of G2,d , and G2,d is just the restricted wreath product G2,d = C2 ≀Vd Aut(Td ) = (⊕Vd C2 ) ⋊ Aut(Td ). Proposition 6.30. The group Gn,d acts by automorphisms on the graph Xn,d by preserving the types of edges. Moreover the action is faithful, continuous, proper and transitive on the set of vertices. Proof. The group Gn,d is a subgroup of the unrestricted permutational wreath product of Sn and Aut(Td ). The latter group has a faithful action on the set of functions Vd → Σn , given by ((σv ), γ) · (fv ) = (σv · fγ −1 v ). (V ) The group Gn,d preserves Σn d because σv fixes 0 almost surely if (σv ) belongs to Un,d . Now the projection from Gn,d onto Aut(Td ) induces an action of Gn,d on the (V ) set Ed , and we will consider the diagonal action of Gn,d on Σn d × Ed . In other words, if g = ((σv ), γ) ∈ Gn,d , and ((fv ), e) ∈ Xn,d , (3) g · ((fv ), e) = (σv · fγ −1 v ), γe . Fix x = ((fv ), e) ∈ Xn,d and g = ((σv ), γ) ∈ Gn,d , and let x′ be a neighbour of x. If x′ is of type 1, then we have x′ = ((fv ), e′ ), where e and e′ share a vertex w in Td . Then γe and γe′ have the vertex γw in common, so that by the formula (3), g · x′ is a neighbour of type 1 of g · x in Xn,d . Now if x′ is of type 2, then we may write x′ = ((fv′ ), e) with fv′ = fv if and only if v 6= w, where w is one of the two vertices of e. It follows that σv · fγ −1 v = σv · fγ′ −1 v if and only if v 6= γw, so that by (3) g · x′ is a neighbour of type 2 of g · x. This shows that the action is by graph automorphisms and preserves the types of edges. Lemma 6.31. Let e ∈ Ed , and let K be the stabilizer of e in Aut(TQ d ). Then the stabilizer of the vertex ((0), e) in Gn,d is the compact open subgroup Sn−1 ⋊ K. Proof. That g = ((σv ), γ) fixes ((0), e) exactly means by (3) that σv fixes 0 for all v and that γ fixes e. So the fact that the action is continuous and proper follows from the lemma, and the transitivity on the set of vertices is an easy verification. Consider now the free product Cd ∗ Cd of two cyclic groups of order d, acting on its Bass-Serre tree Td with one orbit of edges and two orbits of vertices. Denote by Cn the cyclic subgroup of Sn generated by the cycle (0, . . . , n − 1), and set Γn,d := Cn ≀Vd (Cd ∗ Cd ) ≤ Gn,d . Remark that Cd ∗ Cd has a split morphism onto Cd , whose kernel acts on Td with two orbits of vertices and is free of rank d − 1. Therefore Γn,d splits as Γn,d = (Cn2 ≀ Fd−1 ) ⋊ Cd . Lemma 6.32. Γn,d ≤ Gn,d acts freely transitively on the vertices of Xn,d . Proof. This is clear: the image of the vertex ((0), e) by an element ((σv ), γ) is ((σv · 0), γe), so both transitivity and freeness follow from the fact that the actions of Cn on Σn and of Cd ∗ Cd on Ed have these properties. AMENABLE URS’S AND LATTICE EMBEDDINGS 39 We now explain how the groups G(F, F ′ ) act on the graphs Xn,d . In the sequel F, F ′ denote two permutation groups on Ω such that F ≤ F ′ and F ′ preserves the orbits of F , and we denote by n the index of F in F ′ . Fix a bijection between Σn = {0, . . . , n − 1} and F ′ /F , such that 0 is sent to the class F . The action of F ′ on the coset space F ′ /F induces a group homomorphism α : F ′ → Sn , such that α(F ) lies inside Sn−1 . For γ ∈ G(F, F ′ ) and v ∈ Vd , write ργ,v = α(σ(γ, γ −1 v)) ∈ Sn . Note that ργ,v ∈ Sn−1 if and only if σ(γ, γ −1 v) ∈ F . We also denote by ργ = (ργ,v ). Proposition 6.33. Let F ≤ F ′ ≤ Sym(Ω), and n the index of F in F ′ . The map ϕ : G(F, F ′ ) → Gn,d , γ 7→ ϕ(γ) = (ργ , γ), is a well-defined group morphism that is injective, continuous, and with a closed and cocompact image. Proof. The map ϕ is well-defined because ργ,v ∈ Sn−1 for all but finitely many V ,S v, so that we indeed have ργ ∈ Snd n−1 . The fact that ϕ is a group morphism follows from the cocycle identity (1) satisfied by local permutations. Indeed for γ, γ ′ ∈ G(F, F ′ ) we have ϕ(γ)ϕ(γ ′ ) = (ψ, γγ ′ ) with ψv = ργ,v ργ ′ ,γ −1 v = α(σ(γ, γ −1 v))α(σ(γ ′ , γ ′−1 γ −1 v)) = α(σ(γγ ′ , (γγ ′ )−1 v)) = ργγ ′ ,v , so ψ = ργγ ′ and ϕ(γ)ϕ(γ ′ ) = ϕ(γγ ′ ). Injectivity of ϕ is clear since the composition with the projection to Aut(Td ) d ⋊ Aut(Td ) is is injective. The preimage in G(F, F ′ ) of the open subgroup SVn−1 ′ the subgroup U (F ), which is open in G(F, F ) by definition of the topology, so it follows that the map ϕ is continuous. Also the intersection between Im(ϕ) and the d ⋊ Aut(Td ) is ϕ(U (F )), and it is easy to check that the latter open subgroup SVn−1 is indeed a closed subgroup of Gn,d , so it follows that Im(ϕ) is closed in Gn,d . The fact that Im(ϕ) is cocompact will follow from Proposition 6.30 and Proposition 6.34 below. In the sequel for simplicity we will also write G(F, F ′ ) for the image of ϕ : G(F, F ′ ) → Gn,d , γ 7→ ϕ(γ) = (ργ , γ) . In particular when speaking about an action of G(F, F ′ ) on the graph Xn,d , we will always refer to the action defined in Proposition 6.30, restricted to G(F, F ′ ). This means that γ ∈ G(F, F ′ ) acts on (f, e) ∈ Xn,d by γ · (f, e) = (f γ , γe), where (4) (f γ )v = ργ,v · fγ −1 v = α(σ(γ, γ −1 v)) · fγ −1 v . This action should not be confused with the standard action ((fv ), e) 7→ ((fγ −1 v ), γe) coming from the inclusion of G(F, F ′ ) in Aut(Td ). Proposition 6.34. Let F ≤ F ′ ≤ Sym(Ω), and n the index of F in F ′ . 40 ADRIEN LE BOUDEC (a) The group G(F, F ′ )∗ acts cocompactly on Xn,d . When F is transitive on Ω, the group G(F, F ′ )∗ acts transitively on vertices of Xn,d . (b) The stabilizer of a vertex ((0), e) ∈ Xn,d in G(F, F ′ ) is the stabilizer of e in U (F ). In particular the action of G(F, F ′ ) on Xn,d is proper. Therefore when F acts freely transitively on Ω, the group G(F, F ′ )∗ acts freely transitively on the vertices of Xn,d . Proof. We show that for every vertex x = ((fv ), e) of Xn,d , there is g ∈ G(F, F ′ )∗ such that g · x = ((0), e). Since U (F ) preserves the vertices of this form, and since the number of orbits of U (F )∗ ≤ G(F, F ′ )∗ on Ed is finite and is equal to one when F is transitive [BM00a], statement (a) will follow. We argue by induction on the cardinality N of the support of (fv ). There is nothing to show if N = 0. Assume N ≥ 1, and let v0 ∈ Vd with fv0 6= 0 and such that v0 maximizes the distance from e among vertices v such that fv 6= 0. Let e0 be the edge emanating from v0 toward e (if v0 belongs to e then e0 = e), and let a ∈ Ω be the color of e0 . We also denote by T 1 and T 2 the two half-trees defined by e0 , where T 1 contains v0 . For every b ∈ Ω, b 6= a, we denote by e0,b the edge containing v0 and having color c(e0,b ) = b, and by T 1,b the half-tree defined by e0,b not containing v0 . By assumption the permutation group F ′ preserves the F -orbits in Ω, so we have ′ F = F Fa′ . The subgroup α(F ′ ) ≤ Sn being transitive, it follows from the previous decomposition that there exists σ ∈ Fa′ such that α(σ) · fv0 = 0. For every b 6= a, we choose σb ∈ F such that σb (b) = σ(b), and we consider the unique element h ∈ Aut(Td ) whose local permutations are σ(h, v) = 1 if v ∈ T 2 ; σ(h, v0 ) = σ; and σ(h, v) = σb for every v ∈ T 1,b and every b 6= a. It is an easy verification to check that h is a well-defined automorphism of Td , and h ∈ G(F, F ′ ) because all but possibly one local permutations of h are in F . Note that h fixes e by construction. Write h(x) = ((φv ), e). We claim that the support of (φv ) has cardinality N − 1. Since h fixes v0 , by (4) we have φv0 = α(σ(h, v0 )) · fv0 = α(σ) · fv0 = 0. Moreover we also have φv = fv for every v in T 2 because h acts trivially on T 2 . Finally by the choice of v0 we had fv = 0 for every v 6= v0 in T 1 , and since σ(h, v) ∈ F for all these v and α(F ) fixes 0, we still have φv = 0 for every v in T 1 , v 6= v0 . This proves the claim, and the conclusion follows by induction. Statement (b) follows from Lemma 6.31, and the last statement follows from (a) and (b) and the fact that U (F ) acts freely on Ed when F acts freely on Ω. Propositions 6.33-6.34 and Lemma 6.32 imply Theorem 1.4 from the introduction. Note that when F acts freely transitively on Ω, we have an explicit description of a generating subset of the group G(F, F ′ )∗ whose associated Cayley graph is Xn,d . For, fix an edge e0 ∈ Ed , whose color is a ∈ Ω and whose vertices are v0 , v1 , and denote x0 = ((0), e0 ) ∈ Xn,d . For i = 0, 1, let Si be the set of γ ∈ G(F, F ′ ) fixing vi and such that σ(γ, v) ∈ F for every v 6= vi , and σ(γ, vi ) is non-trivial and belongs to F ∪ Fa′ . Then S = S0 ∪ S1 generates G(F, F ′ )∗ , and Cay(G(F, F ′ )∗ , S) → Xn,d , γ 7→ γx0 , is a graph isomorphism. Moreover neighbours of type 1 (resp. type 2) of a vertex of Xn,d are labeled by elements s ∈ Si such that σ(γ, vi ) ∈ F (resp. σ(γ, vi ) ∈ Fa′ ). AMENABLE URS’S AND LATTICE EMBEDDINGS 41 We end the article by observing that there are possible variations in the definition of the graph Xn,d . If Kn is the complete graph on n vertices, let Kn ≀ Td be the wreath product of the graphs Kn and Td (sometimes also called the lamplighter (V ) graph over Td ): the vertex set is Σn d × Vd , and there is an edge between (f, v) and (f ′ , v ′ ) if and only if either f = f ′ and v, v ′ are adjacent in Td , or v = v ′ and f (w) = f ′ (w) if and only if w 6= v. Again if n is the index of F in F ′ , the group G(F, F ′ ) acts on Kn ≀ Td by γ · (f, v) = (f γ , γv), where f γ is given by (4). The previous arguments for the graph Xn,d carry over to this graph, so that we have: Proposition 6.35. Let F ≤ F ′ ≤ Sym(Ω), and n the index of F in F ′ . Then G(F, F ′ ) acts properly and cocompactly on the graph Kn ≀ Td . The reason why we considered the graph Xn,d instead of Kn ≀ Td is to obtain, under the assumption that F acts freely on Ω, a free action of G(F, F ′ )∗ on the set of vertices. In the case of Kn ≀ Td , the stabilizer of a vertex in G(F, F ′ )∗ is finite, but non-trivial. We note that it might be interesting to investigate whether the generalized wreath products of graphs from [Ers06] could provide other kind of interesting groups of automorphisms. Yet another possibility is to take the same vertex set as Xn,d , but declaring that there is an edge between (f, e) and (f ′ , e′ ) if e 6= e′ share a vertex w and fv = fv′ for every v 6= w. This graph Zn,d has larger degree, namely 2(d − 1)n. Again all the results proved above for Xn,d remain true. In the case d = 2, one may check that Zn,2 is the Diestel-Leader graph DL(n, n), so that Zn,d may be thought of as “higher dimensional” versions of these graphs. References [BCGM16] U. Bader, P.E. Caprace, T. Gelander, and S. Mozes, Lattices in amenable groups, arXiv:1612.06220 (2016). [BF14] U. Bader and A. Furman, Boundaries, rigidity of representations, and Lyapunov exponents, Proceedings of ICM (2014). [BFS15] U. Bader, A. Furman, and R. Sauer, On the structure and arithmeticity of lattice envelopes, C. R. Math. Acad. Sci. Paris 353 (2015), no. 5, 409–413. [BKKO14] E. Breuillard, M. Kalantar, M. Kennedy, and N. Ozawa, C ∗ -simplicity and the unique trace property for discrete groups, arXiv:1410.2518v2 (2014). [BM97] M. Burger and Sh. Mozes, Finitely presented simple groups and products of trees, C. R. Acad. Sci. Paris Sér. 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UCLouvain, IRMP, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, Belgium CNRS, Unité de Mathématiques Pures et Appliquées, ENS-Lyon, France E-mail address: [email protected]	4
1 Multisensor Poisson Multi-Bernoulli Filtering with Uncertain Sensor States arXiv:1712.08146v1 [cs.SY] 21 Dec 2017 Markus Fröhle, Christopher Lindberg, Karl Granström, Henk Wymeersch Abstract—In a typical multitarget tracking (MTT) scenario, the sensor state is either assumed known, or tracking is performed based on the sensor’s (relative) coordinate frame. This assumption becomes violated when the MTT sensor, such as a vehicular radar, is mounted on a vehicle, and the target state should be represented in a global (absolute) coordinate frame. Then it is important to consider the uncertain sensor location for MTT. Furthermore, in a multisensor scenario, where multiple sensors observe a common set of targets, state information from one sensor can be utilized to improve the state of another sensor. In this paper, we present a Poisson multi-Bernoulli MTT filter, which models the uncertain sensor state. The multisensor case is addressed in an asynchronous way, where measurements are incorporated sequentially based on the arrival of new sensor measurements. In doing so, targets observed from a well localized sensor reduce the state uncertainty at another poorly localized sensor, provided that a common non-empty subset of features is observed. The proposed MTT filter has low computational demands due to its parametric implementation. Numerical results demonstrate the performance benefits of modeling the uncertain sensor state in feature tracking as well as the reduction of sensor state uncertainty in a multisensor scenario compared to a per sensor Kalman filter. Scalability results display the linear increase of computation time with number of sensors or features present. I. I NTRODUCTION Intelligent transportation systems in general, and autonomous driving (AD) in particular, require accurate position information [1]. Measurements provided by various on-board sensors allow to infer the vehicle state, e.g., position and velocity, as well as information about the surrounding environment. For instance, a global navigation satellite system (GNSS) receiver provides absolute position, whereas a radar sensor provides relative position with respect to (w.r.t.) the sensor origin. Furthermore, vehicles have access to a pre-recorded local dynamic map (LDM) containing static features such as, e.g., landmarks [2]. Dynamic features such as pedestrians, cyclists, etc. are not part of the pre-recorded map. For an AD system to be fully aware of the surrounding environment, dynamic features need to be estimated and tracked over time using the vehicles on-board sensors thus allowing to enrich the vehicle’s LDM. In order to incorporate mobile features into the LDM, which contains map features described in a global coordinate frame, location uncertainty of on-board sensors used to track dynamic features needs to M. Fröhle, K. Granström, and H. Wymeersch are with the Department of Electrical Engineering, Chalmers University of Technology, Gothenburg, Sweden. E-mail: {frohle, karl.granstrom, henkw}@chalmers.se. C. Lindberg is with Zenuity AB, Gothenburg, Sweden. E-mail: [email protected] be considered, i.e., the vehicle’s state uncertainty such as its location and pose. In an ITS, vehicles communicate through the wireless channel with other vehicles (vehicle-to-vehicle (V2V) communication) or with road infrastructure, such as a road side unit (RSU), through vehicle-to-infrastructure (V2I) communication, using IEEE 802.11p or 4G/5G cellular communication. Through information exchange, vehicles can make local LDM information available to their neighbors allowing to enrich their LDMs and their situational awareness. Not only can LDM information be shared, it can also be fused to improve every LDM [3], [4]. For the special case, they observe an overlapping set of dynamic features, information from one vehicle can be utilized to increase location accuracy of other vehicles, and vice versa [5], [6]. Note, in this context no V2V measurements are performed, different to a traditional cooperative localization approach [7], [8]. The problem of vehicular localization using locally observed features with unknown observation to feature correspondence, aggregated at an RSU, can be interpreted as an MTT problem. In MTT, a varying number of mobile features (targets) are tracked, using sensors such as for example radars, LIDARS, or cameras [9]. Thereby, it is typically assumed that the state of the observing sensor is known. Although not true in general, this assumption can be motivated by the fact that sensor state uncertainty is negligible in comparison to the sensors’ measurement accuracy. If sensor state uncertainty is significant, it needs to be modeled in the MTT in order not to have negative impact on feature tracking performance. In this paper, we consider the case of MTT with uncertain sensor state, where there are potentially multiple sensors with varying sensor state uncertainty. To enable accurate feature tracking we model the sensor state uncertainty in the MTT filter. The main contributions are: • • • an asynchronous parametric multitarget-multisensor tracking filter with uncertain sensor state information, fusion of mulitarget-multisensor tracking information with local sensor tracking information, and numerical simulation results demonstrating the performance of the filter in a multisensor vehicular scenario. In the application example, we demonstrate how the proposed filter can be used to transfer location information from a well localized vehicle to a poorly localized vehicle through MTT. Hence, positioning accuracy of the poorly localized vehicle is greatly improved compared to using a local Kalman filter with GNSS measurements alone. 2 Legend Cooperative vehicle Mobile feature RSU Communication link Fig. 1. Urban ITS scenario with two vehicles cooperating through the RSU and six mobile features. A. Motivation In this paper, we consider an urban intelligent transportation system (ITS) scenario, consisting of cooperating vehicles (illustrated in Fig. 1). Each vehicle is equipped with an on-board sensor allowing them to determine their absolute position, e.g., a GNSS receiver, and an on-board sensor to retrieve relative positions of mobile features present in the environment, e.g., a radar. Absolute position measurements are denoted GNSS measurements and measurements taken w.r.t. features are denoted vehicle-to-feature (V2F) measurements. Due to the sensor used to obtain V2F measurements, it is in general not known which feature gave rise to which measurement. A GNSS and a set of V2F measurements from every vehicle are transmitted in a non-time synchronized manner to the RSU, where a centralized filter is run to track the feature as well as the vehicle states. This information can be utilized by the RSU an sent back to the vehicles to increase their situation awareness. B. Related Work 1) MTT with known sensor state: Many MTT filters have been proposed to track mobile features using radar-like sensors when the sensor state is known [9]. The multi-hypothesis tracking (MHT) filter builds a growing hypothesis tree with feature-to-measurements data association (DA), and needs to be pruned to limit computation complexity [10]. The joint probability data association (JPDA) filter finds the most likely DA, where feature state information is reduced after each update step to a single Gaussian per feature. In the last years, MTT filters based on random finite set (RFS) and finite-set statistics (FISST) (which avoid the inherent DA), originally developed by [11], have gained much attention. The probability hypothesis density (PHD) filter propagates the first moment of the RFS density over time [12]–[14]. The Poisson multi-Bernoulli (PMB) filter approximates the global joint DA by the product of (local) marginal DA similar to the JPDA filter [15]. In [16], a derivation of this PMB filter based on standard single target measurement models, without using probability generating functional (p.g.fl.) and functional derivatives, is presented. Furthermore, a connection to the δ-generalized labelled multi-Bernoulli (δ-GLMB) filter [17], [18] is shown, where the δ-GLMB density is a special case of multi-Bernoulli (MB) density for labeled targets and can therefore be seen as a special case of the PMB filter. In [19], a Gaussian mixture (GM) multitarget-multisensor Bernoulli tracker, with known sensor locations, is developed and compared to a particle filter (PF) implementation for multistatic sonobuoy fields. Target state update from multiple sensors was achieved through sequential sensor updates. In [20], [21], a factor graph (FG) based approach, not using FISST, was proposed for a variant of the JPDA filter. A multiscan scenario was considered, and the filter was realized by running loopy belief propagation on the FG containing cycles. In [22], a PF based implementation of the PMB filter [15] was presented. This implementation has been used in [21] as a performance comparison of the FG based MTT. There it was found that the PF based PMB filter implementation scales exponentially with an increasing number of features. In [23], vehicles perform local feature tracking and send track information over the wireless channel to a central fusion center. Then, track-to-track fusion [24], [25] is performed, taking care of out-of-sequence measurements which can arise by utilizing a shared communication media. There the DA problem arises on the decision of which local tracks to fuse. The Mahalanobis distance is employed as DA metric. 2) MTT with uncertain sensor state: In contrast to MTT, simultaneous localization and mapping (SLAM) based methods determine the sensor state while mapping features of the environment [2], [26]. Most of the proposed SLAM methods assume static features such as, e.g., walls and street signs. SLAM methods excel through maintaining the correlation between features. A reduction of complexity was achieved in FastSLAM through Rao-Blackwellization (RB), where the feature state, conditioned on the sensor state, is tracked through a Kalman filter and the sensor state through a PF. In [27], a RFS based approach to the SLAM problem was proposed. There the target state is conditioned on the sensor location and then tracked through a PHD filter following a RB. In [28], an MTT with uncertain (single) sensor location is derived for the SLAM problem using FISST and point process theory. Simulation results are shown with a RB PF implementation. In [29], the problem of sensor uncertainty for Bernoulli filtering [30] of at most one target using a single sensor is addressed. Since the scenario is restricted to a single sensor, a suboptimal approach is presented, where the sensor state is updated only by measurements independent of the target state, i.e., target tracking information is not used to update the sensor location. Similar to [21], a FG based approach was considered in [31] for an urban ITS scenario, where the number of features is assumed a priori known, and in [32] a variant of SLAM was considered for indoor environments using time-of-arrival radio signal measurements. There, features are the (static) source locations of line-of-sight and non line-of-sight signal propagation paths of the transmitted radio wave. C. Notation and Paper Organization Scalars are described by non-bold letters r, vectors by lower-case bold letters x; matrices and sets by upper-case bold 3 letters X. The cardinality of set X is denoted \|X\|. The set operator ] denotes the disjoint set union, i.e., F u ] F d = F means F u ∪ F d = F and F u ∩ F d = ∅. The vehicle state is reserved by letter x, the feature state by letter f , and measurements by letter z. The identity matrix of size n × n is denoted I n . The `2 -norm of vector x is kxk2 . The remainder of this paper is organized as follows. Section II gives some background knowledge on RFS, and Section III introduces the problem formulation and system models. Section IV details the proposed MTT filter with uncertain sensor state, numerical results are given in Section V, and conclusions are drawn in Section VI. II. BACKGROUND ON RFS In this section, we describe some useful properties of an RFS. If not stated otherwise, the source of all these is [15]. A. Random Finite Set Formulation According to [15], RFS based methods have been developed in [11] to conduct statistical inference in problems in which the variables of interest and/or observations form finite sets. In tracking, they address two major challenges of interest: (i) the number of targets present in the scene is unknown, (ii) measurements are invariant to ordering (measurement-totarget correspondence is unknown). An RFS X is a finite-set valued random variable, which can be described by a discrete probability distribution p(n), n ≥ 0 and a family of joint probability densities fn (xπ(1) , . . . , xπ(n) ) yielding X f (x1 , . . . , xn ) = p(n) fn (xπ(1) , . . . , xπ(n) ), (1) π where the sum spans over the n! permutation functions π(·), such that its RFS density f (X) is permutation invariant. The set integral of a real-valued function g(X) of a finite-set variable X is defined as [11, p. 361] Z Z ∞ X 1 g(x1 , . . . , xn )dx1 · · · dxn . g(X)δX , g(∅) + n! n=1 (2) A Bernoulli process X with probability of existence r and existence-conditioned probability density function (PDF) f (x) has RFS density   X=∅ 1 − r, f (X) = r · f (x), X = x (3)   0, otherwise, The RFS density of a MB process for the RFS X = {x1 , . . . , xn } is [11, p. 368] "N # n Y X Y rik f (X) = (1 − ri ) fi (xi ) . 1 − rik k i=1 i≤i1 6=...6=in ≤N k=1 (4) A Poisson point process (PPP) with intensity function λc (y) has RFS density [15] Y f (Y ) = exp(− hλc , 1i) λc (y) (5) y∈Y R with inner product hλc , hi , λc (y)h(y)dy. Remark 1: If X and Y are independent RFSs such that Z = X ] Y , then X fZ (Z) = fX (X)fY (Y ). (6) X]Y =Z Note, for RFS X a MB and RFS Y a PPP (6) is called a PMB density. B. State estimation from RFS density A common way to estimate the set states from a Bernoulli process with RFS density f (X) is by comparing the probability of existence r against an existence threshold rth . For r > rth , the target is said to exist and has PDF f (x) (c.f. (3)). RIts state can then be estimated by the mean of f (x), i.e., x̂ = xf (x)dx. III. P ROBLEM F ORMULATION AND S YSTEM M ODELS Here, we first present the problem formulation, and the vehicle and feature dynamics. This is followed by the GNSS and V2F measurement models, and the communication model. A. Problem Formulation The goal of the filter, which runs on the RSU, is to track the features and the states of all vehicles in every discrete time step t through incorporation of all sensor measurements (GNSS and V2F measurements) up until time step t. We are therefore interested in the joint posterior distribution of the feature and vehicle states at every time step t. B. Vehicle and Feature Dynamics Vehicle state motion follows independent Markovian processes, where the time-varying vehicle state xs,t ∈ RNx of each vehicle s ∈ S at time step t ∈ N0 is statistically modeled as p(xs,t \|xs,t−1 ), with linear state-space model xs,t = As,t xs,t−1 + ws,t , (7) where As,t denotes the state-transition matrix and ws,t ∼ N (0, W s,t ) with error covariance matrix W s,t . A single time-varying feature k ∈ K with state f k,t−1 ∈ RNf survives to the next time step t following an independent identically distributed (IID) Markovian process with survival probability pS (f k,t ). The feature state motion follows IID Markovian processes and is statistically modeled as p(f k,t \|f k,t−1 ) with linear state-space model f k,t = B t f k,t−1 + v k,t , (8) where B t denotes the state-transition model and process noise v k,t ∼ N (0, V t ) with error covariance matrix V t . The statetransition matrices, as well as the error covariance matrices 4 are assumed equal among the features. Note that vehicle and feature state motion is independent of each other.1,2 In the following, we will drop the subscript indexing on states and measurements w.r.t. vehicle/feature/time whenever the context allows. C. Measurement models At time step t, vehicle s ∈ S obtains two different kind of measurements: (i) measurements of the vehicle state x w.r.t. the reference frame, i.e., GNSS-like measurements; and (ii) measurements w.r.t. to features, i.e., from a radar-like (onboard) V2F sensor. Without loss of generality, we assume that the on-board sensors’ state is equal to the vehicle state. Thus an uncertain vehicle location implies an uncertain location for the on-board V2F sensor. 1) GNSS measurement: The GNSS measurement z G ∈ RMG of vehicle s ∈ S at time t is statistically modeled through the likelihood function p(z G \|x) with linear observation model z G = H G x + r, (9) where H G is the linear observation matrix and r ∼ N (0, R) with error covariance matrix R. 2) V2F measurement: Let Z F be a set of measurements from a tracking sensor that is susceptible to measurement noise, missed detections, and false detections. Examples of such sensors include camera, radar and LIDAR. Consequently, Z F = Z FA ] Z D , z = H 1 x + H 2 f + q, (11) where H 1 and H 2 denote observation matrices, and q ∼ N (0, Q) with error covariance matrix Q. Note that the measurement-to-feature state correspondence denoted DA, is in general not known and needs to be inferred from the measurements. Let the measurement likelihood for a single feature RFS F , i.e. \|F \| ≤ 1, be  1 if F = ∅, ZF = ∅   0  if F = ∅, Z F = {z} 1 − pD (x, f ) if F = {f }, Z F = ∅ η(Z F \|x, F ) =  F   pD (x, f )`(z\|x, f ) if F = {f }, Z =F{z} if \|F \| > 1, or \|Z \| > 1, (12) 1 For where 0 < α ≤ 1 is a constant such that it is a valid PDF. Depending on the specific sensor at hand, the FoV may be different and (13) needs to be adapted. D. Communication model We assume that every vehicle is able to communicate all obtained measurements (V2F and GNSS) with the RSU instantaneously and without errors. This implies that at any time t the number of vehicles communicating with the RSU can vary. The incorporation of a realistic V2I channel model and its performance impact is a point for future work. IV. P OISSON M ULTI -B ERNOULLI FILTERING WITH UNCERTAIN SENSOR STATE (10) where Z FA denotes the set of false alarm measurements due to clutter modeled by a PPP with intensity λc (z), and Z D denotes the set of detected features. Let a V2F measurement z ∈ Z D with state dimension RMV2F be obtained through the V2F sensor at vehicle s ∈ S w.r.t. feature k at time t. It is modeled through the likelihood function `(z\|x, f ) with linear observation model 0 where `(z\|x, f ) follows the measurement model (11). Note that due to (11), the probability of detection pD (x, f ) in (12) depends on the vehicle state x as well as on the feature state f . For instance, a limited sensor field-of-view (FoV) affects the probability of feature detection based on the distance between vehicle and feature. Remark 2: For the case the radar-like V2F sensor is able to detect features within a radius rmax , the probability of detection is defined ( α, if kH 1 x + H 2 f k2 ≤ rmax (13) pD (x, f ) = 0, otherwise, the sake of brevity, we present linear system and measurement models. In case the true system dynamics and/or measurement model are (mild) nonlinear, linearization steps can be performed similar to the steps taken in an extended Kalman filter (EKF) or the unscented Kalman filter (UKF) [33], [34]. In doing so, the proposed filter remains valid unaltered. 2 Note that the proposed filter only predicts over small time-horizons in the order of a few tens of milliseconds. Then the assumption on vehicle and feature states evolving independently is reasonable, because there is very little interaction among them within the prediction horizon. In this section, we formulate the proposed PMB filter with uncertain sensor state. We consider a tracking scenario subject to Section I-A, where there may be multiple features. In Section IV-C, we proceed with the asynchronous multisensor case allowing to track multiple features and vehicles with uncertain vehicle state. In Section IV-D, a tractable Gaussian density approximation of the proposed filter is given. The vehicle state PDF at time step t − 1 is indicated by subscript ’−’, i.e., p− (x), the PDF predicted to the current time step t (before updating by a measurement) is indicated by subscript ’+’, i.e., p+ (x), and the posterior PDF is stated without subscript. Similar definitions hold w.r.t. the feature RFS density. The proposed filter is developed within a Bayesian framework with alternating prediction and update steps operating on an RFS X by [11, Ch. 14]. Z f+ (X) = f (X\|X 0 )f− (X 0 )δX 0 , (14) and f (X\|Z) ∝ `(Z\|X)f+ (X), (15) where f− (X 0 ) is the prior RFS density, f (X\|X 0 ) is the RFS transition density, f+ (X) is the predicted density, and `(Z\|X) is the RFS measurement likelihood for measurement set Z. From the problem definition stated in Section III-A, we are interested in the joint posterior density of the vehicle and features considering all measurements up to the current time step t. We now proceed with the development of the proposed filter within this framework. 5 With a single vehicle, the prior joint vehicle-feature density is of form f− (x, F ) = p− (x)f− (F ), (16) where p− (x) is the prior PDF on the vehicle state, and f− (F ) is the prior PMB density. The latter density can be written in u terms of an PPP intensity of undetected features f− (F u ), i.e., features which are hypothesized to exist but have never been detected [15, Def. I], and the prior MB RFS density of detected d features f− (F d ), as [11, p. 484], [15] X u d f− (F u )f− (F d ). (17) f− (F ) = F u ]F d =F In (17), the PPP density of undetected features is Y u u u f− (F u ) = e−hD− ,1i D− (f ), (18) f ∈F u u where D− (f ) is the intensity of undetected features. We are interested in a low computational complexity method to compute the (posterior) joint vehicle-feature density f (x, F ) in every discrete time step t through incorporation of all sensor measurements. In doing so, the posterior density should remain in the same form as the prior joint density (16). A. Prediction step With the vehicle state x, and existing feature RFS F , the predicted joint vehicle-feature density is f+ (x, F ) = p+ (x)f+ (F ). (19) Here, the predicted vehicle state PDF is given by the Chapman-Kolmogorov equation [35] Z p+ (x) = p(x\|x0 )p− (x0 )dx0 , (20) where p(x\|x0 ) is the state transition PDF described by (7), and p− (x0 ) is the prior PDF. Similarly, the predicted feature state PMB density is calculated by Z f+ (F ) = f (F \|F 0 )f− (F 0 )δF 0 , (21) where f (F \|F 0 ) is the transition RFS density, and f− (F 0 ) is the prior PMB density. the predicted intensity of undetected u u features D+ (f ) of the predicted PPP density f+ (F u ) is given by Z u u D+ (f ) = Db (f ) + p(f \|f 0 )pS (f 0 )D− (f 0 )df 0 . (22) Here, the birth intensity is denoted Db (f ), feature transition PDF p(f \|f 0 ) is described by (8), feature survival probability u is denoted pS (f 0 ), and D− (f 0 ) denotes the prior intensity. The MB RFS density of detected features of (21) is X d i (F i ), (23) f+ (F d ) = f+ ]i∈I+ F i =F d where I+ denotes the set of existing features (before measurement update), and  i , Fi = ∅  1 − r+ i i i i f+ (F ) = (24) r p (f ), F i = {f i }  + + 0, otherwise. Here, the predicted PDF of feature i is Z p+ (f i ) = p(f i \|f 0i )p− (f 0i )df 0i , (25) where p− (f 0i ) is the prior PDF of feature i. The probability of existence of feature i is [15, Eqn. (40)] Z i i r+ pS (f 0i )p− (f 0i )df 0i , (26) = r− i where r− denotes the prior probability of existence, p− (·) the prior PDF, and pS (·) the probability of feature survival. B. Measurement update step Updating the joint vehicle-feature density (19) by any of the two types of different measurements, GNSS and V2F measurements, involves the application of Bayes’ theorem. In the following, we describe the update calculations using the different type of measurements. 1) Update with vehicle state measurement: Let z G be a measurement related to the vehicle state x, and unrelated to the set of features F , p(z G \|x, F ) = p(z G \|x). (27) For example, z G could be a GNSS and/or an inertial measurment unit (IMU) measurement. Given a predicted vehiclefeature density (19), and by Bayes’ theorem the updated density is f (x, F \|z G ) = p(x\|z G )f+ (F ). (28) In other words, the vehicle state density is updated with the measurement z G , the feature set density is unaffected by the update, and the independent form is retained (c.f. (16)). Note, this update step can be omitted in the absence of GNSS-like measurements, e.g., in a pure SLAM application [26]. 2) Update with cluttered set of feature measurements: Let the set of measurements Z F subject to the measurement model of Section III-C2 be indexed by M, and let A be the space of all DAs A for the predicted MB. A DA A ∈ A is an assignment of each measurement in Z F to a source, either to the background (clutter or new feature) or to one of the existing features indexed by I. It is therefore a partition of M ∪ I into non-empty disjoint subsets C ∈ A, called index cells3 . Remark 3: Due to the standard MTT assumption that the features generate measurements independent of each other 3 For example, let M = (m , m , m ) and I = (i , i ), i.e., three 1 2 3 1 2 measurements and two features. One valid partition of M ∪ I, i.e., one of the possible associations, is {m1 }, {m2 , i1 }, {m3 }, {i2 }. The meaning of this is that measurement m2 is associated to feature i1 , feature i2 is not detected, and measurements m1 and m3 are not associated to any previously detected feature, i.e., measurements m1 and m3 are either clutter or from new features . 6 [9], an index cell contains at most one feature index, i.e., \|C ∩ I\| ≤ 1 for all C ∈ A. Any association in which there is at least one cell, with at least two feature indices, will have zero likelihood because this violates the independence assumption. Further, due to the point feature assumption, any feature generates at most one measurement in each time step, i.e., \|C ∩ M\| ≤ 1 for all C ∈ A. Any association in which there is at least one cell, with at least two measurement indices, will have zero likelihood because this violates the point feature assumption. If the index cell C contains a feature index, then let iC denote the corresponding feature index. Further, if the index cell C contains a measurement index, then let mC denote the corresponding measurement index. Measurements in C not assigned to any feature are associated to the background. With the help of Bayes’ rule, the updated joint vehicle-feature density is X f u (F u \|x) f (x, F \|Z F ) = × F u ]F d =F X A A w p (x\|Z F )f d,A (F d \|x, Z F ), (29) A∈A P where wA denotes the weight of DA A ∈ A with A∈A wA = F 1, and pA (x\|Z ) denotes the vehicle state posterior stated in Appendix A. For now, let us assume the DA weights are given. The undetected feature density f u (F u \|x) and the detected feature density f d,A (F d \|x, Z F ) are stated in Appendix B. Equation (29) does not factorize as f (x, F \|Z F ) = p(x\|Z F )f (F \|Z F ) where f (F \|Z F ) is a multi-Bernoulli density such that it remains in the same form as the prior (c.f. (19)); on the contrary, there are many dependencies between the feature state RFS and the vehicle state. This means that existing independence-assuming tracking frameworks cannot be applied directly [9], or introduce a significant increase in computational complexity [28]. To overcome this, we approximate f u (F u \|x) ≈ fˆu (F u ), f d,A (F d \|x, Z F ) ≈ f˜d,A (F d \|Z F ), (30) (31) where the functions on the right hand side need to be found. Towards this end, we make the following approximations for the vehicle and feature dependent probability of detection u p̂D , f ∈ F u pD (x, f ) ≈ (32) p̂iD , f ∈ F d , with F u ] F d = F , and where ZZ u p̂uD = pD (x, f )p+ (x)D+ (f )dxdf , ZZ p̂iD = pD (x, f )p+ (x)pi+ (f )dxdf . for p̂iD . Under approximation (32), the updated density (29) becomes X fˆ(x, F \|Z F ) ∝ fˆu (F u ) × F u ]F d =F X A A w p (x\|Z F )fˆd,A (F d \|x, Z F ) (35) A∈A with fˆu (F u ) and fˆd,A (F d \|x, Z F ) given in Appendix C. We observe in (35) that the undetected feature density fˆu (F u ) depends only on the undetected feature RFS F u , and is independent of the other stochastic variables. What remains are dependencies between the detected features, and the vehicle. To remove the dependency on the vehicle state in the detected feature density fˆd,A (F d \|x, Z F ) in (35), we map the vehicle state uncertainty onto the V2F measurement uncertainty. This is done by averaging the V2F measurement likelihood by the vehicle state uncertainty. In doing so, the detected feature density under association A becomes independent of the vehicle state x leading to the approximation (31), where the approximated updated feature set density f˜d,A (F d \|Z F ) is given in Appendix D. The updated joint vehicle-feature density is approximated as X f (x, F \|Z F ) ∝ fˆu (F u ) F u ]F d =F X A A w p (x\|Z F )f˜d,A (F d \|Z F ). × (36) A∈A In this form, the vehicle state PDF is now independent on the feature RFS, and so is the feature density on the vehicle state. This allows to state the weights wA , which are given in Appendix E. Note, (36) is a Poisson multi-Bernoulli mixture (PMBM) density, where each DA A ∈ A denotes a hypothesis on the posterior vehicle state x and the detected feature state F d , weighted by wA . It can be reduced to a PMB density using, e.g., the variational approximation presented in [36], or based on the marginal DA probabilities [15]. We apply the latter approach, where for the reduction of (36) to the form of (16), the track-oriented marginal MeMBer/Poisson (TOMB/P) algorithm (c.f. [15]) is used. This results in a single hypothesis per detected feature described by a Bernoulli process (c.f. (3)), and per vehicle described by its PDF; as well as the intensity of undetected feature described by a PPP (c.f. (5)). This means, the summation over the DA space A has vanished in (36), retaining the form of (16). C. Multi-scan scenario using multiple vehicles with uncertain state (33) (34) Here, (33) is the expected probability of detection for an undetected feature under the predictive distributions for x and f , and (34) is the expected probability of detection for a detected feature. An alternative (and stronger) R approximation for (33) would be p̂uD = pD (x̂, f̂ ) with x̂ = xp+ (x)dx and f̂ the estimated feature state (c.f. Section II-B), and similarly Up to this point, we discussed PMB filtering with a single vehicle and uncertain state, where GNSS and V2F measurements are used. To achieve feature tracking as described in Section I-A, where sensors are mounted on several vehicles, we have to consider the multisensor case. Furthermore, depending on the infrastructure, sensors are time synchronized, i.e., take measurements at the same time step t or are not synchronized, i.e., measurements from a sensor arrives timestamped, but the time the sensor acquires the measurement is independent of other sensors. Let there be a single RFS F 7 modeling the features state, S = \|S\| vehicles with uncertain vehicle state xs with s = 1, 2, . . . S. The set of vehicles taking a measurement at time step t is given by C ⊆ S, where each vehicle c ∈ C provides a vector with a GNSS measurement z c and V2F measurements Z F c . Furthermore, T T T T T T T let x̃ = [xT 1 , x2 , . . . , x\|S\| ] , x̂ = [x1 , x2 , . . . , x\|C\| ] , ẑ G = S F F T T T [z G T 1 , z G 2 , . . . , z G \|C\| ] , and Ẑ = c∈C Z c . In the multisensor case, the PMB filter with uncertain sensor state follows the unisensor case proposed in Section IV: the joint vehicle-feature density is predicted, and updated by the GNSS and the V2F measurements. Thereby, the predicted vehicle-feature density (19) becomes f+ (x̃, F ) = f+ (F )p+ (x̃). (37) Updating the joint density (19) by the GNSS measurement results in (28). In the multisensor case, it becomes f (x̃, F \|ẑ G ) = f+ (F )p(x̃\|ẑ G ). F u ]F d =F X F F wA pA (x̃\|Ẑ )f˜d,A (F d \|Ẑ ), (39) A∈A where F F pA (x̃\|Ẑ ) ∝ p+ (x̃)`A (Ẑ \|x̂), In order to obtain a low complexity implementation, we describe the vehicle state PDF by a Gaussian with PDF p(x) , N (µ, Σ) with mean parameter µ and covariance matrix Σ. Similarly, the RFS density of a MB process of feature i is described by a Bernoulli random variable ri and a Gaussian PDF p(f i ) , N (µf , Σf ) with parameters µf and Σf . Under this description and the system models of Section III, we can express the prediction and update steps of the proposed filter in closed form with low computational complexity, whose steps are described next. 1) Prediction Step: The predicted vehicle state PDF of (19) is (41) p+ (x) = N (µx+ , Σx+ ), where with the use of (7) µx+ = Aµx− , (38) After incorporating the GNSS measurements we proceed with incorporating the V2F measurements. In the unisensor case, this resulted in the approximated joint sensor-feature density (36). In the multisensor case, this density becomes X F f (x̃, F \|Ẑ ) ∝ fˆu (F u ) × D. Gaussian Density Approximation Σx+ = AΣx− A + W . (43) The notation above means that if at time step t − 1 the vehicle state PDF p− (x) = N (µx− , Σx− ), then at time step t, the predicted state PDF (before updating by a measurement) is p+ (x) = N (µx+ , Σx+ ). The intensity of undetected features (22) is modeled by GM consisting of newborn features with weight λb (f ) = λb and PDF p(f ) = N (µb , Σb ), where µb andΣb are the birth mean and covariance matrix; and undetected features survived to the current time step with prior parameters {λu , µf − , Σf − } and predicted parameters (40) F and f˜d,A (F d \|Ẑ ) involves the marginalization over the vehicle prior PDF p+ (x̂), i.e., containing only vehicles which provide a V2F measurement, according to (70). Note that here (38) is used as prior in the joint sensor-feature density (39). Furthermore, the space A of all DA increases with an increase in the number of communicating sensors \|C\| for the predicted MB. In terms of complexity, this increase can be significant, because of the increase of possible feature stateto-measurement associations. Remark 4: Several different approaches exist to tackle this DA problem in a tractable manner. For instance, by employing sequential sensor-by-sensor measurement updates on (36), or by performing variational inference [37], or by solving the DA in parallel on a sensor-by-sensor basis [21]. Here, we employ the sequential measurement update strategy to limit the size of the DA space A. In doing so, subsequent sensors will benefit from updated vehicle and feature information of preceding sensors. In our application example (c.f. Section I-A), this means that an update of the joint vehicle-feature density, with measurements from a well localized vehicle (certain vehicle state), results in an improvement of feature tracking performance when prior information on the features is low. An update of the joint vehicle-feature density, with measurements from a poorly localized vehicle (uncertain vehicle state), allows to reduce the uncertainty of its own vehicle state when prior information on the features is high. (42) T λu+ = hpS , λu i , µf + = Bµf − , (44) (45) Σf + = BΣf − B T + V . (46) The predicted MB density of detected features, stated in (21), i predicted using has single feature Bernoulli parameters r+ i (26), and single feature PDF p+ (f ) calculated similarly to (45) and (46). 2) Update step: The joint vehicle-feature state density (28) is computed by updating the predicted vehicle-feature density (19) with the GNSS measurement z G through the Kalman update step [33], [35], where the vehicle state PDF is given by p(x\|z G ) = N (µx , Σx ). (47) Here, µx = µx+ + K (z G − H G µx+ ), (48) Σx = Σx+ − K H G Σx+ , (49) T K = Σx+ H G S , (50) T S = H G Σx+ H G + R. (51) The matrices H G and R are defined in (9). The updated joint vehicle-feature density (36) is computed by updating the predicted vehicle-feature density (19) with the V2F measurement Z F . Note that depending on the time difference between the GNSS and V2F measurements, (28) may be used instead of (19) as prior on the joint vehicle-feature 8 density. In order to calculate (36), the vehicle measurementstate likelihood `(z\|x) used in (64) and (65), and the feature measurement-state likelihood `(z\|f ) used in (74) are needed. The measurement-feature state likelihood (74) for z given, can be written in terms of f in closed form by `(z\|f ) ∝ N (µz\|f , Σz\|f ) (52) with −1 T T H 2, Σ−1 z\|f = H 2 Q + H 1 Σx+ H 1 µz\|f = H + 2 z − H 1 µx+ . (53) (54) Here, p+ (x) = N (µx+ , Σx+ ), and (·)+ denotes the MoorePenrose pseudo inverse. Proof: See Appendix F. The vehicle measurement-state likelihood (64) for a given z and marginalized over a detected feature, and in (65) marginalized over an undetected feature, can be written in terms of x in closed form by `(z\|x) = N (µz\|x , Σz\|x ) (55) described by a Bernoulli component (c.f. (3)) with Gaussian PDF, has a memory footprint of a Bytes needed to store {r, f , cov(f )}, where f ∈ RNf . Storing the vehicle state requires b Bytes, where the state of each vehicle is described by a Gaussian PDF with parameters {x, cov(x)} with x ∈ RNx . Without pruning of low-probability Bernoulli components in F , the memory footprint of the proposed filter is a\|F \| + b Bytes. In the multisensor approach of Section IV-C, the RSU receives GNSS measurement z G and V2F measurement Z F from a vehicle and then performs the filter update computation. The RSU broadcasts the vehicle state estimates either whenever a new measurement has been processed, or based on a fixed schedule. Should information of detected (and tracked) features be required at the vehicles, then the single-feature pdfs need to be transmitted as well (c.f. Section II-B). V. N UMERICAL R ESULTS We consider a scenario similar to the one outlined in Fig.1, where we apply the proposed multifeature-multisensor state tracking filter presented in Section IV. with −1 T T H 1, Q + H Σ H Σ−1 = H 2 f 2 1 z\|x + µz\|x = H + 1 z − H 2 µf + . (56) (57) Here, (55) equals (64) for feature PDF p(f ) , pi+C (f ) = u N (µf + , Σf + ), and in (65) for feature PDF p(f ) , D+ (f ) = N (µf + , Σf + ). Proof: The proof is analogous to the proof of the feature measurement-state likelihood (52) with the only difference that the unknown is x instead of f . E. Computational complexity, memory footprint, and communication demand Computational complexity is dominated by the matrix inversion needed to update the feature and vehicle densities, and the measurement-to-feature state DA. Updating the joint vehiclefeature density f (x, F \|z G ) using GNSS measurement z G requires a matrix inversion which scales as O(Nx3 ). The update of an MB component in f˜d,A (F \|x, Z F ) by V2F measurement z mC ∈ Z F scales as O(Nf3 ), and consequently as O(\|Z F \|Nf3 ) for the whole measurement set. Computational complexity of DA is O(\|F \|\|Z F \|) [38]. Hence, the update of the joint vehiclefeature density (36) by V2F measurement set Z F scales as O(\|F \|\|Z F \| + Nf3 \|Z F \| + Nx3 \|Z F \|), where the last term comes from vehicle state update of Z F . In each time step t, the size of undetected features RFS F u increases by \|Db (f )\| new born targets. The number of existing feature-tracks increases by \|Z F \|, a new feature-track per measurement [15], using a Bernoulli component per track. For each existing feature-track \|Z F \| + 1 hypotheses (plus one for a missed detection) are computed. The TOMB/P algorithm reduces each feature-track to a single hypothesis track. Pruning of feature-tracks with low probability of existence r allows to keep the number of feature-tracks tractable. Each hypothesis, A. Setup The state of a vehicle at time step t is x = [pT , v T ]T with position p ∈ R2 and velocity v ∈ R2 . Vehicle dynamics follow a linear constant velocity (CV) model described by (7) with 1 Ts A= ⊗ I 2, (58) 0 1 where Ts = 0.5 s, and W =r Ts3 /3 Ts2 /2 Ts2 /2 Ts ⊗ I2 (59) with r = 0.05 m2 , and ⊗ denoting the Kronecker product. The state of a feature at time t, denoted f ∈ R4 , is comprised of Cartesian position and velocity, similar to the vehicle state x. There are maximal five features present, if not noted otherwise. Furthermore, feature dynamics follow the CV model with the same parameters used for the vehicles. To generate a challenging scenario for DA, we initialize the feature states f ∼ N (0, 0.25I 4 ) at t = 175 for all features and run the CV model forward and backward in time similar to [15, Sec. VI]. The first feature enters the scene after t = 0, the second after t = 20 and so on. Once present, features stay alive for the remaining simulation time. Vehicle and feature trajectories are shown in Fig. 2. The observation matrix of the GNSS measurement model (9) is H G = [1 0] ⊗ I 2 , (60) 2 where R = σG I 2 . For vehicle 1, we assume it has low 2 location uncertainty with σG = 0.1 m2 and for vehicle 2 2 high location uncertainty with σG = 10 m2 , corresponding to a vehicle with high quality GNSS receiver and one with low quality GNSS receiver. In the single sensor case, only vehicle 1 is present, and in the multisensor case both vehicles are present, if not noted otherwise. The V2F measurement model follows (11), where H 1 , H G , H 2 , −H G and 9 200 y dimension in m 100 vehicle feature 0 −100 −200 −300 −500 −400 −300 −200 −100 x dimension in m 0 100 Fig. 2. Vehicle and feature trajectories. Fig. 3. V2F measurements in x (top panel) and y dimension (bottom panel) for each time step t. 2 Q = σV2F I 2 . In Fig. 3, V2F measurements are shown for each time step t including clutter measurements. Following u [15], we set the initial undetected feature intensity to D− (f ) = 2 2 T 10N (0, P ), where P = diag([100 , 1, 100 , 1] ) to cover the ranges of interest of the feature state. The feature birth intensity is set to Db (f ) = 0.05N (0, P ), the average number of false alarms per scan to λc = 20, with uniform spatial distribution on [−rmax , rmax ] with parameter rmax = 1000 m. Furthermore, the probability of survival is pS = 0.7 and the probability of detection is pD (x, f ) , pD = 1. To asses feature tracking performance, we use the optimal sub-pattern assignment (OSPA) metric with cut-off parameter c = 20 m and order p = 2 [39]. The vehicle tracking performance is assessed in terms of the root mean square error (RMSE). B. Discussion First, we discuss the impact of an uncertain vehicle state on feature tracking performance using a single vehicle and multiple-features. After that, we consider the multitarget- multisensor case from Section IV-C, and show scaling results in terms of numbers of features and vehicles tracked. 1) Impact of uncertain vehicle state on feature tracking performance: The features and vehicle trajectories are outlined in Fig. 2 with V2F measurements in Fig. 3. In Fig. 4, the feature state OSPA is plot for each time step t. We observe that there are peaks with a high OSPA value when a new feature enters the scene. These peaks are due to a cardinality mismatch between the feature RFS estimate and the true feature set. Furthermore, there is a high OSPA value around time step t = 175. At this point in time, features are closely spaced 2 together w.r.t. the V2F measurement variance σV2F resulting in a challenging scenario for measurement-to-state DA. In Fig. 5 the cardinality of the feature RFS is plotted over time. Around time step t = 175, the filter overestimates the feature RFS cardinality, which may be caused by clutter measurements. Note, different Monte-Carlo (MC) realizations produce a slightly different outcome and feature OSPA value, but with the same tendency. This behavior w.r.t. feature appearance and the effect when they are spatially close agrees with the findings in [15] for a known sensor (vehicle) state. Furthermore, we observe in Fig. 4, that the OSPA is low for time steps where features are spatially separated and already present in the scene. Then the MTT filter is able to produce feature estimates with low error. In Fig. 6, the feature state OSPA averaged over all 351 time steps is plot for different values of GNSS measurement 2 . The increase of GNSS measurement variance variance σG leads to an increased vehicle state uncertainty with the effect of an increase of the average feature OSPA. This OSPA increase consists of two components. First, an increased feature state 2 estimation error due to a higher value of σG . Second, this results in features staying spatially close together w.r.t. the 2 2 together) and σV2F feature state measurement uncertainty (σG for a longer period of time around time step t = 175. Hence, DA is more challenging with the effect of an increased feature OSPA in this regime. In the same figure, the average feature OSPA without modeling the present vehicle state uncertainty is plotted using the conventional PMB filter [15]. We observe that not modeling the present vehicle state uncertainty has a negative effect on feature tracking performance. In Fig. 7, the average feature state OSPA is plotted for 2 different values of V2F measurement noise variance σV2F . We observe, that a higher V2F noise variance leads to an increased OSPA value. This is because the single feature state estimation error increases and DA becomes more challenging. Note, the results of Fig. 6 and Fig. 7 are averages over 10 MC realizations. 2) Multitarget-multisensor tracking performance: The RMSE of the vehicle state is plotted for each time step t for 2 small) in Fig. 8, vehicle 1 with low location uncertainty (σG 2 and in Fig. 9 for vehicle 2 with high location uncertainty (σG high). As a benchmark, results from a centralized Kalman filter (KF) are plot as well, where measurement-to-feature DA is known and where the augmented state vector contains all vehicle and all feature states. Furthermore, the tracking performance using a local KF is plot. The local KF performs filtering only on the individual vehicle state separately using 20 4 15 3 RMSE Feature OSPA 10 10 0 50 100 150 200 time-step t 250 300 0 350 2 = 10−3 m2 and σ 2 2 Fig. 4. Feature OSPA with σG V2F = 0.25 m . cardinality 4 100 150 200 time-step t 250 300 350 0.8 2 0 50 100 150 200 time-step t 250 300 CDF 0.6 350 0.2 0 proposed conventional 15 Vehicle Vehicle Vehicle Vehicle Vehicle Vehicle 0.4 20 Avg. feature OSPA 50 1 estimate true 2 = 10−3 m2 and σ 2 2 Fig. 5. Feature cardinality with σG V2F = 0.25 m . 10 0 0.5 1 1, 1, 1, 2, 2, 2, proposed local KF central KF (known DA) proposed local KF central KF (known DA) 1.5 RMSE in m 2 2.5 3 Fig. 10. CDF plot of vehicle state RMSE. 5 0 2 4 6 8 10 2 σ G in m2 Fig. 6. Average feature OSPA for different values of GNSS measurement 2 . The V2F noise variance is set to σ 2 2 variance σG V2F = 0.25 m . Avg. feature OSPA 4 3 2 1 0 0.2 0.4 2 σ V2F 0.6 in 0.8 1 m2 Fig. 7. Average feature OSPA for different values of V2F measurement 2 2 = 10−3 m2 . variance σV2F . The GNSS noise variance is set to σG 1 local KF central KF (known DA) proposed 0.8 RMSE 0 Fig. 9. Vehicle state RMSE of vehicle 2. 6 0 2 1 5 0 local KF central KF (known DA) proposed 0.6 0.4 0.2 0 0 50 100 150 200 time-step t Fig. 8. Vehicle state RMSE of vehicle 1. 250 300 350 only GNSS measurements and does not estimate feature states. Note, the performance of the local KF can be considered as the worst-case performance on vehicle state estimation, since V2F measurements are not considered at all. We observe from Fig. 8, that for vehicle 1, which has low GNSS measurement noise, all three filter methods deliver a similar performance. The reason for this is that, due to the high accuracy of GNSS measurements, not a lot of information (to improve the vehicle state) is provided from feature tracking, i.e., feature tracking error is high w.r.t. the vehicle state tracking error of vehicle 1 (after updating by the GNSS measurement). In Fig. 10, the cumulative distribution function (CDF) of the RMSE is plot. Here, the low RMSE of vehicle 1 using the three different filters can be observed as well. Moving the focus to vehicle 2, we observe that the RMSE of the local KF is much higher compared to the central KF, which is caused by the high noise in the GNSS measurements. Due to the low RMSE of vehicle 1’s state there is relevant position information in the system, which can be transfered from vehicle 1 to vehicle 2 via the features utilizing the V2F measurements. In 80% of all cases, the RMSE of vehicle 2 is below 0.5 m with the proposed filter, compared to 1.4 m with the local KF. Despite this great improvement of the proposed filter over the local KF, it does not achieve the performance of the central KF, where the RMSE is below 0.3 m. The reason for the difference is that the central KF has knowledge of the correct DA, knows the true number of features present, and ignores clutter V2F measurements. Furthermore, it tracks any present correlations between features and vehicles not modeled 11 0.1 time in s 8 · 10−2 6 · 10−2 4 · 10−2 2 · 10−2 0 0 5 10 No. of features 15 20 Fig. 11. Average computation time per time step for different number of present features. The number of vehicles is set to two. time in s 0.2 0.15 0.1 5 · 10−2 0 0 5 10 No. of vehicles 15 be incorporated either in a time-synchronized manner where sensor measurements are aggregated in a super-sensor state, or in a non time-synchronized manner through asynchronous update steps, executed whenever sensor measurements arrive at the central node. Simulation results showed for a unisensormultitarget tracking scenario with known sensor state that tracking performance assessed by the OSPA distance metric is equivalent to William’s PMB filter. In a scenario with present vehicle state uncertainty, the proposed filter showed superior feature tracking performance over the conventional PMB filter due to the modeling of this type of additional uncertainty. In a multisensor-multitarget tracking scenario, feature information from a well localized vehicle (sensor) allows to significantly reduce the vehicle state uncertainty of (previously) poor localized vehicles. This improvement is possible through joint observation of a subset of the present features and is supported by simulation results. 20 A PPENDIX A. Vehicle State Posterior Fig. 12. Average computation time per time step for different number of vehicles. The number of features is set to five. by the proposed filter. The proposed filter needs to infer the measurement-to-feature DA, estimate the number of features currently present, and needs to appropriately handle clutter in the V2F measurement set Z F . 3) Scaling results: In Fig. 11, the average computation time per time step t is plot for a simulation with two vehicles and different number of present features. We observe that computation time increases linearly as the number of present features increases. This scaling result is different to the PF based implementation of the PMB filter in [21] with a known vehicle state. There, the authors reported an exponential increase of computation time. In Fig. 12, the average computation time per time step t is plot for a simulation with five features and different number of vehicles. Here, computation time increases linearly as the number of vehicles increases. Furthermore, we investigated the average computation time per time step t for 2 different values of the GNSS measurement variance σG and 2 of the V2F measurement variance σV2F . In the simulation scenario with five features and two vehicles, the average computation time remained constant around 2.4 · 10−2 s for 2 10−3 m2 ≤ σG ≤ 10 m2 . For a V2F measurement variance −3 2 2 10 m ≤ σV2F ≤ 10 m2 , the average computation time 2 linearly increased from 2 · 10−2 s to 3 · 10−2 s as σV2F increased. VI. C ONCLUSIONS This paper presented a Poisson multi-Bernoulli filter for multisensor-multitarget tracking with uncertain sensor states. Two different kind of measurements, observations of the sensor state and observations of the features, were used to obtain accurate feature and sensor state tracking. The proposed parametric filter implementation scales linearly with the number of features or sensors. Information from multiple sensors can The vehicle state posterior of (29) and (36) is proportional to the vehicle’s prior PDF p+ (x) times the measurement likelihood `A (Z F \|x) as pA (x\|Z F ) ∝ p+ (x)`A (Z F \|x), (61) where A Y Z F ` (Z \|x) ∝ `(z mC \|x, f )pi+C (f )df C∈A: C∩I6=∅ C∩M6=∅ f ∈F d × Y Z u `(z mC \|x, f )D+ (f )df (62) C∈A: C∩I=∅ C∩M6=∅ f C ∈F u ∝ Y `iC (z mC \|x) C∈A: C∩I6=∅ C∩M6=∅ f ∈F d Y `u (z mC \|x) (63) C∈A: C∩I=∅ C∩M6=∅ f ∈F u with Z iC ` (z\|x) = `u (z\|x) = Z `(z\|x, f )pi+C (f )df , (64) u `(z\|x, f )D+ (f )df . (65) Here, (64) and (65) map the feature uncertainty on the V2F measurement likelihood. B. Updated Undetected and Detected Feature Density The updated joint vehicle-feature density (29) has undetected feature density Y u f u (F u \|x) ∝ (1 − pD (x, f ))D+ (f ) (66) f ∈F u 12 where we average over the predicted vehicle state and 0 marginalize over all subsets F C not equal to F C . Eqn. (70) has Bernoulli parameters [15, Eqns. (45) to (57)]  i i (1−p̂DC )r+C  C ∩ I 6= ∅, C ∩ M = ∅  iC iC  1−p̂D r+ F iC r̃ \|Z = 1 if C ∩ I 6= 0 C ∩ M 6= 0 u  ,D+ p̂u i  D h`z mC E D  C∩I=0 C ∩ M 6= 0, m u u and detected feature density f d,A ∝ d F (F \|x, Z ) X ]C∈A F C =F Y × d Y iC (F C ) η(∅\|x, F C )f+ C∈A: C∩I6=∅ C∩M=∅ iC (F C ) η({z mC }\|x, F C )f+ λc (z C∈A: C∩I6=∅ C∩M6=∅ D `zmC ,D+ (72) Y × C )+p̂ u (F C ). η({z mC }\|x, F C )f+ (67) C∈A: C∩I=∅ C∩M6=∅ p̃iC (f \|Z F ) =  i p+C (f )     `zmC (f )pi+C (f ) Here, the first product considers the cases where no measurement is associated to any of the existing features, the second product considers the cases where a measurement is associated to an existing feature, and the last line considers the case where a measurement is associated to the background (clutter or undetected feature). E i `zmC ,p+C u `zmC (f )D+ (f ) E D u `zmC ,D+ D     if C ∩ I 6= ∅, C∩M=∅ C ∩ I 6= 0 C ∩ M 6= 0 C∩I=0 C ∩ M 6= 0, (73) where Z `z (f ) = `(z\|f ) = `(z\|x, f )p+ (x)dx. (74) E. Approximated Updated Feature Density Weights C. Updated Undetected and Detected Feature Density Approximation The joint vehicle-feature density approximation (35) has undetected feature density Y fˆu (F u ) ∝ (1 − p̂uD )D+ (f ) (68) The weight of a global association hypothesis A ∈ A stated in (36) is [15, Eqn. (67)] Y Y iC iC iC iC p̂D `zmC , pi+C p̂D ) r+ wA ∝ (1 − r+ C∈A: C∩I6=∅ C∩M=∅ f ∈F u × ]C∈A F × C =F Y d Y iC (1 − p̂uD ∆\|F C \| )f+ (F C ) Here, we proof (52). With (11) and p+ (x) = N (µx+ , Σx+ ) known, define iC η({z mC }\|x, F C )f+ (F C ) Y (75) F. Proof of Measurement-Feature State Likelihood C∈A: C∩I6=∅ C∩M=∅ C∈A: C∩I6=∅ C∩M6=∅ × u ). (λc (z mC ) + p̂uD `zmC , D+ C∈A: C∩I=∅ C∩M6=∅ and detected feature density fˆd,A (F d \|x, Z F ) X ∝ Y C∈A: C∩I6=∅ C∩M6=∅ y , z − H 1 x − q, (76) p(y) = N (z − H 1 µx+ , H 1 Σx+ H T 1 ). (77) and consequently u η({z mC }\|x, F C )f+ (F C ), (69) C∈A: C∩I=∅ C∩M6=∅ C where ∆\|F C \| = 1 for F 6= ∅, and zero otherwise. 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Implementation of Tetris as a Model Counter∗ Atri Rudra University at Buffalo, SUNY [email protected] arXiv:1701.07473v1 [cs.DS] 25 Jan 2017 Jimmy Dobler University at Buffalo, SUNY [email protected] Abstract Solving #SAT problems is an important area of work. In this paper, we discuss implementing Tetris, an algorithm originally designed for handling natural joins, as an exact model counter for the #SAT problem. Tetris uses a simple geometric framework, yet manages to achieve the fractional hypertree-width bound. Its design allows it to handle complex problems involving extremely large numbers of clauses on which other state-of-the-art model counters do not perform well, yet still performs strongly on standard SAT benchmarks. We have achieved the following objectives. First, we have found a natural set of model counting benchmarks on which Tetris outperforms other model counters. Second, we have constructed a data structure capable of efficiently handling and caching all of the data Tetris needs to work on over the course of the algorithm. Third, we have modified Tetris in order to move from a theoretical, asymptotic-time-focused environment to one that performs well in practice. In particular, we have managed to produce results keeping us within a single order of magnitude as compared to other solvers on most benchmarks, and outperform those solvers by multiple orders of magnitude on others. ∗ This research was supported in part by grant# NSF CCF-1319402. 1 1 Introduction #SAT is the prototypical #P-complete problem. #SAT (as well as its NP-complete cousin SAT) are not only of great interest in computational complexity but by their completeness turn out to be a great tool to model a wide host of practical problems. This has led to an explosion of SAT solvers that try to solve practical instances of #SAT or SAT by exploiting structure in these instances. For this paper, we will assume the importance of designing #SAT and SAT solvers as a given. We refer the reader to the book chapters by Gomes, Sabharwal and Selman on #SAT solvers (also known as model counters) [15] and by Gomes et al. on SAT solvers [14] for more details. A common technique is the DPLL procedure, a depth-first search procedure where the algorithm makes guesses on the assignments one variable at a time, determines at each stage whether or not this produces a conflict, and uses that information to learn new clauses and get closer to finding the satisfying assignment [17]. Recently in the database literature, the work of Abo Khamis et al. [6] connected the DPLL procedure to computing natural joins. In particular, they presented the Tetris algorithm, which computes the natural join with beyond worst-case theoretical guarantees. As a special case, Tetris also recovers some of the recent worst-case optimal join results [24, 30, 25]. Abo Khamis et al. then showed that Tetris is an DPLL procedure and pointed out how one of the main step in their algorithm is exactly the resolution step that is ubiquitous in DPLL-based SAT solvers. Given the close ties of SAT solvers to DPLL, they left open the following intriguing possibility: Can Tetris be implemented as a SAT solver or model counter that can compete with state-ofthe-art solvers? Our contributions Our main result in this paper is to show that Tetris can indeed be implemented as a model counter that is competitive with state-of-the-art model counters on actual datasets. While [6] presented a nice geometric framework to reason about algorithms to compute the natural join query, some of its simplicity arose from inefficiencies that matter when implementing Tetris as a model counter. Before we present the issues we tackle, we give a quick overview of Tetris. The fundamental idea is that, rather than working to create the output of a join directly, it instead attempts to rule out large sections of the cross product of the joined tables [6]. Initially, Tetris is given a set of sets whose union is the set of all incorrect solutions to the problem Tetris is solving. In other words, any solution to the problem must not be a member of this union. By efficiently querying this set of sets, and by adding to it intelligently at various times (such as by adding a new exclusion whenever an output point is found), Tetris is able to rule out increasingly large sets of potential solutions. Once it has ruled out all possible solutions, it terminates and outputs the list of solutions. We tackle the following three issues with the theoretical presentation of Tetris in [6]: 1. At any point, Tetris needs to keep track of the union of all the potential solutions it has ruled out. To do this, [6] used a simple trie data structure to keep track of the union. However, this loses some poly-logarithmic factors and proves detrimental to practical performance. To deal with this, we design a new data structure that essentially compresses consecutive layers in the traditional trie into one ‘mega layer.’ Inspired by the used of SIMD instructions by EmptyHeaded [5] (to speed up implementations of worst-case optimal join algorithms [25]), we set up the compression in a manner that lends itself to speedup via SIMD instructions. 2 2. The analysis of Tetris in [6] was for data complexity. This implies they could afford to use exponential (in the size of the join query) time algorithm to find an appropriate ordering in which to explore different variables. In #SAT instances, we can no longer assume that the number of variables is a constant, and hence we cannot obtain an optimal ordering using a brute force algorithm. We deal with this by designing heuristics that take the structure of Tetris into account. 3. As mentioned earlier, Tetris (like a DPLL procedure) performs a sequence of resolutions and theoretically, it can store the outcomes of all the resolutions it performs. However, for practical efficiency, we use a heuristic to decide which resolution results to cache and which ones to discard. Our experimental results are promising. On some natural #SAT benchmarks based on counting number of occurrences of small subgraphs in a large graph (which we created), our implementation of Tetris is at least two orders of magnitude (and in most cases more than three orders of magnitude) faster than the standard model counters (sharpSAT [28], Cachet [26] and dSharp [22]). We also compared Tetris with these model counters on standard SAT benchmarks, where Tetris was either comparable or at most 25x slower. Theoretical Implications While this paper deals with an experimental validation of the theoretical result from [6], we believe that it highlights certain theoretical questions that are worth investigating by the database community. We highlight some of our favorite ones that correspond to each of our three main contributions: 1. Extending Tetris beyond join queries. As our work has shown, Tetris can be used to solve problem beyond the original natural join computation. Recently, the worst-case optimal join algorithms were shown to be powerful enough to solve problems in host of other areas such as CSPs (of which MaxSAT is a prominent example), probabilistic graphical models and logic [7]. (Also see the followup work [18].) The beyond worst-case results in [6] have so far seemed more of a theoretical novelty. However, given that this paper demonstrates the viability of Tetris in practice, this work opens up the tantalizing possibility of extending the theoretical results of Tetris to problems captured by [7, 18]. Such a result even for MaxSAT would be of interest in practice. 2. Computing orderings efficiently. As mentioned earlier, the theoretical results for Tetris assumes that the required ordering among variables can be computed in exponential time. However, for applications in SAT (as well as other areas such as probabilistic graphical models, assuming the question in the item above can be answered), we need to compute orderings that are approximately good in polynomial time. Thus, a further avenue of theoretical investigation is to come up with a polynomial time algorithm to compute the ordering, and to prove some guarantees on the loss of performance from the case where Tetris has access to the ‘optimal’ ordering. Some of the heuristics developed in our paper might prove to be good starting points for this investigation. We would like to point that that the importance of efficiently computing variable orderings has been studied a lot in AI and database literature. Some of the very recent work on Generalized Hypertree Decompositions (which are well known to be equivalent to variable elimination orderings) could potentially be useful towards this goal [13]. 3. Time-space tradeoff. Recent results on worst-case optimal algorithms to compute natural joins [24, 30, 25] and to compute joins with functional dependencies [8] all focus exclusively on time complexity. However, as highlighted by our work, being more prudent with space usage in fact benefits 3 actual performance. This point was also indirectly highlighted in [6], where it was shown that resolution schemes that did not cache their intermediate results are strictly less powerful than those that do (in the context of computing the natural join). However, we believe that a systematic theoretical study of the tradeoff between time and space needed to compute the natural join is an attractive route to pursue. We will begin in Section 2 by introducing the fundamental concepts necessary to understand both SAT problems and details on Tetris itself, all while giving a hands-on example of how Tetris would handle a toy example. From there, we will move into Section 3, an in-depth analysis of our major contributions. Afterwards, we will continue with our experimental results in Section 4. Then we will discuss related work in the field in Section 5. 2 Background In this section, we will introduce the concepts necessary to understand how Tetris functions, introduce the concept of resolutions, and walk through how Tetris would handle a simple input. 2.1 SAT and Boxes We begin by defining several key terms and ideas. Recall that a SAT problem consists of a series of Boolean variables, x 1 , x 2 , ..., x n , joined together in a series of AND and OR clauses. Problems are generally presented in the conjunctive normal form (CNF), a simplification wherein the entire formula is written as a series of ANDs over a set of disjunctive clauses. One such example would be (x 1 ∨ x 2 ) ∧ (x 1 ∨ x¯2 ). A solution, or satisfying assignment, to a SAT problem is an assignment of true or false to each of the variables such that the Boolean formula is satisfied; that is, that all clauses are satisfied. Next, we will consider the idea of boxes, which is how our algorithm will interpret SAT problems. Each box is an n-dimensional structure in {0, 1}n , where n is the number of variables in the original SAT problem. We will define this set as the output space; that is, all potential outputs will be elements of this set. Each of our boxes exists within this hypercube, and along each dimension has the value 0, the value 1, or extends along the full length of the edge. The reason for this is simple: 0 corresponds to false, 1 to true, and the length of the edge to both. Henceforth, we will use λ to refer to edges with length 1. We thus form the following definition: Definition 1 (Box Notation). A box takes the form 〈b 1, b 2 , ..., b n 〉, where each b i ∈ {T, F, λ}. Observe that, from these definitions, we can consider every assignment to be a 0-dimensional box; this will be important later. Then, our goal will be to find the set of points within the output space that are not contained (see Definition 2) by any boxes. Any such point will be termed an output point, and the goal of an algorithm working on these boxes is to find all such output points. Definition 2 (Containment). A box b is said to contain another box c if, for all points p ∈ {0, 1}n such that p ∈ c, it is true that p ∈ b. Equivalently, the box 〈b 1, b 2 , ..., b n 〉 contains the box 〈c 1 , c 2 , ..., c n 〉 if, for all i , b i = c i or b i = λ. However, there is one key difference between these two representations. Each clause in a CNF formula is essentially a subproblem wherein at least one variable’s assignment must match its value in the 4 clause for the assignment to possibly be satisfying. But with boxes, the exact opposite is true: if an assignment matches the value for the boxes on all non-λ dimensions, we reject the assignment. In other words, if we consider a geometric visualization of these boxes, any and all assignments that fall within a box are rejected. Hence, our next step is to devise a means by which to convert any given SAT problem in CNF form to the boxes format that Tetris can understand. As follows from our above observation, the most important step is simply the negation of the CNF formula; the rest is all bookkeeping. For the exact algorithm, see Algorithm 1. Algorithm 1 Conversion from CNF to boxes 1: for each CNF clause do 2: Negate the clause 3: Set all x¯i to F, and all x j to T 4: Set all variables not present in the clause to λ. 5: Insert into the database {See Definition 3} Let us consider the following toy example CNF problem: Example 1. (x 1 ∨ x 2 ) ∧ (x 1 ∨ x¯2 ) ∧ (x 2 ∨ x 3 ). Our first step is to negate each clause, which will give us a disjunctive normal form (DNF) co-problem: (x¯1 ∧ x¯2 ) ∨ (x¯1 ∧ x 2 ) ∨ (x¯2 ∧ x¯3 ). Next, we will convert to boxes (by replacing a variable with T and its negation with F ) and add all missing variables (as λ). After conversion, our three clauses become 〈F, F, λ〉, 〈F, T, λ〉, and 〈λ, F, F 〉. 1 x2 1 0 1 x3 1 1 0 x1 (x 1 ∨ x 2 ), 〈F, F, λ〉 x2 1 1 x3 x2 0 x1 1 x3 1 x1 (x 1 ∨ x¯2 ), 〈F, T, λ〉 (x 2 ∨ x 3 ), 〈λ, F, F 〉 Figure 1: Our starting boxes, and the corresponding SAT clauses. While boxes are, technically speaking, strictly the corners of what we depict as the boxes, we depict them with the edges and surfaces drawn for the purpose of visual clarity. At this point, it is time to insert the boxes into our data structure. Let us list the fundamental operations the data structure must be able to perform: Definition 3 (Tetris data structure). The Tetris data structure shall be able to perform the following operations: 1) Insert, input is the box to be inserted, no output 2) Contains, input is the box we are seeing if the structure contains, output is the containing box (see Definition 2) 5 3) GetAllContainingBoxes, input is the box we are seeing if the structure contains, output is the set of all containing boxes We will return to the details of the data structure implementation in Section 3.1. 2.2 Resolution We then come to the concept of resolution, a key aspect of Tetris and most state-of-the-art SAT solvers. Resolution can be defined over both CNF clauses and boxes; let us begin with the former. Let us consider two clauses in our CNF Example 1 once again: specifically, (x 1 ∨x 2 ) and (x 1 ∨ x¯2 ). We see that these are two very similar clauses; they differ only in that the x 2 term is negated in one and not the other. Therefore, we can resolve these two clauses by removing the x 2 term and then taking the OR of all remaining variables. In this case, this gives us (x 1 ). We then remove the original two clauses from the CNF problem and insert this new clause in its place. This is a significant simplification. Similarly, we can resolve any two clauses such that there is exactly one pivot point, by which we mean a variable that appears in both clauses, but is negated in one and not the other. For instance, looking back at our example, we can also resolve (x 1 ∨ x¯2 ) with (x 2 ∨ x 3 ) to form (x 1 ∨ x 3 ) In this case, we would not be able to remove the original two clauses, but we would have gained information. Let us now formally define this process: Definition 4 (Resolution on Clauses). Two clauses (x i 1 ∨ x i 2 ∨...∨ x i m ∨ v) and (y j 1 ∨ y j 2 ∨...∨ y j l ∨ v̄), i ∈ I , j ∈ J , I , J ⊂ [n], can be resolved if and only if there exists exactly one variable v, the pivot point, such that v ∈ x and v̄ ∈ y. The resolution of the two clauses is (x i 1 ∨ x i 2 ∨ ... ∨ x i m ∨ y j 1 ∨ y j 2 ∨ ... ∨ y j m ). Since boxes are simply another representation of the same problem, it follows that resolution can be performed on boxes as well. First, we will require that there must exist exactly one variable on which one box is true and the other box is false.1 Call this the pivot variable. In the output, set this variable to λ. Then, for each other variable, if it is T in one or both boxes, set it to T in the output box; if it is F in one or both boxes, set it to F in the output box; and if both variables are λ, then the resolution of the two is also λ. We see two possible resolutions in our Example 1. The resolution of 〈F, F, λ〉 and 〈F, T, λ〉, two coplanar and parallel edges, is the square 〈F, λ, λ〉, as depicted in Figure 2, and the resolution of the askew edges 〈F, T, λ〉 and 〈λ, F, F 〉 is the edge 〈F, λ, F 〉, as depicted in Figure 3. For a formal definition of the resolution operator, henceforth ⊕, see Definition 5. 1 x2 1 0 1 x3 1 x2 1 0 x1 + 1 1 x3 x2 0 x1 = 1 x3 1 x1 Figure 2: The resolution of 〈F, F, λ〉 and 〈F, T, λ〉 on the vertex x 2 is the square 〈F, λ, λ〉. This is equivalent to (x 1 ∨ x 2 ) resolved with (x 1 ∨ x¯2 ) being the clause (x 1 ). 1 This is exactly analogous to the requirement that we resolve on a pivot point in the clause version. 6 1 x2 1 0 1 x3 1 x1 x2 1 0 + 1 1 x3 x1 x2 0 = 1 x3 1 x1 Figure 3: The resolution of 〈F, T, λ〉 and 〈λ, F, F 〉 on the vertex x 2 is the edge 〈F, λ, F 〉 This is equivalent to (x 1 ∨ x¯2 ) resolved with (x 2 ∨ x 3 ) being the clause (x 1 ∨ x 3 ). Definition 5 (Resolution on Boxes). Two boxes 〈b 1 , b 2 , ..., b n 〉 and 〈c 1 , c 2 , ..., c n 〉 can be resolved if and only if there exists exactly one i such that b i is true and c i is false, or vice-versa. In the resolved box a, a i = λ. Each a j , j 6= i is equal to b j ⊕ c j , where ⊕ is defined as follows: T ⊕T = T F ⊕F = F λ⊕T = T λ⊕F = F T ⊕λ = T F ⊕λ = F λ⊕λ = λ T ⊕ F is undefined. Observe that resolution on boxes and resolution on clauses are identical: Lemma 1. Resolution on boxes, with the additional restriction that exactly one variable must be true in one box and false in the other, is exactly equivalent to resolution on SAT clauses. Proof. Let (x i 1 ∨...∨x i m ) and (y j 1 ∨...∨ y j l ), i ∈ I , j ∈ J , w her e I and J ⊂ [n], be the clauses we are resolving. Assume WLOG that i 1 = j 1 is the pivot point. Then the resolved clause is (x i 2 ∨ ... ∨ x i m ∨ y j 2 ∨ ... ∨ y j l ). The boxes equivalent to our starting to clauses are 〈b 1 , ..., b n 〉 and 〈c 1 , ..., c n 〉, where b k = F if x k ∈ x, b k = T if x¯k ∈ x, and λ otherwise, and c k is defined similarly with respect to y. The resolution of these boxes a is then defined as 〈a 1, ..., a n 〉, where a k = λ if k = i 1 = j 1 , and a k = b k ⊕ c k otherwise. Now, let us calculate the box equivalent of the output of the clause-based resolution, t . It can be shown that t k = λ if k = i 1 = j 1 , t k = x k = y k if k ∈ I , k ∈ J and k 6= i 1 , t k = x k if k ∈ I and k ∉ J , t k = y k if k ∈ J and k ∉ I , and t k = λ if k ∉ I and k ∉ J . Inspection with the above definition of ⊕ reveals that t is exactly equivalent to a. Since the same problem with arbitrary, equivalent inputs produced equivalent outputs, the two operations must be equivalent. Tetris introduces one additional restriction on resolution. Definition 6 (Resolution on Boxes in Tetris). Two boxes b and c can be resolved if and only if there is exactly one spot i such that b i = T and c i = F, or vice-versa, and for all j > i , b j = c j = λ. In other words, we will demand that that the pivot variable be the final non-λ variable. Therefore, while Tetris will perform the resolution of 〈F, F, λ〉 and 〈F, T, λ〉 (Figure 2), it will not perform the resolution of 〈F, T, λ〉 and 〈λ, F, F 〉 (Figure 3). We see, then, that the ordering of the variables determines whether or 7 not a resolution is even possible. This makes determining the global ordering of the variables a key issue, as mentioned earlier as Theoretical Implication 2, which we will address later in Section 3.2. In general, Tetris performs resolution on pairs of recently found boxes. Let k be the location of the last non-λ variable in a box b. Then b k must be either true or false. If it is false, we will store the box for future use. If it is true, then we will take this box b and resolve it with the stored box with the same value for k whose last non-λ variable was false. By doing so, we will guarantee the production of a box where the last n −k +1 variables have the value λ. For more details on how this works, along with the reasoning for why such pairs can always be found, see Section 2.3. 2.3 Tetris For now, let us return to our Example 1. When we last left off, we were just inserting the three clauses into our data structure, which was loosely defined in Definition 3. For a formal definition of the database and details for how it allows the set of boxes Tetris knows about to be quickly and efficiently queried, see Section 3.1; for now, one can simply assume it to be a trie-based structure. Additionally, we can resolve the first two boxes while leaving the third untouched, as in Figure 2; therefore, the database will contain exactly the boxes 〈F, λ, λ〉 and 〈λ, F, F 〉 (see Figure 4). Furthermore, we will prepare an empty array of boxes L of size n, which will be used later. The purpose of this array is to store and retrieve boxes that we wish to resolve with other boxes. Now that we have our database established, it is time to perform Tetris proper. The basic idea here is very simple. We will pick a point P in the output space, which we will call the probe point; recall that this point is itself a 0-dimensional box. We then determine whether or not any box in the database contains this point. If one does, we will store this box in an additional data structure referred to as the cache, which functions identically to the main database, and probe a new point. If no box contains P , we will list the point as a solution and furthermore add this point into the cache. Along the way, we will perform resolution in order to create new and larger boxes. This process continues until the entirety of the output space is covered by a single box, at which point we must have found every output point and are done. Algorithm 3 has the details. It should be noted that this algorithm was originally presented recursively in [6]; here, we present it iteratively both for the purposes of speed and because this allows for non-chronological backtracking; in other words, we can backtrack more than one layer at a time. Algorithm 2 Advance(box b, probe point &p) (note: p is a global variable) 1: while b Contains p do 2: if the last non-λ variable of p is F then 3: Set that variable to T {Return to the previous branching point and take the right, or true, branch} 4: else 5: while the last non-λ variable of p is T do 6: Set that variable to λ {Return to the most recent level where we branched left} 7: Set the last non-λ variable of p to T {Branch right here} 8: Replace all λs after this variable with F {Repeatedly branch left} Now, let us consider how this algorithm behaves with regards to our earlier Example 1. We pick as our first probe point 〈F, F, F 〉, giving us the situation illustrated in Figure 4. We first scan our local cache, C , for any boxes that contain this point; however, since this is the first probe point, the cache is trivially empty. Next, we scan the database D. D contains all the boxes corresponding to the clauses in the original SAT 8 Algorithm 3 General Tetris for SAT 1: Establish variable ordering 2: Build the database D using Algorithm 1 3: C ← ; 4: L ← An empty array of size n {This array is implicit in [6]} 5: p ← 〈F, F, ..., F 〉 6: while 〈λ, λ, ..., λ〉 ∉ C do 7: if (b ← C .Contains(p)) is nonempty then 8: Advance(b, p) {Advance (see Algorithm 2) the probe point past b} 9: else if (A ← D.GetAllContainingBoxes(p)) is nonempty then 10: for all boxes b ∈ A do 11: C .Insert(b) 12: Advance(b, p) {Advance the probe point past b} 13: else 14: Add p to the output{There is no containing box, so p is an output point} 15: C .Insert(p) 16: Advance(p, p) {Advance the probe point past itself} 17: k ← the location of the last non-λ variable in b w.r.t. the variable ordering 18: if b k = F then 19: L[k] = b {Store the most recent left-branching box for a given depth} 20: else 21: r ← b ⊕ L[k] {Resolve this right-branching box with the corresponding left-branching box} 22: C .Insert(r ) problem. The database just so happens to contain two containing boxes; for reasons that will become clear shortly, the operation will choose to output 〈F, λ, λ〉. We insert the box 〈F, λ, λ〉 into C . Our next task is to advance the probe point until it lies beyond our box. To do this, we proceed according to Algorithm 2. The idea is to think of the set of all possible probe points as a tree that we are performing a depth-first search on, with F representing left-branching paths and T representing rightbranching paths. We will continue along this depth-first search until we find a point not covered by the most recently discovered box. This takes us to 〈T, F, F 〉. Note that if the database had fetched 〈F, F, λ〉, we would not have been able to advance the probe point as far. Finally, we insert our containing box into the array L at location 1, since only the first variable is nonλ, for future use. We know to do insertion here, rather than trying to resolve with a non-existent box, because the value of that first non-λ variable is false. This takes us to the situation depicted in Figure 5. Again we scan C , this time for the probe point 〈T, F, F 〉, and again we find no containing box in C . So we scan D once again and find the containing box 〈λ, F, F 〉. We then insert this box into the cache and advance the probe point to 〈T, F, T 〉. This time, although our containing box features a λ at the first location, we determine the location in L into which we will insert based on the location of the last non-λ variable, so we insert it into L[3]. This time, we find no containing boxes in either the cache or the database. Therefore, we have found an output point (see Figure 6 for an illustration). We add 〈T, F, T 〉 to our output set, then add the box 〈T, F, T 〉 to our cache, which marks the point as found. At this juncture, we find that the last non-λ variable is at location 3, but this time, it is true. Therefore, 9 1 x2 1 0 1 1 x3 Probe Point p Cache C Database D x2 1 0 x1 1 1 x3 x2 0 x1 Probe Point Map 1 1 x3 x1 Figure 4: The initial state of the database D, cache C , and the location of the first probe point p, which will search the output space in a depth-first manner that can be tracked using the map on its right. D is simply the union of all the boxes we created from the initial SAT problem, while p is set to an initial value of 〈F, F, F 〉. L is currently empty. 1 x2 1 0 1 x3 1 Probe Point p Cache C Database D x2 1 0 x1 1 x3 1 x2 0 x1 1 x3 Probe Point Map 1 x1 Figure 5: The state of the database, cache, and probe point after the first round of the algorithm. Our probe point found the box 〈F, λ, λ〉, so it added this box to C . Then, p was advanced until it reached a point not contained by this box, which turned out to be 〈T, F, F 〉. L(1) is the box 〈F, λ, λ〉, which is the box corresponding to the orange vertex in the map; the array is empty elsewhere. we will extract the same-length box we stored previously at L[3] and resolve it with this box. We know that this will be a legal resolution because we are scanning the output space in a tree-like fashion. This means that, when retreating from a right branch, the box containing the corresponding left branch must be able to contain the right branch if the final non-λ variable were set to λ instead. It follows that this final non-λ variable must be the one and only pivot point between the two. Therefore, we can and do perform this resolution; in this example, it is 〈T, F, T 〉 resolved with 〈λ, F, F 〉. This outputs the box 〈T, F, λ〉. We furthermore store this box in L at location 2, since this box ends with false at that index. We continue forth with probe points 〈T, T, F 〉 and 〈T, T, T 〉. Neither will be found in either D or C , and are therefore output points. Both again have their final non-λ variables at index 3, with 〈T, T, F 〉 being inserted into L at that index and then 〈T, T, T 〉 recovering that box so it can resolve with it to form the box 〈T, T, λ〉. This time, the output of our resolution ends with true, so we recover the box at index 2 in L, 〈T, F, λ〉, and take the resolution of these two boxes, giving us 〈T, λ, λ〉. Once again this ends in T, so we can resolve it with the box we found back at the beginning that has been waiting in slot 1, 〈F, λ, λ〉, to form the box 〈λ, λ, λ〉. This box completely covers the output space; therefore, the algorithm knows that it has found all possible output points and terminates (see Figure 7 for an illustration). 10 1 x2 1 0 1 1 x3 Probe Point p Cache C Database D x2 1 0 x1 1 1 x3 x2 0 x1 Probe Point Map 1 1 x3 x1 Figure 6: The state of the database, cache, and probe point after the first output point is found at 〈T, F, T 〉. In getting here, we first found the box 〈λ, F, F 〉, and then probed the output point. After finding the output point, that box was resolved with the aforementioned box to produce 〈T, F, λ〉, which was also added to C . Note that, while the box 〈λ, F, λ〉 could be produced at this juncture, Tetris will not do so. L(1) contains 〈F, λ, λ〉 (the orange dot); L(2) contains 〈T, F, λ〉 (the purple dot); L(3) is empty. 1 x2 1 0 1 x3 1 Output Points Cache C Database D x2 1 0 x1 1 x3 1 x2 0 x1 1 x3 Probe Point Map 1 x1 Figure 7: The state of the database, cache, and probe point after the entire output space has been covered. With each further output point discovered, boxes were added to the cache, which produced a chain of resolutions that eventually resulted in the production of the box 〈λ, λ, λ〉. Note that there is no longer a probe point, as there is nothing left to probe. 3 Our Improvements Here, we will discuss the major additions introduced into Tetris in order to handle CNF inputs and increase practical efficiency. These include a new data structure, work on heuristically determining a global variable ordering, and selectively caching only certain boxes. 3.1 Data Structure and Compression For its data structure, the original Tetris paper simply states that a trie will suffice to achieve asymptotic runtime guarantees. While this is true, a simple trie still leaves much to be desired; attempts to implement Tetris in such a simple matter produced a system significantly slower than other state-of-the-art model counters. Our contribution is to design a novel system of tries that takes advantage of the nature of the problem space to improve both runtime and memory usage. 11 3.1.1 Data Structure Description As described above, the database must allow each variable to store three values: false, true, and λ. Therefore, the immediate approach is to use 3-tries as the base data structure. However, when #SAT instances routinely have hundreds of variables, this results in an extremely deep problem space that requires a lot of time to probe. Our next step, then, is to compress multiple layers into a single node that can be queried in a single instruction. To this end, we will first come up with a means to enumerate all possible boxes: Definition 7. Let φ be a bijective function from the set of boxes onto the integers. Example 2. One such way is as follows: Let λ be assigned 1; false, 2; true, 3. Then, for each box 〈b 1 , b 2 , ..., b n 〉, its numerical value is 3n−1 b 1 + 3n−2 b 2 + ... + b n .2 This gives us a bijective ternary numeration. From there, we observe that there is exactly one box for which n is 0 (namely, the empty box 〈〉), three boxes with n equal to 1, nine boxes with n equal to 2, twenty-seven boxes with n equal to 3, and eighty-one boxes with n equal to 4. We will require that a single node within the trie be able to record a box of any of these lengths. Additionally, it must be able to store the children of all possible boxes with n equal to 4. Therefore, by compacting four logical layers into a single layer within the database, the result is a trie that can store 121 possible boxes, the sum of the five aforementioned values, and can have 81 possible children. We will refer to this collection of variables as a cluster. Definition 8 (Cluster). A Cluster is the set of variables that the database handles in a single operation. By default, each cluster contains 4 variables. Of course, this raises a new issue: When checking if the database contains a given input string x, there can exist up to sixteen children of x that must be checked (since each T or F can be replaced by a λ), and an even greater number of boxes that could be contained in this cluster may contain the input string. This creates the need for a way to quickly and efficiently determine if a containing box exists in this cluster and to create the list of children to be searched. 3.1.2 SIMD-based Trie Here, we take inspiration from EmptyHeaded, a relational database engine [5], and utilize SIMD. As an example, let us consider the simplified version of the data structure that contains only two layers, and with n = 3. Suppose that 〈F, λ, λ〉 was known to be a box, and that 〈λ, T, F 〉 and 〈T, T, F 〉 are boxes that, to be found, necessitate traversing into child clusters. We can see this depicted in Figure 8. Now, we will determine whether or not this data structure contains the box 〈F, T, F 〉. Since each cluster contains two layers, clusters with depth 0 will look at only the first two variables in the input box to determine input; therefore, let us consider the sub-box 〈F, T 〉. Using φ, we can find in a lookup table the two 128-bit bitstrings corresponding to this input sub-box. The first lists the set of boxes that, if they exist within the cluster, would contain 〈F, T 〉, and is the second line of Figure 9. The second bitstring does much the same for the set of children that, if truncated to two variables, contain 〈F, T 〉. Additionally, a cluster stores two more 128-bit bitstrings: BOXES, which marks the boxes contained by the cluster; and CHILDREN, which marks the child nodes of the cluster. The BOXES bitstring is specifically the top line of Figure 9. BOXES is marked for the box 〈F 〉 (which is equivalent to 〈F, λ, λ〉), while CHILDREN has bits marked corresponding to the box prefixes 〈F, T 〉 and 〈T, T 〉. 2 This is exactly the mapping used in our implementation. 12 It follows that the intersection of these two pairs of bitstrings is the set of boxes and child prefixes present in the data structure that contain the input; a single AND operation suffices to calculate it. ; Cluster 1 Cluster 0 Layer 0 Layer 1 〈F 〉 〈λ〉 〈T 〉 Layer 2 〈λ, T 〉 〈T, T 〉 Layer 3 〈λ, T, F 〉 〈T, T, F 〉 Layer 4 Figure 8: The SIMD operation, shown as the associated clusters, with each box marked by a blue box. Each layer corresponds to a variable (with Layer 4 being empty because all boxes have only three variables), and checkmarks mark the boxes that are found to be containing boxes. The central branch is created from the box 〈F, λ, λ〉. In the left branch, created by 〈λ, T, F 〉, a child is created corresponding to the sub-box 〈λ, T 〉, and then the box is inserted in the child cluster at 〈λ, T, F 〉. In the right branch, created by 〈T, T, F 〉, we similarly create a child corresponding to the sub-box 〈T, T 〉, and then inserting 〈T, T, F 〉 in the child cluster. We have drawn the database here as a 3-trie; to see it as a single, flattened cluster, see Figure 10. Let us inspect our output. In this particular example, we find that we have matched the box 〈F, λ〉 and the child with prefix 〈λ, T 〉. At this point, we are faced with an interesting choice. Suppose for the sake of argument that the child eventually leads to some containing box for the input. Now, if this is a check for containment, the algorithm can only return the one box that it considers the “best" containing box. Which one, then, should the algorithm choose? In general, it will choose the shortest-length box; in this example, it will choose the box 〈F, λ〉. The reason for this is simple: the more λs in a box, the more space it covers; and the more space it covers, the more output space it can cover. Additionally, when those λs are at the tail end of a box, we know that we can immediately advance the probe point considerably. On the other hand, when the λs are in the middle, the algorithm must re-find these boxes every time it scans a point that is contained within this hypothetical box. This repetition is costly; we would rather avoid it. Now, let us consider how our algorithm would handle this input if we were simply using traditional tries; in other words, consider what would happen if our cluster size were 1, as in a standard 3-trie. This algorithm would query the input box for its first value, find that it is F , and know that it could check both the λ branch and the F branch. Since λs are generally to be preferred, it would take the λ branch. Then, it would match the T value to the T branch, and proceed to the third layer, where it would match the F value to the F branch and find a containing box, which it will return. In this way, it has taken us three comparisons to find a containing box. Furthermore, the box found in the 3-trie version is of lower quality: we would have preferred to find the 〈F, λ, λ〉 box instead. On the other hand, let us consider what occurs when using full, four-variable clusters (Figure 10). In this case, a single operation immediately finds all three containing boxes. Therefore, we have outper- 13 Database’s BOXES: 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 10 11 12 & Input: 〈F, T 〉: 1 1 1 0 1 1 0 1 0 1 0 0 0 0 1 2 3 4 5 6 7 8 9 10 11 12 = Containing Boxes: 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 10 11 12 Figure 9: The algorithm takes the logical AND of the stored bitstrings, top, with the bitstring corresponding to the input, row 2, in order to produce the list of containing boxes and children, bottom. The BOXES bitstring contains a single 1 to mark the box 〈F, λ〉. The second line contains all the potential boxes that would contain 〈F, T 〉; in addition to the boxes that do exist, these include 〈λ〉, 〈F, T 〉, and many more. The output of the operation has bits set corresponding to the box 〈F, λ〉. formed the traditional variation both in terms of number of comparisons and in terms of the quality of the output. In summary, each box is stored as a single bit in a 128-bit vector (with the last 7 bits unused), as is a record of whether or not a given child exists. A lookup table is used to find the 128-bit vectors corresponding to the possible outputs. Then, the former two are concatenated, as are the latter two, and all are compared using a single 256-bit AND operation. It can be seen that the output of this operation must be the intersection of the potential containing boxes and children, found from the lookup table, and the ones that actually exist. Hence, by calculating φ(b) for some box b, we can quickly find a box containing b, if it exists; and if it does not, we can quickly generate the exact list of children to examine. Additionally, in practice, it turns out that there is a certain sub-box value that shows up far more often than any other: the sequence 〈λ, λ, λ, λ〉. Notably, this is the very sequence that accepts every single possible input string. Therefore, in the event that a layer contains only this child and no words, it is in fact possible to skip the entire layer. In practice, this produces significant savings on both computational costs and memory usage, which contributes towards Theoretical Implication 3. Let us now formally define the data structures and its algorithms used in our implementation: Definition 9 (Data Structure). The data structure is a 121-ary trie, where each node of the trie is called a cluster. The top level consists of a pointer to the root cluster, which is the cluster covering the first four variables in the ordering, and can perform the following operations: Insert (Algorithm 4), Contains (Algorithm 6), and GetAllContainingBoxes (Algorithm 8). Definition 10 (Cluster). Each cluster in the database contains two bitstrings, BOXES and CHILDREN, bitstrings identifying the sets of boxes and children, respectively, along with an integer DEPTH that informs the cluster of its depth. The operation .at (i ) on a bitstring calculates the intersection of that bitstring with a bitstring retrieved from a lookup table that lists the set of boxes or children that contain or potentially contain, respectively, the box with value i ; setting .at (i ) to 1 sets the specific bit referring exactly to that box. Each cluster corresponds to four layers of a standard 3-ary trie. Definition 11 (Index of a Box). The index of a box b, i nd ex(b), is the location of the last non-λ variable in that box. 14 Cluster 0 ; 〈F 〉 〈λ, T, F 〉 〈T, T, F 〉 Figure 10: The clusters in Figure 8, flattened out into a single node with four variables to a cluster. This is how the node would be stored in the actual database. Note that it is far more compressed than when drawn out as a trie as in Figure 8. This cluster would have a DEPTH of 0, have three bits set in BOXES, and no bits set in CHILDREN. Example 3. The index of the box b = 〈F, T, λ, F, λ〉, i nd ex(b), is 4. Algorithm 4 Insert 1: Given box b to insert, b = 〈c 1 , c 2 , ..., c n 〉, where each c i here is a cluster consisting of b i 1 ...b i 4 2: k ← i nd ex(b) 3: Call InsertCluster(b, k) (Algorithm 5) called on the root cluster Algorithm 5 InsertCluster 1: Input: A cluster T with depth DEPTHto insert on, box b to insert, b = 〈c 1 , c 2 , ..., c n 〉, where each c i here is a cluster consisting of b i 1 , ..., b i 4 , and the location of the final non-λ cluster k 2: if DEPTH = k then 3: if BOXES.at (φ(c DEPTH )) is nonempty then 4: Return {If a containing box of b is already in the data structure, stop.} 5: else 6: Set BOXES.at(φ(c DEPTH )) to 1 7: else 8: if CHILDREN.at (φ(c DEPTH )) 6= 1 then 9: Create a child cluster at location φ(c DEPTH ) with depth (DEPTH + 1). 10: Set CHILDREN.at (φ(c DEPTH )) to 1 11: Call InsertCluster on the cluster indexed as φ(c DEPTH ). 12: Return In the Insert algorithms, Algorithms 4 and 5, we recursively traverse clusters until we find the appropriate location in the data structure, and then set the appropriate bit to 1. Along the way, we will check for containing boxes and immediately cease operation if one is found. Furthermore, if a child cluster that contains the box we are inserting, or that contains a cluster along the path to that cluster, does not exist, we create it. Algorithm 6 Contains 1: Given box b to find a containing box of, b = 〈c 1 , c 2 , ..., c n 〉, where each c i here is a cluster consisting of b i 1 ...b i 4 2: out put ← ContainsCluster(b) (Algorithm 7) called on the root cluster 3: Return out put 15 Algorithm 7 ContainsCluster 1: Input: Cluster T with depth DEPTHthat is being checked for a containing box, box b to check containment of, b = 〈c 1 , c 2 , ..., c n 〉, where each c i here is a cluster consisting of b i 1 ...b i 4 . 2: if ( f = BOXES.at (φ(c DEPTH ))) is nonempty (i.e., there is at least one box in the intersection) then 3: o = mi n x∈ f I nd ex(x) 4: Return o 5: else if ( f = CHILDREN.at (φ(c DEPTH ))) is nonempty (i.e., there is at least one child in the intersection) then 6: for all Children k ∈ f do 7: a = k.ContainsCluster(b) {Scan these in order of increasing index} 8: if a is nonempty then 9: Return a 10: else 11: Return ; In the Contains algorithms, Algorithms 6 and 7, we check the database to see if it contains any box that contains some box b. Therefore, we traverse along clusters in our path, first checking to see if any containing boxes exist; if we find one, we return that box immediately and cease checking further. If none exists, we will perform a depth-first search of the children that could potentially contain a containing box. This process continues until either a containing box is found, or else the search space is exhausted and it is determined that no containing box exists. Algorithm 8 GetAllContainingBoxes 1: Given box b to find all containing boxes of, b = 〈c 1 , c 2 , ..., c n 〉, where each c i here is a cluster consisting of b i 1 ...b i 4 2: out put ← GetAllContainingBoxesCluster (b) (Algorithm 9) called on the root cluster 3: Return out put Algorithm 9 GetAllContainingBoxesCluster 1: Input: Cluster T with depth DEPTH that is being checked, box b to find all containing boxes of, b = 〈c 1 , c 2 , ..., c n 〉, where each c i here is a cluster consisting of b i 1 ...b i 4 . 2: O ← ; 3: if (F = BOXES.at (φ(c DEPTH ))) is nonempty (i.e., there is at least one box in the intersection) then 4: O = F. 5: else if ( f = CHILDREN.at (φ(c DEPTH ))) is nonempty (i.e., there is at least one child in the intersection) then 6: for all Children k ∈ f do 7: A = k.GetAllContainingBoxesCluster(b) 8: O =O∪A 9: else 10: Return O 16 The GetAllContainingBoxes algorithms, Algorithms 8 and 9, are similar to the Contains algorithms. There are, however, two key differences. First, while Contains terminates as soon as it found a single containing box, GetAllContainingBoxes will continue. Secondly, it returns the set of all containing boxes, rather than just one; hence, the name. In all other regards, it behaves exactly as Contains does. 3.2 Global Variable Ordering Up until this point, we have simply been assuming that all boxes must order their variables in exactly the same order as they appear in the original SAT formulas. In other words, in each box b, b 1 must correspond to x 1 , b 2 must correspond to x 2 , and so on. However, this does not need to be the case. We can reorder the variables, and by doing so can greatly improve the runtime of our system. The original Tetris paper cites the importance of the variable ordering. However, it assumes that there exists an exponential in n time algorithm to compute the optimal variable ordering. While this is justifiable in the context of join problems, where n is small compared to the size of the database, in SAT problems it is unacceptable. Furthermore, as computing the optimal ordering is NP-hard, we cannot hope to improve upon this result. Indeed, even approximating the ordering is intractable [20]. Nevertheless, initial experimentation with Tetris made clear how impactful this choice can be. A slight variation in ordering can result in a large difference in runtime. Thus, we turned to various heuristics and intuitions in order to find a quick and effective means to generate an ordering that works well in practice and thereby contribute to Theoretical Implication 2. First, let us define a few terms that we will use in our discussion of various ordering strategies: Definition 12 (Degree). The degree of a variable is the number of clauses of which the variable is a part. Example 4. In Example 1, (x 1 ∨x 2 )∧(x 1 ∨ x¯2 )∧(x 2 ∨x 3 ), x 1 has degree 2, x 2 has degree 3, and x 3 has degree 1. Definition 13 (Closeness). Variables x i and x j are said to be close if there exists a clause that includes both x i and x j . The fewer terms in the clause, the closer the two variables are said to be. Specifically, the closeness of two variables, θ(x 1 , x 2 ), is equal to 1 divided by the size of the smallest clause containing both variables minus 1. Example 5. In the clause (x 1 ∨ x 2 ), x 1 and x 2 would have a closeness of 1; and in the clause (x 1 ∨ x 2 ∨ x 3 ), x 1 and x 3 would have a closeness of 21 . If both clauses were part of the same SAT problem, x 1 and x 2 would still have a closeness of 1 because the first clause has a smaller size than the second. Definition 14 (Interconnectedness). The interconnectedness of a cluster C , IC (C ), is the sum of the close¡ ¢ ness values for each 42 pairs of variables in the cluster. Example 6. Using Example 5, if x 1 , x 2 , and x 3 compose a cluster, its interconnectedness would be 2, since θ(x 1 , x 2 ) = 1, θ(x 1 , x 3 ) = 12 , and θ(x 2 , x 3 ) = 12 . In general, we note two high-level strategies that we have found improve the performance of a given global variable ordering: a) High-degree First In Tetris, handling high-degree variables early tends to improve performance. To see the reason for this, we must consider the nature of the algorithm. Scattering high-degree variables throughout the ordering forces the algorithm to branch frequently, which means that, when testing for 17 inclusion or for containing boxes, the algorithm must scan all possible branches. This is highly inefficient. Instead, we focus the branches as much as possible to the beginning, with the hope being that, as the algorithm progresses down from there, most layers will have few if any divergent choices. If this is true, then the inclusion check can be handled quickly. Let us proceed to see why this is so. First, we will introduce an example of why placing high-degree variables early in the ordering proves effective. Let us consider the following sample problem: Example 7. Consider the SAT formula (x 1 ∨x 3 )∧(x 2 ∨x 3 ). The equivalent box problem, using the ordering (x 1 , x 2 , x 3 ), would begin with database D containing the boxes {〈F, λ, F 〉, 〈λ, F, F 〉}. x 1 has degree 1, x 2 has degree 1, and x 3 has degree 2. Now, let us consider how the algorithm would attack this problem if we used this naive, low-degree first ordering, which would be (x 1 , x 2 , x 3 ). We immediately note two things. First, there are no boxes that have λ for the final variable. This means that the algorithm will never find a box that allows it to skip multiple probe points unless it can use resolution to create a new box that happens to have that property. In fact, that will not occur. Additionally, we can consider all 8 possible probe points and track how many comparisons the algorithm would need to make on each point, assuming that each cluster only covers a single variable instead of four. We can see that the algorithm will always have to calculate the set intersection for the cluster of depth 1, will have to perform the set intersection for the cluster with depth 2 on 50% of probe points, and will have to perform the set intersection for the cluster with depth 3 on 75% of probe points. Let us contrast this with the high-degree ordering, which moves x 3 to the front and x 1 to the back to give us the ordering (x 3 , x 2 , x 1 ). Now, our database contains the boxes 〈F, λ, F 〉 and 〈F, F, λ〉. This time, we do have a box with a λ at the back; specifically, 〈F, F, λ〉. Therefore, after the probe point 〈F, F, F 〉 finds this box, the algorithm will advance past 〈F, F, T 〉 entirely. Additionally, while we still have to perform a set intersection for the cluster with depth 1 100% of the time, and 50% of the time for the cluster with depth 2, we only have to perform the set intersection at depth 3 25% of the time — and all of these numbers ignore how we skipped one of the probe points entirely. While this is of course a very simple example, this illustrates the principles that cause the strategy to be effective in larger datasets. b) Local Interconnectedness As a direct result of the 121-ary trie-based system described in Section 3.1, if a box has multiple non-λ variables within the same 4-variable cluster, they can all be recovered with a single operation. Therefore, maximizing the interconnectedness of these 4-variable blocks provides an advantage. To illustrate, let us consider the following example: Example 8. (x 1 ∨ x 3 ) ∧ (x 2 ∨ x 4 ), with each cluster containing 2 variables rather than 4. If we use the naive strategy of keeping the variables ordered as-is, the first cluster contains x 1 and x 2 while the second cluster contains x 3 and x 4 . Therefore, both clusters have interconnectedness 0, and we find that all boxes transcend a cluster boundary. In other words, it will always take at least two comparisons to find either of these boxes. However, if we had gone with the ordering (x 1 , x 3 , x 2 , x 4 ) instead, the box corresponding to (x 1 ∨ x 3 ) would be entirely contained within the first cluster, and the box corresponding to (x 2 ∨ x 4 ) would be entirely contained within the second cluster. Therefore, each box would be entirely contained within a cluster, and each cluster would have had an interconnectedness of 1, and each box could have been recovered with only a single comparison. This saves a large number of comparisons over the long term. 18 3.2.1 Ordering Algorithms Two major methods were employed in order to achieve these aims. The first was a descending degree sort. While this only directly achieved goal (a), in practice it did an acceptable job with goal (b). Additionally, we constructed three variations on this method. The first, naive degree descent, is where we simply order the variables according to their degree, using Algorithm 10. Algorithm 10 Naive Degree Descent Ordering 1: Given a set of variables V and, for each variable v, its degree v d : 2: O ← SORT(V on v d , descending) 3: Return O The second, optimally grouped degree descent, forms all possible groups of four variables, finds the greatest possible interconnectivity among these groups, and then selects from all groups with the greatest interconnectivity on the basis of the combined degree of the group of four, using Algorithm 11. While this proved effective, it is a slow algorithm with a runtime of Θ(n 8 ). Algorithm 11 Optimally Grouped Degree Descent Ordering 1: Given a set of variables V and, for each variable v, its degree v d : 2: O ← ; 3: G ← all possible sets of four variables {v i , v j , v k , v l } from V . 4: while G is nonempty do 5: max IC ← max g ∈G IC (g ) {Determine the maximum possible interconnectedness of all remaining groups.} P 6: X ← max g ∈G s.t . IC (g )=maxIC ( v∈g v d ) {Of the groups with max interconnectedness, 7: select the group for which the sum of the degrees of all variables is the greatest.} O ← (O, X ) {Append this group to the ordering.} for v ∈ X do for Y ∈ G do 10: if v ∈ Y then 11: G ← G \ Y {Remove each grouping that contains one of the variables in the group that was selected.} 12: Return O 8: 9: This necessitated the creation of the third subtype, the heuristically grouped degree descent ordering. This ordering works in groups of four. When creating a group, the first node chosen is the highestdegree remaining variable. Then, for each of the remaining three variables, the algorithm picks the variable with the highest interconnectedness to the nodes already chosen for this group of four, breaking inevitable ties based on degree. The result, Algorithm 12, is an algorithm that can compute its ordering significantly faster than the optimal ordering, while Tetris run on this ordering runs is competitive with the optimal ordering. 19 Algorithm 12 Heuristically Grouped Degree Descent Ordering 1: Given a set of variables V and, for each variable v, its degree d v : 2: i ← 0 3: X ← ; 4: O ← ; 5: while V is nonempty do 6: if i = 0 then 7: x ← y where De g r ee(y) = max v∈V (d v ) 8: else P 9: max IC ← max v∈V ( x∈X θ(v, x)) {Calculate which variable has the best interconnectedness with the already chosen variables} 10: x ← y where d y = max v∈V s.t .VIC =maxIC (d v ) {Break ties based on degree} 11: O ← (O, x) 12: X ← X ∪x 13: i ← i +1 14: if i = 3 then 15: i =0 16: x ←; 17: Return O Additionally, we employ the Treewidth tree decomposition, which was introduced in [16]. In essence, the idea here is to minimize the width of the search tree; in our domain, this corresponds to increasing the locality and local interconnectedness of variables. This naturally did a very good job with interconnectivity, while a decent job with placing high-degree variables early. We also experimented with the minfill ordering, described in [12]. This ordering sets the elimination order such that the node to be eliminated is the node whose removal makes the smallest impact on the overall graph. While this ordering has proved effective in similar applications, we found it to perform poorly with Tetris. In Table 1, we can see how these various orderings performed in practice on representative graphbased and non-graph-based benchmarks. For instance, while the Treewidth sort outperformed all others on the AIS8 dataset [3], on the WikiVotes dataset (created using a SNAP [19] dataset; see Section 4.1.1) the ordering caused Tetris to timeout. Notably, we see that the Heuristically Grouped Degree Descent takes only slightly longer to process the input compared to the Naive Degree Descent, but significantly less time than the Optimally Grouped Degree Descent; however, the runtime does not suffer significantly when going from the optimal ordering to the heuristic one. 3.3 Selective Insertion While the original Tetris paper calls for the insertion of every box created through the resolution process to be inserted into the database, this proved to be inefficient in practice. Very frequently, this will result in a huge increase in the number of branches that the algorithm must scan while trying to find the output point without notably improving the quality of the containing boxes found. Therefore, we only insert those boxes that contain a suitably high percentage of λs. The best results generally come from requiring slightly less than 50 percent of the layers to be composed of entirely λs. In Table 2, we have posted the runtime for the AIS10 dataset [3] showing the relationship between the number of λs we require in storing a box and the runtime of Tetris. Performance suffers at extreme 20 Dataset AIS8 WikiVotes RUNTIME WITH D IFFERENT O RDERINGS Ordering Load Time (Seconds) Naive Degree Descent .001 Heuristically Grouped Degree Descent .009 Treewidth .002 Minfill .008 Optimally Grouped Degree Descent 1.629 Naive Degree Descent .927 Heuristically Grouped Degree Descent 1.024 Treewidth 2.349 Minfill 2.032 Optimally Grouped Degree Descent 2.06 Runtime (Seconds) 1.494 2.607 .45 7.005 0.573 34.124 21.509 Timeout 4059.426 23.142 Table 1: The performance of various ordering schemes on two datasets. As one can see, no ordering does best on both datasets; indeed, the best ordering on one is the worst on the other. Insertion Ratio (see Section 3.3) was set to .5 for these tests. settings, with optimal performance resulting from an insertion ratio of close to 21 . Hence, with regard to Theoretical Implication 3, we find that by decreasing space complexity, we furthermore improve runtime. 4 Our Experimental Results Here, we compare CNFTetris — that is, Tetris designed to solve CNF problems — with other model counters in order to compare and contrast their ability to tackle model counting problems. A model counting problem, simply put, is, given a CNF formula, output the number of satisfying solutions to that formula. Since Tetris was originally designed to handle database joins, these are more natural problems for the algorithm to solve than the corresponding SAT problems, which are to simply determine whether or not any solution exists. While the model counting problem allows for a solver to simply find the number of solutions without finding the solutions themselves, CNFTetris does in fact output all of the solutions. While this admittedly poses a disadvantage compared to the solvers we are comparing against, for some datasets, CNFTetris runs faster in spite of this. We compare our results with those of the sharpSAT [28] , dSharp[22], and Cachet[26] model counters due to their recognition as state-of-the-art model counters. All tests were performed using a single thread on an 8-core E5 v3 2.6GHz processor with 64GB of RAM. Additionally, we include two types of datasets. The first is derived from join problems on graphs. These are the sort of problems that Tetris was originally designed to solve; as such, CNFTetris does a very good job on them. The second set is a selection of standard model counting benchmarks from various competitions held over the past several years. Most model counters have been trained to solve such problems, so they serve as an apt second set of benchmarks for CNFTetris to compete against. 21 I NSERTION R ATIO VS . RUNTIME Ratio Time (Seconds) .00 140.999 .05 123.49 .10 79.003 .15 71.229 .20 41.956 .25 36.685 .30 32.064 .35 24.405 .40 22.749 .45 21.784 .50 24.171 .55 25.205 .60 28.157 .65 35.738 .70 44.657 .75 51.809 .80 54.622 .85 66.529 .90 64.16 .95 75.829 1.00 88.001 Table 2: A comparison of insertion ratios with the time to solve the AIS10 dataset [3]. For these tests, we used the Treewidth ordering; similar behavior was observed from all ordering strategies. 4.1 Graph Results Here, we will compare and contrast how various solvers performed on model counting problems created from graphs. 4.1.1 Dataset Generation The CNF graph datasets were created using the publicly available SNAP datasets [19]. These are graph datasets; that is, each consists of a set of vertices and a set of edges connecting those vertices. Each of these datasets is a natural problem; some arose from social networks, while others are anonymized data from other corners on the Internet. We can then use this data to run various queries; for instance, we can determine how many triangles exist in the graph. Our goal, then, is to convert these problems into an equivalent CNF problem so that we can use CNFTetris and the other model counters to solve them. To do this, each vertex is first assigned a unique binary encoding using log(n) bits. We furthermore increase the number of bits such that there are log(n) bits times the size of the data structure being looked for in the graph. For instance, if we are performing the triangle query on the dataset, there will be 3 · log(n) bits used in the encoding. Henceforth, let k represent the size of this query. Each of these bits will correspond to a variable in the CNF encoding of the problem. In essence, each of these repetitions 22 RUNTIME Query 3-clique 2-path Base Graph Wikivotes Facebook Soc-Epinions Wikivotes Facebook Soc-Epinions AIS6 AIS8 AIS10 AIS12 ls8-simplified4 LS5 firstr CNFTetris ON VARIOUS D ATASETS sharpSAT dSharp Cachet Loadtime 1.080 .593 77.420 2.32 .752 6.22 Runtime 21.55 11.22 309.418 35.171 10.236 236.439 Runtime Timeout Timeout Timeout 31801.8 4637.25 Timeout Speedup n/a n/a n/a 904.2 453.03 n/a Runtime Timeout Timeout Timeout 36947.5 3871.86 Timeout Speedup n/a n/a n/a 1050.51 378.26 n/a Runtime Timeout Timeout Timeout 28838.1 2210.42 Timeout Speedup n/a n/a n/a 819.94 215.95 n/a 0.0021 0.0027 .00798 .0198 .00063 .000656 .013 .45 21.55 1732.03 .123 .409 .004 .0466 2.629 124.937 .004933 .0156 .308 0.104 .122 .0721 .0401 .0381 .0074 .3386 14.757 1002.73 .021 .095 .569 0.752 .685 .579 .171 .232 .0135 .300 16.4724 1020.64 .01 .025 1.034 .667 .764 .589 .0813 .0611 Table 3: This table shows the comparative results of various solvers on CNF datasets created using various SNAP graphical datasets and SAT datasets. All runtimes are in seconds; timeout was set at 40,000 seconds. For these tests, we used a insertion ratio of .45 and the Heuristic Degree Descent ordering for CNFTetris. Wikivote [19] contains 39 variables and 745485 clauses; Facebook [19] contains 36 variables and 464234 clauses; and Soc-Epinions [19] contains 51 variables and 4578589 clauses (all clause data is for 3-clique; 2-path has approximately 2/3rd that number of clauses). For the SAT datasets, AIS6 [3] has 61 variables and 581 clauses; AIS8 [3] has 113 variables and 1520 clauses; AIS10 [3] has 181 variables and 3151 clauses; AIS12 [3] has 265 variables and 5666 clauses; ls8-simplified4 [2] has 119 variables and 410 clauses; and LS5-firstr [2] has 125 variables and 529 clauses. In CNFTetris, loadtime refers to the time to determine the variable ordering and insert the boxes into the database, while runtime is the time to find all satisfying solutions. represents a vertex in e.g. the triangle. Next, we will encode each absent edge (i.e., a pair of vertices v 1 and v 2 such that the edge (v 1 , v 2 ) ∉ E , ¡ ¢ where E is the edge set of the graph) as k2 total Boolean formulas for the k-clique query, and k formulas for the k-path query. Each of these formulas corresponds to e.g. one of the three edges of a triangle. Observe that any possible satisfying solution to the SAT problem cannot select an edge that does not exist; therefore, any assignment that matches one of these formulas on all variables must be rejected. Equivalently, any accepting assignment must match at least one variable in the inverse. This naturally leads to a CNF definition, which is what we create. We will repeat this encoding over each possible set of vertex pairings such that the lower-indexed vertex is always written before the higher-indexed vertex, while adding additional clauses to reject all edges that would be from the higher-indexed vertex to the lower-indexed vertex. Simplifying resolutions are also performed where possible. Therefore, we have created a CNF problem where, for each output to the query in the original problem, there exists a solution. We can and do use this problem instance as input for both Tetris and other model counters. Let us examine an example instance of this. Consider the very simple example graph depicted in Figure 11 below, using the triangle query. In order to encode the non-edge (v 2 , v 4 ), we must first calculate the binary encodings of each of these vertices. These are (01) and (11), respectively. We then flip all of the 23 bits, giving us (10) and (00). Then we construct three CNF clauses, each of which corresponds to being the first, the second, or the third edge of the triangle. The first will be (x 1 ∨ x¯2 ∨ x¯3 ∨ x¯4 ), with (x 1 ∨ x¯2 ) corresponding to (10) and (x¯3 ∨ x¯4 ) corresponding to 00. Similarly, the second will be (x 1 ∨ x¯2 ∨ x¯5 ∨ x¯6 ); and the third will be (x 3 ∨ x¯4 ∨ x¯5 ∨ x¯6 ). Additionally, we insert clauses forbidding “bad" orderings of the points; in other words, we are making sure we do not count (v 1 , v 2 , v 3 ) and (v 2 , v 3 , v 1 ) as separate triangles. Then, when Tetris run on this CNF input attempts to recover the number of triangles, when it uses the probe point (assuming naive ordering) 〈T, T, T, F, F, T 〉 — that is, the probe point that corresponds to the inverted binary representations of v 1 , v 2, and v 3 , the three vertices in the top-right triangle — let us consider what happens. Since each of the selected edges does not correspond to the missing edge, we know that all three of those clauses must be satisfied; and since each edge is in the index order, we know that the additional clauses that we added will also accept our input. Therefore, Tetris will add this probe point to the output list. This continues for all other probe points until Tetris has found all triangles. v1 v2 v4 v3 Figure 11: A sample graph on four vertices, used in the above example. We will encode the non-edge (v 2 , v 4 ) as our SAT formula, along with additional clauses that ensure we only count triangles once, which will allow us to run the query as a #SAT problem. 4.1.2 Results Analysis As can be seen in Table 3, while all of the other solvers find these problems to be difficult, CNFTetris solves them quickly. Queries that take seconds on CNFTetris wind up taking hours on the competition, with CNFTetris running nearly a thousand times faster on some problems. This is largely due to the extremely high number of clauses relative to the number of variables, along with the fact that these clauses contain a large number of variables; these factors are not present in many of the standard SAT benchmarks. For instance, while the average clause in many SAT benchmarks contains two or three variables, here the average clause has thirty or more. And while SAT benchmarks rarely have over ten times as many clauses as variables, here the system is forced to tackle an environment where the number of clauses is exponentially larger than the number of variables. Note that all of the solvers we are comparing against use unit propagation techniques in order to count models [9]; see Section 5 for details. Because of this, the increased number of clauses directly corresponds to increased work for these solvers. 4.2 Nongraph Results In this section, we will discuss how CNFTetris performed as compared to other solvers on standard model counting benchmarks. 24 4.2.1 About the Datasets These datasets are a combination of datasets from the SATLIB datasets [3] and the SampleCount benchmarks for model counting taken from International Joint Conference on Artificial Intelligence ’07 [2]. We chose to use the AIS datasets for several reasons. First, each of the datasets terminates in a reasonable amount of time on all solvers, allowing us to find interesting comparisons. Secondly, due to the existence of increasingly-sized versions of this dataset, we can use this as insight into whether or not Tetris is scaling efficiently with the size of the dataset. Additionally, we featured the ls8-simplified4 and LS5firstr datasets, which gave us insights into our implementation’s strengths and weaknesses. 4.2.2 Results Analysis As Table 3 shows, Tetris is competitive with dSharp and Cachet on many of the datasets. Indeed, a factor of 2 separates us from either solver on all of the AIS datasets, a difference that engineering work alone can easily overcome. While there is significantly more space between it and sharpSAT, a factor of 10 on average, we believe the distance is not insurmountable. The largest gaps exist on the ls8-simplified and LS5-firstr datasets. On these, CNFTetris is roughly a factor of 5 off of the worst of the competition, and a factor of over 20 as compared to sharpSAT. The reason for this is simple: both of these datasets contain pure variables. In other words, there exist variables x such that, in all clauses, x̄ never appears, or vice-versa. This is an important piece of information, one that can and must be utilized, but CNFTetris in its current state does not know how to do so. However, since we know what the problem is, we expect to be able to quickly and efficiently attack this issue. 5 Related Work Our work builds on Tetris as developed by Abo Khamis et al. in [6]. In that work, the authors introduced Tetris as a beyond-worst-case algorithm for geometrically solving the database join problem. This in turn built on work on the Minesweeper [23], NPRR [24] and Leapfrog [29] algorithms, of which Tetris is a generalization. Furthermore, Tetris itself is can be considered a version of the DPLL algorithm [10] with clause learning. In DPLL, which is itself an evolution of the earlier DP [11] algorithm, a variable is chosen at every stage and assigned to be either true or false. The algorithm then uses unit propagation in order to simplify clauses under these assumptions. In this techniques, after the solver assigns a value to a variable, every other clause is inspected to see if this assignment creates a unit clause (i.e. a clause with only one variable in it), and to see if resolutions can be performed. This process continues until a conflicting clause (that is, a clause that is violated by the assignments) is found, at which point the algorithm is forced to backtrack. In the clause learning versions, introduced in [27], the solver takes this as an opportunity. It determines where it went astray, adds a new clause to its cache that is the negation of this errant assignment, non-chronologically backtracks to where this decision-making took place, and then proceeds in the opposite direction. The reasoning why CNFTetris is a form of this algorithm follows from the aforementioned method of converting from SAT clauses to boxes, and vice-versa (see Algorithm 1). Since these two representations are exactly equivalent, any operation performed on one representation can be translated into an operation on the other. Hence, every single operation Tetris performs on the boxes over its execution must correspond exactly to a set of operations on the original clauses. For instance, the Contains operation matches up with the idea of a conflicting clause. When a containing box is found, we can consider this as finding a box that rejects the current probe point as a po25 tential output point. Meanwhile, a conflicting clause rejects a potential satisfying assignment in much the same way. Furthermore, over the course of Tetris, the algorithm tentatively assigns a variable to either true or false, and then proceeds along with this assumption until a contradiction is found, all while learning additional clauses where possible through the resolution process. When a containing box is found or synthesized through resolution, and we advance the probe point accordingly, we are in essence backtracking to the earliest decision point and choosing to go in the opposite direction, just as DPLL with clause learning does. Therefore, this is exactly the DPLL algorithm with clause learning, with the added restriction of a fixed global variable ordering [6]. Tetris additionally utilizes a three-value logic system. While similar systems have been utilized in database schemes, such as by Zaniolo in [31], in these systems the three values are true, false, and unknown. Here, however, the three values we are considering can be summarized as true, false, and both. This causes a number of key differences. For instance, true ∧ unknown is equivalent to unknown, while true ∧ both is equivalent to true. Similarly, true ∨ unknown is equivalent to true, while true ∨ both is equivalent to both. Much work has been done in creating SAT solvers. Let us briefly discuss those state-of-the-art solvers we are comparing our work against. First, let us consider Cachet[26]. This solver was originally released in 2005, with minor compatibility updates continuing through the most recent version, which came out in 2015 [1]. Next, there is sharpSAT. First released in 2006, sharpSAT significantly eclipsed contemporary solvers [28]. sharpSAT has been maintained over time, with the most recent release in 2013 [4]. Finally, we come to dSharp. The most recently released of our three competitors, dSharpwas introduced in 2012 in order to efficiently compile CNF problems into the Decomposable Negation Normal Form language [22]. Further work allowed it to function as a model counter, which is how we utilize it. The version we use was released in 2016. What all of these solvers have in common, including CNFTetris, is that they have at their core a form of the DPLL algorithm with clause learning; indeed, almost all modern SAT and #SAT solvers do so [9]. The differences, then, come in terms of efficiency. Each solver uses a different array of techniques in order to effectively cache and recover learned clauses, to determine the variable ordering, and to identify clause conflicts. With Cachet, the authors focused on adding component caching capabilities on top of an existing SAT solver, ZChaff [26], the theoretical grounds for which were themselves introduced in [21]. This caching involved the storing of subproblems in a local cache, so that these clauses would not have to be re-derived by Cachet at a later juncture, thereby reducing redundant calculations over the course of the algorithm. This can be viewed as analogous to how CNFTetris stores learned boxes in a local cache, which it checks for containing boxes before examining the original database. A subproblem, meanwhile, could be thought of as a box with a high percentage of variables set to λ. However, one key difference here is the nature of the cached components. In Cachet, due to how the algorithm functions, it must regularly prune the cache of siblings that would otherwise cause it to undercount the number of models [26]. CNFTetris, in contrast, needs to perform no such pruning; it will naturally determine the exact number of models without any additional work. sharpSAT built on the work in Cachet while adding new ideas of its own [28]. Boolean constraint propagation (also known as the failed literal rule [15]) and unit propagation heuristics are used by sharpSAT to identify failed literals with greater efficiency than was done in Cachet [28]. However, by fixing the variable order, CNFTetris simplifies this process. Ultimately, this means that it finds its conflicting boxes in a fundamentally different manner than sharpSAT does, which provides room for CNFTetris to 26 outperform sharpSAT. dSharp, much like how sharpSAT built on Cachet uses sharpSAT as a core component [22]. The authors perform a DNNF translation, and then use properties of decomposability and determinism to perform model counting [15]. Though these differences do allow it to outperform more pure DPLLbased solvers on some benchmarks [15], since this system still uses sharpSAT as a core component, it still shares many of the same advantages and disadvantages in comparison to CNFTetris. As we have seen, all of the competing solvers can be viewed as evolutions along a single line. While CNFTetris does not throw the baby out with the bathwater — that is, while CNFTetris still continues to implement the classic DPLL algorithm — it does represent a distinct deviation from that line, challenging assumptions such as the necessity of allowing a non-fixed global variable ordering and the much more complex data storage scheme necessary in order to accommodate this. While this has necessitated much work in order to implement, it has also shown vast promise. 6 Acknowledgments We would like to thank Mahmoud Abo Khamis, Hung Q. Ngo, Christopher Ré, and Ce Zhang for very helpful discussions. References [1] Cachet. http://www.cs.rochester.edu/users/faculty/kautz/Cachet/index.htm. cessed: 2016-11-24. [2] International joint conference on artificial intelligence ’07 dataset Ac- collection. 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Neural Affine Grayscale Image Denoising arXiv:1709.05672v1 [cs.CV] 17 Sep 2017 Sungmin Cha, Taesup Moon College of Information and Communication Engineering Sungkyunkwan University, Suwon, Korea 16419 [email protected] Abstract We propose a new grayscale image denoiser, dubbed as Neural Affine Image Denoiser (Neural AIDE), which utilizes neural network in a novel way. Unlike other neural network based image denoising methods, which typically apply simple supervised learning to learn a mapping from a noisy patch to a clean patch, we formulate to train a neural network to learn an affine mapping that gets applied to a noisy pixel, based on its context. Our formulation enables both supervised training of the network from the labeled training dataset and adaptive fine-tuning of the network parameters using the given noisy image subject to denoising. The key tool for devising Neural AIDE is to devise an estimated loss function of the MSE of the affine mapping, solely based on the noisy data. As a result, our algorithm can outperform most of the recent state-of-the-art methods in the standard benchmark datasets. Moreover, our fine-tuning method can nicely overcome one of the drawbacks of the patch-level supervised learning methods in image denoising; namely, a supervised trained model with a mismatched noise variance can be mostly corrected as long as we have the matched noise variance during the fine-tuning step. 1 Introduction Image denoising is one of the oldest problems in image processing and various denoising methods have been proposed over the past several decades, e.g., BM3D [1], wavelet shrinkage [2], field of experts [3], sparse-coding based approach [4], WNNM [5], EPLL [6] and CSF [7], etc. In this paper, we propose a new image denoiser, dubbed as Neural Affine Image Denoiser (Neural AIDE), which utilizes neural network in a novel way. The method is inspired by the recent work in discrete denoising [8], in which a novel “pseudo-labels” were devised to train a denoiser solely based on the noisy data. We extend the approach to the continuous-valued data case and devise a novel estimated loss function based on the noisy data that is an unbiased estimate of the true MSE. By investigating the devised estimated loss function we formulate to train a neural network to learn an affine mapping that gets applied to a noisy pixel, based on its context. Such formulation enables both supervised training of the network from the labeled training dataset and adaptive fine-tuning of the network parameters using the given noisy image subject to denoising. Our experimental results extensively show how we made subtle design choices in developing our algorithm. Furthermore, we show that Neural AIDE significantly outperforms strong state-of-the-art baselines in the standard benchmark test datasets. 2 Notations and Problem Setting We denote xn×n as the clean grascale image, and each pixel xi ∈ {0, . . . , 255} is corrupted by an independent additive noise to result in a noisy pixel Zi , i.e., Zi = xi + Ni , i = 1, . . . , n2 , (1) where the continuous noise variables Ni ’s are independent (not necessarily identically distributed nor Gaussian) over i and E(Ni ) = 0, E(Ni2 ) = σ 2 for all i. Moreover, As in the standard processing in grayscale image denoising, we normalize both xi ’s and Zi ’s with 255 and treat them as real numbers. Importantly, following the universal setting in discrete denoising [9, 8], we treat the clean image xn×n as an individual image without any probabilistic model and only treat Z n×n as random. 2 Generally, a denoiser can be denoted as X̂ n×n = {X̂i (Z n×n )}ni=1 denoting that each reconstruction at location i is a function of the noisy image Z n×n . The standard loss function used for the grayscale image denoising to measure the denoising quality is the mean-squared error (MSE) denoted as 2 n×n ΛX̂ n×n (x ,Z n×n ) n 1 X Λ xi , X̂i (Z n×n ) 2 n i=1 = (2) where Λ(x, x̂) = (x − x̂)2 is the per-symbol squared-error. Conventionally, the MSE is compared in the dB-scale using the Peak Signal-to-Noise-Ratio (PSNR) defined as 10 log10 (1/ΛX̂ n×n (xn×n , Z n×n )). 2.1 Estimated loss function for the affine denoiser In this paper, we consider the denoiser of the form X̂i (Z n×n ) = a(Z \i ) · Zi + b(Z \i ) for each i, in which Z \i stands for the entire noisy image except for Zi . Namely, the reconstruction at location i has the affine function form of the noisy symbol Zi , but the slope and the intercept parameters, i.e., a(Z \i ) and b(Z \i ), of the affine function can be functions of the surrounding pixels. Hence, separete parameters can be learned from data for each location. Before presenting more concrete form of our denoiser, we first consider the following lemma. Lemma 1 Consider a single-symbol case Z = x + N with E(N ) = 0 and E(N 2 ) = σ 2 , and suppose a single-symbol denoiser has the form of X̂(Z) = aZ + b. Then, L(Z, (a, b); σ 2 ) = (Z − (aZ + b))2 + 2aσ 2 (3) is an unbiased estimate of Ex Λ(x, X̂(Z)) + σ 2 , in which Λ(x, x̂) = (x − x̂)2 and Ex (·) notation stands for the expectation over Z given that the clean symbol is x. Remark: Note while the true MSE, Λ(x, X̂(Z)), can be evaluated only when the clean symbol x is known, the estimated loss L(Z, (a, b)) can be evaluated soley with the noisy symbol Z, the affine mapping (a, b) and the noisy variance σ 2 . Thus, L(Z, (a, b)) plays a key role in adaptively learning the neural network-based affine denoiser as shown in the next section. Proof: By simple algebra, we have the following equalities: Ex (x − X̂(Z))2 = Ex (x2 + (aZ + b)2 − 2x(aZ + b)) = Ex (x2 + (aZ + b)2 − 2ax2 − 2bx) 2 2 2 2 (4) 2 = Ex (Z − σ + (aZ + b) − 2a(Z − σ ) − 2bZ) 2 = Ex Z − (aZ + b) + (2a − 1)σ 2 (5) (6) = Ex L(Z, (a, b); σ 2 ) − σ 2 , in which (4) follows from Ex (Z) = x, (5) follows from Ex (Z 2 ) = x2 + σ 2 and replacing x2 with Ex (Z 2 − σ 2 ), and (6) follows from simply rearranging the terms. Thus, we have the lemma. From Lemma 1, we can also show that for the denoisers of the form X̂i (Z n×n ) = a(Z \i )·Zi +b(Z \i ), Exi Λ(xi , X̂i (Z n×n )) Z \i = Exi L(Zi , (a(Z \i ), b(Z \i )); σ 2 )\|Z \i − σ 2 (7) holds since a(Z \i ) and b(Z \i ) become constant given Z \i and the noise is independent over i. The Exi (·\|Z \i ) in (7) stands for the conditional expectation of Zi given the clean symbol xi and the noisy symbols Z \i . Note the estimated loss function similar to (3) has been also used to the filtering problem [10]. 2 3 3.1 Neural AIDE: Neural Affine Image DEnoiser Neural network-based affine denoiser Our proposing Neural Affine Image DEnoiser (Neural AIDE) considers the denoiser of the form \i \i X̂i (Z n×n ) = a(Ck×k ) · Zi + b(Ck×k ), i = 1, . . . , n × n (8) \i in which Ck×k stands for the noisy image patch, or the context, of size k × k surrounding Zi that does not include Zi . Thus, the patch has a hole in the center. Then, we define a neural network g(w, ·) : [0, 1]k that takes the context \i Ck×k 2 −1 → R2+ (9) as input and outputs the slope and intercept parameters \i a(Ck×k ) and \i b(Ck×k ) for each location i. We denote w as the weight parameters of the neural network, which will be learned by the process described in the later sections. As it will get clear in our arguments below, the specific form of our denoiser in (8) enables learning the parameters by both supervised learning with labelled training data and adaptive fine-tuning with the given noisy image. Note in (9), we put a constraint that the slope and intercept of the affine function, i.e., the output of the network, should be nonnegative. While such constraint would appear apparent in our experimental results, it also makes an intuitive sense; the denoiser (8) tries to estimate xi from Zi , which are both in the interval [0, 1], hence, the nonnegative slope and intercept parameters should suffice. The nonnegativity constraint is realized in the neural network by applying f (x) = log(1 + ex ) (10) as the activation function for the final output layer of the neural network. The rest of the network architecture is the ordinary fully-connected neural network with ReLU activation functions, as depicted in Figure 1. There are two sharp differences with our Neural AIDE and other neural network based denoisers, e.g., [11, 12]. First, the other schemes take the full noisy image patch (including the center location) as input to the network, and the network is trained to directly infer the corresponding clean image patches. In contrast, Neural AIDE is trained to first learn an affine mapping based on Figure 1: The arthe noisy image patch with a hole (i.e., the context of Zi ), then the learned chitecture of Neural mapping is applied to Zi to obtain the recostruction X̂i . Such difference AIDE enables the development of the estimated loss function in Lemma 1 and the adaptive training process described in the next section. The principle of learning a mapping first and applying the mapping to the noisy symbol for denoising or filtering has been utilized in [13, 10, 8]. Second, unlike the other schemes, in which the patch-level reconstructions should somehow be aggregated to generate the final denoised image, Neural AIDE simply generates the final pixel-bypixel reconstructions. Thus, there is no need for a step to aggregate multiple number of reconstructed patches, which simplifies the denoising step. Furthermore, since the neural network of Neural AIDE only has to estimate the two parameters of the affine mapping from each context, Neural AIDE can make much more efficient usage of the data with a simpler model compared to the networks in other schemes that need to estimate the full k × k-patch, e.g., [11]. 3.2 Adaptive training with noisy image We first describe how the network parameters w can be adaptively learned from the given noisy image Z n×n without any additional labelled training data. That is, by denoting each output element \i of the neural network g(w, ·) for the context Ck×k as \i \i \i \i g(w, Ck×k )1 , a(Ck×k ) and g(w, Ck×k )2 , b(Ck×k ), we can define an objective function for the neural network to minimize as Ladaptive (w, Z n×n n2 1 X \i \i L Zi , (g(w, Ck×k )1 , g(w, Ck×k )2 ); σ 2 ), 2 n i=1 3 (11) by using the estimated loss function L(Z, (a, b); σ 2 ) defined in Lemma 1. The training process using (11) is identical to the ordinary neural network learning, i.e., start with randomly initiallized w, then use backprogagation and variants of mini-batch SGD for updating the parameters. The formulation (11) may seem similar to training a neural network for a regression problem; namely, 2 \i {(Ck×k , Zi )}ni=1 , which are solely obtained from the noisy image Z n×n , can be analogously thought of as the input-target label pairs for the supervised regression. But, unlike regression, which tries to directly learn a mapping from input to the target label, our network learns the affine mapping for each context and apply it to Zi to estimate the unobserved clean symbol xi . The fact that (11) only depends on the given noisy image Z n×n (and the assumed σ 2 ) makes the learning adaptive. The rationale behind using L(Z, (a, b); σ 2 ) in (11) is the following; as shown in (7), the estimated \i loss is an unbiased estimate of the true expected squared-error given the context Ck×k . Therefore, minimizing (11) may result in the network that produces the slope and intercept parameters that minimize the true MSE for the reconstrunctions of the corresponding affine mappings. This formulation of training neural network parameters solely based on the noisy data is inspired by the recent work in discrete denoising [8]. Once the training is done, we can then denoise the very noisy image Z n×n used for training by applying the affine mapping at each location as (8). That is, by denoting w∗ as the learned parameter by minimizing (11), the reconstruction at location i by Neural AIDE becomes \i \i X̂i,Neural AIDE (Z n×n ) = g(w∗ , Ck×k )1 · Zi + g(w∗ , Ck×k )2 . 3.3 (12) Supervised training and adaptive fine-tuning While the formulation in (11) gives an effective way of adaptively training a denoiser based on the given noisy image Z n×n , the specific form of the denoiser in (8) makes it possible to carry out the supervised pre-training of w before the adaptive training step. That is, we can collect abundant clean images, x̃n×n , from the various image sources (e.g., World Wide Web) and corrupt them with the assumed additive noise with variance σ 2 in (1) to generate the correspoding noisy images, Z̃ n×n , and the labelled training data of size N , D = {(x̃i , C̃i,k×k )}N i=1 . (13) In (13), C̃i,k×k stands for the noisy image patch of size k × k at location i that includes the noisy symbol Z̃i , and x̃i is the clean symbol that correspond to Z̃i . Now, the subtle point is that, unlike the usual supervised learning that may directly learn a mapping from C̃i,k×k to x̃i , we remain in using the neural network defined in (9) and learn w by minimizing N 1 X \i \i Λ x̃i , g(w, C̃k×k )1 · Z̃i + g(w, C̃k×k )2 . Lsupervised (w, D) , N i=1 (14) Note Λ(x, x̂) = (x − x̂)2 as before. The training process of minimizing (14) is again done by the usual backpropagation and the variants of mini-batch SGD. Once the objective function (14) converges after sufficient iteration of weight updates, we denote the converged parameter as w̃. Then, for a given noisy image to denoise, Z n×n , we can further update w̃ adaptively for Z n×n by minimizing Ladaptive (w, Z n×n ) in (11) starting from w̃. That is, we adaptively fine-tune w̃ until Ladaptive (w, Z n×n ) converges, then denoise Z n×n with the converged parameter as (12). This capability of adaptively fine-tuning the supervised trained weight parameter is the unique characteristic of Neural AIDE that differentiates it from other neural network-based denoisers. 4 Experimental Results We compared the denoising performance of the proposed Neural AIDE with several state-of-the-art denoising methods, including BM3D [1], MLP [11], EPLL [6], WNNM [5] and CSF [7]. 4 4.1 Data and experimental setup For the supervised training, we generated the labelled training set using 2000 images available in public datasets. Out of 2000 images, 300 images are taken from train/validation set in the Berkeley Segmentation Dataset and the remaining 1700 images are taken from Pascal VOC 2012 Dataset. For the Pascal VOC images, we resized them to match the resolution of the Berkeley Segmentation Dataset [14], 481 × 321. We corrupted the images with additive Gaussian noise and tested with multiple noise levels, namely, σ = 5, 10, 15, 20, 25. That is, we built separate training set of size 2000 for each noise level. The total number of training data points (i.e., N in (13)) in each dataset was thus about 308 million. We evaluated the performance of the denoisers with 11 standard test images, i.e., {Barbara, Boat, C.man Couple, F.print, Hill, House, Lena, Man, Montage and Peppers}, and 68 standard Berkeley images [3]. Our network had 9 fully connected layers with 512 nodes in each layer, which showed the best result among a few tried models 1 . ReLU was used as activation functions, and we used Adam [15] as the optimizer to train the network. For the supervised training, we trained the network up to 50 epochs and halved the learning rate every 10 epochs starting from 10−4 . For the adaptive fine-tuning, we also trained up to 50 epochs and halved the learning rate every 20 epochs starting from 10−5 . We \i did not use any regularization methods while training. Moreover, for the context data, Ck×k , we subtracted 0.5 from the values to make the input to the network get centered around 0. (Note Zi that the affine mappping gets applied to in (12) still is in the original scale.) For all our experiments, we used Keras (version 1.2.2) with Tensorflow (version 0.11.0) backend and NVIDIA’s GPU (GeForce GTX1080) with CUDA library version 8.0. 4.2 Training Neural AIDE In this section, we systematically show the reasoning behind choosing the context size k, the empirical justification of the nonnegative contraint on the outputs of g(w, ·) and the validity of the combination of the supervised pre-training with adaptive fine-tuning. 4.2.1 Adaptive training with noisy image We first carried out the adaptive training solely with the given noisy image as described in Section 3.2. That is, for each given noisy image, we randomly initialized the weight parameters of the neural network and trained with the objective function (11). After training, the image was denoised as (12). Figure 2(a) shows the PSNR results on the standard 11 test images with varying k values and output activation functions, i.e., Linear (f (x) = x), Positive (f (x) = log(1 + ex ) in (10)) and Sigmoid (f (x) = 1/(1 + e−x )). The noise level was σ = 25. From the figure, we can see that the adaptive training alone can still result in a decent denoiser, although some PSNR gap exists compared to the state-of-the-arts as shown in Table 1. We see that k = 7 tend to be the best context size for adaptive training. Moreover, the choice of the output activation functions turns out to be important, and more discussion is given on the activation function in the next section. 4.2.2 Supervised training and adaptive fine-tuning Since the limitation of the adaptive training alone was apparent, we then carried out the supervised training in Section 3.3. That is, we took the 300 images from the Berkeley Segmentation Dataset and trained the network with varying k values as shown in Figure 2(b). Denoising of the noisy image was done identically as before by applying the learned affine mapping to each noisy pixel. Note in this case, we only carried out the experiments with the Linear activation function. We can see that the supervised training can result in a much higher PSNR values than the adaptive training, already very close to the state-of-the-arts. Also, the performance seems to get saturated around k = 17, so in all our experiments below, we used k = 17. Encouraged by this result, we moved on to adaptively fine-tuning the weight parameters by minimizing the objective function (11) for each image initialized with the parameters learned by supervised learning. This is when the subtle issue regarding the activation function we describe below comes 1 The difference among the models were not huge. 5 (a) Adaptive training (random initialization) (b) Supervised training (300 training images) Figure 2: Adaptive and supervised training results on the standard 11 test images (σ = 25) up. In Figure 3, we trained supervised learning models with Linear and Positive output activation functions using 800 images for σ = 25, then adaptively fine-tuned the parameters for given noisy image (F.print and Montage image). Figure 3(a)-3(d) show the distributions of the slope (a) and intercept (b) paramters that each model outputs for the given image, and 3(i) shows the change of PSNR value in the process of adaptive fine-tuning. From Figure 3(a) and 3(e), we can see that when trained with supervised learning with Linear output activation function, the values of a and b all lie in the interval [0, 1]. However, when fine-tuned for each image, Figure 3(b) and 3(f) show that many negative a values are produced for the Linear activation. This can be readily seen by examining the form of L(Z, (a, b); σ 2 ) in (3), which does not hinder a from having negative values when there is no constraint. As shown in Figure 3(i), such negative a values for the affine mapping sometime does not have big effect on the fine-tuning process and the final denoising performance as in the case of F.print, in which the PSNR increases significantly from the supervised model by fine-tuning. However, as in the case of Montage in Figure 3(i), we suspect such negative a values sometimes hurt the denoising performance greatly. In contrast, when we put the nonnegativitiy contstraint on a and b in the neural network, we observe a stable fine-tuning process, as is observed in Figure 3(d), 3(h) and 3(i). Thus, the results of Neural AIDE from now on all uses the positive activation function. 2 Figure 4 shows the adaptive fine-tuning process of the standard 11 images for σ = 15. The supervised model was trained with the full training set of 2000 images. From the figures, we can see that the learning is done appropriately and the PSNR does improve with fine-tuning. 2 We also tested with the sigmoid activation and the result was more or less the same. 6 (a) F.print(Lin.,s) (b) F.print(Lin.,ft) (e) Montage(Lin.,s) (f) Montage(Lin.,ft) (c) F.print(Pos.,s) (d) F.print(Pos.,ft) (g) Montage(Pos.,s) (h) Montage(Pos.,ft) (i) PSNR values during adaptive fine-tuning. Figure 3: (a-h) Distribution of a and b values for F.print and Montage after supervised training (s) and fine-tuning (ft) for Linear (Lin.) and Positive (Pos.) activation functions. The distributions obtained for fine-tuning are from the models at 50 epoch. (i) PSNR values during fine-tuning. (a) PSNR (b) Objective function (11) Figure 4: PSNR and objective function value during fine-tuning for the standard 11 images (σ = 15) 4.3 4.3.1 Quantitative evaluation Standard 11 images Table 1 summarizes our denoising results compared to the recent state-of-the-arts on the standard 11 images for various noise levels. We show both mean and standard deviation of PSNR values. For the baseline methods, we downloaded the codes from the authors’ webpages and ran the code on the noisy images, thus, the numbers can be compared fairly. (MLP and CSF57×7 could run only on selected noise levels.) N-AIDES stands for the Neural AIDE that is only supervised trained (with 2000 images). N-AIDEfB and N-AIDEfH are fine-tuned models after supervised learning; N-AIDEfB is the best model (in terms of epoch) chosen based on PSNR (thus, not practical) and N-AIDEfH is the model that is chosen with a heuristic rule - i.e., stop fine-tuning when the training loss becomes smaller than σ 2 , otherwise fine-tune until 50 epochs. From the table, we can see that N-AIDEfH significantly outperforms all other baselines on average except for WNNM. The difference of mean PSNR between WNNM and N-AIDEfH is almost 7 negligible and N-AIDEfH tend to have smaller variance in terms of PSNR than WNNM. By comparing N-AIDES and N-AIDEfH , we can definitely see that adaptive fine-tuning is effective. Also, when the noise level is low, the improvement gets larger. Furthermore, by comparing N-AIDES with MLP, which is another neural network based denoiser and uses much more data points (362 million exmample) and larger model, we can confirm that our model more efficiently uses the data. σ 5 10 15 20 25 PSNR Mean Std Mean Std Mean Std Mean Std Mean Std BM3D 38.24 1.24 34.71 1.37 32.76 1.48 31.43 1.50 30.40 1.51 MLP 34.45 1.12 30.24 1.43 EPLL 37.88 1.07 34.27 1.18 32.29 1.35 30.90 1.34 29.81 1.38 WNNM 38.43 1.28 34.95 1.42 32.99 1.54 31.59 1.57 30.51 1.56 CSF57×7 32.40 1.27 29.93 1.41 N-AIDEs 38.14 1.17 34.66 1.31 32.77 1.44 31.38 1.50 30.36 1.53 N-AIDEfB 38.44 1.18 34.92 1.33 32.97 1.42 31.58 1..46 30.51 1.45 N-AIDEfH 38.44 1.18 34.91 1.33 32.96 1.42 31.55 1.44 30.47 1.46 Table 1: PSNR comparsions on the 11 standard benchmark images for σ = 5, 10, 15, 20, 25. Figure 5(a) shows the competitive comparison between N-AIDEfH and the baselines. That is, the figure plots the number of images of which the PSNR of N-AIDEfH is better than the baseline methods. We can see that our method mostly outperforms all baselines competitively, including WNNM. One of the main drawbacks of MLP [11] is that the neural networks have to be trained separately for all noise levels and the mismatch of σ significantly hurts the denoising performance. While the supervised training of Neural AIDE is also done in the similar way, Figure 5(b)-5(c) show that the adaptive fine-tuning can be very effective in overcoming such limitation. Figure 5(b) shows the PSNR results of the mismatched N-AIDEs models before fine-tuning. Each row is normalized with the PSNR of the matched case, i.e., the diagonal element, and the PSNR values are color-coded. We clearly see the sensitivity of PSNR in the mismatch of σ as the off-diagonal values show significant gaps compared to the diagonal values in each row. On the other hand, Figure 5(c) shows the PSNR values of N-AIDEfH ’s that have mismatched supervised models but are adaptively fine-tuned with the correct σ’s. We can clearly see that the PSNR gaps of the mismatched supervised models can be significantly closed by adaptive fine-tuning, which gives a significant edge over MLP in [11]. (a) Competitive comparison (b) PSNR of N-AIDEs (c) PSNR of N-AIDEfH Figure 5: (a) Competitive comparison of N-AIDEfH with baselines (b) PSNR of mismatched N-AIDEs (c) PSNR of N-AIDEfH with mismatched N-AIDEs but fine-tuned with correct σ 4.3.2 Standard 68 Berkeley images Table 2 shows the PSNR results on the 68 standard Berkeley images from [3]. We can clear see that N-AIDEfH again outperforms the baseline state-of-the-art methods, including WNNM, with significant margins. 8 σ 5 10 15 20 25 MLP 33.41 28.73 EPLL 37.50 33.32 31.09 29.60 28.47 WNNM 37.71 33.48 31.18 29.63 28.46 CSF57×7 31.10 28.41 N-AIDEs 37.72 33.62 31.45 29.98 28.93 N-AIDEfB 37.82 33.71 31.52 30.05 28.97 N-AIDEfH 37.79 33.66 31.47 30.00 28.90 Table 2: PSNR comparisons on the 68 standard Berkeley images. 5 Concluding remarks We devised a novel neural network based image denoiser, Neural AIDE. The algorithm is devised with a different principle from the other state-of-the-art methods. As a result, we show that a very simple adaptive affine model, which Neural AIDE learns differently for each pixel, can significantly outperform many strong baselines. Also, the adaptive fine-tuning of Neural AIDE can successfully overcome the σ mismatch problem, which is a serious drawback of other neural network based methods. As a future work, we would like to more thoroughly carry out the experiments in even noisier regime. Also, since our algorithm does not require the noise to be Gaussian (only the additivity of the noise and σ 2 are assumed), we would try to other types of noise, e.g., Laplacian noise. Furthermore, extending our framework to non-additive noise such as multiplicative noise would be another interesting direction. Finally, theoretical anayses of our method based on information theory and learning theory would be another direction worth pursuing. 9 References [1] K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian. Image denoising by sparse 3-d transformdomain collaborative filtering. IEEE Trans. Image Processing, 16(8):2080–2095, 2007. [2] E.P. Simoncelli and E.H. Adelson. Noise removal via bayesian wavelet coring. In ICIP, 1996. [3] S. Roth and M.J Black. Field of experts. IJCV, 82(2):205–229, 2009. [4] J. Mairal, F. Bach, J. Ponce, G. Sapiro, and A. Zisserman. 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Convolutional Neural Networks for Histopathology Image Classification: Training vs. Using Pre-Trained Networks Brady Kieffer1 , Morteza Babaie2 Shivam Kalra1 , and H.R.Tizhoosh1 1 KIMIA Lab, University of Waterloo, ON, CANADA 2 Mathematics and Computer Science Department, Amirkabir University of Technology, Tehran, IRAN arXiv:1710.05726v1 [cs.CV] 11 Oct 2017 e-mails: [email protected], [email protected], [email protected], [email protected] Abstract— We explore the problem of classification within a medical image data-set based on a feature vector extracted from the deepest layer of pre-trained Convolution Neural Networks. We have used feature vectors from several pre-trained structures, including networks with/without transfer learning to evaluate the performance of pre-trained deep features versus CNNs which have been trained by that specific dataset as well as the impact of transfer learning with a small number of samples. All experiments are done on Kimia Path24 dataset which consists of 27,055 histopathology training patches in 24 tissue texture classes along with 1,325 test patches for evaluation. The result shows that pre-trained networks are quite competitive against training from scratch. As well, fine-tuning does not seem to add any tangible improvement for VGG16 to justify additional training while we observed considerable improvement in retrieval and classification accuracy when we fine-tuned the Inception structure. Keywords— Image retrieval, medical imaging, deep learning, CNNs, digital pathology, image classification, deep features, VGG, Inception. I. I NTRODUCTION We are amid a transition from traditional pathology to digital pathology where scanners are replacing microscopes rapidly. Capturing the tissue characteristics in digital formats opens new horizons for diagnosis in medicine. On on hand, we will need to store thousands and thousands of specimens in large physical archives of glass samples. This will be a relief for many hospitals with limited space. On the other hand, acquiring an image from the specimen enables more systematic analysis, collaborations possibilities and, last but not least, the computer-aided diagnosis for pathology, arguable the final frontier of vision-based disease diagnosis. However, like any other technology, digital pathology comes with its own challenges; whole-scan imaging generally generates gigapixel files that also require (digital) storage and are not easy to analyze via computer algorithms. Detection, segmentation, and identification of tissue types in huge digital images, e.g., 50,000×70,000 pixels, appears to be a quite daunting task for computer vision algorithms. Looking at the computer vision community, the emergence of deep learning and its vast possibilities for recognition and classification seems to be a lucky coincidence when we intend to address the above-mentioned obstacles of digital pathology. Diverse deep architectures have been trained with large set of images, e.g., ImageNet project or Faces in the Wild database, to perform difficult tasks like object classification and face recognition. The results have been more than impressive; one may objectively speak of a computational revolution. Accuracy numbers in mid and high 90s have become quite common when deep networks, trained with millions of images, are tested to recognize unseen samples. In spite of all progress, one can observe that the applications of deep learning in digital pathology hast not fully started yet. The major obstacle appears to be the lack of large labelled datasets of histopathology scans to properly train some type of multi-layer neural networks, a requirement that may still be missing for some years to come. Hence, we have to start designing and training deep nets with the available datasets. Training from scratch when we artificially increase the number of images, i.e., data augmentation, is certainly the most obvious action. But we can also use nets that have been trained with millions of (non-medical) images to extract deep features. As a last possibility, we could slightly train (finetune) the pre-trained nets to adjust them to the nature of or data before we use them as feature extractors or classifiers. In this paper, we investigate the usage of deep networks for Kimia Path24 via training from scratch, feature extraction, and fine-tuning. The results show that employing a pre-trained network (trained with non-medical images) may be the most viable option. II. BACKGROUND Over recent years researchers have shown interest in leveraging machine-learning techniques for digital pathology images. These images pose unique issues due to their high variation, rich structures, and large dimensionality. This has lead researchers to investigate various image analysis techniques and their application to digital pathology [1]. For dealing with the large rich structures within a scan, researchers have attempted segmentation on both local and global scales. For example, researchers have conducted works on the segmentation of various structures in breast histopathology images using methods such as thresholding, fuzzy c-means clustering, and adaptive thresholding with varying levels of success [1]–[4]. When applying these methods to histopathological images, it is often desired that a computer aided diagnosis (CAD) method be adopted for use in a content-based image retrieval (CBIR) system. Work has been done to propose various CBIR systems for CAD by multiple groups [5]. Recently, hashing To appear in proceedings of the 7th Intern. Conf. on Image Processing Theory, Tools and Applications (IPTA 2017), Nov 28-Dec 1, Montreal, Canada. methods have been employed for large-scale image retrieval. Among the hashing methods, kernelized and supervised hashing are considered the most effective [5], [6]. More recently Radon barcodes have been investigated as a potential method for creating a CBIR [7]–[9]. Yi et al. utilized CNNs on a relatively small mammography dataset to achieve a classification accuracy of 85% and an ROC AUC of 0.91 whereas handcrafted features were only able to obtain an accuracy of 71% [10]. Currently, there is interest in using pre-trained networks to accomplish a variety of tasks outside of the original domain [11]. This is of great interest for medical tasks where there is often a lack of comprehensive labeled data to train a deep network [12]. Thus, other groups have leveraged networks trained on the ImageNet database which consists of more than 1.2 million categorized images of 1000+ classes [12]–[14]. These groups have reported a general success when attempting to utilize pre-trained networks for medical imaging tasks [12], [14], [15]. In this study we explore and evaluate the performance of a CNN when pre-trained on non-medical imaging data [12], [16]. Specifically, when used as feature extractors with and without fine tuning for a digital pathology task. for the fine-tuning process (we do not use all of them to emulate cases where no large dataset is available; besides more extensive training may destroy what a network has already learned). The values of each patch were subsequently normalized into [0, 1]. The patches were finally downsized to a 224×224 to be fed into the CNN architecture. Following the above steps, we first obtained 27,055 patches from each scan based purely on the homogeneity threshold. Then, randomly sampled 100 patches from each class leading to the much smaller training set of 2,400 patches. A selection of patches from the training set can be viewed within Fig. 1. As Fig. 2 shows the testing samples are relatively balanced in Kimia Path24 dataset, whereas the training set is rather imbalanced. Different size and frequency of specimens are the main reasons for the imbalance. B. Accuracy Calculation The accuracy measures used for the experiments are adopted from [17]. These were chosen so that results between the papers could be compared. There are ntot = 1, 325 testing patches Psj that belong to 24 sets Γs = {Psi \|s ∈ S, i = 1, 2, . . . } with s = 0, 1, 2, . . . , 23 [17]. Looking at the set of retrieved images for an experiment, R, the patch-to-scan accuracy, ηp , can be defined as III. DATA S ET ηp = The data used to train and test the CNNs was the Kimia Path24 consisting of 24 whole scan images (WSIs), manually selected from more than 350 scans, depicting diverse body parts with distinct texture patterns. The images were captured by TissueScope LE 1.0 1 bright field using a 0.75 NA lens. For each image, one can determine the resolution by checking the description tag in the header of the file. For instance, if the resolution is 0.5µm, then the magnification is 20x, and if the resolution is 0.25µm, then the magnification is 40x. The dataset offers 27,055 training patches and 1,325 (manually selected) test patches of size 1000×1000 (0.5mm×0.5mm) [17]. The locations of the test patches in the scans have been removed (whitened) such that they cannot be mistakenly used for training. The color (staining) is neglected in Kimia Path24 dataset; all patches are saved as grayscale images. The Kimia Path24 dataset is publicly available2 . 1 X \|R ∩ Γs \| ntot (1) s∈S The whole-scan accuracy, ηw , can be defined as ηw = 1 X \|R ∩ Γs \| 24 (2) s∈S With the total accuracy is defined as ηtotal = ηp × ηw . By incorporating both the accuracy measurements the resulting problem becomes much more difficult when attempting to obtain acceptable results [17]. IV. M ETHODS Each experiment was run using the architecture for both the VGG16 and Inception-v3 networks as provided in the Keras Python package [18]–[20]. Utilizing a pre-trained network, we then analyze the effectiveness of the network when using it just as a feature extractor, and when transferring the network (some of its weights) to the medical imaging domain. A. Patch Selection To create the Kimia Path24 dataset, each scan is divided into patches that are 1000×1000 pixels in size with no overlap between patches. Background pixels (i.e., very bright pixels) are set to white and ignored using a homogeneity measure for each patch. The homogeneity for selection criterion is that every patch with a homogeneity of less than 99% is ignored. The high threshold ascertains that no patch with significant texture pattern is ignored. From the set of patches each scan had 100 randomly sampled patches are selected to be used A. Fine-Tuning Protocols When fine-tuning a deep network, the optimal setup varies between applications [21]. However, using a pre-trained network and applying it to other domains has yielded better performing models [12]. It was decided that only the final convolutional block (block 5) within VGG16 and the final two inception blocks within Inception-v3 would be re-trained [12], [18], [19], [21]. As in [14] a single fully connected layer 1 http://www.hurondigitalpathology.com 2 http://kimia.uwaterloo.ca 2 To appear in proceedings of the 7th Intern. Conf. on Image Processing Theory, Tools and Applications (IPTA 2017), Nov 28-Dec 1, Montreal, Canada. Fig. 1. A selection of patches from each training scan within the Kimia Path24 dataset. The patches are 1000×1000 pixels in size or 0.5mm×0.5mm. From top left to bottom right: scan/class 0 to scan/class 23. 3 To appear in proceedings of the 7th Intern. Conf. on Image Processing Theory, Tools and Applications (IPTA 2017), Nov 28-Dec 1, Montreal, Canada. Fig. 2. Instance distribution for training set (left) and testing set (right) of Kimia Path24 . of size 256 (followed by an output layer of size 24) was chosen to replace the default VGG16 fully connected layers when fine-tuning. This was found to give better results. The optimizer we used follows the logic from [12], [14] where the learning rate chosen was very small (10−4 ) and the momentum used was large (0.9), both of which were selected to ensure no drastic changes within the weights of the network during training (which would destroy what had been already learned). The Keras data augmentation API was used to generate extra training samples and the network was trained for a total of 200 epochs (after which the accuracy was no longer changing) with a batch size of 32 [20]. softmax classification layer. The fully connected layers were pretrained on bottleneck features and then attached to the convolutional layers and training on the final two inception blocks was then performed. The resulting networks (Transfer Learned VGG16 or TL-VGG16 and TL-Inception-v3) were then used to classify the test patches. The class activation mappings (CAMs) for the fine-tuned Inception-v3 network on randomly selected test patches can be viewed in Fig. 3. V. R ESULTS The results of our experiments are summarized in Table 1. It can be stated the results for VGG16 and CNN1 are quite similar; training from scratch, using a pre-trained network as feature extractor, and fine-tuning a pre-trained network are all delivering comparable results for Kimia Path24 . Whereas the results for Inception-v3 are similar with the transfer-learned model outperforming the feature extractor. As TL-Inceptionv3 produced the best results, ηtotal = 56.98%, and minimally updating the weights of a pre-trained network is not a time consuming task, one may prefer to utilize it. However, one may prefer using Inception-v3 to training from scratch and fine-tuning a pre-trained net as it requires no extra effort and produces similar results with a linear SVM. B. Pre-Trained CNN as a Feature Extractor By using the provided implementation of the specified architectures within Keras, the pre-trained network was first used as a feature extractor without any fine-tuning (Feature Extractor VGG16 or FE-VGG16 and FE-Inception-v3) [20]. The last fully connected layer of the network – prior to classification – was used extracted to be used a feature vector. As pre-trained networks are trained in other domains (very different image categories) and hence cannot be used as classifier, we used the deep features to train a linear Support Vector Machine (SVM) for classification. The Python package scikit-learn as well as LIBSVM were used to train SVM classifiers with a linear kernel [22], [23]. Both NumPy and SciPy were leveraged to manipulate and store data during these experiments [24], [25]. VI. D ISCUSSIONS It was surprising to find out that simply using features from a pre-trained network (trained on non-medical images, see Fig. 4) can deliver results comparable with a network that, with considerable effort and resources, has been trained from scratch for the domain in focus (here histopathology). As well, such simpler approach was even able to achieve a noticeable accuracy increase of ≈ 8.74% in overall performance for Kimia Path24 dataset. Another surprising effect was that transfer learning via fine-tuning for VGG16 was not able to provide any improvement compared to extracting deep features from a pre-trained network without any change in the learned of its weights whereas with Inception-v3 the improvement was immediate. Perhaps the most obvious reaction to this finding is that if we had enough samples, i.e., millions of histopathological images, and if we would use proper computational devices for efficient training, then CNN would perhaps deliver the C. Fine-Tuned CNN as a Classifier The proposed network was then fine-tuned to the Kimia Path24 dataset. Using the Keras library, the convolutional layers were first separated from the top fully connected layers [20]. The training patches were fed through the model to create a set of bottleneck features to initially pre-train the new fullyconnected layers [26]. These features were used to initialize the weights of a fully connected MLP consisting of one 256 dense ReLU layer and a softmax classification layer. Next, the fully connected model was attached to the convolutional layers and training on each convolutional block, except the last block, was performed to adjust classification weights [12], [14]. Similarily, for the Inception-v3 network the fully connected layers were replaced with one 1024 dense ReLU layer and a 4 To appear in proceedings of the 7th Intern. Conf. on Image Processing Theory, Tools and Applications (IPTA 2017), Nov 28-Dec 1, Montreal, Canada. Table 1. Comparing the results training form scratch (CNN1 reported in [17]), using deep features via a pre-trained network with no change (FE-VGG16), and classification after fine-tuning a pre-trained network (TL-VGG16, TL-Inception-v3). The best scores are highlighted in bold. Scheme Train from scratch Pre-trained features Fine-tuning the pre-trained net Pre-trained features Fine-tuning the pre-trained net Approach CNN1 [17] FE-VGG16 TL-VGG16 FE-Inception-v3 TL-Inception-v3 ηp 64.98% 65.21% 63.85% 70.94% 74.87% ηw 64.75% 64.96% 66.23% 71.24% 76.10% ηtotal 41.80% 42.36% 42.29% 50.54% 56.98% Fig. 4. Sample images from ImageNet project. One may object to using features that have been learned from such images in order to classify highly sensitive images of histopathology for medical diagnosis. However, experiments with Kimia Path24 dataset shows that features extracted from these images are expressive enough to compete against networks trained by histopathology images from scratch [Source: http://openai.com/ ]. a deep network, architecture not well suited to the problem, or an overly simplistic fully connected network. However, as previously discussed in [17], the problem given by the Kimia Path24 dataset is indeed a hard problem, most likely due to the high variance between the different patches within a given scan (intra-class variability). This is further validated when looking at the results in Fig. 3. The two columns contain patches that have distinct patterns with their own unique features. The CAM from the first column shows that the network responds strongly to the unique structures within the 4 label (very strongly for the final patch). Whereas when presented with completely different patterns in the second column, the network responds strongly to other areas, typically ones that embody inner edges within the sample. This shows evidence that the model has at the very least begun to learn higher level Fig. 3. Activation maps using randomly selected patches from the Kimia Path24 testing data. The patches within each column are the same class and the labels per column are 4 and 8, respectively. The activation maps are created using the Keras Visualization Toolkit and the Grad-CAM algorithm [27]–[29]. Red areas had more influence on the label prediction [28]. best results clearly better than transfer learning. Although this statement is supported by comparable empirical evidence, it remains speculation for a sensitive field like medical imaging. But why is so difficult to train a CNN for this case? It is most likely due to a number of factors such as a relative lack of image data, the effect of scaling down a patch for use within 5 To appear in proceedings of the 7th Intern. Conf. on Image Processing Theory, Tools and Applications (IPTA 2017), Nov 28-Dec 1, Montreal, Canada. structures within individual patches. Further investigation with different architectures would likely improve upon these results as would more aggressive augmentation. [10] D. Yi, R. L. Sawyer, D. C. III, J. Dunnmon, C. Lam, X. Xiao, and D. Rubin, “Optimizing and visualizing deep learning for benign/malignant classification in breast tumors,” CoRR, vol. abs/1705.06362, 2017. [Online]. Available: http: //arxiv.org/abs/1705.06362 [11] S. J. Pan and Q. Yang, “A survey on transfer learning,” IEEE Transactions on Knowledge and Data Engineering, vol. 22, no. 10, pp. 1345– 1359, Oct 2010. [12] H. C. Shin, H. R. Roth, M. Gao, L. Lu, Z. Xu, I. Nogues, J. Yao, D. Mollura, and R. M. Summers, “Deep convolutional neural networks for computer-aided detection: Cnn architectures, dataset characteristics and transfer learning,” IEEE Transactions on Medical Imaging, vol. 35, no. 5, pp. 1285–1298, May 2016. [13] J. Deng, W. Dong, R. Socher, L.-J. Li, K. Li, and L. Fei-Fei, “Imagenet: A large-scale hierarchical image database,” in Computer Vision and Pattern Recognition, 2009. CVPR 2009. IEEE Conference on. IEEE, 2009, pp. 248–255. [14] R. Girshick, J. Donahue, T. Darrell, and J. Malik, “Region-based convolutional networks for accurate object detection and segmentation,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 38, no. 1, pp. 142–158, Jan 2016. [15] Y. Bar, I. Diamant, L. Wolf, and H. Greenspan, “Deep learning with non-medical training used for chest pathology identification,” in Proc. SPIE, vol. 9414, 2015, p. 94140V. [16] Y. Lecun, L. Bottou, Y. Bengio, and P. Haffner, “Gradient-based learning applied to document recognition,” Proceedings of the IEEE, vol. 86, no. 11, pp. 2278–2324, Nov 1998. [17] M. Babaie, S. Kalra, A. Sriram, C. Mitcheltree, S. Zhu, A. Khatami, S. Rahnamayan, and H. R. Tizhoosh, “Classification and Retrieval of Digital Pathology Scans: A New Dataset.” [Online]. Available: http://arxiv.org/abs/1705.07522 [18] K. Simonyan and A. Zisserman, “Very deep convolutional networks for large-scale image recognition,” CoRR, vol. abs/1409.1556, 2014. [19] C. Szegedy, V. Vanhoucke, S. Ioffe, J. Shlens, and Z. Wojna, “Rethinking the inception architecture for computer vision,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2016, pp. 2818–2826. [20] F. Chollet et al., “Keras,” https://github.com/fchollet/keras, 2015. [21] N. Tajbakhsh, J. Y. Shin, S. R. Gurudu, R. T. Hurst, C. B. Kendall, M. B. Gotway, and J. Liang, “Convolutional neural networks for medical image analysis: Full training or fine tuning?” IEEE Transactions on Medical Imaging, vol. 35, no. 5, pp. 1299–1312, May 2016. [22] F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay, “Scikit-learn: Machine learning in Python,” Journal of Machine Learning Research, vol. 12, pp. 2825–2830, 2011. [23] C.-C. Chang and C.-J. Lin, “LIBSVM: A library for support vector machines,” ACM Transactions on Intelligent Systems and Technology, vol. 2, pp. 27:1–27:27, 2011, software available at http://www.csie.ntu. edu.tw/∼cjlin/libsvm. [24] S. van der Walt, S. C. Colbert, and G. Varoquaux, “The numpy array: A structure for efficient numerical computation,” Computing in Science & Engineering, vol. 13, no. 2, pp. 22–30, 2011. [Online]. Available: http://aip.scitation.org/doi/abs/10.1109/MCSE.2011.37 [25] E. Jones, T. Oliphant, P. Peterson et al., “SciPy: Open source scientific tools for Python,” 2001–, [Online; accessed ¡today¿]. [Online]. Available: http://www.scipy.org/ [26] D. Yu and M. L. Seltzer, “Improved bottleneck features using pretrained deep neural networks,” in Twelfth Annual Conference of the International Speech Communication Association, 2011. [27] R. Kotikalapudi and contributors, “keras-vis,” https://github.com/ raghakot/keras-vis, 2017. [28] R. R. Selvaraju, M. Cogswell, A. Das, R. Vedantam, D. Parikh, and D. Batra, “Grad-cam: Visual explanations from deep networks via gradient-based localization,” See https://arxiv. org/abs/1610.02391 v3, 2016. [29] M. D. Zeiler and R. Fergus, Visualizing and Understanding Convolutional Networks. Cham: Springer International Publishing, 2014, pp. 818–833. [Online]. Available: http://dx.doi.org/10.1007/ 978-3-319-10590-1 53 VII. C ONCLUSIONS Retrieval and classification of histopathological images are useful but challenging tasks in analysis for diagnostic pathology. Whole scan imaging (WSI) generates gigapixel images that are immensely rich in details and exhibit tremendous interand intra-class variance. Both a feature extractor and transferlearned network were able to offer increases in classification accuracy on the Kimia Path24 dataset when compared to a CNN trained from scratch. Comparatively low performance of the latter could be due to the architecture not being well suited for the problem, lack of sufficient number of training images, and/or the inherent difficulty of the classification task for high-resolutional and highly variable histopathology images. Further work would warrant using different architectures for comparison, more aggressive data augmentation, and potentially increasing the size of training samples used from the Kimia Path24 dataset. However, both transfer-learned and feature extractor models were able to compete with the stateof-the-art methods reported in literature [17], and therefore show potential for further improvements. ACKNOWLEDGEMENTS The authors would like to thank Huron Digital Pathology (Waterloo, ON, Canada) for its continuing support. R EFERENCES [1] M. N. Gurcan, L. E. Boucheron, A. Can, A. Madabhushi, N. M. Rajpoot, and B. Yener, “Histopathological image analysis: A review,” IEEE reviews in biomedical engineering, vol. 2, pp. 147–171, 2009. [2] S. Naik, S. Doyle, M. Feldman, J. Tomaszewski, and A. Madabhushi, “Gland segmentation and computerized gleason grading of prostate histology by integrating low-, high-level and domain specific information,” in MIAAB workshop, 2007, pp. 1–8. [3] P. S. Karvelis, D. I. Fotiadis, I. Georgiou, and M. Syrrou, “A watershed based segmentation method for multispectral chromosome images classification,” in Engineering in Medicine and Biology Society, 2006. EMBS’06. 28th Annual International Conference of the IEEE. IEEE, 2006, pp. 3009–3012. [4] S. Petushi, F. U. Garcia, M. M. Haber, C. Katsinis, and A. Tozeren, “Large-scale computations on histology images reveal gradedifferentiating parameters for breast cancer,” BMC medical imaging, vol. 6, no. 1, p. 14, 2006. [5] X. Zhang, W. Liu, M. Dundar, S. Badve, and S. Zhang, “Towards largescale histopathological image analysis: Hashing-based image retrieval,” IEEE Transactions on Medical Imaging, vol. 34, no. 2, pp. 496–506, Feb 2015. [6] W. Liu, J. Wang, R. Ji, Y. G. Jiang, and S. F. Chang, “Supervised hashing with kernels,” in 2012 IEEE Conference on Computer Vision and Pattern Recognition, June 2012, pp. 2074–2081. [7] H. R. Tizhoosh, “Barcode annotations for medical image retrieval: A preliminary investigation,” in Image Processing (ICIP), 2015 IEEE International Conference on. IEEE, 2015, pp. 818–822. [8] H. R. Tizhoosh, S. Zhu, H. Lo, V. Chaudhari, and T. 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arXiv:1709.07929v2 [math.AC] 12 Dec 2017 TRIANGULATED EQUIVALENCES AND RECONSTRUCTION OF CLASSIFYING SPACES HIROKI MATSUI Abstract. In algebra such as algebraic geometry, modular representation theory and commutative ring theory, we study algebraic objects through associated triangulated categories and topological spaces. In this paper, we consider the relationship between such triangulated categories and topological spaces. To be precise, we explore necessary conditions for derived equivalence of Noetherian schemes, stable equivalence of finite groups, and singular equivalence of commutative Noetherian rings by using associated topological spaces. 1. Introduction As is a common approach in many branches of algebra including algebraic geometry, modular representation theory and commutative ring theory, we assign to an algebraic object A (e.g., a scheme X, a finite group G, a commutative Noetherianring R) a triangulated category T (e.g., the perfect derived category Dperf (X), the stable module category mod kG, the singularity category Dsg (R)) and a topological space S (e.g., the underlying topological spaces X, Proj H∗ (G; k), Sing R). By studying such a triangulated category and a topological space, we aim to grasp the structure of the original algebraic object. From this motivation, it is natural to ask what kind of relationship there exists between T and S. algebraic objects A: X, G, R ❖❖❖ ❖❖❖ ❖❖❖ ❖❖' ♥♥ ♥♥♥ ♥ ♥ ♥♥ w ♥♥ ♥ triangulated categories T : Dperf (X), mod kG, Dsg (R) o /o /o /o /o /o /o ??? o/ /o /o /o /o /o /o /o /o / topological spaces S: X, Proj H∗ (G; k), Sing R In this paper, we consider this question, more precisely, the following: Question 1.1. Let A, A′ be algebraic objects, T , T ′ corresponding triangulated categories, and S, S ′ corresponding topological spaces, respectively. Does the implication T ∼ = S′ = T ′ =⇒ S ∼ hold? 2010 Mathematics Subject Classification. 13C14, 13D09, 14F05, 18E30, 19D23, 20C20. Key words and phrases. triangulated category, triangulated equivalence, classifying space, (classifying) support data, quasi-affine scheme, finite p-group, complete intersection. The author is partly supported by Grant-in-Aid for JSPS Fellows 16J01067. 1 2 HIROKI MATSUI We introduce the notion of a classifying space of a triangulated category (see Definition 2.9), and prove the following result, which gives a machinery to answer the above question. Theorem 1.2 (Theorem 3.13). Let T , T ′ be essentially small triangulated categories and S, S ′ classifying spaces for T and T ′ , respectively. Then the implication T ∼ = S′ = T ′ =⇒ S ∼ holds. The key role to prove this theorem is played by the support theory for triangulated categories. For tensor triangulated categories, the support theory has been developed by Balmer [Bal02, Bal05] and is a powerful tool to show such a reconstruction theorem. Since we focus on triangulated categories without tensor structure, we need to invent the support theory without tensor structure. 1.1. Algebraic geometry. Let X be a scheme. The derived category of perfect complexes on X is called the perfect derived category and denoted by Dperf (X). The case where X = Spec R is affine, it is well known that the original scheme is reconstructed from Dperf (R) := Dperf (X). Indeed, for two commutative rings R and S, if the perfect derived categories of R and S are equivalent, then R is isomorphic to S (see [Ric, Proposition 9.2]), and hence Dperf (R) ∼ = Dperf (S) =⇒ Spec R ∼ = Spec S as topological spaces. (∗) However, such a result no longer holds for non-affine schemes. In fact, there exist a lot of non-isomorphic schemes X and Y such that Dperf (X) ∼ = Dperf (Y ); see [Muk, Orl97]. When perf perf ∼ there is a triangulated equivalence D (X) = D (Y ), X and Y are said to be derived equivalent. In section 3, we shall prove that the underlying topological spaces of a certain class of schemes can be reconstructed from their perfect derived categories: Theorem 1.3 (Theorem 3.10). Let X and Y be Noetherian quasi-affine schemes (i.e., open subschemes of affine schemes). Then the implication Dperf (X) ∼ = Dperf (Y ) =⇒ X ∼ = Y as topological spaces holds. This theorem recovers (∗) for Noetherian rings as any affine scheme is quasi-affine. A typical example of a non-affine quasi-affine scheme is the punctured spectrum of a local ring. As an application of this theorem, we obtain that a derived equivalence of X and Y yields the equality of the dimensions of X and Y . 1.2. Modular representation theory. In modular representation theory, finite groups are studied in various contexts. From an algebraic viewpoint, a finite group G has been studied through its group algebra kG and stable module category mod kG, where k is a field whose characteristic divides the order of G. Here, mod kG is a triangulated category consisting of finitely generated kG-modules modulo projectives. On the other hand, the cohomology ring H∗ (G; k) gives an approach to study a finite group G from the topological aspect because it is isomorphic to the cohomology ring of a classifying space BG of G; see [Ben, Chapter 2] for instance. The second main result in section 3 is the following: TRIANGULATED EQUIVALENCES AND RECONSTRUCTION OF CLASSIFYING SPACES 3 Theorem 1.4 (Theorem 3.13). Let k (resp. l) be a field of characteristic p (resp. q), and let G (resp. H) be a finite p-group (resp. q-group). Then the implication mod kG ∼ = mod lH =⇒ Proj H∗ (G; k) ∼ = Proj H∗ (H; l) as topological spaces holds. If there exists a triangulated equivalence mod kG ∼ = mod lH, we say that kG and lH are stably equivalent. As an application of this theorem, we have that a stable equivalence of kG and lH yields that the p-rank of G and the q-rank of H are equal. 1.3. Commutative ring theory. Let R be a left Noetherian ring. The singularity category of R is by definition the Verdier quotient Dsg (R) := Db (modR)/Dperf (R), which has been introduced by Buchweitz [Buc] in 1980s. Here, mod R stands for the category of finitely generated left R-modules and Db (modR) its bounded derived category. The singularity categories have been deeply investigated from algebro-geometric and representation-theoretic motivations [Che, IW, Ste, Tak] and connected to the Homological Mirror Symmetry Conjecture by Orlov [Orl04]. One of the important subjects in representation theory of rings is to classify rings up to certain category equivalence. For example, left Noetherian rings R and S are said to be: • Morita equivalent if mod R ∼ = mod S as abelian categories, • derived equivalent if Db (mod R) ∼ = Db (mod S) as triangulated categories, • singularly equivalent if Dsg (R) ∼ = Dsg (S) as triangulated categories. It is well known that these equivalences have the following relations: Morita equivalence ⇒ derived equivalence ⇒ singular equivalence. Complete characterizations of Morita and derived equivalence have already been obtained in [Mor, Ric], while singular equivalence is quite difficult to characterize even in the case of commutative rings. Indeed, only a few examples of singular equivalences of commutative Noetherian rings are known. Furthermore, for all of such known examples, the singular loci of rings are homeomorphic. Thus, it is natural to ask the following question. Question 1.5. Let R and S be commutative Noetherian rings. Are their singular loci homeomorphic if R and S are singularly equivalent? In section 4, we show that this question is affirmative for certain classes of commutative Noetherian rings. To be precise, we shall prove the following theorem. Theorem 1.6 (Theorem 4.4). Let R and S be commutative Noetherian local rings that are locally hypersurfaces on the punctured spectra. Assume that R and S are either (a) complete intersection rings, or (b) Cohen-Macaulay rings with quasi-decomposable maximal ideal. Then the implication Dsg (R) ∼ = Dsg (S) =⇒ Sing R ∼ = Sing S as topological spaces holds. 4 HIROKI MATSUI Here, we say that an ideal I of a commutative ring R is quasi-decomposable if there is an R-regular sequence x in I such that I/(x) is decomposable as an R-module. Moreover, we prove that singular equivalence localizes by using such a homeomorphism. The organization of this paper is as follows. In section 2, we introduce the notions of a support data and a classifying support data for a given triangulated category and develop the support theory without tensor structure, and finally prove Theorem 1.2. In section 3, we connect the results obtained in section 2 with the support theory for tensor triangulated categories and study reconstructing the topologies of the Balmer spectra without tensor structure. Using this method, we prove Theorem 1.3 and 1.4. In section 4, we prove Theorem 1.6 and give examples of commutative rings which are not singularly equivalent. Throughout this paper, all categories are assumed to be essentially small. For two triangulated category T , T ′ (resp. topological spaces X, X ′ ), the notation T ∼ = T ′ (resp. X∼ = X ′ ) means that T and T ′ are equivalent as triangulated categories (resp. X and X ′ are homeomorphic) unless otherwise specified. 2. The support theory without tensor structure In this section, we discuss the support theory for triangulated categories without tensor structure. Throughout this section, T denotes a triangulated category with shift functor Σ. First of all, let us recall some basic definitions which are used in this section. Definition 2.1. Let X be a topological space and T a triangulated category. (1) We say that X is sober if every irreducible closed subset of X is the closure of exactly one point. (2) We say that X is Noetherian if every descending chain of closed subspaces stabilizes. (3) We say that a subset W of X is specialization-closed if it is closed under specialization, namely if an element x of X belongs to W , then the closure {x} is contained in W . Note that W is specialization-closed if and only if it is a union of closed subspaces of X. (4) We say that a non-empty additive full subcategory X of T is thick if it satisfies the following conditions: (i) closed under taking shifts: ΣX = X . (ii) closed under taking extensions: for a triangle L → M → N → ΣL in T , if L and N belong to X , then so does M. (iii) closed under taking direct summands: for two objects L, M of T , if the direct sum L ⊕ M belongs to X , then so do L and M. For a subcategory X of T , denote by thickT X the smallest thick subcategory of T containing X . We introduce the notion of a support data for a triangulated category. Definition 2.2. Let T be a triangulated category. A support data for T is a pair (X, σ) where X is a topological space and σ is an assignment which assigns to an object M of T a closed subset σ(M) of X satisfying the following conditions: (1) σ(0) = ∅. (2) σ(Σn M) = σ(M) for any M ∈ T and n ∈ Z. (3) σ(M ⊕ N) = σ(M) ∪ σ(N) for any M, N ∈ T . TRIANGULATED EQUIVALENCES AND RECONSTRUCTION OF CLASSIFYING SPACES 5 (4) σ(M) ⊆ σ(L) ∪ σ(N) for any triangle L → M → N → ΣL in T . Support data naturally appear in various areas of algebras. Example 2.3. (1) Let R be a commutative Noetherian ring. For M ∈ Dsg (R), we define the singular support of M by SSupp (M) := {p ∈ Sing R \| Mp ∼ 6= 0 in Dsg (Rp )}. R Then (Sing R, SSuppR ) is a support data for Dsg (R). Indeed, it follows from [AIL, Theorem 1.1] and [BM, Lemma 4.5] that SSuppR (M) is a closed subset of Sing R and that SSuppR satisfies the condition (1) in Definition 2.2. The remained conditions (2)-(4) are clear because the localization functor Dsg (R) → Dsg (Rp ) is exact. Assume that R is Gorenstein. Denote by CM(R) the category of maximal CohenMacaulay R-modules (i.e., modules M satisfying ExtiR (M, R) = 0 for all integers i > 0). Recall that the stable category CM(R) of CM(R) is the category whose objects are the same as CM(R) and the set of morphisms from M to N is given by HomR (M, N) := HomR (M, N)/PR (M, N), where PR (M, N) consists of all R-linear maps from M to N factoring through some free R-module. Then the stable category CM(R) has the structure of a triangulated category; see [Hap]. Moreover, the natural inclusion induces a triangle equivalence ∼ = → Dsg (R) by [Buc]. Thus we obtain the support data (Sing R, SuppR ) for F : CM(R) − CM(R) by using this equivalence. Here, Supp (M) := SSupp (F (M)) = {p ∈ Sing R \| Mp ∼ 6= 0 in CM(Rp )} R R for M ∈ CM(R). (2) Let X be a Noetherian scheme. For F ∈ Dperf (X), we define the cohomological support of F by SuppX (F ) := {x ∈ X \| Fx ∼ 6= 0 in Dperf (OX,x )}. S Then, SuppX (F ) = n∈Z SuppX (Hn (F )) is a finite union of supports of coherent OX modules and hence is a closed subspace of X. Moreover, (X, SuppX ) is a support data for Dperf (X) because the localization is exact. For details, please see [Tho]. (3) Let k be a field of characteristic p > 0 and G a finite group such that p divides the order of G. Then as in the case of Gorenstein rings, we can define the stable category mod kG of mod kG and it is also a triangulated category. We denote by ( ⊕i∈Z Hi (G; k) p = 2 H∗ (G; k) = ⊕i∈2Z Hi (G; k) p : odd the direct sum of cohomologies of G with coefficient k. Then H∗ (G; k) has the structure of a graded-commutative Noetherian ring by using the cup product and we can consider its homogeneous prime spectrum Proj H∗ (G; k). Denote by VG (M) the support variety for a finitely generated kG-module M which is a closed space of Proj H∗ (G; k). Then the pair (Proj H∗ (G; k), VG ) becomes a support data for mod kG. For details, please refer to [Ben, Chapter 5]. Remark 2.4. Actually, the above examples of support data satisfy the following stronger condition: (1′ ) σ(M) = ∅ if and only if M ∼ = 0. 6 HIROKI MATSUI Definition 2.5. Let U be a full subcategory of T . We say that U is a ⊕-ideal if it satisfies M ∈ U, N ∈ T ⇒ M ⊕ N ∈ U. Remark 2.6. U ⊆ T is a ⊕-ideal if and only if T \ U is closed under taking direct summands. Example 2.7. (1) The full subcategory T \ {0} is a ⊕-ideal. (2) The full subcategory T(T ) of test objects (see Definition 4.8 below) of T is a ⊕-ideal. Let us fix the following notations: Notation 2.8. Let T be a triangulated category, U ⊆ T a ⊕-ideal, and X a topological space. Then we set: • Th(T ) := {thick subcategories of T }, • ThU (T ) := {thick subcategories of T containing an object of U}, • Spcl(X) := {specialization closed subsets of X}, • Nesc(X) := {non-empty specialization-closed subsets of X}, • Nec(X) := {non-empty closed subsets of X}, • Irr(X) := {irreducible closed subsets of X}. Let (X, σ) be a support data for T , X a thick subcategory of T , and W S a specializationclosed subset of X. Then one can easily check that fσ (X ) := σ(X ) := M ∈X σ(M) is a specialization-closed subset of X and gσ (W ) := σ −1 (W ) := {M ∈ T \| σ(M) ⊆ W } is a thick subcategory of T . Therefore, we obtain two order-preserving maps with respect to the inclusion relations. Definition 2.9. Let (X, σ) be a support data for T and U ⊆ T a ⊕-ideal. Then we say that (X, σ) is a classifying support data for T with respect to U if (i) X is a Noetherian sober space, and (ii) the above maps fσ and gσ restrict to mutually inverse bijections: ThU (T ) o fσ / gσ Nesc(X). When this is the case, we say that X is a classifying space of T with respect to U. We say simply a classifying support data for T (resp. a classifying space of T ), we mean a classifying support data for T (resp. a classifying space of T ) with respect to T \ {0}. Remark 2.10. A classifying support data (X, σ) for T classifies all thick subcategories of T containing gσ (∅) = σ −1 (∅). Indeed, the map gσ : Nesc(X) → ThU (T ) is injective with image {X ∈ Th(T ) \| X ) σ −1 (∅)}. Thus, we obtain a one-to-one correspondence {X ∈ Th(T ) \| X ⊇ σ −1 (∅)} o fσ gσ / Spcl(X). In particular, if (X, σ) satisfies the condition (1′ ) in Remark 2.4, we obtain a one-to-one correspondence: Th(T ) o fσ gσ / Spcl(X). Every classifying support data automatically satisfies the following realization property. TRIANGULATED EQUIVALENCES AND RECONSTRUCTION OF CLASSIFYING SPACES 7 Lemma 2.11. Let (X, σ) be a classifying support data for T with respect to U. Then for any non-empty closed subset Z of X, there is an object M of U, such that Z = σ(M). Proof. Since X is a Noetherian sober space and σ(M) ∪ σ(N) = σ(M ⊕ N), we may assume that Z = {x} for some x ∈ X. From the assumption, one has Z = fσ gσ (Z) = S M ∈gσ (Z) σ(M). Hence, there is an element x of σ(M) for some M ∈ gσ (Z). Then we obtain x ∈ σ(M) ⊆ Z = {x} and this implies that σ(M) = {x} = Z. By definition of a classifying support data with respect to U, gσ (Z) = {N ∈ T \| σ(N) ⊆ σ(M)} contains a object T of U. We conclude that σ(T ⊕M) = σ(T )∪σ(M) = σ(M) = Z for T ⊕ M ∈ U. Let me give two more notations. Definition 2.12. Let U be a ⊕-ideal of T . (1) We say that a thick subcategory X of T is U-principal if there is an object M of U such that X = thickT M. Denote by PThU (T ) the set of all U-principal thick subcategories of T . (2) We say that a U-principal thick subcategory X of T is U-irreducible if X = thickT (X1 ∪ X2 ) (X1 , X2 ∈ PThU (T )) implies that X1 = X or X2 = X . Denote by IrrU (T ) the set of all U-irreducible thick subcategories of T . The following lemma shows that by using classifying support data with respect to U, we can also classify U-principal thick subcategories and U-irreducible thick subcategories. Lemma 2.13. Let (X, σ) be a classifying support data for T with respect to U, then the one-to-one correspondence fσ ThU (T ) o / Nesc(X) gσ restricts to one-to-one correspondences PThU (T ) o IrrU (T ) o fσ / gσ fσ / gσ Nec(X), Irr(X). Proof. Note that fσ (thickT M) = σ(M) for any M ∈ T . Therefore, the injective map fσ : ThU (T ) → Nesc(X) induces a well defined injective map fσ : PThU (T ) → Nec(X). The surjectivity has already been shown in Lemma 2.11. Next, we show the second one-to-one correspondence. For X1 , X2 ∈ ThU (T ), one has [ (1) fσ (thickT (X1 ∪ X2 )) = σ(M) M ∈thickT (X1 ∪X2 ) = [ σ(M) M ∈X1 ∪X2 =( [ M ∈X1 σ(M)) ∪ ( [ M ∈X2 = fσ (X1 ) ∪ fσ (X2 ). σ(M)) 8 HIROKI MATSUI On the other hand, for Z1 , Z2 ∈ Nesc(X), one has fσ (thickT (gσ (Z1 ) ∪ gσ (Z2 ))) = fσ (gσ (Z1 )) ∪ fσ (σ(Z2 )) = Z1 ∪ Z2 . Applying gσ to this equality, we get (2) thickT (gσ (Z1 ) ∪ gσ (Z2 )) = gσ (Z1 ∪ Z2 ). Let W be an irreducible closed subset of X. Assume gσ (W ) = thickT (X1 ∪ X2 ) for some X1 , X2 ∈ PThU (T ). Then from the above equality (1), we obtain an equality W = fσ (gσ (W )) = fσ (thickT (X1 ∪ X2 )) = fσ (X1 ) ∪ fσ (X2 ). Since W is irreducible, fσ (X1 ) = W or fσ (X2 ) = W and hence X1 = gσ (fσ (X1 )) = gσ (W ) or X2 = gσ (fσ (X2 )) = gσ (W ). This shows that gσ (W ) is U-irreducible. Conversely, take a U-irreducible thick subcategory X of T and assume fσ (X ) = Z1 ∪ Z2 for some non-empty closed subsets Z1 , Z2 of X. From the above equality (2), we get X = gσ (fσ (X )) = gσ (Z1 ∪ Z2 ) = thickT (gσ (Z1 ) ∪ gσ (Z2 )). Since X is U-irreducible, X = gσ (Z1 ) or X = gσ (Z2 ) and therefore, Z1 = fσ (gσ (Z1 )) = fσ (X ) or Z2 = fσ (gσ (Z2 )) = fσ (X ). Thus, fσ (X ) is irreducible. These observations show the second one-to-one correspondence. From this lemma, we can show the following uniqueness result for classifying support data with respect to U. Proposition 2.14. Let (X, σ) and (Y, τ ) be classifying support data for T with respect to a ⊕-ideal U. Then X and Y are homeomorphic. Proof. First note that for a topological space X, the natural map ιX : X → Irr(X), x 7→ {x} is bijective if and only if X is sober. Define maps ϕ : X → Y and ψ : Y → X to be the composites ι gσ fτ ι−1 Y X → Y, Irr(X) −→ IrrU (T ) −→ Irr(Y ) −− ϕ : X −→ ι gτ fσ ι−1 Y X ψ : Y −→ Irr(Y ) − → IrrU (T ) −→ Irr(X) −− → X. Then ϕ and ψ are well defined and mutually inverse bijections by Lemma 2.13. Fix x ∈ X. For x′ ∈ {x}, one has ιX (x′ ) ⊆ ιX (x) and hence {ϕ(x′ )} = ιY (ϕ(x′ )) = fτ (gσ (ιX (x′ ))) ⊆ {ϕ(x)} = ιY (ϕ(x)) = fτ (gσ (ιX (x))). In particular, ϕ(x′ ) belongs to {ϕ(x)}. Therefore, ϕ({x}) ⊆ {ϕ(x)}. Conversely, for y ∈ {ϕ(x)}, the above argument shows ψ(y) ∈ ψ({ϕ(x)}) ⊆ {ψϕ(x)} = {x}. Applying ϕ to this inclusion, we obtain y ∈ ϕ({x}) and therefore, {ϕ(x)} ⊆ ϕ({x}). Thus, we conclude that ϕ({x}) = {ϕ(x)}. Since X is Noetherian, this equation means that ϕ is a closed map. Similarly, ψ is also a closed map. The following theorem is the main result of this section. Theorem 2.15. Consider the following setting: • T and T ′ are triangulated categories. TRIANGULATED EQUIVALENCES AND RECONSTRUCTION OF CLASSIFYING SPACES 9 • U and U ′ are ⊕-ideals of T and T ′ , respectively. • (X, σ) and (Y, τ ) are classifying support data for T and T ′ with respect to U and U ′ , respectively. Suppose that there is a triangle equivalence F : T → T ′ with F (U) = U ′ . Then X and Y are homeomorphic. Proof. From the assumption, F induces a one-to-one correspondence ∼ = → ThU ′ (T ′ ), X 7→ F̃ (X ), F̃ : ThU (T ) − where F̃ (X ) := {N ∈ T ′ \| ∃M ∈ X such that N ∼ = F (M)}. For an object M of T , set τ F (M) := τ (F (M)). Then we can easily verify that the pair (Y, τ F ) is a support data for T . Furthermore, it becomes a classifying support data for T with respect to U. Indeed, for X ∈ ThU (T ) and W ∈ Nesc(Y ), we obtain [ [ [ fτ F (X ) = τ F (M) = τ (F (M)) = τ (N) = fτ (F̃ (X )), M ∈X M ∈X N ∈F̃ (X ) F̃ (gτ F (W )) = F̃ ({M ∈ T \| τ F (M) ⊆ W }) = {N ∈ T ′ \| τ (N) ⊆ W } = gτ (W ). From these equalities, we get equalities fτ F = fτ ◦ F̃ and F̃ ◦ gτ F = gτ and thus fτ F and gτ F give mutually inverse bijections between ThU (T ) and Nesc(Y ). Consequently, we obtain two classifying support data (X, σ) and (Y, τ F ) for T with respect to U, and hence X and Y are homeomorphic by Proposition 2.14. 3. Comparison with tensor triangulated structure In this section, we discuss relation between the support theory we discussed in section 2 and the support theory for tensor triangulated categories. Recall that a tensor triangulated category (T , ⊗, 1) consists of a triangulated category T together with a symmetric monoidal tensor product ⊗ with unit object 1 which is compatible with the triangulated structure of T . For the precise definition, please refer to [HPS, Appendix A]. Example 3.1. (1) Let X be a Noetherian scheme. Then (Dperf (X), ⊗LOX , OX ) is a tensor triangulated category. Here, ⊗LOX denotes the derived tensor product. (2) Let k be a field and G a finite group. Then (mod kG, ⊗k , k) is a tensor triangulated category. Throughout this section, fix a tensor triangulated category (T , ⊗, 1). We begin with recalling some basic definitions which are used in the support theory of tensor triangulated categories. Definition 3.2. (1) A full subcategory X of T is called a thick tensor ideal if it is a thick subcategory of T and is closed under the action of T by ⊗: M ⊗ N ∈ X for any M ∈ X and N ∈ T . For a subcategory X of T , denote by hX i the smallest thick tensor ideal of T containing X . 10 HIROKI MATSUI (2) For a thick subcategory X of T , define its radical by √ X := {M ∈ T \| ∃n > 0 such that M ⊗n ∈ X }. Here, M ⊗n denotes the n-fold tensor product of M. By [Bal05, Lemma 4.2], the radical of a thick subcategory is always a thick tensor ideal. √ A thick tensor ideal X of T is called radical if it satisfies X = X . (3) A thick tensor ideal X of T is called prime if it satisfies M ⊗ N ∈ X ⇒ M ∈ X or N ∈ X . Denote by Spc T the set of all prime thick tensor ideals of T . (4) For M ∈ T , the Balmer support of M is defined as SppM := {P ∈ Spc T \| M ∈ / P}. The set Spc T is a topological space with closed basis {SppM \| M ∈ T } and call it the Balmer spectrum of T . (5) Let X be a topological space. We say that a subset W of X is a Thomason subset if it is a union of closed subsets whose complements are quasi-compact. Denote by Thom(X) the set of all Thomason subsets of X. Note that Thom(X) ⊆ Spcl(X). We say that a support data (X, σ) for T is tensorial if it satisfies: σ(M ⊗ N) = σ(M) ∩ σ(N) for any M, N ∈ T . In [Bal05], tensorial support data are called simply support data. Then gσ (W ) is a radical thick tensor ideal of T for every specialization-closed subset W of X. We say that a tensorial support data (X, σ) is classifying if X is a Noetherian sober space and there is a one-to-one correspondence: {radical thick tensor ideals of T } o fσ / gσ Spcl(X). Balmer showed the following celebrated result: Theorem 3.3. [Bal05, Lemma 2.6, Theorem 4.10] (1) The pair (Spc T , Spp) is a tensorial support data for T . (2) There is a one-to-one correspondence: {radical thick tensor ideals of T } o fSpp gSpp / Thom(Spc T ). Remark 3.4. If a topological space X is Noetherian, then every specialization-closed subset of X is Thomason. Therefore, the above theorem shows that (Spc T , Spp) is a classifying tensorial support data for T provided Spc T is Noetherian. Recall that a tensor triangulated category T is rigid if (1) the functor M ⊗ − : T → T has a right adjoint F (M, −) : T → T for each M ∈ T and (2) every object M is strongly dualizable (i.e., the natural map F (M, 1) ⊗ N → F (M, N) is an isomorphism for each N ∈ T ). If T is rigid, then (Spc T , Spp) satisfies the stronger condition. Lemma 3.5. Assume that T is rigid. Then the support data (Spc T , Spp) satisfies the condition (1′ ) in Remark 2.4. TRIANGULATED EQUIVALENCES AND RECONSTRUCTION OF CLASSIFYING SPACES 11 Proof. Take an object M ∈ T with Spp(M) = ∅. By [Bal05, Corollary 2.4], there is a positive integer n such that M ⊗n ∼ = 0. On the other hand, by [HPS, Lemma A 2.6], M i ⊗ 2i belongs to thickT (M ) for any positive integer since every object is strongly dualizable. Therefore, by using induction, we conclude that M ∼ = 0. Note that a tensorial classifying support data for T is a classifying tensorial support data for T . Indeed, for a tensorial classifying support data (X, σ) for T and X ∈ Th(T ), we obtain an equalities p p X = gσ (fσ (X )) = gσ (fσ ( thick⊗ X )) = thick⊗ X . The following lemma gives a criterion for the converse implication of this fact. Lemma 3.6. Let (X, σ) be a classifying tensorial support data for T . Suppose that T is rigid. Then the following are equivalent: (1) There is a one-to-one correspondence: Th(T ) o fσ gσ / Spcl(X). (2) (X, σ) is a classifying support data for T . (3) Every thick subcategory of T is a thick ⊗-ideal. (4) T = thickT 1. Proof. By Lemma 3.5 and Theorem [Bal05, Theorem 5.2], (X, σ) satisfies the condition (1′ ) in Remark 2.4. Therefore, (1) and (2) means the same conditions from Remark 2.10. (1) ⇒ (3): From the assumption, every thick subcategory X of T is of the form X = gσ (W ) for some specialization-closed subset W of X. On the other hand, gσ (W ) is a radical thick ⊗-ideal as (X, σ) is a tensorial support data. (3) ⇒ (4): By assumption, the thick subcategory thickT 1 is a thick tensor ideal. Thus, for any M ∈ T , M ∼ = M ⊗ 1 belongs to thickT 1. (4) ⇒ (1): Note that 1 is strongly dualizable and the family of all strongly dualizable objects forms a thick subcategory of T by [HPS, Theorem A.2.5 (a)]. Therefore, every object of T = thickT 1 is strongly dualizable. Thus, for any object M ∈ T , M belongs to thick⊗ T (M ⊗ M) by [HPS, Lemma A.2.6]. Then [Bal05, Proposition 4.4] shows that every thick tensor ideal of T is radical. On the other hand, for any thick subcategory X of Y, one can easily verify that the subcategory Y := {M ∈ T \| M ⊗ X ⊆ X } is a thick ⊗-ideal of T containing 1. Thus, we obtain Y = thickT 1 = T and hence X is a thick ⊗-ideal. From these discussion, we conclude that every thick subcategory of T is a radical thick ⊗-ideal and this shows the implication (4) ⇒ (1). The following corollaries are direct consequences of this lemma, Proposition 2.14 and Theorem 2.15. Corollary 3.7. Let T be a rigid tensor triangulated category. Assume that the Balmer spectrum Spc T of T is Noetherian and that T = thickT 1. Then for any classifying support data (X, σ) for T , X is homeomorphic to Spc T . Corollary 3.8. Let T and T ′ be rigid tensor triangulated categories such that 12 HIROKI MATSUI (1) Spc T and Spc T ′ are Noetherian, and (2) T and T ′ are generated by their unit objects. If T and T ′ are equivalent as triangulated categories, then Spc T and Spc T ′ are homeomorphic. Next, we consider applications of these corollaries to tensor triangulated categories appeared in Example 3.1. Thomason showed the following classification theorem of thick tensor ideas of Dperf (X): Theorem 3.9. [Tho, Theorem 3.15] Let X be a Noetherian scheme. Then (X, SuppX ) is a classifying tensorial support data for Dperf (X). As an application of Corollary 3.8, we can reconstruct underlying topological spaces of a certain class of schemes from their perfect derived categories without tensor structure. Theorem 3.10. Let X and Y be Noetherian quasi-affine schemes (i.e., open subschemes of affine schemes). If X and Y are derived equivalent, then X and Y are homeomorphic. In particular, topologically determined properties, such as the dimensions and the numbers of irreducible components of quasi-affine Noetherian schemes are preserved by derived equivalences. Proof. First, let me remark that the functor F ⊗LOX − : Dperf (X) → Dperf (X) has a right adjoint RHomOX (F , −) : Dperf (X) → Dperf (X) for each F ∈ Dperf (X) and moreover Dperf (X) is rigid. Note that a scheme X is quasi-affine if and only if its structure sheaf OX is ample. Thus, every thick subcategory of Dperf (X) is thick tensor ideal by [Tho, Proposition 3.11.1]. Applying Corollary 3.8, we obtain the result. Remark 3.11. Let X and Y be Noetherian schemes. (1) As we have already remarked in the introduction, if X and Y are affine, then a derived equivalence Dperf (X) ∼ = Dperf (Y ) implies that X and Y are isomorphic as schemes. (2) By [Bal02, Theorem 9.7], if Dperf (X) and Dperf (Y ) are equivalent as tensor triangulated categories, then X and Y are isomorphic as schemes. Next consider stable module categories over group rings of finite groups. In this case, the following classification theorem is given by Benson-Carlson-Rickard for algebraically closed field k and by Benson-Iyengar-Krause for general k. Theorem 3.12. [BCR, BIK] Let k be a field of characteristic p > 0 and G a finite group such that p divides the order of G. Then the support data (Proj H∗ (G; k), VG ) is a classifying tensorial support data for mod kG. Applying Corollary 3.8 to this classifying tensorial support data, we obtain the following result: Theorem 3.13. Let k (resp. l) be field of characteristic p (resp. q), G (resp. H) be a finite p-group (resp. q-group). If kG and lH are stably equivalent, then Proj H∗ (G; k) and Proj H∗ (H; l) are homeomorphic. Proof. For each M ∈ mod kG, the functor M ⊗k − : mod kG → mod kG has a right adjoint Homk (M, −) : mod kG → mod kG and in addition mod kG is rigid. Moreover, for a pgroup G, kG has only one simple module k. Therefore, we have mod kG = thickmod kG k. Applying Corollary 3.8, we are done. TRIANGULATED EQUIVALENCES AND RECONSTRUCTION OF CLASSIFYING SPACES 13 Recall that the p-rank of a finite group G is by definition, rp (G) := sup{r \| (Z/p)r ⊆ G}. Quillen [Qui] showed that the dimension of the cohomology ring H ∗ (G; k) is equal to the p-rank of G. Thus, the p-rank is an invariant of stable equivalences: Corollary 3.14. Let k, l, G, H be as in Theorem 3.13. Assume that there is a stable equivalence between kG and lH, then rp (G) = rq (H). Remark 3.15. Let G and H be a p-group and k a field of characteristic p. (1) By [Lin, Corollary 3.6], if there exists a stable equivalence between kG and kH, then \|G\| = \|H\|. (2) By [Lin, Corollary 3.2], if there exists a stable equivalence of Morita type between kG and kH, then G ∼ = H. 4. A necessary condition for singular equivalences Recall that commutative Noetherian rings R and S are said to be singularly equivalent if their singularity categories are equivalent as triangulated categories. The only known examples of singular equivalences are the following: Example 4.1. (1) If R ∼ = Dsg (S). = S, then Dsg (R) ∼ (2) If R and S are regular, then Dsg (R) ∼ = Dsg (S). =0∼ (3) (Knörrer’s periodicity [Yos, Chapter 12]) Let k be an algebraically closed field of characteristic 0. Set R := k[[x0 , x1 , ..., xd ]]/(f ) and S := k[[x0 , x1 , ..., xd , u, v]]/(f +uv). Then Dsg (R) ∼ = Dsg (S). Remark 4.2. All of these singular equivalences, the singular loci Sing R and Sing S are homeomorphic. In fact, the cases (1) and (2) are clear. Consider the case of R := k[[x0 , x1 , ..., xd ]]/(f ) and S := k[[x0 , x1 , ..., xd , u, v]]/(f + uv). Then Sing S = V(∂f /∂x0 , . . . ∂f /∂xd , u, v) ∼ = Spec(S/(∂f /∂x0 , . . . , ∂f /∂xd , u, v)) ∼ = Spec(k[[x0 , x1 , ..., xd , u, v]]/(f + uv, ∂f /∂x0 , . . . , ∂f /∂xd , u, v)) ∼ = Spec(k[[x0 , x1 , ..., xd ]]/(f, ∂f /∂x0 , . . . , ∂f /∂xd ) ∼ = V(∂f /∂x0 , . . . ∂f /∂xd ) = Sing R. Here, the first and the last equalities are known as the Jacobian criterion. Let me give some definitions appearing in the statement of the main theorem of this section. Definition 4.3. Let (R, m, k) be a commutative Noetherian local ring. (1) We say that an ideal I of R is quasi-decomposable if there is an R-regular sequence x of I such that I/(x) is decomposable as an R-module. (2) A local ring R is said to be complete intersection if there is a regular local ring S and an S-regular sequence x such that the completion R̂ of R is isomorphic to S/(x). We say that R is a hypersurface if we can take x to be an S-regular sequence of length 1. (3) A local ring R is said to be locally a hypersurface on the punctured spectrum if Rp is a hypersurface for every non-maximal prime ideal p. 14 HIROKI MATSUI The following theorem is the main result of this section. Theorem 4.4. Let R and S be commutative Noetherian local rings that are locally hypersurfaces on the punctured spectra. Assume that R and S are either (a) complete intersection rings, or (b) Cohen-Macaulay rings with quasi-decomposable maximal ideal. If R and S are singularly equivalent, then Sing R and Sing S are homeomorphic. For a ring R satisfying the condition (b) in Theorem 4.4, Nasseh-Takahashi [NT, Theorem B] shows that (Sing R, SSuppR ) is a classifying support data for Dsg (R). Therefore, the statement of Theorem 4.4 follows from Theorem 2.15. Therefore, the problem is the case of (a). For a ring R satisfying the condition (a) in Theorem 4.4, Takahashi [Tak] classified thick subcategories of Dsg (R) containing the residue field k of R by using the singular locus Sing R and the singular support SSuppR . We would like to apply Theorem 2.15 also for this case. The problem is that whether the condition “containing the residue field k” is preserved by stable equivalences. As we will show later, this condition is actually preserved by singular equivalences for local complete intersection rings. To do this, we discuss replacing the residue field k with some categorically defined object. First of all, let us recall the notion of a test module. Definition 4.5. Let R be a Noetherian ring. We say that a finitely generated R-module T is a test module if for any finitely generated R-module M, TorR n (T, M) = 0 for n ≫ 0 ⇒ pdR M < ∞. Example 4.6. For a Noetherian local ring (R, m, k), the syzygy Ωn k of its residue field is a test module for each n. For commutative Noetherian rings admitting dualizing complexes (e.g., Gorenstein rings), there is another characterization for test modules: Theorem 4.7. [CDT, Theorem 3.2] Let R be a commutative Noetherian ring admitting a dualizing complex. Then, test modules are nothing but finitely generated R-modules T satisfying the following condition: for any finitely generated R-module M, ExtnR (T, M) = 0 for n ≫ 0 ⇒ idR M < ∞. Motivated by this theorem, we introduce the following notion. Definition 4.8. Let T be a triangulated category. We say that T ∈ T is a test object if for any object M of T , HomT (T, Σn M) = 0 for n ≫ 0 ⇒ M = 0. Denote by T(T ) the full subcategory of T consisting of test objects. The following lemma shows that we can consider the notion of a test object is a generalization of the notion of a test module. Lemma 4.9. Let R be a Gorenstein ring. Then one has T(CM(R)) = {T ∈ CM(R) \| T is a test module}. TRIANGULATED EQUIVALENCES AND RECONSTRUCTION OF CLASSIFYING SPACES 15 Proof. By Theorem 4.7, we have only to show ≫0 T(CM(R)) = {T ∈ CM(R) \| all N ∈ mod R with ExtR (M, N) = 0 satisfy idR N < ∞}. Fix a maximal Cohen-Macaulay R-module T and a finitely generated R-module M. Since R is Gorenstein and T is maximal Cohen-Macaulay, one has Ext1R (T, R) = 0. Therefore, we get isomorphisms Exti (T, M) ∼ = Exti+1 (T, ΩR M) ∼ = Exti+2 (T, Ω2 M) ∼ = ··· R R R R for any positive integer i. Therefore, we get isomorphisms Hom (T, Σd+n Ωd M) ∼ = Extn (T, M) = Extd+n (T, Ωd M) ∼ R R R R R for n > 0. Here, d denotes the dimension of R. Thus, we are done since ΩdR M is free if and only if M has finite injective dimension. Let us recall several classes of subcategories of modules. Definition 4.10. (1) An additive subcategory X of mod R is called resolving if it satisfies the following conditions: (i) X is closed under extensions: for an exact sequence 0 → L → M → N → 0 in mod R, if L and N belong to X , then so does M. (ii) X is closed under kernels of epimorphisms: for an exact sequence 0 → L → M → N → 0 in mod R, if M and N belong to X , then so does L. (iii) X contains all projective R-modules. For a finitely generated R-module M, denote by resR (M) the smallest resolving subcategory of mod R containing M. (2) A non-empty additive subcategory X of mod R is called thick if X satisfies 2-out-of3 property: for an exact sequence 0 → L → M → N → 0 in mod R, if 2-out-of {L, M, N} belong to X , then so does the third. For a finitely generated R-module M, denote by thickR (M) the smallest thick subcategory of mod R containing M. Lemma 4.11. Let T be a triangulated category and T an object of T . If thickT (T ) contains a test object of T , then T is also a test object. Proof. Take an object M ∈ T with HomT (T, Σn M) = 0 for n ≫ 0. Set X := {N ∈ T \| HomT (N, Σn M) = 0 for n ≫ 0}. Then one can easily verify that X is a thick subcategory of T . By assumption, X contains a test object as X contains T . Thus, M must be zero and hence T is a test object. The next proposition plays a key role to prove our main theorem. Proposition 4.12. Let (R, m, k) be a d-dimensional local complete intersection ring and T a finitely generated R-module. Then the following are equivalent: (1) T is a test module. (2) ΩdR k ∈ resR (T ). (3) k ∈ thickR (T ⊕ R). (4) k ∈ thickDb (mod R) (T ⊕ R). (5) k ∈ thickDsg (R) (T ). (6) ΩdR k ∈ thickCM(R) (Ωd T ). 16 HIROKI MATSUI Proof. Notice resR (ΩiR T ) ⊆ resR (T ), thickR (T ⊕ R) = thickR (ΩiR T ⊕ R), thickDb (mod R) (T ⊕ R) = thickDb (mod R) (ΩiR T ⊕ R), thickDsg (R) (T ) = thickDsg (R)(ΩiR T ) and T is a test module if and only if so is ΩR T . Hence we may assume that T is maximal Cohen-Macaulay. Then we have resR (T ) ⊆ thickR (T ⊕ R) = thickDb (mod R) (T ⊕ R) ∩ mod R. Here, the first inclusion directly follows from the definition, and the second equality is ∼ = → given by [KS, Theorem 1]. Moreover, the composition functor Db (mod R) → Dsg (R) − CM(R) sends k to ΩdR k[d], and the inverse image of thickCM(R) (T ) is thickDb (mod R) (T ⊕ R). Therefore, the implications (2) ⇒ (3) ⇔ (4) ⇔ (5) ⇔ (6) hold true. Furthermore, by using Lemma 4.9 and Lemma 4.11, the implication (5) ⇒ (1) follows. Thus, it remains to show the implication (1) ⇒ (2). Assume that T is a test module. Recall that the complexity cxR (M) of a finitely generated R-module M is the dimension of the support variety VR (M) associated to M; see [AB] for details. By [CDT, Proposition 2.7], T has maximal complexity, namely cxR (T ) = codim(R) =: c. Thanks to the prime avoidance lemma, we can take an R-regular sequence x of length d from m \ m2 . Set R = R/(x) and T = T /(x). Then R is an Artinian complete intersection ring and cxR (T ) = cxR (T ) = c = codimR = codim(R). Moreover, one has VR (T ) = Acka = VR (k), where k a denotes the algebraic closure of k. This follows from the fact that VR (T ) and VR (k) are c-dimensional closed subvarieties of the c-dimensional affine space Acka . Hence, by [CI, Theorem 5.6], k belongs to thickDb (mod R) (T ). As a result, we get k ∈ thickDb (mod R) (T ) ∩ mod(R) ⊆ thickDb (mod R) (T ⊕ R) ∩ mod(R) = thickR (T ⊕ R). Again, the second equality uses [KS, Theorem 1]. Since thickR (T ⊕ R) = resR (T ) by [DT, Corollary 4.16], we deduce ΩdR k ∈ resR (T ) by using [Tak, Lemma 5.8]. Gathering [Tak, Theorem 6.7], [NT, Theorem B], Lemma 4.9 and Proposition 4.12, we obtain the following proposition. Proposition 4.13. Let R be a Noetherian local ring. (1) If R satisfies the condition (a) in Theorem 4.4, then (Sing R, SSuppR ) is a classifying support data for Dsg (R) with respect to T(Dsg (R)). (2) If R satisfies the condition (b) in Theorem 4.4, then (Sing R, SSuppR ) is a classifying support data for Dsg (R). Now, the proof of Theorem 4.4 has almost been done. Proof of Theorem 4.4. Use Proposition 4.13 and Theorem 2.15. Here, let me remark that test objects are preserved by singular equivalences. Remark 4.14. For a hypersurface ring R, the triangulated category Dsg (R) becomes a pseudo tensor triangulated category (i.e., tensor triangulated category without unit). It is shown by Yu implicitly in the paper [Yu] that for two hypersurfaces R and S, if a singular equivalence between R and S preserves tensor products, then Sing R and Sing S are homeomorphic. Indeed, Sing R is reconstructed from Dsg (R) by using its pseudo tensor triangulated structure. TRIANGULATED EQUIVALENCES AND RECONSTRUCTION OF CLASSIFYING SPACES 17 Since Theorem 4.4 gives a necessary condition for singular equivalences, we can generate many pairs of rings which are not singularly equivalent. Let us start with the following lemma. Lemma 4.15. Let R be a local complete intersection ring with only an isolated singularity and r > 1 an integer. Then the ring R[[u]]/(ur ) is a local complete intersection ring which is locally a hypersurface on the punctured spectrum, and Sing(R[[u]]/(ur )) is homeomorphic to Spec R. Proof. Of course T := R[[u]]/(ur ) is a local complete intersection ring. ∼ = The natural inclusion R → T induces a homeomorphism f : Spec T − → Spec R. Then one can easily check that P = (f (P ), u)T for any P ∈ Spec T and TP ∼ = Rf (P ) [[u]]/(ur ). Therefore, T is locally a hypersurface on the punctured spectrum and Sing T = Spec T . Corollary 4.16. Let R and S be local complete intersection rings which have only isolated singularities. Assume that Spec R and Spec S are not homeomorphic. Then for any integers r, s > 1, one has Dsg (R[[u]]/(ur )) ∼ 6= Dsg (S[[v]]/(v s )). In particular, Dsg (R ∗ R) ∼ 6= Dsg (S ∗ S). Here R ∗ R denotes the trivial extension ring of a commutative ring R. Proof. From the above lemma, we obtain (1) R[[u]]/(ur ) and S[[v]]/(v s ) satisfies the condition (a) in Theorem 4.4, (2) Sing R[[u]]/(ur ) ∼ = Spec S are not homeomorphic. = Spec R and Sing S[[u]]/(v r ) ∼ r ∼ Thus, we conclude Dsg (R[[u]]/(u )) 6= Dsg (S[[v]]/(v s )) by Theorem 4.4. The second statement follows from an isomorphism R ∗ R ∼ = R[[u]]/(u2 ). The following corollary says that a Knörrer-type equivalence fails over a non-regular ring. Corollary 4.17. Let S be a regular local ring. Assume that S/(f ) has an isolated singularity. Then one has Dsg (S[[u]]/(f, u2)) ∼ 6= Dsg (S[[u, v, w]]/(f + vw, u2)). Proof. Sing S[[u]]/(f, u2) ∼ = Spec S[[v, w]]/(f + = Spec S/(f ) and Sing S[[u, v, w]]/(f +vw, u2) ∼ vw) have different dimensions and hence are not homeomorphic. For the last of this paper, we will show that singular equivalence localizes. Lemma 4.18. Let R be a d-dimensional Gorenstein local ring and p a prime ideal of R. Then a full subcategory Xp := {M ∈ Dsg (R) \| Mp ∼ = 0 in Dsg (Rp )} is thick and there is a triangle equivalence Dsg (R)/Xp ∼ = Dsg (Rp ). Proof. By using the triangle equivalence Dsg (R) ∼ = CM(R), we may show the triangle equivalence CM(R)/Xp ∼ = CM(Rp ), ∼ where Xp := {M ∈ CM(R) \| Mp = 0 in CM(Rp )}. Note that the localization functor Lp : CM(R) → CM(Rp ), M 7→ Mp is triangulated. Since Xp = Ker Lp , Xp is a thick subcategory of CM(R) and Lp induces a triangulated 18 HIROKI MATSUI functor Lp : CM(R)/Xp → CM(Rp ). Thus, we have only to verify that Lp is dense and fully faithful. (i): Lp is dense. δ Let U be an Rp -module. Take a finite free presentation Rpn − → Rpm → U → 0 of U. Then δ can be viewed as an m × n-matrix (αij ) with entries in Rp . Write αij = aij /s for some aij ∈ R and s ∈ R \ p. Then the cokernel M := Coker((aij ) : Rn → Rm ) is a finitely generated R-module and Mp ∼ = U. Since Mp is a maximal Cohen-Macaulay Rp -module, we obtain isomorphisms (Ω−d Ωd M)p ∼ =U = Ω−d Ωd Mp ∼ = Mp ∼ R R Rp Rp in CM(Rp ). This shows that the functor Lp is dense. (ii): Lp is faithful. Let α : M → N be a morphism in CM(R)/Xp . Then α is given by a fraction f /s of morphisms f : M → Z and s : N → Z in CM(R) such that the mapping cone C(s) of s belongs to Xp . Assume Lp (α) = Lp (s)−1 Lp (f ) = (sp )−1 fp = 0. Then fp = 0 in HomRp (Mp , Zp ). From the isomorphism HomR (M, Z)p ∼ = HomRp (Mp , Zp), there is a ∈ R\p such that af = 0 in HomR (M, Z). Since a : Zp → Zp is isomorphism, the mapping cone of the morphism a : Z → Z in CM(R) belongs to Xp . Thus, α = f /s = (af )/(as) = 0 in CM(R)/Xp . This shows that Lp is faithful. (iii): Lp is full. Let g : Mp → Np be a morphism in CM(Rp ) where M, N ∈ CM(R). By the isomorphism HomR (M, N)p ∼ = HomRp (Mp , Np ), there is a morphism f : M → N in CM(R) and a ∈ R\p such that g = fp /a. Since the mapping cone of a : N → N is in Xp , we obtain a morphism f /a : M → N in CM(R)/Xp and Lp (f /a) = fp /a = g. This shows that Lp is full. Corollary 4.19. Let R and S be complete intersection rings which are locally hypersurfaces on the punctured spectra. If R and S are singularly equivalent, then there is a homeomorphism ϕ : Sing R → Sing S such that Rp and Sϕ(p) are singularly equivalent for any p ∈ Sing R. Proof. As in Lemma 4.18, we may consider the category CM(R). Let F : CM(R) → CM(S) be a triangle equivalence. Take a homeomorphism ϕ : Sing R → Sing S given in Proposition 2.14 and Theorem 2.15. Then by construction, it satisfies [ SuppS F (M) {ϕ(p)} = M ∈CM(R), SuppR (M )⊆V(p) for each p ∈ Sing R. Moreover, the following diagram is commutative: F̃ ThT(CM(R)) (CM(R)) −−−→ ThT(CM(S)) (CM(S))   fSupp  fSupp y y S R Nesc(Sing R) −−−→ ϕ̃ Nesc(Sing S), where the map F̃ and ϕ̃ are defined by F̃ (X ) := {N ∈ T ′ \| ∃M ∈ X such that N ∼ = F (M)} and ϕ̃(W ) := ϕ(W ), respectively. Let p be an element of Sing R. Set Wp := {q ∈ Sing R \| q 6⊆ p} which is a specializationclosed subset of Sing R. We establish two claims. TRIANGULATED EQUIVALENCES AND RECONSTRUCTION OF CLASSIFYING SPACES 19 Claim 1. gSuppR (Wp ) = Xp . Proof of Claim 1. Let M ∈ Xp . Since Mp = 0 in CM(Rp ), one has p 6∈ SuppR (M). Thus, SuppR (M) ⊆ Wp and hence M ∈ gSuppR (Wp ). Next, take M ∈ gSuppR (Wp ). Then SuppR (M) ⊆ Wp means that p does not belong to SuppR (M). Therefore, Mp = 0 in CM(Rp ) and hence M ∈ Xp . Claim 2. ϕ(Wp ) = Wϕ(p) := {q ∈ Sing S \| q 6⊆ ϕ(p)}. Proof of Claim 2. One can easily check that ϕ is order isomorphism with respect to the inclusion relations. Since Sing R \ Wp has a unique maximal element p, ϕ(Sing R \ Wp ) = Sing S \ ϕ(Wp ) also has a unique maximal element ϕ(p). This shows ϕ(Wp ) = Wϕ(p) . From the above two claims, we obtain F̃ (Xp ) = F̃ (gSuppR (Wp )) = gSuppS (ϕ̃(Wp )) = gSuppS (Wϕ(p) ) = Xϕ(p) , where the second equality comes from the above commutative diagram and the last equality is shown by the same proof as Claim 1. Consequently, the triangle equivalence F induces triangle equivalences: CM(Rp ) ∼ = CM(R)/Xp ∼ = CM(S)/Xϕ(p) ∼ = CM(Sϕ(p) ). Acknowledgments. The author is grateful to his supervisor Ryo Takahashi for many supports and his helpful comments. References [AB] L. L. Avramov; R.-O. Buchweitz, Support varieties and cohomology over complete intersections, Invent. Math. 142 (2000), no. 2, 285–318. [AIL] L. L. Avramov; S. B. Iyengar; J. Lipman, Reflexivity and rigidity for complexes I. Commutative rings, Algebra Number Theory 4 (2010), no. 1, 47–86. [Bal02] P. 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MILP and Max-Clique based heuristics for the Eternity II puzzle arXiv:1709.00252v2 [cs.DS] 5 Oct 2017 Fabio Salassaa , Wim Vancroonenburgb , Tony Wautersb,∗, Federico Della Crocea , Greet Vanden Bergheb a Politecnico di Torino, DIGEP, Corso Duca degli Abruzzi 24, 10129 Torino, Italy b KU Leuven, Department of Computer Science, CODeS & iMinds-ITEC, Gebroeders De Smetstraat 1, 9000 Gent, Belgium Abstract The present paper considers a hybrid local search approach to the Eternity II puzzle and to unsigned, rectangular, edge matching puzzles in general. Both an original mixed-integer linear programming (MILP) formulation and a novel Max-Clique formulation are presented for this NP-hard problem. Although the presented formulations remain computationally intractable for medium and large sized instances, they can serve as the basis for developing heuristic decompositions and very large scale neighbourhoods. As a side product of the Max-Clique formulation, new hard-to-solve instances are published for the academic research community. Two reasonably well performing MILP-based constructive methods are presented and used for determining the initial solution of a multi-neighbourhood local search approach. Experimental results confirm that this local search can further improve the results obtained by the constructive heuristics and is quite competitive with the state of the art procedures. Keywords: Edge matching puzzle, hybrid approach, local search 1. Introduction The Eternity II puzzle (EII) is a commercial edge matching puzzle in which 256 square tiles with four coloured edges must be arranged on a 16 × 16 grid ∗ Corresponding author Email address: [email protected] (Tony Wauters) Preprint submitted to . October 6, 2017 such that all tile edges are matched. In addition, a complete solution requires that the ‘grey’ patterns, which appear only on a subset of the tiles, should be matched to the outer edges of the grid. An illustration of a complete solution for a small size puzzle, 5 × 5, is provided in Figure 1. Figure 1: Solution to an Eternity II-like edge matching puzzle of size 5 × 5 (Image generated with the Eternity II editor, http://sourceforge.net/projects/eternityii/, accessed on 24 January 2014) The EII puzzle was created by Christopher Monckton and released by the toy distributor Tomy UK Ltd. in July 2007. Along with the puzzle release, a large cash prize of 2 million USD was announced to be awarded to the first person who could solve the puzzle. As can be expected, this competition attracted considerable attention. Many efforts were made to tackle this challenging problem, yielding interesting approaches and results. However, no complete solution has ever been generated. Meanwhile, the final scrutiny date for the cash price, 31 December 2010, has passed, leaving the large money prize unclaimed. The EII puzzle belongs to the more general class of Edge Matching Puzzles, which have been shown to be NP-complete [5]. Many approaches to Edge Matching Puzzles are now available in the literature. Constraint programming approaches [2, 11] have been developed, in addition to metaheuristics [4, 11, 13], backtracking [12] and evolutionary methods [10]. Other methods translate the problem into a SAT formulation and then solve it with SAT solvers [1, 7]. An extensive literature overview on the topic is provided in [14], while [8] provides a survey on the complexity of other puzzles. 2 The present paper introduces a novel Mixed-Integer Linear Programming (MILP) model and a novel Max-Clique based formulation for EII-like puzzles of size n × n. Both formulations serve as components of heuristic decompositions used in a multi-neighbourhood local search approach. The remainder of the paper is structured as follows. Section 2 presents the MILP and MaxClique formulations. In Section 3, several hybrid heuristic approaches are introduced. Computational results are presented in Section 4. Final conclusions are drawn in Section 5. 2. Problem formulations 2.1. Mixed integer linear programming formulation A novel mixed-integer linear programming model was developed for the EII puzzle problem. The following notation will be used. The puzzle consists of an n × n square onto which n2 tiles need to be placed. The index t = 1 . . . n2 is used to refer to tiles. The indices r = 1 . . . n, c = 1 . . . n denote the rows, resp. columns, of the puzzle board. The index α = 0 . . . 3 refers to the rotation of the tile, i.e. α = 0 means not rotated, α = 1 means rotated clockwise over 90◦ , etc. The coefficient CTt,α,l (resp. CB, CL, CR) is equal to 1 if tile t has colour l at its top (resp. bottom, left, right) position when rotated by α. The decision variables of the MILP model are defined as follows: ( 1 if tile t is placed in row r, column c with rotation α, xt,r,c,α = 0 otherwise. ( 1 if the right edge of position (r, c) is unmatched , hr,c = 0 otherwise. ( 1 if the bottom edge of position (r, c) is unmatched , vr,c = 0 otherwise. The model is then defined as follows: Min n X n−1 X r=1 c=1 hr,c + n n−1 X X r=1 c=1 s.t. 3 vr,c (1) n X n X 3 X xt,r,c,α = 1 ∀t = 1, . . . , n2 (2) r=1 c=1 α=0 2 n X 3 X xt,r,c,α = 1 ∀r = 1, . . . , n, c = 1, . . . , n (3) t=1 α=0 2 n X 3 X 2 CRt,α,l xt,r,c,α − t=1 α=0 n X 3 X CLt,α,l xt,r,c+1,α ≤ hr,c t=1 α=0 ∀r = 1, . . . , n, c = 1, . . . , n − 1, l = 1, . . . , L 2 − n X 3 X 2 CRt,α,l xt,r,c,α + t=1 α=0 n X 3 X CLt,α,l xt,r,c+1,α ≤ hr,c t=1 α=0 ∀r = 1, . . . , n, c = 1, . . . , n − 1, l = 1, . . . , L CBt,α,l xt,r,c,α − n X 3 X CTt,α,l xt,r+1,c,α ≤ vr,c t=1 α=0 t=1 α=0 ∀r = 1, . . . , n − 1, c = 1, . . . , n, l = 1, . . . , L − t=1 α=0 (6) 2 2 n X 3 X (5) 2 2 n X 3 X (4) CBt,α,l xt,r,c,α + n X 3 X CTt,α,l xt,r+1,c,α ≤ vr,c t=1 α=0 ∀r = 1, . . . , n − 1, c = 1, . . . , n, l = 1, . . . , L (7) n2 3 XX CTt,α,0 · xt,0,c,α = 1 ∀c = 1, . . . , n (8) CBt,α,0 · xt,n−1,c,α = 1 ∀c = 1, . . . , n (9) t=1 α=0 2 n X 3 X t=1 α=0 2 n X 3 X CLt,α,0 · xt,r,0,α = 1 ∀r = 1, . . . , n (10) CRt,α,0 · xt,r,n−1,α = 1 ∀r = 1, . . . , n (11) t=1 α=0 2 n X 3 X t=1 α=0 xt,r,c,α ∈ {0, 1} ∀t = 1, . . . , n2 , r = 1, . . . , n, c = 1, . . . , n, α = 0, . . . , 3 0 ≤ hr,c ≤ 1 ∀r = 1, . . . , n, c = 1, . . . , n − 1 4 (12) (13) 0 ≤ vr,c ≤ 1 ∀r = 1, . . . , n − 1, c = 1, . . . , n (14) The objective function (Expression 1) minimises the number of unmatched edges in the inner region of the puzzle. Constraints (2) indicate that each tile must be assigned to exactly one position, with one rotation. Constraints (3) require that exactly one tile must be assigned to a position. The edge constraints (4) and (5) force the hr,c variables to take on the value 1 if the tiles on positions (r, c) and (r, c+1) are unmatched. Similarly, constraints (6) and (7) do the same for the vertical edge variables vr,c . Finally, constraints (8) - (11) ensure that the border edges are matched to the gray frame (colour l = 0). We point out that constraining the objective function to zero (i.e. no unmatched edges allowed), turns the model into a feasibility problem where every feasible solution is also optimal. However, preliminary testing showed that the latter model is only relevant for very small size problem instances. If the MILP solver needs to be stopped prematurely on the feasibility model, no solution is returned. 2.2. Clique formulation The EII puzzle itself is a decision problem and can be modelled as (i.e. it reduces to) the well known decision version of the clique problem [3] as follows. Given a parameter k and an undirected graph G = (V, E), the clique problem calls for finding a subset of pairwise adjacent nodes, called a clique, with a cardinality greater than or equal to k. Let the nodes of the graph correspond to variables xt,r,c,α from the formulation introduced in Section 2.1. Each node thus represents a tile in a given position on the puzzle and with a given rotation α. The nodes are connected iff there is no conflict between the nodes in the puzzle. Possible causes of conflicts are: • unmatching colors for adjacent positions • same tile assigned to different positions • same tile assigned to the same position with different rotations • different tiles assigned to the same position. The objective is to find a clique of size n2 , where n is the size of the puzzle. 5 Puzzle size Optimal solution Clique Number of nodes Number of edges Graph density q = 1, 000, 000 q = 10, 000, 000 q = 50, 000, 000 MILP Number of variables Number of constraints 1 thread, 3600s time 4 threads, 3600s time 3x3 12 4x4 24 5x5 40 6x6 60 7x7 84 8x8 112 30 291 66.9% E (T ) 12 (0.2) 12 (1.9) 12 (9.3) 138 6707 70.9% E (T ) 24 (0.3) 24 (3.8) 24 (19.3) 478 86184 75.6% E (T ) 40 (0.8) 40 (7.6) 40 (37.8) 1290 674512 81.1% E (T ) 57 (1.4) 60 (13.5) 60 (67.0) 2910 3620565 85.5% E (T ) 78 (2.5) 80 (22.4) 82 (110.5) 5770 14751304 88.6% E (T ) 102 (4.8) 106 (35.7) 106 (172.4) 336 126 E (T ) 12 (0.1) 12 (0.1) 1048 288 E (T ) 24 (0.1) 24 (0.1) 2540 630 E (T ) 40 (16.7) 40 (1.2) 5244 1056 E (T ) 60 (243.5) 60 (158.5) 9688 1638 E (T ) 81 (3600.0) 81 (3600.0) 16496 2624 E (T ) 103 (3600.0) 103 (3600.0) Table 1: Results obtained with the maximum clique formulation solved with the algorithm by [6], and the MILP formulation solved with CPLEX, on small size edge matching puzzle instances. 2.3. Comparison of the MILP model and the Clique model The applicability of the MILP model and the Clique model is investigated in what follows. Initial testing was performed on a set of small puzzle instances, ranging from 3 × 3 up to 8 × 8 (refer to Section 4 for more information on these instances). The MILP model was implemented using CPLEX 12.6. A state of the art heuristic was used for solving the maximum clique problem [6], kindly provided by its authors. The heuristic has only one parameter, i.e. the number of selections q. The computing time of the algorithm is linear with respect to q. We tested the heuristic with q ∈ {1, 000, 000; 10, 000, 000; 50, 000, 000}. Both the MILP model and the Max Clique heuristic were tested on a modern desktop pc1 . Table 1 shows the results obtained with the max clique formulation and the MILP formulation. For each instance, we report for the clique formulation: the number of nodes, the number of edges, the optimal solution (namely the max. number of matching edges), the density, the best number of matching edges E and the average computing time T (in seconds) for 10 runs. The number of variables and constraints is reported for the MILP formulation. The solutions depicted in bold are optimal. The results show that instances up to size 6 × 6 can be easily solved using a state of the art maximum clique 1 Intel Core i7-2600 [email protected] 6 algorithm. Instances of size 7 × 7 could not be solved completely, even when the algorithm was executed with higher values of q and more runs. The MILP is also able to solve up to size 6 × 6. However, the clique formulation is significantly faster from size 6 × 6 upwards. Note that the edge matching puzzles correspond with large, difficult clique instances, for which current state-of-the-art max clique solvers are not able to find the optimal solution. We provide the corresponding max clique instances of the 3 × 3 to 9 × 9 instances to the academic community2 . Larger size instances are hard to manage. The 10 × 10 graph file, for example, is larger than 1 GB. 3. Solution Approaches Both the MILP model and the clique formulation presented in the previous section proved to be computationally intractable for medium sized instances. The size appears to be restricted to 7 × 7 and 8 × 8 when the execution time is limited to one day. The true EII puzzle instance is still far beyond the grasp of these models. However, these models can serve as the basis for some well performing heuristics, presented in the following paragraphs. 3.1. MILP-based greedy heuristic A MILP-based greedy constructive heuristic has been developed for the problem studied here. The heuristic is based on subproblem optimisation. The puzzle is divided in regions, e.g. by considering individual rows/columns or rectangular regions. Regions are then consecutively constructed by employing a variant of the MILP model presented in Section 2. First we introduce the notion of a partial solution S ∗ = T ∗ 7→ R∗ , in which a subset T ∗ of the tiles T = {1, . . . , n2 } have been assigned to a subset R∗ of the positions R = {(r, c)\|r = 1, . . . , n, c = 1, . . . , n}. Given a partial solution S ∗ , model (1)–(14) can be modified such that it only considers the positions in a region R0 ⊆ R\R∗ , and it only aims to assign tiles T 0 ⊆ T \T ∗ (i.e. tiles that have not been assigned elsewhere). In addition, we restrict 0 0 R0 to a rectangular region, denoted by (rmin , c0min ) × (rmax , c0max ) (i.e. the min/max positions of the region). 2 The instances can be downloaded from https://dl.dropbox.com/u/24916303/ Graph_Pieces.zip. A generator for these instances is available upon request from the authors. 7 Figure 2 illustrates how model (1)–(14) can be modified to solve a region (1, 4) × (4, 8), given a partial solution on (1, 1) × (4, 4). In this example, it is required to select 16 of the available 64 − 16 = 48 tiles (16 tiles are already assigned to region (1, 1) × (4, 4)) in such a way that the unmatched edges are minimised. Hence, in order to consider region (1, 4) × (4, 8), the objective function is modified as follows Min 4 X 7 X hr,c + r=1 c=4 3 X 8 X vr,c (15) r=1 c=4 Only 16 of the remaining 48 tiles must be selected and assigned to the region and therefore constraints (2) are modified as follows. Note that the inequality indicates that not all tiles will be selected. 4 X 8 X 3 X xt,r,c,α ≤ 1 ∀t ∈ T \T ∗ (16) r=1 c=4 α=0 Similarly, constraints (3–14) are also suitably modified in order to take into account the specific region to be considered. Note that the edge constraints forcing the values of the hr,c and vr,c variables also hold for rows and columns matching the boundaries of previously solved regions. This enables building a solution with only a few unmatched edges between region boundaries. This partial optimization model can be applied to solve all regions sequentially, thus constructing the final complete solution. Initially R1 is optimised, after which region R2 , disjoint from region R1 , is optimised, and so on. The variables corresponding with the region are then optimally assigned by the MILP solver. Algorithm 1 presents pseudocode of this approach. Algorithm 1 Greedy heuristic Require: R = {R1 , R2 , R3 , . . . RK } . A decomposition of R in K regions Ri ∗ ∗ ∗ T0 ← ∅, R0 ← ∅, S0 ← ∅ . The initial partial solution S ∗ has no tiles assigned for i = 1, . . . , K do ∗ ,T∗ ∗ Si∗ , Ti∗ , Ri∗ ← Apply MILP model to region Ri , given Si−1 i−1 and Ri−1 . ∗ . Get a new partial solution Si , with tiles assigned to Ri end for ∗ return SK For each puzzle’s size, differently sized subsets of tiles have been tested to assess the quality of the approach. Preliminary tests of regions varying 8 Partial solution hrc ∀t,α vrc xt15α xt16α xt17α xt18α xt25α xt26α xt27α xt28α xt35α xt36α xt37α xt38α xt45α xt46α xt47α xt48α Empty region Figure 2: Illustration of how model (1)–(14) can be modified such that it only considers the positions in a region (1, 4) × (4, 8), given a partial solution S ∗ on region (1, 1) × (4, 4). 9 from 2 by 2 tiles (size 4) to 16 by 2 tiles (size 32) have been performed on the EII puzzle instance. This preliminary analysis revealed that the CPU time required at each iteration of the greedy heuristic limits the subset size to 32 tiles. This roughly corresponds to 32500 MILP variables for the first region of the real EII puzzle. Clearly, an increased number of tiles leads to better results. However, more CPU time is needed to compute the optimal solution, limiting the use in any hybrid framework. 3.2. MILP-based backtracking constructive heuristic A backtracking version of the greedy heuristic has also been developed. The main idea, namely building a complete solution by constructing optimal regions, is the same as for the greedy heuristic. The backtracking version, however, restricts the optimal value of each subproblem to zero. All tiles in the region should match both internally and with respect to the tiles outside the region. Whenever a subproblem is determined to be infeasible (i.e. no completely edge matching region can be constructed), the procedure backtracks to the previous region in order to find a new assignment in that region. This may afterwards enable constructing a feasible assignment in the next region. If not, then the process is repeated until the backtracks are sufficient to find a complete solution. Model (1)–(14), suitably modified, is again used to build partial solutions. Let Ri be the current region considered by the procedure and Si∗ the related partial solution once the corresponding MILP model is solved. Whenever the lower bound of the MILP model related to region Ri is detected to be greater than zero, optimisation of region Rk is stopped. Instead, the previous region 0∗ , again with Ri−1 is reconsidered in order to obtain a new partial solution Si−1 ∗ value 0. Let Xi−1 be the set of variables xt,r,c,α having value 1 in solution ∗ ∗ Si−1 . The previous partial solution Si−1 must be cut off when searching for 0∗ solution Si−1 . The following new constraint is added to the model: X ∗ xt,r,c,α ≤ \|Xi−1 \|−1 (17) ∗ t,r,c,α : xt,r,c,α ∈Xi−1 ∗ The rationale is to force at least one of the variables of set Xi−1 to be equal to zero. If no solution of the previous region Ri−1 can lead to a zero lower bound in the current region Ri , the procedure backtracks further and searches for a new solution for region Ri−2 (and so on). 10 Due to the enumerative nature, this procedure can lead to incomplete solutions despite long computation times. We decided to limit the backtracking procedure to a fixed time limit, after which the greedy heuristic continues until a complete solution is generated. This backtracking heuristic is sketched in Algorithm 2, in which a recursive method (BACKT RACKIN G HEU RIST IC) ∗ obtained attempts to solve the current region Ri , given partial solution Si−1 in the previous region Ri−1 . If the lower bound lb of the current region is greater than 0, the method backtracks to the previous level. However, if the lower bound is still 0 (and a perfectly matched assignment is found), the heuristic attempts to solve the next region. This will continue calling recursively until the puzzle is solved, or shown infeasible given the current ∗ ∗ assignments in Si−1 . In the latter case, the current partial solution Si−1 will 0∗ be excluded and a new partial solution Si−1 will be constructed, different ∗ and any other previously excluded partial solution. from Si−1 If a timeout is reached, the method will continue with the best partial solution and solve the remaining regions with the greedy heuristic, discussed in the previous section. 3.3. A multi-neighbourhood local search approach A multi-neighbourhood local search approach has been developed to improve the solutions generated by the constructive heuristics (or any random solution). The key idea is to test, after an initial, complete solution is generated by the heuristics, whether a controlled-size neighbourhood can still improve the current solution. This local search method is a Steepest Descent search that tries to improve a solution with the following neighbourhoods: Border Optimisation, Region Optimisation, Tile Assignment and Tiles Swap and Rotation. We refer to Figure 3 for an illustration of the regions considered by these neighbourhoods. The Border Optimisation (BO) neighbourhood only considers placing tiles in the border, while all the tiles in the inner part are fixed. The decomposition tries to find the optimal border in terms of matching edges, also considering the fixed tiles on the adjacent inner part. Correspondingly, model (1)–(14) is modified in such a way that the inner tile/positions variables are fixed to their current value. Only the border tile/positions variables can change value. This subproblem corresponds to a one-dimensional edgematching problem. Preliminary computational tests indicated that the related MILP model could always be solved. Solutions for the largest instances, 11 Algorithm 2 BACKT RACKIN G HEU RIST IC Require: R = {R1 , R2 , R3 , . . . RK } . A decomposition of R in K regions Ri Require: i . Current recursive level (i = 1, . . . , K) ∗ ∗ 7→ R∗ Require: Si−1 : Ti−1 . Partial solution of the previous level (S0∗ is the i−1 initial empty partial solution) excludedsolutions ← ∅ . Partial solutions not leading to feasible solutions while not timeout do ∗ , R , excludedsolutions) Si∗ ← OP T IM IZE REGION (Si−1 i if lb > 0 then ∗ return Si−1 . No feasible solution in current level. (backtrack) else ∗ Si+1 ← BACKT RACKIN G HEU RIST IC(R, i + 1, Si∗ ) ∗ if Si+1 = Si∗ then . Si∗ does not lead to feasible solution excludedsolutions ← excludedsolutions ∪ Si∗ else ∗ ∗ S ∗ ← Si+1 . Si+1 is the complete solution ∗ return S end if end if end while S ∗ ← GREEDY (Si∗ , Ri∗ ) return S ∗ 12 such as the original EII puzzle, can be generated within little computation time. When the BO neighborhood is considered, the corresponding MILP model is solved, returning a solution at least as good as the current solution and consisting of an optimal border with respect to the (n − 2) × (n − 2) inner region. The Region Optimisation (RO) neighbourhood relates to the optimisation of a smaller region inside the puzzle and only considers the tiles of this region in the puzzle. Correspondingly, given the current solution, model (1)– (14) is suitably modified in such a way that the tile/position variables outside the region are fixed to their current value. Only the region’s tile/position variables can change value. The RO neighborhood can also be tackled by means of the Max-Clique formulation by generating a graph only containing nodes corresponding to tile/position assignments in the specified region. However, only feasible tile/position assignments should be considered and nodes conflicting with assignments adjacent to (but outside) the region should not be added to the graph. We recall that the purpose of the model is to find complete assignments, that is, without any unmatched edges. However, given the tiles in the considered region, it may not be feasible to find such a solution. In this case, holes are left in the region to which the remaining, unassigned tiles should be assigned. The related MILP region model is solved where all assigned tile/position variables are fixed to the value determined by the MaxClique solver. When the RO neighborhood is considered, the local search procedure samples regions of fixed size in the current solution under consideration. For small sizes, the Max-Clique model (heuristically) is solved faster than the MILP model. Therefore, the RO neighborhood is always addressed by means of the Max-Clique formulation where the MILP formulation is only used for completing the solution whenever holes are left in the region. In the Tile Assignment (TA) neighbourhood, k tiles are removed from non-adjacent positions (diagonally adjacent is allowed) and optimally reinserted, thereby minimising the number of unmatched edges. The related subproblem corresponds to a pure bipartite weighted matching problem, which is optimally solvable by e.g. the Hungarian Algorithm [9]. The TA neighbourhood was first introduced by Schaus and Deville [11] who called it a very large neighbourhood. Wauters et al. [14] developed a probabilistic version of the TA neighbourhood that sets a higher probability to selecting tiles with many unmatched edges. The latter TA variant was applied in the present 13 paper. The TA separates the inner and the border moves. It is prohibited to reassign border pieces to the inner region and vice versa. An extention to the TA neighbourhood is also tested. In particular a “checkers” configuration of selected tiles is studied, i.e. all tiles on the board that are diagonally adjacent. We denote this extension Black and White (BW). The local search procedure iterates in this neighbourhood, iteratively changing between “black” and “white” positions and solving the related bipartite weighted matching problem until no more improvements are found. Finally, the Tiles Swap and Rotation (TSR) neighbourhood is a standard local search swap operator, in this case swapping the assignment of two tiles, trying all possible rotations as well. The local search procedure exhaustively searches the neighbourhood until a local optimum is reached. 4. Computational Results This section provides computational results obtained by the multi-neighbourhood local search approach on the Eternity II puzzle as well as the 10×10, 12×12, 14 × 14 and 16 × 16 instances that were used in the META 2010 EII contest3 . The latter instances serve as an interesting test set for comparison, due to the availability of some results from the contest. In addition, to the best of our knowledge, the complete solutions of these instances are not publicly available. The 3 × 3 to 9 × 9 instances used in Section 2 also originate from this set. All tests were performed on a 40 cores Intel Nehalem cluster with 120 GB ram, with each core running @ 3.46 GHz with 8 MB cache. Computational resources provided by DAUIN’s HPC Initiative4 . This cluster was used to solve different instances/runs in parallel in order to reduce the total time required to run all tests. Each individual test was run on a single processing core, thus no parallelism was employed in the algorithms. All MILP models are solved using CPLEX 12.4. 3 rd 3 International Conference on Metaheuristics and Nature Inspired Computing, Djerba Island, Tunisia, October 27th -31th 2010 – Eternity 2 contest http://www2.lifl. fr/META10/pmwiki.php?n=Main.Contest. 4 For more details see http://www.dauin-hpc.polito.it. 14 (a) (b) (d) (c) (e) Figure 3: Illustration of the regions involved in the neighbourhood operations: (a) optimizing the border (BO); (b) optimizing a rectangular region (RO); (c) optimizing nonadjacent tile assignments (TA); (d) optimizing diagonally adjacent tile assignments in a checkers fashion (BW); (e) swapping two tiles and possibly rotating them (TSR). 15 Parameter Name Value T A.K 16 tiles T A.N 1000 it Clique.W 6 cols Clique.H 6 rows Clique.N 10 it Table 2: Parameter Settings Table 2 summarizes the parameter settings of the multi-neighbourhood local search approach. The local search procedure starts either from a random solution or from a solution obtained by the constructive heuristics. The algorithm cycles through the proposed neighbourhoods in the following sequence: TA (for T A.N iterations with sample size T A.K), BO (for one iteration), BW (till local optimum), TSR (till local optimum) and finally RO by means of the Max-Clique formulation (for Clique.N iterations with rectangular sample size Clique.W × Clique.H). This sequence was determined experimentally, though the difference in performance between sequences was very limited. At the end of the multi-neighbourhood step, the final solution is a local minimum with respect to all considered neighbourhoods. Table 3 shows the results for twenty runs of the MILP-based greedy heuristic and the MILP-based backtracking heuristic for different region sizes on all the problem instances. A timeout of 1200 seconds was set for the backtracking heuristic for the 10 × 10 instance, 1800 seconds for the 12 × 12 instance, 2400 seconds for the 14×14 instance and 3600 seconds for the 16×16 and EII instances. The greedy heuristic (executed by itself, or after the backtracking heuristic) is executed until all regions are solved. The table also shows the results of both constructive methods after subsequent optimisation by the local search heuristic. In general, both constructive heuristics generate better results when larger regions are used. This clearly affects the CPU time needed to compute optimal solutions for each region. By comparing the results of the two heuristics (without the local search phase), it seems that the backtracking procedure does not strongly dominate (on maximum, average and minimum values) the greedy one, while consuming all the available time. This dominance tends to be more evident for small puzzles and region sizes, while for larger instances with solutions generated by larger regions, the gap becomes smaller. 16 In almost all cases, the local search procedure manages to improve the results of the constructive heuristics by several units, indicating that these initial solutions are not local optima with respect to the considered neighbourhoods. We conclude that many neighbourhoods in a complex structure are effective for improving these greedy constructive solutions. Table 4 shows the performance of the local search procedure starting from a random initial solution of poor quality. The procedure can achieve good quality results for the 10 × 10 instance, but not for larger instances. This can easily be related to the size of the RO neighborhood. As it is quite large with respect to the puzzle size in the 10 × 10 instance, it is able to optimize a large part of the puzzle. However, this ratio becomes smaller and is thus less effective for larger instances. Finally, Table 5 compares the best published results with the results obtained by the hybrid local search procedure. The CPU times refer to the considered time limits. Table 5 also reports a large test of the procedure, where the best performing configuration (backtracking+LS) was run 100 times within a doubled execution time limit. Larger execution times (using CPLEX 12.4 as ILP solver) do not induce further improvements of the results. We note that some of the entries of Table 5 are missing. Many of the approaches only deal with a subset of the considered instances. Only three studies [4, 13, 14] report results for the 10 × 10 to 16 × 16 instances that were tested in this paper. Some approaches [11, 10, 12] were only applied to the real EII game puzzle. The algorithm reported in [11] was executed on a CPU Intel Xeon(TM) 2.80GHz, with computation time 24 hours. The best score over 20 runs equals 458. [10] obtained a best score of 371. No indication was provided on the computer and the required CPU time. The algorithm of [4] was run on a PC Pentium Core 2 Quad (Q6600), 2.4 GHz, with 8 GB of RAM. It considered EII style problems (but not the real puzzle) with sizes 10 × 10, 12 × 12, 14 × 14 and 16 × 16. The corresponding time limits were 1200, 1800, 2400 and 3600 seconds respectively and the entries of Table 5 report the best solution obtained over 30 runs. The algorithm of [13] addressed the same instances with the same time limits and number of runs as [4]. It was tested on a personal computer with 1.8GHz CPU and 1GB RAM. Time limits and number of runs were the same as in [13]. The tests were performed on an Intel Core 2 Duo @ 3Ghz with 4GB of 17 RAM. The algorithm of [14] was tested on all the instances from [4] and [13] and also on the EII real game puzzle. The entry reports the best result obtained over 30 runs with a time limit of 3600 seconds for EII. Finally, the algorithm of [12] was tested on the EII real game puzzle only running on a grid computing system over a period of several weeks/months not explicitly indicated by the authors. The results show that the algorithm is competitive with the state of the art, obtaining top results for the 10 × 10 instance in a similar time frame as the other algorithms. Most interesting, the initial solution constructed by the greedy and backtracking heuristics is already of high quality, leaving only a limited gap from the optimal solution. Therefore, we expect that these methods may serve as the basis for reaching new top results. The best result for the official EII puzzle instance, 467 obtained using a slipping tile, scanrow backtracking algorithm [12], is still out of our current grasp. However, that algorithm was highly tailored to the EII puzzle instances, used precomputed sequences and was run over the course of several weeks/months (see http://www.shortestpath.se/eii/eii details.html). A direct comparison with the approach presented is partially misleading. Among the other existing approaches, only [14] shows to be slightly superior to our approach. However, our approach should become more competitive along with the expected performance improvement of MILP solvers over the years. Clearly, solving larger subregions in both the constructive heuristics (greedy and backtracking) will lead to better initial solutions. In addition, the effectiveness of the MIP-based local search neighbourhoods is expected to improve when larger regions can be solved. If performance improvements allow ILP solvers to address instances of size 8 × 8 or even 9 × 9 in a reasonable amount of time, it may safely be assumed that the proposed approach will lead to improved results competitive with the other state of the art approaches. 5. Conclusions The present work introduced a hybrid approach to the Eternity II puzzle. A MILP formulation is related to the puzzle’s optimisation version, where the total number of unmatched edges should be minimised. It is shown that the Eternity II puzzle can be modelled as a clique problem, providing, as a byproduct of this work, new hard instances of the maximum clique problem 18 to the community. Preliminary testing revealed it was clear that these models cannot handle large size instances (such as the original EII puzzle), as they quickly become computationally intractable. Therefore, these models were used as the basis for heuristic decompositions, which could then be used in a hybrid approach. A greedy and a backtracking constructive heuristic have been designed, which strongly rely on the capability of optimally solving a specific region of the puzzle. Within a reasonable time limit, high quality solutions can be generated using these heuristics. A multi-neighbourhood local search approach has also been proposed. By applying a set of different neighbourhoods, the local search procedure manages to improve upon the initial solutions generated by the constructive heuristics and reaches solutions competitive to the best available results. These results confirm that a novel and clever use of mathematical models and solvers/heuristics is effective for large size problems, which cannot be solved all in once by the same MILP solver. We believe that hybridizing local search approaches and mathematical programming techniques in a matheuristic context is the key to break up the intractability of hard problems such as the EII puzzle. References [1] Ansótegui C, Béjar R, Fernàndez C, Mateu C. Edge matching puzzles as hard SAT/CSP benchmarks. In: CP ’08 Proceedings of the 14th international conference on Principles and Practice of Constraint Programming, 560-565, 2008. [2] Benoist T, Bourreau E. Fast global filtering for Eternity II. Constraint Programming Letters, 3, 35-50, 2008. [3] Bomze IM, Budinich M, Pardalos PM, PelilloM. The maximum clique problem. In: Du DZ, Pardalos PM (eds) Handbook of combinatorial optimization. Kluwer Academic, Dordrecht, 174, 1999. [4] Coelho I, Coelho B, Coelho V, Haddad M, Souza M, Ochi L. A general variable neighborhood search approach for the resolution of the Eternity II puzzle. In: Proceedings of the 3rd International Conference 19 on Metaheuristics and Nature Inspired Computing, META’10. 2010, http://www2.lifl.fr/META10/proceedings//meta20100 submission 157.pdf. [5] Demaine ED, Demaine ML. Jigsaw puzzles, edge matching, and polyomino packing: Connections and complexity. Graphs and Combinatorics, 23, 195-208, 2007. [6] Grosso A, Locatelli M, Pullan W. Simple ingredients leading to very efficient heuristics for the maximum clique problem. Journal of Heuristics, 14, 587-612, 2008. [7] Heule MJH. Solving edge-matching problems with satisfiability solvers. In: Proceedings of the Second International Workshop on Logic and Search (LaSh 2008), 88-102, 2008. [8] Kendall G, Parkes A, Spoerer K. A survey of NP-Complete puzzles. International Computer Games Association Journal, 31, 13-34, 2008. [9] Kuhn HW. The Hungarian Method for the assignment problem. Naval Research Logistics Quarterly, 2, 83-97, 1955. [10] Muñoz J, Gutierrez G, Sanchis A. Evolutionary genetic algorithms in a constraint satisfaction problem: Puzzle Eternity II. In: Cabestany J, Sandoval F, Prieto A, Corchado J, editors. Bio-Inspired Systems: Computational and Ambient Intelligence; LNCS 5517, 720-727, 2009. [11] Schaus P, Deville Y. Hybridization of CP and VLNS for Eternity II. In: JFPC’08 Quatrième Journées Francophones de Programmation par Contraintes. 2008, http://www.info.ucl.ac.be/∼yde/Papers/JFPC2008 Eternity en.pdf. [12] Verhaard L. http://www.shortestpath.se/eii/index.html; 2008. [13] Wang WS, Chiang TC. Solving eternity-ii puzzles with a tabu search algorithm. In: Proceedings of the 3rd International Conference on Metaheuristics and Nature Inspired Computing. 2010, http://www2.lifl.fr/META10/proceedings//meta20100 submission 161.pdf. [14] Wauters T, Vancroonenburg W, Vanden Berghe G. A guide-and-observe hyper-heuristic approach to the Eternity II puzzle. Journal of Mathematical Modelling and Algorithms, 11, 217-233, 2012. 20 INSTANCE 10x10 12x12 14x14 16x16 EII-instance START greedy greedy+LS greedy greedy+LS backtracking backtracking+LS backtracking backtracking+LS greedy greedy+LS greedy greedy+LS backtracking backtracking+LS backtracking backtracking+LS greedy greedy+LS greedy greedy+LS backtracking backtracking+LS backtracking backtracking+LS greedy greedy+LS greedy greedy+LS backtracking backtracking+LS backtracking backtracking+LS greedy greedy+LS greedy greedy+LS backtracking backtracking+LS backtracking backtracking+LS Region Size 1 x 10 1 x 10 2 x 10 2 x 10 1 x 10 1 x 10 2 x 10 2 x 10 1 x 12 1 x 12 2 x 12 2 x 12 1 x 12 1 x 12 2 x 12 2 x 12 1 x 14 1 x 14 2 x 14 2 x 14 1 x 14 1 x 14 2 x 14 2 x 14 1 x 16 1 x 16 2 x 16 2 x 16 1 x 16 1 x 16 2 x 16 2 x 16 1 x 16 1 x 16 2 x 16 2 x 16 1 x 16 1 x 16 2 x 16 2 x 16 MAX 165 168 170 170 169 172 170 171 245 247 249 250 248 250 249 252 338 339 344 344 342 344 344 344 448 449 454 454 453 454 457 457 449 450 456 457 453 454 457 457 AVG 161.10 164.90 166.35 167.05 165.65 167.75 167.65 168.10 241.20 244.00 247.00 247.50 245.35 247.75 247.60 248.45 333.00 335.56 340.50 340.80 338.25 340.69 342.17 342.33 444.05 446.20 451.38 451.88 448.80 451.53 453.69 454.00 443.75 446.50 451.75 451.95 449.00 451.35 452.80 453.15 MIN 158 161 164 164 162 165 164 165 239 240 244 245 242 246 246 246 329 332 338 338 334 336 340 340 440 443 448 448 444 449 451 451 440 442 450 450 446 447 448 449 Time Avg. (s) 4.29 1204.29 25.93 1225.93 1200.24 2400.24 1207.06 2407.06 6.83 1806.83 101.02 1901.02 1801.51 3601.51 1868.85 3668.85 10.43 2410.43 2335.42 4735.42 2401.36 4801.36 4664.65 7064.65 24.08 3624.08 11188.03 14788.03 3605.68 7205.68 13722.86 17322.86 21.85 3621.85 13835.04 17435.04 3605.08 7205.08 15294.52 18894.52 Table 3: Results for the MILP-based greedy and backtracking heuristics with and without local search, using different region sizes. ObjMax ObjAvg ObjMin Optimum 10 × 10 167 163.2 158 180 12 × 12 238 231.1 223 264 14 × 14 312 299.2 292 364 16 × 16 398 384.3 367 480 EII − instance 391 382 372 480 Table 4: Results for the local search procedure with optimal neighbourhoods starting from a random solution. 21 Present paper (20 runs) Present paper (100 runs) Muñoz et al. [10] Wang and Chiang [13] Coelho et al. [4] Schaus and Deville [11] Wauters et al. [14] Verhaard [12] Optimum 10 × 10 172 (20 min) 172 (40 min) 163 (20 min) 167 (20 min) 172 (20 min) 180 12 × 12 252 (30 min) 252 (60 min) 234 (30 min) 241 (30 min) 254 (30 min) 264 14 × 14 344 (40 min) 347 (80 min) 318 (40 min) 325 (40 min) 348 (40 min) 364 16 × 16 457 (180 min) 457 (360 min) 418 (60 min) 425 (60 min) 460 (60 min) 480 EII − instance 457 (180 min) 458 (360 min) 371 (-) 458 (1440 min) 461 (60 min) 467 (weeks/months) 480 Table 5: Comparison of the best results to other approaches available in the literature. (Execution times presented within parenthesis) 22	8
Mapping Images to Scene Graphs with Permutation-Invariant Structured Prediction Roei Herzig * 1 Moshiko Raboh * 1 Gal Chechik 2 3 Jonathan Berant 1 Amir Globerson 1 arXiv:1802.05451v1 [stat.ML] 15 Feb 2018 Abstract Structured prediction is concerned with predicting multiple inter-dependent labels simultaneously. Classical methods like CRF achieve this by maximizing a score function over the set of possible label assignments. Recent extensions use neural networks to either implement the score function or in maximization. The current paper takes an alternative approach, using a neural network to generate the structured output directly, without going through a score function. We take an axiomatic perspective to derive the desired properties and invariances of a such network to certain input permutations, presenting a structural characterization that is provably both necessary and sufficient. We then discuss graph-permutation invariant (GPI) architectures that satisfy this characterization and explain how they can be used for deep structured prediction. We evaluate our approach on the challenging problem of inferring a scene graph from an image, namely, predicting entities and their relations in the image. We obtain state-of-the-art results on the challenging Visual Genome benchmark, outperforming all recent approaches. 1. Introduction Structured prediction addresses the problem of classification when the label space contains multiple inter-dependent labels. For example, in semantic segmentation of an image, each pixel is assigned a label, while considering the labels of nearby pixels. A similar problem is the task of recognizing multiple entities and their relations in an image, where recognizing one entity affects recognition of the others. Structured prediction has attracted considerable attention because it applies to many learning problems and poses unique theoretical and applied challenges (e.g., see Taskar et al., 2004; Chen et al., 2015; Belanger et al., 2017). * Equal contribution 1 Tel-Aviv University, Israel 2 Google Brain, CA, 94043 3 Gonda brain research institute, Bar-Ilan University, Ramat-Gen 52900, Israel. Correspondence to: Roei Herzig <[email protected]>, Moshiko Raboh <[email protected]>, Gal Chechick <[email protected]>, Jonathan Berant <[email protected]>, Amir Globerson <[email protected]>. Typically, structured prediction models define a score function s(x, y) that quantifies how well a label assignment y is compatible, or consistent, with an input x. In this setup, the inference task amounts to finding the label that maximizes the compatibility score y ∗ = arg maxy s(x, y). This score-based approach separates a scoring component – implemented by a parametric model, from an optimization component – aimed at finding a label that maximizes that score. Unfortunately, for a general scoring function s(·), the space of possible label assignments grows exponentially with input size. For instance, the set of possible pixel label assignments is too large even for small images. Thus, inferring the label assignment that maximizes a scoring function is computationally hard in the general case. An alternative approach to scored-based methods is to map an input x to a structured output y with a “black box” neural network, without explicitly defining a score function. This raises a natural question: what properties and invariances must be satisfied by such a network? We take this axiomatic approach and argue that one important property is invariance to a particular type of input permutation. We then prove that this invariance is equivalent to imposing certain structural constraints on the architecture of the network, and describe architectures that satisfy these constraints, significantly extending the expressive power of current structured prediction approaches. We argue that respecting permutation invariance is important, as otherwise the model would have to spend capacity on learning this invariance at training time. Conceptually, our approach is motivated by recent work on D EEP S ETS (Zaheer et al., 2017), which asked a similar question for black-box functions on sets. To evaluate our approach, we tackle the challenging task of mapping an image to a scene graph, which describes the entities in the image and their relations. We describe a model that satisfies the permutation invariance property, and show that it achieves state-of-the-art results on the competitive Visual Genome benchmark (Krishna et al., 2017), demonstrating the power of our new design principle. In summary, the novel contributions of this paper are: First, we derive sufficient and necessary conditions for a deep structured prediction architecture. Second, we improve the state-of-the-art with this approach in a challenging pixel-tograph problem on a large dataset of complex visual scenes. Mapping Images to Scene Graphs with Permutation-Invariant Structured Prediction 2. Structured Prediction Scored-based methods in structured prediction define a score function s(x, y)1 that reflects the degree to which y is compatible with x, and infer a label by solving y ∗ = arg maxy s(x, y) (e.g., see Lafferty et al., 2001; Taskar et al., 2004; Meshi et al., 2010; Chen et al., 2015; Belanger et al., 2017). Most score functions previously used decompose as a sum over simpler functions, s(x, y) = P i fi (x, y), where solving maxy fi (x, y) can be performed efficiently.2 This local maximization forms the basic building block of algorithms for approximately maximizing s(x, y). One way to achieve this is to restrict fi (x, y) to depend only on a small subset of the y variables. The renewed interest in deep learning led to efforts to integrate deep networks with structured prediction, including modeling the fi functions as deep networks. In this context, the most widely-used score functions are singleton f (yi , x) and pairwise fij (yi , yj , x). Initial work used a two-stage architecture, learning local scores independently of the structured prediction goal (Chen et al., 2014; Farabet et al., 2013). Later works considered end-to-end architectures where the inference algorithm is part of the computation graph (Chen et al., 2015; Pei et al., 2015; Schwing & Urtasun, 2015; Zheng et al., 2015). These studies used standard inference algorithms, such as loopy belief propagation, mean field methods and gradient descent (Belanger et al., 2017). Score-based methods provide several advantages. First, they allow intuitive specification of local dependencies between labels (like pairwise dependencies) and how these translate to global dependencies. Second, when the score function is linear in its parameters (s(x, y; w) is linear in w), the learning problem has natural convex surrogates (e.g., logloss in CRF), making learning efficient. Third, inference in large label spaces is often possible via exact combinatorial algorithms or empirically accurate approximations. However, with the advent of deep scoring functions s(x, y; w), learning is no longer convex. Thus, it is worthwhile to rethink the architecture of structured prediction models, and consider models that map inputs x to outputs y directly without an explicit score function. We want these models to enjoy the expressivity and predictive power of neural networks, while maintaining the ability to specify local dependencies between labels in a flexible manner. In the next section, we present such an approach and consider a natural question: what should be the properties of such a deep neural network used for structured prediction. 1 The term energy function is also used. More precisely, many P message passing algorithms require that functions fi (x, y) + k δk (yk ) can be maximized efficiently. 2 3. Permutation Invariant Structured Prediction We begin with some notation, focusing on structures that consists of pairwise interactions, as these are simpler in terms of notation, and sufficient for describing the structure in many problems. We denote a structured label with n entries by y = [y1 , . . . , yn ]. In a score-based approach, the score is defined via a set of singleton scores fi (yi , x) and pairwise scores fij (yi , yj , x), where the overall score s(x, y) is the sum of these singleton and pair scores. For brevity, we also denote fij = fij (yi , yj , x) and fi = fi (yi , x). An inference algorithm takes as input the set of local scores fi , fij and outputs the assignment maximizing s(x, y). We can therefore abstractly view an inference algorithm as a blackbox that takes as input a set of node- and edge-dependent inputs (i.e., the local scores fi , fij ) and returns a label y, even without an explicit score function s(x, y). While numerous inference algorithms exist for this setup, including belief propagation (BP) and mean field, here we aim to develop a framework for a deep learning labeling algorithm (we avoid the term “inference” since the algorithm does not explicitly maximize a score function). Such an algorithm will be a black-box with the f functions as input and the labels y1 , . . . , yn as output. We next ask what architecture such an algorithm should have. We follow with several definitions. A graph labeling function F : (V, E) → Y is a function whose input is an ordered set of node features V = [z 1 , . . . , z n ] and ordered set of edge features E = [z 1,2 . . . , z i,j , . . . , z n,n−1 ]. For example, the z i ’s can be the array of values fi (yi , x), and the z i,j ’s can be the table of values fi,j (yi , yj , x) . For simplicity, assume z i ∈ Rd and z i,j ∈ Re . The output of F is a set of labels y = [y1 , . . . , yn ], which can be thought of as labeling the nodes. Thus, inference algorithms like BP are graph labeling functions, since they take f as input and output a set of labels. However, graph labeling functions need not correspond to any inference algorithm (i.e., an algorithm that maximizes a score function). A natural requirement is that the algorithm produces the same result when given the same score function. For example, consider a label space containing three variables y1 , y2 , y3 , and assume that the inference algorithm takes as input z = (z 1 , z 2 , z 3 , z 12 , z 13 , z 23 ) = (f1 , f2 , f3 , f12 , f13 f23 ), and outputs a label y = (y1∗ , y2∗ , y3∗ ). When the same algorithm is given an input that is permuted in a consistent way, z 0 = (f2 , f1 , f3 , f21 , f23 f13 ), this defines exactly the same score function as the first scenario. Hence, we would expect it to output the same label, only permuted, namely, it should output y = (y2∗ , y1∗ , y3∗ ). Most inference algorithms, including BP and mean field, satisfy this symmetry requirement by de- Mapping Images to Scene Graphs with Permutation-Invariant Structured Prediction 3.1. Characterizing Permutation Invariance Figure 1. Graph permutation invariance and structured prediction. A graph labeling function F is graph permutation invariant (GPI) if permuting the names of nodes maintains the output. sign. Here we design a deep learning black-box, hence need to guarantee invariance to input permutations. A black-box that does not satisfy this invariance has to waste capacity on learning it at training time. In what follows we use z to denote the joint set of node and edge features. z can be thought of as a container with n + n(n − 1) = n2 elements. We next consider what happens to a graph labeling function, when graph variables are permuted by a permutation σ. Importantly, the edges in this case are also permuted in a way that is consistent with the node permutation σ. Definition 1. Let z be a set of node and edge features. Given a permutation σ of {1, . . . , n}, denote σ(z) to be a new set of node and edge features that are given by: [σ(z)]i = z σ(i) , [σ(z)]i,j = z σ(i),σ(j) . (1) σ(z) has the same elements as in z, but the node elements are permuted according to σ, and the edge elements are permuted accordingly. In what follows, we use the notation σ([y1 , . . . , yn ]) = [yσ(1) , . . . , yσ(n) ], namely, σ applied to a set of labels yields the same labels, only permuted by σ. Next comes our key definition of a function F whose output is invariant to permutations of the input graph. Definition 2. A graph labeling function F is said to be graph-permutation invariant (GPI), if for all permutations σ of {1, . . . , n} and for all z it satisfies: F(σ(z)) = σ(F(z)). (2) Figure 1 illustrates the desired invariance. The above property says that as long as the input to F describes the same node and edge properties, the same labeling will be output. This is indeed a property we would like any such F to have, and we thus turn to characterizing a necessary and sufficient structure for achieving it. Motivated by the above discussion, we ask: what structure is necessary and sufficient to guarantee that F is graphpermutation invariant? Note that a function F takes as input an ordered set z. Therefore its output on z could certainly differ from its output on σ(z). To achieve permutation invariance, F should intuitively contain certain symmetries. For example, one permutation invariant architecture is to define yi = g(z i ) for any function g, but this characterization is too restrictive to cover all permutation invariant functions. The next Theorem provides a complete characterization, while Figure 2 shows the corresponding architecture. Theorem 1. Let F be a graph labeling function. Then F is graph-permutation invariant if and only if there exist functions α, ρ, φ such that for all k = 1, . . . , n: [F(z)]k = ρ(z k , n X α(z i , i=1 X φ(z i , z i,j , z j ))), (3) j6=i where φ : R2d+e → RL and α : Rd+L → RW and ρ : RW +d → R. Proof. First, we show that any F satisfying the conditions of Theorem 1 is GPI. Namely, for any permutation σ, [F(σ(z))]k = [F(z)]σ(k) . To see this, write [F(σ(z))]k using Eq. 3 and Definition 1 as: X X ρ(z σ(k) , α(z σ(i) , φ(z σ(i) , z σ(i),σ(j) , z σ(j) ))). i j6=i The second argument of ρ above is clearly invariant under σ, because the sum considers an index i and all other indices j, hence the same elements are covered under permutation. The expression therefore equals to: X X ρ(z σ(k) , α(z i , φ(z i , z i,j , z j ))) = [F(z)]σ(k) i j6=i where the equality follows from Eq. 3. We thus proved that Eq. 3 implies graph permutation invariance. Next, we prove that any black-box graph-permutation invariant function can be expressed as in Eq. 3. Namely, we show how to define φ, α and ρ that can implement any permutation invariant function F. The key idea is to construct φ, α such that the second argument of ρ in Eq. 3 contains all the information about the graph features z, including the edges they originated from . Then, the function ρ consists of an application of the black box F to this representation, followed by extracting the label yk . To simplify notation assume that edge features are scalar (e = 1). The extension to the vector case is simple, but involves more indexing. We also assume that z k uniquely identifies the node (i.e., no two nodes share the same node Mapping Images to Scene Graphs with Permutation-Invariant Structured Prediction Figure 2. A schematic representation of the GPI architecture in Theorem 1. Singleton features z i are omitted for simplicity. First, the features z i,j are processed element-wise by φ. Next, they are summed to create a vector si , which is concatenated with z i . Third, a representation of the entire graph is created by applying α n times and summing the created vector. The graph representation is then finally processed by ρ together with z k . feature), which can be achieved by adding the index as another feature of z k . Finally, we assume that F is a function only of the pairwise features z i,j . This can be achieved by adding singleton features into the pairwise ones. Let H be a hash function with L buckets mapping node features z i to an index (bucket). Assume that H is perfect (this can be achieved for a large enough L). Define φ to map the pairwise features to a vector of size L. Let 1 [j] be a one-hot vector of dimension RL , with one in the j th coordinate. Recall that we consider scalar z i,j so that φ is indeed in RL , and define φ as: φ(z i , z i,j , z j ) = 1 [H(z j )] zi,j (4) i.e., φ “stores” zi,j in the unique bucket for node j. P Let si = j6=i φ(z i , zi,j , z j ) be the second argument of α in Eq. 3 (si ∈ RL ). Then, since all z j are distinct, si stores all the pairwise features for neighbors of i in unique positions within its L coordinates. Since si (H(z k )) contains the feature zi,k whereas sj (H(z k )) contains the feature z j,k , we cannot simply sum the si , since we would lose the information of which edges the features originated from. Instead, we define α to map si to RL×L such that each feature is mapped to a distinct location. Formally: α(z i , si ) = 1 [H(z i )] sTi . (5) α outputs a matrix that is all zeros except for the features correspondingP to node i that are stored in row H(z i ). The matrix M = i α(z i , si ) (namely, the second argument of ρ in Eq. 3) is a matrix with all the edge features in the graph including the graph structure. Figure 3. Illustration of the proof construction for Theorem 1. Here H is a hash function of size L = 5 such that H(1) = 1, H(3) = 2, H(2) = 4, G is a three-node input graph, and z i,j ∈ R are the pairwise features (in purple) of G. (a) φ is applied to each z i,j . Each application yields a vector in R5 . The three dark yellow columns correspond to φ(z 1,1 ),φ(z 1,2 ) and φ(z 1,3 ). Then, all vectors φ(z i,j ) are summed over j to obtain three si vectors. (b) α’s (blue matrices) are an outer product between 1 [H(z i )] and si (see Eq. 5) resulting in a matrix of zeros except one row. The dark blue matrix corresponds for α(z 1 , s1 ). (c) All α’s are summed to a 5 × 5 matrix, isomorphic to the original z i,j matrix. To complete the construction we set ρ to have the same outcome as F. We first discard rows and columns in M that do not correspond to original nodes (reducing M to dimension n × n). Then, we use the reduced matrix as the input z to the given F. Assume for simplicity that M does not need to be contracted.3 Let the output of F on M be y = y1 , . . . , yn . Then we set ρ(z k , M ) = y H(zk ) . Since F is invariant to permutations this indeed returns the output of F on the original input. General Graphs So far, we discussed complete graphs, where all edges correspond to valid feature pairs. Many graphs however may be sparse and have certain structures. For example, an n-variable chain graph in sequence labeling has only n − 1 edges. For such sparse graphs, the input to F would not be all z i,j pairs but rather only features corresponding to valid edges of the graph, and we are only interested in invariances that preserve the graph structure, namely, the automorphisms of the graph. Thus, the desired invariance is that σ(F(z)) = F(σ(z)) only for automorphisms of the graph. It is easy to see that P Theorem P1 holds in this case, if one replaces the sum j6=i with j∈N (i) , where N (i) are the neighbors of node i in the graph. 3 This merely introduces another indexing step. Mapping Images to Scene Graphs with Permutation-Invariant Structured Prediction 4. Deep Graph Prediction Theorem (1) provides the general requirements for designing an architecture for structured prediction. For a given problem, one has to choose a specific architecture and parameterization for α, φ, ρ. For instance, it is interesting to consider how an algorithm like belief propagation (BP) can be implemented in our framework. Following the proof of Theorem 1, one would use φ, α to aggregate features, and then ρ would apply BP to these features. Our architecture is of course more general by construction. For example, it could use φ and α to “sketch” the input graph, such that labeling can be performed on a reduced representation. We now survey certain architectures consistent with Theorem 1 and discuss their expressive power. Introducing Attention. Attention is a powerful architectural component in deep learning (Bahdanau et al., 2015), but most inference algorithms do not use attention. We now show how attention can be introduced in our framework. Intuitively, attention means that instead of aggregating features of neighbors, a node i weighs neighbors based on their relevance. For example, the label of an entity in an image may depend more strongly on entities that are spatially closer. We now implement attention for the architecture of Eq. 3. Formally, we learn attention weights for the neighbors j of a node i, which scale the features z i,j of that neighbor. We can also learn different attention weights for individual features of each neighbor in a similar way. Let wi,j ∈ R be an attention mask specifying the weight that node i gives to node j: X wi,j (z i , z i,j , z j ) = eβ(zi ,zi,j ,zj ) / eβ(zi ,zi,t ,zt ) , (6) t where β can be any scalar-valued function of its arguments (e.g., a dot product of z i and z j as in standard attention models). To introduce attention we wish α ∈ Re to have the form of weighting P wi,j over neighboring feature vectors z i,j , namely, α = j6=i wi,j z i,j . Figure 4. An image (top) and its scene graph (bottom) from the Visual Genome dataset (Krishna et al., 2017). The scene graph captures the entities in the image (nodes, blue circles) and their pairwise relations (edges, red circles). Example relationships in this graph include: hat, on, dog and dog, on, motorcycle . Using RNNs as Components. Theorem 1 allows arbitrary functions for φ, α and ρ, except for their input dimensionality. Specifically, these functions can involve highly expressive recursive computation, simulate existing message passing algorithms, and new algorithms that are learned from data. This can of course be extended to more elaborate structures like LSTMs (Hochreiter & Schmidhuber, 1997) and Neural Turing Machines (Graves et al., 2014), which we leave for future work. Theorem 1 suggests that any function in the form of F is graph permutation invariant. It is easy to show that composing two functions that are GPI, is also GPI. Therefore, we can run F iteratively by providing the output of one step of F as part of the input to the next step and maintain graph-permutation invariance. This results in a recurrent architecture, which we will employ in the next section to obtain state-of-the-art performance on scene graph prediction. 5. Application - Scene Graph Classification X si,1:e X eβ(zi ,zi,j ,zj ) z i,j P = = wi,j z i,j β(z ,z ,z ) i i,j j si,e+1 j6=i e We demonstrate the benefits of our axiomatic approach in the task of inferring scene graphs from images. In this problem, the input is an image annotated with a set of rectangles that bound entities in the image, known as bounding boxes. The goal is to label each bounding box with the correct entity category, and every pair of entities with their relation, such that they form a coherent graph, known as a scene graph. In a scene graph, nodes correspond to bounding boxes labeled with the entity category and edges correspond to relations among entities, which could be spatial (“on”) or functional (“wearing”). Thus, in an image with 5 bounding boxes there are 5 + 5 × 4 = 25 output variables. A similar approach can be applied over α and ρ to model attention over the outputs of α as well (graph nodes). This concept is illustrated in Figure 4, showing an image of a dog on a motorcycle (top) and the corresponding scene To achieve this form we extend φ by a single entry, defining φ ∈ Re+1 (namely we set L = e+1) as φ1:e (z i , z i,j , z j ) = eβ(zi ,zi,j ,zj ) z i,j (here φ1:e are the first e elements of φ) and φe+1 (z iP , z i,j ; z j ) = eβ(zi ,zi,j ,zj ) . We keep the definition si,1:e of si = j6=i φ(z i , zi,j , z j ). Next, we define α = si,e+1 and substitute si and φ to obtain the desired form as attention weights wi,j over neighboring feature vectors z i,j : α(z i , si ) = j6=i j6=i Mapping Images to Scene Graphs with Permutation-Invariant Structured Prediction graph below. The pink box in the image is labeled “motorcycle” and the white box is labeled “dog”. These two boxes correspond to two nodes (light blue circles in Figure 4 bottom), and their relation “on” corresponds to an edge (red circle labeled “on”). While scene graphs are typically very sparse, one can view a scene graph as complete if each pair of unrelated entities is connected by a ‘null’ edge. A scene graph can be represented as a collection of triplets, each with a relation and two entities, like dog, on, motorcycle . 5.1. Model Our model has two components: A Label Predictor (LP) that takes as input an image with bounding boxes and outputs a distribution over labels for each entity and relation. Then, a Scene Graph Predictor (SGP) that takes all label distributions and predicts more consistent label distributions jointly for all entities and relations. Label Prediction. The LP module (Figure 5) receives an input image x and a set of bounding boxes bb1 , . . . , bbn , corresponding to image entities (as in Figure 4). LP outputs a set of entity label probabilities p(yient \| bbi ) for each box i from a pre-defined set of candidate entity labels P and a set rel of relation probabilities p(yi,j \| bbi , bbj ) from another predefined set of relation labels R. These unary and pairwise potentials are later fed to the SGP module. To predict entity labels, we used R ES N ET 50 (He et al., 2016a;b), taking as input a patch cropped from the full image according to the ith bounding box (Figure 5). We used a second R ES N ET 50 to predict relations, using a 5-channel tensor input: three channels for the RGB image, and two channels for binary masks for the subject entity and object entity bounding boxes (Figure 5). The image patch provided to this network was cropped such that it covers both the subject entity and the object entity. Providing the two binary masks breaks the symmetry of the subject entity and object entity and allow the network to discriminate between triplets like man, wearing, shirt and shirt, on, man ). Scene Graph Prediction. While the LP module described above is trivially GPI, because the output variables rel yient , yi,j are predicted independently, constructing a GPI architecture for a Scene Graph Predictor is harder. We now outline this construction. Entity classification in this module is GPI following Theorem 1, where z i are features for every bounding box and z i,j are features for box pairs. To classify relations, we added a function ρrelation that reuses the GPI representation created during entity classification. Because the input to ρrelation is a GPI representation, it is easy to show that our entire network is GPI. Let z i be the concatenation of z features and z spatial . Where i i features zi is the current label probability for entity i (logits Figure 5. The Label Predictor. (a) Entity recognition network: A network takes an image patch cropped based on a bounding box, and outputs classification probabilities per label. (b) Relation recognition network: A network takes an input tensor, containing the RGB image in the first 3 channels, and two binary masks for the subject and object entities in the remaining two channels. before the final softmax layer) and z spatial is i’s bounding i box given as a (x, y, width, height). In addition, for z i,j , we used the confidences for relation i, j (logits before the final softmax layer). In each step of SGP we apply the function F, which receives all entity features z i and all relation features z i,j , and output updated confidences for entities and relations. Because composing GPI functions is GPI, our SGP module is GPI. We now describe our implementation of the three components of F: φ, α and ρ. (1) φ is a network with two FC-layers. It receives (a) subject features z i (b) relations features z i,j (c) entity features z j and outputs a vector of size 500. Next, for each entity i, we aggregate φ(z i , z i,j , z j ) into si using the attention mechanism described in Section 4. To calculate the weights wi,j , we implement β(·) (Eq. 6) with a FC layer that receives the same input as φ and outputs a scalar. (2) α is a two FC-layer network, receiving entity features z i and context features si . The outputs of α are aggregated with a similar attention mechanism over entities, resulting in a vector g ∈ R500 representing the entire graph. (3) ρ, consists of ρentity , which classifies entities, and ρrelation , which classifies relations. ρentity is a three FC-layer network of size 500. It receives z i , si and g as input, and outputs a vector qi with one scalar per entity class. Unlike Theorem 1, we allow ρ direct access to si , which maintains the GPI property, and improved learning in practice. The final output confidence is a linear interpolation of the current confidence z fi eatures and the new confidence qi , controlled by a learned forget gate, i.e., the output is qi + forget · z fi eatures . ρrelation , the relation classifier, is analogous to the entity classifier, receiving as input z i , z j , the relation features z i,j , and the graph representation g. Mapping Images to Scene Graphs with Permutation-Invariant Structured Prediction We also explored concatenating word embeddings of the most probable entity class to z i . Word vectors were learned with G LOV E (Pennington et al., 2014) from the ground-truth captions of Visual Genome (Krishna et al., 2017). 5.2. Experimental Setup Dataset. We evaluated our approach on the Visual Genome (VG) dataset (Krishna et al., 2017). VG consists of 108,077 images annotated with bounding boxes, entities and relations. Distribution over entity classes and relations is long-tailed with a total of 75,729 unique entity classes and 40,480 unique relations. To allow apple-to-apple comparison with previous studies of this dataset (Xu et al., 2017; Newell & Deng, 2017; Zellers et al., 2017), we used the same preprocessed data, including the train and test splits, as provided by (Xu et al., 2017). This dataset had on average 12 entities and 7 relations per image. For evaluation, we used the same 150 entity categories and 50 relations as in (Xu et al., 2017; Newell & Deng, 2017; Zellers et al., 2017). To tune hyper-parameters, we also split the training data into two by randomly selecting 5K examples, resulting in a final 70K/5K/32K split for train/validation/test sets. Training Procedure. We trained all networks using Adam (Kingma & Ba, 2014); Input images were resized to 224x224 to conform with the R ES N ET architecture. We first trained the LP module, and then trained the SGP module using the best LP model. In what follows, all the particular chosen values were tuned on the validation set. For LP, we trained the relation network with cross-entropy loss and a positive-to-negative ratio of 1:3 (where ‘positive’ refers to a labeled relation and ‘negative’ to unlabeled), and performed early-stopping after 90 epochs. We chose a batch size of 64, and also used data augmentation techniques such as translation and rotation to further improve the results. The loss function for the SGP was the sum of cross entropy losses over all entities and relations in the image. In the loss, we penalized entities 4 times more strongly than relations, and penalized negative relations 10 times more weakly than positive relations. We used batch size 100 and early-stopped after 120 epochs. The recurrent application of F was performed for 2 steps. Evaluation. Xu et al. (2017) defined three different subtasks when inferring scene graphs, and we focus on two: (1) SGCls: Given ground-truth bounding boxes for entities, predict all entity categories and relations categories. (2) PredCls: Given bounding boxes annotated with entity labels, predict all relations. Following (Lu et al., 2016), we used Recall@K as the evaluation metric. It measures the fraction of correct ground-truth triplets that appear within the K most confident triplets proposed by the model. Two evaluation protocols are used in the literature, which differ by whether they enforce graph constraints over model predictions. The first protocol requires that the top-K triplets assign one consistent class per entity and relation. This rules out putting more than one triplet for a pair of bounding boxes. It also rules out inconsistent assignment, like a bounding box that is labeled as one entity in one triplet, and as another entity in another triplet. The second evaluation protocol does not enforce any such constraints. Models and baselines. We compare four variants of our GPI approach with the reported results of four baselines that are currently the state-of-the-art on various scene graph subtasks. All models use the same data split and pre-processing as (Xu et al., 2017): 1. (L U ET AL ., 2016): This work leverages word embeddings to fine-tune the likelihood of predicted relations. 2. (X U ET AL ., 2017): This model passes messages between entities and relations, and iteratively refines the feature map used for prediction. 3. (N EWELL & D ENG , 2017). The P IXEL 2G RAPH model uses associative embeddings (Newell et al., 2017) to produce a full graph from the image. 4. (Z ELLERS ET AL ., 2017) The N EURAL M OTIF method encodes global context for capturing highorder motifs in scene graphs. 5. GPI: N O ATTENTION: Our GPI model, but with no attention mechanism. Instead, following Theorem 1, we simply sum the features. 6. GPI: N EIGHBOR ATTENTION: Our GPI model, using attention over neighbors as described in Section 5.1. 7. GPI: M ULTI ATTENTION: Our GPI model, except that we learn different attention weights per feature. 8. GPI: L INGUISTIC : Same as GPI: M ULTI ATTENTION but also concatenating the word embedding vector for the most probable entity label (see Sec. 5.1). 5.3. Results Table 1 lists recall@50 and recall@100 for four variants of our approach compared with three baselines, evaluating with graph constraints. The GPI approach performs well, and L INGUISTIC outperforms all baselines for both PredCls and SGCls. Table 2 provides a similar comparison when evaluating without graph constraints; again L INGUISTIC performs best. More details in the supplemental material. Figure 6 illustrates the model behavior. Predicting isolated labels with LP (column (c)) mislabels several entities, but these are corrected after joint prediction (column (d)). Column (e) shows that the system learned to attend more to nearby entities (the window and building are closer to the tree), and column (f) shows that stronger attention is learned for the classes bird, presumably because it is usually more informative than common classes like tree. Mapping Images to Scene Graphs with Permutation-Invariant Structured Prediction Figure 6. (a) An input image with bounding boxes from VG. (b) The ground-truth scene graph. (c) The LP fails to recognize some entities (building and tree) and relations (in front of instead of looking at). (d) GPI:L INGUISTIC fixes most incorrect LP predictions. (e) Window is the most significant neighbor of Tree. (f) The entity bird receives substantial attention, while Tree and building are less informative. Table 1. Test set results for graph-constrained evaluation SGC LS R@50 R@100 (L U ET AL ., 2016) (X U ET AL ., 2017) N EURAL M OTIFS N O ATTENTION N EIGHBOR ATTEN . M ULTI ATTENTION L INGUISTIC 11.8 21.7 31.3 28.9 30.6 32.3 32.1 14.1 24.4 32.1 31.0 32.4 34.1 34.1 P RED C LS R@50 R@100 35.0 44.8 65.8 63.0 63.7 65.4 66.0 27.9 53.0 68.0 65.2 66.5 67.8 68.3 Table 2. Test set results for unconstrained evaluation SGC LS R@50 R@100 P IXEL 2G RAPH N O ATTENTION N EIGHBOR ATTEN . M ULTI ATTENTION L INGUISTIC 26.5 35.2 37.6 39.0 39.2 30.0 42.1 43.2 44.5 44.9 P RED C LS R@50 R@100 68.0 70.3 73.7 74.5 75.3 75.2 75.1 79.2 81.0 82.4 6. Related Work There has been significant recent interest in extending deep learning to structured prediction. Much of this work has been on semantic segmentation, where convolutional networks (Shelhamer et al., 2017) became a standard approach for obtaining “singleton scores” and various approaches were proposed for adding structure on top. Most of these approaches used variants of message passing algorithms, unrolled into a computation graph (Xu et al., 2017). Some studies parameterized parts of the message passing algorithm and learned its parameters (Lin et al., 2015). Recently, gradient descent has also been used for maximizing score functions (Belanger et al., 2017; Gygli et al., 2017). An alternative approach for deep structured prediction is via greedy decoding, where one label is inferred at a time, based on previous labels. This has been popular in sequence-based applications like dependency parsing (Chen & Manning, 2014). These works rely on the sequential structure of the Table 3. Recall@5 of PredCls for the 20-top relations ranked by their frequency, as in (Xu et al., 2017) R ELATION ON HAS IN OF WEARING NEAR WITH ABOVE HOLDING BEHIND UNDER SITTING ON IN FRONT OF ATTACHED TO AT HANGING FROM OVER FOR RIDING L U ET AL . (2016) X U ET AL . (2017) L INGUIS - 99.71 98.03 80.38 82.47 98.47 85.16 31.85 49.19 61.50 79.35 28.64 31.74 26.09 8.45 54.08 0.0 9.26 12.20 72.43 99.25 97.25 88.30 96.75 98.23 96.81 88.10 79.73 80.67 92.32 52.73 50.17 59.63 29.58 70.41 0.0 0.0 31.71 89.72 99.4 98.9 96.2 98.2 99.5 95.0 94.2 83.7 95.6 90.6 83.2 90.5 74.7 77.2 81.1 74.7 54.9 43.4 96.2 TIC input, where BiLSTMs can be effectively applied. The concept of architectural invariance was recently proposed in D EEP S ETS (Zaheer et al., 2017). The invariance we consider is much less restrictive (i.e., we do not need to be invariant to all permutations of singleton and pairwise features, just those consistent with a graph re-labeling), and hence results in a substantially different set of architectures. Extracting scene graphs from images provides a semantic representation that can later be used for reasoning, question answering, and image retrieval (Johnson et al., 2015; Lu et al., 2016; Raposo et al., 2017). It is at the forefront of machine vision research, integrating challenges like object detection, action recognition and detection of human-object interactions (Liao et al., 2016; Plummer et al., 2017). Mapping Images to Scene Graphs with Permutation-Invariant Structured Prediction 7. Conclusion We presented a deep learning approach to structured prediction, which constrains the architecture to be invariant to structurally identical inputs. As in score-based methods, our approach relies on pairwise features, capable of describing inter-label correlations, and thus inheriting the intuitive aspect of score-based approaches. However, instead of maximizing a score function (which leads to computationallyhard inference), we directly produce an output that is invariant to equivalent representations of the pairwise terms. The axiomatic approach can be extended in many ways. For image labeling, geometric invariances (shift or rotation) may be desired. In other cases, invariance to feature permutations may be desirable. We leave the derivation of the corresponding architectures to future work. Finally, there may be cases where the invariant structure is unknown and should be discovered from data, which is related to work on lifting graphical models (Bui et al., 2013). It would be interesting to explore algorithms that discover and use such symmetries for deep structured prediction. References Bahdanau, D., Cho, K., and Bengio, Y. Neural machine translation by jointly learning to align and translate. In International Conference on Learning Representations (ICLR), 2015. Belanger, David, Yang, Bishan, and McCallum, Andrew. End-to-end learning for structured prediction energy networks. In Precup, Doina and Teh, Yee Whye (eds.), Proceedings of the 34th International Conference on Machine Learning, volume 70, pp. 429–439. PMLR, 2017. Bui, Hung Hai, Huynh, Tuyen N., and Riedel, Sebastian. Automorphism groups of graphical models and lifted variational inference. In Proceedings of the TwentyNinth Conference on Uncertainty in Artificial Intelligence, UAI’13, pp. 132–141, Arlington, Virginia, United States, 2013. AUAI Press. URL http://dl.acm. org/citation.cfm?id=3023638.3023652. Chen, Danqi and Manning, Christopher. A fast and accurate dependency parser using neural networks. In Proceedings of the 2014 conference on empirical methods in natural language processing (EMNLP), pp. 740–750, 2014. 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Improved Space-efficient Linear Time Algorithms for Some Classical Graph Problems Sankardeep Chakraborty1, Seungbum Jo2 , and Srinivasa Rao Satti3 arXiv:1712.03349v1 [cs.DS] 9 Dec 2017 1 The Institute of Mathematical Sciences, HBNI, Chennai, India. [email protected] 2 University of Siegen, Siegen, Germany. [email protected] 3 Seoul National University, Seoul, South Korea. [email protected] We provide space-efficient linear time algorithms for computing bridges, topological sorting, and strongly connected components improving on several recent results of Elmasry et al. [STACS’15], Banerjee et al. [COCOON’16] and Chakraborty et al. [ISAAC’16]. En route, we also provide another DFS implementation with weaker input graph representation assumption without compromising on the time and space bounds of the earlier results of Banerjee et al. [COCOON’16] and Kammer et al. [MFCS’16]. 1 Introduction Since the early days of designing graph algorithms, researchers have developed several approaches for testing whether a given undirected (or directed) graph G = (V, E) with n vertices and m edges is (strongly connected) biconnected and/or 2-edge connected, and finding cut vertices and/or bridges of G. All of these methods use depth-first search (DFS) as the backbone to design the main algorithm. The classical linear time algorithms due to Tarjan [11, 12] computes the so-called “low-point” values (which are defined in terms of a DFS-tree of G) for every vertex v, and checks some conditions using that to determine whether G has the desired property. There are other linear time algorithms as well for these problems (see [10] and all the references therein). All of these classical algorithms take O(m + n) time and O(n) words (our model of computation is the standard word RAM model with word size w = Ω(lg n) bits) of space. Our aim is to improve the space bounds of these algorithms without increasing the running time. 1.1 Motivation and Related Work Motivated mainly by the “big data” phenomenon among others, recently there has been a surge of interest in improving the space complexity of the fundamental linear time graph algorithms by paying little or no penalty in the running time i.e., reducing the working space of the classical graph algorithms (which generally take O(n lg n) bits) to o(n lg n) bits without compromising on time. Towards this, Elmasry et al. [7] gave, among others, an implementation for DFS taking O(m + n) time and O(n lg lg n) bits of space. For sparse graphs (when m = O(n)), the Time Space (in bits) DFS O(n + m) O(n + m) O(n + m) O(n + m) O(n lg n) O(n + m) O(n lg(m/n)) O(n lg lg n) [5] [1, 9] [3] [7] Testing biconnectivity & reporting cut vertices [11] [1] [3] [9] Testing 2-edge connectivity & reporting bridges [12] [1] [3] This paper Topological sort [5] This paper This paper [7] Testing strong connectivity [11] This paper This paper [7] Table 1: Summary of our results. space bound was improved further to O(n) bits keeping the same linear time in [1]. Banerjee et al. [1] gave, among others, a space efficient implementation for performing BFS using just 2n + o(n) bits of space and linear time, improving upon the result of [7]. Such algorithms for a few other graph problems also have been considered recently [2, 3, 4, 6, 9]. 1.2 Our Results We assume that the input graph G, which is represented using adjacency array [1, 3, 7, 9], i.e., G is represented by an array of length \|V \| where the i-th entry stores a pointer to an array that stores all the neighbors of the i-th vertex, is given in a read-only memory with a limited read-write working memory, and write-only output. We count space in terms of the number of bits in workspace used by the algorithms. Our main goal here is to improve the space bounds of some of the classical and fundamental graph algorithms. We summarize all our main results in Table 1. In this paper, basically we complete the full spectrum of results regarding the space bounds for these problems keeping the running time linear by providing/improving the missing/existing algorithms in the recent space efficient graph algorithm literature. Due to lack of space, we provide only sketches of our proofs. 2 Testing 2-Edge Connectivity and Finding Bridges In an undirected graph G, a bridge is an edge that when removed (without removing the vertices) from a graph creates more components than previously in the graph. A (connected) graph with at least two vertices is 2-edge-connected if and only if it has no bridge. Let T denote the DFS tree of G. Following Kammer et al. [9], we call a tree edge (u, v) of T with u being the parent of v full marked if there is a back edge from a descendant of v to a strict ancestor of u, half marked if it is not full marked and there exists a back edge from a descendant of v to u, and unmarked, otherwise. They use this definition to prove the following: (i) every vertex u (except the root r) is a cut vertex exactly if at least one of the edges from u to one of its children is either an unmarked edge or a half marked edge, and (ii) root r is a cut vertex exactly if it has at least two children in T . Based on the above characterization, they gave O(m + n) time and O(n lg lg n) bits algorithm to test/report if G has any cut vertex. Our main observation is that we can give a similar characterization for bridges in G, and essentially using a similar implementation, we can also obtain O(m + n) time and O(n lg lg n) bits algorithms for testing 2-edge connectivity and reporting bridges of G. We start with the following lemma. Lemma 1. A tree edge e = (u, v) in T is a bridge of G if and only if it is unmarked. Proof sketch: If e is unmarked, then no descendants of v reaches u or any strict ancestor of u, so deleting e would result in disconnected graph, thus e has to be a bridge. On the other direction, it is easy to see that if e is a bridge, it has to be an unmarked edge. Now we state our theorem below. Theorem 2. Given an undirected graph G, in O(m + n) time and O(n lg lg n) bits of space we can determine whether G is 2-edge connected. If G is not 2-edge connected, then in the same amount of time and space, we can compute and output all the bridges of G. Proof sketch: Using Lemma 1 and the similar implementation of using stack compression and other tools of the algorithm provided in Section 3.2 of Kammer et al. [9] with few modifications, we can prove the theorem. Note that the space bound of Theorem 2 improves the results of [1] and [3] for sufficiently dense graphs (when m = ω(n lg lg n) and m = ω(n lgO(1) n) respectively) while keeping the same linear runtime (see Table 1). 3 DFS without Cross Pointers Banerjee et al. [1] and subsequently Kammer et al. [9] gave O(m + n) bits and O(m + n) time implementations of DFS improving on the bounds of [7] for sparse graphs. But both of these DFS implementations assume that the input graph is represented using the adjacency array along with cross pointers i.e., for undirected graphs, every neighbour v in the adjacency array of a vertex u stores a pointer to the position of vertex u in the adjacency array of v. See [7] for detailed definitions for directed graphs. We emphasize that this input assumption can double the space usage, compared to the raw adjacency array in worst case. In what follows, we provide the proof sketch of a DFS implementation taking the same time and space bounds as that of [1, 9] but without using the cross pointers. Our main theorem is as follows. Theorem 3. Given a directed or undirected graph G, represented as adjacency array, we can perform DFS traversal of G using O(m + n) bits and O(m + n) time. Proof sketch: We essentially modify the proof of [1] which uses a bitvector A of length O(m + n) having one to one mapping with the unary encoding of the degree sequence to mark the tree edges, and subsequently uses cross pointers to find the parent of any vertex during backtracking as well as starting with next unvisited vertex after backtracking. We note that we can represent the parents of all the vertices in another bitvector P of length O(m + n) (parallel to A). Now to perform backtracking efficiently, we could use the constant time append only structure (also with constant time rank/select) of Grossi et al. [8] along with the P array. With these modifications, we could get rid of cross pointers without compromising on the running time and space bound of the earlier algorithms. 4 Testing Strong Connectivity and Topological Sorting Towards giving improved space efficient algorithms for strong connectivity (SC) and topological sorting (TS), we first improve Lemma 4.1 of [7] which says the following: if DFS of a directed graph G takes T (n, m) time and S(n, m) space, then we can output the vertices of G in reverse postorder of the DFS tree T of G taking O(T (n, m)) time and O(S(n, m) + n lg lg n) space. Combining this lemma with the classical algorithms for SC and TS [5] they obtained O(n lg lg n) bits and O(m + n) time algorithms for both these problems. We improve these by showing the following, Theorem 4. If DFS of a directed graph G takes T (n, m) time and S(n, m) space, then the vertices of G can be output in reverse postorder with respect to a DFS forest of G taking O(T (n, m)) time and O(S(n, m) + m + n) space. As a result, we can also solve SC and TS in O(m + n) time using O(n + m) bits of space. Proof sketch: We use the DFS algorithm of Theorem 3 to first mark all the tree edges in the array A. Now we start with the rightmost leaf vertex of the DFS tree and use rank/select operations [8] on A and P (as defined in the proof of Theorem 3) carefully to traverse the tree in reverse direction (along with standard DFS backtracking etc) to generate reverse postorder sequence. Now using this as the back bone of the classical algorithms, we obtain O(m + n) bit and O(n + m) time algorithms for SC and TS. Theorem 4 improves the result of [7] for sparse (when m = O(n)) graphs. Now if we use the DFS algorithm of Chakraborty et al. [3] and modify it suitably to perform the traversal of the DFS tree in reverse, we obtain the following result. Theorem 5. If DFS of a directed graph G takes T (n, m) time and S(n, m) space, then the vertices of G can be output in reverse postorder with respect to a DFS forest of G taking O(T (n, m)) time and O(S(n, m) + n lg(m/n)) space. As a result, we can also solve SC and TS using O(m + n) time and O(n lg(m/n)) bits. References [1] N. Banerjee, S. Chakraborty, and V. Raman. Improved space efficient algorithms for BFS, DFS and applications. In 22nd COCOON, volume 9797, pages 119–130. Springer, LNCS, 2016. [2] N. Banerjee, S. Chakraborty, V. Raman, S. Roy, and S. Saurabh. Time-space tradeoffs for dynamic programming in trees and bounded treewidth graphs. In 21st COCOON, volume 9198, pages 349–360. springer, LNCS, 2015. [3] S. Chakraborty, V. Raman, and S. R. Satti. Biconnectivity, chain decomposition and stnumbering using O(n) bits. In 27th ISAAC, volume 64 of LIPIcs, pages 22:1–22:13. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. [4] S. Chakraborty and S. R. Satti. Space-efficient algorithms for Maximum Cardinality Search, Stack BFS, Queue BFS and applications. In 23rd COCOON 2017, Hong Kong, China, August 3-5, 2017, pages 87–98, 2017. [5] T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein. Introduction to Algorithms. [6] S. Chakraborty, V. Raman, and S. R. Satti. Biconnectivity, st-numbering and other applications of DFS using O(n) bits. J. Comput. Syst. Sci., 90:63–79, 2017. [7] A. Elmasry, T. Hagerup, and F. Kammer. Space-efficient basic graph algorithms. In 32nd STACS, pages 288–301, 2015. [8] R. Grossi and G. Ottaviano. The wavelet trie: maintaining an indexed sequence of strings in compressed space. In 31st PODS, pages 203–214, 2012. [9] F. Kammer, D. Kratsch, and M. Laudahn. Space-efficient biconnected components and recognition of outerplanar graphs. In 41st MFCS, 2016. [10] J. M. Schmidt. A simple test on 2-vertex- and 2-edge-connectivity. Inf. Process. Lett., 113(7):241–244, 2013. [11] R. E. Tarjan. Depth-first search and linear graph algorithms. SICOMP, 1(2):146–160, 1972. [12] R. E. Tarjan. A note on finding the bridges of a graph. Inf. Pro. Lett., 2(6):160–161, 1974.	8
arXiv:1704.01458v1 [math.ST] 5 Apr 2017 β-mixing and moments properties of a non-stationary copula-based Markov process Fabio Gobbi Sabrina Mulinacci University of Bologna - Department of Statistics April 6, 2017 Abstract This paper provides conditions under which a non-stationary copula-based Markov process is β-mixing. We introduce, as a particular case, a convolution-based gaussian Markov process which generalizes the standard random walk allowing the increments to be dependent. JEL classification: C22,C10 Mathematics Subject Classification (2010): 62M10, 62H20 Keywords: Markov process, copula, β-mixing, gaussian process. 1 Introduction In this paper we analyze the temporal dependence properties satisfied by a discrete times nonstationary Markov process. Temporal dependence is relevant since it permits to verify of how well theoretical models explain temporal persistency observed in financial data. Moreover, it is also a useful tool to establish large sample properties of estimators for dynamic models. In particular, in this paper we analyze the β-mixing property and we give sufficient conditions that ensure this property be satisfied. In the copula approach to univariate time series modelling, the finite dimensional distributions are generate by copulas. Darsow et al. (1992) provide necessary and sufficient conditions for a copula-based time series to be a Markov process. Recent literature on this topic has mainly focused on the stationary case. Chen and Fan (2006) introduce a copula-based strictly stationary first order Markov process generated from (G0 (·), C(·, ·, α0 )) where G0 (·) is the invariant distribution of Yt and C(·, ·, α0 ) is the parametric copula for (Yt−1 , Yt ). The authors show that the β-mixing temporal dependence measure is purely determined by the properties of copulas and present sufficient conditions to ensure that the process (Yt )t based on gaussian and EFGM copulas are geometric β-mixing. Beare (2010) shows that all Markov models generated via symmetric copulas with positive and square integrable densities are geometric β-mixing. Many commonly used bivariate copulas without tail dependence such as gaussian, EFGM and Frank copulas satisfy this condition. Chen et al. (2009) show that Clayton, Gumbel and Student’s t copula based Markov models are geometrically ergodic which is a stronger condition than the geometric β-mixing. In this paper we focus on Markov processes where some dependence between each state variable and increment is allowed and modeled through a time-invariant copula. In particular, we introduce a gaussian Markov process, which is non-stationary and generalizes the classical gaussian random walk, and we study related moments properties and provide conditions under wich the process is β-mixing. 1 The paper is organized as follows. Section 2 presents a general result on the β-mixing properties satisfied by non-stationary Markov processes. Section 3 restricts the study to the gaussian case. Section 4 concludes. 2 Copula-based Markov processes and β-mixing properties Throughout the paper Y = (Yt )t∈Z is a discrete time Markov process. Thanks to the seminal paper of Darsow et al. (1992), the markovianity of a stochastic process can be characterized through a specific requirement that the copulas, representing the dependence structure of the finite dimensional distributions induced by the stochastic process (for a detailed discussion on copulas see Nelsen (2006), Joe (1997), Cherubini et al. (2012) and Durante and Sempi (2015)), must satisfy. In particular, in Darsow et al (1992) it is proved that the Chapman-Kolmogorov equations for transition probabilities are equivalent to the requirement that, if Ci,j is the copula associated to the vector (Yi , Yj ), then Cs,t (u, v) = Cs,r ∗ Cr,t (u, v) = Z 1 0 ∂ ∂ Cs,r (u, w) Cr,t (w, v) dw, ∀s < r < t. ∂w ∂w As a consequence, since Y is a discrete times Markov process, if we assume that the set of bivariate copulas Ct,t+1 (representing the dependence structure of the stochastic process at two adjacent times) is given for t ∈ Z and k > 0, then necessarily (we remind that the ∗-operator is associative) Ct,t+k (u, v) = Ct,t+k−1 ∗ Ct+k−1,t+k (u, v) = Ct,t+1 ∗ Ct+1,t+2 ∗ · · · ∗ Ct+k−1,t+k (u, v). (1) Notice that, in the stationary case considered in Beare (2010), Ct,t+1 = C for all t ∈ Z, therefore all bivariate copulas Ct,t+k are functions of the copula C and of the lag k and not of the time t. In this paper we extend the study to the more general non-stationary case. In particular we analyze the temporal dependence problem with a special attention to mixing properties. The notion of β-mixing was introduced by Volkonskii and Rozanov (1959 and 1961) and was attribute there to Kolmogorov. Given a (not necessarily stationary) sequence of random variables Y = (Yt )t∈Z , let Ftl be the σ-field Ftl = σ(Yt , t ≤ t ≤ l) with −∞ ≤ t ≤ l ≤ +∞ and let J I +∞ t β̃(F−∞ , Ft+k )= 1 XX \|P(Ai ∩ Bj ) − P(Ai )P(Bj )\|, {Ai },{Bj } 2 i=1 j=1 sup (2) where the second supremum is taken over all finite partitions {A1 , ...AI } and {B1 , ...BJ } of Ω such t ∞ that Ai ∈ F−∞ for each i and Bj ∈ Ft+k for each j. Define the following dependence coefficient +∞ t βk = sup β̃(F−∞ , Ft+k ). t∈Z We say that the sequence (Yt )t∈Z is β−mixing (or absolutely regular) if βk → 0 as k → +∞. In next Theorem we give conditions on the set of copulas Ct,t+1 , t ∈ Z in order to guarantee that the Markov resulting process is β-mixing. These conditions are based on specific requirements on the maximal correlation coefficients of the copulas Ct,t+1 . We remind that the maximal correlation η of a copula C is given by Z Z 1 1 sup f,g f (x)g(x)C(dx, dy) 0 0 R1 R1 R1 R1 where f, g ∈ L2 ([0, 1]), 0 f (x)dx = 0 g(x)dx = 0 and 0 f 2 (x)dx = 0 g 2 (x)dx = 1 and we refer to Beare (2010) and Rényi (1959) for more details. 2 Theorem 2.1. Let Y = (Yt )t∈Z be a Markov process. Let Ct,t+1 be the copula associated to the vector (Yt , Yt+1 ) for t ∈ Z that we assume to be absolutely continuous, with symmetric and square-integrable density ct,t+1 so that (ct,t+1 )t∈Z is uniformly bounded in L2 ([0, 1]). If the maximal correlation coefficients ηt of Ct,t+1 satisfy η̂ = sup ηt < 1, (3) t∈Z then Y is β-mixing. Proof. The proof follows that of Theorem 3.1 in Beare (2010) who proves a similar result for stationary copula-based Markov processes. First of all, since the stochastic process is Markovian, (2) can be rewritten in terms of the cumulative distribution functions of (Yt , Yt+k ), Yt and Yt+k (Ft,t+k , Ft and Ft+k , respectively) and the total variation norm k · kT V (see Bradley, 2007) and then, applying Sklar’s theorem, we can write 1 k Ft,t+k (x, y) − Ft (x)Ft+k (y) kT V = 2 1 = k Ct,t+k (Ft (x), Ft+k (y)) − Ft (x)Ft+k (y) kT V ≤ 2 1 ≤ k Ct,t+k (u, v) − uv) kT V . 2 +∞ t β̃(F−∞ , Ft,t+k )= From (1) it follows that all bivariate copulas of type Ct,t+k for t ∈ Z and k ≥ 1 are absolutely continuous: let us denote their density as ct,t+k . Then +∞ t β̃(F−∞ , Ft+k )≤ 1 1 k ct,t+k (u, v) − 1 kL1 ≤ k ct,t+k (u, v) − 1 kL2 2 2 and 1 sup k ct,t+k (u, v) − 1 kL2 . 2 t∈Z βk ≤ Since ct,t+1 is a symmetric square-integrable joint density with uniform margins, it admits the following series expansion in terms of a complete orthonormal sequence (φi )i≥1 in L2 [0, 1], +∞ X ct,t+1 (u, v) = 1 + λi,t φi (u)φi (v), i=1 where the eigenvalues (λi,t )i form a square-summable sequence of nonnegative real numbers: notice that, as proved in Lancaster(1958) max λi,t = ηt . (4) i≥1 Applying (1), we get ct,t+k (u, v) = 1 + +∞ X i=1 Then, using (4) and (3), we get k ct,t+k (u, v) − 1 kL2 = +∞ X i=1  k−1 Y  j=0  k−1 Y  j=0  λi,t+j  φi (u)φi (v).  λi,t+j  φi (u)φi (v) L2 1/2   k−1 +∞ Y X  λ2i,t+j  = = i=1 j=0   1/2   1/2 +∞ k−1 +∞ k−1 X Y X Y 2  = λ2i,t  λ2i,t+j  ≤ λ2i,t  ηt+j ≤ i=1 ≤ η̂ k−1 j=1 "+∞ X i=1 λ2i,t #1/2 i=1 j=1 = η̂ k−1 k ct,t+1 (u, v) − 1 kL2 . 3 (5) Therefore βk ≤ 1 k−1 η̂ sup k ct,t+1 (u, v) − 1 kL2 2 t∈Z which, since (ct,t+1 )t∈Z is uniformly bounded in L2 ([0, 1]), tends to zero as k → +∞. 3 A gaussian convolution-based Markov process From now on, we assume that the Markov process Y is obtained through Yt = Yt−1 + ξt , Y0 = 0, (6) where (ξt )t≥1 is a sequence of identically distributed random variables such that ξt is dependent on Yt−1 for each t. The dependence structure is modelled by a time-invariant copula function C. The process defined in (6) is not stationary. However, we can determine the distribution of C Yt for each t thanks to the C-convolution operator (denoted by ∗), introduced in Cherubini et al. (2011) as a tool to recover the distribution of the sum of two dependent random variables. As shown in Cherubini et al. (2011, 2012 and 2015) the C-convolution technique may be used in the construction of dependent increments stochastic processes like (6). More precisely, if Ft−1 is the cumulative distribution function of Yt−1 and Ht that of ξt , we may recover the cumulative distribution function of Yt iterating the C-convolution for all t Z 1 C −1 Ft (yt ) = (Ft−1 ∗ Ht )(yt ) = D1 C(w, Ht (yt − Ft−1 (w)))dw, t ≥ 2 (7) 0 while, the copula associated to (Yt−1 , Yt ) is Z u −1 D1 C(w, Ht (Ft−1 (v) − Ft−1 Ct−1,t (u, v) = (w)))dw, t≥2 (8) 0 ∂ C(u, v). where D1 C(u, v) = ∂u Equations (7) and (8) provide the ingredients to construct discrete times Markov processes according to Darsow et al. (1992). Our model (6) is a sort of a modified version of a random walk process where the independence assumption for the innovations (ξt )t≥1 is no longer required: however, its weakness is that in most cases the distribution function cannot be expressed in closed form and it may be evaluated only numerically. From now on we assume that innovations (ξt )t≥1 are gaussian identically distributed with zero mean and standard deviation σξ and that the copula between Yt−1 and ξt is a (stationary) gaussian copula with constant parameter ρ for all t. This way, the distribution of Yt is gaussian for all t and, more specifically, in section 4.3.1 of Cherubini et al. (2016) it is shown that Yt ∼ N (0, Vt2 ), where Vt2 = V ar(Yt ) = V12 + (t − 1)σξ2 + 2ρσξ (9) t−1 X Vt−i , t ≥ 2, (10) i=1 where V12 = σξ2 since by assumption Y1 = ξ1 . Moreover, the copula between Yt and Yt+1 is gaussian with parameters Vt + ρσξ τt,t+1 = , t≥2 Vt+1 since E[Yt Yt+1 ] = Vt2 + ρVt σξ . 4 The limiting behavior of the standard deviation Vt has also been analyzed in Cherubini et al. (2016) where it is proved that σε − 2ρ , if ρ ∈ (−1, 0) lim Vt = t→+∞ +∞, otherwise. Notice that only in case of negative correlation with the increments, the standard deviation of the levels does not explode: in the following we will restrict the analysis to the case ρ ∈ (−1, 0). 3.1 Moments and autocorrelation function In this subsection we study the behavior of moments and autocorrelation functions of the process (Yt )t≥1 when t → +∞. It is just the case to recall that in the standard random walk model the k-th order autocorrelation function of (Yt )t≥1 tends to 1 as t → +∞, for each lag k. In our more general setting, this is no longer true. The limit of the k-th order autocorrelation function of (Yt )t≥1 is a function of k and ρ as the following proposition shows. Proposition 3.1. Let ρ ∈ (−1, 0). The k-th order autocorrelation function of (Yt )t≥1 tends to (1 − 2ρ2 )k for any k ≥ 1 as t → +∞. Proof. As proved in section 4.3.1 in Cherubini et al. (2016), using the fact that the ∗-product of two gaussian copulas has a parameter given by the product of the parameters of the copulas involved in the ∗-product, we have that the copula between Yt and Yt+k is gaussian with parameter τt,t+k = k−1 Y Vt+s + ρσξ . Vt+s+1 s=0 Therefore, since as t → +∞ and for any s ≥ 1 σ − 2ρξ + ρσξ2 Vt+s + ρσξ → = 1 − 2ρ2 , σ Vt+s+1 − 2ρξ (11) we easily get the result. On the other hand, the innovations (ξt )t≥1 are no longer serially independent as in the random walk case and the k-th order autocorrelation function approaches to a limit which again depends on ρ and k. Proposition 3.2. Let ρ ∈ (−1, 0). The k-th order autocorrelation function of (ξt )t≥1 tends to −ρ2 (1 − 2ρ2 )k−1 for any k ≥ 1 as t → +∞. Proof. We compute first the autocovariance of order k, with k ≥ 1, E[ξt ξt+k ]. We have E[ξt ξt+k ] = E[(Yt − Yt−1 )(Yt+k − Yt+k−1 )] = = E[Yt Yt+k ] − E[Yt Yt+k−1 ] − E[Yt−1 Yt+k ] + E[Yt−1 Yt+k−1 ] = = τt,t+k Vt Vt+k − τt,t+k−1 Vt Vt+k−1 − τt−1,t+k Vt−1 Vt+k + τt−1,t+k−1 Vt−1 Vt+k−1 . σ Since for any fixed k ≥ 1, τt,t+k → (1 − 2ρ2 )k and Vt → − 2ρξ as t → +∞ we get E[ξt ξt+k ] → σξ2 4ρ2 (1 − 2ρ2 )k − (1 − 2ρ2 )k−1 − (1 − 2ρ2 )k+1 + (1 − 2ρ2 )k = = −ρ2 σξ2 (1 − 2ρ2 )k−1 , as t → +∞. Moreover, it is immediate to find the statement of the proposition since as t → +∞ corr(ξt ξt+k ) → −ρ2 (1 − 2ρ2 )k−1 . 5 3.2 β-mixing properties In our gaussian framework ct,t+1 is the density of a gaussian copula for which it is well known that the maximal correlation coefficient is equal to the absolute value of the simple correlation coefficient (see Lancaster, 1957). Therefore, according to the notation of Theorem 2.1, for each t, ηt = \|τt,t+1 \|. The following results, which is an application of Theorem 2.1, holds Corollary 3.1. The Markov process defined by (6) with ξt ∼ N (0, σξ ) and ρ ∈ (−1, 0) is β-mixing. Proof. Firstly notice that \|τt,t+1 \| < 1, ∀t. 2 2 In fact this is equivalent to (Vt + ρσξ ) < Vt+1 which is always verified since ρ2 < 1 by assumption. 2 Thanks to (11), since \|1 − 2ρ \| < 1, we have that \|τt,t+1 \| = ηt is bounded by a constant smaller than 1, that is (3) is satisfied. Furthermore, it is not hard to prove that for any t k ct,t+1 (u, v) − 1 kL2 = 2 τt,t+1 η̂ 2 ≤ . 2 1 − τt,t+1 1 − η̂ 2 Thus Theorem 2.1 applies. 4 Concluding remarks In this paper we provide conditions under which a non-stationary copula-based Markov process is βmixing. Our results represent a generalization of those in Beare (2010), where the author considers the stationary case. Our analysis is focused on the particular case of a gaussian Markov process with dependent increments that represents a generalization of the standard gaussian random walk. In this particular non-stationary setting it is proved that the k-th order autocorrelation function of the process does not converge to 1, as in the random walk case, but to a quantity that depends on the lag and the correlation between the state variable and the innovation, which is assumed to be time-invariant. Additionally, it is proved that the process satisfies the conditions required to be β-mixing. References [1] Beare B. (2010): ”Copulas and temporal dependence”, Econometrica, 78(1), 395-410. [2] Bradley R.C. (2007): Introduction to strong mixing conditions, vols. 1-3. Kendrick Press. herber City. [3] Chen X., Fan Y. (2006): ”Estimation of Copula-Based Semiparametric Time Series Models”, Journal of Econometrics, 130, 307335. [4] Chen X., Wu W.B., Yi Y. (2009): ”Efficient estimation of copula-based semiparametric Markov models”, The Annals of Statistics, 37(6B) 42144253. [5] Cherubini U., Gobbi F., Mulinacci S., (2016) ”Convolution Copula Econometrics”, SpringerBriefs in Statistics. [6] Cherubini U., Gobbi F., Mulinacci S., Romagnoli S. (2012): Dynamic Copula Methods in Finance, John Wiley & Sons. [7] Cherubini, U., Mulinacci S., Romagnoli S. (2011) ”A Copula-based Model of Speculative Price Dynamics in Discrete Time”, Journal of Multivariate Analysis, 102, 1047-1063. [8] Darsow W.F. - Nguyen B. - Olsen E.T.(1992): ”Copulas and Markov Processes”, Illinois Journal of Mathematics, 36, 600-642. 6 [9] Durante F., Sempi C. (2015) Principles of copula theory, Boca Raton: Chapman and Hall/CRC [10] Joe H. (1997): Multivariate Models and Dependence Concepts, Chapman & Hall, London [11] Lancaster,H.O. (1957) ”Some properties of the bivariate normal distribution considered in the form of a contingency table”, Biometrika, 44, 289-292. [12] Lancaster H.O. (1958): ”The structure of bivariate distributions”, Annals of Mathematical Statistics, 29, 719-736. [13] Nelsen R.(2006): An Introduction to Copulas, Springer [14] Renyi A. (1959): ”On measures of dependence”, Acta Mathematica Academiae Scientiarum Hungaricae, 10, 441-451. [15] Volkonskii V.A., Rozanov Yu.A. (1959): ”Some limit theorems for random functions I”, Theor. Probab. Appl., 4, 178-197. [16] Volkonskii V.A., Rozanov Yu.A. (1961): ”Some limit theorems for random functions II”, Theor. Probab. Appl., 6, 186-198. 7	10
Pragmatic-Pedagogic Value Alignment arXiv:1707.06354v2 [cs.AI] 5 Feb 2018 Jaime F. Fisac, Monica A. Gates, Jessica B. Hamrick, Chang Liu, Dylan Hadfield-Menell, Malayandi Palaniappan, Dhruv Malik, S. Shankar Sastry, Thomas L. Griffiths, and Anca D. Dragan Abstract As intelligent systems gain autonomy and capability, it becomes vital to ensure that their objectives match those of their human users; this is known as the value-alignment problem. In robotics, value alignment is key to the design of collaborative robots that can integrate into human workflows, successfully inferring and adapting to their users’ objectives as they go. We argue that a meaningful solution to value alignment must combine multi-agent decision theory with rich mathematical models of human cognition, enabling robots to tap into people’s natural collaborative capabilities. We present a solution to the cooperative inverse reinforcement learning (CIRL) dynamic game based on well-established cognitive models of decision making and theory of mind. The solution captures a key reciprocity relation: the human will not plan her actions in isolation, but rather reason pedagogically about how the robot might learn from them; the robot, in turn, can anticipate this and interpret the human’s actions pragmatically. To our knowledge, this work constitutes the first formal analysis of value alignment grounded in empirically validated cognitive models. Key words: Value Alignment, Human-Robot Interaction, Dynamic Game Theory 1 Introduction The accelerating progress in artificial intelligence (AI) and robotics is bound to have a substantial impact in society, simultaneously unlocking new potential in augmenting and transcending human capabilities while also posing significant challenges to safe and effective human-robot interaction. In the short term, integrating robotic systems into human-dominated environments will require them to assess the intentions All authors are with the University of California, Berkeley. e-mail: {jfisac,mgates,jhamrick,changliu,dhm,malayandi,dhruvmalik, shankar_sastry,tom_griffiths,anca}@berkeley.edu 1 2 J. F. Fisac, M. A. Gates, J. B. Hamrick, C. Liu, D. Hadfield-Menell, et al. and preferences of their users in order to assist them effectively, while avoiding failures due to poor coordination. In the long term, ensuring that advanced and highly autonomous AI systems will be beneficial to individuals and society will hinge on their ability to correctly assimilate human values and objectives [1]. We envision the short- and long-term challenges as being inherently coupled, and predict that improving the ability of robots to understand and coordinate with their human users will inform solutions to the general AI value-alignment problem. Successful value alignment requires moving from typical single-agent AI formulations to robots that account for a second agent—the human—who determines what the objective is. In other words, value alignment is fundamentally a multi-agent problem. Cooperative Inverse Reinforcement Learning (CIRL) formulates value alignment as a two-player game in which a human and a robot share a common reward function, but only the human has knowledge of this reward [2]. In practice, solving a CIRL game requires more than multi-agent decision theory: we are not dealing with any multi-agent system, but with a human-robot system. This poses a unique challenge in that humans do not behave like idealized rational agents [3]. However, humans do excel at social interaction and are extremely perceptive of the mental states of others [4, 5]. They will naturally project mental states such as beliefs and intentions onto their robotic collaborators, becoming invaluable allies in our robots’ quest for value alignment. In the coming decades, tackling the value-alignment problem will be crucial to building collaborative robots that know what their human users want. In this paper, we show that value alignment is possible not just in theory, but also in practice. We introduce a solution for CIRL based on a model of the human agent that is grounded in cognitive science findings regarding human decision making [6] and pedagogical reasoning [7]. Our solution leverages two closely related insights to facilitate value alignment. First, to the extent that improving their collaborator’s understanding of their goals may be conducive to success, people will tend to behave pedagogically, deliberately choosing their actions to be informative about these goals. Second, the robot should anticipate this pedagogical reasoning in interpreting the actions of its human users, akin to how a pragmatic listener interprets a speaker’s utterance in natural language. Jointly, pedagogical actions and pragmatic interpretations enable stronger and faster inferences among people [7]. Our result suggests that it is possible for robots to partake in this naturally-emerging equilibrium, ultimately becoming more perceptive and competent collaborators. 2 Solving Value Alignment using Cognitive Models 2.1 Cooperative Inverse Reinforcement Learning (CIRL) Cooperative Inverse Reinforcement Learning (CIRL) [2] formalizes value alignment as a two-player game, which we briefly present here. Consider two agents, a human Pragmatic-Pedagogic Value Alignment 3 H and a robot R, engaged in a dynamic collaborative task involving a (possibly infinite) sequence of steps. The goal of both agents is to achieve the best possible outcome according to some objective θ ∈ Θ . However, this objective is only known to H. In order to contribute to the objective, R will need to make inferences about θ from the actions of H (an Inverse Reinforcement Learning (IRL) problem), and H will have an incentive to behave informatively so that R becomes more helpful, hence the term cooperative IRL. Formally, a CIRL game is a dynamic (Markov) game of two players (H and R), described by a tuple hS, {AH , AR }, T, {Θ , r}, P0 , γi, where S is the set of possible states of the world; AH , AR are the sets of actions available to H and R respectively; T : S × S × AH × AR → [0, 1] a discrete transition measure1 over the next state, conditioned on the previous state and the actions of H and R: T (s0 \|s, aH , aR ); Θ is the set of possible objectives; r : S × AH × AR × Θ → R is a cumulative reward function assigning a real value to every tuple of state and actions for a given objective: r(s, aH , aR ; θ ); P0 : S × Θ → [0, 1] is a probability measure on the initial state and the objective; γ ∈ [0, 1] is a geometric time discount factor making future rewards gradually less valuable. 2.2 Pragmatic Robots for Pedagogic Humans Asymmetric information structures in games (even static ones) generally induce an infinite hierarchy of beliefs: our robot will need to maintain a Bayesian belief over the human’s objectives to decide on its actions. To reason about the robot’s decisions, the human would in principle need to maintain a belief on the robot’s belief, which will in turn inform her decisions, thereby requiring the robot to maintain a belief on the human’s belief about its own belief, and so on [8]. In [2], it was shown that an optimal pair of strategies can be found for any CIRL game by solving a partially observed Markov decision process (POMDP). This avoids this bottomless recursion as long as both agents are rational and can coordinate perfectly before the start of the game. Unfortunately, when dealing with human agents, rationality and prior coordination are nontrivial assumptions. Finding an equivalent tractability result for more realistic human models is therefore crucial in using the CIRL formulation to solve real-world value-alignment problems. We discover the key insight in cognitive studies of human pedagogical reasoning [7], in which a teacher chooses actions or utterances to influence the beliefs of a learner who is aware of the teacher’s intention. The teacher can then exploit the fact that the learner can interpret utterances pragmatically. Infinite recursion is averted by finding a fixed-point relation between the teacher’s best utterance and the learner’s best interpretation, exploiting a common modeling assumption in Bayesian theory of mind: the learner models the teacher as a noisily rational decision maker [9], who will be likelier to choose utterances 1 Note that the theoretical formulation is easily extended to arbitrary measurable sets; we limit our analysis to finite state and objective sets for computational tractability and clarity of exposition. 4 J. F. Fisac, M. A. Gates, J. B. Hamrick, C. Liu, D. Hadfield-Menell, et al. causing the learner to place a high posterior belief on the correct hypothesis, given the learner’s current belief. While in reality, the teacher cannot exactly compute the learner’s belief, the model supposes that she estimates it (from the learner’s previous responses to her utterances), then introduces noise in her decisions to capture estimation inaccuracies. This framework can predict complex behaviors observed in human teaching-learning interactions, in which pedagogical utterances and pragmatic interpretations permit efficient communication [7]. We adopt an analogous modeling framework to that in [7] for value alignment, with a critical difference: the ultimate objective of the human is not to explicitly improve the robot’s understanding of the true objective, but to optimize the team’s expected performance towards this objective. Pedagogic behavior thus emerges implicitly to the extent that a well-informed robot becomes a better collaborator. 2.3 Pragmatic-Pedagogic Equilibrium Solution to CIRL The robot does not have access to the true objective θ , but rather estimates a belief bR over θ . We assume that this belief on θ can be expressed parametrically (this is always true if Θ is a finite set), and define 4Θ to be the corresponding (finitedimensional) parameter space, denoting R’s belief by bR ∈ 4Θ . While in reality the human cannot directly observe bR , we assume, as in [7], that she can compute it or infer it from the robot’s behavior (and model estimation inaccuracies as noise in her policy). We can then let Q : S × 4Θ × AH × AR ×Θ → R represent the state-action value function of the CIRL game for a given objective θ , which we are seeking to compute: if θ ∈ Θ is the true objective known to H, then Q(s, bR , aH , aR ; θ ) represents the best performance the team can expect to achieve if H chooses aH and R chooses aR from state s, with R’s current belief being bR . In order to solve for Q, we seek to establish an appropriate dynamic programming relation for the game, given a well-defined information structure and a model of the human’s decision making. Since it is typically possible for people to predict a robot’s next action if they see its beginning [10], we assume that H can observe aR at each turn before committing to aH . A well-established model of human decision making in psychology and econometrics is the Luce choice rule, which models people’s decisions probabilistically, making high-utility choices more likely than those with lower utility [9]. In particular, we employ a common case of the Luce choice rule, the Boltzmann (or soft-max) noisy rationality model [6], in which the probability of a choice decays exponentially as its utility decreases in comparison to competing options. The relevant utility metric in our case is the sought Q (which captures H’s best expected outcome for each of her available actions aH ). Therefore the probability that H will choose action aH has the form πH (aH \|s, bR , aR ; θ ) ∝ exp β Q(s, bR , aH , aR ; θ ) , (1) Pragmatic-Pedagogic Value Alignment 5 where β > 0 is termed the rationality coefficient of H and quantifies the concentration of H’s choices around the optimum; as β → ∞, H becomes a perfect rational agent, while, as β → 0, H becomes indifferent to Q. The above expression can be interpreted by R as the likelihood of action aH given a particular θ . The evolution of R’s belief bR is then given (deterministically) by the Bayesian update b0R (θ \|s, bR , aR , aH ) ∝ πH (aH \|s, bR , aR ; θ )bR (θ ) , (2) Jointly, (1) and (2) define a fixed-point equation analogous to the one in [7], which states how R should pragmatically update bR based on a noisily rational pedagogic aH . This amounts to a deterministic transition function for R’s belief, b0R = fb (s, bR , aH , aR ). Crucially, however, the fixed-point relation derived here involves Q itself, which we have yet to compute. Unlike H, R is modeled as a rational agent; however, not knowing the true θ , the best R can do is to maximize2 the expectation of Q based on its current belief3 bR : πR∗ (s, bR ) := arg max aR ∑ Q(s, bR , aH , aR ; θ ) · πH (aH \|s, bR , aR ; θ )bR (θ ) . (3) aH ,θ Combining (2) with the state transition measure T (s0 \|s, aH , aR ), we can define the Bellman equation for H under the noisily rational policy πH for any given θ ∈ Θ : h i Q(s, bR , aH , aR ; θ ) = r(s, aH , aR ; θ ) + Es0 ,a0H γ · Q0 s0 , b0R , a0H , πR∗ (s0 , b0R ); θ , (4) where s0 ∼ T (s0 \|s, aH , aR ); b0R = fb (s, bR , aH , aR ); a0H ∼ πH (aH \|s0 , b0R , πR∗ (s0 , b0R ); θ ). Note that H’s next action a0H implicitly depends on R’s action at the next turn. Substituting (1-3) into (4), we obtain the sought dynamic programming relation for the CIRL problem under a noisily rational-pedagogic human and a pragmatic robot. The human is pedagogic because she takes actions according to (1), which takes into account how her actions will influence the robot’s belief about the objective. The robot is pragmatic because it assumes the human is actively aware of how her actions convey the objective, and interprets them accordingly. The resulting problem is similar to a POMDP (in this case formulated in beliefstate MDP form), with the important difference that the belief transition depends on the value function itself. In spite of this complication, the problem can be solved in backward time through dynamic programming: each Bellman update will be based on a pragmatic-pedagogic fixed point that encodes an equilibrium between the Q function (and therefore H’s policy for choosing her action) and the belief transition (that is, R’s rule for interpreting H’s actions). Evidence in [7] suggests that people are proficient at finding such equilibria, even though uniqueness is not guaranteed in general; study of disambiguation is an open research direction. 2 We assume for simplicity that the optimum is unique or a well-defined disambiguation rule exists. Note that this does not imply certainty equivalence, nor do we assume separation of estimation and control: R is fully reasoning about how its actions and those of H may affect its future beliefs. 3 6 J. F. Fisac, M. A. Gates, J. B. Hamrick, C. Liu, D. Hadfield-Menell, et al. 3 A Proof-of-Concept We introduce the benchmark domain ChefWorld, a household collaboration setting in which a human H seeks to prepare a meal with the help of an intelligent robotic manipulator R. There are multiple possible meals that H may want to prepare using the available ingredients, and R does not know beforehand which one she has chosen (we assume H cannot or will not tell R explicitly). The team obtains a reward only if H’s intended recipe is successfully cooked. If H is aware of R’s uncertainty, she should take actions that give R actionable information, particularly the information that she expects will allow R to be as helpful as possible as the task progresses. Our problem has 3 ingredients, each with 2 or 3 states: spinach (absent, chopped), tomatoes (absent, chopped, puréed), and bread (absent, sliced, toasted). Recipes correspond to (joint) target states for the food. Soup requires the tomatoes to be chopped then puréed, the bread to be sliced then toasted, and no spinach. Salad requires the spinach and tomatoes to be chopped, and the bread to be sliced then toasted. H and R can slice or chop any of the foods, while only R can purée tomatoes or toast bread. A simple scenario with the above two recipes is solved using discretized beliefstate value iteration and presented as an illustrative example in Fig 1. R has a wrong initial belief about H’s intended recipe. Under standard IRL, H fails to communicate her recipe. But if R is pragmatic and H is pedagogic, H is able to change R’s belief and they successfully collaborate to make the meal. In addition, we computed the solution to games with 4 recipes through a modification of POMDP value iteration (Table 1). In the pragmatic-pedagogic CIRL equilibrium with β = 5, H and R successfully cook the correct recipe 97% of the time, whereas under the standard IRL framework (with H acting as an expert disregarding R’s inferences) they only succeed 46% of the time—less than half as often. Fig. 1 Simple collaborative scenario with 2 possible objectives. The human H wants soup but the robot R initially believes her goal is salad. Even under a full POMDP formulation, if R reasons “literally” about H’s actions using standard IRL (assuming H behaves as if R knew the true objective), it fails to infer the correct objective. Conversely, under the pragmatic-pedagogic CIRL equilibrium, R views H as incentivized to choose pedagogic actions that will fix R’s belief when needed. Under the pragmatic interpretation, H’s wait action in turn 2 (instead of adding spinach, which would be preferred by a pedagogic H wanting salad) indicates H wants soup. While H’s actions are the same under both solutions, only the pragmatic R achieves value alignment and completes the recipe. Pragmatic-Pedagogic Value Alignment IRL CIRL Boltzmann (β = 1) Boltzmann (β = 2.5) Boltzmann (β = 5) 0.2351 0.3783 0.4555 0.2916 0.7026 0.9727 7 Rational 0.7083 1.0000 Table 1 A comparison of the expected value (or equivalently here, the probability of success) achieved by CIRL and IRL on the ChefWorld domain with four recipes when the robot begins with a uniform belief over the set of recipes. We ran each algorithm across different models of the human’s behavior, namely a rational model and a Boltzmann-rational model with various values of β (a higher β corresponds to a more rational human). When the human is highly irrational (β = 1), both CIRL and IRL unsurprisingly perform rather poorly. However, as the human becomes less noisy (β = 2.5, β = 5), CIRL outperforms IRL by a significant margin; in fact, the pragmaticpedagogic CIRL strategy with a Boltzmann-rational human performs comparably (β = 2.5) or even substantially outperforms (β = 5) the IRL result when the human is perfectly rational. 4 Discussion We have presented here an analysis of the AI value alignment problem that incorporates a well-established model of human decision making and theory of mind into the game-theoretic framework of cooperative inverse reinforcement learning (CIRL). Using this analysis, we derive a Bellman backup that allows solving the dynamic game through dynamic programming. At every instant, the backup rule is based on a pragmatic-pedagogic equilibrium between the robot and the human: the robot is uncertain about the objective and therefore incentivized to learn it from the human, whereas the human has an incentive to help the robot infer the objective so that it can become more helpful. We note that this type of pragmatic-pedagogic equilibrium, recently studied in the cognitive science literature for human teaching and learning [7], may not be unique in general: there may exist two actions for H and two corresponding interpretations for R leading to different fixed points. For example, H could press a blue or a red button which R could then interpret as asking it to pick up a blue or a red object. Although we might feel that blue-blue/red-red is a more intuitive pairing, blue-red/red-blue is valid as well: that is, if H thinks that R will interpret pressing the blue button as asking for the red object then she will certainly be incentivized to press blue when she wants red; and in this case R’s policy should consistently be to pick up the red object upon H’s press of the blue button. When multiple conventions are possible, human beings tend to naturally disambiguate between them, converging on salient equilibria or “focal points” [11]. Accounting for this phenomenon is likely to be instrumental for developing competent human-centered robots. On the other hand, it is important to point out that, although they are computationally simpler than more general multi-agent planning problems, POMDPs are still PSPACE-complete [12], so reducing pragmatic-pedagogic equilibrium computation to solving a modified POMDP falls short of rendering the problem tractable in general. However, finding a POMDP-like Bellman backup does open the door to efficient CIRL solution methods that leverage and benefit from the extensive research on practical algorithms for approximate planning in large POMDPs [13]. 8 REFERENCES We find the results in this work promising for two reasons. First, they provide insight into how CIRL games can be not only theoretically formulated but also practically solved. Second, they demonstrate, for the first time, formal solutions to value alignment that depart from the ideal assumption of a rational human agent and instead benefit from modern studies of human cognition. We predict that developing efficient solution approaches and incorporating more realistic human models will constitute important and fruitful research directions for value alignment. Acknowledgements This work is supported by ONR under the Embedded Humans MURI (N00014-13-1-0341), by AFOSR under Implicit Communication (16RT0676), and by the Center for Human-Compatible AI. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] D. Amodei, J. Steinhardt, D. Man, and P. Christiano. “Concrete Problems in AI Safety”. arXiv preprint (2017). D. Hadfield-Menell, A. Dragan, P. Abbeel, and S. Russell. “Cooperative Inverse Reinforcement Learning”. NIPS (2016). A. Tversky and D. Kahneman. “Judgment under Uncertainty: Heuristics and Biases”. Science 185.4157 (1974). F. Heider and M. Simmel. “An Experimental Study of Apparent Behavior”. The American Journal of Psychology 57.2 (1944). A. N. Meltzoff. “Understanding the intentions of others: Re-enactment of intended acts by 18-month-old children.” Dev. Psych. 31.5 (1995). C. L. Baker and J. B. Tenenbaum. “Modeling Human Plan Recognition Using Bayesian Theory of Mind”. Plan, Activity, and Intent Recognition. 2014. P. Shafto, N. D. Goodman, and T. L. Griffiths. “A rational account of pedagogical reasoning: Teaching by, and learning from, examples”. Cog. Psych. 71 (2014). S. Zamir. “Bayesian games: Games with incomplete information”. Computational Complexity: Theory, Techniques, and Applications (2012). R. D. Luce. Individual Choice Behavior: a Theoretical Analysis. John Wiley and Sons, 1959. A. D. Dragan and S. Srinivasa. “Integrating human observer inferences into robot motion planning”. Autonomous Robots (2014). T. C. Schelling. The strategy of conflict. Harvard University Press, 1960. M. Mundhenk, J. Goldsmith, C. 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arXiv:1606.09481v1 [cs.DS] 30 Jun 2016 Generating massive complex networks with hyperbolic geometry faster in practice Moritz von Looz Mustafa Safa Özdayi Karlsruhe Institute of Technology (KIT), Germany Email: [email protected] Istanbul Technical University, Turkey Email: [email protected] Sören Laue Henning Meyerhenke Friedrich Schiller University Jena, Germany Email: [email protected] Karlsruhe Institute of Technology (KIT), Germany Email: [email protected] Abstract—Generative network models play an important role in algorithm development, scaling studies, network analysis, and realistic system benchmarks for graph data sets. The commonly used graph-based benchmark model R-MAT has some drawbacks concerning realism and the scaling behavior of network properties. A complex network model gaining considerable popularity builds random hyperbolic graphs, generated by distributing points within a disk in the hyperbolic plane and then adding edges between points whose hyperbolic distance is below a threshold. We present in this paper a fast generation algorithm for such graphs. Our experiments show that our new generator achieves speedup factors of 3-60 over the best previous implementation. One billion edges can now be generated in under one minute on a shared-memory workstation. Furthermore, we present a dynamic extension to model gradual network change, while preserving at each step the point position probabilities. 1. Introduction Relational data of complex relationships often take the form of complex networks, graphs with heterogeneous and often hierarchical structure, low diameter, high clustering, and a heavy-tailed degree distribution. Examples include social networks, the graph of hyperlinks between websites, protein interaction networks, and infrastructure routing networks on the autonomous system level [1]. Frequently found properties in generative models for complex network are non-negligible clustering (ratio of triangles to triads), a community structure, and a heavytailed degree distribution [2], such as a power-law. Benchmarks developed to evaluate a system with respect to floating point operations do not represent the requirements of graph algorithms, especially with heterogeneous datasets such as complex networks. The Graph500 benchmark [3] addresses this gap; it is the most widely-used graph benchmark in high-performance computing. It uses the Recursive Matrix (R-MAT) [4] model to generate synthetic networks as benchmark instances. Graphs from this model are efficiently computable, but suffer from drawbacks in terms of realism. For example, even with fixed parameters, the clustering coefficient shrinks with graph size, while the number of connected components increases, which is problematic for scaling studies [5]. An interesting model without this problem are random hyperbolic graphs (RHG), a family of geometric graphs in the hyperbolic plane. Krioukov et al. [6] introduced this graph model and showed how the structure of complex networks naturally develops from the properties of hyperbolic geometry. To generate a RHG, one randomly samples node positions in a hyperbolic disk, then connects two nodes with an edge with a probability depending on their hyperbolic distance. In a special case of this model, an edge between two nodes is added exactly if their distance is below a threshold. This subset of RHG, sometimes called threshold random hyperbolic graphs, is well-analyzed theoretically [7], [8], [9] and could be considered as unitdisk graphs in hyperbolic space. The resulting graphs show a power-law degree distribution with adjustable exponent, provably high clustering [8], and small diameter [9]. Motivation, outline, and contribution. A fast generator implementation that scales to large graph sizes and provides sufficient realism is necessary to create meaningful graph benchmark instances in acceptable time. While our previous work [10] was able to improve the quadratic time complexity of the pairwise probing approach [11] for threshold RHGs, it still has superlinear time complexity. We therefore provide a faster generation algorithm in this paper for threshold random hyperbolic graphs (Section 3), using a new spatial data structure. The key idea is to divide the relevant part of the hyperbolic plane into ring-shaped slabs and use these to bound the coordinates of possible neighbors in each slab. As our experiments (Section 4) show, a network with 10 million vertices and 1G edges can be generated with our shared-memory parallel implementation in under one minute, yielding a speedup factor of up to 60 over the best previous implementation [10]. For a graph with n nodes and m edges, the measurements suggest an O(n log n+m) time on the hyperbolic plane, the distance between them is given by the hyperbolic law of cosines: complexity, but we do not have a proof for this. While an algorithm with optimal expected linear time complexity has been suggested in a theoretical paper [12], our present work provides the fastest implementation to date. The generator code is publicly available in our network analysis toolkit NetworKit [13]. cosh dist(p, q) = cosh rp cosh rq −sinh rp sinh rq cos \|φp − φq \| (3) As mentioned briefly in Section 1, an important special case is T = 0, where an edge is added to a node pair exactly if the hyperbolic distance between the points is below a threshold. This graph family is sometimes called threshold random hyperbolic graphs, hyperbolic unit-disk graphs or (slightly confusingly) just random hyperbolic graphs. While we consider hyperbolic unit-disk graphs to be more precise, we stick with threshold random hyperbolic graphs to avoid name proliferation. Many theoretical results are for this special case [9]. 2. Related Work Generative Models. Due to the growing interest in complex networks, numerous generators for them exist. For a comprehensive overview, which would be outside the scope of this paper, we refer the interested reader to Goldenberg’s survey [14]. None of the models is suitable for all use cases. As mentioned above, the Recursive Matrix (R-MAT) [4] model has received particular attention in the HPC community due to its use in the Graph500 benchmark [3]. RHG Generation Algorithms. Previous generators for random hyperbolic graphs exist, both for the general and special case. Aldecoa et al. [11] present a generator for the general case with quadratic time complexity, calculating distances and sampling edges for all Θ(n2 ) node pairs. Von Looz et al. [10] use polar quadtrees to generate threshold RHGs with a time complexity of O((n3/2 + m) log n) (with high probability). Recently, von Looz and Meyerhenke [17] have extended this approach to generate general RHGs with the same time complexity. Bringmann et al. [12] propose Geometric Inhomogeneous Random Graphs as a generalization of RHGs and describe a generation algorithm with expected linear time complexity. To our knowledge no implementation of this algorithm is available. Hyperbolic Geometry. Hyperbolic space is one of the three isotropic spaces, the other two being the (more common) Euclidean space and spherical space. In contrast to the flat Euclidean geometry and the positively curved spherical geometry, hyperbolic geometry has negative curvature [15]. Among other interesting properties, hyperbolic geometry shows an exponential expansion of space: While the area of a Euclidean circle grows quadratically with the circle radius, the area of a circle on the hyperbolic plane grows exponentially with its radius. In balanced trees, the number of nodes at a certain distance from the root also grows exponentially with said distance, leading to the suggestion that hierarchical complex networks with tree-like structures might be easily embeddable in hyperbolic space [6]. Indeed, Boguñá et al. [16] demonstrate the connection between hyperbolic geometry and complex networks by embedding the autonomous system internet graph in the hyperbolic plane and enabling locally greedy routing. As a generative model, Krioukov et al. [6] introduced random hyperbolic graphs in 2010. To generate a graph, points are first distributed randomly within in a disk DR of radius R in the hyperbolic plane. The probability density functions for the point distributions are given in polar coordinates, the angular coordinate φ is distributed uniformly over [0, 2π], the radial coordinate r is given by [6, Eq. (17)]: f (r) = α sinh(αr) cosh(αR) − 1 3. Algorithm Our main idea is to partition the hyperbolic plane into concentric ring-shaped slabs (Section 3.1) and use them to limit the number of necessary distance calculations during edge creation (Algorithm 1). Point positions are sampled, sorted by their angular coordinates and stored in the appropriate slab as determined by their radial coordinates. To gather the neighborhood of a point v , we then iterate over all slabs and examine possible neighbors within them. Since each slab limits the radial coordinates of points it contains, we can use Eq. (3) to also bound the angular coordinates of possible neighbors in each slab, thus reducing the number of comparisons and running time. (1) The parameter α governs node dispersion, which determines the power-law exponent of the resulting degree distribution. After sampling point positions, edges are then added to each node pair (u, v) with a probability given in [6, Eq. (41)], depending on their hyperbolic distance and parametrized by a temperature T ≥ 0: 1 f (x) = (1/T )·(x−R)/2 (2) e +1 3.1. Data Structure Let C = {c0 , c1 , ...cmax } be a set of log n ordered radial boundaries, with c0 = 0 and cmax = R. We then define a slab Si as the area enclosed by ci and ci+1 . A point p = (φp , rp ) is contained in slab Si exactly if ci ≤ rp < ci+1 . Since slabs are ring-shaped, they partition the hyperbolic disk DR : For α ≥ 21 , the resulting degree distribution follows a power law with exponent γ := 2α + 1 [6, Eq. (29)]. Given two points in polar coordinates p = (φp , rp ), q = (φq , rq ) DR = 2 log [n i=0 Si . φm ax Algorithm 1: Graph Generation 1 2 φ min 3 4 ci ci+1 5 6 7 8 R 9 10 11 Figure 1: Graph in hyperbolic geometry with unit-disk neighborhood. Neighbors of the bold blue vertex are in the hyperbolic circle, marked in blue. 12 13 14 15 The choice of radial boundaries is an important tuning parameter. After experimenting with different divisions, we settled on a geometric sequence with ratio p = 0.9. The relationship between successive boundary values is then: ci+1 − ci = p · (ci − ci−1 ). From c0 = 0 and cmax = R, we derive the value of c1 : log n−1 X 16 17 18 19 20 log n 1−p c1 p = R ⇔ c1 1−p (1 − p)R = R ⇔ c1 = 1 − plog n k=0 (4) The remaining values follow geometrically. Figure 1 shows an example of a graph in the hyperbolic plane, together with slab Si . The neighbors of the bold blue vertex v are those within a hyperbolic circle of radius R (0.2R in this visualization), marked by the blue egg-shaped area. When considering nodes in Si as possible neighbors of v , the algorithm only needs to examine nodes whose angular coordinate is between φmin and φmax . k 21 22 23 24 Input: number of vertices n, average degree k , power-law exponent γ Output: G = (V, E) α = (γ − 1)/2; R = getTargetRadius(n, k, α); V = n vertices; C = {c0 , c1 , ...cmax } set of log n ordered radial coordinates, with c0 = 0 and cmax = R; B = {b0 , b1 , ...bmax } set of log n empty sets; for vertex v ∈ V do in parallel draw φ[v] from U[0, 2π); draw r[v] with density f (r) = α sinh(αr)/(cosh(αR) − 1); insert (φ[v], r[v]) in suitable bi so that ci ≤ r[v] ≤ ci+1 ; end for b ∈ B do in parallel sort points in b by their angular coordinates; end for vertex v ∈ V do in parallel for band bi ∈ B , where ci+1 > r[v] do minφ , maxφ = getMinMaxPhi(φ[v], r[v]), ci , ci+1 , R); for vertex w ∈ bi , where minφ ≤ φ[w] ≤ maxφ do if distH (v, w) ≤ R then add (v, w) to E ; end end end end return G; Algorithm. Algorithm 1 shows the generation of G = (V, E) with average degree k and power-law exponent γ . First, the radius R of the hyperbolic disk is calculated according to desired graph size and density (Line 2). The value of ζ can be fixed while retaining all degrees of freedom in the model [7], we thus assume ζ = 1. We then use binary search with fixed n, α and desired k to find an R that gives us a close approximation of the desired average degree k . Note that the above equation is only an approximation and might give wrong results for extreme values. Our implementation could easily be adapted to skip this step and accept the commonly used [18] parameter C , with R = 2 ln n+C or even accept R directly. For increased usability, we accept the average degree k as a parameter in the default version. getTargetRadius. This function is unchanged from our previous work [10]. For given values of n, α and R, an approximation of the expected average degree k is given by [6, Eq. (22)] and the notation ξ = (α/ζ)/(α/ζ − 1/2): 2 2 k = ξ 2 n · e−ζR/2 + ξ 2 n (5) π π 2 R π ζ ζ · e−αR α − (π − 1) + (π − 2) − 1 2 4 α α (6) Vertex Positions and Bands. After settling the disk boundary, the radial boundaries ci are calculated (Line 4) as defined above, the disk DR is thus partitioned into log n slabs. For each slab Si , a set bi stores the vertices located in the area of Si . These sets bi are initially empty (Line 5). The vertex positions are then sampled randomly in polar coordinates (Lines 7 and 8) and stored in the corresponding set, i.e, vertex v is put into set bi iff ci ≤ r[v] < ci+1 (Line 9). Within each set, vertices are sorted with respect to their angular coordinates (Lines 11 to 13). 3.2. Generation Algorithm 3 outside the hyperbolic disk DR . In this case, the movement is inverted and the node “bounces” off the boundary. The different probability densities in the center of the disk and the outer regions can be translated into movement speed: A node is less likely to be in the center; thus it needs to spend less time there while traversing it, resulting in a higher speed. We implement this movement in two phases: In the initialization, step values τφ and τr are assigned to each node according to the desired movement. Each movement step of a node then consists of a rotation and a radial movement. The rotation step is a straightforward addition of angular coordinates: rotated(φ, r, τφ ) = (φ+τφ ) mod 2π . The radial movement is described in Algorithm 2 and a visualization is shown in Figure 2. getMinMaxPhi. The neighbors of a given vertex v = (φv , rv ) are those whose hyperbolic distance to v is at most R. Let bi be the slab between ci and ci+1 , and u = (φu , ru ) ∈ bi a neighbor of v in bi . Since u is in bi , ru is between ci and ci+1 . With the hyperbolic law of cosines, we can conclude: cosh R ≥ cosh rv cosh ci − sinh rv sinh ci cos \|φu − φv \| ⇔ (7) cosh rv cosh ci − cosh R ≤ sinh rv sinh ci cos \|φu − φv \| ⇔ (8) cosh rv cosh ci − cosh R ⇔ cos \|φu − φv \| ≥ sinh rv sinh ci (9) cosh r cosh c − cosh R v i −1 \|φu − φv \| ≤ cos sinh rv sinh ci (10) Algorithm 2: Radial movement in dynamic model Input: r, τr , R, α. Output: rnew 1) x = sinh(r · α); 2) y = x+τr ; 3) z = asinh(y)/α; 4) Return z To gather the neighborhood of a vertex v = (φv , rv ), we iterate over all slabs Si and compute for each slab how far the angular coordinate φq of a possible neighbor in bi can deviate from φv (Line 16). We call the vertices in bi whose angular coordinates are within these bounds the neighbor candidates for v in bi . Since points are sorted according to their angular coordinates, we can quickly find the leftmost and rightmost neighbor candidate in each slab using binary search. We then only need to check each neighbor candidate (Line 17), compute its hyperbolic distance to v and add an edge if this distance is below R (Lines 18 and 19). Since edges can be found from both ends, we only need to iterate over slabs in one direction; we choose outward in our implementation (Line 15). The process is repeated for every vertex v (Line 14). Not surprisingly the running time of Algorithm 1 is dominated by the range queries (Lines 14-23). Our experiments in Section 4 suggest a running time of O(n log n + m) for the complete algorithm. This should be seen as an empirical observation; we leave a mathematical proof for future work. If the new node position would be outside the boundary (r > R) or below the origin (r < 0), the movement is reflected and τr set to −τr . Theorem 1. Let fr,φ ((pr , pφ )) be the probability density of point positions, given in polar coordinates. Let move((pr , pφ )) be a movement step. Then, the node movement preserves the distribution of angular and radial distributions: fr,φ (move((pr , pφ ))) = fr,φ ((pr , pφ )). Proof. Since the distributions of angular and radial coordinates are independent, we consider them separately: fr,φ (pr , pφ ) = fr (pr ) · fφ (pφ ). As introduced in Eq. (1), the radial coordinate r is sampled from a distribution with density α sinh(αr)/(cosh(αR) − 1). We introduce random variables X, Y, Z for each step in Algorithm 2, each is denoted with the upper case letter of its equivalent. An additional random variable Q denotes the pre-movement radial coordinate. The other variables are defined as X = sinh(Q · α), Y = X + τr and Z = asinh(Y )/α. Let fQ , fX , fY and fZ denote the density functions of these variables: α sinh(αr) fQ (r) = (11) cosh(αR) − 1 asinh(r) αr fX (r) = fQ = (12) α cosh(αR) − 1 αr − τr fY (r) = fX (r − τr ) = (13) cosh(αR) − 1 α sinh(αr) − τr fZ (r) = fY (sinh(r · α)) = (14) cosh(αR) − 1 τr = fQ (r) − (15) cosh(αR) − 1 3.3. Dynamic Model To model gradual change in networks, we design and implement a dynamic version with node movement. While deleting nodes or inserting them at random positions is a suitable dynamic behavior for modeling internet infrastructure with sudden site failures or additions, change in e. g., social networks happens more gradually. A suitable node movement model needs to be consistent: After moving a node, the network may change, but properties should stay the same in expectation. Since the properties emerge from the node positions, the probability distribution of node positions needs to be preserved. In our implementation, movement happens in discrete time steps. We choose the movement to be directed: If a node i moves in a certain direction at time t, it will move in the same direction at t + 1, except if the new position would be 4 to 60 for graphs with 107 nodes and ≈ 4 · 107 edges. Very roughly, the experimental running times fit a complexity of O(n log n + m). While the running times of the faster generator appear to grow more steeply with increasing edge count, this is an artifact of the logarithmic plot: The same constant increase is relatively larger compared to a smaller running time, and thus appears larger in the logarithmic drawing. sinh(αR) R R ⇒ ⇒ 0 1 n = 105 , our impl. n = 106 , our impl. n = 107 , our impl. n = 105 , impl. of [10] n = 106 , impl. of [10] n = 107 , impl. of [10] n = 105 , theoretical fit n = 106 , theoretical fit n = 107 , theoretical fit 0 Figure 2: For each movement step, radial coordinates are mapped into the interval [1, sinh(αR)), where the coordinate distribution is uniform. Adding τr and transforming the coordinates back results in correctly scaled movements. The distributions of Q and Z only differ in the constant addition of τr /(cosh(αR) − 1). Every (cosh(αR) − 1)/τr steps, the radial movement reaches a limit (0 or R) and is reflected, causing τr to be multiplied with -1. On average, τr is thus zero and FQ (r) = FZ (r). A similar argument works for the rotational step: While the rotational direction is unchanged, the change in coordinates is balanced by the addition or subtraction of 2π whenever the interval [0, 2π) is left, leading to an average of zero in terms of change. running time in seconds 103 4. Experimental Evaluation Setup. The generation algorithm is implemented in C++11 and parallelized with OpenMP. Running time measurements were made on a server with 256 GB RAM and 2x8 Intel Xeon E5-2680 cores at 2.7 GHz. With hyperthreading enabled, we use up to 32 threads. For memory allocations, we use the lock-free malloc implementation of Intel’s Threading Building Blocks library. Our code is included in the network analysis toolkit NetworKit [13]. To compare performance, we generate graphs with 105 , 6 10 and 107 nodes and average degrees between 1 and 64, both with the algorithm presented in this work and the implementation of von Looz et al. [10]. To validate the distribution of generated graphs, we compare our implementation with the implementation of Aldecoa et al. [11]. We generate graphs with 104 nodes each for a combination of parameters and calculate several network analytic characteristics, averaging over 100 runs. For the dynamic model, we measure the time required for a movement step and again compare the distributions of network analytic properties. 102 101 100 10−1 105 106 107 edges 108 109 Figure 3: Comparison of running times to generate networks with 104 -107 vertices, α = 1 and varying k . Circles represent running times of our implementation, diamonds the running times of the implementation of [10]. Our running times are fitted with the equation T (n, m) = (7.07 · n log10 n + 2.23 · m + 891) · 10−8 seconds. The scaling behavior for 1 to 32 threads on 16 cores is shown in Figure 4. Considering edge sampling alone, it shows strong scaling up to the number of physical cores, with a speedup of 13.48 for 16 threads. With hyperthreading, the speedup increases to 18.38. Combining the edge lists later on into the NetworKit graph data structure, however, requires coordination and proves to be a bottleneck in parallel. If only edge lists are required, this final step can be omitted – as done for example in the Graph500 benchmark. Running Time. Figure 3 shows the running times to generate graphs with 105 to 107 nodes and 2 · 105 to 128 · 107 edges. The speedup over the previously fastest implementation [10] increases with graph size and sparsity, reaching up Distribution of Generated Graphs. The average degree assortativity, degeneracy, clustering coefficient and size and diameter of largest components of our generator and the 5 20 18.38 total edge sampling speedup factor 15 [3] D. A. Bader, J. Berry, S. Kahan, R. Murphy, E. J. Riedy, and J. Willcock, “Graph 500 benchmark 1 (”search”), version 1.1,” Graph 500, Tech. Rep., 2010. [4] D. Chakrabarti, Y. Zhan, and C. Faloutsos, “R-MAT: A recursive model for graph mining,” in Proc. 4th SIAM Intl. Conf. on Data Mining (SDM). Orlando, FL: SIAM, Apr. 2004. [5] T. G. Kolda, A. Pinar, T. Plantenga, and C. Seshadhri, “A scalable generative graph model with community structure,” SIAM J. Scientific Computing, vol. 36, no. 5, pp. C424–C452, Sep 2014. [6] D. Krioukov, F. Papadopoulos, M. Kitsak, A. Vahdat, and M. Boguñá, “Hyperbolic geometry of complex networks,” Physical Review E, vol. 82, no. 3, p. 036106, Sep 2010. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevE.82.036106 [7] M. Bode, N. Fountoulakis, and T. Müller, “The probability that the hyperbolic random graph is connected,” Random Structures and Algorithms, 2016, to appear. Preprint available at http://www.staff. science.uu.nl/∼muell001/Papers/BFM.pdf. [8] L. Gugelmann, K. Panagiotou, and U. Peter, “Random hyperbolic graphs: Degree sequence and clustering - (extended abstract),” in Automata, Languages, and Programming - 39th International Colloquium, ICALP 2012, Proceedings, Part II, ser. Lecture Notes in Computer Science, A. Czumaj, K. Mehlhorn, A. M. Pitts, and R. Wattenhofer, Eds., vol. 7392. Springer, 2012, pp. 573–585. [Online]. Available: http://dx.doi.org/10.1007/978-3-642-31585-5 51 [9] A. Bonato, “A survey of models of the web graph,” in Combinatorial and Algorithmic Aspects of Networking, ser. Lecture Notes in Computer Science. Springer Berlin Heidelberg, 2005, vol. 3405, pp. 159–172. [Online]. Available: http://dx.doi.org/10.1007/11527954 16 10 5 0 4 8 12 16 20 24 28 32 threads Figure 4: Speedup curves for n = 107 , k = 6, γ = 3 on a machine with 16 physical cores (marked with a vertical line) and hyperthreading. Averaged over 10 runs. one by Aldecoa et al. [11] are shown in Plots 5 and 6 in Appendix A. Averaged over 100 runs, the network analytic properties show a very close match between the distributions of the two generation algorithms. [10] M. von Looz, R. Prutkin, and H. Meyerhenke, “Generating random hyperbolic graphs in subquadratic time,” in ISAAC 2015: Proc. 26th Int’l Symp. on Algorithms and Computation, 2015. Dynamic Model. Our implementation allows updating a graph without rebuilding it from scratch. Moving up to 12% of nodes and updating an existing graph is still faster than a new static generation. The distribution of generated graphs is indistinguishable from the static model (Appendix B). [11] R. Aldecoa, C. Orsini, and D. Krioukov, “Hyperbolic graph generator,” Computer Physics Communications, 2015. [Online]. Available: http://www.sciencedirect.com/science/article/pii/ S0010465515002088 [12] K. Bringmann, R. Keusch, and J. Lengler, “Geometric inhomogeneous random graphs,” arXiv preprint arXiv:1511.00576, 2015. 5. Conclusions [13] C. L. Staudt, A. Sazonovs, and H. Meyerhenke, “NetworKit: A tool suite for large-scale complex network analysis,” Network Science, 2015, to appear. We have provided the fastest implementation so far to generate massive complex networks based on threshold random hyperbolic graphs. The running time improvement is particularly large for graphs with realistic densities. We have also presented a model extension to cover gradual node movement and have proved its consistency regarding the probability densities of vertex positions. Both the static and the dynamic model can serve as complex network generators with reasonable realism and fast generation times even for massive networks. [14] A. Goldenberg, A. X. Zheng, S. E. Fienberg, and E. M. Airoldi, “A survey of statistical network models,” Foundations and Trends R in Machine Learning, vol. 2, no. 2, pp. 129–233, 2010. [15] J. W. Anderson, Hyperbolic geometry; 2nd ed., ser. Springer undergraduate mathematics series. Berlin: Springer, 2005. [16] M. Boguñá, F. Papadopoulos, and D. Krioukov, “Sustaining the internet with hyperbolic mapping,” Nature Communications, no. 62, September 2010. [Online]. Available: http://www.nature.com/ ncomms/journal/v1/n6/abs/ncomms1063.html [17] M. von Looz and H. Meyerhenke, “Querying Probabilistic Neighborhoods in Spatial Data Sets Efficiently,” ArXiv preprint arXiv:1509.01990, Sep. 2015. Acknowledgements. This work is partially supported by German Research Foundation (DFG) grant ME 3619/3-1 (FINCA) and grant GI-711/5-1, both within the Priority Programme 1736 Algorithms for Big Data. [18] M. Kiwi and D. Mitsche, “A bound for the diameter of random hyperbolic graphs,” in 2015 Proceedings of the Twelfth Workshop on Analytic Algorithmics and Combinatorics (ANALCO). SIAM, Jan 2015, pp. 26–39. [Online]. Available: http://epubs.siam.org/doi/abs/ 10.1137/1.9781611973761.3 References [1] M. Newman, Networks: An Introduction. 2010. Oxford University Press, [2] D. Chakrabarti and C. Faloutsos, “Graph mining: Laws, generators, and algorithms,” ACM Computing Surveys (CSUR), vol. 38, no. 1, p. 2, 2006. 6 Appendix A. Comparison tion [11] with Previous Degree Assortativity Degree Assortativity γ = 2.2 γ = 4.6 γ = 7.0 0.4 0.2 0 0.6 degree assortativity degree assortativity 0.6 Implementa- −0.2 γ = 2.2 γ = 4.6 γ = 7.0 0.4 0.2 0 −0.2 101 102 101 k 102 Clustering Coefficient k 0.86 0.84 Degeneracy Degeneracy Clustering Coefficient 0.86 k=4 k = 32 k = 256 0.82 cc 2 k 3 4 5 6 7 2 3 4 γ ·104 101 102 vertices in largest component 102 0.76 0.76 101 101 0.8 0.78 0.78 2 k Figure 5: Comparison of degree assortativity and degeneracy for the implementation of [11] (left) and our implementation (right). Degree assortativity describes whether vertices have neighbors of similar degree. A value near 1 signifies subgraphs with equal degree, a value of -1 star-like structures. k -Cores, in turn, are a generalization of connected components and result from iteratively peeling away vertices of degree k and assigning to each vertex the core number of the innermost core it is contained in. Degeneracy refers to the largest core number. Values are averaged over 100 runs. Size of Largest Component 1 0.8 0.6 0.4 0.2 γ = 2.2 γ = 4.6 γ = 7.0 0 101 ·104 Size of Largest Component 0.6 0.4 0.2 γ = 2.2 γ = 4.6 γ = 7.0 0 101 102 k Diameter of Largest Component diameter of largest component 7 0.8 102 γ = 2.2 γ = 4.6 γ = 7.0 600 400 200 0 102 k 6 1 k 101 5 γ vertices in largest component 101 10 0.8 Diameter of Largest Component diameter of largest component max core number max core number 10 2 cc 0.82 γ = 2.2 γ = 4.6 γ = 7.0 γ = 2.2 γ = 4.6 γ = 7.0 k=4 k = 32 k = 256 0.84 600 γ = 2.2 γ = 4.6 γ = 7.0 400 200 0 101 102 k Figure 6: Comparison of clustering coefficients, size of largest component and diameter of largest components for the implementation of [11] (left) and our implementation (right). Values are averaged over 100 runs. 7 Appendix B. Consistency of Dynamic Model Degree Assortativity Degree Assortativity 0.2 0 −0.2 γ = 2.2 γ = 4.6 γ = 7.0 0.4 0.2 0 −0.2 101 102 101 Clustering Coefficient 102 0.86 k k 0.84 Degeneracy max core number 0.8 102 101 102 k 0.76 0.76 2 3 4 5 6 7 2 3 4 γ ·104 101 101 0.8 0.78 0.78 101 vertices in largest component max core number 102 0.82 cc γ = 2.2 γ = 4.6 γ = 7.0 k=4 k = 32 k = 256 0.84 0.82 cc Degeneracy γ = 2.2 γ = 4.6 γ = 7.0 Clustering Coefficient 0.86 k=4 k = 32 k = 256 102 k Figure 7: Comparison of degree assortativity and degeneracy for graphs with 104 nodes, before and after one movement step. All nodes were moved, with τφ ∈ (−1, 1) and τr ∈ (−10, 1) sampled randomly. Distribution of graphs after node movement are shown left, before node movement right. Values are averaged over 100 runs. Size of Largest Component 1 0.8 0.6 0.4 0.2 γ = 2.2 γ = 4.6 γ = 7.0 0 101 ·104 Size of Largest Component 0.6 0.4 0.2 γ = 2.2 γ = 4.6 γ = 7.0 0 101 102 k Diameter of Largest Component diameter of largest component 7 0.8 102 γ = 2.2 γ = 4.6 γ = 7.0 400 200 0 102 k 6 1 k 101 5 γ vertices in largest component 0.4 0.6 Diameter of Largest Component diameter of largest component γ = 2.2 γ = 4.6 γ = 7.0 degree assortativity degree assortativity 0.6 600 γ = 2.2 γ = 4.6 γ = 7.0 400 200 0 101 102 k Figure 8: Comparison of clustering coefficients, size of largest component and diameter of largest components for graphs with 104 nodes, before and after one movement step. All nodes were moved, with τφ ∈ (−1, 1) and τr ∈ (−10, 1) sampled randomly. Distribution of graphs after node movement are shown left, before node movement right. Values are averaged over 100 runs. 8	8
The (1\|1)-Centroid Problem on the Plane Concerning Distance Constraints ⋆ arXiv:1608.03680v1 [cs.CG] 12 Aug 2016 Hung-I Yu, Tien-Ching Lin, and D. T. Lee Institute of Information Science, Academia Sinica, Nankang, Taipei 115, Taiwan, {herbert,kero,dtlee}@iis.sinica.edu.tw Abstract. In 1982, Drezner proposed the (1\|1)-centroid problem on the plane, in which two players, called the leader and the follower, open facilities to provide service to customers in a competitive manner. The leader opens the first facility, and the follower opens the second. Each customer will patronize the facility closest to him (ties broken in favor of the first one), thereby decides the market share of the two facilities. The goal is to find the best position for the leader’s facility so that its market share is maximized. The best algorithm of this problem is an O(n2 log n)-time parametric search approach, which searches over the space of market share values. In the same paper, Drezner also proposed a general version of (1\|1)centroid problem by introducing a minimal distance constraint R, such that the follower’s facility is not allowed to be located within a distance R from the leader’s. He proposed an O(n5 log n)-time algorithm for this general version by identifying O(n4 ) points as the candidates of the optimal solution and checking the market share for each of them. In this paper, we develop a new parametric search approach searching over the O(n4 ) candidate points, and present an O(n2 log n)-time algorithm for the general version, thereby close the O(n3 ) gap between the two bounds. Keywords: competitive facility, Euclidean plane, parametric search 1 Introduction In 1929, economist Hotelling introduced the first competitive location problem in his seminal paper [13]. Since then, the subject of competitive facility location has been extensively studied by researchers in the fields of spatial economics, social and political sciences, and operations research, and spawned hundreds of contributions in the literature. The interested reader is referred to the following survey papers [3,7,8,9,11,12,17,19]. Hakimi [10] and Drezner [5] individually proposed a series of competitive location problems in a leader-follower framework. The framework is briefly described as follows. There are n customers in the market, and each is endowed ⋆ Research supported by under Grants No. MOST 103-2221-E-005-042, 103-2221-E005-043. 2 Hung-I Yu, Tien-Ching Lin, and D. T. Lee with a certain buying power. Two players, called the leader and the follower, sequentially open facilities to attract the buying power of customers. At first, the leader opens his p facilities, and then the follower opens another r facilities. Each customer will patronize the closest facility with all buying power (ties broken in favor of the leader’s ones), thereby decides the market share of the two players. Since both players ask for market share maximization, two competitive facility location problems are defined under this framework. Given that the leader locates his p facilities at the set Xp of p points, the follower wants to locate his r facilities in order to attract the most buying power, which is called the (r\|Xp )-medianoid problem. On the other hand, knowing that the follower will react with maximization strategy, the leader wants to locate his p facilities in order to retain the most buying power against the competition, which is called the (r\|p)-centroid problem. Drezner [5] first proposed to study the two competitive facility location problems on the Euclidean plane. Since then, many related results [4,5,6,11,14] have been obtained for different values of r and p. Due to page limit, here we introduce only previous results about the case r = p = 1. For the (1\|X1 )-medianoid problem, Drezner [5] showed that there exists an optimal solution arbitrarily close to X1 , and solved the problem in O(n log n) time by sweeping technique. Later, Lee and Wu [14] obtained an Ω(n log n) lower bound for the (1\|X1 )-medianoid problem, and thus proved the optimality of the above result. For the (1\|1)-centroid problem, Drezner [5] developed a parametric search based approach that searches over the space of O(n2 ) possible market share values, along with an O(n4 )-time test procedure constructing and solving a linear program of O(n2 ) constraints, thereby gave an O(n4 log n)-time algorithm. Then, by improving the test procedure via Megiddo’s result [16] for solving linear programs, Hakimi [11] reduced the time complexity to O(n2 log n). In [5], Drezner also proposed a more general setting for the leader-follower framework by introducing a minimal distance constraint R ≥ 0 into the (1\|X1 )medianoid problem and the (1\|1)-centroid problem, such that the follower’s facility is not allowed to be located within a distance R from the leader’s. The augmented problems are respectively called the (1\|X1 )R -medianoid problem and (1\|1)R -centroid problem in this paper. Drezner showed that the (1\|X1 )R medianoid problem can also be solved in O(n log n) time by using nearly the same proof and technique as for the (1\|X1 )-medianoid problem. However, for the (1\|1)R -centroid problem, he argued that it is hard to generalize the approach for the (1\|1)-centroid problem to solve this general version, due to the change of problem properties. Then, he gave an O(n5 log n)-time algorithm by identifying O(n4 ) candidate points on the plane, which contain at least one optimal solution, and performing medianoid computation on each of them. So far, the O(n3 ) bound gap between the two centroid problems remains unclosed. In this paper, we propose an O(n2 log n)-time algorithm for the (1\|1)R centroid problem on the Euclidean plane, thereby close the gap last for decades. Instead of searching over market share values, we develop a new approach based on the parametric search technique by searching over the O(n4 ) candidate points The (1\|1)R -Centroid Problem on the Plane 3 mentioned in [5]. This is made possible by making a critical observation on the distribution of optimal solutions for the (1\|X1 )R -medianoid problem given X1 , which provides us a useful tool to prune candidate points with respect to X1 . We then extend the usage of this tool to design a key procedure to prune candidates with respect to a given vertical line. The rest of this paper is organized as follows. Section 2 gives formal problem definitions and describes previous results in [5,11]. In Section 3, we make the observation on the (1\|X1 )R -medianoid problem, and make use of it to find a “local” centroid on a given line. This result is then extended as a new pruning procedure with respect to any given line in Section 4, and utilized in our parametric search approach for the (1\|1)R -centroid problem. Finally, in Section 5, we give some concluding remarks. 2 Notations and Preliminary Results Let V = {v1 , v2 , · · · , vn } be a set of n points on the Euclidean plane R2 , as the representatives of the n customers. Each point vi ∈ V is assigned with a positive weight w(vi ), representing its buying power. To simplify the algorithm description, we assume that the points in V are in general position, that is, no three points are collinear and no two points share a common x or y-coordinate. Let d(u, w) denote the Euclidean distance between any P two points u, w T ∈ R2 . For any set Z of points on the plane, we define W (Z) = {w(v)\|v ∈ V Z}. Suppose that the leader has located his facility at X1 = {x}, which is shortened as x for simplicity. Due to the minimal distance constraint R mentioned in [5], any point y ′ ∈ R2 with d(y ′ , x) < R is infeasible to be the follower’s choice. If the follower locates his facility at some feasible point y, the set of customers patronizing y instead of x is defined as V (y\|x) = {v ∈ V \|d(v, y) < d(v, x)}, with their total buying power W (y\|x) = W (V (y\|x)). Then, the largest market share that the follower can capture is denoted by the function W ∗ (x) = max y∈R2 ,d(y,x)≥R W (y\|x), which is called the weight loss of x. Given a point x ∈ R2 , the (1\|x)R -medianoid problem is to find a (1\|x)R -medianoid, which denotes a feasible point y ∗ ∈ R2 maximizing the weight loss of x. In contrast, the leader tries to minimize the weight loss of his own facility by finding a point x∗ ∈ R2 such that W ∗ (x∗ ) ≤ W ∗ (x) for any point x ∈ R2 . The (1\|1)R -centroid problem is to find a (1\|1)R -centroid, which denotes a point x∗ minimizing its weight loss. Note that, when R = 0, the two problems degenerate to the (1\|x)-medianoid and (1\|1)-centroid problems. 4 2.1 Hung-I Yu, Tien-Ching Lin, and D. T. Lee Previous approaches In this subsection, we briefly review previous results for the (1\|x)R -medianoid, (1\|1)-centroid, and (1\|1)R -centroid problems in [5,11], so as to derive some basic properties essential to our approach. Let L be an arbitrary line, which partitions the Euclidean plane into two halfplanes. For any point y ∈ / L, we define H(L, y) as the close half-plane including y, and H − (L, y) as the open half-plane including y (but not L). For any two distinct points x, y ∈ R2 , let B(y\|x) denote the perpendicular bisector of the line segment from x to y. Given an arbitrary point x ∈ R2 , we first describe the algorithm for finding a (1\|x)R -medianoid in [5]. Let y be an arbitrary point other than x, and y ′ be some point on the open line segment from y to x. We can see that H − (B(y\|x), y) ⊂ H − (B(y ′ \|x), y ′ ), which implies the fact that W (y ′ \|x) = W (H − (B(y ′ \|x), y ′ )) ≥ W (H − (B(y\|x), y)) = W (y\|x), It shows that moving y toward x does not diminish its weight capture, thereby follows the lemma. Lemma 1 [5] There exists a (1\|x)R -medianoid in {y \| y ∈ R2 , d(x, y) = R}. For any point z ∈ R2 , let CR (x) and Cγ (x) be the circles centered at z with radii R and γ = R/2, respectively. By Lemma 1, finding a (1\|x)R -medianoid can be reduced to searching a point y on CR (x) maximizing W (y\|x). Since the perpendicular bisector B(y\|x) of each point y on CR (x) is a tangent line to the circle Cγ (x), the searching of y on CR (x) is equivalent to finding a tangent line to Cγ (x) that partitions the most weight from x. The latter problem can be solved in O(n log n) time as follows. For each v ∈ V outside Cγ (x), we calculate its two tangent lines to Cγ (x). Then, by sorting these tangent lines according to the polar angles of their corresponding tangent points with respect to x, we can use the angle sweeping technique to check how much weight they partition. Theorem 1 [5] Given a point x ∈ R2 , the (1\|x)R -medianoid problem can be solved in O(n log n) time. Next, we describe the algorithm of the (1\|1)R -centroid problem in [5]. Let S be a subset of V . We define C(S) to be the set of all circles Cγ (v), v ∈ S, and CH(C(S)) to be the convex hull of these circles. It is easy to see the following. Lemma 2 [5] Let S be a subset of V . For any point x ∈ R2 , W ∗ (x) ≥ W (S) if x is outside CH(C(S)). For any positive number W0 , let I(W0 ) be the intersection of all convex hulls CH(C(S)), where S ⊆ V and W (S) ≥ W0 . We have the lemma below. Lemma 3 [5] Let W0 be a positive real number. For any point x ∈ R2 , W ∗ (x) < W0 if and only if x ∈ I(W0 ). The (1\|1)R -Centroid Problem on the Plane 5 Proof. Consider first the case that x ∈ I(W0 ). By definition, x intersects with every CH(C(S)) of subset S ⊆ V with W (S) ≥ W0 . Let S ′ ⊆ V be any of such subsets. Since x ∈ CH(C(S ′ )), for any point y feasible to x, there must exist a point v ∈ S such that v ∈ / H − (B(y\|x), y), implying that no feasible point y can acquire all buying power from customers of S ′ . It follows that no feasible point y can acquire buying power larger than or equal to W0 , i.e., W ∗ (x) < W0 . If x ∈ / I(W0 ), there must exist a subset S ⊆ V with W (S) ≥ W0 , such that x∈ / CH(C(S)). By Lemma 2, W ∗ (x) ≥ W (S) ≥ W0 . ⊓ ⊔ Drezner [5] argued that the set of all (1\|1)R -centroids is equivalent to some intersection I(W0 ) for smallest possible W0 . We slightly strengthen his argument below. Let W = {W (y\|x) \| x, y ∈ R2 , d(x, y) ≥ R}. The following lemma can be obtained. Lemma 4. Let W0∗ be the smallest number in W such that I(W0∗ ) is not null. A point x is a (1\|1)R -centroid if and only if x ∈ I(W0∗ ). Proof. Let WOP T be the weight loss of some (1\|1)R -centroid x∗ . We first show that I(W0 ) is null for any W0 ≤ WOP T . Suppose to the contrary that it is not null and there exists a point x′ in I(W0 ). By Lemma 3, W ∗ (x′ ) < W0 ≤ WOP T , which contradicts the optimality of x∗ . Moreover, since I(W0∗ ) is not null, we have that W0∗ > WOP T . We now show that a point x is a (1\|1)R -centroid if and only if x ∈ I(W0∗ ). If x is a (1\|1)R -centroid, we have that W ∗ (x) = WOP T < W0∗ . By Lemma 3, x ∈ I(W0∗ ). On the other hand, if x is not a (1\|1)R -centroid, we have that W ∗ (x) > WOP T . Since by definition W ∗ (x) ∈ W, we can see that W ∗ (x) ≥ W0∗ . Thus, again by Lemma 3, x ∈ / I(W0∗ ). ⊓ ⊔ Although it is hard to compute I(W0∗ ) itself, we can find its vertices as solutions to the (1\|1)R -centroid problem. Let T be the set of outer tangent lines of all pairs of circles in C(V ). For any subset S ⊆ V , the boundary of CH(C(S)) is formed by segments of lines in T and arcs of circles in C(V ). Since I(W0 ) is an intersection of such convex hulls, its vertices must fall within the set of intersection points between lines in T , between circles in C(V ), and between one line in T and one circle in C(V ). Let T × T , C(V ) × C(V ), and T × C(V ) denote the three sets of intersection points, respectively. We have the lemma below. Lemma 5 [5] There exists a (1\|1)R -centroid in T × T , C(V ) × C(V ), and T × C(V ). Obviously, there are at most O(n4 ) intersection points, which can be viewed as the candidates of being (1\|1)R -centroids. Drezner thus gave an algorithm by evaluating the weight loss of each candidate by Theorem 1. Theorem 2 [5] The (1\|1)R -centroid problem can be solved in O(n5 log n) time. We remark that, when R = 0, CH(C(S)) for any S ⊆ V degenerates to a convex polygon, so does I(W0 ) for any given W0 , if not null. Drezner [5] proved 6 Hung-I Yu, Tien-Ching Lin, and D. T. Lee that in this case I(W0 ) is equivalent to the intersection of all half-planes H with W (H) ≥ W0 . Thus, whether I(W0 ) is null can be determined by constructing and solving a linear program of O(n2 ) constraints, which takes O(n2 ) time by Megiddo’s result [16]. Since \|W\| = O(n2 ), by Lemma 4, the (1\|1)-centroid problem can be solved in O(n2 log n) time [11], by applying parametric search over W for W0∗ . Unfortunately, it is hard to generalize this idea to the case R > 0, motivating us to develop a different approach. Local (1\|1)R -Centroid within a Line 3 In this section, we analyze the properties of (1\|x)R -medianoids of a given point x in Subsection 3.1, and derive a procedure that prunes candidate points with respect to x. Applying this procedure, we study a restricted version of the (1\|1)R centroid problem in Subsection 3.2, in which the leader’s choice is limited to a given non-horizontal line L, and obtain an O(n log2 n)-time algorithm. The algorithm is then extended as the basis of the test procedure for the parametric search approach in Section 4. 3.1 Pruning with Respect to a Point Given a point x ∈ R2 and an angle θ between 0 and 2π, let y(θ\|x) be the point on CR (x) with polar angle θ with respect to x.1 We define M A(x) = {θ \| W (y(θ\|x)\|x) = W ∗ (x), 0 ≤ θ < 2π}, that is, the set of angles θ maximizing W (y(θ\|x)\|x) (see Figure 1). It can be observed that, for any θ ∈ M A(x) and sufficiently small ǫ, both θ + ǫ and θ − ǫ belong to M A(x), because each v ∈ V (y(θ\|x)\|x) does not intersect B(y(θ\|x)\|x) by definition. This implies that angles in M A(x) form open angle interval(s) of non-zero length. To simplify the terms, let W (θ\|x) = W (y(θ\|x)\|x) and B(θ\|x) = B(y(θ\|x)\|x) in the remaining of this section. Also, let F (θ\|x) be the line passing through x and parallel to B(θ\|x). The following lemma provides the basis for pruning. Lemma 6 Let x ∈ R2 be an arbitrary point, and θ be an angle in M A(x). For any point x′ ∈ / H − (F (θ\|x), y(θ\|x)), W ∗ (x′ ) ≥ W ∗ (x). Proof. Since x′ ∈ / H − (F (θ\|x), y(θ\|x)) and y(θ\|x) ∈ H − (F (θ\|x), y(θ\|x)), by the definition of bisectors, the distance between F (θ\|x′ ) and B(θ\|x) is no less than R/2, which implies that H − (B(θ\|x), y(θ\|x)) ⊆ H − (B(θ\|x′ ), y(θ\|x′ )). Therefore, we can derive the following inequality W ∗ (x′ ) ≥ W (θ\|x′ ) = W (H − (B(θ\|x′ ), y(θ\|x′ )) ≥ W (H − (B(θ\|x), y(θ\|x)) = W (θ\|x) = W ∗ (x), 1 We assume that a polar angle is measured counterclockwise from the positive x-axis. The (1\|1)R -Centroid Problem on the Plane 7 Fig. 1. The black arcs represent the intervals of angles in M A(x), whereas the open circles represent the open ends of these intervals. which completes the proof. ⊓ ⊔ This lemma tells us that, given a point x and an angle θ ∈ M A(x), all points not in H − (F (θ\|x), y(θ\|x)) can be ignored while finding (1\|1)R -centroids, as their weight losses are no less than that of x. By this lemma, we can also prove that the weight loss function is convex along any line on the plane, as shown below. Lemma 7 Let x1 , x2 be two arbitrary distinct points on a given line L. For any point x ∈ x1 x2 \{x1 , x2 }, W ∗ (x) ≤ max{W ∗ (x1 ), W ∗ (x2 )}. Proof. Suppose by contradiction that W ∗ (x) > W ∗ (x1 ) and W ∗ (x) > W ∗ (x2 ) for some point x ∈ x1 x2 \{x1 , x2 }. Since W ∗ (x) > W ∗ (x1 ), by Lemma 6 there exists an angle θ ∈ M A(x) such that x1 is included in H − (F (θ\|x), y(θ\|x)). However, since x ∈ x1 x2 \{x1 , x2 }, x1 and x2 locate on different sides of F (θ\|x). It follows that x2 is outside H − (F (θ\|x), y(θ\|x)) and W ∗ (x2 ) ≥ W ∗ (x) by Lemma 6, which contradicts the assumption. Thus, the lemma holds. ⊓ ⊔ We further investigate the distribution of angles in M A(x). Let CA(x) be the minimal angle interval covering all angles in M A(x) (see Figure 2(a)), and δ(CA(x)) be its angle span in radians. As mentioned before, M A(x) consists of open angle interval(s) of non-zero length, which implies that CA(x) is an open interval and δ(CA(x)) > 0. Moreover, we can derive the following. Lemma 8 If δ(CA(x)) > π, x is a (1\|1)R -centroid. Proof. We prove this lemma by showing that W ∗ (x′ ) ≥ W ∗ (x) ∀ x′ 6= x. Let x′ ∈ R2 be an arbitrary point other than x, and θ′ be its polar angle with respect to x. Obviously, any angle θ satisfying x′ ∈ H − (F (θ\|x), y(θ\|x)) is in the open interval (θ′ − π/2, θ′ + π/2), the angle span of which is equal to π. Since 8 Hung-I Yu, Tien-Ching Lin, and D. T. Lee (a) CA(x) (b) W edge(x) Fig. 2. CA(x) and W edge(x). δ(CA(x)) > π, by its definition there exists an angle θ ∈ M A(x) such that x′ ∈ / H − (F (θ\|x), y(θ\|x)). Thus, by Lemma 6, we have W ∗ (x′ ) ≥ W ∗ (x), thereby proves the lemma. ⊓ ⊔ We call a point x satisfying Lemma 8 a strong (1\|1)R -centroid, since its discovery gives an immediate solution to the (1\|1)R -centroid problem. Note that there are problem instances in which no strong (1\|1)R -centroids exist. Suppose that δ(CA(x)) ≤ π for some point x ∈ R2 . Let W edge(x) denote the wedge of x, defined as the intersection of the two half-planes H(F (θb \|x), y(θb \|x)) and H(F (θe \|x), y(θe \|x)), where θb and θe are the beginning and ending angles of CA(x), respectively. As illustrated in Figure 2(b), W edge(x) is the infinite region lying between two half-lines extending from x (including x and the two halflines). The half-lines defined by F (θe \|x) and F (θb \|x) are called its boundaries, and the counterclockwise (CCW) angle between the two boundaries is denoted by δ(W edge(x)). Since 0 < δ(CA(x)) ≤ π, we have that W edge(x) 6= ∅ and 0 ≤ δ(W edge(x)) < π. It should be emphasized that W edge(x) is a computational byproduct of CA(x) when x is not a strong (1\|1)R -centroid. In other words, not every point has its wedge. Therefore, we make the following assumption (or restriction) in order to avoid the misuse of W edge(x). Assumption 1 Whenever W edge(x) is mentioned, the point x has been found not to be a strong (1\|1)R -centroid, either by computation or by properties. Equivalently, δ(CA(x)) ≤ π. The following essential lemma makes W edge(x) our main tool for prune-andsearch. (Note that its proof cannot be trivially derived from Lemma 6, since by definition θb and θe do not belong to the open intervals CA(x) and M A(x).) Lemma 9 Let x ∈ R2 be an arbitrary point. For any point x′ ∈ / W edge(x), W ∗ (x′ ) ≥ W ∗ (x). Proof. By symmetry, suppose that x′ ∈ / H(F (θb \|x), y(θb \|x)). We can further divide the position of x′ into two cases, (1) x′ ∈ H(F (θe \|x), y(θe \|x)) and (2) x′ ∈ / H(F (θe \|x), y(θe \|x)). The (1\|1)R -Centroid Problem on the Plane 9 Consider case (1). The two assumptions ensure that there exists an angle θ′ ∈ (θb , θe ], such that F (θ′ \|x) passes through x′ . Obviously, any angle θ′′ ∈ (θb , θ′ ) satisfies that x′ ∈ / H(F (θ′′ \|x), y(θ′′ \|x)). By the definition of CA(x), there must ′ exist an angle θb ∈ (θb , θ′ ) infinitely close to θb , such that θb′ belongs to M A(x). Thus, by Lemma 6, we have that W ∗ (x′ ) ≥ W ∗ (x). In case (2), for any angle θ′′ ∈ M A(x), we have that x′ ∈ / H(F (θ′′ \|x), y(θ′′ \|x)), ′′ ∗ ′ ∗ since θ is in (θb , θe ). Again, W (x ) ≥ W (x) by Lemma 6. ⊓ ⊔ Finally, we consider the computation of W edge(x). Lemma 10 Given a point x ∈ R2 , M A(x), CA(x), and W edge(x) can be computed in O(n log n) time. Proof. By Theorem 1, we first compute W ∗ (x) and those ordered tangent lines in O(n log n) time. Then, by performing angle sweeping around Cγ (x), we can identify in O(n) time those open intervals of angles θ with W (θ\|x) = W ∗ (x), of which M A(x) consists. Again by sweeping around Cγ (x), CA(x) can be obtained from M A(x) in O(n) time. Now, if we find x to be a strong (1\|1)R -centroid by checking δ(CA(x)), the (1\|1)R -centroid problem is solved and the algorithm can be terminated. Otherwise, W edge(x) can be constructed in O(1) time. ⊓ ⊔ 3.2 Searching on a Line Although computing wedges can be used to prune candidate points, it does not serve as a stable prune-and-search tool, since wedges of different points have indefinite angle intervals and spans. However, Assumption 1 makes it work fine with lines. Here we show how to use the wedges to compute a local optimal point on a given line, i.e. a point x with W ∗ (x) ≤ W ∗ (x′ ) for any point x′ on the line. Let L be an arbitrary line, which is assumed to be non-horizontal for ease of discussion. For any point x on L, we can compute W edge(x) and make use of it for pruning purposes by defining its direction with respect to L. Since δ(W edge(x)) < π by definition, there are only three categories of directions according to the intersection of W edge(x) and L: Upward – the intersection is the half-line of L above and including x; Downward – the intersection is the half-line of L below and including x; Sideward – the intersection is x itself. If W edge(x) is sideward, x is a local optimal point on L, since by Lemma 9 W ∗ (x) ≤ W ∗ (x′ ) ∀ x′ ∈ L. Otherwise, either W edge(x) is upward or downward, the points on the opposite half of L can be pruned by Lemma 9. It shows that computing wedges acts as a predictable tool for pruning on L. Next, we list sets of breakpoints on L in which a local optimal point locates. Recall that T is the set of outer tangent lines of all pairs of circles in C(V ). We define the T -breakpoints as the set L × T of intersection points between L and lines in T , and the C-breakpoints as the set L × C(V ) of intersection points between L and circles in C(V ). We have the following lemmas for breakpoints. 10 Hung-I Yu, Tien-Ching Lin, and D. T. Lee Lemma 11 Let x1 , x2 be two distinct points on L. If W ∗ (x1 ) > W ∗ (x2 ), there exists at least a breakpoint on the segment x1 x2 \{x1 }. Proof. Let θ be an arbitrary angle in M A(x1 ) and S be the subset of V located in the half-plane H − (B(θ\|x1 ), y(θ\|x1 )). By definition, x1 is outside the convex hull CH(C(S)) and W ∗ (x1 ) = W (S). On the other hand, since W ∗ (x2 ) < W ∗ (x1 ) = W (S) by assumption, we have that x2 is inside CH(C(S)) by Lemma 2. Thus, the segment x1 x2 \{x1 } intersects with the boundary of CH(C(S)). Since the boundary of CH(C(S)) consists of segments of lines in T and arcs of circles in C(V ), the intersection point is either a T -breakpoint or a C-breakpoint, thereby proves the lemma. ⊓ ⊔ Lemma 12 There exists a local optimal point x∗L which is also a breakpoint. Proof. Let x∗L be a local optimal point such that W ∗ (x′ ) > W ∗ (x∗L ) for some point x′ adjacent to x∗L on L. Note that, if no such local optimal point exists, every point on L must have the same weight loss and be local optimal, and the lemma holds trivially. If such x∗L and x′ exist, by Lemma 11 there is a breakpoint on x′ x∗L \{x′ }, which is x∗L itself. Thus, the lemma holds. ⊓ ⊔ We remark that outer tangent lines parallel to L are exceptional cases while considering breakpoints. For any line T ∈ T that is parallel to L, either T does not intersect with L or they just coincide. In either case, T is irrelevant to the finding of local optimal points, and should not be counted for defining T -breakpoints. Now, by Lemma 12, if we have all breakpoints on L sorted in the decreasing order of their y-coordinates, a local optimal point can be found by performing binary search using wedges. Obviously, such sorted sequence can be obtained in O(n2 log n) time, since \|L × T \| = O(n2 ) and \|L × C(V )\| = O(n). However, in order to speed up the computations of local optimal points on multiple lines, alternatively we propose an O(n2 log n)-time preprocessing, so that a local optimal point on any given line can be computed in O(n log2 n) time. The preprocessing itself is very simple. For each point v ∈ V , we compute a sequence P (v), consisting of points in V \{v} sorted in increasing order of their polar angles with respect to v. The computation for all v ∈ V takes O(n2 log n) time in total. Besides, all outer tangent lines in T are computed in O(n2 ) time. We will show that, for any given line L, O(n) sorted sequences can be obtained from these pre-computed sequences in O(n log n) time, which can be used to replace a sorted sequence of all T -breakpoints in the process of binary search. For any two points v ∈ V and z ∈ R2 , let T r (z\|v) be the outer tangent line of Cγ (v) and Cγ (z) to the right of the line from v to z. Similarly, let T l (z\|v) be the outer tangent line to the left. (See Figure 3.) Moreover, let trL (z\|v) and tlL (z\|v) be the points at which T r (z\|v) and T l (z\|v) intersect with L, respectively. We partition T into O(n) sets T r (v) = {T r (vi \|v)\|vi ∈ V \{v}} and T l (v) = {T l (vi \|v)\|vi ∈ V \{v}} for v ∈ V , and consider their corresponding T -breakpoints independently. By symmetry, we only discuss the case about L × T r (v). The (1\|1)R -Centroid Problem on the Plane 11 Fig. 3. Outer tangent lines of v. Lemma 13 For each v ∈ V , we can compute O(1) sequences of T -breakpoints on L, which satisfy the following conditions: (a) Each sequence is of length O(n) and can be obtained in O(log n) time. (b) Breakpoints in each sequence are sorted in decreasing y-coordinates. (c) The union of breakpoints in all sequences form L × T r (v). Proof. Without loss of generality, suppose that v is either strictly to the right of L or on L. Note that each point vi ∈ V \{v} corresponds to exactly one outer tangent line T r (vi \|v), thereby exactly one breakpoint trL (vi \|v). Such one-to-one correspondence can be easily done in O(1) time. Therefore, equivalently we are computing sequences of points in V \{v}, instead of breakpoints. In the following, we consider two cases about the relative position between L and Cγ (v), (1) L intersects with Cγ (v) at zero or one point, (2) L intersects with Cγ (v) at two points. Case (1): Let θL be the angle of the upward direction along L. See Figure 4(a). We classify the points in V \{v} by their polar angles with respect to v. Let P1 (v) denote the sequence of those points with polar angles in the interval (θL , θL + π) and sorted in CCW order. Similarly, let P2 (v) be the sequence of points with polar angles in (θL + π, θL ) and sorted in CCW order. Obviously, P1 (v) and P2 (v) together satisfy condition (c). (Note that points with polar angles θL and θL + π are ignored, since they correspond to outer tangent lines parallel to L.) By general position assumption, we can observe that, for any two distinct points vi , vj in P1 (v), trL (vi \|v) is strictly above trL (vj \|v) if and only if vi precedes vj in P1 (v). Thus, the ordering of points in P1 (v) implicitly describes an ordering of their corresponding breakpoints in decreasing y-coordinates. Similarly, the ordering in P2 (v) implies an ordering of corresponding breakpoints in decreasing y-coordinates. It follows that both P1 (v) and P2 (v) satisfy condition (b). As for condition (a), both P1 (v) and P2 (v) are of length O(n) by definition. Also, since we have pre-computed the sequence P (v) as all points in V \{v} 12 Hung-I Yu, Tien-Ching Lin, and D. T. Lee (a) no intersection (b) two intersection points Fig. 4. Two subcases about how Cγ (v) intersects C. sorted in CCW order, P1 (v) and P2 (v) can be implicitly represented as concatenations of subsequences of P (v). This can be done in O(log n) time by searching in P (v) the foremost elements with polar angles larger than θL and θL + π, respectively. Case (2): Suppose that the two intersection points between L and Cγ (v) are c1 and c2 , where c1 is above c2 . Let θ1 = θ1′ + π/2 and θ2 = θ2′ + π/2, in which θ1′ and θ2′ are respectively the polar angles of c1 and c2 with respect to v. See Figure 4(b). By assumption, we have that θL < θ1′ < θL + π/2 < θ2′ < θL + π, which implies that θ1 ∈ (θL , θL + π) and θ2 ∈ (θL + π, θL ). We divide the points in V \{v} into four sequences P1 (v), P2 (v), P3 (v), and P4 (v) by their polar angles with respect to v. P1 (v) consists of points with polar angles in (θL , θ1 ), P2 (v) in [θ1 , θL + π), P3 (v) in (θL + π, θ2 ], and P4 (v) in (θ2 , θL ), all sorted in CCW order. It follows that the four sequences satisfy conditions (c). Condition (a) and (b) hold for P1 (v) and P4 (v) from similar discussion as above. However, for any two distinct points vi , vj in P2 (v), we can observe that trL (vi \|v) is strictly below trL (vj \|v) if and only if vi precedes vj in P2 (v). Similarly, the argument holds for P3 (v). Thus, what satisfy condition (b) are actually the reverse sequences of P2 (v) and P3 (v), which can also be obtained in O(log n) time, satisfying condition (a). ⊓ ⊔ By Lemma 13.(c), searching in L × T r (v) is equivalent to searching in the O(1) sequences of breakpoints, which can be computed more efficiently than the obvious way. Besides, we can also obtain a symmetrical lemma constructing sequences for L × T l (v). In the following, we show how to perform a binary search within these sequences. Lemma 14 With an O(n2 log n)-time preprocessing, given an arbitrary line L, a local optimal point x∗L can be computed in O(n log2 n) time. The (1\|1)R -Centroid Problem on the Plane 13 Proof. By Lemma 12, the searching of x∗L can be done within L × T and L × C(V ). L × T can be further divided into L × T r (v) and L × T l (v) for each v ∈ V . By Lemma 13, these 2n sets can be replaced by O(n) sorted sequences of breakpoints on L. Besides, L × C(V ) consists of no more than 2n breakpoints, which can be computed and arranged into a sorted sequence in decreasing ycoordinates. Therefore, we can construct N0 = O(n) sequences P1 , P2 , · · · , PN0 of breakpoints, each of length O(n) and sorted in decreasing y-coordinates. The searching in the N0 sorted sequences is done by performing parametric search for parallel binary searches, introduced in [1]. The technique we used here is similar to the algorithm in [1], but uses a different weighting scheme. For each sorted sequence Pj , 1 ≤ j ≤ N0 , we first obtain its middle element xj , and associate xj with a weight mj equal to the number of elements in Pj . Then, we compute the weighted defined as the ele0 middle elements, P median [18] of the NP P ment x such that {m \|x is above x} ≥ m /2 and {m \|xj is below x} ≥ j j j j P mj /2. Finally, we apply Lemma 10 on the point x. If x is a strong (1\|1)R centroid, of course it is local optimal. If not, Assumption 1 holds and W edge(x) can be computed. If W edge(x) is sideward, a local optimal point x∗L = x is directly found. Otherwise, W edge(x) is either upward or downward, and thus all breakpoints on the opposite half can be pruned by Lemma 9. The pruning makes a portion of sequences, that possesses over half of total breakpoints by the definition of weighted median, lose at least a quarter of their elements. Hence, at least one-eighths of breakpoints are pruned. By repeating the above process, we can find x∗L in at most O(log n) iterations. The time complexity for finding x∗L is analyzed as follows. By Lemma 13, constructing sorted sequences for L × T r (v) and L × T l (v) for all v ∈ V takes O(n log n) time. Computing and sorting L × C(V ) also takes O(n log n) time. There are at most O(log n) iterations of the pruning process. At each iteration, the N0 middle elements and their weighted median x can be obtained in O(N0 ) = O(n) time by the linear-time weighted selection algorithm [18]. Then, the computation of W edge(x) takes O(n log n) time by Lemma 10. Finally, the pruning of those sequences can be done in O(n) time. In summary, the searching ⊓ ⊔ of x∗L requires O(n log n) + O(log n) × O(n log n) = O(n log2 n) time. We remark that, by Lemma 14, it is easy to obtain an intermediate result for the (1\|1)R -centroid problem on the plane. By Lemma 5, there exists a (1\|1)R centroid in T × T , T × C(V ), and C(V ) × C(V ). By applying Lemma 14 to the O(n2 ) lines in T , the local optimum among the intersection points in T × T and T × C(V ) can be obtained in O(n3 log2 n) time. By applying Theorem 1 on the O(n2 ) intersection points in C(V )× C(V ), the local optimum among them can be obtained in O(n3 log n) time. Thus, we can find a (1\|1)R -centroid in O(n3 log2 n) time, a nearly O(n2 ) improvement over the O(n5 log n) bound in [5]. 4 (1\|1)R -Centroid on the Plane In this section, we study the (1\|1)R -centroid problem and propose an improved algorithm of time complexity O(n2 log n). This algorithm is as efficient as the 14 Hung-I Yu, Tien-Ching Lin, and D. T. Lee best-so-far algorithm for the (1\|1)-centroid problem, but based on a completely different approach. In Subsection 4.1, we extend the algorithm of Lemma 14 to develop a procedure allowing us to prune candidate points with respect to a given vertical line. Then, in Subsection 4.2, we show how to compute a (1\|1)R -centroid in O(n2 log n) time based on this newly-developed pruning procedure. 4.1 Pruning with Respect to a Vertical Line Let L be an arbitrary vertical line on the plane. We call the half-plane strictly to the left of L the left plane of L and the one strictly to its right the right plane of L. A sideward wedge of some point on L is said to be rightward (leftward ) if it intersects the right (left) plane of L. We can observe that, if there is some point x ∈ L such that W edge(x) is rightward, every point x′ on the left plane of L can be pruned, since W ∗ (x′ ) ≥ W ∗ (x) by Lemma 9. Similarly, if W edge(x) is leftward, points on the right plane of L can be pruned. Although the power of wedges is not fully exerted in this way, pruning via vertical lines and sideward wedges is superior than directly via wedges due to predictable pruning regions. Therefore, in this subsection we describe how to design a procedure that enables us to prune either the left or the right plane of a given vertical line L. As mentioned above, the key point is the searching of sideward wedges on L. It is achieved by carrying out three conditional phases. In the first phase, we try to find some proper breakpoints with sideward wedges. If failed, we pick some representative point in the second phase and check its wedge to determine whether or not sideward wedges exist. Finally, in case of their nonexistence, we show that their functional alternative can be computed, called the pseudo wedge, that still allows us to prune the left or right plane of L. In the following, we develop a series of lemmas to demonstrate the details of the three phases. Property 1 Given a point x ∈ L, for each possible direction of W edge(x), the corresponding CA(x) satisfies the following conditions: Upward – CA(x) ⊆ (0, π), Downward – CA(x) ⊆ (π, 2π), Rightward – 0 ∈ CA(x), Leftward – π ∈ CA(x). Proof. When W edge(x) is upward, by definition the beginning angle θb and the ending angle θe of CA(x) must satisfy that both half-planes H(F (θb \|x), y(θb \|x)) and H(F (θe \|x), y(θe \|x)) include the half-line of L above x. It follows that 0 ≤ θb , θe ≤ π, and thus CA(x) ⊆ (0, π). (Recall that θb , θe ∈ / CA(x).) The case that W edge(x) is downward can be proved in a symmetric way. When W edge(x) is rightward, we can see that H(F (θb \|x), y(θb \|x)) must not contain the half-line of L above x, and thus π < θb < 2π. By similar arguments, 0 < θe < π. Therefore, counterclockwise covering angles from θb to θe , CA(x) must include the angle 0. The case that W edge(x) is leftward can be symmetrically proved. ⊓ ⊔ The (1\|1)R -Centroid Problem on the Plane 15 Lemma 15 Let x1 , x2 be two points on L, where x1 is strictly above x2 . For any angle 0 ≤ θ ≤ π, W (θ\|x1 ) ≤ W (θ\|x2 ). Symmetrically, for π ≤ θ ≤ 2π, W (θ\|x2 ) ≤ W (θ\|x1 ). Proof. For any angle 0 ≤ θ ≤ π, we can observe that H − (B(θ\|x1 ), y(θ\|x1 )) ⊂ H − (B(θ\|x2 ), y(θ\|x2 )), since x1 is strictly above x2 . It follows that W (θ\|x1 ) ≤ W (θ\|x2 ). The second claim also holds by symmetric arguments. ⊓ ⊔ Lemma 16 Let x be an arbitrary point on L. If W edge(x) is either upward or downward, for any point x′ ∈ L\W edge(x), W edge(x′ ) has the same direction as W edge(x). Proof. By symmetry, we prove that, if W edge(x) is upward, W edge(x′ ) is also upward for every x′ ∈ L strictly below x. By Property 1, the fact that W edge(x) is upward means that CA(x) ⊂ (0, π) and thus M A(x) ⊂ (0, π). Let x′ be a point on L strictly below x. By Lemma 15, we have that W (θ\|x′ ) ≥ W (θ\|x) for 0 < θ < π and W (θ\|x′ ) ≤ W (θ\|x) for π ≤ θ ≤ 2π. It follows that M A(x′ ) ⊂ (0, π) and CA(x′ ) ⊂ (0, π), so W edge(x′ ) is upward as well. ⊓ ⊔ Following from this lemma, if there exist two arbitrary points x1 and x2 on L with their wedges downward and upward, respectively, we can derive that x1 must be strictly above x2 , and that points with sideward wedges or even strong (1\|1)R -centroids can locate only between x1 and x2 . Thus, we can find sideward wedges between some specified downward and upward wedges. Let xD be the lowermost breakpoint on L with its wedge downward, xU the uppermost breakpoint on L with its wedge upward, and GDU the open segment xD xU \{xD , xU }. (For ease of discussion, we assume that both xD and xU exist on L, and show how to resolve this assumption later by constructing a bounded box.) Again, xD is strictly above xU . Also, we have the following corollary by their definitions. Corollary 17 If there exist breakpoints in the segment GDU , for any such breakpoint x, either x is a strong (1\|1)R -centroid or W edge(x) is sideward. Given xD and xU , the first phase can thus be done by checking whether there exist breakpoints in GDU and picking any of them if exist. Supposing that the picked one is not a strong (1\|1)R -centroid, a sideward wedge is found by Corollary 17 and can be used for pruning. Notice that, when there are two or more such breakpoints, one may question whether their wedges are of the same direction, as different directions result in inconsistent pruning results. The following lemma answers the question in the positive. Lemma 18 Let x1 , x2 be two distinct points on L, where x1 is strictly above x2 and none of them is a strong (1\|1)R -centroid. If W edge(x1 ) and W edge(x2 ) are both sideward, they are either both rightward or both leftward. Proof. We prove this lemma by contradiction. By symmetry, suppose the case that W edge(x1 ) is rightward and W edge(x2 ) is leftward. This case can be further divided into two subcases by whether or not CA(x1 ) and CA(x2 ) intersect. 16 Hung-I Yu, Tien-Ching Lin, and D. T. Lee Fig. 5. CH(C(S)) intersects L between x1 and x2 . Consider first that CA(x1 ) does not intersect CA(x2 ). Because W edge(x1 ) is rightward, 0 ∈ CA(x1 ) by Property 1. Thus, there exists an angle θ, 0 < θ < π, such that θ ∈ M A(x1 ). Since x1 is strictly above x2 , by Lemma 15 we have that W ∗ (x1 ) = W (θ\|x1 ) ≤ W (θ\|x2 ) ≤ W ∗ (x2 ). Furthermore, since W edge(x2 ) is leftward, we can see that x1 ∈ / W edge(x2 ) and therefore W ∗ (x1 ) ≥ ∗ W (x2 ) by Lemma 9. It follows that W (θ\|x2 ) = W ∗ (x2 ) and thus θ ∈ M A(x2 ). By definition, M A(x1 ) ⊆ CA(x1 ) and M A(x2 ) ⊆ CA(x2 ), which implies that CA(x1 ) and CA(x2 ) intersect at θ, contradicting the subcase assumption. When CA(x1 ) intersects CA(x2 ), their intersection must be completely included in either (0, π) or (π, 2π) due to Assumption 1. By symmetry, we assume the latter subcase. Using similar arguments as above, we can find an angle θ′ , where 0 < θ′ < π, such that θ′ ∈ M A(x1 ) and θ′ ∈ M A(x2 ). This is a contradiction, since θ′ ∈ / (π, 2π). Since both subcases do not hold, the lemma is proved. ⊓ ⊔ The second phase deals with the case that no breakpoint exists between xD and xU by determining the wedge direction of an arbitrary inner point in GDU . We begin with several auxiliary lemmas. Lemma 19 Let x1 , x2 be two distinct points on L such that W ∗ (x1 ) = W ∗ (x2 ) and x1 is strictly above x2 . There exists at least one breakpoint in the segment (a) x1 x2 \{x2 }, if M A(x2 ) intersects (0, π) but M A(x1 ) does not, (b) x1 x2 \{x1 }, if M A(x1 ) intersects (π, 2π) but M A(x2 ) does not. Proof. By symmetry, we only show the correctness of condition (a). From its assumption, there exists an angle θ, where 0 < θ < π, such that θ ∈ M A(x2 ). Let T S = V H − (B(θ\|x2 ), y(θ\|x2 )). By definition, we have that W (S) = W (θ\|x2 ) = W ∗ (x2 ) = W ∗ (x1 ) and CH(C(S)) ⊂ H − (F (θ\|x2 ), y(θ\|x2 )), which implies that CH(C(S)) is strictly above F (θ\|x2 ). (See Figure 5.) We first claim that CH(C(S)) intersects L. If not, there must exist an angle θ′ , where 0 < θ′ < π, such that CH(C(S)) ⊂ H − (F (θ′ \|x1 ), y(θ′ \|x1 )), that is, The (1\|1)R -Centroid Problem on the Plane 17 S ⊂ H − (B(θ′ \|x1 ), y(θ′ \|x1 )). By definition, W ∗ (x1 ) ≥ W (θ′ \|x1 ) ≥ W (S). Since W ∗ (x1 ) = W (S), W (θ′ \|x1 ) = W ∗ (x1 ) and thus θ′ ∈ M A(x1 ), which contradicts the condition that M A(x1 ) does not intersect (0, π). Thus, the claim holds. When CH(C(S)) intersects L, x1 locates either inside or outside CH(C(S)). Since x2 locates outside CH(C(S)), in the former case the boundary of CH(C(S)) intersects x1 x2 \{x2 } and forms a breakpoint, thereby proves condition (a). On the other hand, if x1 is outside CH(C(S)), again there exists an angle θ′′ such that CH(C(S)) ⊂ H − (F (θ′′ \|x1 ), y(θ′′ \|x1 )). By similar arguments, we can show that θ′′ ∈ M A(x1 ). By assumption, θ′′ must belong to (π, 2π), which implies that CH(C(S)) is strictly below F (θ′′ \|x1 ). Since CH(C(S)) is strictly above F (θ\|x2 ) as mentioned, any intersection point between CH(C(S)) and L should be inner ⊓ ⊔ to x1 x2 . Therefore, the lemma holds. Lemma 20 Let G be a line segment connecting two consecutive breakpoints on L. For any two distinct points x1 , x2 inner to G, W ∗ (x1 ) = W ∗ (x2 ). Proof. Suppose to the contrary that W ∗ (x1 ) 6= W ∗ (x2 ). By Lemma 11, there exists at least one breakpoint in x1 x2 , which contradicts the definition of G. Thus, the lemma holds. ⊓ ⊔ Lemma 21 When there is no breakpoint between xD and xU , any two distinct points x1 , x2 in GDU have the same wedge direction, if they are not strong (1\|1)R centroids. Proof. Suppose by contradiction that the directions of their wedges are different. By Lemmas 16 and 18, there are only two possible cases. (1) W edge(x1 ) is downward, and W edge(x2 ) is either sideward or upward. (2) W edge(x1 ) is sideward, and W edge(x2 ) is upward. In the following, we show that both cases do not hold. Case (1): Because W edge(x1 ) is downward, we have that CA(x1 ) ⊆ (π, 2π) by Property 1 and thus M A(x1 ) does not intersect (0, π). On the other hand, whether W edge(x2 ) is sideward or upward, we can see that CA(x2 ) and M A(x2 ) intersect (0, π) by again Property 1. Since W ∗ (x1 ) = W ∗ (x2 ) by Lemma 20, the status of the two points satisfies the condition (a) of Lemma 19, so at least one breakpoint exists between x1 and x2 . By definitions of x1 and x2 , this breakpoint is inner to GDU , thereby contradicts the assumption. Therefore, Case (1) does not hold. Case (2): The proof of Case (2) is symmetric to that of Case (1). The condition (b) of Lemma 19 can be applied similarly to show the existence of at least one breakpoint between x1 and x2 , again a contradiction. Combining the above discussions, we prove that the wedges of x1 and x2 are of the same direction, thereby completes the proof of this lemma. ⊓ ⊔ 18 Hung-I Yu, Tien-Ching Lin, and D. T. Lee This lemma enables us to pick an arbitrary point in GDU , e.g., the bisector point xB of xD and xU , as the representative of all inner points in GDU . If xB is not a strong (1\|1)R -centroid and W edge(xB ) is sideward, the second phase finishes with a sideward wedge found. Otherwise, if W edge(xB ) is downward or upward, we can derive the following and have to invoke the third phase. Lemma 22 If there is no breakpoint between xD and xU and W edge(xB ) is not sideward, there exist neither strong (1\|1)R -centroids nor points with sideward wedges on L. Proof. By Lemma 16, this lemma holds for points not in GDU . Without loss of generality, suppose that W edge(xB ) is downward. For all points in GDU above xB , the lemma holds by again Lemma 16. Consider an arbitrary point x ∈ xB xU \{xB , xU }. We first show that x is not a strong (1\|1)R -centroid. Suppose to the contrary that x really is. By definition, we have that δ(CA(x)) > π, and thus CA(x) and M A(x) intersect (0, π). On the other hand, CA(xB ) and M A(xB ) do not intersect (0, π) due to downward W edge(xB ) and Property 1. Since W ∗ (xB ) = W ∗ (x) by Lemma 20, applying the condition (a) of Lemma 19 to xB and x shows that at least one breakpoint exists between them, which contradicts the no-breakpoint assumption. Now that x is not a strong (1\|1)R -centroid, it must have a downward wedge, as xB does by Lemma 21. Therefore, the lemma holds for all points on L. ⊓ ⊔ When L satisfies Lemma 22, it consists of only points with downward or upward wedges, and is said to be non-leaning. Obviously, our pruning strategy via sideward wedges could not apply to such non-leaning lines. The third phase overcomes this obstacle by constructing a functional alternative of sideward wedges, called the pseudo wedge, on either xD or xU , so that pruning with respect to L is still achievable. Again, we start with auxiliary lemmas. Lemma 23 If L is non-leaning, the following statements hold: (a) W ∗ (xD ) 6= W ∗ (xU ), (b) W ∗ (x) = max{W ∗ (xD ), W ∗ (xU )} for all points x ∈ GDU . Proof. We prove the correctness of statement (a) by contradiction, and suppose that W ∗ (xD ) = W ∗ (xU ). Besides, the fact that L is non-leaning implies that no breakpoint exists in GDU . By Lemmas 22 and 21, the wedges of all points in GDU are of the same direction, either downward or upward. Suppose the downward case by symmetry, and pick an arbitrary point in GDU , say, xB . Since W edge(xB ) is downward, we have that M A(xB ) does not intersect (0, π). Oppositely, by definition W edge(xU ) is upward, so CA(xU ) and M A(xU ) are included in (0, π). Because xB is strictly above xU and W ∗ (xD ) = W ∗ (xU ), according to the condition (a) of Lemma 19, there exists at least one breakpoint in xB xU \{xU }, which is a contradiction. Therefore, statement (a) holds. The proof of statement (b) is also done by contradiction. By symmetry, assume that W ∗ (xD ) > W ∗ (xU ) in statement (a). Consider an arbitrary point The (1\|1)R -Centroid Problem on the Plane 19 x ∈ GDU . By Lemma 7, we have that W ∗ (x) ≤ max{W ∗ (xD ), W ∗ (xU )} = W ∗ (xD ). Suppose that the equality does not hold. Then, by Lemma 11, at least one breakpoint exists in the segment xD x\{xD }, contradicting the no-breakpoint fact. Thus, W ∗ (x) = W ∗ (xD ) and statement (b) holds. ⊓ ⊔ Let W1 = max{W ∗ (xD ), W ∗ (xU )}. We are going to define the pseudo wedge on either xU or xD , depending on which one has the smaller weight loss. We consider first the case that W ∗ (xD ) > W ∗ (xU ), and obtain the following. Lemma 24 If L is non-leaning and W ∗ (xD ) > W ∗ (xU ), there exists one angle θ for xU , where π ≤ θ ≤ 2π, such that W (H(B(θ\|xU ), y(θ\|xU ))) ≥ W1 . Proof. We first show that there exists at least a subset S ⊆ V with W (S) = W1 , such that xU locates on the upper boundary of CH(C(S)). Let x be the point strictly above but arbitrarily close to xU on L. By Lemma 23, W ∗ (x) = W1 , hence W ∗ (x) > W ∗ (xU ) by case assumption. It follows that xU ∈ W edge(x) by Lemma 9 and W edge(x) must be downward. By Property 1, we have that CA(x) ⊆ (π, 2π). Thus, there exists an angle θ′ ∈ M A(x), where π < θ′ < 2π, ′ such that W (H − \|x), y(θ′ \|x))) = W (θ′ \|x) = W1 . T(B(θ − Let S = V H (B(θ′ \|x), y(θ′ \|x)). Since W ∗ (xU ) < W1 = W (S), xU is inside CH(C(S)) by Lemma 2. Oppositely, by the definition of S, x is outside the convex hull CH(C(S)). It implies that xU is the topmost intersection point between CH(C(S)) and L, hence on the upper boundary of CH(C(S)). (It is possible that xU locates at the leftmost or the rightmost point of CH(C(S)).) The claimed angle θ is obtained as follows. Since xU is a boundary point of CH(C(S)), there exists a line F passing through xU and tangent to CH(C(S)). Let θ be the angle satisfying that F (θ\|xU ) = F , π ≤ θ ≤ 2π, and CH(C(S)) ⊂ H(F (θ\|xU ), y(θ\|xU )). Obviously, we have that S ⊂ H(B(θ\|xU ), y(θ\|xU )) and thus W (H(B(θ\|xU ), y(θ\|xU ))) ≥ W ∗ (S) = W1 . ⊓ ⊔ Let θU be an arbitrary angle satisfying the conditions of Lemma 24. We apply the line F (θU \|xU ) for trimming the region of W edge(xU ), so that a sideward wedge can be obtained. Let P W (xU ), called the pseudo wedge of xU , denote the intersection of W edge(xU ) and H(F (θU \|xU ), y(θU \|xU )). Deriving from the three facts that W edge(xU ) is upward, δ(W edge(xU )) < π, and π ≤ θU ≤ 2π, we can observe that either P W (xU ) is xU itself, or it intersects only one of the right and left plane of L. In the two circumstances, P W (xU ) is said to be null or sideward, respectively. The pseudo wedge has similar functionality as wedges, as shown in the following corollary. Corollary 25 For any point x′ ∈ / P W (xU ), W ∗ (x′ ) ≥ W ∗ (xU ). Proof. If x′ ∈ / W edge(xU ), the lemma directly holds by Lemma 9. Otherwise, we have that x′ ∈ / H(F (θ\|xU ), y(θ\|xU )) and thus H(B(θ\|x′ ), y(θ\|x′ )) contains H(B(θ\|xU ), y(θ\|xU )). Then, by Lemma 24, W ∗ (x′ ) ≥ W (H(B(θ\|x′ ), y(θ\|x′ ))) ≥ W (H(B(θ\|xU ), y(θ\|xU ))) ≥ W1 , thereby completes the proof. ⊓ ⊔ 20 Hung-I Yu, Tien-Ching Lin, and D. T. Lee By this lemma, if P W (xU ) is found to be sideward, points on the opposite half-plane with respect to L can be pruned. If P W (xU ) is null, xU becomes another kind of strong (1\|1)R -centroids, in the meaning that it is also an immediate solution to the (1\|1)R -centroid problem. Without confusion, we call xU a conditional (1\|1)R -centroid in the latter case. On the other hand, considering the reverse case that W ∗ (xD ) < W ∗ (xU ), we can also obtain an angle xD and a pseudo wedge P W (xD ) for xD by symmetric arguments. Then, either P W (xD ) is sideward and the opposite side of L can be pruned, or P W (xD ) itself is a conditional (1\|1)R -centroid. Thus, the third phase solves the problem of the nonexistence of sideward wedges. Recall that the three phases of searching sideward wedges is based on the existence of xD and xU on L, which was not guaranteed before. Here we show that, by constructing appropriate border lines, we can guarantee the existence of xD and xU while searching between these border lines. The bounding box is defined as the smallest axis-aligned rectangle that encloses all circles in C(V ). Obviously, any point x outside the box satisfies that W ∗ (x) = W (V ) and must not be a (1\|1)R -centroid. Thus, given a vertical line not intersecting the box, the half-plane to be pruned is trivially decided. Moreover, let Ttop and Tbtm be two arbitrary horizontal lines strictly above and below the bounding box, respectively. We can obtain the following. Lemma 26 Let L be an arbitrary vertical line intersecting the bounding box, and x′D and x′U denote its intersection points with Ttop and Tbtm , respectively. W edge(x′D ) is downward and W edge(x′U ) is upward. Proof. Consider the case about W edge(x′D ). As described above, we know the fact that W ∗ (xD ) = W (V ). Let θ be an arbitrary angle with 0 ≤ θ ≤ π. We can observe that H − (F (θ\|x′D ), y(θ\|x′D )) cannot contain all circles in C(V ), that is, V 6⊂ H − (B(θ\|x′D ), y(θ\|x′D )). This implies that W (θ\|x′D ) < W ∗ (x′D ) and θ ∈ / M A(x′D ). Therefore, we have that M A(x′D ) ⊂ (π, 2π) and W edge(x′D ) is downward by Property 1. By similar arguments, we can show that W edge(x′U ) is upward. Thus, the lemma holds. ⊓ ⊔ According to this lemma, by inserting Ttop and Tbtm into T , the existence of xD and xU is enforced for any vertical line intersecting the bounding box. Besides, it is obvious to see that the insertion does not affect the correctness of all lemmas developed so far. Summarizing the above discussion, the whole picture of our desired pruning procedure can be described as follows. In the beginning, we perform a preprocessing to obtain the bounding box and then add Ttop and Tbtm into T . Now, given a vertical line L, whether to prune its left or right plane can be determined by the following steps. 1. If L does not intersect the bounding box, prune the half-plane not containing the box. 2. Compute xD and xU on L. The (1\|1)R -Centroid Problem on the Plane 21 3. Find a sideward wedge or pseudo wedge via three forementioned phases. (Terminate whenever a strong or conditional (1\|1)R -centroid is found.) (a) If breakpoints exist between xD and xU , pick any of them and check it. (b) If no such breakpoint, decide whether L is non-leaning by checking xB . (c) If L is non-leaning, compute P W (xU ) or P W (xD ) depending on which of xU and xD has smaller weight loss. 4. Prune the right or left plane of L according to the direction of the sideward wedge or pseudo wedge. The correctness of this procedure follows from the developed lemmas. Any vertical line not intersecting the bounding box is trivially dealt with in Step 1, due to the property of the box. When L intersects the box, by Lemma 26, xD and xU can certainly be found in Step 2. The three sub-steps of Step 3 correspond to the three searching phases. When L is not non-leaning, a sideward wedge is found, either at some breakpoint between xD and xU in Step 3(a) by Corollary 17, or at xB in Step 3(b) by Lemma 21. Otherwise, according to Lemma 24 or its symmetric version, a pseudo wedge can be built in Step 3(c) for xU or xD , respectively. Finally in Step 4, whether to prune the left or right plane of L can be determined via the just-found sideward wedge or pseudo wedge, by respectively Lemma 9 or Corollary 25. The time complexity of this procedure is analyzed as follows. The preprocessing for computing the bounding box trivially takes O(n) time. In Step 1, any vertical line not intersecting the box can be identified and dealt with in O(1) time. Finding xD and xU in Step 2 requires the help of the binary-search algorithm developed in 3.2. Although the algorithm is designed to find a local optimal point, we can easily observe that slightly modifying its objective makes it applicable to this purpose without changing its time complexity. Thus, Step 2 can be done in O(n log2 n) time by Lemma 14. In Step 3(a), all breakpoints between xD and xU can be found in O(n log n) time as follows. As done in Lemma 14, we first list all breakpoints on L by O(n) sorted sequences of length O(n), which takes O(n log n) time. Then, by performing binary search with the y-coordinates of xD and xU , we can find within each sequence the breakpoints between them in O(log n) time. In Step 3(a) or 3(b), checking a picked point x is done by computing CA(x), that requires O(n log n) time by Lemma 10. To compute the pseudo wedge in Step 3(c), the angle θU satisfying Lemma 24, or symmetrically θD , can be computed in O(n log n) time by sweeping technique as in Lemma 10. Thus, P W (xU ) or P W (xD ) can be computed in O(n log n) time. Finally, the pruning decision in Step 4 takes O(1) time. Summarizing the above, these steps require O(n log2 n) time in total. Since the invocation of Lemma 14 needs an additional O(n2 log n)-time preprocessing, we have the following result. Lemma 27 With an O(n2 log n)-time preprocessing, whether to prune the right or left plane of a given vertical line L can be determined in O(n log2 n) time. 22 4.2 Hung-I Yu, Tien-Ching Lin, and D. T. Lee Searching on the Euclidean Plane In this subsection, we come back to the (1\|1)R -centroid problem. Recall that, by Lemma 5, at least one (1\|1)R -centroid can be found in the three sets of intersection points T × T , C(V ) × T , and C(V ) × C(V ), which consist of total O(n4 ) points. Let L denote the set of all vertical lines passing through these O(n4 ) intersection points. By definition, there exists a vertical line L∗ ∈ L such that its local optimal point is a (1\|1)R -centroid. Conceptually, with the help of Lemma 27, L∗ can be derived by applying prune-and-search approach to L: pick the vertical line L from L with median x-coordinates, determine by Lemma 27 whether the right or left plane of L should be pruned, discard lines of L in the pruned half-plane, and repeat above until two vertical lines left. Obviously, it costs too much if this approach is carried out by explicitly generating and sorting the O(n4 ) lines. However, by separately dealing with each of the three sets, we can implicitly maintain sorted sequences of these lines and apply the prune-and-search approach. Let LT , LM , and LC be the sets of all vertical lines passing through the intersection points in T × T , C(V ) × T , and C(V ) × C(V ), respectively. A local optimal line of LT is a vertical line L∗t such that its local optimal point has weight loss no larger than those of points in T × T . The local optimal lines L∗m and L∗c can be similarly defined for LM and LC , respectively. We will adopt different prune-and-search techniques to find the local optimal lines in the three sets, as shown in the following lemmas. Lemma 28 A local optimal line L∗t of LT can be found in O(n2 log n) time. Proof. Let N1 = \|T \|. By definition, there are (N1 )2 intersection points in T × T and (N1 )2 vertical lines in LT . For efficiently searching within these vertical lines, we apply the ingenious idea of parametric search via parallel sorting algorithms, proposed by Megiddo [15]. Consider two arbitrary lines Tg , Th ∈ T . If they are not parallel, let tgh be their intersection point and Lgh be the vertical line passing through tgh . Suppose that Tg is above Th in the left plane of Lgh . If applying Lemma 27 to Lgh prunes its right plane, Tg is above Th in the remained left plane. On the other hand, if the left plane of Lgh is pruned, Th is above Tg in the remained right plane. Therefore, Lgh can be treated as a “comparison” between Tg and Th , in the sense that applying Lemma 27 to Lgh determines their ordering in the remained half-plane. It also decides the ordering of their intersection points with the undetermined local optimal line L∗t , since the pruning ensures that a local optimal line stays in the remained half-plane. It follows that, by resolving comparisons, the process of pruning vertical lines in LT to find L∗t can be reduced to the problem of determining the ordering of the intersection points of the N1 lines with L∗t , or say, the sorting of these intersection points on L∗t . While resolving comparisons during the sorting process, we can simultaneously maintain the remained half-plane by two vertical lines as its boundaries. Thus, after resolving all comparisons in LT , one of the two boundaries must be a local optimal line. As we know, the most efficient way to The (1\|1)R -Centroid Problem on the Plane 23 obtain the ordering is to apply some optimal sorting algorithm AS , which needs to resolve only O(N1 log N1 ) comparisons, instead of (N1 )2 comparisons. Since resolving each comparison takes O(n log2 n) time by Lemma 27, the sorting is done in O(n log2 n) × O(N1 log2 N1 ) = O(n3 log3 n) time, so is the finding of L∗t . However, Megiddo [15] observed that, when multiple comparisons can be indirectly resolved in a batch, simulating parallel sorting algorithms in a sequential way naturally provides the scheme for batching comparisons, thereby outperform the case of applying AS . Let AP be an arbitrary cost-optimal parallel sorting algorithm that runs in O(log n) steps on O(n) processors, e.g., the parallel merge sort in [2]. Using AP to sort the N1 lines in LT on L∗t takes O(log N1 ) parallel steps. At each parallel step, there are k = O(N1 ) comparisons L1 , L2 , · · · , Lk to be resolved. We select the one with median x-coordinate among them, which is supposed to be some Li . If applying Lemma 27 to Li prunes its left plane, for each comparison Lj to the left of Li , the ordering of the corresponding lines of Lj in the remained right plane of Li is directly known. Thus, the O(k/2) comparisons to the left of Li are indirectly resolved in O(k/2) time. If otherwise the right plane of Li is pruned, the O(k/2) comparisons to its right are resolved in O(k/2) time. By repeating this process of selecting medians and pruning on the remaining elements O(log k) times, all k comparisons can be resolved, which takes O(n log2 n) × O(log k) + O(k + k/2 + k/4 + · · · ) = O(n log3 n) + O(N1 ) = O(n2 ) time. Therefore, going through O(log N1 ) parallel steps of AP requires O(n2 log N1 ) = O(n2 log n) time, which determines the ordering of lines in LT ⊓ ⊔ on L∗t and also computes a local optimal line L∗t . Lemma 29 A local optimal line L∗m of LM can be found in O(n2 log n) time. Proof. To deal with the set LM , we use the ideas similar to the proofs of Lemmas 13 and 14 in order to divide C(V ) × T into sorted sequences of points. Given a fixed circle C = Cγ (u0 ) for some point u0 ∈ V , we show that the intersection points in C × T r (v) and C × T l (v) for each v ∈ V can be grouped into O(1) sequences of length O(n), which are sorted in increasing x-coordinates. Summarizing over all circles in C(V ), there will be total O(n2 ) sequences of length O(n), each of which maps to a sequence of O(n) vertical lines sorted in increasing x-coordinates. Then, finding a local optimal line L∗m can be done by performing prune-and-search to the O(n2 ) sequences of vertical lines via parallel binary searches. The details of these steps are described as follows. First we discuss about the way for grouping intersection points in C × T r (v) and C ×T l (v) for a fixed point v ∈ V , so that each of them can be represented by O(1) subsequences of P (v). By symmetry, only C × T r (v) is considered. Similar to Lemma 13, we are actually computing sequences of points in V \{v} corresponding to these intersection points. For each vi ∈ V \{v}, the outer tangent line T r (vi \|v) may intersect C at two, one, or zero point. Let trC,1 (vi \|v) and trC,2 (vi \|v) denote the first and second points, respectively, at which T r (vi \|v) intersects C along the direction from v to vi . Note that, when T r (vi \|v) intersects C at less than two points, trC,2 (vi \|v) or both of them will be null. In the following, we consider the sequence computation under two cases about the relationship between u0 and v, (1) u0 = v, and (2) u0 6= v. 24 Hung-I Yu, Tien-Ching Lin, and D. T. Lee (a) no intersection (b) two intersection points Fig. 6. Two subcases about how Cγ (v) intersects C. Case (1): Since v coincides with u0 , C × T r (v) is just the set of tangent points trC,1 (vi \|v) for all vi ∈ V \{v}. It is easy to see the the angular sorted sequence P (v) directly corresponds to a sorted sequence of these n − 1 tangent points in CCW order. P (v) can be further partitioned into two sub-sequences P1 (v) and P2 (v), which consist of points in V \{v} with polar angles (with respect to v) in the intervals [0, π) and [π, 2π), respectively. Since they are sorted in CCW order, we have that intersection points corresponding to P2 (v) and to the reverse of P1 (v) are sorted in increasing x-coordinates, as we required. Obviously, P1 (v) and P2 (v) are of length O(n) and can be obtained in O(log n) time. Case (2): Suppose without loss of generality that v locates on the lower left quadrant with respect to u0 , and let θ0 be the polar angle of u0 with respect to v. This case can be further divided into two subcases by whether or not Cγ (v) intersects C at less than two points. Consider first the subcase that they intersect at none or one point (see Figure 6(a).) Let θ3 and θ4 be the angles such that T r (y(θ3 \|v)\|v) and T r (y(θ4 \|v)\|v) are inner tangent to Cγ (v) and C, where θ3 ≤ θ4 (Note that θ3 = θ4 only when the two circles intersect at one point.) For each vi ∈ V \{v}, T r (vi \|v) does not intersect C, if the polar angle of vi with respect to v is neither in [θ0 , θ3 ] nor in [θ4 , θ0 + π]. We can implicitly obtain from P (v) two subsequences P3 (v) and P4 (v), consisting of points with polar angles in [θ0 , θ3 ] and in [θ4 , θ0 + π], respectively. It can be observed that the sequence of points vi listed in P3 (v) corresponds to a sequence of intersection points trC,1 (vi \|v) listed in clockwise (CW) order on C and, moreover, a sequence of trC,2 (vi \|v) listed in CCW order on C. Symmetrically, the sequence of points vj in P4 (v) corresponds to a sequence of trC,1 (vj \|v) in CCW order and a sequence of trC,2 (vj \|v) in CW order. The four implicit sequences of intersection points on C can be further partitioned by a horizontal line Lh passing through The (1\|1)R -Centroid Problem on the Plane 25 its center u0 , so that the resulted sequences are naturally sorted in either increasing or decreasing x-coordinates. Therefore, we can implicitly obtain at most eight sorted sequences of length O(n) in replace of C × T r (v), by appropriately partitioning P (v) in O(log n) time. Consider that Cγ (v) intersects Cγ (u0 ) at two points c5 and c6 , where c5 is to the upper right of c6 (see Figure 6(b).) Let θ5 and θ6 be the angles such that T r (y(θ5 \|v)\|v) and T r (y(θ6 \|v)\|v) are tangent to Cγ (v) at c5 and c6 , respectively. Again, P (v) can be implicitly partitioned into three subsequences P5 (v), P6 (v), and P7 (v), which consists of points with polar angles in [θ0 , θ5 ), [θ5 , θ6 ), and [θ6 , θ0 + π], respectively. By similar observations, P5 (v) corresponds to two sequences of intersection points listed in CW and CCW order, respectively, and P7 (v) corresponds to two sequences listed in CCW and CW order, respectively. However, the sequence of points vi in P6 (v) corresponds to the sequences of trC,1 (vi \|v) and trC,2 (vi \|v) listed in both CCW order. These sequences can also be partitioned by Lh into sequences sorted in x-coordinates. It follows that we can implicitly obtain at most twelve sorted sequences of length O(n) in replace of C × T r (v) in O(log n) time. According to the above discussion, for any two points u, v ∈ V , Cγ (u)×T r (v) and Cγ (u) × T l (v) can be divided into O(1) sequences in O(log n) time, each of which consists of O(n) intersection points on Cγ (u) sorted in increasing xcoordinates. Thus, C(V ) × T can be re-organized as O(n2 ) sorted sequences of length O(n) in O(n2 log n) time, which correspond to O(n2 ) sorted sequences of O(n) vertical lines. Now, we can perform parametric search for parallel binary search to these sequences of vertical lines, by similar techniques used in Lemma 14. For each of the O(n2 ) sequences, its middle element is first obtained and assigned with a weight equal to the sequence length in O(1) time. Then, the weighted median L of these O(n2 ) elements are computed in O(n2 ) time [18]. By applying Lemma 27 to L in O(n log2 n) time, at least one-eighths of total elements can be pruned from these sequences, taking another O(n2 ) time. Therefore, a single iteration of pruning requires O(n2 ) time. After O(log n) such iterations, a local optimal line L∗m can be found in total O(n2 log n) time, thereby proves the lemma. ⊓ ⊔ Lemma 30 A local optimal line L∗c of LC can be found in O(n2 log n) time. Proof. There are at most O(n2 ) points in C(V )×C(V ). Thus, LC can be obtained and sorted according to x-coordinates in O(n2 log n) time. Then, by simply performing binary search with Lemma 27, a local optimal line L∗c can be easily found in O(log n) iterations of pruning, which require total O(n log3 n) time. In summary, the computation takes O(n2 log n) time, and the lemma holds. ⊓ ⊔ By definition, L∗ can be found among L∗t , L∗m , and L∗c , which can be computed in O(n2 log n) time by Lemmas 28, 29, and 30, respectively. Then, a (1\|1)R centroid can be computed as the local optimal point of L∗ in O(n log2 n) time by Lemma 14. Combining with the O(n2 log n)-time preprocessing for computing the angular sorted sequence P (v)s and the bounding box enclosing C(V ), we have the following theorem. 26 Hung-I Yu, Tien-Ching Lin, and D. T. Lee Theorem 3 The (1\|1)R -centroid problem can be solved in O(n2 log n) time. 5 Concluding Remarks In this paper, we revisited the (1\|1)-centroid problem on the Euclidean plane under the consideration of minimal distance constraint between facilities, and proposed an O(n2 log n)-time algorithm, which close the bound gap between this problem and its unconstrained version. Starting from a critical observation on the medianoid solutions, we developed a pruning tool with indefinite region remained after pruning, and made use of it via multi-level structured parametric search approach, which is quite different to the previous approach in [5,11]. Considering distance constraint between facilities in various competitive facility location models is both of theoretical interest and of practical importance. However, similar constraints are rarely seen in the literature. It would be good starting points by introducing the constraint to the facilities between players in the (r\|Xp )-medianoid and (r\|p)-centroid problems, maybe even to the facilities between the same player. References 1. R. Cole, “Slowing down sorting networks to obtain faster sorting algorithms,” Journal of the ACM, vol. 34, no. 1, pp. 200–208, 1987. 2. R. Cole, “Parallel merge sort,” SIAM Journal on Computing, vol. 17, no. 4, pp. 770–785, 1988. 3. A. Dasci, “Conditional Location Problems on Networks and in the Plane,” In: H. A. Eiselt, V. Marianov (eds.), Foundations of Location Analysis, Springer, New York, pp. 179–206, 2011. 4. I. Davydov, Y. Kochetov, and A. Plyasunov, “On the complexity of the (r\|p)centroid problem in the plane,” TOP, vol. 22, no. 2, pp. 614–623, 2013. 5. Z. Drezner, “Competitive location strategies for two facilities,” Regional Science and Urban Economics, vol 12, no. 4, pp. 485–493, 1982. 6. Z. Drezner, and Z. Eitan, “Competitive location in the plane,” Annals of Operations Research, vol. 40, no. 1, pp. 173–193, 1992. 7. H. A. Eiselt and G. Laporte, “Sequential location problems,” European Journal of Operational Research, vol. 96, pp. 217–231, 1997. 8. H. A. Eiselt, G. Laporte, and J.-F. Thisse, “Competitive location models: a framework and bibliography,” Transportation Science, vol. 27, pp. 44–54, 1993. 9. H. A. Eiselt, V. Marianov, Vladimir and T. Drezner, Tammy, “Competitive Location Models,” In: G. Laporte, S. Nickel, F. Saldanha da Gama (eds.), Location Science, Springer International Publishing, pp. 365–398, 2015. 10. S. L. Hakimi, “On locating new facilities in a competitive environment,” European Journal of Operational Research, vol. 12, no. 1, pp. 29–35, 1983. 11. S. L. Hakimi, “Locations with spatial interactions: competitive locations and games,” In: P.B. Mirchandani, R.L. Francis (eds), Discrete location theory, Wiley, New York, pp. 439–478, 1990. 12. P. Hansen, J.-F. Thisse, and R. W. Wendell, “Equilibrium analysis for voting and competitive location problems,” In: P. B. Mirchandani, R. L. Francis RL (eds), Discrete location theory, Wiley, New York, pp 479–501, 1990. The (1\|1)R -Centroid Problem on the Plane 27 13. H. Hotelling, “Stability in competition,” Economic Journal, vol. 39, 41–57, 1929. 14. D. T. Lee, Y. F. Wu, “Geometric complexity of some location problems,” Algorithmica, vol. 1, no. 1, pp. 193–211, 1986. 15. N. Megiddo, “Applying parallel computation algorithms in the design of serial algorithms,” Journal of the ACM, vol. 30, no. 4, pp. 852–865, 1983. 16. N. Megiddo, “Linear-time algorithms for linear programming in R3 and related problems,” SIAM Journal on Computing, vol. 12, no. 4, pp. 759–776, 1983. 17. F. Plastria, “Static competitive facility location: an overview of optimisation approaches.” European Journal of Operational Research, vol. 129, no. 3, pp. 461–470, 2001. 18. A. Reiser, “A linear selection algorithm for sets of elements with weights,” Information Processing Letters, vol. 7, no. 3, pp. 159–162, 1978. 19. D. R. Santos-Peñate, R. Suárez-Vega, and P. Dorta-González, “The leader–follower location model,” Networks and Spatial Economics, vol. 7, no. 1, pp. 45–61, 2007.	8
HYBRID FUEL CELLS POWER FOR LONG DURATION ROBOT MISSIONS IN FIELD ENVIRONMENTS Jekanthan Thangavelautham1, Danielle Gallardo2, Daniel Strawser1, Steven Dubowsky1 1 Mechanical Engineering Department, Massachusetts Institute of Technology 77 Massachusetts Ave., Cambridge, MA, 02139, {jekan, dstrawse, dubowsky}@mit.edu 2 Department of Mechanical, Aerospace and Nuclear Engineering, Rensselaer Polytechnic Institute, 110th 8th Jonsson Engineering Center, Troy New York, 12180 Mobile robots are often needed for long duration missions. These include search rescue, sentry, repair, surveillance and entertainment. Current power supply technology limit walking and climbing robots from many such missions. Internal combustion engines have high noise and emit toxic exhaust while rechargeable batteries have low energy densities and high rates of self-discharge. In theory, fuel cells do not have such limitations. In particular Proton Exchange Membrane (PEMs) can provide very high energy densities, are clean and quiet. However, PEM fuel cells are found to be unreliable due to performance degradation. This can be mitigated by protecting the fuel cell in a fuel-cell battery hybrid configuration using filtering electronics that ensure the fuel cell is isolated from electrical noise and a battery to isolate it from power surges. Simulation results are presented for a HOAP 2 humanoid robot that suggests a fuel cell powered hybrid power supply superior to conventional batteries. I. Introduction Mobile robots, including walking robots are needed to perform long duration missions that are difficult, dangerous and tedious. These include search and rescue, repair, entertainment, sentry, surveillance applications [1, 2]. Continuous operation of these robots, lasting days and weeks (not hours) would be ideal for these applications. Typical power demands for field robots will vary significantly during a mission, often with high peak power demands. These field systems often have constraints on their mass, volume and noise. Current power supply technology is a key limiting factor for long duration field robotic applications. Internal combustion engines can provide high power for long durations but produce toxic exhaust, noise and strong thermal signatures making them inappropriate for many important applications. Current rechargeable batteries have very low energy densities and high rates of selfdischarge, requiring systems to stop and recharge every few hours, making them 1 2 ineffective for continuous long duration missions. Hence, there is a significant need for a power supply that can provide the high total energy required for long duration missions that is quiet and clean. Figure 1: (Left) Boston Dynamics Big Dog, a four-legged supply robot. (Right) Robonaut 2 repair robot. II. Fuel Cell Power for Mobile Robots Fuel cells are high energy sources of power that have been suggested for robots [7, 8]. They are a promising alternative mobile source of power and have the potential to overcome limitations of current batteries and internal combustion engines. They are simple electrochemical devices that convert chemical energy into electricity (Figure 2). Unlike a battery, fuel cells require a constant supply of fuel and oxidant to produce electricity. Proton Exchange Membrane (PEM) fuel cells are particularly attractive for robotics. These devices consist of simple solid state components sandwiched together as shown in Figure 2. They combine hydrogen fuel, and oxygen (from breathing air) through the most energy releasing reaction known, to produce electricity and water. It has been demonstrated; PEM fuel cells can reach 65-70 % or higher operating efficiencies at room temperature and produce clean water exhaust [9]. III. The Challenges of Fuel Cells for Robots While PEM fuel cells are simple and sound great in theory, they have three fundamental problems for practical robotics applications. These problems are storage of hydrogen fuel, long-life reliability of fuel cells and low power. Hydrogen fuel due to its high energy content and low density is difficult to store. In our research, we have developed simple, innovative hydrogen storage 3 technologies that promise energy storage densities better than the best batteries of today [5]. Figure 2: A PEM Fuel Cell Consumes Reactants, Hydrogen and Oxygen to Produce Electricity, Water and Heat. Second, PEM fuel cells have been found to be unreliable [3]. Our studies of PEM fuel cells show that they are delicate and unreliable due to degradation of their components, resulting in short lives and premature failure [4]. However our physical models and experiments suggests that PEM fuel cells controlled to operate within narrow operating can be made robust, have long lives of 3-5 years or more and high operating efficiencies [4]. Among the factors known to degrade fuel cells, are high operating voltages and electrical noise. As discussed below, mobile field robots operating in unstructured environments are subject to very substantial variation, which, without proper control can result in fuel cell degradation that shortens their lives. A solution to this problem is discussed in the section below. A third problem with fuel cells is that while they are high energy devices, they have relatively low power. This is a problem for robotics, where typical power requirements can vary substantially over a mission, with low-power rest periods and short bursts at peak power. These varying power demands are known to stress the fuel cells, resulting in short lives. A solution is to use fuel-cells in a hybrid system for mobile robots that maintains a fuel cell at optimal operating conditions to maximize life and efficiency, by protecting it from external electrical load variations, noises and meeting peak power requirement using a battery (see Figure 3). 4 Fuel cell hybrid systems have been subjected to meet rapid, transient power demands in large and stationary applications, and in robotics, to meet power surges [7, 8]. However these hybrid system designs have not considered the effects of fuel cell degradation. Figure 3: Proposed Fuel Cell-Battery Hybrid Power Supply for Robots. IV. Research The research presented here is focused on developing hybrid system design concept for mobile robots that have energy densities that exceed the best battery technology. The hybrid system is designed to meet the required peak power demands and isolate the fuel cell from degrading stresses of high and low frequency noises generated by conditioning circuits required for battery management. Physical models are used to simulate expected conditions and control systems are developed to demonstrate the concept. It is shown that the results are vast improvement over conventional batteries in terms of life, efficiency, energy density and power density. V. Case Study: Power for Humanoid Walking Robot Here a hybrid fuel cell power supply for a HOAP-2 humanoid walking robot (Figure 4) developed by Fujitsu is presented. The HOAP-2 is a 7.8 kg robot, with a maximum rated power of 250 W. It contains a 1.2 kg Nickel Metal Hydride rechargeable battery pack by default. The system contains 25 servo actuators, 6 for each leg and 5 for each arm, 2 for the head and 1 one for the waist. The robot has onboard computer equivalent to a PC-104 Pentium III system, a vision system consisting of 2 CCD cameras, onboard accelerometers, gyroscope and pressure sensors on each feet. 5 Figure 4: (Right) Fujitsu’s HOAP 2 Robot (Left) Cyberbotics WebotsTM Model of the HOAP-2. A simulation model of HOAP-2 from Cyberbotics WebotsTM [6] is used for our power demand calculations. The robot system consists of three different subsystems for power calculations, namely electro-mechanical system, computer + sensors and power system. For the electro-mechanical system the simulator model provides mechanical power output of the servo motors. The servo motors are assumed to have a 50 % electrical to mechanical efficiency. The computer and sensor system are assumed to be always powered and consume 40 W based on the HOAP-2 specifications. Below, power demand profiles of the robot’s walking behavior is shown (Figure 5). For these scenarios, alternative power sources are compared with the default nickel metal hydride battery packs that weigh 1.2 kg. VI. Fuel Cell Hybrid System The fuel cell hybrid system consists of a fuel cell stack that provides steady power source and rechargeable lithium ion A123 Nanophosphate TM battery that meets peak power demands. The fuel cells within the stack are operated at constant operating voltage of 0.8, providing a 65 % operating efficiency. Our research into the degradation of fuel cells, based on models and experimental results shows that increased operating voltages exponentially decreases the life of the fuel cell [4]. Operating the fuel cell at constant voltage of 0.8 V or less ensures long-life, while providing sufficiently high operating efficiency. The fuel cell trickle charges the battery during idle times, ensuring the battery is fully charged, to meet power peaks. The NanophosphateTM battery handles peak demands and can better handle deep battery discharges compared to 6 conventional lithium ion batteries. Ensuring the battery is nearly fully charged, maximizes its life. The battery for the hybrid system is sized based on its specific power density to meet the maximum possible power requirements of the robot. An oscillation suppression circuit interfaces a fuel cell to the power management system, consisting of power switching circuitry and a DC-DC convertor. The interface circuit effectively extracts the energy from the fuel cell and transfers it into the battery. The oscillation suppression circuits prevents any voltage oscillation from electrical circuits, particularly DC-DC convertors from being noticed by the fuel cell. This ensures the fuel cell operates at steady operating voltage, without any electrical load oscillations. Figure 5: Power Demand of a HOAP-2 Robot, walking at 0.06 m/s. VII. Hybrid System Sizing Based on the power demand profiles (Figure 5), the fuel cell will provide a constant steady source of power of 45 W. While the power peaks will be handled by the battery. For 45 W steady supply of power, a fuel cell stack weighing 150 g will be required. For a 250 W peak required by the robot, a 135 g lithium ion NanophosphateTM battery is required (with specific power of 1,850 W/kg for APR18650 cells) . Another 115 g is allocated for mass of power electronics and other items, leaving 800 g for the Lithium Hydride fuel supply with an energy density of 4,950 Wh/kg. 7 VIII. Power System Comparison Four power supply configurations are compared, including a nickel metal hydride and lithium ion battery system, a fuel cell system and a fuel cell hybrid system. See Table 1. Table 1: Power Supply Comparison for HOAP-2 Humanoid Robot Power Supply FC Stack FC Fuel Energy System RunMass Mass Density Life Time NiMH Battery 40 Wh/kg 0.3 year 3 hours Li Ion Battery 120 Wh/kg 1 year 9 hours Fuel Cell 300 g 900 g 4950 Wh/kg 5 days 99 hours Fuel Cell Hybrid 150 g 800 g 4950 Wh/kg 3 years 88 hours Nickel metal hydride and lithium ion batteries have the lowest energy densities and thus provide short run-times, before requiring recharging. The system life of the batteries is computed based on expected lifetime of 1,000 charge/recharge multiplied by the run-times hours. A direct fuel cell system has the longest runtime. However the life of the system, based on our degradation models is expected to last just 5 days making this option impractical. The fuel cell hybrid system offers a good trade-off between both run-time and system life. IX. Summary and Conclusions Based on these results, the fuel cell hybrid system concept offers high energy density, long-life and would meet the required peak power demands of a battery. The key to our hybrid system concept is effective control and design, where a fuel cell and battery are optimally sized to minimize stress on the fuel cell, while enabling a battery to meet power demand peaks. By minimizing stresses on the fuel cell, the system can be operated for long-lives at high operating efficiencies. X. References 1. 2. 3. 4. Hägele, M., “Contribution to World Robotics 2005,” Technical Report, European Robotics Network, 2005. Asada, H., et al. “A Roadmap to US Robotics: From Internet to Robotics,” Technical Report, Editor: Computing Community Consortium/Computing Research Association, 2009. Rubio, M.A., Urquia, A., and Dormida, S. (2009) “Diagnosis of Performance Degradation Phenomena in PEM Fuel Cells,” International Journal of Hydrogen Energy, pp. 1-5. Thangavelautham, J., Dubowsky, S., (2013) “On the Catalytic Degradation in Fuel Cell Power Supplies for Long-Life Mobile Field Sensors,” Fuel Cells, vol. 13, no. 2, pp. 1-24. 8 5. Thangavelautham, J., Strawser, D., Dubowsky, S., “Lithium hydride powered PEM fuel cells for long-duration small mobile robotic missions,” IEEE International Conference on Robots and Automation, 2012, p. 1-8. 6. Michel, O. / Cyberbotics Ltd. (2004) “WebotsTM: Professional Mobile Robot Simulation,” International Journal of Advanced Robotic Systems, pp. 39-42, Vol. 1 No. 1. Kesner, S. B., Plante, J. S., Boston, P., Fabian, T., Dubowsky, S., “Mobility and Power Feasibility of a Microbot Team System for Extraterrestrial Cave Exploration,” Proceedings of the 2007 IEEE International Conference Robotics and Automation, Rome, Italy, April 2007. Joh, H., et al., (2010) A direct methanol fuel cell system to power a humanoid robot, Journal of Power Sources, Vol. 195, No. 1, pp. 293-298. F. Barbir, PEM Fuel Cells: Theory and Practice, Academic Press, 2005. 7. 8. 9. View publication stats	3
Using a hierarchy of Domain Specific Languages in complex software systems design V. S. Lugovsky <[email protected]> arXiv:cs/0409016v1 [cs.PL] 9 Sep 2004 February 1, 2008 Abstract A new design methodology is introduced, with some examples on building Domain Specific Languages hierarchy on top of Scheme. 1 Introduction nized as “true” macros as they hide an access to the host language. This situation looks like a paradox. On the one hand, industry uses the metaprogramming ideas and tools, and it is easy to imagine how it would suffer without them. On the other hand, industry does not want to hear anything related to the metaprogramming. It does not want people inventing new programming languages — plenty of industry coders barely use only one language and IT managers believe without any reason that they can not be taught to use more [6]. Industry prefers to “re–invent a wheel” and to express any sort of complexity in the form of libraries for static, steady languages. For some strange reason learning complicated libraries for a language which barely fits problem domain needs is preferred to learning a small new language specifically designed for it. In this paper I am trying to advocate the metaprogramming approach as the major design methodology for complex systems. Sounds like another one “silver bullet” invention? There were many methodologies claiming to solve all possible problems of the mankind — RUP, eXtreme programming, etc. Why do we need another one? Simply because the previous approaches did not succeed. They were too tied to particular programming technologies (mostly — OOP varieties), which are definitely not “silver bullets”. Metaprogramming Programs that write programs that write programs (...) Too complicated? A hackers technique which can not be applied to the “real world” problems? This is exactly how IT industry specialists think about metaprogramming. And this is a completely wrong notion! Metaprogramming is the only known way to reduce the complexity significantly. In some areas “programs that write programs” are accepted by the industry due to the enormous level of complexity of the corresponding handwritten code — regular expressions, lexers and parsers generators to name a few, and code wizards and templates in popular “integrated development environments” are also widely used. But this does not help at all in the overall methodology recognition. The industry’s most beloved and widely buzzworded language, Java, does not have even such a rudimentary preprocessor as C does. Very few C++ programmers have an idea on how to use the templates, they just utilize STL without any understanding of the true source of the power. Even in the enlightened world of Lisp programming the misunderstanding is surprisingly wide: almost all of Lisp dialects and Scheme implementations have problems with macros (not so many people are using them), and even the current Scheme standard R5RS contains only hygienic macros that can hardly be recog1 methodology is different, it strongly encourages of domain specific languages is not very tricky. the use of all possible programming technolo- An approach I will describe here is based on metaprogramming techniques. It requires a so gies and to invent the “impossible” ones. called Core Language, on top of which we will build a hierarchy of our domain specific lan2 Domain specific lan- guages. The Core Language should possess the following properties: guages Below I am providing an outline of the proposed methodology. Any problem domain can be best expressed using a language (mathematical, programming, natural, ...) specially designed for it. In most cases there should be one entity in a language for every entity in a problem domain. For example, if a problem domain is the recognition of syntax constructions in a characters stream, the Domain Specific Language should contain characters and characters sets as a primary entity and automata constructions for expressing syntax. That is enough — regular expressions language is designed. It is hard to believe that somebody will ever invent anything better for this purpose than this “most optimal” DSL. If a problem domain is already specified as an algebra, we even do not have to design the DSL: it will be this algebra itself, galvanised with any underlying computational semantics — this is the way SQL was born. If a problem domain is 3D graphics, linear algebra and stereometry should be used. All the languages and data formats dedicated to 3D contain subsets of this formal theories. As it is stated in [1], “The object of a DSL-based software architecture is to minimise the semantic distance between the system’s specification and its implementation.” 3 Core language For any problem it is convenient to have a language that best fits it. There already exist specialized languages for some common problems. But what to do if none is available? The answer is trivial: implement it. Implementation • True macros. That is, we must have an access to a complete programming language (preferably the same as a host language, or a different one) inside the macro definitions. Macros should be real programs which can do anything that the programs written in the host language can do. Macros are producing the code in the host language, in the form of text or directly as an abstract syntax tree. • True runtime eval. Programs that are generated in the runtime should be evaluated. This can be a different language than the host language, or, better, the same one. • Turing-completeness. This should be a real programming language, equivalent in its expressive power to the “general purpose” languages. • Simplicity. It is an extensible core and should not contain any unnecessary complexity that can be later added by a user who really needs it. • Comprehensive and easy to use data types system. If a type system is well suited for expressing any possible abstract syntax trees, the language fits this requirement. On top of the Core Language we have to build functionality that will be needed to implement programming languages. It is lexing, parsing, intermediate languages that fit well computational models different from the model of the Core Language (e.g., if the core language is imperative or an eager functional, we will need a graph reduction engine to implement lazy functional DSLs, or a term unification engine to implement logical languages and a stack machine if we have to go to lower levels). The 2 Core Language enriched with this “Swiss army knife” for programming languages development then becomes a major tool for any project. 4 New methodology The development process must fit in the following chain: • divide the problem into sub–problems, possibly using some object oriented design techniques, or whatever fits better. • formalize each sub–problem. 5 Scheme example A good example of a practical Core Language is Scheme (with addition of Common Lisp–style macros). It uses S–expressions as an AST, and S–expressions composition is very natural. S–expressions are good enough to represent any possible AST (for example, XML is naturally represented as SXML). It provides a true runtime eval hosting the same language as in compile time. There exist some practical and efficient Scheme implementations which provide performance acceptable for most tasks, good FFI, and, thus, integration with legacy libraries. • implement the Domain Specific Language after this formalization, using the Let us start with adding the functionality Core Language and other DSL with the described above to Scheme. First of all we will same semantics. need parsing — not all of our team members • solve the problem using the best possible are fond of parentheses, so we have to implelanguage. ment many complicated syntaxes. The most This way any project will grow into a tree (hier- natural way for a functional programming lanarchy) of domain specific languages. Any lan- guage is to implement a set of parsing combinaguage is a subset or a superset of another lan- tors for building recursive descendant parsers guage in the hierarchy (or, may be, combina- (mostly LL(1), but it is not such a fixed limit as tion of several languages), and the amount of LALR(1) for Yacc–like automata generators). coding for a new language if we already have a deep and comprehensive hierarchy is quite small. A development team working within this methodology should consist of at least one specialist who maintains this hierarchy, an architect who formalizes problems, and a number of coders who specialize in particular problem domains, they even may not be programmers at all — they just have to know well their domains and operate them in terms that are as close as possible to the native problem domain terminology. For example, HTML designer will be happy operating HTML–like tags for his templates (that is why JSP custom tags are so popular); mathematician will find a language modelled after the standard mathematical notation intuitive — for this reason Wolfram Mathematica is so popular among non-programmers; game script writer will operate a language expressing characters, their properties and action rules — stating, not programming. This list can be continued infinitely. Of course we will use metaprogramming wherever possible. All the parsers should be functions which consume a list of tokens (e.g. characters) as an input and return the result in the following form: ((RESULT anyresult) unparsed-input-rest) or ((FAIL reason) input) To access the parsing result we will provide the following macros: (define-macro (success? r ) ‘(not (eq? (caar ,r ) ’FAIL))) And if we are sure that we have some result, we will use the following macro to extract it (otherwise, this will return a fail message): (define-macro (result r ) ‘(cdar ,r )) 3 The last definition looks surprisingly comIn any case, we can access the rest of the pact, thanks to the pselect macro. From this stream after the parsing pass: stage the power of metaprogramming becomes (define-macro (rest r ) more and more obvious. ‘(cdr ,r )) Just as a reference, we will show here the These macros could also be implemented as definition of a choice combinator: functions. But all the macros are available in (define-macro (pOR0 p1 p2 ) the context of macro definitions while functions ‘(λ (l) are not. (let ((r1 (,p1 l))) Almost all of the parsers should fail on the (if (success? r1 ) end of the input, so the following safeguard r1 macro will be extremely useful: (,p2 l))))) (define-macro (parser p) ‘(λ (l) (if (∅? l) ’((FAIL "EMPTY")) (,p l)))) And its nested version is obvious: (define-macro (pOR p1 . p~o ) 0 ‘(pselect pOR ,p1 ,@p~o )) We will skip the rest of the combinators Now this game becomes more interesting. definitions and just show what we gained after Here is a very handy macro that nests a se- all. For example, now to define a floating point quence of applications into the form of (m p1 number recognizer, we can use this definition: (m p2 . . . (m px pn ))): (define parse-num (p+ (define-macro (pselect m p1 . p~o ) (pOR (pcsx-or (#\− #\+)) (if (∅? p~o ) p1 parse-any) (let ((p2 (car p~o )) (pMANY pdigit) (px (cdr p~o ))) OR (p (p+ (pcharx #\.) ‘(,m ,p1 (pselect ,m ,p2 ,@px ))))) (pMANY pdigit)) Sequence parsing combinator with two arparse-any))) guments can be declared as follows: It looks like BNF, but still too Schemish. (define-macro (p+0 p1 p2 ) This is already a Domain Specific Language on ‘(λ (l) top of Scheme, but it does not conform to the (let ((r1 (,p1 l))) perfectionist requirement. However, we can use (if (success? r1 ) this still not perfect parsing engine to imple(let ((r2 (,p2 (rest r1 )))) ment an intermediate regular expressions lan(if (success? r2 ) guage as a macro. Omitting the definitions, we (cons (cons ’RESULT will show the previous recognizer implemented (append in a new way: (result r1 ) (define parse-num (result r2 ))) (regexp (rest r2 )) ((#\− / #\+) / parse-any) + (cons (list ’FAIL "p+" (car r2 )) l))) (pdigit ∗) + r1 )))) (("." + (pdigit ∗)) / And it will be immediately turned into the parse-any))) sequence parsing combinator with an arbitrary This new Domain Specific Language can be number of arguments: used in many ways. For example, we can build (define-macro (p+ p1 . p~o ) ‘(pselect p+0 ,p1 ,@p~o )) a simple infix pre–calculator for constants: (define-macro (exp1 . v ) 4 (defparsers (letrec ((epr (let ((body (regexp (num :−> $ 0) / (lst -> (aprs epr ))))) (regexp ((body + (SCM psym +) + epr ) :−> (list (+ $ 0 $ 2))) / ((body + (SCM psym −) + epr ) :−> (list (− $ 0 $ 2))) / ((body + (SCM psym ∗) + epr ) :−> (list (∗ $ 0 $ 2))) / ((body + (SCM psym / ) + epr ) :−> (list (/ $ 0 $ 2))) / body )))) (car (result (epr v )))))) only languages with a computational model which is close to the model of Scheme (eager dynamically typed functional languages with imperative features), but any possible languages, providing small intermediate DSLs which simulate alternative computational models. For those who need very lowlevel power it is possible to produce an intermediate code in C language (for example, the Bigloo Scheme [3] implementation allows to include C code when compiling through C backend). For implementing complicated runtime models it is easy to produce an intermediate Forth–like DSL on top of Scheme and then use both Scheme and Forth metaprogramming powers. 6 Alternatives To make the picture complete, it is necesAnd then, wherever we want to calculate a sary to mention other possible choices for the numerical constant in the compilation time, we Core Language. The popular programming may use the exp1 macro: language, C++, could become such a Core (exp1 5 + ((10 / 2)−(1 / 5))) Language relatively easily. It has a TuringThis language does not look like Scheme complete macro system, unfortunately, featurany more. And we can go even further, im- ing the language different from the host lanplementing a Pascal (or Rlisp)–like language guage (so only one stage preprocessing is poson top of Scheme, using just the same regexp sible). It lacks a good type system, but it could macro to describe both a lexer and a parser, be simulated on top of the existing lowlevel feaand then to compile the resulting code to the tures. There exist some implementations of the recursive descendant parsing combinators underlying Scheme. for C++ (e.g., Boost Spirit library [2]), implementation of the functional programming (e.g., (pasqualish Boost Lambda [2]), and even Lisp compilers on " top of the C++ template system. The runtime function fac(x) evaluation is available in different ways: using begin pluggable scripting languages other than C++, if (x > 0) then using the C++ interpreter [14]. An interesting x*fac(x - 1) approach is described in [13]. else 1; Another choice is Forth. It is a powerful end metalanguage, but the core language remains ") too lowlevel and unsafe. Forth is often the only No more parenthesis that frighten non–Lisp choice available for the embedded systems with programmers so much! Now even Pascal pro- limited resources. It is worth mentioning modern experimengrammers can use Scheme. The code samples above demonstrate some tal extensions for strictly typed functional lanof the techniques available in this approach. guages: Template Haskell [12] and MetaOCaml The complete implementation can be down- [11]. Both of them conform well to all of loaded from [4]. It is possible to produce not the Core Language requirements. Objective 5 Caml also provides one–stage metaprogram- metaprogramming too. ming using a sophisticated preprocessing engine CamlP4. And OCaml is quite good for Conclusion implementing interpreters using the closure– 7 based technique. Some examples can be found The idea of metaprogramming is not something in [7]. esoteric. Metaprogramming is used widely by No doubt that Common Lisp would also commercial programmers, they just do not rebe a very good platform since it shares almost alize it. The methodology proposed in this paall the features with Scheme with exception of per is an attempt of uncovering all the hidsimplicity. The killing feature of Common Lisp den power of the metaprogramming techniques is advanced runtime compilation in some of the available. major implementations (CMU CL [8] and its The Scheme example presented above is descendant SBCL [9] are good examples), and part of the working project, which already the defmacro is guaranteed to be working in all proved the supremacy of this approach. A subthe implementations available, which is a great set of the Domain Specific Languages hierarchy advantage over Scheme. designed for the WWW data acquiring project For relatively small projects Tcl [10] would is shown on the Fig. 1. be a good choice. Its computational model The subject discussed requires future reis based on rewrites (and primary data struc- search and practical approbation, whose final tures are just the strings of text), which ren- result may be a completely formalized, matheders an extremely powerful metaprogramming matically strict methodology description and a tool. JavaScript language is also based on Core Language which will best fit this methodthe rewrites semantics, so it could be used for ology. 6 References [1] Diomidis Spinellis. Reliable software implementation using domain specific languages. In G. I. Schuëller and P. Kafka, editors, Proceedings ESREL ’99 — The Tenth European Conference on Safety and Reliability, pages 627–631, Rotterdam, September 1999. ESRA, VDI, TUM, A. A. Balkema // [draft] http://www.dmst.aueb.gr/dds/pubs/conf/1999-ESREL-SoftRel/html/dsl.html [2] The Boost project // http://www.boost.org/ [3] The Bigloo Practical Scheme implementation // http://www.bigloo.org/ [4] V. S. Lugovsky, DSLEngine project home // http://dslengine.sourceforge.net/ [5] P. Graham, The Hundred-Year Language // http://www.paulgraham.com/hundred.html [6] P. Graham, The Python Paradox // http://www.paulgraham.com/pypar.html [7] V. S. Lugovsky, publications list // http://ontil.ihep.su/˜vsl [8] CMU Common Lisp // http://www.cons.org/cmucl/ [9] Steel Bank Common Lisp // http://sbcl.sourceforge.net/ [10] Tcl programming language resource // http://tcl.activestate.com/ [11] MetaOCaml project home // http://www.metaocaml.org/ [12] T. Sheard, S. P. Jones, Template metaprogramming for Haskell // http://research.microsoft.com/˜simonpj/papers/meta-haskell/ [13] Tempo project home // http://compose.labri.fr/prototypes/tempo/ [14] CINT project home // http://root.cern.ch/root/Cint.html 7 Core language Parsing combinators Stack machine Lexer Graph machine Unification machine Templates language Parser generator Forth-like Regular expressins Data aquision regexps Pascal-like Rule engine SQL templates Figure 1: A sample DSLs hierarchy subset for the Web crawler project. 8	2
ACD Term Rewriting arXiv:cs/0608016v1 [cs.PL] 3 Aug 2006 Gregory J. Duck, Peter J. Stuckey, and Sebastian Brand NICTA Victoria Laboratory Department of Computer Science & Software Engineering, University of Melbourne, Australia Abstract. In this paper we introduce Associative Commutative Distributive Term Rewriting (ACDTR), a rewriting language for rewriting logical formulae. ACDTR extends AC term rewriting by adding distribution of conjunction over other operators. Conjunction is vital for expressive term rewriting systems since it allows us to require that multiple conditions hold for a term rewriting rule to be used. ACDTR uses the notion of a “conjunctive context”, which is the conjunction of constraints that must hold in the context of a term, to enable the programmer to write very expressive and targeted rewriting rules. ACDTR can be seen as a general logic programming language that extends Constraint Handling Rules and AC term rewriting. In this paper we define the semantics of ACDTR and describe our prototype implementation. 1 Introduction Term rewriting is a powerful instrument to specify computational processes. It is the basis of functional languages; it is used to define the semantics of languages and it is applied in automated theorem proving, to name only a few application areas. One difficulty faced by users of term rewriting systems is that term rewrite rules are local, that is, the term to be rewritten occurs in a single place. This means in order to write precise rewrite rules we need to gather all relevant information in a single place. Example 1. Imagine we wish to “program” an overloaded ordering relation for integers variables, real variables and pair variables. In order to write this the “type” of the variable must be encoded in the term1 as in: int(x) ≤ int(y) → intleq(int(x), int(y)) real(x) ≤ real(y) → realleq(real(x), real(y)) pair(x1 , x2 ) ≤ pair(y1 , y2 ) → x1 ≤ y1 ∨ x1 = y1 ∧ x2 ≤ y2 In a more standard language, the type information for variables (and other information) would be kept separate and “looked up” when required. ⊓ ⊔ 1 Operator precedences used throughout this paper are: ∧ binds tighter than ∨, and all other operators, e.g. ¬, =, bind tighter than ∧. Term rewriting systems such as constraint handling rules (CHRs) [5] and associative commutative (AC) term rewriting [3] allow “look up” to be managed straightforwardly for a single conjunction. Example 2. In AC term rewriting the above example could be expressed as: int (x) ∧ int (y) ∧ x ≤ y → int(x) ∧ int(y) ∧ intleq(x, y) real (x) ∧ real (y) ∧ x ≤ y → real (x) ∧ real (y) ∧ realleq (x, y) pair (x, x1 , x2 ) ∧ pair (y, y1 , y2 ) ∧ x ≤ y → pair (x, x1 , x2 ) ∧ pair (y, y1 , y2 )∧ (x1 ≤ y1 ∨ x1 = y1 ∧ x2 ≤ y2 ) where each rule replaces the x ≤ y by an appropriate specialised version, in the conjunction of constraints. The associativity and commutativity of ∧ is used to easily collect the required type information from a conjunction. ⊓ ⊔ One difficulty remains with both AC term rewriting and CHRs. The “look up” is restricted to be over a single large conjunction. Example 3. Given the term int(x1 ) ∧ int(y1 ) ∧ pair (x, x1 , x2 ) ∧ pair (y, y1 , y2 ) ∧ x ≤ y. Then after rewriting x ≤ y to (x1 ≤ y1 ∨ x1 = y1 ∧ x2 ≤ y2 ) we could not rewrite x1 ≤ y1 since the types for x1 , y1 appear in a different level. In order to push the type information inside the disjunction we need to distribute conjunction over disjunction. ⊓ ⊔ Simply adding distribution rules like A ∧ (B ∨ C) → A ∧ B ∨ A ∧ C A ∧ B ∨ A ∧ C → A ∧ (B ∨ C) (1) (2) does not solve the problem. Rule (1) creates two copies of term A, which increases the size of the term being rewritten. Adding Rule (2) to counter this effect results in a non-terminating rewriting system. 1.1 Conjunctive context We address the non-termination vs. size explosion problem due to distributivity rewrite rules in a similar way to how commutativity is dealt with: by handling distributivity on the language level. We restrict ourselves to dealing with expanding distributivity of conjunction ∧ over any other operator, and we account for idempotence of conjunction.2 Thus we are concerned with distribution rules of the form P ∧ f (Q1 , . . . , Qn ) → P ∧ f (P ∧ Q1 , . . . , P ∧ Qn ). 2 (3) This means that conjunction is distributive over any function f in presence of a redundant copy of P , i.e. P ∧ (P ∧ f (Q1 , . . . , Qn )) → P ∧ f (P ∧ Q1 , . . . , P ∧ Qn ). We use idempotence to simplify the RHS and derive (3). 2 Let us introduce the conjunctive context of a term and its use in rewrite rules, informally for now. Consider a term T and the conjunction C ∧ T modulo idempotence of ∧ that would result from exhaustive application of rule (3) to the superterm of T . By the conjunctive context of T we mean the conjunction C. Example 4. The conjunctive context of the boxed occurrence of x in the term (x = 3) ∧ (x2 > y ∨ ( x = 4) ∧ U ∨ V ) ∧ W, is (x = 3) ∧ U ∧ W . ⊓ ⊔ We allow a rewrite rule P → T to refer to the conjunctive context C of the rule head P . We use the following notation: C \ P ⇐⇒ T. This facility provides ∧-distributivity without the undesirable effects of rule (3) on the term size. Example 5. We can express that an equality can be used anywhere “in its scope” by viewing the equality as a conjunctive context: x = a \ x ⇐⇒ a. Using this rule on the term of Example 4 results in (x = 3) ∧ (32 > y ∨ (3 = 4) ∧ U ∨ V ) ∧ W ⊓ ⊔ without dissolving the disjunction. 1.2 Motivation and Applications Constraint Model Simplification. Our concrete motivation behind associative commutative distributive term rewriting (ACDTR) is constraint model mapping as part of the G12 project [7]. A key aim of G12 is the mapping of solver independent models to efficient solver dependent models. We see ACDTR as the basis for writing these mappings. Since models are not flat conjunctions of constraints we need to go beyond AC term rewriting or CHRs. Example 6. Consider the following simple constraint model inspired by the Social Golfers problem. For two groups g1 and g2 playing in the same week there can be no overlap in players: maxOverlap(g1 , g2 , 0) The aim is to maximise the number of times the overlap between two groups is less than 2; in other words minimise the number of times two players play together in a group. ^ constraint maxOverlap(g1 , g2 , 0) ∀w∈Weeks ∀g1 ,g2 ∈weeks[w] g1 <g2 maximise X holds(maxOverlap(g1 , g2 , 1)) ∀w1 ,w2 ∈Weeks ∀g1 ∈weeks[w1 ] ∀g2 ∈weeks[w2 ] g1 <g2 3 Consider the following ACDTR program for optimising this constraint model. maxOverlap(a, b, c1 ) \ maxOverlap(a, b, c2 ) ⇐⇒ c2 ≥ c1 \| true holds(true) ⇐⇒ 1 holds (false) ⇐⇒ 0 The first rule removes redundant maxOverlap constraints. The next two rules implement partial evaluation of the holds auxiliary function which coerces a Boolean to an integer. By representing the constraint model as a giant term, we can optimise the model by applying the ACDTR program. For example, consider the trivial case with one week and two groups G1 and G2 . The model becomes maxOverlap(G1 , G2 , 0) ∧ maximise(holds (maxOverlap(G1 , G2 , 1))). The subterm holds(maxOverlap(G1 , G2 , 1)) simplifies to 1 using the conjunctive context maxOverlap(G1 , G2 , 0). ⊓ ⊔ It is clear that pure CHRs are insufficient for constraint model mapping for at least two reasons, namely – a constraint model, e.g. Example 6, is typically not a flattened conjunction; – some rules rewrite functions, e.g. rules (2) and (3) rewriting function holds, which is outside the scope of CHRs (which rewrite constraints only). Global Definitions. As we have seen conjunctive context matching provides a natural mechanism for making global information available. In a constraint model, structured data and constraint definitions are typically global, i.e. on the top level, while access to the data and the use of a defined constraint is local, e.g. the type information from Example 1. Another example is partial evaluation. Example 7. The solver independent modelling language has support for arrays. Take a model having an array a of given values. It could be represented as the top-level term array (a, [3, 1, 4, 1, 5, 9, 2, 7]). Deeper inside the model, accesses to the array a occur, such as in the constraint x > y + lookup(a, 3). The following rules expand such an array lookup: array(A, Array) \ lookup(A, Index ) ⇐⇒ list element(Array, Index ) list element([X\|Xs], 0) ⇐⇒ X list element ([X\|Xs], N ) ⇐⇒ N > 0 \| list element (Xs, N − 1) Referring to the respective array of the lookup expression via its conjunctive context allows us to ignore the direct context of the lookup, i.e. the concrete constraint or expression in which it occurs. ⊓ ⊔ 4 Propagation rules. When processing a logical formula, it is often useful to be able to specify that a new formula Q can be derived from an existing formula P without consuming P . In basic term rewriting, the obvious rule P ⇐⇒ P ∧ Q causes trivial non-termination. This issue is recognised in CHRs, which provide support for inference or propagation rules. We account for this fact and use rules of the form P =⇒ Q to express such circumstances. Example 8. The following is the classic CHR leq program reimplemented for ACD term rewriting (we omit the basic rules for logical connectives): leq(X, X) ⇐⇒ true leq(X, Y ) \ leq(Y, X) ⇐⇒ X = Y leq(X, Y ) \ leq(X, Y ) ⇐⇒ true leq(X, Y ) ∧ leq(Y, Z) =⇒ leq(X, Z) (reflexivity) (antisymmetry) (idempotence) (transitivity) These rules are almost the same as the CHR version, with the exception of the second and third rule (antisymmetry and idempotence) which generalise its original by using conjunctive context matching. ⊓ ⊔ Propagation rules are also used for adding redundant information during model mapping. The rest of the paper is organised as follows. Section 2 covers the standard syntax and notation of term rewriting. Section 3 defines the declarative and operational semantics of ACDTR. Section 4 describes a prototype implementation of ACDTR as part of the G12 project. Section 5 compares ACDTR with related languages. Finally, in Section 6 we conclude. 2 Preliminaries In this section we briefly introduce the notation and terminology used in this paper. Much of this is borrowed from term rewriting [3]. We use T (Σ, X) to represent the set of all terms constructed from a set of function symbols Σ and set of variables X (assumed to be countably infinite). We use Σ (n) ⊆ Σ to represent the set of function symbols of arity n. A position is a string (sequence) of integers that uniquely determines a subterm of a term T , where ǫ represents the empty string. We define function T \|p , which returns the subterm of T at position p as T \|ǫ = T f (T1 , . . . , Ti , . . . , Tn )\|ip = Ti \|p We similarly define a function T [S]p which replaces the subterm of T at position p with term S. We define the set Pos(T ) to represent the set of all positions of subterms in T . An identity is a pair (s, t) ∈ T (Σ, X) × T (Σ, X), which is usually written as s ≈ t. Given a set of identities E, we define ≈E to be the set of identities closed under the axioms of equational logic [3], i.e. symmetry, transitivity, etc. 5 We define the congruence class [T ]≈E = {S ∈ T (Σ, X)\|S ≈E T } as the set of terms equal to T with respect to E. Finally, we define function vars(T ) to return the set of variables in T . 3 Syntax and Semantics The syntax of ACDTR closely resembles that of CHRs. There are three types of rules of the following form: (simplification) (propagation) (simpagation) r @ H ⇐⇒ g \| B r @ H =⇒ g \| B r @ C \ H ⇐⇒ g \| B where r is a rule identifier, and head H, conjunctive context C, guard g and body B are arbitrary terms. The rule identifier is assumed to uniquely determine the rule. A program P is a set of rules. We assume that vars(g) ⊆ vars(H) or vars(g) ⊆ vars(H) ∪ vars(C) (for simpagation rules). The rule identifier can be omitted. If g = true then the guard can be omitted. We present the declarative semantics of ACDTR based on equational logic. First we define the set of operators that ACDTR treats specially. Definition 1 (Operators). We define the set of associate commutative operators as AC. The set AC must satisfy AC ⊆ Σ (2) and (∧) ∈ AC. For our examples we assume that AC = {∧, ∨, +, ×}. We also treat the operator ∧ as distributive as explained below. ACDTR supports a simple form of guards. Definition 2 (Guards). A guard is a term. We denote the set of all “true” guards as G, i.e. a guard g is said to hold iff g ∈ G. We assume that true ∈ G and false 6∈ G. We can now define the declarative semantics for ACDTR. In order to do so we employ a special binary operator where to explicitly attach a conjunctive context to a term. Intuitively, the meaning of T where C is equivalent to that of T provided C is true, otherwise the meaning of T where C is unconstrained. For Boolean expressions, it is useful to interpret where as conjunction ∧, therefore where-distribution, i.e. identity (6) below, becomes equivalent to ∧-distribution (3). The advantage of distinguishing where and ∧ is that we are not forced to extend the definition of ∧ to arbitrary (non-Boolean) functions. We denote by B the following set of built-in identities: A◦B ≈B ◦A (A ◦ B) ◦ C ≈ A ◦ (B ◦ C) T ≈ (T where true) A ∧ B ≈ (A where B) ∧ B T where (W1 ∧ W2 ) ≈ (T where W1 ) where W2 f (A1 , ..., Ai , ..., An ) where W ≈ f (A1 , ..., Ai where W, ..., An ) where W 6 (1) (2) (3) (4) (5) (6) for all ◦ ∈ AC, functions f ∈ Σ (n) , and i ∈ {1, . . . , n}. Definition 3 (Declarative Semantics for ACDTR). The declarative semantics for an ACDTR program P (represented as a multiset of rules) is given by the function JK defined as follows: JP K = {Jθ(R)K \| ∀R, θ . R ∈ P ∧ θ(guard(R)) ∈ G} ∪ B JH ⇐⇒ g \| BK = ∃vars(B)−vars(H) (H ≈ B) JC \ H ⇐⇒ g \| BK = ∃vars(B)−vars(C,H) (H where C ≈ B where C) JH =⇒ g \| BK = ∃vars(B)−vars(H) (H ≈ H ∧ B) where function guard(R) returns the guard of a rule. The function JK maps ACDTR rules to identities between the head and the body terms, where body-only variables are existentially quantified.3 Note that there is a new identity for each possible binding of guard(R) that holds in G. A propagation rule is equivalent to a simplification rule that (re)introduces the head H (in conjunction with the body B) in the RHS. This is analogous to propagation rules under CHRs. A simpagation rule is equivalent to a simplification rule provided the conjunctive context is satisfied. The built-in rules B from Definition 3 contain identities for creating/destroying (3) and (4), combining/splitting (5), and distributing downwards/upwards (6) a conjunctive context in terms of the where operator. The set B also contains identities (1) and (2) for the associative/commutative properties of the AC operators. Example 9. Consider the following ACDTR rule and the corresponding identity. JX = Y \ X ⇐⇒ Y K = (Y where X = Y ) ≈ (X where X = Y ) (7) Under this identity and using the rules in B, we can show that f (A)∧(A = B) ≈ f (B) ∧ (A = B), as follows. f (A) ∧ (A = B) ≈(4) (f (A) where (A = B)) ∧ (A = B) ≈(6) (f (A where (A = B)) where (A = B)) ∧ (A = B) ≈(7) (f (B where (A = B)) where (A = B)) ∧ (A = B) ≈(6) (f (B) where (A = B)) ∧ (A = B) ≈(4) f (B) ∧ (A = B) ⊓ ⊔ 3.1 Operational Semantics In this section we describe the operational semantics of ACDTR. It is based on the theoretical operational semantics of CHRs [1,4]. This includes support for identifiers and propagation histories, and conjunctive context matching for simpagation rules. 3 All other variables are implicitly universally quantified, where the universal quantifiers appear outside the existential ones. 7 Propagation history. The CHR concept of a propagation history, which prevents trivial non-termination of propagation rules, needs to be generalised over arbitrary terms for ACDTR. A propagation history is essentially a record of all propagation rule applications, which is checked to ensure a propagation rule is not applied twice to the same (sub)term. In CHRs, each constraint is associated with a unique identifier. If multiple copies of the same constraint appear in the CHR store, then each copy is assigned a different identifier. We extend the notion of identifiers to arbitrary terms. Definition 4 (Identifiers). An identifier is an integer associated with each (sub)term. We use the notation T #i to indicate that term T has been associated with identifier i. A term T is annotated if T and all subterms of T are associated with an identifier. We also define function ids(T ) to return the set of identifiers in T , and term(T ) to return the non-annotated version of T . For example, T = f (a#1, b#2)#3 is an annotated term, where ids(T ) = {1, 2, 3} and term(T ) = f (a, b). Identifiers are considered separate from the term. We could be more precise by separating the two, i.e. explicitly maintain a map between Pos(T ) and the identifiers for T . We do not use this approach for space reasons. We extend and overload all of the standard operations over terms (e.g. from Section 2) to annotated terms in the obvious manner. For example, the subterm relation T \|p over annotated terms returns the annotated term at position p. The exception are elements of the congruence class [T ]≈AC , formed by the AC relation ≈AC , which we assume satisfies the following constraints. A#i ◦ B#j ≈AC B#j ◦ A#i A#i ◦ (B#j ◦ C#k) ≈AC (A#i ◦ B#j) ◦ C#k We have neglected to mention the identifiers over AC operators. These identifiers will be ignored later, so we leave them unconstrained. A propagation history is a set of entries defined as follows. Definition 5 (Entries). A propagation history entry is of the form (r @ E), where r is a propagation rule identifier, and E is a string of identifiers. We define function entry(r, T ) to return the propagation history entry of rule r for annotated term T as follows. entry(r, T ) = (r @ entry(T )) entry(T1 ◦ T2 ) = entry(T1 ) entry(T2 ) entry(f (T1 , ..., Tn )#i) = i entry(T1 ) ... entry(Tn ) ◦ ∈ AC otherwise This definition means that propagation history entries are unaffected by associativity, but are effected by commutativity. Example 10. Consider the annotated term T = f ((a#1 ∧ b#2)#3)#4. We have that T ∈ [T ]≈AC and T ′ = f ((b#2 ∧ a#1)#3)#4 ∈ [T ]≈AC . Although T and T ′ belong to [T ]≈AC they have different propagation history entries, e.g. entry(r, T ) = (r @ (4 1 2)) while entry(r, T ′ ) = (r @ (4 2 1)). ⊓ ⊔ 8 When a (sub)term is rewritten into another, the new term is assigned a set of new unique identifiers. We define the auxiliary function annotate(P, T ) = Ta to map a set of identifiers P and un-annotated term T to an annotated term Ta such that ids(Ta ) ∩ P = ∅ and \|ids(Ta )\| = \|Pos(T )\|. These conditions ensure that all identifiers are new and unique. When a rule is applied the propagation history must be updated accordingly to reflect which terms are copied from the matching. For example, the rule f (X) ⇐⇒ g(X, X) essentially clones the term matching X. The identifiers, however, are not cloned. If a term is cloned, we expect that both copies will inherit the propagation history of the original. Likewise, terms can be merged, e.g. g(X, X) ⇐⇒ f (X) merges two instances of the term matching X. In this case, the propagation histories of the copies are also merged. To achieve this we duplicate entries in the propagation history for each occurrence of a variable in the body that also appeared in the head. Definition 6 (Updating History). Define function update(H, Ha , B, Ba , T0 ) = T1 where H and B are un-annotated terms, Ha and Ba are annotated terms, and T0 and T1 are propagation histories. T1 is a minimal propagation history satisfying the following conditions: – T0 ⊆ T1 ; – ∀p ∈ Pos(H) such that H\|p = V ∈ X (where X is the set of variables), and ∃q ∈ Pos(B) such that B\|q = V , then define identifier renaming ρ such that ρ(Ha \|p ) and Ba \|q are identical annotated terms. Then if E ∈ T0 we have that ρ(E) ∈ T1 . Example 11. Consider rewriting the term Ha = f ((a#1 ∧ b#2)#3)#4 with a propagation history of T0 = {(r @ (1 2))} using the rule f (X) ⇐⇒ g(X, X). The resulting term is Ba = g((a#5 ∧b#6)#7), (a#8 ∧b#9)#10#11 and the new propagation history is T1 = {(r @ (1 2)), (r @ (5 6)), (r @ (8 9))}. ⊓ ⊔ Conjunctive context. According to the declarative semantics, a term T with conjunctive context C is represented as (T where C). Operationally, we will never explicitly build a term containing a where clause. Instead we use the following function to compute the conjunctive context of a subterm on demand. Definition 7 (Conjunctive Context). Given an (annotated) term T and a position p ∈ Pos(T ), we define function cc(T, p) to return the conjunctive context at position p as follows. cc(T, ǫ) = true cc(A ∧ B, 1p) = B ∧ cc(A, p) cc(A ∧ B, 2p) = A ∧ cc(B, p) cc(f (T1 , . . . , Ti , . . . , Tn ), ip) = cc(Ti , p) 9 (f 6= ∧) States and transitions. The operational semantics are defined as a set of transitions on execution states. Definition 8 (Execution States). An execution state is a tuple of the form hG, T, V, Pi, where G is a term (the goal), T is the propagation history, V is the set of variables appearing in the initial goal and P is a set of identifiers. We also define initial and final states as follows. Definition 9 (Initial and Final States). Given an initial goal G for program P , the initial state of G is hGa , ∅, vars(G), ids(Ga )i where Ga = annotate(∅, G). A final state is a state where no more rules are applicable to the goal G. We can now define the operational semantics of ACDTR as follows. Definition 10 (Operational Semantics). hG0 , T0 , V, P0 i ֌ hG1 , T1 , V, P1 i 1. Simplify: There exists a (renamed) rule from P H ⇐⇒ g \| B such that there exists a matching substitution θ and a term G′0 such that – – – – G0 ≈AC G′0 ∃p ∈ Pos(G′0 ) . G′0 \|p = θ(H) θ(g) ∈ G Ba = annotate(P0 , θ(B)) Then G1 = G′0 [Ba ]p , P1 = P0 ∪ ids(G1 ) and T1 = update(H, G′0 \|p , B, Ba , T0 ). 2. Propagate: There exists a (renamed) rule from P r @ H =⇒ g \| B such that there exists a matching substitution θ and a term G′0 such that – – – – – G0 ≈AC G′0 ∃p ∈ Pos(G′0 ) . G′0 \|p = θ(H) θ(g) ∈ G entry(r, G′0 \|p ) 6∈ T0 Ba = annotate(P0 , θ(B)) Then G1 = G′0 [G′0 \|p ∧ Ba ]p , T1 = update(H, G′0 \|p , B, Ba , T0 ) ∪ {entry(r, G′0 \|p )} and P1 = P0 ∪ ids(G1 ). 3. Simpagate: There exists a (renamed) rule from P C \ H ⇐⇒ g \| B such that there exists a matching substitution θ and a term G′0 such that 10 h(leq(X1 , Y2 )3 ∧4 leq(Y5 , Z6 )7 ∧8 ¬9 leq(X10 , Z11 )12 ), ∅i ֌trans h(leq(X1 , Y2 )3 ∧4 leq(Y5 , Z6 )7 ∧13 leq(X15 , Z16 )14 ∧8 ¬9 leq(X10 , Z11 )12 ), T i ֌idemp h(leq(X1 , Y2 )3 ∧4 leq(Y5 , Z6 )7 ∧13 leq(X15 , Z16 )14 ∧8 ¬9 true17 ), T i ֌simplif y h(leq(X1 , Y2 )3 ∧4 leq(Y5 , Z6 )7 ∧13 leq(X15 , Z16 )14 ∧8 f alse18 ), T i ֌simplif y h(leq(X1 , Y2 )3 ∧4 leq(Y5 , Z6 )7 ∧13 f alse19 ), T i ֌simplif y h(leq(X1 , Y2 )3 ∧4 f alse20 ), T i ֌simplif y h(f alse21 ), T i Fig. 1. Example derivation for the leq program. – – – – – G0 ≈AC G′0 ∃p ∈ Pos(G′0 ) . G′0 \|p = θ(H) ∃D.θ(C) ∧ D ≈AC cc(G′0 , p) θ(g) ∈ G Ba = annotate(P0 , θ(B)) Then G1 = G′0 [Ba ]p , T1 = update(H, G′0 \|p , B, Ba , T0 ) and P1 = P0 ∪ ids(G1 ). Example. Consider the leq program from Example 8 with the goal leq(X, Y ) ∧ leq(Y, Z) ∧ ¬leq(X, Z) Figure 1 shows one possible derivation of this goal to the final state representing f alse. For brevity, we omit the V and P fields, and represent identifiers as subscripts, i.e. T #i = Ti . Also we substitute T = {transitivity @ (3 2 1 7 5 6)}. We can state a soundness result for ACDTR. Theorem 1 (Soundness). If hG0 , T0 , V, Pi ֌∗ hG′ , T ′ , V, P ′ i with respect to a program P , then JP K \|= ∃vars(G′ )−V G0 ≈ G′ This means that for all algebras A that satisfy JP K, G0 and G′ are equivalent for some assignment of the fresh variables in G′ . 4 Implementation We have implemented a prototype version of ACDTR as part of the mapping language of the G12 project, called Cadmium. In this section we give an overview of the implementation details. In particular, we will focus on the implementation of conjunctive context matching, which is the main contribution of this paper. Cadmium constructs normalised terms from the bottom up. Here, a normalised term is one that cannot be reduced further by an application of a rule. Given a goal f (t1 , ..., tn ), we first must recursively normalise all of t1 , ..., tn (to say s1 , ..., sn ), and then attempt to find a rule that can be applied to the top-level of f (s1 , ..., sn ). This is the standard execution algorithm used by many TRSs implementations. 11 This approach of normalising terms bottom up is complicated by the consideration of conjunctive context matching. This is because the conjunctive context of the current term appears “higher up” in the overall goal term. Thus conjunctive context must be passed top down, yet we are normalising bottom up. This means there is no guarantee that the conjunctive context is normalised. Example 12. Consider the following ACDTR program that uses conjunctive context matching. X = V \ X ⇐⇒ var(X) ∧ nonvar(V ) \| V. one(X) ⇐⇒ X = 1. not one(1) ⇐⇒ f alse. Consider the goal not one(A)∧one(A), which we expect should be normalised to f alse. Assume that the sub-term not one(A) is selected for normalisation first. The conjunctive context for not one(A) (and its subterm A) is one(A). No rule is applicable, so not one(A) is not reduced. Next the subterm one(A) is reduced. The second rule will fire resulting in the new term A = 1. Now the conjunctive context for the first term not one(A) has changed to A = 1, so we expect that A should be rewritten to the number ⊓ ⊔ 1. However not one(A) has already being considered for normalisation. The current Cadmium prototype solves this problem by re-normalising terms when and if the conjunctive context “changes”. For example, when the conjunctive context one(A) changes to A = 1, the term not one(X) will be renormalised to not one(1) by the first rule. The general execution algorithm for Cadmium is shown in Figure 2. Function normalise takes a term T , a substitution θ, a conjunctive context CC and a Boolean value Ch which keeps track of when the conjunctive context of the current subterm has changed. If Ch = true, then we can assume the substitution θ maps variables to normalised terms. For the initial goal, we assume θ is empty, otherwise if we are executing a body of a rule, then θ is the matching substitution. Operationally, normalise splits into three cases depending on what T is. If T is a variable, and the conjunctive context has changed (i.e. Ch = true), then θ(T ) is no longer guaranteed to be normalised. In this case we return the result of renormalising θ(T ) with respect to CC. Otherwise if Ch = f alse, we simply return θ(T ) which must be already normalised. If T is a conjunction T1 ∧ T2 , we repeatedly call normalise on each conjunct with the other added to the conjunctive context. This is repeated until a fixed point (i.e. further normalisation does not result in either conjunct changing) is reached, and then return the result of apply rule on the which we will discuss below. This fixed point calculation accounts for the case where the conjunctive context of a term changes, as shown in Example 12. Otherwise, if T is any other term of the form f (T1 , ..., Tn ), construct the new term T ′ by normalising each argument. Finally we return the result of apply rule applied to T ′ . The function call apply rule(T ′ ,CC) will attempt to apply a rule to normalised term T ′ with respect to conjunctive context CC. If a matching rule is found, then 12 normalise(T ,θ,CC,Ch) if is var(T ) if Ch return normalise(θ(T ),θ,CC,f alse) else return θ(T ) else if T = T1 ∧ T2 do T1′ := T1 T2′ := T2 T1 := normalise(T1′ ,θ,T2′ ∧ CC,true) T2 := normalise(T2′ ,θ,T1′ ∧ CC,true) while T1 6= T1′ ∧ T2 6= T2′ return apply rule(T1′ ∧ T2′ ,CC) else T = f (T1 , ..., Tn ) T ′ := f (normalise(T1 ,θ,CC,Ch), ..., normalise(Tn ,θ,CC,Ch)) return apply rule(T ′ ,CC) Fig. 2. Pseudo code of the Cadmium execution algorithm. the result of normalise(B,θ,CC,f alse) is returned, where B is the (renamed) rule body and θ is the matching substitution. Otherwise, T ′ is simply returned. 5 Related Work ACDTR is closely related to both TRS and CHRs, and in this section we compare the three languages. 5.1 AC Term Rewriting Systems The problem of dealing with associative commutative operators in TRS is well studied. A popular solution is to perform the rewriting modulo some permutation of the AC operators. Although this complicates the matching algorithm, the problem of trivial non-termination (e.g. by continually rewriting with respect to commutativity) is solved. ACDTR subsumes ACTRS (Associative Commutative TRS) in that we have introduced distributivity (via simpagation rules), and added some “CHR-style” concepts such as identifiers and propagation rules. Given an ACTRS program, we can map it to an equivalent ACDTR program by interpreting each ACTRS rule H → B as the ACDTR rule H ⇐⇒ B. We can now state the theorem relating ACTRS and ACDTR. Theorem 2. Let P be an ACTRS program and T a ground term, then T →∗ S under P iff hTa , ∅, ∅, ids(Ta )i ֌∗ hSa , ∅, ∅, Pi under α(P ) (where Ta = annotate(∅, T )) for some P and term(Sa ) = S. 13 5.2 CHRs and CHR∨ ACDTR has been deliberately designed to be an extension of CHRs. Several CHR concepts, e.g. propagation rules, etc., have been adapted. There are differences between CHRs and ACDTR. The main difference is that ACDTR does not have a “built-in” or “underlying” solver, i.e. ACDTR is not a constraint programming language. However it is possible to encode solvers directly as rules, e.g. the simple leq solver from Example 8. Another important difference is that CHRs is based on predicate logic, where there exists a distinction between predicate symbols (i.e. the names of the constraints) and functions (used to construct terms). ACDTR is based on equational logic between terms, hence there is no distinction between predicates and functions (a predicate is just a Boolean function). To overcome this, we assume the existence of a set Pred, which contains the set of function symbols that are Boolean functions. We assume that AC ∩ Pred = {∧(2) }. The mapping between a CHR program and an ACDTR program is simply α(P ) = P ∪ {X ∧ true ⇐⇒ X}.4 However, we assume program P is restricted as follows: – rules have no guards apart from implicit equality guards; and – the only built-in constraint is true and the initial goal G is also restricted: – G must be of the form G0 ∧ ... ∧ Gn for n > 0; – Each Gi is of the form fi (A0 , ..., Am ) for m ≥ 0 and fi ∈ Pred; – For all p ∈ Pos(Aj ), 0 ≤ j ≤ m we have that if Aj \|p = g(B0 , ..., Bq ) then g (q) 6∈ AC and g (q) 6∈ Pred. These conditions disallow predicate symbols from appearing as arguments in CHR constraints. Theorem 3. Let P be a CHR program, and G an initial goal both satisfying V the above conditions, then hG, ∅, true, ∅iV 1 ֌ h∅, S, true, T ii (for some T , i and V = vars(G)) under the theoretical operational semantics [4] for CHRs iff hGa , ∅, V, ids(Ga )i ֌ hSa , T ′ , V, Pi (for some T ′ , P) under ACDTR, where term(Sa ) = S1 ∧...∧Sn and S = {S1 #i1 , ..., Sn #in } for some identifiers i1 , ..., in . We believe that Theorem 3 could be extended to include CHR programs that extend an underlying solver, provided the rules for handling tell constraints are added to the ACDTR program. For example, we can combine rules for rational tree unification with the leq program from Example 8 to get a program equivalent to the traditional leq program under CHRs. ACDTR generalises CHRs by allowing other operators besides conjunction inside the head or body of rules. One such extension of CHRs has been studied before, namely CHR∨ [2] which allows disjunction in the body. Unlike ACDTR, 4 There is one slight difference in syntax: CHRs use ‘,’ to represent conjunction, whereas ACDTR uses ‘∧’. 14 which manipulates disjunction syntactically, CHR∨ typically finds solutions using backtracking search. One notable implementation of CHR∨ is [6], which has an operational semantics described as an and/or (∧/∨) tree rewriting system. A limited form of conjunctive context matching is used, similar to that used by ACDTR, based on the knowledge that conjunction ∧ distributes over disjunction ∨. ACDTR generalises this by distributing over all functions. 6 Future Work and Conclusions We have presented a powerful new rule-based programming language, ACDTR, that naturally extends both AC term rewriting and CHRs. The main contribution is the ability to match a rule against the conjunctive context of a (sub)term, taking advantage of the distributive property of conjunction over all possible functions. We have shown this is a natural way of expressing some problems, and by building the distributive property into the matching algorithm, we avoid non-termination issues that arise from naively implementing distribution (e.g. as rewrite rules). We intend that ACDTR will become the theoretical basis for the Cadmium constraint mapping language as part of the G12 project [7]. Work on ACDTR and Cadmium is ongoing, and there is a wide scope for future work, such as confluence, termination and implementation/optimisation issues. References 1. S. Abdennadher. Operational semantics and confluence of constraint propagation rules. In Gert Smolka, editor, Proceedings of the Third International Conference on Principles and Practice of Constraint Programming, LNCS 1330, pages 252–266. Springer-Verlag, 1997. 2. S. Abdennadher and H. Schütz. CHR∨ : A flexible query language. In International conference on Flexible Query Answering Systems, number 1495 in LNCS, pages 1–14, Roskilde, Denmark, 1998. Springer-Verlag. 3. F. Baader and T. Nipkow. Term rewriting and all that. Cambridge Univ. Press, 1998. 4. G. Duck, P. Stuckey, M. Garcia de la Banda, and C. Holzbaur. The refined operational semantics of constraint handling rules. In B. Demoen and V. Lifschitz, editors, Proceedings of the 20th International Conference on Logic Programming, LNCS 3132, pages 90–104. Springer-Verlag, September 2004. 5. T. Frühwirth. Theory and practice of constraint handling rules. Journal of Logic Programming, 37:95–138, 1998. 6. L. Menezes, J. Vitorino, and M. Aurelio. A High Performance CHR∨ Execution Engine. In Second Workshop on Constraint Handling Rules, Sitges, Spain, 2005. 7. P.J. Stuckey, M. Garcia de la Banda, M. Maher, K. Marriott, J. Slaney, Z. Somogyi, M. Wallace, and T. Walsh. The G12 project: Mapping solver independent models to efficient solutions. In M. Gabrielli and G. Gupta, editors, Proceedings of the 21st International Conference on Logic Programming, number 3668 in LNCS, pages 9–13. Springer-Verlag, 2005. 15 A Examples A.1 Further Motivating Examples Example 13 (Conjunctive Normal Form). One of the roles of mapping models is to convert a model written in an expressive language into a restricted language which is easy to solve. Many standard approaches to solving propositional formulae require that the formulae are in conjunctive normal form (CNF). Disjunction ∨ is distributive over ∧, which can be used to establish CNF in a direct way, using the oriented rule P ∨ Q ∧ R → (P ∨ Q) ∧ (P ∨ R). CNF conversion based on this rule can exponentially increase the size of the formula. This undesirable circumstance means that in practice CNF conversions are preferred that replace subformulae by new propositional atoms, which increases the formula size at most linearly. Let us formulate this approach in rewrite rules. To keep this example simple, we assume that the non-CNF subformula P ∨ Q ∧ R occurs in a positive context (for example by a preprocessing into negation normal form). We replace Q ∧ R by a new atom s defined by the logical implication s ⇒ (Q ∧ R). In rewrite rule form, we have P ∨ Q ∧ R → (P ∨ s) ∧ (¬s ∨ Q) ∧ (¬s ∨ R). (8) Unit resolution and unit subsumption can be formalised in rewrite rules. Here are two versions, one using conjunctive context and a regular one: with conj. context: regular: P \ P ⇐⇒ true P ∧P → P P ∧ (P ∨ Q) → P P ∧ ¬P → false P \ ¬P ⇐⇒ false P ∧ (¬P ∨ Q) → P ∧ Q We furthermore assume rules eliminating the logical constants true and false from conjunctions and disjunctions in the obvious way. Let us contrast the two rule sets for the formula (a∨b∧(c∨d))∧d. The following is a terminating rewrite history: with conj. context: regular: (a ∨ b ∧ (c ∨ d)) ∧ d (a ∨ b ∧ (c ∨ d)) ∧ d ֌ (a ∨ b ∧ (c ∨ true)) ∧ d ֌ (a ∨ b ∧ true) ∧ d ֌ ֌ (a ∨ s) ∧ (¬s ∨ b) ∧ (¬s ∨ c ∨ d) ∧ d (a ∨ s) ∧ (¬s ∨ b) ∧ true ∧ d ֌ (a ∨ b) ∧ d ֌ (a ∨ s) ∧ (¬s ∨ b) ∧ d 16 To obtain the simple conjunct (a ∨ b) using the regular rule format, a rule expressing binary resolution, i.e. from (P ∨ S) ∧ (¬S ∨ Q) follows (P ∨ Q), would be required. However, such a rule is undesirable as it would create arbitrary binary resolvents, increasing formula size. Moreover, the superfluous atom s remains in the formula. ⊓ ⊔ Example 14 (Type remapping). One of the main model mappings we are interested in expressing is where the type of a variable is changed from a high level type easy for modelling to a low level type easy to solve. A prime example of this is mapping a set variable x ranging over finite subsets of some fixed set s to an array x′ of 0/1 variables indexed by s. So for variable x we have e ∈ x ⇔ x′ [e] = 1. For this example we use the more concrete modelling syntax: t : x indicates variable x has type t, the types we are interested are l..u an integers in the range l to u, set of S a set ranging over elements in S, and array[I] of E an array indexed by set I of elements of type E. We use f orall and sum looping constructs which iterate over sets. This is expressed in ACDTR as follows. set of s : x ⇐⇒ array[s] of 0..1 : x′ ∧ map(x, x′ ) map(x, x′ ) \ x ⇐⇒ x′ array[s] of 0..1 : x \ card(x) ⇐⇒ sum(e in s) x[e] array[s] of 0..1 : x ∧ z :: (array[s] of 0..1 : z ∧ ⇐⇒ array[s] of 0..1 : y \ x ∩ y f orall(e in s) z[e] = x[e] && y[e]) array[s] of 0..1 : x ∧ z :: (array[s] of 0..1 : z ∧ ⇐⇒ array[s] of 0..1 : y \ x ∪ y f orall(e in s) z[e] = x[e] \|\| y[e]) array[s] of 0..1 : x \ x = ∅ ⇐⇒ f orall(e in s) x[e] = 0 card(t :: c) ⇐⇒ card(t) :: c (t1 :: c) ∪ t2 ⇐⇒ t1 ∪ t2 :: c t1 ∪ (t2 :: c) ⇐⇒ t1 ∪ t2 :: c (t1 :: c) ∩ t2 ⇐⇒ t1 ∩ t2 :: c t1 ∩ (t2 :: c) ⇐⇒ t1 ∩ t2 :: c (t1 :: c) = t2 ⇐⇒ t1 = t2 ∧ c t1 = (t2 :: c) ⇐⇒ t1 = t2 ∧ c (t1 :: c) ≤ t2 ⇐⇒ t1 ≤ t2 ∧ c t1 ≤ (t2 :: c) ⇐⇒ t1 ≤ t2 ∧ c (t :: c1 ) :: c2 ⇐⇒ t :: (c1 ∧ c2 ) maxOverlap(x, y, c) ⇐⇒ card(x ∩ y) ≤ c (typec) (vsubs) (card) (cap) (cup) (emptyset) (↑ card) (↑ cupl) (↑ cupr) (↑ capl) (↑ capr) (↑ eql) (↑ eqr) (↑ leql) (↑ leqr) (↑ cc) (maxO) The :: constructor adds some local conjunctive context to an arbitrary term (like where) and the last 11 rules bar 1 move this context outwards to the nearest predicate scope. The last rule defines the maxOverlap predicate. They are used to introduce new variables z and their type and the constraints upon then. As 17 an example, consider the following derivation: ֌maxO ֌typec ֌vsubs ֌typec ֌vsubs ֌cap ֌↑card ֌↑leql set of 1..n : x ∧ set of 1..n : y ∧ maxOverlap(x, y, 1) set of 1..n : x ∧ set of 1..n : y ∧ card(x ∩ y) ≤ 1 array[1..n] of 0..1 : x′ ∧ map(x, x′ ) ∧ set of 1..n : y ∧ card(x ∩ y) ≤ 1 array[1..n] of 0..1 : x′ ∧ map(x, x′ ) ∧ set of 1..n : y ∧ card(x′ ∩ y) ≤ 1 array[1..n] of 0..1 : x′ ∧ map(x, x′ ) ∧ array[1..n] of 0..1 : y ′ ∧ map(y, y ′ ) ∧ card(x′ ∩ y) ≤ 1 array[1..n] of 0..1 : x′ ∧ map(x, x′ ) ∧ array[1..n] of 0..1 : y ′ ∧ map(y, y ′ ) ∧ card(x′ ∩ y ′ ) ≤ 1 array[1..n] of 0..1 : x′ ∧ map(x, x′ ) ∧ array[1..n] of 0..1 : y ′ ∧ map(y, y ′ ) ∧ card(z :: (array[1..n] of 0..1 : z ∧ f orall(e in 1..n) z[e] = x′ [e] && y ′ [e]) ≤ 1 array[1..n] of 0..1 : x′ ∧ map(x, x′ ) ∧ array[1..n] of 0..1 : y ′ ∧ map(y, y ′ ) ∧ card(z) :: (array[1..n] of 0..1 : z ∧ f orall(e in 1..n) z[e] = x′ [e] && y ′ [e]) ≤ 1 array[1..n] of 0..1 : x′ ∧ map(x, x′ ) ∧ array[1..n] of 0..1 : y ′ ∧ map(y, y ′ ) ∧ card(z) ≤ 1 ∧ array[1..n] of 0..1 : z ∧ f orall(e in 1..n) z[e] = x′ [e] && y ′ [e] The final goal is a flat conjunction of constraints and types. It can be similarly translated into a conjunction of pseudo-Boolean constraints that can be sent to a finite domain solver, by unrolling f orall and replacing the arrays by sequences of n variables. ⊓ ⊔ Example 15 (Rational Tree Unification). We can directly express the rational tree unification algorithm of Colmerauer5 as an ACD term rewriting system. f (s1 , . . . sn ) = f (t1 , . . . , tn ) ⇐⇒ s1 = t1 ∧ · · · sn = tn f (s1 , . . . sn ) = g(t1 , . . . , tm ) ⇐⇒ f alse (split) (f ail) The (split) rule must be defined for each constructor f /n and the (fail) rule for each pair of different constructors f /n and g/m. The remaining rules are: x = x ⇐⇒ var(x) \| true t = x ⇐⇒ var(x) ∧ nonvar(t) \| x = t x = s \ x = t ⇐⇒ var(x) ∧ nonvar(s) ∧ size(s) ≤ size(t) \| s = t x = y \ x ⇐⇒ var(x) ∧ var(y) ∧ x 6≡ y \| y (id) (f lip) (tsubs) (vsubs) where size(t) is the size of the term t in terms of number of symbols, and ≡ is syntactic identity. Even though the goals are a single conjunction of constraints, ACD is used for succinctly expressing the (vsubs) rule which replaces one variable by another in any other position. 5 A. Colmerauer. Prolog and Infinite Trees. Logic Programming, APIC Studies in Data Processing (16). Academic Press. 1992 18 The following derivation illustrates the unification process in action. The underlined part show the matching elements ֌f lip ֌vsubs ֌vsubs ֌tsubs ֌split ֌split ֌tsubs ֌split ֌tsubs ֌split ֌id x = y ∧ f (f (x)) = x ∧ y = f (f (f (y))) x = y ∧ x = f (f (x)) ∧ y = f (f (f (y))) x = y ∧ y = f (f (x)) ∧ y = f (f (f (y))) x = y ∧ y = f (f (y)) ∧ y = f (f (f (y))) x = y ∧ y = f (f (y)) ∧ f (f (y)) = f (f (f (y))) x = y ∧ y = f (f (y)) ∧ f (y) = f (f (y)) x = y ∧ y = f (f (y)) ∧ y = f (y) x = y ∧ f (y) = f (f (y)) ∧ y = f (y) x = y ∧ y = f (y) ∧ y = f (y) x = y ∧ y = f (y) ∧ f (y) = f (y) x = y ∧ y = f (y) ∧ f (y) = f (y) x = y ∧ y = f (y) ∧ true ⊓ ⊔ A.2 Expanded Examples The purpose of this section is to show some example derivations under the operational semantics of ACDTR, rather than high-level descriptions. We allow for some shorthand, namely T #i = Ti . Identifiers and conjunctive context. In this section we explain parts of the derivation from Example 15 in more detail. The initial goal is x = y ∧ f (f (x)) = x ∧ y = f (f (f (y))) which corresponds to the initial state: h(((x1 = y2 )3 ∧ (f (f (x4 )5 )6 = x7 )8 )9 ∧ (y10 = f (f (f (y11 )12 )13 )14 )15 )16 , ∅, {x, y}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}i The initial state is a quadruple contained an annotated version of the goal, an empty propagation history, the set of variables in the goal and a set of “used” identifiers. The first derivation step is a Simplify transition with the f lip rule: h(((x1 = y2 )3 ∧ (f (f (x4 )5 )6 = x7 )8 )9 ∧ (y10 = f (f (f (y11 )12 )13 )14 )15 )16 , ∅, {x, y}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}i ֌ h(((x1 = y2 )3 ∧ (x17 = f (f (x18 )19 )20 )21 )9 ∧ (y10 = f (f (f (y11 )12 )13 )14 )15 )16 , ∅, {x, y}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21}i We have replaced the annotated subterm (f (f (x4 )5 )6 = x7 )8 with x17 = f (f (x18 )19 )20 )21 (i.e. flipped the operands to the equality) and reannotated the 19 new term with fresh identifiers. These were also added to the set of used identifiers. Since the propagation history is empty, it remains unchanged. The next derivation step is a Simpagate transition with the vsubs rule. h(((x1 = y2 )3 ∧ (x17 = f (f (x18 )19 )20 )21 )9 ∧ (y10 = f (f (f (y11 )12 )13 )14 )15 )16 , ∅, {x, y}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21}i ֌ h(((x1 = y2 )3 ∧ (y21 = f (f (x18 )19 )20 )21 )9 ∧ (y10 = f (f (f (y11 )12 )13 )14 )15 )16 , ∅, {x, y}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22}i The conjunctive context for subterm x17 is cc(Ga , p) = (x1 = y2 )3 ∧ (y10 = f (f (f (y11 )12 )13 )14 )15 ∧ true where Ga is the current goal and p is the position of x17 . The first conjunct matches the conjunctive context of the vsubs rule, thus subterm x17 is replaced with y21 . Identifier 21 is added to the list of used identifiers. Execution proceeds until the final state h(x = y ∧ y = f (y)) ∧ true, ∅, {x, y}, Pi is reached, for some annotation of the goal and some set of identifiers P. This is a final state because no more rules are applicable to it. AC matching and propagation histories. Consider the propagation rule from the leq program: trans @ leq(X, Y ) ∧ leq(Y, Z) =⇒ X 6≡ Y ∧ Y 6≡ Z \| leq(X, Z) and the initial state hleq(A1 , B2 )3 ∧4 leq(B5 , A6 )7 , ∅, {A, B}, {1, 2, 3, 4, 5, 6, 7}i. We can apply Propagate directly (i.e. without permuting the conjunction) to arrive at the state: h(leq(A1 , B2 )3 ∧4 leq(B5 , A6 )7 ) ∧8 leq(A9 , A10 )11 , {trans @ (3 1 2 7 6 5)}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}i. The propagation history prevents the rule from firing on the same terms again, however we can permute the terms to find a new matching. Namely, we can permute the annotated goal (which we call Ga ) (leq(A1 , B2 )3 ∧4 leq(B5 , A6 )7 ) ∧8 leq(A9 , A10 )11 to (leq(B5 , A6 )7 ∧4 leq(A1 , B2 )3 ) ∧8 leq(A9 , A10 )11 . 20 The latter is an element of [Ga ]AC , and the identifiers have been preserved in the correct way. The entry trans @ (7 6 5 3 1 2) is not in the propagation history, so we can apply Propagate again to arrive at: h((leq(B5 , A6 )7 ∧4 leq(A1 , B2 )3 ) ∧12 leq(B13 , B14 )15 ) ∧8 leq(A9 , A10 )11 , {trans @ (3 1 2 7 6 5), trans @ (7 6 5 3 1 2)}, {1...15}i. Now the propagation history prevents the rule trans being applied to the first two leq constraints. The guard also prevents the trans rule firing on either of the two new constraints,6 thus we have reached a final state. Updating propagation histories. Consider a modified version of the previous example, now with two rules: X ∧ X ⇐⇒ X trans @ leq(X, Y ) ∧ leq(Y, Z) =⇒ leq(X, Z) The first rule enforces idempotence of conjunction. Consider the initial state: hleq(A1 , A2 )3 ∧4 leq(A5 , A6 )7 ∧8 leq(A9 , A10 )11 , ∅, {A}, {1...11}i We apply the trans rule to the first two copies of the leq constraint (with identifiers 3 and 7). hleq(A1 , A2 )3 ∧4 leq(A5 , A6 )7 ∧8 leq(A9 , A10 )11 ∧12 leq(A13 , A14 )15 , {trans @ (3 1 2 7 5 6)}, {A}, {1...15}i Next we apply idempotence to leq constraints with identifiers 7 and 11. hleq(A1 , A2 )3 ∧4 leq(A16 , A17 )18 ∧12 leq(A13 , A14 )15 , {trans @ (3 1 2 7 5 6), trans @ (3 1 2 18 16 17)}, {A}, {1...18}i An extra entry (trans @ (3 1 2 18 16 17)) is added to the propagation history in order to satisfy the requirements of Definition 6. This is because we have replaced the annotated constraint leq(A5 , A6 )7 with the newly annotated term leq(A16 , A17 )18 , which defines an identifier renaming ρ = {5 7→ 16, 6 7→ 17, 7 7→ 18}. Since E = (trans @ (3 1 2 7 5 6)) is an element of the propagation history, we have that ρ(E) = (trans @ (3 1 2 18 16 17)) must also be an element, and hence the history is expanded. 6 Without the guard both ACDTR and CHRs are not guaranteed to terminate. 21	6

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arXiv:1611.04732v1 [math.AC] 15 Nov 2016 BETTI NUMBERS OF CERTAIN SUM IDEALS JOYDIP SAHA, INDRANATH SENGUPTA, AND GAURAB TRIPATHI A BSTRACT. In this paper we compute the Betti numbers for ideals of the form I1 (XY ) + J, where X and Y are matrices and J is the ideal generated by the 2 × 2 minors of the matrix consisting of any two rows of X. 1. I NTRODUCTION Let K be a field and {xij ; 1 ≤ i, j ≤ n}, {yj ; 1 ≤ j ≤ n} be indeterminates over K; n ≥ 2. Let R := K[xij , yj ] denote the polynomial algebra over K. Let X denote an n × n matrix such that its entries belong to the ideal h{xij ; 1 ≤ i ≤ n, 1 ≤ j ≤ n}i. Let Y = (yj )n×1 be the n × 1 column Pnmatrix. Let I1 (XY ) denote the ideal generated by the polynomials gj = i=1 xji yi , j = 1, . . . , n, which are the 1 × 1 minors or entries of the n × n matrix XY . The primality, primary decomposition and Betti numbers of ideals of the form I1 (XY ) have been studied in [9] and [10], with the help of Gröbner bases for I1 (XY ). Ideals of the form I1 (XY ) + J are particularly interesting because they occur in several geometric considerations like linkage and generic residual intersection of polynomial ideals, especially in the context of syzygies. Bruns-Kustin-Miller [1] resolved the ideal I1 (XY )+Imin(m,n) (X), where X is a generic m×n matrix and Y is a generic n×1 matrix. Johnson-McLoud [5] proved certain properties for the ideals of the form I1 (XY ) + I2 (X), where X is a generic symmetric matrix and Y is either generic or generic alternating. We say that I and J intersect transversally if I ∩J = IJ. Suppose that F· resolves R/I and G· resolves R/J minimally. It is interesting to note that if I and J intersect transversally, then the tensor product complex F ⊗R G 2010 Mathematics Subject Classification. Primary 13D02; Secondary 13C40, 13P10, 13D07. Key words and phrases. Gröbner basis, Betti numbers, transversal intersection, mapping cone. The first author thanks UGC for the Senior Research Fellowship. The second author is the corresponding author, who is supported by the the research project EMR/2015/000776 sponsored by the SERB, Government of India. The third author thanks CSIR for the Senior Research Fellowship. 1 2 JOYDIP SAHA, INDRANATH SENGUPTA, AND GAURAB TRIPATHI resolves R/I + J minimally; see Lemma 3.7. Therefore, it is useful to know if two ideals intersect transversally, especially when one is trying to compute minimal free resolutions and Betti numbers for ideals of the form I + J, through iterated techniques; see [4]. 2. N OTATION M AIN T HEOREM xi1 xi2 · · · xin e • If X is generic and i < j; let Xij = . xj1 xj2 · · · xjn • If X is generic symmetric and i < j; let x1i · · · xii · · · xij · · · xin e Xij = . x1j · · · xij · · · xjj · · · xjn AND THE eij . • Let Gij denote the set of all 2 × 2 minors of X eij ) denote the ideal generated Gij . • I2 ( X Our aim in this paper is to prove the following theorem: Theorem 2.1. Let X = (xij ) be either the generic or the generic symmetric matrix of order n. Let 1 ≤ i < j ≤ n. eij ) + hgi , gj i are given by (1) The total Betti numbers for theideal I2 (X n n n b0 = 1, b1 = 2 + 2, b2 = 2 3 + n, bi+1 = i i+1 + (i − 2) ni , for 2 ≤ i ≤ n − 2 and bn = n − 2. (2) Let 1 ≤ k ≤ n−2. Let βk,p denote the p-th total Betti number for the eij )+hgi , gj , gl1 , . . . , gl i, such that 1 ≤ l1 < . . . < lk ≤ n ideal I2 (X k and lt is the smallest in the set {1, 2, . . . , n} \ {i, j, l1 , . . . , lt−1 }, for every 1 ≤ t ≤ k. They are given by βk,0 = 1, βk,p = βk−1,p−1 + βk−1,p for 1 ≤ p ≤ n + k − 1 and βk,n+k = n − 2. eij ) are In particular, the total Betti numbers for the ideal I1 (XY ) + I2 (X βn−2,0 , βn−2,1 , . . . , βn−2,2n−2 . 3. P RELIMINARIES 3.1. Determinantal Ideals. We recall some useful results on determinantal ideals pertaining to our work. We refer to [2], [3], [8] for detailed discussions on these. Theorem 3.1. Let K be a field and let xij : 1 ≤ i ≤ m, 1 ≤ j ≤ n be indeterminates over K. Let A = (xij ) be the m×n matrix of indeterminates and Im (A) denotes the ideal generated by the maximal minors of A. The set of maximal minors of A is a universal Gröbner basis for the ideal Im (A). SUM IDEALS 3 Proof. See [2]. The Eagon-Northcott Complex. We present the relevant portion from the book [3] here. Let F = Rf and G = Rg be free modules of finite rank over the polynomial ring R. The Eagon-Northcott complex of a map α : F −→ G (or that of a matrix A representing α) is a complex df −g df −g+1 EN(α) : 0 → (Symf −g G)∗ ⊗ ∧f F −→ (Symf −g−1 G)∗ ⊗ ∧f −1 F −→ d d2 g ∧g α 3 · · · −→ (Sym2G)∗ ⊗ ∧g+2 F −→ G∗ ⊗ ∧g+1 F ∧ F −→ ∧g G. Here Symk G is the k-th symmetric power of G and M ∗ = HomR (M, R). The map dj are defined as follows. First we define a diagonal map (Symk G)∗ → G∗ ⊗ (Symk−1G)∗ X ′ ′′ ui ⊗ ui u 7→ i as the dual of the multiplication map G ⊗ Symk−1G −→ Symk G in the symmetric algebra of G. Next we define an analogous diagonal map ∧k F −→ F ⊗ ∧k−1 F X ′ ′′ v 7→ vi ⊗ vi i as the dual of the multiplication in the exterior algebra of F ∗ . Theorem 3.2 (Eagon-Northcott). The Eagon-Northcott complex is a free resolution of R/Ig (α) iff grade(Ig (α)) = f − g + 1 where Ig (α) denotes the g × g minors of the matrix A representing α. Proof. See [3]. 3.2. Mapping Cone. We present the relevant portion from the book [8] ′ ′ here. Let R be the polynomial ring. Let φ· : (U· , d· ) → (U· , d· ) be a map of complexes of finitely generated R-modules. The mapping cone of φ· is the ′ complex W· with differential δ· defined as follows. Let Wi = Ui−1 ⊕ Ui , ′ ′ ′ ′ with δ\|Ui−1 = −d + φ : Ui−1 −→ Ui−2 ⊕ Ui−1 and δ\|U ′ = d : Ui → Ui−1 i for each i. Theorem 3.3. Let M be an ideal minimally generated by the polynomials f1 , . . . , fr . Set Mi = hf1 , . . . , fi i, for 1 ≤ i ≤ r. Thus, M = Mr . For each i ≥ 1, we have the short exact sequence fi+1 0 −→ S/(Mi : fi+1 ) −→ S/Mi −→ S/Mi+1 −→ 0. If resolutions of S/Mi and S/(Mi : fi+1 ) are known then we can construct a resolution of S/Mi+1 by the mapping cone construction. 4 JOYDIP SAHA, INDRANATH SENGUPTA, AND GAURAB TRIPATHI Proof. See Construction 27.3 in [8]. 3.3. Gröbner basis and transversal intersection of Ideals. Lemma 3.4. Let h1 , h2 · · · , hn ∈ R be such that with respect to a suitable monomial order on R, the leading terms of them are mutually coprime. Then, h1 , h2 · · · , hn is a regular sequence in R. Proof. . See Lemma 4.3 in [9]. Lemma 3.5. Suppose that X is either generic or generic symmetric. The eij ), with respect a suitable set Gij is a Gröbner basis for the ideal I2 (X monomial order. Proof. We choose the lexicographic monomial order given by the following ′ ′ ordering among the variables: xst > xs′ t′ if (s , t ) > (s, t) and yn > yn−1 > · · · , y1 > xst for all s, t. We now apply Lemma 4.2 in [9] for the matrix X t and for k = 2. Definition 1. Let T ⊂ R be a set of monomials. We define supp(T ) = {(i, j, 0) \| xij divides m for some m ∈ T } ∪ {(0, 0, k) \| yk divides m for some m ∈ T }. If T = {m}, then we write supp(m) instead of supp({m}). Lemma 3.6. Let > be a monomial ordering on R. Let I and J be ideals in R, such that m(I) and m(J) denote unique minimal generating sets for their leading ideals Lt(I) and Lt(J) respectively. Then, I ∩ J = IJ if supp(m(I))∩supp(m(J)) = ∅. In other words, the ideals I and J intersect transversally if the set of variables occurring in the set m(I) is disjointed from the the set of variables occurring in the set m(J). Proof. Let f ∈ I ∩ J; we show that f ∈ IJ. Now f ∈ I ∩ J implies that f ∈ I and therefore Lt(f ) ∈ Lt(I). Hence, mi ∈ m(I) such that mi \| Lt(f ). Similarly, there exists monomial mj ∈ m(J) such that mj \| Lt(f ). Given that mi and mj are of disjoint support, we have mi mj \| Lt(f ) and this (f ) proves that Lt(f ) ∈ Lt(IJ). We replace f by f − Lt . Now the proof mi mj Lt(f ) < Lt(f ). follows by induction since, Lt f − m i mj SUM IDEALS 5 3.4. Homological Lemmas. Lemma 3.7. Let I and J be graded ideals in a graded ring R, such that I ∩ J = I · J. Suppose that F and G are minimal free resolutions of I and J respectively. Then F ⊗ G is a minimal free resolution for the graded ideal I + J. Proof. Consider the short exact sequence 0 −→ I −→ R −→ R/I −→ 0 and tensor it with R/J over R. We get the exact sequence 0 −→ Tor1 (R/I, R/J) −→ I/I · J −→ R/J −→ R/I + J −→ 0. The terms on the left are 0 since R is a flat R module. Moreover, the kernel of the map from I/I · J −→ R/J is I ∩ J/I · J. Therefore Tor1 (R/I, R/J) = 0 if and only if I ∩J = I ·J. By the corollary 1 of theorem 3 proved in [6], Tor1 (R/I, R/J) = 0 implies that Tori (R/I, R/J) = 0 for all i ≥ 1. Therefore, Hi (F ⊗ G ) ≃ Tori (R/I, R/J) ≃ 0 for all i ≥ 1 and H0 (F ⊗ G ) ≃ R/I + J. This proves that F ⊗ G resolves I + J. The resolution is minimal since F and G are minimal. Lemma 3.8. Let A A 1 2 Ra1 −→ Ra2 −→ Ra3 be an exact sequence of free modules. Let Q1 , Q2 , Q3 be invertible matrices of sizes a1 , a2 , a3 respectively. Then, Ra1 Q−1 2 A1 Q 1 Ra2 −→ Q−1 3 A2 Q 2 −→ Ra3 is also an exact sequence of free modules. Proof. The following diagram is a commutative diagram is free modules and the vertical maps are isomorphisms: A1 ROa1 / RaO 2 Q1 Q−1 2 A1 Q 1 Ra1 Therefore, Ra1 Ra3 is exact. Q−1 2 A1 Q 1 −→ Ra2 / R Q−1 3 A2 Q 2 −→ A2 / RaO 3 Q2 Q3 Q−1 A2 Q 2 3 a2 / a3 R A A 1 2 Ra3 is exact since Ra1 −→ Ra2 −→ Corollary 3.9. Let C B A Ra1 −→ Ra2 −→ Ra3 −→ Ra4 be an exact sequence of free modules. Let P1 , P2 , P3 be invertible matrices of sizes a1 , a2 , a3 respectively. Then, P −1 CP1 BP AP −1 3 Ra1 2−→ Ra2 −→2 Ra3 −→ Ra4 6 JOYDIP SAHA, INDRANATH SENGUPTA, AND GAURAB TRIPATHI is also an exact sequence of free modules. C B Proof. Consider the sequence Ra1 −→ Ra2 −→ Ra3 . If we take Q1 = P1 , Q2 = P2 and Q3 = I and apply Lemma ??, we get that the sequence Ra1 P2−1 CP1 −→ BP Ra2 −→2 Ra3 is exact. We further note that the entire se- P −1 CP1 BP A An+1 A quence Ra1 2−→ Ra2 −→2 Ra3 −→ Ra4 is exact as well, since Im(B) = BP Im(BP2 ) and P2 is invertible. Let us now consider the sequence Ra2 −→2 A Ra3 −→ Ra4 . We take Q1 = Q3 = I, Q2 = P3−1 and apply Lemma 2.3 to arrive at our conclusion. Lemma 3.10. Let An−1 n · · · −→ Rβn+1 −→ Rβn −→ Rβn−1 −→ Rβn−2 −→ · · · be an exact sequence of free R modules. Let aij denote the (i, j)-th entry of An . Suppose that alm = ±1 for some l and m, ali = 0 for i 6= m and ′ ajm = 0 for j 6= l. Let An+1 be the matrix obtained by deleting the m-th ′ row from An+1 , An−1 the matrix obtained by deleting the l-th column from ′ An−1 and An the matrix obtained by deleting the l-th row and m-th column from An . Then, the sequence ′ · · · −→ R βn+1 An+1 −→ R ′ ′ βn −1 An −→ R βn−1 −1 An−1 −→ Rβn−2 −→ · · · is exact. Proof. The fact that the latter sequence is a complex is self evident. We need to prove its exactness. By the previous lemma we may assume that l = m = 1, for we choose elementary matrices to permute rows and columns and these matrices are always invertible. Now, due to exactness of the first complex we have An−1 An = 0. This implies that the first col′ umn of An−1 = 0, which implies that Im(An−1 ) = Im(An−1 ). Therefore, the right exactness of An+1 is preserved. By a similar argument we can ′ prove that the left exactness of An+1 is preserved. ′ from R. If (x) ∈ ker(A Let (x) denote a tuple with entries n ), then (0, x) ∈ ker(An ). There exists y ∈ Rβn+1 such that An−1 y = (0, x). ′ ′ It follows that An−1 y = (x), proving the left exactness of An . By a ′ similar argument we can prove the right exactness of An . Lemma 3.11. Let A be q × p matrix over R with Aij = ±1, for some i and j. Let C be a p × s matrix and B a r × q matrix over R. There exist an invertible q × q matrix X and an invertible p × p matrix Y , such that SUM IDEALS 7 (i) (XAY )kj = δki and (XAY )ik = δjk , that is   ··· 0 ···  ··· 0 ···    ..  .. .  . .  .  .   XAY = 0 · · · 0 1 0 · · · 0 ; 1 at the (i, j)−th spot.  . ..  ..  .. .  .    ··· 0 ···  ··· 0 ··· P (ii) (Y −1 C)kl = Ckl for k 6= j and (Y −1 C)jl = Cjl + t6=i (ait )Ctl P (iii) (BX −1 )kl = Bkl for l 6= i and (BX −1 )ki = Bki + t6=i (atj )Bkt . Proof. (i) We prove for aij = 1. The other case is similar. We take Y = Πk6=j Ejk (−aik ) and X = Πk6=i Eki (−akj ), where Ekl (α) denotes the matrix E with Ekl = α, Ett = 1 and Eut = 0 for u 6= t and (u, t) 6= (k, l). (ii) and (iii) are easy to verify. Lemma 3.12. Let A be q × p matrix, C be a p × s matrix and B a r × q matrix over R. The matrices A, B and C satisfy property Pij if they satisfy the following conditions: • Aij = 1 , Aik ∈ m for k 6= j and Akj ∈ m for k 6= i; • Bki ∈ m, for 1 ≤ k ≤ r; • Cjl ∈ m, for 1 ≤ l ≤ s. The matrices XAY , BX −1 and Y −1 C satisfy property Pij , if A, B, C satisfy property Pij . Proof. This follows from the above lemma since aik and akj belong to m. 4. M INIMAL eij ) + hgi , gj i FREE RESOLUTION OF I2 (X Lemma 4.1. Let X be generic or generic symmetric. Let i < j. eij )) = n − 1. (i) ht(I2 (X fij ). (ii) The Eagon-Northcott complex minimally resolves the ideal I2 (X Proof. (i) We show that f1 , . . . , fn−1 , given by fk = xik xj,k+1 − xjk xi,k+1, 1 ≤ k ≤ n − 1 form a regular sequence. Let us first assume that X is generic. We take the lexicographic monomial order induced by the following ordering among the variables: xi1 > e and xi2 > · · · > xin > xj2 > xj3 > · · · > xjn > xj1 not appearing in X the variables yk are smaller than xj1 . Then, Lt(fk ) = xik xj,k+1 and hence 8 JOYDIP SAHA, INDRANATH SENGUPTA, AND GAURAB TRIPATHI gcd(Lt(fk ), Lt(fl )) = 1 for every k 6= l. Therefore, f1 , . . . , fn−1 is a regular e ≥ n − 1. On the other hand, sequence by Lemma 3.4 and hence ht(I2 (X)) e ≤ n − 1, by Theorem [13.10] in [7]. Hence, ht(I2 (X)) e ≤ n − 1. ht(I2 (X)) If X is generic symmetric, then we have to choose the lexicographic monomial order induced by xii > xij > x1i > x2i > · · · > xi−1,i > xi,i+1 > · · · > xin > xjj > · · · > xc ij > · · · > xj−1,j > xj,j+1 > · · · > xjn fij and the variables yp are smaller than and variables xkl not appearing in X xjn . fij ) is n − 1, which is the maximum. Hence, the (ii) The height of I2 (X fij ). Eagon-Northcott complex minimally resolves the ideal I2 (X Lemma 4.2. Let X be generic or generic symmetric. Let i < j. Then, eij ) ∩ hgi i = I2 (X eij ) · hgi i, that is, the ideals I2 (X eij ) and hgi i intersect I2 ( X transversally. Proof. Let X be generic. We choose the lexicographic monomial order ′ ′ given by the following ordering among the variables: xst > xs′ t′ if (s , t ) > (s, t) and yn > yn−1 > · · · > y1 > xst for all s, t. Then, by Lemma 3.5 eij ). the set of all 2 × 2 minors forms a Gröbner basis for the ideal I2 (X eij ))) doesn’t involve the inClearly, the minimal generating set m(Lt(I2 (X determinates xin and yn , whereas Lt(gi ) = xin yn . Hence, the supports of eij ))) and m(Lt(gi )) are disjoint. Therefore, by Lemma 3.6 we m(Lt(I2 (X are done. Let X be generic symmetric. Once we choose the correct monomial order, the rest of the proof is similar to the generic case. Suppose that (i, j) = (n − 1, n). We choose the lexicographic monomial order given by the following ordering among the variables: y1 > yn > yn−1 > · · · > y2 > > > > xn−1,n−1 > xn−1,n > x1,n−1 > x2,n−1 > · · · > xn−2,n−1 xnn x1n > · · · > xn−2,n xst for all other s, t. Suppose that (i, j) 6= (n − 1, n). We choose the lexicographic monomial order given by the following ordering among the variables: yn > yn−1 > · · · > y1 > xii > xij > x1i > x2i > · · · > xi−1,i > xi,i+1 > · · · > xin > xjj > · · · > xj−1,j > xj,j+1 > · · · > xjn > xst for all other s, t. SUM IDEALS 9 eij ) + hgi i : gj ) = Lemma 4.3. Let X be generic and i < j. Then, (I2 (X eij ) + hgi i : hxi1 , . . . , xin i. If X is generic symmetric and i < j, then (I2 (X gj ) = hx1i , . . . , xi−1,i , xii , . . . , xin i. P Proof. X be generic. We have xit gj = xjt gi + nk=1 (xit xjk − xik xjt )yk . fij ) + hgi i : gj ). Moreover, I2 (X eij ) + hgi i ⊆ Hence, hxi1 , · · · , xin i ⊆ (I2 (X hxi1 , · · · , xin i and gj ∈ / hxi1 , · · · , xin i. The ideal hxi1 , · · · , xin i being a eij ) + hgi i : gj ). The prime ideal, it follows that hxi1 , · · · , xin i ⊇ (I2 (X proof for the generic symmetric case is similar. 5. R ESOLUTION OF THE SUM IDEALS eij ) + Our aim is to construct a minimal free resolution for the ideal I2 (X eij ) and hgi i intersect hg1, . . . , gn i. We have proved that the ideals I2 (X eij ) + hgi i can therefore be resolved transversally; see 4.2. The ideal I2 (X eij ) + minimally by Theorem 3.7. We have also proved that the ideal I2 (X hgi i and the ideal hgj i have linear quotient; see 4.3. Therefore, the ideal eij ) + hgi , gj i can be resolved by the mapping cone construction. A I2 ( X minimal free resolution can then be extracted from this resolution by apeij ) + hgi , gj i plying Lemma 3.12. Next, we will show that the ideal I2 (X intersects transversally with the ideal hgl1 i, if l1 is the minimum in the eij ) + set {1, 2, . . . , n} \ {i, j}; see Lemma 5.4. Therefore, the ideal I2 (X hgi , gj , gl1 i can be resolved minimally by Theorem 3.7. Proceeding in this eij )+hgi , gj , gl1 , . . . , gl i manner, we will be able to show that the ideals I2 (X k and hglk+1 i intersect transversally, if 1 ≤ l1 < . . . < lk < lk+1 ≤ n and lk+1 is the smallest in the set {1, 2, . . . , n} \ {i, j, l1 , . . . , lk }; see Lemma eij ) + 5.4. This finally gives us a minimal free resolution for the ideal I2 (X hgi , gj , gl1 , . . . , gln−2 i, with 1 ≤ l1 < . . . < ln−2 ≤ n and lt ∈ / {i, j} for every t. Let us assume that X is generic and i = 1 and j = 2. The proofs for the general i and j, with i < j would be similar according to the aforesaid scheme. The proofs in the case when X is generic symmetric would be similar as well. Comments for general i < j and the symmetric case have been made whenever necessary. 10 JOYDIP SAHA, INDRANATH SENGUPTA, AND GAURAB TRIPATHI e12 ) + hg1 , g2 i. The minimal free 5.1. A minimal free resolution for I2 (X e12 ) is given by the Eagon-Northcott complex, which is resolution of I2 (X the following: δk e −→ 0 Ek−1 −→ · · · E1 −→ E0 −→ R/I2 (X) E : 0 −→ En−1 −→ · · · Ek −→ n where E0 ∼ = R1 , Ek = Rk(k+1) and for each k = 0, . . . , n − 2, the map δk : Ek → Ek−1 is defined as δk (ei1 ∧ · · · ∧ eik+1 ) ⊗ v2k−1 δk (ei1 ∧ · · · ∧ eik+1 ) ⊗ v1k−1 δk (ei1 ∧ · · · ∧ eik+1 ⊗ v1j v2k−j−1) = k+1 X x2s (ei1 ∧ · · · ∧ eˆs ∧ · · · ∧ eik+1 ) ⊗ v2k−2 = k+1 X (−1)s+1 x1s (ei1 ∧ · · · ∧ eˆs ∧ · · · ∧ eik+1 ) ⊗ v1k−2 s=1 s=1 = k+1 X (−1)s+1 x1s (ei1 ∧ · · · ∧ eˆs ∧ · · · ∧ eik+1 ) ⊗ v1j−1v2k−j−1 s=1 + k+1 X x2s (ei1 ∧ · · · ∧ eˆs ∧ · · · ∧ eik+1 ) ⊗ v1j v2k−j−2 s=1 for every ordered k + 1 tuple (i1 , i2 , · · · , ik+1 ), with 1 ≤ i1 < · · · < ik+1 ≤ n and for every j = 1, 2, · · · , k − 2. A minimal resolution of hg1 i is given by: g1 G : 0 −→ R −→ R −→ R/hg1 i −→ 0. e12 ) and hg1 i intersect transversally, by Lemma 4.2. ThereThe ideals I2 (X e12 ) + hg1 i is given fore, by Lemma 3.7, a minimal free resolution for I2 (X by the tensor product complex ψk+1 e12 )+hg1 i → 0 E ⊗G : 0 → En−1 → · · · → Ek+1 ⊕Ek −→ Ek ⊕Ek−1 → · · · → E0 → R/I2 (X such that ψk : Ek ⊕ Ek−1 −→ Ek−1 ⊕ Ek−2 is the map defined as j k−j−1 j k−j−1 ψk ei1 ∧ · · · ∧ eik+1 ⊗ v1 v2 = δk (ei1 ∧ · · · ∧ eik+1 ) ⊗ v1 v2 ψk (ei1 ∧ · · · ∧ eik ) ⊗ v1j v2k−j−2 = (−1)k−1g1 (ei1 ∧ · · · ∧ eik ) ⊗ v1j v2k−j−2 j k−j−2 . + δk−1 (ei1 ∧ · · · ∧ eik ) ⊗ v1 v2 e12 ) + hg1, g2 i by mapping Now we find a minimal free resolution for I2 (X e12 ) + cone. Let Ck := (E· ⊗ G· )k . We have proved in Lemma 4.3 that hI2 (X hg1i : g2 i = hx11 , x12 , · · · , x1n i; which is minimally resolved by the Koszul SUM IDEALS 11 complex. Let us denote the Koszul Complex by (F· ; σk ), where σk is the kth differential. We first construct the connecting map τ· : F· → E· ⊗ G· . Let n n n us write Fk := R(k ) and Ck := Rk(k+1) ⊕R(k−1)( k ) . The map τk : Fk → Ck is defined as: X τk (ei1 ∧ · · · ∧ eik ) = yj (ei1 ∧· · ·∧eik ∧ej )⊗v1k−1 −(ei1 ∧· · ·∧eik )⊗v1k−2 . j Let us choose the lexicographic ordering among the k tuples (i1 , . . . , ik ), n such that 1 ≤ i1 < · · · < ik ≤ n in order to write an ordered basis for R(k ) . We define lexicographic ordering among the tuples (i1 , . . . , ik+1 , k − j, j), for j = 0, . . . , k and k = 1, . . . , n − 1 to order the basis elements for n n n Rk(k+1) . Moreover, in the free module Ck = Rk(k+1) ⊕ R(k−1)( k ) , we order n the basis elements in such a way that those for Rk(k+1) appear first. The matrix representation of τk with respect to the chosen ordered bases is the following:   Ak( n )×(n) 0k( n )×(n) k k+1 k k+1          .  −I(n)×(n)   k k    0    (k−1)(nk)×(nk)   0(k−2)(n)×(n) k k Theorem 5.1. The following diagram commutes for every k = 1, . . . , n−1: τk FO k / σk+1 CO k ψk+1 Fk+1 τk+1 / Ck+1 Proof. It suffices to prove the statement for a basis element (ei1 ∧· · ·∧eik+1 ) of Fk+1 . Without loss of generality we consider (e1 ∧ · · · ∧ ek+1 ). We first compute (τk ◦ σk+1 )(e1 ∧ · · · ∧ ek+1 ). σk+1 (e1 ∧ · · · ∧ ek+1 ) 7−→ k+1 X (−1)j+1 x1j (e1 ∧ · · · ∧ eˆj ∧ · · · ∧ ek+1 ) j=1 τk 7−→ k+1 X (−1) j=1 − k+1 X j=1 j+1 n X x1j [ ys (e1 ∧ e2 ∧ · · · ∧ eˆj ∧ · · · ∧ ek+1 ∧ es ) ⊗ v1k−1] s=1 (−1)j+1 x1j (e1 ∧ e2 ∧ · · · ∧ eˆj ∧ · · · ∧ ek+1 ) ⊗ v1k−2 . 12 JOYDIP SAHA, INDRANATH SENGUPTA, AND GAURAB TRIPATHI We now compute (ψk+1 ◦ τk+1 )(e1 ∧ · · · ∧ ek+1 ). n τk+1 X ys (e1 ∧ · · · ∧ ek+1 ∧ es ) ⊗ v1k (e1 ∧ · · · ∧ ek+1 ) 7−→ s=1 −(e1 ∧ · · · ∧ ek+1 ) ⊗ v1k−1 n X ψk+1 X 7−→ [ (−1)j+1 ys x1j (e1 ∧ · · · ∧ eˆj ∧ · · · ∧ ek+1 ∧ es ) ⊗ v1k−1 s=1 j=1,2,··· ,k+1,s −(−1)k g1 (e1 ∧ · · · ∧ eˆj ∧ · · · ∧ ek+1 ∧ ej ) ⊗ v1k−1 − k+1 X (−1)j+1 x1j (e1 ∧ · · · ∧ eˆj ∧ · · · ∧ ek+1 ∧ ej ) ⊗ v1k−2 j=1 = n k+1 X X [x1j ys (−1)j+1 (e1 ∧ · · · ∧ eˆj ∧ · · · ∧ ek+1 ∧ es ) ⊗ v1k−1] j=1 s=1 n X (−1)s+1 ys x1s (e1 ∧ · · · ∧ eˆs ∧ · · · ∧ ek+1 ∧ es ) ⊗ v1k−1 + s=1 −(−1)k g1 (e1 ∧ · · · ∧ eˆj ∧ · · · ∧ ek+1 ∧ ej ) ⊗ v1k−1 − k+1 X (−1)j+1 x1j (e1 ∧ · · · ∧ eˆj ∧ · · · ∧ ek+1 ∧ ej ) ⊗ v1k−2 j=1 = k+1 X n X [x1j ys (−1)j+1 (e1 ∧ · · · ∧ eˆj ∧ · · · ∧ ek+1 ∧ es ) ⊗ v1k−1 ] j=1 s=1 n X (−1)s+1 (−1)k+1−s ys x1s (e1 ∧ · · · ∧ eˆs ∧ · · · ∧ ek+1 ∧ es ) ⊗ v1k−1 + s=1 −(−1)k g1 (e1 ∧ · · · ∧ eˆj ∧ · · · ∧ ek+1 ∧ ej ) ⊗ v1k−1 − k+1 X (−1)j+1 x1j (e1 ∧ · · · ∧ eˆj ∧ · · · ∧ ek+1 ∧ ej ) ⊗ v1k−2 j=1 = k+1 X n X [x1j ys (−1)j+1 (e1 ∧ · · · ∧ eˆj ∧ · · · ∧ ek+1 ∧ es ) ⊗ v1k−1 ] j=1 s=1 +(−1)k g1 (e1 ∧ · · · ∧ eˆj ∧ · · · ∧ ek+1 ∧ ej ) ⊗ v1k−1 −(−1)k g1 (e1 ∧ · · · ∧ eˆj ∧ · · · ∧ ek+1 ∧ ej ) ⊗ v1k−1 − k+1 X (−1)j+1 x1j (e1 ∧ e2 ∧ · · · ∧ eˆj ∧ · · · ∧ ek+1 ∧ ej ) ⊗ v1k−2 j=1 = k+1 X n X [x1j ys (−1)j+1 (e1 ∧ · · · ∧ eˆj ∧ · · · ∧ ek+1 ∧ es ) ⊗ v1k−1 ] j=1 s=1 − k+1 X j=1 (−1)j+1 x1j (e1 ∧ · · · ∧ eˆj ∧ · · · ∧ ek+1 ∧ ej ) ⊗ v1k−2. SUM IDEALS 13 e12 )+ Hence the mapping cone M(E· ⊗G· ; F· ) gives us the resolution for I2 (X hg1, g2 i as described in 3.2. However, this resolution is not minimal. We now construct a minimal free resolution from M(E· ⊗ G· ; F· ). e12 ) + hg1 , g2 i has been constructed in A free resolution for the ideal I2 (X 3.1, which is given by dn+2 dk+1 d k Dk−1 · · · −→ D1 −→ D0 −→ 0, 0 −→ Dn+2 −→ Dn+1 · · · −→ Dk −→ n n n such that Dk = Fk−1 ⊕ Ck = R(k−1) ⊕ (Rk(k+1) ⊕ R(k−1)( k ) ) and dk = (−σk−1 + τk−1 , ψk ). Let us recall that the map ψ is the differential in the e12 ) + hg1 i, the map σ is the differential in the Koszul free resolution for I2 (X resolution for hx11 , x12 , . . . , x1n i and τ is the connecting homomorphism between the complexes defined in 3.1. Let us order bases for Fk−1 and Ck with respect to the lexicographic ordering. Finally we order basis for Dk in such a way that the basis elements for Fk−1 appear first, followed by the basis elements for Ck . Therefore, the matrix representation for the differential map dk is given by   −σk−1    τk−1 −σk−1        0 A     =         ψk τk−1 =     0   0 0       −I         0         .   ψk      The entries in the matrices representing σk−1 and ψk can only belong to the maximal ideal hxij , yj i, since both are differentials of minimal free resolutions. The block matrix A has also elements in the maximal ideal hxij , yj i. The only block which has elements outside the maximal ideal hxij , yj i is in the identity block appearing in τk−1 . Therefore, it is clear from the matrix representation of the map dk that we can apply Lemma 3.12 repeatedly to get rid of non-minimality. Hence, we get a minimal free resolution and the 14 JOYDIP SAHA, INDRANATH SENGUPTA, AND GAURAB TRIPATHI e12 ) + hg1 , g2 i are total Betti numbers for the ideal I2 (X b0 = 1, n + 2, b1 = 2 n + n, b2 = 2 · 3 n n n n n − − + + (k − 1) bk+1 = k k k−1 k−1 k k+1 n n , for 2 ≤ k ≤ n − 1, + (k − 2) = k k k+1 bn = n − 2. eij ) + hg1 , . . . , gn i. 5.2. A minimal free resolution for I2 (X Lemma 5.2. Let Gk = G12 ∪ {g1 , g2 , . . . , gk }, 1 ≤ k ≤ n, where G12 is the e12 defined in the list of notations in section 2. The set of all 2×2 minors of X e12 ) + hg1 , . . . , gk i with respect set Gk is a Gröbner basis for the ideal I2 (X to a suitable monomial order. Proof. We take the lexicographic monomial ordering in R induced by the following ordering among the indeterminates: xnn > · · · > xtt > · · · > x33 > y1 > · · · > yn > x11 > · · · > x1n > x21 > · · · > x2n > xst for other s, t. Then, we observe that for every s ≥ 3, Lt(gs ) is coprime with Lt(gt ) for every 1 ≤ t ≤ k; t 6= s and also coprime with Lt(h) for every h ∈ G. e12 ). Therefore, Moreover, by Lemma 3.5, G12 is a Gröbner basis for I2 (X we only have test the S-polynomials S(g1, g2 ), S(g1 , h) and S(g2 , h), for h ∈ G. Pn We can write S(g1 , g2 ) = k=1 [12\|1k]yk and note that Lt([12\|1k]) ≤ Lt(S(g1 , g2 ) for every 1 ≤ k ≤ n. Hence, S(g1 , g2 ) →G ′ 0. We note that, if i 6= 1 then the leading terms of g1 and [12\|st] are mutually coprime and therefore P S(g1 , [12\|st]) →Gk 0. Next, the expression S(g1 , [12\|1t]) = x1t g2 + s6=t [12\|st]ys shows that S(g1 , [12\|1t]) →Gk 0. Similarly, if s 6= 1 SUM IDEALS 15 then the leading terms of g2 and [12\|st] are mutually coprime and therefore S(g2 , [12\|st]) →Gk 0. The proof for S(g2 , [12\|1t]) is similar to that of S(g1, [12\|st]). Remark. The corresponding result for i < j in general would be the following: Lemma 5.3. Let Gi,j,k = Gij ∪ {gi , gj , gl1 , . . . , glk−2 }, 1 ≤ k ≤ n, 1 ≤ l1 < · · · < lk−2 ≤ n and lt is the smallest in the set {1, 2, . . . , n} \ eij defined {i, j, l1 , . . . , lt−1 }; Gij denotes the set of all 2 × 2 minors of X in the list of notations in section 2. The set Gi,j,k is a Gröbner basis for the eij ) + hg1 , . . . , gk i with respect to a suitable monomial order. ideal I2 (X Proof. While proving this statement with i < j arbitrary, we have to choose the following monomial orders. The rest of the proof remains similar. Suppose that X is generic, we choose the lexicographic monomial ordering in R induced by the following ordering among the indeterminates: xnn > · · · > xc cii > · · · > x11 > jj > · · · > x > > > y1 > · · · > yn xi1 > · · · > xin xj1 > · · · > xjn xst for all other s, t. If X is generic symmetric, we choose the lexicographic monomial ordering in R induced by the following ordering among the indeterminates: xnn > · · · > xc cii > · · · > x11 > y1 > · · · > yn jj > · · · > x > xii > xij > x1i > x2i > · · · > xi−1,i > xi,i+1 > · · · > xin > xjj > · · · > xj−1,j > xj,j+1 > · · · > xjn > xst for all other s, t. e12 )+hg1 , . . . , gk i and hgk+1i intersect transverLemma 5.4. The ideals I2 (X sally, for every 2 ≤ k ≤ n − 1. e12 ) + hg1 , . . . , gk i such Proof. Suppose not, then, there exists hk+1 ∈ / I2 ( X e that hk+1 gk+1 ∈ I2 (X12 ) + hg1, . . . , gk i. Let us choose the same monomial order on R as defined in Lemma 5.2. Upon division by elements of Gk , we may further assume that Lt(h) ∤ Lt(hk+1 ) for every h ∈ Gk , since Gk e12 ) + hg1, . . . , gk i by Lemma 5.2. On is a Gröbner basis for the ideal I2 (X e12 ) + hg1 , . . . , gk i and therefore Lt(h) \| the other hand hk+1 gk+1 ∈ I2 (X Lt(hk+1 ), for some h ∈ Gk , since Lt(h) and Lt(gk+1 ) are mutually coprime, - a contradiction. 16 JOYDIP SAHA, INDRANATH SENGUPTA, AND GAURAB TRIPATHI Remark. The corresponding result for i < j in general would be the eij ) + hgi , gj , gl1 , . . . , gl i and hgl i intersect following: The ideals I2 (X k k+1 transversally, if 1 ≤ l1 < . . . < lk < lk+1 ≤ n and lk+1 is the smallest in the set {1, 2, . . . , n} \ {i, j, l1 , . . . , lk }, for every 1 ≤ k ≤ n − 3. The proof is essentially the same as above after we use the Lemma 5.3. Proof of Theorem 2.1. Part (1) of the theorem has been proved in 5.1. We now prove part (2) under the assumption i = 1, j = 2. Let the minimal e12 ) + hg1 , g2 , · · · , gk i be (L , δ ). By Lemma 5.4 free resolution of I2 (X e12 ) + hg1 , . . . , gk+1i is and Lemma 3.7, the minimal free resolution of I2 (X gk+1 given by the tensor product of (L , δ ) and 0 −→ R −→ R −→ 0, and that is precisely (K , ∆ ), with Kp = Lp ⊕Lp−1 and ∆p = (λp , (−1)p gk+1 + λp−1 ). Let βk,p , 0 ≤ p ≤ n + k, denote the p-th total Betti number for the ideal e12 ) + hg1 , . . . , gk i. Then, the total Betti numbers βk+1,p , 0 ≤ p ≤ I2 ( X e12 ) + hg1 , . . . , gk+1i are given by βk+1,0 = 1, n + k + 1 for the ideal I2 (X βk+1,p = βk,p−1 + βk,p for 1 ≤ p ≤ n + k and βk+1,n+k+1 = n − 2. The proof for general i < j follows similarly according to the strategy discussed in the beginning of section 5. eij )+hg1 , . . . , gn i In particular, the total Betti number βn−2,p for the ideal I2 (X are given by βn−2,0 = 1, βn−2,p = βn−3,p−1 + βn−3,p for 1 ≤ p ≤ 2n − 3 and βn−2,2n−2 = n − 2. Example. We show the Betti numbers at each stage for n = 4 and n = 5. 1 6 1 7 n=4: 1 8 1 9 1 10 1 1 1 n=5: 1 1 1 10 11 12 13 14 15 8 3 14 11 3 12 7 2 20 19 9 2 29 39 28 11 2 20 5 4 30 25 9 4 25 25 14 3 37 50 39 17 3 50 87 89 56 20 3 64 137 176 145 76 23 3 R EFERENCES [1] W. Bruns, A.R. Kustin, M. Miller, The Resolution of the Generic Residual Intersection of a Complete Intersection, Journal of Algebra 128 (1990) 214-239. [2] A. Conca, Emanuela De Negri, Elisa Gorla, Universal Gröbner bases for Maximal Minors, International Mathematics Research Notices 11(2015) 3245-3262. SUM IDEALS 17 [3] D. Eisenbud, Geometry of Syzygies, Springer-Verlag, NY, 2005. [4] P. Gimenez, I. Sengupta and H. Srinivasan, Minimal graded free resolution for monomial curves defined by arithmetic sequences, Journal of Algebra 388 (2013) 294-310. [5] M.R., Johnson, J. McLoud-Mann, On equations defining Veronese Rings, Arch. Math. (Basel) 86(3)(2006) 205-210. [6] S.Lichtenbaum, On the vanishing of Tor in regular local rings, Illinois J.Math. 10: 220- 226,1966. [7] H. Matsumura, Commutative Ring Theory, Cambridge University Press, NY, 1986. [8] I. Peeva, Graded Syzygies, Springer-Verlag London Limited, 2011. [9] J. Saha, I. Sengupta, G. Tripathi, Ideals of the form I1 (XY ), arXiv:1609.02765 [math.AC] 2016. [10] J. Saha, I. Sengupta, G. Tripathi, Primality of certain Determinanatal ideals, arXiv:1610.00926 [math.AC] 2016. Department of Mathematics, RKM Vivekananda University, Belur Math, Howrah 711202, India. E-mail address: [email protected] Discipline of Mathematics, IIT Gandhinagar, Palaj, Gandhinagar, Gujarat 382355, INDIA. E-mail address: [email protected] Department of Mathematics, Jadavpur University, Kolkata, WB 700 032, India. E-mail address: [email protected]	0
arXiv:1711.07737v1 [math.GT] 21 Nov 2017 SUPERRIGIDITY OF ACTIONS ON FINITE RANK MEDIAN SPACES ELIA FIORAVANTI Abstract. Finite rank median spaces are a simultaneous generalisation of finite dimensional CAT(0) cube complexes and real trees. If Γ is an irreducible lattice in a product of rank one simple Lie groups, we show that every action of Γ on a complete, finite rank median space has a global fixed point. This is in sharp contrast with the behaviour of actions on infinite rank median spaces. The fixed point property is obtained as corollary to a superrigidity result; the latter holds for irreducible lattices in arbitrary products of compactly generated groups. We exploit Roller compactifications of median spaces; these were introduced in [Fio17a] and generalise a well-known construction in the case of cube complexes. We provide a reduced 1-cohomology class that detects group actions with a finite orbit in the Roller compactification. Even for CAT(0) cube complexes, only second bounded cohomology classes were known with this property, due to [CFI16]. As a corollary, we observe that, in Gromov’s density model, random groups at low density do not have Shalom’s property HF D . Contents 1. Introduction. 2. Preliminaries. 2.1. Median spaces and median algebras. 2.2. Bridges. 2.3. The Haagerup class. 3. Haagerup class and elementarity of actions. 3.1. The main statement. 3.2. Elementarity and Shalom’s property HF D . 4. Superrigidity. 4.1. The superrigidity result. 4.2. Homomorphisms to coarse median groups. Appendix A. Structure of UBS’s. References 1 2 9 9 20 21 23 23 26 27 27 34 35 41 2 ELIA FIORAVANTI 1. Introduction. A metric space X is median if, for any three points x1 , x2 , x3 ∈ X, there exists a unique point m ∈ X such that d(xi , m) + d(m, xj ) = d(xi , xj ) for all 1 ≤ i < j ≤ 3. Simple examples are provided by real trees and Rn with the `1 metric. Under a finite dimensionality assumption, to each connected median space b The spaces X and X b are biX corresponds a canonical CAT(0) space X. b Lipschitz equivalent and isometries of X induce isometries of X [Bow16]. For instance, to Rn with the `1 metric we associate Rn with its euclidean distance. More elaborate examples of median spaces are provided by simply connected cube complexes satisfying Gromov’s link condition; in this case we b obtain a median space X by endowing each cube with the `1 metric and X is the corresponding CAT(0) cube complex. Median spaces generally display wilder features than cube complexes, as, like real trees, they can be essentially non-discrete objects. Note in this regard that the class of median spaces is closed under ultralimits; these also preserve a notion of dimension, usually called rank (see Section 2.1 for a precise definition). Despite non-discreteness, finite rank median spaces retain many of the good combinatorial properties of cube complexes. In addition to the CAT(0) metric, there is a notion of boundary compatible with the median property [Fio17a] and many groups of isometries contain free non-abelian subgroups [CS11, Fio17b]. We would expect many known results for CAT(0) cube complexes to extend to finite rank median spaces without significant complications, for instance [BCG+ 09, GH10, NS13, Fer15, CFI16, KS16] to name a few. There is however a notable exception to this pattern: general group actions on median spaces do not have a clear connection with the existence of codimension one subgroups [Sag95, Ger97, Fio17b]. Such close similarities between cube complexes and general median spaces can be ascribed to the existence of a collection W of walls. These encode the geometry of the space in the same way as hyperplanes do in CAT(0) cube complexes. The set W should not be thought of as discrete and needs to be endowed with a measure µ that encodes the “thickness” of sets of walls [CDH10, Fio17a]. Indeed, the concept of median space is in a certain sense dual to the notion of space with measured walls [CMV04, dCTV08, CDH10]; these extend spaces with walls [HP98], which are the dual viewpoint on CAT(0) cube complexes. Our main theorem is a superrigidity result for irreducible lattices Γ in products G = G1 × ... × G` of locally compact, compactly generated groups. Namely, under weak assumptions of non-elementarity, every action of Γ on a finite rank median space X essentially arises from continuous actions Gi y Yi on median spaces of lower rank. For cube complexes, this was known due to [CFI16]. SUPERRIGIDITY OF ACTIONS ON FINITE RANK MEDIAN SPACES 3 In the more general context of CAT(0) spaces, similar results were obtained long ago in [Mon06, CM09]. Unfortunately, applying these to a median space X only provides actions of the factors Gi on CAT(0) subspaces b these subspaces might bear no relation to the median structure on Zi ⊆ X; X. This might seem like an irrelevant subtlety, but it is, on the contrary, key to the fixed point properties that this paper provides. As an illustration of this, consider an irreducible lattice Γ < SL2 R×SL2 R. It acts properly and cocompactly on the CAT(0) space H2 ×H2 . In particular, it is quasi-isometric to a product of surface groups, which can easily be recognised as cubulated. It was shown in [CD17] that Γ acts properly and coboundedly on an infinite rank median space. The group Γ is moreover coarse median in the sense of [Bow13a]. Thus, it should appear particularly striking that every action of Γ on a connected finite rank median space fixes a point; this follows from our superrigidity result, see Corollary D below. The proof of our superrigidity theorem follows a very similar outline to Monod’s [Mon06]. This is mostly hidden in our application of Theorem 4.1 from [Sha00], thus we believe it is important to highlight the analogy here. We have already mentioned that, to each finite rank median space X, one b However, one can also can associate a finite dimensional CAT(0) space X. consider the infinite dimensional CAT(0) spaces L2 (W , µ) and L2 (H , νb); here (H , νb) is a close relative of (W , µ) and will be introduced below. Retracing the proof of Monod Superrigidity in our context, we would first induce a continuous action of G on the infinite dimensional CAT(0) space b (see [Mon06] for a definition). Then, we would prove that a L2 (G/Γ, X) b splits as a product Z1 × ... × Zk , where the action subspace of L2 (G/Γ, X) of G on the i-th factor only depends on the projection to Gi . Finally, we b the information gained about Γ y L2 (G/Γ, X). b would carry back to Γ y X Our application of Shalom’s machinery instead constructs a continuous action of G on the infinite dimensional CAT(0) space L2 (G/Γ, L2 (H , νb)). Again, one then proves a splitting theorem for this action and carries the gained insight back to Γ y L2 (H , νb). Our main contribution lies in transferring information back and forth between the actions Γ y L2 (H , νb) and Γ y X. Indeed, Shalom’s machinery can only be set in motion once we have a nonvanishing reduced cohomology class for Γ y L2 (H , νb); similarly, once we have a superrigidity statement for Γ y L2 (H , νb), we need to translate it back to Γ y X. We now describe our results in greater detail. 1.1. A cohomological characterisation of elementary actions. Each median space X has a distinguished collection H of subsets called halfspaces; every wall gives rise to two halfspaces and each halfspace arises from a wall. The collection H is equipped with a measure νb, see [Fio17a]. In the case of cube complexes, one recovers the usual notion of halfspace and νb is simply the counting measure. 4 ELIA FIORAVANTI Given a topological group G and an isometric action G y X, one naturally obtains a unitary representation ρ : G → U (L2 (H , νb)) and a cocycle b : G → L2 (H , νb) which will be referred to as the Haagerup cocycle. This construction is well-known and appears for instance in [CMV04, dCTV08, CDH10, FV16]. If G y X has continuous orbits, ρ and b are continuous; thus b induces a reduced continuous cohomology class [b] ∈ Hc1 (G, ρ). In [Fio17b], we introduced a notion of elementarity for actions on median spaces; namely, we say that G y X is Roller elementary if G has at least one finite orbit within a certain compactification X, the Roller compactification. Roller elementarity implies the existence of a finite orbit in the visual b compactification of the CAT(0) space X. If G y X is an isometric action with continuous orbits, Roller elementarity can be described in terms of the reduced cohomology class [b]. Theorem A. Let X be a complete, finite rank median space. The Haagerup class [b] ∈ Hc1 (G, ρ) vanishes if and only if G y X is Roller elementary. Theorem A extends various known results. In the case of simplicial trees, it appears in [FV16]. For CAT(0) cube complexes, the implication “Roller nonelementary ⇒ [b] 6= 0” is implicit in [DP16]. In [CFI16], the authors construct a family of bounded cohomology classes detecting Roller elementarity in CAT(0) cube complexes. We remark that Theorem A equally holds if we replace L2 (H , νb) with any Lp (H , νb), 1 ≤ p < +∞, although it is slightly simpler to exploit the richer structure of Hilbert spaces in its proof. Our superrigidity result only relies on the implication of Theorem A that yields [b] 6= 0, but we believe the full statement of Theorem A to be of independent interest. The proof of the other implication turns out to be quite technical and requires a careful study of the structure of UBS’s in median spaces; these are a generalisation of the simplices in Hagen’s simplicial boundary of a CAT(0) cube complex [Hag13]. Most of these details will be relegated to the appendix. 1.2. Superrigidity of actions. Once we have a nontrivial reduced cohomology class, as provided by Theorem A, we can apply well-established machinery (namely Theorem 4.1 in [Sha00]) to obtain superrigidity results. Let X be a complete, finite rank median space. Its Roller compactification X is partitioned into components [Fio17a]. The subset X ⊆ X forms a full component; every other component is a complete median space of strictly lower rank. This aspect of X shares some similarities with refined boundaries of CAT(0) spaces [Lee00, Cap09]. Given a component Z ⊆ X, a median subalgebra of Z is a subset Y ⊆ Z that is itself a median space with the restriction of the median metric of Z; equivalently, the median map m : Z 3 → Z takes Y 3 into Y . We are now ready to state our main superrigidity result. SUPERRIGIDITY OF ACTIONS ON FINITE RANK MEDIAN SPACES 5 Theorem B. Let X be a complete, finite rank median space. Let Γ be a uniform, irreducible lattice in a product G = G1 × ... × G` of compactly generated, locally compact groups with ` ≥ 2. Suppose Γ y X is a Roller nonelementary action. There exist a finite index subgroup Γ0 ≤ Γ, a Γ0 invariant component Z ⊆ X and a Γ0 -invariant closed median subalgebra Y ⊆ Z where the action Γ0 y Y extends to a continuous action G0 y Y , for some open finite index subgroup G0 ≤ G. Remarks. (1) Theorem B also applies to nonuniform lattices, as long as they are square-integrable; this is a well-known technical condition that implies finite generation and ensures that Theorem 4.1 in [Sha00] still holds. Irreducible lattices in G1 × ... × G` are square-integrable if each Gi is the group of ki -rational points of a semisimple, almost ki simple, ki -isotropic linear algebraic group defined over some local field ki [Sha00]. Further examples of nonuniform square-integrable lattices include minimal Kac-Moody groups over sufficiently large finite ground fields; these can be regarded as irreducible lattices in the product of the closed automorphism groups of the associated buildings [Rém99, Rém05]. (2) Theorem B should be compared to Shalom’s superrigidity result for actions on simplicial trees, Theorem 0.7 in [Sha00]. If X is a simplicial tree, Y is always a subcomplex of X and Γ0 = Γ, G0 = G. The complications in the statement of Theorem B reflect phenomena that do not happen in the world of trees. However, as soon as we leave the context of rank one median spaces, our result is optimal even if one restricts to CAT(0) square complexes; see Examples 4.7 and 4.8. We remark that, when X is a general CAT(0) square complex, the median algebra Y might not be a subcomplex of X or Z. (3) We can take Γ0 = Γ, G0 = G and Z = X as long as X does not split as a nontrivial product of median spaces and G has no finite b see Theorem 4.4 below. orbits in the visual compactification of X; However, even in this case, the action in general extends only to a proper median subalgebra of X. (4) For CAT(0) cube complexes, the superrigidity result of [CFI16] is slightly more general than Theorem B as it applies to all nonuniform lattices. This is due to the use of bounded cohomology (namely Theorem 16 in [BM02]) rather than reduced cohomology. Our strategy of proof was hinted at on page 9 of [Sha00]. 1.3. Fixed point properties for irreducible lattices. Unlike automorphism groups of CAT(0) cube complexes, the isometry group of a median space needs not be totally disconnected. Still, it is possible to exploit Theorem B to derive a fixed point property for irreducible lattices in connected groups. 6 ELIA FIORAVANTI Given a locally compact topological group Q, we denote the connected component of the identity by Q0 . We say that Q satisfies condition (∗) if Q/Q0 is amenable or has Shalom’s property HF D (see Section 1.5 for a definition). In particular, all almost-connected and connected-by-(T) groups satisfy condition (∗). Theorem C. Let X be a complete, finite rank median space. Let Γ be a square-integrable, irreducible lattice in a product G1 × ... × G` with ` ≥ 2. Suppose that every Gi is compactly generated and satisfies condition (∗). (1) Every action Γ y X is Roller elementary. (2) If Γ does not virtually map onto Z, every action Γ y X has a finite orbit within X. If moreover X is connected, every action Γ y X has a global fixed point. When X is a real tree, Theorem C also follows from Theorem 6 in [Mon06]. We remark that every group that virtually maps onto Z admits a Roller elementary action on Rn with unbounded orbits, for some n ≥ 1. Corollary D. Let X be a complete, connected, finite rank median space. Let Γ be an irreducible lattice in a connected, higher rank, semisimple Lie group G. Every action Γ y X fixes a point. An analogous result for CAT(0) cube complexes was proved in [CFI16]. If each simple factor of G has rank at least two, then Γ has property (T) and Corollary D follows from Theorem 1.2 in [CDH10]. The assumption that X have finite rank is essential for Corollary D to hold. If at least one simple factor Gi < G is locally isomorphic to O(n, 1) or U (n, 1), n ≥ 2, the lattice Γ admits an action on an infinite rank median space with unbounded orbits [CDH10]. Moreover, if all Gi ’s are locally isomorphic to O(ni , 1), ni ≥ 2, then Γ even admits a proper and cobounded action on an infinite rank median space [CD17]. 1.4. Homomorphisms to coarse median groups. Coarse median spaces were introduced in [Bow13a] as an attempt to formulate a coarse notion of nonpositive curvature. They have recently received a lot of attention [Bow13b, Hae16a, Zei16, SW17, ANWZ17, NWZ17] and proved instrumental to striking results such as [Hae16b, BHS17b]. A group is said to be coarse median if its Cayley graphs are coarse median spaces. Examples of finite rank coarse median groups include hyperbolic groups, cubulated groups [HS16], fundamental groups of closed irreducible 3-manifolds not modelled on Nil or Sol, mapping class groups and, more generally, all groups that are HHS [Bow15, BHS17a, BHS15]. We will be mainly interested in equivariantly coarse median groups. If we view coarse median groups as a generalisation of groups that are HHS, equivariantly coarse median groups generalise hierarchically hyperbolic groups (HHG). In particular, hyperbolic groups, cubulated groups and mapping class groups also are equivariantly coarse median of finite rank. SUPERRIGIDITY OF ACTIONS ON FINITE RANK MEDIAN SPACES 7 More precisely, we say that a group H is equivariantly coarse median if it is equipped with a finite generating set S ⊆ H and a coarse median µ : H 3 → H [Bow13a] such that dS (µ(ha, hb, hc), hµ(a, b, c)) ≤ C < +∞ for all elements h, a, b, c ∈ H; here dS denotes the word metric induced by S. Note that this definition does not depend on the choice of S. Equivariantly coarse median groups have already been considered in [Zei16] under a different name. If H is a coarse median group of finite rank, every asymptotic cone of H is endowed with a bi-Lipschitz equivalent median metric [Zei16]. As an example, in asymptotic cones of mapping class groups the median geodesics are limits of hierarchy paths [BDS11b, BDS11a]. When H is equivariantly coarse median, the median metric on each asymptotic cone is preserved by the action of the ultrapower of H. Given a group Γ and an infinite sequence of pairwise non-conjugate homomorphisms Γ → H, we can apply the Bestvina-Paulin construction [Bes88, Pau91]. The result is an isometric action on a median space X with unbounded orbits; this is obtained as the canonical median space bi-Lipschitz equivalent to an asymptotic cone of H. Along with Theorem C, this implies the following. Corollary E. Let H be an equivariantly coarse median group of finite rank. Let Γ be as in the second part of Theorem C. There exist only finitely many pairwise non-conjugate homomorphisms Γ → H. Corollary E applies in particular to the case when Γ is an irreducible lattice in a connected, higher rank, semisimple Lie group. When Γ has property (T), this result already appears in [Zei16] and, if in addition H is HHG, a much stronger statement is provided by Corollary D of [Hae16b]. Note that Haettel’s method cannot be applied in our context as lattices in products of rank one groups do admit nonelementary actions on hyperbolic spaces. Also compare Theorem 4.1 in [BF14] for the case of hyperbolic H and an arbitrary product G of locally compact, second countable groups. We remark that a stronger conclusion can be reached if H acts freely on a complete, finite rank median space. Indeed, the following is an immediate consequence of Theorem F in [Fio17b]. Proposition F. Let H be a group admitting a free action on a complete, finite rank median space X. Suppose that every action Γ y X is Roller elementary. Then every homomorphism Γ → H factors through a virtually abelian subgroup of H. Proposition F applies for instance to the case when Γ has no non-abelian free subgroups, has property HF D or satisfies the hypotheses of the first part of Theorem C. In particular, if Γ is an irreducible lattice in a connected, higher rank, semisimple Lie group, every homomorphism Γ → H has finite image. This should motivate a certain interest in groups acting freely on complete, finite rank median spaces. If a group acts freely on a finite dimensional 8 ELIA FIORAVANTI CAT(0) cube complex, it clearly falls into this class; however, it is unclear at this stage whether these are the only finitely generated examples. See [CRK15] for partial results in this direction. Note that the infinitely generated group Q even admits a proper action on a rank two median space, which splits as a product of a simplicial tree and the real line; see Example II.7.13 in [BH99]. However, since Q is a divisible group, all its elements must act elliptically on any (possibly infinite-dimensional) CAT(0) cube complex [Hag07]. Even within finitely generated groups, actions on median spaces tend to be more flexible than actions on CAT(0) cube complexes. For every group H, we can consider dimf m H, i.e. the minimum rank of a complete median space X admitting a free action of H; if H does not act freely on any complete median space, we set dimf m H := −1. Restricting to CAT(0) cube complexes, we can similarly define dimf c H and, if we only consider (metrically) proper actions, we obtain dimpm H and dimpc H. Thus, dimpc Q = dimf c Q = −1, while dimpm Q = 2 and dimf m Q = 1. We remark that 1 ≤ dimf m H < dimcm H and 1 ≤ dimpm H < dimpm H for many finitely generated groups H. For instance, dimf c H = 1 if and only if H is free. On the other hand, by work of E. Rips, dimf m H = 1 if and only if H is a free product of free abelian and surface groups (excluding a few nonorientable surfaces); see e.g. Theorem 9.8 in [BF95]. One can use the same observation to construct free actions of various RAAGs on median spaces of rank strictly lower than the dimension of the Salvetti complex. Considering more general actions, we mention that there exist finitely generated groups admitting actions on real trees with unbounded orbits, but whose actions on simplicial trees (and in fact even finite dimensional CAT(0) cube complexes) must have a global fixed point [Min16]. 1.5. Shalom’s property HF D and random groups. Theorem A also allows us to prove that various (non-amenable) groups do not have property HF D . The latter was introduced in [Sha04]: a topological group G has property HF D if every unitary representation π with Hc1 (G, π) 6= 0 has a finite dimensional subrepresentation. Property HF D is trivially satisfied by every locally compact group with property (T) [Del77], but also, at the opposite end of the universe of groups, by a large class of amenable groups. This includes polycyclic groups, lamplighter groups and all connected, locally compact, amenable groups [Sha04, Mar06]. An example of an amenable group without HF D is provided by the wreath product Z o Z [Sha04]. We prove the following; see Proposition 3.7 below for a more general result. Corollary G. Let Γ be a discrete group with property HF D . If Γ acts freely and cocompactly on a CAT(0) cube complex X, then Γ is virtually abelian. Property HF D has been studied almost exclusively within the class of amenable groups, where it happens to be a quasi-isometry invariant [Sha04]. It was a key ingredient (implicitely, or explicitely) in recent more elementary SUPERRIGIDITY OF ACTIONS ON FINITE RANK MEDIAN SPACES 9 proofs of Gromov’s theorem on groups of polynomial growth [Kle10, Oza16]. It has moreover interesting applications to the study of quasi-isometric embeddings into Hilbert spaces [dCTV07]. Property HF D is inherited by uniform lattices and it is stable under direct products and central extensions [Sha04]. Being satisfied by groups that fall into two extremely different classes, namely amenable and Kazhdan groups, it is reasonable to expect a wide variety of groups with property HF D . However, it seems that no answer is known to the following question. Question. Does every finitely generated group with property HF D virtually split as a direct product of an amenable group and finitely many groups with property (T)? Does every word hyperbolic group with property HF D also satisfy property (T)? Corollary G and the results of [OW11] imply that random groups at low density do not satisfy HF D . Corollary H. With overwhelming probability, random groups at density d < 16 in Gromov’s density model do not have property HF D . Note however that, at density d > 13 , random groups are Kazhdan [Ż03, KK13], hence satisfy property HF D . Acknowledgements. The author warmly thanks Brian Bowditch, Pierre-Emmanuel Caprace, Indira Chatterji, Yves Cornulier, Thomas Delzant, Mark Hagen, Masato Mimura, Narutaka Ozawa, Romain Tessera, Pierre Pansu, Alain Valette for helpful conversations. The author expresses special gratitude to Cornelia Druţu and Talia Fernós for contributing many of the ideas of this paper. This work was undertaken at the Mathematical Sciences Research Institute in Berkeley during the Fall 2016 program in Geometric Group Theory, where the author was supported by the National Science Foundation under Grant no. DMS-1440140 and by the GEAR Network. Part of this work was also carried out at the Isaac Newton Institute for Mathematical Sciences, Cambridge, during the programme “Non-positive curvature, group actions and cohomology” and was supported by EPSRC grant no. EP/K032208/1. The author was also supported by the Clarendon Fund and the Merton Moussouris Scholarship. 2. Preliminaries. 2.1. Median spaces and median algebras. Let X be a metric space. Given points x, y ∈ X, the interval I(x, y) is the set of points z ∈ X that lie between x and y, i.e. that satisfy d(x, y) = d(x, z)+d(z, y). We say that X is a median space if for all x, y, z ∈ X there exists a unique point m(x, y, z) that lies in I(x, y)∩I(y, z)∩I(z, x). The median map m : X 3 → X that we obtain this way endows X with a structure of median algebra. Most definitions in the theory of median spaces can also be given for arbitrary median algebras; 10 ELIA FIORAVANTI we will follow this approach in introducing the necessary notions. The reader can consult e.g. [Rol98, Nic08, CDH10, Bow13a, Bow16, Fio17a, Fio17b] for more background on median spaces and algebras. In a median space, I(x, y) = {z ∈ I(x, y) \| z = m(x, y, z)}; this can be taken as a definition of intervals in general median algebras. If (M, m) is a median algebra, we say that a subset C ⊆ M is convex if I(x, y) ⊆ C whenever x, y ∈ C. The intersection of a finite family of pairwise intersecting convex sets is always nonempty; this is known as Helly’s Theorem, see Theorem 2.2 in [Rol98]. A subset h ⊆ M is a halfspace if both h and h∗ := M \ h are convex; we will denote the set of halfspaces of M by H (M ), or simply by H when there is no ambiguity. Halfspaces h, k are said to be transverse if no two distinct elements of the set {h, h∗ , k, k∗ } are comparable in the poset (H , ⊆). Equivalently, the intersections h ∩ k, h ∩ k∗ , h∗ ∩ k, h∗ ∩ k∗ are all nonempty. Given A ⊆ H , we write A∗ for {h ∈ H \| h∗ ∈ A}. A subset σ ⊆ H is said to be an ultrafilter if any two halfspaces in σ intersect and H = σ t σ ∗ . For instance, for each x ∈ M the set σx := {h ∈ H \| x ∈ h} is an ultrafilter. Given subsets A, B ⊆ M , we write H (A\|B) := {h ∈ H \| B ⊆ h, A ⊆ h∗ } and σA := H (∅\|A); we refer to sets of the form H (x\|y), x, y ∈ M , as halfspace intervals. If C, C 0 ⊆ M are disjoint and convex, the set H (C\|C 0 ) is nonempty, see Theorem 2.7 in [Rol98]. In particular σx = σy if and only if the points x, y ∈ M coincide. A subset Ω ⊆ H is inseparable if, whenever j ∈ H satisfies h ⊆ j ⊆ k for h, k ∈ Ω, we have j ∈ Ω. Given a subset A ⊆ H , its inseparable closure is the smallest inseparable subset of H that contains A; it coincides with the union of the sets H (k∗ \|h), for h, k ∈ A. A wall is a set of the form w = {h, h∗ }, with h ∈ H ; we say that h and ∗ h are the sides of w. The wall w separates subsets A, B ⊆ M if either h or h∗ lies in H (A\|B); we denote by W (A\|B) = W (B\|A) the set of walls separating A and B and by W (M ), or simply W , the set of all walls of the median algebra M . A wall is contained in a halfspace k if one of its sides is; a wall w is contained in disjoint halfspaces k1 , k2 if and only if k2 = k∗1 and w = {k1 , k2 }. If a side of the wall w1 is transverse to a side of the wall w2 , we say that w1 and w2 are transverse. The rank of the median algebra M is the maximum cardinality of a set of pairwise transverse walls; various alternative (and equivalent) definitions of the rank can be found in Proposition 6.2 of [Bow13a]. We remark that M has rank zero if and only if it consists of a single point. If X is a median space of finite rank r, the topological dimension of every locally compact subset of X is bounded above by r, see Theorem 2.2 and Lemma 7.6 in [Bow13a]. If moreover X is complete and connected, X is b [Bow16]. The visual bi-Lipschitz equivalent to a canonical CAT(0) space X b is finite dimensional by Proposition 2.1 in [CL10]. boundary of X SUPERRIGIDITY OF ACTIONS ON FINITE RANK MEDIAN SPACES 11 b yielding a homomorEvery isometry of X extends to an isometry of X b Every convex subset of X is also convex in X; b phism Isom X ,→ Isom X. the converse is not true: the euclidean convex hull of the points (1, 0, 0), (0, 1, 0) and (1, 1, 1) in [0, 1]3 is not even a median subalgebra. Halfspaces in finite rank median spaces are fairly well-behaved [Fio17a]: Proposition 2.1. Let X be a complete median space of finite rank r. Every halfspace is either open or closed (possibly both). Moreover, if h1 ) ... ) hk is a chain of halfspaces with h∗1 ∩ hk 6= ∅, we have k ≤ 2r. The following is a simple but extremely useful observation: given ultrafilters σ1 , σ2 ⊆ H (M ) and h, k ∈ σ1 \ σ2 , we either have h ⊆ k, or k ⊆ h, or h and k are transverse. Along with Dilworth’s Theorem [Dil50] this yields the following. Lemma 2.2. Let M be a median algebra of finite rank r and let σ1 , σ2 ⊆ H be ultrafilters. (1) We can decompose σ1 \ σ2 = C1 t ... t Ck , where k ≤ r and each Ci is nonempty and totally ordered by inclusion. (2) Every infinite subset of σ1 \ σ2 contains an infinite subset that is totally ordered by inclusion. If C ⊆ M is a subset and x ∈ M , a gate for (x, C) is a point y ∈ C such that y ∈ I(x, z) for every z ∈ C; gates are unique when they exist. If a gate exists for every point of M , we say that C is gate-convex ; in this case we can define a gate-projection πC : M → C by associating to each point of M the unique gate. The gate-projection to C is always a morphism of median algebras and satisfies W (x\|C) = W (x\|πC (x)) for every x ∈ M . Gate-convex subsets are always convex, but the converse is not always true. For every y, z ∈ M , the interval I(y, z) is gate-convex with gateprojection x 7→ m(x, y, z). A proof of the following statements can be found in [Fio17a]. Proposition 2.3. Let C, C 0 ⊆ M be gate-convex. −1 (1) The sets {h ∈ H (M ) \| h∩C 6= ∅, h∗ ∩C 6= ∅}, {πC (h) \| h ∈ H (C)} and H (C) are all naturally in bijection. (2) There exists a pair of gates, i.e. a pair (x, x0 ) of points x ∈ C and x0 ∈ C 0 such that πC (x0 ) = x and πC 0 (x) = x0 . In particular, we have H (x\|x0 ) = H (C\|C 0 ). (3) The set πC (C 0 ) is gate-convex with gate-projection πC ◦ πC 0 . Moreover, πC ◦ πC 0 ◦ πC = πC ◦ πC 0 . (4) If C ∩ C 0 6= ∅, we have πC (C 0 ) = C ∩ C 0 and πC ◦ πC 0 = πC 0 ◦ πC . In particular, if C 0 ⊆ C, we have πC 0 = πC 0 ◦ πC . A median algebra (M, m) endowed with a Hausdorff topology is said to be a topological median algebra if the median map m : M 3 → M is continuous; here we equip M 3 with the product topology. Median spaces always provide 12 ELIA FIORAVANTI topological median algebras; indeed, the median map m is 1-Lipschitz in that case. In compact median algebras and complete median spaces, a subset is gateconvex if and only if it is closed and convex; moreover, gate-projections are continuous. In median spaces, gate-projections are even 1-Lipschitz. Let X be a complete, finite rank median space. In [Fio17a] we endowed b and a measure νbX (usually denoted just νb). the set H with a σ-algebra B b as measurable sets. Unlike [Fio17a], here we simply refer to the elements of B The map ∗ : H → H sending each halfspace to its complement is measure preserving. Every inseparable subset of H is measurable; in particular, all ultrafilters are measurable and, for all x, y ∈ X, we have νb(σx 4σy ) = d(x, y). Almost every halfspace h ∈ H is thick, i.e. both h and h∗ have nonemtpy b and νb differ from their counterparts in interior. Note that in general B [CDH10]. Proposition 2.4. Let X be a complete, finite rank median space and σ ⊆ H an ultrafilter such that νb(σ4σx ) < +∞ for some x ∈ X. There exists y ∈ X such that νb(σ4σy ) = 0. Thus X can be equivalently described as the collection of all ultrafilters on H that satisfy νb(σ4σx ) < +∞ for some x ∈ X; we identify ultrafilters whose symmetric difference is νb-null. Considering the space of all ultrafilters on H we obtain a set X in which X embeds. A structure of median algebra can be defined on X by setting m(σ1 , σ2 , σ3 ) := (σ1 ∩ σ2 ) ∪ (σ2 ∩ σ3 ) ∪ (σ3 ∩ σ1 ). We endow X with a topology such that ultrafilters σn ⊆ H converge to σ ⊆ H if and only if lim sup(σn 4σ) is νb-null. We refer to X as the Roller compactification of X [Fio17a]. Proposition 2.5. The Roller compactification X is a compact topological median algebra. The inclusion X ,→ X is a continuous morphism of median algebras with dense, convex image. In general, X is not open in X and the inclusion X ,→ X is not a homeomorphism onto its image. The Roller boundary is defined as ∂X := X \ X. A point of X can in general be represented by several distinct ultrafilters with null symmetric differences. However, for each ξ ∈ X there is a unique preferred ultrafilter σξ representing ξ [Fio17a]; this should be seen as a generalisation of the ultrafilters σx when x ∈ X. We can extend each halfspace h of X to a halfspace e h of X such that e e h ∩ X = h; indeed, it suffices to define h := {ξ ∈ X \| h ∈ σξ }. When ξ, η ∈ X, we save the notation H (ξ\|η) for the set ση \ σξ ⊆ H (X), instead of the analogous subset of H (X). If Y ⊆ X is a closed median subalgebra, the restriction of the metric of X turns Y into a complete median space with rank(Y ) ≤ rank(X); moreover: SUPERRIGIDITY OF ACTIONS ON FINITE RANK MEDIAN SPACES 13 Lemma 2.6. There is a canonical morphism of median algebras ιY : Y ,→ X. Proof. We write HY := {h ∈ H (X) \| h ∩ Y 6= ∅, h∗ ∩ Y 6= ∅}; intersecting with Y gives a map p : HY → H (Y ). Lemma 6.5 in [Bow13a] implies that p is surjective. Thus, for every ultrafilter σ ⊆ H (Y ), there is a unique ultrafilter σ 0 ⊆ H (X) such that σY ⊆ σ 0 and p(σ 0 ∩ HY ) = σ. Applying this to canonical ultrafilters yields the required embedding. Given ultrafilters σ1 , σ2 ⊆ H , we set d(σ1 , σ2 ) := νb(σ1 4σ2 ). We refer to d as the extended metric on X as it satisfies all the axioms of a metric, even though the value +∞ is allowed. Note that for points of X this is the same as the original median metric on X. A component Z ⊆ X is a maximal set of points having finite pairwise distances; components are convex subsets of X. One component always coincides with X ⊆ X; all other components are contained in ∂X. The following appears in [Fio17a]. Proposition 2.7. Let X be a complete median space of finite rank r. Let Z ⊆ ∂X be a component and let d denote the extended metric on X. (1) The metric space (Z, d) is a complete median space of rank at most r − 1. (2) Every thick halfspace of Z is of the form e h ∩ Z for a unique h ∈ H . If C ⊆ X is closed and convex, the closure of C in X is gate-convex and naturally identified with the Roller compactification of C; thus the notation C is not ambiguous. We denote by πC : X → C the corresponding gateprojection; it extends the usual gate-projection X → C. If σ ⊆ H (X) is an ultrafilter representing the point ξ ∈ X, the set σ ∩ H (C) is an ultrafilter on H (C) and represents πC (ξ). Similarly, if Z ⊆ ∂X is a component, the closure of Z in X is gateconvex and naturally identified with the Roller compactification Z. The gate-projection πZ : X → Z satisfies πZ (X) ⊆ Z. In terms of ultrafilters, πZ takes the point of X represented by σ ⊆ H (X) to the point of Z represented by σ ∩ H (Z) ⊆ H (Z). The intersection makes sense as, by part 2 of Proposition 2.7, almost every halfspace of Z arises from a halfspace of X. Let Γ be a group. An isometric action Γ y X is said to be Roller elementary if there exists a finite orbit within X. The action is Roller minimal if rank(X) ≥ 1 and Γ does not preserve any proper, closed, convex subset C ⊆ X. Roller elementarity implies – but is in general much stronger than – the b When X is existence of a finite orbit in the visual compactification of X. a CAT(0) cube complex, an action is Roller minimal if and only if, in the terminology of [CS11], it is essential and does not fix any point in the visual b boundary of X. Neither Roller elementarity, nor Roller minimality implies the other one. However, Roller minimal actions naturally arise from Roller nonelementary ones [Fio17b]: 14 ELIA FIORAVANTI Proposition 2.8. Let X be a complete, finite rank median space with an isometric action Γ y X. Either Γ y X fixes a point or there exist a Γinvariant component Z ⊆ X and a Γ-invariant, closed, convex subset C ⊆ Z such that Γ y C is Roller minimal. A Γ-invariant, closed, convex subset C ⊆ Z ⊆ X always gives rise to a ∗ ); here we introduced the measurable decomposition H = HC t (σC ∪ σC sets HC := {h ∈ H \| C ∩ e h 6= ∅, C ∩ e h∗ 6= ∅} and σC := {h ∈ H \| C ⊆ e h}. Note that, by part 2 of Proposition 2.7, the measure spaces (HC , νbX ) and (H (C), νbC ) are isomorphic. We say that an action Γ y X is without wall inversions if there do not exist g ∈ Γ and h ∈ H such that gh = h∗ . By Proposition 2.1, any action of a connected, complete, finite rank median space is without wall inversions. The following was proved in [Fio17b]; compare with [CS11]. Proposition 2.9. Let X be a complete, finite rank median space with thick halfspaces h ⊆ k. Let Γ y X be a Roller minimal action without wall inversions. (1) There exists g ∈ Γ such that gh∗ ( h and d (gh∗ , h∗ ) > 0. (2) There exists g ∈ Γ such that gk ( h ⊆ k and d(gk, h∗ ) > 0. When G is a topological group, all isometric actions G y X will be implicitely required to have continuous orbit maps. Equivalently, the homomorphism G → Isom X is continuous, where we endow Isom X with the topology of pointwise convergence. We remark that Isom X is a Hausdorff, sequentially complete topological group as soon as X is complete. If (M1 , m1 ) and (M2 , m2 ) are median algebras, the product median algebra is defined as (M1 × M2 , m), where m = (m1 ◦ p1 , m2 ◦ p2 ); here pi denotes the projection onto the i-th factor. If (X1 , d1 ) and (X2 , d2 ) are median spaces, we endow the product X1 ×X2 with the `1 metric, namely d = d1 ◦p1 +d2 ◦p2 . The median algebra associated to the median space (X1 × X2 , d) is just the product median algebra arising from X1 and X2 . A median space X is said to be irreducible if it is not isometric to any nontrivial product X1 × X2 . Proposition 2.10. Let X1 , ..., Xk be irreducible, complete, finite rank median spaces; consider the product X = X1 × ... × Xk . (1) We have a measurable partition H (X) = H1 t ... t Hk , where each Hi is canonically identified with H (Xi ). If h ∈ Hi and k ∈ Hj with i 6= j, the halfspaces h and k are transverse. (2) Every isometry of X permutes the members of the partition. The product Isom X1 × ... × Isom Xk sits inside Isom X as an open, finite index subgroup. (3) Every closed, convex subset C ⊆ X is of the form C1 × ... × Ck , where each Ci is a closed convex subset of Xi . (4) The Roller compactification X is naturally identified with the product median algebra X1 × ... × Xk with the product topology. SUPERRIGIDITY OF ACTIONS ON FINITE RANK MEDIAN SPACES 15 Proof. For part 1 and the first half of part 2, see Corollary 2.9 and Proposition 2.11 in [Fio17b]. We conclude the proof of part 2 by showing that Isom X1 × ... × Isom Xk is open in Isom X. Choose points xi , yi ∈ Xi for each i and a real number > 0 such that 2 < d(xi , yi ) for all i; denote by pi : X → Xi the projection onto the i-th factor. Let x ∈ X be the point with coordinates xi ; we also consider the points zi ∈ X such that pj (zi ) = xj for all i 6= j and pi (zi ) = yi . Suppose that F ∈ Isom X is such that d(F (x), x) < and d(F (zi ), zi ) < for all i. We claim that F (Hi ) = Hi for all i; this implies that the product of the isometry groups of the factors is open in Isom X. Suppose for the sake of contradiction that, for indices i 6= j, we have F (Hi ) = Hj . This induces an isometry f : Xi → Xj such that pj ◦ F = f ◦ pi , see [Fio17a]. In particular, we have d(xj , f (xi )) ≤ d(x, F (x)) < and d(xj , f (yi ) ≤ d(zi , F (zi )) < ; thus, d(xi , yi ) = d(f (xi ), f (yi )) < 2, a contradiction. Irreducibility of the factors plays no role in part 3, so it suffices to consider the case k = 2. Let C1 and C2 be the projections of C to X1 and X2 . If x ∈ C1 and y ∈ C2 , there exist u ∈ X1 and v ∈ X2 such that the points x := (x, v) and y := (u, y) lie in C. It is immediate to observe that the point (x, y) lies in I(x, y) ⊆ C. Finally, part 4 is Lemma 2.10 in [Fio17b]. Halfspaces h1 , ..., hn form a facing n-tuple, n ≥ 2, if they are pairwise disjoint. We say that h, k ∈ H are strongly separated if h ∩ k = ∅ and no j ∈ H is transverse to both h and k. See [Fio17b] for the following. Proposition 2.11. Let X be an irreducible, complete, finite rank median space; let h be a thick halfspace. (1) If X admits a Roller minimal action without wall inversions, there exist thick halfspaces h0 ⊆ h ⊆ h00 such that h0 and h00∗ are strongly separated. (2) If X admits a Roller nonelementary, Roller minimal action without walls inversions, h is part of a facing n-tuple of thick halfspaces for every n ≥ 2. Every complete, finite rank median space can be isometrically embedded into its barycentric subdivision X 0 . This is a complete median space of the same rank, see [Fio17b]; when X is the 0-skeleton of a CAT(0) cube complex with the `1 metric, the space X 0 is given by the 0-skeleton of the customary barycentric subdivision. We have a natural homomorphism Isom X ,→ Isom X 0 . Given any isometric action Γ y X, the induced action Γ y X 0 is without wall inversions. We write (H 0 , ν 0 ) instead of (H (X 0 ), νX 0 ). There is an inclusion preserving map p : H 0 → H ; it is surjective, (Isom X)-equivariant and its fibres have cardinality at most two. We have #(p−1 (h)) = 2 if and only if {h} is an atom for νb; in this case, we refer to each element of p−1 (h) as a hemiatom. See [Fio17b] for the following lemma. 16 ELIA FIORAVANTI Lemma 2.12. Let X be a complete, finite rank median space. An action Γ y X is Roller elementary if and only if the induced action Γ y X 0 is. The sets {−1, 1} and {−1, 0, 1} inherit a median algebra structure from the median space R; in particular, we can consider the product median algebras {−1, 1}k and {−1, 0, 1}k for every k ≥ 0. For every point x ∈ X 0 \ X, b there exists a canonical, gate-convex subset C(x) ⊆ X 0 ; it is isomorphic k to {−1, 0, 1} , for some k ≥ 1, via an isomorphism that takes x to the b centre (0, ..., 0). The intersection C(x) := C(x) ∩ X is gate-convex in X k and corresponds to the subset {−1, 1} ⊆ {−1, 0, 1}k . For x ∈ X, we set b C(x) = C(x) = {x}. See [Fio17b] for more details. Lemma 2.13. Let X be a complete, finite rank median space. Every infinite, convex subset C ⊆ X 0 intersects X. Proof. Let x ∈ C be a point minimising R := rank(C(x)); if R = 0 we have b x ∈ C ∩ X. Otherwise, there exists a point y ∈ C that does not lie in C(x); b b the gate-projection z of y to C(x) lies in C(x) \ {x}. In particular C(z) is contained in a face of C(x) and has strictly lower rank, a contradiction since z ∈ I(x, y) ⊆ C. In particular, we can obtain the following extension of Lemma 2.12. Lemma 2.14. If Γ y X is Roller nonelementary and Roller minimal, so is the action Γ y X 0 . Proof. Suppose for the sake of contradiction that C ⊆ X 0 is a nonempty, Γ-invariant, closed, convex subset. By Corollary 4.31 in [Fio17a] there exists a Γ-invariant component W ⊆ X 0 with W ∩ C 6= ∅; note that C ∩ W is unbounded by Corollary 2.16 in [Fio17b], since Γ y X 0 is Roller nonelementary by Lemma 2.12. The component W is the barycentric subdivision of a component Z ⊆ X and Lemma 2.13 implies that C ∩ Z 6= ∅. Since Γ y X is Roller minimal, we must have Z = X and C ∩ X = X; hence X 0 ⊆ C by part 2 of Proposition 2.14 in [Fio17b]. We conclude that C = X 0 . Let us now fix a basepoint x ∈ X, where X is a complete finite rank median space; the following discussion is independent of our choice of x. A diverging chain of halfspaces is a sequence (hn )n≥0 such that d(x, hn ) → +∞ and hn+1 ⊆ hn for each n ≥ 0; we use the same terminology for the set {hn \| n ≥ 0}. Given ξ ∈ ∂X, a UBS for ξ is an inseparable subset Ω ⊆ σξ \σx containing a diverging chain of halfspaces. Given UBS’s Ω1 , Ω2 ⊆ σξ \ σx , we say that Ω1 is almost contained in Ω2 if the halfspaces in Ω1 \ Ω2 lie at uniformly bounded distance from x; this is denoted by Ω1 Ω2 . If Ω1 Ω2 and Ω2 Ω1 , the UBS’s are equivalent and we write Ω1 ∼ Ω2 . We denote the equivalence class of Ω ⊆ H by [Ω] and the set of all equivalence classes of UBS’s for ξ by U(ξ); the relation descends to a partial order on U(ξ). A UBS Ω is said to be minimal if SUPERRIGIDITY OF ACTIONS ON FINITE RANK MEDIAN SPACES 17 [Ω] is a minimal element of (U(ξ), ). A minimal UBS is equivalent to the inseparable closure of any diverging chain that it contains. We define a directed graph G(ξ) as follows [Hag17, Fio17b]. The vertex set of G(ξ) is identified with the set of minimal elements of (U(ξ), ). Given diverging chains (hm )m≥0 and (kn )n≥0 in Ω1 and Ω2 respectively, we draw an oriented edge from [Ω1 ] to [Ω2 ] if almost every hm is transverse to almost every kn , but not vice versa; this is independent of the choices involved. A subset A ⊆ G(ξ)(0) is said to be inseparable if every directed path between vertices in A only crosses vertices in A. The following can be found in [Fio17b]. Proposition 2.15. Let X be a complete median space of finite rank r and ξ ∈ ∂X. (1) The graph G(ξ) has at most r vertices and contains no directed cycles. (2) The poset (U(ξ), ) is isomorphic to the poset of inseparable subsets of G(ξ)(0) , ordered by inclusion. The isomorphism maps [Ω] ∈ U(ξ) to the set of equivalence classes of minimal UBS’s almost contained in Ω. In particular, the set U(ξ) is finite. (3) Given a UBS Ω and a set {Ω1 , ..., Ωk } of representatives of all equivalence classes of minimal UBS’s almost contained in Ω, we have sup d(x, h) < +∞. h∈Ω4(Ω1 ∪...∪Ωk ) If Ω is a minimal UBS, we denote by Ω0 the subset of halfspaces k ∈ Ω that are not transverse to any diverging chain of halfspaces in Ω. We say that Ω is reduced if Ω = Ω0 . We say that Ω is strongly reduced if we can write Ω = C1 t ... t Ck for some k ≥ 1, where each Ci is totally ordered by inclusion and contains a diverging chain of halfspaces. Consider the median spaces in Figures 1 and 2; both are subsets of R2 with the restriction of the `1 metric. In both cases, the UBS Ω = σξ \ σx is minimal. Figure 1 shows that Ω can be reduced, but not strongly reduced. In Figure 2 the UBS Ω is strongly reduced and exhibits how the decomposition Ω = C1 t ... t Ck can require k ≥ 2. Lemma 2.16. Let X be a complete, finite rank median space. Consider x ∈ X and ξ ∈ ∂X; let Ω ⊆ σξ \ σx be a minimal UBS. (1) The subset Ω0 is a reduced UBS equivalent to Ω. (2) There exists a strongly reduced UBS contained in Ω; all its sub-UBS’s are strongly reduced. (3) If Ω is strongly reduced, it is reduced. Proof. If h ⊆ k lie in σξ \ σx and k is transverse to a diverging chain of halfspaces, then h is transverse to an infinite subchain. This implies that Ω0 is inseparable; moreover, Ω \ Ω0 cannot contain a diverging chain or Ω would contain two inequivalent UBS’s. This proves part 1. To obtain part 2, we decompose Ω = C1 t ... t Ck as in Lemma 2.2; let A be the union of the sets Ci that do not contain a diverging chain. 18 ELIA FIORAVANTI Figure 1 Figure 2 SUPERRIGIDITY OF ACTIONS ON FINITE RANK MEDIAN SPACES 19 There exists D < +∞ such that d(x, h) ≤ D for every h ∈ A. The set {h ∈ Ω \| d(x, h) > D} is a strongly reduced UBS and so are its sub-UBS’s. Regarding part 3, decompose Ω = C1 t ... t Ck , where each Ci is totally ordered by inclusion and contains a diverging chain. If there existed k ∈ Ω \ Ω0 , we would have k ∈ Ci for some i; in particular, k would not be transverse to a diverging chain in Ci . Since Ω is minimal, k would not be transverse to any diverging chain in Ω, a contradiction. Given ξ ∈ ∂X, we denote by Isomξ X the group of isometries that fix ξ. Let Kξ be the kernel of the action of Isomξ X on U(ξ); it is a finite-index subgroup of Isomξ X. If Ω ⊆ σξ \ σx is a UBS, we denote by KΩ the subgroup of Isomξ X that fixes [Ω]. We can define χΩ : KΩ → R by the for a transfer−1character −1 mula χΩ (g) = νb g Ω \ Ω − νb Ω \ g Ω . This is a homomorphism and only depends on the equivalence class [Ω]. If Ω1 , ..., Ωk are a set of representatives for G(ξ)(0) , we can also consider the full transfer homomorphism χξ = (χΩ1 , ..., χΩk ) : Kξ → Rk . Proposition 2.17. Let X be a complete, finite rank median space; consider ξ ∈ ∂X. The subgroup Kξ is open in Isomξ X and the full transfer homomorphism χξ : Kξ → Rk is continuous. Every finitely generated subgroup of ker χξ has a finite orbit in X; if X is connected, every finitely generated subgroup of ker χξ fixes a point. Before proving the proposition, we will need to obtain the following lemma. Note that, for every point ξ ∈ ∂X and every halfspace h ∈ σξ , the set H (h∗ \|ξ) = {k ∈ σξ \| k ⊆ h} is a UBS. Lemma 2.18. For every thick halfspace h ∈ σξ and every > 0, there exists a neighbourhood U of the identity in Isomξ X such that H (h∗ \|ξ) ∼ H (gh∗ \|ξ) and νb (H (h∗ \|ξ)4H (gh∗ \|ξ)) < for all g ∈ U . Proof. Pick a point x ∈ X with d(x, h) > 0; in a neighbourhood of the identity of Isomξ X, we have x ∈ gh∗ . If k ∈ H (h∗ \|ξ) \ H (gh∗ \|ξ) and y is the gate-projection of x to k, we have y ∈ gh∗ by part 4 of Proposition 2.3, since k ∩ gh∗ 6= ∅ and x ∈ gh∗ ; thus d(y, g −1 y) ≥ d(k, h∗ ). We conclude that, for every k ∈ H (h∗ \|ξ) with d(k, h∗ ) > 0, there exists a neighbourhood Vk of the identity in Isomξ X such that k ∈ H (gh∗ \|ξ) for all g ∈ Vk . Decompose H (h∗ \|ξ) = C1 t ... t Ck as in Lemma 2.2. Let ki be the union of all k ∈ Ci with d(h∗ , k) ≥ 2k ; if nonempty, it is a halfspace and d(h∗ , ki ) ≥ 2k . ∗ Elements of H (h \|ξ) not contained in any k form a subset of measure at i P most νb(H (h∗ \|ki )) ≤ 2 . Let V be the intersection of the Vki for ki 6= ∅; if g ∈ V , the set H (h∗ \|ξ) \ H (gh∗ \|ξ) has measure at most 2 and consists of halfspaces at uniformly bounded distance from x ∈ X. It now suffices to consider U := V ∩ V −1 . Proof of Proposition 2.17. We only need to prove that Kξ is open and that χξ is continuous; the rest of the statement is contained in Theorem F of [Fio17b]. 20 ELIA FIORAVANTI For every v ∈ G(ξ)(0) , there exists hv such that the vertices w ∈ G(ξ)(0) with w [H (h∗v \|ξ)] are precisely v and those that are at the other end of an incoming edge at v. Indeed, given a diverging chain in a UBS representing v, almost every halfspace in the chain can be chosen as hv . Let Ai be the set of vertices v ∈ G(ξ)(0) such that there exists no directed path of length ≥ i in G(ξ) that ends at v. Note that, by Proposition 2.15, we have ∅ = A0 ( A1 ( ... ( Ak = G(ξ)(0) for some k ≤ r. We will show that the subgroup Ki ≤ Isomξ X that fixes Ai ⊆ G(ξ)(0) pointwise is open in Isomξ X and that, for every [Ω] ∈ Ai , the homomorphism χΩ : Ki → R is continuous. We proceed by induction on i, setting K0 := Isomξ X. The base step is trivial. If i ≥ 1, let Ai \ Ai−1 = {v1 , ..., vs } and consider the halfspaces hv1 , ..., hvs . Setting Ξj := H (h∗vj \|ξ), Lemma 2.18 provides a neighbourhood U of id in Isomξ X such that gΞj ∼ Ξj for all j and all g ∈ U . A minimal UBS almost contained in Ξj projects to an element of Ai−1 or to vj ; hence, we have gvj = vj for every g ∈ U ∩ Ki−1 . Since, by the inductive hypothesis, Ki−1 is open, so is Ki . Continuity of the transfer characters is obtained with a similar argument. 2.2. Bridges. Let a median algebra (M, m) and two gate-convex subsets C1 , C2 be fixed throughout this section. All the following results have analogues in Section 2.G of [CFI16]. Denote by πi : M → Ci the gate-projection to Ci . We will refer to the sets S1 := {x1 ∈ C1 \| ∃x2 ∈ C2 s.t. (x1 , x2 ) are gates for (C1 , C2 )} , S2 := {x2 ∈ C2 \| ∃x1 ∈ C1 s.t. (x1 , x2 ) are gates for (C1 , C2 )} , as the shores of C1 and C2 , respectively. By part 3 of Proposition 2.3, these coincide with π1 (C2 ) and π2 (C1 ), hence they are gate-convex. The map π2 \|S1 : S1 → S2 is a bijection with inverse π1 \|S2 ; if M arises from a median space X, this is an isometry as gate-projections are 1-Lipschitz. The bridge is the set G G (∗) B := I (x1 , π2 (x1 )) = I (π1 (x2 ), x2 ) . x1 ∈S1 x2 ∈S2 The union is disjoint because, if (x1 , x2 ) is a pair of gates for (C1 , C2 ), we have πi (I(x1 , x2 )) = {xi } for i = 1, 2; this follows from part 4 of Proposition 2.3 and the observation that I(x1 , x2 ) ∩ Ci = {xi }. Proposition 2.19. The bridge B is gate-convex and W (B) = (W (C1 ) ∩ W (C2 )) t W (C1 \|C2 ). For any pair of gates (x1 , x2 ), the bridge is canonically isomorphic to the product S1 × I(x1 , x2 ); this is an isometry if M arises from a median space. Proof. Pick a pair of gates (x1 , x2 ), set I := I(x1 , x2 ) and consider the morphism of median algebras φ := πS1 × πI : M → S1 × I. If (x01 , x02 ) is another SUPERRIGIDITY OF ACTIONS ON FINITE RANK MEDIAN SPACES 21 pair of gates, the projection πI provides an isomorphism I(x01 , x02 ) → I mapping each x0i to xi . This observation and the decomposition (∗) above imply that the restriction φ\|B is bijective. The map πB = (φ\|B )−1 ◦φ : M → B is surjective and it is a gate-projection by Proposition 2.1 in [Fio17a]. By part 1 of Proposition 2.3 and the discussion above, every wall of B arises either from a wall of M cutting S1 or from a wall of M cutting I; the latter correspond to W (C1 \|C2 ) by part 2 of Proposition 2.3, so we are left to show that W (S1 ) = W (C1 ) ∩ W (C2 ). This follows from the fact that S1 = π1 (C2 ). When M arises from a median space X, the measure νb induces a measure µ b on the set W [Fio17a]. In this case, the fact that φ\|B is an isometry follows from the decomposition of W (B) above and the observation that µ b(W (x\|y)) = d(x, y) for all x, y ∈ X. We can extend the notion of strong separation to arbitrary gate-convex subsets of median algebras. We say that C1 and C2 are strongly separated if they are disjoint and W (C1 ) ∩ W (C2 ) = ∅. Note that the condition W (C1 ) ∩ W (C2 ) = ∅ alone already implies that C1 ∩ C2 consists of at most one point. In a median space, two halfspaces are strongly separated in the sense of Section 2.1 if and only if their closures are strongly separated according to the definition above; see Lemma 2.21 below for a stronger result. Proposition 2.19 implies that two disjoint, gate-convex sets are strongly separated if and only if their shores are singletons; this yields the following result. Corollary 2.20. Let C1 , C2 ⊆ M be strongly separated. There exists a unique pair of gates (x1 , x2 ) and π1 (C2 ) = {x1 }, π2 (C1 ) = {x2 }. We will also need the following: Lemma 2.21. Let h, k ∈ H be strongly separated as halfspaces. The closures H, K of e h, ek in X are strongly separated as subsets of X. Proof. Since h, k have disjoint closures in X, the sets H and K are disjoint (see e.g. the proof of Theorem 5.1 in [Fio17b]). By Proposition 2.19, it then suffices to prove that the shore S ⊆ H is a singleton. Suppose for the sake of contradiction that S contains distinct points ξ, η and let ξ 0 , η 0 ∈ K be their projections to K; in particular, ση \ σξ = ση0 \ σξ0 . Given j ∈ ση \ σξ , the argument at the beginning of the proof shows that the closures of j and h inside X intersect nontrivially; by Lemma 3.6 in [Fio17a], almost every j ∈ ση \ σξ intersects h. Considering complements, we conclude that almost every j ∈ ση \σξ is transverse to h and, similarly, to k. Since d(ξ, η) > 0, there exists such a j, contradicting the fact that h, k are strongly separated. 2.3. The Haagerup class. Let X be a median space, G a topological group and p ∈ [1, +∞). Given a Banach space E, we denote by U (E) the group of linear isometries of E. 22 ELIA FIORAVANTI An isometric action G y X corresponds to a measure preserving action G y (H , νb). We obtain a continuous representation ρp : G → U (Lp (H , νb)); when p = 2, we simply write ρ := ρ2 . We will use interchangeably the notations Hc1 (G, ρp ) and Hc1 (G, Lp (H , νb)) to denote continuous cohomology. Given x ∈ X, we can consider the continuous 1-cocycle bx : G → Lp (H , νb) defined by bx (g) := g · 1σx − 1σx ; it satisfies kbx (g)kp = (2 · d(x, gx))1/p . We will refer to bx as a Haagerup cocycle. The cohomology class [bx ] ∈ Hc1 (G, ρp ) does not depend on the point x and we will simply denote it by [b]. The action G y X has bounded orbits if and only if the affine action G y Lp (H , νb) induced by bx fixes a point; this follows for instance from the Ryll-Nardzewski Theorem for p > 1 and from Theorem A in [BGM12] in the case p = 1. Thus, we have [b] = 0 if and only if the action G y X has bounded orbits. We can also consider the projection [b] of [b] to reduced continuous cohomology, which carries more interesting geometrical information (see Theorem A in the introduction). We will refer to [b] ∈ Hc1 (G, ρ) as the Haagerup class of G y X. The choice of p = 2 here is not particularly relevant and the same discussion could be equally carried out for any other p ∈ [1, +∞) with few complications; see Remark 3.6. We conclude this section by collecting a few straightforward lemmata for later use. Let H be a Hilbert space and G → U (H) a continuous unitary representation. Lemma 2.22. If H ≤ G is open and finite-index, functoriality induces an injective map Hc1 (G, H) ,→ Hc1 (H, H). Lemma 2.23. Given a G-invariant decomposition H = H1 ⊥ H2 , the projections onto the two factors induce Hc1 (G, H) ' Hc1 (G, H1 ) ⊕ Hc1 (G, H2 ). Recall that we denote by X 0 the barycentric subdivision of X and by i : X ,→ X 0 the standard inclusion; we write (H 0 , ν 0 ) instead of (H (X 0 ), νbX 0 ). Every isometric action G y X also induces a continuous representation ρ0 : G → U (L2 (H 0 , ν 0 )). Lemma 2.24. Let X be complete and finite rank and let G y X be an isometric action. The projection p : H 0 → H induces an isometric embedding p∗ : L2 (H , νb) ,→ L2 (H 0 , ν 0 ) and a monomorphism p∗ : Hc1 (G, ρ) ,→ Hc1 (G, ρ0 ) taking the Haagerup class of G y X to the Haagerup class of G y X 0 . Proof. The fact that p∗ is an isometric embedding follows from the observation that p∗ ν 0 = νb. Injectivity of p∗ follows from Lemma 2.23 applied to L2 (H , νb) and its orthogonal complement. Finally, if x ∈ X and bx is the corresponding Haagerup cocycle for G y X, the cocycle p∗ ◦ bx is the Haagerup cocycle for G y X 0 relative to the point i(x) ∈ X 0 . SUPERRIGIDITY OF ACTIONS ON FINITE RANK MEDIAN SPACES 23 3. Haagerup class and elementarity of actions. 3.1. The main statement. Let X be a complete, finite rank median space and let G y X be an isometric action of a topological group G. The goal of this section is to prove Theorem A. By Lemmata 2.12 and 2.24, it suffices to consider the case when G y X is without wall inversions; this will be a standing assumption throughout the rest of the section. Lemma 3.1. Suppose that X is irreducible and that G y X is Roller minimal and Roller nonelementary. There exists a non-abelian free subgroup H ≤ G such that ρ has no H-almost-invariant vectors and H y X has unbounded orbits. Proof. Part 2 of Proposition 6.4 in [Fio17b] provides a non-abelian free subF group H ≤ G and a measurable, ∗-invariant partition H = h∈H Hh with gHh = Hgh for all g, h ∈ H. It is immediate from the construction of H that it acts on X with unbounded orbits. If there existed a sequence of almost invariant vectors (Fn )n≥0 in L2 (H , νb), say with kFn k2 = 1, we could define functions fn ∈ `2 (H) by fn (h) := kFn 1Hh k2 . It is immediate to check that kfn k2 = 1 for every n ≥ 0 and, for every g ∈ H, 2 X kgfn − fn k22 = kFn 1Hg−1 h k2 − kFn 1Hh k2 = h∈H = X k(gFn ) 1Hh k2 − kFn 1Hh k2 2 ≤ h∈H ≤ X k(gFn ) 1Hh − Fn 1Hh k22 = kgFn − Fn k22 −−−−−→ 0. h∈H n→+∞ Thus, the regular representation of H would contain almost-invariant vectors, implying amenability of H (see e.g. Theorem G.3.2 in [BdlHV08]). This is a contradiction. We can already prove the “only if” half of Theorem A. Proposition 3.2. If G y X is Roller nonelementary, we have [b] 6= 0. Proof. Note that, by functoriality of reduced cohomology, it suffices to consider the case when G has the discrete topology; thus, we do not need to worry about continuity issues. We proceed by induction on r = rank(X). When r = 0, all actions are Roller elementary; assume for the rest of the proof that r ≥ 1. We can also assume that G y X is Roller minimal. Indeed, if C ⊆ Z ⊆ X are the subsets provided by Proposition 2.8, the action G y C is again without wall inversions and rank(C) ≤ r, by Proposition 2.7. The G-equivariant, ∗ ) induces an orthogonal decommeasurable partition H = HC t (σC ∪ σC 2 position of L (H , νb) and a G-equivariant splitting ∗ Hc1 G, L2 (H , νb) = Hc1 G, L2 (H (C), νbC ) ⊕ Hc1 G, L2 (σC ∪ σC , νb) . 24 ELIA FIORAVANTI If p1 and p2 are the orthogonal projections of L2 (H , νb) onto the two direct summands, we can write [bx ] = [p1 bx ] + [p2 bx ] for every x ∈ X. Note that the gate-projection πC : X → C maps x to a point ξ ∈ C. The cocycle p1 bx is precisely the Haagerup cocycle bξ for the action G y C, by part 2 of Proposition 2.7. In particular, if [bξ ] 6= 0 we can conclude that [b] 6= 0. We thus assume in the rest of the proof that X = C. If X is irreducible, Lemma 3.1 provides i : H ,→ G such that H y X has unbounded orbits and ρ has no H-almost-invariant vectors. The first condition implies that i∗ [b] 6= 0; the second condition and Theorem 1 in [Gui72] thus yield i∗ [b] 6= 0. In particular, [b] 6= 0. If instead X splits as a nontrivial product X1 × X2 and j : G0 ,→ G is the finite-index subgroup preserving this decomposition, it suffices to show that j ∗ [b] 6= 0. Writing νbi instead of νbXi , Proposition 2.10 and Lemma 2.23 imply: Hc1 G0 , L2 (H , νb) = Hc1 G0 , L2 (H (X1 ), νb1 ) ⊕ Hc1 G0 , L2 (H (X2 ), νb2 ) ; hence, if x = (x1 , x2 ), we have [bx ] = [bxX11 ] + [bxX22 ]. The action G0 y X is Roller nonelementary, since G0 is finite-index in G; thus, up to exchanging the two factors, G0 y X1 is Roller nonelementary. Since rank(X1 ) < r, the inductive hypothesis guarantees that [bxX11 ] 6= 0 and this concludes the proof. Before proving the rest of Theorem A we need to obtain a few more results. Lemma 3.3. If the G-orbit of ξ ∈ X is finite, the G-stabiliser of ξ is open. Proof. Suppose that Gξ = {ξ} t {ξ1 , ..., ξk } and d(ξ, ξi ) ≥ > 0 for all i, where d is the extended metric on X. By Proposition 4.24 in [Fio17a], we can find xi , yi ∈ X such that, denoting by πi the projection to Ii := I(xi , yi ), we have d(πi (ξ), πi (ξi )) > 34 . In a neighbourhood U ⊆ G of the identity element we have d(gxi , xi ) < 41 , d(gyi , yi ) < 41 and d(gπi (ξ), πi (ξ)) < 41 , for all i. If g ∈ U , we have d(πgIi (ξi ), πi (ξi )) ≤ d(gxi , xi ) + d(gyi , yi ) < 12 ; if in addition we had gξ = ξi , we would have πgIi (ξi ) = gπi (ξ). As a consequence, d(πi (ξ), πi (ξi )) ≤ d(πi (ξ), gπi (ξ)) + d(gπi (ξ), πi (ξi )) < 34 , which would contradict our choice of Ii . We conclude that U is contained in the stabiliser of ξ, which must be open in G. The proof of the following fact is rather lengthy and technical; it will be carried out in Appendix A. Proposition 3.4. Let ξ ∈ ∂X and K ⊆ Isomξ X be a compact set of isometries acting trivially on U(ξ). There exists a point xK ∈ X such that σξ \σxK coincides, up to a null set, with ΩK := Ω1K t ... t ΩkK , where • each ΩiK is a strongly reduced, minimal UBS; SUPERRIGIDITY OF ACTIONS ON FINITE RANK MEDIAN SPACES 25 • if g ∈ K we have gΩiK ⊆ ΩiK whenever χΩi (g) ≥ 0 and gΩiK ⊇ ΩiK K whenever χΩi (g) ≤ 0; K • if i 6= j and g ∈ K, we have ΩiK ∩ gΩjK = ∅. Given points ξ ∈ ∂X, x ∈ X and a UBS Ω ⊆ σξ \ σx , we can define a function αΩ : Ω → R as αΩ (h) := νb (H (x\|h) ∩ Ω). The dependence on the point x is not particularly relevant, so we do not record it in our notation. We can consider the sets Ωc := {h ∈ Ω \| αΩ (h) ≤ c}. In Appendix A we will obtain the following result (see Lemma A.9). Proposition 3.5. Suppose that Ω is minimal and strongly reduced. Let K ⊆ Isomξ X be a compact set of isometries such that gΩ ⊆ Ω for all g ∈ K. As c → +∞, the functions i h αΩ · 1Ωc (g − id) · − 1 − c converge to 1Ω\gΩ in L2 (H , νb), uniformly in g ∈ K. If instead gΩ ⊇ Ω for all g ∈ K, they converge to the function −1gΩ\Ω . We are finally ready to complete the proof of Theorem A. Proof of Theorem A. By Proposition 3.2, it suffices to consider the case when G has a finite orbit in X and, by Lemmata 2.22 and 3.3, we can actually assume that G fixes a point ξ ∈ X. If ξ ∈ X, we have [b] = 0; suppose instead that ξ ∈ ∂X. By Proposition 2.17, an open, finite-index subgroup G0 ≤ G acts trivially on U(ξ); by Lemma 2.22, it suffices to consider the case G = G0 . Fix x ∈ X. For every > 0 and every compact subset K ⊆ G, we need to construct a function ψ ∈ L2 (H , νb) such that kbx (g) − (gψ − ψ)k2 < for all g ∈ K. Considering the point xK ∈ X provided by Proposition 3.4, it suffices to find a function φ ∈ L2 (H , νb) such that kbxK (g) − (gφ − φ)k2 < for all g ∈ K and then set ψ := φ + 1σx − 1σxK . If g ∈ K, considering all equalities up to null sets, we have σgxK \ σxK = [(σgxK \ σxK ) ∩ σξ ] t (σgxK \ σxK ) ∩ σξ∗ = = [(σξ \ σxK ) \ (σξ \ σgxK )] t [(σξ \ σgxK ) \ (σξ \ σxK )]∗ = = (ΩK \ gΩK ) t (gΩK \ ΩK )∗ . In particular, since by construction ΩiK ∩ gΩjK = ∅ whenever i 6= j, σgxK \ σxK = k G ∗ ΩiK \ gΩiK t gΩiK \ ΩiK . i=1 Introducing the notation 2A := 1A − 1A∗ for subsets A ⊆ H , we can rewrite X X bxK (g) = 2σgxK \σxK = 2Ωi \gΩi − 2gΩi \Ωi . K χΩi (g)>0 K K K χΩi (g)<0 K K 26 ELIA FIORAVANTI For c > 0, consider the function Gc := k X i=1 αΩi K 2(ΩiK ) . − 1− c c Proposition 3.5 shows that it suffices to take φ = Gc for large c. Remark 3.6. Theorem A also holds for the analogous class in Hc1 (G, ρp ), for every p ∈ [1, +∞). Indeed, Lemma 2.23 applies to any decomposition of a Banach space into closed subspaces. In the proof of Lemma 2.24, a closed complement to Lp (H , νb) within Lp (H 0 , ν 0 ) is always provided by the subspace of functions on H 0 that take opposite values on hemiatoms. Theorem 1 in [Gui72] holds for representations in general Banach spaces. Finally, if F is a non-abelian free group, `p (F ) has no F -almost-invariant vectors for every p ∈ [1, +∞). The value of p also has little importance for most of the material in Appendix A. Note however that Proposition 3.5 fails for p = 1; one needs to consider functions with a quicker decay in that case. 3.2. Elementarity and Shalom’s property HF D . Let X be a complete, finite rank median space and let G be a topological group. Our main result on property HF D is the following. Proposition 3.7. If G has property HF D , every isometric action G y X is Roller elementary. We will need the following lemma. Lemma 3.8. Suppose that X is irreducible and let G y X be a Roller nonelementary, Roller minimal action. Let E ⊆ H be a measurable subset such that νb(gE4E) = 0 for all g ∈ G. Then νb(E) is either 0 or +∞. Proof. Without loss of generality we can assume that G y X is without wall inversions, as Lemma 2.14 allows us to pass to the barycentric subdivision X 0 if necessary. Now, suppose that E is such a set and 0 < νb(E) < +∞. Since X has finite rank, we can find a thick halfspace h such that, replacing E with E ∗ if necessary, the set Eh := E ∩ {k ∈ H \| k ⊆ h} satisfies a := νb(Eh ) > 0. By part 2 of Proposition 2.11, the halfspace h is part of a facing n-tuple with n > a1 · νb(E). By Proposition 2.9, there exist g2 , ..., gn ∈ G such that h, g2 h, ..., gn h are a facing n-tuple. The sets Eh , g2 Eh , ..., gn Eh are pairwise disjoint and contained in E, up to null sets. However, their union has measure na > νb(E), a contradiction. Proof of Proposition 3.7. Suppose for the sake of contradiction that G y X is Roller nonelementary. Without loss of generality, we can assume that X has minimal rank r among complete median spaces admitting Roller nonelementary actions of G. In particular, X must be irreducible, see the proof of Proposition 3.2. By Proposition 2.8, we can also assume that G y X is Roller minimal. Theorem A guarantees that Hc1 (G, ρ) 6= {0} and, since SUPERRIGIDITY OF ACTIONS ON FINITE RANK MEDIAN SPACES 27 G has property HF D , there exists a finite dimensional subrepresentation V < L2 (H , νb). We will construct a measurable G-invariant subset of H with positive, finite measure, thus violating Lemma 3.8. Let f1 , ..., fk be measurable functions on H whose equivalence classes in L2 (H , νb) form an orthonormal basis of V . Define, for c > 0, n o Ec := h ∈ H \| ∃α = (α1 , ..., αk ) ∈ Sk−1 s.t. \|(α1 f1 + ... + αk fk )(h)\| > c . Since in the definition of Ec it suffices to look at α’s lying in a countable dense subset of Sk−1 , we conclude that Ec is measurable. If h ∈ Ec , we must have \|fi (h)\| > kc for some i, hence νb(Ec ) < +∞; if c is sufficiently small, we have νb(Ec ) > 0. Given g ∈ G, there exist real numbers P αij , 1 ≤ i, j ≤ k, such that, outside a measure zero set, we have fi = j αij (gfj ) for every i. P If h ∈ Ec \ gEc , we must have fi (h) 6= j αij (gfj )(h) for some i; we conclude that νb(gEc 4Ec ) = 0 for all g ∈ G. Corollary 3.9. Let Γ be a discrete group with property HF D . If Γ acts freely and cocompactly on a CAT(0) cube complex X, then Γ is virtually abelian. Proof. Cocompactness of the action implies that X is finite dimensional. By Propositions 2.17 and 3.7, there exists a finite-index subgroup Γ0 ≤ Γ and a normal subgroup N C Γ0 consisting of elliptic elements, such that Γ0 /N is abelian. Since Γ acts freely, N is trivial. Recall that, in Gromov’s density model, random groups at density d < 12 are nonelementary hyperbolic with overwhelming probability [Gro93, Oll04]. Together with Theorem 10.4 in [OW11], Corollary 3.9 then immediately implies the following result. Corollary 3.10. With overwhelming probability, random groups at density d < 61 in Gromov’s density model do not have property HF D . 4. Superrigidity. 4.1. The superrigidity result. Let X be a complete median space of finite rank r and Γ y X an action by isometries of a discrete group Γ. Lemma 4.1. Suppose h1 , h2 , h3 ∈ H form a facing triple; let ki ∈ H and h∗i be strongly separated for i = 1, 2, 3. There exists a point z ∈ X such that m(ξ1 , ξ2 , ξ3 ) = z whenever ξi ∈ kei . Proof. Let C be the intersection of the closures of e h∗i inside X; it is nonempty, closed and convex. Given points ξi ∈ kei , set m := m(ξ1 , ξ2 , ξ3 ). By convexity we have I(ξ2 , ξ3 ) ⊆ e h∗1 , hence m ∈ e h∗1 ; permuting the indices, we obtain m ∈ C. In particular, denoting by πC the gate projection X → C, we have m = πC m(ξ1 , ξ2 , ξ3 ) = m(πC ξ1 , πC ξ2 , πC ξ3 ). The closures of e h∗i and eki in X are strongly separated by Lemma 2.21; let {xi } be the shore of h∗i and set z := m(x1 , x2 , x3 ). By Corollary 2.20, 28 ELIA FIORAVANTI we have πC (ξi ) = xi , hence m = z no matter which points ξi ∈ kei we have chosen. Lemma 4.2. Suppose that X is irreducible and assume that Γ y X is Roller nonelementary and Roller minimal. Given 0 6= f ∈ L2 (H , νb), consider N S (f ) := (gn ) ∈ Γ \| gn g · f −−−−−→ g · f, ∀g ∈ Γ . n→+∞ There exists z ∈ X such that, for every (gn ) ∈ S (f ), there exists N ≥ 0 such that gn · z = z for all n ≥ N . Proof. By part 1 of Proposition 2.10, the barycentric subdivision X 0 of X is an irreducible median space of the same rank. The action Γ y X 0 is without wall inversions, Roller nonelementary and Roller minimal by Lemma 2.14. As usual, we write (H 0 , ν 0 ) for (H (X 0 ), νbX 0 ). The function f ∈ L2 (H , νb) induces f 0 ∈ L2 (H 0 , ν 0 ) with S (f ) ⊆ S (f 0 ). We approximate f 0 by a linear combination F of characteristic functions of halfspace intervals H 0 (x1 \|y1 ), ..., H 0 (xk \|yk ) such that kF − f 0 k < 31 kf 0 k. Proposition 2.9 implies that ν 0 (H 0 ) = +∞; all halfspace intervals have finite measure, so there exists h0 ∈ H 0 such that xi , yi ∈ h0 for all i. Propositions 2.9 and 2.11 provide a thick halfspace h ∈ H 0 such that h∗ and h0 are strongly separated; in particular, h contains every wall in the set W 0 (x1 \|y1 ) ∪ ... ∪ W 0 (xk \|yk ). Propositions 2.9 and 2.11 also provide γ1 ∈ Γ such that h and γ1 h are strongly separated and γ2 ∈ Γ such that γ1 h∗ and γ2 h are strongly separated. We can assume without loss of generality that d(γ1 h∗ , γ2 h) ≥ 1, as by Proposition 2.1 the quantity d(γ1 h∗ , γ2n h) diverges as n goes to infinity. Thus, we can choose elements γm ∈ Γ such that h∗ ) γ1 h ) γ2 h ) ... and, for all i ≥ 1, the halfspaces γi h∗ and γi+1 h are strongly separated and at distance at least 1. We denote by S ⊆ H 0 the support of the function F and by P the set of all points xi , yi . Let D be the maximum distance from h∗ of a point of P and d := dD + 2e. Let us fix an integer m > d and g ∈ Γ such that kgγm f 0 − γm f 0 k < 31 kf 0 k and kgf 0 − f 0 k < 13 kf 0 k. We prove that gγm h ⊆ γm−d h. A straightforward repeated application of the triangle inequality yields kgF − F k < 2kF k and kgγm F − γm F k < 2kF k; thus, νb(S ∩ gS) > 0 and νb(γm S ∩ gγm S) > 0. Let w be a wall corresponding to a halfspace in γm S ∩ gγm S. Since w is contained in both γm h and gγm h and Γ y X 0 is without wall inversions, we conclude that γm h ∩ gγm h 6= ∅. A similar argument shows that h ∩ gh 6= ∅. Let u be the wall corresponding to gγm h. If u is contained in γm−1 h, we either have gγm h ⊆ γm−1 h or γm−1 h∗ ∩ gγm h∗ = ∅. The former case immediately yields gγm h ⊆ γm−d h, while the latter leads to a contradiction as h ⊆ γm−1 h∗ and gh ⊆ gγm h∗ intersect. SUPERRIGIDITY OF ACTIONS ON FINITE RANK MEDIAN SPACES 29 If instead u is not contained in γm−1 h, it is contained in γm h∗ , by strong separation. Let 1 ≤ l ≤ m be minimum such that γl h∗ contains u. We have γl h ⊆ gγm h, since γl h ∩ gγm h ⊇ γm h ∩ gγm h 6= ∅. Let k be the side of w that is contained in γm h. Since either k or k∗ lies in gγm S, there exists q ∈ P such that gγm q ∈ k ⊆ γm h. Hence, m − l ≤ d(γm h, γl h∗ ) ≤ d(gγm q, gγm h∗ ) = d(q, h∗ ) ≤ D and m − l + 2 ≤ D + 2 ≤ d. By strong separation and minimality of l, the wall u is contained in γl−2 h ⊆ γm−d h. Hence gγm h ⊆ γm−d h, since otherwise gγm h∗ ∩ γm−d h∗ = ∅ would again violate the fact that h ∩ gh 6= ∅. Now consider the intersection C of the closures in X 0 of the halfspaces γm e h. It consists of a single point ξ since any j ∈ H 0 with ej ∩ C 6= ∅ and ej∗ ∩ C 6= ∅ would have to be transverse to almost all γm h, violating strong separation. Strong separation also implies that ξ actually lies in X ⊆ X 0 . Given (gn ) ∈ S (f 0 ) we can assume, removing a finite number of elements if necessary, that kgn f 0 − f 0 k < 13 kf 0 k for all n. Let N (m) be a natural number such that kgn γm f 0 − γm f 0 k < 13 kf 0 k for all n ≥ N (m). When n ≥ N (m), we have shown that gn γm h ⊆ γm−d h; thus we have gn ξ ∈ gn γme h ⊆ γm−de h. In this case, strong separation implies that σgn ξ 4σξ consists only of halfspaces whose corresponding walls are contained in γm−d−1 h. This shows that lim sup σgn ξ 4σξ = ∅; we conclude that gn ξ → ξ in the topology of X, for every (gn ) ∈ S (f 0 ). We finally construct the point z ∈ X. Let j, m be thick halfspaces of X 0 so that m∗ and j are strongly separated and ξ ∈ ej. Part 2 of Proposition 2.11 provides a facing triple consisting of m, m1 , m2 . We choose thick halfspaces j1 , j2 ∈ H 0 such that m∗i and ji are strongly separated for i = 1, 2; by Proposition 2.9 we can find hi ∈ Γ such that hi j ⊆ ji . Let z ∈ X 0 be the point provided by Lemma 4.1 applied to j, j1 , j2 and m, m1 , m2 ; in particular, we have z = m(ξ, h1 ξ, h2 ξ), hence z ∈ X. Since the set S (f 0 ) is closed under conjugation by elements of Γ, we have gn hi ξ → hi ξ for all (gn ) ∈ S (f 0 ). Hence, given (gn ) ∈ S (f 0 ), there exists N ≥ 0 such that for every n ≥ N we have gn ξ ∈ ej and gn hi ξ ∈ eji , for i = 1, 2. Thus, gn z = gn m(ξ, h1 ξ, h2 ξ) = m(gn ξ, gn h1 ξ, gn h2 ξ) = z. In the rest of the section, we consider a locally compact group G and a lattice Γ < G. Any Borel fundamental domain U ⊆ G defines a cocycle α : G × U → Γ such that gu ∈ α(g, u) · U . We say that Γ is square-integrable if Γ is finitely generated and U can be chosen so that Z \|α(g, u)\|2S du < +∞, ∀g ∈ G; U here du is the Haar measure on U and \| · \|S denotes the word length with respect to a finite generating set S ⊆ Γ. Integrability does not depend 30 ELIA FIORAVANTI on the choice of S. Uniform lattices are always square-integrable and a few nonuniform examples were mentioned in the introduction; see [Sha00, Rém99, Rém05, CR09, CR10] for more details and examples. We assume that G splits as a product G1 × ... × G` , where each Gi is compactly generated and ` ≥ 2. We also require the lattice Γ < G to be irreducible, i.e. to project densely into each factor Gi . Consider a unitary representation π : Γ → U (H); we denote by H0 the subspace of invariant vectors. Let c : Γ → H be a 1-cocycle for π. We will make use of the following result of Y. Shalom in an essential way; see page 14 and Theorem 4.1 in [Sha00] for a proof. Theorem 4.3. Suppose that Γ is square-integrable and that H0 = {0}. There exist Γ-invariant closed subspaces Hi ≤ H, i = 1, ..., `, where the restriction πi : Γ → U (Hi ) extends to a continuous representation πi : G → U (Hi ) that factors through the projection pi : G → Gi . Furthermore, there are cocycles ci : Γ → Hi , i = 1, ..., `, such that c and c1 + ... + c` represent the same class in H 1 (Γ, π). The following is a version of our Theorem B under stronger hypotheses. Theorem 4.4. Suppose that Γ is square-integrable and X is irreducible; let Γ y X be Roller nonelementary and Roller minimal. There exists a Γ-invariant, closed median subalgebra Y ⊆ X where Γ y Y extends to a continuous action G y Y . Moreover, G y Y factors through a projection p i : G → Gi . Proof. We have H 1 (Γ, ρ) 6= {0} by Theorem A. Lemma 3.8 implies that ρ has no nonzero invariant vectors; thus, Theorem 4.3 provides a Γ-invariant subspace {0} = 6 Hi ⊆ L2 (H , νb) where the action of Γ extends to a continuous action of G factoring through a projection pi : G → Gi . Pick any 0 6= f ∈ Hi and consider the set S (f ) introduced in Lemma 4.2. Any sequence (gn ) ∈ ΓN with pi (gn ) → id lies in S (f ). Thus, Lemma 4.2 implies that the Γ-invariant set n o Y := x ∈ X \| ∀(gn ) ∈ ΓN s.t. pi (gn ) → id, we have gn x → x is nonempty. Note that Y is a median subalgebra of X, thus the restriction of the metric of X gives Y a structure of median space. Since Y is a closed subset of X, it is a complete median space. Finally, Proposition 4.3 in [Sha00] provides a continuous extension to G of Γ y Y and it factors through pi . The assumption that Γ y X be Roller minimal and Roller nonelementary can be replaced with the (stronger) requirement that Γ have no finite orbit in b see Proposition 5.2 in [Fio17b]. the visual boundary of the CAT(0) space X; The homomorphism φ : G → Isom Y provided by Theorem 4.4 is continuous with respect to the topology of pointwise convergence. We remark however that φ remains continuous even if we endow Isom Y with the topology mentioned in Remark 4.5 below; this will be a key point in our proof of Theorem C. SUPERRIGIDITY OF ACTIONS ON FINITE RANK MEDIAN SPACES 31 Remark 4.5. In the proof of Theorem 4.4, Lemma 4.2 actually yields that the smaller set n o Y0 := x ∈ X \| ∀(gn ) ∈ ΓN s.t. pi (gn ) → id, ∃N ≥ 0 s.t. gn x = x, ∀n ≥ N is nonempty. Thus, φ : G → Isom Y is continuous with respect to the topology on Isom Y that is generated by stabilisers of points of Y0 . In the statement of Theorem 4.4, we can always take Y to be the closure of Y0 in X. This topology on Isom Y might seem a lot finer than the topology of pointwise convergence. To clarify this phenomenon, we mention the following fact, without proof. Let W be an irreducible, complete, finite rank median space admitting a Roller nonelementary, Roller minimal action; then there exists a dense, convex subset C ⊆ W such that, for every x ∈ C, the stabiliser of x is open for the topology of pointwise convergence on Isom W . This essentially follows from Lemma 4.1. Relaxing the hypotheses of Theorem 4.4, we obtain Theorem B for all square-integrable lattices: Corollary 4.6. Suppose that Γ is square-integrable; let Γ y X be Roller nonelementary. There exist a finite index subgroup Γ0 ≤ Γ, a Γ0 -invariant component Z ⊆ X and a Γ0 -invariant closed median subalgebra Y ⊆ Z where the action Γ0 y Y extends to a continuous action G0 y Y , for an open finite index subgroup G0 ≤ G. Proof. We proceed by induction on rank(X); when the rank is zero there is nothing to prove, so we assume that the statement holds for all median spaces of rank at most r − 1. By Proposition 2.8, there exists a Γ-invariant, closed, convex subset D of a component W ⊆ X such that Γ y D is Roller minimal and Roller nonelementary. If W ⊆ ∂X, we have rank(D) < r and we conclude by the inductive hypothesis; thus we can assume that W = X. Let D = D1 × ... × Dk be the splitting of D into irreducible factors; if k = 1, the result follows from Theorem 4.4. If k ≥ 2, let Γ1 ≤ Γ be a finite index subgroup preserving the splitting of D; up to permuting the factors, we can assume that Γ1 y Di is Roller nonelementary for 1 ≤ i ≤ s and Roller elementary for i > s. A further finite index subgroup Γ2 ≤ Γ1 fixes a point ξi ∈ Di for each i > s; we denote by Zi ⊆ Di the component containing ξi . Note that Γ2 is a square-integrable, irreducible lattice in an open, finite index subgroup of G. Since rank(Di ) < r, for each i ≤ s the inductive hypothesis yields a finite index subgroup Γ0i ≤ Γ2 , an open finite index subgroup G0i ≤ G and a Γ0i -invariant, closed median subalgebra Yi of a component Zi ⊆ Di where the action of Γ0i extends to a continuous action of G0i . Let Γ0 be the intersection of all Γ0i and G0 be the intersection of all G0i , for i ≤ s. The set Y := Y1 × ... × Ys × {ξs+1 } × ... × {ξk } ⊆ D is a closed median subalgebra of Z1 × ... × Zk , which is a component of D; in particular, Z1 × ... × Zk is a closed, convex subset of a component Z ⊆ X. The action Γ0 y Y trivially extends to a continuous action G0 y Y . 32 ELIA FIORAVANTI Figure 3 We now describe two examples that illustrate how: • in Theorem 4.4 the space Y cannot be taken to coincide with X, nor with a convex subset (Example 4.7); • in Corollary 4.6 it cannot be avoided to pass to the finite index subgroup Γ0 , even when the action is Roller minimal (Example 4.8). The actions that we consider are actually on CAT(0) square complexes. Since Burger-Mozes groups play an important role in the construction of the two examples, we briefly recall a few facts regarding their construction. Given an integer n ≥ 3, we denote by Tn the n-regular tree and by An the group of even permutations on n elements. We fix a legal colouring on Tn , i.e. a way of associating an integer in {1, ..., n} to every edge of Tn so that we see all n integers around each vertex; in particular, we have a bijection iv : lk(v) → {1, ..., n} for every vertex v. Let U (An ) ≤ Isom Tn be the subgroup of isometries g such that igv ◦ g ◦ i−1 v ∈ An for every vertex v of Tn ; we denote by U (An )+ the intersection of U (An ) with the subgroup of Isom Tn generated by edge stabilisers. If n ≥ 4, the subgroup U (An )+ has index 2 in U (An ), see Proposition 3.2.1 in [BM00a]. The subgroup U (An ) is closed in Isom Tn ; in particular, it is locally compact, second countable and compactly generated (e.g. by Theorem 4.C.5 and Proposition 5.B.5 in [CdlH16]). By Theorem 6.3 in [BM00b], there exists a uniform irreducible lattice Λ ≤ U (A2k ) × U (A2k ) for every integer k ≥ 19. For the next two examples, we fix such a lattice Λ. Let p1 , p2 : Λ → U (A2k ) + be the projections into the two factors and set Λ0 := p−1 1 (U (A2k ) ); this is + an irreducible lattice in the open, index 2 subgroup U (A2k ) × U (A2k ) of U (A2k ) × U (A2k ). Let τ : Λ → Z/2Z be the homomorphism with kernel Λ0 . Example 4.7. Given any tree T , we can blow up every edge to a square as in Figure 3, thus obtaining a “tree of squares” T. Adjacent squares only share a vertex; if T has no leaves, each square has a pair of opposite vertices that are shared with other squares and a pair of opposite vertices that are not shared. The space T is a complete, rank two median space in which T embeds as a median subalgebra; edges of T correspond to diagonals joining shared pairs of vertices of a square. SUPERRIGIDITY OF ACTIONS ON FINITE RANK MEDIAN SPACES 33 We can embed Isom T ,→ Isom T by extending each isometry of T so that the restriction to each square is orientation preserving. Let σ ∈ Isom T be the isometry that fixes pointwise the image of the embedding T ,→ T and acts on each square as a reflection in a diagonal; we have σ 2 = id and Isom T × hσi ,→ Isom T. Viewing p2 : Λ → U (A2k ) as a homomorphism into Isom T2k we can define a homomorphism Λ → Isom T2k × hσi by λ 7→ (p2 (λ), τ (λ)). We denote by ψ the composition of this map with the embedding Isom T2k × hσi ,→ Isom T2k . The action Λ y T2k induced by ψ is Roller nonelementary and Roller minimal since the action Λ y T2k induced by p2 is. As T2k is irreducible, Theorem 4.4 guarantees a continuous extension of Λ y Y to U (A2k ), for some Λ-invariant median subalgebra of T2k . Indeed, one can take Y to be the image of T2k ,→ T2k . However, Y cannot be taken to be a convex subspace (or even a subcomplex) of T2k . Indeed, Y would be forced to be the whole T2k , as this is the only Λ-invariant convex subset of T2k . The action Λ y T2k does not extend to U (A2k ) × U (A2k ) by factoring via p1 ; this is because, whenever elements gn ∈ Λ satisfy p1 (gn ) → id, the sequence (p2 (gn )) must diverge. However, Λ y T2k also does not extend by factoring through p2 : we have p2 (Λ0 ) = p2 (Λ) = U (A2k ), but ψ(Λ0 ) is contained in the closed subgroup Isom T2k < Isom T2k and ψ(Λ) is not. Example 4.8. Choose an element g ∈ Λ \ Λ0 and consider the action Λ0 y T2k × T2k given by λ · (x, y) = (p2 (λ) · x, p2 (g −1 λg) · y). Since the action U (A2k ) y T2k does not preserve any proper closed subtree, the same holds for the action of p2 (Λ0 ). Part 3 of Proposition 2.10 then implies that Λ0 does not leave any proper, closed, convex subset of T2k × T2k invariant. Note that no component of ∂(T2k × T2k ) = (∂T2k × T2k ) ∪ (T2k × ∂T2k ) is preserved by Λ0 , as this would correspond to a fixed point for p2 (Λ0 ) y T2k , hence to a fixed point for U (A2k ) y T2k . We conclude that Λ0 y T2k × T2k is Roller minimal and the same argument also shows that it is Roller nonelementary. One can easily check that Λ0 y T2k × T2k can be extended to an action of the whole Λ by setting λg · (x, y) = (p2 (λg 2 ) · y, p2 (g −1 λg) · x) for all λ ∈ Λ0 . This action also is Roller minimal and Roller nonelementary. We will show, however, that there exists no Λ-equivariant isometric embedding j : Y ,→ T2k × T2k of a median space Y such that the action on Y extends continuously to U (A2k ) × U (A2k ) by factoring through one of the factors. Let j : Y ,→ T2k × T2k be a Λ-equivariant embedding; note that j(Y ) is entirely contained in a Λ-invariant component Z ⊆ T2k × T2k . In particular, the previous discussion shows that Z = T2k × T2k . By Lemma 6.5 in [Bow13a], each wall of j(Y ) arises from a wall of T2k ×T2k , i.e. a wall of one of the two factors, see part 1 of Proposition 2.10. Since the two factors are exchanged by g ∈ Λ, we conclude that Y splits as Y1 × Y2 , with Λ0 y Y preserving this decomposition and g exchanging Y1 and Y2 . 34 ELIA FIORAVANTI Suppose for the sake of contradiction that Λ y Y extends to an action of U (A2k ) × U (A2k ) by factoring through one of the two factors. As in Example 4.7, we see that the extension cannot factor via p1 . However, since p2 (Λ0 ) is dense in U (A2k ) and Λ0 preserves the splitting Y = Y1 × Y2 , part 2 of Proposition 2.10 implies that an extension factoring through p2 would also preserve the splitting Y = Y1 × Y2 . This contradicts the fact that g exchanges Y1 and Y2 . We conclude the section by proving Theorem C. Proof of Theorem C. We begin by observing that part 2 follows from part 1 and Proposition 2.17. Now, suppose for the sake of contradiction that Γ admits a Roller nonelementary action on X. As in the proof of Proposition 3.7, we can assume that X is irreducible and that Γ y X is Roller minimal. Theorem 4.4 then yields a factor Gi , a closed median subalgebra Y ⊆ X and actions Gi y Y and Γ y Y . Without loss of generality, we can assume that Y is the closure of Y0 inside X, as in Remark 4.5. Stabilisers of points of Y0 are open in Gi , thus the identity component G0i must fix Y0 pointwise. As Y0 is dense in Y , the entire action G0i y Y vanishes and Gi y Y descends to an action of the group Gi /G0i . Since Gi satisfies condition (∗), Proposition 3.7 above and Corollary 6.5 in [Fio17b] imply that the action Gi /G0i y Y is Roller elementary. However, by Lemma 2.6, the actions Γ y Y and Gi y Y are Roller nonelementary, a contradiction. 4.2. Homomorphisms to coarse median groups. We defined equivariantly coarse median groups in the introduction. Here we simply prove Corollary E. Proof of Corollary E. Fix a non-principal ultrafilter ω on N and let Hω be the corresponding ultrapower of H. We endow H with a word metric dS arising from a finite generating set S ⊆ H. Given λ = (λn ) ∈ RN + , we denote by Coneω (H, λ) the asymptotic cone obtained by taking all basepoints at the identity and λ as sequence of scaling factors. Let d denote the metric that dS induces on Coneω (H, λ); it is a geodesic metric and it is preserved by the natural action Hω y Coneω (H, λ). If λn → +∞, the coarse median on H induces a structure of finite rank median algebra on Coneω (H, λ), see Section 9 in [Bow13a]; we denote by m the corresponding median map. The action Hω y Coneω (H, λ) is by automorphisms of the median algebra structure. By Propositions 3.3 and 5.1 in [Zei16], we can endow Coneω (H, λ) with a median metric dm that is biLipschitz equivalent to d and preserved by the Hω -action; furthermore, the median algebra structure associated to dm is given by the map m. Now suppose for the sake of contradiction that there exist pairwise nonconjugate homomorphisms φn : Γ → H, for n ≥ 0; these correspond to a homomorphism Γ → Hω , hence to an action on every asymptotic cone of H that preserves the median metric dm . The Bestvina-Paulin construction SUPERRIGIDITY OF ACTIONS ON FINITE RANK MEDIAN SPACES 35 [Bes88, Pau91] provides us with a sequence µn → +∞ such that, modifying each φn within its conjugacy class if necessary, the induced action Γ y Coneω (H, µ) has no global fixed point. This, however, contradicts Theorem C. Appendix A. Structure of UBS’s. Let X be a complete median space of finite rank r. We fix points x ∈ X and ξ ∈ ∂X. Let Ω ⊆ σξ \ σx be a minimal, reduced UBS. Lemma A.1. Let T g ∈ Isomξ X be an isometry satisfying gΩ ∼ Ω. Consider the UBS Ω1 := 0≤i≤r g −i Ω ⊆ Ω. (1) If χΩ (g) ≥ 0, we have g r! Ω1 ⊆ Ω1 and g r! h ⊆ h for all h ∈ Ω1 . (2) If χΩ (g) = 0, we have g r! Ω1 = Ω1 and g r! h = h for all h ∈ Ω1 . Proof. Observe that Ω1 ∼ Ω; we fix a diverging chain (kn )n≥0 in Ω1 . If h ∈ Ω1 , the halfspaces g i h with 0 ≤ i ≤ r all lie in Ω ⊆ σξ \ σx and cannot be pairwise transverse; thus, there exists 0 ≤ i ≤ r such that either g i h ⊆ h, or g i h ⊇ h. Hence, for every h ∈ Ω1 we either have g r! h ⊆ h or g r! h ⊇ h. Suppose that g r! h0 ( h0 for some h0 ∈ Ω1 ; set hk := g kr! h0 . For each k ≥ 0, a cofinite subchain of {g −kr! kn }n≥0 is a diverging chain in Ω1 , as gΩ1 ∼ Ω1 . Hence, for each k ≥ 0 we have h0 ⊇ g −kr! kn if n is sufficiently large, since Ω is reduced; in particular h0 ⊇ hk ⊇ kn . We conclude that each hk lies in Ω1 ; Proposition 2.1 guarantees that (hk )k≥0 is a diverging chain. Let Ξ ⊆ Ω1 be the inseparable closure of {hk }k≥0 ; it is a UBS equivalent to Ω and it satisfies g r! Ξ ⊆ Ξ. Observe that, for each k ≥ 0, kr! · χΩ (g) = χΩ (g kr! ) = χΞ (g kr! ) = νb(Ξ \ g kr! Ξ) ≥ d(hk , h∗0 ); since d(hk , h∗0 ) > 0 for some k ≥ 0, we conclude that χΩ (g) > 0. The same argument applied to g −1 shows that χΩ (g) < 0 if there exists h0 ∈ Ω1 with g r! h0 ) h0 . This proves part 2 and shows that g r! h ⊆ h for all h ∈ Ω1 if χΩ (g) > 0. In the latter case, for every h ∈ Ω1 we have h ⊇ g r! h ⊇ kn for sufficiently large n, since Ω is reduced; thus, g r! h ∈ Ω1 and g r! Ω1 ⊆ Ω1 . Corollary A.2. Let g ∈ Isomξ X be an isometry satisfyingTgΩ ∼ Ω. Define Ω1 as in the previous lemma and consider the UBS Ω2 := 1≤i≤r! g i−1 Ω1 . (1) If χΩ (g) ≥ 0, we have gΩ2 ⊆ Ω2 . (2) If χΩ (g) = 0, we have gΩ2 = Ω2 . In the rest of the appendix, we also consider a compact subset K ⊆ Isomξ X such that gΩ ∼ Ω for every g ∈ K. Lemma A.3. (1) There exists a constant C1 = C1 (Ω, K) such that every h ∈ Ω with d(x, h) > C1 lies in gΩ for every g ∈ K. (2) If Ξ ⊆ σξ \ σx is a minimal, reduced UBS such that Ξ 6∼ Ω and gΞ ∼ Ξ for all g ∈ K, there exists a UBS Ω ⊆ Ω that is disjoint from gΞ for all g ∈ K. 36 ELIA FIORAVANTI Proof. Let (hn )n≥0 be a diverging chain in Ω with h0 thick. For every g ∈ K, a cofinite subchain of (g −1 hn )n≥0 is contained in Ω, as gΩ ∼ Ω; since Ω is reduced, there exists n(g) ≥ 0 so that g −1 hn(g) ⊆ h0 . By Proposition 2.1 we can assume that νb(H (gh∗0 \|hn(g) )) > 0 and Lemma 2.18 provides a neighbourhood U (g) of g in K such that H (γh∗0 \|ξ)∩H (gh∗0 \|hn(g) ) 6= ∅ for all γ ∈ U (g); in particular, hn(g) ⊆ γh0 for all γ ∈ U (g). There exist g1 , ..., gk ∈ K such that K = U (g1 ) ∪ ... ∪ U (gk ); if N is the maximum of the n(gi ), we have hN ⊆ gh0 for all g ∈ K. Let ΩN be the inseparable closure of {hn }n≥N . If m ≥ N and g ∈ K, we have gh0 ⊇ hm ⊇ ghn for every sufficiently large n, since Ω is reduced. This shows that ΩN is contained in gΩ for all g ∈ K. Since ΩN ∼ Ω, there exists a constant C1 such that every h ∈ Ω with d(x, h) > C1 lies in ΩN . To prove part 2, let C be the supremum of distances d(x, h) for h ∈ Ω ∩ Ξ; we have C < +∞ since Ω 6∼ Ξ. Let M be the maximum distance d(x, gx) for g ∈ K and consider C 0 := max{C, C1 (Ξ, K −1 ) + M }; we define Ω to be the set of h ∈ Ω with d(x, h) > C 0 . If there existed h ∈ Ω ∩ gΞ for some g ∈ K, we would have g −1 h ∈ Ξ and d(x, g −1 h) > C1 (Ξ, K −1 ); thus, part 1 implies that g −1 h ∈ g −1 Ξ, i.e. h ∈ Ξ, and this contradicts the fact that h ∈ Ω and d(x, h) > C. Recall that we have introduced the function αΩ : H → R, defined by the formula αΩ (h) := νb (H (x\|h) ∩ Ω) and the sets Ωc := {h ∈ Ω \| αΩ (h) ≤ c}. Observe that αΩ (k) ≤ αΩ (h) whenever h ⊆ k; in particular αΩ is measurable. We have αΩ (h) ≤ νb(H (x\|h)) = d(x, h) for all h ∈ H . We say that Ω is small if νb(Ω) < +∞; otherwise, Ω is large. If X is a CAT(0) cube complex, every UBS is large; an example of a small UBS in a rank two median space appears in Figure 3 of [Fio17a]. If Ω is small, we have χΩ (h) = 0 for every isometry h fixing [Ω]. Note that the supremum of αΩ is precisely νb(Ω). Lemma A.4. Let hn ∈ Ω be halfspaces with hn+1 ⊆ hn for all n ≥ 0. Then, αΩ (hn ) → νb(Ω) if and only if (hn )n≥0 is a diverging chain of halfspaces. Proof. The fact that αΩ (hn ) → νb(Ω) if (hn )n≥0 is a diverging chain follows from the fact that Ω is reduced. For the other implication, let (km )m≥0 be a diverging chain in Ω. Since αΩ (h) ≤ d(x, h), it suffices to consider the case when Ω is small. For every m ≥ 0, the set {j ∈ Ω \| j ⊆ km } has measure am > 0 by Proposition 2.1. For large n we have αΩ (hn ) > νb(Ω) − am , hence there exists j ⊆ km such that j ∈ H (x\|hn ); in particular, hn ⊆ km . Since m is arbitrary, this shows that (hn )n≥0 is a diverging chain. Lemma A.5. (1) For every 0 ≤ c < νb(Ω), the set Ω \ Ωc is a UBS. (2) For all c ≥ 0, we have νb(Ωc ) ≤ rc. Proof. Since Ω is reduced, any h ∈ Ω \ Ωc contains almost every halfspace in any diverging chain in Ω; this provides a diverging chain in Ω \ Ωc . Inseparability follows from the monotonicity of αΩ . SUPERRIGIDITY OF ACTIONS ON FINITE RANK MEDIAN SPACES 37 To prove part 2, we decompose Ωc = C1 t ... t Ck as in Lemma 2.2. If h, k ∈ Ci and h ⊆ k, we have c ≥ αΩ (h) ≥ νb (H (k∗ \|h)). Hence, Lemma 2.27 in [Fio17a] implies that the inseparable closure of Ci has measure at most c. We conclude that νb(Ωc ) ≤ kc ≤ rc. Lemma A.6. Assume that gΩ ⊆ Ω for all g ∈ K. (1) For all h ∈ Ω and g ∈ K, we have −d(x, gx) − χΩ (g) ≤ αΩ (g −1 h) − αΩ (h) ≤ d(x, gx); in particular kgαΩ − αΩ k∞ ≤ C2 for some constant C2 = C2 (Ω, K). (2) For every c > 0, we have gΩc ⊆ Ωc+C2 . Proof. Indeed, αΩ (g −1 h) ≤ νb H (x\|g −1 x) + νb H (g −1 x\|g −1 h) ∩ Ω = = d(x, g −1 x) + νb (H (x\|h) ∩ gΩ) ≤ d(x, gx) + αΩ (h), and αΩ (h) ≤ νb (H (x\|gx)) + νb (H (gx\|h) ∩ Ω) = = d(x, gx) + νb H (x\|g −1 h) ∩ g −1 Ω ≤ ≤ d(x, gx) + νb H (x\|g −1 h) ∩ Ω + νb g −1 Ω \ Ω = = d(x, gx) + αΩ (g −1 h) + χΩ (g). We then take C2 to be the maximum of d(x, gx) + χΩ (g) for g ∈ K; this exists due to Proposition 2.17. Regarding part 2, observe that, if h ∈ Ωc , αΩ (gh) = αΩ (h) + (αΩ (gh) − gαΩ (gh)) ≤ c + C2 . Lemma A.7. Assume that Ω is strongly reduced. (1) For every d ≥ 0, there exists a constant C3 = C3 (Ω, d) such that h ⊆ k for all h, k ∈ Ω with d(x, h) > C3 and d(x, k) ≤ d. (2) If Ξ ⊆ Ω is a UBS and d(x, k) ≤ d for all k ∈ Ω \ Ξ, we have αΩ (h) − αΞ (h) = νb(Ω \ Ξ) for all h ∈ Ω with d(x, h) > C3 (Ω, d). Proof. Decompose Ω = C1 t ... t Ck , where each Ci is totally ordered by inclusion and contains a diverging chain. Pick halfspaces ki ∈ Ci with d(x, ki ) > d. By part 3 of Lemma 2.16, the halfspace ki cannot be transverse to a diverging chain in Ω; thus, part 2 of Lemma 2.2 guarantees that the halfspaces of Ω that are transverse to ki lie at uniformly bounded distance from x. We conclude that there exists C3 such that every h ∈ Ω with d(x, h) > C3 is contained in each ki , hence also in each k ∈ Ω with d(x, k) ≤ d. This proves part 1; part 2 is an immediate consequence. In the rest of the section, we assume that Ω is minimal and strongly reduced. Lemma A.8. Suppose that χΩ (g) ≥ 0 for all g ∈ K. 38 ELIA FIORAVANTI (1) There exists a constant C4 = C4 (Ω, K) such that, for every halfspace h ∈ Ω with d(x, h) > C4 and every g ∈ K, we have αΩ (gh) ≥ αΩ (h). (2) There exists a constant C5 = C5 (Ω, K) < νb(Ω) such that we have g(Ω \ Ωc ) ⊆ Ω \ Ωc and Ωc \ gΩc = Ω \ gΩ whenever c > C5 and g ∈ K. Proof. We first observe that, given a minimal UBS Ξ ⊆ σξ \ σx and an isometry g ∈ Isomξ X with gΞ ⊆ Ξ, we have αΞ (gh) − αΞ (h) = νb H (g −1 x\|h) ∩ g −1 Ξ − νb (H (x\|h) ∩ Ξ) = = νb H (g −1 x\|h) ∩ Ξ − νb (H (x\|h) ∩ Ξ) + νb H (g −1 x\|h) ∩ (g −1 Ξ \ Ξ) = = −b ν H (x\|g −1 x, h) ∩ Ξ + νb H (g −1 x\|h) ∩ (g −1 Ξ \ Ξ) , since H (g −1 x\|x) ∩ Ξ = ∅, as Ξ ⊆ σξ \ σx . Note that H (x\|g −1 x) ∩ gΞ = g −1 H (gx\|x) ∩ g 2 Ξ ⊆ g −1 (H (gx\|x) ∩ Ξ) = ∅. Thus αΞ (gh) − αΞ (h) equals − νb H (x\|g −1 x, h) ∩ (Ξ \ gΞ) + νb H (g −1 x\|h) ∩ (g −1 Ξ \ Ξ) ≥ ≥ −b ν (σx∗ ∩ (Ξ \ gΞ)) + νb (H (x\|gh) ∩ (Ξ \ gΞ)) ≥ ≥ −b ν ((σx∗ \ H (x\|gh)) ∩ (Ξ \ gΞ)) ≥ −b ν ({k ∈ Ξ \ gΞ \| gh 6⊆ k}) . T Now, consider for each g ∈ K the set Ω2 (g) := −r+1≤i≤r! g i−1 Ω as in Corollary A.2; we have gΩ2 (g) ⊆ Ω2 (g) and, by part 1 of Lemma A.3, there exists a constant C1 such that each h ∈ Ω with d(x, h) > C1 lies in gΩ2 (g) for every g ∈ K. By part 1 of Lemma A.7, if d(x, gh) > C3 (Ω, C1 ) we have αΩ2 (g) (gh) − αΩ2 (g) (h) ≥ −b ν ({k ∈ Ω2 (g) \ gΩ2 (g) \| gh 6⊆ k}) = 0. By part 2 of Lemma A.7, if d(x, h) > C3 (Ω, C1 ) and d(x, gh) > C3 (Ω, C1 ), we have αΩ (gh) − αΩ (h) = αΩ2 (g) (gh) − αΩ2 (g) (h) ≥ 0. We conclude that αΩ (gh) ≥ αΩ (h) whenever d(x, h) > C4 := C3 (Ω, C1 ) + M , where M is the maximum of the distances d(x, gx) for g ∈ K. We now prove part 2. Let C4 be as in part 1. Part 1 of Lemma A.3 provides a constant C10 such that each h ∈ Ω with d(x, h) > C10 lies in gΩ ∩ g −1 Ω for every g ∈ K. Let C5 be the supremum of the values αΩ (h) for h ∈ Ω with d(x, h) ≤ max{C4 , C10 }; by Lemma A.4 we have C5 < νb(Ω). If c > C5 , any h ∈ Ω \ Ωc satisfies d(x, h) > max{C4 , C10 }, hence αΩ (gh) ≥ αΩ (h) > c and gh ∈ Ω; in particular g(Ω \ Ωc ) ⊆ Ω \ Ωc . Observe that (Ω \ gΩ) ∩ gΩc ⊆ g(Ωc \ Ω) = ∅ and Ω \ gΩ ⊆ Ωc , by our choice of the constant C10 ; thus, Ω \ gΩ ⊆ Ωc \ gΩc . Conversely, it is clear that Ωc \ gΩc ⊆ Ω and (Ωc \ gΩc ) ∩ gΩ = Ωc ∩ g (Ω \ Ωc ) ⊆ Ωc ∩ (Ω \ Ωc ) = ∅. Consider now, for c > 0, the functions Fc,Ω := − 1 − αΩ c 1Ωc in L2 (H , νb). SUPERRIGIDITY OF ACTIONS ON FINITE RANK MEDIAN SPACES 39 Lemma A.9. Assume that gΩ ⊆ Ω for all g ∈ K. For every , there exists a constant C = C (Ω, K) < +∞ such that k(g − id)Fc,Ω − 1Ω\gΩ k2 < for all g ∈ K and all c ≥ C . If instead gΩ ⊇ Ω for all g ∈ K, we have k(g − id)Fc,Ω + 1gΩ\Ω k2 < for c ≥ C . Proof. Observe that gαΩ αΩ (g − id)Fc,Ω = − 1 − 1gΩc + 1 − 1Ωc = c c gαΩ αΩ gαΩ − αΩ =− 1− 1gΩc \Ωc + 1 − 1Ωc \gΩc + 1gΩc ∩Ωc . c c c We will analyse the three summands separately. By Lemma A.6, we have gαΩ (h) αΩ (h) − C2 C2 αΩ (h) + C2 2C2 ≤ 1− ≤ 1− < ≤ 1− , − c c c c c for each h ∈ gΩc \ Ωc ⊆ Ωc+C2 \ Ωc . By part 2 of Lemma A.5, gαΩ 2C2 (r(c + C2 ))1/2 2C2 1− 1gΩc \Ωc ≤ ·b ν (gΩc \Ωc )1/2 ≤ −−−−→ 0. c→+∞ c c c 2 By part 1 of Lemma A.3, there exists a constant C1 such that each h ∈ Ω with d(x, h) > C1 lies in gΩ for every g ∈ K; in particular, if k ∈ Ω \ gΩ for some g ∈ K, then αΩ (k) ≤ d(x, k) ≤ C1 . By part 2 of Lemma A.8, if c ≥ C5 we have Ωc \ gΩc = Ω \ gΩ and αΩ αΩ 1Ωc \gΩc − 1Ω\gΩ = 1Ωc \gΩc − 1Ω\gΩ − 1 = 1− c c Ωc \gΩc 2 2 αΩ C1 M 1/2 C1 = 1Ω\gΩ ≤ · νb (Ω \ gΩ)1/2 ≤ −−−−→ 0, c→+∞ c c c 2 where M is the maximum of χΩ (g) for g ∈ K, which exists by Proposition 2.17. Finally, by part 2 of Lemma A.5 and part 1 of Lemma A.6, gαΩ − αΩ 1gΩc ∩Ωc c ≤ 2 C2 C2 (cr)1/2 · νb(Ωc )1/2 ≤ −−−−→ 0. c→+∞ c c If instead gΩ ⊇ Ω for all g ∈ K, the previous discussion shows that, for large c, we have k(g −1 − id)Fc,Ω − 1Ω\g−1 Ω k2 < ; the conclusion follows by applying g. Lemma A.10. Let Ξ ⊆ σξ \ σx be a UBS and let Ξ1 , ..., Ξk ⊆ Ξ be pairwise inequivalent, reduced UBS’s representing all minimal equivalence classes of UBS’s almost contained in Ξ; for every i, set σi := νb(Ξi ). There exist (i) increasing sequences (cn )n≥0 such that (i) • cn → σi for all i =1, ..., k; 1 1 • Ξ \ Ξ (1) ∪ ... ∪ Ξk \ Ξk(k) is inseparable for all n ≥ 0. cn cn Proof. We proceed by induction on k. If k = 1, the lemma is immediate. Suppose that k ≥ 2; without loss of generality, we can assume that Ξk corresponds to a vertex with no incoming edges in the full subgraph of G(ξ) 40 ELIA FIORAVANTI with vertices [Ξ1 ], ..., [Ξk ]. Fix > 0; we will construct c(i) > σi − satisfying the inseparability condition. (i) (k) Pick a diverging chain (hn )n≥0 in each Ξi ; up to replacing (hn )n≥0 with a cofinite subchain, we can assume that, for each i ≤ k − 1 and each m ≥ 0, (k) (i) the halfspace hm is transverse to hn for almost every n. By Lemma A.4, there exists c(k) > σk − such that Ξk \ Ξkc(k) is contained in the inseparable (k) closure of {hn }n≥0 . As a consequence, for every h ∈ Ξk \ Ξkc(k) and every (i) i ≤ k − 1, the halfspaces h and hn are transverse for almost every n. Halfspaces lying in the inseparable closure of Ξ1 ∪ ... ∪ Ξk , but in neither of the Ξi , are at uniformly bounded distance from x, by part 3 of Proposition 2.15; say that these distances are bounded above by M < +∞. Enlarge M so that all j ∈ Ξk with d(x, j) > M lie in Ξk \ Ξkc(k) . By part 4 of Proposition 2.15 there exists a UBS contained in Ξ such that [Ξ1 ], ..., [Ξk−1 ] are all the equivalence classes of minimal UBS’s almost contained in Ξ. The inductive hypothesis and Lemma A.4 imply that we can find c(i) > σi − so that Ξ1 \ Ξ1c(1) ∪ ... ∪ Ξk−1 \ Ξk−1 is inseparable and d(x, h) > M for c(k−1) all h ∈ Ξi \ Ξic(i) with i ≤ k − 1. Now, if Ξ1 \ Ξ1c(1) ∪ ... ∪ Ξk \ Ξkc(k) were not inseparable, there would exist j ∈ H such that ku ⊆ j ⊆ kv , for halfspaces ku ∈ Ξu \ Ξuc(u) and kv ∈ Ξv \ Ξvc(v) , but j would not lie in any of the Ξi \ Ξic(i) . In particular, u 6= v and k ∈ {u, v}; observe that v 6= k, otherwise kv ∈ Ξk \ Ξkc(k) would not be transverse to diverging chains in Ξu . Thus, u = k; moreover, for all (i) i ≤ k − 1, the halfspace j must be transverse to hn for almost every n, since v ≤ k − 1 and j does not lie in Ξ1 \ Ξ1c(1) ∪ ... ∪ Ξk−1 \ Ξck−1 (k−1) , which is inseparable. Since d(x, j) ≥ d(x, kv ) > M , we have j ∈ Ξ1 ∪ ... ∪ Ξk ; the fact that each Ξi is reduced implies that j ∈ Ξk . By our choice of M , we have j ∈ Ξk \ Ξkc(k) , a contradiction. Lemma A.11. There exists a UBS Ξξ ⊆ σξ \σx such that every UBS Ξ ⊆ Ξξ with Ξ ∼ Ξξ is of the form σξ \ σy for some y ∈ X, up to a null set. Proof. Let Ξ1 , ..., Ξk ⊆ σξ \σx be pairwise inequivalent minimal UBS’s representing all minimal elements of U(ξ); we can assume that they are all reduced by Lemma 2.16. Halfspaces in σξ \σx that are transverse to a diverging chain in each Ξi lie at uniformly bounded distance from x, by part 3 of Proposition 2.15; say that these distances are bounded above by M < +∞. Let Ξξ consist of all h ∈ σξ \ σx with d(x, h) > M ; it is a UBS equivalent to σξ \ σx . Observe that, if Ξ ⊆ Ξξ is a UBS equivalent to Ξξ and ξ ∈ e h for halfspaces h ⊆ k ∈ Ξ, then h ∈ Ξ. Indeed, otherwise h would not contain any halfspace of Ξ, by inseparability, and it would therefore be transverse to a diverging chain in each of the Ξi ; hence d(x, k) ≤ d(x, h) ≤ M , a contradiction. SUPERRIGIDITY OF ACTIONS ON FINITE RANK MEDIAN SPACES 41 Now, given Ξ ⊆ Ξξ , consider the set σ := (σξ \ Ξ) t Ξ∗ ⊆ H . Observe that σ is an ultrafilter. Indeed, since σξ \ Ξ ⊆ σξ and Ξ∗ ⊆ σx , it suffices to check that h ∩ k 6= ∅ whenever h ∈ σξ \ Ξ and k ∈ Ξ∗ . If such halfspaces were disjoint, we would have ξ ∈ e h and h ⊆ k∗ ∈ Ξ, contradicting the observation we made above since h 6∈ Ξ. Finally, by Lemma 4.3 in [Fio17b], we can decompose σξ \ σx = Ξ t Σ for some set Σ with νb(Σ) < +∞; in particular, σx \ σξ = Ξ∗ t Σ∗ . Since Ξ and σx are disjoint, we have σx \ σ = σx \ (σξ ∪ Ξ∗ ) = Σ∗ , hence νb(σx 4σ) < +∞. Proposition 2.4 implies that there exists y ∈ X such that σ4σy is null; thus, up to measure zero, σξ \ σy = σξ \ σ = Ξ. We are now ready to prove Proposition 3.4. Proof of Proposition 3.4. Let Ξ1 , ..., Ξk ⊆ σξ \ σx be pairwise inequivalent UBS’s representing all minimal elements of the poset U(ξ), ; by Lemma 2.16, we can assume that each Ξi is strongly reduced. Up to replacing each Ξi with a smaller UBS, part 2 of Lemma A.3 guarantees that we can assume that Ξi ∩ Ξj = ∅ and Ξi ∩ gΞj for all i 6= j and g ∈ K. By part 2 of Lemma A.8, we have g(Ξi \ Ξic ) ⊆ Ξi \ Ξic for all C5 (Ξi , K) ≤ c < νb(Ξi ) and all g ∈ K with χΞi (g) ≥ 0, while we have g(Ξi \ Ξic ) ⊇ Ξi \ Ξic if χΞi (g) ≤ 0. Lemma A.10 provides constants C5 (Ξi , K) ≤ ci < νb(Ξi ) such that ΩK := Ξ1 \ Ξ1c1 ∪ ... ∪ Ξk \ Ξkck is inseparable. Thus ΩK is a UBS equivalent to σξ \ σx by part 3 of Proposition 2.15. We conclude by Lemma A.11, enlarging the constants ci if necessary, so that ΩK ⊆ Ξξ ; this is possible by Lemma A.4. References [ANWZ17] Goulnara Arzhantseva, Graham A. Niblo, Nick Wright, and Jiawen Zhang. A characterization for asymptotic dimension growth. arXiv:1612.06638v2, 2017. 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Massive MIMO Performance Comparison of Beamforming and Multiplexing in the Terahertz Band Sayed Amir Hoseini∗† , Ming Ding† and Mahbub Hassan∗† arXiv:1710.09031v2 [cs.IT] 26 Oct 2017 ∗ School of Computer Science and Engineering, University of New South Wales, Sydney, Australia † Data61, CSIRO, Sydney, Australia Email: [email protected], [email protected], [email protected] Abstract—In this paper, we compare the performance of two main MIMO techniques, beamforming and multiplexing, in the Terahertz (THz) band. The main problem with the THz band is its huge propagation loss, which is caused by the tremendous signal attenuation due to molecule absorption of the electromagnetic wave. To overcome the path loss issue, massive MIMO has been suggested to be employed in the network and is expected to provide Tbps for a distance within a few meters. In this context, beamforming is studied recently as the main technique to take advantage of MIMO in THz and overcome the very high path loss with the assumption that the THz communication channel is Line-of-Sight (LoS) and there are not significant multipath rays. On the other hand, recent studies also showed that the well-known absorbed energy by molecules can be reradiated immediately in the same frequency. Such re-radiated signal is correlated with the main signal and can provide rich scattering paths for the communication channel. This means that a significant MIMO multiplexing gain can be achieved even in a LoS scenario for the THz band. Our simulation results reveal a surprising observation that the MIMO multiplexing could be a better choice than the MIMO beamforming under certain conditions in THz communications. I. I NTRODUCTION To respond to the huge increasing demand for the wireless data traffic, recently the terahertz (THz) band (0.1-10 THz) is envisioned to make Tbps wireless link feasible [1]. In spite of the wide unused bandwidth in this spectrum, the high propagation loss is the main issue of using such spectrum. Thus, the potential applications of the THz link are limited to short range communications such as nanosensors [2], wireless on-chip communications and wireless personal area networks [3]. Moreover, part of the radio signal attenuation at the THz frequencies is due to molecular absorption which is frequency selective and increases the total loss to more than 200 dB for some frequencies at 10-meters distance. Basically, to overcome the very high path loss the transmit power could be largely increased. Unfortunately, this is not feasible with the current technology and it is limited to a few of mW [3]. Alternately, channel gain can be significantly improved by means of the multi-antenna beamforming technique. Indeed, Due to the very small footprint of a large number of antennas at the THz band, beamforming using very large scale Multiple Input Multiple Output (MIMO) systems has been considered in the field as a practical solution which can provide up to 55 dB channel gain at 1 THz [1]. However, beamforming comes at the cost of system complexity and signaling overhead where the transmitter should receive the channel state information continuously and align the beam to the receiver. On the other hand, to achieve a significant MIMO beamforming gain in high frequency spectrum the beam would become very narrow which is sometimes described as a pencil beam. This makes beamforming vulnerable to any transmitter/receiver mobility because it is difficult to perform beam re-alignment in a very short time interval. Another approach to take advantage of MIMO is the MIMO multiplexing technique. While the beamforming technique strives to focus the transmission energy and achieve a large channel gain in a specific direction, the multiplexing technique builds it strength on creating parallel information channels. However, the multiplexing gain is significant only when there are enough non-negligible multipath signal components in a rich scattering environment. Because of the huge path loss, THz communication is usually assumed to be applied in as a Line-of-Sight (LoS) dominant channel and thus, the research focus has been on beamforming rather than multiplexing. However, recent studies show that in the channel medium, molecules absorb and re-radiate the electromagnetic energy in THz band [3–6], which transforms the LoS channel into a rich-scattering environment. The re-radiation is usually considered as noise but the theorical model shows it is highly correlated to main signal [6]. In this paper, we will theoretically investigate the THz channel capacity for both cases of beamforming and multiplexing in a MIMO set-up. We find that the multiplexing technique can provide a considerable capacity gain in comparison with the beamforming technique on certain conditions. Also, in some other conditions where the beamforming yields a higher capacity, the multiplexing technique is still preferable choice due to its easier implementation. Note that in this work we assume a multiplexing technique using a blind precoding scheme without channel state information (CSI). In contrast, the beamforming technique always requires accurate CSI to smartly direct its energy in the spatial domain. The rest of the paper is structured as follows. In Section II, we present the molecular absorption model for the calculation of channel transfer function, Section III analyzes the MIMO channel model considering the molecular re-radiation, followed by simulation results in Section IV. Finally, we conclude the paper in Section V. II. C HANNEL MODEL AND MIMO CAPACITY The molecular absorption model defines how different species of molecules in a communication channel absorb energy from the electromagnetic signals and how they re-radiate them back to the environment. This section first explains the concept of absorption coefficient used to characterize the absorption capacity of a given molecule species, followed by the attenuation and re-radiation models that are built upon this coefficient. A. Molecular absorption coefficient The medium absorption coefficient, k(f ), at frequency f is a weighted sum of the molecular absorption coefficients in the medium [5], which can be formulated as k(f ) = N X mi ki (f ), (1) i=1 where ki (f ) is the molecular absorption coefficient of species Si on condition of temperature T and and pressure P. ki (f ) can be obtained from HITRAN [7]. In this work, to get the values of k(f ), we will use some predefined standard atmosphere conditions and their corresponding ratio of molecules in the air, which are tabulated in [7]. B. Attenuation of radio signal The attenuation of the radio signal at the THz frequencies is due to spreading and molecular absorption. In more detail, the spreading attenuation is given by C. Molecular re-radiation The existing molecules in communication medium will be excited by electromagnetic waves at specific frequencies. The excitement is temporary and the vibrational-rotational energy level of molecules will come back to a steady state and the absorbed energy will be re-radiated in the same frequency. These re-radiated waves are usually considered as noise in the literature [3]. Molecular absorption is not white and its power spectral density (PSD) is not flat because of the different resonant frequencies of various species of molecules. The PSD of the molecular absorption noise that affects the transmission B of a signal, SNabs , is contributed by the atmospheric noise SN X and the self-induced noise SN as addressed in [5]: B X SNabs (f, d) = SN (f, d) + SN (f, d), c 2 B , SN (f, d) = limd→∞ (kB T0 (1 − e−k(f )d )) √ 4πf c 2 X , SN (f, d) = Pt (f )(1 − e−k(f )d ) 4πf d (4) (5) (6) where k(f ) is the absorption coefficient of the medium at frequency f , T0 is the reference temperature (296K), kB is the Boltzmann constant, Pt (f ) is the power spectral density of the transmitted signal and c is the speed of light. The first term in (4), which is called sky noise and defined in (5) is independent of the signal wave. However, the self-induced noise in (6) is highly correlated with the signal wave [6], and can be considered as a distorted copy of the signal wave. Thus, equation (6) can be revised as the received power of the reradiated signal by molecules at the receiver by c 2 Pr,a (f, d) = Pt (f )(1 − e−k(f )d ) . (7) 4πf d Since the phase of the re-radiated wave depends on the phase of molecular vibration, which varies from molecules to molecules [8], the received power in this case is affected by a large number of phase-independent re-radiated photons. Thus, we assume a uniformly distributed random phase for the received signal, with its power given by (7). D. Channel Transfer Function (3) The channel transfer function for a single LoS channel is given by s 2 d c e−k(f )×d × ej2π λ h̃LoS (f, d) = 4πf d (8) d c j2π −k(f )× d 2 × e λ. = e 4πf d where k(f ) is the absorption coefficient of the medium at frequency f . Thus, the line-of-sight (LoS) received power at the receiver becomes 2 c Pr,LoS (f, d) = Pt (f ) × × e−k(f )×d . 4πf d Then, the partial channel transfer function resulted from the molecular absorption and excluding the LoS component can be represented by s c 2 × ej2πβrandom h̃a (f, d) = (1 − e−k(f )d ) 4πf d (9) c j2πβrandom −k(f )d 21 ×e . = (1 − e ) 4πf d Aspread (f, d) = 4πf d c 2 , (2) where c is the speed of light. The attenuation due to molecular absorption is characterized as Aabs (f, d) = ek(f )×d , Hence, the total channel transfer function is the superposition of the partial channel transfer functions, which is written as h̃(f, d) = h̃LoS (f, d) + h̃a (f, d), h̃(f, d) = c 4πf d (10) d d e−k(f )× 2 × ej2π λ c 1 × ej2πβrandom . (11) +(1 − e−k(f )d ) 2 4πf d E. MIMO channel model and capacity In this paper, we consider a MIMO system that is consisted of nt transmitting antennas and nr receiving ones. The received signal vector y at nr receiving antennas can be formulated as y = H̃x + n, (12) where x is the transmitted signal vector form nt transmitting antennas, and n is an nr ×1 vector with zero-mean independent noises with variance σ 2 . H̃ is the channel matrix where each of its elements, h̃ij , is a complex value denoting the transfer coefficient associated with the jth transmitter antenna and the ith receiver antenna. Note that h̃ij can be obtained from (11) for frequency f and distance dij . The capacity of MIMO channel can be written as C = log2 det(Inr + P H̃H̃† ), nt σ 2 (13) where P is total transmitting power, and I is the identity matrix Since the determinant of (Inr + ntPσ2 H̃H̃† ) can be computed by the product of the eigenvalues of the matrix H̃H̃† , the MIMO capacity can thus be written in the form of a product of non-zero eigenvalues as [9] C= κ X i=1 log2 (1 + P λ2i ), κσ 2 (14) where λi denotes singular values of the matrix H̃, and hence the squared singular values λ2i denotes the eigenvalues of the matrix H̃H̃† . Each of the λ2i characterize an equivalent P λ2 information channel where kσ2i is the corresponding signalto-noise ratio (SNR) of the channel at the receiver. Note that κ denotes the number of non-zero λ2i , which for beamforming technique it is equal to one and in multiplexing technique it could be the rank of H̃ with κ ≤ min(nr , nt ) [9]. However, because we use blind precoding and uniform power allocation for multiplexing technique κ = nt . Therefore, equation (14) is valid for uniform power allocation at the transmitter. P λ2 Furthermore, the equivalent channel SNR, κσ2i , should meet a minimum receiver threshold to be reliably detectable by the receiver. In this paper, we assumed 0 dB as the SNR threshold and uniform power allocation at the transmitter. The main difference between beamforming and multiplexing techniques is how to tune or exploit the eigenvalue distribution. In more details, beamforming technique aims to maximize λ1 to improve the channel SNR for a single data stream while in the multiplexing technique, a uniform eigenvalue distribution is preferable. In this way, multiplexing technique can utilize parallel data streams through MIMO and maximize the data rate. The complexity of beamforming comes from eigenvalues tuning because it means the channel state information (CSI) should be measured and sent back to the transmitter periodically for optimum precoding. This also results in a protocol overhead in the channel. On the other hand, multiplexing gain can take advantage of eigenvalue value distribution even with a blind precoding. This is more beneficial when there is a rich scattering environment in the channel. In next section, we will discuss how the re-radiation can provide a rich scattering environment. III. A NALYSIS ON THE CHANNEL WITH MOLECULAR ABSORPTION To analyze the MIMO channel capacity and characterize the scattering richness of channel quantitatively, lets decompose and normalize channel transfer function H̃ as H(f, d) = r K HLoS (f, d) + K +1 r 1 Ha (f, d), (15) K +1 where H, HLoS and Ha are normalized with corresponding channel gain. Because of uniformly distributed random phase of received re-radiated signal, elements of Ha are independent and identically distributed (i.i.d) complex Gaussian random variables with zero mean and unit magnitude variance. K is the ratio of powers of the LoS signal and the re-radiated components and if we assume the channel distance is much longer than antenna space, it can be obtained by K= e−k(f )d Pr,LoS (f, d) . = Pr,a (f, d) 1 − e−k(f )d (16) This is same as the well-known Rician channel model where the K is called Rician K-factor. Equivalently, K-factor shows how much channel is rich in term of scattering and multipath rays. Equation (16) shows K is a function of absorption coefficient of channel medium k(f ) and the distance between transmitter and receiver d so that a longer distance and a higher absorption result smaller K, as shown in Figure 1a. The capacity of MIMO channel considering Rician K-factor is studied in several works [10, 11]. Authors in [11] showed the lower bound of Rician channel expected capacity for large number of antennas is the expected capacity of channel considering only NLoS component, r 1 (17) E(C(H)) ≥ E(C( Ha )), K +1 p =⇒ E(C(H)) ≥ E(C( 1 − e−k(f )d Ha )), (18) where E(.) denotes the expectation. It is clear that the lower band is a increasing function of absorption coefficient, ∀f1 , f2 such that k(f2 ) ≥ k(f1 ), Emin (C(f2 )) ≥ Emin (C(f1 )). (a) K-factor (b) MIMO capacity using beamforming (c) MIMO capacity using multiplexing Fig. 1: K-factor is an increasing function of distance and absorption coefficient. For both multiplexing and beamforming techniques, the performance gain is affected by K-factor. The capacity is calculated for 225x225 MIMO system. IV. S IMULATION AND DISCUSSION A. Simulation set-up In this section, to evaluate the molecular absorption impact on THz MIMO capacity, we consider a simple n × n MIMO system with a square uniform Arrays, where at both transmitter and receiver, the inter-element spacing s is equal to half of the wavelength and the channel distance is d. Moreover, we consider uniform power allocation to transmitter arrays operating in an open-space LoS scenario. The default values of the parameters are listed in Table I, and different values will be explained when necessary. Since we apply random phases on NLoS components created by molecular re-radiation, we conduct the evaluation of the MIMO capacity with molecular re-radiation for 1000 times and show the average result. We use the online browsing and plotting tools1 , which is based on HITRAN databases [7] to generate absorption coefficients for different single gas or some predefined standard gas mixture of the atmosphere at sea level, as shown in Table II. Since the water molecules play main roles in a normal air environment at THz bands we use the highest and lowest water ratio in Table II, i.e., the ”USA model, high latitude, winter” and ”USA model, tropics”. The corresponding absorption coefficients in THz bands have been shown in Figure 2a for an ambient temperature of 273 K and a sea level pressure of 1 atm. For a tropic atmosphere, the water ratio is higher than that of the winter atmosphere, and thus we can see a significant increase in the absorption coefficient among these two gas mixtures. In our simulation, we assume a constant transmit power over the entire frequency spectrum and display the MIMO capacity in bps/Hz for THz bands. We consider a MIMO set-up with 225 antennas at each side in a uniform square planar array. Our aim is to compare the beamforming and multiplexing 1 http://hitran.iao.ru/gasmixture/simlaunch TABLE I: Simulation parameters Transmitter and receiver distance (d) Inter-element spacing (s) Transmitter arrays angle (φ) Receiver arrays angle (θ) Number of arrays on each side (n) Transmit power Noise power 0.1, 1, 10 m 0.5λ (wave length) 90◦ 90◦ 225 0, 10 dBm −80 dBm techniques in different channel conditions. First, we calculate the channel capacity for beamforming while the re-radiation is totally ignored in the channel. Next, the beamforming capacity is re-calculated when the re-radiation is taken into account. Finally, the multiplexing gain is calculated with and without the consideration of re-radiation. In all scenarios, capacity is obtained by 14. In the first step, the simulation is run at 500 GHz with the practical range of absorption coefficient (10−5 ∼ 10+3 ) over the THz spectrum, as shown in Figure 1. It should be noted that the actual value of absorption coefficient at 500 GHz is shown in Figure 2a. The beamforming and multiplexing techniques capacity is calculated for a range of 0.1∼10 m distance and a 1 mW transmit power. Secondly, the channel is simulated for two different transmit power and three distances with realistic absorption coefficients. Our assumption on the transmit power is based on current technology [3] and a previous work on THz massive MIMO [1]. Furthermore, distances have been chosen to cover various application scenarios. For example, THz nanosensors are considered to communicate in a very short distance in the order of 0.1-10 cm or less, while THz communications are also nominated to provide terabit per second ultra high video communication link at around 1 m distance for home entrainment devices like TV or virtual reality (VR). In addition, longer distances to a few meters characterize wireless personal or local networks. Simulation results are presented in Figure 2. B. The MIMO Capacity vs. the K-factor Figure 1 illustrates how the channel is transformed from a LoS dominant channel to a Rayleigh channel and how it effects on the MIMO beamforming and multiplexing capacity gain. As can be seen in Figure 1b, the beamforming gain is decreasing when the absorption coefficient increases which is because in the very high absorption, the channel is not LoS dominant anymore and there is significant NLoS signal component generated by molecule re-radiation or equivalently lower K-factor. In contrast, Figure 1c shows the multiplexing technique takes advantage of higher absorption to reach a huge data rate. However, the low SNR limit the multiplexing gain in longer distances so that it drops sharply to zero beyond 2 m. In Figure 2, more results for the THz spectrum with realistic absorption coefficients will be presented. TABLE II: Atmosphere standard gas mixture ratio in percentage for different climates [7] USA USA USA USA USA model, model, model, model, model, mean latitude, summer, H=0 mean latitude, winter, H=0 high latitude, summer, H=0 high latitude, winter, H=0 tropics, H=0 H2O: H2O: H2O: H2O: H2O: 1.860000 0.432000 1.190000 0.141000 2.590000 CO2: CO2: CO2: CO2: CO2: 0.033000 0.033000 0.033000 0.033000 0.033000 O3: O3: O3: O3: O3: 0.000003 0.000003 0.000002 0.000002 0.000003 C. The MIMO Capacity vs. the Transmit Power and Distance The channel attenuation including molecular attenuation in (3) and spreading attenuation in (2) is illustrated in Figure 2b. While the spreading attenuation is increasing linearly in dB with distance and frequency, the molecular attenuation is also increasing with distance but is frequency selective. For example, while the total loss at 10m is 107 dB for 500 GHz, the total attenuation at 550 GHz is 86 dB at 1 m and it grows to 220 dB at 10 m which is mostly because of very high absorption of water molecules in the channel medium at this frequency. Note that the channel atmosphere for this case is from tropic data where the ratio of water molecules in the air is more than 0.02, as shown in Table II. Figure 2c and 2d illustrate the capacity of the investigated transmission techniques for a 10 cm distance. The transmit power is increased from 1 mW in Figure 2c to 10 mW in Figure 2d. It can be seen that a huge performance difference exists between multiplexing and beamforming, thanks to the tremendous multiplexing gain provided by the rich scattering environment due to molecule re-radiation. Furthermore, in very high absorption frequencies which existing studies consider as infeasible windows for THz communications, a significant capacity improvement can be observed. This is because more absorption leads to more re-radiation, which transforms a LoS dominant channel to Rayleigh channel. The details can be found in Section III, where we have discussed about how the re-radiation decreases the K-factor and creates a rich scattering environment. To sum up, the re-radiation improves the multiplexing gain which is fundamentally supported by a better eigenvalue distribution and channel matrix rank in mathematical analysis. In Figure 2e and 2f, the distance is increased to 1 m. With a relatively large distance for THz communications, it can be seen the beamforming gain is comparable with the multiplexing gain. However, we can see the multiplexing gain in high absorption windows, such as 540-580 GHz, is significantly higher than the rest of spectrum for a 10 mW transmit power. It is a different story for a 1 mW transmit power where the capacity drops to zero in high P λ2i kσ2 absorption windows because the equivalent SNR, ( ), of most parallel channels created by the multiplexing technique is less than 0 dB and practically such parallel channels are useless because the receiver can not reliably detect the received signals. Such results are not surprising since it has been shown in several works on conventional communication band [12] that the multiplexing performance drops dramatically in low SNR. However, considering the implementation challenges of beamforming, the multiplexing technique might still be a preferable choice for frequency up to 1 THz. For example, it can be observed in Figure 2e at 0.9 THz, the capacity is 4 and N2O: N2O: N2O: N2O: N2O: 0.000032 0.000032 0.000031 0.000032 0.000032 CO: CO: CO: CO: CO: 0.000015 0.000015 0.000015 0.000015 0.000015 CH4: CH4: CH4: CH4: CH4: 0.000170 0.000170 0.000170 0.000170 0.000170 O2: O2: O2: O2: O2: 20.900001 20.900001 20.900001 20.900001 20.900001 N2: N2: N2: N2: N2: 77.206000 78.634779 77.876781 78.925780 76.476779 11.7 bps/Hz for the multiplexing and beamforming techniques, respectively. Finally, Figures 2g and 2h present the results for a 10 m distance. For such a distance, path loss leads to a very low reception SNR and thus the beamforming performance is significantly better than the multiplexing performance. It is wellknown that beamforming technique is not very effective where there are strong multipath rays [12]. Thus, it is observed that in very high absorption frequency windows, the beamforming performance drops sharply. It is not only because of receiving strong NLoS rays caused by molecule re-radiation but also due to LoS signal attenuation. Note that the multiplexing technique can take advantage of same windows in high SNR as we discussed above for Figure 2f. V. C ONCLUSION In this paper, we compared the beam forming and multiplexing techniques of MIMO in the terahertz band. We showed in high SNR, high transmit power or lower distance, the multiplexing technique can provide a considerable capacity gain compared with beamforming. However, for beyond a few meters such as 10 meters, there should be enough transmitting power possibility to use multiplexing technique, otherwise the capacity drops to zero where the beamforming technique can still provide effective spectrum efficiency at the cost of complexity and protocol overhead. Our theoretical model also showed re-radiation of molecules in the THz band can be helpful for massive MIMO system to improve the channel performance using multiplexing technique. The re-radiation can provide significantly strong multipath components to achieve a full spatial multiplexing gain where the receiver is in an enough SNR coverage. It means some very high absorption frequency windows which have been formerly pointed as not feasible for communication might be more preferable choices for MIMO in some certain applications. R EFERENCES [1] I. F. Akyildiz and J. M. Jornet, “Realizing ultra-massive mimo (10241024) communication in the (0.0610) terahertz band,” Nano Communication Networks, vol. 8, pp. 46 – 54, 2016. [2] E. Zarepour, M. Hassan, C. T. Chou, and A. A. Adesina, “Semon: Sensorless event monitoring in self-powered wireless nanosensor networks,” ACM Transactions on Sensor Networks (TOSN), vol. 13, no. 2, p. 15, 2017. [3] I. F. Akyildiz, J. M. Jornet, and C. 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Akyildiz, “Femtosecond-Long Pulse-Based Modulation for Terahertz Band Communication in Nanonetworks,” IEEE Transactions on Communications, vol. 62, no. 5, pp. 1742–1754, May 2014. 10 4 US Model tropic - High H 2 O Absorption Coefficient (m -1 ) US Model High latitude winter - low H 2 O 10 2 10 0 10 -2 10 -4 10 -6 10 2 10 3 10 4 Frequency (GHz) (a) absorption coefficient, T= 273 K, P= 1 atm (b) Signal Attenuation 1200 600 Beamforming w re-radiation Beamforming w/o re-radiation Multiplexing - w re-radiation Multiplexing - w/o re-radiation Beamforming w re-radiation Beamforming w/o re-radiation Multiplexing - w re-radiation Multiplexing - w/o re-radiation 1000 400 800 Capacity (bps/Hz) Capacity (bps/Hz) 500 300 200 600 400 100 200 0 10 2 10 3 0 10 2 10 4 Frequency (GHz) 10 3 10 4 Frequency (GHz) (c) distance=0.1m, transmit power=1mW (d) distance=0.1m, transmit power=10mW 20 250 Beamforming w re-radiation Beamforming w/o re-radiation Multiplexing - w re-radiation Multiplexing - w/o re-radiation 18 16 Beamforming w re-radiation Beamforming w/o re-radiation Multiplexing - w re-radiation Multiplexing - w/o re-radiation 200 Capacity (bps/Hz) Capacity (bps/Hz) 14 12 10 8 150 100 6 4 50 2 0 10 2 10 3 0 10 2 10 4 Frequency (GHz) (e) distance=1m, transmit power=1mW 10 4 (f) distance=1m, transmit power=10mW 12 15 Beamforming w re-radiation Beamforming w/o re-radiation Multiplexing - w re-radiation Multiplexing - w/o re-radiation 10 Beamforming w re-radiation Beamforming w/o re-radiation Multiplexing - w re-radiation Multiplexing - w/o re-radiation 8 Capacity (bps/Hz) Capacity (bps/Hz) 10 3 Frequency (GHz) 6 4 10 5 2 0 10 2 10 3 10 4 Frequency (GHz) (g) distance=10m, transmit power=1mW 0 10 2 10 3 10 4 Frequency (GHz) (h) distance=10m, transmit power=10mW Fig. 2: 225x225 MIMO channel performance in tropic atmosphere. [6] J. M. Jornet Montana, “Fundamentals of electromagnetic nanonetworks in the terahertz band,” Ph.D. dissertation, Georgia Institute of Technology, 2013. [7] L. Rothman, I. Gordon, Y. Babikov, A. Barbe et al., “The hitran2012 molecular spectroscopic database,” Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 130, pp. 4 – 50, 2013. [8] L. D. Barron, Molecular light scattering and optical activity. Cambridge University Press, 2004. [9] D. Tse and P. Viswanath, “Chapter 07: MIMO I : spatial multiplexing and channel modeling,” Fundamentals of Wireless Communication, pp. 290–331, 2005. [10] F. R. Farrokhi, G. J. Foschini, A. Lozano, and R. A. Valenzuela, “Link-optimal space-time processing with multiple transmit and receive antennas,” IEEE Communications Letters, vol. 5, no. 3, pp. 85–87, 2001. [11] G. Lebrun, M. Faulkner, M. Shafi, and P. J. Smith, “Mimo ricean channel capacity: an asymptotic analysis,” IEEE Transactions on Wireless Communications, vol. 5, no. 6, pp. 1343–1350, 2006. [12] D. Gesbert, M. Shafi, D.-s. Shiu, P. J. Smith et al., “From theory to practice: An overview of mimo space-time coded wireless systems,” IEEE Journal on selected areas in Communications, vol. 21, no. 3, pp. 281–302, 2003.	7
An Emptiness Algorithm for Regular Types with Set Operators arXiv:cs/9811015v1 [cs.LO] 11 Nov 1998 Lunjin Lu and John G. Cleary Department of Computer Science University of Waikato Hamilton, New Zealand Phone: +64-838-4627/4378 {lunjin,jcleary}@cs.waikato.ac.nz Abstract. An algorithm to decide the emptiness of a regular type expression with set operators given a set of parameterised type definitions is presented. The algorithm can also be used to decide the equivalence of two regular type expressions and the inclusion of one regular type expression in another. The algorithm strictly generalises previous work in that tuple distributivity is not assumed and set operators are permitted in type expressions. Keywords: type, emptiness, prescriptive type 1 Introduction Types play an important role in programming languages [6]. They make programs easier to understand and help detect errors. Types have been introduced into logic programming in the forms of type checking and inference [5,9,12,26,32] or type analysis [25,33,17,19,13,22,7,23] or typed languages [16,21,28,31]. Recent logic programming systems allow the programmer to declare types for predicates and type errors are then detected either at compile time or at run time. The reader is referred to [27] for more details on types in logic programming. A type is a possibly infinite set of ground terms with a finite representation. An integral part of any type system is its type language that specifies which sets of ground terms are types. To be useful, types should be closed under intersection, union and complement operations. The decision problems such as the emptiness of a type, inclusion of a type in another and equivalence of two types should be decidable. Regular term languages [14,8], called regular types, satisfy these conditions and have been used widely used as types [29,25,33,9,17,21,28,31,12,32,19,13,22,7,23]. Most type systems use tuple distributive regular types which are strictly less powerful than regular types [29,25,33,17,21,28,31,12,32,19,13,22,7,23]. Tuple distributive regular types are regular types closed under tuple distributive closure. Intuitively, the tuple distributive closure of a set of terms is the set of all terms constructed recursively by permuting each argument position among all terms that have the same function symbol [32]. This paper gives an algorithm to decide if a type expression denotes an empty set of terms. The correctness of the algorithm is proved and its complexity is analysed. The algorithm works on prescriptive types [28]. By prescriptive types, we mean that the meaning of a type is determined by a given set of type definitions. We allow parametric and overloading polymorphism in type definitions. Prescriptive types are useful both in compilers and other program manipulation tools such as debuggers because they are easy to understand for programmers. Type expressions may contain set operators with their usual interpretations. Thus, the algorithm can be used to decide the equivalence of two type expressions and the inclusion of one type expression in another. The introduction of set operators into type expressions allows concise and intuitive representation of regular types. Though using regular term languages as types allow us to make use of theoretical results in the field of tree automata [14], algorithms for testing the emptiness of tree automata cannot be applied directly as type definitions may be parameterised. For instance, in order to decide the emptiness of a type expression given a set of type definitions, it would be necessary to construct a tree automaton from the type expression and the set of type definitions before an algorithm for determining the emptiness of an tree automaton can be used. When type definitions are parameterised, this would make it necessary to construct a different automaton each time the emptiness of a type expression is tested. Thus, an algorithm that works directly with type definitions is desirable as it avoids this repeated construction of automata. Attempts have been made in the past to find algorithms for regular types [25,12,32,33,31,10,9]. To our knowledge, Dart and Zobel’s work [10] is the only one to present decision algorithms for emptiness and inclusion problems for prescriptive regular types without the tuple distributive restriction. Unfortunately, their decision algorithm for the inclusion problem is incorrect for regular types in general. See [24] for a counterexample. Moreover, the type language of Dart and Zobel is less expressive than that considered in this paper since it doesn’t allow set operators and parameterised type definitions. Set constraint solving has also been used in type checking and type inference [3,2,20,18,11]. However, set constraint solving methods are intended to infer descriptive types [28] rather than for testing emptiness of prescriptive types [28]. Therefore, they are useful in different settings from the algorithm presented in this paper. Moreover, algorithms proposed for set constraint solving [3,4,2,1] are not applicable to the emptiness problem we considered in this paper as they don’t take type definitions into account. The remainder of this paper is organised as follows. Section 2 describes our language of type expressions and type definitions. Section 3 presents our algorithm for testing if a type expression denotes an empty set of terms. Section 4 addresses the of the algorithm. Section 5 presents the complexity of the algorithm and section 6 concludes the paper. Some lemmas are presented in the appendix. 2 Type Language Let Σ be a fixed ranked alphabet. Each symbol in Σ is called a function symbol and has a fixed arity. It is assumed that Σ contains at least one constant that is a function symbol of arity 0. The arity of a symbol f is denoted as arity(f ). Σ may be considered as the set of function symbols in a program. Let T (Φ) be the set of all terms over Φ. T (Σ) is the set of all possible values that a program variable can take. We shall use regular term languages over Σ as types. A type is represented by a ground term constructed from another ranked alphabet Π and {⊓, ⊔, ∼, 1, 0}, called type constructors. It is assumed that (Π ∪ {⊓, ⊔, ∼, 1, 0}) ∩ Σ = ∅. Thus, a type expression is a term in T (Π ∪ {⊓, ⊔, ∼, 1, 0}). The denotations of type constructors in Π are determined by type definitions whilst ⊓, ⊔, ∼, 1 and 0 have fixed denotations that will be given soon. Several equivalent formalisms such as tree automata [14,8], regular term grammars [14,10,8] and regular unary logic programs [32] have been used to define regular types. We define types by type rules. A type rule is a production rule of the form c(ζ1 , · · · , ζm ) → τ where c ∈ Π, ζ1 , · · · , ζm are different type parameters and τ ∈ T (Σ ∪ Π ∪ Ξm ) where Ξm = {ζ1 , · · · , ζm }. The restriction that every type parameter in the righthand side of a type rule must occur in the lefthand side of the type rule is often referred to as type preserving [30] and has been used in all the type definition formalisms. Note that overloading of function symbols is permitted as a function symbol can appear in the righthand sides of many type rules. We def S denote by ∆ the set of all type rules and define Ξ = c∈Π Ξarity(c) . hΠ, Σ, ∆i is a restricted form of context-free term grammar. Example 1. Let Σ = {0, s(), nil, cons(, )} and Π = {Nat, Even, List()}. ∆ defines natural numbers, even numbers, and lists where    Nat   → 0 \| s(Nat),  ∆ = Even → 0 \| s(s(Even)),    List(ζ) → nil \| cons(ζ, List(ζ))  where, for instance, Nat → 0 \| s(Nat) is an abbreviation of two rules Nat → 0 and Nat → s(Nat). ∆ is called simplified if τ in each production rule c(ζ1 , · · · , ζm ) → τ is of the form f (τ1 , · · · , τn ) such that each τj , for 1 ≤ j ≤ n, is either in Ξm or of the form d(ζ1′ , · · · , ζk′ ) and ζ1′ , · · · , ζk′ ∈ Ξm . We shall assume that ∆ is simplified. There is no loss of generality to use a simplified set of type rules since every set of type rules can be simplified by introducing new type constructors and rewriting and adding type rules in the spirit of [10]. Example 2. The following is the simplified version of the set of type rules in example 1. Σ = {0, s(), nil, cons(, )}, Π = {Nat, Even, Odd, List()} and ∆= ( Nat → 0 \| s(Nat), Even → 0 \| s(Odd), Odd → s(Even), List(ζ) → nil \| cons(ζ, List(ζ)) ) A type valuation φ is a mapping from Ξ to T (Π ∪{⊓, ⊔, ∼, 1, 0}). The instance φ(R) of a production rule R under φ is obtained by replacing each occurrence of each type parameter ζ in R with φ(ζ). E.g., List(Nat⊓(∼Even)) → cons(Nat⊓(∼Even), List(Nat⊓(∼Even))) is the instance of List(ζ) → cons(ζ, List(ζ)) under a type valuation that maps ζ to Nat⊓(∼Even). Let def ground(∆) = {φ(R) \| R ∈ ∆ ∧ φ ∈ (Ξ 7→ T (Π ∪ {⊓, ⊔, ∼, 1, 0}))} ∪ {1 7→ f (1, · · · , 1) \| f ∈ Σ} ground(∆) is the set of all ground instances of grammar rules in ∆ plus rules of the form 1 → f (1, · · · , 1) for every f ∈ Σ. Given a set ∆ of type definitions, the type denoted by a type expression is determined by the following meaning function. def [1]]∆ = T (Σ) def [0]]∆ = ∅ def [E1 ⊓E2]∆ = [E1]∆ ∩ [E2]∆ def [E1 ⊔E2]∆ = [E1]∆ ∪ [E2]∆ def [∼E]]∆ = T (Σ) − [E]]∆ def [ω]]∆ = [ {f (t1 , · · · , tn ) \| ∀1 ≤ i ≤ n. ti ∈ [Ei]∆ } (ω→f (E1 ,···,En ))∈ground(∆) [·]]∆ gives fixed denotations to ⊓, ⊔, ∼, 1 and 0. ⊓, ⊔ and ∼ are interpreted by [·]]∆ as set intersection, set union and set complement with respect to T (Σ). 1 denotes T (Σ) and 0 the empty set. Example 3. Let ∆ be that in example 2. We have [Nat]]∆ = {0, s(0), s(s(0)), · · ·} [Even]]∆ = {0, s(s(0)), s(s(s(s(0)))), · · ·} [Nat⊓∼Even]]∆ = {s(0), s(s(s(0))), s(s(s(s(s(0))))), · · ·} [List(Nat⊓∼Even)]]∆ = {cons(s(0), nil), cons(s(s(s(0))), nil), · · ·} The lemma 5 in the appendix states that every type expression denotes a regular term language, that is, a regular type. We extend [·]]∆ to sequences θ of type expressions as follows. def [ǫ]]∆ = {ǫ} def [hEi • θ′]∆ = [E]]∆ × [θ′]∆ where ǫ is the empty sequence, • is the infix sequence concatenation operator, hEi is the sequence consisting of the type expression E and × is the Cartesian product operator. As a sequence of type expressions, ǫ can be thought of consisting of zero instance of 1. We use Λ to denote the sequence consisting of zero instance of 0 and define [Λ]]∆ = ∅. We shall call a sequence of type expressions simply a sequence. A sequence expression is an expression consisting of sequences of the same length and ⊓, ⊔ and ∼. The length of the sequences in a sequence expression θ is called the dimension of θ and is denoted by kθk. Let θ, θ1 and θ2 be sequence expressions of the same length. def [θ1 ⊓θ2]∆ = [θ1]∆ ∩ [θ2]∆ def [θ1 ⊔θ2]∆ = [θ1]∆ ∪ [θ2]∆ def [∼θ]]∆ = T (Σ) × · · · × T (Σ) −[[θ]]∆ \| {z kθk times } A conjunctive sequence expression is a sequence expression of the form γ1 ∧ · · · ∧ γm where γi for, 1 ≤ i ≤ m, are sequences. 3 Emptiness Algorithm This section presents an algorithm that decides if a type expression denotes the empty set with respect to a given set of type definitions. The algorithm can also be used to decide if (the denotation of) one type expression is included in (the denotation of) another because E1 is included in E2 iff E1 ⊓∼E2 is empty. We first introduce some terminology and notations. A type atom is a type expression of which the principal type constructor is not a set operator. A type literal is either a type atom or the complement of a type atom. A conjunctive type expression C is of the form ⊓i∈I li with li being a type literal. Let α be a type atom. F (α) defined below is the set of the principal function symbols of the terms in [α]]∆ . def F (α) = {f ∈ Σ \| ∃ζ1 · · · ζk .((α → f (ζ1, · · · , ζk )) ∈ ground(∆))} Let f ∈ Σ. Define def Afα = {hα1 , · · · , αk i \| (α → f (α1 , · · · , αk )) ∈ ground(∆)} We have [Afα]∆ = {ht1 , · · · , tk i \| f (t1 , · · · , tk ) ∈ [α]]∆ }. Both F (α) and Afα are finite even though ground(∆)) is usually not finite. The algorithm repeatedly reduces the emptiness problem of a type expression to the emptiness problems of sequence expressions and then reduces the emptiness problem of a sequence expression to the emptiness problems of type expressions. Tabulation is used to break down any possible loop and to ensure termination. Let O be a type def expression or a sequence expression. Define empty(O) = ([[O]]∆ = ∅). 3.1 Two Reduction Rules We shall first sketch the two reduction rules and then add tabulation to form an algorithm. Initially the algorithm is to decide the validity of a formula of the form empty(E) (1) where E is a type expression. The first reduction rule rewrites a formula of the form (1) into a conjunction of formulae of the following form. Reduction Rule One. empty(σ) (2) where σ is a sequence expression where ∼ is applied to type expressions but not to any sequence expression. It is obvious that a type expression has a unique (modulo equivalence of denotation) disjunctive normal form. Let DNF(E) be the disjunctive normal form of E. empty(E) can written into ∧C∈DNF(E) empty(C). Each C is a conjunctive type expression. We assume that C contains at least one positive type literal. This doesn’t cause any loss of generality as [1⊓C]]∆ = [C]]∆ for any conjunctive type expression C. We also assume that C doesn’t contain repeated occurrences of the same type literal. Let C = ⊓1≤i≤m ωi ⊓ ⊓1≤j≤n ∼τj where ωi and τj are type atoms. def The set of positive type literals in C is denoted as pos(C) = {ωi \| 1 ≤ i ≤ m} while the set of complemented type atoms are denoted as def neg(C) = {τj \| 1 ≤ j ≤ n}. lit(C) denotes the set of literals occurring in C. By lemma 3 in the appendix, empty(C) is equivalent to ∀f ∈ ∩α∈pos(C) F (α). (3) empty((⊓ω∈pos(C) (⊔Afω ))⊓(⊓τ ∈neg(C) ∼(⊔Afτ ))) The intuition behind the equivalence is as follows. [C]]∆ is empty iff, for every function symbol f , the set of the sequences ht1 , · · · , tk i of terms such that f (t1 , · · · , tk ) ∈ [C]]∆ is empty. Only the function symbols in ∩α∈pos(C) F (α) need to be considered. We note the following two special cases of the formula (3). (a) If ∩α∈pos(C) F (α) = ∅ then the formula (3) is true because ∧∅ = true. In particular, F (0) = ∅. Thus, if 0 ∈ pos(C) then ∩α∈pos(C) F (α) = ∅ and hence the formula (3) is true. (b) If Afτ = ∅ for some τ ∈ neg(C) then ⊔Afτ = h0, · · · , 0i and ∼(⊔Afτ ) = h1, · · · , 1i. Thus, τ has no effect on the subformula for f when Afτ = ∅. In order to get rid of complement operators over sequence subexpressions, the complement operator in ∼(⊔Afτ ) is pushed inwards by the function push defined in the following. def push(∼(⊔i∈I γi )) = ⊓i∈I push(∼γi ) def push(∼hE1 , E2 , · · · , Ek i) = ⊔1≤l≤k h 1, · · · , 1, ∼El , 1, · · · , 1 i def push(∼ǫ) = Λ \| {z l−1 } \| {z k−l for k ≥ 1 } It follows from De Morgan’s law and the definition of [·]]∆ that [push(∼(⊔Afτ ))]]∆ = [∼(⊔Afτ )]]∆ . Substituting push(∼(⊔Afτ )) for ∼(⊔Afτ ) in the formula (3) gives rise to a formula of the form (2). The second reduction rule rewrites a formula of the form 2 to a conjunction of disjunctions of formulae of the form 1. Formula 2 is written into a disjunction of formulae of the form. empty(Γ ) Reduction Rule Two. where Γ be a conjunctive sequence expression. In the case kΓ k = 0, by lemma 4 in the appendix, empty(Γ ) can be decided without further reduction. If Λ ∈ Γ then empty(Γ ) is true because [Λ]]∆ = ∅. Otherwise, empty(Γ ) is false because [Γ ]∆ = {ǫ}. In the case kΓ k = 6 0, empty(Γ ) is equivalent to ∨1≤j≤kΓ k empty(Γ↓j) def where, letting Γ = γ1 ⊓ · · · ⊓γk , Γ↓j = ⊓1≤i≤k γij with γij being the j th component of γi . Note that Γ↓j is a type expression and empty(Γ↓j) is of the form 1. 3.2 Algorithm The two reduction rules in the previous section form the core of the algorithm. However, they alone cannot be used as an algorithm as a formula empty(E) may reduce to a formula containing empty(E) as a sub-formula, leading to nontermination. Suppose Σ = {f (), a}, Π = {Null} and ∆ = {Null → f (Null)}. Clearly, empty(Null) is true. However, by the first reduction rule, empty(Null) reduces to empty(hNulli) which then reduces to empty(Null) by the second reduction rule. This process will not terminate. The solution, inspired by [10], is to remember in a table a particular kind of formulae of which truth is being tested. When a formula of that kind is tested, the table is first looked up. If the formula is implied by any formula in the table, then it is determined as true. Otherwise, the formula is added into the table and then reduced by a reduction rule. The emptiness algorithm presented below remembers every conjunctive type expression of which emptiness is being tested. Thus the table is a set of conjunctive type expressions. Let C1 and C2 be def conjunctive type expressions. We define (C1 C2 ) = (lit(C1 ) ⊇ lit(C2 )). Since Ci = ⊓l∈lit(Ci ) l, C1 C2 implies [C1]∆ ⊆ [C2]∆ and hence (C1 C2 ) ∧ empty(C2 ) implies empty(C1 ). Adding tabulation to the two reduction rules, we obtain the following algorithm for testing the emptiness of prescriptive regular types. Let BCf = (⊓ω∈pos(C) (⊔Afω ))⊓(⊓τ ∈neg(C) push(∼(⊔Afτ ))). def etype(E) = etype(E, ∅) def etype(E, Ψ ) = ∀C ∈ DNF(E).etype conj(C, Ψ ) (4) (5) def etype conj(C, Ψ ) =  if pos(C) ∩ neg(C) 6= ∅,  true, true, if ∃C ′ ∈ Ψ.C C ′ ,  ∀f ∈ ∩ f otherwise. α∈pos(C) F(α).eseq(BC , Ψ ∪ {C}), def eseq(Θ, Ψ ) = ∀Γ ∈ DNF(Θ).eseq conj(Γ, Ψ ) def eseq conj(Γ, Ψ ) = ( true if kΓ k = 0 ∧ Λ ∈ Γ , false if kΓ k = 0 ∧ Λ 6∈ Γ , ∃1 ≤ j ≤ kΓ k.etype(Γ↓j, Ψ ) if kΓ k = 6 0. (6) (7) (8) Equation 4 initialises the table to the empty set. Equations 5 and 6 implement the first reduction rule while equations 7 and 8 implement the second reduction rule. etype(, ) and etype conj(, ) test the emptiness of an arbitrary type expression and that of a conjunctive type expression respectively. eseq(, ) tests emptiness of a sequence expression consisting of sequences and ⊓ and ⊔ operators while eseq conj(, ) tests the emptiness of a conjunctive sequence expression. The expression of which emptiness is to be tested is passed as the first argument to these functions. The table is passed as the second argument. It is used in etype conj(, ) to detect a conjunctive type expression of which emptiness is implied by the emptiness of a tabled conjunctive type expression. As we shall show later, this ensures the termination of the algorithm. Each of the four binary functions returns true iff the emptiness of the first argument is implied by the second argument and the set of type definitions. Tabling any other kind of expressions such as arbitrary type expressions can also ensure termination. However, tabling conjunctive type expressions makes it easier to detect the implication of the emptiness of one expression by that of another because lit(C) can be easily computed given a conjunctive type expression C. In an implementation, a conjunctive type expression C in the table can be represented as lit(C). The first two definitions for etype conj(C, Ψ ) in equation 6 terminates the algorithm when the emptiness of C can be decided by C and Ψ without using type definitions. The first definition also excludes from the table any conjunctive type expression that contains both a type atom and its complement. 3.3 Examples We now illustrate the algorithm with some examples. Example 4. Let type definitions be given as in example 2. The tree in figure 1 depicts the evaluation of etype(Nat⊓∼Even⊓∼Odd) by the algorithm. Nodes are labeled with function calls. We will identity a node with its label. Arcs from a node to its children are labeled with the number of the equation that is used to evaluate the node. Abbreviations used in the labels are defined in the legend to the right of the tree. Though [A]]∆ = [B]]∆ , A and B are syntactically different type expressions. The evaluation returns true, verifying [Nat⊓∼Even⊓∼Odd]]∆ = ∅. Consider etype conj(B, {A}). We have B A as lit(A) = lit(B). Thus, by equation 6, etype conj(B, {A}) = true. etype(A) 3 etype(A, ∅) 4 etype conj(A, ∅) ∧ 5 5 eseq(C, {A}) eseq(ǫ⊓Λ, {A}) 6 6 eseq conj(ǫ⊓Λ, {A}) eseq conj(C, {A}) 7 7 true etype(B, {A}) 4 etype conj(B, {A}) 5 true Legend: A = N at⊓∼Even⊓∼Odd B = N at⊓∼Odd⊓∼Even C = hN ati⊓h∼Oddi⊓h∼Eveni Fig. 1. Evaluation of etype(N at⊓∼Even⊓∼Odd)) Example 5. Let type definitions be given as in example 2. The tree in figure 2 depicts the evaluation of etype(List(Even⊓∼Nat)) by the algorithm. The evaluation returns false, verifying [List(Even⊓∼Nat)]]∆ 6= ∅. Indeed, [List(Even⊓∼Nat)]]∆ = {nil}. The rightmost node is not evaluated as its sibling returns false, which is enough to establish the falsity of their parent node. etype(A) (3) etype(A, ∅) (4) etype conj(A, ∅) ∧ (5)/nil (5)/cons(,) eseq(ǫ, {A}) eseq(hB, Ai, {A}) (6) eseq conj(ǫ, {A}) (7) false Legend: A = List(Even⊓∼N at) B = Even⊓∼N at Fig. 2. Evaluation of etype(List(Even⊓∼N at)) Example 6. The following is a simplified version of the type definitions that is used in [24] to show the incorrectness of the algorithm by Dart and Zobel for testing inclusion of one regular type in another [10]. Let Π = {α, β, θ, σ, ω, ζ, η}, Σ = {a, b, g(), h(, )} and ∆= ( α → g(ω), β → g(θ) \| g(σ), θ → a \| h(θ, ζ), σ → b \| h(σ, η), ω → a \| b \| h(ω, ζ) \| h(ω, η), ζ → a, η→b ) Let t = g(h(h(a, b), a)). t ∈ [α]]∆ and t 6∈ [β]]∆ , see example 3 in [24] for more details. So, [α]]∆ 6⊆ [β]]∆ . This is verified by our algorithm as follows. Let Ψ1 = {α⊓∼β} and Ψ2 = Ψ1 ∪ {ω⊓∼θ⊓∼σ}. By applying equations 4, 5, 6, 7, 8 and 5 in that order, we have etype(α⊓∼β) = etype conj(ω⊓∼θ⊓∼σ, Ψ1 ). By equation 6, we have etype(α⊓∼β) = eseq(ǫ⊓Λ⊓ǫ, Ψ2 ) ∧ eseq(ǫ⊓ǫ⊓Λ, Ψ2 ) ∧ eseq(Θ, Ψ2) where Θ = (hω, ζi⊔hω, ηi)⊓(h∼θ, 1i⊔h1, ∼ζi)⊓(h∼σ, 1i⊔h1, ∼ηi). We choose not to simplify expressions such as ǫ⊓ǫ⊓∼Λ so as to make the example easy to follow. By applying equations 7 and 8, we have both eseq(ǫ⊓Λ⊓ǫ, Ψ2 ) = true and eseq(ǫ⊓ǫ⊓Λ, Ψ2 ) = true. So, etype(α⊓∼β) = eseq(Θ, Ψ2). Let Γ = hω, ζi⊓h∼θ, 1i⊓h1, ∼ηi. To show etype(α⊓∼β) = false, it suffices to show eseq conj(Γ, Ψ2 ) = false by equation 7 because Γ ∈ DNF(Θ) and etype(α⊓∼β) = eseq(Θ, Ψ2). Figure 3 depicts the evaluation of eseq conj(Γ, Ψ2). The node that is linked to its parent by a dashed line is not evaluated because one of its siblings returns false, which is sufficient to establish the falsity of its parent. It is clear from the figure that etype conj(Θ, Ψ2) = false and hence etype(α⊓∼β) = false. 4 Correctness This section addresses the correctness of the algorithm. We shall first show that tabulation ensures the termination of the algorithm because the table can only be of finite size. We then establish the partial correctness of the algorithm. etype conj(Γ, Ψ2 ) ∨ 7 7 etype(ζ⊓∼η, Ψ2 ) etyp(ω⊓∼θ, Ψ2 ) 4 4 etyp conj(ω⊓∼θ, Ψ2 ) etype conj(ζ⊓∼η, Ψ2 ) ∧ 5/b 5/a 5/h(,) 5 eseq(ǫ⊓Λ, Ψ3 ) eseq(ǫ⊓ǫ, Ψ4 ) eseq(ǫ⊓ǫ, Ψ3 ) eseq(Θ1 , Ψ3 ) 6 6 6 eseq conj(ǫ⊓Λ, Ψ3 ) eseq conj(ǫ⊓ǫ, Ψ3 ) eseq conj(ǫ⊓ǫ, Ψ4 ) 7 7 7 true false false Legend: Θ1 = (hω, ζi⊔hω, ηi)⊓(h∼θ, 1i⊔h1, ∼ζi) Ψ3 = Ψ2 ∪ {ω⊓∼θ} Ψ4 = Ψ2 ∪ {ζ⊓∼η} Γ = hω, ζi⊓h∼θ, 1i⊓h1, ∼ηi Fig. 3. Evaluation of etype conj(Γ, Ψ2 ) 4.1 Termination Given a type expression E, a top-level type atom in E is a type atom in E that is not a sub-term of any type atom in E. The set of top-level type atoms in E is denoted by TLA(E). For instance, letting E = ∼List(Nat)⊔T ree(Nat⊓∼Even), TLA(E) = {List(Nat), T ree(Nat⊓∼Even)}. We extend TLA(·) to sequences def S by TLA(hE1 , E2 , · · · , Ek i) = 1≤i≤k TLA(Ei ). Given a type expression E0 , the evaluation tree for etype(E0 ) contains nodes of the form etype(E, Ψ ), etype conj(C, Ψ ), eseq(Θ, Ψ ) and eseq conj(Γ, Ψ ) in addition to the root that is etype(E0 ). Only nodes of the form etype conj(C, Ψ ) add conjunctive type expressions to the table. Other forms of nodes only pass the table around. Therefore, it suffices to show that the type atoms occurring in the first argument of the nodes are from a finite set because any conjunctive type expression added into the table is the first argument of a node of the form etype conj(C, Ψ ). The set RTA(E0 ) of type atoms relevant to a type expression E0 is the smallest set of type atoms satisfying – TLA(E0 ) ⊆ RTA(E0 ), and – if τ is in RTA(E0 ) and τ → f (τ1 , τ2 , · · · , τk ) is in ground(∆) then TLA(τi ) ⊆ RTA(E0 ) for 1 ≤ i ≤ k. The height of τi is no more than that of τ for any τ → f (τ1 , τ2 , · · · , τk ) in ground(∆). Thus, the height of any type atom in RTA(E0 ) is finite. There are only a finite number of type constructors in Π. Thus, RTA(E0 ) is of finite size. It follows by examining the algorithm that type atoms in the first argument of the nodes in the evaluation tree for etype(E0 ) are from RTA(E0 ) which is finite. Therefore, the algorithm terminates. 4.2 Partial Correctness The partial correctness of the algorithm is established by showing etype(E0 ) = true iff empty(E0 ). Let Ψ be a set of conjunctive type def expressions. Define ρΨ = ∧C∈Ψ empty(C). The following two lemmas form the core of our proof of the partial correctness of the algorithm. Lemma 1. Let Ψ be a set of conjunctive type expressions, E a type expression, C a conjunctive type expression, Θ a sequence expression and Γ a conjunctive sequence expression. (a) (b) (c) (d) If If If If ρΨ ρΨ ρΨ ρΨ \|= empty(C) \|= empty(E) \|= empty(Γ ) \|= empty(Θ) then then then then etype conj(C, Ψ ) = true, and etype(E, Ψ ) = true, and etype(Γ, Ψ ) = true, and etype(Θ, Ψ ) = true. Proof. The proof is done by induction on the size of the complement of Ψ with respect to the set of all possible conjunctive type expressions in which type atoms are from RTA(E0 ) where E0 is a type expression. Basis. The complement is empty. Ψ contains all possible conjunctive type expressions in which type atoms are from RTA(E0 ). We have C ∈ Ψ and hence etype conj(C, Ψ ) = true by equation 6. Therefore, (a) holds. (b) follows from (a) and equation 5. (c) follows from (b), equation 8 and lemma 4 in the appendix, and (d) follows from (c) and equation 7. Induction. By lemma 3 in the appendix, ρΨ \|= empty(C) implies ρΨ \|= empty(BCf ) for any f ∈ ∩α∈pos(C) F (α). Thus, ρΨ∪{C} \|= empty(BCf ). The complement of Ψ ∪ {C} is smaller than the complement of Ψ . By the induction hypothesis, we have eseq(BCf , Ψ ∪{C}) = true. By equation 6, etype conj(C, Ψ ) = true. Therefore, (a) holds. (b) follows from (a) and equation 5. (c) follows from (b), equation 8 and lemma 4 in the appendix and (d) follows from (c) and equation 7. This completes the proof of the lemma. Lemma 1 establishes the completeness of etype(, ), etype conj(, ), eseq(, ) and eseq conj(, ) while the following lemma establishes their soundness. Lemma 2. Let Ψ be a set of conjunctive type expressions, E a type expression, C a conjunctive type expression, Θ a sequence expression and Γ a conjunctive sequence expression. (a) (b) (c) (d) ρΨ ρΨ ρΨ ρΨ \|= empty(C) \|= empty(E) \|= empty(Γ ) \|= empty(Θ) if if if if etype conj(C, Ψ ) = true, and etype(E, Ψ ) = true, and etype(Γ, Ψ ) = true, and etype(Θ, Ψ ) = true. Proof. It suffices to prove (a) since (b),(c) and (d) follow from (a) as in lemma 1. The proof is done by induction on dp(C, Ψ ) the depth of the evaluation tree for etype conj(C, Ψ ). Basis. dp(C, Ψ ) = 1. etype conj(C, Ψ ) = true implies either (i) pos(C) ∩ neg(C) 6= ∅ or (ii) ∃C ′ ∈ Ψ.C C ′ . In case (i), empty(C) is true and ρΨ \|= empty(C). Consider case (ii). By the definition of and ρΨ , we have etype conj(C, Ψ ) = true implies ρΨ \|= empty(C). Induction. dp(C, Ψ ) > 1. Assume etype conj(C, Ψ ) = true and ρΨ \|= ¬empty(C). By lemma 3, there is f ∈ ∩α∈pos(C) F (α) such that ρΨ \|= ¬empty(BCf ). We have ρΨ∪{C} \|= ¬empty(BCf ). dp(BCf , Ψ ∪ {C}) < dp(C, Ψ ). By the induction hypothesis, we have etuple(BCf , Ψ ∪ {C}) = false for otherwise, ρΨ∪{C} \|= BCf . By equation 6, etype conj(C, Ψ ) = false which contradicts etype conj(C, Ψ ) = true. So, ρΨ \|= empty(C) if etype conj(C, Ψ ) = true. This completes the induction and the proof of the lemma. The following theorem is a corollary of lemmas 1 and 2. Theorem 1. For any type expression E, etype(E) = true iff empty(E). Proof. By equation 4, etype(E) = etype(E, ∅). By lemma 1.(b) and lemma 2.(b), we have etype(E, ∅) = true iff ρ∅ \|= empty(E). The result follows since ρ∅ = true. 5 Complexity We now address the issue of complexity of the algorithm. We only consider the worst-case time complexity of the algorithm. The time spent on evaluating etype(E0 ) for a given type expression E0 can be measured in terms of the number of nodes in the evaluation tree for etype(E0 ). The algorithm cycles through etype(, ), etype conj(, ), eseq(, ) and eseq conj(, ). Thus, children of a node of the form etype(E, Ψ ) can only be of the form etype conj(C, Ψ ), and so on. Let \|S\| be the number of elements in a given set S. The largest possible table in the evaluation of etype(E0 ) contains all the conjunctive type expressions of which type atoms are from RTA(E0 ). Therefore, the table can contain at most 2\|RTA(E0 )\| conjunctive type expressions. So, the height of the tree is bounded by O(2\|RTA(E0 )\| ). We now show that the branching factor of the tree is also bounded by O(2\|RTA(E0 )\| ). By equation 5, the number of children of etype(E, Ψ ) is bounded by two to the power of the number of type atoms in E which is bounded by \|RTA(E0 )\| because E can only contain type atoms from RTA(E0 ). By equation 6, the number of children of etype conj(C, Ψ ) is bounded by \|Σ\|. The largest number of children of a node eseq(Θ, Ψ ) is bounded by two to the power of the number of sequences in Θ where Θ = BCf . For each τ ∈ neg(C), \|push(∼(⊔Afτ ))\| is O(arity(f )) and \|C\| < \|RTA(E0 )\|. Thus, the number of sequences in Θ is O(arity(f ) ∗ \|RTA(E0 )\|) and hence the number of children of eseq(Θ, Ψ ) is O(2\|RTA(E0 )\| ) since arity(f ) is a constant. By equation 8, the number of children of eseq conj(Γ, Ψ ) is bounded by maxf ∈Σ arity(f ). Therefore, the branching factor of the tree is bounded by O(2\|RTA(E0 )\| ). The above discussion leads to the following conclusion. Proposition 1. The time complexity of the algorithm is O(2\|RTA(E0 )\| )). The fact that the algorithm is exponential in time is expected because the complexity coincides with the complexity of deciding the emptiness of any tree automaton constructed from the type expression and the type definitions. A deterministic frontier-to-root tree automaton recognising [E0]∆ will consist of 2\|RTA(E0 )\| states as observed in the proof of lemma 5. It is well-known that the decision of the emptiness of the language of a deterministic frontier-to-root tree automaton takes time polynomial in the number of the states of the tree automaton. Therefore, the worst-case complexity of the algorithm is the best we can expect from an algorithm for deciding the emptiness of regular types that contain set operators. 6 Conclusion We have presented an algorithm for deciding the emptiness of prescriptive regular types. Type expressions are constructed from type constructors and set operators. Type definitions prescribe the meaning of type expressions. The algorithm uses tabulation to ensure termination. Though the tabulation is inspired by Dart and Zobel [10], the decision problem we consider in this paper is more complex as type expressions may contain set operators. For that reason, the algorithm can also be used for inclusion and equivalence problems of regular types. The way we use tabulation leads to a correct algorithm for regular types while the Dart-Zobel algorithm has been proved incorrect for regular types [24] in general. To the best of our knowledge, our algorithm is the only correct algorithm for prescriptive regular types. In addition to correctness, our algorithm generalises the work of Dart and Zobel [10] in that type expressions can contain set operators and type definitions can be parameterised. Parameterised type definitions are more natural than monomorphic type definitions [12,26,32] while set operators makes type expressions concise. The combination of these two features allows more natural type declarations. For instance, the type of the logic program append can be declared or inferred as append(List(α), List(β), List(α⊔β)). The algorithm is exponential in time. This coincides with deciding the emptiness of the language recognised by a tree automaton constructed from the type expression and the type definitions. However, the algorithm avoids the construction of the tree automaton which cannot be constructed a priori when type definitions are parameterised. Another related field is set constraint solving [3,2,20,18,11]. However, set constraint solving methods are intended to infer descriptive types [28] rather than for testing the emptiness of a prescriptive type [28]. Therefore, they are useful in different settings from the al- gorithm presented in this paper. In addition, algorithms proposed for solving set constraints [3,4,2,1] are not applicable to the emptiness problem we considered in this paper. Take for example the constructor rule in [3,2] which states that emptiness of f (E1 , E2 , · · · , Em ) is equivalent to the emptiness of Ei for some 1 ≤ i ≤ m. However, empty(List(0)) is not equivalent to empty(0). The latter is true while the former is false since [List(0)]]∆ = {nil}. The constructor rule doesn’t apply because it deals with function symbols only but doesn’t take the type definitions into account. References 1. A. Aiken, D. Kozen, M. Vardi, and E. Wimmers. The complexity of set constraints. In Proceedings of 1993 Computer Science Logic Conference, pages 1–17, 1992. 2. A. Aiken and T.K. Lakshman. Directional type checking of logic programs. In B. Le Charlier, editor, Proceedings of the First International Static Analysis Symposium, pages 43–60. Springer-Verlag, 1994. 3. A. Aiken and E. Wimmers. Solving systems of set constraints. In Proceedings of the Seventh IEEE Symposium on Logic in Computer Science, pages 329–340. The IEEE Computer Society Press, 1992. 4. A. Aiken and E. Wimmers. Type inclusion constraints and type inference. In Proceedings of the 1993 Conference on Functional Programming Languages and Computer Architecture, pages 31–41, Copenhagen, Denmark, June 1993. 5. C. Beierle. Type inferencing for polymorphic order-sorted logic programs. In L. Sterling, editor, Proceedings of the Twelfth International Conference on Logic Programming, pages 765–779. The MIT Press, 1995. 6. L. Cardelli and P. Wegner. On understanding types, data abstraction, and polymorphism. ACM computing surveys, 17(4):471–522, 1985. 7. M. Codish and V. Lagoon. Type dependencies for logic programs using aciunification. In Proceedings of the 1996 Israeli Symposium on Theory of Computing and Systems, pages 136–145. IEEE Press, June 1996. 8. H. Comon, M. Dauchet, R. Gilleron, D. Lugiez, S. Tison, and M. Tommasi. Tree Automata Techniques and Applications. Draft, 1998. 9. P.W. Dart and J. Zobel. Efficient run-time type checking of typed logic programs. 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Gécseg and M. Steinby. Tree languages. In G. Rozenberg and A. Salomma, editors, Handbook of Formal Languages, pages 1–68. Springer-Verlag, 1996. 16. M. Hanus. Horn clause programs with polymorphic types: semantics and resolution. Theoretical Computer Science, 89(1):63–106, 1991. 17. N. Heintze and J. Jaffar. A finite presentation theorem for approximating logic programs. In Proceedings of the seventh Annual ACM Symposium on Principles of Programming Languages, pages 197–209. The ACM Press, 1990. 18. N. Heintze and J. Jaffar. A decision procedure for a class of set constraints. Technical Report CMU-CS-91-110, Carnegie-Mellon University, February 1991. (Later version of a paper in Proc. 5th IEEE Symposium on LICS). 19. N. Heintze and J. Jaffar. Semantic types for logic programs. In Frank Pfenning, editor, Types in Logic Programming, pages 141–155. The MIT Press, 1992. 20. N. Heintze and J. Jaffar. Set constraints and set-based analysis. 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Furukawa, editor, Logic Programming. Proceedings of the Eighth International Conference, pages 379–93. The MIT Press, 1991. 32. E. Yardeni and E. Shapiro. A type system for logic programs. Journal of Logic Programming, 10(2):125–153, 1991. 33. J. Zobel. Derivation of polymorphic types for Prolog programs. In J.-L. Lassez, editor, Logic Programming: Proceedings of the fourth international conference, pages 817–838. The MIT Press, 1987. Appendix Lemma 3. Let C be a conjunctive type expression. empty(C) iff ∀f ∈ ∩α∈pos(C) F (α). empty((⊓ω∈pos(C) (⊔Afω ))⊓(⊓τ ∈neg(C) ∼(⊔Afτ ))) Proof. Let t be a sequence of terms and f a function symbol. By the definition of [·]]∆ , f (t) ∈ [C]]∆ iff f ∈ ∩α∈pos(C) F (α) and t ∈ [⊓ω∈pos(C) (⊔Afω ))]]∆ \ [(⊔τ ∈neg(C) (⊔Afτ ))]]∆ . t ∈ [⊓ω∈pos(C) (⊔Afω ))]]∆ \ [(⊔τ ∈neg(C) (⊔Afτ ))]]∆ iff t ∈ [(⊓ω∈pos(C) (⊔Afω ))⊓(⊓τ ∈neg(C) ∼(⊔Afτ ))]]∆ . Thus, empty(C) iff empty((⊓ω∈pos(C) (⊔Afω ))⊓(⊓τ ∈neg(C) ∼(⊔Afτ ))) for each f ∈ ∩α∈pos(C) F (α). Lemma 4. Let Γ be a conjunctive sequence expression. Then empty(Γ ) iff ⊔1≤jkΓ kempty(Γ↓j) Proof. Let kΓ k = Tn and Γ = γ1 ⊓γ2 ⊓ · · · ⊓γm with γi = hγi,1 , γi,2, · · · , γi,n i. We have [Γ ]∆ = 1≤j≤m [γj]∆ . We have Γ↓j = γ1,j ⊓γ2,j ⊓ · · · ⊓γm,j . T ∃1 ≤ j ≤ n.empty(Γ↓j) iff ∃1 ≤ j ≤ n. 1≤i≤m [γi,j]∆ = ∅ iff [Γ ]∆ = ∅ iff empty(Γ ). Lemma 5. [M]]∆ is a regular term language for any type expression M. Proof. The proof is done by constructing a regular term grammar for M [14]. We first consider the case M ∈ T (Π ∪ {1, 0}). Let R = hRTA(M), Σ, ∅, Υ, Mi with Υ = {(α → f (α1 , · · · , αk )) ∈ ground(∆) \| α ∈ RTA(M)} R is a regular term grammar. It now suffices to prove that t ∈ [M]]∆ iff M ⇒∗R t. – Sufficiency. Assume M ⇒∗R t. The proof is done by induction on derivation steps in M ⇒∗R t. • Basis. M ⇒R t. t must be a constant and M → t is in Υ which implies M → t is in ground(∆). By the definition of [·]]∆ . t ∈ [M]]∆ . (n−1) • Induction. Suppose M ⇒ f (M1 , · · · , Mk ) ⇒R t. Then ni t = f (t1 , · · · , tk ) and Mi ⇒R t with ni ≤ (n − 1). By the induction hypothesis, ti ∈ [Mi]∆ and hence t ∈ [M]]∆ by the definition of [·]]∆ . – Necessity. Assume t ∈ [M]]∆ . The proof is done by the height of t, denoted as height(t). • height(t) = 0 implies that t is a constant. t ∈ [M]]∆ implies that M → t is in ground(∆) and hence M → t is in Υ . Therefore, M ⇒R t. • Let height(t) = n. Then t = f (t1 , · · · , tk ). t ∈ [M]]∆ implies that (M → f (M1, · · · , Mk )) ∈ ground(∆) and ti ∈ [Mi]∆ . By the definition of Υ , we have (M → f (M1 , · · · , Mk )) ∈ Υ . By the definition of RTA(·), we have Mi ∈ RTA(M). By the induction hypothesis, Mi ⇒∗R ti . Therefore, M ⇒R f (M1 , · · · , Mk ) ⇒∗R f (t1 , · · · , tk ) = t. Now consider the case M ∈ T (Π ∪ {⊓, ⊔, ∼, 1, 0}). We complete the proof by induction on the height of M. – height(M) = 0. Then M doesn’t contain set operator. We have already proved that [M]]∆ is a regular term language. – Now suppose height(M) = n. If M doesn’t contain set operator then the lemma has already been proved. If the principal type constructor is one of set operators then the result follows immediately as regular term languages are closed under union, intersection and complement operators [14,15,8]. It now suffices to prove the case M = c(M1 , · · · , Ml ) with c ∈ Π. Let N = c(X1 , · · · , Xl ) where each Xj is a different new type constructor of arity 0. Let Π ′ = Π{X1, · · · , Xl }, Σ ′ = Σ ∪ {x1 , · · · , xl } and ∆′ = ∆ ∪ {Xj → xj \|1 ≤ j ≤ l}. [N ]∆′ is a regular term language on Σ ∪ {x1 , · · · , xl } because N doesn’t contain set operators. By the induction hypothesis, [Mj]∆ is a regular term language. By the definition of [·]]· , we have [M]]∆ = [N ]∆′ [x1 := [M1]∆ , · · · , xl := [Ml]∆ ] which is a regular term language [14,15,8]. S[y1 := Sy1 , · · · , ] is the set of terms each of which is obtained from a term in S by replacing each occurrence of yj with a (possibly different) term from Syj . This completes the induction and the proof. The proof also indicates that a non-deterministic frontier-to-root tree automaton that recognises [M]]∆ has \|RTA(M)\| states and that a deterministic frontier-to-root tree automaton that recognises [M]]∆ has O(2\|RTA(M)\| ) states.	6
1 A Study of the Allan Variance for Constant-Mean Non-Stationary Processes arXiv:1702.07795v3 [math.ST] 22 Jun 2017 Haotian Xu, Stéphane Guerrier, Roberto Molinari & Yuming Zhang Abstract—The Allan Variance (AV) is a widely used quantity in areas focusing on error measurement as well as in the general analysis of variance for autocorrelated processes in domains such as engineering and, more specifically, metrology. The form of this quantity is widely used to detect noise patterns and indications of stability within signals. However, the properties of this quantity are not known for commonly occurring processes whose covariance structure is non-stationary and, in these cases, an erroneous interpretation of the AV could lead to misleading conclusions. This paper generalizes the theoretical form of the AV to some non-stationary processes while at the same time being valid also for weakly stationary processes. Some simulation examples show how this new form can help to understand the processes for which the AV is able to distinguish these from the stationary cases and hence allow for a better interpretation of this quantity in applied cases. Index Terms—Metrology, Sensor Calibration, Bias-Instability, Longitudinal Studies, Haar Wavelet Variance, Heteroscedasticity. I. I NTRODUCTION The Allan Variance (AV) is a widely used quantity in areas going from engineering to physics where there is an interest in studying the stochastic stability of error measurements from various instruments such as, among others, clocks and oscillators. Its usefulness resides in the fact that it provides an extremely informative summary on the variance of time series or, more generally, of autocorrelated processes, especially when these are non-stationary and with infinite variance. Indeed, [8] underlined how the AV is a better measure of uncertainty compared to standard methods (e.g. moving average variance) for processes such as random walks and non-stationary Fractional Autoregressive-Moving Average (ARFIMA) models, while being considerably useful also for stationary processes. For these processes the AV has a well known form which can help detect the kind of process, for example, from the log-log plot of the AV of an observed signal. The behaviour and forms of the AV for stationary and some non-stationary processes was studied in [8] where the AV is used to detect and understand the process underlying a signal issued from different voltage measurements. However, there are many other applications for which the AV is of interest H. Xu is a PhD student, Geneva School of Economics and Management, University of Geneva, 1211, Switzerland (E-mail: [email protected]). S. Guerrier is Assistant Professor, Department of Statistics, Pennsylvania State University, PA, 16801, USA. R. Molinari is Visiting Assistant Professor, Department of Statistics & Applied Probability, University of California, Santa Barbara, CA, 93117, USA. Y. Zhang is a graduate student, Department of Statistics, Pennsylvania State University, PA, 16801, USA. such as the detection of noise terms characterising inertial sensors [see 2] and many others [see 5, for an overview]. However, although the AV is extremely useful in the above settings, it is not known how it behaves when in the presence of other types of processes and whether it is able to distinguish between them. In this paper we intend to investigate the form of the AV for a particular class of processes which includes all those processes that have a constant mean but have a time-varying variance-covariance structure such as, for example, the Generalized AutoRegressive Conditional Heteroscedasticity (GARCH) models [see 1], while processes with specific forms of mean or higher-order non-stationarity are not considered since they can either be dealt with through statistical regression techniques or simply cannot be detected by the AV. In particular, we focus on those processes which are characterized by a dependence structure by blocks since they are common in settings such as longitudinal studies or sensor calibration for navigation engineering. In the latter cases, the AV is often approximated by that of other known stationary processes such as, for example, the bias-instability process whose AV is often approximated by that of a firstorder autoregressive process [see, for example, 7]. Moreover, it is not clear whether the AV can actually help to distinguish between these processes and those processes for which its form is currently known. The latter aspect is of particular relevance since it could lead to an erroneous interpretation of the observed process, for example assuming stationarity when this is not the case and reaching false conclusions. In order to deal with the above mentioned non-stationary processes, this paper intends to study the theoretical form of the AV when the covariance structure is non-stationary. The consequent advantage of this study is that, by considering the varying covariance structure in the AV definition, it extends the applicability of those approaches that make use of the AV and raises awareness on its limitations, and inappropriate interpretation, in distinguishing and identifying these processes from stationary ones. With this in mind, Section II briefly defines the AV and describes its theoretical form for those processes which have been considered up to now. Section III introduces the new theoretical form of both the overlapping and non-overlapping AV for processes whose covariance structure is non-stationary and shows how the form of the AV for stationary processes is a special case of this new form. In the same section, three case studies are presented which highlight the importance of these findings in order to better interpret processes through the AV. Finally Section IV concludes. 2 II. OVERVIEW OF THE A LLAN VARIANCE γ(h) = cov(Xt+h , Xt ), mean to estimate different kind of processes [see for example 4]. However, there are many commonly encountered processes whose AV is not known and it is unclear to what point these can actually be distinguished from stationary processes. The next section delivers a more general form of the AV which includes these processes and studies if this quantity can actually be helpful in detecting them. which depends solely on h, the distance between observations, 2 with σX = γ(0) being the process variance. We can consequently define the autocorrelation function as III. A LLAN VARIANCE FOR C ONSTANT-M EAN N ON -S TATIONARY P ROCESSES To introduce the AV, let us first define (Xt )t=1,...,T as a weakly stationary, discrete time and regularly spaced stochastic process with a constant mean µ (i.e. E[Xt ] = µ) and an autocovariance function defined as follows ρ(h) = γ(h) . γ(0) We consider the AV computed at dyadic scales (τ ) starting from local averages of the process which can be denoted as n (n) X̄t ≡ 1X Xt−n+i , n i=1 (1) where n ∈ {x ∈ N : 1 ≤ x < log2 (T ) − 1} therefore determines the number of consecutive observations considered for the average. If the process has constant mean µ, this implies (n) that X̄t also has the same mean and, based on these averages and following [6], the maximum-overlapping AV (MOAV) T 2 1 X (n) (n) E X̄k − X̄k−n . (2) AVarn (Xt ) ≡ 2m? As underlined in the previous sections, the AV is particularly useful for measuring uncertainty in non-stationary processes, especially when these have infinite variance. Nevertheless, there are other forms of non-stationarity for which the properties of the AV are unknown and these consist in those processes (Xt ) with a constant mean µ (independent of time) but a non-stationary covariance structure. This implies that the covariance function between observations at distance h is also a function of time t and can therefore be denoted as γ(h, t). This type of process is very common in different areas going from engineering [see 2] to economics [see 3]. To study the theoretical form of the AV for this class of (n) processes, let us first define Xt as being the following vector of n consecutive observations starting at t − n + 1 (n) Xt T ≡ [Xt−n+1 · · · Xt ] . k=2n ? where m = T − 2n + 1 and whose corresponding estimator is given by [ n (Xt ) = AVar T 2 1 X (n) (n) x̄k − x̄k−n . ? 2m (3) k=2n (n) (n) where x̄t denotes the sample equivalent of X̄t based on a realization of the process (Xt ). Another version of the AV is the non-overlapping AV (NOAV) whose estimator however is not statistically as efficient as that of the MOAV (see App. B for more details). Keeping in mind the above definitions of the MOAV, [8] delivered a general theoretical form of this quantity when applied to weakly stationary processes which is given by 2 σX AVarn (Xt ) = n [1 − ρ(n)] (4) n2 n−1 X + i [2ρ(n − i) − ρ(i) − ρ(2n − i)] . i=1 Based on the above equation, the exact form of the AV for different stationary processes, such as the general class of AutoRegressive Moving Average (ARMA) models, can be derived. Moreover, [8] provided the theoretical AV for nonstationary processes such as the random walk and ARFIMA models for which the AV, as mentioned earlier, represents a better measure of uncertainty compared to other methods. Using the known theoretical forms of the AV, it is therefore possible to detect and distinguish different processes based on the pattern of their AV. Due to this, this quantity (or similar quantities such as the Haar wavelet variance) can be used as a which contains the observations used to build the average in (1). Using the above vector, for t = n, ..., T , we can then define the matrix (n) (n) Σt ≡ var Xt , (5) and, for t = 2n, ..., T , we define the matrix (n) (n) (n) , Γt ≡ cov Xt−n , Xt (6) These matrices represent the covariance matrices of the obser(n) vations contained in each consecutive average. Indeed, Σt represents the covariance matrix of the observations within the (n) average X̄t which is used in definitions (D-1) and (2) while (n) Γt represents the cross-covariance between these two sets of observations. A visual representation of these quantities is given in App. A. In this section we will only consider the non-stationary MOAV while the form of the non-stationary NOAV is given in App. B. Based on the above matrices, we can also define different quantities according to the matrix of reference and the lags between observations. More specifically, let us first consider the case in which we are interested in lags h such that 0 ≤ h < n. Because of the overlapping nature of the AV, the observations at these lags can belong to the sets of (n) (n) observations within both the matrix Σt and the matrix Γt (n) and, for these sets of observations within the matrix Σt , we can define the following quantity γ e(h) ≡ T n−h−1 X X 1 cov(Xt−n−s−h , Xt−n−s ) ? 2m (n − h) t=2n s=0 + cov(Xt−s−h , Xt−s ). 3 If, however, the observations at the considered lags are among (n) the set of observations only within the matrix Γt , we define the quantity below T h 1 XX γ e (h) ≡ ? cov (Xt−n+s−h , Xt−n+s ) . m h t=2n s=1 ∗ Finally, when considering lags h such that n ≤ h ≤ 2n − 1, the set of observations at these lags can only be considered (n) within the matrix Γt and, for this final case, we define the quantity γ e(h) ≡ T 2n−h−1 X X 1 cov (Xt−s−h , Xt−s ) . ? m (2n − h) t=2n s=0 The above definitions can be seen as generalized definitions of the autocovariance which consists in the average autocovariance for a given lag h. Because of this, it must be underlined that these definitions are not at all equivalent to the covariance function γ(h, t) but correspond to an average of this function over all times t. Having specified this, we can now provide the following lemma. L EMMA 1: The non-stationary MOAV is given by ( " Pn−1 1 ? ? e(0) + 2 e(h) AVarn = 2m? n2 2nm γ h=1 2m (n − h) γ #) −m? h γ e∗ (h) − P2n−1 h=n m? (2n − h) γ e(h) . The proof of this lemma is given in App. C. Considering this expression, an aspect that must be underlined is that the definitions of the functions γ e(h) and γ e∗ (h) given earlier simplify to the autocovariance function γ(h) when dealing with a weakly stationary process. In the latter case, the form of the non-stationary AV consequently reduces to the expression in (4) for which a more detailed discussion is given in App. ??. As a final note to this result it should also be highlighted that, in some of the considered non-stationary cases, the estimators of the MOAV defined in (3) and of the NOAV (see App. B) do not necessarily have the same expectation. Having underlined these points and with the general definition of the AV given in Lemma 1, we can now investigate its properties when assuming the process of interest is within the class of processes treated in this paper. The next sections report some simulation studies regarding some of these processes in attempt to understand also whether the AV is a useful quantity to detect them and distinguish them from weakly stationary processes. In all cases, the simulated process is of length T = 1, 000 (except for the bias-instability process where T = 2, 500) and is simulated 50 times. The estimated AV is represented in plots along with the theoretical stationary and non-stationary forms of the AV in order to understand its behaviour under these different assumptions. 1) Non-Stationary White Noise: The first process we study is the non-stationary white noise by which we intend, without loss of generality, all those zero-mean processes whose variance changes with time. The evolution of the variance in time can either be completely random or can follow a specific Fig. 1. Logarithm of the MOAV of the non-stationary white noise process for scales τ = 2n . Estimated MOAV (light-blue lines); theoretical non-stationary MOAV (black line with dots) and theoretical stationary AV based on the average variance (red line with triangles). parametric model such as, for example, a GARCH process. The goal of studying the AV for these types of processes is to understand whether it is able to detect such a structure in a time series and if it can be distinguished from a stationary white noise process. For this purpose, the true non-stationary process considered in the simulation study is generated from the following model Xt ∼ N (0, σt2 ), where σt2 = t, with t = 1, . . . , T . The theoretical stationary form is based on the average of the variances used to simulate the processes (i.e. σ̄ 2 = (1+T )/2 in this example). Fig. 1 represents the estimated AVs along with the theoretical forms (stationary and non-stationary). In this case, it can be seen how both theoretical forms correspond and the estimated AVs closely follow these quantities. This example confirms that the AV is therefore unable to distinguish between a stationary white noise process and a white noise process whose secondorder behaviour is non-stationary. 2) Bias-Instability: The bias-instability process is a commonly known process in the engineering domain, specifically for inertial sensor calibration for navigation. The characteristic of this process is that it consists in different concatenated sequences (blocks) where, within each block, the realization of a random variable is repeated (i.e. constant). More formally, let bi , i = 1, . . . , B, represent the set of time indices belong iid to the ith block within the time series, and let Ci ∼ N (0, σ 2 ). We can then define this process as Xt = ci if t ∈ bi . One realization of the bias-instability process is illustrated in the top panel Fig. 2 where the length of block bi is 10 for all i = 1, . . . , B, and B = 250. Since the theoretical form of the AV for this process is not known exactly, it is often approximated by the AV of a First-Order AutoRegressive (AR1) process. Although this approximation can be useful, it is nevertheless still an approximation and, using the form given in Lemma 1, we can now obtain a theoretical form for the AV of this process which is represented in the bottom panel of Fig. 4 Fig. 2. Top: Realization of the bias-instability process with σ 2 = 1 and the length of block bi = 10, ∀i = 1, . . . , B, and B = 250. Bottom: Logarithm of the MOAV of the bias-instability process for scales τ = 2n . Estimated MOAV (light-blue lines); theoretical non-stationary MOAV (black line with dots) and theoretical stationary MOAV of an AR1 approximating the biasinstability MOAV (red line with triangles). Fig. 3. Top: Realization of the block-structure autoregressive process with φ = 0.9, σ 2 = 1 and the length of block bi = 10, ∀i = 1, . . . , B, and B = 100. Bottom: Logarithm of the MOAV of the block-structure firstorder autoregressive process for scales τ = 2n . Estimated MOAV (light-blue lines); theoretical non-stationary MOAV (black line with dots) and theoretical stationary MOAV assuming no block structure (red line with triangles). process 2. Indeed the latter plot shows that the estimated AVs closely follow the theoretical non-stationary form given earlier. The red line represents the AV of a stationary AR1 process which is supposed to approximate the true AV of bias-instability. The latter is the result of the averaging of the theoretical AV for a stationary AR1 process estimated via maximum-likelihood on each of the simulated processes. It is clear how, although close over some scales, this approximation is not good enough when considering the logarithmic representation of the AV. Therefore, knowing the exact form of the AV for this process would allow to better interpret the signals characterised by bias-instability. 3) Block-Structure Autoregressive Processes: As a final example we consider a block-structure AR1 process. Similarly to the bias-instability process, within this paper, we define a block-structure process as a process whose parameters are fixed but is made by concatenated time periods (blocks) where observations within each block are generated independently from those in the other blocks. An example is given by the settings of longitudinal studies in which each subject can be measured over time and, although the subjects are independent from each other, these measurements can be explained by an autocorrelated process within each subject. To define this (i) process formally, let Xt ∼ Fθ denote the following AR1 (i) Xt (i) = φXt−1 + t with parameter vector θ = [φ σ 2 ]T where φ ∈ (−1, 1) and iid t ∼ N (0, σ 2 ). If again we let bi denote the ith block, then the block-structure AR1 process can be defined as (i) Xt = Xt (i) if t ∈ bi . (j) where Xt is independent of Xt , ∀ i 6= j. By defining φ = 0.9 and σ 2 = 1 for the simulation study, the top panel of Fig. 3 shows a realization of this process while the bottom panel of Fig. 3 illustrates the results of the simulations for this particular process. As for bias-instability, it can be observed how the stationary form of the AV (that does not consider the block structure) is not close to the estimated AVs while the non-stationary form provided in this paper adequately represents this process and can therefore allow to distinguish between a stationary autoregressive process and a block-structure one. IV. C ONCLUSIONS Within this paper we wanted to underline an issue concerning the AV which had not yet been studied. Indeed, the behaviour of the AV in commonly occurring settings where the covariance structure of the processes is non-stationary was 5 unknown and, in many cases, was either ignored or dealt with through approximations. The consequence of the latter approaches would probably consist in erroneous interpretations and conclusions drawn from an AV analysis. For this reason, this paper studied the form of the AV for this class of processes thereby generalizing its form also for weakly stationary processes. Based on this, several examples were provided in which the properties of the AV were studied, highlighting its ability to detect these processes and to eventually distinguish them from stationary ones, making researchers and practitioners more aware of issues related to the interpretation and use of this quantity in more general and common settings. Γi Γ 2n Σn Σ 2n Σ i− ΓT n ΣT Σi −n Γi Γ 2n ΣT ΓT trix ce Ma an i var Co R EFERENCES [1] Bollerslev, T.: Generalized autoregressive conditional heteroskedasticity. Journal of econometrics 31(3), 307–327 (1986) [2] El-Sheimy, N., Hou, H., Niu, X.: Analysis and modeling of inertial sensors using allan variance. IEEE Transactions on instrumentation and measurement 57(1), 140–149 (2008) [3] Gallegati, M., Semmler, W.: Wavelet applications in economics and finance, vol. 20. Springer (2014) [4] Guerrier, S., Skaloud, J., Stebler, Y., Victoria-Feser, M.P.: Wavelet-variance-based estimation for composite stochastic processes. Journal of the American Statistical Association 108(503), 1021–1030 (2013) [5] Percival, D.B.: A wavelet perspective on the allan variance. IEEE transactions on ultrasonics, ferroelectrics, and frequency control 63(4), 538–554 (2016) [6] Percival, D.B., Guttorp, P.: Long-memory processes, the allan variance and wavelets. Wavelets in geophysics 4, 325–344 (1994) [7] Unsal, D., Demirbas, K.: Estimation of deterministic and stochastic imu error parameters. In: Position Location and Navigation Symposium (PLANS), 2012 IEEE/ION, pp. 862–868. IEEE (2012) [8] Zhang, N.F.: Allan variance of time series models for measurement data. Metrologia 45(5), 549 (2008) A PPENDIX A G RAPHICAL ILLUSTRATION OF MOAV To graphically illustrate the quantities defined in Section III, Fig. 4 represents the true covariance matrix for a given process and highlights how the AV is related to this matrix by overlapping square matrices along the diagonal, each of which is composed of the quantities defined in Eq. (5) and Eq. (6). (n) Fig. 4. Graphical illustration of matrices Σt k=1 This estimator is less efficient than the MOAV, mainly because it is based on fewer averages and therefore on a smaller sample size. To define the theoretical form of the NOAV for the non(n) stationary processes of interest, we first define the vector Xj of n consecutive observations starting at (j − 1)n + 1, i.e. T (n) Xj ≡ X(j−1)n+1 · · · Xjn . Using the above, for k = 1, ..., m, we define the matrices (n) (n) (n) Σ2k , Σ2k−1 and Γk as follows: (n) (n) (n) (n) Σ2k ≡ var X2k , Σ2k−1 ≡ var X2k−1 (n) (n) (n) and Γk ≡ cov X2k−1 , X2k . As in Appendix A, the above matrices are graphically represented in Fig. 5 where, as opposed to the MOAV, these matrices do not overlap along the diagonal of the covariance (n) (n) (n) matrix of the process. We then let σ̄2k , σ̄2k−1 and γ̄k (n) (n) (n) denote the averages of the matrices Σ2k , Σ2k−1 and Γk , respectively, i.e. n n 1 X X (n) (n) σ̄2k ≡ 2 Σ2k , n i=1 j=1 i,j (n) A PPENDIX B T HEORETICAL FORM OF THE NOAV FOR N ON -S TATIONARY P ROCESSES k=1 (n) γ̄k (D-1) for the MOAV. T where m = b 2n c. The corresponding estimator for this quantity is given by m 2 X (n) (n) ] n (Xt ) = 1 AVar x̄2k − x̄2k−1 . 2m σ̄2k−1 ≡ The non-overlapping AV (NOAV) is defined as: m 2 1 X (n) (n) E X̄2k − X̄2k−1 , AVarn (Xt ) ≡ 2m (n) and Γt n n 1 X X (n) Σ2k−1 , n2 i=1 j=1 i,j n n 1 X X (n) ≡ 2 Γk . n i=1 j=1 i,j We further define σ̄ (n) and γ̄ (n) as follows: m m i 1 X h (n) 1 X (n) (n) σ̄ (n) ≡ σ̄2k + σ̄2k−1 and γ̄ (n) ≡ γ̄k . 2m m k=1 k=1 6 L EMMA 2: We define the Non-stationary NOAV as ( " Pn−1 1 AVarn = 2mn2 2mn γ e(h) e(0) + 2 h=1 2m(n − h) γ Γ1 Σ1 Σ2 Γ1 Σ 2i− Γi 1 Σ 2i Γi Σ 2m #) Γm −1 ∗ Σ 2m −mh γ e (h) − Γm ce h=n m(2n − h) γ e(h) . Proof of Lemma 2: The proof of Lemma 2 is direct from the above definitions. Indeed, we have 2 (n) (n) (n) (n) E X̄2k − X̄2k−1 = var X̄2k + var X̄2k−1 (n) (n) −2 cov X̄2k−1 , X̄2k trix an i var Co P2n−1 Ma (n) (n) (n) = σ̄2k + σ̄2k−1 − 2γ̄k . (n) (n) (n) Fig. 5. Graphical illustration of matrices Σ2k , Σ2k−1 and Γk NOAV. for the Then, using Eq. (D-1), we obtain 2 Pm (n) (n) 1 E X̄ − X̄ AVarn = 2m k=1 2k 2k−1 1 2m Pm (n) (n) (n) σ̄ + σ̄ − 2γ̄ 2k(n) 2k−1 (n) k 1 2m σ̄ − 2m γ̄ = ( 2m " Pn−1 1 2mn γ e(0) + 2 = 2mn e(h) 2 h=1 2m(n − h) γ = k=1 #) Based on the earlier defined matrices, as for the MOAV, we can also define different quantities according to the matrix of reference and the lags between observations. More specifically, let us first consider the case in which we are interested in lags h such that 0 ≤ h < n. The observations at these lags can belong to the sets of observations within the matrix (n) (n) (n) Σ2k , Σ2k−1 and Γk and, for these sets of observations (n) (n) within matrices Σ2k and Σ2k−1 , we can define the following quantity 2m n−h XX 1 γ e(h) ≡ cov X(k−1)n+s , X(k−1)n+s+h . 2m(n − h) s=1 k=1 If, however, the observations at the considered lags are among (n) the set of observations only within the matrix Γk , we define the quantity below m γ e∗ (h) ≡ h 1 XX cov X(2k−1)n+s−h , X(2k−1)n+s . mh s=1 k=1 Finally, when considering lags h such that n ≤ h ≤ 2n − 1, the set of observations at these lags can only be considered (n) within the matrix Γt and, for this final case, we define the quantity ∗ −mh γ e (h) − P2n−1 h=n m(2n − h) γ e(h) , which concludes the proof. A PPENDIX C (n) Proof of Lemma 1: In order to prove Lemma 1 let σ̄t (n) (n) (n) and γ̄t denote the averages of the matrices Σt and Γt , respectively, i.e. n n 1 X X (n) (n) Σt , σ̄t ≡ 2 n i=1 j=1 i,j (n) γ̄t n n 1 X X (n) ≡ 2 Γt . n i=1 j=1 i,j Further, we define σ̄ (n) and γ̄ (n) as follows: σ̄ (n) ≡ T X 1 (n) (n) σ̄ + σ̄t−n 2(T − 2n + 1) t=2n t γ̄ (n) ≡ T X 1 (n) γ̄ . T − 2n + 1 t=2n t m 2n−h X X 1 cov X(k−1)n+s , X(k−1)n+s+h . m(2n − h) Based on these definitions we have k=1 s=1 2 (n) (n) (n) (n) As for the MOAV case, the above definitions can be seen as E X̄k − X̄k−n = var X̄k + var X̄k−n generalized definitions of the autocovariance which consists in (n) (n) the average autocovariance for a given lag h. Using the above −2 cov X̄k , X̄k−n notations and definitions, we can provide the following result. (n) (n) (n) = σ̄k + σ̄k−n − 2γ̄k . γ e(h) ≡ 7 Then, using Eq. (2), we obtain AVarn 1 2m? = = 1 2m? = ( = 1 2m? n2 PT k=2n E (n) X̄k (n) k=2n σ̄k + ? (n) PT 1 2m∗ (n) − X̄k−n (n) 2 (n) σ̄k−n − 2γ̄k − 2m? γ̄ (n) 2m σ̄ " ? 2nm γ e(0) + 2 Pn−1 h=1 2m? (n − h) γ e(h) #) ? ? −m h γ e (h) − which concludes the proof. P2n−1 h=n ? m (2n − h) γ e(h) ,	10
ACCEPTED BY IEEE TRANSACTIONS ON CYBERNETICS 1 Learning Domain-Invariant Subspace using Domain Features and Independence Maximization arXiv:1603.04535v2 [cs.CV] 22 Jun 2017 Ke Yan, Lu Kou, and David Zhang, Fellow, IEEE Abstract—Domain adaptation algorithms are useful when the distributions of the training and the test data are different. In this paper, we focus on the problem of instrumental variation and time-varying drift in the field of sensors and measurement, which can be viewed as discrete and continuous distributional change in the feature space. We propose maximum independence domain adaptation (MIDA) and semi-supervised MIDA (SMIDA) to address this problem. Domain features are first defined to describe the background information of a sample, such as the device label and acquisition time. Then, MIDA learns a subspace which has maximum independence with the domain features, so as to reduce the inter-domain discrepancy in distributions. A feature augmentation strategy is also designed to project samples according to their backgrounds so as to improve the adaptation. The proposed algorithms are flexible and fast. Their effectiveness is verified by experiments on synthetic datasets and four realworld ones on sensors, measurement, and computer vision. They can greatly enhance the practicability of sensor systems, as well as extend the application scope of existing domain adaptation algorithms by uniformly handling different kinds of distributional change. Index Terms—Dimensionality reduction, domain adaptation, drift correction, Hilbert-Schmidt independence criterion, machine olfaction, transfer learning I. I NTRODUCTION I N many real-world machine learning problems, the labeled training data are from a source domain and the test ones are from a target domain. Samples of the two domains are collected under different conditions, thus have different distributions. Labeling samples in the target domain to develop new prediction models is often labor-intensive and time-consuming. Therefore, domain adaptation or transfer learning is needed to improve the performance in the target domain by leveraging unlabeled (and maybe a few labeled) target samples [1]. This topic is receiving increasing attention in recent years due to its broad applications such as computer vision [2], [3], [4] and text classification [5], [6]. It is also important in the field of sensors and measurement. Because of the variations in the The work is partially supported by the GRF fund from the HKSAR Government, the central fund from Hong Kong Polytechnic University, the NSFC fund (61332011, 61272292, 61271344), Shenzhen Fundamental Research fund (JCYJ20150403161923528, JCYJ20140508160910917), and Key Laboratory of Network Oriented Intelligent Computation, Shenzhen, China. K. Yan is with the Department of Electronic Engineering, Graduate School at Shenzhen, Tsinghua University, Shenzhen 518055, China (e-mail: [email protected]). L. Kou is with the Department of Computing, The Hong Kong Polytechnic University, Kowloon, Hong Kong (e-mail: [email protected]). D. Zhang is with the Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen 518055, China, and also with the Department of Computing, Biometrics Research Centre, The Hong Kong Polytechnic University, Kowloon, Hong Kong (e-mail: [email protected]). fabrication of sensors and devices, the responses to the same signal source may not be identical for different instruments, which is known as instrumental variation. Furthermore, the sensing characteristics of the sensors, the operating condition, or even the signal source itself, can change over time, which leads to complex time-varying drift. As a result, the prediction model trained with the samples from the initial device in an earlier time period (source domain) is not suitable for new devices or in a latter time (target domains). A typical application plagued by this problem is machine olfaction, which uses electronic noses (e-noses) and pattern recognition algorithms to predict the type and concentration of odors [7]. The applications of machine olfaction range from agriculture and food to environmental monitoring, robotics, biometrics, and disease analysis [8], [9], [10], [11]. However, owing to the nature of chemical sensors, many e-noses are prone to instrumental variation and time-varying drift mentioned above [12], [13], which greatly hamper their usage in real-world applications. Traditional methods dealing with these two kinds of drift (“drift correction” methods hereinafter) require a set of transfer samples, which are predefined gas samples needed to be collected with each device and in each time period [12], [10], [14], [15]. They are often used to learn regression models to map the features in the target domain to the source domain [10], [14]. Nevertheless, collecting transfer samples repeatedly is a demanding job especially for nonprofessional e-nose users. In such cases, domain adaptation techniques with unlabeled target samples are desirable. An intuitive idea is to reduce the inter-domain discrepancy in the feature level, i.e. to learn domain-invariant feature representation [5], [16], [17], [3], [18], [19], [20], [21], [22]. For example, Pan et al. [5] proposed transfer component analysis (TCA), which finds a latent feature space that minimizes the distributional difference of two domains in the sense of maximum mean discrepancy. More related methods will be introduced in Section II-A. When applied to drift correction, however, existing domain adaptation algorithms are faced with two difficulties. First, they are designed to handle discrete source and target domains. In time-varying drift, however, samples come in a stream, so the change in data distribution is often continuous. One solution is to split data into several batches, but it will lose the temporal order information. Second, because of the variation in the sensitivity of chemical sensors, the same signal in different conditions may indicate different concepts. In other words, the conditional probability P (Y \|X) may change for samples with different backgrounds, where “background” means when and with which device a sample was collected. Methods like ACCEPTED BY IEEE TRANSACTIONS ON CYBERNETICS TCA project all samples to a common subspace, hence the samples with similar appearance but different concepts cannot be distinguished. In this paper, we present a simple yet effective algorithm called maximum independence domain adaptation (MIDA). The algorithm first defines “domain features” for each sample to describe its background. Then, it finds a latent feature space in which the samples and their domain features are maximally independent in the sense of Hilbert-Schmidt independence criterion (HSIC) [23]. Thus, the discrete and continuous change in distribution can be handled uniformly. In order to project samples according to their backgrounds, feature augmentation is performed by concatenating the original feature vector with the domain features. We also propose semi-supervised MIDA (SMIDA) to exploit the label information with HSIC. MIDA and SMIDA are both very flexible. (1) They can be applied in situations with single or multiple source or target domains thanks to the use of domain features. In fact, the notion “domain” has been extended to “background” which is more informative. (2) Although they are designed for unsupervised domain adaptation problems (no labeled sample in target domains), the proposed methods naturally allow both unlabeled and labeled samples in any domains, thus can be applied in semi-supervised (both unlabeled and labeled samples in target domains) and supervised (only labeled samples in target domains) problems as well. (3) The label information can be either discrete (binary- or multi-class classification) or continuous (regression). To illustrate the effect of our algorithms, we first evaluate them on several synthetic datasets. Then, drift correction experiments are performed on two e-nose datasets and one spectroscopy dataset. Note that spectrometers suffer the same instrumental variation problem as e-noses [24]. Finally, a domain adaptation experiment is conducted on a well-known object recognition benchmark: Office+Caltech [25]. Results confirm the effectiveness of the proposed algorithms. The rest of the paper is organized as follows. Related work on unsupervised domain adaptation and HSIC is briefly reviewed in Section II. Section III describes domain features, MIDA, and SMIDA in detail. The experimental configurations and results are presented in Section IV, along with some discussions. Section V concludes the paper. 2 information between all samples and their binary domain labels, which can be viewed as a primitive version of the domain features used in this paper. They also minimized the negated mutual information between the target samples and their cluster labels to reduce the expected classification error. The low-rank transfer subspace learning (LTSL) algorithm presented in [19] is a reconstruction guided knowledge transfer method. It aligns source and target data by representing each target sample with some local combination of source samples in the projected subspace. The label and geometry information can be retained by embedding different subspace learning methods into LTSL. Another class of methods first project the source and the target data into separate subspaces, and then build connections between them [17], [25], [26], [3]. Fernando et al. [17] utilized a transformation matrix to map the source subspace to the target one, where a subspace was represented by eigenvectors of PCA. The geodesic flow kernel (GFK) method [25] measures the geometric distance between two different domains in a Grassmann manifold by constructing a geodesic flow. An infinite number of subspaces are combined along the flow in order to model a smooth change from the source to the target domain. Liu et al. [26] adapted GFK to correct timevarying drift of e-noses. A sample stream is first split into batches according to the acquisition time. The first and the latest batches (domains) are then connected through every intermediate batch using GFK. Another improvement of GFK is domain adaptation by shifting covariance (DASC) [3]. Observing that modeling one domain as a subspace is not sufficient to represent the difference of distributions, DASC characterizes domains as covariance matrices and interpolates them along the geodesic to bridge the domains. B. Hilbert-Schmidt Independence Criterion (HSIC) HSIC is used as a convenient method to measure the dependence between two sample sets X and Y . Let kx and ky be two kernel functions associated with RKHSs F and G, respectively. pxy is the joint distribution. HSIC is defined as the square of the Hilbert-Schmidt norm of the cross-covariance operator Cxy [23]: HSIC(pxy , F, G) = kCxy k2HS =Exx0 yy0 [kx (x, x0 )ky (y, y 0 )] + Exx0 [kx (x, x0 )]Eyy0 [ky (y, y 0 )] II. R ELATED W ORK A. Unsupervised Domain Adaptation Two good surveys on domain adaptation can be found in [1] and [2]. In this section, we focus on typical methods that extract domain-invariant features. In order to reduce the inter-domain discrepancy while preserving useful information, researchers have developed many strategies. Some algorithms project all samples to a common latent space [5], [16], [19]. Transfer component analysis (TCA) [5] tries to learn transfer components across domains in a reproducing kernel Hilbert space (RKHS) using maximum mean discrepancy. It is further extended to semi-supervised TCA (SSTCA) to encode label information and preserve local geometry of the manifold. Shi et al. [16] measured domain difference by the mutual − 2Exy [Ex0 [kx (x, x0 )]Ey0 [ky (y, y 0 )]]. Here Exx0 yy0 is the expectation over independent pairs (x, y) and (x0 , y 0 ) drawn from pxy . It can be proved that with characteristic kernels kx and ky , HSIC(pxy , F, G) is zero if and only if x and y are independent [27]. A large HSIC suggests strong dependence with respect to the choice of kernels. HSIC has a biased empirical estimate. Suppose Z = X × Y = {(x1 , y1 ), . . . , (xn , yn )}, Kx , Ky ∈ Rn×n are the kernel matrices of X and Y , respectively, then [23]: HSIC(Z, F, G) = (n − 1)−2 tr(Kx HKy H), (1) n×n where H = I − n−1 1n 1T is the centering matrix. n ∈R Due to its simplicity and power, HSIC has been adopted for feature extraction [28], [5], [29] and feature selection ACCEPTED BY IEEE TRANSACTIONS ON CYBERNETICS 3 [27]. Researchers typically use it to maximize the dependence between the extracted/selected features and the label. However, to our knowledge, it has not been utilized in domain adaptation to reduce the dependence between the extracted features and the domain features. III. P ROPOSED M ETHOD A. Domain Feature We aim to reduce the dependence between the extracted features and the background information. A sample’s background information should (1) naturally exist, thus can be easily obtained; (2) have different distributions in training and test samples; (3) correlate with the distribution of the original features. The domain label (which domain a sample belongs) in common domain adaptation problems is an example of such information. According to these characteristics, the information clearly interferes the testing performance of a prediction model. Thus, minimizing the aforementioned dependence is desirable. First, a group of new features need to be designed to describe the background information. The features are called “domain features”. From the perspective of drift correction, there are two main types of background information: the device label (with which device the sample was collected) and the acquisition time (when the sample was collected). We can actually encode more information such as the place of collection, the operation condition, and so on, which will be useful in other domain adaptation problems. Formally, if we only consider the instrumental variation, the following one-hot coding scheme can be used. Suppose there are ndev devices, which result in ndev different but related domains. The domain feature vector is thus d ∈ Rndev , where dp = 1 if the sample is from the pth device and 0 otherwise. If the time-varying drift is also considered, the acquisition time can be further added. If a sample was collected from the pth device at time t, then d ∈ R2ndev , where   1, q = 2p − 1, (2) dq = t, q = 2p,   0, otherwise. According to (1), the kernel matrix Kd of the domain features needs to be computed for HSIC. We apply the linear kernel. Suppose D = [d1 , . . . , dn ] ∈ Rmd ×n , md is the dimension of a domain feature vector. Then Kd = DT D. (3) Note that in traditional domain adaptation problems with several discrete domains, the one-hot coding scheme can be applied to construct domain features, because the problems are similar to instrumental variation. B. Feature Augmentation Feature augmentation is used in this paper to learn background-specific subspaces. In [30], the author proposed a feature augmentation strategy for domain adaptation by replicating the original features. However, this strategy requires that data lie in discrete domains and cannot deal with timevarying drift. We propose a more general and efficient feature augmentation strategy: concatenating the original features and the domain features, i.e. x x̂ = ∈ Rm+md . (4) d The role of this strategy can be demonstrated through a linear dimensionality reduction example. Suppose a projection matrix W ∈ R(m+md )×h has been learned for the augmented feature vector. h is the dimension of the subspace. W has two Wx parts: W = , Wx ∈ Rm×h , Wd ∈ Rmd ×h . The embedWd ding of x̂ can be expressed as W T x̂ = WxT x + WdT d ∈ Rh , which means that a background-specific bias (WdT d)i has been added to each dimension i of the embedding. From another perspective, the feature augmentation strategy maps the samples to an augmented space with higher dimension before projecting them to a subspace. It will be easier to find a projection direction in the augmented space to align the samples well in the subspace. Take machine olfaction for example, there are situations when the conditional probability P (Y \|X) changes along with the background. For instance, the sensitivity of chemical sensors often decays over time. A signal that indicates low concentration in an earlier time actually suggests high concentration in a later time. In such cases, feature augmentation is important, because it allows samples with similar appearance but different concepts to be treated differently by the background-specific bias. The strategy also helps to align the domains better in each projected dimension. Its effect will be illustrated on several synthetic datasets in Section IV-A and further analyzed in the complementary materials. C. Maximum Independence Domain Adaptation (MIDA) In this section, we introduce the formulation of MIDA in detail. Suppose X ∈ Rm×n is the matrix of n samples. The training and the test samples are pooled together. More importantly, we do not have to explicitly differentiate which domain a sample is from. The feature vectors have been augmented, but we use the notations X and m instead of X̂ and m + md for brevity. A linear or nonlinear mapping function Φ can be used to map X to a new space. Based on the kernel trick, we need not know the exact form of Φ, but the inner product of Φ(X) can be represented by the kernel matrix Kx = Φ(X)T Φ(X). Then, a projection matrix W̃ is applied to project Φ(X) to a subspace with dimension h, leading to the projected samples Z = W̃ T Φ(X) ∈ Rh×n . Similar to other kernel dimensionality reduction algorithms [31], [32], the key idea is to express each projection direction as a linear combination of all samples in the space, namely W̃ = Φ(X)W . W ∈ Rn×h is the projection matrix to be actually learned. Thus, the projected samples are Z = W T Φ(X)T Φ(X) = W T Kx . (5) Intuitively, if the projected features are independent of the domain features, then we cannot distinguish the background ACCEPTED BY IEEE TRANSACTIONS ON CYBERNETICS 4 of a sample by its projected features, suggesting that the interdomain discrepancy is diminished in the subspace. Therefore, after omitting the scaling factor in (1), we get the expression to be minimized: tr(Kz HKd H) = tr(Kx W W T Kx HKd H), where Kz is the kernel matrix of Z. In domain adaptation, the goal is not only minimizing the difference of distributions, but also preserving important properties of data, such as the variance [5]. It can be achieved by maximizing the trace of the covariance matrix of the project samples. The covariance matrix is cov(Z) = cov(W T Kx ) = W T Kx HKx W, (6) where H = I − n−1 1n 1T n is the same as that in (1). An orthonormal constraint is further added on W . The learning problem then becomes max W − tr(W T Kx HKd HKx W ) + µ tr(W T Kx HKx W ), s.t. W T W = I, (7) where µ > 0 is a trade-off hyper-parameter. Using the Lagrangian multiplier method, we can find that W is the eigenvectors of Kx (−HKd H + µH)Kx corresponding to the h largest eigenvalues. Note that a conventional constraint is requiring W̃ to be orthonormal as in [29], which will lead to a generalized eigenvector problem. However, we find that this strategy is inferior to the proposed one in both adaptation accuracy and training speed in practice, so it is not used. When computing Kx , a proper kernel function needs to be selected. Common kernel functions include linear (k(x, y) = xT y), polynomial (k(x, y) = (σxT y + 1)d ), Gaussian radial 2 basis function (RBF, k(x, y) = exp( kx−yk 2σ 2 )), and so on. Different kernels indicate different assumptions on the type of dependence in using HSIC [27]. According to [27], the polynomial and RBF kernels map the original features to a higher or infinite dimensional space, thus are able to detect more types of dependence. However, choosing a suitable kernel width parameter (σ) is also important for these more powerful kernels [27]. The maximum mean discrepancy (MMD) criterion is used in TCA [5] to measure the difference of two distributions. Song et al. [27] showed that when HSIC and MMD are both applied to measure the dependence between features and labels in a binary-class classification problem, they are identical up to a constant factor if the label kernel matrix in HSIC is properly designed. However, TCA is feasible only when there are two discrete domains. On the other hand, MIDA can deal with a variety of situations including multiple domains and continuous distributional change. The stationary subspace analysis (SSA) algorithm [33] is able to identify temporally stationary components in multivariate time series. However, SSA only ensures that the mean and covariance of the components are stationary, while they may not be suitable for preserving important properties in data. Concept drift adaptation algorithms [34] are able to correct continuous time-varying drift. However, most of them rely on newly arrived labeled data to update the prediction models, while MIDA works unsupervisedly. D. Semi-supervised MIDA (SMIDA) MIDA aligns samples with different backgrounds without considering the label information. However, if the labels of some samples are known, they can be incorporated into the subspace learning process, which may be beneficial to prediction. Therefore, we extend MIDA to semi-supervised MIDA (SMIDA). Since we do not explicitly differentiate the domain labels of the samples, both unlabeled and labeled samples can exist in any domain. Similar to [28], [5], [29], [27], HSIC is adopted to maximize the dependence between the projected features and the labels. The biggest advantage of this strategy is that all types of labels can be exploited, such as the discrete labels in classification and the continuous ones in regression. The label matrix Y is defined as follows. For c-class classification problems, the one-hot coding scheme can be used, i.e. Y ∈ Rc×n , yi,j = 1 if xi is labeled and belongs to the jth class; 0 otherwise. For regression problems, the target values can be centered first. Then, Y ∈ R1×n , yi equals to the target value of xi if it is labeled; 0 otherwise. The linear kernel function is chosen for the label kernel matrix, i.e. Ky = Y T Y. (8) The objective of SMIDA is max W s.t. tr(W T Kx (−HKd H + µH + γHKy H)Kx W ), W T W = I, (9) where γ > 0 is a trade-off hyper-parameter. Its solution is the eigenvectors of Kx (−HKd H + µH + γHKy H)Kx corresponding to the h largest eigenvalues. The outline of MIDA and SMIDA is summarized in Algorithm III.1. The statements in brackets correspond to those specialized for SMIDA. Algorithm III.1 MIDA [or SMIDA] Input: The matrix of all samples X and their background information; [the labels of some samples]; the kernel function for X; h, µ, [and γ]. Output: The projected samples Z. 1: Construct the domain features according to the background information, e.g. Section III-A. 2: Augment the original features with domain features (4). 3: Compute the kernel matrices Kx , Kd (3), [and Ky (8)]. 4: Obtain W , namely the eigenvectors of Kx (−HKd H + µH)Kx [or Kx (−HKd H + µH + γHKy H)Kx ] corresponding to the h largest eigenvalues. 5: Z = W T Kx . Besides variance and label dependence, another useful property of data is the geometry structure, which can be preserved by manifold regularization (MR) [35]. MR can be conveniently incorporated into SMIDA. In our experiments, adding MR generally increases the accuracy slightly with the cost of three more hyper-parameters. Consequently, it is not adopted in this paper. ACCEPTED BY IEEE TRANSACTIONS ON CYBERNETICS IV. E XPERIMENTS In this section, we first conduct experiments on some synthetic datasets to verify the effect of the proposed methods. Then, drift correction experiments are performed on two enose datasets and a spectroscopy dataset. To show the universality of the proposed methods, we further evaluate them on a visual object recognition dataset. Comparison is made between them and recent unsupervised domain adaptation algorithms that learn domain-invariant features. A. Synthetic Dataset In Fig. 1, TCA [5] and MIDA are compared on a 2D dataset with two discrete domains. The domain labels were used to construct the domain features in MIDA according to the one-hot coding scheme introduced in Section III-A. The similar definition was used in synthetic datasets 3 and 4. For both methods, the linear kernel was used on the original features and the hyper-parameter µ was set to 1. In order to quantitatively assess the effect of domain adaptation, logistic regression models were trained on the labeled source data and tested on the target data. The accuracies are displayed in the caption, showing that the order of performance is MIDA > TCA > original feature. TCA aligns the two domains only on the first projected dimension. However, the two classes have large overlap on that dimension, because the direction for alignment is different from that for discrimination. Incorporating the label information of the source domain (SSTCA) did no help. On the contrary, MIDA can align the two domains well in both projected dimensions, in which the domainspecific bias on the second dimension brought by feature augmentation played a key role. A 3D explanation is included in the supplementary materials. Thus, good accuracy can be obtained by using the two dimensions for classification. In Fig. 2, SSA [33] and MIDA are compared on a 2D dataset with continuous distributional change, which resembles timevarying drift in machine olfaction. Samples in both classes drift to the upper right. The chronological order of the samples was used to construct the domain features in MIDA, i.e. d = 1 for the first sample, d = 2 for the second sample, etc. The parameter setting of MIDA was the same with that in Fig. 1, whereas the number of stationary components in SSA was set to 1. The classification accuracies were obtained by training a logistic regression model on the first halves of the data in both classes, and testing them on the last halves. SSA succeeds in finding a direction (z1 ) that is free from time-varying drift. However, the two classes cannot be well separated in that direction. In plot (c), the randomly scattered colors suggest that the time-varying drift is totally removed in the subspace. MIDA first mapped the 2D data into a 3D space with the third dimension being time, then projected them to a 2D plane orthogonal to the direction of drift in the 3D space. No label information was used in the last two experiments. If keeping the label dependence in the subspace is a priority, SMIDA can be adopted instead of MIDA. In the 3D synthetic dataset in Fig. 3, the best direction (x3 ) to align the two domains also mixes the two classes, which results in the output of MIDA in plot (b). The labels in the source domain 5 were used when learning the subspace. From plot (c), we can observe that the classes are separated. In fact, class separation can still be found in the third dimension of the space learned by MIDA. However, for the purpose of dimensionality reduction, we generally hope to keep the important information in the first few dimensions. Nonlinear kernels are often applied in machine learning algorithms when data is not linearly separable. Besides, they are also useful in domain adaptation when domains are not linearly “alignable”, as shown in Fig. 4. In plot (a), the inter-domain changes in distributions are different for the two classes. Hence, it is difficult to find a linear projection direction to align the two domains, even with the domainspecific biases of MIDA. Actually, domain-specific rotation matrices are needed. Since the target labels are not available, the rotation matrices cannot be obtained accurately. However, a nonlinear kernel can be used to map the original features to a space with higher dimensions, in which the domains may be linearly alignable. We applied an RBF kernel with width σ = 10. Although the domains are not perfectly aligned in plot (c), the classification model trained in the source domain can be better adapted to the target domain. A comparison on different kernel and kernel parameters on two synthetic datasets is included in the supplementary materials. B. Gas Sensor Array Drift Dataset The gas sensor array drift dataset1 collected by Vergara et al. [36] is dedicated to research in drift correction. A total of 13910 samples were collected by an e-nose with 16 gas sensors over a course of 36 months. There are six different kinds of gases at different concentrations. They were split into 10 batches by the authors according to their acquisition time. Table A in the supplementary material details the dataset. We aim to classify the type of gases, despite their concentrations. Similar to [36], [26], we took the samples in batch 1 as labeled training samples, whereas those in batches 2–10 are unlabeled test ones. This evaluation strategy resembles the situation in real-world applications. In the dataset, each sample is represented by 128 features extracted from the sensors’ response curves [36]. Each feature was first normalized to zero mean and unit variance within each batch. The timevarying drift of the preprocessed features across batches can be visually inspected in Fig. 5. It is obvious that samples in different batches have different distributions. Next, the labeled samples in batch 1 were adopted as the source domain and the unlabeled ones in batch b (b = 2, . . . , 10) as the target domain. The proposed algorithms together with several recent ones were used to learn domain-invariant features based on these samples. Then, a logistic regression model was trained on the source domain and tested on each target one. For multiclass classification, the one-vs-all strategy was utilized. As displayed in Table I, the compared methods include kernel PCA (KPCA), transfer component analysis (TCA), semi-supervised TCA (SSTCA) [5], subspace alignment (SA) 1 http://archive.ics.uci.edu/ml/datasets/ Gas+Sensor+Array+Drift+Dataset+at+Different+Concentrations ACCEPTED BY IEEE TRANSACTIONS ON CYBERNETICS 2 6 Pos. source data 15 Neg. source data Pos. target data 10 Neg. target data 1.5 1 2 1.5 1 5 0.5 0.5 x z2 z2 2 0 0 0 −5 −0.5 −0.5 −10 −1 −1.5 −15 (a) −2 −2 −1 0 x1 1 2 −1 −1.5 (b) −20 −10 −5 0 z1 5 (c) −2 −10 10 −5 0 z1 5 10 Fig. 1. Comparison of TCA and MIDA in a 2D synthetic dataset. Plots (a)-(c) show data in the original space and projected spaces of TCA and MIDA, respectively. The classification accuracies are 53%, 70% (only using the first projected dimension z1 ), and 88%. 16 10 20 14 15 8 12 10 10 6 5 z2 z2 x 2 8 4 0 6 4 −5 2 Pos. old data Neg. old data Pos. new data Neg. new data 2 0 (a) −2 −2 0 2 x1 4 −10 0 −15 (b) −2 −4 6 −2 0 z1 2 (c) −20 −80 4 −60 −40 −20 z1 0 20 40 Fig. 2. Comparison of SSA and MIDA in a 2D synthetic dataset. Plots (a)-(c) show data in the original space, projected spaces of SSA and MIDA, respectively. The chronological order of a sample is indicated by color. The classification accuracies are 55%, 74% (only using the first projected dimension z1 ), and 90%. 3 3 Pos. source data Neg. source data Pos. target data 2 Neg. target data (a) 2 1 5 4 (b) (c) 3 1 0 z2 z2 x3 2 0 0 −1 −1 1 −1 −2 −2 −2 0 2 1.5 2 −0.5 0 0.5 x1 1 x −2 −3 −10 −1 −5 0 −3 −4 5 −2 0 z1 2 4 6 z1 Fig. 3. Comparison of MIDA and SMIDA in a 3D synthetic dataset. Plots (a)-(c) show data in the original space and projected spaces of MIDA and SMIDA, respectively. The classification accuracies are 50%, 55%, and 82%. 4 Pos. source data Neg. source data Pos. target data Neg. target data 3 5 1.9 4 1.8 3 1.7 2 1.6 1 1.5 z2 x 2 1 0 z2 2 0 1.4 −1 1.3 −1 −2 −2 −3 −4 −3 (a) −2 0 2 x 1 4 6 −4 −60 1.2 1.1 (b) −40 −20 0 z 1 20 40 60 1 (c) 9 9.5 10 10.5 z 1 Fig. 4. Comparison of different kernels in a 2D synthetic dataset. Plots (a)-(c) show data in the original space and projected spaces of MIDA with linear and RBF kernels, respectively. The classification accuracies are 50%, 57%, and 87%. ACCEPTED BY IEEE TRANSACTIONS ON CYBERNETICS 7 75 Average classification accuracy (%) 5 PC2 0 −5 Batch 1 Batch 3 Batch 5 Batch 7 Batch 9 −10 −15 −30 −20 −10 0 10 70 65 60 55 50 20 PC1 Fig. 5. Scatter of ethanol (dots) and acetone (plus signs) samples in batches 1,3,5,7,9 in the gas sensor array drift dataset. Samples are projected to a 2D subspace using PCA. Different colors indicate different batches. [17], geodesic flow kernel (GFK) [25], manifold regularization with combination GFK (ML-comGFK) [26], informationtheoretical learning (ITL) [16], structural correspondence learning (SCL) [20], and marginalized stacked denoising autoencoder (mSDA) [21]. For all methods, the hyper-parameters were tuned for the best accuracy. In KPCA, TCA, SSTCA, and the proposed MIDA and SMIDA, the polynomial kernel with degree 2 was used. KPCA learned a subspace based on the union of source and target data. In TCA, SSTCA, MIDA, and SMIDA, eigenvalue decomposition needs to be done on kernel matrices. In order to reduce the computational burden, we randomly chose at most nt samples in each target domain when using these methods, with nt being twice the number of the samples in the source domain. GFK used PCA to generate the subspaces in both source and target domains. The subspace dimension of GFK was determined according to the subspace disagreement measure in [25]. The results of ML-comGFK are copied from [26]. In SCL, the pivot features were binarized before training pivot predictors using logistic regression. We also compared several variants of our methods. In Table I, the notation “(discrete)” means that two discrete domains (source and target) were used in MIDA and SMIDA, which is similar to other compared methods. The domain feature vector of a sample was thus [1, 0]T if it was from the source domain and [0, 1]T if it was from the target. However, this strategy cannot make use of the samples in intermediate batches. An intuitive assumption is that the distributions of adjacent batches should be similar. When adapting the information from batch 1 to b, taking samples from batches 2 to b − 1 into consideration may improve the generalization ability of the learned subspace. Concretely, nt samples were randomly selected from batches 2 to b instead of batch b alone. For each sample, the domain feature was defined as its batch index, which can be viewed as a proxy of its acquisition time. MIDA and SMIDA then maximized the independence between KPCA TCA SSTCA SA ITL MIDA (continuous) SMIDA (continuous) 0 20 40 60 # Projected dimensions (h) 80 100 Fig. 6. Performance comparison on the gas sensor array drift dataset with respect to the subspace dimension h. the learned subspace and the batch indices. The results are labeled as “(continuous)” in Table I. Besides, the accuracies of continuous SMIDA without feature augmentation (no aug.) are also shown. From Table I, we can find that as the batch index increases, the accuracies of all methods generally degrade, which confirms the influence of the time-varying drift. Continuous SMIDA achieves the best average domain adaptation accuracy. The continuous versions of MIDA and SMIDA outperform the discrete versions, proving that the proposed methods can effectively exploit the chronological information of the samples. They also surpass ML-comGFK which uses the samples in intermediate batches to build connections between the source and the target batches. Feature augmentation is important in this dataset, since removing it in continuous SMIDA causes a drop of four percentage points in average accuracy. In Fig. 6, the average classification accuracies with varying subspace dimension are shown. MIDA and SMIDA are better than other methods when more than 30 features are extracted. C. Breath Analysis Dataset As a noninvasive approach, disease screening and monitoring with e-noses is attracting more and more attention [8], [11]. The concentration of some biomarkers in breath has been proved to be related to certain diseases, which makes it possible to analyze a person’s health state with an e-nose conveniently. For example, the concentration of acetone in diabetics’ breath is often higher than that in healthy people [11]. However, the instrumental variation and time-varying drift of e-noses hinder the popularization of this technology in real-world applications. Unsupervised domain adaptation algorithms can be applied to solve this problem. We have collected a breath analysis dataset in years 2014– 2015 using two e-noses of the same model [11]. In this paper, samples of five diseases were selected for experiments, including diabetes, chronical kidney disease (CKD), cardiopathy, lung cancer, and breast cancer. They have been proved to be related to certain breath biomarkers. We performed ACCEPTED BY IEEE TRANSACTIONS ON CYBERNETICS 8 TABLE I C LASSIFICATION ACCURACY (%) ON THE GAS SENSOR ARRAY DRIFT DATASET. B OLD VALUES INDICATE THE BEST RESULTS . Batch 2 3 4 5 6 7 8 9 10 Average 80.47 79.26 69.57 77.16 77.39 64.21 52.04 47.87 48.78 66.30 KPCA 75.88 69.04 49.07 57.87 62.65 52.26 37.07 47.66 49.97 55.72 TCA [5] 82.96 81.97 65.22 76.14 89.09 58.98 49.32 66.17 49.50 68.82 SSTCA [5] 84.57 80.90 80.12 75.63 87.26 66.37 54.76 61.28 54.44 71.70 SA [17] 80.79 80.01 71.43 75.63 78.35 64.68 52.04 48.51 49.58 66.78 GFK [25] 77.41 80.26 71.43 76.14 77.65 64.99 36.39 47.45 48.72 64.49 ML-comGFK [26] 80.25 74.99 78.79 67.41 77.82 71.68 49.96 50.79 53.79 67.28 ITL [16] 76.85 79.45 59.63 96.45 78.00 60.95 49.32 77.02 48.58 69.58 SCL [20] 77.57 82.03 68.32 82.74 77.22 65.18 53.74 48.51 48.08 67.04 mSDA [21] 73.87 79.19 65.84 80.20 76.39 65.90 51.70 48.51 48.92 65.61 MIDA (discrete) 81.03 85.62 60.25 75.63 87.61 62.44 48.30 67.87 48.36 68.57 SMIDA (discrete) 80.47 87.07 65.22 75.63 90.04 59.20 50.00 62.77 44.81 68.36 MIDA (continuous) 84.32 81.59 68.32 75.63 91.74 63.13 78.91 62.34 45.14 72.35 SMIDA (no aug.) 82.23 83.17 67.70 75.13 85.22 61.67 51.02 61.49 54.61 69.14 SMIDA (continuous) 83.68 82.28 73.91 75.63 93.00 63.49 79.25 62.34 45.50 73.23 1 0.8 (a) 0.6 0.4 0.2 0 0 Sensor response (Volt) five binary-class classification tasks to distinguish samples with one disease from the healthy samples. Each sample was represented by the steady state responses of nine gas sensors in the e-nose. When a gas sensor is used to sense a gas sample, its response will reach a steady state in a few minutes. The steady state response has a close relationship with the concentration of the measured gas. Therefore, the 9D feature vector contains most information needed for disease screening. To show the instrumental variation and time-varying drift in the dataset, we draw the steady state responses of two sensors of the CKD samples in Fig. 7. Each data point indicates a breath sample. In plot (a), the sensitivity of the sensor in both devices gradually decayed as time elapsed. In plot (b), the aging effect was so significant that we had to replace the sensors in the two devices with new ones on about day 200. In this case, a signal at 0.3 V will suggest low concentration on day 0 but high concentration on day 150. In addition, the responses in different devices are different (e.g. plot (b), after day 200). The numbers of samples in the six classes (healthy and the five diseases mentioned above) are 125, 431, 340, 97, 156, and 215, respectively. We chose the first 50 samples collected with device 1 in each class as labeled training samples. Among the other samples, 10 samples were randomly selected in each class for validation, the rest for testing. The hyper-parameters were tuned on the validation sets. Logistic regression was adopted as the classifier, with F-score as the accuracy criterion. Results are compared in Table II. In KPCA, TCA, SSTCA, MIDA, and SMIDA, the RBF kernel was used. Because methods other than stationary subspace analysis (SSA) [33], MIDA, and SMIDA are not capable of handling the chronological information, we simply regarded each device as a discrete domain and learned device-invariant features with them. The same strategy was used in discrete MIDA and SMIDA. In continuous MIDA and SMIDA, the Sensor response (Volt) Original feature 50 100 150 200 250 300 Acquisition time (day) 350 400 450 Device 1 Device 2 1.5 (b) 1 0.5 0 0 50 100 150 200 250 300 Acquisition time (day) 350 400 450 Fig. 7. Illustration of the instrumental variation and time-varying drift in the breath analysis dataset. Plots (a) and (b) show the steady state responses of the CKD samples of sensors 2 and 7, respectively. domain features were defined according to (4), where t was the exact acquisition time converted to years and the number of devices ndev = 2. SSA naturally considers the chronological information by treating the sample stream as a multivariate time series and identifying temporally stationary components. However, SSA cannot deal with time series with multiple sources, such as the multi-device case in this dataset. Thus, the samples were arranged in chronological order despite their device labels. From Table II, we can find that the improvement made by SSA is little, possibly because the stationary criterion is not suitable for preserving important properties in data. For example, the noise in data can also be stationary [5]. MIDA and SMIDA achieved obviously better results than other methods. They can address both instrumental variation and ACCEPTED BY IEEE TRANSACTIONS ON CYBERNETICS 9 TABLE II C LASSIFICATION ACCURACY (%) ON THE BREATH ANALYSIS DATASET. B OLD VALUES INDICATE THE BEST RESULTS . Task 1 2 3 4 5 Average Original feature 34.34 63.67 73.71 43.17 42.93 51.57 KPCA 58.05 72.58 84.78 44.95 42.60 60.59 TCA [5] 67.19 68.31 59.93 67.08 68.17 66.14 SSTCA [5] 67.01 68.06 74.14 68.31 67.36 68.97 SA [17] 29.95 72.42 72.74 42.19 44.54 52.37 GFK [25] 41.49 68.50 58.96 75.63 70.16 62.95 ITL [16] 68.59 66.53 74.75 66.67 68.03 68.91 SSA [33] 49.77 72.10 33.49 52.64 55.38 52.68 SCL [20] 32.52 61.16 75.43 35.35 51.86 51.26 mSDA [21] 36.86 69.51 76.69 35.51 50.49 53.81 MIDA (discrete) 62.17 71.74 84.21 67.05 67.06 70.45 SMIDA (discrete) 80.16 84.18 88.47 68.45 52.41 74.73 MIDA (continuous) 68.30 67.54 74.01 73.04 69.63 70.50 SMIDA (no aug.) 82.80 72.57 72.61 80.33 70.05 75.67 SMIDA (continuous) 85.29 80.18 91.67 74.28 66.55 79.59 time-varying drift. With the background-specific bias brought by feature augmentation, they can compensate for the change in conditional probability in this dataset. SMIDA is better than MIDA because the label information of the first 50 samples in each class was better kept. D. Corn Dataset Similar to e-noses, data collected with spectrometers are one-dimensional signals indicating the concentration of the analytes. Instrumental variation is also a problem for them [24]. In this section, we test our methods on the corn dataset2 . It is a spectroscopy dataset collected with three near-infrared spectrometers designated as m5, mp5, and mp6. The moisture, oil, protein, and starch contents of 80 corn samples were measured by each device, with ranges of the measured values as [9.377, 10.993], [3.088, 3.832], [7.654, 9.711], and [62.826, 66.472], respectively. Each sample is represented by a spectrum with 700 features. This dataset resembles traditional domain adaptation datasets because there is no time-varying drift. Three discrete domains can be defined based on the three devices. We adopt m5 as the source domain, mp5 and mp6 as the target ones. In each domain, samples 4, 8, . . . , 76, 80 were assigned as the test set, the rest as the training set. For hyper-parameter tuning, we applied a three-fold crossvalidation on the training sets of the three domains. After the best hyper-parameters were determined for each algorithm, a regression model was trained on the training set from the source domain and applied on the test set from the target domains. The regression algorithm was ridge regression with the L2 regularization parameter λ = 1. Table III displays the root mean square error (RMSE) of the four prediction tasks and their average on the two target domains. We also plot the overall average RMSE of 2 http://www.eigenvector.com/data/Corn/ the two domains with respect to the subspace dimension h in Fig. 8. ITL was not investigated because it is only applicable in classification problems. In KPCA, TCA, SSTCA, MIDA, and SMIDA, the RBF kernel was used. For the semisupervised methods SSTCA and SMIDA, the target values were normalized to zero mean and unit variance before subspace learning. The domain features were defined according to the device indices using the one-hot coding scheme. We can find that when no domain adaptation was done, the prediction error is large. All domain adaptation algorithms managed to significantly reduce the error. KPCA also has good performance, which is probably because the source and the target domains have similar principal directions, which also contain the most discriminative information. Therefore, source regression models can fit the target samples well. In this dataset, different domains have identical data composition. As a result, corresponding data can be aligned by subspaces alignment, which explains the small error of SA. However, this condition may not hold in other datasets. MIDA and SMIDA obtained the lowest average errors in both target domains. Aiming at exploring the prediction accuracy when there is no instrument variation, we further trained regression models on the training set of the two target domains and tested on the same domain. The results are listed as “train on target” in Table III. It can be found that SMIDA outperforms these results. This could be attributed to three reasons: (1) The inter-domain discrepancy in this dataset is relatively easy to correct; (2) The use of RBF kernel in SMIDA improves the accuracy; (3) SMIDA learned the subspace on the basis of both training and test samples. Although the test samples were unlabeled, they can provide some information about the distribution of the samples to make the learned subspace generalize better, which can be viewed as the merit of semi-supervised learning. To testify this assumption, we conducted another experiment with multiple target domains. The training samples from the source domain and the test ones from both target domains were leveraged together for subspace learning in MIDA and SMIDA. The average RMSE for the two target domains are 0.209 and 0.217 for MIDA, and 0.208 and 0.218 for SMIDA. Compared with the results in Table III with single target domain, the results have been further improved, showing that incorporating more unlabeled samples from target domains can be beneficial. E. Visual Object Recognition Dataset In [25], Gong et al. evaluated domain adaptation algorithms on four visual object recognition datasets, namely Amazon (A), Caltech-256 (C), DSLR (D), and Webcam (W). Ten common classes were selected from them, with 8 to 151 samples per class per domain, and 2533 images in total. Each image was encoded with an 800-bin histogram using SURF features. The normalized histograms were z-scored to have zero mean and unit variance in each dimension. Following the experimental setting provided in the sample code from the authors of [25], experiments were conducted in 20 random trials for each pair of domains. For each unsupervised trail, 20 (for A, C, W) or 8 (for D) labeled samples per class ACCEPTED BY IEEE TRANSACTIONS ON CYBERNETICS 10 TABLE III R EGRESSION RMSE ON THE CORN DATASET. B OLD VALUES INDICATE THE BEST RESULTS . Mp5 as target domain Mp6 as target domain Moisture Oil Moisture Oil Protein Starch Average Original feature 1.327 0.107 1.155 2.651 1.310 1.433 0.101 1.413 2.776 1.431 KPCA 0.477 0.165 0.215 0.315 0.293 0.396 0.164 0.238 0.290 0.272 TCA [5] 0.539 0.322 0.217 0.402 0.370 0.398 0.145 0.259 0.572 0.343 SSTCA [5] 0.343 0.093 0.140 0.366 0.235 0.367 0.088 0.186 0.318 0.240 SA [17] 0.302 0.094 0.186 0.351 0.233 0.324 0.079 0.158 0.390 0.238 GFK [25] 0.267 0.197 0.342 0.621 0.357 0.263 0.189 0.264 0.485 0.301 SCL [20] 0.283 0.115 0.249 0.619 0.316 0.311 0.108 0.257 0.683 0.340 mSDA [21] 0.264 0.107 0.211 0.446 0.257 0.285 0.097 0.198 0.471 0.263 MIDA 0.317 0.078 0.141 0.378 0.228 0.317 0.084 0.158 0.352 0.228 SMIDA 0.287 0.072 0.143 0.339 0.210 0.316 0.073 0.152 0.325 0.217 Train on target 0.176 0.094 0.201 0.388 0.215 0.182 0.108 0.206 0.414 0.228 0.5 KPCA TCA SSTCA SA MIDA SMIDA Average regression RMSE 0.45 0.4 0.35 0.3 0.25 0.2 0 10 Protein Starch Average 20 30 40 # Projected dimensions (h) 50 60 Fig. 8. Performance comparison on the corn dataset with respect to the subspace dimension h. were randomly chosen from the source domain as the training set (other samples were used unsupervisedly for domain adaptation), while all unlabeled samples in the target domain made up the test set. In semi-supervised trails, three labeled samples per class in the target domain were also assumed to be labeled. Averaged accuracies on each pair of domains as well as standard errors are listed in Tables IV and V. For GFK, low-rank transfer subspace learning (LTSL), domain adaptation by shifting covariance (DASC), and a recent method called integration of global and local metrics for domain adaptation (IGLDA), we copied the best results reported in the original papers [18], [19], [3], [22]. For other methods tested, the hyper-parameters were tuned for the best accuracy. Logistic regression was adopted as the classifier. The polynomial kernel with degree 2 was used in KPCA, TCA, SSTCA, MIDA, and SMIDA. The domain features were defined according to the domain labels using the onehot coding scheme. MIDA and SMIDA achieve the best average accuracies in both unsupervised and semi-supervised visual object recognition experiments. We observe that TCA and SSTCA have comparable performance with MIDA and SMIDA, which may be explained by the fact that the HSIC criterion used in MIDA and MMD used in TCA are identical under certain conditions when there are one source and one target domain [27]. Besides, the feature augmentation strategy in MIDA is not crucial in this dataset because there is no change in conditional probability. On the other hand, TCA and SSTCA can only handle one source and one target domains. SSTCA uses the manifold regularization strategy to preserve local geometry information, hence introduces three more hyper-parameters than SMIDA. Moreover, computing the data adjacency graph in SSTCA and the matrix inversion operation in TCA and SSTCA make them slower than MIDA and SMIDA. We compared their speed on the domain adaptation experiment C → A. They were run on a server with Intel Xeon 2.00 GHz CPU and 128 GB RAM. No parallel computing was used. The codes of the algorithms were written in Matlab R2014a. On average, the running times of each trial of MIDA, SMIDA, TCA, and SSTCA were 2.4 s, 2.5 s, 3.0 s, and 10.2 s, respectively. Therefore, MIDA and SMIDA are more practical to use than TCA and SSTCA. Besides, they were initially designed for drift correction. This dataset is used to show their universality. V. C ONCLUSION In this paper, we introduced maximum independence domain adaptation (MIDA) to learn domain-invariant features. The main idea of MIDA is to reduce the inter-domain discrepancy by maximizing the independence between the learned features and the domain features of the samples. The domain features describe the background information of each sample, such as the domain label in traditional domain adaptation problems. In the field of sensors and measurement, the device label and acquisition time of the each collected sample can be expressed by the domain features, so that unsupervised drift correction can be achieved by using MIDA. The feature augmentation strategy proposed in this paper adds domainspecific biases to the learned features, which helps MIDA to align domains. ACCEPTED BY IEEE TRANSACTIONS ON CYBERNETICS 11 TABLE IV U NSUPERVISED DOMAIN ADAPTATION ACCURACY (%) ON THE VISUAL OBJECT RECOGNITION DATASET. B OLD VALUES INDICATE THE BEST RESULTS . X → Y MEANS THAT X IS THE SOURCE DOMAIN AND Y IS THE TARGET ONE . C→A Ori. ft. D→A W→A A→C D→C W→C A→D C→D W→D A→W C→W D→W Average 43.2±2.2 34.9±1.1 36.8±0.6 38.5±1.6 31.7±1.2 32.7±0.9 37.3±3.1 40.5±3.6 80.6±2.0 37.5±2.9 37.1±3.6 76.7±2.0 43.97 KPCA 27.4±2.0 27.0±1.6 27.6±1.2 25.8±1.3 22.0±2.4 23.7±1.6 29.2±2.9 27.6±2.7 55.1±1.8 29.0±2.7 27.1±3.2 50.3±2.6 30.97 TCA [5] 49.8±2.7 38.6±1.4 39.2±1.0 42.8±2.1 35.3±1.5 35.7±0.8 42.8±3.2 45.9±3.8 83.9±1.7 41.7±3.3 42.8±5.4 81.5±2.1 48.35 SSTCA [5] 50.5±2.8 39.3±1.6 40.5±0.7 42.4±1.8 36.1±1.5 35.9±0.8 42.8±3.1 46.6±3.5 80.6±2.3 42.5±2.7 42.8±4.7 81.8±1.9 48.48 ITL [16] 41.2±3.0 35.7±2.0 38.4±1.1 34.8±1.8 28.7±1.5 31.4±1.6 34.5±3.0 32.3±3.8 67.0±2.7 31.8±3.6 36.4±4.2 71.1±3.2 40.27 SA [17] 48.4±2.9 36.3±2.6 37.3±1.7 38.5±2.1 33.4±1.9 35.1±0.9 35.0±3.6 39.7±5.0 63.2±2.8 36.7±4.4 41.3±5.5 70.3±2.5 42.93 GFK [18] 40.4±0.7 36.2±0.4 35.5±0.7 37.9±0.4 32.7±0.4 29.3±0.4 35.1±0.8 41.1±1.3 71.2±0.9 35.7±0.9 35.8±1.0 79.1±0.7 42.50 LTSL [19] 50.4±0.4 40.2±0.6 44.1±0.3 38.6±0.3 35.3±0.3 37.4±0.2 38.3±1.1 53.7±0.9 79.8±0.4 38.8±1.3 47.0±1.0 72.8±0.7 48.03 DASC [3] 39.1±0.3 39.3±0.8 37.7±0.7 49.8±0.4 48.5±0.8 45.4±0.9 36.5±0.3 35.6±0.3 88.3±0.4 36.3±0.4 33.3±0.3 79.8±0.9 47.47 SCL [20] 43.5±2.2 34.7±1.3 36.9±0.7 38.5±1.5 31.7±1.3 33.2±0.9 37.8±3.0 40.6±3.6 81.1±1.8 37.6±3.3 37.1±4.0 76.7±2.0 44.10 mSDA [21] 45.3±1.9 37.4±1.2 38.3±0.7 40.3±1.8 33.7±1.3 35.5±1.1 38.1±2.8 40.4±4.0 82.0±1.8 38.5±3.4 37.8±3.7 79.0±2.1 45.51 IGLDA [22] 51.0±2.7 38.4±1.9 38.6±1.2 41.5±1.8 36.4±2.2 34.2±1.5 38.9±2.5 45.1±2.4 82.6±1.8 40.0±3.4 42.2±3.6 82.4±2.4 47.61 MIDA 50.3±2.5 39.2±1.9 39.8±1.0 42.7±1.8 35.5±1.1 35.7±0.7 42.3±2.8 45.7±3.6 82.2±2.0 42.8±2.8 43.6±5.0 82.4±2.0 48.51 SMIDA 50.5±2.4 39.1±1.8 39.8±1.1 42.7±2.0 35.5±1.2 35.4±0.8 42.4±2.6 45.8±3.3 82.5±2.1 42.9±2.8 43.4±5.1 81.9±2.0 48.49 TABLE V S EMI - SUPERVISED DOMAIN ADAPTATION ACCURACY (%) ON THE VISUAL OBJECT RECOGNITION DATASET. B OLD VALUES INDICATE THE BEST RESULTS . C→A D→A W→A A→C D→C W→C A→D C→D W→D A→W C→W D→W Average Ori. ft. 48.8±1.8 44.5±1.6 43.5±1.4 41.6±1.9 36.7±2.1 37.3±1.4 48.1±4.2 49.3±3.2 81.7±2.4 51.0±3.4 50.9±4.4 80.5±2.2 51.17 KPCA 53.3±2.4 46.2±1.7 43.2±1.1 44.1±1.4 39.1±1.8 37.8±1.1 47.3±3.3 53.9±3.4 81.5±2.9 49.7±2.7 54.0±4.1 81.1±2.2 52.60 TCA [5] 55.3±2.2 48.6±1.8 45.7±1.4 46.1±2.0 40.3±2.1 39.7±1.4 52.1±3.0 56.3±4.5 83.7±2.9 55.4±3.5 58.3±4.5 84.2±1.9 55.46 SSTCA [5] 55.3±2.2 48.6±1.8 45.6±1.4 46.0±2.0 40.3±2.1 39.7±1.3 52.1±3.1 56.3±4.6 83.7±2.9 55.4±3.5 58.4±4.5 84.2±1.9 55.47 ITL [16] 51.5±3.1 47.7±2.5 44.1±2.1 40.0±2.2 36.8±3.2 36.6±2.2 44.4±4.1 48.2±4.0 59.7±2.6 51.5±4.5 54.9±3.8 68.5±3.3 48.65 SA [17] 51.8±2.3 47.6±2.8 44.7±1.7 42.6±1.7 36.9±2.9 36.8±2.0 45.3±4.2 49.0±3.9 71.0±2.9 47.2±2.9 49.0±3.6 76.1±2.5 49.82 GFK [18] 46.1±0.6 46.2±0.6 46.2±0.7 39.6±0.4 33.9±0.6 32.3±0.6 50.9±0.9 55.0±0.9 74.1±0.9 56.9±1.0 57.0±0.9 80.2±0.4 51.53 LTSL [19] 50.4±0.5 47.4±0.5 47.8±0.4 39.8±0.4 36.7±0.4 38.5±0.3 59.1±0.7 59.6±0.6 82.6±0.5 59.5±1.1 59.5±0.8 78.3±0.4 54.93 SCL [20] 48.8±1.7 45.0±1.4 43.4±1.3 41.3±1.8 36.3±2.2 37.4±1.4 48.9±4.4 49.3±3.5 81.8±2.5 51.9±3.7 52.0±4.4 81.0±2.2 51.42 mSDA [21] 50.4±2.2 48.1±1.6 45.6±1.6 43.6±1.9 38.9±2.2 39.3±1.5 48.9±4.5 49.2±4.9 82.3±2.5 52.9±3.9 52.1±4.4 81.6±2.1 52.75 MIDA 55.2±2.2 48.6±1.8 45.6±1.4 46.1±2.0 40.4±2.2 39.7±1.4 52.1±3.0 56.4±4.6 83.7±2.8 55.3±3.4 58.5±4.6 84.2±1.8 55.48 SMIDA 55.2±2.2 48.6±1.8 45.7±1.4 46.1±2.0 40.3±2.1 39.7±1.4 52.2±3.1 56.3±4.6 83.7±2.9 55.4±3.5 58.5±4.5 84.3±1.9 55.49 MIDA and SMIDA are flexible algorithms. 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IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, SUBMITTED 1 Large-Scale Low-Rank Matrix Learning with Nonconvex Regularizers arXiv:1708.00146v2 [cs.LG] 23 Sep 2017 Quanming Yao, Member IEEE, James T. Kwok, Fellow IEEE, Taifeng Wang, Member IEEE, and Tie-Yan Liu, Fellow IEEE Abstract—Low-rank modeling has many important applications in computer vision and machine learning. While the matrix rank is often approximated by the convex nuclear norm, the use of nonconvex low-rank regularizers has demonstrated better empirical performance. However, the resulting optimization problem is much more challenging. Recent state-of-the-art requires an expensive full SVD in each iteration. In this paper, we show that for many commonly-used nonconvex low-rank regularizers, the singular values obtained from the proximal operator can be automatically threshold. This allows the proximal operator to be efficiently approximated by the power method. We then develop a fast proximal algorithm and its accelerated variant with inexact proximal step. A convergence rate of O(1/T ), where T is the number of iterations, can be guaranteed. Furthermore, we show the proposed algorithm can be parallelized, and the resultant algorithm achieves nearly linear speedup w.r.t. the number of threads. Extensive experiments are performed on matrix completion and robust principal component analysis. Significant speedup over the state-of-the-art is observed. Index Terms—Low-rank matrix learning, Nonconvex regularization, Proximal algorithm, Parallel algorithm, Matrix completion, Robust principle component analysis F 1 I NTRODUCTION L OW -rank matrix learning is a central issue in many machine learning and computer vision problems. For example, matrix completion [1], which is one of the most successful approaches in collaborative filtering, assumes that the target rating matrix is low-rank. Besides collaborative filtering, matrix completion has also been used on tasks such as video and image processing [2], [3], [4], [5], [6]. Another important use of low-rank matrix learning is robust principal component analysis (RPCA) [7], which assumes that the target matrix is low-rank and also corrupted by sparse noise. RPCA has been popularly used in computer vision applications such as shadow removal, background modeling [7], [8], [9], and robust photometric stereo [10]. Besides, low-rank matrix learning has also been used in face recognition [11] and subspace clustering [12]. However, minimization of the matrix rank is NP-hard [1]. To alleviate this problem, a common approach is to use a convex surrogate such as the nuclear norm (which is the sum of singular values of the matrix). It is known that the nuclear norm is the tightest convex lower bound of the rank. Though the nuclear norm is non-smooth, the resultant optimization problem can be solved efficiently using modern tools such as the proximal algorithm [13], [14], [15], Frank-Wolfe algorithm [16], and active subspace selection method [17]. Despite the success of the nuclear norm, recently there have been numerous attempts to use nonconvex surrogates that better approximate the rank function. The key idea is • • Q. Yao and J. Kwok are with the Department of Computer Science and Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong. E-mails: {qyaoaa, jamesk}@cse.ust.hk T. Wang and T. Liu are with Microsoft Research Asia, Beijing, China 100010. E-mails: {taifengw, tyliu}@microsoft.com that the larger, and thus more informative, singular values should be less penalized. Example nonconvex low-rank regularizers include the capped-`1 penalty [18], log-sum penalty (LSP) [19], truncated nuclear norm (TNN) [3], [9], smoothly clipped absolute deviation (SCAD) [20], and minimax concave penalty (MCP) [21]. They have been applied on various computer vision tasks, such as image denoising [6] and background modeling [9]. Empirically, these nonconvex regularizers achieve better recovery performance than the convex nuclear norm regularizer. Recently, theoretical results have also been established [22]. However, the resultant nonconvex optimization problem is much more challenging. Most existing optimization algorithms that work with the nuclear norm cannot be applied. A general approach that can still be used is the concave-convex procedure [23], which decomposes the nonconvex regularizer into a difference of convex functions [3], [18]. However, a sequence of relaxed optimization problems have to be solved, and can be computationally expensive [24], [25]. A more efficient approach is the recently proposed iteratively re-weighted nuclear norm (IRNN) algorithm [5]. It is based on the observation that existing nonconvex regularizers are concave with non-increasing super-gradients. Each IRNN iteration only involves computing the supergradient of the regularizer and a singular value decomposition (SVD). However, performing SVD on a m × n matrix takes O(mn2 ) time (assuming m ≥ n), and can be expensive on large matrices. Recently, the proximal algorithm has been used for nonconvex low-rank matrix learning [3], [5], [9], [26]. However, it requires the full SVD to solve the proximal operator, which can be expensive. In this paper, we observe that for the commonly-used nonconvex low-rank regularizers [3], [9], [18], [19], [20], [21], the singular values obtained from the corresponding proximal operator can be automatically IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, SUBMITTED thresholded. One then only needs to find the leading singular values/vectors in order to generate the next iterate. Moreover, instead of computing the proximal operator on a large matrix, one only needs to use the matrix projected onto its leading subspace. The matrix size is significantly reduced and the proximal operator can be made much more efficient. Besides, by using the power method [27], a good approximation of this subspace can be efficiently obtained. While the proposed procedure can be readily used with the standard proximal algorithm, its convergence properties are not directly applicable as the proximal step here is only approximately solved. In the sequel, we will show that inexactness on the proximal step can be controlled, and a O(1/T ) convergence rate can still be guaranteed. Moreover, the algorithm can be further speeded up using acceleration. Effectiveness of the proposed algorithms is demonstrated on two popular low-rank matrix learning applications, namely matrix completion and robust principal component analysis (RPCA). For matrix completion, we show that additional speedup is possible by exploring the problem’s “sparse plus low-rank” structure; whereas for RPCA, we extend the proposed algorithm so that it can handle the two parameter blocks involved in the RPCA formulation. With the popularity of multicore shared-memory platforms, we parallelize the proposed algorithms so as to handle much larger data sets. We will show that they can achieve almost linear speedup w.r.t. the number of threads. Experiments are performed on both synthetic and realworld data sets. Results show that the proposed nonconvex low-rank matrix learning algorithms can be several orders faster than the state-of-the-art, and outperform other approaches including factorization and the use of nuclear norm regularization. Preliminary results of this paper have been reported in [28]. In this full version, we speed up the algorithm with acceleration, and demonstrate how it can be applied to two important instances of low-rank matrix learning problems, namely matrix completion and RPCA. Besides, we show how the proposed algorithms can be parallelized. More extensive empirical evaluations are also performed on both the sequential and parallel versions of the algorithms. Notation: In the sequel, vectors are denoted by lowercase boldface, matrices by uppercase boldface, and the transpose by the superscript (·)> . For a square matrix X, tr(X) is its trace. For a rectangle matrix X,P kXkF = tr(X> X) is its Frobenius norm, and kXk∗ = i σi (X), where σi (X) is the ith leading singular value of X, is the nuclear norm. Given x = [xi ] ∈ Rm , Diag(x) constructs a m × m diagonal matrix whose ith diagonal element is xi . I denotes the identity matrix. For a differentiable function f , we use ∇f for its gradient. For a nonsmooth function, we use ∂f for its subdifferential, i.e., ∂f (x) = s : f (y) ≥ f (x) + s> (y − x) . 2 2.1 where f is a smooth loss, r is a nonsmooth low-rank regularizer, and λ is a regularization parameter. We make the following assumptions on f . A1. A2. f is not necessarily convex, but is differentiable with ρ-Lipschitz continuous gradient, i.e., k∇f (X1 ) − ∇f (X2 )kF ≤ ρkX1 − X2 kF . Without loss of generality, we assume that ρ ≤ 1. f is bounded below, i.e., inf f (X) > −∞, and limkXkF →∞ f (X) = ∞. In recent years, the proximal algorithm [29] has been popularly used for solving (1). At iteration t, it produces Xt+1 = prox λ r (Xt − τ 1 ∇f (Xt )), τ (2) where τ > ρ is the stepsize, and 1 prox λ r (Z) ≡ arg min kX − Zk2F + λr(X) (3) τ X 2 is the proximal operator [29]. The proximal step in (2) can also be rewritten as Xt+1 = arg minY tr(∇f (Xt )> (Y − Xt )) + τ2 kY − Xt k2F + λr(Y). When f and r are convex, the proximal algorithm converges to the optimal solution at a rate of O(1/T ), where T is the number of iterations. This can be further accelerated to the rate of O(1/T 2 ), by replacing Xt in (2) with a proper linear combination of Xt and Xt−1 [30]. Recently, the accelerated proximal algorithm has been extended to problems where f or r may be nonconvex [25], [31]. The state-ofthe-art is the nonmonotone accelerated proximal gradient (nmAPG) algorithm [25] (Algorithm 1). Each iteration may perform two proximal steps (steps 4 and 8). Acceleration is performed in step 3. The objective is then checked to determine whether Xat+1 is accepted (step 5). As the problem is nonconvex, its convergence rate is still open. However, empirically it is much faster. Algorithm 1 Nonmonotone APG (nmAPG) [25]. Input: choose τ > ρ, δ > 0 and η ∈ [0, 1); 1: initialize X0 = X1 = Xa 1 = 0, α0 = α1 = 1, b1 = F (X1 ) and q1 = 1; 2: for t = 1, 2, . . . , T do −1 3: Yt = Xt + ααt−1 (Xat − Xt−1 ) + αt−1 (Xt − Xt−1 ); αt t 1 a 4: Xt+1 = prox λ r (Yt − τ ∇f (Yt )); τ 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: BACKGROUND 2 if F (Xat+1 ) ≤ bt − 2δ kXat+1 − Yt k2F then Xt+1 = Xat+1 ; else Xpt+1 = prox λ r (Xt − τ1 ∇f (Xt )); ( τ Xat+1 if F (Xat+1 ) ≤ F (Xpt+1 ) Xt+1 = ; Xpt+1 otherwise end if p αt+1 = 21 ( 4αt2 + 1 + 1); 1 bt+1 = qt+1 (ηqt bt + F (Xt+1 )) where qt+1 = ηqt + 1; end for return XT +1 . Proximal Algorithm In this paper, we consider low-rank matrix learning problems of the form min F (X) ≡ f (X) + λr(X), X (1) 2.2 Nonconvex Low-Rank Regularizers For the proximal algorithm to be successful, the proximal operator has to be efficient. The following shows that the IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, SUBMITTED proximal operator of the nuclear norm k · k∗ has a closedform solution. > Proposition 2.1 ([32]). proxµk·k∗ (X) = U (Σ − µI)+ V , where UΣV> is the SVD of X, and (A)+ = [max(Aij , 0)]. While the (convex) nuclear norm makes low-rank optimization easier, it may not be a good enough approximation of the matrix rank [3], [5], [6], [9], [26]. As mentioned in Section 1, a number of nonconvex surrogates have been recently proposed. In this paper, we make the following assumption on the low-rank regularizer r in (1), which is satisfied by all nonconvex low-rank regularizers in Table 1. A3. r is possibly non-smooth and nonconvex, and of the Pm form r(X) = i=1 r̂(σi (X)), where r̂(α) is concave and non-decreasing for α ≥ 0 and r̂(0) = 0. TABLE 1 r̂’s for some popular nonconvex low-rank regularizers. For the TNN regularizer, θ ∈ {1, . . . , n} is the number of leading singular values that are not penalized; for SCAD, θ > 2; and for the others, θ > 0. µr̂(σi (X)) capped-`1 [18] µ min(σi (X), θ) 1 LSP [19] (µ log θ σi (X) + 1 µσi (X) 0 TNN [3], [9] SCAD [20] MCP [21]   µσ (X)   i2 if i > θ otherwise −σi (X)+2θµσi (X)−µ2 2(θ−1)    (θ+1)µ2 2 ( σ 2 (X) µσi (X) − i2θ θµ2 2 if σi (X) ≤ µ if µ < σi (X) ≤ θµ otherwise if σi (X) ≤ θµ otherwise Recently, the iteratively reweighted nuclear norm (IRNN) algorithm [5] has been proposed to handle this nonconvex low-rank matrix optimization problem. In each iteration, it solves a subproblem in which the original nonconvex regularizer is approximated by a weighted version of the nuclear Pm norm kXkw = i=1 wi σi (X) and 0 ≤ w1 ≤ · · · ≤ wm . The subproblem has a closed-form solution, but SVD is needed which takes O(mn2 ) time. Other solvers that are designed for specific nonconvex low-rank regularizers include [8] (for capped-`1 ), [3], [9] (for TNN), and [21] (for MCP). All these (including IRNN) perform SVD in each iteration, which takes O(mn2 ) time and are slow. While the proximal algorithm has mostly been used on convex problems, recently it is also applied to nonconvex problems [3], [5], [6], [8], [9], [26]. The generalized proximal gradient (GPG) algorithm [26] is the first proximal algorithm which can handle all the above nonconvex regularizers. In particular, its proximal operator can be computed as follows. Proposition 2.2 (Generalized singular value thresholding (GSVT) [26]). For any r satisfying assumption A3, proxµr (Z) = UDiag(y∗ )V> , where UΣV> is the SVD of Z, and y∗ = [yi∗ ] with 1 2 (4) yi∗ ∈ arg min (yi − σi (Z)) + µr̂(yi ). yi ≥0 2 In [26], problem (4) is solved by fixed-point iteration. However, closed-form solutions indeed exist for regularizers in Table 1 [24]. Nevertheless, Proposition 2.2 still involves SVD, which takes O(mn2 ) time. 3 3 P ROPOSED A LGORITHM In this section, we show how the proximal algorithm can be made much faster by using approximate GSVT. 3.1 Automatic Thresholding of Singular Values The following Proposition shows that yi∗ in (4) becomes zero when σi (Z) is smaller than a regularizer-specific threshold. Proof can be found in Appendix C.1. Proposition 3.1. There exists a threshold γ > 0 such that yi∗ = 0 when σi (Z) ≤ γ . Together with Proposition 2.2, solving the proximal operator (3) only needs the leading singular values/vectors of Z. For the nonconvex regularizers in Table 1, simple closed-form solutions of γ can be obtained by examining the optimality conditions of (4). Proof can be found in Appendix C.2. Corollary 3.2. The γ values for the following regularizers are: √ • Capped-`1 : γ = min 2θµ, µ ; • LSP: γ = min µθ , θ ; • TNN: γ = max (µ, σθ+1 (Z)); • SCAD: γ =√µ; • MCP: γ = θµ if 0 < θ < 1, and µ otherwise. This can also be used with the nuclear norm. It can be shown that γ = λτ , and yi∗ = max σi (A) − λτ , 0 . However, since our focus is on nonconvex regularizers, the case for nuclear norm will not be further pursued in the sequel. 3.2 Approximate GSVT Proposition 2.2 computes the proximal operator by exact SVD. In this section, we show that one can use approximate SVD, which is more efficient. 3.2.1 Reducing the Size of SVD Assume that Z has k̂ singular values larger than γ , then we only need to a rank-k SVD on Z with k ≥ k̂ . Let the rank-k̂ SVD of Z be Uk̂ Σk̂ Vk̂> . The following Proposition shows that proxµr (Z) can be obtained from the proximal operator on a smaller matrix.1 The proof can be found in Appendix C.3. Proposition 3.3. Assume that Q ∈ Rm×k , where k ≥ k̂ , is orthogonal and span(Uk̂ ) ⊆ span(Q). Then, proxµr (Z) = Q proxµr (Q> Z). 3.2.2 Obtaining an Approximate GSVT To obtain such a Q, we use the power method [27] (Algorithm 2) which has been recently used to approximate the SVT in nuclear norm minimization [15], [17]. As in [17], we set the number of power iterations to 3. Warm-start can be used via matrix R in Algorithm 2. This is particularly useful because of the iterative nature of proximal algorithm. Obtaining an approximate Q using Algorithm 2 takes O(mnk) time. As in [14], [34], [35], the PROPACK 1. We noticed a similar result in [33] after the conference version of this paper [28] has been accepted. However, [33] only considers the case where r is the nuclear norm regularizer. IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, SUBMITTED 4 algorithm [36] can also be used to obtain Q in O(mnk) time. However, it finds the Q exactly and cannot benefit from warm-start. Hence, though it has the same time complexity as power method, empirically it is much less efficient [15]. Xt for the proximal gradient algorithm in (2)). An approximate proximal step solution X̃p is generated in step 4, and we try to ensure Algorithm 2 Powermethod(Z, R). (note that this is less stringent than where c1 = τ −ρ 4 the condition in Lemma 3.4). If (5) holds, we accept X̃p ; otherwise, we improve X̃p by using Ṽp−1 to warm-start the next iterate. The following Proposition shows convergence of Algorithm 4. The proof can be found in Appendix C.4. m×n n×k Input: Z ∈ R , R ∈ R and the number of power iterations J = 3. 1: Y1 = ZR; 2: for j = 1, 2, . . . , J do 3: Qj = QR(Yj ); // QR decomposition (returning only the Q matrix) 4: Yj+1 = Z(Z> Qj ); 5: end for 6: return QJ . The approximate GSVT procedure is shown in Algorithm 3. Step 1 uses the power method to efficiently obtain an orthogonal matrix Q that approximates span(Uk̂ ). Step 2 performs a small SVD. Though this SVD is still exact, Q> Z is much smaller than Z (k × n vs m × n), and SVD(Q> Z) takes only O(nk 2 ) time. In step 3, the singular values Σii ’s are thresholded using Corollary 3.2. Steps 6-8 obtains an (approximate) proxµr (Z) using Proposition 2.2. The time complexity for GSVT is reduced from O(mn2 ) to O(mnk). F (X̃p ) ≤ F (X) − c1 kX̃p − Xk2F , (5) Proposition 3.5. If k ≥ k̂A , where k̂A is the number of singular values in A larger than γ , then limp→∞ X̃p = prox λ r (A). τ Algorithm 4 Inexact proximal step: InexactPS(X, R). Input: X ∈ Rm×n , and R ∈ Rn×k for warm-start; 1: A = X − τ1 ∇f (X); 2: Ṽ0 = R; 3: for p = 1, 2, . . . do 4: [X̃p , Ṽp ] = ApproxGSVT (A, Ṽp−1 , λτ ); 5: if F (X̃p ) ≤ F (X) − c1 kX̃p − Xk2F then 6: break; 7: end if 8: end for 9: return X̃p . Algorithm 3 Approximate GSVT: ApproxGSVT(Z, R, µ). Input: Z ∈ Rm×n and R ∈ Rn×k for warm-start; 1: Q = PowerMethod(Z, R); 2: [U, Σ, V] = SVD(Q> Z); 3: a = number of Σii ’s that are > γ in Corollary 3.2; 4: Ua = a leading columns of U; 5: Va = a leading columns of V; 6: for i = 1, 2, . . . , a do 7: obtain yi∗ from (4); 8: end for 9: return low-rank components of X̃ (QUa , Diag([y1∗ , . . . , ya∗ ]) and Va> ), and V. 3.3 Inexact Proximal Step In this section, the proximal step will be inexact, and so it can utilize the approximate GSVT in Algorithm 3. Inexact proximal step has been considered in [37], [38]. However, r in (1) is assumed to be convex in [38]. Attouch et al. [37] considered nonconvex r, but they require a difficult and expensive condition to control inexactness (an example is provided in Appendix B). Let A = X − τ1 ∇f (X). The following shows that the objective F is always decreased (as τ > ρ) after an exact proximal step. Lemma 3.4 ( [24], [37]). F prox λ r (A) ≤ F (X) − τ −ρ (A) 2 kprox λ τr − Xk2F . τ Motivated by this Lemma, we propose to control the proximal step’s inexactness by Algorithm 4 (note that X = 3.4 The Complete Procedure The complete procedure for solving (1) is shown in Algorithm 5, and will be called FaNCL (Fast NonConvex Lowrank). Similar to [15], [17], we perform warm-start using the column spaces of the previous iterates (Vt and Vt−1 ). For further speedup, we employ a continuation strategy at step 3 as in [5], [14], [34]. Specifically, λt is initialized to a large value and then decreases gradually. Algorithm 5 FaNCL (Fast NonConvex Low-rank) algorithm. Input: choose τ > ρ, λ0 > λ and ν ∈ (0, 1); 1: initialize V0 , V1 ∈ Rn as random Gaussian matrices and X1 = 0; 2: for t = 1, 2, . . . T do 3: λt = (λt−1 − λ)ν t + λ; 4: Rt = QR([Vt , Vt−1 ]); // warm start 5: Xt+1 = InexactPS(Xt , Rt ); 6: end for 7: return XT +1 . Assume that evaluations of f and ∇f take O(mn) time, which is valid for many applications such as matrix completion and RPCA. Let rt be the rank of Xt at the tth iteration, and kt = rt + rt−1 . In Algorithm 5, step 4 takes O(nkt2 ) time; and step 5 takes O(mnpkt ) time as Rt has kt columns. The iteration time complexity is thus O(mnpkt ). In the experiment, we set p = 1, which is enough to guarantee (5) empirically. The iteration time complexity of Algorithm 5 thus reduces to O(mnkt ). In contrast, exact GSVT takes O(mn2 ) time, and is much slower as kt n. Besides, the space complexity of Algorithm 5 is O(mn). IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, SUBMITTED 5 Proposition 3.6. r can be decomposed as r̆ − r̃, where r̆ and r̃ are convex. O(mnkt ). Besides, its space complexity is O(mn), which is the same as Algorithm 5. There are several major differences between Algorithm 6 and nmAPG. First, the proximal step of Algorithm 6 is only inexact. To make the algorithm more robust, we do not allow nonmonotonous update (i.e., F (Xt+1 ) cannot be larger than F (Xt )). Moreover, we use a simpler acceleration scheme (step 4), in which only Xt and Xt−1 are involved. On matrix completion problems, this allows using the “sparse plus low-rank” structure [14], [15] to greatly reduce the iteration complexity (Section 4.1). Finally, we do not require extra comparison of the objective at step 10. This further reduces the iteration complexity. Based on this decomposition, we introduce the definition of critical point. Algorithm 6 Accelerated FaNCL algorithm (FaNCL-acc). 3.5 Convergence Analysis The inexact proximal algorithm is first considered in [38], which assumes r to be convex. The nonconvex extension is considered in [37]. However, as discussed in Section 3.3, they use an expensive condition to control inexactness of the proximal step. Thus, their analysis cannot be applied here. It is known that r̂ in Assumption A3 can be decomposed as a difference of convex functions [24]. The following Proposition shows that r also admits such a decomposition. The proof is in Appendix C.5. Definition 1 ([39]). If 0 ∈ ∇f (X)+λ (∂ r̆(X) − ∂ r̃(X)), then X is a critical point of F . The following Proposition shows that Algorithm 5 generates a bounded sequence. The proof is in Appendix C.6. Proposition 3.7. The sequence {Xt } generated Algorithm 5 is bounded, and has at least one limit point. from Let G λ r (Xt ) = Xt − prox λ r (Xt − τ1 ∇f (Xt )), which τ τ is known as the proximal mapping of F at Xt [29]. If G λ r (Xt ) = 0, Xt is a critical point of (1) [24], [37]. This τ motivates the use of kG λ r (Xt )k22 to measure convergence τ in [31]. However, kG λ r (Xt )k22 cannot be used here as r τ is nonconvex and the proximal step is inexact. As Proposition 3.7 guarantees the existence of limit points, we use kXt+1 − Xt k2F instead to measure convergence. If the proximal step is exact, kG λ r (Xt )k2F = kXt+1 − Xt k2F . The τ following Corollary shows convergence of Algorithm 5. Its proof can be found in Appendix C.7. Corollary 3.8. mint=1,...,T kXt+1 − Xt k2F ≤ F (X1 )−inf F c1 T . The following Theorem shows that any limit point is also a critical point. The proof is in Appendix C.8. Theorem 3.9. Assume that Algorithm 4 returns X only when X = prox λ r (X− τ1 ∇f (X)) (i.e., the input is returned as output τ only if it is a limit point). Let {Xtj } be a subsequence of {Xt } generated by Algorithm 5 such that limtj →∞ Xtj = X∗ . Then, X∗ is a critical point of (1). 3.6 Acceleration In convex optimization, acceleration has been commonly used to speed up convergence of proximal algorithms [30]. Recently, it has also been extended to nonconvex optimization [25], [31]. A state-of-the-art algorithm is the nmAPG [25] (Algorithm 1). In this section, we integrate nmAPG with FaNCL. The whole procedure is shown in Algorithm 6. The accelerated iterate is obtained in step 4. If the resultant inexact proximal step solution can achieve a sufficient decrease (step 7) as in (5), this iterate is accepted (step 8); otherwise, we choose the inexact proximal step solution obtained with the nonaccelerated iterate Xt (step 10). Note that step 10 is the same as step 5 of Algorithm 5. Thus, the iteration time complexity of Algorithm 6 is at most twice that of Algorithm 5, and still Input: choose τ > ρ, λ0 > λ, δ > 0 and ν ∈ (0, 1); 1: initialize V0 , V1 ∈ Rn as random Gaussian matrices, X0 = X1 = 0 and α0 = α1 = 1; 2: for t = 1, 2, . . . T do 3: λt = (λt−1 − λ)ν + λ; −1 4: Yt = Xt + αt−1 (Xt − Xt−1 ); αt 5: Rt = QR([Vt , Vt−1 ]); // warm start 6: Xat+1 = InexactPS(Yt , Rt ); 7: if F (Xat+1 ) ≤ F (Xt ) − 2δ kXat+1 − Yt k2F then 8: Xt+1 = Xat+1 ; 9: else 10: Xt+1 = InexactPS(Xt , Rt ); 11: end if p 12: αt+1 = 12 ( 4αt2 + 1 + 1); 13: end for 14: return XT +1 . The following Proposition shows that Algorithm 6 generates a bounded sequence. Proof can be found in Appendix C.9. Proposition 3.10. The sequence {Xt } generated Algorithm 6 is bounded, and has at least one limit point. from In Corollary 3.8, kXt+1 − Xt k2F is used to measure progress before and after the proximal step. In Algorithm 6, the proximal step may use the accelerated iterate Yt or the non-accelerated iterate Xt . Hence, we use kXt+1 − Ct k2F , where Ct = Yt if step 8 is performed, and Ct = Xt otherwise. Similar to Corollary 3.8, the following shows a O(1/T ) convergence rate. Proof can be found in Appendix C.10. Corollary 3.11. For Algorithm 6, mint=1,...,T kXt+1 −Ct k2F ≤ F (X1 )−inf F min(c1 ,δ/2)T . On nonconvex optimization problems, the optimal convergence rate for first-order methods is O(1/T ) [31], [40]. Thus, the convergence rate of Algorithm 6 (Corollary 3.11) cannot improve that of Algorithm 5 (Corollary 3.8). However, in practice, acceleration can still significantly reduce the number of iterations on nonconvex problems [25], [31]. On the other hand, as Algorithm 6 may need a second proximal step (step 10), its iteration time complexity can be higher than that of Algorithm 5. However, this is much compensated by the speedup in convergence. As will be demonstrated in Section 6.1, empirically Algorithm 6 is much faster. IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, SUBMITTED The following Theorem shows that any limit point of the iterates from Algorithm 6 is also a critical point. Proof can be found in Appendix C.11. Theorem 3.12. Let {Xtj } be a subsequence of {Xt } generated by Algorithm 6 such that limtj →∞ Xtj = X∗ . With the assumption in Theorem 3.9, X∗ is a critical point of (1). 4 A PPLICATIONS In this section, we consider two important instances of problem (1), namely, matrix completion [1] and robust principal component analysis (RPCA) [7]. As the accelerated FaNCL algorithm (Algorithm 6) is usually faster than its nonaccelerated variant, we will only consider the accelerated variant here. For matrix completion (Section 4.1), we will show that Algorithm 6 can be made even faster and require much less memory by using the “sparse plus low-rank” structure of the problem. In Section 4.2, we show how Algorithm 6 can be extended to deal with the two parameter blocks in RPCA. 4.1 Matrix Completion Matrix completion attempts to recover a low-rank matrix O ∈ Rm×n by observing only some of its elements [1]. Let the observed positions be indicated by Ω ∈ {0, 1}m×n , such that Ωij = 1 if Oij is observed, and 0 otherwise. Matrix completion can be formulated as an optimization problem in (1), with F (X) = 1 kPΩ (X − O)k2F + λr(X), 2 (6) where [PΩ (A)]ij = Aij if Ωij = 1 and 0 otherwise. In the following, we show that the time and space complexities of Algorithm 6 can be further reduced. 4.1.1 Utilizing the Problem Structure First, consider step 7, which checks the objectives. Computing F (Xt ) relies only on the observed positions in Ω and the singular values of Xt . Hence, instead of explicitly constructing Xt , we maintain the SVD Ut Σt Vt> of Xt and a sparse matrix PΩ (Xt ). Computing F (Xt ) then takes O(kΩk1 rt ) time. Computing F (Xat+1 ) takes O(kΩk1 kt ) time, as Rt has rank kt . Next, since Yt is a linear combination of Xt and Xt−1 in step 4, we can use the above SVD-factorized form and compute kXat+1 − Yt k2F in O((m + n)kt2 ) time. Thus, step 7 then takes O(kΩk1 kt + (m + n)kt2 ) time. Steps 6 and 10 perform inexact proximal step. For the first proximal step (step 6), Yt (defined in step 4) can be rewritten as (1+βt )Xt −βt Xt−1 , where βt = (αt−1 −1)/αt . When it calls InexactPS, step 1 of Algorithm 4 has 1 A = Yt + PΩ (Yt − O) τ = (1 + βt )Xt − βt Xt−1 + 1 PΩ (Yt − O). τ (7) The first two terms involve low-rank matrices, while the last term involves a sparse matrix. This special “sparse plus low-rank” structure [14], [15] can speed up matrix 6 multiplications. Specifically, for any V ∈ Rn×k , AV can be obtained as > AV =(1 + βt )Ut Σt (Vt> V) − βt Ut−1 Σt−1 (Vt−1 V) 1 + PΩ (O − Yt )V. (8) τ Similarly, for any U ∈ Rm×k , U> A can be obtained as > U> A =(1 + βt )(U> Ut )Σt Vt> −βt (U> Ut−1 )Σt−1 Vt−1 1 + U> PΩ (O − Yt ). (9) τ Both (8) and (9) take O((m + n)kt k + kΩk1 k) time (instead of O(mnk)). As Rt in step 5 of Algorithm 6 has kt columns, each call to approximate GSVT takes O((m+n)kt2 +kΩk1 kt ) time [15] (instead of O(mnkt )). Finally, step 5 in Algorithm 4 also takes O((m + n)kt2 + kΩk1 kt ) time. As a result, step 6 of Algorithm 6 takes a total of O((m + n)kt2 + kΩk1 kt ) time. Step 10 is slightly cheaper (as no Xt−1 is involved), and its time complexity is O((m + n)rt kt + kΩk1 rt ). Summarizing, the iteration time complexity of Algorithm 6 is O((m + n)kt2 + kΩk1 kt ). (10) Usually, kt n and kΩk1 mn [1], [14]. Thus, (10) is much cheaper than the O(mnkt ) complexity of standard FaNCL-acc (Section 3.6). The space complexity is also reduced. We only need to store the low-rank factorizations of Xt and Xt−1 , and the sparse matrices PΩ (Xt ) and PΩ (Xt−1 ). These take a total of O((m+n)kt +kΩk1 ) space (instead of O(mn) in Section 3.6). These techniques can also be used on Algorithm 5. It can be easily shown that its iteration time complexity is O((m + n)rt kt + kΩk1 rt ), and its space complexity is O((m + n)rt + kΩk1 ) (as no Xt−1 is involved). 4.1.2 Comparison with Existing Algorithms Table 2 compares the convergence rates iteration time complexities, and space complexities of various matrix completion algorithms that will be empirically compared in Section 6. Overall, the proposed algorithms (Algorithms 5 and 6) enjoy fast convergence, cheap iteration complexity and low memory cost. While Algorithms 5 and 6 have the same convergence rate, we will see in Section 6.1 that Algorithm 6 (which uses acceleration) is significantly faster. 4.2 Robust Principal Component Analysis (RPCA) Given a noisy data matrix O ∈ Rm×n , RPCA assumes that O can be approximated by the sum of a low-rank matrix X plus some sparse noise S [7]. Its optimization problem is: min F (X, S) ≡ f (X, S) + λr(X) + υg(S), X,S (11) where f (X, S) = 21 kX+S−Ok2F , r is a low-rank regularizer, and g is a sparsity-inducing regularizer. Here, we allow both r and g to be nonconvex and nonsmooth. Thus, (11) can be seen as a nonconvex extension of RPCA (which uses the nuclear norm regularizer for r and `1 -regularizer for g ). Some examples of nonconvex r are shown in Table 1, and examples of nonconvex g include the `1 -norm, capped-`1 norm [18] and log-sum-penalty [19]. IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, SUBMITTED 7 TABLE 2 Comparison of the iteration time complexities, convergence rates and space complexity of various matrix completion solvers. Here, kt = rt + rt−1 , ν ∈ (0, 1) and integer Ta > 0 are constants. For the active subspace selection method (active) [17], Ts is the number of inner iterations required. regularizer (convex) nuclear norm fixed-rank factorization nonconvex method APG [13], [34] active [17] AIS-Impute [15] LMaFit [41] ER1MP [42] IRNN [5] GPG [26] FaNCL FaNCL-acc convergence rate O(1/T 2 ) O(ν T −Ta ) O(1/T 2 ) — O(ν T ) — — O(1/T ) O(1/T ) While (11) involves two blocks of parameters (X and S), they are not coupled together. Thus, we can use the separable property of proximal operator [29]: proxλr+υg ([X, S]) = [proxλr (X), proxυg (S)]. For many popular sparsity-inducing regularizers, computing proxυg (S) takes only O(mn) time [24]. For P example, when g(S) = = i,j \|Sij \|, [proxυg (S)]ij sign(Sij ) max(\|Sij \| − υ, 0), where sign(x) is the sign of x. However, directly computing proxλr (X) requires O(mn2 ) time and is expensive. To alleviate this problem, Algorithm 5 can be easily extended to Algorithm 7. The iteration time complexity, which is dominated by the inexact proximal steps in steps 6 and 13, is reduced to O(mnkt ). Algorithm 7 FaNCL-acc algorithm for RPCA. Input: choose τ > ρ, λ0 > λ, δ > 0, ν ∈ (0, 1) and c1 = τ −ρ 4 ; 1: initialize V0 , V1 ∈ Rn as random Gaussian matrices, X0 = X1 = 0, S0 = S1 = 0 and α0 = α1 = 1; 2: for t = 1, 2, . . . T do 3: λ (λt−1 − λ)ν + λ ; t = YtX , YtS = Xt , St 4: −1 Xt , St − Xt−1 , St−1 ; + αt−1 αt 5: Rt = QR([Vt , Vt−1 ]); // warm start 6: Xat+1 = InexactPS(YtX , Rt ); 7: Sat+1 = prox υ g (YtS − τ1 ∇S f (YtX , YtS )); τ 8: ∆t = kXat+1 − YtS k2F + kSat+1 − YtS k2F ; 9: if F (Xat+1 , Sat+1 ) ≤ F (Xt , St ) − 2δ ∆t then 10: Xt+1 = Xat+1 ; 11: St+1 = Sat+1 ; 12: else 13: Xt+1 = InexactPS(Xt , Rt ); 14: St+1 = prox υ g (St − τ1 ∇S f (Xt , St )); τ 15: end if p 16: αt+1 = 12 ( 4αt2 + 1 + 1); 17: end for 18: return XT +1 and ST +1 . iteration time complexity O(mnrt ) O(kΩk1 kt Ts ) O(kΩk1 kt + (m + n)kt2 ) O(kΩk1 rt + (m + n)rt2 ) O(kΩk1 ) O(mn2 ) O(mn2 ) O (kΩk1 rt + (m + n)rt kt ) O kΩk1 kt + (m + n)kt2 space complexity O(mn) O((m + n)kt + kΩk1 ) O((m + n)kt + kΩk1 ) O((m + n)rt + kΩk1 ) O((m + n)rt + kΩk1 ) O(mn) O(mn) O((m + n)rt + kΩk1 ) O((m + n)kt + kΩk1 ) Theorem 4.3. Let [Xtj , Stj ] be a subsequence of {[Xt , St ]} generated by Algorithm 6 such that limtj →∞ Xtj = X∗ and limtj →∞ Stj = S∗ With the assumption in Theorem 3.9, [X∗ , S∗ ] is a critical point of (11). 5 PARALLEL FA NCL FOR M ATRIX C OMPLETION In this section, we show how the proposed algorithms can be parallelized. We will only consider the matrix completion problem in (6). Extension to other problems, such as RPCA in Section 4.2, can be similarly performed. Moreover, for simplicity of discussion, we focus on the simpler FaNCL algorithm (Algorithm 5). Its accelerated variant (Algorithm 6) can be similarly parallelized and is shown in Appendix A. Parallel algorithms for matrix completion have been proposed in [43], [44], [45]. However, they are based on stochastic gradient descent and matrix factorization, and cannot be directly used here. 5.1 Proposed Algorithm Convergence results in Section 3.6 can be easily extended to this RPCA problem. Proofs of the following can be found in Appendices C.12, C.13, and C.14. Operations on a matrix X are often of the form: (i) multiplications U> X and XV for some U, V (e.g., in (8), (9)); and (ii) element-wise operation (e.g., evaluation of F (X) in (5)). A popular scheme in parallel linear algebra is block distribution [46]. Assume that there are q threads for parallelization. Block distribution partitions the rows and columns of X into q parts, leading to a total of q 2 blocks. Figure 1 shows how computations of XV, U> X and element-wise operation can be easily parallelized. In Algorithm 5, the most important variables are the low-rank factorized form Ut Σt Vt> of Xt , and the sparse matrices PΩ (Xt ), PΩ (O). Using block distribution, they are thus partitioned as in Figure 2. The resultant parallelized version of FaNCL is shown in Algorithm 8. Steps that can be parallelized are marked with “B”. Two new subroutines are introduced, namely, IndeSpan-PL (step 6) which replaces QR factorization, and ApproxGSVT-PL (step 9) which is the parallelized version of Algorithm 3. They will be discussed in more detail in the following Sections. Note that Algorithm 8 is equivalent to Algorithm 5 except that it is parallelized. Thus, the convergence results in Section 3.5 still hold. Proposition 4.1. The sequence {[Xt , St ]} generated from Algorithm 7 is bounded, and has at least one limit point. Corollary 4.2. Let Ct = YtX , YtS if steps 10 and 11 are performed, and Ct = [Xt , St ] otherwise. Then, (X1 ,S1 )−inf F mint=1,...,T k[Xt+1 , St+1 ] − Ct k2F ≤ Fmin(c . 1 ,δ/2)T 5.1.1 Identifying the Span (Step 5) In step 4 of Algorithm 5, QR factorization is used to find the span of matrix [Vt , Vt−1 ]. This can be parallelized with the Householder transformation and Gaussian elimination [46], which however is typically very complex. The following IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, SUBMITTED 8 (a) U> X. (b) XV. (c) Element-wise operation. Fig. 1. Parallelization of different matrix operations. Here, the number of threads q is equal to 3. Each dotted path denotes operation of a thread. Moreover, though step 3 uses SVD it only takes O(k 3 ) time. Algorithm 9 Parallel algorithm to identify the span of A: IndeSpan-PL(A). (a) UΣV> . (b) O. Fig. 2. Partitioning of variables UΣV> and O, and three threads are used (q = 3). Input: matrix A ∈ Rn×k ; 1: B B = A> A; 2: [U, Σ, V] = SVD(B); 3: construct w as in Proposition 5.1; 4: V = VDiag(w); 5: B Q = AV; > > 6: return Q. // Q = [Q> 1 , . . . , Qp ] Algorithm 8 FaNCL in parallel: FaNCL-PL. Input: choose τ > ρ, λ0 > λ and ν ∈ (0, 1); 1: initialize V0 , V1 ∈ Rn as random Gaussian matrices and X1 = 0; 2: partition X1 , PΩ (X1 ) and PΩ (O); 3: start q threads for parallelization; 4: for t = 1, 2, . . . T do 5: λt = (λt−1 − λ)ν t + λ; 6: B Rt = IndeSpan-PL ([Vt , Vt−1 ]); 7: B At = Xt − τ1 PΩ (Xt − O); 8: for p = 1, 2, . . . do 9: B [X̃p , Rt ] = ApproxGSVT-PL(At , Rt , λτ ); 10: B ap = F (X̃p ); 11: B at = F (Xt ); 12: B aF = kX̃p − Xt k2F ; 13: if ap ≤ at − c1 aF then 14: break; 15: end if 16: end for 17: B Xt+1 = X̃p ; 18: end for 19: return XT +1 . Proposition proposes a simpler method to identify the span of a matrix. Proof can be found in Appendix C.15. Proposition 5.1. Given a matrix A, let the SVD of A> A be VΣV> , w = [wi ] where wi = Σii if Σii > 0 and 1 otherwise. −1 Then, AV (Diag(w)) 2 is orthogonal and contains span(A). The resultant parallel algorithm is shown in Algorithm 9. Its time complexity is O(( nq + q)k 2 + k 3 ). Algorithm 8 calls Algorithm 9 with input [Vt , Vt−1 ], and thus takes O(( nq + q)kt2 + kt3 ) time, where kt = rt + rt−1 . We do not parallelize steps 2-4, as only k × k matrices are involved and k is small. 5.1.2 Approximate GSVT (Step 8) The key steps in approximate GSVT (Algorithm 3) are the power method and SVD. The power method can be parallelized straightforwardly as in Algorithm 10, in which we also replace the QR subroutine with Algorithm 9. Algorithm 10 Parallel power method: PowermethodPL(Z, R). Input: matrix Z ∈ Rm×n , R ∈ Rn×k . 1: B Y1 = ZR; 2: for j = 1, 2, . . . , J do 3: B Qj = IndeSpan-PL(Yj ); 4: B Yj+1 = Z(Z> Qj ); 5: end for 6: return QJ . As for SVD, multiple QR factorizations are usually needed for parallelization [46], which are complex as discussed in Section 5.1.1. The following Proposition performs it in a simpler manner. Proof can be found in Appendix C.16. Proposition 5.2. Given a matrix B ∈ Rn×k , let P ∈ Rn×k be orthogonal and equals span(B), and the SVD of P> B be UΣV> . Then, the SVD of B is (PU)ΣV. The resultant parallelized procedure for approximate GSVT is shown in Algorithm 11. At step 5, a small SVD is performed (by a single thread) on the k × k matrix P> B. At step 8 of Algorithm 8, X̃p is returned from Algorithm 11, and we keep X̃p in its low-rank factorized form. Besides, when Algorithm 11 is called, Z = At and has the “sparse plus low-rank” structure mentioned earlier. Hence, (8) and (9) can be used to speed up matrix multiplications2 . As Rt has kt columns in Algorithm 8, 2. As no acceleration is used, βt is equal to 0 in these two equations. IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, SUBMITTED 2 PowerMethod-PL in step 1 takes O( kqt kΩk1 + m+n q kt ) time, n 2 3 steps 2-6 take O(( q + q)kt + kt ) time, and the rest takes O(kt ) time. The total time complexity for Algorithm 11 is 2 2 O( kqt kΩk1 + m+n q kt + (q + kt )kt ). Algorithm 11 Approximate GSVT in parallel: ApproxGSVTPL(Z, R, µ). Input: partitioned matrix Z ∈ Rm×n and R ∈ Rn×k ; 1: B Q = PowerMethod-PL(Z, R); 2: B B = Z> Q; // B ∈ Rn×k 3: B P = Iden-Span(B); 4: B A = P> B; 5: [U, Σ, V] = SVD(A); // U, Σ, V, A ∈ Rk×k 6: B U = PU; 7: a = number of Σii ’s that are > γ in Corollary 3.2; 8: B Ua = a leading columns of U; 9: B Va = a leading columns of V; 10: for i = 1, 2, . . . , a do 11: obtain yi∗ from (4); 12: end for 13: return the low-rank components of X̃ (QUa , Diag([y1∗ , . . . , ya∗ ]) and Va> ), and V. 5.1.3 Checking of Objectives (steps 9-11) As shown in Figures 1(c), computation of kPΩ (Xt − O)k2F in F (·) can be directly parallelized and takes O( p1 kΩk1 ) time. As r only relies on Σt , only one thread is needed to evaluate r(Xt ). Thus, computing F (Xt ) takes O( rqt kΩk1 ) time. Similarly, computing F (X̃p ) takes O( kqt kΩk1 ) time. > > As kX̃p − Xt k2F = tr(X̃> p X̃p − 2X̃p Xt − Xt Xt ), the lowrank factorized forms of X̃p and Xt can be utilized. Based 2 on Figures 1(a) and 1(b), it can be performed in O( m+n q kt ) time. Thus, the time complexity for steps 9-11 in Algorithm 8 2 is O( kqt kΩk1 + m+n q kt ). The iteration time complexity for Algorithm 8 is thus O( 1q ((m + n)kt2 + kΩk1 kt ) + (q + kt )kt2 ). Compared with (10), the speedup w.r.t. the number of threads q is almost linear. 6 E XPERIMENTS In this section, we perform experiments on matrix completion, RPCA and the parallelized variant of Algorithm 6 (Appendix A). Experiments are performed on a Windows server 2013 system with Intel Xeon E5-2695-v2 CPU (12 cores, 2.4GHz) and 256GB memory. All the algorithms in Sections 6.1 and 6.2 are implemented in Matlab. For Section 6.3, we use C++, the Intel-MKL package3 for matrix operations, and the standard thread library4 for multi-thread programming. 6.1 Matrix Completion We compare a number of low-rank matrix completion solvers, including models based on (i) the commonly used (convex) nuclear norm regularizer; (ii) fixed-rank factorization models [41], [42], which decompose the observed 3. https://software.intel.com/en-us/intel-mkl 4. http://www.cplusplus.com/reference/thread/thread/ 9 matrix O into a product of rank-k matrices U and V. Its optimization problem can be written as: minU,V 12 kPΩ (UV − O)k2F + λ2 (kUk2F + kVk2F ); and (iii) nonconvex regularizers, including √ the capped-`1 (with θ in Table 1 set to 2λ), LSP (with θ = λ), and TNN (with θ = 3). The nuclear norm minimization algorithms to be compared include: 1) 2) 3) Accelerated proximal gradient (APG) algorithm [13], [34], with the partial SVD by PROPACK [36]; AIS-Impute [14], an inexact and acceleration proximal algorithm. The “sparse plus low-rank” structure of the matrix iterate is utilized to speed up computation (Section 4.1); and Active subspace selection (denoted “active”) [17], which adds/removes rank-one subspaces from the active set in each iteration. The nuclear norm optimization problem is then reduced to a smaller problem defined only on this active set. We do not compare with the Frank-Wolfe algorithm [16] and stochastic gradient descent [47], as they have been shown to be less efficient [15], [17]. For the fixed-rank factorization models (where the rank is tuned by the validation set), we compare with the two state-of-the-art algorithms: 1) 2) Low-rank matrix fitting (LMaFit) algorithm [41]; and Economical rank-one matrix pursuit (ER1MP) [42], which pursues a rank-one basis in each iteration. We do not compare with the concave-convex procedure [3], [18], since it has been shown to be inferior to IRNN [24]. For models with nonconvex low-rank regularizers, we compare with the following solvers: 1) 2) 3) Iterative reweighted nuclear norm (IRNN) [5]; Generalized proximal gradient (GPG) algorithm [26], with the underlying problem (4) solved using the closed-form solutions in [24]; and The proposed FaNCL algorithm (Algorithm 5) and its accelerated variant FaNCL-acc (Algorithm 6). We set J = 3 and p = 1. All the algorithms are stopped when the difference in objective values between consecutive iterations becomes smaller than 10−5 . 6.1.1 Synthetic Data The observed m × m matrix is generated as O = UV + G, where the elements of U ∈ Rm×k , V ∈ Rk×m (with k = 5) are sampled i.i.d. from the standard normal distribution N (0, 1), and elements of G sampled from N (0, 0.1). A total of kΩk1 = 2mk log(m) random elements in O are observed. Half of them are used for training, and the rest as validation set for parameter tuning. Testing is performed on the unobserved elements. For performance evaluation, we use (i) the normalized mean squared error NMSE = kPΩ⊥ (X − UV)kF /kPΩ⊥ (UV)kF , where X is the recovered matrix and Ω⊥ denotes the unobserved positions; (ii) rank of X; and (iii) training CPU time. We vary m in the range {500, 1000, 2000}. Each experiment is repeated five times. IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, SUBMITTED 10 TABLE 3 Matrix completion performance on the synthetic data. Here, NMSE is scaled by 10−2 , CPU time is in seconds and the number in brackets is the data sparsity. The best results (according to the pairwise t-test with 95% confidence) are highlighted. m = 500 (12.43%) m = 1000 (6.91%) m = 2000 (3.80%) NMSE rank time NMSE rank time NMSE rank time nuclear norm APG 4.26±0.01 50 12.6±0.7 4.27±0.01 61 99.6±9.1 4.13±0.01 77 1177.5±134.2 AIS-Impute 4.11±0.01 55 5.8±2.9 4.01±0.03 57 37.9±2.9 3.50±0.01 65 338.1±54.1 active 5.37±0.03 53 12.5±1.0 6.63±0.03 69 66.4±3.3 6.44±0.10 85 547.3±91.6 fixed rank LMaFit 3.08±0.02 5 0.5±0.1 3.02±0.02 5 1.3±0.1 2.84±0.03 5 4.9±0.3 ER1MP 21.75±0.05 40 0.3±0.1 21.94±0.09 54 0.8±0.1 20.38±0.06 70 2.5±0.3 capped `1 IRNN 1.98±0.01 5 14.5±0.7 1.99±0.01 5 146.0±2.6 1.79±0.01 5 2759.9±252.8 GPG 1.98±0.01 5 14.8±0.9 1.99±0.01 5 144.6±3.1 1.79±0.01 5 2644.9±358.0 FaNCL 1.97±0.01 5 0.3±0.1 1.98±0.01 5 1.0±0.1 1.79±0.01 5 5.0±0.4 FaNCL-acc 1.97±0.01 5 0.1±0.1 1.95±0.01 5 0.5±0.1 1.78±0.01 2.3±0.2 LSP IRNN 1.96±0.01 5 16.8±0.6 1.89±0.01 5 196.1±3.9 1.79±0.01 5 2951.7±361.3 GPG 1.96±0.01 5 16.5±0.4 1.89±0.01 5 193.4±2.1 1.79±0.01 5 2908.9±358.0 FaNCL 1.96±0.01 5 0.4±0.1 1.89±0.01 5 1.3±0.1 1.79±0.01 5 5.5±0.4 FaNCL-acc 1.96±0.01 5 0.2±0.1 1.89±0.01 5 0.7±0.1 1.77±0.01 2.4±0.2 TNN IRNN 1.96±0.01 5 18.8±0.6 1.88±0.01 5 223.1±4.9 1.77±0.01 5 3220.3±379.7 GPG 1.96±0.01 5 18.0±0.6 1.88±0.01 5 220.9±4.5 1.77±0.01 5 3197.8±368.9 FaNCL 1.95±0.01 5 0.4±0.1 1.88±0.01 5 1.4±0.1 1.77±0.01 5 6.1±0.5 FaNCL-acc 1.96±0.01 5 0.2±0.1 1.88±0.01 5 0.8±0.1 1.77±0.01 2.9±0.2 Results are shown in Table 3. As can be seen, nonconvex regularization (capped-`1 , LSP and TNN) leads to much lower NMSE’s than convex nuclear norm regularization and fixed-rank factorization. Moreover, the nuclear norm and ER1MP output much higher ranks. In terms of speed among the nonconvex low-rank solvers, FaNCL is fast, and FaNCLacc is the fastest. The larger the matrix, the higher is the speedup of FaNCL and FaNCL-acc over GPG and IRNN. 6.1.2 MovieLens Experiment is performed on the popular MovieLens data set (Table 4), which contain ratings of different users on movies. We follow the setup in [42], and use 50% of the observed ratings for training, 25% for validation and the rest for testing. For performance evaluation, we use q the root mean squared (a) capped-`1 . error on the test set Ω̄: RMSE = kPΩ̄ (X − O)k2F /kΩ̄k1 , where X is the recovered matrix. The experiment is repeated five times. TABLE 4 Recommendation data sets used in the experiments. #users #movies #ratings MovieLens 100K 943 1,682 100,000 1M 6,040 3,449 999,714 10M 69,878 10,677 10,000,054 netflix 480,189 17,770 100,480,507 yahoo 249,012 296,111 62,551,438 (b) LSP. Fig. 3. Objective vs CPU time for the capped-`1 and LSP on MovieLens100K. The plot of TNN is similar and thus not shown. Results are shown in Table 5. Again, nonconvex regularizers lead to the lowest RMSE’s. Moreover, FaNCLacc is also the fastest among nonconvex low-rank solvers, even faster than the state-of-the-art GPG. In particular, FaNCL and its accelerated variant FaNCL-acc are the only solvers (for nonconvex regularization) that can be run on the MovieLens-1M and 10M data sets. Figure 3 compares the objectives vs CPU time for the nonconvex regularization solvers on MovieLens-100K. As can be seen, FaNCL and FaNCL-acc decrease the objective and RMSE much faster than the others. Figure 4 shows the testing RMSEs on MovieLens-10M. Again, FaNCL-acc is the fastest. use 50% of the observed ratings for training, 25% for validation and the rest for testing. Each experiment is repeated five times. Results are shown in Table 6. APG, GPG and IRNN cannot be run as the data set is large. AIS-Impute has similar running time as LMaFit but inferior performance, and thus is not compared. Again, the nonconvex regularizers converge faster, yield lower RMSE’s and solutions of much lower ranks. Figure 5 shows the objectives and RMSE vs time, and FaNCL-acc is the fastest.5 6.1.3 Netflix and Yahoo Next, we perform experiments on two very large recommendation data sets, Netflix and Yahoo (Table 4). We randomly 5. On these two data sets, ER1MP easily overfits as the rank increases. Hence, the validation set selects a smaller rank (relative to that obtained by the nuclear norm) and ER1MP stops earlier. However, as can be seen, its RMSE is much worse. IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, SUBMITTED 11 TABLE 5 Matrix completion results on the MovieLens data sets (time is in seconds). The best results (according to the pairwise t-test with 95% confidence) are highlighted. MovieLens-100K MovieLens-1M MovieLens-10M RMSE rank time RMSE rank time RMSE rank time nuclear APG 0.877±0.001 36 47.1±7.6 0.818±0.001 67 2174.4±117.3 — — > 105 norm AIS-Impute 0.878±0.002 36 1.2±0.1 0.819±0.001 67 12.4±0.9 0.813±0.001 100 540.6±107.9 active 0.878±0.001 36 2.7±0.3 0.820±0.001 67 82.6±5.2 0.814±0.001 100 2338.3±304.7 fixed LMaFit 0.865±0.002 2 1.2±0.2 0.806±0.003 6 24.8±1.3 0.792±0.001 9 501.7±95.2 rank ER1MP 0.917±0.003 5 0.1±0.1 0.853±0.001 13 1.3±0.1 0.852±0.002 22 62.7±17.8 capped `1 LSP TNN IRNN GPG FaNCL FaNCL-acc IRNN GPG FaNCL FaNCL-acc IRNN GPG FaNCL FaNCL-acc 0.854±0.003 0.855±0.002 0.855±0.003 0.860±0.009 0.856±0.001 0.856±0.001 0.856±0.001 0.853±0.001 0.854±0.004 0.853±0.005 0.865±0.016 0.861±0.009 3 3 3 3 2 2 2 2 3 3 3 3 289.9±60.6 223.6±49.7 0.5±0.2 0.3±0.1 286.6±48.1 277.5±56.9 0.5±0.1 0.2±0.1 133.8±30.7 591.3±127.2 1.2±0.2 0.4±0.1 — — 0.788±0.002 0.791±0.001 — — 0.786±0.001 0.787±0.001 — — 0.786±0.001 0.786±0.001 — — 5 5 — — 5 5 — — 5 5 > 104 > 104 16.8±2.9 4.7±0.5 > 104 > 104 24.6±1.5 5.4±0.5 > 104 > 104 23.8±0.9 5.4±0.3 — — 0.783±0.001 0.778±0.001 — — 0.779±0.001 0.779±0.001 — — 0.780±0.001 0.778±0.001 — — 8 8 — — 9 9 — — 8 9 > 105 > 105 341.6±58.5 98.4±14.7 > 105 > 105 641.4±82.6 209.3±42.5 > 105 > 105 712.0±142.6 207.6±56.9 TABLE 6 Results on the netflix and yahoo data sets (CPU time is in minutes). The best results (according to the pairwise t-test with 95% confidence) are highlighted. netflix yahoo RMSE rank time RMSE rank time fixed rank LMaFit 0.811±0.001 15 116.4±12.2 0.666±0.001 10 229.3±62.8 ER1MP 0.862±0.006 25 7.1±1.1 0.810±0.003 77 27.9±7.5 capped-`1 FaNCL 0.798±0.001 13 220.0±24.0 0.656±0.001 8 333.0±113.9 FaNCL-acc 0.795±0.001 13 92.8±13.8 0.651±0.001 8 90.2±16.2 LSP FaNCL 0.794±0.001 15 145.5±10.2 0.652±0.001 9 339.8±93.9 FaNCL-acc 0.792±0.001 15 69.6±21.1 0.650±0.001 8 76.1±33.2 TNN FaNCL 0.797±0.001 13 275.1±16.7 0.657±0.001 7 321.2±69.1 FaNCL-acc 0.795±0.001 13 104.3±17.1 0.650±0.001 7 72.9±16.1 (a) netflix. Fig. 4. RMSE vs CPU time on the MovieLens-10M data sets. 6.2 6.2.1 Robust Principal Component Analysis Synthetic Data In this section, we first perform experiments on a synthetic data set. The observed m × m matrix is generated as O = UV+ S̃+G, where elements of U ∈ Rm×k , V ∈ Rk×m (with k = 0.01m) are sampled i.i.d. from N (0, 1), and elements of G are sampled from N (0, 0.1). Matrix S̃ is sparse, with 1% of its elements randomly set to 5kUVk∞ or −5kUVk∞ with equal probabilities. The whole data set is then randomly split into training and test sets of equal size. The standard `1 regularizer is used as the sparsity regularizer g in (11), and different convex/nonconvex lowrank regularizers are used as r. Hyperparameters λ and υ in (11) are tuned using the training set. For performance evaluation, we use (i) NMSE = k(X + S) − (UV + S̃)kF /kUV + S̃kF , where X and S are the recovered low-rank and sparse components, respectively; (ii) (b) yahoo. Fig. 5. RMSE vs CPU time on the netflix and yahoo data sets. accuracy on locating the sparse support of S̃ (i.e., percentage of entries that S̃ij and Sij are nonzero or zero together); (iii) the recovered rank and (iv) CPU time. We vary m in {500, 1000, 2000}. Each experiment is repeated five times. Note that IRNN and active subspace selection cannot be IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, SUBMITTED 12 TABLE 7 RPCA performance on synthetic data. NMSE is scaled by 10−3 , and CPU time is in seconds. The best results (according to the pairwise t-test with 95% confidence) are highlighted. m = 500 m = 1000 m = 2000 NMSE rank time NMSE rank time NMSE rank time nuclear norm APG 4.88±0.17 5 4.3±0.2 3.31±0.06 10 24.5±1.0 2.40±0.05 20 281.2±26.7 capped-`1 GPG 4.51±0.16 5 8.5±2.6 2.93±0.07 10 42.9±6.6 2.16±0.05 20 614.1±64.7 FaNCL-acc 4.51±0.16 5 0.8±0.2 2.93±0.07 10 2.8±0.1 2.16±0.05 20 24.9±2.0 LSP GPG 4.51±0.16 5 8.3±2.3 2.93±0.07 10 42.6±5.9 2.16±0.05 20 638.8±72.6 FaNCL-acc 4.51±0.16 5 0.8±0.1 2.93±0.07 10 2.9±0.1 2.16±0.05 20 26.6±4.1 TNN GPG 4.51±0.16 5 8.5±2.4 2.93±0.07 10 43.2±5.8 2.16±0.05 20 640.7±59.1 FaNCL-acc 4.51±0.16 5 0.8±0.1 2.93±0.07 10 2.9±0.1 2.16±0.05 20 26.9±2.7 TABLE 8 PSNR (in dB) and CPU time (in seconds) on the video background removal experiment. The PSNRs for all the input videos are 16.47dB. bootstrap campus escalator hall PSNR time PSNR time PSNR time PSNR time nuclear norm APG 23.07±0.02 524±84 22.47±0.02 101±6 24.01±0.01 594±86 24.25±0.03 553±85 capped-`1 GPG 23.81±0.01 3122±284 23.21±0.02 691±43 24.62±0.02 5369±238 25.22±0.03 4841±255 FaNCL-acc 24.05±0.01 193±18 23.24±0.02 53±5 24.68±0.02 242±22 25.22±0.03 150±10 LSP GPG 23.93±0.03 1922±111 23.61±0.02 324±27 24.57±0.01 5053±369 25.37±0.03 2889±222 FaNCL-acc 24.30±0.02 189±15 23.99±0.02 69±8 24.56±0.01 168±15 25.37±0.03 144±9 TNN GPG 23.85±0.03 1296±203 23.12±0.02 671±21 24.60±0.01 4091±195 25.26±0.04 4709±367 FaNCL-acc 24.12±0.02 203±11 23.14±0.02 49±5 24.66±0.01 254±30 25.25±0.06 148±11 used here. Their objectives are of the form “smooth function plus low-rank regularizer”, but RPCA also has a nonsmooth `1 regularizer. Similarly, AIS-Impute is only for matrix completion. Moreover, FaNCL, which has been shown to be slower than FaNCL-acc, will not be compared. Results are shown in Table 7. The accuracies on locating the sparse support are always 100% for all methods, and thus are not shown. Moreover, while both convex and nonconvex regularizers can perfectly recover the matrix rank and sparse locations, the nonconvex regularizers have lower NMSE’s. As in matrix completion, FaNCL-acc is much faster; the larger the matrix, the higher the speedup. 6.2.2 1 [6]: PSNR = −10 log10 ( mn kX − Ok2F ) where X ∈ Rm×n is the recovered video, and O ∈ Rm×n is the ground-truth. Results are shown in Table 8. As can be seen, the nonconvex regularizers lead to better PSNR’s than the convex nuclear norm. Moreover, FaNCL-acc is much faster than GPG. Figure 7 shows PSNR vs CPU time on the bootstrap and campus data sets. Again, FaNCL-acc converges to higher PSNR much faster. Results on hall and escalator are similar. Background Removal on Videos In this section, we use RPCA for background removal in videos. Four benchmark videos in [7], [8] are used (Table 9), and example frames are shown in Figure 6. As in [7], the image background is considered low-rank, while the foreground moving objects contribute to the sparse component. TABLE 9 Videos used in the experiment. bootstrap campus escalator #pixels / frame 19,200 20,480 20,800 total #frames 9,165 4,317 10,251 (a) bootstrap. (b) campus. (c) escalator. (a) bootstrap. hall 25,344 10,752 (d) hall. Fig. 6. Example image frames in the videos. Given a video with n image frames, each m1 × m2 frame is first reshaped as a m-dimensional column vector (where m = m1 m2 ), and then all the frames are stacked together to form a m × n matrix. The pixel values are normalized to [0, 1], and Gaussian noise from N (0, 0.15) is added. The experiment is repeated five times. For performance evaluation, we use the commonly used peak signal-to-noise ratio (b) campus. Fig. 7. PSNR vs CPU time on the bootstrap and campus videos. 6.3 Parallel Matrix Completion In this section, we experiment with the proposed parallel algorithm in Section 5 on the Netflix and Yahoo data sets (Table 4). We do not compare with factorization-based IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, SUBMITTED algorithms [44], [45], as they have inferior performance (Section 6.1). The machine has 12 cores, and we use one thread for each core. As suggested in [44], we randomly shuffle all the matrix columns and rows √ before partitioning. We use the LSP penalty (with θ = λ) and fix the total number of iterations to 250. The hyperparameters are the same as in Section 6.1.3. Experiments are repeated five times. Convergence of the objective for a typical run is shown in Figure 8. As we have multiple threads running on a single CPU, we report the clock time instead of CPU time. As can be seen, the accelerated algorithms are much faster than the non-accelerated ones, and parallelization provides further speedup. (a) netflix. (b) yahoo. Fig. 8. Objective value vs clock time for the sequential/parallel versions of FaNCL on the netflix and yahoo data sets. Figure 9 shows the speedup with different numbers of threads. As can be seen, the parallelized variants scale well with the number of threads. In particular, scaling is better on yahoo. The observed entries in its partitioned data submatrices are distributed more evenly, which improves performance of parallel algorithms [48]. Another observation is that the speedup can be larger than one. As discussed in [49], in performing multiplications with a large sparse matrix, a significant amount of time is spent on indexing its nonzero elements. When the matrix is partitioned, each submatrix becomes smaller and easier to be indexed. Thus, the memory cache also becomes more effective. 7 C ONCLUSION In this paper, we considered the challenging problem of nonconvex low-rank matrix optimization. The key observations are that for the popular low-rank regularizers, the singular values obtained from the proximal operator can 13 Fig. 9. Speedup vs the number of threads for parallel FaNCL. The red dashed line indicates linear speedup. be automatically thresholded, and the proximal operator can be computed on a smaller matrix. This allows the proximal operator to be efficiently approximated by the power method. We extended the proximal algorithm in this nonconvex optimization setting with acceleration and inexact proximal step. We further parallelized the proposed algorithm, which scales well w.r.t. the number of threads. Extensive experiments on matrix completion and RPCA show that the proposed algorithm is much faster than the state-of-the-art. 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Dhillon, “Scalable coordinate descent approaches to parallel matrix factorization for recommender systems,” in Proceedings of the 12nd International Conference on Data Mining, 2012, pp. 765–774. [45] B. Recht and C. Ré, “Parallel stochastic gradient algorithms for large-scale matrix completion,” Mathematical Programming Computation, vol. 5, no. 2, pp. 201–226, 2013. [46] J. Demmel, M. Heath, and H. Van Der Vorst, “Parallel numerical linear algebra,” Acta Numerica, vol. 2, pp. 111–197, 1993. [47] H. Avron, S. Kale, V. Sindhwani, and S. Kasiviswanathan, “Efficient and practical stochastic subgradient descent for nuclear norm regularization,” in Proceedings of the 29th International Conference on Machine Learning, 2012, pp. 1231–1238. [48] Y. Zhuang, W.-S. Chin, Y.-C. Juan, and C.-J. Lin, “A fast parallel SGD for matrix factorization in shared memory systems,” in Proceedings of the 7th ACM Conference on Recommender Systems, 2013, pp. 249–256. [49] G. Goumas, K. Kourtis, N. 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IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, SUBMITTED A PPENDIX A PARALLEL FA NCL- ACC Algorithm 12 shows the parallel version of FaNCL-acc. Acceleration is performed at step 6. The first inexact proximal step is performed at steps 8-18. Step 19 checks whether the accelerated iterate is accepted. If the condition fails, a second inexact proximal step is performed at steps 22-32. Note that the algorithm is equivalent to Algorithm 6, and thus the convergence analysis in Section 3.6 still holds. Algorithm 12 FaNCL-acc in parallel: FaNCL-acc-PL. Input: choose τ > ρ, λ0 > λ, δ > 0 and ν ∈ (0, 1); 1: initialize V0 , V1 ∈ Rn as random Gaussian matrices, X0 = X1 = 0 and α0 = α1 = 1; 2: partition X0 , X1 , PΩ (X0 ), PΩ (X1 ) and PΩ (O); 3: start q threads for parallelization; 4: for t = 1, 2, . . . T do 5: λt = (λt−1 − λ)ν t + λ; α −1 6: B Yt = Xt + t−1 (Xt − Xt−1 ); αt 7: B Rt = IndeSpan-PL ([Vt , Vt−1 ]); 8: B Aat = Yt − τ1 PΩ (Yt − O); 9: for p = 1, 2, . . . do 10: B [X̃p , Rt ] = ApproxGSVT-PL(Aat , Rt , λτ ); 11: B ap = F (X̃p ); 12: B at = F (Xt ); 13: B aF = kX̃p − Xt k2F ; 14: if ap ≤ at − c1 aF then 15: break; 16: end if 17: end for 18: B Xat+1 = X̃p ; 19: if F (Xat+1 ) ≤ F (Xt ) − 2δ kXat+1 − Yt k2F then 20: B Xt+1 = Xat+1 ; 21: else 22: B At = Xt − τ1 PΩ (Xt − O); 23: for p = 1, 2, . . . do 24: B [X̃p , Rt ] = ApproxGSVT-PL(At , Rt , λτ ); 25: B bp = F (X̃p ); 26: B bt = F (Xt ); 27: B bF = kX̃p − Xt k2F ; 28: if bp ≤ bt − c1 bF then 29: break; 30: end if 31: end for 32: B Xt+1 = X̃p ; 33: end if p 34: αt+1 = 12 ( 4αt2 + 1 + 1); 35: end for 36: return XT +1 . 15 example. Using Proposition 3.6, we can decompose r(X̃p ) as r̆(X̃p ) + r̃(X̃p ), where r̆(X̃p ) r̃(X̃p ) 1 kX̃p k∗ , θ " !# n X σi (X̃p ) σi (X̃p ) . = − log 1 + θ θ i=1 = Let the SVD of X̃p be UΣV> . Assume that X̃p has k singular values larger than 0. Let Uk (resp. Vk ) be the matrix containing the first k columns of U (resp. V). Then, 1 ∂ r̆(X̃p ) = Uk Vk> + B , θ where B ∈ {C : U> k C = 0, CVk = 0, and σ1 (C) ≤ 1}. Let c = [ci ] with ci = θ1 − σi (X1p )+θ . Then, ∂ r̃(X̃p ) = UDiag(c)V> . Thus, a full SVD on X̃p is needed, which is expensive and impractical for large matrices. A PPENDIX C P ROOFS C.1 Proposition 3.1 For simplicity of notations, we write σi (Z) as σi . First, we introduce the definition of super-gradient for a concave function and two lemmas. Definition 2 ( [51]). For a concave function f , its super-gradient ˆ ≡ ∂ (−f ). is given by g ∈ ∂f Lemma C.1 ( [51]). (i) inf g∈∂ˆr̂(y) g ≥ 0; (ii) Assume that yj ≥ yi ≥ 0. Then, supgj ∈∂ˆr̂(yj ) gj ≤ inf gi ∈∂ˆr̂(yi ) gi . Lemma C.2. (i) yi∗ − max (σi − µgi , 0) = 0, where gi ∈ ∂ˆr̂(yi∗ ); (ii) if yi∗ > 0, then yi∗ increases with σi . Proof. (Part (i)): Let gi ∈ ∂ˆr̂(yi∗ ). From the first-order optimality condition of (4), consider the two possibilities: (a) (b) σi + µgi ≤ 0: In other words, the optimal solution is achieved at the boundary, and yi∗ = 0. σi + µgi > 0: We have 0 = yi∗ − σi + µgi , and yi∗ > 0. Combining these two cases, the relationship between yi∗ and σi can be expressed as yi∗ = max (σi − µgi , 0) . (12) (Part (ii)): Assume that yi∗ > 0. Then, (12) becomes yi∗ = σi − µgi . (13) Let σi becomes larger as σj . according to (13), we have two possibilities for its corresponding yj∗ , i.e., A PPENDIX B T HE C HECKING C ONDITION IN [37] The condition in [37] to accept an approximate X̃p is: ∃A ∈ ∂ r̆(X̃p ) − ∂ r̃(X̃p ) where r̃ and r̆ are two convex functions, such that kA + ∇f (X̃p )k2F ≤ bkX̃p − Xk2F for some constant b > 0. Using the LSP regularizer as an • • yj∗ > yi∗ : Then, supgj ∈∂ˆr̂(y∗ ) gj ≤ inf gi ∈∂ˆr̂(y∗ ) gi from j i Lemma C.1. Together with the fact that σj > σi , there exists a yj∗ which is not smaller than yi∗ to make (12) hold. yj∗ ≤ yi∗ : Then, inf gj ∈∂ˆr̂(y∗ ) gj ≥ supgi ∈∂ˆr̂(y∗ ) gi from j i Lemma C.1. However, such a solution may not exist (e.g., when r̂(α) = α). IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, SUBMITTED Thus, while there can be multiple solutions to ensure (13), the first case must exist. We take the largest solution of all possible candidates. Thus, if σi gets larger, yi∗ also becomes larger. Proof of Proposition 3.1. From Lemma C.2, we have and   if 0 ≤ σi ≤ µ 0 p∗i = σi − µ if µ < σi ≤ µ + θ .   θ otherwise qi∗ 0 = yi∗ − max (σi − µgi , 0) . We can see that y ∗ = 0 once σi − µgi ≤ 0. However, if σi becomes smaller, σi − µgi will reach 0 before σi reaches zero. This comes from two facts. First, yi∗ becomes smaller as σi gets smaller (Lemma C.2), but inf gi ∈∂ˆr̂(y∗ ) gi i will not become smaller (Lemma C.1). Second, we have limy→0+ inf g∈∂ˆr̂(y) g > 0. An illustration of the relationships among σi , yi∗ and gi is shown in the following figure. Thus, there exists γ > 0 such that once σi ≤ γ , σi − µgi ≤ 0, and yi∗ becomes 0. 16 (14) Let qi∗ = arg minqi >θ h2 (qi ). As h2 is also quadratic and cannot be θ, we have ( no solution if 0 ≤ σi ≤ θ ∗ qi = . (15) σi otherwise Note that when σi ∈ [0, θ], there is no solution to qi∗ , as qi∗ can arbitrarily close to θ. Since h1 (θ) = h2 (θ), the possibility for θ = arg minyi ≥0 h(yi ) is covered by arg min0≤pi ≤θ h1 . Thus, (14) and (15) have covered all possibilities of yi∗ . Using them, we have 1) If θ ≤ µ, then  h1 (0)    min (h (0), h (σ )) 1 2 i min h = min (h1 (σi − µ), h2 (σi )) yi ≥0    min (h1 (θ), h2 (σi )) 0 ≤ σi ≤ θ θ < σi ≤ µ . µ < σi ≤ µ+θ σi > µ+θ In order to get yi∗ = 0, we need min (h1 (0), h2 (σi )) = h1 (0), which leads to p σi ≤ 2µθ. √ Thus, if 0 ≤ σi ≤ min 2µθ, µ , then yi∗ = 0. C.2 2) Corollary 3.2 In this section, we show how to derive the threshold γ for the capped-`1 penalty. Derivations for the other penalties can be obtained similarly. C.2.1 Capped-`1 Penalty Proof. Note that problem (4) considers each singular value separately. For simplicity of notations, let σi denote σi (Z). For the ith singular value, let h(yi ) ≡ 1 2 (yi − σi ) + µ min (yi , θ) . 2 Thus, Finally, combining the above two cases, we can con√ clude that√ once σi ≤ min( 2θµ, µ), then yi∗ = 0. Thus, γ = min( 2θµ, µ). Proposition 3.3 Proof. First, we introduce the following theorem. , where 1 (pi − σi )2 +µpi , 2 1 h2 (qi ) = (qi − σi )2 + µθ. 2 Note that h1 is quadratic. There are only three possibilities for p∗i = arg min0≤pi ≤θ h1 (pi ), i.e.,  1 2  if p∗i = 0  2 σi 1 2 min h1 (pi ) = µσi − 2 µ if p∗i = σi − µ ,  0≤pi ≤θ 1 2 ∗ 2 (θ − σi ) + µσi if pi = θ h1 (pi ) = 0 ≤ σi ≤ µ µ < σi ≤ θ . θ < σi ≤ µ+θ σi > µ+θ Thus, if 0 ≤ σi ≤ µ, then we have yi∗ = 0. C.3 ( arg min0≤yi ≤θ h1 (yi ) arg min h(yi ) = yi ≥0 arg minyi >θ h2 (yi ) If θ > µ, then   h1 (0)  h (σ − µ) 1 i min h =  yi ≥0 min (h1 (σi − µ), h2 (σi ))    min (h1 (θ), h2 (σi )) Theorem C.3 (Separation theorem [52]). Let A ∈ Rm×n and B ∈ Rm×r with B> B = I. Then σi B> A ≤ σi (A), for i = 1, . . . , min(r, n). Let the SVD of Z be UΣV> . Z can then be rewritten as Σk̂ Z = [Uk̂ ; U⊥ ] [Vk̂ ; V⊥ ]> , (16) Σ⊥ where Uk̂ contains the k̂ leading columns of U, and U⊥ the remaining columns. Similarly, Σk̂ (resp. Vk̂ ) contains the k̂ leading eigenvalues (resp. columns) of Σ (resp. V). Hence, σi Q> Z = max ũ> Q> Z ṽi . (17) i ũi ,ṽi IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, SUBMITTED Let ũi = Q> ui and ṽi = vi , (18) u> i Zvi σi (Z), > p > > Then, kBp B> p − Uk Uk kF ≤ η kB0 B0 − Uk Uk kF , where η ∈ (0, 1) is a constant. Proof. At the pth iteration of Algorithm 4, inside Algorithm 2 (step 3), since Z = A and R = Ṽp−1 , we have where ui (resp. vi ) is the ith column of U (resp. V). Q> Z ṽi = u> QQ> Zvi ũ> i i = = 17 Q1 = QR(AṼp−1 ) = Bp−1 . (19) (20) where (19) is due to span(Uk̂ ) ⊆ span(Q). From Theorem C.3, by substituting Q = B and A = Z, we have σi (Q> Z) ≤ σi (Z). Combining with (20), we obtain that (18) is the optimal solution of (17). Thus, the rank-k̂ SVD of Q> Z is (Q> Uk̂ )Σk̂ Vk̂> , with the corresponding left and right singular vectors contained in Q> Uk̂ and Vk̂ , respectively. Again, by Theorem C.3, we have σk̂+1 Q> Z ≤ σk̂+1 (Z) ≤ γ. Besides, using (16), > ṽi . σi Q> Z =max ũ> Q> Uk̂ Σk̂ Vk̂> +Q> U⊥ Σ⊥ V⊥ i Then, for Ṽp , inside Algorithm 3 (step 2), we have span(Ṽp ) = span(A> Q). Thus, span(AṼp ) = span(A(A> Q)) = span(A(A> QJ )) = span(YJ+1 ), = QR(AṼp ) = Bp . > > > kBp B> p − Uk Uk kF = kQJ+1 QJ+1 − Uk Uk kF > ≤ αJ kQ1 Q> 1 − Uk Uk kF > proxµr (Q Z) = > proxµr (Q Uk̂ Σk̂ Vk̂> + Q> U⊥ Σ⊥ V⊥ ) > > > > ) proxµr (Q Uk̂ Σk̂ Vk̂ ) + proxµr (Q U⊥ Σ⊥ V⊥ > > proxµr (Q Uk̂ Σk̂ Vk̂ ). > = ηkBp−1 B> p−1 − Uk Uk kF , > (22) where η = αJ ∈ (0, 1). Thus, > p > > kBp B> p − Uk Uk kF ≤ η kB0 B0 − Uk Uk kF . (23) where (22) follows from that Q> Uk̂ (resp. Vk̂ ) is orthogonal to QU⊥ (resp. V⊥ ). (21) shows that there are only k̂ singular values in Q> Z larger than γ . Thus, > proxµr (Q> U⊥ Σ⊥ V⊥ ) = 0 and we get (23). Finally, Qproxµr (Q> Z) = Q Q> Uk̂ proxµr (Σk̂ )Vk̂> = Uk̂ proxµr (Σk̂ )Vk̂> (24) = proxµr (Z), (25) where (24) comes from span(Uk̂ ) ⊆ span(Q); (25) comes from that rank-k̂ SVD of Z is Uk̂ Σk̂ Vk̂> and Z only has k̂ singular values larger than γ . C.4 (30) Note that C = Q1 in Lemma C.4. Together with (27) and (30), we have Then, = (29) QJ+1 = QR(YJ+1 ) ũ,ṽ = (28) where (28) comes from the fact that Q is returned after J iterations of Algorithm 2; and (29) from the definition of Yj+1 at step 4 in Algorithm 2. Thus, ũi ,ṽi The first k̂ singular values are from the term Q> Uk̂ Σk̂ Vk̂ . Hence, > ṽ ≤ γ. (21) σk̂+1 Q> Z = max ũ> Q> U⊥ Σ⊥ V⊥ (27) Proposition 3.5 Proof. First, we introduce the following Lemmas. Lemma C.4 ( [27], [53]). In Algorithm 2, let the SVD of Z be ŪΣ̄V̄> , and Ūk contain the first k columns of Ū. We have > j−1 kQj Q> kCC> − Ūk Ū> j − Ūk Ūk kF ≤ α k kF , where α = σk+1 (Z)/σk (Z) ∈ (0, 1) and C = QR(ZR). Lemma C.5. In Algorithm 4, let the rank-k SVD of A be Uk Σk Vk> , and Bp = QR(AṼp ). (26) (Proof of Proposition 3.5) For Bp in (26), we have limp→∞ Bp = Uk from Proposition C.5 where Uk comes from rank-k SVD of A. As k ≥ k̂A , span(Uk̂A ) ⊆ span(Uk ). Then, from Proposition 3.3, we have Uk prox λ r (U> k A) = prox λ r (A). τ τ Thus, limp→∞ X̃p = prox λ r (A). τ C.5 Proposition 3.6 Proof. First, we introduce Lemma C.6. Pm Lemma C.6 ( [54]). Let φ(X) = i=1 f (σi (X)). If f is convex, φ is also convex on X. For r̂ in Assumption A3, it can be rewritten as r̂(α) = r̂1 (α) − r̂2 (α), where r̂1 (α) = κα (for some constant κ) and r̂2 (α) = κα − r̂(α). Obviously, both r̂1 and r̂2 are convex. Define m m X X r̆(X) = r̂1 (σi (X)), and r̃(X) = r̂2 (σi (X)). i=1 i=1 From Lemma C.6, both r̆ and r̃ are convex. Thus, r can also be written as a difference of convex functions: r(X) = r̆(X) − r̃(X). IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, SUBMITTED C.6 C.9 Proposition 3.7 Proof. From step 5 of Algorithm 5 (which ensures (5)), we have F (Xt+1 ) ≤ F (Xt ) − c1 kXt+1 − Xt k2F . Proposition 3.10 Proof. Consider the two cases: 1) Step 8 in Algorithm 6 is performed: Then, 2) δ F (Xt+1 ) ≤ F (Xt ) − kXt+1 − Yt k2F . 2 Step 10 is performed: Then, Summing this from t = 1 to T , we have c1 T X kXt+1 − Xt k2F ≤ F (X1 ) − F (XT +1 ) F (Xt+1 ) ≤ F (Xt ) − c1 kXt+1 − Xt k2F . t=1 ≤ F (X1 ) − inf F. (31) As F is bounded from below (Assumption A2), a1 ≡ F (X1 ) − inf F is a positive constant. Let T → ∞, we have ∞ X kXt+1 − Xt k2F ≤ t=1 a1 . c1 (32) lim f (X) → ∞, F (X1 ) − F (XT +1 ) X δ X ≥ kXt+1 − Yt k2F + c1 kXt+1 − Xt k2F . 2 1 2 t∈ΩT Corollary 3.8 Proof. Combining (31) and (32), we have min kXt+1 − Xt k2F t=1,...,T ≤ T 1X kXt+1 − Xt k2F T t=1 ≤ ∞ 1X kXt+1 − Xt k2F T t=1 ≤ F (X1 ) − inf F . c1 T lim kXkF →∞ As {Xtj } is a subsequence of {Xt } with limit point X∗ , lim Xtj +1 = InexactPS lim Xtj , lim Rtj , (33) tj →∞ lim kXt+1 − Xt k2F = 0, X 3) Both \|Ω1∞ \| and \|Ω2∞ \| are infinite: As in the above two cases, {Xt } is bounded when either of \|Ω1∞ \| and \|Ω2∞ \| is infinite. Combining the above, {Xt } generated from Algorithm 6 is bounded and has at least one limit point. t→∞ which implies tj →∞ \|Ω1∞ \| is infinite but \|Ω2∞ \| is finite: For tj ∈ Ω1∞ , note that, F (Xtj +1 ) ≤ F (Xtj ) due to (35) and (36). From Assumption A2, then the sequence {F (Xtj )} is bounded. Again, from Assumption A2, we have (40), then {Xtj } is also bounded which has at least one limit point [51]. From (32), we have lim Xtj +1 = lim Xtj = X∗ , (40) inf F (X) > −∞, Lemma C.7 ( [24], [37]). If X = prox λ r (X − τ1 ∇f (X)), then τ X is a critical point of (1). tj →∞ f (X) = ∞, which indicates that maxtj =1,...,∞ kXtj kF < ∞. Together with (39), the sequence {Xt } is bounded, which has at least one limit point [51]. Proof. First, we introduce Lemma C.7. tj →∞ \|Ω1∞ \| is finite but \|Ω2∞ \| is infinite: For tj ∈ Ω2∞ , we have from (38) X a1 kXtj +1 − Xtj k2F ≤ . (39) c1 2 From Assumption A2, we also have Theorem 3.9 tj →∞ (38) t∈Ω∞ tj ∈Ω∞ 2) C.8 (37) where a1 = F (X1 ) − inf F > 0 is a constant. Consider the three cases: 1) C.7 (36) t∈ΩT t∈Ω∞ which implies that maxt=1,...,∞ kXt kt < ∞. Together with (32), {Xt } is a bounded sequence with at least one limit point [51]. (35) Partition the iterations {1, . . . , T } into two sets Ω1T and Ω2T , such that t ∈ Ω1T if step 8 is performed, and t ∈ Ω2T if step 10 is performed. Sum (35) and (36) from t = 1 to T , As F is bounded from below (Assumption A2), X Xδ kXt+1 − Yt k2F + c1 kXt+1 − Xt k2F ≤ a1 . 2 2 1 From Assumption A2, we also have kXkF →∞ 18 (34) for {Xtj }. Combining (33) and (34), we have lim X∗ = InexactPS lim Xtj , lim Rtj , tj →∞ tj →∞ tj →∞ = InexactPS X∗ , lim Rtj . C.10 Corollary 3.11 Proof. Let c2 = min(δ/2, c1 ). From (37), we have F (X1 ) − F (XT +1 ) X X ≥ c2 ( kXt+1 − Yt k2F + kXt+1 − Xt k2F ) t∈Ω1T tj →∞ Thus, X∗ = prox λ r (X∗ − τ1 ∇f (X∗ )) holds by the assumpτ tion. From Lemma C.7, X∗ is a critical point of (1). = c2 T X t=1 kXt+1 − Ct k2F . t∈Ω2T (41) IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, SUBMITTED Thus, min kXt+1 − Ct k2F ≤ T 1X kXt+1 − Ct k2F T t=1 ≤ ∞ 1X kXt+1 − Ct k2F T t=1 t=1,...,T Combining (45) and (46), we have lim Xtj +1 = InexactPS lim Ytj , lim Rtj tj →∞ tj →∞ tj →∞ = InexactPS X∗ , lim Rtj tj →∞ = X∗ . F (X1 ) − inf F ≤ , c2 T Thus, by the assumption, we also have 1 ∇f (X∗ )). τ From Lemma C.7, X∗ is also a critical point of (1). X∗ = prox λ r (X∗ − where the last inequity comes from (41). C.11 τ Theorem 3.12 Proof. Partition the iterations {1, . . . , ∞} into two sets Ω1∞ and Ω2∞ , such that t ∈ Ω1∞ if step 8 is performed, and t ∈ Ω2∞ if step 10 is performed. Consider the three cases: 1) \|Ω1∞ \| is finite but \|Ω2∞ \| is infinite: Let {Xtj } be a subsequence of {Xt } where t ∈ Ω2∞ , and limtj →∞ Xtj = X∗ . From (39), we have lim Xtj = lim Xtj +1 = X∗ . tj →∞ tj →∞ (42) Besides, lim Xtj +1 = InexactPS tj →∞ 3) C.12 1) Steps 10 and 11 are performed: Then, 2) F (Xt+1 , St+1 ) ≤ F (Xt , St ) (47) δ − (kXt+1 − YtX k2F + kSt+1 − YtS k2F ). 2 Steps 13 and 14 are performed: Then, (43) tj →∞ = X∗ . Proposition 4.1 Proof. Consider the two cases: tj →∞ Combining (42) and (43), we have lim Xtj +1 = InexactPS lim Xtj , lim Rtj tj →∞ tj →∞ tj →∞ = InexactPS X∗ , lim Rtj Both \|Ω1∞ \| and \|Ω2∞ \| are infinite: From the above two cases, we can see that the limit point X∗ is also a critical point of (1) when either \|Ω1∞ \| or \|Ω2∞ \| is infinite. Thus, limit points of {Xt } are also critical points of (1). lim Xtj , lim Rtj . tj →∞ F (Xt+1 , St+1 ) ≤ F (Xt , St ) −(c1 kXt+1 − ΘT = 1 X∗ = prox λ r (X∗ − ∇f (X∗ )). τ τ Xδ t∈Ω1T + From Lemma C.7, X∗ is also a critical point of (1). \|Ω1∞ \| is infinite but \|Ω2∞ \| is finite: Let {Xtj } be a subsequence of {Xt } where t ∈ Ω1∞ , and limtj →∞ Xtj = X∗ . From (37), we have X δ kXt+1 − Yt k2F < ∞ 2 1 2 X τ −ρ (c1 kXt+1 − Xt k2F + kSt+1 − St k2F ). 2 2 t∈ΩT Summing (47) and (48) from t = 1 to T , we have F (X1 , S1 ) − F (XT +1 , ST +1 ) ≥ ΘT . (49) As F is bounded from below (Assumption A2), we have Θ∞ ≤ a2 , which indicates (50) where a2 = F (X1 , S1 ) − inf F . We consider the three cases: lim Xtj +1 − Ytj = 0. tj →∞ (44) From (44), we have lim Ytj = lim Xtj +1 = X∗ . tj →∞ (45) Besides, lim Xtj +1 = InexactPS 1) \|Ω1∞ \| is finite but \|Ω2∞ \| is infinite; For tj ∈ Ω2∞ , X a2 kXtj +1 − Xtj k2F ≤ , c1 tj ∈Ω2∞ X 2a2 kStj +1 − Stj k2F ≤ . τ −ρ 2 tj ∈Ω∞ tj →∞ (48) −ρ 2 kSt+1 − St kF ). 2 (kXt+1 − YtX k2F + kSt+1 − YtS k2F ) t∈Ω∞ tj →∞ τ Xt k2F + Partition {1, . . . , T } into two sets as Ω1T and Ω2T , where t ∈ Ω1T if steps 10-11 are performed; otherwise, t ∈ Ω2T (and steps 13-14 are performed). Let Thus, by the assumption, we also have 2) 19 lim Ytj , lim Rtj . tj →∞ tj →∞ (46) Again from Assumption A2, we have lim kXkF →∞ or kSkF →∞ f (X, S) → ∞. (51) IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, SUBMITTED Thus, maxtj =1,...,∞ k[Xtj , Stj ]kF < ∞, and the sequence {[Xtj , Stj ]} is bounded with at least one limit point [51]. 2) 3) \|Ω1∞ \| is infinite but \|Ω2∞ \| is finite: For tj ∈ Ω1∞ , from (47) and (48), we have F (Xtj +1 , Stj +1 ) ≤ F (Xtj , Stj ). As, F is bounded from below (Assumption A2), the sequence {F ([Xtj , Stj ])} must be bounded. Besides, again from Assumption A2, we have (51), which indicates the sequence [Xtj , Stj ]) must be bounded with at least one limit point [51]. \|Ω1∞ \| 20 Partition {1, . . . , ∞} into two sets as Ω1∞ and Ω2∞ , where t ∈ Ω1∞ if steps 10-11 are performed; otherwise, t ∈ Ω2∞ (and steps 13-14 are performed), we consider three cases here. 1) \|Ω1∞ \| is finite but \|Ω2∞ \| is infinite: Let {[Xtj , Stj ]} be a subsequence of {[Xt , St ]} where t ∈ Ω2∞ , and lim Xtj , Stj = [X∗ , S∗ ] . tj →∞ From (49), we have ∞ X \|Ω2∞ \| Both and are infinite: As in the above two cases, {[Xt , St ]} is bounded with at least one limit point once \|Ω1∞ \| or \|Ω2∞ \| is infinite. kXt+1 − Xt k2F < ∞, t=1 ∞ X kSt+1 − St k2F < ∞. t=1 Thus, the sequence {[Xt , St ]} generated from Algorithm 6 is bounded and has at least one limit point. These indicate lim Xtj +1 − Xtj = 0, (53) lim Stj +1 − Stj = 0. (54) tj →∞ C.13 Corollary 4.2 tj →∞ Proof. Let c2 = min(δ/2, c1 ). First, we have Xδ t∈Ω1T + 2 (kXt+1 − YtX k2F + kSt+1 − From (53), we have YtS k2F ) lim Xtj +1 = lim Xtj = X∗ . tj →∞ X τ −ρ (c1 kXt+1 − Xt k2F + kSt+1 − St k2F ) 2 2 T X k[Xt+1 , St+1 ] − Ct k2F . (52) t=1 tj →∞ Together with (49) and (50), we have = X∗ . min k[Xt+1 , St+1 ] − Ct k2F Thus, t=1,...,T 1 ∇X f (X∗ , S∗ )) (56) τ holds by the assumption. Then, the proximal operator is always exact for S. Using (54), we have T 1X ≤ k[Xt+1 , St+1 ] − Ct k2F T t=1 ≤ ∞ X X∗ = prox λ r (X∗ − τ k[Xt+1 , St+1 ] − Ct k2F lim Stj +1 = lim prox µ g (Stj − t=1 tj →∞ F (X1 , S1 ) − inf F , ≤ c2 T τ Definition 3 ( [39]). If X and S satisfy 0 ∈ ∇X f (X, S)+λ (∂ r̆(X)−∂ r̃(X)) , 0 ∈ ∇S f (X, S) + λ (∂ğ(S) − ∂g̃(S)) , 1 ∇S f (Xtj , Stj )) τ 1 ∇S f (X∗ , S∗ )) τ (57) Combining with (56) and (57), [X∗ , S∗ ] is a critical point of (11) by using Lemma C.8. 2) \|Ω1∞ \| is infinite but \|Ω2∞ \| is finite: Let {[Xtj , Stj ]} be a subsequence of {[Xt , St ]} where t ∈ Ω1∞ , and lim Xtj , Stj = [X∗ , S∗ ] . tj →∞ From (49), we have X kXtj +1 − YtXj k2F ≤ ∞, tj ∈Ω2∞ then [X, S] is a critical point of F . X Lemma C.8 ( [37]). If X and S satisfy 1 X = prox λ r (X− ∇X f (X, S)), τ τ 1 S = prox ν g (S − ∇S f (X, S)), τ τ then [X, S] is a critical point of F . τ = S∗ Theorem 4.3 Proof. Let g = ğ + g̃ be the difference of convex decomposition of g . As two blocks of variables are involved, its critical points are defined as follows. tj →∞ = prox µ g (S∗ − where the last inequality comes from (52). C.14 (55) Combing (53) and (55), we have lim Xtj +1 = InexactPS lim Xtj , lim Rtj tj →∞ tj →∞ tj →∞ = InexactPS X∗ , lim Rtj t∈ΩT ≥ c2 tj →∞ kStj +1 − YtSj k2F ≤ ∞, tj ∈Ω2∞ and then lim Xtj +1 − YtXj = 0, (58) lim Stj +1 − YtSj = 0. (59) tj →∞ tj →∞ IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, SUBMITTED C.16 Thus, lim Xtj +1 = lim YtXj = X∗ . tj →∞ (60) tj →∞ Combing (58) and (60), we have lim Xtj +1 = InexactPS lim YtXj , lim Rtj tj →∞ tj →∞ tj →∞ = InexactPS X∗ , lim Rtj tj →∞ 1 ∇X f (X∗ , S∗ )) (61) τ τ holds by the assumption. Then, the proximal operator is always exact for S. Using (59), X∗ = prox λ r (X∗ − lim Stj +1 = lim prox µ g (YtSj τ t →∞ t →∞ j j = prox µ g (S∗ − τ 1 − ∇S f (YtXj , YtSj )) τ 1 ∇S f (X∗ , S∗ )) τ = S∗ (62) Combining with (61) and (62), [X∗ , S∗ ] is a critical point of (11) by using Lemma C.8. Both \|Ω1∞ \| and \|Ω2∞ \| are infinite: As above, either \|Ω1∞ \| or \|Ω2∞ \| is infinite, a limit point [X∗ , S∗ ] is a critical point of (11). Thus, the limit points of the sequence {[Xt , St ]} are also critical points of (11). Proposition 5.1 Proof. As the SVD of A> A is VΣV> , the SVD of A can 1 be written as UΣ 2 V> where U is an orthogonal matrix containing the span of A. From the construction of w, we have AV (Diag(w)) − 21 1 = UΣ 2 (V> V) (Diag(w)) 1 = UΣ 2 (Diag(w)) − 12 − 12 . Consider the two cases. 1) 2) Proof. Let the SVD of B be ŪΣ̄V̄> . Then, P> B = (P> Ū)Σ̄V̄> . Note that (P> Ū)> P> Ū = Ū> (PP> )Ū = Ū> (ŪŪ> )Ū = I, span(P) = span(Ū). Thus, C.15 Proposition 5.2 where the second equality comes from = X∗ . 3) 21 A is of full column rank: Then, 1 1 1 −1 UΣ 2 (Diag(w)) 2 = U Σ 2 Σ− 2 = U, which contains the span of A. Assume that A has k columns and its rank is k̄ < k : Then, 1 −1 UΣ 2 (Diag(w)) 2 1 1 2 = UDiag Σ11 , . . . , Σk̄2k̄ , 0, . . . , 0 −1 −1 Diag Σ112 ,. . ., Σk̄k̄2 , 1,. . ., 1 = [Uk̄ , 0] , where Uk̄ contains the first k̄ columns of U. As A is 1 −1 only of rank k̄ , UΣ 2 (Diag(w)) 2 again covers the span of A. The Proposition then follows. > > (63) Thus, the SVD of P B is (P Ū)Σ̄V̄. As a result, we have V = V̄, Σ = Σ̄. Finally, from U = P> Ū, we have PU = PP> Ū = Ū(Ū> Ū) = Ū, where the second equality again comes from (63).	2
When 3D-Aided 2D Face Recognition Meets Deep Learning: An extended UR2D for Pose-Invariant Face Recognition arXiv:1709.06532v1 [cs.CV] 19 Sep 2017 Xiang Xu, Pengfei Dou, Ha A. Le, Ioannis A. Kakadiaris∗ Computational Biomedicine Lab University of Houston 4800 Calhoun Rd. Houston, TX, USA Abstract Most of the face recognition works focus on specific modules or demonstrate a research idea. This paper presents a pose-invariant 3D-aided 2D face recognition system (UR2D) that is robust to pose variations as large as 90◦ by leveraging deep learning technology. The architecture and the interface of UR2D are described, and each module is introduced in detail. Extensive experiments are conducted on the UHDB31 and IJB-A, demonstrating that UR2D outperforms existing 2D face recognition systems such as VGG-Face, FaceNet, and a commercial off-the-shelf software (COTS) by at least 9% on the UHDB31 dataset and 3% on the IJB-A dataset on average in face identification tasks. UR2D also achieves state-of-the-art performance of 85% on the IJB-A dataset by comparing the Rank-1 accuracy score from template matching. It fills a gap by providing a 3D-aided 2D face recognition system that has compatible results with 2D face recognition systems using deep learning techniques. Keywords: Face Recognition, 3D-Aided 2D Face Recognition, Deep Learning, Pipeline 2010 MSC: 00-01, 99-00 Preprint submitted to SI on Biometrics in the wild, Image and Vision Computing September 20, 2017 Figure 1: Depiction of existing pose problem from selected samples. Distribution of yaw angles are from −90◦ to +90◦ in (T) constrained dataset UHDB31 [1] and in (B) the wild dataset IJB-A [2]. 1. Introduction Face recognition is an application in which the computer either classifies human identity according to the face (face identification) or verifies whether two images belong to the same subject (face verification). A common face recognition system has two steps: enrollment and matching. Specifically, in the enrollment stage, features are obtained from a facial image or a set of images to obtain a signature or a template for each subject. The enrollment usually has three steps: (i) face detection, (ii) face alignment, and (iii) signature generation. In the matching stage, these signatures are compared to obtain a distance for the identification or verification problem. Recently, face recognition technology has significantly advanced by the deployment of deep learning technology, especially using Convolutional Neural Networks (CNN). Pure 2D face recognition (2D-FR) systems have achieved human performance or even better. DeepFace, proposed by Taigman et al. [3], first reported performance on the Labeled Faces in the Wild (LFW) standard benchmark [4] that was better than human efforts. FaceNet, proposed by Schroff et al. [5], used triplet loss to train a deep neural ∗ Corresponding author Email address: [email protected] (Ioannis A. Kakadiaris) 2 network using 200 million labeled faces, and obtained a performance of 99.63% verification accuracy on the same dataset. The success of deep learning techniques in face recognition indeed relies on the following four aspects: (i) a large amount of data either from public datasets such as WebFace [6] and Ms-Celeb-1M [7], or private datasets, (ii) advanced network architecture such as VGG [8] and ResNet [9], (iii) discriminative learning approaches such as Triplet Loss [5], Center Loss [10], Range Loss [11], SphereFace [12], and (iv) regularization methods such as Noisy Softmax [13]. However, face recognition is still not a solved problem in real-world conditions. Some datasets, such as LFW, use Viola-Jones face detector, which is not designed to work in the whole pose distribution from −90◦ to +90◦ . In an unconstrained scenario, especially using surveillance camera, there is a plethora of images with large variations in head pose, expression, illumination, and occlusions. To overcome these challenges, a 3D face model can be applied to assist a 2D face recognition. A 3D facial model is intrinsically invariant to pose and illumination. To use a 3D face model, a model should is fitted on the facial images and a 3D-2D projection matrix is estimated. With the help of a projection matrix and fitted 3D model, it is easy to rotate the face out-plane and align the input images from any arbitrary large pose positions to the frontal position for the feature extraction and signature matching. In the last few years, researchers focused on the 2D face recognition from pure 2D image view and have developed numerous loss function approaches to learn the discriminative features from the different poses. A limited number of 3D-aided 2D face recognition systems (3D2D-FR) have been developed using the 3D model to help align 2D images. Kakadiaris et al. [14] proposed a pose and illumination invariant system which frontalized the face image using annotated face model (AFM). Hu et al. [15] proposed a unified 3D morphable model (U-3DMM) which has additional PCA subspace for perturbation. To address the problem mentioned above, this paper presents a 3D-aided 2D face recognition system called UR2D which significantly improves face recognition performance using the AFM and deep learning technology, especially in large pose scenarios. There is enormous demand [16] for pose-invariant face recognition systems because frontal face recognition can be considered as a solved problem. 3 UR2D consists of several independent modules: face detection, landmark detection, 3D model reconstruction, pose estimation, lifting texture, signature generation, and signature/template matching. Despite face detection methods, all other methods are developed in the Computational Biomedicine Lab. It provides sufficient tools and interfaces to use different sub-modules designed in the system. The core code is written in efficient C++, which provides bindings to Python. The system leverages several open-sourced libraries such as OpenCV [17], glog [18], gflags [19], pugixml [20], JSON for modern C++ [21], and Caffe [22]. In UR2D, after detecting the face and 2D landmarks from image, a 3D model is constructed from a 2D image or several 2D images. By estimating the 3D-2D projection matrix, the correspondence between the 3D model and 2D image can be computed. Then, a 3D model is used to help frontalize the face. The pose-robust features and occlusion encodings are extracted to represent the face. For matching, we use cosine similarity to compute the similarity between two signature vectors. In summary, this paper extends Xu et al. [23] and make the following contributions: • A brief survey of recent face recognition pipeline and each module are summarized; • A pose-invariant 3D-aided 2D face recognition system using deep learning is developed. The intrinsic value of a 3D model is explored to frontalize the face, and the pose-invariant features are extracted for representation. We demonstrate results that a 3D-aided 2D face recognition system exhibits a performance that is comparable to a 2D only FR system. Our face recognition results outperform the VGG-Face, FaceNet, and COTS by at least 9% on the UHDB31 dataset and 3% on the IJB-A dataset on average. In addition, we demonstrate that UR2D can generate template signatures from multiple images and achieve state-of-the-art performance of 85% on the IJB-A dataset. The rest of the paper is organized as follows: modern face recognition systems are reviewed in Sec. 2. In Sec. 3, the architecture of UR2D and its functionalities are discussed. In Sec. 4, each module separately is introduced in detail. Detailed evaluations on the indoor and in-the-wild datasets are reported in Sec. 5. 4 2. Related work We divide the current existing face-related work into two categories: In Sec. 2.1, we discuss detailed recent related work for each module in the common face recognition pipeline from an academic view. System level papers about the implementation are discussed in Sec. 2.2 . 2.1. Modules Face Detection: Face detection is the first step, as well as the most studied topic, in the face recognition domain. Zefeiriou et al. [24] presented a comprehensive survey on this topic. They divided the approaches into two categories: rigid template-based methods, and deformable-parts-models-based methods. In addition to the methods summarized in [24], the approaches of object detection under the regions with a convolutional neural network (R-CNN) framework [25] have been well developed. Some techniques can be directly integrated to face detection [26]. Li et al. [27] used a 3D mean face model and divided the face into ten parts. They joined face proposals into a single R-CNN model. The approach proposed by Hu and Ramanan [28] explored context and resolution of images to fine-tune the residual networks (ResNet) [29], which was demonstrated to detect a face as small as three pixels. Despite the two-stage face detectors above using proposal and classification technique, single-stage detectors have also been developed. SSD [30] and YOLO [31] classify a fixed grid of boxes and learn regression functions to map to the objects simultaneously. Lin et al. [32] address the issue that the performance of single-stage detectors are not as strong as two-stage detectors because of unbalanced positive and negative samples. With focal loss, they also trained state-of-the-art single-stage object detector. Very recently, SSH has been proposed by Najibi et al. [33] using multi-task loss for both classification and regression in the network. Face Alignment: Face alignment refers to aligning the face image to a specific position. Usually, researchers include landmark detection in this topic. Jin and Tan [34] summarized the categories of popular approaches for this task. Cascaded regression was a major trend in this topic and classification frameworks tend to be popular recently. Zhu et al. [35] searched for similar shapes from exemplars and regressed the 5 shapes by using SIFT features and updating the probability of shapes. An ensemble of random ferns [36] are used to learn the local binary discriminative features. Xu and Kakadiaris [37] proposed to jointly learn head pose estimation and face alignment tasks in a single framework (JFA) using global and local CNN features. Some researchers treat the face alignment task as a classification problem. KEPLER [38] joined CNN features from different layers and captured the response map to localize the landmarks. Wu et al. [39] proposed the GoDP algorithm to localize landmarks under a fully convolutional network (FCN) framework by exploring two-pathway information. Some recent works use generative adversarial networks (GAN) to frontalize the face [40, 41, 42]. Huang et al. [40] used two-pathway GAN (TP-GAN) for photorealistic frontal synthesis images, but kept identity and details of texture. Yin et al. [41] incorporated a 3D model with GAN to frontalize faces for large poses in the wild. DRGAN was proposed by Tran et al. [42] to generate the frontalized face from face images under different poses. They also demonstrated the usage of GAN in face recognition. Signature Generation: An emerging topic in face recognition research is generating a discriminative representation for a subject. When training with millions of face images using deep learning technology, many feature descriptors have been proposed recently. Parkhi et al. [8] proposed the VGG-Face descriptor within VGG-Very-Deep architectures. Triplet loss was proposed by Schroff et al. [5] to train a deep neural network using 200 million labeled faces from Google. Masi et al. [43] developed face recognition for unconstrained environments by fine-tuning the ResNet and VGG-Face on 500K 3D rendering images. In addition to frontalizing the face, they also rendered face images to half-profile 40◦ , and full-profile (75◦ ). Masi et al. [44] addressed the question of whether we need to collect millions of faces for training a face recognition system. They argued that we can use synthesized images instead of real images to train the model and still obtain comparable results. Despite triplet loss, many other loss functions have been proposed recently. Center loss was added by Wen et al. [10] alongside cross entropy loss to obtain more discriminative features for deep face recognition. Range loss [11] was designed by Zhang et al. to train deep neural networks with a long tail distribution. A-Softmax Loss [12] was used in SphereFace and demonstrated efficiency in learning the discriminative features. Marginal Loss [45] was proposed to 6 Name Category Core Detection Alignment Representation Matching OpenBR [46] 2D C++ X X X X FaceID1-3 [47] 2D - X X X DeepFace [3] 2D - X X X FaceNet [5] 2D - X X X VGG-Face [8] 2D - X X X OpenFace [48] 2D Torch X X X U-3DMM [15] 3D2D - X X X UR2D 3D2D C++ X X X X X X X Modern Active X Table 1: Comparison of recent existing 2D face recognition pipelines. We employ the same definition of modern and active made by Klontz et al. [46] (“-” means that this information is not provided in the paper). enhance the discriminative ability by maximizing the inter-class distances from large scale training data. 2.2. System OpenCV and OpenBR are some well known open-source computer vision and pattern recognition libraries. However, the eigenface algorithm in OpenCV is out-ofdate. OpenBR has not been updated since 9/29/2015. Both libraries only support nearly frontal face recognition, since the face detector can only detect the frontal face. OpenFace is an open-source implementation of FaceNet [5] by Amos et al. [48] using Python and Torch, which provides four demos for usage. OpenFace applied Dlib face detector and landmark detector to do the pre-processing, which is better than OpenBR. There is another official Tensorflow implementation of FaceNet in which the authors use MTCNN [49] to detect and align face, which boosts performance speed and detection accuracy. To the best of our knowledge, there is a limited amount of well-designed system papers. Most face-related papers focus on different sub-modules or the research of face representations. The comparison of recent existing 2D face recognition systems is presented in Tab. 1 including the research on face representation. 7 3. System Design UR2D is a 3D-aided 2D face recognition system designed for pose-invariant face recognition. Moreover, this system is suitable for face-related research, and can fast pre-process images, provide baselines, plot the results, and support further development. 3.1. System requirements UR2D is written in clean and efficient C++, which is developed on a Linux platform (Ubuntu system). It requires GCC 4.9 or above for compilation. It leverages a list of open-sourced libraries and tools such as CMake, Boost, OpenCV, gflags, glog, puxixml, JSON, and Caffe. Most of the dependencies are available in the Ubuntu repository except Caffe. Therefore, to install dependencies, it only requires installing Caffe manually. 3.2. Architecture Overview Figure 2 illustrates the architecture of UR2D, which explicitly illustrates modules and functionality. The blue blocks are external shared libraries. The other three components belong to our system. As a base of the software, green blocks provide the basic functions. The algorithm modules are constructed as high-level APIs. The applications and GUIs are the top of the software and are built by combining these APIs. The users can directly call these applications and obtain the results. The advantages of this architecture are that it is simple and well-structured. With full development of libraries, the system can use CPUs/GPU and other features easily. 3.3. Data Structures In UR2D, the basic element is File on the disk. All operations or algorithms are based on the files. The basic data structure is Data, which is a hash table with pairs of keys and values. Both keys and values are in string type. Unlike OpenBR [46], to avoid saving giant data in the memory, we only keep the file path in the memory. 8 GUI Applications Detection … File System External Core Caffe Utility Matching Logging Dataset Utility IO GLog Caffe GFlag Modules Internal QT OpenCV Apps Figure 2: Depiction of UR2D’s architecture. In addition to external libraries, it includes some other base libraries to process files, use CUDA, manage the data files, etc. Based on these basic libraries, high level APIs were implemented by calling function from each module. Based on UR2D’s SDK, it is easier to write various applications for different purposes. Also, we created the GUIs to demonstrate our UR2D. 3.4. Configuration We have two approaches to run UR2D. The first one is defining the configuration file (JSON format), which points out the datasets, input files, output directories, involving modules and their model locations, and evaluation. Attribute dataset contains the information of input dataset including the name and path. Attribute input contains the list of galleries and probes. Attribute output defines the output directories. Attribute pipelines defines the modules used in the pipeline. The pip command line application only accepts the argument of the configuration file, which will parse the configuration file, load the models, and run defined modules. The advantages of this approach are simplicity and flexibility. Unlike the OpenBR framework, it does not require a detailed understanding of the option or input long arguments in the command line. The users only need to change some values in the attributes dataset and 9 input (e.g., set dataset directory and file to enroll), and program pip will generate the output they defined in this configuration file. 3.5. Command Line Interface To make full use of SDK of UR2D, we created some corresponding applications to run each module. All applications accept the file list (text or csv file by default, which includes tag at the top line), a folder, or a single image. The IO system will load the data in the memory and process the data according to the data list. The arguments specify the location of the input file/directory and where the output should be saved. UR2D’s enrollment is executed and generates signatures to the output directory. The path of the signature is recorded in the Data. By calling the API from IO system, the list of Data will be written to the file (default is in .csv format). 4. Face Recognition Figure 3 depicts the overview of enrollment in the UR2D, which contains face detection, face alignment, 3D face reconstruction, pose estimation, texture lifting, and signature generation. 4.1. Face Detection A serious problem in OpenBR [46], OpenFace [48], and even the commercial offthe-shelf face recognition software (COTS) is the face detection rate. OpenBR only supports OpenCV frontal face detector. OpenFace also supports Dlib [50] face detector. However, in recent years, many face detection algorithms have been developed [51, 27, 28] with deep learning technology to support multi-view images. To detect the face in multi-view poses, some modern detectors such as Headhunter [52] and DDFD [51], and Dlib-DNN face detector are supported in our system. Mathias et al. [52] trained Headhunter by using multi-scale templates. DDFD face detector is proposed by Farfade et al. [51] by fine-tuning AlexNet [53] and using non-maximum suppression (NMS-max, NMS-avg). To support different face detectors for downstream modules, we perform the bounding box regression on detected bounding box to reduce the variations of the bounding 10 Generate Bounding box Face Detection Gallery Lift textures from original image according to 3D-2D correspondence Reconstruct 3D face model from a single image 3D Reconstruction Euclidean Distance Texture Lifting Refinement Landmark Detection Pose Estimation Representation Localize landmarks based on the response map Compute 3D-2D projection matrix Residual network to learn pose-invariant features Probe Cosine Similarity 𝑃"" 𝑃"# 𝑃"$ 𝑃"& 𝑃#" 𝑃## 𝑃#$ 𝑃#& 𝑃$" 𝑃$# 𝑃$$ 𝑃$& Feature & Occlusion Encoding Images Enrollment Matching Figure 3: Depiction of the whole pipeline (follow the arrow in the middle) of UR2D. The rounded rectangles represent different modules. Dashed arrows represent the workflow. The enrollment encompasses the modules listed. A face is first detected and then transferred to localize landmarks. A 3D model is constructed directly from a 2D image with a bounding box. With 2D landmarks and a 3D model, a 3D-2D projection matrix can be estimated. The frontalized image and occlusion map are generated according to the 3D model and projection matrix. The pose robust features are extracted from these images along with occlusion encoding. The matching step computes features from visible parts and outputs a similarity score. box. The first advantage of this approach is that we do not need to re-train or fine-tune the models for downstream modules after switching the face detector. The second advantage is that this approach provides a more robust bounding box for the landmark localization module. 4.2. Landmark Localization To detect face landmarks, we use GoDP proposed by Wu et al. [39], which is demonstrated to be robust to pose variations. GoDP landmark detector relies on confidence maps generated by a fully convolutional network. A confidence map is generated for each landmark to indicate the possibility of a landmark appearing at a specific location in the original image. The prediction is made by simply selecting the location that has the maximum response in the confidence map. This winner-take-all strategy helps to suppress false alarms generated by background regions and improves the robustness 11 of the algorithm under large head pose variations. Compared to other confidence-mapbased landmark detectors, the novel architecture of GoDP merges the information of the deep and shallow layers based on a new loss function, increases the resolution and discrimination of the confidence maps, and achieves state-of-the-art results on multiple challenging face alignment databases. 4.3. 3D Reconstruction of Facial Shape To reconstruct the 3D facial shape of the input 2D image, we integrate into our pipeline the E2FAR algorithm proposed by Dou et al. [54]. It uses a subspace model to represent a 3D AFM as a parameter vector and employs CNN to estimate the optimal parameter values from a single 2D image. To train the deep neural network, a large set of synthetic 2D and 3D data has been created using the 3D rendering of randomly generated AFMs. To improve the robustness to illumination variation, the deep neural network is pre-trained on real facial images and fine-tuned on the synthetic data. Compared with existing work, it is more efficient due to its end-to-end architecture, which requires a single feed-forward operation to predict the model parameters. Moreover, it only relies on face detection to localize the facial region of interest on the image. As a result, compared with landmark-based approaches, it is more robust to the pose variation that can degrade landmark detection accuracy. 4.4. Pose estimation Given 2D landmarks X2D obtained from landmark detection and 3D landmarks X3D obtained from a 3D model, the transformation matrix P can be estimated by solving a least-squares problem as follows: min \|\|X2D − P X3D \|\|22 + λ\|\|P \|\|22 . P (1) In our implementation, we use the Levenberg-Marquardt algorithm, also known as DLS, to solve this equation. 4.5. Texture Lifting Facial texture lifting is a technique first proposed by Kakadiaris et al. [14], which lifts the pixel values from the original 2D images to a UV map. Given the 3D-2D 12 projection matrix P , 3D AFM model M , and original image I, it first generates the geometry image G, each pixel of which captures the information of an existing or interpolated vertex on the 3D AFM surface. With G, a set of 2D coordinates referring to the pixels on an original 2D facial image is computed. In this way, the facial appearance is lifted and represented into a new texture image T . A 3D model M and Z-Buffer technique are used to estimate the occlusion status for each pixel. This process generates an occlusion mask Z. This module has the following two advantages: It generates the frontal normalized face images, which is convenient for feature extraction and comparison. Second, it generates occlusion masks, which identify the parts of the face images that are occluded, providing the evidence to exclude the face regions. 4.6. Signatures To improve the performance of face recognition in matching non-frontal facial images, we integrate into our pipeline the algorithm proposed by Dou et al. [55] for extracting Pose-Robust Face Signature (PRFS), a part-based face representation with discriminative local facial features and explicit pose and self-occlusion encoding. The facial texture T and the self-occlusion mask Z are first divided into multiple local patches. Then, on each local patch, discriminative features are extracted and selfocclusion encoding is computed. The ensemble of local features, each enhanced by the self-occlusion encoding, forms the pose-robust face signature. We use two types of local features, namely the DFD feature proposed by Lei et al.[56] and a deep feature we trained by following Wen et al.[10] using center loss. To train the DFD feature, we use a small subset of the FRGC2 database that consists of 907 frontal facial images of 109 subjects. We divide the facial texture into 64 non-overlapping patches and train a DFD feature extractor for each local patch separately. To train the deep feature, the CASIA WebFace dataset [6] is used as training data. We divide the facial texture into 8 partially-overlapping patches and train a deep neural network for each local patch separately. In this paper, we call the face signature with the DFD feature PRFS, and the face signature with the deep feature DPRFS. 13 5. Experiments In this section, we provide a systematical and numerical analysis on two challenging datasets in both constrained and in-the-wild scenarios. First, the datasets used to verify UR2D are introduced. Then, a fair comparison of UR2D with VGG face descriptor (VGG-Face) and a commercial face recognition software (COTS) on these two challenging datasets is conducted for the image matching. In the end, the template matching experiments on IJB-A dataset was performed. Dataset Images Subjects Environment UHDB31 IJB-A 24,255 77 Constrained 25,808 500 In-the-wild Poses Illuminations Usage 21 3 2D-2D, 3D-2D, 3D-3D face recognition Various Various 2D unconstrained face recognition Table 2: Comparison of datasets: UHDB31 [1] and IJB-A [2]. Both are challenging due to pose variations, illumination, and resolution. 5.1. Datasets UHDB31 [1] was created in a controlled lab environment, which allows facerelated research on pose and illumination issues. In addition to 2D images, it also provides the corresponding 3D model of subjects. An interesting fact of this dataset is that pose follows the uniform distribution on three dimensions: pitch, yaw, and roll. For each subject, a total of 21 high-resolution 2D images from different views and 3D data are collected at the same time. Then, a 3D model is registered from the 3D data from different poses to generate a specific 3D face model. In addition to three illuminations, the resolutions are downsampled to 128, 256, and 512 from the original size. IJB-A [2] is another challenging dataset which consists of images in the wild. This dataset was proposed by IARPA and is managed by NIST. This dataset merges images and frames together and provides evaluations on the template level. A template contains one or several images/frames of a subject. According to the IJB-A protocol, it splits galleries and probes into 10 folders. In our experiment, we modify this protocol to use it for close-set face identification. The details will be introduced in Sec. 5.4. A summary of these two datasets is presented in Tab. 2. Our system provides dataset utility to parse and load the data from these two datasets. 14 Yaw −90◦ −60◦ −30◦ 0◦ +30◦ +60◦ +90◦ 14/11/58/ 69/32/95/ 94/90/100/ 99/100/100/ 95/93/99/ 79/38/92/ 19/7/60/ 47/82 90/99 100/100 100/100 100/99 95/99 47/75 22/9/84/ 88/52/99/ 100/99/100/ 100/100/100/ 94/73/99/ 27/10/91/ 81/96 100/100 100/100 100/100 100/100 84/96 8/0/44/ 2/19/80/ 91/90/99/ 96/99/99/ 96/98/97/ 52/15/90/ 9/3/35/ 44/74 90/97 99/100 100/100 99/100 95/96 58/78 Pitch +30◦ 0◦ −30◦ - Table 3: Comparison of Rank-1 percentage of different systems on UHDB31.R128.I03. The methods are ordered as VGG-Face, COTS v1.9, FaceNet, UR2D-PRFS, and UR2D-DPRFS. The index of poses are ordered from the left to right and from the top to bottom (e.g., pose 3 is pitch −30◦ and yaw −90◦ , pose 11 is pitch 0◦ and yaw 0◦ ). The frontal face is gallery while the other poses are probes. In all cases, our system achieves the best performance compared with the state-of-the-art. 5.2. Baselines To perform a fair comparison with current state-of-the-art face recognition systems, we choose VGG-Face and COTS v1.9 as baselines. The VGG-Face descriptor was developed by Parkhi et al. [8]. The original release contains a Caffe model and a MATLAB example. We re-used their model, implemented their embedding method on multi-scaled images, and fused the features in C++. In our implementation, we tried different combinations of descriptor and matching methods. We found that embedding features with cosine similarity metrics works the best for the VGG-Face. In our experiment, we use VGG-Face to represent the embedding features with matching using a cosine similarity metric. As in the baseline module, UR2D provides API to obtain the features. The FaceNet algorithm was proposed by Schroff et al. [5]. We use a personal implemented FaceNet from GitHub 1 trained using WebFace [6] and MS-Celeb-1M [7]. They first use MTCNN [49] to align face and extract 128 dimensions features. They provide pre-trained models that achieves 99.20% ± 0.30% accuracy on the LFW dataset. The accuracy is a little bit lower than the original paper, but still can be con1 https://github.com/davidsandberg/facenet 15 sidered state-of-the-art. COTS is a commercial software developed for scalable face recognition. It provides SDK and applications which can be used directly. In our experiments, we used version 1.9 to compare with our system. This version is considered to be a significant boost compared with previous versions. In our experiment, we report the performance using both PRFS and DPRFS features. The summary of software configuration is reported in Tab. 4. We compute the Rank-1 identity accuracy from successfully enrolled signatures. System Features Dims Metric VGG-Face Embedding 4096 Cosine COTS v1.9 - - - FaceNet - 128 Cosine UR2D PRFS 64 × 1024 Cosine UR2D DPRFS 8 × 1024 Cosine Table 4: Comparison of systems configuration used in our experiments. 5.3. UHDB31: Pose-Invariant Face Recognition In this experiment, we chose a configuration from the UHDB31 dataset named UHDB31.R0128.I03. This is a subset in which all images are down-sampled to the size 153 × 128 in the neutral illumination. This subset was chosen to demonstrate that our system, UR2D, is robust to different poses. Therefore, we use this configuration to exclude the other variations such as illumination, expressions, etc, but only keep the pose variations. We treated the frontal face images (pose 11) as gallery and images from the other 20 poses (poses 1 − 10, 12 − 21) as probes, independently. Both the gallery and the probe contain 77 images, each of which belongs to a subject. The face identification experiment was performed using 20 pairs of sigsets. Table 3 depicts the comparison of Rank-1 accuracy among 20 poses (except pose 11, which is used for gallery), which indicates that UR2D is robust to the different poses 16 compared with other systems. We observed the VGG-Face and COTS v1.9 algorithms cannot generalize all pose distributions. FaceNet works better than VGG-Face and COTS v1.9 on the extreme poses. One possible answer is that this model is trained from the most available datasets using Ms-Celeb-1M and WebFace, which provide more extreme pose cases. However in cases such as pose 3 (−30◦ , −90◦ ) and pose 21 (−30◦ , −90◦ ) in Tab. 3, the performance of 2D only face recognition pipelines still has significant room for improvement. On the other hand, with the help of the 3D model, our system keeps the consistent and symmetric performance among the different poses. Even in the cases with yaw −90◦ or +90◦ , our system can tolerate the pose variations, and achieves around 80% Rank-1 identity accuracy with DPRFS features and around 50% Rank-1 identity accuracy with PRFS features on average. 5.4. IJB-A: In-the-Wild Face Recognition However, in a real-world case, a face recognition system does not suffer only from pose variations. In this experiment, we want to explore whether our system is can also be used in an in-the-wild environment. We designed a different protocol for face identification experiments based on the original 10 splits. Unlike the original templatelevel comparison, we conducted an image pairs comparison. First, we removed some samples in the IJB-A splits to make 10 close-set comparison pairs. Then, we cropped the face according to the annotations. Image thumbnails with resolution below 50 were up-sampled, while those with resolution larger than 1000 were down-sampled. Herein, we do not compare with FaceNet since there are overlapping samples between the training set and IJB-A dataset. Method Split-1 Split-2 Split-3 Split-4 Split-5 Split-6 Split-7 Split-8 Split-9 Split-10 Avg. VGG-Face 76.18 74.37 24.33 47.67 52.07 47.11 58.31 54.31 47.98 49.06 53.16 COTS v1.9 75.68 76.57 73.66 76.73 76.31 77.21 76.27 74.50 72.52 77.88 75.73 UR2D-PRFS 47.61 49.27 47.71 47.71 48.97 44.83 52.98 44.14 43.40 49.02 47.56 UR2D-DPRFS 78.20 76.97 77.31 79.00 78.01 79.00 81.15 78.40 74.97 78.57 78.16 Table 5: Comparison of Rank-1 percentage of different systems on 10 splits of IJB-A. UR2D achieves the best performance with DPRFS features on each split. Table 5 depicts the rank-1 identification rate with different methods on IJB-A 17 dataset. Our system UR2D with DPRFS reports better performance compared with VGG-Face and COTS v1.9. Also, our system results are consistent on 10 splits, which indicates that our system is robust. Why do PRFS features in our system not perform well on the IJB-A dataset? One possible answer is that PRFS features are trained on the FRGC dataset, which has notably fewer variations of pose, illumination, and resolution problems. The current PRFS features cannot generalize on these images with large variances. The corresponding solution is retraining the PRFS feature model on the in-the-wild dataset. Third, COTS performs well on this challenging dataset, since it is designed for the real scenario. By comparing the experiment in Sec. 5.3, we are left with the question why does our system perform only slightly better than baselines? We argue that in in-the-wild scenarios there are complicated combinations of pose variations, illumination, expression, and occlusions. A robust face recognition system should take all cases into consideration. In addition, COTS dropped hard samples and enrolled fewer signatures than ours, which would boost the performance to some extent. Method Rank-1 (%) OpenBR [46] 24.60 ± 1.10 Wang et al. [57] 82.20 ± 2.30 DCNN [58] 85.20 ± 1.80 PAM [43] 77.10 ± 1.60 DR-GAN [42] 85.50 ± 1.50 UR2D 85.65 ± 1.74 Table 6: Comparison of Rank-1 percentage of different systems on 10 splits of IJB-A. UR2D achieves the best performance with DPRFS features on each split. We extended UR2D to enroll the several images for a subject to generate a template. The template is an average of signatures computed by generating a unified 3D model from several 2D images. Here we use the results from [42] to do the comparison. Table 6 lists the average Rank-1 identification accuracy for each method. UR2D achieved 18 Method Split-1 Split-2 Split-3 Split-4 Split-5 Split-6 Split-7 Split-8 Split-9 Split-10 Avg. VGG-Face 74.44 74.26 70.68 73.96 69.60 72.64 72.91 70.03 72.25 71.78 72.25 UR2D(DPRFS) 87.22 86.82 83.68 86.05 83.52 88.22 85.14 83.59 84.86 87.38 85.65 Table 7: Detailed Rank-1 percentage of different systems on 10 splits of IJB-A. the best performance. The detailed comparison of Rank-1 identification accuracy with VGG-Face is summarized in Tab. 7 for 10 splits in the IJB-A dataset. 5.5. Memory Usage and Running Time We conducted the analysis of UR2D in terms of both memory and time. Cafferelated implementation runs on GPU (GTX TITAN X). COTS v1.9 makes full use of eight CPUs. Table 8 summarizes the system run-times for different systems. Some modules in our implementation or external libraries run on CPU, such as face detection, pose estimation, text-lifting, and PRFS feature extraction. Therefore, the time used by PRFS features takes 1.5 s more than DPRFS features. Due to loading several large models, DPRFS requires more memory. The user can define the suitable feature extractors according to their needs. Since we optimized Memory for DPRFS, it shares the memory block in the GPU. The memory cost is reduced to the same level as PRFS. We use DPRFS by default. System GPU Memory (GB) Time (s) VGG-Face Full 1.2 0.9 COTS v1.9 No 0.1 0.5 UR2D (PRFS) Partial 2.4 2.5 UR2D (DPRFS) Partial 2.4 1.0 Table 8: Comparison of system run-times. “Partial” in GPU column denotes part of the code does not support GPU acceleration. Time means the average enrollment time for a single image. 19 6. Conclusion In this paper, a well-designed 3D-aided 2D face recognition system (UR2D) that is robust to pose variations as large as 90◦ using deep learning technology has been presented. An overview of the architecture, interface, and each module in UR2D are introduced i detailed. Extensive experiments are conducted on UHDB31 and IJB-A to demonstrate that UR2D is robust to the pose variations, and it outperforms existing 2D-only face recognition systems such as VGG face descriptor, FaceNet, and a commercial face recognition software by at least 9% on UHDB31 dataset and 3% on IJB-A dataset in average. And the system achieves the state-of-the-art performance of 85% in template matching on IJB-A dataset. 7. Acknowledgment This material is based upon work supported by the U.S. Department of Homeland Security under Grant Award Number 2015-ST-061-BSH001. This grant is awarded to the Borders, Trade, and Immigration (BTI) Institute: A DHS Center of Excellence led by the University of Houston, and includes support for the project “Image and Video Person Identification in an Operational Environment” awarded to the University of Houston. 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arXiv:1611.05067v1 [math.AC] 15 Nov 2016 On S-coherence Driss Bennis1,a and Mohammed El Hajoui1,b 1. Department of Mathematics, Laboratory of Analysis, Algebra and Decision Support, Faculty of Sciences, B.P. 1014, Mohammed V University in Rabat, Rabat, Morocco a. [email protected]; driss [email protected] b. [email protected] Abstract. Recentely, Anderson and Dumitrescu’s S-finiteness has attracted the interest of several authors. In this paper, we introduce the notions of Sfinitely presented modules and then of S-coherent rings which are S-versions of finitely presented modules and coherent rings, respectively. Among other results, we give an S-version of the classical Chase’s characterization of coherent rings. We end the paper with a brief discussion on other S-versions of finitely presented modules and coherent rings. We prove that these last S-versions can be characterized in terms of localization. Key Words. S-finite, S-finitely presented, S-coherent modules, S-coherence rings. 2010 Mathematics Subject Classification. 13E99. 1 Introduction Throughout this paper all rings are commutative with identity; in particular, R denotes such a ring, and all modules are unitary. S will be a multiplicative subset of R. We use (I : a), for an ideal I and an element a ∈ R, to denote the quotient ideal {x ∈ R; xa ∈ I}. According to [1], an R module M is called S-finite if there exists a finitely generated submodule N of M such that sM ⊆ N for some s ∈ S. Also, from [1], an R-module 1 2 D. Bennis and M. El Hajoui M is called S-Noetherian if each submodule of M is S-finite. In particular, R is said to be an S-Noetherian ring, if it is S-Noetherian as an R-module; that is, every ideal of R is S-finite. It is clear that every Noetherian ring is S-Noetherian. The notions of S-finite modules and of S-Noetherian rings were introduced by Anderson and Dumitrescu motivated by the works done in [8] and [2]. They succeeded to generalize several well-known results on Noetherian rings including the classical Cohen’s result and Hilbert basis theorem under an additional condition. Since then the S-finiteness has attracted the interest of several authors (see for instance [6, 7, 10, 11, 12, 14]). Recentely, motivated by the work of Anderson and Dumitrescu, S-versions of some classical notions have been introduced (see for instance [6, 10]). In this paper we are inerested in S-versions of finitely presented modules and coherent rings. Actually, there are two possibilities which could be considered as S-versions of finitely presented modules which lead to two S-versions of coherent rings. We prove that the S-version of coherent rings defined by one of them has a characterization similar to the classical one given by Chase for coherent rings [3, Theorem 2.2]. This is why we adopt this notion as the suitable S-version of finitely presented modules. However, it seems not evident to characterize this notion in terms of localization. We prove that indeed it is the other S-version, which is briefly studied at the end of the paper, has a characterization in terms of localization. The organization of the paper is as follows: In Section 2, we introduce and study an S-version of finitely presented modules. We call it an S-finitely presented module (see Definition 2.1). Then, we study the behavior of S-finiteness in short exact sequences (see Theorem 2.5). We end Section 2 with some change of rings results (see Proposition 2.7 and Corollary 2.8). Section 3 is devoted to the S-version of coherent rings which are called S-coherent rings (see Definition 3.3). Our main result represents the S-counterpart of Chase’s result [3, Theorem 2.2] (see Theorem 3.8). Also an S-version of coherent modules is introduced (see Definition 3.1 and Proposition 3.2). We end the paper with a short section which presents the other Sversion of S-finiteness (see Definitions 4.1 and 4.4). We prove that these notions can be characterized in terms of localization (see Proposition 4.3 and Theorem 4.7). We end the paper with results which relate S-finiteness with the notion of S-saturation (see Propositions 4.9 and 4.8 and Corollary 4.10). 3 On S-coherence 2 S-finitely presented modules In this section, we introduce and investigate an S-version of the classical finitely presented modules. Other version is discuted in Section 4. Definition 2.1 An R-module M is called S-finitely presented, if there exists an exact sequence of R-modules 0 −→ K −→ F −→ M −→ 0, where K is S-finite and F is a finitely generated free R-modules. Clearly, every finitely presented module is S-finitely presented. However, the converse does not hold in general. For that, it suffices to note that when R is a nonNoetherian S-Noetherian ring, then there is an S-finite ideal I which is not finitely generated. Then, the R-module R/I is S-finitely presented but it is not finitely presented. Also, it is evident that every S-finitely presented module is finitely generated. To give an example of a finitely generated module which is not S-finitely presented, it suffices to consider an ideal I which is not S-finite and then use Proposition 2.4 given hereinafter. One could remark that in Definition 2.1 we assume that the free module F is finitely generated rather than S-finite. In fact, because of the following result both of notions coincide for free modules. Proposition 2.2 Every S-finite free R-module is finitely generated. Proof. Let M = L Rei be an S-finite free R-module, where (ei )i∈I is a basis of M i∈I and I is an index set. Then, there exist a finitely generated R-module N and an s ∈ S such that sM ⊆ N ⊆ M . Then, N = Rm1 + · · · + Rmn for some m1 , ..., mn ∈ M (n > 0 is an integer). For every k ∈ {1, ..., n}, there exists a finite subset Jk of I n S P Jk . Then, the finitely generated R-module λkj ej . Let J = such that mk = j∈Jk k=1 L Rej contains N . We show that M ′ = M . Deny. There exists an i0 ∈ I\J M′ = j∈J P ′ λj ej for some λ′j ∈ R. such that ei0 ∈ / M ′ . But sei0 ∈ N ⊆ M ′ and so sei0 = j∈J This is impossible since (ei )i∈I is a basis. Remark 2.3 Similarly to the proof of Proposition 2.2 above, one can prove that any S-finite torsion-free module cannot be decomposed into an infinite direct sum of non-zero modules. This shows that any S-finite projective module is countably 4 D. Bennis and M. El Hajoui generated by Kaplansky [9, Theorem 1]. Then, naturaly one would ask of the existence of S-finite projective module which is not finitely generated. For this, consider ∞ Q the Boolean ring R = ki , where ki is the field of two elements for every i ∈ N. i=1 Consider the projective ideal M = ∞ L ki and the element e = (1, 0, 0, ...) (see [4, i=1 Example 2.7]). Then, S = {1, e} is a multiplicative subset of R. Since eM = k1 is a finitely generated R-module, M is the desired example of S-finite projective module which is not finitely generated. However, determining rings over which every S-finite projective module is finitely generated could be of interest. It is worth noting that rings over which every projective module is a direct sum of finitely generated modules satisfy this condition. These rings were investigated in [13]. Next result shows that, as in the classical case [5, Lemma 2.1.1], an S-finitely presented module does not depend on one specific short exact sequence of the form given in Definition 2.1. Proposition 2.4 An R-module M is S-finitely presented if and only of M is finitely f generated and, for every surjective homomorphism of R-modules F −→ M −→ 0, where F is a finitely generated free R-module, ker f is S-finite. Proof. (⇐) Obvious. (⇒) Since M is S-finitely presented, there exists an exact sequence of R-modules 0 −→ K −→ F ′ −→ M −→ 0, where K is S-finite and F ′ is finitely generated and free. Then, by Schanuel’s lemma, K ⊕ F ∼ = ker f ⊕ F ′ , then ker f is S-finite. The following result represnts the behavior of S-finiteness in short exact sequences. It is a generalization of [5, Theorem 2.1.2] for modules with λ-dimension at most 1. Note that one can give an S-version of the classical λ-dimension (see [5, page]). However, here we prefer to focus on the notion of S-finitely presented modules, and a discussion on the suitable S-version of the λ-dimension could be the subject of a further work. f g Theorem 2.5 Let 0 −→ M ′ −→ M −→ M ′′ −→ 0 be an exact sequence of Rmodules. The following assertions hold: 1. If M ′ and M ′′ are S-finite, then M is S-finite. In particular, every finite direct sum of S-finite modules is S-finite. 2. If M ′ and M ′′ are S-finitely presented, then M is S-finitely presented. In particular, every finite direct sum of S-finitely presented modules is Sfinitely presented. 5 On S-coherence 3. If M is S-finite, then M ′′ is S-finite. In particular, a direct summand of an S-finite module is S-finite 4. If M ′ is S-finite and M is S-finitely presented, then M ′′ is S-finitely presented. 5. If M ′′ is S-finitely presented and M is S-finite, then M ′ is S-finite. Proof. 1. Since M ′′ is S-finite, there exist a finitely generated submodule N ′′ of n P M ′′ and an s ∈ S such that sM ′′ ⊆ N ′′ . Let N ′′ = Rei for some ei ∈ M ′′ and i=1 n ∈ N. Since g is surjective, there exists an mi ∈ M such that g(mi ) = ei for every i ∈ {1, ..., n, }. Let x ∈ M , so sx ∈ N = g −1 (N ′′ ). Then g(sx) ∈ g(N ) = N ′′ , and n n n n P P P P so g(sx) = αi ei = αi g(mi ) = g( αi mi ). Then, g(sx − αi mi ) = 0. Thus, (sx − n P i=1 i=1 i=1 i=1 αi mi ) ∈ ker g = Imf which is S-finite. So there exist a finitely generated i=1 submodule N ′ of Imf and an s′ ∈ S such that s′ Imf ⊆ N ′ . Then, s′ sx ∈ N ′ + and so s′ sM is a submodule of N ′ + n P n P Rmi i=1 Rmi which is a finitely generated submodule i=1 of M . Therefore, M is S-finite. 2. Since M ′ and M ′′ are S-finitely presented, there exist two shorts exacts sequences: 0 −→ K ′ −→ F ′ −→ M ′ −→ 0 and 0 −→ K ′′ −→ F ′′ −→ M ′′ −→ 0, with K ′ and K ′′ are S-finite R-modules and F ′ and F ′′ are finitely generated free R-modules. Then, by Horseshoe Lemma, we get the following diagram 0O 0O 0O 0 / M′ O /M O / M ′′ O /0 0 / F′ O / F ′ ⊕ F ′′ O / F ′′ O /0 0 / K′ O /K O / K ′′ O /0 0 0 0 By the first assertion, K is S-finite. Therefore, M is S-finitely presented. 3. Obvious. 6 D. Bennis and M. El Hajoui 4. Since M is S-finitely presented, there exists a short exact sequence of R-modules 0 −→ K −→ F −→ M −→ 0, where K is S-finite and F is a finitely generated free R-module. Consider the following pullback diagram 0O 0O 0 / M′ O /M O / M ′′ /0 0 /D O /F O / M ′′ /0 KO KO 0 0 By (1), D is S-finite. Therefore, M ′′ is S-finitely presented. 5. Since M ′′ is S-finitely presented, there exists a short exact sequence 0 −→ K −→ F −→ M ′′ −→ 0 where K is S-finite and F is a finitely generated free R-module. Consider the following pullback diagram 0O 0O 0 / M′ /M O / M ′′ O /0 0 / M′ /D O /F O /0 KO KO 0 0 Since F is free, D ∼ = M ′ ⊕ F , and so D is S-finite (since M ′ and F are S-finite). ′ Therefore, M is S-finite. As a simple consequence, we get the following result which extends [5, Corollary 2.1.3]. Corollary 2.6 Let N1 and N2 be two S-finitely presented submodules of an Rmodule. Then, N1 + N2 is S-finitely presented if only if N1 ∩ N2 is S-finite. 7 On S-coherence Proof. Use the short exact sequence of R-modules 0 −→ N1 ∩ N2 −→ N1 ⊕ N2 −→ N1 + N2 −→ 0. We end this section with the following change of rings results. The following result extends [5, Theorem 2.1.7]. Proposition 2.7 Let A and B be rings, let φ : A −→ B be a ring homomorphism making B a finitely generated A-module and let V be a multiplicative subset of A such that 0 6∈ φ(V ). Every B-module which is V -finitely presented as an A-module it is φ(V )-finitely presented as a B-module. Proof. Let M be a B-module which is V -finitely presented as an A-module. Then M is a finitely generated A-module. Then, M is a finitely generated B-module. Thus there is an exact sequence of B-modules 0 −→ K −→ B n −→ M −→ 0, where n > 0 is an integer. This sequence is also an exact sequence of A-modules. Since M is an V -finitely presented A-module and B n is a finitely generated A-module (since B is a finitely generated A-module), K is an V -finite A-module, and so K is a φ(V )-finite B-module. Therefore, M is a φ(V )-finitely presented B-module. The following result extends [5, Theorem 2.1.8 (2)]. Proposition 2.8 Let I be an ideal of R and let M be an R/I-module. Assume that I ∩ S = ∅ so that T := {s + I ∈ R/I; s ∈ S} is a multiplicative subset of R/I. Then, 1. M is an S-finite R-module if and only if M is a T -finite R/I-module. 2. If M is an S-finitely presented R-module, then M is a T -finitely presented R/I-module. The converse holds when I is an S-finite ideal of R. Proof. 1. Easy. 2. Use the canonical ring surjection R −→ R/I and Proposition 2.7. Conversely, if M is a T -finitely presented R/I-module. Then, there is an exact sequence of R/I-modules, and then of R-modules 0 −→ K −→ (R/I)n −→ M −→ 0, where n > 0 is an integer and K is a T -finite R/I-module. By the first assertion, K is also an S-finite R-module. And since I is an S-finite ideal of R, (R/I)n is an S-finitely presented R-module. Therefore, by Theorem 2.5 (4), M is an S-finitely presented R-module. 8 3 D. Bennis and M. El Hajoui S-coherent rings Before giving the definition of S-coherent rings, we give, following the calssical case, the definition of S-coherent modules. Definition 3.1 An R-module M is said to be S-coherent, if it is finitely generated and every finitely generated submodule of M is S-finitely presented. Clearly, every coherent module is S-coherent. However, using Proposition 3.2(1) below, one can show that, for an S-finite ideal I of R which is not finitely generated, the R-module R/I is S-coherent but it is not coherent. The reason of why we consider finitely generated submodules rather than S-finite submodules is explained in assertion (4) of Remark 3.4. The following result studies the behavior of S-coherence of modules in short exact sequences. It generalizes [5, Theorem 2.2.1]. f g Proposition 3.2 Let 0 −→ P −→ N −→ M −→ 0 be an exact sequence of Rmodules. The following assertions hold: 1. If P is S-finite and N is S-coherent, then M is S-coherent. 2. If M and P are S-coherent, then so is N . In particular, every finite direct sum of S-coherent modules is S-coherent. 3. If N is S-coherent and P is finitely generated, then P is S-coherent. Proof. 1. It is clear that M is finitely generated. Let M ′ be a finitely generated submodule of M . Then, f (P ) ⊆ g −1 (M ′ ), so there exist two shorts exacts sequences of R-modules 0 −→ K −→ Rn −→ P −→ 0 and 0 −→ K ′ −→ Rm −→ M ′ −→ 0, where n and m 9 On S-coherence are two positive integers. Then, by Horseshoe Lemma, we get the following diagram 0O 0O 0O 0 /P O / g −1 (M ′ ) O / M′ O /0 0 / Rn O / Rn+m O / Rm O /0 0 /K O / K ′′ O / K′ O /0 0 0 0 Since g −1 (M ′ ) is a finitely generated submodule of the S-coherent module N , g −1 (M ′ ) is S-finitely presented. Then, using Theorem 2.5 (5), K ′′ is S-finite, and so K ′ is S-finite. Therefore, M ′ is S-finitely presented. 2. Clearly N is finitely generated. Let N ′ be a finitely generated submodule of f g N . Consider the exact sequence 0 −→ Ker(g/N ′ ) −→ N ′ −→ g(N ′ ) −→ 0. Then, g(N ′ ) is a finitely generated submodule of the S-coherent module M . Then, g(N ′ ) is S-finitely presented. Then, Ker(g/N ′ ) is finitely generated by Theorem 2.5 (5), and since P is S-coherent, Ker(g/N ′ ) is S-finitely presented. Therefore, by (2) of Theorem 2.5, N ′ is S-finitely presented. 3. Evident since a submodule of P can be seen as a submodule of N . Now we set the definiton of S-coherent rings. Definition 3.3 A ring R is called S-coherent, if it is S-coherent as an R-module; that is, if every finitely generated ideal of R is S-finitely presented. Remark 3.4 1. Note that every S-Noetherian ring is S-coherent. Indeed, this follows from the fact that when R is S-Noetherian, every finitely generated free R-module is S-Noetherian (see the discussion before [1, Lemma 3]). Next, in Example 3.6, we give an example of an S-coherent ring which is not S-Noetherian. 2. Clearly, every coherent ring is S-coherent. The converse is not true in general. As an example of an S-coherent ring which is not coherent, we consider the trivial extension A = Z ⋉ (Z/2Z)(N) and the multiplicative set 10 D. Bennis and M. El Hajoui V = {(2, 0)n ; n ∈ N}. Since (0 : (2, 0)) = 0 ⋉ M is not finitely generated, T is not coherent. Now, for every ideal I of A, (2, 0)I is finitely generated; in fact, (2, 0)I = 2J ⋉ 0, where J = {a ∈ Z; ∃b ∈ (Z/2Z)(N) , (a, b) ∈ I}. Since J is an ideal of Z, J = aZ for some element a ∈ Z. Then, (2, 0)I = 2J ⋉ 0 = (2a, 0)A. This shows that A is V -Noetherian and so V -coherent. 3. It is easy to show that, if M is an S-finitely presented R-module, then MS is a finitely presented RS -module. Thus, if R is a S-coherent ring, RS is a coherent ring. However, it seems not evident to give a condition so that the converse holds, as done for S-Noetherian rings (see [1, Proposition 2 (f)]). In Section 4, we give another S-version of coherent rings which can be characterized in terms of localization. 4. One would propose for an S-version of coherent rings, the following condition “S-C: every S-finite ideal of R is S-finitely presented”. However, if R satisfies the condition S-C, then in particular, every S-finite ideal of R is finitely generated. So, every S-finite ideal of R is finitely presented; in particular, R is coherent. This means that the notion of rings with the condition S-C cannot be considered as an S-version of the classical coherence. Nevertheless, these rings could be of particular interest as a new class of rings between the class of coherent rings and the class of Noetherian rings. To give an example of a coherent ring which does not satisfy the condition ∞ Q S-C, one could consider the Boolean ring B = ki , where ki is the field of i=1 two elements for every i ∈ N, and the multiplicative subset V = {1, e} of B, ∞ L where e = (1, 0, 0, ...) ∈ B. Indeed, the ideal B = ki is V -finite but not i=1 finitely generated. Also, note that the following condition “S-c: every S-finite ideal of R is finitely generated” could be of interest. Indeed, clearly one can show the following equivalences: (a) A ring R satisfies the condition S-C if and only if R is coherent and satisfies the condition S-c. (b) A ring R is coherent if and only if R is S-coherent and satisfies the condition S-c. (c) A ring R is Noetherian if and only if R is S-Noetherian and satisfies the condition S-c. To give an example of an S-coherent ring which is not S-Noetherian, we use the following result. 11 On S-coherence Proposition 3.5 Let R = S = n Y n Y Ri be a direct product of rings Ri (n ∈ N) and i=1 Si be a cartesian product of multiplicative sets Si of Ri . Then, R is S- i=1 coherent if and only if Ri is Si -coherent for every i ∈ {1, ..., n}. Proof. The result is proved using standard arguments. Example 3.6 Consider the ring A given in Remark 3.4 (2). Let B be a coherent ring which has a multiplicative set W such that BV is not Noetherian. Then, A × B is V × W -coherent (by Proposition 3.5), but it is not V × W -Noetherian (by [1, Proposition 2 (f )]). Now, we give our main result. It is the S-counterpart of the classical Chase’s result [3, Theorem 2.2]. We mimic the proof of [5, Theorem 2.3.2]. So we use the following lemma. Lemma 3.7 ([5], Lemma 2.3.1) Let R be a ring, let I = (u1 , u2 , ..., un ) be a finitely generated ideal of R (n ∈ N) and let a ∈ R. Set J = I + Ra. Let F be f a free module on generators x1 , x2 , ..., xr+1 and let 0 −→ K −→ F −→ J −→ 0, be an exact sequence with f (xi ) = ui (1 ≤ i ≤ r) and f (xr+1 ) = a. Then there exists n L g an exact sequence 0 −→ K ∩ F ′ −→ K −→ (I : a) −→ 0, where F ′ = Rxi . i=1 Theorem 3.8 The following assertions are equivalent: 1. R is S-coherent. 2. Every S-finitely presented R-module is S-coherent. 3. Every finitely generated R-submodule of a free R-module is S-finitely presented. 4. (I : a) is an S-finite ideal of R, for every finitely generated ideal I of R and a ∈ R. 5. (0 : a) is an S-finite ideal of R for every a ∈ R and the intersection of two finitely generated ideals of R is an S-finite ideal of R. 12 D. Bennis and M. El Hajoui Proof. The proof is similar to that of [3, Theorem 2.2] (see also [5, Theorem 2.3.2]). However, for the sake of completeness we give its proof here. (1⇒2) Follows from Proposition 3.2 (1). (2⇒1) Obvious. (1⇒3) Let N be a finitely generated submodule of a free R-module F . Hence, there exists a finitely generated free submodule F ′ of F containing N . Then, by (1), F ′ is S-coherent. Therefore, N is S-finitely presented. (3⇒1) Trivial. (1⇒4) Let I be a finitely generated ideal of R. Then, I is S-finitely presented. Consider J = I + Ra, where a ∈ R. Then, J is finitely generated, and so it is S-finitely presented. Thus, there exists an exact sequence 0 −→ K −→ Rn+1 −→ J −→ 0, where K is S-finite. By Lemma 3.7, there exists a surjective homomorphism g : K −→ (I : a) which shows that (I : a) is S-finite. (4 ⇒ 1) This is proved by induction on n, the number of generators of a finitely generated ideal I of R. For n = 1, use assertion (4) and the exact sequence 0 −→ (0 : I) −→ R −→ I −→ 0. For n > 1, use assertion (4) and Lemma 3.7. (1 ⇒ 5) Since R is S-coherent, Proposition 2.4 applied on the exact sequence 0 −→ (0 : a) −→ R −→ aR −→ 0 shows that the ideal (0 : a) is S-finite. Now, Let I and J be two finitely generated ideals of R. Then, I + J is finitely generated and so Sfinitely presented. Then, applying Theorem 2.5 (5) on the short the exact sequence 0 −→ I ∩ J −→ I ⊕ J −→ I + J −→ 0, we get that I ∩ J is S-finite. (5 ⇒ 1) This is proved by induction on the number of generators of a finitely generated ideal I of R, using the two short exact sequences used in 1 ⇒ 5. It is worth noting that, in Chase’s paper [3], coherent rings were characterized using the notion of flat modules. Then, naturaly one can ask of an S-version of flatness that characterizes S-coherent rings similarly to the classical case. We leave it as an interesting open question. We end this section with some change of rings results. The following results extends [5, Theorem 2.4.1]. Proposition 3.9 Let I be an S-finite ideal of R. Assume that I ∩ S = ∅ so that T := {s + I ∈ R/I; s ∈ S} is a multiplicative subset of R/I. Then, an R/I-module M is T -coherent if only if it is an R-module S-coherent. In particular, the following assertions hold: 1. If R is an S-coherent ring, then R/I is a T -coherent ring. 2. If R/I is a T -coherent ring and I is an S-coherent R-module, then R is an S-coherent ring. On S-coherence 13 Proof. Use Proposition 2.8. Next result generalizes [5, Theorem 2.4.2]. It studies the transfer of S-coherence under localizations. Lemma 3.10 Let f : A → B be a ring homomorphism such that B is a flat Amodule, and let V be a multiplicative set of A. If an A-module M is V -finite (resp., a V -finitely presented), then M ⊗A B is an f (V )-finite (resp., f (V )-finitely presented) B-module. Proof. Follows using the fact that flatness preserves injectivity. Proposition 3.11 If R is S-coherent, then RT is an ST -coherent ring for every multiplicative set T of R. Proof. Let J be a finitely generated ideal of RT . Then, there is a finitely generated ideal I of R such that J = IT . Since R is S-coherent, I is S-finitely presented. Then, using Lemma 3.10, the ideal J = I ⊗R RT of RT is ST -finitely presented, as desired. 4 Other S-version of finiteness In this short section, we present another S-version of S-finiteness and we prove that this notion can be characterized in terms of localization. The following definition gives another S-version of finitely presented modules. Definition 4.1 An R module M is called c-S-finitely presented, if there exists a finitely presented submodule N of M such that sM ⊆ N ⊆ M for some s ∈ S. Remark 4.2 1. Clearly, every finitely presented module is c-S-finitely presented. However, the converse does not hold in general. For that it suffices to consider a coherent ring which has an S-finite module which is not finitely generated. An example of a such ring is given in Remark 3.4 (4). 2. The inclusions in Definition 4.1 complicate the study of the behavior of of cS-finitely presented modules in short exact sequences as done in Theorem 2.5. This is why we think that c-S-finitely presented modules will be mostly used by commutative rings theorists rather than researchers interested in notions of homological algebra. This is the reason behind the use of the letter “c” in “c-S-finitely presented”. 14 D. Bennis and M. El Hajoui 3. It seems that there is not any relation between the two notions of c-S-finitely presented and S-finitely presented modules. Nevertheless, we can deduce that in a c-S-coherent ring (defined below), every S-finitely presented ideal is c-Sfinitely presented. It is well-known that if, for an R-module M , MS is a finitely presented RS -module, then there is a finitely presented R-module N such that MS = NS . Nevertheless, what doest not make things work with respect to localization for S-finitely presented modules is the fact that the module N which satisfies MS = NS is not necessarily a submodule of M . For c-S-finitely presented modules we give the following result. Proposition 4.3 1. If an R-module M is c-S-finitely presented, then MS is a finitely presented RS -module. 2. A finitely generated R-module M is c-S-finitely presented if and only if there is a finitely presented submodule N of M such that MS = NS . Proof. 1. Obvious. 2. (⇒) Clear. (⇐) Since M is finitely and MS = NS , there is an s ∈ S such that sM ⊆ N , as desired. Now we define the other S-version of the classical coherence of rings. Definition 4.4 A ring R is called c-S-coherent, if every S-finite ideal of R is Sfinitely presented. Clearly, every coherent ring is c-S-coherent. The converse is not true in general. The ring given in Example 3.4 (2) can be used as an example of a c-S-coherent ring which is not coherent. Also, it is evident that every S-Noetherian ring is c-S-coherent. As done in Example 3.6, we use the following result to give an example of a c-S-coherent ring which is not S-Noetherian. Proposition 4.5 Let R = S = n Y n Y Ri be a direct product of rings Ri (n ∈ N) and i=1 Si be a cartesian product of multiplicative sets Si of Ri . Then, R is c- i=1 S-coherent if and only if Ri is c-Si -coherent for every i ∈ {1, ..., n}. Proof. The result is proved using standard arguments. On S-coherence 15 Example 4.6 Consider a c-S-coherent ring A which is not coherent. Let B be a coherent ring which has a multiplicative set W such that BV is not Noetherian. Then, A×B is c-V ×W -coherent (by Proposition 4.5), but it is not V ×W -Noetherian (by [1, Proposition 2 (f )]). The follwoing result characterizes c-S-coherent rings can be characterized in terms of localization. Theorem 4.7 The following assertions are equivalent: 1. R is c-S-coherent. 2. Every finitely generated ideal of R is c-S-finitely presented. 3. For every finitely generated ideal I of R, there is a finitely presented ideal J ⊆ I such that IS = JS . In particular, RS is a coherent ring. Proof. (1⇒ 2 ⇒ 3 ) Straightforward. (3⇒1) Let I be an S-finite ideal of R. Then, there exist an s ∈ S and a finitely generated ideal J of R such that sI ⊆ J ⊆ I. By assertion (3), there is a finitely presented ideal K ⊆ J such that KS = JS . Then, there is a t ∈ S such that tJ ⊆ K. Therefore, tsI ⊆ K ⊆ I, as desired. We end the paper with a result which relates c-S-coherent rings with the notion of S-saturation. In [1], the notion of S-saturation is used to characterize S-Noetherian rings. Assume that R is an integral domain. Let SatS (I) denotes the S-saturation of an ideal I of R; that is, SatS (I) := IRS ∩ R. In [1, Proposition 2 (b)], it is proved that if SatS (I) is S-finite, then I is S-finite and SatS (I) = (I : s) for some s ∈ S. This fact was used to prove that a ring R is S-Noetherian if and only if RS is Noetherian and, for every finitely generated ideal of R, SatS (I) = (I : s) for some s ∈ S (see [1, Proposition 2 (f)]). The following result shows that the implication of [1, Proposition 2 (b)] is in fact an equivalence in more general context. Consider N ⊆ M an inclusion of R-modules. Let f : M → MS be the canonical R-module homomorphism. Denote by f (N )RS the RS -submodule of MS generated by f (N ). We set SatS,M (N ) := f −1 (f (N )RS ) and (N :M s) := {m ∈ M ; sm ∈ N }. Proposition 4.8 Let N be an R-submodule of an R-module M . SatS,M (N ) is Sfinite if and only if N is S-finite and SatS,M (N ) = (N :M s) for some s ∈ S. 16 D. Bennis and M. El Hajoui Proof. (⇒) Set K = SatS,M (N ). Since K is S-finite, there exist an s ∈ S and a finitely generated R-module J such that sK ⊆ J ⊆ K. Thus, sN ⊆ sK ⊆ J. We can write J = Rx1 + Rx2 + · · · + Rxn for some x1 , x2 , ..., xn ∈ J. For each xi , there n Q exists a ti ∈ S such that ti xi ∈ N . We set t = ti . Then, tsN ⊆ tsK ⊆ tJ ⊆ N . i=1 Then, N is S-finite. On the other hand, since sK ⊆ tJ ⊆ N ⊆ K, K ⊆ (N :M s). Conversely, let x ∈ (N :M s). Then, sx ∈ N , so x ∈ K, as desired. (⇐) Since N is S-finite, there exist a t ∈ S and a finitely generated R-module J such that tN ⊆ J ⊆ N . On the other hand, since K = (N : s) for some s ∈ S, sK ⊆ N . Consequently, tsK ⊆ tN ⊆ J ⊆ N ⊆ K. Therefore, K is S-finite. The following result is proved similarly to the proof of Proposition 4.8. However, to guarantee the preservation of finitely presented modules when multiplying by elements of S, we assume that S does not contain any zero-divisor of R. Proposition 4.9 Assume that every element of S is regular. Let N be an Rsubmodule of an R-module M . SatS,M (N ) is c-S-finitely presented if and only if N is c-S-finitely presented and SatS,M (N ) = (N :M s) for some s ∈ S. Corollary 4.10 Assume that every element of S is regular. The following assertions are equivalent: 1. For every finitely generated ideal I of R, SatS (I) is c-S-finitely presented. 2. R is c-S-coherent and, for every finitely generated ideal I of R, SatS (I) = (I : s) for some s ∈ S. Acknowledgement. A part of this work was presented by the second author at the Scientific day of Algebra “JA GRAAF 2016” held in Faculty of Sciences of Rabat (May 17, 2016). The authors would like to thank Professor Zine El Abidine Abdelali for his helpful comments during the preparation of this paper. References [1] D. D. Anderson and T. Dumitrescu, S-Noetherian rings, Comm. Algebra, 30 (2002), 4407–4416. [2] D. D. Anderson, D. J. Kwak and M. Zafrullah, Agreeable domains, Comm. Algebra, 23 (1995), 4861–4883. On S-coherence 17 [3] S. U. Chase, Direct products of modules, Trans. Amer. Math. Soc. 97 (1960), 457–473. [4] D. Costa, Parameterizing families of non-Noetherian rings, Comm. Algebra 22 (1994), 3997–4011. [5] S. Glaz, Commutative coherent rings, Lecture Notes in Math., Springer-Verlag, Berlin, 1989. [6] A. Hamed and S. Hizem, Modules Satisfying the S-Noetherian Property and S-ACCR, Comm. Algebra 44 (2016), 1941–1951. [7] A. Hamed and S. Hizem, S-Noetherian rings of the forms A[X] and A[[X]], Comm. Algebra 43 (2015), 3848–3856. [8] E. Hamann, E. Houston and J. L. Johnson, Properties of uppers to zero in R[X], Com. Alg. 23 (1995), 4861–4883. [9] I. Kaplansky, Projective modules, Ann. Math. 68 (1958), 372–377. [10] H. Kim, M. O. Kim and J. W. Lim, On S-strong Mori domains, J. Algebra, 416 (2014), 314–332. [11] J. W. Lim and D. Y. Oh, S-Noetherian properties on amalgamated algebras along an ideal, J. Pure Appl. Algebra 218 (2014), 1075–1080. [12] J. W. Lim and D. Y. Oh, S-Noetherian properties of composite ring extensions, Comm. Algebra 43 (2015), 2820–2829. [13] W. W. McGovern, G. Puninski and P. Rothmaler, When every projective module is a direct sum of finitely generated modules, J. 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Fast construction of efficient composite likelihood arXiv:1709.03234v1 [math.ST] 11 Sep 2017 equations Zhendong Huang and Davide Ferrari ∗ School of Mathematics and Statistics, The University of Melbourne Abstract Growth in both size and complexity of modern data challenges the applicability of traditional likelihood-based inference. Composite likelihood (CL) methods address the difficulties related to model selection and computational intractability of the full likelihood by combining a number of low-dimensional likelihood objects into a single objective function used for inference. This paper introduces a procedure to combine partial likelihood objects from a large set of feasible candidates and simultaneously carry out parameter estimation. The new method constructs estimating equations balancing statistical efficiency and computing cost by minimizing an approximate distance from the full likelihood score subject to a `1 -norm penalty representing the available computing resources. This results in truncated CL equations containing only the most informative partial likelihood score terms. An asymptotic theory within a framework where both sample size and data dimension grow is developed and finite-sample properties are illustrated through numerical examples. Keywords: Composite likelihood estimation, likelihood truncation, `1 -penalty. ∗ Corresponding author: Davide Ferrari, School of Mathematics and Statistics, The University of Mel- bourne, Parkville, VIC 3010, Australia. E-mail: [email protected]. 1 1 Introduction Since the idea of likelihood was fully developed by Fisher (1922), likelihood-based inference has played a role of paramount importance in statistics. The complexity of modern data, however, poses nontrivial challenges to traditional likelihood methods. One issue is related to model selection, since the full likelihood function can be difficult or impossible to specify in complex multivariate problems. Another difficulty concerns computing and the necessity to obtain inferences quickly. These challenges have motivated the development of composite likelihood (CL) methods, which avoid intractable full likelihoods by compounding a set of low-dimensional likelihood objects. Besag (1974) pioneered CL inference in the context of spatial data; Lindsay (1988) developed CL inference in its generality. Due to its flexible framework and established theory, the CL framework has become a popular tool in many areas of applied statistics; see Varin et al. (2011) for an overview of CL inference and common applications. Consider n independent observations on the d × 1 random vector X = (X1 , . . . , Xd )T with pdf in the parametric family {f (x; θ), x ∈ X , θ ∈ Θ ⊆ Rp }, where θ∗ ∈ Θ denotes the true parameter. In this paper, we are mainly concerned with large data sets where both the data dimension d and the sample size n are large. Given i.i.d. observaP tions X (1) , . . . , X (n) on X, we write EFn (g) = n−1 i≤n g(X (i) ) for the empirical mean of P the function g, where Fn (x) = n−1 i≤n I(X (i) ≤ x) is the empirical cdf, and use E(g) to denote its expected value. The operator “∇” denotes differentiation with respect to θ. In the CL setting, the maximum likelihood score uM L (·, θ) = ∇ log f (·, θ) and the associated estimating equations EFn uM L (θ) = 0 are intractable due to difficulties in computing or specifying the full d-dimensional density f (·; θ). Suppose, however, that one can obtain m tractable pdfs f1 (s1 ; θ), . . . , fm (sm ; θ) for sub-vectors S1 , . . . , Sm of X, where each Sj has dimension much smaller than d. For example, S1 could represent a single element of X like X1 , a variable pair like (X1 , X2 ), or a conditional sub-vector like (X1 , X2 )\|X1 . Typically, the total number of sub-models m grows quickly with d; for instance, taking all variable pairs 2 in X results in m = d(d − 1)/2 candidate sub-likelihoods. The specific choice for the set of pdfs {fj , j = 1, . . . , m} is sometimes referred to as CL design (Lindsay et al., 2011) and is typically specified by the practitioner . For simplicity, here the CL design is treated as given, and we assume f1 = · · · = fm , as it is often the case in applications. b We focus on the maximum composite likelihood estimator (MCLE), θ(w), defined as the solution to the CL estimating equations 0 = EFn [u(θ, w)] = EFn [w1 u1 (θ) + · · · + wm um (θ)], (1) where uj (·, θ) = ∇ log{fj (·; θ)} is the jth partial score (sub-likelihood score) associated with the jth subset Sj of X. Here w ∈ Rm is a given vector of coefficients to be determined, which we refer to as composition rule. In addition to well-known computational advantages compared to MLE and flexible modeling, the MCLE enjoys first-order properties analogous to those of the maximum likelihood estimator (MLE). Since the partial scores commonly define unbiased estimating equations (i.e. Euj (θ) = 0 at θ = θ∗ , for all 1 ≤ j ≤ m), the CL b score u(θ, w) in (1) is also unbiased, a property leading to consistency of θ(w). Unfortunately, the MCLE does not have the same second-order properties as the MLE since the asymptotic b variance of θ(w) is generally different from the inverse of Fisher information −E[∇uM L (θ∗ )], with the two coinciding only in special families of models. The choice of the composition rule w is crucial in determining both efficiency and computb ing cost associated with θ(w). Established theory of unbiased estimating equations prescribes b to find w so to minimize the asymptotic variance of θ(w) (Heyde, 2008, Chapter 2), given by the inverse of the p × p Godambe information matrix G(θ, w) = E{∇u(θ, w)} var {u(θ, w)}−1 E{∇u(θ, w)}. (2) Although theoretically appealing, this is a notoriously difficult task due the well-known instability of common estimators of the term var {u(θ, w)} in G(θ, w) (Lindsay et al., 2011). 3 On the other hand, the common practice of retaining all terms in (1) by choosing fixed wj 6= 0 for all j ≥ 1 (e.g. wj = 1, j ≥ 1) is undesirable from both computational and statistical efficiency perspectives, especially when the partial scores uj exhibit pronounced correlation. Cox and Reid (2004) discuss the detrimental effect caused by the presence of b many correlated scores on the variance of θ(w) when n is small compared to m in pairwise likelihood estimation. In the most serious case where the correlation between scores is overwhelming, keeping all the terms in (1) may lead to lack of consistency for the implied b MCLE θ(w). Motivated by the above considerations, we introduce a new method called sparse composite likelihood estimation and selection (SCLE) consisting of two main steps: a truncation Step (T-Step) and an estimation Step (E-Step). In the T-Step, the composition rule w is obtained by minimizing an approximate distance between the unknown full likelihood score uM L (θ) and the CL score u(θ, w), subject to a `1 -norm constraint on w. This step may be viewed as maximizing statistical accuracy for given afforded computing. Alternatively it may be interpreted as minimizing the computing cost for given level of statistical efficiency. Due to the geometry of the `1 -norm, the resulting composition rule, say w, b contains a number of non-zero elements (see Lemma 3.1). While the most useful terms for improving MCLE’s statistical accuracy are retained, the noisy sub-likelihoods contributing little or no improvement are dropped. In the E-step, we solve the estimating equations (1) with w = w b and find b w). the final estimator θ( b Compared to traditional CL estimation, the main advantage of our approach is to reduce the computational burden, while retaining relatively high efficiency in large data sets. The reduced number of terms in the estimating equations (1) translates into fast computing and enhanced stability for the final estimator at a relatively small cost in terms of statistical efficiency. The remainder of this paper is organized as follows. In Section 2, we describe the main methodology for simultaneous likelihood truncation and parameter estimation. In Section 3, we study the properties of the truncated composition rule and for the implied estimator 4 within a framework where both the sample size n and the data dimension d are allowed to diverge. Section 4 illustrates the properties of our methodology in the context of estimation of location and scale for multivariate normal models. In Section 5, we study the trade-off between computational and statistical efficiency in finite samples through numerical simulations. Section 6 concludes with final remarks. Technical lemmas used in our main results are deferred to the appendix. 2 Main methodology Throughout the paper, we consider unbiased partial scores {uj (θ), 1 ≤ j ≤ m} satisfying Euj (θ) = 0, for all 1 ≤ j ≤ m, (3) when θ = θ∗ and assume that θ∗ is the unique solution for all the equations in (3). The approach described in this section is applicable to problems with arbitrary sample size n and data dimension d, but we are mainly concerned with the situation where the data dimension d (and number of available sub-likelihood objects m) is large compared to the sample size n. Although we focus on log-likelihood partial scores for concreteness, our methodology and the properties in Section 3 remain essentially unchanged if uj (θ) is any arbitrary unbiased M-estimating equation. For instance, when θ is a location parameter, a more appropriate choice in the presence of outliers may be the Huber-type partial score uj (θ) = ψ(sj − θ), where ψ(z) = −k if z ≤ k, ψ(z) = z if \|z\| ≤ k and ψ(z) = k if z ≥ k, with k > 0. Another suitable choice in the same setting is the Lq-likelihood estimating equation of Ferrari and Yang (2010) defined by uj (θ) = ∇ logq {fj (sj ; θ)}, where logq (z) = log(u) if q = 1, and logq (z) = (z 1−q − 1)/(1 − q) if q 6= 1. In the rest of the paper we use U (θ) to denote the p × m matrix with column vectors u1 (θ), . . . , um (θ) and define the m × m matrix S(θ) = U (θ)T U (θ) with (jk)th entry {S(θ)}j,k = uj (θ)T uk (θ). We write UA (θ) for the sub-matrix of U (θ) with columns corre5 sponding to A ⊆ {1, . . . , m}, while U\A (θ) denotes the sub-matrix containing the remaining columns. Accordingly, we define the \|A\| × \|A\| matrix SA (θ) = U (θ)TA UA (θ) and use wA to denote the sub-vector of w with elements {wj , j ∈ A}, while w\A represents the vector containing all the elements in w not in wA . 2.1 Sparse and efficient estimating equations Our main objective is to solve the CL estimating equations 0 = EFn u(θ, w) defined in (1) with respect to θ using coefficients w = wλ (θ) obtained by minimizing the ideal criterion m X 1 wj uj (θ) Qλ (θ, w) = E uM L (θ) − 2 j=1 2 +λ 2 m X αj \|wj \| , (4) j=1 where k · k2 denotes the Euclidean norm, λ ≥ 0 is a given constant, and the αj s are pre-set constants not depending on the data. For clarity of exposition, we set αj = 1 for all j ≥ 1 in the remainder of the paper. The optimal solution wλ (θ) is interpreted as one that maximizes the statistical accuracy of the implied CL estimator, subject to a given level of computing. Alternatively, wλ (θ) may be viewed as to minimize the complexity of the CL equations, subject to given efficiency compared to MLE. The tuning constant λ balances the trade-off between statistical efficiency and computational burden The first term in Qλ (θ, w) aims to obtain efficient estimating equations by finding a CL score close to the ML score. When λ = 0 and θ = θ∗ , the composition rule w0∗ = w0 (θ∗ ) is optimal in the sense that the score function u(θ, w0∗ ) is closest to the MLE score uM L (θ). Although this choice gives estimators with good statistical efficiency, it offers no control for the CL score complexity since all the partial likelihood scores are included in the final P estimating equation. The second term λ m j=1 αj \|wj \| in (4) is a penalty discouraging overly complex estimating equations. In Section 3.1, we show that typically this form of penalty implies a number of elements in wλ (θ) exactly zero for any λ > 0. For relatively large λ, many elements in wλ (θ) are exactly zero, thus simplifying considerably the CL estimating 6 equations 0 = EFn u(θ, wλ (θ)). When a very large fraction of such elements is zero, we say that wλ (θ) and the CL equations 0 = EFn u(θ, wλ (θ)) are sparse. Sparsity is a key advantage of our approach to reduce the computational burden when achievable without loosing much statistical efficiency. On the other hand, if λ is too large, one risks to miss the information in some useful data subsets which may otherwise improve statistical accuracy. 2.2 Empirical criterion and one-step estimation Obvious difficulties related to direct minimization of the ideal criterion Qλ (θ, w) are the presence of the intractable likelihood score uM L and the expectation depending on the unknown parameter θ∗ . To address these issues, first note that, up to a negligible term not depending on w, Criterion (4) can be written as m X 1 E wj uj (θ) 2 j=1 2 − 2 m X m X ML T wj E u (θ) uj (θ) + λ αj \|wj \| . j=1 (5) j=1 If θ = θ∗ , we have E[uM L (θ)uj (θ)T ] = E[uj (θ)uj (θ)T ]. To see this, recall that partial scores are unbiased and differentiate both sides of 0 = Euj (θ) under appropriate regularity conditions. This result is used to eliminate the explicit dependency on the score uM L . Finally, replacing expectations in (5) by empirical averages leads to the following empirical objective: m X 1 b Qλ (θ, w) = EFn wj uj (θ) 2 j=1 2 − 2 m X j=1 m X T wj EFn uj (θ) uj (θ) + λ αj \|wj \| . (6) j=1 Under appropriate regularity conditions, the empirical criterion (6) estimates consistently the population criterion (4) up to an irrelevant constant not depending on w, with the caveat that θ must be close to θ∗ . These considerations motivate the following estimation strategy: b compute the truncated 1) T-Step. Given a preliminary root-n consistent estimator θ, 7 composition rule w bλ by solving b w). bλ (θ, w bλ = argmin Q (7) w∈Rm 2) E-Step. Update the parameter estimator by the one-step Newton-Raphson iteration h i−1 b b b bw θλ = θ − EFn ∇u(θ, w bλ ) EFn u(θ, bλ ). (8) Theorem 3.2 shows that the convex minimization problem in the T-Step has unique solution. Particularly, let Eb ⊆ {1, . . . , m} is the subset of partial scores such that n o T b b EFn uj (θ) rj (θ, w bλ ) ≥ λ, (9) where rj is the pseudo-residual defined by rj (θ, w) = uj (θ) − u(θ, w) and and write \Eb for b Then the solution of the T-Step is the set {1, . . . m} \ E. o o−1 n n b − λ sign(w b bλ,Eb) , w bλ,\Eb = 0, diag{EFn SEb(θ)} w bλ,Eb = EFn SEb(θ) (10) b sign(w) is the vector where: SEb = UEbT UEb and UEb is a matrix with column vectors {uj , j ∈ E}; sign function with jth element taking values −1, 0 and 1 if wj < 0, wj = 0 and wj > 0, respectively; and diag(A) denotes the diagonal of the square matrix A. More insight on the meaning of (9) may be useful. Differentiating (5) in wj 6= 0 and then expanding around θ∗ under Conditions C.1 and C.2 in Section 3.1 gives n o b T rj (θ, b w) = E uj (θ∗ )T uM L (θ∗ ) − u(θ∗ , w) + op (1). EFn uj (θ) (11) Combining (9) and (11) highlights that the jth partial likelihood score uj (θ) is selected when it is sufficiently correlated with the residual difference uM L (θ) − u(θ, w). Hence, our 8 criterion retains only those uj s which are maximally useful to explain the gap between the full likelihood score uM L (θ) and the CL score u(θ, w), while it drops the remaining scores. When λ = 0, we have Eb = {1, . . . , m} meaning that the corresponding composition rule w b0 does not contain zero elements. From (10) for λ = 0 it is required that the empirical b is non-singular which is violated when n < m. covariance matrix for all partial scores EFn S(θ) b may be nearly singular due to the presence of largely Even for n > m, however, EFn S(θ) correlated partial scores. On the other hand, setting λ > 0 always gives a non-singular b and guarantees existence of w matrix EFn SEb(θ) bλ,Eb. The proposed approach requires an initial root-n consistent estimator, which is often easy to obtain when the partial scores are unbiased. One simple option entails solving EFn u(w, θ) = 0 with w = (1, . . . , 1)T . If m is large, one may choose w by the stochastic CL strategy of Dillon and Lebanon (2010), where the elements of w may be set as either 0 or 1 randomly according to some user-specified scheme. Although the initial estimator θb could be quite inefficient, the one-step update (8) improves upon this situation. Moreover, the estimator θbλ and coefficients w bλ can be refined by iterating the T-Step (with θb = θbλ ) and the E-Step a few times. 2.3 Computational aspects: LARS implementation and selection of λ The empirical composition rule w bλ in (7) cannot be computed using derivative-based apb w). To address this issue, we propose an bλ (θ, proaches due to non-differentiability of Q implementation based on the least-angle regression (LARS) algorithm of Efron et al. (2004) originally developed for sparse parameter estimation in the context of linear regression modb our implementation of LARS minimizes Q b w) by including one score bλ (θ, els. For given θ = θ, b at the time in the composite likelihood score u(θ, b w). In each step, the score with the uj (θ) b − u(θ, b w) is included, largest correlation with the currently available residual difference uj (θ) followed by an adjustment step on w. The numerical examples in Section 5, suggest that 9 our implementation of the LARS algorithm for CL selection is very fast. In at most m × p steps, it returns a path of estimated composition rules w bλ1 , . . . , w bλm , where λj here is the value of the tuning constant λ in (6) at which the jth partial score enters the CL estimating equation. Selection of λ is of practical importance since it balances the trade-off between statistical and computational efficiency. For a given budget on afforded computing, say λ∗ , we include one partial score at the time, for example using the LARS approach above, and stop when b = max{λ : φ(λ) > τ }, for some user-specified 0 < τ ≤ 1, where we reach λ φ(λ) = b tr{EFn Sλ (θ)} I(λ > λ∗ ). b tr{EFn S(θ)} (12) Here EFn Sλ = EFn UλT Uλ denotes the empirical covariance matrix for the selected partial scores indexed by the set Ebλ = {j : w bλ,j 6= 0}. The criterion φ(λ) can be viewed as the proportion of score variability explained by currently selected partial scores. In practice, we choose τ close to 1, such as τ = 0.9, 0.95 or 0.99. If the computing budget is reached, we set b = λ∗ . In analogy with principal component analysis, the selected combination of scores λ accounts for the largest variability in the collection of empirical scores. 3 Properties This section investigates the asymptotic behavior of the sparse composition rule w bλ and the corresponding SCLE θbλ defined in (8) within a setting where m – the number of candidate partial likelihoods – is allowed to grow with the sample size n. We use m∗ = EkuM L (θ∗ )k22 to denote the trace of Fisher information based on the full likelihood. Here m∗ may be interpreted as the maximum knowledge about θ if the full likelihood score uM L were available. Although m∗ can grow with m, reflecting the rather natural notion the one can learn more about the true model as the overall data size increases, it is not allowed to grow as fast as n; e.g., m∗ = o(log n). This is a rather common situation in CL estimation occuring, for 10 instance, when the sub-likelihood scores are substantially correlated or they are independent but with heterogeneous and increasing variances (see examples in Section 4.1). 3.1 Sparsity and optimality of the composition rule In this section, we give conditions ensuring uniqueness of the empirical composition rule w bλ and weak convergence to its population counterpart wλ∗ . To this end, we work θ within the root-n neighborhood of θ∗ , Θn = {θ : kθ − θ∗ k < c0 n−1/2 }, for some c0 > 0, and assume the following regularity conditions on S(θ): C.1 There exist positive constants c1 , c2 > 0 such that E{supθ∈Θn S(θ)j,k } < c1 , and V ar{supθ∈Θn S(θ)j,k } < c2 , for all j, k ≥ 1. C.2 Each element ES(θ)j,k is continuous with uniformly bounded first and second order derivatives on Θn . Our analysis begins by deriving the Karush-Kuhn-Tucker Condition (KKTC) (Kuhn, 2014) for the population objective Qλ (θ∗ , w) defined in (4). The KKTC characterizes the amount of sparsity – and, the computational complexity – associated with the selected estimating equations depending on the value of the tuning constant λ. Let c(θ, w) = diag(S(θ))−S(θ)w, where S(θ) = U (θ)U (θ)T is as defined in Section 2 Lemma 3.1 (KKTC). Under Condition C.1, the minimizer wλ∗ of Qλ (θ∗ , w) defined in (4) satisfies E\|{c(θ∗ , wλ∗ )j }\| = λ · γj , j = 1, . . . , m, ∗ ∗ ∗ where γj ∈ {1} if wλ,j > 0, γj ∈ {−1} if wλ,j < 0, and γj ∈ [−1, 1] if wλ,j = 0; c(·, ·)j is the jth element of vector c(·, ·). ∗ Proof. Let dj = −Ec(θ∗ , wλ∗ )j +λ·sign(wλ,j ) and note that the Tayor expansion of Qλ (θ∗ , wλ∗ ) ∗ around wλ,j 6= 0 is Qλ (θ∗ , wλ∗ + ) = Qλ (θ∗ , wλ∗ ) + dj + 11 2 tr{Ij (θ∗ )}, 2 (13) where = (0, . . . , j , . . . , 0)T , and Ij (θ∗ ) = E uj (θ∗ )uj (θ∗ )T is the p × p Fisher information matrix for the jth likelihood component, and tr{Ij (θ∗ )} < c1 by Condition C.1. ∗ 6= 0, we have dj = 0. Otherwise, if dj 6= 0, choosing j such that sign(j ) = If wλ,j −sign(dj ) and \|j \| < 2\|dj \|/c1 , implies Qλ (θ∗ , wλ∗ + ) < Qλ (θ∗ , wλ∗ ), but this is a contra∗ = 0, we need to show \|Ec(θ∗ , wλ∗ )j \| ≤ λ. diction since wλ∗ minimizes Qλ (θ∗ , ·). If wλ,j Assume \|Ec(θ∗ , wλ∗ )j \| > λ and take j such that sign(j ) = sign(Ec(θ∗ , wλ∗ )j ) and \|j \| < 2\|Ec(θ∗ , wλ∗ )j − λ\|/\|c1 \|. Then dj + 2 tr{I(θ∗ )}/2 < −\|j \|(\|Ec(θ∗ , wλ∗ )j \| − λ) + 2 c1 /2 < 0, which implies Qλ (θ∗ , wλ∗ +) < Qλ (θ∗ , wλ∗ ). But this is contradicted by wλ∗ being the minimizer of Qλ . Hence, Ec(θ∗ , wλ∗ )j = λ · γj , for all j = 1, . . . , m. An argument analogous to that used in the proof of Lemma 3.1 leads to the KKTC for b w). Specifically, for w bw b θ, w bλ , the minimizer of the empirical loss Q( bλ we have EFn c(θ, bλ )j = λ·γ bj , j = 1, . . . , m, where γ bj ∈ {1} if w bλ,j > 0, γ bj ∈ {−1} if w bλ,j < 0, and γ bj ∈ [−1, 1] if w bλ,j = 0. Lemma 3.1 has important implications in our current setting, since it relates λ to the size of the covariance between the jth sub-likelihood score uj (θ) and the residual difference uM L (θ) − u(θ, w) at θ = θ∗ . Particularly, if such a covariance is sufficiently small, i.e. λ > E{c(θ∗ , wλ∗ )j } = \|E {uj (θ∗ ) [u(θ∗ , wλ∗ ) − uj (θ∗ )]}\| = E uj (θ∗ ) u(θ∗ , wλ∗ ) − uM L (θ∗ ) , ∗ then the correspondent coefficient is wλ,j = 0. Thus, the tuning parameter λ controls the level of sparsity of the composite score u(θ∗ , wλ∗ ) by forcing the weights of those non-important score components with small pseudo-covariance c(θ∗ , wλ∗ )j to be exactly zero. For uniqueness of wλ∗ and w bλ , a simple condition is that the partial scores cannot replace each other, i.e. we require that the scores are in general position. Specifically, we say that the score components u1 , . . . , um are in general position if any affine subspace L ⊂ Rm of dimension l < m contains at most l + 1 elements of {±u1 , ..., ±um } excluding antipodal pairs of points. 12 C.3 The partial scores uj (x, θ), j ≥ 1, are continuous and in general position with probability 1 for all θ ∈ Θn . Theorem 3.2. Under Conditions C.1-C.3 the solution of the T-Step, w bλ , defined in (7) is unique and is given by (10) for any λ > 0. Moreover, w bλ contains at most np ∧ m non-zero elements. Proof. Let Eb = {j ∈ {1, . . . , m} : \|b γj \| = 1} to be the index set of non-zero elements of w bλ where where γj is as defined after Lemma 3.1. First note that the composite likelihood score bw b Tw b ·) defined in (6), due bλ (θ, u(θ, bλ ) = U (θ) bλ is unique for all solutions w bλ which minimize Q b w). Uniqueness of u(θ, bw bλ (θ, to strict convexity of Q bλ ) implies that γ b and the corresponding index set Eb are unique by Lemma 3.1. b we first note that the square matrix Next, to show uniqueness of w bλ,j for all j ∈ E, b has full rank. Otherwise, some row the matrix can be written as a linear b T U b(θ)] EFn [UEb(θ) E b b = P aj EFn [uj (θ) b T U b(θ)]. b T U b(θ)] b i.e. EFn [uk (θ) combination of other rows in the set E, k6=j E E P b 2 −P aj EFn uj (θ) b 2 = λb Then Lemma 3.1 implies also the event EFn uk (θ) γk − j6=k aj λb γj for j6=k the same set of coefficients aj s, which has probability equals to 0 since each uj is continuous b b(θ) b T ] has full rank, meaning that the size of Eb satisfies \|E\| b ≤ and random. Thus, E[UEb(θ)U E b T ] implies strict convexity of b b(θ) b full rank of E[U b(θ)U np ∧ m. For fixed wj = 0, j ∈ \E, E E b w b) where w b is the sub-vector of w containing elements indexed by E. bλ (θ, b Hence, w Q bλ,Eb λ,E λ,E is unique. The arguments in Theorem 3.2 go through essentially unchanged for the population composition rule wλ∗ by showing the full rank of E[UE (θ∗ )T UE (θ∗ )] using Condition C.3 and Lemma 3.1, where E is the index set of non-zero elements in wλ∗ . This implies also uniqueness of wλ∗ . Next, we turn to convergence of the empirical composition rule w bλ to wλ∗ , thus showing bλ (θ, w) (6) is a suitable replacement for the intractable criterion Qλ (θ, w) that the objective Q (4). Since criterion b w) = wT EFn {S(θ)}w/2 b b T w + λkwk1 bλ (θ, Q − EFn {diag(S(θ))} 13 is used as an approximation of the population criterion Qλ (θ∗ , w) defined in (4), clearly the b and ES(θ∗ ) affects the accuracy of such an approximation. Let distance between EFn S(θ) b and r1 = supθ∈Θn kEFn S(θ) − ES(θ∗ )k2 be the supreme variation between matrices EFn S(θ) ES(θ∗ ), where kAk2 is the matrix induced 2-norm for matrix A. As n → ∞, the rate at which r1 goes to 0 depends mainly on the number of partial scores m and the behavior of the random elements in S, which can vary considerably in different models. For example, when the elements of S(θ) are sub-Gaussian, one needs only log(m)/n = o(1) (Cai et al., 2010). In more general cases, m4 /n = o(1) suffices to ensure r1 = op (1) (Vershynin, 2012). Next we investigate how m and m∗ should increase compared to r1 when λ → 0 as n → ∞ to ensure a suitable behavior for w bλ . To obtain weak convergence of w bλ to wλ∗ , we introduce the additional requirement that the covariance matrix of the partial scores ES(θ∗ ) does not shrink to zero too fast. 2 C.4 There exists a sequence cn , such that cn r1λm∗2 → ∞ and xT ES(θ∗ )x ≥ cn kxk1 for any x ∈ Rm , as n → ∞. Condition C.4 is analogous to the compatibility condition in `1 -penalized least-squares estimation for regression (Bühlmann and Van De Geer, 2011), where it ensures a good behavior of the observed design matrix of regressors. Differently from the sparse regression setting, where Condition C.4 is applied to the set of true nonzero regression coefficients, here no sparsity assumption on the composition rule w is imposed. P Theorem 3.3. Under Conditions C.1-C.4, if r1 m∗ 2 λ−2 = op (1) then kw bλ − wλ∗ k1 → 0, as n → ∞. b w Proof. From Lemma 7.3, EFn kU (θ)( bλ − wλ∗ )k22 = op (1). Note that b w b w EFn kU (θ)( bλ − wλ∗ )k22 = (w bλ − wλ∗ )T EFn S(θ)( bλ − wλ∗ ) n o ∗ ∗ T ∗ ∗ ∗ T b =(w bλ − wλ ) ES(θ )(w bλ − wλ ) + (w bλ − wλ ) EFn S(θ) − ES(θ ) (w bλ − wλ∗ ), 14 and the second term of the last equality is Op (r1 m∗ 2 λ−2 ) by Lemma 7.2. Thus, (w bλ − P wλ∗ )T ES(θ∗ )(w bλ −wλ∗ ) = Op (r1 m∗ 2 λ−2 ), which implies kw bλ −wλ∗ k1 → 0 by Condition C.4. Corollary 3.4. Let λ be a sequence such that λ → 0 as n → ∞. Under Conditions C.1-C.4, if r1 m∗ 2 λ−2 = op (1), we have P sup EFn ku(θ, w bλ ) − u(θ, wλ∗ )k2 → 0, as n → ∞. (14) θ∈Θn bw b w∗ )k2 = op (1). The result follows by noting Proof. From Lemma 7.3, EFn ku(θ, bλ ) − u(θ, λ 2 bw b w∗ )k2 = that for any θ ∈ Θn , the difference EFn ku(θ, w bλ ) − u(θ, wλ∗ )k22 − EFn ku(θ, bλ ) − u(θ, λ 2 b w bλ − wλ∗ ) is op (1) according to Conditions C.1, C.2 and Theorem (w bλ − wλ∗ )T EFn [S(θ) − S(θ)]( 3.3 Corollary 3.4 states that the composite likelihood score u(θ, w bλ ) is a reasonable approximation to u(θ, wλ∗ ). Particularly, even for λ close to zero, the composite score u(θ, w bλ ) still b uses a fraction \|E\|/m of sub-likelihood components. At the same time u(θ, w bλ ) is near the optimal score u(θ, w0∗ ), where w0∗ is the composition rule yielding the closest CL score u(θ) to the maximum likelihood score uM L (θ). Moreover, the implied Godambe information G(θ, w bλ ) = E{∇u(θ, w bλ )}var{∇u(θ, w bλ )}−1 E∇{u(θ, w bλ )} is expected to be close to G(θ, w) with w = w0∗ . However, while the MCLE based on w0∗ (or other choice of wj 6= 0, j ≥ 1) may be unavailable or computationally intractable due to common difficulties in estimating var{∇u(θ, w0∗ )} (Lindsay et al., 2011; Varin et al., 2011), our truncated composition rule w bλ implies a more stable estimation of G(θ, w bλ ) by requiring only a fraction of scores. 15 3.2 Asymptotic behavior of the one-step SCLE In this section, we show consistency and give the asymptotic distribution for the SCLE bw θbλ = θ( bλ ) defined in the E-Step (8). One advantage of one-step estimation is that consistency and asymptotic normality are treated separately. The one-step estimator θbλ inherits b under standard rethe properties leading to consistency from the preliminary estimator θ, quirements on S(θ). For normality, additional conditions on the sub-likelihood scores are needed. Let H(θ) be the p×mp matrix obtained by stacking all the p×p sub-matrices ∇uj (θ). Let r2 = supθ∈Θn maxj,k \|EFn H(θ)j,k −EH(θ∗ )j,k \| be the maximum variation between the emb − Euj (θ∗ )k1 be pirical and the optimal Hessian matrices. Let r3 = supθ∈Θn maxj kEFn uj (θ) the supreme variation between empirical scores and their expected value around Θn . In the rest of this section, we use Jλ∗ = Cov [u(θ∗ , wλ∗ )] and Kλ∗ = −E∇u(θ∗ , wλ∗ ) to denote the population variability and sensitivity p × p matrices, respectively, both depending implicitly on n. We further assume: C.5 There exist positive constants c5 and c6 such that E[supθ∈Θn H(θ)j,k ] < c5 , and V ar[supθ∈Θn H(θ)j,k ] < c6 , for all j, k ≥ 1. C.6 Each element EH(θ)j,k , j, k ≥ 1 of the matrix EH(θ) is continuous with uniformly bounded first and second derivatives on θ ∈ Θn . Theorem 3.5. Suppose there exist N > 0 such that Kλ∗ is non-singular with all eigenvalues bounded away from 0 for all n > N . Under Conditions C.1 - C.6, if r1 m∗ 2 λ−2 = op (1), √ r2 m∗ λ−1 = op (1) and r3 m∗ λ−1 = op (1), then as n → ∞ we have P (i) kθbλ − θ∗ k1 → 0, and (ii) √ 1 D nJλ∗ − 2 Kλ∗ (θbλ − θ∗ ) → Np (0, I), where Jλ∗ = Cov [u(θ∗ , wλ∗ )] and Kλ∗ = −E∇u(θ∗ , wλ∗ ) denote p × p population variability and sensitivity matrices. 16 Proof. Without loss of generality, we only prove the case p = 1. Since p is fixed, the bλ = proof can be easily generalized to the case p > 1 without additional conditions. Let K bw −EFn ∇u(θ, bλ ) be the empirical sensitivity matrix. Then θbλ can be written as θbλ = θb + bw b −1 EFn u(θ, K bλ ), with θb being a consistent preliminary estimator. Note that Eu(θ∗ , wλ∗ ) = 0 λ and bw bw b w∗ )k1 kEFn u(θ, bλ ) − Eu(θ∗ , wλ∗ )k1 ≤ kEFn u(θ, bλ ) − EFn u(θ, λ (15) b w∗ ) − EFn u(θ∗ , w∗ )k1 + kEFn u(θ∗ , w∗ ) − Eu(θ∗ , w∗ )k1 . + kEFn u(θ, λ λ λ λ The first term on the right hand side of (15) is op (1) by Lemma 7.3. The second term is op (1) b − EFn uj (θ∗ )\|kw∗ k1 , which converges b w∗ ) − EFn u(θ∗ , w∗ )k1 ≤ maxj \|EFn uj (θ) since kEFn u(θ, λ λ λ to 0 by Theorem’s assumptions and Lemma 7.2. The last term of (15) is also op (1) by the P bw Law of Large Numbers. This shows that EFn u(θ, bλ ) → 0. Moreover, from Lemma 7.4, b λ − K ∗ k1 = op (1). Since K ∗ has all eigenvalues bounded away from 0 for large n, we have kK λ λ P P P bw bw b −1 EFn u(θ, b −1 EFn u(θ, K bλ ) → 0. Since θb → θ∗ , we have θbλ = θb + K bλ ) → θ∗ , which shows λ λ part (i) of the theorem. bw b −1 EFn u(θ, To show normality in (ii), re-arrange θbλ = θb + K bλ ) and obtain λ bw b λ (θbλ − θ∗ ) = K b λ (θb − θ∗ ) + EFn u(θ, bλ − K e λ ](θb − θ∗ ) K bλ ) = EFn u(θ∗ , w bλ ) + [K bλ − K e λ ](θb − θ∗ ), =EFn u(θ∗ , wλ∗ ) + [EFn u(θ∗ , w bλ ) − EFn u(θ∗ , wλ∗ )] + [K (16) ew e λ = −EFn ∇u(θ, where K bλ ) and θe is some value between θb and θ∗ . The second equality b For the first term in (16), we follows from the first-order expansion of EFn u(θ∗ , w) b at θ. √ 1 D have nJλ∗ − 2 EFn u(θ∗ , wλ∗ ) → Np (0, I), since the Lindeberg-Feller Central Limit Theorem applies to u(θ∗ , wλ∗ ) by Lemma 7.6. By Lemma 7.5 Jλ∗ = O(m∗ ), so the first term in (16) is p Op ( m∗ /n). The second term in (16) EFn u(θ∗ , w bλ ) − EFn u(θ∗ , wλ∗ ) = EFn U (θ∗ )T (w bλ − wλ∗ ) P is of smaller order compared to first term EFn u(θ∗ , wλ∗ ) = EFn U (θ∗ )T wλ∗ since kw bλ − wλ∗ k → 0 17 by Theorem 3.3. For the last term in (16), we have b − E∇uj (θ∗ ) bλ − K e λ )(θb − θ )\|1 ≤ max EFn ∇uj (θ) \|(K ∗ j ∗ e − E∇uj (θ ) + max EFn ∇uj (θ) j kw bλ k1 \|θb − θ∗ \| ≤2r2 kw bλ k1 \|θb − θ∗ \|. (17) and the last expression in (17) is op (r2 m∗ n−1/2 λ−1 ) by Lemma 7.2. Theorem’s assumption that r1 m∗ 2 λ−2 = op (1) implies that the last term in (16) is of smaller order compared to the b λ − K ∗ k1 = op (1) according to Lemma 7.4, Slutsky’s Theorem first term. Finally, since kK λ implies the desired result. Consistency and asymptotic normality for the one-step estimator θbλ follow mainly from w bλ converging in probability to the target composition rule wλ∗ . Since each sub-likelihood score is unbiased and asymptotically normal, their linear combination is also normally dis√ 1 tributed. The overall convergence rate is given by k nJλ∗ − 2 Kλ∗ k1 which is of order between √ √ n and nm. The actual order depends on the underlying correlation between partial √ scores u1 , . . . , um . While the optimal rate nm is achieved when the scores are perfectly independent, combining highly correlated scores into the final estimating equation will give √ rates closer to n. 4 Examples for special families of models In this section, we illustrate the SCLE through estimation of location and scale estimation for special multivariate normal models. 4.1 Estimation of common location for heterogeneous variates Let X ∼ Nm (θ1m , Σ), where the m × m covariance matrix Σ has off-diagonal elements σjk (j 6= k) and diagonal elements σk2 (j = k). Computing the MLE of θ requires Σ−1 and in 18 b = EFn X T X. When n < m, Σ b is singular and the MLE practice Σ is replaced by the MLE Σ of θ is not available in practice, whilst CL estimation is still feasible. The jth partial score is uj (θ) = (Xj − θ)/σj2 and the CL estimating equation (1) based on the sample X (1) , . . . , X (n) is 0 = EFn u(θ, w) = m n X wj X (i) (Xj − θ), 2 nσ j j=1 i=1 leading to the profiled MCLE b θ(w) = m X ! wj σj−2 X j j=1 / m X ! wk σj−2 , (18) j=1 which is a weighted average of marginal sample means X j = n−1 Pn i=1 (i) Xj , j ≥ 1. In this example, one can work out directly the optimal composition rule wλ∗ and no estimation is required. Particularly, it is useful to inspect the special case where X has independent components (σjk = 0 for all j 6= k). This corresponds to the fixed-effect meta-analysis model where estimators from m independent studies are combined to improve accuracy. Under independence, we have the explicit solution ∗ wλ,j = (1 − σj2 λ)I σj2 < λ−1 , 1 ≤ j ≤ m, which highlights that overly noisy data subsets with variance σj2 ≥ λ−1 are dropped and thus do not influence the final estimator (18). The number of non-zero elements in wλ∗ is Pm 2 −1 j=1 I(σj < λ ). Note that when λ = 0, we have uniform weights w0∗ = (1, . . . , 1)T and the corresponding b ∗) MCLE is the usual optimal meta-analysis solution. Although the implied estimator θ(w 0 has minimum variance, it offers no control for the overall computational cost since all m sub-scores are selected. On the other hand, choosing judiciously λ > 0 may lead to low computational burden with negligible loss for the resulting estimator. For instance, assuming 19 σj2 = j 2 , for θ ∈ Θn , a straightforward calculation shows E [u(θ, wλ∗ ) − u(θ, w0∗ )]2 ≤ λ2 X j∈E Since the number of the non-zero scores Pm j=1 j2 + X j −2 + o(1), (19) j ∈E / 1 I (j 2 < λ−1 ) = bλ− 2 c, the first term the mean squared difference between u(θ, wλ∗ ) and the optimal score u(θ, w0∗ ) is bounded by 1 1 λ2 λ−1 λ− 2 = λ 2 , up to a vanishing term. Thus, if λ = o(1), the composite score u(θ, wλ∗ ) converges to the optimal composite score u(θ, w0∗ ). Particularly, if λ decreases at a sufficiently slow rate, the truncated score u(θ, wλ∗ ) can still contain a relatively small number of terms, b ∗ ) is approximately equal the optimal estimator θ(w b ∗) while the correspondent estimator θ(w 0 λ in terms of statistical accuracy. If the elements of X are correlated (σjk 6= 0 for j 6= k), the partial scores contain overlapping information on θ. In this case, tossing away some highly correlated partial scores improves computing while maintaining satisfactory statistical efficiency for the final estimator. Figure 1 shows the solution path of wλ∗ and the asymptotic relative efficiency of b ∗ ) compared to MLE for different values of λ. When m is large the corresponding SCLE θ(w λ (e.g. m = 1000), the asymptotic relative efficiency drops gradually until a few scores are left. This example illustrates that a relatively high efficiency can be achieved by our truncated CL equations, when a few partial scores already contains the majority of information about θ. In such cases, the final SCLE with a sparse composition rule is expected to achieve a good trade-off between computational cost and statistical efficiency. 4.2 Location estimation in exchangeable normal variates In our second example we consider exchangeable variables with X ∼ Nm (θ1m , Σ) with Σ = (1 − ρ)Im + ρ1m 1Tm , 0 < ρ < 1. The marginal scores uj (θ) = Xj − θ are identically distributed and exchangeable with equal correlation. Differently from Example 4.1, the 20 ρ = 0, m = 20 Number of sub-likelihoods 1 200 0 −2 log(λ) 0 −7 −1 0.8 0.0 0.4 ARE 0.8 0 −5 ^ log(λ) 0.0 0.4 ARE 0.8 ^ log(λ) 0.0 −2 log(λ) −3 log(λ) 0.0 −4 ^ log(λ) −4 1 0.4 −2 log(λ) 10 wλ,j 0.0 wλ,j 0.0 wλ,j −4 ARE 3 1.0 11 ρ = 0.5, m = 1000 Number of sub-likelihoods 1.0 3 1.0 14 ρ = 0.5, m = 20 Number of sub-likelihoods −4 −2 log(λ) 0 −8 −4 log(λ) 0 Figure 1: Top Row: Solution paths for the minimizer wλ∗ of Criterion Q(θ∗ , w) in (4) for different values of λ with corresponding number of sub-likelihoods. Bottom Row: Asymptotic b ∗ ) compared to MLE. The vertical dashed lines on relative efficiency (ARE) of the SCLE θ(w λ b selected by Criterion (12) with τ = 0.9. Results correspond to the the bottom represent λ common location model X ∼ Nm (θ∗ 1m , Σ)√with jth diagonal element of Σ equal to j, and (jk)th off-diagonal element of Σ equal to ρ jk. 21 solution wλ∗ to Criterion (4) has equal elements ∗ = wλ,j 1−λ I(λ < 1), 1 ≤ j ≤ m, ρ(m − 1) + 1 b ∗ ) = Pm X j /m regardless of the value of λ. so the optimal parameter estimator is θ(w λ j=1 The first eigenvalue of ES(θ) is θ(m − 1) + 1, whilst the remaining m − 1 eigenvalues are all equal to 1 − θ, suggesting that the first score contains a relatively large information on b ∗ )} = θ compared to the other scores. When m is much larger than n, we have var{θ(w 0 [ρ2 (m − 1) + 1]/(mn) ρ2 /n. The trade-off between statistical and computational efficiency may be measured by the ratio of estimator’s variance with m = ∞ compared to that with m < ∞. This ratio is t(m) = ρ2 m/{ρ2 (m − 1) + 1}, which increases quickly for smaller m and much slower for larger m (e.g., t(5) = 0.83, t(9) = 0.90 and t(50) = 0.98, if ρ = 0.75). Thus, although all the elements in wλ∗ are nonzero, a few partial scores contain already the majority of the information on θ. This suggests that in practice taking a sufficiently large value for λ, so that the sparse empirical solution w bλ contains only a few of zero elements, bw already ensures a relatively high statistical efficiency for the corresponding MLCE θ( bλ ). 4.3 Exponentially decaying covariances Let X ∼ Nd (0, Σ(θ)), where the jkth element of Σ(θ) is σjk (θ) = exp{−θd(j, k)}. The quantity d(j, k) may be regarded as the distance between spatial locations j and k. Evaluating the ML score in this example is computationally expensive when d is large, since it requires computing the inverse of Σ(θ), a task involving O(d3 ) operations. On the other hand, the CL score is obtained by inverting 2 × 2 covariance matrices, thus requiring at most b O(d2 ) operations. Given i.i.d. observations X (1) , . . . , X (n) on X, the MCLE θ(w) solves the 22 equation 0= X wjk j<k n X (i) (i) ujk (θ, Xj , Xk ) i=1   (i) 2 (i) 2 (i) (i) n X σjk (θ){Xj + Xk − 2Xj Xk σjk (θ)}  σjk (θ)d(j, k)  = wjk 2 }2 {1 − σ (θ) jk i=1 j<k # " (i) (i) n X X σjk (θ) + Xj Xk − σjk (θ)d(j, k), wjk 1 − σjk (θ)2 i=1 j<k X where ujk corresponds to the score of a bivariate normal distribution for the pair (Xj , Xk ). Figure 2 shows the analytical solution path of the minimizer wλ∗ of Criterion (4) for b ∗ ) compared to different values of λ, and the asymptotic relative efficiency of the SCLE θ(w λ MLE. We consider a number of pairs ranging from m = 45 to m = 1225 for various choices of θ. When λ = 0, the SCLE has relatively high asymptotic efficiency. Interestingly, efficiency remains steady around 90% until only a few sub-likelihoods are left. This suggests again that a very small proportion of partial-likelihood components contains already the majority of the information about θ. In such cases, the SCLE reduces dramatically the computing burden while retaining satisfactory efficiency for the final estimator. 5 Numerical examples In this section, we study the finite-sample performance of the SCLE in terms by assessing its mean squared error and computing cost when the data dimension d increases. As a b preliminary estimator, we use the MCLE θ(w) with w = (1, . . . , 1)T , which is perhaps the most common choice for w in CL applications (Varin et al., 2011). 5.1 Example 1 We generate samples of size 50 from X ∼ Nm (θ1m , Σl ), l = 1, . . . , 4. We specify the following covariance structures: Σ1 = Im ; Σ2 is diagonal with kth diagonal elements σk2 = k; Σ3 has 23 458 200 wλ,j 0 −4 0 −5 −2 log(λ) 0 −1 −3 log(λ) −1 0.8 ARE 0.0 0.0 0.4 ARE 0.4 0.0 −4 −3 log(λ) ^ log(λ) 0.8 ^ log(λ) 0.8 ^ log(λ) −2 log(λ) 0.4 −2 log(λ) 60 −0.4 −0.4 0.2 wλ,j 0.2 −0.4 wλ,j −4 ARE 18 0.6 44 θ = 0.6, d = 50, m = 1225 Number of sub-likelihoods 0.6 18 0.6 44 θ = 0.6, d = 10, m = 45 Number of sub-likelihoods 0.2 θ = 0.4, d = 10, m = 45 Number of sub-likelihoods −4 −2 log(λ) 0 −5 Figure 2: Top Row: Solution paths for the minimizer wλ∗ of Criterion Q(θ∗ , w) defined in (4) for different values of λ with corresponding number of sub-likelihoods reported. Botb ∗ ) compared to MLE. The tom Row: Asymptotic relative efficiency (ARE) of SCLE θ(w λ b selected by Criterion (12) with vertical dashed lines on the bottom row correspond to λ τ = 0.9. Results correspond to the model X ∼ Nd (0, Σ(θ)) with (j, k)th element of Σ equal p to exp{−θ 2\|j − k\|}. 24 unit diagonal elements with the first 10 elements of X uncorrelated with any other element while the other elements in X have pairwise correlations 0.8\|j−k\| (10 < j < k < d); Σ4 has unit diagonal elements and a block diagonal structure with independent blocks of six elements each and within-block correlation of 0.6. Figure 3 (left), shows the relative mean squared error of the SCLE θbλ compared to that of the MLE for a moderate data dimension (d = m = 30). The points in the trajectories correspond to inclusion of a new sub-likelihood component according to the the least-angle algorithm described in Section 2.3. The SCLE θbλ achieves more than 90% efficiency compared to MLE for all the covariance structures considered, always before all the candidate partial likelihoods are included. The advantage of SCLE becomes evident when the sub-likelihood scores exhibit relatively strong correlation. For example, for Σ = Σ4 where sub-likelihoods are independent between blocks, the maximum efficiency is achieved when only a few representative partial scores are selected from each block. Figure 3 (right) shows the ratio between mean squared error of the SCLE compared and that of the MLE for a relatively large data dimension (d = m = 1000) compared to the sample size (n = 50). Although here the MLE is used as a theoretical benchmark, in practice such an estimator is not available as m is larger than the sample size n. Interestingly, when the sample size n is fixed, including all the sub-likelihoods eventually leads to substantial loss of efficiency. In this examples, selecting too many sub-likelihoods not only wastes computing resources but also implies estimators with larger errors . On the other hand, a proper choice of the tuning constant λ (corresponding to about 20 selected sub-likelihoods) can balance computational and statistical efficiency. 5.2 Example 2 In our second numerical example, we consider covariance estimation for the model X ∼ Nd (0, Σ(θ)) with Σ(θ)j,k = exp{−θ2(j − k)2 }. Here the covariance between components Xj and Xk in the random vector X decreases rapidly as the distance (j − k)2 between 25 Σ=Σ1 Σ=Σ2 Σ=Σ3 Σ=Σ4 0 5 MSEMLE MSESCLE MSEMLE MSESCLE 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 10 15 20 25 30 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 m=50 m=100 m=200 m=500 m=1000 0 Number of partial scores included 10 20 30 40 50 Number of partial scores included Figure 3: Monte-Carlo estimate of the mean square error of the MLE (MSEMLE ) divided by that of the SCLE (MSESCLE ), for the model X ∼ Nm (θ1m , Σ). Each trajectory is based on 1000 Monte-Carlo samples of size n = 50. Each point in the trajectories correspond to inclusion of a new sub-likelihood component based on the least-angle algorithm described in Section 2.3. Left: Different specifications for Σ detailed in Section 5 with m = 30. Right: Covariance Σ = Σ2 with m ranging from 50 to 1000. components of X increases. Figure 4 shows Monte-Carlo estimates for the mean square bw error of the SCLE θ( bλ ) compared to that of the MCLE with uniform composition rule (i.e. b θ(w) with w = (1, . . . , 1)T ), for θ = 0.2 and 0.4. Each point in the trajectories correspond to inclusion of a new sub-likelihood component using the least-angle algorithm described in Section 2.3. The SCLE is already more efficient than the uniform MCLE when a handful of partial scores are selected. For example if θ = 0.2 and m = 1035, selecting ten sub-likelihoods already ensures 1.5 times the accuracy of the uniform MCLE. Since the uniform MCLE uses all the m = 1035 pairs of sub-likelihoods, the SCLE obtains more accurate results at a much lower computing cost. 26 θ=0.2 m=45 m=435 m=1035 1.0 0.5 1.5 MSEUnif MSESCLE MSEUnif MSESCLE 1.5 θ=0.4 m=45 m=435 m=1035 1.0 0.5 0.0 0.0 0 10 20 30 40 50 60 Number of partial scores included 0 10 20 30 40 50 60 Number of partial scores included Figure 4: Monte-Carlo estimate of the mean square error of the MCLE with w = (1, . . . , 1)T (MSEUnif ) divided by that for the SCLE (MSESCLE ). Each point in the trajectories corresponds to the inclusion of a new sub-likelihood component based on the least-angle algorithm described in Section 2.3. Results are based on 1000 Monte Carlo samples of size n = 50 from the model X ∼ Nd (0, Σ(θ)) with Σ(θ)j,k = exp{−2θ(j − k)2 }. Trajectories correspond to θ = 0.2 (left) and θ = 0.4 (right) for different numbers of sub-likelihoods, m, ranging from 45 to 1035. 6 Conclusion and final remarks In recent years, inference for complex and large data sets has become one of the most active research areas in statistics. In this context, CL inference has played an important role in applications as a remedy to the drawbacks of traditional likelihood approaches. Despite the popularity of CL methods, how to address the trade-off between computational parsimony and statistical efficiency in CL inference from a methodological perspective remains a largely unanswered question. Motivated by this gap in the literature, we introduced a new likelihood selection methodology which is able truncate quickly overly complex CL equations potentially encompassing many terms, while attaining relatively low mean squared error for the implied estimator. This is achieved by selecting CL estimating equations satisfying a `1 -constraint on the CL complexity while minimizing an approximate `2 -distance from the full-likelihood score. Inference based on statistical objective functions with `1 -penalties on the parameter 27 θ is not new in the statistical literature (e.g., see Giraud (2014) for a book-length exposition on this topic). Note, however, that differently from existing approaches the main goal here is to reduce the computational complexity of the overall CL estimating equations regardless of the model parameter θ, which is viewed as fixed in size. Accordingly, our `1 -penalty involves only the composition rule w, but not the model parameter θ. In the future, developing approaches for simultaneous penalization on θ and w may be useful to deal with situations where both the data dimension and the size of the parameter space increase. Two main perks of the proposed approach make it an effective alternative to traditional CL estimation from practitioner’s perspective. The first advantage is that the SCLE methodology constructs CL equations and returns inferences very quickly. Theorem 3.2 shows that for any λ > 0 the empirical composition rule w bλ retains at most np ∧ m non-zero elements. This is an important feature of our method, which reduces – sometimes dramatically – the bw amount of computing needed to obtain the implied MCLE θ( bλ ) and its standard error. Lemma 3.1 highlights that the non-zero elements in w bλ correspond to partial scores maximally correlated with the residual difference r(θ, w) = uM L (θ) − u(θ, w). This means that our approach constructs estimators with relatively high efficiency by dropping only those uj s contributing the least in the CL equations for approximating uM L (θ). The second desirable feature of our method concerns model selection and the ability to reduce the complexity of large data sets. In essence, the truncation step (T-Step) described in (7) is a dimensionreduction step: starting from observations on a possibly large the d-dimensional vector X, our method generates a collection of lower-dimensional subsets Sλ = {Sj , j ∈ Ebλ , λ > 0} where Ebλ = {j : w bλ,j 6= 0}. While individually the selected data subsets in Sλ are of size much smaller than d, collectively they contain most of the information on θ for a given level of computing represented by λ. From a theoretical perspective, little work has been devoted to study the properties of CL estimators when the number of sub-likelihoods m diverges. Cox and Reid (2004) discuss estimators based CL equations with m = d2 + d terms by taking all pairwise and marginal 28 scores for the d-dimensional vector X. They take non-sparse and more rigid composition rules compared to ours with wjk = 1 for all pairs (j 6= k) and wjj = −a × d for all marginals (j = k), where a is a tuning constant used to increase efficiency. To our knowledge, the current paper is the first studying the behavior of more flexible sparsity-inducing composition rules and implied CL estimating equations in the setting where both m and n grow. Theorem 3.3 and Corollary 3.4 provide us with guidance on when the selected score u(θ, w bλ ) is a meaningful approximation to the unknown ML score in the sense of the objective (4). A first requirement is that the total information on θ available if the full likelihood were actually known, m∗ = kuM L (θ∗ )k22 , is not overwhelming compared to the sample size n. If X ∼ Nm (θ1m , Σ), we require m∗ /n = tr{Σ−1 }/n → 0. This condition is very mild when relatively few elements X contain a strong signal on θ, whilst the remaining elements are noisy and with heterogeneous variances. In Section 4.1, we illustrate this by taking Σ diagonal with increasing diagonal elements. A second requirement is that the tuning √ √ constant λ dominates asymptotically m∗ r1 , where r1 represents the convergence rate of the empirical covariance of scores EFn {S(θ)}. For instance, if the elements of S(θ) are subGaussian, we have r1 = op (log(m)/n), meaning that λ should be asymptotically larger than p m∗ log(m)/n. Finally, we show that statistical optimality and computationally parsimony can co-exhist within the same selection procedure when λ is judiciously selected. If λ → 0 at the rate described in Theorem 3.3, the truncated composition rule w bλ with \|Ebλ \| scores approximates the optimal composition rule w0∗ consisting of m nonzero terms. Accordingly, Corollary 3.4 suggests that the implied truncated CL score function u(θ, w bλ ) approximates the optimal score u(θ, w bλ ), uniformly on a neighborhood of θ∗ . Extending this type of result and developing further theoretical insight on the interplay between the type of penalty and the MCLE accuracy beyond the current i.i.d. setting would represent another exciting future research direction. For example, findings would be particularly valuable in spatial statistics where often the number of sub-likelihood components is overwhelming and poses serious challenges 29 to traditional CL methods. Appendix In this section, we show technical lemmas required by the main results in Section 3. Lemma 7.1. kw bλ k1 and kwλ∗ k1 are decreasing in λ. Proof. Denote the first term of Criterion Qλ (θ, w) defined in (4) (without the penalty term) by Q1 (θ, w). Suppose λ1 > λ2 , and let w1 , w2 be the minimizers of Qλ1 (θ∗ , w), Qλ2 (θ∗ , w) respectively. Then, Q1 (w1 )+λ1 kw1 k1 ≤ Q1 (w2 )+λ1 kw2 k1 and Q1 (w1 )+λ2 kw1 k1 ≥ Q1 (w2 )+ λ2 kw2 k1 . Subtracting the last two inequalities gives (λ1 − λ2 )kw1 k1 ≤ (λ1 − λ2 )kw2 k1 . Since λ1 > λ2 , we have kw1 k1 ≤ kw2 k1 . An analogous argument shows that kw bλ k1 is decreasing. Lemma 7.2. Under Conditions C.1 - C.3, if r1 /λ = op (1), then kwλ∗ k1 = O(m∗ /λ) and kw bλ k1 = Op (m∗ /λ). Proof. For wλ∗ , note that Qλ (θ∗ , wλ∗ ) = EkuM L (θ∗ ) − u(θ∗ , wλ∗ )k22 /2 + λkwλ∗ k1 ≤ Qλ (θ∗ , 0) = m∗ /2, or λkwλ∗ k1 ≤ m∗ /2. Hence, kwλ∗ k1 = O(m∗ /λ). bw b θ, For w bλ , we have Q( bλ ) = bw b Tw b 0) = 0. Since EuM L (θ∗ )T uj (θ∗ ) = b θ, w bλT EFn S(θ) bλ /2 − diag(EFn S(θ)) bλ + λkw bλ k1 ≤ Q( Euj (θ∗ )T uj (θ∗ ), we have b Tw b − ES(θ∗ ))T w λkw bλ k1 ≤ diag(EFn S(θ)) bλ = diag(EFn S(θ) bλ + EuM L (θ∗ )T u(θ∗ , w bλ ), b − ES(θ∗ ))T w b − ES(θ∗ ))T \|j,k kw with diag(EFn S(θ) bλ ≤ maxj,k \|EFn S(θ) bλ k1 ≤ r1 kw bλ k1 and EuM L (θ∗ )T u(θ∗ , w bλ ) = Op (m∗ ) Hence, kw bλ k1 = Op (m∗ /λ). 30 Lemma 7.3. Let λ → 0 as n → ∞. Under Conditions C.1-C.3, if r1 m∗ 2 λ−2 = op (1), we bw b w∗ ) have EFn u(θ, bλ ) − u(θ, λ P 2 → 0, as n → ∞, where θb is the preliminary root-n consistent estimator used to compute w bλ in the T-Step (7). Proof. Note that r1 m∗2 λ2 = op (1) implies r1 /λ = op (1). Therefore, we have kw bλ − wλ∗ k1 = bw b w∗ ) gives bλ (θ, bλ (θ, Op (m∗ /λ) by Lemma 7.2. Moreover, re-arranging Q bλ ) ≤ Q λ 1 b T (w b ∗ } ≤ EFn {U 2 (θ) bw bλ k1 + λkwλ∗ k1 . bλ − wλ∗ )} − λkw bλ − wλ∗T S(θ)w EF {w bT S(θ) λ 2 n λ b (θ) b T w∗ }T (w Subtracting EFn {U (θ)U bλ − wλ∗ ) from both sides gives λ 1 b T (w b w∗ )}T (w EFn kU (θ) bλ − wλ∗ )k22 ≤ EFn {c(θ, bλ − wλ∗ ) − λkw bλ k1 + λkwλ∗ k1 λ 2 h iT h iT b w∗ ) − Ec(θ∗ , w∗ ) w b w∗ ) − Ec(θ∗ , w∗ ) w∗ = EFn c(θ, bλ − EFn c(θ, λ λ λ λ λ + Ec(θ∗ , wλ∗ )T w bλ − λkw bλ k1 − Ec(θ∗ , wλ∗ )T wλ∗ − λkwλ∗ k1 h iT ∗ ∗ ∗ b ≤ EFn c(θ, wλ ) − Ec(θ , wλ ) (w bλ − wλ∗ ) h iT ∗ ∗ b b = diag EFn S(θ) − ES(θ ) − EFn S(θ)) − E(S(θ ) wλ∗ (w bλ − wλ∗ ), where the inequality is implied by Lemma 3.1. The last expression is op (1), since r1 m∗ 2 /λ2 = op (1) and kw bλ −wλ∗ k1 = Op (m∗ /λ) by Lemma 7.2, and the matrix maximum norm is bounded by matrix 2-norm. √ Lemma 7.4. If r1 m∗ 2 λ−2 = op (1) and r2 m∗ λ−1 = op (1), then under conditions C.1-C.6, b λ − K ∗ k1 = op (1). kK λ P b λ − K ∗ k1 ≤ r2 kw Proof. This is a direct result since kK b − w∗ k1 , r2 → 0 according to lemma λ P assumption and kw b − w∗ k1 → 0 by Theorem 3.3. Lemma 7.5. Under Conditions C.1-C.6, Eku(θ∗ , wλ∗ )k22 = O(m∗ ). Proof. Note that EkuM L (θ∗ )−u(θ∗ , wλ∗ )k22 ≤ EkuM L (θ∗ )−u(θ∗ , wλ∗ )k22 +λkwλ∗ k1 ≤ EkuM L (θ∗ )k22 = 31 m∗ . Expanding EkuM L (θ∗ ) − u(θ∗ , wλ∗ )k22 gives Eku(θ∗ , wλ∗ )k22 ≤ 2EuM L (θ∗ )T u(θ∗ , wλ∗ ) q √ q ≤ 2 EkuM L (θ∗ )k22 · Eku(θ∗ , wλ∗ )k22 = 2 m∗ Eku(θ∗ , wλ∗ )k22 . Re-arranging gives Eku(θ∗ , wλ∗ )k22 ≤ 4m∗ . Lemma 7.6. Assume Conditions C.1-C.6. For every > 0, we have n o p 1 X n ∗ ∗ 2 ∗ ∗ ∗ ) → 0, E u (θ , w ) I(\|u(θ , w )\| ≥ nJ i λ λ λ nJλ∗ i=1 where ui (θ, w) = Pm j=1 as n → ∞, (i) wj ∇ log fj (Xj ; θ) is the composite likelihood score corresponding to the ith observation. Proof. Without loss of generality, assume p = 1. Recall that Jλ∗ = Eu(θ∗ , wλ∗ )2 . For every > 0, and constants a, b > 1 such that 1/a + 1/b = 1 n o p 1 X n ∗ ∗ 2 ∗ ∗ ∗ E u (θ , w ) I(\|u(θ , w )\| ≥ nJ ) i λ λ λ nJλ∗ i=1 o p 1 n = ∗ E ui (θ∗ , wλ∗ )2 I(\|u(θ∗ , wλ∗ )\| ≥ nJλ∗ ) Jλ 1 1 1 b (J ∗ )a−1 ∗ ∗ 2a ≤ ∗ E \|ui (θ , wλ )\| · 2 = 2λ 1/b , Jλ n ( n) (20) where the inequality follows by applying Hölder’s and Chebyshev’s inequalities. By the assumption at the beginning of Section 3 that m∗ = o(log(n)), Lemma 7.5 implies Jλ∗ = o(log(n)). Hence, (20) converges to 0 as n → ∞, which proves the desired result. References J. Besag. Spatial interaction and the statistical analysis of lattice systems. Journal of the Royal Statistical Society. Series B (Methodological), pages 192–236, 1974. 32 P. Bühlmann and S. Van De Geer. Statistics for high-dimensional data: methods, theory and applications. Springer Science & Business Media, 2011. T. T. Cai, C.-H. Zhang, H. H. Zhou, et al. Optimal rates of convergence for covariance matrix estimation. 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Noname manuscript No. (will be inserted by the editor) Identifying Hazardousness of Sewer-Pipeline Gas-Mixture using Classification Methods A Comparative Study arXiv:1707.00561v1 [cs.NE] 16 May 2017 Varun Kumar Ojha · Parmartha Dutta · Atal Chaudhuri Received: date / Accepted: date Abstract In this work, we formulated a real-world problem related to sewerpipeline gas detection using the classification-based approaches. The primary goal of this work was to identify the hazardousness of sewer-pipeline to offer safe and non-hazardous access to sewer-pipeline workers so that the human fatalities, which occurs due to the toxic exposure of sewer gas components, can be avoided. The dataset acquired through laboratory tests, experiments, and various literature-sources were organized to design a predictive model that was able to identify/classify hazardous and non-hazardous situation of sewer-pipeline. To design such prediction model, several classification algorithms were used and their performances were evaluated and compared, both empirically and statistically, over the collected dataset. In addition, the performances of several ensemble methods were analyzed to understand the extent of improvement offered by these methods. The result of this comprehensive study showed that the instance-based-learning algorithm performed better than many other algorithms such as multi-layer perceptron, radial basis function network, support vector machine, reduced pruning tree, etc. Similarly, it was observed that multi-scheme ensemble approach enhanced the performance of base predictors. V. K. Ojha IT4Innovations, VŠB Technical University of Ostrava, Ostrava, Czech Republic and Dept. of Computer Science & Engineering, Jadavpur University, Kolkata, India E-mail: [email protected] P. Dutta Dept. of Computer & System Sciences, Visva-Bharati University, India E-mail: [email protected] A Chaudhuri Dept. of Computer Science & Engineering, Jadavpur University, Kolkata, India E-mail: [email protected] Neural Computing and Applications DOI: 10.1007/s00521-016-2443-0 2 Varun Kumar Ojha et al. Keywords Sewer gas detection · Neural network · Classification · KS test 1 Introduction This is in the view of providing a solution to a real-world problem using technology, where the human fatalities need to be avoided. Hence, the technology should be as simple as possible. In this work, we addressed a complex realworld problem related to sewer-pipeline gas detection, where sewer-pipeline safety detection (in terms of non-toxic environment) was required to allow maintenance and cleaning of the pipeline. The sewer gas detection is a highly complex problem because of the presence of several toxic gases in a mixture form, and a single gas detector may not offer reliable solution. Therefore, we studied the complexity of this problem in terms of gas mixture. The primary goal was to offer a simple solution with a high accuracy so that it was easy to categorize the hazardous situation in straightforward way such as “hazardous” or “non-hazardous.” To meet this simplicity, we formulated sewer-pipeline gas detection problem as a classification problem. Sewer-pipeline contains a mixture of several toxic gases such as hydrogen sulphide (H2 S), ammonia (NH3 ), methane (CH4 ), carbon dioxide (CO2 ), nitrogen oxides (NOx ), etc., [1, 2, 3]. Usually, this mixture is generated due to the biodegradation of the waste and the sewage into the sewer-pipeline. Such toxic gas-mixture is fatal for those who come to the proximity/exposure of these gases. Following this, an alarming number of human fatalities are reported each year by the newspapers and the other agencies [4, 5,6]. The authorities those are responsible for maintaining and cleaning of the sewer pipeline provides various electronic portable gas detectors available in the market to the employed persons so that they can determine the safeness of the sewer-environment before physically get involve into the maintenance work. However, the available electronic portable gas detectors are not providing satisfactory results. It is evident from the recent comments from the judiciary to these authorities. In a judgment to a civil appeal number 5322 of 2011, the Supreme Court of India stated, “the State and its agencies/instrumentalities cannot absolve themselves of the responsibility to put in place effective mechanism for ensuring safety of the workers employed for maintaining and cleaning the sewage system [7].” Similarly, in another judgment, the Supreme Court of India stated, “...entering sewer lines without safety gears should be made a crime even in emergency situations... [8, 9].” This motivated us to carry out our research in this domain and to come out with a simple solution so that without having the minimum knowledge of the technicalities of gas composition and safety limits, a person is able to understand the environment of a sewer system before entering. To ensure the simplicity in model, we collected and preprocessed data to realize sewer gas-detection as a binary class classification problem. However, in this work, apart from the objective of constructing a prediction model, we set a secondary objective, which was to analyze the performances of the classifiers, Identifying Hazardousness of Sewer-Pipeline 3 both empirically and statistically. To meet these objectives, we used 12 base predictors from four different categories such as neural network based classifiers, tree based classifiers, instance based classifiers, and rule based classifiers. The algorithms were applied over the collected dataset and the performance of the algorithms were collected in terms of the accuracy. The collected results were then used for analyzing the performance superiority of the one algorithm over another or the one category of algorithms over another. We observed that the performance of the algorithms were independent of the category they belong to. For example, the performance of instance based k-nearest neighbor, logistic model tree, and support vector machine came from three different categories, but they had a very competitive performance. However, we must consider the “No-free-lunch theorem” that suggests that some algorithms perform better on some problem and some on another [10]. Therefore, to find out which predictor performs best in this case, we used 12 base predictors and nine ensemble methods. Rest of the article is organized as follows. A background study is provided in Section 2.1, which leads to setting ground for describing our contribution to the sewer-pipeline gas detection. In Section 2.2, we provide a detailed description of the data collection and preprocessing mechanisms, which constitute the core and significant part for formulating gas detection problem as a binary classification problem. Section 2.3 deals with the brief descriptions of the classifiers/algorithms and methods used for constructing the prediction model. The design of comprehensive experiment set for the evaluation of the classifiers is reported in Section 3. Whereas, Section 3 describes empirical and statistical evaluation of the classifiers, discussions and conclusion are reported in Sections 4 and 5, respectively. 2 Methodology In this Section, we put together the background study, the data collection mechanisms, and the classification methods definitions. The background study describes the significance of the sewer-pipeline gas detection problem and the data collection mechanism describes the formulation of gas-detection as a classification problem. 2.1 Background Study Literature review was conducted in the perspective of electronic-nose (ENOSE) and gas-detection-system to cover a broad area of research in the field of gas detection and modeling using intelligent computing techniques/algorithms. Although not much work specifically on sewer gas-mixture-detection was reported in the past, few notable contributions were observed. Li et al. [11] reported a noticeable research work on the development and design of an electronic nose (E-NOSE) and gas detection system, where a neural network 4 Varun Kumar Ojha et al. (NN)-based mixed gas (NOx, and CO) measurement system was developed. On the other hand, Sirvastava et al. [12, 13] proposed a design of intelligent ENOSE system using backpropagation (BP) and neuro-genetic approach. Llobet et al. [14] presented a pattern recognition approach, based on the wallet transformation for gas mixture analysis using single tin-oxide sensor. Liu et al. [13] addressed a genetic-NN algorithm to recognize patterns of mixed gases (a mixture of three component gases) using infrared gas sensor. Lee et al. [15] illustrated uses of micro gas sensor array (GSA) combined with NN for recognizing combustible leakage gases. Ambard et al. [16] have demonstrated use of NN for gas discrimination using a tin-oxide GSA for the gases H2 , CO and CH4 . In [17], authors have illustrated a NN-based technique for developing a gas sensory system for sensing gases in a dynamic environment. Pan et al. [18] have shown several applications of E-NOSE. Wongchoosuka et al. [19] have proposed an E-NOSE detection system based on carbon nanotube-SnO2 gas sensors for detecting methanol. Zhang et al. [20] developed a knowledge-based genetic algorithm for detecting mixed gas in mines. Won et al. [21] proposed a system for estimation of hazardous gas release rate using optical sensor and NN-based technique. The following salient points came out of the above mentioned articles: – Mainly, BP and NN-based approaches were studied so far for detecting gas-mixtures. – Mostly, the E-Nose systems reported in the past were developed for the gas-mixtures of only two or three gases and the sensors of the gases used were less cross-sensitive to the other gases in mixtures. – Cross-sensitivity during sensing is an important factor in gas detection system, which was least reported in literature as yet. However, Ojha et al. [22, 23, 24, 25,26, 27] offered a few methods such as neuro-genetic, neuro-swarm, ant-colony-based, neuro-simulated annealing, etc., where cross-sensitivity factor has been addressed to some extent. However, these works were primarily related to regression modeling. – The impact of humidity and temperature on sensors remained ignored so far. – The gas detection system or E-Nose was viewed only in the framework of regression problems and not classification problem. Classification based approach led us to determine the hazardous and nonhazardous situation of a sewer-pipeline. In addition, the collection, organization, and the preprocessing of the collected data enabled us to address the cross-sensitivity issue firmly. The cross-sensitivity issue occurs because of the sensitivity of one gas-sensor towards multiple gases. So was our case, where a semiconductor-based GSA was designed using five gas-sensors. Each gas-sensor was typically meant for detecting its respective target gas. Hence, when the GSA was used for collecting data for a mixture of gases, the crosssensitivity in the sensed values (collected data) became inevitable. Therefore, rather than considering pure results of the respective gases, we registered the cross-sensitive results as a part of our-collected data. Since a computation- Identifying Hazardousness of Sewer-Pipeline 5 ally intelligent model learned from the data and also maintained the crosssensitivity patterns registered in terms of data values itself, a learned model accurately predicts an unknown gas mixture. 2.2 Equipment and Data Collection Mechanism (A/trained/Classifier/embedded/into/ electronic/(EEPROM)./ (Buzzer/light)/ Intelligent/unit./ Output/Unit (Figure/2) Gas/Sensor/ Array/ Before explaining the details of data collection and equipment, we need to explain the basic design and the purpose of our work, which is to offer an intelligent gas detection system (an electronic portable gas detector) that will be a result of embedding learned-predictor (trained-classifier) into an electronic system. The data flow into our developed intelligent system is shown in Fig. 1, which describes the entire process of the intelligent system design, which is divided into three phases: 1) The data acquisition unit, which consists of gas suction-motor chamber, GSA, and data acquisition-cum data-preprocessor block; 2) An intelligent unit (classifier unit), which receives data from dataacquisition unit and classifying the acquired data patterns; 3) The output unit, which prompts the result in terms of colored light and buzzer. Hence, our objective here was limited to only train a classifier using the collected data. We describe the data collection process as follows. Fig. 1 Block diagram of intelligent system design (real time data flow process) At first, we collected the data samples from the data-sheets, literature, and laboratories test of the collected gas mixture samples from sewer-pipelines. Second, we designed our own metal oxide semiconductor (MOS) gas sensors array (GSA) that was used for verifying the literature and laboratory data and for generating the data samples for the purpose experiments. Our designed GSA consists of five gas-sensors for sensing five different gases. They include hydrogen sulphide (H2 S), ammonia (NH3 ), methane (CH4 ), carbon dioxide (CO2 ), and nitrogen oxides (NOx ). Typically, MOS sensors are resistance-type electrical sensors, where responses are change in circuit resistance proportional to gas concentration, A resistance type sensor responds to change in resistance due to change in the concentration of gases. The change in resistance is given as δRs /R0 , where δRs is change in MOS sensor resistance and R0 is base resistance or the sensing resistance at a specifics gas concentration in clean air [19]. The R0 of the sensors MiCS - 4514, MQ - 7, MQ - 136, MQ - 135, and MQ - 4 is 0.25 ppm, 100 ppm, 10 ppm, 100 ppm and 1000 ppm, respectively. Here, ppm is the unit for measuring concentration of gas into air which is defined as follows: 1 ppm is equal to 1 volume of a gas into 106 volume of air. 6 Varun Kumar Ojha et al. A typical arrangement of a gas sensor array is shown in Fig. 2. The circuitry shown in Fig. 2 (left) was developed in our laboratory. Here, the fabricated and installed sensors were MiCS - 4514, MQ - 7, MQ - 136, MQ - 135, and MQ - 4 for gases NO2 , CO, H2 S, NH3 , and CH4 , respectively [28, 29]. The gas sensors used were sensitive to not only their target gases, but they were sensitive also to other gases in the gas-mixture [30, 31]. Hence, crosssensitivity effect over MOS sensors was confirmed [32]. It was moreover confirmed that the sensor responses were noisy and accordingly the pattern of such noise were considered and recorded as an instance into our dataset. Hence, a non-intelligent use of raw values of sensor response for hazardousness prediction may be misleading in operating (real-world) environment. Therefore, a training electronic portable gas detector may be used to predict sewer hazardousness, accurately. So was the effort in this work to provide a classifier. Data collection had vital role in training of a classifier. Data samples were collected as per the following steps. At first, several manhole samples collected from the Kolkata, India municipal area were tested in laboratory to identify the presence of several toxic gases such as nitrogen dioxide (NO2 ), carbon monoxide (CO), hydrogen sulphide (H2 S), ammonia (NH3 ), methane (CH4 ), and carbon dioxide (CO2 ). Secondly, gas sensors were identified for each of the respective gases. As a result we came out with the procurement of gas sensor MiCS - 4514, MQ - 7, MQ - 136, MQ - 135, and MQ - 4 for NO2 , CO, H2 S, NH3 , and CH4 , respectively. We collected data sheets form the companies for the respective sensors. In the third step, a laboratory was setup for the verification and collection of the sensor response of the respective gas sensors in certain range of their concentration. Specifically, the concentration range in ppm laid down in sensor manuals of sensors MiCS - 4514, MQ - 7, MQ - 136, MQ - 135, and MQ - 4 are [0.25 - 5], [20 - 1000], [1 - 100], [10 - 300] and [300 - 10000] of the gases NO2 , CO, H2 S, NH3 , and CH4 , respectively. In addition, the lab was setup (see Fig. 2 [right]), where gas cylinders were connected to a gas concentration measuring unit called mass flow controller (MFC), which was further connected to a gas chamber, where each gas was allowed to pass in a specific concentration over an array of gas sensor. More specifically, the behavior of each of the gas sensors was recorded. Fig. 2 Laboratory-scale gas sensor array (GSA) [28, 29] Identifying Hazardousness of Sewer-Pipeline 7 The following steps were used for preparing data sample for the classifiers’ training. First, hazardous (safety) limits of the component gases of manhole gas mixture were collected. Secondly, three different levels, (i) above safetylimit, (ii) at safety-limit, and (iii) below safety-limit for each manhole gas were recognized. Thirdly, gases were mixed in different combination to prepare several mixture sample that were used to pass over GSA. Table 1 indicates few examples of such mixture of gases in different combinations. For example, when we mix five gases each of which has three different recognized concentration levels, we get 243 different combinations (35 ). In addition, we considered the role of humidity and temperature to influence the sensor’s behavior. Accordingly, the data values were recorded. Hence, our collected dataset contained seven input features and an output class. Each sample was labeled with “0” for safe sample (if the responses of all five sensors were under the maximum safety limit) or “1” for unsafe sample (if the responses of any among the five sensors were above the maximum safety limit). The safety limits of the manhole gases are as follows: safety limit of NH3 is between 25 ppm and 40 ppm [33], CO is in between 35 ppm and 100 ppm [34], H2 S is in between 50 ppm and 100 ppm [35], CO2 is in between 5000 ppm and 8000 ppm [36] and CH4 is in between 5000 ppm and 10000 ppm [37]. Table 2 illustrates a fraction of the collected data samples. 2.3 Classification Based Approach We categorized the classifiers in the four different groups of classifiers. Each category of classifiers contains three classifiers. Table 1 Samples of gas-mixture in different concentration. Concentration of gases in ppm # Humidity Temperature NO2 CO H2S NH CH Class Status 1 2 3 : 7535 7536 7537 : 16036 16037 16038 65 65 65 20 20 20 0 0 0 10 10 10 10 10 10 20 20 20 2000 5000 10000 0 0 1 safe safe unsafe 65 65 65 30 30 30 0 0 0 10 10 10 10 10 10 20 20 20 2000 5000 10000 0 1 0 safe unsafe safe 75 75 75 50 50 50 20 20 20 50 50 50 50 50 50 50 100 100 10000 2000 5000 1 1 1 unsafe unsafe unsafe 8 Varun Kumar Ojha et al. 2.3.1 Network Based Classifiers Multi-layer perceptron (MLP) is a computational model that imitates human brain, and learn from environment, i.e., data. In our work, we used threelayered MLP, where layers are input layer, hidden layer, and output layer [38]. Radial Basis Function Network (RBF) is a special class of MLP, where inputs are mapped onto a hidden layer that consists of radial basis function, which does the non-linear mapping of input to a hidden layer [39]. Support vector machine (SVM) is a supervised learning computational model that maps input to a high dimension feature space using kernel trick. Hence, non-linear separable patterns in input space are linearly classified on a high dimensional feature space [40]. 2.3.2 Tree Based Classifiers Reduced pruning tree (REP) is a tree based classifier method, where a treelike structure is designed for predicting target class based on the input variables [41, 42]. More specifically, the leaves of tree offers decision of the class based on the conjunction of the input feature represented by the branches of the tree. REP tree is a decision tree, where the tree size is reduced by pruning inefficient branches [43]. Naive Bayes tree (NBT) is a special class of decision tree, where the leaf nodes of decision tree that offer decision on the class is replaced by a Naive Bayes classifier, which decides the class label, based on the features and learned threshold [44]. Table 2 Samples of calibrated sensor responses based on the knowledge gathered from literature, data-sheets, lab tests, and scaling process. Sensors response (δRs /R0 ) # Humidity Temperature InNO2 InCO InH2S InNH InCH Class 1 2 3 : 7535 7536 7537 : 16036 16037 16038 65 65 65 20 20 20 0.813 1.301 1.035 6.929 7.521 6.658 5.938 5.525 5.841 3.433 3.521 3.633 3.985 2.178 1.620 0 0 1 65 65 65 30 30 30 1.038 1.054 0.642 7.565 6.694 7.210 5.658 5.745 5.819 3.228 3.692 1.326 2.275 1.268 3.530 0 1 0 75 75 75 50 50 50 4.645 4.712 4.911 2.764 2.985 2.433 0.608 0.641 0.381 2.709 1.228 0.937 0.499 0.450 0.481 1 1 1 Identifying Hazardousness of Sewer-Pipeline 9 Logistic Model Trees (LMT) is similar to NBT that does the transformation of leaves of a decision tree into a logistic regression node. A logistic regression maps independent variables to categorical dependent variables using a logistic function [45, 46]. Hence, LMT is a simple idea, where nodes of a decision/classification tree are replaced by logistic regression model [47]. 2.3.3 Rule Based Classifiers Decision Table (DT) is a simple representation of data into a table based system, where the decision is made based on the features matching or searched into a decision table. On a successful search, the majority class label is returned, otherwise the majority class label of the entire dataset is returned as a decision for an unlabeled data [48]. PART is a rule based classification method based on partial decision tree that generates a list of rules, used subsequently for making prediction of unknown data instance. The rules are generated based on the partial decision tree, which splits dataset into subsets until the entire dataset gets exhausted to form nodes and leaf nodes of the tree [49]. Majority Predictor (Zero R) is the simplest possible form of classification method. It is based on the majority of class label into a dataset. In simple words, it always predicts the majority class. 2.3.4 Instance Based Classifiers Instance-Based Learning (IBK) provides the concept description which is the primary output of an IBK algorithm. It is a function that maps an instance to a category (class label). The concept description function is updated based on training procedure that involves two functions similarity and classification. The similarity function computes the similarity between the training instances and the pre-stored instances, and returns a numeric-value. Then, the classification function provides class label to the instances based on the results of similarity function. Accordingly, the concept description is updated [50]. K∗ (K Star) is an instance-based learner that uses an entropy-based similarity matching function for searching/matching test instances to the learned instances [51]. Locally Weighted Learning (LWL). In a locally weighted learning, the prediction models are allowed to create at local points in a dataset or the specific point of interest rather than creating model for entire dataset. Hence, a linear regression or naive Bayes classifier or any other classifier may be used to create local models. In this case, we use Decision Stamp, which is a single level decision tree model for prediction [52, 53]. 10 Varun Kumar Ojha et al. 2.3.5 Ensemble Methods In this work, we tried to exploit different method of making ensemble. For an ensemble to perform well, we need to take into account two things which are accuracy of predictors and diversity among the predictors [54]. For example, Bagging maintains diversity by bootstrapping dataset, AddBoost combines several weak predictors, Random Subspace maintains diversity by splitting feature space, Random committee maintains diversity by creating predictors using different random seeds, and Rotation forest maintains diversity by splitting and extracting feature subspace using principal component analysis. Similarly, in multi-scheme and voting scheme, we combine several predictors to maintain diversity. Here, we describe the ensemble methods as follows. Bagging. In Bagging, several copies of same predictor is created. Each copy of the predictor learns a different replicate of learning set created from the complete training set using bootstrapping. Finally, the predictor’s decision is combined using plurality voting method [55]. Adaptive Boosting (AdaBoost) is an ensemble technique that combines several weak predictors and inaccurate rules to create an accurate predictor [56]. Random Subspace (Random SUB). In random subspace ensemble method feature space is divided into several feature subset. Hence, predictors are constructed for each feature subset. Finally, the decision of each constructed predictors are combined using voting method [57]. Random Committee (Random COM): In a random committee ensemble, several predictors are constructed over similar dataset, but they use different random seeds to maintain diversity in the ensemble. Rotation Forest (Rotation FRST). In this approach, training set for the predictors are created by splitting feature set into K subsets, and Principal Component Analysis is applied to extract all the principle components [58]. Hence, diversity among the predictors are maintained by K axis rotation to form new feature set for training [58]. Ensemble Selection (Ensemble SEL). In the ensemble selection approach, the ensemble starts with an empty bag, and the predictors (chosen from a library of trained predictors) maximizing the performance of ensemble are added to the bag one by one to compute the decision of ensemble by using voting method [59]. Voting Scheme (Vote). The voting scheme combines probability distribution of several chosen predictors/classifiers (or predictors available in a bag for making ensemble) using majority voting combination method [60]. Identifying Hazardousness of Sewer-Pipeline 11 Multi-Scheme (Multi). The multi-scheme ensemble approach uses a bag of predictors and selects the output class by selecting a predictor from the bag of predictors based on cross-validation performance of the predictors [60]. Weighted Predictor Ensemble (WPE). In this scheme of ensemble, the weight of predictors were determined. Subsequently, the ensemble output of k many predictors were computed as follows: c y = arg max j=1 k X wj I (Pj = ωj ) , j=1 where c is the number of classes (here it is two), I (Pj = ωj ) is a function that returns value one for the predicted class ωj . 3 Experimental Framework and Results Our aim in the experiment design was to obtain a highly accurate model for predicting hazardousness of the environment in a sewer pipeline. The sewerpipeline environment was represented by the collected dataset. The second objective of the experiment design was to obtain results for analyzing the classifiers (predictors). Accordingly, the results of the classifiers were collected. Table 3 represents the parameter setting of the chosen classifiers. For the evaluation of the classifiers, we repeated our experiments 10 times. Finally, the results were compared based on empirical and statistical (Kolmogorov– Smirnov test) evaluation. We used WEKA [61] and MATLAB tools [62] for the purpose of our experiments. We organized the experimental results into three parts as reflected in Table 4. The first part in the table describes the category wise performance of classifier. Hence, the performance of the category of classifiers was evaluated. We represented the performance of the classifiers as per their training and test accuracy. An accuracy close to 1.0 indicates 100% classification accuracy. Accordingly, the standard deviation (std) of training and test accuracies were reported for understanding the consistency of the classifiers’ performance. In Table 4, the performance of the classifiers were arranged as follows. The category is arranged in the ascending order of their average accuracy over 10-fold CV test set, i.e., better performing classifier to the less performing classifier. The dataset was portioned into 10 equal sets and each time 9 sets were used for training and one set for testing. This process was repeated 10 times and each time a unique test set was used. In the second part, we organized the results according to rank of the classifiers’ performance over 10-fold test set. It may please be noted that for each classifier, we collected 10 instances of 10-fold CV training and test results. Hence, the results in Table 5 reflect averaged training and test accuracy of the classifiers. However, ranking the classifiers based only on the average results does not say much about the quality of the classifier. Hence, in the 12 Varun Kumar Ojha et al. Table 3 Parameter setting of different classifiers category Network-Based Classifiers (F1) Tree-Based Classifiers (F2) Instance-Based Classifiers (F3) Rule-Based Classifiers (F4) Ensemble Classifiers (E1) Ensemble Classifiers (E2) Classifiers Parameters MLP Learning rate: 0.3, momentum factor: 0.2, iteration: 500, nodes in hidden layer: 100 Kernel: Gaussian basis function. Kernel: Radial basis function Minimum no. of instance per leaf: 2, split proportion: 0.001 Leaf node: nave Bayes classifier. Node: logistic function, Number of instance per node for splitting: 15 Similarity function: linear nearest neighbor search, neighbor size: 1 Similarity function: entropy distance measure. Similarity function: linear nearest neighbor search, Weight function: Linear, Classifier: Decision Stamp. Evaluation metric: accuracy, Search method: best first Confidence threshold for pruning: 0.25 Ensemble size: 10. Classifier: REP Tree Ensemble size: 10. Classifier: Decision Stamp. Ensemble size: 10. Classifier: REP Tree Ensemble size: 10. Classifier: Random Tree Ensemble size: 10. Classifier: Random Tree Ensemble size: 10. Classifier: REP Tree Ensemble size: 12. Classifiers: F1, F2, F3 and F4 Ensemble size: 12. Classifiers: F1, F2, F3 and F4 Ensemble size: 12. Classifiers: F1, F2, F3 and F4 RBF SVM REP NBT LMT IBK K Star LWL DT PART Zero R Bagging AdaBoost Random SEL Random COM Rotation FRST Ensemble SEL Vote Multi Scheme WPE third part of the results, we used pairwise comparison of the classifiers using Kolmogorov–Smirnov (KS) test, which ascertains whether the supremacy of one classifier over the other is statistically significant or not. A comprehensive matrix of the pairwise KS test results are presented in Table 6. The KS Test is a non-parametric statistical test that determines the difference between the cumulative frequency distribution (cfd) of two samples. In other words, it indicates whether the empirical cfd of one sample is equal “=”, larger “”, or smaller “≺” than the other. It tells whether two dataset A and B are statistically similar “A=B”, dissimilar “A≺B”, where A being statistically dominated by B, or dissimilar “AB”, where A being statistically dominant over B. In our experiments, the KS test was evaluated with 5% significance level, i.e., with 95% confidence. 4 Discussions Since the developed electronic portable gas detector shall be used by naive persons who are engaged in maintaining sewer-pipeline, we are looking for binary answer. Hence, our objective is to search for classification accuracy and Identifying Hazardousness of Sewer-Pipeline 13 Table 4 Experimental Results of Classifiers over 10 fold cross validation error category NN-Based Classifiers Tree-Based Classifiers Instance-Based Classifiers Rule-Based Classifiers Ensemble Classifiers Classifiers Training avg. accuracy std Test avg. accuracy std SVM MLP RBF LMT REP Tree NB Tree IBK K Star LWL PART Decision Table Zero R Multi Rotation FRST Random COM Bagging WPE Ensemble SEL Vote Random SUB AdaBoostM1 0.9407 0.8681 0.8064 0.9697 0.9528 0.9064 1.0000 0.9997 0.7613 0.9275 0.8672 0.7613 1.0000 1.0000 1.0000 0.9728 0.9635 0.9577 0.9423 0.9160 0.7613 0.9340 0.8664 0.8051 0.9360 0.9265 0.8898 0.9671 0.9638 0.7613 0.9062 0.8553 0.7613 0.9672 0.9622 0.9549 0.9395 0.9356 0.9330 0.9214 0.8720 0.7613 0.0041 0.0081 0.0187 0.0023 0.0067 0.0418 0.0023 0.0036 0.0128 0.0103 0.0154 0.0091 0.0035 0.0036 0.0073 0.0077 0.0056 0.0077 0.0143 0.0165 0.0091 0.0008 0.0029 0.0090 0.0039 0.0025 0.0469 0.0000 0.0001 0.0014 0.0097 0.0049 0.0010 0.0000 0.0000 0.0000 0.0008 0.0006 0.0009 0.0128 0.0103 0.0010 Table 5 Ranking algorithms according to their performance on test set (10 Fold CV). Rank category Classifiers Training Test 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 E2 F3 F3 E1 E1 E1 E2 F2 E1 F1 F2 E2 F4 F2 F1 E1 F4 F1 F3 F4 E1 Multi IBK KStar Rotation FRST Random COM Bagging WPE LMT Ensemble SEL SVM REPTree Vote PART NBTree MLP Random SUB DT RBF LWL ZeroR AdaBoost 100.0000 100.0000 99.9653 100.0000 100.0000 97.2874 96.3564 96.7454 95.7173 94.0837 95.2469 93.9593 92.7331 89.4770 86.8566 90.9650 86.6874 80.7795 76.1285 76.1285 76.1302 96.8060 96.7945 96.4677 96.2725 95.7737 94.1025 93.9865 93.4674 93.3778 93.3647 92.4733 92.1314 90.8675 88.0795 86.6336 86.4397 85.4790 80.7571 76.1451 76.1451 76.1302 14 Varun Kumar Ojha et al. RBF REPTree SVM ZeroR Multi-Scheme Vote AdaBoost Bagging Ensemble SEL Random COM ≺ ≺ . . . . . . . . . . . . . . . . ≺ ≺ ≺ . . . . . . . . . . . . . . . ≺ ≺ ≺ ≺ . . . . . . . . . . . . . . ≺ . . . . . . . . . . . . . ≺ ≺ ≺ ≺ ≺ ≺ . . . . . . . . . . . . ≺ ≺ ≺ ≺ ≺ ≺ ≺ . . . . . . . . . . . = . . . . . . . . . . ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ . . . . . . . . . ≺ ≺ ≺ ≺ ≺ ≺ ≺ . . . . . . . . = = . . . . . . . ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ . . . . . . ≺ ≺ ≺ ≺ ≺ ≺ ≺ = ≺ ≺ ≺ . . . . . ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ . . . . WPE PART . . . . . . . . . . . . . . . . . Random SUB NBTree ≺ . . . . . . . . . . . . . . . . . . Rotation FRST MLP ≺ . . . . . . . . . . . . . . . . . . . LMT ≺ . . . . . . . . . . . . . . . . . . . . LWL . . . . . . . . . . . . . . . . . . . . . IBK DT IBK Kstar LMT LWL MLP NBTree PART RBF REPTree SVM ZeroR Multi-Scheme Vote AdaBoost Bagging Ensemble SEL Random COM Random SUB Rotation FRST WPE Kstar Classifiers DT Table 6 Ranking algorithms according to their performance on test set (10 Fold CV). ≺ ≺ = ≺ ≺ ≺ . . . ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ . . ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ ≺ . the model (weights) with the highest accuracy so that such a combination may be implemented into electronic portable gas detector form. Moreover, it is also a difficult task to be certain with the accuracy of an implemented electronic portable gas detector because the toxic exposure of a gas is also proportional to the time and not only its safety limit. However, with a real-time monitoring and requisite maintenance involved, the accuracy of detector may be relaxed and hence, we resorted to choose 90% accuracy as the accuracy for our developed detector. So, the classifier’s performance was compared with a threshold setting of 90% accuracy. First, let us discuss on the obtained results. For the classifiers belonging to network-based category F1, the classifier SVM performs better than its counterparts MLP and RBF both in terms of high accuracy (test accuracy 0.93403) and high consistency (std on test accuracy 0.0041). On the other hand, the performance of MLP was reported next to SVM with high consistency. The performance of the RBF was found to be inconsistent and poorer in comparison to its counterparts. In the tree based category F2, the performance of LMT and REPTree was comparable to whereas, NBTree has shown poor performance compared to its counterparts. Identifying Hazardousness of Sewer-Pipeline 15 In instance-base category F3, the performance of IBK and K Star was comparative with a high accuracy and high consistency. LWL performed poor with a very low accuracy. When it came to the category of rule based classifier F4, PART has outperformed others in its category, but the consistency was not as high as the consistency of the other well performing classifiers IBK, SVM, MLP, etc. The classifier ZeroR consistently performed poor in comparison to all other classifiers. In the ensemble category E1 and E2, the multi-scheme, Random COM, Rotation FRST, Bagging, WPE, and Ensemble SEL performed with high accuracies (over 90%) and consistency. However, the performance of the ensembles Random Forest, Vote, and AdaBoost were not as satisfactory as compared to the other ensembles. One of the reason behind poor performance of Random SUB was the usage of subset of the features. Therefore, the feature selection may not help in case of this dataset because of the high correlation maintained by each of the features with the output feature. Similarly, Voting used probability measures to combine the predictors and AddBoost combined weak predictors, whereas, the entirely better performing ensemble exploited the best predictors. Hence, they performed better in this scenario. Considering the assumption of 90% accuracy being a good predictor for implementation as gas detector, we can figure out from Table 5 that the classifiers belong to category F3 (exception of the classifier LWL) had performed better than the classifiers of other categories. However, the instance based classifier IBK is not suitable for the implementation as electronic gas detector since it required a large memory for its computation for saving all the instances of the training set. IBK prediction is computed based on all training samples. Hence, it takes long time to compute the output, which is unacceptable in real time. The next category whose performance was found close to IBK were the classifiers of category F2 (tree based classifier). Two classifies, LMT and REP Tree qualified the 90% accuracy threshold. On the contrary, two classifiers from each F1 and F4 had performed lower than 90% accuracy. However, SVM performed significantly well with a very high accuracy 93.36%. Similarly, classifier PART from category F4 had an accuracy of 90.86%. However, since the SVM produced less number of parameters than the tree based predictor and it robustly accommodates the noisy attributes, it was recommended from these experiments that SVM is a proper choice for the implementation of the proposed gas detector. 5 Conclusion In this work, we explored a real world problem in the context of classification, where we simplified the approach by offering binary decision to the problem. We explored the problem related to the detection of hazardousness of a sewer pipeline environment. This is very crucial problem since it is related to the safety of the persons who have to work under the toxic environment of 16 Varun Kumar Ojha et al. the sewer-pipeline. Usually, a sewer-pipeline environment contains mixture of toxic gases. Hence, we collected samples from sewer pipelines from different locations. Then we examined those samples to identify data samples for our experiments. We prepared a large dataset by collecting gas sensor responses from laboratory tests, literature and scaled the collected gas sensor responses to form a dataset where non-hazardous samples were labeled 0 and hazardous samples were labeled 1. Finally, we applied 21 different classifiers over the identified dataset and their empirical and statistical performance were evaluated. We discovered that for this problem, the instance based classifier performed best followed by the performance of tree based classifiers. However, we found that the performance of the classifiers were dependent on the ability and mechanism of the classifiers themselves and not on the information regarding which category they belong to. Acknowledgements This work was supported by the IPROCOM Marie Curie Initial Training Network, funded through the People Programme (Marie Curie Actions) of the European Unions Seventh Framework Programme. References 1. J. Whorton, ““the insidious foe”–sewer gas,” Western Journal of Medicine, vol. 175, no. 6, pp. 427–428, Dec. 2001. 2. R. J. Lewis, Sax’s Dangerous Properties of Industrial Materials, 12th ed. Wiley, 2010. 3. N. Gromicko, “Sewer gases in the home,” 2006, http://www.nachi.org/sewer-gaseshome.html. 4. T. Hindu, “Deaths in the drains,” 2014, http://www.thehindu.com/opinion/oped/deaths-in-the-drains/article5868090.ece?homepage=true., Accessed on 15 Dec 2015. 5. 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The Generic Model of Computation Nachum Dershowitz School of Computer Science Tel Aviv University Tel Aviv, Israel [email protected] Over the past two decades, Yuri Gurevich and his colleagues have formulated axiomatic foundations for the notion of algorithm, be it classical, interactive, or parallel, and formalized them in the new generic framework of abstract state machines. This approach has recently been extended to suggest a formalization of the notion of effective computation over arbitrary countable domains. The central notions are summarized herein. 1 Background Abstract state machines (ASMs), invented by Yuri Gurevich [24], constitute a most general model of computation, one that can operate on any desired level of abstraction of data structures and native operations. All (ordinary) models of computation are instances of this one generic paradigm. Here, we give an overview of the foundational considerations underlying the model (cobbled together primarily from [18, 3, 12]).1 Programs (of the sequential, non-interactive variety) in this formalism are built from three components: • There are generalized assignments f (s1 , . . . , sn ) := t, where f is any function symbol (in the vocabulary of the program) and the si and t are arbitrary terms (in that vocabulary). • Statements may be prefaced by a conditional test, if C then P or if C then P else Q, where C is a propositional combination of equalities between terms. • Program statements may be composed in parallel, following the keyword do, short for do in parallel. An ASM program describes a single transition step; its statements are executed repeatedly, as a unit, until no assignments have their conditions enabled. (Additional constructs beyond these are needed for interaction and large-scale parallelism, which are not dealt with here.) As a simple example, consider the program shown as Algorithm 1, describing a version of selection sort, where F(0), . . . , F(n − 1) contain values to be sorted, F being a unary function symbol. Initially, n ≥ 1 is the quantity of values to be sorted, i is set to 0, and j to 1. The brackets indicate statements that are executed in parallel. The program proceeds by repeatedly modifying the values of i and j, as well as of locations in F, referring to terms F(i) and F( j). When all conditions fail, that is, when j = n and i + 1 = n, the values in F have been sorted vis-à-vis the black-box relation “>”. The program halts, as there is nothing left to do. (Declarations and initializations for program constants and variables are not shown.) This sorting program is not partial to any particular representation of the natural numbers 1, 2, etc., which are being used to index F. Whether an implementation uses natural language, or decimal numbers, 1 For a video lecture of Gurevich’s on this subject, see http://www.youtube.com/v/7XfA5EhH7Bc. E. Kashefi, J. Krivine, F. van Raamsdonk (Eds.) DCM 2011 EPTCS 88, 2012, pp. 59–71, doi:10.4204/EPTCS.88.5 c N. Dershowitz This work is licensed under the Creative Commons Attribution License. 60 Generic Model of Computation Algorithm 1 An abstract-state-machine program for sorting.   i := i + 1 if j = n then if i + 1 6= n then do  j := i + 2     F(i) := F( j)    if F(i) > F( j) then do  F( j) := F(i) else do     j := j + 1 Algorithm 2 An abstract-state-machine program for bisection search.   if sgn f ((a + b)/2) = sgn f (a) then a := (a + b)/2 if \|b − a\| > ε then do  if sgn f ((a + b)/2) = sgn f (b) then b := (a + b)/2 or binary strings is immaterial, as long as addition behaves as expected (and equality and disequality, too). Furthermore, the program will work regardless of the domain from which the values of F are drawn (be they integers, reals, strings, or what not), so long as means are provided for evaluating the inequality (>) relation. Another simple ASM program is shown in Algorithm 2. This is a standard bisection search for the root of a function, as described in [22, Algorithm #4]. The point is that this abstract formulation is, as the author of [22] wrote, “applicable to any continuous function” over the reals—including ones that cannot be programmed. What is remarkable about ASMs is that this very simple model of computation suffices to precisely capture the behavior of the whole class of ordinary algorithms over any domain. The reason is that, by virtue of the abstract state machine (ASM) representation theorem of [25] (Theorem 2 below), any algorithm that satisfies three very natural “Sequential Postulates” can be step-by-step, state-for-state emulated by an ASM. Those postulates, articulated in Section 2, formalize the following intuitions: (I) an algorithm is a state-transition system; (II) given the algorithm, state information determines future transitions and can be captured by a logical structure; and (III) state transitions are governed by the values of a finite and input-independent set of terms. The significance of the Sequential Postulates lies in their comprehensiveness. They formalize which features exactly characterize a classical algorithm in its most abstract and generic manifestation. Programs of all models of effective, sequential computation satisfy the postulates, as do idealized algorithms for computing with real numbers (e.g. Algorithm 2), or for geometric constructions with compass and straightedge (see [34] for examples of the latter). Abstract state machines are a computational model that is not wedded to any particular data representation, in the way, say, that Turing machines manipulate strings using a small set of tape operations. The Representation Theorem, restated in Section 3, establishes that ASMs can express and precisely emulate any and all algorithms satisfying the premises captured by the postulates. For any such algorithm, there is an ASM program that describes precisely the same state-transition function, state after state, as does the algorithm. In this sense, ASMs subsume all other computational models. It may be informative to note the similarity between the form of an ASM, namely, a single repeated loop of a set of generalized assignments nested within conditionals with the “folk theorem” to the effect N. Dershowitz 61 that any flowchart program can be converted to a single loop composed of conditionals, sequencing, and assignments, with the aid of some auxiliary variables (see [29]). Parallel composition gives ASMs the ability to perform multiple actions sans extra variables, and to capture all that transpires in a single step of any algorithm. This versatility of ASMs is what makes them so ideal for both specification and prototyping. Indeed, ASMs have been used to model all manner of programming applications, systems, and languages, each on the precise intended level of abstraction. See [13] and the ASM website (http://www.eecs.umich. edu/gasm) for numerous exemplars. ASMs provide a complete means of describing algorithms, whether or not they can be implemented effectively. On account of their abstractness, one can express generic algorithms, like our bisection search for arbitrary continuous real-valued functions, or like Gaussian elimination, even when the field over which it is applied is left unspecified. AsmL [26], an executable specification language based on the ASM framework, has been used in industry, in particular for the behavioral specification of interfaces (see, for example, [1]). Church’s Thesis asserts that the recursive functions are the only numeric functions that can be effectively computed. Similarly, Turing’s Thesis stakes the claim that any function on strings that can be mechanically computed can be computed, in particular, by a Turing machine. More generally, one additional natural hypothesis regarding the describability of initial states of algorithms, as explained in Section 5, characterizes the effectiveness of any model of computation, operating over any (countable) data domain (Theorem 4). On account of the ability of ASMs to precisely capture single steps of any algorithm, one can infer absolute bounds on the complexity of algorithms under arbitrary effective models of computation, as will be seen (Theorem 6) at the end of Section 5. 2 Sequential Algorithms The Sequential Postulates of [25] regarding algorithmic behavior are based on the following key observations: • A state should contain all the relevant information, apart from the algorithm itself, needed to determine the next steps. For example, the “instantaneous description” of a Turing machine computation is just what is needed to pick up a machine’s computation from where it has been left off; see [38]. Similarly, the “continuation” of a Lisp program contains all the state information needed to resume its computation. First-order structures suffice to model all salient features of states. Compare [32, pp. 420–429]. • The values of programming variables, in and of themselves, are meaningless to an algorithm, which is implementation independent. Rather, it is relationships between values that matter to the algorithm. It follows that an algorithm should work equally well in isomorphic worlds. Compare [19, p. 128]. An algorithm can—indeed, can only—determine relations between values stored in a state via terms in its vocabulary and equalities (and disequalities) between their values. • Algorithms are expressed by means of finite texts, making reference to only finitely many terms and relations among them. See, for example, [31, p. 493]. The three postulates given below (from [25], modified slightly as in [4, 5, 6, 3]) assert that a classical algorithm is a state-transition system operating over first-order structures in a way that is invariant under isomorphisms. An algorithm is a prescription for updating states, that is, for changing some of the interpretations given to symbols by states. The essential idea is that there is a fixed finite set of terms Generic Model of Computation 62 that refer (possibly indirectly) to locations within a state and which suffice to determine how the state changes during any transition. 2.1 Sequential Time To begin with, algorithms are deterministic state-transition systems. Postulate I (Sequential Time) An algorithm determines the following: • A nonempty set2 S of states and a nonempty subset S0 ⊆ S of initial states. • A partial next-state transition function τ : S ⇀ S . Terminal states S‡ ⊆ S are those states X for which no transition τ (X ) is defined. Having the transition depend only on the state means that states must store all the information needed to determine subsequent behavior. Prior history is unavailable to the algorithm unless stored in the current state. State-transitions are deterministic. Classical algorithms in fact never leave room for choices, nor do they involve any sort of interaction with the environment to determine the next step. To incorporate nondeterministic choice, probabilistic choice, or interaction with the environment, one would need to modify the above notion of transition. This postulate is meant to exclude formalisms, such as [21, 33], in which the result of a computation—or the continuation of a computation—may depend on (the limit of) an infinite sequence of preceding (finite or infinitesimal) steps. Likewise, processes in which states evolve continuously (as in analog processes, like the position of a bouncing ball), rather than discretely, are eschewed. Though it may appear at first glance that a recursive function does not fit under the rubric of a state-transition system, in fact the definition of a traditional recursive function comes together with a computation rule for evaluating it. As Rogers [36, p. 7] writes, “We obtain the computation uniquely by working from the inside out and from left to right”. 2.2 Abstract State Algorithm states are comprehensive: they incorporate all the relevant data (including any “program counter”) that, when coupled with the program, completely determine the future of a computation. States may be regarded as structures with (finitely many) functions, relations, and constants. To simplify matters, relations will be treated as truth-valued functions and constants as nullary functions. So, each state consists of a domain (base set, universe, carrier) and interpretations for its symbols. All relevant information about a state is given explicitly in the state by means of its interpretation of the symbols appearing in the vocabulary of the structure. The specific details of the implementation of the data types used by the algorithm cannot matter. In this sense states are “abstract”. This crucial consideration leads to the second postulate. Postulate II (Abstract State) The states S of an algorithm are (first-order) structures over a finite vocabulary F , such that the following hold: • If X is a state of the algorithm, then any structure Y that is isomorphic to X is also a state, and Y is initial or terminal if X is initial or terminal, respectively. • Transitions preserve the domain; that is, Dom τ (X ) = Dom X for every non-terminal state X . 2 Or class; the distinction is irrelevant for our purposes. N. Dershowitz 63 • Transitions respect isomorphisms, so, if ζ : X ∼ = Y is an isomorphism of non-terminal states X ,Y , then also ζ : τ (X ) ∼ = τ (Y ). State structures are endowed with Boolean truth values and standard Boolean operations, and vocabularies include symbols for these. As a structure, a state interprets each of the function symbols in its vocabulary. For every k-ary symbol f in the vocabulary of a state X and values a1 , . . . , ak in its domain, some domain value b is assigned to the location f (a1 , . . . , ak ), for which we write f (ā) 7→ b. In this way, X assigns a value [[t]]X in Dom X to (ground) terms t. Vocabularies are finite, since an algorithm must be describable in finite terms, so can only refer explicitly to finitely many operations. Hence, an algorithm can not, for instance, involve all of Knuth’s arrow operations, ↑, ↑↑, ↑↑↑, etc. Instead one could employ a ternary operation λ x, y, z. x ↑z y. This postulate is justified by the vast experience of mathematicians and scientists who have faithfully and transparently presented every kind of static mathematical or scientific reality as a logical structure. In restricting structures to be “first-order”, we are limiting the syntax to be first-order. This precludes states with infinitary operations, like the supremum of infinitely many objects, which would not make sense from an algorithmic point of view. This does not, however, limit the semantics of algorithms to first-order notions. The domain of states may have sequences, or sets, or other higher-order objects, in which case, the state would also need to provide operations for dealing with those objects. Closure under isomorphism ensures that the algorithm can operate on the chosen level of abstraction. The states’ internal representation of data is invisible and immaterial to the program. This means that the behavior of an algorithm, in contradistinction with its “implementation” as a C program—cannot, for example, depend on the memory address of some variable. If an algorithm does depend on such matters, then its full description must also include specifics of memory allocation. It is possible to liberalize this postulate somewhat to allow the domain to grow or shrink, or for the vocabulary to be infinite or extensible, but such “enhancements” do not materially change the notion of algorithm. An extension to structures with partial operations is given in [3]; see Section 4. 2.3 Effective Transitions The actions taken by a transition are describable in terms of updates of the form f (ā) 7→ b, meaning that b is the new interpretation to be given by the next state to the function symbol f for values ā. To program such an update, one can use an assignment f (s̄) := t such that [[s̄]]X = ā and [[t]]X = b. We view a state X as a collection of the graphs of its operations, each point of which is a location-value pair also denoted f (ā) 7→ b. Thus, we can define the update set ∆(X ) as the changed points, τ (X ) \ X . When X is a terminal state and τ (X ) is undefined, we indicate that by setting ∆(X ) = ⊥. The point is that ∆ encapsulates the state-transition relation τ of an algorithm by providing all the information necessary to update the interpretation given by the current state. But to produce ∆(X ) for a particular state X , the algorithm needs to evaluate some terms with the help of the information stored in X . The next postulate will ensure that ∆ has a finite representation and its updates can be determined and performed by means of only a finite amount of work. Simply stated, there is a fixed, finite set of ground terms that determines the stepwise behavior of an algorithm. Postulate III (Effective Transitions) 3 For every algorithm, there is a finite set T of (ground) critical terms over the state vocabulary, such that states that agree on the values of the terms in T also share the same update sets. That is, ∆(X ) = ∆(Y ), for any two states X ,Y such that [[t]]X = [[t]]Y for all t ∈ T . In particular, if one of X and Y is terminal, so is the other. 3 Or Bounded Exploration. Generic Model of Computation 64 The intuition is that an algorithm must base its actions on the values contained at locations in the current state. Unless all states undergo the same updates unconditionally, an algorithm must explore one or more values at some accessible locations in the current state before determining how to proceed. The only means that an algorithm has with which to reference locations is via terms, since the values themselves are abstract entities. If every referenced location has the same value in two states, then the behavior of the algorithm must be the same for both of those states. This postulate—with its fixed, finite set of critical terms—precludes programs of infinite size (like an infinite table lookup) or which are input-dependent. A careful analysis of the notion of algorithm in [25] and an examination of the intent of the founders of the field of computability in [18] demonstrate that the Sequential Postulates are in fact true of all ordinary, sequential algorithms, the (only) kind envisioned by the pioneers of the field. In other words, all classical algorithms satisfy Postulates I, II, and III. In this sense, the traditional notion of algorithm is precisely captured by these axioms. Definition 1 (Classical Algorithm) An object satisfying Postulates I, II, and III shall be called a classical algorithm. 2.4 Equivalent Algorithms It makes sense to say that two algorithms have the same behavior, or are behaviorally equivalent, if they operate over the same states and have the same transition function. Two algorithms are syntactically equivalent if their states are the same up to renaming of symbols (α -conversion) in their vocabularies, and if transitions are the same after renaming. For a wide-ranging discussion of algorithm equivalence, see [2]. 3 Abstract State Machines Abstract state machines (ASMs) are an all-powerful description language for the classical algorithms we have been characterizing. 3.1 Programs The semantics of the ASM statements, assignment, parallel composition, and conditionals, are as expected, and are formalized below. The program, as such, defines a single step, which is repeated forever or until there is no next state. For convenience, we show only a simple form of ASMs. Bear in mind, however, that much richer languages for ASMs are given in [24] and are used in practice [27]. Programs are expressed in terms of some vocabulary. By convention, ASM programs always include symbols for the Boolean values (true and false), undef for a default, “undefined” value, standard Boolean operations (¬, ∧, ∨), and equality (=, 6=). The vocabulary of the sorting program, for instance, contains F = {1, 2, +, >, F, n, i, j} in addition to the standard symbols. Suppose that its states have integers and the three standard values for their domain. The nullary symbols 0 and n are fixed programming constants and serve as bounds of F. The nullary symbols i and j are programming “variables” and are used as array indices. All its states interpret the symbols 1, 2, +, >, as well as the standard symbols, as usual. Unlike i, j, and F, these are static; their interpretation will never be changed by the program. Initial states have n ≥ 0, i = 0, j = 1, some integer values for F(0), . . . , F(n − 1), plus undef for all other points N. Dershowitz 65 States X such that Update set ∆(X ) 0 [[ j]] = [[n]] = [[i]] + 1 ⊥ 1 [[ j]] = [[n]] 6= [[i]] + 1 i 7→ [[i]] + 1, j 7→ [[i]] + 2 2 [[ j]] 6= [[n]] , [[F(i)]] > [[F( j)]] F([[i]]) 7→ [[F( j)]] , F([[ j]]) 7→ [[F(i)]] , j 7→ [[ j]] + 1 3 [[ j]] 6= [[n]] , [[F(i)]] 6> [[F( j)]] j 7→ [[ j]] + 1 Table 1: Update sets for sorting program. of F. This program always terminates successfully, with j = n = i + 1 and with the first n elements of F in nondecreasing order. There are no hidden variables in ASMs. If some steps of an algorithm are intended to be executed in sequence, say, then the ASM will need to keep explicit track of where in the sequence it is up to. 3.2 Semantics Unlike algorithms, which are observed to either change the value of a location in the current state, or not, an ASM might “update” a location in a trivial way, giving it the same value it already has. Also, an ASM might designate two conflicting updates for the same location, what is called a clash, in which case the standard ASM semantics are to cause the run to fail (just as real-world programs might abort). An alternative semantics is to imagine a nondeterministic choice between the competing values. (Both were considered in [24].) Here, we prefer to ignore both nondeterminism and implicit failure, and tacitly presume that an ASM never involves clashes, albeit this is an undecidable property. To take the various possibilities into account, a proposed update set ∆+ P (X ) (cf. [4]) for an ASM P may be defined in the following manner: ∆+f (s1 ,...,sn ):=t (X ) = { f ([[s1 ]]X , . . . , [[sn ]]X ) 7→ [[t]]X } + ∆+ (X ) ∪ · · · ∪ ∆+ Pn (X ) do {P1 ···Pn } (X ) = ∆ (P1 + ∆P (X ) if X \|= C ∆+ (X ) = if C then P else Q ∆+ (X ) otherwise ( Q ∆+ P (X ) if X \|= C ∆+ (X ) = if C then P ∅ otherwise . Here X \|= C means, of course, that Boolean condition C holds true in X . When the condition C of a conditional statement does not evaluate to true, the statement does not contribute any updates. When ∆+ (X ) = ∅ for ASM P, its execution halts with success, in terminal state X . (Since no confusion will arise, we are dropping the subscript P.) Otherwise, the updates are applied to X to yield the next state by replacing the values of all locations in X that are referred to in ∆+ (X ). So, if the latter contains only trivial updates, P will loop forever. For terminal states X , the update set ∆(X ) is ⊥, to signify that there is no next state. For non-terminal X , ∆(X ) is the set of non-trivial updates in ∆+ (X ). The update sets for the sorting program (Algorithm 1) are shown in Table 1, with the subscript in [[·]]X omitted. For example, if state X is such that n = 2, i = 0, Generic Model of Computation 66 j = 1, F(0) = 1, and F(1) = 0, then (per row 2) ∆+ (X ) = {F(0) 7→ 0, F(1) 7→ 1, j 7→ 2}. For this X , ∆(X ) = ∆+ (X ), and the next state X ′ = τ (X ) has i = 0 (as before), j = 2, F(0) = 0 and F(1) = 1. After one more step (per row 1), in which F is unchanged, the algorithm reaches a terminal state, X ′′ = τ (X ′ ), with j = n = i + 1 = 2. Then (by row 0), ∆+ (X ′′ ) = ∅ and ∆(X ′′ ) = ⊥. 4 The Representation Theorem Abstract state machines clearly satisfy the three Sequential Postulates: ASMs define a state-transition function; they operate over abstract states; and they depend critically on the values of a finite set of terms appearing in the program (and on the unchanging values of parts of the state not modified by the program). For example, the critical terms for our sorting ASM are all the terms appearing in it, except for the left-hand sides of assignments, which contribute their proper subterms instead. These are j 6= n, ( j = n) ∧ (i + 1 6= n), F(i) > F( j), i + 2, j + 1, and their subterms. Only the values of these affect the computation. Thus, any ASM describes a classical algorithm over structures with the same vocabulary (similarity type). The converse is of greater significance: Theorem 2 (Representation [25, Theorem 6.13]) Every classical algorithm, in the sense of Definition 1, has a behaviorally equivalent ASM, with the exact same states and state-transition function. The proof of this representation theorem constructs an ASM that contains conditions involving equalities and disequalities between critical terms. Closure under isomorphisms is an essential ingredient for making it possible to express any algorithm in the language of terms. A typical ASM models partial functions (like division or tangent) by using the special value, undef, denoting that the argument is outside the function’s domain of definition, and arranging that most operations be strict, so a term involving an undefined subterm is likewise undefined. The state of such an ASM would return true when asked to evaluate an expression c/0 = undef, and it can, therefore, be programmed to work properly, despite the partiality of division. In [3], the analysis and representation theorem have been refined for algorithms employing truly partial operations, operations that cause an algorithm to hang when an operation is attempted outside its domain of definition (rather than return undef). The point is that there is a behaviorally equivalent ASM that never attempts to access locations in the state that are not also accessed by the given algorithm. Such partial operations are required in the next section. 5 Effective Algorithms The Church-Turing Thesis [30, Thesis I† ] asserts that standard models capture effective computation. Specifically: All effectively computable numeric (partial) functions are (partial) recursive. All (partial) string functions can be computed by a Turing machine. We say that an algorithm computes a partial function f : Dk ⇀ D if there are input states I ⊆ S0 , with particular locations for input values, such that running the algorithm results in the correct output values of f . Specifically: • The domain of each input state is D. There are k terms such that their values in input states cover all tuples in Dk . Other than that, input states all agree on the values of all other terms. N. Dershowitz 67 • For all input values ā, the corresponding input state leads, via a sequence of transitions τ , to a terminal state in which the value of a designated term t (in the vocabulary of the algorithm) is f (ā) whenever the latter is defined, and leads to an infinite computation whenever it is not. To capture what it is that makes a sequential algorithm mechanically computable, we need for input states to be finitely representable. Accordingly, we insist that they harbor no information beyond the means to reach domain values, plus anything that can be derived therefrom. We say that function symbols C construct domain D in state X if X assigns each value in D to exactly one term over C , so restricting X to C gives a free Herbrand algebra. For example, the domain of the sorting algorithm, consisting of integers and Booleans, can be constructed from 0, true, false, undef, and a “successor” function (call it c) that takes non-negative integers (n) to the predecessor of their negation (−n − 1) and negative integers (−n) to their absolute value (n). Postulate III ensures that the transition function is describable by a finite text, and—in particular–by the text of ASM. For an algorithm to be effective, its states must also be finitely describable. Definition 3 (Effectiveness) 1. A state is effective if it includes constructors for its domain, plus operations that are almost everywhere the same, meaning that all but finitely-many locations (these can hold input values) have the same default value (such as undef ). 2. A classical algorithm is effective if its initial states are. 3. Moreover, effective algorithms can be bootstrapped: A state is effective also if its vocabulary can be enriched to C ⊎ G so that C constructs its domain, while every (total or partial) operation in G is computed by an effective algorithm over those constructors. 4. A model (of computation), that is, a set of algorithms with shared domain(s), is effective if all its algorithms are, via the same constructors. This effectiveness postulate excludes algorithms with ineffective oracles, such as the halting function. Having only free constructors at the foundation precludes the hiding of potentially uncomputable information by means of equalities between distinct representations of the same domain element. This is the approach to effectiveness advocated in [11], extended to include partial functions in states, as in [3]. For each n ≥ 1, our sorting algorithm is effective in this sense, since addition (+) of the natural numbers and comparisons (>) of integers, operations that reside in its initial states, can be programmed from the above-mentioned constructors (0, true, false, undef, c). In particular, partial-recursion for natural numbers and Turing machines for strings form effective models [11]. Furthermore, it is shown in [12] that three prima facie different definitions of effectiveness over arbitrary domains, as proposed in [11, 18, 35], respectively, comprise exactly the same functions, strengthening the conviction that the essence of the underlying notion of computability has in fact been captured. Theorem 4 (Church-Turing Thesis [11]) For every effective model, there is a representation of its domain values as strings, such that its algorithms are each simulated by some Turing machine. Call an effective computational model maximal if adding any function to those that it computes results in a set of functions that cannot be simulated by any effective model. Remarkably (or perhaps not), there is exactly one such model: Theorem 5 (Effectiveness [12, Theorem 4]) The set of partial recursive functions (and likewise the set of Turing-computable string functions) is the unique maximal effective model, up to isomorphism, over any countable domain. Generic Model of Computation 68 We have recently extended the proof of the Church-Turing Thesis and demonstrated the validity of the widely believed Extended Church-Turing Thesis: Theorem 6 (Extended Church-Turing Thesis [17]) Every effective algorithm can be polynomially simulated by a Turing machine. 6 Conclusion We have dealt herein with the classical type of algorithms, that is to say, with the “small-step” (meaning, only bounded parallelism) “sequential-time” (deterministic, no intra-step interaction with the outside world) case. Abstract state machines can faithfully emulate any algorithm in this class, as we have seen in Theorem 2. Furthermore, we have characterized the distinction between effective algorithms and their more abstract siblings in Theorem 4. There are various “declarative” styles of programming for which the state-transition relation is implicit, rather than explicit as it is for our notion of algorithm. For such programs to be algorithms in the sense of Definition 1, they would have to be equipped with a specific execution mechanism, like the one for recursion mentioned above. For Prolog, for example, the mechanism of unification and the mode of search would need to be specified [14]. The abstract-state-machine paradigm can be extended to handle more modern notions: • When desired, an algorithm can make an explicit distinction between successful and failing terminal states by storing particular values in specific locations of the final state. Alternatively, one may declare failure when there is a conflict between two or more enabled assignments. See [24]. • There is no difficulty in allowing for nondeterminism, that is, for a multivalued transition function. If the semantics are such that a choice is made between clashing assignment statements, then transitions are indeed nondeterministic. See [24, 28]. • More general forms of nondeterminism can be obtained by adding a choice command of some sort to the language. See [24]. • Nothing needs to be added to the syntax of ASMs to apply to cases for the environment provides input incrementally. One need only imagine that the environment is allowed to modify the values of some (specified) set of locations in the state between machine steps. See [24]. • In [4, 5, 6], the analysis of algorithms was extended to the case when an algorithm interacts with the outside environment during a step, and execution waits until all queries of the environment have been responded to. • In [8, 9], all forms of interaction are handled. • In [7], the analysis was extended to massively parallel algorithms. • Distributed algorithms are handled in [24, 20]. • The fact that ASMs can emulate algorithms step-for-step facilitates reasoning about the complexity of algorithms, as for Theorem 6 above. 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Commonsense L OCATED N EAR Relation Extraction arXiv:1711.04204v2 [cs.CL] 16 Nov 2017 Frank F. Xu∗, Bill Y. Lin∗ and Kenny Q. Zhu {frankxu, yuchenlin}@sjtu.edu.cn, [email protected] Department of Computer Science and Engineering Shanghai Jiao Tong University, Shanghai, China 1 Introduction Artificial Intelligent systems can benefit from incorporating commonsense knowledge as background, such as ice is cold (H AS P ROPERTY), chewing is a sub-event of eating (H AS S UBEVENT), chair and table are typically found near each other (L OCATED N EAR), etc. This kind of commonsense facts have been utilized in many downstream tasks, such as textual entailment [4, 1] and visual recognition tasks [29]. The commonsense knowledge is often represented as relation triples in commonsense knowledge bases, such as ConceptNet by MIT [20], one of the largest commonsense knowledge graph available today. However, this kind of commonsense knowledge bases are usually manually curated or crowd-sourced by community efforts and thus do not scale well. This paper aims at automatically extracting the commonsense L OCATED N EAR relation between physical objects from textual corpora, which is defined as two objects typically found near each other in real life. We focus on L OCATED N EAR relation for these reasons: (i) L OCATED N EAR facts are helpful prior knowledge for object detection in complex image scenes; Figure 1 illustrates two motivating examples; (ii) such commonsense knowledge can potentially benefit general reasoning in reading comprehension, question answering as well as many other AI tasks; (iii) existing knowledge bases have very few facts for this relation (ConceptNet 5 has only 49 triples of L OCATED N EAR). Figure 1: L OCATED N EAR relation facts assist the detection of vague objects: in a dimly lit room with settings shown in the left sub-figure, if a bright laptop is present on a table, one may guess that a lamp, a photo frame or books maybe nearby. Similarly in the right sub-figure, if a set of knife, fork and plate is on the table, one may believe there could be a glass beside based on the commonsense, even though these objects are hardly visible due to low light. We propose two novel tasks in extracting L OCATED N EAR relation from textual corpora. One is a binary relation classification problem which judges whether or not a sentence is describing two objects physically close by. The other task is to produce a ranked list of L OCATED N EAR facts with the given classified results of large number of sentences. We believe both two tasks can help the community further automatically complete and populate existing commonsense knowledge bases. ∗ The first two authors contribute equally. 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. Additionally, we also create two benchmark datasets for evaluating L OCATED N EAR relation extraction systems on the two tasks: one is 5,000 sentences each describing a scene of two physical objects and with a label indicating if the two objects are co-located in the scene; the other consists of 500 pairs of objects with human-annotated scores indicating confidences that a certain pair of objects are commonly located near in the real life. We propose several methods to solve the tasks including feature-based and LSTM-based neural architecture. The proposed neural architecture compares favorably with the current state-of-the-art method for general-purpose relation classification problem. From our relatively smaller proposed datasets, we extract in total 2,067 new L OCATED N EAR triples that are not in ConceptNet. 2 Sentence-level L OCATED N EAR Relation Classification Given a sentence s mentioning a pair of physical objects <ei , ej >, we call <s, ei , ej > an instance. In this section, we aim to determine whether ei and ej are located near each other in a physical scene described in the sentence s. For example, suppose ei is “dog", ej is “cat”, and s = “The King puts his dog and cat on the table.”. As it is true that the two objects are located near in this sentence, a successful classification model is expected to label this instance as True. While if s2 = “My dog is older than her cat.”, then the answer to the instance <s2 , ei , ej > is False, for s2 is just talking about a general comparison. In the following subsections, we present two different kinds of baseline methods for this binary classification task: feature-based methods and LSTM-based neural architectures. 2.1 Feature-based Methods Our first baseline is an SVM classifier based on following features. We claim that such semantic and syntactic features are widely utilized among existing relation classification models [2, 6, 28, 17]. Note that we put special focus on adverbs and prepositions based on the assumption that these lexical units describing directions and positions in physical world will help identify L OCATED N EAR relations. Proposed features: - Bag of Words (BW) The set of words that ever appeared in the sentence. - Bag of Path Words (BPW) The set of words that appeared on the shortest dependency path between objects ei and ej in the dependency tree of the sentence s, plus the words in the two subtrees rooted at ei and ej in the parse tree. - Bag of Adverbs and Prepositions (BAP) The existence of adverbs and prepositions in the sentence as binary features. - Global Features (GF) The length of the sentence, the number of nouns, verbs, adverbs, adjectives, determiners, prepositions and punctuations in the whole sentence. - Shortest Dependency Path Features (SDP) From the dependency parse tree of the sentence and the shortest path between the two objects ei and ej . - Semantic Similarity Features (SS) The cosine similarity between the pre-trained GloVe word embeddings [16] of the two object words. Obtaining such features for every instances, we then feed processed data into a SVM classifier. We evaluate linear and RBF kernels with different parameter settings, and the RBF kernel with {C = 100, γ = 10−3 } performs the best overall. 2.2 LSTM-based Neural Architectures Long Short Term Memory based recurrent neural architectures (LSTMs) [8] are widely used in relation classification [19, 5, 22, 24]. We observe that the existence of L OCATED N EAR relation in an instance <s,e1 ,e2 > depends on two major information sources: one is from the semantic and syntactical features of sentence s and the other is from the object pair <e1 ,e2 >. By this intuition, we design our LSTM-based model with two parts, shown in Figure 2. The left part is for encoding the syntactical and semantic information of the sentence s, while the right part is encoding the semantic similarity between the pre-trained word embeddings of e1 and e2 . 2 Output confidence σ LSTM Dense Layer Token Vector Representation Position Normalized Sequence Original Sentence DT lead#s lead DT 𝐸1 into PR JJ 𝐸2 -4 -3 -2 -1 0 1 2 3 4 -8 -7 -6 -5 -4 -3 -2 -1 0 The king Token Pre-trained word vectors of 𝐸1 and 𝐸2 Position led the dog into his nice garden . dog garden Figure 2: The proposed LSTM-based model 2.2.1 Sentence Normalization Using the original word sequence as of a sentence s as input has two problems: (i) the irrelevant words in the sentence can take noise into model; (ii) the large vocabulary of original words induce too many parameters, which may cause over-fitting. For example, given two sentences “The king led the dog into his nice garden.” and “A criminal led the dog into a poor garden.”. The object pair is <dog, garden> in both sentences. The two words “lead” and “into” are essential for determining whether the object pair is located near, but they are not given more bias than other words. Also, the semantic differences between irrelevant words, such as “king” and “criminal”, “beautiful” and “poor”, are not useful to the co-location relation between the “dog” and “garden”, and thus tends to act as noise. Level Objects Lemma Dependency Role POS Tag Examples E1 , E2 open, lead, into, ... open#s, open#o, into#o, ... DT, PR, CC, JJ, ... Table 1: Examples of four types of tokens during sentence normalization. (#s represents the subject of given verb or preposition, and #o represents the object) Considering above problems, we propose utilizing POS (Part-of-Speech) tags instead to capture more syntactical information and reduce the vocabulary size. However, solely doing this loses too much semantic dependency between the words. Thus, we propose a normalized sentence representation method merging the three most important and relevant kinds of information about each instance: lemma, POS tags and dependency role 2 . We first replace the two nouns in the object pair as E1 and E2 , keep the lemmatized form of the original words for all the verbs, adverbs and prepositions, which are highly relevant to describing physical scenes. Then, we replace the subjects and direct objects of the verbs and prepositions (nsubj, dobj for verbs and case for prepositions in dependency parse tree) with special tokens indicating their dependency roles. For the remaining words, we simply use their POS tags to replace the originals. The four kinds of tokens are illustrated in Table 1. Table 2 is a real example of our normalized sentence representation, where the object pair of interest is <dog, garden>. The DT king open#s opened open the DT door open#o and CC led lead the DT dog E1 into into his PR Table 2: Sentence Normalization Example 2 We utilize Stanford CoreNLP tool: https://stanfordnlp.github.io/CoreNLP/ 3 nice JJ garden. E2 . 2.2.2 Model Training As shown in Figure 2, the bottom of the figure shows the original sentence, which is transformed to normalized sequence described above. Apart from the normalized tokens of the original sequence, to capture more structural information, we also encode the distance from each token to E1 and E2 . Such word position embeddings (position/distance features) are proposed by [27] with the intuition that information needed to determine the relation between two target nouns normally comes from words which are close to the target nouns. Then, we leverage LSTM to encode the whole sequence of the tokens of normalized representation plus position embedding. In the meantime, two pretrained GloVe word embeddings [16] of the original two physical object words are fed into a hidden dense layer. Finally, we concatenate both outputs and then use sigmoid activation function to obtain the final prediction. We choose to use the widely-used standard binary cross-entropy as our loss function, and RMSProp [7] is used as optimizer. Following [26], we add 0.5 dropout in LSTM as well as embedding layer, and utilize batch normalization [10, 3] for overfitting problem due to relatively small dataset. 3 L OCATED N EAR Relation Extraction Figure 3 shows the overall workflow of our automatic framework to mine LocatedNear relations from raw text. We first construct a vocabulary of physical objects and generate all candidate instances. For each sentence in the corpus, if a pair of physical objects ei and ej appear as nouns in a sentence s, then we apply our L OCATED N EAR relation classifier on this instance. The relation classifier yields a probabilistic score s indicating the confidence of the existence of L OCATED N EAR relation. Finally, all scores of <s,ei ,ej > instances from the corpus are grouped by the object pairs and aggregated, where each object pair is associated with a final score. Such mined physical pairs with scores can easily be integrated into existing commonsense knowledge base. More specifically, for each object pair <ei , ej >, we find all the m sentences in our corpus mentioning both objects. We classify the m instances with the sentence-level relation classifier and get confidences for each instance, feed them into a function f to obtain the final score of the object pair. There are five variants of the scoring functions: m m X 1 X f0 = m, f1 = conf(sk , ei , ej ), f2 = conf(sk , ei , ej ) m k=1 f3 = m X k=1 m 1 X f4 = 1{conf(sk ,ei ,ej )>0.5} m 1{conf(sk ,ei ,ej )>0.5} , k=1 Object Pairs < 𝑠𝑚 , 𝑒𝑖 , 𝑒𝑗 > ... 𝑓 ... 𝑐𝑜𝑛𝑓 < 𝑠𝑚 , 𝑒𝑖 , 𝑒𝑗 > ... 𝑐𝑜𝑛𝑓 < 𝑠1 , 𝑒𝑖 , 𝑒𝑗 > Object co-location classifier ... Corpus Classification Confidence .. .. .. .. < 𝑒𝑖 , 𝑒𝑗 > ... ... < 𝑠1 , 𝑒𝑖 , 𝑒𝑗 > k=1 𝑠𝑐𝑜𝑟𝑒 < 𝑒𝑖 , 𝑒𝑗 > LocatedNear Relation Scores Figure 3: Computing the L OCATED N EAR scores of object pairs 4 Datasets Our proposed vocabulary of single-word physical objects is constructed by the intersection of all entities that belong to “physical object” class in Wikidata and all ConceptNet concepts. We then manually filtered out some words that have the meaning of an abstract concept, which results in 1169 physical objects in total. Afterwards, we utilize a cleaned subset of the Project Gutenberg corpus [11], which contains 3,036 English books written by 142 authors. An assumption here is that sentences in fictions are more 4 Acc. P R F1 Random 0.500 0.551 0.500 0.524 Majority 0.551 0.551 1.000 0.710 SVM 0.584 0.606 0.702 0.650 SVM(-BW) 0.577 0.579 0.675 0.623 SVM(-BPW) 0.556 0.567 0.681 0.619 SVM(-BAP) 0.563 0.573 0.811 0.672 Acc. P R F1 SVM(-SDP) 0.579 0.597 0.728 0.656 SVM(-SS) 0.584 0.605 0.708 0.652 DRNN [22] 0.635 0.658 0.702 0.679 LSTM+Word 0.637 0.635 0.800 0.708 LSTM+POS 0.641 0.650 0.751 0.697 LSTM+Norm 0.653 0.654 0.784 0.713 SVM(-GF) 0.605 0.616 0.751 0.677 Table 3: Performance of baselines on co-location classification task with ablation. (Acc.=Accuracy, P=Precision, R=Recall, “-” means without certain feature) likely to describe real life scenes. We sample and investigate the density of L OCATED N EAR relations in Gutenberg with other widely used corpora, namely Wikipedia, used by Mintz et al. (2009) and New York Times corpus, created by Riedel et al. (2010) and used by Lin et al. (2016), Hoffmann et al. (2011), Surdeanu et al. (2012). In the English Wikipedia dump, out of all sentences which mentions at least two physical objects, 32.4% turn out to be positive. In the New York Times corpus, the percentage of positive sentences is only 25.1%. In contrast, that percentage in the Gutenberg corpus is 55.1%, much higher than the other two corpora, making it a good choice for L OCATED N EAR relation extraction. From this corpus, we identify 15,193 pairs that co-occur in more than 10 sentences. Among these pairs, we randomly select 500 object pairs and 10 sentences with respect to each pair for annotators to label their commonsense L OCATED N EAR. Each instance is labeled by at least three annotators who are college students and proficient with English. The final truth label of a sentence is decided by a majority vote from the four annotators. The Cohen’s Kappa among the three annotators is 0.711 which suggests substantial agreement. We randomly choose 4000 instances as the training set and 1000 as the test set for evaluating the first sentence-level relation classification task. For the second task, we further ask the annotators to label whether each pair of objects are likely to locate near each other in the real world. Majority votes determine the final truth labels. The inter-annotator agreement here is 0.703. Both datasets are made publicly available.3 5 Evaluation 5.1 Sentence-level L OCATED N EAR Relation Classification We evaluate the proposed methods against the state-of-the-art general domain relation classification model (DRNN) [23]. The results are shown in Table 3. For feature-based SVM, we do feature ablation on each of the 6 feature types (Section 2.1). For LSTM-based model, we experiment on variants of input sequence of original sentence. “LSTM+Word” uses the original words as the input tokens, while “LSTM+POS” uses just the POS tag sequence as the input tokens. “LSTM+Norm” uses the tokens of sequence after sentence normalization. 4 From the results, we find that the SVM model without the Global Features performs best, which indicates that bag-of-word features benefit more in shortest dependency paths than on the whole sentence. We find that DRNN performs best (0.658) on precision but not significantly higher than LSTM+Norm (0.654). The experiment also shows that LSTM+Word enjoys the highest recall score. In terms of the overall performance, LSTM+Norm is the best one. One possible reason is that our proposed the normalization representation reduces input sequences’ token vocabulary size, while preserving important syntactical and semantic information. While LSTM+POS also reduces the vocabulary size, it loses too much information. Another reason is that L OCATED N EAR relation are described in sentence mostly with the prepositions/adverbs decorating them, which are the descendants of object word in the dependency tree, other than words merely along the shortest dependency path. Thus, DRNN cannot capture the information from the words belonging to the descendants of the two object words in the tree, while this 3 https://adapt.seiee.sjtu.edu.cn/~frank/location_relation_data.zip Besides, we added two naive baselines: “Random” baseline classifies the instances into two classes with equal probability; “Majority” baseline considers all the instances to be positive. 4 5 f f0 f1 f2 f3 f4 MAP 0.42 0.58 0.48 0.59 0.56 P@50 0.40 0.70 0.56 0.68 0.40 P@100 0.44 0.60 0.52 0.63 0.48 P@200 0.42 0.53 0.49 0.55 0.50 P@300 0.38 0.44 0.42 0.44 0.42 Table 4: Ranking performances of the 5 scoring methods. information is captured by LSTM+Norm. For the rest of the experiments, we will use LSTM+Norm as the classifier of our choice. 5.2 L OCATED N EAR Relation Extraction Once we have classified the sentences using LSTM+Norm, we can extract L OCATED N EAR relation using the four scoring functions in Section 3. We first present the quantitative results. We use each of the scoring functions to rank the 500 commonsense L OCATED N EAR object pairs described in Section 3. Table 4 shows the ranking results using Mean Average Precision (MAP) and Precision at K as metric. Accumulative scores (f1 and f3 ) generally do better. (door, room) (ship, sea) (fire, wood) (fire, smoke) (book, table) (boy, girl) (house, garden) (house, fire) (door, hall) (fruit, tree) (cup, tea) (arm, leg) (horse, saddle) (door, street) (table, chair) Table 5: Top object pairs returned by best performing scoring function f3 Qualitatively, we show 15 object pairs with some of the highest f3 scores in Table 5. Setting a threshold of 40.0 for f3 , which is the minimum non-zero f3 score for all true object pairs in the L OCATED N EAR object pairs data set (500 pairs), we obtain a total of 2,067 L OCATED N EAR relations, with a precision of 68% by human inspection. 6 Related Work Classifying relations between entities in a certain sentence plays a key role in NLP applications and thus has been a hot research topic recently. Feature-based methods [6] and neural network techniques [19, 5] are most common. Xu et al. (2015) introduce multi-channel SDP-based LSTM model to classify relations incooperating several different kinds of information of a sentence improved by Xu et al. (2016), which performed best on SemEval-2010 Task 8 and is one of our baseline methods. The most related work to ours is the extraction of visual commonsense knowledge by Yatskar et al. (2016). This work learns the textual representation of seven types of fine-grained visual relations using textual caption for the image in MS-COCO dataset [13]. Another important related work is from Li et al. (2016), which enriches several popular relations in ConceptNet with little textual information from real large corpora. However, L OCATED N EAR relation was not studied in this work, while this relation is extremely scarce in ConceptNet and has its own distinctiveness. 7 Conclusion We presented a novel study on enriching L OCATED N EAR relationship from textual corpora. Based on our two newly-collected benchmark datasets, we proposed several methods to solve the sentence-level relation classification problem. We showed that existing methods do not work as well on this task and discovered that LSTM-based model does not have significant edge over simpler feature-based model. Whereas, our multi-level sentence normalization turns out to be useful. 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Continuous-time GARCH process driven by semi-Lévy process arXiv:1803.00733v1 [math.ST] 2 Mar 2018 M. Mohammadi ∗ S. Rezakhah∗ N. Modarresi† March 5, 2018 Abstract In this paper we study the simple semi-Lévy driven continuous-time generalized autoregressive conditionally heteroscedastic (SS-COGARCH) process. The statistical properties of this process are characterized. This process has the potential to approximate any semi-Lévy driven COGARCH processes. We show that the state representation of such SS-COGARCH process can be described by a random recurrence equation with periodic random coefficients. The almost sure absolute convergence of the state process is proved. The periodically stationary solution of the state process is shown which cause the volatility to be periodically stationary under some suitable conditions. Also it is shown that the increments with constant length of such SS-COGARCH process is itself a periodically correlated (PC) process. Finally, we apply some test to investigate the PC behavior of the increments (with constant length) of the simulated samples of proposed SS-COGARCH process. Keywords: Continuous-time GARCH process; Semi-Lévy process; Periodically correlated; Periodically stationary. 1 Introduction Many financial data and indices have heteroscedastic structure. Examples of this kind are stocks returns, network traffic and natural data, see [4, 18, 16]. Popular model for these data are autoregressive conditionally heteroscedastic (ARCH) model proposed by Engle [13] and generalized ARCH (GARCH), Bollerslev [3]. The GARCH type processes have become the most popular tools to model heteroscedasticity in discrete time. ∗ Faculty of Mathematics and Computer Science, Amirkabir University of Technology, 424 Hafez Avenue, Tehran 15914, Iran. E-mail: [email protected](M. Mohammadi) and [email protected](S. Rezakhah). † Department of Mathematics and computer science, Allameh Tabatabai University, Tehran, Iran. E-mail: [email protected](N. Modarresi). 1 In practice, for various reasons such as high-frequency data, many time series are irregularly spaced and this has created a demand for continuous-time models, [8]. For the first time, Kluppelberg et al. [17] introduced a continuous-time version of the GARCH(1,1) (COGARCH(1,1)) process, which preserves the essential features of the discrete-time GARCH(1,1) processes. They replaced the noise of the discrete-time GARCH(1,1) process with the increments of some Lévy process. The volatility of this process satisfies a stochastic differential equation. They proved the stationarity property and also second order properties under some regularity conditions on the corresponding Lévy process. Brockwell et al. [8] generalized the Lévy driven COGARCH(1,1) process to the Lévy driven COGARCH(p, q) process for q ≥ p ≥ 1 when its volatility is a continuous-time ARMA (CARMA) process [7]. They showed that the state representation of the volatility can be expressed as a stochastic recurrence equation with random coefficients. Periodic behavior is common in many real-world time series such as power market prices, car accident claims for an insurance company and sales with seasonal interest. The term periodically correlated (PC) was introduced by Gladyshev [14], but the same property was introduced by Bennett [1] who called them cyclostationary ([15]). Properties of PC processes are studies by Hurd and Miamee [15]. Bibi and Lescheb [2] studied the class of bilinear processes with periodic time-varying coefficients of periodic ARMA and periodic GARCH models. Lévy processes introduced by Lévy have stationary and independent increments and right continuous paths with left limits [21]. Such processes have potential to be applied to financial data following stochastic volatility structure. A generalization of Lévy process is semi-Lévy process, that has periodically stationary increments, studied by Maejima and sato [19]. We considered this process as the underlying process in CARMA [7] and COGARCH [8, 17] processes that can be applied when there is evident that the underlying process has PC increments. The observations of such processes have significant dependency to the ones of previous periods. So semi-Lévy process are more prominent than Lévy processes in such cases. In this paper we introduce a COGARCH process driven by some simple semi-Lévy process, which we call SS-COGARCH process. The simple semi-Levy process is defined as a compound Poisson process with periodic time-varying intensity with period τ . This process enables us to provide the statistical properties of the SS-COGARCH process. Moreover, we find a random recurrence equation with periodic random coefficients for the state representation of such process. By some regularity condition we show the absolute convergence of the state equation. We also show that the volatility of the SS-COGARCH process is strictly periodically stationary. The increments of the SS-COGARCH process with constant length h = τ /% where % is some integer is a discrete-time PC process with period %. Such SS-COGARCH process has the potential to provide an approximation for every semi-Lévy driven COGARCH process. Finally, we investigate the theoretical results concerning PC structure of the increment process by simulation. We show that the increments of the SS-COGARCH process with length h is PC with some period % and the support of the squared coherence statistics consists of lines parallel to the main 2 diagonal and having spacing of 2π/%. This paper is organized as follows. In section 2 we introduce the simple semi-Levy driven COGARCH processes. For this, we present the simple semi-Levy process and obtain the characteristic function of it. Section 3 is devoted to some sufficient conditions which make the volatility process strictly periodically stationary. We obtain the mean, covariance function of the state process and volatility process in section 4. We also investigate second order properties of the squared increments of the COGARCH process in this section. In section 5 we illustrate the results with simulations. All proofs are contained in Section 6. 2 Simple semi-Lévy driven COGARCH processes In this section we study the preliminaries such as the additive processes and their characteristic functions and semi-Lévy process in subsection 2.1. We also describe the structure of simple semi-Lévy process and characteristics it in subsection 2.2. Then we introduce the simple semi-Lévy driven COGARCH (SS-COGARCH) process in subsection 2.3. 2.1 Preliminaries Let (Ω, F, (Ft )t≥0 , P) be a filtered probability space, where Ft is the smallest rightcontinuous filtration such that F0 contains all the P-null sets of F. A process (Xt )t≥0 defined on the probability space (Ω, F, (Ft )t≥0 , P) is called an additive process if X0 = 0 a.s., it is stochastically continuous, it has independent increments and its sample paths are right-continuous and have left limits in t > 0. Further, if (Xt )t≥0 has stationary increments, it is a Lévy process [11, 21]. The characteristic function of the additive process (Xt )t≥0 has a following Lévy-Khinchin representation [21, Theorems 9.1-9.8]. Theorem 2.1 Let (Xt )t≥0 be an additive process on Rd . Then (Xt )t≥0 has infinitely divisible distribution for t ≥ 0. The law of (Xt )t≥0 is uniquely determined by its spot characteristic triplet (Γt , Πt , ψt )t≥0 E[ei<w,Xt > ] = eϕt (w) , 1 ϕt (w) = i < w, Γt > − < w, Πt w > + 2 Z w ∈ Rd , (ei<w,x> − 1 − i < w, x > I{\|\|x\|\|≤1} )ψt (dx). Rd where < ·, · > is inner product and \|\| · \|\|Ris Euclidean vector norm. The spot Lévy measure ψt satisfies the integrability condition Rd min{1, \|\|x\|\|2 }ψt (dx) < ∞ for t ≥ 0. Remark 2.1 By [11, p.458-459], the spot characteristic triplet (Γt , Πt , ψt )t∈[0,T ] can be 3 defined by Z t Γt = γs ds 0 Z t Πt = σs2 ds Z0 t υs (B)ds, ψt (B) = ∀B ∈ Rd , 0 where Rd is σ−field on the Rd . The triplet (γt , σt2 , υt )t∈[0,T ] is called the local characteristic triplet of (Xt )0≤t≤T which satisfy the following conditions: • γt : [0, T ] → Rd is a deterministic function with finite variation. • σt : [0, T ] → Md×d (R) is a symmetric, continuous and matrix valued function which RT verifies 0 σt2 dt < ∞. • (υt )t∈[0,T ] is a family of Lévy measures which verifies Z TZ min{1, \|\|x\|\|2 }υt (dx)dt < ∞. 0 Rd As an extension of Lévy process, we present the definition of semi-Lévy processes [19]. Definition 2.1 A subclass of additive processes (Xt )t≥0 is called semi-Lévy process with period τ > 0, if for any 0 ≤ s ≤ t, d Xt − Xs = Xt+τ − Xs+τ d where = denotes the equality in distributions. 2.2 Structure of simple semi-Lévy process For describing the structure of the simple semi-Lévy process, we define the general structure of the intensities function of the Poisson process with periodically stationary increments. We also characterize this pure jump process by representation the characteristic function and introduce the corresponding semi-Lévy measure. Definition 2.2 : Poisson process with periodically stationary increment A process N (t) t≥0 is a Poisson process with periodically stationary increment where E N (t) = Λ(t), Z t Λ(t) = λ(u)du (2.1) 0 and the intensity λ(·) is a periodic non-negative function with some period τ > 0, so λ(t) = λ(t + kτ ) for t ≥ 0, k ∈ N. 4 Definition 2.3 : Simple compound Poisson process Let 0 = t0 < t1 < · · · be a partition of positive real line. Also assume that Aj = Pl [tj−1 , tj ), j ∈ N, and \|Aj \| = \|Aj+l \| for some integer l ∈ N and τ = j=1 \|Aj \|. Let N (t) t≥0 be a Poisson process which has periodically stationary increments with period τ > 0 and intensity function Λ(t) defined by (2.1). Then the simple compound Poisson process (St )t≥0 is defined as S t = Dt + N (t) X Zn (2.2) n=1 P S where Zn = lj=1 Znj I{Υn ∈Dj } , Υn is the arrival time of nth jump Zn , Dj = ∞ A and R 2 k=0 j+kl j Zn are independent and have distribution Fj , j = 1, · · · , l, such that R z Fj (dz) < ∞ for j = 1, · · · , l. Also Dt , t > 0, is a deterministic drift function with period τ , say Dt = Dt+τ , and D0 = 0. One can easily verify that (St )t≥0 has independent increment. Now we find characteristic function of the simple compound Poisson process (St )t≥0 by the following Lemma. Lemma 2.2 Let (N (t))t≥0 be a Poisson process with periodically stationary increment and mean Λ(t), defined by (2.1). Then the process (St )t≥0 defined by (2.2) has the following characteristic function for t ≥ 0 E[eiwSt ] = eϕt (w) , Z ϕt (w) = iwΓt + (eiwz − 1 − iwz I{\|z \|≤1} )ψt (dz ), R where Γt = Dt + m−1 l Z XX k=0 r=1 + j−1 Z X Zr=1 z Λ(tkl+r ) − Λ(tkl+r−1 ) Fr (dz) \|z\|≤1 z Λ(tml+r ) − Λ(tml+r−1 ) Fr (dz) \|z\|≤1 + z Λ(t) − Λ(tml+j−1 ) Fj (dz), \|z\|≤1 and ψt (dz) = m−1 l XX Λ(tkl+r ) − Λ(tkl+r−1 ) Fr (dz) k=0 r=1 j−1 + X Λ(tml+r ) − Λ(tml+r−1 ) Fr (dz) r=1 + Λ(t) − Λ(tml+j−1 ) Fj (dz), where m = [ τt ] and (t − mτ ) ∈ Aj for some j = 1, · · · , l. 5 Proof: see Appendix, P1. Remark 2.2 By Remark 2.1, Lemma 2.2 and (2.1), the spot characteristic triplet of process (St )0≤t≤T , (Γt , 0, ψt )t∈[0,T ] , have the local characteristic triplet (γs , 0, υs )s∈[0,T ] which has the following form m−1 l Z XX γs = dDs + zλ(s)I[tkl+r−1 ,tkl+r ) (s)Fr (dz) \|z\|≤1 k=0 r=1 + j−1 Z X r=1 zλ(s)I[tml+r−1 ,tml+r ) (s)Fr (dz) \|z\|≤1 Z zλ(s)I[tml+j−1 ,t] (s)Fj (dz), + \|z\|≤1 and υs (dz) = m−1 l XX λ(s)I[tkl+r−1 ,tkl+r ) (s)Fr (dz) k=0 r=1 j−1 + X λ(s)I[tml+r−1 ,tml+r ) (s)Fr (dz) r=1 + λ(s)I[tml+j−1 ,t] (s)Fj (dz). (2.3) It follows from definition 2.3 and Remark 2.2 that the family (υs )s∈[0,T ] of semi-Lévy measures verify Z TZ \|z\|2 υs (dz)ds < ∞. (2.4) 0 R This implies that (St )t≥0 is semi-martingale, so it has Lévy-Ito decomposition and has quadratic variation process [11, p.459-460]. Corollary 2.3 By lemma 2.2, the stochastic process (St )t≥0 defined by (2.2) is a semiLévy process with period τ. Proof: see Appendix, P2. 2.3 Structure of simple semi-Lévy driven COGARCH process Let (St )t≥0 be a simple semi-Lévy process with period τ defined by (2.2). Process (Gt )t≥0 with parameters α0 > 0, α1 , · · · , αp ∈ R, β1 , · · · , βq ∈ R, αp 6= 0, βq 6= 0, and αp+1 = · · · αq = 0 is a simple semi- Lévy √ driven COGARCH(p,q) process (SS-COGARCH(p,q)), q ≥ p ≥ 1, defined by dGt = Vt dSt or equivalently Z tp Gt = Vu dSu , t > 0, G0 = 0, (2.5) 0 6 in which the left-continuous volatility process (Vt )t≥0 is defined by Vt = α0 + a0 Yt− , t > 0, V0 = α0 + a0 Y0 , (2.6) where the state process (Yt )t≥0 is the unique càdlàg solution of the stochastic differential equation dYt = BYt− dt + e(α0 + a0 Yt− )d[S, S]t , t > 0, (2.7) d denotes differentiation with respect to t. The initial value Y0 is F0 -measurable and independent of the driving semi-Lévy process (St )t≥0 , and       0 1 0 ··· 0 0 α1  0  0 0 1 · · · 0   α2     ..    . . . . . . . . . B= . (2.8)  , a =  ..  , e =  ..  . . . . .     .    0  0 0 ··· 1 0 αq −βq −βq−1 −βq−2 · · · −β1 1 3 Periodic stationarity conditions In this section we provide some conditions to prove that the volatility process (Vt )t≥0 defined by (2.6) is strictly periodically stationary with period τ . As a result of main theorem, we prove that the increments with constant length of process (Gt )t≥0 is itself a periodically correlated (PC) process which is the mian aim of this paper. We also give a sufficient an necessary condition by which we can determine the volatility is non-negative. In the following theorem in (b) a Lr −matrix norm of the (q × q)-matrix C is defined as kCkr = kCckr . c∈Cq \{0} kckr sup Theorem 3.1 (a) Let (Yt )t≥0 be the state process of the SS-COGARCH(p,q) process with parameters B, a and α0 defined by (2.5). Suppose that (St )t≥0 be a simple semi-Lévy process defined by (2.2). Then for all 0 ≤ s ≤ t Yt = Js,t Ys + Ks,t , (3.1) where (Js,t , Ks,t )0≤s≤t is a family of random (q × q)−matrix Js,t and random vector Ks,t in q R . In addition, Js+kτ,t+kτ , Ks+kτ,t+kτ k∈N0 are independent and identically distributed. (b) Let ηi , i = 1, · · · , q, be the eigenvalues of invertible matrix B which have strictly negative real parts. Also suppose that exists one r ∈ [1, ∞] such that Z 1 log 1 + \|\|P −1 ea0 P \|\|r z 2 dνt (z) < − νt (R)ητ, ∀t ∈ [0, τ ), (3.2) Λ(t + τ ) − Λ(t) R 7 where P is a matrix in which P −1 BP is diagonal and η := maxi=1,··· ,q ηi and (νt )t≥0 is semi-Lévy measure defined by (2.3). Then Yt+mτ converges in distribution to a finite random vector U(t) for fixed t ∈ [0, τ ), as m goes to infinity. The distribution of the vector U(t) is the unique solution of the random equation d where U(t) U(t) = Jt,t+τ U(t) + Kt,t+τ , is independent of Jt,t+τ , Kt,t+τ . (3.3) d (c) Let the conditions of (b) hold and Y0 = U(0) , Then (Yt )t≥0 and (Vt )t≥0 are strictly periodically stationary with period τ . In the other hands, for any s1 , s2 , · · · , sn ≥ 0 and Borel sets E1 , E2 , · · · , En of Rd and Borel sets J1 , J2 , · · · , Jn of R and k ∈ N, P Ys1 ∈ E1 , Ys2 ∈ E2 , · · · , Ysn ∈ En = P Ys1 +kτ ∈ E1 , Ys2 +kτ ∈ E2 , · · · , Ysn +kτ ∈ En , and P Vs1 ∈ J1 , Vs2 ∈ J2 , · · · , Vsn ∈ Jn = P Vs1 +kτ ∈ J1 , Vs2 +kτ ∈ J2 , · · · , Vsn +kτ ∈ Jn . Proof: see Appendix P3. In the following remark we describe the non-negativity of the Lyapunov exponent which leads to the absolutely convergence of the state process (Yt )t≥0 in Theorem 3.1. Remark 3.1 (a) The proof of Theorem 3.1 will be based on the use of the general theory of multivariate random recurrence equations, as discussed by Bougerol and Picard [5], Brandt [6] and Vervaat [22] in the one dimensional case. The state vector (Yt )t≥0 defined by (2.7) satisfies multivariate random recurrence equation. (b) The condition (3.2) which provides the stability of the model based on the existence of a vector norm \|\| · \|\|r such that Jt,t+τ and Kt,t+τ for all t ∈ [0, τ ) satisfy the conditions E log + \|\|Kt,t+τ \|\|r < ∞ (3.4) E log\|\|Jt,t+τ \|\|r < 0, where log + (x) = log(max{1, x}). E log\|\|Jt,t+τ \|\| < 0 is equivalent to the assertion that r the Lyapunov exponent of the Jt+kτ,t+(k+1)τ k∈N0 is strictly negative almost surely. i.e. 1 lim sup log\|\|Jt,t+τ · · · Jt+kτ,t+(k+1)τ \|\|r < 0, k−→∞ k a.s. (c) The conditions of Theorem 3.1 imply (3.4) with the natural matrix norm \|\|A\|\|B,r = \|\|P −1 AP \|\|r , for some matrix A, which corresponds to the following the natural vector norm \|\|c\|\|B,r := \|\|P −1 c\|\|r , where P is a matrix in which P −1 AP is diagonal. 8 c ∈ Cq Corollary 3.2 If (Vt )t≥0 is a strictly periodically stationary process with period τ , then increments with constant length of the process (Gt )t≥0 make a PC process. In the other words, for any t ≥ 0 and h ≥ p > 0 and k ∈ N, (p) (p) E Gt = E Gt+kτ , (p) (p) (p) (p) cov Gt , Gt+h = cov Gt+kτ , Gt+h+kτ , (p) where Gt := R t+p √ t Vs dSs . Proof: see Appendix P4. Theorem 3.3 Let (Yt )t≥0 be the state process of the SS-COGARCH(p,q) process (Gt )t≥0 with parameters B, a and α0 > 0. Suppose that γ ≥ −α0 is a real constant and the following two conditions hold: a0 eBt e ≥ 0 ∀t ≥ 0, a0 eBt Y0 ≥ γ a.s. ∀t ≥ 0. (3.5) (3.6) Then, with probability one, V t ≥ α0 + γ ≥ 0 ∀t ≥ 0. Conversely, if either (3.6) fails, or (3.6) holds with γ > −α0 and (3.5) fails, then there exists a simple semi-Lévy process (St )t≥0 and t0 ≥ 0 such that P (Vt0 < 0) > 0. The proof of the non-negativity volatility process (Vt )t≥0 is similar to the proof of Theorem 5.1 in [8] for Lévy process. 4 Characterization of the state process The aim of this section is to study expected value and covariance function of the state process {Yt : t ≥ 0} and volatility process {Vt : t ≥ 0}. First, we prove that by some sufficient conditions the expected value and covariance Yt exist. Then, by presenting the first and second moments of the random vector U (0) , we find the expected value and covariance function of the state process. Furthermore, a closed form for square increments of the COGARCH process is characterized. c Lemma 4.1 Let the assumptios of Theorem 3.1 hold. If E \|\|Y0 \|\|r < ∞, for c = 1, 2, then (a) If E(St2 ) < ∞, then E(Yt ) < ∞ and E(U(0) ) < ∞. (b) If E(St4 ) < ∞, then cov(Yt ) < ∞ and cov(U(0) ) < ∞. where {St : t ≥ 0} is the simple semi-Levy process. 9 Proof: see Appendix P5. Remark 4.1 By Theorem 3.1(b), (a) we find that E(U(0) ) is the solution of the following random equation I − E(J0,τ ) E U(0) = E(K0,τ ), and (b) E (U(0) )(U(0) )0 is the solution of the following equation Iq2 − E(J0,τ ⊗ J0,τ ) vec E (U(0) )(U(0) )0 = E(K0,τ ⊗ J0,τ ) + E(J0,τ ⊗ K0,τ ) E(U(0) ) + vec E(K0,τ K00,τ ) , where ⊗ is the Kronecker product of two matrices and for a matrix C, vec(C) is the 2 column vector in Cq which is constructed by stacking the columns of matrix A in a vector. The following lemmas establish the the mean and covariance function of the state process. Lemma 4.2 Suppose that {Yt : t ≥ 0} be the state process and the conditions of Theorem 0 3.1 and lemma 4.1 hold. Then for t, h ≥ 0, there exists m, n ∈ N and t1 , t2 ∈ [0, τ ) such that t ∈ mτ, (m + 1)τ , t + h ∈ nτ, (n + 1)τ , t = t1 + mτ and t + h = t2 + nτ and E(Yt ) = E(J0,t1 )E(U) + E(K0,t1 ), (4.1) m−n−1 h cov(Yt , Yt+h ) = E(J0,t2 ) E(J0,τ ) E(J0,τ E(UU0 )J00,t1 ) − E(J0,τ )E(U)E(U0 )E(J00,t1 ) + E(J0,τ E(U)K00,t1 ) − E(J0,τ )E(U)E(K00,t1 ) 0 + E(K0,τ E(U0 )J00,t1 ) − E(K0,τ )E(U )E(J00,t1 ) i 0 0 + E(K0,τ K0,t1 ) − E(K0,τ )E(K0,t1 ) . (4.2) Proof: see Appendix P6. Corollary 4.3 Let {Vt : t ≥ 0} be the volatility process. Then for t, h ≥ 0, expected value and covariance function of Vt have the following forms. E(Vt ) = α0 + a0 E(Yt ) cov(Vt , Vt+h ) = a0 cov(Yt , Yt+h )a. Proof: see Appendix P7. In financial time series, the returns have negligible correlation while the squared returns are significantly correlated, therefore we investigate the behavior of the second-order properties of the increments of the COGARCH process. We assume that volatility process is strictly periodically stationary and non-negative. (p) Now we present the first and second orders of the increment process Gt that in defined in Corollary 4.2. 10 Proposition 4.4 Let G be a zero mean simple semi-Levy driven COGARCH process. Then for t ≥ 0 and h ≥ p > 0, (a) (p) E(Gt = 0, (p) (p) Gt , Gt+h cov (4.3) = 0. (4.4) (b) There exist m, m0 ∈ N0 where t ∈ Aml+i , i = 1, · · · , l and t + p ∈ A(m+m0 )l+i0 , i0 = i, · · · , l, then Z Z t+p tml+i (p) 2 2 2 E Vs λ(s)ds E Vs λ(s)ds + E (Zi0 ) E (Gt ) = E (Zi ) t(m+m0 )l+i0 −1 t m0 l+i0 −2 + X 2 E (Zr+1 ) Z tml+r+1 E Vs λ(s)ds. (4.5) tml+r r=i Moreover, there exist n, n0 ∈ N0 (n ≥ m and n0 ≥ m0 ) where t + h ∈ Anl+j , j = 1, · · · , l and t + h + p ∈ A(n+n0 )l+j 0 , j 0 = j, · · · , l, then cov (p) (p) (Gt )2 , (Gt+h )2 2 0 Z tnl+j = E (Zj ) a (p) λ(s)E(Jt+p,s− )cov (Gt )2 , Yt+p ds t+h + E (Zj 0 )2 a0 Z t+h+p (p) λ(s)E(Jt+p,s− )cov (Gt )2 , Yt+p ds t(n+n0 )l+j 0 −1 n0 l+j 0 −2 + X 2 0 Z tnl+r+1 E (Zr+1 ) a (p) λ(s)E(Jt+p,s− )cov (Gt )2 , Yt+p ds. tnl+r r=j (4.6) Proof: see Appendix P8. Remark 4.2 For s ≥ 0, if we assume that s ≥ 0, R R z 3 νs (dz) = 0, Then (p) cov (Gt )2 , Yt+p = 2E(It+p Yt+p ) − 2E(Jt,t+p )E(It Yt ) Z t+p − cov(Yt+p ) + cov(Yt+p , Yt ) − cov(Yt+p , Ys− )dsB 0 e, t+ where It := Rt 0 √ Gs− Vs dSs and Z E(It Yt ) = B t Z tZ E(Is Ys )ds + α0 0 0 11 R E(Is Ys )z 2 νs (dz)ds. 5 Simulation In this section we simulate the simple semi-Lévy process defined by (2.2). This process is a compound Poisson process with time-varying arrival rate Λ(t) defined by (2.1). Then we verify the theoretical results concerning PC structure of the increments of the SSCOGARCH(p,q) process (Gt )t≥0 defined by (2.5) by simulation. For this we simulate the state process (Yt )t≥0 defined by (2.7) at jump time points and non-jump time points using its random recurrence equation (3.1). Then we evaluate the discretized version of the volatility process (Vt )t≥0 defined by (2.6) and corresponding the SS-COGARCH(p,q) process (Gt )t≥0 . Finally, we verify the PC structure of the increments of the SS-COGARCH process by following the method of [12]. For simulating the simple semi-Lévy process defined by (2.2) with the underlying Poisson process N (t) t≥0 , we consider T1 as the time of the first jump and Tn , n = 2, 3, · · · the P time intervals between the (n − 1)th and nth jumps. Then Υn = nj=1 Tj , n ∈ N, are the arrival times and Υ0 = 0. Therefore for j = 1, 2, · · · FTsn (x) := P Tn ≤ x Υn−1 = s = 1 − P N (s + x) − N (s) = 0 = 1 − e−Λ(s+x)+Λ(s) , (5.1) where Λ(t) defined by (2.1). The arrival times Υ1 , Υ2 , · · · are generated by the following algorithm. 1. Generate the independent and identically distributed (iid) sequence U1 , U2 , · · · from Uniform (0,1). Then by (5.1) as Υ0 = 0 the first arrival time Υ1 = T1 has distribution FΥ0 1 (x) = 1 − e−Λ(x) . Therefore d Λ(Υ1 ) = −ln(1 − U ), where U denotes a Uniform (0,1). So by generating U1 , Υ1 = Λ−1 − ln(1 − U1 ) can be considered as a generated sample for the first arrival time. If Υn−1 , n = 2, 3, · · · , Υ is the (n − 1)th evaluated arrival time, then by (5.1) Tn has distribution FTnn−1 (x) = 1 − e−Λ(x+Υn−1 )+Λ(Υn−1 ) . Therefore d Λ(Υn ) = Λ(Υn−1 ) − ln(1 − U ). So by generating Un , Υn = Λ−1 Λ(Υn−1 ) − ln(1 − Un ) is a generated sample for the nth arravial time. Thus applying the iid sample U1 , U2 , · · · we can evaluate successively the nth arrival time by the (5.1), for the details see [10, p.99]. So by having the periodic intensity function λ(u) in (2.1), one can evaluate Λ−1 (·) by available software. 2. Consider some periodic drift function Ht and as the successive jump size Zn generate independently and has distribution Fn (·) if corresponding arrival time belongs to Dj = S∞ A , j = 1, 2, · · · , l. Now evaluate the simple semi-Lévy process (St )t≥0 from (2.2) k=0 j+kl 12 as St = Ht + N (t) l X X Znj I{Υn ∈Dj } . (5.2) n=1 j=1 Now we consider the following steps for the simulation of the SS-COGARH(p,q) process defined by (2.5)-(2.7). 1. Consider p and q as some integer such that q ≥ p ≥ 1. 2. Choose real parameters β1 , · · · , βq and α1 , · · · , αp and α0 > 0 such that the eigenvalues of the matrix B defined by (2.8) have strictly negative real parts and conditions (3.2), (3.5) and (3.6) are satisfied. 3. Having evaluated arrival times Υn by the above algorithm, generate the state process (YΥn )n∈N by the following the recurrence equation after assuming some initial value for YΥ0 2 B Υn −Υn−1 0 B Υn −Υn−1 YΥn = e YΥn−1 + e α0 + a e YΥn−1 Zn , n ∈ N. This recurrence equation obtained by replacing s = Υn−1 and t = Υn in (3.1). The jump size Zn can be simulated by (5.2) for predefined distributions µ1 , µ2 , · · · , µl . 4. As the simple semi-Lévy process (St )t≥0 (2.2) has no jump over [Υn−1 , Υ− n ], n ∈ N, − ]. Therefore for t ∈ [Υ it follows from (2.7) that dYt = BYt dt for t ∈ [Υn−1 , Υ− n−1 , Υn ] n e−Bt dYt = Be−Bt Yt dt so that d e−Bt Yt = 0. From this follows that for t ∈ [Υn−1 , Υ− n] Z t d e−Bu Yu = 0 Υn−1 hence Yt = eB(t−Υn−1 ) YΥn−1 . (5.3) By (2.6) and (5.3), the discrete-time version of the process (Vt )t≥0 is as VΥn = α0 + a0 YΥ−n = α0 + a0 eB(Υn −Υn−1 ) YΥn−1 , 13 (5.4) and using that the process (St )t≥0 (2.2) has one jump at time Υn over [Υn−1 , Υn ] it follows from (2.5) that Z Υn p Z Υn−1 p GΥn − GΥn−1 = Vu dSu − Vu dSu 0 0 Z Υn p Vu dSu = Υn−1 p = VΥn Zn . (5.5) 5. Having evaluated values of the process (YΥn )n∈N and G0 = 0, generate the process (VΥn )n∈N by (5.4) and corresponding the process (GΥn )n∈N by (5.5). 6. Finally, using the values of VΥn and GΥn provided by previous step, evaluate the sampled processes (Vih )i∈N and (Gih )i∈N for some h > 0 by the followings: (i) Suppose that ih ∈ [Υn−1 , Υn ), for i, n ∈ N. Since the simple semi-Lévy process (St )t≥0 (2.2) has no jump over [Υn−1 , Υn ), it follows from (2.6) and (5.3) that for ih ∈ [Υn−1 , Υn ) Vih = α0 + a0 Yih− = α0 + a0 eB(ih−Υn−1 ) YΥn−1 , note that if ih = Υn−1 , then it follows from step 4 that Vih = α0 + a0 YΥ−n−1 = α0 + a0 eB(Υn−1 −Υn−2 ) YΥn−2 . (ii) Using that the process (St )t≥0 (2.2) has no jump over [Υn−1 , ih] it follows from (2.5) that Z ih Gih − GΥn−1 = Z p Vu dSu − 0 Z Υn−1 p Vu dSu 0 ih = p Vu dSu = 0 Υn−1 hence Gih = GΥn−1 . 5.1 Test for the PC Structure of the increments process To detect the PC structure of a process, Hurd and Miamee [15] and Dudek et al. [12] showed that their proposed spectral coherence can be used to test whether a discrete-time 14 process is PC. Their method is based on the fact that the support of the spectral coherence of a PC process with period % is contained in the subset of parallel lines λs = λr + 2jπ/% for j = −(% − 1), · · · , −1, 0, 1, · · · , (% − 1). The squared coherence statistic for the series X1 , X2 , · · · , XN is computed as follows PM −1 X̃N (λr−M/2+m )X̃N (λs−M/2+m )\|2 \|γ̂(λr , λs , M )\| = PM −1 PM −1 2 2 m=1 \|X̃N (λr−M/2+m )\| m=1 \|X̃N (λs−M/2+m )\| 2 \| m=1 P −iλj k where X̃N (λj ) = N is discrete Fourier transform of Xk for j = 0, 1, · · · , N −1, k=1 Xk e λj = 2πj/N and λj ∈ (0, 2π]. This statistic satisfies 0 ≤ \|γ̂(λr , λs , M )\|2 ≤ 1. Under the null hypothesis that X̃N (λj ) are complex Gaussian with uncorrelated real and imaginary parts for each j, squared coherence statistic has probability density, [15] p(\|γ\|2 ) = (M − 1) 1 − \|γ\|2 M −2 , 0 ≤ \|γ\|2 ≤ 1. For type I error α, the squared coherence α-threshold is determined, [15] xα := \|γ\|2α = 1 − elog(α)/(M −1) . The values of statistic \|γ̂(λr , λs , M )\|2 are computed for all r and s that pair (λr , λs ) ∈ [0, 2π) × [0, 2π). By plotting the values of statistic that exceed the α−threshold, if there are some significant values of statistic that lie along the parallel equally spaced diagonal lines, then Xk is PC. The graph of these significant values indicates the presence of the subset of parallel lines s = r + jN/% for j = −(% − 1), · · · , −1, 0, 1, · · · , (% − 1). To ensure that periodic structure of the series Xk is not a consequence of a periodic mean, it is recommended to remove the periodic mean from this series first. Example 5.1 Let (St )t≥0 be a simple semi-Lévy process by the rate function λ(t) = −cos( π6 t) +4, for t ≥ 0. Furthermore, τ = 12, l = 5 and the length of the successive partitions of each period intervals are 2, 2, 2, 3, 3. Moreover, the distribution of jumps size on these subintervals are assumed to be N (3, 1), N (0, 1), N (1.25, 1.25), N (4, 1), and N (0, 1.5), where N (µ, σ 2 ) denotes a Normal distribution with mean µ and variance σ 2 . In this example we consider SS-COGARCH(1,3) process with parameters of α0 = 1, α1 = 0.03, β1 = 5, β2 = 9 and β3 = 5. Thus, the matrix B is   0 1 0 0 1 B= 0 −5 −9 −5 and conditions (3.2), (3.5) and (3.6) are satisfy. For such the SS-COGARCH process we simulate GΥn for the duration of 40 period intervals with the parameters specified above, Y0 = (8.3580, 2.3377, 0.9040)0 and G0 = 0. Then, using step 6, we sample from this process in equally space partition with distance one (h=1). So we get 480 discretized 15 samples of this 40 period intervals. Then we follow to verify that the increments of the sampled process are a PC process. Figure 1: Top: the increments of the simulated process {Gi : i ∈ N} of size 480; bottom left: the sample autocorrelation (1) plot of Gt ; bottom right: the significant values of the sample spectral coherence with α = 0.05. In figure 1 graph of the increments of the sampled process of size 480 (top) with the sample autocorrelation graph of this process (bottom left) are presented. The bottom right graph shows the sample coherent statistics values for a specified collection of pairs (λr , λs ) ∈ [0, 2π) × [0, 2π) and M = 240 that exceed the threshold corresponding to α = 0.05. The parallel lines for the sample spectral coherence confirm the increments of the sampled process are PC. Also in this graph, the significant off-diagonal is at \|r−s\| = 40 which verifies the first peak at 40 and shows that there is a second order periodic structure with period % = 480/40 = 12. 16 Table 1: Some different values of \|γ̂(λr , λs , M )\|2 for some values r, s with α = 0.05 and xα = 0.0125. The some different values of sample coherence statistics for the test that the increments of the sampled process have period 12 are presented in Table 1. As the corresponding α = 0.05 threshold shows the test is significant on the corresponding parallel lines of Figure 1. 6 Appendix P1: Proof of Lemma 2.2 For any t ≥ 0 there exist j = 1, · · · , l and m ∈ N0 such that t ∈ Aj+ml . Thus, using the Definition 2.2, Definition 2.3 and the fact that (St )t≥0 has independence increments we have PN (t) PN (t) iw Dt + n=1 Zn iwSt = eiwDt E eiw n=1 Zn E e =E e PN (t) PN (t1 ) 1 PN (t ) j 2 iw n=N (t Zn iw n=N2 (t )+1 Zn ml+j−1 )+1 1 = eiwDt E eiw n=1 Zn × e × ··· × e iwDt =e l m−1 YY iw E e k=0 r=1 PN (t) iw n=N (t × E e PN (tkl+r ) n=N (tkl+r−1 )+1 j Zn ml+j−1 )+1 r Zn × j−1 Y r=1 . 17 iw E e PN (tml+r ) n=N (tml+r−1 )+1 r Zn Since for r = 1, · · · , l, Znr are independent and have distribution Fr , it follows from Definition 2.2 and conditional expected value for k = 0, · · · , m and r = 1, · · · , l that PN (t ) PN (t )−N (tkl+r−1 ) r iw n=Nkl+r Zr Zn iw n=1kl+r (tkl+r−1 )+1 n \|N (tkl+r ) − N (tkl+r−1 ) = N E e =E E e ∞ X = E eiw PN n=1 r Zn P N (tkl+r ) − N (tkl+r−1 ) = N N =0 ∞ X = N =0 = e− N e r N Λ(tkl+r ) − Λ(tkl+r−1 ) E(eiwZn ) N! hR iN iwz X ∞ Λ(t ) − Λ(t ) F (dz) e kl+r kl+r−1 r R Λ(tkl+r )−Λ(tkl+r−1 ) N! N =0 R =e − Λ(tkl+r )−Λ(tkl+r−1 ) iwz −1) R (e Λ(tkl+r )−Λ(tkl+r−1 ) Fr (dz) Therefore R E e iwSt iwDt =e e h iwz −1) R (e iw Dt + =e h R \|z\|≤1 z + R ×e r=1 k=0 + Pm−1 Pl Pj−1 Pm−1 Pl r=1 k=0 r=1 i Λ(tml+r )−Λ(tml+r−1 ) Fr (dz)+ Λ(t)−Λ(tml+j−1 ) Fj (dz) r=1 Pj−1 Λ(tkl+r )−Λ(tkl+r−1 ) Fr (dz) Λ(tkl+r )−Λ(tkl+r−1 ) Fr (dz) i Λ(tml+r )−Λ(tml+r−1 ) Fr (dz)+ Λ(t)−Λ(tml+j−1 ) Fj (dz) h iwz −1−iwzI {\|z \|≤1} ) R (e Pm−1 Pl r=1 k=0 + Pj−1 r=1 Λ(tkl+r )−Λ(tkl+r−1 ) Fr (dz) i Λ(tml+r )−Λ(tml+r−1 ) Fr (dz)+ Λ(t)−Λ(tml+j−1 ) Fj (dz) . P2: Proof of Corollary 2.3 It is sufficient to prove that for any 0 ≤ s < t and K ∈ N, d St − Ss = St+Kτ − Ss+Kτ . For any 0 ≤ s < t there exist m, m0 ∈ N0 and j, j 0 = 1, · · · , l such that s ∈ Aj+ml and t ∈ Aj 0 +(m+m0 )l . Thus PN (t) PN (t) PN (s) iw St −Ss iw Dt −Ds + n=1 Zn − n=1 Zn iw Dt −Ds E e =E e =e E eiw n=N (s)+1 Zn PN (tml+j ) j PN (t) j0 iw n=N (t Zn iw n=N (s)+1 Zn iw Dt −Ds 0 )l+j 0 −1 )+1 (m+m =e E e × ··· × e 18 iw(Dt −Ds ) =e ×E e PN (tml+j ) iw N (s)+1 j Zn × l Y E e iw PN (tml+r ) n=N (tml+r−1 )+1 r Zn r=j+1 m+m0 −1 × l Y Y iw PN (tkl+r ) E e n=N (tkl+r−1 )+1 r Zn k=m+1 r=1 × 0 −1 jY iw E e PN (t(m+m0 )l+r ) n=N (t(m+m0 )l+r−1 )+1 r Zn ×E e iw 0 PN (t) n=N (t(m+m0 )l+j 0 −1 )+1 j Zn . r=1 By similar method in the Proof of Lemma 2.2, we have h R iwz (e −1) Λ(t )−Λ(s) Fj (dz)+···+ j+ml R E eiw St −Ss = eiw Dt −Ds e P 0 P 0 + m −1 k=1 Λ(t(m+k)l+1 )−Λ(t(m+k)l ) F1 (dz)+···+ m −1 k=1 Λ(t(m+1)l )−Λ(t(m+1)l−1 ) Fl (dz) Λ(t(m+k+1)l )−Λ(t(m+k+1)l−1 ) Fl (dz) i + Λ(t(m+m0 )l+1 )−Λ(t(m+m0 )l )F1 (dz) +···+ Λ(t)−Λ(t(m+m0 )l+j 0 −1 )Fj 0 (dz) . Since s + Kτ ∈ Aj+(m+K)l and t + Kτ ∈ Aj 0 +(m+m0 +K)l , it follows the same method used in the computation of the characteristic function of St − Ss that h R iwz (e −1) Λ(tj+(m+K)l )−Λ(s+Kτ ) Fj (dz)+··· R iw Dt+Kτ −Ds+Kτ iw St+Kτ −Ss+Kτ e =e E e + Λ(t(m+K+1)l )−Λ(t(m+K+1)l−1 ) Fl (dz) + Pm0 −1 + Pm0 −1 k=1 k=1 Λ(t(m+K+k)l+1 )−Λ(t(m+K+k)l ) F1 (dz)+··· Λ(t(m+K+k+1)l )−Λ(t(m+K+k+1)l−1 ) Fl (dz) + Λ(t(m+m0 +K)l+1 )−Λ(t(m+m0 +K)l )F1 (dz) +··· i + Λ(t+Kτ )−Λ(t(m+m0 +K)l+j 0 −1 )Fj 0 (dz) . By Definition 2.2 and Definition 2.3, we have for partition 0 ≤ ti < tj and K ∈ N Λ(tj ) − Λ(ti ) = Λ(tj + Kτ ) − Λ(ti + Kτ ) = Λ(tj+Kl ) − Λ(ti+Kl ) (6.1) and Dt = Dt+Kτ for t ≥ 0. Thus iw St −Ss E e =E e iw St+Kτ −Ss+Kτ . P3: Proof of Theorem 3.1 Proof a : Let (S)t≥0 be the simple semi-Lévy process defined by (2.2) and Zi be ith jump 19 size. Furthermore T1 denote the time that the first jump occurs P and Tj , j = 2, 3, · · · , be th th the time intervals between the (j − 1) and j jumps and Υn := nj=1 Tj , for n ∈ N, and Υ0 = 0. For n ∈ N Qn := I + (Zn )2 ea0 eB(Υn −Υn−1 ) , Rn := α0 (Zn )2 e. It follows from [8] that Yt satisfies in Yt = Js,t Ys + Ks,t , 0≤s≤t (6.2) where, Js,t = eB(t−ΥN (t) ) QN (t) · · · QN (s)+2 I + (ZN (s)+1 )2 ea0 eB(ΥN (s)+1 −s) , Ks,t = eB(t−ΥN (t) ) RN (t) + QN (t) RN (t)−1 + · · · + QN (t) · · · QN (s)+2 RN (s)+1 . In order to prove that the sequence Js+kτ,t+kτ , Ks+kτ,t+kτ k≥0 is independently identically distributed, let mτ ≤ s ≤ t ≤ (m + 1)τ such that m ∈ N0 . We define (m+1) := ΥN (s)+1 − s, Υ1 (m+1) Z1 := ZN (s)+1 , .. . (m+1) ΥN (t)−N (s) := ΥN (t) − s, (m+1) ZN (t)−N (s) := ZN (t) . (6.3) (m+1) (m+1) (m+1) , , · · · , ΥN (t)−N (s) , Z1 Therefore, Js,t , Ks,t is function of random vector N (t)−N (s), Υ1 (m+1) · · · , ZN (t)−N (s) . Using the fact that increments of the Poisson process {N (t) : t ≥ 0} are independent and the density function of random vector (X1 , · · · , Xn ) can be computed as follows: P x < X ≤ x + δ , · · · , x < X ≤ x + δ 1 1 1 1 n n n n (x1 , · · · , xn ) = f lim , X1 ,··· ,Xn δ1 ,··· ,δn →∞ δ1 · · · δn (m+1) (m+1) , · · · , Υn \|N (t) − N (s) = n as follows: we can give the conditional density of Υ1 f ) (s1 , · · · , sn ) = n!λ(s1 ) × · · · × λ(s n n , Λ(t) − Λ(s) (m+1) (m+1) Υ1 ,··· ,Υn \|N (t)−N (s)=n 0 < s1 < · · · < sn < τ. Since the increment process N (t) − N (s) is a Poisson process with mean Λ(t) − Λ(s) such that Λ(t) − Λ(s) = Λ(t + kτ ) − Λ(s + kτ ), for all k ∈ N0 , it follows from Definition 2.1 and conditional density that d Js,t , Ks,t = Js+kτ,t+kτ , Ks+kτ,t+kτ . 20 The independence of the sequence Js+kτ,t+kτ , Ks+kτ,t+kτ is clear, since Js+kτ,t+kτ and Ks+kτ,t+kτ are constructed only from the segment Su , (s + kτ ) ≤ u ≤ (t + kτ ), of the semi-Levy process S. If 0 ≤ s≤ t, there are n, m ∈ N0 such that s ∈ nτ, (n + 1)τ and t ∈ (n + m)τ, (n + m + 1)τ . By iterating (3.1) we obtain h Yt = J(n+m)τ,t J(n+m−1)τ,(n+m)τ · · · J(n+1)τ,(n+2)τ Js,(n+1)τ Ys + K(n+m)τ,t i + J(n+m)τ,t K(n+m−1)τ,(n+m)τ + · · · + J(n+m)τ,t J(n+m−1)τ,(n+m)τ · · · J(n+1)τ,(n+2)τ Ks,(n+1)τ . It follows from [8] and (6.2) that Js,t = J(n+m)τ,t J(n+m−1)τ,(n+m)τ · · · J(n+1)τ,(n+2)τ Js,(n+1)τ Ks,t = K(n+m)τ,t + J(n+m)τ,t K(n+m−1)τ,(n+m)τ + · · · + J(n+m)τ,t · · · J(n+1)τ,(n+2)τ Ks,(n+1)τ , therefore, for all k ∈ N0 , d Js,t , Ks,t = Js+kτ,t+kτ , Ks+kτ,t+kτ . (b) By iterating (3.1) we obtain h Yt+mτ = Jt+(m−1)τ,t+mτ · · · Jt,t+τ Yt + Kt+(m−1)τ,t+mτ + Jt+(m−1)τ,t+mτ Kt+(m−2)τ,t+(m−1)τ i + · · · + Jt+(m−1)τ,t+mτ · · · Jt+τ,t+2τ Kt,t+τ . Since Jt+(m−1)τ,t+mτ , Kt+(m−1)τ,t+mτ follows immediately d Yt+mτ = m Y m∈N , are independent and identically distributed it Jt+(k−1)τ,t+kτ Yt + Kt,t+τ + m−1 X Jt,t+τ · · · Jt+(k−1)τ,t+kτ Kt+kτ,t+(k+1)τ . k=1 k=1 Note that the Kt,t+τ + infinite series (t) U Pm−1 k=1 Jt,t+τ · · · Jt+(k−1)τ,t+kτ Kt+kτ,t+(k+1)τ is the partial sums of the := Kt,t+τ + ∞ X Jt,t+τ · · · Jt+(k−1)τ,t+kτ Kt+kτ,t+(k+1)τ . (6.4) k=1 Thus, using the general theory of random recurrence equations (see Bougerol and Picard [5], Brandt [6] and Vervaat [22]) and condition (3.2), we prove the almost sure absolute convergence of the series (6.4). Let P be such that ∆ := P −1 BP is diagonal. Then we have for t ≥ 0, \|\|eBt \|\|B,r = \|\|P e∆t P −1 \|\|B,r = \|\|P −1 P e∆t P −1 P \|\|B,r = \|\|e∆t \|\|r = eηt . 21 Using (6.2), (6.3) and condition (3.2) show (2) (2) E log\|\|Jt,t+τ \|\|B,r ≤ E ητ + log 1 + (ZN (t+τ )−N (τ ) )2 \|\|ea0 \|\|B,r + · · · + log 1 + (Z1 )2 \|\|ea0 \|\|B,r (1) (1) + log 1 + (ZN (τ )−N (t) )2 \|\|ea0 \|\|B,r + · · · + log 1 + (Z1 )2 \|\|ea0 \|\|B,r < 0 and it follows from [8], (6.2) and (2.4) E log\|\|Kt,t+τ \|\|B,r )−N (t) h N (t+τ i X ≤E log(1 + (ZN (t+τ )−j+1 )2 \|\|ea0 \|\|B,r ) + (ZN (t+τ )−j+1 )2 j=1 + log(α0 \|\|e\|\|B,r ) < ∞. Hence, the strong law of large numbers yield k 1 X log\|\|Jt+(j−1)τ,t+jτ \|\|B,r + log\|\|Kt+kτ,t+(k+1)τ \|\|B,r < 0 a.s., lim sup k→∞ k j=1 i.e. k1 lim sup \|\|Jt,t+τ · · · Jt+(k−1)τ,t+kτ Kt+kτ,t+(k+1)τ \|\|B,r < 1 a.s. k→∞ From Cauchy’s root criterion follows that series (6.4) is almost sure absolute convergence. Since the state process Y has cadlag paths, it follows that \|\|Yt \|\|B,r is almost surely finite. Therefore (t) \|\|Yt+mτ − U \|\|B,r ≤ \|\| m Y \|\| Jt+(k−1)τ,t+kτ \|\|B,r \|\|Yt − U(t) m B,r −→ 0 a.s. k=1 P (t) where Um := Kt+mτ,t+(m+1)τ + ∞ k=m Jt+mτ,t+(m+1)τ · · · Jt+kτ,t+(k+1)τ Kt+(k+1)τ,t+(k+2)τ . It follows from [9] that Yt+mτ converges in distribution to U(t) , for fixed t ∈ [0, τ ). That U(t) satisfies (3.3) and is the unique solution is clear by the general theory of random recurrence equations. (c) It suffices to show that for any s1 , s2 , · · · , sn and k ∈ N0 d Ys1 , Ys2 , · · · , Ysn = Ys1 +kτ , Ys2 +kτ , · · · , Ysn +kτ . Using the recursion equation (6.2) and analysis is used in (a) we obtain above relation. We give the proof for s1 ∈ [0, τ ) and s2 ∈ [τ, 2τ ). The general case is similar. Therefore Ys2 = Jτ,s2 Js1 ,τ Ys1 + Kτ,s2 + Jτ,s2 Ks1 ,τ , and Ys1 = J0,s1 Y0 + K0,s1 . 22 The random vector Ys1 , Ys2 is function from J0,s1 , K0,s1 , Js1 ,τ , Ks1 ,τ , Jτ,s2 , Kτ,s2 , Y0 and with similar argument also shows that the random vector Ys1 +kτ , Ys2 +kτ is function from Jkτ,s1 +kτ , Kkτ,s1 +kτ , Js1 +kτ,(k+1)τ , Ks1 +kτ,(k+1)τ , J(k+1)τ,s2 +kτ , K(k+1)τ,s2 +kτ , Ykτ . Using d d (a) and assumption Y0 = U(0) , it follows that Ys1 , Ys2 = Ys1 +kτ , Ys2 +kτ . P4: Proof of Corollary 3.2 Since for all s ∈ [t, t + p], the process dSs is independent of Fs , it follows from (2.5) and Corollary 3. that Z t+p p (p) E Gt = E( Vs )E(Ss+ds − Ss ) t Z t+p p d (p) E( Vs+kτ )E(Ss+ds+kτ − Ss+kτ ) = E Gt+kτ . = t (p) In order to prove that the covariance function of Gt is periodic, it suffices to show that (p) (p) (p) (p) E Gt Gt+h = E Gt+mτ Gt+h+mτ . Let Et+p denote conditional expectation with respect to the σ-algebra Ft+p . Since the increments of S on the interval (t + h, t + h + p] are independent of Ft+p and the increment (p) process Gt is measurable Ft+p , we have (p) (p) (p) (p) E Gt Gt+h = E Gt Et+p Gt+h Z t+h+p Z t+p p p Vs Vu dSu E dSs . = E t+h t √ √ Since Vs Vu dSu is function of Ju+du,s− , Ku+du,s− , Ju,u+du , Ku,u+du , Ju− ,u , Ku− ,u , Jt,u− , Kt,u− , − +kτ , Ju+kτ,u+du+kτ , Yt and this vector has the same distribution with Ju+du+kτ,s− +kτ , Ku+du+kτ,s − − − − Ku+kτ,u+du+kτ , Ju +kτ,u+kτ , Ku +kτ,u+kτ , Jt+kτ,u +kτ , Kt+kτ,u +kτ , Yt+kτ , it follows that (p) (p) (p) (p) cov Gt , Gt+h = cov Gt+kτ , Gt+h+kτ . P5: Proof of Lemma 4.1 (a) Let Ỹt be state process of semi Levy driven COGARCH(1,1) process. Then Ỹt = J̃0,t Ỹ0 + K̃0,t , where J̃0,t = exp ηt + N (t) X log(1 + \|\|ea0 \|\|B,r Zi2 ) , i=1 23 (6.5) K̃0,t = α0 \|\|e\|\|B,r × exp N (t) X log(1 + \|\|ea 0 N (t) X \|\|B,r Zi2 ) i=1 Zi2 . i=1 It follows from [8] that for all t ≥ 0 \|\|J0,t \|\|B,r ≤ exp ηt + N (t) X log(1 + \|\|ea 0 \|\|B,r Zi2 ) , (6.6) i=1 \|\|K0,t \|\|B,r ≤ α0 \|\|e\|\|B,r × exp N (t) X log(1 + \|\|ea 0 \|\|B,r Zi2 ) N (t) X i=1 Zi2 . (6.7) i=1 Now define a cadlag process {Xt : t ≥ 0} by Xt = −ηt − N (t) X log(1 + \|\|ea0 \|\|B,r Zi2 ), t ≥ 0. i=1 Then, Xt is a negative simple pure jump semi Levy procress. It follows from Definition 2.1 and Remark 3.2 that E e −cXt = E exp cηt + c N (t) X log(1 + \|\|ea0 \|\|B,r Zi2 ) i=1 Z tZ = exp cηt + 0 (1 + \|\|ea0 \|\|B,r z 2 )c − 1 νs (dz)ds . R Using a similar analysis is used in proof of Proposition 3.2 [17], it follows from [8] that Z t h i 0 −1 −Xt K̃0,t = \|\|ea \|\|B,r α0 \|\|e\|\|B,r e −η e−(Xt −Xu ) du − 1 . 0 It follows from (6.5), (6.6) and (6.7) that \|\|Yt \|\|B,r < Ỹt , for all t ≥ 0. Thus E(St2 ) < ∞ and E(St4 ) < ∞ imply E(Yt ) < ∞ and cov(Yt ) < ∞, respectively. In the proof of Theorem 3.1(a) we have seen that (3.2) implies that the sequence Ỹmτ converges in distribution to a finite random vector Ũ which of the vector Ũ is the unique solution of the random equation d Ũ = J̃0 Ũ + K̃0 , d where J̃0 , K̃0 = J̃0,τ , K̃0,τ and Ũ is independent of J̃0 , K̃0 . It follows from (6.4), (6.6) and (6.7) that U ≤ Ũ. Thus E(St2 ) < ∞ and E(St4 ) < ∞ imply E(U) < ∞ and cov(U) < ∞, respectively. 24 P6: Proof of Lemma 4.2 Using (5.1) and independence Ymτ and (Jmτ,s1 +mτ , Kmτ,s1 +mτ ), we obtain E(Yt ) = E(Jmτ,s1 +mτ )E(Ymτ ) + E(Kmτ,s1 +mτ ) = E(J0,s1 )E(U) + E(K0,s1 ) d where the last equality follows from that (Jmτ,s1 +mτ , Kmτ,s1 +mτ ) = (J0,s1 , K0,s1 ) and assumption of section (c) of the Theorem 3.1. For computing cov(Yt , Yt+h ) it is sufficient to obtain E(Yt+h Yt0 ). It will therefore be followed from recursion equations which used in the proof of Theorem 4.1 that Yt = Jmτ,t Ymτ + Kmτ,t , Yt+h = Jnτ,t+h J(n−1)τ,nτ · · · Jmτ,(m+1)τ Ymτ h + Knτ,t+h + Jnτ,t+h K(n−1)τ,nτ + n−m−1 X Jnτ,t+h · · · J(n−i)τ,(n−i+1)τ K(n−i−1)τ,(n−i)τ i i=1 The relation (4.2) follows from independence the sequence (Jkτ,s+kτ , Kkτ,s+kτ ) for any k ∈ N0 and s ∈ [0, τ ] and also independence Ymτ from this sequence for any k ≥ m. P7: Proof of Corollary 4.3 Since for fixed t, almost surely Vt = Vt+ = α0 + a0 Yt , we have the expected value and covariance function volatility process from (2.6). P8: Proof of Proposition 4.4 (a) We imitate the proof of Theorem 6.1 of Brockwell, Chadraa, and Lindner [8]. Since S is a martingale with zero mean, we have (4.3). It follows from Ito isometry for square integrable martingales as integrators (e.g. [20], IV 27) that Z t+h+p (p) (p) Vs I[t,t+p) (s)I[t+h,t+h+p (s)d[S, S]s = 0, E Gt Gt+h = E 0 and hence (4.4) follows. (b) It follows from partial integration that Z t+p (p) 2 (Gt ) = 2 Gs− dGs + [G, G]t+p t+ t+ Z t+p X p =2 Gs− Vs dSs + Vs (∆Ss )2 . t t<s≤t+p By similar analysis is used in (a), the compensation formula and [11] we have Z t+p Z X (p) 2 2 E (Gt ) = E Vs (∆Ss ) = E(Vs )z 2 νs (dz)ds t t<s≤t+p 25 R (6.8) From Remark 3.2 the relation (4.5) follows. For proof of (4.6), Since the increments of S on the interval (t, t + p] are independent of Ft+p and S has expectation 0, it follows that Z t+p p Et+p Gs− Vs dSs = 0. t Thus it follows from the compensation formula and (6.2) that X (p) Et+p (Gt+h )2 = Et+p α0 + a0 Jt+p,s− Yt+p + a0 Kt+p,s− (∆Ss )2 t+h<s≤t+h+p Z t+h+p Z = t+h α0 + a0 E(Jt+p,s− )Yt+p + a0 E(Kt+p,s− ) z 2 νs (dz)ds, R therefore (p) (p) (p) (p) (p) (p) cov (Gt )2 , (Gt+h )2 = E (Gt )2 Et+p (Gt+h )2 − E (Gt )2 E (Gt+h )2 , (p) and by Remark 3.2 and (2.6) we have (4.6). To calculate cov Yt+p , (Gt )2 , partial integration (6.8) to get Z t+p Z t+p p (p) 2 Vs d[S, S]s . cov Yt+p , (Gt ) = 2cov Yt+p , Gs− Vs dS̄s + cov Yt+p , t+ t Rt √ To calculate the first term, let It := 0 Gs− Vs dSs . We know E(It ) = 0 for all t ≥ 0. Therefore Z t+p p cov Yt+p , Gs− Vs dSs = E It+p Yt+p − E(Jt,t+p )E It Yt − E(It )E(Kt,t+p ). t From [8], partial integration and substituting dVt+ = a0 BYt dt + α0 Vt d[S, S]t it follows that Z t Z t E(It Vt+ ) = E Is− dVs+ + E Vs dIs + E [V+ , I]t 0 0 Z t Z tZ 0 =aB E(Is− Ys )ds + αq E(Is− Vs )z 2 νs (dz)ds 0 Z0 t ZR t p p +E Gs− Vs Vs dSs + αq E Gs− Vs Vs dMs , 0 0 where Ms := )3 is a locally integrable martingale, with mean zero as a 0<u≤s (∆SsR result of assumption that R z 3 νs (dz) = 0, for all s ≥ 0. Thus, using the fact that Rt √ E 0 Gs− Vs Vs dSs = 0, E(It Vt+ ) = a0 E(It Yt ) and that Is Ys = Is− Ys = Is− Ys− almost surely for fixed s, so we have Z t Z tZ 0 0 0 a E(It Yt ) = a B E(Is Ys )ds + α0 a E(Is Ys )z 2 νs (dz)ds, P 0 0 26 R The equality holds for any vector a, hence Z t Z tZ E(It Yt ) = B E(Is Ys )ds + α0 E(Is Ys )z 2 νs (dz)ds. 0 0 R To calculate the second term of the covariance, it follows from [8] and (2.7) that Z t+p Z t+p 0 0 Ys− dsB 0 )e cov Yt+p , Vs d[S, S]s = cov Yt+p , (Yt+p − Yt − t+ t+ Z t+p = cov(Yt+p ) − cov(Yt+p , Yt ) − cov(Yt+p , Ys− )dsB 0 e. t+ References [1] W. R. Bennett (1958). Statistics of regenerative digital transmission. Bell System Technical Journal, 37, 1501-1542. [2] A. Bibi and I. Lescheb (2012). On general periodic time-varying bilinear processes. Economics Letters, 114, 353357. [3] T. Bollerslev (1986). Generalized Autoregressive Conditional Heteroskedasticity. Journal of Econometrics, 31, 307327. [4] T. Bollerslev, A.J. Patton and W. 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AdaDNNs: Adaptive Ensemble of Deep Neural Networks for Scene Text Recognition Chun Yang† , Xu-Cheng Yin†∗ , Zejun Li† , Jianwei Wu† , arXiv:1710.03425v1 [cs.CV] 10 Oct 2017 † Chunchao Guo‡ , Hongfa Wang‡ , and Lei Xiao‡ Department of Computer Science and Technology, University of Science and Technology Beijing, Beijing, China ‡ TEG, Tencent Co. LTD, Shenzhen, China ∗ Corresponding author: [email protected] Abstract Recognizing text in the wild is a really challenging task because of complex backgrounds, various illuminations and diverse distortions, even with deep neural networks (convolutional neural networks and recurrent neural networks). In the end-to-end training procedure for scene text recognition, the outputs of deep neural networks at different iterations are always demonstrated with diversity and complementarity for the target object (text). Here, a simple but effective deep learning method, an adaptive ensemble of deep neural networks (AdaDNNs), is proposed to simply select and adaptively combine classifier components at different iterations from the whole learning system. Furthermore, the ensemble is formulated as a Bayesian framework for classifier weighting and combination. A variety of experiments on several typical acknowledged benchmarks, i.e., ICDAR Robust Reading Competition (Challenge 1, 2 and 4) datasets, verify the surprised improvement from the baseline DNNs, and the effectiveness of AdaDNNs compared with the recent state-ofthe-art methods. Scene text is widely used as visual indicators for navigation and notification, and text recognition from scene images and videos is one key factor for a variety of practical applications with reading in the wild (Ye and Doermann 2015; Yin et al. 2016; Tian et al. 2017), such as assisting for visually impaired people (Goto and Tanaka 2009; Sanketi, Shen, and Coughlan 2011), real-time translation (Shi and Xu 2005; Fragoso et al. 2011), user navigation (Minetto et al. 2011), driving assistance systems (Wu, Chen, and Yang 2005), and autonomous mobile robots (Létourneau et al. 2003). Scene text (cropped word) recognition methods can be generally grouped into segmentation-based word recognition and holistic word recognition. Typical segmentationbased approaches over-segment the word image into small segments, combine adjacent segments into candidate characters, classify them using convolutional neural networks (CNNs) or gradient feature-based classifiers, and find an approximately optimal word recognition result (Bissacco et al. 2013; Jaderberg, Vedaldi, and Zisserman 2014). Because of complex backgrounds and diverse distortions, character segmentation is another more challenging task. Thereby, holistic word recognition approaches with deep neural networks are more impressive for text reading in the wild. Copyright c 2017-2018, All rights reserved. Word spotting, the direct holistic approach, usually calculates a similarity measure between the candidate word image and a query word (Jaderberg et al. 2016; Gordo 2015). Sequence matching, the indirect holistic approach, recognizes the whole word image by embedding hidden segmentation strategies. For example, Shi et al. constructed an endto-end training deep neural network for image-based sequence recognition (scene text recognition) (Shi, Bai, and Yao 2017). However, there are a variety of grand challenges for scene text recognition (see samples in Fig. 1), even with recent deep neural networks (DNNs), where additional characters will be probably identified for text distortions and complex backgrounds, some characters are wrongly recognized for changing illuminations and complex noises, and characters are sometimes missed for low resolutions and diverse distortions. Figure 1: Some challenging examples (from 2015 ICDAR Robust Reading Competition Challenge 4 dataset) of scene text images which are incorrectly recognized by the baseline DNNs (see related descriptions in Experiments). The captions show the recognized text (left) versus the ground truth (right): additional characters, wrong characters and missing characters in target words. Stochastic Gradient Descent (SGD) (Bottou 2010) and its variants have become the defacto techniques for optimizing DNNs, where SGD always leads to local minima, even though the popularity of SGD can be attributed to its ability to avoid spurious saddle-points and local minima (Dauphin et al. 2014). There are a plenty number of (more than million) possible local minima in DNNs (Kawaguchi 2016), and local minima with flat basins are supposed to generalize better in the learning system (Keskar et al. 2017). As a result, although different local minima often have similar error rates, the corresponding neural networks in DNNs tend to make different mistakes. This diversity and complementarity can be exploited via classifier ensemble (Huang et al. 2017). There are two major ways for ensemble of deep neural networks. On the one hand, different learning systems with DNNs are first trained independently, and then the final system is a trivial ensemble of these different deep learning architectures via majority voting or averaging. For example, most high profile competitions in ImageNet 1 and Kaggle 2 are won by such ensemble techniques. Because of the huge computation complexity, this ensemble becomes uneconomical and impossible for most researchers in the universities and even in the small companies. On the other hand, one learning system with DNNs is first trained, and then the final ensemble selects and combines neural network components 3 in this only one system without incurring any additional training cost. Huang et al. proposed such an ensemble technique, called as Snapshot Ensembling, where a specific optimization strategy is designed to train DNNs and “model snapshots” (neural network components) in all cycles are combined for the final ensemble in the learning procedure (Huang et al. 2017). However, how to design the specific and effective optimization algorithms for DNNs is also a challenge. In this paper, we propose a new and adaptive ensemble of deep neural networks (AdaDNNs) in the most simplest way, i.e., given trained neural networks (of all iterations) from a learned DNNs system 4 , a subset of neural network components are simply selected and adaptively combined to perform the final predictions. And the ensemble is formally formulated as a Bayesian framework for classifier weighting and combination. We argue that because of the diversity and complementarity in DNNs with SGD, AdaDNNs via ensembling with diversity can improve robust performance of the final learning system. On the same time, because of the high accuracy of components in DNNs, AdaDNNs via combination with accurate neural network components can improve precision performance of the final classification system. A variety of experiments on several acknowledged benchmarks, i.e., ICDAR Robust Reading Competition (Challenge 1, 2 and 4) datasets, have shown that the simple but effective AdaDNNs improves largely from the baseline DNNs. Moreover, our proposed approach has the 1 www.image-net.org. www.kaggle.com. 3 The neural network component means the resulting DNN of each iteration in the whole training procedure. 4 Here, the DNNs system can be trained with conventional optimization algorithms (Bottou, Curtis, and Nocedal 2016), or even with the specific algorithms, e.g., Snapshot Ensembling (Huang et al. 2017). 2 top performance compared with the latest state-of-the-art methods. Related Work Recognizing text in scene videos attracts more and more interests in the fields of document analysis and recognition, computer vision, and machine learning. The existing methods for scene text (cropped word) recognition can be grouped into segmentation-based word recognition and holistic word recognition. In general, segmentation-based word recognition methods integrate character segmentation and character recognition with language priors using optimization techniques, such as Markov models (Weinman et al. 2014) and CRFs (Mishra, Alahari, and Jawahar 2012; Shi et al. 2013). In recent years, the mainstream segmentationbased word recognition techniques usually over-segment the word image into small segments, combine adjacent segments into candidate characters and classify them using CNNs or gradient feature-based classifiers, and find an approximately optimal word recognition result using beam search (Bissacco et al. 2013), Hidden Markov Models (Alsharif and Pineau 2014), or dynamic programming (Jaderberg, Vedaldi, and Zisserman 2014). Word spotting (Manmatha, Han, and Riseman 1996), a direct holistic word recognition approach, is to identify specific words in scene images without character segmentation, given a lexicon of words (Wang and Belongie 2010). Word spotting methods usually calculate a similarity measure between the candidate word image and a query word. Impressively, some recent methods design a proper CNN architecture and train CNNs directly on the holistic word images (Jaderberg et al. 2014; Jaderberg et al. 2016), or use label embedding techniques to enrich relations between word images and text strings (Almazan et al. 2014; Gordo 2015). Sequence matching, an indirect holistic word recognition approach, recognizes the whole word image by embedding hidden segmentation strategies. Shi et al. constructed an endto-end train deep neural network for image-based sequence recognition (scene text recognition), where a convolutional recurrent neural networks framework (CRNN) is designed and utilized (Shi, Bai, and Yao 2017). In this paper, a similar CRNN architecture is used in AdaDNNs for recognizing scene text sequently and holistically. Classifier ensemble can be mainly divided into two categories. The first one aims at learning multiple classifiers at the feature level, where multiple classifiers are trained and combined in the learning process, e.g., Boosting (Freund and Schapire 1997), Bagging (Breiman 1996), and Rotation Forest (Rodriguez, Kuncheva, and Alonso 2006). The second tries to combine classifiers at the output level, where the results of multiple available classifiers are combined to solve the targeted problem, e.g., multiple classifier systems (classifier combination) (Zhou 2012; Yin et al. 2014). AdaDNNs in this paper follows the second one. Namely, given multiple classifiers (neural network components sequently learned in DNNs), AdaDNNs is constructed by combining intelligently these component classifiers within a Bayesian-based formulation framework. Adaptive Ensemble of Deep Neural Networks As we have known, both SGD and batch optimization can lead to different local minima in DNNs, and neural network components are always with diversity and complementarity. Conventionally, there are tens of thousands of iterations and also neural network components in the learning system of DNNs. Considering the acceptable computation complexity in the testing procedure, one thing is to quickly select a small subset of neural network components in different training iterations. At the same time, considering the high accuracy requirement, another thing is to adaptively combine this subset of neural network components and construct a final classification system. In the following, the unified framework of AdaDNNs is first formulated. Next, the detail procedure of AdaDNNs is then described. There are two key issues for optimizing Eq. 3. The first one is the calculation of W (y, hi (x)). As mentioned above, P (y\|hi , x) is the distribution of describing the correlation between decision y and hi (x). Thus, W (y, hi (x)) can be derived from y, hi (x) and the distance between y and hi (x). Here, W (y, hi (x)) is assumed to be computed as W (y, hi (x)) = I(y = hi (x)) + U (y) ∗ V (y, hi (x)) (4) where both U (∗) and V (∗, •) are functions. I(y = hi (x)) returns 1 when y = hi (x); otherwise, I(y = hi (x)) = 0. For the scene text recognition task, on the one hand, with a given dictionary 5 , U (y) can be calculated as U (y) = { 1 0 y ∈ Dict y∈ / Dict (5) Unified Framework To formulate the ensemble decision, the individual classifier decisions can be combine by majority voting, which sums the votes for each class and selects the class that receives most of the votes. While the majority voting is the most popular combination rule, a major limitation of majority voting is that only the decision of each model is taken into account without considering the distribution of decisions. In particular, all the possible models in the hypothesis space could be exploited by considering their individual decisions and the correlations with other hypotheses. Here, we use a Bayesian-based framework to combine classifiers. Given a sample x and a set H of independent classifiers, the probability of label y can be estimated by a Bayesian Model (BM) as X P (y\|H, x) = P (y\|hi , x)P (hi \|x) (1) On the other hand, the correlation between y and hi (x) can be assumed by the function V of Cost Levenshtein Distance (CLD). In the traditional Levenshtein Distance, the cost of any two different characters is always 1. However, in spelling correction, the cost of two characters with similar shape tends to have a smaller distance. In this paper, we statistics the frequencies of different character pairs at the same location from the label and the hypothesis on the validation set (bootstrapped from the training set in Experiments), and calculate the cost of two different characters (a and b) as cost(a, b) = 1 − P (a\|b) (6) Note that if both y and hi (x) are from the given dictionary, then they will have a competitive relationship with each other. Thus, V (y, hi (x)) can be calculated with hi ∈H where P (y\|hi , x) is the distribution of describing the correlation between decision y and hi (x), and P (hi \|x) denotes the posterior probability of model hi . The posterior P (hi \|x) can be computed as P (hi \|x) = P P (D\|hi )P (hi ) hi ∈H P (D\|hi )P (hi ) (2) where P (hi ) is the prior probability of classifier hi and P (D\|hi ) isP the model likelihood on the training set D. Here, P (hi ) and hi ∈H P (D\|hi )P (hi ) are assumed to be a constant in Eq. 2. Therefore, BM assigns the optimal label y to y ∗ according to the following decision rule, i.e., P (y ∗ ) = argmaxy PP(y\|H, x) = argmaxy Phi ∈H P (y\|hi , x)P (hi \|x) = argmaxy P hi ∈H P (y\|hi , x)P (D\|hi ) = argmaxy 2P hi ∈H P (y\|hi , x)P (D\|hi ) − P (D) = argmaxy Phi ∈H (2P (y\|hi , x) − 1)P (D\|hi ) = argmaxy hi ∈H W (y, hi (x))P (D\|hi ) (3) where W (y, hi (x)) is a function of y and hi (x). By multiplying the scaling factor λ > 0, W (y, hi (x)) can have a different range in R. hi (x) ∈ Dict hi (x) ∈ / Dict (7) where F is a function of the CLD between y and hi (x). By a heuristic approach, the values of F can be empirically assigned at the multiple integral points, and the values at other points can be calculated by the piecewise linear interpolation. An example of F is shown in Fig. 2. In general, F has a small range, e.g., [1.5, 1.5] in Fig. ??. So, the obtained weights from Eq. 4 are convenient for linear combination of classifiers. The second issue is about generating voting candidates (more probable labels of the hypotheses). Obviously, the ground truth doesn’t always appear in the decisions made by H. It is necessary to find an effective way to generate good candidates from all the decisions, i.e., to find a more probable label yi (x) from the existed initial label yi0 (x) of hypothesis hi . Generally speaking, a good candidate means it has a small edit distance with most of the hypotheses. Following this idea, we propose an algorithm to semantically generate voting candidates (see Algorithm 1). V (y, hi (x)) = { F (−CLD(y, hi (x))) F (CLD(y, hi (x))) 5 In our experiments, a 90k word dictionary from (Jaderberg et al. 2016) is used as the given dictionary. batch orderings will converge to different solutions. Those snapshots often have the similar error rates, but make different mistakes. This diversity can be exploited by ensembling, in which multiple snapnots are average sampling and then combined with majority voting. Focusing on scene text recognition, the CRNN model (Shi, Bai, and Yao 2017) is used to generate base classifiers (neural network components) as our text recognizer. CRNN uses CTC (Graves et al. 2006) as its output layer, which estimates the sequence probability conditioned on the input image, i.e. P (h\|x), where x is the input image and h represents a character sequence. Figure 2: An example of F of describing the relationship between V (in Y-axis) and the Cost Levenshtein Distance (in X-axis). Algorithm 1: Generating Voting Candidates. Input: H = {h1 , h2 , ..., hL }: the base classifier set, \|H\| = L. Y0 : the initial decisions made by H. ED: the measurement function of the pairwise distance. θ: the upper bound of the distance between the candidate and the hypothesis. Output: Y : the voting candidate set. Parameter: H ? : a subset of H, ∀h?i , h?j ∈ H ? , ED(h?i , h?j ) ≤ 2θ. Procedure: 1: Y = ∅. 2: For each H ? ⊂ H; 3: For each y ∈ Y0 : 4: If maxh?i ∈H ? ED(y, h? (x)) ≤ θ: 5: Y = Y ∪ {y}. 6: End 7: End In Algorithm 1, the searching process of H ? is an implicit computational way for P (D\|hi ). In our experiments, a special simple case of algorithm 1 is used, where during the voting candidates generation process, Y0 is initialized only by H, the upper bound is set from θ to inf, and P (D\|hi ) is assumed to be a constant. AdaDNNs Algorithm Within the above framework, the procedure of AdaDNNs for scene text recognition includes three major steps, i.e., base classifiers generation, classifier combination, and ensemble pruning. Base Classifiers Generation Ensembles work best if the base models have high accuracy and do not overlap in the set of examples they misclassify. Deep neural networks (DNNs) are naturally used as a base classifier generator for ensembles. On the one hand, DNNs have dramatically improved the state-of-the-art in many domains, such as speech recognition, visual object recognition and object detection, by being composed of multiple processing layers to learn representations of data with multiple levels of abstraction. On the other hand, during the training phase of one individual deep neural network, two snapshots with different mini- Classifier Combination for AdaDNNs The core of AdaDNNs is to calculate y ∗ (by Eq. 3), i.e., the calculation of F , which is a function of distance between y and hi (x). Here, F is represented by the set of values at the multiple integral points. These values are assigned with the highest recognition rate on the validation set. The detail procedure of the AdaDNNs ensemble is shown in Algorithm 2. Algorithm 2: AdaDNNs (classifier combination). Input: H = {h1 , h2 , ..., hL }: the base classifier set, \|H\| = L. Dict: the given dictionary. F : a function of distance between y and hi (x). Parameter: Y : the voting candidates set generated by Algorithm 1. Output: y ∗ : the label of prediction. Procedure: 1: Initialize Y by H and Dict. 3: For y ∈ Y : 4: Calculate P (y\|H, x) through Eq. 2. 5: End 6: Calculate y ∗ through Eq. 3. AdaDNNs Pruning In classifier ensemble, pruning can generally improve the ensemble performance. Here, we use Genetic Algorithm (GA) to pruning the ensemble. GA is a meta heuristic inspired by the process of natural selection that belongs to the larger class of evolutionary algorithms. GAs are commonly used to generate high-quality solutions for optimization and search problems by relying on bioinspired operators such as mutation, crossover and selection. In AdaDNNs pruning, firstly, a population of binary weight vectors is randomly generated, where 1 means the classifier is remained. Secondly, the population is iteratively evolve where the fitness of a vector w is measured on the V validation set V , i.e., f (w) = Rw (R stands for the recognition rate). Finally, the ensemble is correspondingly pruned by the evolved best weight vector w∗ . Experiments To evaluate the effectiveness of the proposed AdaDNNs method, a variety of experiments for text (cropped word) recognition are conducted on acknowledged benchmark datasets. We first focused on the most challenging task, i.e., incidental scene text recognition (ICDAR Robust Reading Competition Challenge 4), trained our AdaDNNs learning system (on both the synthetic dataset from (Jaderberg et al. 2016) and the training set of Challenge 4), and performed comparative experiments. Then, we also conducted experiments of this learned AdaDNNs on other text recognition tasks, i.e., focused scene text recognition and borndigital text recognition (ICDAR Robust Reading Competition Challenge 1 and 2), and checked the generalization of AdaDNNs. Here, the baseline DNNs model, CRNN, is same to the one in (Shi, Bai, and Yao 2017). The official metrics in ICDAR 2011/2013/2015 Robust Reading Competition (Shahab, Shafait, and Dengel 2011; Karatzas et al. 2013; Karatzas et al. 2015) are used. Figure 3: Challenging samples of scene text from COCO-text which are correctly recognized (with C.R.W. upper) by AdaDNNs: “GEMS”, “mgennisgal”, “RGAO”, “RAILROAD”, “UNITED”, “Kappa”, “XMAS”, “ZOOM”, “YouTube”, “YORK”, “WALK”, “WPRD”, “WHEN”, “YEAR”, and “WISCONSIN”. Experiments with Incidental Scene Text Recognition The ICDAR 2015 Robust Reading Competition Challenge 4 database (Karatzas et al. 2015) is a widely used and highly competitive benchmark database for scene text recognition within complex situations in the recent 3 years. The public dataset includes a training set of 1, 000 images and a test set of 500, with more than 10, 000 annotated text regions (cropped words). Because of complex backgrounds, various illuminations and diverse distortions, this incidental scene text recognition topic is a very challenging task. In our experiments, a variety of methods are conducted and compared, i.e., the baseline DNNs, AdaDNNs, AdaDNNs pruning, the winning participation method in the official competition (marked as bold words), and the latest top submissions of the Robust Reading Competition (RRC) website 6 in 2017 (marked as italic words). Table 1: Comparative results on 2015 ICDAR Challenge 4 dataset (incidental scene text recognition), where the comparative results are from the RRC website. Date Method T.E.D 2017/7/6 2017/7/6 2017/6/29 2015/4/1 - Baidu IDL v3 HIK OCR v3 HKU-VisionLab MAPS Baseline DNNs AdaDNNs AdaDNNs Pruning 211.59 191.25 258.59 1,128.01 384.76 251.98 224.7 C.R.W (%) 80.02 78.29 72.03 32.93 60.18 76.31 79.78 T.E.D. (upper) 171.15 158.84 212.17 1,068.72 303.77 185.36 147.11 C.R.W. (upper) 82.33 80.12 74.19 33.90 64.90 80.55 84.21 As can be seen from Table 1, our proposed AdaDNNs is much better than the baseline DNNs. For example, for the measure of “C.R.W (upper)”, AdaDNNs has a surprised improvement, i.e., from 64.90% to 80.55%. That is to say, the adaptive ensemble of DNNs in a simple but effective strategy can largely improved the performance from the original baseline DNNs. Moreover, compared with the latest top submissions (e.g., “Baidu IDL v3” and “HIK OCR v3”), our method, AdaDNNs Pruning, has the best performance with “C.R.W (upper)”, i.e., 84.21%. We also perform experiments on the COCO-text dataset (Veit et al. 2016), a similar challenging but largescale incidental scene text dataset. Images in this dataset are from the MS COCO dataset that contain text (63, 686 images with 173, 589 text regions). ICDAR2017 Robust Reading Challenge on COCO-Text is holding and will be released 6 http://rrc.cvc.uab.es. in ICDAR 2017. So, the comparative results of AdaDNNs, AdaDNNs Pruning and the baseline DNNs are only on the validation set; they are 58.08%, 66.07%, and 66.27%, respectively. Some scene text recognition samples for COCOtext are shown in Fig. 3. Experiments with Focused Scene Text Recognition and Born-Digital Text Recognition In order to investigate the generalization of AdaDNNs, we directly use the trained AdaDNNs system above (for 2015 ICDAR Challenge 4), and perform experiments on 2013 ICDAR Challenge 2 (cropped word recognition) dataset. The Challenge 2 dataset contains 1, 015 ground truths cropped word images. In our experiments, a variety of methods are conducted and compared, i.e., the baseline DNNs, AdaDNNs, AdaDNNs pruning, the winning participation method in the official competition (marked as bold words), the top three results in published papers, and the latest top submissions of the RRC website in 2017 (marked as italic words). Table 2: Comparative results on 2013 ICDAR Challenge 2 dataset (focused scene text recognition), where the comparative results without publications are from the RRC website. Date Method T.E.D 2017/8/14 2017/7/28 2017/2/24 2016 2016 2017 2013/4/6 - TencentAILab Tencent Youtu HIK OCR CNN (Jaderberg et al. 2016) RARE (Shi et al. 2016) CRNN (Shi, Bai, and Yao 2017) PhotoOCR Baseline DNNs AdaDNNs AdaDNNs Pruning 42 48.12 64.95 – – – 122.75 306.43 193.51 182.7 C.R.W (%) 95.07 92.42 90.78 – – – 82.83 75.34 83.20 85.21 T.E.D. (upper) 39.35 40.37 42.31 – – – 109.9 282.35 170.13 164.38 C.R.W. (upper) 95.34 93.42 93.33 90.8 88.6 89.6 85.30 78.63 86.67 88.13 Similarly, AdaDNNs is much better than the baseline DNNs, e.g., the measure of “C.R.W (upper)” increases from 78.63% to 86.67%. Surprisedly, only trained for another task (Challenge 4), the AdaDNNs (AdaDNNs Pruning) has a competitive performance on a new dataset (Challenge 2 dataset), compared with the recent published methods (e.g., CRNN (Shi, Bai, and Yao 2017)), and even with the latest submission results. Apart from the above experiments on text recognition from scene images (ICDAR Robust Reading Competition Challenge 2 and 4), we also directly perform the learned AdaDNNs on the born-digital images track (Challenge 1). Though born-digital images are not scene images, they have similar challenging issues for text recognition, e.g., complex backgrounds, low resolution and various colors. We also compare AdaDNNs (AdaDNNs Pruning) with the baseline DNNs, the winning participation method in the official competition (marked as bold words), and the latest top submissions of the RRC website (marked as italic words). The similar conclusions are drawn. Firstly, AdaDNNs improves largely compared with the baseline DNNs (from 84.50 to 92.22 for “C.R.W (upper)”). Secondly, AdaDNNs has a comparative performance with the latest submission results (e.g., “Dahua OCR v1” with 92.49% in 2017/9/1). Table 3: Comparative results on ICDAR Challenge 1 dataset (born-digital text recognition), where the comparative results are from the RRC website. Date Method T.E.D 2017/8/22 2017/7/21 2017/9/1 2013/4/6 - Tecent Youtu TecentAILab Dahua OCR v1 PhotoOCR Baseline DNNs AdaDNNs AdaDNNs Pruning 17.51 18.77 57.47 103.41 87.7 55.44 55.64 C.R.W (%) 96.80 96.18 91.31 82.21 82.42 89.58 89.53 T.E.D. (upper) 13.67 12.91 42.87 87.19 72.32 39.61 39.81 C.R.W. (upper) 97.29 97.22 92.49 85.41 84.50 92.22 92.17 We fully believe that if AdaDNNs (AdaDNNs Pruning) performs re-training on ICDAR Challenge 1 and Challenge 2 datasets, the performance will correspondingly be improved and obtain a more impressive results compared with the latest submission systems. This is also a near issue for our future work. Conclusion and Discussion A variety of DNNs based methods have been proposed and are still being investigated in the literature for scene text recognition because of the grand challenges, e.g., complex backgrounds, various illuminations and diverse distortions. In order to fully take advantage of the complementary diversity and the high accuracy of neural network components in DNNs, an adaptive ensemble of deep neural networks (AdaDNNs) is proposed to simply select and adaptively combine neural networks in the whole training procedure. Comparative experiments of scene text (cropped word) recognition showed that AdaDNNs achieves a remarkable increase in the final performance (more than 10%) compared with the baseline DNNs. Note that the DNNs methods have dramatically improved the state-of-the-art in object detection, object recognition, speech recognition and many other domains. 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Yang et al. BMC Systems Biology 2014, 8(Suppl 4):S7 http://www.biomedcentral.com/1752-0509/8/S4/S7 RESEARCH Open Access Microbial community pattern detection in human body habitats via ensemble clustering framework Peng Yang1, Xiaoquan Su2, Le Ou-Yang3, Hon-Nian Chua1, Xiao-Li Li1, Kang Ning2* From Asia Pacific Bioinformatics Network (APBioNet) Thirteenth International Conference on Bioinformatics (InCoB2014) Sydney, Australia. 31 July - 2 August 2014 Abstract Background: The human habitat is a host where microbial species evolve, function, and continue to evolve. Elucidating how microbial communities respond to human habitats is a fundamental and critical task, as establishing baselines of human microbiome is essential in understanding its role in human disease and health. Recent studies on healthy human microbiome focus on particular body habitats, assuming that microbiome develop similar structural patterns to perform similar ecosystem function under same environmental conditions. However, current studies usually overlook a complex and interconnected landscape of human microbiome and limit the ability in particular body habitats with learning models of specific criterion. Therefore, these methods could not capture the real-world underlying microbial patterns effectively. Results: To obtain a comprehensive view, we propose a novel ensemble clustering framework to mine the structure of microbial community pattern on large-scale metagenomic data. Particularly, we first build a microbial similarity network via integrating 1920 metagenomic samples from three body habitats of healthy adults. Then a novel symmetric Nonnegative Matrix Factorization (NMF) based ensemble model is proposed and applied onto the network to detect clustering pattern. Extensive experiments are conducted to evaluate the effectiveness of our model on deriving microbial community with respect to body habitat and host gender. From clustering results, we observed that body habitat exhibits a strong bound but non-unique microbial structural pattern. Meanwhile, human microbiome reveals different degree of structural variations over body habitat and host gender. Conclusions: In summary, our ensemble clustering framework could efficiently explore integrated clustering results to accurately identify microbial communities, and provide a comprehensive view for a set of microbial communities. The clustering results indicate that structure of human microbiome is varied systematically across body habitats and host genders. Such trends depict an integrated biography of microbial communities, which offer a new insight towards uncovering pathogenic model of human microbiome. Background Metagenomic background The human body is a content that complex microbial communities are living inside and on. This microbiome occupies body habitats and endows us with ecosystem functions, such as nutrition, pathogen resistance and * Correspondence: [email protected] 2 Computational Biology Group of Single Cell Center, Shandong Key Laboratory of Energy Genetics and CAS Key Laboratory of Biofuels, Qingdao Institute of Bioenergy and Bioprocess Technology, Chinese Academy of Science, Qingdao 266101, China Full list of author information is available at the end of the article immune system development [1,2], to help maintain our health. Hence systematically defining the “normal” states of human microbiome is an important step towards understanding role of microbiota in pathogenesis [3]. However, the majority of microbiomes have been poorly investigated. To understand the principle of human microbiome, prior research concentrated on particular body habitats [3-8]. For example, Turnbaugh et al. [9] investigated the gut microbiome in obese and lean twins to address how host, environmental condition and diet influence the © 2014 Yang et al.; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The Creative Commons Public Domain Dedication waiver (http:// creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated. Yang et al. BMC Systems Biology 2014, 8(Suppl 4):S7 http://www.biomedcentral.com/1752-0509/8/S4/S7 microbial components. Grice et al. [10] targeted human skin microbiome to characterize its topological and personal variations within multiple sites. Bik et al. [11]’s research indicated the distinctness of microbial structure on oral cavity and tongue. However, human microbial habitats are not isolated with one another; instead they reveal community structure correlation across body habitats [12]. In this case, ensemble of different habitat samples could bring global and full-scale insights into microbiome. Recent studies had aggregated microbial samples from different body habitats to perform a comprehensive study. Costello et al. [13] surveyed the microbiomes that were gathered from 27 body habitats of nine adults. Mitreva [12] carried out the extensive sampling on 18 body habitats from 242 individuals. In order to establish a global insight of human microbiome, they built a “whole-body” microbial similarity network where the nodes were consisted of metagenomic samples from multiple human body sites and the edges as pair-wise phylogenetic similarity of samples were measured in terms of their shared evolutionary history. Clustering approaches [14] had been applied on this large-scale similarity network to group samples that shared more similar phylogenetic structures with each other within the clusters than other ones. From these clusters, researchers could infer how microbial patterns were affected by body habitat, host gender and environmental condition with time. Costello et al. [13] proposed a hierarchical clustering algorithm on a microbial community network and found out personal microbiota relatively stable within habitats over time. Turnbaugh et al. [9] identified two distinct functional modules on gut microbiome via principal components analysis (PCA) and hierarchical clustering algorithm, and experimental results disclosed that microbiome within same clusters carried out similar ecosystem-level functions. Mitreva [12] adopted a centroid-based clustering algorithm and discovered the covariation and co-exclusion of microbiome between different habitats. Current Limitations Clustering approach aims to group metagenomic samples with similar phylogenetic patterns. It can be achieved by various algorithms that differ significantly in terms of computational principles and measures, by which each generated clustering results can be viewed as taking a different “look” through data (as shown in Table 1). However, most of prior studies employ one particular clustering approach, by which the clustering outputs tend to be specific towards the criterion of the proposed approach. For example, density-based clustering algorithm groups samples that are densely connected in similarity network. However, true microbial Page 2 of 12 communities are not limited to densely connected structures; samples with sparsely microbial structure widely exist in the lake [15]. Graph partition-based clustering such as MCL [16] and K-means clustering [17] explores the best partition of a network. But these algorithms do not allow the overlaps between clusters. Therefore, they are unable to discover shared microbe between two communities, such as some species that could adapt in multi-environmental conditions like microbial mats and biofilms. Hierarchical clustering algorithm [18] learns the hierarchical structure of a network, which has been used in [13], but hierarchical structure is determined by local optimization criterion as such there is no global objective function, which might lead to small clusters with only part of similar samples. Distribution-based clustering approach, like expectation-maximization (EM) [19], identifies the clusters that follow statistical Condorcet criteria. But statistical model for microbial community remains rarely known and therefore it is difficult to evaluate reliability of the results. Advantage of proposed Ensemble clustering framework Ideally, a clustering algorithm should be able to exploit clustering patterns as comprehensive as possible. However, as we have mentioned above, few algorithms are capable of taking into consideration all factors. Different clustering algorithms may produce different partitions of the network. Given multiple clustering results, we need to explore their information and output more robust results that can exploit the complementary nature of these patterns. Ensemble clustering was proposed recently which has been successfully used to solve many community detection problems [20-23]. Thus, we use ensemble clustering framework to integrate the various kinds of clusters (here we call them base clustering results) and output more comprehensive results. In this study, we first construct a consensus matrix which measures similarity of samples based on co-occurrence of samples in base clustering results [24]. Next we apply Symmetric Nonnegative Matrix Factorization (NMF) [25] on the consensus matrix to derive clusters. Symmetric NMF provides a lower rank approximation of a nonnegative matrix, which could be easily related to the clustering of the nonnegative data. As mentioned in [25], the factorization of the consensus matrix will generate a clustering assignment matrix that could capture the cluster structure inherent in the network. Unlike prior researches that applied single cluster algorithm on particular habitat microbiome, our framework assembled clustering algorithms of different human microbiome in different body habitats. We carried out our experiments to demonstrate its capability in capturing the microbial community. Experimental Yang et al. BMC Systems Biology 2014, 8(Suppl 4):S7 http://www.biomedcentral.com/1752-0509/8/S4/S7 Page 3 of 12 Table 1 Summary of four particular clustering approaches Clustering Approaches Density-based clustering Characteristics Limitations on microbial pattern Clusters are defined as connected dense regions in the network True microbial community are not limited to densely connected structures; sparsely microbial structure still exists Graph partition- Clusters are generated via graph partitioning techniques based clustering Partition based algorithms do not allow the overlaps between clusters. Therefore, they are unable to discover shared microbe among clusters, such as some species that could adapt in multi-environmental conditions like microbial mats and biofilms Hierarchical clustering Clusters are built based on an agglomerative clustering model that shows relations between the members and groups Hierarchical structure is determined by local optimization criterion as such there is no global objective function, which might lead to small clusters with only part of similar samples Distributionbased clustering Clusters are modelled using statistical distributions Statistical models of microbial communities are still unknown and need to be further explored results showed that predicted clusters were capable of revealing the spatial and gender roles of human microbiota and eventually elaborated human microbiome biogeography, which provided new insights about disease pathogenesis of human microbiome [9,12,13]. Material and methods In this section, we first briefly introduced the experimental data, the similarity measurements of metagenomic samples and GPU based fast similarity matrix computing. Then we described the schema of ensemble clustering framework and its phases to structure microbial community. Experimental data In this work, we used 1920 metagenomic samples from the project “Moving pictures of human microbiome” [26] to build the microbial matrix and similarity network (refer to section “Similarity measurements of metagenomic samples” for details). A sample of metagenomic matrix and network were illustrated in Figure 1 and the similarity matrices of all datasets were shown in Additional file 2: Table S1. GPU-Meta-Storms [27] were performed to measure structural similarity of metagenomic samples (Efficiency of GPU-Meta-Storms is shown in Additional file 2: Figure S1). Metagenomic samples were annotated by two meta-labels: Habitat = {gut, skin, oral cavity} defined human body habitat the samples live in, while Gender = {male, female} defined the gender of host the samples inhabit. Combining the two meta-labels, each sample was partition into one of six meta-classes, they were {male & gut, male & skin, male & oral cavity, female & gut, female & skin, female & oral cavity}. Table 2 summarized the distribution of 1920 metagenomic samples on three body habitats and two host genders. Similarity measurements of metagenomic samples The scoring function of Meta-Storms [27] compared two microbial samples’ structure by calculating the maximum common component of their common phylogenetic tree Figure 1 An example of (A) similarity matrix and (B) its similarity network. In the matrix, each tile indicates a similarity value between 2 samples by colour gradient from red (high) to green (low). In the network, each node represents a sample, and edges represent similarity values in the matrix. Yang et al. BMC Systems Biology 2014, 8(Suppl 4):S7 http://www.biomedcentral.com/1752-0509/8/S4/S7 Page 4 of 12 Table 2 1920 microbial samples on six human body habitats Gut Skin Oral 1395 Male 331 698 366 Female 130 262 133 525 Total 461 960 499 1920 Total considering the b-diversity, phylogenetic distance and abundance of each species (Formula 1). The scoring function first evaluated the common abundance of each species on the leaf node, which was considered as the smaller abundance value in two samples. These abundance values were propagated to their ancestors iteratively, and the accumulative common abundance values at the root node reflected the overall similarity between the two metagenomic samples, which could be computed using Similarity (Root) defined in Formula 1. ⎧ ⎪ ⎪ Common Abundance(X) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ Similarity(X) = If X is a leaf node Common Abundance(X) If X is an internal node ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ +Similarity(X.Left) ⎪ ⎪ ⎩ +Similarity(X.Right) (1) Then we constructed the similarity matrix based on the pair-wised similarity among all sample pair (Figure 1 (A)). Exploiting the multi-core architecture of the GPU [28], Formula 1 could be invoked in parallel using a large number of threads to compute similarity between different pairs of metagenomic samples. To compute the pair-wise similarity matrix for N samples, we spawned N * N threads in the GPU such that each similarity value in the matrix was processed by an independent thread. Figure 2 Overview of the GPU based similarity matrix computing. Figure 2 illustrated the GPU computing workflow: to build the common phylogenetic tree, we first loaded and initialize abundant specie data from the file system to main memory; this data was then reloaded to the GPU for computing. When all threads of the GPU kernel had been completed (Figure 1, step 3, the key step), these values were returned back to RAM to populate the similarity matrix, which was then stored in the file system. Ensemble clustering framework In this subsection, we proposed a novel ensemble clustering framework, namely Meta-EC, to perform microbial community pattern detection. The framework consisted of two stages: a generation phase where a consensus matrix was constructed based on base clustering results and an identification phase in which a symmetric NMF-based clustering was used to detect reliable clusters from the consensus matrix. The schema of our Meta-EC algorithm was presented in Figure 3. Terminology: After computing the pair-wise similarity matrix of the metagenomic samples, we used it to construct the microbial similarity network that was reformatted as a simple undirected graph G = (V, E), where V defined a vertex set which containeed \|V\| = N vertices, and E an edge set. A vertex v ∈ V represented a metagenomic sample and a weighted edge e ∈ E represented the polygenetic structure similarity of two samples (Figure 1(B)). A cluster Ci = Vci , Eic was a subnetwork of G such that Vci ⊂ V and Eic was the set of edges induced by Vci from G. A microbial community of G was a set of predicted microbial clusters, defined as {C1,...,Cm}. Generation phase: When the similarity network was ready, a set of base clustering results were calculated by Yang et al. BMC Systems Biology 2014, 8(Suppl 4):S7 http://www.biomedcentral.com/1752-0509/8/S4/S7 Page 5 of 12 Figure 3 The schema of Meta-EC algorithm. applying four clustering algorithms (base clustering algorithms) on the similarity network with different initializations, as shown in Figure 3(A). The base clustering algorithms included EM algorithm, K-mean clustering, hierarchical clustering and density-based clustering, as present in Table 1 and Additional file 1: Section 1. A consensus matrix W was introduced to measure the co-occurrence of samples in clusters of the base clustering results. Each Wij indicated the number of base clustering results in which sample i and sample j were assigned to the same cluster, divided by the total number of base clustering results. Therefore, matrix W took into consideration all generated clusters and reflected the co-clusters similarity between each pair of samples based on different clustering criterions. The higher the value of Wij, the more likely sample i and sample j belonged to the same cluster. Identification phase: When the consensus matrix was constructed, we applied a symmetric NMF-based clustering algorithm on this matrix to derive the clusters. The flowchart of this algorithm was shown in Figure 3 (B). The main idea of this algorithm was outlined as follows: The symmetric NMF defined in Equation (2) was suitable for network clustering based on similarity matrix W: minH≥0 D W\|HHT (2) Here D(W\|HH T ) was a predefined cost function K and K was the predefined number of clusters. H was a cluster indicator matrix in which each entry hi,k denoted the real-valued membership of sample i belonging to cluster k. So we could easily infer the clustering assignment of sample i from the i-th row of H. In this study, we used Kullback-Leibler (KL) divergence [28] as the cost function, which could be represented as: D W HHT = DKL W HHT = N i,j=1 Wij Wij log HHT ij − Wij + HHT ij (3) We chose KL-divergence as the cost function since it was free of noise parameter and had been widely used in NMF. A sample may belong to more than one cluster, but it seldom belonged to all clusters. Thus, the cluster indicator matrix H should be sparse. To achieve sparsity of Yang et al. BMC Systems Biology 2014, 8(Suppl 4):S7 http://www.biomedcentral.com/1752-0509/8/S4/S7 Page 6 of 12 the solution of H, a L1-norm regularization for H was integrated. Neglecting constants and adding the L1norm regularization for H, the modified formulation was as follows: min − H≥0 N N i=1 j=1 Wi,j log HHT i,j − HHT i,j + N K i=1 z=1 βhi,z (4) where the hyper-parameter b > 0 controlled the sparsity of H, and H ≥ 0 was the cluster indicator matrix. Solution to NMF-based Ensemble Clustering: Minimization of the cost function in equation (4) with constraints formed a constrained nonlinear optimization problem. Similar to [29,30], we adopted the multiplicative update rule [31] to estimate H, which was widely accepted as a useful algorithm in solving nonnegative matrix factorization problem. By the multiplicative update rule, we obtained the following update rules for hi,z: \|V\| j=1 hi,z ← Wi,j hj,z K l=1 hi,l hj,l hi,z 1 + hi,z \|V\| 2 2 j=1 hj,z + 0.5β (5) We iteratively updated H according to the updating rule (5) until they satisfied a stopping criterion. Let Hl be the cluster indicator matrix at iteration time l (l > 1). The algorithm was stopped whenever \|\|Hl - Hl-1\|\|1 < r, where r was a predefined tolerance parameter. Here we set r = 10-6 as the default value of tolerance parameter. In addition, the maximum of iteration time was limited to 200 iterations if the stopping criteria r was unsatisfied. In order to avoid local minimum, for random initialization, we repeated the algorithm 10 times with random initial conditions and chose the results with lowest value of the cost function (4). From cluster indicator matrix to microbial clusters: Similar to [32], we obtained microbial clusters from cluster indicator matrix H by taking the threshold τ to assign a sample to a cluster when its weight for the cluster exceeded τ. In this way, we obtained theresultant samplecluster membership matrix H∗ = h∗i,z , where h∗i,z = 1 if hi,z ≥ τ and h∗i,z = 0 if hi,z < τ . Here, h∗i,z = 1 mean sample i was assigned to detected cluster z and h* mean the final output of h. After completing these steps, we obtained the refined clusters EK that satisfied the following conditions: EK = {C1 , ..., CK } : vi ∈ CZ , if .h∗i,z = 1 (6) where i = 1,...,N and z = 1,...,K. We summarized the whole algorithm in Figure 4. Results In this section, we focused on evaluating the effectiveness of Meta-EC algorithm. Before presenting the experimental results, we first introduced our experiment design: evaluation metrics and experimental settings in Figure 4 The algorithm of Meta-EC for microbial community pattern detection. our study. Then we conducted experimental comparison between Meta-EC and base clustering approaches, and comparison between constructed consensus network and original metagenomic similarity network. Finally, from clustering results, we investigated how human microbial community was influenced by body habitat and host gender. Evaluation metrics In this work, we evaluated the effectiveness of clustering algorithms by observing how well detected clusters corresponded to the sampling information of habitats and genders (six meta-classes, refer to subsection Terminology for details). Since the true number of cluster patterns for habitat and gender was unknown, and there were no literature references to clearly mention how to determine the number of cluster patterns in either body habitat or host gender, we empirically defined reference clusters based on six meta-classes. Assuming that metagenomic samples with identical meta-classes were likely to have similar microbial structures [13], we bring the metagenomic samples with identical meta-classes into one reference cluster. Typically, the quality of the predicted clusters could be evaluated by following three quantity measures, f-measure [33], PR metrics and F-score, which could measure how well the detected clusters corresponded to reference clusters. Among these three measures, f-measure which was the harmonic mean of Precision and Recall, aimed at assessing how well the detected clusters matched reference Yang et al. BMC Systems Biology 2014, 8(Suppl 4):S7 http://www.biomedcentral.com/1752-0509/8/S4/S7 Page 7 of 12 ones at cluster level (Precision measured what fraction of the detected clusters were matched with reference ones and Recall measured what fraction of reference clusters were matched to detected clusters). PR-based metric took into account the overlap between detected and reference clusters. F-score focused on measuring whether samples within identical habitats were grouped together in the detected clusters. The value of each measure varied from 0 to 1, and the higher value indicated better match. For more details of f-measure, PR metrics and F-score, please refer to Additional file 1: Section 2. And the parameter setting in the experiments is introduced in Additional file 1: Section 3. Evaluation of clustering results generated by Meta-EC algorithm In this subsection, to evaluate the performance of MetaEC algorithm, we presented performance comparison of proposed Meta-EC algorithm with base clustering approaches and comparison of constructed consensus matrix with original microbial similarity matrix. Comparison against four base clustering approaches: To evaluate the performance of ensemble clustering approach, the accuracy of the clustering results derived from our proposed approach was compared with the ones derived from these base clustering algorithms. Figure 5 illustrated the performance of different clustering algorithms in terms of three metrics (PR, f-measure and F-score) with respect to the reference clusters. From Figure 5, we could observe that our ensemble-based approach had competitive performance compared with the base clustering algorithms as regard to all three measures. Among the base clustering algorithms, K-means with cluster number set to 6 had better performance in terms of PR, while K-means and Density-based clustering with cluster number set to 6 had better performance in terms of f-measure, and Hierarchical clustering with cluster number set to 9 and 10 had comparable performance with K-means and Density-based clustering with cluster number set to 6 in terms of F-score. But none of them could have superior performance than others as regard to all three measures. However, our ensemble-based approach obtained the best performance in terms of all the three measures. This may be owing to the fast that ensemble-based approach could make use of clusters derived from different base clustering algorithms and extract more reliable results. In addition, we conducted sensitivity study of phylogenetic structure similarity on microbial network. We ran algorithm Meta-EC with threshold value of metagenomic similarity in matrix tuning from 0.7 to 0.9 with 0.1 as step size, the results in Additional file 2: Figure S4 showed Meta-EC outperformed other state-of-art clustering techniques in the wide range of edge threshold, indicating that our Figure 5 Performance comparison of ensemble clustering framework to base clustering algorithms with respect to fmeasure, PR and F-score. Note that ensemble-based approach with random initialization is denoted as “Ensemble_random”, while ensemble-based approach with a base clustering result as initial input is denoted as “Ensemble_initial”. The result of “Ensemble_random” is obtained with β = 1. algorithm is robust and insensitive to the similarity network noisy and data coverage. In addition, we have compared the computational time with base clustering approaches in Table 3 and results show that Meta-EC Yang et al. BMC Systems Biology 2014, 8(Suppl 4):S7 http://www.biomedcentral.com/1752-0509/8/S4/S7 Page 8 of 12 Table 3 Comparison with bases clustering approaches on computational time Method EM Time(S) 221.17 K-Means Hierarchical Density-based Meta-EC 11.57 55.04 12.24 72.8 exactly spend more time than that by K-mean and Hierarchical clustering but less than EM clustering, so the total time cost of MetaEC is the sum of all base clustering algorithms plus 72.8 seconds. With rapid development of computational capability, we could improve the time efficiency on large amount of operations. Comparison of constructed consensus network with original similarity network: To demonstrate the benefits of combining different base clustering results, we applied symmetric NMF on original metagenomic similarity network and evaluated its performance. To be fair, the results of symmetric NMF on original metagenomic similarity network were obtained over the best tuned parameter. The comparison of the two tested similarity network is present in Figure 6 as regard to F-measure, PR and f-measure. The results in Figure 6 showed that applying symmetric NMF on consensus matrix achieved better performance than that on the original similarity network. These results demonstrated the benefits of combining different base clustering results. If the similarity matrix was well constructed (each element reflected the cocluster similarity), the factorization of the similarity matrix would generate a clustering assignment matrix Figure 6 Performance comparison of Bayesian NMF based clustering algorithm applied on ensemble clustering similarity network and original microbial similarity network. Additive values of three measures are present for each data source. For random initialization case, the value of β is set to 1 and the result corresponds to “Original_random”. We also choose the base clustering results which presents the best performance as the initial input of symmetric NMF and the result corresponds to “Original_initial”. that could well capture the cluster structure inherent in the network representation. However, the original network weighted the interaction via measuring the phylogenetic structure of samples. In this way, metagenomic samples with higher phylogenetic similarity were more likely to be involved in one cluster. If the actual microbial pattern was uncorrelated with phylogenetic similarity, the community detected by symmetric NMF may be unreliable. In ensemble clustering framework, we generated a consensus matrix that integrated the clustering results derived from different clustering algorithms. Each element in consensus matrix indicated the frequency of the corresponding sample pair being clustered together in these base clustering results. Thus, applying symmetric NMF on consensus matrix could take into consideration the co-cluster strength of multiple clustering patterns and output a more comprehensive and robust result. Interpretation of Microbial community patterns on human body habitats based on clustering results Recall that metagenomic samples were clustered in terms of co-occurrence frequency in base clustering results. Hence the final output clusters assembled samples to represent unique microbial patterns that are the consensus from base clustering approaches. Next, from the clustering results, we infer how microbial pattern was influenced by body habitats and host genders. Structural variation across body habitats: Through analyzing the enrichment of body habitat and host gender over six predicted clusters, the results in Figure 7 revealed a stronger coherence by body habitat than host gender. These clusters dominated by particular body habitats inferred that these body habitats harboured distinctive microbial patterns, which was also observed in base clustering results in Additional file 2: Figure S3. Although four base clustering algorithms generate clustering patterns with different criterions, most clusters in Additional file 2: Table S2 were enriched with particular habitats. Meanwhile, we observed that microbial communities at different body habitat exhibited different degree of compositional structure variation. Figure 7 showed that microbial structure remained relatively stable in oral cavity, compared with diverse microbial structures harboured in skin. It was biologically reasonable to detect diverse patterns on skin, since there were quite different places where skin microbial communities could be sampled. Different extend of habitat structural variation were also observed in base clustering results. In Additional file 2: Figure S2, gut and oral cavity microbial community patterns were only fit with one clustering criterion, gut consistent with K-means and oral cavity with Yang et al. BMC Systems Biology 2014, 8(Suppl 4):S7 http://www.biomedcentral.com/1752-0509/8/S4/S7 Page 9 of 12 Figure 7 Sample distribution on predicted clusters with respect to body habitat and host gender. hierarchical clustering. Contrary to gut and oral cavity, skin-enriched cluster could be recognized by four clustering criterions in all experimental settings, inferring skin samples have many cluster patterns with diverse microbial structures. Note that the proposed Meta-EC generates a more comprehensive community patterns with respect to meta-data since our result is an agreement by consensus of multiple base clustering approaches. For example, compared to Hierarchical and EM clustering results in Additional file 2: Figure S3 that only capture male-gut cluster, ensemble clustering is able to uncover femalegut specific clusters (shown in Figure 7), indicating that Meta-EC could reveal degree of structural variation over body habitat more comprehensively than base clustering results. Structural variation across host gender: We further assessed microbial structure variation with respect to host gender. Meta-Storm [27] was used to measure similarity of two metagenomic samples. The results in Figure 8 indicated that over all habitats, variation was significantly less within same gender samples than between opposite gender samples. However, these habitats perform different degree of structural variation with respect to host gender. Oral cavity microbiome exhibited a stable structure both among same and opposite gender individuals (both above 92% phylogenetic structure similarity). And skin communities had no unique structural variation patterns regarding to host gender. Gut community structure was highly variable between samples from opposite gender hosts (less than 90% similarity value for opposite gender samples of gut cluster 3), but exhibited strong coherence to same gender hosts. On the other hand, the enrichment study in Figure 7 showed that two gut clusters were distinct with host gender, indicating that opposite sexual individuals may exhibit a distinct microbial composition in gut. Microbial interconnection over habitats: Although microbial communities reflected unique structures (distributions) over body habitats, the interconnected microbial components among the body habitats were still observed in the clustering results. For example, cluster 1 in Figure 7 contained 10 skin samples that shared similar microbial compositions with oral cavity communities, while skin cluster 2, 4 and 6 harboured 6, 15 and 2 oral cavity samples respectively. Since skin microbial pattern was closely associated with external Yang et al. BMC Systems Biology 2014, 8(Suppl 4):S7 http://www.biomedcentral.com/1752-0509/8/S4/S7 Figure 8 Structural variation over host gender in oral cavity, gut and skin-dominated clusters. environment [34] and oral cavity was an open system where microbiome from external environment was imported by breathing, eating food and drinking water [35], oral cavity and skin would respond to outside environmental conditions, and gradually evolve similar microbiomes. Conclusions and discussions The human microbiomes are microbiomes that are hosted in gut, oral mucosa and multi-layer of skin etc. These organisms perform ecosystem-level functions that are useful for human host to maintain healthy, yet detailed factors that attribute the microbial community structures in human body habitats and host gender remain poorly conceptualized. To fully understand the roles of human microbiome in disease and health, prior studies focus on particular body habitats of health individuals with specific clustering approaches, based on the assumption that metagenomic samples of same body habitats would develop similar microbial structure patterns. However, human habitats are not isolated; they are interacted and correlated to form an integrated and complex system. And identified structures might be unsuccessful due to noisy sample similarity and specific topological structure within metagenomic network. Hence, single clustering algorithm rarely achieves optimal outcome. To uncover a global and comprehensive landscape of human microbiome, we perform an ensemble clustering framework Meta-EC on large- Page 10 of 12 scale metagenomic samples. In this study, our proposed Meta-EC algorithm has four main advantages on microbial pattern detection: (1) Meta-EC could effectively identify more reliable microbial communities via integrating many base clustering results, (2) As regard to the modularity of microbial communities, defined as the clustering of microbial communities (modularity) according to the effects of their related environments or treatments (meta-data), the consensus clustering network is much clearer at showing such modularity property (such as how environments (or meta-data) shape microbial communities in body habitat, which is critical to healthcare and diognosis) than the original metagenomic similarity network [25], (3) Ensemble framework is robust for the coverage of metagenomic similarity network (as shown in Additional file 2: Figure S4), and (4) Compared to base clustering results in Additional file 2: Figure S3, Meta-EC algorithm could reveal the spatial and gender patterns of microbiome (as shown in Figure 7) more comprehensively, as the ensemble clustering result is a general agreement by multiple base clustering approaches. Nevertheless, it should be acknowledged that the performance of our algorithm depends on the base clustering results and quality of original metagenomic similarity network. If all these base results were generated by poor clustering algorithms, the ensemble outputs would be far from real microbial community similarity patterns. If the original similarity network is unreliable to capture the modularity of metagenomic samples, none of clustering approaches could work. To address this problem, we have to integrate more base clustering approaches with diverse optimization criterions and pattern assumptions, to reduce the bias generated by base approaches. We assume these algorithms can capture a wide variety of clustering patterns in similarity network to alleviate the effect of unreliable clustering results. On the other hand, the proposed NMF based mode, which could be used in association study of bioinformatics domain [36-39], is a more complex method to implement, and convergence could be slow, as shown in Table 3. With rapid development of computational capability, we could improve the time efficiency on large amount of operations. And the nonnegative constraints on cluster indicator matrix H may be an insufficient condition for achieving sparseness in some cases [20]. Then one may set appropriate thresholds to enforce sparseness. In summary, Meta-EC is an ensemble clustering framework for large-scale metagenomic data analysis and microbial community pattern detection. In the future, NMF based model could be exploited to offer potential applications on bipartite model of drug-target association [40] and disease gene prediction [41]. Yang et al. BMC Systems Biology 2014, 8(Suppl 4):S7 http://www.biomedcentral.com/1752-0509/8/S4/S7 Availability The data sets and supporting experimental results of this article are available for download from http:// datam.i2r.a-star.edu.sg/MetaEC. Page 11 of 12 5. 6. 7. Additional material Additional file 1: Experimental Design. The file show the experimental design in this paper, including: (1) introductory of four base clustering approaches; (2) evaluation of microbial clusters; (3) parameter setting. Additional file 2: Supplementary Material. The file presents several figures, tables and additional experimental results mentioned in this paper, including: (1) the efficiency of GPU-meta-storm algorithm; (2) evaluation of four base clustering results; (3) sensitivity study of phylogenetic structure similarity on microbial network. 8. 9. 10. 11. Competing interests The authors declare that they have no competing interests. Authors’ contributions Conceptualized and designed the method and drafted manuscript: PY KN. Responsible for the implementation: PY XS LOY. Provided raw data: KN XS. Participated in discussion and improved the method as well as revised the draft: XS LOY H-NC X-LL. Read and approved the manuscript: PY XS LOY HNC X-LL KN. Acknowledgements This work is supported in part by Chinese Academy of Sciences’ e-Science grant INFO-115-D01-Z006, Ministry of Science and Technology’s high-tech (863) grant 2009AA02Z310 and 2014AA21502, as well as National Science Foundation of China grant 61103167 and 31271410. Declarations Publication costs for this article were partially funded by Chinese Academy of Sciences’ e-Science grant INFO-115-D01-Z006, Ministry of Science and Technology’s high-tech (863) grant 2009AA02Z310 and 2014AA21502, as well as National Science Foundation of China grant 61103167 and 31271410 and by the Institute for Infocomm Research, Agency for Science, Technology & Research (ASTAR), Singapore. This article has been published as part of BMC Systems Biology Volume 8 Supplement 4, 2014: Thirteenth International Conference on Bioinformatics (InCoB2014): Systems Biology. The full contents of the supplement are available online at http://www.biomedcentral.com/bmcsystbiol/supplements/ 8/S4. Authors’ details Institute for Infocomm Research, Agency for Science, Technology & Research (ASTAR), Singapore, 138632, Singapore. 2Computational Biology Group of Single Cell Center, Shandong Key Laboratory of Energy Genetics and CAS Key Laboratory of Biofuels, Qingdao Institute of Bioenergy and Bioprocess Technology, Chinese Academy of Science, Qingdao 266101, China. 3Center for Computer Vision and Department of Mathematics, Sun Yat-Sen University, Guangzhou, 510275, China. 1 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. Published: 8 December 2014 References 1. Wilson M: Bacteriology of humans: an ecological perspective. John Wiley & Sons 2009. 2. Dethlefsen L, McFall-Ngai M, Relman DA: An ecological and evolutionary perspective on human-microbe mutualism and disease. Nature 2007, 449(7164):811-818. 3. Turnbaugh PJ, Ley RE, Hamady M, Fraser-Liggett CM, Knight R, Gordon JI: The human microbiome project: exploring the microbial part of ourselves in a changing world. 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Dynamic Loop Parallelisation Adrian Jackson∗ and Orestis Agathokleous∗ ∗ EPCC, arXiv:1205.2367v1 [cs.PL] 10 May 2012 The University of Edinburgh, Kings Buildings, Mayfield Road, Edinburgh, EH9 3JZ, UK Abstract—Regions of nested loops are a common feature of High Performance Computing (HPC) codes. In shared memory programming models, such as OpenMP, these structure are the most common source of parallelism. Parallelising these structures requires the programmers to make a static decision on how parallelism should be applied. However, depending on the parameters of the problem and the nature of the code, static decisions on which loop to parallelise may not be optimal, especially as they do not enable the exploitation of any runtime characteristics of the execution. Changes to the iterations of the loop which is chosen to be parallelised might limit the amount of processors that can be utilised. We have developed a system that allows a code to make a dynamic choice, at runtime, of what parallelism is applied to nested loops. The system works using a source to source compiler, which we have created, to perform transformations to user’s code automatically, through a directive based approach (similar to OpenMP). This approach requires the programmer to specify how the loops of the region can be parallelised and our runtime library is then responsible for making the decisions dynamically during the execution of the code. Our method for providing dynamic decisions on which loop to parallelise significantly outperforms the standard methods for achieving this through OpenMP (using if clauses) and further optimisations were possible with our system when addressing simulations where the number of iterations of the loops change during the runtime of the program or loops are not perfectly nested. I. I NTRODUCTION High Performance Computing (HPC) codes, and in particular scientific codes, require parallel execution in order to achieve a large amount of performance increase. Depending on the underlying parallel platform which is used, programmers use different programming models in order to achieve parallel execution. In distributed memory systems, the message passing programming model is the most commonly used approach for applying parallelism in the codes. In shared memory systems however, an attractive choice for parallel programming is through OpenMP[16]. The parallelisation of codes with OpenMP is often achieved with loop parallelisation. As long as the iterations of a loop are independent, they can be distributed to the available processors of the system in order to execute them in parallel. A programmer is required to specify a loop that can be parallelised by placing compiler directives before the loop, resolving any dependency issues between the iterations beforehand. HPC codes often consist of regions with nested loops of multiple levels. In order to parallelise these regions, a choice must be made on how parallelism should be applied on the loops. Even though OpenMP supports a variety of strategies for parallelising nested loops, only a single one can be used to parallelise the code. A static choice however, cannot exploit any runtime characteristics during the execution of the program. Changes in the input parameters of the executable which affect the iterations of the loops may render the parallelisation decision suboptimal. In addition to this, the iterations of a loop can change at runtime due to the nature of the code. A common feature of HPC codes is to organise the data into hierarchies, for example blocks of multi-dimensional arrays. Depending on the problem, the blocks can have different shapes and sizes. These parameters affect the loops that are responsible for accessing this data. In some situations, a static decision has the potential to impose a limitation on the amount of processors that can be used for the parallel execution of the loops. With the current trend of chip manufactures to increase the number of cores in the processors in each generation leading to larger and larger shared memory system being readily available to computational scientists on the desktop and beyond, a more dynamic approach must be considered for taking such decisions. This report outlines our investigations into various strategies that can be applied at runtime in order to make a dynamic decision on how to parallelise a region with nested loops. Our approach is to try to automatically perform modifications to users code before compilation in order to enable the code to make these decisions dynamically at runtime. Specifically, we investigated the possibility of having multiple versions of a loop within a region of nested loops in order to make a dynamic choice on whether a loop should be execute sequentially or in parallel. II. O PEN MP OpenMP [1] is, arguably, the dominant parallel programming model currently used for writing parallel programs for used on shared memory parallel systems. Now at version 3.1, and supported by C and FORTRAN, OpenMP operates using compiler directives. The programmer annotates their code specifying how it should be parallelised. The compiler then transforms the original code into a parallel version when the code is compiled. By providing this higher level of abstraction, OpenMP codes tend to be easier to develop, debug and maintain. Moreover, with OpenMP it is very easy TABLE I S TRATEGIES FOR PARALLELISING NESTED LOOP REGIONS Name Description Outermost Inner Loop Nested Loop Parallelisation of the outermost loop Parallelisation of one of the inner loops Parallelisation of multiple loops with nested parallel regions Collapsing the loops into a single big loop Loop Collapsing Loop Selection Runtime loop selection using if clauses to develop the parallel version of a serial code without any major modifications. Whilst there are a number of different mechanisms that OpenMP provides for adding parallel functionality to programs, the one that is generally used most often is loop parallelisation. This involves taking independent iterations of loops and distributing them to a group of threads that perform these sets of independent operations in parallel. Since each of the threads can access shared data, it is generally straightforward to parallelise any loop with no structural changes to the program. III. N ESTED L OOPS HPC codes, and particularly scientific codes, deal with numerical computations based on mathematical formulas. These formulas are often expressed in the form of nested loops, where a set of computations is applied to a large amount of data (generally stored in arrays) and parallelisation can be applied to each loop individually. The arrays often consist of multiple dimensions and the access on the data is achieved with the presence of nested loops. Furthermore it is not uncommon that the arrangement of the data is done in multiple hierarchies, most commonly in blocks with multidimensional arrays, where additional loops are require in order to traverse all the data. When such code is presented, a choice must be made on which loop level to parallelise (where the parallelisation should occur) [2]. A summary of the available strategies is presented in Table I A. Outermost loop The most commonly used approach is to parallelise the outermost loop of a nested loop region, as shown in Listing 1. Using this strategy, the iterations of the loop are distributed to the members of the thread team. The threads operate in parallel by executing the portion of iterations they are assigned to them individually. The nested loops of the parallel region are executed in a sequential manner. 1 2 3 4 5 6 #pragma omp parallel for private (j) for(i = 0; i < I; i++){ for(j = 0; j < J; j++){ work(); } } Listing 1. Outer loop parallelisation of a nested loop region Parallelising the outermost loop is often a good choice, as it minimises the parallel overheads of the OpenMP implementation (such as the initialisation of the parallel region, the scheduling of loop iterations to threads and the synchronisation which takes place at the end of the Parallel loops). More extensive work on the overheads of various OpenMP directives can be found in [3]. Despite the advantages of the Outermost Loop parallelisation strategy in this context, there are drawbacks of this choice. The maximum amount of available parallelism is limited by the number of iterations of the outerloop loop. Considering the example code in Listing 1, it is only possible to have I tasks being executed in parallel. This restricts the number of threads the code can utilise upon execution, and therefore the number of processors or cores that can be exploited. B. Inner loop This is a variant on the outermost loop strategy, with the difference that one of the inner loops of the region is chosen to be parallelised. This approach will only be required or beneficial if the outer loop does not have enough iterations to parallelise efficiently as this variant on the parallelisation strategy introduces parallelisation overheads by requiring the parallelisation to be performed for each loop of the outerloop rather than once for all the loops (as shown in Listing 2). Further nesting of the parallelisation (at deeper loop levels) will further increase the performance problems; the parallel overheads appear a lot more times, whereas the amount of work of each iteration becomes finer. 1 2 3 4 5 6 for(i = 0; i < I; i++){ #pragma omp parallel for shared (i) for(j = 0; j < J; j++){ work(); } } Listing 2. Inner loop parallelisation of a nested loop region Another issue with this strategy is the scenario where loops are not perfectly nested. In this situation, when there are computations in-between the loops, as shown in Listing 3, parallelising a loop of a deeper level will result in sequential execution of that work. Depending on the amount of the execution time which is now serialised, this approach has the potential to increase the execution time of the code. 1 2 3 4 5 6 for(i = 0; i < I; i++){ somework(); for(j = 0; j < J; j++){ otherwork(); } } Listing 3. Poorly nested loop region example C. Nested The Nested parallelisation strategy exploits the fact that more than one loop can be executed in parallel. By opening multiple nested parallel regions at different levels of loops, as presented in Listing 4, more threads can be utilised during the parallel execution of the code. Unlike the Outermost Loop and the Inner Loop approaches, which can only utilise as many threads as the iterations of the loop with the biggest number of iterations, this strategy can exploit further parallelisation opportunities. Other studies have shown that nested parallelism can give good results on systems with a large number of processors [4] [5]. 1 2 3 4 5 6 7 #pragma omp parallel for private (j) for(i = 0; i < I; i++){ #pragma omp parallel for shared (i) for(j = 0; j < J; j++){ work(); } } Listing 4. Nested loop parallelisation of a nested loop region D. Loop Collapsing The loop collapsing strategy takes a different approach for exposing additional parallelism within nested loop regions. By performing code transformations, multiple nested loops are combined, or collapsed, into a single loop. The newly created loop has a larger amount of iterations, which can be distributed to the threads. As of version 3.0, OpenMP supports loop collapsing by using the COLLAPSE clause in the Loop Construct, requiring the programmer to provide the number of loop levels to collapse. To be able to use the COLLAPSE clause the loops have to be perfectly nested (i.e. no code between the loops) and the number of loop iterations (when multiplied together) need to be able to be regularly divided. Loop collapsing can produce better results than both the inner loop and nested loop strategies, since the parallel overheads are minimal, however it is not always available, either because not all compilers support OpenMP version 3.0, or because the conditions outlined above cannot be met. 1 2 3 4 5 6 7 8 #pragma omp parallel for collapse (3) for(i = 0; i < I; i++){ for(j = 0; j < J; j++){ for(k = 0; k < K; k++){ work(); } } } parallel region is always created in either case. The presence of the if clause only affects the number of threads that get assigned to the parallel region. When sequential execution is triggered, the code is only executed by the master thread, for parallel execution all threads execute the code. Furthermore, with the if clause, programmers are still required to manually write code which makes the decision, construct sensible scalar-expressions to be evaluated, and manually parallelise each loop that is a potential target for parallelisation. IV. DYNAMIC LOOP One of the motivators for this work was a parallelisation that was undertaken of a finite-volume cell-centred structured Navier-Stokes code for undertaking Computational Fluid Dynamics (CFD) simulation. It is a structured mesh, multigrid, code which works with multiblock grids, and includes a range of CFD solvers including; steady state, time-domain dual time-stepping, frequency-domain harmonic balance, and timedomain Runge-Kutta. The general pattern for the computations within the code is shown in Listing 6. Whilst this type of computational pattern is not uncommon for scientific codes one of the challenges in the parallelisation is that as the code can use a range of different methods, as previously outlined, the range of these loops can vary. For instance when performing a time domain simulation the harmonic loop has a single iteration. However, when performing a harmonic balance simulation it can have a range of values, generally between 2 and 16. Furthermore, it is not uncommon to run large simulations with a single block, or a small number of blocks, meaning that the block loop has a very small number of iterations. Finally, each block in the simulation can have different values for its dimensions. In theory, the loop collapsing strategy would be ideal for this type of simulation code as this would enable parallelisation without having to deal with the varying sizes of the nested loops. However, it cannot be guaranteed that for all input datasets the loop iterations can be regularly divided, and there are also particular areas of the code where the loops are not perfectly nested. 1 2 3 4 5 Listing 5. Parallelisation of a nested loop region with loop collapsing 6 7 8 E. Loop Selection OpenMP already provides a way of forcing a parallel region to execute sequentially with the use of the if clause on OpenMP directives. The if clause, of the following form, if (scalar − expression), is used to determine at runtime whether the code enclosed in the parallel region should execute sequentially or in parallel. When the scalar expression of the clause evaluates to 0, the region is executed sequentially. Any other value will result in parallel execution. However, a new PARALLELISATION 9 10 11 for(iter = 0; iter < n_iters; iter++){ for(block = 0; block < n_blocks; block++){ for(harmonics = 0; harmonics < n_harmonics; harmonics++){ for(j_cell = 0; j_cell < n_cells_j; j_cell++){ for(i_cell = 0; i_cell < n_cells_i; i_cell++){ perform computations; } } } } } Listing 6. Example scientific code loops Given the different techniques that can be used to parallelise nested loops, the occurrence of nested loops in many scientific simulation codes, and the fact that the loop iterations of nested loops can change for different input datasets of a code or when performing different functions with a code, we wanted a system that enabled the selection of different parallelisation choices to be available to code at runtime when the specific ranges of the nested loops are known. Our strategy for providing this functionality is to create code, based on the provided user code, that can perform a parallelisation of any of the nested loops and add decision making algorithms to dynamically choose, at runtime, which parallelisation is used. Specifically, we have created tools that create multiple versions of a loop within a region of nested loops in order to make a dynamic choice on whether a loop should execute sequentially or in parallel In general, code duplication is considered bad programming practice as it can, amongst other issues, lead to update anomalies (where not all instances of the functionality are modified when modifications occur) and thus damage the maintainability of the code. However, if the duplicate code (in our instance the serial and parallel versions of each loop in the nested loop structure) can be generated automatically for standard user code then it will not adversely affect the maintainability of the user program. We created a source-to-source compiler that recognises compiler directives within user’s source code and uses them to pre-process the source code and generate a program that has the alternative parallelisation strategies encapsulated within it. By exposing a simple interface to the programmers through compiler directives, which are similar to the already familiar OpenMP compiler directives, we can automatically provide the dynamic parallelisation functionality for users without requiring significant changes to the original source code. Furthermore, this approach provides the users the choice of enabling or disabling our functionality with minimum effort. To complement the code duplication we have also implemented functionality (in a small runtime library) that produces the code which is responsible for deciding what parallelisation to perform automatically. The decision functionality considers the number of iterations of a loop in order to chose a parallelisation strategy that makes best use of the processors or cores available. Our implementation is currently limited to parallelising a single loop of a nested loop region, taking advantage only the Outermost and Inner loop strategies. Other authors [2] have already taken a similar approach by modifying the OpenMP runtime library in order to make these decisions dynamically. However, applying this logic in the OpenMP runtime library would have limited the implementation to a specific compiler. Using our source-to-source compile approach we are aiming to transfer the logic in user code in order to maintain the portability of our solution. In addition to simple heuristics, we also explored the idea of a profile-based approach at runtime in order to detect the best possible parallelisation strategy with time measurements. A heuristics based approach alone cannot capture any information on the amount of the actual computations when making a decision on parallelising a loop. Whilst this is generally irrelevant for perfectly nested loops (as all the work is in the lowest loop), it may have more of an impact where there is work between the different loops as well. There may also Fig. 1. Compilation process using the source-to-source compiler be situations where a different inner loop has slightly more iterations than an outer loop so could be chosen by a simple heuristic as the place where the parallelisation occurs but the overheads associated with parallelising that inner loop actually make this a suboptimal choice. Providing a profiling based decision mechanism may help with both these scenarios, and enable us to identify situations where, for instance, using less threads to parallelise an outer loop might provide a better execution time. The idea of an auto tuning code has already been proposed by other compiler-related researches [6] [7] for producing optimised code, we apply similar logic. V. S OURCE - TO - SOURCE COMPILER Our source-to-source compiler acts as a preprocessor to C code which can contain OpenMP directives, as well as our own directives. The compiler parses the code, and creates an internal representation of the code in the form of an Abstract Syntax Tree (AST). The regions of the input code that contain our directives are translated into the semantics of the C programming language and OpenMP directives during the parse phase, and appropriate nodes for these regions are placed in the AST. The created AST is then translated back to C code with OpenMP directives. This generated code is then compiled using a standard, OpenMP enabled, C compiler to produce a parallel executable (this process is illustrated in Figure 1. Our compiler, implemented using the Lua [8] programming language along with the Lpeg [9] parsing library, recognises a number of our own bespoke compiler directives of the form #pragma preomp. A loop that is preceded by a #pragma preomp f or directive is considered by our compiler as a suitable candidate for applying parallelisation. When such a loop is found, our compiler performs the necessary code transformations so that a decision can be made at runtime whether the loop should run sequentially or in parallel (and to ensure that both the sequential and parallel versions of the loop are available in the executable at runtime). In addition to this, a simple analysis of the loop is performed in order to facilitate the computation of a loops iterations during the making of the decision. An example of such a code is presented in Listing 7. 1 2 3 4 5 6 7 #pragma preomp parallel for private(j) for(i=0; i<I; i++){ #pragma preomp parallel for shared(i) for(j=0; j<J; j++){ work(); } } Listing 7. A nested loop region with preomp Furthermore we also extend the grammar to support an additional clause, the parallel threshold(expression) clause. This is optional, and when it is not present the compiler will assume a default value of 1.0. This clause is used to allow control over when a loop is parallelised, and will be discussed further in Section VI. A. Code Duplication The main function of the source-to-source compiler is to take the original user code and duplicate the loops to be parallelised so that there are both serial and parallel versions of those loops that can be selected at runtime. As previously mentioned our system only allows one loop to be parallelised at any given time (although which loop is parallelised can change over the runtime of a program as the parameters of the loop change), but both the serial and parallel versions of all the loops to be parallelised must appear in the executable to enable a selection at runtime to take place. When a loop is preceded by a #pragma preomp f or directive, the loop is duplicated and wrapped in a normal if − else statement which evaluates a decision function from our runtime library and selects the if or else branch based on the outcome of the evaluation. B. OpenMP if As a comparison to our code duplication approach we also implemented the same functionality uses the existing if clause of the OpenMP Parallel Construct. Our custom directive is translated into an OpenMP Parallel For directive, with an attached if clause in order to decide whether to execute the loop in parallel or not (rather than a serial and parallel version of the loop). The expression of the if clause consists of a call to a decision function of our runtime library, which takes the evaluated expressions of the loops information in order to make a decision. This functionality was included to allow a comparison of our approach to the standard method that developers could currently use to provide dynamic selection of parallelism with OpenMP. However, a major drawback of this approach (and the reason we do not uses it for our functionality) is that a parallel region will be created regardless of whether a loop is parallelised or not. Considering the example in Figure 2, parallelising the outer loop of two nested loops with two threads will result in three parallel regions. Each thread of the outer region will Fig. 2. An example of using the if clause to parallelise (a) the outer and (b) the inner loop of two nested loops with two threads create a new parallel region and become its master. In the case of the inner loop being parallelised, two parallel regions are created. For nested regions with a larger number of loops this method has the potential to produce excessive parallel overheads. VI. D ECISION FUNCTIONS AND THE RUNTIME LIBRARY The runtime library implements the logic for deciding which version of a loop is chosen during execution. Once a code has been processed by the source-to-source compiler it must then be linked with our runtime library to enable this functionality to be used. A. Decision Based On Heuristics Here we use heuristics, based on information collected at runtime, to decide whether a loop should execute sequentially or in parallel. The idea of this approach is to look for the first loop that has enough iterations to utilise all of the available threads, based on the assumption that parallelising outer loops is more efficient than parallelising inner loops as the amount of parallel overheads should be lower (as the OpenMP parallel regions are encountered less frequently). Before the execution of a loop, the decider checks whether a loop of an outer level is already running in parallel. If this condition is met, then the loop is serialised. In the case that no outer loop is running in parallel the number of iterations of the loop is calculated and it is divided with the available number of threads. If this results in a value that is greater than or equal to a specified threshold, then the parallel version of a loop is chosen, otherwise the loop is serialised. As discussed in Section V, the default value of the threshold is 1 (there must be no idle threads) although this can be controlled by the user. The calculations of the iterations is based on the parameters of the loop which are extracted by the source to source compiler and are provided as arguments to the decision function. In the case that the original code of the loop uses variables for its boundaries, any change in their value will also be captured by the decision function during the calculation. This design allows constant monitoring of any changes in the iterations of the loops which also results in dynamic adaptation of the parallelisation strategy during the execution of the program. The algorithm is very simple and with minimum overheads. Moreover, there is no need to maintain any state for the loops. However, the logic which is used by the function is of a program the profiling overhead will only be imposed in the first few iterations of the program. Figure 3 outlines this with an example of three nested loops. VII. P ERFORMANCE E VALUATION Fig. 3. An example of the Heuristics With Profiling Decider on three loops based on optimism. It only considers the amount of parallelism exposed by the loop regardless of whether the amount of work of the loop is big enough to justify any overheads of the parallelisation or whether there is any work between loops. To evaluate the performance of our new functionality we aimed to benchmark it against standard, static, OpenMP parallelisations with a range of different configurations. In particular, we focussed on varying the number of loop iterations, the amount of work between and within loops, and the number of changes that occur to loop bounds during execution to evaluate whether and when our approach is beneficial compared to a static parallelisation. To undertake these benchmarks we used two different codes. The first is a synthetic, configurable, benchmark C code, shown in Listing 8, which we constructed for this evaluation. The number of iterations of each loop can be configured, as can the amount of work that is simulated (by calling the delay function) between the second and third loops, and within the third loop. 1 B. Decision Based On Heuristics With Profiling 2 To address the potential issue with the basic decision based on heuristics previously discussed we also implemented a more complex decision function based on both the size of loops and some evaluation of the work in the loops. In the same manner as the heuristics decider, it uses the same information extracted by the source to source compiler in order to determine whether the loop should be parallelised or not. However, if a loop does not meet the conditions, then the function reverts to a profiling mode in order to decide which version of the loop, serial or parallel, to choose from based on timings. The first time a loop is executed, the heuristics decider determines if the loop should be parallelised. If the conditions are not met, the sequential version of the loop is chosen and profiling is enabled for this loop. At the next execution of the loop, the evaluation of the heuristics is still performed. If the conditions are still not met (for example there where no changes in the iterations of the loop), the loop is now parallelised since at this point we only have timing information for the serial version. Consecutive executions of the loop will first check the heuristics conditions, falling back to profiling mode if the condition is not satisfied. However, the function will detect that timings for both versions are available and utilise the information gathered from profiling to decide what loop to parallelise (providing the number of iterations of the loop have not changed), with the fastest version chosen as the final decision. In contrast to this, if the amount of work is not the same (i.e. the number of loop iterations has changed) the timings get invalidated, and profiling is re-initiated. To implement this functionality requires additional code, when compared to the basic heuristic decision function. This will impose an extra overhead to the produced program, although if the loop iterations are static throughout the run 4 3 5 6 7 8 for(i=0; i<num_iters; i++){ for(j=0; j<outer_iters; j++){ delay(outer_delayreps); for(k=0; k<inner_iters; k++){ delay(inner_delayreps); } } } Listing 8. Synthetic benchmark code The second benchmark code was an extract from the CFD code outlined in Listing 6. This code is more complex than the synthetic benchmark and more representative of realistic scientific simulation codes. This code is used to explore the performance of our solution when the loop iterations vary and when the bounds of loops are dynamic during the course of the execution of the benchmark (i.e. one or more loops change their loop bound as the outer loops are progressed). A. Benchmark Environment The platform used to evaluate the dynamic loop parallelisation functionality was Ness [10], at EPCC. The system is composed by two parts, a front-end for development and job submission and a back-end for job execution. The management of the two parts is handled by the Sun Grid Engine which allows submission of jobs from the front-end that must be executed on the back-end nodes in isolation. The back-end part of the system is composed by two SUN X4600 Shared Memory nodes. The central processing unit (CPU) of each node is an AMD Opteron processor of 16 2.6GHz processing cores and 32 GB or main memory. Each core has 64K of L1 cache for data and 64K L1 cache for instructions. In addition there is also 1 MB of L2 available to each core (combined for data and instructions). We used the Portland Group (PGI) C compiler for the majority of the benchmarks, with the following compiler flags: -O4,-c99,-mp. For the benchmarking involving the OpenMP if functionality we used the GNC C compiler instead as the version of the PGI compiler we used does not support a thread team of a nested parallel region to have more than one threads when an outer region is serialised with the if clause (this seems contrary to the OpenMP specification where the if clause only affects the number of threads that get assigned to a particular parallel region, not the thread teams of its nested regions). When using the GNU C compiler we used the following compiler flags: -O3,-stf=c99,-fopenmp. Timing information was collected using the omp get wtime() function, with each benchmark executed three times and the worst time taken (since this is the limiting factor for the execution time). (a) Outer Loop Work : 0s (b) Outer Loop Work : 0.022s (c) Outer Loop Work : 0.079s (d) Outer Loop Work : 0.15s B. Synthetic benchmark results If we consider the example code in Listing 8, the execution time of the code of the two internal nested loops when only the outer loop is parallelised with a certain amount of threads (outer threads) can be calculated as shown in Equation 1. TpOuter is the execution time when parallelising the outer loop, Touter work is the time needed for the work in-between the loops and Tinner work is the time needed for the amount of work within the innermost loop. basicstyle= Tp Outer = outer_iters ∗ (Touter_work + (inner_iters ∗ Tinner_work )) (1) outer_threads In a similar fashion, when parallelising the inner loop using inner threads, the execution time of the loops is shown in Equation 2. basicstyle= Tp Inner = outer_iters ∗ (Touter_work + inner_iters ∗ Tinner_work ) (2) inner_threads If we want to have a reduction in the overall execution time by parallelising the inner loop, the constraint TpInner < TpOuter must be satisfied. Solving this constraint in terms of Touter work we can get the maximum allowed threshold of the execution time for the work of the outer loop as shown in Equation 3. It is worth mentioning that this model is an ideal performance model, where the work is evenly distributed to the threads. In reality, the time of Touter work might be affected by the presence of parallel overheads. basicstyle= Touter_work < 1 1 inner_iters ∗ Tinner_work ∗ ( − ) outer_threads inner_threads 1 1− outer_threads (3) In order to test our hypothesis, we measured the amount of time which is required by the delay function for various values, with the results shown in Figure 4. The graphs in Figure 4 show the performance of four different parallelisation strategies. OpenM P Outer(1) and OpenM P Inner(2) are the results from manual, static, parallelisations of the individual loops in the benchmark. Heuristics are the results from our basic decision function using a value of one (i.e. only parallelise the loop if there are more iterations than threads available), and Heuristic P rof iler are the results from our system using the profiling functionality where appropriate. Fig. 4. Synthetic Benchmark results with varying levels of work between the loops From the results it is evident that when the loops are perfectly nested, and regular (i.e. the loop bounds are not changing), then there is no benefit from using the profiling functionality. The basic heuristics will choose the optimal loop to parallelise apart from when we are using 6 threads. The variation in outcomes for 6 threads is is a consequence of the number of loop iterations chosen for the benchmark (8 iterations of the outer loop and 16 iterations of the inner loop). The distribution of 8 iterations to 6 threads results in all of the threads to get assigned 1 iteration of the outer loop each, and 2 of the threads get and extra iteration. The total execution time in this case is limited by the slowest threads, which is the time of 32 iterations; 2 iterations of the outer loop multiplied by 16 iterations of the inner one. Parallelising the inner loop with 6 threads however, 2 of the thread get from 2 iterations whereas the rest the threads get 3 iterations each. In this case, the total execution time of the parallel loops is the amount of time required for 24 iterations; 3 iterations of the inner loop multiplied by 8 iterations of the outer loop. Since both decision functions only utilise the heuristics decision (when the number of threads is less than the number of iterations) they cannot exploit this opportunity as no profiling is actually performed in this case. This could be altered by setting the decision heuristic to a value other than 1 (i.e setting the heuristic to 1.5). From the graphs we can observe that our threshold value calculations hold. For the parameters we used for this benchmark the calculated threshold value is approximately Touter work < 0.0468 seconds. When the work of the outer work is less than the calculated threshold (Figures 4a 4b) parallelising the inner loop with 16 threads is still faster than parallelising the outer loop with 8 threads. As the amount of work increases, the impact on the execution time when parallelising the inner loop TABLE II L OOP PARAMETERS USED FOR THE CFD CODE BENCHMARKING Parameter Value iters n cell j n cell i 500 2496 or 8 8 or 2496 is increased, since more work is now being serialised. In these cases, the heuristics decider makes the wrong choice (Figures 4c and 4d) since its decision only concerns the amount of iterations of the loops and the available threads. In contrast to this, when profiling is used in the decision function, it correctly detected that the fastest execution time is achieved by not parallelising the inner loop. In the case, where the amount of work of the outer loop exceeds the calculated threshold, parallelising the inner loop, even with 16 threads, increases the total execution time. The benefit from using 16 threads to parallelise the inner loop is not enough to justify the work that is serialised. C. CFD benchmarking results The first benchmark that we performed using the extract from the CFD code was to compare the OpenMP if clause with our basic heuristic functionality. We used, as a reference, the timings of the manually parallelised the n blocks, n harmonics and n cell j loops and compare the execution time of the heuristics decision function for the two code generation modes of our compiler. In order to avoid cases of the iterations not being evenly distributed to the threads, we only consider cases of 2, 4, 8, 12 and 16 threads. The parameters used for the loop iterations are shown in Table II, with varying amount of work in the inner loop. We also consider cases where blocks do not have the same shape by altering the values of the n cell j and n cell i loops. No alterations indicate that all of the blocks have a grid shape of 2496x8(j cell x i cell). An alteration of 2 means that the first and third blocks have a grid shape of 8x2496 whereas the second and fourth blocks have a shape of 2496x8. The performance results shown in Figure 5 highlight the fact that there is a significant difference between our implemented functionality and that provided by OpenMP (the if clause). Not only is the if clause slower than the basic OpenMP parallelisation, but it also increases the overall execution time of the code. For Figure 5a, where 2 and 4 threads are available, only the loop of the outer level is parallelised in both code generation modes. However, the if clause mode produces a slower execution time than the code duplication mode. When more than 4 threads are used, the parallelisation is applied on the n cell j loop. In contrast to the code duplication mode which produces an execution time similar to the case of statically parallelising the loop, the if clause mode is still slower. A similar performance pattern is seen at 16 threads. Moreover, in the presence of alterations in the shape of the blocks, as shown in Figures 5b and 5d, the if clause mode produces an even slower execution time. On the other hand, (a) Small work, no alterations (b) Small work, 2 alterations (c) Large work, no alterations (d) Large work, 2 alterations Fig. 5. CFD benchmark with n blocks = 4 and n harmonics = 4 with varied alterations in the i and j cell loops and varied amount of work in the inner loop the code duplication mode can exploit this opportunity in order to utilise all of the available threads by applying parallelism on the n cell i loop. Increasing the amount of work in the core calculation has a positive effect on the if clause code generation mode. We can observe from Figure 5c that compared to Figure 5a the difference between using the if clause and the static parallelisation is not as large for small numbers of threads. This is likely to be because the performance cost of executing the if clause is proportionally smaller compared to the overall execution time. However, the same performance degradation is still observed when increasing the number of threads. The execution times of the code using the OpenMP if clause raised some concerns over whether the code was operating correctly. After extensive testing and verification we ascertained that both versions of the code (the if clause and code duplication) were correct and producing the same behaviour. Therefore, we investigated the parallel overheads of the OpenMP runtime library of the GCC compiler. Other authors [11] have already studied the overheads of nested parallelism on various compilers, including a more recent version of the GCC compiler than the one used in this work. Their findings suggest that the implementation of nested parallel regions of the GCC compiler has significant overheads. What is not presented in their work is whether or not the use of the if clause on nested parallel regions produces the same overheads. In order to ensure that the behaviour we observed in our results is the cause of nested parallel regions and not the presence of the if clause, we have constructed a simple micro benchmark. TABLE III M ICRO BENCHMARK RESULTS OF GNU’ S C COMPILER ’ S IMPLEMENTATION OF NESTED PARALLELISM Parallel loop Execution time (seconds) Outer Inner Nested (with if clause) Nested (with num threads clause) 38.845619 153.06809 163.05681 162.85479 (a) n blocks = 4, n harmonic = 8,(b) n blocks = 4, n harmonic = 8, no alterations 2 alterations D. Nested parallel micro benchmark We created four versions of a benchmark code with three nested loops and the delay function of the EPCC Microbenchmark Suite in the block of the innermost loop. The first version of the benchmark creates a parallel region on the loop of the second level. The second version performs the same operation on the innermost loop. The third version uses the if clause on both loops by serialising the outer loop with a value of 0 and parallelising the inner loop with a value of 1. Finally, the last version creates a parallel region on both of these loops, however we force the number of threads on the thread team of the outer loop to 1 using the num threads clause. Through this we manage to reproduce the same behaviour as with the if clause code case when the inner loop is parallelised. The number of iterations of the parallel loops are the same as the number of available threads. Table III presents the execution times of each case. We can see that parallelising the inner loop with nested parallel regions takes 10 seconds longer than parallelising the inner loop manually, even for this small and simple benchmark. Moreover, the two versions that contain nested parallel regions achieve very similar execution times. From this test we can concluded that it is likely that the behaviour we observed from the if clause code generation mode is affected by the overheads of the implementation of the GCC compiler for nested parallel regions. E. Decision function benchmarking Finally, we investigated the performance of our profiling decision functionality for the CFD extract code. This code is perfectly nested, so the basic heuristic decision function should be optimal here as it should chose the best loop to parallelise with very little overheads, whereas the profiling function has extra functionality and therefore imposes extra overheads on the performance of the code. The results from our experiments are shown in Figure 6. We can observe from Figures 6a that both the decision functions make the correct choice of parallelisation strategy up to 12 threads. However, the overheads of the profiling functionality have a negative impact on the overall execution time. Even when profiling is not actually being performed, the functions which are inserted before and after the execution of each loop to count the amount of work performed at each loop level increase the overall time. Moreover, we can observe that at 16 threads the profiler actually chooses to parallelise the harmonics loop, whereas the heuristics decider produces (c) n blocks = 8, n harmonic = 4,(d) n blocks = 8, n harmonic = 4, no alterations 4 alterations Fig. 6. CFD benchmark with varied alterations in the i and j cell loops and a large amount of work in the inner loop the correct behaviour of profiling the n cell loop. The timings which are performed for each loop version during the profiling mode are sensitive to the presence of any overheads which ultimately affect the decision of the function (such as the overhead of taking the timings). When alterations are present in the shape of the loops, as shown in Figures 6b and 6d, the heuristics decider manages to adapt its behaviour, parallelising the innermost loop in order to utilise more threads, and can significantly out perform the static parallelisation. In all of the test cases, the decision function which is based on profiling provides slower execution times than the decision function which is based on heuristics. Moreover, the additional logic which is included in the decision function with profiling caused a suboptimal decision to be made in some situations. VIII. I MPROVED PROFILING DECISIONS The results from the previous benchmarks lead to considerations of the reasons behind the poor execution of the decision function which performs profiling. Comparing the functionality of this function with the simple case of the heuristics decision function there are two sources of additional overheads. The first one is the logic of profiling each version of a loop. In order to make a choice between the two versions of a loop, the slow version must also be executed. However, if an actual simulation code runs for a significant amount of time this overhead should be negligible (providing the loop bounds do not alter and trigger the profiling functionality too many times) as it should only be incurred infrequently. (a) Small work, n blocks = n harmonic = 4, 2 alterations 4,(b) Large work, n blocks = n harmonic = 4, 4 alterations 8, Fig. 7. CFD benchmark with varied alterations in the i and j cell loops and a large amount of work in the inner loop The second source of overheads is the inclusion of additional function calls before and after each loop in order to measure the time of the execution and count the amount of work performed. The elimination of the functionality for taking the slow path is not possible since this is the essence of profiling. Both versions of a loop must be executed in order to make a comparison between their execution time. However, we can relax the conditions on the validity of the timings. If we only consider the number of the iterations of the specific loop which is being profiled, then we can eliminate all the logic that performs the counting of the work for the internal loops. When the decision function decides that a version of a loop should be profiled (after the failure of the heuristics conditions) the number of the iterations of the version of the loop that is going to be executed is saved in the state of the loop at that point. This way, the code of the function calls which are placed before and after each loop remains simple, only adjusting the loop level counter of each thread as well as marking the starting and ending times of the execution of a loop which is being profiled, rather than counting the iterations of internal loops as the initial profiling functionality does. In order to test our theory we have created a new version of the runtime library which includes the above modifications, called the relaxed profiler. From the graphs in Figure 7 we can see that the removal of the additional logic which performs the counting benefits the decision function with profiling. When no profiling is performed (2 and 4 threads), the relaxed version of the decision function is faster than the accurate version, and the same performance pattern holds when the profiling is performed (8 threads and more for Figure 7a and 12 threads and more for Figure 7b). Comparing the execution time of the new version of the decision function with profiling to the execution time of the heuristics decision function, the latter still produces a faster execution time, however the difference is not large. This behaviour is expected, since the presence of profiling introduces additional computations within the code itself from the functions which are placed before and after each loop. Moreover, in the cases where the parallelisation is applied on a nested loop, the decision function must execute both versions of a loop, one of them being the slow version, in order to make a decision. Finally, we can see that the relaxed decision function rectifies the problem with the original profiling decision function of it choosing the wrong option in some cases. For Figure 7a we can see that at 12 and 16 threads the relaxed profiler makes the correct choice, and the same for Figure 7b at 16 threads (where the performance of the relaxed profile decision function is comparable to the heuristics decision function). IX. C ONCLUSION The main focus of this work was to investigate the possibility of dynamically choosing at runtime the best loop of a nested loop region which best utilises the available threads. We have successfully created a source-to-source compiler and a runtime library in order to automatically allow a dynamic choice to be made at runtime. As our solution uses a directives based approach, similar to OpenMP, we requires minimum effort and code change from the users point of view. We have discovered that the current mechanism users can exploit to perform this, the OpenMP if clause, does not perform efficiently (at least for the implementation we tested). Despite the fact that this behaviour is the result of the inefficient implementation of the GCC compiler which was used in this work, the same compiler with the code duplication mode was able to provide additional speedup in the execution time of the code. From this we conclude that by relying on the OpenMP runtime library to perform loop nesting, the execution time is limited by the compilers implementation of nested parallel regions. Although code duplication is considered to be a bad programming practice, when it is done automatically, it can eliminate unnecessary parallel overheads. 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Avoiding Your Teacher’s Mistakes: Training Neural Networks with Controlled Weak Supervision Mostafa Dehghani1 Aliaksei Severyn2 Sascha Rothe2 Jaap Kamps1 1 University of Amsterdam 2 Google Research [email protected], [email protected], [email protected], [email protected] arXiv:1711.00313v2 [cs.LG] 7 Dec 2017 Abstract In this paper, we propose a semi-supervised learning method where we train two neural networks in a multi-task fashion: a target network and a confidence network. The target network is optimized to perform a given task and is trained using a large set of unlabeled data that are weakly annotated. We propose to weight the gradient updates to the target network using the scores provided by the second confidence network, which is trained on a small amount of supervised data. Thus we avoid that the weight updates computed from noisy labels harm the quality of the target network model. We evaluate our learning strategy on two different tasks: document ranking and sentiment classification. The results demonstrate that our approach not only enhances the performance compared to the baselines but also speeds up the learning process from weak labels. 1 Introduction Deep neural networks have shown impressive results in a lot of tasks in computer vision, natural language processing, and information retrieval. However, their success is conditioned on the availability of exhaustive amounts of labeled data, while for many tasks such a data is not available. Hence, unsupervised and semi-supervised methods are becoming increasingly attractive. Using weak or noisy supervision is a straightforward approach to increase the size of the training data. For instance in web search, for the task of ranking, the ideal training data would be rankings of documents ordered by relevance for a large set of queries. However, it is not practical to collect such a data in large scale and only a small set of judged query-document pairs is available. However, for this task, the output of heuristic methods (Dehghani et al., 2017c) or clickthrough logs (Joachims, 2002) can be used as weak or noisy signals along with a small amount of labeled data to train learning to rank models. This is usually done by pre-training the network on weak data and fine-tuning it with true labels (Dehghani et al., 2017c; Severyn and Moschitti, 2015a). However, these two independent stages do not leverage the full capacity of information from true labels. For instance, in the pet-raining stage there is no handle to control the extent to which the data with weak labels contribute in the learning process, while they can be of different quality. In this paper, we propose a semi-supervised method that leverages a small amount of data with true labels along with a large amount of data with weak labels. Our proposed method has three main components: A weak annotator, which can be a heuristic model, a weak classifier, or even human via crowdsourcing and it is employed to annotate massive amount of unlabeled data, a target network which uses a large set of weakly annotated instances by weak annotator to learn the main task, and a confidence network which is trained on a small human-labeled set to estimate confidence scores for instances annotated by weak annotator. We train target network and confidence network in a multi-task fashion. In a joint learning process, target network and confidence network try to learn a suitable representation of the data and this layer is shared between them as a two-way communication channel. The target network tries to learn to predict the label of the given input under the supervision of the weak annotator. In the same time, the output of confidence network, which are the confidence scores, define the magnitude of the weight updates to the target network with respect to the loss computed based on labels from weak annotator, during the back- propagation phase of the target network. This way, confidence network helps target network to avoid mistakes of her teacher, i.e. weak annotator, by down-weighting the weight updates from weak labels that do not look reliable to confidence network. From a meta-learning perspective (Dehghani et al., 2017b), the goal of the confidence network trained jointly with the target network is to calibrate the learning rate for each instance in the batch. I.e., the weights w of the target network fw at step t+1 are updated as follows: wt − w t+1 =w lt b wt ) (1) ∑cθ (τi ,ỹi )∇L(fwt (τi ),y˜i )+∇R(w b i=1 where lt is the global learning rate, b is the batch size, L(⋅) the loss of predicting ŷ = fw (τ ) for an input τ when the target label is ỹ; cθ (⋅) is a scoring function learned by the confidence network taking input instance τ i and its noisy label ỹi and R(.) is the regularization term. Thus, we can effectively control the contribution to the parameter updates for the target network from weakly labeled instances based on how reliable their labels are according to the confidence network (learned on a small supervised data). Our setup requires running a weak annotator to label a large amount of unlabeled data, which is done at pre-processing time. For many tasks, it is possible to use a simple heuristic, or implicit human feedback to generate weak labels. This set is then used to train the target network. In contrast, a small expert-labeled set is used to train the confidence network, which estimates how good the weak annotations are, i.e. controls the effect of weak labels on updating the parameters of the target network. Our method allows learning different types of neural architectures and different tasks, where a meaningful weak annotator is available. In this paper, we study the performance of our proposed model by focusing on two applications in information retrieval and natural language processing: document ranking and sentiment classification. Whilst these two applications differ considerably, as do the exact operationalization of our model to these cases, there are also some clear similarities. First, in both cases the human gold standard data is based on a cognitively complex, or subjective, judgments causing high interrater variation, increasing both the cost of obtaining labels as the need for larger sets of labels. Second, also in both cases, the weak supervision signal is more systemic or objective, which facilitates the learning of the data representation. Our experimental results suggest that the proposed method is more effective in leveraging large amounts of weakly labeled data compared to traditional fine-tuning in both tasks. We also show that explicitly controlling the weight updates in the target network with the confidence network leads to faster convergence since the filtered supervision signals are more solid and less noisy. In the following, in Section 2, we introduce the general architecture of our model and explain the training process. Then, we describe the details of the applications to which we apply our model in Section 3. In Section 4 we present the experimental setups for each of the tasks along with its results and analysis. We then review related works and conclude the paper. 2 The Proposed Method In the following, we describe our recipe for semi-supervised learning of neural networks, in a scenario where along with a small human-labeled training set a large set of weakly labeled instances is leveraged. Formally, given a set of unlabeled training instances, we run a weak annotator to generate weak labels. This gives us the training set U . It consists of tuples of training instances τi and their weak labels ỹi , i.e. U ={(τi ,ỹi ),...}. For a small set of training instances with true labels, we also apply the weak annotator to generate weak labels. This creates the training set V , consisting of triplets of training instances τj , their weak labels ỹj , and their true labels yj , i.e. V = {(τj , ỹj ,yj ),...}. We can generate a large amount of training data U at almost no cost using the weak annotator. In contrast, we have only a limited amount of data with true labels, i.e. ∣V ∣<<∣U ∣. 2.1 General Architecture In our proposed framework we train a multi-task neural network that jointly learns the confidence score of weak training instances and the main task using controlled supervised signals. The high-level representation of the model is shown in Figure 1: it comprises a weak annotator and two neural networks, namely the confidence network and the target network. The goal of the weak annotator is to provide weak labels ỹi for all the instances τi ∈ U ∪V . We have this assumption that ỹi provided by the weak annotator are imperfect estimates of true labels yi , where yi are available for set V , but not for set U . Prediction loss wrt. the weak labels Supervision Layer Prediction loss wrt. the weak labels Confidence Network Supervision Layer Goodness of instances Representation Learning Weak Annotator Goodness of instances Representation Learning True Labels (a) Full Supervision Mode: Training on batches of data with true labels. Confidence Network Weak Annotator True Labels (b) Weak Supervision Mode: Training on batches of data with weak labels. Figure 1: Learning from controlled weak supervision: Our proposed multi-task network for learning a target task in a semi-supervised fashion, using a large amount of weakly labeled data and a small amount of data with true labels. Faded parts of the network are disabled during the training in the corresponding mode. Red-dotted arrows show gradient propagation. Parameters of the parts of the network in red frames get updated in the backward pass, while parameters of the network in blue frames are fixed during the training. The goal of the confidence network is to estimate the confidence score c̃j of training instances. It is learned on triplets from training set V : input τj , its weak label ỹj , and its true label yj . The score c̃j is then used to control the effect of weakly annotated training instances on updating the parameters of the target network in its backward pass during backpropagation. The target network is in charge of handling the main task we want to learn, or in other words, approximating the underlying function that predicts the correct labels. Given the data instance, τi and its weak label ỹi from the training set U , the target network aims to predict the label ŷi . The target network parameter updates are based on noisy labels assigned by the weak annotator, but the magnitude of the gradient update is based on the output of the confidence network. Both networks are trained in a multi-task fashion alternating between the full supervision and the weak supervision mode. In the full supervision mode, the parameters of the confidence network get updated using batches of instances from training set V . As depicted in Figure 1b, each training instance is passed through the representation layer mapping inputs to vectors. These vectors are concatenated with their corresponding weak labels ỹj generated by the weak annotator. The confidence network then estimates c̃j , which is the probability of taking data instance j into account for training the target network. In the weak supervision, mode the parameters of the target network are updated using training set U . As shown in Figure 1a, each training instance is passed through the same representation learning layer and is then processed by the supervision layer which is a part of the target network predicting the label for the main task. We also pass the learned representation of each training instance along with its corresponding label generated by the weak annotator to the confidence network to estimate the confidence score of the training instance, i.e. c̃i . The confidence score is computed for each instance from set U . These confidence scores are used to weight the gradient updating target network parameters or in other words the step size during back-propagation. It is noteworthy that the representation layer is shared between both networks, so besides the regularization effect of layer sharing which leads to better generalization, sharing this layer lays the ground for the confidence network to benefit from the largeness of set U and the target network to utilize the quality of set V . 2.2 Model Training Our optimization objective is composed of two terms: (1) the confidence network loss Lc , which captures the quality of the output from the confidence network and (2) the target network loss Lt , which expresses the quality for the main task. Both networks are trained by alternating between the weak supervision and the full supervision mode. In the full supervision mode, the parameters of the confidence network are updated using training instance drawn from training set V . We use cross-entropy loss function for the confidence network to capture the difference between the predicted confidence score of instance j, i.e. c̃j and the target score cj : Ranker Lc = ∑ −cj log(c̃j )−(1−cj )log(1−c̃j ), (2) j∈V The target score cj is calculated based on the difference of the true and weak labels with respect to the main task. In the weak supervision mode, the parameters of the target network are updated using training instances from U . We use a weighted loss function, Lt , to capture the difference between the predicted label ŷi by the target network and target label ỹi : Lt = ∑ c̃i Li , Compositionality Embedding Weights (3) i∈U where Li is the task-specific loss on training instance i and c̃i is the confidence score of the weakly annotated instance i, estimated by the confidence network. Note that c̃i is treated as a constant during the weak supervision mode and there is no gradient propagation to the confidence network in the backward pass (as depicted in Figure 1a). We minimize two loss functions jointly by randomly alternating between full and weak supervision modes (for example, using a 1:10 ratio). During training and based on the chosen supervision mode, we sample a batch of training instances from V with replacement or from U without replacement (since we can generate as much train data for set U ). Since in our setups usually ∣U ∣ >> ∣V ∣, the training process oversamples the instance from V . The key point here is that the “main task” and “confidence scoring” task are always defined to be close tasks and sharing representation will benefit the confidence network as an implicit data augmentation to compensate the small amount of data with true labels. Besides, we noticed that updating the representation layer with respect to the loss of the other network acts as a regularization for each of these networks and helps generalization for both target and confidence network since we try to capture all tasks (which are related tasks) and less chance for overfitting. We also investigated other possible setups or training scenarios. For instance, we tried updating the parameters of the supervision layer of the target network using also data with true labels. Or instead of using alternating sampling, we tried training the target network using controlled weak supervision signals after the confidence network is fully trained. As shown in the experiments the architecture and training strategy described above provide the best performance. Figure 2: The target network for the document ranking. 3 Applications In this section, we apply our semi-supervised method to two different tasks: document ranking and sentiment classification. For each task, we start with an introduction of the task, followed by the setup of the target network, i.e. description of the representation learning layer and the supervision layer. 3.1 Document Ranking This task is the core information retrieval problem which is challenging as it needs to capture the notion of relevance between query and documents. We employ a state-of-the-art pairwise neural ranker architecture as target network (Dehghani et al., 2017c). In this setting, each training instance τ consists of a query q, and two documents d+ and d− . The labels, ỹ and y, are scalar values indicating the probability of d+ being ranked higher than d− with respect to q. The general schema of the target network is illustrated in Figure 2. The Representation Learning Layer is a setup proposed in (Dehghani et al., 2017c). This layer is a function ψ, which learns the representation of the input data instances, i.e. (q,d+ ,d− ), and consists of three components: (1) an embedding function ε ∶ V → Rm (where V denotes the vocabulary set and m is the number of embedding dimensions), (2) a weighting function ω ∶ V → R, and (3) a compositionality function ⊙ ∶ (Rm , R)n → Rm . More formally, the function ψ is defined as: ∣q∣ ψ(q,d+ ,d− )=[⊙i=1 (ε(tqi ),ω(tqi )) ∣∣ ∣d+ ∣ + + ∣d− ∣ − − ⊙i=1 (ε(tdi ),ω(tdi )) ∣∣ ⊙i=1 (ε(tdi ),ω(tdi )) ], (4) where tqi and tdi denote the ith term in query q respectively document d. The embedding function ε maps each term to a dense m- dimensional real value vector, which is learned during the training phase. The weighting function ω assigns a weight to each term in the vocabulary. The compositionality function ⊙ projects a set of n embedding-weighting pairs to an m- dimensional representation, independent from the value of n: n ⊙(ε(ti ),ω(ti ))= i=1 ∑ni=1 exp(ω(ti ))⋅ε(ti ) , ∑nj=1 exp(ω(tj )) (5) which is in fact the normalized weighted elementwise summation of the terms’ embedding vectors. It has been shown that having global term weighting function along with embedding function improves the performance of ranking as it simulates the effect of inverse document frequency (IDF), which is an important feature in information retrieval (Dehghani et al., 2017c). In our experiments, we initialize the embedding function ε with word2vec embeddings (Mikolov et al., 2013) pre-trained on Google News and the weighting function ω with IDF. The Supervision Layer receives the vector representation of the inputs processed by the representation learning layer and outputs a prediction ỹ. We opt for a simple fully connected feed-forward network with l hidden layers followed by a softmax. Each hidden layer zk in this network computes zk = α(Wk zk−1 +bk ), where Wk and bk denote the weight matrix and the bias term corresponding to the k th hidden layer and α(.) is the non-linearity. These layers follow a sigmoid output. We employ the weighted cross entropy loss: Lt = ∑ c̃i [−ỹi log(ŷi )−(1− ỹi )log(1− ŷi )], (6) i∈BU where BU is a batch of instances from U , and c̃i is the confidence score of the weakly annotated instance i, estimated by the confidence network. The Weak Annotator is BM25 (Robertson et al., 2009) which is a well-performing unsupervised retrieval method. In the pairwise documents ranking setup, ỹi for a given instance τj =(q,d+ ,d− ) is the probability of document d+ being ranked higher than d− , based on the scores obtained from the annotator: ỹi =Pq,d+ ,d− = sq,d+ , sq,d+ +sq,d− (7) whereas sq,d is the score obtained from the weak annotator. To train the confidence network, the target label cj is calculated using the absolute difference of the true label and the weak label: cj =1−∣yj − ỹj ∣, where yj is calculated similar to ỹi , but sq,d comes from true labels created by humans. 3.2 Sentiment Classification This task aims to identify the sentiment (e.g., positive, negative, or neutral) underlying an individual sentence. Our target network is a convolutional model similar to (Deriu et al., 2017; Severyn and Moschitti, 2015a,b; Deriu et al., 2016). Each training instance τ consists of a sentence s and its sentiment label ỹ. The architecture of the target network is illustrated in Figure 3 The Representation Learning Layer learns a representation for the input sentence s and is shared between the target network and confidence network. It consists of an embedding function ε ∶ V → Rm , where V denotes the vocabulary set and m is the number of embedding dimensions. This function maps the sentence to a matrix S ∈ Rm×∣s∣ , where each column represents the embedding of a word at the corresponding position in the sentence. Matrix S is passed through a convolution layer. In this layer, a set of f filters is applied to a sliding window of length h over S to generate a feature map matrix O. Each feature map oi for a given filter F is generated by oi = ∑k,j S[i ∶ i+h]k,j Fk,j , where S[i∶i+h] denotes the concatenation of word vectors from position i to i+h. The concatenation of all oi produces a feature vector o∈R∣s∣−h+1 . The vectors o are then aggregated over all f filters into a feature map matrix O ∈Rf ×(∣s∣−h+1) . We also add a bias vector b ∈ Rf to the result of a convolution. Each convolutional layer is followed by a non-linear activation function (we use ReLU(Nair and Hinton, 2010)) which is applied element-wise. Afterward, the output is passed to the max pooling layer which operates on columns of the feature map matrix O returning the largest value: pool(oi ) ∶ R1×(∣s∣−h+1) → R (see Figure 3). This architecture is similar to the state-of-the-art model for Twitter sentiment classification from Semeval 2015 and 2016 (Severyn and Moschitti, 2015b; Deriu et al., 2016). We initialize the embedding matrix with word2vec embeddings (Mikolov et al., 2013) pretrained on a collection of 50M tweets. The Supervision Layer is a feed-forward neural Classifier Pooled Repr. Conv. Feature Map Embedding Embedding Figure 3: The target network for the sentiment classification. network similar to the supervision layer in the ranking task (with different width and depth) but with softmax instead of sigmoid as the output layer which returns ŷi , the probability distribution over all three classes. We employ the weighted cross entropy loss: Lt = ∑ c̃i ∑ −ỹik log(ŷik ), i∈BU (8) k∈K where BU is a batch of instances from U , and c̃i is the confidence score of the weakly annotated instance i, and K is a set of classes. The Weak Annotator for the sentiment classification task is a simple unsupervised lexicon-based method (Hamdan et al., 2013; Kiritchenko et al., 2014). We use SentiWordNet03 (Baccianella et al., 2010) to assign probabilities (positive, negative and neutral) for each token in set U . Then a sentencelevel distribution is derived by simply averaging the distributions of the terms, yielding a noisy label ỹi ∈ R∣K∣ , where ∣K∣ is the number of classes, i.e. ∣K∣=3. We empirically found that using soft labels from the weak annotator works better than assigning a single hard label. The target label cj for the confidence network is calculated by using the mean absolute difference of the true label and the weak 1 label: cj =1− ∣K∣ ∑k∈K ∣yjk − ỹjk ∣, where yj is the onehot encoding of the sentence label over all classes. 4 Experiments and Results Here we first describe baselines. Afterward, we present the experimental setups for each of our tasks along with their results and analysis. 4.1 Baselines and General Setups For both tasks, we evaluate the performance of our method compared to the following baselines: • (1.WA) Weak Annotator, i.e. the unsupervised method that we used for annotating the unlabeled data. • (2.WSO) Weak Supervision Only, i.e. the target network trained only on weakly labeled data. • (3.FSO) Full Supervision Only, i.e. the target network trained only on true labeled data. • (4.WS+FT) Weak Supervision + Fine Tuning, i.e. the target network trained on the weakly labeled data and fine-tuned on true labeled data. • (5.WS+SFT) Weak Supervision + Supervision Layer Fine-Tuning, i.e. the target network trained only on weakly labeled data and the supervision layer is fine-tuned on true labeled data while the representation learning layer is kept fixed. • (6.WS+RFT) Weak Supervision + Representation Fine Tuning, i.e. WS+SFT, except the supervision layer is kept fixed during fine tuning. • (7.NLI) New Label Inference (Veit et al., 2017) is similar to our proposed neural architecture inspired by the teacher-student paradigm (Hinton et al., 2015; Romero et al., 2014), but instead of having the confidence network to predict the “confidence score” of the training instance, there is a label generator network which is trained on set V to map the weak labels of the instances in U to the new labels. The new labels are then used as the target for training the target network. • (8.CWSJT ) Controlled Weak Supervision with Joint Training is our proposed neural architecture in which we jointly train the target network and the confidence network by alternating batches drawn from sets V and U (as explained in Section 2.2). • (9.CWSJT+ ) Controlled Weak Supervision + Full Supervision with Joint Training is the same as CWSJT , except that parameters of the supervision layer in target network are also updated using batches from V , with regards to the true labels. Additionally, we compare the performance of CWSJT , with other possible training setups: • (a.CWSST ) Separate Training, i.e. we consider the confidence network as a separate network, without sharing the representation learning layer, and train it on set V . We then train the target network on the controlled weak supervision signals. • (b.CWSCT ) Circular Training, i.e. we train the target network on set U . Then the confidence network is trained on data with true labels, and the target network is trained again but on controlled weak supervision signals. • (c.CWSPT ) Progressive Training is the mixture of the two previous baselines. Inspired by (Rusu et al., 2016), we transfer the learned information from the converged target network to the confidence network using progressive training. We then train the target network again on the controlled weak supervision signals. The proposed architectures are implemented in TensorFlow (Tang, 2016; Abadi et al., 2015). We use the Adam optimizer (Kingma and Ba, 2014) and the back-propagation algorithm. Furthermore, to prevent feature co-adaptation, we use dropout (Srivastava et al., 2014) as a regularization technique in all models. In our setup, the confidence network to predict c̃j is a fully connected feed forward network. Given that the confidence network is learned only from a small set of true labels and to speed up training we initialize the representation learning layer with pre-trained parameters, i.e., pre-trained word embeddings. We use ReLU (Nair and Hinton, 2010) as a non-linear activation function α in both target network and confidence network. In the following, we describe task-specific setups and the experimental results. 4.2 Document Ranking Setup & Results Collections. We use two standard TREC collections for the task of ad-hoc retrieval: The first collection (Robust04) consists of 500k news articles from different news agencies as a homogeneous collection. The second collection (ClueWeb) is ClueWeb09 Category B, a large-scale web collection with over 50 million English documents, which is considered as a heterogeneous collection. Spam documents were filtered out using the Waterloo spam scorer 1 (Cormack et al., 2011) with the default threshold 70%. Data with true labels. We take query sets that contain human-labeled judgments: a set of 250 queries (TREC topics 301–450 and 601–700) for the Robust04 collection and a set of 200 queries (topics 1-200) for the experiments on the ClueWeb collection. For each query, we take all documents judged as relevant plus the same number of documents judged as non-relevant and form pairwise combinations among them. Data with weak labels. We create a query set Q using the unique queries appearing in the AOL 1 http://plg.uwaterloo.ca/˜gvcormac/clueweb09spam/ query logs (Pass et al., 2006). This query set contains web queries initiated by real users in the AOL search engine that were sampled from a three-month period from March 2006 to May 2006. We applied standard pre-processing (Dehghani et al., 2017c,a) on the queries. We filtered out a large volume of navigational queries containing URL substrings (“http”, “www.”, “.com”, “.net”, “.org”, “.edu”). We also removed all non-alphanumeric characters from the queries. For each dataset, we took queries that have at least ten hits in the target corpus using our weak annotator method. Applying all these steps, We collect 6.15 million queries to train on in Robust04 and 6.87 million queries for ClueWeb. To prepare the weakly labeled training set U , we take the top 1000 retrieved documents using BM25 for each query from training query set Q, which in total leads to ∼∣Q∣×106 training instances. Parameters and Settings. We conducted a nested 3-fold cross validation with 80/20 training/validation split in each fold. All hyperparameters of all models and baselines were tuned individually on the validation set using batched GP bandits with an expected improvement acquisition function (Desautels et al., 2014). The size and number of hidden layers for the ranker and the confidence network were separately selected from {64, 128, 256, 512} and {1, 2, 3, 4}, respectively. The initial learning rate and the dropout parameter were selected from {10−3 ,10−5 } and {0.0,0.2,0.5}, respectively. We considered embedding sizes of {300,500}. The batch size in our experiments was set to 128. In all experiments, the parameters of the network are optimized employing the Adam optimizer (Kingma and Ba, 2014) and using the computed gradient of the loss to perform the back-propagation algorithm. At inference time, for each query, we take the top 2000 retrieved documents using BM25 as candidate documents and re-rank them using the trained models. We use the Indri2 implementation of BM25 with default parameters (i.e., k1 =1.2, b=0.75, and k3 =1000). Results and Discussions. We evaluate on set V and report two standard evaluation metrics: mean average precision (MAP) of the top-ranked 1000 documents and normalized discounted cumulative gain calculated for the top 20 retrieved documents (nDCG@20). Statistical significant differences of MAP and nDCG@20 values are determined using 2 https://www.lemurproject.org/indri.php Table 1: Performance of the proposed method and baseline models on different datasets. (Ĳ or Ź indicates that the improvements or degradations are statistically significant, at the 0.05 level using the paired two-tailed t-test. For all model, the improvement/degradations is with respect to the “weak supervision only” baseline (WSO). For CWSJT , the improvement over all baselines is considered and the Bonferroni correction is applied on the significant tests.) Method 1 WABM25 2 3 WSO 4 5 6 WS+FT 7 8 9 NLI FSO WS+SFT WS+RFT CWSJT CWS+JT Robust04 Table 2: Performance of the variants of the proposed method on different datasets. (Ĳ orŹ indicates that the improvements or degradations are statistically significant, at the 0.05 level using the paired two-tailed t-test. For all model, the improvement/degradations is with respect to the “weak supervision only” baseline (WSO on Table 1) . For CWSJT , the improvement over all baselines is considered and the Bonferroni correction is applied on the significant tests.) ClueWeb MAP nDCG@20 MAP nDCG@20 0.2503 0.4102 0.1021 0.2070 0.2702 0.1790Ź 0.4290 0.3519Ź 0.1297 0.0782Ź 0.2201 0.1730Ź 0.2830Ĳ 0.2711 0.2810Ĳ 0.4355Ĳ 0.4203 0.4316 0.1346Ĳ 0.1002Ź 0.1286 0.2346Ĳ 0.1940Ź 0.2240 0.2421Ź 0.3024Ĳ 0.2786Ĳ 0.4092Ź 0.4507Ĳ 0.4367Ĳ 0.1010Ź 0.1372Ĳ 0.1310 0.2004Ź 0.2453Ĳ 0.2244 the two-tailed paired t-test with p value < 0.05, with Bonferroni correction. Table 1 shows the performance on both datasets. Based on the results, CW S JT provides a significant boost on the performance over all datasets. There are two interesting points we want to highlight. First, among the fine-tuning experiments, updating all parameters of the target network is the best fine tuning strategy. Updating only the parameters of the representation layer based on the true labels works better than updating only parameters of the supervision layer. This supports our designed choice of a shared embedding layer which gets updated on set V . Second, while it seems reasonable to make use of true labels for updating all parameters of the target network, CWS+JT achieves no better results than CWSJT . It also performs mostly even worse than WS+FT. This is because during training, the direction of the parameter optimization is highly affected by the type of supervision signal and while we control the magnitude of the gradients, we do not change their directions, so alternating between two sets with different label qualities (different supervision signal types, i.e. weak and string) confuses the supervision layer of the target network. In fine tinning, we don not have this problem since we optimize the parameters with respect to the supervision from these two sets in two separate stages. It is noteworthy that we have also tried CWS+JT with another objective function for the target network taking both weak and true labels into account which was slightly better, but gives no Method a b c CWSST CWSCT CWSPT CWSJT Robust04 ClueWeb MAP nDCG@20 MAP nDCG@20 0.2716 0.2961Ĳ 0.2784Ĳ 0.3024Ĳ 0.4237 0.4440Ĳ 0.4292 0.4507Ĳ 0.1320 0.1378Ĳ 0.1314 0.1372Ĳ 0.2213 0.2431Ĳ 0.2207 0.2453Ĳ improvement over CWSJT . In the ranking task, the target network is designed in particular to be trained on weak annotations (Dehghani et al., 2017c), hence training the network only on weak supervision performs better than FSO. This is due to the fact that ranking is a complex task requiring many training instances, while relatively few true labels are available. The performance of NLI is worse than CWSJT as learning a mapping from imperfect labels to accurate labels and training the target network on new labels is essentially harder than learning to filter out the noisy labels, hence needs a lot of supervised data. The reason is that for the ranking, due to a few training instances with regards to the task complexity, NLI fails to generate better new labels, hence it directly misleads the target network and completely fails to improve the performance. Table 2 shows the performance of different training strategies. As shown, CWSJT and CWSCT perform better than other strategies. CWSCT is to let the confidence network to be trained separately, while still being able to enjoy shared learned information from the target network. However, it is less efficient as we need two rounds of training on weakly labeled data. CWSST performs poorly since the training data V is too small to train a high-quality confidence network without taking advantage of the vast amount of weakly annotated data in U . We also noticed that this strategy leads to a slow convergence compared to WSO. Also transferring learned information from target network to confidence network via progressive training, i.e. CWSPT , performs no better than full sharing of the representation learning layer. Table 3: Performance of the baseline models as well as the proposed method on different datasets. (Ĳ orŹ indicates that the improvements or degradations are statistically significant, at the 0.05 level using the paired two-tailed t-test. For all model, the improvement/degradations is with respect to the “weak supervision only” baseline (WSO). For CWSJT , the improvement over all baselines is considered and the Bonferroni correction is applied on the significant tests.) Method SemEval-14 SemEval-15 1 WALexicon 0.5141 0.4471 2 3 WSO 0.6719 0.6307 0.5606 0.5811 4 5 6 WS+FT 0.7080Ĳ 0.6875 0.6932 0.6441Ĳ 0.6193Ĳ 0.6102Ĳ 7 8 9 NLI 0.7113Ĳ 0.7362Ĳ 0.7310Ĳ 0.7162Ĳ 0.6433Ĳ 0.6626Ĳ 0.6551Ĳ 0.6618Ĳ FSO WS+SFT WS+RFT CWSJT CWS+JT SemEval1th Table 4: Performance of the variants of the proposed method for sentiment classification task, on different datasets. (Ĳ orŹ indicates that the improvements or degradations are statistically significant, at the 0.05 level using the paired two-tailed t-test. For all model, the improvement/degradations is with respect to the “weak supervision only” baseline (WSO on Table 3) . For CWSJT , the improvement over all baselines is considered and the Bonferroni correction is applied on the significant tests.) Method a b c CWSST CWSCT CWSPT CWSJT 4.3 SemEval-14 Ĳ 0.7183 0.7363Ĳ 0.7009Ĳ 0.7362Ĳ SemEval-15 0.6501Ĳ 0.6667Ĳ 0.6118Ĳ 0.6626Ĳ Sentiment Classification Setup & Results Collections. We test our model on the twitter message-level sentiment classification of SemEval-15 Task 10B (Rosenthal et al., 2015). Datasets of SemEval-15 subsume the test sets from previous editions of SemEval, i.e. SemEval-13 and SemEval-14. Each tweet was preprocessed so that URLs and usernames are masked. Data with true labels. We use train (9,728 tweets) and development (1,654 tweets) data from SemEval13 for training and SemEval-13-test (3,813 tweets) for validation. To make our results comparable to the official runs on SemEval we use SemEval-14 (1,853 tweets) and SemEval-15 (2,390 tweets) as test sets (Rosenthal et al., 2015; Nakov et al., 2016). Data with weak labels. We use a large corpus containing 50M tweets collected during two months for both, training the word embeddings and creating the weakly annotated set U using the lexicon-based method explained in Section 3.2. Parameters and Settings. Similar to the docu- ment ranking task, we tuned the hyper-parameters for each model, including baselines, separately with respect to the true labels of the validation set using batched GP bandits with an expected improvement acquisition function (Desautels et al., 2014). The size and number of hidden layers for the classifier and the confidence network were separately selected from {32,64,128} and {1,2,3}, respectively. We tested the model with both, 1 and 2 convolutional layers. The number of convolutional feature maps and the filter width is selected from {200, 300} and {3, 4, 5}, respectively. The initial learning rate and the dropout parameter were selected from {1E − 3,1E − 5} and {0.0,0.2,0.5}, respectively. We considered embedding sizes of {100,200} and the batch size in these experiments was set to 64. Results and Discussion. We report the performance of our model and the baseline models in terms of official SemEval metric, Macro-F1, in Table 3. We have also report statistical significance of F1 improvements using two-tailed paired t-test with p value < 0.05, with Bonferroni correction. Our method is the best performing among all the baselines. Unlike the ranking task, training the network only on data with true labels, i.e. TSO, performs rather good. In the sentiment classification task, learning representation of input which is a sentence (tweet) is simpler than the ranking task in which we try to learn representation for query and long documents. Consequently, we need fewer data to be able to learn a suitable representation and with the amount of available data with true labels, we can already capture a rather good representation without helps of weak data, while it was impossible in the ranking task. However, as the results suggest, we can still gain improvement using fine-tuning. In this task, behaviors of different fine-tuning experiments are similar to the ranking task. Furthermore, updating parameters of the supervision layer, with respect to the true labels, i.e. CWS+JT model, does not perform better than CWSJT , which again supports our choice of updating just the representation learning layer with respect to the signals from data with true labels. In the sentiment classification task, the performance of NLI is acceptable compared to the ranking task. This is first of all because generating new classification labels is essentially simpler. Secondly, in this task, we need to learn to represent a simpler input, and learn a simpler function to predict the labels, but a relatively bigger set of supervised data which helps to generate new labels. However, the performance of NLI is still lower than CWSJT . We can argue that CWSJT is a more conservative approach. It is in fact equipped with a soft filter that decreases the effect of noisy training examples from set U on parameter updates during training. This is a smoother action as we just down-weight the gradient, while NLI might change the direction of the gradient by generating a completely new label and consequently it is prone to more errors, especially when there is not enough high-quality training data to learn to generate better labels. In the sentiment classification task, besides the general baselines, we also report the best performing systems, which are also convolution-based models (Rouvier and Favre (2016) on SemEval-14; Deriu et al. (2016) on SemEval-15). Our proposed model outperforms the best system on both datasets. Table 4 also presents the results of different training strategies for the sentiment classification task. As shown, similar to the ranking task, CWSJT and CWSCT perform better than other strategies. Although CWSCT is slightly better (not statistically significant) in terms of effectiveness compared to CWSJT , it is not as efficient as CWSJT during training. Compared to the ranking task, for sentiment classification, it is easier to estimate the confidence score of instances with respect to the amount of available supervised data. Therefore, CWSST is able to improve the performance over WSO significantly. Moreover, CWSPT fails compared to the strategies where the representation learning layer is shared between the target network and the confidence network. 4.4 Faster Learning Pace Controlling the effect of supervision to train neural networks not only improves the performance, but also provides the network with more solid signals which speeds up the learning process. Figure 4 illustrates the training/validation loss for both networks, compared to the loss of training the target network with weak supervision, along with their performance on test sets, with respect to different amounts of training data for the sentiment classification task3 . As shown, in the training, the loss of the target network in our model, i.e. Lt is higher than the loss of the network which is trained only on weakly 3 We have observed similar speed-up in the learning process of the ranking task, however we skip bringing its plots due to space limit since we have nested cross-validation for the ranking task and a set of plots for each fold. Figure 4: Loss of the target network (Lt ) and the confidence network (Lc ) compared to the loss of WSO (LWSO ) on training/validation set and performance of CWS, WSO, and WA on test sets with respect to different amount of training data on sentiment classification. supervised data, i.e. LWSO . However, since these losses are calculated with respect to the weak labels (not true labels), having very low training loss can be an indication of overfitting to the imperfection in the weak labels. In other words, regardless of the general problem of lack of generalization due to overfitting, in the setup of learning from weak labels, predicting labels that are similar to train labels (very low training loss) is not necessarily a desirable incident. In the validation set, however, Lt decreases faster than LWSO , which supports the fact that LWSO overfits to the imperfection of weak labels, while our setup helps the target network to escape from this imperfection and do a good job on the validation set. In terms of the performance, compared to WSO, the performance of CWS on both test sets increases very quickly and CWS is able to pass the performance of the weak annotator by seeing much fewer instances annotated by the weak annotator. 5 Related Work Learning from weak or noisy labels has been studied in the literature (Frénay and Verleysen, 2014). We briefly review research most relevant to our work. There are semiSemi-supervised learning. supervised learning algorithms (Zhu, 2005) developed to utilize weakly or even unlabeled data. Self-training (Rosenberg et al., 2005) or pseudo-labeling (Lee, 2013) tries to predict labels of unlabeled data. This unlabeled data is provided additionally. In particular for neural networks, methods use greedy layer-wise pre-training of weights using unlabeled data alone followed by supervised fine-tuning (Deriu et al., 2017; Severyn and Moschitti, 2015b,a; Go et al., 2009). Other methods learn unsupervised encodings at multiple levels of the architecture jointly with a supervised signal (Ororbia II et al., 2015; Weston et al., 2012). Meta-learning. From the meta-learning perspective, our approach is similar to Andrychowicz et al. (2016) where a separate recurrent neural network called optimizer learns to predict an optimal update rule for updating parameters of the target network. The optimizer receives a gradient from the target network and outputs the adjusted gradient matrix. As the number of parameters in modern neural networks is typically on the order of millions the gradient matrix becomes too large to feed into the optimizer, so the approach of Andrychowicz et al. (2016) is applied to very small models. In contrast, our approach leverages additional weakly labeled data where we use the confidence network to predict per-instance scores that calibrate gradient updates for the target network. Direct learning with weak/noisy labels. Many studies tried to address learning in the condition of imperfect labels. Some noise cleansing methods were proposed to remove or correct mislabeled instances (Brodley and Friedl, 1999). Other studies showed that weak or noisy labels can be leveraged by employing a particular architecture or defining a proper loss function to avoid overfitting the training data imperfection (Dehghani et al., 2017c; Patrini et al., 2016; Beigman and Klebanov, 2009; Zeng et al., 2015; Bunescu and Mooney, 2007). Modeling imperfection. There is also research trying to model the pattern of the noise or weakness in the labels. Some methods leverage generative models to denoise weak supervision sources that a discriminative model can learn from (Ratner et al., 2016; Rekatsinas et al., 2017; Varma et al., 2017). Other methods aim to capture the pattern of the noise by inserting an extra layer or a separated module (Sukhbaatar et al., 2014; Veit et al., 2017), infer better labels from noisy labels and use them to supervise the training of the network. This is inspired by the teacher-student paradigm (Hinton et al., 2015; Romero et al., 2014; Xiao et al., 2015) in which the teacher generates a new label given the training instance with its corresponding weak or noisy label. However, as we show in our experiments, this approach is not sufficient when the amount of supervised data is not enough to generate better labels. 6 Conclusion and Future Directions Training neural networks using large amounts of weakly annotated data is an attractive approach in scenarios where an adequate amount of data with true labels is not available. In this paper, we propose a multi-task neural network architecture that unifies learning to estimate the confidence score of weak annotations and training neural networks to learn a target task with controlled weak supervision, i.e. using weak labels to updating the parameters, but taking their estimated confidence scores into account. 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Machine Learning Application in the Life Time of Materials Xiaojiao Yu Abstract: Materials design and development typically takes several decades from the initial discovery to commercialization with the traditional trial and error development approach. With the accumulation of data from both experimental and computational results, data based machine learning becomes an emerging field in materials discovery, design and property prediction. This manuscript reviews the history of materials science as a disciplinary the most common machine learning method used in materials science, and specifically how they are used in materials discovery, design, synthesis and even failure detection and analysis after materials are deployed in real application. Finally, the limitations of machine learning for application in materials science and challenges in this emerging field is discussed. Keywords: Machine learning, Materials discovery and design, Materials synthesis, Failure detection 1. Introduction Materials science has a long history that can date back to the Bronze age 1. However, only until the 16th century, first book on metallurgy was published, marking the beginning of systematic studies in materials science 2. Researches in materials science were purely empirical until theoretical models were developed. With the advent of computers in the last century, numerical methods to solve theoretical models became available, ranging from DFT (density functional theory) based quantum mechanical modeling of electronic structure for optoelectronic properties calculation, to continuum based finite element modeling for mechanical properties 3-4. Multiscale modeling that bridge various time and spatial scales were also developed in the materials science to better simulate the real complex system 5. Even so, it takes several decades from materials discovery to development and commercialization 6-7 . Even though physical modeling can reduce the amount of time by guiding experiment work. The limitation is also obvious. DFT are only used for functional materials optoelectronic property calculation, and that is only limited to materials without defect 8 . The assumption itself is far off from reality. New concept such as multiscale modeling is still far away from large scale real industrial application. Traditional ways of materials development are impeding the progress in this field and relevant technological industry. With the large amount of complex data generated by experiment, especially simulation results from both published and archived data including materials property value, processing conditions, and microstructural images, analyzing them all becoming increasingly challenging for researchers. Inspired by the human genome initiative, Obama Government launched a Materials Genome Initiative hoping to reduce current materials development time to half 9. With the increase of computing power and the development of machine learning algorithms, materials informatics has increasingly become another paradigm in the field. Researchers are already using machine learning method for materials property prediction and discovery. Machine learning forward model are used for materials property prediction after trained on data from experiments and physical simulations. Bhadeshia et al. applied neural network(NN) technique to model creep property and phase structure in steel 10-11. Crystal structure prediction is another area of study for machine learning thanks to the large amount of structural data in crystallographic database. K -nearest- neighbor’s method was used to identify materials’ structure type based on its neighbors’ structure types 12-13 . Machine learning is also applied for materials discovery by searching compositional, structural space for desired properties, which is essentially solving a constrained optimization problem. Baerns et al. was able to find an effective multicomponent catalyst for low-temperature oxidation of lowconcentration propane with a genetic algorithm and neural network 14. There are a few reviews on machine learning application in materials science already. Dane Morgan and Gerbrand Ceder reviewed the data mining methods in materials development 15. Tim Mueller, Aaron Gilad Kusne, and Rampi Ramprasad also reviewed the progress and application of machine learning in materials science, more specifically in phase diagram, crystal structural and property prediction 16. However, their reviews are mostly based on applications in fundamental of materials science. Here, we are taking a more practical approach of reviewing machine learning application in material design, development and stages after deployment. We first discuss data problems specifically in materials science. Then, machine learning concept and most widely used methods are introduced. Up-to-date reviews on machine leaning application in materials discovery, design, development, deployment and recall is conducted. The relation between data driven research and traditional experimental and physical modeling is discussed afterwards. Finally, challenges and future endeavors of machine learning based materials science research is pointed out for researchers in this niche area. 2.1 Data Problem in Materials Science The successful application of informatics in biology, astronomy and business has inspired similar application in materials science. However, materials science differs from other subjects due to its unique characteristics. Some researchers are debating whether there is a big data problem in materials science, after all the size of materials data is nothing comparable to biology data. The largest existing database based on experimental results from materials has 5x105 data records 17. However, the rapid progress in computational science and microscopy techniques is resulting in enormous amounts of output data 18 . Furthermore, Materials science data tends to be complex and heterogeneous in terms of their sources and types ranging from discrete numerical values to qualitative descriptions of materials behavior and imaging data 19. Data in materials science also exhibit the Veracity characteristics of big data problem, by that we acknowledge the practical reality of data missing and uncertainties with the data 19 . According to the 4V (volume, variety, velocity, veracity) characteristics of big data, materials science does have a big data problem 19 . With the emergence of this big data in materials science, how to extract hidden information from the complex data and interpret resulted information is becoming increasingly important for materials design and development. 2.2 Machine Learning Methods Machine learning, a branch of artificial intelligence, is about computer learning from existing data without being explicitly programmed and make predictions on new data by building a model from input samples. Depending on the assigned task, machine learning can be classified into three categories: supervised learning, machine learning algorithms are trained with a set of input value and labeled output value first, then they are used to predict output values for corresponding unseen input values; unsupervised learning, where there is no labelled output value for training data and machine learning algorithm is used to discover patterns in the input value; reinforcement learning (program interact with environment dynamically to maximize accumulated rewards). Reinforcement learning is not used in materials science field; hence it is not introduced in detail in this manuscript. Supervised learning can either be a classification problem or a regression problem depends on the whether the output value is discrete or continuous. 2.3 Method Workflow Machine learning method typically comprise several steps including raw data collection, data preprocessing (filling in missing data, handling outliers, data transformation), feature engineering for feature selection and extraction (principle component analysis), model selection, training, validations and testing. A detailed workflow is presented in Fig 1. To select the best algorithm for a particular task, model evaluation is important. Different algorithms are evaluated with different metrics. For instance, a classifier’s evaluation metrics include confusion matrix, AUC (area under curve), precision recall, F measure, Kolomogorov Smirnov chart (K-S). Confusion matrix is a 2X2 matrix with four elements: true positive (TP), true negative(TN), false positive (FP), false negati ve(FN) 20. Other accuracy measures are 𝑇𝑃 𝑇𝑁 sensitivity (True Positive Rate= ), specificity (True negative rate= ). AUC is the area under 𝑇𝑃+𝐹𝑁 𝑇𝑁+𝐹𝑃 ROC curve, which consider the relation between sensitivity and specificity. The greater the area under the curve, the more accurate is the model. Precision is 𝑇𝑃 , recall is the true positive rate defined as 𝑇𝑃+𝐹𝑃 above. Precision-recall shows the fraction of predictions that are false positive 21. F measure is also a measure of the model accuracy and is defined as the weighted harmonic mean of the precision and recall of the test. F is the balance between precision and recall 22 . K-S evaluate how the model separates between the positive and negative distributions. Higher KS value means better separation 23 . For regression algorithms, evaluation metric includes mean absolute error, (root) mean squared error (RMSE = √∑𝑁 ̂𝑖 ) 2 𝑖=1(𝑦𝑖 −𝑦 𝑟𝑒𝑔𝑟𝑒𝑠𝑠𝑖𝑜𝑛 𝑣𝑎𝑟𝑖𝑎𝑡𝑖𝑜𝑛 𝑁 𝑡𝑜𝑡𝑎𝑙 𝑣𝑎𝑟𝑖𝑎𝑡𝑖𝑜𝑛 ), coefficient of determination (𝑅 2 ). 𝑅 2 = = ̂ ̅ 2 ∑𝑁 𝑖=1(𝑌𝑖−𝑌) ∑𝑁 (𝑌−𝑌̅)2 measures the percent of total variability that is explained by the regression model 24. Fig. 1. Flowchart of a typical machine learning method 2.4 Method Comparison 𝑖=1 Some of the most common machine learning algorithms are SVM (support vector machine), ANN (artificial neural network), logistic regression, decision trees. Support vector machine algorithms are used to find the hyperplane that separate different classes with highest margin 25. The advantage of SVM is that the solution is global and unique. Computation complexity of SVM does not depend on the dimension of the input space and is less prone to overfitting 26-27 . However, SVM does not work well on unbalanced data 26 . Artificial neural network is inspired by biological brain, where artificial neurons are connected to mimic the connection of neurons in the brain 28. Multiple hidden layers and neurons can add to the complexity of the neuron network architecture. The strength of ANN is that they are flexible and can represent any nonlinear and linear function. However, it needs large amount of training data and is prone to overfitting. Hyperparameter tuning is tedious and troublesome for ANN. Decision tree is another commonly used basis classification algorithm, which comprises a root node, internal node, branch, leaf node, and depth 29. Decision tree progressively splits the tested data based on input feature value, decision process follows the branch, which is the collection between an internal node and its parent node, until it reaches a leaf node 30. Ensemble methods such as random forest and adaboost which are based on constructing a large number of trees with bootstrap samples and iteratively build an ensemble of weak learners, in an attempt to generate a strong overall model. Ensemble methods usually perform better than basic machine learning algorithms in terms of reducing variance and bias 31. 3.1 Machine Learning Application in Materials Discovery and Design An important concept in materials science field is structural-property-performance relationship 32. Developing materials that meet the required performance and property goes back to control processing conditions, structural and compositions of the materials. Hence, understanding how processing condition, structural and compositions affect materials property and performance is the first step towards materials design. Traditionally, controlled experiments are conducted to isolate the effect of one variable. However, variables often are correlated with each other. It is infeasible to isolate some variable for experimental testing 33. Data mining can help revealing hidden relations between large amount of materials parameters, processing conditions and their re lations with dependent materials properties 33. Traditional ways of materials development can be disrupted and reshaped by making the use of available data. 3.1.1 Materials Property Prediction Materials design first of all requires understanding of how de sired properties such as materials’ yield strength, toughness, ultimate tensile strength, and fatigue life etc. are affected by intrinsic microstructure, chemical composition, crystal structure, and external processing, loading conditions and temperatures. Machine learning algorithm can derive the quantitative relation between the independent and dependent variables and hence make prediction with enough training data when physical model does not exist or is too complicated to apply 33. Neural network algorithm has been used in ferritic steel welds toughness prediction due to their ability to handle complex models 33. Toughness was studied as a function of chemical composition, microstructure, welding process and testing temperature. Their influence on toughness was shown in Fig 2. The interaction between different variables can also be predicted with neural network algorithm as shown in Fig 3. The cross of the two toughness curves as a function of temperature and manganese compositions indicates at higher temperatures the influence of manganese on toughness was not only reduced but also negative. Fig 2. Bar chart showing a measure of the model-perceived significance of each of the input variable in influencing toughness. 33 Fig 3. Variation in the normalized toughness as a function of the manganese concentration and the test temperature. 33 ANN can also be used to predict constitutive relations. For instance, the constitutive flow behavior of 42CrMo Steel is predicted with strain, log strain rate and temperature as input and flow stress as output. Predicted results show good correlation with experimental value, indicating excellent capacity of the developed model in predicting flow stress, Fig 4 34. Austenite Stainless Steel grade 304L and 316L ultimate tensile strength, yield strength, tensile elongation rate, strain hardening exponent and strength coefficient were also able to be predicted by ANN with a function of temperature and strain rate. The optimum architecture is [2-6-5] for ASS 304L and [2-17-5] for ASS 316L using feed forward back propagation learning. Model accuracy is verified with correlation coefficient, average absolute error and its standard deviation 35 . Fatigue properties has always been among the most difficult ones to predict due to the high cost and long time for fatigue testing and the prevalence of structural failure caused by fatigue 36-37. Existing physical models are either lacking of generality or fail to give quantitative indications 38. Agrawal et al. predicted the fatigue strength of steel using data from the Japan National Institute of Materials Science (NIMS) MatNavi database 39-40. They used 12 regression-based predictive model, among them, neural network, decision tree and multivariate polynomial regression were able to achieve a high R 2 value of >0.97. Fig 4. Comparison between experimental value and predicted flow stress of 42CrMo steel using BP ANN. 34 (a) Predicted training data (b) Predicted testing data 3.1.2 Inversed Design of Materials Understanding how mechanical properties are influenced by materials internal and external factors help reducing searching space in the inversed materials design task. However, the inverse problem is more challenging because of the possibility of multiple solutions and the enormous structural dimension 41 . Machine learning application has shown promise in inversed materials discovery and design by reducing searching path and searching region. Ruoqian Liu et al. developed a machine learning method for the inverse design of Fe-Ge alloy microstructure with enhanced elastic, plastic and magnetostrictive properties 41. A systematic approach consisting of random data generation, feature selection, and classification was developed. Firstly, features that can quantitatively describe microstructures and properties were developed. Then, randomly generated structural and properties pairs were simulated to form the most desired and least desired classes. Two crucial steps, search path refinement and search space reduction are conducted prior to the actual searching to find the most efficient orders of features in search and the most promising search regions of features. This method was validated with five design problems, which involves identification of microstructures that satisfy both linear and nonlinear property constraints. This framework shows supremacy comparing with traditional optimization methods in reducing as much as 80% of running time and achieving optimality that would not be attained. 3.2 Machine learning application in materials processing and synthesis Design of materials can be facilitated with the data driven machine learning approach, however the commercialization of materials is still impeded by the availability to synthesize them. To disrupt the trial and error synthesis methods, Olivetti group in MIT is working on creating a predictive synthesis system for advanced materials processing. They are building a curated database of solid state materials and their synthesis methods compiled from thousands of materials synthesis journal articles. The database also contains algorithms developed through machine learning approaches, which are capable of predicting synthesis routes for novel materials based on chemical formulae and other known physical input data.  Even failed experiments can be used by the machine learning algorithm for materials discovery and synthesis which truly shows the power of data mining and machine learning. After all, onl y a small amount of information is published in the research work, most of the data are archived and not been used to its full potential. Paul Raccuglia, et al. trained a machine learning model based on failed hydrothermal syntheses data to predict reaction outcomes under different conditions such as temperature, concentration, reactant quantity and acidity 42 . The model was validated and tested with previously untested data and shown better performance than human researchers who have 10 years’ experience. It was able to predict conditions for new organically templated inorganic product formation with a success rate of 89%. 3.3 Machine learning application in microstructure recognition and failure analysis Microstructure damage and failure pre-detection is another area that machine learning find its applications. Traditionally, materials scientist examines the SEM, OPM images of samples for failure analysis similar to medical doctors analyze X-Ray images of patients. With the increasing penetration of machine learning methods in medical imaging analysis, the same kind of application in materials imaging is expect to happen as well 43. In fact, there are already reports on machine learning and computer vision researches on materials microstructure automatic recognition. Aritra et al. applied computer vision methods to identify images that contain dendritic morphology and then classify whether the dendrites are al ong the longitudinal direction or traverse direction if they do exist in the image. To extract features and reduce feature dimensions, they used visual bag of words, texture and shape statistics, and pre -trained convolutional neural network. Classification was conducted using support vector machine, nearest neighbors and random forest models 44. It was shown that pre-trained convolutional neural network performs best in terms of micrograph recognition and feature extraction, which confirmed with other reports 45-46. Classification methods were able to reach great accuracy for both task. Another example is the automatic measurement of ferrite volume fraction from the ferrite-austenite binary phase structures based on GPF (Graph Processing Framework) algorithm developed by Hafiz Muhammad Tanveer, et al 47. Machine learning algorithm can also be used in failure detection by examining microstructure images. Matthias Demant et al. introduced an enhanced machine learning algorithm for crack detection in photoluminescence (PL) images of as-cut wafers. The detection algorithm is based on a classification of cracks due to the comparison of the crack descriptions with previous trained crack data. Crack centers are identified by detecting features appearing as star or line-like structure. Grain boundary information is extracted from additional images in the visible range to avoid false detections. Support vector machine is used to train labelled data for crack and non-crack structures classification 48. The algorithm is able to achieve a high precision of 91.1% and sensitivity of 80.4% for crack length greater than 3 mm. Elaheh Rabiei et al. developed a dynamic Bayesian network(DBN) based on the variation of modulus of elasticity to estimate damages from a prognostic approach when crack is not observable yet. Various sources of information were taken into account to reduce uncertainties. DBN was applied to relate the variables and their causal or correlation relationship. Degradation model parameters are learned with joint particle filtering technique. Support vector regression models was applied to define unknown nonparametric and nonlinear correlation between the input variables. More precise damage estimation and crack initiation prediction in a metallic alloy under fatigue was confirmed by experimental observations 49. This method is different from traditional empirical damage models (Paris law) since direct damage indicators such as crack is not required to predict damage stage. Thus, underling damages can be monitored at an earlier stage. It is easy to imagine manufacturing companies such as GE can monitor their jet engine data to predict whether it needs inspection or maintenance. Fig. 5. Overview of the crack detection algorithm 48 . 4. limitations of machine learning in materials science applications Although machine learning has been widely used in a lot of fields and increasingly been used in materials science, machine learning is by no means a panacea. Without understanding its limitations and blindly apply it to every possible area can lead to wrongful predictions and a waste of time and effort. First of all, machine learning system are opaque, making them very hard to debug. Machine learning prediction heavily relies on training data. Machine learning often have overfitting or overfitting problems that needs to be concerned when taking their prediction results into consideration. Input data quality needs to be ensured. Interpolation and extrapolation can lead to problems when training data is not sufficient in the interpolated or extrapolated regime or when training data is noisy. Hence, error bar prediction is needed for evaluating prediction accuracy. Machine learning does not explain the results from the physics point of view. Materials scientists often are interested in understanding the mechanism of certain phenomena. Machine learnin g cannot elucidate the mechanism since it works on data driven model training and prediction. Interpretation of the machine learning results needs domain knowledge. Without understanding the underline physics, nonsense predictions can’t be recognized. Even in the process of feature selection, a good understanding of the causal relationship between these variable and dependent properties can be helpful for selecting most effective features and build less complicated models. Machine learning is also inseparable from experiment and physical simulation. 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SELF-LEARNING TO DETECT AND SEGMENT CYSTS IN LUNG CT IMAGES WITHOUT MANUAL ANNOTATION Ling Zhang1 , Vissagan Gopalakrishnan2 , Le Lu1 , Ronald M. Summers1 , Joel Moss2 , Jianhua Yao1 arXiv:1801.08486v1 [cs.CV] 25 Jan 2018 1 Radiology and Imaging Sciences Department, 2 Cardiovascular and Pulmonary Branch, NHLBI, National Institutes of Health (NIH), Bethesda MD ABSTRACT Image segmentation is a fundamental problem in medical image analysis. In recent years, deep neural networks achieve impressive performances on many medical image segmentation tasks by supervised learning on large manually annotated data. However, expert annotations on big medical datasets are tedious, expensive or sometimes unavailable. Weakly supervised learning could reduce the effort for annotation but still required certain amounts of expertise. Recently, deep learning shows a potential to produce more accurate predictions than the original erroneous labels. Inspired by this, we introduce a very weakly supervised learning method, for cystic lesion detection and segmentation in lung CT images, without any manual annotation. Our method works in a self-learning manner, where segmentation generated in previous steps (first by unsupervised segmentation then by neural networks) is used as ground truth for the next level of network learning. Experiments on a cystic lung lesion dataset show that the deep learning could perform better than the initial unsupervised annotation, and progressively improve itself after self-learning. Index Terms— Convolutional neural networks, weakly supervised learning, medical image segmentation, graph cuts 1. INTRODUCTION Image segmentation is a fundamental problem in medical image analysis. Classic segmentation algorithms [1] are usually formulated as optimization problems relying on cues from low-level image features. In recent years, deep learning has made much progress on image segmentation tasks (e.g., FCN [2], HED [3]), achieved dominant performances on many medical image segmentation benchmarks, e.g., UNet [4] is competitive enough for many applications. The success of deep learning based segmentation requires supervised learning on large manually annotated data. However, expert annotations on big medical datasets are expensive to obtain This research was supported - in part - by the Intramural Research Program of the National Institutes of Health Clinical Center. The authors thank Dr. Li Zhang from Beijing Institute of Big Data Research for his inspiring discussion, and Nvidia for the TITAN X Pascal GPU donation. Mild Moderate Severe Fig. 1. Examples of the cystic lung lesions with different severity levels in CT image and their manual annotation (red). or even unavailable. For example, manual annotation of hundreds of cysts in CT volume dataset (examples shown in Fig. 1) is not feasible for a recent large-sized clinical study of lymphangioleiomyomatosis (LAM) [5]. To alleviate the annotation burden, researchers exploit weakly supervised methods for deep learning based segmentation. One direction is to reduce the effort (e.g., time, expertise) for annotation. By combining FCN and active learning, 50% training data is needed to train a model with comparable performance as training on all data [6]. Another direction applies image-level annotation by incorporating FCN in a multiple instance learning framework [7]. However, expertise from physicians are still needed, such as assigning imagelevel annotations and estimating the lesion size. Recently, deep learning has shown a potential to beat the teacher (i.e., perform better than the training data labels) [8, 9] or even self-learn to be an expert without human knowledge in AlphaGo Zero [10]. Specifically, for some classification [8] and semantic segmentation [9] tasks, when provided with data labels with certain amount of errors, deep learning could produce lower errors than the original erroneous labels. In addition, with assisting by domain-specific algorithm (e.g., Monte Carlo tree search in Go game [10]; GrabCut in image segmentation [9]), training samples/labels can be generated to iteratively or recursively update the neural network parameters to achieve better performance. transfer Unsupervised Segmentation Segmentation Net – level 1 transfer Segmentation Net – level 2 … Segmentation Net – level n Data stream Annotation stream Fig. 2. Learning to segment medical images without manual annotation. Segmentation networks (level 1 – level n) are recursively trained with the previous network segmentation as training labels. In this paper, we propose a very weakly supervised approach for LAM cyst detection and segmentation. As shown in Fig. 1, the detection and segmentation of cysts is a challenging task due to the large number of cysts, greatly variation of cyst sizes, severe touching of cysts, inconsistent image quality, and image noise and motion artifact, etc. Moreover, it is infeasible to obtain manual segmentation on LAM studies. Our method, differs from weakly supervised methods, can automatically learn from medical images without any manual pixel- [6], sparse- [9], or image-level [7] annotation and without pre-training a segmentation network on other labeled datasets [9]. Starting from classic segmentation techniques, specifically unsupervised K-means clustering with spatial information followed by graph cuts [11] refinement, the initial annotation is generated and serves as labels for a segmentation network (UNet [4] in this paper) learning. New networks are then recursively trained with the previous network predictions as training labels. An improved segmentation network could be trained under two hypotheses: 1) deep learning might generate better predictions than the training data labels [8], and 2) better training data labels produce better predictions [10]. Note that the value of K in K-means clustering is the only value provided to the framework. 2. METHODS Given a medical image dataset without manual annotation, our method works in a self-learning manner (Fig. 2), where the previously generated (first by unsupervised segmentation then by segmentation networks) pixel-level annotations serve as inputs for the next level of network learning. 2.1. Unsupervised Segmentation K-means clustering is an unsupervised segmentation approach. By involving pixel intensity, average and median pixel intensities of a local window into a feature space, a spatial K-means [12] classifies the image by grouping similar pixels in the feature space into clusters. The number of clus- ters K needs to be manually set in different applications. For the cyst segmentation in CT images, we set K = 3 to obtain three clusters. c1 , c2 , and c3 are the cluster centers, indicating cyst, lung tissue, and others, respectively. With c1 , c2 , and c3 , we construct a three-terminal graph with the energy function consisting of a data term and a pixel continuity term as in [11]. The data term is assigned as the squared intensity differences between pixels and the cluster centers; The pixel continuity term is 0 when two neighboring pixels values are the same, and δ otherwise (δ = 0.003 through empirical evaluation on our data). Then, max-flow algorithm [11] is used to optimize the energy function, and the global optimal pixel labels are obtained. 2.2. Segmentation Network After obtaining the initial annotation for all the images in the dataset by using spatial K-means graph cuts, UNet is used as the network architecture to learn a better segmentor because of its efficiency and accuracy for medical image segmentation [4]. UNet is constituted of four layers of contraction (pooling) and four layers of expansion (up-convolution). Skip connections from contracting path to expansive path strengthen context information in higher resolution layers. During UNet training, the inputs are raw CT images with original resolution, and the outputs are 1-channel annotations (cross-entropy loss is utilized). The training focuses on distinguishing between cysts and lung tissues and ignoring background labels. One critical problem in training UNet for medical images is that the label/class distribution can be highly imbalanced, e.g., much more positive samples than negative or vice versa. In our experiments, we use the distribution of cysts and lung tissues in the image to balance the positive and negative classes in loss function as in [3]. We also avoid sampling empty CT slices (no cyst in the slice) in the training. 2.3. Recursive Learning The trained UNet will become its own teacher – it is applied to segment all the CT images in training set to generate a new set of pixel-level cyst labels, which will be used as the new ground truth to train a next level UNet. The network parameters of the previous UNet are transferred to initialize the next network, and a lower learning rate is used to train the next network. The self-learning terminates when the similarity between successive segmentation is larger than a threshold. 3. EXPERIMENTAL METHODS In this study, we evaluated our method on a LAM dataset. A total of 183 CT volumes from patients with LAM in a natural history protocol were studied. High resolution CT scans of the chest were obtained. The scans contained 9-13 slices and the slice thickness ranged from 1 to 1.25 mm at 3-cm intervals. Each CT slice is with 512×512 pixels. The UNet is implemented using Caffe [13]. We train the UNet model from scratch. Three UNet models are trained progressively in the recursive framework, named as UNetlevel1, UNet-level2, and UNet-level3, respectively. The initial learning rate is 1×10−7 for UNet-level1 and decreases by a factor of 10 for every next level thanks to transfer learning from previous level. Each UNet-level is trained for 50k iterations. Mini-batch of 1 image since it provides better performance (than 5, 10, etc.) in a preliminary experiment. The proposed method is tested on a DELL TOWER 7910 workstation with 2.40 GHz Xeon E5-2620 v3 CPU, 32 GB RAM, and a Nvidia TITAN X Pascal GPU of 12 GB of memory. Our model is trained on 166 CT volumes. The remaining 17 volumes including 5 mild, 6 moderate, and 6 severe cases are left out as unseen testing data. To evaluate the segmentation performance, a medical student manually detect and segment one slice from each of the 17 testing volumes. The manual segmentation was tedious that it took 4 working days. Quantification metrics include Dice coefficient and absolute difference of cyst scores (ADCS). Cyst score is defined as the percentage of lung region occupied by cysts, which is a critical clinical factor in LAM assessment [5]. It’s worth mentioning that differing from traditional concept of training set, our model does not learn from any manual annotation from the 166 training data (which is not available), therefore, these data can also be seen as testing data for performance evaluation. Six images (from 6 CT volumes) with large ADCS between unsupervised segmentation results and UNet results are additionally selected from the 166 dataset. Manual segmentation is then conducted on these slices for evaluation of the progressive improvement of our framework. In addition, we compare our method with the cyst segmentation method in [5] where semi-automated thresholding followed by some postprocessing techniques were used. 4. RESULTS Table 1 shows the performance on unseen images. 15 out of the 17 images are with good image quality while 2 are noisy. Table 1. Performance comparison on 17 unseen CT images. SK-GC: spatial K-means graph cuts; ADCS: absolute difference of cyst scores. Bold indicates the best results. Dice (%) ADCS (%) Semi-automated [5] 62.64 8.34 SK-GC (teacher) 74.67 3.71 UNet-LV.1 (student) 75.41 3.65 UNet-LV.2 (student) 75.87 3.38 UNet-LV.3 (student) 74.94 4.56 Table 2. Performance comparison on 6 CT images with large ADCS between SK-GC and UNet from learning set. SK-GC: spatial K-means graph cuts; ADCS: absolute difference of cyst scores. Bold indicates the best results. Dice (%) ADCS (%) Semi-automated [5] 79.05 5.19 SK-GC (teacher) 70.39 11.25 UNet-LV.1 (student) 82.25 2.89 UNet-LV.2 (student) 82.65 1.98 UNet-LV.3 (student) 81.93 3.42 Student (i.e., UNet) learning could achieve higher segmentation accuracy than its teacher (i.e., spatial K-means graph cut, SK-GC), but the self-improvement seems to stop at level 3. The same trends could be observed in Table 2, where the performance on images from the learning set is shown. In these 6 CT images with large ADCS between SK-GC and UNet, compared to manual annotation, UNet learning performs substantially better than SK-GC. The lower Dice of UNet in Table 1 compared to which in Table 2 is mainly caused by the lower Dice values from the 5 mild cases, where both SK-GC and UNet have Dice values around 60%. Our proposed self-learning method is also more accurate than the semi-automated method [5]. Three examples in Fig. 3 show how the proposed strategy recursively improves the segmentation performance itself. Given inaccurate segmentation provided by SK-GC, one level of UNet learning (UNet-LV.1) can already correct most oversegmentation and undersegmentation of cysts, thus achieve both higher sensitivity and higher specificity. Higher levels of UNet tend to obtain more accurate cyst boundaries especially for the overlapping cysts. The whole training process takes about 17 hours and testing is 0.13 sec./slice. 5. CONCLUSIONS We report the first results of very weakly supervised learning to detect and segment cysts in lung CT images without manual annotation. By first learning from classic unsupervised segmentation, deep learning shows its potential to perform even better after a few levels of self-learning. In future work, we will extend this method to segment other medical images. CT slice SK-GC UNet-LV.1 UNet-LV.2 Manual annotation Fig. 3. Three examples (2 good image quality and 1 noisy) show segmentation results obtained by SK-GC, UNet-level1 and UNet-level2, given manual annotation as reference. UNet-level3 is not shown due to space constraint. 6. REFERENCES [1] M. Sonka, V. Hlavac, and R. Boyle, Image Processing, Analysis, and Machine Vision, Cengage Learning, 2014. [2] J. Long, E. Shelhamer, and T. Darrell, “Fully convolutional networks for semantic segmentation,” in CVPR, 2015, pp. 3431–3440. [3] S. Xie and Z. Tu, “Holistically-nested edge detection,” in ICCV, 2015, pp. 1395–1403. [4] O. Ronneberger, P. Fischer, and T. Brox, “U-Net: Convolutional networks for biomedical image segmentation,” in MICCAI, 2015, pp. 234–241. [5] J. Yao, A.M. Taveira-DaSilva, A.M Jones, P. JulienWilliams, M. Stylianou, and J. Moss, “Sustained effects of sirolimus on lung function and cystic lung lesions in lymphangioleiomyomatosis,” Am. J. Respir. Crit. Care Med., vol. 190, no. 11, pp. 1273–1282, 2014. [6] L. Yang, Y. Zhang, J. Chen, S. Zhang, and D.Z. Chen, “Suggestive annotation: A deep active learning framework for biomedical image segmentation,” in MICCAI, 2017. [7] Z. Jia, X. Huang, E. I. Chang, and Y. Xu, “Constrained deep weak supervision for histopathology image segmentation,” IEEE TMI, 2017. [8] M.Y. Guan, V. Gulshan, A.M. Dai, and G.E. Hinton, “Who said what: Modeling individual labelers improves classification,” arXiv preprint arXiv:1703.08774, 2017. [9] A. Khoreva, R. Benenson, J. Hosang, M. Hein, and B. Schiele, “Simple does it: Weakly supervised instance and semantic segmentation,” in CVPR, 2017. [10] D. Silver, J. Schrittwieser, K. Simonyan, I. Antonoglou, A. Huang, A. Guez, T. Hubert, L. Baker, M. Lai, A. Bolton, Y. Chen, T. Lillicrap, F. Hui, L. Sifre, G. Van Den Driessche, T. Graepel, and D. Hassabis, “Mastering the game of Go without human knowledge,” Nature, vol. 550, pp. 354–359, 2017. [11] Y. Boykov and V. Kolmogorov, “An experimental comparison of min-cut/max-flow algorithms for energy minimization in vision,” TPAMI, vol. 26, no. 9, pp. 1124– 1137, 2004. [12] K. Li, Z. Lu, W. Liu, and J. Yin, “Cytoplasm and nucleus segmentation in cervical smear images using radiating gvf snake,” Pattern Recognition, vol. 45, no. 4, pp. 1255–1264, 2012. [13] Y. Jia, “Caffe: An open source convolutional architecture for fast feature embedding,” http://caffe.berkeleyvision.org/, 2013.	1
arXiv:1801.07743v1 [cs.IR] 23 Jan 2018 Entity Retrieval and Text Mining for Online Reputation Monitoring Pedro dos Santos Saleiro da Cruz Departamento de Engenharia Informática Faculdade de Engenharia da Universidade do Porto In partial fulfillment of requirements for the degree of Doctor in Informatics Engineering FEUP 2017 Supervisor: Dr. Carlos Soares Co-Supervisor: Dr. Eduarda Mendes Rodrigues Faculdade de Engenharia Universidade do Porto Rua Dr. Roberto Frias, s/n 4200-465 Porto, Portugal Copyright © 2017 by Pedro Saleiro Doctoral Committee: Dr. Eugénio Oliveira, Full Professor at FEUP, University of Porto Dr. Mark Carman, Senior Lecturer at Monash University Dr. Bruno Martins, Assistant Professor at IST, University of Lisbon Dr. Luís Torgo, Associate Professor at FCUP, University of Porto Dr. Carlos Soares, Associate Professor at FEUP, University of Porto Esta tese é dedicada à minha Mãe, Maria de Lurdes, pelo seu amor e dedicação constantes. Acknowledgements First, I would like to thank everybody that contributed somehow to this work, from co-authors to reviewers, colleagues and faculty staff. I will probably forget to mention someone in particular and I sincerely apologize for that. This work was funded by SAPO Labs, FCT and Microsoft Research. Without their financial support, I would not be able to conclude this thesis. I am deeply grateful to my supervisor, Carlos Soares. Although I was not working in meta-learning :) Carlos always showed a genuine enthusiasm about this work. Thank you for giving me the freedom to grow independently as a researcher and to pursuit my own ideas, even in the moments I was not delivering at the rhythm you expected. I believe you made me more pragmatic after all. We had really interesting and thorough discussions during the last 5 years. We have not tried all the cool ideas but I hope we will do it someday. Last, I would like to thank you for all the support and protection, as well as, for believing that I should expand my horizons to the US! I will send you a postcard from Chicago! I am also thankful to Eduarda Mendes Rodrigues, co-supervisor of this work. It all started with you. Thank you for receiving me at FEUP back in October 2012. I still remember the day we drew the first draft of our framework for ORM. You always have encouraged me and your positive feedback was a source of inspiration and motivation. Even at distance you have always been available when I needed. I must also mention your decisive role in helping me pursuing a Summer internship in a top notch place such as Microsoft Research. Being a graduate student is an opportunity to collaborate with new and inspirational people and that was what happened when I had the chance to start working with Natasa Milic-Frayling at Microsoft Research. When we are around Natasa we believe we can make things happen. Thank you for your patience, support and motivation. I hope we can keep our collaboration for many years. I show my gratitude to Prof. Eugénio Oliveira who helped me with a smooth transition to LIACC and always had the door open to discuss my issues. Furthermore, Prof. Eugénio was very enthusiastic about my work even not being my supervisor and I iv sincerely appreciate that. Thank you for your advices and I will miss our conversations about the past and future of AI. I wish to thank Luís Sarmento for introducing me to the world of Data Science, and Text Mining in particular. I just regret we did not had the chance to collaborate more often. I must thank two special friends that are also graduate students, Jorge Teixeira and Damião Rodrigues. Jorge, you were my “brother in arms” throughout this journey and I will always be thankful. Damião, my friend and colleague for more than 10 years, is the friend I searched for sharing the ups and downs of being graduate student. Thank you for your support and motivation. I would like to thank Cristina Ribeiro for the administrative support regarding my funding throughout these years. I must also address Rosaldo Rossetti for believing in my abilities and for starting our productive collaboration, and of course, for being a really enjoyable and funny colleague. Arian Pasquali also deserves a personal mention for all the support, as well as, Luís Gomes, Luís Rei, Sílvio Amir, Tiago Cunha and Gustavo Laboreiro. I would like also to thank all the members of the POPSTAR project, specially Pedro Magalhães. Nesta hora não poderia deixar de estender os agradecimentos aos amigos e à família. Em especial queria referir o grupo do Rumo ao Penta onde todos os passarões me proporcionam grandes momentos de boa disposição sem os quais seria impossível abstrair-me dos problemas do dia a dia. Um grande abraço para o André, João Miguel, Jorge e Márcio. Queria também deixar aqui uma palavra para a minha prima Xana pela amizade desde sempre. Como é óbvio tenho imenso a agradecer à minha Mãe, a quem dedico esta tese. Obrigado pelo amor, dedicação e pela liberdade que sempre me deste em todas as minhas escolhas. Como não deixaria de ser, sempre foste uma entusiasta deste meu desafio que agora chega ao fim. Parabéns a ti! Deixo também um beijinho à minha irmã Guida, ao querido Tomás e ao bebé Diogo que ainda não conheci mas a vida de emigrante tem destas coisas. E por fim, deixo o meu sentido agradecimento à minha namorada, Maria. Foste crucial nesta caminhada. Ter-te ao meu lado deu-me força todos os dias para avançar mais um pouco. Sei que este trabalho comprometeu muito o nosso tempo a dois mas agradeço-te a compreensão e os incentivos constantes para levar isto até ao fim. Prometo compensar no futuro :) Ah e claro tenho que agradecer ao Bobby pela companhia que me fez durante a escrita e por me conseguir fazer sorrir mesmo quando a vida é madrasta. Abstract Online Reputation Monitoring (ORM) is concerned with the use of computational tools to measure the reputation of entities online, such as politicians or companies. In practice, current ORM methods are constrained to the generation of data analytics reports, which aggregate statistics of popularity and sentiment on social media. We argue that this format is too restrictive as end users often like to have the flexibility to search for entity-centric information that is not available in predefined charts. As such, we propose the inclusion of entity retrieval capabilities as a first step towards the extension of current ORM capabilities. However, an entity’s reputation is also influenced by the entity’s relationships with other entities. Therefore, we address the problem of Entity-Relationship (E-R) retrieval in which the goal is to search for multiple connected entities. This is a challenging problem which traditional entity search systems cannot cope with. Besides E-R retrieval we also believe ORM would benefit of text-based entity-centric prediction capabilities, such as predicting entity popularity on social media based on news events or the outcome of political surveys. However, none of these tasks can provide useful results if there is no effective entity disambiguation and sentiment analysis tailored to the context of ORM. Consequently, this thesis address two computational problems in Online Reputation Monitoring: Entity Retrieval and Text Mining. We researched and developed methods to extract, retrieve and predict entity-centric information spread across the Web. We proposed a new probabilistic modeling of the problem of E-R retrieval together with two fusion-based design patterns for creating representations of both entities and relationships. Furthermore, we propose the Entity-Relationship Dependence Model, a novel early-fusion supervised model based on the Markov Random Field framework for Retrieval. Together with a new semi-automatic method to create test collections for E-R retrieval, we released a new test collection for that purpose that will foster research in this area. We performed experiments at scale with results showing that it is possible to perform E-R retrieval without using fix and pre-defined entity and relationship types, enabling a wide range of queries to be addressed. vi We tackled Entity Filtering and Financial Sentiment Analysis using a supervised learning approach and studied several possible features for that purpose. We participated in two well known external competitions on both tasks, obtaining state-of-the-art performance. Moreover, we performed analysis of the predictive power of a wide set of signals extracted from online news to predict the popularity of entities on Twitter. We also studied several sentiment aggregate functions on Twitter to study the feasibility of using entity-centric sentiment on social media to predict political opinion polls. Finally, we created and released an adaptable Entity Retrieval and Text Mining framework that puts together all the building blocks necessary to perform ORM and can be reused in multiple application scenarios, from computational journalism to politics and finance. This framework is able to collect texts from online media, identify entities of interest, perform entity and E-R retrieval as well as classify sentiment polarity and intensity. It supports multiple data aggregation methods together with visualization and modeling techniques that can be used for both descriptive and predictive analytics. Resumo A Monitorização da Reputação Online (MRO) consiste na utilização de ferramentas computacionais para medir a reputação de entidades online, como por exemplo, políticos ou empresas. Na prática, os métodos actuais de MRO estão restringidos à produção de relatórios constituídos por análises de dados, tais como estatísticas agregadas da popularidade e do sentimento nos media sociais. Consideramos que esta prática é demasiado restritiva uma vez que os utilizadores finais das plataformas MRO desejam frequentemente ter a flexibilidade que lhes permita pesquisar por informação centrada nas entidades que vai além da disponibilizada nos gráficos pré-definidos. Por conseguinte, propomos a inclusão da capacidade de recuperação de entidades como um primeiro passo no sentido de estender as o estado atual das ferramentas de MRO. No entanto, a reputação de uma dada entidade também é influenciada pelas relações desta com outras entidades. Neste sentido, propomo-nos a tratar do problema de recuperação de entidade-relações (E-R) onde o objectivo consiste na pesquisa por múltiplas entidades relacionadas entre si. Trata-se de um desafio que os sitemas tradicionais de recuperação de entidades ainda não são capazes de lidar. Para além da recuperação E-R, também acreditamos que a MRO iria beneficiar da capacidade de efectuar previsões baseadas em texto e centradas nas entidades, como por exemplo a previsão da popularidade de entidades nos media sociais utilizando eventos retratados nas notícias ou o resultado de sondagens. No entanto, nenhuma destas tarefas terá sucesso e utilidade se não houver a capacidade efetiva de desambiguar entidades mencionadas nos textos, assim como uma análise de sentimento específica para o contexto da MRO. Consequentemente, esta tese trata dois problemas computacionais da Monitorização da Reputação Online: Recuperação de Entidades e Prospeção de Texto. Investigámos e desenvolvemos métodos para extrair, recuperar e prever informação centrada em entidades e espalhada pela Internet. Propomos um novo modelo probabilístico do problema de recuperação E-R conjuntamente com dois padrões de desenho baseados em fusão de texto para criar representações de entidades e relações. Propomos também o Modelo de Dependência viii Entitdade-Relação (MDER), um novo modelo supervisionado de fusão antecipada baseado no Campo Aleatório de Markov para a Recuperação de Informação. Conjutamente com um novo método semi-automático de geração de coleções de teste para recuperação E-R, lançamos uma nova coleção de teste com esse propósito que irá fomentar a investigação nesta área. Efetuamos experiências de grande escala e os resultados mostram que é possível realizar recuperação E-R sem utilizar tipos fixos e pré-definidos de entidades e relações, o que permite atuar sobre o conjunto alargado de pesquisas. Tratamos também das tarefas de Filtragem de Entidades e Análise de Sentimento Financeiro utilizando uma abordagem de aprendizagem supervisionada em que estudamos várias características para esse fim. Participámos em duas competições exterrnas em ambas as tarefas, atingindo resultados ao nível do estado da arte. Além disso, realizámos uma análise do poder preditivo de um grande conjunto de sinais extraídos das notícias online para parever a popularidade de entidades no Twitter. Assim como, um estudo de várias funções de agregação de sentimento do Twitter para estudar a praticabilidade de utilizar informação de sentimento nos media sociais para prever sondagens eleitorais. Finalmente, criámos e disponibilizámos uma plataforma de recuperação de entidades e prospeção de texto que conjuga todos os blocos necessários para a realização de MRO. Pode ser reutilizada em diversos cenários de aplicação, desde o jornalismo computacional à política e finança. Esta plataforma é capaz de recolher textos dos media online, identificar entidades alvo, efectuar recuperação de entidades e relaçãos, assim como classificar sentimento e intensidade associada. Suporta vários métodos de agregação de dados e juntamente com métodos de visualização e previsão pode ser utilizada tanto para análises descritivas como preditivas. Table of contents List of figures xiii List of tables xv 1 Introduction 1.1 Thesis Statement . . . . . . . . 1.2 Objectives . . . . . . . . . . . . 1.3 Research Methodology . . . . . 1.4 Contributions and Applications 1.5 Foundations . . . . . . . . . . . 1.6 Thesis Outline . . . . . . . . . . . . . . . . 1 2 5 7 8 10 12 . . . . . . . . . . . 13 13 14 15 19 20 22 23 24 26 28 29 . . . . . . . . . . . . . . . . . . 2 Background and Related Work 2.1 Online Reputation Monitoring . . . . 2.1.1 Related Frameworks . . . . . 2.2 Entity Retrieval and Semantic Search 2.2.1 Markov Random Field for IR 2.2.2 Sequential Dependence Model 2.2.3 MRF for Entity Retrieval . . 2.3 Named Entity Disambiguation . . . . 2.4 Sentiment Analysis . . . . . . . . . . 2.5 Word Embeddings . . . . . . . . . . 2.6 Predicting Collective Attention . . . 2.7 Political Data Science . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Entity Retrieval for Online Reputation 3.1 Entity-Relationship Retrieval . . . . . 3.1.1 E-R Queries . . . . . . . . . . . 3.1.2 Modeling E-R Retrieval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Monitoring 33 . . . . . . . . . . . . . . . . . . 34 . . . . . . . . . . . . . . . . . . 35 . . . . . . . . . . . . . . . . . . 36 Table of contents x 3.2 3.3 3.4 Design Patterns for Entity-Relationship 3.2.1 Early Fusion . . . . . . . . . . . 3.2.2 Association Weights . . . . . . 3.2.3 Early Fusion Example . . . . . 3.2.4 Late Fusion . . . . . . . . . . . 3.2.5 Late Fusion Example . . . . . . 3.2.6 Implementation . . . . . . . . . Entity-Relationship Dependence Model 3.3.1 Graph Structures . . . . . . . . 3.3.2 Feature Functions . . . . . . . . 3.3.3 Ranking . . . . . . . . . . . . . 3.3.4 Discussion . . . . . . . . . . . . Summary of the Contributions . . . . . Retrieval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Entity-Relationship Retrieval over a Web Corpus 4.1 RELink Query Collection . . . . . . . . . . . . . . 4.1.1 Tabular Data and Entity Relationships . . . 4.1.2 Selection of Tables . . . . . . . . . . . . . . 4.1.3 Formulation of Queries . . . . . . . . . . . . 4.1.4 Collection Statistics . . . . . . . . . . . . . . 4.2 Experimental Setup . . . . . . . . . . . . . . . . . . 4.2.1 Data and Indexing . . . . . . . . . . . . . . 4.2.2 Retrieval Method and Parameter Tuning . . 4.2.3 Test Collections . . . . . . . . . . . . . . . . 4.3 Results and Analysis . . . . . . . . . . . . . . . . . 4.4 Summary of the Contributions . . . . . . . . . . . . 5 Entity Filtering and Financial Sentiment 5.1 Entity Filtering . . . . . . . . . . . . . . 5.1.1 Task Overview . . . . . . . . . . 5.1.2 Pre-processing . . . . . . . . . . . 5.1.3 Features . . . . . . . . . . . . . . 5.1.4 Experimental Setup . . . . . . . . 5.1.5 Results . . . . . . . . . . . . . . . 5.2 Financial Sentiment Analysis . . . . . . 5.2.1 Task Overview . . . . . . . . . . 5.2.2 Financial Word Embeddings . . . Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 41 43 44 46 48 51 52 53 55 60 61 62 . . . . . . . . . . . 63 64 65 65 67 67 69 69 70 71 72 77 . . . . . . . . . 79 80 81 81 81 82 84 86 87 87 Table of contents 5.3 5.2.3 Approach . . . . . . . 5.2.4 Experimental Setup . . 5.2.5 Results and Analysis . 5.2.6 Concluding Remarks . Summary of the Contributions xi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Text-based Entity-centric Prediction 6.1 Exploring Online News for Reputation Monitoring 6.1.1 Approach . . . . . . . . . . . . . . . . . . 6.1.2 Experimental Setup . . . . . . . . . . . . . 6.1.3 Results and Discussion . . . . . . . . . . . 6.2 Predicting Political Polls using Twitter Sentiment . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Methodology . . . . . . . . . . . . . . . . 6.2.2 Data . . . . . . . . . . . . . . . . . . . . . 6.2.3 Experimental Setup . . . . . . . . . . . . . 6.2.4 Results and Discussion . . . . . . . . . . . 6.2.5 Feature Importance . . . . . . . . . . . . . 6.2.6 Outlook . . . . . . . . . . . . . . . . . . . 6.3 Summary of the Contributions . . . . . . . . . . . 7 A Framework for Online Reputation Monitoring 7.1 Framework Overview . . . . . . . . . . . . . . . . 7.1.1 RELink . . . . . . . . . . . . . . . . . . . 7.1.2 TexRep . . . . . . . . . . . . . . . . . . . 7.2 RELink Use Case . . . . . . . . . . . . . . . . . . 7.2.1 News Processing Pipeline . . . . . . . . . . 7.2.2 Demonstration . . . . . . . . . . . . . . . 7.3 TexRep Use Case . . . . . . . . . . . . . . . . . . 7.3.1 Data Aggregation . . . . . . . . . . . . . . 7.3.2 Visualization . . . . . . . . . . . . . . . . 7.4 Learning Word Embeddings for ORM . . . . . . . 7.4.1 Neural Word Embedding Model . . . . . . 7.4.2 Experimental Setup . . . . . . . . . . . . . 7.4.3 Results and Analysis . . . . . . . . . . . . 7.4.4 Concluding Remarks . . . . . . . . . . . . 7.5 Summary of the Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 90 90 93 93 on Twitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 96 97 101 103 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 107 110 112 113 116 117 118 . . . . . . . . . . . . . . . 119 119 120 122 127 128 129 130 132 134 134 136 137 140 144 145 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii Table of contents 8 Conclusions 147 8.1 Summary and Main Contributions . . . . . . . . . . . . . . . . . . . . . 147 8.2 Limitations and Future Work . . . . . . . . . . . . . . . . . . . . . . . 154 References 157 List of figures 1.1 Entity Retrieval and Text Mining as computational problems of ORM. 3 2.1 Markov Random Field document and term dependencies. . . . . . . . . 19 3.1 3.2 3.3 Bayesian networks for E-R Retrieval with queries of different lengths. . Markov Random Field dependencies for E-R retrieval, \|Q\| = 3. . . . . . Markov Random Field dependencies for E-R retrieval, \|Q\| = 5. . . . . . 39 53 54 4.1 4.2 4.3 4.4 Example of Wikipedia table row. . . . . . . . . . . Example of metadata provided to editors. . . . . . Illustration of E-R indexing from a web corpus. . . ′ ′ Values of λ for ERDM: (a) all λ, (b) λE , (c) λR . obtained using sum normalization. . . . . . . . . . . . . . . . . . . . . . were . . . . 67 67 69 5.1 Results grouped by entity’s category using Run 2. . . . . . . . . . . . . 85 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 Daily popularity on Twitter of entities under study. . . . . . . . . . . Training and testing sliding window - first 2 iterations. . . . . . . . . Individual feature type F1 score for tp = 12 at k = 0.5. . . . . . . . . Negatives share (berminghamsovn) of political leaders in Twitter. . . Representation of the monthly poll results of each political candidate Error predictions for polls results. . . . . . . . . . . . . . . . . . . . . Error predictions for polls results variation. . . . . . . . . . . . . . . . Mean absolute error buzz vs sentiment. . . . . . . . . . . . . . . . . . Aggregate functions importance in the Random Forests models. . . . . . . . . . . . . 102 102 105 110 112 114 114 115 117 7.1 7.2 7.3 7.4 High-level overview on the ORM framework. . . . . . . RELink Framework architecture overview. . . . . . . . Architecture and data flows of the TexRep framework. News processing pipeline. . . . . . . . . . . . . . . . . . . . . . 120 121 124 128 . . . . . . . . . . . . . . . (b) and . . . . . . . . . . . . . . . . . . . . . . . (c) . . . . . . . . . . . . . . . . . . . . . . 76 xiv 7.5 7.6 7.7 List of figures Cristiano Ronaldo egocentric network. . . . . . . . . . . . . . . . . . . 129 Twitter buzz share of political leaders. . . . . . . . . . . . . . . . . . . 133 Continuous line represents loss in the training data while dashed line represents loss in the validation data. Left side: effect of increasing \|V \| using 100% of training data. Right side: effect of varying the amount of training data used with \|V \| = 32768. . . . . . . . . . . . . . . . . . . . 142 List of tables 3.1 3.2 3.3 3.4 3.5 E-R retrieval definitions. . . . . . . . . . . . . . . . . . . . . . . . . . Illustrative example of the entity index in Early Fusion. . . . . . . . . Illustrative example of the relationship index in Early Fusion. . . . . Illustrative example of the document index in Late Fusion. . . . . . . Clique sets and associated feature functions by type and input nodes. . . . . . 34 44 46 49 56 4.1 4.2 4.3 4.4 4.5 4.6 Examples of query annotations. . . . . . . . . . . . . . . . RELink collection statistics. . . . . . . . . . . . . . . . . . ClueWeb09-B extractions statistics. . . . . . . . . . . . . . Description of query sets used for evaluation. . . . . . . . . Early Fusion and ERDM comparison using LM and BM25. Results of ERDM compared with three baselines. . . . . . . . . . . . . . . . . . . . . . . . 68 68 70 71 73 74 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 RepLab 2013 Filtering Task dataset description. . . . . . . . . . . . Entity filtering versions description. . . . . . . . . . . . . . . . . . . Official results for each version plus our validation set accuracy. . . Training set examples for both sub-tasks. . . . . . . . . . . . . . . . Microblog results with all features on validation and test sets. . . . Features performance breakdown on test set using RF. . . . . . . . News Headlines results with all features on validation and test sets. Features performance breakdown on test set using MLP. . . . . . . . . . . . . . . . . . . . . . . 83 84 84 87 90 91 92 92 6.1 6.2 Summary of the four type of features we consider. . . . . . . . . . . . . 100 F1 score of popularity high as function of tp and k equal to 0.5, 0.65 and 0.8 respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Distribution of positive, negative and neutral mentions per political party110 6.3 7.1 . . . . . . . . . . . . . . . . . . . . . . . . Number of 5-grams available for training for different sizes of target vocabulary \|V \| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 xvi List of tables Overall statistics for 12 combinations of models learned varying \|V \| and volume of training data. Results observed after 40 training epochs. . . 141 7.3 Evaluation of resulting embeddings using Class Membership, Class Distinction and Word Equivalence tests for different thresholds of cosine similarity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 7.2 Chapter 1 Introduction Nowadays, people have pervasive access to connected devices, applications and services that enable them to obtain and share information almost instantly, on a 24/7 basis. With Social Media growing at an astonishing speed, user opinions about people, companies and products quickly spread over large communities. Consequently, companies and personalities are under thorough scrutiny, with every event and every statement potentially observed and evaluated by a global audience, which reflects one’s perceived reputation. Van Riel and Fombrun [1] define reputation as the “overall assessment of organizations by their stakeholders.” The authors use the term organization in the definition, but it may as well apply to individuals (e.g. politicians) or products (e.g. mobile phone brands). A stakeholder is someone who has some relationship with the organization, such as employees, customers or shareholders. This definition and other similar ones [2], focus on the perspective that reputation represents perceptions that others have on the target entity. However, the rise of Social Media and online news publishing has brought about wider public awareness about the entities’ activities, influencing people’s perceptions about their reputation. While traditional reputation analysis is mostly manual and focused on particular entities, with online media it is possible to automate much of the process of collecting, preparing and understanding large streams of content, to identify facts and opinions about a much wider set of entities. Online Reputation Monitoring (ORM) addresses this challenge: the use of computational tools to measure the reputation of entities from online media content. Early ORM started with counting occurrences of a brand name in Social Media as a channel to estimate the knowledge/reach of a brand. Introduction 2 There are several challenges to collect, process and mine online media data for these purposes [3]. Social Media texts are short, informal, with many abbreviations, slang, jargon and idioms. Often, the users do not care about the correct use of grammar and therefore the text tends to have misspellings, incomplete and unstructured sentences. Furthermore, the lack of context poses a very difficult problem for tasks relevant in the context of Text Mining, such as Named Entity Disambiguation or Sentiment Analysis. Once we classify the sentiment polarity of a given document (e.g. tweet or news title), it is necessary to aggregate several document scores to create meaningful hourly/daily indicators. These tasks are technically complex for most of the people interested in tracking entities on the web. For this reason, most research has focused on investigating parts of this problem leading to the development of tools that only address sub-tasks of this endeavor. Text data usually includes a large number of entities and relationships between them. We broadly define an entity to be a thing or concept that exists in the world, such as a person, a company, organization, an event or a film. Entities exist as mentions across documents and in external knowledge resources. In recent years, entities have gained increased importance as the basic unit of information to answer particular information needs, instead of entire documents or text snippets [4, 5]. The volume of entity-centric data is rapidly increasing on the Web, including RDF and Linked Data, Schema.org, Facebook’s Open Graph, and Google’s Knowledge Graph, describing entities (e.g., footballers and coaches) and relationships between them (e.g., “manages”). These developments have a great impact in Online Reputation Monitoring as it is mainly focused on entities. More specifically, the ORM process consists in searching and tracking an entity of interest: the personality, the company, organization or brand/product under analysis. On the other hand, news stories, topics and events discussed in the news or Social Media usually contain mentions of entities or concepts represented in a Knowledge Base. Thus, we can say that entities are the gravitational force that drives the Online Reputation Monitoring process. 1.1 Thesis Statement The ultimate goal of ORM is to track everything that is said on the Web about a given target entity and consequently, to assess/predict the impact on its reputation. From our perspective, this goal is very hard to achieve for two reasons. The first reason has to do with the difficulty of computationally processing, interpreting and accessing the huge amount of information published online everyday. The second 1.1 Thesis Statement 3 reason is inherent to the definition of reputation as being intangible but having tangible outcomes. More specifically, Fombrun and Van Riel [6] and later Stacks [7] found a correlation between several indicators, such as reputation or trust, and financial indicators, such as sales or profits. However, this finding does not imply causality, as financial indicators can be influenced by many factors, besides stakeholders’ perceived reputation. In conclusion, there is no consensus on how to measure reputation, neither intrinsically nor extrinsically. To the best of our knowledge, current ORM is still very limited and naive. The most standard approach consists in counting mentions of entity names and applying sentiment analysis to produce descriptive reports of aggregated entity popularity and overall sentiment. We propose to make progress in ORM by tackling two computational problems: Entity Retrieval and Text Mining (Figure 1.1). Online Reputation Monitoring Text and Entities Entity Retrieval Text Mining Fig. 1.1 Entity Retrieval and Text Mining as computational problems of ORM. We believe that a ORM platform, besides providing aggregated statistics and trends about entity popularity and sentiment on the news and social media, would benefit from providing entity retrieval capabilities. End users often like to have the flexibility to search for specific information that is not available in predefined charts. However, ORM has some specificities that traditional entity search systems cannot cope with. More specifically, an entity’s reputation is also influenced by the entity’s relationships with other entities. 4 Introduction For instance, the reputation of Apple Inc. was severely damaged with the so called “Apple Foxconn scandal”. Foxconn was one of the several contractor companies in Apple’s supply chain that was accused of exploiting Chinese workers. Although the facts were not directly concerned with Apple itself, its relationship with Foxconn triggered bad public opinion about Apple. The same happened recently with the “Weinstein sex scandal”, as accusations of sexual harassment aimed at Harvey Weinstein created a wave of damage to companies and personalities associated with the disgraced Hollywood producer. Therefore, a ORM platform should provide entity-relationship search capabilities. Entity-Relationship (E-R) Retrieval is a complex case of entity retrieval where the goal is to search for multiple unknown entities and relationships connecting them. Contrary to traditional entity queries, E-R queries expect tuples of connected entities as answers. For instance, “US technology companies contracts Chinese electronics manufacturers" can be answered by tuples <Apple, Foxconn>, while “Companies founded by disgraced Hollywood producer" is expecting tuples <Miramax, Harvey Weinstein>. In essence, an E-R query can be decomposed into a set of sub-queries that specify types of entities and types of relationships between entities. On the other hand, ORM requires accurate and robust text processing and data analysis methods. Text Mining plays an essential enabling role in developing better ORM. There are several challenges with collecting and extracting relevant entity-centric information from raw text data. It is necessary to filter noisy data otherwise downstream processing tasks, such as sentiment analysis, will be compromised. More specifically, it is essential to develop named entity disambiguation approaches that can distinguish relevant text passages from non-relevant. Named entities are often ambiguous, for example, the word “bush” is a surface form for two former U.S. presidents, a music band and a shrub. The ambiguity of named entities is particularly problematic in social media texts, where users often mention entities using a single term. ORM platforms would be even more useful if they would be able to predict if social media users will talk a lot about the target entities or not. For instance, on April 4th 2016, the UK Prime-minister, David Cameron, was mentioned on the news regarding the Panama Papers story. He did not acknowledge the story in detail on that day. However, the news cycle kept mentioning him about this topic in the following days and his mentions on social media kept very high. He had to publicly address the issue on April 9th, when his reputation had already been severely damaged, blaming himself for not providing further details earlier. Thus we also want to study the feasibility of 1.2 Objectives 5 using entity-centric knowledge extracted from Social Media and online news to predict real world surveys results, such as political polls. 1.2 Objectives The work reported on this dissertation aimed to understand, formalize and explore the scientific challenges inherent to the problem of using unstructured text data from different Web sources for Online Reputation Monitoring. We now describe the specific research challenges we proposed to overcome. Entity-Relationship Retrieval: Existing strategies for entity search can be divided in IR-centric and Semantic-Web-based approaches. The former usually rely on statistical language models to match and rank co-occurring terms in the proximity of the target entity [8]. The latter consists in creating a SPARQL query and using it over a structured knowledge base to retrieve relevant RDF triples [9]. Neither of these paradigms provide good support for entity-relationship (E-R) retrieval. Recent work in Semantic-Web search tackled E-R retrieval by extending SPARQL to support joins of multiple query results and creating an extended knowledge graph [10]. Extracted entities and relationships are typically stored in a knowledge graph. However, it is not always convenient to rely on a structured knowledge graph with predefined and constraining entity types. In particular, ORM is interested in transient information sources, such as online news or social media. General purpose knowledge graphs are usually fed with more stable and reliable data sources (e.g. Wikipedia). Furthermore, predefining and constraining entity and relationship types, such as in Semantic Web-based approaches, reduces the range of queries that can be answered and therefore limits the usefulness of entity search, particularly when one wants to leverage free-text. To the best of our knowledge, E-R retrieval using IR-centric approaches is a new and unexplored research problem within the Information Retrieval research community. One of the objectives of our research is to explore to what degree we can leverage the textual context of entities and relationships, i.e., co-occurring terminology, to relax the notion of an entity or relationship type. Instead of being characterized by a fixed type, e.g., person, country, place, the entity would be characterized by any contextual term. The same applies to the relationships. Traditional knowledge graphs have fixed schema of relationships, e.g. child of, created by, works for while our approach relies on contextual terms in the text proximity of 6 Introduction every two co-occurring entities in a raw document. Relationships descriptions such as “criticizes”, “hits back”, “meets” or “interested in” would be possible to search for. This is expected to significantly reduce the limitations which structured approaches suffer from, enabling a wider range of queries to be addressed. Entity Filtering and Sentiment Analysis: Entity Filtering is a sub-problem of Named Entity Disambiguation (NED) in which we have a named entity mention and we want to classify it as related or not related with the given target entity. This is a relatively easy problem in well formed texts such as news articles. However, social media texts pose several problems to this task. We are particularly interested in Entity Filtering of tweets and we aim to study a large set of features that can be generated to describe the relationship between a given target entity and a tweet, as well as exploring different learning algorithms to create supervised models for this task. Sentiment Analysis has been thoroughly studied in the last decade [11]. There have been several PhD thesis entirely dedicated to this subject. It is a broad problem with several ramifications depending on the text source and specific application. Within the context of ORM, we will focus in a particular domain: finance. Sentiment Analysis on financial texts has received increased attention in recent years [12]. Neverthless, there are some challenges yet to overcome [13]. Financial texts, such as microblogs or newswire, usually contain highly technical and specific vocabulary or jargon, making the development of specific lexical and machine learning approaches necessary. Text-based Entity-centric Prediction: We hypothesize that for entities that are frequently mentioned on the news (e.g. politicians) it is possible to establish a predictive link between online news and popularity on social media. We cast the problem as a supervised learning classification approach: to decide whether popularity will be high or low based on features extracted from the news cycle. We aim to assess if online news are valuable as source of information to effectively predict entity popularity on Twitter. More specifically, we want to find if online news carry different predictive power based on the nature of the entity under study and how predictive performance varies with different times of prediction. We propose to explore different text-based features and how particular ones affect the overall predictive power and specific entities in particular. On the other hand, we will study if it is possible to use knowledge extracted from social media texts to predict the outcome of public opinion surveys. The automatic content analysis of mass media in the social sciences has become necessary and possible 1.3 Research Methodology 7 with the rise of social media and computational power. One particularly promising avenue of research concerns the use of sentiment analysis in microblog streams. However, one of the main challenges consists in aggregating sentiment polarity in a timely fashion that can be fed to the prediction method. A Framework for ORM: The majority of the work in ORM consists in ad-hoc studies where researchers collect data from a given social network and produce their specific analysis or predictions, often unreproducible. The availability of open source platforms in this area is scarse. Researchers typically use specific APIs and software modules to produce their studies. However, there has been some effort among the research community to address these issues through open source research platforms. We therefore aim to create an adaptable text mining framework specifically tailored for ORM that can be reused in multiple application scenarios, from politics to finance. This framework is able to collect texts from online media, such as Twitter, and identify entities of interest and classify sentiment polarity and intensity. The framework supports multiple data aggregation methods, as well as visualization and modeling techniques that can be used for both descriptive analytics, such as analyze how political polls evolve over time, and predictive analytics, such as predict elections. 1.3 Research Methodology We adopted distinct research methodologies in the process of developing the research work described in this thesis. The origin of this work was the POPSTAR project. POPSTAR (Public Opinion and Sentiment Tracking, Analysis, and Research) was a project that developed methods for the collection, measurement and aggregation of political opinions voiced in microblogs (Twitter), in blogs and online news. A first prototype of the framework for ORM was implemented and served as the backend of the POPSTAR website (http://www.popstar.pt/). The ground work concerned with the development of a framework for ORM was carried in the scope of the project. Therefore, the POPSTAR website served as use case for validating the effectiveness and adaptability of the framework. The Entity Filtering and Sentiment Analysis modules of the framework were evaluated using well known external benchmarks resulting in state-of-the-art performance. We participated in RepLab 2013 Filtering Task and evaluated our Entity Filtering method using the dataset created for the competition. One of our submissions obtained the first place at the competition. We also participated in SemEval 2017 Task 5: Introduction 8 Fine-grained Sentiment Analysis on Financial Microblogs and News. We were ranked 4th using one of the metrics at the sub-task 5.1 Microblogs. We performed two experiments regarding the text-based entity centric predictions. For predicting entity popularity on Twitter based on the news cycle we collected tweets and news articles from Portugal using the SocialBus twitter collector and online news from 51 different news outlets collected by SAPO. We used the number of entity mentions on Twitter as target variable and we extracted text-based features from the news datasets. Both datasets were aligned in time. We used the same Twitter dataset for studying different sentiment aggregate functions to serve as features for predicting political polls of a private opinion studies company, Eurosondagem. Improvements of Entity-Relationship (E-R) retrieval techniques have been hampered by a lack of test collections, particularly for complex queries involving multiple entities and relationships. We created a method for generating E-R test queries to support comprehensive E-R search experiments. Queries and relevance judgments were created from content that exists in a tabular form where columns represent entity types and the table structure implies one or more relationships among the entities. Editorial work involved creating natural language queries based on relationships represented by the entries in the table. We have publicly released the RELink test collection comprising 600 queries and relevance judgments obtained from a sample of Wikipedia List-of-lists-of-lists tables. We evaluated the new methods proposed for E-R retrieval using the RELink query collection together with two other smaller query collections created by research work in Semantic Web-based E-R retrieval. We used a large web corpus, the ClueWeb-09B containing 50 million web pages for creating E-R retrieval tailored indexes for running our experiments. Moreover, we implemented a demo using a large news collection of 12 million Portuguese news articles, resulting in the best demo award at ECIR 2016. 1.4 Contributions and Applications This work resulted in the following contributions: 1. A Text Mining framework that puts together all the building blocks required to perform ORM. The framework is adaptable and can be reused in different application scenarios, such as finance and politics. The framework provides entityspecific Text Mining functionalities that enable the collection, disambiguation, sentiment analysis, aggregation, prediction and visualization of entity-centric information from heterogeneous Web data sources. Furthermore, given that it is 1.4 Contributions and Applications 9 built using a modular architecture providing abstraction layers and well defined interfaces, new functionalities can easily be integrated. 2. Generalization of the problem of entity-relationship search to cover entity types and relationships represented by any attribute and predicate, respectively, rather than a pre-defined set. 3. A general probabilistic model for E-R retrieval using Bayesian Networks. 4. Proposal of two design patterns that support retrieval approaches using the E-R model. 5. Proposal of a Entity-Relationship Dependence model that builds on the basic Sequential Dependence Model (SDM) to provide extensible entity-relationship representations and dependencies, suitable for complex, multi-relations queries. 6. An Entity-relationship indexing and retrieval approach including learning to rank/data fusion methods that can handle entity and relationships ranking and merging of results. 7. The proposal of a method and strategy for automatically obtaining relevance judgments for entity-relationship queries. 8. We make publicly available queries and relevance judgments for the previous task. 9. Entity Filtering and Financial Sentiment Analysis methods tailored for Twitter that is able to cope with short informal texts constraints. 10. Analysis of the predictive power of online news regarding entity-centric metrics on Twitter, such as popularity or sentiment. 11. Analysis of how to combine entity-centric knowledge obtained from heterogeneous sources for survey-like prediction tasks. We believe this work can be useful in a wide range of applications from which we highlight six: Reputation Management is concerned with influencing and controlling company or individual reputation and consequently tracking what is said about entities online is one of the main concerns of this area. For instance, knowing if a given news article will have a negative impact on entity’s reputation would be crucial for damage control. Introduction 10 Digital Libraries are special libraries comprising a collection of digital objects (e.g. text or images) stored in a electronic media format. They are ubiquitous nowadays, from academic repositories, to biomedical databases, law enforcement repositories, etc. We believe the contributions we make to the Entity-Relationship Retrieval research problem can be applied to any digital library enabling a new wide range of search capabilities. Fraud Detection and inside trading detection is an area where information about entities (individuals and companies) and relationships between entities is very useful to discover hidden relationships and contexts of entities that might represent conflicts of interests or even fraud. Journalism, or more specifically, computational journalism would benefit of a powerful entity-relationship search tool in which journalists could investigate how entities were previously mentioned on the Web, including online news through time, as well as relationships among entities and their semantics. Political Science has given a lot of attention to Social Media in recent years due to the sheer amount of people reactions and opinions regarding politically relevant events. Being able to analyze the interplay between online news and Social Media from a political entity perspective can be very interesting for political scientists. On the other hand, it is becoming increasingly difficult to obtain pollsresponses via telephone and it is necessary to start testing alternative approaches. Social Media Marketing focuses on communicating through social networks with company potential and effective customers. Evaluating the success of a given campaign is a key aspect of this area. Therefore assessing the volume and polarity of mentions of a given company before and after a campaign would be very useful. 1.5 Foundations Most of the material of this thesis was previously published in journal, conference and workshop publications: • P.Saleiro, E. M. Rodrigues, C. Soares, E. Oliveira, “TexRep: A Text Mining Framework for Online Reputation Monitoring”, New Generation Computing, Volume 35, Number 4 2017 [14] 1.5 Foundations 11 • P. Saleiro, N. Milic-Frayling, E. M. Rodrigues, C. Soares, “RELink: A Research Framework and Test Collection for Entity-Relationship Retrieval”, 40th International ACM SIGIR Conference on Research and Development in Information Retrieval (SIGIR 2017) [15] • P. Saleiro, N. Milic-Frayling, E. M. Rodrigues, C. Soares, “Early Fusion Strategy for Entity-Relationship Retrieval”, The First Workshop on Knowledge Graphs and Semantics for Text Retrieval and Analysis (KG4IR@SIGIR 2017) [16] • P. Saleiro, L. Sarmento, E. M. Rodrigues, C. Soares, E. Oliveira, “Learning Word Embeddings from the Portuguese Twitter Stream: A Study of some Practical Aspects”, Progress in Artificial Intelligence (EPIA 2017) [17] • P. Saleiro, E. M. Rodrigues, C. Soares, E. Oliveira, “FEUP at SemEval-2017 Task 5: Predicting Sentiment Polarity and Intensity with Financial Word Embeddings”, International Workshop on Semantic Evaluation (SemEval@ACL 2017) [18] • P. Saleiro and C. Soares, “Learning from the News: Predicting Entity Popularity on Twitter” in Advances in Intelligent Data Analysis XV (IDA 2016) [19] • P. Saleiro, J. Teixeira, C. Soares, E. Oliveira, “TimeMachine: Entity-centric Search and Visualization of News Archives” in Advances in Information Retrieval: 38th European Conference on IR Research (ECIR 2016) [20] • P. Saleiro, L. Gomes, C. Soares, “Sentiment Aggregate Functions for Political Opinion Polling using Microblog Streams” in International C* Conference on Computer Science and Software Engineering (C3S2E 2016) [21] • P. Saleiro, S. Amir, M. J. Silva, C. Soares , “POPmine: Tracking Political Opinion on the Web” in IEEE International Conference on Computer and Information Technology; Ubiquitous Computing and Communications; Dependable, Autonomic and Secure Computing; Pervasive Intelligence and Computing (IUCC 2015) [22] • P. Saleiro, L. Rei, A. Pasquali, C. Soares, et al., “POPSTAR at RepLab 2013: Name ambiguity resolution on Twitter” in Fourth International Conference of the CLEF initiative (CLEF 2013) [23] Introduction 12 1.6 Thesis Outline In Chapter 2 we discuss related work to this thesis. In Chapter 3 we present a formalization of the problem of E-R retrieval using a IR-centric approach. We provide two design patterns for fusion-based E-R retrieval: Early Fusion and Late Fusion. We end the chapter by introducing a new supervised early fusion-based Entity Relationship Dependence Model (ERDM) that can be seen as an extension of the MRF framework for retrieval adapted to E-R retrieval. In Chapter 4 we describe a set of experiments on E-R retrieval over a Web corpus. First we introduce a new query collection, RELink QC, specifically tailored to this problem. We developed a semi-automatic approach to collect relevance judgments from tabular data and the editorial work consisted in creating E-R queries answered by those relevance judgments. We run experiments using the ClueWeb09-B as dataset and provide evaluation results for the new proposed methods for E-R retrieval. Chapter 5 is dedicated to Entity Filtering and Financial Sentiment Analysis. We evaluate our approaches using well known external benchmarks, namely, RepLab 2013 and SemEval 2017. In Chapter 6, we present two experiments of text-based entity-centric predictions. In the first experiment, we try to predict the popularity of entities on social media using solely features extracted from the news cycle. On the second experiment, we try to assess which sentiment aggregate functions are useful in predicting political polls results. In Chapter 7, we present an unified framework of ORM. The framework is divided in two major containers: RELink (Entity Retrieval) and TexRep (Text Mining). We present the data flow within the framework and how it can be used as a reference open source framework for researching in ORM. We also present some case studies of using this framework. We end this thesis with Chapter 8 which is dedicated to the conclusions. Chapter 2 Background and Related Work This chapter introduces an overview of the background concepts and previous research work on the tasks addressed in this dissertation. We start by presenting a brief description of the task of Online Reputation Monitoring (ORM), including related frameworks for ORM. We then survey previous research work in Entity Retrieval and Semantic Search, including a detailed explanation of the Markov Random Field model for retrieval and its variations. We describe the tasks of Named Entity Disambiguation, Sentiment Analysis and previous work on training word embeddings. We end this chapter by providing an overview of related work on text-based predictions, including predicting social media attention or the outcome of political elections. 2.1 Online Reputation Monitoring The reputation of a company is important for the company itself but as well for the stakeholders. More specifically, stakeholders make decisions about the company and its products faster if they are aware of the image of the company [24]. From the company perspective, reputation is an asset as it attracts stakeholders and it can represent economic profit at the end [25, 6] In 2001, Newell and Goldsmith used questionnaire and survey methodologies to introduce the first standardized and reliable measure of credibility of companies from a consumer perspective [26]. There have been also studies that find a correlation between company indicators such as reputation, trust and credibility, and financial indicators, such as sales and profits [6, 7]. These studies found that although reputations are intangible they influence tangible assets. Following this reasoning, Fombrum created a very successful measurement framework, named RepTrak [27]. Background and Related Work 14 A different methodology compared to questionnaires is media analysis (news, TV and radio broadcasts). Typically, the analysis involves consuming and categorizing media according to stakeholder and polarity (positive, negative) towards the company. Recently, Social Media analysis is becoming an important proxy of people opinion, originating the field of Online Reputation Monitoring [28]. While traditional reputation monitoring is mostly manual, online media pose the opportunity to process, understand and aggregate large streams of facts about about a company or individual. ORM requires some level of continuous monitoring [29]. It is crucial to detect early the changes in the perception of a company or personality conveyed in Social Media. Online buzz may be good or bad and consequently, companies must react and address negative trends [30, 31]. It also creates an opportunity to monitor the reputation of competitors. In this context, Text Mining plays a key, enabling role as it offers methods for deriving high-quality information from textual content [32]. For instance, Gonzalo [31] identifies 5 different Text Mining research areas relevant to ORM: entity filtering, topic tracking, reputation priority detection, user profiling and automatic reporting/summarization. Social Media as a new way of communication and collaboration is an influence for every stakeholder of society, such as personalities, companies or individuals [33]. Social Media users share every aspect of their lives and that includes information about events, news stories, politicians, brands or organizations. Companies have access to all this sharing which opens new horizons for obtaining insights that can be valuable to them and their online reputation. Companies also invest a big share of their public relations on Social Media. Building a strong reputation can take long time and effort but destroying it can take place overnight. Therefore, as the importance of Social Media increased, so did the importance of having powerful tools that deal with this enormous amount of data. 2.1.1 Related Frameworks The great majority of work in ORM consists in ad-hoc studies and platforms for ORM are usually developed by private companies that do not share internal information. However, there are some open source research projects that can be considered as related frameworks to this work. Trendminer [34] is one of such platforms that enables real time analysis of Twitter data, but has a very simple sentiment analysis using word counts and lacks flexibility in order to support entity-centric data processing. A framework for ORM should be 2.2 Entity Retrieval and Semantic Search 15 entity-centric, i.e., collect, process and aggregate texts and information extracted from those texts in relation to the entities being monitored. conTEXT [35] addresses adaptability and reusability by allowing a modular interface and allowing plugin components to extend their framework, specially from the perspective of the data sources and text analysis modules. For instance, it does not support Sentiment Analysis module by default but it could be plugged in. Neverthless, conTEXT does not support the plugin of aggregation and prediction modules which makes it not suitable for ORM. The FORA framework [30] is specifically tailored for ORM. It creates an ontology based on fuzzy clustering of texts but it is only concerned with extracting relevant linguistic units regarding the target entities and does not include automatic sentiment analysis and it does not allow the plugin of new modules. POPmine [36] was the first version of our Text Mining framework for ORM and it was developed specifically in the context of a project in political data science. It comprises a richer set of modules, including cross media data collection (Twitter, blog posts and online news) and real-time trend analysis based on entity filtering and sentiment analysis modules. In fact, our current version of TexRep, our Text Mining framework for ORM, can be seen as an extension of the POPmine architecture by creating a more general purpose framework for ORM which is not restricted to political analysis. While it would be possible to adapt POPmine’s entity disambiguation and sentiment analysis modules, its aggregations are specific to the political scenarios. On the other hand, TexRep supports users to define and plug custom-specific aggregate functions. Moreover, POPmine has limited user configurations (e.g. lacks support for pre-trained word embeddings) and does not include predictive capabilities. 2.2 Entity Retrieval and Semantic Search Information Retrieval deals with the “search for information”. It is defined as the activity of finding relevant information resources (usually documents) that meet an information need (usually a query), from within a large collection of resources of an unstructured nature (usually text) [37]. In early boolean retrieval systems, documents were retrieved if the exact query term was present and they were represented as a list of terms [37]. With the introduction of the Vector Space Model, each term represents a dimension in a multi-dimensional space, and consequently, each document and query are represented as vectors [38]. Values of each dimension of the document vector correspond to the term frequency 16 Background and Related Work (TF) of the term in the document. Therefore, the ranking list of documents is produced based on their spatial distance to the query vector. The concept of inverse document frequency (IDF) was later introduced to limit the effect of common terms in a collection [39]. A term that occurs in many documents of the collection has a lower IDF than terms that occur less often. The combination TF-IDF and variants, such as BM25 [40], became commonly used weighting statistics for Vector Space Model. Recently, it has been observed that when people have focused information needs, entities better satisfy those queries than a list of documents or large text snippets [5]. This type of retrieval is called Entity Retrieval or Entity-oriented retrieval and includes extra Information Extraction tasks for processing documents, such as Named Entity Recognition (NER) and Named Entity Disambiguation (NED). Entity Retrieval is closely connected with Question answering (QA) though, QA systems focus on understanding the semantic intent of a natural language query and deciding which sentences represent the answer to the user. Considering the query “British politicians in Panama papers”, the expected result would be a list of names rather than documents related to British politics and the “Panama Papers” news story. There are two search patterns related to Entity Retrieval [4]. First, the user knows the existence of a certain entity and aims to find related information about it. For example, a user searching for product related information. Second, the user defines a predicate that constrains the search to a certain type of entities, e.g. searching for movies of a certain genre. Online Reputation Monitoring systems usually focus on reporting statistical insights based on information extracted from Social Media and online news mentioning the target entity. However, this kind of interaction limits the possibility of users to explore all the knowledge extracted about the target entity. We believe Entity Retrieval could enhance Online Reputation Monitoring by allowing free text search over all mentions of the target entity and, consequently, allow users to discover information that descriptive statistical insights might not be able to identify. Entity Retrieval differs from traditional document retrieval in the retrieval unit. While document retrieval considers a document as the atomic response to a query, in Entity Retrieval document boundaries are not so important and entities need to be identified based on occurrence in documents [41]. The focus level is more granular as the objective is to search and rank entities among documents. However, traditional Entity Retrieval systems does not exploit semantic relationships between terms in the 2.2 Entity Retrieval and Semantic Search 17 query and in the collection of documents, i.e. if there is no match between query terms and terms describing the entity, relevant entities tend to be missed. Entity Retrieval has been an active research topic in the last decade, including various specialized tracks, such as Expert finding track [42], INEX entity ranking track [43], TREC entity track [44] and SIGIR EOS workshop [45]. Previous research faced two major challenges: entity representation and entity ranking. Entities are complex objects composed by a different number of properties and are mentioned in a variety of contexts through time. Consequently, there is no single definition of the atomic unit (entity) to be retrieved. Additionally, it is a challenge to devise entity rankings that use various entity representations approaches and tackle different information needs. There are two main approaches for tackling Entity Retrieval: “profile based approach” and “voting approach” [46]). The “profile based approach” starts by applying NER and NED in the collection in order to extract all entity occurrences. Then, for each entity identified, a meta-document is created by concatenating every passage in which the entity occurs. An index of entity meta-documents is created and a standard document ranking method (e.g. BM25) is applied to rank meta-documents with respect to a given query [47, 48]. One of the main challenges of this approach is the transformation of original text documents to an entity-centric meta-document index, including pre-processing the collection in order to extract all entities and their context. In the “voting approach”, the query is processed as typical document retrieval to obtain an initial list of documents [46, 49]. Entities are extracted from these documents using NER and NED techniques. Then, score functions are calculated to estimate the relation of entities captured and the initial query. For instance, counting the frequency of occurrence of the entity in the top documents combined with each document score (relevance to the query) [46]. Another approach consists in taking into account the distance between the entity mention and the query terms in the documents [50]. Recently, there is an increasing research interest in Entity Search over Linked Data, also referred as Semantic Search, due to the availability of structured information about entities and relations in the form of Knowledge Bases [51–53]. Semantic Search exploits rich structured entity related in machine readable RDF format, expressed as a triple (entity, predicate, object). There are two types of search: keyword-based and natural language based search [54, 55]. Regardless of the search type, the objective is to interpret the semantic structure of queries and translate it to the underlying schema of the target Knowledge Base. Most of the research focus is on interpreting the query intent [54, 55] while others focus on how to devise a ranking framework that deals with 18 Background and Related Work similarities between different attributes of the entity entry in the KB and the query terms [53] Relationship Queries: Li et al. [56] were the first to study relationship queries for structured querying entities over Wikipedia text with multiple predicates. This work used a query language with typed variables, for both entities and entity pairs, that integrates text conditions. First it computes individual predicates and then aggregates multiple predicate scores into a result score. The proposed method to score predicates relies on redundant co-occurrence contexts. Yahya et al. [10] defined relationship queries as SPARQL-like subject-predicateobject (SPO) queries joined by one or more relationships. The authors cast this problem into a structured query language (SPARQL) and extended it to support textual phrases for each of the SPO arguments. Therefore it allows to combine both structured SPARQL-like triples and text simultaneously. It extended the YAGO knowledge base with triples extracted from ClueWeb using an Open Information Extraction approach [57]. In the scope of relational databases, keyword-based graph search has been widely studied, including ranking [58]. However, these approaches do not consider full documents of graph nodes and are limited to structured data. While searching over structured data is precise it can be limited in various respects. In order to increase the recall when no results are returned and enable prioritization of results when there are too many, Elbassuoni et al. [59] propose a language-model for ranking results. Similarly, the models like EntityRank by Cheng et al. [60] and Shallow Semantic Queries by Li et al. [56], relax the predicate definitions in the structured queries and, instead, implement proximity operators to bind the instances across entity types. Yahya et al. [10] propose algorithms for application of a set of relaxation rules that yield higher recall. Entity Retrieval and proximity: Web documents contain term information that can be used to apply pattern heuristics and statistical analysis often used to infer entities as investigated by Conrad and Utt [61], Petkova and Croft [50], Rennie and Jaakkola [62]. In fact, early work by Conrad and Utt [61] demonstrates a method that retrieves entities located in the proximity of a given keyword. They show that using a fixed-size window around proper-names can be effective for supporting search for people and finding relationship among entities. Similar considerations of the co-occurrence statistics have been used to identify salient terminology, i.e. keyword to include in the document index [50]. 2.2 Entity Retrieval and Semantic Search 2.2.1 19 Markov Random Field for IR In this section we detail the generic Markov Random Field (MRF) model for retrieval and its variation, the Sequential Dependence Model (SDM). As we later show, this model is the basis for our entity-relationship retrieval model. The Markov Random Field (MRF) model for retrieval was first proposed by Metzler and Croft [63] to model query term and document dependencies. In the context of retrieval, the objective is to rank documents by computing the posterior P (D\|Q), given a document D and a query Q: P (D\|Q) = P (Q, D) P (Q) (2.1) For that purpose, a MRF is constructed from a graph G, which follows the local Markov property: every random variable in G is independent of its non-neighbors given observed values for its neighbors. Therefore, different edge configurations imply different independence assumptions. Fig. 2.1 Markov Random Field document and term dependencies. Metzler and Croft [63] defined that G consists of query term nodes qi and a document node D, as depicted in Figure 2.1. The joint probability mass function over the random variables in G is defined by: PG,Λ (Q, D) = 1 Y ψ(c; Λ) ZΛ c∈C(G) (2.2) where Q = q1 , ...qn are the query term nodes, D is the document node, C(G) is the set of maximal cliques in G, and ψ(c; Λ) is a non-negative potential function over clique P Q configurations. The parameter ZΛ = Q,D c∈C(G) ψ(c; Λ) is the partition function that normalizes the distribution. It is generally unfeasible to compute ZΛ , due to the exponential number of terms in the summation, and it is ignored as it does not influence ranking. Background and Related Work 20 The potential functions are defined as compatibility functions between nodes in a clique. For instance, a tf-idf score can be measured to reflect the “aboutness” between a query term qi and a document D. Metzler and Croft [63] propose to associate one or more real valued feature function with each clique in the graph. The non-negative potential functions are defined using an exponential form ψ(c; Λ) = exp[λc f (c)], where λc is a feature weight, which is a free parameter in the model, associated with feature function f (c). The model allows parameter and feature functions sharing across cliques of the same configuration, i.e. same size and type of nodes (e.g. 2-cliques of one query term node and one document node). For each query Q, we construct a graph representing the query term dependencies, define a set of non-negative potential functions over the cliques of this graph and rank documents in descending order of PΛ (D\|Q): rank PΛ (D\|Q) = log PΛ (D\|Q) rank = log PΛ (Q, D) − log PΛ (Q) rank = X log ψ(c; Λ) (2.3) c∈C(G) rank = X log exp[λc f (c)] c∈C(G) rank = X λc f (c) c∈C(G) (2.4) Metzler and Croft concluded that given its general form, the MRF can emulate most of the retrieval and dependence models, such as language models [64]. 2.2.2 Sequential Dependence Model The Sequential Dependence Model (SDM) is the most popular variant of the MRF retrieval model [63]. It defines two clique configurations represented in the following potential functions ψ(qi , D; Λ) and ψ(qi , qi+1 , D; Λ). Basically, it considers sequential dependency between adjacent query terms and the document node. The potential function of the 2-cliques containing a query term node and a document node is represented as ψ(qi , D; Λ) = exp[λT fT (qi , D)]. The clique configuration containing contiguous query terms and a document node is represented by two real valued functions. The first considers exact ordered matches of the 2.2 Entity Retrieval and Semantic Search 21 two query terms in the document, while the second aims to capture unordered matches within N fixed window sizes. Consequently, the second potential function is ψ(qi , qi+1 , D; Λ) = exp[λO fO (qi , qi+1 , D) + λU fU (qi , qi+1 , D)]. Replacing ψ(c; Λ) by these potential functions in Equation 3.38 and factoring out the parameters λ, the SDM can be represented as a mixture model computed over term, phrase and proximity feature classes: rank X P (D\|Q) = λT fT (qi , D) + qi ∈Q X λO fO (qi , qi+1 , D) + qi ,qi+1 ∈Q X λU fU (qi , qi+1 , D) qi ,qi+1 ∈Q where the free parameters λ must follow the constraint λT +λO +λU = 1. Coordinate Ascent was chosen to learn the optimal λ values that maximize mean average precision using training data [65]. Considering tf the frequency of the term(s) in the document D, cf the frequency of the term(s) in the entire collection C, the feature functions in SDM are set as: " # cfq tfqi ,D +µ \|C\|i fT (qi , D) = log (2.5) \|D\|+µ  fO (qi , qi+1 , D) = log  tf#1(qi ,qi+1 ),D +µ \|D\|+µ  fU (qi , qi+1 , D) = log  cf#1(q ,q i i+1) \|C\| tf#uwN (qi ,qi+1 ),D +µ cf#uwN (q ,q i i+1) \|C\| \|D\|+µ   (2.6)   (2.7) where µ is the Dirichlet prior for smoothing, #1(qi , qi+1 ) is a function that searches for exact matches of the phrase “qi qi+1 ” and #uwN (qi , qi+1 ) is a function that searches for co-occurrences of qi and qi+1 within a window of fixed-N terms (usually 8 terms) across document D. SDM has shown state-of-the-art performance in ad-hoc document retrieval when compared with several bigram dependence models and standard bag-ofwords retrieval models, across short and long queries [66]. Background and Related Work 22 2.2.3 MRF for Entity Retrieval The current state-of-the-art methods in ad-hoc entity retrieval from knowledge graphs are based on MRF [53, 67]. The Fielded Sequential Dependence Model (FSDM) [53] extends SDM for structured document retrieval and it is applied to entity retrieval from knowledge graphs. In this context, entity documents are composed by fields representing metadata about the entity. Each entity document has five fields: names, attributes, categories, similar entity names and related entity names. FSDM builds individual language models for each field in the knowledge base. This corresponds to replacing SDM feature functions with those of the Mixture of Language Models [68]. The feature functions of FSDM are defined as: f˜T (qi , D) = log F X   tfqi ,Dj + µj wjT  j f˜O (qi , qi+1 , D) = log F X f˜U (qi , qi+1 , D) = log \|Dj \| + µj  tf#1(qi ,qi+1 ),Dj + µj O wj   (2.8)   cf#1(qi ,qi+1 ),j \|Cj \| \|Dj \| + µj j F X cfqi ,j \|Cj \|  tf#uwN (qi ,qi+1 ),Dj + µj U  wj  cf#uwN (qi ,qi+1 ),j \|Dj \| + µj j \|Cj \|    (2.9)    (2.10) where µj are the Dirichlet priors for each field and wj are the weights for each field P and must be non-negative with constraint Fj wj = 1. Coordinate Ascent was used in two stages to learn wj and λ values [53]. The Parameterized Fielded Sequential Dependence Model (PFSDM) [67] extends the FSDM by dynamically calculating the field weights wj to different query terms. Part-of-speech features are applied to capture the relevance of query terms to specific fields of entity documents. For instance, NNP feature is positive if query terms are proper nouns, therefore the query terms should be mapped to the names field. Therefore, the field weight contribution of a given query term qi and a query bigram qi ,qi+1 in a field j are a linear weighted combination of features: wqi ,j = X U αj,k ϕk (qi , j) (2.11) k wqi ,qi+1 ,j = X k B αj,k ϕk (qi , qi+1 , j) (2.12) 2.3 Named Entity Disambiguation 23 U where ϕk (qi , j) is the k feature function of a query unigram for the field j and αj,k is its respective weight. For bigrams, ϕk (qi , qi+1 , j) is the k feature function of a query B bigram for the field j and αj,k is its respective weight. Consequently, PFSDM has F ∗ U + F ∗ B + 3 total parameters, where F is the number of fields, U is the number of field mapping features for unigrams, B is the number of field mapping features for bigrams, plus the three λ parameters. Their estimation is performed in a two stage optimization. First α parameters are learned separately for unigrams and then bigrams. This is achieved by setting to zero the corresponding λ parameters. In the second stage, the λ parameters are learned. Coordinate Ascent is used in both stages. The ELR model exploits entity mentions in queries by defining a dependency between entity documents and entity links in the query [69]. 2.3 Named Entity Disambiguation Given a mention in a document, Named Entity Disambiguation (NED) or Entity Linking aims to predict the entity in a reference knowledge base that the string refers to, or NIL if no such entity is available. Usually the reference knowledge base (KB) includes a set of documents, where each document describes one specific entity. Wikipedia is by far the most popular reference KB [70]. Previous research typically performs three steps to link an entity mention to a KB: 1) representation of the mention, i.e. extend the entity mention with relevant knowledge from the background document, 2) candidate generation, i.e. find all possible KB entries that the mention might refer to and their representation 3) disambiguation, by computing the similarity between the represented mention and the candidate entities. Entity Filtering, or targeted entity disambiguation, is a special case of NED in which there is only one candidate entity, i.e. the entity that is being monitored. There is an increasing interest in developing Entity Filtering methods for Social Media texts, considering its specificities and limitations [71, 72]. These approaches focus on finding relevant keywords for positive and negative cases using co-occurrence, web and collection based features. Another line of work creates topic-centric entity extraction systems where entities belong to a certain topic and are used as evidence to disambiguate the short message given its topic [73]. Similarly, Hangya et al. [74] create features representing topic distributions over tweets using Latent Dirichlet Allocation (LDA). The majority of research work in NED is usually applied to disambiguate entities in reasonably long texts as news or blog posts. In recent years, there has been an increasing interest in developing NED methods for Social Media texts and its specificities and Background and Related Work 24 limitations [75–78]. A survey and evaluation of state-of-the-art NER and NED for Tweets concluded that current approaches do not perform robustly on “ill-formed, terse, and linguistically compressed” microblog texts [79]. Some Twitter-specific methods reach F1 measures of over 80%, but are still behind the state-of-the-art results obtained on well-formed news texts. Social Media texts are too short to provide sufficient information to calculate context similarity accurately [76, 80, 78, 77, 81]. In addition, most of state-of-theart approaches leverage on neighboring entities in the documents but, once again, tweets are short and do not have more than one or two entities mentioned. Most of them [82, 77, 81] extract information obtained from other tweets, and disambiguate entity mentions in these tweets collectively. The assumption is that Twitter users are content generators and tend to scatter their interests over many different messages they broadcast, which is not necessarily true [83]. Entity Filtering has also been studied in the context of real-time classification. Davis et al. [81] propose a pipeline containing three stages. Clearly positive examples are exploited to create filtering rules comprising collocations, users and hashtags. The remaining examples are classified using a Expectation-Maximization (EM) model trained using the clearly positive examples. Recently, Habib et al. [84] proposed an hybrid approach where authors first query Google to retrieve a set of possible candidate homepages and then enrich the candidate list with text from the Wikipedia. They extract a set of features for each candidate, namely, a language model and overlapping terms between tweet and document, as well as URL length and mention-URL string similarity. In addition, a prior probability of the mention corresponding to a certain entity on the YAGO [85] knowledge base is also used. Recent work in NED or Entity Linking includes graph based algorithms for collective entity disambiguation, such as TagMe[86], Babelfy [87] and WAT [88]. Word and entity embeddings have been also used for entity disambiguation [89–91]. More specifically, Fang [90] and Moreno [91] propose to learn an embedding space for both entities and words and then compute similarity features based on the combined representations. 2.4 Sentiment Analysis In the last decade, the automatic processing of subjective and emotive text, commonly known as Sentiment Analysis, has triggered huge interest from the Text Mining research community [92]. A typical task in Sentiment Analysis is text polarity classification and in the context of this work can be formalized as follows: given a text span that mentions 2.4 Sentiment Analysis 25 a target entity, decide whether it conveys positive, negative or neutral sentiment towards the target. With the rise of Social Media, research on Sentiment Analysis shifted towards Twitter. New challenges have risen, including slang, misspelling, emoticons, poor grammatical structure [92]. A number of competitions were organized, such as SemEval [93], leading to the creation of resources for research [94]. There are two main approaches to sentiment polarity classification: lexicon-based using a dictionary of terms and phrases with annotated polarity – or supervised learning – building a model of the differences in language associated with each polarity, based on training examples. In the supervised learning approach, a classifier is specifically trained for a particular type of text (e.g. tweets about politics). Consequently, it is possible to capture peculiarities of the language used in that context. As expected, this reduces the generality of the model, as it is biased towards a specific domain. Supervised learning approaches require training data. In Twitter, most of previous work obtained training data by assuming that emoticons represent the tweet polarity (positive, negative, neutral) [95], or by using third party software, such as the Stanford Sentiment Analyzer [96]. Lexicon-based approaches have shown to work effectively on conventional text [97] but tend to be ill suited for Twitter data. With the purpose of overcoming this limitation, an algorithm that uses a human-coded lexicon specifically tailored to Social Media text was introduced [98]. SentiStrength has become a reference in recent years due to its relatively good performance and consistent performance on polarity classification of Social Media texts. Nevertheless, it is confined to a fixed set of words and it is context independent. The recent interest in deep learning led to approaches that use deep learned word embeddings as features in a variety of Text Mining tasks [99, 100]. In Sentiment Analysis, recent work integrated polarity information of text into the word embedding by extending the probabilistic document model obtained from Latent Dirichlet Allocation [101]. While others learned task-specific embeddings from an existing embedding and sentences with annotated polarity [102]. Or learning polarity specific word embeddings from tweets collected using emoticons [103] and directly incorporating the supervision from sentiment polarity in the loss functions of neural networks [104]. Background and Related Work 26 2.5 Word Embeddings The most popular and simple way to model and represent text data is the Vector Space Model [105]. A vector of features in a multi-dimensional feature space represents each lexical item (e.g. a word) in a document and each item is independent of other items in the document. This allows to compute geometric operations over vectors of lexical items using well established algebraic methods. However, the Vector Space Model faces some limitations. For instance, the same word can express different meanings in different contexts - the polysymy problem - or different words may be used to describe the same meaning - the synonymy problem. Since 2000, a variety of different methods (e.g. LDA [106]) and resources (e.g. DBpedia [107]) have been developed to try to assign semantics, or meaning, to concepts and parts of text. Word embedding methods aim to represent words as real valued continuous vectors in a much lower dimensional space when compared to traditional bag-of-words models. Moreover, this low dimensional space is able to capture lexical and semantic properties of words. Co-occurrence statistics are the fundamental information that allows creating such representations. Two approaches exist for building word embeddings. One creates a low rank approximation of the word co-occurrence matrix, such as in the case of Latent Semantic Analysis [108] and GloVe [109]. The other approach consists in extracting internal representations from neural network models of text [110, 111, 100]. Levy and Goldberg [112] showed that the two approaches are closely related. Although, word embedding research goes back several decades, it was the recent developments of Deep Learning and the word2vec framework [100] that captured the attention of the NLP community. Moreover, Mikolov et al. [113] showed that embeddings trained using word2vec models (CBOW and Skip-gram) exhibit linear structure, allowing analogy questions of the form “man:woman::king:??.” and can boost performance of several text classification tasks. In this context, the objective is to maximize the likelihood that words are predicted given their context. word2vec has two models for learning word embeddings, the skip-gram model (SG) and the continuous-bag-of-word model (CBOW). Here we focus on CBOW. More formally, every word is mapped to a unique vector represented by a column in a projection matrix W ∈ Rd×V with d as embedding dimension and V as the total number of words in the vocabulary. Given a sequence of words w−2 , w−1 , wt , w1 , w2 , ..., wT , the objective is to maximize the average log probability: V X 1X log P (wt \|wt+j ) T t=1 −c≤j≤c,j̸=0 (2.13) 2.5 Word Embeddings 27 where c is the size of the context window and wt+j is a word in the context window of the center word wt . The context vector is obtained by averaging the embeddings of each word w−c≤j≤c,j̸=0 and the prediction of the center word wt is performed using a softmax multiclass classifier over all vocabulary V : eywt P (wt \|wt+j ) = P yw e i (2.14) Each of yi is un-normalized log-probability for each output word i. After training, a low dimensionality embedding matrix E encapsulating information about each word in the vocabulary and its surrounding contexts is learned, transforming a one-hot sparse representation of words into a compact real valued embedding vector of size d × 1. This matrix can then be used as input to other learning algorithms tailored for specific tasks to further enhance performance. For large vocabularies it is unfeasible to compute the partition function (normalizer) of softmax therefore Mikolov [100] proposes to use the hierarchical softmax objective function or to approximate the partition function using a technique called negative sampling. Stochastic gradient descent is usually applied for training the softmax where the gradient is obtained via backpropagation. There are several approaches to generating word embeddings. One can build models that explicitly aim at generating word embeddings, such as Word2Vec or GloVe [100, 109], or one can extract such embeddings as by-products of more general models, which implicitly compute such word embeddings in the process of solving other language tasks. One of the issues of recent work in training word embeddings is the variability of experimental setups reported. For instance, in the paper describing GloVe [109] the authors trained their model on five corpora of different sizes and built a vocabulary of 400K most frequent words. Mikolov et al. [113] trained with 82K vocabulary while Mikolov et al. [100] was trained with 3M vocabulary. Recently, Arora et al. [114] proposed a generative model for learning embeddings that tries to explain some theoretical justification for nonlinear models (e.g. word2vec and GloVe) and some hyper parameter choices. The authors evaluated their model using 68K vocabulary. SemEval 2016-Task 4: Sentiment Analysis in Twitter organizers report that participants either used general purpose pre-trained word embeddings, or trained from Tweet 2016 dataset or “from some sort of dataset” [115]. However, participants neither report the size of vocabulary used neither the possible effect it might have on the task specific results. 28 Background and Related Work Recently, Rodrigues et al. [116] created and distributed the first general purpose embeddings for Portuguese. Word2vec gensim implementation was used and authors report results with different values for the parameters of the framework. Furthermore, authors used experts to translate well established word embeddings test sets for Portuguese language, which they also made publicly available and we use some of those in this work. 2.6 Predicting Collective Attention Online Reputation Monitoring systems would be even more useful if they would be able to know in advance if social media users will talk a lot about the target entities or not. In recent years, a number of research works have studied the relationship and predictive behavior of user response to the publication of online media items, such as, commenting news articles, playing Youtube videos, sharing URLs or retweeting patterns [117–120]. The first attempt to predict the volume of user comments for online news articles used both metadata from the news articles and linguistic features [119]. The prediction was divided in two binary classification problems: if an article would get any comments and if it would be high or low number of comments. Similarly, other studies found that shallow linguistic features (e.g. TF-IDF or sentiment) and named entities have good predictive power [121, 122]. Research work more in line with ours, tries to predict the popularity of news articles shares (url sharing) on Twitter based on content features [117]. The authors considered the news source, the article’s category, the article’s author, the subjectivity of the language in the article, and number of named entities in the article as features. Recently, there was a large study of the life cycle of news articles in terms of distribution of visits, tweets and shares over time across different sections of the publisher [123]. Their work was able to improve, for some content type, the prediction of web visits using data from social media after ten to twenty minutes of publication. Other lines of work, focused on temporal patterns of user activities and have consistently identified broad classes of temporal patterns based on the presence of a clear peak of activity [124–126, 118]. Classes differentiate by the specific amount and duration of activity before and after the peak. Crane and Sornette [124] define endogenous or exogenous origin of events based on being triggered by internal aspects of the social network or external, respectively. They find that hashtag popularity is mostly influenced by exogenous factors instead of epidemic spreading. Other work [125] extend these classes by creating distinct clusters of activity based on the distributions 2.7 Political Data Science 29 in different periods (before, during and after the peak) that can be interpreted based on semantics of hashtags. Consequently, the authors applied text mining techniques to semantically describe hashtag classes. Yang and Leskovec [118] propose a new measure of time series similarity and clustering. The authors obtain six classes of temporal shapes of popularity of a given phrase (meme) associated with a recent event, as well as the ordering of media sources contribution to its popularity. Recently, Tsytsarau et al. [127] studied the time series of news events and their relation to changes of sentiment time series expressed on related topics on social media. The authors proposed a novel framework using time series convolution between the importance of events and media response function, specific to media and event type. Their framework is able to predict time and duration of events as well as shape through time. 2.7 Political Data Science Content analysis of mass media has an established tradition in the social sciences, particularly in the study of effects of media messages, encompassing topics as diverse as those addressed in seminal studies of newspaper editorials [128], media agenda-setting [129], or the uses of political rhetoric [130], among many others. By 1997, Riffe and Freitag [131], reported an increase in the use of content analysis in communication research and suggested that digital text and computerized means for its extraction and analysis would reinforce such a trend. Their expectation has been fulfilled: the use of automated content analysis has by now surpassed the use of hand coding [132]. The increase in the digital sources of text, on the one hand, and current advances in computation power and design, on the other, are making this development both necessary and possible, while also raising awareness about the inferential pitfalls involved [133, 134]. One avenue of research that has been explored in recent years concerns the use of social media to predict present and future political events, namely electoral results [135– 143]. Although there is no consensus about methods and their consistency [144, 145]. Gayo-Avello [146] summarizes the differences between studies conducted so far by stating that they vary about period and method of data collection, data cleansing and pre-processing techniques, prediction approach and performance evaluation. One particular challenge when using sentiment is how to aggregate opinions in a timely fashion that can be fed to the prediction method. Two main strategies have been used to predict elections: buzz, i.e., number of tweets mentioning a given candidate or 30 Background and Related Work party and the use of sentiment polarity. Different computational approaches have been explored to process sentiment in text, namely machine learning and linguistic based methods [147–149]. In practice, algorithms often combine both strategies. Johnson et al. [150] concluded that more than predicting elections, social media can be used to gauge sentiment about specific events, such as political news or speeches. Defending the same idea, Diakopoulos el al. [151] studied the global sentiment variation based on Twitter messages of an Obama vs McCain political TV debate while it was still happening. Tumasjan et al. [140] used Twitter data to predict the 2009 Federal Election in Germany. They stated that “the mere number of party mentions accurately reflects the election result”. Bermingham et al. [135] correctly predicted the 2011 Irish General Elections also using Twitter data. Gayo-Avello et al. [145] also tested the share of volume as predictor in the 2010 US Senate special election in Massachusetts. On the other hand, several other studies use sentiment as a polls result indicator. Connor et al. [142] used a sentiment aggregate function to study the relationship between the sentiment extracted from Twitter messages and polls results. They defined the sentiment aggregate function as the ratio between the positive and negative messages referring an specific political target. They used the sentiment aggregate function as predictive feature in the regression model, achieving a correlation of 0.80 between the results and the poll results, capturing the important large-scale trends. Bermingham et al. [135] also included in their regression model sentiment features. Bermingham et al. introduced two novel sentiment aggregate functions. For inter-party sentiment, they modified the share of volume function to represent the share of positive and negative volume. For intra-party sentiment , they used a log ratio between the number of positive and negative mentions of a given party. Moreover, they concluded that the inclusion of sentiment features augmented the effectiveness of their model. Gayo-Avello et al. [145] introduced a different aggregate function. In a two-party race, all negative messages on party c2 are interpreted as positive on party c1, and vice-versa. In summary, suggestions for potentially independent or in other words predictive metrics appear in a wide variety of forms: the mention share that a party received within all party mentions during a given time-span [135, 152–155, 140], the mention share of political candidates [156–159, 153], the share of positive mentions a party received [135, 160], the positive mention share of candidates [142, 161, 158], the share of users commenting on a candidate or party [155], the share of mentions for a candidate followed by a word indicative of electoral success or failure [162], the relative increase of positive mentions of a candidate [163] or simply a collection of various potentially 2.7 Political Data Science 31 politically relevant words identified by their statistical relationship with polls or political actors in the past [164–167]. Suggestions for the dependent variable, metrics of political success, show a similar variety. They include the vote share that a party received on election day [135, 163, 152– 154], the vote share of a party adjusted to include votes only for parties included in the analysis [140], the vote share of candidates on election day [157–159, 162, 153], campaign tracking polls [164, 165, 158, 166, 142, 161, 160], politicians’ job approval ratings [167, 142], and the number of seats in parliament that a party received after the election [155]. Chapter 3 Entity Retrieval for Online Reputation Monitoring We start by presenting a formal definition of E-R queries and how can we model the E-R retrieval problem from a probabilistic perspective. We assume that a E-R query can be formulated as a sequence of individual sub-queries each targeting a specific entity or relationship. If we create specific representations for entities (e.g. context terms) as well as for pairs of entities, i.e. relationships then we can create a graph of probabilistic dependencies between sub-queries and entity/relationship representations. We show that these dependencies can be depicted in a probabilistic graphical model, i.e. a Bayesian network. Therefore, answering an E-R query can be reduced to a computation of factorized conditional probabilities over a graph of sub-queries and entity/relationship documents. However, it is not possible to compute these conditional probabilities directly from raw documents in a collection. Such as with traditional entity retrieval, documents serve as proxies to entities (and relationships) representations. It is necessary to fuse information spread across multiple documents. We propose two design patterns inspired from Model 1 and Model 2 of Balog et al. [46] to create entity/relationship centric and document centric representations. The first design pattern - Early Fusion - consists in aggregating context terms of entity and relationship occurrences to create two dedicated indexes, the entity index and the relationship index. Then it is possible to use any retrieval method to compute the relevance score of entity and relationship documents given the E-R sub-queries. The second design pattern - Late Fusion - can be applied on top of a standard document index alongside a set of entity occurrences in each document. First we compute the relevance score of documents given a E-R sub-query, then based on Entity Retrieval for Online Reputation Monitoring 34 the entity occurrences of the top k results we compute individual entity or relationship scores. Once again any retrieval method can be used to score documents. When combined with traditional retrieval methods (e.g. Language Models or BM25) these design patterns can be used to create unsupervised baselines for E-R retrieval. Finally, we follow a recent research line in entity retrieval [53, 69, 67] which exploits term dependencies using the Markov Random Field (MRF) framework for retrieval[63]. We introduce the Entity-Relationship Dependence Model (ERDM), a novel supervised Early Fusion-based model for E-R retrieval that creates a MRF to compute term dependencies of E-R queries and entity/relationship documents. 3.1 Entity-Relationship Retrieval E-R retrieval is a complex case of entity retrieval. E-R queries expect tuples of related entities as results instead of a single ranked list of entities as it happens with general entity queries. For instance, the E-R query “Ethnic groups by country" is expecting a ranked list of tuples <ethnic group, country> as results. The goal is to search for multiple unknown entities and relationships connecting them. Table 3.1 E-R retrieval definitions. Q QEi QRi−1,i DEi DRi−1,i QE QR DE DR \|Q\| TE E-R query (e.g. “congresswoman hits back at US president”). Entity sub-query in Q (e.g. “congresswoman”). Relationship sub-query in Q (e.g. “hits back at”). Term-based representation of an entity (e.g. <Frederica Wilson> = {representative, congresswoman}). We use the terminology representation and document interchangeably. Term-based representation of a relationship (e.g. <Frederica Wilson, Donald Trump> = {hits,back}). We use the terminology representation and document interchangeably. The set of entity sub-queries in a E-R query (e.g. {“congresswoman”,“US president” }). The set of relationship sub-queries in a E-R query. The set of entity documents to be retrieved by a E-R query. The set of relationship documents to be retrieved by a E-R query. E-R query length corresponding to the number of entity and relationship sub-queries. The entity tuple to be retrieved (e.g. <Frederica Wilson, Donald Trump>). 3.1 Entity-Relationship Retrieval 35 In this section, we present a definition of E-R queries and a probabilistic formulation of the E-R retrieval problem from an Information Retrieval perspective. Table 3.1 presents several definitions that will be used throughout this chapter. 3.1.1 E-R Queries E-R queries aim to obtain a ordered list of entity tuples TE = <E1 , E2 , ..., En > as a result. Contrary to entity search queries where the expected result is a ranked list of single entities, results of E-R queries should contain two or more entities. For instance, the complex information need “Silicon Valley companies founded by Harvard graduates” expects entity-pairs (2-tuples) <company, founder> as results. In turn, “European football clubs in which a Brazilian player won a trophy" expects triples (3-tuples) <club, player, trophy> as results. Each pair of entities Ei−1 , Ei in an entity tuple is connected with a relationship R(Ei−1 , Ei ). A complex information need can be expressed in a relational format, which is decomposed into a set of sub-queries that specify types of entities E and types of relationships R(Ei−1 , Ei ) between entities. For each relationship sub-query there must be two sub-queries, one for each of the entities involved in the relationship. Thus a E-R query Q that expects 2-tuples, is mapped into a triple of sub-queries Q = {QE1 , QR1,2 , QE2 }, where QE1 and QE2 are the entity attributes queried for E1 and E2 respectively, and QR1,2 is a relationship attribute describing R(Ei , Ei+1 ). If we consider a E-R query as a chain of entity and relationship sub-queries Q = {QE1 , QR1,2 , QE2 , ..., QEn−1 ,QRn−1,n , QEn } and we define the length of a E-R query \|Q\| as the number of sub-queries, then the number of entity sub-queries must be \|Q\|+1 and the number of relationship sub-queries equal to \|Q\|−1 . Consequently, the 2 2 size of each entity tuple TE to be retrieved must be equal to the number of entity sub-queries. For instance, the E-R query “soccer players who dated a top model” with answers such as <Cristiano Ronaldo, Irina Shayk>) is represented as three sub-queries QE1 = {soccer players}, QR1,2 = {dated}, QE2 = {top model}. Automatic mapping of terms from a E-R query Q to sub-queries QEi or QRi−1,i is out of the scope of this work and can be seen as a problem of query understanding [168, 54, 169]. We assume that the information needs are decomposed into constituent entity and relationship sub-queries using Natural Language Processing techniques or by user input through an interface that enforces the structure Q = {QE1 , QR1,2 , QE2 , ..., QEn−1 ,QRn−1,n , QEn }. 36 3.1.2 Entity Retrieval for Online Reputation Monitoring Modeling E-R Retrieval Our approach to E-R retrieval assumes that we have a raw document collection (e.g. news articles) and each document Dj is associated with one or more entities Ei . In other words, documents contain mentions to one or more entities that can be related between them. Since our goal is to retrieve tuples of related entities given a E-R query that expresses entity attributes and relationship attributes, we need to create term-based representations for both entities and relationships. We denote a representation of an entity Ei as DEi . In E-R retrieval we are interested in retrieving tuples of entities TE = <E1 , E2 , ..., En > as a result. The number of entities in each tuple can be two, three or more depending on the structure of the particular E-R query. When a E-R query aims to get tuples of more than two entities, we assume it is possible to combine tuples of length two. For instance, we can associate two tuples of length two that share the same entity to retrieve a tuple of length three. Therefore we create representations of relationships as pairs of entities. We denote a representation of a relationship R(Ei−1 , Ei ) as DRi−1,i . Considering the example query “Which spiritual leader won the same award as a US vice president?” it can be formulated in the relational format as QE1 = {spiritual leader}, QR1,2 = {won}, QE2 = {award}, QR2,3 = {won}, QE3 = {US vice president}. Associating the tuples of length two <Dalai Lama, Nobel Peace Prize> and <Nobel Peace Prize, Al Gore> would result in the expected 3-tuple <Dalai Lama, Nobel Peace Prize, Al Gore>. For the sake of clarity we now consider an example E-R query with three sub-queries (\|Q\| = 3). This query aims to retrieve a tuple of length two, i.e. a pair of entities connected by a relationship. Based on the definition of a E-R query, each entity in the resulting tuple must be relevant to the corresponding entity sub-queries QE . Moreover, the relationship between the two entities must also be relevant to the relationship sub-queries QR . Instead of calculating a simple posterior P (D\|Q) as with traditional information retrieval, in E-R retrieval the objective is to rank tuples based on a joint posterior of multiple entity and relationship representations given a E-R query, such as P (DE2 , DE1 , DR1,2 \|Q) when \|Q\| = 3. E-R queries can be seen as chains of interleaved entity and relationship subqueries. We take advantage of the chain rule to formulate the joint probability P (DE2 , DE1 , DR1,2 , Q) as a product of conditional probabilities. Formally, we want to rank entity and relationship candidates in descending order of the joint posterior P (DE2 , DE1 , DR1,2 \|Q) as: 3.1 Entity-Relationship Retrieval 37 P (DE2 , DE1 , DR1,2 , Q) P (Q) E2 E1 R1,2 , Q).P (DE1 \|DR1,2 , Q).P (DR1,2 \|Q).P (Q) rank P (D \|D , D = P (Q) rank P (DE2 , DE1 , DR1,2 \|Q) = rank = P (DE2 \|DR1,2 , Q).P (DE1 \|DR1,2 , Q).P (DR1,2 \|Q) rank ∝ P (DE2 \|DR1,2 , QE2 ).P (DE1 \|DR1,2 , QE1 ).P (DR1,2 \|QR1,2 ) (3.1) (3.2) We consider conditional independence between entity representations within the joint posterior, i.e., the probability of a given entity representation DEi being relevant given a E-R query is independent of knowing that entity DEi+1 is relevant as well. As an example, consider the query “action movies starring a British actor”. Retrieving entity representations for “action movies” is independent of knowing that <Tom Hardy> is relevant to the sub-query “British actor”. However, it is not independent of knowing the set of relevant relationships for sub-query “starring”. If a given action movie is not in the set of relevant entity-pairs for “starring” it does not make sense to consider it as relevant. Consequently, P (DE2 \|DE1 , DR1,2 , Q) = P (DE2 \|DR1,2 , Q). Since E-R queries can be decomposed in constituent entity and relationship subqueries, ranking candidate tuples using the joint posterior P (DE2 , DE1 , DR1,2 \|Q) is rank proportional to the product of conditional probabilities on the corresponding entity and relationship sub-queries QE2 , QE1 and QR1,2 . We now consider a longer E-R query aiming to retrieve a triple of connected entities. This query has three entity sub-queries and two relationship sub-queries, thus \|Q\| = 5. As we previously explained, when there are more than one relationship sub-queries we need to join entity-pairs relevant to each relationship sub-query that have one entity in common. From a probabilistic point of view this can be seen as conditional dependence from the entity-pairs retrieved from the previous relationship sub-query, i.e. P (DR2,3 \|DR1,2 , Q) ̸= P (DR2,3 \|Q). To rank entity and relationship candidates we need to calculate the following joint posterior: 38 Entity Retrieval for Online Reputation Monitoring rank P (DE3 , DE2 ,DE1 , DR2,3 , DR1,2 \|Q) = P (DE3 \|DE2 , DE1 , DR2,3 , DR1,2 , Q). P (DE3 \|DE2 , DR2,3 , DR1,2 , Q).P (DE1 \|DR2,3 , DR1,2 , Q). P (DR2,3 \|DR1,2 , Q).P (DR1,2 \|Q) rank = P (DE3 \|DR2,3 , Q).P (DE2 \|DR2,3 , DR1,2 , Q). P (DE1 \|DR1,2 , Q).P (DR2,3 \|DR1,2 , Q).P (DR1,2 \|Q) rank ∝ P (DE3 \|DR2,3 , QE3 ).P (DE2 \|DR2,3 , DR1,2 , QE2 ). P (DE1 \|DR1,2 , QE1 ).P (DR2,3 \|DR1,2 , QR2,3 ).P (DR1,2 \|QR1,2 ) (3.3) (3.4) When compared to the previous example, the joint posterior for \|Q\| = 5 shows that entity candidates for DE2 are conditional dependent of both DR2,3 and DR1,2 . In other words, entity candidates for DE2 must belong to entity-pairs candidates for both relationships representations that are connected with E2 , i.e. DR2,3 and DR1,2 . We are now able to make a generalization of E-R retrieval as a factorization of conditional probabilities of a joint probability of entity representations DEi , relationship representations DRi−1,i , entity sub-queries QEi and relationship sub-queries QRi−1,i . These set of random variables and their conditional dependencies can be easily represented in a probabilistic directed acyclic graph,i.e. a Bayesian network [170]. In Bayesian networks, nodes represent random variables while edges represent conditional dependencies. Every other nodes that point to a given node are considered parents. Bayesian networks define the joint probability of a set of random variables as a factorization of the conditional probability of each random variable conditioned Q on its parents. Formally, P (X1 , . . . , Xn ) = ni=1 P (Xi \|pai ), where pai represents all parent nodes of Xi . Figure 3.1 depicts the representation of E-R retrieval for different query lengths \|Q\| using Bayesian networks. We easily conclude that graphical representation contributes to establish a few guidelines for modeling E-R retrieval. First, each sub-query points to the respective document node. Second, relationship document nodes always point to the contiguous entity representations. Last, when there are more than one relationship sub-query, relationship documents also point to the subsequent relationship document. Once we draw the graph structure for the number of sub-queries in Q we are able to compute a product of conditional probabilities of each node given its parents. Adapting 3.1 Entity-Relationship Retrieval 39 (a) \|Q\| = 3 (b) \|Q\| = 5 (c) \|Q\| = 7 Fig. 3.1 Bayesian networks for E-R Retrieval with queries of different lengths. the general joint probability formulation of Bayesian networks to E-R retrieval we come up with the following generalization: \|Q\|−1 2 \|Q\|+1 2 rank Y P (DE , DR \|Q) = P (DEi \|DRi−1,i , DRi,i+1 , QEi ) i=1 P (DRi,i+1 \|DRi−1,i , QRi,i+1 ) Y i=1 (3.5) We denote D as the set of all candidate relationship documents in the graph and DE the set of all candidate entity documents in the graph. In Information Retrieval is often convenient to work in the log-space as it does not affect ranking and transforms the product of conditional probabilities in a summation, as follows: R rank P (DE , DR \|Q) = log P (DE , DR \|Q) \|Q\|+1 2 rank X = i=1 \|Q\|−1 2 logP (DEi \|DRi−1,i , DRi,i+1 , QEi ) + X logP (DRi,i+1 \|DRi−1,i , QRi,i+1 ) i=1 (3.6) (3.7) 40 Entity Retrieval for Online Reputation Monitoring We now present two design patterns to compute each conditional probability for every entity and relationship candidate documents. 3.2 Design Patterns for Entity-Relationship Retrieval Traditional ad-hoc document retrieval approaches create direct term-based representations of raw documents. A retrieval model (e.g. Language Models) is then used to match the information need, expressed as a keyword query, against those representations. However, E-R retrieval requires collecting evidence for both entities and relationships that can be spread across multiple documents. It is not possible to create direct term-based representations. Raw documents serve as proxy to connect queries with entities and relationships. Abstractly speaking, entity retrieval can be seen as a problem of object retrieval in which the search process is about fusing information about a given object, such as in the case of verticals (e.g. Google Finance). Recently, Zhang and Balog [171] presented two design patterns for fusion-based object retrieval. The first design pattern – Early Fusion – is an object-centric approach where a termbased representation of objects is created earlier in the retrieval process. First, it creates meta-documents by aggregating term counts across the documents associated with the objects. Later, it matches queries against these meta-documents using standard retrieval methods. The second design pattern - Late Fusion - is a document-centric approach where relevant documents to the query are retrieved first and then later in the retrieval process, it ranks objects associated with top documents. These design patterns represent a generalization of Balog’s Model 1 and Model 2 for expertise retrieval [46]. In essence, E-R retrieval is an extension, or a more complex case, of object-retrieval where besides ranking objects we need to rank tuples of objects that satisfy the relationship expressed in the E-R query. This requires creating representations of both entities and relationships by fusing information spread across multiple raw documents. We propose novel fusion-based design patterns for E-R retrieval that are inspired from the design patterns presented by Zhang and Balog [171] for single object-retrieval. We extend those design patterns to accommodate the specificities of E-R retrieval. We hypothesize that it should be possible to generalize the term dependence models to represent entity-relationships and achieve effective E-R retrieval without entity or relationship type restrictions (e.g. categories) as it happens with the Semantic Web based approaches. 3.2 Design Patterns for Entity-Relationship Retrieval 3.2.1 41 Early Fusion The Early Fusion strategy presented by Zhang and Balog [171] consists in creating a term-based representation for each object under retrieval, i.e., a meta-document containing all terms in the proximity of every object mention across a document collection. As described in previous section, E-R queries can be formulated as a sequence of multiple entity queries QE and relationship queries QR . In a Early Fusion approach, each of these queries should match against a previously created term-based representation. Since there are two types of queries, we propose to create two types of term-based representations, one for entities and other for relationships. Our Early Fusion design pattern is similar to Model 1 of Balog et al. [46]. It can be thought as creating two types of meta-documents DE and DR . A meta-document DEi is created by aggregating the context terms of the occurrences of Ei across the raw document collection. On the other hand, for each each pair of entities Ei−1 and Ei that co-occur close together across the raw document collection we aggregate context terms that describe the relationship to create a meta-document DRi−1,i In our approach we focus on sentence level information about entities and relationships although the design pattern can be applied to more complex segmentations of text (e.g. dependency parsing). We rely on Entity Linking methods for disambiguating and assigning unique identifiers to entity mentions on raw documents D. We collect entity contexts across the raw document collection and index them in the entity index. The same is done by collecting and indexing entity pair contexts in the relationship index. We define the (pseudo) frequency of a term t for an entity meta-document DEi as follows: f (t, DEi ) = n X f (t, Ei , Dj )w(Ei , Dj ) (3.8) j=1 where n is the total number of raw documents in the collection, f (t, Ei , Dj ) is the term frequency in the context of the entity Ei in a raw document Dj . w(Ei , Dj ) is the entity-document association weight that corresponds to the weight of the document Dj in the mentions of the entity Ei across the raw document collection. Similarly, the term (pseudo) frequency of a term t for a relationship meta-document DRi−1,i is defined as follows: f (t, D Ri−1,i )= n X j=1 f (t, Ri−1,i , Dj )w(Ri−1,i , Dj ) (3.9) 42 Entity Retrieval for Online Reputation Monitoring where f (t, Ri−1,i , Dj is the term frequency in the context of the pair of entity mentions corresponding to the relationship Ri−1,i in a raw document Dj and w(Ri−1,i , Dj ) is the relationship-document association weight. In this work we use binary associations weights indicating the presence/absence of an entity mention in a raw document, as well as for a relationship. However, other weight methods can be used. The relevance score for an entity tuple TE can then be calculated using the posterior P (DE , DR \|Q) defined in previous section (equation 3.6). We calculate the individual conditional probabilities as a product of a retrieval score with an association weight. Formally we consider: logP (DEi \|DRi−1,i , DRi,i+1 , QEi ) = score(DEi , QEi )w(Ei , Ri−1,i , Ri,i+1 ) logP (DRi,i+1 \|DRi−1,i , QRi,i+1 ) = score(DRi,i+1 , QRi,i+1 )w(Ri,i+1 , Ri−1,i ) (3.10) (3.11) where score(DRi,i+1 , QRi,i+1 ) represents the retrieval score resulting of the match of the query terms of a relationship sub-query QRi,i+1 and a relationship meta-document DRi,i+1 . The same applies to the retrieval score score(DEi , QEi ) which corresponds to the result of the match of an entity sub-query QEi with a entity meta-document DEi . For computing both score(DRi,i+1 , QRi,i+1 ) and score(DEi , QEi ) any retrieval model can be used. Different scoring functions will be introduced below. We use a binary association weight for w(Ei , Ri−1,i , Ri,i+1 ) which represents the presence of a relevant entity Ei to a sub-query QEi in its contiguous relationships in the Bayesian network, i.e. Ri−1,i and Ri,i+1 which must be relevant to the sub-queries QRi−1,i and QRi,i+1 . This entity-relationship association weight is the building block that guarantees that two entities relevant to sub-queries QE that are also part of a relationship relevant to a sub-query QR will be ranked higher than tuples where just one or none of the entities are relevant to the entity sub-queries QE . On the other hand, the entity-relationship association weight w(Ri,i+1 , Ri−1,i ) guarantees that consecutive relationships share one entity between them in order to create triples or 4-tuples of entities for longer E-R queries (\|Q\| > 3). The relevance score of an entity tuple TE given a query Q is calculated by summing individual relationship and entity relevance scores for each QRi−1,i and QEi in Q. We define the score for a tuple TE given a query Q as follows: 3.2 Design Patterns for Entity-Relationship Retrieval E R rank P (D , D \|Q) = 43 \|Q\|+1 2 score(DEi , QEi )w(Ei , Ri−1,i , Ri,i+1 )+ X i=1 (3.12) \|Q\|−1 2 X score(DRi,i+1 , QRi,i+1 )w(Ri,i+1 , Ri−1,i ) i=1 (3.13) Considering Dirichlet smoothing unigram Language Models (LM) the constituent retrieval scores can be computed as follows:  scoreLM (DRi,i+1 , QRi,i+1 ) = log  t∈DRi,i+1 ∩QRi,i+1  scoreLM (DEi , QEi ) = X t∈DEi ∩QEi  f (t,C R ) R µ  \|C R \|  + µR Ri,i+1 )+  f (t, D X \|DRi,i+1 \|  f (t,C E ) E  f (t, D ) + \|C E \| µ  log   \|DEi \| + µE Ei (3.14) (3.15) where t is a term of a sub-query QEi or QRi,i+1 , f (t, DEi ) and f (t, DRi,i+1 ) are the (pseudo) frequencies defined in equations 3.8 and 3.9. The collection frequencies f (t, C E ), f (t, C R ) represent the frequency of the term t in either the entity index C E or in the relationship index C R . \|DEi \| and\|DRi,i+1 \| represent the total number of terms in a meta-document while \|C R \| and \|C E \| represent the total number of terms in a collection of meta-documents. Finally, µE and µR are the Dirichlet prior for smoothing which generally corresponds to the average document length in a collection. 3.2.2 Association Weights Both Early Fusion and Late Fusion share three components: w(Ri,i+1 , Dj ), w(Ei , Dj ) and w(Ei , Ri,i+1 ). The first two represent document associations which determine the weight a given raw document contributes to the relevance score of a particular entity tuple TE . The last one is the entity-relationship association which indicates the strength of the connection of a given entity Ei within a relationship Ri,i+1 . In our work we only consider binary association weights but other methods could be used. According to the binary method we define the weights as follows: w(Ri,i+1 , Dj ) = 1 if R(Ei , Ei+1 ) ∈ Dj , 0 otherwise (3.16) Entity Retrieval for Online Reputation Monitoring 44 w(Ei , Dj ) = 1 if Ei ∈ Dj , 0 otherwise (3.17) w(Ei , Ri−1,i , Ri,i+1 ) = 1 if Ei ∈ DRi−1,i and Ei ∈ DRi,i+1 , 0 otherwise (3.18) w(Ri,i+1 , Ri−1,i ) = 1 if Ei ∈ DRi−1,i and Ei ∈ DRi,i+1 , 0 otherwise (3.19) Under this approach the weight of a given association is independent of the number of times an entity or a relationship occurs in a document. A more general approach would be to assign real numbers to the association weights depending on the strength of the association [8]. For instance, uniform weighting would be proportional to the inverse of the number of documents where a given entity or relationship occurs. Other option could be a TF-IDF approach. 3.2.3 Early Fusion Example Let us consider an illustrative example of the Early Fusion design pattern for E-R retrieval using unigram Language Models and the E-R query Q = {soccer players who dated a top model}. This query can be decomposed in three sub-queries, QEi = { soccer players}, QEi+1 = { top model} and QRi,i+1 = { dated }. The first two sub-queries target the entity index and the last targets the relationship index. Table 3.2 presents a toy entity index with 3 entities as example for each of the two entity sub-queries, including the term frequency f (t, DEi ) for each sub-query term. Table 3.2 Illustrative example of the entity index in Early Fusion. Ei <Tom Brady> <Cristiano Ronaldo> <Lionel Messi> <Luís Figo> <Gisele Bundchen> <Irina Shayik> <Helen Svedin> ... f (t, DEi ) soccer:0 player:600 soccer:800 player:800 soccer:700 player:700 soccer:200 player:200 top:400 model:400 top:300 model:300 top:150 model:150 ... \|DEi \| 3000 5000 4000 800 3000 2000 600 ... 3.2 Design Patterns for Entity-Relationship Retrieval 45 Considering the remaining variables required to calculate the scoreLM (DEi , QEi ): \|C E \| = 100000 \|µE \| = 1500 f (soccer, C E ) = 3000 f (player, C E ) = 8000 f (top, C E ) = 8000 f (model, C E ) = 4000 We calculate the scoreLM (DEi , QEi ) for the respective entities and sub-queries. For the first entity query – “soccer players” – the ranked list of relevant entities and the respective LM score would be the following: 1. <Lionel Messi>: -1.6947 2. <Cristiano Ronaldo>: -1.7351 3. <Luís Figo>: -1.8291 4. <Tom Brady>: -2.7958 For the second entity query – “top models”: 1. <Gisele Bundchen>: -1.6295 2. <Irina Shayik>: -1.7093 3. <Helen Svedin>: -1.9698 Table 3.3 shows 3 relationships, i.e. entity pairs, relevant to the sub-query “dated” and the respective term frequency f (t, DRi,i+1 ). Considering the remaining variables required to calculate the scoreLM (DRi,i+1 , QRi,i+1 ): \|C R \| = 20000 \|µR \| = 500 f (dated, C R ) = 5000 Entity Retrieval for Online Reputation Monitoring 46 Table 3.3 Illustrative example of the relationship index in Early Fusion. Ri,i+1 <Gisele Bundchen, Tom Brady> <Irina Shayik, Cristiano Ronaldo> <Helen Svedin, Luís Figo> ... f (t, DRi,i+1 ) dated:500 dated:300 dated:100 ... \|DRi,i+1 \| 800 600 200 ... We calculate the scoreLM (DRi,i+1 , QRi,i+1 )for the respective relationship and the sub-query and we obtain the following ranked list: 1. <Gisele Bundchen, Tom Brady>: -0.3180 2. <Irina Shayik, Cristiano Ronaldo>: -0.4130 3. <Helen Svedin, Luís Figo>: -0.4929 We can now sum up individual scores for each sub-query and calculate the final score for the early fusion design pattern score(TE , Q) using the equation 3.12. The final ranked list of tuples is the following: 1. <Irina Shayik, Cristiano Ronaldo>: -3.8575 2. <Helen Svedin, Luís Figo>: -4.2919 3. <Gisele Bundchen, Tom Brady>: -4.6977 The entity tuple <Irina Shayik, Cristiano Ronaldo> is the most relevant to the query “soccer players who dated a top model”. Although <Gisele Bundchen, Tom Brady> has higher individual scores in two sub-queries (“top model” and “dated”) it ranks last due to the poor relevance of Tom Brady to the sub-query “soccer player”. The entity <Lionel Messi> is the most relevant entity to the sub-query “soccer player” but it is not relevant to the relationship sub-query, therefore it is excluded from the final ranked list of entity tuples. 3.2.4 Late Fusion The Late Fusion design pattern presented by Zhang and Balog [171] is a documentcentric strategy, i.e. first we query raw individual documents then we aggregate the associated objects with the relevant documents. Instead of creating term-based representations of entities and relationships (pairs of entities), in late fusion we use the 3.2 Design Patterns for Entity-Relationship Retrieval 47 raw documents as hidden variables, separating the E-R query from the relevant entity tuples to be retrieved. Our vision of ORM implies processing raw documents to detect entities occurrences and extract sentence level information that will be used in downstream Entity Retrieval and Text Mining tasks. Therefore, we are not interested in applying a Late Fusion strategy in this work. However, we believe it makes sense to present a theoretical formulation of a Late Fusion design pattern for E-R retrieval. We leave the practical experiments with Late Fusion for future work in the context of generic E-R retrieval. The process of retrieving entity tuples using our late fusion strategy consists in processing each sub-query independently, as in the early fusion strategy, but in this case, we use a single index comprising a term based representation of the collection of raw documents. A retrieval model is used to calculate a relevance score of each individual raw document and a given sub-query. Once we have the relevant documents we use entity linking to extract the entities that are mentioned in each relevant raw document. Following this strategy we calculate aggregated counts of entity occurrences weighted by the individual relevance score of the individual raw documents. At the end, we join the results of each sub-query and calculate the overall relevance score of the entity tuples. Formally, we define the relevance score of an entity tuple TE given a query Q as follows: rank P (DE , DR \|Q) = \|Q\|+1 2 n X X score(Dj , QEi )w(Ei , Dj )w(Ei , Ri−1,i , Ri,i+1 )+ i=1 j=1 (3.20) \|Q\|−1 2 n X X score(Dj , QRi,i+1 )w(Ri,i+1 , Dj )w(Ri,i+1 , Ri−1,i ) i=1 j=1 (3.21) where score(Dj , QRi,i+1 ) represents the retrieval score resulting of the match of the query terms of a relationship sub-query QRi,i+1 and a raw document Dj . The same applies to the retrieval score score(Dj , QEi ) which corresponds to the result of the match of an entity sub-query QEi with a raw document Dj . The weights w(Ri,i+1 , Dj ) and w(Ei , Dj ) represent association weights between relationships and raw documents, and entities and raw documents, respectively. We use binary association weights in this work but other weights can be used. We also use a binary association weight Entity Retrieval for Online Reputation Monitoring 48 for w(Ei , Ri−1,i , Ri,i+1 ) and w(Ri,i+1 , Ri−1,i ) which represent the entity-relationship association weights, similarly to what happens with the case of Early Fusion. For computing both score(Dj , QRi,i+1 ) and score(Dj , QEi ) any retrieval model can be used. Considering BM25 the scores can be computed as follows: scoreBM 25 (Dj , QRi,i+1 ) = X log t∈Dj ∩QRi,i+1 N − n(t) + 0.5 f (t, Dj )(K1 + 1) . \|Dj \| n(t) + 0.5 f (t, Dj ) + K1 (1 − b + b avg(\|D\|) ) (3.22) f (t, Dj )(K1 + 1) N − n(t) + 0.5 . \|Dj \| n(t) + 0.5 ) f (t, Dj ) + K1 (1 − b + b avg(\|D\|) t∈Dj ∩QEi (3.23) Ei Ri,i+1 where t is a term of a sub-query Q or Q and f (t, Dj ) is the query term frequency in a raw document Dj . The inverse document frequency, IDF (t), is computed −n(t)+0.5 as log Nn(t)+0.5 with N as the number of documents on the collection and n(t) the number of documents where the term occurs.\|Dj \| is the total number of terms in a raw document Dj and avg(\|D\|) is the average document length. K1 and b are free parameters usually chosen as 1.2 and 0.75, in the absence of specific optimization. scoreBM 25 (Dj , QEi ) = 3.2.5 X log Late Fusion Example Considering the same toy example query introduced in the previous sub-section, we now have a single index, the document index, as illustrated in Table 3.4. The remaining parameters required for calculating the scoreBM 25 (Dj , QEi ) and scoreBM 25 (Dj , QRi,i+1 ) are the following: • N=2000 • n(soccer)=100 • n(player)=130 • n(dated)=60 • n(top)=250 • n(model)=80 • avg(\|D\|)=120 3.2 Design Patterns for Entity-Relationship Retrieval 49 Table 3.4 Illustrative example of the document index in Late Fusion. Dj docid-1 docid-2 docid-3 docid-4 docid-5 docid-6 docid-7 docid-8 docid-9 ... f (t, Dj ) soccer:10 player:10 soccer:5 player:5 soccer:5 player:5 top:4 model:4 \|Dj \| dated:5 top:6 model:6 model:4 dated:2 player:2 dated:3 top:2 model:2 dated:2 soccer:2 player:2 ... 200 Ei <Cristiano Ronaldo> <Lionel Messi> 150 <Cristiano Ronaldo> 100 <Luís Figo> 150 <Gisele Bundchen> 80 <Gisele Bundchen> <Tom Brady> 100 <Irina Shayik> 100 120 <Gisele Bundchen> <Adriana Lima> <Tom Brady> <Irina Shayik> <Cristiano Ronaldo> 150 <Luís Figo> <Helen Svedin> ... ... For the first entity sub-query, “soccer players”, the relevant documents ranked by the scoreBM 25 (Dj , QEi ) are the following: 1. docid-1 (<Cristiano Ronaldo>, <Lionel Messi>): 4.7606 2. docid-3 (<Luís Figo>): 4.6426 3. docid-2 (<Cristiano Ronaldo>): 4.3716 4. docid-9 (<Luís Figo>, <Helen Svedin>): 3.2803 5. docid-7 (<Gisele Bundchen>, <Adriana Lima>, <Tom Brady>): 1.8418 For the second entity sub-query, “top model”: 1. docid-6 (<Irina Shayik>): 3.1618 50 Entity Retrieval for Online Reputation Monitoring 2. docid-4 (<Gisele Bundchen>): 2.7393 3. docid-9 (<Luís Figo>, <Helen Svedin>): 2.1694 4. docid-7 (<Gisele Bundchen>, <Adriana Lima>, <Tom Brady>): 1.4714 For the relationship sub-query, “dated”: 1. docid-5 (<Gisele Bundchen>, <Tom Brady>): 2.8081 2. docid-8 (<Irina Shayik>, <Cristiano Ronaldo>): 2.3668 3. docid-7 (<Gisele Bundchen>, <Adriana Lima>, <Tom Brady>): 2.1728 4. docid-9 (<Luís Figo>, <Helen Svedin>): 1.9349 Since in Late Fusion there is no relationship meta-documents that could be used directly as entity tuples, we need to extract the candidate tuples from the raw documents retrieved using the relationship sub-query. When there are more than two entity associations in a relevant document we combine entities to create tuples. For instance, docid-7 has three entity associations therefore we extract three candidate tuples: <Gisele Bundchen, Tom Brady>, <Gisele Bundchen, Adriana Lima> and <Adriana Lima, Tom Brady>. For each candidate tuple we sum up scoreBM 25(Dj , QRi,i+1 )w(Ri,i+1 , Dj ) over every relevant document Dj for the relationship sub-query that is associated with each entity tuple. The same applies to individual entities from the candidate tuples that are associated with relevant documents for each entity sub-query. For instance, for the entity sub-query “soccer players” we sum score(Dj , QEi )w(Ei , Dj )w(Ei , Ri,i+1 ) over the relevant documents that mentioned an entity that belongs to a candidate tuple. When both entities of the candidate tuple are mentioned in relevant documents for both entity sub-queries, e.g. <Helen Svedin, Luís Figo>, we assign each entity to the sub-query that maximizes the final score score(TE , Q), i.e., we use the scores of the entity sub-query “soccer player” for <Luís Figo> and the entity sub-query “top model” for <Helen Svedin>. The final ranked list of entity tuples is the following: 1. <Irina Shayik, Cristiano Ronaldo>: 14.0443 2. <Helen Svedin, Luís Figo>: 12.5970 3. <Gisele Bundchen, Tom Brady>: 10.9784 3.2 Design Patterns for Entity-Relationship Retrieval 51 4. <Gisele Bundchen, Adriana Lima>: 9.3459 5. <Adriana Lima, Tom Brady>: 6.9245 Once again <Lionel Messi> is excluded from the final ranked list of entity tuples because he is not associated with any document relevant to the relationship sub-query “dated”. On the other hand, <Adriana Lima> is included in the final ranking although it is not true that she has dated either <Tom Brady> or <Gisele Bundchen>. In this example, the top three entity tuples are ranked in the same order as in the Early Fusion strategy example. 3.2.6 Implementation In this section we proposed two design patterns for E-R retrieval: Early Fusion (EF) and Late Fusion (LF). Both can be seen as a flexible framework for ranking tuples of entities given a E-R query expressed as a sequence of entity and relationship sub-queries. This framework is flexible enough to allow using any retrieval method to compute individual retrieval scores between document and query nodes in a E-R graph structure. When using Language Models (LM) or BM25 as scoring functions, these design patterns can be used to create unsupervised baseline methods for E-R retrieval (e.g. EF-LM, EF-BM25, LF-LM, LF-BM25, etc.). In the case of Early Fusion there is some overhead over traditional document search, since we need to create two E-R dedicated indexes that will store entity and relationship meta-documents. The entity index is created by harvesting the context terms in the proximity of every occurrence of a given entity across the raw document collection. This process must be carried for every entity in the raw document collection. A similar process is applied to create the relationship index. For every two entities occurring close together in a raw document we extract the text between both occurrences as a term-based representation of the relationship between the two. Once again, this process must be carried for every pair of co-occurring entities in sentences across the raw document collection. Late Fusion requires less overhead and can be implemented on top of a web search engine with reduced effort. We only need to have a list of entity occurrences alongside each document. Therefore there is no need to create a separate index(es). On the other hand, it requires more processing on query time since we need to first rank raw documents for each sub-query and then aggregate entity occurrences at the top k documents retrieved. Moreover, it does not contain any proximity-based information 52 Entity Retrieval for Online Reputation Monitoring on the entity occurrences, so two entities occurring very far in the text might be considered as relationship candidates. It might be prone to a higher false positive rate. One advantage of Early Fusing lies in its flexibility as we need to create two separate indexes for E-R retrieval it is possible to combine data from multiple sources in seamless way. For instance, one could use a well established knowledge base (e.g. DBpedia) as entity index and use a specific collection, such as a news collection or a social media stream, for harvesting relationships having a more transient nature. Common to both design patterns is a challenge inherent to the problem of E-R retrieval: the size of the search space. Although the E-R problem is formulated as a sequence of independent sub-queries, the results of those sub-queries must be joined together. Consequently, we have a multi-dimensional search space in which we need to join results based on shared entities. This problem becomes particularly hard when sub-queries are short and contain very popular terms. Let us consider “actor” as QEi , there will be many results to this sub-query, probably thousands. There is a high probability that will need to process thousands of sub-results before finding one entity that is also relevant to the relationship sub-query QRi−1,i . If at the same time we have computational power constraints, we will probably apply a strategy of just considering top k results for each sub-query which can lead to reduced recall in the case of short sub-queries with popular terms. 3.3 Entity-Relationship Dependence Model In this section we present the Entity-Relationship Dependence Model (ERDM), a novel supervised Early Fusion-based model for E-R retrieval. Recent approaches to entity retrieval [53, 67, 69] have demonstrated that using models based on Markov Random Field (MRF) framework for retrieval [63] to incorporate term dependencies can improve entity search performance. This suggests that MRF could be used to model E-R query term dependencies among entities and relationships documents. One of the advantages of the MRF framework for retrieval is its flexibility, as we only need to construct a graph G representing dependencies to model, define a set of non-negative potential functions ψ over the cliques of G and to learn the parameter vector Λ to score each document D by its unique and unnormalized joint probability with Q under the MRF [63]. The non-negative potential functions are defined using an exponential form ψ(c; Λ) = exp[λc f (c)], where λc is a feature weight, which is a free parameter in the model, 3.3 Entity-Relationship Dependence Model 53 associated with feature function f (c). Learning to rank is then used to learn the feature weights that minimize the loss function. The model allows parameter and feature functions sharing across cliques of the same configuration, i.e. same size and type of nodes (e.g. 2-cliques of one query term node and one document node). 3.3.1 Graph Structures The Entity-Relationship Dependence Model (ERDM) creates a MRF for modeling implicit dependencies between sub-query terms, entities and relationships. Each entity and each relationship are modeled as document nodes within the graph and edges reflect term dependencies. Contrary to traditional ad-hoc retrieval using MRF (e.g. SDM), where the objective is to compute the posterior of a single document given a query, the ERDM allows the computation of a joint posterior of multiple documents (entities and relationships) given a E-R query which consists also of multiple sub-queries. Fig. 3.2 Markov Random Field dependencies for E-R retrieval, \|Q\| = 3. The graph structures of the ERDM for two E-R queries, one with \|Q\| = 3 and other with \|Q\| = 3 are depicted in Figure 3.2 and Figure 3.3, respectively. Both graph structures contain two different types of query nodes and document nodes: entity query and relationship query nodes, QE and QR , plus entity and relationship document nodes, DE and DR . Within the MRF framework, DE and DR are considered “documents” but they are not actual real documents but rather objects representing an entity or a relationship between two entities. Unlike real documents, these objects do not have direct and explicit term-based representations. Usually, it is necessary to gather evidence across multiple real documents that mention the given object, in order to be able to match them against keyword queries. Therefore, ERDM can be seen as Early Fusion-based retrieval model. The existence of two different types of documents implies two different indexes: the entity index and the relationship index. 54 Entity Retrieval for Online Reputation Monitoring Fig. 3.3 Markov Random Field dependencies for E-R retrieval, \|Q\| = 5. The relationship-specific dependencies of ERDM are found in the 2-cliques formed by one entity document and one relationship document: DEi−1 - DRi−1,i , DEi - DRi−1,i and for \|Q\| = 5, DEi - DRi,i+1 and DRi−1,i - DRi,i+1 . The graph structure does not need to assume any explicit dependence between entity documents given a relationship document. They have an implicit connection through the dependencies with the relationship document. The likelihood of observing an entity document DEi given a relationship document DRi−1,i is not affected by the observation of any other entity document. Explicit dependence between the two entity documents could be used to represent the direction of the relationship between the two entities. To support this dependence, relationship documents would need to account the following constraint: R(Ei−1 , Ei ) ̸= R(Ei , Ei−1 ), ∀ DRi−1,i ∈ C R , with C R representing the relationship index. Then, we would compute an ordered feature function between entities in a relationship, similar to the ordered bigram feature function in SDM. In this work, we do not explicitly model asymmetric relationships. For instance, if a user searches for the relationship entity A “criticized” entity B but was in fact entity B who criticized entity A we assume that the entity tuple <entity A, entity B> is still relevant for the information need expressed in the E-R query. ERDM follows the SDM [63] dependencies between query terms and documents due to its proved effectiveness in multiple contexts. Therefore, ERDM assumes a dependence between neighboring sub-query terms: Ei Ei P (qjEi \|DEi , qj̸E=i l ) = P (qjEi \|DEi , qj−1 , qj+1 ) (3.24) 3.3 Entity-Relationship Dependence Model R R 55 R R R i−1,i i−1,i i−1,i Ei P (qj i−1,i \|DRi−1,i , qj̸=i−1,i \|DRi−1,i , qj−1 , qj+1 ) l , D ) = P (qj (3.25) MRF for retrieval requires the definition of the sets of cliques (maximal or nonmaximal) within the graph that one or more feature functions is to be applied to. The set of cliques in ERDM containing at least one document are the following: • T E - set of 2-cliques containing an entity document node and exactly one term in a entity sub-query. • OE - set of 3-cliques containing an entity document node and two ordered terms in a entity sub-query. • T R - set of 2-cliques containing a relationship document node and exactly one term in a relationship sub-query. • OR - set of 3-cliques containing a relationship document node and two ordered terms in a relationship sub-query. • S ER - set of 2-cliques containing one entity document node and one relationship document node. • S RER - set of 3-cliques containing one entity document node and two consecutive relationship document nodes. The joint probability mass function of the MRF is computed using the set of potential functions over the configurations of the maximal cliques in the graph [63]. Non-negative potential functions are constructed from one or more real valued feature functions associated with the respective feature weights using an exponential form. 3.3.2 Feature Functions ERDM has two types of feature functions: textual and non-textual. Textual feature functions measure the textual similarity between one or more sub-query terms and a document node. Non-textual feature functions measure compatibility between entity and relationship documents, i.e., if they share a given entity. Table 3.5 presents an overview of the feature functions associated with clique sets and the type of input nodes. Although we could define a wide set of different feature functions, we decided to adapt SDM textual feature functions to ERDM clique configurations. Therefore we define unigram based feature functions fTE and fTR to Entity Retrieval for Online Reputation Monitoring 56 Table 3.5 Clique sets and associated feature functions by type and input nodes. Clique Set TE OE TR OR S ER S RER Feature Functions fTE fOE and fUE fTR fOR and fUE fSER fSRER Type Textual Textual Textual Textual Non-textual Non-textual Input Nodes {qjEi , DEi } Ei {qjEi , qj+1 , DEi } Ri−1,i {qj , DRi−1,i } R Ri−1,i {qj i−1,i , qj+1 , DRi−1,i } {DEi , DRi−1,i } {DEi , DRi−1,i , DRi,i+1 } 2-cliques containing a single sub-query term and a entity or relationship document node. For 3-cliques containing consecutive sub-query terms and a document node, we define two feature functions. One considers consecutive sub-query terms and matches ordered bigrams with entity or relationship documents. This feature function is denoted as fOE and fOR , depending if the clique is OE or OR . The second feature function matches bigrams with documents using an unordered window of 8 terms (uw8), i.e., it matches bigrams with documents if the two terms of the bigram occur with a maximum of 6 other terms between each other. This feature function is denoted as fUE and fUR , depending if the clique is OE or OR . For each textual feature function we decided to use two variants: Dirichlet smoothing Language Models (LM) and BM25. We now present the summary of the textual feature functions used in this work. LM-T-E  E (qjEi , DEi ) = log  fT,LM  E f (q i ,C E ) E f (qj i ,DEi )+ j E µE \|C \| \|DEi \|+µE    LM-O-E E E f#1 (q i ,q i ,C E ) E Ei j j+1 f#1 (qj i ,qj+1 µE ,DEi )+ \|C E \| \|DEi \|+µE  E E f#uw8 (q i ,q i ,C E ) E Ei j j+1 µE f#uw8 (qj i ,qj+1 ,DEi )+ E \|C \| E E \|D i \|+µ   Ei E fO,LM (qjEi , qj+1 , DEi ) = log     LM-U-E  Ei E fU,LM (qjEi , qj+1 , DEi ) = log     3.3 Entity-Relationship Dependence Model 57 LM-T-R  Ri−1,i  f (qj R R (qj i−1,i , DRi−1,i ) = log  fT,LM Ri−1,i R ,C ) j µR \|C R \| Ri−1,i \|D \|+µR ,DRi−1,i )+ f (q    LM-O-R Ri−1,i Ri−1,i R f#1 (q ,q ,C ) j j+1 µR \|C R \| Ri−1,i \|D \|+µR  Ri−1,i Ri−1,i R f#uw8 (q ,q ,C ) Ri−1,i R j j+1 ,DRi−1,i )+ f#uw8 (qj i−1,i ,qj+1 µR \|C R \| Ri−1,i R   R Ri−1,i  f#1 (qj R i−1,i R (qj i−1,i , qj+1 , DRi−1,i ) = log  fO,LM R i−1,i ,qj+1 ,DRi−1,i )+   LM-U-R  R R i−1,i R fU,LM (qj i−1,i , qj+1 , DRi−1,i ) = log   \|D \|+µ R Here, f (qjEi , DEi ) and f (qj i−1,i , DRi−1,i ) represent the sub-query term frequencies in a entity document and relationship document, respectively. The collection frequencies R f (qjEi , C E ), f (qj i−1,i , C R ) represent the frequency of sub-query term in either the entity index C E or in the relationship index C R . The variants of these functions f#1 and f#uw8 represent ordered and unordered bigram matching frequency. \|DEi \| and\|DRi,i+1 \| represent the total number of terms in a meta-document while \|C R \| and \|C E \| represent the total number of terms in a collection of meta-documents. Finally, µE and µR are the Dirichlet prior for smoothing which generally corresponds to the average document length in a collection. BM25-T-E E Ei E Ei fT,BM 25 (qj , D ) BM25-O-E = log N E −n(qj i )+0.5 E n(qj i )+0.5 E . f (qj i ,DEi )(K1 +1) E f (qj i ,DEi )+K1 (1−b+b \|D Ei \| ) avg(\|D E \|)   Entity Retrieval for Online Reputation Monitoring 58 Ei Ei E Ei fO,BM 25 (qj , qj+1 , D ) =log Ei N E − n#1 (qjEi , qj+1 ) + 0.5 · Ei Ei n#1 (qj , qj+1 ) + 0.5 (3.26) Ei f#1 (qjEi , qj+1 , DEi )(K1 + 1) Ei \|D \| Ei f#1 (qjEi , qj+1 , DEi ) + K1 (1 − b + b avg(\|D E \|) ) (3.27) BM25-U-E Ei E Ei fU,BM 25 (qj , D ) =log Ei N E − n#uw8 (qjEi , qj+1 ) + 0.5 · Ei Ei n#uw8 (qj , qj+1 ) + 0.5 (3.28) Ei , DEi )(K1 + 1) f#uw8 (qjEi , qj+1 Ei \|D \| Ei , DEi ) + K1 (1 − b + b avg(\|D f#uw8 (qjEi , qj+1 E \|) ) (3.29) BM25-T-R R Ri−1,i R fT,BM , DRi−1,i ) 25 (qj =log N R − n(qj i−1,i ) + 0.5 R n(qj i−1,i ) + 0.5 · (3.30) R f (qj i−1,i , DRi−1,i )(K1 + 1) R Ri−1,i \| \|D f (qj i−1,i , DRi−1,i ) + K1 (1 − b + b avg(\|D R \|) ) (3.31) BM25-O-R 3.3 Entity-Relationship Dependence Model 59 R R R i−1,i i−1,i R fO,BM , qj+1 , DRi−1,i ) =log 25 (qj R i−1,i N R − n#1 (qj i−1,i , qj+1 ) + 0.5 R R i−1,i n#1 (qj i−1,i , qj+1 ) + 0.5 R · R i−1,i f#1 (qj i−1,i , qj+1 , DRi−1,i )(K1 + 1) R Ri−1,i R \|D \| i−1,i f#1 (qj i−1,i , qj+1 , DRi−1,i ) + K1 (1 − b + b avg(\|D R \|) ) (3.32) (3.33) BM25-U-R R R Ri−1,i Ri−1,i R fU,BM , qj+1 , DRi−1,i ) 25 (qj =log i−1,i ) + 0.5 N R − n#uw8 (qj i−1,i , qj+1 R R i−1,i ) + 0.5 n#uw8 (qj i−1,i , qj+1 R · R i−1,i , DRi−1,i )(K1 + 1) f#uw8 (qj i−1,i , qj+1 R Ri−1,i R \| \|D i−1,i , DRi−1,i ) + K1 (1 − b + b avg(\|D f#uw8 (qj i−1,i , qj+1 R \|) ) (3.34) (3.35) Here, N E and N R represent the total number of documents in the entity index and relationship index, respectively. The document frequency of unigrams and bigrams is represented using n(),n#1 () and n#uw8 (). \|DEi \| and \|DRi−1,i \| are the total number of terms in a entity or relationship document while avg(\|DE \|) and avg(\|DR \|) are the average entity or relationship document length. K1 and b are free parameters usually chosen as 1.2 and 0.75, in the absence of specific optimization. We define two non-textual features in ERDM. The first one, fTER is assigned to 2-cliques composed by one entity document and one relationship document and it is inspired in the feature function fE of Hasibi and Balog’s ELR model [69]. It is defined as follows: h i) fSER (DEi , DRi−1,i ) = (1 − α)f (DEi , DRi−1,i ) + α n(E NR i (3.36) where the linear interpolation implements the Jelinek-Mercer smoothing method with α ∈ [0, 1] and f (DEi , DRi−1,i ) = {0, 1} which measures if the entity Ei represented Entity Retrieval for Online Reputation Monitoring 60 in DEi belongs to the relationship R(Ei−1 , Ei ) represented in DRi−1,i . The background model employs the notion of entity popularity within the collection of relationship documents. n(DEi ) represents the number of relationship documents DR that contain the entity Ei and N R represents the total number of relationship documents in the relationship index. For E-R queries with more than one relationship sub-query, we draw an edge between consecutive relationship documents within the ERDM graph. This edge creates a 3-clique containing two relationship documents and one entity document. The feature function fSRER measures if a given entity Ei is shared between consecutive relationship documents within the graph. We opted to define a simple binary function: fSER (DEi , DRi−1,i , DRi,i+1 ) = 1 if Ei ∈ DEi ∩ DRi−1,i ∩ DRi,i+1 , 0 otherwise (3.37) In summary, we described the set of feature functions associated with each clique configuration within the ERDM graph. We leave for future work the possibility of exploring other type of features to describe textual similarity and compatibility between different nodes in the ERDM graph, such as neural language models. 3.3.3 Ranking We have defined the set of clique configurations and the real valued feature functions that constitute the non-negative potential functions over the cliques in the graph of ERDM. We can now formulate the calculation of the posterior P (DE , DR \|Q using the probability mass function of the MRF, as follows: 3.3 Entity-Relationship Dependence Model rank PΛ (DE , DR \|Q) = X 61 λc f (c) c∈C(G) rank E = λT XX λE O XX λE U XX λR T X X fTE (qjEi , DEi )+ E QEi Ei fOE (qjEi , qj+1 , DEi )+ E QEi Ei , DEi )+ fUE (qjEi , qj+1 E QEi R fTR (qj i−1,i , DRi−1,i )+ (3.38) R QRi,j λR O X X λR U X X R R R R i−1,i , DRi−1,i )+ fOR (qj i−1,i , qj+1 R QRi,j i−1,i , DRi−1,i )+ fUR (qj i−1,i , qj+1 R QRi,j λER S XX R λRER S fSER (DEi , DRi−1,i )+ E XX R fSER (DEi , DRi−1,i , DRi,i+1 ) E (3.39) In essence, E-R retrieval using the ERDM corresponds to ranking candidate entity tuples using a linear weighted sum of the feature functions over the cliques in the graph. Therefore, we can apply any linear learning to rank algorithm to optimize the ranking with respect to the vector of feature weights Λ. Given a training set T composed by relevance judgments, a ranking of entity tuples RΛ and an evaluation function E(RΛ ; T ) that produces a real valued output, our objective is to find the values of the vector Λ that maximizes E. As explained in [65], we require E to only consider the ranking produced and not individual scores. This is the standard characteristic among information retrieval evaluation metrics (e.g. MAP or NDCG). 3.3.4 Discussion In this section we introduced the Entity-Relationship Dependence Model (ERDM), a novel supervised Early Fusion-based model for E-R retrieval. Inspired by recent work in entity retrieval we believe that modeling term dependencies between sub-queries and entity/relationship documents can increase search performance. Entity Retrieval for Online Reputation Monitoring 62 ERDM can be seen as an extension of the SDM model [63] for ad-hoc document retrieval in a way that besides modeling query term dependencies we create graph structures that depict dependencies between entity and relationship documents. Consequently, instead of computing a single posterior P (D\|Q) we propose to use the MRF for retrieval for computing a joint posterior of multiple entity and relationship documents given a E-R query, P (DE , DR \|Q). Moreover, since ERDM is a supervised model, we believe that tuning weights of feature functions, besides optimizing search performance, can also help to explain the inter-dependencies between sub-query terms and the respective documents, but also how entity documents and relationship documents contribute to the overall relevance of entity tuples given a E-R query. 3.4 Summary of the Contributions In this chapter we present several contributions to the problem of entity-relationship retrieval from a IR perspective: • Generalization of the problem of entity-relationship search to cover entity types and relationships represented by any attribute and predicate, respectively, rather than a predefined set. • A general probabilistic model for E-R retrieval using Bayesian Networks. • Proposal of two design patterns that support retrieval approaches using the E-R model. • Proposal of a Entity-Relationship Dependence model that builds on the basic Sequential Dependence Model (SDM) to provide extensible entity-relationship representations and dependencies, suitable for complex, multi-relations queries. Chapter 4 Entity-Relationship Retrieval over a Web Corpus We start this chapter by presenting a new semi-automatic method for generating E-R test collections together with a new E-R test collection, the RELink Query Collection comprising 600 E-R queries. We leverage web tabular data containing entities and relationships among them as they share the same row in a table. We exploit the Wikipedia Lists-of-lists-of-lists tree of articles containing lists of Entities in the form of tables. We developed a table parser that extracts tuples of entities from these tables together with associated metadata. This information is then provided to editors that create E-R queries fulfilled by the extracted tuples. We then report a set of evaluations of the ERDM model using four different query sets. In order to leverage information about entities and relations in a corpus, it is necessary to create a representation of entity related information that is amenable to ER search. In our approach we focus on sentence level information about entities although the method can be applied to more complex segmentation of text. Our experiments are based on the ClueWeb-09-B data set with FACC1 text annotation that refer to entities found in the text, including the variances of their surface forms. Each entity is designated by its unique ID and for each unique entity instance we created ’entity documents’ comprising a collection of sentences that contain the entity. These context documents are indexed, comprising the entity index. The same is done by creating entity pair documents and the entity pair index. These two indexes enable us to execute E-R queries using different retrieval models, including the ERDM that models the dependence between entities. Entity-Relationship Retrieval over a Web Corpus 64 4.1 RELink Query Collection 1 Improvements of entity-relationship (E-R) search techniques have been hampered by a lack of test collections, particularly for complex queries involving multiple entities and relationships. In this section we describe a method for generating E-R test queries to support comprehensive E-R search experiments. Queries and relevance judgments are created from content that exists in a tabular form where columns represent entity types and the table structure implies one or more relationships among the entities. Editorial work involves creating natural language queries based on relationships represented by the entries in the table. We have publicly released the RELink test collection comprising 600 queries and relevance judgments obtained from a sample of Wikipedia List-of-lists-of-lists tables. The latter comprise tuples of entities that are extracted from columns and labelled by corresponding entity types and relationships they represent. Improvement of methods for both extraction and search is hampered by a lack of query sets and relevance judgments, i.e., gold standards that could be used to compare effectiveness of different methods. In this section we introduce: 1. A low-effort semi-automatic method for acquiring instances of entities and entity relationships from tabular data. 2. RELink Query Collection (QC) of 600 E-R queries with corresponding relevance judgments Essential to our approach is the observation that tabular data typically includes entity types as columns and entity instances as rows. The table structure implies a relationship among table columns and enables us to create E-R queries that are answered by the entity tuples across columns. Following this approach, we prepared and released the RELink QC comprising 600 E-R queries and relevance judgments based on a sample of Wikipedia List-of-lists-of-lists tables. The query collection and the research framework are publicly available2 , enabling the community to expand the RELink Framework with additional document collections and alternative indexing and search methods. It is important to maintain and enhance the RELink QC by providing updates to the existing entity types and creating new queries and relevant instances from additional tabular data. 1 The material contained in this section was published in P. Saleiro, N. Milic-Frayling, E. M. Rodrigues, C. Soares, “RELink: A Research Framework and Test Collection for Entity-Relationship Retrieval”[15]. 2 https://sigirelink.github.io/RELink/ 4.1 RELink Query Collection 4.1.1 65 Tabular Data and Entity Relationships Information that satisfies complex E-R queries is likely to involve instances of entities and their relationships dispersed across Web documents. Sometimes such information is collected and published within a single document, such as a Wikipedia page. In such cases, traditional search engines can provide excellent search results without applying special E-R techniques or considering entity and relationship types. Indeed, the data collection, aggregation, and tabularization has been done by a Wikipedia editor. That also means that a tabular Wikipedia content, comprising various entities, can be considered as representing a specific information need, i.e., the need that motivated editors to create the page in the first place. Such content can, in fact, satisfy many different information needs. We focus on exploiting tabular data for exhaustive search for pre-specified E-R types. In order to specify E-R queries, we can use column headings as entity types. All the column entries are then relevance judgments for the entity query. Similarly, for a given pair of columns that correspond to distinct entities, we formulate the implied relationship. For example the pair <car, manufacturing plant> could refer to “is made in” or “is manufactured in” relationships. The instances of entity pairs in the table then serve as evidence for the specific relationship. This can be generalized to more complex information needs that involve multiple entity types and relationships. Automated creation of E-R queries from tabular content is an interesting research problem. For now we asked human editors to provide natural language and structured E-R queries for specific entity types. Once we collect sufficient amounts of data from human editors we will be able to automate the query creation process with machine learning techniques. For the RELink QC we compiled a set of 600 queries with E-R relevance judgments from Wikipedia lists about 9 topic areas. 4.1.2 Selection of Tables Wikipedia contains a dynamic index “The Lists of lists of lists”3 which represents the root of a tree that spans curated lists of entities in various domains. We used a Wikipedia snapshot from October 2016 to traverse “The Lists of lists of lists” tree starting from the root page and following every hyperlink of type “List of ” and their children. This resulted in a collection of 95,569 list pages. While most of the pages contain tabular data, only 18,903 include tables with consistent column and row structure. As in [172], we restrict content extraction to wikitable HTML class that 3 http://en.wikipedia.org/wiki/List_of_lists_of_lists 66 Entity-Relationship Retrieval over a Web Corpus typically denotes data tables in Wikipedia. We ignore other types of tables such as infoboxes. In this first instance, we focus on relational tables, i.e., the tables that have a key column, referring to the main entity in the table [173]. For instance, the ”List of books about skepticism” contains a table “Books” with columns “Author”, “Category” and “Title”, among others. In this case, the key column is “Title” which contains titles of books about skepticism. We require that any relationship specified for the entity types in the table must contain the “Title” type, i.e., involve the “Title” column. In order to detect key columns we created a Table Parser that uses the set of heuristics adopted by Lehmberg et al. [173], e.g., the ratio of unique cells in the column or text length. Once the key column is identified, the parser creates entity pairs consisting of the key column and one other column in the table. The content of the column cells then constitutes the set of relevant judgments for the relationship specified by the pair of entities. For the sake of simplicity we consider only those Wikipedia lists that contain a single relational table. Furthermore, our goal is to create queries that have verifiable entity and entity pair instances. Therefore, we selected only those relational tables for which the key column and at least one more column have cell content linked to Wikipedia articles. With these requirements, we collected 1795 tables. In the final step, we selected 600 tables by performing stratified sampling across semantic domains covered by Wikipedia lists. For each new table, we calcuated the Jaccard similarity scores between the title of the corresponding Wikipedia page and the titles of pages associated with tables already in the pool. By setting the maximum similarity threshold to 0.7 we obtained a set of 600 tables. The process of creating RELink queries involves two steps: (1) automatic selection of tables and columns within tables and (2) manual specification of information needs. For example, in the table “Grammy Award for Album of the Year” the columns “winner”, “work” were automatically selected to serve as entity types in the E-R query (Figure 4.1). The relationship among these entities is suggested by the title and we let a human annotator to formulate the query. The RELink query set was created by 6 annotators. We provided the annotators with access to the full table, metadata (e.g., table title or the first paragraph of the page) and entity pairs or triples to be used to specify the query (Figure 4.2). For each entity pair or triple the annotators created a natural language information need and an E-R query in the relational format Q = {QEi−1 , QRi−1,i , QEi }, as shown in Table 4.1. 4.1 RELink Query Collection 67 Fig. 4.1 Example of Wikipedia table row. Fig. 4.2 Example of metadata provided to editors. 4.1.3 Formulation of Queries The relational query format is introduced to support a variety of experiments with E-R queries. In essence, a complex information need is decomposed into a set of subqueries that specify types of entities E and types of relationships R(Ei−1 , Ei ) between entities. For each relationship query there is one query for each entity involved in the relationship. Thus a query Q that expects a pair of entities for a given relationship, is mapped into three sub-queries (QEi−1 , QRi−1,i , QEi ), where QEi−1 and QEi are the entity types for Ei−1 and Ei respectively, and QRi−1,i is a relationship type describing R(Ei−1 , Ei ). 4.1.4 Collection Statistics RELink QC covers 9 thematic areas from the Lists-of-Lists-of-Lists in Wikipedia: Mathematics and Logic, Religion and Belief Systems, Technology and Applied Sciences, Miscellaneous, People, Geography and Places, Natural and Physical Sciences, General Reference and Culture and the Arts. The most common thematic areas are Culture and the Arts with 70 queries and Geography and Places with 67 queries. In Table 4.2 we show the characteristics of the natural language and relational queries. Among 600 E-R queries, 381 refer to entity pairs and 219 to entity triples. As Entity-Relationship Retrieval over a Web Corpus 68 Table 4.1 Examples of query annotations. ID NL Query Relational Format RELink_P_164 What are the regiments held by the Indian Army? {regiment, held by, Indian Army} RELink_T_071 In which seasons NHL players scored more than 50 goals and the team they represented? {NHL season, scored more than 50 goals in, NHL player, played for, NHL team } Table 4.2 RELink collection statistics. Total queries Avg. queries length Avg. QE length Avg. QR length # uniq. entity attributes (QE ) # uniq. relationships (QR ) Avg. # relevant judgments 2-entity 381 56.5 20.9 11.8 679 145 67.9 3-entity 219 83.8 20.9 12.6 592 205 41.8 All 600 66.5 20.9 12.3 1251 317 58.5 expected, natural language descriptions of 3-entity queries are longer (on average 83.8 characters) compared to 2-entity queries (56.5 characters). We further analyze the structure of relational queries and their components, i.e., entity queries QE that specify the entity type and relationship queries QR that specify the relationship type. Across 600 queries, there are 1251 unique entity types QE (out of total 1419 occurrences). They are rather unique across queries: only 65 entity types occur in more than one E-R query and 44 occur in exactly 2 queries. The most commonly shared entity type is “country”, present in 9 E-R queries. In the case of relationships, there are 317 unique relationship types QR (out of 817 occurrences) with a dominant type “located in” that occurs in 140 queries. This is not surprising since in many domains the key entity is tied to a location that is included in one of the columns. Nevertheless, there are only 44 relationship types QR occurring more than once implying that RELink QC is a diverse set of queries, including 273 relationship types occurring only once. 4.2 Experimental Setup 4.2 69 Experimental Setup In this section we detail how we conducted our experiments in E-R retrieval. Since we only have access to test collections comprising general purpose E-R queries we decided to use a Web corpus as dataset, more precisely ClueWeb-09-B4 .The ClueWeb09 dataset was created to support research on information retrieval and related human language technologies and contains 1 billion web pages. The part B is a subset of the most popular 50 million English web pages, including the Wikipedia. Part B was created as a resource for research groups without processing power for processing the all ClueWeb09 collection. We used the ClueWeb-09-B Web collection with FACC1 text span annotations linked to Wikipedia entities to show how RELink can be used for E-R retrieval over Web content. We developed our prototype using Apache Lucene for indexing and search. We used a specific Python library (PyLucene) that allowed our customized implementation tailored for E-R retrieval. 4.2.1 Data and Indexing E1 E3 E4 E3 E2 E1E3 E3E4 E3 E1 E2 E3E4 D1 D2 D3 D4 E1 E1 E2 E2 E1E3 E3 E3 E3 E3E4 E3E4 E4 Entity Index Relationship Index Fig. 4.3 Illustration of E-R indexing from a web corpus. As a text corpus, we use ClueWeb-09-B combined with FACC1 text span annotations with links to Wikipedia entities (via Freebase). The entity linking precision and recall in FACC1 is estimated to be 80-85% and 70-85%, respectively [174]. For our experiments we created two main indexes: one for entity extractions and one for entity pairs 4 https://lemurproject.org/clueweb09/ Entity-Relationship Retrieval over a Web Corpus 70 (relationships) extractions. We extract entity and pairs occurrences using an Open Information Extraction method like OLLIE [57] over the annotated ClueWeb-09-B corpus as follows. For each entity annotation, we extract the sentence where it occurred as an entity context. For pairs of entities, we look for co-occurring entities in the same sentence and we extract the separating string, i.e., the context of the relationship connecting them. Figure 4.3 illustrates the indexing process adopted in this work. We obtained 476 million entity extractions and 418 million entity pairs extractions, as described in Table 4.3. In order to compute \|DEi \| and \|DRi−1,i \| we incrementally updated two auxiliary indices, containing the number of terms per entity and per entity pair, respectively. We ran our experiments using Apache Lucene and made use of GroupingSearch for grouping extractions by entity and entity pair at query time. To get the statistics for ordered and unordered bigrams we made use of SpanNearQuery. Table 4.3 ClueWeb09-B extractions statistics. Entities Entity pairs 4.2.2 Total 476,985,936 418,079,378 Unique 1,712,010 71,660,094 Avg. doc. len. 9977 138 Retrieval Method and Parameter Tuning For experiments using ERDM we adopted a three stage retrieval method. First, queries QEi−1 ,QEi are submitted against the entity index and QRi−1,i is submitted against the entity-pair index. Initial sets of top 20000 results grouped by entity or entity-pairs, respectively, are retrieved using Lucene’s default search settings. Second, the feature functions of the specific retrieval model are calculated for each set, using an in-house implementation. This process is easily parallelized. The final ranking score for each entity-pair is then computed using the learned λ weights. Evaluation scores are reported on the top 100 entity-pair results. Parameter tuning for ERDM and baselines was directly optimized with respect to the Mean Average Precision (MAP). We make use of the RankLib’s implementation of the coordinate ascent algorithm under the sum normalization and non-negativity constraints with 3 random restarts. Coordinate ascent is a commonly used optimization technique [65] that iteratively optimizes a single parameter while holding all other parameters fixed. Parameters are estimated using 5-fold cross validation for each of the 4 query sets separately. To be able to use the same train and test folds throughout all experiments, 4.2 Experimental Setup 71 we first randomly create fixed train and test folds from the initial result set, for each query set. All reported evaluation metrics were macro-averaged over 5 folds. We do not optimize the Dirichlet priors µE and µR in language models and set them equal to the traditional average document length, i.e., the average entity and entity pairs extractions length, respectively. The unordered window size N for fUE and fUR is set to be 8, as suggested in [63]. 4.2.3 Test Collections We ran experiments with a total of 548 E-R queries. We decided to just perform experiments using queries aiming 2-tuples of entities. We leave for future work the evaluation of queries aiming at triples. Besides RELink QC we used other 3 relationshipcentric query sets, with pairs of Wikipedia entities as answers, i.e., relevance judgments. The query sets cover a wide range of domains as described in Table 4.4. Query sets for entity-relationship retrieval are scarce. Generally entity retrieval query sets are not relationship-centric [10]. Table 4.4 Description of query sets used for evaluation. Query Set QALD-2 Count 79 Domains Geography and places, Politics and society, Culture and the Arts, Technology and science ERQ 28 COMPLEX 60 RELink 381 Award, City, Club, Company, Film, Novel, Person, Player, Song, University Cinema, Music, Books, Sports, Computing, Military conflicts General Reference, Culture and the Arts, Geography and places, Mathematics and logic, Natural and physical Sciences, People, Religion and belief systems, Society and social sciences, Technology and applied science Total 548 One exception is the QALD-2 query set used in the DBpedia-entity collection [175]. It contains a subset of relational queries, e.g.“Who designed the Brooklyn Bridge?”. Most of relational queries in QALD-2 have a fixed relevant entity, e.g., “Brooklyn Bridge” and can be easily transformed from single entity relevance judgments into pairs. From the 79 relational queries in QALD-2, we identified 6 with no fixed relevant entity 72 Entity-Relationship Retrieval over a Web Corpus in the query (e.g. “Give me the capitals of all countries in Africa.”). In these cases, for provided single entity relevance judgment we needed to annotate the missing entity manually to create a pair. For instance, given a capital city in Africa we identified the corresponding African country. In addition, we used two benchmarks created in previous work using SemanticWeb-based approaches: ERQ [56] and COMPLEX [10]. Neither ERQ nor COMPLEX provide complete relevance judgments and consequently, we manually evaluated each answer in our experiments. ERQ consists of 28 queries that were adapted from INEX17 and OWN28 [56]. However, 22 of the queries have a given fixed entity in the query (e.g. “Find Eagles songs”). Only 6 queries are asking for pairs of unknown entities, such as “Find films starring Robert De Niro and please tell directors of these films.”. COMPLEX queries were created with a semi-automatic approach [10]. It contains 70 queries from which we removed 10 that expect 3-tuples of entities. This query set consists of pure relationship-centric queries for unknown pairs of entities, such as “Currency of the country whose president is James Mancham “Kings of the city which led the Peloponnesian League.” and “Who starred in a movie directed by Hal Ashby?”. We used four different retrieval metrics, Mean Average Precision at 100 results (MAP), precision at 10 (P@10), mean reciprocal rank (MRR) and normalized discounted cumulative gain at 20 (NDCG@20). 4.3 Results and Analysis We start by performing a simple experiment for comparing Early Fusion and ERDM using both Language Models (LM) and BM25 as retrieval functions. Since we are only interested in comparing relative performance we opted to scale down our experimental setup. Instead of computing the term frequency for every extraction for a given entity or relationship we cap to 200 the number for each group of documents retrieved in the first passage. We tried several different values and for values below 200 extraction the performance reduced significantly. For 200, while the performance reduces it is not dramatic. This setup reduces the experimental runtime and since we had limited resources this proved to be useful. Table 4.5 depicts the results for this comparative evaluation. We decided to only use the three test collections specifically tailored for relationship retrieval. As we can see the results are very similar between EF and ERDM for both LM and BM25 variants. In the three test collections ERDM presents slightly better performance than the corresponding EF variant (e.g. BM25). However when performing statistical 4.3 Results and Analysis 73 significance tests we obtained p-values above 0.05 when comparing EF and ERDM. This is very interesting as it shows that for general purpose E-R evaluation the overhead of computing sequential dependencies does not carry significant improvements. Table 4.5 Early Fusion and ERDM comparison using LM and BM25. EF-LM EF-BM25 ERDM-LM ERDM-BM25 EF-LM EF-BM25 ERDM-LM ERDM-BM25 EF-LM EF-BM25 ERDM-LM ERDM-BM25 ERQ MAP P@10 MRR NDCG@20 0.251 0.15 0.3408 0.3508 0.1939 0.1423 0.1783 0.2861 0.2611 0.1615 0.3151 0.3589 0.2106 0.1462 0.2839 0.3257 COMPLEX MAP P@10 MRR NDCG@20 0.1703 0.0596 0.1839 0.2141 0.1855 0.0719 0.1907 0.2454 0.1719 0.0789 0.2466 0.2492 0.1955 0.0772 0.2257 0.248 RELink(381 queries) MAP P@10 MRR NDCG@20 0.0186 0.0063 0.0192 0.0249 0.0203 0.0071 0.0227 0.0259 0.0213 0.0058 0.0273 0.0255 0.0213 0.0061 0.0265 0.0275 On the other hand, we detect sensitivity to the retrieval function used. In ERQ, both ERDM-LM and EF-LM outperform BM25 but the opposite happens for COMPLEX and RELink. This sensitivity means that we cannot generalize the assumption that one of the retrieval functions is more adequate for E-R retrieval. Another important observation has to do with the overall lower results on the RELink test collection in comparison with ERQ and COMPLEX. Contrary to our expectations ClueWeb-09B has very low coverage of entity tuples relevant to the RELink test collection. We now present the results of comparing ERDM with three baselines using sequential dependence to evaluate the impact of modeling dependencies between query terms. The first baseline method, BaseEE, consists in submitting two queries against the entity index: QEi−1 + QRi−1,i and QRi−1,i + QEi . Entity-pairs are created by cross product of the two entity results set retrieved by each query. For each method we compute the Sequential Dependence Model(SDM) [63] scores. Entity-Relationship Retrieval over a Web Corpus 74 The second baseline method, BaseE, consists in submitting again a single query Q towards the entity index used in ERDM. Entity-pairs are created by cross product of the entity results set with itself. The third baseline method, BaseR, consists in submitting a single query Q towards an entity-pair index. This index is created using the full sentence for each entity-pair co-occurrence in ClueWeb-09-B, instead of just the separating string as in ERDM. This approach aims to capture any entity context that might be present in a sentence. ERDM relies on the entity index for that purpose. In this evaluation we decided to not cap the number of extractions to compute term frequencies inside each group of results returned from the first passage with Lucene GroupingSearch. Due to the low coverage of ClueWeb for the entire RELink collection, we decided to just perform the evaluation using the top 100 queries with highest number of relevance judgments in our indexes. We also include results for the adapted QALD-2 test collection. Table 4.6 Results of ERDM compared with three baselines. BaseEE BaseE BaseR ERDM MAP 0.0087 0.0306 0.0872 0.1520 BaseEE BaseE BaseR ERDM MAP 0.0085 0.0469 0.1041 0.3107 BaseEE BaseE BaseR ERDM MAP 0.0035 0.0264 0.0585 0.2879 BaseEE BaseE BaseR ERDM MAP 0.03 0.0395 0.0451 0.1249 QALD-2 P@10 MRR NDCG@20 0.0027 0.0093 0.0055 0.004684 0.0324 0.0363 0.01678 0.0922 0.0904 0.0405 0.1780 0.1661 ERQ P@10 MRR NDCG@20 0.004 0.00730 0.0030 0.01086 0.0489 0.038 0.05086 0.1089 0.1104 0.1903 0.37613 0.3175 COMPLEX P@10 MRR NDCG@20 0 0.00430 0 0.005 0.03182 0.1223 0.01836 0.0748 0.0778 0.1417 0.32959 0.3323 RELink(100 queries) P@10 MRR NDCG@20 0.01 0.0407 0.02946 0.019 0.0679 0.03948 0.021 0.0663 0.07258 0.048 0.1726 0.1426 4.3 Results and Analysis 75 Table 4.6 presents the results of our experiments on each query set. We start by comparing the three baselines among each other. As follows from Table 4.6, BaseR baseline outperforms BaseEE and BaseE on all query sets, while BaseEE is the worst performing baseline. The BaseR retrieval is the only relationship-centric approach from the three baselines, as its document collection comprises entity-pairs that co-occurred in ClueWeb-09-B corpus. BaseEE and BaseE retrieve entity pairs that are created in a post-processing step which reduces the probability of retrieving relevant results. This results shows the need for a relationship-centric document collection when aiming to answer entity-relationship queries. ERDM significantly outperform all baselines on all query sets. We performed statistical significance testing of MAP using ERDM against each baseline obtaining p-values below 0.05 on all the query sets. This results show that our Early Fusion approach using two indexes (one for entities and other for relationships) is adequate and promising. We believe this approach can become a reference for future research in E-R retrieval from an IR-centric perspective. Nevertheless, based on the absolute results obtained on each evaluation metric and for each query set we can conclude that E-R retrieval is still very far from being a solved problem. There is room to explore new feature functions and retrieval approaches. This is a very difficult problem and the methods we proposed are still far from optimal performance. Queries such as “Find world war II flying aces and their services” or “Which mountain is the highest after Annnapurna?” are examples of queries with zero relevant judgments returned. On the other hand, ERDM exhibits interesting performance in some queries with high complexity, such as “Computer scientists who are professors at the university where Frederick Terman was a professor.” We speculate about some aspects that might influence performance. One aspect has to do with the lack of query relaxation in our experimental setup. The relevant entity tuples might be in our indexes but if the query terms used to search for entity tuples do not match the query terms harvested from ClueWeb-09B it is not possible to retrieve those relevant judgments. Query relaxation approaches should be tried in future work. More specifically, with the recent advances in word embeddings it is possible to expand queries with alternative query terms that are in the indexes. On the other hand, we adopted a very simple approach for extracting entities and relationships. The use of dependency parsing and more complex methods of relation extraction would allow to filter out noisy terms. We also leave this for future work. Moreover, to further assess the influence of the extraction method we propose to use Entity-Relationship Retrieval over a Web Corpus 76 selective text passages containing the target entity pairs and the query terms associated as well. Then different extraction methods could be tried and straightforward evaluation of their impact. (a) (b) (c) ′ ′ Fig. 4.4 Values of λ for ERDM: (a) all λ, (b) λE , (c) λR . (b) and (c) were obtained using sum normalization. To understand how much importance is attributed to the different types of clique sets, we plot the values of the lambda parameters: λE parameters represent the feature importance of the set of functions targeting the dependence between entity query terms and the entity documents in overall ranking score for entity-pairs; λR represent the importance of the feature functions of the relationship type queries and finally, the value for λER which is assigned to the feature function that evaluates if each entity retrieved from both entity type queries belongs to the entity-pair retrieved from the relationship type query. 4.4 Summary of the Contributions 77 We plot the feature weights learned on each query set, as depicted in Figure 4.4. We see that λER and λE T (weight for the unigram language model in the entity type queries) dominate the ranking function. We further evaluated the relative weights for each one of the three SDM-like functions using a sum normalization of the three weights for both entity documents and entity-pair documents. We observe that λE T dominates on R every query set, however the same does not happen with λT . For relationship type queries the bigram features have higher values for COMPLEX and RELink. 4.4 Summary of the Contributions In this chapter we presented the following contributions to the E-R retrieval research area: 1. Indexing method that supports generalization of entity types and entity-relationships to any attribute and predicate, respectively 2. A semi-automatic method for generating E-R test collections, which resulted in the RELink Query Collection comprising 600 E-R queries. 3. Results of experiments at scale, with a comprehensive set of queries and corpora. Chapter 5 Entity Filtering and Financial Sentiment Analysis In this chapter we present the work developed to tackle two fundamental Text Mining problems in ORM: Entity Filtering and Sentiment Analysis. We start by describing our participation at the Filtering task of RepLab 2013 [32]. We developed a supervised method to classify tweets as relevant or non-relevant to given target entity. This method obtained the first place at the competition. Entity Filtering can be seen as target based Named Entity Disambiguation (NED). Given a target entity under study, we need to develop a binary classifier to filter out tweets that are not talking about the target entity. This task is fundamental in ORM as downstream tasks such as Sentiment Analysis or entity-centric predictions would produce misleading results if noisy signals were used. Sentiment Analysis has been widely studied over the last decade. It is a research area with several ramifications as it is dependent on the type of texts and the objective of the analysis. We decided to focus our efforts in a not so well explored sub-area of Sentiment Analysis. SemEval 2017 Task 5 focused on fine-grained sentiment analysis of financial news and microblogs. As one of the use cases of ORM is to track the online reputation of companies and try to assess its impact on the stock market we decided it was a specific task within Sentiment Analysis in which we could make a contribution. We obtained the fourth place in the Microblogs sub-task using one of the evaluation metrics. The task consisted in predicting a real continuous variable from -1.0 to +1.0 representing the polarity and intensity of sentiment concerning companies/stocks mentioned in short texts. We modeled it as a regression analysis problem. Entity Filtering and Financial Sentiment Analysis 80 5.1 Entity Filtering1 The relationship between people and public entities has changed with the rise of social media. Online users of social networks, blogs and micro-blogs are able to directly express and spread opinions about public entities, such as politicians, artists, companies or products. Online Reputation Monitoring (ORM) aims to automatically process online information about public entities. Some of the common tasks within ORM consist in collecting, processing and aggregating social network messages to extract opinion trends about such entities. Twitter, one of the most used online social networks, provides a search system that allows users to query for tweets containing a set of keywords. ORM systems often use Twitter as a source of information when monitoring a given entity. However, search results are not necessarily relevant to that entity because keywords can be ambiguous. For instance, a tweet containing the word “columbia” can be related with several entities, such as a federal state, a city or a university. Furthermore, tweets are short which results in a reduced context for entity disambiguation. When monitoring the reputation of a given entity on Twitter, it is first necessary to guarantee that all tweets are relevant to that entity. Consequently, other processing tasks, such as sentiment analysis will benefit from filtering out noise in the data stream. In this work, we tackle the aforementioned problem by applying a supervised learning approach. Given a set of entities E = {e1 , e2 , ..., ei , ...}, a stream of texts S = {s1 , s2 , ..., si , ...} (e.g. tweets), we are interested in monitoring the mentions of an entity ei on the stream S, i.e. the discrete function fm (ei , S). We cast the prediction of fm as a supervised learning classification problem, in which we want to infer the target variable fˆm (ei , S) ∈ {0, 1} We implemented a large set of features that can be generated to describe the relationship between an entity representation and a text mention. We use metadata (e.g. entity names, category) provided in the user configurations, text represented with TF-IDF, similarity between texts and Wikipedia, Freebase entities disambiguation, feature selection of terms based on frequency and feature matrix transformation using SVD. The learning algorithms from scikit-learn Python library that were tested for Entity Filtering include Naive Bayes, SVM, Random Forests, Logistic Regression and MultiLayer Perceptron. 1 Most of the material contained in this section was published in P.Saleiro, E. M. Rodrigues, C. Soares, E. Oliveira, “TexRep: A Text Mining Framework for Online Reputation Monitoring” [14] 5.1 Entity Filtering 5.1.1 81 Task Overview RepLab 2013 [32] focused on monitoring the online reputation of entities on Twitter. The Filtering task consisted in determining which tweets are relevant to each entity. The corpus consists of a collection of tweets obtained by querying the Twitter Search API with 61 entity names during the period from the June 2012 until the December 2012. The corpus contains tweets both in English and Spanish. The balance between both languages varies for each entity. Tweets were manually annotated as “Related” or “Unrelated” to the respective target entity. The data provided to participants consists in tweets and a list of 61 entities. For each tweet in the corpus we have the target entity id, the language of the tweet, the timestamp and the tweet id. The content of each URL in the tweets is also provided. Due to Twitter’s terms of service, the participants were responsible to download the tweets using the respective id. The data related with entities contain the query used to collect the tweets (e.g. “BMW”), the official name of the entity (e.g. “Bayerische Motoren Werke AG”), the category of the entity (e.g. “automotive”), the content of its homepage and both Wikipedia articles in English and Spanish. 5.1.2 Pre-processing The Entity Filtering module includes methods to normalize texts by removing all punctuation, converting text to lower case, removing accents and converting non-ASCII characters to their ASCII equivalent. Lists of stop words for several languages are also available, which are used to filter out non relevant words. We rely on the Natural Language Toolkit (NLTK) to provide those lists. Contrary to other types of online texts (e.g. news or blog posts) tweets contain informal and non-standard language including emoticons, spelling errors, wrong letter casing, unusual punctuation and abbreviations. Therefore, when dealing with tweets, the Entity Filtering module uses a tokenizer [176] optimized for segmenting words in tweets. After tokenization we extract user mentions and URLS and hashtags textual content. 5.1.3 Features Many different types of features can be used to optimize relevance classification, including language models, keyword similarities between tweets and entities as well as external resources projections. We implemented a large number of those. We assume that future users of our framework for ORM will provide entity-specific data (e.g. 82 Entity Filtering and Financial Sentiment Analysis homepage/Wikipedia content) prior to training and configuring the Entity Filtering module. Language Model: text is encapsulated in a single feature to avoid high dimensionality issues when adding other features. A TF-IDF representation of unigrams, bigrams and trigrams for training a text classifier which calculates the probability of a text being related to the expected entity. The output probabilities of the classifier are used as a feature. Keyword similarity: similarity scores between metadata and the texts, obtained by calculating the ratio of the number of common terms in the texts and the terms of query and entity name. Similarities at character level are also available in order to include possible spelling errors in the text. Web similarity: similarity between the text and the normalized content of the entity’s homepage and normalized Wikipedia articles are also available. The similarity value is the number of common terms multiplied by logarithm of the number of terms in tweet. Freebase: For each keyword of the entity’s query that exists in the text, two bigrams are created, containing the keyword and the previous/subsequent word. These bi-grams are submitted to the Freebase Search API and the list of retrieved entities are compared with the id of the target entity on Freebase. A Freebase score is computed by using the inverse position of the target entity in the list of results retrieved. If the target entity is the first result, the score is 1, if it is the second, the score is 0.5, and so on. If the target entity is not in the results list, the score is zero. The feature corresponds to the maximum score of the extracted bigrams of each text. Category classifier: a sentence category classifier is created using the Wikipedia articles of each entity. Each sentence of the Wikipedia articles is annotated with the category of the corresponding entity. TF-IDF for unigrams, bigrams and trigrams are calculated and a multi-class classifier (SVM) is trained to classify each text. The feature is the probability of the text being relevant to its target class. 5.1.4 Experimental Setup The dataset used for the competition consists of a collection of tweets both in English and Spanish, possibly relevant to 61 entities from four domains: automotive, banking, 5.1 Entity Filtering 83 Dataset Related Unrelated Total Training Development Validation Test 33,193 26,534 6,659 75,470 10,389 8,307 2,082 21,378 43,582 34,841 8,741 96,848 Table 5.1 RepLab 2013 Filtering Task dataset description. universities and music.The dataset consists of a collection of tweets obtained by querying the Twitter Search API with 61 entity names during the period from the June 2012 until the December 2012. The balance between both languages varies for each entity. The complementary data about each target entity is the following: • query used to collect the tweets (e.g. “BMW”) • official name of the entity (e.g. “Bayerische Motoren Werke AG”) • category of the entity (e.g. “automotive”) • content of entity homepage • Wikipedia article both in English and Spanish Tweets were manually annotated as “Related” or “Unrelated” to the respective target entity. The dataset is divided in training, test and development (Table 5.1). The training set consists in a total of 45,671 tweets from which we were able to download 43,582. Approximately 75% of tweets in the training set are labeled as “Related”. We split the training dataset into a development set and a validation set, containing 80% and 20% of the original, respectively. We adopted a randomly stratified split approach per entity, i.e., we group tweets of each target entity and randomly split them preserving the balance of “Related”/“Unrelated” tweets. The test dataset consists of 90,356 tweets from which we were able to download 88,934. We used the development set for trying new features and test algorithms. We divided the development set in 10 folds generated with the randomly stratified approach. We used the validation set to validate the results obtained in the development set. The purpose of this validation step is to evaluate how well the Entity Filtering classifier generalizes from its training data to the validation data and thus estimate how well it will generalize to the test set. It allows us to spot overfitting. After validation, we trained the classifier using all of the data in the training dataset and evaluated in the test set. Entity Filtering and Financial Sentiment Analysis 84 5.1.5 Results We created different classifier runs using different learners, features and we also created entity specific models as explained in Table 5.2, [177]. We applied selection of features based on frequency and transformation of content representation using SVD. The learners tested include Naive Bayes (NB), SVM, Random Forests (RF), Logistic Regression (LR) and MultiLayer Perceptron (MLP). The evaluation measures used are accuracy and the official metric of the competition, F-measure which is the harmonic mean of Reliability and Sensitivity [178]. We present results for the top 4 models regarding the F-measure. We replicated the best system at RepLab 2013 in the run 1. Run Learner Features No. of models 1 2 3 SVM RF RF All All All global global per entity Table 5.2 Entity filtering versions description. Table 5.3 shows the results of top performing runs and the official baseline of the competition. This baseline classifies each tweet with the label of the most similar tweet of target entity in the training set using Jaccard similarity coefficient. The baseline results were obtained using 99.5% of the test set. Run Acc. (Val. Set) 1 0.944 2 0.945 3 0.948 Official Baseline Best RepLab - Acc. R S F-measure 0.906 0.908 0.902 0.8714 0.908 0.759 0.729 0.589 0.4902 0.729 0.428 0.470 0.451 0.488 0.444 0.448 0.3199 0.3255 0.451 0.488 Table 5.3 Official results for each version plus our validation set accuracy. Based on the results achieved we are able to conclude that the models of our classifier are able to generalize successfully. Results obtained in the validation set are similar to those obtained in the test set. During development, solutions based on one model per entity were consistently outperformed by solutions based on global models. We also noticed during development that language specific models (English and Spanish) did not exhibit improvements in global accuracy, therefore we opted to use language as a feature. Results show that the best model uses the Random Forests 5.1 Entity Filtering 85 classifier with 500 estimators for training a global model. Though, the Language Modeling feature encapsulates text by using a specific model trained just with TF-IDF of n-grams of tweets. We performed a “break down” analysis for each one of the four categories of RepLab 2013 using Run 2 model, as depicted in Figure 5.1. We observe that University, Banking and Automotive categories exhibit similar average F-measure results, all above 0.50. In contrast, results for Music shows it is a rather difficult category of entities to disambiguate (achieving F-measure of 0.39). In fact, some of the entity names of this category contain very ambiguous tokens, such as “Alicia Keys”, “U2”, “The Wanted” or “The Script”. Fig. 5.1 Results grouped by entity’s category using Run 2. The main goal of this task was to classify tweets as relevant or not to a given target entity. We have explored several types of features, namely similarity between keywords, language models and we have also explored external resources such as Freebase and Wikipedia. Results show that it is possible to achieve an Accuracy over 0.90 and an F-measure of 0.48 in a test set containing more than 90000 tweets of 61 entities. In future work, we expect to include the possibility of using entity-specific embedding to learn a joint embedding space of entities and words, similar to [91]. Entity Filtering and Financial Sentiment Analysis 86 5.2 Financial Sentiment Analysis2 Sentiment Analysis on financial texts has received increased attention in recent years [12]. Nevertheless, there are some challenges yet to overcome [13]. Financial texts, such as microblogs or newswire, usually contain highly technical and specific vocabulary or jargon, making the development of specific lexical and machine learning approaches necessary. Most of the research in Sentiment Analysis in the financial domain has focused in analyzing subjective text, labeled with explicitly expressed sentiment. However, it is also common to express financial sentiment in an implicit way. Business news stories often refer to events that might indicate a positive or negative impact, such as in the news title “company X will cut 1000 jobs”. Economic indicators, such as unemployment and change over time such as drop or increase can also provide clues on the implicit sentiment [179]. Contrary to explicit expressions (subjective utterances), factual text types often contain objective statements that convey a desirable or undesirable fact [92]. Recent work proposes to consider all types of implicit sentiment expressions [180]. The authors created a fine grained sentiment annotation procedure to identify polar expressions (implicit and explicit expressions of positive and negative sentiment). A target (company of interest) is identified in each polar expression to identify the sentiment expressions that are relevant. The annotation procedure also collected information about the polarity and the intensity of the sentiment expressed towards the target. However, there is still no automatic approach, either lexical-based or machine learning based, that tries to model this annotation scheme. In this work, we propose to tackle the aforementioned problem by taking advantage of unsupervised learning of word embeddings in financial tweets and financial news headlines to construct a domain-specific syntactic and semantic representation of words. We combine bag-of-embeddings with traditional approaches, such as pre-processing techniques, bag-of-words and financial lexical-based features to train a regressor for sentiment polarity and intensity. We study how different regression algorithms perform using all features in two different sub-tasks at SemEval-2017 Task 5: microblogs and news headlines mentioning companies/stocks. Moreover, we compare how different combinations of features perform in both sub-tasks. The system source code and word embeddings developed for the competition are publicly available.3 2 The material contained in this section was published in P. Saleiro, E. M. Rodrigues, C. Soares, E. Oliveira, “FEUP at SemEval-2017 Task 5: Predicting Sentiment Polarity and Intensity with Financial Word Embeddings” [18] 3 https://github.com/saleiro/Financial-Sentiment-Analysis 5.2 Financial Sentiment Analysis 5.2.1 87 Task Overview The task 5 of SemEval 2017 [181] consisted of fine-grained sentiment analysis of financial short texts and it was divided in two sub-tasks based on the type of text. Sub-task 5.1 – Microblogs – consisted of stocktwits and tweets focusing on stock market events and assessments from investors and traders. Companies/stocks were identified using stock symbols, the so called cashtags, e.g.“$AMZN” for the company Amazon.com, Inc. Sub-task 5.2 – News Headlines – consisted of sentences extracted from Yahoo Finance and other financial news sources on the Internet. In this case, companies/stocks were identified using their canonical name and were previously annotated by the task organizers. Sub-task 5.1 - Microblogs 5.2 - Headlines Company JPMorgan Glencore Text Span Sentiment Score “its time to sell banks" -0.763 “Glencore’s annual results +0.900 beat forecasts" Table 5.4 Training set examples for both sub-tasks. The goal of both sub-tasks was the following: predict the sentiment polarity and intensity for each of the companies/stocks mentioned in a short text instance (microblog message or news sentence). The sentiment score is a real continuous variable in the range of -1.0 (very negative/bearish) to +1.0 (very positive/bullish), with 0.0 designating neutral sentiment. Table 5.4 presents two examples from the training set. Task organizers provided 1700 microblog messages for training and 800 messages for testing in sub-task 5.1, while in sub-task 5.2, 1142 news sentences were provided for training and 491 for testing. Submissions were evaluated using the cosine similarity [181]. 5.2.2 Financial Word Embeddings Mikolov et al. [182] created word2vec, a computationally efficient method to learn distributed representation of words, where each word is represented by a distribution of weights (embeddings) across a fixed set of dimensions. Furthermore, Mikolov et al. [100] showed that this representation is able to encode syntactic and semantic similarities in the embedding space. The training objective of the skip-gram model, defined by Mikolov et al. [100], is to learn the target word representation (embeddings) that maximize the prediction of its surrounding words in a context window. Given the wt word in a vocabulary the objective is to maximize the average log probability: Entity Filtering and Financial Sentiment Analysis 88 T X 1X log P (wt+j \|wt ) T t=1 −c≤j≤c,j̸=0 (5.1) where c is the size of the context window, T is the total number of words in the vocabulary and wt+j is a word in the context window of wt . After training, a low dimensionality embedding matrix E encapsulates information about each word in the vocabulary and its use (surrounding contexts). We used word2vec to learn word embeddings in the context of financial texts using unlabeled tweets and news headlines mentioning companies/stocks from S&P 500. Tweets were collected using the Twitter streaming API with cashtags of stocks titles serving as request parameters. Yahoo Finance API was used for requesting financial news feeds by querying the canonical name of companies/stocks. The datasets comprise a total of 1.7M tweets and 626K news titles. We learned separate word embeddings for tweets and news headlines using the skip-gram model. We tried several configurations of word2vec hyperparameters. The setup resulting in the best performance in both sub-tasks was skip-gram with 50 dimensions, removing words occurring less than 5 times, using a context window of 5 words and 25 negative samples per positive example. Even though the text collections for training embeddings were relatively small, the resulting embedding space exhibited the ability to capture semantic word similarities in the financial context. We performed simple algebraic operations to capture semantic relations between words, as described in Mikolov et al. [113]. For instance, the skip-gram model trained on tweets shows that vector (“bearish”) - vector(“loss”) + vector(“gain”) results in vector (“bullish”) as most similar word representation. 5.2.3 Approach In this section we describe the implementation details of the proposed approach. Pre-Processing A set of pre-processing operations are applied to every microblog message and news sentence in the training/test sets of sub-tasks 5.1 and 5.2, as well as in the external collections for training word embeddings: • Character encoding and stopwords: every message and headline was encoded in UTF-8. Standard english stopword removal is also applied. 5.2 Financial Sentiment Analysis 89 • Company/stock and cash obfuscation: both cashtags and canonical company names strings were replaced by the string _company_. Dollar or Euro signs followed by numbers were replaced by the string _cash_amount_. • Mapping numbers and signs: numbers were mapped to strings using bins (0-10, 10-20, 20-50, 50-100, >100). Minus and plus signs were coverted to minus and plus, “B” and “M” to billions and millions, respectively. The % symbol was converted to percent. Question and exclamation marks were also converted to strings. • Tokenization, punctuation, lowercasing: tokenization was performed using Twokenizer [183], the remaining punctuation was removed and all characters were converted to lowercase. Features We combined three different group of features: bag-of-words, lexical-based features and bag-of-embeddings. • Bag-of-words: we apply standard bag-of-words as features. We tried unigrams, bi-grams and tri-grams with unigrams proving to obtain higher cosine similarity in both sub-tasks. • Sentiment lexicon features: we incorporate knowledge from manually curated sentiment lexicons for generic Sentiment Analysis as well as lexicons tailored for the financial domain. The Laughran-Mcdonald financial sentiment dictionary [184] has several types of word classes: positive, negative, constraining, litigious, uncertain and modal. For each word class we create a binary feature for the match with a word in a microblog/headline and a polarity score feature (positive - negative normalized by the text span length). As a general-purpose sentiment lexicon we use MPQA [185] and created binary features for positive, negative and neutral words, as well as, the polarity score feature. • Bag-of-Embeddings: we create bag-of-embeddings by taking the average of word vectors for each word in a text span. We used the corresponding embedding matrix trained on external Twitter and Yahoo Finance collections for sub-task 5.1 and sub-task 5.2, respectively. Entity Filtering and Financial Sentiment Analysis 90 5.2.4 Experimental Setup In order to avoid overfitting we created a validation set from the original training datasets provided by the organizers. We used a 80%-20% split and sampled the validation set using the same distribution as the original training set. We sorted the examples in the training set by the target variable values and skipped every 5 examples. Results are evaluated using Cosine similarity [181] and Mean Average Error (MAE). The former gives more importance to differences in the polarity of the predicted sentiment while the latter is concerned with how well the system predicts the intensity of the sentiment. We opted to model both sub-tasks as single regression problems. Three different regressors were applied: Random Forests (RF), Support Vector Machines (SVM) and MultiLayer Perceptron (MLP). Parameter tuning was carried using 10 fold cross validation on the training sets. 5.2.5 Results and Analysis In this section we present the experimental results obtained in both sub-tasks. We provide comparison of different learning algorithms using all features, as well as, a comparison of different subsets of features, to understand the information contained in each of them and also how they complement each other. Task 5.1 - Microblogs Table 5.5 presents the results obtained using all features in both validation set and test sets. Results in the test set are worse than in the validation set with the exception of MLP. The official score obtained in sub-task 5.1 was 0.6948 using Random Forests (RF), which is the regressor that achieves higher cosine similarity and lower MAE in both training and validation set. Regressor Set RF Val RF Test SVR Val SVR Test MLP Val MLP Test Table 5.5 Microblog results with all Cosine MAE 0.7960 0.1483 0.6948 0.1886 0.7147 0.1944 0.6227 0.2526 0.6720 0.2370 0.6789 0.2132 features on validation and test sets. 5.2 Financial Sentiment Analysis 91 We compared the results obtained with different subsets of features using the best regressor, RF, as depicted in Table 5.6. Interestingly, bag-of-words (BoW) and bag-of-embeddings (BoE) complement each other, obtaining better cosine similarity than the system using all features. Financial word embeddings (BoE) capture relevant information regarding the target variables. As a single group of features it achieves a cosine similarity of 0.6118 and MAE of 0.2322. It is also able to boost the overall performance of BoW with gains of more than 0.06 in cosine similarity and reducing MAE more than 0.03. The individual group of features with best performance is Bag-of-words while the worst is a system trained using Lex (only lexical-based features). While Lex alone exhibits poor performance, having some value but marginal, when combined with another group of features, it improves the results of the latter, as in the case of BoE + Lex and BoW + Lex. Features Cosine MAE Lex 0.3156 0.3712 BoE 0.6118 0.2322 BoW 0.6386 0.2175 BoE + Lex 0.6454 0.2210 Bow + Lex 0.6618 0.2019 Bow + BoE 0.7023 0.1902 All 0.6948 0.1886 Table 5.6 Features performance breakdown on test set using RF. Task 5.2 - News Headlines Results obtained in news headlines are very different from the ones of the previous sub-task, proving that predicting sentiment polarity and intensity in news headlines is a completely different problem compared to microblogs. Table 5.7 shows that MLP obtains the best results in the test set using both metrics while SVR obtains the best performance in the validation set. The best regressor of sub-task 5.1, RF is outperformed by both SVR and MLP. The official result obtained at sub-task 5.2 was a cosine similarity of 0.68 using MLP. Table 5.8 shows the results of the different groups of features in sub-task 5.2 for MLP regressor. The most evident observation is that word embeddings are not effective in this scenario. On the other hand, lexical based features have significantly better performance in news headlines than in microblogs. Despite this, the best results are obtained using all features. Entity Filtering and Financial Sentiment Analysis 92 Regressor Set RF Val RF Test SVR Val SVR Test MLP Val MLP Test Table 5.7 News Headlines results with Features BoE Lex BoW BoE + Lex BoW + Lex BoW + BoE All Table 5.8 Features performance Cosine MAE 0.5316 0.2539 0.6562 0.2258 0.6397 0.2422 0.6621 0.2424 0.6176 0.2398 0.6800 0.2271 all features on validation and test sets. Cosine MAE 0.0383 0.3537 0.5538 0.2788 0.6420 0.2364 0.5495 0.2830 0.6733 0.2269 0.6417 0.2389 0.6800 0.2271 breakdown on test set using MLP. Analysis Financial word embeddings were able to encapsulate valuable information in sub-task 5.1 - Microblogs but not so much in the case of sub-task 5.2 - News Headlines. We hypothesize that as we had access to a much smaller dataset (∼ 600K) for training financial word embeddings for news headlines, this resulted in reduced ability to capture semantic similarities in the financial domain. Other related works in Sentiment Analysis usually take advantage of a much larger dataset for training word embeddings [186]. On the other hand, lexical features showed poor performance in microblog texts but seem to be very useful using news headlines. The fact that microblogs have poor grammar, slang and informal language reveals that financial lexicons created using well written and formal financial reports, result better in news headlines rather than in microblog texts. After inspecting microblog texts and headlines in which our models showed poor performance we believe it would be important to also encapsulate syntactic and semantic dependencies in our models. For instance, our model predicted a sentiment score of -0.467 for the microblog message “was right to reject the offer” while the true value is 0.076. Similar examples include “Glencore shares in record crash as profit fears grow” and “I would rather be a buyer at these levels then trying to sell”, in which our models 5.3 Summary of the Contributions 93 has absolute errors around 0.5. Other type of errors have to do with intensity of the sentiment in which our model correctly predicts the polarity but still has a large error. 5.2.6 Concluding Remarks Work reported here reported is concerned with the problem of predicting sentiment polarity and intensity of financial short texts. Previous work showed that sentiment is often depicted in an implicit way in this domain. We created financial-specific continuous word representations in order to obtain domain specific syntactic and semantic relations between words. We combined traditional bag-of-words and lexicalbased features with bag-of-embeddings to train a regressor of both sentiment polarity and intensity. Results show that different combination of features attained different performances on each sub-task. Future work will consist on collecting larger external datasets for training financial word embeddings of both microblogs and news headlines. We also have planned to perform the regression analysis using Deep Neural Networks. 5.3 Summary of the Contributions In this chapter we present some contributions to two fundamental Text Mining problems in ORM. • A supervised learning approach for Entity Filtering on tweets, achieving state-ofthe-art performance using a relatively small training set. • Created and made available word embeddings trained from financial texts. • A supervised learning approach for fine-grained sentiment analysis of financial texts. Chapter 6 Text-based Entity-centric Prediction In this chapter we explore the predictive power of entity-centric information in online news and social media in the context of ORM. We address two different predictive tasks. The first is concerned with predicting entity popularity on Twitter based on signals extracted from the news cycle. We aim to study different sets of signals extracted from online news mentioning specific entities that could influence or at least are correlated with future popularity of those entities on Twitter. We know that entity popularity on social media can be influenced by several factors but we are only interested in exploring the interplay between online news and social media for entities that are frequently mentioned on the news cycle such as politicians or footballers. This could be particularly interesting for anticipating public relations damage control once a polemic news article is published. Or even for editorial purposes to maximize buzz on social media. The second predictive task consists in using entity-centric sentiment polarity extracted from tweets to predict political polls. There has been several research work trying to assess the predictive power of social media to predict the outcome of political opinion surveys or elections. However, each study proposes its own method of aggregating polarity scores over time, however, there is not a consensus on which sentiment aggregate function is the most adequate for this problem. We propose to use and contrast several sentiment aggregate functions reported in the literature, by assessing their predictive power on a specific case comprising data collected during the Portuguese bailout (2011-2013). Text-based Entity-centric Prediction 96 6.1 Exploring Online News for Reputation Monitoring on Twitter 1 Online publication of news articles has become a standard behavior of news outlets, while the public joined the movement either using desktop or mobile terminals. The resulting setup consists of a cooperative dialog between news outlets and the public at large. Latest events are covered and commented by both parties in a continuous basis through the social media, such as Twitter. When sharing or commenting news on social media, users tend to mention the most predominant entities mentioned in the news story. Therefore, entities, such as public figures, organizations, companies or geographic locations, can act as latent connections between online news and social media. Online Reputation Monitoring (ORM) focuses on continuously tracking what is being said about entities on social media and online news. Automatic collection and processing of comments and opinions on social media is now crucial to understand the reputation of individuals and organizations and therefore to manage their public relations. However, ORM systems would be even more useful if they would be able to know in advance if social media users will talk a lot about the target entities or not. We hypothesize that for entities that are frequently mentioned on the news (e.g. politicians) it is possible to establish a predictive link between online news and popularity on social media. We cast the problem as a supervised learning classification approach: to decide whether popularity will be high or low based on features extracted from the news cycle. We define four set of features: signal, textual, sentiment and semantic. We aim to respond to the following research questions: • Is online news a valuable source of information to effectively predict entity popularity on Twitter? • Do online news carry different predictive power based on the nature of the entity under study? • How do different thresholds for defining high and low popularity affect the effectiveness of our approach? • Does the performance remain stable for different prediction times? • What is the most important feature set for predicting entity popularity on Twitter based on the news cycle? 1 The material contained in this section was published in P. Saleiro and C. Soares, “Learning from the News: Predicting Entity Popularity on Twitter” [19] 6.1 Exploring Online News for Reputation Monitoring on Twitter 97 • Do individual sets of features exhibit different importance for different entities? 6.1.1 Approach The starting point of our hypothesis is that for entities that are frequently mentioned on the news (e.g. politicians) it is possible to predict popularity on social media using signals extracted from the news cycle. The first step towards a solution requires the definition of entity popularity on social media. Entity Popularity There are different ways of expressing the notion of popularity on social media. For example, the classical way of defining it is through the number of followers of a Twitter account or the number of likes in a Facebook page. Another notion of popularity, associated with entities, consists on the number of retweets or replies on Twitter and post likes and comments on Facebook. We define entity popularity based on named entity mentions in social media messages. Mentions consist of specific surface forms of an entity name. For example, “Cristiano Ronaldo” might be mentioned also using just “Ronaldo” or “#CR7”. Given an set of entities E = {e1 , e2 , ..., ei , ...}, a daily stream of social media messages S = {s1 , s2 , ..., si , ...} and a daily stream of online news articles N = {n1 , n2 , ..., ni , ...} we are interested in monitoring the mentions of an entity ei on the social media stream St , i.e. the discrete function fm (ei , St ). Let T be a daily time frame T = [tp , tp+h ], where the time tp is the time of prediction and tp+h is the prediction horizon time. We want to learn a target popularity function fp on social media stream S as a function of the given entity ei , the online news stream N and the time frame T : t=tp+h fp (ei , N, T ) = X fm (ei , St ) t=tp which corresponds to integrating fm (ei , S) over T . Given a day di , a time of prediction tp , we extract features from the news stream N until tp and predict fp until the prediction horizon tp + h. We measure popularity on a daily basis, and consequently, we adopted tp+h as 23:59:59 everyday. For example, if tp equals to 8 a.m, we extract features from N until 07:59:59 and predict fp in the interval 08:00 - 23:59:59 on day di . In the case of tp equals to midnight, we extract Text-based Entity-centric Prediction 98 features from N on the 24 hours of previous day di−1 to predict fp for the 24 hours of di . We cast the prediction of fp (ei , N, T ) as a supervised learning classification problem, in which we want to infer the target variable fˆp (ei , N, T ) ∈ {0, 1} defined as: fˆp =   0(low), if P (fp (ei , N, T ) ≤ δ) = k  1(high), if P (fp (ei , N, T ) > δ) = 1 − k where δ is the inverse of cumulative distribution function at k of fp (ei , N, T ) as measured in the training set, a similar approach to Tsagkias et al. [119]. For instance, k = 0.5 corresponds to the median of fp (ei , N, T ) in the training set and higher values of k mean that fp (ei , N, T ) has to be higher than k examples on the training set to consider fˆp = 1, resulting in a reduced number of training examples of the positive class high. News Features Previous work has focused on the influence of characteristics of the social media stream S in the adoption and popularity of memes and hashtags [126]. In contrast, the main goal of this work is to investigate the predictive power of the online news stream N . Therefore we extract four types of features from N which we label: (i) signal, (ii) textual, (iii) sentiment and (iv) semantic, as depicted in Table 6.1. One important issue is how can we filter relevant news items to ei . There is no consensus on how to link a news stream N with a social media stream S. Some works use URLs from N , shared on S, to filter simultaneously relevant news articles and social media messages [117]. As our work is entity oriented, we select news articles with mentions of ei as our relevant N . Signal Features - This type of features depict the “signal” of the news cycle mentioning ei and we include a set of counting variables as features, focusing on the total number of news mentioning ei in specific time intervals, mentions on news titles, the average length of news articles, the different number of news outlets that published news mentioning ei as well as, features specific to the day of the week to capture any seasonal trend on the popularity. The idea is to capture the dynamics of news events, for instance, if ei has a sudden peak of mentions on N , a relevant event might have happened which may influence fp . Textual features - To collect textual features we build a daily profile of the news cycle by aggregating all titles of online news articles mentioning ei for the daily time frame [0, tp ] in di . We select the top 10,000 most frequent terms (unigrams and bi-grams) in 6.1 Exploring Online News for Reputation Monitoring on Twitter 99 the training set and create a document-term matrix R. Two distinct methods were applied to capture textual features. The first method is to apply TF-IDF weighting to R. We employ Singular Value Decomposition (SVD) to capture similarity between terms and reduce dimensionality. It computes a low-dimensional linear approximation σ. The final set of features for training and testing is the TF-IDF weighted term-document matrix R combined with σR which produces 10 real valued latent features. When testing, the system uses the same 10,000 terms from the training data and calculates TF-IDF using the IDF from the training data, as well as, σ for applying SVD on test data. The second method consists in applying Latent Dirichlet allocation (LDA) to generate a topic model of 10 topics (features). The system learns a topic-document distribution θ and a word distribution over topics φ using the training data for a given entity ei . When testing, the system extracts the word distribution of the news title vector r on a test day d′i . Then, by using φ learned on training data, it calculates the probability of r belonging to one of the 10 topics learned before. The objective of extracting this set of features is to create a characterization of the news stream that mentions ei , namely, which are the most salient terms and phrases on each day di as well as the latent topics associated with ei . By learning our classifier we hope to obtain correlations between certain terms and topics and fp . Sentiment features - We include several types of word level sentiment features. The assumption here is that subjective words on the news will result in more reactions on social media, as exposed in [187]. Once again we extract features from the titles of news mentioning ei for the daily time frame [0, tp ]. We use a sentiment lexicon as SentiWordNet to extract subjective terms from the titles daily profile and label them as positive, neutral or negative polarity. We compute count features for number of positive, negative, neutral terms as well as difference and ratio of positive and negatives terms. Similar to textual features we create a TFIDF weighted term-document matrix R using the subjective terms from the title and apply SVD to compute 10 real valued sentiment latent features. Semantic features - We use the number of different named entities recognized in N on day di until tp, as well as, the number of distinct news category tags extracted from the news feeds metadata. These tags, common in news articles, consist of author annotated terms and phrases that describe a sort of semantic hierarchy of news categories, topics and news stories (e.g. “european debt crisis”). We create a TF-IDF weighted entity-document and TF-IDF tag-document matrices and applied SVD to each of them to reduce dimensionality to 10. The idea is to capture interesting entity Text-based Entity-centric Prediction 100 Table 6.1 Summary of the four type of features we consider. Number Signal 1 2 3 4 5 6 7 8 Textual 9-18 19-28 Sentiment 29 30 31 32 33 34 35-44 Semantic 45 46 47-56 57-66 Feature Description news news di−1 news total di−1 news titles avg content sources weekday is weekend number of news mentions of ei in [0, tp ] in di number of news mentions of ei in [0, tp ] in di−1 number of news mentions of ei in [0, 24[ in di−1 number of title mentions in news of ei in [0, tp ] in di average content length of news of ei in [0, tp ] in di number of different news sources of ei in [0, tp ] in di day of week true if weekend, false otherwise tfidf titles LDA titles TF-IDF of news titles [0, tp ] in di LDA-10 of news titles [0, tp ] in di pos neg neu ratio diff subjectivity tfidf subj number of positive words in news titles [0, tp ] in di number of negative words in news titles [0, tp ] in di number of neutral words in news titles [0, tp ] in di positive/negative positive − negative P (positive + negative + neutral)/ words TF-IDF of subjective words (pos, neg and neu) entities tags tfidf entities tfidf tags number of entities in news [0, tp ] in di number of tags in news [0, tp ] in di TF-IDF of entities in news [0, tp ] in di TF-IDF of news tags [0, tp ] in di co-occurrences as well as, news stories that are less transient in time and might be able to trigger popularity on Twitter. Learning Framework Let x be the feature vector extracted from the online news stream N on day di until tp. We want to learn the probability P (fˆp = 1\|X = x). This can be done using the inner product between x and a weighting parameter vector w ∈ R, w⊤ x. Using logistic regression and for binary classification one can unify the definition of ˆ p(fp = 1\|x) and p(fˆp = 0\|x) with 6.1 Exploring Online News for Reputation Monitoring on Twitter p(fˆp \|x) = 101 1 1+ e−fˆp w⊤ x Given a set of z instance-label pairs (xi ,fˆp i ), with i = 1, ..., z and fˆp i ∈ {0, 1} we solve the binary class L2 penalized logistic regression optimization problem, where C>0 n X 1 ˆ ⊤ min w⊤ w + C log(1 + e−fp i w xi ) w 2 i=1 We apply this approach following an entity specific basis, i.e. we train an individual model for each entity. Given a set of entities E to which we want to apply our approach and a training set of example days D = {d1 , d2 , ..., di , ...}, we extract a feature vector xi for each entity ei on each training day di . Therefore, we are able to learn a model of w for each ei . The assumption is that popularity on social media fp is dependent of the entity ei and consequently we extract entity specific features from the news stream N . For instance, the top 10,000 words of the news titles mentioning ei are not the same for ej . 6.1.2 Experimental Setup This work uses Portuguese news feeds and tweets collected from January 1, 2013 to January 1, 2016, consisting of over 150 million tweets and 5 million online news articles2 . To collect and process raw Twitter data, we use a crawler, which recognizes and disambiguates named entities on Twitter [188]. News data is provided by a Portuguese online news aggregator3 . This service handles online news from over 60 Portuguese news outlets and it is able to recognize entities mentioned on the news. We choose the two most common news categories: politics and football and select the 3 entities with highest number of mentions on the news for both categories. The politicians are two former Prime-ministers, José Sócrates and Pedro Passos Coelho and the incumbent, António Costa. The football entities are two coaches, Jorge Jesus and José Mourinho, and the most famous Portuguese football player, Cristiano Ronaldo. Figure 7.5 depicts the behavior of daily popularity of the six entities on the selected community stream of Twitter users for each day from July 2014 until July 2015. As expected, it is easily observable that in some days the popularity on Twitter exhibits 2 3 Dataset is available for research purposes. Access requests via e-mail. http://www.sapo.pt Text-based Entity-centric Prediction 102 6000 5000 Pedro Passos Coelho José Sócrates António Costa Jorge Jesus Cristiano Ronaldo José Mourinho 4000 3000 2000 1000 0 Aug Sep Oct Nov Dec Jan 2015 Feb Mar Apr May Jun Fig. 6.1 Daily popularity on Twitter of entities under study. Training Iteration 1 Jan 2013 Feb 2013 Test ... Dec 2014 Jan 2015 Training Iteration 2 Jan 2013 Feb 2013 Mar 2013 Feb 2015 Test ... Jan 2015 Feb 2015 Fig. 6.2 Training and testing sliding window - first 2 iterations. bursty patterns. For instance, when José Sócrates was arrested in November 21st 2014 or when Cristiano Ronaldo won the FIFA Ballon d’Or in January 12th 2015. We defined the years of 2013 and 2014 as training set and the whole year of 2015 as test set. We applied a monthly sliding window setting in which we start by predicting entity popularity for every day of January 2015 (i.e. the test set) using a model trained on the previous 24 months, 730 days (i.e. the training set). Then, we use February 2015 as the test set, using a new model trained on the previous 24 months. Then March and so on, as depicted in Figure 6.2. We perform this evaluation process, rolling the training and test set until December 2015, resulting in 365 days under evaluation. 6.1 Exploring Online News for Reputation Monitoring on Twitter 103 The process is applied for each one of the six entities, for different time of predictions tp and for different values of the decision boundary k. We test tp = 0, 4, 8, 12, 16, 20 and k = 0.5, 0.65, 0.8. Therefore, we report results in Section 6.2.4 for 18 different experimental settings, for each one of the six entities. The goal is to understand how useful the news cycle is for predicting entity popularity on Twitter for different entities, at different hours of the 24 hours cycle and with different thresholds for considering popularity as high or low. 6.1.3 Results and Discussion Results are depicted in Table 6.2. We report F1 on positive class since in online reputation monitoring is more valuable to be able to predict high popularity than low. Nevertheless, we also calculated overall Accuracy results, which were better than the F1 reported here. Consequently, this means that our system is fairly capable of predicting low popularity. We organize this section based on the research questions we presented in the beginning of this section. Is online news valuable as source of information to effectively predict entity popularity on Twitter? Do online news carry different predictive power based on the nature of the entity under study? Results show that performance varies with each target entity ei . In general, results are better in the case of predicting popularity of politicians. In the case of football public figures, Jorge Jesus exhibits similar results with the three politicians but José Mourinho and especially Cristiano Ronaldo represent the worst results in our setting. For instance, when Cristiano Ronaldo scores three goals in a match, the burst on popularity is almost immediate and not possible to predict in advance. Further analysis showed that online news failed to be informative of popularity in the case of live events covered by other media, such as TV. Interviews and debates on one hand, and live football games on the other, consist of events with unpredictable effects on popularity. Cristiano Ronaldo can be considered a special case in our experiments. He is by far the most famous entity in our experiments and in addition, he is also an active Twitter user with more than 40M followers. This work focus on assessing the predictive power of online news and its limitations. We assume that for Cristiano Ronaldo, endogenous features from the Twitter itself would be necessary to obtain better results. Text-based Entity-centric Prediction 104 Table 6.2 F1 score of popularity high as function of tp and k equal to 0.5, 0.65 and 0.8 respectively. Entity \tp (hour) k = 0.50 António Costa José Sócrates Pedro Passos Coelho Cristiano Ronaldo Jorge Jesus José Mourinho k = 0.65 António Costa José Sócrates Pedro Passos Coelho Cristiano Ronaldo Jorge Jesus José Mourinho k = 0.80 António Costa José Sócrates Pedro Passos Coelho Cristiano Ronaldo Jorge Jesus José Mourinho 0 4 8 12 16 20 0,76 0,77 0,72 0,35 0,73 0,62 0,67 0,66 0,63 0,41 0,68 0,46 0,74 0,73 0,70 0,45 0,69 0,51 0,77 0,75 0,70 0,37 0,68 0,56 0,75 0,75 0,74 0,35 0,69 0,55 0,72 0,75 0,71 0,32 0,70 0,45 0,61 0,63 0,58 0,29 0,63 0,56 0,60 0,57 0,57 0,35 0,61 0,39 0,66 0,62 0,65 0,42 0,63 0,48 0,64 0,66 0,67 0,41 0,59 0,56 0,60 0,64 0,67 0,36 0,62 0,47 0,60 0,62 0,65 0,30 0,64 0,38 0,48 0,48 0,47 0,14 0,50 0,32 0,51 0,42 0,46 0,29 0,48 0,32 0,55 0,47 0,56 0,31 0,51 0,36 0,53 0,53 0,56 0,26 0,48 0,41 0,44 0,47 0,52 0,20 0,57 0,41 0,49 0,35 0,54 0,21 0,56 0,36 How do different thresholds for defining high and low popularity affect the effectiveness of our approach? Our system exhibits top performance with k = 0.5, which corresponds to balanced training sets, with the same number of high and low popularity examples on each training set. Political entities exhibit F1 scores above 0.70 with k = 0.5. On the other hand, as we increase k, performance deteriorates. We observe that for k = 0.8, the system predicts a very high number of false positives. It is very difficult to predict extreme values of popularity on social media before they happen. We plan to tackle this problem in the future by also including features about the target variable in the current and previous hours, i.e., time-series auto-regressive components. Does performance remain stable for different time of predictions? Results show that time of prediction affects the performance of the system, specially for the political entities. In their case, F1 is higher when time of prediction is noon 6.1 Exploring Online News for Reputation Monitoring on Twitter 105 Fig. 6.3 Individual feature type F1 score for tp = 12 at k = 0.5. and 4 p.m. which is an evidence that in politics, most of the news events that trigger popularity on social media are broadcast by news outlets in the morning. It is very interesting to compare results for midnight and 4 a.m./8 a.m. The former use the news articles from the previous day, as explained in Section 6.1.1, while the latter use news articles from the first 4/8 hours of the day under prediction. In some examples, Twitter popularity was triggered by events depicted on the news from the previous day and not from the current day. What is the most important feature set for predicting entity popularity on Twitter based on the news cycle? Do individual set of features exhibit different importance for different entities? Figure 6.3 tries to answer these two questions. The first observation is that the combination of all groups of features does not lead to substantial improvements. Semantic features alone achieve almost the same F1 score as the combination of all features. However in the case of Mourinho and Ronaldo, the combination of all features lead to worse F1 results than the semantic set alone. Text-based Entity-centric Prediction 106 Sentiment features are the second most important for all entities except José Mourinho. Signal and Textual features are less important and this was somehow a surprise. Signal features represent the surface behavior of news articles, such as the volume of news mentions of ei before tp and we were expecting an higher importance. Regarding Textual features, we believe that news articles often refer to terms and phrases that explain past events in order to contextualize a news article. In future work, we consider alternative approaches for predicting future popularity of entities that do not occur everyday on the news, but do have social media public accounts, such as musicians or actors. In opposition, entities that occur often on the news, such as economics ministers and the like, but do not often occur in the social media pose also a different problem. 6.2 Predicting Political Polls using Twitter Sentiment4 Surveys and polls using the telephone are widely used to provide information of what people think about parties or political entities [150]. Surveys randomly select the electorate sample, avoiding selection bias, and are designed to collect the perception of a population regarding some subject, such as in politics or marketing. However this method is expensive and time consuming [150]. Furthermore, over the years it is becoming more difficult to contact people and persuade them to participate in these surveys [189]. On the other hand, the rise of social media, namely Twitter and Facebook, has changed the way people interact with news. This way, people are able to react and comment any news in real time [135]. One challenge that several research works have been trying to solve is to understand how opinions expressed on social media, and their sentiment, can be a leading indicator of public opinion. However, at the same time there might exist simultaneously positive, negative and neutral opinions regarding the same subject. Thus, we need to obtain a value that reflects the general image of each political target in social media, for a given time period. To that end, we use sentiment aggregate functions. In summary, a sentiment aggregate function calculates a global value based on the number of positive, negative, and neutral mentions of each political target, in a given period. We conducted an exhaustive study and collected and implemented several sentiment aggregate functions from the state of the art [135–143]. 4 The material contained in this section was published in P. Saleiro, L. Gomes, C. Soares, “Sentiment Aggregate Functions for Political Opinion Polling using Microblog Streams” [21] 6.2 Predicting Political Polls using Twitter Sentiment 107 Thus, the main objective of our work is to study and define a methodology capable of successfully estimating the poll results, based on opinions expressed on social media, represented by sentiment aggregators. We applied this problem to the Portuguese bailout case study, using Tweets from a sample of the Portuguese Tweetosphere and Portuguese polls as gold standard. Given the monthly periodicity of polls, we needed to aggregate the data by month. This approach allows each aggregate value to represent the monthly sentiment for each political party. Due to the absence of a general sentiment aggregate function suitable for different case studies, we decided to include all aggregate functions as features of the regression model. Therefore the learning algorithm is able to adapt to the most informative aggregate functions through time. 6.2.1 Methodology To collect and process raw Twitter data, we use an online reputation monitoring platform [36] which can be extended by researchers interested in tracking political opinion on the web. It collects tweets from a predefined sample of users, applies named entity disambiguation [177] and generates indicators of both frequency of mention and polarity (positivity/negativity) [190] of mentions of entities over time. In our case, tweets are collected from the stream of 100 thousand different users, representing a sample of the Portuguese community on Twitter. This sample was obtained by expanding a manually annotated seed set of 1000 users using heuristics such as, as language of posts, language of followers posts or geo-location [188]. The platform automatically classifies each tweet according to its sentiment polarity. If a message expresses a positive, negative or neutral opinion regarding an entity (e.g. politicians), it is classified as positive, negative or neutral mention, respectively. The sentiment classifier uses a corpus of 1500 annotated tweets as training set and it has achieved an accuracy over 80% using 10-fold cross validation. These 1500 tweets were manually annotated by 3 political science students. Mentions of entities and respective polarity are aggregated by counting positive, negative, neutral and total mentions for each entity in a given period. Sentiment aggregate functions use these cumulative numbers as input to generate a new value for each specific time period. Since we want to use sentiment aggregate functions as features of a regression model to produce an estimate of the political opinion, we decided to use traditional poll results as gold standard. Text-based Entity-centric Prediction 108 Sentiment Aggregate Functions Let Mei be a mention on Twitter of an entity ei , then Me+i , Me∗i and Me−i are positive, neutral and negative classified mentions of entity ei on Twitter. Therefore, given a time frame T (e.g. a month), sentiment aggregate functions applied to the aggregated data between polls are the following: P • entitybuzz: T Mei , the sum of the number of mentions (buzz) of a given entity in the time frame T . • entitypositives: T Me+i , sum of the positively classified mentions of a given entity in the time frame T . P • entityneutrals: T Me∗i , the sum of the neutral classified mentions of a given entity in a time frame T . P • entitynegatives: T Me−i , the sum of the negatively classified mentions of a given entity in a time frame T . P M + +M − P ei T P ei • entitysubjectivity: , the ratio of positive and negative classified M e i T mentions of entity ei over its buzz in a time frame T . M+ P • entitypolarity: PT Me−i , the ratio of positive over negative classified mentions in ei T a time frame T . P M− ei T • berminghamsovn: P P , the ratio of the negative classified mentions of Me−i T E entity ei over the total number of negative mentions of all entities in time frame T. P Me+i +1 T P P log10 Me− +1 - bermingham [135]: T - berminghamsovp [135]: T E Me+i P M+ PT e−i - connor [191]: T P Mei Me−j Ej̸=i + Mei +Me−i E Me+i + P T P P - gayo [141]: T - polarity: E P Me+i T P P P T Me+i − - polarityON eutral: P Me−i P Me+i − T Me−i P Me0i T T T P i 6.2 Predicting Political Polls using Twitter Sentiment P - polarityOT otal: P P T P - subjOT otal: - subjN euv: T M ei Me+i + T Me−i P M ei T P T P 109 Me−i T Me+i − T Me+i + T Me−i P Me0i T P P + P − E ei ei M + M - subjSoV : PT Pei M +T+Me−i T - subjV ol: P T - share [135]: Me+i + Me−i P M ei T P P T E M ei - shareOf N egDistribution: P − M P T ei Mei T P P Me− T i P E T in the poll , where n is the number of political entities M ei + P M - normalized_positive: PT Meei i T - normalized_negative: P M− PT e−i T - normalized_neutral: Mei P M0 PT ei T M ei - normalized_bermingham: log10 - normalized_connor: normalized_positives+1 normalized_negatives+1 normalized_positives normalized_negatives - normalized_gayo: normalized_positives+normalized_others_negatives normalized_total_positives+normalized_total_negatives - normalized_polarity: normalized_positives − normalized_negatives The sentiment aggregate functions are used as features in the regression models. Text-based Entity-centric Prediction 110 Fig. 6.4 Negatives share (berminghamsovn) of political leaders in Twitter. 6.2.2 Data The data used in this work consists of tweets mentioning Portuguese political party leaders and polls from August 2011 to December 2013. This period corresponds to the Portuguese bailout when several austerity measures were adopted by the incumbent right wing governmental coalition of the PSD and CDS parties. Twitter Table 6.3 Distribution of positive, negative and neutral mentions per political party PSD PS CDS CDU BE Negative 69 723 28 660 41 935 2 445 9 603 Positive 121 225 51 79 306 Neutral 37 133 15 326 17 554 5 604 4 214 Total Mentions 106 977 44 211 59 540 8 128 14 123 The Twitter data set contains 232,979 classified messages, collected from a network of 100 thousand different users classified as Portuguese. Table 6.3 presents the distribution of positive, negative, and neutral mentions of the political leaders of the 5 most voted political parties in Portugal (PSD, PS, CDS, PCP and BE). The negative mentions represent the majority of the total mentions, except for CDU where the number of negative mentions is smaller than the neutral ones. The positive mentions represent less than 1% of the total mentions of each party, except for BE where they represent 6.2 Predicting Political Polls using Twitter Sentiment 111 2% of the total mentions. The most mentioned parties are PS, PSD and CDS. The total mentions of these three parties represent 90% of the data sample total mentions. Figure 6.4 depicts the time series of the berminghamsovn (negatives share) sentiment aggregate function. The higher the value of the function the higher is the percentage of negative tweets mention a given political entity in comparison with the other entities. As expected, Pedro Passos Coelho (PSD) as prime-minister is the leader with the higher score throughout the whole time period under study. Paulo Portas (CDS) leader of the other party of the coalition, and also member of the government is the second most negatively mentioned in the period, while António José Seguro (PS) is in some periods the second higher. PSD and CDS are the incumbent parties while PS is the main opposition party in the time frame under study. PSD and CDS as government parties were raising taxes and cutting salaries. PS was the incumbent government during the years that led to the bailout and a fraction of the population considered responsible for the financial crisis. The bailout and the consequent austerity measures could explain the overwhelming percentage of negative mentions although we verified that in other time periods the high percentage of negatives mentions remains. We can say that Twitter users of this sample when mentioning political leaders on their tweets tend to criticize them. Political Opinion Polls The polling was performed by Eurosondagem, a Portuguese private company which collects public opinion. This data set contains the monthly polls results of the five main Portuguese parties, from June 2011 to December 2013. Figure 6.5 represents the evolution of Portuguese polls results. We can see two main party groups: The first group, where both PSD and PS are included, has a higher value of vote intention (above 23%). PSD despite starting as the preferred party in vote intention, has a downtrend along the time, losing the leadership for PS in September 2012. On the other hand, PS has in general an uptrend. The second group, composed by CDS, PCP and BE, has a vote intention range from 5% to 15%. While CDS has a downtrend in public opinion, PCP has an ascendant one. Although the constant tendencies (up and down trends), we noticed that the maximum variation observed between two consecutive months is 3%. In June 2013 there was political crises in the government when CDS threaten to leave the government coalition due to the austerity measures being implemented and corresponds to the moment when PS takes the lead in the polls. Text-based Entity-centric Prediction 112 Fig. 6.5 Representation of the monthly poll results of each political candidate 6.2.3 Experimental Setup We defined the period of 2011 to December 2012 as training set and the whole year of 2013 as test set. We applied a sliding window setting in which we predict the poll results of a given month using the previous 16 months as training set: • Training set – containing the monthly values of the aggregators (both sentiment and buzz aggregator) for 16 months prior the month intended to be predicted. • Test set - containing the values of the aggregators (both sentiment and buzz aggregator) of the month intended to be predicted. We start by predicting the poll results of January 2013 using the previous 16 months as training set: 1. We select the values of the aggregators of the 16 months prior January 2013 (September 2011 to December 2012). 2. We use that data to train our regression model. 3. Then we input the aggregators’ values of January 2013 - the first record of the test set - in the the trained model, to obtain the poll results prediction. 6.2 Predicting Political Polls using Twitter Sentiment 113 4. We select the next month of the test set and repeat the process until all months are predicted. The models are created using two regression algorithms: a linear regression algorithm (Ordinary Least Squares - OLS) and a non-linear regression algorithm (Random Forests - RF). We also run an experiment using the derivative of the polls time series as gold standard, i.e., poll results variations from poll to poll. Thus, we also calculate the variations of the aggregate functions from month to month as features. Furthermore, we repeat each experiment including and excluding the lagged self of the polls, i.e., the last result of the poll for a given candidate (yt−1 ) or the last polls result variation (∆yt−1 ) when predicting polls variations. We use Mean Absolute Error (MAE) as evaluation measure, to determine the absolute error of each prediction. Then, we calculate the average of the twelve MAE’s so we could know the global prediction error of our model. Pn \|fi − yi \| (6.1) n n is the number of forecasts, fi is the model’s forecast and yi the real outcome. M AE = 6.2.4 i=1 Results and Discussion In this section we explain in detail the experiments and their results. We perform two different experiments: (1) using absolute values and (2) using monthly variations. Predicting Polls Results In this experiment, the sentiment aggregators take absolute values in order to predict the absolute values of polls results. Mathematically speaking, this experiment can be seen as: y ← {yt−1 , buzzAggregators, sentimentAggregators}. In Figure 6.6 we see the global errors we obtained. The results show that we obtain a MAE for the 5 parties poll results over 12 months of 6.55% using Ordinary Least Squares and 3.1% using Random Forests. The lagged self of the polls, i.e., assuming the last known poll result as prediction results in a MAE of 0.61 which was expectable since the polls exhibit slight changes from month to month. This experiment shows that the inclusion of the lagged self (yt−1 ) produces average errors similar to the lagged self. Text-based Entity-centric Prediction 114 Fig. 6.6 Error predictions for polls results. Fig. 6.7 Error predictions for polls results variation. Predicting Polls Results Variation According to our exploratory data analysis, the polls results have a small variation between two consecutive months. Thus, instead of predicting the absolute value of poll results, we tried to predict the variation, ∆y ← {∆(yt−1 ), ∆buzzAggregators, ∆sentimentAggregators} In this particular experiment, the inclusion of the ∆yt−1 as feature in the regression model has not a determinant role (Figure 6.7). Including that feature we could not obtain lower MAE than excluding it. It means that the real monthly poll variation is not constant over the year. In general, using a non-linear regression algorithm we obtain lower MAE. The results show that when leading with polls results with slight 6.2 Predicting Political Polls using Twitter Sentiment 115 Fig. 6.8 Mean absolute error buzz vs sentiment. changes from poll to poll it makes sense to transform the dataset by taking differences between consecutive time-steps. Buzz and Sentiment Several studies state that the buzz has predictive power and reflects correctly the public opinion on social media. Following that premise, we trained our models with buzz and sentiment aggregators separately to predict polls variations: • ∆y ← {∆(yt−1 ), ∆buzzAggregators} • ∆y ← {∆(yt−1 ), ∆sentimentAggregators} This experiment allowed us to compare the behavior of buzz and sentiment aggregators. According to Figure 6.8, buzz and sentiment aggregators have similar results. Although the OLS algorithm combined only with buzz aggregators has a slightly lower error than the other models, it is not a significant improvement. These results also show that Random Forests algorithm performs the best when combined only with sentiment aggregators. Feature Selection One of the main goals of our work is to understand which aggregator (or group of aggregators) better suits our case study. According to the previous experiments, we can achieve lower prediction errors when training our model with buzz and sentiment aggregators separately. However, when training our model with these two kinds of aggregators separately, we are implicitly performing feature selection. We only have Text-based Entity-centric Prediction 116 two buzz features (share and total_mentions). Due to that small amount of features, it was not necessary to perform any feature selection technique within buzz features. Thus, we decided to apply a feature selection technique to the sentiment aggregators, in order to select the most informative ones to predict the monthly polls results variation. We use univariate feature selection, selecting 10% of the sentiment features (total of 3 features). Using this technique, the Random Forests’ global error rose from 0.65 to 0.73. However, OLS presents an MAE drop from 0.72 to 0.67. Another important fact to notice is that if we perform univariate feature selection to all aggregators (buzz and sentiment), we will achieve the same MAE value that when applied only to sentiment aggregators. It means that buzz aggregators are discarded by the feature selection technique. We try a different approach and perform a recursive feature elimination technique. In this technique, features are eliminated recursively according to a initial score given by the external estimator. This method allows us to determine the number of features to select. Thus, also selecting 3 features, the OLS’ MAE drops to 0.63. Once again, none of the buzz features were selected. Furthermore, both feature selection techniques select different features for each monthly prediction. 6.2.5 Feature Importance We select the Random Forest model of monthly variations to study the features importance as depicted in Figure 6.9. The higher the score, the more important the feature is. The importance of a feature is computed as the (normalized) total reduction of the criterion brought by that feature. It is also known as the Gini importance. Values correspond to the average of the Gini importance over the different models trained in the experiments. The single most important feature is the bermingham aggregate function, followed by neutrals. It is important to notice that when combining all the aggregate functions as features in a single regression model, the buzz does not comprise a high Gini importance, even though when used as a single feature it produces similar results to the sentiment aggregate functions. In general, the standard deviation of the Gini importance is relatively high. This has to do with our experimental setup, as the values depicted in the bar chart correspond to the average of the Gini importance over 12 different models (12 months of testing set). Therefore, feature importances vary over time while the MAE tends to remain unchanged. We can say that different features have different informative value over time and consequently it is useful to combine all the sentiment aggregation functions as features of the regression models over time. 6.2 Predicting Political Polls using Twitter Sentiment 117 Fig. 6.9 Aggregate functions importance in the Random Forests models. 6.2.6 Outlook We studied a large set of sentiment aggregate functions to use as features in a regression model to predict political opinion poll results. The results show that we can estimate the polls results with low prediction error, using sentiment and buzz aggregators based on the opinions expressed on social media. We introduced a strong baseline for comparison, the lagged self of the polls. In our study, we built a model where we achieve the lowest MAE using the linear algorithm (OLS), combined only with buzz aggregators, using monthly variations. The model has an MAE of 0.63%. We performed two feature selection techniques: (1) univariate feature selection and (2) recursive feature elimination. Applying the recursive technique to the sentiment features, we can achieve an MAE of 0.63, matching our best model. Furthermore, the chosen features are not the same in every prediction. Regarding feature importance analysis 118 Text-based Entity-centric Prediction our experiments showed that bermingham aggregate function represents the highest Gini importance in the Random Forests model. 6.3 Summary of the Contributions In this chapter we presented research work about entity-centric text-based prediction for ORM, making the following contributions: • Analysis of the predictive power of online news regarding entity popularity on Twitter for entities that are frequently mentioned on the news. • Analysis of how to combine different sentiment aggregate functions to serve as features for predicting political polls. Chapter 7 A Framework for Online Reputation Monitoring In this chapter, we present a framework that puts together all the building blocks required to perform ORM. The framework is divided in two distinct components, one is dedicated to Entity Retrieval and the other to Text Mining. In practice these two components can act as two separate frameworks. Both are adaptable and can be reused in different application scenarios, from computational journalism to finance or politics. We start with a framework overview description and then we focus specifically on each of the two components. The first component is RELink, a research framework for E-R retrieval. We carried the experiments on E-R retrieval, described in Chapter 4 using RELink. Furthermore, since we did not have access to training data based on news articles, we describe a case study of using RELink for entity retrieval from a large news collection. We then describe the TexRep framework which is responsible for Text Mining related tasks for ORM, such as Entity Filtering, Sentiment Analysis or Predictive tasks. The experiments described in both Chapter 5 and Chapter 6 were carried out using TexRep. We also provide further detail how TexRep was used as backend of the POPSTAR project. Finally, we perform an independent study of practical aspects of general purpose word embeddings from the Twitter stream to serve as resource for future users of TexRep. 7.1 Framework Overview The framework provides Entity Retrieval and Text Mining functionalities that enable the collection, disambiguation, retrieval of entities and relationships, sentiment analysis, data aggregation, prediction and visualization of entity-centric information from A Framework for Online Reputation Monitoring 120 heterogeneous Web data sources. Furthermore, given that both components are built using modular architectures providing abstraction layers and well defined interfaces, new functionalities or methods can be easily integrated. The framework is divided in two components: RELink and TexRep. Both can work as independently dedicated frameworks using specific data sources or can be put together in a unifying setup for ORM. As depicted in Figure 7.1, when working together, RELink and TexRep are connected through the Entity Occurrences Warehouse. This is the central module of our framework for ORM. The Entity Occurrences Warehouse contains extractions from occurrences of the entities of interest across the Web data sources. RELink (Entity Retrieval) ENTITY OCCURRENCES WAREHOUSE TexRep (Text Mining) Fig. 7.1 High-level overview on the ORM framework. The data flow starts with TexRep collecting data from Web text data sources, extraction of text passages containing entity mentions and disambiguation. Entitycentric text passages are then stored in the Entity Occurrences Warehouse. This data can then be used for E-R retrieval indexing using RELink or for downstream Text Mining tasks (e.g. Sentiment Analysis) using other modules of TexRep. We now describe RELink and TexRep architectures and internal data flow. 7.1.1 RELink The RELink framework is designed to facilitate experiments with E-R Retrieval query collections. The formulation of E-R queries in natural language and relational format (QEi−1 , QRi−1,i , QEi ) provide opportunities to define and explore a range of query formulations and search algorithms. Although, RELink provides support for Late Fusion design patterns, it is mostly tailored for Early Fusion approaches where it is necessary to create entity and relationship representations at indexing time. A typical Early Fusion E-R retrieval experimental setup would involve search over a free-text collection to extract relevant instances of entity tuples and then verify their correctness against the relevance judgments. The key enabling components therefore are: (1) test collections of documents with annotated entity instances that could be 7.1 Framework Overview 121 extracted during E-R search, (2) an indexing facility, and (3) a retrieval module to process queries and rank results. Fig. 7.2 RELink Framework architecture overview. Figure 7.2 depicts the architecture of RELink used in the experiments described in Chapter 4. We include the modules responsible for deriving relevance judgments from Wikipedia. The Table Parser module is described in Section 4.1.2 in Chapter 4. Currently, the RELink Framework includes the ClueWeb-09-B1 collection combined with FACC1[174] text span annotations with links to Wikipedia entities (via Freebase). The entity linking precision and recall in FACC1 are estimated at 85% and 70-85%, respectively [174]. The RELink Extractor, part of E-R Indexer, applies an Open Information Extraction method [57] over the annotated ClueWeb-09-B corpus. The two additional components are Corpus E-R Index and E-R Retrieval, both depicted in Figure 7.2. The implementation of all modules in E-R Retrieval and the Indexer 1 http://www.lemurproject.org/clueweb09/ A Framework for Online Reputation Monitoring 122 module in Corpus E-R Index are based on Apache Lucene and the Letor module serves as a wrapper for RankLib2 . Indexing and Retrieval Based on the ClueWeb-09-B collection we create two essential resources: entity index and entity pair relationship index for the entities that occur in the corpus. For a given entity instance, the ER Indexer identifies co-occuring terms within the same sentence and considers them as entity types for the observed entity instance. Similarly, for a given pair of entities, the ER Indexer verifies whether they occur in the same sentence and extracts the separating string. That string is considered a context term for the entity pair that describes their relationship type. We obtain 476M entity and 418M entity pair extractions with corresponding sentences that are processed by the Indexer. Once the inverted index (ER Index) is created, any instance of an entity or entity pair can be retrieved in response to the contextual terms, i.e., entity types and relationship types, specified by the users. Search Process The E-R retrieval process is managed by the RELinker module (Figure 7.2). The Query Analyzer module processes information requests and passes queries in the structured format to the Retriever. Query search is performed in stages to allow for experimentation with different methods and parameter settings. First, the Retriever provides an initial set of results using Lucene’s default search settings and groups them by entity or entity pairs on query time using the Lucene’s GroupingSearch. The Scorer then generates and applies feature functions of specific retrieval models with required statistics. Currently, the Scorer has implementations for Early Fusion variants EF-LM, EF-BM25 and ERDM. The RELinker is responsible for re-ranking and providing final results based on the scores provided by the Scorer and the parameter weights learned by Letor. 7.1.2 TexRep TexRep is a research framework that implements Text Mining techniques to perform Online Reputation Monitoring (ORM) in various application domains, such as computational social sciences, political data science, computational journalism, computational finance or online marketing. 2 http://www.lemurproject.org/ranklib.php 7.1 Framework Overview 123 TexRep was designed with two main challenges in mind: 1) it should be able to cope with the Text Mining problems underlying ORM and 2) it should be flexible, adaptable and reusable in order to support the specificities of different application scenarios. We define that a Text Mining based system for Online Reputation Monitoring must follow a set of technical and operational requirements: • Batch and real-time operation: such a system must naturally be able to operate in real-time, i.e. collecting data as it is generated, processing it and updating indicators. However, it is also important to be able to operate in batch mode, in which it collects specific data from a period indicated by the user, if available, and then processes it. The system should use a distributed approach to deal with great volumes of data, (e.g. Hadoop). It should also be able to operate autonomously for long periods of time, measured in months. • Adaptability: the system should be able to adapt its models (e.g. polarity classification) through time as well as across different applications. Updating models often requires manually annotated data (e.g. NED). Therefore the system should provide a flexible annotation interface. • Modularity: researchers should be able to plug in specific modules, such as a new data source and respective crawler or a different visualization. The system interfaces should use REST APIs and JSON data format, which allow users to add new modules that interact with other data sources (e.g. Wikipedia or Facebook). • Reusability: the system should enable repeatability of all experiments to allow the research community to obtain equal results. We will make the software package of a prototype publicly available as well as the data sources and configuration parameters used in experiments. • Language independence: each component of the system should apply a statistical language modeling completely agnostic to the language of the texts. We decompose the use of Text Mining for ORM into four distinct but interconnected tasks: Data Collection, Entity Filtering, Sentiment Analysis and Analytics. Each task is accomplished by one or more software modules. For instance, Analytics tasks usually involves the use of the Aggregation, Prediction and Visualization modules. Figure 7.3 presents the TexRep architecture, including the data flow between modules. A Framework for Online Reputation Monitoring 124 PIPELINE MANAGER SENTIMENT ANALYSIS CONFIGURATIONS TRAINING DATA DATA COLLECTION SERVER DATA COLLECTION CLIENT VISUALIZATION DATA COLLECTION CLIENT ENTITY OCCURRENCES WAREHOUSE ENTITY FILTERING AGGREGATION PREDICTION TEXREP DATA SOURCES KNOWLEDGE BASE Fig. 7.3 Architecture and data flows of the TexRep framework. Entity Filtering and Sentiment Analysis represent the most challenging Text Mining problems tackled in the TexRep framework. When tracking what is being said online about the target entities it is necessary to disambiguate mentions. When this is done incorrectly, the knowledge obtained by the other modules is negatively affected. Consequently, other Text Mining tasks, such as Sentiment Analysis, will benefit from filtering non relevant texts. The current implementation of the Entity Filtering module uses the scikit-learn Python library as the Machine Learning library interface, providing access to TexRep users to the most suitable learning algorithm and parameter tuning for their specific needs. We studied a large set of features that describe the relationship between the target entity representation and a given text and we tried several different supervised learning algorithms that are available through the framework, such as Support Vector Machines (SVM) and Random Forests (RF). The Sentiment Analysis module also uses scikit-learn implementation of supervised learning algorithms in order to predict sentiment polarity and intensity in short texts 7.1 Framework Overview 125 using regression analysis. We use unsupervised learning of word embeddings [182] in short texts to construct syntactic and semantic representations of words. The Sentiment Analysis module combines word embeddings with traditional approaches, such as pre-processing techniques, bag-of-words and lexical-based features to train a classifier for sentiment polarity and a regressor for sentiment intensity. Analytics modules include Aggregation, Visualization and Prediction. These modules are application specific and depend on user configurations. For instance, in the political domain it is common to create aggregate functions that represent relative popularity indicators between political parties or candidates. These indicators are then used to predict elections. On the other hand, if we consider the financial domain, due to its high volatility, aggregation is usually performed with lower granularity (minutes instead of days) and target prediction variables are individual stock prices or variations. TexRep implements various aggregation functions and allows custom plug-in of tailored prediction models based on each application. Therefore, TexRep is able to adapt itself to the specificities of different application scenarios by implementing a modular and flexible design through user configurations and abstraction layers. Data Collection depends on the specified data sources, thus TexRep decouples client-side implementations from the data collection process management using a REST API. If a user needs a different Data Collection Client from the ones provided by default, she is able to implement a specific client that is easily integrated into the framework. The same applies to the Analytics modules which are extensible by loading user-implemented methods through an abstraction layer. Furthermore, if users wish to extend TexRep with Topic Modeling, they only need to plug-in the new module and write topic assignments through the Entity Occurrences Warehouse. New aggregation functions could be implemented that use the topic of each mention as input in order to create entity-centric topic trends visualizations. The framework can be fully configured using configuration files that are processed in the Pipeline Manager, which is the module responsible for forwarding specific parameterization to the other modules. It is possible to specify the entities of interest, data sources, aggregate functions and prediction time windows. Module specific configurations are also specified in this module, such as which training data should be used by the modules that rely on machine learning. As explained, TexRep addresses the two aforementioned challenges of developing a Text Mining framework for ORM. The current version of the framework is implemented in Python, uses MongoDB as NoSQL database and implements the MapReduce paradigm for aggregations. The external and pluggable resources used are the scikit- 126 A Framework for Online Reputation Monitoring learn library and the matplotlib for visualization, though users can replace these two resources by others of their preference. We provide the implementations of each module that we believe are the most generic as possible within the context of ORM. Nevertheless, users are also able to extend each module with the methods they see fit, such as, new features or data pre-processing steps. We now describe in detail how the different modules interact with each other, as well as, a detailed explanation of the current implementation of the Entity Filtering, Sentiment Analysis and Analytics modules. Data Flow TexRep collects data continuously and performs mini-batch processing and analytics tasks. The standard data flow is organized as follows. First the user defines the entities of interest in the configurations files, including canonical and alternative names. These configurations are processed by the Pipeline Manager and forwarded to the Data Collection clients to search for texts (e.g. news articles and tweets) using entity names as queries on each data source-specific API. The Data Collection Clients implement source-specific API clients, such as the case of Twitter and Yahoo Finance, for instance. If the user is interested in collecting RSS feeds of news outlets, then the Data Collection Client can be adapted to subscribe to those feeds and process them accordingly. Once collected, texts are stored in the Entity Occurrences Warehouse. Entity Filtering classifies each text as relevant or not for each target entity using a supervised learning approach. A knowledge base (e.g. Freebase) is used to extract target entity representations and to compute similarity features with extracted mentions contexts. Once the non-relevant texts are filtered, Sentiment Analysis takes place. The framework implements both polarity classification and sentiment regression for sentiment intensity detection. Then, Analytics modules are able to aggregate and create visualizations of trends in data or predictions of application specific dependent variables. Data Collection The Data Collection Server communicates with each Data collection Client using a REST API and therefore it allows modularity and a plugin approach for adapting to specific data sources. The task of data collection is based on user-defined entity configurations containing the list of entities under study. Each data source has specific web interfaces (e.g. RSS feeds, Yahoo Finance API or Twitter API). The Data Collection Server manages the Data Collection Clients through specific interfaces (plugins) that are adequate for the corresponding source. For instance, collecting data 7.2 RELink Use Case 127 from Twitter poses some challenges, namely due to the limits on the amount of data collected. We opted to create by default a Data Collection Client for SocialBus [192], a distributed Twitter client that enables researchers to continuously collect data from particular user communities or topics, while respecting the established limits. Some data sources allow query by topics (e.g. entity names) while others do not (e.g. RSS feeds). Moreover, in the case of Twitter, we might be interested in continuously monitoring a fixed group of Twitter users (e.g., the accounts of the entities of interest). In such cases, when we cannot search directly by entity name in the specific data source, we use the list of entity names to process collected texts that might be relevant. The Data Collection Server applies a sequential classification approach using a prefix tree to detect mentions. This method can be seen as first step of filtering but it is still prone to noisy mentions. For instance, a tweet with the word “Cameron” can be relative to several entities, such as a former UK prime minister, a filmmaker or a company. Consequently, this problem is later tackled by the Entity Filtering module. Collected texts (e.g. news or tweets) are stored in a centralized document-oriented NoSQL database (e.g. MongoDB), the Entities Occurrence Warehouse. This setup provides modularity and flexibility, allowing the possibility of developing specific data collection components tailored to specific data sources and is completely agnostic to the data format retrieved from each data source. The Data Collection Server annotates each text with the target entity which will be used by the Entity Filtering module to validate that annotation. 7.2 RELink Use Case In this section we present a use case of the RELink framework in the context of ORM applied to computational journalism. Never before has computation been so tightly connected with the practice of journalism. In recent years, the computer science community has researched [193–196, 36, 197–199] and developed3 new ways of processing and exploring news archives to help journalists perceiving news content with an enhanced perspective. We created a demo the TimeMachine, that brings together a set of Natural Language Processing, Text Mining and Information Retrieval technologies to automatically extract and index entity related knowledge from the news articles [177, 200, 36, 201, 197–199]. It allows users to issue queries containing keywords and phrases about news stories or events, and retrieves the most relevant entities mentioned in the news articles through 3 NewsExplorer (IBM Watson): http://ibm.co/1OsBO1a 128 A Framework for Online Reputation Monitoring time. TimeMachine provides readable and user-friendly insights and a temporal perspective of news stories and mentioned entities. It visually represents relationships among public figures co-mentioned in news articles as a social network graph, using a force atlas algorithm layout [202] for the interactive and real-time clustering of entities. 7.2.1 News Processing Pipeline The news processing pipeline, depicted in Figure 7.4, starts with a news cleaning module which performs the boilerplate removal from the raw news files (HTML/XML). Once the news content is processed we apply the NERD module which recognizes entity mentions and disambiguates each mention to an entity using a set of heuristics tailored for news, such as job descriptors (e.g. “Barack Obama, president of USA”) and linguistic patterns well defined for the journalistic text style. We use a bootstrap approach to train the NER system [201]. Our method starts by annotating entity names on a dataset of 50,000 news items. This is performed using a simple dictionarybased approach. Using such training set we build a classification model based on Conditional Random Fields (CRF). We then use the inferred classification model to perform additional annotations of the initial seed corpus, which is then used for training a new classification model. This cycle is repeated until the NER model stabilizes. The Fig. 7.4 News processing pipeline. entity snippet extraction consists of collecting sentences containing mentions to a given entity. All snippets are concatenated generating an entity document, which is then indexed in the entity index. The entity index represents the frequency of co-occurrence of each entity with each term that it occurs with in the news. Therefore, by relying on the redundancy of news terms and phrases associated with an entity we are able to retrieve the most relevant entity to a given input keyword or phrase query. As we also index the snippet datetime it is possible to filter query results based on a time span. For instance, the keyword “corruption” might retrieve a different entity list results in different time periods. Quotations are typically short and very informative sentences, which may directly or indirectly quote a given entity. Quotations are automatically 7.2 RELink Use Case 129 extracted (refer to "Quotations Extraction" module) using linguistic patterns, thus enriching the information extracted for each entity. Finally, once we have all mentioned entities in a given news articles we extract entity tuples representing co-occurrences of entities in a given news article and update the entity graph by incrementing the number of occurrences of a node (entity) and creating/incrementing the number of occurrences of the edge (relation) between any two mentions. 7.2.2 Demonstration The setup for demonstration uses a news archive of Portuguese news. It comprises two different datasets: a repository from the main Portuguese news agency (1990-2010), and a stream of online articles provided by the main web portal in Portugal (SAPO) which aggregates news articles from 50 online newspapers. The total number of news articles used in this demonstration comprises over 12 million news articles. The system is working on a daily basis, processing articles as they are collected from the news stream. TimeMachine allows users to explore its news archive through an entity search box or by selecting a specific date. Both options are available on the website homepage and in the top bar on every page. There are a set of “stories” recommendations on the homepage suited for first time visitors. The entity search box is designed to be the main entry point to the website as it is connected to the entity retrieval module of TimeMachine. Fig. 7.5 Cristiano Ronaldo egocentric network. Users may search for surface names of entities (e.g. “Cristiano Ronaldo”) if they know which entities they are interested to explore in the news, although the most 130 A Framework for Online Reputation Monitoring powerful queries are the ones containing keywords or phrases describing topics or news stories, such as “Eurozone crisis” or “Ballon d’Or nominees”. When selecting an entity from the ranked list of results, users access the entity profile page which contains a set of automatically extracted entity specific data: name, profession, a set of news articles, quotations from the entity and related entities. An entity timeline is also provided to allow users to navigate entity specific data through time. By selecting a specific period, different news articles, quotations and related entities are retrieved. Furthermore, users have the option of “view network” which consists in a interactive network depicting connections among entities co-mentioned in news articles for the selected time span. An example of such visualization is depicted in Figure 7.5, and it is implemented using the graph drawing library Sigma JS, together with "Force Atlas" algorithm for the clustered layout of entities. Nodes consist of entities and edges represent a co-occurrence of mentioned entities in the same news articles. The size of the nodes and the width of edges is proportional to the number of mentions and co-occurrences, respectively. Different node colors represent specific news topics where entities were mentioned. By selecting a date interval on the homepage, instead of issuing a query, users get a global interactive network of mentions and co-occurrences of the most frequent entities mentioned in the news articles for the selected period of time. 7.3 TexRep Use Case This section describes the design and implementation of the POPmine system, an use case of the proposed framework, developed in the scope of the POPSTAR project. It is an open source platform which can be used and extended by researchers interested in tracking reputation of political entities on the Web. POPmine operates either in batch or online mode and is able: to collect texts from web-based conventional media (news items in mainstream media sites) and social media (blogs and Twitter); to process those texts, recognizing topics and political entities; to analyze relevant linguistic units; to generate indicators of both frequency of mention and polarity (positivity/negativity) of mentions to political entities across sources, types of sources, and across time. As a proof of concept we present these indicators in a web application tailored for tracking political opinion in Portugal, the POPSTAR website. The system is available as an open source software package that can be used by other researchers from social sciences but also from any other area that is interested in tracking public opinion on the web. 7.3 TexRep Use Case 131 We opted to use data from news articles, tweets and blog posts and each of these data sources requires its specific crawler. News articles and blog posts are collected using RSS feeds which eases the implementations of a specific crawler. Collecting data from Twitter poses some challenges. The need for large amounts of data, coupled with Twitter’s imposed limits demand for a distributed system. We opted to use SocialBus4 which enables researchers to continuously collect data from particular user communities, while respecting Twitter’s imposed limits. The data collection components crawl data from specific data sources which implement specific web interfaces (e.g. RSS feeds, Twitter API). Each data source must have its own data collection module which in turn connects to the POPmine system using REST services. POPmine stores data collected in a document oriented NoSQL database (MongoDB). This configuration allows modularity and flexibility, allowing the possibility of developing specific data collection components tailored to specific data sources. The default setting of data collection modules comprise the following components: • News: Data from online news are provided by the service Verbetes e Notícias from Labs Sapo. This service handles online news from over 60 Portuguese news sources and is able to recognize entities mentioned in the news. • Blogs: Blog posts are provided by the blogs’ monitoring system from Labs Sapo, which includes all blogs with domain sapo.pt, blogspot.pt (Blogger) and Wordpress (blogs written in Portuguese). • Twitter: Tweets are collected using the platform SocialBus, responsible for the compilation of messages from 100.000 Portuguese users of Twitter. Tweets are collected in real time and submitted to a language classification. In our experiments we opted to collect the tweets written in Portuguese. The information extraction component comprises a knowledge base containing metadata about entities, e.g., names or jobs. Using a knowledge base is crucial to filter relevant data mentioning politicians, such as news, tweets and blog posts. In our application scenario, we opted to use Verbetes, a knowledge base which comprises names, alternative names, and professions of Portuguese people mentioned often in news articles. The Information Extraction components address two tasks: Named Entity Recognition and Named Entity Disambiguation. We envision an application scenario where we 4 http://reaction.fe.up.pt/socialbus/ A Framework for Online Reputation Monitoring 132 need to track political entities. Usually this type of entities are well known therefore we opted to use a knowledge base to provide metadata about the target entities, namely the most common surface forms of their names. Once we had the list of surface forms to search for we applied a sequential classification approach using a prefix tree to detect mentions. This method is very effective on news articles and blog posts but can result in noisy mentions when applied to Twitter. For instance, a tweet containing the word “Cameron” can be related with several entities, such as the former UK prime minister, a filmmaker or a company. Furthermore, tweets are short which results in a reduced context for entity disambiguation. We then apply the Entity Filtering approach of TexRep. The opinions warehouse contains the messages filtered by the information extraction component and applies polarity classification to those messages using an external resource - the Opinionizer classifier [203]. One of the requirements of the Opinionizer is to use manually labeled data to train the classifier. We developed an online annotation tool for that effect. We create opinion and polls indicators using the aggregator which is responsible to apply aggregation functions and smoothing techniques. Once we obtain the aggregated data we make available a set of web services that can be consumed by different applications such as the POPSTAR website or other research experiences, such as polls predictions using social media opinions. 7.3.1 Data Aggregation Buzz is the daily frequency with which political leaders are mentioned by Twitter users, bloggers and online media news. We use two types of indicators. The first type is the relative frequency with which party leaders are mentioned by each medium (Twitter, Blogs and News), on each day. This indicator is expressed, for each leader of each party, as a percentage relative to the total number of mentions to all party leaders. The second indicator is the absolute frequency of mentions, a simple count of citations for each political leader. To estimate trends in Buzz, we use the Kalman Filter. We allow users to choose the smoothing degree for each estimated trend. Users can choose between three alternatives: a fairly reactive one, where trend is highly volatile, allowing close monitoring of dayby-day variations; a very smooth one, ideal to capture long term trends; and an intermediate option, displayed by default. After identifying the polarity in each of the tweets, there are several ways to quantify the overall sentiment regarding political leaders. We can, for instance, look at each 7.3 TexRep Use Case 133 target independently or in relative terms, compare positive with negative references or simply look at one side of the polarity, or look at daily, weekly or monthly data records. In this first prototype we opted to present two separate indicators and their evolution across time, using in both cases the day as reference period. The fist indicator is the logarithm of the ratio of positive and negative tweets by political leader (party leaders and the president). In other words, a positive sign means that the political leader under consideration received more positive than negative tweets that day, while a negative result means that he received more negative than positive tweets. In mathematical notation: logsentimenti = log( positivesi + 1 ) negativesi + 1 The second approach is to simply look at the negative tweets (the vast majority of tweets in our base classifier) and calculate their relative frequency for each leader. In this way it is possible to follow each day which party leaders were, in relative terms, more or less subject to tweets with negative polarity. In mathematical notation: negativesi,d negativesshare = P negativesd Fig. 7.6 Twitter buzz share of political leaders. A Framework for Online Reputation Monitoring 134 7.3.2 Visualization We created a website5 to allow interactive visualization of the data collected and processed in real time by the POPmine platform. The site was developed within the scope of the POPSTAR project (Public Opinion and Sentiment Tracking, Analysis, and Research) and presents the following data: a) mentions to Portuguese party leaders in Twitter, in the blogosphere and in online news; b) sentiment conveyed through tweets regarding party leaders, c) voting intentions for the main political parties, measured by traditional polls; and d) evaluation of the performance of said party leaders, measured by polls. An example chart is depicted in Figure 7.6. Besides providing our indicators in the form of charts, the website also has a dashboard offering a more compact view of trends across indicators for all politicians. 7.4 Learning Word Embeddings for ORM6 Word embeddings have great practical importance since they can be used as precomputed high-density features to ML models, significantly reducing the amount of training data required in a variety of Text Mining tasks. We aim to provide general purpose pre-trained word embeddings for the Text Mining tasks in ORM. We are particularly interested in learning word embeddings from the Twitter stream due to the specificities of user generated content. It is relatively easy to get access to word embeddings trained from well formed texts such as Wikipedia or online news. However, to the best of our knowledge there are no publicly available word embeddings learned from the Portuguese Twitter stream. There are several inter-related challenges with computing and consistently distributing word embeddings concerning the: • intrinsic properties of the embeddings. How many dimensions do we actually need to store all the “useful" semantic information? How big should the embedded vocabulary be to have practical value? How do these two factors interplay? • type of model used for generating the embeddings. There are multiple possible models and it is not obvious which one is the “best", both in general or in the context of a specific type of applications. 5 6 http://www.popstar.pt The material contained in this section was published in P. Saleiro, L. Sarmento, E. M. Rodrigues, C. Soares, E. Oliveira, “Learning Word Embeddings from the Portuguese Twitter Stream: A Study of some Practical Aspects” [17]. 7.4 Learning Word Embeddings for ORM 135 • the size and properties of training data: What is the minimum amount of training data needed? Should we include out of vocabulary words in the training? • optimization techniques to be used, model hyperparameter and training parameters. Not only the space of possibilities for each of these aspects is large, there are also challenges in performing a consistent large-scale evaluation of the resulting embeddings [204]. This makes systematic experimentation of alternative word-embedding configurations extremely difficult. In this work, we make progress in trying to find good combinations of some of the previous parameters. We focus specifically in the task of computing word embeddings for processing the Portuguese Twitter stream. User-generated content (such as twitter messages) tends to be populated by words that are specific to the medium, and that are constantly being added by users. These dynamics pose challenges to NLP systems, which have difficulties in dealing with out of vocabulary words. Therefore, learning a semantic representation for those words directly from the user-generated stream - and as the words arise - would allow us to keep up with the dynamics of the medium and reduce the cases for which we have no information about the words. Starting from our own implementation of a neural word embedding model, which should be seen as a flexible baseline model for further experimentation, our research tries to answer the following practical questions: • how large is the vocabulary the one can realistically embed given the level of resources that most organizations can afford to buy and to manage (as opposed to large clusters of GPU’s only available to a few organizations)? • how much data, as a function of the size of the vocabulary we wish to embed, is enough for training meaningful embeddings? • how can we evaluate embeddings in automatic and consistent way so that a reasonably detailed systematic exploration of the previously describe space of possibilities can be performed? By answering these questions based on a reasonably small sample of Twitter data (5M), we hope to find the best way to proceed and train embeddings for Twitter vocabulary using the much larger amount of Twitter data available (300M), but for which parameter experimentation would be unfeasible. This work can thus be seen as a preparatory study for a subsequent attempt to produce and distribute a large-scale database of embeddings for processing Portuguese Twitter data. 136 7.4.1 A Framework for Online Reputation Monitoring Neural Word Embedding Model The neural word embedding model we use is the Continuous Bag-of-words (CBOW) [182]. Given a sequence of 5 words - wi−2 wi−1 wi wi+1 wi+2 , the task the model tries to perform is that of predicting the middle word, wi , based on the two words on the left - wi−2 wi−1 - and the two words on the right - wi+1 wi+2 : P (wi \|wi−2 , wi−1 , wi+1 , wi+2 ). This should produce embeddings that closely capture distributional similarity, so that words that belong to the same semantic class, or which are synonyms and antonyms of each other, will be embedded in “close” regions of the embedding hyper-space. The neural model is composed of the following layers: • a Input Word Embedding Layer, that maps each of the 4 input words represented by a 1-hot vectors with \|V \| dimensions (e.g. 32k) into a low dimension space (64 bits). The projections matrix - Winput - is shared across the 4 inputs. This is not be the embedding matrix that we wish to produce. • a Merge Layer that concatenates the 4 previous embeddings into a single vector holding all the context information. The concatenation operation ensures that the rest of the model has explicit information about the relative position of the input words. Using an additive merge operation instead would preserve information only about the presence of the words, not their sequence. • a Intermediate Context Embedding Dense Layer that maps the preceding representation of 4 words into a lower dimension space, still representing the entire context. We have fixed this context representation to 64 dimensions. This ultimately determines the dimension of the resulting embeddings. This intermediate layer is important from the point of view of performance because it isolates the still relatively high-dimensional input space (4 x 64 bits input word embeddings) from the very high-dimensional output space. • a final Output Dense Layer that maps the takes the previous 64-bit representation of the entire input context and produces a vector with the dimensionality of the word output space (\|V \| dimensions). This matrix - Woutput - is the one that stores the word embeddings we are interested in. • A Softmax Activation Layer to produces the final prediction over the word space, that is the P (wi \|wi−2 , wi−1 , wi+1 , wi+2 ) distribution 7.4 Learning Word Embeddings for ORM 137 All neural activations in the model are sigmoid functions. The model was implemented using the Syntagma7 library which relies on Keras [205] for model development, and we train the model using the built-in ADAM [206] optimizer with the default parameters. 7.4.2 Experimental Setup We are interested in assessing two aspects of the word embedding process. On one hand, we wish to evaluate the semantic quality of the produced embeddings. On the other, we want to quantify how much computational power and training data are required to train the embedding model as a function of the size of the vocabulary \|V \| we try to embed. These aspects have fundamental practical importance for deciding how we should attempt to produce the large-scale database of embeddings we will provide in the future. All resources developed in this work are publicly available8 . Apart from the size of the vocabulary to be processed (\|V \|), the hyperparamaters of the model that we could potentially explore are i) the dimensionality of the input word embeddings and ii) the dimensionality of the output word embeddings. As mentioned before, we set both to 64 bits after performing some quick manual experimentation. Full hyperparameter exploration is left for future work. Our experimental testbed comprises a desktop with a nvidia TITAN X (Pascal), Intel Core Quad i7 3770K 3.5Ghz, 32 GB DDR3 RAM and a 180GB SSD drive. Training Data We randomly sampled 5M tweets from a corpus of 300M tweets collected from the Portuguese Twitter community [192]. The 5M comprise a total of 61.4M words (approx. 12 words per tweets in average). From those 5M tweets we generated a database containing 18.9M distinct 5-grams, along with their frequency counts. In this process, all text was down-cased. To help anonymizing the n-gram information, we substituted all the twitter handles by an artificial token “T_HANDLE". We also substituted all HTTP links by the token “LINK". We prepended two special tokens to complete the 5-grams generated from the first two words of the tweet, and we correspondingly appended two other special tokens to complete 5-grams centered around the two last tokens of the tweet. Tokenization was perform by trivially separating tokens by blank space. No linguistic pre-processing, such as for example separating punctuation from words, was made. We 7 8 https://github.com/sarmento/syntagma https://github.com/saleiro/embedpt A Framework for Online Reputation Monitoring 138 Table 7.1 Number of 5-grams available for training for different sizes of target vocabulary \|V \| \|V \| # 5-grams 2048 2,496,830 8192 6,114,640 32768 10,899,570 opted for not doing any pre-processing for not introducing any linguistic bias from another tool (tokenization of user generated content is not a trivial problem). The most direct consequence of not performing any linguistic pre-processing is that of increasing the vocabulary size and diluting token counts. However, in principle, and given enough data, the embedding model should be able to learn the correct embeddings for both actual words (e.g. “ronaldo") and the words that have punctuation attached (e.g. “ronaldo!"). In practice, we believe that this can actually be an advantage for the downstream consumers of the embeddings, since they can also relax the requirements of their own tokenization stage. Overall, the dictionary thus produced contains approximately 1.3M distinct entries. Our dictionary was sorted by frequency, so the words with lowest index correspond to the most common words in the corpus. We used the information from the 5-gram database to generate all training data used in the experiments. For a fixed size \|V \| of the target vocabulary to be embedded (e.g. \|V \| = 2048), we scanned the database to obtain all possible 5-grams for which all tokens were among the top \|V \| words of the dictionary (i.e. the top \|V \| most frequent words in the corpus). Depending on \|V \|, different numbers of valid training 5-grams were found in the database: the larger \|V \| the more valid 5-grams would pass the filter. The number of examples collected for each of the values of \|V \| is shown in Table 7.1. Since one of the goals of our experiments is to understand the impact of using different amounts of training data, for each size of vocabulary to be embedded \|V \| we will run experiments training the models using 25%, 50%, 75% and 100% of the data available. Metrics Related with the Learning Process We tracked metrics related to the learning process itself, as a function of the vocabulary size to be embedded \|V \| and of the fraction of training data used (25%, 50%, 75% 7.4 Learning Word Embeddings for ORM 139 and 100%). For all possible configurations, we recorded the values of the training and validation loss (cross entropy) after each epoch. Tracking these metrics serves as a minimalistic sanity check: if the model is not able to solve the word prediction task with some degree of success (e.g. if we observe no substantial decay in the losses) then one should not expect the embeddings to capture any of the distributional information they are supposed to capture. Tests and Gold-Standard Data for Intrinsic Evaluation Using the gold standard data (described below), we performed three types of tests: • Class Membership Tests: embeddings corresponding to members of the same semantic class (e.g. “Months of the Year", “Portuguese Cities", “Smileys") should be close, since they are supposed to be found in mostly the same contexts. • Class Distinction Test: this is the reciprocal of the previous Class Membership test. Embeddings of elements of different classes should be different, since words of different classes ere expected to be found in significantly different contexts. • Word Equivalence Test: embeddings corresponding to synonyms, antonyms, abbreviations (e.g. “porque" abbreviated by “pq") and partial references (e.g. “slb and benfica") should be almost equal, since both alternatives are supposed to be used be interchangeable in all contexts (either maintaining or inverting the meaning). Therefore, in our tests, two words are considered: • distinct if the cosine of the corresponding embeddings is lower than 0.70 (or 0.80). • to belong to the same class if the cosine of their embeddings is higher than 0.70 (or 0.80). • equivalent if the cosine of the embeddings is higher that 0.85 (or 0.95). We report results using different thresholds of cosine similarity as we noticed that cosine similarity is skewed to higher values in the embedding space, as observed in related work [207, 208]. We used the following sources of data for testing Class Membership: • AP+Battig data. This data was collected from the evaluation data provided by [116]. These correspond to 29 semantic classes. 140 A Framework for Online Reputation Monitoring • Twitter-Class - collected manually by the authors by checking top most frequent words in the dictionary and then expanding the classes. These include the following 6 sets (number of elements in brackets): smileys (13), months (12), countries (6), names (19), surnames (14) Portuguese cities (9). For the Class Distinction test, we pair each element of each of the gold standard classes, with all the other elements from other classes (removing duplicate pairs since ordering does not matter), and we generate pairs of words which are supposed belong to different classes. For Word Equivalence test, we manually collected equivalente pairs, focusing on abbreviations that are popular in Twitters (e.g. “qt" ≃ “quanto" or “lx" ≃ “lisboa" and on frequent acronyms (e.g. “slb" ≃ “benfica"). In total, we compiled 48 equivalence pairs. For all these tests we computed a coverage metric. Our embeddings do not necessarily contain information for all the words contained in each of these tests. So, for all tests, we compute a coverage metric that measures the fraction of the goldstandard pairs that could actually be tested using the different embeddings produced. Then, for all the test pairs actually covered, we obtain the success metrics for each of the 3 tests by computing the ratio of pairs we were able to correctly classified as i) being distinct (cosine < 0.7 or 0.8), ii) belonging to the same class (cosine > 0.7 or 0.8), and iii) being equivalent (cosine > 0.85 or 0.95). It is worth making a final comment about the gold standard data. Although we do not expect this gold standard data to be sufficient for a wide-spectrum evaluation of the resulting embeddings, it should be enough for providing us clues regarding areas where the embedding process is capturing enough semantics, and where it is not. These should still provide valuable indications for planning how to produce the much larger database of word embeddings. 7.4.3 Results and Analysis We run the training process and performed the corresponding evaluation for 12 combinations of size of vocabulary to be embedded, and the volume of training data available that has been used. Table 7.2 presents some overall statistics after training for 40 epochs. The average time per epoch increases first with the size of the vocabulary to embed \|V \| (because the model will have more parameters), and then, for each \|V \|, with the volume of training data. Using our testbed (Section 7.4.2), the total time of learning in our experiments varied from a minimum of 160 seconds, with \|V \| = 2048 and 25% 7.4 Learning Word Embeddings for ORM 141 Table 7.2 Overall statistics for 12 combinations of models learned varying \|V \| and volume of training data. Results observed after 40 training epochs. Embeddings # Training Data Tuples Avg secs/epoch Training loss Validation loss \|V \| = 2048 561,786 (25% data) 4 3.2564 3.5932 \|V \| = 2048 1,123,573 (50% data) 9 3.2234 3.4474 \|V \| = 2048 1,685,359 (75% data) 13 3.2138 3.3657 \|V \| = 2048 2,496,830 (100% data) 18 3.2075 3.3074 \|V \| = 8192 1,375,794 (25% data) 63 3.6329 4.286 \|V \| = 8192 2,751,588 (50% data) 151 3.6917 4.0664 \|V \| = 8192 4,127,382 (75% data) 187 3.7019 3.9323 \|V \| = 8192 6,114,640 (100% data) 276 3.7072 3.8565 \|V \| = 32768 2,452,402 (25% data) 388 3.7417 5.2768 \|V \| = 32768 4,904,806 (50% data) 956 3.9885 4.8409 \|V \| = 32768 7,357,209 (75% data) 1418 4.0649 4.6 \|V \| = 32768 10,899,570 (100% data) 2028 4.107 4.4491 of data, to a maximum of 22.5 hours, with \|V \| = 32768 and using 100% of the training data available (extracted from 5M tweets). These numbers give us an approximate figure of how time consuming it would be to train embeddings from the complete Twitter corpus we have, consisting of 300M tweets. We now analyze the learning process itself. We plot the training set loss and validation set loss for the different values of \|V \| (Figure 7.7 left) with 40 epochs and using all the available data. As expected, the loss is reducing after each epoch, with validation loss, although being slightly higher, following the same trend. When using 100% we see no model overfitting. We can also observe that the higher is \|V \| the higher are the absolute values of the loss sets. This is not surprising because as the number of words to predict becomes higher the problem will tend to become harder. Also, because we keep the dimensionality of the embedding space constant (64 dimensions), it becomes increasingly hard to represent and differentiate larger vocabularies in the same hyper-volume. We believe this is a specially valuable indication for future experiments and for deciding the dimensionality of the final embeddings to distribute. On the right side of Figure 7.7 we show how the number of training (and validation) examples affects the loss. For a fixed \|V \| = 32768 we varied the amount of data used for training from 25% to 100%. Three trends are apparent. As we train with more data, we obtain better validation losses. This was expected. The second trend is that by using less than 50% of the data available the model tends to overfit the data, as indicated by the consistent increase in the validation loss after about 15 epochs (check 142 A Framework for Online Reputation Monitoring Fig. 7.7 Continuous line represents loss in the training data while dashed line represents loss in the validation data. Left side: effect of increasing \|V \| using 100% of training data. Right side: effect of varying the amount of training data used with \|V \| = 32768. dashed lines in right side of Figure 7.7). This suggests that for the future we should not try any drastic reduction of the training data to save training time. Finally, when not overfitting, the validation loss seems to stabilize after around 20 epochs. We observed no phase-transition effects (the model seems simple enough for not showing that type of behavior). This indicates we have a practical way of safely deciding when to stop training the model. Intrinsic Evaluation Table 7.3 presents results for the three different tests described in Section 7.4.2. The first (expected) result is that the coverage metrics increase with the size of the vocabulary being embedded, i.e., \|V \|. Because the Word Equivalence test set was specifically created for evaluating Twitter-based embedding, when embedding \|V \| = 32768 words we achieve almost 90% test coverage. On the other hand, for the Class Distinction test set - which was created by taking the cross product of the test cases of each class in Class Membership test set - we obtain very low coverage figures. This indicates that it is not always possible to re-use previously compiled gold-standard data, and that it will be important to compile gold-standard data directly from Twitter content if we want to perform a more precise evaluation. The effect of varying the cosine similarity decision threshold from 0.70 to 0.80 for Class Membership test shows that the percentage of test cases that are classified as correct drops significantly. However, the drop is more accentuated when training with 7.4 Learning Word Embeddings for ORM 143 only a portion of the available data. The differences of using two alternative thresholds values is even higher in the Word Equivalence test. The Word Equivalence test, in which we consider two words equivalent word if the cosine of the embedding vectors is higher than 0.95, revealed to be an extremely demanding test. Nevertheless, for \|V \| = 32768 the results are far superior, and for a much larger coverage, than for lower \|V \|. The same happens with the Class Membership test. On the other hand, the Class Distinction test shows a different trend for larger values of \|V \| = 32768 but the coverage for other values of \|V \| is so low that it would not make sense to hypothesize about the reduced values of True Negatives (TN) percentage obtained for the largest \|V \|. It would be necessary to confirm this behavior with even larger values of \|V \|. One might hypothesize that the ability to distinguish between classes requires larger thresholds when \|V \| is large. Also, we can speculate about the need of increasing the number of dimensions to be able to encapsulate different semantic information for so many words. Table 7.3 Evaluation of resulting embeddings using Class Membership, Class Distinction and Word Equivalence tests for different thresholds of cosine similarity. Embeddings \|V \|, %data Class Membership coverage 2048, 25% 2048, 50% 12.32% Acc. Acc. @0.70 Class Distinction Word Equivalence TN TN @0.80 @0.70 30.71% 4.94% 29.13% 12.69% coverage 1.20% Acc. Acc. @0.80 @0.85 @0.95 100% 100% 26.67% 2.94% 100% 100% coverage 31.25% 26.67% 2.94% 2048, 75% 29.13% 18.12% 100% 100% 33.33% 2.94% 2048, 100% 32.28% 26.77% 100% 100% 33.33% 6.67% 8192, 25% 14.17% 4.94% 100% 100% 14.71% 2.94% 99% 100% 8192, 50% 29.60% 22.41% 12.69% 6.54% 70.83% 20.59% 2.94% 8192, 75% 27.51% 18.12% 99% 100% 20.59% 2.94% 8192, 100% 33.77% 21.91% 97% 100% 29.41% 5.88% 32768, 25% 17.73% 5.13% 98% 100% 16.28% 2.33% 83% 98% 32768, 50% 47.79% 52.30% 21.06% 18.31% 89.58% 34.88% 9.30% 32768, 75% 85.15% 49.41% 44% 88% 58.14% 23.26% 32768, 100% 95.59% 74.80% 13% 57% 72.09% 34.88% Further Analysis regarding Evaluation Metrics Despite already providing interesting practical clues for our goal of trying to embed a larger vocabulary using more of the training data we have available, these results also 144 A Framework for Online Reputation Monitoring revealed that the intrinsic evaluation metrics we are using are overly sensitive to their corresponding cosine similarity thresholds. This sensitivity poses serious challenges for further systematic exploration of word embedding architectures and their corresponding hyper-parameters, which was also observed in other recent works [208]. By using these absolute thresholds as criteria for deciding the similarity of words, we create a dependency between the evaluation metrics and the geometry of the embedded data. If we see the embedding data as a graph, this means that metrics will change if we apply scaling operations to certain parts of the graph, even if its structure (i.e. relative position of the embedded words) does not change. For most practical purposes (including training downstream ML models) absolute distances have little meaning. What is fundamental is that the resulting embeddings are able to capture topological information: similar words should be closer to each other than they are to words that are dissimilar to them (under the various criteria of similarity we care about), independently of the absolute distances involved. It is now clear that a key aspect for future work will be developing additional performance metrics based on topological properties. We are in line with recent work [209], proposing to shift evaluation from absolute values to more exploratory evaluations focusing on weaknesses and strengths of the embeddings and not so much in generic scores. For example, one metric could consist in checking whether for any given word, all words that are known to belong to the same class are closer than any words belonging to different classes, independently of the actual cosine. Future work will necessarily include developing this type of metrics. 7.4.4 Concluding Remarks Producing word embeddings from tweets is challenging due to the specificities of the vocabulary in the medium. We implemented a neural word embedding model that embeds words based on n-gram information extracted from a sample of the Portuguese Twitter stream, and which can be seen as a flexible baseline for further experiments in the field. Work reported in this paper is a preliminary study of trying to find parameters for training word embeddings from Twitter and adequate evaluation tests and gold-standard data. Results show that using less than 50% of the available training examples for each vocabulary size might result in overfitting. The resulting embeddings obtain reasonable performance on intrinsic evaluation tests when trained a vocabulary containing the 32768 most frequent words in a Twitter sample of relatively small size. Nevertheless, results exhibit a skewness in the cosine similarity scores that should be further explored 7.5 Summary of the Contributions 145 in future work. More specifically, the Class Distinction test set revealed to be challenging and opens the door to evaluation of not only similarity between words but also dissimilarities between words of different semantic classes without using absolute score values. Therefore, a key area of future exploration has to do with better evaluation resources and metrics. We made some initial effort in this front. However, we believe that developing new intrinsic tests, agnostic to absolute values of metrics and concerned with topological aspects of the embedding space, and expanding gold-standard data with cases tailored for user-generated content, is of fundamental importance for the progress of this line of work. Furthermore, we plan to make public available word embeddings trained from a large sample of 300M tweets collected from the Portuguese Twitter stream. This will require experimenting with and producing embeddings with higher dimensionality (to avoid the cosine skewness effect) and training with even larger vocabularies. Also, there is room for experimenting with some of the hyper-parameters of the model itself (e.g. activation functions, dimensions of the layers), which we know have impact on final results. 7.5 Summary of the Contributions The work reported in this chapter makes the following contributions: • A framework that supports research in Entity Retrieval and Text Mining tasks in the context of Online Reputation Monitoring. This framework is composed by two major components that can act as independent frameworks: RELink and TexRep. • The RELink framework that supports comprehensive research work in E-R retrieval, supporting the semi-automatic creating of test queries, as well as, Early Fusion based approaches for E-R retrieval. • The TexRep framework that is able to collect texts from online media, such as Twitter or online news, and identify entities of interest, classify sentiment polarity and intensity. The framework supports multiple data aggregation methods, as well as visualization and modeling techniques that can be used for both descriptive analytics, such as analyze how political polls evolve over time, and predictive analytics, such as predict elections. 146 A Framework for Online Reputation Monitoring • A study of some practical aspects, namely vocabulary size, training data size and intrinsic evaluation for the training and publishing word embeddings from the Portuguese Twitter stream that can be later used for ORM related tasks. Chapter 8 Conclusions In this thesis we have addressed two computational problems in Online Reputation Monitoring: Entity Retrieval and Text Mining. Entities are the gravitational force that drives the ORM process and consequently the work reported in this thesis gravitates around entities and their occurrences across the Web. We researched and developed methods for text-based extraction, entity-relationship retrieval, analysis and prediction of entity-centric information spread across the Web. The main objectives of this thesis were achieved resulting in several contributions to the problem of Online Reputation Monitoring. Several competitive baselines were developed which we believe represent significant progress in a research area where open source work is scarce. However, there are still many issues to be addressed in the future. Recent developments in Deep Neural Networks create opportunities to improve performance in several tasks we addressed in this thesis. Once we have access to larger quantities of training data it will be possible to easily adapt our research framework to include these techniques. 8.1 Summary and Main Contributions Entity-Relationship Retrieval We have established that ORM benefits from entity retrieval capabilities and should not be constrained to classic data analytics reports. Users ought to be able to search for entity-centric information from Social Media and online news. Furthermore, reputation is not an isolated asset and depends also of the reputation of “neighboring” entities. We studied the problem of Entity-Relationship Retrieval using a IR-centric perspective and we made several contributions to this line of research: 148 Conclusions • Generalization of the problem of entity-relationship search to cover entity types and relationships represented by any attribute and predicate, respectively, rather than a predefined set. • A general probabilistic model for E-R retrieval using Bayesian Networks. • Proposal of two design patterns that support retrieval approaches using the E-R model. • Proposal of a Entity-Relationship Dependence model that builds on the basic Sequential Dependence Model (SDM) to provide extensible entity-relationship representations and dependencies, suitable for complex, multi-relations queries. • Proposal of an indexing method that supports a retrieval approach to the above problem. • A semi-automatic method for generating E-R test collections, which resulted in the RELink Query Collection comprising 600 E-R queries. • Results of experiments at scale, with a comprehensive set of queries and corpora. Entity-Relationship (E-R) Retrieval is a complex case of Entity Retrieval where the goal is to search for multiple unknown entities and relationships connecting them. Contrary to entity retrieval from structured knowledge graphs, IR-centric approaches to E-R retrieval are more adequate in the context of ORM. This happens due to the dynamic nature of the data sources which are much more transient than other more stable sources of information (e.g Wikipedia) used in general Entity Retrieval. Consequently, we developed E-R retrieval methods that do not rely on fixed and predefined entity types and relationships, enabling a wider range of queries compared to Semantic Web-based approaches. We started by presenting a formal definition of E-R queries where we assume that a E-R query can be decomposed as a sequence of sub-queries each containing keywords related to a specific entity or relationship. Then we adopted a probabilistic formulation of the E-R retrieval problem. When creating specific representations for entities (e.g. context terms) and for pairs of entities (i.e. relationships) it is possible to create a graph of probabilistic dependencies between sub-queries and entity plus relationship representations. We use a Bayesian network to depict these dependencies in a probabilistic graphical model. To the best of our knowledge this represents the first probabilistic model of E-R retrieval. 8.1 Summary and Main Contributions 149 However, these conditional probabilities cannot be computed directly from raw documents in a collection. In fact, this is a condition inherent to the problem of Entity Retrieval. Documents serve as proxies to entities and relationship representations and consequently, we need to fuse information spread across multiple documents to be able to create those representations. We proposed two design patterns, Early Fusion and Late Fusion, inspired from Model 1 and Model 2 of Balog et al. [46]. However, in the context of ORM, we are only interested in Early Fusion. Early Fusion aggregates context terms of entity and relationship occurrences to create two dedicated indexes, the entity index and the relationship index. Once we have the two indexes it is possible to apply any retrieval method to compute the relevance scores of entity and relationship documents (i.e. representations) given the E-R sub-queries. The joint probability to retrieve the final entity tuples is computed using a factorization of the conditional probabilities, i.e., the individual relevance scores. On the other hand, Late Fusion consists in matching the E-R sub-queries directly on a standard document index alongside a set of entity occurrence in each document. Once we compute the individual relevance scores of each document given a E-R sub-query, we then aggregate the entity occurrences of the top k results to compute the final joint probability. When using traditional retrieval models, such as Language Models or BM25, these design patterns can be used to create unsupervised baselines for E-R retrieval. Since our objective was to explore an Early Fusion approach to E-R retrieval we developed a novel supervised Early Fusion-based model for E-R retrieval, the EntityRelationship Dependence Model (ERDM). It uses Markov Random Field to model term dependencies of E-R sub-queries and entity/relationship documents. ERDM can be seen as an extension of the Sequential Dependence Model (SDM) [63] for ad-hoc document retrieval in a way that it relies on query term dependencies but creates a more complex graph structure that connects terms of multiple (sub-)queries and multiple documents to compute the probability mass function under the MRF. One of the difficulties we faced while researching E-R retrieval was the lack of test collections. We therefore decided to contribute to this research problem by creating a semi-automatic method for creating test collections. We realized that web tabular data often include implicit relationships between entities that belong to the same row in a table. We developed a table parser that extracts tuples of related entities from Wikipedia Lists-of-lists-of-lists tables. We then extract metadata, such as table title or column name, and provide it to editors, together with the list of entity tuples. We 150 Conclusions asked editors to create E-R queries in which the list of entity tuples could serve as relevance judgments. This process resulted in the creation and publication of the RELink Query Collection comprising 600 E-R queries. We believe RELink QC will foster research work in E-R retrieval. We performed experiments at scale using the ClueWeb-09B Web corpus from which we extracted and indexed more than 850 million entity and relationship occurrences. We evaluated our methods using four different query sets comprising a total of 548 E-R queries. As far as we know, this is the largest experiment in E-R retrieval, considering the size of the query set and the data collection. Results show consistently better performance of the ERDM model over three proposed baselines. When comparing Language Models and BM25 as feature functions we observed variance on the performance depending on the query set. Furthermore, using unsupervised Early Fusion proved to be very competitive when compared to ERDM, suggesting that it can be used in some application scenarios where the overhead of computing sequential dependencies might be unfeasible. Entity Filtering and Sentiment Analysis Entity Filtering and Sentiment Analysis are two fundamental Text Mining problems in ORM. We participated in two well known external benchmark competitions in both tasks resulting in state-of-the-art performance. We made the following contributions to these two problems: • A supervised learning approach for Entity Filtering on tweets, achieving state-ofthe-art performance using a relatively small training set. • Created and made available word embeddings trained from financial texts. • A supervised learning approach for fine-grained sentiment analysis of financial texts. Entity Filtering can be seen as targeted named entity disambiguation. We developed a supervised method that classifies tweets as relevant or non-relevant to a given target entity. This task is fundamental in ORM as downstream tasks, such as prediction, can be highly affected by noisy input data. We implemented a large set of features that can be generated to describe the relationship between a tweet mentioning a entity and a reference entity representation. 8.1 Summary and Main Contributions 151 We relied on metadata, such as entity categories, text represented with TF-IDF, similarity between tweets and Wikipedia entity articles, Freebase entities disambiguation, feature selection of terms based on frequency and feature matrix transformation using SVD. Although our approach can be perceived as relatively simple and low cost, we achieved first place with an Accuracy over 0.90 at the Filtering Task of RepLab 2013, in a test set containing more than 90 thousand tweets and 61 different target entities. Regarding Sentiment Analysis, we decided to focus our efforts in a not so well explored sub-area, namely financial texts. We participated in SemEval 2017 Task 5 which focused on fine-grained sentiment analysis of financial news and microblogs. The task consisted in predicting a real continuous variable from -1.0 to +1.0 representing the polarity and intensity of sentiment concerning companies/stocks mentioned in short texts. We modeled it as a regression analysis problem. Previous work in this domain showed that financial sentiment is often depicted in an implicit way. We created financial-specific word embeddings in order to obtain domain specific syntactic and semantic relations between words in this context. We combined traditional bag-of-words, lexical-based features and bag-of-embeddings to train a regressor of both sentiment and intensity. Results showed that different combination of features attained different performances on each sub-tasks. Nevertheless, we were able to obtain cosine similarities above 0.65 in both sub-tasks and mean average errors below 0.2 in a scale range of 2.0, representing less than 10% of the maximum possible error. Text-based Entity-centric Prediction We explored two text-based prediction problems in the context of ORM, performing analysis of the predictive power of entity-centric information on the news to predict entity popularity on Twitter, as well, as a study of sentiment aggregate functions to predict political opinion. We made the following contribution in this research area: • Analysis of the predictive power of online news regarding entity popularity on Twitter for entities that are frequently mentioned on the news. • Analysis of how to combine different sentiment aggregate functions to serve as features for predicting political polls. We are aware that entity popularity on social media can be influenced by endogenous and exogenous factors but we are only interested in exploring the interplay between 152 Conclusions online news and social media reactions. This could be useful for anticipating public relations damage control or even for editorial purposes to maximize attention and consequently revenue. We explored different sets of signal extracted from online news mentioning entities that are frequently mentioned on the news such as politicians of footballers. These signals could influence or at least are correlated with future popularity of those entities on Twitter. Results show that performance varies depending on the target entity. In general, results are better in the case of predicting popularity of politicians, due to the high unpredictability of live events associated with sports. This is a general conclusion of this study as online news do not have predictive power for live events as Twitter reactions happen quickly than the publication of the news for such cases. Results also show that the time of prediction affects the performance of the models. For instance, in the case of politicians F1 score is higher when time of prediction occurs after lunch time, which is an evidence that in politics most of the news events that trigger social media reactions are reported in the morning news. The second predictive studied we carried out consisted in using entity-centric sentiment polarity extracted from tweets to predict political polls. There is no consensus on previous research work on what sentiment aggregate functions is more adequate to predict political results. We explored several sentiment aggregate functions described in the literature to assess which one or combination would be more effective on predicting polls during the Portuguese bailout (2011-2013). In our study, we achieved the lowest mean average error using a combination of buzz aggregation functions to predict monthly poll variations instead of absolute values. On the other hand, the most important individual feature was an aggregate function consisting on the logarithm of the ration positive and negative classified tweets. A Framework for ORM We also created a framework specifically tailored for ORM that puts together the sub-tasks we tackled throughout this thesis. We believe this framework represents a significant contribution and paves the way to future research in the computational problems inherent to the process of monitoring reputation online. More precisely we make the following contributions: • A framework that supports research in Entity Retrieval and Text Mining tasks in the context of Online Reputation Monitoring. This framework is composed by two major components that can act as independent frameworks: RELink and TexRep. 8.1 Summary and Main Contributions 153 • The RELink framework that supports comprehensive research work in E-R retrieval, supporting the semi-automatic creating of test queries, as well as, Early Fusion based approaches for E-R retrieval. • The TexRep framework that is able to collect texts from online media, such as Twitter or online news, and identify entities of interest, classify sentiment polarity and intensity. The framework supports multiple data aggregation methods, as well as visualization and modeling techniques that can be used for both descriptive analytics, such as analyze how political polls evolve over time, and predictive analytics, such as predict elections. • A study of some practical aspects, namely vocabulary size, training data size and intrinsic evaluation for the training and publishing word embeddings from the Portuguese Twitter stream that can be later used for ORM related tasks. The framework is divided in two distinct components, one is dedicated to Entity Retrieval and the other to Text Mining. In practice these two components can act as two separate frameworks. Both are adaptable and can be reused in different application scenarios, from computational journalism to finance or politics. RELink framework is designed to facilitate experiments with E-R Retrieval query collections. TexRep was designed with two main challenges in mind: 1) it should be able to cope with the Text Mining problems underlying ORM and 2) it should be flexible, adaptable and reusable in order to support the specificities of different application scenarios. We also presented two use cases of our framework for ORM. In the first we use RELink in the context of computational journalism while in the second we described the design and the implementation of the POPmine system, an use case of the proposed framework in the scope of the POPSTAR project. Furthermore, we presented a study of the practical aspects of learning word embeddings from the Twitter stream. Our goal was to try to assess the feasibility of producing and publishing general purpose word embeddings for ORM. Results showed that using less than 50% of the available training examples for each vocabulary size might result in over-fitting. We obtained interesting performance on intrinsic evaluation when trained a vocabulary containing 32768 most frequent words in a Twitter sample of relatively small size. We proposed a set of gold standard data for intrinsic evaluation of word embeddings from user generated content. Nevertheless, we realized that evaluation metrics using absolute values as thresholds might not be suitable due to the cosine skewness effect on large dimensional embedding spaces. We propose to develop topological intrinsic evaluation metrics in future work. Conclusions 154 8.2 Limitations and Future Work One of the major obstacles we faced during the course of this thesis was the limited availability of labeled data for training and evaluation of the different tasks we tackled. This is a common limitation in the scope of Online Reputation Monitoring. Due to this obstacle we did not have the chance to perform extensive experimentation using more than one data source and language for each task. This aspect reduces the generalization of the results obtained since they might be biased towards the available datasets we had access to. Therefore, we leave for future work experimentation on each task with multiple datasets using different data sources and languages to perform comparable evaluations. We also recognize that we tried to address many different tasks which reduced our capability of addressing every task with the same level of depth. Nevertheless, we believe that exploring several new tasks in the scope of ORM constitutes a strong contribution to foster future research work in this area. During the course of this thesis, we did not have the possibility of performing user studies to assess the global usefulness of our framework for ORM. We would like to leave that as future work. While we had the objective of applying E-R retrieval in online news and social media which represent the natural data sources for ORM, it was not possible to evaluate our approaches using these type of data sources. Research work in E-R retrieval is still in its early stages and we believed it was necessary to first contribute to general E-R retrieval and leave for future work specific evaluation in the context of ORM. We implemented and created a demo of the Early Fusion approach since it is unsupervised. However, it was not possible to apply ERDM to online news due to the lack of training queries and relevance judgments for parameter tuning. In either cases, we aim to conduct an user experience in a near future to collect queries and relevance judgments in the context of ORM. Recent work in Deep Neural Networks makes the opportunity to beat the baselines we created in this thesis however, most of the tasks we addressed do not have enough labeled data to use these techniques. One of the most interesting avenues we would like to explore would be the use of neural networks as feature functions of the ERDM model. Since we have a dataset of more than 850 million entity and relationship extractions this represents an ideal scenario for Deep Learning. We propose to use a window based prediction task similar to the CBOW model for training word embeddings. Given a fixed window size, one would learn a neural network that would provide a ranked list of entities/relationships given an input query. We believe this approach would reduce 8.2 Limitations and Future Work 155 the computational costs of the current ERDM feature functions since we would not need to keep two huge indexes at query time. We would like also to explore different priors in entity and relationship documents within ERDM. For instance, creating source and time sensitive rankings would be useful when using transient information sources. 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International Journal of Computing and Business Research (IJCBR) ISSN (Online) : 2229-6166 Volume 3 Issue 3 September 2012 TIME EFFICIENT APPROACH TO OFFLINE HAND WRITTEN CHARACTER RECOGNITION USING ASSOCIATIVE MEMORY NET Tirtharaj Dash B.Tech Final Year Student, Department of Information Technology National Institute of Science and Technology Berhampur-761008, India Abstract: In this paper, an efficient Offline Hand Written Character Recognition algorithm is proposed based on Associative Memory Net (AMN). The AMN used in this work is basically auto associative. The implementation is carried out completely in ‘C’ language. To make the system perform to its best with minimal computation time, a Parallel algorithm is also developed using an API package OpenMP. Characters are mainly English alphabets (Small (26), Capital (26)) collected from system (52) and from different persons (52). The characters collected from system are used to train the AMN and characters collected from different persons are used for testing the recognition ability of the net. The detailed analysis showed that the network recognizes the hand written characters with recognition rate of 72.20% in average case. However, in best case, it recognizes the collected hand written characters with 88.5%. The developed network consumes 3.57 sec (average) in Serial implementation and 1.16 sec (average) in Parallel implementation using OpenMP. Keywords: Offline; Hand Written Character; Associative Memory Net; OpenMP; Serial; Parallel. 1. Introduction In the recent years, Hand Written Character Recognition has been a challenging and interesting research area in the field of pattern recognition and image processing (Impedovo et al., 1991; Mori et al., 1992). It contributes mainly to the Human-Computer interaction and improves the interface between the two (Pradeep et al., 2011). Other International Journal of Computing and Business Research (IJCBR) ISSN (Online) : 2229-6166 Volume 3 Issue 3 September 2012 human cognition methods viz. face, speech, thumb print recognitions are also being great area of research (Imtiaz and Fattah, 2011; Khurana and Singh, 2011; Kurian and Balakriahnan, 2012). Generally, character recognition can be broadly characterized into two types (i) Offline and (ii) Online. In offline method, the pattern is captured as an image and taken for testing purpose. But in case of online approach, each point of the pattern is a function of time, pressure, slant, strokes etc. Both the methods are best based on their application in the field. Yielding best accuracy with minimal cost of time is a crucial precondition for pattern recognition system. Therefore, hand written character recognition is continuously being a broad area of research. In this work, an approach for offline character recognition has been proposed using Associative Memory Network (AMN). In fact, to make it time efficient, a parallel algorithm has been developed for the implementation of AMN using OpenMP (Open Multiprocessing) (www.openmp.org). AMN is a neural network which can store patterns as memories. When the network is being tested with a key pattern, it corresponds by producing one of the stored patterns, which closely resembles to key pattern. Based on the testing pattern, AMN can be of two types (i) auto-associative memory net or (ii) hereto-associative memory net. Both the networks contains two layers (a) input layer and (b) output layer. In case of auto-associative memory net, the input and target pattern are same (Sivanandam and Deepa, 2011). But, in case of hetero-associative memory net the two patterns are different. This work uses the auto-AMN, as the character to be tested is same as the stored character. The characters considered in this work are English alphabets (both small and capital letters). This paper is organized as follows. Section 1 presented a general introduction to the character recognition systems and methods. Section 2 gives a brief literature review of some methods proposed for character recognition. Section 3 describes the proposed methodology of this work. Section 4 is a result and discussion section which gives a detailed analysis of the work. The paper is concluded in section 5 with a note to future works. 2. Literature Review International Journal of Computing and Business Research (IJCBR) ISSN (Online) : 2229-6166 Volume 3 Issue 3 September 2012 Available literatures convey that various algorithms and techniques have been used in order to accomplish the task of character recognition. Some studies are described below. Source of the literature are Google scholar, Scopus and IEEE library. Neural Network (NN) has been a backend of character classification in most of the methods. This is due to its faster and reliable computation. The methods used in front end could be (a) statistical approaches (b) kernel methods (c) support methods or (d) hybrid of fuzzy logic controllers. Multilayer Perceptron (MLP) was used for ‘Bangla’ alphabet recognition by Basu et al., 2005. The accuracy achieved in this work was 86.46% and 75.05% on the samples of training and testing respectively. Manivannan and Neil, 2010 proposed and demonstrated an optical correlator-neural network architecture for pattern recognition. English alphabet used as patterns for the training and testing process. (Pal and Singh, 2010) proposed NN based English character recognition system. In this work, MLP with one hidden layer was used. About 500 testing were carried out to test the performance of the design. The best case accuracy obtained in this work was 94%. (Perwej and Chaturvedi, 2011) worked on English alphabet recognition using NN. In this work, binary pixels of the alphabets were used train the NN. The accuracy achieved was found to be 82.5%. (Pal et al., 2007) proposed a modified quadratic classifier approach for handwritten numerals of six popular Indian scripts with high level of recognition accuracy. (Dinesh et al., 2007) used horizontal and vertical strokes and end points as feature for handwritten numerals. This method reported an accuracy rate of 90.5% in best case. However, this method used a thinning method resulting in loss of features. Yanhua and Chuanjun, 2009 recommended a novel Chinese character recognition algorithm which was based on minimum distance classifier. The algorithm attempted to work with two classes of feature extraction-structure and statistics. The statistic feature decided the primary class and the structure feature used to identify Chinese characters. A good method of character recognition was proposed by (Huiqin et al, 2011). In this work, they proposed a distribution based algorithm based on image segmentation and International Journal of Computing and Business Research (IJCBR) ISSN (Online) : 2229-6166 Volume 3 Issue 3 September 2012 distribution of pixels. Deflection Correction method was adopted for flexibility as well as reduction of matching error. This work avoided the burden of extracting the skeleton from the character. The method gave excellent result and was robust. 3. Methodology A step-wise methodology has been proposed which is demonstrated in Figure 1. Figure 1: Proposed Methodology Step-1: Collection of English alphabets (both small and capital) from (i) system and (ii) persons (Hand Written) Step-2: Extraction of pixels from the characters Step-3: Implementation of auto AMN: (i) Training and (ii) Testing using both (a) serial and (b) parallel algorithms. Step-4: Comparison of results from serial and parallel processing with respect to time of execution 3.1 Generation of English alphabets English alphabets (both small and capital) are designed in the system using MS Paint version 6.1 in Arial font size-28 (No Bold) in BMP file format. The dimension of the bmp file is 31×39, with bit depth of 4. Some alphabets are given in Figure 2. International Journal of Computing and Business Research (IJCBR) ISSN (Online) : 2229-6166 Volume 3 Issue 3 September 2012 Figure 2: English alphabets of the system Hand written English alphabets are collected, each one from different persons. The characters are given in Figure 3. Figure 3: English alphabets collected from different persons 3.2 Extraction of Pixel from the characters Pixels are extracted from the character images (bitmap files) using a standard image function of MATLAB version 10. The function is imread(‘filename.bmp’). The function extracts the decimal values associated with each pixel. The pixels are then stored in a text (.txt) file for the experiment purpose. 3.3 Auto-associative memory net (Auto-AMN) implementation 3.3.1 Serial algorithm INITIALIZE weight (W) to 0 SET the target pattern as the system’s pattern International Journal of Computing and Business Research (IJCBR) ISSN (Online) : 2229-6166 Volume 3 Issue 3 September 2012 INPUT the Handwritten pattern to the first layer of AMN FOR i=1 to n DO FOR j=1 to n DO CALCULATE the weight as Wij(new)=W ij (old) + INPUTi×TARGETj END END FOR i=1 to n DO FOR j=1 to n DO CALCULATE the net input to each output node as, n Yinj= ∑ x i Wij i =1 IF(Yinj>0) Yj=+1; ELSE Yj=-1; END END 3.3.2 Parallel algorithm INITIALIZE weight (W) to 0 SET the target pattern as the system’s pattern INPUT the Handwritten pattern to the first layer of AMN #pragma omp paralle shared(W,Yin,chunk,p) private(tid,i,j) DO #pragma omp for schedule (static, chunk) FOR i=1 to n DO FOR j=1 to n DO CALCULATE the weight as Wij(new)=W ij (old) + INPUTi×TARGETj END END International Journal of Computing and Business Research (IJCBR) ISSN (Online) : 2229-6166 Volume 3 Issue 3 September 2012 #pragma omp for schedule (static, chunk) FOR i=1 to n DO FOR j=1 to n DO CALCULATE the net input to each output node n Yinj= ∑ x i Wij i =1 IF(Yinj>0) Yj=+1; ELSE Yj=-1; END END END 3.3.3 System Specification A computer system having 1 GB RAM and Four processors is used for the complete work. The operating system is Ubuntu 10.04 (Linux). However, for auto optimization by the compiler, ‘–g’ tag is used in the compilation command. 4. Results and Discussion The contribution of this work is a detailed analysis on the recognition accuracy for all the handwritten English alphabets (Total 52). Time of computation has been noted for both serial and parallel algorithm to compare the decision making speed. 4.1 Recognition accuracy Table 1 shows a result of the testing the developed AMN for a set of hand written characters. It should be noted that the network is trained with the machine’s alphabets and tested with the hand written alphabet. However, for reliability issue, a hand written character is checked for 5 times and the matching percentage is the average of the 5 results. Table 1: Recognition accuracy of AMN for offline Hand written character recognition International Journal of Computing and Business Research (IJCBR) ISSN (Online) : 2229-6166 Volume 3 Issue 3 September 2012 System’s Alphabet A B C D E F G H I J K L M N O P Q R S T U V W X Y Z a b c d e f g h i j Hand Written Alphabet for which highest match is achieved A B C O F F G H I J K L H H O P O R S T U V U X Y Z o b e d e p y b i j Recognition Accuracy (%) 66.56 56.80 67.19 60.88 62.85 68.34 58.78 70.73 85.74 86.28 64.85 85.16 70.38 71.13 64.17 67.74 61.42 61.16 63.20 80.18 71.46 73.69 73.13 73.39 76.67 64.01 69.49 65.35 73.96 67.26 67.47 73.54 69.36 70.52 83.62 88.50 Time of Computation (sec.) Serial Parallel 3.00 0.99 3.11 0.98 2.98 0.74 4.60 1.55 3.77 2.01 4.65 2.32 3.12 1.87 3.67 0.99 4.01 1.03 4.32 1.93 4.17 1.05 3.04 0.87 4.12 1.21 4.61 1.35 3.02 0.76 4.00 1.04 2.98 0.83 3.41 0.99 4.04 1.12 2.87 0.77 2.76 0.76 2.78 0.87 3.05 1.00 3.83 1.43 4.05 1.45 4.00 1.45 3.77 1.88 3.78 1.43 3.17 1.42 3.77 1.54 3.18 1.31 3.17 1.22 3.18 1.03 4.01 1.12 3.19 1.27 4.01 1.23 International Journal of Computing and Business Research (IJCBR) ISSN (Online) : 2229-6166 Volume 3 Issue 3 September 2012 k l m n o p q r s t u v w x y z K l m n o p q r s t u v w x y z 69.44 83.60 62.62 72.97 70.03 68.61 65.45 82.90 68.66 79.92 78.54 82.52 67.16 78.55 78.27 70.08 3.66 3.75 2.76 2.89 2.75 3.78 3.89 4.05 3.00 2.01 4.89 4.02 3.96 2.95 3.81 3.99 1.44 1.03 1.05 0.75 0.76 0.94 0.94 1.11 0.77 0.52 1.33 1.04 1.21 0.76 0.98 1.23 Table-1 can be viewed as a detailed analysis of performance of the developed auto AMN for offline hand written English alphabet recognition. The network recognizes the handwritten character ‘j’ with highest matching of 88.50%. However, the network doesn’t recognize some alphabets like ‘D’, ‘E’, ‘M’, ‘N’, ‘Q’, ‘W’, ‘a’, ‘c’, ‘f’, ‘g’, ‘h’, ‘j’ and ‘k’ and these alphabets are recognized as ‘O’, ‘F’, ‘H’,’H’, ‘O’, ‘U’, ‘o’, ‘e’, ‘p’, ‘y’, ‘b’, ‘I’ and ‘K’ respectively with some matching error. 4.2 Level of matching of each alphabet A plot has been given in Figure 4 to view the level up to which each English alphabet is matched by the AMN. The alphabets which are not recognized are awarded with 0% matching. International Journal of Computing and Business Research (IJCBR) ISSN (Online) : 2229-6166 Volume 3 Issue 3 September 2012 90 80 Level of Matching (%) 70 60 50 40 30 20 10 0 A B C D E F G H I J K L M N O P Q R S T U VW X Y Z a b c d e f g h i j k l m n o p q r s t u v w x y z English Alphabet Figure 4 This plot shows level of matching of each alphabet 4.3 Time Efficiency As it is already mentioned that the network is developed with two algorithms, (i) serial and (ii) parallel; it will be a good idea to check the timing variation in both the cases. A plot given in Figure 5, shows speed up after achieved after the execution of parallel algorithm. International Journal of Computing and Business Research (IJCBR) ISSN (Online) : 2229-6166 Volume 3 Issue 3 September 2012 5 Serial Parallel 4.5 Decision making speed (sec.) 4 3.5 3 2.5 2 1.5 1 0.5 0 5 10 15 20 25 30 35 Alphabet Serial Number (1-52) 40 45 50 Figure 5 Decision making speed by Serial and Parallel algorithm 5. Conclusion In this paper, an offline English character recognition system has been proposed. The system is developed using auto Associative Memory Net. To make the developed system faster and reliable, a parallel algorithm has been developed and tested successfully. Experimental study showed that, the system recognizes characters with average recognition rate of 72.20%. Character ‘j’ is recognized with highest accuracy rate of 88.5%. The average time required by the serial algorithm to recognize a character is 3.57 sec where as the parallel algorithm takes only 1.16 sec on an average. However, automatic checking of a sequence of character by the network will play a great role in the world of character recognition. The author is currently working on this issue. References 1. Basu, S. et al. (2005) “Handwritten ‘Bangla’ alphabet recognition using an MLP based classifier,” Proceeding of 2nd National Conference on Computer Processing of Bangla, pp. 285-291. 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arXiv:1512.03987v3 [math.ST] 8 Oct 2016 On the Finite-Sample Analysis of Θ-estimators Yiyuan She Department of Statistics Florida State University, Tallahassee, FL 32306 Abstract In large-scale modern data analysis, first-order optimization methods are usually favored to obtain sparse estimators in high dimensions. This paper performs theoretical analysis of a class of iterative thresholding based estimators defined in this way. Oracle inequalities are built to show the nearly minimax rate optimality of such estimators under a new type of regularity conditions. Moreover, the sequence of iterates is found to be able to approach the statistical truth within the best statistical accuracy geometrically fast. Our results also reveal different benefits brought by convex and nonconvex types of shrinkage. 1 Introduction Big data naturally arising in machine learning, biology, signal processing, and many other areas, call for the need of scalable optimization in computation. Although for low-dimensional problems, Newton or quasi-Newton methods converge fast and have efficient implementations, they typically do not scale well to high dimensional data. In contrast, first-order optimization methods have recently attracted a great deal of attention from researchers in statistics, computer science and engineering. They iterate based on the gradient (or a subgradient) of the objective function, and have each iteration step being cost-effective. In high dimensional statistics, a first-order algorithm typically proceeds in the following manner β (t+1) = P ◦ (β (t) − α∇l(β (t) )), 1 (1) where P is an operator that is easy to compute, ∇l denotes the gradient of the loss function l, and α gives the stepsize. Such a simple iterative procedure is suitable for large-scale optimization, and converges in arbitrarily high dimensions provided α is properly small. P can be motivated from the perspective of statistical shrinkage or regularization and is necessary to achieve good accuracy when the dimensionality is moderate or high. For example, a proximity operator (Parikh and Boyd, 2014) is associated with a convex penalty function. But the problems of interest may not always be convex. Quite often, P is taken as a certain thresholding rule Θ in statistical learning, such as SCAD (Fan and Li, 2001). The resulting computation-driven estimators, which we call Θ-estimators, are fixed points of β = Θ(β − ∇l(β); λ). To study the non-asymptotic behavior of Θ-estimators (regardless of the sample size and dimensionality), we will establish some oracle inequalities. During the last decade, people have performed rigorous finite-sample analysis of many high-dimensional estimators defined as globally optimal solutions to some convex or nonconvex problems—see Bunea et al. (2007), Zhang and Huang (2008), Bickel et al. (2009), Lounici et al. (2011), Zhang and Zhang (2012), She (2014), among many others. Θ-estimators pose some new questions. First, although nicely, an associated optimization criterion can be constructed for any given Θ-estimator, the objective may not be convex, and the estimator may not correspond to any functional local (or global) minimum. Second, there are various types of Θ-estimators due to the abundant choices of Θ, but a comparative study regarding their statistical performance in high dimensions is lacking in the literature. Third, Θ-estimators are usually computed in an inexact way on big datasets. Indeed, most practitioners (have to) terminate (1) before full computational convergence. These disconnects between theory and practice when using iterative thresholdings motivate our work. The rest of the paper is organized as follows. Section 2 introduces the Θestimators, the associated iterative algorithm–TISP, and some necessary notation. Section 3 presents the main results, including some oracle inequalities, and sequential analysis of the iterates generated by TISP. Section 4 provides proof details. 2 2 Background and Notation 2.1 Thresholding functions Definition 1 (Thresholding function). A thresholding function is a real valued function Θ(t; λ) defined for −∞ < t < ∞ and 0 ≤ λ < ∞ such that (i) Θ(−t; λ) = −Θ(t; λ); (ii) Θ(t; λ) ≤ Θ(t′ ; λ) for t ≤ t′ ; (iii) limt→∞ Θ(t; λ) = ∞; (iv) 0 ≤ Θ(t; λ) ≤ t for 0 ≤ t < ∞. A vector version of Θ (still denoted by Θ) is defined componentwise if either t or λ is replaced by a vector. From the definition, Θ−1 (u; λ) := sup{t : Θ(t; λ) ≤ u}, ∀u > 0 (2) must be monotonically non-decreasing and so its derivative is defined almost everywhere on (0, ∞). Given Θ, a critical number LΘ ≤ 1 can be introduced such that dΘ−1 (u; λ)/ du ≥ 1 − LΘ for almost every u ≥ 0, or LΘ := 1 − ess inf{ dΘ−1 (u; λ)/ du : u ≥ 0}, (3) where ess inf is the essential infimum. For the perhaps most popular soft-thresholding and hard-thresholding functions ΘS (t; λ) = sgn(t)(\|t\| − λ)+ , ΘH (t; λ) = t1\|t\|≥λ , LΘ equals 0 and 1, respectively. For any arbitrarily given Θ, we construct a penalty function PΘ (t; λ) as follows PΘ (t; λ) = Z 0 \|t\| −1 (Θ (u; λ) − u) du = Z 0 \|t\| (sup{s : Θ(s; λ) ≤ u} − u) du (4) for any t ∈ R. This penalty will be used to make a proper objective function for Θ-estimators. The threshold τ (λ) := Θ−1 (0; λ) may not equal λ in general. For ease in notation, in writing Θ(·; λ), we always assume that λ is the threshold parameter, i.e., λ = τ (λ), unless otherwise specified. Then an important fact is that given λ, any thresholding rule Θ satisfies Θ(t; λ) ≤ ΘH (t; λ), ∀t ≥ 0, due to property (iv), from which it follows that PΘ (t; λ) ≥ PH (t; λ), 3 (5) where PH (t; λ) = Z 0 \|t\| 2 2 (Θ−1 H (u; λ) − u) du = (−t /2 + λ\|t\|)1\|t\|<λ + (λ /2)1\|t\|≥λ . (6) 2 In particular, PH (t; λ) ≤ P0 (t; λ) := λ2 1t6=0 and PH (t; λ) ≤ P1 (t; λ) := λ\|t\|. When Θ has discontinuities, such as t = ±λ in ΘH (t; λ), ambiguity may arise in definition. To avoid the issue, we assume the quantity to be thresholded never corresponds to any discontinuity of Θ. This assumption is mild because practically used thresholding rules have few discontinuity points and such discontinuities rarely occur in real applications. 2.2 Θ-estimators We assume a model y = Xβ ∗ + ǫ, (7) where X is an n × p design matrix, y is a response vector in Rn , β ∗ is the unknown coefficient vector, and ǫ is a sub-Gaussian random vector with mean zero and scale bounded by σ, cf. Definition 2 in Section 4 for more detail. Then a Θ-estimator β̂, driven by the computational procedure (1), is defined as a solution to the Θ-equation ρβ = Θ(ρβ + X T y/ρ − X T Xβ/ρ; λ), (8) where ρ, the scaling parameter, does not depend on β. Having ρ appropriately large is crucial to guarantee the convergence of the computational procedure. All popularly used penalty functions are associated with thresholdings, such as the ℓr (0 < r ≤ 1), ℓ2 , SCAD (Fan and Li, 2001), MCP (Zhang, 2010a), capped ℓ1 (Zhang, 2010b), ℓ0 , elastic net (Zou and Hastie, 2005), Berhu (Owen, 2007; He et al., 2013), ℓ0 + ℓ2 (She, 2009), to name a few. Table 1 lists some examples. From a shrinkage perspective, thresholding rules usually suffice in statistical learning. Equation (8) can be re-written in terms of the scaled deign X̃ = X/ρ and the corresponding coefficient vector β̃ = ρβ β̃ = Θ(β̃ + X̃ T y − X̃ T X̃ β̃; λ). 4 (9) We will show that the λ in the scaled form does not have to adjust for the sample size, which is advantageous in regularization parameter tuning. A simple iterative procedure can be defined based on (8) or (9): β̃ (t+1) = Θ(β̃ (t) + X̃ T y − X̃ T X̃ β̃ (t) ; λ), β (t+1) = β̃ (t+1) /ρ, (10) which is called the Thresholding-based Iterative Selection Procedure (TISP) (She, 2009). From Theorem 2.1 of She (2012), given an arbitrary Θ, TISP ensures the kXk2 following function-value descent property when ρ ≥ 2−L : Θ f (β (t+1) ; λ) ≤ f (β (t) ; λ). (11) Here, the energy function (objective function) is constructed as p X 1 f (β; λ) = kXβ − yk22 + P (ρ\|βj \|; λ), 2 j=1 (12) where the penalty P can be PΘ as defined in (4), or more generally, P (t; λ) = PΘ (t; λ) + q(t; λ), (13) with q an arbitrary function satisfying q(t, λ) ≥ 0, ∀t ∈ R and q(t; λ) = 0 if t = Θ(s; λ) for some s ∈ R. Furthermore, we can show that when ρ > kXk2 /(2 − LΘ ), any limit point of β (t) is necessarily a fixed point of (8), and thus a Θ-estimator. See She (2012) for more detail. Therefore, f is not necessarily unique when Θ has discontinuities—for example, penalties like the capped ℓ1 , 2 P0 (t; λ) = λ2 1t6=0 and PH are all associated with the same ΘH . Because of the many-to-one mapping from penalty functions to thresholding functions, iterating (1) with a well-designed thresholding rule is perhaps more convenient than solving a nonconvex penalized optimization problem. Indeed, some penalties (like SCAD) are designed from the thresholding viewpoint. The following theorem shows thatPthe set of Θ-estimators include all locally optimal solutions of 12 kXβ − yk22 + pj=1 PΘ (\|βj \|; λ) =: fΘ (β). Theorem 1. Let β̂ be a local minimum point (or a coordinate-wise minimum point) of fΘ (·). If Θ is continuous at β̂ + X T y − X T X β̂, β̂ must satisfy β = Θ(β + X T y − X T Xβ; λ). The converse is not necessarily true. Namely, Θ-estimators may not guarantee functional local optimality, let alone global optimality. This raises difficulties in statistical analysis. We will give a novel and unified treatment which can yield nearly optimal error rate for various thresholdings. 5 Table 1: Some examples of thresholding functions and their associated quantities. Θ LΘ soft (t − λsgn(t))1\|t\|>λ 0 ridge −η PΘ λ\|t\| η 2 t 2 hard t1\|t\|>λ 1 ( − 21 t2 + λ\|t\|, 1 2 λ , 2 t 1+η if \|t\| < λ if \|t\| ≥ λ 2 P elastic net (η ≥ 0) berhu (η ≥ 0)  if \|t\| < λ 0  t − λsgn(t) if λ ≤ \|t\| ≤ λ + λ/η   t if \|t\| > λ + λ/η 1+η 0 ( λ\|t\| if \|t\| ≤ λ/η min(λ\|t\|, λ2 ) (‘capped ℓ1 ’), hard-ridge (η ≥ 0) λ2 1 2 t6=0 Θ t−λsgn(t) 1\|t\|≥λ 1+η LΘ −η PΘ λ\|t\| + 21 ηt2 P (‘ℓ0 + ℓ2 ’) scad (a > 2) mcp (γ ≥ 1)   0, if \|t\| ≤ λ   if \|t\| < λ   t − λ sgn(t), 0, if λ < \|t\| ≤ 2λ t−λsgn(t) , if λ ≤ \|t\| < γλ (a−1)t−aλ sgn(t) 1−1/γ   , if 2λ < \|t\| ≤ aλ   a−2   t, if \|t\| ≥ γλ t, if \|t\| > aλ 1/(a − 1) 1/γ  ( 2 if \|t\| ≤ λ λ sgn(t), t  + λ\|t\|, if \|t\| < γλ − 2γ aλ sgn(t)−t dP = = γ1 PH (t; γλ) , if λ < \|t\| ≤ aλ dt a−1 γλ2   , if \|t\| ≥ γλ 2 0, if \|t\| > aλ lr (0 < r < 1, ζ ≥ 0) ( 0, if \|t\| ≤ ζ 1/(2−r) (2 − r)(2 − 2r)(r−1)/(2−r) sgn(t) max{ζ 1/(2−r) [r(1 − r)]1/(2−r) ≤ θ ≤ \|t\| : θ + ζrθ r−1 = \|t\|}, otherwise.(The set is a singleton.) 1 ζ\|t\|r Θ LΘ PΘ Θ LΘ P ηt2 2 + 2 λ 2η if \|t\| > λ/η. t 1 1+η \|t\|>λ 1 ( − 21 t2 + λ\|t\|, 2 1 2 λ ηt + 12 1+η , 2 η 2 1 λ2 1 + 2t 2 1+η t6=0 if \|t\| < if \|t\| ≥ λ 1+η λ . 1+η 3 Main Results To address the problems in arbitrary dimensions (with possibly large p and/or n), we aim to establish non-asymptotic oracle inequalities (Donoho and Johnstone, 1994). For any β = [β1 , . . . , βp ]T , define J (β) = {j : βj 6= 0}, J(β) = \|J (β)\| = kβk0 . 2 (14) Recall P1 (t; λ) = λ\|t\|, P0 (t; λ) = λ2 1t6=0 , PH (t; λ) = (−t2 /2 + λ\|t\|)1\|t\|<λ + (λ2 /2)1\|t\|≥λ. For convenience, we use P1 (β; λ) to denote λkβk1 when there is no ambiguity. P0 (β; λ) and PH (β; λ) are used similarly. We denote by . an inequality that holds up to a multiplicative constant. Unless otherwise specified, we study scaled Θ-estimators satisfying equation 6 (9), where β̃ = ρβ, X̃ = X/ρ, and ρ ≥ kXk2 (and so kX̃k2 ≤ 1). By abuse of notation, we still write β for β̃, and X for X̃. As mentioned previously, we always assume that Θ is continuous at β̂ + X T y − X T X β̂ in Sections 3.1 & 3.2; similarly, Section 3.3 assumes that Θ is continuous at β (t) + X T y − X T Xβ (t) . The past works on the lasso show that a certain incoherence requirement must be assumed to obtain sharp error rates. In most theorems, we also need to make similar assumptions to prevent the design matrix from being too collinear. We will state a new type of regularity conditions, which are called comparison regularity conditions, under which oracle inequalities and sequential statistical error bounds can be obtained for any Θ. 3.1 PΘ -type oracle inequalities under R0 In this subsection, we use PΘ to make a bound of the prediction error of Θestimators. Our regularity condition is stated as follows. A SSUMPTION R0 (δ, ϑ, K, β, λ) Given X, Θ, β, λ, there exist δ > 0, ϑ > 0, K ≥ 0 such that the following inequality holds for any β ′ ∈ Rp ϑPH (β ′ − β; λ) + ≤ LΘ ′ kβ − βk22 2 2−δ kX(β ′ − β)k22 + PΘ (β ′ ; λ) + KPΘ (β; λ). 2 (15) Roughly, (15) means that 2kX(β ′ − β)k22 can dominate LΘ kβ ′ − βk22 with the help from PΘ (β ′ ; λ) and KPΘ (β; λ) for some K > 0. T T Theorem 2. Let p β̂ be any Θ-estimator satisfying β = Θ(β + X y − X Xβ; λ) with λ = Aσ log(ep) and A a constant. Then for any sufficiently large A, the following oracle inequality holds for β ∈ Rp E[kX β̂ − Xβ ∗ k22 ] . kXβ − Xβ ∗ k22 + PΘ (β; λ) + σ 2 , (16) provided R0 (δ, ϑ, K, β, λ) is satisfied for some constants δ > 0, ϑ > 0, K ≥ 0. Theorem 2 is applicable to any Θ. Let’s examine two specific cases. First, consider LΘ ≤ 0, which indicates that PΘ is convex. Because PH ≤ PΘ and PH is sub-additive: PH (t + s) ≤ PH (t) + PH (s) due to its concavity (Zhang and Zhang, 2012), R0 (δ, ϑ, K, β, λ) is always satisfied (for any δ ≤ 2, 0 < ϑ ≤ 1, K ≥ ϑ). 7 Corollary 1. Suppose Θ satisfies LΘ ≤ 0. Then, (16) holds for all corresponding Θ-estimators, without requiring any regularity condition. In the case of hard-thresholding or SCAD thresholding, PΘ (β; λ) does not depend on the magnitude of β, and we can get a finite complexity rate in the oracle inequality. Also, R0 can be slightly relaxed, by replacing KPΘ (β; λ) with KP0 (β; λ) in (15). We denote the modified version by R′0 (δ, ϑ, K, β, λ). Corollary 2. Suppose that Θ corresponds to a bounded nonconvex penalty satisfying PΘ (t; λ) ≤ Cλ2 , ∀t ∈ R, for some constant C > 0. Then in the setting of Theorem 2, E[kX β̂ − Xβ ∗ k22 ] . kXβ − Xβ ∗ k22 + σ 2 J(β) log(ep) + σ 2 , (17) provided R′0 (δ, ϑ, K, β, λ) is satisfied for some constants δ > 0, ϑ > 0, K ≥ 0. Remark 1. The right-hand side of the oracle inequalities involves a bias term kXβ − Xβ ∗ k22 and a complexity term PΘ (β; λ). Letting β = β ∗ in, say, (16), the bias vanishes, and we obtain a prediction error bound of the order σ 2 J ∗ log(ep) (omitting constant factors), where J ∗ denotes the number of nonzero components in β ∗ . On the other hand, the existence of the bias term ensures the applicability of our results to approximately sparse signals. For example, when β ∗ has many small but nonzero components, we can use a reference β with a much smaller support than J (β ∗ ) to get a lower error bound, as a benefit from the bias-variance tradeoff. Remark 2. When R0 holds with δ > 1, the proof of Theorem 2 shows that the multiplicative constant for kXβ − Xβ ∗ k22 can be as small as 1. The corresponding oracle inequalities are called ‘sharp’ in some works (Koltchinskii et al., 2011). This also applies to Theorem 3. Our proof scheme can also deliver highprobability form results, without requiring an upper bound of kXk2 . Remark 3. Corollary 2 applies to all “hard-thresholding like” Θ, because when Θ(t; λ) = t for \|t\| > cλ, PΘ (t; λ) ≤ c2 λ2 . It is worth mentioning that the error rate of σ 2 J ∗ log(ep) cannot be significantly improved in a minimax sense. In fact, under the Gaussian noise contamination and some regularity conditions, there exist constants C, c > 0 such that inf β̌ supβ∗ :J(β∗ )≤J E[kX(β̌ − β ∗ )k22 )/(CPo (J))] ≥ c > 0, where β̌ denotes an arbitrary estimator of β ∗ and Po (J) = σ 2 {J + J log(ep/J)}. See, e.g., Lounici et al. (2011) for a proof. The bound in (17) achieves the minimax optimal rate up to a mild logarithm factor for any n and p. 8 3.2 P0 -type oracle inequalities under R1 This part uses P0 instead of PΘ to make an oracle bound. We will show that under another type of comparison regularity conditions, all thresholdings can attain the essentially optimal error rate given in Corollary 2. We will also show that in the case of soft-thresholding, our condition is more relaxed than many other assumptions in the literature. A SSUMPTION R1 (δ, ϑ, K, β, λ) Given X, Θ, β, λ, there exist δ > 0, ϑ > 0, K ≥ 0 such that the following inequality holds for any β ′ ∈ Rp ϑPH (β ′ − β; λ) + LΘ ′ kβ − βk22 + PΘ (β; λ) 2 (18) 2−δ ′ 2 ′ 2 kX(β − β)k2 + PΘ (β ; λ) + Kλ J(β). ≤ 2 p Theorem 3. Let β̂ be a Θ-estimator and λ = Aσ log(ep) with A a sufficiently large constant. Then E[kX β̂ − Xβ ∗ k22 ] . kXβ − Xβ ∗ k22 + λ2 J(β) + σ 2 holds for any β ∈ Rp if R1 (δ, ϑ, K, β, λ) is satisfied for some constants δ > 0, ϑ > 0, K ≥ 0. Remark 4. Some fusion thresholdings, like those associated with elastic net, Berhu and Hard-Ridge (cf. Table 1), involve an additional ℓ2 shrinkage. In the situation, the complexity term in the oracle inequality should involve both J(β) and kβk22 . We can modify our regularity conditions to obtain such ℓ0 +ℓ2 bounds using the same proof scheme. The details are however not reported in this paper. In addition, our results can be extended to Θ-estimators with a stepsize parameter. Given λ > 0 and 0 < α ≤ 1, suppose λα is introduced such that αPΘ (t; λ) = PΘ (t; λα ) for any t. Then, for any β̂ as a fixed point of β = Θ(β −αX T Xβ +αX T y; λα ), an analogous result can be obtained (the only change is that LΘ is replaced by LΘ /α). To give some more intuitive regularity conditions, we suppose PΘ is concave on [0, ∞). Examples include ℓr (0 ≤ r ≤ 1), MCP, SCAD, and so on. The concavity implies PΘ (t + s) ≤ PΘ (t) + PΘ (s), and so PΘ (βJ′ ; λ) − PΘ (βJ ; λ) ≤ PΘ ((β ′ − β)J ; λ) and PΘ (βJ′ c ; λ) = PΘ ((β ′ − β)J c ; λ), where J c is the complement of J and βJ is the subvector of β indexed by J . Then R1 is implied by R′1 below for given J = J (β). 9 A SSUMPTION R′1 (δ, ϑ, K, J , λ) Given X, Θ, J , λ, there exist δ > 0, ϑ > 0, K ≥ 0 such that for any ∆ ∈ Rp , PΘ (∆J ; λ) + ϑPH (∆J ; λ) + LΘ k∆k22 2 2−δ kX∆k22 + Kλ2 J + PΘ (∆J c ; λ) − ϑPH (∆J c ; λ), ≤ 2 (19) or (1 + ϑ)PΘ (∆J ; λ) + LΘ 2−δ k∆k22 ≤ kX∆k22 + Kλ2 J + (1 − ϑ)PΘ (∆J c ; λ). 2 2 (20) When Θ is the soft-thresholding, it is easy to verify that a sufficient condition for (20) is √ (1 + ϑ)k∆J k1 ≤ K J kX∆k2 + k∆J c k1 , (21) for some ϑ > 0 and K ≥ 0. (21) has a simper form than R1 . In the following, we give the definitions of the RE and the compatibility condition (Bickel et al., 2009; van de Geer and Bühlmann, 2009) to make a comparison to (21). A SSUMPTION RE(κRE , ϑRE , J ). Given J ⊂ [p], we say that X ∈ Rn×p satisfies RE(κRE , ϑRE , J ), if for positive numbers κRE , ϑRE > 0, JkX∆k22 ≥ κRE k∆J k21 , (22) kX∆k22 ≥ κRE k∆J k22 , (23) (1 + ϑRE )k∆J k1 ≥ k∆J c k1 . (24) or more restrictively, for all ∆ ∈ Rp satisfying Assume RE(κRE , ϑRE , J ) holds. When (1 + ϑRE )k∆J k1 ≤ k∆J c k√1 , (21) holds trivially with ϑ = ϑRE ; otherwise, (22) indicates (1+ϑ)k∆J k1 ≤ K JkX∆k2 √ with K = (1 + ϑRE )/ κRE . So intuitively, we have the following relationship: (23) + (24) ⇒ (22) + (24) ⇒ (21) ⇒ (20) ⇒ (19) ⇒ (18). 10 In particular, R1 is less demanding than RE. Next, let’s compare the regularity conditions required by ΘS and ΘH to achieve the nearly optimal error rate. Recall R1 (δ, ϑ, K, β, λ) and R′0 (δ, ϑ, K, β, λ) in Theorem 3 and Corollary 2, respectively 2−δ kX(β ′ − β)k22 + λkβ ′ k1 + Kλ2 J, 2 2−δ 1 kX(β ′ − β)k22 + PH (β ′ ; λ) + Kλ2 J. ϑPH (β ′ − β; λ) + kβ ′ − βk22 ≤ 2 2 ϑPH (β ′ − β; λ) + λkβk1 ≤ R′0 (δ, ϑ, K, β, λ) implies R1 (δ, ϑ, K + 1, β, λ). Indeed, for ∆ = β ′ − β, λkβk1 − λkβ ′ k1 ≤ λk∆J k1 − λkβJ′ c k1 1 1 ≤ λ2 J + k∆J k22 − PH (βJ′ c ; λ) 2 2 1 1 2 ≤ λ J + k∆J k22 − PH (β ′ ; λ) + PH (βJ′ ; λ) 2 2 1 1 2 ≤ λ J + k∆J k22 − PH (β ′ ; λ) + P0 (βJ′ ; λ) 2 2 1 ≤ λ2 J + k∆J k22 − PH (β ′ ; λ). 2 On the other hand, Corollary 2 studies when all ΘH -estimators have the optimal performance guarantee, while practically, one may initialize (10) with a carefully chosen starting point. Theorem 4. Given any Θ, there exists a Θ-estimator (which minimizes (12)) such that (16) holds without requiring any regularity condition. In particular, if Θ corresponds to a bounded nonconvex penalty as described in Corollary 2, then there exists a Θ-estimator such that (17) holds free of regularity conditions. Theorem 4 does not place any requirement on X. So it seems that applying ΘH may have some further advantages in practice. (How to efficiently pick a ΘH estimator to completely remove all regularity conditions is however beyond the the scope of the current paper. For a possible idea of relaxing the conditions, see Remark 6.) Finally, we make a discussion of the scaling parameter ρ. Our results so far are obtained after performing X ← X/ρ with ρ ≥ kXk2 . The prediction error is invariant to the transformation. But it affects the regularity conditions. 11 Seen from (8), 1/ρ2 is related to the stepsize α appearing in (1), also known as the learning rate in the machine learning literature. From the computational results in Section 2.2, ρ must be large enough to guarantee TISP is convergent. The larger the value of ρ is, the smaller the stepsize is (and so the slower the convergence is). Based on the machine learning literature, slow learning rates are always recommended when training a nonconvex learner (e.g., artificial neural networks). Perhaps interestingly, in addition to computational efficiency reasons, all our statistical analyses caution against using an extremely large scaling when LΘ > 0. For example, R′0 (δ, ϑ, K, β, λ) for an unscaled X reads ϑPH (ρ(β ′ − β); λ) + ρ2 kβ ′ − βk22 /2 ≤ (2 − δ)kX(β ′ − β)k22 /2 + PH (ρβ ′ ; λ) + Kλ2 J, which becomes difficult to hold when ρ is very large. This makes the statistical error bound break down easily. Therefore, a good idea is to have ρ just appropriately large (mildly greater than kXk2 ). The sequential analysis of the iterates in the next part also supports the point. 3.3 Sequential Algorithmic Analysis We perform statistical error analysis of the sequence of iterates defined by TISP: β (t+1) = Θ(β (t) + X T y − X T Xβ (t) ; λ), where kXk2 ≤ 1 and β (0) is the starting point. The study is motivated from the fact that in large-scale applications, Θ-estimators are seldom computed exactly. Indeed, why bother to run TISP till computational convergence? How does the statistical accuracy improve (or deteriorate) at t increases? Lately, there are some key advances on the topic. For example, Agarwal et al. (2012) showed that for convex problems (not necessarily strongly convex), proximal gradient algorithms can be geometrically fast to approach a globally optimal solution β̂ within the desired statistical precision, under a set of conditions. We however care about the statistical error between β (t) and the genuine β ∗ in this work. We will introduce two comparison regularity conditions (analogous to R0 and R1 ) to present both PΘ -type and P0 -type error bounds. Hereinafter, denote (β T Aβ)1/2 by kβkA , where A is a positive semi-definite matrix. A SSUMPTION S0 (δ, ϑ, K, β, β ′ , λ) Given X, Θ, β, β ′ , λ, there exist δ > 0, ϑ > 0, K ≥ 0 such that the following inequality holds LΘ + δ ′ kβ − βk22 2 ≤ kX(β ′ − β)k22 + PΘ (β ′ ; λ) + KPΘ (β; λ). ϑPH (β ′ − β; λ) + 12 (25) A SSUMPTION S1 (δ, ϑ, K, β, β ′ , λ) Given X, Θ, β, β ′ , λ, there exist δ > 0, ϑ > 0, K ≥ 0 such that the following inequality holds LΘ + δ ′ kβ − βk22 + PΘ (β; λ) 2 ≤ kX(β ′ − β)k22 + PΘ (β ′ ; λ) + (K + 1)λ2 J(β). ϑPH (β ′ − β; λ) + (26) (25) and (26) require a bit more than (15) and (18), respectively, due to kXk2 ≤ 1. The theorem and the corollary below perform sequential analysis of the iterates and reveal the explicit roles of δ, ϑ, K (which can often be treated as constants). ∗ (t+1) Theorem 5. Suppose S0 (δ, ϑ, pK, β , β p , λ) is satisfied for some δ > 0, ϑ > 0, K ≥ 0, then for λ = Aσ log(ep)/ (δ ∧ ϑ)ϑ with A sufficiently large, the 2 following error bound holds with probability at least 1 − Cp−cA : 1 + δ (t+1) 1 kβ − β ∗ k2(I−X T X) ≤ kβ (t) − β ∗ k2(I−X T X) + (K + 1)PΘ (β ∗ ; λ), 2 2 (27) where C, c are universal positive constants. Similarly, under the same choice of regularity parameter, if S1 (δ, ϑ, K, β ∗ , β (t) , λ) is satisfied for some δ > 0, ϑ > 0, K ≥ 0, (28) is true with probability at least 2 1 − Cp−cA : 1 + δ (t+1) 1 kβ − β ∗ k2(I−X T X) ≤ kβ (t) − β ∗ k2(I−X T X) + (K + 1)λ2 J ∗ . 2 2 (28) Corollary 3. In the setting of Theorem 5, for any initial point β (0) ∈ Rp , we have κ K ′ PΘ (β ∗ ; λ), 1−κ κ K ′ λ2 J ∗ , ≤ κt kβ (0) − β ∗ k2(I−X T X) + 1−κ kβ (t) − β ∗ k2(I−X T X) ≤ κt kβ (0) − β ∗ k2(I−X T X) + (29) kβ (t) − β ∗ k2(I−X T X) (30) under S0 (δ, ϑ, K, β ∗ , β (s) , λ) and S1 (δ, ϑ, K, β ∗ , β (s) , λ), 0 ≤ s ≤ t − 1, respec2 tively, with probability at least 1 − Cp−cA . Here, κ = 1/(1 + δ), K ′ = 2(K + 1). Remark 5. We can get some sufficient conditions for S0 and S1 , similar to the discussions made in Section 3.2. When kXk2 is strictly less than 1, (25) can be relaxed to ϑPH (β ′ − β; λ) + (LΘ + δ)kβ ′ − βk22 /2 ≤ (2 + δ)kX(β ′ − β)k22 /2 + 13 PΘ (β ′ ; λ) + KPΘ (β; λ) for some δ > 0. The proof in Section 4.4 also gives expectation-form results, with an additional additive term Cσ 2 /(δ ∧ ϑ) in the upper bounds. Similar to Remark 4, we can also study Θ-iterates with stepsize α, in which case the weighting matrix in (27)-(30) changes from I −X T X to I/α − X T X, and the factor (LΘ + δ)/2 in (25) and (26) is replaced by (LΘ + δ)/(2α). Remark 6. Theorem 5 still applies when δ, ϑ, K and λ are dependent on t. For example, if we use a varying threshold sequence, i.e., β (t+1) = Θ(β (t) + X T y − X T Xβ (t) ; λ(t) ), then (30) becomes kβ (t) − β ∗ k2(I−X T X) t ≤ κ kβ (0) − β ∗ k2(I−X T X) ′ +K J ∗ t−1 X κt−s λ2s . s=0 This allows for much larger values of λs to be used in earlier iterations to attain the same accuracy. It relaxes the regularity condition required by applying a fixed threshold level. At the end, we re-state some results under ρ > kXk2 , to get more intuition and implications. For a general X (unscaled), (30) reads kβ (t) − β ∗ k2(ρ2 I−X T X) ≤ κt kβ (0) − β ∗ k2(ρ2 I−X T X) + κ K ′ σ 2 λ2 J ∗ . 1−κ Set ρ to be a number slightly larger than kXk2 , i.e., ρ = (1+ǫ)kXk2 , ǫ > 0. Then, we know that the prediction error kXβ (t) − Xβ ∗ k22 decays geometrically fast to O(σ 2J ∗ log(ep)) with high probability, when ǫ, δ, ϑ, K are viewed as constants; a similar conclusion is true for the estimation error. This is simply due to ρ2 − kXk22 (t) kβ −β ∗ k2X T X ≤ (ρ2 −kXk22 )kβ (t) −β ∗ k22 ≤ kβ (t) −β ∗ k2(ρ2 I−X T X) . 2 kXk2 Accordingly, there is no need to run TISP till convergence—one can terminate the algorithm earlier, at, say, tmax = log{ρ2 kβ (0) − β ∗ k2 /(Kσ 2 λ2 J ∗ )} /log(1/κ), without sacrificing much statistical accuracy. The formula also reflects that the quality of the initial point affects the required iteration number. There are some related results in the literature. (i) As mentioned previously, in a broad convex setting Agarwal et al. (2012) proved the geometric decay of the optimization error kβ (t) − β̂k to the desired statistical precision, where β̂ is the convergent point. Loh and Wainwright (2015) extended the conclusion to a family of nononvex optimization problems, and they showed that when some 14 regularity conditions hold, every local minimum point is close to the authentic β ∗ . In comparison, our results are derived toward the statistical error between β (t) and β ∗ directly, without requiring all local minimum points to be statistically accurate. (ii) Zhang (2010b) showed a similar fast-converging statistical error bound for an elegant multi-stage capped-ℓ1 regularization procedure. However, the procedure carries out an expensive ℓ1 optimization at each step. Instead, (10) involves a simple and cheap thresholding, and our analysis covers any Θ. Acknowledgement The author would like to thank the editor, the associated editor and two anonymous referees for their careful comments and useful suggestions that improve the quality of the paper. The author also appreciates Florentina Bunea for the encouragement. This work was supported in part by NSF grant DMS-1352259. 4 Proofs Throughout the proofs, we use C, c, L to denote universal non-negative constants. They are not necessarily the same at each occurrence. Given any matrix A, we use R(A) to denote its column space. Denote by PA the orthogonal projection matrix onto R(A), i.e., PA = A(AT A)+ AT , where + stands for the MoorePenrose pseudoinverse. Let [p] := {1, · · · , p}. Given J ⊂ [p], we use XJ to denote a column submatrix of X indexed by J . Definition 2. ξ is called a sub-Gaussian random variable if there exist constants 2 C, c > 0 such that P{\|ξ\| ≥ t} ≤ Ce−ct , ∀t > 0. The scale (ψ2 -norm) for ξ is defined as σ(ξ) = inf{σ > 0 : E exp(ξ 2/σ 2 ) ≤ 2}. ξ ∈ Rp is called a sub-Gaussian random vector with scale bounded by σ if all one-dimensional marginals hξ, αi are sub-Gaussian satisfying khξ, αikψ2 ≤ σkαk2 , ∀α ∈ Rp . Examples include Gaussian random variables and bounded random variables such as Bernoulli. Note that the assumption that vec (ǫ) is sub-Gaussian does not imply that the components of ǫ must be i.i.d. We begin with two basic facts. Because they are special cases of Lemma 1 and Lemma 2 in She (2012), respectively, we state them without proofs. 15 Lemma 1. Given an arbitrary thresholding rule Θ, let P be any function satisR \|θ\| fying P (θ; λ) − P (0; λ) = PΘ (θ; λ) + q(θ; λ) where PΘ (θ; λ) , 0 (sup{s : Θ(s; λ) ≤ u} − u) du, q(θ; λ) is nonnegative and q(Θ(t; λ)) = 0 for all t. Then, β̂ = Θ(y; λ) is always a globally optimal solution to minβ 12 ky − βk22 + P (\|β\|; λ). It is the unique optimal solution provided Θ(·; λ) is continuous at \|y\|. Lemma 2. Let Q0 (β) = ky − βk22 /2 + PΘ (\|β\|; λ). Denote by β̂ the unique minimizer of Q0 (β). Then for any δ, Q0 (β̂ + δ) − Q0 (β̂) ≥ (1 − LΘ )kδk22 /2. 4.1 Proof of Theorem 1 Let s(u; λ) := Θ−1 (u; λ) − u for u ≥ 0. Assume β̂ is a local minimum point (the proof for a coordinate-wise minimum point follows the same lines). We write fΘ as f for simplicity. Let δf (β; h) denote the Gateaux differential of f at β with (β) . By the definition of PΘ , δf (β, h) increment h: δf (β; h) = limǫ→0+ f (β+ǫh)−f ǫ 1 p exists for any h ∈ R . Let l(β) = 2 kXβ − yk22 . We consider the following directional vectors: dj = [d1 , · · · , dp ]T with dj = ±1 and dj ′ = 0, ∀j ′ 6= j. Then for any j, δl(β; dj ) = dj xTj (Xβ − y), ( s(\|βj \|)sgn(βj )dj , δPΘ (β; dj ) = s(\|βj \|), (31) if βj = 6 0, if βj = 0. (32) Due to the local optimality of β̂, δf (β̂; dj ) ≥ 0, ∀j. When β̂1 6= 0, we obtain T x1 (X β̂−y)+s(\|β̂1\|; λ)sgn(β̂1 ) = 0. When β̂1 = 0, xT1 (X β̂−y)+s(\|β̂1 \|; λ) ≥ 0 and −xT1 (X β̂−y)+s(\|β̂1 \|; λ) ≥ 0, i.e., \|xT1 (X β̂−y)\| ≤ s(\|β̂1 \|; λ) = Θ−1 (0; λ). To summarize, when f achieves a local minimum or a coordinate-wise minimum (or more generally, a local coordinate-wise minimum) at β̂, we have β̂j 6= 0 ⇒ Θ−1 (\|β̂j \|; λ)sgn(β̂j ) = β̂j − xTj (X β̂ − y) β̂j = 0 ⇒ Θ(xTj (Xβ − y); λ) = 0 (33) (34) When Θ is continuous at β̂j − xTj (X β̂ − y), (33) implies that β̂j = Θ(β̂j − xTj (X β̂ − y); λ). Hence β̂ must be a Θ-estimator satisfying β = Θ(β + X T y − X T Xβ; λ). 16 4.2 Proofs of Theorem 2 and Theorem 3 Given Θ, let β̂ be any Θ-estimator, β be any p-dimensional vector (non-random) and ∆ = β̂ − β. The first result constructs a useful criterion for β̂ on basis of Lemma 1 and Lemma 2. Lemma 3. Any Θ-estimator β̂ satisfies the following inequality for any β ∈ Rp 1 1 kX(β̂ − β ∗ )k22 + ∆T (X T X − LΘ I)∆ 2 2 1 ∗ 2 ≤ kX(β − β )k2 + PΘ (β; λ) − PΘ (β̂; λ) + hǫ, X∆i, 2 (35) where ∆ = β̂ − β. To handle hǫ, X∆i, we introduce another lemma. p Lemma 4. Suppose kXk2 ≤ 1 and let λo = σ log(ep). Then there exist universal constants A1 , C, c > 0 such that for any constants a ≥ 2b > 0, the following event √ 1 1 sup {2hǫ, Xβi − kXβk22 − [PH (β; abA1 λo )]} ≥ aσ 2 t a b β∈Rp (36) 2 occurs with probability at most C exp(−ct)p−cA1 , where t ≥ 0. The lemma plays an important role in bounding the last stochastic term in (35). Its proof is based on the following results. Lemma 5. Suppose kXk2 ≤ 1. There exists a globally optimal solution β o to minβ 12 ky − Xβk22 + PH (β; λ) such that for any j : 1 ≤ j ≤ p, either βjo = 0 or \|βjo\| ≥ λ. p Lemma 6. Given X ∈ Rn×p and J : 1 ≤ J ≤ p, define Γ′J = {α ∈ R : kαk2 ≤ p ′ 2 1, α ∈ R(XJ ) for some J : \|J \| = J}. Let Po (J) = σ {J + log J }. Then for any t ≥ 0, ! p (37) P sup hǫ, αi ≥ tσ + LPo′ (J) ≤ C exp(−ct2 ), α∈Γ′J where L, C, c > 0 are universal constants. 17 √ 1 1 Let R = sup1≤J≤p sup∆∈ΓJ {hǫ, X∆i − 2b PH (∆; abA1 λo ) − 2a kX∆k22 }, o with λ , A1 given in Lemma 4. (The starting value of J is 1 because when J(∆) = 0, hǫ, X∆i = 0.) Substituting it into (35) gives 1 1 kX(β̂ − β ∗ )k22 + ∆T (2X T X − LΘ I)∆ 2 2 √ 1 1 ∗ 2 ≤ kX(β − β )k2 + PΘ (β; λ) − PΘ (β̂; λ) + PH (∆; abA1 λo ) 2 2b 1 1 2 2 + kX∆k2 + kX∆k2 + R 2a 2 √ 1 1 ∗ 2 ≤ kX(β − β )k2 + PΘ (β; λ) − PΘ (β̂; λ) + PH (∆; abA1 λo ) 2 2b 1 1 2 + (1 + )kX∆k2 + R. 2 a Because P(R ≥ aσ 2 t) ≤ C exp(−ct), we know E[R] . aσ 2 . √ Let λ = Aλo with A = A1 ab and set b ≥ 1/(2ϑ). The regularity condition R0 (δ, ϑ, K, β, λ) implies that LΘ 2−δ 1 PH (∆; λ) + k∆k22 ≤ kX∆k22 + PΘ (β̂; λ) + KPΘ (β; λ). (38) 2b 2 2 Choose a to satisfy a > 1/δ, a ≥ 2b. Combining the last two inequalities gives E[kX(β̂ − β ∗ )k22 ] ≤kX(β − β ∗ )k22 + 2(K + 1)PΘ (β; λ) + E[(1 + .kX(β − β ∗ )k22 + PΘ (β; λ) + σ 2 , 1 − δ)kX∆k22 ] + 2 E[R] a (39) with the last inequality due to kX∆k22 ≤ (1+1/c)kX(β−β ∗ )k22 +(1+c)kX(β̂− β ∗ )k22 for any c > 0. The proof of Theorem 3 follows the lines of the proof of Theorem 2, with (38) replaced by 1 LΘ 2−δ PH (∆; λ) + k∆k22 + PΘ (β; λ) ≤ kX∆k22 + PΘ (β̂; λ) + Kλ2 J(β), 2b 2 2 and (39) replaced by E[kX(β̂ − β ∗ )k22 ] ≤kX(β − β ∗ )k22 + 2Kλ2 J(β) + E[(1 + .kX(β − β ∗ )k22 + λ2 J(β) + σ 2 . 18 1 − δ)kX∆k22 ] + 2 E[R] a The details are omitted. 4.3 Proof of Theorem 4 From the proof of Lemma 5, there exists a Θ-estimator β̂ which minimizes f (β) = l(β) + PΘ (β; λ). This means that the term 12 ∆T (X T X − LΘ I)∆ can be dropped from (35). Following the lines of Section 4.2, (17) holds under a modified version of R0 (δ, ϑ, K, β, λ), which replaces (15) with ϑPH (β ′ − β; λ) ≤ 1−δ kX(β ′ − β)k22 + PΘ (β ′ ; λ) + KPΘ (β; λ). 2 (40) Using the sub-additivity of PH , we know that any design matrix satisfies (40) for any 0 < ϑ ≤ 1, δ ≤ 1, K ≥ ϑ. 4.4 Proof of Theorem 5 and Corollary 3 Let f (β) = l(β) + PΘ (β; λ) where l(β) = 21 kXβ − yk22 . Lemma 7. Let β (t+1) = Θ(β (t) + X T y − X T Xβ (t) ; λ). Then the following ‘triangle inequality’ holds for any β ∈ Rp 1 1 − LΘ (t+1) kβ − βk22 + kβ (t+1) − β (t) k2I−X T X 2 2 1 (t) ≤ kβ − βk2I−X T X + f (β) − f (β (t+1) ). 2 Letting β = β ∗ in the lemma, we have 1 (t+1) 1 kβ − β ∗ k2X T X+(1−LΘ )I + kβ (t+1) − β (t) k2I−X T X + PΘ (β (t+1) ; λ) 2 2 1 (t) ∗ 2 (t+1) ≤ kβ − β kI−X T X + hǫ, X(β − β ∗ )i + PΘ (β ∗ ; λ). 2 Moreover, under S0 (δ, ϑ, K, β ∗ , β ′ , λ) with β ′ = β (t+1) , 1 + δ (t+1) kβ − β ∗ k22 − KPΘ (β ∗ ; λ) 2 1 (t+1) 1 ∗ 2 ≤ kβ − β kX T X+(1−LΘ )I + PΘ (β (t+1) ; λ) + kβ (t+1) − β ∗ k2X T X . 2 2 ϑPH (β (t+1) − β ∗ ; λ) + 19 Combining the last two inequalities gives 1 + δ (t+1) 1 kβ − β ∗ k2I−X T X + kβ (t+1) − β (t) k2I−X T X 2 2 δ (t+1) − β ∗ k2X T X + ϑPH (β (t+1) − β ∗ ; λ) + kβ 2 1 ≤ kβ (t) − β ∗ k2I−X T X + (K + 1)PΘ (β ∗ ; λ) + hǫ, X(β (t+1) − β ∗ )i. 2 p Let ΓJ = {β ∈ Rp : J(β) = J}, λo = σ log(ep). We define an event E with its complement given by √ 1 1 E c , {sup{2hǫ, Xβi − kXβk22 − [PH (β; abA1 λo )]} ≥ 0}. a b β By Lemma 4, there exists a universal constant L such that for any A21 ≥ L, a ≥ 2 2b > 0, P (E c) ≤ Cp−cA1 . Clearly, E implies √ 1 1 kβ (t+1) − β ∗ k2X T X + PH (β (t+1) − β ∗ ; abA1 λo ). 2a 2b (41) √ o √ Take b = 1/(2ϑ), a = 1/(δ ∧ ϑ), A1 ≥ L, and λ = A1 abλ . Then, on E we get the desired statistical accuracy bound hǫ, X(β (t+1) − β ∗ )i ≤ 1 + δ (t+1) 1 kβ − β ∗ k2I−X T X ≤ kβ (t) − β ∗ k2I−X T X + (K + 1)PΘ (β ∗ ; λ). 2 2 The bound under S1 can be similarly proved. Noticing that (41) holds for any t, Corollary 3 is immediately true. 4.5 Proofs of Lemmas 4.5.1 Proof of Lemma 3 Let f (β) = l(β) + PΘ (β; λ) with l(β) = 12 kXβ − yk22. Define 1 g(β, γ) = l(β) + h∇l(β), γ − βi + kγ − βk22 + PΘ (γ; λ). 2 Given β, g(β, γ) can be expressed as 1 kγ − (β − ∇l(β))k22 + PΘ (γ; λ) + c(β), 2 20 (42) where c(β) depends on β only. Let β̂ be a Θ-estimator satisfying β̂ = Θ(β̂ − X T X β̂ + X T y; λ). Based on Lemma 1 and Lemma 2, we have g(β̂, β̂ + ∆) − g(β̂, β̂) ≥ 1 − LΘ k∆k22 , 2 from which it follows that 1 f (β̂ + ∆) − f (β̂) ≥ ∆T (X T X − LΘ I)∆. 2 This holds for any ∆ ∈ Rp . 4.5.2 Proof of Lemma 4. Let 1 lH (β) = 2hǫ, Xβi − kXβk22 − a 1 l0 (β) = 2hǫ, Xβi − kXβk22 − a √ 1 [PH (β; abA0 λo )] b √ 1 [P0 (β; abA0 λo )], b and EH = {supβ∈Rp lH (β) ≥ atσ 2 }, and E0 = {supβ∈Rp l0 (β) ≥ atσ 2 }. Because P0 ≥ PH , E0 ⊂ EH . We prove that EH = E0 . The occurrence of EH implies that lH (β o ) ≥ atσ 2 for any β o defined by √ 1 1 β o ∈ arg min kXβk22 − 2hǫ, Xβi + [PH (β; abA0 λo )], β a b With a ≥ 2b > 0, Lemma exists at least one global minimizer √ 5 states that there √ β oo satisfying PH (β oo ; abA1 λo ) = P0 (β oo ; abA1 λo ) and thus lH (β oo ) = l0 (β oo ). This means that sup l0 (β) ≥ l0 (β oo ) = lH (β oo ) ≥ atσ 2 . So EH ⊂ E0 , and it suffices to prove E0c occurs with high probability, or more specifically, P(E0 ) ≤ 2 C exp(−ct)p−cA1 . Given 1 ≤ J ≤ p, define ΓJ = √{β ∈ Rp : J(β) = J}. Let R = 1 1 P0 (β; abA1 λo ) − 2a kXβk22 }. We will use sup1≤J≤p supβ∈ΓJ {hǫ, Xβi − 2b Lemma 6 to bound its tail probability. Let Po′ (J) = σ 2 {J + log Jp }. We claim that P[ sup {hǫ, Xβi − β∈ΓJ 1 kXβk22 − aLPo′ (J)} > atσ 2 ] ≤ C exp(−ct). 2a 21 (43) Indeed, 1 2hǫ, Xβi − kXβk22 − 2aLPo′ (J) a p 1 ≤2hǫ, Xβ/kXβk2 ikXβk2 − 2kXβk2 LPo′ (J) − kXβk22 2a p 1 =2kXβk2 hǫ, Xβ/kXβk2 i − LPo′ (J) − kXβk22 2a p 1 ≤2kXβk2 hǫ, Xβ/kXβk2 i − LPo′ (J) − kXβk22 2a + 2 p ≤2a hǫ, Xβ/kXβk2 i − LPo′ (J) , (44) + where the last inequality is due to Cauchy-Schwarz inequality. (43) now follows from Lemma 6. √ Set A1 ≥ 4 L. We write P0 (β; λo ) with β ∈ ΓJ as P0 (J; λo ). Noticing ′ some basic facts that ≤ CP0p (J; λo ) due to Stirling’s p (i) Po (J) ≤ CJ log(ep) p approximation, (ii) (A21 /2)P0 (J; λo ) ≥ LPo′ (J) + cA21 P0 (J; λo ) for some c > 0, and (iii) J log(ep) ≥ log p + J for any J ≥ 1, we get P(R ≥ aσ 2 t) ! 2 p q X P a sup hǫ, Xβ/kXβk2 i − (A21 /2)P0 (J; λo ) ≤ ≥ aσ 2 t J=1 p β∈ΓJ + q X √ P( sup hǫ, αi − (A21 /2)P0(J; λo ) ≥ σ t) = J=1 p ≤ X ≤ X J=1 p α∈Γ′J q p √ ′ P( sup hǫ, αi − LPo (J) ≥ tσ + cA21 P0 (J; λo )) α∈Γ′J C exp(−ct) exp{−cA21 (J + log(p))} J=1 ≤C exp(−ct) p X exp(−cA21 log p) exp(−cA21 J) J=1 −cA21 ≤C exp(−ct)p , where the last inequality due to the sum of geometric series. 22 4.5.3 Proof of Lemma 5. Similar to the proof of Lemma 3, we set fH (β) = l(β) + PH (β; λ) with l(β) = 1 kXβ − yk22 and construct gH (β, γ) = fH (γ) + 12 kγ − βk22 − (l(γ) − l(β) − 2 h∇l(β), γ − βi). Under kXk2 ≤ 1, for any (β, γ), 1 gH (β, γ) − fH (γ) = (γ − β)T (I − X T X)(γ − β) ≥ 0. 2 o Let β be a globally optimal solution to minβ fH (β). Then γ o := ΘH (β o − X T Xβ o + X T y; λ) gives fH (γ o ) ≤ gH (β o , γ o ) ≤ gH (β o , β o ) = fH (β o ), with the second inequality due to Lemma 1. Therefore, γ o must also be a global minimizer of fH , and by definition, γ o demonstrates a threshold gap as desired. 4.5.4 Proof of Lemma 6. By definition, {hǫ, αi : α ∈ Γ′J } is a stochastic process with sub-Gaussian increments. The induced metric on Γ′J is Euclidean: d(α1 , α2 ) = σkα1 − α2 k2 . To bound the metric entropy log N (ε, Γ′J , d), where N (ε, Γ′J , d) is the smallest cardinality of an ε-net that covers Γ′J under d, we notice that α is in a Jdimensional ball in Rp . The number of such balls {PXJ ∩ Bp (0, 1) : J ⊂ [p]} is at most Jp , where Bp (0, 1) denotes the unit ball in Rp . By a standard volume argument (see, e.g., Vershynin (2012)), p p Cσ J ′ + J log(Cσ/ε), (45) ) = log ( log N (ε, Γr,J , d) ≤ log ε J J where C is a universal constant. The conclusion follows from Dudley’s integral bound (Talagrand, 2005). 4.5.5 Proof of Lemma 7 We use the notation in the proof of Lemma 3 with g defined in (42). By Lemma Θ 1 and Lemma 2, we obtain g(β (t) , β) − g(β (t) , β (t+1) ) ≥ 1−L kβ (t+1) − βk22 , 2 namely, 1 h∇l(β (t) ), β − β (t+1) i + PΘ (β) − PΘ (β (t+1) ) + kβ − β (t) k22 2 1 (t) 1 − L Θ − kβ − β (t+1) k22 ≥ kβ (t+1) − βk22 . 2 2 23 To cancel the first-order term, we give two other inequalities based on secondorder lower/upper bounds: 1 l(β) − l(β (t) ) − h∇l(β (t) ), β − β (t) i ≥ kβ (t) − βk2X T X , 2 1 l(β (t) ) + h∇l(β (t) ), β (t+1) − β (t) i − l(β (t+1) ) ≥ − kβ (t+1) − β (t) k2X T X . 2 Adding the three inequalities together gives the triangle inequality. References Agarwal, A., Negahban, S., and Wainwright, M. J. (2012). Fast global convergence of gradient methods for high-dimensional statistical recovery. Ann. Statist., 40(5):2452–2482. Bickel, P. J., Ritov, Y., and Tsybakov, A. B. (2009). 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arXiv:1709.07708v1 [math.GR] 22 Sep 2017 COMPATIBLE ACTIONS AND NON-ABELIAN TENSOR PRODUCTS VALERIY G. BARDAKOV AND MIKHAIL V. NESHCHADIM Abstract. For a pair of groups G, H we study pairs of actions G on H and H on G such that these pairs are compatible and non-abelian tensor products G ⊗ H are defined. 1. Introduction R. Brown and J.-L. Loday [1, 2] introduced the non-abelian tensor product G ⊗ H for a pair of groups G and H following works of C. Miller [6], and A. S.-T. Lue [5]. The investigation of the non-abelian tensor product from a group theoretical point of view started with a paper by R. Brown, D. L. Johnson, and E. F. Robertson [3]. The non-abelian tensor product G ⊗ H depends not only on the groups G and H but also on the action of G on H and on the action of H on G. Moreover these actions must be compatible (see the definition in Section 2). In the present paper we study the following question: what actions are compatible? The paper is organized as follows. In Section 2, we recall a definition of non-abelian tensor product, formulate some its properties and give an answer on a question of V. Thomas, proving that there are nilpotent group G and some group H such that in G ⊗ H the derivative subgroup [G, H] is equal to G. In the Section 3 we study the following question: Let a group H acts on a group G by automorphisms, is it possible to define an action of G on H such that this pair of actions are compatible? Some necessary conditions for compatibility of actions will be given and in some cases will be prove a formula for the second action if the first one is given. In the Section 4 we construct pairs compatible actions for arbitrary groups and for 2-step nilpotent groups give a particular answer on the question from Section 3. In Section 5 we study groups of the form G ⊗ Z2 and describe compatible actions. 2. Preliminaries In this article we will use the following notations. For elements x, y in a group G, the conjugation of x by y is xy = y −1 xy; and the commutator of x and y is [x, y] = x−1 xy = x−1 y −1 xy. We write G′ for the derived subgroup of G, i.e. G′ = [G, G]; Gab for the abelianized group G/G′ ; the second hypercenter ζ2 G of G is the subgroup of G such that ζ2 G/ζ1 G = ζ1 (G/ζ1 G), Date: March 24, 2018. 2010 Mathematics Subject Classification. Primary 20E22; Secondary 20F18, 20F28. Key words and phrases. tensor product; compatible action, nilpotent group. 1 2 V. G. BARDAKOV AND M. V. NESHCHADIM where ζ1 G = Z(G) is the center of a group G. Recall the definition of the non-abelian tensor product G ⊗ H of groups G and H (see [1, 2]). It is defined for a pair of groups G and H where each one acts on the other (on right) G × H −→ G, (g, h) 7→ g h ; H × G −→ H, (h, g) 7→ hg and on itself by conjugation, in such a way that for all g, g1 ∈ G and h, h1 ∈ H, −1 g h1 h g1 h1 g1−1 (hg1 ) . and h(g ) = hh1 g = g In this situation we say that G and H act compatibly on each other. The non-abelian tensor product G ⊗ H is the group generated by all symbols g ⊗ h, g ∈ G, h ∈ H, subject to the relations gg1 ⊗ h = (gg1 ⊗ hg1 )(g1 ⊗ h) and g ⊗ hh1 = (g ⊗ h1 )(gh1 ⊗ hh1 ) for all g, g1 ∈ G, h, h1 ∈ H. In particular, as the conjugation action of a group G on itself is compatible, then the tensor square G⊗G of a group G may always be defined. Also, the tensor product G ⊗ H is defined if G and H are two normal subgroups of some group M and actions are conjugations in M . The following proposition is well known. We give a proof only for fullness. Proposition 2.1. 1) Let G and H be abelian groups. Independently on the action of G on H and H on G, the group G ⊗ H is abelian. 2) (See [2, Proposition 2.4]) Let G and H be arbitrary groups. If the actions of G on H and H on G are trivial, then the group G ⊗ H ∼ = Gab ⊗Z H ab is the abelian tensor product. Proof. 1) We have the equality (g ⊗ h)g1 ⊗h1 = g [g1 ,h1] ⊗ h[g1 ,h1 ] , where g[g1 ,h1 ] is the action of the commutator [g1 , h1 ] ∈ G by conjugation on g, but G is abelian and g [g1 ,h1 ] = g. Analogously, h[g1 ,h1] = h. Hence, G ⊗ H is abelian. 2) From the previous formula and triviality actions we have −1 g1 h1 −1 g1 h1 −1 h−1 1 g 1 h1 g 1 h1 [g1 ,h1 ] g1−1 h−1 1 = g h1 = g. = g g1 = g g1 g =g = g g1 Analogously, h[g1 ,h1] = h. Hence, G ⊗ H is abelian. Remind presentation of non-abelian tensor product as a central extension (see [4]). The derivative subgroup of G by H is called the following subgroup DH (G) = [G, H] = hg −1 g h \| g ∈ G, h ∈ Hi. The map κ : G ⊗ H −→ DH (G) defined by κ(g ⊗ h) = g −1 gh is a homomorphism, its kernel A = ker(κ) is the central subgroup of G ⊗ H and G acts on G ⊗ H by the rule (g ⊗ h)x = gx ⊗ hx , x ∈ G, i.e. there exists the short exact sequence 1 −→ A −→ G ⊗ H −→ DH (G) −→ 1. COMPATIBLE ACTIONS AND NON-ABELIAN TENSOR PRODUCTS 3 In this case A can be viewed as Z[DH (G)]-module via conjugation in G ⊗ H, i.e. under the action induced by setting a · g = x−1 ax, a ∈ A, x ∈ G ⊗ H, κ(x) = g. The following proposition gives an answer on the following question: is there nonabelian tensor product G ⊗ H such that [G, H] = G? which of V. Thomas formulated in some letter to the authors. Proposition 2.2. Let G = Fn /γk Fn , k ≥ 2, be a free nilpotent group of rank n ≥ 2 and H = Aut(G) is its automorphism group. Then DH (G) = [G, H] = G. Proof. Let Fn be a free group of rank n ≥ 2 with the basis x1 , . . . , xn , G = Fn /γk Fn be a free k − 1-step nilpotent group for k ≥ 2. Let G acts trivially on H and elements of H act by automorphisms on G. It is easy to see that these actions are compatible. Let us show that in this case [G, H] = G. To do it, let us prove that x1 lies in [G, H]. Take ϕ1 ∈ H = Aut(G), which acts on the generators of G by the rules: ϕ1 ϕ1 ϕ1 1 xϕ 1 = x1 , x2 = x2 x1 , x3 = x3 , . . . , xn = xn . Then ϕ1 −1 ϕ1 −1 ϕ1 −1 ϕ1 x−1 1 x1 = 1, x2 x2 = x1 , x3 x3 = 1, . . . , xn xn = 1. Hence the generator x1 lies in [G, H]. Analogously, x2 , x3 , . . . , xn lie in [G, H]. This completes the proof. 3. What actions are compatible? In this section we study Question 1. Let a group H acts on a group G by automorphisms. Is it possible to define an action of G on H such that this pair of actions are compatible? Consider some examples. Example 3.1. Let us take G = {1, a, a2 } ∼ = Z3 , H = {1, b, b2 } ∼ = Z3 . In dependence on actions we have three cases. 1) If the action of H on G and the action of G on H are trivial, then by the second part of Proposition 2.1 G ⊗ H = Z3 ⊗Z Z3 ∼ = Z3 is abelian tensor product. 2) Let H acts non-trivially on G, i.e. ab = a2 and the action G on H is trivial. It is not difficult to check that G and H act compatibly on each other. To find DH (G) = [G, H] we calculate [a, b] = a−1 ab = a2 a2 = a. Hence, DH (G) = G. But DG (H) = 1. By the definition, G ⊗ H is generated by elements a ⊗ b, a2 ⊗ b, a ⊗ b2 , a2 ⊗ b2 . Using the defining relations: gg1 ⊗ h = (g g1 ⊗ hg1 )(g1 ⊗ h), g ⊗ hh1 = (g ⊗ h1 )(gh1 ⊗ hh1 ), 4 V. G. BARDAKOV AND M. V. NESHCHADIM we find a2 ⊗b = (aa ⊗ba )(a⊗b) = (a⊗b)2 , a⊗b2 = (a⊗b)(ab ⊗bb ) = (a⊗b)(a2 ⊗b) = (a⊗b)3 . On the other side 1 = a2 a ⊗ b = (a2 ⊗ ba )(a ⊗ b) = (a ⊗ b)3 . Hence, a ⊗ b2 = a2 ⊗ b2 = 1 and in this case we have the same result: Z3 ⊗ Z3 = Z3 . 3) Let H acts non-trivially on G, i.e. ab = a2 and G acts non-trivially on H. In this case G and H act non-compatibly on each other. Indeed, a) a(b but Hence, the equality a a−1 b a a 2 = ab = (a2 )b = a, (ba ) does not hold. = (ab )a = (a2 )2 = a2 . b a a−1 a = Let G, H be some groups. Actions of G on H and H on G are defined by homomorphisms β : G → Aut(H), α : H → Aut(G), and by definition gh = gα(h) , hg = hβ(g) , g ∈ G, h ∈ H. The actions (α, β) are compatible, if g 1 −1 α(h) β(g1 ) )= g g1 gα(h and β (g α(h1 ) ) h = h−1 1 h β(g) h1 for all g, g1 ∈ G, h, h1 ∈ H. In this case we will say that the pair (α, β) is compatible. Rewrite these equalities in the form (1) α hβ(g1 ) = gb1 −1 α(h)gb1 and c1 −1 β(g)c h1 , β gα(h1 ) = h (2) where b g is the inner automorphism of G which is induced by conjugation of g, i.e. gb : g1 7→ g−1 g1 g, g, g1 ∈ G, and analogously, b h is the inner automorphism of H which is induced by the conjugation of h, i.e. b h : h1 7→ h−1 h1 h, h, h1 ∈ H. COMPATIBLE ACTIONS AND NON-ABELIAN TENSOR PRODUCTS 5 Theorem 3.2. 1) If the pair (α, β) defines compatible actions of H on G and G on H, then the following inclusions hold NAut(G) (α(H)) ≥ Inn(G), NAut(H) (β(G)) ≥ Inn(H). Here Inn(G) and Inn(H) are the subgroups of inner automorphisms. 2) If α : H → Aut(G) is an embedding and NAut(G) (α(H)) ≥ Inn(G), then defining β : G → Aut(H) by the formula β(g) : h 7→ α−1 gb−1 α(h)b g , h ∈ H, we get the compatible actions (α, β). Proof. The first claim immediately follows from the relations (1), (2). To prove the second claim it is enough to check (2), or that is equivalent, the equality β(g) h1 β (g α(h1 ) ) h−1 1 . (3) h = h Using the definition β, rewrite the left side of (3): −1 α(h1 ) \ ) = α−1 g\ α(h1 ) α(h)g α(h1 ) . hβ (g (4) Rewrite the right side of (3): β(g) h1 −1 β(g) −1 −1 h−1 = h−1 h1 = h−1 g α(h1 hh−1 g )h1 . h 1 1 (h1 hh1 ) 1 α (b 1 )b (5) From (4) and (5): −1 α α(h1 ) g\ −1 α(h1 ) α(h)g\ −1 −1 = h−1 g α(h1 hh−1 g )h1 . 1 α (b 1 )b Using the homomorphism α: α(h1 ) g\ −1 α(h1 ) = α h−1 α−1 (b g )h1 = α(h)g\ g−1 α(h1 hh−1 1 )b 1 = α(h1 )−1 gb−1 α(h1 hh−1 g α(h1 ) = 1 )b α(h1 ) = α(h1 )−1 gb−1 α(h1 )α(h)α(h1 )−1 )b g α(h1 ) = g\ −1 α(h1 ) . α(h)g\ In the last equality we used the formula Hence, the equality (3) holds. α(h1 ) . α(h1 )−1 gbα(h1 ) = g\ Question 2. Are the inclusions NAut(G) (α(H)) ≥ Inn(G), NAut(H) (β(G)) ≥ Inn(H) sufficient for compatibility of the pare (α, β)? 6 V. G. BARDAKOV AND M. V. NESHCHADIM 4. Compatible actions for nilpotent groups At first, recall the following definition. Definition 4.1. Let G and H be groups and G1 E G, H1 E H are their normal subgroups. We will say that G is comparable with H with respect to the pare (G1 , H1 ), if there are homomorphisms ϕ : G −→ H, ψ : H −→ G, such that x ≡ ψϕ(x)(mod G1 ), for all x ∈ G, y ∈ H, i.e. y ≡ ϕψ(y)(mod H1 ) x−1 · ψϕ(x) ∈ G1 , y −1 · ϕψ(y) ∈ H1 . Note that if G1 = 1, H1 = 1, then ϕ, ψ are mutually inverse isomorphisms. The following theorem holds. Theorem 4.2. Let G, H be groups and there exist homomorphisms ϕ : G −→ H, ψ : H −→ G, such that x ≡ ψϕ(x)(mod ζ2 G), y ≡ ϕψ(y)(mod ζ2 H) for all x ∈ G, y ∈ H. Then the action of G on H and the action of H on G by the rules xy = ψ(y)−1 xψ(y), y x = ϕ(x)−1 yϕ(x), x ∈ G, y ∈ H, are compatible, i.e. the following equalities hold x(y x1 ) = ((xx1 )y )x1 , −1 y (x y1 ) = ((y y1 )x )y1 , −1 x, x1 ∈ G, y, y1 ∈ H. Proof. Let us prove that the following relation holds x(y x1 ) = ((xx1 )y )x1 . −1 For this denote the left hand side of this relation by L and transform it: L = x(y x1 ) = xϕ(x1 ) −1 yϕ(x 1) = ψ(ϕ(x1 )−1 y −1 ϕ(x1 ))xψ(ϕ(x1 )−1 yϕ(x1 )) = = (ψϕ(x1 ))−1 ψ(y)−1 (ψϕ(x1 ))x(ψϕ(x1 ))−1 ψ(y)(ψϕ(x1 )) = −1 −1 −1 = (c(x1 )−1 x−1 1 ψ(y) x1 c(x1 ))x(c(x1 ) x1 ψ(y)x1 c(x1 )). Here ψϕ(x1 ) = x1 c(x1 ), c(x1 ) ∈ ζ2 G. Since c(x1 ) ∈ ζ2 G, then the commutator [x−1 1 ψ(y)x1 , c(x1 )] lies in the center of G. Hence L = xx 1 −1 ψ(y)x1 . Denote the right hand side of this relation by R and transform it: R = ((xx1 )y )x1 = ((xx1 )ψ (y))x1 = xx1 −1 −1 −1 ψ(y)x1 . We see that L = R, i.e. the first relation from the definition of compatible action holds. The checking of the second relation is the similar. COMPATIBLE ACTIONS AND NON-ABELIAN TENSOR PRODUCTS 7 From this theorem we have particular answer on Question 1 for 2-step nilpotent groups. Corollary 4.3. If G, H are 2-step nilpotent groups, then any pare of homomorphisms ϕ : G −→ H, ψ : H −→ G define the compatible action. Problem 1. Let G and H be free 2-step nilpotent groups. By Corollary 4.3, any pair of homomorphisms (ϕ, ψ), where ϕ ∈ Hom(G, H), ψ ∈ Hom(H, G) defines a tensor product M (ϕ, ψ) = G ⊗ H. Give a classification of the groups M (ϕ, ψ). Note that for arbitrary groups Corollary 4.3 does not hold. Indeed, let G = hx1 , x2 i, H = hy1 , y2 i be free groups of rank 2. Define the homomorphisms ϕ : G −→ H, ψ : H −→ G by the rules ϕ(x1 ) = y1 , ϕ(x2 ) = y2 , ψ(y1 ) = ψ(y2 ) = 1. Then ϕ(x1 ) y2x1 = y2 = y2y1 6= y2 , i.e. the conditions of compatible actions does not hold. 5. Tensor products G ⊗ Z2 Note that the group Aut(Z2 ) is trivial and hence, any group G acts on Z2 only trivially. This section is devoted to the answer on the following question. Question 3. Let G be a group and ψ ∈ Aut(G) be an automorphism of order 2. Let Z2 = hϕi and α : Z2 −→ Aut(G) such that α(ϕ) = ψ. Under what conditions the pare (α, 1) is compatible? If ψ ∈ Aut(G) is trivial automorphism, then by the second part of Proposition 2.1 G ⊗ Z2 = Gab ⊗Z Z2 is an abelian tensor product. In the general case we have Proposition 5.1. Let 1) G be a group, 2) Z2 = hϕi be a cyclic group of order two with the generator ϕ, 3) α : Z2 −→ Aut(G) be a homomorphism, β = 1 : G → Aut(Z2 ) be the trivial homomorphism, Then the pare of actions (α, β) is compatible if and only if for any g ∈ G holds g α(ϕ) = gc(g), where c(g) is a central element of G such that c(g)α(ϕ) = c(g)−1 . In particular, if the center of G is trivial, then G ⊗ Z2 = Gab ⊗Z Z2 . 8 V. G. BARDAKOV AND M. V. NESHCHADIM Proof. Since Inn(G) normalizes α(Z2 ), then for every g ∈ G holds gb−1 α(ϕ)b g = α(ϕ). Using this equality for arbitrary element x ∈ G we get g−1 gα(ϕ) xα(ϕ) (g−1 gα(ϕ) )−1 = xα(ϕ) . Since xα(ϕ) is an arbitrary element of G, then c(g) is a central element of G. Applying α(ϕ) to the equality g α(ϕ) = gc(g) we have 2 g = g α(ϕ) = gα(ϕ) c(g)α(ϕ) = gc(g)c(g)α(ϕ) , that is c(g)α(ϕ) = c(g)−1 . For an arbitrary abelian group A we know that A ⊗Z Z = A. The following proposition is some analog of this property for non-abelian tensor product. Proposition 5.2. Let A be an abelian group, Z2 = hϕi is the cyclic group of order 2 and ϕ acts on the elements of A by the following manner aϕ = a−1 , a ∈ A. Then the non-abelian tensor product A ⊗ Z2 is defined and there is an isomorphism A ⊗ Z2 ∼ = A. Proof. It is not difficult to check that defined actions are compatible. Since A acts on Z2 trivially and A is abelian, then the defining relations of the tensor product: aa1 ⊗ h = (aa1 ⊗ ha1 )(a1 ⊗ h), a, a1 ∈ A, h ∈ Z2 , have the form aa1 ⊗ h = (a ⊗ h)(a1 ⊗ h) = (a1 ⊗ h)(a ⊗ h). (1) The relations a ⊗ hh1 = (a ⊗ h1 )(ah1 ⊗ hh1 ), a ∈ A, h, h1 ∈ Z2 , give only one non-trivial relation 1 = a ⊗ ϕ2 = (a ⊗ ϕ)(a−1 ⊗ ϕ), a ∈ A, which follows from (1). Since the set of relations (1) is a full system of relations for A ⊗ Z2 , then there exists the natural isomorphism of A ⊗ Z2 on A that is defined by the formular a ⊗ ϕ 7→ a, a ∈ A. Acknowledgement. The authors gratefully acknowledge the support from the RFBR-16-01-00414 and RFBR-15-01-00745. Also, we thank S. Ivanov, A. Lavrenov and V. Thomas for the interesting discussions and useful suggestions. COMPATIBLE ACTIONS AND NON-ABELIAN TENSOR PRODUCTS 9 References [1] R. Brown, J.-L. Loday, Excision homotopique en basse dimension, C. R. Acad. Sci. Paris Ser. I Math. 298 (15) (1984), 353–356. [2] R. Brown, J.-L. Loday, Van Kampen theorems for diagrams of spaces, Topology 26 (3) (1987), 311–335, with an appendix by M. Zisman. [3] R. Brown, D. L. Johnson, E. F. Robertson, Some computations of non-abelian tensor products of groups, J. Algebra, 1987, 111, 177–202. [4] G. Donadze, M. Larda, V. Thomas, More on the non-abelian tensor product and the Bogomolov multiplier, Preprint, 2015, 16 pp. [5] A. S.-T. Lue, The Ganea map for nilpotent groups, J. London Math. Soc. 14, 309–312, (1976). [6] C. Miller, The second homology group of a group; relations among commutators, Proceedings AMS 3, 588–595, (1952). Sobolev Institute of Mathematics, Novosibirsk 630090, Russia, Novosibirsk State University, Novosibirsk 630090, Russia, Novosibirsk State Agrarian University, Dobrolyubova street, 160, Novosibirsk, 630039, Russia, E-mail address: [email protected] Sobolev Institute of Mathematics and Novosibirsk State University, Novosibirsk 630090, Russia, E-mail address: [email protected]	4
Spanning Tree Congestion and Computation of Generalized Győri-Lovász Partition L. Sunil Chandran arXiv:1802.07632v1 [cs.DS] 21 Feb 2018 1 ?1 , Yun Kuen Cheung ??2 , and Davis Issac2 Department of Computer Science and Automation, Indian Institute of Science, India. [email protected] 2 Max Planck Institute for Informatics, Saarland Informatics Campus, Germany. [email protected] , [email protected] Abstract. We study a natural problem in graph sparsification, the Spanning Tree Congestion (STC) problem. Informally, the STC problem seeks a spanning tree with no tree-edge routing too many of the original edges. The root of this problem dates back to at least 30 years ago, motivated by applications in network design, parallel computing and circuit design. Variants of the problem have also seen algorithmic applications as a preprocessing step of several important graph algorithms. For any general connected graph with n vertices and m edges, we show that √ its STC is at most O( mn), which is asymptotically optimal since we also √ demonstrate graphs with STC at least Ω( mn). We present a polynomial√ time algorithm which computes a spanning tree with congestion O( mn · log n). We also present another algorithm for computing a spanning tree √ with congestion O( mn); this algorithm runs in sub-exponential time 2 when m = ω(n log n). For achieving the above results, an important intermediate theorem is generalized Győri-Lovász theorem, for which Chen et al. [14] gave a non-constructive proof. We give the first elementary and constructive proof by providing a local search algorithm with running time O∗ (4n ), which is a key ingredient of the above-mentioned sub-exponential time algorithm. We discuss a few consequences of the theorem concerning graph partitioning, which might be of independent interest. We also show that for any graph which satisfies certain expanding properties, its STC is at most O(n), and a corresponding spanning tree can be computed in polynomial time. We then use this to show that a random graph has STC Θ(n) with high probability. 1 Introduction Graph Sparsification/Compression generally describes a transformation of a large input graph into a smaller/sparser graph that preserves certain feature (e.g., ? ?? This work was done while this author was visiting Max Planck Institute for Informatics, Saarbrücken, Germany, supported by Alexander von Humboldt Fellowship. Part of the work done while this author was a visitor at the Courant Institute, NYU. The visit was funded in part by New York University. 2 L. Sunil Chandran, Yun Kuen Cheung, and Davis Issac distance, cut, congestion, flow) either exactly or approximately. The algorithmic value is clear, since the smaller graph might be used as a preprocessed input to an algorithm, so as to reduce subsequent running time and memory requirement. In this paper, we study a natural problem in graph sparsification, the Spanning Tree Congestion (STC) problem. Informally, the STC problem seeks a spanning tree with no tree-edge routing too many of the original edges. The problem is well-motivated by network design applications, where designers aim to build sparse networks that meet traffic demands, while ensuring no connection (edge) is too congested. Indeed, the root of this problem dates back to at least 30 years ago under the name of “load factor” [8,36], with natural motivations from parallel computing and circuit design applications. The STC problem was formally defined by Ostrovskii [30] in 2004, and since then a number of results have been presented. The probabilistic version of the STC problem, coined as probabilistic capacity mapping, also finds applications in several important graph algorithm problems, e.g., the Min-Bisection problem. Two canonical goals for graph sparsification problems are to understand the trade-off between the sparsity of the output graph(s) and how well the feature is preserved, and to devise (efficient) algorithms for computing the sparser graph(s). These are also our goals for the STC problem. We focus on two scenarios: (A) general connected graphs with n vertices and m edges, and (B) graphs which exhibit certain expanding properties: √ – For (A), we show that the p spanning tree congestion (STC) is at most O( mn), which is a factor of Ω( m/n) better than the trivial bound of m. We present a polynomial-time algorithm which computes a spanning tree with congestion √ O( mn · log n). We also √ present another algorithm for computing a spanning tree with congestion O( mn); this algorithm runs in sub-exponential time when m = ω(n log2 n). For almost all ranges √ of average degree 2m/n, we also demonstrate graphs with STC at least Ω( mn). – For (B), we show that the expanding properties permit us to devise polynomialtime algorithm which computes a spanning tree with congestion O(n). Using this result, together with a separate lower-bound argument, we show that a random graph has Θ(n) STC with high probability. For achieving the results for (A), an important intermediate theorem is generalized Győri-Lovász theorem, which was first proved by Chen et al. [14]. Their proof uses advanced techniques in topology and homology theory, and is non-constructive. Definition 1.1. In a graph G = (V, E), a k-connected-partition is a k-partition of V into ∪kj=1 Vj , such that for each j ∈ [k], G[Vj ] is connected. Theorem 1.2 ([14, Theorems 25, 26]). Let G = (V, E) be a k-connected 3 graph. Let w be a weight function w : V → R+ . For any U ⊂ V , let w(U ) := P v∈U w(v). Given any k distinct terminal vertices t1 , · · · , tk , and k positive 3 For brevity, we say “k-connected” for “k-vertex-connected” henceforth. Spanning Tree Congestion 3 Pk integers T1 , · · · , Tk such that for each j ∈ [k], Tj ≥ w(tj ) and i=1 Ti = w(V ), there exists a k-connected-partition of V into ∪kj=1 Vj , such that for each j ∈ [k], tj ∈ Vj and w(Vj ) ≤ Tj + maxv∈V w(v) − 1. One of our main contributions is to give the first elementary and constructive proof by providing a local search algorithm with running time O∗ (4n ):4 Theorem 1.3. (a) There is an algorithm which given a k-connected graph, computes a k-connected-partition satisfying the conditions stated in Theorem 1.2 in time O∗ (4n ). (b) If we need a (bk/2c + 1)-partition instead of k-partition (the input graph remains assumed to be k-connected), the algorithm’s running time improves to O∗ (2O((n/k) log k) ). We make three remarks. First, the O∗ (2O((n/k) log k) )-time algorithm is a key√ingredient of our algorithm for computing a spanning tree with congestion O( mn). Second, since Theorem 1.2 guarantees the existence of such a partition, the problem of computing such a partition is not a decision problem but a search problem. Our local search algorithm shows that this problem is in the complexity class PLS [20]; we raise its completeness in PLS as an open problem. Third, the running times do not depend on the weights. The STC Problem, Related Problems and Our Results. Given a connected graph G = (V, E), let T be a spanning tree. For an edge e = (u, v) ∈ E, its detour with respect to T is the unique path from u to v in T ; let DT(e, T ) denote the set of edges in this detour. The stretch of e with respect to T is \|DT(e, T )\|, the length of its detour. The dilation of T is maxe∈E \|DT(e, T )\|. The edge-congestion of an edge e ∈ T is ec(e, T ) := \|{f ∈ E : e ∈ DT(f, T )}\|, i.e., the number of edges in E whose detours contain e. The congestion of T is cong(T ) := maxe∈T ec(e, T ). The spanning tree congestion (STC) of the graph G is STC(G) := minT cong(T ), where T runs over all spanning trees of G. We note that there is an equivalent cut-based definition for edge-congestion, which we will use in our proofs. For each tree-edge e ∈ T , removing e from T results in two connected components; let Ue denote one of the components. Then ec(e, T ) := \|E(Ue , V \ Ue )\|. Various types of congestion, stretch and dilation problems are studied in computer science and discrete mathematics. In these problems, one typically seeks a spanning tree (or some other structure) with minimum congestion or dilation. We mention some of the well-known problems, where minimization is done over all the spanning trees of the given graph: 1. The Low Stretch Spanning Tree (LSST) problem is to find a spanning tree which minimizes the total stretch of all the edges of G. [3] It is easy to see that minimizing the total stretch is equivalent to minimizing the total edge-congestion of the selected spanning tree. 2. The STC problem is to find a spanning tree of minimum congestion. [30] 4 O∗ notation hides all polynomial factors in input size. 4 L. Sunil Chandran, Yun Kuen Cheung, and Davis Issac 3. Tree Spanner Problem is to find a spanning tree of minimum dilation. [13] The more general Spanner problem is to find a sparser subgraph of minimum distortion. [4] There are other congestion and dilation problems which do not seek a spanning tree, but some other structure. The most famous among them is the Bandwidth problem and the Cutwidth problem; see the survey [34] for more details. Among the problems mentioned above, several strong results were published in connection with the LSST problem. Alon et al. [3] had shown a lower bound of Ω(max{n log n, m}). Upper bounds have been derived and many efficient algorithms have been devised; the current best upper bound is Õ(m log n). [3,15,1,22,2] Since total stretch is identical to total edge-congestion, the best upper bound for the LSST problem automatically implies an Õ( m n log n) upper bound on the average edge-congestion. But in the STC problem, we concern the maximum edge-congestion; as we shall p see, for some graphs, the maximum edge-congestion has to be a factor of Ω̃( n3 /m) larger than the average edge-congestion. In comparison, there were not many strong and general results for the STC Problem, though it was studied extensively in the past 13 years. The problem was formally proposed by Ostrovskii [30] in 2004. Prior to this, Simonson [36] had studied the same parameter under a different name to approximate the cut width of outer-planar graph. A number of graph-theoretic results were presented on this topic [31,25,24,23,10]. Some complexity results were also presented recently [29,9], but most of these results concern special classes √ of graphs. The most general result regarding STC of general graphs is an O(n n) upper bound by Löwenstein, Rautenbach and Regen in 2009 [27], and a matching lower bound by Ostrovskii in 2004 [30]. Note that the above upper bound is not interesting when the graph is sparse, since there is also a trivial upper bound of m. In this paper we come up with a strong improvement to these bounds after 8 years: Theorem (informal): For a connected graph √ G with n vertices and m edges, its spanning tree congestion is at most O( mn). p In terms of average degree davg = 2m/n, we can state this upper bound as O(n davg ). There is a matching lower bound. √ Our proof for achieving the O( mn) upper bound is constructive. It runs in exponential time in general; for graphs with m = ω(n log2 n) edges, it runs in sub-exponential time. By using an algorithm of Chen et al. [14] for computing single-commodity confluent flow from single-commodity splittable flow, we improve the running time to polynomial, but with a slightly worse upper bound guarantee √ of O( mn · log n). Motivated by an open problem raised by Ostrovskii [32] concerning STC of random graphs, we formulate a set of expanding properties, and prove that for any graph satisfying these properties, its STC is at most O(n). We devise a polynomial time algorithm for computing a spanning tree with congestion O(n) for such graphs. This result, together with a separate lower-bound argument, n permit us to show that for random graph G(n, p) with 1 ≥ p ≥ c log for some n Spanning Tree Congestion 5 small constant c > 1,5 its STC is Θ(n) with high probability, thus resolving the open problem raised by Ostrovskii completely. Min-Max Graph Partitioning and the Generalized Győri-Lovász Theorem. It looks clear that the powerful Theorem 1.2 can make an impact on graph partitioning. We discuss a number of its consequences which might be of wider interest. Graph partitioning/clustering is a prominent topic in graph theory/algorithms, and has a wide range of applications.A popular goal is to partition the vertices into sets such that the number of edges across different sets is small. While the min-sum objective, i.e., minimizing the total number of edges across different sets, is more widely studied, in various applications, the more natural objective is the min-max objective, i.e., minimizing the maximum number of edges leaving each set. The min-max objective is our focus here. Depending on applications, there are additional constraints on the sets in the partition. Two natural constraints are (i) balancedness: the sets are (approximately) balanced in sizes, and (ii) induced-connectivity: each set induces a connected subgraph. The balancedness constraint appears in the application of domain decomposition in parallel computing, while the induced-connectivity constraint is motivated by divide-and-conquer algorithms for spanning tree construction. Imposing both constraints simultaneously is not feasible for every graph; for instance, consider the star graph with more than 6 vertices and one wants a 3-partition. Thus, it is natural to ask, for which graphs do partitions satisfying both constraints exist. Theorem 1.2 implies a simple sufficient condition for existence of such partitions. By setting the weight of each vertex in G to be its degree, and using the elementary fact that the maximum degree ∆(G) ≤ n ≤ 2m/k for any k-connected graph G on n vertices and m edges, we have Proposition 1.4. If G is a k-connected graph with m edges, then there exists a k-connected-partition, such that the total degree of vertices in each part is at most 4m/k. Consequently, the min-max objective is also at most 4m/k. Due to expander graphs, this bound is optimal up to a small constant factor. This proposition (together with Lemma 4.1) implies the following crucial lemma for achieving some of our results. Lemma 1.5. Let G be a k-connected graph with m edges. Then STC(G) ≤ 4m/k. Proposition 1.4 can be generalized to include approximate balancedness in terms of number of vertices. By setting the weight of each vertex to be cm/n plus its degree in G, we have Proposition 1.6. Given any fixed c > 0, if G is a k-connected graph with m edges and n vertices, then there exists a k-connected-partition such that the total 5 Note that the STC problem is relevant only for connected graphs. Since the threshold function for graph connectivity is logn n , this result applies for almost all of the relevant range of values of p. 6 L. Sunil Chandran, Yun Kuen Cheung, and Davis Issac degree of vertices in each part is at most (2c + 4)m/k, and the number of vertices n in each part is at most 2c+4 c · k. Further Related Work. Concerning STC problem, Okamoto et al. [29] gave an O∗ (2n ) algorithm for computing the exact STC of a graph. The probabilistic version of the STC problem, coined as probabilistic capacity mapping, is an important tool for several graph algorithm problems, e.g., the Min-Bisection problem. Räcke [33] showed that in the probabilistic setting, distance and capacity are interchangeable, which briefly says a general upper bound for one objective implies the same general upper bound for the other. Thus, due to the above-mentioned results on LSST, there is an upper bound of Õ(log n) on the maximum average congestion. Räcke’s result also implies an O(log n) approximation algorithm to the Min-Bisection problem, improving upon the O(log3/2 n) approximation algorithm of Feige and Krauthgamer [16]. However, in the deterministic setting, such interchanging phenomenon does not hold: there is a √ simple tight bound Θ(n) for dilation, but for congestion it can be as high as Θ(n n). For the precise definitions, more background and key results about the concepts we have just discussed, we recommend the writing of Andersen and Feige [5]. Graph partitioning/clustering is a prominent research topic with wide applications, so it comes no surprise that a lot of work has been done on various aspects of the topic; we refer readers to the two extensive surveys by Schaeffer [35] and by Teng [41]. Kiwi, Spielman and Teng [21] formulated the min-max k-partitioning problem and gave bounds for classes of graphs with small separators, which are then improved by Steurer [38]. On the algorithmic side, many of the related problems are NP-hard, so the focus is on devising approximation algorithms. Sparkled by the seminal work of Arora, Rao and Vazirani [6] on sparsest cut and of Spielman and Teng [37] on local clustering, graph partitioning/clustering algorithms with various constraints have attracted attention across theory and practice; we refer readers to [7] for a fairly recent account of the development. The min-sum objective has been extensively studied; the min-max objective, while striking as the more natural objective in some applications, has received much less attention. The only algorithmic work on this objective (and its variants) are Svitkina and Tardos [40] and Bansal et al. [7]. None of the above work addresses the induced-connectivity constraint. The classical version of Győri-Lovász Theorem (i.e., the vertex weights are uniform) was proved independently by Győri [17] and Lovász [26]. Lovász’s proof uses homology theory and is non-constructive. Győri’s proof is elementary and is constructive implicitly, but he did not analyze the running time. Polynomial time algorithms for constructing the k-partition were devised for k = 2, 3 [39,42], but no non-trivial finite-time algorithm was known for general graphs with k ≥ 4.6 Recently, Hoyer and Thomas [19] provided a clean presentation of Győri’s proof 6 In 1994, there was a paper by Ma and Ma in Journal of Computer Science and Technology, which claimed a poly-time algorithm for all k. However, according to a recent study [18], Ma and Ma’s algorithm can fall into an endless loop. Also, Győri said the algorithm should be wrong (see [28]). Spanning Tree Congestion 7 by introducing their own terminology, which we use for our constructive proof of Theorem 1.2. Notation. Given a graph G = (V, E), an edge set F ⊆ E and 2 disjoint vertex subsets V1 , V2 ⊂ V , we let F (V1 , V2 ) := { e = {v1 , v2 } ∈ F \| v1 ∈ V1 and v2 ∈ V2 }. 2 Technical Overview To prove the generalized Győri-Lovász theorem constructively, we follow the same framework of Győri’s proof [17], and we borrow terminology from the recent presentation by Hoyer and Thomas [19]. But it should be emphasized that proving our generalized theorem is not straight-forward, since in Győri’s proof, at each stage a single vertex is moved from one set to other to make progress, while making sure that the former set remains connected. In our setting, in addition to this we also have to ensure that the weights in the partitions do not exceed the specified limit; and hence any vertex that can be moved from one set to another need not be candidate for being transferred. The proof is presented in Section 3. As discussed, a crucial ingredient for our upper bound results is Lemma 1.5, which is a direct corollary of the generalized Győri-Lovász theorem. The lemma takes care of the highly-connected cases; for other cases we provide a recursive way to construct a low congestion spanning tree; see Section 4 for details. For showing our lower bound for general graphs, the challenge is to maintain high congestion while keeping density small. To achieve this, we combine three expander graphs with little overlapping between them, and we further make those overlapped vertices of very high degree. This will force a tree-edge adjacent to the centroid of any spanning tree to have high congestion; see Section 5 for details. We formulate a set of expanding properties which permit constructing a spanning tree of better congestion guarantee in polynomial time. The basic idea is simple: start with a vertex v of high degree as the root. Now try to grow the tree by keep attaching new vertices to it, while keeping the invariant that the subtrees rooted at each of the neighbours of v are roughly balanced in size; each such subtree is called a branch. But when trying to grow the tree in a balanced way, we will soon realize that as the tree grow, all the remaining vertices may be seen to be adjacent only to a few number of “heavy” branches. To help the balanced growth, the algorithm will identify a transferable vertex which is in a heavy branch, and it and its descendants in the tree can be transferred to a “lighter” branch. Another technique is to use multiple rounds of matching between vertices in the tree and the remaining vertices to attach new vertices to the tree. This will tend to make sure that all subtrees do not grow uncontrolled. By showing that random graph satisfies the expanding properties with appropriate parameters, we show that a random graph has STC of Θ(n) with high probability. 3 Generalized Győri-Lovász Theorem We prove Theorem 1.3 in this section. Observe that the classical Győri-Lovász Theorem follows from Theorem 1.2 by taking w(v) = 1 for all v ∈ V and Tj = nj 8 L. Sunil Chandran, Yun Kuen Cheung, and Davis Issac for all j ∈ [k]. We note that a perfect generalization where one requires that w(Vj ) = Tj is not possible — think when all vertex weights are even integers, while some Tj is odd. Let G = (V, E) be a k-connected graph on n vertices and m P edges, and w : V → R+ be a weight function. For any subset U ⊆ V , w(U ) := u∈U w(u). Let wmax := maxv∈V w(v). 3.1 Key Combinatorial Notions We first highlight the key combinatorial notions used for proving Theorem 1.3; see Figures 1 and 2 for illustrations of some of these notions. Fitted Partial Partition. First, we introduce the notion of fitted partial partition (FPP). An FPP A is a tuple of k subsets of V , (A1 , . . . , Ak ), such that the k subsets are pairwise disjoint, and for each j ∈ [k]: 1. tj ∈ Aj , 2. G[Aj ] is connected and 3. w(Aj ) ≤ Tj + wmax − 1 (we say the set is fitted for satisfying this inequality). We say an FPP is a Strict Fitted Partial Partition (SFPP) if A1 ∪ · · · ∪ Ak is a proper subset of V . We say the set Aj is light if w(Aj ) < Tj , and we say it is heavy otherwise. Note that there Pk exists at least one light set in any SFPP, for otherwise w(A1 ∪ · · · ∪ Ak ) ≥ j=1 Tj = w(V ), which means A1 ∪ · · · ∪ Ak = V . Also note that by taking Aj = {tj }, we have an FPP, and hence at least one FPP exists. Configuration. For a set Aj in an FPP A and a vertex v ∈ Aj \ {tj }, we define the reservoir of v with respect to A, denoted by RA (v), as the vertices in the same connected component as tj in G[Aj ] \ {v}. Note that v ∈ / RA (v). For a heavy set Aj , a sequence of vertices (z1 , . . . , zp ) for some p ≥ 0 is called a cascade of Aj if z1 ∈ Aj \ {tj } and zi+1 ∈ Aj \ RA (zi ) for all 1 ≤ i < p. The cascade is called a null cascade if p = 0, i.e., if the cascade is empty. Note that for light set, we do not need to define its cascade since we do not use it in the proof. (See Figure 1.) A configuration CA is defined as a pair (A, D), where A = (A1 , · · · , Ak ) is an FPP, and D is a set of cascades, which consists of exactly one cascade (possibly, a null cascade) for each heavy set in A. A vertex that is in some cascade of the configuration is called a cascade vertex. Given a configuration, we define rank and level inductively as follows. Any vertex in a light set is said to have level 0. For i ≥ 0, a cascade vertex is said to have rank i + 1 if it has an edge to a level-i vertex but does not have an edge to any level-i0 vertex for i0 < i. A vertex u is said to have level i, for i ≥ 1, if u ∈ RA (v) for some rank-i cascade vertex v, but u ∈ / RA (w) for any cascade vertex w such that rank of w is less than i. A vertex that is not in RA (v) for any cascade vertex v is said to have level ∞. A configuration is called a valid configuration if for each heavy set Aj , rank is defined for each of its cascade vertices and the rank is strictly increasing in the Spanning Tree Congestion 9 z3 z2 RA(z3) RA(z2) z1 RA(z1) tj Fig. 1. Given a configuration (A, D) and a heavy set Aj in A, the figure shows a cascade (z1 , z2 , z3 ) for the heavy set Aj and several reservoirs of the cascade vertices. For any z` , z` ∈ / RA (z` ). A cascade vertex z` is a cut-vertex of G[Aj ], i.e., G[Aj \ {z` }] is disconnected. The removal of z` from Aj will lead to at least two connected components in G[Aj \ {z` }], and the connected component containing tj is the reservoir of z` . We identify tj = z0 , but we clarify that a terminal vertex is never in a cascade. Each epoch between z` and z`+1 , and also the epoch above z3 , is a subset of vertices B ⊂ Aj , where B 3 z` and G[B] is connected. Note that in general, it is possible that there is no vertex above the last cascade vertex. cascade, i.e., if {z1 , . . . , zp } is the cascade, then rank(z1 ) < · · · < rank(zp ). Note that by taking Aj = {tj } and taking the null cascade for each heavy set (in this case Aj is heavy if w(tj ) = Tj ), we get a valid configuration. (See Figure 2.) Configuration Vectors and Their Total Ordering. For any vertex, we define its neighborhood level as the smallest level of any vertex adjacent to it. A vertex v of level ` is said to satisfy maximality property if each vertex adjacent on it is either a rank-(` + 1) cascade vertex, has a level of at most ` + 1, or is one of the terminals tj for some j. For any ` ≥ 0, a valid configuration is called an `-maximal configuration if all vertices having level at most ` − 1 satisfy the maximality property. Note that by definition, any valid configuration is a 0-maximal configuration. For a configuration CA = ((A1 , . . . , Ak ) , D), we define SA := V \(A1 ∪· · ·∪Ak ). An edge uv is said to be a bridge in CA if u ∈ SA , v ∈ Aj for some j ∈ [k], and level(v) 6= ∞. 10 L. Sunil Chandran, Yun Kuen Cheung, and Davis Issac L=∞ z12, rank = 2 L=2 L=∞ z22, rank = 5 L=5 z11, rank = 1 z32, rank = 4 L=4 z21, rank = 3 z31, rank = 1 L=1 L=3 L=1 t1 t2 t3 Vertices in all light sets Level = 0 Fig. 2. An instance of a valid configuration. Every blue segment/curve represent an edge from a cascade vertex to a vertex in some reservoir or light set. Every cascade vertex connected to a light set has rank 1, and all vertices in the epoch immediately below a rank 1 cascade vertex are of level 1. Inductively, every cascade vertex connected to a vertex of level i has rank i + 1, and all vertices in the epoch immediately below a rank i cascade vertex are of level i. All vertices above the last cascade vertex of each cascade has level ∞. A valid configuration CA is said to be `-good if the highest rank of a cascade vertex in CA is exactly ` (if there are no cascade vertices, then we take the highest rank as 0), CA is `-maximal, and all bridges uv in CA (if any) are such that u ∈ SA and level(v) = `. Note that taking Aj = {tj } and taking the null cascade for each heavy set gives a 0-good configuration. For each configuration CA = (A, D), we define a configuration vector as below: ( LA , NA0 , NA1 , NA2 , . . . , NAn ), where LA is the number of light sets in A, and NA` is the total number of all level-` vertices in CA . Next, we define ordering on configuration vectors. Let CA and CB be configurations. We say CA >0 CB if Spanning Tree Congestion 11 – LA < LB , or – LA = LB , and NA0 > NB0 . We say CA =0 CB if LA = LB and NA0 = NB0 . We say CA ≥0 CB if CA =0 CB or 0 0 CA >0 CB . We say CA =` CB if LA = LB , and NA` = NB` for all `0 ≤ `. For 1 ≤ ` ≤ n, we say CA >` CB if – CA >`−1 CB , or – CA =`−1 CB , and NA` > NB` . We say CA ≥` CB if CA =` CB or CA >` CB . We say CA > CB (CA is strictly better than CB ) if CA >n CB . 3.2 Proof of Theorem 1.3 We use two technical lemmas about configuration vectors and their orderings to prove Theorem 1.3(a). The proof of Theorem 1.3(b) follows closely with the proof of Theorem 1.3(a), but makes use of an observation about the rank of a vertex in the local search algorithm, to give an improved bound on the number of configuration vectors navigated by the algorithm. Lemma 3.1. Given any `-good configuration CA = (A = (A1 , . . . , Ak ), DA )) that does not have a bridge, we can find an (` + 1)-good configuration CB = (B = (B1 , B2 , . . . , Bk ) , DB ) in polynomial time such that CB > CA . Proof. Since CA is `-maximal, any vertex that is at level `0 < ` satisfies maximality property. So, for satisfying (` + 1)-maximality, we only need to worry about the vertices that are at level `. Let Xj be the set of all vertices x ∈ Aj such that x is adjacent to a level-` vertex, level(x) ≥ ` + 1 (i.e., level(x) = ∞ as the highest rank of any cascade vertex is `), x = 6 tj , and x is not a cascade vertex of rank `. We claim that there exists at least one j for which Xj is not empty. If that is not the case, then we exhibit a cut set of size at most k − 1. For each j such that Aj is a heavy set with a non-null cascade, let yj be the highest ranked cascade vertex in Aj . For each j such that Aj is a heavy set with a null cascade, let yj be tj . Let Y be the set of all yj such that Aj is a heavy set. Note that \|Y \| ≤ k − 1 as A is an SFPP and hence has at least one light set. Let Z∞ be the set of all vertices in V \ Y that have level ∞ and Z be the remaining vertices in V \ Y . Since A is an SFPP, SA 6= ∅, and since all vertices in SA have level ∞, we have that Z∞ 6= ∅. Z is not empty because there exists at least one light set in A and the vertices in a light set have level 0. We show that there is no edge between Z∞ and Z in G. Suppose there exists an edge uv such that u ∈ Z∞ and v ∈ Z. If u ∈ SA , then uv is a bridge which is a contradiction by our assumption that CA does not have a bridge. Hence u ∈ Aj for some j ∈ [k]. Note that Aj has to be a heavy set, otherwise u has level 0. We have that u is not a cascade vertex (as all cascade vertices with level ∞ are in Y ) and u 6= tj (as all tj such that level(tj ) = ∞ are in Y ). Also, v is not of level ` as otherwise, u ∈ Xj but we assumed Xj is empty. But then, v has level at most ` − 1, u has level ∞, and there 12 L. Sunil Chandran, Yun Kuen Cheung, and Davis Issac is an edge uv. This means that CA was not `-maximal, which is a contradiction. Thus, there exists at least one j for which Xj is not empty. For any j such that Xj 6= ∅ , there is at least one vertex xj such that Xj \ {xj } ⊆ RA (xj ). Now we give the configuration CB as follows. We set Bj = Aj for all j ∈ [k]. For each heavy set Aj such that Xj 6= ∅, we take the cascade of Bj as the cascade of Aj appended with xj . For each heavy set Aj such that Xj = ∅, we take the cascade of Bj as the cascade of Aj . It is easy to see that CB is (` + 1)-maximal as each vertex that had an edge to level-` vertices in CA is now either a rank ` + 1 cascade vertex or a level-(` + 1) vertex or is tj for some j. Also, notice that all the new cascade vertices that we introduce (i.e., the xj ’s) have their rank as ` + 1 and there is at least one rank ` + 1 cascade vertex as Xj is not empty for some j. Since there were no bridges in CA , all bridges in CB has to be from SB to a vertex having level ` + 1. Hence, CB is (` + 1)-good. All vertices that had level at most ` in CA retained their levels in CB . And, at least one level-∞ vertex of CA became a level-(` + 1) vertex in CB because the cascade vertex that was at rank ` becomes level-(` + 1) vertex now in at least one set. Since CA had no level-(` + 1) vertices, this means that CB > CA . Lemma 3.2. Given an `-good configuration CA = (A = (A1 , . . . , Ak ), DA ) having a bridge, we can find in polynomial time a valid configuration CB = (B = (B1 , . . . , Bk ) , DB ) such that one of the following holds: – CB >` CA , and CB is an `-good configuration, or – CB ≥`−1 CA , there is a bridge u0 v 0 in CB such that u0 ∈ SB and level(v 0 ) ≤ ` − 1, and CB is an (` − 1)-good configuration. Proof. Let uv be a bridge where u ∈ SA . Let Aj ∗ be the set containing v. Note that level(v) = ` because CA is `-good. We keep Bj = Aj for all j 6= j ∗ . But we modify Aj ∗ to get Bj ∗ as described below. We maintain that if Aj is a heavy set then Bj is also a heavy set for all j, and hence maintain that LB ≤ LA . Case 1: Aj ∗ is a light set (i.e., when ` = 0). We take Bj ∗ = Aj ∗ ∪ {u}. For all j such that Bj is a heavy set, cascade of Bj is taken as the null cascade. We have w(Aj ∗ ) ≤ Tj − 1 because Aj ∗ is a light set. So, w(Bj ∗ ) = w(Aj ∗ ) + w(u) ≤ (Tj − 1) + wmax , and hence Bj ∗ is fitted. Also, G[Bj ∗ ] is connected and hence (B1 , . . . , Bk ) is an FPP. We have CB >0 CA because either Bj ∗ became a heavy set in which case LB < LA , or it is a light set in which case LB = LA and NB0 > NA0 . It is easy to see that CB is 0-good. Case 2: Aj ∗ is a heavy set i.e., when ` ≥ 1. Case 2.1: w(Aj ∗ ∪ {u}) ≤ Tj + wmax − 1. We take Bj ∗ = Aj ∗ ∪ {u}. For each j such that Bj is a heavy set (Aj is also heavy set for such j), the cascade of Bj is taken as the cascade of Aj . G[Bj ∗ ] is clearly connected and Bj ∗ is fitted by assumption of the case that we are in. Hence B is indeed an FPP. Observe that all vertices that had level `0 ≤ ` in CA still has level `0 in CB . Since level(v) was ` in CA by `-goodness of CA , u also has level ` in CB ; and u had level ∞ in CA . Hence, CB >` CA . It is also easy to see that CB remains `-good. Case 2.2: w(Aj ∗ ∪ {u}) ≥ Tj + wmax . Let z be the cascade vertex of rank ` in Aj ∗ . Note that Aj ∗ should have such a cascade vertex as v ∈ Aj ∗ has level `. Let Spanning Tree Congestion 13 R̄ be Aj ∗ \ (RA (z) ∪ z), i.e., R̄ is the set of all vertices in Aj ∗ \ {z} with level ∞. We initialize Bj ∗ := Aj ∗ ∪ {u}. Now, we delete vertices one by one from Bj ∗ in a specific order until Bj ∗ becomes fitted. We choose the order of deleting vertices such that G[Bj ∗ ] remains connected. Consider a spanning tree τ of G[R̄ ∪ {z}]. τ has at least one leaf, which is not z. We delete this leaf from Bj ∗ and τ . We repeat this process until τ is just the single vertex z or Bj ∗ becomes fitted. If Bj ∗ is not fitted even when τ is the single vertex z, then delete z from Bj ∗ . If Bj ∗ is still not fitted then delete u from Bj ∗ . Note that at this point Bj ∗ ⊂ Aj ∗ and hence is fitted. Also, note that G[Bj ∗ ] remains connected. Hence (B1 , . . . , Bk ) is an FPP. Bj ∗ does not become a light set because Bj became fitted when the last vertex was deleted from it. Before this vertex was deleted, it was not fitted and hence had weight at least Tj ∗ + wmax before this deletion. Since the last vertex deleted has weight at most wmax , Bj ∗ has weight at least Tj ∗ and hence is a heavy set. Now we branch into two subcases for defining the cascades. Case 2.2.1: z ∈ Bj∗ (i.e, z was not deleted from Bj ∗ in the process above). For each j such that Bj is a heavy set, the cascade of Bj is taken as the cascade of Aj . Since a new ` level vertex u is added and all vertices that had level at most ` retain their level, we have that CB >` CA . It is also easy to see that CB remains `-good. Case 2.2.2: z ∈ / Bj ∗ (i.e, z was deleted from Bj ∗ ). For each j such that Bj is a heavy set, the cascade of Bj is taken as the cascade of Aj but with the rank ` cascade vertex (if it has any) deleted from it. CB ≥`−1 CA because all vertices that were at a level of `0 = ` − 1 or smaller, retain their levels. Observe that there are no bridges in CB to vertices that are at a level at most ` − 2, all vertices at a level at most ` − 2 still maintain the maximality property, and we did not introduce any cascade vertices. Hence, CB is (` − 1)-good. It only remains to prove that there is a bridge u0 v 0 in CB such that level(v 0 ) ≤ ` − 1. We know z ∈ SB . Since z was a rank ` cascade vertex in CA , z had an edge to z 0 such that z 0 had level ` − 1 in CA . Observe that level of z 0 is at most ` − 1 in CB as well. Hence, taking u0 v 0 = zz 0 completes the proof. Proof of Theorem 1.3(a): . We always maintain a configuration CA = (A, DA ) that is `-good for some ` ≥ 0. If the FPP A is not an SFPP at any point, then we are done. So assume A is an SFPP. We start with the 0-good configuration where Aj = {tj } and the cascades of all heavy sets are null cascades. If our current configuration CA is an `-good configuration that has no bridge, then we use Lemma 3.1 to get a configuration CB such that CB > CA and B is (` + 1)-good. We take CB as the new current configuration CA . If our current configuration CA is an `-good configuration with a bridge, then we get an `0 -good configuration CB for some `0 ≥ 0 such that CB > CA by repeatedly applying Lemma 3.2 at most ` times. So in either case, we get a strictly better configuration that is `0 -good for some `0 ≥ 0 in polynomial time. We call this an iteration of our algorithm. Notice that the number of iterations possible is at most the number of distinct configuration vectors possible. It is easy to see that the number of distinct configuration vectors with highest rank at most r is at most n+r−1 . Since rank n 14 L. Sunil Chandran, Yun Kuen Cheung, and Davis Issac of any point is at most n, the number of iterations of our algorithm is at most n (k + 1) · 2n n , which is at most n · 4 . Since each iteration runs in polynomial time as guaranteed by the two lemmas, the required running time is O∗ (4n ). When the algorithm terminates, the FPP given by the current configuration is not an SFPP and this gives the required partition. Proof of Theorem 1.3(b): . Since any k-connected graph is also (bk/2c + 1)−vertex connected, the algorithm will give the required partition due to Theorem 1.3(a). We only need to prove the better running time claimed by Theorem 1.3(b). For this, we show that the highest rank attained by any vertex during the algorithm is at most 2n/(k − 2). Since the number of distinct config uration vectors with highest rank r is at most n+r−1 , we then have that the n 2n −1 ∗ O((n/k) log k) running time is O∗ n+ k−2 , which is O (2 ), as claimed. Hence, it n only remains to prove that the highest rank is at most 2n/(k − 2). For this, observe that in an `-good configuration, for each 0 ≤ i < `, the union of all vertices having level i and the set of (bk/2c + 1) terminals together forms a cutset. Since the graph is k-connected, this means that for each 0 ≤ i < `, the number of vertices having level i is at least k/2 − 1. The required bound on the rank easily follows. 4 Upper Bounds for Spanning Tree Congestion We first state the following easy lemma, which together with Proposition 1.4, implies Lemma 1.5. Lemma 4.1. In a graph G = (V, E), let t1 be a vertex, and let t2 , · · · , t` be any (` − 1) neighbours of t1 . Suppose that there exists a `-connected-partition ∪`j=1 V` such that for all j ∈ `, tj ∈ Vj , and the sum of degree of vertices in each Vj is at most D. Let τj be an arbitrary spanning tree of G[Vj ]. Let ej denote the edge S {t1 , tj }. Let τ be the spanning tree of G defined as τ := ∪`j=1 τj ∪`j=2 ej . Then τ has congestion at most D. Theorem 4.2. For any connected graph G = (V, E), there is an algorithm which √ √ O n log n/ m/n ∗ computes a spanning tree with congestion at most 8 mn in O 2 time. Theorem 4.3. For any connected graph G = (V, E), there is a polynomial time √ algorithm which computes a spanning tree with congestion at most 16 mn · log n. The two algorithms follow the same framework, depicted in Algorithm 1. It is a recursive algorithm; the parameter m̂ is a global parameter, which is the number of edges in the input graph G in the first level of the recursion; let n̂ denote the number of vertices in this graph. The only difference between the two algorithms is in Line 15 on how this step is executed, with trade-off between the running time of the step T (m̂, nH , mH ), and the guarantee D(m̂, nH , mH ). For proving Theorem 4.2, we use Theorem 1.3(b), Spanning Tree Congestion 15 p Proposition 1.4 andLemma 4.1, yielding D(m̂, nH , mH ) ≤ 8mH nH /m̂ and √ T (m̂, nH , mH ) = O∗ 2O(nH log nH / m̂/nH ) . For proving Theorem 4.3, we make p use of an algorithm in Chen et al. [14], which yields D(m̂, nH , mH ) ≤ 16mH nH /m̂· log nH and T (m̂, nH , mH ) = poly(nH , mH ). Algorithm 1: FindLCST(H, m̂) Input : A connected graph H = (VH , EH ) on nH vertices and mH edges Output : A spanning tree τ of H p 1 if mH ≤ 8 m̂nH then 2 return an arbitrary spanning tree of H 3 end lp m 4 k ← m̂/nH 5 6 7 8 9 10 11 12 13 14 15 Y ← a global minimum vertex cut of H if \|Y \| < k then X ← the smallest connected component in H[VH \ Y ] (See Figure 3) Z ← VH \ (X ∪ Y ) τ1 ← FindLCST( H[X], m̂ ) τ2 ← FindLCST( H[Y ∪ Z], m̂); (H[Y ∪ Z] is connected as Y is a global min cut) return τ1 ∪ τ2 ∪ (an arbitrary edge between X and Y ) else t1 ← an arbitrary vertex in VH Pick bk/2c neighbours of t1 in the graph H; denote them by t2 , t3 , · · · , tbk/2c+1 . Let ej denote edge t1 tj for 2 ≤ j ≤ bk/2c + 1. (See Figure 4) Compute a (bk/2c + 1)-connected-partition of H, denoted by bk/2c+1 16 17 18 ∪j=1 Vj , such that for each j ∈ [bk/2c + 1], tj ∈ Vj , and the total degree (w.r.t. graph H) of vertices in each Vj is at most D(m̂, nH , mH ). Let the time needed be T (m̂, nH , mH ). For eachj ∈ [bk/2c + 1], τj← an arbitrary spanning tree of G[Vj ] S bk/2c+1 bk/2c+1 return ∪j=1 τj ∪j=2 ej end In the rest of this section, we first discuss the algorithm in Chen et al., then we prove Theorem 4.3. The proof of Theorem 4.2 is almost identical, and is deferred to Appendix A.2. Single-Commodity Confluent Flow and The Algorithm of Chen et al. In a single-commodity confluent flow problem, the input includes a graph G = (V, E), a demand function w : V → R+ and ` sinks t1 , · · · , t` ∈ V . For each v ∈ V , a flow of amount w(v) is routed from v to one of the sinks. But there is a restriction: at every vertex u ∈ V , the outgoing flow must leave u on at most 1 edge, i.e., the outgoing flow from u is unsplittable. The problem is to seek a flow satisfying the demands which minimizes the node congestion, i.e., the maximum 16 L. Sunil Chandran, Yun Kuen Cheung, and Davis Issac Fig. 3. The scenario in Algorithm 1 when the graph has low connectivity. The vertex set Y is a global minimum vertex cut of the graph. The vertex set X is the smallest connected component after the removal of Y , and Z is the union of all the other connected components. Fig. 4. The scenario in Algorithm 1 when the graph has high connectivity. incoming flow among all vertices. Since the incoming flow is maximum at one of the sinks, it is equivalent to minimize the maximum flow received among all sinks. (Here, we assume that no flow entering a sink will leave.) Single-commodity splittable flow problem is almost identical to single-commodity confluent flow problem, except that the above restriction is dropped, i.e., now the outgoing flow at u can split along multiple edges. Note that here, the maximum incoming flow might not be at a sink. It is known that single-commodity splittable flow can be solved in polynomial time. For brevity, we drop the phrase “single-commodity” from now on. Theorem 4.4 ([14, Section 4]). Suppose that given graph G, demand w and ` sinks, there is a splittable flow with node congestion q. Then there exists a Spanning Tree Congestion 17 polynomial time algorithm which computes a confluent flow with node congestion at most (1 + ln `)q for the same input. Corollary 4.5. Let G be a k-connected graph with m edges. Then for any ` ≤ k and for any ` vertices t1 , · · · , t` ∈ V , there exists a polynomial time algorithm which computes an `-connected-partition ∪`j=1 V` such that for all j ∈ `, tj ∈ Vj , and the total degrees of vertices in each Vj is at most 4(1 + ln `)m/`. Corollary 4.5 follows from Theorem 4.4 and Proposition 1.4. See Appendix A.1 for details. Congestion Analysis. We view the whole recursion process as a recursion tree. There is no endless loop, since down every path in the recursion tree, the number of vertices in the input graphs are strictly decreasing. On the other hand, note that the leaf of the recursion tree pis resulted by either (i) when the input graph H to that call satisfies mH ≤ 8 m̂nH , or (ii) when Lines 13–17 are executed. An internal node appears only when the vertex-connectivity of the input graph H is low, and it makes two recursion calls. We prove the following statement by induction from bottom-up: for each graph which is the input to some call in thep recursion tree, the returned spanning tree of that call has congestion at most 16 m̂nH log nH . We first handle the two basis cases (i) and (ii). In case (i), FindLCST p returns an arbitrary spanning tree, and the congestion is bounded by mH ≤ 8 m̂nH . In case (ii), by Corollaryp4.5 and Lemma 4.1,pFindLCST returns a tree with congestion at most 16mH nH /m̂ · log nH ≤ 16 m̂nH · log nH . Next, let H be the input graph to a call which is represented by an internal node of the recursion tree. Recall the definitions of X, Y, Z, τ1 , τ2 in the algorithm. Let \|X\| = x. Note that 1 ≤ x ≤ nH /2. Then by induction hypothesis, the congestion of the returned spanning tree is at most max{ congestion of τ1 in H[X] , congestion of τ2 in H[Y ∪ Z] } + \|X\| · \|Y \| p p ≤ 16 m̂(nH − x) log(nH − x) + m̂/nH + 1 · x. (1) Viewing x as a real variable, by taking derivative, it is easy to see that the above expression is maximized at x = 1. Thus the congestion is at most p p p 16 m̂(nH − 1) log(nH −1)+ m̂/nH +1 ≤ 16 m̂nH log nH , as desired by Theorem 4.3. Runtime Analysis. At every internal node of the recursion tree, the algorithm makes two recursive calls with two vertex-disjoint and strictly smaller (w.r.t. vertex size) inputs. The dominating knitting cost is in Line 5 for computing a global minimum vertex cut, which is well-known that it can be done in polynomial time. Since at every leaf of the recursion tree the running time is polynomial, by standard analysis on divide-and-conquer algorithms, the running time of the whole algorithm is polynomial, which completes the proof of Theorem 4.3. 18 5 L. Sunil Chandran, Yun Kuen Cheung, and Davis Issac Lower Bound for Spanning Tree Congestion Here, we give a lower bound on spanning tree congestion which matches our upper bound. Theorem 5.1. For any sufficiently large n, and for any m satisfying n2 /2 ≥ m ≥ max{16n log n, 100n}, there exists a connected graph with N = (3 − o(1))n vertices √ and M ∈ [m, 7m] edges, for which the spanning tree congestion is at least Ω ( mn). We start with the following lemma, which states that for a random graph G(n, p), when p is sufficiently large, its edge expansion is Θ(np) with high probability. The proof of the lemma uses only fairly standard arguments and is deferred to Appendix A.3. Lemma 5.2. For any integer n ≥ 4 and 1 ≥ p ≥ 32 · logn n , let G(n, p) denote the random graph with n vertices, in which each edge occurs independently with probability p. Then with probability at least 1 − O(1/n), (i) the random graph is connected, (ii) the number of edges in the random graph is between pn2 /4 and pn2 , and (iii) for each subset of vertices S with \|S\| ≤ n/2, the number of edges leaving S is at least p2 · \|S\| · (n − \|S\|). In particular, for any sufficiently large integer n, when n2 /2 ≥ m ≥ 16n log n, by setting p = 2m/n2 , there exists a connected graph with n vertices and [m/2, 2m] edges, such that for each subset of vertices S with \|S\| ≤ n/2, the m number of edges leaving S is at least 2n · \|S\| = Θ(m/n) · \|S\|. We denote such a graph by H(n, m). We discuss our construction here (see Figure 5) before delving into the proof. The vertex set V is the union of three vertex subsets V1 , V2 , V3 , such that p \|V1 \| = \|V2 \| = \|V3 \| = n, \|V1 ∩ V2 \| = \|V2 ∩ V3 \| = m/n, and V1 , V3 are disjoint. In each of V1 , V2 and V3 , we embed H(n, m). The edge sets are denoted E1 , E2 , E3 respectively. Up to this point, the construction is similar to that of Ostrovskii [30], except that we use H(n, m) instead of a complete graph. The new component in our construction is adding the following edges. For each vertex v ∈ V1 ∩ V2 , add an edge between v and every vertex in (V1 ∪ V2 ) \ {v}. The set of these edges are denoted F1 . Similarly, for each vertex v ∈ V3 ∩ V2 , add an edge between v and every vertex in (V3 ∪ V2 ) \ {v}. The set of these edges are denoted F3 . This new component is crucial: without it, we could only prove a √ √ √ lower bound of Ω(m/ n) = Ω( mn · nm ). Proof of Theorem 5.1: . p Let G = (V, E) be the graph constructed as above. The whole graph has 3n − 2 m/n vertices. The number m p of edges is at least √ (due to edges in E1 and E3 ), and is at most 6m + 2 m/n · 2n = 6m + 4 mn, which is at most 7m for all sufficiently large n. It is well known that for any tree on n vertices, there exists a vertex x called a centroid of the tree such that, removing x decomposes the tree into connected components, each of size at most n/2. Now, consider any spanning tree of the Spanning Tree Congestion 19 V2 H(n, m) V1 V3 v3 v1 H(n, m) H(n, m) Fig. 5. Our lower-bound construction for spanning tree congestion. V1 , V2 , V3 are three vertex subsets of the same size. In each of the subsets, we embed expander H(n, m). There is a small overlap between V2 and V1 , V3 , while V1 , V3 are disjoint. For any vertex v1 ∈ V1 ∩ V2 , we add edges between it and any other vertex in V1 ∪ V2 ; similarly, for any vertex v3 ∈ V3 ∩ V2 , we add edges (not shown in figure) between it and any other vertex in V3 ∪ V2 . given graph, let u be a centroid of the tree. Without loss of generality, we can assume that u ∈ / V1 ; otherwise we swap the roles of V1 and V3 . The removal of u (and its adjacent edges) from the tree decomposes the tree into a number of connected components. For any of these components which intersects V1 , it must contain pat least one vertex of V1 ∩ V2 , thus the number of such components is at most m/n, and hence there exists one of them, denoted by Uj , such that p p b1 := \|Uj ∩ V1 \| ≥ n/( m/n) = n n/m. Let ej denote the tree-edge that connects u to Uj . Then there are three cases: p p Case 1: n n/m ≤ b1 ≤ n − n n/m. Due to the property √ of H(n, m), the congestion of ej is at least Θ(m/n) · min{b1 , n − b1 } ≥ Θ( mn). p p Case 2: b1 > n − n n/m and \|Uj ∩ V1 ∩ V2 \| ≤ 21 · m/n. Let W := p (V1 ∩ V2 ) \ Uj . Note that by this case’s assumption, \|W1 \| ≥ 12 · m/n. Due to the edge subset F1 , the congestion of ej is at least √ 1 p n F1 (W , V1 \ W ) ≥ · m/n · = Θ mn . 2 2 p p Case 3: b1 > n − n n/m and \|Uj ∩ V1 ∩ V2 \| > 21 · m/n. Let W 0 := Uj ∩ V1 ∩ V2 , and let Z := (V2 \ V1 ) ∩ Uj . p Note that b1 > n − n n/m ≥ 9n/10. Suppose \|Z\| ≥ 6n/10, then \|Uj \| > 9n/10 + 6n/10 > \|V \|/2, a contradiction to the assumption that u is a 20 L. Sunil Chandran, Yun Kuen Cheung, and Davis Issac centroid. Thus, \|Z\| < 6n/10. Due to the edge subset F2 , the congestion of ej is at least F2 (W 0 ∪ Z , V2 \ (W 0 ∪ Z)) ≥ \|W 0 \| · (n − \|W 0 \| − \|Z\|) p √ 1 p 6n ≥ · m/n · n − m/n − = Θ( mn). 2 10 6 Graphs with Expanding Properties For any vertex subset U, W ⊂ V , let NW (U ) denote the set of vertices in W which are adjacent to a vertex in U . Let N (U ) := NV \U (U ). Definition 6.1. A graph G = (V, E) on n vertices is an (n, s, d1 , d2 , d3 , t)expanding graph if the following four conditions are satisfied: (1) for each vertex subset S with \|S\| = s, \|N (S)\| ≥ d1 n; (2) for each vertex subset S with \|S\| ≤ s, \|N (S)\| ≥ d2 \|S\|; (3) for each vertex subset S with \|S\| ≤ n/2 and for any subset S 0 ⊂ S, \|NV \S (S 0 )\| ≥ \|S 0 \|− t. (4) For each vertex subset S, \|E(S, V \ S)\| ≤ d3 \|S\|. Theorem 6.2. For any connected graph G which is an (n, s, d1 , d2 , d3 , t)-expanding graph, there is a polynomial time algorithm which computes a spanning tree with congestion at most " # log(2−δ) 2 3d1 n 1 t d3 · 4 · max s + 1 , . · + t , where δ = d2 2d1 d1 n Next, we present the polynomial time algorithm in Theorem 6.2 and its analysis. Algorithm. Let G be an (n, s, d1 , d2 , d3 , t)-expanding graph. By Condition (2), every vertex has degree at least d2 . Let v0 be a vertex of degree d ≥ d2 , and let v1 , · · · , vd be its d neighbours. We maintain a tree T rooted at v0 such that T = T1 ∪ T2 ∪ · · · ∪ Td ∪ {v0 v1 , v0 v2 , . . . , v0 vd } where T1 , T2 , · · · , Td are trees rooted at v1 , v2 , . . . , vd respectively. We call the Ti0 s as branches. (See Figure 6). We start with each branch Ti = vi . In order to minimize congestion, we grow T in a balanced way, i.e., we maintain that the Tin’s are roughlyoof the same size. A branch is saturated if it contains at least max s + 1 , 3dd12n vertices. At any point of time, let VT be the set of vertices in T and VT be the vertices not in T . Often, we will move a subtree of a saturated branch Ti to an unsaturated branch Tj to ensure balance. For any x ∈ VT , let Tx denote the subtree of T rooted at x. A vertex x of a saturated branch Ti is called transferable (to branch Tj ) if x has a neighbour y in Tj and the tree Tj ∪ {xy} ∪ Tx is unsaturated. (See Figure 7.) Spanning Tree Congestion 21 v0 v1 vd v2 T1 T2 Td VT Fig. 6. The tree T and its branches v0 Ti∗ v0 Tj Ti∗ Tj y x y x Fig. 7. Transfer of a subtree from a saturated branch to an unsaturated branch The algorithm is divided into two phases which are described below. Throughout the algorithm, whenever a branch Ti gets modified, T gets modified accordingly, and whenever T gets modified VT and VT gets modified accordingly. Phase 1: Repeatedly do one of the following two actions, until \|VT \| ≥ d1 n: (We will prove that the precondition of at least one of the actions is satisfied if \|VT \| < d1 n) 1. If there exists a b ∈ VT such that b has a neighbour a in some unsaturated branch Ti : Add the vertex b and the edge ab to branch Ti . 2. If there exists at least one transferable vertex: (see Figure 7) Find the transferable vertex x such that Tx is the smallest. Let Ti∗ be the 22 L. Sunil Chandran, Yun Kuen Cheung, and Davis Issac branch currently containing x, Tj be a branch to which it is transferable, and y be an arbitrarily chosen neighbour of x in Tj . (a) Remove the subtree Tx from Ti∗ and add it to Tj with x as a child of y. (b) Pick a b ∈ VT that has a neighbour a (arbitrarily chosen, if many) either in Ti∗ or in Tj . (We will show in the analysis that such b exists). We add vertex b and edge ab to the branch containing a (i.e. to Ti∗ or Tj ). Phase 2: While VT 6= ∅, repeat: Find a maximum matching of G[VT , VT ], the bipartite graph formed by edges of G between VT and VT . Let M be the matching. Add all edges of M to T . In the analysis below, we say that a tree is saturated if it contains at least A vertices; we will determine its appropriate value by the end of the analysis. Analysis of Phase 1. We claim that during Phase 1, i.e. if \|VT \| < d1 n, the precondition of either step 1 or step 2 is satisfied. We also show the existence of a vertex b as specified in step 2b, whenever step 2b is reached. Given these and the fact that a vertex in VT is moved to VT (either in step 1 or in step 2b) during each round of Phase 1, we have that Phase 1 runs correctly and terminates after a linear number of rounds. During Phase 1, we will also maintain the invariant that each branch has at most A vertices; thus, each saturated branch has exactly A vertices. We call this invariant the balancedness. Note that balancedness is not violated due to step 1, as the new vertex is added to an unsaturated branch. It is not violated during step 2 as the branches Ti∗ and Tj (as defined in step 2) become unsaturated at the end of the step. We define the hidden vertices of T (denoted by H ≡ HT ) as follows: they are the vertices which are not adjacent to any vertices outside the tree, i.e., to any vertex in VT . If there is an unsaturated branch with a non-hidden vertex, clearly the precondition of step 1 is satisfied. So, let us assume that all the vertices in all unsaturated branches are hidden. In such a case, we show that the precondition of step 2 is satisfied if \|VT \| < d1 n. We argue that in this case \|H\| ≤ s: otherwise, take a subset H 0 ⊂ H of cardinality s, then by condition (2), N (H 0 ), which is contained in VT , has cardinality at least d1 n, a contradiction. Since \|VT \| < d1 n, the number of saturated branches is at most d1 n/A. To ensure that at least one unsaturated branch exists, we set A such that d1 n/A < d2 . Let U denote the set of vertices in all unsaturated branches. Since all vertices in U are hidden vertices, \|U \| ≤ s. Then by condition (2), \|N (U )\| ≥ d2 \|U \|. Note that the vertices in N (U ) are all in the saturated branches. By the pigeon-hole principle, there exists a saturated branch containing at least N (U )/(d1 n/A) ≥ Ad2 \|U \| d1 n vertices of N (U ). By setting A ≥ 3dd12n , the above calculation guarantees the existence of a saturated branch containing at least 3\|U \| ≥ \|U \| + 2 vertices of N (U ); let Ti be such a branch. Spanning Tree Congestion 23 In Ti , pick a vertex x ∈ Ti ∩ N (U ) such that Tx does not contain any vertex in N (U ), except x. Then the size of Tx is at most A − \|N (U )∩Ti \|+1 ≥ A −(\|U \|+1). Let y ∈ U be a vertex which is adjacent to x and Tj be the branch containing y. Since Tj has at most \|U \| vertices, x is a transferable vertex (to Tj ). Thus precondition of step 2 is satisfied. We further set A > s so that in each saturated branch, there is at least one unhidden vertex. In particular, Ti has an unhidden vertex, which is adjacent to some b ∈ VT . The vertex b is either adjacent to a vertex in Tx , or a vertex in Ti \ Tx as required in step 2b. Analysis of Phase 2. Since G is connected, M is non-empty in each iteration of Phase 2, and hence Phase 2 terminates in linear number of rounds. At the end of Phase 2, since VT is empty, T is clearly a spanning tree. It only remains to estimate the congestion of this spanning tree. Towards this, we state the following modified Hall’s theorem, which is an easy corollary of the standard Hall’s theorem. Lemma 6.3. In a bipartite graph (L, R) with \|L\| ≤ \|R\|, for any vertex w ∈ L, let R(w) denote the neighbours of w in R; then for any W ⊂ L, let R(W ) := ∪w∈W R(w). Suppose that there exist t ≥ 0 such that for any W ⊂ L, we have \|R(W )\| ≥ \|W \| − t. Then the bipartite graph admits a matching of size at least \|L\| − t. Recall that Phase 2 consists of multiple rounds of finding a matching between VT and VT . As long as \|VT \| ≤ n/2, condition (3) (with S = VT ) plus the modified Hall’s theorem (with L = VT and R = VT ) guarantees that in each round, at least t \|VT \| − t ≥ 1− · \|VT \| =: (1 − δ)\|VT \| d1 n m l number of vertices in VT are matched. Thus, after at most log(2−δ) 2d11 rounds of matching, \|VT \| ≥ n/2. After reaching \|VT \| ≥ n/2, condition (3) (with S = VT ) plus the modified Hall’s theorem (with L = VT and R = VT ) guarantees that after one more round of matching, all but t vertices are left in VT . By the end of Phase 1, each branch had at most A vertices. After each round of matching, the cardinality of each branch is doubled at most. Thus, the maximum possible number of vertices in each branch after running the whole algorithm is at most log(2−δ) 2 l m 1 log(2−δ) 2d1 +1 1 + t ≤ 4A · A·2 + t. 2d1 and hence the STC is at most " d3 · 4A · 1 2d1 log(2−δ) 2 # +t . Recall that we need A to satisfy d1 n/A < d2 , A ≥ n l mo set A := max s + 1 , 3dd12n . 3d1 n d2 and A > s. Thus we 24 6.1 L. Sunil Chandran, Yun Kuen Cheung, and Davis Issac Random Graph Let G ∈ G(n, p) where p ≥ c0 log n/n, and c0 = 64. The following lemmas show that with high probability G is an (n, s, d1 , d2 , d3 , t)-expanding graph with s = Θ(1/p), d1 = Θ(1), d2 = Θ(np), d3 = Θ(np), t = Θ(1/p) (and hence δ = o(1)). The proof of the lemmas are deferred to Appendix B.2. Lemma 6.4. For any S ⊆ V (G) such that \|S\| = d1/pe, we have \|N (S)\| ≥ c2 n with probability at least 1 − e−n/16 , where c2 = 1/25. Lemma 6.5. For any S ⊆ V (G) such that \|S\| ≤ 1/p, we have \|N (S)\| ≥ c3 np\|S\| with probability at least 1 − O(1/n2 ), where c3 = 1/16. Lemma 6.6. For all A ⊆ V (G) such that \|A\| ≤ n/2, and for all S ⊆ A, with probability at least 1 − e−n , S has at least \|S\| − c4 /p neighbors in V (G) \ A, where c4 = 12. Lemma 6.7. For all S ⊆ V (G), the cut size \|E(S, V (G) \ S)\| is at most np\|S\| with probability at least 1 − n−c0 /4 . Plugging the bounds from above lemmas into Theorem 6.2, together with a separate lower bound argument (Theorem B.2 in Appendix B.1), we have the following theorem; in Appendix B.1, we also present a non-algorithmic proof of this theorem. Theorem 6.8. If G ∈ G(n, p) where p ≥ 64 log n/n, then with probability at least 1 − O(1/n), its STC is Θ(n). 7 Discussion and Open Problems In this paper, we provide thorough understanding, both combinatorially and algorithmically, on the spanning tree congestion of general graphs and random graphs. On course of doing so, we also provide the first constructive proof for the generalized Győri-Lovász theorem, which might be of independent interest. Following are some natural open problems: – Finding the spanning tree with minimum congestion is NP-hard; indeed, Bodlaender et al. [9] showed a (9/8 − )-approximation NP-hardness for the STC problem. Does a constant or a poly-logarithmic factor approximation polynomial time algorithm exist? – We present √ an algorithm for computing a spanning tree achieving congestion at most O( mn). The algorithm runs in sub-exponential time when m = ω(n log2 n). Is there a polynomial time algorithm for constructing such a spanning tree? – For a k-connected graph, a connected k-partition where all parts are of size at most O((n/k) log k) can be found in polynomial time due to an algorithm of Chen et al. [14]. Can we improve the sizes of parts to O(n/k)? – Is finding Győri-Lovász partition PLS-complete? If not, is it polynomial time solvable? Spanning Tree Congestion 25 References 1. Ittai Abraham, Yair Bartal, and Ofer Neiman. Nearly tight low stretch spanning trees. 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Efficient algorithms for tripartitioning triconnected graphs and 3-edge-connected graphs. In Graph-Theoretic Concepts in Computer Science, 19th International Workshop, WG ’93, Utrecht, The Netherlands, June 16-18, 1993, Proceedings, pages 132–143, 1993. Spanning Tree Congestion A A.1 27 Missing Proofs in Sections 4 and 5 Proof of Corollary 4.5 First of all, we set the demand of each vertex in the flow problem to be the the degree of the vertex in G, and t1 , · · · , t` as the sinks in the flow problem. By Proposition 1.4, there exists an `-connected-partition ∪`j=1 U` such that for all j ∈ [`], tj ∈ Uj , and the total degrees of vertices in each Uj is at most 4m/`. With this, by routing the demand of a vertex in Uj to tj via an arbitrary path in G[Uj ] only, we construct a splittable flow with node congestion at most 4m/`. By Theorem 4.4, one can construct a confluent flow with node congestion at most 4(1 + ln `)m/` in polynomial time. Obviously, in the confluent flow, all the flow originating from one vertex goes completely into one sink. Set Vj to be the set of vertices such that the flows originating from these vertices go into tj . It is then routine to check that ∪`j=1 V` is our desired `-connected-partition. A.2 Proof of Theorem 4.2 Instead of giving the full proof, we point out the differences from the proof of Theorem 4.3. First, in handling the basis case (ii), by Theorem 1.3(b), Proposition 1.4 and Lemma 4.1, we havep an improvedp upper bound on the congestion of the returned tree, which is 8mH / m̂/nH ≤ 8 m̂nH . Thus, (1) can be improved to s p 8 m̂(nH − x) + m̂ · x. nH Again, by viewing x as a real variable and taking derivative, it is easy to see that the above expression is maximized at x = 1. So the above bound is at most s p 8 m̂(nH − 1) + m̂ nH p ≤ 8 m̂nH , as desired. Concerning the running time, it is clear that in the worst case, it is dominated by some calls to the algorithm in Theorem 1.3(b). Note that the number of such calls is at most n̂, since each call to the algorithm is on a disjoint set of vertices. There remains one concern, which is the connectedness of H[Y ∪ Z]. Suppose the contrary that H[Y ∪ Z] is not connected. Let C be one of its connected components, so that it contains the least number of vertices from Y . Then C contains at most b\|Y \|/2c vertices from Y , i.e., \|C ∩ Y \| < \|Y \|. Note that C ∩ Y is a vertex cut set of the graph H, thus contradicting that Y is a global minimum vertex cut set. 28 A.3 L. Sunil Chandran, Yun Kuen Cheung, and Davis Issac Proof of Lemma 5.2 It is well known that the requirements (i) and (ii) are satisfied with probability 1 − o(1/n). [11] For each subset S with \|S\| ≤ n/2, by the Chernoff bound, h i p P E(S, V \ S) ≤ · \|S\| · (n − \|S\|) ≤ e−p\|S\|(n−\|S\|)/8 ≤ e−pn\|S\|/16 . 2 Since p ≥ 32 · logn n , the above probability is at most n−2\|S\| . Then by a union bound, the probability that (iii) is not satisfied is at most bn/2c X s=1 B B.1 n · n−2s ≤ s bn/2c X s=1 bn/2c ns · n−2s ≤ X s=1 n−s ≤ 2 . n Spanning Tree Congestion of Random Graphs Non-Algorithmic Proof of Theorem 6.8 We first present a simple non-algorithmic proof that random graph has STC Θ(n) with high probability. Theorem B.1 gives the upper bound and Theorem B.2 gives the lower bound. The proof of Theorem B.1 uses Lemma 1.5 and the fact that for random graphs, vertex-connectivity and minimum degree are equal with high probability. Theorem B.1 does not give an efficient algorithm. Theorem B.1. If G ∈ G(n, p) where p ≥ 8 log n/n, then the spanning tree congestion of G is at most 16n with probability at least 1 − o(1/n). Proof. It is known that the threshold probability for a random graph being k-connected is same as the threshold probability for it having minimum degree at least k [12]. Since p ≥ 8 log n/n, using Chernoff bound and taking union bound over all vertices gives that G has minimum degree at least np/2 with probability at least 1 − o(1/n). Hence G is (np/2)-connected with probability at least 1 − o(1/n). We also have that the number of edges in G is at most 2n2 p with probability at least 1 − o(1/n). Now, by using Lemma 1.5, we have that with probability at least 1 − o(1/n), the spanning tree congestion is at most 16n. Theorem B.2. If G ∈ G(n, p) where p ≥ 32 log n/n, then the spanning tree congestion of G is Ω(n) with probability 1 − O(1/n). Proof. By using Chernoff Bounds and applying union bound, it is easy to show that with probability 1 − o(1/n), every vertex of G has degree at most c1 np for a sufficiently large constant c1 . Also, by Lemma 5.2, with probability 1 − O(1/n), properties (i) and (iii) of that lemma holds. In the proof below, we conditioned on the above mentioned highly probable events. Take a spanning tree T of G which gives the minimum congestion. Let u be a centroid of the tree T , i.e., each connected component of T \ {u} has at most n/2 vertices. If there is a connected component with number of vertices at least Spanning Tree Congestion 29 n/4, then define this connected component as T 0 . Else, all connected components have at most n/4 vertices. In this case, let T 0 be the forest formed by the union of a minimum number of connected components of T \ {u} such that \|T 0 \| ≥ n/4. It is easy to see that \|T 0 \| ≤ n/2. Also, the number of edges in T from V (T 0 ) to V (T ) \ V (T 0 ) is at most degG (u), which is at most c1 np. By property (iii) of Lemma 5.2, the number of edges between V (T 0 ) and V (G)\V (T 0 ) is Ω(n2 p). Each of these edges in G between V (T 0 ) and V (G)\V (T 0 ) have to contribute to the congestion of at least one of the edges in T between V (T 0 ) and V (G) \ V (T 0 ). Now since T 0 sends at most c1 np tree edges to other parts of T , it follows that there exists one edge in T with congestion at least Ω(n2 p)/(c1 np) = Ω(n), as claimed. B.2 Random Graph Satisfies Expanding Properties Constants. For easy reference, we list out the constants used. c0 = 64, c2 = 1/25, c3 = 1/16, c4 = 12 Proof of Lemma 6.4: . Let S = V (G) \ S. The probability that a fixed vertex in S does not have edge to S is at most (1 − p)\|S\| ≤ (1 − p)1/p ≤ e−1 . Since \|S\| ≥ n − 2/p ≥ n − 2n/(c0 log n) ≥ 31n/32, the expected value of \|N (S)\| is at least (31/32) n(1 − e−1 ) ≥ n/2. Hence, using Chernoff bound, the probability that \|N (S)\| < c2 n = n/25 is at most e−n/8 . Since the number of such S is at most n2/p = 22n/c0 ≤ 2n/32 , we have the lemma by applying union bound. Proof of Lemma 6.5: . Let S = V (G) \ S. Since \|S\| ≤ 1/p ≤ n/ log n, we have \|S\| ≥ n/2 for sufficiently large n. Divide S into groups of size d1/(p\|S\|)e. The probability that such a group does not have edge to S is at most (1 − p)\|S\|(1/(p\|S\|)) ≤ 1/e. The expected number of groups having edge to S is at least (np\|S\|/2)(1 − 1/e) ≥ np\|S\|/4. Thus, by Chernoff bound, the probability that \|N (S)\| ≤ np\|S\|/16 is at most e−np\|S\|/16 ≤ 2−c0 \|S\| log n/16 ≤ 2−4\|S\| log n . The number of sets of size \|S\| is at most 2\|S\| log n . Hence, taking union bound over all S with \|S\| ≤ 1/p, we get the required lemma. Proof of Lemma 6.6: . First, we prove that for all C, D ⊆ V (G) such that \|C\| ≥ n/4,\|D\| ≥ c4 /p, and C ∩ D = ∅, there exist at least one edge between C and D with high probability. The probability that there is no edge between such a fixed C and D is at most (1 − p)(n/4)(c4 /p) ≤ e−c4 n/4 . The number of pairs of such C and D is at most 22n . Hence, by taking union bound, the probability that for all C and D, the claim holds is at least 1 − e2n−(c4 n/4) ≥ 1 − e−n . Using the above claim, we prove that for all S ⊆ A, S has at least \|S\| − c4 /p neighbors in A := V (G) \ A with high probability. Suppose there is an S which violates the claim. Note that we can assume \|S\| ≥ c4 /p, because otherwise the claim is vacuously true. Let B := A \ N (S). There cannot be any edges between S and B. Also, \|B\| ≥ (n/2) − (\|S\| − (c4 /p)). So, \|B\| is at least c4 /p and when \|B\| < n/4, \|S\| is at least n/4. Hence, using the previous claim, there is an edge between S and B with probability at least 1 − e−n . Hence, we get a contradiction, and hence our claim is true with probability at least 1 − e−n . 30 L. Sunil Chandran, Yun Kuen Cheung, and Davis Issac Proof of Lemma 6.7: . Let C(S) denote \|E(S, V (G) \ S)\|. For a fixed vertex subset S, the expected value of C(S) is at most np\|S\|. Therefore, probability that C(S) > np\|S\| ≥ c0 \|S\| log n is at most n−c0 \|S\|/2 using Chernoff bounds. The probability that C(S) ≤ np\|S\| for all sets S of size k is at least 1−n−c0 k/2+k ≥ 1− n−c0 /2+1 using union bound and using k ≥ 1. The probability that C(S) ≤ np\|S\| for all vertex subsets S is at least 1 − n−c0 /2+2 ≥ using union bound over all k ∈ [n].	8
arXiv:1711.08848v2 [cs.CV] 29 Nov 2017 Real-Time Seamless Single Shot 6D Object Pose Prediction Bugra Tekin EPFL Sudipta N. Sinha Microsoft Research Pascal Fua EPFL [email protected] [email protected] [email protected] Abstract detection pipeline made of one CNN to coarsely segment the object and another to predict the 2D locations of the projections of the object’s 3D bounding box given the segmentation, which are then used to compute the 6D pose using a PnP algorithm [16]. The method is effective but slow due to its multi-stage nature. SSD-6D [10] is a different pipeline that relies on the SSD architecture [19] to predict 2D bounding boxes and a very rough estimate of the object’s orientation in a single step. This is followed by an approximation to predict the object’s depth from the size of its 2D bounding box in the image, to lift the 2D detections to 6D. Both BB8 and SSD-6D require a further pose refinement step for improved accuracy, which increases their running times linearly with the number of objects being detected. We propose a single-shot approach for simultaneously detecting an object in an RGB image and predicting its 6D pose without requiring multiple stages or having to examine multiple hypotheses. Unlike a recently proposed single-shot technique for this task [10] that only predicts an approximate 6D pose that must then be refined, ours is accurate enough not to require additional post-processing. As a result, it is much faster – 50 fps on a Titan X (Pascal) GPU – and more suitable for real-time processing. The key component of our method is a new CNN architecture inspired by [27, 28] that directly predicts the 2D image locations of the projected vertices of the object’s 3D bounding box. The object’s 6D pose is then estimated using a PnP algorithm. For single object and multiple object pose estimation on the L INE M OD and O CCLUSION datasets, our approach substantially outperforms other recent CNN-based approaches [10, 25] when they are all used without postprocessing. During post-processing, a pose refinement step can be used to boost the accuracy of these two methods, but at 10 fps or less, they are much slower than our method. In this paper, we propose a single-shot deep CNN architecture that takes the image as input and directly detects the 2D projections of the 3D bounding box vertices. It is end-to-end trainable and accurate even without any a posteriori refinement. And since, we do not need this refinement step, we also do not need a precise and detailed textured 3D object model that is needed by other methods [10, 25]. We only need the 3D bounding box of the object shape for training. This can be derived from other easier to acquire and approximate 3D shape representations. We demonstrate state-of-the-art accuracy on the L INE M OD dataset [8], which has become a de facto standard benchmark for 6D pose estimation. However, we are much faster than the competing techniques by a factor of more than five, when dealing with a single object. Furthermore, we pay virtually no time-penalty when handling several objects and our running time remains constant whereas that of other methods grow proportional to the number of objects, which we demonstrate on the O CCLUSION dataset [1]. 1. Introduction Real-time object detection and 6D pose estimation is crucial for augmented reality, virtual reality, and robotics. Currently, methods relying on depth data acquired by RGBD cameras are quite robust [1, 3, 4, 11, 13]. However, active depth sensors are power hungry, which makes 6D object detection methods for passive RGB images more attractive for mobile and wearable cameras. There are many fast keypoint and edge-based methods [21, 31, 35] that are effective for textured objects. However, they have difficulty handling weakly textured or untextured objects and processing low-resolution video streams, which are quite common when dealing with cameras on wearable devices. Therefore, our contribution is an architecture that yields a fast and accurate one-shot 6D pose prediction without requiring any post-processing. It extends single shot CNN architectures for 2D detection in a seamless and natural way to the 6D detection task. Our implementation is based on YOLO [28] but the approach is amenable to other singleshot detectors such as SSD [19] and its variants. Deep learning techniques have recently been used to address these limitations [10, 25]. BB8 [25] is a 6D object 1 2. Related Work We now review existing work on 6D pose estimation ranging from classical feature and template matching methods to newer end-to-end trainable CNN-based methods. Classical methods. Traditional RGB object instance recognition and pose estimation works used local keypoints and feature matching. Local descriptors needed by such methods were designed for invariance to changes in scale, rotation, illumination and viewpoints [21, 31, 35]. Such methods are often fast and robust to occlusion and scene clutter. However, they only reliably handle textured objects in high resolution images [15]. Other related methods include 3D model-based registration [17, 20], Hausdorff matching [9], oriented Chamfer matching for edges [18] and 3D chamfer matching for aligning 3D curve-based models to images [26]. RGB-D methods. The advent of commodity depth cameras has spawned many RGB-D object pose estimation methods [1, 3, 4, 11, 14, 23, 32, 38]. For example, Hinterstoisser et al. proposed template matching algorithms suitable for both color and depth images [7, 8]. Rios et al. [30] extended their work using discriminative learning and cascaded detections for higher accuracy and efficiency respectively. RGB-D methods were used on indoor robots for 3D object recognition, pose estimation, grasping and manipulation [3, 4, 5, 13, 14, 39]. Brachmann et al. [1] proposed using regression forests to predict dense object coordinates, to segment the object and recover its pose from dense correspondences. They also extended their method to handle uncertainty during inference and deal with RGB images [2]. Zach et al. [37] explored fast dynamic programming based algorithms for RGB-D images. CNN-based methods. In recent years, research in most pose estimation tasks has been dominated by CNNs. Techniques such as Viewpoints and Keypoints [34] and Render for CNN [33] cast object categorization and 3D pose estimation into classification tasks, specifically by discretizing the pose space. In contrast, PoseNet [12] proposes using a CNN to directly regress from a RGB image to a 6D pose, albeit for camera pose estimation, a slightly different task. Since PoseNet outputs a translational and a rotational component, the two associated loss terms have to be balanced carefully by tuning a hyper-parameter during training. To avoid this problem, the newer PoseCNN architecture [36] is trained to predict 6D object pose from a single RGB image in multiple stages, by decoupling the translation and rotation predictors. A geodesic loss function more suitable for optimizing over 3D rotations have been suggested in [22]. Another way to address this issue has recently emerged. In [10, 25], the CNNs do not directly predict object pose. Instead, they output 2D coordinates, 2D masks, or discrete orientation predictions from which the 6D pose can be inferred. Because all the predictions are in the 2D image, the problem of weighting different loss terms goes away. Also training becomes numerically more stable, resulting in better performance on the L INE M OD dataset [8]. We also adopt this philosophy in our work. In parallel to these developments, on the 2D object detection task, there has been a progressive trend towards single shot CNN frameworks as an alternative to two-staged methods such as Faster-RCNN [29] that first find a few candidate locations in the image and then classifies them as objects or background. Recently, single shot architectures such as YOLO [27, 28] and SSD [19] have been shown to be fast and accurate. SSD has been extended to predict the object’s identity, its 2D bounding box in the image and a discrete estimate of the object’s orientation [10, 24]. In this paper, we go beyond such methods by extending a YOLO-like architecture [28] to directly predict a few 2D coordinates from which the full 6D object pose can be accurately recovered. 3. Approach With our goal of designing an end-to-end trainable network that predicts the 6D pose in real-time, we were inspired by the impressive performance of single shot 2D object detectors such as YOLO [27, 28]. This led us to design the CNN architecture [27, 28] shown in Fig. 1. We designed our network to predict the 2D projections of the corners of the 3D bounding box around our objects. The main insight was that YOLO was originally designed to regress 2D bounding boxes and to predict the projections of the 3D bounding box corners in the image, a few more 2D points had to be predicted for each object instance in the image. Then given these 2D coordinates and the 3D ground control points for the bounding box corners, the 6D pose can be calculated algebraically with an efficient PnP algorithm [16]. BB8 [25] takes a similar approach. However, they first find a 2D segmentation mask around the object and present a cropped image to a second network that predicts the eight 2D corners in the image. We now describe our network architecture and explain various aspects of our approach in details. 3.1. Model We formulate the 6D pose estimation problem in terms of predicting the 2D image coordinates of virtual 3D control points associated with the 3D models of our objects of interest. Given the 2D coordinate predictions, we calculate the object’s 6D pose using a PnP algorithm. We parameterize the 3D model of each object with 9 control points. For these control points, we select the 8 corners of the tight 3D bounding box fitted to the 3D model, similar to [25]. In addition, we use the centroid of the object’s 3D model as the 9th point. This parameterization is general and can be S S (9x2+1+C) Figure 1. Overview: (a) The proposed CNN architecture. (b) An example input image with four objects. (c) The S × S grid showing cells responsible for detecting the four objects. (d) Each cell predicts 2D locations of the corners of the projected 3D bounding boxes in the image. (e) The 3D output tensor from our network, which represents for each cell a vector consisting of the 2D corner locations, the class probabilities and a confidence value associated with the prediction. used for any rigid 3D object with arbitrary shape and topology. In addition, these 9 control points are guaranteed to be well spread out in the 2D image and could be semantically meaningful for many man-made objects. Our model takes as input a single full color image, processes it with a fully-convolutional architecture shown in Figure 1(a) and divides the image into a 2D regular grid containing S × S cells as shown in Figure 1(c). In our model, each grid location in the 3D output tensor will be associated with a multidimensional vector, consisting of predicted 2D image locations of the 9 control points, the class probabilities of the object and an overall confidence value. At test time, predictions at cells with low confidence values, ie. where the objects of interest are not present, will be pruned. The output target values for our network are stored in a 3D tensor of size S × S × D visualized in Fig. 1(e). The target values for an object at a specific spatial cell location i ∈ S × S is placed in the i-th cell in the 3D tensor in the form of a D dimensional vector vi . When N objects are present in different cells, we have N such vectors, v1 , v2 , . . . , vn in the 3D tensor. We train our network to predict these target values. The 9 control points in our case are the 3D object model’s center and bounding box corners but could be defined in other ways as well. To train our net- work, we only need to know the 3D bounding box of the object, not a detailed mesh or an associated texture map. As in YOLO, it is crucial that a trained network is able to predict not only the precise 2D locations but also high confidence values in regions where the object is present and low confidence where it isn’t present. In case of 2D object detection, YOLO uses for its confidence values, an intersection over union (IoU) score associated with the predicted (and true 2D rectangles) in the image. In our case, the objects are in 3D and to compute an equivalent IoU score with two arbitrary cuboids, we would need to calculate a 3D convex hull corresponding to their intersections. This would be tedious and would slow down training. Therefore, we take a different approach. We model the predicted confidence value using a confidence function shown in Figure 2. The confidence function, c(x), returns a confidence value for a predicted 2D point denoted by x based on its distance DT (x) from the ground truth i.e. target 2D point. Formally, we define the confidence function c(x) as follows: ( c(x) = α(1− e 0 DT (x) dth ) , if DT (x) < dth otherwise (1) The distance DT (x) is defined as the 2D Euclidean distance in the image space. To achieve precise localization points, we do not constrain the network’s output as those points should be allowed to fall outside the cell. The predicted control point (gx , gy ) is defined as Confidence 1 0 0 Distance Figure 2. Confidence c(x) as a function of the distance DT (x) between a predicted point and the true point. with this function, we choose a sharp exponential function with a cut-off value dth instead of a monotonically decreasing linear function. The sharpness of the exponential function is defined by the parameter α. In practice, we apply the confidence function to all the control points and calculate the mean value and assign it as the confidence. As mentioned earlier, we also predict C conditional class probabilities at each cell. The class probability is conditioned on the cell containing an object. Overall, our output 3D tensor depicted in Figure 1(e) has dimension S × S × D, where the 2D spatial grid corresponding to the image dimensions has S × S cells and each such cell has a D dimensional vector. Here, D = 9×2+C +1, because we have 9 (xi , yi ) control points, C class probabilities and one confidence value. Our network architecture follows the fully convolutional YOLO v2 architecture [28]. Thus, our network has 23 convolutional layers and 5 max-pooling layers. Similar to YOLO v2, we choose S = 13 and have a 13 × 13 2D spatial grid on which we make our predictions. We also allow higher layers of our network to use fine-grained features by adding a passthrough layer. Specifically, we bring features from an earlier layer at resolution 26 × 26, apply batch normalization and resize the input image during training onthe-fly. As the network downsamples the image by a factor of 32, we change the input resolution to a multiple of 32 randomly chosen from the set {320, 352, . . . , 608} to be robust to objects of different size. 3.2. Training Procedure Our final layer outputs class probabilities, (x, y) coordinate locations for the control points, and the overall confidence score. During training, this confidence value is computed on the fly using the function defined in Eq. 1 to measure the distance between the current coordinate predictions and the ground-truth, DT (x). We predict offsets for the 2D coordinates with respect to (cx , cy ), the top-left corner of the associated grid cell. For the centroid, we constrain this offset to lie between 0 and 1. However, for the corner gx = f (x) + cx (2) gy = f (y) + cy (3) where f (·) is chosen to be a 1D sigmoid function in case of the centroid and the identity function in case of the eight corner points. This has the effect of forcing the network to first find the approximate cell location for the object and later refine its eight corner locations. We minimize the following loss function to train our complete network. L = λpt Lpt + λconf Lconf + λid Lid (4) Here, the terms Lpt , Lconf and Lid denote the coordinate loss, confidence loss and the classification loss respectively. We use mean-squared error for the coordinate and confidence losses, and cross entropy for the classification loss. As suggested in [27, 28], to improve model stability, we downweight the confidence loss for cells that don’t contain objects by setting λconf to 0.1. For cells that contain objects, we set λconf to 5.0. When multiple objects are located close to each other in the 3D scene, they are more likely to appear close together in the images or be occluded by each other. In these cases, certain cells might contain multiple objects. To be able to predict the pose of such multiple objects that lie in the same cell, we allow up to 5 candidates per cell and therefore predict five sets of control points per cell. Similarly to [28], we precompute with k-means, five anchor boxes that define the size, ie. the width and height of a 2D rectangle tightly fitted to a masked region around the object in the image. During training, we assign whichever anchor box has the most similar size to the current object as the responsible one to predict the 2D coordinates for that object. 3.3. Pose Prediction We detect and estimate the pose of objects in 6D by invoking our network only once. At test time, we estimate the class-specific confidence scores for each object by multiplying the class probabilities and the score returned by the confidence function. Each grid cell produces predictions in one network evaluation and cells with predictions with low confidence are pruned using a confidence threshold. For large objects and objects whose projections lie at the intersection of two cells, multiple cells are likely to predict highly confident detections. To obtain a more robust and well localized pose estimate, we inspect the cells in the 3×3 neighborhood of the cell which has the maximum confidence score. We combine the individual corner predictions of these adjacent cells by computing a weighted average of the individual detections, where the weights used are the confidence scores of the associated cells. At run-time, the network gives the 2D projections of the object’s centroid and corners of its 3D bounding box along with the object identity. We estimate the 6D pose from the correspondences between the 2D and 3D points using a Perspective-n-Point (PnP) pose estimation method [16]. In our case, PnP uses only 9 such control point correspondences and provides an estimate of the 3D rotation R and 3D translation t of the object in camera coordinates. 4. Implementation Details We initialize the parameters of our network by training the original network on the ImageNet classification task. As the pose estimates in the early stages of training are inaccurate, the confidence values computed using Eq. 1 are initially unreliable. To remedy this, we pretrain our network parameters by setting the regularization parameter for confidence to 0. Subsequently, we train our network by setting λconf to 5 for the cells that contain an object, and to 0.1 otherwise, to have more reliable confidence estimates in the early stages of the network. In practice, we set the sharpness of the confidence function α to 2 and the distance threshold to 30 pixels. We use stochastic gradient descent for optimization. We start with a learning rate of 0.001 and divide the learning rate by 10 at every 100 epochs. To avoid overfitting, we use extensive data augmentation by randomly changing the hue, saturation and exposure of the image by up to a factor of 1.2. We also randomly scale and translate the image by up to a factor of 20% of the image size. Our implementation is based on PyTorch. We will make our code publicly available for the sake of reproducibility. 5. Experiments We first evaluate our method for estimating the 6D pose of single objects and then we evaluate it in the case where multiple objects are present in the image. We use the same datasets and evaluation protocols as in [2, 10, 25], which we review below. We then present and compare our results to the state of the art methods. 5.1. Datasets We test our approach on two datasets that were designed explicitly to benchmark 6D object pose estimation algorithms. We describe them briefly below. LineMod [8] has become a de facto standard benchmark for 6D object pose estimation of textureless objects in cluttered scenes. The central object in each RGB image is assigned a ground-truth rotation, translation, and ID. A full 3D mesh representing the object is also provided. Method Object Ape Benchvise Cam Can Cat Driller Duck Eggbox Glue Holepuncher Iron Lamp Phone Average w/o Refinement Brachmann BB8 OURS [2] [25] 95.3 92.10 80.0 95.06 80.9 93.24 84.1 97.44 97.0 97.41 74.1 79.41 81.2 94.65 87.9 90.33 89.0 96.53 90.5 92.86 78.9 82.94 74.4 76.87 77.6 86.07 69.5 83.9 90.37 w/ Refinement Brachmann BB8 [2] [25] 85.2 96.6 67.9 90.1 58.7 86.0 70.8 91.2 84.2 98.8 73.9 80.9 73.1 92.2 83.1 91.0 74.2 92.3 78.9 95.3 83.6 84.8 64.0 75.8 60.6 85.3 73.7 89.3 Table 1. Comparison of our approach with state-of-the-art algorithms on LineMod in terms of 2D reprojection error. We report percentages of correctly estimated poses. Bold face numbers denote the best overall methods, bold italic numbers denote the best methods among those that do not use refinement as opposed to the ones that use, if different. Note that even though we do not rely on the knowledge of a detailed 3D object model our method consistently outperforms the baselines. OCCLUSION [1] is a multi-object detection and pose estimation dataset that contains additional annotations for all objects in a subset of the LineMod images. As its name suggests, several objects in the images are severely occluded due to scene clutter, which makes pose estimation extremely challenging. With the exception of [10, 25], it has primarily been used to test algorithms that require depth images. 5.2. Evaluation Metrics We use three standard metrics to evaluate 6D pose accuracy, namely – 2D reprojection error, average 3D distance of model vertices (referred to as ADD metric), and IoU score as in [2, 10, 25]. In all cases, we calculate the accuracy as the percentage of correct pose estimates for certain error thresholds. When using the reprojection error, we consider a pose estimate to be correct when the mean distance between the 2D projections of the object’s 3D mesh vertices using the estimate and the ground truth pose is less than 5 pixels [2]. This measures the closeness of the true image projection of the object to that obtained by using the estimated pose. This metric is suitable for augmented reality applications. When comparing 6D poses using the ADD metric, we take a pose estimate to be correct if the mean distance between the true coordinates of 3D mesh vertices and those estimated given the pose is less than 10% of the object’s diameter [8]. For most objects, this is approximately a 2cm threshold but for smaller objects, such as ape, the threshold drops to about 1cm. For rotationally symmetric objects whose pose can only be computed up to one degree of rotational freedom, we modify slightly the metric as in [2, 8] Method and compute s= 1 X min k(Rx + t) − (R̂x + t̂)k , M \|M\| (5) x1 ∈M where (R, t) are the ground-truth rotation and translation, (R̂, t̂) the predicted ones, and M the vertex set of the 3D model. We use this metric when evaluating the pose accuracy for the rotationally invariant objects, eggbox and glue as in [2, 8]. To compute the IoU metric, we measure the overlap between the projections of the 3D model given the ground-truth and predicted pose and accept a pose as correct if the overlap is larger than 0.5. 5.3. Single Object Pose Estimation We first estimate the 6D pose of the central object in the RGB only LineMod images, without reference to the depth ones. We compare our approach to those of [2, 10, 25], which operate under similar conditions. In this dataset, the training images are selected such that the relative orientation between corresponding pose annotations are larger than a threshold. To avoid being influenced by the scene context, we segment the training images using the segmentation masks provided with the dataset and replace the background by a random image from the PASCAL VOC dataset [6]. We use exactly the same training/test splits as in [25]. We report our results in terms of 2D reprojection error in Table 1 and 6D pose error in Table 2. We provide example pose predictions of our approach in Figure 3. 5.3.1 w/o Refinement Brachmann BB8 SSD-6D Object [2] [25] [10] Ape 27.9 0 Benchvise 62.0 0.18 Cam 40.1 0.41 Can 48.1 1.35 Cat 45.2 0.51 Driller 58.6 2.58 Duck 32.8 0 Eggbox 40.0 8.9 Glue 27.0 0 Holepuncher 42.4 0.30 Iron 67.0 8.86 Lamp 39.9 8.20 Phone 35.2 0.18 Average 32.3 43.6 2.42 w/ Refinement OURS Brachmann BB8 SSD-6D [2] [25] [10] 21.62 33.2 40.4 65 81.80 64.8 91.8 80 36.57 38.4 55.7 78 68.80 62.9 64.1 86 41.82 42.7 62.6 70 63.51 61.9 74.4 73 27.23 30.2 44.3 66 69.58 49.9 57.8 100 80.02 31.2 41.2 100 42.63 52.8 67.2 49 74.97 80.0 84.7 78 71.11 67.0 76.5 73 47.74 38.1 54.0 79 55.95 50.2 62.7 79 Table 2. Comparison of our approach with state-of-the-art algorithms on LineMod in terms of ADD metric. We report percentages of correctly estimated poses. Bold face numbers denote the best overall methods, bold italic numbers denote the best methods among those that do not use refinement as opposed to the ones that use, if different. Threshold Object Ape Benchvise Cam Can Cat Driller Duck Eggbox Glue Holepuncher Iron Lamp Phone Average [10] 0 0.18 0.41 1.35 0.51 2.58 0 8.9 0 0.30 8.86 8.20 0.18 2.42 10% OURS 21.62 81.80 36.57 68.80 41.82 63.51 27.23 69.58 80.02 42.63 74.97 71.11 47.74 55.95 30% [10] OURS 5.62 70.67 2.07 91.07 34.52 81.57 61.43 99.02 36.87 90.62 56.01 99.01 5.56 70.70 24.61 81.31 14.18 89.00 18.23 85.54 59.26 98.88 57.64 98.85 35.55 91.07 31.65 88.25 50% [10] OURS 19.95 88.10 10.62 98.85 63.54 94.80 85.49 99.90 64.04 98.80 84.86 99.80 32.65 89.39 48.41 98.31 26.94 97.20 38.75 96.29 88.31 99.39 81.03 99.62 61.22 98.85 54.29 96.78 Table 3. Comparison of our approach with SSD-6D [10] without refinement using different thresholds for the 6D pose metric. Comparative Accuracy 6D Accuracy in terms of projection error. In Table 1, we compare our results to those of Brachmann et al. [2] and to BB8 [25]. Both of these competing methods involve a multi-stage pipeline that comprises a 2D detection step followed by pose prediction and refinement. Since we do not have a refinement stage, we show in the table their results without and with it. In both cases, we achieve better 6D pose estimation accuracies. In Table 4, we perform a similar comparison with SSD6D [10], whose authors report their projection accuracy in terms of the IoU metric. That method also requires a posteriori refinement and our results are again better in both cases, even though SSD-6D relies on a large training set of rendered images that are sampled over a wide range of viewpoints and locations. the competing methods. Before refinement, we outperform all the methods by a significant margin of at least 12%. After refinement, our pose estimates are still better than Brachmann et al. [2]. By assuming the additional knowledge of a full 3D CAD model and using it to further refine the pose, BB8 1 and SSD-6D 2 boost their pose estimation accuracy. Without any bells and whistles, our approach achieves state-of-the-art pose estimation accuracy in all the metrics without refinement. When compared against methods that rely on the additional knowledge of full 3D CAD models and pose refinement, it still achieves state-of-the-art performance in 2D projection error and IoU metrics and yields comparable accuracy in the ADD metric. Our approach could be used in conjunction with such refinement strategies to further increase the accuracy however this comes at a heavy computational cost as we describe below. 6D Accuracy in terms of the ADD metric. In Tables 2 and 3, we compare our methods against the other in terms of the average of the 3D distances, as described in Section 5.2. In Table 2, we give numbers before and after refinement for 1 The authors do not report results without refinement, however they provided us with the accuracy numbers reported in Table 2. 2 The authors were not able to provide their accuracy numbers without refinement for this metric, but made their code publicly available. We ran their code with provided pretrained models to obtain the 6D pose errors. Method Object Ape Benchvise Cam Can Cat Duck Glue Holepuncher Iron Lamp Phone Average Driller Eggbox w/o Refinement SSD-6D OURS [10] 98.46 99.81 100 99.90 99.53 100 100 99.81 99.34 99.90 99.04 100 97.24 99.81 98.95 99.90 99.65 100 99.38 100 99.91 100 99.22 99.92 100 99.91 w/ Refinement SSD-6D [10] 99 100 99 100 99 98 98 99 99 99 100 99.4 99 99 Table 4. Comparison of our approach against [10] on LineMod using IoU metric. The authors of [10] were able to provide us the results of our approach w/o the refinement. 5.3.2 Accuracy / Speed Trade-off In Table 5, we report the computational efficiency of our approach for single object pose estimation in comparison to the state-of-the-art approaches [2, 10, 25]. Our approach runs at real-time performance in contrast to the existing approaches which fall short of it. In particular, our algorithm runs at least 5 times faster than the state-of-the-art techniques for single object pose estimation. As can be seen in Table 2, pose refinement in Brachmann et al. increase the accuracy significantly by 17.9% at an additional run-time of 100 miliseconds per object. BB8 also gets a substantial improvement of 19.1% in accuracy at an additional run-time of 21 miliseconds per object. Even without correcting for the pose error, our approach outperforms Brachmann et al. and yields close accuracy to BB8 while being 16 times faster for single object pose estimation. As discussed also in [10], the unrefined poses computed from the bounding boxes of the SSD 2D object detector, are rather approximate. We confirmed this by running their publicly available code with the provided pretrained models. We report the accuracy numbers without the refinement using the ADD metric in Table 3 for different thresholds. While providing a good initialization for the subsequent pose processing, the pose estimates of SSD-6D without refinement are much less accurate than our approach. The further refinement increases the pose estimation accuracy significantly, however st of a computational time of 24 miliseconds per object. Moreover, in contrast to our approach, the refinement requires the knowledge of the full 3D object CAD model. In Figure 3, we show example results of our method on the L INE M OD. We include more visual results of our method in the supplementary material. Method Overall speed for 1 object Refinement runtime 2 fps 3 fps 10 fps 50 fps 100 ms/object 21 ms/object 24 ms/object - Brachmann et al. [2] Rad & Lepetit [25] Kehl et al. [10] OURS Table 5. Comparison of the overall computational runtime of our approach for a single object in comparison to [2, 10, 25]. We further provide the computational runtime induced by the pose refinement stage of [2, 10, 25] and report pose estimation accuracy as in [25]. The identity of the objects cannot be assumed to be known a priori and has to be guessed. To this end, the method of [25] assumes that it has access to image crops based on the ground-truth 2D bounding boxes. 3 We make no such assumptions. Instead, we jointly detect the object in 2D, estimate its identity and predict its 6D pose. We generate our training images with the approach explained in Section 5.2. We further augment the LineMod training data by adding into the images objects extracted from other training sequences. We report our pose estimation accuracy in Figure 4 and demonstrate that even without assuming ground-truth information as in the case of [25], our method yields satisfactory pose accuracy in the case of severe occlusions. For object detection purposes, we consider an estimate to be correct if its detection IoU is larger than 0.5. Note that here the detection IoU corresponds to the overlap of the 2D bounding boxes of the object, rather than the overlap of the projected masks as is the case for the IoU metric defined in Sec 5.2. In Table 6, we report a mean average precision (MAP) of 0.48 which is similar to the accuracy reported by [2] and outperforms the ones reported by [7, 10]. Method MAP Hinterstoisser et al. [7] Brachmann et al. [2] Kehl et al. [10] OURS 0.21 0.51 0.38 0.48 Table 6. The detection experiment on the Occlusion dataset [2]. (Left) Precision-recall plot. (Right) 5.4. Multiple Object Pose Estimation Our approach provides accurate 6D poses with real-time performance. Upon one network invocation, our only computational overhead is an efficient PnP algorithm which operates on just 9 points per object. Furthermore we do not require full 3D colored object models to further refine our initial pose estimates. Our approach is therefore scalable to handle multiple objects as shown in Figure 5 and has only a negligible computational overhead of PnP (0.2 miliseconds/object) while the competing approaches [10] have a linear runtime growth. We use the OCCLUSION dataset to compare our approach to Brachmann et al. [2] for multi-object detection 3 This it is not explicitly stated in [25], but the authors confirmed this to us in private email communication. Figure 3. Pose estimation results of our approach. Note that our method can recover the 6D pose in these challenging scenarios, which involve significant amounts of clutter, occlusion and orientation ambiguity. In the last column, we show failure cases due to motion blur, severe occlusion and specularity (this figure is best viewed on a computer screen). tions. With only 1-2 % decrease in accuracy we can reach to a runtime of 94 fps and the runtime virtually remains the same for estimating the pose of multiple objects. Method 416 × 416 480 × 480 544 × 544 608 × 688 Figure 4. Percentage of correctly estimated poses as a function of the projection error for different objects of the Occlusion dataset [2]. 500 Kehl et al. `17 OURS Runtime in miliseconds 400 89.71 90.00 90.37 90.65 94 fps 67 fps 50 fps 43 fps Table 7. Accuracy/speed trade-off of our method on the L INE M OD dataset. Accuracy reported is the percentage of correctly estimated poses with respect to the 2D projection error. The same network model is used for all four input resolutions. Timings are on a . NVIDIA Titan X (Pascal) GPU. 6. Conclusion 300 200 100 0 2D projection metric Speed 0 5 10 15 20 Number of objects Figure 5. The runtime of our approach with increasing number of objects as compared to that of [10]. We also evaluated the accuracy and speed of our approach for different input resolutions. As explained in Section 3.1, we adopt a multi-scale training procedure and change the input resolution during training randomly as in [28]. This allows us to be able to change the input resolution at test-time and predict from images with higher resolution. This is especially useful for predicting the pose of small objects more robustly. As we do not have an initial step for 2D object detection and produce image crops which are then resized to higher resolutions for pose prediction as in [25], our approach requires better handling of the small objects. In Table 7, we compare the accuracy and computational efficiency of our approach for different input resolu- We have proposed a new CNN architecture for fast and accurate single-shot 6D pose prediction that naturally extends the single shot 2D object detection paradigm to 6D object detection. Our network predicts 2D locations of the projections of the objects 3D bounding box corners which involves predicting just a few more 2D points than for 2D bounding box regression. Given the predicted 2D corner projections, the 6D pose is computed via an efficient PnP method. For high accuracy, existing CNN-based 6D object detectors all refine their pose estimates during postprocessing, a step that requires an accurate 3D object model and also incurs a runtime overhead per detected object. In contrast, our single shot predictions are very accurate which alleviates the need for refinement. Due to this, our method is not dependent on access to 3D object models and there is virtually no overhead when estimating the pose of multiple objects. Our method is real-time; it runs at 50 – 94 fps depending on the image resolution. This makes it substantially faster than existing methods. Acknowledgements. We would like to thank Mahdi Rad and Vincent Lepetit for fruitful discussions and providing the results of their method in Table 2. Also, we thank Wadim Kehl, Fabian Manhardt and Slobodan Ilic for helpful discussions and for their help in evaluating their algorithm without postprocessing in Table 4. References [1] E. Brachmann, A. Krull, F. Michel, S. Gumhold, J. Shotton, and C. Rother. Learning 6D Object Pose Estimation Using 3D Object Coordinates. In ECCV, 2014. 1, 2, 5, 10 [2] E. Brachmann, F. Michel, A. Krull, M. Ying Yang, S. Gumhold, et al. Uncertainty-Driven 6D Pose Estimation of Objects and Scenes from a Single RGB Image. In CVPR, 2016. 2, 5, 6, 7, 8 [3] C. Choi and H. I. Christensen. 3D Textureless Object Detection and Tracking: An Edge-Based Approach. In IROS, 2012. 1, 2 [4] C. Choi and H. I. Christensen. RGB-D Object Pose Estimation in Unstructured Environments. Robotics and Autonomous Systems, 2016. 1, 2 [5] A. Collet, M. Martinez, and S. S. Srinivasa. The MOPED Framework: Object Recognition and Pose Estimation for Manipulation. The International Journal of Robotics Research, 2011. 2 [6] M. Everingham, L. Van Gool, C. K. I. Williams, J. Winn, and A. Zisserman. The PASCAL Visual Object Classes (VOC) Challenge. IJCV, 2010. 6, 10 [7] S. Hinterstoisser, S. Holzer, C. Cagniart, S. Ilic, K. Konolige, N. Navab, and V. Lepetit. Multimodal Templates for Real-Time Detection of Texture-less Objects in Heavily Cluttered Scenes. In ICCV, 2011. 2, 7 [8] S. Hinterstoisser, V. Lepetit, S. Ilic, S. Holzer, G. Bradski, K. Konolige, and N. Navab. Model Based Training, Detection and Pose Estimation of Texture-less 3D Objects in Heavily Cluttered Scenes. In ACCV, 2012. 1, 2, 5, 6, 10 [9] D. P. Huttenlocher, G. A. Klanderman, and W. J. Rucklidge. Comparing Images using the Hausdorff Distance. TPAMI, 1993. 2 [10] W. Kehl, F. Manhardt, F. Tombari, S. Ilic, and N. Navab. SSD-6D: Making RGB-Based 3D Detection and 6D Pose Estimation Great Again. In ICCV, 2017. 1, 2, 5, 6, 7, 8 [11] W. Kehl, F. Milletari, F. Tombari, S. Ilic, and N. Navab. Deep Learning of Local RGB-D Patches for 3D Object Detection and 6D Pose Estimation. In ECCV, 2016. 1, 2 [12] A. Kendall, M. Grimes, and R. Cipolla. PoseNet: A Convolutional Network for Real-Time 6-DOF Camera Relocalization. In ICCV, 2015. 2 [13] K. Lai, L. Bo, X. Ren, and D. Fox. A Large-Scale Hierarchical Multi-View RGB-D Object Dataset. In ICRA, 2011. 1, 2 [14] K. Lai, L. Bo, X. Ren, and D. Fox. A Scalable Tree-Based Approach for Joint Object and Pose Recognition. In AAAI, 2011. 2 [15] V. Lepetit and P. Fua. Monocular Model-Based 3D Tracking of Rigid Objects: A Survey. Foundations and Trends in Computer Graphics and Vision, 2005. 2 [16] V. Lepetit, F. Moreno-Noguer, and P. Fua. EPnP: An Accurate O(n) Solution to the PnP problem. IJCV, 2009. 1, 2, 5 [17] Y. Li, L. Gu, and T. Kanade. Robustly Aligning a Shape Model and Its Application to Car Alignment of Unknown Pose. TPAMI, 2011. 2 [18] M.-Y. Liu, O. Tuzel, A. Veeraraghavan, and R. Chellappa. Fast Directional Chamfer Matching. In CVPR, 2010. 2 [19] W. Liu, D. Anguelov, D. Erhan, C. Szegedy, S. Reed, C.-Y. Fu, and A. C. Berg. SSD: Single Shot MultiBox Detector. In ECCV, 2016. 1, 2 [20] D. G. Lowe. Fitting Parameterized Three-Dimensional Models to Images. TPAMI, 1991. 2 [21] D. G. Lowe. Object Recognition from Local Scale-Invariant Features. In ICCV, 1999. 1, 2 [22] S. Mahendran, H. Ali, and R. Vidal. 3D Pose Regression using Convolutional Neural Networks. CVPRW, 2017. 2 [23] F. Michel, A. Kirillov, E. Brachmann, A. Krull, S. Gumhold, B. Savchynskyy, and C. Rother. Global Hypothesis Generation for 6D Object Pose Estimation. In CVPR, 2017. 2 [24] P. Poirson, P. Ammirato, C.-Y. Fu, W. Liu, J. Kosecka, and A. C. Berg. Fast Single Shot Detection and Pose Estimation. In 3DV, 2016. 2 [25] M. Rad and V. Lepetit. BB8: A Scalable, Accurate, Robust to Partial Occlusion Method for Predicting the 3D Poses of Challenging Objects without Using Depth. In ICCV, 2017. 1, 2, 5, 6, 7, 8, 11, 12 [26] K. Ramnath, S. N. Sinha, R. Szeliski, and E. Hsiao. Car Make and Model Recognition using 3D Curve Alignment. In WACV, 2014. 2 [27] J. Redmon, S. Divvala, R. Girshick, and A. Farhadi. You Only Look Once: Unified, Real-Time Object Detection. In CVPR, 2016. 1, 2, 4 [28] J. Redmon and A. Farhadi. YOLO9000: Better, Faster, Stronger. CVPR, 2017. 1, 2, 4, 8 [29] S. Ren, K. He, R. Girshick, and J. Sun. Faster R-CNN: Towards Real-Time Object Detection with Region Proposal Networks. In NIPS, 2015. 2 [30] R. Rios-Cabrera and T. Tuytelaars. Discriminatively Trained Templates for 3d Object Detection: A Real Time Scalable Approach. In ICCV, 2013. 2 [31] F. Rothganger, S. Lazebnik, C. Schmid, and J. Ponce. 3D Object Modeling and Recognition using Local Affine-Invariant Image Descriptors and Multi-View Spatial Constraints. IJCV, 2006. 1, 2 [32] J. Sock, S. H. Kasaei, L. S. Lopes, and T.-K. Kim. Multi-view 6D Object Pose Estimation and Camera Motion Planning using RGBD Images. In ICCV, 2017. 2 [33] H. Su, C. R. Qi, Y. Li, and L. J. Guibas. Render for CNN: Viewpoint Estimation in Images Using CNNs trained with Rendered 3D Model Views. In ICCV, 2015. 2 [34] S. Tulsiani and J. Malik. Viewpoints and Keypoints. In CVPR, 2015. 2 [35] D. Wagner, G. Reitmayr, A. Mulloni, T. Drummond, and D. Schmalstieg. Pose Tracking from Natural Features on Mobile Phones. In ISMAR, 2008. 1, 2 [36] Y. Xiang, T. Schmidt, V. Narayanan, and D. Fox. PoseCNN: A Convolutional Neural Network for 6D Object Pose Estimation in Cluttered Scenes. arXiv preprint arXiv:1711.00199, 2017. 2 [37] C. Zach, A. Penate-Sanchez, and M.-T. Pham. A Dynamic Programming Approach for Fast and Robust Object Pose Recognition from Range Images. In CVPR, 2015. 2 [38] H. Zhang and Q. Cao. Combined Holistic and Local Patches for Recovering 6D Object Pose. In ICCV, 2017. 2 [39] M. Zhu, K. G. Derpanis, Y. Yang, S. Brahmbhatt, M. Zhang, C. Phillips, M. Lecce, and K. Daniilidis. Single Image 3D Object Detection and Pose Estimation for Grasping. In ICRA, 2014. 2 Supplemental Material: “Real-Time Seamless Single Shot 6D Object Pose Prediction” In the supplemental material, we provide details on how the training images were prepared and on the proposed confidence weighted prediction step. We also present qualitative results on O CCLUSION [1] and L INE M OD [8]. Training Images. As discussed in the main paper, we segment the foreground object in the images in the training set, using the segmentation masks provided and paste the segmented image over a random image taken from the PASCAL VOC dataset [6]. Examples of such images, which are given as input to the network at training time are shown in Figure 6. This operation of removing the actual background prevents the network from learning the scene context and is essential in order to achieve proper generalization. Confidence-weighted prediction. In the final step of our method, we compute a weighted sum of multiple sets of predictions for the corners and the centroid, using associated confidence values as weights. On L INE M OD, this gave a 1–2% improvement in accuracy with the 2D projection metric. The first step involves scanning the full 17×17 grid to find the cell with the highest confidence for each potential object. We then consider a 3 × 3 neighborhood around it on the grid and prune the cells with confidence values lower than the detection threshold of 0.5. On the remaining cells, we compute a confidence-weighted average of the associated predicted 18-dimensional vectors, where the eight corner points and the centroid have been stacked to form the vector. The averaged coordinates are then used in the PnP method. This sub-pixel refinement on the grid usually improves the pose of somewhat large objects that occupy several adjoining cells in the grid. Figure 7 shows an example where the ape object lies between two adjoining cells and the confidence weighting improves the pose accuracy. Figure 7. (Left) The 17×17 grid on a 544×544 image. (Middle) Confidence values for predictions of the ape object on the grid. (Right) Cropped view of our pose estimate (shown in blue) and the ground truth (shown in green). Here, three cells next to the best cell have good predictions and their combination gives a more accurate pose than the best prediction alone (best viewed in color). Figure 6. (Top) Using segmentation masks given in L INE M OD, we extract the foreground objects in our training images and composite them over random images from PASCAL VOC [6]. (Bottom) We also augment the training set by combining images of multiple objects taken from different training images. Qualitative Results. We show qualitative results from the O CCLUSION [1] and L INE M OD [8] datasets in Figures 8 to 13. These examples show that our method is robust to severe occlusions, rotational ambiguities in appearance, reflections, viewpoint change and scene clutter. (a) (b) (c) Figure 8. Results on the O CCLUSION dataset. Our method is quite robust against severe occlusions in the presence of scene clutter and rotational pose ambiguity for symmetric objects. (a) Input images, (b) 6D pose predictions of multiple objects, (c) A magnified view of the individual 6D pose estimates of six different objects is shown for clarity. In each case, the 3D bounding box is rendered on the input image. The following color coding is used – A PE (gold), B ENCHVISE (green), C AN (red), C AT (purple), D RILLER (cyan), D UCK (black), G LUE (orange), H OLEPUNCHER (blue). In addition to the objects from the O CCLUSION dataset, we also visualize the pose predictions of the Benchvise object from the L INE M OD dataset. As in [25], we do not evaluate on the Eggbox object, as more than 70% of close poses are not seen in the training sequence. This image is best viewed on a computer screen. (a) (b) (c) Figure 9. Results on the O CCLUSION dataset. Our method is quite robust against severe occlusions in the presence of scene clutter and rotational pose ambiguity for symmetric objects. (a) Input images, (b) 6D pose predictions of multiple objects, (c) A magnified view of the individual 6D pose estimates of six different objects is shown for clarity. In each case, the 3D bounding box is rendered on the input image. The following color coding is used – A PE (gold), B ENCHVISE (green), C AN (red), C AT (purple), D RILLER (cyan), D UCK (black), G LUE (orange), H OLEPUNCHER (blue). In addition to the objects from the O CCLUSION dataset, we also visualize the pose predictions of the Benchvise object from the L INE M OD dataset. As in [25], we do not evaluate on the Eggbox object, as more than 70% of close poses are not seen in the training sequence. This image is best viewed on a computer screen. Figure 10. Example results on the L INE M OD dataset: (left) A PE, (middle) B ENCHVISE, (right) C AM. The projected 3D bounding boxes are rendered over the image and they have been cropped and resized for ease of visualization. The blue cuboid is rendered using our pose estimate whereas the green cuboid is rendered using the ground truth object pose. Note that the input image dimension is 640 × 480 pixels and the objects are often quite small. Noticeable scene clutter and occlusion makes these examples challenging. Figure 11. Example results on the L INE M OD dataset: (left) C AN, (middle) C AT, (right) D RILLER. The projected 3D bounding boxes are rendered over the image and they have been cropped and resized for ease of visualization. The blue cuboid is rendered using our pose estimate whereas the green cuboid is rendered using the ground truth object pose. Note that the input image dimension is 640 × 480 pixels and the objects are often quite small. Noticeable scene clutter and occlusion makes these examples challenging. Figure 12. Example results on the L INE M OD dataset: (left) D UCK, (middle) E GGBOX, (right) G LUE. The projected 3D bounding boxes are rendered over the image and they have been cropped and resized for ease of visualization. The blue cuboid is rendered using our pose estimate whereas the green cuboid is rendered using the ground truth object pose. Note that the input image dimension is 640 × 480 pixels and the objects are often quite small. Noticeable scene clutter and occlusion makes these examples challenging. Figure 13. Example results on the L INE M OD dataset: (left) H OLE P UNCHER, (middle) I RON, (right) L AMP and P HONE. The projected 3D bounding boxes are rendered over the image and they have been cropped and resized for ease of visualization. The blue cuboid is rendered using our pose estimate whereas the green cuboid is rendered using the ground truth object pose. Note that the input image dimension is 640 × 480 pixels and the objects are often quite small. Noticeable scene clutter and occlusion makes these examples challenging.	1
Deep Residual Text Detection Network for Scene Text Xiangyu Zhu, Yingying Jiang, Shuli Yang, Xiaobing Wang, Wei Li, Pei Fu, Hua Wang and Zhenbo Luo Machine Learning Lab Samsung R&D Institute of China, Beijing Beijing, China {xiangyu.zhu, yy.jiang} @samsung.com Abstract—Scene text detection is a challenging problem in computer vision. In this paper, we propose a novel text detection network based on prevalent object detection frameworks. In order to obtain stronger semantic feature, we adopt ResNet as feature extraction layers and exploit multi-level feature by combining hierarchical convolutional networks. A vertical proposal mechanism is utilized to avoid proposal classification, while regression layer remains working to improve localization accuracy. Our approach evaluated on ICDAR2013 dataset achieves 0.91 F-measure, which outperforms previous state-ofthe-art results in scene text detection. Keywords—Scene text detection; Deep CTPN Residual Networks; I. INTRODUCTION Text detection is an important part of text content analysis, especially for reading natural text in the wild. Scene text detection is becoming increasing attractive to researchers, with the development of smart phone and tremendous demands of text recognition in Augmentation Reality (AR). Unlike traditional documental text, detecting scene text seems to be a much more challenging task due to illuminations, perspective distortion and complex background. In the last few decades, series of methods [1, 2, 3] had been proposed to deal with this problem, which achieved considerable performance. Those methods can be categorized into Sliding Window based methods and Connected Component (CC) based methods. Sliding Window based method utilizes sliding windows to search the image densely for candidate text regions, and classifies text/non-text regions by traditional machine learning tools. This kind of method can be quite slow as a consequence of densely search and multiscale windows. Comparing to the previous method, CC base method draws more attention until recently. It involved several steps, typically three. First, CCs are extracted from images as character candidates. Second, a character classifier is trained to remove the non-text CCs. Finally, remained CCs are going to be grouped into text-lines by clustering or rules. Maximally Stable Extremal Regions (MSER) is one of the most popular CC-based methods. It had been reported outstanding performance in ICDAR2013 benchmark [4]. However, the following limitations constrain its further improvement in performance. Words constituent of single character are ignored by grouping rules for the sake of precision, characters in low color contrast can not be extracted by MSER, and another disadvantage is the complex post-processing. Convolutional Neural Networks (CNN) approach has led to a great breakthrough in object detection. Region proposal CNN (R-CNN) [5] was the first attempt to classify proposals by CNN. Then Faster R-CNN [6] was proposed, where a subnetwork named RPN was designed to generate proposals autonomously by feature maps and a few additional convolution layers. Faster R-CNN used VGG-16 [7] as baseline for feature map extraction and proposal classification until deep residual network (ResNet) [8] was presented. ResNet was reported better performance in PASCAL VOC 2007 [9] and ILSVRC 2016 comparing to VGG16 and GoogLeNet [10, 11]. Moreover, the structure of ResNet was designed fully convolutional, without heavy fully connected layers. The ResNet version of Faster R-CNN was observed better performance. Inspired by the great progress in object detection, a few CNN based methods [12, 13, 14, 15, 16, 17] had been proposed to address scene text detection. The Connectionist Text Proposal Network (CTPN) [12] is a novel framework based on Faster R-CNN, which benefits from an additional recurrent neural network and vertical proposal mechanism. In this paper, we came up with a framework called Residual Text detection Network (RTN). RTN were inspired by ResNet and CTPN vertical proposal mechanism. First, ResNet was used to generate strong semantic feature instead of traditional networks like VGG-16. Rather than a naively layer replacement, we combine multi-level features to produce hierarchy residual feature. The outstanding performance was mainly contributed by this stronger semantic feature. Second, vertical proposal mechanism was adopted and an additional regression part was used to improve localization accuracy, this step was implemented by a two stage training strategy. It achieved 91.54% F-measure on ICDAR2013. II. RELATED WORK A. Object detection With the success of deep convolutional network in image recognition, R-CNN was inspired to classify region proposal via CNN. After R-CNN was proposed, the related object detection approaches had been developed rapidly, such as SPPnet, Fast R-CNN, Faster R-CNN, R-FCN. Faster R-CNN is a mature prevalent framework that trained and tested from end to end. The framework constitutes of three parts: (1) Feature map generation. Feature maps representing semantic information were extracted by deep convolutional network, VGG-16 was used in Faster R-CNN. (2) Proposal generation. A simple ResNet Vertical mechanism RPN Conv 1x1 BLSTM 2k box coordinates regression Hierarchy residual feature PSROI Pooling pool Fig.1. Architecture of Residual Text detection Network (RTN) convolutional network name Region Proposal Networks (RPN) was designed to generate candidate regions with the input of feature maps. (3) Region classification and regression. By sharing features regions proposals were projected to the location in feature maps, then a following Fast R-CNN structure outputted final results by classification and regression. Influenced by the latest progress in image recognition, other deeper convolutional networks were transplant to this framework instead of VGG-16 [7], including GoogLeNet [10, 11] and ResNet [8]. ResNet was proved to be a superior convolutional network than GoogLeNet and VGG16 in ImageNet classification task. R-FCN [18] is a completely fully convolutional architecture that combines ResNet and Faster RCNN together. FPN (Feature Pyramid Network) [19] exploits multi-scale pyramid of ResNet, and the framework using FPN won the champion of COCO [20] detection challenge 2016. Besides Faster R-CNN based pipeline, Single Shot MultiBox Detector (SSD) [21] and You Look Only Once (YOLO) [22] are two representative and promising works. SSD is one of the first attempts to utilizing multi-level convolutional networks, while YOLO is extremely faster than all the methods mentioned above. However, they do not get a superior performance with significant margin comparing to Faster RCNN pipeline. B. CNN based text detection General object detection pipeline can be transplant to text detection realm barrier free. CNN based text detection gradually becomes the most promising approach. Zhang [14] proposed a fully convolutional network for text detection in arbitrary orientation instead of semantic segmentation. It achieved an F-measure of 0.74 on ICDAR2013. DeepText [13] proposed a Inception-RPN and multi-level region-of-interest pooling based on the framework of Faster R-CNN. It achieved 0.85 F-measure on ICDAR2013. Inspired by SSD, Liao [15] presented a approach called TextBoxes, multi-level jointly predictions and word recognition were utilized. CTPN [12] is a unique network abandoned Fast R-CNN classification and regression, which can be treated as a novel individual RPN with Recurrent Neural Network (RNN). It achieved previous state-of-the-art on ICDAR2013 as 0.88 Fmeasure among published papers. Nevertheless, it was just a prototype for detection using RNN and fixed width proposal is harmful for localization accuracy. III. RESIDUAL TEXT DETECTION NETWORK The architecture of this Residual Text detection Network (RTN) is shown in Fig. 1. It consists of three parts: hierarchy residual feature map for feature extraction, vertical mechanism RPN for proposal prediction and bounding box regression part for higherlocalization accuracy. A. Hierarchy residual feature map In our framework, we use ResNet to derive feature map from original images. The feature map is a serial of features in 2D formation, similar to handcraft feature. It is fed to RPN and regression part. ResNet consists of 5 concatenate blocks (i.e., conv1, conv2_x, conv3_x, conv4_x and conv5_x). Conv4_x have the same stride as VGG-16 output (16 pixels). In RFCN[18], region proposals were predicted by conv4_x. They believed the conv4_x feature maps were semantic strong enough and comparable to VGG-16 feature maps. VGG-16 differs from ResNet in structure. Thus, a simple replacement, from VGG-16 to ResNet, would not work properly. Unlike VGG-16, typical ResNet based detection does not share the same feature map between RPN and regression parts. Conv4_x is utilized to generate proposals in RPN while conv5_x for regression. In this kind of methods, RPN is unable to use a deeper semantic feature. By visualizing feature maps of conv3_x, conv4_x, conv5_x and VGG-16, we find out conv3_x contains too many low level features, while conv4_x and conv5_x are competitive to VGG-16 on the first glance. We have carried out series of experiments on Faster R-CNN baseline using conv3_x, conv4_x and conv5_x respectively. Framework using conv3_x detected edges and lines instead of objects and required much more computation due to larger feature map sizes. It was a strong evidence that conv3_x contained too many low level features to be used directly. On the contrary, baselines using conv4_x and conv5_x detected text correctly. However, framework using conv5_x fails on detecting small text due to coarse resolution feature maps. Although conv5_x represents deeper feature, the resolution is half comparing to conv4_x. Even we adopt the “ à trous algorithm” [23] to compensate stride difference, the performance is still unsatisfactory. Using conv5_x as the only feature maps might be insufficient for text detection, but abandon deeper representations seems to be an unwise choice. We believe using conv5_x in a proper way will contribute to proposal prediction. It is rational to come up with a naive idea that predicting multi-scale proposals on conv4_x and conv5_x respectively, like previous approaches did, such as SSD and TextBoxes. In this way, not only we can detect fine scale text and robust to scale invariance, but also utilizing deeper feature representations. Nevertheless, it is inconvenient to identify reliability from multi-scale proposals without an additional classification, as we introduced vertical mechanism to RPN, it seems to be a rather complicated problem. To deal with that, we combine the hierarchy feature maps (conv4_x and conv5_x) together to produce a new hierarchy feature map. In this way, we can use both conv4_x and con5_x feature maps simultaneously, and the task to identify which feature maps are more reliable is assigned to convolution layers. As shown in Fig 2, the input size of original images is 224  224 , after several convolutional layers, conv4_x and conv5_x get feature maps in size 14  14 and 7  7 , corresponding to 16 pixels and 32 pixels stride. A deconvolution layer was used to upsample conv5_x, make sure the shapes of conv5_x (res5c) match conv4_x (res4b22) exactly. We attach a convolution layer with kernel 1  1 , which aim to work as learnable weights for combining conv5_x and conv4_x. Our experiment shows hierarchy feature lead to an improvement on both precision and recall. 7x7 Conv1x1 14 x 14 Conv5 14 x 14 deconv Conv4 Conv1x1 14 x 14 hierarchy feature 224 x 224 Fig.2. Hierarchy residual network architecture. First, conv5_x was upsampled to make sure its shapes match conv4_x. Second ,we attached convlutional layers with kernal to both conv5_x and conv4_x. Finally, hierarchy feature was produced by element wise addition. B. Vertical mechanism RPN In Faster R-CNN, a serial of CNN is used to classify proposals. The structure is called Fast R-CNN [24]. However, CTPN abandoned Fast R-CNN structure, namely RPN output vertical proposals directly without classification and regression. As we know, RPN can be treated as a general object detection system. If the detection task is to distinguish only one category from background (two categories in total), it seems that RPN is already competent for text detection. Depending on vertical proposal mechanism and recurrent neural network, CTPN [12] was able to detect text without Fast-RCNN. That mechanism makes the final model much smaller. In this approach, we adopt this vertical mechanism to RPN. Anchors and ground truth are divided into fixed width (16 pixels) boxes, shown in Fig 3. Particularly, spaces between ground truths are treated as negative samples. This enable the method to output result in word level. Sequences of vertical proposals will be predicted by RPN. A threshold is applied to remove non-text vertical proposals, therefore remained adjacent text proposals can be connected together to produce text line proposals. Fig.3 Yellow box: ground truth of vertical proposals. Green box: space between words which are treated as negative samples C. Bounding box regression By connecting vertical proposals, we will obtain text-line proposals as result. Nevertheless, fixed width proposal might lead to inaccurate localization, when the beginning and the end of vertical proposals are not exactly fit text. In small text case, the problem becomes more serious. Unlike general object detection, this inaccuracy will influence recognition tremendously. If parts of the characters are not included in bounding box, they might be omitted or wrongly recognized. On the contrary, a loose bounding box contains much background, and that could be recognized as additional characters. In conclusion, a tight and exact bounding box is significant for text detection and recognition. To achieve this goal, we introduce bounding box regression to get exact coordinates, just as Faster R-CNN and R-FCN did in their framework. In this paper, we refer to Fast R-CNN structure. As text line proposals are obtained in section B, bounding box offset of every proposal were calculated. However, classification is not contained in this part, only regression is remained. A further classification is unnecessary, and experiments show it is harmful for performance. This is because recurrent neural networks we adopted in RPN have a tendency to connect words into text lines. After we set word level as network learning goal, text line level proposal might be classified as negative result. The bounding box regression loss is defined as: 𝐿𝑙𝑜𝑐 (𝑡 𝑢 , 𝑣) = ∑ 𝑠𝑚𝑜𝑜𝑡ℎ𝐿1 (𝑡𝑖𝑢 − 𝑣𝑖 ) 𝑖∈{𝑥,𝑤} 𝑠𝑚𝑜𝑜𝑡ℎ𝐿1 (𝑥) = { 0.5𝑥 2 \|𝑥\| − 0.5 𝑖𝑓\|𝑥\| < 1 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 In this functions, 𝑣 = (𝑣𝑥 , 𝑣𝑤 , ) is the ground truth of bounding box, 𝑡 𝑢 = (𝑡𝑥 , 𝑡𝑤 , ) is predicted coordinates, 𝑥 and 𝑤 stand for x coordinate and width. 𝐿1 smooth function is used for regression. This loss function is almost the same as what used in Fast R-CNN except that two coordinates (𝑥, 𝑤) offsets are predicted instead of four coordinates (𝑥, 𝑦, ℎ, 𝑤) . It is unnecessary to regression y coordinate (𝑦) and height (ℎ) which was done in RPN layers for every single vertical proposal. We develop a two stage training strategy to implement this further regression:  Stage one. Hierarchy residual feature and vertical mechanism RPN were trained; the learning rates of regression parts were set to 0.  Stage two. Regression parts were trained individually, the learning rates of ResNet, hierarchy residual feature and RPN were set to 0. A normal RPN presented in Faster R-CNN is used to generate anchors and train regression parts and will not be used in test model. D. Training and testing details Our model was trained on 15,000 natural images collected and labeled by ourselves. These images were labeled in word level and resized to (600, 1000) scale. There is no overlap between these images or any kind of public dataset available on the internet. On the condition of the extremely similarity between ICDAR2013 training set and testing set, ICDAR2013 training set was not included to prevent over-fitting ICDAR2013 testing set. Training ground truth is labeled in word level, and then divided into vertical ground truth by a fixed width (16 pixels) in proposal layer, corresponding to vertical proposals mentioned above. The space between words were labeled as negative samples, anchor has an IoU> 0.5 overlap with space samples were signed as negative label. About 10% of negative samples are space. By adding space sample, the networks tend to output word level proposal rather than text-line level. IV. EXPERIMENTS A. Evaluation of hierarchy residual feature In CTPN, 3,000 natural images were collected and label for training, much less than ours. In order to prove the improvement is a consequence of stronger semantic feature map rather than much more training data, we implement our own version of CTPN and training on 15,000 images. All the experiments carried out below were trained on the same amount of images. In this experiment, we used VGG-16 and ResNet-101 as backbone for feature extraction. Feature map generated by different layers were evaluated, including conv5 of VGG-16, conv4_x (res4b22) of ResNet-101 and hierarchy residual feature map (res4b22 + res5c) used in RTN. Table 1 shows the performances on ICDAR2013, we use CTPN framework as baseline and different feature maps mentioned above are evaluated, all the parameters and following processing are the same. We evaluated these methods on two scales respectively, namely (600, 1000) and (960, 1280). Scale (600, 1000) means the shortest side of images is no more than 600 pixels and the longest side can not exceed 1000 pixels, so does to scale (960, 1280). One observation is that all these feature maps are competitive in scale (600, 1000). However, when it comes to scale (960, 1280), margins between these methods becoming considerable. We had run the open source test code 2 provided by the author of CTPN, marked as CTPN-tianzhi. Larger scale did not benefit performance. On the contrary, the F-score degraded. Moreover, our CTPN implementation with VGG-16 improved slightly on F-scores. In conclusion, larger test scale does not always helpful for detection and localization. Nevertheless, by simply replacing VGG-16 to ResNet-101 conv4_x (res4b22), F-score improved to from 88.75% to 90.32%, it proves ResNet-101 has a superior feature representation comparing to VGG-16 as other papers [8, 18] mentioned. Furthermore, baseline with hierarchy residual feature map (res4b22 + res5c) achieved the best performance with F-score=91.17%, which improve 5 points on recall comparing to the original CTPN. The results shows baseline with hierarchy residual feature achieves the best performance on both recall and precision on scale (960, 1280), which could be a convincing evidence for stronger semantic feature. We evaluated RTN on ICDAR 2013 benchmarks. It consists of 233 focused text images taken in the wild. The evaluation criteria are provided by the ICDAR2015 Robust Reading Competition website1 as previous works did. First, the effectiveness of hierarchy residual feature map was verified comparing to other prevalent feature extraction layers. Then, additional regression layers were proved to be helpful for localization accuracy. Finally, this method was compared to other published methods, and it achieved state-ofthe-art performance. TABLE 1. EVALUATING BASELINE WITH DIFFERENT FEATURE MAP ON ICDAR2013 Method CTPN-Tianzhi CTPN-Tianzhi CTPN CTPN CTPN CTPN RTN RTN  1. http://rrc.cvc.uab.es 2. https://github.com/tianzhi0549/CTPN/ Backbone VGG-16 VGG-16 VGG-16 VGG-16 ResNet-101 ResNet-101 ResNet-101 ResNet-101 Scale (600,1000) (960,1280) (600,1000) (960,1280) (600,1000) (960,1280) (600,1000) (960,1280) Feature map conv5 conv5 conv5 conv5 res4b22 res4b22 res4b22+res5c res4b22+res5c Precision 92.98% 91.02% 92.56% 91.52% 93.38% 93.62% 93.65% 93.64% Recall 82.98% 82.98% 83.96% 86.14% 82.76% 88.09% 83.14% 88.82% F score 87.69% 86.81% 88.06% 88.75% 87.75% 90.32% 88.08% 91.17% Fig.4. Example detection results of our RTN on the ICDAR2013 benchmark. The first row of images is the result before connection and regression. The second row of images is the result after vertical proposal connection and regression. TABLE 2 REGRESSION IMPROVEMENT BY ADDITIONAL Convolutional layers RTN_no_regression RTN_regression REGRESSION Precision Recall 93.64% 88.82% 94.20% 89.02% TABLE 3 COMPARISON WITH STATE-OF-THE-ART PUBLICATIONS ON ICDAR2013 Method Yin [1] Faster R-CNN baseline[6] R-FCN[18] Multi-OrientedFCN[14] SegLink[17] DeepText[13] TextBoxes[15] CCTN[16] CTPN[12] Proposed RTN F score 91.17% 91.54% B. Regression improvement Proposals connected by fixed width vertical proposals are inaccurate on both beginning and end sides. Moreover, the evaluation criteria are extremely strict. Detection bounding box can be judged as false positive sample if its boundary exceed ground truth slightly. It means this inaccuracy can degrade performance on both recall and precession, even if the texts are detected correctly. Through bounding box regression, we are able to deal with this problem properly. As shown in Table 2, RTN with regression improved 0.4% on F-scores, both recall and precision benefit from this additional regression. C. Evaluation of RTN on ICDAR2013 After proving the effectiveness of hierarchy residual feature and additional regression, we compare RTN with other published methods on ICDAR2013. This single model approach did not utilize multi-scale training and multi-scale testing. Running time of each image is about 0.8s with GPU. Fig4 shows examples of detection results on ICDAR2013. First, we compared RTN with methods mentioned in recent publications. CNN based text detection methods were compared, including Textboxes, DeepText, Multi-oriented FCN, CCTN, SegLink and CTPN. The prevalent object detection frameworks like Faster R-CNN and R-FCN are also evaluated. Table 3 shows RTN achieved the best performance with great margin. Second, we submitted our results to ICDAR2015 Robust Reading Competition website and compared RTN with other competitors on CHALLENGE 2.1. This task is also evaluated on ICDAR2013 dataset. RTN with single model ranked third performance with slightly margin (F-scores=0.3%) compared to “Tencent Youtu” and “NLPR-CASIA”. Precision 0.88 Recall 0.66 F score 0.76 0.86 0.75 0.80 0.90 0.76 0.83 0.88 0.78 0.83 0.87 0.87 0.89 0.90 0.93 0.94 0.83 0.83 0.83 0.83 0.83 0.89 0.85 0.85 0.86 0.86 0.88 0.91 V. CONCLUSIONS In this paper, a deep residual text detection network is proposed based on the prevalent object detection framework. First, stronger semantic feature is obtained by using deep residual networks and combining multi-level feature from different convolutional networks. Then, a vertical proposal mechanism is introduced inRPN inspired by CTPN. At last, an additional regression system is used to improve localization accuracy. 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Fiducial, confidence and objective Bayesian posterior distributions for a multidimensional parameter arXiv:1612.01882v1 [math.ST] 6 Dec 2016 Piero Veronese and Eugenio Melilli Bocconi University, Milano, Italy Abstract We propose a way to construct fiducial distributions for a multidimensional parameter using a step-by-step conditional procedure related to the inferential importance of the components of the parameter. For discrete models, in which the nonuniqueness of the fiducial distribution is well known, we propose to use the geometric mean of the “extreme cases” and show its good behavior with respect to the more traditional arithmetic mean. Connections with the generalized fiducial inference approach developed by Hannig and with confidence distributions are also analyzed. The suggested procedure strongly simplifies when the statistical model belongs to a subclass of the natural exponential family, called conditionally reducible, which includes the multinomial and the negative-multinomial models. Furthermore, because fiducial inference and objective Bayesian analysis are both attempts to derive distributions for an unknown parameter without any prior information, it is natural to discuss their relationships. In particular, the reference posteriors, which also depend on the importance ordering of the parameters are the natural terms of comparison. We show that fiducial and reference posterior distributions coincide in the location-scale models, and we characterize the conditionally reducible natural exponential families for which this happens. The discussion of some classical examples closes the paper. Keywords: Confidence distribution, Jeffreys prior, Location-scale parameter model, Multinomial model, Natural exponential family, Reference prior. 1 Introduction Fiducial distributions, after having been introduced by Fisher (1930, 1935) and widely discussed (and criticized) in the subsequent years, have been de facto brushed aside for a long time and only recently they have obtained new vitality. The original idea of Fisher was to construct a distribution for a parameter which includes all the information given by the data, without resorting to the Bayes theorem. This is obtained by transferring the randomness from the observed quantity given by the statistical model to the parameter. Originally Fisher considered a continuous sufficient statistic S with distribution function Fθ , depending on a real parameter θ. Let qα (θ) denote the quantile of order α of Fθ and 1 let s be a realization of S. If qα (θ) is increasing in θ (i.e., Fθ is decreasing in θ), the statement s < qα (θ) is equivalent to θ > qα−1 (s) and thus Fisher assumes qα−1 (s) as the quantile of order 1 − α of a distribution which he names fiducial. The set of all quantiles qα−1 (s), α ∈ (0, 1), establishes the fiducial distribution function Hs (θ) so that Hs (θ) = 1 − Fθ (s) and hs (θ) = ∂ ∂ Hs (θ) = − Fθ (s). ∂θ ∂θ (1) Of course Hs , and its density hs , must be properly modified if Fθ is increasing in θ. Fisher (1973, cap.VI) also provides some examples of multivariate fiducial distributions obtained by a “step-by-step” procedure, but he never develops a general and rigorous theory. This fact, along with the problem to cover discrete models, the presence of some inconsistencies of the fiducial distribution (e.g. the marginalization paradox, see Dawid & Stone, 1982), and the difficulties in its interpretation, gave rise to a quite strong negative attitude towards Fisher proposal. In the renewed interest for the fiducial approach a relevant role is played by the generalized fiducial inference introduced and developed by Hannig (2009, 2013), see also Hannig et al. (2016) for a review. He provides a formal and mathematically rigorous definition which has a quite general applicability. The crucial element of his definition is a data-generating equation X = G(U, θ), which links the unknown parameter θ and the observed data X through a random element U having a known distribution. Roughly speaking, by shifting the randomness of U from X to θ (inverting G with respect to θ after having fixed X = x), the distribution given by the statistical model leads to a distribution for the parameter θ. Contrary to the original idea of Fisher, the generalized fiducial distribution is non-unique and Hannig widely discusses this point. Applications to different statistical models can be found for instance in Hannig et al. (2007), Hannig & Iyer (2008) and Wandler & Hannig (2012). Other recent contributions to the topic of fiducial distributions are given by Taraldsen & Lindqvist (2013), Martin & Liu (2013) and Veronese & Melilli (2015), henceforth V&M (2015). In this last paper the authors derive fiducial distributions for a parameter in a discrete or continuous real natural exponential family (NEF), and discuss some of their properties with particular emphasis on the frequentist coverage of the fiducial intervals. In the past fiducial distributions have often been associated with confidence distributions even if these latter have a different meaning. A modern definition of confidence distribution is given in Schweder & Hjort (2002) and Singh et al. (2005), see the book by Schweder & Hjort (2016) for a complete and updated review on confidence distributions and their connections with fiducial inference. It is important to emphasize that a confidence distribution must be regarded as a function of the data with reasonable properties from a purely frequentist point of view. A confidence distribution is conceptually similar 2 to a point estimator: as there exist several unbiased estimators, several confidence distributions can be provided for the same parameter and choosing among them can be done resorting to further optimality criteria. Thus the confidence distribution theory allows to compare, in a quite general setting, formal distributions for the parameter derived by different statistical procedures. In this paper we suggest a way to construct a unique distribution for a multidimensional parameter, indexing discrete and continuous models, following a step-by-step procedure similar to that used by Fisher (1973) in some examples. We call it fiducial distribution, but we look at it simply as a distribution on the parameter space in the spirit of the confidence distribution theory. The key-point of the construction is the procedure by conditioning: the distribution of the data is factorized as a product of one-dimensional laws and, for each of these, the fiducial density for a real parameter component, possibly conditional on other components, is obtained. The joint fiducial density for the parameter is then defined as the product of the (conditional) one-dimensional fiducial densities. It is well known that Fisher’s fiducial argument presents several drawbacks in higher dimensions, essentially because one cannot recover the fiducial distribution for a function of the parameters starting from the joint fiducial distribution, see Schweder & Hjort (2016, Ch. 6 and 9). Our approach, when it can be applied, presents the advantage to construct sequentially the fiducial distribution directly on the parameters of interest and different fiducial distributions can be obtained focusing on different parameters of interest. Also, it should be noticed that a general definition of confidence distribution for a multidimensional parameter does not exist and more attention is given to the construction of approximate confidence curves for specific nested families of regions, see Schweder & Hjort (2016, Ch. 9 and Sec. 15.4). Interestingly, our joint fiducial distribution coincides in many cases with the Bayesian posterior obtained using the reference prior. This fact motivates the second goal of the paper: to investigate the relationships between the objective Bayesian posteriors and the suggested fiducial distributions. Objective Bayesian analysis, see e.g. Berger (2006), essentially studies how to perform a good Bayesian inference, especially for moderate sample size, when one is unwilling or unable to assess a subjective prior. Under this approach, the prior distribution is derived directly from the model and thus it is labeled as objective. The reference prior, introduced by Bernardo (1979) and developed by Berger & Bernardo (1992), is the most successful default prior proposed in the literature. For a multidimensional parameter the reference prior depends on the grouping and ordering of its components and, in general, no longer coincides with the Jeffreys prior. This is the reference prior only for a real parameter and it is unsatisfactory otherwise, as well known. 3 Lindley (1958) was the first to discuss the connections between fiducial and posterior distributions for a real parameter, when a real continuous sufficient statistic exists. V&M (2015) extend this result to real discrete NEFs, characterizing all families admitting a fiducial prior, i.e. a prior leading to a posterior coinciding with the fiducial distribution. This prior is strictly related to the Jeffreys prior. We show here that when the parameter is multidimensional this relationship no longer holds and a new one is established with the reference prior. In particular we prove results for location-scale parameter models and conditionally reducible NEFs, a subclass of NEFs defined in Consonni & Veronese (2001). The paper is structured as follows. Section 2 reviews some basic facts on fiducial and confidence distributions for real NEFs and on generalized fiducial distributions. The proposal for constructing a step-by-step multivariate fiducial distribution is presented in Section 3, which also discusses: the relationships with confidence distributions (Section 3.1), the use of the geometric mean of fiducial densities for solving the non-uniqueness problem in discrete models (Section 3.2), the connections with the generalized fiducial inference and the consistency with the sufficiency principle (Section 3.3). Section 3.4 studies the fiducial distributions for conditionally reducible NEFs and provides their explicit expression for a particular subclass which includes the multinomial and the negativemultinomial model. Section 4 analyzes the relationships between the fiducial distributions and the reference posteriors, in particular for location-scale parameter models (Section 4.1) and NEFs (Section 4.2), characterizing those which admit the fiducial prior. Section 5 discusses further examples in which fiducial and reference posteriors coincide. Section 6 concludes the paper presenting some possible asymptotic extensions. Finally, Appendix A1 collects some useful technical results on conditionally reducible NEFs, while Appendix A2 includes the proofs of all the results stated in the paper. 2 Preliminary results The modern definition of confidence distribution for a real parameter φ of interest, see Schweder & Hjort (2002, 2016) and Singh et al. (2005), can be formulated as follows: Definition 1. Let {Fφ,λ , φ ∈ Φ ⊆ R, λ ∈ Λ} be a parametric model for data X ∈ X ; here φ is the parameter of interest and λ is a nuisance parameter. A function C : X × Φ → R is a confidence distribution for φ if C(x, ·) is a distribution function for each x ∈ X and C(X, φ0 ) has a uniform distribution in (0, 1) under Fφ0 ,λ0 , where (φ0 , λ0 ) is the true parameter value. The relevant requirement in the previous definition is the uniformity of the distribution, which ensures the correct coverage of the confidence intervals. As seen in Section 4 1, the confidence distribution theory must be placed in a purely frequentist context and allows to compare distributions on the parameter space, obtained using different approaches. Finally, the definition of confidence distribution can be generalized by requiring that the uniformity assumption holds only asymptotically. Strictly linked to the notion of confidence distribution is that of confidence curve, defined, for each observed X = x, as the function φ → cc(φ) = \|1 − 2C(x, φ)\|; see Schweder & Hjort (2016). This function gives the extremes of equal-tail confidence inter- vals for any level 1 − α, allowing a fast and clear comparison of confidence distributions with respect to their interval length. When the parameter of interest is multidimensional, how to extend the definitions of confidence distribution and confidence curve is much less clear and various proposals have been made, see Schweder & Hjort (2002, 2016) and Singh et al. (2005). As detailed in Section 1, Hannig (2009) has proposed the notion of generalized fiducial distribution, which is based on a data-generating equation X = G(θ, U). Because several functions G can generate the same statistical model, and not all the resulting fiducial distributions are reasonable in terms of properties or computational tractability, Hannig (2013, Sec. 5) gives some hints on the choice of a default function G. In particular, if X = (X1 , . . . , Xn ) is an independent identically distributed (i.i.d.) random sample from an (absolutely) continuous distribution function Fθ , with density fθ , θ ∈ Rd , he suggests to use Xi = Fθ−1 (Ui ), i = 1, . . . , n, where Ui are i.i.d. uniform random variables on (0, 1) and Fθ−1 is the inverse (or generalized inverse) of Fθ . If other regularity assumptions are satisfied, the generalized fiducial distribution for θ can be written as r(θ) = R fθ (x)J(x, θ) , Θ fθ (x)J(x, θ)dθ (2) where the expression of J(x, θ), given in Hannig (2013, formula (3.7)), is J(x, θ) = X det {(i1 ,...,id ):1≤i1 <···<id ≤n} d dθ (Fθ (xi1 ), . . . , Fθ (xid )) Qd j=1 fθ (xij ) . (3) In (3) the numerator of the ratio is the determinant of the matrix whose kj-entry is ∂Fθ (xij )/∂θk . This procedure leads to the Fisher definition of fiducial density (1) when n = d = 1. Hannig (2013, Example 4) explicitly recognizes the advise of Wilkinson (1977) that the choice of a fiducial distribution should depend on the parameter of interest and uses the well known example of d independent normal distributions N(µi , 1), in which the P parameter of interest is θ = ( di=1 µ2i )1/2 . He shows that the default data-generating equations Xi = µi + Ui , i = 1, . . . , d, lead to a fiducial distribution which has good frequentist properties for inference on the µ’s, but very bad ones when the interest is on 5 θ, as already recognized by Stein (1959). Thus Hannig suggests an ad hoc alternative equation, which leads to a better solution. Notice that our general procedure, suggested in the next section, constructs a fiducial distribution starting directly from the parameter of interest and do not required the choice a priori of a data generating function. Fiducial distributions and their properties, with particular emphasis on the frequentist coverage of the fiducial intervals, for a discrete or a continuous real regular NEF, are discussed in V&M (2015). More specifically, consider the sufficient statistic S associated with a sample of size n and denote by S its support. Let Fθ (s) be the distribution function of S and pθ (s) = exp {θs − nM (θ)} the corresponding density (with respect to a measure ν). Let a = inf S, b = sup S and define S ∗ = [a, b) if ν(a) > 0, otherwise S ∗ = (a, b). Then, for s ∈ S ∗ , Petrone & Veronese (2010) have proved that   θ ≤ inf Θ   0 Hs (θ) = 1 − Fθ (s) inf Θ < θ < sup Θ    1 θ ≥ sup Θ (4) is a fiducial distribution function for the natural parameter θ. It follows that the fiducial density of θ is ∂ ∂ hs (θ) = Hs (θ) = − Fθ (s) = ∂θ ∂θ Z (−∞,s] (nM ′ (θ) − t)pθ (t)dν(t). (5) It is important to underline, and simple to verify, that the distribution function Hs is also a confidence distribution (only asymptotically, in the discrete case), according to Definition 1. Notice that, for discrete NEFs, Fθ (s) = Prθ {S ≤ s} and Prθ {S < s} do not coincide and thus, besides Hs in (4), one could define a left fiducial distribution as Hsℓ (θ) = 1 − Prθ {S < s}. (6) For convenience, sometimes Hs will be called right fiducial distribution. A standard way to overcome this non-uniqueness is referring to the half-correction device (see Schweder & Hjort, 2016, pag. 62) which amounts to consider the mixture HsA (θ) = (Hs (θ) + Hsℓ (θ))/2 = Prθ {S > s} + Prθ {S = s}/2, whose density is the arithmetic mean of hs (θ) and hℓs (θ). Instead, we will suggest to average hs and hℓs using their geometric mean hG s (suitably normalized) and show that it presents better properties than hA s (Section 3.2) and a more direct connection with objective Bayesian inference (Section 4.2), even if, operationally, the difference is usually not particularly big. Table 1 provides the fiducial distributions obtained in V&M (2015) for some important discrete and continuous NEFs, which will be used in the forthcoming examples. It also establishes the abbreviations used in the paper for the standard distributions. 6 Table 1: Fiducial distributions for some real NEFs N(µ, σ 2 ) Sufficient Fiducial statistic P Hs (µ) : N(s/n, σ 2 /n) S= distributions i Xi (σ 2 known) N(µ, σ 2 ) S= P S= P i Xi Hs (λ) : Ga(nα, s) S= P i log(Xi /x0 ) Hs (λ) : Ga(n, s) S= P i Xic Hs (λ) : Ga(n, s) S= P i Xi Hs (p) : Be(s + 1, nm − s) i (Xi − µ)2 Hs (σ 2 ): In-Ga(n/2, s/2) (µ known) Ga(α, λ) (α known) Pa(λ, x0 ) (x0 known) We(λ, c) (c known) Bi(m, p) Hsℓ (p) : Be(s, nm − s + 1) (m known) HsG (p) : Be(s + 1/2, nm − s + 1/2) Po(µ) S= P i Xi Hs (µ) : Ga(s + 1, n) Hsℓ (µ) : Ga(s, n) HsG (µ) : Ga(s + 1/2, n) Ne-Bi(m, p) S= P i Xi Hs (p) : Be(nm, s + 1) Hsℓ (p) : Be(nm, s) (m known) HsG (p) : Be(nm, s + 1/2) The following notations are used: Ga(α, λ) for a gamma distribution with shape α and mean α/λ; In-Ga(α, λ) for an inverse-gamma distribution (if X ∼ Ga(α, λ) then 1/X ∼ In-Ga(α, λ)); Be(α, β) for a beta distribution with parameters α and β; Bi(m, p) for a binomial distribution with m trials and success probability p; Ne-Bi(m, p) for a negative-binomial with m successes and success probability p; Po(µ) for the Poisson distribuition with mean µ; Pa(λ, x0 ) for a Pareto distribution with density λxλ0 x−λ−1 , x > x0 > 0, λ > 0; We(λ, c) for a Weibull distribution with density cλxc−1 exp(−λxc), x, λ, c > 0. 7 3 Fiducial distributions for multidimensional parameters A natural way to construct a suitable fiducial distribution for a multidimensional parameter is to follow the step-by-step procedure used by Fisher (1973) in some examples. The key-point of our proposal stems on the factorization of the sampling distribution as a product of one-dimensional conditional laws. For each of these the fiducial density for a real component of the parameter, possibly conditional on other components, is defined. It is well known that different factorizations of sampling distributions can produce different joint fiducial distributions, see e.g. Dempster (1963). However, we do not consider this aspect a drawback of the procedure if it is linked to the inferential importance ordering of the parameter components implied by the factorization. For example, if a parameter θ ∈ R2 is transformed in such a way that φ is the parameter of interest and λ the nuisance, the obvious ordering is (φ, λ) and a suitable factorization must be defined accordingly, see Example 4 (ctd.) in this section for an illustration. The crucial role played by the ordering of the parameters accordingly to their inferential importance is widely acknowledged by objective Bayesian inference, in which reference priors are different for different orderings, see Section 4. In order to construct a fiducial distribution, we consider two basic transformations: one involving the sample data X = (X1 , . . . , Xn ), having a distribution parameterized by θ = (θ1 , . . . , θd ), d ≤ n, and one involving θ. Given X, consider a statistic T = (T1 , . . . , Tm ), d ≤ m ≤ n, with density pθ (t), which summarizes X without losing information on θ. T can be a sufficient statistic or a one-to-one transformation of X. Split T in (T[d] , T−[d] ), where T[d] = (T1 , . . . , Td ) and T−[d] = (Td+1 , . . . , Tm ), and suppose that T−[d] is ancillary for θ. As a consequence pθ (t) = pθ (t[d] \|t−[d] )p(t−[d] ) and all the information on θ provided by X are included in the conditional distribution of T[d] given T−[d] . Assume now that there exists a one-to-one smooth reparameterization from θ to φ, with φ1 , . . . , φd ordered with respect to their importance, such that pφ (t[d] \|t−[d] ) = d Y k=1 pφd−k+1 (tk \|t[k−1] , t−[d] ; φ[d−k] ). (7) The density pφd−k+1 (tk \|t[k−1] , t−[d] ; φ[d−k] ), with the corresponding distribution function Fφd−k+1 (tk \|t[k−1] , t−[d] ; φ[d−k] ), must be interpreted as the conditional distribution of Tk given (T[k−1] = t[k−1], T−[d] = t−[d] ), parameterized by φd−k+1 , assuming φ[d−k] known. In the following, we will always assume that all the one-dimensional conditional distribution functions Fφj ’s involved in the analysis are monotone and differentiable in φj and have limits 0 and 1 when φj tends to the boundaries of its domain. Notice that this is always true if Fφj belongs to a NEF, see (4). Under these assumptions, the joint fiducial 8 density of φ is obtained as ht (φ) = d Y k=1 ht[k] ,t−[d] (φd−k+1 \|φ[d−k] ), (8) ∂ Fφ (tk \|t[k−1] , t−[d] ; φ[d−k] ) . ∂φd−k+1 d−k+1 (9) and ht[k] ,t−[d] (φd−k+1 \|φ[d−k] ) = Several applications of this procedure to well known models will be provided in Section 5. Here we illustrate some interesting features of the fiducial distribution (8). i) The existence of an ancillary statistic is not necessary if there exists a sufficient statistic with the same dimension of the parameter (m = d). An important case is m = d = 1 so that formula (8) and (9) reduce to ht (φ) = \|∂Fφ (t)/∂φ\|, the original formula suggested by Fisher (1930). ii) If one is only interested in φ1 , it follows from (7) that it is enough to consider ht (φ1 ) = ∂ Fφ (td \|t[d−1] , t−[d] ) , ∂φ1 1 which, depending on all observations, does not lose any sample information. A typical choice for Td is given by the maximum likelihood estimator φb1 of φ1 and thus, when φb1 is not sufficient, we have to consider the distribution of φb1 given the ancillary statistic t−[d] . Similarly, if one is interested in φ1 , φ2 , it is enough to consider ht (φ1 ) · ht[d−1] ,t−[d] (φ2 \|φ1 ), and so on. iii) When an ancillary statistic T−[d] is needed, the fiducial distribution (8) is invariant with respect to any one-to-one transformation of T−[d] . All the sampling distributions are conditional on it and thus any transformation establishes the same constraints; see Section 4.1 for an example. iv) The construction by successive conditioning makes the fiducial distribution invariant under the so called one-to-one lower triangular transformation of T[d] , for fixed T−[d] . More precisely, we consider a transformation T∗ = (T∗[d] , T−[d] ) such that Tk∗ = gk (T[k] , T−[d] ), for k = 1, . . . , d. To see this, assuming for instance t∗k = gk (t[k] , t−[d] ) increasing in tk , it is sufficient to show that Prφd−k+1 (Tk∗ ≤ t∗k \| T∗[k−1] = t∗[k−1] , T−[d] = t−[d] ; φ[d−k] ) = Prφd−k+1 (gk (T[k] , T−[d] ) ≤ gk (t[k] , t−[d] ) \| T∗[k−1] = t∗[k−1] , T−[d] = t−[d] ; φ[d−k] ) = Prφd−k+1 (Tk ≤ tk \| T[k−1] = t[k−1] , T−[d] = t−[d] ; φ[d−k] ). 9 It follows immediately that T and T∗ lead to the same fiducial distribution. v) If (T[k−1] , T−[d] ) is sufficient for φ[d−k] , for each k, then the conditional distribution of Tk given (T[k−1] = t[k−1] , T−[d] = t−[d] ) does not depend on φ[d−k] and the fiducial distribution (8) becomes the product of the “marginal” fiducial distributions of the φk ’s. As a consequence, (9) can be used alone to make inference on φd−k+1 and the fiducial distribution does not depend on the inferential ordering of the parameters. An important case in which this happens will be discussed in Section 3.4. We close this section establishing the invariance property of the fiducial distribution ht (φ) under a lower triangular transformation, i.e. a transformation from φ to λ = (λ1 , . . . , λd ), say, which maintains the same decreasing ordering of importance in the components of the two vectors. Proposition 1. If φ = φ(λ) is a one-to-one lower triangular continuously differentiable function from Λ to Φ, then the fiducial distribution hφ t (φ), obtained applying (8) to the model pφ (t), and the fiducial distribution hλ t (λ), obtained applying (8) to the model pλ (t) = pφ(λ) (t), are such that, for each measurable A ⊂ Φ, Z Z φ hλ ht (φ)dφ = t (λ)dλ. A 3.1 (10) λ−1 (A) Relationships with confidence distributions Given a real NEF, Hs (θ) in (4) is an exact or approximate confidence distribution if the observations are continuous or discrete, respectively. It is possible to verify that the same is true for the marginal fiducial distribution of the main parameter of interest φ1 in the more general definition (8). Indeed, the distribution function of φ1 is Ht (φ1 ) = 1 − Fφ1 (td \|t[d−1] , t−[d] ), so that the first requirement in Definition 1 is clearly satisfied, thanks to the assumption on the distribution function given after formula (7). For what concerns the uniformity condition, assuming that Fφ1 is decreasing in φ1 (if it is increasing, replace 1 − Fφ1 with Fφ1 ), we have, for u ∈ (0, 1) and arbitrary φ, Prφ Ht[d] ,t−[d] (φ1 ) ≤ u = 1 − Prφ Fφ1 (td \|T[d−1] , T−[d] ) < 1 − u = =1− Z Prφ Fφ1 (td \|t[d−1] , t−[d] ) < 1 − u dFφ (t[d−1] , t−[d] ) = u because, by construction, the integrand is equal to 1 − u for all fixed (t[d−1] , t−[d] ). 3.2 The discrete case: the geometric mean of the left and right fiducial densities As mentioned in Section 2, for a discrete statistic S with distribution depending on a real parameter θ, we suggest to use the geometric mean of the right and left fiducial 10 −1 ℓ 1/2 , where c is the normalizing constant, instead of densities, hG s (θ) = c (hs (θ)hs (θ)) their arithmetic mean hA s (θ). A first justification of the use of the geometric mean of densities is suggested by Berger et al. (2015) who mention its property to be the density “closest” to hs and hℓs with respect to the the Kullback-Leibler divergence, as specified in the following proposition. We give a simple proof of this fact, without resorting to the calculus of variations. Recall that, given two densities p and q, having the same support and the same dominating measure ν, the Kullback-Leibler divergence of p from q is defined as R KL(q\|p) = q(x) log(q(x)/p(x))dν(x). Proposition 2. Consider two densities p1 and p2 with the same support. The density q which minimizes KL(q\|p1 ) + KL(q\|p2 ) is given by q = pG ∝ (p1 p2 )1/2 , which is the (normalized) geometric mean of p1 and p2 . Furthermore, Krishnamoorthy & Lee (2010) observe that a distribution for θ, whose aim is to give a synthesis of two fiducial distributions, should “stochastically” lie between them. In our setting, the extreme distributions are Hs and Hsℓ . This property is surely satisfied by the arithmetic mean, because Hs (θ) < HsA (θ) < Hsℓ (θ) uniformly with respect to θ, for each s belonging to the set S0 for which both Hs and Hsℓ can be defined. The same inequalities are true for HsG under mild assumptions. As usual, here we assume that Hs (θ) is defined as 1 − Fθ (s). Proposition 3. Let pθ , θ ∈ Θ ⊆ R, be the probability mass function of a real observation S, having a continuous derivative with respect to θ. For each s ∈ S0 , assume that the function γs (θ) = ∂pθ (s) ∂θ ∂pθ (s) ∂Fθ (s) / − = /hs (θ) ∂θ ∂θ is decreasing on Θ. Then Hs (θ) < HsG (θ) < Hsℓ (θ) uniformly on Θ. The assumptions required in the previous proposition are satisfied by many important models. For example we have the following Corollary 1. If pθ is the probability mass function of a real NEF, then Hs (θ) < HsG (θ) < Hsℓ (θ) uniformly on Θ. We now discuss the relationship between HsG and HsA . Proposition 4. Let pθ , θ ∈ Θ ⊆ R, be the probability mass function of a real observation S, satisfying the following assumptions in addition to those stated in Proposition 3: lim γs (θ) = +∞; θ→inf Θ lim θ→sup Θ 11 γs (θ) = −1. Then, for each s ∈ S0 , there exists θ ∗ ∈ Θ (depending on s) such that HsG (θ) < HsA (θ) for θ < θ ∗ and HsG (θ) > HsA (θ) for θ ≥ θ ∗ . The result in Proposition 4 is important in connection with confidence intervals, because it shows that HsG gives, for a fixed level, a confidence interval smaller than that obtained from HsA (θ); see Figure 1 (graph 2) for an example. Notice that the assumptions on γs (θ) in Proposition 4 are fulfilled by a real NEF with natural parameter space Θ = R, as it occurs in the binomial and Poisson models. However, these assumptions are not necessary to ensure the stated behavior of HsG and HsA , that we conjecture to be quite general, as the following example shows. Example 1. Consider an i.i.d. sample of size n from a logarithmic distribution with parameter θ ∈ (0, 1) with probability mass function pθ (x) = The sufficient statistic T = Pn i=1 Xi pθ (t) = θx I (x). −x log(1 − θ) {1,2,...} is distributed as n!\|s(t, n)\|θ t I (t), t!(− log(1 − θ))n {n,n+1,...} where s(t, n) is the Stirling number of the first kind with arguments t and n, see Johnson et al. (2005). The distribution of T belongs to a real NEF with Fθ (t) decreasing in θ, so that the fiducial distribution function Ht , for t = n, n + 1, . . . and θ ∈ (0, 1), is Ht (θ) = 1 − Fθ (t) = 1 − t X j=n n!\|s(j, n)\|θ j . j!(− log(1 − θ))n For this model ∂pθ (s) ∂Fθ (s) \|s(t, n)\|θ t−1 (nθ + t(1 − θ) log(1 − θ)) γt (θ) = − / = − Pt . ∂θ ∂θ t! j=n \|s(j, n)\|θ j−1 (nθ + j(1 − θ) log(1 − θ))/j! It can be seen that, for each t ≥ n, γt is decreasing in θ and lim γt (θ) = +∞, θ→0+ −1  t X \|s(j, n)\| t!  lim γt (θ) = −  ∈ (−1, 0). − \|s(t, n)\| j! θ→1 j=n Nevertheless, the fiducial distributions HtG and HtA behave as stated in Proposition 4, see Figure 1 (graph 1). Finally, we justify our preference for HsG versus HsA showing that its confidence risk under quadratic penalty, as defined in Schweder & Hjort (2016, Sec. 5.3), is uniformly better for all the important discrete models reported in Table 1. The confidence risk 12 Figure 1: Graph 1: Fiducial distributions for a sample from the logarithmic distribution G (n = 10, t = 12): hℓt (θ) (red), hA t (θ) (green), ht (θ) (yellow), ht (θ) (blue). Graph 2: G Confidence curves for hA t (θ) (green) and ht (θ) (yellow)). R(µ, Hs ) for the mean parameter µ and a confidence (or fiducial) distribution Hs under quadratic penalty is R(µ, Hs ) = Z (µ′ − µ)2 dHs (µ′ ) = Eµ (VarHs (µ)) + Eµ (µ̂ − µ)2 , where VarHs (µ) denotes the variance of µ under Hs , Eµ the expected value with respect to the distribution of S and µ̂ = E Hs (µ) is the mean of µ under Hs . Now, recalling that for the binomial and the negative-binomial distribution in Table 1, assuming m = 1 for simplicity, we have µ = p and µ = (1 − p)/p, it is easy to verify that for both these models and the Poisson model µ̂ is the same under HsG and HsA . As a consequence, A G R(µ, HsA )−R(µ, HsG ) = Eµ (VarHs (µ))−Eµ (VarHs (µ)) which becomes (4(n+1)(n+2))−1 , (4(n − 1)(n − 2)−1 and (4n2 )−1 , for the three models above, respectively. All these values are strictly positive for each n uniformly in µ. Let us now consider the fiducial distribution for a multivariate parameter defined in (8). For each discrete component of the product, starting from Prφd−k+1 {Tk ≤ tk \|T[k−1] = t[k−1] , T−[d] = t−[d] ; φ[d−k] } = Fφd−k+1 (tk \|t[k−1] , t−[d] ; φ[d−k] ) and Prφd−k+1 {Tk < tk \|T[k−1] = t[k−1] , T−[d] = t−[d] ; φ[d−k] }, it is possible to define a right and a left fiducial distribution, respectively, and hence their geometric and arithmetic means. Notice that each compo- nent of (8) involves a one-dimensional parameter and a real observation (the remaining quantities being fixed), so that the Propositions 3 and 4 can be applied. Multivariate fiducial distributions for discrete observations can thus be obtained combining in the various possible way these univariate distributions. In particular, we will consider Ht (φ), obtained as the product of all the right univariate conditional fiducial distributions, Htℓ (φ), obtained as the product of all the left univariate conditional fiducial distributions, HtA (φ), defined as the product of the d mixtures HtA[k] ,t−[d] = (Ht[k] ,t−[d] + Htℓ[k] ,t−[d] )/2 and finally 13 HtG (φ), corresponding to the density hG t (φ) obtained as the product of the d geometric 1/2 . Notice that hG (φ) coincides with the geometric ℓ means hG t t[k] ,t−[d] ∝ (ht[k] ,t−[d] ·ht[k] ,t−[d] ) mean of all the 2d fiducial densities derived as described above. 3.3 Fiducial inference and the sufficiency principle The step-by-step procedure introduced at the beginning of Section 3 gives a generalized fiducial distribution, according to Hannig (2009), if one considers as data-generating equation T = G(φ, U) with ( Gk (φ, U[k] , U−[d] ) k = 1, . . . , d , Tk = Uk k = d + 1, . . . , m where U is a random vector with a completely known distribution. The functions Gk can be explicitly obtained iteratively as follows: T1 = G1 φ, U1 , U−[d] = Fφ−1 U \|, U ; φ 1 −[d] [d−1] d T2 = G2 φ, U1 , U2 , U−[d] = Fφ−1 U \|G (φ, U , U ), U ; φ 2 1 1 −[d] −[d] [d−2] d−1 and so on. It is interesting to observe that the generalized fiducial distribution r(θ) given in (2) does not necessarily satisfy the sufficiency principle. This can be verified immediately looking at the Example 2 in Hannig (2013), in which a uniform distribution on (θ, θ 2 ) is considered and r(θ) does not depend on the Xi ’s only through the sufficient statistic S = (X(1) , X(n) ), where X(i) denotes the i-th order statistic. Despite its simple form, this model is highly irregular, but the inconsistency with the sufficiency principle of the generalized fiducial distribution r(θ) can also occur for more standard models. In particular, if a real continuous sufficient statistic S for a real parameter exists, one could derive two different fiducial distributions starting from S or from the whole sample. A simple example of this issue can be easily constructed considering a beta model with parameters 2 and θ. Another interesting example is the following. Example 2. Let X = (X1 , . . . , Xn ) be an i.i.d. sample from a truncated exponential density pθ (x) = θe−θx /(1 − e−θ ), 0 < x < 1, θ ∈ R − {0}. This density is not defined for θ = 0, but it can be completed by continuity setting p0 (x) = 1. The distribution function of Xi is Fθ (xi ) = (1 − e−θxi )/(1 − e−θ ), 0 < xi < 1, so that from (3) we have n J(x, θ) = where s = Pn i=1 xi . X s e−θ (1 − e−θxi ), + −θ θ θ(1 − e ) i=1 Thus, using (2), we obtain θ n−1 e−θs r(θ) ∝ (1 − e−θ )n+1 ! n X (1 − e−θxi ) , s(1 − e−θ ) + e−θ i=1 14 θ ∈ R, (11) Figure 2: Graph 1: Fiducial densities: r(θ; x1 = 0.05, x2 = 0.95) (red); r(θ; x1 = 0.5, x2 = 0.5) (green); h(θ; s = 1) (blue). Graph 2: Fiducial densities r(θ; x1 = 0.02, x2 = 0.48) (red); r(θ; x1 = 0.2, x2 = 0.3) (green); h(θ; s = 0.5) (blue). Figure 3: Confidence curves for r(θ; x1 = 0.5, x2 = 0.5) (red) and h(θ; s = 1) (blue). which depends on the values of the specific xi ’s. Consider now the sufficient statistic P S = ni=1 Xi and, for simplicity, assume n = 2. The density of S is  θ2  e−ts s 0<s≤1 (1−e−θ )2 pθ (s) = . 2 θ −ts  (2 − s) 1 < s < 2 −θ 2 e (1−e ) and the generalized fiducial density (11) reduces to hs (θ) = ∂Fθ (s)/∂θ. In Figure 2 we report the fiducial densities r and hs for different values of (x1 , x2 ) and s = x1 + x2 . For s = 1 all densities are symmetric with the mode in 0, while the dispersion is increasing in \|x1 − x2 \|, so that the more concentrated fiducial density is obtained for x1 = x2 = 0.5. However, for s 6= 1, the densities have different modes and are shifted to the left when x1 increases. In all cases the fiducial density hs is in the middle of the various cases. Notice that hs has all the good properties discussed in V&M (2015) and, in particular, it is a confidence distribution because the model belongs to a NEF. The confidence intervals corresponding to hs (θ) are slightly smaller than those corresponding to r(θ), as can be seen from the confidence curves reported in Figure 3. For instance, when x1 = x2 = 0.5, the 95% confidence intervals are (-4.191,4.191) and (-4.399,4.399), for hs (θ) and r(θ), respectively. 15 The computation of the fiducial distribution Ht defined in (8) is greatly simplified starting with the sufficient statistic instead of the whole sample. However, when both the alternatives are feasible, they seem to lead to the same result. In particular, the following proposition states that the sufficiency principle is always satisfied by Ht when there exists a complete sufficient statistic for the parameter. Proposition 5. Consider the fiducial distribution Ht (φ), defined in (8) and (9), with T a one-to-one transformation of the data X = (X1 , . . . , Xn ). If S = S(T) is a complete and sufficient statistic of dimension d for φ, such that S = g(T[d] , T−[d] ) is a one-toone lower triangular transformation of T[d] for fixed T−[d] , then the fiducial distribution Hs (φ) for φ, obtained using S instead of T in (8) and (9), coincides with Ht (φ). Notice that the completeness of S is not necessary to satisfy the sufficiency principle, as the following example shows. Example 3. Given an i.i.d. sample X of size n from a uniform distribution on (θ, θ + 1), it is immediate to verify that the sufficient statistic S = (X(1) , X(n) ) is not complete. Because θ is a location parameter, Z = X(n) − X(1) is an ancillary statistic and the fiducial distribution for θ can be obtained starting from the distribution function of X(n) given Z, which is Fθ (x(n) \|z) = (x(n) − z − θ)/(1 − z), x(n) − 1 < θ < x(n) − z = x(1) . Thus hs (θ) = − ∂ ∂ x(n) − z − θ 1 Fθ (x(n) \|z) = − = , ∂θ ∂θ 1−z 1−z x(n) − 1 < θ < x(n) − z = x(1) .(12) If we start directly with X, we can consider the distribution function of Xn given Z = (Z1 , . . . , Zn−1 ), where Zi = Xn − Xi . Omitting tedious calculations, we have Fθ (xn \| z) = xn − θ − max(zi , 0) , 1 + min(zi , 0) − max(zi , 0) θ + max(zi , 0) < xn < θ + 1 + min(zi , 0), and thus, for xn − 1 − min(zi , 0) < θ < xn − max(zi , 0), hx (θ) = − ∂ 1 Fθ (xn \|z) = . ∂θ 1 + min(zi , 0) − max(zi , 0) Observing that min(zi , 0) = zi unless xn = x(n) and recalling that zi = xn − xi , i = 1, . . . , n−1, it follows that xn −1−min(zi , 0) = x(n) −1 and similarly xn −max(zi , 0) = x(1) , so that hx (θ) coincides with hs (θ) given in (12). 3.4 Conditionally reducible natural exponential families Consider a multivariate natural exponential family whose density, with respect to a fixed σ-finite positive measure ν, is given by ( d ) X pθ (x) = exp θk xk − M (θ) , θ = (θ1 . . . , θd ) ∈ Θ, x = (x1 , . . . , xd ) ∈ Rd . k=1 16 (13) A NEF is d-conditionally reducible (in the sequel cr-NEF) if its joint density (13) can be factorized as a product of d conditional densities each belonging to a real exponential family. More precisely if pφ (x\|φ(θ)) = d Y k=1 pφk (xk \|x[k−1] ; φk (θ)) = d Y k=1 exp φk (θ)xk − Mk (φk (θ); x[k−1] ) , (14) where φ = (φ1 , . . . , φd ) is a one-to-one function from Θ onto φ(Θ) = Φ. Furthermore, it can be shown that Φ = Φ1 × · · · × Φd , with φk ∈ Φk , k = 1, . . . , d, so that the φk ’s are variation independent. Notice that φk is the natural parameter of the k-th conditional distribution. For details on these families, with emphasis on enriched conjugate priors and on reference Bayesian analysis, see Consonni & Veronese (2001) and Consonni et al. (2004), respectively. Both these papers deal in particular with the families having simple quadratic variance function, named NEF-SQVFs, which include, as most interesting cases, the multinomial and negative-multinomial models, see Casalis (1996) and Appendix A1. Example 4 (Multinomial model). Consider a random vector X distributed according to a multinomial distribution and denote by pk the probability of the k-th outcome Xk , k = P P 1, . . . , d, with dk=1 xk ≤ N , and dk=1 pk ≤ 1. It is well know that the conditional disPk−1 Pk−1 pj )), xj , pk /(1− j=1 tribution of Xk given X[k−1] = x[k−1] , k = 2, . . . , d, is Bi(N − j=1 whereas the marginal distribution of X1 is Bi(N, p1 ). Since a binomial distribution is a real NEF, one can factorize the multinomial distribution as in (14) with   k−1 X pk φk = log xj  log(1 + eφk ), φk ∈ R. (15) , Mk (φk ; x[k−1] ) = N − P 1 − kj=1 pj j=1 For models belonging to a cr-NEF, the construction of the fiducial distribution proposed in Section 3 drastically simplifies. The existence of a sufficient statistic of the same dimension of the parameter makes the ancillary statistic not necessary, while the φ-parameterization, indexing each conditional distribution with a real parameter, implies the independence of the φk ’s under the fiducial distribution. Proposition 6. Let S be the sufficient statistic distributed according to a regular crNEF on Rd , parameterized by φ ∈ Φ = Φ1 × · · · × Φd , with Φk coinciding with the natural parameter space of the k-th conditional distribution. Then, for S = s, with sk , k = 1, . . . , d, satisfying conditions similar to those given before (4), Hs (φ) = d Y k=1 Hs[k] (φk ) = d Y (1 − Fφk (sk \|s[k−1] ) k=1 17 (16) is a fiducial distribution function on φ with density hs (φ) = d Y hs[k] (φk ), where hs[k] (φk ) = k=1 ∂ ∂ Hs[k] (φk ) = − Fφ (sk \|s[k−1] ). ∂φ ∂φk k (17) The φk ’s are independent under Hs (φ) and thus their importance ordering is irrelevant. This fact also justifies the simplification in the index notation adopted in (16). Notice, however, that the definition and the interpretation of the φk ’s depend on the particular ordering considered for the Xk ’s, as seen in Example 4. As recalled in Section 2, a general definition of multi-dimensional confidence distribution does not exist. However, in our context, since Hs (φ) is constructed as a product of marginal confidence distributions, it can be considered as a multivariate (possibly asymptotic) confidence distribution for φ. Some of the examples of Section 5 can be reconnected with this framework, but here we consider in specific the NEF-SQVFs, whose variance function is given in (35). For this class, with the exclusion of the negative-multinomial/hyperbolic secant distribution, it is possible to give a simple explicit expression of the fiducial density of φ, recalling the definition of Bk (φk ) given in (31) and setting zk = zkk in (35). The specifications of zkk and q, appearing in (35), and of Bk (φk ) can be found in Appendix A1. Proposition 7. Consider a sample of size n from a Poisson/normal, a multinomial, or a negative-multinomial family on Rd . If S denotes the sufficient statistic, then the (right) fiducial distribution for φ has density      d d k−1   Y Y X hs (φ) = hs[k] (φk ) ∝ sj − 1 Bk (φk ) , (18) exp φk (sk + zk ) − n + q    k=1 j=1 k=1 while for the negative-multinomial/gamma/normal family, with an m dimensional nega- tive multinomial component, the (right) fiducial distribution is given by hs (φ) = d Y k=1 h[sk ] (φk ) ∝ m Y k=1 exp{φk (sk + 1)}(1 − exp(φk ))n/q+ × exp {φm+1 sm+1 } (−φm+1 )n/q+ × d Y k=m+2 Pk−1 j=1 sj −1 Pm j=1 sj −1 exp φk sk − n sm+1 φ2k /2 . (19) Notice that the discrete components of a basic NEF-SQVF are integer-valued with zk = 1, so that the left fiducial distribution is obtained by the previous formulas replacing the term (sk + 1) by sk in (18) and (19). Thus it follows that the geometric mean hG s has the same structure in (18) and (19) with (sk + 1/2) instead of (sk + 1). 18 Example 4 (ctd.). For the multinomial family, because q = −1/N , zk = 1 and Bk (φk ) = N log(1 + eφk ), k = 1, . . . , d, it easily follows from formula (18) that ( ) 1 d d Y Y 1 eφk (sk + 2 ) G G hs[k] (φk ) = hs (φ) = ,(20) Pk−1 Pk 1 1 φk )nN − j=1 sj +1 B(s + , nN − s + ) j k (1 + e j=1 2 2 k=1 k=1 where B(·, ·) denotes the beta function. The fiducial distribution for φ not always is of particular interest in itself, but it can be used as a starting point for the construction of the fiducial distribution for alternative and more relevant parameters. We consider here the mean-parameter µ, which is a lower triangular transformation of φ, see (32), so that its fiducial distribution can be directly obtained from that of φ thanks to Proposition 1. Corollary 2. The (right) fiducial distribution for the mean parameter µ, relative to the ordering µ1 , . . . , µd , for the following NEF-SQVFs on Rd , has density: • Poisson/normal family (with m Poisson components) hs (µ) ∝ m Y s µkk−1 exp(−nµk ) k=1 d Y k=m+1 n n o exp − 2 (µ2 − 2µk sk /n) , 2σ (21) which corresponds to the product of m densities Ga(sk , n), k = 1, . . . , m and (d−m) densities N(sk /n, σ 2 /n), k = m + 1, . . . , d. • Multinomial family hs (µ) ∝ d Y k=1  µskk N − k X j=1 γk µj  where γk = −1 for k = 1, . . . , d − 1 and γd = N n − 1 − , (22) Pd j=1 sj . • Negative-multinomial family, with R occurrences in the (d + 1)-th cell γk  k d X Y µj  , µskk R + hs (µ) ∝ (23) j=1 k=1 where γk = −1 for k = 1, . . . , d − 1 and γd = −Rn − 1 − Pd j=1 sj . • Negative-multinomial/gamma/normal family (with an m-dimensional negative - 19 multinomial component with R occurrences in the (m + 1)-th cell)  γk m k Y X hs (µ) ∝ µj  µskk R + j=1 k=1  × R + × d Y m X j=1 Rn+Pm j=1 sj µj  −(d−m−1) µm+1 k=m+2 −(Rn+ µm+1 (24)   Pm   R + µ s m+1 j=1 j j=1 sj )−1 exp −   µm+1 Pm sk nsm+1 2 , µk − 2 µk µm+1 exp − 2 n 2µm+1 P where γk = −1 for k = 1, . . . , m − 1 and γm = −Rn − 1 − m j=1 sj . Notice that the Pm P density of µm+1 given µ[m] is an In-Ga(Rn + j=1 sj , sm+1 (R + m j=1 µj )), while the density of µk given µ[k−1] , k = m + 2, . . . , d, is a N(µm+1 sk /n, µ2m+1 /(nsm+1 )) depending only on µm+1 . Example 4 (ctd.). Inference for the multinomial distribution is usually performed for the cell-probabilities parameter p = (p1 . . . , pd ). Since pk = µk /N , the fiducial distribution hG for p is easily derived from (22), noting that the left fiducial density can be obtained replacing (sk + 1) by sk in (18), (and not in (22), which is derived aggregating the hyperparameters). It follows that the geometric mean hG s (p) is given by  γk d k d Y X X sk −1/2  G  hs (p) ∝ pj pk 1− , pk = 1, 0 < pk < 1, k=1 j=1 with γk = −1/2 for k = 1, . . . , d − 1 and γd = N n − 1/2 − Dirichlet distribution. Clearly, hG s (p) (25) k=1 Pd j=1 sj . This is a generalized in (25) refers to the specific order of importance p1 , p2 , . . . , pd . If we change this order, the fiducial distribution will change accordingly. Similarly, for the negative-multinomial model, with R occurrences in the (d + 1)th cell, hG s (φ) can be easily computed from (18) observing that zk = 1, q = 1/R and Bk (φk ) = −R log(1 − exp(φk )). 4 Connections with objective Bayesian inference As mentioned in Section 1, if we look at fiducial inference as a way to obtain a distribution on the parameter space of the model without any prior information, it appears natural to compare it with objective Bayesian inference. Recall that when a fiducial distribution coincides with a posterior, the corresponding prior is called fiducial prior. The step-by-step construction of the fiducial distribution ht (φ) defined in (8) is based on the inferential importance ordering of the parameter components φ1 , . . . , φd . 20 This aspect is also crucial in the procedure adopted to construct reference priors, see Bernardo & Smith (1994, Sec. 5.4.5). The reference prior π R for a parameter φ is generated by successive conditioning, established by the importance ordering of its compoQ nents, as π R (φ) = dk=1 π R (φd−k+1 \|φ[d−k] ). It is widely recognized that the dependence of the reference prior on the choice of the parameter of interest is necessary to obtain good frequentist properties such as coverage and consistency. For a one-dimensional parameter φ the reference prior coincides with the Jeffreys prior π J (φ) ∝ I(φ)1/2 , where I(φ) denotes the Fisher information. While the Jeffreys prior is invariant under a reparameterization of the model, the reference prior (and thus the reference posterior) is generally not invariant unless the transformation from φ to λ = (λ1 , . . . , λd ) is lower triangular, see Datta & Ghosh (1996). Thus the reference posterior has the same invariance property of the fiducial distribution proved in Proposition 1. Recently Berger et al. (2015) recognize the existence of situations in which one is interested simultaneously in all the parameter components of the model, or in none of them but a prior (and thus a posterior) distribution is necessary to perform other inferences such as predictions. In these cases an overall prior is needed. Its determination is an open problem but, as they highlight, when there exists a “common reference prior for all parameters”, this is the natural choice for the overall prior. A similar problem occurs in our context and we will comment on this aspect in the following sections. Notice that here the fiducial distribution (2) suggested by Hannig can be a good choice. 4.1 Location-scale parameter models For location-scale parameter models the fiducial prior exists and coincides with the reference prior. Assume first that only one parameter, θ, is unknown. In this case the model admits an ancillary statistic Z and, in particular, we take Zi = Xi − X1 or Zi = Xi /X1 , i = 2, . . . , n, if θ is a location or a scale parameter, respectively. Proposition 8. Let X = (X1 , . . . , Xn ) be an i.i.d. sample from a density pθ , θ ∈ Θ ⊆ R. If θ is a location or a scale parameter, then the fiducial distribution coincides with the Bayesian posterior obtained with the Jeffreys prior π J (θ) ∝ 1 or π J (θ) ∝ 1/θ, respectively. Example 5. Let X be an i.i.d. sample from the uniform distribution on (0, θ), θ > 0, so that θ is a scale parameter. First notice that S = X(n) is a sufficient statistic for θ and thus we can obtain directly the fiducial distribution ∂ nsn ∂ ∂ s n hs (θ) = = n+1 , Hs (θ) = − Fθ (s) = − ∂θ ∂θ ∂θ θ θ θ > s. (26) However the same result can be obtained without resorting to the sufficient statistic. Set w = max(z2 , . . . , zn ) and consider the distribution function of X1 given the ancillary 21 statistic Z = (X2 /X1 , . . . , Xn /X1 ) ( n Fθ (x1 \|z) = x1 θ x1 w n θ 0 < x1 < θ, 0 < x1 < θ, 0<w≤1 w>1 . (27) Now, because w ≤ 1 means x1 = max(x1 , . . . , xn ), while for w > 1 we have x1 w = max(x2 , . . . , xn ), expression (27), as a function of θ, is equivalent to Fθ (s) appearing in (26) and thus provides the same fiducial distribution. It is immediate to verify that it coincides with the Jeffreys posterior. A case in which the sufficient statistic is not one-dimensional and thus it is necessary to use an ancillary statistic can be found in the previous Example 3. Trivially hs (θ) given in (12) coincides with the Bayesian posterior obtained by π J (θ) ∝ 1. Consider now a model with a location parameter θ and a scale parameter σ, both unknown. Given an i.i.d. sample of size n, an ancillary statistic is, for example, Z = (Z3 , . . . , Zn ), with Zj = (Xj − X1 )/Z2 , j = 3, . . . , n, where Z2 = X2 − X1 is marginally ancillary for θ. Then, the one-to-one transformation from X to (X1 , Z2 , Z) allows to write the sampling distribution as pσ (z2 \|z)pθ (x1 \|z2 , z; σ)p(z). Note that in specific contexts other transformations could be more appropriate. For example, in a normal model one P P could use (X̄ = ni=1 Xi /n, S 2 = ni=1 (Xi − X̄)2 , Z) with Zj = (Xj − X̄)/S, j = 3, . . . , n, so that the factorization becomes pσ (s2 \|z)pθ (x̄\|s2 , z; σ)p(z). Proposition 9. Let X = (X1 , . . . , Xn ) be an i.i.d. sample from a density pθ,σ , where θ and σ are a location and a scale parameter, respectively. Then the fiducial distribution hx (σ, θ) for (σ, θ) coincides with the Bayesian posterior obtained with the reference prior R (σ, θ) ∝ 1/σ. πσ,θ Notice that π R (σ, θ) ∝ 1/σ is different from π J (σ, θ) ∝ 1/σ 2 obtained by the Jeffreys rule which, as already recalled, is not suitable for multidimensional parameters. Furthermore, while π R does not depend on the ordering of θ and σ, the step-by-step fiducial distribution is in general not allowable if the ordering is reversed. However, hx (σ, θ) coincides with the fiducial distribution obtained through other “symmetric” approaches, see Hannig (2009) and Fraser (1961). Thus the inferential ordering of importance seems irrelevant for this model and hx (σ, θ) can be assumed as an overall fiducial distribution. 4.2 Exponential families Lindley (1958) was the first to study the existence of a fiducial prior, analyzing in particular the case of continuous real NEFs and proving that it exists only for gaussian (with known variance) and gamma (with known shape) models. A full characterization of the real NEFs which admit a fiducial prior is given in V&M (2015). The following proposition summarizes their results. 22 Proposition 10. Let F be a real NEF with natural parameter θ. i) A fiducial prior exists if and only if F is an affine transformation of one of the following families: normal with known variance, gamma with known shape param- eter, binomial, Poisson and negative-binomial. For the three discrete families, the fiducial prior exists for all Hs , Hsℓ and HsG . ii) When a fiducial prior exists, it belongs to the family of conjugate distributions. Moreover, it coincides with the Jeffreys prior for continuous NEFs and for discrete NEFs too if we choose HsG as the fiducial distribution. iii) The fiducial distribution Hs (or HsA in the discrete case) and the Bayesian posterior distribution corresponding to the Jeffreys prior have the same Edgeworth’s expansion up to the term of order n−1 . The previous results establish a strong connections between Jeffreys posteriors and fiducial distributions for real NEFs, and thus the two different approaches lead, in some sense, to the same objective inference. A discussion about the coverage of the fiducial and the Jeffreys intervals and their good frequentist properties, in particular when compared with the standard Wald intervals, is given in V&M (2015, Section 5). Consider now a cr-NEF. It is easy to verify that the fiducial distribution hs (φ) in (18) belongs to the enriched conjugate family defined in Consonni & Veronese (2001, Section 4.3). This fact is the key-point to prove the following proposition. Proposition 11. Let S be a sufficient statistic distributed according to a cr-NEF on Rd , parameterized by φ = (φ1 , . . . , φd ). Then a fiducial prior for φ exists if and only if the conditional distribution of Sk given S[k−1] = s[k−1] is an affine transformation of one of the following families: normal with known variance, gamma with known shape parameter, binomial, Poisson and negative-binomial. In particular, all basic NEF-SQVFs, with the exclusion of the Negative-multinomial/ hyperbolic secant, admit a fiducial prior, which belongs to the enriched conjugate family. Moreover, if for the discrete components of these models we consider the geometric mean hG s[k] , then the product of the Jeffreys priors computed from the conditional distribution of Sk given S[k−1] = s[k−1] , the reference prior and the fiducial prior are all equal. Example 4 (ctd.). The multinomial distribution is a basic NEF-SQVF and thus from Proposition 11, setting sk = n = 0 in hG s (φ) given in (20), we obtain the fiducial prior π(φ) ∝ d Y eφk /2 /(1 + eφk /2 ), k=1 23 (28) which coincides with the reference prior and with the product of the Jeffreys priors for φk , k = 1, . . . , d, computed on the distribution of Xk given X[k−1] = x[k−1] . Finally, we observe that the fiducial distribution (17) is always an overall fiducial distribution for φ. However, the φ-parameterization is often not interesting in itself even if in some cases it is strictly related with a more relevant one. For example, following Berger et al. (2015), consider a multinomial model applied to directional data, as it happens for outcomes from an attitude survey. In this case the cells are naturally ordered, so that it is meaningful to reparameterize the model in terms of the conditional probabilities p∗k = exp(φk )/(1 + exp(φk )), k = 1, . . . , d. Then π(φ) in (28) induces on p∗ = (p∗1 , . . . , p∗d ) an overall fiducial prior which is a product of independent Be(1/2,1/2) distributions coinciding with the overall reference prior. 5 Further examples 5.1 Examples concerning normal models i) Difference of means. Consider two independent normal i.i.d. samples, each of size n, with known common variance σ 2 and means µ1 and µ2 , respectively. The sufficient statistics are the sample sums S1 and S2 , with Si ∼ N(nµi , nσ 2 ), i = 1, 2. If the parameter of interest is φ1 = µ2 − µ1 , we can reparameterize the joint density of (S1 , S2 ) in (φ1 = µ2 −µ1 , φ2 = µ1 ), so that the conditional distribution of S2 given S1 +S2 , being N((nφ1 + s1 + s2 )/2, nσ 2 /2), depends only on φ1 . From Table 1, the fiducial distribution of φ1 /2 + (s1 + s2 )/(2n) is N(s2 /n, σ 2 /(2n)), and thus φ1 is N(x̄2 − x̄1 , 2σ 2 /n), where x̄i = si /n. Because S1 + S2 is N(φ1 + 2φ2 , 2nσ), arguing as before, the fiducial distribution of φ2 given φ1 is N((x̄1 + x̄2 −φ1 )/2, σ 2 /(2n)), so that hS1 ,S2 (φ1 , φ2 ) = hS1 ,S2 (φ1 )hS1 +S2 (φ2 \|φ1 ). Notice that the same joint fiducial distribution is obtained if we consider the ordering (φ2 , φ1 ) or even if we compute the (marginal) fiducial distributions of µ1 and µ2 and obtain that of (φ1 , φ2 ) through the change-of-variable rule. Thus the ordering of the parameter is irrelevant and hS1 ,S2 (φ1 , φ2 ) is an overall fiducial distribution. Furthermore, it coincides with the reference posterior obtained with a constant prior and the marginal distribution of φ1 and φ2 are both confidence distributions. ii) Many normal means (Neyman Scott Problem). Consider n samples of size two (Xi1 , Xi2 ), with each Xij independently distributed according to a N(µi , σ 2 ), i = 1, . . . , n P and let X̄i = (Xi1 + Xi2 )/2 and W = ni=1 (Xi1 − Xi2 )2 . The aim is to make inference on the common variance σ 2 , with nuisance parameter µ = (µ1 , . . . , µn ). This well known example is used to show that the maximum likelihood estimator σ̂ 2 = W/(4n) of σ 2 is inconsistent, because W/(4n) → σ 2 /2, n → ∞. To obtain the fiducial distribution of σ 2 , first notice that the joint distribution of the sufficient statistics X̄ = (X̄1 , . . . , X̄n ) 24 and W can be factorized as Qn i=1 pµi ,σ2 (x̄i ) pσ2 (w), for the independence of X̄ and W , with W ∼ Ga(n/2, 1/(4σ 2 )). Using Table 1 one can easily obtain from pσ2 (w) the fiducial distribution for 1/(4σ 2 ), and hence that for σ 2 which is In-Ga(n/2, w/4), while that of each µi given σ 2 , derived from pµi ,σ2 (x̄i ), is N(x̄i , σ 2 /2). As a consequence Qn 2 2 hx̄,w (σ 2 , µ) = i=1 hx̄i (µi \|σ ) hw (σ ). This distribution coincides with the posterior obtained from the order invariant reference prior π R (σ 2 , µ1 , . . . , µn ) ∝ 1/σ 2 and does not present the inconsistency of the likelihood estimator, which instead occurs for the posterior distribution obtained from the Jeffreys prior π J (σ 2 , µ1 , . . . , µn ) ∝ 1/σ n+2 . 5.2 Comparison of two Poisson rates The comparison of Poisson rates µ1 and µ2 is a classical problem arising in many contexts, see for example Lehmann & Romano (2005) for a discussion on an unbiased uniformly most powerful test for the ratio φ1 = µ2 /µ1 . Given two i.i.d. samples of size n from two independent Poisson distributions, the sufficient statistics are the sample sums S1 and S2 , with Si ∼ Po(nµi ), i = 1, 2. Reparameterizing the joint density of (S1 , S2 ) in (φ1 = µ2 /µ1 , φ2 = µ1 + µ2 ), we have that the conditional distribution of S2 given S1 + S2 is Bi(s1 + s2 , φ1 /(1 + φ1 )) and the marginal distribution of S1 + S2 is Po(nφ2 ). Thus the sampling distribution is a cr-NEF and we can apply (16). Using Table 1, the fiducial density for φ1 /(1 + φ1 ), derived from the conditional distribution of S2 given S1 + S2 is Be(s2 + 1/2, s1 + 1/2) which implies hG s1 ,s2 (φ1 ) = 1 s −1/2 φ12 (1 + φ1 )−s1 −s2 −1 , B(s2 + 1/2, s1 + 1/2) φ1 > 0. (29) From the marginal distribution of S1 + S2 and using again Table 1, it follows that G G G hG s1 +s2 (φ2 ) is Ga(s1 +s2 +1/2, n) and thus hs1 ,s2 (φ1 , φ2 ) = hs1 ,s2 (φ1 )hs1 +s2 (φ2 ). This joint fiducial distribution is order-invariant, coincides with the reference posterior according to Proposition 11, and is an overall distribution for (φ1 , φ2 ). Notice that hG s1 ,s2 (φ1 ) is a confidence distribution and that it differs from the fiducial distribution induced on φ1 by the two independent marginal fiducial densities for µ1 and µ2 . 5.3 Bivariate binomial A Bayesian analysis for the bivariate binomial model has been discussed by Crowder & Sweeting (1989) in connection with a microbiological application. Consider m spores, each with a probability p to germinate, and denote by R the random number of germinating spores, so that R is Bi(m, p). If q is the probability that one of the latter spores bends in a particular direction and S is the random number of them, the probability distribution of S given R = r is Bi(r, q). The joint distribution of R and S is called bivariate binomial. Crowder and Sweeting observe that the Jeffreys prior π J (p, q) ∝ p−1 (1 − p)−1/2 q −1/2 (1 − q)−1/2 25 is not satisfactory for its asymmetry in p and 1 − p, while Polson & Wasserman (1990) show that this fact does not occur using the order-invariant reference prior π R (p, q) ∝ p−1/2 (1 − p)−1/2 q −1/2 (1 − q)−1/2 which is the product of the two independent Jeffreys priors. G The joint fiducial density hG r,s (q, p) can be obtained as the product of hr,s (q), derived from the conditional model Bi(r, q) of S given R = r, and hG r (p\|q), derived from the marginal model Bi(m, p) of R, which does not depend on q. Thus p and q are independent under hG r,s so that it is an overall fiducial distribution. Because for the binomial model the fiducial prior is equal to the Jeffreys prior, see Proposition 10, it follows immediately that hG r,s (q, p) coincides with the reference posterior. All previous conclusions hold even if we consider the alternative parametrization (η = pq, λ = p(1 − q)/(1 − pq)). 5.4 Ratio of parameters of a trinomial distribution Bernardo & Ramon (1998) perform the Bayesian reference analysis for the ratio of two multinomial parameters presenting some applications. In particular they discuss the case of (X1 , X2 ) distributed according to a trinomial distribution with parameters n and p = (p1 , p2 ), and provide the joint reference prior for (φ1 = p1 /p2 , φ2 = p2 ), with φ1 the parameter of interest. Then they derive the marginal reference posterior for φ1 which is x −1/2 π R (φ1 \|x1 , x2 ) ∝ φ1 1 (1 + φ1 )−x1 −x2 −1 . (30) To find the fiducial distribution of φ1 , we reparameterize the trinomial model in (φ1 , φ2 ). The conditional distribution of X1 given T = X1 + X2 = t is Bi(t; φ1 /(1 + φ1 )), so that, by Table 1, the fiducial density for φ1 /(1 + φ1 ) is Be(x1 + 1/2, t − x1 + 1/2) and hG x1 ,t (φ1 ) coincides with (30). From the marginal distribution of T , which is Bi(n; φ2 (1 + φ1 )), it is possible to derive the fiducial density hG t (φ2 \|φ1 ) = Γ(n + 1) t−1/2 (1 + φ1 )t+1/2 φ2 (1 − (1 + φ1 )φ2 )n−t−1/2 Γ(t + 1/2)Γ(n − t + 1/2) G G so that the joint fiducial density is hG x1 ,x2 (φ1 , φ2 ) = hx1 ,t (φ1 )ht (φ2 \|φ1 ) which coincides with the joint reference posterior. 6 Conclusions and final remarks We have suggested a way to construct a fiducial distribution which depends on the inferential importance ordering of the parameter components. Our proposal appears to be quite simple to apply and, even if it is not so general as the theory suggested by Hannig, has some advantages in connection with the modern confidence distribution theory and it is strictly related to objective Bayesian analysis. 26 In complex models an exact analysis is generally not possible, but approximate results can be derived working with asymptotic distributions. In V&M (2015), starting from the sufficient statistic, an expansion up to the first order of the fiducial distribution for the mean parameter of a real NEF is provided. This result can be extended to arbitrary regular models starting from the maximum likelihood estimator of the parameter. When the maximum likelihood estimator is not sufficient a better fiducial distribution can be obtained using an ancillary statistic, as suggested in Section 3. To this aim the magic formula p∗ given by Barndorff-Nielsen (1983), which provides an approximation of the conditional distribution of the maximum likelihood estimator given an ancillary statistic, can be fruitfully adopted. Furthermore, these asymptotic results appear to be strictly connected with the theory of matching priors, i.e. priors that ensure approximate frequentist validity of posterior credible set. Notice that also these priors crucially depend on the inferential ordering of the parameters, see Tibshirani (1989) and Datta & Mukerjee (2004). However, a normal approximation of the fiducial distribution, when it can be established and is enough for the analysis, can be proved to be order-invariant. These type of results will be discussed in a forthcoming paper. Acknowledgements This research was supported by grants from Bocconi University. Appendix A1: Useful results on cr-NEFs Some technical aspects related to cr-NEFs are the following. 1. A NEF is a cr-NEF if and only if the principal k × k matrix of the variance function does not depend on µk+1 , . . . µd , for k = 1, . . . , d − 1. 2. The Fisher information matrix relative to the φ-parametrization is diagonal with the kk-th element depending only on φ[k] . 3. The cumulant transform Mk (φk ; x[k−1] ) of the k-th conditional density is given by Mk (φk ; x[k−1] ) = k−1 X Akj (φk )xj + Bk (φk ), (31) j=1 for some functions Akj and Bk . 4. The conditional expectation of Xk given X[k−1] = x[k−1] is linear in x[k−1] , because it is the gradient of (31). 27 5. The parameter µk depends on φ only through φ[k] , because from (31) µk = k−1 X ∂Akj (φk ) j=1 ∂φk µj + ∂Bk (φk ) . ∂φk (32) 6. Using (14) and (31), it can be checked that θk = φk − d X Auk (φu ), and M (θ) = d X Bk (φk (θ)). (33) k=1 u=k+1 As a consequence of the first part of (33), there exists a function gk such that φk = θk + gk (θk+1 , . . . , θd ). (34) Of course all the previous formulas hold for k = 1, . . . , d, with the understanding that components that lose meaning for a specific k are set to zero. A NEF has a simple quadratic variance function (SQVF) if the ij-th element of its variance-covariance matrix, seen as a function of the mean parameter µ = (µ1 , . . . , µd ), P (k) can be written as Vij (µ) = qµi µj + dk=1 µk Lij + Cij , where q is a real constant and L(k) , k = 1, . . . , d and C are constant d × d symmetric matrices. Any NEF-SQVF can be obtained, via a nonsingular affine transformation, from one of the basic families: Poisson/normal (q = 0), multinomial (q = −1/N , N positive integer), negative-multinomial (q = 1/R, R positive integer), negative-multinomial/gamma/normal (q = 1/R) and negative-multinomial/hyperbolic-secant (q = 1/R), see Casalis (1996) for a detailed description of these distributions. The ij-th element of the variance function V (µ) of a basic NEF-SQVF is Vij (µ) = qµi µj + d X zik µk + Cij , (35) k=1 where zij = zji , i, j ∈ {1, . . . , d}, are constants. The values of zii for the basic NEFSQVFs, together with other technical details, are given in the proof of Corollary 2. A2: Proofs Proof of Proposition 1. By the standard change-of-variable rule applied to the first integral in (10), it is enough to show that λ hφ t (φ(λ))\|Jφ (λ)\| = ht (λ), 28 (36) where Jφ (λ) is the Jacobian of the transformation from φ to λ. Now, from (8) we have hφ t (φ(λ)) = = d Y ht[k] ,t−[d] (φd−k+1 (λ[d−k+1] )\|φ[d−k] (λ[d−k] )) k=1 d Y k=1 ∂ Fφ (tk \|t[k−1] , t−[d] ; φ[d−k] ) ∂φd−k+1 d−k+1 φ[d−k+1] =φ[d−k+1] (λ[d−k+1] ) while Jφ (λ) = d Y ∂φd−k+1 (λ[d−k+1] ) , ∂λd−k+1 k=1 because the transformation from φ to λ is lower triangular. It follows from the last two formulas and the chain rule that hφ t (φ(λ))\|Jφ (λ)\| = d Y k=1 ∂ ∂λd−k+1 Fλd−k+1 (tk \|t[k−1] , t−[d] ; λ[d−k] ) , where Fλd−k+1 is the distribution function of Tk given (T[k−1] = t[k−1], T[−d] = t[−d] ) in the λ parameterization. The equality (36) follows by applying (9) to the model parameterized by λ. ⋄ Proof of Proposition 2. Let pG (x) = c−1 (p1 (x)p2 (x))1/2 , where c = constant. Then Z R (p1 (x)p2 (x))1/2 dν(x) is the normalizing Z q(x) q(x) KL(q\|p1 ) + KL(q\|p2 ) = log q(x)dν(x) + log q(x)dν(x) p1 (x) p2 (x) Z Z q(x) q 2 (x) q(x)dν(x) = 2 log q(x)dν(x) = log p1 (x)p2 (x) CpG (x) Z q(x) = 2 log G q(x)dν(x) − 2 log c = 2KL(q\|pG ) − 2 log c. (37) p (x) Because c does not depend on q, it follows that the functional in (37) achieves its minimum (equal to −2 log c) if and only if KL(q\|pG ) = 0, i.e. q = pG . ⋄ Proof of Proposition 3. We only prove that Hs (θ) < HsG (θ); the other inequality can be shown in the same way. Using (5) and (6), we can write hG s (θ) hs (θ) = = 1 c 1 c p s hs (θ)hℓs (θ) hs (θ) 1+ 1 = c ∂ ∂θ pθ (s) hs (θ) = 29 q hs (θ)(hs (θ) + ∂ ∂θ pθ (s)) hs (θ) 1p 1 + γs (θ). c (38) By hypothesis γs (θ) is decreasing and thus, from (38), hG s (θ)/hs (θ) is also decreasing on Θ. This is a sufficient condition for Hs (θ) < HsG (θ), see Shaked & Shanthikumar (2007, Theorem 1.C.1). ⋄ Proof of Corollary 1. Let pθ (s) = exp(θs − nM (θ) be the probability mass function (with respect to a measure ν) of a real NEF, with θ the natural parameter. Fixing s ∈ S0 , we can write ∂pθ (s) ∂ (s − nM ′ (θ)) exp(θs − nM (θ)) γs (θ) = (1 − Fθ (s)) = P+∞ / ′ ∂θ ∂θ t=s+1 (t − nM (θ)) exp(θt − nM (θ)) !−1 +∞ X t − nM ′ (θ) . (39) exp{(t − s)θ} = s − nM ′ (θ) t=s+1 The elements in the sum (39) are continuous and increasing functions of θ in both intervals for which θ < θbs and θ > θbs , where θbs = (M ′ )−1 (s/n). Thus γs (θ) is decreasing in these intervals. Moreover, γs (θ) is equal to zero for θ = θbs , positive for θ < θbs and negative for θ > θbs , because the denominator of γs (θ) is hs (θ), which is positive. Then γs (θ) is decreasing over all Θ and from Proposition 3 the result follows. ⋄ Proof of Proposition 4. In order to prove the proposition, it is sufficient to show that there exist θ1 and θ2 in Θ, G A G θ1 < θ2 , such that hA s (θi ) = hs (θi ), i = 1, 2, with hs (θ) < hs (θ) for θ1 < θ < θ2 and G hA s (θ) > hs (θ) otherwise, see Shaked & Shanthikumar (2007, proof of Theorem 3.A.44). G Thus we analyze the sign of hA s (θ) − hs (θ). We can write q 1 1 G ℓ (θ) = hA (θ) − h (h (θ) + h (θ)) − hs (θ)hℓs (θ) s s s s 2 c 1p 1 1 + γs (θ) , (2 + γs (θ)) − = hs (θ) 2 c G so that the sign of the difference hA s (θ)−hs (θ) is a function of γs (θ) only. First notice that, Rp by a standard property of the arithmetic and geometric means, c = hs (θ)hℓs (θ)dθ < R (hs (θ) + hℓs (θ))/2 dθ = 1 for all θ. After some straightforward algebra, it can be seen √ −2 2 G 2 that hA s (θ) − hs (θ) = 0 when (and only when) γs (θ) = 2c ((1 − c ) − 1 − c ) = k1 or √ γs (θ) = 2c−2 ((1 − c2 ) + 1 − c2 ) = k2 , with k1 ∈ (−1, 0) and k2 > 0. Moreover we have G A G hA s (θ) < hs (θ) for k1 < γs (θ) < k2 and hs (θ) > hs (θ) for γs (θ) < k1 or γs (θ) > k2 . By assumption, γs (θ) is decreasing on Θ from +∞ to -1, so that there exist θ1 and θ2 , with γs (θ1 ) = k2 and γs (θ2 ) = k1 , satisfying the sufficient condition stated at the beginning of the proof. ⋄ 30 Proof of Proposition 5. First notice that if we use for constructing the fiducial distribution the sufficient statistic S, which has the same dimension of the parameter, we do not need an ancillary statistic. Furthermore, T is a one-to-one transformation of X, and thus S is a function of T = (T [d] , T −[d] ) but, since S is complete, it is stochastically independent of T −[d] by Basu’s Theorem. As a consequence, S[k] is also independent of T −[d] and thus Prφd−k+1 (Sk ≤ sk \| S[k−1] = s[k−1] ; φ[d−k] ) = Prφd−k+1 (Sk ≤ sk \| S[k−1] = s[k−1] , T−[d] = t−[d] ; φ[d−k] ). (40) From the one-to-one lower triangular transformation s = g(t[d] , t−[d] ), we have that sk = gk (tk , t[k−1] , t−[d] ), with gk invertible with respect to tk , so that (assuming gk increasing) (40) becomes Prφd−k+1 (gk (Tk , T[k−1] , T −[d] ) ≤ sk \| T[k−1] = t[k−1], T−[d] = t−[d] ; φ[d−k] ) = Prφd−k+1 (Tk ≤ gk−1 (sk , T[k−1] , T −[d] ) \| T[k−1] = t[k−1], T−[d] = t−[d] ; φ[d−k] ) = Prφd−k+1 (Tk ≤ tk \| T[k−1] = t[k−1], T−[d] = t−[d] ; φ[d−k] ), which proves the proposition. ⋄ Proof of Proposition 6. Because each conditional distribution of Xk given X[k−1] = x[k−1] belongs to a NEF with natural parameter φk , using (4) we have that Hs[k] (φk ) is a distribution function for φk . The result follows from the postulated independence among the φk ’s. ⋄ Proof of Proposition 7. Formulas (18) and (19) derive by a direct application of (17) to the conditional distributions of the different families. For a detailed description of the cr-NEFs involved, see Consonni & Veronese (2001, proof of Theorem 3). ⋄ Proof of Corollary 2. First notice that the fiducial distribution hs (µ) can be more easily obtained via a double transformation, namely θ hµ s (µ) = hs (θ(µ))\|Jφ (θ(µ))\|\|Jθ (µ)\|, where the Jacobian \|Jφ (θ)\| = 1 for (34), and ) ( d X θk (µ)zk − q(d + 1)M (θ(µ)) , \|Jθ (µ)\| ∝ det{V (µ)}−1 = exp − k=1 31 (41) see Gutiérrez-Peña & Smith (1997, pag. 34) for the proportionality relationship and Consonni et al. (2004, Prop. 1) for the equality. We consider now each family. Poisson/normal family with m Poisson components. We have q = 0, φk = θk ; zk = 1, θk = log(µk ) and Bk (φk ) = exp(φk ) for k = 1, . . . , m, while zk = 0, θk = µk /σ 2 and Bk (φk ) = σ 2 φ2k /2 for k = m + 1, . . . , d, where σ 2 is the known variance of the normal Q −1 components. Then from (41), it follows that \|Jφ (θ(µ))\| = m k=1 µk , and thus using (18), the result (21) follows. Multinomial family. Using the relationships in Example 1, (41) gives \|Jφ (θ(µ))\| = (N − Pd Qd −1 −1 k=1 µk ) k=1 µk and using (8) we obtain (22). Negative-multinomial family. We have q = 1/R, R > 0, zk = 1, φk (θ) = θk − log(1 − Pd Pd φk θu j=1 µj ), and Bk (φk ) = −R log(1 − e ) for k = u=k+1 e ), θk = log(µk ) − log(R + Pd Q 1, . . . , d. Then from (41), it follows that \|Jφ (θ(µ))\| = (R + k=1 µk )−1 dk=1 µ−1 k , and thus using (18), the result (23) follows. Negative-multinomial/gamma/normal family (with an m dimensional negative-multinomial P component). We have q = 1/R, R > 0; zk = 1 and φk = log(µk /(R + kj=1 µj )), P k = 1, . . . , m; zm+1 = 0 and φm+1 = −(R + m j=1 µj )/µm+1 ; zk = 0 and φk = µk /µm+1 , k = m+2, . . . , d. In this case it is convenient to compute the fiducial density of µ directly from (19). Observing that the Jacobian of the transformation from φ to µ is  !−1 m ! m m X Y X −d+m+1 R + \|Jφ (µ)\| = R + µk µj  µ−2 µ−1 m+1 µm+1 k k=1 j=1 k=1 and using the previous expression of φk , the density (24) follows. ⋄ Proof of Proposition 8. Let X be an i.i.d. sample of size n, with Xi ∼ pθ (xi ) = f (xi − θ), i.e. θ is a location parameter, and consider the transformation Z1 = X1 , Zi = Xi − X1 , i = 2, . . . , n, whose Jacobian is one. Then, setting z = (z1 , . . . , zn ) R +∞ Hx (θ) = Hz (θ) = 1 − Fθ (z1 \|z2 , . . . , zn ) = z1 f (t R +∞ −∞ f (t − θ) − θ) Qn i=2 f (t Qn i=2 f (t + zi − θ)dt + zi − θ)dt . Using now the substitution m = −t + θ + z1 in the previous two integrals, and recalling that π J (θ) ∝ 1 we obtain Rθ Qn i=2 f (z1 + zi − m)dm −∞ f (z1 − m) R +∞ Qn i=2 f (z1 + zi − m)dm −∞ f (z1 − m) = = Rθ Qn J i=1 f (xi − m)π (m)dm −∞ R +∞ Qn J i=1 f (x1 − w)π (m)dm −∞ Z θ J π (m\|x)dm. −∞ 32 The result relative to the scale parameter follows recalling that the model pθ (x) = f (x/θ)/θ can be transformed in a model with location parameter µ setting y = log(x) and µ = log(θ). In this case a constant prior on µ is equivalent to a prior on θ proportional ⋄ to 1/θ. Proof of Proposition 9. Let X be an i.i.d. sample of size n, with Xi ∼ pθ,σ (xi ) = f ((xi − θ)/σ)/σ, i = 1, . . . , n and notice that the absolute value of the Jacobian of the transformation from x to (x1 , z2 , z), with z2 = x2 − x1 , z = (z3 , . . . , zn ) and zj = (xj − x1 )/z2 , j = 3, . . . , n, is \|z2 \|n−2 . Furthermore, the reference prior π R (θ, σ) is order-invariant and can be written as π R (θ\|σ)π R (σ), where π R (θ\|σ) ∝ 1 and π(σ) ∝ 1/σ, see Fernández & Steel (1999). Working conditionally on σ we can thus apply Proposition 8 to conclude that the reference posterior and the fiducial distribution for θ given σ coincide. It remains to show that Z ∞ R R p θ,σ (x1 , z2 , z)π R (θ\|σ)π R (σ)dθ (42) π (σ\|x) = π (σ\|x1 , z2 , z) ∝ ∝ Z ∞ 1 −∞ σ n+1 f −∞ x1 − θ σ n z2 zi + x1 − θ z2 + x1 − θ Y f f dθ σ σ i=3 corresponds to the fiducial density hz2 ,z (σ). We have Z R +∞ R +∞ −∞ z2 = +∞ Z +∞ X ,Z2 \|Z (t, w\|z)/pZ (z)dtdw z2 −∞ \|w\|n−2 wzi +t−θ t−θ w+t−θ Qn dtdw f f f i=3 σn σ σ σ Hz2 ,z (σ) = 1 − Fσ (z2 \|z) = 1 p θ,σ pZ (z) (43) , where the density of pZ (z) does not depend on the parameters because z is ancillary. Assuming z2 > 0, and using the transformation m = x1 − v(t − θ)/σ, v = z2 σ/w, which implies t = σ(x1 − m)/v + θ, w = z2 σ/v, with Jacobian z2 σ 2 /v 3 , the fiducial distribution Hz2 ,z (σ) in (43) becomes R σ R +∞ z2n−1 0 −∞ vn+1 f x1 −m v f z2 +x1 −m v pZ (z) Qn i=3 f z2 zi +x1 −m v dmdv . Taking the derivative with respect to σ, it is immediate to see that the fiducial density for σ coincides with the posterior distribution given in (42). If z2 < 0, and applying to the integral the same transformation used in the previous case, we have Fσ (z2 \|z) = R z2 R +∞ −∞ −∞ = − R σ R +∞ 0 −∞ \|w\|n−2 σn f t−θ σ (−z2 )n−1 f vn+1 f w+t−θ σ Z p (z) x1 −m v 33 f Qn i=3 f z2 +x1 −m v pZ (z) wzi +t−θ σ Qn i=3 f dtdw z2 zi +x1 −m v dmdv , so that again the derivative with respect to σ of Hz2 ,z (σ) = 1 − Fσ (z2 \|z) leads to (42). ⋄ The following lemma will be used in the proof of Proposition 11. Lemma 1. Consider a cr-NEF on Rd , with the k-th diagonal element in the Fisher information matrix given by Ikk (φ) = ak (φk )bk (φ[k−1] ). Then the d-group (order-invariant) reference prior π R for φ = (φ1 , . . . , φd ) is π R (φ) = d Y k=1 πkJ (φk ) ∝ d Y (ak (φk ))1/2 , (44) k=1 where πkJ (φk ) is the Jeffreys prior obtained from the conditional distribution of Xk given X[k−1] = x[k−1] . Proof of Lemma 1 First observe that µ[k] is a one-to-one transformation of φ[k] and that the information matrix I(φ) of a cr-NEF is diagonal, see Appendix A1 (points 2 and 5). From (14) and (31), the kk-th element of I is 2 ∂ X Ikk (φ[k] ) = −Eφ log pφk (xk \|x[k−1] ; φk ) = −EφX (Mk′′ (φk ; x[k−1] )) ∂φ2k = k−1 X A′′kj (φk )µj (φ[j] ) + Bk′′ (φk ). j=1 Under the assumption in the proposition, we can write Ikk (φ[k] ) = ak (φk )b∗k (µ[k−1] (φ[k−1] )). From Datta & Ghosh (1995), it follows that the reference prior on φ is order-invariant and is given by the last product in (44). Consider now the Jeffreys prior on φk obtained from pφk (xk \|x[k−1] ). This is propor- tional to the square root of −E Xk \|x[k−1] (Mk′′ (φk ; x[k−1] )) = k−1 X A′′kj (φk )xj + Bk′′ (φk ) = ak (φk )b∗k (x[k−1] ), j=1 where again the last equality holds by the assumption in the proposition. Thus the ⋄ product of the d Jeffreys priors is equal to (44) and the result holds. Proof of Proposition 11. Due to the independence of the φk ’s, a fiducial prior for φ exists if and only if there exists a fiducial prior for each φk . Because the conditional distribution of Sk given S[k−1] = s[k−1] belongs to a real NEF with natural parameter φk , the result of the first part of the proposition follows from Proposition 10. The first statement of the second part of the proposition follows checking directly the form of the conditional distributions of the basic NEF-SQVFs and using again Proposition 10. 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Source Forager: A Search Engine for Similar Source Code Vineeth Kashyap∗, David Bingham Brown† , Ben Liblit† , David Melski∗ , and Thomas Reps∗† ∗ GrammaTech, arXiv:1706.02769v1 [cs.SE] 8 Jun 2017 Inc., Ithaca, New York, USA Email: {vkashyap,melski}@grammatech.com † University of Wisconsin–Madison, USA Email: {bingham,liblit,reps}@cs.wisc.edu Abstract—Developers spend a significant amount of time searching for code—e.g., to understand how to complete, correct, or adapt their own code for a new context. Unfortunately, the state of the art in code search has not evolved much beyond text search over tokenized source. Code has much richer structure and semantics than normal text, and this property can be exploited to specialize the code-search process for better querying, searching, and ranking of code-search results. We present a new code-search engine named Source Forager. Given a query in the form of a C/C++ function, Source Forager searches a pre-populated code database for similar C/C++ functions. Source Forager preprocesses the database to extract a variety of simple code features that capture different aspects of code. A search returns the k functions in the database that are most similar to the query, based on the various extracted code features. We tested the usefulness of Source Forager using a variety of code-search queries from two domains. Our experiments show that the ranked results returned by Source Forager are accurate, and that query-relevant functions can be reliably retrieved even when searching through a large code database that contains very few query-relevant functions. We believe that Source Forager is a first step towards muchneeded tools that provide a better code-search experience. Index Terms—code search, similar code, program features. I. Introduction In this age of software proliferation, it is useful to be able to search large source-code corpora effectively for code with desired properties.1 Developers routinely use code search as a learning and debugging tool for tasks such as looking for existing functionality in a code base, determining how to use an API or library, gathering information about what code is intended to do, etc. [1]. Text-based search techniques are not always precise enough for code because they focus purely on strings in the code: Supported, in part, by a gift from Rajiv and Ritu Batra; by AFRL under DARPA MUSE award FA8750-14-2-0270, and by the UW–Madison Office of the Vice Chancellor for Research and Graduate Education with funding from the Wisconsin Alumni Research Foundation. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors, and do not necessarily reflect the views of the sponsoring agencies. T. Reps has an ownership interest in GrammaTech, Inc., which has licensed elements of the technology reported in this publication. 1 In this paper, the term “search” is used in the sense of Google search— namely, to retrieve documents that are related to a specified query. “Search” is not used in the sense of finding an occurrence of a user-specified string or pattern in a given document. comments, complete or partial names of functions and variables, and so on. Text search largely ignores code structure and semantics (i.e., what the code does and how it does it). A text-based approach can cause searching to be imprecise: relevant code fragments may be missed, while many spurious matches may be returned. Recent search techniques allow users to specify certain aspects of code semantics in addition to the textual query [2]–[8]. Some techniques allow users to specify structural requirements, such as that the search target should have nested loops. Others specify context, such as that the search target should implement a particular interface. Yet others specify sets of input/output pairs. Additional semantic information can improve search accuracy. However, existing techniques share the following shortcomings: • The techniques do not provide a unified way of specifying semantics for the search query. Each technique has its own ad-hoc specification of the semantic aspects of the code that it uses. • Each technique is closely married to its chosen semantic aspect, which is deeply ingrained into the implementation of the search technique. This tight coupling makes it hard to extend these techniques to model additional semantic aspects. We propose a search technique for finding similar source code that addresses these shortcomings: • • Unified Query Specification. Our code-search mechanism takes code fragments as queries. Various kinds of semantic information can be extracted from the query and used by the search. This approach provides a unified mechanism for code search: searching code using code fragments. Moreover, the same techniques for extracting semantic information are used on both queries and elements of the corpus being searched, leading to greater consistency. Extensibility. Our code-search technique uses a vector of feature-observations extracted from elements in the corpus. Feature-observations capture various aspects of the syntax and semantics of a program (each such aspect is called a feature-class), and provide a unified interface for querying. This approach also makes our search technique extensible: it is easy to introduce more feature-classes that model additional aspects of the code. int binsearch (int x, int v[], int n) { int low , high , mid; low = 0; high = n - 1; while (low <= high) { mid = (low + high) / 2; if (x < v[mid ]) { high = mid - 1; } else if (x > v[mid]) { low = mid + 1; /* found match / } else { return mid; } } / no match / return -1; } feature-observations of various feature-classes weights similar-code results feature-class weight determination similarity-based neighbor search query ... corpus ... program elements feature extraction engine code database Fig. 2. Example program that implements a binary search over a sorted integer array Fig. 1. Overview of the Source Forager architecture In addition to being useful on its own right as a developer A. Offline Phase: Population of the Source Forager Database tool, similar-code search can serve as an important building In this phase, Source Forager analyzes a given code corpus, block for automated program repair and program synthesis. and populates a code database with rich information about The ability to find other code similar to a query can help each of the functions in the code corpus. Source Forager automated tools learn from the similar code, and fix bugs or extracts several different kinds of information about each perform code completion tasks on the query. function; we refer to each of the different kinds of information The main contributions of Source Forager are: as a feature-class. §III describes our different feature-classes in • The ability to perform C/C++ code searches using code detail. A feature-observation is some specific value observed fragments as queries. The searches and answers of Source for a given feature-class. Thus, each function has one featureForager are both based on a query formalism that is close observation for each feature-class. For example, one of our to the concepts that developers are already familiar with. feature-classes is Numeric Literals. The corresponding feature• A code-search architecture that uses multiple code featureobservation is the set of all the numeric constants used in classes simultaneously. The architecture is extensible, althe function. For the binary-search implementation code given lowing easy addition of new code feature-classes, which in Fig. 2, the Numeric Literals feature-observation is the set enhances the dimensions along which code is searched. {−1, 0, 1, 2}. • A mechanism for automatically selecting useful code featureA feature extraction engine consists of several feature extracclasses to be employed in code search of a given query, given tors, which collect a given function’s feature-observations into no a priori domain information about the query. a feature-vector. Note that the elements of the feature-vector • A supervised-learning technique to pre-compute the relative importance of different feature-classes, when it is known can be non-numeric, such as sets, multisets, trees, maps, etc. that a query belongs to a specific domain for which suitable The number of feature-classes determines the length of the feature-vector. training data is available. Organization: The remainder of the paper is organized The feature extractors operate on a code corpus, and popinto four sections: §II gives an overview of our approach and ulate a code database. Each element of the code database algorithms. §III describes the methods in detail. §IV presents consists of a C/C++ function from the corpus along with its our experimental results. §V discusses related work. extracted feature-vector. If Numeric Literals is employed as one of the feature-classes, then one element of a function’s II. Overview feature-vector is the set of numeric constants. Source Forager is a search engine for finding similar source The code database also has access to several similarity code. It takes an input query as C/C++ source text, then functions, one for each feature-class. The similarity function searches a pre-populated database for similar C/C++ code, for a given feature-class takes any two feature-observations returning a ranked list of results. The units of code about belonging to that feature-class and returns a value between 0.0 which Source Forager can reason about are called program and 1.0. A higher value indicates greater similarity between elements. In its current incarnation, program elements are two feature-observations. For example, the similarity function C/C++ functions; that is, both queries and results are C/C++ for Numeric Literals is the Jaccard index. Given two sets S1 functions. and S2 , the Jaccard index is given by: Fig. 1 provides an architectural overview of Source Forager. Source Forager has two stages: an offline phase to populate its \|S1 ∩ S2 \| . (1) simJacc(S1,S2 ) = code database, and an online query-search phase. \|S1 ∪ S2 \| 2 The second implementation integrates our infrastructure with Pliny-DB [10], which is an in-memory object-store database implemented in C++. The feature-observations in feature-vectors are serialized into efficient in-memory data structures by Pliny-DB. Pliny-DB has access to similarity functions implemented in C++ for all feature-classes. It implements the search for the k functions most similar to the query by (1) scanning all the feature-vectors in the database, (2) comparing each of them to the query feature-vector, and (3) maintaining a priority queue of size k that keeps track of the k most-similar feature-vectors. Given a query feature-vector and relative weights for different feature-classes, Pliny-DB can find the 10 most-similar functions in a code database containing 500,000 functions in under 2 seconds on a single machine with 8 Intel i7 3.6 GHz cores and 16 GB RAM. Effort is underway by the developers of Pliny-DB to make a distributed version, which would allow Source Forager to search large code databases without taking a big performance hit: a large code database can be split into p smaller units that can each be searched in parallel, and the sorted k most-similar results from each of the p units can be merged using a multi-way merge algorithm. int bins(int key ,int array [],int min ,int max) { if (max < min) { return KEY_NOT_FOUND; } else { int midpoint = (int)floor (( min+max )/2); if( array[ midpoint ] < key) { return bins(key , array , midpoint +1, max); } else if( array [midpoint ] > key) { return bins(key , array , min , midpoint -1); } else { return midpoint ; } } } Fig. 3. Example Source Forager code-search result for the query in Fig. 2. This result is a recursive implementation of binary search. B. Online Phase: Search for Similar Code In the online search phase, Source Forager takes a query and uses the same feature-extraction infrastructure to obtain the feature-vector that corresponds to the query. This infrastructure reuse creates a consistent representation and view of code throughout the code-search infrastructure. For each featureclass in the feature-vector, a weight is assigned to determine the importance of that feature-class. This feature-class weight determination is based on which configuration Source Forager is run with; sections III-B, III-C and IV-B provide an overview of the different configurations. A combined similarity function is defined on any two feature-vectors by combining the per-feature-class similarity functions with the per-feature-class weight assignment using a weighted average. That is, Íncl sim ( A® , B® ) · wc ® B) ® = c=1 Ícn c c simcombined ( A, , (2) cl w c=1 c C. Extensible Architecture Source Forager’s architecture allows for easy extension. To add a new feature-class, one implements (1) a feature extractor that determines the feature-observation for any given function, and (2) a corresponding similarity function. We currently implement our feature extractors using CodeSonar®. However, Source Forager is not tightly coupled with CodeSonar: any C/C++ processing tool can be used to implement a feature extractor. The feature-observations for all existing feature-classes are represented with well-known container data structures, such as lists, maps, and trees; all similarity functions work at the level of container data structures, and thus are available to be reused with any additional user-supplied feature extractors. Furthermore, Source Forager is not tied to having functions as the only kind of program element. The underlying architecture is also not limited to C/C++, and thus Source Forager can be re-targeted to perform code searches of programs written in other languages. where A® and B® are two feature-vectors; ncl is the total number of feature-classes (i.e, the length of each feature-vector); simc is the similarity function for feature-class c; A®c and B®c ® are the feature-observations for feature-class c in A® and B, respectively; and wc is the weight assigned to feature-class c. The feature-vector of the query is compared with each of the feature-vectors in the code database using this combined similarity function, and the k most-similar functions (that is, with the highest similarity scores to the query) are returned as results (for some configurable limit k). Fig. 3 shows an example Source Forager code-search result when the code in Fig. 2 is used as query. We have two implementations of Source Forager. The first one is a slower-performing version, in which the code database is implemented as a large, in-memory JSON [9] object, and the various similarity functions and the algorithm for k-mostsimilar function-search are implemented in Python. This implementation allows for easier and quicker experimentation with new ideas. We use this version for the experiments reported in §IV. III. Code Search In this section, we first describe the different feature-classes and the accompanying similarity functions that are employed in Source Forager. We then describe two configurations of Source Forager. The first configuration (dyn-select) selects a subset of the feature-classes on a per-query basis for performing code search: this configuration is useful when no additional information is available regarding a code query. The second configuration (svm-weights) pre-computes the relative importance of feature-classes for a specific domain ahead of time using supervised-learning techniques. This configuration is useful when the domain of the code query is known. 3 Seq TABLE I A brief overview of the different feature-classes employed in Source Forager. The marked feature-classes all use Jaccard index (Eq. (1)) as the similarity function. The similarity functions used for the remaining feature-classes accompany their descriptions in §III-A. − negate Loop Seq Feature-Class Brief Description Type–Operation Coupling* types used and operations performed on the types Skeleton Tree structure of loops and conditionals Decorated Skeleton Tree structure of loops, conditionals, and operations Weighted NL Terms processed natural language terms in code 3 Graph CFG BFS CFG subgraphs of size 3, BFS used for generating subgraphs 4 Graph CFG BFS CFG subgraphs of size 4, BFS used for generating subgraphs 3 Graph CFG DFS CFG subgraphs of size 3, DFS used for generating subgraphs 4 Graph CFG DFS CFG subgraphs of size 4, DFS used for generating subgraphs Modeled Library Calls* calls made to modeled libraries Unmodeled Library Calls* calls made to unmodeled libraries User-Defined Library Calls* calls made to user-defined libraries Type Signature input types and the return type Local Types* types of local variables Numeric Literals* numeric data constants used String Literals* string data constants used Comments* associated comment words Seq Loop <= / + Seq Cond Seq Cond < − Seq Cond Seq Cond > + (a) Skeleton Tree (b) Decorated Skeleton Tree Fig. 4. Tree-structured feature-observations for the example program in Fig. 2. The AST is further abstracted by retaining only the loops (for, while, do. . . while) and conditionals (if. . . else, switch). Operationally, the feature extractor can be realized as a tree transducer that drops all AST nodes that are not loops or conditionals. Sequences of loops or conditionals are encapsulated within a sequence node, and empty sequences are dropped from the feature-observation. The intuition behind using this feature-class for code search is that similar functions tend to have similar loop and conditional structures. Fig. 4a shows the Skeleton Tree feature-observation for the example code in Fig. 2. The similarity function used for Skeleton Tree featureobservations is based on tree edit distances. Let dr be a rough approximation of the distance between two trees, only based on their sizes: A. Feature-Classes and Similarity Functions Table I summarizes Source Forager’s feature-classes. Below, we further describe these feature-classes and their associated similarity functions. Type–Operation Coupling: The feature-observation for this feature-class consists of the types of variables operated on in the function, coupled with the operations performed on those types. The feature-observation is a set of (type, operation) pairs. Primitive types are paired with the builtin arithmetic, logical, and relational operations, for example, (int, >=). User-defined types such as C++ classes are paired with the user-defined operations on them, including direct and indirect field accesses and method calls. For example, the pair (Bar, .foo) indicates that the field foo of an aggregate data type Bar is accessed. The intuition behind including this feature-class is that similar functions tend to use similar type–operation pairs. For the example in Fig. 2, the Type– Operation Coupling feature-observation extracted is the set {(int, unary-), (int, /), (int, +), (int, >), (int, +), (int, <=), (int, -), (int, <)}. Skeleton Tree: The feature-observation for this featureclass is based on the abstract syntax tree (AST) of a function. dr (T1, T2 ) = \|size(T1 ) − size(T2 )\| max(size(T1 ), size(T2 )) Further, let DT be a fixed distance threshold (which we set to 0.5). We obtain an approximate distance between two trees, dt as follows:  dr (T1, T2 )   !     ed(pre(T1 ), pre(T2 )), dt (T1, T2 ) = max  ed(post(T1 ), post(T2 ))      max(size(T1 ), size(T2 )) if dr (T1, T2 ) ≥ DT otherwise Here pre(T ) is the sequence obtained by performing a pre-order traversal of the tree T , post(T ) is the sequence obtained by performing a post-order traversal of the tree T , and ed(S1, S2 ) is the word edit distance between the sequences S1 and S2 . The similarity function used for Skeleton Tree feature-observations is then computed as: simtree (T1, T2 ) = 1 − dt (T1, T2 ) 4 (3) An exact tree-edit-distance computation [11] has quartictime complexity in the size of the trees being compared. We instead use a fast under-approximation of edit distance [12] that gives our similarity function quadratic-time complexity overall. Note that we also use a further rough approximation based on just the size of the trees, if one of the two trees being compared is at least twice as large as the other. We found that using these approximations as opposed to the exact tree-editdistance based similarity made no discernible difference in the quality of the final search results obtained, but made a big difference in performance: more than 6× faster in our tests. Decorated Skeleton Tree: This feature-class is similar to the Skeleton Tree, except that instead of retaining just the loop and conditional structure in the feature-observations, most operations (e.g., +, -, and <) are also retained from the AST. We discard some common operations, such as assignment (=) and address-of (&), because they cause excessive bloat. The intuition behind including this feature-class is that similar functions use similar operations in structurally similar locations. Fig. 4b shows the Decorated Skeleton Tree featureobservation for the example code in Fig. 2. The similarity function used is simtree from Eq. (3). Weighted NL Terms: The feature-observations for this feature-class consist of various natural-language (NL) terms in source code, such as function name, comments, local variable names, and parameter names of a function. Such NL terms, after extraction, are subjected to a series of standard NL preprocessing steps, such as splitting words with under_scores or CamelCase, stemming, lemmatization, and removing singlecharacter strings and stop-words. Stop-word removal discards both typical English stop words such as “the”, “and”, and “is” [13], as well as stop words specialized for code, such as “fixme”, “todo”, and “xxx”. Additionally, we use a greedy algorithm [14] for splitting terms into multiple words based on dictionary lookup. This splitting is to handle the case where programmers choose identifiers that combine multiple words without under_scores or CamelCase. After NL pre-processing, we compute a term frequencyinverse document frequency (TF-IDF) score for each NL term. We consider each function as a document, and compute the TF-IDF per C/C++ project. We give function-name terms an inflated score (5× more than other terms) because these often provide significant information about functions’ purposes. The intuition behind including this feature-class is that similar functions tend to have similar natural-language vocabulary. The feature-observation for the example in Fig. 2 is {“bin”: 0.65, “search”: 0.65, “high”: 0.13, “low”: 0.13, “found”: 0.13, “mid”: 0.13, “match”: 0.13}. The similarity function for two observations of Weighted NL Terms uses cosine similarity: Ín i=1 Ai Bi simnl (A, B) = q q Ín 2 Ín B 2 i=1 Ai i=1 i A B C D A B C D A 0 0 0 0 B 1 0 0 0 C 0 1 0 0 D 0 1 0 0 Fig. 5. An example 4-graph and its corresponding adjacency matrix. Serializing the adjacency matrix entries yields binary digits “0100 0011 0000 0000”, or 17,152 in decimal. Node ordering in the adjacency matrix is the traversal order. K-Subgraphs of CFG: We implement multiple featureclasses based on k-sized subgraphs of the control flow graph (CFG) of a function. Given the CFG of a function, we begin either a breadth-first-search (BFS) traversal or a depth-first search (DFS) traversal at a node until k nodes are traversed; a subgraph of the CFG involving these k nodes is extracted. If fewer than k nodes are reachable from a node (including itself), then such a sub-graph is thrown away. We repeat this process for every node in the CFG, extracting at most n subgraphs of size k, where n is the size of the CFG. We represent a graph of size k as a k 2 -bit integer, which is a 1-D representation of a 2-D adjacency-matrix representation of the graph, obtained by concatenating each of the matrix rows in order. Thus, from each function’s CFG, we extract a multiset of k-graph shapes. Fig. 5 shows an example of converting a 4-graph into a 16-bit integer in this manner. We implement the following four feature-classes based on the value of k and the traversal strategy chosen: 3 Graph CFG BFS: k = 3, traversal strategy is BFS. 4 Graph CFG BFS: k = 4, traversal strategy is BFS. 3 Graph CFG DFS: k = 3, traversal strategy is DFS. 4 Graph CFG DFS: k = 4, traversal strategy is DFS. For the example in Fig. 2, the feature-observation extracted for the feature-class “4 Graph CFG BFS” is the multiset {134, 134, 134, 194, 194, 194, 194, 194, 2114, 2114, 2114}. The intuition behind including these feature-classes is that similar functions tend to have similar control-flow structures [15]. The similarity function used for these feature-class is based on the generalized Jaccard index between two multisets O1 and O2 : Í min(O1i, O2i ) simGen-Jacc(O1,O2 ) = Í i (4) i max(O1i, O2i ) Here, i iterates over all the unique elements in O1 ∪ O2 , and O1i is the number of times i appeared in the multiset O1 . Calls to Library Functions: We implement three featureclasses that extract calls to various kinds of library functions: Modeled Library Calls: CodeSonar models a large range of library functions for performing static analysis on C/C++ code. For this feature-class, calls made to any of these modeled library functions are extracted. Unmodeled Library Calls: Calls made to any unmodeled library functions are extracted for this feature-class—that is, calls to a function not modeled by CodeSonar, and whose definition is not available in the source code. Here n is the total number of words in the universe, A and B are vectors with TF-IDF scores, and the i th index Ai is the TF-IDF value for the i th word. 5 User-Defined Library Calls: For this feature-class, calls to B. Dynamic Feature-Class Selection functions whose definitions are available in a directory difCombining feature-classes can be beneficial for code search, ferent from the caller function are extracted. We use such however, the feature-classes that are useful for performing a functions as a heuristic for identifying user-defined libraries. code search may vary from one query to another. For example, The intuition behind including the above three featureconsider a query function containing of just straight-line code. classes is that similar code tends to call the same library A significant number of functions in our code-database are functions. For each of these three feature-classes, the featuredevoid of loops and conditionals,2 and all such functions look values are sets of library functions called. A library function identical to the query function with respect to the Skeleton is represented as tuple: it includes the name of the function Tree feature-class. Thus, performing a code search with this together with the file name containing the function’s declaraquery by including the Skeleton Tree feature-class can lead to tion. For example, if a function calls strcpy and strncpy, lower-quality results. On the other hand, if a query function then the feature-observation corresponding to Modeled Library has an unusual loop and conditional structure that is idiomatic Calls for that function is {(strcpy, string.h), (strncpy, to the computation being performed, then the Skeleton Tree string.h)}. feature-class would be useful in code search: other instances Type Signature: For this feature-class, the featureof the same distinctive structure from the code database would observations consist of the type signature of the function: i.e., have high similarity scores to the query function. the argument types and the return type of a function. Together, Thus, it is useful to select feature-classes automatically on a the argument types and the return type form a multiset of per-query basis for code search. This configuration of Source types. For the example code in Fig. 2, the feature-observation Forager is called dyn-select. Intuitively, a feature-class for a corresponding to Type Signatures is {int, int, int, int}. given query is selected for code search if the corresponding Type signatures define a function’s interface for interaction feature-observation is sufficiently discriminatory/unique with with the rest of the code. Similar code tends to have similar respect to the overall feature-observation distribution for that interfaces, and therefore type signatures could help with code feature-class. search. To prepare for the dynamic feature-class selection on a perThe generalized Jaccard index (Eq. (4)) is used as the query basis, we take following steps offline: similarity function for this feature-class. • From the code database, we retrieve a random sample S Local Types: For this feature-class, the featureof feature-vectors. Random sampling gives an inexpensive observations consist of the set of types of all the local variables. estimate of feature-observation distributions across the entire The intuition behind using local variable types in code search code database. is that similar code creates and operates on variables of similar types. For the example code in Fig. 2, the Local Types feature- • We calculate a similarity threshold for each feature-class c by (1) computing pairwise similarity scores on the featureobservation is {int}. observations for c in S and (2) taking the sum of means Constants: We implement two feature-classes that extract and standard deviations of the similarity scores. Two featureconstants from a function: observations for c are considered similar if their similarity Numeric Literals: This feature-class is described in §II-A. score is above the similarity threshold for c. String Literals: For this feature-class, a feature-observation Online, when a query is posed, we take the following steps is the set of all the literal strings used in a function. The intuition behind using sets of constants in code search is for each feature-class c (which can be performed in parallel): • We compare the query’s feature-observation for c with all that similar code typically uses similar constants. other feature-observations for c in sample S (of size nsamp ), Comments: For this feature-class, the featureand count the number of similar feature-observations nsim-c. observations consist of the comments associated with a function. The comments are represented as a set of words. • We select the feature-class c for code search if it is not too < tuniq . Here tuniq is a threshold common, that is, if nnsim-c The intuition behind using comments in code search is that samp the comments in similar pieces of code are likely to use that indicates a feature-observation is sufficiently unique in a similar vocabulary. For the example code in Fig. 2, the the sample. For example, tuniq = 0.15 indicates that any Comments feature-observation is {“found”, “match”, “no”}. feature-observation that is similar to less than 15% of the Combining Feature-Classes: Using several featuresample feature-observations is considered distinctive enough classes in combination allows Source Forager to obtain good to warrant inclusion. code-search results in a fairly robust manner by using different Each feature-class is assigned a weight of exactly 1.0 or dimensions of the code. For example, consider the binary- exactly 0.0, based on whether the feature-class is selected search implementation in Fig. 2. We see that variables named in the above process. These weights are used for combining mid, low, high are used; that there are two conditionals feature-class similarities for code search (Eq. (2)), and the knested inside a single loop; and that an integer division and most-similar-function search is carried out between the query integer less-than-or-equal-to operation is performed. When put together, these observations are hallmarks of a binary-search 2We did a brief study of feature-observation distributions for the Skeleton Tree feature-class over our corpus, which revealed this data point. implementation. 6 function and the functions in the code database as described in §II-B, to obtain the k functions most similar to the query. TABLE II Task categories used for code-search queries in algo-qs. “#Similar” gives the number of similar functions that were manually found for a given task category. “Partial” reports how many function pairs MOSS considered to be potential clones. “Significant” reports the % of function pairs with at least 50% code overlap. C. SVM-Guided Feature-Class Weight Generation Note that dyn-select does not need any additional knowledge about the query. However, if we know ahead of time that a query belongs to a specific domain, and we have ground-truth information available regarding what constitutes similar code in that domain, then we can use supervised-learning techniques to learn good feature-class weights (for Eq. (2)) for that domain ahead of time, and use these weights for code search with all future queries in that domain. Given a particular ground-truth data set with labeled similar code, we generate fine-tuned weights by training a binaryclassification support vector machine (SVM). We do not train using raw code text, or even raw sets of feature-observations. Because we use the SVM training process to generate relative weights for feature-class similarity scores in Eq. (2), we train the SVM on these similarity scores directly. The similarity scores for all feature-classes between two functions are assembled into a similarity vector. The SVM is then trained on examples of similarity vectors for both similar and dissimilar functions, each labeled accordingly. This technique allows us to optimize ahead of time how these feature-classes are relatively weighted in a code search, by using the same similarity functions that are employed in code search of a query. Our SVM uses a linear classifier, which allows a convenient interpretation of internal weights [16]. The final pre-processing step is to extract these internal weights and normalize them relative to the sum of their magnitudes, truncating negative weights. These normalized weights are then used directly as feature-class weights in Eq. (2). §IV-B provides more details about the corpus and training process. Of course, it is not obvious that weights obtained by training for classification purposes are useful in ranking results for code-search queries. §IV-C measures the effectiveness of this strategy in practice. MOSS Detected Task Category Binary Search Edit Distance Insertion Sort Knapsack Modular Exponentiation Non Recursive Depth First Search Red Black Tree Left Rotate #Similar Partial Significant 5 5 5 5 6 5 6 20% 40% 30% 10% 0% 0% 40% 10% 0% 30% 0% 0% 0% 13% A code-search task involves searching for relevant documents from a group of documents that include both relevant and non-relevant documents. (In the case of Source Forager, “documents” are C/C++ functions.) Non-relevant documents are also known as distractors, which leads naturally to the following question: RQ4 How much does Source Forager’s code-search performance degrade as we increase the number of distractors in the code base being searched? B. Experimental Setup and Methodology Source Forager uses CodeSonar, an industrial-strength C/C++ static-analysis engine, to analyze C/C++ corpora and implement feature extractors. CodeSonar handles real-world C/C++ projects with tens of millions of lines of code. CodeSonar also exposes a wealth of information about a program through well-defined APIs. Source Forager’s feature extractors are implemented as CodeSonar plugins that use these APIs. Consequently, Source Forager inherits CodeSonar’s requirement that programs must be compilable to be analyzable. Code-Search Tasks: Our experiments assess Source Forager’s performance under various configurations. Code-search tasks are set up as follows. For each query function, there is a set of known relevant functions that are similar to the query. The relevant functions are treated as ground truth. The relevant functions are then mixed with many non-relevant functions as distractors, and together they form the code database used in the experiment. Source Forager then searches the code database for similar functions. We compute informationretrieval statistics based on the ranking of the known-relevant functions in the returned results. Queries: We use two query ground-truth sets for the code-search tasks, representing two domains. One, called algo-qs, represents “algorithmic” code queries. For algo-qs, we created seven tasks, outlined in Table II, and manually curated a total of thirty-eight functions that each accomplish one of the seven tasks. The functions were mostly obtained from GitHub, and were written by a variety of programmers, none of whom are authors of this paper. The functions that accomplish a specific task have been manually vetted to be IV. Experimental Evaluation This section outlines the research questions we seek to answer through experiments (§IV-A); describes the setup and methodology used in the experiments (§IV-B); and presents the results of the experiments (§IV-C). A. Research Questions Our experiments were designed to answer the following research questions: RQ1 How do the individual feature-classes described in §III perform in code-search tasks relative to each other? RQ2 Does combining feature-classes using per-query dynamic feature-class selection (§III-B) improve Source Forager’s performance? RQ3 Does combining feature-classes using supervised learning (§III-C) further improve Source Forager’s performance, when the query domain is known? 7 similar to each other. We thus have a total of thirty-eight base queries. We use these sets of real-world functions as queries (and the desired search results), and consider them to be an appropriate proxy for the code-search queries performed (and search results expected) by users in the algorithm domain. We have made the labeled queries available for inspection.3 To make sure that the similar functions we found were not all clones of each other, we ran them through the MOSS software-plagiarism detector [17]. Given a group of programs, MOSS reports program pairs that may be clones, along with an overlap percentage. Table II reports MOSS’s findings, run using default settings. In this table, partial overlap represents any pair that MOSS reports as possible clones, while significant overlap counts only possible clones with at least 50% overlap. Observe that many function pairs marked manually as being similar are not just MOSS-detectable clones of each other. Thus, recognizing similar function pairs in this corpus is a nontrivial challenge. The second query ground-truth set we use is called libc-qs, and represents code queries from systems programming. We looked at three implementations of the standard C library: musl libc [18], diet libc [19], and uClibc [20]. From these we define 88 function categories corresponding to 88 functions that all three implementations provide. We assume that within the same function category, the three libc implementations are “similar.” For this domain, we have 88×3 queries. For example, musl libc’s sprintf is labeled to be similar to diet libc’s sprintf and uClibc’s sprintf, and dissimilar to everything else. Distractor Functions: The distractor functions have been taken from the openly available MUSE corpus [21], and mainly consist of code from Fedora source packages (SRPMs). Our feature extractors currently require compilable code, which Fedora SRPMs provide. Due to the large size of the distractorfunction corpus (over 200,000), we have not manually vetted all of the distractor functions to be sure that they are irrelevant to the queries issued. It is possible that some distractor functions are indeed relevant to some queries, so our retrieval statistics are under-approximations. With the exception of the experiments reported in Fig. 7, all experiments use 10,000 distractors. Retrieval Statistics: We compute Mean Average Precision (MAP) as the retrieval statistic, as is common in information retrieval. MAP is typically used to measure the quality of ranked retrieval results, because MAP takes into account the rank of the relevant documents in the retrieved results. MAP provides a measure of quality across all recall levels. MAP is the mean of the average precision computed for each query. The average precision (AP) for each query is Í given by nk=1 P(k) · r(k) /R, where n is the total number of documents searched; R is the number of documents marked relevant to the query; P(k) is the precision when k documents are requested; and r(k) is 1 when the k th retrieved document is relevant, and 0 otherwise. That is, AP is the average precision at all the points when a new relevant document is retrieved in a ranked result list. The best MAP score that can be achieved is 1.0, when for each query, the R relevant documents appear as the top R search results. SVM-Guided Weights: We applied the techniques discussed in §III-C on algo-qs and libc-qs to provide labeled data-sets on which to train an SVM. Each instance in our training set is generated by comparing two functions a and b, yielding a single similarity vector that consists of similarity scores for each feature-class. The binary classification for each training instance is 1.0 if a and b are implementations of the same function, 0.0 otherwise. We use LIBLINEAR [22] to train the SVM to classify these function comparisons; this process takes roughly twenty milliseconds. Using this technique, we are able to achieve over 98% accuracy under ten-fold cross-validation. Once the SVM is trained, we extract and normalize its internal weights for use in code search. For the svm-weights configuration described below, within each domain, the dataset is divided into multiple folds of training-set and test-set pairs. The weights extracted from the training set are used to obtain MAP scores on the test set. That is, weights are trained on a subset of a given domain (algo-qs or libc-qs) and tested using queries from a different subset of the same domain. For the cross-weights configuration described below, algo-qs is used to train weights for queries from libc-qs, and vice-versa. Source Forager Configurations: Our experiments run Source Forager under many configurations. Each configuration is defined by the weight wc assigned to each of the featureclasses c given in Table I. These weights are used in Eq. (2) for performing the code search. solo-c: For each query, the weight wc corresponding to feature-class c is 1.0. Weights corresponding to all other feature-classes are set to 0.0. equal-all: For each query, for all feature-classes c, wc = 1.0, giving equal importance to all feature-classes for all queries. dyn-select: For each query, a subset of feature-classes are selected and given equal weights, as described in §III-B. The dynamic selection of feature-classes adds a small run-time overhead to each query.4 rand-select: For each query, a new random configuration is used as follows: a random subset of the feature-classes is selected, and the selected feature-classes are given equal weights. Repeat this process 10 times with different random selections, and report mean results over these 10 trials. svm-weights: For each query, use weights learned for the domain that the query belongs to, as described in §III-C and above. cross-weights: For each query, use weights learned for the domain that the query does not belong to. 4In our naive Python implementation, dyn-select adds an average run-time overhead of 2.1 seconds per query for dynamic selection of feature-classes. Currently, the selection decision on each feature-class is done sequentially, instead of in parallel as suggested in §III-B. 3Available at the URL: http://tinyurl.com/source-forager-algo-benchmarks. 8 Note that, unlike the other configurations, the svm-weights and cross-weights configurations permit weights to give different (non-zero) importance levels to different feature-classes. RQ3 Finding: When the domain of a query is known, and training data is available, combining multiple featureclasses using weights derived from supervised learning (§III-C) is the most effective strategy for code search. C. Results and Discussion The cross-weights (0.74, 0.85) configuration tests whether the weights learned from one domain are useful in a different The left side of Fig. 6 shows how each individual featuredomain. The rightmost two bars in Fig. 6 show that it is class performs on the code-search tasks in isolation. This hard to derive a single set of relative feature-class weights that experiment addresses RQ1. The solo feature-class Weighted work well for queries in both domains. Thus, in the absence of NL Terms (0.70, 0.86)5 performs the best individually on both domain information about the query, dyn-select is preferred. algo-qs and libc-qs. Thus: Fig. 7 shows how Source Forager’s result quality scales with increasing distractor-set sizes. This experiment addresses RQ1 Finding: If we were to drive Source Forager using RQ4. Source Forager is used in the dyn-select and svm-weights only one feature-class, Weighted NL Terms is the best configurations for this experiment. As one would expect, MAP option. However, Fig. 6 shows that the performance of the scores decline as distractors proliferate. However, consider different feature-classes varies considerably depending on that relevant sets contain just 2 to 6 items competing against the query ground-truth set. This variance suggests that distractor sets that are up to five orders of magnitude larger. different feature-classes are important for different kinds of queries. RQ4 Finding: Resilient MAP scores indicate that Source Forager returns high-quality results even when distractors outnumber relevant items by several orders of magnitude. RQ2 asks whether multiple feature-classes can be usefully combined, and whether dyn-select is a good way to do such a combination. A straight-forward manner in which featureclasses can be combined is the equal-all configuration, which represents a baseline to compare against other configurations. The dyn-select configuration selects different subsets of the feature-classes on a per-query basis (§III-B). As a sanity check for the selections performed by dyn-select, we also compare it with the rand-select configuration, which randomly selects feature-class subsets for every query. The right side of Fig. 6 shows that dyn-select (0.84, 0.89) performs better on both algo-qs and libc-qs when compared to equal-all (0.67, 0.73) and rand-select (0.57, 0.63). dyn-select also outperforms each of the solo configurations from the left side of Fig. 6. D. Threats to Validity The issue of whether evaluation benchmarks are appropriate is a potential threat to the validity of any information retrieval system. We mitigate this threat for Source Forager in several ways. First, we use benchmark queries from two different domains, algo-qs and libc-qs. Second, we use the MOSS plagiarism detector to show that our manually labeled set of relevant functions in algo-qs are not trivial clones of each other. Third, we draw the algo-qs and libc-qs data sets from real-world code written by arbitrary programmers, not artificial programs written by us. Feature-classes can be combined in various ways to perform code searches. We have explored part of the vast space of all such combinations, and our results speak only to those we have tried. We find that the MAP scores of the configuration dynselect on both algo-qs and libc-qs are good. We designed the experiments with equal-all and rand-select configurations to test whether the selections made by dyn-select are indeed necessary and useful, and find that they are. RQ2 Finding: In the absence of any additional information about the query, combining multiple feature-classes and dynamically selecting feature-classes on a per-query basis (§III-B) is the most effective strategy for code search. RQ3 addresses the scenario where the domain of a query is known, and additional information is available regarding that domain (as described in §III-C). The svm-weights configuration tests Source Forager under this scenario. Pre-learning the relative importance of feature-classes for a given domain (in the form of weights wc for each feature-class) also makes code search more efficient by eliminating any run-time overhead in feature-class selection. The right side of Fig. 6 shows that svmweights (0.86, 0.95) outperforms all other configurations. V. Related Work Code-search engines: Several popular text-based codesearch tools “grep” over tokenized source code: GitHub, SearchCode, Open HUB, etc. While these tools are useful, they fall short in many use cases, as they do not exploit the rich semantics of code. For example, the top search results for the term “dfs” on C code projects in GitHub yields function declarations, macro names, and #include directives that mention “dfs”, but that are not actually useful. The Sourcerer code-search engine [2] combines text-based search techniques with information about relations among programming “entities” like packages, classes, methods, and fields. 5Pairs of numbers following a configuration in this section indicates the MAP scores of that configuration on algo-qs and libc-qs, respectively. 9 0.95 0.85 0.74 0.40 0.09 0.00 0.00 0.03 0.09 0.01 0.04 0.00 0.07 User-Defined Library Calls 0.18 0.01 0.00 Unmodeled Library Calls 0.04 0.04 0.03 0.06 0.03 0.21 0.25 0.07 0.12 0.04 0.12 0.14 0.01 0.2 0.31 0.4 0.86 0.84 0.89 0.67 0.73 0.58 0.70 0.45 0.6 0.19 MAP score 0.8 algo-qs libc-qs 0.57 0.63 0.86 1 cross-weights svm-weights dyn-select rand-select equal-all Comments String Literals Numeric Literals Local Types Type Signatures Modeled Library Calls 4 Graph CFG DFS 3 Graph CFG DFS 4 Graph CFG BFS 3 Graph CFG BFS Weighted NL Terms Decorated Skeleton Tree Skeleton Tree Type–Operation Coupling 0 Fig. 6. Information retrieval performance with 10,000 distractors. The left side of the plot, from “Type–Operation Coupling” through “Comments”, uses the solo-c configuration with the given feature-class as the only non-zero-weighted feature. The right side of the plot, from “equal-all” through “cross-weights”, uses the various other Source Forager configurations that leverage multiple feature-classes simultaneously. Although no MAP score is exactly zero, several are below 0.005 and therefore round to “0.00” in the data-point labels above. 0.87 0.70 0.91 0.93 0.75 0.94 0.84 0.79 0.96 0.97 0.88 0.86 0.98 0.99 0.90 0.99 0.95 0.93 0.98 0.99 0.95 0.92 1.00 0.96 100 200 400 800 1,600 3,200 6,400 12,800 25,600 51,200 102,400 204,800 MAP score Strathcona [8] returns relevant Java code examples to developers learning to use complex object-oriented frameworks. It 1 uses several heuristics based on class-inheritance hierarchies, method calls, and type uses. Source Forager could also use the 0.8 applicable heuristics from Sourcerer and Strathcona as featureclasses, but additionally demonstrates how to search using more complex structures, such as decorated skeleton trees and 0.6 CFG-subgraphs. CodeGenie [7], [23] proposes test-driven code search, in which the user supplies a set of unit tests for the code compo0.4 nent they want to find. CodeGenie leverages Sourcerer [2] to svm-weights with libc-qs perform keyword-based search; test cases refine these results. dyn-select with libc-qs Source Forager could be used as a replacement for Sourcerer 0.2 svm-weights with algo-qs to perform similar code search in CodeGenie. dyn-select with algo-qs Stollee et al. [4], [24] perform code search based on logical 0 characterizations of programs’ I/O behaviors, obtained via symbolic execution. A query consists of concrete I/O pairs for the desired code fragment. While this approach precisely captures the semantics of the corpus elements, it does not imNumber of Distractor Functions mediately handle some common programming constructs, such as loops and global variables. It also restricts the size of the Fig. 7. Impact of the number of distractor functions on MAP scores using dynprogram elements in the corpus, because symbolic execution select and svm-weights for all algo-qs and libc-qs queries. The horizontal axis is on a log scale. of larger elements may lead to path explosion. Source Forager can easily be extended to use I/O pairs as an additional featureclass in scenarios where the above restrictions are acceptable. Sourcerer also uses fingerprints that capture some light-weight XSnippet [5] and ParseWeb [25] are specialized code-search structural information about the code, such as depth of loop engines: XSnippet looks specifically for code that instantiates nesting and presence or absence of certain language constructs. objects of given type in a given context, ParseWeb has a similar Queries in Sourcerer are text-based and are powered by Lucene focus on code sequences that instantiate objects. Codify [6] (http://lucene.apache.org), as opposed to the code-based search extracts and stores a large amount of metadata for each symbol by Source Forager. in a program, and provides a user interface for querying that 10 metadata. Codify aids in understanding and browsing code. The goal of Source Forager’s code search is different from the above, i.e., to find source code similar to a query. Code-clone detection: Source Forager’s code searches differ from the typical clone detection problem in that we are interested in finding code that has both semantic and syntactic similarity. Therefore, we use a range of feature-classes that span from syntactic to semantic. Source Forager’s notion of similarity does not neatly fall into any of the definitions of standard clone types 1–4 [26]. Similar-machine-code search: Finding similar machine code [15], [27]–[29] is useful in finding known vulnerabilities in third-party code for which source code is not available. The primary difference in code search at the source-level and machine-level is that machine code has poorer syntactic, semantic, and structural information available compared to source code. As a result, while there is some overlap between techniques, research on machine-code search is focused on tackling different problems, such as how to do similar-machinecode search across different CPU architectures, compiler optimizations, compilers, operating systems, etc. SVM-based code-classification: Rosenblum et al. [30] train SVMs with features extracted from source code in the attempt to classify programs by author. Source Forager builds on this idea by training an SVM with similarity scores derived from feature-observations, and then extracting internal weights from the trained SVM to strengthen the combined similarity function used for code search. References [1] C. Sadowski, K. T. Stollee, and S. G. Elbaum, “How developers search for code: a case study,” in Found. of Softw. Eng., 2015. [2] E. Linstead, S. K. Bajracharya, T. C. Ngo, P. Rigor, C. V. Lopes, and P. Baldi, “Sourcerer: Mining and searching internet-scale software repositories.” Data Mining and Knowledge Discovery, vol. 18, no. 2, pp. 300–336, Apr. 2009. [3] S. P. Reiss, “Semantics-based code search,” in Int. Conf. on Softw. Eng., 2009, pp. 243–253. [4] K. T. Stollee, S. G. Elbaum, and D. Dobos, “Solving the search for source code,” Trans. on Softw. Engineering and Methodology, vol. 23, no. 3, May 2014. [5] N. Sahavechaphan and K. T. Claypool, “XSnippet: mining for sample code,” in Conf. on Object-Oriented Prog. Systems, Languages, and Applications, 2006, pp. 413–430. [6] A. Begel, “Codifier: A programmer-centric search user interface,” in Workshop on Human-Computer Interaction and Inf. Retrieval, 2007. [7] O. Lemos, B. K. Bajracharya, J. Ossher, R. S. Morla, P. C. Masiero, P. Baldi, and C. V. Lopes, “Codegenie: using test-cases to search and reuse source code,” in Int. Conf. on Automated Softw. Eng., 2007. [8] R. Holmes and G. C. Murphy, “Using structural context to recommend source code examples,” in Int. Conf. on Softw. Eng., 2005. [9] D. Crockford, “Introducing JSON,” Apr. 2016. [Online]. Available: http://json.org/ 11 [10] C. Jermaine, “The pliny database,” Aug. 2016. [Online]. Available: http://cmj4.web.rice.edu/PlinyDBSlides.pdf [11] K. Zhang and D. Shasha, “Simple fast algorithms for the editing distance between trees and related problems,” SIAM J. Comput., vol. 18, no. 6, Dec. 1989. [12] S. Guha, H. Jagadish, N. Koudas, D. Srivastava, and T. Yu, “Approximate XML joins,” in Int. Conf. on Management of Data. ACM, 2002, pp. 287–298. [13] NLTK Project, “Stopwords corpus,” Mar. 2016. [Online]. Available: http://www.nltk.org/nltk_data [14] H. Feild, D. Binkley, and D. Lawrie, “An empirical comparison of techniques for extracting concept abbreviations from identifiers,” in Proc. IASTED Int. Conf. on Software Engineering and Applications (SEA). Citeseer, 2006. [15] W. M. Khoo, A. Mycroft, and R. 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This is a pre-print of the conference paper accepted at the IEEE Winter Conference on Applications of Computer Vision (WACV) 2018. Towards Robust Deep Neural Networks with BANG Andras Rozsa, Manuel Günther, and Terrance E. Boult Vision and Security Technology (VAST) Lab University of Colorado, Colorado Springs, USA arXiv:1612.00138v3 [cs.CV] 30 Jan 2018 {arozsa,mgunther,tboult}@vast.uccs.edu Abstract Machine learning models, including state-of-the-art deep neural networks, are vulnerable to small perturbations that cause unexpected classification errors. This unexpected lack of robustness raises fundamental questions about their generalization properties and poses a serious concern for practical deployments. As such perturbations can remain imperceptible – the formed adversarial examples demonstrate an inherent inconsistency between vulnerable machine learning models and human perception – some prior work casts this problem as a security issue. Despite the significance of the discovered instabilities and ensuing research, their cause is not well understood and no effective method has been developed to address the problem. In this paper, we present a novel theory to explain why this unpleasant phenomenon exists in deep neural networks. Based on that theory, we introduce a simple, efficient, and effective training approach, Batch Adjusted Network Gradients (BANG), which significantly improves the robustness of machine learning models. While the BANG technique does not rely on any form of data augmentation or the utilization of adversarial images for training, the resultant classifiers are more resistant to adversarial perturbations while maintaining or even enhancing the overall classification performance. (a) MNIST Samples and Their Distortions Yielding Misclassifications (b) CIFAR-10 Samples and Their Distortions Yielding Misclassifications Figure 1: I MPROVING ROBUSTNESS VIA BANG. This figure demonstrates the enhanced robustness against perturbations generated via the non-gradient-based hot/cold adversarial generation method on MNIST digits and CIFAR-10 samples displayed in top rows of (a) and (b). Underneath the raw test images, we show their distorted versions formed by the smallest perturbations that change the correctly classified class labels of the test samples. The second rows of (a) and (b) present perturbations that we obtained on regularly trained learning models, while the last rows show examples that we generated on networks trained via our Batch Adjusted Network Gradients (BANG) approach. As indicated by most of the perturbations being highly perceptible, the learning models trained with BANG have become more robust to adversarial perturbations. 1. Introduction Machine learning is broadly used in various real-world vision applications and recent advances in deep learning have made deep neural networks the most powerful learning models that can be successfully applied to different vision problems [27, 25, 7, 28, 20, 14, 18, 17, 29]. The recent performance gain is mainly the result of improvements in two fields, namely, building more powerful learning models [25, 7] and designing better strategies to avoid overfitting [24]. These advancements are then leveraged by the use of larger datasets and massive GPU-enhanced computing. Although deep neural networks (DNNs) achieve state- of-the-art performance in a wide range of tasks, the generalization properties of these learning models were questioned by Szegedy et al. [26] when the existence of adversarial examples was revealed. DNNs are capable of learning high-level feature embeddings that enable them to be successfully adapted to different problems. They were generally considered to generalize well and, hence, expected to be robust to moderate distortions to their inputs. Surprisingly, adversarial examples formed by applying imperceptible perturbations to otherwise correctly recognized inputs can lead machine learning models – including state-of-the- art DNNs – to misclassify those samples, often with high confidence. This highly unexpected and intriguing property of machine learning models highlights a fundamental problem that researchers have been trying to solve. To explain why adversarial examples exist, several controversial explanations were proposed. As hypothesized in [4, 2], adversarial instability exists due to DNNs acting as high-dimensional linear classifiers that allow even imperceptibly small, well-aligned perturbations applied to inputs to spread among higher dimensions and radically change the outputs. This belief was challenged in [19], where – by analyzing and experimenting with DNNs trained to recognize objects in more unconstrained conditions – it was demonstrated that those classifiers are only locally linear to changes on the recognized object, otherwise DNNs act nonlinearly. After performing various experiments, Gu et al. [6] concluded that adversarial instability is rather related to “intrinsic deficiencies in the training procedure and objective function than to model topology.” The problem addressed in this paper is not only about preventing attacks via adversarial examples, the focus is on the overall robustness and generalizability of DNNs. This fundamental problem of deep learning has recently received increasing attention by researchers [3, 8, 21]. Considering state-of-the-art learning models applied to computer vision tasks, the classification of many incorrectly or uncertainly recognized inputs can be corrected and improved by small perturbations [30, 22], so this is a naturally occurring problem for learning-based vision systems. In this paper, we introduce our theory on the instability of machine learning models and the existence of adversarial examples: evolutionary stalling. During training, network weights are adjusted using the gradient of loss, evolving to eventually classify examples correctly. Ideally, we prefer broad flat regions around samples to achieve good generalization [11] and adversarial robustness [2]. However, after a training sample is correctly classified, its contribution to the loss and, thus, on forming the weight updates is reduced. As the evolution of the local decision surface stalls, the correctly classified samples cannot further flatten and extend their surroundings to improve generalization. Therefore, as the contributions of those correctly classified training samples to boundary adjustments are highly decreased compared to other batch elements, samples can end up being stuck close to decision boundaries and, hence, susceptible to small perturbations flipping their classifications. To mitigate evolutionary stalling, we propose our Batch Adjusted Network Gradients (BANG) training algorithm. We experimentally evaluate robustness using a combination of gradient- and non-gradient-based adversarial perturbations, and random distortions. The paper explores the impact of BANG parameters and architectural variations, such as Dropout [24], on instability and adversarial robust- ness. In conclusion, we validate our theory by experimentally demonstrating that BANG significantly improves the robustness of deep neural networks optimized on two small datasets while the trained learning models maintain or even improve their overall classification performance. 2. Related Work Deep neural networks (DNNs) achieve high performance on various tasks as they are able to learn non-local generalization priors from training data. Counter-intuitively, Szegedy et al. [26] showed that machine learning models can misclassify samples that are formed by slightly perturbing correctly recognized inputs. These so-called adversarial examples are indistinguishable from their originating counterparts to human observers, and their unexpected existence itself presents a problem. The authors introduced the first technique that is capable of reliably finding adversarial perturbations and claimed that some adversarial examples generalize across different learning models. A computationally cheaper adversarial example generation algorithm, the Fast Gradient Sign (FGS) method, was presented by Goodfellow et al. [4]. While this approach also uses the inner state of DNNs, it is more efficient as FGS requires the gradient of loss to be calculated only once. The authors demonstrated that by using adversarial examples generated with FGS implicitly in an enhanced objective function, both accuracy and robustness of the trained classifiers can be improved. In their paper focusing on adversarial machine learning, Kurakin et al. [13] proposed new algorithms extending the FGS method to target a specific class and to calculate and apply gradients iteratively instead of a single gradient calculation via FGS. The authors compared the effect of different types of adversarial examples used for implicit adversarial training and found that the results vary based upon the type of the applied adversarial examples. Rozsa et al. [23] introduced the non-gradient-based hot/cold approach, which is capable of efficiently producing multiple adversarial examples for each input. They demonstrated that using samples explicitly with higher magnitudes of adversarial perturbations than the sufficient minimal can outperform regular adversarial training. The authors also presented a new metric – the Perceptual Adversarial Similarity Score (PASS) – to better measure the distinguishability of original and adversarial image pairs in terms of human perception. As the commonly used L2 or L∞ norms are very sensitive to small geometric distortions that can remain unnoticeable to us, PASS is more applicable to quantify similarity and the quality of adversarial examples. Although adversarial training, both implicit and explicit, was demonstrated to decrease the instability of learning models, forming those examples is still computationally expensive, which limits the application of such techniques. Furthermore, considering the various adversarial generation techniques, utilizing certain types of those samples might not lead to improved robustness to adversarial examples of other techniques. Alternatively, Zheng et al. [30] proposed their stability training as a lightweight and still effective method to stabilize DNNs against naturally occurring distortions in the visual input. The introduced training procedure uses an additional stability objective that makes DNNs learn weights that minimize the prediction difference of original and perturbed images. In order to obtain general robustness and not rely on any class of perturbations, the authors applied Gaussian noise to distort the training images. Gu et al. [6] conducted experiments with different network topologies, pre-processing, and training procedures to improve the robustness of DNNs. The authors proposed the Deep Contractive Network (DCN), which imposes a layerwise contractive penalty in a feed-forward DNN. The formulated penalty aims to minimize output variances with respect to perturbations in inputs, and enable the network to explicitly learn flat, invariant regions around the training data. Based on positive initial results, they concluded that adversarial instability is rather the result of the intrinsic deficiencies in the training procedure and objective function than of model topologies. Luo et al. [19] proposed a foveation-based technique that selects and uses only a sub-region of the image during classification. As the authors demonstrated, the negative effect of foveated perturbations to the classification scores can be significantly reduced compared to entire perturbations. Graese et al. [5] showed that transformations of the normal image acquisition process can also negate the effect of the carefully crafted adversarial perturbations. While these preprocessing techniques can alleviate the problem posed by adversarial images, they do not solve the inherent instability of DNNs. In other words, these methods treat the symptoms and not the disease. In summary, a wide variety of more or less efficient approaches were proposed in the literature that all aim at improving the robustness and generalization properties of DNNs, but none of those proved to be effective enough. 3. Approach In this section, we first briefly describe our intuition about why the unexpected adversarial instability exists in machine learning models. Afterwards, we present our simple and straightforward modification in the training procedure that aims to optimize weights in a way that the resulting DNNs become more robust to distortions of their inputs. 3.1. Intuition During training, some inputs in the batch are correctly and others are incorrectly classified. In general, the calculated loss and, thus, the gradient of loss for the misclassified ones are larger than for the correctly classified inputs of the same batch. Therefore, in each training iteration most of the weight updates go into learning those inputs that are badly predicted. On the other hand, the correctly classified samples do not have a significant impact on advancing decision boundaries and can remain in the positions close to what they obtained when becoming correctly classified. Due to this evolutionary stalling, samples with low gradients cannot form a flatter, more invariant region around themselves. Consequently, samples of those regions remain more susceptible to adversarial perturbations – even a small perturbation can push them back into an incorrect class. By increasing the contribution of the correctly classified examples in the batch on the weight updates, and forcing them to continue improving decision boundaries, it is reasonable to think that we can flatten the decision space around those training samples and train more robust DNNs. 3.2. Implementation The core concept of our Batch Adjusted Network Gradients (BANG) approach is a variation of batch normalization [9]. However, rather than trying to balance the inputs of the layers, we seek to ensure that the contributions on the weight updates are more balanced among batch elements by scaling their gradients. Let us dive into the details and introduce our notations we use to formulate BANG. In short, we scale the gradients of batch elements that will be used to compute the weight updates in each training iteration. Let us consider a network fw with weights w in a layered structure having layers y (l) where l ∈ [1, L], with their respective weights w(l) : fw (xi ) = y (L) y (L−1) . . . y (1) (xi ) . . . . (1) For a given input xi , the partial derivatives of the loss E(fw , xi ) with respect to the output of layer y (l) are: (l) κi = κ(l) (xi ) = ∂Ei (l) . (2) ∂yi For simplicity, we leave out the structure of the weights w(l) in layers and the structure of the layer outputs which can be either one-dimensional for fully connected layers or threedimensional for convolutional layers. With BANG, our goal is to balance gradients in the batch by scaling up those that have lower magnitudes. In order to do so, we determine the highest gradient for the batch having N inputs xi , i ∈ {1, . . . , N } at given layer y (l) in terms of L2 norm. We use that as the basis for balancing the magnitudes of gradients in the batch. Weight updates are calculated after scaling each derivative κi in the batch with the element-wise learning rate:  ρi(l) (l) max kκ 0 k i  i0 ∈[1,N ]  (l) ηi =  (3)  (l) kκi k where:    (l) ρi = (l) 1 − (l) kκi k max 0 i ∈[1,N ]  (l)  kκi0 k . (4) As a key parameter for our approach, (l) specifies the degree of gradient balancing among batch elements. While the (l) exponent ρi might appear a little complex and ambiguous, its sole purpose is to scale up gradients with small magnitudes more than others having larger L2 norms. Assuming that the regular backward pass combines the gradients of the batch elements by calculating: ∇fw(l) N N 1 X (l) ∂y (l) 1 X ∂Ei = κ = N i=1 ∂w(l) N i=1 i ∂w(l) (5) which is normally scaled with the learning rate and then used to update weights (after combining with the previous weight update scaled with momentum), BANG produces: ∇fw(l) = β (l) N 1 X (l) ∂Ei η , N i=1 i ∂w(l) (6) where β (l) is the second (set of) parameter(s) of our approach used for scaling. In general, β (l) acts as a local learning rate that can play a more important role in future work. Throughout our experiments, we keep BANG parameters fixed for all layers: (l) = and β (l) = β (which will actually just modify the original learning rate η). Note that although our approach changes the actual calculation of weight updates for the layers, there is no impact on the backpropagation of the original gradient down the network. Finally, we implemented BANG by applying small modifications to the regular training procedure with negligible computational overhead. 4. Experiments To evaluate our approach, we conducted experiments on the slightly modified versions of LeNet [16] and “CIFAR10 quick” models distributed with Caffe [10]. Namely, after running preliminary experiments with BANG, we added a Dropout layer [24] to both model architectures that serves multiple purposes. We observed that BANG tends to cause overfitting on the trained LeNet networks, and the resultant models made very confident classifications – even when they misclassified the test images. While the additional Dropout layer alleviates both problems, the adjusted network architectures also result in improved classification performances with both regular and BANG training. After obtaining learning models with regular and BANG training, we assess and compare the robustness of those classifiers in two ways. It is important to note that we do not select the best training models based on their performance on the validation set for these evaluations, but we simply use the models obtained at the last training iteration. As our primary goal is to measure the evolving robustness, we believe that this decision leads to a fairer comparison, however, the classification performance of the selected models are not optimal. Finally, we would like to mention that we conducted experiments to discover the effectiveness of BANG used for fine-tuning regularly trained models, and found that the robustness of the resultant networks are not even comparable to those that we trained from scratch. First, we evaluate the adversarial vulnerability against two adversarial example generation methods: the gradientbased Fast Gradient Sign (FGS) method [4] and the nongradient-based hot/cold approach [23]. Although the latter is capable of forming multiple adversarial perturbations for each input, we only target the most similar class with the hot/cold approach, referred to as HC1. We aim to form adversarial perturbations for every correctly classified image from the MNIST [15] or CIFAR10 [12] test set, respectively. We consider an adversarial example generation attempt successful, if the direction specified by either FGS or HC1 leads to a misclassification, where the only constraint is that the discrete pixel values are in [0, 255] range. Of course, this limitation means that the formed perturbations may or may not be adversarial in nature as they can be highly perceptible to human observers. We compare the adversarial robustness of classifiers by collecting measures to quantify the quality of the produced adversarial examples. For this purpose, we calculate the Perceptual Adversarial Similarity Score (PASS) [23] of original and adversarial image pairs, and we also determine the L∞ norms of adversarial perturbations. Although the L∞ norm is not a good metric to quantify adversarial quality in terms of human perception, it can demonstrate how far the actual perturbed image is from the original sample. Second, we quantify how the robustness of the learning models evolve during training by applying a more general approach. For a given pair of classifiers where one was regularly trained while the other was obtained by BANG training, we add a certain level of random noise to 100 test images from each class that are correctly classified by both networks at all tested stages and compute the proportion of perturbed images that are classified differently than the originating one. While the previously described test assessing the adversarial vulnerability explores only two directions – specified by the FGS method and the HC1 approach – applying 1000 random distortions to each inspected image for every noise level gives us a more general evaluation. Although experimenting with random noise is more universal as it does not rely on any specific adversarial generation technique, small random perturbations that cause misclassifications are hard to find [23] and, hence, the collected Table 1: L E N ET T RAINING. This table highlights the difference between LeNet models obtained by using regular (R0-R1) and BANG training (B0-B5). Accuracy on the MNIST test set, the achieved success rates of FGS and HC1 adversarial example generation methods with PASS scores and L∞ norms of the produced examples on the MNIST test set are listed. ID β Accuracy FGS-Rate FGS-PASS FGS-L∞ HC1-Rate HC1-PASS HC1-L∞ R0 - - 99.16% 90.33% 0.4072 ± 0.1081 40.51 ± 15.72 99.53% 0.7535 ± 0.1143 122.60 ± 49.46 R1 - - 99.15% 91.41% 0.4072 ± 0.1065 40.70 ± 15.88 99.77% 0.7517 ± 0.1160 122.16 ± 49.07 B0 1.00 0.785 99.16% 3.51% 0.6806 ± 0.1457 8.34 ± 04.32 95.13% 0.5359 ± 0.2023 187.86 ± 63.66 B1 1.00 0.815 99.22% 1.68% 0.7638 ± 0.1367 5.52 ± 02.86 94.19% 0.4880 ± 0.2110 201.84 ± 62.51 B2 1.20 0.810 99.31% 2.13% 0.7579 ± 0.1452 5.57 ± 03.05 94.56% 0.5129 ± 0.2015 186.10 ± 63.77 B3 1.35 0.780 99.25% 3.86% 0.6763 ± 0.1471 8.28 ± 04.26 94.73% 0.5709 ± 0.2127 178.11 ± 65.76 B4 1.50 0.840 99.11% 1.52% 0.8220 ± 0.1310 4.19 ± 03.01 97.68% 0.4669 ± 0.1881 203.50 ± 58.97 B5 1.60 0.780 99.32% 4.45% 0.6771 ± 0.1487 8.20 ± 04.40 98.95% 0.6376 ± 0.1829 146.96 ± 61.97 (a) Accuracy (b) FGS Success Rate (c) HC1 PASS Figure 2: L E N ET M ODELS T RAINED WITH BANG. These plots summarize our results on LeNet models trained with BANG using combinations of β and . We tested a grid of those two parameters where β ∈ [1.0, 1.6] with step size 0.05, and ∈ [0.78, 0.84] with step size 0.005. We trained a single model with each combination and show (a) the obtained accuracy on the MNIST test set, (b) the achieved success rates by using FGS and (c) the mean PASS score of HC1 adversarial examples on the MNIST test images. Each solid green line represents the level of regularly trained learning models. For better visual representation we applied interpolation. results are qualitatively not as good as explicitly forming adversarial perturbations. Furthermore, in order to evaluate the stability of the trained classifiers, we distorted the images with Gaussian noise far beyond the noise level that can be considered imperceptible or adversarial. 4.1. LeNet on MNIST We commenced our experiments by evaluating BANG on the LeNet model optimized on the MNIST dataset. MNIST contains 70k images overall: 50k used for training, 10k for validation, and the remaining 10k for testing. The tested network originally has four layers (two convolutional and two fully connected) – extended with one additional Dropout layer – that we optimize without changing the hyperparameters distributed with Caffe. The learning model is trained with a batch size of 64 for 10k iterations using the inverse decay learning rate policy with an initial learning rate of 0.01. Since our training procedure has two parameters, β defined in Equation (6) and introduced in Equation (4), we trained LeNet models with parameter combinations from a grid, and evaluated the accuracy and adversarial vulnerability of the trained classifiers. The results of the conducted experiments are visualized in Figure 2, we also show accuracies and metrics indicating adversarial robustness in Table 1 for some models obtained with regular training (R0R1) and optimized with BANG training (B0-B5). As we can see in Table 1, FGS success rates achieved by regular training can be dramatically decreased by BANG: the rate drops from above 90% to below 2%. Almost every single failed adversarial example generation attempt is due to blank gradients – the gradient of loss with respect to the original image and its ground-truth label contains only zeros – which means that methods utilizing that gradient of loss cannot succeed. As we increase , or in other words, as we balance the contributions of batch elements more by scaling up gradients with lower magnitudes, the resultant classifiers become more resistant to gradient-based adversarial generation methods. Although the success rates obtained by the HC1 method remain relatively high, the qual- (a) Regular Training (b) BANG Training (c) Absolute Improvement Figure 3: L E N ET: ROBUSTNESS TO R ANDOM D ISTORTIONS. These plots show the evolving robustness of LeNet models: (a) obtained with regular training (R0 from Table 1), (b) trained with BANG (B1 from Table 1), and (c) displays the improvement. After identifying 100 test images per class that are correctly classified by both networks at every 500 iterations, we perturb each 1000 times by adding the level of Gaussian noise specified by the standard deviation, and test the networks at several stages of training. The plots show the percentage of distortions yielding misclassifications. For better visual representation we applied interpolation. ities of HC1 examples degrade significantly on LeNet models trained with BANG compared to the regular training as displayed in Figure 2(c). This degradation is highlighted by both decreasing PASS scores and by the significantly increased L∞ norms of perturbations listed in Table 1. With respect to the achieved classification performances, we find that there can be a level of degradation depending on the selected values for β and . This phenomenon can be seen in Figure 2(a); it is partially due to random initializations and can be the result of overfitting or our decision to evaluate all networks at 10k training iterations. Still, we can observe that BANG can yield improved classification performance over regular training paired with improved robustness as listed in Table 1. Additionally, we conducted experiments to quantify and compare how the robustness to random perturbations evolves during training. For this general approach, we selected to test two classifiers from Table 1: R0 optimized with regular training and B1 trained with BANG. We can see in Figure 3(a) that the regularly trained model is initially highly susceptible to larger distortions, but as the training progresses it becomes more stable, and settles at approximately 20% with respect to the strongest class of Gaussian noise that we formed by using standard deviation of 100 pixels. Contrarily, the classifier trained with BANG maintains significantly lower rates throughout the whole training as shown in Figure 3(b), and after 10k iterations only 3% of the strongest distortions can alter the original classification. The absolute improvements are displayed in Figure 3(c). 4.2. CIFAR-10 We also evaluated training with BANG on the so-called “CIFAR-10 quick” model of Caffe trained on the CIFAR10 dataset. CIFAR-10 consists of 60k images, 50k training images, and 10k images used for both validation and testing purposes. The network architecture originally has five layers (three convolutional and two fully connected) that we extended with one Dropout layer, and the learning model is trained with a batch size of 100 for 20k iterations (40 epochs). We use a fixed learning rate of 0.001 that we decrease by a factor of 10 after 36 epochs, and once again after another 2 epochs. Due to the different nature of CIFAR-10 training, we slightly adjusted BANG parameters. Specifically, as the classification performance is significantly worse than achieved by LeNet on MNIST yielding proportionately more incorrectly classified samples in each mini-batch, we applied lower local learning rates (β) and higher values for scaling (). Furthermore, we found that scaling incorrectly classified inputs less than correct ones has beneficial effects on robustness, hence, we applied 50% of the specified values on the incorrectly classified batch elements. Similarly to our conducted experiments on LeNet, we trained classifiers on CIFAR-10 with all possible combinations of β and parameters of a grid and then measured the accuracy and adversarial vulnerability of each of those networks. The results are visualized in Figure 4, and for some models obtained with regular training (R0-R1) and optimized with BANG training (B0-B5), we show accuracies and metrics indicating adversarial robustness in Table 2. As we can see in Table 2, FGS success rates achieved by regular training are significantly decreased by BANG: the rate drops from approximately 96% to 34% where, again, the majority of the failed adversarial example generation attempts are due to blank gradients. Figure 4(b) shows that as we increase , the classifiers become more resistant to gradient-based adversarial generation methods. The higher levels of success rates in comparison to LeNet might sim- Table 2: CIFAR-10 T RAINING. This table shows the difference between classifiers obtained using regular (R0-R1) and BANG training (B0-B5). The accuracy on the CIFAR-10 test set, the achieved success rates of FGS and HC1 adversarial example generation methods with PASS scores and L∞ norms of the formed examples on the CIFAR-10 test images are listed. ID β Accuracy FGS-Rate FGS-PASS FGS-L∞ HC1-Rate HC1-PASS HC1-L∞ R0 - - 79.59% 96.52% 0.9553 ± 0.0969 4.08 ± 06.40 98.97% 0.9669 ± 0.1005 18.15 ± 29.80 R1 - - 79.55% 96.71% 0.9513 ± 0.1057 4.43 ± 07.05 98.91% 0.9557 ± 0.1332 22.16 ± 39.77 B0 0.40 0.855 79.26% 34.27% 0.9511 ± 0.1302 4.11 ± 10.31 95.94% 0.8712 ± 0.1649 55.52 ± 49.98 B1 0.45 0.805 80.43% 45.94% 0.9818 ± 0.0548 2.04 ± 02.49 96.20% 0.7966 ± 0.2438 77.34 ± 71.20 B2 0.75 0.800 79.74% 41.71% 0.9828 ± 0.0586 1.94 ± 03.03 98.34% 0.8362 ± 0.2195 64.26 ± 63.57 B3 0.75 0.845 79.41% 35.00% 0.9526 ± 0.1266 3.94 ± 08.71 96.54% 0.8603 ± 0.1981 59.83 ± 58.28 B4 0.95 0.840 79.30% 34.88% 0.9575 ± 0.1236 3.61 ± 09.60 96.87% 0.8994 ± 0.1487 48.44 ± 47.35 B5 1.00 0.800 79.22% 41.34% 0.9803 ± 0.0722 2.03 ± 03.64 98.17% 0.8586 ± 0.1948 61.14 ± 61.23 (a) Accuracy (b) FGS Success Rate (c) HC1 PASS Figure 4: BANG CIFAR-10 M ODELS. These plots summarize our results on CIFAR-10 models trained with BANG using combinations of β and . We tested a grid of those two parameters where β ∈ [0.4, 1.0] with step size 0.05, and ∈ [0.80, 0.86] with step size 0.005. We trained a single model with each combination and show (a) the obtained accuracy on the CIFAR-10 test set, (b) the achieved success rates by FGS, and the (c) mean PASS score of HC1 adversarial examples on the CIFAR-10 test images. Each solid green line represents the level of regularly trained learning models. For better visual representation we applied interpolation. ply be due to the fact that the classifiers trained on CIFAR10 are less accurate, therefore, learning the incorrect samples of the batch still has a large contribution on weight updates. While the success rates achieved by HC1 remain high, the quality of HC1 adversarial examples degrades significantly compared to regular training. This degradation is highlighted by both decreasing PASS scores shown in Figure 4(c) and by the significantly increased L∞ norms of adversarial perturbations listed in Table 2. Finally, as shown in Table 2, we can train classifiers with BANG that slightly outperform models of regular training in terms of classification accuracy. Of course, the achieved overall performance depends on the chosen parameters as depicted in Figure 4(a). Finally, we ran experiments to better quantify and compare how the robustness of the trained classifiers to random perturbations evolves during training. Similarly to our experiments on LeNet, we selected two classifiers from Table 2 for testing: R0 trained regularly and B0 optimized with BANG. We can see in Figure 5(a) that the regularly trained R0 model is highly susceptible to larger distortions, its robustness does not improve during training, and finally achieves 46.0% with respect to the strongest class of Gaussian noise that we formed by using standard deviation of 40 pixels. Contrarily, the B0 model trained with BANG remains more robust throughout training epochs as shown in Figure 5(b) and at the end 39.1% of the strongest distortions change the original classification. The absolute improvements are visualized in Figure 5(c). We can conclude that although BANG enhanced robustness to random perturbations, the results are less impressive in comparison to LeNet – at least, with respect to the strongest distortions. 5. Conclusion In this paper, we introduced our theory to explain an intriguing property of machine learning models. Namely, the regular training procedure can prevent samples from forming flatter and broader regions around themselves. This evolutionary stalling yields samples remaining close to de- (a) Regular Training (b) BANG Training (c) Absolute Improvement Figure 5: CIFAR-10: ROBUSTNESS TO R ANDOM D ISTORTIONS. These plots show the evolving robustness of CIFAR-10 models: (a) obtained with regular training (R0 from Table 2), (b) trained with BANG (B0 from Table 2), and (c) displays the improvement. After identifying 100 test images per class that are correctly classified by both networks at every second epoch, we perturb each 1000 times with the level of Gaussian noise specified by the standard deviation, and test the networks at different stages of training. The plots show the percentage of distortions yielding misclassifications. For better visual representation we applied interpolation. cision boundaries and, hence, being susceptible to imperceptibly small perturbations causing misclassifications. To address this problem, we proposed a novel approach to improve the robustness of Deep Neural Networks (DNNs) by slightly modifying the regular training procedure. Our approach does not require additional training data – neither adversarial examples nor any sort of data augmentation – to achieve improved robustness, while the overall performance of the trained network is maintained or even enhanced. We experimentally demonstrated that optimizing DNNs with our Batch Adjusted Network Gradient (BANG) technique leads to significantly enhanced stability in general. By balancing the contributions of batch elements on forming the weight updates, BANG allows training samples to form flatter, more invariant regions around themselves. The trained classifiers become more robust to random distortions, and as we demonstrated with the gradient-based Fast Gradient Sign (FGS) method and the non-gradient-based hot/cold approach where we targeted the closest scoring class (HC1), they are also less vulnerable to adversarial example generation methods. To visualize the advancement achieved by BANG training in terms of improved adversarial robustness, in Figure 1 correctly classified MNIST and CIFAR-10 test images are presented along with adversarial examples formed via the HC1 approach on DNNs trained regularly and with BANG. While BANG helps to mitigate adversarial instability, learning models can maintain or even improve their overall classification performance. Our proposed approach achieves these results with negligible computational overhead over the regular training procedure. Although we managed to achieve good results on two DNNs trained on different datasets, we found that BANG parameters needed to be adjusted to these problems. To obtain better results, exploring the effect of different pa- rameters on different layers, and changing the contributions of correctly and incorrectly classified batch elements can be considered. Future work will focus on having a better understanding of BANG, enhancing the algorithm to be more self-adaptive, and exploring its application for training DNNs on real-world datasets. While some might argue that a similar balancing effect can be achieved by distillation, Carlini et al. [1] demonstrated that defensive distillation is not effective to improve adversarial robustness. The effectiveness of BANG to adversarial perturbations obtained via various adversarial example generation techniques likely varies – as Kurakin et al. [13] observed for adversarial training – and further research needs to explore that. In summary, we can conclude that the adversarial instability of DNNs is closely related to the applied training procedures – as was claimed by Gu et al. [6] – and there is a huge potential in this research area to further advance the generalization properties of machine learning models and their overall performances as well. Acknowledgments This research is based upon work funded in part by NSF IIS-1320956 and in part by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via IARPA R&D Contract No. 2014-14071600012. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the ODNI, IARPA, or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon. References [1] N. 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Neural Domain Adaptation for Biomedical Question Answering Georg Wiese1,2 , Dirk Weissenborn2 and Mariana Neves1 1 Hasso Plattner Institute, August Bebel Strasse 88, Potsdam 14482 Germany 2 Language Technology Lab, DFKI, Alt-Moabit 91c, Berlin, Germany [email protected], [email protected], [email protected] Abstract arXiv:1706.03610v2 [cs.CL] 15 Jun 2017 Factoid question answering (QA) has recently benefited from the development of deep learning (DL) systems. Neural network models outperform traditional approaches in domains where large datasets exist, such as SQuAD (≈ 100, 000 questions) for Wikipedia articles. However, these systems have not yet been applied to QA in more specific domains, such as biomedicine, because datasets are generally too small to train a DL system from scratch. For example, the BioASQ dataset for biomedical QA comprises less then 900 factoid (single answer) and list (multiple answers) QA instances. In this work, we adapt a neural QA system trained on a large open-domain dataset (SQuAD, source) to a biomedical dataset (BioASQ, target) by employing various transfer learning techniques. Our network architecture is based on a state-of-theart QA system, extended with biomedical word embeddings and a novel mechanism to answer list questions. In contrast to existing biomedical QA systems, our system does not rely on domain-specific ontologies, parsers or entity taggers, which are expensive to create. Despite this fact, our systems achieve state-of-the-art results on factoid questions and competitive results on list questions. 1 Introduction Question answering (QA) is the task of retrieving answers to a question given one or more contexts. It has been explored both in the opendomain setting (Voorhees et al., 1999) as well as domain-specific settings, such as BioASQ for the biomedical domain (Tsatsaronis et al., 2015). The BioASQ challenge provides ≈ 900 factoid and list questions, i.e., questions with one and several answers, respectively. This work focuses on answering these questions, for example: Which drugs are included in the FEC-75 regimen? → fluorouracil, epirubicin, and cyclophosphamide. We further restrict our focus to extractive QA, i.e., QA instances where the correct answers can be represented as spans in the contexts. Contexts are relevant documents which are provided by an information retrieval (IR) system. Traditionally, a QA pipeline consists of namedentity recognition, question classification, and answer processing steps (Jurafsky, 2000). These methods have been applied to biomedical datasets, with moderate success (Zi et al., 2016). The creation of large-scale, open-domain datasets such as SQuAD (Rajpurkar et al., 2016) have recently enabled the development of neural QA systems, e.g., Wang and Jiang (2016), Xiong et al. (2016), Seo et al. (2016), Weissenborn et al. (2017), leading to impressive performance gains over more traditional systems. However, creating large-scale QA datasets for more specific domains, such as the biomedical, would be very expensive because of the need for domain experts, and therefore not desirable. The recent success of deep learning based methods on open-domain QA datasets raises the question whether the capabilities of trained models are transferable to another domain via domain adaptation techniques. Although domain adaptation has been studied for traditional QA systems (Blitzer et al., 2007) and deep learning systems (Chen et al., 2012; Ganin et al., 2016; Bousmalis et al., 2016; Riemer et al., 2017; Kirkpatrick et al., 2017), it has to our knowledge not yet been applied for end-to-end neural QA systems. To bridge this gap we employ various do- main adaptation techniques to transfer knowledge from a trained, state-of-the-art neural QA system (FastQA, Weissenborn et al. (2017)) to the biomedical domain using the much smaller BioASQ dataset. In order to answer list questions in addition to factoid questions, we extend FastQA with a novel answering mechanism. We evaluate various transfer learning techniques comprehensively. For factoid questions, we show that mere fine-tuning reaches state-of-the-art results, which can further be improved by a forgetting cost regularization (Riemer et al., 2017). On list questions, the results are competitive to existing systems. Our manual analysis of a subset of the factoid questions suggests that the results are even better than the automatic evaluation states, revealing that many of the ”incorrect” answers are in fact synonyms to the gold-standard answer. 2 Related Work Traditional Question Answering Traditional factoid and list question answering pipelines can be subdivided into named-entity recognition, question classification, and answer processing components (Jurafsky, 2000). Such systems have also been applied to biomedical QA such as the OAQA system by Zi et al. (2016). Besides a number of domain-independent features, they incorporate a rich amount of biomedical resources, including a domain-specific parser, entity tagger and thesaurus to retrieve concepts and synonyms. A logistic regression classifier is used both for question classification and candidate answer scoring. For candidate answer generation, OAQA employs different strategies for general factoid/list questions, choice questions and quantity questions. Neural Question Answering Neural QA systems differ from traditional approaches in that the algorithm is not subdivided into discrete steps. Instead, a single model is trained end-to-end to compute an answer directly for a given question and context. The typical architecture of such systems (Wang and Jiang, 2016; Xiong et al., 2016; Seo et al., 2016) can be summarized as follows: 1. Embedding Layer: Question and context tokens are mapped to a high-dimensional vector space, for example via GloVe embeddings (Pennington et al., 2014) and (optionally) character embeddings (Seo et al., 2016). 2. Encoding Layer: The token vectors are processed independently for question and context, usually by a recurrent neural network (RNN). 3. Interaction Layer: This layer allows for interaction between question and context representations. Examples are Match-LSTM (Wang and Jiang, 2016) and Coattention (Xiong et al., 2016). 4. Answer Layer: This layer assigns start and end scores to all of the context tokens, which can be done either statically (Wang and Jiang, 2016; Seo et al., 2016) or by a dynamic decoding process (Xiong et al., 2016). FastQA FastQA fits into this schema, but reduces the complexity of the architecture by removing the interaction layer, while maintaining state-of-the-art performance (Weissenborn et al., 2017). Instead of one or several interaction layers of RNNs, FastQA computes two simple wordin-question features for each token, which are appended to the embedding vectors before the encoding layer. We chose to base our work on this architecture because of its state-of-the-art performance, faster training time and reduced number of parameters. Unsupervised Domain Adaptation Unsupervised domain adaptation describes the task of learning a predictor in a target domain while labeled training data only exists in a different source domain. In the context of deep learning, a common method is to first train an autoencoder on a large unlabeled corpus from both domains and then use the learned input representations as input features to a network trained on the actual task using the labeled source domain dataset (Glorot et al., 2011; Chen et al., 2012). Another approach is to learn the hidden representations directly on the target task. For example, domain-adversarial training optimizes the network such that it computes hidden representations that both help predictions on the source domain dataset and are indistinguishable from hidden representations of the unlabeled target domain dataset (Ganin et al., 2016). These techniques cannot be straightforwardly applied to the question answering task, because they require a large corpus of biomedical question-context pairs (albeit no answers are required). Supervised Domain Adaptation In contrast to the unsupervised case, supervised domain adaptation assumes access to a small amount of labeled training data in the target domain. The simplest approach to supervised domain adaptation for neural models is to pre-train the network on data from the source domain and then fine-tune its parameters on data from the target domain. The main drawback of this approach is catastrophic forgetting, which describes the phenomenon that neural networks tend to ”forget” knowledge, i.e., its performance in the source domain drops significantly when they are trained on the new dataset. Even though we do not directly aim for good performance in the source domain, measures against catastrophic forgetting can serve as a useful regularizer to prevent over-fitting. Progressive neural networks combat this issue by keeping the original parameters fixed and adding new units that can access previously learned features (Rusu et al., 2016). Because this method adds a significant amount of new parameters which have to be trained from scratch, it is not well-suited if the target domain dataset is small. Riemer et al. (2017) use fine-tuning, but add an additional forgetting cost term that punishes deviations from predictions with the original parameters. Another approach is to add an L2 loss which punishes deviation from the original parameters. Kirkpatrick et al. (2017) apply this loss selectively on parameters which are important in the source domain. 3 Model Our network architecture is based on FastQA (Weissenborn et al., 2017), a state-of-the-art neural QA system. Because the network architecture itself is exchangeable, we treat it as a black box, with subtle changes at the input and output layer as well as to the decoding and training procedure. These changes are described in the following. See Figure 3 for an overview of the system. 3.1 Input Layer In a first step, words are embedded into a highdimensional vector space. We use three sources of embeddings, which are concatenated to form a single embedding vector: • GloVe embeddings: 300-dimensional GloVe vectors (Pennington et al., 2014). These are Start Probabilities pstart End Probabilitiesp(e\|s) p(e\|s) End End Probabilities Probabilities pend sigmoid softmax EndScores Scoresee(s)(s) End End Scores yend Start Scores ystart Extractive QA System Biomedical Embeddings ... ... GloVe Embeddings Character Embeddings Question Type Features Context Embeddings Question Embeddings Figure 1: Network architecture of our system for biomedical question answering. At its core, it uses an extractive neural QA system as a black box (we use FastQA (Weissenborn et al., 2017)). The embedding layer is modified in order to include biomedical word embeddings and question type features. The output layer is adjusted to add the ability to answer list questions in addition to factoid questions. open-domain word vectors trained on 840 billion tokens from web documents. The vectors are not updated during training. • Character embeddings: As used in FastQA (Weissenborn et al., 2017) and proposed originally by Seo et al. (2016), we employ a 1-dimensional convolutional neural network which computes word embeddings from the characters of the word. • Biomedical Word2Vec embeddings: 200dimensional vectors trained using Word2Vec (Mikolov et al., 2013) on about 10 million PubMed abstracts (Pavlopoulos et al., 2014). These vectors are specific to the biomedical domain and we expect them to help on biomedical QA. As an optional step, we add entity tag features to the token embeddings via concatenation. Entity tags are provided by a dictionary-based entity tagger based on the UMLS Metathesaurus. The entity tag feature vector is a 127-dimensional bit vector that for each of the UMLS semantic types states whether the current token is part of an entity of that type. This step is only applied if explicitly noted. Finally, a one-hot encoding of the question type (factoid or list) is appended to all the input vectors. With these embedding vectors as input, we invoke FastQA to produce start and end scores for each of the n context tokens. We denote start i scores by ystart and end scores conditioned on a i,j predicted start at position i by yend , with start index i ∈ [1, n] and end index j ∈ [i, n]. 3.2 Output Layer In our adapted output layer, we convert the start and end scores to span probabilities. The computation of these probabilities is independent of the question type. The interpretation, however, depends on the question type: While for factoid questions, the list of answer spans is interpreted as a ranked list of answer candidates, for list questions, answers above a certain probability threshold are interpreted as the set of answers to the question. 1 n Given the start scores ystart , ..., ystart and end i,n i,1 scores yend , ..., yend , we compute the start and end probabilities as follows: i pistart = σ(ystart ) i,· pi,· end = softmax(yend ) (1) (2) where σ(x) is the sigmoid function. As a consequence, multiple tokens can be chosen as likely start tokens, but the network is expected to select a single end token for a given start token, hence the softmax function. Finally, the probability that a given span (i, j) answers the question i,j i is pi,j span = pstart · pend . This extension generalizes the FastQA output layer such that multiple answer spans with different start positions can have a high probability, allowing us to retrieve multiple answers for list questions. 3.3 Decoding Given a trained model, start probabilities can be obtained by running a forward pass and computing the start probability as in Equation 1. For the top 20 starts, we compute the end probabilities as given by Eq. 2. From the start and end probabilities, we extract the top 20 answer spans ranked by pi,j span . As a simple post-processing step, we remove duplicate strings and retain only those with the highest probability. For factoid questions, we output the 5 most likely answer spans as our ranked list of answers. For list questions, we learn a probability cutoff threshold t that defines the set of list answers A = {(i, j)\|pi,j span ≥ t}. We choose t to be the threshold that optimizes the list F1 score on the respective development set. 3.4 Domain Adaptation Fine-tuning Our training procedure consists of two phases: In the pre-training phase, we train the model on SQuAD, using a token F1 score as the training objective as by Weissenborn et al. (2017). We will refer to the resulting parameters as the base model. In the fine-tuning phase, we initialize the model parameters with the base model and then continue our optimization on the BioASQ dataset with a smaller learning rate. Forgetting Cost Regularization To avoid catastrophic forgetting during fine-tuning as a means to regularize our model, we optionally add an additional forgetting cost term Lf c , as proposed by Riemer et al. (2017). It is defined as the cross-entropy loss between the current predictions and the base model’s predictions. L2 Weight Regularization We also add an L2 loss term Ll2 which penalizes deviations from the base model’s parameters. Note that a more advanced approach would be to apply this loss selectively on weights which are particularly important in the source domain (Kirkpatrick et al., 2017). The final loss is computed as Lf inal = Loriginal + Cf c · Lf c + Cl2 · Ll2 where Cf c and Cl2 are hyperparameters which are set to 0 unless otherwise noted. 4 4.1 Experimental Setup Datasets SQuAD SQuAD (Rajpurkar et al., 2016) is a dataset of ≈ 100, 000 questions with relevant contexts and answers that sparked research interest into the development of neural QA systems recently. The contexts are excerpts of Wikipedia articles for which crowd-source workers generated questions-answer pairs. Because of the large amount of training examples in SQuAD, it lends itself perfectly as our source dataset. BioASQ The BioASQ challenge provides a biomedical QA dataset (Tsatsaronis et al., 2015) consisting of questions, relevant contexts (called snippets) from PubMed abstracts and possible answers to the question. It was carefully created with the help of biomedical experts. In this work, we focus on Task B, Phase B of the BioASQ challenge, in which systems must answer questions from gold-standard snippets. These questions can be either yes/no questions, summary questions, factoid questions, or list questions. Because we employ an extractive QA system, we restrict this study to answering factoid and list questions by extracting answer spans from the provided contexts. The 2017 BioASQ training dataset contains 1, 799 questions, of which 413 are factoid and 486 are list questions. The questions have ≈ 20 snippets on average, each of which are on average ≈ 34 tokens long. We found that around 65% of the factoid questions and around 92% of the list questions have at least one extractable answer. For questions with extractable answers, answers spans are computed via a simple substring search in the provided snippets. All other questions are ignored during training and treated as answered incorrectly during evaluation. 4.2 Training We minimize the cross-entropy loss for the gold standard answer spans. However, for multiple answer spans that refer to the same answer (e.g. synonyms), we only minimize the loss for the span of the lowest loss. We use the ADAM (Kingma and Ba, 2014) for optimization on SQuAD with a learning rate starting at 10−3 which is halved whenever performance drops between checkpoints. During the fine-tuning phase, we continue optimization on the BioASQ dataset with a smaller learning rate starting at 10−4 . During both phases, the model is regularized by variational dropout of rate 0.5 (Gal and Ghahramani, 2015). 4.3 Evaluation The official evaluation measures from BioASQ are mean reciprocal rank (MRR) for factoid questions and F1 score for list questions 1 . For factoid questions, the list of ranked answers can be at most five entries long. The F1 score is measured on the gold standard list elements. For both measures, 1 The details can be found at http:// participants-area.bioasq.org/Tasks/b/ eval_meas/ case-insensitive string matches are used to check the correctness of a given answer. A list of synonyms is provided for all gold-standard answers. If the system’s response matches one of them, the answer counts as correct. For evaluation, we use two different finetuning datasets, depending on the experiment: BioASQ3B, which contains all questions of the first three BioASQ challenges, and BioASQ4B which additionally contains the test questions of the fourth challenge. BioASQ4B is used as the training dataset for the fifth BioASQ challenge whereas BioASQ3B was used for training during the fourth challenge. Because the datasets are small, we perform 5fold cross-validation and report the average performance across the five folds. We use the larger BioASQ4B dataset except when evaluating the ensemble and when comparing to participating systems of previous BioASQ challenges. All models were implemented using TensorFlow (Abadi et al., 2016) with a hidden size of 100. Because the context in BioASQ usually comprises multiple snippets, they are processed independently in parallel for each question. Answers from all snippets belonging to a question are merged and ranked according to their individual probabilities. 5 5.1 Results Domain Adaptation In this section, we evaluate various domain adaptation techniques. The results of the experiments are summarized in Table 1. Baseline As a baseline without transfer learning, Experiment 1 trains the model on BioASQ only. Because the BioASQ dataset by itself is very small, a dropout rate of 0.7 was used, because it worked best in preliminary experiments. We observe a rather low performance, which is expected when applying deep learning to such a small dataset. Fine-tuning Experiments 2 and 3 evaluate the pure fine-tuning approach: Our base model is a system trained on SQuAD only and tested on BioASQ (Experiment 2). For Experiment 3, we fine-tuned the base model on the BioASQ4B training set. We observe that performance increases significantly, especially on list questions. This increase is expected, because the network is trained Experiment Factoid MRR List F1 (1) Training on BioASQ only 17.9% 19.1% (2) Training on SQuAD only (3) Fine-tuning on BioASQ 20.0% 24.6% 8.1% 23.6% (4) Fine-tuning on BioASQ w/o biomedical embeddings (5) Fine-tuning on BioASQ w/ entity features 21.3% 23.3% 22.4% 23.8% (6) Fine-tuning on BioASQ + SQuAD (7) Fine-tuning on BioASQ w/ forgetting cost (Cf c = 100.0) (8) Fine-tuning on BioASQ w/ L2 loss on original parameters (Cl2 = 0.3) 23.9% 26.2% 22.6% 23.8% 21.1% 20.4% Table 1: Comparison of various transfer learning techniques. In Experiment 1, the model was trained on BioASQ only. In Experiment 2, the model was trained on SQuAD and tested on BioASQ. We refer to it as the base model. In Experiment 3, the base model parameters were fine-tuned on the BioASQ training set. Experiments 4-5 evaluate the utility of domain dependent word vectors and features. Experiments 6-8 address the problem of catastrophic forgetting. All experiments have been conducted with the BioASQ4B dataset and 5-fold cross-validation. on biomedical- and list questions, which are not part of the SQuAD dataset, for the first time. Overall, the performance of the fine-tuned model on both question types is much higher than the baseline system without transfer learning. Features In order to evaluate the impact of using biomedical word embeddings, we repeat Experiment 3 without them (Experiment 4). We see a factoid and list performance drop of 3.3 and 1.2 percentage points, respectively, showing that biomedical word embeddings help increase performance. In Experiment 5, we append entity features to the word vector, as described in Section 3.1. Even though these features provide the network with domain-specific knowledge, we found that it actually harms performance on factoid questions. Because most of the entity features are only active during fine-tuning with the small dataset, we conjecture that the performance decrease is due to over-fitting. Catastrophic Forgetting We continue our study with techniques to combat catastrophic forgetting as a means to regularize training during fine-tuning. In Experiment 6 of Table 1 we fine-tune the base model on a half-half mixture of BioASQ and SQuAD questions (BioASQ questions have been upsampled accordingly). This form of joint training yielded no significant performance gains. Experiment 7 regularizes the model via an additional forgetting cost term, as proposed by Riemer et al. (2017) and explained in Section 3.4. We generally found that this technique only increases performance for factoid questions where the performance boost was largest for Cf c = 100.0. The fact that the forgetting loss decreases performance on list questions is not surprising, as predictions are pushed more towards the predictions of the base model, which has very poor performance on list questions. Experiment 8 adds an L2 loss which penalizes deviations from the base model’s parameters. We found that performance decreases as we increase the value of Cl2 which shows that this technique does not help at all. For the sake of completeness we report results for Cl2 = 0.3, the lowest value that yielded a significant drop in performance. 5.2 Ensemble Model ensembles are a common method to tweak the performance of a machine learning system. Ensembles combine multiple model predictions, for example by averaging, in order to improve generalization and prevent over-fitting. We evaluate the utility of an ensemble by training five models on the BioASQ3B dataset using 5-fold crossvalidation. Each of the models is evaluated on the 4B test data, i.e., data which is not included in BioASQ3B. During application, we run an ensemble by averaging the start and end scores of individual models before they are passed to the sigmoid / softmax functions as defined in Eq. 1 and 2. In Table 2 we summarize the average performance of Experiment Factoid MRR List F1 Average Best Ensemble 23.4% 24.3% 27.3% 24.0% 27.7% 28.6% Table 2: Performance of a model ensemble. Five models have been trained on the BioASQ3B dataset and tested on the 4B test questions. We report the average and best single model performances, as well as the ensemble performance. the five models, the best performance across the five models, and the performance of the ensemble. We observe performance gains of 3 percentage points on factoid questions and a less than 1 percentage point on list questions, relative to the best single model. This demonstrates a small performance gain that is consistent with the literature. 5.3 Comparison to competing BioASQ systems Because the final results of the fifth BioASQ challenge are not available at the time of writing, we compare our system to the best systems in last year’s challenge 2 . For comparison, we use the best single model and the model ensemble trained on BioASQ3B (see Section 5.2). We then evaluate the model on the 5 batches of last year’s challenge using the official BioASQ evaluation tool. Each batch contains 100 questions of which only some are factoid and list questions. Note that the results underestimate our system’s performance, because our competing system’s responses have been manually evaluated by humans while our system’s responses are evaluated automatically using string matching against a potentially incomplete list of synonyms. In fact, our qualitative analysis in Section 5.4 shows that many answers are counted as incorrect, but are synonyms of the gold-standard answer. The results are summarized in Table 3 and compared to the best systems in the challenge in each of the batches and question type categories. With our system winning four out of five batches on factoid questions, we consider it stateof-the-art in biomedical factoid question answering, especially when considering that our results might be higher on manual evaluation. The results on list questions are slightly worse, but still very 2 Last year’s results are available at http: //participants-area.bioasq.org/results/ 4b/phaseB/ competitive. This is surprising, given that the network never saw a list question prior to the finetuning phase. Due to small test set sizes, the sampling error in each batch is large, causing the single model to outperform the model ensemble on some batches. 5.4 Qualitative Analysis In order to get a better insight into the quality of the predictions, we manually validated the predictions for the factoid questions of batch 5 of the fourth BioASQ challenge as given by the best single model (see Table 3). There are in total 33 factoid questions, of which 23 have as the gold standard answer a span in one of the contexts. According to the official BioASQ evaluation, only 4 questions are predicted correctly (i.e., the gold standard answer is ranked highest). However, we identified 10 rank-1 answers which are not counted as correct but are synonyms to the gold standard answer. Examples include ”CMT4D disease” instead of ”Charcot-Marie-Tooth (CMT) 4D disease”, ”tafazzin” instead of ”Tafazzin (TAZ) gene”, and ”β-glucocerebrosidase” instead of ”Beta glucocerebrosidase”. In total, we labeled 14 questions as correct and 24 questions as having their correct answer in the top 5 predictions. In the following, we give examples of mistakes made by the system. Questions are presented in italics. In the context, we underline predicted answers and present correct answers in boldface. We identified eight questions for which the semantic type of the top answer differs from the question answer type. Some of these cases are completely wrong predictions. However, this category also includes subtle mistakes like the following: In which yeast chromosome does the rDNA cluster reside? The rDNA cluster in Saccharomyces cerevisiae is located 450 kb from the left end and 610 kb from the right end of chromosome XII... Here, it predicted a yeast species the rDNA cluster is located in, but ignored that the question is asking for a chromosome. Another type of mistakes is that the top answer is somewhat correct, but is missing essential information. We labeled four predictions with this category, like the following example: Batch Factoid MRR Best Participant Single Ensemble Best Participant List F1 Single 1 2 3 4 5 12.2% (fa1) 22.6% (LabZhu-FDU) 24.4% (oaqa-3b-3) 32.5% (oaqa-3b-4) 28.5% (oaqa-3b-5) 25.2% 16.4% 24.7% 34.0% 23.7% 29.2% 24.2% 20.6% 40.3% 23.2% 16.8% (fa1) 15.5% (LabZhu-FDU) 48.3% (oaqa-3b-3) 31.2% (oaqa-3b-4) 29.0% (oaqa-3b-5) 29.1% 25.8% 31.8% 29.0% 23.5% 27.9% 20.8% 33.3% 24.1% 26.1% Avg. 24.0% 24.8% 27.5% 28.1% 27.8% 26.5% Ensemble Table 3: Comparison to systems on last year’s (fourth) BioASQ challenge for factoid and list questions. For each batch and question type, we list the performance of the best competing system, our single model and ensemble. Note that our qualitative analysis (Section 5.4) suggests that our factoid performance on batch 5 would be about twice as high if all synonyms were contained in the gold standard answers. How early during pregnancy does non-invasive cffDNA testing allow sex determination of the fetus? Gold Standard Answer: "6th to 10th week of gestation" or "first trimester of pregnancy" Given Top Answer: "6th-10th" In summary, to our judgment, 14 of 33 questions (42.4%) are answered correctly, and 24 of 33 questions (72.7%) are answered correctly in one of the top 5 answers. These are surprisingly high numbers considering low MRR score of 23.7% of the automatic evaluation (Table 3). poor prior to fine-tuning which is due to the lack of list questions in SQuAD. We believe that large scale open-domain corpora for list questions would enhance performance further. Unsupervised domain adaptation could be an interesting direction for future work, because the biomedical domain offers large amounts of textual data, some of which might even contain questions and their corresponding answers. We believe that leveraging these resources holds potential to further improve biomedical QA. 6 In this paper, we described a deep learning approach to address the task of biomedical question answering by using domain adaptation techniques. Our experiments reveal that mere fine-tuning in combination with biomedical word embeddings yield state-of-the-art performance on biomedical QA, despite the small amount of in-domain training data and the lack of domain-dependent feature engineering. Techniques to overcome catastrophic forgetting, such as a forgetting cost, can further boost performance for factoid questions. Overall, we show that employing domain adaptation on neural QA systems trained on large-scale, open-domain datasets can yield good performance in domains where large datasets are not available. Discussion and future work The most significant result of this work is that state-of-the-art results in biomedical question answering can be achieved even in the absence of domain-specific feature engineering. Most competing systems require structured domain-specific resources, such as biomedical ontologies, parsers, and entity taggers. While these resources are available in the biomedical domain, they are not available in most domains. Our system, on the other hand, requires a large open-domain QA dataset, biomedical word embeddings (which are trained in an unsupervised fashion), and a small biomedical QA dataset. This suggests that our methodology is easily transferable to other domains as well. Furthermore, we explored several supervised domain adaptation techniques. In particular, we demonstrated the usefulness of forgetting cost for factoid questions. 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1 Uncertainty Marginal Price, Transmission Reserve, and Day-ahead Market Clearing with Robust Unit Commitment arXiv:1507.01540v3 [math.OC] 2 Aug 2016 Hongxing Ye, Member, IEEE, Yinyin Ge, Mohammad Shahidehpour, Fellow, IEEE, Zuyi Li, Senior Member, IEEE Abstract—The increasing penetration of renewable energy in recent years has led to more uncertainties in power systems. These uncertainties have to be accommodated by flexible resources (i.e. upward and downward generation reserves). In this paper, a novel concept, Uncertainty Marginal Price (UMP), is proposed to price both the uncertainty and reserve. At the same time, the energy is priced at Locational Marginal Price (LMP). A novel market clearing mechanism is proposed to credit the generation and reserve and to charge the load and uncertainty within the Robust Unit Commitment (RUC) in the Day-ahead market. We derive the UMPs and LMPs in the robust optimization framework. UMP helps allocate the cost of generation reserves to uncertainty sources. We prove that the proposed market clearing mechanism leads to partial market equilibrium. We find that transmission reserves must be kept explicitly in addition to generation reserves for uncertainty accommodation. We prove that transmission reserves for ramping delivery may lead to Financial Transmission Right (FTR) underfunding in existing markets. The FTR underfunding can be covered by congestion fund collected from uncertainty payment in the proposed market clearing mechanism. Simulations on a six-bus system and the IEEE 118-bus system are performed to illustrate the new concepts and the market clearing mechanism. Index Terms—Uncertainty Marginal Price, Cost Causation, Robust Unit Commitment, Financial Transmission Right, Generation Reserve, Transmission Reserve N OMENCLATURE Indices i, l, t m, n mi k indices for generators, lines, and time intervals index for buses index of bus where unit i is located index of the worst point for uncertainty Functions and sets ˆ F U CiP (·), CiI (·) L(·) G(m) K symbol for the optimal value of a variable feasible set for UC and dispatch uncertainty set cost related to dispatch and UC for unit i Lagrangian function set of units located at bus m set of the indices for ˆk up down Km,t , Km,t set of indices k for upward and downward UMPs at bus m time t Constants ND , NT number of buses and time intervals dm,t aggregated equivalent load Fl transmission line flow limit Γl,m shift factor for line l with respect to bus m Pimin , Pimax minimum and maximum generation outputs riu , rid ramping-up/down limits between sequential intervals Riu , Rid ramping-up/down limits for uncertainty accommodation um,t bound for uncertainty ˆ ˆk is the k th worst uncertainty vector in K, ˆk ∈ RND NT , ˆkm,t ∈ R FTRm→n FTR amount from bus m to n Variables Ii,t unit on/off status indicators yi,t , zi,t unit start-up and shut-down indicators Pi,t generation dispatch inj Pm,t net power injection m,t uncertainty at bus m time t Z optimal value of problem (SP) given (x, y, z, I, P ) ∆Pi,t generation re-dispatch pos ∆fl,t transmission capacity reserve in positive direction neg ∆fl,t transmission capacity reserve in negative direction inj ∆Pm,t net power injection change λt , λkt Lagrangian multipliers α, β, η non-negative Lagrangian multipliers u,k e πm,t marginal prices. πm,t for energy price; πm,t is the u,up UMP for kth uncertainty point; πm,t for upward u,down UMP; πm,t for downward UMP up Qi,t , Qdown i,t upward and downward generation reserves Ψm,t charge for uncertainty source T ΘG i,t , Θl,t credits to generation reserve for unit i and transmission reserve for line l at time t This work is supported by the U.S. National Science Foundation Grant I. I NTRODUCTION ECCS-1549937. The early version of this work was available on arXiv July 06, 2015, titled “Market Clearing for Uncertainty, Generation Reserve, and N modern power systems, uncertainties grow significantly Transmission Reserve”. The authors are with the Galvin Center for Electricity with the increasing penetration of Renewable Energy Innovation at Illinois Institute of Technology, Chicago, IL 60616, USA. (email: [email protected]; [email protected]; [email protected]; [email protected]). Source (RES), such as wind power generation. They pose 0885-8950 c 2016 IEEE. http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=7524711, Citation information: DOI 10.1109/TPWRS.2016.2595621, IEEE Transactions on Power Systems I 2 new challenges for the operation of electricity markets. In the Day-ahead market (DAM), the Unit Commitment (UC) and Economic Dispatch (ED) problems considering uncertainties have become a focus of research in recent years. The objective of the UC problem is to find the least cost UC solution for the second day while respecting both system-wide and unit-wise constraints. By fixing the UC variables, the ED problem is established. The Locational Marginal Price (LMP) and reserve price are then obtained as byproducts of the ED problem [1], [2]. When considering the uncertainties, the generation from uncontrollable RES are uncertain parameters in the optimization problem. Recently, Robust UC (RUC) is proposed to address the issues of uncertainty [3]–[7]. The largest merit is that the UC solution can be immunized against all the uncertainties in predefined set. The key idea of the two-stage RUC is to determine the optimal UC in the first stage which leads to the least cost for the worst scenario in the second stage. However, this approach is conservative and the Robust ED (RED) is absent. Authors in [8] combined the stochastic and robust approach using a weight factor in the objective function to address the conservativeness issue. [9], [10] employed the Affine Policy (AP) to formulate and solve the RED problem. A Multi-stage RUC is proposed to incorporate the latest information in each stage [11], where AP is also used to overcome the computational challenge. Recently, we reported a new approach which tries to bridge the gap of RUC and RED [7], [12]. In DAM, the main difficulty for pricing is that RED is absent in the traditional RUC [4], [5], [8]. On the other hand, a large number of works on pricing reserves exists within the UC framework considering contingencies [2], [13], [14] and stochastic security [15]. They are normally modeled as co-optimization problem. In [2], the reserve is cleared on zonal levels. Instead of countable contingency scenarios [15] or single additional scenario for reserve [16], the infinite continuous uncertainties are considered in the RUC, and the reserves are fully deliverable in infinite scenarios. In this paper, we propose a novel mechanism to price the energy, uncertainty, and flexibility simultaneously based on the RUC in [7]. An explicit price signal is derived for pricing the uncertainty. As the ED solution obtained is robust [7], both marginal impacts of the uncertainty and flexibility are reflected in these prices. In the proposed mechanism, reserve costs are allocated to uncertainty sources. Generation reserves, also called flexibilities in this paper, are the key factor for the robust optimization approaches. They are entitled to proper credits based on their contribution to uncertainty management. According to the market equilibrium analysis, market participants (price takers) can get the maximal profit by following the ISO/RTO’s dispatch instruction. The generation reserve and its deliverability are the main focus in [7]. The definition of LMP in [17] are employed to derive the energy price. The new concept, Uncertainty Marginal Price (UMP), is proposed to define the marginal cost of immunizing the next increment of uncertainty at a specific location. Load and generation are a pair, and they are priced at LMP. Uncertainty and flexibility (i.e., generation reserve) are another pair, and they are priced at UMP. Both LMPs and UMPs may vary with the locations due to transmission congestions. Limited by the transmission capacity and power flow equations, sometimes the uncertainties at certain buses cannot be mitigated by the system-wide cheapest generation reserve, and expensive generation reserve, which is deliverable, has to be kept in the system. Therefore, uncertainty sources are charged and generation reserves are credited based on UMPs at the corresponding buses. As the transmission reserve is kept within the RUC framework, the congestion component may exist in both the energy price and reserve price even if the physical limit of the line is not reached yet in the base case scenario. LMP congestion costs are distributed to Financial Transmission Right (FTR) holders in the existing market according to the LMP difference and the FTR amount. The revenue inadequacy occurs when the LMP congestion cost collected is smaller than the credit distributed to FTR holders, which is also called FTR underfunding. This has been a serious issue in recent years in the industry [18], [19]. We reveal that transmission reserve will be another reason for FTR underfunding when physical transmission limit is adopted in Simultaneous Feasibility Test (SFT) for FTR market [18], [20], [21]. This conclusion is applicable to any robust UC framework for DAM. The main contributions of this paper are listed as follows. 1) The novel UMP for uncertainties and generation reserves, as well as LMP for energy, are derived within a robust UC framework. The derivation is for uncertainties set with interval and budget constraints. The general concepts still apply when other uncertainty sets are modeled. 2) It is revealed that transmission capacities have to be reserved for uncertainty accommodation and the transmission reserves may cause FTR underfunding because of the deficiency of energy congestion revenues based on existing market rules. 3) A new market clearing mechanism is proposed to credit the generation and reserve and to charge the load and uncertainty. The payment collected from uncertainty sources can exactly cover the credits to generation reserves and transmission reserves, effectively resolving the FTR underfunding issue. The rest of this paper is organized as follows. Derivation of the LMP and UMP is presented in Section II, so is the market clearing mechanism for charge and credit based on LMP and UMP. Case studies are presented in Section III. Section IV concludes this paper. II. RUC AND M ARKET C LEARING One motivation of this work is to price the uncertainty, and allocate the cost of uncertainty accommodation to the uncertainty source. As the uncertainty source is charged the uncertainty payment, it has the incentive to reduce the uncertainty. With UMP, we can follow the cost causation principle, which is normally required in the market design, to charge the uncertainty sources. Cost causation principle is described as “require that all approved rates reflect to some degree the costs 3 actually caused by the customer who must pay them” in KN Energy, Inc. V. FERC, 968 F.2d 1295, 1300 (D.C. Cir. 1992). Another important motivation is to provide a theory that supports the application of the RUC in the DAM clearing. Although the RUC/RED are studied extensively, the only application of the RUC now is for the Reliability Assessment Commitment (RAC) in the DAM. There are several reasons why they are not applied in the DAM clearing. First, the computation burden of RUC is much larger than the standard UC. Second, as the objective is the cost of the worst-case scenario [3], [5], the solution is criticized on over conservatism. Third, no economic dispatch and prices are available within the RUC framework. Recently, with the new achievements in the algorithms, models, and high-performance computing application [6]–[8], [22]–[24], the first two obstacles are being addressed with great promises. This paper tries to clear the last obstacle with the new model [7]. Adopting RUC in the market clearing can give clear price signals for the uncertainties and reserves. On the other side, it is also easier for the solution to pass the robustness test, which is a RUC, in RAC. To our best knowledge, this is the first work on pricing energy, uncertainties, and reserves within the robust optimization framework in DAM. Hence, we focus on illustrating the concept with the following assumptions. • Network loss is ignored. Shift factor matrix is constant. • Uncertainty is from load and RES. Contingency is ignored. • The uncertainty budget set can be truly formulated by the ISO/RTO. A. RUC and RED ISOs/RTOs desire to get the optimal UC and ED solution in the base-case scenario. They can re-dispatch the flexible resources, such as adjustable load demands and generators with fast ramping capabilities, to follow the load when deviation occurs (or uncertainty is revealed). Consistent with the robust literature [4], [5], the uncertainty set is modeled as U := { ∈ RND NT : −um,t ≤ m,t ≤ um,t , ∀m, t X \|m,t \| ≤ Λ∆ t , ∀t} u m,t m (RUC) min C I (x) + C P (p) s.t. Ax + Bp ≤ b n F := (x, p) : ∀ ∈ U, ∃∆p such that o Cx + Dp + G∆p ≤ d + E . (1) (2) The basic idea of the above model is to find a robust UC and ED for the base-case scenario. The UC x and dispatch p are immunized against any uncertainty ∈ U. When uncertainty occurs, it is accommodated by the generation adjustment ∆p. Please refer to Appendix A for the detailed formulation. (x,p) s.t. Ax + Bp ≤ b Cx + Dp + G∆pk ≤ d + Eˆ k , ∀k ∈ K (3a) and Z := max min 1> s ∈U (s,∆p)∈R() n R() := (s, ∆p) : s ≥ 0 (SP) (4a) (4b) o G∆p − s ≤ d − Cx − Dp + E (4c) where K is the index set for uncertainty points ˆ which are dynamically generated in (SP) with iterations. Please refer to Appendix B for the detailed formulation. It should be noted that ˆk is the extreme point of U. Variable ∆pk is associated with ˆk . The objective function in (SP) is to find the worst point in U given (x, p). The procedure is 1: K ← ∅, k ← 1, Z ← +∞, define feasibility tolerance δ 2: while Z ≥ δ do 3: Solve (MP), obtain optimal (x̂, p̂). 4: Solve (SP) with x = x̂, p = p̂, get solution (Z, ˆk ) 5: K ← K ∪ k, k ← k + 1 6: end while Once the procedure is converged, we also get the optimal UC and ED solution by solving (MP). Similar to traditional LMP calculation, we fix the binary variables as x̂. Then a convex linear programming problem (RED) can be formed as XX CiP (Pi,t ) (RED) min (5) P,∆P s.t. (λt ) X (β̄i,t ) (αi,t ) ¯ (η̄l,t ) (ηl,t ) ¯ k (β̄i,t ) k (βi,t ) ¯k (ᾱi,t ) k (αi,t ) t i X dm,t , ∀t, m Pi,t ≤ Iˆi,t Pimax , ∀i, t −Pi,t ≤ −Iˆi,t Pimin , ∀i, t Pi,t − Pi,t−1 ≤ riu (1 − ŷi,t ) + Pimin ŷi,t , ∀i, t −Pi,t + Pi,t−1 ≤ rid (1 − ẑi,t ) + Pimin ẑi,t , ∀i, t X inj Γl,m Pm,t ≤ Fl , ∀l, t m X inj − Γl,m Pm,t ≤ Fl , ∀l, t m X X k ∆Pi,t = ˆkm,t , ∀t, ∀k ∈ K m i k Pi,t + ∆Pi,t ≤ Iˆi,t Pimax , ∀i, t, ∀k ∈ K k −Pi,t − ∆Pi,t ≤ −Iˆi,t Pimin , ∀i, t, ∀k ∈ K k ∆Pi,t ≤ Riu (1 − ŷi,t ), ∀i, t, ∀k ∈ K k −∆Pi,t ≤ Rid (1 − ẑi,t+1 ), ∀i, t, ∀k ∈ K Pi,t = i (λkt ) (x,p)∈F min C I (x) + C P (p) (MP) (βi,t ) ¯ (ᾱi,t ) is the budget parameter and assumed as an integer [3]. It is noted that all the flexible resources are modeled as generators. In this paper, the RUC is formulated according to the model in [7]. Λ∆ t Column and Constraint Generation (CCG) based method is used to solve the above model [6]. Problem (MP) and (SP) are established as follows. ¯ X inj inj,k k (η̄l,t ) Γl,m (Pm,t + ∆Pm,t ) ≤ Fl , ∀l, t, ∀k ∈ K (6a) (6b) (6c) (6d) (6e) (6f) (6g) (7a) (7b) (7c) (7d) (7e) (7f) m k (ηl,t ) − ¯ X m inj inj,k Γl,m (Pm,t + ∆Pm,t ) ≤ Fl , ∀l, t, ∀k ∈ K,(7g) 4 where (6a)-(6g) are the constraints for the base ED, and (7a)(7g) are constraints for different extreme points in U. (7a) denotes the load balance after re-dispatch. The generation adjustments respects capacity limits (7b)(7c) and ramping limits (7d)(7e). Network constraints are denoted by (7f)(7g). inj inj,k The Pm,t and ∆Pm,t are defined as inj Pm,t := X i∈G(m) and inj,k ∆Pm,t := X i∈G(m) Pi,t − dm,t , ∀m, t, k ∆Pi,t − ˆkm,t , ∀m, t, k, respectively. B. Marginal Prices In this section, marginal prices for the energy, uncertainty, and generation reserve are derived based on the Lagrangian function. Denote the Lagrangian function for (RED) as L(P, ∆P, λ, α, β, η), which is shown in Appendix C. According to the definition of marginal price [17], the LMP for energy at bus m is ∂L(P, ∆P, λ, α, β, η) (8) ∂dm,t XX X k k =λt − Γl,m η̄l,t − ηl,t − Γl,m η̄l,t − ηl,t ¯ ¯ l l k∈K e πm,t = It is observed that the impact of the uncertainty is also reflected in the LMP. The new concept, UMP for DAM, is defined as the marginal cost of immunizing the next unit increment of uncertainty. For ˆk , an extreme point of U, the UMP is X ∂L(P, ∆P, λ, α, β, η) k k = λkt − Γl,m η̄l,t − ηl,t = k ∂ˆ m,t ¯ l (9) u,k Both the uncertainty and generation reserve are priced at πm,t . u,k In the derivation of πm,t , the worst point ˆk is the only concern. Therefore, the general principles in this paper still work when U is replaced with other sets. It should be noted that (9) is intermediate price signals. In order to get the aggregated UMPs, the following new sets are u,k defined based on the sign of πm,t . u,k πm,t up u,k Km,t := {k : πm,t ≥ 0}; u,k down Km,t := {k : πm,t < 0} (10) The aggregated upward and downward UMPs are defined as u,up πm,t := X up k∈Km,t u,k πm,t ; u,down πm,t := X u,k πm,t (11) down k∈Km,t respectively. In the following context, we will show how the aggregated UMPs are used. C. Market Clearing Mechanism With LMP and UMP, the charges and credits for the market participants become clear and fair in the DAM. Energy clearing is straightforward. The basic principle related to uncertainty and flexibility is that those who cause uncertainties (uncertainty sources), such as RES, pay based on UMP and those who contribute to the management of uncertainties (uncertainty mitigators), such as generators or storage with ramping capabilities, get paid. 1) Energy Payment and Credit: LSEs pay based on the amount of the load and LMP. The energy payment from the e LSE at Bus m at t is πm,t dm,t . It should be noted that RES is entitled to the credit due to the negative load modeled in RUC. Generator i, located at Bus mi , is entitled to the credit e πm P for energy production. i ,t i,t 2) Charge to Uncertainty Source: The uncertainty source can be charged as X u,k Ψm,t = πm,t ˆkm,t (12) k∈K The uncertainty source pays based on the marginal price and the worst point ˆk . The uncertainty source is charged only u,k when πm,t is non-zero, and it may have to pay more when the uncertainty becomes larger. The uncertainty point ˆkm,t can be upward (i.e. ˆkm,t ≥ 0) or downward (i.e. ˆkm,t ≤ 0). We have the following lemma regarding the relation between the u,k signs of πm,t and ˆkm,t . u,k Lemma 1. If ˆkm,t > 0, then πm,t ≥ 0. If ˆkm,t < 0, then u,k πm,t ≤ 0. Please check Appendix D-A for the proof. When the budget set is adopted, the extreme point ˆkm,t ∈ {−um,t , 0, um,t } [4], [7], so the uncertainty charge in (12) can also be written as (13) according to Lemma 1 and (11). u,up u,down Ψm,t = πm,t um,t + πm,t (−um,t ) (13) Thus, upward and downward uncertainties are charged separately. It should be noted that we still need to use (12) when other uncertainty sets are used. 3) Credit to Generation Reserve: Only resources that can provide deliverable generation reserve are entitled to credits. If i ∈ G(m), then the credits can be formulated as X u,k k ΘG πm,t ∆Pi,t . (14) i,t = k∈K In other words, generation reserve is paid the UMP at the bus u,k where it is located. If πm,t = 0, then the associated credit k is zero no matter what the value of ∆Pi,t is. Similar to the uncertainties, the generation reserves can be in either upward or downward direction. Denote the upward generation reserve down as Qup i,t and the downward generation reserve as Qi,t n o ˆi,t Pimax − Pi,t , Riu (1 − ŷi,t ) , (15) Qup := min I i,t n o down Qi,t := max Iˆi,t Pimin − Pi,t , −Rid (1 − ẑi,t+1 ) . (16) We also have the following lemma regarding the relation down k between Qup and ∆Pi,t . i,t , Qi,t 5 k to Lemma 2. If i ∈ G(m), then the optimal solution ∆Pi,t problem (RED) is ( u,k Qup if πm,t >0 k i,t , ∆Pi,t = u,k Qdown , if π m,t < 0 i,t and u,k k k k k πm,t = β̄i,t − βi,t + ᾱi,t − αi,t , (17) ¯ ¯ Please check Appendix D-B for the proof. The credit to generation reserve i located at bus m (14) can be rewritten as (18) according to Lemma 2 and (11). u,up up u,down down ΘG i,t = πm,t Qi,t + πm,t Qi,t (18) (18) shows that the upward and downward generation reserves are credited separately. Flexible resources may receive credits for both the upward and downward generation reserves simultaneously. (18) always holds even if other uncertainty sets are modeled in RUC. D. Transmission Reserve and Revenue Adequacy Some transmission capacities are reserved according to the solution to (RED). These transmission reserves are used to ensure the ramping deliverability when the uncertainty is revealed, as shown in (7f) and (7g). It is noted that they are determined automatically in (RED), and kept explicitly without explicit transmission reserve requirement constraints. Just like the “scheduled” generation reserve, the “scheduled” transmission reserves in positive direction and negative direction are X pos inj , (19) ∆fl,t := Fl − Γl,m Pm,t neg ∆fl,t := Fl + m X inj , Γl,m Pm,t (20) m respectively. They are always non-negative. An important issue related to the transmission reserve is the credit entitled to the Financial Transmission Right (FTR) holders. FTR is a financial instrument used to hedge congestion cost in the electricity market, where participants are charged or credited due to the transmission congestion [21], [25]. Within the robust framework, the effective transmission capacity for base-case scenario is different from the physical limit, which is used in the Simultaneous Feasibility Test (SFT) for FTR market [18], [20], [21]. In the existing market, the FTR credit is funded by the energy congestion cost, which is the net payment of energy. However, the energy congestion cost may not be sufficient to fund the FTR credit [18], [19]. We argue that the transmission reserve becomes a new reason for FTR underfunding in any framework to guarantee the ramping deliverability. pos neg Theorem 1. If transmission reserve ∆fl,t and ∆fl,t are kept for line l at time t in DAM, then the maximum FTR underfunding associated with line l at time t is X pos neg k k η̄l,t ∆fl,t + ηl,t ∆fl,t (21) ¯ k∈K due to the deficiency of energy congestion cost. Uncertainty Payment Gen. Res. Credit Energy Payment Trans. Res. Credit LMP Cong. Cost Energy Credit FTR Credit Fig. 1. Money flow of the proposed market clearing mechanism, where uncertainty sources make the uncertainty payment, and LSEs make the energy payment. Please check Appendix D-C for the proof. From the FTR holder’s point of view, (21) is the credit due to the transmission reserve. Therefore, we also call (21) transmission reserve credit, and denote it as X pos neg k k η̄l,t ∆fl,t ∆fl,t ΘTl,t := + ηl,t . (22) ¯ k∈K k k At most one of η̄l,t and ηl,t is non-zero for transmission l. ¯ The credit to positivePtransmission reserve is zero for line l at pos k time t, when either k∈K η̄l,t = 0 or ∆fl,t = 0. Theorem 2. If (RED) is feasible, then uncertainty payment can exactly cover generation reserve credit and transmission reserve credit, and the revenue adequacy is always guaranteed in the proposed market clearing mechanism. Please check Appendix D-D for the proof. Theorem 1 reveals that FTR underfunding issue can occur within the existing market structures as long as the transmission reserve is non-zero, even if the LMPs are calculated based on other approaches. Theorem 2 shows that the new market clearing mechanism overcomes the FTR underfunding issue. The money flow of the proposed market clearing mechanism is depicted in Fig.1. Energy payment collected based on LMP is distributed to FTR holders as LMP congestion cost and generators as energy credit. On the other hand, the payment collected based on UMP is distributed to FTR holders as transmission reserve credit and flexible resources as generation reserve credit. The LMP congestion cost and transmission reserve credit can exactly cover the FTR credit, which is calculated based on the LMP difference and FTR amount. E. Market Equilibrium In this section, we characterize the competitive market equilibrium model. In the electricity industry, the partial market equilibrium model is often employed [1], [13], [26], where market participants are price takers [27]. The energy is cleared according to (6a). Uncertainty and generation reserve are cleared according to (7a). Without loss of generality, consider unit i located at bus m. Its profit maximization problem can be formulated as ! u,up up u,down down e X Pi,t πm,t + πm,t Qi,t + πm,t Qi,t (PMPi )max Pi,t −CiP (Pi,t ) t s.t. (6b) − (6e), (15) − (16) 6 where the decision variable is Pi,t given the price signal u,up u,down e (πm,t , πm,t , πm,t ). As proved in Appendix D-E, unit i is not inclined to change its power output level as it can obtain the maximum profit by following the ISO’s dispatch e instruction P̂i,t . Price signal πm,t provides the incentives for u,k unit i to dispatch power output to P̂i,t , and price πm,t gives incentives for unit i to maintain the generation reserve for uncertainty. Hence, the dispatch instruction P̂i,t and price u,up u,down e signal (πm,t , πm,t , πm,t ) constitute a competitive partial equilibrium [27]. F. Discussions down k As the Pi,t and ∆Pi,t (or (Qup i,t , Qi,t )) are coupled by k k (7b) and (7c), the opportunity cost (β̄i,t − βi,t ) is enough to ¯ provide the incentives for i to keep the generation level at P̂i,t . k k Including ᾱi,t − αi,t in the generation reserve price has several ¯ benefits. Firstly, generation reserves provided by different units are priced fairly. Generation reserve prices are the same for the units at the same bus, and they may vary with locations if line congestions exist. Secondly, higher generation reserve price attracts long-term investment for flexible resources. Thirdly, it is consistent with the existing reserve pricing practice [2], [28]. In fact, generation reserve price is consistent with the UMP. Therefore, the uncertainties and flexibilities are also treated fairly at the same bus. The upward and downward UMPs are obtained according to (11), respectively. The uncertainty sources are charged according to (13). The generation reserves are credited according to u,k k defined in (9) and re-dispatch ∆Pi,t (18). The price signal πm,t are intermediate variables for market clearing. The proposed UMP may be non-zero even if the uncertainty at a bus is zero. This is similar to the LMP, which may also be non-zero for the bus without load. The market clearing mechanism proposed in this paper follows the cost causation principle for the cost allocation. In reality, it may be controversial to allocate the reserve cost to uncertainty sources. However, we argue that it would be fair and must be done when the RES penetration level is high. An extreme case is when the loads are all supplied by RES. There has been study showing it is possible that the increasing RES penetration can cause higher system operation cost. This issue cannot be handled by the existing market clearing mechanism, in which loads pay for the additional system reserve that is required to accommodate the uncertainty from RES. In other words, loads are actually providing subsidies to RES. When the RES penetration level is low, the subsidies can help the growth of the RES. However, when the RES penetration level is high, these growing subsidies will cause serious fairness issue. On the other hand, with UMP as the stimulating price signals, RES will have the incentives to improve its forecast techniques and reduce its uncertainty. In the ideal case when its uncertainty approaches zero, RES will no longer pay. Following the existing practice, the UC variables are fixed during the marginal price derivation. Hence, the uplift issue, which exists in the real market, still remains in the proposed market clearing mechanism. Although the UC variables are fixed, the LMP and reserve price in the real market can provide effective signals for the long-term investment of generation and transmission as well as consumption strategy of electricity. Similarly, the uncertainty impact is not only reflected in UC, but also in the ED within the RUC model in this paper. Hence, the proposed LMP and UMP can also provide signals for the long-term investment of flexibilities (i.e. generation, transmission, and demand). The pricing for uncertainties proposed in this paper is not in conflict with the pricing for traditional reserves, which are mainly prepared for the contingencies. The traditional reserve prices can be derived in the framework by adding extra traditional reserve constraints, and the corresponding reserve costs can still be allocated to LSEs. It is observed that the credit in (14) is the sum of credits for all extreme points. That is because the related constraints may be binding for multiple extreme points, and the dual variables (shadow prices) for these constraints work together in the dual problem. The traditional price for energy and reserve also has similar form when multiple contingencies are modeled. Although only one scenario will happen in reality, we still need to consider the worst scenario defined in uncertainty set and keep enough reserves in DAM. That is because DAM is a financial market, and the LMP and UMP are the financially binding prices. This is similar to the existing market model considering contingencies. Even if the contingency seldom occurs, they are still modeled for market clearing, and the contingencies are reflected in LMP and reserve price. The issue of price multiplicity still exists in the proposed model [29] because problem (RED) is a linear programming (LP) problem. However, the price is unique with the nondegeneracy assumption. For simplicity, we have considered a single-sided auction in the proposed model. By introducing the demand bids, we can formulate a double-sided auction and the general principles in this paper will still apply. III. C ASE S TUDY A six-bus system and the IEEE 118-Bus system are simulated to illustrate the proposed market clearing mechanism. In the six-bus system, the basic ideas of UMP are presented within the robust optimization framework. FTR underfunding issue is illustrated and a comparison between the UMP and traditional reserve price are presented. In the IEEE 118-Bus system, the UMP related products are presented for different uncertainty levels. The behaviors and impacts of flexible sources are analyzed by an energy storage example. A. Six-bus System A six-bus system is studied in this section. The one-line diagram is shown in Fig. 2. The unit data and line data are shown in Table I and Table II, respectively. Table III presents the load and uncertainty information. Column “Base Load” shows the hourly forecasted load. Assume that the load distributions are 20%, 40%, and 40% for Bus 3, Bus 4, and Bus 5, respectively. ū1,t and ū3,t in Table III are the bounds of the uncertainties at Bus 1 and Bus 3, respectively. The uncertainty bounds at other buses are 0. 7 1 G1 G2 2 L1 TABLE IV M ARGINAL C OSTS AT D IFFERENT G ENERATION L EVELS ($/MW H ) 3 Gen. 1 P1w 6 5 4 L3 L2 ¯ 100 124 148 172 196 G3 Fig. 2. One-line diagram for 6-bus system. TABLE I U NIT DATA FOR THE 6- BUS S YSTEM # P min P max P0 1 100 2 10 6 10 a b c 4 3 1 4 2 1 4 3 −2 P min ,P max ,P0 : min/max/initial generation level (MW); fuel cost ($): aP 2 + bP + c ; Ru ,Rd : ramping up/down rate (MW/h); Cu ,Cd : startup/shutdown cost ($); T on ,T off ,T0 : min on/min off/initial time (h) It is assumed that the relative forecasting errors increase with hours. Uncertainty 1,t and 3,t also respect (23a) −Λ · ūm,t ≤ m,t ≤ Λ · ūm,t , ∀t, m X \|m,t \| ≤ Λ∆ , ∀t, ūm,t m (23b) where (23a) denotes the uncertainty interval at a single bus, and (23b) represents the system-wide uncertainty [5]. The Λ and Λ∆ are the budget parameters for the single bus and system, respectively. 1) LMP and UMP: Consider the case where Λ = 1, Λ∆ = 2. The CCG based approach converges after 2 iterations. TABLE II L INE DATA FOR THE 6- BUS S YSTEM from 1 1 2 5 3 2 4 to 2 4 4 6 6 3 5 x(p.u.) 0.17 0.258 0.197 0.14 0.018 0.037 0.037 capacity(MW) 200 100 100 100 100 200 200 TABLE III L OAD AND U NCERTAINTY DATA FOR THE 6- BUS S YSTEM (MW) Time (h) Base Load 1 2 3 4 5 6 7 8 9 10 11 12 175.19 165.15 158.67 154.73 155.06 160.48 173.39 177.6 186.81 206.96 228.61 236.1 ū1,t 1.09 2.06 2.98 3.87 4.85 6.02 7.59 8.88 10.51 12.94 15.72 17.71 ū3,t Time (h) Base Load 0.29 0.55 0.79 1.03 1.29 1.6 2.02 2.37 2.8 3.45 4.19 4.72 13 14 15 16 17 18 19 20 21 22 23 24 242.18 243.6 248.86 255.79 256 246.74 245.97 237.35 237.31 232.67 195.93 195.6 mar. cost 124 148 172 196 220 14.396 14.588 14.78 14.972 15.164 P2w P̄2w ¯ 10 28 28 46 46 64 64 82 82 100 Gen. 3 mar. cost 32.638 32.674 32.71 32.746 32.782 P3w ¯ 10 12 14 16 18 P̄3w mar. cost 12 14 16 18 20 17.71 17.73 17.75 17.77 17.79 TABLE V G ENERATION AND R ESERVE (Λ = 1, Λ∆ = 2, MW) Ru Rd Cu Cd T on T off T0 220 120 0.004 13.5 176.9 24 24 180 50 100 50 0.001 32.6 129.9 12 12 360 40 20 0 0.005 17.6 137.4 5 5 60 0 Gen. 2 P̄1w ū1,t ū3,t 19.68 21.32 23.33 25.58 27.2 27.76 29.21 29.67 31.15 31.99 28.16 29.34 5.25 5.68 6.22 6.82 7.25 7.4 7.79 7.91 8.31 8.53 7.51 7.82 T P1 P2 P3 up up up Q1 Qdown Q2 Qdown Q3 1 2 21 195.19 25.58 16.54 24 −24 12 −12 3.46 Qdown 3 −5 Hence, K = {1, 2}. Given the UC solutions, the problem (RED) can be solved by commercial LP solver. The marginal prices are then obtained as byproducts. The generation outputs are presented in Table V at Hours 21. It can be observed that G1 supplies most of the loads at Hour 21, which is 195.19 MW. According to the bid information in Table IV, G2 is much more expensive than G1 and G3. Hence, the output of G2 is relatively small and at the low level of its capacity. The upward and downward generation reserves provided by the three units are also listed in Table V. These data can be obtained directly from Eqs. (15) and (16) given the generation output Pi,t . Although the remaining generation capacity of G1 is 220 − 195.19 = 24.62 MW, the upward reserve is limited by its upward ramping rate 24 MW. In the meantime, the upward reserve provided by G3 is limited by its generation capacity although it has more remaining ramping capacity (i.e. min{20 − 16.54, 5} = 3.46 MW). Table VI shows the extreme points obtained in the CCGbased approach. The intermediate price signals for these points u,k πm,t are also presented. It can be observed that the worst point is always obtained at the extreme point of the uncertainty set. For example, at Hour 21, the ˆ11,1 is 31.15 MW. It is exactly the upper bound of the uncertainty at Hour 21 at Bus 1. The data in Table VI also verifies Lemma 1. The intermediate UMPs u,k πm,t have the same sign as the uncertainties ˆkm,t at the same bus. The LMPs, aggregated upward UMPs, and aggregated downward UMPs at Hour 21 are shown in Table VII. It is noted that UMPs still exist at buses without uncertainties (i.e., Buses 2,4,5,6). This is similar to LMPs, which also exist at buses where net power injections are 0. The LMPs vary with locations, which indicates that the line congestion exists. TABLE VI E XTREME P OINTS OF U NCERTAINTY S ET k=1 t ˆ11,t ˆ13,t u,1 π1,t k=2 u,1 π3,t ˆ21,t ˆ23,t u,2 π1,t u,2 π3,t 21 31.15 8.31 14.87 14.87 −31.15 8.31 −17.67 1.77 8 TABLE VII LMP AND UMP AT H OUR 21 (Λ = 1, Λ∆ = 2) Bus1 Bus2 Bus3 Bus4 Bus5 Bus6 πe π u,up 14.97 14.87 -17.67 32.64 14.87 0 34.4 16.63 0 43.71 25.94 0 41.94 24.17 0 35.26 17.49 0 π u,down 10 16 Price $/MW Price Price $/MW 16.5 15.5 15 0 2 4 Bus The load at Bus 4 has to pay the highest LMP $43.71/MWh. The UMPs are also different at various locations. The highest upward UMP at Hour 21 is also located at Bus 4. With these prices, the market participants can be paid and credited. The LMP paid to G3 is $35.26/MWh on Bus 6, which is $17.49/MWh larger than its marginal cost. In the same time, The upward UMP is $17.49/MWh on Bus 6, which is exactly the difference between the LMP and G3’s marginal cost. Hence, G3 is the UMP setter on Bus 6. The UMPs provide important price signals on the planning of renewable energy sources and storages. For example, the UMP at Bus 2 is relatively small, so it is an ideal location for renewable energy sources in terms of payment for uncertainties. In contrast, the UMP at Bus 4 is large, which may attract the long-term investment for storages or generation plants with large ramping rates. 2) Comparison with Existing LMPs and Reserve Prices: The motivation of this part is to compare the proposed clearing scheme with the existing one. However, as the reserve is not robust in the traditional scheme, we cannot compare them fairly. With the observation that the transmission constraints are the most challenging one in the robust UC framework, we drop these constraints in this subsection and add reserve constraints as follows. Ii,t Pimin ≤ Qdown i,t + Pi,t , max Qup , ∀i, t (24a) i,t + Pi,t ≤ Ii,t Pi u −Rid Ii,t ∆T ≤ Qdown Qup i,t , i,t ≤ Ri Ii,t ∆T, ∀i, t X X up Qdown Qi,t ≥ R̄t , ∀t, i,t ≤ Rt , ¯ i i (24b) (24c) down where Qup are the largest upward and downward i,t and Qi,t reserves, respectively. Rt and R̄t are system-wide reserve ¯ requirements. Refer to [2], [30] for more details on the reserve formulations. In the experiment, ∆T is set to 1 and Λ = 0.8, Λ∆ = 2. The reserve requirements Rt and R̄t are set ¯ to the lower and upper bounds of the system-wide uncertainty in (23b), respectively. The results are as expected. The optimal solutions of the RUC and the standard UC with explicit reserve constraints are the same. LMPs calculated in the RUC and UC also have the same values. The UMPs calculated in the proposed mechanism are also exactly the same as the reserve prices in standard UC. Two things are verified with these results. First, without transmission constraints, the solution to standard UC can easily be robust by adding reserve constraints. Second, the proposed LMPs and UMPs are consistent with LMPs and reserve prices in the existing market when the transmission constraints are dropped. When considering transmission constraints, the generation reserve cannot be guaranteed at bus levels in the traditional 5 (a) Hour 20 6 2 4 6 Bus (b) Hour 21 Fig. 3. Upward UMP (blue bar) and reserve price (red bar) with network constraint. UC model. For simplicity, we assume that the 6 buses are in a zone. Consider the case where Λ = 0.8, Λ∆ = 2. The upward UMP and reserve price at Hour 20 are depicted in Fig. 3a. It is observed UMPs at Bus 1 and Bus 2 are lower than the traditional reserve prices. In the same time, the UMPs at Bus 4 and Bus 5 is higher than the traditional reserve prices. The differences are caused by the congestion of Line 1-4 for reserve delivery. It is worth mentioning that the LMP differences in two models are within 1% at Hour 20. The prices illustrated in Fig. 3b reveals another trends that the UMP may be higher than the traditional reserve prices. At Hour 21, the zonal reserve price is 0 while the UMPs are nonzeros at bus 4, 5, and 6. Because the constraint related with reserves in the RUC is stronger than the one in traditional UC model. Consequently, more expensive resources are used in RUC, which also generally leads to higher UMPs. 3) FTR Underfunding: When Λ∆ = 2, Λ = 1, the generation schedules at Hour 21 are 195.193MW, 25.577MW, and 16.54MW. The power flow of Line 2 is 97.63MW, which is 2.47MW smaller than its physical limit of 100MW. The transmission reserve 2.47 MW is kept to guarantee the delivery of the generation reserve. The binding constraint for Line 2 causes LMP differences. Hence, the FTR holder gets credits. Consider a set of FTR amounts [202.3429, 23.2771, −55.772, −94.924, −94.924, 20]. It can be verified that the FTR amounts satisfy the SFT in the FTR market. Then the total credit for the FTR holders is $5,554.77. However, the congestion cost in the DAM is $5,422.87. It means that the LMP congestion cost collected is not enough to cover the FTR credit. The FTR underfunding value is $5554.77 − $5422.87 = $131.90. The revenue residues after UMP settlement is $131.9. It exactly covers the FTR underfunding in this scenario. Therefore, the revenue is adequate at Hour 21. B. IEEE 118-Bus System The simulations are performed for the IEEE 118-bus system with 54 thermal units and 186 branches in this section. The peak load is 6600MW. The detailed data including generator parameters, line reactance and ratings, and load profiles can be found at http://motor.ece.iit.edu/Data/RUC118UMP.xls. Two cases are studied in this section. 1) The uncertainty levels and load levels are changed to analyze the simulation results in the system level. The impact of transmission line capacity on prices is also studied. 9 ·104 TABLE VIII O PERATION C OST AND UMP PAYMENT ($,Λ∆ = 10) ·106 UP GRC 2 OC Op. Cost Un. Payment Gen. Res. Credit Rev. Res. 0.2 0.25 0.3 1,866,023 1,871,364 1,877,471 11,043 20,044 30,879 10,560 19,209 28,658 483 835 2,221 1.9 3 OC ($) Λ UP and GRC ($) 4 2 1.8 2) An energy storage is installed at a specified bus with high UMP to show the potential application of UMPs. 95 100 105 Load Level (%) Fig. 5. Uncertainty payment (UP), generation reserve Credit (GRC), and operation cost (OC) with different load levels (Λ = 0.25) Upward UMP LMP 50 30 40 Price $/MW Price $/MW 1) Case 1: We assume that the uncertainty sources are located at buses (11, 15, 49, 54, 56, 59, 60, 62, 80, 90). The budget parameter Λ∆ is set to 10 in this section. The buslevel uncertainty budget parameter Λ changes from 0.2 to 0.3, and the bound of the uncertainty is the base load. The simulation results are shown in Table VIII. It can be observed that the total operation cost increases with increasing Λ. It indicates that a larger uncertainty level may increase the operation cost. The columns “Un. Payment” and “Gen. Res. Credit” denote the total payment from uncertainty sources and credit to generation reserves, respectively. The lowest payment is $11,043 and the highest one is $30,879. On the other hand, the credit entitled to the generation reserves is also a monotonically increasing function of Λ. When Λ = 0.3, the generation reserves have the highest credit. The last column “Rev. Res.” shows that the revenue residues related to UMPs. It can be observed that the residue is always positive. Fig. 4a in the next page depicts the heat map for the upward UMPs from Bus 80 to Bus 100 in 24 hours. The xaxis represents time intervals and the y-axis represents bus numbers. The color bar on the right shows different colors for various UMP values. For example, the $0/MWh is denoted by the blue color at the bottom, and the $18/MWh is represented by the dark red color on the top of the color bar. It can be observed that the uncertainty sources have system-wide unique UMPs at some intervals, such as Hours 8, 13, 15, and so on. It indicates that there is no transmission reserve in these hours. On the other hand, the UMPs at Hour 11 vary dramatically with different locations. The highest upward UMP is around $18/MWh, and the lowest one is around $2/MWh. According to the data shown in Fig. 4a, the high UMP at Bus 94 may attract investment of flexible resources, such as energy storages, in terms of generation reserve credit, and Bus 100 is an attractive location for the investment of renewable energy sources in terms of uncertainty payments. Fig. 5 shows the uncertainty payment and generation reserve credit with respect to load levels. The base load level is set at 100%. Higher loads in general lead to more uncertainty payments and generation reserve credits. It is also consistent with the heat map of UMPs in Fig. 4a, where UMPs at peak load hours are high. It suggests that the generation reserves also become scarce resources when load levels are high. The transmission line capacity plays an important role in the price calculation. Fig. 6 shows the LMPs and upward UMPs at Hour 11 with respect to increasing capacity of Line 94100. The prices at Buses 88, 94, and 100 are depicted. When the line capacity increases from 165MW to 175MW, LMP at 1 30 20 10 20 0 170 180 170 Line Capacity 180 Line Capacity Bus 88 Bus 94 Bus 100 Fig. 6. LMP (left) and upward UMP (right) at Hour 11 with respect to increasing capacity of Line 94-100 Bus 94 decreases from $47.92/MWh to $35.84/MWh and that at Bus 88 also drops to $30.58/MWh from $38.78/MWh. The upward UMPs at Bus 94 and Bus 88 also drop by $8.20/MWh and $12.08/MWh, respectively. In contrast, the LMP and upward UMP at Bus 100, which is connected to Line 94-100, remain at $19.42/MWh and $1.64/MWh, respectively. It shows that the change of line capacity may only have impacts on the prices at some buses. When the line capacity further increases to 185MW from 175MW, the changes of LMPs and UMPs at Bus 94 and Bus 88 are within $0.1/MWh, and there is still no change at Bus 100. It means that the additional 10MW cannot help deliver cheaper energy and reserves to Bus 94 and Bus 88. These results are also consistent with the analysis of the traditional LMPs [31]. 2) Case 2: As discussed in Case 1, the upward UMP on Bus 94 is high at Hour 11. Assume that an energy storage (8MW/30MWh) is installed at Bus 94. A simple model for the energy storage is formulated as follows. Et = Et−1 + ρd PtD + ρc PtC , ∀t 0 ≤ Et ≤ E max , ∀t 0 ≤ −PtD ≤ ItD RD , ∀t 0 ≤ PtC ≤ ItC RC , ∀t ItD + ItC ≤ 1, ∀t ENT = E0 , where Et denotes the energy level, PtD and PtC represent the discharging and charging rates, and ItD and ItC are the indicators of discharging and charging. As the UMP is the major concern in this section, we use simplified parameters for 10 18.00 18.00 Bus 82 Bus 82 16.00 16.00 Bus 84 Bus 84 14.00 14.00 Bus 86 Bus 86 12.00 12.00 Bus 88 Bus 88 10.00 Bus 90 8.00 Bus 92 10.00 Bus 90 8.00 Bus 92 6.00 Bus 94 6.00 Bus 96 4.00 Bus 96 4.00 Bus 98 2.00 Bus 98 2.00 Bus 94 Bus 100 Hour 5 Hour 10 Hour 15 Hour 20 0.00 Bus 100 Hour 5 Hour 10 Hour 15 Hour 20 0.00 (b) With Storage at Bus 94 (a) Without Storage at Bus 94 Fig. 4. Heat Map for Upward UMPs (Λ = 0.3). Different colors represent various UMPs. Figure (a) depicts the UMPs from Bus 80 to Bus 100 in 24 hours without the energy storage at Bus 94. Figure (b) depicts the new UMPs after the energy storage is sited at Bus 94. storage. The discharging efficiency ρd and charging efficiency ρc are set to 100%. The capacity E max and initial energy level E0 are set to 30 MWh and 15 MWh, respectively. The maximal charging rate RD and discharging rate RC are set to 8 MW/h. By siting the energy storage, we can lower the new operation cost to $1,875,211 from $1,877,471. The payment collected from the uncertainty sources becomes $27,473, and the credit to generation reserves decreases to $24,289. Compared to the data in Table VIII, the energy storage also helps to reduce the payment related to UMPs. The storage is entitled to $1326 generation reserve credit. Fig. 4b depicts the new upward UMPs after the installation of the energy storage. Compared to that in Fig. 4a, the upward UMP for Hour 11 at Bus 94 decreases a lot. The UMPs for Hour 10 and 12 are also lower. It suggests that sitting the energy storage at Bus 94 effectively lower the generation reserve price. The simulation results demonstrate that flexible resources can lower the UMPs, and UMPs provide the investment signal at locations where generation reserves are scarce resources. prices within the new market scheme, as the reserve fees are paid by uncertainty sources. Many potential applications on UMPs are open. As UMPs are unified prices of uncertainties and reserves, it is interesting to investigate the optimal strategy for the one who is the uncertainty source as well as reserve provider in the market (e.g. the wind generation company with energy storages). The UMPs derived in this paper also provide an important price signal for the long-term investment of flexible resources. When the upward UMP or downward UMP at a bus is high, the investor can get more return in terms of generation reserves. Another potential future research on UMP is to study how to determine the budget uncertainty set in the market. Modeling the traditional spinning reserve for the contingency [13], [30] is also our future work. In this paper, the demand bids are not considered. We have forecasted load, forecasted RES, and uncertainty of load and RES for market clearing with a singlesided model. In an extended double-sided model, we can have demand bids, forecasted RES, and uncertainty of load and RES for market clearing. The forecasted load, forecasted RES, uncertainty of load and RES can be used in RAC. IV. C ONCLUSIONS A novel market model in this paper clears uncertainty, energy, and generation reserve simultaneously within the RUC framework in DAM. The uncertainty sources are charged and the generator reserve providers are credited based on the proposed UMP. The UMP formulation is derived within a robust optimization framework. We also characterize the market equilibrium for the new market clearing mechanism. As the market clearing mechanism is established within the robust optimization framework, the robustness of the dispatch is guaranteed. The optimal reserves for uncertainty accommodation are obtained in the model. The UMP proposed in this paper can effectively address the issue on how to charge and credit the uncertainties and generation reserve fairly in the market with RES. Our study also shows that traditional pricing mechanism within RUC framework may lead to FTR underfunding. The proposed market clearing mechanism can address this issue. Our study shows load serving entities can have lower energy A PPENDIX A D ETAILED F ORMULATION FOR P ROBLEM (RUC) (RUC) s.t. min (x,y,z,I,P )∈F X Pi,t = X Ii,t Pimin m t X m i −Fl ≤ XX i CiP (Pi,t ) + CiI (Ii,t ) dm,t , ∀t.  Γl,m  X i∈G(m) (25a) (25b)  Pi,t − dm,t  ≤ Fl , ∀l, t (25c) ≤ Pi,t ≤ Ii,t Pimax , ∀i, t Pi,t − Pi,(t−1) ≤ riu (1 − yi,t ) + Pimin yi,t , ∀i, t −Pi,t + Pi,(t−1) ≤ rid (1 − zi,t ) + Pimin zi,t , ∀i, t minimum on/off time limit (26a) (26b) (26c) 11 inj ∆Pm,t = and n F := (x, y, z, I, P ) : ∀ ∈ U, ∃∆P such that X X ∆Pi,t = m,t , ∀t, i∈G(m) (27a) m i Ii,t Pimin ≤ Pi,t + ∆Pi,t ≤ Ii,t Pimax , ∀i, t (27b) −Rid (1 − zi,t+1 ) ≤ ∆Pi,t ≤ Riu (1 − yi,t ), ∀i, t (27c) X inj ∆Pm,t = ∆Pi,t − m,t , ∀m, t (27d) i∈G(m) −Fl ≤ o inj inj Γl,m (Pm,t + ∆Pm,j ) ≤ Fl , , ∀l, t . (27e) X m The basic idea of the above model is to find a robust UC and dispatch for the base-case scenario. In the base-case scenario, (25b) denotes the load balance constraint; (25c) represents the transmission line constraint; (26a) denotes the unit capacity limit constraint; (26b)-(26c) denote the unit ramping up/down limits. Ii,t , yi,t , and zi,t are the indicators of the unit being on, started-up, and shutdown, respectively. Units also respect the minimum on/off time constraints which are related to these binary variables [1]. The UC and dispatch solution are immunized against any uncertainty ∈ U. When uncertainty occurs, it is accommodated by the generation adjustment ∆Pi,t (27a). Generation dispatch is also enforced by the capacity limits (27b). (27c) models the ramping rate limits of generation adjustment ∆Pi,t . In fact, the right and left hand sides of (27c) can correspond to a response time ∆T , which is similar to the 10-min or 30-min reserves in the literatures [30]. (27e) stands for the network constraint after uncertainty accommodation. A PPENDIX B D ETAILED F ORMULATION FOR P ROBLEM (MP) AND (SP) (MP) S.T. X i min (x,y,z,I,P,∆P ) XX t CiP (Pi,t ) + CiI (Ii,t ) i (25b), (25c), (26a)-(26c), minimum on/off time limit X k ∆Pi,t = km,t , ∀t, ∀k ∈ K (28a) m k Ii,t Pimin ≤ Pi,t + ∆Pi,t ≤ Ii,t Pimax , ∀i, t, ∀k ∈ K (28b) k −∆Pi,t (28d) k ∆Pi,t ≤ Riu (1 − yi,t ), ∀i, t, ∀k ∈ K −Fl ≤ Rid (1 ≤ X m inj,k ∆Pm,t = − zi,t+1 ), ∀i, t, ∀k ∈ K (28e) k ∆Pi,t − km,t , ∀m, t, ∀k ∈ K, (28f) i∈G(m) and (28c) inj inj,k Γl,m (Pm,t + ∆Pm,t ) ≤ Fl , ∀k ∈ K, ∀l, t X (SP) max ∈U min (s+ ,s− ,∆P )∈R() XX − (s+ m,t + sm,t ) (29a) m t n + − R():= (s , s , ∆P ) : X X − ∆Pi,t = (m,t + s+ m,t − sm,t ), ∀m, t i −Fl ≤ X m m X (29b) (29c) inj inj + ∆Pm,t ≤ Fl , ∀l, t (29d) Γl,m Pm,t − ∆Pi,t − (m,t + s+ m,t − sm,t )(29e) − s+ m,t , sm,t ≥ 0, ∀m, t o (27b), (27c) (29f) where K is the index set for uncertainty points ˆ which are dynamically generated in (SP) with iterations. It should be k noted that ˆk is the extreme point of U. Variable ∆Pi,t is k associated with ˆ . The objective function in (SP) is the − summation of non-negative slack variables s+ m,t and sm,t , which evaluates the violation associated with the solution − (x, y, z, I, P ) from (MP). s+ m,t and sm,t are also explained as un-followed uncertainties (generation or load shedding) due to system limitations. A PPENDIX C L AGRANGIAN F UNCTION FOR P ROBLEM (RED) Please check equation (30) in the next page. A PPENDIX D P ROOFS FOR LEMMAS AND THEOREMS A. Proof of Lemma 1 u,k < 0. With a small perturbation Proof. Consider ˆkm,t > 0, πm,t δ > 0 to ˆkm,t , we replace ˆkm,t with ˆkm,t − δ in (RED). u,k As the πm,t < 0 , then the optimal value to problem (RED) increases. It means that there are violations for the original optimal solution Pi,t to problem (RED) with ˆkm,t − δ. Hence, the optimal solution Pi,t to problem (RED) cannot be immunized against the uncertainty ˆkm,t −δ. It contradicts with the robustness of the solution Pi,t . Therefore, if ˆkm,t > 0, then u,k u,k ≤ 0. πm,t ≥ 0. Similarly, if ˆkm,t < 0, then πm,t B. Proof of Lemma 2 Proof. Assume i ∈ G(m), according to the KKT condition ∂L(P, ∆P, λ, α, β, η) =0 k ∂∆Pi,t (31) at the optimal point, we have k k k k β̄i,t − βi,t + ᾱi,t − αi,t − λkt + ¯ ¯ X l k k (η̄l,t − ηl,t )Γl,m = 0. (32) ¯ u,k k k Then (17) holds. If πm,t > 0, then β̄i,t + ᾱi,t > 0 as k k k k β̄i,t , βi,t , ᾱi,t , and αi,t are non-negative. According to the ¯ ¯ complementary conditions for (7b) and (7d), at least one of (7b) and (7d) is binding. Hence, n o k ∆Pi,t = min Iˆi,t Pimax − Pi,t , Riu (1 − ŷi,t ) u,k holds. Similarly, the other equation holds when πm,t < 0. 12 L(P, ∆P, λ, α, β, η) XX XX X X X CiP (Pi,t ) + λt dm,t − Pi,t + β̄i,t (Pi,t − Iˆi,t Pimax ) + βi,t (Iˆi,t Pimin − Pi,t ) ¯ t t m t i i i XX min min + ᾱi,t Pi,t − Pi,t−1 − riu (1 − ŷi,t ) − Pi,t ŷi,t + αi,t Pi,t−1 − Pi,t − rid (1 − ẑi,t ) − Pi,t ŷi,t ¯ t i X X X X XX X X inj inj k + η̄l,t Γl,m Pm,t − Fl − ηl,t Γl,m Pm,t + Fl + λkt km,t − ∆Pi,t ¯ t m m m i l k∈K t X XX k k max k ˆ min k ˆ + β̄i,t (Pi,t + ∆Pi,t − Ii,t Pi ) + βi,t (Ii,t Pi − Pi,t − ∆Pi,t ) ¯ i k∈K t X XX k k k k ∆Pi,t + Rid (1 − ẑi,t ) ∆Pi,t − Riu (1 − ŷi,t ) − αi,t + ᾱi,t ¯ i k∈K t X X XX X inj inj,k inj inj,k k k + η̄l,t Γl,m (Pm,t + ∆Pm,t ) − Fl − ηl,t Γl,m (Pm,t + ∆Pm,t ) + Fl ¯ m m k∈K t l = C. Proof of Theorem 1 Proof. The energy congestion cost at t is X X e e πm,t dm,t − πm,t Pi,t m = X (33) i∈G(m) m XX X X k η̄l,t k ηl,t − ηl,t − ¯ m l k∈K k∈K ¯ ! X X pos k = η̄l,t + η̄l,t Fl − ∆fl,t = Γl,m l − = = η̄l,t + k∈K X X l X k ηl,t ηl,t + ¯ k∈K ¯ ! X neg ∆fl,t − Fl X k k η̄l,t + ηl,t + ηl,t ¯ ¯ l k∈K k∈K XX pos neg k k − η̄l,t ∆fl,t + ηl,t ∆fl,t ¯ l k∈K X pos neg − η̄l,t ∆fl,t + ηl,t ∆fl,t ¯ l X η̄l,t + X k k η̄l,t + ηl,t + ηl,t ¯ ¯ l k∈K k∈K XX pos neg k k − η̄l,t ∆fl,t + ηl,t ∆fl,t ¯ l k∈K η̄l,t + X ! inj Pm,t ! ! Fl Fl The first equality holds following the definition of net power injection. The second equality holds according to (8) and P inj P following (19) and m m,t = 0. The third equality Pholds k (20). The sign change of ηl,t and l ηl,t in the third equality ¯ direction. According is because of the definition¯ of power flow to the complementary conditions, the third term in the fourth equality must be zero based on the following three cases. pos ∆fl,t 3) η̄l,t = 0 and ηl,t = 0. ¯ The second term in the last equality corresponds to (21). The credits to FTR holders can be written as X e e (πm,t − πn,t )FTRm→n (34) (m→n) inj e πm,t Pm,t 1) If η̄l,t 6= 0, then = 0, and ηl,t = 0. neg 2) If η̄l,t = 0 and ηl,t 6= 0, then ∆f¯l,t = 0. ¯ (30) X  Γl,m (η̄l,t − ηl,t )   ¯ l   XX  k k  Γl,m (η̄l,t − ηl,t ) − X  ¯  l k∈K   X =  FTRm→n   −λt + Γ (η̄ − η ) l,n l,t l,t  (m→n)  ¯   l   XX  k k  + Γl,n (η̄l,t − ηl,t ) ¯ l k∈K X   k (Γl,n − Γl,m )(η̄l,t + η̄l,t ) X X  k∈K   FTRm→n = X  k  −(Γl,n − Γl,m )(ηl,t + ηl,t ) (m→n) l ¯ k∈K ¯ ! X X X k k ≤ η̄l,t + η̄l,t + ηl,t + ηl,t Fl ¯ l k∈K k∈K ¯  λt − The first equality holds according to (8). The inequality is true as the amount of FTRm→n respects X −Fl ≤ (Γl,m − Γl,n )FTRm→n ≤ Fl . m→n according to the SFT for FTR market [18], [20], [21]. The right-hand-side of the inequality is the first term in the last equality of (33). Based on (33) and (34), the maximum difference between the FTR credit and the energy congestion cost is equal to the transmission reserve credit. That is, the maximum FTR underfunding is (21). D. Proof of Theorem 2 Proof. According to Theorem 1, the FTR underfunding value is (21) due to the deficiency of the energy congestion cost. 13 Therefore, we need to prove that the money collected from uncertainty sources can cover the FTR underfunding and credits to generation reserve. Without loss of generality, we consider the payment collected from uncertainty sources at time t for ˆk X u,k πm,t ˆkm,t m λkt − m = l XX m = X k ∆Pi,t + l i∈G(m) l k k Γl,m η̄l,t ( ∆Pi,t ) i∈G(m) XX m i∈G(m) k Γl,m η̄l,t l X P P X − k η̄l,t X X m l inj − Fl Γl,m Pm,t pos k η̄l,t ∆fl,t l k k Γl,m ηl,t ˆm,t can be reformulated similarly. Hence, ¯ the second equality holds. The third equality holds from (9). Therefore, XX XX XX Ψm,t = ΘG ΘTl,t i,t + m l m t i t l t holds. That is, the uncertainty payment covers the generation reserve credit and transmission reserve credit. Then, following the energy congestion cost shown in (33), XX XX e πm,t dm,t + Ψm,t ≥ m t X X i + e πm P + i ,t i,t t X X t m→n e (πm,t − m t X X ΘG i,t t i e πn,t )FTRm→n holds. That is, the total payments collected from loads and uncertainty sources can cover the total credits to energy, generation reserve, and FTR holders. So, the revenue adequacy of the proposed market clearing mechanism is guaranteed. E. Proof of Competitive Equilibrium down Proof. Pi,t and (Qup i,t , Qi,t ) are coupled by constraints (15) and (16). According to (17), we can rewrite generation reserve credit as X u,k u,up up u,down down k πm,t Qi,t + πm,t Qi,t = πm,t ∆Pi,t k∈K k k k k k (β̄i,t − βi,t + ᾱi,t − αi,t )∆Pi,t ¯ ¯ k∈K ! k ˆ k β̄i,t (Ii,t Pimax − Pi,t ) + βi,t (Pi,t − Iˆi,t Pimin ) = (35) ¯k d k +ᾱi,t (Riu (1 − ŷi,t )) + αi,t Ri (1 − ẑi,t+1 ) k∈K ¯ Substituting (35) into problem (PMPi ), we can decouple Pi,t down and (Qup i,t , Qi,t ). In fact, we also get all terms related to Pi,t in Lagrangian L(P, λ, α, β, η) for problem (RED). Since the problem (RED) is a linear programming problem, the saddle point P̂i,t , which is the optimal solution to (RED), is also the optimal solution to (PMPi ). Consequently, unit i is not inclined to deviate its output level as it can obtain the maximum profit by following the ISO’s dispatch instruction u,k e P̂i,t . Therefore, dispatch P̂i,t and price signal (πm,t , πm,t ) constitute a competitive partial equilibrium [27]. X The first equalityPholds to (9). According to (7a), P according k k (7f), and (7g), the m l Γl,m η̄l,t ˆm,t in the second line can be rewritten as X X XX k k k Γl,m η̄l,t (∆Pi,t + Pi,t ) − dm,t − η̄l,t Fl X X k k k Γl,m η̄l,t ˆm,t − ηl,t ¯ m l X XX X k k k k k ( ∆Pi,t = ∆Pi,t λt − Γl,m η̄l,t − ηl,t ) ¯ m i l i∈G(m) X pos neg k k η̄l,t + ∆fl,t + ηl,t ∆fl,t ¯ l X X u,k X pos neg k k k η̄l,t = πm,t ∆Pi,t + ∆fl,t + ηl,t ∆fl,t ¯ m i∈G(m) l X u,k X pos neg k k k πmi ,t ∆Pi,t η̄l,t = + ∆fl,t + ηl,t ∆fl,t ¯ i l = X = R EFERENCES [1] M. Shahidehpour, H. Yamin, and Z. Li, Market Operations in Electric Power Systems: Forecasting, Scheduling, and Risk Management, 1st ed. Wiley-IEEE Press, 2002. [2] T. Zheng and E. Litvinov, “Contingency-Based Zonal Reserve Modeling and Pricing in a Co-Optimized Energy and Reserve Market,” IEEE Trans. Power Syst., vol. 23, no. 2, pp. 277–286, May 2008. [3] R. Jiang, J. Wang, and Y. Guan, “Robust unit commitment with wind power and pumped storage hydro,” IEEE Trans. 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Hongxing Ye (S’14-m’16) received his B.S. degree in Information Engineering, in 2007, and M.S. degree in Systems Engineering, in 2011, both from Xi’an Jiaotong University, China, and the Ph.D. degree in Electrical Engineering from the Illinois Institute of Technology, Chicago in 2016. His research interests include large-scale optimization in power systems, electricity market, renewable integration, and cyber-physical system security in smart grid. He is “Outstanding Reviewer” for IEEE Transactions on Power Systems and IEEE Transactions on Sustainable Energy in 2015. He received Sigma Xi Research Excellence Award at Illinois Institute of Technology in 2016. Yinyin Ge (S’14) received the B.S. degree (2008) in Automation and M.S. degree (2011) in Systems Engineering from Xian Jiaotong University, China. She also received Ph.D. degree (2016) in Electrical Engineering at Illinois Institute of Technology, Chicago. Her research interests are power system optimization and modeling; PMU applications in Smart Grid; monitoring, visualization, and state estimation for distribution systems. Mohammad Shahidehpour (F’01) received his Ph.D. degree from the University of Missouri in 1981 in electrical engineering. He is currently the Bodine Chair Professor and Director of the Robert W. Galvin Center for Electricity Innovation at the Illinois Institute of Technology, Chicago. He is the founding Editor-in-Chief of IEEE Transactions on Smart Grid. He is a member of US National Academy of Engineering (NAE). Zuyi Li (SM’09) received the B.S. degree from Shanghai Jiaotong University, Shanghai, China, in 1995, the M.S. degree from Tsinghua University, Beijing, China, in 1998, and the Ph.D. degree from the Illinois Institute of Technology (IIT), Chicago, in 2002, all in electrical engineering. Presently, he is a Professor in the Electrical and Computer Engineering Department at IIT. His research interests include economic and secure operation of electric power systems, cyber security in smart grid, renewable energy integration, electric demand management of data centers, and power system protection.	3
QUIVER MUTATIONS AND BOOLEAN REFLECTION MONOIDS arXiv:1711.09995v3 [math.RA] 21 Feb 2018 BING DUAN, JIAN-RONG LI, AND YAN-FENG LUO Abstract. In 2010, Everitt and Fountain introduced the concept of reflection monoids. The Boolean reflection monoids form a family of reflection monoids (symmetric inverse semigroups are Boolean reflection monoids of type A). In this paper, we give a family of presentations of Boolean reflection monoids and show how these presentations are compatible with mutations of certain quivers. A feature of the quivers in this paper corresponding to presentations of Boolean reflection monoids is that the quivers have frozen vertices. Our results recover the presentations of Boolean reflection monoids given by Everitt and Fountain and the presentations of symmetric inverse semigroups given by Popova. Surprisingly, inner by diagram automorphisms of irreducible Weyl groups or Boolean reflection monoids can be constructed by sequences of mutations preserving the same underlying diagrams. As an application, we study the cellularity of semigroup algebras of Boolean reflection monoids and construct new cellular bases of such cellular algebras using presentations we obtained and inner by diagram automorphisms of Boolean reflection monoids. Key words: Boolean reflection monoids; presentations; mutations of quivers; inner by diagram automorphisms; cellular semigroups; cellular basis 2010 Mathematics Subject Classification: 13F60; 20M18; 16G20; 20F55; 51F15 1. Introduction In their influential work on cluster algebras, Fomin and Zelevinsky associated mutations of skew-symmetrizable matrices [20, Definition 4.2] with mutations of quivers [21, Proposition 8.1]. The quivers whose underlying graphs are Dynkin diagrams play an important role in the cluster algebra theory, as they appear in the finite type classification [21]. It is well known that a finite irreducible crystallographic reflection group W or a finite irreducible Weyl group W can be classified by Dynkin diagrams, whose vertex set is in one-to-one correspondence with a family S of simple reflections and for which there is an edge labeled 1 (respectively, 2, 3) between vertices i and j if and only if (si sj )3 = e (respectively, (si sj )4 = e, (si sj )6 = e) where si , sj ∈ S, e is the identity element of W , see [2, 4, 6, 28]. Let Γ be a Dynkin diagram and a Γ quiver be a quiver whose underlying diagram is Γ. In [2], Barot and Marsh gave presentations of the reflection group WΓ determined by Γ and showed that these presentations are compatible with mutation of Γ quivers. More precisely, Barot and Marsh introduced some additional relations (cycle relations) corresponding to chordless cycles arising in quivers of finite type. For each quiver Q mutation equivalent to a Γ quiver, they first defined an abstract group W (Q) by generators (corresponding to vertices of Q) and relations, and then proved that W (Q) ∼ = WΓ . Motivated 1 2 BING DUAN, JIAN-RONG LI, AND YAN-FENG LUO by Barot and Marsh’s work, the similar presentations of affine Coxeter groups, braid groups, Artin groups, and Weyl groups of Kac-Moody algebras, have been considered in [17, 18, 25, 30, 38], respectively. Let V be a Euclidean space with standard orthonormal basis {v1 , v2 , . . . , vn } and Φ ⊆ V an irreducible crystallographic root system which is in turn classified by Dynkin diagrams. In [11], Everitt and Fountain introduced the concept of reflection monoids. The Boolean reflection monoid M (Φ, B) of type Φ, formed from the Weyl group W (Φ) for classical root system Φ and the Boolean system B, is a family of reflection monoids. Symmetric inverse semigroups are Boolean reflection monoids of type A. Note that the root systems of types Bn and Cn give rise to the same Weyl group. So we only concern the classical Weyl group W (Bn ). In [12], Everitt and Fountain provided a presentation of the Boolean reflection monoid M (Φ, B) for Φ = An−1 , Bn , or Dn . One of the aims in present paper is to obtain new presentations of Boolean reflection monoids and show how these presentations are compatible with mutation of certain quivers. Let Aεn−1 (respectively, Bnε , Dnε ) be the Dynkin diagram with n (respectively, n + 1, n + 1) vertices, where the first n − 1 (respectively, n, n) vertices are mutable vertices and the n-th (respectively, n + 1-th, n + 1-th) vertex ε is a frozen vertex, which is shown in the 4-th column of Table 1. In practice the label is left on an edge only if its weight is greater than 2, and the edge is left unlabelled if its weight is 1. Let ∆ ∈ {Aεn−1 , Bnε , Dnε } and Q any quiver mutation equivalent to a ∆ quiver. We define an inverse monoid M (Q) from Q, see Section 4.2, and then we show that M (Q) ∼ = M (Φ, B), see Theorem 4.7 and Proposition 4.8. This implies that Boolean reflection monoids can also be classified by ∆, see Table 1. In [2, 17, 25, 30], the diagrams corresponding to generators of irreducible Weyl groups, affine Coxeter groups, braid groups, Artin groups have no frozen vertices. In present paper, the diagrams corresponding to generators of Boolean reflection monoids have frozen vertices. Type of Φ Boolean reflection monoids Generators An−1 (n ≥ 2) M (An−1 , B) {s1 , . . . , sn−1 , sε } Bn (n ≥ 2) M (Bn , B) {s0 , . . . , sn−1 , sε } ∆ = Φε 1 2 n−1 ε 1 n−1 ε 2 n−1 ε 2 0 1 Dn (n ≥ 4) M (Dn , B) {s0 , . . . , sn−1 , sε } 0 Table 1. Boolean reflection monoids and Dynkin diagrams Aεn−1 , Bnε , Dnε . In Proposition 3.1 of [11], Everitt and Fountain proved that the symmetric inverse semigroup In is isomorphic to the Boolean reflection monoid of type An−1 . So we recover the presentation of the symmetric inverse semigroup In defined in [10, 34]. The presentation corresponds exactly to the presentation determined by Dynkin diagram QUIVER MUTATIONS AND BOOLEAN REFLECTION MONOIDS 3 Aεn−1 . Moreover, we also recover Everitt and Fountain’s presentations of Boolean reflection monoids defined in Section 3 of [12]. These presentations can be obtained from any ∆ quiver by a finite sequence of mutations. We show in Theorem 4.10 that the inner automorphism group of Boolean reflection monoid M (Φ, B) is naturally isomorphic to W (Φ)/Z(W (Φ)). We further study the actions of (inward) mutations. Surprisingly, inner by diagram automorphisms of finite irreducible Weyl groups and Boolean reflection monoids can be constructed by a sequence of mutations preserving the same underlying diagrams, see Theorem 3.7 and Theorem 4.11 respectively. As an application, we study the cellularity of semigroup algebras of Boolean reflection monoids. It is well known that Hecke algebras of finite type, q-Schur algebras, the Brauer algebra, the Temperley-Lieb algebras and partition algebras are cellular, see [22–24, 42, 43]. Recently, the cellularity of semigroup algebras is investigated by East [8], Wilox [41], Guo and Xi [26], and Ji and Luo [32] respectively. By applying Geck’s and East’s results, we show that semigroup algebras of Boolean reflection monoids are cellular algebras, see Proposition 4.12. Moreover, we construct new cellular bases of such cellular algebras by presentations we obtained and inner by diagram automorphisms of Boolean reflection monoids. The results and methods of this paper have applications in several lines of research which will be studied in future work, including automorphisms of Boolean reflection monoids [39], Hecke algebras of Boolean reflection monoids [27, 31, 36], Coxeter arrangement monoids [7, 11, 12, 14, 15], Braid inverse monoids [9, 13, 25], and algebraic monoids. The paper is organized as follows. In Section 2, we recall some notations and background knowledge which will be useful to us. In Section 3, building on Barot and Marsh’s work, we further study inner by diagram automorphisms of irreducible Weyl groups (Theorem 3.7) and cellular basis of group algebras of irreducible Weyl groups. In Section 4, we state our main results, Theorem 4.7 and Proposition 4.8, which show that presentations of Boolean reflection monoids are compatible with mutations of ∆ quivers. We recover the presentations of Boolean reflection monoids given by Everitt and Fountain and the presentations of symmetric inverse semigroups given by Popova. Moreover, we characterize inner by diagram automorphisms of Boolean reflection monoids by the method of mutations, Theorem 4.11. Furthermore, we study the cellularity of semigroup algebras of Boolean reflection monoids and give new cellular bases of such cellular algebras. In Section 5, we consider the way of mutations of ∆ quivers and the oriented cycles appearing in them. In Section 6, we find an efficient subset of the relations sufficient to define the inverse monoid M (Q). The last section, Section 7, we prove our main result, Theorem 4.7. 2. Preliminaries 2.1. Mutation of quivers. Let Q be a quiver with finitely many vertices and finitely many arrows that have no loops or oriented 2-cycles. Given a quiver Q, let I be the set of its vertices and Qop its opposite quiver with the same set of vertices but with the reversed orientation for all the arrows. If there are q arrows pointing from a vertex i to a vertex j, then we draw an arrow from i to j with a weight wij = q. We will frequently draw an arrow with no label if wij = 1. 4 BING DUAN, JIAN-RONG LI, AND YAN-FENG LUO For each mutable vertex k of Q, one can define a mutation of Q at k, due to Fomin and Zelevinsky [21]. This produces a new quiver denoted by µk (Q) which can be obtained from Q in the following way: (i) The orientations of all edges incident to k are reversed and their weights intact. (ii) For any vertices i and j which are connected in Q via a two-edge oriented path going through k, the quiver mutation µk affects the edge connecting i and j in the way shown in Figure 1, where the weights c and c′ are related by √ √ √ ± c ± c′ = ab, √ √ where the sign before c (resp., c′ ) is “+” if i, j, k form an oriented cycle in Q (resp., in µk (Q)), and is “−” otherwise. Here either c or c′ may be equal to 0, which means no arrows between i and j. i k ✂@ ❂❂❂ ✂ ❂❂b a ✂✂ ❂❂ ✂✂ ❂ ✂ ✂ c µ k ←→ j i k ✂ ^❂❂❂ ✂ ❂❂b a ✂✂ ❂❂ ✂✂ ❂ ✂ ✂ c′ j Figure 1. Quiver mutation (iii) The rest of the edges and their weights in Q remain unchanged. Two quivers Q1 and Q2 are said to be mutation equivalent if there exists a finite sequence of mutations taking one to the other. We write Q1 ∼mut Q2 to indicate that Q1 is mutation equivalent to Q2 . The underlying diagram of a quiver Q is a undirected diagram obtained from Q by forgetting the orientation of all the arrows. We call a quiver Q connected if its underlying diagram is connected (every node is “reachable”). It is obvious that Dynkin quivers are connected quivers. It was shown in [21, Theorem 1.4] that there are only finitely many quivers in the mutation classes of Dynkin quivers. We call a cycle in the underlying diagram of a quiver a chordless cycle if no two vertices of the cycle are connected by an edge that does not itself. As shown in [21, Proposition 9.7] (or see [2, Proposition 2.1]), all chordless cycles are oriented in the mutation classes of Dynkin quivers. 2.2. Cellular algebras and cellular semigroups. Let us first recall the basic definition of cellular algebras introduced by Graham and Lehrer [24]. Let R be a commutative ring with identity. Definition 2.1. An associative R-algerba A is called a cellular algebra with cell datum (Λ, M, C, i) if the following conditions are satisfied: (C1) Λ is a finite partially ordered set. Associated with each λ ∈ Λ there is a finite set λ \| λ ∈ Λ; S, T ∈ M (λ)} of A; M (λ) of indices and there exists an R-basis {CS,T 2 λ λ ; (C2) i is an R-linear anti-automorphism of A with i = i, which sends CS,T to CT,S (C3) For each λ ∈ Λ, S, T ∈ M (λ), and each a ∈ A, X λ aCS,T ≡ ra (S ′ , S)CSλ′ ,T (mod A(< λ)), S ′ ∈M (λ) QUIVER MUTATIONS AND BOOLEAN REFLECTION MONOIDS 5 where ra (S ′ , S) ∈ R is independent of T and A(< λ) is the R-submodule of A generated by {CSµ′′ ,T ′′ \| µ < λ, S ′′ , T ′′ ∈ M (µ)}. Cellular algebras provide a general framework to studying the representation theory of many important classes of algebras including Hecke algebras of finite type, q-Schur algebras, the Brauer algebra, the Temperley-Lieb algebras and partition algebras, see [22–24, 42, 43]. Recently, the cellularity of semigroup algebras is investigated by East [8], Wilox [41], Guo and Xi [26], and Ji and Luo [32] respectively. In the following, we shall recall some basic notions and facts from the theory of semigroups. Let S be a semigroup. For any a, b ∈ S, define a L b ⇔ S 1 a = S 1 b, a R b ⇔ aS 1 = bS 1 , a J b ⇔ S 1 aS 1 = S 1 bS 1 , H = L ∩ R and D = L ∨ R = L ◦ R = R ◦ L, where S 1 is the monoid obtained from S by adding an identity if necessary. A semigroup S is said to be inverse if for each element s ∈ S, there exists a unique inverse s−1 ∈ S such that ss−1 s = s and s−1 ss−1 = s−1 . If S is a finite inverse semigroup, then D = J . For any s, t ∈ S, define Ds ≤ Dt if and only if s ∈ S 1 tS 1 . Let S be an inverse semigroup with the set E(S) of idempotents. Let D be a D-class of S. Suppose that e1 , . . . , ek ∈ D ∩ E(S). Choose a1 = e1 , . . . , ak ∈ Le1 such that aj R ej for each j. Then D = Re1 ∪ Ra2 ∪ · · · ∪ Rak . Put HD = He1 and by Green’s Lemma, for each x ∈ D we have x = ai ga−1 j for unique 1 ≤ i, j ≤ k and g ∈ HD . Using East’s symbol in [8], let [ei , ej , g]D be the element x. Then x−1 = [ej , ei , g−1 ]D . For more detail knowledge of semigroups, the reader is referred to [8, 29]. A semigroup S is said to be cellular if its semigroup algebra R[S] is a cellular algebra. In [8], East proved the following theorem: Theorem 2.2. [8, Theorems 15] Let S be a finite inverse semigroup with the set E(S) of idempotents. If S satisfies the following conditions: (1) For each D-class D, the subgroup HD is cellular with cell datum (ΛD , MD , CD , iD ); (2) The map i : R[S] → R[S] sending [e, f, g]D to [f, e, iD (g)]D is an R-linear antihomomorphism. Then S is a cellular semigroup with cell datum (Λ, M, C, i), where Λ = {(D, λ) \| D ∈ S/D, λ ∈ ΛD } with partial order defined by (D, λ) ≤ (D ′ , λ′ ) if D < D ′ or D = D ′ and λ ≤ λ′ , M (D, λ) = {(e, s) \| e ∈ E(S) ∩ D, s ∈ MD (λ)} for (D, λ) ∈ Λ, and (D,λ) λ ] \| (D, λ) ∈ Λ; (e, s), (f, t) ∈ M (D, λ)} for (D, λ) ∈ Λ and C = {C(e,s),(f,t) = [e, f, Cs,t D (e, s), (f, t) ∈ M (D, λ). By the definition of cellular algebras, we have the following corollary. Corollary 2.3. Suppose that A is a cellular algebra over R with cell datum (Λ, M, C, i) λ λ ) for any λ ∈ Λ, (S, T ) ∈ and ϕ is an R-linear automorphism of A. Let C S,T = ϕ(CS,T M (λ) × M (λ), and i = ϕiϕ−1 . Then (Λ, M, C, i) is a cellular basis of A. λ Proof. Since {CS,T \| λ ∈ Λ, (S, T ) ∈ M (λ) × M (λ)} is an R-basis of A and ϕ is an λ R-linear automorphism of A, {C S,T \| λ ∈ Λ, (S, T ) ∈ M (λ) × M (λ)} is also an R-basis 2 λ λ of A. It follows from the definition of i that i = i and i(C S,T ) = C T,S . 6 BING DUAN, JIAN-RONG LI, AND YAN-FENG LUO For each λ ∈ Λ, S, T ∈ M (λ), and each a ∈ A, X λ aCS,T ≡ ra (S ′ , S)CSλ′ ,T (mod A(< λ)), S ′ ∈M (λ) where ra (S ′ , S) ∈ R is independent of T and A(< λ) is the R-submodule of A generated by {CSµ′′ ,T ′′ \| µ < λ, S ′′ , T ′′ ∈ M (µ)}. For each a ∈ A, there exists a b ∈ A such that a = ϕ(b). Then X λ λ λ ) = ϕ(bCS,T )≡ rb (S ′ , S)ϕ(CSλ′ ,T ) (mod A(< λ)) aC S,T = ϕ(b)ϕ(CS,T S ′ ∈M (λ) ≡ X λ rb (S ′ , S)C S ′ ,T (mod A(< λ)), S ′ ∈M (λ) where rb (S ′ , S) ∈ R is independent of T . Therefore (Λ, M, C, i) is a cellular basis of A, as required. 3. Some new results of irreducible Weyl groups Let V be a Euclidean space with a standard orthonormal basis {v1 , v2 , . . . , vn }. Let Φ ⊆ V be a root system and Π the set of simple roots in Φ. For each αi ∈ Π, the associated simple reflection is si . Then the finite irreducible Weyl group W (Φ) can be generated by S = {si \| αi ∈ Π} and the number of reflections in W (Φ) is equal to the number of positive roots in Φ. We refer the reader to [3, 19, 28, 35] for more information about Weyl groups, root systems and reflection groups. 3.1. Barot and Marsh’s results. It is well known that the finite irreducible crystallographic reflection groups or the irreducible Weyl groups have been classified by Dynkin diagrams, see [28]. Let Γ be a Dynkin diagram and I the set of its vertices. Let WΓ be the finite irreducible Weyl group determined by Γ. We say that a Γ quiver is a quiver whose underlying diagram is Γ. In [2], Barot and Marsh gave presentations of WΓ . The construction works as follows: Let Q be a quiver mutation equivalent to a Γ quiver. Barot and Marsh defined an inward mutation at vertex k as follows: ( sk si sk if there is an arrow i → k in Q (possibly weighted), ti = (3.1) si otherwise. For two vertices i, j of Q, one defines   2 i and j are not connected,  3 i and j are connected by an edge with weight 1, mij =  4 i and j are connected by an edge with weight 2,    6 i and j are connected by an edge with weight 3. (3.2) Definition 3.1. Let W (Q) be the group with generators si , i ∈ I, subjecting to the following relations: (1) s2i = e for all i; (2) (si sj )mij = e for all i 6= j; QUIVER MUTATIONS AND BOOLEAN REFLECTION MONOIDS 7 (3) For any chordless cycle C in Q: ω ω ωd−1 ω 1 2 0 i0 −→ i1 −→ · · · −→ id−1 −→ i0 , where either all of the weights are 1, or ω0 = 2, we have: (si0 si1 · · · sid−2 sid−1 sid−2 · · · si1 )2 = e; where e is the identity element of W (Q). One of Barot and Marsh’s main results in [2] is stated as follows. Theorem 3.2 ([2, Theorem A]). The group W (Q) does not depend on the choice of a quiver in the mutation class of Q. In particular, W (Q) ∼ = WΓ for each quiver Q mutation equivalent to a Γ quiver. 3.2. Inner by diagram automorphisms of irreducible Weyl groups. Let W be a Coxeter group defined by a set S of generators and relations (1) and (2) of Definition 3.1. We call the pair (W, S) a Coxeter system. In what follows, given any two Coxeter systems (W, S1 ) and (W, S2 ), we say that there exists an automorphism of W , we mean that there is an automorphism α ∈ Aut(W ) such that α(S1 ) = S2 . If such an automorphism can always be chosen from Inn(W ), the group of inner automorphisms of W , then W is called strongly rigid. In case W is strongly rigid, the group Aut(W ) has a very simple structure (see Corollary 3.2 of [5]): Aut(W ) = Inn(W ) × Diag(W ), where Diag(W ) consists of diagram automorphisms of the unique Coxeter diagram corresponding to W . The following lemma is well known. Lemma 3.3. Let W be a finite group generated by a finite set S of simple reflections. Then the set of all reflections in W is {wsw−1 \| w ∈ W, s ∈ S}. In [1, Table I], Bannai computed the center Z(W (Φ)) of an irreducible Weyl group W (Φ). The longest element w0 in W (Φ) is a central element of W (Φ) except for Φ = An (n ≥ 2), D2k+1 (k ≥ 2), E6 . The following important notation was introduced by Franzsen in [16]. Definition 3.4. [16, Definition 1.36] An inner by diagram automorphism is an automorphism generated by some inner and diagram automorphisms in Aut(W ). The subgroup of inner automorphisms is a normal subgroup of Aut(W ), therefore any inner by diagram automorphism can be written as the product of an inner and a diagram automorphism. The following two lemmas collect together some facts from [16] which will be useful later. ∼ Lemma 3.5. [16, Proposition 2.4] If W is Weyl group of type An , then Aut(W (An )) = ∼ W (An ) if n 6= 5, while Aut(W (A5 )) = W (A5 ) ⋊ Z2 . So, for n 6= 5, any automorphism of Weyl group of type An maps reflections to reflections, furthermore any automorphism of A5 that does preserve reflection is inner. Lemma 3.6. [16, Proposition 1.44, Propositions 2.8–2.10, Proposition 2.13] Let W (Φ) be Weyl group of type Φ. 8 BING DUAN, JIAN-RONG LI, AND YAN-FENG LUO (a) Any automorphism of W (Bn ) for n > 2 that does preserve reflections must be inner. (b) For k ≥ 2, Aut(W (D2k+1 )) ∼ = W (D2k+1 ), that is, all automorphisms of W (D2k+1 ) map reflections to reflections. (c) All automorphisms of W (E6 ) or W (E7 ) are inner. (d) All automorphisms of Weyl groups that preserve reflections are inner by diagram automorphisms. The following theorem reveals a connection between inner by diagram automorphisms of irreducible Weyl groups and quiver mutations. Theorem 3.7. Let Q be a Γ quiver and W (Q) the corresponding Weyl group generated by a set S of simple reflections. Then α is an inner by diagram automorphism of W (Q) if and only if there exists a sequence of mutations preserving the underlying diagram Γ such that α(S) can be obtained from Q by mutations. In particular, all the reflections in W (Q) can be obtained from Q by mutations. Proof. Through observation, every variable obtained by mutations must be some reflection of the corresponding Weyl group. The sufficiency follows from the fact that all automorphisms of Weyl groups that preserve reflections are inner by diagram automorphisms, see Lemmas 3.5 and 3.6. To prove necessity, assume without loss of generality that the vertex set of Q is {1, 2, . . . , n} and α is an inner by diagram automorphism of W (Q). Note that diagram automorphisms of any Dynkin diagram keep the underlying Dynkin diagram. Then relabelling the vertices of Q if necessary, there exists an inner automorphism α of W (Q) such that α(S) = α(S). It is sufficient to prove that α(S) can be obtained from Q by mutations and the sequence of mutations preserves the underlying diagram Γ. Let g = si1 si2 · · · sir ∈ W (Q) be a reduced expression for g, where sik ∈ S, 1 ≤ k ≤ r. We assume that α(S) = gSg −1 . In the following we shall use induction to prove that gSg−1 can be obtained from Q by a sequence of mutations preserving the underlying diagram Γ. Step 1. We mutate firstly Q at the vertex i1 twice. Then we get a quiver Q1 , which has the same underlying diagram with Q. Moreover, the set S becomes si1 Ssi1 = {si1 s1 si1 , si1 s2 si1 , . . . , si1 sn−1 si1 , si1 sn si1 }. We keep vertices of Q1 having the same label as vertices of Q. Step 2. We then mutate Qt−1 , t = 2, 3, · · · , r at the vertex it twice. Note that the variable corresponding to the vertex it of Qt−1 is si1 . . . sit−1 sit sit−1 . . . si1 . Then we get a quiver Qt , which has the same underlying diagram with Q, Q1 , . . . , Qt−1 . Moreover, the set si1 . . . sit−1 Ssit−1 . . . si1 of generators in Qt−1 becomes si1 · · · sit−1 sit sit−1 · · · si1 (si1 · · · sit−1 Ssit−1 · · · si1 )si1 · · · sit−1 sit sit−1 · · · si1 = si1 · · · sit−1 sit Ssit sit−1 · · · si1 . We keep vertices of Qt having the same label as vertices of Qt−1 . Step 3. We repeat Step 2 until we get the quiver Qr . By induction, it is not difficult to show that the set of generators in Qr is si1 si2 · · · sir Ssir · · · si2 si1 = gSg −1 = α(S). QUIVER MUTATIONS AND BOOLEAN REFLECTION MONOIDS 9 Finally, every reflection in W (Φ) is conjugate to a simple reflection by Lemma 3.3. So we assume that a reflection is of the form gsi g−1 , where g = si1 si2 · · · sik ∈ W (Φ) is a reduced expression for some subset {i1 , i2 , . . . , ik } ⊆ {1, 2, . . . , n}. By the same arguments as before, we mutate the sequence i1 , i1 , i2 , i2 , . . . , ik , ik starting from Q, we can obtain the reflection gsi g−1 . Remark 3.8. (1) In types An , Bn (n > 2), D2k+1 , E6 , E7 , and E8 , all inner by diagram automorphisms of the corresponding Weyl groups are inner automorphisms. W (Φ) is strongly rigid for Φ = An (n 6= 5), D2k+1 , E6 , E7 . (2) If (W, S) is a Coxeter system of W , then for any inner by diagram automorphism α of W , (W, α(S)) is also a Coxeter system of W . (3) Suppose that Q is a quiver mutation equivalent to a Γ quiver. Let W (Q) be the corresponding Weyl group with a set S of generators. Then Theorem 3.7 holds for Q. 3.3. Cellular basis of group algebras of irreducible Weyl groups. In [23], Geck proved Hecke algebras of finite type are cellular algebras. Let ϕ be any inner by diagram automorphism of irreducible Weyl groups, see Theorem 3.7. By Lemma 2.3, we can obtain new cellular basis of group algebras of irreducible Weyl groups. 4. Main results of Boolean reflection monoids Let Aεn−1 (respectively, Bnε , Dnε ) be the Dynkin diagram with n (respectively, n + 1, n + 1) vertices, where the first n − 1 (respectively, n, n) vertices are mutable vertices and the n-th (respectively, n + 1-th, n + 1-th) vertex ε is a frozen vertex, which is shown in Figure 2. The label is left on an edge only if its weight is greater than 2, and the edge is left unlabelled if its weight is 1. We shall always assume that ∆ is one of Aεn−1 , Bnε and Dnε . A quiver is said to be a ∆ quiver if the underlying diagram of such quiver is ∆. Aεn−1 Bnε 1 2 3 n−1 ε 1 2 n−1 ε 2 3 n−1 ε 2 0 1 Dnε 0 Figure 2. Classical Dynkin diagrams Aεn−1 , Bnε and Dnε with a frozen vertex ε. 10 BING DUAN, JIAN-RONG LI, AND YAN-FENG LUO 4.1. Boolean reflection monoids. In 2010, Everitt and Fountain introduced reflection monoids, see [11]. The Boolean reflection monoids are a family of reflection monoids (symmetric inverse semigroups are Boolean reflection monoids of type A). Let V be a Euclidean space with standard orthonormal basis {v1 , v2 , . . . , vn } and let Φ ⊆ V be a root system and W (Φ) the associated Weyl group of type Φ. A partial linear isomorphism of V is a vector space isomorphism Y → Z for two vector subspaces Y , Z of V . Any partial linear isomorphism of V can be realized by restricting a full isomorphism to some subspace. We will write gY for the partial isomorphism with domain Y and effect that of restricting g to Y . We denote by M (V ) (respectively, GL(V )) the general linear monoid (respectively, general linear group) on V consisting of partial linear isomorphisms (respectively, linear isomorphisms) of V . In M (V ), gY = hZ if and only if Y = Z and gh−1 is in the isotropy group GY = {g ∈ GL(V )\|gv = v for all v ∈ Y }. Moreover, gY hZ = (gh)Y ∩g−1 Z and (gY )−1 = (g −1 )gY . Let us recall the notation “system” in V for a group G ⊆ GL(V ) introduced in [11]. Definition 4.1 ([11, Definition 2.1]). Let V be a real vector space and G ⊆ GL(V ) a group. A collection S of subspaces of V is called a system in V for G if and only if (1) V ∈ S, (2) GS = S, that is, gX ∈ S for any g ∈ G and X ∈ S, (3) if X, Y ∈ S, then X ∩ Y ∈ S. For J ⊆ X = {1, 2, . . . , n}, let X(J) = M j∈J Rvj ⊆ V, and B = {X(J) : J ⊆ X} with X(∅) = 0. Then B is a Boolean system in V for W (Φ), where Φ = An−1 , Bn /Cn , or Dn . For example, the Weyl group W (An−1 )-action on the subspaces X(J) ∈ B is just g(π)X(J) = X(πJ), where g(π) 7→ π induces an isomorphism between W (An−1 ) and the symmetric group Sn on the set X. Note that B is not a system for any of the exceptional W (Φ). Definition 4.2 ([11, Definition 2.2]). Let G ⊆ GL(V ) be a group and S the system in V for G. The monoid of partial linear isomorphisms given by G and S is the submonoid of M (V ) defined by M (G, S) := {gX : g ∈ G, X ∈ S}. If G is a reflection group, then M (G, S) is called a reflection monoid. Let G be the reflection group W (Φ) for Φ = An−1 , Bn /Cn , or Dn , and S the Boolean system in V for G, then M (G, S) is called a Boolean reflection monoid. In general, we write M (Φ, B) instead of M (W (Φ), B), and call M (Φ, B) the Boolean reflection monoid of type Φ. Recall that Everitt and Fountain gave a presentation of the Boolean reflection monoid M (Φ, B) for Φ = An−1 , Bn , or Dn in Section 4 of [12]. QUIVER MUTATIONS AND BOOLEAN REFLECTION MONOIDS 11 Lemma 4.3. Everitt and Fountain’s presentations of Boolean reflection monoids are shown as follows: M (An−1 , B) = hs1 , s2 , . . . , sn−1 , sε \| (si sj )mij = e for 1 ≤ i, j ≤ n − 1, s2ε = sε , si sε = sε si for i 6= 1, sε s1 sε s1 = s1 sε s1 sε = sε s1 sε i . M (Bn , B) = hs0 , s1 , . . . , sn−1 , sε \| (si sj )mij = e for 0 ≤ i, j ≤ n − 1, s2ε = sε , s0 s1 sε s1 = s1 sε s1 s0 , s0 sε = sε , si sε = sε si for i 6= 1, s1 sε s1 sε = sε s1 sε s1 = sε s1 sε i. M (Dn , B) = hs0 , s1 , . . . , sn−1 , sε \| (si sj )mij = e for 0 ≤ i, j ≤ n − 1, s2ε = sε , si sε = sε si for i > 1, sε s1 sε s1 = s1 sε s1 sε = sε s1 sε , s0 sε s0 = s1 sε s1 , s0 s2 s1 sε s1 s2 = s2 s1 sε s1 s2 s0 i . Here mij is defined in (3.2). 4.2. Inverse monoids determined by quivers. Let I ∪ {ε} be the set of vertices of a quiver Q with a frozen vertex ε. For any i, j ∈ I and ε, define   2 if i and j are not connected,  3 if i and j are connected by an edge of weight 1, mij =  4 if i and j are connected by an edge of weight 2,    6 if i and j are connected by an edge of weight 3.   2 if ε and j are not connected, mεj = 3 if ε and j are connected by an edge of weight 1,   1 if ε and j are connected by an edge of weight 2.   2 if ε and j are not connected, mjε = 4 if ε and j are connected by an edge of weight 1,   2 if ε and j are connected by an edge of weight 2. Let mii = 1 for any i ∈ I ∪ {ε}. Then (mij )i,j∈I is a Coxeter matrix and (mij )i,j∈I∪{ε} is a generalized Coxeter matrix, see [40]. To illustrate, generalized Coxeter matrices corresponding to a Aεn−1 quiver and a Bnε quiver are respectively     1 3 2 ··· ··· ··· 2 1 4 2 ··· ··· ··· 2 3 1 4 1 3 2 · · · · · · 2 3 2 · · · · · · 2     2 3   1 3 2 · · · 2 1 3 2 · · · 2  2 3   .. . . . . . . . . . . ..   .. . . . . . . . . . . ..  and . .   . . . . . . . . . . . .  .  2 · · · 2 2 · · · 2 3 1 3 2 3 1 3 2     2 · · · · · · 2   3 1 4 2 ··· ··· 2 3 1 4 2 ··· ··· ··· 2 3 1 n×n 2 ··· ··· ··· 2 3 1 (n+1)×(n+1) Let (i1 , . . . , ε) be an ordered tuple such that the subquiver of Q on the vertices i1 , . . . , ε contains only one underlying subdiagram or and does not contain one 12 BING DUAN, JIAN-RONG LI, AND YAN-FENG LUO . Such an ordered tuple (i1 , . . . , ε) is called a shortest path underlying subdiagram tuple if it is the shortest path from i1 to ε in Q. For any shortest path tuple (i1 , . . . , ε), we denote by P (si1 , sε ) the word si1 . . . sε . Denote by e the identity element of an inverse monoid and denote by (aba . . .)m an alternating product of m terms. Definition 4.4. Let Q be any quiver mutation equivalent to a ∆ quiver. Define an inverse monoid M (Q) with generators si , i ∈ I ∪ {ε} and relations: (R1) s2i = e for i ∈ I, s2ε = sε ; (R2) (si sj )mij = e for i, j ∈ I and (sε sj sε · · · )mεj = (sj sε sj · · · )mjε = (sε sj sε · · · )mεj +1 for any j ∈ I; (R3) (i) for every chordless oriented cycle C in Q: w w w 1 2 0 i0 −→ i1 −→ · · · → id−1 −→ i0 , where id ∈ I for d = 0, 1, . . . , d−1, either all of the weights are 1, or w0 = 2, we have: (si0 si1 · · · sid−2 sid−1 sid−2 · · · si1 )2 = e. (ii) for every chordless oriented cycle C in Q: ε −→ i1 −→ · · · → id−1 −→ ε, where id ∈ I for d = 1, . . . , d − 1, we have: sε si1 · · · sid−2 sid−1 sid−2 · · · si1 = si1 · · · sid−2 sid−1 sid−2 · · · si1 sε . (iii) for every chordless oriented cycle C in Q: w 2 w 1 2 ε −→ i1 −→ i2 −→ ε, where i1 , i2 ∈ I, if w1 = 1 and w2 = 2, we have sε si1 si2 si1 = si1 si2 si1 sε ; if w1 = 2 and w2 = 1, we have si1 si2 sε si2 = si2 sε si2 si1 . (R4) (path relations) for every underlying subdiagram of Q of the form shown in the first column of Table 2, we take path relations listed in the second column of Table 2. Remark 4.5. (1) In (R2), if mjε = 2 then mεj = 2. In this case the equation (sε sj sε . . .)mεj = (sj sε sj . . .)mjε = (sε sj sε . . .)mεj +1 can be reduced to be sε sj = sj sε . (2) For (R3) (ii) relation, though in this paper we only use the case d = 3, 4, the defined relation for arbitrary d is still meaningful, see our unpublished paper [7]. The following lemma is well known and easily verified. Lemma 4.6. If two quivers Q1 and Q2 both have the same underlying diagram G and G is a tree, then Q1 ∼mut Q2 . It follows from the connectivity and finiteness of ∆ that any two ∆ quivers are mutation-invariance. QUIVER MUTATIONS AND BOOLEAN REFLECTION MONOIDS Subdiagrams of Q 2 ε 0 i1 Path relations P (s0 ,sε )=P (si1 ,sε ), P (sε ,s0 )=P (sε ,si1 ) 2 0 P (s0 ,sε )=P (si1 ,sε ), P (sε ,s0 )=P (sε ,si1 ) 2 ε i1 i3 i4 13 i2 C ε i1 id−1 P (si2 ,sε )P (sε ,si2 )=si3 si4 ···sid P (si1 ,sε )P (sε ,si1 )sid ···si4 si3 for d ≥ 4 id i2 i3 i1 ε i1 ε P (si2 ,sε )P (sε ,si2 )=P (si4 ,sε )P (sε ,si4 ) i4 i2 i3 P (si2 ,sε )P (sε ,si2 )=P (si4 ,sε )P (sε ,si4 ) i4 i2 i1 ε P (si2 ,sε )P (sε ,si2 )=P (si3 ,sε )P (sε ,si3 ) i3 Table 2. Path relations of underlying subdiagrams of Q, where C stands for a chordless cycle. Now we are ready for our main results in this section. Theorem 4.7. Let ∆ ∈ {Aεn−1 , Bnε , Dnε } and Q0 be a ∆ quiver. If Q ∼mut Q0 then M (Q) ∼ = M (Q0 ). We will prove Theorem 4.7 in Section 7. Up to the above isomorphism, we denote by M (∆) the inverse monoid determined by any quiver appearing in the mutation class of ∆ quivers. When we say that we mutate a sequence (n, n − 1, · · · , 2, 1) of vertices of a quiver we mean that we first mutate the n-th vertex of the quiver, then we mutate the (n − 1)-th vertex, and so on until the first vertex. The following proposition shows that Everitt and Fountain’s presentations of Boolean reflection monoids can be obtained from any ∆ quiver by mutations. Proposition 4.8. Let Φ = An−1 , Bn or Dn . Then M (Φ, B) = M (Φε ). Proof. All ∆ quivers are mutation-invariance, so any ∆ quiver can be viewed as an initial quiver we mutate. 14 BING DUAN, JIAN-RONG LI, AND YAN-FENG LUO We mutate the sequence (n − 1, n − 2, · · · , 2, 1) of vertices of the following Aεn−1 quiver ◦ /◦ 1 / ◦ ❴ ❴ ❴/ ◦ 2 3 n−1 /•, ε we obtain the quiver Q1 : /◦ ◦ ε / ◦ ❴ ❴ ❴/ ◦ 1 2 / • . n−1 n−2 Then by Definition 4.4 M (Q1 ) = s1 , s2 , . . . , sn−1 , sε \| (si sj )mij = e for 1 ≤ i, j ≤ n − 1, s2ε = sε , si sε = sε si for i 6= 1, sε s1 sε s1 = s1 sε s1 sε = sε s1 sε i , where   1 if i = j, mij = 3 if \|i − j\| = 1,   2 otherwise. By Lemma 4.3, we deduce that M (Q1 ) = M (An−1 , B). Mutating a sequence (n − 1, n − 2, · · · , 1, 0) of vertices of the following Bnε quiver 2 ◦ /◦ / ◦ ❴ ❴ ❴/ ◦ 1 0 2 n−1 /•, ε we get obtain quiver Q2 : ε ◦ ☎☎ 2 ☎☎ ☎ ☎☎ ☎ ☎ • \✿ 2 0 ✿✿ ✿✿ ✿✿ ✿✿ /◦ 1 . / ◦ ❴ ❴ ❴/ ◦ 2 n−2 / ◦ n−1 Then by Definition 4.4 M (Q2 ) = hs0 , s1 , . . . , sn−1 , sε \| (si sj )mij = e for 0 ≤ i, j ≤ n − 1, s2ε = sε , s0 s1 sε s1 = s1 sε s1 s0 , s0 sε = sε s0 = sε , si sε = sε si for i 6= 1, s1 sε s1 sε = sε s1 sε s1 = sε s1 sε i, where  1    2 mij =  3    4 if if if if i = j, i and j are not connected, i and j are connected by an edge with weight 1, i and j are connected by an edge with weight 2. By Lemma 4.3, we have M (Q2 ) = M (Bn , B). QUIVER MUTATIONS AND BOOLEAN REFLECTION MONOIDS 15 By mutating a sequence (n − 1, n − 2, · · · , 2, 0) of vertices of the following Dnε quiver 0 ◦ / ◦2 O / ◦3 ❴ ❴ ❴/ n−1 ◦ / •ε , ◦ 1 we obtain the quiver Q3 : 0 ◦ ✆B ✾✾✾ ✆ ✾✾ ✆ ✾✾ ✆✆ ✆ ✾ ✆ ε ✆ 2 • \✾ ◦ ✾✾ ✆✆ ✾✾ ✆✆ ✾✾ ✆ ✾✾ ✆✆✆ ✆ . / ◦3 ❴ ❴ ❴/ n−2 ◦ / n−1 ◦ ◦ 1 Then by Definition 4.4 M (Q3 ) = s0 , s1 , . . . , sn−1 , sε \| (si sj )mij = e for 0 ≤ i, j ≤ n − 1, s2ε = sε , si sε = sε si for i > 1, sε sj sε sj = sj sε sj sε = sε sj sε for j = 0, 1, s0 sε s0 = s1 sε s1 , s0 s2 s1 sε s1 s2 = s2 s1 sε s1 s2 s0 i , where   1 mij = 2   3 if i = j, if i and j are not connected, if i and j are connected by an edge with weight 1. We claim that M (Q3 ) = M (Dn , B), which follows from Lemma 6.2 (3). Suppose that a quiver Q is mutation equivalent to a ∆ quiver, then by Theorem 4.7 and Proposition 4.8, M (Q) gives a presentation of the Boolean reflection monoid M (∆). In [11], Everitt and Fountain proved that the Boolean reflection monoid M (An−1 , B) (respectively, M (Bn , B)) is isomorphic to the symmetric inverse semigroup In (respectively, the monoid I±n of partial signed permutations). Hence our results recover the presentation of the symmetric inverse semigroup In defined in [10, 34], that is, such presentation is exactly the presentation of M (Q) for a Aεn−1 quiver Q. The following example is given to explain Theorem 4.7. Example 4.9. We start with a Aε3 quiver Q0 which is shown in Figure 3 (a). Let Q1 = µ2 (Q0 ) be the quiver obtained from Q0 by a mutation at 2. 16 BING DUAN, JIAN-RONG LI, AND YAN-FENG LUO 1 (a) ◦ / ◦2 / •ε µ2 −→ 1 (b) ◦ o 2 ◦o ε • Figure 3. (a) A Aε3 quiver Q0 ; (b) The quiver Q1 = µ2 (Q). It follows from Definition 4.4 that M (Q0 ) = s1 , s2 , sε \| s21 = s22 = e, s2ε = sε , s1 s2 s1 = s2 s1 s2 , s1 sε = sε s1 , s2 sε s2 sε = sε s2 sε s2 = sε s2 sε i , M (Q1 ) = t1 , t2 , tε \| t21 = t22 = e, t2ε = tε , t1 t2 t1 = t2 t1 t2 , t1 tε t1 tε = tε t1 tε t1 = tε t1 tε , t2 tε t2 tε = tε t2 tε t2 = tε t2 tε , tε t2 t1 t2 = t2 t1 t2 tε i . Then ϕ : M (Q0 ) → M (Q1 ) is an inverse monoid isomorphism defined by ( t2 ti t2 if i = 1, ϕ(si ) = ti otherwise. 4.3. Inner by diagram automorphisms of Boolean reflection monoids. We first consider inner automorphisms of Boolean reflection monoids. It is well known that for any group G, Inn(G) ∼ = G/Z(G), where Inn(G) is the inner automorphism group of G, Z(G) is the center of G. It has been shown in [33,39] that automorphisms of the Boolean reflection monoid M (An−1 , B) are inner: for every automorphism α of M (An−1 , B), there exists a uniquely determined element g ∈ W (An−1 ) of the Weyl group W (An−1 ) such that α(t) = gtg −1 for all t ∈ M (An−1 , B). In other words, the automorphism group of M (An−1 , B) is naturally isomorphic to W (An−1 )/Z(W (An−1 )) for n ≥ 3 and the automorphism group of M (A1 , B) is naturally isomorphic to W (A1 ). As a generalization of the above result, we have the following theorem. Theorem 4.10. The inner automorphism group of M (Φ, B) is naturally isomorphic to W (Φ)/Z(W (Φ)), where Φ = An−1 (≥ 3), Bn (n ≥ 2), Dn (≥ 4). Proof. Let α be an inner automorphism of M (Φ, B). Since W (Φ) ⊆ M (Φ, B) is the unique unit group of M (Φ, B), α\|W (Φ) ∈ Inn(W (Φ)) = W (Φ)/Z(W (Φ)). In the following we prove the cases of Φ = Bn (n ≥ 2), Dn (≥ 4). Let Φε be one of Bnε , and Dnε shown in Figure 2. Suppose that Λ = {s0 , s1 , . . . , sn−1 , sε } is a set of generators of M (Φ, B). For any element g ∈ W (Φ), we claim that the set gΛg−1 is still a set of generators of M (Φ, B). Firstly, it is obvious that (gsi g−1 )2 = e and (gsε g−1 )2 = gsε g−1 . Nextly we will prove that gΛg−1 satisfies (R2)–(R4) in Definition 4.4. Case 1. There is no edge between i and j. Then gsi g−1 gsj g−1 = gsi sj g−1 = gsj si g−1 = gsj g −1 gsi g−1 . Case 2. There is an edge labeled by 1 between i and j. Then gsi g−1 gsj g−1 gsi g−1 = gsi sj si g−1 = gsj si sj g−1 = gsj g−1 gsi g −1 gsj g−1 . Case 3. There is an edge labeled by 2 between i and j. Then gsi g−1 gsj g−1 gsi g−1 gsj g−1 = gsi sj si sj g−1 = gsj si sj si g −1 = gsj g−1 gsi g−1 gsj g−1 gsi g−1 . QUIVER MUTATIONS AND BOOLEAN REFLECTION MONOIDS 17 Case 4. There is no edge between i and ε. Then gsi g−1 gsε g−1 = gsi sε g−1 = gsε si g−1 = gsε g−1 gsi g −1 . Case 5. There is an edge labeled by 1 between i and ε. Then gsi g−1 gsε g−1 gsi g−1 gsε g−1 = gsi sε si sε g−1 = gsε si sε si g−1 = gsε si sε g −1 = gsε g−1 gsi g−1 gsε g−1 gsi g −1 = gsε g−1 gsi g−1 gsε g−1 . Case 6. In type Bn , gs0 g −1 gs1 g −1 · · · gsε g−1 = gs0 s1 · · · sε g−1 = gs1 · · · sε g−1 = gs1 g−1 · · · gsε g−1 , gsε g−1 · · · gs1 g−1 gs0 g−1 = gsε · · · s1 s0 g−1 = gsε · · · s1 g−1 = gsε g−1 · · · gs1 g−1 . Case 7. In type Dn , gs0 g−1 gs2 g−1 · · · gsε g −1 · · · gs2 g−1 gs0 g −1 = gs0 s2 · · · sε · · · s2 s0 g−1 = gs1 s2 · · · sε · · · s2 s1 g−1 = gs1 g−1 gs2 g−1 · · · gsε g−1 · · · gs2 g −1 gs1 g −1 . Finally, we shall show that g1 sε g1−1 = g2 sε g2−1 for any g1 , g2 ∈ Z(W (Φ)). It suffices to prove that sε = w0 sε w0 , where w0 is the longest element in W (Φ) and w0 is an involution. In Section 1.2 of [16], we have w0 = wn wn−1 · · · w1 , where wi = si−1 · · · s1 s0 s1 · · · si−1 in type Φ = Bn , and wi = si−1 · · · s3 s2 s1 s0 s2 s3 · · · si−1 for i ≥ 3 and w1 = s0 , w2 = s1 in type Φ = Dn . Then by (R1), (R2), and (R4) of Definition 4.4, w0 sε w0 = wn wn−1 · · · w1 sε w1 · · · wn−1 wn = wn sε wn ( sn−1 · · · s1 (s0 s1 · · · sn−1 sε sn−1 · · · s1 s0 )s1 · · · sn−1 = sε = sn−1 · · · s3 s2 s1 (s0 s2 s3 · · · sn−1 sε sn−1 · · · s3 s2 s0 )s1 s2 s3 · · · sn−1 = sε in type Bn , in type Dn . Therefore the inner automorphism group of M (Φ, B) is isomorphic to W (Φ)/Z(W (Φ)) for Φ = An−1 (≥ 3), Bn (n ≥ 2), Dn (≥ 4). Let ∆ be one of Aεn−1 , Bnε , and Dnε shown in Figure 2. Let Q be a ∆ quiver. Let I ∪ {ε} be the set of vertices of Q and Q′ = µk (Q) the quiver obtained by a mutation of Q at a mutable vertex k. Following Barot and Marsh’s work [2], one can define variables 18 BING DUAN, JIAN-RONG LI, AND YAN-FENG LUO ti for i ∈ I, and tε in M (Q′ ) as follows: ( sk si sk if there is an arrow i → k in Q (possibly weighted), ti = si otherwise, ( sk sε sk if there is an arrow ε → k in Q (possibly weighted), tε = sε otherwise. (4.1) From Lemma 3.3 and Equation (4.1), it follows that new elements ti , i ∈ I, appearing in the procedure of mutations of quivers, must be some reflections of Weyl groups. By our Theorem 4.7 and Proposition 4.8, up to isomorphism, Boolean reflection monoids are encoded by their (generalized) Coxeter diagrams, see Figure 2. In the following theorem, we show that inner by diagram automorphisms of Boolean reflection monoids can be constructed by a sequence of mutations preserving the same underlying diagrams. Theorem 4.11. Let Q be a ∆ quiver and M (Q) the corresponding Boolean reflection monoid generated by a set S consisting of simple reflections and sε . Then α is an inner by diagram automorphism of M (Q) if and only if there exists a sequence of mutations preserving the underlying diagram ∆ such that α(S) can be obtained from Q by mutations. In particular, all reflections in W (Q\{ε}) and gsε g −1 for g ∈ W (Q\{ε}) can be obtained from Q by mutations. Proof. Let ∆ be one of Aεn−1 , Bnε , and Dnε shown in Figure 2. Suppose that S = {s1 , s2 , . . . , sn−1 , sε } for ∆ = Aεn−1 or S = {s0 , s1 , . . . , sn−1 , sε } for ∆ = Bnε , Dnε . In the case of type An , all automorphisms of M (Φ, B) are inner, see [33, 39]. A sequence µ of mutations preserving the underlying diagram of Q induces to an inner automorphism of M (Φ, B). All automorphisms of Weyl groups that preserve reflections are inner by diagram automorphisms, see Lemmas 3.5 and 3.6. So we assume without loss of generality that M (µ(Q)) = ht0 , t1 , . . . , tn−1 , tε i, where ti = gsi g−1 for 0 ≤ i ≤ n − 1, g ∈ W (Bn ) (respectively, g ∈ W (Dn )). We claim that tε = gsε g −1 . Firstly, if tε = gsε g−1 , then {t0 , t1 , . . . , tn−1 , tε } is a set of generators of M (µ(Q)) and the generalized Coxeter diagram corresponding to {t0 , t1 , . . . , tn−1 , tε } preserves the underlying diagram ∆. Since tε ti = ti tε for 0 ≤ i ≤ n − 2, we have tε ∈ Z(W (Bn−1 )) (respectively, tε ∈ Z(W (Dn−1 ))), where W (Bn−1 ) = htε t0 , tε t1 , . . . , tε tn−2 i (respectively, W (Dn−1 ) = htε t0 , tε t1 , . . . , tε tn−2 i). The variable tε must be of the form g′ sε g′ −1 for some g′ ∈ W (Bn ) (respectively, g′ ∈ W (Dn )). So tε is not the longest word w0 in W (Bn−1 ) (respectively, W (Dn−1 )). Therefore tε must be the unique identity element in W (Bn−1 ) (respectively, W (Dn−1 )) and hence tε = gsε g−1 . Conversely, for each inner automorphism α of M (Q), by Theorem 4.10, there exists an element g ∈ W (Q\{ε}) of the Weyl group W (Q\{ε}) ⊆ M (Q) such that α(t) = gtg −1 for all t ∈ M (Q). The remainder proof of the necessity is similar to the proof of the necessity of Theorem 3.7. Every reflection in W (Q\{ε}) is of the form gsi g−1 , where g = si1 si2 · · · sik ∈ W (Q\{ε}) is a reduced expression for g. By the same arguments as before, we mutate the sequence i1 , i1 , i2 , i2 , . . . , ik , ik starting from Q, we get gsi g −1 and gsε g−1 . QUIVER MUTATIONS AND BOOLEAN REFLECTION MONOIDS 19 4.4. Cellularity of semigroup algebras of Boolean reflection monoids. In this section, we show that semigroup algebras of Boolean reflection monoids are cellular algebras. We use these presentations we obtained to construct new cellular bases of such cellular algebras. Let R be a commutative ring with identity. Recall that a semigroup S is said to be cellular if its semigroup algebra R[S] is a cellular algebra. Proposition 4.12. The Boolean reflection monoid M (Φ, B) for Φ = An−1 , Bn , or Dn is a cellular semigroup. Proof. All maximal subgroups of the Boolean reflection monoid M (Φ, B) are finite reflection groups. It has been shown in [23] that any finite reflection group W (Φ) is cellular with respect to which the anti-involution is inversion. Therefore, for each D-class D of M (Φ, B), the subgroup HD ⊆ M (Φ, B) is cellular with cell datum (ΛD , MD , CD , iD =−1 ), which satisfies East’s first assumption, see Theorem 15 in [8] or Theorem 2.2. We define a map i : R[M (Φ, B)] → R[M (Φ, B)] X X rj gj 7→ rj gj−1 , j j where rj ∈ R, gj ∈ M (Φ, B). The map i is an R-linear anti-homomorphism and −1 i([e, f, g]D ) = [e, f, g]−1 D = [f, e, g ]D = [f, e, iD (g)]D for any g ∈ HD , e D f in M (Φ, B). From Theorem 19 in [8] or Theorem 2.2, it follows that the Boolean reflection monoid M (Φ, B) is a cellular semigroup, as required. Remark 4.13. The case that a finite inverse semigroup whose maximal subgroups are direct products of symmetric groups has been considered by East, see Theorem 22 of [8]. The Boolean reflection monoid of type An−1 is isomorphic to the symmetric group Sn of degree n. Maximal subgroups of the Boolean reflection monoid of type Bn are finite reflection groups of type Br , r ≤ n, which are isomorphic to (Z2 × Z2 . . . × Z2 ) ⋊ Sr . Let ∆ be one of Aεn−1 , Bnε , and Dnε shown in Figure 2. For two quivers with the same underlying diagrams appearing in the mutation class of ∆ quivers, we always use their presentations to construct inner by diagram automorphisms of Boolean reflection monoids, see Theorem 4.11 and then we extend it an R-linear automorphism of semigroup algebras of Boolean reflection monoids. By Corollary 2.3, we obtain new cellular bases of semigroup algebras of Boolean reflection monoids. 4.5. An example. Let In be the symmetric inverse semigroup on [n] = {1, 2, . . . , n}. Let w be a partial permutation on a set A ⊆ [n] and denote the image of i ∈ A under the map w by wi and the image of i 6∈ A under the map w by wi = ∅. We denote w by the sequence (w1 w2 . . . wn ). For example, (3 − 2) is the partial permutation with domain {1, 3} and range {2, 3} under which 1 → 3, 3 → 2. The following example gives new cellular bases of R[I3 ] by the method of quiver mutations. Example 4.14. Let Q0 be the quiver in Example 4.9 and by the results of preceding sections, M (Q0 ) ∼ = I3 a Boolean reflection monoid. We have M (Q0 )/D = {D0 < D1 < 20 BING DUAN, JIAN-RONG LI, AND YAN-FENG LUO D2 < D3 }, where each Di is the set of all elements of M (Q0 ) of rank i, and idempotents in each Di are the partial identity permutation on i-subsets of [3]. Let A be an i-subset of [3]. As shown in Example 23 of [8], the H-class containing the idempotent idA is the subgroup {x ∈ I3 \| im(x) = dom(x) = A} ∼ = S\|A\| . It is well known that the group algebra R[Sn ] has cellular bases with respect to which the anti-involution is inversion. Indeed the Khazdan-Luzstig bases and the Murphy basis both have this property (see, Example (1.2) of [24], Example (2.2) of [36] or Section 4 of [37]). Take s1 = (2 1 3), s2 = (1 3 2), and sε = (1 2 −). By mutating Q0 , we obtain the following isomorphic quivers (Theorems 4.10 and 4.11): / s◦2 s1 (a) ◦ s1 (b) ◦ (c) s1 s2 s1 s2 s1 s2 / s◦2 ◦ s2 / • ◦ / / s◦1 ◦ s2 s2 sε s2 / s1 s2 s1 s2 s1 s2 (f ) ◦ / s•ε ◦ / (d) ◦ (e) / s•ε / s◦1 / / s1 s2 sε s2 s1 • s2 sε s2 • s1 s2 sε s2 s1 • From Theorem 4.7 and Proposition 4.8, it follows that the inverse monoids determined by quivers (a)–(f ) are isomorphic to the symmetric inverse semigroup I3 , respectively. The presentation of I3 determined by the quiver (a) admits an initial cellular bases by Theorem 19 of [8] or Theorem 2.2. We can construct an R-linear automorphism of R[I3 ] using these presentations corresponding to quivers (a)–(f ), and then by Corollary 2.3, we obtain new cellular bases of R[I3 ]. 5. Mutations of quivers of finite type Throughout this section, let as before ∆ be one of Aεn−1 , Bnε , and Dnε in Figure 2, we consider the way of mutations of ∆ quivers and the oriented cycles appearing in them, refer to [2, 21]. A quiver without no loops and no 2-cycles is said to be of finite type if it is mutation equivalent to a Dynkin quiver. A chordless cycle is a cycle such that no two vertices of the cycle are connected by an edge that does not itself. One can show [21, Proposition 9.7] (or [2, Proposition 2.1]) that all chordless cycles are oriented in the mutation classes of Dynkin quivers. We extend the results in [21] and [2, Corollary 2.3] to the case of ∆ quivers. Lemma 5.1. Let Q be a ∆ quiver and k a mutable vertex of Q. Suppose that k has two neighbouring vertices. Then the induced subquivers of Q containing vertex k and its neighbours are shown in Figure 4. The effect of the mutation of Q at k is shown in each case. QUIVER MUTATIONS AND BOOLEAN REFLECTION MONOIDS µ k (a) ◦ ✆B \✾✾✾ ✾✾ ✆✆ ✆ ✾✾ ✆✆ ✾✾ ✆ ✆✆ ◦ i ◦ ◦ B ◦ \✾ ✆✆ ✾✾✾ ✆ ✾✾2 ✆✆ ✾✾ ✆✆ ✾ ✆ ✆ i ◦ ◦ ◦ i i µk ←→ ◦ j k (a’) ◦ B ◦ \✾ ✆✆ ✾✾✾ ✆ ✾✾ ✆✆ ✾✾ ✆✆ ✾ ✆✆ • (c’) ◦ ◦ ✆ \✾✾✾ ✾✾ ✆✆ ✆ ✾✾ ✆✆ ✾✾ ✆ ✆ ✆ ←→ • ε i µ k (e’) B◦✾ ✆✆ ✾✾✾ ✆ ✾✾ ✆✆ ✾✾ ✆✆ ✾ ✆ ✆ k ←→ 2 ◦ (g’) • k ◦ \✾ ✆✆ ✾✾✾ ✆ 2 ✆ ✾✾2 ✆ ✾✾ ✆✆ ✾ ✆ ✆ /• ◦ i i ◦ ◦ ✆B ✾✾✾ ✾✾ ✆✆ ✆ ✾✾ ✆✆ ✾✾ ✆ ✆✆ • ◦o (d’) k (f’) ε i µk ←→ 2 ε ←→ • ◦ i j k ◦ \✾ ✆✆ ✾✾✾ ✆ ✾✾ ✆✆ ✾✾ ✆✆ ✾ ✆ ✆ /• ◦ ε k ◦ ✆ ✾✾✾ ✾✾ ✆✆ ✆ ✾✾ ✆✆ ✾✾ ✆ ✆ ✆ 2 • ◦ µ k ←→ • ε i 2 ◦ ◦ \✾ ✆✆ ✾✾✾ ✆ 2 ✆ ✾✾2 ✆ ✾✾ ✆✆ ✾ ✆ ✆ /◦ ◦ ←→ k ◦ \✾ ✆✆ ✾✾✾ ✆ ✾✾ ✆✆ ✾✾ ✆✆ ✾ ✆ ✆ j k i ε i 2 µk k B ◦ \✾ ✆✆ ✾✾✾ 2 ✆✆ ✾✾ ✆ ✾✾ ✆✆ ✾ ✆ ✆ ◦ \✾ ✆✆ ✾✾✾ ✆ 2 ✆ ✾✾ ✆ ✾✾ ✆✆ ✾ ✆ ✆ /◦ ◦ i µk ε i ε ←→ j B◦✾ ✆✆ ✾✾✾ ✆ ✾✾ ✆✆ ✾✾ ✆✆ ✾ ✆✆ j k i µk k 2 ε i ◦ B◦✾ ✆✆ ✾✾✾ ✆ 2 ✆ ✾✾2 ✆ ✾✾ ✆✆ ✾ ✆ ✆ ◦o ◦ • ◦ \✾ ✆✆ ✾✾✾ ✆ ✾✾ ✆✆ ✾✾ ✆✆ ✾ ✆ ✆ /• ◦ ←→ j (b’) k i B◦✾ ✆✆ ✾✾✾ ✆ 2 ✆ ✾✾ ✆ ✾✾ ✆✆ ✾ ✆ ✆ k ◦ ✆ \✾✾✾ ✾✾ ✆✆ ✆ ✾✾ ✆✆ ✾✾ ✆ ✆ ✆ /◦ ◦ i µk k ε i µk k ◦✾ ✆✆ ✾✾✾ ✆ ✾✾ ✆✆ ✾✾ ✆✆ ✾ ✆✆ ◦ j i j k ◦ ε i 2 i ←→ ◦ j (f) k ←→ k ◦ ◦ \✾ ✆✆ ✾✾✾ ✆ ✾✾2 ✆✆ ✾✾ ✆✆ ✾ ✆ ✆ /◦ ◦ ◦ ✆B ✾✾✾ ✾✾ ✆✆ ✆ ✾✾ ✆✆ ✾✾ ✆ ✆✆ i (d) k µk ◦ ◦ j ◦✾ ✆✆ ✾✾✾ ✆ ✾✾2 ✆✆ ✾✾ ✆✆ ✾ ✆ ✆ µ k (b) k ←→ j B◦✾ ✆✆ ✾✾✾ ✆ ✾✾2 ✆✆ ✾✾ ✆✆ ✾ ✆ ✆ ◦ ✆ ✾✾✾ ✾✾ ✆✆ ✆ ✾✾ ✆✆ ✾✾ ✆ ✆ ✆ i µk k (e) ◦ j k (c) k k ←→ 21 k B◦✾ ✆✆ ✾✾✾ ✆ ✾✾ ✆✆ ✾✾ ✆✆ ✾ ✆ ✆ o ◦ • 2 • ε i 2 ε k B◦✾ ✆✆ ✾✾✾ ✆ 2 ✆ ✾✾2 ✆ ✾✾ ✆✆ ✾ ✆ ✆ • ◦o i ε Figure 4. Subquivers of mutations of Q. In a diagram, a vertex is said to be connected to another if there is an edge between them. Let Q be a quiver mutation equivalent to a Dynkin quiver. In Lemma 2.4 of [2], Barot and Marsh have described the way vertices in Q can be connected to a chordless cycle: A vertex is connected to at most two vertices of a chordless cycle, and if it is connected to two vertices, then the two vertices must be adjacent in the cycle. The following lemma is a generalization of Barot and Marsh’s results [2, Lemma 2.5]. Lemma 5.2. Let Q′ = µk (Q) be the mutation of Q at vertex k. We list various types of induced subquivers in Q and corresponding cycles in Q′ . Then every chordless cycle in Q′ arises in such a way. 22 BING DUAN, JIAN-RONG LI, AND YAN-FENG LUO µ k (a) ◦ ◦ ✂ ]❁❁❁ ❁❁ ✂✂ ✂ ❁❁ ✂✂ ❁ ✂ ✂ µ ◦ ◦ ✂ ]❁❁❁ ❁❁2 ✂✂ ✂ ❁❁ ✂✂ ❁ ✂ ✂ k (a’) ◦ k −→ ◦ ◦ ✆B ✾✾✾ ✾✾ ✆✆ ✆ ✾✾ ✆✆ ✾✾ ✆ ✆✆ • µ k B◦✾ ✆✆ ✾✾✾ ✆ ✾✾ ✆✆ ✾✾ ✆✆ ✾ ✆ ✆ k −→ 2 ◦ (e’) ◦ \✾ ✆✆ ✾✾✾ ✆ 2 ✆ ✾✾2 ✆ ✾✾ ✆✆ ✾ ✆ ✆ /• ◦ (b’) k (d’) 2 µk ←→ ◦ d❏❏ (h’) ◦k O k k−1 2 6 ◦ ❴ ❴ ❴/ ◦ / ◦1 k ←→ k ◦ o❴ ❴ ❴ ◦ o d−2 k−1 2 6 ◦ ❴ ❴ ❴/ ◦ ◦ k k ◦S ◦ o❴ ❴ ❴ ◦ o k+1 d−1 ◦ d i ε i • ε ε j / ◦1 ◦ o❴ ❴ ❴ ◦ o k+1 µ ◦ d−2 2 ◦O ❴ ❴ ❴/ ◦ k−1 k ←→ k d ◦ ( •L k ◦ v ε ε j ✉: ◦ ✉✉ ✉✉ ✉ ✉ ✉✉ ✉✉ ✉✉✉ ◦o • 2 ◦O ❴ ❴ ❴/ ◦ d 2 i µk k−1 d−1 / ◦1 i • ε j ◦ B◦✾ ✆✆ ✾✾✾ ✆ ✾✾ ✆✆ ✾✾ ✆✆ ✾ ✆ ✆ o ◦ • µk / ◦i ←→ ◦k o ◦o µ ε k / ◦ ←→ ◦ o❏ ◦O O ❏❏ ❏❏ ❏❏ ❏❏ ❏❏ ❏❏ ❏$ • ◦o • i ε k ◦ ✆B ✾✾✾ ✾✾ ✆✆ ✆ ✾✾ ✆✆ ✾✾ ✆ ✆✆ ◦o • 2 k (f’) ◦ O / ◦i ◦ ◦S (i’) j ◦o ε k −→ ε i ε • k+1 ◦ \✾ ✆✆ ✾✾✾ ✆ ✾✾ ✆✆ ✾✾ ✆✆ ✾ ✆ ✆ ◦ ✂A ❁❁❁ ❁❁2 ✂✂ ✂ ❁❁ ✂✂ ❁ ✂ ✂ o ◦ ◦ i µ k ◦ B◦✾ ✆✆ ✾✾✾ ✆ 2 ✆ ✾✾2 ✆ ✾✾ ✆✆ ✾ ✆ ✆ ◦o • j • 2 k i µ k −→ ε i ε ❏❏ ❏❏ ❏❏ ❏❏ ❏❏ ❏❏ /• k ◦ ◦ \✾ ✆✆ ✾✾✾ ✆ ✾✾ ✆✆ ✾✾ ✆✆ ✾ ✆ ✆ /• ◦ µk / ◦i ←→ ◦o ◦ ✆ \✾✾✾ ✾✾ ✆✆ ✆ ✾✾ ✆✆ ✾✾ ✆ ✆ ✆ k 2 k ε i ε (g’) ◦j O (e) • k i k k ◦ ✂A ❁❁❁ ❁❁ 2 ✂✂✂ ❁❁ ✂✂ ❁ ✂ ✂ ◦ ◦o 2 µ k ←→ ◦ ✂ ]❁❁❁ ❁❁2 ✂✂ ✂ ❁❁ ✂✂ ❁ ✂ ✂ /◦ ◦ 2 ε i ◦ 2 ◦ ✆ \✾✾✾ ✾✾ ✆✆ ✆ ✾✾ ✆✆ ✾✾ ✆ ✆ ✆ /• ◦ i k −→ k (d) ◦ ✂A ❁❁❁ ❁❁2 ✂✂ ✂ ❁❁ ✂✂ ❁ ✂ ✂ 2 o ◦ ◦ ε i (c’) µ k −→ ◦ ✂ ]❁❁❁ ❁❁ 2 ✂✂✂ ❁❁ ✂✂ ❁ ✂ ✂ ◦ k µ k (b) ◦ ✂A ❁❁❁ ❁❁ ✂✂ ✂ ❁❁ ✂✂ ❁ ✂ ✂ ◦ ◦o ◦ k (c) k k −→ v d−1 / ◦1 i ε ◦ ( •L ◦ o❴ ❴ ❴ ◦ o k+1 d−1 ◦ d (j’) The vertex k does not connect to an oriented chordless cycle C in Q. Then C is the corresponding cycle in Q′ . QUIVER MUTATIONS AND BOOLEAN REFLECTION MONOIDS 23 (k’) The vertex k connects to one vertex of an oriented chordless cycle C in Q (via an edge of unspecified weight). Then C is the corresponding cycle in Q′ . By Lemmas 5.1 and 5.2, we have the following corollary. Corollary 5.3. Let Q be a quiver in the mutation class of a ∆ quiver. Then the frozen vertex ε in Q has one neighbour or two neighbours and if it has two neighbours, then ε must be in an oriented cycle. 6. Cycle relations and path relations In this section, we find an efficient subset of the relations sufficient to define Boolean reflection monoids, which generalizes Barot and Marsh’s results, Lemmas 4.1, 4.2, 4.4 and Proposition 4.6 in [2]. Lemma 6.1 ([2, Lemmas 4.1, 4.2 and 4.4]). Let Q be a Dynkin quiver and W (Q) the reflection group determined by Q, see Section 3. (1) If Q contains a chordless cycle Cd , see Figure 5 (1), d ≥ 3, then the following are equivalent: (a) (sa sa+1 · · · sa+d−1 sa+d−2 · · · sa+1 )2 = e (with subscripts modulo d) for a single fixed value of a, 0 ≤ a ≤ d − 1; (b) (sa sa+1 · · · sa+d−1 sa+d−2 · · · sa+1 )2 = e (with subscripts modulo d) for any a = 0, 1, · · · , d − 1. (2) If Q contains a chordless 3-cycle C3 , see Figure 5 (2), then the following are equivalent: (a) (s1 s2 s3 s2 )2 = e; (b) (s2 s3 s1 s3 )2 = e. Furthermore, if one of the above holds, then the following holds: (c) (s3 s1 s2 s1 )3 = e. (3) If Q contains a chordless 4-cycle C4 , see Figure 5 (3), then the following are equivalent: (a) (s1 s2 s3 s4 s3 s2 )2 = e; (b) (s3 s4 s1 s2 s1 s4 )2 = e. Furthermore, if one of the above holds, then the following holds: (c) (s2 s3 s4 s1 s4 s3 )3 = e; (d) (s4 s1 s2 s3 s2 s1 )3 = e. (1) 0 ◦O ◦ o d−1 / ◦1 .. (2) . 2 ◦ ✆B ✾✾✾ ✆ ✾✾ 2 ✆✆ ✾✾ ✆ ✆ ✾✾ ✆ ✆ ✆ o ◦ ◦ 1 2 3 1 (3) ◦O / ◦2 2 2 ◦o 4 ◦ 3 Figure 5. (1) A chordless d-cycle Cd , (2) a chordless 3-cycle C3 , and (3) a chordless 4-cycle C4 . 24 BING DUAN, JIAN-RONG LI, AND YAN-FENG LUO Let ∆ be one of Aεn−1 , Bnε , and Dnε in Figure 2. Suppose that Q is any quiver mutation equivalent to a ∆ quiver. The following lemma shows an efficient subset of relations (R3) and (R4) in Definition 4.4, which generalizes the above lemma. Lemma 6.2. Let M (Q) be an inverse monoid with generators subjecting to relations (R1), (R2) in Definition 4.4. (1) If Q contains a chordless cycle C3′ , see Figure 6 (1), then the following statements are equivalent: (a) sε s1 s2 s1 = s1 s2 s1 sε ; (b) s1 s2 sε s2 = s2 sε s2 s1 . Furthermore, if one of the above holds, then the following statements are equivalent: (c) sε s1 sε = sε s1 sε s1 = s1 sε s1 sε ; (d) sε s2 sε = sε s2 sε s2 = s2 sε s2 sε . (2) If Q contains a chordless cycle C3′′ , see Figure 6 (2), then s1 s2 sε s2 = s2 sε s2 s1 . (3) If Q contains a chordless cycle C4′ , see Figure 6 (3), then the following statement holds: (a) sε s1 s2 s3 s2 s1 = s1 s2 s3 s2 s1 sε ; (b) s1 s2 s3 sε s3 s2 = s2 s3 sε s3 s2 s1 . Furthermore, if one of the above holds, then the following statements are equivalent: (c) sε s1 sε = sε s1 sε s1 = s1 sε s1 sε ; (d) sε s3 sε = sε s3 sε s3 = s3 sε s3 sε . (4) If Q contains a subquiver Cd′ , see Figure 6 (4), then the following statements are equivalent: (a) sa sa+1 · · · sd P (s1 , sε )P (sε , s1 )sd · · · sa+1 sa = sa−1 sa−2 · · · s1 s1 P (s1 , sε ) P (sε , s1 )s1 s1 · · · sa−2 sa−1 for a single fixed value of a, 2 ≤ a ≤ d; (b) sa sa+1 · · · sd P (s1 , sε )P (sε , s1 )sd · · · sa+1 sa = sa−1 sa−2 · · · s1 s1 P (s1 , sε ) P (sε , s1 )s1 s1 · · · sa−2 sa−1 for any a = 2, · · · , d. (1) 2 ◦ ✆B ✾✾✾ ✆ ✾✾ ✆✆ ✾✾ ✆✆ ✾✾ ✆ ✆✆ o • ◦ ε 1 (3) 2 ◦ ✆B ✾✾✾ ✆ ✾✾ 2 ✆✆ ✾✾ ✆✆ ✾✾ ✆ ✆✆ o • ◦ (2) 2 ◦O / ◦3 ◦o • 1 ε 2 1 ε 2 ◦O (4) ✝ ✝✝ ✝ ✝ ✝✝ ✝ ✝ /◦o •❴ ❴ ❴◦ ε 1 d / ◦3 .. . Figure 6. (1) A chordless 3-cycle C3′ , (2) a chordless 3-cycle C3′′ , (3) a chordless 4-cycle C4′ , and (4) a subquiver Cd′ . (see Lemma 6.2) QUIVER MUTATIONS AND BOOLEAN REFLECTION MONOIDS 25 Proof. For (1), the equivalence of (a) and (b) follows from: s1 s2 sε s2 = s2 (s1 s2 s1 sε )s2 , s2 sε s2 s1 = s2 (sε s1 s2 s1 )s2 , using (R2). Suppose that (a) and (b) hold. Then by (R1), (R2), (a), and (b), the equivalence of (c) and (d) follows from: s1 s2 s1 (sε s1 sε )s1 s2 s1 = sε s1 s2 s1 s1 s1 s2 s1 sε = sε s2 sε , s1 s2 s1 (sε s1 sε s1 )s1 s2 s1 = sε (s1 s2 sε s2 )s1 = sε s2 sε s2 , s1 s2 s1 (s1 sε s1 sε )s1 s2 s1 = s1 (s2 sε s2 s1 )sε = s2 sε s2 sε . For (3), the equivalence of (a) and (b) follows from: s1 s2 s3 sε s3 s2 = s2 s3 (s1 s2 s3 s2 s1 sε )s3 s2 , s2 s3 sε s3 s2 s1 = s2 s3 (sε s1 s2 s3 s2 s1 )s3 s2 , using first s1 s2 s3 s2 s1 = s3 s2 s1 s2 s3 and then (R1). Suppose that (a) and (b) hold. Using first s2 sε = sε s2 and then (a), we have: s1 s2 s3 s2 s1 s2 (sε s1 sε )s2 s1 s2 s3 s2 s1 = (s1 s2 s3 s2 s1 sε )s2 s1 s2 (sε s1 s2 s3 s2 s1 ) = (sε s1 s2 s3 s2 s1 )s2 s1 s2 (s1 s2 s3 s2 s1 sε ) = sε s1 s2 s3 s2 s3 s2 s1 sε = sε s3 sε , (by (R2)) where in the last equation we used that s1 and s3 commute. Using (R1), (R2), (a), and (b), by a similar argument, we have s1 s2 s3 s2 s1 s2 (s1 sε s1 sε )s2 s1 s2 s3 s2 s1 = s1 s2 s3 s1 s2 sε s1 s2 s1 s2 s3 s2 s1 sε , = s1 s2 s1 s3 sε s1 s3 s2 s1 sε , = s2 (s1 s2 s3 sε s3 s2 )s1 s2 sε , = s2 (s2 s3 sε s3 s2 s1 )s1 s2 sε , = s3 sε s3 sε . s1 s2 s3 s2 s1 s2 (sε s1 sε s1 )s2 s1 s2 s3 s2 s1 = (s1 s2 s3 s2 s1 sε )s2 s1 sε s2 s1 s3 s2 s1 = sε s1 s2 s3 s2 s1 s2 s1 s2 sε s3 s1 s2 s1 = sε s2 (s1 s2 s3 sε s3 s2 )s1 s2 = sε s2 (s2 s3 sε s3 s2 s1 )s1 s2 = sε s3 sε s3 . Therefore (c) and (d) are equivalent. For (4), using (R1), it is obvious. At an end of this section, we show that M (Q) could be defined using only the underlying unoriented weighted diagram of Q, by taking relations (R1)–(R4) corresponding to both Q and Qop as the defining relations. Our result can be viewed as a generalization of Proposition 4.6 of [2]. Proposition 6.3. Let M (Φ, B) be a Boolean reflection monoid with generators si , i ∈ I ∪ {ε}. Then the generators satisfy (R1)–(R4) with respect to Q if and only if they satisfy (R1)–(R4) with respect to Qop . 26 BING DUAN, JIAN-RONG LI, AND YAN-FENG LUO Proof. We assume that generators si , i ∈ I ∪ {ε} satisfy relations (R1)–(R4) with respect to Q, and show that these generators satisfy relations (R1)–(R4) with respect to Qop . The converse follows by replacing Q with Qop . Since (R1) and (R2) do not depend on the orientation of Q, generators si , i ∈ I ∪ {ε} satisfy relation (R1) and (R2) with respect to Qop . The cases of chordless cycles appearing in quivers of finite type have been proved in Proposition 4.6 of [2]. The remaining needed to check the cases are C3′ , C3′′ , C4′ , and Cd′ shown in Figure 6. Case 1. In C3′ , we have sε s2 s1 s2 = sε s1 s2 s1 = s1 s2 s1 sε = s2 s1 s2 sε , s2 s1 sε s1 = s1 s2 (s1 s2 sε s2 )s2 s1 = s1 s2 (s2 sε s2 s1 )s2 s1 = s1 sε s1 s2 . Case 2. In C3′′ , we have sε s2 s1 s2 = s2 s1 (s1 s2 sε s2 )s1 s2 = s2 s1 (s2 sε s2 s1 )s1 s2 = s2 s1 s2 sε . Case 3. In C4′ , note that s1 s2 s3 s2 s1 = s3 s2 s1 s2 s3 . We have sε s3 s2 s1 s2 s3 = sε s1 s2 s3 s2 s1 = s1 s2 s3 s2 s1 sε = s3 s2 s1 s2 s3 sε , s3 s2 s1 sε s1 s2 = s2 s1 s3 s2 (s1 s2 s3 sε s3 s2 )s2 s3 s1 s2 = s2 s1 s3 s2 (s2 s3 sε s3 s2 s1 )s2 s3 s1 s2 = s2 s1 sε s1 s2 s3 . Case 4. In Cd′ , it follows from Lemma 6.2 (4) that (R4) do not depend on the orientation of chordless cycles in Cd′ . Since every chordless cylce in Qop corresponds to a chordless cycle in Q, the result holds. 7. The proof of Theorem 4.7 In this section, we give the proof of Theorem 4.7. Let ∆ be one of Aεn−1 , Bnε , and Dnε in Figure 2. We fix a ∆ quiver Q. Let Q′ = µk (Q) be the mutation of Q at vertex k, k ∈ I. Throughout the section, we will write si and ri for the generators corresponding to vertex i ∈ I ∪ {ε} of M (Q) and M (Q′ ) respectively. Similar to [2], we define elements ti , i ∈ I, and tε in M (Q) as follows: ( sk si sk if there is an arrow i → k in Q (possibly weighted), ti = si otherwise, ( (7.1) sk sε sk if there is an arrow ε → k in Q (possibly weighted), tε = sε otherwise. Then ( (sk si sk )(sk si sk ) = e if there is an arrow i → k in Q (possibly weighted), t2i = s2i = e otherwise, ( (sk sε sk )(sk sε sk ) = tε if there is an arrow ε → k in Q (possibly weighted), t2ε = otherwise. s2ε = tε (7.2) QUIVER MUTATIONS AND BOOLEAN REFLECTION MONOIDS 27 In order to prove Theorem 4.7, we need the following proposition, which we will prove in Section 7.2. Proposition 7.1. For each i ∈ I ∪ {ε}, the map Φ : M (Q′ ) −→ M (Q) ri 7−→ ti , is an inverse monoid homomorphism. 7.1. Proof Theorem 4.7. For each vertex i ∈ I ∪ {ε} of Q define the elements t′i in M (Q′ ) as follows: ( rk ri rk if there is an arrow k → i in Q′ , t′i = ri otherwise, ( rk rε rk if there is an arrow k → ε in Q′ , t′ε = rε otherwise. We claim that these elements t′i , for each vertex i ∈ I ∪ {ε}, satisfy the relations (R1)– (R4) defining M (Q). This follows from Proposition 7.1 by interchanging Q and Q′ and using the fact that the definition of M (Q) is unchanged under reversing the orientation of all the arrows in Q (see Lemma 6.3). Therefore there is an inverse monoid homomorphism Θ : M (Q) → M (Q′ ) such that Θ(si ) = t′i for each i. If there is no arrow i → k in Q, then there is also no arrow k → i in Q′ and consequently Θ ◦ Φ(ri ) = Θ(si ) = ri . If there is an arrow i → k in Q, then there is an arrow k → i in Q′ and therefore Θ ◦ Φ(ri ) = Θ(sk si sk ) = Θ(sk )Θ(si )Θ(sk ) = rk (rk ri rk )rk = ri . So Θ ◦ Φ = idM (Q′ ) , and, similarly, Φ ◦ Θ = idM (Q) , and hence Θ and Φ are isomorphisms. 7.2. The proof of Proposition 7.1. We will prove Proposition 7.1 by showing that the elements ti , i ∈ I ∪ {ε} satisfy the (R1)–(R4) relations in M (Q′ ). We denote by m′ij the value of mij for Q′ . By Equation (7.2), (R1) is obvious. In the sequel, the proof that the elements ti , i ∈ I ∪ {ε} satisfy (R2) in M (Q′ ) follows from Lemma 7.2 and the rest of proof is completed case by case. Lemma 7.2. The elements ti , for i a vertex of Q, satisfy the following relations. ′ (1) If i = k or j = k and i, j 6= ε, then (ti tj )mij = e. (2) If at most one of i, j is connected to k in Q (or, equivalently, in Q′ ) and i, j 6= ε, ′ then (ti tj )mij = e. (3) Let i be in I. Then   if ε, i are not connected in Q′ ,  ti tε = tε ti tε ti tε = tε ti tε ti = ti tε ti tε if ε, i are connected by an edge with weight 1 in Q′ ,   tε = ti tε = tε ti if ε, i are connected by an edge with weight 2 in Q′ . Proof. In Lemma 5.1 of [2], Barot and Marsh proved the parts (1) and (2). We only need to prove the part (3). 28 BING DUAN, JIAN-RONG LI, AND YAN-FENG LUO Suppose without loss of generality that i = k. The only nontrivial case is when there is an arrow ε → k = i with a weight q in Q. If q = 1, then tε tk tε tk = (sk sε sk )sk (sk sε sk )sk = sk sε sk sε = sε sk sε sk = sk (sk sε sk )sk (sk sε sk ) = tk tε tk tε , = sε sk sε = sk sε sk sε sk = (sk sε sk )sk (sk sε sk ) = tε tk tε . If q = 2, note that sε sk = sk sε = sε , we have tk tε = sk (sk sε sk ) = sε sk = sk sε = sε = tε tk = tε . In the following, suppose that i 6= k. We divide this proof into three cases. Case 1. There are no arrows from i, ε to k, then ti = si , tε = sε hold (3). Case 2. There are arrows from one of i, ε to k and there are no arrows from the other of i, ε to k in Q, then we assume that there are arrows from ε to k and there are no arrows from i to k in Q. If ε, i are not connected in Q, we have ti tε = si (sk sε sk ) = (sk sε sk )si = tε ti . If ε, i are connected by an edge with weight 1 in Q, then tε ti tε = (sk sε sk )si (sk sε sk ) = sk sε si sε sk = sk sε si sε sk si = (sk sε sk )si (sk sε sk )si = tε ti tε ti = si sk sε si sε sk = si (sk sε sk )si (sk sε sk ) = ti tε ti tε . That ε, i are connected by an edge with weight 2 and there are no arrows from i to k in Q is impossible, because of the fact that there is only 3-chordless cycle in the mutation class of Bnε quivers and Corollary 5.3. Case 3. There are arrows from i, ε to k. The possibilities for the subquivers induced by i, ε, and k are enumerated in (a’)–(g’) of Figure 4. We show that ti and tε satisfy (3) by checking each case. Within each case, subcase (i) is when the subquiver of Q is the diagram on the left, and subcase (ii) is when the subquiver of Q is the diagram on the right. (a′ )(i) We have ti tε = (sk si sk )(sk sε sk ) = sk si sε sk = sk sε si sk = (sk sε sk )(sk si sk ) = tε ti . (a′ )(ii) We have ti tε = si sε = sε si = tε ti . (b′ ) (i) We have tε ti tε = sε (sk si sk )sε = sε si sk si sε = si (sε sk sε )si , ( s i s ε s k s ε s k s i = s ε s i s k s i s ε s i s k s i = s ε s k s i s k s ε s k s i s k = tε ti tε ti , = si sk sε sk sε si = si sk si sε si sk si sε = (sk si sk )sε (sk si sk )sε = ti tε ti tε . (b′ ) (ii) We have ti tε = si (sk sε sk ) = sk (sk si sk sε )sk = sk (sε sk si sk )sk = (sk sε sk )si = tε ti . (c′ ) (i) We have tε ti tε = (sk sε sk )si (sk sε sk ) = sk sε si sk si sε sk = sk si sε sk sε si sk , ( sk si sε sk sε sk si sk = sk sε si sk si sε sk si = (sk sε sk )si (sk sε sk )si = tε ti tε ti , = sk si sk sε sk sε si sk = si sk sε si sk si sε sk = si (sk sε sk )si (sk sε sk ) = ti tε ti tε . (c′ ) (ii) We have ti tε = (sk si sk )sε = si sk si sε = sε si sk si = sε sk si sk = tε ti . (d′ ) (i) We have ti tε = (sk si sk )(sk sε sk ) = sk si sε sk = sk sε si sk = (sk sε sk )(sk si sk ) = tε ti . QUIVER MUTATIONS AND BOOLEAN REFLECTION MONOIDS 29 (d′ )(ii) We have ti tε = si sε = sε si = tε ti . (e′ )(i) Note that si sk sε = sk sε and sε sk si = sε sk . We have ti tε = (sk si sk )sε = sk (sk sε ) = sε = tε , tε ti = sε (sk si sk ) = (sε sk )sk = sε = tε . (e′ )(ii) We have ti tε = si (sk sε sk ) = si sk (sε sk si sk )sk si = si sk (sk si sk sε )sk si = sk sε sk si = tε ti . (f ′ )(i) Note that si sk sε = sk sε and sε sk si = sε sk . We have ti tε = si (sk sε sk ) = sk sε sk = tε , tε ti = (sk sε sk )si = sk sε sk = tε . (f ′ )(ii) We have ti tε = (sk si sk )sε = sk (si sk sε sk )sk = sk (sk sε sk si )sk = sε sk si sk = tε ti . (g′ )(i) Note that sk sε = sε sk = sε . We have ( s i s ε s i s ε = ti tε ti tε , tε ti tε = (sk sε sk )si (sk sε sk ) = sε si sε = s ε s i s ε s i = tε ti tε ti . (g′ )(ii) Note that sk sε = sε sk = sε . We have ( sε si sε sk = sε si sε si sk = sε (sk si sk )sε (sk si sk ) = tε ti tε ti , tε ti tε = sε (sk si sk )sε = sk sε si sε = sk si sε si sε = (sk si sk )sε (sk si sk )sε = ti tε ti tε . The possibilities for chordless cycles in mutation classes of ∆ quivers are enumerated in Lemma 5.2. For (R3), Barot and Marsh proved in [2] that (R3) (i) holds for (a)–(e). We show that (R3) (ii) and (R3) (iii) hold by checking (a′ )–(k′ ). In each case, we need to check that the corresponding cycle relations hold. Within each case, subcase (i) is when the subquiver of Q is the diagram on the left, and subcase (ii) is when the subquiver of Q is the diagram on the right. In the sequel, we frequently use (R1) and (R2) without comment. (a′ )(i) We have tε tk ti tk = sε sk (sk si sk )sk = sε si = si sε = sk (sk si sk )sk sε = tk ti tk tε . (b′ )(i) We have tε ti tk ti = (sk sε sk )si sk si = sk sε si sk = sk si sε sk = si sk si (sk sε sk ) = ti tk ti tε . (c′ )(i) We have tε tk ti tk = sε sk (sk si sk )sk = sε si = si sε = sk (sk si sk )sk sε = tk ti tk tε . (d′ )(i) We have ti tk tε tk = si sk (sk sε sk )sk = si sε = sε si = sk (sk sε sk )sk si = tk tε tk ti . (e′ )(i) Note that sk sε = sε sk = sε and sk si sε si = si sε si sk . We have tε ti tk ti = (sk sε sk )si sk si = sε si sk si = si sk (sk si sε si )sk si = si sk (si sε si sk )sk si = si sk si sε = si sk si (sk sε sk ) = ti tk ti tε . 30 BING DUAN, JIAN-RONG LI, AND YAN-FENG LUO (e′ )(ii) Note that sk sε = sε sk = sε and sε si sk si = si sk si sε . We have tk ti tε ti = sk (sk si sk )sε (sk si sk ) = si sε si sk = si (sε si sk si )si = si (si sk si sε )si = sk si sε si = (sk si sk )sε (sk si sk )sk = ti tε ti tk . (f ′ ) (i) Note that sj sε = sε sj and sε si sj sk sj si = si sj sk sj si sε . We have tε tk tj tk = sε sk (sk sj sk )sk = sε sj = sj sε = sk (sk sj sk )sk sε = tk tj tk tε , tε ti tj ti = sε si (sk sj sk )si = sε si sj sk sj si = si sj sk sj si sε = si (sk sj sk )si sε = ti tj ti tε . (f ′ ) (ii) Note that sk si = si sk and sε si sj si = si sj si sε .We have tε ti tj tk tj ti = (sk sε sk )si sj sk sj si = sk sε si sk sj sk sj si = sk (sε si sj si )sk = sk (si sj si sε )sk = si sj sk sj si (sk sε sk ) = ti tj tk tj ti tε . (g′ ) (i) Note that sj sε = sε sj and sε sk sj si sj sk = sk sj si sj sk sε . We have tε tj tk tj = (sk sε sk )sj sk sj = sk sε sj sk = sk sj sε sk = sj sk sj (sk sε sk ) = tj tk tj tε , tε tj ti tj = (sk sε sk )sj si sj = sj si sj sk sε sk = tj ti tj tε . (g′ ) (ii) Note that sk si = si sk and sε sj si sj = sj si sj sε . We have tε tk tj ti tj tk = sε sk (sk sj sk )si (sk sj sk )sk = sε sj sk si sk sj = sε sj si sj = sj si sj sε = sk (sk sj sk )si (sk sj sk )sk sε = tk tj ti tj tk tε . (h′ ) (i) Note that si sj = sj si , sε sk = sk sε , and sε sj sk si sk sj = sj sk si sk sj sε . We have ti tk tj tk = si sk (sk sj sk )sk = si sj = sj si = sk (sk sj sk )sk si = tk tj tk ti , tε tj ti tj = sε (sk sj sk )si (sk sj sk ) = sk (sε sj sk si sk sj )sk = sk (sj sk si sk sj sε )sk = (sk sj sk )si (sk sj sk )sε = tj ti tj tε . (h′ ) (ii) Note that sε sj si sj = sj si sj sε . We have tε tj tk ti tk tj = sε sj sk (sk si sk )sk sj = sε sj si sj = sj si sj sε = sj sk (sk si sk )sk sj sε = tj tk ti tk tj tε . Case (i′ ) follows from either Barot and Marsh’s result or (a′ ) or (b′ ) or (h′ ). Case is trivial and Case (k′ ) follows from the commutative property of tk and ti for each vertex i in C. For (R4), by Lemma 6.2, we prove the following several cases, where in each case we number the vertices 0, 1, . . ., d, ε of these subquivers for convenience. Within each case, subcase (i) is when the subquiver of Q is the diagram on the left, and subcase (ii) is when the subquiver of Q is the diagram on the right. In the sequel, we frequently use (R1) and (R2) without comment. (1) (j ′ ) ◦ ]❁ ✂✂ ❁❁❁ 2 ✂✂ ❁❁ ❁❁ ✂✂ ✂ ✂ / ◦ ❴ ❴ ❴/ ◦ ◦ 0 2 1 k−1 µ k ←→ /◦ k /• ε k ◦ \✾ ◦ ✂ \✾✾✾ ✆✆ ✾✾✾ ✾✾ ✂✂ 2 ✆✆ ✾ ✂ ✾✾ ✾✾ ✂ ✆ ✾✾ ✾✾ ✂✂ ✆✆ ✆ ✂ ✆ ✂ / ◦ ❴ ❴ ❴/ ◦ /• ◦ 0 2 1 k−1 ε QUIVER MUTATIONS AND BOOLEAN REFLECTION MONOIDS 31 (2) 0 / ◦d ❴ ❴ ❴/ ◦4 ε • / ◦3 0 µ2 ←→ B◦ ✆✆ ✆ ✆✆ ✆✆ ✆ / ◦2 ✾✾ ✾✾ ✾✾ ✾✾ ✾ ε • B◦✾ ✆✆ ✾✾✾ ✆ ✾✾ ✆✆ ✾✾ ✆✆ ✆ 2 / ◦3 o B◦ ✾✾ ✾✾ ✆✆ ✆ ✾✾ ✆ ✾✾ ✆✆ ✾ ✆✆✆ / ◦d ❴ ❴ ❴/ ◦4 ◦ ◦ 1 1 (3) 0 ε • / ◦d ❴ ❴ ❴/ ◦4 0 µ0 ◦ \✾ ←→ ✆✆ ✾✾✾ ✆ ✾✾ ✆✆ ✾✾ ✆✆ ✆ 1 / ◦3 ◦ ✾✾ ✆B ✾✾ ✆ ✾✾ ✆✆ ✾✾ ✆✆ ✾ ✆✆✆ ε • B◦✾ ✆✆ ✾✾✾ ✆ ✾✾ ✆✆ ✾✾ ✆✆ 1 ✆ / ◦3 o B◦ ✾✾ ✾✾ ✆✆ ✆ ✾✾ ✆ ✾✾ ✆✆ ✾ ✆✆✆ / ◦d ❴ ❴ ❴/ ◦4 ◦ ◦ 2 2 (4) 1 ◦O / ◦2 ❴ ❴ ❴/ k−1 ◦ ε 0 • ❴ ❴ ❴/ ◦ k ◦ )◦o d 1 µ k ←→ ◦O ε 0 • ❴ ❴ ❴/ ◦ v ◦ o❴ ❴ ❴ ◦ d−1 / ◦2 ❴ ❴ ❴/ k−1 ◦ h k+1 k ◦K )◦o d ◦ o❴ ❴ ❴ ◦ d−1 (1) (i) Note that sk sk−1 sk = sk−1 sk sk−1 and sk−1 sε = sε sk−1 . We have P (t0 , tε ) = t0 P (t1 , tk−1 )tε = s0 P (s1 , sk−1 )sk sk−1 sε = s0 P (s1 , sk−1 )sk sε sk−1 = P (s1 , sk−1 )sk sε sk−1 = P (s1 , sk−1 )sk sk−1 sε = P (t1 , tε ), P (tε , t1 ) = tε P (tk−1 , t1 ) = sε sk−1 sk P (sk−1 , s1 ) = sk−1 sε sk P (sk−1 , s1 ) = sk−1 sε sk P (sk−1 , s1 )s0 = sε sk−1 sk P (sk−1 , s1 )s0 = P (tε , t0 ). (1) (ii) We have P (t0 , tε ) = P (t0 , tk−1 )tk tε = P (s0 , sk−1 )sk (sk sε sk ) = P (s0 , sk−1 )sε sk = P (s1 , sk−1 )sk (sk sε sk ) = P (t1 , tε ), P (tε , t1 ) = (sk sε sk )sk P (sk−1 , s1 ) = sk sε P (sk−1 , s1 )s0 = sk sε sk sk P (sk−1 , s1 )s0 = P (tε , t0 ). k+1 32 BING DUAN, JIAN-RONG LI, AND YAN-FENG LUO (2) (i) We have P (t0 , tε )P (tε , t0 ) = t0 t3 P (t4 , tε )P (tε , t4 )t3 t0 = s0 (s2 s3 s2 )P (s4 , sε )P (sε , s4 )(s2 s3 s2 )s0 = s0 s2 s3 P (s4 , sε )P (sε , s4 )s3 s2 s0 = s1 s2 s3 P (s4 , sε )P (sε , s4 )s3 s2 )s1 = s1 (s2 s3 s2 )P (s4 , sε )P (sε , s4 )(s2 s3 s2 )s1 = t1 t3 P (t4 , tε )P (tε , t4 )t3 t1 = P (t1 , tε )P (tε , t1 ). (2) (ii) We have P (t0 , tε )P (tε , t0 ) = t0 t2 P (t3 , tε )P (tε , t3 )t2 t0 = (s2 s0 s2 )s2 P (s3 , sε )P (sε , s3 )s2 (s2 s0 s2 ) = s2 P (s0 , sε )P (sε , s0 )s2 = s2 P (s1 , sε )P (sε , s1 )s2 = (s2 s1 s2 )s2 P (s3 , sε )P (sε , s3 )s2 (s2 s1 s2 ) = t1 t2 P (t3 , tε )P (tε , t3 )t2 t1 = P (t1 , tε )P (tε , t1 ). (3) (i) We have P (t0 , tε )P (tε , t0 ) = P (s0 , sε )P (sε , s0 ) = P (s2 , sε )P (sε , s2 ) = P (t2 , tε )P (tε , t2 ). (3) (ii) We have P (t2 , tε )P (tε , t2 ) = s2 (s0 s3 s0 )P (s4 , sε )P (sε , s4 )(s0 s3 s0 )s2 , = s2 (s0 s3 P (s4 , sε )P (sε , s4 )s3 s0 )s2 = P (s3 , sε )P (sε , s3 ) = s3 s0 P (s4 , sε )P (sε , s4 )s0 s3 = s0 (s0 s3 s0 )P (s4 , sε )P (sε , s4 )(s0 s3 s0 )s0 , = P (t0 , tε )P (tε , t0 ). (4) (i) We have t2 · · · tk−1 tk+1 · · · td P (t0 , tε )P (tε , t0 )td · · · tk+1 tk−1 · · · t2 = s2 · · · (sk sk−1 sk )sk+1 · · · sd P (t0 , tε )P (tε , t0 )sd · · · sk+1 (sk sk−1 sk ) · · · s2 = s2 · · · (sk−1 sk sk−1 )sk+1 · · · sd P (t0 , tε )P (tε , t0 )sd · · · sk+1 (sk−1 sk sk−1 ) · · · s2 = s2 · · · sk−1 sk sk+1 · · · sd P (t0 , tε )P (tε , t0 )sd · · · sk+1 sk sk−1 · · · s2 = s1 P (t0 , tε )P (tε , t0 )s1 = t1 P (t0 , tε )P (tε , t0 )t1 . QUIVER MUTATIONS AND BOOLEAN REFLECTION MONOIDS 33 (4) (ii) We have t2 · · · tk tk+1 · · · td P (t0 , tε )P (tε , t0 )td · · · tk+1 tk · · · t2 = s2 · · · sk (sk sk+1 sk ) · · · sd P (t0 , tε )P (tε , t0 )sd · · · (sk sk+1 sk )sk · · · s2 = s2 · · · sk+1 sk · · · sd P (t0 , tε )P (tε , t0 )sd · · · sk sk+1 · · · s2 = s2 · · · sk+1 · · · sd P (t0 , tε )P (tε , t0 )sd · · · sk+1 · · · s2 = s1 P (t0 , tε )P (tε , t0 )s1 = t1 P (t0 , tε )P (tε , t0 )t1 . Acknowledgements B. Duan would like to express his gratitude to B. Everitt, W. N. Franzsen, R. Schiffler, C. C. Xi for helpful discussions. B. Duan was supported by China Scholarship Council to visit Uconn Department of Mathematics and he would like to thank Uconn Department of Mathematics for hospitality during his visit. This work was partially supported by the National Natural Science Foundation of China (no. 11371177, 11501267, 11401275). The research of J.-R. Li on this project is supported by the Minerva foundation with funding from the Federal German Ministry for Education and Research. References [1] E. Bannai, Automorphisms of irreducible Weyl groups, J. Fac. Sci. Univ. Tokyo Sect. I 16 (1969), 273–286. [2] M. Barot and R. J. Marsh, Reflection group presentations arising from cluster algebras, Trans. Amer. Math. Soc. 367 (2015), no. 3, 1945–1967. [3] N. Bourbaki, Lie groups and Lie algebras, Chapters 4–6, Springer-Verlag, Berlin, 2002. [4] H. S. M. Coxeter, The complete enumeration of finite groups of the form ri2 = (ri rj )kij = 1, J. London Math. 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[43] , Cellular algebras, available at https://webusers.imj-prg.fr/~ bernhard.keller/ictp2006/lecturenotes/x Bing Duan: School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, P. R. China. E-mail address: [email protected] Jian-Rong Li: Dept. of Mathematics, The Weizmann Institute of Science, Rehovot 7610001, Israel; school of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, P. R. China. E-mail address: [email protected] QUIVER MUTATIONS AND BOOLEAN REFLECTION MONOIDS 35 Yan-Feng Luo: School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, P. R. China. E-mail address: [email protected]	4
arXiv:1709.02152v1 [math.GR] 7 Sep 2017 THE CONJUGACY RATIO OF GROUPS LAURA CIOBANU, CHARLES GARNET COX, AND ARMANDO MARTINO Abstract. In this paper we introduce and study the conjugacy ratio of a finitely generated group, which is the limit at infinity of the quotient of the conjugacy and standard growth functions. We conjecture that the conjugacy ratio is 0 for all groups except the virtually abelian ones, and confirm this conjecture for certain residually finite groups of subexponential growth, hyperbolic groups, right-angled Artin groups, and the lamplighter group. 1. Introduction In this paper we introduce and study the conjugacy ratio of a group, which is the limit of the quotient of two functions naturally associated to any finitely generated group: conjugacy growth and standard growth. More precisely, if G is generated by the finite set X, let BG,X (n) denote the ball of radius n with respect to X, and let CG,X (n) denote the set of conjugacy classes of G which have a representative in BG,X (n). Then the conjugacy ratio of G with respect to X is: (1) crX (G) = lim sup n→∞ \|CG,X (n)\| . \|BG,X (n)\| The motivation of this paper is twofold. On one hand, the conjugacy ratio of a finite group H is equal to the degree of commutativity dc(H) of H, which measures the probability that two elements of the group commute, and is defined as: \|{(x, y) ∈ H × H : xy = yx}\| (2) . dc(H) = \|H\|2 The degree of commutativity of a group has received a lot of attention recently, as its definition was extended to finitely generated infinite groups in [AMV17] to be dcX (G) = lim sup n→∞ \|{(x, y) ∈ BG,X (n)2 : ab = ba}\| . \|BG,X (n)\|2 As raised in [Cox16], it is natural to explore whether the degree of commutativity and the conjugacy ratio are related for infinite groups as well. Our second motivation comes from the fact that very few quantitative results comparing standard and conjugacy growth in groups exist in the literature. While in any group there are fewer conjugacy classes than elements, the gap between these two functions has not been been explored in detail, and it is worth investigating. For example, the standard and conjugacy growth rates (i.e. taking the limit of the nth root of the function at n) are equal in some of the most frequently encountered families of infinite groups: hyperbolic groups [AC17], graph products [CHM17], Date: March 19, 2018. 2010 Mathematics Subject Classification. 20P05, 20F69. Key words and phrases. Conjugacy growth, degree of commutativity, polynomial growth, RAAGs, hyperbolic groups, wreath products. 1 2 LAURA CIOBANU, CHARLES GARNET COX, AND ARMANDO MARTINO many wreath products [Mer17]; thus in these examples the quotient of the two functions, as a function of n, must be at most subexponential, and if the conjugacy ratio is 0, the convergence to 0 will not be very fast. Our starting point is the following conjecture, inspired by [AMV17, Conj. 1.6]. Conjecture 1.1. Let G be a group generated by a finite set X. Then crX (G) > 0 if and only if G is virtually abelian. Our results on the conjugacy ratio in several families of groups support Conjecture 1.1. In Section 3 we investigate groups of stable subexponential growth (Definition 3.1). We first show that any virtually abelian group has crX (G) > 0 for any finite generating set X. We then show that, if N is a normal, finite index subgroup of G, then (for any finite generating set X of G) crX (G) 6 dc(G/N ). This allows us to apply a technique from [AMV17] to show that any residually finite group G of stable subexponential growth which is not virtually abelian has crX (G) = 0 for any finite generating set X. We also show in Theorem 3.9 that if G is a finitely generated virtually abelian group, with finite generating sets X and Y , then crX (G) = crY (G). We say that a group, G, with generating set X has stable subexponential growth \|BG,X (n+1)\| = 1, Definition 3.1. This includes all finitely generated if limn→∞ \|B G,X (n)\| virtually-nilpotent groups. Since all finitely generated virtually-nilpotent groups are residually finite, the theorem below means that Conjecture 1.1 is true for all groups of polynomial growth. Theorem 3.7. The conjugacy ratio for all finitely generated, residually finite groups of stable subexponential growth that are not virtually abelian is zero, with respect to all finite generating sets. The proof of Theorem 3.7 cannot be generalised to groups of exponential growth, but we provide independent arguments for several important classes of groups of exponential growth. Theorem 4.1. Let G be a non-elementary hyperbolic group. Then crX (G) = 0 for any finite generating set X. Theorem 4.3. Let G be the lamplighter group, that is, the wreath product C2 ≀ Z. Then crX (G) = 0 for the standard generating set X (defined in (12)). Theorem 4.12. Let G = (GV , XV ) be a right-angled Artin group (RAAG) based on a graph Γ = (V, E) with generating set XV . Then crXV (G) = 0 unless G is free abelian, in which case crXV (G) = 1. We may also consider the strict or spherical conjugacy ratio, where the counting is done in the sphere of radius n rather than the ball of radius n, that is, we may take the ratio of the strict conjugacy growth function over the spherical growth function. More precisely, let SG,X (n) be the sphere of radius n in the group G s with respect to finite generating set X, and let CG,X (n) be the conjugacy classes that intersect SG,X (n) but not BG,X (n − 1), that is, those conjugacy classes with a minimal length representative in SG,X (n). The spherical conjugacy ratio is then (3) crsX (G) = lim sup n→∞ s \|CG,X (n)\| . \|SG,X (n)\| THE CONJUGACY RATIO OF GROUPS 3 Remark 1.2. By the Stolz-Cesàro theorem, anytime the spherical conjugacy ratio turns out to be a limit, the conjugacy ratio will be equal to this limit. In particular, if the spherical conjugacy ratio is 0, then the conjugacy ratio is 0. 2. Preliminaries Recall that for a finitely generated group, G, with generating set X, the exponential growth rate of G with respect to X is: q (4) ExpX (G) = lim n \|BG,X (n)\|. n→∞ Definition 2.1. A group, G, with finite generating set X, is said to have exponential growth if ExpX (G) > 1 and subexponential growth if ExpX (G) = 1. This does not depend on the generating set, X. Additionally, for any ǫ > 0, if λ = ExpX (G), then for sufficiently large n, λn ≤ \|BG,X (n)\| ≤ (λ + ǫ)n . Moreover, if we replace balls with spheres, we get the same limit and inequality. We collect below a few results on convergence of series that will be relevant later. Theorem 2.2 (Stolz-Cesàro). Let an , bn , n ≥ 1 be two sequences with bn strictly increasing and divergent. If the lefthandside limit exists, an+1 − an an lim = l =⇒ lim = l. n→∞ bn+1 − bn n→∞ bn Proposition 2.3 is a partial converse to the Stolz-Cesàro theorem. It implies that for groups of exponential growth, if the conjugacy ratio is a limit and the ratio of sizes of consecutive balls has a limit, then the spherical conjugacy ratio is equal to the conjugacy ratio. Proposition 2.3. Let an , bn , n ≥ 1 be two sequences with bn strictly increasing and divergent, such that the lefthandside limit exists and limn→∞ bn+1 bn 6= 1. Then an an+1 − an = l =⇒ lim = l. n→∞ bn n→∞ bn+1 − bn lim Proposition 2.4. Let an , bn , cn , dn , n ≥ 0 be monotonically increasing sequences of positive integers. Define the sequences b cn and dbn as b c0 := c0 , db0 := d0 , and cn := cn − cn−1 and dbn := dn − dn−1 , for n ≥ 1. b Suppose that (i) an ≤ bn and b cn ≤ dbn for all n, an cn (ii) bn → 0 and dn → 0 as n → ∞. Then Pn ai b cn−i = 0. lim Pi=0 n n→∞ bi dbn−i i=0 Proof. Given ǫ > 0, fix an N such that abnn < ǫ for all n ≥ N . Next choose an M ≥ N such that dcnn < aǫN for all n ≥ M . Then, for n ≥ M ≥ N , n n n X X X bi dbn−i . ai b cn−i < ǫ bi b cn−i ≤ ǫ i=N i=N i=0 4 LAURA CIOBANU, CHARLES GARNET COX, AND ARMANDO MARTINO Thus, for n ≥ M , Pn PN PN Pn cn−i cn−i ai b cn−i cn−i i=N +1 ai b i=0 ai b i=0 ai b = Pn + Pn < Pni=0 + ǫ. Pn b b b bi dn−i bi dn−i bi dn−i bi dbn−i i=0 i=0 i=0 i=0 Now we obtain the result by using the fact that for n ≥ M PN PN cn−i b cn−i cn i=0 ai b ≤ aN Pni=0 ≤ aN < ǫ. Pn dn bi dbn−i dbn−i i=0 i=0 Proposition 2.5. Let an , bn , cn , dn , n ≥ 0, be sequences of positive integers satisfying the following properties: (i) an , bn are monotone sequences, (ii) an ≤ bn and cn ≤ dn for all n, (iii) abnn → 0 as n → ∞, (iv) dbnn ≤ δ n for all sufficiently large n, and for some 0 < δ < 1. Then, Pn ai cn−i = 0. lim Pi=0 n n→∞ i=0 bi dn−i Proof. Given ǫ > 0, fix an N such that n X ai cn−i < ǫ′ i=N an bn n X i=N < ǫ′ < ǫ for all n ≥ N . Then, for n ≥ N , bi cn−i ≤ ǫ′ n X bi dn−i . i=0 Thus, for n ≥ N , Pn Pn PN PN ai cn−i ai cn−i ai cn−i i=N +1 ai cn−i i=0 i=0 Pn = Pn + Pn < Pni=0 + ǫ′ b d b d b d i=0 i n−i i=0 i n−i i=0 i n−i i=0 bi dn−i and so it suffices to show that PN PN ai cn−i i=0 ai cn−i = 0. lim Pi=0 ≤ lim n n→∞ n→∞ bn d0 i=0 bi dn−i Now PN ai cn−i 6 bn d0 i=0 PN N N X cn−i X dn−i ai cn−i ≤ aN ≤ aN . bn b b i=0 n−i i=0 n−i i=0 Using hypothesis (iv), there is a sufficiently large n such that N N X X dn−i δ n 1 − δ N +1 aN 1 aN δ n−i = aN N ≤ δn < ǫ − ǫ′ . ≤ aN N (1 − δ) b δ 1 − δ δ i=0 n−i i=0 3. Results for groups of stable subexponential growth Definition 3.1. A group G, with finite generating set, X, is said to be of stable G.X (n+1)\| = 1. subexponential growth if limn→∞ \|B\|B G,X (n)\| THE CONJUGACY RATIO OF GROUPS 5 Note that being of stable subexponential growth, implies that ExpX (G) = 1, and hence that the group has subexponential growth. By the celebrated result of Gromov, every finitely generated group of polynomial growth - where BG,X (n) is bounded above by a polynomial function - is virtually nilpotent. All these groups are of stable subexponential growth since, by a result of Bass, [BASS72], if G is a finitely generated, virtually nilpotent group, and X is any finite generating set, then, for some exponent d, and constants, A, B: (5) And ≤ \|BG,X (n)\| ≤ Bnd . The exponent d is calculated explicitly in [BASS72]; for a virtually abelian group it is equal to the rank of a finite index free abelian subgroup. From (5) we get that for any positive integer, k, (6) lim n→∞ \|BG,X (n + k)\| = 1. \|BG,X (n)\| The main result which we require for this class is the following. Proposition 3.2. [BV02]. Let G be a finitely generated group with stable subexponential growth, and finite generating set X. For every finite index subgroup H 6 G and every g ∈ G, we have \|gH ∩ BG,X (n)\| 1 \|Hg ∩ BG,X (n)\| lim = lim = . n→∞ n→∞ \|BG,X (n)\| \|BG,X (n)\| [G : H] Furthermore, if H is an infinite index subgroup of G then both limits are zero for any coset of H. Remark 3.3. The last statement does not appear explicitly in [BV02], but follows easily from their arguments. Alternatively, one could prove this via the construction of an invariant mean which requires the choice of an ultrafilter. The stable subexponential condition ensures that any ultrafilter will do, and hence that all limit points of the sequences above are equal. From now on, whenever there is no ambiguity concerning the group and its generating set, we will write C(n) instead of CG,X (n) and B(n) instead of BG,X (n). Proposition 3.4. Suppose that G is a finitely generated, virtually abelian group. Then, for any finite generating set X of G, we have that crX (G) > 0. More precisely, if [G : A] = m where A is abelian, then crX (G) ≥ 1/m2 . Proof. Let [G : A] = m, where A is abelian. We note that G acts by multiplication on the right cosets of A. If g and h lie in the same right coset, then h = αg for some α ∈ A, so for any a ∈ A, h−1 ah = (αg)−1 a(αg) = g −1 ag since A is abelian. Thus there are at most m conjugates of each element a ∈ A and so, for all n ∈ N, 1 . Now we have that \|C(n) ∩ A\| > \|B(n) ∩ A\| · m \|C(n) ∩ A\| \|B(n) ∩ A\| \|C(n) ∩ A\| \|B(n) ∩ A\| 1 \|C(n)\| > = · > · \|B(n)\| \|B(n)\| \|B(n) \|B(n) ∩ A\| \|B(n)\| m which tends to 1/m2 by Proposition 3.2. Lemma 3.5. Let G be a group of stable subexponential growth with finite generating set X, let g ∈ G and let H be a finite index subgroup of G. For d ∈ N we have 1 \|gH ∩ BG,X (n + d)\| = . lim n→∞ \|BG,X (n)\| [G : H] 6 LAURA CIOBANU, CHARLES GARNET COX, AND ARMANDO MARTINO Proof. This follows from writing lim n→∞ \|B(n + d)\| \|gH ∩ B(n + d)\| \|gH ∩ B(n + d)\| = lim n→∞ \|B(n)\| \|B(n)\| \|B(n + d)\| together with Proposition 3.2 and (6). Proposition 3.6. Let G be a finitely generated group of stable subexponential growth and N a subgroup of finite index in G. Then crX (G) ≤ dc(G/N ) for any finite generating set X of G. Proof. Let [G : N ] = m, so that G = g1 N ⊔g2 N ⊔. . .⊔gm N for some g1 , . . . , gm ∈ G. Let d := max{\|gi \|X : i = 1, . . . , m}. Now consider if xN ∼ yN (in G/N ). Then yN = g −1 xgN = g −1 xgg −1 N g = −1 g xN g for some g ∈ G. Moreover, since x and y are conjugate in G/N , we may choose g from {g1 , . . . , gm } and so \|g\|X 6 d. Now let yk1 ∈ B(n). We know there must exist some xk2 ∈ xN such that g −1 (xk2 )g = yk1 . But then xk2 = gyk1 g −1 , and so xk2 ∈ B(n + 2d). Hence, for every n ∈ N, each element in B(n) ∩ yN is conjugate to some element in B(n + 2d) ∩ xN . Let x1 , . . . xk ∈ {g1 , . . . , gm } be the representatives of the conjugacy classes in G/N . For every i ∈ N and every j ∈ Zk , we will assume that there are \|B(n) ∩ xj N \| conjugacy classes in B(n) ∩ xj N . Hence Pk \|xi N ∩ B(n + 2d)\| \|C(n)\| 6 i=1 \|B(n)\| \|B(n)\| which tends to k/m by the previous lemma. Theorem 3.7. Conjecture 1.1 is true for all finitely generated, residually finite groups of stable subexponetial growth. Proof. Proposition 3.4 states that, if a finitely generated group G is virtually abelian, then, for any finite generating set X, crX (G) > 0. For the other direction we apply the method of [AMV17, Proof of Thm. 1.3] by using Proposition 3.6. For completeness we will describe their argument. It requires the following result from [Gal70]: if F is a finite group and N E F , then (7) dc(F ) 6 dc(F/N ) · dc(N ). Our hypotheses are that G is: finitely generated, residually finite, of stable subexponential growth, and not virtually abelian. We wish to show that crX (G) = 0 for any finite generating set X. We will work with finite quotients and will build a chain of normal subgroups. Since G is finitely generated we may choose these subgroups to be characteristic, and will do this because being characteristic is transitive. Since G is not virtually abelian, choose g1 , g2 ∈ G that do not commute and, using the residually finite assumption, let [g1 , g2 ] 6∈ K1 where K1 is a characteristic and finite index subgroup of G. Hence G/K1 is non-abelian, and by Gustafson’s result we have that dc(G/K1 ) 6 5/8. Now, since the properties of G which we have used also apply to finite index subgroups, this argument also applies to K1 . Hence we may construct a descending chain of characteristic finite index subgroups . . . 6 Ki 6 Ki−1 6 . . . 6 K2 6 K1 6 K0 = G THE CONJUGACY RATIO OF GROUPS 7 where, for every i ∈ N, dc(Ki−1 /Ki ) 6 5/8. Moreover (G/Ki )/(Ki−1 /Ki ) = G/Ki−1 and so, from (7), dc(G/Ki ) 6 dc(G/Ki−1 ) · dc(Ki−1 /Ki ) 6 5/8 · dc(G/Ki−1 ). By induction dc(G/Ki ) 6 (5/8)i and so, by Proposition 3.6, for any finite generating set X of G, we have that crX (G) 6 dc(G/Ki ) 6 (5/8)i . Since this holds for every i ∈ N, we obtain that crX (G) = 0. Corollary 3.8. Conjecture 1.1 is true for all finitely generated, virtually nilpotent groups, or equivalently, all groups of polynomial growth. 3.1. Virtually Abelian groups. The goal of this section is to prove: Theorem 3.9. Let G be a finitely generated, virtually abelian group, and X, Y be finite generating sets for G. Then crX (G) = crY (G). It will be useful to have the following shorthand: Definition 3.10. Let G be generated by the finite set X. A subset, S, of G is \|S∩BG,X (n)\| generic if lim supn→∞ \|BG,X (n)\| = 1, and negligible if the limit is 0. Given a group, G, with finite generating set X, a finitely generated subgroup, H, of G is said to be undistorted if any word metric on H is bi-Lipschitz equivalent to any word metric on G, when restricted to H. This makes sense since any two finite generating sets on a group induce bi-Lipschitz equivalent word metrics. It is easy to see that a finite index subgroup is always undistorted, and that a subgroup H is undistorted if and only if it has an undistorted subgroup of finite index. Retracts are also undistorted (recall that a retract of G is the image of an endomorphism ρ : G → G such that ρ2 = ρ). We now collect the following facts: Proposition 3.11. Suppose that G is a finitely generated virtually abelian group, with finite generating set X, having a subgroup of finite index isomorphic to Zd . (i) Every subgroup of G is both finitely generated and undistorted. (ii) Let H be an infinite subgroup of G. Let T (n) = TH,X (n) (for transversal) be the number of cosets of H that have a representative in BG,X (n). Then, lim n→∞ T (n) = 0. \|BG,X (n)\| Proof. (i) Let H ≤ G. It is well known that H is finitely generated, as this fact is true in the case where G is virtually polycyclic, which includes the finitely generated virtually nilpotent (and abelian) case. However, the fact that H is undistorted is not true more generally, and follows from the fact that every subgroup of a finitely generated free abelian group has finite index in a direct summand. In our case, H has a finite index subgroup which is a retract of a finite index subgroup of G, and is therefore undistorted in G. (ii) From above, H is finitely generated and undistorted. Since H is infinite, it must contain an element of infinite order, so there exists an ǫ > 0 such that \|H ∩ BG,X (n)\| ≥ ǫn. More precisely, \|H ∩ BG,X (n)\| will have polynomial bounds of degree e, d ≥ e ≥ 1. 8 LAURA CIOBANU, CHARLES GARNET COX, AND ARMANDO MARTINO Let A, B, d be the constants in (5). Then Hence, B2d nd ≥ \|BG,X (2n)\| ≥ T (n)\|H ∩ BG,X (n)\| ≥ T (n)ǫn. T (n) B2d nd−1 = 0. ≤ lim n→∞ \|BG,X (n)\| n→∞ ǫAnd 0 ≤ lim From now on, we let G be an infinite, finitely generated, virtually abelian group. Let A be a normal, finite index, free abelian subgroup, and B be the centraliser of A in G. Note that A is a subgroup of B, which therefore has finite index. Proposition 3.12. Let G be a finitely generated, virtually abelian group and X any finite generating set for G. Let A be a normal, finite index, free abelian subgroup, and B be the centraliser of A in G. Then the set of minimal length G-conjugacy representatives in G \ B is negligible. Proof. Let y 6∈ B be an element of G and denote by CyA (n) the number of conjugacy classes which have a representative in BG,X (n) ∩ yA. Then we claim that lim n→∞ CyA (n) = 0. \|BG,X (n)\| For each conjugacy class with a representative in BG,X (n) ∩ yA, choose a shortest such representative, and denote this set of representatives Z = {yai : ai ∈ A}. From these, extract the set U = {ai }, rewriting the ai as geodesics if required. Note that, for some fixed k (the length of y), we have CyA (n) = \|Z ∩ BG,X (n)\| ≤ \|U ∩ BG,X (n + k)\|. Now let My denote the automorphism of A induced by conjugation with y, which we think of as a matrix. For any a1 , a2 ∈ A we have that: a−1 2 (ya1 )a2 = y(a1 + (I − My )a2 ), if we switch to an additive notation in A. Let H be the image of (I − My ) in A, that is, H = h[a, y] : a ∈ Ai. Since y 6∈ B, we can conclude that H is a non-trivial subgroup of A and is therefore infinite. Moreover, the elements of U are all in distinct cosets of H. Hence, by Proposition 3.11 part (ii), we may conclude that: \|Z∩BG,X (n)\| \|BG,X (n)\| ≤ \|U∩BG,X (n+k)\| \|BG,X (n)\| = TH,X (n+k) \|BG,X (n+k)\| \|BG,X (n+k)\| \|BG,X (n)\| → 0. Proposition 3.12 shows that the only elements of G that contribute to the conjugacy ratio are the elements of B. (The representative of a conjugacy class might not have a shortest representative in our particular coset yA, but varying y we see that we have an overcount of the number of conjugacy classes in the complement of B, which nonetheless gives 0.) Thus the strategy for proving Theorem 3.9 is the following. First note that each element of B has finite conjugacy class in G. We split the elements in B into those which centralise elements from outside of B and those whose centraliser is completely in B. Proposition 3.14 shows the former ones form a negligible set, and THE CONJUGACY RATIO OF GROUPS 9 the latter ones a generic set of B (Corollary 3.15); moreover, for the latter ones the size of the G-conjugacy class is the index of the B-centraliser, which is constant for elements in the same A-coset. Therefore, each coset (or, rather, conjugacy class of cosets) of A contributes a fixed amount to the conjugacy ratio, which is algebraically determined. We use the notation ZK (g) for the K-centraliser of g ∈ G, that is, ZK (g) = {k ∈ K : k −1 gk = g}. Lemma 3.13. Let x ∈ G. Then ZB (x) = ZB (xa) for any a ∈ A. Moreover, [G : ZB (x)] < ∞ if x ∈ B. S Proposition 3.14. The set y6∈B ZB (y) is a finite union of infinite index subgroups of G. Hence this set is negligible with respect to any finite generating set. Proof. Since ZB (y) = ZB (ya) for any a ∈ A, this is a finite union. So it is enough to show that each ZB (y) has infinite index. In fact, it is sufficient to show that ZA (y) = ZB (y) ∩ A is an infinite index subgroup of A. However, ZA (y), is a pure subgroup of A; that is, if am ∈ ZA (y) and m 6= 0 then a ∈ ZA (y). This implies that ZA (y) is a direct summand of A. But since y 6∈ B, this direct summand cannot be the whole of A and is therefore an infinite index subgroup of A as required. Corollary 3.15. There is a generic set of elements of B (with respect to any generating set) whose centraliser lies entirely in B. Proof. If for some b ∈ B there exists t ∈ / B such that [t, b] = 1, then b ∈ ZB (t) ⊂ S y6∈B ZB (y), which is negligible by Proposition 3.14. Proof of Theorem 3.9: For each r, let Ar be the elements b ∈ B for which ZB (b) has index r in G (and therefore conjugacy class size r in G), and let N = {b ∈ B : ZB (b) 6⊂ B}, that is,S N is the set of elements of B whose centraliser does not fully lie in B. Then N = y6∈B ZB (y) and so by Corollary 3.15 it is a negligible set. Since A ≤ ZB (b) ≤ G for any b ∈ B and A has finite index in G, there are only finitely many values for the index of ZB (b) in G, and thus finitely many r for which Ar is non-empty. Moreover, since ZB (y) = ZB (ya) for any y ∈ B \ A and a ∈ A, if y ∈ Ar , then ya ∈ Ar , so each non-empty Ar is a union of A-cosets and thus (8) \|Ar ∩ BG,X (n)\| = δ, n→∞ \|BG,X (n)\| lim where δ is 1/[G : A] times the number of A-cosets in Ar , so is independent of X. It is easy to see that there is an integer, k, such that if two elements of B are conjugate in G, then they are conjugate by an element of length at most k; the same holds for Ar as Ar ⊂ B. Moreover, since B is normal in G, it is easy to see that G acts on Ar by conjugation; G acts by conjugation on N , and hence on Ar \ N , as well. Let Cn be the number of conjugacy classes of G which meet BG,X (n) and are contained in Ar \ N . Then, (9) \|(Ar \ N ) ∩ BG,X (n)\| ≤ rCn ≤ \|(Ar \ N ) ∩ BG,X (n + 2k)\|. The first inequality comes from the fact that each element of Ar \ N has r conjugates in G, and the second from the fact that each of the conjugates can be obtained from a conjugator of length at most k. 10 LAURA CIOBANU, CHARLES GARNET COX, AND ARMANDO MARTINO Now Cn \|CG,X (n) ∩ Ar \| Cn \|N ∩ BG,X (n)\| ≤ ≤ + , \|BG,X (n)\| \|BG,X (n)\| \|BG,X (n)\| \|BG,X (n)\| and by (8), (9) and Corollary 3.15 Cn = lim lim n→∞ n→∞ \|BG,X (n)\| \|N ∩ BG,X (n)\| Cn + \|BG,X (n)\| \|BG,X (n)\| = δ , r so we get lim n→∞ δ \|CG,X (n) ∩ Ar \| = . \|BG,X (n)\| r Hence the number of conjugacy classes of G that meet Ar is independent of the generating set. Summing over the finitely many r gives the result. Remark 3.16. The same ideas as those just presented can be used to show that, if G is a finitely generated, virtually abelian group, and X is any finite generating set, then crX (G) = inf N Ef G cr(G/N ). That is, the conjugacy ratio is equal to the infimum of conjugacy ratios of the finite quotients. Hence, if one were to measure the conjugacy ratio using invariant means, one would get the same numerical value. Unpublished results indicate that this is the same as the degree of commutativity. For similar reasons, the same is true whenever G is a finitely generated virtually nilpotent group, the virtually abelian case being the key one. 4. Results for other families of groups 4.1. Hyperbolic groups. In this section we prove Conjecture 1.1 for non-elementary hyperbolic groups. We will write f (n) ∼ g(n) to mean f (n)/g(n) → 1 as n → ∞. Theorem 4.1. Let G be a non-elementary hyperbolic group. Then crX (G) = 0 for any finite generating set X. Proof. Let G be a non-elementary hyperbolic group with finite generating set X. Then by a result of Coornaert (see [Cor93]) there are positive constants A0 ,B0 , and integer n0 , such that for all n ≥ n0 (10) A0 enh ≤ \|BG,X (n)\| ≤ B0 enh , where h = ExpX (G). By Theorem 1.2 in [AC17], there are positive constants A1 , B1 and n1 such that (11) A1 enh enh ≤ \|CG,X (n)\| ≤ B1 n n for all n ≥ n1 . Thus from (10) and (11) we get \|CG,X (n)\| B1 6 \|BG,X (n)\| A0 n for all n ≥ max(n0 , n1 ), and by taking the limit we obtain that crX (G) = 0. THE CONJUGACY RATIO OF GROUPS 11 4.2. The lamplighter group. We follow the notation in [Mer17]. Let I be a L non-empty set. For η ∈ i∈I G we write η(i) for the ith component of η, and if L moreover I is a group and x ∈ I, we define η x ∈ i∈I G by η x (i) = η(x−1 i), and say that η x is the left translate of η by x. Definition 4.2. Consider groups H and L with symmetric generating sets A and B, and neutral elements e and e′ , respectively. The wreath product of G by L, written G ≀ L, is defined as M H ≀ L := H ⋊ L, i∈L where for (η, m), (θ, n) ∈ H ≀ L, (η, m)(θ, n) = (ηθm , mn). L For h ∈ H, let ~h ∈ i∈L H be such that ~h(e′ ) = h and ~h(i) = e for i 6= e′ . Then (12) X := {(~e, a) : a ∈ A} ∪ {(~b, e′ ) : b ∈ B}. generates H ≀ L. For the lamplighter group G = C2 ≀ Z we let A := {a}, where a is the non-trivial element of C2 , and let B be the standard generating set of Z. Theorem 4.3. Let G be the lamplighter group, that is, the wreath product C2 ≀ Z. Then crX (G) = 0 for the standard generating set X. Proof. The statement follows immediately from [Mer17, Example 5.0.3], where n it √ √ n 2 1+ 5 1+ 5 s is shown that \|CG,X (n)\| ∼ n , and the fact that \|SG,X (n)\| ∼ by 2 2 [Par92]. 4.3. Right-Angled Artin Groups. Let Γ = (V, E) be a simple graph (i.e. a non-oriented graph without loops or multiple edges) with vertex set V and edge set E. For each vertex v of Γ, let Gv be a group. The graph product of the groups Gv with respect to Γ is defined to be the quotient of their free product by the normal closure of the relators [gv , gw ] for all gv ∈ Gv , gw ∈ Gw for which {v, w} is an edge of Γ. Here we consider right-angled Artin groups (RAAGs), which are graph products with all Gv = Z, and denote by (GV , XV ) the RAAG based on the graph Γ with generating set XV (in bijection to V ). Conjugacy representatives in a RAAG come, to a large extent, from taking one word out of each cyclic permutation class, so we first establish the asymptotics of the language of cyclic representatives in a rather general setting. Example 4.4. In a free group on the free generating basis, counting the conjugacy classes with a minimal representative of length n is equivalent to counting the number of cyclically reduced words of length n, up to cyclic permutation. 4.3.1. Cyclic representatives of languages. We follow the notation in [CHM17, Section 2.3]. Let L be a language over a finite alphabet X, that is, L ⊆ X ∗ , and let L(n) denote the set of√words of length ≤ n in L. For n ≥ 1, n ∈ N, let Ln := {wn \| w ∈ L} and n L = {v \| v n ∈ L}. Define Prim(L) := {w ∈ L \| ∄k > 1, v ∈ L such that v k = w} to be the language of primitive words in L. Suppose L is closed under cyclic permutations; then we construct a language CycRep(L) of cyclic representatives of L out of the words wc , where wc the word that is least lexicographically among all cyclic permutations of w, for w ∈ L: CycRep(L) := {wc \| w ∈ L}. 12 LAURA CIOBANU, CHARLES GARNET COX, AND ARMANDO MARTINO Proposition 4.5 (see also Lemma 2.10 (4), [CHM17]). Let L be an exponential growth language closed under cyclic permutations. Furthermore assume that Lk ⊆ L √ k and L ⊆ L for all k ≥ 1. Then lim n→∞ \|CycRep(L)(n)\| = 0. \|L(n)\| Proof. For simplicity of notation let a(n) := \|Ls (n)\|, p(n) := \|Prim(L)s (n)\| and c(n) := \|CycRep(L)s (n)\|, that is, we consider the numbers of words of length exactly n in each language.S Write L as L = k≥1 Primk (L), and notice that the number of cyclic representatives of length n in Prim(L) is p(n)/n, and the number of cyclic representatives of P P length nk in Primk (L) is also p(n)/n. Thus a(n) = d/n p(d) and c(n) = d/n p(d) d . Let µ(n) and φ(n) be the standardPnumber theoretic Möbius and Euler functions. Then by Möbius inversion p(n) = d/n µ( nd )a(n) and so P X X l/(n/d) µ(l) φ(n/d) l a(d) a(d) = , c(n) = d n d/n d/n P P φ(n) which follows from d/n φ(d) = n and d/n µ(d) d = n . Since a(n) is exponential, only the last term in the sum above is of the same magnitude as a(n), so (13) c(n) ∼ \|CycRep(L)s (n)\| a(n) =⇒ lim = 0. n→∞ n \|Ls (n)\| By Stolz-Cesàro we obtain the result. 4.3.2. Conjugacy representatives in RAAGs. We first establish a result about the conjugacy ratio of direct products. Lemma 4.6. Let H and K be two groups with finite generating sets X and Y , respectively. If either (i) crX (H) = crY (K) = 0 or, (ii) crX (H) = 0 and ExpX (H) > ExpY (K), then crX∪Y (H × K) = 0. Proof. We calculate the conjugacy ratio with respect to balls in H × K. To do this we use balls in H and spheres in K. Let an := \|CH,X (n)\|, bn := \|BH,X (n)\|, s tn := \|CK,Y (n)\| and sn := \|SK,Y (n)\|. Then Pn ai tn−i . crX∪Y (H × K) = lim sup Pni=0 n→∞ i=0 bi sn−i If crX (H) = crY (K) = 0, then by Proposition 2.4 (putting tn = cbn , sn = dc n ) we get that crX∪Y (H × K) = 0. Similarly, if crX (H) = 0 and ExpX (H) > ExpY (K) then Proposition 2.5 (putting cn = tn , dn = sn ) states that this limit is zero, so crX∪Y (H × K) = 0. Since RAAGs interpolate between free and free abelian groups, the presence of commutativity does not allow us to simply consider cyclically reduced words up to permutation, as in free groups. We need to single out the words for which taking cyclic representatives produces conjugacy representatives, and use Crisp, Godelle and Wiest’s approach from [CGW], which was further developed in [CHM17]. THE CONJUGACY RATIO OF GROUPS 13 Definition 4.7 (Def 2.19, [CGW]). Let V = {a1 , . . . , aN } and set the total order −1 a1 < a−1 1 < a2 < a2 < . . . . A cyclically reduced word w is in cyclic normal form if it is in the shortlex language SL(GV , XV ) of GV with respect to XV and all its cyclic conjugates are in SL(GV , XV ) as well. Not all elements posses a cyclic normal form. For example, if [a1 , a2 ] = 1, the word a1 a2 is in SL(GV , XV ), but its cyclic permutation a2 a1 is not. To deal with this situation, [CGW] divides the words over XV into split and non-split. Definition 4.8 (Definition 2.13, [CGW]). Let w be a cyclically reduced word over XV and denote by ∆(w) the full subgraph spanned by Supp(w). Let ∆(w) be the graph complement of ∆(w). (i) The word w is split if ∆(w) is disconnected, which amounts to being able to write w as a product of commuting subwords (or blocks). (ii) The word w is non-split if ∆(w) is connected. (iii) Let CycSL(GV , XV ) denote the set of all non-trivial cyclic normal forms corresponding to non-split words in GV . We say that a group element is non-split (split) if it can be represented by a cyclically reduced word which is non-split (split). Proposition 4.9 (Prop. 2.21, [CGW]). Two cyclic normal forms represent conjugate elements if and only if they are equal up to a cyclic permutation. Proposition 4.10 (Remark 2.14, [CGW]). Let w and v be two cyclically reduced split words. Then they are conjugate if and only if ∆(w) = ∆(v) and the words corresponding to the commuting blocks are conjugate, respectively. Lemma 4.11. [CHM17] Let CycSL(GV , XV ) be the set of cyclic normal forms in GV . The following hold: (1) CycSLk (GV , XV ) ⊆ CycSL(GV , XV ) for all k ≥ 1, and (2) CycSL(GV , XV )) is closed under cyclic permutations. Theorem 4.12. Let G = (GV , XV ) be a right-angled Artin group (RAAG) based on a graph Γ = (V, E) with generating set XV . Then crXV (G) = 0 unless G is free abelian, in which case crXV (G) = 1. Proof. We use induction on the number of vertices. Let n := \|V \|. The result is trivial for n = 1. If G is a direct product, then we get cr(G) = 0 if at least one of the factors has cr = 0; this follows from Lemma 4.6(i) if both factors have cr = 0 and from Lemma 4.6(ii) if, say, the first factor has cr = 0, as the second is by induction free abelian, and of strictly smaller growth rate than the first. We get cr(G) = 1 when each factor is free abelian. So suppose G is not a direct product. We split the conjugacy classes CGV ,XV of G into two types: those which have a shortest length representative with support XU , where U ( V , and denote these by CGV ,≤XV , and those which have a shortest length representative with support exactly XV , and denote these by CGV ,=XV . By Propositions 4.9 and 4.10, this is well defined. Moreover, by Propositions 4.9 and 4.10, two cyclically reduced words w1 , w2 with support XU are conjugate in GV if and only if they are conjugate in GU (note that if a word w ∈ CycSL(GV , XV )∩XU∗ , where U ( V , then w ∈ CycSL(GU , XU )). 14 LAURA CIOBANU, CHARLES GARNET COX, AND ARMANDO MARTINO Thus we can write CGV ,≤XV ⊆ CGV ,XV ⊆ (14) Then (14) implies that (15) \|CGV ,XV (n)\| ≤ \|BGV ,XV (n)\| Now for U ( V so P S CGU ,XU and express the above as: [ CGV ,=XV . CGU ,XU U(V [ U(V XU ,U(V \|CGU ,XU (n)\| + \|CGV ,=XV (n)\| \|BGV ,XV (n)\| . \|CGU ,XU (n)\| \|CGU ,XU (n)\| \|BGU ,XU (n)\| = , \|BGV ,XV (n)\| \|BGU ,XU (n)\| \|BGV XV (n)\| \|BGU ,XU (n)\| \|CGU ,XU (n)\| ≤ crXU (GU ) lim sup . (n)\| \|B \|B n→∞ n→∞ GV ,XV GV ,XV (n)\| The right hand side is equal to 0 since either (i) crXU (GU ) = 0 by induction, or (ii) GU is free abelian (so of polynomial growth); if (ii), since G itself if not a direct product by assumption, it is of exponential growth, and the last fraction is 0. \|C (n)\| V It remains to find lim supn→∞ \|BGGV ,=X (n)\| , the second part of the right hand V ,XV side of (15). Since G is not a direct product, all conjugacy representatives with support exactly XV are non-split, so it suffices to consider cyclic normal forms up to cyclic permutations, that is lim sup \|CGV ,=XV (n)\| \|CycRep(CycSL(GV , XV )(n)\| ≤ = \|BGV ,XV (n)\| \|SL(G, XV )(n)\| \|CycRep(CycSL(GV , XV )(n)\| \|CycSL(GV , XV )(n)\| , \|CycSL(G, XV )(n)\| \|SL(G, XV )(n)\| and by Proposition 4.5 applied to the language CycSL(GV , XV ) (which satisfies the hypothesis of Proposition 4.5 by Lemma 4.11) lim n→∞ This proves the result. \|CycRep(CycSL(GV , XV )(n)\| = 0. \|CycSL(G, XV )(n)\| 5. Reflections and open questions Our results on the conjugacy ratio values are essentially identical to those on the degree of commutativity in [AMV17, Cox16, Val17]. That is, the two quantities are equal for all the classes of groups we studied. However, we could not establish a direct general link between them. Question 1. Is the limsup in the definition of the conjugacy ratio a limit? Question 2. What are the groups for which dcX (G) ≤ crX (G) (or vice versa)? They are equal in the virtually nilpotent case, in the hyperbolic group case and in many more. As is the case for the degree of commutativity, we do not know whether the conjugacy ratio might be influenced by a change of generators. Question 3. Does there exist a group G with finite generating sets X and Y such that crX (G) 6= crY (G)? THE CONJUGACY RATIO OF GROUPS 15 Finally, it would be interesting to unify the proofs confirming our conjecture for larger classes of groups, such as all groups of exponential growth, for example. References [AC17] Y. Antolı́n and L. Ciobanu, Formal conjugacy growth in acylindrically hyperbolic groups, Int. Math. Res. Notices, 1 (2017), 121–157. [AMV17] Y. Antolı́n, A. Martino, and E. Ventura, Degree of commutativity of infinite groups, Proceedings of the American Mathematical Society (Feb. 2017). [BASS72] H. 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Acad. Sci. Hungar 19 (1968), 413–435. MR 0232833 [Gal70] P.X. Gallagher, The number of conjugacy classes in a finite group, Math Z. 118 (1970), 175-179. [Gus73] W. H. Gustafson, What is the probability that two group elements commute?, Amer. Math. Monthly 80 (1973), 1031–1034. MR 0327901 [Mer17] V. Mercier, Conjugacy growth series in some wreath products, Preprint 2017, https://arxiv.org/abs/1610.07868. [Par92] W. Parry. Growth series of some wreath products. Trans. Amer. Math. Soc., 331 (1992), 2:751–759. [Val17] M. Valiunas. Degree of commutativity for right-angled Artin groups. Preprint 2017, https://arxiv.org/abs/1701.04374 Heriot-Watt University, Edinburgh, EH14 4AS, UK E-mail address: [email protected] URL: http://www.macs.hw.ac.uk/~lc45/ University of Bath, BA2 7AY, UK E-mail address: [email protected] Mathematical Sciences, University of Southampton, SO17 1BJ, UK E-mail address: [email protected] URL: http://www.personal.soton.ac.uk/am1t07/	4
1 Complex Systems Science meets 5G and IoT Nicola Marchetti, Irene Macaluso, Nicholas Kaminski, Merim Dzaferagic, M. Majid Butt, Marco Ruffini, Saul Friedner, Julie Bradford, Andrea Zanella, arXiv:1710.11548v1 [cs.NI] 31 Oct 2017 Michele Zorzi, and Linda Doyle Abstract We propose a new paradigm for telecommunications, and develop a framework drawing on concepts from information (i.e., different metrics of complexity) and computational (i.e., agent based modeling) theory, adapted from complex system science. We proceed in a systematic fashion by dividing network complexity understanding and analysis into different layers. Modelling layer forms the foundation of the proposed framework, supporting analysis and tuning layers. The modelling layer aims at capturing the significant attributes of networks and the interactions that shape them, through the application of tools such as agent-based modelling and graph theoretical abstractions, to derive new metrics that holistically describe a network. The analysis phase completes the core functionality of the framework by linking our new metrics to the overall network performance. The tuning layer augments this core with algorithms that aim at automatically guiding networks toward desired conditions. In order to maximize the impact of our ideas, the proposed approach is rooted in relevant, near-future architectures and use cases in 5G networks, i.e., Internet of Things (IoT) and self-organizing cellular networks. Index Terms Complex systems science, Agent-based modelling, Self-organization, 5G, Internet of Things. Nicola Marchetti ([email protected]), Irene Macaluso, Nicholas Kaminski, Merim Dzaferagic, M. Majid Butt, Marco Ruffini, and Linda Doyle are with CONNECT / The Centre for Future Networks and Communications, Trinity College, The University of Dublin, Ireland. Saul Friedner and Julie Bradford are with Real Wireless, UK. Andrea Zanella and Michele Zorzi are with the University of Padova, Italy. This material is based upon works supported by the Science Foundation Ireland under Grants No. 13/RC/2077 and 10/CE/i853. 2 I. I NTRODUCTION The transition of humanity into the Information Age has precipitated the need for new paradigms to comprehend and overcome a new set of challenges. Specifically, the telecommunication networks that underpin modern societies represent some of the largest scale construction and deployment efforts ever attempted by humanity, with renovations occurring nearly continuously over the course of decades. This results in networks that consist of numerous subsections, each following its own trajectory of development, commingled into a complex1 cacophony. A few emerging trends confirm the picture just drawn. Mobile and wireless networks are getting denser and more heterogeneous in nature. Nodes in the network vary hugely in form and functionality - ranging from tiny simple sensors to sophisticated cognitive entities. There is a wider range of node and network-wide parameters to set, many of which are interdependent and which impact heavily on network performance. Networks are becoming more and more adaptive and dynamic, and many parameters are set during run-time in response to changing contexts. As networks evolve, all of the above issues become more exaggerated - e.g., 5G networks will see more antennas, more base stations and devices, more modes of operation, more variability, and more dynamism. In a world like that, there is no way to systematically capture network behaviour. There is no straightforward network theory or information theoretic approach that can be used to describe the overall network or the interplay between the different networks. We propose to tackle this by studying wireless networks from the perspective of Complex Systems Science (CSS), developing complexity metrics and relating them to more traditional measures of network performance. One of the key questions in CSS relates to the degree of 1 With the term ’complexity’ we refer to a specific set of complex systems science quantities, related to the interactions between network entities (rather than to entities themselves) and between networks. As the current and future trend is towards more diverse networks coexisting and more entities (e.g., within IoT, or ultra dense small cell networks), the amount of interactions will increase, leading to an increase in complexity (in the meaning given to the word by complex systems science). 3 organization of a system [1], in terms of both the difficulty in describing its organizational structure (A), and the amount of information shared between the parts of the system as a result of the organizational structure (B). For example, the measure of excess entropy [2] (type (A)) can be used to describe the behaviour of a collection of self-organising networks [3], [4]; while the signalling complexity associated with future network resource management can be analyzed through a type (B) measure, i.e., functional complexity, introduced in [5], [6]. The above conceptual structure based on complexity informs an agent-based modelling (ABM) paradigm to examine the interactions between the different entities that shape a network. ABM provides a method of modelling complex systems from the ground up, which allows for a deeper investigation of the interactions that shape the ultimate system performance. ABM provides powerful modelling of entities in a variety of areas and contexts [7]–[9]. The attributes of ABM can be applied to inform communication networks’ decision making; in particular, ABM can be used to investigate the impact of several Medium Access Control (MAC) component technologies on the Key Performance Indicators (KPI) of both telecom networks and applications, for example in the case of a wireless sensor network aiding an Internet of Things (IoT) system [10]. In summary, we propose a new paradigm for telecommunications, drawing on concepts of a complex systems science nature, to understand and model the behaviour of highly heterogeneous networks and systems of networks. We also employ our framework to create new technologies for supporting network operation. II. M OTIVATION We propose the development of a conceptual framework as a means of exploring a broad range of possibilities in wireless networks, including a vast array of 5G technological possibilities. This framework for thought applies concepts from complex systems science [11], [12] to provide a means to understand wireless networks holistically on a variety of scales. Specifically, we 4 consider the communication patterns that enable network functions, by capturing all nodes necessary to perform a given function; then by drawing connections between these nodes we highlight their functional dependencies. We call a graph obtained in this way functional topology. This approach allows us to analyze the communication patterns on multiple scales. The lowest scale models the communication between individual devices/nodes. In other words, the lowest scale focuses on the communication between a node and all the immediate neighbors of this node in the functional topology. The second scale models the communication between a node, all its immediate neighbors and all neighbors of its neighbors. The increasing scale size moves the focus away from the communication between individual nodes, and allows us to analyze communication patterns between groups of nodes (i.e., functional entities/groups). Considering the high degree of heterogeneity and dense interplay of network elements in proposed 5G and IoT systems, achieving a holistic understanding of network operation is poised to become an even more challenging prospect in the near future. To address these challenges, we demonstrate the power of our framework for the modeling and analysis of relevant 5G scenarios, i.e., self-organizing cellular and IoT networks. While our framework supports innovation beyond these concepts, we feel these scenarios adequately represent the near-future applications of our work. The development of our concept is organized in a layered fashion, with a modelling layer forming the foundation of the framework and supporting analysis and tuning layers. The main aspects of our framework are represented in Fig. 1 and will be discussed in detail in the remainder of the paper. As compared to the CSS literature addressing communication systems [13]–[17], we study wireless networks from the infrastructure perspective. As a simple example, in [3], [4] excess entropy is used to measure complexity and, in combination with entropy, leads to an understanding of the structure emerging in a lattice of self-organising networks. The self-organising systems 5 Modelling Analysis Communication metrics CSS metrics Functional topology graphs Links between: 1. Communication & CSS metrics constraints 2. Technological behaviours 3. ABM parameters and range values Tuning Guidelines for tuning 5G Network Local rules  Global fitness Adaptive multi-dimensional 5G resource allocation Fig. 1: Our complex systems science based layered approach to 5G networks. Functional topology graphs are abstracted from the network, and are then used to compute complexity and telecom metrics, and find their relations. The understanding of such relations will then feed an ABM approach to network tuning. studied in [3], [4] exhibit a complex behaviour and this relates to robustness against changes in the environment; in particular, exploring frequency planning from a complex systems perspective leads to conclude that future networks shall eschew any current frequency planning approaches and instead determine frequency of operation on the fly. This has enormous implications for design and roll-out of networks, deployment of small cells, and network operation. III. M ETHODOLOGY Significant impacts have been made by CSS in a wide range of areas including physics, biology, economics, social sciences, computer sciences, and various engineering domains. We claim that the CSS perspective provides the necessary means to redefine the general understanding of telecommunication networks. We draw on concepts from information theory and ABM; each 6 concept augmenting and developing the understanding of wireless networks. We will now briefly review some of the most important tools and concepts we use in our studies. In order to specify and analyse the complexity of a network function, [5] introduced a framework representing an abstraction of a telecommunication network, by modelling its operation and capturing all elements, i.e., nodes and connections, necessary to perform a given function. Our framework includes functional topologies, i.e., graphs created based on the functional connectivity between system entities (see Fig. 1). A node in our topology represents a functional entity of a network node or any information source that is part of the given network function. The links indicate dependencies between nodes. The definition of functional topologies allows us to visualise the relationships between system entities, and enables the systematic study of interactions between them. Based on these topologies one can define CSS inspired metrics such as functional complexity [5], which quantifies the variety of structural patterns and roles of nodes in the functional topology, or other information theoretical-inspired metrics. Agent-based modelling (ABM) is a useful method to model networks. In [3], [4], [10] ABM was used to investigate the impact of several MAC component technologies, in terms of both telecom and IoT application’s Key Performance Indicators (KPI). This is key for our framework’s analysis and tuning layers. Our framework enables multi-scale modelling, analysis and tuning of wireless networks, in which changes in the 5G networks domain can be analysed and assessed. Indeed, in order to maximize the impact of our framework, our proposed approach is rooted in relevant, near-future architectures and use cases in 5G networks, such as self-organizing cellular and IoT networks. The use cases define the expected parameters, types of users/devices and environments; a general set of possible scenarios we could investigate using our framework is shown in Table I. 7 TABLE I: Possible use cases. Parameters Type of users Environments Low latency, High throughput, High reliability, Extensive coverage, Energy efficiency Typical mobile broadband, Healthcare, Automotive, Home/industrial automation, Wearable devices Busy train station, Emergency/disaster location, Busy office complex/campus, Large utility/manufacturing plant A. Solution Approach Our framework is based around the idea of using concepts, tools and measures of a complex systems science nature. The framework is based on a modelling layer which supports the analysis and tuning layers (see Fig. 1). 1) Modeling Layer: The modelling phase focuses on developing techniques to capture the significant attributes of networks and the interactions that shape them. Along with the traditional attributes used to characterize networks (e.g. coverage and throughput), the modelling phase develops new complexity metrics and investigates their relation to telecom KPIs. These metrics shall be developed distinctly for each application, based on existing and new concepts we draw from CSS. The modelling component of the framework develops appropriate abstractions and formalisms to enable metric calculation. To this end, we produce a multi-scale abstraction for networks. The first level or device level of this abstraction focuses on individual elements within a network, targeting the interplay that results from information being collected and used locally by a single entity. Interference and stability of the connection (e.g., as a function of power available at the node) between nodes in a network are two examples of notions studied at the device scale. Available local information may (as in the case of the interference perceived at a certain network node) or may not (battery level) result from the actions of other nodes. That is, the device scale typically models the implicit exchange of information, where nodes infer information for each other’s actions without directly exchanging messages, such as the interaction-through-interference 8 paradigm of a distributed Time Division Multiple Access (TDMA) system. The higher scales model the explicit exchange of information between groups of nodes in the network; at this level (interaction scale) the nodes act on the basis of information provided by some other node directly, as occurs for example when assigning a slot in a centralized TDMA system. The interactions that shape the network formation and operation are directly modelled using ABM. Our model considers the interactions between the interests of different network operators. These agents operate in a hierarchical fashion (see Fig. 2) with network operator agents who, in turn, contain sub-agents that determine specific aspects of the network, based on technical behaviours. Anything that makes decisions in a network can be viewed as an agent, and ABM is applied to model interactions between agents. For example, IoT agents may attempt to use the infrastructure provided by operator agents, as shown in Fig. 2. To capture the range of possibilities, we can use nested subagents, in which major agents might represent a whole network with subagents representing individual cells. ABM allows conversion of experience with detailed processes (micro-level behaviours) into knowledge about complete systems (macrolevel outcomes). In general we can consider several radio resources in our ABM model, e.g., resources belonging to frequency, power and space domains. Several alternative techniques and technologies can be applied within each domain, which entails a wide set of resources and related modes of utilisation. 2) Analysis Layer: In the analysis layer, the models are reviewed to determine the representative power and meaning of the metrics developed by linking: (i) the operator behaviours with our new CSS metrics; (ii) the operator behaviours with network KPIs (fitness); (iii) our new CSS metrics and KPIs. As an example, we could analyse the relationship between operator decisions on the amount of shared resources (infrastructure and/or spectrum) and the resulting network characteristics. For each scenario, measures of network performance can be identified, including standard 9 Network Operator Agent Cellular network agent - Cell Agent - Access Point Agent IoT Agent Network Operator Agent Cellular network agent Network Operator Agent WiFi network agent WiFi network agent WiFi network agent Cellular network agent IoT Agent IoT Agent Fig. 2: Agent organization. Our agent model is hierarchical, with major agents representing a whole network, and subagents representing IoT agents or individual cells and access points agents. operator KPIs such as cell edge, peak and mean throughput, spectrum utilisation relative to available bandwidth, network reliability, and coverage. For each type of the above mentioned relations (i), (ii), (iii), we can determine the most promising pairing of elements (i.e., operator behaviour and CSS metric, or CSS metric and KPI) within each scale and between scales for determining connections. In particular, we can identify which behaviours correlate to specific network performance measures on each scale, and to what extent and how our CSS metrics describe these relationships. Further, we can investigate how a certain CSS metric-KPI relation at a certain scale affects another CSS metric-KPI relation at a different scale (e.g. a strategy leading to throughput maximisation at the device level might compromise the fairness objective of the resource allocation scheduler at the interaction level). This process involves assessing the ability of the CSS metrics to describe the impact of operator behaviours, analysing the effect of these behaviours on the network KPIs, and finally describing the network KPIs in terms of the CSS metrics. Determining the link between network CSS metrics and KPIs would allow us to attempt to answer fundamental questions such as whether one needs a minimum complexity for achieving a given level of KPI (fitness), and what excess 10 complexity implies in terms of adaptivity and robustness vs. cost. In summary, the analysis layer completes the development of the core of our framework, by establishing a compact representation of the networks by linking complexity metrics to network performance. 3) Tuning Layer: The tuning layer augments the framework with algorithms that automatically guide the operation and management behaviours of relevant agents to achieve desired network properties. This tuning approach utilizes the holistic information encoded into the complexity based quantities, to select appropriate parameters and constraints for the behaviours of the agents. The developed tuning approach can be based on the application of multi-objective optimization techniques; the algorithms to be developed within this paradigm might apply multi-objective optimization algorithms (e.g., NSGA-II, PGEN, SMS-EMOA, successive Pareto optimization) to determine the Pareto fronts for the state spaces of the agent behaviours on the basis of achieving desirable CSS metrics values. These Pareto fronts provide the parameters and constraints of the operator behaviours, allowing operators to further optimize for specific differentiations while maintaining desired holistic properties. A particular solution may be selected from the Pareto front on the basis of agent preferences, such as a preference for high adaptivity and robustness or low complexity, without compromising the overall quality of the solution. IV. A PPLICATIONS OF THE P ROPOSED F RAMEWORK A. Modeling Layer 1) Agent-Based Modelling of the Internet of Things: We employ an instance of our framework concept to investigate the tightened coupling between operative reality and information transfer precipitated by IoT. As such, this investigation resides primarily in the modelling phase with some extension into the analysis phase. Within this work, we apply the tool of ABM to study the impact of communications technologies within the scope of IoT [10]. An automatic traffic management system is considered, where for the purposes of illustrating 11 DM Fig. 3: Single Intersection Diagram. Sensors are deployed alongside the roads and are represented as dots. Inactive sensors are depicted as black dots; sensors detecting moving and static cars are shown as orange and purple dots respectively. the nature of our ABM approach, a single intersection is assumed, depicted in Fig. 3, controlled with traffic lights, in which the avenue of the cross-road is observed by sensor nodes. A processing unit, here denoted as the decision maker (DM), serves as the sink of sensor information and the source of light control commands. Sensor nodes mark the advancement of cars, here portrayed as yellow squares and proceeding on the left side of the roadway, toward the intersection. Two MAC protocols (CSMA and Aloha) are investigated, for communication between the sensors and the DM. The DM applies the resultant information from this process to govern vehicular progress through the coloration of traffic signals. Notably, the semantics of communications greatly impact the operation of the physical system. Fig. 4 exemplifies this notion through a depiction of the difference between the actual number of cars waiting at a traffic light and the perceived number of cars known to the DM component of the system. As revealed by ABM, the minor difference of pre-sensing a channel (CSMA) or not (Aloha) causes either an over- or an under-estimation of the actual number of vehicles by the controlling element in the system. As such, the application of ABM techniques allows the development of an understanding of the various inter-relationships that direct the behavior of a complete telecommunication system. 12 Fig. 4: Impact of MAC on Perception of Situation on a single-intersection scenario. Vehicles always travel in a straight line at constant speed, unless they need to stop due to traffic lights or other cars. At each iteration the probability of a new car arriving at one of the four edges of the grid and travelling in the corresponding direction is 50%. 2) Functional Complexity: As another example of work at the modelling layer, we have developed a metric to capture the amount of information shared between elements of a network (as a result of the organization of the network) in support of a network function. This analytical approach to quantify the complexity of a functional topology provides us with the means to capture the signaling complexity of functional operations within a network, such as handover or frequency assignment. That is, our complexity metric provides a new method of describing the functional operation of telecommunication networks. Our complexity metric is built upon the concept of Shannon entropy (Hr (xn )). We employ the Bernoulli random variable xn to model the potential of a node to interact with other nodes. The probability of interaction pr (xn = 1) is defined as the reachability of a node n (pr (xn = 1) = inr /j, where inr is the number of nodes that can reach node n and j is the number of nodes for the given subgraph). The definition of reachability, in terms of the number of hops allowed between two nodes in the functional topology, enables the analysis of complexity on multiple scales (r). The one hop reachability represents the lowest possible scale (r = 1), where 13 each node interacts only with its immediate neighbors. The increasing number of allowed hops between the nodes brings the nodes closer to each other in terms of interactions, and moves the focus from interactions among nodes to interactions among groups of nodes, i.e., analysis of higher scales. The total amount of information of the k th subgraph with j nodes for scale r is calculated as Ir (Λjk ) = X Hr (xn ), (1) n∈Λjk where Λjk is the k th subgraph with j nodes. The total amount of information represents the total uncertainty which is related to the actual roles of nodes that appear within a subgraph and different subgraph patterns. Our complexity metric, which is calculated with Eq. (2), quantifies the amount of order and structure in a system that is seemingly disordered. R−1 N 1 X X r+1−j CF = \|hIr (Λj )i − Ir (ΛN )\| R − 1 r=1 j=1+r r+1−N (2) where R is the maximum scale size, which is defined as the diameter of the functional topology, N is the number of nodes in the functional topology, ΛN is the whole functional graph, and hIr (Λj )i is the average amount of information for a given subgraph size j. We call the metric in Eq. (2) functional complexity. Our approach holistically gauges the functional organization of a network by first describing the interactions necessary to perform a given function topologically. Within this representation, we capture the network elements involved in performing some function and the interactions that support the operation of the function. Our quantification of networks in terms of their functional relationships provides a wholly new approach to understanding the operation of networks. As corroborated by Fig. 5, more typical metrics for network topology do not capture the notions represented by our complexity metric (in fact, the correlation of our complexity metric with other traditional metrics is lower than 14 Average Path Length, Clustering Coefficient, Complexity 0.28 Average Degree, Complexity Average Path Length, Complexity Average Path Length, Average Degree, Complexity 0.47 -0.42 0.5 Clustering Coefficient, Average Degree, Complexity 0.15 0.3 Clustering Coefficient, Complexity Fig. 5: Correlation between the proposed complexity metric and the three most used measures of network topology (i.e., average path length, average degree distribution, clustering coefficient). 0.5 in all the cases we consider); this complexity metric thus provides an alternative method of describing network operation. The above functional topology and complexity framework can be applied for instance to understand the underlying mechanisms that lead to certain network properties (i.e., scalability, energy efficiency) in Wireless Sensor Networks (WSN) as the result of different clustering algorithms [6]. B. Analysis Layer In the context of the analysis layer of our framework, we focus on a cellular network that selforganises from a frequency perspective to understand the collective behaviour of the network. We calculate the excess entropy EC = ∞ X (h(M ) − h) (3) M =1 to measure complexity, and the entropy h = lim h(M ), M →∞ (4) where h(M ) is the entropy of the target cell X conditioned on M surrounding cells. By measuring EC and h we gain an understanding of the structure emerging in the lattice for a self-organising 15 network. Based on Eqs. (3) and (4), in [4] one shows that a self-organising cellular network can exhibit a complex behaviour, and that it can be robust against changes in the environment. In more detail, a self-organised and a centralised channel allocation are analyzed, with respect to their robustness to local changes in the environment. In order to compare the stability of the two types of channel allocation, 102 instances of the self-organising frequency allocation algorithm are run using 102 × 102 lattices. Then, for each resulting channel allocation, all possible cells n are considered, and for each cell all possible frequencies are in turn considered. Then the optimal minimum distance c to an interference-free channel allocation is computed (we define the distance between two channel allocations as the number of changes that are necessary to move from one configuration to the other). We found that the locally perturbed channel allocation matrices resulting from self-organisation are more stable than those resulting from a centralized frequency planner. What we know so far is that there is a relation between some complexity metrics and some telecom KPIs (i.e., between excess entropy and robustness to changes [4], and between functional complexity and the trade-off scalability-energy efficiency [6]). The complexity metrics we introduced have shed some new light on very relevant telecom KPIs/properties in the context of 5G networks, i.e., excess entropy can measure self-organization capabilities in the frequency allocation context and functional complexity can measure scalability in WSN. As widely acknowledged, self-organization and scalability are very important properties of 5G systems (e.g., for IoT and dense small cell deployments). In the future we plan to improve and expand such understanding to all the most prominent 5G network technologies and KPIs. C. Tuning Layer ABM rules will choose the technological behaviour options that maximize the targeted communication network KPI, subject to constraints defined by the correlation between CSS metrics 16 Set of available parameters - Complexity – robustness Complexity – energy efficiency Complexity - resilience Fitness functions Waveforms MIMO Frequency reuse Duplexing Tuning Layer - MIMO multi-user scheme OFDMA Full duplex Frequency assignment algorithm Network configuration parameters Fig. 6: The adaptation of network configuration parameters in the tuning layer. The set of available parameters represents a virtual pool of all the available network resources. The fitness functions depict a relationship between different network KPIs and complexity metrics which are calculated upon the set of available parameters. and other telecom KPIs. Local decisions will be based on only a few CSS metrics, and will lead to desired global behaviours/KPIs of the network. The local decisions are made according to ABM rules, by exploring and selecting the fittest behaviours (where by behaviour we mean some algorithm or policy acting on some radio resources). Our goal, for different services (e.g., mobile broadband, M2M) is to choose behaviours that allow the network to achieve satisfactory KPIs, in terms of, e.g., delay, throughput, coverage, energy efficiency, out-of-band emission, etc. The question is whether we can keep achieving globally satisfactory KPIs just by changing ABM rules in a distributed fashion at different nodes. Such adaptation will act within a certain resource allocation domain (e.g., picking among different massive MIMO schemes) or between performance-equivalent allocations using resources from different domains (e.g., spectrum or infrastructure). The main ideas behind the tuning layer of our framework are exemplified in Fig. 6. Although our own work on the tuning layer is still in the initial phase, from a substantial amount of literature, we can gather evidence that different physical layer (PHY) and Radio 17 Resource Management (RRM) techniques in the 5G domain should be chosen depending on environmental conditions and network requirements, i.e., we are potentially in a situation where our tuning layer is relevant and beneficial. We give a brief account of such evidence next. In [18], it is shown that for a massive MIMO system, the sum-rate has a linear or sublinear behaviour with respect to the number of base station antennas, depending on the spatial richness of the environment; related work on adaptive precoding for distributed MIMO is explored in [19]. Several works investigate the coexistence of various waveforms in terms of cross-waveform leakage interference [20], [21] and possible implications for the waveform selection [22]. The fraction of cells that have full duplex base stations can be used as a design parameter, to target an optimal trade-off between area spectral efficiency and outage in a mixed full/half duplex cellular system [23], [24]. In [25], it is shown that increasing the frequency reuse can improve the throughput-coverage trade-off for ultra-dense small cell deployments, while a lower frequency reuse should be favoured if the target is maximizing throughput given a certain BS density. In summary, we plan to use the above understanding of the benefit of adaptation at PHY and MAC layers in 5G networks, and extend it as needed in terms of technology components, KPIs and adaptation criteria, to inform our framework, and show its immediate benefit in understanding, operating and designing 5G systems. V. O PEN C HALLENGES Several more 5G component technologies, in addition to those considered in this paper, can enrich the set of possible choices used to model, analyse and tune the network, including (massive) co-located or distributed multiple antenna arrays; different waveforms and multiple access schemes; different duplexing schemes; novel spectrum sharing schemes such as License Assisted Access (LAA); and different frequency reuse schemes including probabilistic ones for ultra-dense networks. 18 What we know so far is that there is a relation between some complexity metrics and some telecom KPIs (i.e., between excess entropy and robustness to changes, and between functional complexity and the trade-off scalability-energy efficiency). In the future the aim is to improve and expand such understanding to all the most prominent technologies and KPIs for 5G networks. In particular it is still an open question how to achieve the desired network tuning properties within a large optimization space encompassing many different network resources, KPI objectives and constraints, many different heterogeneous co-existing networks and a very large number of nodes and decision points. We conjecture ABM can help us achieve such ambitious goal, as a key tool to engineer desired emergent properties in such future challenging networks. As the network graph representations discussed in the proposed framework might dynamically change according to the different radio resource domains and related techniques used, one open area of investigation is to study how the complexity metrics can be calculated and how they evolve over time for such dynamic multi-dimensional resource allocation; and then use such metrics to analyse and tune the network behaviour taking into account robustness, resilience, network utilization, and other time-dependent network characteristics. VI. C ONCLUSION Current complex systems science literature focusing on communication systems draws on network science, studying applications and traffic modelling, but lacks considerations of architecture, infrastructure, and technology. We instead apply complex systems science to wireless networks from the functional perspective, drawing on concepts from information (i.e., different metrics of complexity) and computational (i.e., agent based modeling) theory, adapted from complex system science. Since complex systems science metrics are currently absent from the quantities considered when operating and designing communication networks, by introducing our proposed framework we initiate a completely new way to model, analyse and engineer networks, founding a new 19 theory and practice of telecommunications not previously anticipated. As a simple example, our work on exploring frequency planning from a complex systems perspective leads us to conclude that future networks shall eschew any current frequency planning approaches and instead determine frequency of operation on the fly, with enormous implications for design, rollout and operation of networks. We believe such distributed decision making paradigm is likely going to be the way forward for many of the future 5G and IoT resource allocation problems. In particular, we have reasons to believe that complex systems science provides the key to unlock the full potential of self-organization in telecom systems. R EFERENCES [1] S. Lloyd, “Measures of complexity: A nonexhaustive list,” IEEE Control Systems Magazine, vol. 21, no. 4, pp. 7–8, Aug. 2001. [2] D.P. Feldman, J.P. Crutchfield, “Structural information in two-dimensional patterns: Entropy convergence and excess entropy,” Physical Review E, 2003. [3] I. Macaluso, H. Cornean, N. Marchetti, L. 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Analysis of Unprotected Intersection Left-Turn Conflicts based on Naturalistic Driving Data arXiv:1702.00135v2 [cs.SY] 3 Apr 2017 Xinpeng Wang1 , Ding Zhao2 , Huei Peng3 and David J. LeBlanc2 Abstract— Analyzing and reconstructing driving scenarios is crucial for testing and evaluating highly automated vehicles (HAVs). This research analyzed left-turn / straight-driving conflicts at unprotected intersections by extracting actual vehicle motion data from a naturalistic driving database collected by the University of Michigan. Nearly 7,000 left turn across path - opposite direction (LTAP/OD) events involving heavy trucks and light vehicles were extracted and used to build a stochastic model of such LTAP/OD scenario, which is among the top priority light-vehicle pre-crash scenarios identified by National Highway Traffic Safety Administration (NHTSA). Statistical analysis showed that vehicle type is a significant factor, whereas the change of season seems to have limited influence on the statistical nature of the conflict. The results can be used to build testing environments for HAVs to simulate the LTAP/OD crash cases in a stochastic manner. I. INTRODUCTION Before highly automated vehicles (HAVs) can be released to the general public, a well-defined process for testing and evaluating them must be established. The Google self-driving car project experienced its first shared-responsibility crash in February 2016 [1]. Moreover, Tesla Autopilot failed to detect a semi-truck in its first fatal crash happened in May 2016 and was criticized for using the consumers as beta testers [2]. Fig. 1 briefly demonstrates how this crash happened, with the red sedan representing the Tesla. The National Highway Traffic Safety Administration (NHTSA) is now considering the possibility of putting a pre-market approval process into place [3], in addition to a rigorous self-certification process still anticipated from the vehicle manufacturers. A key factor in HAV testing is the test scenarios and behaviors of other road users, particularly those of other vehicles. The test conditions need to be not only realistic but also feasible for repeated safety tests. Test scenario models can be divided into two types. The first type has fixed scenarios, such as the tests of lane support systems (LSS) [4] and autonomous emergency braking (AEB) [5] launched by The European New Car Assessment Programme (EURO NCAP). A major advantage of this type is that it is repeatable. However, it is hard to use this type of models to * This work is funded by the Mobility Transformation Center Denso Tailor Project at the University of Michigan with grant No. N020210. 1 X. Wang is with the Department of Automation, Tsinghua University, Beijing, China, 100084, and now he is a visiting scholar at the University of Michigan, Ann Arbor, MI 48109, U.S. 2 D. Zhao (corresponding author: [email protected]) and D. J. LeBlanc are with the University of Michigan Transportation Research Institute, Ann Arbor, MI 48109, U.S. 3 H. Peng is with Department of Mechanical Engineering at the University of Michigan Transportation Research Institute, Ann Arbor, MI 48109, U.S. Fig. 1. A brief description of the Tesla accident represent the highly complex and variable nature of the human driving environment. Moreover, HAVs could be adjusted to pass certain fixed scenarios while their performance under broad conditions might not be well assessed. To overcome these drawbacks, we proposed a second type of models. In our previous works [6]–[8], we proposed a stochastic test method and built a test environment for car-following and lane-change scenarios. In this paper, we will focus on the intersection scenario. The intersection has been one of the most challenging scenarios for HAVs, due to the variety of road users, complexity of traffic flow and the unpredictability of vehicles and pedestrians. According to [9], crashes at intersections took up a major portion, about 44 %, of all the traffic crashes in the US. Among all kinds of scenarios with potential risks at an intersection, unprotected left turn across path opposite direction (LTAP/OD) is a typical one. This scenario is ranked second among 10 priority V2V light-vehicle precrash scenarios [10]. In an LTAP/OD scenario, two vehicles are considered: the turning vehicle (TV) and the straightdriving vehicle (SdV). Although a lot of research, such as [11], [12], has been conducted on traffic conflict analysis of LTAP/OD scenario, the factor of vehicle type has not been widely investigated. Now that the crash of the Tesla with Autopilot system has been attributed to its failure to detect the truck turning ahead [13], it is crucial that more attention should be paid to scenarios involving heavy trucks. Moreover, there has been insufficient research on the influence of season change on driving behaviors at intersections. As extreme weather such as storm and fog has a strong impact on the driving behaviors of human drivers, we propose that they are possibly influential to HAVs as well. This research focused on two major tasks: first, it built a stochastic model of traffic conflicts in the LTAP/OD scenario TABLE I INTRODUCTION TO IVBSS DATABASE Vehicle type Distance Time Trips Vehicles / Drivers Type of front radar Light vehicle 213, 309 mi Apr. 2009 - Apr. 2010 22,657 16 sedans / 108 drivers Bosch LRR2 Heavy truck 601, 994 mi Feb. 2009 - Dec. 2009 22,724 10 tractors / 20 drivers TRW AC20 (a) Light vehicle Fig. 2. based on naturalistic driving data. Events involving both light vehicles (LVs) and heavy trucks (HTs) as the SdV were extracted from the database, reconstructed into realistic trajectories of TVs and SdVs, and finally described with several key variables. Secondly, the influence of vehicle type of the SdV and the season factor to the driving behavior of the TV was analyzed by comparing the distribution of these key variables between LVs and HTs as well as between summer and winter. II. DATA S OURCE The data source for this research is the Integrated VehicleBased Safety Systems (IVBSS) database [14], which was collected from 2009 to 2010 and maintained by the University of Michigan Transportation Research Institute (UMTRI). The database consists of two parts: LV platform and HT platform. It comes from a naturalistic field operational test (N-FOT) which is to assess the potential safety benefits and driver acceptance associated with a prototype integrated crash warning system [15], [16]. As the system incorporates forward crash warning (FCW), lateral drift warning (LDW), lane-change/merge warning (LCM) and curve speed warning (CSW), non of the functions are designed to deal with LTAP/OD scenario. Thus, it is assumed in this research that whether the warning system is enabled will not affect driver behavior at LTAP/OD scenario. For LV platform [14], 16 identical prototype vehicles were driven by 108 drivers for their own personal use for over six weeks. On each test vehicle, there is one long-distance 77-GHz radar that looks forward and six 24-GHz radars that cover the adjacent lanes as well as the area behind the vehicle. In addition, there is a vision system, an automotivegrade non-differential global positioning system (GPS) and an on-board digital map. Around 700 different channels of signals have been collected. For the HT platform [17], 18 male commercial truck drivers from Con-way Freight drove 10 equipped Class 8 tractors for 10 months. There are eight radars, three exterior cameras and several interior cameras on the test truck, recording over 500 channels of data including the driving environment, drivers activity, system behaviors and vehicle kinematics. Basic information about the LV and HT platforms in the IVBSS database is listed in Table I; the configuration of on-board sensors on each platform is shown in Fig. 2. The test area covered by IVBSS N-FOT is primarily in (b) Heavy truck Sensor configuration of IVBSS test vehicles the Detroit area of the U.S. Most of the HT trips took place in the lower peninsula of Michigan (63 %) and Ohio (33 %) [16]. most of the LV trips fell within a similar region [15]. The database provides adequate information for this research. For event extraction, data from the on-board GPS sensor is used to locate the instrumented vehicle; data from the front long-range radar is used to reconstruct the trajectory of target vehicles; video recordings from vision cameras around the vehicles are used as a supplemental tool for event screening. In addition, as the IVBSS test lasted approximately one year, driving data under a variety of weather conditions throughout the year were covered, enabling us to uncover the influence of season factors. III. E XTRACTION OF L EFT T URN S CENARIO In order to extract eligible left-turn events from the database for both the LV and the HV platform, three major tasks were performed. First, we processed data from radar for further use. Second, we searched the database for all left-turn events that meet our criteria. Finally, data points in each event were interpreted into trajectories of the TV and the SdV. A. Target Association of Truck Data For radar data from the HT platform, we need to associate and mark data points together that belong to the same target in order to screen out unfit targets and create a trajectory for every eligible TV. To cluster points of interests, we apply the following criteria to processing HT data: • Only objects (TVs) that move in the opposite direction are detected. (vT V <-0.3 m/s). • Only points with small azimuth angle (\|α\| < 5.5°) are considered, as the effective detecting range of the radar is 11°. • When the cluster with point i is expended, only data points within a small time slot ([t(i), t(i) + 0.85 s]) are considered. • Only neighbor points that satisfy the following rules are grouped: 1) Strong correspondence between range, range rate and time difference: δtpred \| < 0.3 (1) \|1 − t(j) − t(i) where δtpred = 2 ∗ r(j) − r(i) rr(j) + rr(i) (a) Range Fig. 3. (b) Transversal Fig. 4. Configuration of the instrumented vehicle and the target vehicle for event extraction in LTAP/OD scenario An exemplary result of target association Here, r(i) is the range of point i, and rr(i) is the range rate of point i 2) Reasonable difference in transversal. tr(j) − tr(i) < 20m/s t(j) − t(i) (2) Here, tr(i) is the transversal of data point i, and t(i) is the time. Fig. 3 shows an example of data points that are associated and divided into different groups in one event. The dots with the same color show trajectories of targets, while red dots do not belong to any group and are seen as noise. In such a typical LTAP/OD scenario, a vehicle is turning in front of the instrumented truck. Fig. 3(a) shows how the range of target points change over time. As target vehicles cross the intersection when the instrumented truck is moving forward at a steady speed, the ranges to different targets are decreasing linearly. Moreover, Fig. 3(b) shows that the transversal of multiple targets is increasing from negative to positive, indicating that they cross from left to right in the view of the instrumented truck. Once data points from each target are clustered, the HT platform can be used for further event extraction for eligible LTAP/OD scenarios. B. Event Screening An unprotected LTAP/OD scenario can be recorded by either the SdV or the TV. In this paper, we use only the scenarios recorded by SdVs. Fig. 4 demonstrates the configuration of the instrumented vehicle, i.e., the SdV and the target vehicle, i.e., the TV for event extraction in LTAP/OD scenarios. For both the LV and HT platforms, eligible leftturn events are queried based on the following criteria: • The intersection has a stop sign or a set of signal lights. Although there will be protected LTAP/OD events retrieved with this criterion, they can be screened out by the following conditions, such as the constraint on the velocity of the TV and the SdV. • The instrumented vehicle is moving straight (speed larger than 3 m/s & change of heading angle smaller than 10°). • The target vehicle is moving towards the instrumented vehicle (the longitudinal projection of speed smaller than -0.5 m/s) and moving from left to right (due to the difference in radars, transversal goes from positive to negative for LVs, and from negative to positive for HTs). Fig. 5. • • Procedure and interim results for event extraction Time duration of the event is adequate (more than 1.5 s). The maximum of time difference between two consecutive points (defined as δt) in an event should be small enough to be seen as points of the same target (max{δt} < 1 s). Event extraction follows a similar procedure for LV and HT. For the LV platform, we first select all straight-driving occurrence at intersections; we then extract those with leftturn objects from the opposite direction. These tasks are completed in the Microsoft SQL Server Management Studio (SSMS). Afterwards, the extracted events are exported to MATLAB for the last round of screening, which guarantees reasonable speed, targets, and time duration. For the HT platform, the only difference is that after retrieving all the occurrences of straight-driving at intersections, we export data from the database server directly into MATLAB for target association and the following extraction tasks. The diagram in Fig. 5 illustrates the procedure and interim results of each phase for event extraction. Finally, HT has 5,780 eligible LTAP/OD events, whereas LV has 1,055. The location of these events is shown in Fig. 6. C. Trajectory Reconstruction Once all eligible events have been selected, the trajectories of both TV and SdV in each event are then reconstructed. The exact position of SdV comes from the on-board GPS sensor; the data from the front long-range radar are used to extract the relative position of TV in the coordinate of SdV. After synchronization on GPS and radar data, the trajectories of TV and SdV are generated. Fig. 7 shows the reconstructed trajectories for SdV and TV in one event. Here, dots with the same color represent the position of TV and SdV at the same moment. The TV crossed intersection before the SdV in this example. (a) Light vehicle Fig. 6. (b) Heavy truck Fig. 8. The location of extracted events Time to the conflict point of the SdV in a LTAP/OD scenario vSdV : Speed of the SdV at Tx vT V : Speed of the TV at Tx First, we demonstrate an example of conflict analysis on a single LTAP/OD event. Here we use the aforementioned occurrence, where the TV crossed the intersection before the SdV did. Fig. 8 uses Tcp to demonstrate how the SdV and the TV interacted in one real LTAP/OD event. The vertical axis indicates predicted time to the point of conflict of the SdV, whereas the horizontal axis shows the real elapsed time relative to the moment when the TV crosses the intersection, that is, Tx . In this scenario, time to the conflict point decreased linearly over time, indicating the margin between the TV and the SdV was large enough for the SdV to maintain a nearly constant speed when the TV was crossing. When the TV reached the conflict point, there was a 2.3-second margin for SdV, that is, Tcp , which is demonstrated by the red dot. Here, Tcp described the essence of this interaction between the SdV and the TV. Then, the following modeling and analysis will ignore the detailed interaction of the TV and the SdV, paying attention only to the four aforementioned variables in each event. We use all events we retrieved in the previous section from both the HT and LV platforms as the source for modeling. • • Fig. 7. Example of reconstructed trajectory IV. C ONFLICT A NALYSIS AND C OMPARISON A. Definition and Metrics of Conflicts In this section, conflict is used to describe risky events in traffic. According to [18], conflict is defined as an observational situation in which two or more road users approach each other in space and time to such an extent that a collision is imminent if their movements remain unchanged. Many conflict metrics have been used for measuring the level of safety for an LTAP/OD event, including post-encroachment time (PET), leading buffer (LB) and trailing buffer (TB) used by [19], and gap time (GT) used by [20]. For this paper, as the goal is to construct a stochastic model, we choose only the most representative time slice in each event to model LTAP/OD conflicts. The heading angle of the SdV is taken as constant during each event, with any small deviation being ignored. Thus, a conflict point is naturally defined as the location of the TV when its transversal in the radar of SdV crosses zero, and this exact moment is regarded as the representative moment of this conflict, defined as Tx . Consequently, four variables at Tx are chosen to model the conflict, including two modified conflict metrics: time to the conflict point (Tcp ) and distance to the conflict point (Dcp ): • Dcp : Distance to the conflict point for the SdV at Tx Dcp = dist(PSdV (Tx ), PT V (Tx )) • In this section, the effect of vehicle type on traffic conflict in LTAP/OD scenarios is discussed. Distributions of variables for LVs and HTs are compared. As events with smaller Dcp and Tcp are more dangerous, we generated the distributions of the reciprocal of Dcp and Tcp to put these risky but rare events in the tail, as shown in (3) Here PSdV and PT V are the positions of SdV and TV Tcp : Time to the conflict point for the SdV at Tx Tcp = Dcp /vSdV B. Comparison between Light Vehicles and Heavy Trucks (4) −1 (a) Distribution of Dcp −1 (b) Distribution of Tcp −1 −1 Fig. 9. Comparison of Dcp and Dcp between heavy trucks and light vehicles Light vehicle Heavy truck (a) Speed distribution of straight- (b) Speed distribution of turning driving heavy trucks and straight- heavy trucks and turning light vehidriving light vehicles cles Fig. 10. Comparison of the speed between heavy trucks and light vehicles Fig. 9. The dots and bars at the top of figures show the mean value and the standard deviation of each empirical −1 distribution. From Fig. 9, we can see that when Dcp or −1 Tcp increases, there are fewer points of data, giving rise to a shape with a long tail. Moreover, events with an HT as −1 −1 SdV tend to have both smaller Dcp and smaller Tcp than with an LV, indicating less severe conflicts. Fig. 10 shows the distributions of vSdV and vT V . The distribution of vSdV for HT and LV platforms both have a triangular shape. Most vSdV ranges from 12 to 20 m/s, whereas most vT V is less than 10 m/s at the conflict point. Though there is no obvious difference with the distribution of vSdV between events with LV and HT as the SdV, the vT V tends to be significantly higher when the SdV is an LV than is an HT. Combined with the previous results of Dcp and Tcp , we can conclude that for left-turn conflicts where HTs are SdVs, conflict metrics have significantly higher value, and TVs tend to turn with less aggressive speed. This means that when the TV chooses the time of turning and commences turning action, it behaves more conservatively when confronted by an HT coming from the opposite direction than an LV. The difference in vehicle type does influence the driving behavior of the TV and the severity of the conflict. C. Analysis of Season Factor In this section, we uncover the influence of season factor on behaviors of SdVs and TVs in LTAP/OD scenarios. During the test, 7 % of driving for HTs [16] and 15 % [15] for LVs took place in freezing temperature. The months with events that took place in freezing temperatures are defined as winter, which includes December through March of the following year. This period also coincides with the time when the average snowfall in Ann Arbor is over 8 inches. On the other hand, summer is defined as being from June to August. We have retrieved 272 events in summer and 391 events in winter for LVs, whereas the numbers for HTs are 1818 and −1 −1 844 respectively. Tcp , Dcp , vSdV and vT V are compared for summer and winter driving. Mann-Whitney-Wilcoxon (MWW) test [21] is a nonparametric hypothesis test of the null hypothesis that two populations are the same against an alternative hypothesis. Here, we used it to determine whether the conflict metrics differ between summer and winter. Fig. 11. Comparison between events in summer and winter Fig. 11 shows the result of the comparison. It can be concluded that for both LV and HT platforms, the mean values for summer and winter of all four variables that describe the conflict at LTAP/OD for both SdVs and TVs are very close. As the p-value from the MWW test is large (p-value > 0.6) for all the eight distributions, we are not able to distinguish between the left-turn pattern in summer and −1 −1 , vT V and vSdV . This result in winter in terms of Dcp , Tcp indicates that despite a large difference in climate, there is no significant difference between the way people drive in winter and in summer at LTAP/OD scenarios in the Great Lakes area. This conclusion has its significance for designing and testing of HAVs. V. CONCLUSION In this research, traffic conflicts of TVs and SdVs in LTAP/OD scenarios are modeled and analyzed based on nearly 7,000 left-turn events extracted and reconstructed from the naturalistic database. The two modified conflict metrics, Tcp and Dcp are used to model turning behavior of the TV. This stochastic model can be further used for developing simulation tools for evaluating HAVs. The significance of vehicle type and season are also addressed in the research. In general, when the SdV is an HT, the driver of the TV tends to turn in a more conservative fashion with a wider margin. Surprisingly, despite prevailing snow and freezing weather in the winter of Michigan, driver behavior at LTAP/OD scenarios during the N-FOT test did not differ significantly between summer and winter. These two conclusions can be useful for designing automated driving algorithms and for establishing regulations and policies for HAVs. In the following research, we will improve the accuracy of trajectory reconstruction by conducting sensor fusion to the GPS and yaw rate sensor, and by re-synchronizing data from different channels. Moreover, we will further investigate the reasons behind the conclusion on the similarity of driver behavior between summer and winter. Possible causes could be: snow on the road was shoveled promptly in winter thus normal driving was almost unaffected; the N-FOT trips avoided extreme weather in winter so that the data was biased. Besides, We will also facilitate the model to build a stochastic simulation environment for the testing and evaluation of HAVs. D ISCLAIMERS This work was funded in part by the University of Michigan Mobility Transformation Center Denso Pool project. The findings and conclusions in the report are those of the authors and do not necessarily represent the views of the MTC or Denso. R EFERENCES [1] “Google Self-Driving Car Project Monthly Report (February 2016),” pp. 14–16, 2016. [2] O. Solon, “Should Tesla be ’beta testing’ autopilot if there is a chance someone might die?” [Online]. Available: https://www.theguardian.com/technology/2016/jul/ 06/tesla-autopilot-fatal-crash-public-beta-testing [3] U.S. Department of Transportation and National Highway Traffic Safety Administration, “Federal Automated Vehicles Policy,” Tech. Rep. September, 2016. [Online]. Available: https://www.transportation. gov/sites/dot.gov/files/docs/AVpolicyguidancePDF.pdf [4] “European New Car Assessment Programme - TEST PROTOCOL - Lane Support Systems,” 2015. [Online]. 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A Revised Incremental Conductance MPPT Algorithm for Solar PV Generation Systems Meng Yue and Xiaoyu Wang Sustainable Energy Technologies Department Brookhaven National Laboratory Upton, NY 11973, USA [email protected], [email protected] Abstract—A revised Incremental Conductance (IncCond) maximum power point tracking (MPPT) algorithm for PV generation systems is proposed in this paper. The commonly adopted traditional IncCond method uses a constant step size for voltage adjustment and is difficult to achieve both a good tracking performance and quick elimination of the oscillations, especially under the dramatic changes of the environment conditions. For the revised algorithm, the incremental voltage change step size is adaptively adjusted based on the slope of the power-voltage (P-V) curve. An accelerating factor and a decelerating factor are further applied to adjust the voltage step change considering whether the sign of the P-V curve slope remains the same or not in a subsequent tracking step. In addition, the upper bound of the maximum voltage step change is also updated considering the information of sign changes. The revised MPPT algorithm can quickly track the maximum power points (MPPs) and remove the oscillation of the actual operation points around the real MPPs. The effectiveness of the revised algorithm is demonstrated using a simulation. Index Terms—IncCond MPPT algorithm, fractional opencircuit/short-circuit MPPT algorithm, P&O MPTT algorithm, solar PV generation. I. INTRODUCTION As one of the most promising renewable energy technologies, the installed capacity of the solar photovoltaic (PV) generation has increased dramatically in recent years. Although the cost of PV generation continues to drop, the economic competitiveness of solar PV energy is still low compared to the traditional energy sources, even with various local and federal policy instruments [1]. While it is desirable to further lower the cost and increase the efficiency of solar energy systems including both the solar panels and the power electronic devices, increasing the efficiency of the installed PV energy systems by simply improving the existing control algorithms should also be pursued. One way of achieving this is to modify the existing MPPT algorithms to extract more solar energy under various environmental conditions. Many different types of MPPT algorithms have been proposed in the literature. As summarized in [2], different algorithms have their own pros and cons, in terms of complexity, accuracy and convergence speed, etc. Among the commonly used algorithms, the hill-climbing/perturbation and observation (P&O) method [3-6] is easy to implement using either analog or digital circuits. It periodically perturbs either the duty ratio of the converter or the the PV array operating voltage even when the MPP is achieved. Further, the “true” MPPT cannot be achieved using the P&O method since the operation point of the PV system is oscillating around the MPP. Under the conditions of continuous fast changing irradiance, the operating point might continuously deviate from the MPPs and eventually the optimal operation points cannot be achieved at all. These issues degrade the performance of the solar generation system. The fractional open-circuit voltage (or short-circuit current) method [7-9] needs only to sense one voltage (or current) parameter to approximate the MPP by using empirical parameters. The major issue related to this method is that the PV circuit has to be periodically operated in open-circuit (or short-circuit) conditions and may have significant impact on the grid operation. Other types of algorithms, such as those based on fuzzy logic control and neural network may accurately track the MPPs under different environmental conditions [2]. The MPPT performance, however, is not guaranteed since they both rely heavily on the algorithm developers and/or a significant volume of field data under all kind of conditions for the design and implementation of such algorithms. The IncCond method (see, e.g., [10-17]) appears to be the most popular one in practice due to its medium complexity and the relatively good tracking performance. One of the major difficulties implementing the IncCond method is the selection of the (fixed) voltage change step size for simultaneously satisfying the tracking speed and maintaining the MPP. A large step size of voltage change helps the system rapidly approach the MPPs. On the other hand, this large value generally induces persisting oscillations around the MPP if no other special countermeasures were taken. The issues with using a small step size of voltage change are the opposite. A simple and effective revised IncCond algorithm is proposed in this paper. An adaptive voltage step change scheme is first adopted based on the slope where the operation point locates on the P-V curve. An accelerating factor and a decelerating factor are then applied to further adjust the voltage step change considering whether the sign of the P-V curve slope remains the same or changes in a subsequent tracking step. The same information of sign changes is also used to update the upper bound of the maximum voltage step change. The adaptive voltage step change enables the PV system to quickly track the environment condition variations, i.e. reach and stay at the MPPs. In this way, more solar energy generation can be harvested from the PV energy systems. These improvements enable the quick response to the environment condition changes and rapid landing on the MPP. The revised method is easy to implement since it does not require knowledge of the I-V characteristics of specific PV panels and the parameters are easy to tune. The revised IncCond algorithm is described in detail in Section II with an overview of various modified IncCond methods. Modeling of generic PV generation systems is presented in Section III for simulation purposes. Simulation results using the proposed MPPT algorithm will be shown in Section IV and concluding remarks are given in Section V. II. A REVISED INCCOND MPPT METHOD The MPP is achieved by adjusting the terminal output voltage of a solar array through controlling the converter duty ratio. While the cell temperature can be easily measured, the irradiance is difficult to measure accurately, and the desired voltage at the MPP is hard to know exactly. Therefore, a test condition needs to be developed in order to determine whether the current operating point is the MPP or not without measuring the temperature and irradiance. For a solar panel, there is only one maximum power point for a given irradiance level and cell temperature. Note, the presence of a partial shading condition of a panel may cause multiple local maxima and is not considered in this paper, although the revised algorithm can be used together with the two-staged methods proposed in [16, 17]. The IncCond method uses the information of the solar P-V curve, i.e., at the left hand side of the MPP the slope is greater than zero, at the right hand side of the MPP the slope is less than zero, and the slope is zero exactly at the MPP. Therefore, the solar array terminal voltage needs to be increased when the slope is positive and decreased when the slope is negative. The slope dP/dV can be calculated as, dP d ( IV ) dI = = I +V dV dV dV (1) with dI/dV ≈ ∆I/∆V (i.e., the incremental conductance) in an implementation. Under the MPP condition, i.e., dP/dV = 0, the following relationship holds, ∆I I =− ∆V V (2) The major difficulty with the IncCond method is the selection of the incremental step size of the duty ratio for adjusting the solar terminal output voltage. A fixed incremental step size of the duty ratio in general will not bring the array to the MPP because the operating point will oscillate around the MPP, i.e., either on the left or the right of the MPP. Ref. [14] divided the entire I-V (current-voltage) curves into two domains using "square root" functions with all of the MPPs contained in only one of them. Therefore, the first step of performing MPPT is to bring the operating point to the domain that contains all of the MPPs. This method, however, requires a good understanding of the PV panel I-V characteristics that are panel specific. In [15], a so-called "Van Allen's oscillator" was added between the solar panel and the inverter for a purpose of balancing the power source and the load that continues changing. A simple proportional integral (PI) controller was developed to track the MPPs based on this configuration. It is intuitively easy to avoid a fixed voltage change step size by adjusting the increment proportionally to the steepness of the slope and eventually the increment of the duty ratio will become zero at the MPP, where the slope is zero, similar to the PI controller proposed in [15]. The implementation, however, appears to be very difficult because (1) the P-V curve steepness around the MPP can be very different for different operating conditions (i.e., the P-V curve for a lower irradiance level can be more flat) and (2) a sudden change in the operating condition of the solar array may produce a very large numerical difference in calculating the slope when the change occurs. Since the duty ratio is between 0 and 1, this may cause unacceptable change in the solar terminal output voltage and make it very difficult to bring the voltage back to normal. Note, refs [16] and [17] proposed twostage methods mainly to avoid the local maxima caused by the non-uniform insolation experienced by the solar panels. The traditional IncCond method was still used after bringing the operating point close to the global MPP by using, e.g., monitoring cells in [17]. In this section, a simple and effective modified IncCond method is proposed based on observations that (1) in two consecutive tracking steps, a changing in sign of the slope (i.e., from positive to negative or from negative to positive) indicates that the increment step size is too large (otherwise, it may land on the MPP or the same side on the P-V curve) of the duty ratio; and (2) the same sign of the slope in two consecutive tracking steps indicates that the increment step size is too small (otherwise, the operating point may land on the MPP or the other side on the P-V curve). Based on these observations, the strategy proposed here is to (1) adjust the incremental step size considering the steepness of the slope; and (2) further adjust the incremental step size comparing the sign of slopes in two consecutive tracking steps, i.e., decrease the incremental size in the former case (e.g., by multiplying the incremental size by a factor DEACC such as 0.7) and to increase the incremental size in the latter case (e.g., by multiplying the increment by a factor ACC such as 1.2). By applying this improved strategy, the solar array will approach the MPP in an accelerating manner after a change in the operating condition(s), and the magnitude of oscillation around the MPP may rapidly decrease until the test condition is considered to be satisfied. After landing onto the MPP, the duty ratio will not be adjusted until the operating condition changes again. In the implementation, the upper- and lower-bounds for the incremental step size need to be defined to avoid extremely drastic changes in the duty ratio. However, the upper-bound is generally fixed and needs to be large enough to permit the rapid tracking of the MPP for a sudden change of the operation condition. The issue with a fixed upper bound is when the array starts tracking the new MPP and it quickly approaches the MPP and lands on the other side of the MPP, the duty ratio needs to be adjusted in the reverse direction using the incremental step, which could be large (due to the factor ACC) and remain large for some time (although the factor DEACC has been applied). At this point, having a large incremental step does not help because it may cause very large fluctuations or overshoot of the voltage before the MPP is achieved. Therefore, the second proposed improvement is, when the sign of the slope changes, the upper-bound is also decreased together with the incremental size. It is also preferred to maintain the upper-bound small nearby the MPP until the MPP is reached. Note, in the implementation of the algorithm, a nominal incremental step size is pre-selected. After the test condition of the slope is considered to be satisfied, the duty ratio will not be changed but the incremental step size might become very small now, which, if not corrected, will cause the very slow response in the beginning of attempting to track the next MPP under a different operating condition. A simple solution is to reset the step size to the nominal value without adjusting the duty ratio when the MPP is considered to be reached. ∆I I ∆I I or δd (k ) = δd (k − 1) × DEACC× \| + \| + \| ∆V V ∆V V if the slope is not small enough (i.e., greater than a preselected constant ε and equation (2) is still not satisfied), and the duty ratio d (k ) = d (k − 1) + δd (k ) . × ACC× \| δd (0) indicates the initial incremental size of the duty ratio, δd max (0) the initial upper bound of the incremental size. δd (k ) and δd max (k ) are the updated (based on conditions discussed above, as indicated in Fig. 1) increment size and the upper boundary of the incremental size. III. MODELING OF PV ENERGY SYSTEMS A. Solar Array In general, a solar array consists of many solar modules connected in series and/or parallel, each module being manufactured by serially connecting a certain number of solar cells. A solar cell is essentially represented by an equivalent electrical circuit as shown in Fig. 2. For illustration purposes, modeling of a solar cell is briefly summarized in this part. Interested readers can find details in other references, e.g., [18]. Note also, the PV terminal output voltage may be very sensitive to the duty ratio, especially for duty ratio close to 0.0 or 1.0. The dc-dc converter input and output voltages should thus be selected such that the duty ratio is in the middle of the duty ratio range. Fig. 2. An equivalent electrical circuit of a solar cell. For the solar cell model in Fig. 2, the following equation can be derived, (3) I PV = I Ph − I d − (VPV + I PV × RS ) / RP where IPV and VPV represent the solar cell terminal output current and voltage, respectively. IPh is the photon current source, Id the diode current. The series resistance RS and shunt resistance RP are used to represent the power losses while the latter can generally be neglected. It is noted in equation (3) that the photon current and the diode current are temperature and irradiance dependent. For given panel temperature T (Kelvin) and irradiance level G (W/m2), IPh and Id can be calculated using the following equations, G G  I Ph \|T ,G I= I SC \|Tref ,Gref [1 + α (T − Tref )] = Ph \|T ,Gref Gref Gref   \|T ,G I 0 \|T ,G ×[exp(q (V + I × RS ) / (nkT )) − 1] Id =   3 −qEg 1 1 T n  ) exp( ( − )) I 0 \| T ,G = I 0 \|Tref ,G × (  Tref nk T Tref Fig. 1. Flowchart of the revised IncCond MPPT algorithm.  I 0 \|Tref ,G I SC \|Tref ,Gref /[exp(qVOCref / (nkTref )) − 1] =  Based on the above discussions, the flow chart of the  proposed modified IncCond algorithm is shown in Fig. 1. In (4) Fig. 1, V(·) and I(·) are the terminal output voltage and current and G represent the reference cell In equation (4), T ref ref of the solar array, which will be adjusted according to the slope calculated by the dc-dc converter, δd (k ) = δd (k − 1) temperature (Tref = 225˚C, i.e., 298 K) and reference irradiance (Gref = 1,000 W/m ) under the standard condition. α is the (5) dV \|V can be obtained from the manufacturers' data dI OCref sheet. After substituting the above parameters into equation (3), the I-V characteristics of the solar cell can be numerically computed for any given cell temperature and irradiance level and then used to represent a solar array consisting of interconnected modules and cells. where A buck-boost converter is used to step up the output dc voltage of a solar array such that a bulky step-up transformer can be avoided and perform the MPPT by controlling the duty ratio of the converter. See, e.g., [19] for more details. IV. SIMULATION RESULTS The revised IncCond algorithm was implemented in an integrated Matlab-based power system simulation software EPTOOL that was developed based on the Power System Toolbox [20]. EPTOOL can be used to perform a transient analysis of the grid under faulted conditions and the solar irradiance and temperature changes. Only MPPT simulation results are presented here to validate the effectiveness of the revised algorithm using a hypothetical solar irradiance profile as the input to solar plant, as tabulated in Table I. The other input, the panel temperature is assumed constant of 25˚C during the cloud transients. A centralized solar PV plant consists of 17170000 solar BP SX 150 panels [21]. The capacity of the PV plant is 433 MW or 4.33 pu (100 base MVA and under standard environmental conditions). A 50-machine-145-bus system was used as an example for carrying out the simulation purpose only. TABLE I: VARIATION OF IRRADIANCE AT A SOLAR PLANT DURING A CLOUD TRANSIENT 0.0 0.2 0.7 0.9 1.2 Time (s) Irradiance ( W / m 2 ) 1000 20 200 300 400 1.5 1.9 2.5 3.0 4.0 Time (s) 500 650 850 990 150 Irradiance ( W / m 2 ) 4.2 4.3 4.4 4.5 4.8 Time (s) Irradiance ( W / m 2 ) 120 20 210 330 340 4.9 Time (s) Irradiance ( W / m 2 ) 350 The conventional IncCond algorithm with a fixed incremental step size (0.001) of duty ratio is first applied and the MPPT is performed every 10 ms. Other parameters are selected as the following: δdmax = 0.01, and ε = 5E-4. PV Output Voltage (kV) qVOCref dV RS = − \|VOC −nkTref / [ I 0 \|Tref ,G q exp( )] ref dI nkTref Simulation results for the solar array terminal output voltage and the deviation of the actual dc output power from the calculated maximum power points are shown in Fig. 3, from which one can observe the persisting oscillations around the MPPs in most of the time, i.e., the MPPs were not truly achieved. The reason for the oscillations is that, as implied in the algorithm description of Section II, the terminal voltage continues to be adjusted. Fig. 3 also indicates that the conventional IncCond algorithm is not able to track the MPP for a rapid variation of the irradiance since the output power deviations from the actual solar power generation are significant for the large change in irradiance at 0.2 s, although for slow variations it can provide acceptable performance. This significant power deficiency around 0.2 s is caused by an inability to adjust the panel voltage rapidly enough to compensate for the large decrement of irradiance level, as can be seen by comparing the top curves in Fig. 3 with those given in Fig. 4-5. This highlights the inefficiencies of selecting control parameters such as the incremental step size and the upper-bound in accordance with conventional MPPT algorithms. These inefficiencies related to the conventional IncCond method can be addressed by the modified algorithm proposed in this paper. 0.65 0.6 0.55 0.5 0.45 PV DC Power Deviation (p.u.) temperature coefficient and ISC is the short-circuit current of the solar cell. These are both constants that can be obtained from manufacturers’ data sheets. I0\|T,G is the reverse saturation current of the diode, n the diode ideality factor, q = 1.602e-19 C the Coulomb constant, k = 1.38e-23 J/K the Boltzmann constant. Eg is band-energy gap (eV) and is given as Eg=1.16–0.000702T2/(T-1108). VOC is the open-circuit voltage of the solar cell. The series resistance can be solved using parameters at reference temperature and irradiance, i.e., 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 Time(s) 3 3.5 4 4.5 5 2 0 -2 -4 -6 Fig. 3. Terminal voltage variation and deviated power output of the PV plant using conventional IncCond algorithm. For the modified MPPT algorithm proposed in Section II, the deceleration and acceleration factors are applied in the modified IncCond algorithm with DEACC = 0.8 and ACC = 1.2 while other parameters remain the same. In the first scenario, the upper bound of the incremental step size of the duty ratio is fixed, i.e., δdmax(k) = 0.01, k = 0, 1, …. As shown in Fig. 4, the oscillations are eliminated quickly as the irradiance changes. Fig. 4 also shows that the revised algorithm with the fixed upper-bound of the step size can quickly make the operating point reach the MPP, as the voltage level becomes quickly stable even after the sudden change of irradiance at time 0.2s. However, relatively large terminal voltage overshoot at changing points of irradiance is now introduced and must be addressed. Fig. 5 shows further improvement of the tracking performance for a second scenario where an adaptive upperbound of the incremental step is used. The overshoot at the change points of the irradiance have been significantly decreased. The output power deviation is also reduced. Simulation also shows that the tracking performance is not sensitive to the associated parameters (e.g., δd), which makes parameter tuning very easy and the modified IncCond algorithm very robust. [3] 0.6 [5] 0.55 0.5 0.45 0.4 PV DC Power Deviation (p.u.) PV Output Voltage (kV) [4] 0.65 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 [7] -2 -4 -6 [8] 0 0.5 1 1.5 2 2.5 Time(s) 3 3.5 4 4.5 5 Fig. 4. Terminal voltage and deviated power output of the array using the modified IncCond algorithm (fixed upper bound of the incremental step size of duty ratio). [9] 0.65 [10] 0.6 0.55 0.5 0.45 0.4 PV DC Power Deviation (p.u.) PV Output Voltage (kV) [6] 2 [11] 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 2 0 [12] -2 -4 -6 0 0.5 1 1.5 2 2.5 Time(s) 3 3.5 4 4.5 5 [13] Fig. 5. Terminal voltage and deviated power output of the array using the modified IncCond algorithm (with adaptive upper bound of the incremental step size of duty ratio). V. CONCLUSIONS A revised IncCond algorithm was presented in this paper for PV generation systems. Compared with traditional IncCond methods, the voltage step change is adaptively determined based on the slope of the P-V curve and the location of the operating points in two consecutive tracking steps such that the PV system can track the rapid change in environmental conditions while the oscillation of the PV system operating points around the MPP can be avoided. In addition, the upper bound of the voltage step change is assigned a factor, DEACC (less than 1) to constrain the step change when a change in the sign of slope is detected.. The simulation results demonstrate the effectiveness of the proposed algorithm. 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Focus: Querying Large Video Datasets with Low Latency and Low Cost Kevin Hsieh†§ Ganesh Ananthanarayanan§ Peter Bodik§ Paramvir Bahl§ Matthai Philipose§ Phillip B. Gibbons† Onur Mutlu∗† † Carnegie Mellon University § Microsoft ∗ ETH Zürich Focus-Opt-Query Focus-Balance Query-all Large volumes of videos are continuously recorded from cameras deployed for traffic control and surveillance with the goal of answering “after the fact” queries: identify video frames with objects of certain classes (cars, bags) from many days of recorded video. While advancements in convolutional neural networks (CNNs) have enabled answering such queries with high accuracy, they are too expensive and slow. We build Focus, a system for lowlatency and low-cost querying on large video datasets. Focus uses cheap ingestion techniques to index the videos by the objects occurring in them. At ingest-time, it uses compression and video-specific specialization of CNNs. Focus handles the lower accuracy of the cheap CNNs by judiciously leveraging expensive CNNs at query-time. To reduce query time latency, we cluster similar objects and hence avoid redundant processing. Using experiments on video streams from traffic, surveillance and news channels, we see that Focus uses 58× fewer GPU cycles than running expensive ingest processors and is 37× faster than processing all the video at query time. Normalized Query Latency arXiv:1801.03493v1 [cs.DB] 10 Jan 2018 Abstract Focus-Opt-Ingest Ingest-all 1 0.8 0.6 0.4 0.2 0 0.03 0.02 0.01 (I=141X, Q=46X) (I=26X, (I=86X, Q=63X) Q=56X) 0 0 0.2 0.4 0.6 0.8 1 Normalized Ingest Cost 0 0.02 0.04 0.06 Normalized Ingest Cost Figure 1: Effectiveness of Focus at reducing both ingest cost and query latency, for an example traffic video. We compare against two baselines: “Ingest-all” that runs ResNet152 on all video frames during ingest, and “Query-all” that runs ResNet152 on all the video frames at query time. By zooming in, we see that Focus (the Focus-Balance point) is simultaneously 86× cheaper than Ingest-all in its GPU consumption and 56× faster than Query-all in query latency, all the while achieving at least 95% precision and recall. (Also shown are two alternatives offering slightly different trade-offs.) ResNet152), using them for video analytics queries is both expensive and slow. Using the ResNet152 classifier at query-time to identify video frames with cars on a month-long traffic video requires 280 GPU hours and costs $250 in the Azure cloud. The latency for running queries is also high. To achieve a query latency of one minute on 280 GPU hours of work would require tens of thousands of GPUs classifying the frames of the video in parallel, which is many orders of magnitude more than what is typically provisioned (few tens or hundreds) by traffic jurisdictions or retail stores. Note that the above cost and latency values are after using motion detection techniques to exclude frames with no moving objects. We believe that enabling low-latency and low-cost querying over large video datasets will make video analytics more useful and open up many new opportunities. A natural approach to enabling low latency querying is doing all classifications with ResNet152 at ingest-time, i.e., on the live videos, and store the results in an index of object classes to video frames. Any queries for specific classes (e.g., cars) will thus involve only a simple index lookup at query-time. There are, however, at least two problems with this approach. First, the cost to index all the video at ingest-time, e.g., $250/month/stream in the 1. Introduction Cameras are ubiquitous, with millions of them deployed by government and private entities at traffic intersections, enterprise offices, and retail stores. Videos from these cameras are continuously recorded [2,7]. One of the main purposes for recording the videos is answering “after-thefact” queries: identify video frames with objects of certain classes (like cars or bags) over many days of recorded video. As results from these queries are used by analysts and investigators, achieving low query latencies is crucial. Advances in convolutional neural networks (CNNs) backed by copious training data and hardware accelerators (e.g., GPUs [13]) have led to high accuracy in the computer vision tasks like object detection and object classification. For instance, the ResNet152 object classifier CNN [38] won the ImageNet challenge that evaluates classification accuracy on 1, 000 classes using a public image dataset with labeled ground truths [63]. For each image, these classifiers return a ranked list of 1, 000 classes in decreasing order of confidence. Despite the accuracy of image classifier CNNs (like 1 above example, is prohibitively high. Second, most of this ingest-time cost is wasteful because typically only a small fraction of recorded videos get queried [16]. Following a theft, the police would query a few days of video from a handful of surveillance cameras, but not all the videos. We present Focus, a system to support low-latency low-cost querying on large video datasets. To address the above drawbacks, Focus has the following goals: (1) low cost indexing of video at ingest-time, (2) providing high accuracy and low latency for queries, and (3) allowing trade offs between the cost at ingest-time against the latency at query-time. As input, the user specifies the ground-truth CNN (or “GT-CNN”, e.g., the ResNet152 classifier) and the desired accuracy of results that Focus needs to achieve relative to the GT-CNN. Focus uses four key techniques – cheap CNNs for ingest, using top-K results from the ingest-time CNN, clustering similar objects, and judicious selection of system and model parameters. First, to make video ingestion cheap, Focus uses compressed and specialized versions of CNNs, to create an ingest-time index of object classes to frames. CNN compression (e.g., [66]) creates new CNNs with fewer convolutional layers and smaller input images. Specialization [35, 65] trains those CNNs on a smaller set of object classes specific to each video stream so that those cheaper CNNs can classify these video-specific objects more accurately. Together, these techniques result in highly efficient CNNs for video indexing. Second, the cheap ingest CNNs, however, are also less accurate than the expensive GT-CNN (like ResNet152), measured in terms of recall and precision. Recall is the fraction of frames in the video that contained objects of the queried class that were actually returned in the query’s results. Precision, on the other hand, is the fraction of frames in the query’s results that contained objects of the queried class. To increase recall, Focus relies on an empirical observation: while the top-most (i.e., most confident) classification results of the cheap and expensive CNNs may not always match, the top-most result of the expensive CNN falls within the top-K results of the cheap CNN. Therefore, at ingest-time, Focus indexes each object with the “top-K” results of the cheap CNN (instead of just the top-most). To increase precision, at query-time, we first filter the objects from the top-K index and then classify the filtered objects with the expensive GT-CNN. Third, to reduce the query-time latency of using the expensive GT-CNN, Focus relies on the significant similarity between objects in videos. For example, a car moving across an intersection will look very similar in consecutive frames. Focus leverages this similarity by clustering the objects at ingest-time, classifying only the cluster centroids with the expensive GT-CNN at query-time, and assigning the same class to all objects in the cluster, thus considerably reducing query latency. In a nutshell, Focus’s ingest-time and query-time operations are as follows. At ingest-time, it classifies the detected objects using a cheap CNN, clusters similar objects, and indexes each cluster centroid using the top-K classification results. At query-time, when the user queries for class X, Focus looks up the ingest index for centroids that match class X and classifies them using the GT-CNN. For centroids that were classified as class X, it returns all objects from the corresponding clusters to the user. Finally, Focus smartly chooses the ingest-time CNN and its parameters to meet user-specified targets on precision and recall. Among the choices that meet the accuracy targets, it allows the user to trade off between the ingest cost and query latency. For example, using a cheaper ingest CNN reduces the ingest cost but increases the query latency as Focus needs to use a larger K for the top-K index to retain the accuracy targets. Focus identifies the “sweet spot” in parameters that sharply improve one of ingest cost or query latency for a small worsening of the other. We built Focus and evaluated it on thirteen 12-hour videos from three domains – traffic cameras, surveillance cameras, and news channels. We compare against two baselines: “Ingest-all” that runs GT-CNN on all video frames during ingest, and “Query-all” that runs GT-CNN on all the video frames at query time. We use ResNet152 as GT-CNN and augment both baselines with motion detection to remove frames with no objects, which is one of the core techniques in a recent prior work, NoScope [44]. Figure 1 shows a representative result, for a traffic video from a commercial intersection. On average, Focus is 58× (up to 98×) cheaper than Ingest-all and 37× (up to 57×) faster than Query-all. This leads to the cost of ingestion coming down from $250/month/stream to $4/month/stream, and the latency to query a 24 hour video dropping from 1 hour to under 2 minutes. See §6 for the full details. We make the following contributions. 1. We formulate the problem of querying video datasets by showing the trade-offs between query latency, ingest cost, and accuracy (precision and recall) of results. 2. We propose techniques to ingest videos with low cost by leveraging compressed and video-specific specialization of CNNs, while retaining high accuracy targets by creating approximate (top-K) indexes. 3. We identify and leverage similarity between objects in a video to cluster them using CNN features and significantly speeding up queries. 4. We propose and build a new end-to-end system to support low-latency, low-cost querying on large video datasets. We show that our system offers new trade-off options between ingestion cost and query latency, as it is significantly cheaper than analyzing all videos frames at ingest time and significantly faster than analyzing queried video frames at query time. 2 2. Background and Motivation can only process 77 images/second even with a high-end GPU (NVIDIA K80 [13]). This makes querying on large video datasets using these CNNs to be slow and costly. There are at least two recent techniques designed to reduce the cost of CNNs. First, compression is a set of techniques aiming to reduce the cost of CNN inference (classification) at the expense of reduced accuracy. Such techniques include removing some expensive convolutional layers [66], matrix pruning [31, 37], and others [42, 62] and can dramatically reduce the classification cost of a CNN. For example, ResNet18, which is a ResNet152 variant with only 18 layers is 8× cheaper. Second, a more recent technique is CNN specialization [35], where the CNNs are trained on a subset of a dataset specific to a particular context, also making them much cheaper. Using the combination of cheap and expensive CNNs is a key facet of our solution, described in §4. We first provide a brief overview of convolutional Neural Networks (CNN), the state-of-the-art approach to detecting and classifying objects in images (§2.1). We then discuss new observations we made about real-world videos, which motivate the design of our techniques (§2.2). 2.1. Convolutional Neural Networks A Convolution Neural Network (CNN) [47] is a specific class of neural networks that works by extracting the visual features in images. During image classification, or “inference”, a CNN takes an input image and outputs the probability of each class (e.g., dog, flower, or car). CNNs are the state-of-the-art method used for many computer vision tasks, such as image classification (e.g., [38,45,71]) and face recognition (e.g., [46, 64]). Input Image Pooling Layers Convolutional + Rectification Layers Fully-Connected Layer Prob.(Apple) ✓ Prob.(Car) Prob.(Orange) Prob.(Cat) … Prob.(Flower) Prob.(Dog) . . . . . . Extracted . . Features . 2.2. Characterizing Real-world Videos We aim to support queries of the form, find all frames in the video that contain objects of class X. We identify some key characteristics of real-world videos towards supporting these queries: (1) large portions of videos can be excluded (§2.2.1), (2) only a limited set of object classes occur in each video (§2.2.2), and (3) objects of the same class have similar feature vectors (§2.2.3). The design of Focus is based on these characteristics. We have analyzed 12 hours of video from six video streams each. The six video stream span across traffic cameras, surveillance cameras, and news channels. (§6.1 provides the details.) We detect the objects in each frame of these videos (using background subtraction [43]), and classify each object with the ResNet152 CNN [38] among the supported 1, 000 object classes. In this paper, we use results from the costly ResNet152 CNN as ground truth. 2.2.1. Excluding large portions of videos. We find considerable potential to avoid processing large portions of videos at query-time. Significant portions of video streams either have no objects at all (as in a garage camera at night) or the objects are stationary (like parked cars). We find that in our video sets, one-third to one-half of the frames fall in these categories. Therefore, queries to any object class would benefit from pre-processing filters applied to exclude these portions of the videos. Even among the frames that do contain objects, not all of them are relevant to a query because each query only looks for a specific class of objects. In our video sets, an object class occurs in only 0.01% of the frames on average, and even the most frequent object classes occur in no more than 16% − 43% of the frames in the different videos. This is because while there are usually some dominant classes (e.g., cars in a traffic camera, people in a news channel), most other classes are rare. Since queries are for specific object classes, there is considerable poten- Figure 2: Architecture of an image classification CNN. Figure 2 illustrates the architecture of an image classification CNN. Broadly, almost all CNNs consist of three key types of network layers: (1) convolutional and rectification layers, which detect visual features from input pixels, (2) pooling layers, which down-sample the input by merging neighboring pixel values, and (3) fully-connected layers, which provide the reasoning to classify the input object based on the outputs from previous layers. The outputs of an image classification CNN are the the probabilities of all object classes, and the class with the highest probability is the predicted class for the input image. The output of the penultimate (i.e., previous-to-last) layer can be considered as “representative features” of the input image [45]. The features are a real-valued vector, with lengths between 512 and 4096 in state-of-the-art classifier CNNs (e.g., [38, 45, 66, 71]). It has been shown that images with similar feature vectors (i.e., small Euclidean distances) are visually similar [22, 23, 45, 58]. The high accuracy of CNNs comes at a cost: inferring (or classifying) using state-of-the-art CNNs to classify objects in images requires significant computational resources. This is because the higher accuracy of CNNs comes from using deeper architectures (i.e., more layers) to obtain better visual features. For instance, ResNet152 [38], the winner of the ImageNet competition [63] in 2015, has been trained to classify across 1000 classes from the ImageNet dataset using 152 layers, but 3 CDF (number of objects) 1 0.9 since they are specifically trained to extract visual features for classification. We verify the robustness of feature vectors using the following analysis. In each video, for each object i, we find its nearest neighbor j using feature vectors from the cheap ResNet18 CNN and compute the fraction of object pairs that belong to the same class. This fraction is over 99% in each of our videos, which shows using feature vectors from cheap CNNs can potentially help identify duplicate objects. 95% of objects 0.8 Auburn Lausanne CNN 0.7 0.6 Jackson Hole Sittard MSNBC 0.5 0% 2% 4% 6% 8% Percentage of ResNet152's 1000 classes 10% Figure 3: CDF of frequency of object classes. The x-axis is the fraction of classes out of the 1000 recognized by ResNet152 (truncated to 10%). 3. Overview of Focus The goal of Focus is to index live video streams by the object classes occurring in them and enable answering “after-the-fact” queries later on the stored videos of the form find all frames that contain objects of class X. Optionally, the query can be restricted to a subset of cameras and a time range. Such a query formulation is the basis for many widespread applications and could be used either on its own (such as for detecting all cars or bicycles in the video) or used as a basis for further processing (e.g., finding all collisions between cars and bicycles). Focus is designed to work with a wide variety of current and future CNNs. At system configuration time, the user (system administrator) provides a ground-truth CNN (GT-CNN), which serves as the accuracy baseline for Focus, but is far too costly to run on every video frame. Through a sequence of techniques, Focus provides nearlycomparable accuracy but at greatly reduced cost. By default, and throughout this paper, we use the ResNet152 image classifier as the GT-CNN. Because the acceptable target accuracy is applicationdependent, Focus permits the user to specify the target, while providing reasonable defaults. Accuracy is specified in terms of precision, i.e., fraction of frames output by the query that actually contain an object of class X according to GT-CNN, and recall, i.e., fraction of frames that contain objects of class X according to GT-CNN that were actually returned by the query. The lower the target, the greater the cost-savings provided by Focus. Even for high targets such as 95%–99%, Focus is able to achieve order-of-magnitude or more cost savings. Figure 4 presents the design of Focus. • At ingest-time (left part of Figure 4), Focus classifies objects in the incoming video frames and extracts their feature vectors. To make this step cheap, it uses a highly compressed and specialized version of the GT-CNN model (IT1 in Figure 4). Focus then clusters objects based on their feature vectors (IT2 ) and assign to each cluster the top K most likely classes these objects belong to (based on classification confidence of the ingest CNN); (IT3 ). It creates a top-K index, which maps each class to the set of object clusters (IT4 ). The top-K index is the output of Focus’ ingest-time processing of videos. • At query-time (right part of Figure 4), when the user tial in indexing frames by the classes of objects. 2.2.2. Limited set of object classes in each video. We next focus on the classes of objects that occur in each of the videos and the disparity in frequency among them. Most video streams have a limited set of objects because each video has its own context (e.g., traffic cameras can have automobiles, pedestrians or bikes but not airplanes). It is rare that a video stream contains objects of all the classes recognized by state-of-the-art classifier CNNs. Figure 3 shows the cumulative distribution function (CDF) of the frequency of object classes in our videos (as classified by ResNet152). We make two observations. First, objects of only 22% − 33% (not graphed) of the 1, 000 object classes occur in the less busy videos (Auburn, Jackson Hole, Lausanne, and Sittard). Even in the busier videos (CNN, and MSNBC), objects of only 50% − 69% of the classes appear. Also, there is little overlap between the classes of objects among the different videos. On average, the Jaccard indexes [72] (i.e., intersection over union) between the videos based on their object classes is only 0.46. Second, even among the object classes that do occur, a small fraction of classes disproportionately dominate. Figure 3 shows that 3% − 10% of the most frequent object classes cover ≥ 95% of the objects in each video stream. This suggests that for each video stream, we can automatically (i) determine its most frequently occurring classes and (ii) train efficient CNNs specialized for classifying these classes (§2.1). 2.2.3. Feature vectors for finding duplicate objects. Objects moving in the video often stay in the frame for several seconds; for example, a pedestrian might take a minute to cross a street. Instead of classifying each instance of the same object across the frames, we would like to inexpensively find duplicate objects and only classify one of them using a CNN (and apply the same label to all duplicates). Thus, given n duplicate objects, this requires only one CNN classification operation instead of n. Comparing pixel values across frames is an obvious choice to identify duplicate objects, however, they turn out to be highly sensitive to even small changes in the camera’s real-time view of an object. Instead, feature vectors extracted from the CNNs are much more robust 4 Frames Frames Frames Object feature vectors Objects IT1 Specialized, Compressed CNN Query-time Ingest-time CNN specialization IT3 IT2 QT1 Object clusters Object top-K classes Frames with objects of class X Querying for class X QT4 Matching clusters for X Centroid objects IT4 Top-K index QT2 QT3 GT-CNN Figure 4: Overview of Focus. queries for a certain class X (QT1 ), Focus retrieves the matching clusters from the top-K index (QT2 ), runs the centroids of the clusters through GT-CNN (QT3 ), and returns all frames from the clusters whose centroids were classified by GT-CNN as class X (QT4 ). The top-K ingest index is a mapping between the object class to the clusters. Specifically, object class → hcluster IDi cluster ID → [centroid object, hobjectsi in cluster, hframe IDsi of objects] We next explain how Focus’ key techniques keep ingest cost and query latency low while also meeting the userspecified accuracy targets. 1) Cheap Ingest-time CNN: Focus makes indexing at ingest-time cheap by compressing and specializing the GT-CNN model for each video stream. (i) Compression of CNN models [31, 37, 42, 62, 66] uses fewer convolutional layers and other approximation techniques (§2.1). (ii) Specialization of CNNs [35, 65] uses the observation that a specific video stream contains only a small number of object classes and their appearance is more constrained than in a generic video (§2.2.2). Both techniques are done automatically and together result in ingest-time CNN models that are up to 98× cheaper than GT-CNN. 2) Top-K ingest index: The cheap ingest-time CNNs are less accurate, i.e., their top-most results do not often match the top-most classifications of GT-CNN. Therefore, to keep the recall high, Focus associates each object with the top-K classification results of the cheap CNN, instead of just its top-most result. Increasing the K increases recall because the top-most result of GT-CNN often falls within the ingest-time CNN’s top-K results. At querytime, Focus uses the GT-CNN to remove objects in this larger set that do not match the class, to regain precision lost by including all the top-K. 3) Clustering similar objects: A high value of K at ingest-time increases the work to do at query time, thereby increasing query latency. To reduce this overhead, Focus clusters similar objects at ingest-time using feature vectors from the ingest-time CNN. In each cluster, at querytime, we run only the cluster centroid through GT-CNN and apply the classified result from the GT-CNN to all objects in the cluster. Thus, if the objects are not tightly clustered, clustering can reduce precision and recall. 4) Trading off ingest vs. query costs: Focus automatically chooses the cheap CNN, its K, and specialization and clustering parameters to achieve the desired precision and recall targets. These choices also help Focus trade off between the work done at ingest-time and query-time. For instance, to save ingest work, Focus can select a cheaper ingest-time CNN, and then counteract the resultant loss in accuracy by running the expensive GT-CNN on more objects at query time. Focus chooses its parameters so as to offer a sharp improvement in one of the two costs for a small degradation in the other cost. Because the desired trade-off point is application-dependent, Focus provides users with a choice of three options: ingest-optimized, query-optimized, and balanced (the default). Note that while our explanation is anchored on image classification CNNs, the architecture of Focus is generally applicable to all existing CNNs (e.g., face recognition). Techniques that we use for CNN compression [42, 62] and specialization [35], and feature extraction from the CNNs are all broadly applicable to all CNNs. 4. Video Ingest & Querying Techniques In this section, we describe the main techniques used in Focus: using cheap CNN models at ingest-time (§4.1), identifying similar objects and frames to save on redundant CNN processing (§4.2), and specializing the CNNs to the specific videos that are being analyzed (§4.3). §4.4 describes setting parameters in Focus. 4.1. Cheap Ingestion Focus indexes the live videos at ingest-time to reduce the query-time latency. We perform object detection on each frame, typically an inexpensive operation, and then will classify the extracted objects using ingest-time CNNs that are far cheaper than the ground-truth GT-CNN. We use these classifications to index objects by class. Cheap ingest-time CNN: As noted earlier, the user provides Focus with a GT-CNN. Optionally, the user can also provide other classifier architectures to be used in Focus’ search for cheap CNNs, such as AlexNet [45] and 5 CheapCNN 1 (7X) 100% CheapCNN 2 (28X) CheapCNN 3 (58X) class. The selection of the cheap ingest-time CNN model (CheapCNNi ) and the K value (for the top-K results) has a significant influence on the recall of the outputs produced. Lower values of K reduce recall, i.e., Focus will miss returning frames that contain the queried objects. At the same time, higher values of K increase the number of objects to classify with GT-CNN at query time to keep precision high, and hence adds to the latency. We defer to §4.4 on how Focus sets these parameters as they have to be jointly set with other parameters in §4.2 and §4.3. Recall 80% 60% 40% 20% 0% 10 20 60 100 Number of selected results (K) 200 Figure 5: Effect of K on recall for three cheap CNNs. The number within the parenthesis indicates how much cheaper the model is compared to our GT-CNN, ResNet152. VGG [66] (which vary in their resource costs and accuracies). Starting from these user-provided CNNs, Focus applies various levels of compression, such as removing convolutional layers and reducing the input image resolution (§2.1). This results in a large set of CNN options for ingestion, {CheapCNN1 , . . . , CheapCNNn }, with a wide range of costs and accuracies. Top-K Ingest Index: To keep recall high, Focus indexes each object using the top K object classes from CheapCNNi ’s output, instead of using just the top-most class. Recall from §2.1 that the output of the CNN is a list of object classes in descending order of confidence. We empirically observe that the top-most output of the expensive GT-CNN is often contained within the top-K classes output by the cheap CNN (for a small value of K relative to the 1, 000 classes recognized by the CNNs). Figure 5 plots the effect of K on recall on one of our video streams, lausanne (see §6.1). The three models in the figure are ResNet18 [38], and ResNet18 with 3 and 5 layers removed; additionally, the input images were rescaled to 224, 112, and 56 pixels, respectively. All models were retrained on their original training data (ImageNet [63]). We make two observations. First, we observe steady increase in recall with increasing K, for all three CheapCNNs. As the figure shows, CheapCNN1 , CheapCNN2 , and CheapCNN3 reach 90% recall when K ≥ 60, K ≥ 100, and K ≥ 200, respectively. Note that all these models recognize 1000 classes, so even K = 200 represents only 20% of the possible classes. Second, there is a trade-off between different models – the cheaper they are, the lower their recall with the same K. Overall, we conclude that by selecting the appropriate K, Focus can achieve the target recall. Focus creates the top-K index of an object’s top-K classes output by CheapCNNi at ingest-time. While filtering for objects of the queried class X using the top-K index (with the appropriate K) will have a high recall, it will have very low precision. Since we associate each object with K classes (while it has only one true class), the average precision is only 1/K. Thus, at query time, to keep the precision high, Focus determines the actual class of objects from the top-K index using the expensive GT-CNN and only return objects that match the queried 4.2. Redundancy Elimination At query time, Focus retrieves the objects likely matching the user-specified class from the top-K index and infers their actual class using the GT-CNN. This would ensure precision of 100%, but could cause significant latency at query time. Even if this inference is parallelized across many GPUs, it would still incur a large cost. Focus uses the following observation to reduce this cost: if two objects are visually similar, their feature vectors would be closely aligned and they would likely be classified as the same class (e.g., “cars”) by the GT-CNN model (§2.2.3). Focus clusters objects that are similar, invokes the expensive GT-CNN only on the cluster centroids, and assigns the centroid’s label to all objects in each cluster. Doing so dramatically reduces the work done by the GTCNN classifier at query time. Focus uses the feature vector output by the previous-to-last layer of the cheap ingest CNN (see §2.1) for clustering. Note that Focus clusters the objects in the frames and not the frames as a whole. The key questions regarding clustering are how do we cluster (algorithm) and when do we cluster (system). We discuss both these key questions below. Clustering Heuristic: We require two properties in our clustering technique. First, given the high volume of video data, it should be a single-pass algorithm to keep the overhead low, as the complexities of most clustering algorithms are quadratic. Second, it should make no assumption on the number of clusters and adapt to outliers in data points on the fly. Given these requirements, we use the following simple approach for incremental clustering, which has been well-studied in the literature [27, 55]. We put the first object into the first cluster c1 . To cluster a new object i with a feature vector fi , we assign it to the closest cluster c j if c j is at most distance T away from fi . However, if none of the clusters are within a distance T , we create a new cluster with centroid at fi , where T is a distance threshold. We measure distance as the L2 norm [10] between cluster centroid and object feature vector. We keep the number of clusters at a constant M by removing the smallest ones and storing their data in the top-K index. Using this algorithm, we can keep growing the popular clusters (such as similar cars), while keeping 6 the complexity as O(Mn), which is linear to n, the total number of objects. Clustering can reduce both precision and recall depending on parameter T . If the centroid is classified by GT-CNN as the queried class X but the cluster contains another object of a different class, it reduces precision. If the centroid is classified as a class different than X but the cluster has an object of class X, it reduces recall. We discuss setting T in §4.4. Clustering at Ingest vs. Query Time: Focus clusters the objects at ingest-time rather than at query-time. Clustering at query-time would involve storing all feature vectors, loading them for objects filtered from the ingest index and then clustering them. Instead, clustering at ingest time creates clusters right when the feature vectors are created and only stores the cluster centroids in the top-K index. This makes the query-time latency much lower and also reduces the size of the top-K index. We observe that the ordering of indexing and clustering operations is mostly commutative in practice and has little impact on result accuracy (we do not present these results due to space constraints). We therefore use ingest-time clustering due to its latency and storage benefits. Pixel Differencing of Objects: While clustering primarily reduces work done at query-time (number of objects to be classified by the GT-CNN), Focus also employs pixel differencing among objects in adjacent incoming frames to reduce ingest cost. Specifically, if two objects have very similar pixel values, it only runs the cheap CNN on one of them and assign them both to the same cluster in our top-K index. curacy on video streams, while removing 1/3 of the convolutional layers and making the input image 4× smaller in resolution. This leads to the specialized CheapCNNi being 10× cheaper than even the generic CheapCNNi . Since the specialized CNN classifies across fewer classes, they are more accurate, which allows Focus to select a much smaller K (for the top-K ingest index) to meet the desired recall. We find that specialized models can use K = 2 or 4, much smaller than the typical K = 60 ~ 200 for the generic cheap CNNs (Figure 5). Smaller K directly translates to fewer objects that have to be classified by GT-CNN at query time, thus reducing latency. Model Retraining: On each video stream Focus periodically obtains a small sample of video frames and classifies their objects using GT-CNN to estimate the ground truth of distribution of object classes for the video (similar to Figure 3). From this distribution, Focus selects the most frequently occurring Ls object classes to retrain new specialized models. As we saw in §2.2.2, there is usually a “power law” in the distribution of classes – just a handful of classes account for a dominant majority of the objects – thus, low values of Ls usually suffice.1 Specialization is also based off a family of CNN architectures (such as ResNet [38], AlexNet [45], and VGG [66]) with different number of convolution layers, similar to §4.1. Specialization adds to the set of options available for ingest CNNs ({CheapCNN1 , ..., CheapCNNn } in §4.1), and Focus picks the best model (CheapCNNi ) and the corresponding K for the index. “OTHER” class: While Focus specializes the CNN towards the most frequently occurring Ls classes, we also want to support querying of the less frequent classes. For this purpose, Focus includes an additional class called “OTHER” in the specialized model.2 Being classified as OTHER simply means not being one of the Ls classes. At query time, if the queried class is among the OTHER classes of the ingest CNN’s index, Focus extracts all the clusters that match the OTHER class and classifies their centroids through the GT-CNN model. The parameter Ls (for each stream) exposes the following trade-off. Using a small Ls allows us to train a simpler model with cheaper ingest cost and lower query-time latency for the popular classes, however, it also leads to a larger fraction of objects falling in the OTHER class; querying for them will be expensive because all those objects will have to be classified by the GT-CNN. Using a larger value of Ls , on the other hand, leads to a more expensive ingest and query-time models, but cheaper querying for the OTHER classes. We select Ls next in §4.4. 4.3. Video-specific Specialization of CNNs Recall from §4.1 that Focus uses a cheap ingest-time CNN, CheapCNNi to index object classes. Focus further reduces its cost by specializing the ingest-time CNN model to each video stream. Model specialization benefits from two properties of objects in each video stream. First, while object classification models are trained to differentiate between thousands of object classes, many video streams contain only a small number of classes (§2.2.2). Second, objects in a specific stream are often visually more constrained than objects in general (say, compared to the ImageNet [63] dataset). The cars and buses that occur in a specific traffic camera have much less variability, e.g., they have very similar angle, distortion and size, than a generic set of vehicles. Instead of trying to differentiate among thousands of object classes, differentiating among just (say) fifty classes and in a specific camera’s video is a much simpler task, requiring simpler image features and smaller image resolutions. As a result, the specialized models are smaller and more accurate [35]. For example, by retraining a stream-specific CheapCNNi , we can achieve similar ac- 1 Specialized CNNs can be retrained quickly on a small dataset. Retraining is relatively infrequent and done once every few days. 2 Since there will be considerably fewer objects in the video belonging to the OTHER class, we proportionally re-weight the training data to contain equal number of objects of all the classes. 7 Normalized Query Latency 4.4. Balancing Accuracy, Latency, and Cost Focus’ goals of high accuracy, low ingest cost and low query latency are impacted by the parameters in Focus’ techniques – K, the number of top results from the ingesttime CNN to index an object; Ls , the number of popular object classes we use to create a specialized model; CheapCNNi , the specialized ingest-time cheap CNN; and T , the distance threshold for clustering objects. The effect of these four parameters is intertwined. All the four parameters impact ingest cost, query latency, and recall, but only T impacts precision. This is because we apply the cluster centroid’s classification by GT-CNN to all the objects in its cluster. Thus, if the clustering is not tight (i.e., high value of T ), we lose precision. Parameter Selection: Focus selects parameter values per video stream. It samples a representative fraction of frames of the video stream and classifies them using GT-CNN for the ground truth. For each combination of parameter values, Focus computes the expected precision and recall (using the ground truths generated by GT-CNN) that would be achieved for each of the object classes. To navigate the combinatorial space of options for these parameters, we adopt a two-step approach. In the first step, Focus chooses CheapCNNi , Ls and K using only the recall target. In the next step, Focus iterates through the values of T , the clustering distance threshold, and only select values that meet the precision target. Trading off Ingest Cost and Query Latency: Among the combination of values that meet the precision and recall targets, the selection is based on balancing the ingest- and query-time costs. For example, picking a model CheapCNNi that is more accurate will have higher ingest cost, but lower query cost because we can use a lower K. Using a less accurate CheapCNNi will have the opposite effect. Focus identifies “intelligent defaults” that sharply improve one of the two costs for a small worsening of the other cost. Figure 6 illustrates the parameter selection based on the ingest cost and query latency for one of our video streams (auburn_c). The figure plots all the viable “configurations” (i.e., set of parameters that meet the precision and recall target) based on their ingest cost (i.e., cost of CheapCNNi ) and query latency (i.e., the number of clusters according to K, Ls , T ). We first draw the Pareto boundary [17], which is the set of configurations that cannot improve one metric without worsening the other. Focus can discard all the other configurations because at least one point on the Pareto boundary is better than them in both metrics. Focus balances between the ingest cost and query latency (Balance in Figure 6) by selecting the configuration that minimizes the sum of ingest and query cost (measured in total GPU cycles). Focus also allows for other configurations based on the application’s preferences and query rates. Opt-Ingest 0.035 0.030 0.025 0.020 0.015 0.010 Opt-Ingest Opt-Query Balance 0.00 0.05 0.10 Normalized Ingest Cost 0.15 Figure 6: Parameter selection based on trading off ingest cost and query latency. The ingest cost is normalized to ingesting all objects with ResNet152, while the query latency is normalized to the time for querying all objects with ResNet152. The dashed line is the Pareto boundary. minimizes the ingest cost and is applicable when the application expects most of the video streams to not get queried (such as a surveillance cameras), as this policy also minimizes the amount of wasted ingest work. On the other hand, Opt-Query minimizes query latency even if it incurs a heavy ingest cost. Such flexibility allows Focus to fit different applications. 5. Implementation Details We describe the key aspects in Focus’s implementation. Worker Processes. Focus’s ingest-time work is distributed across many machines, with each machine running one worker process for each video stream’s ingestion. The ingest worker receives the live video stream, and extracts the moving objects (using background subtraction [81]); it is extensible to plug in any other object detector. The detected objects are sent to the ingest-time CNN to infer the top-K classes and the feature vectors. The ingest worker uses the features to cluster objects in its video stream and stores the top-K index in MongoDB [12] for efficient retrieval at query-time. Worker processes also serve queries by fetching the relevant frames off the top-K index database and classifying the objects with GT-CNN. We parallelize a query’s work across many worker processes if resources are idle. GPUs for CNN classification. The cheap CNNs and GTCNN execute on GPUs (or other hardware accelerators for CNNs) which could either be local on the same machine as the worker processes or “disaggregated” on a remote cluster. This detail is abstracted away from our worker process and it seamlessly works with both designs. Dynamically adjusting K at query-time. As an enhanced technique, we can select a new Kx ≤ K at query time and only extract clusters where class X appears among the top-Kx classes; this will result in fewer clusters and thus also lower query-time latency. This technique is useful in two scenarios: 1) some classes might be very accurately classified by the cheap CNN; using a lower Kx will still meet the user-specified accuracy, yet will result in much lower latency; 2) if we want to retrieve only some objects of class X, we can use very low Kx to quickly retrieve them. If more objects are required, we can increase 8 Table 1: Video dataset characteristics Kx to extract a new batch of results. Type Description A commercial area intersection in the City of Auburn [6] A residential area intersection auburn_r AL, USA in the City of Auburn [5] A downtown intersection in city_a_d USA City A3 Traffic A residential area intersection city_a_r USA in City A3 A road-side camera in the City bend OR, USA of Bend [8] A busy intersection (Town jacksonh WY, USA Square) in Jackson Hole [9] A video stream rotates among church_st VT, USA cameras in a shopping mall (Church Street Marketplace) [3] A pedestrian plazalatency (Place de Surveillance lausanne Switzerland la Palud) in Lausanne [11] A bookshop street in the oxford England University of Oxford [15] sittard Netherlands A market square in Sittard [4] News channel cnn USA News News channel foxnews USA News channel msnbc USA 6. Evaluation We evaluate the Focus prototype with more than 150 hours of videos from 13 real video streams that span across traffic cameras, surveillance cameras, and news channels. Our highlights are: 1. On average, Focus is simultaneously 58× (up to 98×) cheaper than the Ingest-all baseline in its GPU consumption and 37× (up to 57×) faster than the Query-all baseline in query latency, all the while achieving at least 95% precision and recall (§6.2, §6.3). 2. Focus provides a rich trade-off space between ingest cost and query latency. Among the video streams, the ingest cost is up to 141× cheaper than the Ingestall baseline (and reduces query latency by 46×) if optimizing for low-cost ingest. The query latency is reduced by up to 66× (with 11× cheaper ingest) if optimizing for query latency (§6.4). 3. Focus is effective under broad conditions such as high accuracy targets (one order-of-magnitude savings even for 99% accuracy target, §6.5) and various frame sampling rates (30 fps-1 fps, §6.6). Name Location auburn_c AL, USA in a one-second segment of video if the GT-CNN reports such class in 50% of the frames in that segment. We use this criteria as our ground truth because our GT-CNN (ResNet152) sometimes gives different answers to the exact same object in consecutive frames, and this criteria can effectively eliminate these random, erroneous results. We set our default accuracy target as 95% recall and 95% precision. We also evaluate the results with other accuracy targets such as 97%, 98% and 99% (§6.5). Note that in most practical cases, only one of the two metrics (recall or accuracy) needs to be high. For example, an investigator cares about high recall, and looking through some irrelevant results is an acceptable trade-off. By setting both targets high, we are lower bounding the performance improvements that Focus can achieve. 6.1. Setup Software Tools. We use OpenCV 3.2.0 [14] to decode the videos into frames, and then use the built-in background subtraction algorithm [43] in OpenCV to extract moving objects from video frames. We use background subtraction instead of object detector CNNs (e.g., YOLOv2 [59] or Faster R-CNN [60]) to detect objects because: (1) running background subtraction is orders of magnitude faster than running these CNNs, and (2) background subtraction can detect moving objects more reliably, while object detector CNNs usually have difficulties on small objects [52]. Nonetheless, our system can seamlessly use object detector CNNs as well. We run and train CNNs with Microsoft Cognitive Toolkit 2.1 [54], an open-source deep learning system. Video Datasets. We evaluate 13 live video streams that span across traffic cameras, surveillance cameras, and news channels. We evaluate each video stream for 12 hours, which evenly cover day time and night time. Table 1 summarizes the video characteristics. By default, we evaluate each video at 30 fps and also evaluate the sensitivity to other frame rates (§6.6). In some figures we only show a representative sample of 9 cameras to improve legibility. Accuracy Target. We use ResNet152, a state-of-the-art CNN, as our ground-truth CNN (GT-CNN). We evaluate all extracted objects with the GT-CNN and use the results as the correct answers. We define a class present Baselines. We use two baselines for comparisons: (1) Ingest-all, the baseline system that uses GT-CNN to analyze all objects at ingest time, and stores the inverted index for query; and (2) Query-all, the baseline system that simply extracts objects at ingest time, and uses GT-CNN to analyze all the objects that fall into the query interval at query time. Note that we strengthen both baselines with basic motion detection (background subtraction). Therefore, the baselines do not run any GT-CNN on the frames that have no moving objects. Note that not running GTCNN on frames with no moving objects is one of the core techniques in the recent NoScope work [44]. Metrics. We use two performance metrics. The first metric is ingest cost, which is the GPU time to ingest each video. The second metric is query latency, which is the latency for an object class query. Specifically, for each video stream, we evaluate all dominant object classes and take the average of their latencies. (Querying for non-dominant “OTHER” classes is much cheaper than querying popular classes, and would skew the results because there are far more such classes; thus, we focus on 3 The video streams are obtained from real and operational traffic cameras in a city. We mask the city name for anonymity. 9 Traffic Surveillance News 33X Avg 37X Surveillance msnbc cnn foxnews sittard 11X oxford 22X (auburn_c, city_a_d and jacksonh), normal intersections or roads (auburn_r and city_a_r, bend), rotating cameras (church_st), busy plazas (lausanne and sittard), a university street (oxford), and different news channels (cnn, foxnews, and msnbc). Among these videos, the gains in query latency are smaller for relatively less busy videos (auburn_r, bend, lausanne, and oxford). This is because these videos are dominated by fewer object classes, and Focus has more work (i.e., analysis using GT-CNN) to do at query time for these classes. We conclude that the core techniques of Focus are general and effective on a variety of real-world videos. 56X 58X msnbc 43X 57X 57X 55X lausanne jacksonh 41X church_st bend 24X 30X 64X cnn sittard oxford lausanne jacksonh city_a_d 52X 52X city_a_r 31X city_a_d 56X auburn_r 80 60 40 20 0 auburn_c Faster than Query-all by (factor) Traffic 52X 48X church_st auburn_r bend 44X 44X 44X 53X city_a_r 58X foxnews 98X 94X 86X auburn_c Cheaper than Ingest-all by (factor) 100 80 60 40 20 0 News 6.3. Effect of Different Focus Components Figure 8 shows the breakdown of ingest-time cost and query latency across different design points of Focus: (1) Compressed model, which applies a generic compressed model for indexing at ingest time, (2) Compressed + Specialized model, which uses a per-stream specialized and compressed model for indexing, and (3) Compressed + Specialized model + Clustering, which adds feature-based clustering at ingest time to reduce redundant work at query time. All of the above include the top-K index and using GT-CNN at query-time, and achieve the same accuracy of 95%. Three main observations are in order. First, generic compressed models provide benefits for both ingest cost and query latency, but they are not the major source of improvement. This is because the accuracy of a generic compressed model degrades significantly when we remove convolutional layers. In order to retain the accuracy target, we need to choose relatively expensive compressed models (CheapCNNi ) and a larger K, which incur higher ingest cost and query latency. Second, specializing the model (in addition to compressing it) greatly reduces ingest cost and query latency. Because of fewer convolutional layers and smaller input resolution, our specialized models are 7× to 71× cheaper than the GT-CNN, while retaining the accuracy target for each video streams. Running a specialized model at ingest time speeds up query latency by 5× to 25× (Figure 8b). Third, clustering is a very effective technique to further reduce query latency with unnoticeable costs at ingest time. As Figure 8b shows, using clustering (on top of a specialized compressed model) reduces the query latency by up to 56×, significantly better than just running a specialized model at ingest time. This gain comes with a negligible cost (Figure 8a), because we run our clustering algorithm (§4.2) on the CPUs of the ingest machine, which is fully pipelined with the GPUs that run the specialized CNN model. Avg Figure 7: (Top) Focus ingest cost compared to Ingest-all. (Bottom) Focus query latency compared to Query-all. the popular ones.) Both metrics include only GPU time spent classifying images and exclude other (CPU) time spent decoding video frames, detecting moving objects, recording and loading video, and reading and writing to the top-K index. We focus solely on GPU time because when the GPU is involved, it is the bottleneck resource. The query latency of Ingest-all is 0 and the ingest cost of Query-all is 0. Experiment Platform. We run the experiments on our local cluster. Each machine in the cluster is equipped with a state-of-the-art GPU (NVIDIA GTX Titan X), 16-core Intel Xeon CPU (E5-2698), 64 GB RAM, a 40 GbE NIC, and runs 64-bit Ubuntu 16.04 LTS. 6.2. End-to-End Performance We first show the end-to-end performance of Focus by showing its ingest cost and query latency when Focus aims to balance these two metrics (§4.4). Figure 7 compares the ingest cost of Focus with Ingest-all and the query latency of Focus with Query-all. We make two main observations. First, Focus significantly improves query latency with a very small ingest cost. Focus makes queries by an average of 37× faster than Query-all with a very small cost at ingest time (an average of 58× cheaper than Ingest-all). With a 10-GPU cluster, the query latency on a 24-hour video goes down from one hour to less than two minutes. The processing cost of each video stream also goes down from $250/month to $4/month. This shows that Focus can strike a very good balance between these two competing goals very effectively. Second, Focus is effective across different video streams with various characteristics. It makes queries 11× to 57× faster with a very small ingest time cost (48× to 98× cheaper) across busy intersections 6.4. Ingest Cost vs. Query Latency Trade-off One of the interesting features of Focus is the flexibility to tune its system parameters to achieve different application 10 (a) Ingest cost 0 cnn Opt-I Opt-Q Opt-I Opt-Q Opt-I auburn_c city_a_r jacksonh church_st lausanne sittard Opt-Q Opt-I Opt-Q Opt-I Opt-Q Opt-I Opt-Q Opt-I Opt-Q 0 Opt-I 20 50 Opt-Q 40 100 Opt-I 60 Ingest Cheaper by Query Faster by 150 Opt-Q Improvements (factor) Compressed model + Specialized model + Clustering 80 auburn_c city_a_r jacksonh church_st lausanne sittard cnn foxnews msnbc Avg auburn_c city_a_r jacksonh church_st lausanne sittard cnn foxnews msnbc Avg Faster than Query-all by (factor) Cheaper than Ingest-all by (factor) 100 80 60 40 20 0 Compressed model + Specialized model + Clustering foxnews msnbc Figure 9: Trade-offs between ingest cost and query latency three higher targets, 97%, 98%, and 99%. As the figures show, with higher accuracy targets, the ingest costs are about the same, and the improvement of query latency decreases. Focus keeps the ingest cost similar (62× to 64× cheaper than the baseline) because it still runs the specialized and compressed CNN at ingest time. However, when the accuracy targets are higher, Focus needs to select more top-K classification results, which increases the work at query time. On average, the query latency of Focus is faster than Query-all by 15×, 12×, and 8× with respect to 97%, 98%, and 99% accuracy targets. We conclude that the techniques of Focus can achieve higher accuracy targets with significant improvements on both ingest cost and query latency. (b) Query latency Figure 8: Effect of different Focus components Ingest cheaper by (factor) goals (§4.4). Figure 1 from §1 depicted three alternative settings for Focus that illustrate the trade-off space between ingest cost and query latency, using the auburn_c video stream: (1) Focus-Opt-Query, which optimizes for query latency by increasing ingest cost, (2) Focus-Balance, which is the default option that balances these two metrics (§4.4), and (3): Focus-Opt-Ingest, which is the opposite of Focus-Opt-Query. The results are’shown relative to the two baselines. The chart at the right of the figure is the zoomed-in region that covers the three settings of Focus, and each data label (I, Q) indicates its ingest cost is I× cheaper than Ingest-all, while its query latency is Q× faster than Query-all. As Figure 1 shows, Focus offers very good options in the trade-off space between ingest cost and query latency. Focus-Opt-Ingest achieves 141× cheaper cost than Ingestall to ingest the video stream, and makes the query 46× faster than doing nothing at ingest (Query-all). On the other hand, Focus-Opt-Query reduces query latency by 63× with a relatively higher ingest cost, but it is still 26× cheaper than Ingest-all. As they are all good options compared to the baselines, such flexibility allows a user to tailor Focus for different contexts. For example, a traffic camera that requires fast turnaround time for queries can use FocusOpt-Query, while a surveillance video stream that will be queried very rarely would choose Focus-Opt-Ingest to reduce the amount of wasted ingest cost. Figure 9 shows the (I, Q) values for both Focus-OptIngest (Opt-I) and Focus-Opt-Query (Opt-Q) for the representative videos. As the figure show, the trade-off flexibility exists among all the other videos. On average, Focus-Opt-Ingest spends only 95× cheaper ingest cost to provide 35× query latency reduction. On the other hand, Focus-Opt-Query makes queries 49× faster with a higher ingest cost (15× cheaper than Ingest-all). We conclude that Focus provides good flexibility between ingest cost and query latency, and makes it a better fit in different contexts. 100 95% 97% 98% 99% 10 1 Query faster by (factor) Figure 10: Ingest cost sensitivity to accuracy target 100 95% 97% 98% 99% 10 1 Figure 11: Query latency sensitivity to accuracy target 6.6. Sensitivity to Frame Sampling A common approach to reduce the video processing time is to use frame sampling (i.e., periodically select a frame to process). However, not all applications can use frame sampling because it can miss objects that show up and disappear within a frame sampling window. As the frame sampling rate is an application dependent choice, we study the sensitivity of Focus’s performance to different frame rates. Figures 12 and 13 show the ingest cost and query latency of Focus at different frame rates (i.e., 30 fps, 10 fps, 5 fps, and 1 fps) compared to Ingest-all and Query-all, respectively. We make two observations. First, the ingest cost reduction is roughly the same across the different frame rates. On average, the ingest 6.5. Sensitivity to Accuracy Target Figures 10 and 11 illustrate the improvements of ingest cost and query latency of Focus compared to the baselines under different accuracy targets. Other than the default 95% accuracy target (recall and precision), we evaluate 11 Ingest cheaper by (factor) 100 30fps 10fps 5fps 1fps 7. Related Work 10 To our best knowledge, Focus is the first system that offers low-cost, low-latency,and high-accuracy video queries by balancing between ingest-time cost and query latency. We now discuss work related to our key techniques. 1) Cascaded classification. Various works in vision research propose speeding up classification by cascading a series of classifiers. Viola et al. [75] is the earliest work which cascades a series of classifiers (from the simplest to the most complicated) to quickly disregard regions in an image. Many improvements followed (e.g., [50, 76, 77]). CNNs are also cascaded (e.g., [26, 35, 49, 70]) to reduce object detection latency. Our work is different in two major ways. First, we decouple the compressed CNN from the GT-CNN, which allows us to choose from a wider range for ingest-time CNNs and allows for better trade-offs between ingest cost and query latency, a key aspect of our work. Second, we cluster similar objects using CNN features to eliminate redundant work, which is a new and effective technique for video streams. 2) Neural network compression. Recent work proposes various techniques to reduce the running time of CNNs. These techniques include shallow models [21], predicting weights [33], matrix pruning [31, 37], model quantization [36], and others (e.g., [20, 34, 39, 41, 42, 57, 62]). Our work is largely orthogonal to these, in that our system is not tied to a specific model compression technique, and we can employ any of these techniques. 3) Context-specific model specialization. Contextspecific specialization of models can improve accuracy [53] or reduce running time [35, 44, 65]. Among these, the closest to our work is Kang et al.’s proposal, NoScope [44], which aims to optimize CNN-based video queries. A few key differences stand out. First, NoScope applies all the optimizations at query-time, while Focus adopts a different architecture by splitting work between ingest- and query-time. Thus, Focus trades off higher ingest cost for even lower query latency. Second, NoScope optimizes CNNs for a single class, while we optimize ingest CNNs for all frequent classes in the stream and allow queries even for the rare – OTHER – classes. Finally, we use the object feature vectors to cluster similar objects and create an index to map classes to clusters; this allows us to efficiently query across all classes, while NoScope has to redo all query-time work, including training specialized CNNs, for each query. 4) Stream processing systems. Systems for general stream data processing (e.g., [1,18,19,24,28,29,51,56,73, 74, 79]) and specific to video analytics (e.g., [80]) mainly focus on the general stream processing challenges such as load shedding, fault tolerance, distributed execution, or limited network bandwidth. In contrast, our work is specific for querying on recorded video data with ingest and query trade-offs, thus it is mostly orthogonal to these. 1 Query faster by (factor) Figure 12: Ingest cost sensitivity to frame sampling 100 30fps 10fps 5fps 1fps 10 1 Figure 13: Query latency sensitivity to frame sampling cost of Focus is 62× cheaper than Ingest-all at 30 fps, and it is 64× to 58× cheaper at lower frame rates. This is because the major ingest cost saving comes from the specialized and compressed CNN models (§6.3), which are orthogonal to frame sampling rates. Second, the query latency improvement of Focus degrades with lower frame rates. This is expected because one of our key techniques to reduce query latency is redundancy elimination, especially clustering similar objects using CNN feature vectors. At lower frame rates, the benefit of this technique reduces because there are fewer redundancies. Nonetheless, on average, Focus is still one order of magnitude faster than Query-all at a very low frame rate (1 fps). 6.7. Applicability with Different Query Rate There are two factors that can affect the applicability of Focus: 1) the number of classes that get queried over time and 2) the fraction of videos that get queried. In the first extreme case where all the classes and all the videos are queried, Ingest-all could be a good option because its cost is amortized among all the queries. In our study, even in such an extreme case, the overall cost of Focus is still 4× cheaper than Ingest-all on average (up to 6× cheaper) because we run a very cheap CNN at ingest time, and we run GT-CNN per object cluster only once (§5), so the overall cost is still cheaper than Ingest-all. The second extreme case is only a tiny fraction of videos gets queried. While Focus can save the ingest cost by up to 141× (§6.4), it can be more costly than Query-all if the fraction of videos gets queried is less than 1 141 = 0.7%. In such a case, we can choose to do nothing at ingest time and run all the techniques of Focus only at query time when we know the fraction of videos that get queried. While this approach increases query latency, it still reduces the query latency by a average of 22× (up to 34×) than Query-all in our evaluation. We conclude that Focus is still better than both baselines even under extreme query rates. 12 We can integrate Focus with one of these general stream processing system to build a more fault tolerable system. 5) Video indexing and retrieval. A large body of works in multimedia and information retrieval research propose various content-based video indexing and retrieval techniques to facilitate queries on videos (e.g., [40,48,68,69]). Among them, most works focus on indexing videos for different types of queries such as shot boundary detection [78], semantic video search [30], video classification [25], or spatio-temporal information-based video retrieval [61]. Some works (e.g., [32, 67]) focus on the query interface to enable query by keywords, concepts, or examples. These works are largely orthogonal to our work because we focus on the cost and latency of video queries, not query types or interfaces. We believe our idea of splitting ingest-time and query-time work is generic for videos queries, and can be extended to different types of queries. [5] “City of Auburn North Ross St and East Magnolia Ave.” [Online]. Available: https://www.youtube.com/watch?v= cjuskMMYlLA [6] “City of Auburn Toomer’s Corner Webcam.” [Online]. Available: https://www.youtube.com/watch?v=yJAk_ FozAmI [7] “Genetec,” https://www.genetec.com/. [8] “Greenwood Avenue Bend, Oregon.” [Online]. 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arXiv:1605.03662v2 [math.ST] 21 Jan 2018 Subspace Perspective on Canonical Correlation Analysis: Dimension Reduction and Minimax Rates Zhuang Ma and Xiaodong Li Abstract Canonical correlation analysis (CCA) is a fundamental statistical tool for exploring the correlation structure between two sets of random variables. In this paper, motivated by the recent success of applying CCA to learn low dimensional representations of high dimensional objects, we propose two losses based on the principal angles between the model spaces spanned by the sample canonical variates and their population correspondents, respectively. We further characterize the non-asymptotic error bounds for the estimation risks under the proposed error metrics, which reveal how the performance of sample CCA depends adaptively on key quantities including the dimensions, the sample size, the condition number of the covariance matrices and particularly the population canonical correlation coefficients. The optimality of our uniform upper bounds is also justified by lower-bound analysis based on stringent and localized parameter spaces. To the best of our knowledge, for the first time our paper separates p1 and p2 for the first order term in the upper bounds without assuming the residual correlations are zeros. More significantly, our paper derives p1 ´ λ2k qp1 ´ λ2k`1 q{pλk ´ λk`1 q2 for the first time in the nonasymptotic CCA estimation convergence rates, which is essential to understand the behavior of CCA when the leading canonical correlation coefficients are close to 1. 1 Introduction Canonical correlation analysis (CCA), first introduced by Hotelling (1936), is a fundamental statistical tool to characterize the relationship between two groups of random variables and finds a wide range of applications across many different fields. For example, in genome-wide association study (GWAS), CCA is used to discover the genetic associations between the genotype data of single nucleotide polymorphisms (SNPs) and the phenotype data of gene expression levels (Witten et al., 2009; Chen et al., 2012). In information retrieval, CCA is used to embed both the search space (e.g. images) and the query space (e.g. text) into a shared low dimensional latent space such that the similarity between the queries and the candidates can be quantified (Rasiwasia et al., 2010; Gong et al., 2014). In natural language processing, CCA is applied to the word co-occurrence matrix and learns vector representations of the words which capture the semantics (Dhillon et al., 2011; Faruqui and Dyer, 2014). Other applications, to name a few, include fMRI data analysis (Friman et al., 2003), computer vision (Kim et al., 2007) and speech recognition (Arora and Livescu, 2013; Wang et al., 2015). The enormous empirical success motivates us to revisit the estimation problem of canonical correlation analysis. Two theoretical questions are naturally posed: What are proper error metrics to quantify the discrepancy between population CCA and its sample estimates? And under such metrics, what are the quantities that characterize the fundamental statistical limits? 1 The justification of loss functions, in the context of CCA, has seldom appeared in the literature. From first principles that the proper metric to quantify the estimation loss should depend on the specific purpose of using CCA, we find that the applications discussed above mainly fall into two categories: identifying variables of interest and dimension reduction. The first category, mostly in genomic research (Witten et al., 2009; Chen et al., 2012), treats one group of variables as responses and the other group of variables as covariates. The goal is to discover the specific subset of the covariates that are most correlated with the responses. Such applications are featured by low signal-to-noise ratio and the interpretability of the results is the major concern. In contrast, the second category is investigated extensively in statistical machine learning and engineering community where CCA is used to learn low dimensional latent representations of complex objects such as images (Rasiwasia et al., 2010), text (Dhillon et al., 2011) and speeches (Arora and Livescu, 2013). These scenarios are usually accompanied with relatively high signalto-noise ratio and the prediction accuracy, using the learned low dimensional embeddings as the new set of predictors, is of primary interest. In recent years, there has been a series of publications establishing fundamental theoretical guarantees for CCA to achieve sufficient dimension reduction (Kakade and Foster (2007); Foster et al. (2008); Sridharan and Kakade (2008); Fukumizu et al. (2009); Chaudhuri et al. (2009) and many others). In this paper, we aim to address the problems raised above by treating CCA as a tool for dimension reduction. 1.1 Population and Sample CCA Suppose x “ rX1 , . . . , Xp1 sJ P Rp1 and y “ rY1 , . . . , Yp2 sJ P Rp2 are two sets of variates with the joint covariance matrix ˆ„ ˙  „ Σx Σxy x Cov . (1.1) “ Σ :“ Σy ΣJ y xy For simplicity, we assume EpXi q “ 0, i “ 1, . . . , p1 , EpYj q “ 0, j “ 1, . . . , p2 . On the population level, CCA is designed to extract the most correlated linear combinations between two sets of random variables sequentially: The ith pair of canonical variables Ui “ φJ i x and Vi “ ψiJ y maximizes λi “ CorrpUi , Vi q such that Ui and Vi have unit variances and they are uncorrelated to all previous pairs of canonical variables. Here pφi , ψi q is called the ith pair of canonical loadings and λi is the ith canonical correlation. It is well known in multivariate statistical analysis that the canonical loadings can be found recursively by the following criterion: pφi , ψi q “ arg max φJ Σxy ψ subject to φJ Σx φ “ 1, ψ J Σy ψ “ 1; J J φ Σx φj “ 0, ψ Σy ψj “ 0, @ 1 ď j ď i ´ 1. 2 (1.2) Although this criterion is a nonconvex optimization, it can be obtained easily by spectral methods: Define Φ :“ rφ1 , ¨ ¨ ¨ , φp1 ^p2 s, Ψ :“ rψ1 , ¨ ¨ ¨ , ψp1 ^p2 s and Λ :“ diagpλ1 , ¨ ¨ ¨ , λp1 ^p2 q. Then ´1{2 ´1{2 1{2 1{2 λ1 , . . . , λp1 ^p2 are singular values of Σx Σxy Σy , and Σx Φ, Σy Ψ are actually left and right ´1{2 ´1{2 singular vectors of Σx Σxy Σy , respectively. 1.2 Canonical variables versus canonical loadings pi , ψ pi quk , the For any given estimates of the leading k canonical loadings, denoted by tpφ i“1 corresponding estimates for the canonical variables can be represented by pJ x, pi “ φ U i pJ y, Vpi “ ψ i i “ 1 . . . , p1 ^ p2 . To quantify the estimation loss, generally speaking, we can either focus on measuring the difference pi , ψ pi quk or measuring the difference between between the canonical loadings tpφi , ψi quki“1 and tpφ i“1 pi , Vpi quk . Here x, y in the definition of tpUi , Vi quk the canonical variables tpUi , Vi quki“1 and tpU i“1 i“1 pi , ψ pi quk are constructed. pi , Vpi quk are independent of the samples based on which tpφ and tpU i“1 i“1 Therefore, for the discrepancy between the canonical variables, there is an extra layer of randomness. As discussed above, in modern machine learning applications such as natural language processing and information retrieval, the leading sample canonical loadings are used for dimension reduction, i.e., for a new observation px0 , y0 q, ideally we hope to use the corresponding values of k J k the canonical variables pui “ φJ i x0 qi“1 and pvi “ ψi y0 qi“1 to represent the observation in a low pJ x0 qk dimension space. Empirically, the actual low dimensional representations are pûi “ φ i i“1 pJ y0 qk . Therefore, the discrepancy between the ideal dimension reduction and and pv̂i “ ψ i i“1 pi , Vpi quk approximate tpUi , Vi quk . actual dimension reduction should be explained by how well tpU i“1 i“1 Consequently, we choose to quantify the difference between the sample and population canonical variables instead of the canonical loadings. 1.3 Linear span However, there are still many options to quantify how well the sample canonical variables approximate their population correspondents. To choose suitable losses, it is convenient to come back to specific applications to get some inspiration. Motivated by applications in natural language processing and information retrieval, the model of multi-view sufficient dimension reduction has been studied in Foster et al. (2008). Roughly speaking, a statistical model was proposed by Foster et al. (2008) to study how to predict Z using two sets of predictors denoted by x “ rX1 , . . . , Xp1 sJ and y “ rY1 , . . . , Yp2 sJ , where the joint covariance of pZ, x, yq is ¨» ﬁ˛ » ﬁ Σx Σxy σxz x Σy σyz ﬂ . Cov ˝– y ﬂ‚ “ –ΣJ xy J J Z σz2 σxz σyz It was proven in Foster et al. (2008) that under certain assumptions, the leading k canonical variables U1 , . . . Uk are sufficient dimension reduction for the linear prediction of Z; That is, the best linear predictor of Z based on X1 , . . . , Xp1 is the same as the best linear predictor based on 3 U1 , . . . Uk . (Similarly, the best linear predictor of Z based on Y1 , . . . , Yp2 is the same as the best linear predictor based on V1 , . . . Vk .) Notice that the best linear predictor is actually determined by the set of all linear combinations of U1 , . . . , Uk (referred to as the “model space” in the literature of linear regression for prediction), which we denote as spanpU1 , . . . , Uk q. Inspired by Foster et al. (2008), we propose to quantify the pi uk by the discrepancy between the corresponding subspaces discrepancy between tUi uki“1 and tU i“1 p1 , . . . , U pk q and spanpU1 , . . . , Uk q (and similarly measure the difference between tVi uk and spanpU i“1 tVpi uki“1 by the distance between spanpVp1 , . . . , Vpk q and spanpV1 , . . . , Vk q). 1.4 Hilbert spaces and principal angles xpU,kq “ spanpU p1 , . . . , U pk q and MpU,kq “ In this section, we define the discrepancy between M spanpU1 , . . . , Uk q by introducing a Hilbert space. Noting that for any given sample tpxi , yi quni“1 , xpU,kq and MpU,kq are composed by linear combinations of X1 , . . . , Xp . Denote the set of all both M 1 possible linear combinations as H “ spanpX1 , . . . , Xp1 q. (1.3) Moreover, for any X1 , X2 P H, we define a bilinear function xX1 , X2 y :“ CovpX1 , X2 q “ EpX1 X2 q. It is easy to show that x¨, ¨y is an inner product and pH, x¨, ¨yq is a p1 -dimensional Hilbert space, which is isomorphic to Rp1 . xpU,kq and MpU,kq are With the natural covariance-based inner product, we know both M subspaces of H, so it is natural to define their discrepancy based on their principal angles π 2 ě θ1 ě . . . ě θk ě 0. In the literature of statistics and linear algebra, two loss functions are usually used p1 , . . . , U pk q, spanpU1 , . . . , Uk qq “ sin2 pθ1 q Lmax pspanpU and 1 psin2 pθ1 q ` . . . ` sin2 pθk qq k In spite of a somewhat abstract definition, we have the following clean formula for these two losses: p1 , . . . , U pk q, spanpU1 , . . . , Uk qq “ Lave pspanpU Theorem 1.1. Suppose for any p1 ˆ k matrix A, PA represents the orthogonal projector onto the column span of A. Assume the observed sample is fixed. Then › ›2 › p1 , . . . , U pk q, spanpU1 , . . . , Uk qq “ 1 ››P 1{2 ´ P Lave pspanpU › 1{2 p 1:k Σx Φ1:k F 2k Σx Φ ›2 ¯ 1 ››´ › “ › Ip1 ´ PΣ1{2 Φ PΣ1{2 Φ (1.4) › p x x 1:k 1:k k F “ ‰ 1 p J Q}22 “ min E }uJ ´ u k QPRkˆk p 1:k q :“ Lave pΦ1:k , Φ 4 and › ›2 › p1 , . . . , U pk q, spanpU1 , . . . , Uk qq “ ››P 1{2 ´ P Lmax pspanpU › 1{2 p 1:k Σx Φ1:k Σx Φ ›2 ›´ ¯ › › PΣ1{2 Φ “ › Ip1 ´ PΣ1{2 Φ › p x x 1:k 1:k ”`` ˘ ˘2 ı pJ Q g “ max min E uJ ´ u (1.5) gPRk QPRkˆk p 1:k q. :“ Lmax pΦ1:k , Φ Here Φ1:k “ rφ1 , . . . , φk s is a p1 ˆ k matrix consisting of the leading k population canonical p 1:k is its estimate based on a given sample. Moreover uJ :“ pU1 , . . . , Uk q and loadings for x, and Φ p1 , . . . , U pk q. ûJ :“ pU 1.5 Uniform upper bounds and minimax rates The most important contribution of this paper is to establish sharp upper bounds for the p 1:k q and estimation/prediction of CCA based on the proposed subspace losses Lmax pΦ1:k , Φ p Lave pΦ1:k , Φ1:k q. It is noteworthy that both upper bounds hold uniformly for all invertible Σx , Σy provided n ą Cpp1 ` p2 q for some numerical constant C. Furthermore, in order to justify the sharpness of these bounds, we also establish minimax lower bounds under a family of stringent and localized parameter spaces. These results will be detailed in Section 2. 1.6 Notations and the Organization Throughout the paper, we use lower-case and upper-case non-bolded letters to represent fixed and random variables, respectively. We also use lower-case and upper-case bold letters to represent vectors (which could be either deterministic or random) and matrices, respectively. For any matrix U P Rnˆp and vector u P Rp , }U }, }U }F denotes operator (spectral) norm and Frobenius norm respectively, }u} denotes the vector l2 norm, U1:k denotes the submatrix consisting of the first k columns of U , and PU stands for the projection matrix onto the column space of U . Moreover, we use σmax pU q and σmin pU q to represent the largest and smallest singular value of U respectively, and κpU q “ σmax pU q{σmin pU q to denote the condition number of the matrix. We use Ip for the identity matrix of dimension p and Ip,k for the submatrix composed of the first k columns of Ip . Further, Opm, nq (and simply Opnq when m “ n) stands for the set of m ˆ n matrices with orthonormal columns and Sp` denotes the set of p ˆ p strictly positive definite matrices. For a random vector x P Rp , spanpxJ q “ txJ w, w P Rp u denotes the subspace of all the linear combinations of x. Other notations will be specified within the corresponding context. In the following, we will introduce our main upper and lower bound results in Section 2. To highlight our contributions in the new loss functions and theoretical results, we will compare our results to existing work in the literature in Section 3. All proofs are deferred to Section 4. 2 Theory In this section, we introduce our main results on non-asymptotic upper and lower bounds for estimating CCA under the proposed loss functions. It is worth recalling that λ1 , . . . , λp1 ^p2 are ´1{2 ´1{2 singular values of Σx Σxy Σy . 5 It is natural to estimate population CCA by its sample counterparts. Similar to equation (1.2), the sample canonical loadings are defined recursively by pi , ψ pi q “ arg max pφ p xy ψ φJ Σ p x φ “ 1, ψ J Σ p y ψ “ 1; subject to φJ Σ p x φj “ 0, ψ J Σ p y ψj “ 0, @ 1 ď j ď i ´ 1. φJ Σ (2.1) p x, Σ p y, Σ p xy are the sample covariance matrices. The sample canonical variables are where Σ defined as the following linear combinations by the sample canonical loadings: pJ x, pi “ φ U i pJ y, Vpi “ ψ i i “ 1 . . . , p1 ^ p2 . We prove the following upper bound for the estimate based on sample CCA. „  x Theorem 2.1. (Upper bound) Suppose „ N p0, Σq where Σ is defined as in (1.1). Assume y Σx and Σy are invertible. Moreover, assume λk ą λk`1 for some predetermined k. Then there exist universal positive constants γ, C, C0 such that if n ě Cpp1 ` p2 q, the top-k sample canonical p 1:k satisfies coefficients matrix Φ ﬀ « ” ı 2 p1 ´ λ2k qp1 ´ λ2k`1 q p1 pp ` p q 1 2 p 1:k q ď C0 ` 2 ` e´γpp1 ^p2 q E Lmax pΦ1:k , Φ pλk ´ λk`1 q2 n n pλk ´ λk`1 q4 « ﬀ ” ı p1 ´ λ2k qp1 ´ λ2k`1 q p1 ´ k pp1 ` p2 q2 ´γpp1 ^p2 q p E Lave pΦ1:k , Φ1:k q ď C0 ` 2 `e pλk ´ λk`1 q2 n n pλk ´ λk`1 q4 p 1:k can be obtained by switching p1 and p2 . The upper bounds for Ψ Since we pursue a nonasymptotic theoretical framework for CCA estimates, and the loss functions we propose are nonstandard in the literature, the standard minimax lower bound results in parametric maximum likelihood estimates do not apply straightforwardly. Instead, we turn to the nonparametric minimax lower bound frameworks, particularly those in PCA and CCA; See, e.g., Vu et al. (2013); Cai et al. (2013); Gao et al. (2015). Compared to these existing works, the technical novelties of our results and proofs are summarized in Sections 3.3 and 6. We define the parameter space Fpp1 , p2 , k, λk , λk`1 , κ1 , κ2 q as the collection of joint covariance matrices Σ satisfying 1. κpΣx q “ κ1 and κpΣy q “ κ2 ; 2. 0 ď λp1 ^p2 ď ¨ ¨ ¨ ď λk`1 ă λk ď ¨ ¨ ¨ ď λ1 ď 1. We deliberately set κpΣx q “ κ1 , κpΣy q “ κ2 to demonstrate that the lower bound is independent of the condition number. For the rest of the paper, we will use the shorthand F to represent this parameter space for simplicity. 6 Theorem 2.2. (Lower bound) There exists a universal constant c independent of n, p1 , p2 and Σ such that ¸ + #˜ ” ı p1 ´ λ2k qp1 ´ λ2k`1 q p1 ´ k p ´ k 1 2 p 1:k q ě c ^1^ inf sup E Lmax pΦ1:k , Φ pλk ´ λk`1 q2 n k p 1:k ΣPF Φ #˜ ¸ + ” ı p1 ´ λ2k qp1 ´ λ2k`1 q p1 ´ k p ´ k 1 2 p 1:k q ě c inf sup E Lave pΦ1:k , Φ ^1^ . pλk ´ λk`1 q2 n k p 1:k ΣPF Φ p 1:k can be obtained by replacing p1 with p2 . The lower bounds for Ψ Corollary 2.3. When p1 , p2 ě p2kq _ Cplog nq and něC pp1 ` p2 qp1 ` p2 {p1 q pλk ´ λk`1 q2 p1 ´ λ2k qp1 ´ λ2k`1 q (2.2) for some universal positive constant c, the minimax rates can be characterized by 3 ” ı p1 ´ λ2 qp1 ´ λ2 q p 1 k k`1 p 1:k q — , inf sup E Lmax pΦ1:k , Φ 2 pλ ´ λ q n p k k`1 Φ1:k ΣPF ” ı p1 ´ λ2 qp1 ´ λ2 q p 1 k`1 k p 1:k q — . inf sup E Lave pΦ1:k , Φ 2 pλk ´ λk`1 q n p 1:k ΣPF Φ Related Work and Our Contributions Recently, the non-asymptotic rate of convergence of CCA has been studied by Gao et al. (2015, 2017) under a sparse setup and by Cai and Zhang (2017) under the usual non-sparse setup. Cai and Zhang (2017) appeared on arXiv almost at the same time as the first version of our paper was posted. In this section, we state our contributions by detailed comparison with these works. 3.1 Novel loss funcitons We proposed new loss functions based on the principal angles between the subspace spanned by the population canonical variates and the subspace spanned by the estimated canonical variates. In contrast, Gao et al. (2017) proposed and studied the loss Lave ; Cai and Zhang (2017) proposed Lmax and studied both Lave and Lmax , where ˇ ” ı p 1:k q “ min E }xJ Φ1:k ´ xJ Φ p 1:k Q}2 ˇˇ Φ p 1:k , Lave pΦ1:k , Φ 2 QPOpk,kq „´´  ¯ ¯2 ˇ ˇ J J p 1:k Q g p 1:k q “ max p 1:k . x Φ1:k ´ x Φ Lmax pΦ1:k , Φ min E ˇΦ gPRk ,\|g\|“1 QPOpk,kq Lave and Lmax resemble our loss functions Lave and Lmax respectively. By Theorem 1.1, we also have ˇ ” ı p 1:k q “ 2 min E }xJ Φ1:k ´ xJ Φ p 1:k Q}2 ˇˇ Φ p 1:k Lave pΦ1:k , Φ 2 QPRkˆk  „´´ ¯ ¯2 ˇ ˇ p J Jp p x Φ1:k ´ x Φ1:k Q g Lmax pΦ1:k , Φ1:k q “ max min E ˇ Φ1:k gPRk ,\|g\|“1 QPRkˆk 7 By these two expressions, we can easily obtain p 1:k q p 1:k q ď 2Lave pΦ1:k , Φ Lave pΦ1:k , Φ p 1:k q ď Lmax pΦ1:k , Φ p 1:k q Lmax pΦ1:k , Φ (3.1) p 1:k q and Lave pΦ1:k , Φ p 1:k q are not equivalent up to a constant. Neither are However, Lave pΦ1:k , Φ p p Lmax pΦ1:k , Φ1:k q and Lmax pΦ1:k , Φ1:k q. In fact, we can prove that as long as n ą maxpp1 , p2 q, if λk “ 1 ą λk`1 , then p 1:k q “ Lmax pΦ1:k , Φ p 1:k q “ 0, Lave pΦ1:k , Φ p 1:k q ‰ 0 and Lmax pΦ1:k , Φ p 1:k q ‰ 0. while almost surely Lave pΦ1:k , Φ To illustrate this comparison, very simple simulation: Suppose „ we can consider „the following  „  1 0 1 0 1 0 p1 “ p2 “ 2, n “ 3 and Σx “ and Σy “ and Σxy “ . In this setup, we 0 1 0 1 0 0.5 know the population canonical are λ1 “ 1 and λ2 “ 0.5, and the leading  „  correlation„coefficients 1 1 . In our simulation, we generated the following data and ψ1 “ canonical loadings are φ1 “ 0 0 matrices » ﬁ 0.0736 1.5496 X “ –1.5390 ´0.0415ﬂ 0.9331 ´0.4776 and » ﬁ 0.0736 2.8982 Y “ –1.5390 ´1.2214ﬂ . 0.9331 2.5931 p1 “ 1 and λ p Furthermore, we can obtain the sample canonical correlations λ „  „  2 “ 0.5210, as well as ´0.9616 ´0.9616 p1 “ p1 “ p1 q “ the leading sample canonical loadings φ and ψ . Then Lave pφ1 , φ 0 0 p1 q “ 0 while Lave pφ1 , φ p1 q ‰ 0, Lmax pφ1 , φ p1 q ‰ 0. Lmax pφ1 , φ This numerical example clearly shows that the sample CCA can exactly identify that among all linear combinations of X1 and X2 and all linear combinations of Y1 and Y2 , aX1 and bY1 are mostly correlated. Our loss functions Lave and Lmax do characterize this exact identification, whereas Lave and Lmax do not. Moreover, the following joint loss was studied in Gao et al. (2015): „› ›2  ´ ´ ¯¯ › J J p 1:k Ψ p ´ Φ1:k Ψ ›› . p 1:k , Ψ p 1:k Ljoint pΦ1:k , Ψ1:k q , Φ “ E ›Φ 1:k 1:k F ´ ´ ¯¯ p 1:k , Ψ p 1:k ‰ 0 almost surely under the special case λk “ 1 ą Similarly, Ljoint pΦ1:k , Ψ1:k q , Φ λk`1 . 3.2 Sharper upper bounds Regardless of loss functions, we explain in the following why Theorem 2.1 implies sharper upper bounds than the existing rates in Gao et al. (2015), Gao et al. (2017) and Cai and Zhang (2017) 8 under the nonsparse case. Our discussion is focused on Lave in the following discussion while the discussion for Lmax is similar. Notice that if we only apply Wedin’s sin-theta law, i.e., replacing the fine bound Lemma 5.4 with the rough bound Lemma 5.2 (also see Gao et al. (2015) for similar ideas), we can obtain the following rough bound: „  ” ı p1 ` p2 p 1:k q ď C0 E Lave pΦ1:k , Φ . (3.2) npλk ´ λk`1 q2 p 1:k from p2 , both Gao et al. (2017) and In order to decouple the estimation error bound of Φ Cai and Zhang (2017) assume the residual canonical correlations are zero, i.e., λk`1 “ . . . “ λp1 ^p2 “ 0. This assumption is essential for proofs in both Gao et al. (2017) and Cai and Zhang (2017) under certain sample size conditions. We got rid of this assumption by developing new proof techniques and these techniques actually work for Lave , Lmax as well. A detailed comparison between our result and that in Cai and Zhang (2017) is summarized in Table 3.2 (The results of Gao et al. (2017) in the non-sparse regime can be implied by Cai and Zhang (2017) under milder sample size conditions). Loss function Sample size Cai and Zhang 2016 L Lave q ˆave pě ˙ ? p1 ` p1 p2 p2 nąC ` 4{3 λ2 Upper Bound Rates n ą Cpp1 ` p2 q λk k λk`1 “ ¨ ¨ ¨ “ λp1 “ 0 Our work Lave Yes p1 nλ2k ` No p1´λ2k qp1´λ2k`1 q p1 ´k n pλk ´λk`1 q2 p1 p2 n2 λ4k ` pp1 `p2 q2 2 n pλk ´λk`1 q4 ` e´γpp1 ^p2 q Perhaps the most striking contribution of our upper bound is that we first derive the factors p1 ´ λ2k q and p1 ´ λ2k`1 q in the literature of nonasymptotic CCA estimate. We now explain why these factors are essential when leading canonical correlation coefficients are close to 1. Example 1: λk “ 1 and λk`1 “ 0 Consider the example that k “ 1, p1 “ p2 :“ p " log n, λ1 “ 1 and λ2 “ 0. Then our bound rates p1´λ2k qp1´λ2k`1 q p1 ´k n pλk ´λk`1 q2 ` pp1 `p2 q2 n2 pλk ´λk`1 q4 ` e´γpp1 ^p2 q actually imply that 2 p1 q ď C p , ELave pφ1 , φ n2 while the rates in Gao et al. (2017) and Cai and Zhang (2017) imply that p1 q ď 2ELave pφ1 , φ p1 q ď C p . ELave pφ1 , φ n p1 q, our result could This shows that even under the condition λk`1 “ 0, under our loss Lave pφ1 , φ imply sharper convergence rates than that in Gao et al. (2017) and Cai and Zhang (2017) if λk “ 1. 9 p1 q “ 0 through Notice that as aforementioned, when λk “ 1, we can actually prove ELave pφ1 , φ a separate argument. How to improve Theorem 2.1 to imply this result is an open problem for future research. Example 2: Both λk and λk`1 are close to 1 Consider the example that k “ 1, p1 “ p2 :“ p " log n, λ1 “ 1 ´ our bound rates p1´λ2k qp1´λ2k`1 q p1 ´k n pλk ´λk`1 q2 ` q2 pp1 `p2 n2 pλk ´λk`1 q4 b 4 p n and λ2 “ 1 ´ 2 ` e´γpp1 ^p2 q actually imply that b 4 p n. Then p1 q ď C p , ELave pφ1 , φ n while the rough rates (3.2) by Wedin’s sin-theta law implies c p p ELave pφ1 , φ1 q ď C . n This shows that our upper bound rates could be much sharper than the rough rates (3.2) when both λk and λk`1 are close to 1. New proof techniques and connection to asymptotic theory To the best of our knowledge, none of the analysis in Gao et al. (2015), Gao et al. (2017), Cai and Zhang (2017) can be used to obtain the multiplicative factor p1´λ2k qp1´λ2k`1 q{pλk ´λk`1 q2 in the first order term of the upper bound, even under the strong condition that λk`1 “ ¨ ¨ ¨ “ λp1 ^p2 “ 0. Following a different path, we do careful non-asymptotic entry-wise perturbation analysis of the estimating equations of CCA to avoid the loss of precision caused by applying matrix inequalities in the early stage of the proof. The main challenge is to analyze the properties of matrix hardmard products, especially to derive tight operator norm bounds for certain hardmard products. We are particularly luckily to find a divide-and-conquer approach (λk ě 21 and λk ă 21 in the proof of Lemma 5.4) to decompose the target matrices into simple-structure matrices where we can apply the tools developed in Lemma 5.6. pi , ψ pi qup1 ^p2 has been studied by The asymptotic distribution of the canonical loadings tpφ i“1 Anderson (1999) under the assumption that all the canonical correlations are distinct and λ1 ‰ 1. Since we focus on subspaces, we only require λk ą λk`1 for the given k. Both Anderson (1999) and our work are based on analyzing the estimating equations ((5.5)) of CCA. Our analysis is more involved because completely novel techniques are required to obtain the factor p1 ´ λ2k qp1 ´ λ2k`1 q in the nonasymptotic framework. 3.3 Sharper lower bounds under parameter spaces with fixed λk and λk`1 The minimax lower bounds for the estimation rates of CCA were first established by Gao et al. (2015, 2017) under the losses Ljoint and Lave . However, the parameter space discussed in Gao et al. (2017) requires λk`1 “ 0. Moreover, the parameter space in Gao et al. (2015) is parameterized by λ satisfying λk ě λ, but λk`1 is not specified. In fact, they also constructed the hypothesis class with λk`1 “ 0 and the resulting minimax lower bound is proportional to λ12 . 10 However, this minimax lower bound b is not sharp when λk and λk`1 are close. Suppose p1 “ 1 1 p2 :“ p, k “ 1, λ1 “ 2 and λ2 “ 2 ´ np . Our minimax lower bound in Theorem 2.2 leads to ” ı p 1:k q ě Op1q. inf sup E Lave pΦ1:k , Φ p 1:k ΣPF Φ In contrast, to capture the fundamental limit of CCA estimates in this scenario under the framework of Gao et al. (2015), one needs to choose λ to capture both λk and λk`1 , i.e., λk ď λ ď λk`1 and hence λ « 1{2. Then the resulting minimax lower bound rate will be nλp 2 “ Op np q, which is much looser than Op1q. Technically speaking, we follow the analytical framework of Gao et al. (2015) and Gao et al. (2017), but the hypothesis classes construction requires any given λk`1 ą 0 instead of λk`1 “ 0, and this brings in new technical challenges. More detailed technical discussions are deferred to Section 6. 4 Proof of Theorem 1.1 Suppose the observed sample of px, yq is fixed and consider the correlation between p1 , . . . , U pk q. Let the two subspaces of H (defined in (1.3)): spanpU1 , . . . , Uk q and spanpU x x x pW1 , W1 q, pW2 , W2 q, . . . , pWk , Wk q be the first, second, ..., and kth pair of canonical p1 , . . . , U pk . variates between U1 , . . . , Uk and U Then spanpW1 , . . . , Wk q “ spanpU1 , . . . , Uk q, x x p p xj y “ xW xi , W xj y “ 0, for any i ‰ j and spanpW1 , . . . , Wk q “ spanpU1 , . . . , Uk q and xWi , Wj y “ xWi , W xi q “ 1, for i “ 1, . . . , k. VarpWi q “ VarpW xi q is actually the ith principal angle By the definition of principal angles, we know =pWi , W p p xi q. This implies that between spanpU1 , . . . , Uk q and spanpU1 , . . . , Uk q, i.e., θi :“ =pWi , W p 1:k q :“ Lave pΦ1:k , Φ k ÿ i“1 2 sin θi “ k ˆ ÿ i“1 ˇA Eˇ2 ˙ ˇ ˇ x 1 ´ ˇ Wi , Wi ˇ . p1 , . . . , U pk are linear combinations of X1 , . . . , Xp , we can denote Since U1 , . . . , Uk , U 1 p x1 , . . . , W xk q “ xJ Σx´1{2 B, w J :“ pW1 , . . . , Wk q “ xJ Σx´1{2 B, and ŵ J :“ pW p :“ rp where B :“ rb1 , . . . , bk s, B b1 , . . . , p bk s P Rpˆk . By the definition of w, we have Ik “ Covpwq “ B J Σx´1{2 CovpxqΣx´1{2 B “ B J B p J B. p Then B, B p are p ˆ k basis matrices. Moreover, we have bJp and similarly Ik “ B i bj “ xj y “ 0, for all i ‰ j. Moreover, we have xWi , W p “ B J B. p Diagpcospθ1 q, . . . , cospθk qq “ Covpw, ŵq “ B J Σx´1{2 CovpxqΣx´1{2 B Notice that spanpU1 , . . . , Uk q “ spanpW1 , . . . , Wk q, pU1 , . . . , Uk q “ xJ Φ1:k , and pW1 , . . . , Wk q “ ´1{2 xJ Σx B. Then Φ1:k “ Σx´1{2 BC ñ Σ1{2 x Φ1:k “ BC 11 1{2 for some nonsingular k ˆ k matrix C. This implies that B and Σx Φ1:k have the same column space. Since B P Rpˆk is a basis matrix, we have BB J “ PΣ1{2 Φ . x Similarly, we have 1:k pB p J “ P 1{2 . B p Σ Φ x 1:k Straightforward calculation gives › ›2 ¯ ´ › pB pJ ´ B pB p J BB J ` B pB p JB pB pJ pB p J ›› “ trace BB J BB J ´ BB J B ›BB J ´ B F pB p J Bq “ 2k ´ 2tracepB J B “ 2k ´ 2tracepDiagpcos2 pθ1 q, . . . , cos2 pθk qqq p 1:k q “ 2psin2 pθ1 q ` . . . ` sin2 pθk qq “ 2kLave pΦ1:k , Φ and ›` ›2 ´` ˘ ˘ ` ˘¯ › pB p JB pB p J ›› “ trace Ip ´ BB J B pB p J Ip ´ BB J › Ip1 ´ BB J B 1 1 F pB p J Bq “ k ´ tracepB J B p 1:k q. “ kLave pΦ1:k , Φ The above equalities yield the first two equalities in (1.4). Notice that both U1 , . . . , Uk and W1 , . . . Wk are both orthonormal bases of spanpU1 , . . . , Uk q. p1 , . . . , U pk and W x1 , . . . W xk are both orthonormal bases of spanpU p1 , . . . , U pk qq.) Then we (Similarly, U J J have u “ w R where R is a k ˆ k orthogonal matrix. Then min E}uJ ´ ûJ Q}22 “ min E}uJ ´ ŵ J Q}22 “ min E}w J R ´ ŵ J Q}22 QPRkˆk QPRkˆk QPRkˆk J “ min E}w ´ ŵ QPRkˆk “ “ “ E min i“1 k ÿ qi PRk , i“1,...,k i“1 i“1 QRJ }22 “ min E}w J ´ ŵ J Q}22 QPRkˆk pWi ´ ŵ J qi q2 min qi PRk , i“1,...,k k ÿ k ÿ J EpWi ´ ŵ J qi q2 min EpWi ´ ŵ J qi q2 qi PRk Notice that minqi PRk EpWi ´ ŵ J qi q2 is obtained by the best linear predictor, so min EpWi ´ ŵ J qi q2 “ VarpWi q ´ Covpŵ, Wi qJ Cov´1 pŵqCovpŵ, Wi q qi PRk “ 1 ´ cos2 θi “ sin2 θi . 12 Therefore, min E}uJ ´ ûJ Q}22 “ QPRkˆk k ÿ i“1 p 1:k q, sin2 θi “ kLave pΦ1:k , Φ which implies the third equality in (1.4). Similarly, max min E gPRk ,}g}“1 QPRkˆk “ “ “ “ “ `` J ˘ ˘2 u ´ ûJ Q g max min E max min E gPRk ,}g}“1 QPRkˆk gPRk ,}g}“1 max gPRk ,}g}“1 max gPRk ,}g}“1 max gPRk ,}g}“1 2 “ sin θ1 QPRkˆk min E QPRkˆk qi `` J ˘ ˘2 w R ´ ŵ J Q RJ g `` J ˘ ˘2 w ´ ŵ J Q g min PRk , k ÿ `` J ˘ ˘2 u ´ ŵ J Q g E i“1,...,k k ÿ i“1 gi2 pWi ´ ŵ J qi q2 gi2 sin2 θi i“1 Finally, we prove (1.5). By Wedin (1983), we have › ›2 ›` ›2 ›` ˘ ˘ ››2 › pB p J ›› “ ›› Ip ´ BB J B pB p J ›› “ ›› Ip ´ BB J B p› ›BB J ´ B 1 1 ´ ` ˘ ` ˘ ¯ p J Ip ´ BB J J Ip ´ BB J B p “ λmax B 1 1 ` ˘ “ λmax Ik ´ Diagpcos2 pθ1 q, . . . , cos2 pθk qq p 1:k q, “ 1 ´ cos2 pθ1 q “ sin2 pθ1 q “ Lmax pΦ1:k , Φ which implies the the equalities in (1.5). 5 Proof of Upper Bound Throughout this proof, we denote ∆ :“ λk ´ λk`1 . 5.1 Linear Invariance Without loss of generality, we assume p2 ě p1 :“ p. By the definition of canonical variables, we know that U1 , . . . , Up and V1 , . . . , Vp are only determined by spanpX1 , . . . , Xp1 q and spanpY1 , . . . , Yp2 q. In other words, for any invertible C1 P Rp1 ˆp1 and C2 P Rp2 ˆp2 , the canonical pairs of pX1 , . . . , Xp1 qC1 and pY1 , . . . , Yp2 qC2 are still pU1 , V1 q, . . . , pUp1 , Vp1 q. Therefore, we can consider the following orthonormal bases U1 , . . . , Up1 P spanpX1 , . . . , Xp1 q 13 and V1 , . . . , Vp1 , Vp1 `1 , . . . , Vp2 P spanpY1 , . . . , Yp2 q. Here pV1 , . . . , Vp1 , Vp1 `1 , . . . , Vp2 q is an orthonormal extension of V1 , . . . , Vp1 . Therefore, we know that pU1 , V1 q, . . . , pUp1 , Vp1 q are also the the canonical pairs between U1 , . . . , Up1 and V1 , . . . , Vp2 . Similarly, for a fixed sample of the variables of x and y, the sample canonical pairs p pp , Vpp q are also sample canonical pairs of the corresponding sample of pU1 , Vp1 q, . . . , pU 1 1 pX1 , . . . , Xp1 qC1 and pY1 , . . . , Yp2 qC2 . This can be easily seen from the concept of sample p1 and Vp1 are respectively the linear combinations of canonical variables. For example, U X1 , . . . , Xp1 and Y1 , . . . , Yp1 , such that their corresponding sample variance are both 1 and sample correlation is maximized. If we replace pX1 , . . . , Xp1 q and pY1 , . . . , Yp1 q with pX1 , . . . , Xp1 qC1 and pY1 , . . . , Yp2 qC2 respectively and seek for the first sample canonical pair, the constraints (linear combinations of the two sets of variables and unit sample variances) and the objective p1 , Vp1 q is still the answer. Similarly, (sample correlation is maximized) are the same as before, so pU p p p p pU1 , V1 q, . . . , pUp1 , Vp1 q are the sample canonical pairs of pX1 , . . . , Xp1 qC1 and pY1 , . . . , Yp2 qC2 . In particular, they are the sample canonical pairs of U1 , . . . , Up1 and V1 , . . . , Vp2 . The above argument gives the following convenient fact: In order to bound p1 , . . . , U pk q, spanpU1 , . . . , Uk qq Lave{max pspanpU we can replace X1 , . . . , Xp1 , Y1 , . . . , Yp2 with U1 , . . . , Up1 , V1 , . . . , Vp2 . In other words, we can assume x and y satisfy the standard form r Σx “ Ip1 , Σy “ Ip2 , Σxy “ rΛ, 0p1 ˆpp2 ´p1 q s :“ Λ where Λ “ Diagpλ1 , λ2 , . . . , λp1 q P Rp1ˆp1 . Moreover Φ1:p1 “ Ip1 , Ψ1:p1 “ which implies that Φ1:k “ 5.2 „ Ik 0pp1 ´kqˆk  „ Ip1 0pp2 ´p1 qˆp1 , Ψ1:k “ „  , Ik 0pp2 ´kqˆk  . Upper Bound Under the Standard Form Under the standard form, by (1.4) and (1.5), we have and ›2 › p1 , . . . , U pk q, spanpU1 , . . . , Uk qq “ 1 ››pIp ´ PΦ q P p ›› Lave pspanpU 1 1:k Φ1:k F k ›2 › › › p p Lmax pspanpU1 , . . . , Uk q, spanpU1 , . . . , Uk qq “ ›pIp1 ´ PΦ1:k q PΦ p 1:k › . 14 (5.1) (5.2) p 1:k Denote Φ « ﬀ pu Φ p u and Φ p l are the upper k ˆ k and lower pp1 ´ kq ˆ k sub“ p l1:k where Φ 1:k 1:k Φ1:k p 1:k respectively. Then matrices of Φ ›2 › ´ ¯ › › J p ´1 p J p p “ trace pI ´ P q Φ p Φ Φ q Φ pI ´ P q , › ›pIp1 ´ PΦ1:k q PΦ p Φ p Φ 1:k 1:k p 1:k 1 1 1:k 1:k 1:k 1:k F › ›2 ¯ ´ › › ´1 p J p 1:k pΦ pJ Φ p ›pIp1 ´ PΦ1:k q PΦ › “ λmax pIp1 ´ PΦ1:k q Φ p 1:k 1:k q Φ1:k pIp1 ´ PΦ1:k q 1:k Since ´1 p J p 1:k pΦ pJ Φ p pIp1 ´ PΦ1:k q Φ 1:k 1:k q Φ1:k pIp1 ´ PΦ1:k q  „ ı 0kˆk ” 1 1 J p p l qJ , p pIp1 ´ PΦ1:k q Φ1:k Φ1:k pIp1 ´ PΦ1:k q “ ĺ 0 p Φ kˆk pl 1:k p 1:k q p 1:k q Φ σk2 pΦ σk2 pΦ 1:k we have ›2 › › › pI ´ P q P › p1 Φ1:k p 1:k › ď trace Φ F and ›2 › › › ›pIp1 ´ PΦ1:k q PΦ p 1:k › ď λmax ˜ ˜ ¸  „ ı p l }2 }Φ 1 0kˆk ” 1:k F pJ “ , 0 Φ kˆk l 1:k p 2 2 p p 1:k q Φ σk pΦ1:k q σk pΦ 1:k ¸  „ ı p l }2 }Φ 0kˆk ” 1 1:k J p . “ Φ 0 kˆk 1:k pl 2 pΦ p 1:k q Φ p 1:k q σk2 pΦ σ 1:k k (5.3) (5.4) p l }2 , as well as a lower bound of p l }2 and }Φ Therefore, it suffices to give upper bounds of }Φ 1:k 1:k F p 1:k q. σk2 pΦ 5.3 Basic bounds Recall that Then and r Σx “ Ip1 , Σy “ Ip2 , Σxy “ rΛ, 0p1 ˆpp2 ´p1 q s :“ Λ. « ﬀ ˆ„ ˙ r x Ip1 Λ Cov :“ Σ “ r J y Λ Ip2 « ﬀ ˆ„ ˙ px Σ p xy x Σ y p “ Cov :“ Σ py . p yx Σ y Σ p 2p as the left upper p2p1 q ˆ p2p1 q principal submatrix of Σ. p We can Moreover, we can define Σ 1 similarly define Σ2p1 . Lemma 5.1. There exist universal constants γ, C and C0 such that when n ě C0 p1 , then with probability at least 1 ´ e´γp1 , the following inequalities hold c › › p1 › › p 1{2 p p . }Σ2p1 ´ Σ2p1 }, }Ip1 ´ Σx }, ›Σx ´ Ip1 › ď C n 15 Proof. It is obvious that }Σ2p1 } ď 2. By Lemma 5.9, there exist constants γ, C0 and C1 , such that when n ě C0 p1 , with probability at least 1 ´ e´γp1 there holds c p1 p }Σ2p1 ´ Σ2p1 } ď C1 . n b p x } ď C1 p1 . Moreover, As submatrices, we have }Ip1 ´ Σ n p x } “ }pIp ´ Σ p 1{2 qpIp ` Σ p 1{2 q} ě σmin pIp ` Σ p 1{2 q}Ip ´ Σ p 1{2 } ě }Ip ´ Σ p 1{2 }, }Ip1 ´ Σ 1 x 1 x 1 x 1 x 1 x b p1 `p2 p 1{2 which implies }Ip1 ´ Σ x } ď C1 n . Lemma 5.2. There exist universal constants c, C and C0 such that when n ě C0 pp1 ` p2 q, then with probability at least 1 ´ e´cpp1 `p2 q , the following inequalities hold c › › p1 ` p2 › › p 1{2 p p p }Σ ´ Σ}, }Ip2 ´ Σy }, }Σxy ´ Σxy }, ›Σy ´ Ip2 › ď C , n c p1 ` p2 ´1{2 p ´1{2 p p p , }Λ ´ Λ} ď }Σx Σxy Σy ´ Σxy } ď C n p 1:k }2 ď 3 , σ 2 pΨ p 1:k q ě 1 , }Ψ p 1:k }2 ď 3 , p 1:k q ě 1 , }Φ σk2 pΦ k 2 2 2 2 c C p ` p 1 2 pl } ď p l }, }Ψ }Φ , 1:k 1:k ∆ n where ∆ “ λk ´ λk`1 is the eigen-gap. The proof is deferred to Section 5.7. 5.4 p l }2 Estimating Equations and upper bound of }Φ 1:k p l }2 . Notice that we have already In this section, we aim to give a sharp upper bound for }Φ 1:k established an upper bound in Lemma 5.2, where Wedin’s sin θ law plays the essential role. However, this bound is actually too loose for our purpose. Therefore, we need to develop new techniques to sharpen the results. p P Rp1 ˆp1 , Ψ p P Rp2 ˆp1 consist of the sample canonical coefficients. By definition, Recall that Φ p x1{2 Φ p and the sample canonical coefficients satisfy the following two estimating equations (because Σ 1{2 ´1{2 ´1{2 py Ψ p are left and right singular vectors of Σ px Σ p xy Σ py Σ respectively), If we define define Λ“ „ Λ1 Λ2  p xy Ψ p “Σ p xΦ pΛ p Σ p yx Φ p “Σ p yΨ p Λ. p Σ PR p1 ˆp1 p “ , Λ 16 « p1 Λ (5.5) ﬀ P Rp1 ˆp1 , p Λ2 (5.6) p 1 are k ˆ k diagonal matrices while Λ2 , Λ p 2 are pp1 ´ kq ˆ pp1 ´ kq diagonal matrices. where Λ1 , Λ Then (5.5) imply p xy Ψ p 1:k “ Σ p xΦ p 1:k Λ p1 Σ (5.7) p yx Φ p 1:k “ Σ p yΨ p 1:k Λ p 1. Σ Divide the matrices into blocks, ﬀ ﬀ « ﬀ « ﬀ « « p 11 Σ p 12 p 11 Σ p 12 p 11 Σ p 12 p 11 Σ p 12 Σ Σ Σ Σ xy xy yx yx y y x x py “ p xy “ p px “ , Σ , Σ Σ p 22 p 21 p 22 p 21 p 22 , Σyx “ Σ p 21 p 22 p 21 Σ Σ Σ Σ Σ x y y xy Σxy yx Σyx x kˆk , Ψ p l P Rpp2 ´kqˆk in p 11 p 11 p 11 pu p 11 where Σ x , Σy , Σxy , Σyx are k ˆ k matrices. Finally, we define Ψ1:k P R 1:k pu ,Φ p l . With these blocks, (5.7) can be rewritten as the same way as Φ 1:k 1:k p 21 p u p p 22 p l p p 22 p l p 21 Ψ pu Σ xy 1:k ` Σxy Ψ1:k “ Σx Φ1:k Λ1 ` Σx Φ1:k Λ1 , p 21 Φ pu p 22 p l p 21 p u p p 22 p l p Σ yx 1:k ` Σyx Φ1:k “ Σy Ψ1:k Λ1 ` Σy Ψ1:k Λ1 , p 11 Φ pu Λ p1 ` Σ p 12 Φ pl Λ p 1, p 12 Ψ pl “ Σ p 11 Ψ pu ` Σ Σ xy 1:k p pu Σ11 yx Φ1:k Define the zero-padding of Λ2 : ` xy 1:k p pl Σ12 yx Φ1:k “ x 1:k 11 p u p p Σy Ψ1:k Λ1 ` (5.8) (5.9) (5.10) x 1:k 12 p l p p Σy Ψ1:k Λ1 . (5.11) r 2 :“ rΛ2 , 0s “ Σ22 P Rpp1 ´kqˆpp2 ´kq . Λ xy The above equations imply the following lemma: Lemma 5.3. The equality (5.7) gives the following result pu pl p l Λ2 ´ Λ2 Φ Φ 2 1:k “ B Φ1:k ` R 1:k 1 r r 2 pΣ p 21 ´ Σ pu ` R p 21 Λ1 qΦ p 21 ´ Σ p 21 Λ1 qΨ p u Λ1 ` Λ “ pΣ yx y xy x 1:k 1:k where (5.12) (5.13) p 21 Λ1 ` Λ r 2Σ p 21 ´ Σ p 21 Λ2 ´ Λ r 2Σ p 21 Λ1 , B :“ Σ xy yx x 1 y 21 21 r p r p R :“ pΣ R1 ´ R3 qΛ1 ´ Λ2 pΣ R2 ` R4 q, x y r ´ pΣ p 21 ´ Σ p 21 Λ1 qR2 . R :“ R x xy and p 12 p l p 1 ´ Λ1 q ` pΣ p 11 ´ Ik qΦ pu Λ p p 12 p l p p 11 pu p u pΛ R1 :“ Φ x 1:k 1 ` Σx Φ1:k Λ1 ´ pΣxy ´ Λ1 qΨ1:k ´ Σxy Ψ1:k , 1:k p 12 Φ pl , pu ´ Σ pu Λ p1 ` Σ p 12 Ψ pl Λ p 1 ´ pΣ p 11 ´ Λ1 qΦ p 1 ´ Λ1 q ` pΣ p 11 ´ Ik qΨ p u pΛ R2 :“ Ψ R3 :“ R4 :“ 1:k 1:k y 1:k yx y 1:k 22 l 22 l l 21 u r p p p 1 ´ Λ1 q ` pΣ p Φ p Λ p p p Φ p pΛ Σ x x 1:k 1 ´ Φ1:k Λ1 q ´ pΣxy ´ Λ2 qΨ1:k , 1:k p 22 r J p l p 22 p l p pl p 21 pu p Σ y Ψ1:k pΛ1 ´ Λ1 q ` pΣy Ψ1:k Λ1 ´ Ψ1:k Λ1 q ´ pΣyx ´ Λ2 qΦ1:k . 17 yx 1:k The proof is deferred to Section 5.7. By Lemma 5.2, one can easily obtain that }R1 }, }R2 } ď C c p1 ` p2 . n Recall that p 21 Φ p u pΛ p 1 ´ Λ1 q ` pΣ p 22 Φ pl Λ p pl p 22 r p l R3 :“ Σ x 1:k x 1:k 1 ´ Φ1:k Λ1 q ´ pΣxy ´ Λ2 qΨ1:k By Lemma 5.2, we have and p 21 Φ p u pΛ p 1 ´ Λ1 q} ď C p1 ` p2 , }pΣ p 22 ´ Λ r 2 qΨ p l } ď C p1 ` p2 , }Σ x xy 1:k 1:k n ∆n p 22 Φ pl Λ p pl p 22 pl p pl p }Σ x 1:k 1 ´ Φ1:k Λ1 } ď }pΣx ´ Ip1 ´k qΦ1:k Λ1 ` Φ1:k pΛ1 ´ Λ1 q} p 22 ´ Ip ´k qΦ pl Λ p pl p ď }pΣ x 1:k 1 } ` }Φ1:k pΛ1 ´ Λ1 q} ď C 1 p1 ` p2 . ∆n `p2 `p2 . Similarly, }R4 } ď C p1∆n . Therefore, we get }R3 } ď C p1∆n Combined with Lemma 5.2, we have and r “ }pΣ p 21 R1 ´ R3 qΛ1 ´ Λ r 2 pΣ p 21 R2 ` R4 q} ď C p1 ` p2 }R} x y ∆n r ` }Σ p 21 ´ Σ p 21 Λ1 }}R2 } ď C p1 ` p2 . }R} ď }R} xy x ∆n The proof of the following lemma is deferred to Section 5.7: Lemma 5.4. If n ě C0 pp1 ` p2 q, then with probability 1 ´ c0 expp´γp1 q, »d ﬁ 2 qp1 ´ λ2 q p p1 ´ λ pp ` p q 1 1 2 ﬂ k k`1 pl } ď C – }Φ ` . 1:k n∆2 n∆2 5.5 Upper bounds of risks Notice that the inequality (5.4) yields ›2 › p l }2 }Φ › › 1:k . ›pIp1 ´ PΦ1:k q PΦ p 1:k › ď 2 p 1:k q σk pΦ By Lemma 5.4 and Lemma 5.2, we know on an event G with probability at least 1 ´ Ce´γp1 , ﬀ « › ›2 p1 p1 ´ λ2k qp1 ´ λ2k`1 q pp1 ` p2 q2 › › ` . ›pIp1 ´ PΦ1:k q PΦ p 1:k › ď C n∆2 n 2 ∆4 18 ›2 › › › Moreover, since ›pIp1 ´ PΦ1:k q PΦ p 1:k › ď 1, by (5.2), we have « ﬀ › ›2 p1 p1 ´ λ2k qp1 ´ λ2k`1 q pp1 ` p2 q2 › › p 1:k q “ E ›pIp ´ PΦ q P p › ď C ELmax pΦ1:k , Φ ` ` e´γp1 . 1 1:k Φ1:k n∆2 n 2 ∆4 Since pIp1 ´ PΦ1:k q PΦ p 1:k is of at most rank-k, we have › ›2 ›2 1 ›› › › › ď pI ´ P q P › ›pIp1 ´ PΦ1:k q PΦ › p1 Φ1:k p 1:k › p 1:k Φ k F Then by (5.1) and the previous inequality, we have › ›2 › › p ELave pΦ1:k , Φ1:k q “ E ›pIp1 ´ PΦ1:k q PΦ p 1:k › ›2 1 ›› › “ E ›pIp1 ´ PΦ1:k q PΦ p 1:k › k F ›2 › › › ď E ›pIp1 ´ PΦ1:k q PΦ p 1:k › ﬀ « p1 p1 ´ λ2k qp1 ´ λ2k`1 q pp1 ` p2 q2 ` ` e´γp1 . ďC n∆2 n 2 ∆4 In fact, the factor p1 in the main term can be reduced to p1 ´ k by similar arguments as done for the operator norm. The Frobenius norm version of Lemma 5.4 is actually much simpler. We omit the proof to avoid unnecessary redundancy and repetition. 5.6 Supporting lemmas in linear algebra and probability Definition 5.5. (Hadamard Operator Norm) For A P Rmˆn , define the Hadamard operator norm as ( \|\|\|A\|\|\| “ sup }A ˝ B} : }B} ď 1, B P Rmˆn Let α1 , ¨ ¨ ¨ , αm and β1 , ¨ ¨ ¨ , βn be arbitrary positive numbers lower bounded by a positive constant δ. n mˆn , Lemma 5.6. Let tαi um i“1 and tβi ui“1 be two sequences of positive numbers. for any X P R there hold › ›« a ﬀ › 1 › α β i j › › ˝ X › ď }X}, (5.14) › › 2 › αi ` βj and ›„ ›  › minpαi , βj q › 1 › ˝ X ›› ď }X}, › αi ` βj 2 ›„ ›  › maxpαi , βj q › 3 › ˝ X ›› ď }X}. › αi ` βj 2 (5.15) Proof. The proof of (5.14) can be found in “Norm Bounds for Hadamard Products and an Arithmetic-Geometric Mean Inequality for Unitarily Invariant Norms” by Horn. Denote  „  minpαi , βj q maxpαi , βj q , G2 “ G1 “ αi ` βj αi ` βj „ The proof of (5.15) relies on the following two results. 19 Lemma 5.7. (Theorem 5.5.18 of Hom and Johnson (1991)) If A, B P Rnˆn and A is positive semidefinite. Then, ˙ ˆ }A ˝ B} ď max Aii }B}, 1ďiďn where } ¨ } is the operator norm. Lemma 5.8. (Theorem 3.2 of Mathias (1993)) The symmetric matrix ´ minpa , a q ¯ i j ai ` aj 1ďi,jďn is positive semidefinite if ai ą 0, 1 ď i ď n. Define γi “ βi , 1 ď i ď n and γi “ αi´n , n ` 1 ď i ď m ` n. Define M P Rpm`nqˆpm`nq by Mij “ mintγi , γj u . γi ` γj By Lemma 5.8, M is also positive semidefinite. Again, apply Lemma 5.7 and notice that G2 is the lower left sub-matrix of M , It is easy to obtain \|\|\|G2 \|\|\| ď \|\|\|M \|\|\| ď 1 . 2 Finally, since G1 ˝ B “ B ´ G2 ˝ B for any B, we have }G1 ˝ B} ď }B} ` }G2 ˝ B}, which implies, 3 \|\|\|G1 \|\|\| ď 1 ` \|\|\|G2 \|\|\| ď . 2 Lemma 5.9. (Covariance Matrix Estimation, Remark 5.40 of Vershynin (2010)) Assume A P Rnˆp has independent sub-gaussian random rows with second moment matrix Σ. Then there exists universal constant C such that for every t ě 0, the following inequality holds with probability at 2 least 1 ´ e´ct , c 1 J t p 2 } A A ´ Σ} ď maxtδ, δ u}Σ} δ“C `? . n n n Lemma 5.10. (Bernstein inequality, Proposition 5.16 of Vershynin (2010)) Let X1 , ¨ ¨ ¨ , Xn be independent centered sub-exponential random variables and K “ maxi }Xi }ψ1 . Then for every a “ pa1 , ¨ ¨ ¨ , an q P Rn and every t ě 0, we have + # " ˆ ˙* n ÿ t t2 , ai Xi \| ě t ď 2exp ´c min . P \| K 2 }a}22 K}a}8 i“1 20 Lemma 5.11. (Hanson-Wright inequality, Theorem 1.1 of Rudelson and Vershynin (2013)) Let x “ px1 , ¨ ¨ ¨ , xp q be a random vector with independent components xi which satisfy Exi “ 0 and }xi }ψ2 ď K, Let A P Rpˆp . Then there exists universal constant c such that for every t ě 0, ˙* " ˆ ( t t2 , . P \|xJ Ax ´ ExJ Ax\| ě t ď 2exp ´c min K 4 }A}2F K 2 }A} Lemma 5.12. (Covering Number of the Sphere, Lemma 5.2 of Vershynin (2010)). The unit Euclidean sphere Sn´1 equipped with the Euclidean metric satisfies for every ǫ ą 0 that 2 \|N pSn´1 , ǫq\| ď p1 ` qn , ǫ where N pSn´1 , ǫq is the ǫ-net of Sn´1 with minimal cardinality. The following variant of Wedin’s sin θ law (Wedin, 1972) is proved in Proposition 1 of Cai et al. (2015). p “ A ` E, define the singular value decompositions of A Lemma 5.13. For A, E P Rmˆn and A p and A as p“U pD p Vp J . A “ U DV J , A Then the following perturbation bound holds, › › › › › › › › ›pI ´ PU1:k q PUp 1:k › “ ›PU1:k ´ PUp 1:k › ď 2}E} , σk pAq ´ σk`1 pAq where σk pAq, σk`1 pAq are the kth and pk ` 1qth singular values of A. 5.7 5.7.1 Proofs of key lemmas Proof of Lemma 5.2 (1) The proof of c › › p1 ` p2 › › p 1{2 p p p }Σ ´ Σ}, }Ip2 ´ Σy }, }Σxy ´ Σxy }, ›Σy ´ Ip2 › ď C n is exactly the same as that of Lemma 5.1. (2) Observe that p x´1{2 Σ p xy Σ p y´1{2 ´ Σxy “ pIp ´ Σ p x1{2 qΣ p x´1{2 Σ p xy Σ p y´1{2 Σ 1 p 1{2 Σ p ´1{2 Σ p xy Σ p ´1{2 pIp ´ Σ p 1{2 q ` pΣ p xy ´ Σxy q. `Σ x x y p1 ď 1. Then p x´1{2 Σ p xy Σ p y´1{2 } “ λ and }Σ 2 y p ´1{2 Σ p xy Σ p ´1{2 ´ Σxy } ď }Ip ´ Σ p 1{2 } ` }Σ p x }}Ip ´ Σ p 1{2 } ` }Σ p xy ´ Σxy }. }Σ x y x y 1 2 21 p and Λ are singular values of Σ p x´1{2 Σ p xy Σ p y´1{2 and Σxy respectively. Hence by the Notice that Λ famous Weyl’s inequality for singular values, p ´ Λ} ď }Σ p ´1{2 Σ p xy Σ p ´1{2 ´ Σxy } }Λ x y p p x }}Ip ´ Σ p 1{2 } ` }Σ p xy ´ Σxy } ď }Ip1 ´ Σx } ` }Σ y 2 ¸ ˜ c c c p1 ` p2 p1 ` p2 p1 ` p2 C1 ď C2 . ď 3 ` C1 n n n Ip1 p 1{2 p p ´1{2 Σ p xy Σ p y´1{2 , we have }Σ p 1{2 p pJp p (3) Since Σ x Φ are left singular vectors of Σx x Φ} “ 1, Φ Σx Φ “ p JΦ p ´ Ip “ ´Φ p J pΣ p x ´ Ip qΦ. p Then we have, and Φ 1 1 p JΦ p ´ Ip } “ }Φ p J pΣ p x ´ Ip qΦ} p ď }Φ p JΣ p 1{2 }}Σ p ´1{2 pΣ p x ´ Ip qΣ p ´1{2 }}Σ p 1{2 Φ} p }Φ 1 1 x x 1 x x p ´1{2 pΣ p x ´ Ip qΣ p ´1{2 }. “ }Σ x 1 x As a submatrix, pJ Φ p p ´1{2 pΣ p x ´ Ip qΣ p ´1{2 } ď }Σ p ´1 }}Σ p x ´ Ip } }Φ x x 1:k 1:k ´ Ik } ď }Σx 1 1 p 1 p x ´ Ip } ď }Σ ´ Σ} ď 1 ď }Σ 1 p x ´ Ip } p ´ Σ} 2 1 ´ }Σ 1 ´ }Σ 1 as long as n ě C0 pp1 ` p2 q for sufficiently large C0 . In this case, By the same argument, (4) Recall that p 1:k q ě 1{2, }Φ p 1:k }2 ď 3{2. σk2 pΦ p 1:k q ě 1{2, }Ψ p 1:k }2 ď 3{2. σk2 pΨ Φ1:k “ „ Ik 0pp1 ´kqˆk  , Ψ1:k “ „ Ik 0pp2 ´kqˆk  . p 1{2 p The last inequality in the lemma relies on the fact that Σ x Φ1:k and Φ1:k are leading k singular ´1{2 p ´1{2 p p vectors of Σx Σxy Σy and Σxy respectively. By a variant of Wedin’s sin θ law as stated in Lemma 5.13, c › 2}Σ › p x´1{2 Σ p xy Σ p y´1{2 ´ Σxy } 2C2 p1 ` p2 › › pIp1 ´ PΦ1:k q› ď ď . ›P Σ p p 1{2 x Φ1:k ∆ ∆ n On the other hand, › › › › › › p 1{2 p › › 1{2 p J p pIp1 ´ PΦ1:k q› “ ›Σx Φ1:k pΣx Φ1:k q pIp1 ´ PΦ1:k q› ›PΣ p p 1{2 x Φ1:k › › › › p 1{2 p J Φ q pI ´ P q “ ›pΣ p1 Φ1:k › 1:k x › › › p 1{2 p l› Φ q “ ›pΣ 1:k › , x 22 p 1{2 p Here the second equality is due to the fact that Σ x Φ1:k has orthonormal columns. Moreover, 1{2 px Φ p 1:k ql denotes the lower pp1 ´ kq ˆ k sub-matrix of Σ p 1{2 p pΣ x Φ1:k . Again, by triangle inequality, › › › ´ ¯l ›› › p l › ›› p 1{2 p l 1{2 p p ›Φ1:k › “ ›pΣx Φ1:k q ´ pΣx ´ Ip1 qΦ1:k ›› › › › › ›› › p 1{2 p › p 1{2 ››p › l› Φ ď ›pΣ Φ q ` Σ ´ I q ›p x p1 › › 1:k › 1:k › x c c c c 2C2 p1 ` p2 C3 p 1 ` p 2 3 p1 ` p2 ď ` C1 ď . ∆ n 2 n ∆ n The last inequality is due to ∆ ď 1. Let C “ maxpC1 , C2 , C3 q, the proof is done. 5.7.2 Proof of Lemma 5.3 The equality (5.10) implies pu ´ Φ p u Λ1 “ Φ p u pΛ p 1 ´ Λ1 q ` pΣ p 11 ´ Ik qΦ pu Λ p p 12 p l p Λ1 Ψ 1:k 1:k 1:k x 1:k 1 ` Σx Φ1:k Λ1 p 12 p l :“ R1 . p 11 pu ´ Σ ´ pΣ xy Ψ xy ´ Λ1 qΨ (5.16) p 1 ´ Λ1 q ` pΣ p 11 pu p p 12 p l p p u pΛ p u Λ1 “ Ψ pu ´ Ψ Λ1 Φ y ´ Ik qΨ1:k Λ1 ` Σy Ψ1:k Λ1 1:k 1:k 1:k p 12 Φ p l :“ R2 . p 11 ´ Λ1 qΦ pu ´ Σ ´ pΣ (5.17) 1:k 1:k Similarly, (5.11) implies yx yx 1:k 1:k The equality (5.8) is equivalent to p 21 p u p p 21 p u p 22 r p l r pl p 21 Ψ pu Σ xy 1:k ` Λ2 Ψ1:k ` pΣxy ´ Λ2 qΨ1:k “ Σx Φ1:k Λ1 ` Σx Φ1:k pΛ1 ´ Λ1 q p 22 pl p pl p l Λ1 ` pΣ `Φ x Φ Λ1 ´ Φ Λ1 q, 1:k 1:k 1:k which can be written as p 21 p u p pl r pl p 21 p u p 21 pu Σ xy Ψ1:k ` Λ2 Ψ1:k ´ Σx Φ1:k Λ1 ´ Φ1:k Λ1 “ Σx Φ1:k pΛ1 ´ Λ1 q p 22 Φ pl Λ p1 ´ Φ p l Λ1 q ´ pΣ p 22 ´ Λ r 2 qΨ p l :“ R3 . ` pΣ (5.18) p 21 Φ pu rJ pl p 21 p u pl p 21 p u p Σ yx 1:k ` Λ2 Φ1:k ´ Σy Ψ1:k Λ1 ´ Ψ1:k Λ1 “ Σy Ψ1:k pΛ1 ´ Λ1 q p 22 r J p l :“ R4 . p 22 pl p pl ` pΣ y Ψ Λ1 ´ Ψ Λ1 q ´ pΣyx ´ Λ2 qΦ (5.19) x 1:k 1:k xy 1:k Apply the same argument to (5.9), we obtain 1:k 1:k 1:k r 2 ˆ (5.19), then Consider (5.18) ˆ p´Λ1 q ´ Λ that is r p 21 p u r p 21 p u p 21 p u 2 p 21 p u pl p l Λ2 ´ Λ2 Φ Φ 2 1:k ` Σx Φ1:k Λ1 ´ Σxy Ψ1:k Λ1 ´ Λ2 Σyx Φ1:k ` Λ2 Σy Ψ1:k Λ1 1:k 1 r 2 R4 q, “ ´pR3 Λ1 ` Λ p l Λ2 ´ Λ2 Φ pl p 21 p u r p 21 p u Φ 1:k 1 2 1:k “Σxy Ψ1:k Λ1 ` Λ2 Σyx Φ1:k r 2 R4 q. r 2Σ p 21 Ψ p u Λ1 ´ pR3 Λ1 ` Λ p 21 Φ p u Λ2 ´ Λ ´Σ y x 1:k 1:k 1 23 (5.20) Combined with (5.16) and (5.17), p l Λ2 ´ Λ2 Φ pl p 21 p u r p 21 p u p 21 p u p 21 Φ 2 1:k “ Σxy Ψ1:k Λ1 ` Λ2 Σyx Φ1:k ´ Σx Λ1 Ψ1:k Λ1 ` Σx R1 Λ1 1:k 1 r 2Σ p 21 r 2Σ pu ´ Λ p 21 r ´Λ y Λ1 Φ y R2 ´ pR3 Λ1 ` Λ2 R4 q 1:k p 21 ´ Σ p 21 Λ1 qΨ p u Λ1 ` Λ r 2 pΣ p 21 ´ Σ p 21 Λ1 qΦ pu “ pΣ xy x yx y 1:k 1:k 21 21 p r p ` pΣx R1 ´ R3 qΛ1 ´ Λ2 pΣy R2 ` R4 q. This finishes the proof of (5.13). Plug (5.17) into (5.21), we get p l Λ2 ´ Λ r 2Φ pl p 21 p 21 pu r p 21 p 21 pu r Φ 1:k 1 2 1:k “ pΣxy ´ Σx Λ1 qpΛ1 Φ1:k ´ R2 q ` Λ2 pΣyx ´ Σy Λ1 qΦ1:k ` R r ´ pΣ p 21 ´ Σ p 21 Λ1 qR2 q. p u ` pR “ BΦ xy 1:k x This finishes the proof of (5.12). 5.7.3 Proof of Lemma 5.4 First, we discuss two quite different cases: λk ě Case 1: λk ě 1 2 and λk ă 21 . 1 2 Let 1 δ :“ λ2k ´ λ2k`1 “ pλk ´ λk`1 qpλk ` λk`1 q ě ∆. 2 Define the pp1 ´ kq ˆ k matrices A by b b λ2j ´ λ2k ` 2δ λ2k`1 ´ λ2k`i ` 2δ Aij “ , 1 ď i ď p1 ´ k, 1 ď j ď k λ2j ´ λ2k`i By (5.12) in Lemma 5.3, there holds where and By Lemma 5.6, we have p u D2 q ` A ˝ pD1 RD2 q, p l “ A ˝ pD1 B Φ Φ 1:k 1:k ¨ 1 D1 “ diag ˝ b , ¨ ¨ ¨ , b ¨ δ 2 D2 “ diag ˝ b 1 λ2k`1 1 λ21 ´ λ2k ` δ 2 ´ λ2p1 ` δ 2 ˛ ˛ ‚ 1 , ¨ ¨ ¨ , b ‚. δ 2 p u D2 } ` 1 }pD1 RD2 q} p l } ď 1 }D1 B Φ }Φ 1:k 1:k 2 2 1 1 u p ď }D1 B}}Φ1:k }}D2 } ` }D1 }}R}}D2 }. 2 2 24 (5.21) p u } ď }Φ p 1:k } ď Recall that }Φ 1:k b 3 2 and it is obvious that }D1 }, }D2 } ď b 2 δ. Moreover, in the 1 `p2 q previous section, we also have shown that }R} ď Cppn∆ . It suffices to bound }D1 B} and to this end we apply the standard covering argument. Step 1. Reduction. Denote by Nǫ pSd q the d-dimensional unit ball surface. For ǫ ą 0 and any pair of vectors u P Rp1 ´k , v P Rk , we can choose uǫ P Nǫ pSp1 ´k´1 q, vǫ P Nǫ pSk´1 q such that }u ´ uǫ }, }v ´ vǫ } ď ǫ. Then J J J uJ D1 Bv “ uJ D1 Bv ´ uJ ǫ D1 Bv ` uǫ D1 Bv ´ uǫ D1 Bvǫ ` uǫ D1 Bvǫ J ď }u ´ uǫ }}D1 Bv} ` }uJ ǫ D1 B}}v ´ vǫ } ` uǫ D1 Bvǫ ď 2ǫ}D1 B} ` uJ ǫ D1 Bvǫ ď 2ǫ}D1 B} ` max uJ ǫ D1 Bvǫ . uǫ ,vǫ Maximize over u and v, we obtain }D1 B} ď 2ǫ}D1 B} ` max uJ ǫ D1 Bvǫ . uǫ ,vǫ Therefore, }D1 B} ď p1 ´ 2ǫq´1 max uJ ǫ D1 Bvǫ . Let ǫ “ 1{4. Then it suffices to give an upper uǫ ,vǫ bound max uJ ǫ D1 Bvǫ with high probability. uǫ ,vǫ Step 2. Concentration. 1 ď i ď p1 ´ k and 1 ď j ď k rD1 Bsi,j “b 1 λ2k`1 ´ λ2k`i ` δ 2 Y ´λl Xl ? 2 for all 1 ď α ď n and 1 ď l ď p1 . Then for Let Zα,l “ α,l 1´λl n 1 ÿ pλj Xα,k`i Yα,j ´ λ2j Xα,k`i Xα,j ` λk`i Yα,k`i Xα,j ´ λk`i λj Yα,k`i Yα,j q n α“1 n 1 ÿ! p1 ´ λ2j qλk`i λj Xα,k`i Xα,j ´ λ2j pYα,k`i ´ λk`i Xα,k`i qpYα,j ´ λj Xα,j q δ n 2 2 λk`1 ´ λk`i ` 2 α“1 ) ` p1 ´ λ2j qλj pYα,k`i ´ λk`i Xα,k`i qXα,j ` p1 ´ λ2j qλk`i pYα,j ´ λj Xα,j qXα,k`i . “b 1 n 1 ÿ! p1 ´ λ2j qλk`i λj Xα,k`i Xα,j δ n 2 2 λk`1 ´ λk`i ` 2 α“1 b b b ´ λ2j 1 ´ λ2k`i 1 ´ λ2j Zα,k`i Zα,j ` p1 ´ λ2j qλj 1 ´ λ2k`i Zα,k`i Xα,j b ) ` p1 ´ λ2j qλk`i 1 ´ λ2k`i Xα,k`i Zα,j . “b 1 In this way, tXα,k`i , Zα,k`i , 1 ď i ď p1 , 1 ď α ď nu are mutually independent standard gaussian 25 random variables. For any given pair of vectors u P Rp1 ´k , v P Rk , uJ D1 Bv “ n p1 ´k ÿ k ! ui vj 1 ÿ ÿ b p1 ´ λ2j qλk`i λj Xα,k`i Xα,j n α“1 i“1 j“1 λ2 ´ λ2 ` δ k`1 k`i 2 b b b ´ λ2j 1 ´ λ2k`i 1 ´ λ2j Zα,k`i Zα,j ` p1 ´ λ2j qλj 1 ´ λ2k`i Zα,k`i Xα,j b ) ` p1 ´ λ2j qλk`i 1 ´ λ2k`i Xα,k`i Zα,j n . 1 ÿ J “ w Aα wα , n α“1 α where J wαJ “ rxJ α , zα s “ rXα,1 , . . . , Xα,p1 , Zα,1 , . . . , Zα,p1 s and Aα P Rp2p1 qˆp2p1 q is symmetric and determined by the corresponding quadratic form. This yields }Aα }2F “ p1 ´k ÿ k ! u2i vj2 1 ÿ p1 ´ λ2j q2 λ2k`i λ2j ` λ4j p1 ´ λ2k`i qp1 ´ λ2j q 2 i“1 j“1 λ2k`1 ´ λ2k`i ` 2δ ) ` p1 ´ λ2j q2 λ2j p1 ´ λ2k`i q ` p1 ´ λ2j q2 λ2k`i p1 ´ λ2k`i q p1 ´k ÿ k ` ˘` ˘ u2i vj2 1 ÿ 1 ´ λ2j λ2k`i ` λ2j ´ 2λ2k`i λ2j δ 2 2 2 i“1 j“1 λk`1 ´ λk`i ` 2 ¸ ˜p ´k ¸ ˜ k 1 ÿ p1 ´ λ2j qpλ2k`i ` λ2j ´ 2λ2k`i λ2j q 1 ÿ max vj2 u2i ď 1ďiďp1 ´k 2 i“1 λ2k`1 ´ λ2k`i ` 2δ j“1 “ ď p1 1 max 2 1ďiďp1 ´k 1ďjďk ď p1 ´ λ2k q max 1ďiďp1 ´k δ 1ďjďk 2 ď p1 ´ λ2k q ď 2p1 ´ 1ďjďk 2 2 ´ λk qp2λj ´ 2λ2k`i λ2j q λ2k`1 ´ λ2k`i ` 2δ max λ2j p1 ´ λ2k`i q ` λ2k`1 ´ λ2i`k p1 ´ λ2k`1 q 1ďiďp1 ´k 1ďjďk 2 λk qp1 ´ λ2k`1 q δ δ 2 . “ K 2, where the second last inequality is due to the facts that λj ď 1 and δ 2 p1 ´ λ2k`i q ` λ2k`1 ´ λ2i`k ď p1 ´ λ2k`1 q δ 2 Moreover }Aα }22 ď }Aα }2F ď K 2 . 26 p7 δ ` λ2k`1 ă λ2k ď 1q. 2 Now define w J :“ rw1J , . . . , wnJ s and » A1 — A2 — A“— – Then we have }A} ď max }Aα } ď K, 1ďαďn .. . }A}2F and ﬁ An ď ﬃ ﬃ ﬃ. ﬂ n ÿ α“1 }Aα }2F ď nK 2 1 J w Aw, where w P N2p1 n p0, I2p1 n q. n Therefore, By the classic Hanson-Wright inequality (Lemma 5.11), there holds " ˆ 2 ˙* ( t t J P n\|u D1 Bv\| ě t ď 2 exp ´c0 min , nK 2 K uJ D1 Bv “ for some numerical constant c0 ą 0. Without loss of generality, we can also assume c0 ď 1. Let ? t “ c40 np1 K. By n ě p1 , straightforward calculation gives " * 4? J P n\|u D1 Bv\| ě np1 K ď 2e´4p1 . c0 Step 3. Union Bound. By Lemma 5.12, we choose 1{4-net such that , $ d / ’ ˆ ? ˙c & 2 2 p1 p1 ´ λk qp1 ´ λk`1 q . 4 2 J P max uǫ D1 Bvǫ ě / ’ c0 n δ %uǫ PNǫ pSp1 ´k´1q vǫ PNǫ pSk´1 q 3 ď 9p1 ´k 9k ˆ 2e´4p1 ď 2e´ 2 p1 . 3 In other words, with probability at least 1 ´ 2e´ 2 p1 , we have ˆ ? ˙c p1 8 2 J ´1 }D1 B} ď p1 ´ 2ǫq max uǫ D1 Bvǫ ď uǫ ,vǫ c0 n d p1 ´ λ2k qp1 ´ λ2k`1 q . δ In summary, we have as long as n ě C0 pp1 ` p2 q, with probability 1 ´ c0 expp´γp1 q, »d ﬁ 2 2 p1 p1 ´ λk qp1 ´ λk`1 q pp1 ` p2 q pl } ď C – ﬂ }Φ ` 1:k nδ2 n∆δ ﬁ »d p1 p1 ´ λ2k qp1 ´ λ2k`1 q pp1 ` p2 q ﬂ. ` ďC– n∆2 n∆2 Here the last inequality is due to δ “ pλk ` λk`1 q∆ ě 21 ∆. Here C0 , C, c0 , γ are absolute constants. 27 Case 2: λk ď 1 2 By (5.13), we have p l “ GΛ1 ` Λ2 F , p l Λ21 ´ Λ22 Φ Φ 1:k 1:k where and p 21 R1 ´ R3 q p 21 ´ Σ p 21 Λ1 qΨ p u ` pΣ G :“ pΣ x xy x 1:k ” ı p 21 ´ Σ p 21 Λ1 qΦ p u ´ pΣ p 21 R2 ` R4 q . F :“ rIp1 , 0p1 ˆpp2 ´p1 q s pΣ yx y 1:k y p 21 and Σ p 21 are submatrices of Σ p 2p . By Lemma 5.1, we have Notice that Σ xy x 1 Moreover, by }R1 } ď C b p1 `p2 n , p 21 ´ Σ p 21 Λ1 } ď C }Σ xy x c p1 . n `p2 }R3 } ď C p1n∆ and Lemma 5.2, there holds }G} ď C ˆc p1 p1 ` p2 ` n n∆ ˙ . p 2p . By a similar p 21 are submatrices of Σ p 21 and rIp , 0p ˆpp ´p q sΣ Similarly, rIp1 , 0p1 ˆpp2 ´p1 q sΣ x 1 yx 1 1 2 1 argument, ˙ ˆc p1 p1 ` p2 ` . }F } ď C n n∆ Then pl “ Φ 1:k „  „  „  „  λj 1 1 λk`i ˝ ˝ ˝G` ˝F λk`i ` λj λj ´ λk`i λk`i ` λj λj ´ λk`i Here 1 ď i ď p1 ´ k and 1 ď j ď k. By Lemma 5.6, there holds for any X, ›„  › ›„  › › › › maxpλk`i , λj q › 3 λj › ›“› X X ›› ď }X} › λk`i ` λj › › λk`i ` λj 2 and Finally, for any X, where ›„  › ›„  › › › minpλk`i , λj q › 1 › λ k`i › › › X ›› ď }X}. › λk`i ` λj X › “ › λk`i ` λj 2 „  1 X “ A ˝ pD1 XD2 q λj ´ λk`i »b A :“ – λj ´ λk ` ∆ 2 b λk`1 ´ λk`i ` λj ´ λk`i 28 ∆ 2 ﬁ ﬂ, ¨ 1 D1 “ diag ˝ b , ¨ ¨ ¨ , b and ¨ Since }D1 }, }D2 } ď b In summary, we have Since 1 2 2 ∆, D2 “ diag ˝ b λk`1 ´ λp1 ` 1 λ1 ´ λk ` ∆ 2 ∆ 2 ˛ ‚, 1 , ¨ ¨ ¨ , b ‚. ∆ 2 by Lemma 5.6, ›„  › › › 1 1 1 › › › λj ´ λk`i X › ď 2 }D1 XD2 } ď ∆ }X}. pl } ď C }Φ 1:k ě λk ě λk`1 , there holds »d pl } ď C – }Φ 1:k 6 ∆ 2 1 ˛ ˆc p1 p1 ` p2 ` 2 n∆ n∆2 ˙ . ﬁ p1 p1 ´ λ2k qp1 ´ λ2k`1 q pp1 ` p2 q ﬂ. ` n∆2 n∆2 Lower Bound: Proof of Theorem 2.2 To establish the minimax lower bounds of CCA estimates for our proposed losses, we follow the analytical frameworks in the literature of PCA and CCA, e.g., Vu et al. (2013); Cai et al. (2013); Gao et al. (2015), where the calculation is focused on the construction of the hypothesis class to which the packing lemma and Fano’s inequality are applied. However, since we fix both λk and λk`1 in the localized parameter spaces, new technical challenges arise and consequently we construct hypothesis classes based on the equality (6.1). In this section we also denote ∆ :“ λk ´ λk`1 . 6.1 On Kullback-Leibler Divergence The following lemma can be viewed as an extension of Lemma 14 in Gao et al. (2015) from λk`1 “ 0 to arbitrary λk`1 . The proof of the lemma can be found in Section 6.4. “ ‰ “ ‰ Lemma 6.1. For i “ 1, 2 and p2 ě p1 ě k, let Upiq , Wpiq P Opp1 , p1 q, Vpiq , Zpiq P Opp2 , p1 q where Upiq P Rp1 ˆk , Vpiq P Rp2 ˆk . For 0 ď λ2 ă λ1 ă 1, let ∆ “ λ1 ´ λ2 and define « ﬀ 1{2 J ` λ W Z J qΣ1{2 Σx Σx pλ1 Upiq Vpiq y 2 piq piq Σpiq “ i “ 1, 2, 1{2 J ` λ Z W J qΣ1{2 Σy pλ1 Vpiq Upiq Σy x 2 piq piq Let Ppiq denote the distribution of a random i.i.d. sample of size n from N p0, Σpiq q. If we further assume « Jﬀ « Jﬀ Vp1q Vp2q rUp1q , Wp1q s “ rU , W s , (6.1) p2q p2q J J Zp1q Zp2q 29 Then one can show that DpPp1q \|\|Pp2q q “ n∆2 p1 ` λ1 λ2 q J 2 }U V J ´ Up2q Vp2q }F . 2p1 ´ λ21 qp1 ´ λ22 q p1q p1q Remark 6.2. The conditon in (6.1) is crucial for obtaining the eigen-gap factor 1{∆2 in the lower bound and is the key insight behind the construction of the hypothesis class in the proof. Gao et al. (2015) has a similar lemma but only deals with the case that the residual canonical correlations are zero. To the best of our knowledge, the proof techniques in Gao et al. (2015, 2017) cannot be directly used to obtain our results. 6.2 Packing Number and Fano’s Lemma The following result on the packing number is based on the metric entropy of the Grassmannian manifold Gpk, rq due to Szarek (1982). We use the version adapted from Lemma 1 of Cai et al. (2013) which is also used in Gao et al. (2015). Lemmaa6.3. For any fixed U0 P Opp, kq and Bǫ0 “ tU P Opp, kq : }U U J ´ U0 U0J }F ď ǫ0 u with ǫ0 P p0, 2rk ^ pp ´ kqs q. Define the semi-metric ρp¨, ¨q on Bǫ0 by ρpU1 , U2 q “ }U1 U1J ´ U2 U2J }F . Then there exists universal constant C such that for any α P p0, 1q, the packing number MpBǫ0 , ρ, αǫ0 q satisfies ˙ ˆ 1 kpp´kq . MpBǫ0 , ρ, αǫ0 q ě Cα The following corollary is used to prove the lower bound. Corollary 6.4. If we change the set in Lemma 6.3 to Brǫ0 “ tU P Opp, kq : }U ´ U0 }F ď ǫ0 u, then we still have ˙ ˆ 1 kpp´kq r . MpBǫ0 , ρ, αǫ0 q ě Cα Proof. Apply Lemma 6.3 to Bǫ0 , there exists U1 , ¨ ¨ ¨ , Un with n ě p1{Cαqkpp´kq such that }Ui UiJ ´ U0 U0J }F ď ǫ0 , 1 ď i ď n, }Ui UiJ ´ Uj UjJ }F ě αǫ0 , 1 ď i ď j ď n. r i “ arg Define U min U PtUi Q, QPOpkqu }U ´ U0 }F , by Lemma 6.5, r i ´ U 0 }F ď }U riU r iJ ´ U0 U0J }F ď ǫ0 . }U r1 , ¨ ¨ ¨ , U rn P Brǫ and Therefore, U 0 which implies, riU r iJ ´ U rj U r jJ }F “ }Ui UiJ ´ Uj UjJ }F ě αǫ0 . }U MpBrǫ0 , ρ, αǫ0 q ě n ě 30 ˆ 1 Cα ˙kpp´kq . Lemma 6.5. For any matrices U1 , U2 P Opp, kq, inf QPOpk,kq }U1 ´ U2 Q}F ď }PU1 ´ PU2 }F Proof. By definition }U1 ´ U2 Q}2F “ 2k ´ 2trpU1J U2 Qq Let U1J U2 “ U DV J be the singular value decomposition. Then V U J P Opk, kq and inf QPOpk,kq }U1 ´ U2 Q}2F ď 2k ´ 2trpU1J U2 V U J q “ 2k ´ 2trpU DU J q “ 2k ´ 2trpDq. On the other hand, }PU1 ´ PU2 }2F “ }U1 U1J ´ U2 U2J }2F “ 2k ´ 2trpU1 U1J U2 U2J q “ 2k ´ 2trpU1J U2 U2J U1 q “ 2k ´ 2trpD 2 q. Since U1 , U2 P Opp, kq, }U1J U2 } ď 1 and therefore all the diagonal elements of D is less than 1, which implies that trpDq ě trpD 2 q and inf QPOpk,kq }U1 ´ U2 Q}2F ď }PU1 ´ PU2 }2F . Lemma 6.6 (Fano’s Lemma Yu (1997)). Let pΘ, ρq be a (semi)metric space and tPθ : θ P Θu a collection of probability measures. For any totally bounded T Ă Θ, denote MpT, ρ, ǫq the ǫ-packing number of T with respect to the metric ρ, i.e. , the maximal number of points in T whoese pairwise minimum distance in ρ is at least ǫ. Define the Kullback-Leibler diameter of T by dKL pT q “ sup DpPθ \|\|Pθ1 q. θ,θ 1 PT Then, ¯ ” ı 2´ p θq ě sup sup ǫ 1 ´ dKL pT q ` log 2 inf sup Eθ ρ2 pθ, log MpT, ρ, ǫq T ĂΘ ǫą0 4 θp θPΘ 31 6.3 Proof of Lower Bound ‰ ‰ “ “ For any fixed Up0q , Wp0q P Opp1 , p1 q and Vp0q , Zp0q P Opp2 , p1 q where Up0q P Rp1ˆk , Vp0q P Rp2 ˆk , Wp0q P Rp1 ˆpp1 ´kq , Vp0q P Rp2 ˆpp2 ´kq , define !` ˘ “ ‰ “ ‰ Hǫ0 “ U , W , V , Z : U , W P Opp1 , p1 q with U P Rp1 ˆk , V , Z P Opp2 , p1 q « ﬀ « Jﬀ J Vp0q ) V with V P Rp2 ˆk , }U ´ Up0q }F ď ǫ0 , rU , W s “ rU , W s . p0q p0q J ZJ Zp0q For any fixed Σx P Sp`1 , Σy P Sp`2 with κpΣx q “ κx , κpΣy q “ κy , consider the parametrization Σxy “ Σx ΦΛΨJ Σy , for 0 ď λk`1 ă λk ă 1, define « ﬀ 1{2 1{2 ! Σx Σx pλk U V J ` λk`1 W Z J qΣy Tǫ0 “ Σ “ , 1{2 1{2 Σy Σy pλk V U J ` λk`1 ZW J qΣx ) ` ˘ Φ “ Σx´1{2 rU , W s, Ψ “ Σy´1{2 rV , Zs, U , W , V , Z P Hǫ0 . It is straightforward to verify that Tǫ0 Ă Fpp1 , p2 , k, λk , λk`1 , κx , κy q. For any Σpiq P Tǫ0 , i “ 1, 2, they yield to the parametrization, « ﬀ 1{2 1{2 J `λ J Σx Σx pλk Upiq Vpiq k`1 Wpiq Zpiq qΣy Σpiq “ , 1{2 1{2 J `λ J Σy pλk Vpiq Upiq Σy k`1 Zpiq Wpiq qΣx ˘ ` ´1{2 piq piq where Upiq , Wpiq , Vpiq , Zpiq P Hǫ0 and the leading-k canonical vectors are Φ1:k “ Σx Upiq , Ψ1:k “ ´1{2 Σy Vpiq . We define a semi-metric on Tǫ0 as › › › › › › › › ρpΣp1q , Σp2q q “ ›PΣ1{2 Φp1q ´ PΣ1{2 Φp2q › “ ›PUp1q ´ PUp2q › . x x 1:k 1:k F F By Lemma 6.1, DpPΣ1 \|\|PΣ2 q “ n∆2 p1 ` λk λk`1 q J 2 }U V J ´ Up2q Vp2q }F . 2p1 ´ λ2k qp1 ´ λ2k`1 q p1q p1q Further by the definition of dKL pT q, dKL pT q “ n∆2 p1 ` λk λk`1 q J J 2 sup }Up1q Vp1q ´ Up2q Vp2q }F . 2p1 ´ λ2k qp1 ´ λ2k`1 q Σp1q ,Σp2q PTǫ0 (6.2) To bound the Kullback-Leibler diameter, for any Σp1q , Σp2q P Tǫ0 , by definition, « Jﬀ « Jﬀ Vp1q Vp2q rUp1q , Wp1q s “ rUp2q , Wp2q s , J J Zp1q Zp2q which implies that they are singular value decompositions of the same matrix. Therefore, there exists Q P Opp1 , p1 q such that rUp2q , Wp2q s “ rUp1q , Wp1q sQ , rVp2q , Zp2q s “ rVp1q , Zp1q sQ. 32 (6.3) Decompose Q into four blocks such that „  Q11 Q12 Q“ . Q21 Q22 Substitute into (6.3), Up2q “ Up1q Q11 ` Wp1q Q21 , Vp2q “ Vp1q Q11 ` Zp1q Q21 . Then, }Up2q ´ Up1q }2F “ }Up1q pQ11 ´ Ik q ` Wp1q Q21 }2F “ }Up1q pQ11 ´ Ik q}2F ` }Wp1q Q21 }2F “ }Q11 ´ Ik }2F ` }Q21 }2F . The second equality is due to the fact that Up1q and Wp1q have orthogonal column space and the third equality is valid because Up1q , Wp1q P Opp1 , kq. By the same argument, we will have }Vp2q ´ Vp1q }2F “ }Q11 ´ Ik }2F ` }Q21 }2F . Notice that J J 2 }Up1q Vp1q ´ Up2q Vp2q }F “ }pUp1q ´ Up2q qVp1q ` Up2q pVp1q ´ Vp2q q}2F ď 2}Up1q ´ Up2q }2F ` 2}Vp1q ´ Vp2q }2F “ 4}pUp1q ´ Up2q q}2F ` ˘ ď 8 }pUp1q ´ Up0q q}2F ` }pUp0q ´ Up2q q}2F ď 16ǫ20 . Then, substitute into (6.2) 8n∆2 p1 ` λk λk`1 q 2 ǫ . p1 ´ λ2k qp1 ´ λ2k`1 q 0 dKL pT q ď (6.4) J ´ Let Bǫ0 “ tU P Opp1 , kq : }U ´ Up0q }F ď ǫ0 u. Under the semi-metric ρrpUp1q , Up2q q “ }Up1q Up1q J } , we claim that the packing number of H is lower bounded by the packing number of B . Up2q Up2q F ǫ0 ǫ0 To prove this claim, it suffices to show that for any U P Bǫ0 , there exists corresponding W , V , Z such that pU , W , V , Zq P Hǫ0 . First of all, by definition, }U ´U0 }F ď ǫ0 . Let W P Opp1 , p1 ´kq be the orthogonal complement of U . Then rU , W s P Opp1 , p1 q and therefore there exists Q P Opp1 , p1 q such that rU , W s “ rUp0q , Wp0q sQ. Set rV , Zs “ rVp0q , Zp0q sQ P Opp2 , p1 q, then rU , W s « VJ ZJ ﬀ “ rUp0q , Wp0q s 33 « J Vp0q J Zp0q ﬀ , which implies pU , W , V , Zq P Hǫ0 . Let ¨ ˛ d a p1 ´ λ2k qp1 ´ λ2k`1 q ǫ “ αǫ0 “ c ˝ k ^ pp1 ´ kq ^ kpp1 ´ kq‚, n∆2 p1 ` λk λk`1 q a where c P p0, 1q depends on α and is chosen small enough such that ǫ0 “ ǫ{α P p0, 2rk ^ pp1 ´ kqs q. By Corollary 6.4, ˆ ˙ 1 kpp1 ´kq MpTǫ0 , ρ, αǫ0 q “ MpHǫ0 , ρr, αǫ0 q ě MpBǫ0 , ρr, αǫ0 q ě . Cα Apply Lemma 6.6 with Tǫ0 , ρ, ǫ, ˜ ¸ „ › ›2  8c2 kpp1 ´ kq ` log2 ǫ2 › › 1´ . inf sup E ›PΣ1{2 Φ › ě sup sup p 1:k ´ PΣ1{2 1 x Φ1:k F x p 1:k ΣPF kpp1 ´ kqlog Cα T ĂΘ ǫą0 4 Φ Choose α small enough such that 1´ 1 8c2 kpp1 ´ kq ` log2 ě . 1 2 kpp1 ´ kqlog Cα Then the lower bound is reduced to # + „ › ›2  c2 p1 ´ λ2 qp1 ´ λ2 q › › k`1 k kpp1 ´ kq ^ k ^ pp1 ´ kq inf sup E ›PΣ1{2 Φ › ě p 1:k ´ PΣ1{2 x Φ1:k F x 8 n∆2 p1 ` λk λk`1 q p 1:k ΣPF Φ ¸ + #˜ p1 ´ λ2k qp1 ´ λ2k`1 q p1 ´ k p1 ´ k 2 ^1^ ěC k ∆2 n k By symmetry, „ › › inf sup E ›PΣ1{2 Ψ p p 1:k ΣPF Ψ y 1:k ´ PΣ1{2 Ψ y 1:k ›2  › › ě C 2k F #˜ p1 ´ λ2k qp1 ´ λ2k`1 q p1 ´ k ∆2 n ¸ p1 ´ k ^1^ k + The lower bound for operator norm error can be immediately obtained by noticing that PΣ1{2 Ψ p 1:k ´ y PΣ1{2 Ψ has at most rank 2k and y 1:k › › ›PΣ1{2 Φ p x 6.4 1:k Proof of Lemma 6.1 ›2 ›2 1 ›› › › ´ PΣ1{2 Φ › ě ´ PΣ1{2 Φ › ›PΣ1{2 Φ p x x x 1:k 1:k 1:k 2k F By simple algebra, the Kullback-Leibler divergence between two multivariate gaussian distributions satisfies ) ¯ n ! ´ ´1 ´1 DpPΣp1q \|\|PΣp2q q “ Tr Σp2q pΣp1q ´ Σp2q q ´ log detpΣp2q Σp1q q . 2 Notice that ﬀ « ﬀ « 1{2 1{2 Σx Σx Σpiq “ 1{2 , 1{2 Ωpiq Σy Σy 34 where Ωpiq ﬀ J ` λ W ZJ Ip1 λ1 Upiq Vpiq 2 piq piq . “ J ` λ Z WJ λ1 Vpiq Upiq Ip2 2 piq piq « Then, DpPΣp1q \|\|PΣp2q q “ Also notice that Ωpiq “ „ Ip1 ) n! ´1 Ω q . Ω q ´ pp ` p q ´ log detpΩ TrpΩ´1 1 2 p1q p1q p2q p2q 2  „ ı λ „ U ” ı λ1 Upiq ” J 1 J J J piq U V Upiq ´Vpiq ´ ` piq piq Ip2 2 Vpiq 2 ´Vpiq ”  „ „ ı ı λ2 Wpiq λ2 Wpiq ” J J J J Wpiq Zpiq Wpiq ´Zpiq ` ´ . 2 Zpiq 2 ´Zpiq  Therefore Ωp1q , Ωp2q share the same set of eigenvalues: 1 ` λ1 with multiplicity k, 1 ´ λ1 with multiplicity k, 1 ` λ2 with multiplicity p1 ´ k, 1 ´ λ2 with multiplicity p1 ´ k and 1 with multiplicity 2pp2 ´ p1 q. This implies log detpΩ´1 Ω qq “ 0. On the other hand, by block inversion formula, we p2q p1q can compute ﬁ » 2 λ21 λ1 J ` λ2 W W J J ´ λ2 W Z J U U U V ´ I ` p 2 2 p2q p2q p2q p2q p2q p2q 1´λ2 p2q 1´λ2 p2q ﬂ 1´λ1 – 1 1´λ1 . Ω´1 λ21 λ22 p2q “ λ2 λ1 J J J J Ip2 ` 1´λ2 Vp2q Vp2q ` 1´λ2 Zp2q Zp2q ´ 1´λ2 Vp2q Up2q ´ 1´λ2 Zp2q Wp2q 1 1 Divide Ω´1 p2q Ωp1q into blocks such that Ω´1 p2q Ωp1q “ „  J11 J12 where J11 P Rp1 ˆp1 , J22 P Rp2 ˆp2 , J21 J22 and λ22 λ21 J J J J J pU U ´ U V V U q ` pWp2q Wp2q ´ Wp2q Zp2q Zp1q Wp1q q p2q p2q p2q p2q p1q p1q 1 ´ λ21 1 ´ λ22 λ1 λ2 λ1 λ2 J J J J ´ pUp2q Vp2q Zp1q Wp1q q´ pWp2q Zp2q Vp1q Up1q q 1 ´ λ21 1 ´ λ22 λ22 λ21 J J J J J pVp2q Vp2q pZp2q Zp2q ´ Zp2q Wp2q ´ Vp2q Up2q Wp1q Zp1q Up1q Vp1q q q` “ 2 1 ´ λ1 1 ´ λ22 λ1 λ2 λ1 λ2 J J J J ´ pVp2q Up2q pZp2q Wp2q Wp1q Zp1q Up1q Vp1q q´ q. 2 1 ´ λ1 1 ´ λ22 J11 “ J22 We spell out the algebra for trpJ11 q, and trpJ22 q can be computed in exactly the same fashion. 1 J J J J J J J J J Vp2q Up2q ` Up1q Vp1q Vp1q Up1q ´ 2Up2q Vp2q Vp1q Up1q q trpUp2q Up2q ´ Up2q Vp2q Vp1q Up1q q “ trpUp2q Vp2q 2 1 J “ }Up1q Vp1q ´ Up2q Vp2q }2F . 2 Similarly, J J trpWp2q Wp2q ´ Wp2q Zp2q Zp1q Wp1q q“ 35 1 J }Wp1q Zp1q ´ Wp2q Zp2q }2F . 2 J ` W Z J “ U V J ` W Z J , we have By the assumption (6.1), i.e., Up1q Vp1q p1q p1q p2q p2q p2q p2q 1 J J J trpWp2q Wp2q ´ Wp2q Zp2q Zp1q Wp1q q “ }Up1q Vp1q ´ Up2q Vp2q }2F . 2 Further, ¯ ´ J J J J J J J trpUp2q Vp2q Zp1q Wp1q q “ tr Up2q Vp2q pUp2q Vp2q ` Wp2q Zp2q ´ Up1q Vp1q q ¯ ´ J J J J “ tr Up2q Vp2q pUp2q Vp2q ´ Up1q Vp1q q 1 J ´ Up2q Vp2q }2F , “ }Up1q Vp1q 2 and by the same argument, 1 J J J trpWp2q Zp2q Vp1q Up1q q “ }Up1q Vp1q ´ Up2q Vp2q }2F . 2 Sum these equations, * λ22 λ1 λ2 λ1 λ2 λ21 J ` ´ ´ }Up1q Vp1q ´ Up2q Vp2q }2F 1 ´ λ21 1 ´ λ22 1 ´ λ21 1 ´ λ22 ∆2 p1 ` λ1 λ2 q J “ }Up1q Vp1q ´ Up2q Vp2q }2F . 2p1 ´ λ21 qp1 ´ λ22 q 1 trpJ11 q “ 2 " Repeat the argument for J22 , one can show that trpJ22 q “ trpJ11 q “ ∆2 p1 ` λ1 λ2 q J }Up1q Vp1q ´ Up2q Vp2q }2F . 2p1 ´ λ21 qp1 ´ λ22 q Therefore, n n Ωp1q q “ ptrpJ11 q ` trpJ22 qq trpΩ´1 p2q 2 2 n∆2 p1 ` λ1 λ2 q }U V J ´ Up2q Vp2q }2F . “ 2p1 ´ λ21 qp1 ´ λ22 q p1q p1q DpPΣp1q \|\|PΣp2q q “ References Anderson, T. 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Using Deep Convolutional Networks for Gesture Recognition in American Sign Language Vivek Bheda and N. Dianna Radpour Department of Computer Science, Department of Linguistics State University of New York at Buffalo {vivekkan, diannara}@buffalo.edu Abstract In the realm of multimodal communication, sign language is, and continues to be, one of the most understudied areas. In line with recent advances in the field of deep learning, there are far reaching implications and applications that neural networks can have for sign language interpretation. In this paper, we present a method for using deep convolutional networks to classify images of both the the letters and digits in American Sign Language. 1. Introduction Sign Language is a unique type of communication that often goes understudied. While the translation process between signs and a spoken or written language is formally called ‘interpretation,’ the function that interpreting plays is the same as that of translation for a spoken language. In our research, we look at American Sign Language (ASL), which is used in the USA and in English-speaking Canada and has many different dialects. There are 22 handshapes that correspond to the 26 letters of the alphabet, and you can sign the 10 digits on one hand. Figure 1. American Sign Language Alphabet Figure 2. American Sign Language Numbers One of the nuances in sign language is how often fingerspelling is used. Fingerspelling is a method of spelling words using only hand gestures. One of the reasons the fingerspelling alphabet plays such a vital role in sign language is that signers used it to spell out names of anything for which there is not a sign. People's names, places, titles, brands, new foods, and uncommon animals or plants all fall broadly under this category, and this list is by no means exhaustive. Due to this reason, the recognition process for each individual letter plays quite a crucial role in its interpretation. 2. Related Work Convolutional Neural Networks have been extremely successful in image recognition and classification problems, and have been successfully implemented for human gesture recognition in recent years. In particular, there has been work done in the realm of sign language recognition using deep CNNs, with input-recognition that is sensitive to more than just pixels of the images. With the use of cameras that sense depth and contour, the process is made much easier via developing characteristic depth and motion profiles for each sign language gesture [5]. The use of depth-sensing technology is quickly growing in popularity, and other tools have been incorporated into the process that have proven successful. Developments such as custom-designed color gloves have been used to facilitate the recognition process and make the feature extraction step more efficient by making certain gestural units easier to identify and classify [8]. Until recently, however, methods of automatic sign language recognition weren’t able to make use of the depth-sensing technology that is as widely available today. Previous works made use of very basic camera technology to generate datasets of simply images, with no depth or contour information available, just the pixels present. Attempts at using CNNs to handle the task of classifying images of ASL letter gestures have had some success [7], but using a pre-trained GoogLeNet architecture. 3. Method Our overarching approach was one of basic supervised learning using mini-batch stochastic gradient descent. Our task was that of classification using deep convolutional neural networks to classify every letter and the digits, 0-9, in ASL. The inputs were fixed size high-pixel images, 200 by 200 or 400 by 400, being padded and resized to 200 by 200. 3.1 Architecture Most implementations surrounding this task have attempted it via transfer learning, but our network was trained from scratch. Our general architecture was a fairly common CNN architecture, consisting of multiple convolutional and dense layers. The architecture included 3 groups of 2 convolutional layers followed by a max-pool layer and a dropout layer, and two groups of fully connected layer followed by a dropout layer and one final output layer. 4. Data We initially trained and tested on a self-generated dataset of images we took ourselves. This dataset was a collection of 25 images from 5 people for each alphabet and the digits 1-9. Since our dataset was not constructed in a controlled setting, it was especially prone to differences in light, skin color, and other differences in the environment that the images were captured in, so we also used a premade dataset to compare our dataset’s performance with [3]. Additionally, a pipeline was developed that can be used so people are able to generate and continue adding images to this dataset. 4.1 Preprocessing For generating our own dataset, we captured the images for each sign, then removed the backgrounds from each of the images using background-subtraction techniques. When we initially split the dataset into two for training and validation, the validation accuracy showed to be high. However, when we used datasets from two different sources, i.e. training on ours and testing on the premade and vice versa, the validation accuracy drastically decreased. Since training on one dataset and validating on another was not yielding as accurate of results, we used the premade dataset for the different gestures to train the network which yielded the following results. Figure 4. Training Accuracy on ASL Alphabets Figure 3. Network Architecture Figure 5. Training Accuracy on ASL Digits Figure 6. Training Loss on ASL Alphabet Figure 8. Validation Accuracy on ASL Letters Figure 7. Training Loss on ASL Digits 4.2 Data Augmentation We saw the performances improve differently in our two datasets via data augmentation. By transforming our images just a few pixels (rotating by 20 degrees, translating by 20% on both axes) there was an increased accuracy of approximately 0.05. We also flipped the images horizontally as we can sign using both hands. While it wasn’t extremely effective, we saw that with better and more representative initial training data, augmenting improved the performance more drastically. This was observed after augmentation of the premade dataset, which improved the performance by nearly 20%. 5. Results We observed 82.5% accuracy on the alphabet gestures, and 97% validation set accuracy on digits, when using the NZ ASL dataset. On our self-generated dataset, we observed much lower accuracy measures, as was expected since our data was less uniform than that which was collected under studio settings with better equipment. We saw 67% accuracy on letters of the alphabet, and 70% accuracy on the digits. In terms of time complexity, gestures of the letters converged in approximately 25 minutes, and the digits converged in nearly 10 minutes. Figure 9. Validation Accuracy on ASL Digits Figure 10. Validation Loss on ASL Letters Figure 11. Validation Loss on ASL Digits 5.1 Evaluation We trained with a categorical cross entropy loss function for both our datasets. It is a fairly common loss function used along with image classification problems. Initially, we observed low accuracy measures when testing on the validation set of the self-generated data, which we accounted largely to the lighting and skin tone variations in the images. The higher accuracy measure for the digits was expected, since the gestures for the digits are much more distinguishable and easier to classify. Compared to previous methods working on this same task, our network performed quite well, considering RF-JA were using both a color glove and depth-sensing Kinect camera. The cause of higher accuracy than Stanford’s method was likely due to their lack of background-subtraction for the images, since they used a large dataset from ILSVRC2012 as part of a competition. Method Accuracy deepCNN (our method) 82.5 Stanford deepCNN [7] 72 RF-JA+C(h-h) [8] 90 RF-JA+C(l-o-o) [8] 70 Figure 12. Comparison of previous methods with ours; Stanford didn’t use background subtraction, RF-JA(h-h) split the training and validation set 50-50, (l-o-o) omitted specific data. 6. Conclusions and Future Work In this paper, we described a deep learning approach for a classification algorithm of American Sign Language. Our results and process were severely affected and hindered by skin color and lighting variations in our self-generated data which led us to resort to a pre-made professionally constructed dataset. With a camera like Microsoft’s Kinect that has a depth sensor, this problem is easy to solve [5]. However, such cameras and technology are not widely accessible, and can be costly. Our method shows to have potential in solving this problem using a simple camera, if enough substantial training data is provided, which can be continuously done and added via the aforementioned processing pipeline. Since more people have access to simple camera technologies, this could contribute to a scalable solution. In recognizing that classification is a limited goal, we plan on incorporating structured PGMs in future implementations of this classification schema that would describe the probability distributions of the different letters’ occurrences based on their sequential contexts. We think that by accounting for how the individual letters interact with each other directly (e.g. the likelihood for the vowel ‘O’ to proceed the letter ‘J’), the accuracy of the classification would increase. This HMM approach with sequential pattern boosting (SP-boosting) has been done with the actual gesture units that occur in certain gestures’ contexts, i.e. capturing the upper-arm movements that precede a certain letter to incorporate that probability weight into the next unit’s class, [6] and processing sequential phonological information in tandem with gesture recognition [4], but not for part-of-word tagging with an application like what we hope to achieve. We also recognize that the representation itself makes a huge difference in the performance of algorithms like ours, so we hope to find the best representation of our data, and building off our results from this research, incorporate it into a zero-shot learning process. We see zero-shot learning as having the potential to facilitate the translation process from American Sign Language into English. Implementing one-shot learning for translating the alphabet and numbers from American Sign Language to written English, and comparing it with a pure deep learning heuristic could be successful and have the potential to benefit from error correction via language models. Recent implementations of one-shot adaptation have also had success in solving real world computer vision tasks, and effectively trained deep convolutional neural networks using very little domain-specific data, even as limited as single-image datasets. We ultimately aim to create a holistic and comprehensive representation learning system for which we have designed a set of features that can be recognized from simple gesture images that will optimize the translation process. 7. References [1] X. Chen and A. Yuille. Articulated pose estimation by a graphical model with image dependent pairwise relations. In Advances in Neural Information Processing Systems (NIPS), 2014. [2] T. Pfister, J. Charles, and A. Zisserman. Flowing convnets for human pose estimation in videos. In IEEE International Conference on Computer Vision, 2015. [3] Barczak, A.L.C., Reyes, N.H., Abastillas, M., Piccio, A., Susnjak, T. (2011), A new 2D static hand gesture colour image dataset for ASL gestures, Research Letters in the Information and Mathematical Sciences, 15, 12-20 [4] Kim, Taehwan & Livescu, K & Shakhnarovich, Greg. (2012). American sign language fingerspelling recognition with phonological feature-based tandem models. In IEEE Spoken Language Technology Workshop (SLT), 119-124. [5] Agarwal, Anant & Thakur, Manish. Sign Language Recognition using Microsoft Kinect. In IEEE International Conference on Contemporary Computing, 2013. [6] Cooper, H., Ong, E.J., Pugeault, N., Bowden, R.: Sign language recognition using sub-units. The Journal of Machine Learning Research, 13(1), 2205–2231, 2012. [7] Garcia, Brandon and Viesca, Sigberto. Real-time American Sign Language Recognition with Convolutional Neural Networks. In Convolutional Neural Networks for Visual Recognition at Stanford University, 2016. [8] Cao Dong, Ming C. Leu and Zhaozheng Yin. American Sign Language Alphabet Recognition Using Microsoft Kinect. In IEEE International Conference on Computer Vision and Pattern Recognition Workshops, 2015.	1
1 arXiv:1705.10091v1 [cs.IT] 29 May 2017 Rate (n − 1)/n Systematic MDS Convolutional Codes over GF(2m) Ángela Barbero Universidad de Valladolid 47011 Valladolid, Spain Email: [email protected] Øyvind Ytrehus Simula@UiB and University of Bergen N-5020 Bergen, Norway Email: [email protected] Abstract A systematic convolutional encoder of rate (n − 1)/n and maximum degree D generates a code of free distance at most D = D + 2 and, at best, a column distance profile (CDP) of [2, 3, . . . , D]. A code is Maximum Distance Separable (MDS) if it possesses this CDP. Applied on a communication channel over which packets are transmitted sequentially and which loses (erases) packets randomly, such a code allows the recovery from any pattern of j erasures in the first j n-packet blocks for j < D, with a delay of at most j blocks counting from the first erasure. This paper addresses the problem of finding the largest D for which a systematic rate (n − 1)/n code over GF(2m ) exists, for given n and m. In particular, constructions for rates (2m − 1)/2m and (2m−1 − 1)/2m−1 are presented which provide optimum values of D equal to 3 and 4, respectively. A search algorithm is also developed, which produces new codes for D for field sizes 2m ≤ 214 . Using a complete search version of the algorithm, the maximum value of D, and codes that achieve it, are determined for all code rates ≥ 1/2 and every field size GF(2m ) for m ≤ 5 (and for some rates for m = 6). I. I NTRODUCTION 1 In many practical communication applications, such as multimedia transmission over packet erasure channels, on-time delivery is an important quality-of-service criterion. Traditional ARQ systems, for example the one used by TCP for transport layer unicast service, suffer from long delays due to erasures when the round-trip time is large. This has led to an increased interest in the design and analysis of systems based on packet-level error correcting codes. Such coded schemes are also known to be beneficial in other transport layer models, for example in the multi-path case. Two main approaches to this coding problem have been discussed in the literature. The deterministic approach [1], [2], [3] is to send packets using a fixed 2m -ary convolutional code with a good column distance profile. This approach is discussed in Subsection II-B. Random coding was proposed as a solution in [4], [5]. In these schemes, the sender transmits k uncoded information packets, followed by n − k parity check packets formed by random linear combinations of all information packets that have not been acknowledged by the receiver so far. Subsection II-C describes this approach, and also discusses a hybrid approach that combines deterministic and random coding. A. Contributions We present new codes in Section III. In Section III-A we present two new, general, and optimum constructions of MDS convolutional codes. In the literature, there exist only a few general constructions of high-rate convolutional codes: As far as we know, only the Wyner-Ash code [6] and their binary generalizations ([7], and Thms. 7.10 and 7.13 in [8]). We present a simple (but as far as we can see, not previously described in the literature) distance-3 construction. This code has the same rate and Viterbi complexity as the binary Wyner-Ash code, but has a better column distance profile. We also present a much more interesting algebraic distance-4 construction in Proposition 3. In Section III-B we describe a search algorithm and in Section III-C we present the codes found by the algorithm. For most parameters, these codes are better (in a sense which will be made more precise) than previously known codes. Further, we present simple upper bounds in Section IV. By convention we will call a convolutional code systematic if is it has a systematic encoder; i.e. one that preserves all information symbols and obtains redundancy by extra parity symbols. 2m -ary systematic rate (n − 1)/n convolutional encoders are useful in order to obtain fast recovery of packet erasures in the common case of channels with moderate erasure rates, and we will focus only on this class of codes. 1 This work is supported by Ministerio de Economı́a, Industria y Competitividad, Gobierno de España, through project MTM2013-46949-P, the Estonian Research council through project EMP133, the Norwegian Research Council through the SARDS project. 2 II. BACKGROUND A. Notation For a thorough introduction to convolutional codes, please see [9]. In the following we will describe the concept of a 2m -ary MDS convolutional code in a way which is convenient for our purposes in this paper. Let m ≥ 1, n ≥ 2, k = n − 1 be integers, F = GF(2m ), and define the matrices and vectors R0 = (r0,1 , . . . , r0,k ) ∈ Fk where Fk is the k-dimensional space of row vectors over F, H0 = (R0 \|1) ∈ Fn , where Fr×c denotes the space of matrices with r rows and c columns over F. For i > 1 define Hi−1 Ri = (ri,1 , . . . , ri,k \|0) ∈ Fk , Hi = ∈ F(i+1)×n Ri and, for an integer L ≥ 2, let H (L) 0(L−1)×n 01×n = (HL , ,..., ) ∈ F(L+1)×n(L+1) , HL−1 H0 (1) where 0r×c is all-zero matrix with r rows and c columns. Then H (L) is the parity check matrix for the Lth truncated block code C (L) of a (systematic) convolutional code C , thus any vector of length (l + 1)n for l ≤ L, (0) (0) (1) (1) (l) (l) [v]l = (v1 , . . . , vn , v1 , . . . , vn , . . . , v1 , . . . , vn ) ∈ F(l+1)n is a codeword in C (l) if and only if the syndrome > (l+1)×1 H (l) [v]> . l = (0, . . . , 0) ∈ F A systematic encoder for the code C (L) is represented  G0 G1  G0  G(L) =   by ··· ··· .. .  GL GL−1   k(L+1)×n(L+1) ..  ∈ F .  (2) G0 where k×n k×n G0 = (Ik \|R> , Gi = (0k \|R> for i > 0, 0)∈F i )∈F > and Ik and 0k are the k ×k identity and zero matrices, respectively. It is straightforward to verify that G(L) ×H (L) = 0k(L+1)×(L+1) . Example 1. Let F = GF(23 ) with primitive element α defined by  1 1 1 0 H (2) =  1 α 0 1 α3 1 0 1 and α 3 + α + 1 = 0. Then the parity and generator matrices  0 0 0 0 0 1 1 0 0 0 α 0 1 1 1  1 0 1 0 0 0 1 1 0 0  0 0 0 1 0 (2) G = 0 0 0 0 1  0 0 0 0 0 0 0 0 0 0 1 α 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1  α3 1  1  α  1 1 define a truncated code C (2) , which is rate 6/9 block code over F. Note that the matrices are completely determined by the parity check coefficients ri, j , i = 0, . . . , L, j = 1, . . . , k. In the conventional polynomial notation of convolutional codes [9], the parity check matrix can be described as D D H(x) = ( ∑ ri,1 xi , . . . , ∑ ri,k xi , 1) ∈ F[x]. i=0 In Example 1, H(x) = (1 + x + α 3 x2 , 1 + αx + x2 , 1). i=0 Similarly, the corresponding polynomial generator matrix is 1 0 1 + x + α 3 x2 G(x) = 0 1 1 + αx + x2 3 B. MDS convolutional codes constructed from superregular matrices In the deterministic approach [1], [2], [3], the goal is to design codes with an optimum column distance profile, which we will define below. The l-th column distance dl = dl (C ) of a convolutional code C is the minimum Hamming weight of any truncated (0) (0) codeword [c]l with the first block (c1 , . . . , cn ) nonzero, and the column distance profile (CDP) is the non-decreasing sequence (d0 , d1 , d2 , . . . , dD = D, D, D, . . .), where D is the free distance of the code and D is the index for which the CDP reaches D. The CDP was originally studied for its significance on the performance of sequential decoding (please see Ch. 13 of [9].) Recently the CDP has received renewed attention in the context of 2m -ary codes, due to its importance for fast recovery from losses of symbols in an erasure channel. Recall that we consider only convolutional codes of rate k/n = (n − 1)/n that have a systematic encoder. In this case, by the Singleton bound for truncated block codes, d0 ≤ 2, and by similar linear algebra arguments, dl ≤ dl−1 + 1 for l > 0. Moreover, dl = dl−1 for l > D. So the best column distance profile one can hope to find in a code with a systematic encoder is d0 = 2, d1 = 3, . . . , d j = j + 2, . . . , dD = D + 2 = D. (3) By an MDS convolutional code, in this paper we will mean a code with a CDP as in (3). Remark 1. The concept of Strongly-MDS codes was introduced in [2]. This concept takes into account that for some codes that do not possess a systematic encoder, the free distance may grow beyond δ + 2, where δ is the memory of a minimal encoder. In order not to complicate the notation, and since Viterbi complexity is not an issue in this paper, we omit the details. Definition 1. Consider a lower triangular matrix     SR =    r0 r1 r2 .. . 0 r0 r1 .. . 0 0 r0 .. . ··· ··· ··· .. . 0 0 0 .. . rL rL−1 rL−2 ··· r0        where each element ri ∈ F. Consider a square submatrix P of size p of SR, formed by the entries of SR in the rows with indices 1 ≤ i1 < i2 < · · · < i p ≤ (L + 1) and columns of indices 1 ≤ j1 < · · · < j p ≤ (L + 1). P, and its corresponding minor, are proper if jl ≤ il for all l ∈ {1, . . . , p}. SR is superregular if all its proper p × p minors are non singular for any p ≤ L + 1. When matrix SR is upper triangular the definition of proper submatrices is analogous. A γ × γ superregular matrix can be used to construct a rate 1/2 code in two ways: (1) [1] a systematic MDS convolutional code with CDP as in (3) with D = D + 2 = γ + 1, (2) [2] when γ = 2δ + 1, a strongly-MDS code (in general nonsystematic) with a parity check matrix of max degree δ and the same CDP as for the systematic codes in case (1). While superregular matrices are known to exist for all dimensions if the field is large enough, general efficient constructions are not known, and for γ & 10 the minimum field size for which a γ × γ superregular matrix exists is not known. Another problem with the deterministic approach is that the existing design methods do not allow a simple construction of codes of high rate and/or high degree. Codes of higher rates (which are desirable in many practical cases) can also be constructed from these superregular matrices, but this involves deleting columns, so that the conditions on a superregular matrix are too strict. This means that in practice only simple codes can be constructed in this way. Since superregular matrices are so hard to construct, the reduction to the superregular matrix problem blocks the code construction. Therefore we generalize Definition 1 as follows: Definition 2. Consider an s-lower triangular matrix (where s is a positive integer)  r0,1 ··· r0,s 0 ··· 0 0 ··· 0  r1,1 · · · r r · · · r 0 · · · 0 1,s 0,1 0,s   r2,1 · · · r r · · · r r · · · r 2,s 1,1 1,s 0,1 0,s  SSR =  .. .. .. .. .. .. .. .. ..  . . . . . . . . .   rL−1,1 · · · rL−1,s rL−2,1 · · · rL−2,s rL−3,1 · · · rL−3,s rL,1 · · · rL,s rL−1,1 · · · rL−1,s rL−2,1 · · · rL−2,s ··· ··· ··· 0 0 0 .. . ··· ··· 0 · · · r0,1 ··· ··· ··· .. . ··· ··· 0 0 0 .. .         0  (4) r0,s Consider a square submatrix P of size p of SSR, formed by the entries of SSR in the rows with indices 1 ≤ i1 < i2 < · · · < i p ≤ (L + 1) and columns of indices 1 ≤ j1 < · · · < j p ≤ s(L + 1). P, and its corresponding minor, are proper if jl ≤ s · il for all l ∈ {1, . . . , p}. The matrix SSR is called s-superregular iff all of its proper p × p minors, for any p ≤ L + 1, are nonsingular. 4 Rate 1/2 1/2 1/2 1/2 2/3 2/3 3/4 Field size 4 8 8 32 16 64 16 D 4 6 6 8 5 5 3 Table I Description [11] [11] [2] superregular [2] superregular [2] ad hoc [2] superregular [2] superregular S OME RATE (n − 1)/n MDS CODES ( NOT NECESSARILY SYSTEMATIC ) DESCRIBED IN THE LITERATURE . The following lemma is a restatement of Theorem 1 in [1], using the terminology of this section. Lemma 1. Let H (D) be the parity check matrix of the D − th truncation of a systematic convolutional code, given by  r0,1 r1,1 r2,1 .. .     (D) H =    rD−1,1 rD,1 ··· ··· ··· .. . r0,k r1,k r2,k .. . 1 0 0 .. . ··· ··· rD−1,k rD,k 0 0 r0,1 r1,1 .. . ··· ··· ··· .. . rD−2,1 rD−1,1 ··· ··· 0 0 r0,k r1,k .. . 0 1 0 .. . rD−2,k rD−1,k 0 0 r0,1 .. . ··· ··· ··· .. . r0,k .. . 0 0 1 .. . rD−3,1 rD−2,1 ··· ··· rD−3,k rD−2,k 0 0 0 0 0 0 ··· ··· ··· 0 0 0 .. . ··· ··· 0 · · · r0,1 ··· ··· ··· .. . 0 0 0 .. . ··· ··· 0 r0,k 0 0 0 .. .         0  1 (5) and let H 0(D) be the matrix obtained from H (D) by removing the columns in positions (k + 1), 2(k + 1), 3(k + 1), . . . , (D + 1)(k + 1), that is   r0,1 ··· r0,k 0 ··· 0 0 ··· 0 ··· 0 ··· 0  r1,1 ··· r1,k r0,1 ··· r0,k 0 ··· 0 ··· 0 ··· 0     r2,1 ··· r2,k r1,1 ··· r1,k r0,1 ··· r0,k ··· 0 ··· 0    0(D) (6) H = .. .. .. .. .. .. .. ..  .. .. .. ..   . . . . . . . . . . · · · . .    rD−1,1 · · · rD−1,k rD−2,1 · · · rD−2,k rD−3,1 · · · rD−3,k · · · 0 · · · 0  rD,1 · · · rD,k rD−1,1 · · · rD−1,k rD−2,1 · · · rD−2,k · · · r0,1 · · · r0,k Then the CDP of the convolutional code given by H (D) is (2, 3, . . . , D + 2) if and only if H 0(D) is a k-superregular matrix. Theorem 1 in [1] is stated without proof. For reference, we include a formal proof in Appendix A. Definition 3. Let ∆(2m , n) be the largest free distance D such that there exists a rate (n − 1)/n systematic MDS convolutional code over GF(2m ) with column distance profile as in (3). The main problem that we address in this paper is to determine exact values, or constructive lower bounds, for ∆(2m , n). Please note that there is no restriction of the degree D in Definition 3. There are few known code constructions in the literature, beyond those based on superregular matrices. Table I contains the current world records with respect to rate (n − 1)/n MDS codes, to the best of our knowledge. We will describe new codes in Section III. Although this paper focuses on rate (n − 1)/n MDS codes, we observe that the following lemma, that follows directly from Theorem 2 in [1], implies that our results will also provide rate 1/n MDS codes. Lemma 2. If a systematic rate (n − 1)/n MDS code of memory D and free distance D + 2 exists, then its dual code is equivalent to a systematic rate 1/n MDS code of memory D and free distance (n − 1)(D + 1) + 1. C. Random convolutional codes In the terminology of this paper, the random approach [4], [5] consists of selecting the coefficients of ri j independently at random. The advantage of this is that one can pick codes with large degrees, and that over large fields the expected performance is “reasonably good”, although the exact loss compared to optimum average performance or optimum guaranteed worst case performance remains to be determined. Coefficients need to be transmitted in the headers of the data packets, but this represents only a small rate loss when large packets are transmitted. Proposition 1. Consider a hybrid scheme where the first blocks of coefficients ri, j (until time i = D) are selected fixed, and subsequent random coefficients ri, j for i > D are selected at random. Thus the parity check equation will be on the form H(x) = HCDP (x) + HRandom (x) (7) 5 where D D HCDP (x) = ( ∑ ri,1 xi , . . . , ∑ ri,k xi , 1) i=0 and i=0 ? HRandom (x)( ∑ i=D+1 ? Ri,1 xi , . . . , ∑ Ri,k xi , 0) i=D+1 where all Ri, j are nonzero randomly selected coefficients and where the degree of the random polynomials does not need to be fixed (except by the application protocol). Then the initial CDP (until time D) is not affected by the random part of the code construction. Proof. Obvious: Only the first component HCDP (x) of the parity check matrix determines the initial part of the CDP. Our suggestion is to use such hybrid codes, i. e. codes where the terms of degree 0, . . . , D of the parity check polynomials are preselected constants yielding an optimum initial column distance profile, while subsequent random parity checks are added as needed. This guarantees optimum recovery for the simplest and most likely erasure patterns, and hence better performance than random codes for light to moderate erasure patterns, while still allowing the degree to grow if required by the application. III. N EW CODES Gluesing-Luerssen et. al. [2] use superregular matrices to design codes. However, the authors also give examples of codes that are better than the ones constructed from superregular matrices, and note that ”..the abundance of (small) examples suggests that such a construction might be possible and might lead to smaller alphabets for given parameters than the construction ,[...] We will leave this as an open question for future research.” So here comes the future research. In this section we present constructions and a new search algorithm that, in combination, improve our knowledge of ∆(2m , n) for almost all sets of parameters, with respect to what we find in the literature. A. Codes with free distance D ∈ {3, 4} We present two optimum constructions, for D ∈ {3, 4}. For D = 3 the construction is simple, but we have not seen it presented in prior literature. We have tacitly assumed the following fact for the constant terms. Here comes the justification. Lemma 3. We can w.l.o.g assume r0,1 = · · · = r0,n = 1. Proof. If there is a r0, j equal to zero, then d0 < 2. We don’t want that. Then assume some nonzero r0, j 6= 1. If we multiply −1 the corresponding column of G(D) by r0, j , we obtain a new code with the same CDP and weight structure. Proposition 2. ∆(qm , qm ) = 3 for q prime and m ≥ 0. Proof. Select r0,i = 1 and r1,i , i = 1, . . . , qm − 1 as the qm − 1 distinct nonzero elements of GF(qm ). Without loss of generality, the parity check matrix of (1) takes the form 1 1 ··· 1 1 0 ··· 0 0 (1) H = 1 2 · · · qm − 1 0 1 · · · 1 1 1 1 ··· 1 0 ··· 0 H 0(1) = 1 2 · · · qm − 1 1 · · · 1 is qm -superregular because it is obvious that all the proper minors of sizes 1 and 2 are nonsingular. Clearly, d0 = 2 and d1 = 3. Remark 2. It is instructive to compare the construction of Proposition 2 with the binary Wyner-Ash codes [6]. Wyner-Ash codes were considered for digital media transmission already in 1974 [10]. The Wyner-Ash code of length 4 has the binary polynomial parity check matrix HWA = 1 + x + x2 1 + x 1 + x2 1 . It is easy to see that the CDP of the Wyner-Ash code is [2, 2, 3], i. e. this is not an MDS code. The construction of Proposition 2 can be considered as a qm -ary generalization of the Wyner-Ash code, of memory 2, but this code is an MDS code, with CDP [2, 3]. For D = 4, we present an optimum construction in Proposition 3. Complete computer searches for m ≤ 5 indicate that the construction is unique and, in a sense, much better than what can be achieved through other choices of the set of first degree coefficients {r1,i }. 6 Lemma 4. For a code with a CDP of [2, 3, 4], its parity check matrix H (2) must satisfy (i) ri,s 6= 0 for i = 1, 2, s = 1, . . . , k, (ii) ri,s 6= ri,t for i = 1, 2, 1 ≤ s < t ≤ k, (iii) r1,t 6= r2,s /r1,s for 1 ≤ s,t ≤ k, (iv) r2,s /r1,s 6= r2,t /r1,t for 1 ≤ s < t ≤ k, (v) r2,s − r2,t 6= r1,u (r1,s − r1,t ) for 1 ≤ s < t ≤ k, 1 ≤ u ≤ k, (vi) r2,s 6= (r1,s (r2,u − r2,t ) − r1,t r2,u + r1,u r2,t )/(r1,u − r1,t ) for 1 ≤ s < t < u ≤ k. Proof. From Lemma 1 we need H 0(2) to be k-superregular. That all 1 × 1 proper minors of H 0(2) are non singular is equivalent to condition (i). Proper minors of size 2 × 2 are of the following types 1 ri,s 1 0 , ri,s r1,t 1 0 , r2,s 1 r 1 , 1,s r2,s ri,t r 1 , 1,s r2,s r1,t r1,t r2,t The first type are trivially non zero, the second type are non zero when condition (i) is satisfied. The third type being nonsingular is equivalent to condition (ii), the fourth type is guaranteed to be non zero if and only if condition (iii) is satisfied and the fifth type being nonsingular is equivalent to condition (iv). Finally, 3 × 3 proper minors can be of four different types: 1 r1,s r2,s 0 1 r2,t 1 0 0 , r1,s r2,s 1 1 r1,t r2,t 1 0 0 , r1,s r2,s 1 1 r1,t r2,t 0 1 r1,u 1 , r1,s r2,s 1 r1,t r2,t 1 r1,u r2,u Those of the first type are trivially nonsingular. Condition (ii) takes care of those in the second type to be nonsingular. Those in the third type are nonsingular if and only if condition (v) is satisfied. The fifth type are non singular if and only if condition (vi) is satisfied. Example 2. Consider the code in Example 1. By checking conditions (i)-(vi) in Lemma 4 we observe that the code has CDP equal to [2,3,4]. Proposition 3. ∆(2m , 2m−1 ) = 4. Proof. Let F = GF(2m ). (≥:) The following construction gives a code that meets the requirements: The trace function [12] is defined by Trm () : F x → GF(2) 2i → Trm (x) = ∑m−1 i=0 x . Consider the set Hβ = {x ∈ F\|Trm (β x) = 0}. When F is regarded as an m-dimensional vector space over GF(2), the set Hβ is a hyperplane (an (m − 1)-dimensional linear subspace) of F. Let k = 2m−1 − 1, select β as an arbitrary nonzero field element, select c as an arbitrary constant in F \ Hβ . Then select a1 , . . . , ak := r1,1 , . . . , r1,k as all distinct nonzero elements in Hβ , and set bs := r2,s = as (as + c) = r1,s (r1,s + c) for s = 1, . . . , k. We need to verify that this construction satisfies the conditions in Lemma 4. (i) This holds because bs = as (as + c) is a product of two nonzeros. (ii) All as ’s are distinct. Assume that bs = bt , s 6= t. Then 0 = as (as + c) = at (at + c) = (as + at )c + a2s + at2 = (as + at )c + (as + at )2 = (as + at )(c + as + at ). The first factor is nonzero since as 6= at . The second factor is also nonzero since as + at ∈ Hβ (because Hβ is closed under addition) while c 6∈ Hβ , a contradiction. (iii) Assume that as at = bs . Then as at = as (as + c) ⇒ at = as + c, a contradiction, since at ∈ Hβ and as + c 6∈ Hβ . (iv) Assume that bs /as = bt /at , s 6= t. Then as + c = at + c ⇒ as = at , a contradiction. (v) bs + bt + au (as + at ) = as (as + c) + at (at + c) + au (as + at ) = a2s + at2 + (as + at )(c + au ) = (as + at )2 + (as + at )(c + au ) = (as + at )(as + at + c + au ) 7 which again is a product of nonzero factors, because c 6∈ Hβ and as + at + au ∈ Hβ , and hence nonzero. (vi) bs + as (bt + bu ) + au bt + at bu at + au as (at (at + c) + au (au + c)) + au at (at + c) + at au (au + c) at + au as (at + au )2 + as c(at + au ) + at au (at + au ) = as (as + c) + at + au = as (as + c) + as (at + au ) + as c + at au = as (as + c) + = a2s + as at + as au + at au = (as + at )(as + au ) 6= 0. (≤:) This follows from Theorem 1 in Section IV. Remark 3. By Theorem 1 later, the construction in Proposition 3 is optimum not only in the sense that it offers the maximum distance 4 for the given field size and code rate, but also it offers the minimum field size for a code of the given rate and distance 4, and the maximum code rate given the field size and distance 4. Moreover, complete computer search for field sizes 2m ≤ 32 show that the construction is unique for these parameters. B. Computer search algorithm The goal of the search algorithm is to select the coefficients ri, j successively, ordered first on i and then reversely on j, in such a way that the conditions on the minors are met. 1) Some useful facts: First, as for the constructions in Section III-A, we use Lemma 3 in order to set r0,1 = · · · = r0,k = 1. In order to simplify the search we apply the following results. Lemma 5. We can w.l.o.g assume r1,i < r1,i+1 , i = 1, . . . , k − 1 for any choice of ordering <. Lemma 6. Consider an MDS convolutional code C with polynomial parity check matrix D D H(x) = (1 + ∑ ri,1 xi , . . . , 1 + ∑ ri,k xi , 1) ∈ F[x]. i=1 i=1 Then the code Cc with parity check matrix D D Hc (x) = (1 + ∑ ci ri,1 xi , . . . , 1 + ∑ ci ri,k xi , 1) ∈ F[x] i=1 i=1 is also MDS for any c ∈ F \ {0}. D i i > > Proof. Let v(x) = (v1 (x), . . . , vn (x)) = (∑D i=0 v1,i x , . . . , ∑i=0 vn,i x ). Then v(x)H(x) = 0 iff vc (x)Hc (x) = 0 for D D i=0 i=0 vc (x) = ( ∑ c−i v1,i xi , . . . , ∑ c−i vn,i xi ). Corollary 1. If a systematic MDS convolutional code exists, we can w.l.o.g. assume that it has a parity check matrix with r1,k = 1. Proof. Assume that a systematic MDS convolutional code exists with a parity check matrix with r1,k = a ∈ F. Then apply Lemma 6 with c = a−1 . Lemma 7. Let M be a k-superregular matrix over GF(qm ), with q a prime. Raising each element of M to power q yields another k-superregular matrix. Proof. Given any square matrix  a1,1  .. A= . an,1  · · · a1,n ..  .  · · · an,n with ai, j ∈ GF(qm ), by definition det(A) = ∑ s(σ )a1,σ (1) · · · an,σ (n) σ ∈Sn where Sn is the group of permutations of n elements and s(σ ) is the sign of each permutation σ . 8 Also we have (x1 + x2 + · · · + xn )q = c(q1 , . . . , qn )x1q1 x2q2 · · · xnqn ∑ q1 + · · · + qn = q 0 ≤ qi ≤ q q−q1 −···−qn−1 1 . and c(q1 , . . . , qn ) = qq1 q−q qn q2 · · · When q is prime, q is a divisor of coefficient c(q1 , . . . , qn ) except in the cases qi = q, q j = 0 for j 6= i, and this for each i ∈ {1, . . . , n}. Therefore, in characteristic q we have (x1 + x2 + · · · + xn )q = x1q + x2q + · · · + xnq Back to the definition of determinant, in characteristic q we have !q (det(A))q = ∑ s(σ )a1,σ (1) · · · an,σ (n) = ∑ σ ∈Sn σ ∈Sn s(σ )q aq1,σ (1) · · · aqn,σ (n) Finally, s(σ ) is either 1 or -1, so in case q = 2 then s(σ ) = 1 for any σ ∈ Sn , and s(σ )2 = s(σ ), and in case q is odd we have s(σ ) = s(σ )q . This gives (det(A))q = ∑ σ ∈Sn s(σ )q aq1,σ (1) · · · aqn,σ (n) = ∑ σ ∈Sn s(σ )aq1,σ (1) · · · aqn,σ (n) = det(A◦q ) where A◦q denotes the q-th Hadamard or Schur power of A, that is, the matrix whose entries are the the entries of A raised to q. Now it is clear that given any proper minor P of size p of matrix M in GF(qm ), P◦q is the corresponding proper minor in ◦q M and P is nonsingular if and only if P◦q is nonsingular, so M is k-superregular if and only if M ◦q is k-superregular. Corollary 2. In particular, let M be a k-superregular matrix over GF(2m ). Squaring each element of M yields another k-superregular matrix. Proof. This is just the particular case for q = 2 of the Lemma above. Hence we also have: Corollary 3. Assume that the values for r0,i , i = 1, . . . , k and for r1,k are all fixed to 1, as allowed by Lemma 3 and Corollary 1. Then, for r1,k−1 , it suffices to consider one representative of each cyclotomic coset. Proof. Consider any minor of M. Squaring all coefficients in M will not change the values of r0,i , i = 1, . . . , k or r1,k . Thus if there is a k-superregular matrix with r1,k−1 = v, then there is also a k-superregular matrix with r1,k−1 = v2 . The search can be simplified (by constant factors O((2m − 1)n · (2m − 1) · (n − 2)!) by use of Lemma 3, Corollary 1, and Lemma 5, respectively. Corollary 3 reduces complexity by an extra factor of approximately log2 (2m ) = m, but this reduction is not entirely independent of the other reductions. In summary, the search algorithm is highly exponential in complexity, but the tricks allow a deeper search than would otherwise be possible. The search algorithm is sketched in Algorithm 1. The trickier steps are explained in some detail in Remark 4. Remark 4. Here we explain the steps of Algorithm 1. (i) In essence, the algorithm runs through a search tree. At each depth of the tree, ρ points to one of the variables ri, j in (6). Abusing notation, we will also say that ρ points to the current depth. Throughout the course of the algorithm, ρ goes back and forth along r1,k−1 , . . . , r1,1 , r2,k , . . . , r2,1 , r3,k , . . ., i. e. along the values of the last row of (6) in reverse order starting at r1,k−1 (since r1,k , r0,1 , . . . , r0,k can be assumed to be all equal to 1 by Lemma 3 and Corollary 1.) We will in this context use the ordering “<” to refer to the reverse order of the last row of (6), and addition and subtraction on ρ moves ρ left and right, respectively, on this row. (ii) Line 3: Let ρ refer to one element ri, j in the last row of (6). Then Mρ is the set of (formal) proper submatrices of SrD0 ,1 (6) which have ρ in its left lower corner. If all matrices in M = ρ=r Mρ for some D0 ≤ D are nonsingular, where 0,k ∗ D = D − 2 is the target maximum degree, then the submatrix of (6) with rD0 ,1 in the lower left corner is k-superregular. The set Mρ can be found by recursion. (The number of proper submatrices M is related to the Catalan numbers. We omit the details.) (iii) Line 5: At depth ρ, values have already been assigned for each depth ρ 0 < ρ. Hence, by keeping track of subdeterminants already computed, for each determinant corresponding to a proper submatrix in Mρ , the value in GF(2m ) that would 9 Algorithm 1: A computer search algorithm Result: finds good 2m -ary MDS codes of rate (n − 1)/n Input : Field size 2m , target distance D ∗ , code length n Data: ρ points to current position 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 initialization; value(r0,i ) := 1, i = 1, . . . , k, value(r1,k ) = 1; SrD ∗ −2,1 Precompute the set of proper submatrices M = ρ=r Mρ ; 1,k−1 ρ := r1,k−1 ; Precompute the set of legal values L (ρ); while ρ ≤ rD ∗ −2,1 and more coefficient values to check for ρ do if more coefficient values to check for ρ then assign next value to coefficient at ρ; update determinants needed for Mρ+1 , and L (ρ + 1); if deepest level so far then record selected values of coefficients; end ρ = ρ + 1; else ρ = ρ − 1; end end make the determinant zero can be obtained in constant time. In other words, going once through Mρ , we can identify the set L (ρ) of all illegal values for coefficient ρ. (iv) Line 6: (a) A complete search version of the algorithm will successively try all values in L (ρ). For a faster but incomplete search, the algorithm may be set to skip an arbitrary subset of values in L (ρ) at each depth ρ. (b) The target distance D ∗ is an input parameter for the algorithm. In order to determine that a code has a maximum distance D, it is necessary to verify that a complete search version of the algorithm will pass depth ρ = rD,1 , but not depth ρ = rD+1,1 . (v) Line 9: Using the set of values currently assigned to coefficients at all depths ρ 0 ≤ ρ, compute subdeterminants that will be useful for computing determinants in Mρ+1 , and initialize the set of legal values L (ρ + 1) for the next depth. (vi) Complexity: The assumptions enabled by the Lemmas of this section, together with the efficient computation of the determinants, allow a deeper search than would be possible with a naı̈ve search. However, the depth of the search tree that finds a code of degree D is (n − 1)D, and for many of the “early depths”, a complete search needs to go through almost 2m values. The size of the set of proper submatrices also grows exponentially with n and D. So the overall ∗ complexity is at least O(2mn·(D −2) · \|M \| · wd ) where \|M \| is the number of proper submatrices. C. Codes found by computer search with D ≥ 5 Here we present codes found from computer search, for field sizes of characteristic 2 ranging from 8 to 16384, and free distances D ≥ 5. Exact values of ∆(qm , qm ) = 3 and ∆(qm , qm−1 ) = 4 are provided by Propositions 2 and 3. In Tables II–XIII, each row summarizes what we have discovered about rate k/n = (n − 1)/n codes. The ∆ column lists the maximum value of D for which we have found a code with CDP [2, 3, . . . , D]. The absence of a ≥ sign in this column indicates that we have established, through an exhaustive search, that this value of D is indeed maximum for this rate and field size. The Coefficients column presents one encoder that possesses this CDP, in terms of logα () of the coefficients (r1,k , . . . , r1,1 ), (r2,k , . . . , r2,1 ), (r3,k , . . . , r3,1 ), . . ., where α is the primitive element of the field. Note that the degree zero terms, (r0,1 , . . . , r0,k ) are suppressed since they are assumed to be identically 1 = α 0 . The R column contains the rareness of the code, which will be explained in Section IV-B. In the Reference column we include references in the few cases where “similar codes” (i. e. codes over the same field which have the same CDP, but that do not necessarily possess a systematic encoder) have previously been described in the literature. We do not list encoders found by the search if we also found codes with the same set of parameters (rate, CDP) over a smaller field. Also, due to Lemma 8, we do not list codes of rate (n − 2)/(n − 1) if there exist codes of rate (n − 1)/n with the same CDP: Lemma 8. If a systematic MDS code with free distance D and rate k/(k + 1) exists for k > 1, then there is also a systematic MDS code with free distance D and rate (k − 1)/k. Proof. Shorten the k-superregular matrix H 0(D) by selecting any j0 ∈ {1, . . . , k} and removing columns j0 , j0 + k, j0 + 2k, . . . in H 0(D) . 10 Coefficients R Remark 0, 1, 4, 3 0.035 [11] Table II TABLE OF BOUNDS ON ∆(23 , n) FOR THE FIELD DEFINED BY 1 + α + α 3 = 0. n 2 ∆ 6 Coefficients R Remark 0, 1, 4, 3, 0 0.024 0 1, 4 0, 1 7 0.014 [2] Table III TABLE OF BOUNDS ON ∆(24 , n) FOR THE FIELD DEFINED BY 1 + α + α 4 = 0. P LEASE ALSO SEE E XAMPLE 3. n 2 3 ∆ 7 5 Example 3. According to Table III, for the finite field GF(24 ) defined by 1+α +α 4 = 0, there exists a systematic code of rate 2/3 and with CDP= [2, 3, 4, 5]. An example of such a code is represented by (r1,2 , r1,1 ), (r2,2 , r2,1 ), (r3,2 , r3,1 ) = (α 0 , α 1 ), (α 4 , α 0 ), (α 1 , α 7 ) = (1, α), (α 4 , 1), (α, α 7 ), and (implicitly) (r0,1 , . . . , r0,k ) =(1, . . . , 1). Thus the code has a polynomial parity check matrix H(x) = (1 + αx + x2 + α 7 x3 , 1 + x + α 4 x2 + αx3 , 1) and encoder/generator matrix G(x) = 1 0 1 + αx + x2 + α 7 x3 0 1 1 + x + α 4 x2 + αx3 Obviously, G(x)H(x)> = (0, 0). The absence of a ≥ symbol in the ∆ column in Table III indicates that a complete search of all systematic MDS codes of rate 2/3 reveals that D = 5 is maximum. The R column, as will be explained later, indicates that one in seventy random assignments of nonzero values for (r1,1 , r1,2 ), (r2,1 , r2,2 ), (r3,1 , r3,2 ) will give a code with the same CDP, i. e. codes with these parameters are not very rare. A nonsystematic code over GF(24 ) with degree δ = 2 and CDP= [2, 3, 4, 5] was presented in [2]. IV. U PPER BOUNDS AND CODE ASSESSMENT It would be useful to determine upper bounds on ∆(qm , n) in order to assess how good the codes from random search are with respect to optimum. The Heller bound [13] relates convolutional codes with a given free distance D with its truncated block codes, and uses known bounds on block codes to determine convolutional code parameters that cannot be achieved. Unfortunately the Heller bound is of limited use in our case, since the truncated code will actually have a much lower minimum distance than D when viewed as a block code, and also since exact bounds on block codes in the range of parameters that we are interested in here are not well known. Moreover, the approach of sphere packing for binary codes [14] cannot be easily adapted to the current case, since the structure of optimum nonbinary codes turns out to be quite different from that of optimum binary codes2 . A simple bound is described in the next subsection. In Subsection IV-B we present an alternative way of describing how great our codes are, through the concept of rareness. Coefficients R 0, 1, 19, 5, 24, 15, 0 3.4 · 10−8 0 1, 11 28, 21 6, 24 11 4.4 · 10−5 0 1 18 2, 5 8 17 25, 3 2 13 18 5.2 · 10−11 Table IV TABLE OF BOUNDS ON ∆(25 , n) FOR THE FIELD DEFINED BY 1 + α 2 + α 5 = 0. n 2 3 5 ∆ 9 6 5 Coefficients R 0, 1, 6, 61, 60, 46, 28, 23 1.2 · 10−10 0 1, 6 0, 2 37, 21 44, 55 28 4.1 · 10−11 0 1 6, 2 6 26, 13 61 38, 30 33 60 1.4 · 10−11 0 1 6 2 12 3, 14 36 26 25 51 13, 19 60 16 62 5 58 3.2 · 10−20 Table V TABLE OF BOUNDS ON ∆(26 , n) FOR THE FIELD DEFINED BY 1 + α + α 6 = 0. n 2 3 4 7 ∆ 10 7 ≥6 ≥5 2 Optimum binary convolutional codes tend to require parity check matrices with many r 1, j = 0, whereas we have seen that in the nonbinary case, all degree one coefficients r1, j are nonzero. These differences impose different combinatorial constraints in the binary and the nonbinary case. 11 Coefficients R 0 1 31 2, 62 103 64 125, 51 57 19 110, 11 39 43 114 8 · 10−18 0 1 31 2 62 32 103, 3 31 15 0 7 1 63, 6.4 · 10−16 8 94 119 51 41 10 17 Table VI TABLE OF BOUNDS ON ∆(27 , n) FOR THE FIELD DEFINED BY 1 + α 3 + α 7 = 0. n 5 8 n 2 3 4 11 ∆ ≥6 ≥5 Coefficients R 0, 1, 25, 3, 0, 198, 152, 56, 68 2.2 · 10−7 0 1, 25 0, 1 238, 100 106, 195 245, 37 33 2.0 · 10−12 0 1 25, 2 25 198, 1 14 228, 113 74 214, 21 250 172 ≈ 2 · 10−17 0 96 95 176 156 169 160 81 11 245, 107 5 223 167 7 177 98 238 93 53, ≈ 3 · 10−28 37 208 233 89 75 74 184 31 119 100 Table VII TABLE OF BOUNDS ON ∆(28 , n) FOR THE FIELD DEFINED BY 1 + α 2 + α 3 + α 4 + α 8 = 0. ∆ ≥11 ≥8 ≥7 ≥5 Coefficients R 0, 54, 91, 181, 267, 291, 379, 28, 95, 143 1.4 · 10−11 0 280 362 276 426, 206 155 326 324 360, 3.9 · 10−11 356 447 507 312 144, 224 375 236 55 448 13 ≥5 0 19 325 321 356 397 317 455 98 130 149 413, 8.4 · 10−27 48 101 120 272 209 188 405 352 46 343 289 152, 318 80 256 98 255 274 147 340 392 453 30 451 Table VIII TABLE OF BOUNDS ON ∆(29 , n) FOR THE FIELD DEFINED BY 1 + α 4 + α 9 = 0. n 2 6 n 3 5 8 17 n 2 4 ∆ ≥13 ≥8 9 ≥6 n 2 6 ∆ ≥14 ≥7 11 ∆ ≥12 ≥6 Coefficients R 0 603, 246 106, 115 693, 483 544, 603 152, 815 788, 984 721 ≈ 10−15 0 498 997 964, 560 214 101 723, 453 111 370 54, 5 · 10−18 455 17 625 509, 904 431 926 856 ≥6 0 322 804 12 140 1004 384, 778 916 786 247 586 698 294, 3 · 10−24 379 7 784 239 817 284 398, 178 588 110 41 425 976 393 ≥5 0 1 77 2 154 78 956 3 10 155 325 79 618 957 231 4, 4 · 10−39 308 0 4 77 11 1 200 10 80 3 24 155 87 325 619 618, 958 768 255 404 577 976 368 374 709 33 530 109 677 594 652 226 Table IX TABLE OF BOUNDS ON ∆(210 , n) FOR THE FIELD DEFINED BY 1 + α 3 + α 10 = 0. ∆ ≥9 ≥7 Coefficients 0, 1992, 813, 1890, 440, 630, 1947, 1574, 1356, 234, 1266 0 1809 1118, 2027 1610 539, 1042 7 1730, 2020 591 1459, 902 899 1584, 172 1192 513 0 1999 762 1845 1102 1115 1014 328, 1349 345 498 1561 27 987 1300 1793, 1728 562 488 304 43 71 1911 1140, 1524 660 465 327 322 748 1574 1414 Table X TABLE OF BOUNDS ON ∆(211 , n) FOR THE FIELD DEFINED BY 1 + α 2 + α 11 = 0. R 1.0 · 10−9 5.6 · 10−15 2.0 · 10−22 Coefficients R 0, 3294, 1040, 448, 3624, 2406, 826, 1122, 587, 1034, 342, 4037 < 10−15 0 3202 2711 92 2688, 3908 1649 1252 3897 1604, 3687 3602 1603 2339 1350, 1.2 · 10−14 1700 2969 104 3406 2679, 1345 919 3302 2116 810 ≥6 0 669 4050 4007 745 3863 324 1617 3951 1343, 3 · 10−31 703 1123 782 3343 1919 3177 1839 1006 2183 426, 2139 2050 1676 1187 3222 467 1764 2387 2868 641, 2564 2249 3187 3114 3228 743 443 1220 3540 2620 Table XI TABLE OF BOUNDS ON ∆(212 , n) FOR THE FIELD DEFINED BY 1 + α 3 + α 4 + α 7 + α 12 = 0. 12 Coefficients R 0 337, 7672 6843, 3625 3361, 7970 7490, 3.6 · 10−11 5531 2322, 5227 5758, 133 2290, 1453 189 5 ≥8 0 441 2192 3413, 3222 7502 7405 4155, 88 5939 343 6171, ≈ 5 · 10−21 1082 8149 2823 7269, 8022 6454 4999 3373, 3518 442 710 6968 7 ≥7 0 5160 5711 7681 748 5319, 2131 6233 723 4539 7315 5654, 2 · 10−19 5126 7465 3577 6826 5553 1131, 4954 6763 6593 1568 7157 8112, 1961 4310 877 2927 7197 2672 13 ≥6 0 5645 7651 3109 2678 802 6934 1946 5589 2833 5821 38, ≈ 8 · 10−37 5394 2500 5877 3141 4724 3374 5191 7218 4844 423 822 6875, 5712 6619 3935 6414 8025 1422 4391 5698 5481 6850 2635 4786, 556 2558 1063 5172 566 7978 3664 5848 3859 6905 6434 71 Table XII TABLE OF BOUNDS ON ∆(213 , n) FOR THE FIELD DEFINED BY 1 + α + α 3 + α 4 + α 13 = 0. n 3 n 4 ∆ ≥9 8 ≥7 15 ≥6 ∆ ≥10 Coefficients R 0 61 9533, 1260 4487 6469, 3689 8777 4510, 11257 13252 1239, 3 · 10−14 15121 10306 11679, 9618 13110 4549, 12420 5210 13006 0 14132 6404 8841 7620 6707 1150, 1.4 · 10−22 14939 8238 9174 9560 1677 4156 11112, 11424 2037 7827 4640 11071 14007 6628, 13374 10684 2080 14648 1097 14383 1198, 10966 15875 9746 9595 13007 4019 1354 0 15439 10581 4136 503 11096 5590 8608 16006 8229 562 15423 14311 16137, 2 · 10−38 5899 1875 8985 16334 15293 13429 5172 5303 9128 109 10068 1358 7752 6288, 13251 13386 11513 2438 443 15582 4641 2845 3509 12593 6608 14686 11470 15578, 8683 12489 444 8891 4727 12844 12383 5530 4478 9079 9226 5886 6790 8363 Table XIII TABLE OF BOUNDS ON ∆(214 , n) FOR THE FIELD DEFINED BY 1 + α + α 11 + α 12 + α 14 = 0. A. A simple bound The following simple bound is tight for D ≤ 4. Theorem 1. For rate (n − 1)/n codes over GF(qm ) with CDP = [2, 3, . . . , D], n − 1 ≤ (qm − 1)/(D − 2). Proof. For D = 3 the result follows from Proposition 2. Assume that D = 4. Recall that all coefficients are nonzero. Consider the 2 × 2 minors of type 1 r1,s 1 r = r1,s + r1,t , 1,s r1,t r2,s r r1,t = r1,s r2,t + r1,t r2,s , and 1,s r2,s r2,t 1 = r2,s + r1,s r1,t . r1,t (8) From the conditions on the 2 × 2 proper minors, since all those minors have to be nonzero, it follows that in order to have D > 3, the values in the sets {r1,1 , . . . , r1,k } and {r2,1 /r1,1 , . . . , r2,k /r1,k } must be 2k distinct values in GF(qm ) \ {0}. Now consider a code with D > 4. Then the minors r2,s r3,s r2,t r = r2,s r3,t + r2,t r3,s , 2,s r3,t r3,s 1 r = r2,s r1,t + r3,s , and 2,s r1,t r3,s r1,t = r2,s r2,t + r1,t r3,s . r2,t (9) Again they all have to be nonzero, and this implies that the set {r3,1 /r2,1 , . . . , r3,k /r2,k } is a new set of k different values, and they are all different from the values in the sets {r1,1 , . . . , r1,k } and {r2,1 /r1,1 , . . . , r2,k /r1,k }. So in order to have D ≥ 5 we need to have at least 3k different non zero elements in the field. Generalizing the argument, it follows that all ri,t /ri−1,t for 1 ≤ i ≤ D − 2, 1 ≤ t ≤ k are distinct nonzero values. B. Rareness In this section we address the probability that a randomly generated convolutional code over GF(2m ) of rate (n − 1)/n will be an MDS code with CDP of [2, . . . , D]. By “randomly generated” code we will mean one generated by a random systematic encoder, where each coding coefficient ri, j is selected independently and uniformly in GF(2m ) \ {0}. We define this probability as the rareness of the parameter pair (n, D). For small values of n and D, the exact value of the rareness can be determined as a by-product of a complete code search. Since for large parameters it quickly becomes intractable to determine the best codes, it also quickly turns difficult to compute exact results for rareness. However, it is possible to obtain estimates of rareness, as described below. 13 First assume that a complete search is applied. This will determine the set G (ρ ∗ , n, m) of distinct sequences r1,k−1 , . . . , r1,1 , r2,k , . . . , ρ ∗ over GF(2m ) for which all proper submatrices in Mρ 0 , ρ 0 ≤ ρ ∗ are nonsingular. Thus the probability that a given randomly selected sequence corresponds to a path in the search tree that satisfies the conditions at depth ρ ∗ is PR (ρ ∗ , n, m) = \|G (ρ ∗ , n, m)\| ∗ . (2m − 1)\|ρ \| For \|ρ ∗ \| > 1, define PR (ρ ∗ , n, m\|ρ ∗ − 1) = \|L ρ ∗ \| PR (ρ ∗ , n, m) = Avg( m ) ∗ PR (ρ − 1, n, m) 2 −1 (10) where Avg() is the average computed over the complete search. PR (ρ ∗ , n, m\|ρ ∗ − 1) is the average conditional probability that a random generator which satisfies depth ρ ∗ − 1 in the search tree also satisfies depth ρ ∗ . For large parameters we are not able to carry out a complete search. However, we can perform deep but incomplete searches, which also provide estimates of the conditional probabilities PR (ρ ∗ , n, m\|ρ ∗ − 1) in (10) as . These estimates will be quite accurate especially for the first depths, and hence they can be changed together to obtain an estimate for PR (ρ ∗ , n, m). As long as there is a substantial number of different search tree paths leading to depth ρ ∗ − 1, the estimate PR (ρ ∗ , n, m\|ρ ∗ − 1) should be reasonably good. Hence we can also estimate PR (ρ ∗ , n, m\|ρ ∗ − 1)) as ˜ P̃R (ρ ∗ , n, m\|ρ ∗ − 1)) = Avg( \|L ρ ∗ \| 2m − 1 ) ˜ where Avg() is the (weighted) average computed over the incomplete search, and we can then estimate PR (ρ ∗ , n, m) as ρ∗ P̃R (ρ ∗ , n, m) = ∏ P̃R (ρ − 1, n, m\|ρ − 1) ρ=(1,k−1) where for ρ = (1, k − 1), P̃R (ρ − 1, n, m\|ρ − 1) = 1. In Tables II–XIII, we include the exact rareness in cases where we can perform a complete search, and otherwise we include the estimate. We concede that this approach is not foolproof. For example, the construction in Proposition 3, is unique at least for field sizes up to 32. For other choices for the first layer of coefficients r1,1 , . . . , r1,2m−1 than indicated in the proof of Proposition 3, it appears that the search tree ends up being considerably shallower. The rareness of the construction in Proposition 3, i. e. the m probability that a random sequence will match that construction exactly, is 2m−1 · (2m−1 − 1)!/(2m − 1)2 −3 . Already for m = 5 −30 −393 the rareness is about 10 , for m = 8 less than 10 . Hence, if for an arbitrary set of search parameters there exists a very rare construction that is not caught by the incomplete search, the estimates for the deepest values of ρ ∗ may be unprecise. However, we do believe that our estimates of PR (ρ ∗ , n, m) provide some intuition about the difficulty of reaching a certain depth in the search tree with a random path, and in the cases where we are able to carry out a complete search, we also note that the estimates as described here are pretty accurate with a modest non-exhaustive search effort. Figure 1 contains exact values (for n = 2, 3) and estimates (for n = 4, 7) of PR (ρ ∗ , n, 6). Please see the figure caption for explanations. We have also include rareness estimates in Tables II–XIII. V. C ONCLUSION AND OPEN PROBLEMS Motivated by the practical problem of fast recovery of a coded packet-erasure channel, we have studied systematic MDS convolutional codes over GF(2m ). We have characterized them in terms of k-superregularity of a certain matrix. We have presented new optimum constructions for free distances D ≤ 4, tables of new codes found by computer search, and a combinatorial upper bound which is tight in the case of small free distances. In order to assess how “good” a code is, we have also introduced the concept of rareness. It would be interesting to establish upper bounds that are tight also for larger free distances. Another issue would be to study whether there exist general algebraic constructions, similar to the one in Proposition 3, for systematic MDS codes of free distance D ≥ 5. It would also be of some theoretical interest to optimize the CDP of strongly-MDS codes over GF(2m ) under an additional constraint on the degree δ of their minimal encoders. We have not considered this problem since the complexity of Viterbi decoding of such codes is prohibitive for all but small values of the product m · δ (and since it seems difficult). R EFERENCES [1] E. M. Gabidulin, “Convolutional codes over large alphabets,” in Proc. Int. Workshop on Algebraic Combinatorial and Coding Theory, Varna, Bulgaria, 1988, pp. 80—84. [2] Heide Gluesing-Luerssen, Joachim Rosenthal, and Roxana Smarandache, “Strongly-MDS Convolutional Codes,” IEEE Transactions on Information Theory, vol. 52, no. February 2006, 584–598. [3] Paulo Almeida, Diego Napp, and Raquel Pinto, “A new class of superregular matrices and MDP convolutional codes,” 14 1 4 10−2 10−6 10−8 10−10 10−12 10−14 10−16 log10 𝑃( random code is MDS) 10−4 9 6 10 5 7 6 Rate 1/2 Rate 2/3 10−18 Rate 3/4 Rate 6/7 5 Number of coefficients 𝜌 (starting with 𝑟1,𝑘−1 ) Figure 1. Rareness PR (ρ, n, 6) of codes for GF(64) for n ∈ {2, 3, 4, 7}: Exact rareness PR (ρ, n, 6) for ρ ≤ 7, estimates P̃R (ρ, n, 6) for n > 7. In the figure, the search depth ρ is measured in terms of number of coefficients. In order to construct a rate 6/7 encoder of distance D = 5, it is necessary to find a sequence of 17 coefficients r1,5 , . . . , r1,1 , r2,6 , . . . , r3,1 . To get an encoder with distance D = 4, it suffices with 11 coefficients. Similar for the other cases. [4] Pierre Ugo Tournoux, Emmanuel Lochin, Jérôme Lacan, Amine Bouabdallah, and Vincent Roca, “On-the-Fly Erasure Coding for Real-Time Video Applications,” IEEE Transactions on Multimedia, vol. 17, no. 4, (2011), pp. 797–812. [5] M. Kim, J. Cloud, A. Parandeh Gheibi, L. Urbina, K. Fouli, D. J. Leith, and M. Médard, “Network Coded TCP (CTCP),” http://arxiv.org/abs/1212.2291. [6] A. D. Wyner and R. B. Ash, “Analysis of recurrent codes,” IEEE Transactions on Information Theory,Vol. 9, Issue: 3, Jul. 1963, pp. 143 – 156. [7] Øyvind Ytrehus, “Ascetic convolutional codes,” Proc. 33rd. Allerton Conference on Communications, control, and Computing (October 1995), 382–390. [8] Robert J. McEliece, The Algebraic Theory of Convolutional Codes,, in: Handbook of Coding Theory, Eds. V. S. Pless and W. C. Huffman, pp. 1065–1138, North-Holland, 1998. [9] S. Lin and D. Costello, Error Control Coding, 2nd. Ed., Prentice-Hall, 2004. [10] J. H. Stott, A. Oliphant, D. W. Osborne, “Digital video: Error correcting codes and a practical study of a Wyner-Ash error corrector,” Techn. Report, British Broadcasting Corporation, December 1974. [11] J. Justesen and L. Hughes, “On maximum-distance-separable convolutional codes (Corresp.),” in IEEE Transactions on Information Theory, vol. 20, no. 2, pp. 288–288, Mar 1974. [12] F. J. MacWilliams and N. J. A. Sloane,, The Theory of Error-Correcting Codes, Elsevier, 1977. [13] J. A. Heller, “Sequential decoding: Short constraint length convolutional codes,” Space Programs Summary JPL, Pasadena, CA, 1968. [14] Eirik Rosnes and Øyvind Ytrehus, “Sphere-packing bounds for convolutional codes,” IEEE Transactions on Information Theory,Vol. 50, Issue: 11, Nov. 2004, pp. 2801 – 2809. A PPENDIX A : P ROOF OF L EMMA 1 Proof. Before starting we will set some notations. In H (D) for each s = 1, . . . , (D + 1) let Cs be the set of column indices Cs = {(s − 1)(k + 1) + 1, (s − 1)(k + 1) + 2, . . . , s(k + 1)}. Taking into account the way H (D) is constructed it is clear that for any s ≤ D, the submatrix of H (D) formed by the first s + 1 rows and the columns in C1 , . . . ,Cs+1 is H (s) (the parity-check matrix of the s-th truncation). Also the submatrix formed by the last s + 1 rows and the columns in CD−s+1 , . . . ,CD+1 is H (s) . For each set of column indices Cs the last index is s(k + 1) and the corresponding column in H (D) is the s-th column of the identity ID+1 . In an analogous way we will call Cs0 the set of column indices Cs0 = {(s − 1)k + 1, (s − 1)k + 2, . . . , sk} in the matrix H 0(D) . 15 In what follows we will use the same name for a square submatrix and for the corresponding minor since it will create no confusion. Now we start the proof. 1) Assume that the CDP is (d0 = 2, d1 = 3, . . . , dD = D + 2). In particular d0 = 2 implies that all the entries r0, j for j ∈ C1 are non zero. Let M 0 be a proper minor of H 0(D) of size p × p formed by the entries of H 0(D) in rows with indices 1 ≤ i01 < i02 < · · · < i0p ≤ s + 1 and columns with indices 1 ≤ j10 < j20 < · · · < j0p ≤ k(D + 1). Since M 0 is proper we have jl0 ≤ ki0l for l = 1, . . . , p. Let F 0 = {i01 , . . . , i0p } be the set of row indices in M 0 . From M 0 we construct a (D + 1) × (D + 1) minor M in H (D) by doing the following: • The row indices are {1, . . . , D + 1}. • For each s ∈ {1, . . . , D + 1} we define the column index js as follows: – If s ∈ F 0 then there exists a unique l(s) ∈ {1, . . . , p} such that i0l(s) = s (note that l(s) ≤ s). Considering the 0 corresponding column index in M 0 we have jl(s) = ql(s) k + rl(s) with 0 ≤ ql(s) ≤ D and 1 ≤ rl(s) ≤ k unique. (We 0 note that l is an increasing function of s and also that jl(s) ≤ ki0l(s) = ks, which implies ql(s) < s). 0 0 Then define js = ql(s) (k +1)+rl(s) . Clearly jl(s) ∈ Cql(s) +1 and jl(s) ∈ Cql(s) +1 and actually the corresponding columns are identical. – If s ∈ / F 0 then js = s(k + 1), so the corresponding column is the last in the block with column indices Cs . Let us note that 1 ≤ j1 , . . . , jD+1 ≤ (D + 1)(k + 1) but those column indices are not guaranteed to be ordered in increasing order as the js0 were. The added columns will form a submatrix which is ID+1−p in the rows that were not in F 0 , and we have ID+1−p ? M= 0 M0 therefore the value of minor M 0 is the same as the value of M and in order to see that H 0 (D) is k-superregular we just need to check that M 6= 0. We will proceed in a recursive way using that each s-th truncation will provide minimum distance ds = s + 2 for each s ≤ D. • M has at least one column index in C1 . Proof: If there are no columns in C1 it means that 1 ∈ F 0 (otherwise j1 = 1(k + 1), which is in C1 , would be in M). 1 ∈ F 0 implies i01 = 1, hence l(1) = 1 and j10 ≤ ki01 = k, that is, j10 = 0 · k + r1 , and j1 = 0(k + 1) + r1 < (k + 1). This means j1 ∈ C1 which would contradict the assumption. • If M has exactly 1 column in C1 then the other D columns have indices in C2 ∪ · · · ∪CD+1 and all have 0 in the first position, so we have r0, j1 0···0 M= ∗ M2...D+1 • where M2...D+1 is the submatrix of M formed by the last D rows and the last D columns. Since r0, j1 6= 0 we have M 6= 0 if and only if M2...D+1 6= 0, and we can proceed working with M2...D+1 in H (D−1) in the same way. If M has at least two columns in C1 . Suppose that s is the first index for which we have that at least two columns of M are in C1 , at least 3 are in C1 ∪C2 , . . . , at least s + 1 columns are in C1 ∪C2 ∪ · · · ∪Cs but there are no s + 2 columns in C1 ∪C2 ∪ · · · ∪Cs ∪Cs+1 . This clearly implies that there are no column of M in Cs+1 . Now let us consider each t ∈ {1, . . . , s + 1}. – If t ∈ F 0 there exists l(t) ≤ t ≤ s + 1 such that i0l(t) = t. 0 = q k+r 0 jl(t) l(t) l(t) ≤ kil(t) = kt, therefore ql(t) ≤ t − 1, and from here jt = ql(t) (k + 1) + rl(t) ≤ (t − 1)(k + 1) + rl(t) . So jt ∈ C1 ∪C2 ∪ · · · ∪Ct ⊆ C1 ∪C2 ∪ · · · ∪Cs+1 , but it cannot be in Cs+1 , then jt ∈ C1 ∪C2 ∪ · · · ∪Cs . – If t ∈ / F 0 . Note that in this case t ≤ s since s + 1 ∈ / F 0 implies column (s + 1)(k + 1) is in M and in Cs+1 , contradicting that there were no columns in Cs+1 Then jt = t(k + 1) ∈ C1 ∪C2 ∪ · · · ∪Cs . We have proven that even though indices j1 , . . . js+1 are not ordered in increasing order, we have that they are all in C1 ∪C2 ∪ · · · ∪Cs . On the other hand, index js+2 ∈ / C1 ∪C2 ∪ · · · ∪Cs+1 . Hence, M can be decomposed as M1...s+1 0 M= ∗ Ms+2...D+1 16 where M1...s+1 is the part of M corresponding to the first s + 1 rows and columns and we have proven it is contained in the submatrix of H (D) formed by the first s + 1 rows and the first (s + 1)(k + 1) columns, which actually is H (s) and it is guaranteed to be non zero because ds = s + 2 and the minor satisfies the condition that it has at least 2 columns among the first k + 1 columns of H (s) , at least three among the first 2(k + 1), . . ., and at least s + 1 among the first s(k + 1). Minor Ms+2...D+1 is formed by the last D − s rows and columns of M and it is contained in the submatrix of H (D) formed by the last D − s rows and the last (D − s)(k + 1) columns, which is H (D−s−1) and the same argument used so far can be used to prove that is is non zero by decomposing it further into blocks; each of them nonzero. Finally, we can note that M will have at most one column index in CD+1 . Having at least two would imply that M 0 has 0 also at least two columns in CD+1 and this would contradict the condition of M 0 being proper since i0p−1 ≤ D, j0p−1 ∈ 0 0 CD+1 implies j p−1 ≥ kD + 1 > kD = ki0p−1 . 2) Suppose now that H 0(D) is k-superregular. Consider a minor M of size (D + 1) in H (D) formed by the columns in positions 1 ≤ j1 < j2 < · · · < jD+1 ≤ (D + 1)(k + 1) and assume that j2 ≤ (k + 1), j3 ≤ 2(k + 1), . . . , jD+1 ≤ D(k + 1). We construct a minor M 0 by removing from M any column which is in position s(k + 1) and the corresponding row s. As before, it is clear that ID+1−p ? M= 0 M0 where D + 1 − p is the number of removed columns and p is the size of the remaining minor M 0 . With a careful analysis similar to the one done in the reciprocal part of the proof, one can prove that M 0 is a proper minor in H 0 (D) and hence non zero. For this we will continue using the same notations as in the demonstration of the reverse. Consider that the rows remaining in M 0 are 1 ≤ i01 < i02 < · · · i0p = D + 1. We call this set of indices F 0 as before. The other rows correspond to the identity columns that have been suppressed. The corresponding column indices in M 0 are 1 ≤ j10 < j20 < · · · < jD+1 , and each of those columns jt0 is a copy of a column 0 . Using the same notations as in the j f (t) in M and it is clear that j f (t) ∈ Cb(t) for some b(t) ≤ D + 1, implies jt0 ∈ Cb(t) other part of the proof, if j f (t) = q f (t) (k + 1) + r f (t) with 0 ≤ q f (t) ≤ D and 1 ≤ r f (t) ≤ k, then jt = q f (t) k + r f (t) , so they will be in the same block of column indices; b(t) = q f (t) + 1. Note that r(t) ≤ k since column indices that are multiples of k + 1 will be removed and will never turn into columns in M 0 . • First we observe that i p = D + 1 because there were no columns of M in the block with indices in CD+1 , hence the last column of ID+1 cannot be removed (it was never there) and row D + 1 remains in F 0 . The corresponding column j0p will be a copy of some column j f (p) ≤ jD+1 ∈ C1 ∪ · · · ∪CD , so j0p ∈ C10 ∪ · · · ∪CD0 , hence j0p ≤ Dk < (D + 1)k ≤ i0p k. So the proper condition is satisfied for the last index. • In general when we consider row index in position p − s we have i0p−s = D + 1 − s − r where r is the number of identity columns after the D + 1 − s that have been removed. Column j0p−s is copy of column j f (p−s) and f (p−s) ≤ D+1−s−r (s columns after it have been already considered and r have been removed. From here we have j f (p−s) ≤ jD+1−s−r ∈ C1 ∪ · · · ∪CD−s−r and this implies j0p−s ≤ (D − s − r)k < i0p−s k. • A final observation is that j10 is always in block C10 (because block C1 contained at least two columns of M, so even is one is removed there will always be at least one column remaining in that first block. On the other hand i01 ≥ 1 and we have j10 ≤ k ≤ ki01 . The first and last observations are not necessary but they help to understand the general case. We have proven that minor M 0 is proper and therefore cannot be singular.	7
FUSION SYSTEMS WITH SOME SPORADIC J-COMPONENTS arXiv:1605.04615v3 [math.GR] 7 Jun 2017 JUSTIN LYND AND JULIANNE RAINBOLT Abstract. Aschbacher’s program for the classification of simple fusion systems of “odd” type at the prime 2 has two main stages: the classification of 2-fusion systems of subintrinsic component type and the classification of 2-fusion systems of J-component type. We make a contribution to the latter stage by classifying 2-fusion systems with a J-component isomorphic to the 2-fusion systems of several sporadic groups under the assumption that the centralizer of this component is cyclic. 1. Introduction The Dichotomy Theorem for saturated fusion systems [AKO11, II 14.3] partitions the class of saturated 2-fusion systems into the fusion systems of characteristic 2-type and the fusion systems of component type. This is a much cleaner statement than the corresponding statement for finite simple groups, and it has a much shorter proof. In the last few years, M. Aschbacher has begun work on a program to give a classification of a large subclass of the 2-fusion systems of component type. A memoir setting down the outline and first steps of such a program is forthcoming [Asc16], but see [Asc15] for a survey of some of its contents. The immediate goal is to give a simpler proof of roughly half of the classification of the finite simple groups by carrying out most of the work in the category of saturated 2-fusion systems. Let F be a saturated fusion system over a finite 2-group S, of which the standard example is the fusion system FS (G), where G is a finite group and S is a Sylow 2-subgroup of G. A component is a subnormal, quasisimple subsystem. The system is said to be of component type if some involution centralizer in F has a component. The 2-fusion systems of odd type consist of those of subintrinsic component type and those of J-component type. This is a proper subclass of the 2-fusion systems of component type. In focusing attention on this restricted class, one is expected to avoid several difficulties in the treatment of standard form problems like the ones considered in this paper. By carrying out the work in fusion systems, it is expected that certain difficulties within the classification of simple groups of component type can be avoided, including the necessity of proving Thompson’s B-conjecture. We refer to [Asc16] for the definition of a fusion system of subinstrinsic component type, as it is not needed in this paper. The fusion system F is said to be of J-component type if it is not of subintrinsic component type, and there is a (fully centralized) involution x ∈ S such that the 2-rank of CS (x) is equal to the 2-rank of S, and CF (x) has a component. We shall call such a component in an involution centralizer a J-component. Date: March 13, 2018. Key words and phrases. fusion systems, sporadic groups, involution centralizer, components. The research of the first author was partially supported by NSA Young Investigator Grant H98230-14-10312 and was supported by an AMS-Simons grant which allowed for travel related to this work. 1 In this paper, we classify saturated 2-fusion systems having a J-component isomorphic to the 2-fusion system of M23 , J3 , McL, or Ly under the assumption that the centralizer of the component is a cyclic 2-group. A similar problem for the fusion system of L2 (q), q ≡ ±1 (mod 8) was treated in [Lyn15] under stronger hypotheses. Theorem 1.1. Let F be a saturated fusion system over the finite 2-group S. Suppose that x ∈ S is a fully centralized involution such that F ∗ (CF (x)) ∼ = Q × K, where K is the 2-fusion system of M23 , J3 , McL, or Ly, and where Q is a cyclic 2-group. Assume further that m(CS (x)) = m(S). Then K is a component of F . Here F ∗ (CF (x)) is the generalized Fitting subsystem of the centralizer CF (x) [Asc11], and m(S) := m2 (S) is the 2-rank of S – that is, the largest rank of an elementary abelian 2-subgroup of S. We mention that any fusion system having an involution centralizer with a component isomorphic to McL or Ly is necessarily of subintrinsic component type by [Asc16, 6.3.5]. This means that, when restricted to those components, Theorem 1.1 gives a result weaker than is needed to fit into the subintrinsic type portion of Aschbacher’s program. However, we have included McL and Ly here because our arguments apply equally well in each of the four cases. There is no almost simple group with an involution centralizer having any of these simple groups as a component, but the wreath product G = (K1 × K2 )hxi with K1x = K2 always has CG (x) = hxi × K with K a component that is diagonally embedded in K1 × K2 . The strategy for the proof of Theorem 1.1 is to locate a suitable elementary abelian subgroup F in the Sylow 2-subgroup of K, and then to show that the normalizer in S of E := hxiF has at least twice the rank as that of F . Thus, the aim is to force a resemblance with the wreath product, in which NG (hxiF ) modulo core is an extension of F1 × F2 (with Fi the projection of F onto the ith factor) by hxi × AutK (F ). Lemma 3.2 is important for getting control of the extension of E determined by NF (E) in order to carry out this argument. Acknowledgements. We would like to thank the Department of Mathematics and Statistics at Saint Louis University and the Departments of Mathematics at Rutgers University and Ohio State University for their hospitality and support during mutual visits of the authors. We would also like to thank R. Solomon and R. Lyons for helpful discussions, and an anonymous referee for their comments and suggestions. 2. Background on fusion systems We assume some familiarity with notions regarding saturated fusion systems as can be found in [AKO11] or [Cra11], although some items are recalled below. Most of our notation is standard. Whenever G is a group, we write G# for the set of nonidentity elements of G. If we wish to indicate that G is a split extension of a group A P G by a group B, then we will write G = A · B. For g ∈ G, denote by cg the conjugation homomorphism cg : x 7→ xg and its restrictions. Morphisms in fusion systems are written on the right and in the exponent. That is, we write xϕ (or P ϕ ) for the image of an element x (or subgroup P ) of S under a morphism ϕ in a fusion system, by analogy with the more standard exponential notation for conjugation in a group. 2 2.1. Terminology and basic properties. Throughout this section, fix a saturated fusion system F over the p-group S. We will sometimes refer to S as the Sylow subgroup of F . For a subgroup P ≤ S, we write AutF (P ) for HomF (P, P ), and OutF (P ) for AutF (P )/ Inn(P ). Whenever two subgroups or elements of S are isomorphic in F , we say that they are F conjugate. Write P F for the set of F -conjugates of P . If E is a subsystem of F on the subgroup T ≤ S and α : T → S is a morphism in F , the conjugate of E by α is the subsystem E α over T α with morphisms ϕα := α−1 ϕα for ϕ a morphism in E. We first recall some of the terminology for subgroups and common subsystems in a fusion system. Definition 2.1. Fix a saturated fusion system over the p-group S, and let P ≤ S. Then • P is fully F -centralized if \|CS (P )\| ≥ \|CS (Q)\| for all Q ∈ P F , • P is fully F -normalized if \|NS (P )\| ≥ \|NS (Q)\| for all Q ∈ P F , • P is F -centric if CS (Q) ≤ Q for all Q ∈ P F , • P is F -radical if Op (OutF (P )) = 1, • P is weakly F -closed if P F = {P }, • the centralizer of P in F is the fusion system CF (P ) over CS (P ) with morphisms those ϕ ∈ HomF (Q, R) such that there is an extension ϕ̃ ∈ HomF (P Q, P R) that restricts to the identity on P , • the normalizer of P in F is the fusion system NF (P ) over NS (P ) with morphisms those ϕ ∈ HomF (Q, R) such that there is an extension ϕ̃ : P Q → P R in F such that P ϕ̃ = P . We write F f and F c for the collections of fully F -normalized and F -centric, respectively, and we write F f c for the intersection of these two collections. Sometimes we refer to an element x of S as being fully F -centralized, when we actually mean that the group hxi is fully F -centralized, especially when x is an involution. For example, this was done in the the statement of the theorem in the introduction. Whenever P ≤ S, we write A(P ) for the set of α ∈ HomF (NS (P ), S) such that P α is fully F -normalized. Lemma 2.2. For each P ≤ S, A(P ) is not empty. Moreover, for each Q ∈ P F ∩ F f , there is α ∈ A(P ) with P α = Q. Proof. This is [BLO03, A.2(b)], applied with K = Aut(P ). By a result of Puig, the centralizer CF (P ) is saturated if P is fully F -centralized, and the normalizer NF (P ) is saturated if P is fully F -normalized. We write Op (F ) for the (unique) largest subgroup P of S satisfying NF (P ) = F , and Z(F ) for the (unique) largest subgroup P of S satisfying CF (P ) = F . We note that if F = FS (G) for some finite group G with Sylow p-subgroup S, then Op (G) ≤ S is normal in F so that Op (G) ≤ Op (F ), but the converse does not hold in general. 2.2. The Model Theorem. A subgroup P ≤ S is F -centric if and only if CS (P ) ≤ P and P is fully F -centralized [AKO11, I.3.1]. If P is F -centric and fully F -normalized, then the normalizer fusion system M := NF (P ) is constrained – that is, Op (M) is M-centric. By the Model Theorem [AKO11, Proposition III.5.10], there is then a unique finite group M up to isomorphism having Sylow p-subgroup NS (P ) and such that Op′ (M) = 1, Op (M) = Op (M), and FNS (P ) (M) ∼ = M. Then M is said to be a model for M in this case. 3 2.3. Tame fusion systems. The main hypothesis of Theorem 1.1 is that the generalized Fitting subsystem of the involution centralizer C is the fusion system of a finite group Q×K, where Q is a cyclic 2-group, and K is simple. In this situation, CF (x) is itself the fusion system of a finite group C with F ∗ (C) = Q × K, where K ∼ = M23 , McL, J3 , or Ly, since each of these simple groups tamely realizes its 2-fusion system [AOV12, Oli16b]. Roughly, a finite group tamely realizes its fusion system if every automorphism of its fusion system is induced by an automorphism of the group. Moreover, a fusion system is said to be tame if there is some finite group that tamely realizes it. We refer to [AOV12] for more details. The importance of tameness in the context of standard form problems was pointed out in [Lyn15, §§1.5]. The discussion there is centered around the notion of strong tameness, which was needed for proofs of the results of [AOV12], but the contents of [Oli13, GL16] imply that a fusion system is tame if and only if it is strongly tame. Recently, Oliver has established the following useful corollary of the results in [AOV12], which we state for our setup here. Theorem 2.3 ([Oli16a, Corollary 2.5]). Let C be a saturated fusion system over a 2-group. Assume that F ∗ (C) = O2 (C)K, where K is simple and tamely realized by a finite simple group K. Then C is tamely realized by a finite group C such that F ∗ (C) = O2 (C)K. Note that, upon application of Theorem 2.3 to the involution centralizer C = CF (x) in Theorem 1.1, we have O2 (C) = Q = O2 (C). Indeed, O2 (C) ≤ Q since O2 (C) is normal in C, and one sees that Q = CS (K) by combining Lemma 1.12(c) of [Lyn15] with Lemma 3.7(c) below. However, O2 (C) is normal and self-centralizing in CC (K) by properties of the generalized Fitting subgroup, so that CC (K)/O2 (C) is a group of outer automorphisms of the cyclic 2-group O2 (C), and so is itself a 2-group. It follows that CC (K) = CS (K) is a normal 2-subgroup of C (since K E C), and hence Q = CS (K) ≤ O2 (C). Thus, the effect of Theorem 2.3 for our purposes is that we may work in the group C, where Q is a normal subgroup. In particular, in the setup of the Theorem 1.1, the quotient C/Q is isomorphic to a subgroup of Aut(K) containing Inn(K), where K is one of the simple groups appearing in Theorem 1.1. 3. Structure of the components In this section, we recall some properties of the simple systems appearing in Theorem 1.1 that are required for the remainder. Lemma 3.1. Let G be A7 or GL2 (4), and V a faithful F2 [G]-module of dimension 4. Then (a) G acts transitively on the nonzero vectors of V , (b) CGL(V ) (G) ≤ G, (c) H 1 (G, V ) = 0, and (d) if G acts on a homocyclic 2-group Y with Ω1 (Y ) = V , then Y = V . Proof. In each case, V is irreducible. There is a unique such module for GL2 (4) ∼ = C3 × A5 , namely the natural F4 [G]-module considered as a module over F2 , and thus (a) holds in this case. The module for A7 is unique up to taking duals; clearly points (a) and (b) are independent of the choice between these two modules, and (c) is independent of such a choice by [Asc00, §17]. Note that A7 acts transitively on V # , which can be seen by noting that a Sylow 7-subgroup acts with exactly one fixed point on V # , and a Sylow 5-subgroup acts with no fixed points. Point (b) holds for G = A7 by absolute irreducibility. Similarly for 4 G = GL2 (4), one has that CGL(V ) (G) = EndF2 [G] (V )× = F× 4 , and so (b) follows in this case as Z(G) ∼ C . Point (c) for G = GL (4) holds because, by coprime action, Z(G) (and so = 3 2 G) has a fixed point on any 5-dimensional module containing V as a submodule (see [Asc00, §17]). Point (c) for G = A7 holds, for example, by applying [AG72] with L = {L0 , L1 , L2 }, where L1 = CG ((1, 2, 3)), L2 = CG ((4, 5, 6)), and L0 = L1 ∩ L2 . The Li indeed satisfy the hypotheses of that theorem: H 0 (Li , V ) = 0 because each Sylow 3-subgroup of GL4 (2) ≥ G has no nontrivial fixed point on V , and H 1 (Li , V ) = 0 using a similar argument via coprime action as above. We now turn to (d), which follows from a special case of a result of G. Higman [Hig68, Theorem 8.2]. This says that if SL2 (4) acts faithfully on a homocyclic 2-group Y in which an element of order 3 acts without fixed points on Y # , then Y is elementary abelian. In case G = GL2 (4), V is the natural module for G and certainly SL2 (4) ≤ G has (with respect to an appropriate basis) a diagonal element of order 3 acting without fixed points. In the case G = A7 , we have G ≤ A8 ∼ = GL4 (2), and the action of G on V is the restriction of the natural (or dual) action of GL4 (2). Restriction of either one to GL2 (4) ∼ = C3 × SL2 (4) ∼ = C3 × A5 shows that SL2 (4) is embedded in G as an A5 moving 5 points in the natural permutation action. So SL2 (4) is contained in G up to conjugacy, and as before, it has an element of order 3 acting without fixed points on V # . Hence, (d) holds in this case as well by Higman’s Theorem. For a vector space E over the field with two elements, the next lemma examines under rather strong hypotheses the structure of extensions of E by certain subgroups of the stabilizer in GL(E) of a hyperplane. Lemma 3.2. Let E ∼ = E2n , P = NA (V ), and = E2n+1 (n ≥ 3), A = Aut(E), V ≤ E with V ∼ U = O2 (P ). Let L be a complement to U in P acting decomposably on E, x ∈ E − V the fixed point for the action of L, and G ≤ L. Let H be an extension of E by UG with the given action and let X be the preimage in H of U under the quotient map H → UG. Assume that (a) G acts transitively on V # ; (b) CGL(V ) (G) ≤ G; and (c) H 1 (G, V ) = 0. Then there is a subgroup Y of X that is elementary abelian or homocyclic of order 22n and a G-invariant complement to hxi in X. Proof. Let X̄ = X/V . Since the commutator map (3.3) [x, −] determines a G-equivariant linear isomorphism X/E → V , G is transitive on the nonzero vectors of X/E by (a), and hx̄i ≤ Z(X̄). Hence if X̄ is not elementary abelian, then it is extraspecial with center hx̄i, and G preserves the squaring map X̄ → Z(X̄). This is not the case, because G is transitive on the nonzero vectors of X/E and n ≥ 3. Therefore, (3.4) X̄ is elementary abelian. Assumption (c) now yields that there is a G-invariant complement Ȳ to hx̄i. Let Y be the preimage of Ȳ in X. We claim that Y is abelian; assume on the contrary. Then [Y, Y ] and ✵1 (Y ) are contained in V since Ȳ is elementary abelian, and by assumption, neither of 5 these are trivial. Similarly, V is contained in Z(Y ), which is not Y . Therefore, V = [Y, Y ] = Φ(Y ) = ✵1 (Y ) = Z(Y ), (3.5) by (a) and (3.3). √ By (3.5), the squaring map Ȳ → V is G-equivariant linear isomorphism; let − be its √ inverse. Then the map ξ : V → V given by ξ(v) = [x, v] is a linear isomorphism commuting with the action of G, and so ξ ∈ ρ(G) by (b), where ρ : G → GL(V ) is the structure map. Let g ∈ G map to ξ −1 under ρ. Then y 7→ [x, y g ] is the squaring map. This means that for each y ∈ Y , we have y gx = y 2 y g . Hence, for each pair w, y ∈ Y , w 2 y 2 w g y g = w gx y gx = (wy)gx = (wy)2w g y g which gives w 2 y 2 = (wy)2 = w 2 y 2 [y, w]. Thus Y is abelian after all. It follows that Ω1 (Y ) = V or Y by (a), and this completes the proof of the lemma. We now examine the structure of the simple systems occupying the role of K in Theorem 1.1. Let T0 be a 2-group isomorphic to a Sylow 2-subgroup of L3 (4). This is generated by involutions t1 , t2 , a1 , a2 , b1 , and b2 such that Z(T0 ) = ht1 , t2 i and with additional defining relations: [a1 , a2 ] = [b1 , b2 ] = 1, [a1 , b1 ] = [a2 , b2 ] = t1 , [a2 , b1 ] = t2 , [a1 , b2 ] = t1 t2 . A Sylow subgroup of M23 or McL is isomorphic to a Sylow 2-subgroup of an extension of L3 (4) by a field automorphism; this is a semidirect product T0 hf i with f 2 = 1, [a1 , f ] = [a2 , f ] = a1 a2 , [b1 , f ] = [b2 , f ] = b1 b2 , [t2 , f ] = t1 . A Sylow subgroup of J3 is isomorphic to a Sylow 2-subgroup of L3 (4) extended by a unitary (i.e., graph-field) automorphism; this is a semidirect product T0 hui with u2 = 1, [a1 , u] = [u, b1 ] = a1 b1 , [a2 , u] = [u, b2 ] = a2 b2 , [t2 , u] = t1 . A Sylow subgroup of Ly is isomorphic to a Sylow 2-subgroup of Aut(L3 (4)); this is a semidirect product T0 hf, ui with [f, u] = 1 and the relations above. Denote by T1 a 2-group isomorphic to one of T0 hf i, T0 hui, or T0 hf, ui. Recall that the Thompson subgroup J(P ) of a finite p-group P is the subgroup generated by the elementary abelian subgroups of P of largest order. Lemma 3.6. Let K be M23 , McL, J3 , or Ly, with Sylow 2-subgroup T1 as above. Then (a) Z(T1 ) = ht1 i is of order 2; (b) A(T1 ) = {F1 , F2 } where F1 = ht1 , t2 , a1 , a2 i and F2 = ht1 , t2 , b1 , b2 i, so that J(T1 ) = T0 . Also, after suitable choice of notation, one of the following holds: (i) K = M23 , McL, or Ly, and AutK (F1 ) ∼ = A7 , or (ii) K = J3 and AutK (F1 ) ∼ = GL2 (4). (c) There is F ∈ A(T1 ) such that the pair (AutK (F ), F ) satisfies assumptions (a)-(c) of Lemma 3.2 in the role of (G, V ). (d) All involutions in ht1 , t2 i are AutK (J(T1 ))-conjugate. Proof. Point (a) holds by inspection of the relations above. Now F1 and F2 are the elementary abelian subgroups of T0 of maximal rank, and so to prove (b), it suffices to show that each elementary abelian subgroup of maximal rank in T1 is contained in T0 . Set L := L3 (4), and identify Inn(L) with L. Write Inndiag(L) ≥ L for the group of inner-diagonal automorphisms 6 of L. Then Inndiag(L) contains L with index 3, corresponding to the size of the center of the universal version SL3 (4) of L [GLS98, Theorem 2.5.12(c)]. Also, Aut(L) is a split extension of of Inndiag(L) by ΦL ΓL = ΦL × ΓL , where ΦL = hϕi ∼ = C2 is generated by a C is generated by a graph automorphism [GLS98, field automorphism of L, and ΓL = hγi ∼ = 2 Theorem 2.5.12]. By [GLS98, Theorems 4.9.1, 4.9.2], each involution of Aut(L)−L is Aut(L)conjugate to ϕ, ϕγ, or γ, and the centralizers in L of these automorphisms are isomorphic to L3 (2), U3 (2) ∼ = (C3 × C3 )Q8 , and Sp2 (4) ∼ = A5 , respectively, again by those theorems. These centralizers have 2-ranks 2, 1, and 2, respectively. Since T0 has 2-rank 4, this shows that J(T1 ) ≤ T0 . From the relations used in defining T0 , each involution in T0 is contained in one of F1 or F2 , so we conclude that A(T1 ) = A(T0 ) = {F1 , F2 }. The description of the automizers in (b) follows from [Fin76a, Table 1] for M23 and McL, [Fin76b, Lemma 3.7] for J3 , and [Wil84] for Ly. Now point (c) follows from (b) and Lemma 3.1, and point (d) follows from (c) and Burnside’s fusion theorem (i.e. the statement that the automizer of a weakly K-closed subgroup of T1 (which is J(T1 ) in this case) controls the K-conjugacy in its center). Lemma 3.7. Let K be one of the sporadic groups M23 , McL, J3 , or Ly, and let T1 be a Sylow 2-subgroup of K. Then (a) Out(K) = 1 if K ∼ = C2 otherwise; and = M23 or Ly, and Out(K) ∼ (b) for each involution α ∈ Aut(K) − Inn(K), (i) CK (α) ∼ = McL, = M11 if K ∼ ∼ (ii) CK (α) ∼ L (17) if K = J3 ; and = 2 (c) each automorphism of K centralizing a member of A(T1 ) is inner. Proof. Points (a) and (b) follow by inspection of Table 5.3 of [GLS98]. By Lemma 3.6(b), the 2-rank of K is 4, while each of the centralizers in (b)(i-ii) is of 2-rank 2, so (c) holds. 4. Preliminary lemmas We now begin in this section the proof of Theorem 1.1, and so we fix the notation and hypotheses that will hold throughout the remainder of the paper. Let F be a saturated fusion system over the 2-group S, and let x ∈ S be an involution. Assume that hxi is fully F -centralized, that m(CS (x)) = m(S), and that F ∗ (CF (x)) = Q×K, where Q is cyclic. Set C = CF (x) and T = CS (x), so that C is a saturated fusion system over T by the remark just after Lemma 2.2. Let T1 be the Sylow subgroup of K, and set R := Q × T1 ≤ T. Assume that K is the fusion system of one of the sporadic groups K = M23 , J3 , McL, or Ly. Since K tamely realizes K in each case, the quotient (4.1) T /R induces a 2-group of outer automorphisms of K by Theorem 2.3. Arguing by contradiction, we assume K is not a component of F. We fix the presentation in Section 3 for T1 in whichever case is applicable, and we note that Ω1 (Q) = hxi by assumption on Q. Lemma 4.2. Notation may be chosen so that T and J(T ) are fully F -normalized. 7 Proof. We repeatedly use Lemma 2.2. Let α ∈ A(T ). Then as \|CS (xα )\| ≥ \|T α \| = \|T \| = \|CS (x)\|, we have that xα is still fully F -centralized. Thus we may assume that T is fully F normalized after replacing x, T , and J(T ) by their conjugates under α. Now let β ∈ A(J(T )). Then as NS (T ) = NNS (J(T )) (T ), it follows that \|NS (T β )\| = \|NNS (J(T )β ) (T β )\| ≥ \|NNS (J(T ))β (T β )\| = \|NNS (J(T )) (T )β \| = \|NS (T )β \| = \|NS (T )\| and so equality holds because T is fully F -normalized. Hence T β is still fully F -normalized. As before, xβ is still fully F -centralized. Lemma 4.3. hxi is not weakly F -closed in T . Proof. Assume on the contrary, in which case T = S, otherwise NS (T ) contains T properly and moves x. It follows that x ∈ Z(S), which is contained in the center of every F -centric subgroup. Hence x ∈ Z(F ) by Alperin’s fusion theorem [BLO03, Theorem A.10], and we conclude that K is a component of C = F , contrary to hypothesis. Lemma 4.4. The following hold. (a) J(T ) = J(R) = hxi × J(T1 ); and (b) CT (T1 ) = Qht1 i. Proof. Suppose (a) does not hold. Choose A ∈ A(T ) with A hxi × J(T1 ). Then A R by the structure of R, and A acts nontrivially on K by (4.1). In particular, Out(K) = 2 and m(CR (A)) ≤ 3 by Lemma 3.7. Hence, m(A) ≤ 4 while m(R) = 5. This contradicts the choice of A and establishes (a). By Lemma 3.6(a), CR (T1 ) = Qht1 i. Also, CT (T1 ) = CR (T1 ) by part (a), so (b) is also established. Lemma 4.5. If T = S, then Ω1 Z(S) = hx, t1 i. If T < S, then Ω1 (Z(S)) = ht1 i. Proof. Note first that Ω1 (Z(S)) ≤ T . By Lemma 3.7(c) and (4.1), Ω1 (Z(S)) ≤ R, and so Ω1 Z(S) ≤ Ω1 (Z(R)) = hx, t1 i by Lemma 3.6(a). Thus, the lemma holds in case T = S. In case T < S, let a ∈ NS (T )−T with a2 ∈ T . Note that [J(T ), J(T )] = [J(T1 ), J(T1 )] = ht1 , t2 i by Lemma 4.4(a). So a normalizes Ω1 Z(T ) = hx, t1 i and Ω1 Z(T ) ∩ [J(T ), J(T )] = hx, t1 i ∩ ht1 , t2 i = ht1 i, but a does not centralize x. Thus, Ω1 Z(S) = ht1 i as claimed. Lemma 4.6. The following hold. (a) x is not F -conjugate to t1 ; and (b) x is conjugate to xt1 if and only if T < S, and in this case x is NS (T )-conjugate to xt1 . Proof. For part (a), let ϕ ∈ F with xϕ ∈ Z(T ). Assume first that T = S. By the extension axiom, ϕ extends to ϕ̃ ∈ AutF (T ), which restricts to an automorphism of J(T ). Lemma 4.4(a) shows that x ∈ / [J(T ), J(T )], while t1 ∈ [J(T ), J(T )] from Lemma 3.6(d). Hence, xϕ 6= t1 ; this shows x is not F -conjugate to t1 in this case. Now assume that T < S. Then x ∈ / Z(S), whereas Z(S) = ht1 i by Lemma 4.5. Since hxi is fully F -centralized by assumption, we conclude that x is not F -conjugate to t1 in this case either. This completes the proof of (a). If T < S, then x is NS (T )-conjugate to xt1 by (a), while if T = S, then (a) and Burnside’s fusion theorem imply that hxi is weakly F -closed in Ω1 (Z(S)) = hx, t1 i. Thus, (b) holds. 8 5. The 2-central case In this section it is shown that T < S; that is, x is not 2-central. We continue the notation set at the beginning of Section 4. Lemma 5.1. If T = S, then hxi is weakly F -closed in R. Proof. Assume T = S. Then Ω1 (Z(S)) = hx, t1 i by Lemma 4.5. Using Burnside’s fusion theorem and assumption, we see from Lemma 4.6 that (5.2) hxi is weakly F -closed in hx, t1 i. By inspection of [GLS98, Table 5.3], K has one class of involutions. Thus, there are exactly three C-classes of involutions, namely {x}, (xt1 )C , and tC1 . The lemma therefore holds by (5.2). Lemma 5.3. If T = R, then T < S. In particular, T < S in case K is the fusion system of M23 or Ly. Proof. Assume T = R, and also to the contrary that T = S. By Lemma 5.1, hxi ≤ Z(S) is weakly F -closed, and so is fixed by each automorphism of each F -centric subgroup. Therefore, hxi ≤ Z(F ), and so K is a component of C = F , contrary to assumption. The last statement follows then follows from Lemma 3.7(a). Lemma 5.4. Assume K is the fusion system of McL or J3 . Then T < S. Proof. Assume T = S. Then R < T by Lemma 5.3. Fix f ∈ xF ∩ (T − R). The extension Khf i := KT1 hf i of K is defined by [Asc11, §8], and Khf i/Q is the 2-fusion system of Aut(McL) or Aut(J3 ) by Theorem 2.3. Thus, \|T : R\| = 2 by Lemma 3.7(b). Conjugating in Khf i if necessary, we may assume hf i is fully Khf i-centralized. By Lemma 3.7(b), all involutions of T1 hf i−T1 are Khf i-conjugate, and CKhf i (f ) ∼ = hf i×F2 (M11 ) or hf i × F2 (L2 (17)). In particular, CT1 (f ) is semidihedral or dihedral, respectively, of order 16 and with center ht1 i. Fix a four subgroup V ≤ CT1 (f ). Then f is conjugate to f t1 (for example, by an element in the normalizer of CT1 hf i (f ) in T1 hf i), and hence is F -conjugate to each element of f V by the structures of F2 (M11 ) and F2 (L2 (17)). Fix α ∈ HomF (hf i, hxi). By the extension axiom, α extends to a morphism, which we also call α, defined on hf iV ≤ CT (f ). Therefore, x is F -conjugate to each element in xV α . Now the intersection xV α ∩ R = V α ∩ R is nontrivial because \|T : R\| = 2, so as x is not itself in V α , we see that x has a distinct conjugate in R. This contradicts Lemma 5.1 and completes the proof. 6. Proof of Theorem 1.1 Continue the notation and hypotheses set at the beginning of Section 4. In addition, we fix F ∈ A(T1 ) satisfying assumptions (a)-(c) of Lemma 3.2, as guaranteed by Lemma 3.6(c), and set E := hxiF . Then E ∈ A(T ) by Lemma 4.4(a), and so (6.1) m(T ) = 5. In this section, we finish the proof of Theorem 1.1, by showing that the hypotheses of Lemma 3.2 hold for a model of the normalizer in F of an appropriate F -conjugate of E. Via Lemma 3.1(d), this forces the 2-rank of S to be at least 8, contrary to the hypothesis that m(S) = m(T ). 9 By Lemmas 5.3 and 5.4, T < S. Lemma 6.2. \| AutF (T ) : AutC (T )\| = 2. Proof. Represent AutF (T ) on Ω1 Z(T ) = hx, t1 i and apply Lemma 4.6. Lemma 6.3. The following hold. (a) \| AutF (J(T )) : AutC (J(T ))\| = 4 and (b) xAutF (E) = xF , and so \| AutF (E) : AutC (E)\| = 16. Proof. Represent AutF (J(T )) on Ω1 Z(J(T )) = hx, t1 , t2 i. Now AutC (J(T )) = CAutF (J(T ) (x), and the former is transitive on Ω1 ZJ(T1 )# by Lemma 3.6(d). Also, since x is NS (T )conjugate to xt1 , we conclude from Lemma 4.6 that xAutF (J(T )) = xZJ(T1 ) is of size 4. Thus, (a) holds. Similarly to (a), we have AutC (E) = CAutF (E) (x), and the former is transitive on F # by choice of F . From Lemma 4.4(a) and Lemma 3.6(a), \|A(T )\| = 2. By part (a) and Lemma 4.2, \|NS (J(T )) : T \| = 4. Representing NS (J(T )) on A(T ), we see that the kernel has index at most 2, so there is an element of NS (J(T ))−T that normalizes E. In particular, NT (E) < NS (E), and so x is AutF (E)-conjugate to a member of xF # . Now by choice of F , another appeal to Lemma 4.6 yields that xAutF (E) = F x has size 16, which establishes (b). Lemma 6.4. The following hold: (a) Q = hxi. (b) E is F -centric. Proof. Suppose on the contrary that Q > hxi and choose w ∈ Q with w 2 = x. Fix a ∈ NS (T ) − T such that a2 ∈ T . Then xa = xt1 and also (w a )2 = xt1 . Further hw a i is normal in T since hwi is. Thus, [hw a i, T1 ] ≤ hw a i ∩ T1 = 1, whereas CT (T1 ) = Qht1 i by Lemma 4.4(b). It follows that hxt1 i = ✵1 (hw a i) ≤ Ω1 ✵1 (Qht1 i) = hxi, a contradiction that establishes (a). Let E0 be one of the two elementary abelian subgroups of rank 5 in T , and set F0 = E0 ∩ J(T1 ). Then E0 = hxiF0 contains x, and so CS (E0 ) = CT (E0 ). By Lemma 3.7(c), CT (E0 ) = CR (E0 ) = QF0 . Hence CS (E0 ) = QF0 = E0 by part (a). We can now prove (b). Fix α ∈ A(E). Since hxi is fully centralized, the restriction of α−1 to hxα i has an extension β : CS (xα ) → CS (x) = T , which is defined on CS (E α ). Thus, setting E0 := CS (E α )β ≤ T , we see from the previous paragraph that \|CS (E)\| = \|CS (E0 )\| = \|E0 \| = \|CS (E α )\|, so that CS (E α ) = E α . As E α is fully F -normalized and contains its centralizer in S, this means that E is F -centric, as claimed. Since we will be working in NF (E) for the remainder, we may assume, after replacing E by an F -conjugate if necessary, that E is fully F -normalized. Hence E ∈ F f c by Lemma 6.4(b). Fix a model H for NF (E) (cf. §§2.2). Lemma 6.5. H satisfies the hypotheses of Lemma 3.2. Proof. Set G̃ = AutC (E) and observe that AutF (E) ∼ = H/E by Lemma 6.4(b). Thus G̃ contains G := A7 or GL2 (4) with index 1 or 2. As G acts transitively on F # and centralizes x, it follows from Lemma 6.3(b) that xF and F # are the orbits of AutF (E) on E # . Hence 10 AutF (E) ≤ NAut(E) (F ), a nontrivial split extension of an elementary abelian 2-group U of order 16 by GL(F ) with the standard action. We claim that U ≤ AutF (E). Suppose that this is not the case. Now G acts transitively on F # and the commutator map [x, −] defines an isomorphism of G-modules from U to F , so G acts transitively on U # . Since U is normalized by AutF (E), we see that AutF (E) ∩ U = 1. In particular, AutF (E) embeds into GL(F ). Now G ∼ = A7 or GL2 (4) in the cases under consideration, and by Lemma 6.3(b), AutF (E) is therefore a subgroup of GL(F ) containing G with index 16 or 32. However, A7 has index 8 in GL4 (2), and GL2 (4) is contained with index 2 in a unique maximal subgroup of GL4 (2), a contradiction. Therefore, U ≤ AutF (E) as claimed. It has thus been shown that AutF (E) contains a subgroup with index 1 or 2 that is a split extension of U = O2 (NAut(E) (F )) by G. Thus, H has a subgroup of index 1 or 2 that is an extension of E by UG. Assumptions (a)-(c) of Lemma 3.2 hold via Lemma 3.6(c) by the choice of F . Proof of Theorem 1.1. Keep the notation of the proof of Lemma 6.5. By that lemma and Lemma 3.2, there is a G-complement Y to hxi in O2 (H) that is homocyclic of order 28 with Ω1 (Y ) = F , or elementary abelian of order 28 . Now G is isomorphic to A7 or GL2 (4) with faithful action on F , so the former case is impossible by Lemma 3.1. Hence, m2 (T ) = 5 < 8 ≤ m2 (S), contrary to hypothesis. References [AG72] [AKO11] [AOV12] [Asc00] [Asc11] [Asc15] [Asc16] [BLO03] [Cra11] [Fin76a] [Fin76b] [GL16] [GLS98] J. L. Alperin and Daniel Gorenstein, A vanishing theorem for cohomology, Proc. Amer. Math. Soc. 32 (1972), 87–88. MR 0291293 Michael Aschbacher, Radha Kessar, and Bob Oliver, Fusion systems in algebra and topology, London Mathematical Society Lecture Note Series, vol. 391, Cambridge University Press, Cambridge, 2011. MR 2848834 Kasper K. S. Andersen, Bob Oliver, and Joana Ventura, Reduced, tame and exotic fusion systems, Proc. Lond. Math. Soc. (3) 105 (2012), no. 1, 87–152. MR 2948790 Michael Aschbacher, Finite group theory, second ed., Cambridge Studies in Advanced Mathematics, vol. 10, Cambridge University Press, Cambridge, 2000. MR 1777008 (2001c:20001) , The generalized Fitting subsystem of a fusion system, Mem. Amer. Math. Soc. 209 (2011), no. 986, vi+110. MR 2752788 , Classifying finite simple groups and 2-fusion systems, ICCM Not. 3 (2015), no. 1, 35–42. 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MR 755827 Institute of Mathematics, University of Aberdeen, Fraser Noble Building, Aberdeen AB24 3UE E-mail address: [email protected] Department of Mathematics and Statistics, Saint Louis University, 220 North Grand Blvd., Saint Louis, MO 63103 E-mail address: [email protected] 12	4
Prediction with a Short Memory arXiv:1612.02526v3 [cs.LG] 9 Nov 2017 Sham Kakade University of Washington [email protected] Percy Liang Stanford University [email protected] Vatsal Sharan Stanford University [email protected] Gregory Valiant Stanford University [email protected] Abstract We consider the problem of predicting the next observation given a sequence of past observations, and consider the extent to which accurate prediction requires complex algorithms that explicitly leverage long-range dependencies. Perhaps surprisingly, our positive results show that for a broad class of sequences, there is an algorithm that predicts well on average, and bases its predictions only on the most recent few observation together with a set of simple summary statistics of the past observations. Specifically, we show that for any distribution over observations, if the mutual information between past observations and future observations is upper bounded by I, then a simple Markov √ model over the most recent I/ observations obtains expected KL error —and hence `1 error —with respect to the optimal predictor that has access to the entire past and knows the data generating distribution. For a Hidden Markov Model with n hidden states, I is bounded by log n, a quantity that does not depend on the mixing time, and we show that the trivial prediction algorithm based on the empirical frequencies of length O(log n/) windows of observations achieves this error, provided the length of the sequence is dΩ(log n/) , where d is the size of the observation alphabet. We also establish that this result cannot be improved upon, even for the class of HMMs, in the following two senses: First, for HMMs with n hidden states, a window length √ of log n/ is information-theoretically necessary to achieve expected KL error , or `1 error . Second, the dΘ(log n/) samples required to accurately estimate the Markov model when observations are drawn from an alphabet of size d is necessary for any computationally tractable learning/prediction algorithm, assuming the hardness of strongly refuting a certain class of CSPs. 1 Memory, Modeling, and Prediction We consider the problem of predicting the next observation xt given a sequence of past observations, x1 , x2 , . . . , xt−1 , which could have complex and long-range dependencies. This sequential prediction problem is one of the most basic learning tasks and is encountered throughout natural language modeling, speech synthesis, financial forecasting, and a number of other domains that have a sequential or chronological element. The abstract problem has received much attention over the last half century from multiple communities including TCS, machine learning, and coding theory. The fundamental question is: How do we consolidate and reference memories about the past in order to effectively predict the future? Given the immense practical importance of this prediction problem, there has been an enormous effort to explore different algorithms for storing and referencing information about the sequence. These efforts have led to recurrent neural networks [1]—which encode the past as a real vector of fixed length that is updated after every observation—and specific classes of such networks, such as Long Short-Term Memory (LSTM) networks [2, 3]. Other recently popular models that have explicit notions of memory include neural Turing machines [4], memory networks [5], differentiable neural computer [6], etc. These models have been quite successful (see e.g. [7, 8]); nevertheless, they seem largely unable to consistently learn long-range dependencies, which are crucial in many settings including language. In parallel to these efforts to design systems that explicitly use memory, there has been much effort from the neuroscience community to understand how humans and animals are able to make accurate predictions about their environment. Many of these efforts also attempt to understand the computational mechanisms behind the formation of memories (memory “consolidation”) and retrieval [9, 10, 11]. Despite the long history of studying sequential prediction, many fundamental questions remain: • How much memory is necessary to accurately predict future observations, and what properties of the underlying sequence determine this requirement? • Must one remember significant information about the distant past or is a short-term memory sufficient? • What is the computational complexity of accurate prediction? • How do answers to the above questions depend on the metric that is used to evaluate prediction accuracy? Aside from the intrinsic theoretical value of these questions, their answers could serve to guide the construction of effective practical prediction systems, as well as informing the discussion of the computational machinery of cognition and prediction/learning in nature. In this work, we provide insights into the first three questions. We begin by establishing the following proposition, which addresses the first two questions with respect to the pervasively used metric of average prediction error: Proposition 1. Let M be any distribution over sequences with mutual information I(M) between the past observations . . . , xt−2 , xt−1 and future observations xt , xt+1 , . . .. The best `-th order Markov model, which makes predictions based only on the most recent ` observations, predicts the p distribution of the next observation with average KL error I(M)/` or average `1 error I(M)/`, with respect to the actual conditional distribution of xt given all past observations. The intuition behind the statement and proof of this general proposition is the following: at time t, we either predict accurately and are unsurprised when xt is revealed to us, or if we predict poorly and are surprised by the value of xt , then xt must contain a significant amount of information about 1 the history of the sequence, which can then be leveraged in our subsequent predictions of xt+1 , xt+2 , etc. In this sense, every timestep in which our prediction is ‘bad’, we learn some information about the past. Because the mutual information between the history of the sequence and the future is bounded by I(M), if we were to make I(M) consecutive bad predictions, we have captured nearly this amount of information about the history, and hence going forward, as long as the window we are using spans these observations, we should expect to predict well. This general proposition, framed in terms of the mutual information of the past and future, has immediate implications for a number of well-studied models of sequential data, such as Hidden Markov Models (HMMs). For a HMM with n hidden states, the mutual information of the generated sequence is trivially bounded by log n, which yields the following corollary to the above proposition. We state this proposition now, as it provides a helpful reference point in our discussion of the more general proposition. Corollary 1. Suppose observations are generated by a Hidden Markov Model with at most n hidden states. The best log n -th order Markov model, which makes predictions based only on the most recent log n predicts the distribution of the next observation with average KL error ≤ or `1 observations, √ error ≤ , with respect to the optimal predictor that knows the underlying HMM and has access to all past observations. In the setting where the observations are generated according to an HMM with at most n hidden states, this “best” `th order Markov model is easy to learn given sufficient data, and corresponds to the naive “empirical” `-gram model based on the previous observations. Specifically, this is the model that, given xt−` , xt−`+1 , . . . , xt−1 , outputs the observed (empirical) distribution of the observation that has followed this length ` sequence. (To predict what comes next in the phrase “. . . defer the details to the ” we look at the previous occurrences of this subsequence, and predict according to the empirical frequency of the subsequent word.) The following theorem makes this claim precise. Theorem 1. Suppose observations are generated by a Hidden Markov Model with at most n hidden states, and output alphabet of size d. For > 1/n there exists a window length ` = O( log n ) and absolute constant c such that for any T ≥ dc` , if t ∈ {1, 2, . . . , T } is chosen uniformly at random, then the expected `1 distance between the true distribution of xt given the entire history (and knowledge of the HMM), and the distribution predicted by the naive “empirical” `-th order √ Markov model based on x0 , . . . , xt , is bounded by . The above theorem states that the window length necessary to predict well is independent of the mixing time of the HMM in question, and holds even if the model does not mix. While the amount of data required to make accurate predictions using length ` windows scales exponentially in `—corresponding to the condition in the above theorem that t is chosen uniformly between 0 and T = dO(`) —our lower bounds, discussed in Section 1.3, argue that this exponential dependency is unavoidable. 1.1 Interpretation of mutual information of past and future While the mutual information between the past observations and the future observations is an intuitive parameterization of the complexity of a distribution over sequences, the fact that it is the right quantity is a bit subtle. It is tempting to hope that this mutual information is a bound on the amount of memory that would be required to store all the information about past observations that is relevant to the distribution of future observations. Consider the following setting: Given a 2 joint distribution over random variables P AST and F U T , suppose we wish to define a function f that maps P AST to a binary “advice”/memory string f (P AST ), possibly of variable length, such that F U T is independent of P AST , given f (P AST ). As is shown in Harsha et al. [12], there are joint distributions over (P AST, F U T ) such that even on average, the minimum length of the advice/memory string necessary for the above task is exponential in the mutual information I(P AST ; F U T ). This setting can also be interpreted as a two-player communication game where one player generates P AST and the other generates F U T given limited communication (i.e. the ability to communicate f (P AST )). 1 Given the fact that this mutual information is not even an upper bound on the amount of memory that an optimal algorithm (computationally unbounded, and with complete knowledge of the distribution) would require, Proposition 1 might be surprising. 1.2 Implications of Proposition 1 and Corollary 1. These results show that a Markov model—a model that cannot capture long-range dependencies or structure of the data—can predict accurately on any data-generating distribution, provided the order of the Markov model scales with the complexity of the distribution, as parameterized by the mutual information between the past and future. Strikingly, this parameterization is indifferent to whether the dependencies in the sequence are relatively short-range as in an HMM that mixes quickly, or very long-range as in an HMM that mixes slowly or does not mix at all. Independent of the nature of these dependencies, provided the mutual information is small, accurate prediction is possible based only on the most recent few observation. (See Figure 1 for a concrete illustration of this result in the setting of an HMM that does not mix and has long-range dependencies.) Figure 1: A depiction of a HMM on n states, that repeats a given length n binary sequence of outputs, and hence does not mix. Corollary 1 and Theorem 1 imply that accurate prediction is possible based only on short sequences of O(log n) observations. At a time where increasingly complex models such as recurrent neural networks and neural Turing machines are in vogue, these results serve as a baseline theoretical result. They also help explain the practical success of simple Markov models, such as “Kneser-Ney” smoothing [13, 14], which are crucial components in state-of-the-art machine translation and speech recognition systems. Although recent recurrent neural networks have yielded empirical gains (see e.g. [7, 8]), current models still seem largely incapable of successfully capturing long-range dependencies.2 In 1 It is worth noting that if the advice/memory string s is sampled first, and then P AST and F U T are defined to be random functions of s, then the length of s can be related to I(P AST ; F U T ) (see [12]). This latter setting where s is generated first corresponds to allowing shared randomness in the two-player communication game; however, this is not relevant to the sequential prediction problem. 2 One amusing example is the recent sci-fi short film Sunspring whose script was automatically generated by an LSTM. Locally, each sentence of the dialogue (mostly) makes sense, though there is no cohesion over longer time frames, and no overarching plot trajectory (despite the brilliant acting). 3 some settings, such as natural language, capturing such long-range dependencies seems crucial for achieving human-level results. Indeed, the main message of a narrative is not conveyed in any single short segment. More generally, higher-level intelligence seems to be about the ability to judiciously decide what aspects of the observation sequence are worth remembering and updating a model of the world based on these aspects. Thus, for such settings, Proposition 1, can be interpreted as a negative result—that average error is not a good metric for training and evaluating models. It is important to note that average prediction error is the metric that ubiquitously used in practice, both in the natural language processing domain and elsewhere. Our results suggest that a different metric might be essential to driving progress towards systems that attempt to capture long-range dependencies and leverage memory in meaningful ways. We discuss this possibility of alternate prediction metrics more in Section 1.4. For many other settings, such as financial prediction and lower level language prediction tasks such as those used in OCR or speech recognition, average prediction error is the most meaningful metric. For these settings, the result of Proposition 1 is extremely positive: no matter the nature of the dependencies in the financial markets, it is sufficient to learn a Markov model. As one obtains more and more data, one can learn a higher and higher order Markov model, and average prediction accuracy should continue to improve. For these applications, the question now becomes a computational question: the naive approach to learning an `th-order Markov model in a domain with an alphabet of size d might require Ω(d` ) space to store, and data to learn. From a computational standpoint, is there a better algorithm? What properties of the underlying sequence imply that such models can be learned, or approximated more efficiently or with less data? Our computational lower bounds, described below, provide some perspective on these computational considerations. 1.3 Lower bounds Our positive results show that accurate prediction is possible via an algorithmically simple model— a Markov model that only depends on the most recent observations—which can be learned in an algorithmically straightforward fashion by simply using the empirical statistics of short sequences of examples, compiled over a sufficient amount of data. Nevertheless, the Markov model has d` parameters, and hence requires an amount of data that scales as Ω(d` ) to learn, where d is a bound on the size of the observation alphabet. This prompts the question of whether it is possible to learn a successful predictor based on significantly less data. We show that, even for the special case where the data sequence is generated from an HMM over n hidden states, this is not possible in general, assuming a natural complexity-theoretic assumption. A HMMs with n hidden states and an output alphabet of size d is defined via only O(n2 + nd) parameters and O (n2 + nd) samples are sufficient, from an information theoretic standpoint, to learn a model that will predict accurately. While learning an HMM is computationally hard (see e.g. [15]), this begs the question of whether accurate (average) prediction can be achieved via a computationally efficient algorithm and and an amount of data significantly less than the dΘ(log n) that the naive Markov model would require. Our main lower bound shows that there exists a family of HMMs such that the Ω(dlog n/ ) sample complexity requirement is necessary for any computationally efficient algorithm that predicts accurately on average, assuming a natural complexity-theoretic assumption. Specifically, we show that this hardness holds, provided that the problem of strongly refuting a certain class of CSPs is 4 hard, which was conjectured in [16] and studied in related works [17] and [18]. See Section 5 for a description of this class and discussion of the conjectured hardness. Theorem 2. Assuming the hardness of strongly refuting a certain class of CSPs, for all sufficiently large n and any ∈ (1/nc , 0.1) for some fixed constant c, there exists a family of HMMs with n hidden states and an output alphabet of size d such that any polynomial time algorithm that achieves average error (with respect to the optimal predictor) for a random HMM in the family must observe dΘ(log n/) observations from the HMM. As the mutual information of the generated sequence of an HMM with n hidden states is bounded by log n , Theorem 2 directly implies that there are families of data-generating distributions M with mutual information I(M) and observations drawn from an alphabet of size d such that any computationally efficient algorithm requires dΩ(I(M)/) samples from M to achieve average error . The above bound holds when d is large compared to log n or I(M), but a different but equally relevant regime is where the alphabet size d is small compared to the scale of dependencies in the sequence (for example, when predicting characters [19]). We show lower bounds in this regime of the same flavor as those of Theorem 2 except based on the problem of learning a noisy parity function; the (very slightly) subexponential algorithm of Blum et al. [20] for this task means that we lose at least a superconstant factor in the exponent in comparison to the positive results of Proposition 1. Proposition 2. Let f (k) denote a lower bound on the amount of time and samples required to learn parity with noise on uniformly random k-bit inputs. For all sufficiently large n and ∈ (1/nc , 0.1) for some fixed constant c, there exists a family of HMMs with n hidden states such that any algorithm that achieves average prediction error (with respect to the optimal predictor) for a random HMM in the family requires at least f (log n/) time or samples. Finally, we also establish the information theoretic optimality of the results of Proposition 1, in the sense that among (even computationally unbounded) prediction algorithms that predict based only on the most recent ` observations, an average KL prediction error of Ω(I(M)/`) and `1 error Ω(I(M)/2 ) with respect to the optimal predictor, is necessary. Proposition 3. There is an absolute constant c < 1 such that for all 0 < < 1/4 and sufficiently large n, there exists an HMM with n hidden states such that it is not information-theoretically √ possible to obtain average KL prediction error less than or `1 error less than (with respect to the optimal predictor) while using only the most recent c log n/ observations to make each prediction. 1.4 Future Directions As mentioned above, for the settings in which capturing long-range dependencies seems essential, it is worth re-examining the choice of “average prediction error” as the metric used to train and evaluate models. One possibility, that has a more worst-case flavor, is to only evaluate the algorithm at a chosen set of time steps instead of all time steps. Hence the naive Markov model can no longer do well just by predicting well on the time steps when prediction is easy. In the context of natural language processing, learning with respect to such a metric intuitively corresponds to training a model to do well with respect to a question answering task instead of a language modeling task. A fertile middle ground between average error (which gives too much reward for correctly guessing common words like “a” and “the”), and worst-case error might be a re-weighted prediction error that provides more reward for correctly guessing less common observations. It seems possible, however, that the techniques used to prove Proposition 1 can be extended to yield analogous statements for such error metrics. 5 Given the many settings for which average error is the most natural metric of prediction accuracy, and the upper bounds of Proposition 1, it is natural to consider what additional structure might be present that avoids the (conditional) computational lower bounds of Theorem 2. One possibility is a robustness property—for example the property that a Markov model would continue to predict well even when each observation were obscured or corrupted with some small probability. The lower bound instances in Theorem 2 and Proposition 2 rely on parity based constructions and hence are very sensitive to noise and corruptions. For learning over product distributions, there are well known connections between noise stability and approximation by low-degree polynomials [21, 22]. Additionally, low-degree polynomials can be learned agnostically over arbitrary distributions via polynomial regression [23]. It is tempting to hope that this thread could be made rigorous, by establishing a connection between natural notions of noise stability over arbitrary distributions, and accurate low-degree polynomial approximations. Such a connection could lead to significantly better sample complexity requirements for prediction on such “robust” distributions of sequences, perhaps requiring only poly(d, I(M), 1/) data. Additionally, such sample-efficient approaches to learning succinct representations of large Markov models may inform the many practical prediction systems that currently rely on Markov models. 1.5 Related Work Parameter Estimation. It is interesting to compare using a Markov model for prediction with methods that attempt to properly learn an underlying model. For example, method of moments algorithms [24, 25] allow one to estimate a certain class of Hidden Markov model with polynomial sample and computational complexity. These ideas have been extended to learning neural networks [26] and input-output RNNs [27]. Using different methods, Arora et al. [28] showed how to learn certain random deep neural networks. Learning the model directly can result in better sample efficiency, and also provide insights into the structure of the data. The major drawback of these approaches is that they usually require the true data-generating distribution to be in (or extremely close to) the model family that we are learning. This is a very strong assumption that often does not hold in practice. Universal Prediction and Coding Theory. On the other end of the spectrum is the class of no-regret online learning methods which assume that the data generating distribution can even be adversarial [29]. However, the nature of these results are fundamentally different from ours: whereas we are comparing to the perfect model that can look at the infinite past, online learning methods typically compare to a fixed set of experts, which is much weaker. There is much work on sequential prediction based on KL-error from the information theory and statistics communities. The philosophy of these approaches are often more adversarial, with perspectives ranging from minimum description length [30, 31] and individual sequence settings [32], where no model of the data distribution process is assumed. With regards to worst case guarantees (where there is no data generation process), and regret as the notion of optimality, there is a line of work on both minimax rates and the performance of Bayesian algorithms, the latter of which has favorable guarantees in a sequential setting. With regards to minimax rates, [33] provides an exact characterization of the minimax strategy, though the applicability of this approach is often limited to settings where the strategies available to the learner is relatively small (i.e., the normalizing constant in [33] must exist). More generally, there has been considerable work on the regret in information-theoretic and statistical settings, such as the works in [32, 34, 35, 36, 37, 38, 39, 40]. With regards to log-loss more broadly, there is considerable work on information consistency (convergence in distribution) and minimax rates with regards to statistical estimation in parametric and non-parametric families [41, 42, 43, 44, 45, 46]. In some of these settings, e.g. minimax risk in 6 parametric, i.i.d. settings, there are characterizations in terms of mutual information [42]. There is also work on universal lossless data compression algorithm, such as the celebrated Lempel-Ziv algorithm [47]. Here, the setting is rather different as it is one of coding the entire sequence (in a block setting) rather than prediction loss. Sequential Prediction in Practice. Our work was initiated by the desire to understand the role of memory in sequential prediction, and the belief that modeling long-range dependencies is important for complex tasks such as understanding natural language. There have been many proposed models with explicit notions of memory, including recurrent neural networks [48], Long Short-Term Memory (LSTM) networks[2, 3], attention-based models[49], neural Turing machines [4], memory networks [5], differentiable neural computers [6], etc. While some of these models have been quite successful in practice (see e.g. [7, 8]), they still largely fail to capture many long-range dependencies–in the case of LSTMs, for example, it is not difficult to show that they forget the past exponentially quickly if they are “stable” [1]. To gain more insight into this problem, we began by analyzing the simplest Markov predictor, and found to our surprise that it performed nearly as well as one could hope. 2 Proof Sketch of Theorem 1 We provide a sketch of the proof of Theorem 1, which is stronger than Proposition 1 but applies specifically to sequences generated from a Hidden Markov Model. The core of this proof is the following lemma that guarantees that the Markov model that knows the true marginal probabilities of all short sequences, will end up predicting well. Additionally, this good expected prediction will hold with respect to only the randomness of the HMM during the short window, as opposed to over the randomness of when the window begins (as in our more general results). For settings such as financial forecasting, this additional guarantee is particularly pertinent; you do not need to worry about the possibility of choosing an “unlucky” time to begin your trading regime, as long as you plan to trade for a duration that spans an entire short window. Beyond the extra strength of this result for HMMs, the proof approach is intuitive and pleasing, in comparison to the more direct proof of Proposition 1. We first state the lemma and sketch its proof, and then conclude the section by describing how this yields Theorem 1. Lemma 4. Consider an HMM with n hidden states, let the hidden state at time s = 0 be chosen according to an arbitrary distribution π, and denote the observation at time s by xs . Let OP Ts denote the conditional distribution of xs given observations x0 , . . . , xs−1 , and knowledge of the hidden state at time s = 0. Let Ms denote the conditional distribution of xs given only x0 , . . . , xs−1 , which corresponds to the naive sth order Markov model that knows only the joint probabilities of sequences of the first s observations. Then with probability at least 1 − 1/nc−1 over the choice of initial state, for ` = c log n/2 , `−1 hX i E kOP Ts − Ms k1 ≤ 4`, s=0 where the expectation is with respect to the randomness in the outputs x0 , . . . , x`−1 . The proof of the this lemma will hinge on establishing a connection between OP Ts —the Bayes optimal model that knows the HMM and the initial hidden state h0 , and at time s predicts the true distribution of xs given h0 , x0 , . . . , xs−1 —and the naive order s Markov model Ms that knows the joint probabilities of sequences of s observations (given that the initial state is drawn according to π), and predicts accordingly. This latter model is precisely the same as the model that knows 7 the HMM and distribution π (but not h0 ), and outputs the conditional distribution of xs given the observations. To relate these two models, we proceed via a martingale argument that leverages the intuition that, at each time step either OP Ts ≈ Ms , or, if they differ significantly, we expect the sth observation xs to contain a significant amount of information about the hidden state at time zero, h0 , which will then improve Ms+1 . Our submartingale will precisely capture the sense that for any s where there is a significant deviation between OP Ts and Ms , we expect the probability of the initial state being h0 conditioned on x0 , . . . , xs , to be significantly more than the probability of h0 conditioned on x0 , . . . , xs−1 . More formally, let H0s denote the distribution of the hidden state at time 0 conditioned on x0 , . . . , xs and let h0 denote the true hidden state at time 0. We show that the following expression is a submartingale: s P r[H0s = h0 ] 1X log − kOP Ti − Mi k21 . 1 − P r[H0s = h0 ] 2 i=0 The fact that this is a submartingale is not difficult: Define Rs as the conditional distribution of xs given observations x0 , · · · , xs−1 and initial state drawn according to π but not being at hidden state h0 at time 0. Note that Ms is a convex combination of OP Ts and Rs , hence kOP Ts − Ms k1 ≤ kOP Ts − Rs k1 . To verify the submartingale property, note that by Bayes Rule, the change in the LHS at any time step s is the log of the ratio of probability of observing the output xs according to the distribution OP Ts and the probability of xs according to the distribution Rs . The expectation of this is the KL-divergence between OP Ts and Rs , which can be related to the `1 error using Pinsker’s inequality. At a high level, the proof will then proceed via concentration bounds (Azuma’s inequality), to 2 show that, with high probability, if the error from the first ` = c log n/ timesteps is large, then P r[H0`−1 =h0 ] is also likely to be large, in which case the posterior distribution of the hidden 1−P r[H0`−1 =h0 ] state, H0`−1 will be sharply peaked at the true hidden state, h0 , unless h0 had negligible mass (less than n−c ) in distribution π. log There are several slight complications to this approach, including the fact that the submartingale we construct does not necessarily have nicely concentrated or bounded differences, as the first term in the submartingale could change arbitrarily. We address this by noting that the first term should not decrease too much except with tiny probability, as this corresponds to the posterior probability of the true hidden state sharply dropping. For the other direction, we can simply “clip” the deviations to prevent them from exceeding log n in any timestep, and then show that the submartingale property continues to hold despite this clipping by proving the following modified version of Pinsker’s inequality: Lemma 1. (Modified Pinsker’s inequality) For any two distributions h µ(x)nand ν(x) oidefined on µ(x) x ∈ X, define the C-truncated KL divergence as D̃C (µ k ν) = Eµ log min ν(x) , C for some fixed C such that log C ≥ 8. Then D̃C (µ k ν) ≥ 12 kµ − νk21 . Given Lemma 4, the proof of Theorem 1 follows relatively easily. Recall that Theorem 1 concerns the expected prediction error at a timestep t ← {0, 1, . . . , dc` }, based on the model Memp corresponding to the empirical distribution of length ` windows that have occurred in x0 , . . . , xt ,. The connection between the lemma and theorem is established by showing that, with high probability, Memp is close to Mπ̂ , where π̂ denotes the empirical distribution of (unobserved) hidden states h0 , . . . , ht , and Mπ̂ is the distribution corresponding to drawing the hidden state h0 ← π̂ and then generating x0 , x1 , . . . , x` . We provide the full proof in Appendix A. 8 3 Definitions and Notation Before proving our general Proposition 1, we first introduce the necessary notation. For any random variable X, we denote its distribution as P r(X). The mutual information between two random variables X and Y is defined as I(X; Y ) = H(Y ) − H(Y \|X) where H(Y ) is the entropy of Y and H(Y \|X) is the conditional entropy of Y given X. The conditional mutual information I(X; Y \|Z) is defined as: I(X; Y \|Z) = H(X\|Z) − H(X\|Y, Z) = Ex,y,z log P r(X\|Y, Z) = Ey,z DKL (P r(X\|Y, Z) k P r(X\|Z)) P r(X\|Z) P p(x) where DKL (p k q) = x p(x) log q(x) is the KL divergence between the distributions p and q. Note that we are slightly abusing notation here as DKL (P r(X\|Y, Z) k P r(X\|Z)) should technically be DKL (P r(X\|Y = y, Z = z) k P r(X\|Z = z)). But we will ignore the assignment in the conditioning when it is clear from the context. Mutual information obeys the following chain rule: I(X1 , X2 ; Y ) = I(X1 ; Y ) + I(X2 ; Y \|X1 ). Given a distribution over infinite sequences, {xt } generated by some model M where xt is random variable denoting the output at time t, we will use the shorthand xji to denote the collection of random variables for the subsequence of outputs {xi , · · · , xj }. The distribution of {xt } is stationary if the joint distribution of any subset of the sequence of random variables {xt } is invariant with respect to shifts in the time index. Hence P r(xi1 , xi2 , · · · , xin ) = P r(xi1 +l , xi2 +l , · · · , xin +l ) for any l if the process is stationary. We are interested in studying how well the output xt can be predicted by an algorithm which only looks at the past ` outputs. The predictor A` maps a sequence of ` observations to a predicted distribution of the next observation. We denote the predictive distribution of A` at time t as QA` (xt \|xt−1 t−` ). We refer to the Bayes optimal predictor using only windows of length ` as P` , hence the prediction of P at time t is P r(xt \|xt−1 t−` ). P` is just the naive `-th order Markov predictor provided with the true distribution of the data. Let the Bayes optimal predictor looking at the entire history of the model be P∞ , the prediction of P∞ at time t is P r(xt \|xt−1 −∞ ). We will evaluate the predictions of A` and P` with respect to P∞ over a long time window [0 : T − 1]. The crucial property of the distribution that is relevant to our results is the mutual information between past and future observations. For a stochastic process {xt } generated by some model M we define the mutual information I(M) of the model M as the mutual information between the past and future, averaged over the window [0 : T − 1]. T −1 1 X ∞ I(M) = I(xt−1 −∞ ; xt ) T (3.1) t=0 ∞ If the process {xt } is stationary, then I(xt−1 −∞ ; xt ) is the same for all time steps hence I(M) = ∞ I(x−1 −∞ ; x0 ). We compare the prediction of the predictor P` and A` with respect to P∞ . Let F (P, Q) be some measure of distance between two predictive distributions. In this work, we consider the KL-divergence, `1 distance and the relative zero-one loss between the two distributions. The KLdivergence and `1 distance between two distributions are defined in the standard way. We define the relative zero-one loss as the difference between the zero-one loss of the optimal predictor P∞ and the algorithm A` . We define the expected loss of any predictor A` with respect to the optimal 9 predictor P∞ and a loss function F as follows: (t) δF (A` ) i h t−1 ) ), QA` (xt \|xt−1 = Ext−1 F (P r(xt \|x−∞ t−` , −∞ T −1 1 X (t) δF (A` ) = δF (A` ) T t=0 (t) δ̂F (A` ) We also define and δ̂F (A` ) for the algorithm A` in the same fashion as the error in estimating P (xt \|xt−1 ), the true conditional distribution of the model M. t−` i h (t) t−1 )) , δ̂F (A` ) = Ext−1 F (P r(xt \|xt−` ), QA` (xt \|xt−1 t−` t−` 4 δ̂F (A` ) = T −1 1 X (t) δ̂F (A` ) T t=0 Predicting Well with Short Windows To establish our general proposition, which applies beyond the HMM setting, we provide an elementary and purely information theoretic proof. Proposition 1. For any data-generating distribution M with mutual information I(M) between past and future observations, the best `-th order Markov model P` obtains average KL-error, δKL (P` ) ≤ I(M)/` with respect to the optimal predictor with access to the infinite history. Also, any predictor A` with δ̂KL (A` ) average KL-error in estimating the joint probabilities over windows of length ` gets average error δKL (A` ) ≤ I(M)/` + δ̂KL (A` ). Proof. We bound the expected error by splitting the time interval 0 to T − 1 into blocks of length `. Consider any block starting at time τ . We find the average error of the predictor from time τ to τ + ` − 1 and then average across all blocks. To begin, note that we can decompose the error as the sum of the error due to not knowing the past history beyond the most recent ` observations and the error in estimating the true joint (t) distribution of the data over a ` length block. Consider any time t. Recall the definition of δKL (A` ). i h (t) t−1 δKL (A` ) = Ext−1 DKL (P r(xt \|xt−1 )) ) k Q (x \|x t A −∞ ` t−` −∞ h i h i t−1 t−1 t−1 = Ext−1 DKL (P r(xt \|xt−1 −∞ ) k P (xt \|xt−` )) + Ext−1 DKL (P r(xt \|xt−` ) k QA` (xt \|xt−` )) −∞ = −∞ (t) δKL (P` ) + (t) δ̂KL (A` ) (t) t−`−1 Therefore, δKL (A` ) = δKL (P` ) + δ̂KL (A` ). It’s easy to verify that δKL (P` ) = I(x−∞ ; xt \|xt−1 t−` ). This relation expresses the intuition that the current output (xt ) has a lot of extra information t−`−1 about the past (x−∞ ) if we cannot predict it as well using the ` most recent observations (xt−1 t−` ) t−1 as can be done by using the entire past (x−∞ ). We will now upper bound the total error for the −1 ∞ window [τ, τ + ` − 1]. We expand I(xτ−∞ ; xτ ) using the chain rule, ∞ τ +`−1 X X τ −1 ∞ τ −1 −1 t−1 I(x−∞ ; xτ ) = I(x−∞ ; xt \|xτ ) ≥ I(xτ−∞ ; xt \|xt−1 τ ). t=τ t=τ (t) −1 t−`−1 Note that I(xτ−∞ ; xt \|xτt−1 ) ≥ I(x−∞ ; xt \|xt−1 t−` ) = δKL (P` ) as t−` ≤ τ and I(X, Y ; Z) ≥ I(X; Z\|Y ). The proposition now follows from averaging the error across the ` time steps and using Eq. 3.1 to average over all blocks of length ` in the window [0, T − 1], τ +`−1 1 X (t) 1 I(M) −1 ∞ δ (P` ) ≤ I(xτ−∞ ; xτ ) =⇒ δKL (P` ) ≤ ` t=τ KL ` ` 10 Proposition 1 also directly gives guarantees for the scenario where the task is to predict the distribution of the next block of outputs instead of just the next immediate output, because KLdivergence obeys the chain rule. The following easy corollary, relating KL error to `1 error yields the following statement, which also trivially applies to zero/one loss with respect to that of the optimal predictor, as the expected relative zero/one loss at any time step is at most the `1 loss at that time step. Corollary 2. For any data-generating distribution M with mutual information I(M) between past and p future observations, the best `-th order Markov model P` obtains average `1 -error, δ`1 (P` ) ≤ I(M)/2` with respect to the optimal predictor with access to the infinite history. Also, any predictor A p` with δ̂`1 (A` ) average `1 -error in estimating the joint probabilities gets average error δ`1 (A` ) ≤ I(M)/2` + δ̂`1 (A` ). Proof. We again decompose the error as the sum of the error in estimating P̂ and the error due to not knowing the past history using the triangle inequality. h i (t) )k1 δ`1 (A` ) = Ext−1 kP r(xt \|xt−1 ) − QA` (xt \|xt−1 −∞ t−` −∞ i i h h t−1 t−1 t−1 t−1 ≤ Ext−1 kP r(xt \|x−∞ ) − P r(xt \|xt−` )k1 + Ext−1 kP r(xt \|xt−` ) − QA` (xt \|xt−` )k1 −∞ −∞ = (t) δ`1 (P` ) + (t) δ̂`1 (A` ) (t) Therefore, δ`1 (A` ) ≤ δ`1 (P` ) + δ̂`1 (A` ). By Pinsker’s inequality and Jensen’s inequality, δ`1 (A` )2 ≤ (t) δKL (A` )/2. Using Proposition 1, T −1 1 X (t) I(M) δKL (A` ) = δKL (A` ) ≤ T ` t=0 Therefore, using Jensen’s inequality again, δ`1 (A` ) ≤ 5 p I(M)/2`. Lower Bound for Large Alphabets Our lower bounds for the sample complexity in the large alphabet case leverage a class of Constraint Satisfaction Problems (CSPs) with high complexity. A class of (Boolean) k-CSPs is defined via a predicate—a function P : {0, 1}k → {0, 1}. An instance of such a k-CSP on n variables {x1 , · · · , xn } is a collection of sets (clauses) of size k whose k elements consist of k variables or their negations. Such an instance is satisfiable if there exists an assignment to the variables x1 , . . . , xn such that the predicate P evaluates to 1 for every clause. More generally, the value of an instance is the maximum, over all 2n assignments, of the ratio of number of satisfied clauses to the total number of clauses. Our lower bounds are based on the presumed hardness of distinguishing random instances of or a certain class of CSP, versus instances of the CSP with high value. There has been much work attempting to characterize the difficulty of CSPs—one notion which we will leverage is the complexity of a class of CSPs, first defined in [16] and studied in [17]: Definition 1. The complexity of a class of k-CSPs defined by predicate P : {0, 1}k → {0, 1} is the largest r such that there exists a distribution supported on the support of P that is (r − 1)-wise independent (i.e. “uniform”), and no such r-wise independent distribution exists. 11 Example 1. Both k-XOR and k-SAT are well-studied classes of k-CSPs, corresponding, respectively, to the predicates PXOR that is the XOR of the k Boolean inputs, and PSAT that is the OR of the inputs. These predicates both support (k − 1)-wise uniform distributions, but not k-wise uniform distributions, hence their complexity is k. In the case of k-XOR, the uniform distribution over {0, 1}k restricted to the support of PXOR is (k − 1)-wise uniform. The same distribution is also supported by k-SAT. A random instance of a CSP with predicate P is an instance such that all the clauses are chosen uniformly at random (by selecting the k variables uniformly, and independently negating each variable with probability 1/2). A random instance will have value close to E[P ], where E[P ] is the expectation of P under the uniform distribution. In contrast, a planted instance is generated by first fixing a satisfying assignment σ and then sampling clauses that are satisfied, by uniformly choosing k variables, and picking their negations according to a (r−1)-wise independent distribution associated to the predicate. Hence a planted instance always has value 1. A noisy planted instance with planted assignment σ and noise level η is generated by sampling consistent clauses (as above) with probability 1 − η and random clauses with probability η, hence with high probability it has value 1 − η + ηE[P ]. Our hardness results are based on distinguishing whether a CSP instance is random or has a high value. As one would expect, the difficulty of distinguishing random instances from noisy planted instances, decreases as the number of sampled clauses grows. The following conjecture of Feldman et al. [16] asserts a sharp boundary on the number of clauses, below which this problem becomes computationally intractable, while remaining information theoretically easy. The notation is made more explicit in Appendix B. Conjectured CSP Hardness [Conjecture 1] [16]: Let Q be any distribution over k-clauses and n variables of complexity r and 0 < η < 1. Any polynomial-time (randomized) algorithm that, given access to a distribution D that equals either the uniform distribution over k-clauses Uk or a (noisy) planted distribution Qησ = (1 − η)Qσ + ηUk for some σ ∈ {0, 1}n and planted distribution Qσ , decides correctly whether D = Qησ or D = Uk with probability at least 2/3 needs Ω̃(nr/2 ) clauses. Feldman et al. [16] proved the conjecture for the class of statistical algorithms 3 . Recently, Kothari et al. [18] showed that the polynomial time Sum-of-Squares (SOS) algorithm requires Ω̃(nr/2 ) clauses to refute random instances of a CSP with complexity r, hence proving Conjecture 1 for any polynomial-size semidefinite programming relaxation for refutation. Note that Ω̃(nr/2 ) is tight, as Allen et al. [17] give a SOS algorithm for refuting random CSPs beyond this regime. Other recent papers such as Daniely and Shalev-Shwartz [51] and Daniely [52] have also used presumed hardness of strongly refuting random k-SAT and random k-XOR instances with a small number of clauses to derive conditional hardness of learning results. A first attempt to encode a k-CSP as a sequential model is to construct a model which outputs k randomly chosen literals for the first k time steps 0 to k −1, and then their (noisy) predicate value for the final time step k. Clauses from the CSP correspond to samples from the model, and the algorithm would need to solve the CSP to predict the final time step k. However, as all the outputs up to the final time step are random, the trivial prediction algorithm that guesses randomly and does not try to predict the output at time k, would be near optimal. To get strong lower bounds, 3 Statistical algorithms are an extension of the statistical query model, these are algorithms that do not access samples from the distribution but instead have access to estimates of the expectation of any bounded function of a sample through an oracle. Feldman et al. [50] point out that almost all algorithms that work on random data also work with this limited access to samples, refer to Feldman et al. [50] for more details and examples. 12 we will output m > 1 functions of the k literals after k time steps, while still ensuring that all the functions remain collectively hard to invert without a large number of samples. We use elementary results from the theory of error correcting codes to achieve this, and prove hardness due to a reduction from a specific family of CSPs to which Conjecture 1 applies. By choosing k and m carefully, we obtain the near-optimal dependence on the mutual information and error —matching the upper bounds implied by Proposition 1. We provide a short outline of the argument, followed by the detailed proof in the appendix. 5.1 Sketch of construction and proof We construct a sequential model M such that making good predictions on the model requires distinguishing random instances of a k-CSP C on n variables from instances of C with a high value. The output alphabet of M is {ai } of size 2n. We choose a mapping from the 2n characters {ai } to the n variables {xi } and their n negations {x̄i }. For any clause C and planted assignment σ to the CSP C, let σ(C) be the k-bit string of values assigned by σ to literals in C. The model M randomly uniformly outputs k characters from time 0 to k − 1, which correspond to literals in the CSP C, hence the k outputs correspond to a clause C of the CSP. For some m to be specified later, we will construct a binary matrix A ∈ {0, 1}m×k , which will correspond to a good error-correcting code. For the time steps k to k + m − 1, with probability 1 − η the model outputs y ∈ {0, 1}m where y = Av mod 2 and v = σ(C) with C being the clause associated with the outputs of the first k time steps. With the remaining probability, η, the model outputs m uniformly random bits. Note that the mutual information I(M) is at most m as only the outputs from time k to k + m − 1 can be predicted. We claim that M can be simulated by a HMM with 2m (2k + m) + m hidden states. This can be done as follows. For every time step i from 0 to k − 1, we maintain 2m hidden states corresponding to vi = 0 and 2m hidden states corresponding to vi = 1. Each of these 2m states stores the current value of the m bits of y. This takes a total of k2m+1 hidden states. We use 2m hidden states for each time step k through k + m − 1 for the k output bits. Finally, we need an additional m hidden states to output m uniform random bits from time k to k + m − 1 with probability η. This accounts for a total of k2m+1 + 2m + m hidden states. Note that the larger m is with respect to k, the higher the cost (in terms of average prediction error) of failing to correctly predict the outputs from time k to k + m − 1. Tuning k and m allows us to control the number of hidden states or the mutual information, and average error incurred by a computationally constrained predictor. We define the CSP C in terms of a collection of predicates P (y) for each y ∈ {0, 1}m . While Conjecture 1 does not directly apply to C, as it is defined by a collection of predicates instead of a single one, we will later show a reduction from a related CSP C0 defined by a single predicate for which Conjecture 1 holds. For each y, the predicate P (y) of C is the set of v ∈ {0, 1}k which satisfy y = Av mod 2. Hence each clause has an additional label y which determines the satisfying assignments, this label is just the output of our sequential model M from time k to k + m − 1. Hence for any planted assignment σ, the set of satisfying clauses C of the CSP C are all clauses such that Av = y mod 2 where y is the label of the clause and v = σ(C). We define a (noisy) planted distribution over clauses Qησ by first uniformly randomly sampling a label y, and then sampling a consistent clause with probability (1 − η), otherwise with probability η we sample a uniformly random clause. Let Uk be the uniform distribution over all k-clauses with uniformly chosen labels y. We will show that Conjecture 1 implies that distinguishing between the distributions Qησ and Uk is hard without sufficiently many clauses. This gives us the hardness results we desire for our sequential model M: if an algorithm obtains low prediction error on the outputs from time k 13 through (k + m − 1), then it can be used to distinguish between instances of the CSP C with a high value and random instances, as no algorithm obtains low prediction error on random instances. Hence hardness of strongly refuting the CSP C implies hardness of making good predictions on M. We now sketch the argument for why Conjecture 1 implies the hardness of strongly refuting the CSP C. We define another CSP C0 which we show reduces to C. The predicate P of the CSP C0 is the set of all v ∈ {0, 1}k such that Av = 0 mod 2. Hence for any planted assignment σ, the set of satisfying clauses of the CSP C0 are all clauses such that v = σ(C) is in the nullspace of A. As before, the planted distribution over clauses is uniform on all satisfying clauses with probability (1 − η), with probability η we add a uniformly random k-clause. For some γ ≥ 1/10, if we can construct A such that the set of satisfying assignments v (which are the vectors in the nullspace of A) supports a (γk − 1)-wise uniform distribution, then by Conjecture 1 any polynomial time algorithm cannot distinguish between the planted distribution and uniformly randomly chosen clauses with less than Ω̃(nγk/2 ) clauses. We show that choosing a matrix A whose null space is (γk − 1)-wise uniform corresponds to finding a binary linear code with rate at least 1/2 and relative distance γ, the existence of which is guaranteed by the Gilbert-Varshamov bound. We next sketch the reduction from C0 to C. The key idea is that the CSPs C0 and C are defined by linear equations. If a clause C = (x1 , x2 , · · · , xk ) in C0 is satisfied with some assignment t ∈ {0, 1}k to the variables in the clause then At = 0 mod 2. Therefore, for some w ∈ {0, 1}k such that Aw = y mod 2, t + w mod 2 satisfies A(t + w) = y mod 2. A clause C 0 = (x01 , x02 , · · · , x0k ) with assignment t + w mod 2 to the variables can be obtained from the clause C by switching the literal x0i = x̄i if wi = 1 and retaining x0i = xi if wi = 0. Hence for any label y, we can efficiently convert a clause C in C0 to a clause C 0 in C which has the desired label y and is only satisfied with a particular assignment to the variables if C in C0 is satisfied with the same assignment to the variables. It is also not hard to ensure that we uniformly sample the consistent clause C 0 in C if the original clause C was a uniformly sampled consistent clause in C0 . We provide a small example to illustrate the sequential model constructed above. Let k = 3, m = 1 and n = 3. Let A ∈ {0, 1}1×3 . The output alphabet of the model M is {ai , 1 ≤ i ≤ 6}. The letter a1 maps to the variable x1 , a2 maps to x̄1 , similarly a3 → x2 , a4 → x̄2 , a5 → x3 , a6 → x̄3 . Let σ be some planted assignment to {x1 , x2 , x3 }, which defines a particular model M. If the output of the model M is a1 , a3 , a6 for the first three time steps, then this corresponds to the clause with literals, (x1 , x2 , x̄3 ). For the final time step, with probability (1 − η) the model outputs y = Av mod 2, with v = σ(C) for the clause C = (x1 , x2 , x̄3 ) and planted assignment σ, and with probability η it outputs a uniform random bit. For an algorithm to make a good prediction at the final time step, it needs to be able to distinguish if the output at the final time step is always a random bit or if it is dependent on the clause, hence it needs to distinguish random instances of the CSP from planted instances. We re-state Theorem 2 below, deferring its proof to Appendix B. Theorem 2. Assuming Conjecture 1, for all sufficiently large T and 1/T c < ≤ 0.1 for some fixed constant c, there exists a family of HMMs with T hidden states and an output alphabet of size n such that, any polynomial time prediction algorithm that achieves average KL-error, `1 error or relative zero-one error less than with probability greater than 2/3 for a randomly chosen HMM in the family needs requires nΘ(log T /) samples from the HMM over any window length which the algorithm uses for prediction. 14 6 Lower Bound for Small Alphabets Our lower bounds for the sample complexity in the binary alphabet case are based on the average case hardness of the decision version of the parity with noise problem, and the reduction is straightforward. In the parity with noise problem on n bit inputs we are given examples v ∈ {0, 1}n drawn uniformly from {0, 1}n along with their noisy labels hs, vi+ mod 2 where s ∈ {0, 1}n is the (unknown) support of the parity function, and ∈ {0, 1} is the classification noise such that P r[ = 1] = η where η < 0.05 is the noise level. Let Qηs be the distribution over examples of the parity with noise instance with s as the support of the parity function and η as the noise level. Let Un be the distribution over examples and labels where each label is chosen uniformly from {0, 1} independent of the example. The strength of of our lower bounds depends on the level of hardness of parity with noise. Currently, the fastest algorithm for the problem due to Blum et al. [20] runs in time and samples 2n/ log n . We define the function f (n) as follows– Definition 2. Define f (n) to be the function such that for a uniformly random support s ∈ {0, 1}n , with probability at least (1 − 1/n2 ) over the choice of s, any (randomized) algorithm that can distinguish between Qηs and Un with success probability greater than 2/3 over the randomness of the examples and the algorithm, requires f (n) time or samples. Our model will be the natural sequential version of the parity with noise problem, where each example is coupled with several parity bits. We denote the model as M(Am×n ) for some A ∈ {0, 1}m×n , m ≤ n/2. From time 0 through (n − 1) the outputs of the model are i.i.d. and uniform on {0, 1}. Let v ∈ {0, 1}n be the vector of outputs from time 0 to (n − 1). The outputs for the next m time steps are given by y = Av + mod 2, where ∈ {0, 1}m is the random noise and each entry i of is an i.i.d random variable such that P r[i = 1] = η, where η is the noise level. Note that if A is full row-rank, and v is chosen uniformly at random from {0, 1}n , the distribution of y is uniform on {0, 1}m . Also I(M(A)) ≤ m as at most the binary bits from time n to n + m − 1 can be predicted using the past inputs. As for the higher alphabet case, M(Am×n ) can be simulated by an HMM with 2m (2n + m) + m hidden states (see Section 5.1). We define a set of A matrices, which specifies a family of sequential models. Let S be the set of all (m × n) matrices A such that the sub-matrix of A corresponding to all rows but only the first 2n/3 columns is full row rank. We need this restriction to lower bound I(M(A)), as otherwise there could be small or no dependence of the parity bits on the inputs from time 0 to 2n/3 − 1. We denote R as the family of models M(A) for A ∈ S. Lemma 2 shows that with high probability over the choice of A, distinguishing outputs from the model M(A) from random examples Un requires f (n) time or examples. Lemma 2. Let A be chosen uniformly at random from the set S. Then, with probability at least (1 − 1/n) over the choice A ∈ S, any (randomized) algorithm that can distinguish the outputs from the model M(A) from the distribution over random examples Un with success probability greater than 2/3 over the randomness of the examples and the algorithm needs f (n) time or examples. The proof of Proposition 2 follows from Lemma 2 and is similar to the proof of Theorem 2. 15 Proposition 2. With f (T ) as defined in Definition 2, For all sufficiently large T and 1/T c < ≤ 0.1 for some fixed constant c, there exists a family of HMMs with T hidden states such that any algorithm that achieves average relative zero-one loss, average `1 loss, or average KL loss less than with probability greater than 2/3 for a randomly chosen HMM in the family needs, requires f (log T /) time or samples samples from the HMM over any window length which the algorithm uses for prediction. 7 Information Theoretic Lower Bounds We show that information theoretically, windows of length cI(M)/2 are necessary to get expected relative zero-one loss less than . As the expected relative zero-one loss is at most the `1 loss, which can be bounded by the square of the KL-divergence, this automatically implies that our window length requirement is also tight for `1 loss and KL loss. In fact, it’s very easy to show the tightness for the KL loss, choose the simple model which emits uniform random bits from time 0 to n − 1 and repeats the bits from time 0 to m − 1 for time n through n + m − 1. One can then choose n, m to get the desired error and mutual information I(M). To get a lower bound for the zero-one loss we use the probabilistic method to argue that there exists an HMM such that long windows are required to perform optimally with respect to the zero-one loss for that HMM. We state the lower bound and a rough proof idea, deferring the details to Appendix D. Proposition 3. There is an absolute constant c such that for all 0 < < 0.5 and sufficiently large n, there exits an HMM with n states such that it is not information theoretically possible to get average relative zero-one loss or `1 loss less than using windows of length smaller than c log n/2 , and KL loss less than using windows of length smaller than c log n/. We illustrate the construction in Fig. 2 and provide the high-level proof idea with respect to Fig. 2 below. Figure 2: Lower bound construction, n = 16 We want show that any predictor P using windows of length ` = 3 cannot make a good prediction. The transition matrix of the HMM is a permutation and the output alphabet is binary. Each state is assigned a label which determines its output distribution. The states labeled 0 emit 0 with probability 0.5 + and the states labeled 1 emit 1 with probability 0.5 + . We will randomly and uniformly choose the labels for the hidden states. Over the randomness in choosing the labels for the permutation, we will show that the expected error of the predictor P is large, which means that there must exist some permutation such that the predictor P incurs a high error. The rough 16 proof idea is as follows. Say the Markov model is at hidden state h2 at time 2, this is unknown to the predictor P. The outputs for the first three time steps are (x0 , x1 , x2 ). The predictor P only looks at the outputs from time 0 to 2 for making the prediction for time 3. We show that with high probability over the choice of labels to the hidden states and the outputs (x0 , x1 , x2 ), the output (x0 , x1 , x2 ) from the hidden states (h0 , h1 , h2 ) is close in Hamming distance to the label of some other segment of hidden states, say (h4 , h5 , h6 ). Hence any predictor using only the past 3 outputs cannot distinguish whether the string (x0 , x1 , x2 ) was emitted by (h0 , h1 , h2 ) or (h4 , h5 , h6 ), and hence cannot make a good prediction for time 3 (we actually need to show that there are many segments like (h4 , h5 , h6 ) whose label is close to (x0 , x1 , x2 )). The proof proceeds via simple concentration bounds. A Proof of Theorem 1 Theorem 1. Suppose observations are generated by a Hidden Markov Model with at most n hidden states, and output alphabet of size d. For > 1/n there exists a window length ` = O( log n ) and absolute constant c such that for any T ≥ dc` , if t ∈ {1, 2, . . . , T } is chosen uniformly at random, then the expected `1 distance between the true distribution of xt given the entire history (and knowledge of the HMM), and the distribution predicted by the naive “empirical” `-th order √ Markov model based on x0 , . . . , xt , is bounded by . Proof. Let πt be a distribution over hidden states such that the probability of the ith hidden state under πt is the empirical frequency of the ith hidden state from time 1 to t − 1 normalized by t − 1. For 0 ≤ s ≤ ` − 1, consider the predictor Pt which makes a prediction for the distribution of observation xt+s given observations xt , . . . , xt+s−1 based on the true distribution of xt under the HMM, conditioned on the observations xt , . . . , xt+s−1 and the distribution of the hidden state at time t being πt . We will show that in expectation over t, Pt gets small error averaged across the time steps 0 ≤ s ≤ ` − 1, with respect to the optimal prediction of the distribution of xt+s which knows the hidden state ht at time t. In order to show this, we need to first establish that the true hidden state ht at time t does not have very small probability under πt , with high probability over the choice of t. Lemma 3. With probability 1 − 2/n over the choice of t ∈ {0, . . . , T }, the hidden state ht at time t has probability at least 1/n3 under πt . Proof. Consider the ordered set Si of time indices t where the hidden state ht = i. For the sets corresponding to hidden states j which have probability less than 1/n2 under πT , the cardinality \|Sj \| ≤ T /n2 . The sum of the cardinality of all such small sets is at most T /n, and hence the probability that a uniformly random t ∈ {1, . . . , T } lies in one of these sets is at most 1/n. Now consider the set of time indices Si corresponding to some hidden state i which has probability at least 1/n2 under πT . For all t which are not among the first T /n3 time indices in this set, the hidden state i has probability at least 1/n3 under πt . As the fraction of the “bad” time steps t corresponding to any hidden state which has probability at least 1/n2 under πT is at most 1/n, the total fraction of these “bad” time steps t is at most 1/n. Therefore using a union bound, with failure probability 2/n, the hidden state ht at time t has probability at least 1/n3 under πt . Consider any time index t, for simplicity assume t = 0, and let OP Ts denote the conditional distribution of xs given observations x0 , . . . , xs−1 , and knowledge of the hidden state at time s = 0. Let Ms denote the conditional distribution of xs given only x0 , . . . , xs−1 , given that the hidden state at time 0 has the distribution π0 . 17 Lemma 4. For > 1/n, if the true hidden state at time 0 has probability at least 1/nc under π0 , then for ` = c log n/2 , `−1 h1 X i E kOP Ts − Ms k1 ≤ 4, ` s=0 where the expectation is with respect to the randomness in the outputs from time 0 to ` − 1. By Lemma 3, for a randomly chosen t ∈ {1, . . . , T } the probability that the hidden state i at time 0 has probability less than 1/n3 in the prior distribution πt is at most 2/n. Hence using Lemma 4, the expected average error of the predictor Pt across all t is at most 4 + 2/n ≤ 6 for ` = 3 log n/2 . Now consider the predictor P̂t which for 0 ≤ s ≤ ` − 1 predicts xt+s given xt , . . . , xt+s−1 according to the empirical distribution of xt+s given xt , . . . , xt+s−1 , based on the observations up to time t. We will now argue that the predictions of P̂t are close in expectation to the predictions of Pt . Recall that prediction of Pt at time t + s is the true distribution of xt under the HMM conditioned on the observations xt , . . . , xt+s−1 and the distribution of the hidden state at time t being drawn from πt . For any s < `, let P1 refer to the prediction of P̂t at time t + s and P2 refer to the prediction of Pt at time t + s. We will show that kP1 − P2 k1 is small in expectation. We do this using a martingale concentration argument. Consider any string r of length s. Let Q1 (r) be the empirical probability of the string r up to time t and Q2 (r) be the true probability of the string r given that the hidden state at time t is distributed as πt . Our aim is to show that \|Q1 (r) − Q2 (r)\| is small. Define the random variable Yτ = P r[[xτ : xτ +s−1 ] = r\|hτ ] − I([xτ : xτ +s−1 ] = r), P where I denotes the indicator function and Y0 is defined to be 0. We claim that Zτ = τi=0 Yi is a martingale with respect to the filtration {φ}, {h1 }, {h2 , x1 }, {h3 , x2 }, . . . , {ht+1 , xt }. To verify, note that, E[Yτ \|{h1 }, {h2 , x1 }, . . . , {hτ , xτ −1 }] = P r[[xτ : xτ +s−1 ] = r\|hτ ] − E[I([xτ : xτ +s−1 ] = r)\|{h1 }, {h2 , x1 }, . . . , {xτ −1 , hτ }] = P r[[xτ : xτ +s−1 ] = r\|hτ ] − E[I([xτ : xτ +s−1 ] = r)\|hτ ] = 0 Therefore E[Zτ \|{h1 }, {h2 , x1 }, . . . , {hτ , xτ −1 }] = Zτ −1 , and hence Zτ is a martingale. Also, note that \|Zτ − Zτ −1 \| ≤ 1 as 0 ≤ P r[[xτ : xτ +s−1 ] = r\|hτ ] ≤ 1 and 0 ≤ I([xτ : xτ +s−1 ] = r) ≤ 1. Hence using Azuma’s inequality (Lemma 8), P r[\|Zt−s \| ≥ K] ≤ 2e−K 2 /(2t) Note that Zt−s /(t − s) = Q2 (r) − Q1 (r). By Azuma’s inequality and doing a union bound over all ds ≤ d` strings r of length s, for c ≥ 4 and t ≥ T /n2 = dc` /n2 ≥ dc`/2 , kQ1 − Q2 k1 ≤ √ 1/dc`/20 with failure probability at most 2d` e− t/2 ≤ 1/n2 . Similarly, for all strings of length s + 1, the estimated probability of the string has error at most 1/dc`/20 with failure probability 1/n2 . As the conditional distribution of xt+s given observations xt , . . . , xt+s−1 is the ratio of the joint distributions of {xt , . . . , xt+s−1 , xt+s } and {xt , . . . , xt+s−1 }, therefore as long as the empirical distributions of the length s and length s + 1 strings are estimated with error at most 1/dc`/20 and the string {xt , . . . , xt+s−1 } has probability at least 1/dc`/40 , the conditional distributions P1 and P2 satisfy kP1 − P2 k1 ≤ 1/n2 . By a union bound over all ds ≤ d` strings and for c ≥ 100, 18 the total probability mass on strings which occur with probability less than 1/dc`/40 is at most 1/dc`/50 ≤ 1/n2 for c ≥ 100. Therefore kP1 − P2 k1 ≤ 1/n2 with overall failure probability 3/n2 , hence the expected `1 distance between P1 and P2 is at most 1/n. By using the triangle inequality and the fact that the expected average error of Pt is at most 6 for ` = 3 log n/2 , it follows that the expected average error of P̂t is at most 6 + 1/n ≤ 7. Note that the expected average error of P̂t is the average of the expected errors of the empirical s-gram Markov models for 0 ≤ s ≤ ` − 1. Hence for ` = 3 log n/2 there must exist at least some s < ` such that the s-gram Markov model gets expected `1 error at most 7. A.1 Proof of Lemma 4 Let the prior for the distribution of the hidden states at time 0 be π0 . Let the true hidden state h0 at time 0 be 1 without loss of generality. We refer to the output at time t by xs . Let H0s (i) = P r[h0 = i\|xs0 ] be the posterior probability of the ith hidden state at time 0 after seeing the observations xs0 up to time t and having the prior π0 on the distribution of the hidden states at time 0. For convenience, denote us = H0s (1) and vs = 1 − us . Define Pis (j) = P r[xs = j\|xs−1 0 , h0 = i] as the distribution of the output at time t conditioned on the hidden state at time 0 being i and s observations xs−1 0 . Note that OP Ts = P1 . As before, define Rs as the conditional distribution of xs given observations xP 0 , · · · , xs−1 and initial distribution π but not being at hidden state h0 at time 0 i.e. Rs = (1/vs ) ni=2 H0s (i)Pis . Note that Ms is a convex combination of OP Ts and Rs , i.e. Ms = us OP Ts + vs Rs . Hence kOP Ts − Ms k1 ≤ kOP Ts − Rs k1 . Define δs = kOP Ts − Ms k1 . Our proof relies on a martingale concentration argument, and in order to ensure that our martingale has bounded differences we will ignore outputs which cause a significant drop in the posterior of the true hidden state at time 0. Let B be theP set of all outputs j at some time t P 4 j∈B Rs (j) Ts (j) 4 4 such that OP ≤ clog j∈B OP Ts (j) ≤ clog n n . Hence by a union Rs (j) ≤ clog n . Note that, 4 Ts (j) bound, with failure probability at most 2 any output j such that OP Rs (j) ≤ clog n is not emitted in a window of length clog n/2 . Hence we will only concern ourselves with sequences of outputs such Ts (j) 4 that the output j emitted at each step satisfies OP Rs (j) ≤ clog n , let the set of all such outputs be S1 , note that P r(xs0 ∈ / S1 ) ≤ 2 . Let ES1 [X] be the expectation of any random variable X conditioned on the output sequence being in the set S1 . Consider the sequence of random variables Xs = log us − log vs for t ∈ [−1, ` − 1] defining X−1 = log(π1 ) − log(1 − π1 ). Let ∆s+1 = Xs+1 − Xs be the change in Xs on seeing the output xs+1 at time s + 1. Let the output at time s + 1 be j. We will first find an expression for ∆s+1 . The posterior probabilities after seeing the (s + 1)th output get updated according to Bayes rule – P r[h0 = 1\|xs0 ]P r[x[s + 1] = j\|h0 = 1, xs0 ] P r[x[s + 1] = j\|xs0 ] us OP Ts+1 (j) = P r[x[s + 1] = j\|xs0 ] H0s+1 (1) = P r[h0 = 1\|xs0 , x[s + 1] = j] = =⇒ us+1 Let P r[x[s + 1] = j\|xs0 ] = dj . Note that H0s+1 (i) = H0s (i)Pis+1 (j)/dj if the output at time s + 1 is j. We can write – Rs+1 = n X H0s (i)Pis+1 /vs i=2 vs+1 = n X H0s+1 (i) = n X i=2 i=2 19 H0s (i)Pis+1 (j) /dj = vs Rs+1 (j)/dj Therefore we can write ∆s+1 and its expectation E[∆s+1 ] as – ∆s+1 = log =⇒ E[∆s+1 ] = X OP Ts+1 (j) Rs+1 (j) OP Ts+1 (j) log j OP Ts+1 (j) = D(OP Ts+1 k Rs+1 ) Rs+1 (j) ˜ s+1 as ∆ ˜ s+1 := min{∆s+1 , log log n} to keep martingale differences bounded. E[∆ ˜ s+1 ] We define ∆ then equals a truncated version of the KL-divergence which we define as follows. Definition 3. hFor any two n distributions oiµ(x) and ν(x), define the truncated KL-divergence as D̃C (µ k ν) = E log min µ(x)/ν(x), C for some fixed C. We are now ready to define our martingale. Consider the sequence of random variables X̃s := Pn 2 ˜ X̃s−1 + ∆s for t ∈ [0, ` − 1], with X̃−1 := X−1 . Define Z̃s := s=1 X̃s − X̃s−1 − δs /2 . Note that ˜ s =⇒ Xs ≥ X̃s . ∆s ≥ ∆ is with respect Lemma 5. ES1 [X̃s − X̃s−1 ] ≥ δs2 /2, where the expectation to the output at time t. Ps Hence the sequence of random variables Z̃s := i=0 X̃s − X̃s−1 − δs2 /2 is a submartingale with respect to the outputs. ˜ s and E[∆ ˜ s ] = D̃C (OP Ts k Rs ), C = log n. By taking an Proof. By definition X̃s − X̃s−1 = ∆ expectation with respect to only sequences S1 instead of all possible sequences, we are removing ˜ s ], hence events which have a negative contribution to E[∆ ˜ s ] ≥ E[∆ ˜ s ] = D̃C (OP Ts k Rs ) ES1 [∆ We can now apply Lemma 6. Lemma 6. (Modified Pinsker’s inequality) For any two distributions h µ(x)nand ν(x) oidefined on µ(x) x ∈ X, define the C-truncated KL divergence as D̃C (µ k ν) = Eµ log min ν(x) , C for some fixed C such that log C ≥ 8. Then D̃C (µ k ν) ≥ 12 kµ − νk21 . ˜ s ] ≥ 1 kOP Ts − Rs k2 . Hence ES [X̃s − X̃s−1 ] ≥ δ 2 /2. Hence ES1 [∆ s 1 1 2 We now claim that our submartingale has bounded differences. √ Lemma 7. \|Z̃s − Z̃s−1 \| ≤ 2 log(cγ 4 log n). 2 )/2 can be at most 2. Z − Z ˜ ˜ Proof. Note that (δs2 − δs−1 s s−1 = ∆s . By definition ∆s ≤ log(log n). ˜ s ≥ − log(clog n/4 ) as we restrict ourselves to sequences in the set S1 . Hence \|Z̃s − Z̃s−1 \| ≤ Also, ∆ √ log(clog n/4 ) + 2 ≤ 2 log(clog n/4 ). We now apply Azuma-Hoeffding – Lemma 8. (Azuma-Hoeffding 2 inequality) Let Zi be a submartingale with \|Zi − Zi−1 \| ≤ C. Then −λ P r[Zs − Z0 ≤ −λ] ≤ exp 2tC 2 20 Applying Lemma 8 we can show, P r[Z̃`−1 − Z̃0 ≤ −log n] ≤ exp −log n ≤ 2 4c(1/)2 log2 (clog n/4 ) (A.1) We now bound the average error in the window 0 to ` − 1. With failure probability at most over the randomness in the outputs, Z̃`−1 − Z̃0 ≥ −log n by Eq. A.1. Let S2 be the set of all sequences in S1 which satisfy Z̃`−1 − Z̃0 ≥ −log n. Note that X0 = X̃0 ≥ log(1/π1 ). Consider the last point after which vs decreases below 2 and remains below that for every subsequent step in the window. Let this point be τ , if there is no such point define τ to be ` − 1. The total contribution of the error at every step after the τ th step to the average error is a 2 term as the error after this step is 2 . Note that Xτ ≤ log(1/)2 =⇒ X̃τ ≤ log(1/)2 as X̃s ≤ Xs . Hence for all sequences in S2 – 2 X̃τ ≤ log(1/)2 =⇒ X̃τ − X̃−1 ≤ log(1/)2 + log(1/π1 ) τ X =⇒ 0.5 δs2 ≤ log(1/)2 + log(1/π1 ) + log n =⇒ s=0 P`−1 2 s=0 δs c log n/2 (By Eq. A.1) ≤ 62 (as log(1/π1 ) ≤ c log n) ≤ 3 (By Jensen’s inequality) P`−1 =⇒ s=0 δs c log n/2 P As the total probability of sequences outside S2 is at most 22 , E[ `−1 s=0 δs ] ≤ 4, whenever the hidden state i at time 0 has probability at least 1/nc in the prior distribution π0 . A.2 Proof of modified Pinsker’s inequality (Lemma 6) Lemma 6. (Modified Pinsker’s inequality) For any two distributions h µ(x)nand ν(x) oidefined on µ(x) x ∈ X, define the C-truncated KL divergence as D̃C (µ k ν) = Eµ log min ν(x) , C for some fixed C such that log C ≥ 8. Then D̃C (µ k ν) ≥ 12 kµ − νk21 . Proof. We rely on the following Lemma which bounds the KL-divergence for binary distributionsLemma 9. For every 0 ≤ q ≤ p ≤ 1, we have 2 1. p log pq + (1 − p) log 1−p 1−q ≥ 2(p − q) 2 2. 3p + (1 − p) log 1−p 1−q ≥ 2(p − q) Proof. For the second result, first observe that log(1/(1 − q)) ≥ 0 and (p − q) ≤ p as q ≤ p. Both the results then follow from standard calculus. Let A := {x ∈ X : µ(x) ≥ ν(x)} and B := {x ∈ X : µ(x) ≥ Cν(x)}. Let µ(A) = p, µ(B) = δ, ν(A) = q and ν(B) = . Note that kµ − νk1 = 2(µ(A) − ν(A)). By the log-sum inequality– D̃C (µ k ν) = X x∈B µ(x) log X X µ(x) µ(x) µ(x) + µ(x) log + µ(x) log ν(x) ν(x) ν(x) x∈A−B 21 x∈X−A = δ log C + (p − δ) log 1. Case 1 : δ p p−δ 1−p + (1 − p) log q− 1−q ≥ 0.5 p 1−p log C + (1 − p) log 2 1−q 1 ≥ 2(p − q)2 = kµ − νk21 2 D̃C (µ k ν) ≥ 2. Case 2 : δ p < 0.5 p δ 1−p + (p − δ) log 1 − + (1 − p) log q− p 1−q p 2δ 1−p ≥ δ log C + (p − δ) log − (p − δ) + (1 − p) log q p 1−q p 1−p ≥ δ(log C − 2) + (p − δ) log + (1 − p) log q 1−q D̃C (µ k ν) = δ log C + (p − δ) log (a) Sub-case 1 : log pq ≥ 6 p 1−p + (1 − p) log q 1−q 1−p ≥ 3p + (1 − p) log 1−q 1 ≥ 2(p − q)2 = kµ − νk21 2 D̃C (µ k ν) ≥ (p − δ) log (b) Sub-case 2 : log pq < 6 p p 1−p D̃C (µ k ν) ≥ δ(log C − 2 − log ) + p log + (1 − p) log q q 1−q 1 ≥ 2(p − q)2 = kµ − νk21 2 B B.1 Proof of Lower Bound for Large Alphabets CSP formulation We first go over some notation that we’ll use for CSP problems, we follow the same notation and setup as in Feldman et al. [16]. Consider the following model for generating a random CSP instance on n variables with a satisfying assignment σ. The k-CSP is defined by the predicate P : {0, 1}k → {0, 1}. We represent a k-clause by an ordered k-tuple of literals from {x1 , · · · , xn , x̄1 , · · · , x̄n } with no repetition of variables and let Xk be the set of all such k-clauses. For a k-clause C = (l1 , · · · , lk ) let σ(C) ∈ {0, 1}k be the k-bit string of values assigned by σ to literals in C, that is {σ(l1 ), · · · , σ(lk )} where σ(li ) is the value of the literal li in assignment σ. In the planted model we draw clauses with probabilities that depend on the value of σ(C). Let Q : {0, 1}k → 22 P R+ , t∈{0,1}k Q(t) = 1 be some distribution over satisfying assignments to P . The distribution Qσ is then defined as followsQ(σ(C)) Qσ (C) = P (B.1) 0 C 0 ∈Xk Q(σ(C )) Recall that for any distribution Q over satisfying assignments we define its complexity r as the largest r such that the distribution Q is (r − 1)-wise uniform (also referred to as (r − 1)-wise independent in the literature) but not r-wise uniform. Consider the CSP C defined by a collection of predicates P (y) for each y ∈ {0, 1}m for some m ≤ k/2. Let A ∈ {0, 1}m×k be a matrix with full row rank over the binary field. We will later choose A to ensure the CSP has high complexity. For each y, the predicate P (y) is the set of solutions to the system y = Av mod 2 where v = σ(C). For all y we define Qy to be the uniform distribution over all consistent assignments, i.e. all v ∈ {0, 1}k satisfying y = Av mod 2. The planted distribution Qσ,y is defined based on Qy according to Eq. B.1. Each clause in C is chosen by first picking a y uniformly at random and then a clause from the distribution Qσ,y . For any planted σ we define Qσ to be the distribution over all consistent clauses along with their labels y. Let Uk be the uniform distribution over k-clauses, with each clause assigned a uniformly chosen label y. Define Qησ = (1 − η)Qσ + ηUk , for some fixed noise level η > 0. We consider η to be a small constant less than 0.05. This corresponds to adding noise to the problem by mixing the planted and the uniform clauses. The problem gets harder as η becomes larger, for η = 0 it can be efficiently solved using Gaussian Elimination. We will define another CSP C0 which we show reduces to C and for which we can obtain hardness using Conjecture 1. The label y is fixed to be the all zero vector in C0 . Hence Q0 , the distribution over satisfying assignments for C0 , is the uniform distribution over all vectors in the null space of A over the binary field. We refer to the planted distribution in this case as Qσ,0 . Let Uk,0 be the uniform distribution over k-clauses, with each clause now having the label 0. For any planted assignment σ, we denote the distribution of consistent clauses of C0 by Qσ,0 . As before define Qησ,0 = (1 − η)Qσ,0 + ηUk,0 for the same η. Let L be the problem of distinguishing between Uk and Qησ for some randomly and uniformly chosen σ ∈ {0, 1}n with success probability at least 2/3. Similarly, let L0 be the problem of distinguishing between Uk,0 and Qησ,0 for some randomly and uniformly chosen σ ∈ {0, 1}n with success probability at least 2/3. L and L0 can be thought of as the problem of distinguishing random instances of the CSPs from instances with a high value. Note that L and L0 are at least as hard as the problem of refuting the random CSP instances Uk and Uk,0 , as this corresponds to the case where η = 0. We claim that an algorithm for L implies an algorithm for L0 . Lemma 10. If L can be solved in time t(n) with s(n) clauses, then L0 can be solved in time O(t(n) + s(n)) and s(n) clauses. Let the complexity of Q0 be γk, with γ ≥ 1/10 (we demonstrate how to achieve this next). By Conjecture 1 distinguishing between Uk,0 and Qησ,0 requires at least Ω̃(nγk/2 ) clauses. We now discuss how A can be chosen to ensure that the complexity of Q0 is γk. B.2 Ensuring high complexity of the CSP Let N be the null space of A. Note that the rank of N is (k − m). For any subspace D, let w(D) = (w1 , w2 , · · · , wk ) be a randomly chosen vector from D. To ensure that Q0 has complexity 23 γk, it suffices to show that the random variables w(N ) = (w1 , w2 , · · · , wk ) are (γk − 1)-wise uniform. We use the theory of error correcting codes to find such a matrix A. A binary linear code B of length k and rank m is a linear subspace of Fk2 (our notation is different from the standard notation in the coding theory literature to suit our setting). The rate of the code is defined to be m/k. The generator matrix of the code is the matrix G such that B = {Gv, v ∈ {0, 1}m }. The parity check matrix of the code is the matrix H such that B = {c ∈ {0, 1}k : Hc = 0}. The distance d of a code is the weight of the minimum weight codeword and the relative distance δ is defined to be δ = d/k. For any codeword B we define its dual codeword B T as the codeword with generator matrix HT and parity check matrix GT . Note that the rank of the dual codeword of a code with rank m is (k − m). We use the following standard result about linear codes– Fact 1. If B T has distance l, then w(B) is (l − 1)-wise uniform. Hence, our job of finding A reduces to finding a dual code with distance γk and rank m, where γ = 1/10 and m ≤ k/2. We use the Gilbert-Varshamov bound to argue for the existence of such a code. Let H(p) be the binary entropy of p. Lemma 11. (Gilbert-Varshamov bound) For every 0 ≤ δ < 1/2, and 0 < ≤ 1 − H(δ), there exists a code with rank m and relative distance δ if m/k = 1 − H(δ) − . Taking δ = 1/10, H(δ) ≤ 0.5, hence there exists a code B whenever m/k ≤ 0.5, which is the setting we’re interested in. We choose A = GT , where G is the generator matrix of B. Hence the null space of A is (k/10 − 1)-wise uniform, hence the complexity of Q0 is γk with γ ≥ 1/10. Hence for all k and m ≤ k/2 we can find a A ∈ {0, 1}m×k to ensure that the complexity of Q0 is γk. B.3 Sequential model of CSP and sample complexity lower bound We now construct a sequential model which derives hardness from the hardness of L. Here we slightly differ from the outline presented in the beginning of Section 5 as we cannot base our sequential model directly on L as generating random k-tuples without repetition increases the mutual information, so we formulate a slight variation L0 of L which we show is at least as hard as L. We did not define our CSP instance allowing repetition as that is different from the setting examined in Feldman et al. [16], and hardness of the setting with repetition does not follow from hardness of the setting allowing repetition, though the converse is true. B.3.1 Constructing sequential model Consider the following family of sequential models R(n, Am×k ) where A ∈ {0, 1}m×k is chosen as defined previously. The output alphabet of all models in the family is X = {ai , 1 ≤ i ≤ 2n} of size 2n, with 2n/k even. We choose a subset S of X of size n, each choice of S corresponds to a model M in the family. Each letter in the output alphabet is encoded as a 1 or 0 which represents whether or not the letter is included in the set S, let u ∈ {0, 1}2n be the vector which stores this encoding so ui = 1 whenever the letter ai is in S. Let σ ∈ {0, 1}n determine the subset S such that entry u2i−1 is 1 and u2i is 0 when σ i is 1 and u2i−1 is 0 and u2i is 1 when σ i is 0, for all i. We choose σ uniformly at random from {0, 1}n and each choice of σ represents some subset S, and hence some model M. We partition the output alphabet X into k subsets of size 2n/k each so the first 2n/k letters go to the first subset, the next 2n/k go to the next subset and so on. Let the ith 24 subset be Xi . Let Si be the set of elements in Xi which belong to the set S. At time 0, M chooses v ∈ {0, 1}k uniformly at random from {0, 1}k . At time i, i ∈ {0, · · · , k−1}, if vi = 1, then the model chooses a letter uniformly at random from the set Si , otherwise if vi = 0 it chooses a letter uniformly at random from Xi − Si . With probability (1 − η) the outputs for the next m time steps from k to (k + m − 1) are y = Av mod 2, with probability η they are m uniform random bits. The model resets at time (k + m − 1) and repeats the process. Recall that I(M) is at most m and M can be simulated by an HMM with 2m (2k + m) + m hidden states (see Section 5.1). B.3.2 Reducing sequential model to CSP instance We reveal the matrix A to the algorithm (this corresponds to revealing the transition matrix of the underlying HMM), but the encoding σ is kept secret. The task of finding the encoding σ given samples from M can be naturally seen as a CSP. Each sample is a clause with the literal corresponding to the output letter ai being x(i+1)/2 whenever i is odd and x̄i/2 when i is even. We refer the reader to the outline at the beginning of the section for an example. We denote C 0 as the CSP C with the modification that the ith literal of each clause is the literal corresponding to a letter in Xi for all 1 ≤ i ≤ k. Define Q0σ as the distribution of consistent clauses for the CSP C 0 . Define Uk0 as the uniform distribution over k-clauses with the additional constraint that the ith literal of each clause is the literal corresponding to a letter in Xi for all 1 ≤ i ≤ k. Define 0 0 Qση = (1 − η)Q0σ + ηUk0 . Note that samples from the model M are equivalent to clauses from Qση . We show that hardness of L0 follows from hardness of L– Lemma 12. If L0 can be solved in time t(n) with s(n) clauses, then L can be solved in time t(n) with O(s(n)) clauses. Hence if Conjecture 1 is true then L0 cannot be solved in polynomial time with less than Ω̃(nγk/2 ) clauses. We can now prove the Theorem 2 using Lemma 12. Theorem 2. Assuming Conjecture 1, for all sufficiently large T and 1/T c < ≤ 0.1 for some fixed constant c, there exists a family of HMMs with T hidden states and an output alphabet of size n such that, any polynomial time prediction algorithm that achieves average KL-error, `1 error or relative zero-one error less than with probability greater than 2/3 for a randomly chosen HMM in the family needs requires nΘ(log T /) samples from the HMM over any window length which the algorithm uses for prediction. Proof. We describe how to choose the family of sequential models R(n, Am×k ) for each value of and T . Recall that the HMM has T = 2m (2k + m) + m hidden states. Let T 0 = 2m+2 (k + m). Note that T 0 ≥ T . Let t = log T 0 . We choose m = t − log(1/) − log(t/5), and k to be the solution of m t = m + log(k + m) + 2, hence k = t/(5) − m − 2. Note that for ≤ 0.1, k ≥ m. Let 0 = 29 k+m . 0 We claim ≤ . To verify, note that k + m = t/(5) − 2. Therefore, 0 = 2m 10(t − log(1/) − log(t/5)) = ≥ , 9(k + m) 9t(1 − 10/t) for sufficiently large t and ≥ 2−ct for a fixed constant c. Hence proving hardness for obtaining error 0 implies hardness for obtaining error . We choose the matrix Am×k as outlined earlier. For 25 each vector σ ∈ {0, 1}n we define the family of sequential models R(n, A) as earlier. Let M be a randomly chosen model in the family. We first show the result for the relative zero-one loss. The idea is that any algorithm which does a good job of predicting the outputs from time k through (k + m − 1) can be used to distinguish between instances of the CSP with a high value and uniformly random clauses. This is because it is not possible to make good predictions on uniformly random clauses. We relate the zero-one error from time k through (k + m − 1) with the relative zero-one error from time k through (k + m − 1) and the average zero-one error for all time steps to get the required lower bounds. Let ρ01 (A) be the average zero-one loss of some polynomial time algorithm A for the output 0 (A) be the average relative zero-one loss of A for the outtime steps k through (k + m − 1) and δ01 put time steps k through (k + m − 1) with respect to the optimal predictions. For the distribution Uk0 it is not possible to get ρ01 (A) < 0.5 as the clauses and the label y are independent and y is 0 chosen uniformly at random from {0, 1}m . For Qση it is information theoretically possible to get ρ01 (A) = η/2. Hence any algorithm which gets error ρ01 (A) ≤ 2/5 can be used to distinguish be0 tween Uk0 and Qση . Therefore by Lemma 12 any polynomial time algorithm which gets ρ01 (A) ≤ 2/5 with probability greater than 2/3 over the choice of M needs at least Ω̃(nγk/2 ) samples. Note that 0 (A) = ρ (A) − η/2. As the optimal predictor P δ01 01 ∞ gets ρ01 (P∞ ) = η/2 < 0.05, therefore 0 (A) ≤ 1/3 =⇒ ρ (A) ≤ 2/5. Note that δ (A) ≥ δ 0 (A) m . This is because δ (A) is the δ01 01 01 01 01 k+m average error for all (k +m) time steps, and the contribution to the error from time steps 0 to (k −1) m 0 (A) < 1 =⇒ ρ (A) ≤ 2/5. is non-negative. Also, 13 k+m > 0 , therefore, δ01 (A) < 0 =⇒ δ01 01 3 Hence any polynomial time algorithm which gets average relative zero-one loss less than 0 with probability greater than 2/3 needs at least Ω̃(nγk/2 ) samples. The result for `1 loss follows directly from the result for relative zero-one loss, we next consider the KL loss. 0 (A) be the average KL error of the algorithm A from time steps k through (k + m − 1). Let δKL 0 (A) ≤ 2/9 =⇒ δ 0 (A) ≤ 1/3. By application of Jensen’s inequality and Pinsker’s inequality, δKL 01 0 (A) < 2/9 needs Ω̃(nγk/2 ) samTherefore, by our previous argument any algorithm which gets δKL 0 (A) ≤ 2/9. Hence any polynomial time algorithm which ples. But as before, δKL (A) ≤ 0 =⇒ δKL succeeds with probability greater than 2/3 and gets average KL loss less than 0 needs at least Ω̃(nγk/2 ) samples. We lower bound k by a linear function of log T / to express the result directly in terms of log T /. We claim that log T / is at most 10k. This follows because– log T / ≤ t/ = 5(k + m) + 10 ≤ 15k Hence any polynomial time algorithm needs nΘ(log T /) samples to get average relative zero-one loss, `1 loss, or KL loss less than on M. B.4 Proof of Lemma 10 Lemma 10. If L can be solved in time t(n) with s(n) clauses, then L0 can be solved in time O(t(n) + s(n)) and s(n) clauses. Proof. We show that a random instance of C0 can be transformed to a random instance of C in time s(n)O(k) by independently transforming every clause C in C0 to a clause C 0 in C such that 26 C is satisfied in the original CSP C0 with some assignment t to x if and only if the corresponding clause C 0 in C is satisfied with the same assignment t to x. For every y ∈ {0, 1}m we pre-compute and store a random solution of the system y = Av mod 2, let the solution be v(y). Given any clause C = (x1 , x2 , · · · , xk ) in C0 , choose y ∈ {0, 1}m uniformly at random. We generate a clause C 0 = (x01 , x02 , · · · , x0k ) in C from the clause C in C0 by choosing the literal x0i = x̄i if vi (y) = 1 and x0i = xi if vi (y) = 0. By the linearity of the system, the clause C 0 is a consistent clause of C with some assignment x = t if and only if the clause C was a consistent clause of C0 with the same assignment x = t. We next claim that C 0 is a randomly generated clause from the distribution Uk if C was drawn from Uk,0 and is a randomly generated clause from the distribution Qσ if C was drawn from Qσ,0 . By our construction, the label of the clause y is chosen uniformly at random. Note that choosing a clause uniformly at random from Uk,0 is equivalent to first uniformly choosing a k-tuple of unnegated literals and then choosing a negation pattern for the literals uniformly at random. It is clear that a clause is still uniformly random after adding another negation pattern if it was uniformly random before. Hence, if the original clause C was drawn to the uniform distribution Uk,0 , then C 0 is distributed according to Uk . Similarly, choosing a clause uniformly at random from Qσ,y for some y is equivalent to first uniformly choosing a k-tuple of unnegated literals and then choosing a negation pattern uniformly at random which makes the clause consistent. As the original negation pattern corresponds to a v randomly chosen from the null space of A, the final negation pattern on adding v(y) corresponds to the negation pattern for a uniformly random chosen solution of y = Av mod 2 for the chosen y. Therefore, the clause C 0 is a uniformly random chosen clause from Qσ,y if C is a uniformly random chosen clause from Qσ,0 . Hence if it is possible to distinguish Uk and Qησ for some randomly chosen σ ∈ {0, 1}n with success probability at least 2/3 in time t(n) with s(n) clauses, then it is possible to distinguish between Uk,0 and Qησ,0 for some randomly chosen σ ∈ {0, 1}n with success probability at least 2/3 in time t(n) + s(n)O(k) with s(n) clauses. B.5 Proof of Lemma 12 Lemma 12. If L0 can be solved in time t(n) with s(n) clauses, then L can be solved in time t(n) with O(s(n)) clauses. Hence if Conjecture 1 is true then L0 cannot be solved in polynomial time with less than Ω̃(nγk/2 ) clauses. Proof. Define E to be the event that a clause generated from the distribution Qσ of the CSP C has the property that for all i the ith literal belongs to the set Xi , we also refer to this property of the clause as E for notational ease. It’s easy to verify that the probability of the event E is 1/k k . We claim that conditioned on the event E, the CSP C and C 0 are equivalent. This is verified as follows. Note that for all y, Qσ,y and Q0σ,y are uniform on all consistent clauses. Let U be the set of all clauses with non-zero probability under Qσ,y and U 0 be the set of all clauses with non-zero probability under Q0σ,y . Furthermore, for any v which satisfies the constraint that y = Av mod 2, let U(v) be the set of clauses C ∈ U such that σ(C) = v. Similarly, let U 0 (v) be the set of clauses C ∈ U 0 such that σ(C) = v. Note that the subset of clauses in U(v) which satisfy E is the same as the set U 0 (v). As this holds for every consistent v and the distributions Q0σ,y and Qσ,y are uniform on all consistent clauses, the distribution of clauses from Qσ is identical to the distribution of clauses Q0σ conditioned on the event E. The equivalence of Uk and Uk0 27 conditioned on E also follows from the same argument. Note that as the k-tuples in C are chosen uniformly at random from satisfying k-tuples, with high probability there are s(n) tuples having property E if there are O(k k s(n)) clauses in C. As the problems L and L0 are equivalent conditioned on event E, if L0 can be solved in time t(n) with s(n) clauses, then L can be solved in time t(n) with O(k k s(n)) clauses. From Lemma 10 and Conjecture 1, L cannot be solved in polynomial time with less than Ω̃(nγk/2 ) clauses. Hence L0 cannot be solved in polynomial time with less than Ω̃(nγk/2 /k k ) clauses. As k is a constant with respect to n, L0 cannot be solved in polynomial time with less than Ω̃(nγk/2 ) clauses. C C.1 Proof of Lower Bound for Small Alphabets Proof of Lemma 2 Lemma 2. Let A be chosen uniformly at random from the set S. Then, with probability at least (1 − 1/n) over the choice A ∈ S, any (randomized) algorithm that can distinguish the outputs from the model M(A) from the distribution over random examples Un with success probability greater than 2/3 over the randomness of the examples and the algorithm needs f (n) time or examples. Proof. Suppose A ∈ {0, 1}m×n is chosen at random with each entry being i.i.d. with its distribution uniform on {0, 1}. Let A0 be the sub-matrix of A corresponding to the first 2n/3 columns and all the m rows. Recall that S is the set of all (m × n) matrices A such that the sub-matrix A0 is full row-rank. We claim that P (A ∈ S) ≥ 1 − m2−n/6 . To verify, consider the addition of each row one by one to A0 . The probability of the ith row being linearly dependent on the previous (i − 1) rows is 2i−1−2n/3 . Hence by a union bound, A0 is full row-rank with failure probability at most m2m−2n/3 ≤ m2−n/6 . From Definition 2 and a union bound over all the m ≤ n/2 parities, any algorithm that can distinguish the outputs from the model M(A) for uniformly chosen A from the distribution over random examples Un with probability at least (1 − 1/(2n)) over the choice of A needs f (n) time or examples. As P (A ∈ S) ≥ 1 − m2−n/6 for a uniformly randomly chosen A, with probability at least (1 − 1/(2n) − m2−n/6 ) ≥ (1 − 1/n) over the choice A ∈ S any algorithm that can distinguish the outputs from the model M(A) from the distribution over random examples Un with success probability greater than 2/3 over the randomness of the examples and the algorithm needs f (n) time or examples. C.2 Proof of Proposition 2 Proposition 2. With f (T ) as defined in Definition 2, For all sufficiently large T and 1/T c < ≤ 0.1 for some fixed constant c, there exists a family of HMMs with T hidden states such that any algorithm that achieves average relative zero-one loss, average `1 loss, or average KL loss less than with probability greater than 2/3 for a randomly chosen HMM in the family needs, requires f (log T /) time or samples samples from the HMM over any window length which the algorithm uses for prediction. Proof. We describe how to choose the family of sequential models Am×n for each value of and T . Recall that the HMM has T = 2m (2n + m) + m hidden states. Let T 0 = 2m+2 (n + m). Note that T 0 ≥ T . Let t = log T 0 . We choose m = t − log(1/) − log(t/5), and n to be the solution of m . t = m + log(n + m) + 2, hence n = t/(5) − m − 2. Note that for ≤ 0.1, n ≥ m. Let 0 = 29 n+m 28 We claim ≤ 0 . To verify, note that n + m = t/(5) − 2. Therefore, 0 = 2m 10(t − log(1/) − log(t/5)) = ≥ , 9(n + m) 9t(1 − 10/t) for sufficiently large t and ≥ 2−ct for a fixed constant c. Hence proving hardness for obtaining error 0 implies hardness for obtaining error . We choose the matrix Am×n as outlined earlier. The family is defined by the model M(Am×n ) defined previously with the matrix Am×n chosen uniformly at random from the set S. Let ρ01 (A) be the average zero-one loss of some algorithm A for the output time steps n through 0 (A) be the average relative zero-one loss of A for the output time steps n through (n+m−1) and δ01 (n + m − 1) with respect to the optimal predictions. For the distribution Un it is not possible to get ρ01 (A) < 0.5 as the clauses and the label y are independent and y is chosen uniformly at random from {0, 1}m . For Qηs it is information theoretically possible to get ρ01 (A) = η/2. Hence any algorithm which gets error ρ01 (A) ≤ 2/5 can be used to distinguish between Un and Qηs . Therefore by Lemma 2 any algorithm which gets ρ01 (A) ≤ 2/5 with probability greater than 2/3 over the choice 0 (A) = ρ (A) − η/2. As the optimal of M(A) needs at least f (n) time or samples. Note that δ01 01 0 (A) ≤ 1/3 =⇒ ρ (A) ≤ 2/5. Note that predictor P∞ gets ρ01 (P∞ ) = η/2 < 0.05, therefore δ01 01 0 (A) m . This is because δ (A) is the average error for all (n + m) time steps, and the δ01 (A) ≥ δ01 01 n+m m contribution to the error from time steps 0 to (n − 1) is non-negative. Also, 31 n+m > 0 , therefore, 1 0 0 δ01 (A) < =⇒ δ01 (A) < 3 =⇒ ρ01 (A) ≤ 2/5. Hence any algorithm which gets average relative zero-one loss less than 0 with probability greater than 2/3 over the choice of M(A) needs f (n) time or samples. The result for `1 loss follows directly from the result for relative zero-one loss, we next consider the KL loss. 0 (A) be the average KL error of the algorithm A from time steps n through (n + m − 1). Let δKL 0 (A) ≤ 2/9 =⇒ δ 0 (A) ≤ 1/3. By application of Jensen’s inequality and Pinsker’s inequality, δKL 01 0 (A) < 2/9 needs f (n) samples. Therefore, by our previous argument any algorithm which gets δKL 0 (A) ≤ 2/9. Hence any algorithm which gets average KL loss But as before, δKL (A) ≤ 0 =⇒ δKL 0 less than needs f (n) time or samples. We lower bound n by a linear function of log T / to express the result directly in terms of log T /. We claim that log T / is at most 10n. This follows because– log T / ≤ t/ = 5(n + m) + 10 ≤ 15n Hence any algorithm needs f (log T /)) samples and time to get average relative zero-one loss, `1 loss, or KL loss less than with probability greater than 2/3 over the choice of M(A). D Proof of Information Theoretic Lower Bound Proposition 3. There is an absolute constant c such that for all 0 < < 0.5 and sufficiently large n, there exits an HMM with n states such that it is not information theoretically possible to get average relative zero-one loss or `1 loss less than using windows of length smaller than c log n/2 , and KL loss less than using windows of length smaller than c log n/. 29 Proof. Consider a Hidden Markov Model with the Markov chain being a permutation on n states. The output alphabet of each hidden state is binary. Each state i is marked with a label li which is 0 or 1, let G(i) be mapping from hidden state hi to its label li . All the states labeled 1 emit 1 with probability (0.5 + ) and 0 with probability (0.5 − ). Similarly, all the states labeled 0 emit 0 with probability (0.5 + ) and 1 with probability (0.5 − ). Fig. 3 illustrates the construction and provides the high-level proof idea. Figure 3: Lower bound construction, ` = 3, n = 16. A note on notation used in the rest of the proof with respect to this example: r(0) corresponds to the label of h0 , h1 and h2 and is (0, 1, 0) in this case. Similarly, r(1) = (1, 1, 0) in this case. The segments between the shaded nodes comprise the set S1 and are the possible sequences of states from which the last ` = 3 outputs could have come. The shaded nodes correspond to the states in S2 , and are the possible predictions for the next time step. In this example S1 = {(0, 1, 0), (1, 1, 0), (0, 1, 0), (1, 1, 1)} and S2 = {1, 1, 0, 0}. Assume n is a multiple of (` + 1), where (` + 1) = c log n/2 , for a constant c = 1/33. We will regard as a constant with respect to n. Let n/(` + 1) = t. We refer to the hidden states by hi , 0 ≤ i ≤ (n − 1), hji refers to the sequence of hidden states i through j. We will show that a model looking at only the past ` outputs cannot get average zero-one loss less than 0.5 − o(1). As the optimal prediction looking at all past outputs gets average zero-one loss 0.5 − + o(1) (as the hidden state at each time step can be determined to an arbitrarily high probability if we are allowed to look at an arbitrarily long past), this proves that windows of length ` do not suffice to get average zero-one error less than − o(1) with respect to the optimal predictions. Note that the Bayes optimal prediction at time (` + 1) to minimize the expected zero-one loss given outputs from ` = s` ) where s` is the sequence of time 1 to ` is to predict the mode of the distribution P r(x`+1 \|xP 1 1 1 outputs from time 1 to `. Also, note that P r(x`+1 \|x`1 = s`1 ) = i P r(hi` =i \|x`1 = s`1 )P r(x`+1 \|hi` =i ) where hi` is the hidden state at time `. Hence the predictor is a weighted average of the prediction of each hidden state with the weight being the probability of being at that hidden state. We index each state hi of the permutation by a tuple (f (i), g(i)) = (j, k) where j = i mod (` + 1) i and k = b `+1 c hence 0 ≤ j ≤ `, 0 ≤ k ≤ (t − 1) and i = k(` + 1) + j. We help the predictor to make the prediction at time (` + 1) by providing it with the index f (i` ) = i` mod (` + 1) of the true hidden state hi` at time `. Hence this narrows down the set of possible hidden states at time ` (in Fig. 3, the set of possible states given this side information are all the hidden states before the shaded states). The Bayes optimal prediction at time (` + 1) given outputs s`1 from time 1 to ` and index f (hi` ) = j is to predict the mode of P r(x`+1 \|x`1 = s`1 , f (hi` ) = j). Note that by the definition of Bayes optimality, the average zero-one loss of the prediction using P r(x`+1 \|x`1 = s`1 , f (hi` ) = j) cannot be worse than the average zero-one loss of the prediction using P r(x`+1 \|x`1 = s`1 ). Hence 30 we only need to show that the predictor with access to this side information is poor. We refer to this predictor using P r(x`+1 \|x`1 = s`1 , f (hi` ) = j) as P. We will now show that there exists some permutation for which the average zero-one loss of the predictor P is 0.5 − o(1). We argue this using the probabilistic method. We choose a permutation uniformly at random from the set of all permutations. We show that the expected average zero-one loss of the predictor P over the randomness in choosing the permutation is 0.5 − o(1). This means that there must exist some permutation such that the average zero-one loss of the predictor P on that permutation is 0.5−o(1). To find the expected average zero-one loss of the predictor P over the randomness in choosing the permutation, we will find the expected average zero-one loss of the predictor P given that we are in some state hi` at time `. Without loss of generality let f (i` ) = 0 and g(i` ) = (` − 1), hence we were at the (` − 1)th hidden state at time `. Fix any sequence of labels for the hidden states `−1 h`−1 emitted by the hidden states h`−1 from time 0 to ` − 1, let E[δ(s0`−1 )] 0 . For any string s0 0 be the expected average zero-one error P over the randomness in the rest of the P of the predictor `−1 permutation. Also, let E[δ(h`−1 )] = s`−1 E[δ(s0 )]P r[s`−1 0 ] be the expected error averaged across 0 all outputs. We will argue that E[δ(h`−1 )] = 0.5 − o(1). The set of hidden states hi with g(i) = k k(`+1)−2 defines a segment of the permutation, let r(k) be the label G(h(k−1)(`+1) ) of the segment k, excluding its last bit which corresponds to the predictions. Let S1 = {r(k), ∀ k 6= 0} be the set of all the labels excluding the first label r(0) and S2 = {G(hk(`+1)+` ), ∀ k} be the set of all the predicted bits (refer to Fig. 3 for an example). Consider any assignment of r(0). To begin, we show that with `−1 `−1 of high probability over the output s`−1 0 , the Hamming distance D(s0 , r(0)) of the output s0 ` from r(0) is at least the set of hidden states h`−1 − 2`. This follows directly from Hoeffding’s 0 2 inequality as all the outputs are independent conditioned on the hidden state4 – 2 −2` P r[D(s`−1 ≤ n−2c 0 , r(0)) ≤ `/2 − 2`] ≤ e (D.1) We now show that for any k 6= 0, with decent probability the label r(k) of the segment k is closer than r(0). Then we argue that with high probability in Hamming distance to the output s`−1 0 in Hamming distance than r(0). Hence there are many such segments which are closer to s`−1 0 these other segments are assigned as much weight in predicting the next output as r(0), which means that the output cannot be predicted with a high accuracy as the output bits corresponding to different segments are independent. We first find the probabilityp that the segment corresponding to some k with label r(k) has a Hamming distance less than 2` − ` log t/8 from any fixed binary string x of length `. Let F (l, m, p) be the probability of getting at least l heads in m i.i.d. trails with each trial having probability p of giving a head. F (l, m, p) can be bounded below by the following standard inequality– l 1 F (l, m, p) ≥ √ exp − mDKL p m 2m h 1−q where DKL (q k p) = q log pq + (1 − q) log 1−p . We can use this to lower bound P r D(r(k), x) ≤ i p `/2 − ` log t/8 , h i p p P r D(r(k), x) ≤ `/2 − ` log t/8 = F (`/2 + ` log t/8, `, 1/2) 4 For n independent random variables {Xi } lying in the interval [0, 1] with X̄ = −2nt2 e . In our case t = and n = `. 31 1 n P i Xi , P r[X ≤ E[X̄] − t] ≤ 1 1 ≥ √ exp − `DKL + 2 2` r log t 1 8` 2 Note that DKL ( 12 + v k 12 ) ≤ 4v 2 by using the inequality log(1 + v) ≤ v. We can simplify the KL-divergence using this and write– h i p √ P r D(r(k), x) ≤ `/2 − ` log t/8 ≥ 1/ 2`t (D.2) p Let D be the set of all k 6= 0 such that D(r(k), x) ≤ 2` − ` log t/8 for some fixed x. We argue that with high probability over the randomness of the permutation \|D\| is large. This follows from Eq. D.2 and the Chernoff bound as the labels for all segments r(k) are chosen independently5 – h i √ p 1 P r \|D\| ≤ t/(8`) ≤ e− 8 t/(2`) q p 1 0.25 Note that t/(8`) ≥ n . Therefore for any fixed x, with probability 1−exp(− 8 2`t ) ≥ 1−n−0.25 q t 0.25 segments in a randomly chosen permutation which have Hamming disthere are 8` ≥ n p p tance less than `/2 − ` log t/8 from x. Note that by our construction 2` ≤ ` log t/8 because log(` + 1) ≤ (1 − 32c) log n. Hence the segments in D are closer in Hamming distance to the output if D(s`−1 s`−1 0 , r(0)) > `/2 − 2`. 0 Therefore if D(s`−1 0 , r(0)) > `/2 − 2`, then with high probability over randomly choosing the segments S1 there is a subset D of segments in S1 with \|D\| ≥ n0.25 such that all of the `−1 `−1 such that segments in D have Hamming distance less than D(s`−1 0 , r(0)) from s0 . Pick any s0 `−1 D(s0 , r(0)) > `/2 − 2`. Consider any set of segments S1 which has such a subset D with respect to the string s`−1 0 . For all such permutations, the predictor P places at least as much weight on the hidden states hi with g(i) = k, with k such that r(k) ∈ D as the true hidden state h`−1 . The prediction for any hidden state hi is the corresponding bit in S2 . Notice that the bits in S2 are independent and uniform as we’ve not used them in any argument so far. The average correlation of an equally weighted average of m independent and uniform random bits with any one of the √ random bits is at most 1/ m. Hence over the randomness of S2 , the expected zero-one loss of the predictor is at least 0.5 − n−0.1 . Hence we can writep −0.1 E[δ(s`−1 )P r[\|D\| ≥ t/(8`)] 0 )] ≥ (0.5 − n 0.25 ≥ (0.5 − n−0.1 )(1 − e−n ) ≥ 0.5 − 2n−0.1 By using Equation D.1, for any assignment r(0) to h`−1 0 h i h i `−1 `−1 E[δ(h`−1 )] ≥ P r D(s`−1 0 , r(0)) > `/2 − 2` E δ(s0 ) D(s0 , r(0)) > `/2 − 2` ≥ (1 − n−2c )(0.5 − 2n−0.1 ) = 0.5 − o(1) As this is true for all assignments r(0) to h`−1 and for all choices of hidden states at time `, using 0 linearity of expectations and averaging over all hidden states, the expected average zero-one loss of 5 For independent random variables {Xi } lying in the interval [0, 1] with X = p P r[X ≤ (1 − )µ] ≤ exp(−2 µ/2). In our case = 1/2 and µ = t/(2`). 32 P i Xi and µ = E[X], the predictor P over the randomness in choosing the permutation is 0.5 − o(1). This means that there must exist some permutation such that the average zero-one loss of the predictor P on that permutation is 0.5 − o(1). Hence there exists an HMM on n states such that is not information theoretically possible to get average zero-one error with respect to the optimal predictions less than − o(1) using windows of length smaller than c log n/2 for a fixed constant c. Therefore, for all 0 < < 0.5 and sufficiently large n, there exits an HMM with n states such that it is not information theoretically possible to get average relative zero-one loss less than /2 < − o(1) using windows of length smaller than c−2 log n. The result for relative zero-one loss follows on replacing /2 by 0 and setting c0 = c/4. The result follows immediately from this as the expected relative zero-one loss is less than the expected `1 loss. 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An Optimal Algorithm for Range Search on Multidimensional Points T.Hema ∗and K.S. Easwarakumar† Department of Computer Science & Engineering Anna University, Chennai 600 025, INDIA. arXiv:1607.00208v1 [cs.CG] 1 Jul 2016 Abstract This paper proposes an efficient and novel method to address range search on multidimensional points in θ(t) time, where t is the number of points reported in <k space. This is accomplished by introducing a new data structure, called BITS k d-tree. This structure also supports fast updation that takes θ(1) time for insertion and O(log n) time for deletion. The earlier best known algorithm for this problem is O(logk n + t) time [5, 15] in the pointer machine model. Keywords: BITS k d-tree, Threaded Trie, Range Search. 1 Introduction k d-trees introduced by J.L.Bentley [4, 6] are multidimensional binary search trees commonly used for storing k dimensional points. They are also used to perform search operations such as exact match, partial match and range queries. Range queries are mostly used in GIS applications to locate cities within a certain region in a map. Similarly, in the geometrical view of a database, one can use orthogonal range search to perform a query. Generally, k d-trees with n nodes have a height n and hence the complexity for insertion and search are high. Although many multi-dimensional search structures are found in the literature [2, 8, 20, 22, 23] they differ from the standard k d-trees mainly in the space-partitioning methods used. Recall that a 2-d tree stores two-dimensional point data of the form (x, y). A 2-d tree splits primarily on the x coordinate of a point at even level and then on the corresponding y coordinate at the odd level, and so on. Hence, the trees are unbalanced and are not efficient for√ search operations. Also, the worst case time complexity for range search on a 2-d tree is O( n + t), where t is the number of points reported and for k dimensions it is O(n1−1/k + t)[4, 16]. In general, most of the k d-tree variants get unbalanced when the data is clustered thereby affecting query operations. P R k -d tree, Bucket P R k -d tree [19], P M R k -d trees [21] and Path level compressed P R k -d trees[18] are some of the trie-based kd trees used to store point data. However, these trees are not always balanced, especially when the data is clustered. One of the dynamic versions of k -d tree is the divided k -d trees [25] for which the range query time is O(n1−1/k log1/k n + t). The best known dynamically balanced tree uses bitwise interlaced data [24] over k d-trees mapping k dimensions to one dimension. Although their search time is O(k(log n + t)) for reporting t points, bitwise interlacing leads to discarded areas during range search. In the case of squarish k d-trees [12], an x, y discriminant is based on the longest side of rectangle enclosing the problem space instead of alternating the keys. Recently, hybrid versions of squarish k d-tree, relaxed k d-tree and median k d-trees [11] have overcome the problem of height balancing. An amortized worst case efficiency of range search for the ∗ Email: [email protected] Author.Email: [email protected] † Corresponding 1 hybrid squarish k d-trees, relaxed and median trees for k -dimensional partial match queries are 1.38628 log2 n, 1.38629 log2 n and 1.25766 log2 n respectively. Their experimental results match the aforementioned theoretical results, where they show that the hybrid median trees outperform the other variants. However, as far as query handling is concerned, these structures perform only partial match queries for two dimensions efficiently. The most recent work in the pointer machine model is an orthogonal range reporting data structure with O(n(log n/ log logn)d ) space that address range queries in O(log n(log n/log log n) time, where d ≥ 4 [1]. Range trees of Bentley and Maurer [6, 5] are yet another class of balanced binary search trees used for rectangular range search which showed improvement in the query time of O(logk n + t) over O(n1−1/k + t) of kd-trees, where k is the dimension for a set of n points and k is the number of reported points. This was later improved to O(logk−1 n + t) using fractional cascading in layered range trees[13] but the space requirements are relatively high of O(n logk−1 n). A k d-Range DSL-tree performs k-dimensional range search in O(logk n+t) time was proposed in [15]. Recently, Chan et.al [10] have proposed two data structures for 2d orthogonal range search in the word RAM model. The first structure takes O(n lg lg n) space and O(lg lg n) query time. They show improved performance over previous results [3] of which O(n lg n) space and O(lg lg n) query time, or with O(n lg lg n) space and O(lg 2 lg n) query time. The second data strucure is based on O(n) space and answers queries in O(lg n) time that outperforms previous O(n) space data structure [17], answers queries in O(lg n/lg lg n) time. Furthermore, they also propose an efficient data structure for 3-d orthogonal range reporting with O(n lg 1+ + n) space and O(lg lg n + k) query time for points in rank space where > 0. This improves their previous results [9] with O(n lg 2 n) space and O(lg lg n+k) query time, or with O(n lg 1+ n) space and O(lg 2 lg n + k) query time, where k points are reported. Finally they have extended range search to higher dimensions also. Since such range queries are common among multi-dimensional queries in database applications, we have mainly considered an orthogonal range search on multi-dimensional points. 2 Our Contributions In this work, we make use of the BIT S-tree [14], a segment tree variant that performs stabbing and range queries on segments efficiently in logarithmic time. Most importantly, the distribution of the data points (uniform or skewed) does not affect the height of the BIT S-tree and in turn facilitates faster search time. Here, we actually use the BIT S-tree structure to store points related to each dimension and thereby form a multi-level tree, called BIT S k d-tree. In addition, certain nodes of the BIT S-tree associate to a variant of the trie data structure, called threaded trie, to facilitate fetching a required node in constant time. Unlike k -d trees, it does not associate co-ordinate axis, level wise, for comparison to locate or insert a point. Instead, the tree at the first level has nodes with a key on only distinct values of first co-ordinate of the points. Therefore, this tree corresponds to the one dimensional data. This tree is then augmented with another tree at second level and there in key values of the nodes associated with distinct first two co-ordinates of the points. In general, ith tree corresponds to the distinct first i co-ordinates of the set of points given. Moreover, in each tree, the inorder sequence provides the sorted sequence. That is, BIT S k -d trees is a multi-level tree, and its construction is illustrated in the subsequent sections. 2.1 BIT S-Trees Originally, the BIT S-tree (Balanced Inorder Threaded Segment Tree) [14] is a dynamic structure that stores segments, and also answers both stabbing and range queries efficiently. Unlike segment trees, it also permits insertion of segment with any interval range. 2 Figure 1: (a) Set of segments.(b) BIT S-Tree for the given segments. Definition 1 A BIT S-tree is a height balanced two-way inorder-threaded binary tree T that satisfies the following properties. 1. Each node v of T is represented as v([a, b], L), where [a, b] is the range associated with the node v, and L is the list of segments containing the range [a, b], i.e if [c, d] ∈ L then [a, b] ⊆ [c, d]. 2. Given v1 ([a1 , b1 ], L1 ) 6= v2 ([a2 , b2 ], L2 ), then   [b1 ] if a2 = b1 [b2 ] if a1 = b2 [a1 , b1 ] ∩ [a2 , b2 ] =  φ otherwise i.e ranges can either overlap only at end points or do not overlap at all. 3. Suppose v1 ([a1 , b1 ], L1 ) appears before v2 ([a2 , b2 ], L2 ) in the inorder sequence, then b1 ≤ a2 . 4. It has a special node, called dummy node denoted by D, with range and list as φ(empty). 5. Suppose v1 ([a1 , b1 ], L1 ) and vn ([an , bn ], Ln ) are the first and last nodes of the inorder sequence respectively, then InP red(v1 ) = InSucc(vn ) = D, and the range, say [a, b], of any node contained in [a1 , bn ], i.e [a, b] ⊆ [a1 , bn ]. Here, the functions InPred() and InSucc() respectively returns inorder predecessor and successor. A sample BIT S-tree is shown in Figure 1. Note that the dangling threads actually point to a dummy node, which is not shown in the figure. The BIT S-tree is originally developed for storing segments, but we use this for a different purpose of storing points. Thus, we modify this structure to suit our requirement as described below. 1. Each node v([a, b], L) is replaced by v(p, L0 , T ), where p is a point in <k , k ≥ 1, and L0 is a pointer to the list of collinear points in dimension k + 1, having p for the first k co-ordinates. However, this list is maintained in the tree at the next level, which is described in section 2.2. Now, T is either null or a pointer to a threaded trie, which is elaborated in the next section. 2. For any two points p1 and p2 stored in a tree, p1 6= p2 . 3. Suppose v1 (p1 , L01 , T1 ) appears before v2 (p2 , L02 , T2 ) in the inorder sequence, then p1 < p2 as per the following definition. Definition 2 Let p1 = (x11 , x12 , . . . x1k ) and p2 = (x21 , x22 , . . . x2k ) be two points in a kdimensional space, then a. p1 = p2 implies x1j = x2j for each j=1, 2,. . . k. b. p1 < (or >)p2 implies head(p1 , j) = head(p2 , j) and x1j+1 < (or >) x2j+1 for some j. 3 In the subsequent sections, for better clarity, we use hyphen(−) for a certain parameter of a node to denote that the particular parameter is irrelevant with respect to the context. For instance, (p, −, T ) denotes that the list contents are irrelevant for that point p at this time. 2.2 Threaded Trie Figure 2: A sample threaded trie Threaded tries are variants of tries that consists of two types of nodes viz. trie node and data node. For instance, in Figure 2, A, B and C are trie nodes and the rest are data nodes. Unlike in tries, the trie node here does not have a field for blank (6b). However, each of these trie nodes contain two segments. One is the index pointer and the other is a tag value, which is either 0 or 1, where 0 denotes the corresponding index point in a thread and otherwise it will be 1. Here, all null pointers are replaced by threaded pointers, which point to the next valid node, if one exists. For instance, the thread pointers of 1, 2, 3 of node A points to the node C, as this is the next valid node. Similarly, thread pointers of 0 and 1, in C points to the data node 42. Note here that ordering on the nodes provides the sorted sequence. Also, data nodes appear at the same level. This is accomplished by having uniform width for all data. For instance, the data 8 is treated as 08 in Figure 2. 2.3 Construction of Multi-level BIT S-Tree Multi-level BIT S-trees are constructed using a collection of BIT S-trees one at each level, and interlinking the trees of two consecutive levels in a specified manner, which are due to the following definitions. These multi-level BIT S-trees are termed here as BIT S-k d trees. Definition 3 Given a point p = (x1 , x2 , . . . xk ) and an integer l ≤ k, the head of p and tail of p are defined respectively as head(p, l) = (x1 , x2 , . . . xl ) and tail(p, l) = (xk−l+1 , xk−l+2 , . . . xk ). Also, having (head(p, l), y1 , y2 , . . . ym ) = (x1 , x2 , . . . xl , y1 , y2 , . . . ym ), leads to (head(p, l), tail(p, k − l)) = p. Definition 4 Given S as the set of points in <k and \|S\|=n, the set S1 is defined as, Sn S1 = i=1 {(x) \| p ∈ S and head(p, Sn 1) = (x)}. That is, S1 is the set of distinct x values of the points in S. In general, Sj = i=1 {(x1 , x2 , . . . xj ) \| p ∈ S and head(p, j) = (x1 , x2 , . . . xj )}, where 1 ≤ j ≤ k. Definition 5 For a point p = (x1 , x2 , . . . xj ) in Sj , the term xj is said to be the dimensional value of p as the set of points in Sj is used to construct j th level BIT S-tree. 4 Figure 3: A BIT S 2d-tree: (a) Spatial representation of points. (b) BIT S-2d tree for points shown in (a). Definition 6 A BIT S kd-tree is a multi-level tree, which is constructed as follows. 1. Create separate BIT S trees, Tj for each Sj , 1 ≤ j ≤ k. 2. Let Xj = (x1 , x2 , . . . xj ). Now, for each node, say vj = (Xj , L, −), of Tj , 1 ≤ j < k, the list L points to the node vj+1 = ((Xj , x0j+1 ), −, −) of Tj+1 , where x0j+1 = min{xj+1 \| head(p, j) = Xj and (head(p, j), xj+1 ) ∈ Sj+1 }. We term these links as cross links, and the node vj+1 as a cross link node in Tj+1 . 3. In T1 , there is only one cross link node, which is the first node in the in order sequence, and the tree pointer always points to this node. 4. For each node v = ((Xj−1 , x0j ), −, T ) in Tj , 1 ≤ j ≤ k, T is pointer to the threaded tree if v is a cross link node, and otherwise T is set to be null. 5. For every cross link node vj = ((Xj−1 , x0j ), −, T ) in Tj , the data node of T for a key, say k 0 , points to the node ((Xj−1 , k 0 ), −, −) in Tj . That is, T provides links to the nodes in {((Xj−1 , xj ), −, −)\|(Xj−1 , xj ) ∈ Sj } and these links are termed as trie links. A BIT S 2d-tree for the sample points in Figure 3(a) is shown in Figure 3(b). Since, BIT S k d-trees are multi-level trees with binary inorder threaded search trees at each level, the height of the trees at each level is O(log n). Also, each node in Ti−1 has a cross link to a node in Ti , which has the least value for the ith co-ordinate with respect to the head value of the node in Ti−1 . Note here that at least one such point exists. This link is useful to locate a list of collinear points in the ith dimension, associated with a point in Ti−1 . Also, the trie links are useful to locate a point in a given range window in constant time. The cross link and trie links also make the structure much suitable to address range queries efficiently. Normally, k d-trees perform insertion by a simple comparison between the respective coordinates at each level. However, deletion is tedious due to candidate replacement. This is because, candidate for replacement can be anywhere in the subtree. Also, it requires a little more work when the right subtree is empty. Now, to find a candidate for replacement, it is required to find the smallest element from the left subtree to avoid violation of the basic 5 rules of k d-trees and then it is required to perform a swap of left and right subtrees, as many possible candidate keys exist in the left subtree. To handle such a situation, we make use of a collection of BIT S-trees, one for each dimension. Here, deleting a point may or may not require a replacement, but if so, it is only the inorder successor and that can be located in θ(1) time as inorder links exist for each node. Also, the cross links that exist between two consecutive levels, practically provide a faster search on next level trees. Another advantage of this structure is that when a node is pruned out at a particular level, it need not be considered in the subsequent levels. That is, nodes that have head values as these will be ignored in the subsequent levels. To the best of our knowledge, there is no such structure using multi-levels of balanced binary search trees, with two-way threads introduced in this work, for storing point data and to perform range search efficiently. 2.4 2d-Range Search for Window Query Given a rectangular range in the form of a window, a range query finds all points lying within this window. Let [x1 : x2 ] × [y1 : y2 ] be a given query range. First, we use a trie stored in the first node, which is the only cross link node, in the tree at level 1 to find the smallest point larger than or equal to x1 in S1 . The non-existence of such a point is determined from the trie itself. On the other hand, once such a point p is located, subsequent points that fall within [x1 : x2 ] can be determined using the inorder threads as the inorder sequence is in sorted order. Let us say that the reported set of points as S 0 . However, if the dimensional value of the point p is greater than x2 , it implies absence of required candidates. Now, using cross links of the node in T1 , that corresponds to each point in S 0 , further search is performed at T2 in a similar fashion. Note that each cross link node in T2 has a trie structure that supports quick access to a node in T2 , where the dimensional value is in [y1 : y2 ]. In case, the dimensional value of the cross link node is within [y1 : y2 ], the respective trie structure need not be looked into, instead the inorder threads are used to find the remaining candidates. Example: 1 For instance, let us consider Figure 3 with search range [1 : 8] × [5 : 7]. First, we use the only cross link node present in T1 . As its dimension value, ie. 2 lies within the range [1 : 8], we do not use the respective trie. Instead, we use the inorder threads to identify the candidate points, which are 2,6 and 8. Now, for each of these candidates, further search is continued respectively from (2, 2),(6, 2) and (8, 10) in T2 , as these are the corresponding cross link nodes. Now, by looking at the tries of these cross link nodes we find a point whose dimensional value is the smallest one is [5 : 7]. Thus, tries of (2, 2) yields (2, 6), (6, 2) yields (6, 6) and (8, 10) yields nothing. Further, by performing inorder traversal from (2, 6) and (6, 6), the final reported points for Q1 are E(2, 6) and C(6, 6). Also, for Q2 i.e, ([5 : 8] × [12 : 14]), no points will be reported. Notice that one can stop the search at T1 without traversing T2 if there is no candidate node in T1 within the given range. This is also applicable in k-d trees because if there is no candidate node in the higher tree, the lower level trees need not be searched. Thus, this structure prunes the search in some cases and thereby practically reduces the time for reporting a query. 2.5 k -d Range Search A range search on k-dimensional points can be performed by extending the search on T3 , T4 , . . . Tk , similar to that of T2 as in the case of 2d range search. However in T1 and T2 , we need to perform the search as described for 2d range search. That is, when we take the query range as [x1 : x01 ] × [x2 : x02 ] × . . . [xk : x0k ], the search is performed to find candidates within the range of [x1 : x01 ] in T1 , [x2 : x02 ] in T2 , [x3 : x03 ] in T3 , and so on. Finally, the points reported from Tk will be in Q. It is important to note that the search requires comparison of keys within the given range of the particular co-ordinate dimension in each of T1 , T2 , . . . Tk . This simplifies subsequent searches at the next level. 6 3 3.1 Implementation Details Two Dimensions Given a set of two dimensional points in <2 , a two-level tree (BIT S2d-tree) is constructed in O(n) time as a point may require at most two insertions, one at T1 and the other at T2 . But the position at which insertion is to be made in T1 and T2 could be determined in constant time as described in the proof of Lemma 5. Thus, to insert n nodes requires O(n) time. Also, it may be required to create a cross link for each node of T1 in the case of BIT S 2d-tree. Since, T1 cannot have more than n points, the number of cross links created cannot exceed n. Also, the number of trie links created cannot exceed the number of nodes in T1 and T2 , which is O(n). Moreover, construction of a trie requires only constant time as the height of the trie is constant due to fixed size of the key. Thus, all these factors lie within O(log n) for each insertion. Regarding space requirements in a BIT S 2d-tree, it is O(n), as the second tree is the one that contains all the n points, and fewer or equal number of points in the first tree. Also, the number of trie nodes is O(n) as the height of a trie is constant which is due to the size of(number of digits) of the key. Thus, we obtain the following lemma. Lemma: 1 Construction of BIT S 2d-tree for n points requires O(n) time and O(n) space. Now, searching a candidate node in T1 is done through the trie in T1 and that requires only constant time as the height of the trie is fixed. Once such a point is identified, subsequent points are identified through inorder threads. Thus. for identifying candidate points, it takes only θ(t1 ) time, if there are t1 candidate points in T1 . Now, using cross links of each of these nodes, we can locate the required tries in constant time and further search is to be done in a similar fashion as described earlier. Thus, it leads to the following lemma. Lemma: 2 Range search for window query using BIT S 2d-tree can be addressed in θ(t) time, where t stands for the number of points reported. 3.2 Higher Dimensions A straight forward extension of BIT S 2d-tree to k dimensions is made easy by connecting (cross links) to the corresponding nodes in the tree at next level. Unlike range trees [7] which build another range tree at a given node from the main tree, we maintain the trees T1 , T2 , . . . Tk , dimension-wise such that the inorder traversal provides an ordered sequence of points stored in the tree. This definitely reduces the overall time taken for range search across k dimensions. As described in the previous section, the time required to find a candidate point in any Ti , 1 ≤ i ≤ k, is only a constant. Thus, it leads to the following lemma. Lemma: 3 Let S be a set of points in k-dimensional space, k ≥ 1. A range search on BIT S kd-tree reports all points that lie within the rectangular query range in θ(t) time, where t is the number of points reported. Lemma: 4 Given a set of n points, a BIT S kd-tree can be constructed in O(n) time and O(n) space. Proof: Since we construct T1 , T2 . . . Tk , such that Tk at level k has at most n nodes, it follows that N (Ti ) ≤ N (Ti+1 ), 1 ≤ i < k and N (Tk )=n, where N (Ti ) is the number of nodes in Ti . Note that levels correspond to dimensions and hence may be used interchangeably. Also, the number of trie nodes is O(n) as its height is constant. Therefore for k levels, a BIT S k d-tree uses O(n) storage in the worst case as k is a constant. Now, construction of BIT S k d-tree is considered as a sequence of insertions. Each insertion, may or may not alter Ti , 1 ≤ i ≤ k, a BIT S tree of a particular level. However, if a BIT S-tree Tj is altered, due to insertion, all trees Tj+1 , Tj+2 , . . . Tk will be altered. Let j be the least index such that 7 Table 1: Theoretical comparison of k d-trees, divided k -d trees, range trees, k -d Range DSLtrees, layered range trees and the proposed BIT S k d-trees. Description k d-Trees [4] Divided k -d trees [25] Range Trees[7] k d-Range DSL-Trees [15] Layered Range Trees[13] BITS k d-Trees Storage O(n) O(n) O(n logk−1 n) O(n logk−1 n) O(n logk−1 n) O(n) Construction O(n log n) O(n log n) O(n logk−1 n) O(n logk n) O(n logk−1 n) O(n) Update O(logk n) O(logk−1 n) O(logk n) O(n logk−1 n) O(logk n) Ins. θ(1) Del. O(log n) Range Search O(n1−1/k + t) O(n1−1/k log1/k n + t) O(logk n + t) O(logk n + t) O(logk−1 n + t) θ(t) n-number of points, k-dimensions, t-number of points reported. the tree Tj is altered. Thus, for T1 , T2 , . . . Tj−1 , with trie links and cross links, one can determine that the required values are already stored in those trees within constant time. Now, from a particular cross link in Tj−1 followed by a trie link in Tj , one can find a position for the new value in Tj . This requires only constant time. Then, while inserting the value if the tree is unbalanced, atmost one rotation is required to balance the tree. So, for Tj too, it requires constant time. Let nj be the new node inserted in Tj . Now, by taking cross link of inorder successor of nj , one can determine the position of the new node in Tj+1 , and that as inorder predecessor of cross link node of inorder successor of nj . This new node in Tj+1 need to have a trie, which again be created in constant time. Then, the process is to be continued for Tj+2 . . . Tk . Here, updation in each Ti , 1 ≤ i ≤ k, takes only constant time and hence each insertion takes θ(1) time. So, construction of BIT S k d-tree for n points requires O(n) time. Lemma: 5 Insertion and Deletion of a point in a BIT S kd-tree can be respectively done in θ(1) and O(log n) time. Proof: As per the description given in the proof of Lemma 4, insertion of a point in BITS k d-tree takes only θ(1) time. But for deletion, finding a node to be removed from a BIT Stree requires only constant time. However, if that node is not a leaf node a cascading replacement with inorder successor is required until reaching a leaf node to be removed physically. Certainly, the number of such replacements to be done cannot exceed O(log n). After that it may require a sequence of rotations on the path from the physically removed leaf to the root, and that too in at most O(log n) rotations. So, deletion of a point in BIT S k d-tree requires O(log n) time. 4 Performance Table 1 summarizes the performance of k d-trees, divided k -d trees, range trees, k d-range DSL-trees and the BIT Sk d- tree proposed in this work. Furthermore, our theoretical comparison of the BIT S k d-tree is made with k d-trees adapted for internal memory(pointer machine model) and not with any of the other bulk loading k d-trees(RAM model). The results give an θ(t) query time using the BIT S k d-tree that shows a reduction in time as compared to the existing bounds. Since we try to capitalize on the efficiency of balanced search trees at all the levels by using cross links and trie links, we ensure that the number of nodes visited during a range query is considerably reduced in BIT S k d-tree. Observe that the storage is increased from O(n) in k d-trees to O(n logk−1 n) in range trees while BIT S k d-tree still maintains an O(n). Notice that the update time for BITSk d-tree has been reduced considerably. To summarize, although the storage requirements of BIT S k dtree are comparable to k -d trees, divided k -d trees, the construction and update time are improved considerably. Moreover, the overall query time is improved to θ(t) time where t is the number of points reported as it prunes points falling outside the query region for each dimension. 8 5 Conclusion A BIT S k d-tree for storing k-dimensional points having update and query operations efficiently than kd-trees is proposed. The main advantage of this tree is that it effectively handles the collinear points. As a result, number of nodes visited during search is much less compared to other k d-tree variants that are either not height balanced or update operation is complex. In the case of height balanced k d-trees, having better search efficiency, insertion is tedious. A k -d range DSL tree gives a logarithmic amortized worst case search time with efficient updates mainly for partial match queries and not for window queries. In BIT S k d-tree, overall insertion time is θ(1). Moreover, points can be dynamically updated at each level. Since co-ordinate dimensions at each level are distributed and using threaded tries, we quickly find points falling within the query range. Also, points falling above and below the search range are pruned efficiently using cross links to the next level and inorder threads similar to the BIT S-tree. In addition, threaded tries introduced in this work link the node, having cross link, by means of trie links to find the points within the given range in constant time. Therefore, range search for points in a rectangular region using BIT S k -d tree takes θ(t) time where t is the number of points reported, and therefore the logarithmic factor in earlier worst case bounds is reduced. Hence it is definitely a remarkable improvement over O(n1−1/k + t) of k d-trees and O(logk n + t) time of k -d range DSL trees. References [1] P. Afshani, L. Arge, and K.G. Larsen. Higher-dimensional orthogonal range reporting and rectangle stabbing in the pointer machine model. 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Overmars. Computational Geometry: algorithms and applications. Springer-Verlag, New York,USA, third edition, 2008. [9] T.M. Chan. Persistent predecessor search and orthogonal point location on the word ram. In Proceedings of the Twenty-second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA ’11, pages 1131–1145. SIAM, 2011. [10] T.M. Chan, K.G. Larsen, and M. Pătraşcu. Orthogonal range searching on the ram, revisited. In Proceedings of the Twenty-seventh Annual Symposium on Computational Geometry, SoCG ’11, pages 1–10, New York, NY, USA, 2011. ACM. 9 [11] M.M.P. Crespo. Design, Analysis and Implementation of New Variants of Kd-trees. Master Thesis, Universitat Politecnica de Catalunya,Departament de Llenguatges i Sistemes Informatics, 2010. [12] L. Devroye, J. Jabbour, and C. Zamora-Cura. Squarish kd-trees. SIAM Journal of Computing, 30:1678–1700, 2000. [13] D.Willard. New data structures for orthogonal queries. Harvard University, TR:22–78, 1978. [14] K.S. Easwarakumar and T. Hema. BITS-Tree-An efficient data structure for segment storage and query processing. International Journal of Computers and Technology, 11(10):3108–3116, December 2013. [15] M.G. Lamoureux and B.G. Nicolson. Determinisitic skip lists for k-dimensional range search. Technical Report(TR95-098), pages 1–95, Novemeber 1995. [16] D.T. Lee and C. K. Wong. Worst-case analysis for region and partial region searches in multidimensional binary search trees and balanced quad trees. Acta Informatica, pages 23–29, 1977. [17] Y. Nekrich. Orthogonal range searching in linear and almost-linear space. Computational Geometry, 42(4):342–351, 2009. [18] S. Nilsson and M.Tikkanen. An experimental study of compression methods for dynamic tries. Algorithmica, 33(1):19–33, 2002. [19] J.A. Orienstein. Multidimensional tries used for associative searching. Information Processing Letters, 14(4):150–157, June 1982. [20] F.P. Preparata and M.L. Shamos. Computational Geometry: An Introduction. 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Slow links, fast links, and the cost of gossip arXiv:1611.06343v2 [cs.DC] 14 Dec 2017 Suman Sourav National University of Singapore [email protected] Peter Robinson Royal Holloway, University of London [email protected] Seth Gilbert National University of Singapore [email protected] Abstract Consider the classical problem of information dissemination: one (or more) nodes in a network have some information that they want to distribute to the remainder of the network. In this paper, we study the cost of information dissemination in networks where edges have latencies, i.e., sending a message from one node to another takes some amount of time. We first generalize the idea of conductance to weighted graphs by defining φ∗ to be the “critical conductance” and `∗ to be the “critical latency”. One goal of this paper is to argue that φ∗ characterizes the connectivity of a weighted graph with latencies in much the same way that conductance characterizes the connectivity of unweighted graphs. We give near tight lower and upper bounds on the problem of information dissemination, up to polylogarithmic factors. Specifically, we show that in a graph with (weighted) diameter D (with latencies as weights) and maximum degree ∆, any information dissemination algorithm requires at least Ω(min(D+ ∆, `∗ /φ∗ )) time in the worst case. We show several variants of the lower bound (e.g., for graphs with small diameter, graphs with small max-degree, etc.) by reduction to a simple combinatorial game. We then give nearly matching algorithms, showing that information dissemination can be solved in O(min((D + ∆) log3 n, (`∗ /φ∗ ) log n) time. This is achieved by combining two cases. We show that the classical push-pull algorithm is (near) optimal when the diameter or the maximum degree is large. For the case where the diameter and the maximum degree are small, we give an alternative strategy in which we first discover the latencies and then use an algorithm for known latencies based on a weighted spanner construction. (Our algorithms are within polylogarithmic factors of being tight both for known and unknown latencies.) While it is easiest to express our bounds in terms of φ∗ and `∗ , in some cases they do not provide the most convenient definition of conductance in weighted graphs. Therefore we give a second (nearly) equivalent characterization, namely the average conductance φavg . 1 1 Introduction Consider the problem of disseminating information in a large-scale distributed system: nodes in the network have information that they want to share/aggregate/reconcile with others. Real world network communication often has a time delay, which we model here as edges with latencies. The latency of an edge captures how long communication takes, i.e., how many rounds it takes for two neighbors to exchange information. Low latency on links imply faster message transmission whereas higher latency implies longer delays. In the case of unweighted graphs, all edges are considered the same and are said to have unit latencies. However, this is not true in real life and link latencies can vary greatly. In fact, even if nodes are connected directly it might not be the fastest route for communication due to large latency of the link (which might arise due to poor connection quality, hardware or software restrictions etc.); often choosing a multi-hop lower latency path leads to faster distribution of information. For unweighted graphs (without latencies), there exists a significant amount of literature, characterizing the connectivity of a graph (referred as the conductance of a graph) which exactly indicates how efficient information dissemination will be. We would like to do the same for graphs with latencies, however, due to the presence of latencies, not all edges can be regarded as the same; and therefore connectivity alone is no longer enough. Thus, we introduce a new notion of the critical conductance φ∗ that generalizes the notion of classical conductance. Using φ∗ , we give nearly tight lower and upper bounds for information dissemination. For some cases, φ∗ might not be the most convenient definition of conductance in weighted graphs. Alternatively, we give a (nearly) equivalent characterization, namely the average conductance φavg . Model. We model the network as a connected, undirected graph G = (V, E) with n = \|V \| nodes. Each node knows the identities of its neighbors and a polynomial upper bound on the size of the network. Nodes communicate bidirectionally over the graph edges, and communication proceeds in synchronous rounds. An edge is said to be activated whenever a node sends any message over the edge. Latencies occur in the communication channel and not on the nodes. For simplicity, we assume that each edge latency is an integer. (If not, latencies can be scaled and rounded to the nearest integer.) Also, the edge latencies here are symmetric. Problems for non-symmetric arbitrarily large latencies are at least as hard as directed unweighted networks (for which many tasks are impossible to achieve efficiently). Let D be the (weighted) diameter of the graph (with latencies as weights), and let `max be the maximum edge latency. We consider both cases where nodes know the latencies of adjacent edges (Section 5) and cases where nodes do not know the latencies of adjacent edges (the rest of the paper). Nodes do not know D or `max .1 In each round, each node can choose one neighbor to exchange information with: it sends a message to that neighbor and (automatically) receives a response.2 If the edge has latency `, then this round-trip exchange takes time `. This model is, within constant factors, equivalent to a more standard model in which a round-trip involves first sending a message with latency `, receiving it at the other end, and then sending a response at a cost of latency `. Notice that each node can initiate a new exchange in every round, even if previous messages have not yet been delivered, i.e., communication is non-blocking. Information dissemination. We focus in this paper on one-to-all information dissemination. A designated source node begins with a message (the rumor) and, when the protocol completes, every node should have received the message. Classic examples include distributed database replication, sensor network data aggregation, and P2P publish-subscribe systems. This fundamental problem has been widely studied under various names, e.g., information dissemination (e.g., [5]), rumor spreading (e.g., [8]), global broadcast (e.g., [18]), one-to-all 1 In real world settings, nodes are often aware of their neighbors. However, due to fluctuations in network quality (and hence latency), a node cannot necessarily predict the latency of a connection. 2 Notice that this model of communication is essentially equivalent to the traditional push-pull where each node can either push data to a neighbor or pull data from a neighbor; here we assume a node always does both simultaneously. Without the ability to pull data, it is easy to see that information exchange takes Ω(nD) time, e.g., in a star. Simple flooding matches this lower bound. 2 multicast, and information spreading (e.g., [6]). As a building block, we look at local broadcast, i.e., the problem of every node distributing a message to all of its neighbors. Conductance in weighted graphs. Our goal in this paper is to determine how long it takes to disseminate information in a graph with latencies. Clearly the running time will depend on the (weighted) diameter D of the graph. Typically, such algorithms also depend on how well connected the graph is, and this is normally captured by the conductance φ. Unfortunately, conductance is no longer a good indicator of connectivity in a graph with latencies, as slow edges (with large weights) are much worse than fast edges.3 We begin by generalizing the idea of conductance to weighted graphs. We give two (nearly) equivalent definitions of conductance in weighted graphs, which we refer to as the critical conductance φ∗ (Definition 2) and the average conductance φavg (Definition 4). While they give (approximately) the same value for every graph, there are times when one definition is more convenient than the other. In fact, we show that the values φ∗ of φ∗ and φavg are closely related; as in 2` < φavg < φ`∗∗ dlog(`max )e (c.f. Theorem 5). We compare these ∗ definitions further in Section 2.3. We use φ∗ in determining the lower and upper bounds for information dissemination as it makes our analysis simpler and then use the above relation to determine the bounds for φavg . A core goal of this paper is to argue that the notion of φ∗ (and φavg ) defined herein well captures the connectivity of weighted graphs, and may be useful for understanding the performance of other algorithms. Lower bounds. These constitute some of the key technical contributions of this paper. For a graph G, with diameter D, maximum degree ∆, critical conductance φ∗ , and critical latency `∗ , we show that any information dissemination algorithm requires Ω(min(D +∆, `∗ /φ∗ )) rounds. That is, in the worst case it may take time D + ∆ to distribute information. However, if the graph is well connected, then we may do better and the time is characterized by the critical conductance. We show that this lower bound holds even in various special cases, e.g., for graphs with small diameter, or with small max-degree, etc. By the relation provided in Theorem 5, we determine the lower bound in terms of average conductance as Ω(min(D + ∆, 1/φavg )). The main technique we use for showing our lower bounds is a reduction to a simpler combinatorial guessing game. (See [25] for a demonstration of how other variants of guessing games can be used to prove lower bounds for radio networks.) We first show that the guessing game itself takes a large number of rounds. Thereafter we reduce the problem of solving the game to that of solving information dissemination via a simulation. Upper bounds. We then show nearly matching upper bounds, i.e., algorithms for solving information dissemination. In this regard, we differentiate our model into two cases. For the case where nodes are not aware of the adjacent edge latencies, we show that the classical push-pull random phone call algorithm [22] in which each node initiates a connection with a randomly chosen neighbor in each round, completes in O((`∗ /φ∗ ) log n) rounds. By using the relationship between φ∗ and φavg , we give a O((log(`max )/φavg ) log n) upper bound in terms of φavg . For the case where nodes do know the latencies of the incident edges, we obtain nearly tight bounds that are independent of ∆ and φ∗ : we give a O(D log3 n)-time algorithm (which is within polylogarithmic factors of the trivial Ω(D) lower bound). The key idea of the algorithm is to build a (weighted) spanner (based on that in [2]). This spanner is then used to distribute information. This algorithm, however, requires knowledge of a polynomial upper bound on n; hence for completeness we also provide an alternate algorithm in Appendix 5.4 that does not require the knowledge of n but takes an additional log D factor (instead of log n), making it unsuitable for graphs with large diameters. Finally, we observe that we can always discover the latencies of the “important” adjacent edges in 3 Notice you might model an edge with weight w as a path of w edges with weight 1. If you calculate the conductance of the resulting graph, you do not get a good characterization of the connectivity of the original graph for a few different reasons. For e.g., consider the ability of the imaginary nodes on the edge to pull data from the endpoints. 3 Õ(D + ∆) time4 , after which we can use the algorithm that works when latencies are known. Hence, even if latencies are unknown, combining the various algorithms, we can always solve the information dissemination in O(min((D + ∆) log3 n, (`∗ /φ∗ ) log n) time (or O(min((D + ∆) log3 n, (log(`max )/φavg ) log n) time), matching the lower bounds up to polylogarithmic factors (with respect to the critical conductance). Summary of our contributions. To the best of our knowledge, this work provides a first ever characterization of conductance in graphs with latencies. In this regard, we provide two different parameters namely φ∗ and φavg . Note that, we provide the summary here only in terms of φ∗ , however, for each case there exists an alternate version in terms of φavg . For lower bounds, we show that there exists graphs with (a) O(log n) diameter with maximum degree ∆ where local broadcast requires Ω(∆) rounds; (b) O(`∗ ) diameter with critical conductance φ∗ where local broadcast requires Ω(1/φ∗ + `∗ ) rounds; (c) Θ(1/φ∗ ) diameter where information dissemination requires Ω(min(D + ∆, `∗ /φ∗ )) rounds; showing the trade-off among the various parameters affecting information dissemination. For upper bounds on information dissemination, we show that (d) the push-pull algorithm takes O(`∗ log(n)/φ∗ ) rounds; (e) a spanner-based algorithm takes O((D + ∆) log3 n) rounds. We view our results as a step towards a more accurate characterization of connectivity in networks with delays and we believe that the metrics φ∗ and φavg can prove useful in solving other graph problems. Prior work. There is a long history studying the time and message complexity of disseminating information when all the links have the same latency. It is interesting to contrast what can be achieved in the weighted case with what can be achieved in the unweighted case. The classic model for studying information dissemination is the random phone call model, introduced by [10]: in each round, each node communicates with a single randomly selected neighbor; if it knows the rumor, then it “pushes” the information to its neighbor; if it does not know the rumor, then it “pulls” it from its neighbor (see, e.g., [13], [23], [16]). An important special case is when the graph is a clique: any pair of nodes can communicate directly. In a seminal paper, Karp et al. [22] show that a rumor can be disseminated in a complete graph in O(log n) rounds with O(n log log n) message complexity. Fraigniaud and Giakkoupis [14] show how to simultaneously achieve optimal communication complexity (except for extremely small rumor sizes). When the graph is not a clique, the performance of the classical push-pull protocol, wherein a node exchanges information with a random neighbor in each round, typically depends on the topology of the graph, specifically, how well connected the graph is. An exciting sequence of papers (see [7, 8, 16, 24] and references therein) eventually showed that rumor spreading in this manner takes time O( logφ n ), where φ is the conductance of the graph. The question that remained open was whether a more careful choice of neighbors lead to faster information dissemination. In a breakthrough result, Censor-Hillel et al. [5] gave a randomized algorithm for solving information dissemination in any (unweighted) graph in time O(D + polylogn), where D here is the nonweighted diameter of the graph. Of note, the protocol has no dependence on the conductance of the graph but only on the diameter (which is unavoidable). There were two key ingredients to their solution: first, they gave a “local broadcast” protocol where each node exchanges information with all its neighbors in O(log3 n) time; second, as a by-product of this protocol they obtain a spanner which they use in conjunction with a simulator (defined therein) to achieve information dissemination in O(D + polylogn) time. Haeupler [18] then showed how local broadcast could be achieved in O(log2 n) time using a simple deterministic algorithm. The conclusion, then, is that in an unweighted graph (with unit latency edges), information dissemination can be achieved in time O(D + polylogn) or in time O(log(n)/φ). 4 The notation Õ hides polylogarithmic factors, which arise due to D and ∆ being unknown. 4 Other related works. The problem has been well researched in several other settings as well. For graphs modeling social networks Doerr et al. [11, 12] show a Θ(log n) time bound for solving broadcast. For the case of direct addressing, Haeupler and Malkhi [19] show that broadcast can be performed optimally in O(log log n) rounds. Information dissemination in random geometric graphs has been studied by Bradonjić et al. [4], in wireless sensor networks and adhoc networks by Boyd et al. [3], Sarwate and Dimakis [26] and Gandhi et al. [15], Giakkoupis et al. [17] study the problem in dynamic graphs. 2 Conductance in Weighted Graphs In this section, we provide two different approaches to characterize conductance in weighted graphs namely the critical conductance and the average conductance and show the relationship between them. In the sections that follow, where we determine the bounds on information dissemination, we use the critical conductance as it makes our analysis simpler. Corresponding bounds for average conductance are obtained by the application of the given relationship in Theorem 5. 2.1 Critical Conductance We now define the critical conductance of a graph, generalizing the classical notion of conductance. For a given graph G = (V, E), and for a set of edges S ⊆ E, we define E` (S) to be the subset of edges of S that have latency 6 `. For a set of nodes U ⊆ V and cut C = (U, V \ U ), we define P E` (C) to be the subset of edges across the cut C with latency 6 `, and we define the volume Vol(U ) = v∈U degv , where degv refers to the degree of node v. We first define the critical conductance of a cut for a given latency `, and then define the weight-` conductance as the minimum critical conductance across all cuts. Definition 1 (Weight-` Conductance). Consider a graph G = (V, E). For any cut C in the set of all possible cuts (C̃) of the graph G and an integer `, we define φ` (C) = \|E` (C)\| . min{Vol(U ), Vol(V \ U )} The weight-` conductance is given by φ` (G) = min{φ` (C) \| C ∈ C̃}. Definition 2 (Critical Conductance). We define the critical conductance φ∗ (G) as φ` (G) is maximum for any ` ∈ (1, `max ) . φ∗ (G) = φ` (G) ` We call `∗ the critical latency for G if `∗ = ` and φ∗ (G) = φ` (G). We simply write φ∗ (or φ` ) instead of φ∗ (G) (or φ` (G)) when graph G is clear from the context. If all edges have latency 1, then φ∗ is exactly equal to the classical graph conductance [21]. 2.2 Average Conductance For a given graph G = (V, E), we first define dlog(`max )e different latency classes, where the first class contains all the edges of latency < 2 and the subsequent ith latency class consists of all the edges in the latency range of (2i−1 , 2i ]. For a set of nodes U ⊆ V and the cut C = (U, V \ U ), we define ki (C) to be the subset of edges across the cut C belonging to latency class i (i.e. all cut edges of latency > 2i−1 and 6 2i ). For a cut C, we first define the average cut conductance as φavg (C), and then define the average conductance as the minimum average cut conductance across all cuts. 5 Definition 3 (Average Cut Conductance). Consider a graph G = (V, E), a set of nodes U ⊆ V and the cut C = (U, V \ U ). Let S be the min{Vol(U ), Vol(V \ U )}. 1 S φavg (C) = dlog(`max )e X i=1 \|ki (C)\| 2i Definition 4 (Average Conductance). Let C̃ be the set of all possible cuts of the graph G. We define the average conductance as φavg (G) = min{φavg (C) \| C ∈ C̃}. We simply write φavg instead of φavg (G) when graph G is clear from the context. If all edges have latency 1, then φavg is exactly equal to the classical graph conductance [21]. 2.3 Comparing Critical and Average Conductances Conductance, in general, is a characterization of the “bottleneck in communication” of a graph. For unweighted graphs, the only bottleneck in communication can be the connectivity of the graph, however, for weighted graphs the bottleneck can arise either due to the graph connectivity or due to the edge latency (even if the nodes are directly connected by a slow edge, there might exist a different multi-hop faster path). Our aim is to capture both aspects of this bottleneck in communication. Having good connectivity facilitates faster communication whereas large latencies result in slow-downs. Ideally, we would want the best connectivity along with the least slowdown for faster communication. We obtain the definition of φ∗ by directly optimizing these orthogonal parameters. The connectivity that maximizes this ratio is defined as the critical conductance φ∗ and the corresponding latency is defined as the critical latency `∗ . In other words, φ∗ captures the bottleneck due to connectivity whereas `∗ captures the bottleneck due to latency. The definition of the average conductance φavg is inspired by the classical notion of conductance. Each cut edge’s contribution towards the overall connectivity is normalized by dividing it with its latency (rounded to the upper bound of its latency class), so as to account for the slow-down. Surprisingly, we see that φ∗ and φavg are closely related and to show the relationship between them, we first define L as the number of non-empty latency classes in the given graph G. Latency class i is said to be non-empty if there is at least one edge in the graph G that has a latency > 2i−1 and 6 2i . The maximum value that L can take is dlog(`max )e which is the total number of possible latency classes. Theorem 5. φ∗ /2`∗ < φavg < Lφ∗ /`∗ . Proof. Consider any weighted graph G that has critical conductance φ∗ and critical latency as `∗ . We first show the upper bound. Let C be the cut from which φ∗ was obtained and let S be the minimum volume among either side of the cut. By the definition of weight-` conductance, Pi φ2i (C) = j=1 \|kj (C)\| S and from the definition of φ∗ , we know that ⇒ φ∗ `∗ φ2i (C) 2i is > ∀i ∈ (1, dlog(`max )e) φ` ` Pi = j=1 \|kj (C)\|/2i S > \|ki (C)\|/2i S for any `, which implies φ i (C) \|ki (C)\|/2i φ∗ > 2 i > `∗ 2 S Note that, in the definition of φavg , the terms corresponding to the empty latency classes becomes zero. We replace each remaining term in the definition of φavg (C) by φ`∗∗ and using the above inequality, we get φavg (C) 6 φ`∗∗ L. Combining with the fact that φavg is the minimum average cut conductance, we obtain 6 φavg 6 φavg (C) 6 φ∗ L `∗ (1) Next we show the lower bound, for this we consider the cut C 0 that determines φavg and let S 0 be the minimum volume among either side of the cut. On this cut C 0 consider the latency class of the critical latency `∗ ; say `∗ lies in the latency class x, which implies that 2x−1 < `∗ 6 2x . From the definition of weight-` conductance, we get φ`∗ (C 0 ) \|k1 (C 0 )\| + \|k2 (C 0 )\| + · · · + \|kx (C 0 )\| 6 . 2`∗ 2`∗ S 0 Rewriting φavg as (from definition) φavg \|kdlog(`max )e (C 0 )\| \|k1 (C 0 )\| \|k2 (C 0 )\| , + + ··· + = 2S 0 22 S 0 2dlog(`max )e S 0 and comparing the first x terms of φavg to that of φ`∗ (C 0 )/2`∗ , we observe that each term in the expression of φavg is at least as large as the corresponding term in the above upper bound on φ`∗ (C 0 )/2`∗ . Also there are some additional positive terms in φavg . Combining this with the fact that φ`∗ /2`∗ 6 φ`∗ (C 0 )/2`∗ (as by definition φ` is chosen as the minimum value among all possible cuts), we obtain φ` ∗ φ` (C 0 ) 6 ∗ 6 φavg 2`∗ 2`∗ (2) This proves the lower bound and completes the proof. 3 Lower Bounds We proceed to lower bound the time for completing information dissemination. The main goal of this section (as found in Theorems 9, 10, and 13) is to show that every gossip algorithm requires Ω(min{∆ + D, `∗ /φ∗ }) on graphs with diameter D, max-degree ∆, critical conductance φ∗ , and critical latency `∗ . Throughout this section, we assume that nodes do not know the latencies of their adjacent links (when nodes do know the latencies, the trivial lower bound of Ω(D) is sufficient). We begin by defining a combinatorial guessing game (a similar approach as in [25]) and show a lower bound for it.5 We then construct several different worst-case graphs and reduce the guessing game to solving information dissemination on these graphs, thereby showing our lower bound. 3.1 The Guessing Game We define a guessing game played by Alice against an oracle. Conceptually, the game is played on a bipartite graph of 2m nodes. The oracle selects a subset of the edges as the target. In each round, Alice guesses a set of at most 2m edges, and the oracle reveals any target edges that have been hit. At the same time, if any edge (u, v) in the target set is guessed by Alice, then all adjacent edges (x, v) in the target set are removed from the target set. Fix an integer m. Let A and B be two disjoint sets of m integers each, i.e., the left and right group of nodes in the bipartite graph. The winning condition of the game depends on a predicate P , which returns a subset of edges from A × B. For example, P = Randomp returns a subset T that contains elements of A × B, where each element is chosen with probability p or discarded with probability 1 − p. 5 The results of [25] do not apply directly to our setting, as their “proposal set” of the player must intersect the target set in exactly 1 element. By contrast, the guessing game here requires us to discover sufficiently many target elements such that every element in the target set occurs at least once. 7 We now define the game Guessing(2m, P ), which begins when Alice receives two disjoint sets A and B. The oracle chooses a target set T1 ⊆ A × B returned by the predicate. Throughout, we assume that Alice has access to a source of unbiased random bits. Alice’s goal is to eliminate all the elements in the target set. In each round r > 1, Alice submits a set Xr ⊆ A × B of size at most 2m as her round r guesses to the oracle. The oracle replies by revealing the items she guessed correctly, i.e., Xr ∩ Tr . The oracle then computes the round r + 1 target set Tr+1 by removing the items that Alice hit, i.e., all the items in Tr that have the same B-component as an item in Xr ∩ Tr : Tr+1 = Tr \ TrA × TrB ∩ XrB . (3) This concludes round r and the next round begins. The game is solved in the first round r0 , where Alice’s guesses result in an empty target set; at this point, the oracle answers halt. In other words, the game ends in round r0 if, for every b ∈ T1B , there was some a0 ∈ A such that (a0 , b) ∈ Xr ∩ Tr , in some r ∈ [1, r0 ]. Alice’s aim is to minimize the number of rounds until the target set becomes empty. We say that a protocol Π solves Guessing(2m, P ) with probability 1 − in r rounds, if Π always terminates within r rounds, and Tr+1 = ∅ with probability > 1 − , for any target set T . In this case, we call Π an -error protocol. 3.2 Guessing by Gossiping Our lower bound results use variants of an n-node distributed network that has a guessing game gadget of 2m nodes embedded as a subgraph. In our gadget construction, we use predicate P , to specify a set of hidden low latency edges, which we call fast edges. We show that, the execution of a gossip algorithm on an n-node network can be simulated by Alice when playing the guessing game Guessing(2m, P ), where n > 2m. We use the notation id(v) to denote the ID of a vertex v, which, by construction is unique. For a given instance of the guessing game, Alice creates a set of nodes L = {v1 , . . . , vm } where id(vi ) = ai ∈ A for i = 1, . . . , m and, similarly, maps the integers in B to the IDs of the vertex set R = {u1 , . . . , um } in a one-to-one fashion. Next, Alice creates a complete bipartite graph on sets L and R by adding m2 cross edges and adds a clique on the vertices in L where all clique edges are considered to have latency 1. For given integer parameters lo and hi, we construct the network in a way that only some cross edges in the target set are useful to the algorithm by giving them a low latency lo whereas all other cross edges are assigned a large latency value hi. Formally, the latencies of a cross edge e = (vi , uj ) is lo iff (id(vi ), id(uj )) ∈ P ; otherwise e has latency hi. We denote this constructed gadget as G(2m, lo, hi, P ), where the parameters refer to the size of the gadget (i.e. 2m), the low latency value lo, the high latency value hi, and the predicate P respectively. We also consider a symmetric variant, called Gsym (2m, lo, hi, P ), where Alice creates a clique on R in addition to the one on L. See figure 1. Since Alice does not know the target set T in advance, she also does not know when a cross edge should have latency lo or latency hi. Nevertheless, implicitly these latency assignments are fixed a priori by the target set (unknown to Alice) which in turn depends on the predicate P . Whenever a cross edge e is activated in our simulation, Alice submits the ID pair of the vertices of e as a guess to the oracle, whose answer reveals the target set membership and hence also the latency of e. Lemma 6 (Gossip Protocol Simulation). Suppose that there is a t-round -error algorithm A that solves local broadcast on a given n-node network H that contains G(2m, 1, h, P ) or Gsym (2m, 1, h, P ) such that the cross edges of the gadget form a cut of H, for h > t, n > 2m, and a predicate P . Then there is an -error protocol Π for Guessing(2m, P ) that terminates in 6 t rounds. Proof. We argue that Alice can simulate the execution of A on network H and, in particular, on the subgraph G(2m, 1, h, P ), until the gossip algorithm A terminates or the oracle answers halt. (It is straightforward to 8 v1 u1 v1 u1 v2 u2 v2 u2 vm-1 um-1 vm-1 vm L R vm um (a) L R um (b) Figure 1: Guessing Game Gadgets. Red edges correspond to “fast” links whereas the blue edges are “slow” links with high latency. extend the argument to a subgraph Gsym (2m, 1, h, P ).) At the same time, Alice can use the behavior of A on the subgraph G(2m, 1, h, P ) to derive a protocol for Guessing(2m, P ). For a given instance of the guessing game, Alice creates the network H by first assigning all edges in the subgraph H \ G(2m, 1, h, P ) a latency of 1. Moreover, she creates the edges of the subgraph G(2m, 1, h, P ) as described in Section 3.2; we will see below that the latency of a cross edges is only set when it is first activated. If an non-cross edge (vi , vj ) (i.e. a clique edge on L or an edge in E(H \ G(2m, 1, h, P ))) is activated by the algorithm, Alice locally simulates the bidirectional message exchange by updating the state of nodes vi and vj accordingly. In each round r of the gossip algorithm, a set of at most 2m cross edges is activated by the vertices simulated by Alice’s. For each activated cross edge (vi , uj ), Alice uses (id(vi ), id(uj )) as one of her round r guesses. Consider some round r > 1, and suppose the oracle returns the empty set. For each one of Alice’s submitted round r guess (ai , bj ) that was not contained in the oracle’s answer, Alice sets the latency of (ai , bj ) to h by updating the local state of ai . Here ai = id(vi ) and bj = id(uj ) that are chosen in round r, for some vi ∈ L and uj ∈ R. It follows by a simple inductive argument that the state of every vertex in the simulation is equivalent to executing the algorithm on the network. We now argue that the above simulation of a t-round gossip algorithm for local broadcast solves the game Guessing(2m, P ) in at most t rounds with probability > 1 − , for any predicate P . Recall that the guessing game ends if T becomes empty, which happens when Alice’s correct guesses have included every b ∈ T B at least once. By the premise of the lemma, the cross edges of G(2m, 1, h, P ) form a cut of H, which tells us that A cannot solve local broadcast without using the cross edges between L × R. Since every such b ∈ R is a neighbor of a node in L, the only way it can receive a local broadcast message is via a fast cross-edge in T . Hence, if the local broadcast algorithm terminates, we know that b was hit by one of Alice’s guesses. 3.3 Guessing Game Lower Bounds The following lemma is instrumental for showing the Ω(∆) lower bound of Theorem 9, which holds when there are no other assumptions on the critical conductance of the graph. Lemma 7. Let Guessing(2m, \|T \| = 1) be the guessing game where the target set is a single pair chosen uniformly at random from A × B. If protocol Π is an -error protocol for Guessing(2m, \|T \| = 1) where < 1, then the number of rounds until Π terminates is at least Ω(m). Proof. For the sake of a contradiction, suppose that Π solves Guessing(2m, \|T \| = 1) in t < m 2 − 1 rounds. We define Time to be the random variable of the number of rounds until termination in a given execution of Π. 9 Consider a round r > 1 of the protocol and suppose that the game has not yet ended, i.e., Alice has not yet guessed all of T correctly and has made at most 2m(r − 1) (incorrect) guesses in the previous rounds. Let Xr denote the (at most 2m) pairs from A × B chosen by Alice in round r. Since from Alice’s point of view, the adversary has chosen the single element of T uniformly at random from the m2 elements in A × B, 2 2m 6 m−2r the probability that Alice guesses the element of T in round r is at most m2 −2m(r−1) . Let Correct denote the event that protocol Π correctly solves the game. It follows that Pr Time = r Correct = Pr Tr ⊆ Xr Correct 6 2 m−2r . (4) In the remainder of the proof, we will lower bound the probability of event {Time > t}. Observe that Pr[Time > t] > Pr Time > t Correct Pr[Correct]. If Time > t, then none of the rounds 1, . . . , t guesses of Alice were successful, i.e., Q Pr Time > t Correct > ti=1 1 − Pr Time = i Correct . Applying (4) to each round i 6 t, we get Pr[Time > t] > Qt i=1 1− 2 m−2i (1 − ) > 1− 1 m 2 −t t (1 − ). Since the running time of Π never exceeds t rounds, i.e., Pr[Time > t] = 0 and < 1, we get a contradiction to t < m 2 − 1. The next lemma bounds the number of guesses required when the target set is less restricted and its edges form a random subset of the cross edges between A × B. This allows us to derive a lower bound on the local broadcast time complexity in terms of the critical conductance in Theorem 10. Lemma 8. For the guessing game input sets A and B, let Randomp be the predicate that defines the target 1 set T by adding each element of A × B to T with probability p, for some p > Ω( m ). Then, any protocol that solves Guessing(2m, Randomp ) requires Ω(1/p) rounds in expectation. On the other hand, if Alice uses the protocol where she submits her 2m guesses in each round by choosing, for each a ∈A, anelement b0 ∈ B uniformly at random, and, for each b ∈ B, an a0 ∈ A uniformly at random, then Ω logpm rounds are required in expectation. Proof. Recall that the game ends when the guesses of Alice have hit each element in T B ⊆ B at least once, whereas T B is itself a random variable. Let Y be the maximum number of guesses required by Alice’ protocol Π. For the sake of our analysis, we will consider Alice’s guesses as occurring sequentially and hence we can assume that elements of T B are discovered one by one. For each j > 1, we define Zj to denote the number of guesses required to guess the j-th element of T B , after having already guessed j − 1 elements. We will first consider general protocols. Considering that each edge is in the target set with probability p, we can assume that the target membership of an edge e is determined only at the point when Alice submits e as a guess. Recalling that Alice has full knowledge of the remaining elements in T B that she still needs to guess (cf. (3)), we can assume that her guess is successful with probability p (as she will only guess edges that potentially discover a new element in T B ). For this guessing strategy, this remains true independently of the current target set and the set of previously discovered elements (which we denote by Dj ). Formally, Pr[Zj \| Dj , T ] = Pr[Zj ] and hence E[Zj \| Dj , T ] = E[Zj ] = d1/pe. Note that any b ∈ B will be part of some target edge in T , i.e., b ∈ T B , with probability > 1 − (1 − p)m = Ω(1), since p = Ω(1/m), and therefore E[\|T \|] = Ω(m). It follows that hP i hP i \|T \| \|T \| m E[Y ] = E[E[Y \| Dj , T ]] = E E[Z \| D , T ] = E E[Z ] i j i = Ω( p ). i=1 i=1 10 Considering that Alice can guess up to 2m elements per round, it follows that the time is Ω( p1 ), which completes the proof for general algorithms. Now consider the case where Alice uses the protocol where she submits her 2m guesses in each round by choosing, for each a ∈ A, an element b0 ∈ B uniformly at random, and, for each b ∈ B, an a0 ∈ A uniformly at random. Note that this process of selecting her guesses is done obliviously of her (correct and incorrect) guesses so far. Observe that Zj depends on a random variable Fj , which is the size of T after the (j − 1)-th successful guess. Since Zj is the number of times that the protocol needs to guess until a new element in T B is discovered, the distribution of Zj corresponds to a geometric distribution. According to Alice’s protocol, the 2 F B probability of guessing a new element is given by mj2 and hence E[Zj \| Fj ] > m Fj . Let U = \|T1 \|; i.e., U is the number of all elements in B that are part of an edge in T initially. We have m E Y \|U > m 2 = E E[Y \| Fi ] \| U > 2 hP i U m E[Z \| F ] \| U > =E i i i=1 2 bm/2c > X E E[Zi \| Fi ] \| U > m 2 i=1 bm/2c > X i=1 m2 E Fi \| U > m 2 , where the last inequality follows from E[1/X] > 1/E[X], for any positive random variable X, due to Jensen’s Inequality. Since Alice has already correctly guessed i − 1 elements from T B , we discard all elements that “intersect” with successful guesses when updating the target set at the end of each round, according to (3). It can happen that the protocol discovers multiple elements of T B using the round r guesses (which we have assumed to happen sequentially in this analysis). In that case, the target set is not updated in-between guesses. However, it is easy to see that this does not increase the probability of guessing a new element of T B . We get E Fi \| U > m 2 6 (m − i)mp, and thus E Y \|U > m 2 > bm/2c m X 1 . p m−i i=1 This sum is the harmonic number Hbm/2c−1 , which is Θ(log m), for sufficiently large m, and hence E Y \|U > m 2 m log m . >Ω p By the law of total expectation it follows that E[Y ] > E Y \| U > m 2 Pr U > m 2 . Finally, a standard probability calculation shows that U > m 2 happens with large probability, assuming that c p > m for a sufficiently large constant c > 0. The time bound follows since Alice can submit 2m guesses per round. 11 3.4 Lower Bounds for Information Dissemination In this section we show three different lower bounds. Together, these show what properties cause poor performance in information dissemination protocols: in some graphs, high degree is the cause of poor performance (Theorem 9); in other graphs, poor connectivity is the cause of poor performance (Theorem 10). And finally, we give a family of graphs where we can see the trade-off between D, ∆, and φ∗ (Theorem 13). We begin with a result showing that Ω(∆) is a lower bound: Theorem 9. For any ∆ ∈ (Θ(1), dn/2e), there is an n-node network that has a weighted diameter of O(log n), and a maximum node degree Θ(∆), where any algorithm requires Ω(∆) rounds to solve local broadcast with constant probability. Proof. Consider the network H of n nodes that consists of the guessing game gadget Gsym (2∆, 1, ∆, P ), where predicate P returns an arbitrary singleton target set, combined with a constant degree regular expander [20] of n − 2∆ vertices (if any) of which any one node is connected to all the vertices on the left side of the gadget; all the edges of, and connected to the expander have latency 1 and the latencies of the edges in the gadget are assigned as in Lemma 6. Clearly, the weighted diameter of H is O(log n) (diameter of the expander [20]). By Lemma7, we know that any guessing game protocol on Guessing(2∆, \|T \| = 1) requires Ω(∆) rounds for the predicate that returns exactly 1 pair as the target set. Lemma 6 tells us that any gossip algorithm that solves local broadcast in H, must require Ω(∆) rounds. We next show that every local broadcast algorithm requires time at least Ω(1/φ∗ + `∗ ). Note that, we get this Ω(1/φ∗ ) lower bound just for local broadcast and not information dissemination, which is in contrast to the results in the unweighted case. The following result is given in terms of the weight-` conductance, for any `, and thus also holds for φ∗ and `∗ . In the proof, we construct a network that corresponds to the bipartite guessing game graph with a target set where each edge is fast with probability φ∗ . That way, we obtain a network with critical conductance Θ(φ∗ ), hop diameter O(1), and a weighted diameter of O(`∗ ). The guessing game lower bound of Lemma 8 tells us that the cost of information dissemination still depends on φ∗ . Theorem 10. For any ` ∈ [1, n] and φ` where Ω(log(n)/n) 6 φ` 6 1/2, there is a network of 2n nodes that has a weighted diameter O(`) (w.h.p.), and critical conductance Θ(φ` ) (w.h.p.), such that any gossip algorithm requires Ω((1/φ` ) + `) rounds for solving local broadcast in expectation. Also, solving local broadcast using push-pull requires Ω((log n/φ` ) + `) rounds in expectation. Proof. Our goal is to reduce the game Guessing(2n, Randomφ` ) to local broadcast, hence we consider the 2n-node graph G(2n, `, n2 , Random φ` ) as our guessing game gadget defined in Section 3.2. Since we want to show the time bound of t = Ω log n φ` + ` rounds (for push-pull), for the high latency edges we can use n the value n2 > log φ` + ` (as Ω(log(n)/n) 6 φ` and ` 6 n). We assign each cross edge latency ` independently with probability φ` and latency n2 with probability 1 − φ` . The fast cross edges have the same distribution as the target set implied by the predicate Randomφ` , which we have used to show a lower bound of Ω( φ1` ) for general protocols on Guessing(2n, Randomφ` ) in n Lemma 8, and also a stronger lower bound of Ω( log φ` ) for “random guessing” protocols, which choose a random edge for each vertex as their guesses. It is straightforward to see that push-pull gossip corresponds exactly to this random guessing game strategy. Applying Lemma 6, this means that local broadcast requires n in expectation Ω( φ1` ) time for general algorithms and Ω( log φ` ) time for push-pull. The additional term of Ω(`) in the theorem statement is required to actually send the broadcast over the latency ` edge once it is discovered. 12 Since each edge of L × R is assigned latency ` with probability φ` = Ω(log(n)/n), it follows that each u ∈ R is connected by a latency ` edge to some node in L with high probability. Hence, the weighted diameter of G(2n, `, n2 , Randomφ` ) is O(`) with high probability. In the remainder of the proof, we show that G(2n, `, n2 , Randomφ` ) has a conductance of Θ(φ` ) with high probability. We point out that several previous works prove bounds on the network expansion (e.g., [27] and [1]). However, as these results were shown for random graphs, we cannot employ these results directly and thus need to adapt these proof techniques to show a conductance of Θ(φ` ) for our guessing game gadget. We assume that there is an integer-valued function f = f (n), such that nf = φ` , noting that this assumption does not change the asymptotic behavior of our bounds. For readability, we only consider ` = 1 and note that the extension to the general case is straightforward. By construction, G(2n, 1, n2 , Randomf /n ) consists of edges with latencies 1 or n2 and we have 1 φ1 φn 2 6 2 6 , n2 n 1 where the last inequality follows from the assumption φ1 > Ω logn n . Thus, we know that φ∗ = φ1 and hence we need to prove φ1 = Θ(f /n). Consider a set S ⊆ L ∪ R of at most n vertices and let l = \|S ∩ L\| and r = \|S ∩ R\|. We first assume that l > r, since the number of latency 1 cross edges is symmetric for vertices in L and R; subsequently, we will remove this assumption by a union bound argument. For vertex sets A and B, let E1 (A, B) be the set of the (randomly sampled) latency 1 edges in the cut (A, B) and define e1 (A, B) = \|E1 (A, B)\|. Given the set S, our goal is to show that many latency 1 edges originating in S ∩ L have their other endpoint in R \ S, assuming that there are sufficiently many latency 1 cross edges to begin with. In other words, we need to bound from above the probability of the event e1 (S ∩ L, S ∩ R) > Ω(f l) conditioned that there are sufficiently many latency 1 cross edges. Claim 11 (Sufficiently many latency 1 cross edges). There exist constants c, c0 > 0, such that events LR := {∀S, \|S\| 6 n : (e1 (S ∩ L, R) > cf l) ∧ (e1 (S ∩ R, L) > cf r)}, − 0 0 LR := {∀S, \|S\| 6 n : (e1 (S ∩ L, R) 6 c f l) ∧ (e1 (S ∩ R, L) 6 c f r)} (5) (6) occur with high probability. Proof. According to the construction of G(2n, 1, n2 , Randomf /n ), the latency 1 cross edges are chosen independently each with probability f /n. Note that each cross edges is assigned latency 1 independently with probability f /n = Ω( logn n ). Thus, for each node v, the expected number of cross edges is f = Ω(log n) and, by a standard Chernoff bound, we know that the number of latency 1 cross edges to v is in [c1 f, c2 f ] with high probability, for suitable constants c2 > c1 > 0. After taking a union bound over all nodes in V (G), we can conclude that the claim holds for any set S ⊆ V (G). Conditioning on LR is equivalent with choosing a subset of (at least) cf l edges among all possible edges in the cut E1 (S ∩ L, R) uniformly at random and assigning them latency 1. Consider an edge (v, u) ∈ E1 (S ∩ L, R). It follows that u ∈ S ∩ R (and hence (v, u) ∈ E1 (S ∩ L, S ∩ R)), with probability r n and we need to exclude the event Bad(S) := {e1 (S ∩ L, S ∩ R) > 45 cf l}, cf l subsets of latency 1 edges incident to vertices in S ∩ L. In addition, we need to bound the probability that Bad(S) happens, for S chosen in any of the nl ways of choosing S that satisfy \|S ∩ L\| = l. for all 4 cf l 5 13 Claim 12. Pr ∃S : Bad(S) LR 6 n−Ω(1) . Proof of claim. Combining the above observations, we get n cf l r 45 cf l . Pr ∃S : Bad(S) LR 6 4 n l 5 cf l (7) First, we assume that r and l are both large, i.e., l > r > c0 n, for a sufficiently small positive constant k m·H2 ( m ) 4 c0 < 5e . Then, we can apply Stirling’s approximation of the form m , where H2 (x) = k ≈ 2 −x log2 (x) − (1 − x) log(1 − x) is the binary entropy function. Thus, for sufficiently large n, we get r 4 cf l 5 n·H2 ( nl )+cf lH2 ( 45 ) Pr ∃S : Bad(S) LR 6 2 · n n+cf l·H2 ( 45 )− 45 cf l , 62 (8) where, to derive the second inequality, we have used the facts that H2 ( nl ) 6 1 and nr 6 12 , since r + l 6 n and r 6 l. By the premise of the theorem, f = Ω(log n), which implies c f l = Ω(n log n). Together with the fact that H2 ( 45 ) < 54 , this means that the term (− 45 c f l) dominates in the exponent of (8) and hence Pr ∃S : Bad(S) LR 6 2−Θ(n log n) . em k Next, we consider the case where r 6 l < c0 n. Applying the upper bound of the form m to (7), k k 6 tells us that 4 en l 5e r 5 cf l Pr ∃S : Bad(S) LR 6 l 4n en l 5c0 e 45 cf l 6 , l 4 since r < c0 n. We get 4 5 0 Pr ∃S : Bad(S) LR 6 exp l 1 + log n − log l + cf l log c e 5 4 4 5 6 exp l 1 + log n + cf l log c0 e . 5 4 4 By assumption, c0 < 5e and hence the term 45 c f l log 45 c0 e in the exponent is negative. Moreover, recall that f = Ω(log n) and thus we can assume that 45 c f l log 54 c0 e 6 −c00 log n, for a sufficiently large constant c00 > 0. This term dominates the other terms in the exponent, thereby completing the proof of the claim. cf \|S\| Considering that l > \|S\| 2 , the above bound implies that at least b 10 c latency 1 edges incident to S are connected to nodes outside in S, with probability at least 1 − n−Ω(1) . Taking a union bound over all possible choices for the values of l and r adhering to r 6 l and r + l 6 n = \|V (G)\| 2 , shows that h i Pr ∀S, \|S\| 6 n : e1 (S ∩ L, R \ S) > cf10\|S\| LR > 1 − n−Ω(1) . Observe that the latency 1 cross edges are constructed symmetrically for the left and right side of the bipartite graph G and thus we can apply the above argument in a similar manner for a set S where r > l, conditioned on e1 (S ∩ R, L) > cf l. Thus, we can conclude that h i Pr ∀S, \|S\| 6 n : e1 (S, V (G) \S) > cf10\|S\| LR > 1 − n−Ω(1) . (9) 14 We can remove the conditioning in (9) by virtue of Claim (11), since h i h Pr ∀S, \|S\| 6 n : e1 (S, V (G) \S) > cf10\|S\| > Pr ∀S, \|S\| 6 n : e1 (S, V (G) \S) > cf \|S\| 10 i LR Pr[LR] > 1 − n−Ω(1) . To upper bound Vol(S) for any set S, we take into account the n cross edges of each node in S. Also, if v ∈ L, then we need to account for the n − 1 incident clique edges of v, yielding Vol(S) 6 2\|S\|n. Considering the upper bound on the number of latency 1 cross edges given by (6), we have φ∗ = min φ1 (S) = min S S e1 (S, V (G) \ S) cf \|S\| > min = Ω( nf ), S 20\|S\|n Vol(S) where the inequality is true with high probability. To see that this bound istight, observe that φ∗ 6 φ1 (L). By (5) and (6), we know that e1 (L, R) = Θ(f n) and hence φ1 (L) = Θ nf with high probability, as required. This completes the proof of Theorem 10. Finally, we give a family of graphs that illustrate the trade-off among the parameters. Intuitively, when the edge latencies are larger, it makes sense to search for the best possible path and the lower bound is Ω(D + ∆); when the edge latencies are smaller, then we can simply rely on connectivity and the lower bound is Ω(`/φ` ). Note that, we can individually obtain a lower bound of Ω((`/φ) log n), using the technique in [7] where we show that there exists a graph with diameter (`/φ) log n. Unlike here, that lower bound is simply D. Theorem 13. For a given α ∈ [Ω(1/n), O(1)] and any integer ` ∈ [1, O(n2 α2 )], there is a class of networks of 2n nodes, critical conductance φ∗ = φ` = Θ(α), maximum degree ∆ = Θ(αn), and weighted diameter D = Θ(1/φ` ), such that any gossip algorithm that solves broadcast with at least constant probability, requires Ω(min{∆ + D, `/φ` }) rounds. Proof. We create a network G consisting of a series of k node layers V1 , . . . , Vk that are wired together asa q 2 8 ring, using the guessing game gadgets introduced above. We define k = cα where c = 34 + 41 9 − nα . This implies that 1 6 c < 3/2 as α ∈ [Ω(1/n), O(1)]. Each layer consists of s = cnα nodes. As it does not change our asymptotic bounds, we simplify the notation by assuming that 2/cα and cnα are integers. V(k/2)+1 Vk/2 V3k/4 V(k/4)+1 V(3k/4)+1 Vk/4 Vk V1 Figure 2: Guessing Game Gadgets wired together as a ring. For each pair Vi and V(i+1) mod k (0 6 i 6 k − 1), we construct the symmetric guessing game gadget Gsym (2cnα, 1, `, P ) (in Section 3.2), for simulating a gossip algorithm to solve the game Guessing(2cnα, \|T \| = 15 1). That is, we create a complete bipartite graph on Vi and V(i+1) mod k and form cliques on Vi and V(i+1) mod k (see Figure 2). We assign latency ` to every cross edge between Vi and V(i+1) mod k , except for a uniformly at random chosen edge that forms the singleton target set, which we assign latency 1. Observe that the weight-j conductance φj cannot be maximal for any j other than 1 or `. Observation 14. Let s = cnα. Graph G is (3s − 1)-regular. Proof. For a layer Vi , we call V(i−1) mod k the predecessor layer and V(i+1) mod k the successor layer. The size of a layer is s = cnα. Each node has 2s edges to its neighbors in the predecessor resp. successor layer and s − 1 edges to nodes in its own layer. This means that G is a (3s − 1)-regular graph. We define a cut C that divides the ring into two equal halves such that none of the internal clique edges are cut edges. By a slight abuse of notation, we also use C to denote the set of vertices present in the smaller side of the partition created by the cut C (ties broken arbitrarily). Lemma 15. φ` (C) = α. Proof. Since C partitions G into two sets of identical size, the volume can be determined by considering either partition of size n, thus we focus on the node set C. Also, by Observation 14 we know that G is (3s − 1)-regular. The volume of C can be calculated to be n(3cnα − 1). The number of cut edges of latency 6 ` is 2(cnα)2 (by the construction of C). According to Definition 1, the `-weight conductance is given by 2(cnα)2 φ` (C) = n(3cnα−1) . By plugging in the value of c, we can verify that φ` (C) is exactly equal to α. Using the conductance bound of Lemma 15 for cut C, we know that φ` 6 α. In the proof of the next lemma, we show that φ` = Ω(α). Lemma 16. The weight-` conductance of the constructed ring network is φ` = Θ(α). Proof. By Lemma 15, we know that φ` 6 α as the actual graph conductance is always 6 to any cut conductance. We will now show φ` = Ω(α) as well. By Observation 14 we know that G is (3s − 1)-regular and therefore for a set of nodes U the volume Vol(U ) is exactly equal to (3s − 1)\|U \|. This clearly implies that for any two sets U and V , Vol(U ) 6 Vol(V ) if and only if \|U \| 6 \|V \|. Now, consider an arbitrary cut (U, V (G) \ U ) of G and suppose that U contains at most half of the nodes of G, i.e., \|U \| 6 n, since G has 2n nodes. If there are at least Θ(s2 ) cut edges, then, using the fact that \|U \| 6 n, we get φ` (U ) > Θ(s2 /s\|U \|) = Θ(s/\|U \|) > Θ(s/n) > Θ(α), and we are done. In the remainder of the proof, we will show that there are Θ(s2 ) cut edges. We distinguish two cases: 1. \|U \| > 3s/4: We classify each node in U either as good if it has at least s/4 adjacent edges across the cut (U, V \ U ) and as bad otherwise. Thus, our goal is to identify Θ(s) good nodes, which in turn implies Θ(s2 ) cut edges. Let S be an arbitrary subset of 3s/4 nodes in U . If all nodes in S are good, we are done. Otherwise, let x ∈ S be a bad node. It is important to note that the following properties are true for every bad node: (a) Node x is in a layer in G which contains at least 3s/4 nodes inside U . (b) The successor layer from x has at least 3s/4 nodes inside U . To see why (a) holds, assume that it was not true. Then, x would have at least s/4 neighbors in its own layer across the cut, contradicting the assumption that x is bad. Similarly, if (b) was false, x would be connected to at least s/4 nodes in the successor layer outside U . (This is true of the predecessor layer too.) Let A be the successor layer to the layer containing x. We now run the following procedure: 16 Invariant: A contains at least 3s/4 nodes in U . If at least half of the nodes in A are good, we are done. Terminate and claim Θ(s2 ) cut edges. Otherwise, let y be a bad node in A. Let A0 be the successor layer of the layer A. Then, start again at Step (1) with A = A0 and y = x. From the assertion (b), A0 contains at least 3s/4 nodes in S. If this procedure ever terminates in Step (2), we are done. Otherwise, it continues around until every layer has been explored. In that case, the invariant implies that every layer contains at least 3s/4 nodes in U . This implies that > 1/2 of the nodes of G are in U , which contradicts the choice of U . Thus, the procedure does terminate, which means there must be at least Θ(s2 ) cut edges, implying φ` > α. 2. \|U \| < 3s/4: Let m be the number of nodes in U . Since G is (3s − 1)-regular, the volume of U is m(3s − 1). Each node in U now contains at least s/4 neighbors outside of U (since it has > s neighbors and there are only < 3s/4 other nodes in U ), so the cut size is at least sm/4. Thus, the conductance of this graph (sm/4) φ` > m(3s−1) = Ω(1) > Θ(α). Since, φ` 6 α and φ` > Θ(α), it is clearly the case that φ` = Θ(α), which is what we wanted to prove. (1) (2) (3) (4) Combining Lemmas 15 and 16 (and again using cut C), we argue that the critical latency is `. Lemma 17. For any ` 6 O(cnα)2 , φ∗ = φ` = Θ(α). Proof. To prove that φ∗ is in fact φ` , which by Lemma 16 is Θ(α), we need to show that (φ` /`) > (φ1 /1) = φ1 . To this end, let us consider the cut C defined above. We will show that φ`` > φ1 (C) > φ1 , and since weight-j conductance φj (cf. Definition 1), cannot be maximal for any j other than 1 or `, we get φ∗ = φ` . There are two latency 1 cross edges in the cut C and the volume of C can be calculated as in the proof of Lemma 15 to be n(3cnα − 1). Thus, we need to show that φ` Θ(α) 2 = > . ` ` (3cnα − 1)n As c is a constant, the above inequality is true as long as ` = O(α2 n2 ), which is ensured by the premise of the theorem. The weighted diameter of the network D = Θ(k/2), since each pair of adjacent node layers is connected by a latency 1 edge and, internally, each layer forms a latency 1 clique. Using the fact that c ∈ [1, 32 ), it can be shown that (2/3α) < D 6 (1/α), implying that D = Θ(1/φ` ) (by lemma 16). Now, consider a source node in layer V1 that initiates the broadcast of a rumor. Each node can either spend time in finding the required fast edge (which we assume can be done in parallel) or, instead, it can instantly use an edge of latency ` to forward the rumor. Lemma 7 tells us that finding the single latency 1 cross edge with constant probability, for the guessing game gadget corresponding to any pair of node layers, requires Ω(∆) rounds, and then forwarding the rumor takes Ω(D) additional rounds. Alternatively, the algorithm can forward the rumor along latency ` edges across node layers and spread the rumor using the latency 1 edges within each clique. It follows that the required time for broadcast is Ω(min{∆ + D, `/φ` }). We obtain the following corollary that gives a lower bound on information dissemination in terms of φavg , either by a similar analysis as above, or by the application of Theorem 5. Corollary 18. For a given α ∈ [Ω(1/n), O(1)] and any integer ` ∈ [1, O(n2 α2 )], there is a class of networks of 2n nodes, average conductance φavg = Θ(α/`), maximum degree ∆ = Θ(αn), and weighted diameter D = Θ(1/`φavg ), such that any gossip algorithm that solves broadcast with at least constant probability, requires Ω(min{∆ + D, 1/φavg }) rounds. 17 Proof. Observe that in the given graph, there exists edges with latency either 1 or `, and as such the number of non-empty latency classes here is 2. Now, Theorem 5 reduces to φ∗ /2`∗ < φavg < 2φ∗ /`∗ . This implies that for this case φavg = Θ(φ∗ /`∗ ). Alternatively, φ∗ = `φavg (as in this case ` = `∗). Replacing this value of φ∗ in Theorem 13 gives us the above required corollary. 4 Algorithms for Unknown Latencies We divide the upper bounds on information dissemination into two sub-components and later combine them n ), to obtain a unified result. First, we analyze classical push-pull, showing that it completes in time O( `∗ φlog ∗ which is optimal when D + ∆ is large. Alternatively for graphs where D + ∆ is small, we give an algorithm wherein each node first spends Õ(D + ∆) time discovering the neighboring latencies after which nodes use the local information to build a spanner, across which data can be distributed in Õ(D) time. 4.1 Push-Pull To show the time required for information dissemination in a weighted graph G using push-pull, we define E` as the set of all edges of latency 6 `, Eu as the set of incident edges of vertex u and Eu,` := E` ∩ Eu . n ) rounds in a network G, where Theorem 19. The push-pull protocol achieves broadcast w.h.p. in O( `∗ φlog ∗ φ∗ is the critical conductance of G and `∗ is the corresponding critical latency. Proof. We construct a strongly edge-induced graph G` , which is a generalization of the strongly (vertex) induced subgraph defined in [5] and which has the same vertex set as G. The edges of G` have a multiplicity6 defined by the edge multiplicity function µ, given by   if (u, v) ∈ E` 1 µ(u, v) = \|Eu \| − \|Eu,` \| if u = v (10)   0 otherwise It is easy to see that the (unweighted) conductance φ(G` ) corresponds to φ` (G), as a self-loop at node u is counted as µ(u, u) edges when computing the volume. We also define another unweighted graph G0 that is derived from G by dropping all edge latencies. Now, we consider the Markov chain process describing the informed node set, i.e., the vertex set that is in possession of some message m originating from a vertex v when running push-pull. Formally, the state space of the Markov chain consists of all possible informed node sets. Only paths that correspond to monotonically growing informed node sets have nonzero probability. We argue that this process on G0 (resp. G) dominates the respective process in the graph G` . We observe that each node v selects an incident edge in E` from G` in the push-pull protocol with the same probability as in G0 . The probability of choosing an edge ∈ Eu \ E` P (i.e., a self loop in case of G` ) is µ(u, u)/ v∈V µ(u, v) in both graphs. Clearly, choosing a self loop of a node u cannot help in the propagation of the message in G` , but choosing the corresponding edge in G0 might. It follows that the Markov process of reaching any informed node set S in G0 dominates over the one in G` , i.e., the probability of reaching any informed node set S by using the Markov chain in G0 is at least as large as the probability of reaching the same set S by using the Markov chain for G` . To translate this result back to our actual network G (with weighted edges), we charge each round of push-pull in G` to ` rounds in G. With similar arguments, it follows that the Markov process of the informed node set given by considering ` consecutive rounds of push-pull in G at a time, dominates the one in G` . 6 The “multiplicity of an edge” is called “edge weight” in [5]. We use a different terminology here to avoid confusion with the latencies of edges and consider “edge weight” as a synonym to edge latency instead. 18 From [16] and [5] it is known that O(log(n)/φ(G` )) rounds suffice w.h.p. to solve broadcast in G` . Hence, achieving broadcast in G requires O(` log(n)/φ` (G)) rounds. Since the above analysis applies for any ` > 1, and in particular for the critical latency `∗ , the theorem follows. We combine Theorem 19 with Theorem 5 to obtain the following corollary that gives the upper bound on information dissemination using push-pull in terms of φavg . n Corollary 20. The push-pull protocol achieves broadcast w.h.p. in O( Lφlog ) rounds in a network G, where avg φavg is the average conductance of G and L is the number of non-empty latency classes in G. 4.2 An Õ(D + ∆) Algorithm In Section 5.1 we provide an algorithm that solves all-to-all information dissemination when each node knows the latencies of all its adjacent edges. The same algorithm can be naturally extended for the case where nodes do not know the adjacent latencies by first discovering the edge latencies and then running the algorithm as such. When both D and ∆ are known: for ∆ rounds, each node broadcasts a request to each neighbor (sequentially) and then waits up to D rounds for a response to determine the adjacent edge’s latency. If both or either values are unknown, the guess and double strategy (described in Section 5.3) can be used, as we can efficiently detect when information dissemination has completed correctly. By similar arguments as in Section 5.3 we obtain an algorithm that solves information dissemination in O((D + ∆) log3 n) time. 5 Algorithms for Known Latencies In this section, we discuss the case where each node knows the latencies of the adjacent edges. We focus on the problem of all-to-all information dissemination (instead of one-to-all information dissemination), as it will simplify certain issues to solve the seemingly harder problem. (Of course, all-to-all information dissemination also solves one-to-all information dissemination. And most one-to-all information dissemination algorithms can be used to solve all-to-all information dissemination by using them to collect and disseminate data.) In Section 5.1, we use the fact that nodes know a polynomial upper bound on the network size (and this is the only place where we rely on that assumption). When edge latencies are known, the spanner algorithm (described below) solves all-to-all information dissemination in O(D log3 n) which differs from the trivial lower bound of Ω(D) by only polylog factors. 5.1 Spanner Algorithm Preliminaries We initially assume that the weighted diameter (D) is known to all nodes; later (in Section 5.3), we do away with the assumption via a guess-and-double technique. It is assumed w.l.o.g. that every edge has latency 6 D: clearly we do not want to use any edges with latency > D. Local broadcast. An important building block of our algorithms is local broadcast. For unweighted graphs, the (randomized) Superstep algorithm by Censor-Hillel et al. [5] and the Deterministic Tree Gossip (DTG) algorithm by Haeupler [18] solve this problem. We make use of the DTG algorithm, which runs in O(log2 n) rounds on unweighted graphs. See [18] and Appendix A.1 for details. Observe that for the unweighted case, if any algorithm solves local broadcast in O(t) rounds, it obtains a t-spanner as a direct consequence, which thereafter can be used for propagating information. However, for graphs with latencies, just solving local broadcast might take O(D) time, resulting in a O(D)-spanner (and leading to an O(D2 ) solution for information dissemination). Recall that a subgraph S = (V, E 0 ) of a graph G = (V, E) is called an α-spanner if any two nodes u, v with distance ` in G have distance at most α` in S. 19 For weighted graphs, we are mainly interested in the `-local broadcast problem in which each node disseminates some information to all its neighbors that are connected to it by edges of latency 6 `. While DTG assumes edges to be unweighted (uniform weight), we can execute the same protocol in a graph with non-uniform latencies simply by ignoring all edges with a latency larger than ` and simulating 1 round of the DTG protocol as ` rounds in our network. We refer to this protocol as the `-DTG protocol. It follows immediately that within O(` log2 n) time, the `-DTG protocol ensures that each node has disseminated the information to all its neighbors connected to it with edges of latency 6 `. Note that we can trivially solve the all-to-all information dissemination problem in O(D2 log2 n) time using `-DTG protocol (if D were known) by simply repeating it D times with ` = D. The challenge now, given the restriction that finding neighbors by a direct edge might be costly, is to somehow find sufficiently short paths to all of them. We show here that with sufficient exploration of the local neighborhood up to O(log n) steps and using only favorable weights, we are able to obtain a global spanner. An intermediate goal of our algorithm is to construct an O(log n)-spanner and to obtain an orientation of the edges such that each node has a small, i.e., O(log n), out-degree.7 Once we have such a structure, we achieve all-to-all information dissemination by using a flooding algorithm that repeatedly activates the out-edges in round-robin order. 5.2 Spanner Construction and Broadcast In a seminal work, Baswana and Sen [2] provide a spanner construction algorithm for weighted graphs (where weights did not correspond to latency) in the LOCAL model of communication. As our goal here is to find a low stretch, low out-degree spanner, we modify the algorithm of [2] by carefully associating a direction with every edge that is added to a spanner such that each node has w.h.p. O(log n) out-degree. To deal with latencies, we choose to locally simulate the algorithm on individual nodes after obtaining the log n-hop neighborhood information by using the `-DTG protocol. We show that this log n-hop neighborhood information is sufficient for obtaining the required spanner. The algorithm in [2] also assumes distinct edge weights. We can ensure this by using the unique node IDs to break ties. We first show that the size of the obtained spanner does not increase significantly when running the algorithm of [2] with an estimate of n (namely n̂). 5.2.1 Spanner Construction Algorithm Each node v executes a set of rules for adding edges (explained below) and each time one of these rules is triggered, v adds some of its incident edges to the spanner while assigning them as outgoing direction. This way, we obtain a low stretch spanner (undirected stretch) where nodes also have a low out-degree, which we leverage in the subsequent phases of our algorithm. For a given parameter k, the algorithm computes a (2k − 1)-spanner by performing k iterations. At the beginning of the i-th iteration, for 1 6 i 6 k − 1, every node that was a cluster center in the previous iteration, chooses to become an active cluster with probability n̂−1/k , for some n 6 n̂ 6 poly(n); note that for i = 1, every node counts as a previously active center. Then, every active center c broadcasts this information to all cluster members. As a cluster grows by at most 1 hop in each round, this message needs to be disseminated throughout the i-neighborhood of c.8 Then, every cluster member broadcasts its membership information to all its neighbors to ensure that every node is aware of its adjacent active clusters. For adding edges to the spanner, nodes also remember its set of incident clusters Ci−1 that were active in iteration i − 1. With this information in hand, every node u adds some of its incident edges to its set of spanner edges Hu , and also (permanently) discards some edges, as follows: 7 8 It is clearly impossible to guarantee small degree in an undirected sense, for example, if the original graph is a star. By slight abuse of notation, we use c to denote cluster centres and the cluster itself when the distinction is clear from the context. 20 (Rule 1) If none of u’s adjacent clusters in Ci−1 were sampled in iteration i, then u adds its least weight edge to cluster c as an outgoing edge to Hu and discards all other edges to nodes in c, for every c ∈ Ci−1 . (Rule 2) If u has active adjacent clusters, then u will add the edge ev to some cluster c with the minimum weight among all these clusters and, for each adjacent cluster c0 ∈ Ci−1 that has a weight less than ev , node u also adds one outgoing edge to the respective node in c0 . All other edges from v to nodes in clusters c and c0 are discarded. In the k-th iteration, every vertex v adds the least weight edge to each adjacent cluster in Ck−1 to Hv . Lemma 21. Consider a synchronous network of n nodes where nodes know only n̂, where n 6 n̂ 6 nc , for some constant c > 1. For any k > c, there’s a distributed algorithm (based on [2]) that computes a spanner and terminates in O(k) rounds in the LOCAL model and each node’s out-degree is O(nc/k log n) w.h.p. Proof. Note that the running time of the algorithm is O(k 2 ) rounds if used with a restricted message size of O(log n). Inspecting the algorithm reveals that the computation at each node only depends on its k-hop neighborhood in the graph. Also, because the decision to remove an edge (u, v) can be taken by either node u or v, each node needs to simulate the running of the algorithm at all its neighbors (to know when to remove the edge (u, v) from consideration) and hence we can simulate the execution of the algorithm locally by first collecting this information regarding (k + 1)-hop neighborhood in k + 1 rounds in the LOCAL model. We now analyse the difference when running the algorithm with n̂ instead of n. First, we observe that sampling clusters with probability n̂(−1/k) does not affect the stretch guarantee. For the sake of our analysis we assume that the spanner is directed: we count every incident edge of v that it adds to its set of spanner edges Hv as an outgoing edge of v. The degree bound will follow by showing an upper bound on the number of outgoing edges of each node. Consider any iteration i in Phase 1 of the algorithm, i.e., 1 6 i < k. We call a cluster sampled in iteration i if it is among the sampled clusters in all iterations 1, . . . , i. Every cluster that was sampled in the previous iteration is sampled again with probability n̂−1/k . (In the very first iteration, every node counts as a previously sampled cluster.) To bound the number of edges that contribute to the out-degree of a node v, we consider the clusters adjacent to v that were sampled in iteration i − 1 and order them as c1 , . . . , cq in increasing order of the weight of their least weight edge incident to v. Let Ai be the event that v adds at least l edges to its outdegree in iteration i. Note that Ai occurs if and only if (1) none of the clusters c1 , . . . cl is sampled in iteration i and (2) there are at least l active clusters in iteration i − 1. By the description of Phase 1 (first k − 1 iterations) of the algorithm, we only add an edge from v to a node in cluster cj in iteration i if Ai does not happen. We have Pr[Ai ] 6 (1−n−c/k )l and taking a union bound over the first k − 1 iterations and over all n nodes, it follows that the probability of any node adding more than l edges to the spanner in any of the first k − 1 iterations is at most exp(−n−c/k l + log k + log n). By choosing l > Ω(n1/k (log n + log k)), this probability is 6 n−Ω(1) as required. In Phase 2 (final iteration), every vertex u adds a least weight (outgoing) edge to every cluster that was sampled in iteration k − 1. Let Xv be the indicator random variable that vertex v is the center of a cluster sampled in iteration k − 1 that is incident to u. We have Pr[Xv ] 6 n− Setting X = P v:(u,v)∈G Xv , c(k−1) k c = n−c+ k . it follows that c c E[X] 6 n1−c+ k 6 n k , since c > 1. Since each cluster is sampled independently all Xv are independent, we can apply a standard Chernoff bound to show that, for some sufficiently large constant c1 depending on c, it holds that h i c c/k Pr X > c1 n k log n 6 e−Θ(n log n) 6 n−Ω(1) . 21 By taking a union bound over all vertices, we can see the number of edges that each vertex adds to the spanner c in Phase 2 is at most O(n k log n) with high probability. Combining this with the bound that we have derived for Phase 1 completes the proof. Theorem 22. There is an O(D log3 n) time algorithm A in the gossip model that yields an O(log n)-spanner that has O(n log n) edges (w.h.p.). Moreover, A also computes an orientation of the edges that guarantees that each node has an out-degree of O(log n) (w.h.p.). Proof. To convert the classic synchronous algorithm for the local model assumed in Lemma 21 to an algorithm that works in the gossip model with latencies, we use the `-DTG protocol and simulate each of the k = log n iterations of the spanner algorithm by first discovering the log n-hop neighborhood. The neighborhood discovery takes O(D log3 n) rounds in our model and then all computations are done locally. To broadcast on this directed spanner we use the RR broadcast algorithm, which is a deterministic round-robin-style exchange of information among nodes. Each node sends all the rumors known to it to all its 1-hop neighbors one by one in a round robin fashion. The algorithm with a parameter k is run on the directed spanner of the graph Gk (G without edges of latency > k). RR Broadcast (k) 1: for each vertex v in parallel do 2: for iteration i equals 1 to (k∆out + k) do 3: propagate rumor set Rv along the out-edges of length 6 k one-by-one in a round robin fashion 4: add all received rumors to Rv Algorithm 1: RR Broadcast u ∆-1 edges k1 u1 ∆-2 edges u2 k2 ∆-2 edges kh v ∆-1 edges Figure 3: An example of message propagation from node u to node v. Lemma 23. After the execution of RR Broadcast algorithm with a parameter k on the directed spanner of graph Gk , any two nodes u and v at a distance 6 k in G have exchanged rumors with one another in O(k∆out + k) rounds, where ∆out is the maximum out-degree of any node in Gk . Proof. Consider a path from a node u to another node v at a distance k or less from it. Clearly, all edges in this path would have a weight of 6 k. Therefore, we can work on Gk (G without edges of latency > k) as well without affecting the correctness of the algorithm. Also, let us assume that the number of hops between u and v to be h which again would be 6 k, since there are no fractional weights. Let the latency between each hop be denoted by ki as shown in Figure 3. Messages reach the next node when either of the nodes initiate a bidirectional exchange. For example, u’s rumor could reach node u1 either by a request initiated by node u or by u1 , depending upon the direction of the edge uu1 . In the worst case nodes have to try all other ∆out − 1 links before initiating a connection along the required edge where ∆out is the maximum out-degree of any node. After a connection is initialized it takes k1 time to exchange rumors. By generalization, we 22 observe that in the non-blocking model, the delay that can be incurred before rumor exchange among any two adjacent nodes ui and ui−1 can be ∆out + ki in the worst case. In this way u’s rumor proceeds towards v in individual steps, each step incurring a maximum cost of ∆out + ki . A node might receive multiple rumors to propagate in the next round, which its adds to its rumor set and forwards to its neighbors in a round robin fashion. As such, the total worst case delay in rumor exchange among node u and v would be represented by h X (∆out + ki ) = h∆out + i=1 h X ki . i=1 Ph But we know that both h and i=1 ki can have a maximum value equal to k . Therefore, we conclude that for any two nodes v and u in Gk , v’s rumor would have reached u and u’s rumor would have reached v if all nodes forward rumors in a round robin fashion for (k∆out + k) rounds. Here, on the created spanner with stretch of O(log n), the maximum distance between any two nodes can be O(D log n). Since the maximum out-degree (∆out ) is O(log n) w.h.p., we get the following corollary. Corollary 24. The RR broadcast algorithm on the constructed spanner takes O(D log2 n) time and solves all-to-all information dissemination w.h.p. We combine all the previously defined techniques to a single algorithm called Efficient Information Dissemination or EID. EID (D) 1: for each vertex v in parallel do 2: for iteration i = 1 to O(log n) do 3: Perform D-DTG 4: call Spanner Construction algorithm 5: call algorithm RR Broadcast (O(D log n)) /* to gain neighborhood information / / executed locally */ Algorithm 2: Efficient information dissemination Lemma 25. For a graph G with diameter D, Efficient Information Dissemination (EID) algorithm takes O(D log3 n) time for solving all-to-all information dissemination w.h.p. when D is known to all the nodes. 5.3 Unknown Diameter For unknown diameter, we apply the standard guess-and-double strategy: begin with an initial guess of 1 for D. Try the algorithm and see if it succeeds. If so, we terminate. Otherwise, double the estimate and repeat. The challenge here is to correctly determine the termination condition i.e. how does a particular node determine whether information dissemination has been achieved for all other nodes. Early termination might lead to partial dissemination whereas late termination might cause the time complexity to increase. The critical observation is as follows: if two nodes u and v cannot communicate in one execution of all-to-all information dissemination (protocol RR Broadcast) for a given estimate of the diameter, then there must be some edge (w, z) on the path from u to v where, in one execution: u is able to communicate with w but not with z. There are two cases: If w is not able to communicate with z, then it is aware that it has an unreachable neighbor and can flag the issue; the next time that u and w communicate, node u learns of the problem. Otherwise, if w can communicate with z, then the next time that u and w communicate, node u learns that there was a node it did not hear from previously. In either case, u knows that the estimate of D was not correct and should continue. Each node also checks whether it has heard from all of its neighbors, 23 and raises an error flag if not. We then repeat all-to-all broadcast so that nodes can check if everyone has the same “rumor set” and that no one has raised an error flag. In total, checking termination has asymptotic complexity of O(D log2 n). The Termination_Check algorithm checks for every node that v contacts or is contacted by (either directly or indirectly) whether that node has (i) exactly the same rumor set as v and (ii) the value 0 as its flag bit. The flag bit of a node is set to 1 if a neighbor of that node is not present in its rumor set or if the node has not yet exchanged all the rumors known to it presently with all of its neighbors in G that are at a distance 6 to the current estimate of D (say k): this condition is easily checked by either doing an additional k-DTG (which does not affect the complexity) or can be checked in parallel with the execution of RR Broadcast. If both of the above conditions are not met, then node v sets its status to “failed” and v uses a broadcast algorithm for propagating the “failed” message. Any broadcast algorithm that, given a parameter k, is able to broadcast and collect back information from all nodes at a distance 6 k from v, can be used. It is easily seen that RR Broadcast satisfies this criteria and can be used in this case. Note that broadcast is achieved here (for General_EID algorithm) by execution of RR Broadcast, however when Path_Discovery algorithm (described later) invokes Termination_Check, broadcast is achieved by execution of the sequence T (k) (also described later). Here, the rumor set known to a particular vertex v is denoted by Rv , Γ(v) represents all its neighbors in G whereas k-neighbors refers to only those nodes that are connected with v with an edge of latency k or less. Also, initially node_status of all nodes is set to “default”. Termination_Check (k) 1: if (node w ∈ Γ(v) and w ∈ / Rv ) or (node v has not exchanged rumors with all k-neighbors) then 2: set flag bit, vf lag = 1 3: else set flag bit, vf lag = 0 4: 5: 6: 7: 8: 9: broadcast and gather all responses from any node u in v’s k-distance neighborhood if ∃ any u such that (Rv 6= Ru ) or (uf lag = 1) then set node_status = “failed” broadcast “failed” message to the k-distance neighborhood if received message = “failed” then set node_status = “failed” Algorithm 3: Termination_Check We prove the following regarding the termination detection: Lemma 26. No node terminates until it has exchanged rumors with all other nodes. Moreover, all nodes terminate in the exact same round. Proof. Suppose that a node v terminates without having exchanged rumors with some other node w. Considering any path from node v to node w, let u be the farthest node (in hop distance) with which v has exchanged rumors with and let x be the next node in the path. Case 1 : u has exchanged rumors with x. It implies that v has also exchanged rumors with x, from the condition that all nodes that exchange rumors with one another have the same rumor set. Thus, contradicting the fact that u is the farthest node on the path that v has exchanged rumors with. Case 2 : u has not exchanged rumors with x. If u had not exchanged rumors with x, then u would have set its flag bit as 1, which would have been detected by v during the broadcast and it would not have terminated. This also gives us a contradiction. Thus, no such node w exists and v terminates only after it has exchanged rumors with all the other nodes. For the second part of the proof, let consider u and v to be nodes such that v is set for termination and has not set its status to “failed” in the Termination_Check algorithm, whereas, in the same iteration, node u 24 has set its status to “failed” and hence is set to continue. We show that there cannot be two such nodes in the same round. The node v did not set its status to “failed” implying all the nodes that it exchanged rumors with had exactly the same set of rumors, none of the nodes had set its flag bit as 1 and in addition it did not receive a “failed” message from any other node. From the first part, we know that the set of nodes that v exchanged rumors with is the entire vertex set of the graph G. That implies, v has also exchanged rumors with u: node u also has the exact set of rumors (which essentially is all the rumors from all the nodes) and does not have a set flag bit. So in the current iteration, if any other node broadcasted a “failed” message both v and u would have received it resulting in both nodes to set their status as “failed”. Again, since the rumor sets of both nodes are identical, both nodes would observe the same flag bits of all the nodes. Then node u will also not satisfy the termination condition and will not set its status as “failed”. This gives us a contradiction that completes the proof. General_EID (k) k=1 2: repeat 3: call algorithm EID (k) 4: call algorithm Termination_Check (k) 5: if node_status = “failed” then 6: k = 2k 7: set node_status to “default” 8: else terminate 1: Algorithm 4: General_EID; code for vertex v. Combining the all-to-all dissemination protocol with the termination detection, we get the following: Theorem 27. There exists a randomized gossip algorithm that solves the all-to-all information dissemination problem w.h.p. and terminates in O(D log3 n) rounds. 5.4 An Alternative All-to-All Information Dissemination Algorithm We propose an alternate algorithm to solve all-to-all information dissemination without any global knowledge (polynomial upper bound of n need not be known) that takes O(D log2 n log D) time. This algorithm works even when nodes cannot initiate a new exchange in every round, and wait till the acknowledgement of the previous message, i.e., communication is blocking. The algorithm involves repeatedly invoking the `-DTG algorithm with different parameters determined by a particular pattern. The intuition behind the choice of the pattern is to make minimal use of the heavier latency edges by collecting as much information as possible near the heavier latencies before making use of that edge. The pattern for k is derived according to a sequence T (k) that is recursively defined as follows: T (1) = 1-DTG T (2) = T (1) · 2-DTG · T (1) T (4) = T (2) · 4-DTG · T (2) .. . T (k) = T (k/2) · k-DTG · T (k/2) We show that, when the above sequence is run for the particular pattern for length k, it guarantees that any node u and v in the graph G, at a distance of 6 k, have exchanged their rumors with one another. Overall, 25 the pattern of values of the parameter ` is 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, . . . , k, . . . , 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, and, for each value `, we perform the `-DTG protocol. That is, T (k) is a sequence of calls to `-DTG with varying parameters according to a known pattern. Lemma 28. After the execution of T (k), any node in the weighted graph G (V,E) has exchanged rumors with all other nodes that are at distance k or less from it. Proof. We proceed by induction over the path length k. For the base case, recall from [18] that, after running T (1) on G1 , i.e., the subgraph of G induced by edges with latency 6 1, any node v has exchanged rumors with all its distance 1 neighbors. For the inductive step, suppose that the claim is true for T (k), i.e. after running the sequence, any node v has exchanged rumors with all other nodes at a weighted distance 6 k. To prove the claim for T (2k) (i.e. T (k) · 2k-DTG · T (k)), we consider the various possibilities of forming a path of length 2k. Case 1: The path consists only of edges with latencies 6 k. Here we distinguish two sub-cases: Case 1a: There exists a node m which is equidistant from both end points u and v (see Figure 4). By the induction hypothesis, both nodes u and v would have exchanged rumors with node m in the initial T (k). In the next T (k), node m propagates all rumors that it received from u to v and vice-versa. path of length k path of length k u m v Figure 4: Case 1a Case 1b: No such node middle exists as depicted in Figure 5. Then, after the initial T (k), node u must have exchanged rumors with m1 and node v with m2 , due to the induction hypothesis. In the invocation of the 2k-DTG, node m1 propagates all rumors gained from u to m2 , and m2 also propagates all rumors gained from v to m1 . This information then travels from m1 to u and from m2 to v in the final T (k). path of length k or less u edge of length k-1 or less m1 path of length k or less m2 v Figure 5: Case 1b Case 2: There exists at most one edge e with latency value in between [k + 1, 2k]. This situation can yield one of the following two sub-cases: Case 2a: Edge e is located at one end of the path (see Figure 6). By the induction hypothesis, node v would have exchanged rumors with m in the initial T (k). In the 2k-DTG, u gets to know this (and other) rumors from m and m also gets to know u’s rumors. In the next T (k), node m propagates all rumors gained from u to v. path of length k-1 or less edge of length [k +1, 2k] u m Figure 6: Case 2a 26 v Case 2b: The edge is located between two inner nodes on the path (see Figure 7). In this case, by the induction hypothesis, node u has exchanged rumors with m1 , whereas node v has exchanged rumors with node m2 in the initial T (k). In the 2k-DTG, node m1 propagates all rumors gained from u to m2 . Moreover, m2 propagates all rumors gained from v to m1 . These rumors then propagate from m1 to u and from m2 to v in the final T (k). path of length k-1 or less u path of length k-1 or less m1 m2 v Figure 7: Case 2b Lemma 29. For known diameter, solving all-to-all information dissemination by executing the sequence T (D), takes O(D log2 n log D) time. Proof. From the way the sequence is constructed, we observe the recurrence relation T (k) = 2T (k/2) + k log2 n. Using standard methods to solve the recurrence completes the proof. When the graph diameter is known to all nodes, nodes can just invoke T (D) to solve all-to-all information dissemination. For completeness, we also present an algorithm called Path_Discovery that uses the sequence of invocations of `-DTG to solve all-to-all information dissemination, when the graph diameter is unknown. This algorithm is similar in flavour to that of the General_EID algorithm described in Section 5.3 and also makes use of the Termination_Check algorithm, albeit with a different broadcasting technique (calling T (k) rather than RR Broadcast). Path_Discovery (k) 1: k=1 2: repeat 3: execute sequence T (k) 4: call algorithm Termination_Check (k) 5: if node_status = “failed” then 6: k = 2k 7: set node_status to “default” 8: else terminate Algorithm 5: Path_Discovery; code for vertex v. Lemma 30. Path_Discovery algorithm takes O(D log2 n log D) time to solve all-to-all information dissemination. Applying techniques similar to section 5.3, the complexity can be easily shown for the case with unknown diameter as well. 6 Unified Upper Bounds Combining the results, we can run both push-pull and the spanner algorithm in parallel to obtain unified upper bounds for both the known and the unknown latencies cases. However, we point out that, for single source broadcast, push-pull works with small message sizes whereas the spanner algorithm does not (because of its 27 reliance on DTG). Also, exchanging messages with the help of the spanner does not have good robustness properties whereas push-pull is inherently quite robust. Theorem 31. There exists randomized gossip algorithms that solves the all-to-all information dissemination problem in O(min((D+∆) log3 n, (`∗ /φ∗ ) log n) time when latencies are unknown and in O(min(D log3 n, (`∗ /φ∗ ) log n)) time when latencies are known. Corollary 32. There exists randomized gossip algorithms that solves the all-to-all information dissemination problem in O(min((D + ∆) log3 n, (L/φavg ) log n) time when latencies are unknown and in O(min(D log3 n, (L/φavg ) log n)) time when latencies are known. 7 Conclusion We have presented two different new concepts, namely the critical conductance and the average conductance, that characterize the bottlenecks in communication for weighted graphs. We believe that these parameters will be useful for a variety of applications that depend on connectivity. A question that remains is whether the running time of O(D log3 n) for information dissemination can be improved, e.g., using better spanner constructions or more efficient local broadcast to save the polylogarithmic factors. (Recall that in the unweighted case, there are information dissemination protocols that run in O(D + polylogn) time.) Another interesting direction would be the development of reliable robust fault-tolerant algorithms in this regard. Another issue is whether we can reduce the number of incoming messages in a round; recently, Daum et al. [9] have considered such a more restricted model, yielding interesting results. It would also be interesting to look at the bounds where each node is only allowed O(1) connections per round, whether initiated by the node itself or by its neighbor. Acknowledgment We thank George Giakkoupis for the helpful conversations and useful ideas. A A.1 Appendix The DTG Local Broadcast Protocol In this section, we describe in more detail the DTG protocol that was originally developed in [18] as well as the `-DTG algorithm. It is clear that the algorithm solves local broadcast because it keeps on contacting new neighbors until it has exchanged rumors with all of its neighbors. The author [18] makes use of binomial trees to derive the time complexity and better explain the working of the algorithm. The key idea used for deriving the time complexity is to show that when information is propagated in a pipelined manner along the binomial trees (created on-the-fly), then for any node that is still active in the ith iteration, it has a binomial tree of order 2i (i-tree of depth i: see Figure 8) rooted at it. Furthermore, it is shown that for any two different nodes that are still active in iteration i, their i-trees are vertex disjoint. Since an i-tree is formed by joining two (i − 1)-trees, the growth rate of an i-tree is exponential which limits the number of iterations to O(log n). Also, each node on an average needs to contact O(log n) nodes (O(i) nodes in the ith round). Thus, the overall complexity of the algorithm becomes O(log2 n). In our case, for `-DTG, the additional waiting time of ` increases the time complexity to O(` log2 n). 28 0-tree 1-tree 2-tree 3-tree Figure 8: i-trees for i ∈ 0, 1, 2, 3 The i-tree can be seen as witness structures that provides an explanation as to why a node was active in that particular iteration. The i-tree rooted at a particular node is built recursively as the rounds progress and essentially store the information about which other nodes communicated with one another in which particular round as viewed from the root node. For example, in Figure 9 , the labels on the edges denote the time in which the node of the higher level contacted the lower level node (as observed by the root node). The root contacts the nodes in first level in rounds according to their label, the nodes on the first level similarly contact the nodes in the second level in rounds according to their label and so on. This observation also helps in the realization of the key idea of a node being active in the ith round having an i-tree rooted at it. The nodes in the first level did not contact the root previously as they were busy contacting the nodes of the second level, the nodes of the second level did not contact nodes on the first level as they were busy contacting the nodes in the third level and so on. Figure 9: 5-tree with edge labels As shown in the pseudo code, in the initial PUSH sequence, the message is propagated in a decreasing order of connection round number (as observed by the root node: given by the labels on the edges of Figure 8), helping in pipe-lining the roots message to all other nodes of the i-tree. Similarly, during the initial PULL sequence the message from the nodes is pipelined up to the root. The subsequent PULL-PUSH sequence helps in maintaining the symmetry of the algorithm such that if node u learns about node v, then node v also learns about node u. Finally, the collection of rumors R is updated to the union of rumors collected in the aforementioned sequences. For ` being an integer > 1, we run the modified DTG algorithm on a sub-graph of G, G` , rather than on G, where G` contains only the edges of length up to `. Lets denote this algorithm as `-DTG. The algorithm is presented below and each node v belonging to G` runs it in parallel. Γ(v) can be considered as the neighborhood of v comprising of set of nodes that are node v’s 1-hop neighbors. 29 `-DTG (`) 1: R = v 2: for i = 1 UNTIL Γ(v)\R = φ do 3: link to any new neighbor ui ∈ Γ(v) 4: R0 = v 5: PUSH : 6: for j = i downto 1 do 7: send rumors in R0 to uj 8: wait for ` time to receive uj ’s rumors 9: add all received rumors to R0 10: PULL: 11: for j = 1 to i do 12: send rumors in R0 to uj 13: wait for ` time to receive uj ’s rumors 14: add all received rumors to R0 15: R00 = v 16: perform PULL, PUSH with R00 17: R = R0 ∪ R00 Algorithm 6: `-DTG References [1] John Augustine, Gopal Pandurangan, Peter Robinson, Scott Roche, and Eli Upfal. Enabling robust and efficient distributed computation in dynamic peer-to-peer networks. In IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS Berkeley, USA, pages 350–369, 2015. [2] Surender Baswana and Sandeep Sen. A simple and linear time randomized algorithm for computing sparse spanners in weighted graphs. 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Cooperative control of multi-agent systems to locate source of an odor arXiv:1711.03819v1 [cs.SY] 10 Nov 2017 Abhinav Sinha, Rishemjit Kaur, Ritesh Kumar and Amol P. Bhondekar Abstract—This work targets the problem of odor source localization by multi-agent systems. A hierarchical cooperative control has been put forward to solve the problem of locating source of an odor by driving the agents in consensus when at least one agent obtains information about location of the source. Synthesis of the proposed controller has been carried out in a hierarchical manner of group decision making, path planning and control. Decision making utilizes information of the agents using conventional Particle Swarm Algorithm and information of the movement of filaments to predict the location of the odor source. The predicted source location in the decision level is then utilized to map a trajectory and pass that information to the control level. The distributed control layer uses sliding mode controllers known for their inherent robustness and the ability to reject matched disturbances completely. Two cases of movement of agents towards the source, i.e., under consensus and formation have been discussed herein. Finally, numerical simulations demonstrate the efficacy of the proposed hierarchical distributed control. Index Terms—Odor source localization, multi-agent systems (MAS), sliding mode control (SMC), homogeneous agents, cooperative control. I. I NTRODUCTION A. Overview Inspiration of odor source localization problem stems from behavior of biological entities such as mate seeking by moths, foraging by lobsters, prey tracking by mosquitoes and blue crabs, etc., and is aimed at locating the source of a volatile chemical. These behaviors have long been mimicked by autonomous robot(s). Chemical source tracking has attracted researchers around the globe due to its applications in both civilian and military domains. A plethora of applications are possible, some of which include detection of forest fire, oil spills, release of toxic gases in tunnels and mines, gas leaks in industrial setup, search and rescue of victims and clearing leftover mine after an armed conflict. A plume containing filaments, or odor molecules, is generally referred to the downwind trail formed as a consequence of mixing of contaminant molecules in any kind of movement of air. The dynamical optimization problem of odor source localization can be effectively solved using multiple robots working in cooperation. The obvious advantages of leveraging multiagent systems (MAS) are increased probability of success, A. Sinha is with School of Mechatronics & Robotics, Indian Institute of Engineering Science and Technology; and Central Scientific Instruments Organization (CSIR- CSIO), India. email: [email protected] R. Kaur, R. Kumar & A. P. Bhondekar are with CSIR- CSIO. emails: [email protected],[email protected], [email protected] redundancy and improved overall operational efficiency and spatial diversity in having distributed sensing and actuation. B. Motivation Odor source localization is a three stage problem– sensing, maneuvering and control. Some of reported literature on odor source localization date back to 1980s when Larcombe et al. [1] discussed such applications in nuclear industry by considering a chemical gradient based approach. Other works in 1990s [2]–[6] relied heavily on sensing part using techniques such as chemotaxis [7], infotaxis [8], anemotaxis [9], [10] and fluxotaxis [11]. The efficiency of such algorithms was limited by the quality of sensors and the manner in which they were used. These techniques also failed to consider turbulence dominated flow and resulted in poor tracking performance. Bio-inspired algorithms have been reported to maneuver the agents, some of which include Braitenberg style [12], E. coli algorithm [13], Zigzag dung beetle approach [14], silkworm moth style [15]–[17] and their variants. A tremendous growth of research attention towards cooperative control has been witnessed in the past decade [18], [19] but very few have addressed the problem of locating source of an odor. Hayes et al. [20] proposed a distributed cooperative algorithm based on swarm intelligence for odor source localization and experimental results proved multiple robots perform more efficiently than a single autonomous robot. A Particle Swarm Optimization (PSO) algorithm [21] was proposed by Marques et al. [22], [23] to tackle odor source localization problems. To avoid trapping into local maximum concentrations, Jatmiko et al. [23] proposed modified PSO algorithms based on electrical charge theory, where neutral and charged robots has been used. Lu et al. [24] proposed a distributed coordination control protocol based on PSO to address the problem. It should be noted that simplified PSO controllers are a type of proportional-only controller and the operating region gets limited between global and local best. This needs complicated obstacle avoidance algorithms and results in high energy expenditure. Lu et al. [25] also proposed a cooperative control scheme to coordinate multiple robots to locate odor source in which a particle filter has been used to estimate the location of odor source based on wind information, a movement trajectory has been planned, and finally a cooperative control scheme has been proposed to coordinate movement of robots towards the source. Motivated by these studies, we have implemented a robust and powerful hierarchical cooperative control strategy to tackle the problem. First layer is the group level in which the information about the source via instantaneous sensing and swarm intelligence is obtained. Second layer is designed to maneuver the agents via a simplified silkworm moth algorithm. Third layer is based on cooperative sliding mode control and the information obtained in the first layer is passed to the third layer as a reference to the tracking controller. C. Contributions Major contributions of this paper are summarized below. 1) As opposed to existing works on cooperative control to locate source of odor, we have considered a more general formulation by taking nonlinear dynamics of MAS into account. When the uncertain function is zero, the problem reduces to stabilizing integrator dynamics. 2) The control layer is designed on the paradigms of sliding mode, a robust and powerful control with inherent robustness and disturbance rejection capabilities. The reaching law, as well as the sliding manifold in this study are nonlinear and novel resulting in smoother control and faster reachability to the manifold. Use of sliding mode controller also helps in achieving a finite time convergence as opposed to asymptotic convergence to the equilibrium point. The proposed control provides stability and ensures robustness even in the presence of bounded disturbances and matched uncertainties. 3) Odor propagation is non-trivial, i.e., odor arrives in packets, leading to wide fluctuations in measured concentrations. Plumes are also dynamic and turbulent. As odor tends to travel downwind, direction of the wind provides an effective information on relative position of the source. Hence, we have used wind information based on a measurement model describing movement of filaments and concentration information from swarm intelligence to locate the source of odor. 4) Formation keeping of agents to locate source of odor has also been demonstrated in this work. D. Paper Organization After introduction to the study in section I, remainder of this work in organized as follows. Section II provides insights into preliminaries of spectral graph theory and sliding mode control. Section III presents dynamics of MAS and mathematical problem formulation, followed by hierarchical distributed cooperative control scheme in section IV. Results and discussions have been carried out in section V, followed by concluding remarks in section VI. II. P RELIMINARIES A. Spectral Graph Theory for Multi-Agent Systems A directed graph, also known as digraph is represented throughout in this paper by G = (V, E, A). V is the nonempty set in which finite number of vertices or nodes are contained such that V = {1, 2, ..., N }. E denotes directed edge and is represented as E = {(i, j) ∀ i, j ∈ V & i 6= j}. A is the weighted adjacency matrix such that A = a(i, j) ∈ RN×N . The possibility of existence of an edge (i, j) occurs iff the vertex i receives the information supplied by the vertex j, i.e., (i, j) ∈ E. Hence, i and j are termed neighbours. The set Ni contains labels of vertices that are neighbours of the vertex i. For the adjacency matrix A, a(i, j) ∈ R+ 0 . If (i, j) ∈ E ⇒ a(i, j) > 0. If (i, j) ∈ / E or i = j ⇒ a(i, j) = 0. The Laplacian matrix L [26] is central to the consensus problem and is given by L = D − A where degree matrix, D is a diagonal matrix, Pn i.e, D = diag(d1 , d2 , ..., dn ) whose entries are di = j=1 a(i, j). A directed path from vertex j to vertex i defines a sequence comprising of edges (i, i1 ), (i1 , i2 ), ..., (il , j) with distinct vertices ik ∈ V, k = 1, 2, 3, ..., l. Incidence matrix B is also a diagonal matrix with entries 1 or 0. The entry is 1 if there exists an edge between leader agent and any other agent, otherwise it is 0. Furthermore, it can be inferred that the path between two distinct vertices is not uniquely determined. However, if a distinct node in V contains directed path to every other distinct node in V, then the directed graph G is said to have a spanning tree. Consequently,the matrix L + B has full rank [26]. Physically, each agent has been modelled by a vertex or node and the line of communication between any two agents has been modelled as a directed edge. B. Sliding Mode Control Sliding Mode Control (SMC) [27] is known for its inherent robustness. The switching nature of the control is used to nullify bounded disturbances and matched uncertainties. Switching happens about a hypergeometric manifold in state space known as sliding manifold, surface, or hyperplane. The control drives the system monotonically towards the sliding surface, i.e, trajectories emanate and move towards the hyperplane (reaching phase). System trajectories, after reaching the hyperplane, get constrained there for all future time (sliding phase), thereby ensuring the system dynamics remains independent of bounded disturbances and matched uncertainties. In order to push state trajectories onto the surface s(x), a proper discontinuous control effort uSM (t, x) needs to be synthesized satisfying the following inequality. sT (x)ṡ(x) ≤ −ηks(x)k, (1) with η being positive and is referred as the reachability constant. ∂s ∂s ẋ = f (t, x, uSM ) (2) ∵ ṡ(x) = ∂x ∂x ∂s ∴ sT (x) f (t, x, uSM ) ≤ −ηks(x)k. (3) ∂x The motion of state trajectories confined on the manifold is known as sliding. Sliding mode exists if the state velocity vectors are directed towards the manifold in its neighbourhood. Under such consideration, the manifold is called attractive, i.e., trajectories starting on it remain there for all future time and trajectories starting outside it tend to it in an asymptotic manner. Hence, in sliding motion, ∂s f (t, x, uSM ) = 0. (4) ∂x uSM = ueq is a solution, generally referred as equivalent control is not the actual control applied to the system but can be thought of as a control that must be applied on an average to maintain sliding motion and is mainly used for analysis of sliding motion. ṡ(x) = III. DYNAMICS OF M ULTI -AGENT S YSTEMS & P ROBLEM F ORMULATION Consider first order homogeneous MAS interacting among themselves and their environment in a directed topology. Under such interconnection, information about the predicted location of source of the odor through instantaneous plume sensing is not available globally. However, local information is obtained by communication among agents whenever at least one agent attains some information of interest. The governing dynamics of first order homogeneous MAS consisting of N agents is described by nonlinear differential equations as ẋi (t) = f (xi (t)) + uSMi (t) + ςi ; i ∈ [1, N ], (5) where f (·) : R+ × X → Rm is assumed to be locally Lipschitz over some fairly large domain DL with Lipschitz constant L̄, and denotes the uncertain nonlinear dynamics of each agent. Also X ⊂ Rm is a domain in which origin is contained. xi and uSMi are the state of ith agent and the associated control respectively. ςi represents bounded exogenous disturbances that enter the system from input channel, i.e., kςi k ≤ ςmax < ∞. The problem of odor source localization can be viewed as a cooperative control problem in which control laws uSMi need to be designed such that the conditions limt→∞ kxi − xj k = 0 and limt→∞ kxi −xs k ≤ θ are satisfied. Here xs represents the probable location of odor source & θ is an accuracy parameter. IV. H IERARCHICAL D ISTRIBUTED C OOPERATIVE C ONTROL S CHEME In order to drive the agents towards consensus to locate the source of odor, we propose the following hierarchy. A. Group Decision Making This layer utilizes both concentration and wind information to predict the location of odor source. Then, the final probable position of the source can be described as ψ(tk ) = c1 pi (tk ) + (1 − c1 )qi (tk ), (6) with pi (tk ) as the oscillation centre according to a simple Particle Swarm Optimization (PSO) algorithm and qi (tk ) captures the information of the wind. c1 ∈ (0, 1) denotes additional weighting coefficient. Remark 1. The arguments in (6) represent data captured at t = tk instants (k = 1, 2, ...) as the sensors equipped with the agents can only receive data at discrete instants. It should be noted that ψ is the tracking reference that is fed to the controller. Now, we present detailed description of obtaining pi (tk ) and qi (tk ). Simple PSO algorithm that is commonly used in practice has the following form. vi (tk+1 ) = ωvi (tk ) + uPSO (tk ), (7) xi (tk+1 ) = xi (tk ) + vi (tk+1 ). (8) Here ω is the inertia factor, vi (tk ) and xi (tk ) represent the respective velocity and position of ith agent. This commonly used form of PSO can also be used as a proportional-only type controller, however for the disadvantages mentioned earlier, we do not use PSO as our final controller. PSO control law uPSO can be described as uPSO = α1 (xl (tk ) − xi (tk )) + α2 (xg (tk ) − xi (tk )). (9) In (9), xl (tk ) denotes the previous best position and xg (tk ) denotes the global best position of neighbours of ith agent at time t = tk , and α1 & α2 are acceleration coefficients. Since, every agent in MAS can get some information about the magnitude of concentration via local communication, position of the agent with a global best can be easily known. By the idea of PSO, we can compute the oscillation centre pi (tk ) as pi (tk ) = α1 xl (tk ) + α2 xg (tk ) , α1 + α2 (10) where xl (tk ) = arg xg (tk ) = arg max {g(xl (tk−1 )), g(xi (tk ))}, 0<t<tk−1 (11) max {g(xg (tk−1 )), max aij g(xj (tk ))}. 0<t<tk−1 j∈N (12) Thus, from (9), (10) uPSO (tk ) = (α1 + α2 ){pi (tk ) − xi (tk )}, (13) which is clearly a proportional-only controller with proportional gain α1 + α2 , as highlighted earlier. In order to compute qi (tk ), movement process of a single filament that consists several order molecules has been modelled. If xf (t) denotes position of the filament at time t, v̄a (t) represent mean airflow velocity and n(t) be some random process, then the model can be described as ẋf (t) = v̄a (t) + n(t). (14) Without loss of generality, we shall regard the start time of our experiment as t = 0. From (14), we have Z t Z t xf (t) = v̄a (τ )dτ + n(τ )dτ + xs (0). (15) 0 0 xs (0) denotes the real position of the odor source at t = 0. Assumption IV.1. We assume the presence of a single, stationary odor source. Thus, xs (t) = xs (0). Implications from remark 1 require (15) to be implemented at t = tk instants. Hence, xf (tk ) = t X v̄a (τm )∆t + m=0 t X n(τm )∆t + xs (tk ), m=0 ? xf (tk ) = xs (tk ) + v̄a? (tk ) + w (tk ). In (17), w? (tk ). Pt m=0 v̄a (τm )∆t = v̄a? (tk ) and (16) (17) Pt m=0 n(τm )∆t = Remark 2. In (17), the accumulated average of v̄a? (tk ) and w? (tk ) can also be considered ∀ possible filament releasing time. From (17), xf (tk ) − v̄a? (tk ) = xs (tk ) + w? (tk ). (18) C. Distributed Control In the control layer, we design a robust and powerful controller on the paradigms of sliding mode. It is worthy to mention that based on instantaneous sensing and swarm information, at different times, each agent can take up the role of a virtual leader whose opinion needs to be kept by other agents. ψ from (6) has been provided to the controller as the reference to be tracked. The tracking error is formulated as ei (t) = xi (t) − ψ(tk ) ; t ∈ [tk , tk+1 [. (23) In terms of graph theory, we can reformulate the error variable as i (t) = (L + B)ei (t) = (L + B)(xi (t) − ψ(tk )). (24) The above relationship, (18) can be viewed as the information about xs (tk ) with some noise w? (tk ). Hence, From this point onward, we shall denote L + B as H. Next, we formulate the sliding manifold qi (tk ) = xs (tk ) + w? (tk ). si (t) = λ1 tanh(λ2 i (t)), (19) Therefore, ψ in (6) can now be constructed from (10) & (19). B. Path Planning Since, detection of information of interest is tied to the threshold value defined for the sensors, the next state is updated taking this threshold value into account. Thus, the blueprints of path planning can be described in terms of three types of behavior. 1) Surging: If the ith agent receives data well above threshold, we say that some clues about the location of the source has been detected. If the predicted position of the source at t = tk as seen by ith agent be given as xsi (tk ), then the next state of the agent is given mathematically as xi (tk+1 ) = xsi (tk ). (20) 2) Casting: If the ith agent fails to detect information at any particular instant, then the next state is obtained using the following relation. xi (tk+1 ) = kxi (tk ) − xsi (tk )k + xsi (tk ). 2 which is a nonlinear sliding manifold offering faster reachability to the surface. λ1 ∈ R+ represents the speed of convergence to the surface, and λ2 ∈ R+ denotes the slope of the nonlinear sliding manifold. These are coefficient weighting parameters that affect the system performance. The forcing function has been taken as ṡi (t) = −µ sinh−1 (m + w\|si (t)\|)sign(si (t)). (26) In (26), m is a small offset such that the argument of sinh−1 function remains non zero and w is the gain of the controller. The parameter µ facilitates additional gain tuning. In general, m << w. This novel reaching law contains a nonlinear gain and provides faster convergence towards the manifold. Moreover, this reaching law is smooth and chattering free, which is highly desirable in mechatronic systems to ensure safe operation. Theorem IV.1. Given the dynamics of MAS (5) connected in a directed topology, error candidates (23, 24) and the sliding manifold (25), the stabilizing control law that ensures accurate reference tracking under consensus can be described as (21) uSMi (t) = − (ΛH)−1 µ sinh−1 (m + w\|si (t)\|)sign(si (t))Γ−1 3) Search and exploration: If all the agents fail to detect odor clues for a time segment [tk , tk+l ] > δ0 for some l ∈ N and δ0 ∈ R+ being the time interval for which no clues are detected or some constraint on wait time placed at the start of the experiment, then the next state is updated as xi (tk+1 ) = xsi (tk ) + zφσ . (25) (22) In (22), zφσ is some random parameter with σ as its standard deviation and φ as its mean. + (f (xi (t)) − ψ̇(tk )) (27) where Λ = λ1 λ2 , Γ = 1 − tanh2 (λ2 i (t)), w > supt≥0 {kςi k} & µ > sup{kΛHςi Γk}. Remark 3. As mentioned earlier, λ1 , λ2 ∈ R+ . This ensures Λ 6= 0 and hence its non singularity. The argument of tanh is always finite and satisfies λ2 i (t) 6= πι(κ + 1/2) for κ ∈ Z, thus Γ is also invertible. Moreover the non singularity of H can be established directly if the digraph contains a spanning tree with leader agent as a root. Proof. From (24) and (25), we can write ṡi (t) = λ1 {λ2 ˙i (t)(1 − tanh2 (λ2 i (t)))} (28) 2 = λ1 λ2 ˙i (t) − λ1 λ2 ˙i (t) tanh (λ2 i (t)) (29) 2 = λ1 λ2 ˙i (t){1 − tanh (λ2 i (t))} (30) = ΛH(ẋi (t) − ψ̇(tk ))Γ (31) with Λ & Γ as defined in Theorem IV.1. From (5), (31) can be further simplified as ṡi (t) = ΛH(f (xi (t)) + uSMi (t) + ςi − ψ̇(tk ))Γ. (32) Using (26), the control that brings the state trajectories on to the sliding manifold can now be written as uSMi (t) = − (ΛH)−1 µ sinh−1 (m + w\|si (t)\|)sign(si (t))Γ−1 + (f (xi (t)) − ψ̇(tk )) . (33) Thus, the derivative of Lyapunov function candidate is negative definite confirming stability in the sense of Lyapunov. Since, µ > 0, ksi k > 0 and sinh−1 (·) > 0 due to the nature of its arguments. Therefore, (37) and (26) together provide implications that ∀si (0), si ṡi < 0 and the surface is globally attractive. This ends the proof. V. R ESULTS AND DISCUSSIONS Interaction topology of the agents represented as a digraph has been shown here in figure 1. The associated graph matrices have been described below. The computer simulation has been performed assuming that agent 1 appears as virtual leader to all other agents, making the topology fixed and directed for this study. It should be noted that, the theory developed so far can be extended to the case of switching topologies and shall be dealt in future. This concludes the proof. 1 Remark 4. The control (27) can be practically implemented as it does not contain the uncertainty term. It is crucial to analyze the necessary and sufficient conditions for the existence of sliding mode when control protocol (27) is used. We regard the system to be in sliding mode if for any time t1 ∈ [0, ∞[, system trajectories are brought upon the manifold si (t) = 0 and are constrained there for all time thereafter, i.e., for t ≥ t1 , sliding motion occurs. Theorem IV.2. Consider the system described by (5), error candidates (23, 24), sliding manifold (25) and the control protocol (27). Sliding mode is said to exist in vicinity of sliding manifold, if the manifold is attractive, i.e., trajectories emanating outside it continuously decrease towards it. Stating alternatively, reachability to the surface is ensured for some reachability constant η > 0. Moreover, stability can be guaranteed in the sense of Lyapunov if gain µ is designed as µ > sup{kΛHςi Γk}. Proof. Let us take into account, a Lyapunov function candidate Vi = 0.5s2i . (34) Taking derivative of (34) along system trajectories yield V̇i = si ṡi = si ΛH(f (xi (t)) + uSMi (t) + ςi − ψ̇(tk ))Γ . 5 Fig. 1: Topology in which agents are connected  0 0 A= 0 0 0 0 1 0 1 0 0 1  1 0 L = D−A =  0 0   1 0 0 0 , B =  0 0 0 0 0 0 −1 0 0 1 0 0 0 0 0 0   1 0 0 0 , D =  0 0 0 0   2 −1 0 0 0 0 ,L + B =  0 1 0 0 −1 1 0 1 −1 0  0 0 0 0 0 0 , 0 1 0 0 0 1 (39)  −1 0 0 0  1 0 −1 1 (40) Agents have the following dynamics. (35) ẋ2 = 0.1 sin(x2 ) + cos(2πt) + uSM2 (t) + ς2 , (42) (36) ẋ3 = 0.1 sin(x3 ) + cos(2πt) + uSM3 (t) + ς3 , (43) ẋ4 = 0.1 sin(x4 ) + cos(2πt) + uSM4 (t) + ς4 , (44) ẋ5 = 0.1 sin(x5 ) + cos(2πt) + uSM5 (t) + ς5 . (45) (37) where η = µ sinh−1 (m + w\|si \|) − ΛHςi Γ > 0 is called reachability constant. For µ > sup{kΛHςi Γk}, we have V̇i < 0. 3 (41) = −µ sinh−1 (m + w\|si \|)ksi k + ΛHςi Γksi k = − µ sinh−1 (m + w\|si \|) + ΛHςi Γ ksi k = −ηksi k, 4 ẋ1 = 0.1 sin(x1 ) + cos(2πt) + uSM1 (t) + ς1 , Substituting the control protocol (27) in (36), we have V̇i = si − µ sinh−1 (m + w\|si \|)sign(si ) + ΛHςi Γ 2 (38) In this study, advection model given in [28] has been used to simulate the plume with both additive and multiplicative disturbances. The initial conditions for simulation are taken to be large values, i.e., far away from the equilibrium point. Time varying disturbance has been taken as ςi = 0.3 sin(π 2 t2 ), accuracy parameter θ = 0.001 and maximum mean airflow velocity v̄amax = 1 m/s. Other key design parameters are mentioned in table 1. Agents progressing towards the source 11 x1 x2 x3 x4 x5 source info position of agents (m) 2 Direction of movement of agents towards the source 1.5 10.9 10.8 10.7 10.6 Direction of movement of filaments released from the source 1 10.5 10.4 0.5 10.3 True Odor Source 10.2 0 10.1 10 12 -0.5 0 2 4 6 8 10 time progression for agents (sec) Fig. 2: Agents in consensus to locate source of odor Agents progressing towards the source in parallel formation 11 position of agents (m) 3 Formation gap 2 x2 10.8 x4 10.7 10.6 10.5 1 10.4 True odor source 0 x1 10.9 10.3 x3 x5 source info (movement of ﬁlaments away from source) odor source location 4 agent 1 initial point agent 2 initial point agent 3 initial point agent 4 initial point agent 5 initial point agent 1 terminal point 10.2 -1 agent 2 terminal point agent 3 terminal point 10.1 2 4 6 8 agent 5 terminal point 10 12 -2 0 agent 4 terminal point 10 time progression for agents (sec) Fig. 3: Agents in formation to locate source of odor Tracking errors Control signals 8 e1 e2 e3 e4 e5 1 u1 u2 u3 u4 u5 6 4 2 u i (t) Norm of error variables e i (t) 1.5 0 -2 0.5 -4 -6 -8 0 -10 0 1 2 3 4 5 6 7 time (sec) Fig. 4: Norm of tracking errors 8 9 10 0 1 2 3 4 5 6 7 8 time (sec) Fig. 5: Control signals during consensus 9 10 position of the source 2.5 Sliding manifolds 1.5 s1 s2 s3 s4 s5 surface variables s i (t) 1 0.5 0 -0.5 -1 -1.5 0 1 2 3 4 5 6 7 8 9 10 time (sec) Fig. 6: Sliding manifolds during consensus TABLE I: Values of the design parameters used in simulation c1 ωmax α1 α2 λ1 λ2 µ m w 0.5 2 rad/s 0.25 0.25 1.774 2.85 5 10−3 2 Figure 2 shows agents coming to consensus in finite time to locate the source of odor and figure 3 shows agents moving in parallel formation to locate the odor source. 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Coresets for Dependency Networks Alejandro Molina FIRST. LAST @ TU - DORTMUND . DE CS Department TU Dortmund, Germany Alexander Munteanu FIRST. LAST @ TU - DORTMUND . DE arXiv:1710.03285v2 [cs.AI] 16 Oct 2017 CS Department TU Dortmund, Germany Kristian Kersting LAST @ CS . TU - DARMSTADT. DE CS Dept. and Centre for CogSci TU Darmstadt, Germany Abstract Many applications infer the structure of a probabilistic graphical model from data to elucidate the relationships between variables. But how can we train graphical models on a massive data set? In this paper, we show how to construct coresets—compressed data sets which can be used as proxy for the original data and have provably bounded worst case error—for Gaussian dependency networks (DNs), i.e., cyclic directed graphical models over Gaussians, where the parents of each variable are its Markov blanket. Specifically, we prove that Gaussian DNs admit coresets of size independent of the size of the data set. Unfortunately, this does not extend to DNs over members of the exponential family in general. As we will prove, Poisson DNs do not admit small coresets. Despite this worst-case result, we will provide an argument why our coreset construction for DNs can still work well in practice on count data. To corroborate our theoretical results, we empirically evaluated the resulting Core DNs on real data sets. The results demonstrate significant gains over no or naive sub-sampling, even in the case of count data. 1. Introduction Artificial intelligence and machine learning have achieved considerable successes in recent years, and an ever-growing number of disciplines rely on them. Data is now ubiquitous, and there is great value from understanding the data, building e.g. probabilistic graphical models to elucidate the relationships between variables. In the big data era, however, scalability has become crucial for any useful machine learning approach. In this paper, we consider the problem of training graphical models, in particular Dependency Networks Heckerman et al. (2000), on massive data sets. They are cyclic directed graphical models, where the parents of each variable are its Markov blanket, and have been proven successful in various tasks, such as collaborative filtering Heckerman et al. (2000), phylogenetic analysis Carlson et al. (2008), genetic analysis Dobra (2009); Phatak et al. (2010), network inference from sequencing data Allen and Liu (2013), and traffic as well as topic modeling Hadiji et al. (2015). Specifically, we show that Dependency Networks over Gaussians—arguably one of the most prominent type of distribution in statistical machine learning—admit coresets of size independent of the size of the data set. Coresets are weighted subsets of the data, which guarantee that models fitting 1 them will also provide a good fit for the original data set, and have been studied before for clustering Badoiu et al. (2002); Feldman et al. (2011, 2013); Lucic et al. (2016), classification Har-Peled et al. (2007); Har-Peled (2015); Reddi et al. (2015), regression Drineas et al. (2006, 2008); Dasgupta et al. (2009); Geppert et al. (2017), and the smallest enclosing ball problem Badoiu and Clarkson (2003, 2008); Feldman et al. (2014); Agarwal and Sharathkumar (2015); we refer to Phillips (2017) for a recent extensive literature overview. Our contribution continues this line of research and generalizes the use of coresets to probabilistic graphical modeling. Unfortunately, this coreset result does not extend to Dependency Networks over members of the exponential family in general. We prove that Dependency Networks over Poisson random variables Allen and Liu (2013); Hadiji et al. (2015) do not admit (sublinear size) coresets: every single input point is important for the model and needs to appear in the coreset. This is an important negative result, since count data—the primary target of Poisson distributions—is at the center of many scientific endeavors from citation counts to web page hit counts, from counts of procedures in medicine to the count of births and deaths in census, from counts of words in a document to the count of gamma rays in physics. Here, modeling one event such as the number of times a certain lab test yields a particular result can provide an idea of the number of potentially invasive procedures that need to be performed on a patient. Thus, elucidating the relationships between variables can yield great insights into massive count data. Therefore, despite our worst-case result, we will provide an argument why our coreset construction for Dependency Networks can still work well in practice on count data. To corroborate our theoretical results, we empirically evaluated the resulting Core Dependency Networks (CDNs) on several real data sets. The results demonstrate significant gains over no or naive sub-sampling, even for count data. We proceed as follows. We review Dependency Networks (DNs), prove that Gaussian DNs admit sublinear size coresets, and discuss the possibility to generalize this result to count data. Before concluding, we illustrate our theoretical results empirically. 2. Dependency Networks Most of the existing AI and machine learning literature on graphical models is dedicated to binary, multinominal, or certain classes of continuous (e.g. Gaussian) random variables. Undirected models, aka Markov Random Fields (MRFs), such as Ising (binary random variables) and Potts (multinomial random variables) models have found a lot of applications in various fields such as robotics, computer vision and statistical physics, among others. Whereas MRFs allow for cycles in the structures, directed models aka Bayesian Networks (BNs) required acyclic directed relationships among the random variables. Dependency Networks (DNs)—the focus of the present paper—combine concepts from directed and undirected worlds and are due to Heckerman et al. (2000). Specifically, like BNs, DNs have directed arcs but they allow for networks with cycles and bi-directional arcs, akin to MRFs. This makes DNs quite appealing for many applications because we can build multivariate models from univariate distributions Allen and Liu (2013); Yang et al. (2015); Hadiji et al. (2015), while still permitting efficient structure learning using local estimtatiors or gradient tree boosting. Generally, if the data are fully observed, learning is done locally on the level of the conditional probability distributions for each variable mixing directed and indirected as needed. Based on these local distributions, samples from the joint distribution are obtained via Gibbs sampling. Indeed, the Gibbs sampling neglects the question of a consistent joint probability distribution and instead makes only use of 2 local distributions. The generated samples, however, are often sufficient to answer many probability queries. Formally, let X = (X (1) , . . . , X (d) ) denote a random vector and x its instantiation. A Dependency Network (DN) on X is a pair (G, Ψ) where G = (V, E) is a directed, possibly cyclic, graph where each node in V = [d] = {1, . . . , d} corresponds to the random variable X (i) . In the set of directed edges E ⊆ V × V \ {(i, i) \| i ∈ [d]}, each edge models a dependency between variables, i.e., if there is no edge between i and j then the variables X (i) and X (j) are conditionally independent given the other variables X \i,j indexed by [d] \ {i, j} in the network. We refer to the nodes that have an edge pointing to X (i) as its parents, denoted by pai = {X (j) \| (j, i) ∈ E}. Ψ = {pi \| i ∈ [d]} is a set of conditional probability distributions associated with each variable X (i) ∼ pi , where pi = p(x(i) \| pai ) = p(x(i) \| x\i ) . As example of such a local model, consider Poisson conditional probability distributions as illustrated in Fig. 1 (left): (i) λi (x\i )x −λi (x\i ) p(x(i) \| pai ) = e . x(i) ! Here, λi (x\i ) highlights the fact that the mean can have a functional form that is dependent on X (i) ’s parents. Often, we will refer to it simply as λi . The construction of the local conditional probability distribution is similar to the (multinomial) Bayesian network case. However, in the case of DNs, the graph is not necessarily acyclic and p(x(i) \| x\i ) typically has an infinite range, and hence cannot be represented using a finite table of probability values. Finally, the full joint distribution is simply defined as the product of local distributions: Y p(x) = p(x(i) \| x\i ) , i∈[d] also called pseudo likelihood. For the Poisson case, this reads (i) p(x) = Y i∈[d] λix −λi . e x(i) ! Note, however, that doing so does not guarantee the existence of a consistent joint distribution, i.e., a joint distribution of which they are the conditionals. Bengio et al. (2014), however, have recently proven the existence of a consistent distribution per given evidence, which does not have to be known in closed form, as long as an unordered Gibbs sampler converges. 3. Core Dependency Networks As argued, learning Dependency Networks (DNs) amounts to determining the conditional probability distributions from a given set of n training instances xi ∈ Rd representing the rows of the data matrix X ∈ Rn×d over d variables. Assuming that p(x(i) \| pai ) is parametrized as a generalized linear model (GLM) McCullagh and Nelder (1989), this amounts to estimating the parameters γ (i) of the GLM associated with each variable X (i) , since this completely determines the local distributions, but p(x(i) \| pai ) will possibly depend on all other variables in the network, and these dependencies define the structure of the network. This view of training DNs as fitting d GLMs to the data allows us to develop Core Dependency Networks (CDNs): Sample a coreset and train a DN over certain members of the GLM family on the sampled corest. 3 Relative Frequency 0.5 fit data 0.4 data fit 0.3 0.2 X (0) 0.1 0.0 X (1) 0 2 4 6 X (2) 8 Number of Goals Figure 1: Illustration of Dependency Networks (DNs) using Poissons. (left) The number of goals scored in soccer games follows a Poisson distribution. The plot shows the distribution of home goals in the season 2012/13 of the German Bundesliga by the home team. The home team scored on average λ = 1.59 goals per game. (right) Example structure of a Poisson DN. The conditional distribution of each count variable given its neighbors is a Poisson distribution. Similar to a Bayesian network a Poisson DN is directed, however, it also contains cycles. (Best viewed in color) A coreset is a (possibly) weighted and usually considerably smaller subset of the input data that approximates a given objective function for all candidate solutions: Definition 1 (ε-coreset) Let X be a set of points from a universe U and let Γ be a set of candidate solutions. Let f : U × Γ → R≥0 be a non-negative measurable function. Then a set C ⊂ X is an ε-coreset of X for f , if ∀γ ∈ Γ : \|f (X, γ) − f (C, γ)\| ≤ ε · f (X, γ). We now introduce the formal framework that we need towards the design of coresets for learning dependency networks. A very useful structural property for `2 based objective (or loss) functions is the concept of an ε-subspace embedding. Definition 2 (ε-subspace embedding) An ε-subspace embedding for the columnspace of X is a matrix S such that ∀γ ∈ Rd : (1 − ε)kXγk2 ≤ kSXγk2 ≤ (1 + ε)kXγk2 We can construct a sampling matrix S which forms an ε-subspace embedding with constant probabilty in the following way: Let U be any orthonormal basis for the columnspace of X. This basis can be obtained from the singular value decomposition (SVD) X = U ΣV T of the data matrix. Now let ρ = rank(U ) = rank(X) and define the leverage scores li = kUi∗ k2 /kU k2F = kUi∗ k2 /ρ for i ∈ [n]. Now we fix a sampling size parameter k = O(ρ log(ρ/ε)/ε2 ), sample the input points one-by-one with probability qi = min{1, k · li } and reweight their contribution to the loss function by wi = 1/qi . Note that, for the sum of squares loss, this corresponds to defining a diagonal 4 √ (sampling) matrix S by Sii = 1/ qi with probability qi and Sii = 0 otherwise. Also note, that the expected number of samples is k = O(ρ log(ρ/ε)/ε2 ), which also holds with constant probability by Markov’s inequality. Moreover, to give an intuition why this works, note that for any fixed γ ∈ Rd , we have X X xi γ 2 2 E kSXγk = qi = (xi γ)2 = kXγk2 . √ qi The significantly stronger property of forming an ε-subspace embedding, according to Definition 2, follows from a matrix approximation bound given in Rudelson and Vershynin (2007); Drineas et al. (2008). Lemma 3 Let X be an input matrix with rank(X) = ρ. Let S be a sampling matrix constructed as stated above with sampling size parameter k = O(ρ log(ρ/ε)/ε2 ). Then S forms an ε-subspace embedding for the columnspace of X with constant probability. Proof Let X = U ΣV T be the SVD of X. By Theorem 7 in Drineas et al. (2008) there exists an absolute constant C > 1 such that r T T log k E kU S SU − U T U k ≤ C kU kF kU k k r log k √ ≤ C ρ ≤ ε, k √ where we used the fact that kU kF = ρ and kU k = 1 by orthonormality of U . The last inequality holds by choice of k = Dρ log(ρ/ε)/ε2 for a large enough absolute constant D > 1 such that 1+log D < 4C1 2 , since D log k log(Dρ log(ρ/ε)/ε2 ) 2ε2 log(Dρ log(ρ/ε)/ε) = ≤ k Dρ log(ρ/ε)/ε2 Dρ log(ρ/ε) 2 2 4ε 1 + log D ε2 4ε (log(ρ/ε) + log D) ≤ < 2 . ≤ Dρ log(ρ/ε) ρ D C ρ By an application of Markov’s inequality and rescaling ε, we can assume with constant probability kU T S T SU − U T U k ≤ ε. (1) We show that this implies the ε-subspace embedding property. To this end, fix γ ∈ Rd . \| kSXγk2 − kXγk2 \| = kγ T X T S T SXγ − γ T X T Xγk = kγ T V ΣU T S T SU ΣV T γ − γ T V ΣU T U ΣV T γk = kγ T V Σ (U T S T SU − U T U ) ΣV T γk ≤ kU T S T SU − U T U k · kΣV T γk2 ≤ kU T S T SU − U T U k · kXγk2 ≤ εkXγk2 , The first inequality follows by submultiplicativity, and the second from rotational invariance of the spectral norm. Finally we conclude the proof by Inequality (1). 5 The question arises whether we can do better than O(ρ log(ρ/ε)/ε2 ). One can show by reduction from the coupon collectors theorem that there is a lower bound of Ω(ρ log ρ) matching the upper bound up to its dependency on √ ε. The hard instance is a dm ×d, m ∈ N orthonormal matrix in which the scaled canonical basis Id / dm−1 is stacked dm−1 times. The leverage scores are all equal to 1/dm , implying a uniform sampling distribution with probability 1/d for each basis vector. Any rank ρ = d preserving sample must comprise at least one of them. This is exactly the coupon collectors theorem with d coupons which has a lower bound of Ω(d log d) Motwani and Raghavan (1995). The fact that the sampling is without replacement does not change this, since the reduction holds for arbitrary large m creating sufficient multiple copies of each element to simulate the sampling with replacement Tropp (2011). Now we know that with constant probability over the randomness of the construction algorithm, S satisfies the ε-subspace embedding property for a given input matrix X. This is the structural key property to show that actually SX is a coreset for Gaussian linear regression models and dependency networks. Consider (G, Ψ), a Gaussian dependency network (GDN), i.e., a collection of Gaussian linear regression models Ψ = {pi (X (i) \|X \i , γ (i) ) = N (X \i γ (i) , σ 2 ) \| i ∈ [d]} on an arbitrary digraph structure G Heckerman et al. (2000). The logarithm of the (pseudo-)likelihood Besag (1975) of the above model is given by Y X ln L (Ψ) = ln pi = ln pi . A maximum likelihood estimate can be obtained by maximizing this function with respect to γ = (γ (1) , . . . , γ (d) ) which is equivalent to minimizing the GDN loss function X fG (X, γ) = kX \i γ (i) − X (i) k2 . Theorem 4 Given S, an ε-subspace embedding for the columnspace of X as constructed above, SX is an ε-coreset of X for the GDN loss function. Proof Fix an arbitrary γ = (γ (1) , . . . , γ (d) ) ∈ Rd(d−1) . Consider the affine map Φ : Rd−1 × \i [d] → Rd , defined by Φ(γ (i) ) = Id γ (i) − ei . Clearly Φ extends its argument from d − 1 to d dimensions by inserting a −1 entry at position i and leaving the other entries in their original order. Let β (i) = Φ(γ (i) ) ∈ Rd . Note that for each i ∈ [d] we have Xβ (i) = XΦ(γ (i) ) = X \i γ (i) − X (i) , (2) and each β (i) is a vector in Rd . Thus, the triangle inequality and the universal quantifier in Definition 2 guarantee that X X \| kSXβ (i) k2 − kXβ (i) k2 \| X = \| (kSXβ (i) k2 − kXβ (i) k2 ) \| X ≤ \|kSXβ (i) k2 − kXβ (i) k2 \| X X ≤ εkXβ (i) k2 = ε kXβ (i) k2 . 6 The claim follows by substituting Identity (2). It is noteworthy that computing one single coreset for the columnspace of X is sufficient, rather than computing d coresets for the d different subspaces spanned by X \i . From Theorem 4 it is straightforward to show that the minimizer found for the coreset is a good approximation of the minimizer for the original data. Corollary 5 Given an ε-coreset C of X for the GDN loss function, let γ̃ ∈ argminγ∈Rd(d−1) fG (C, γ). Then it holds that fG (X, γ̃) ≤ (1 + 4ε) min fG (X, γ). γ∈Rd(d−1) Proof Let γ ∗ ∈ argminγ∈Rd(d−1) fG (X, γ). Then fG (X, γ̃) ≤ ≤ 1 1 fG (C, γ̃) ≤ fG (C, γ ∗ ) 1−ε 1−ε 1+ε fG (X, γ ∗ ) ≤ (1 + 4ε)fG (X, γ ∗ ). 1−ε The first and third inequalities are direct applications of the coreset property, the second holds by optimality of γ̃ for the coreset, and the last follows from ε < 12 . Moreover, the coreset does not affect inference within GDNs. Recently, it was shown for (Bayesian) Gaussian linear regression models that the entire multivariate normal distribution over the parameter space is approximately preserved by ε-subspace embeddings Geppert et al. (2017), which generalizes the above. This implies that the coreset yields a useful pointwise approximation in Markov Chain Monte Carlo inference via random walks like the pseudo-Gibbs sampler in Heckerman et al. (2000). 4. Negative Result on Coresets for Poisson DNs Naturally, the following question arises: Do (sublinear size) coresets exist for dependency networks over the exponential family in general? Unfortunately, the answer is no! Indeed, there is no (sublinear size) coreset for the simpler problem of Poisson regression, which implies the result for Poisson DNs. We show this formally by reduction from the communication complexity problem known as indexing. To this end, recall that the negative log-likelihood for Poisson regression is McCullagh and Nelder (1989); Winkelmann (2008) X `(γ) := `(γ\|X, Y ) = exp(xi γ) − yi · xi γ + ln(yi !). Theorem 6 Let ΣD be a data structure for D = [X, Y ] that approximates likelihood queries ΣD (γ) for Poisson regression, such that ∀γ ∈ Rd : η −1 · `(γ\|D) ≤ ΣD (γ) ≤ η · `(γ\|D). If η < exp( n ) 4 2n2 then ΣD requires Ω(n) bits of storage. 7 Proof We reduce from the indexing problem which is known to have Ω(n) one-way randomized communication complexity Jayram et al. (2008). Alice is given a vector b ∈ {0, 1}n . She produces for every i with bi = 1 the points xi = (r·ω i , −1) ∈ R3 , where ω i , i ∈ {0, . . . , n−1} denote the nth 3 unit roots in the plane, i.e., the vertices of a regular n-polygon of radius r = n/(1 − cos( 2π n )) ≤ n in canonical order. The corresponding counts are set to yi = 1. She builds and sends ΣD of size 3 s(n) to Bob, whose task is to guess the bit bj . He chooses to query γ = (ω j , r · cos( 2π n )) ∈ R . Note j that this affine hyperplane separates r · ω from the other scaled unit roots since it passes exactly through r · ω (j−1) mod n and r · ω (j+1) mod n . Also, all points are within distance 2r from each other by construction and consequently from the hyperplane. Thus, −2r ≤ xi γ ≤ 0 for all i 6= j. If bj = 0, then xj does not exist and the cost is at most `(γ) = X exp(xi γ) − yi · xi γ + ln(yi !) ≤ X 1 + 2r + 1 ≤ 2n + 2nr ≤ 4n4 . If bj = 1 then xj is in the expensive halfspace and at distance exactly 2π j T j xj γ = (rω ) ω − r · cos n 2π = r · 1 − cos = n n So the cost is bounded below by `(γ) ≥ exp(n) − n + 1 ≥ exp( n2 ). exp( n ) Given η < 2n24 , Bob can distinguish these two cases based on the data structure only, by deciding whether ΣD (γ) is strictly smaller or larger than exp( n4 ) · 2n2 . Consequently s(n) = Ω(n), since this solves the indexing problem. Note that the bound is given in bit complexity, but restricting the data structure to a sampling based coreset and assuming every data point can be expressed in O(d log n) bits, this means we still have a lower bound of k = Ω( logn n ) samples. Corollary 7 Every sampling based coreset for Poisson regression with approximation factor η < exp( n ) 4 as in Theorem 6 requires at least k = Ω( logn n ) samples. 2n2 At this point it seems very likely that a similar argument can be used to rule out any o(n)-space constant approximation algorithm. This remains an open problem for now. 5. Why Core DNs for Count Data can still work So far, we have a quite pessimistic view on extending CDNs beyond Gaussians. In the Gaussian setting, where the loss is measured in squared Euclidean distance, the number of important points, i.e., having significantly large leverage scores, is bounded essentially by O(d). This is implicit in the original early works Drineas et al. (2008) and has been explicitly formalized later Langberg and Schulman (2010); Clarkson and Woodruff (2013). It is crucial to understand that this is an inherent property of the norm function, and thus holds for arbitrary data. For the Poisson GLM, in contrast, we have shown that its loss function does not come with such properties from scratch. We 8 constructed a worst case scenario, where basically every single input point is important for the model and needs to appear in the coreset. Usually, this is not the case with statistical models, where the data is assumed to be generated i.i.d. from some generating distribution that fits the model assumptions. Consider for instance a data reduction for Gaussian linear regression via leverage score sampling vs. uniform sampling. It was shown that given the data follows the model assumptions of a Gaussian distribution, the two approaches behave very similarly. Or, to put it another way, the leverage scores are quite uniform. In the presence of more and more outliers generated by the heavier tails of tdistributions, the leverage scores increasingly outperform uniform sampling Ma et al. (2015). The Poisson model yi ∼ Poi(λi ), λi = exp(xi γ). (3) though being the standard model for count data, suffers from its inherent limitation on equidispersed data since E [yi \|xi ] = V [yi \|xi ] = exp(xi γ). Count data, however, is often overdispersed especially for large counts. This is due to unobserved variables or problem specific heterogeneity and contagion-effects. The log-normal Poisson model is known to be inferior for data which specifically follows the Poisson model, but turns out to be more powerful in modeling the effects that can not be captured by the simple Poisson model. It has wide applications for instance in econometric elasticity problems. We review the log-normal Poisson model for count data Winkelmann (2008) yi ∼ Poi(λi ), λi = exp(xi γ)ui = exp(xi γ + vi ), vi = ln ui ∼ N (µ, σ) . 2 A natural choice for the parameters of the log-normal distribution is µ = − σ2 in which case we have E [yi \|xi ] = exp(xi γ + µ + σ 2 /2) = exp(xi γ) , V [yi \|xi ] = E [yi \|xi ] + (exp(σ 2 ) − 1)E [yi \|xi ]2 . It follows that V [yi \|xi ] = exp(xi γ) + Ω(exp(xi γ)2 ) > exp(xi γ), where a constant σ 2 that is independent of xi , controls the amount of overdispersion. Taking the limit for σ → 0 we arrive at the simple model (3), since the distribution of vi = ln ui tends to δ0 , the deterministic Dirac delta distribution which puts all mass on 0. The inference might aim for the log-normal Poisson model directly as in Zhou et al. (2012), or it can be performed by (pseudo-)maximum likelihood estimation of the simple Poisson model. The latter provides a consistent estimator as long as the log-linear mean function is correctly specified, even if higher moments do not possess the limitations inherent in the simple Poisson model Winkelmann (2008). Summing up our review on the count modeling perspective, we learn that preserving the loglinear mean function in a Poisson model is crucial towards consistency of the estimator. Moreover, modeling counts in a log-normal model gives us intuition why leverage score sampling can capture the underlying linear model accurately: In the log-normal Poisson model, u follows a log-normal distribution. It thus holds for ln λ = Xγ + ln u = Xγ + v, that 2 σ 2 v ∼ N − · 1, σ In 2 9 3.58 3.63 3.58 3.59 3.58 CDN Uniform Full 3.6×107 3.55×107 10% 3.48 20% 30% 40% Training data (Sample size in percentage) 3.58 3.35 3.36 3.33 3.31 3.31 3.29 3.6×106 100% 3.26 CDN Uniform Full 3.5×106 3.4×106 3.3×106 3.2×10 6 10% 20% 30% 40% Training data (Sample size in percentage) Negative Log Pseudo Likelihood 100% −2.5 −2.6 −2.55 −2.64 −2.62 −2.65 −2.65 −2.66 −2.66 −2.67 CDN Uniform Full −2.55 −2.6 −2.65 −2.7 2.2 3.5 3.0 1.72 1.76 2.38 2.46 2.79 2.9 3.08 3.18 4.0 CDN Uniform Full 2.5 2.0 1.5 10% 1.55 20% 30% 40% Training data (Sample size in percentage) 1.68 1.4 1.48 1.36 1.42 1.35 1.4 1.8 1.0 100% 1.39 CDN Uniform Full 2.0 1.6 1.4 1.2 1.0 4.0 Log Time (in hours) 3.59 3.0 2.5 Log Time (in minutes) 3.58 Log RMSE (Root Mean Square Error) 5.23 Log RMSE (Root Mean Square Error) 3.7×10 3.59 7 3.65×107 Negative Poisson Pseudo Log–Likelihood Negative Gaussian Pseudo Log–Likelihood 3.75×107 2.0 10% 20% 30% 40% Training data (Sample size in percentage) 0.29 −0.25 0.48 0.09 0.81 0.54 1.05 0.86 100% 2.56 CDN Uniform Full 1.5 1.0 0.5 0.0 10% 20% 30% 40% Training data (Sample size in percentage) 100% RMSE −0.5 10% 20% 30% 40% Training data (Sample size in percentage) 100% Training Time Figure 2: (Q1) Performance (the lower, the better) of Gaussian CDNs on MNIST (upper row) and Poisson CNDs on the traffic dataset (lower row) 10-fold cross-validated. Shown are the negative log pseudo likelihood (left), the squared error loss (middle, in log-space) as well as the training time (right, in log-space) on the y-axis for different proportions of the data sampled (x axis). Please note the jump in the x-axis after 40%. As one can see, CDNs (blue) quickly approach the predictive performance of the full dataset (Full, black). Uniform sampling (Uniform, red) does not perform as well as CDNs. Moreover, CDNs can be orders of magnitude faster than DNs on the full dataset and scale similar to uniform sampling. This is also supported by the vertical lines. They denote the mean performances (the more to the left, the better) on the top axes. (Best viewed in color) by independence of the observations, which implies σ2 2 ln λ ∼ N Xγ − · 1, σ In . 2 2 Omitting the bias µ = − σ2 in each intercept term (which can be cast into X), we notice that this yields again an ordinary least squares problem kXγ − ln(λ)k2 defined in the columspace of X. There is still a missing piece in our argumentation. In the previous section we have used that the coreset construction is an ε-subspace embedding for the columnspace of the whole data set including the dependent variable, i.e., for [X, ln(λ)]. We face two problems. First, λ is only implicitly given in the data, but is not explicitly available. Second, λ is a vector derived from X \i in our setting and might be different for any of the d instances. Fortunately, it was shown via more complicated arguments Drineas et al. (2008), that it is sufficient for a good approximation, if the sampling is done obliviously to the dependent variable. The intuition comes from the fact that the loss of any point in the subspace can be expressed via the projection of ln(λ) onto the subspace spanned by X, and the residual of its projection. A good approximation of the subspace implicitly approximates the projection of any fixed vector, which is then applied to the residual vector of the orthogonal projection. This solves the first problem, since it is only necessary to have a subspace embedding for X. The second issue can be addressed by increasing the sample size by a factor of O(log d) for boosting the error probability to O(1/d) and taking a union bound. 10 Sample portion 10% 20% 30% 40% MNIST GCDN 18.03% 0.57% 0.01% 0.01% GUDN 11162.01% 13.86% 13.33% 2.3% Traffic PCDN 6.81% 2.9% 2.04% 1.59% PUDN 9.6% 3.17% 1.68% 0.99% Table 1: (Q1) Comparison of the empirical relative error (the lower, the better). Best results per dataset are bold. Both Gaussian (GCDNs) and Poisson (PCDNs) CDNs recover the model well, with a fraction of the training data. Uniformly sampled DNs (UDNs) lag behind as the sample size drops. 6. Empirical Illustration Our intention here is to corroborate our theoretical results by investigating empirically the following questions: (Q1) How does the performance of CDNs compare to DNs with access to the full training data set and to a uniform sample from the training data set? and how does the empirical error behave according to the sample sizes? (Q2) Do coresets affect the structure recovered by the DN? To this aim, we implemented (C)DNs in Python calling R. All experiments ran on a Linux machine (56 cores, 4 GPUs, and 512GB RAM). Benchmarks on MNIST and Traffic Data (Q1): We considered two datasets. In a first experiment, we used the MNIST1 data set of handwritten labeled digits. We employed the training set consisting of 55000 images, each with 784 pixels, for a total of 43,120,000 measurements, and trained Gaussian DNs on it. The second data set we considered contains traffic count measurements on selected roads around the city of Cologne in Germany Ide et al. (2015). It consists of 7994 timestamped measurements taken by 184 sensors for a total of 1,470,896 measurements. On this dataset we trained Poisson DNs. For each dataset, we performed 10 fold cross-validation for training a full DN (Full) using all the data, leverage score sampling coresets (CDNs), and uniform samples (Uniform), for different sample sizes. We then compared the predictions made by all the DNs and the time taken to train them. For the predictions on the MNIST dataset, we clipped the predictions to the range [0,1] for all the DNs. For the Traffic dataset, we computed the predictions bxc of every measurement x rounded to the largest integer less than or equal to x. Fig. 2 summarizes the results. As one can see, CDNs outperform DNs trained on full data and are orders of magnitude faster. Compared to uniform sampling, coresets are competitive. Actually, as seen on the traffic dataset, CDNs can have more predictive power than the “optimal” model using the full data. This is in line with Mahoney (2011), who observed that coresets implicitly introduce regularization and lead to more robust output. Table 1 summarizes the empirical relative errors \|f (X, γ̃) − f (X, γ ∗ )\|/f (X, γ ∗ ) between (C/U)DNs γ̃ and DNs γ ∗ trained on all the data. CDNs clearly recover the original model, at a fraction of training data. Overall, this answers (Q1) affirmatively. Relationship Elucidation (Q2): We investigated the performance of CDNs when recovering the graph structure of word interactions from a text corpus. For this purpose, we used the NIPS2 bag1. http://yann.lecun.com/exdb/mnist/ 2. https://archive.ics.uci.edu/ml/datasets/bag+of+words 11 skill item loss component rotation direction tree cell digit map skill set document item disparity object pca oscillator neuron distance pyramid tangent loss component estimator dialogue routing saliency road iiii policy context fuzzy option wavelet speaker user control call star letter delay building attractor controller subscriber block eeg neural evidence potential classifier spike channel instruction rule stress rules chip net lesion circuit light rotation direction tree cell digit map skill set document item disparity object pca oscillator neuron distance pyramid tangent estimator dialogue routing saliency road iiii policy context fuzzy option wavelet speaker user control call star letter delay building attractor controller subscriber block eeg neural evidence potential classifier spike channel instruction rule stress rules chip net lesion circuit light analog set document loss component disparity object estimator pca oscillator neuron dialogue distance routing pyramid saliency tangent road iiii policy context fuzzy option wavelet speaker user control call star letter delay building attractor controller subscriber block eeg neural evidence potential classifier spike channel instruction rule stress rules chip net lesion circuit light rotation direction tree cell digit map analog analog Gaussian CDN Poisson CDN skill item loss component rotation direction tree cell road digit map routing policy object pca neuron pyramid skill set document item disparity estimator oscillator distance dialogue saliency tangent context user iiii fuzzy option wavelet speaker control call star letter delay building attractor controller subscriber block eeg neural evidence potential classifier spike channel instruction rule stress rules chip net lesion circuit light analog loss component rotation direction tree cell road digit map routing policy object pca neuron pyramid skill set document item disparity estimator oscillator distance dialogue saliency tangent context user iiii fuzzy option wavelet speaker control call star letter delay building attractor controller subscriber block eeg neural evidence potential classifier spike channel instruction rule stress rules chip net lesion circuit light analog loss component rotation direction tree cell road digit map routing policy set document object pca neuron pyramid disparity estimator oscillator distance dialogue saliency tangent context user iiii fuzzy option wavelet speaker control call star letter delay building attractor controller subscriber block eeg neural evidence potential classifier spike channel instruction rule stress rules chip net lesion circuit light analog Figure 3: (Q2) Elucidating the relationships between random variables. Shown are the (positive) dependency structures of Gaussian (top) and Poisson (bottom) CDNs on NIPS and different learning sampling sizes: using 40% (Left) , 70% (Middle) and 100% (Right). The edges show the 70 top thresholded positive coefficients of the GLMs. The colors of the edges represent modularity. As one can see, CDNs elucidate relationships among the words that make semantically sense and approach the structure learned using the full dataset. For a quantitative assessment, see Tab. 2. (Best viewed in color) of-words dataset. It contains 1,500 documents with a vocabulary above 12k words. We considered the 100 most frequent words. Fig. 3 illustrates the results qualitatively. It shows three CDNs of sampling sizes 40%, 70% and 100% for Gaussians (top) after a log(x+1) transformation and for Poissons (bottom): CDNs capture well the gist of the NIPS corpus. Table 2 confirms this quantitatively. It shows the Frobenius norms between the DNs: CDNs capture the gist better than naive, i.e., uniform sampling. This answers (Q2) affirmatively. To summarize our empirical results, the answers to questions (Q1) and (Q2) show the benefits of CDNs. 7. Conclusions Inspired by the question of how we can train graphical models on a massive dataset, we have studied coresets for estimating Dependency networks (DNs). We established the first rigorous guarantees for obtaining compressed ε-approximations of Gaussian DNs for large data sets. We proved worstcase impossibility results on coresets for Poisson DNs. A review of log-normal Poisson modeling of counts provided deep insights into why our coreset construction still performs well for count data in practice. 12 Sample portion 40% 70% UDN Gaussian 9.0676 4.8487 CDN Poisson 6.4042 1.6262 Gaussian 3.9135 2.6327 Poisson 0.6497 0.3821 Table 2: (Q2) Frobenius norm of the difference of the adjacency matrices (the lower, the better) recovered by DNs trained on the full data and trained on a uniform subsample (UDN) resp. coresets (CDNs) of the training data. The best results per statiscal type (Gaussian/Poisson) are bold. CDNs recover the structure better than UDNs. Our experimental results demonstrate, the resulting Core Dependency Networks (CDNs) can achieve significant gains over no or naive sub-sampling, even in the case of count data, making it possible to learn models on much larger datasets using the same hardware. CDNs provide several interesting avenues for future work. The conditional independence assumption opens the door to explore hybrid multivariate models, where each variable can potentially come from a different GLM family or link function, on massive data sets. This can further be used to hint at independencies among variables in the multivariate setting, making them useful in many other large data applications. Generally, our results may pave the way to establish coresets for deep models using the close connection between dependency networks and deep generative stochastic networks Bengio et al. (2014), sum-product networks Poon and Domingos (2011); Molina et al. (2017), as well as other statistical models that build multivariate distributions from univariate ones Yang et al. (2015). Acknowledgements: This work has been supported by Deutsche Forschungsgemeinschaft (DFG) within the Collaborative Research Center SFB 876 ”Providing Information by Resource-Constrained Analysis”, projects B4 and C4. 13 References Pankaj K. Agarwal and R. 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PROC. OF THE 6th EUR. CONF. ON PYTHON IN SCIENCE (EUROSCIPY 2013) 15 CATOS: Computer Aided Training/Observing System Jinook Oh∗† arXiv:1404.6384v1 [cs.CE] 25 Apr 2014 F Abstract—In animal behavioral biology, there are several cases in which an autonomous observing/training system would be useful. 1) Observation of certain species continuously, or for documenting specific events, which happen irregularly; 2) Longterm intensive training of animals in preparation for behavioral experiments; and 3) Training and testing of animals without human interference, to eliminate potential cues and biases induced by humans. The primary goal of this study is to build a system named CATOS (Computer Aided Training/Observing System) that could be used in the above situations. As a proof of concept, the system was built and tested in a pilot experiment, in which cats were trained to press three buttons differently in response to three different sounds (human speech) to receive food rewards. The system was built in use for about 6 months, successfully training two cats. One cat learned to press a particular button, out of three buttons, to obtain the food reward with over 70 percent correctness. Index Terms—animal training, animal observing, automatic device 1 I NTRODUCTION It is often the case in animal behavioral biology that a large amount of human resources, time, and data storage (such as video recordings) are required in animal observation and training. Some representative examples of these cases are: • Observation of certain species continuously or monitoring for specific events, which occur irregularly, when behavior of certain species during any time period or specific time period, such as nocturnal behaviors, are investigated. • Certain experiments require a prolonged training period, sometimes over a year. This type of experiment requires reliable responses, which may not correspond to usual behavior patterns, from animals in tasks. Therefore, training may require a long period of time until the subject is ready to be tested. Additionally, long periods of human supervised training can introduce unintended cues and biases for animals. In the first case, an autonomous system for observing animals can save human resources and reduce the amount of data storage. The reduced amount of data can also conserve other types of human resources such as investigation and maintenance of large-scale data. There have been attempts to build autonomous observing or surveillance systems in the fields of biology, such as Kritzler et al. [Kri08]’s work, and * Corresponding author: [email protected] † Cognitive Biology Dept., University of Vienna c 2014 Jinook Oh. This is an open-access article disCopyright ○ tributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. http://creativecommons.org/licenses/by/3.0/ security systems, such as Belloto et al. [Bel09], Vallejo et al. [Val09], for instance. There are also commercial products for surveillance systems with various degrees of automation, or incorporating artificial intelligence. However, the intelligence of each system is case-specific and it is difficult to apply these specific systems to novel situations without considerable adjustments. In the second case, an autonomous system for prolonged, intensive training can also save human resources and eliminate potential cues and biases caused by humans. Training with an autonomous system is an extension of traditional operant conditioning chambers and many modern and elaborated versions have been developed and used, such as in Markham et al. [Mar96], Takemoto et al. [Tak11], Kangas et al. [Kan12], Steurer et al. [Ste12], and Fagot & Bonte [Fag09]. However, many of the previous devices use commercial software. Also, they do not possess the observational features developed in the current project. It would be useful to have an open-source, relatively low-budget, and modularized system which could be customized for the observation, training and the experimentation on animal subjects of various species. CATOS, the system built in the present study, fulfills these necessities. The difference between the previous systems and CATOS (Computer Aided Training/Observing System) in the present work is that the animals do not have to be captured or transported to a separated space at a specific time in order to be trained. The disadvantages of separating animals (e.g., primates) are well-known, and include stress on animals separated from their group or moved from their usual confines, the risky catching procedure for both animal and human (cf. Fagot & Bonte [Fag09]). Similar arguments apply to most animal species, especially when they are social. The automatic learning device for monkeys (ALDM) described in Fagot & Bonte [Fag09] is very similar to the trainer aspect of CATOS described in the present work, but CATOS is different in following features. First of all, it aimed to be opensource based and more modular so that it can be more easily adjusted and adopted to different species and experiments. Another feature is that CATOS is equipped with various observational features, including visual and auditory recording and recognition through video camera and microphone, which make the system able to interact with the subjects, such as reacting immediately to a subject with a motion detection from a camera or a sound recognition from a microphone. CATOS should offer the following advantages. • The system should be flexible in terms of its adjustability and the extendibility to various projects and species. The 16 PROC. OF THE 6th EUR. CONF. ON PYTHON IN SCIENCE (EUROSCIPY 2013) • • • • software should be open-source, and both software and hardware components should be modularized as much as possible, thus the system reassembly for researchers in animal behavioral biology is practical. The system should have various observational features applicable to a broad range of animal species and observational purposes. The system should perform continuous monitoring, and it should record video and/or sound only when a set of particular conditions is fulfilled. This would reduce the amount of data produced during the procedure. The system should have actuators to react in certain situations, which allows it to act as a trainer/experimenter. The human trainer/experimenter designs the procedure by adjusting parameters and modules, but the actual performance should be done by the system. In this way, the system could help reducing the amount of time required for training, and eliminating cues/biases which might be induced by the human interferences. With this system, the animal should not have to be transported to a certain space, or separated from its group, for training. The animals should be able to choose when to start a trial on their own. Two CATOS prototypes have been built during this study. The first build of CATOS has 3 pushbuttons as a main input device for cats and the second build has a touch-screen as a main input device. The first build was an initial attempt to build and test such a system. The second build is the final product of the study. The basic structures of these two builds are more or less the same. The differences are that the second version has improved functions and it uses the touchscreen instead of pushbuttons. The first build of CATOS was tested with domestic cats (Felis catus) to train them to press three different buttons differently depending on the auditory stimuli (three different human speech sounds). The final goal of this training is to investigate human speech perception in cats. There is no doubt in that many animal species can recognize some words in human speech. The examples of speech perception in dogs and chimpanzees can be found in the work of Kaminski et al. [Kam04] and Heimbauer et al. [Hei11] respectively. In some cases, animals can even properly produce words with specific purposes. An example of speech perception and production in a parrot can be found in the work of Pepperberg [Pep87]. Despite these findings, there is ongoing debate about whether the same perceptual mechanisms are used in speech recognition by humans and animals (Fitch [Fit11]). To investigate this issue, animals have to be trained to show different and reliable responses to different human speech sounds. Then, we can test which features of human speech are necessary for different animal species to understand it. Thus, the final aim of the training in this study would be to obtain cats showing different responses to different human speech sounds with statistical significance (over 75 percent). Before reaching this final goal, several smaller steps and goals are required. Fig. 1: Overall system diagram. 2 B RIEF DESCRIPTION OF CATOS (C OMPUTER A IDED T RAINING /O BSERVING S YSTEM ) The overall system is composed of a combination of software and hardware components. The software components are mainly composed of the Python script named as ’AA.<version>.py’ and the program for the microcontroller. The ’AA’ runs all of the necessary processes and communicates with the microcontroller program. The microcontroller program operates sensors and actuators as it communicates with the ’AA’ program. The hardware components are composed of various devices, some of which are directly connected to the computer via USB cables. Some other devices only have GPIO (General Purpose Input Output) pins; therefore they are connected to the microcontroller. The microcontroller itself is connected to the computer via a USB cable. The hardware devices, which are directly connected via USB cables, can be accessed using various software modules, which are imported into the ’AA’ program. The access to other devices only using GPIO pins is performed in the microcontroller and the ’AA’ program simply communicates with the microcontroller program via a serial connection for sending commands to actuators and receiving values from sensors. The software for this system is called AA (Agent for Animals). This software was build with helps of many external libraries such as OpenCV [Bra00], and NumPy/SciPy [Jon01]. Once it starts, seven processes were launched using multiprocessing package of Python and it runs until the user terminates the program. The multiprocessing was used because the heavy calculation for image processing from multiple webcams were concerned. The number of processes can be changed as some of them can be turned on or off. These processes include a video-in process for each camera, a video-out process, an audio-in process, an audio-out process, a schema process, and a message-board process. Figure 1. Even though some of these processes have quite simple tasks, they were separated in order to prevent them from interfering with each other and/or becoming the bottleneck. The system has to process the visual, auditory, and other sensory and motor information simultaneously to recognize the change of the environment and CATOS: COMPUTER AIDED TRAINING/OBSERVING SYSTEM 17 Fig. 2: AA_DataViewer respond to it properly. The output data such as captured video input images, recorded WAV files, movement-records, CSV files for trial results, and the log file are temporarily stored in the ’output’ folder. After the daily session is finished, all of these output files go through an archiving process which can include, but is not restricted to, generating movies, generating images with the movement analysis, labeling sound files, and moving different types of files into the categorized subfolders of an archiving folder named with a timestamp. Besides combining all the above modules and implementing some common functions, one more Python program was implemented to facilitate the process of analyzing the recorded data. The program is called ”AA DataViewer”, which is based on wxPython GUI toolkit and Matplotlib [Hun07] for drawing graphs. Figure 2. It loads the log file, the result CSV (comma separated values) file containing the results of the trial, the movement-record CSV files, the MP4 movie files, and the WAV files from one folder containing all data collected for one session (day). For each video clip, there is a JPEG image showing the movements of the blobs. The circles in the image represent the positions of the blobs and their color represents the time-flow, with the black corresponding to the beginning of the movie, and the white to the end of the movie. A line connecting multiple circles means that those blobs occurred at the same time. Another feature of this program is its ability to generate a graph with selected sessions. In the ’archive’ folder, there are sub-folders, each of which contains all the data for a session. When the ’select sessions’ button is clicked, a pop-up window appears for selecting multiple folders. The result data from these selected sub-folders of ’archive’ folder is drawn as a graph using Matplotlib [Hun07]. By visualizing the data for certain period, it helps the trainer or experimenter quickly assess the current status of the training procedure. The two feeders used in this study is a device mainly comprising the Arduino microcontroller; (refer http://www. arduino.cc/), a motor-shield for the microcontroller, a servomotor, and a frame encasing the whole feeder. Both Feeder variants work in a similar way, by rotating the servomotor by a certain number of degrees, although the second feeder shows better performance in terms of consistent amount of Fig. 3: Automatic feeder Fig. 4: Circuit with a microcontroller food released, due to the usage of an Archimedes’ screw. Initially, an estimate of the amount of food left in the food container was obtained using an IR distance sensor, but this feature was discarded in the second build since the distance information from the IR sensor was not accurate enough for this application. The second feeder confirms the emission of a food reward via the piezoelectric sensor, which is positioned right below the Archimedes’ screw. Figure 3. Communication between the Arduino chip and the main computer was accomplished by using the Arduino module of the ’AA’ program. In the circuit, Figure 4, • • • • The temperature sensor measures the temperature inside of the protective wooden platform. The photocell sensor measures the ambient light level. The light bulb can be turned on when the photocell sensor indicates the ambient light level is below a user-defined threshold. Two fans are turned on when the temperature sensor indicates the temperature is too high in the platform. 18 PROC. OF THE 6th EUR. CONF. ON PYTHON IN SCIENCE (EUROSCIPY 2013) • • The piezoelectric sensor is read while the servomotor is actuating, in order to confirm the occurrence of the food reward. This sensor reading is required because occasionally the food dispensing fails due to the combination of the short motor activation time (<0.5 seconds) and the shape of the dry food pieces (which can fit into other pieces easily and then fail to emerge). The servomotor is responsible for the food dispense by turning the Archimedes’ screw back and forth. 3 R ESULTS OF BUILDING CATOS AND ITS TESTING ON 2 DOMESTICATED CATS The hardware and software were built and tested. The software is available at https://github.com/jinook0707/CATOS_alpha with GNU General Public License, version 3. Both hardware and software are curretnly in its alpha stage. Although its potential to be used to train and test animal cognition was tested and its usage seemed promising to save human resources in certain situations, both hardware and software should be developed further to be practically used for experimenting animal cognition. The two web-cams observed the experimental area for 8 to 12 hours per day for about 5 months (from the middle of October 2012 to the middle of March 2013). The movement records, MP4 movie files, JPEG image files, and WAV sound files generated during this period took 37.35 Giga bytes of storage. To obtain a rough idea of the degree of reduction in data storage that was achieved using the system, the number of recorded frames in the video recording was assessed. Data for 15 days were taken to calculate it. The total observation period was 406138 seconds, corresponding to 112.8 hours. The number of frames recorded was 206024 and the average FPS(Frame Per Second) was 7.5, therefore, approximately, the video recordings were stored for 27470 seconds (=7.6 hours), which is about 6.7 percent of entire observation period. These specific numbers are not very meaningful since they can fluctuate with the increase or decrease of the subject’s movements, but the point is that the most of the meaningless recordings were successfully filtered out by CATOS. Human presence during session is not necessary. Data transfer from one computer to another, maintenance, or modification of the system requires human interaction, but no time and effort is required concerning the training and testing sessions. Because no one attends the sessions, a periodic analysis of the animal’s performance with the system is required. A simple assessment of how much food the animals took, or more specifically, how many correct and incorrect trials occurred, can be done quickly since this information is already stored in result CSV file displaying the number of correct and incorrect trials generated with timestamps at the end of each session. Also, the data-viewer utility program displays all the timestamps and its JPEG image, which presents a brief report on the movement detected in the recorded video-clip. Thus, simply browsing the JPEG images is often enough to assess the session. If it is not enough, then one can obtain a more detailed assessment by playing the video-clips recorded around the trial times. Fig. 5: Recent performance of the trained cat on three human speech discrimination task. Two domesticated cats were trained for testing the system. Both cats learned that approaching the feeder on a playback sound could lead to a food reward. Then one cat further learned that pressing one out of three buttons could lead to a food reward. The training of the association between three different sound stimuli and three different buttons is an ongoing process. The most recent performance data Figure 5 shows over 70 percent of overall performance and also the performance on each button is significantly higher than 33.3 percent of chance level. R EFERENCES [Bel09] N. Bellotto, E. Sommerlade, B. Benfold, C. Bibby, I. Reid, D. Roth, C. Fernandez, L.V. Gool and J. Gonzalez. A distributed camera system for multi-resolution surveillance, Proc. of the third ACM/IEEE Int. Conf. on Distributed Smart Cameras (ICDSC), 2009. [Bra00] G. Bradski. The OpenCV Library, Dr. Dobb’s Journal of Software Tools, 25(11):122-125, Nov 2000. [Jon01] E. Jones, T. Oliphant, P. Peterson, and others, SciPy: Open source scientific tools for Python, 2001. [Fag09] J. Fagot and D. Paleressompoulle. Automatic testing of cognitive performance in baboons maintained in social groups, Behavior Research Methods, 41(2):396-404, May 2009. [Fag10] J. Fagot and E. Bonte. Automated testing of cognitive performance in monkeys: Use of a battery of computerized test systems by a troop of semi-free-ranging baboons (Papio papio), Behavior Research Methods, 42(2):507-516, May 2010. [Fit11] T. Fitch. (2011). Speech perception: a language-trained chimpanzee weighs in, Current Biology, 21(14):R543-R546, July 2011. [Hei11] L.A. Heimbauer, M.J. Beran and M.J. Owren. A chimpanzee recognizes synthetic speech with significantly reduced acoustic cues to phonetic content, Current Biology, 21(14):1210-1214, June 2011. [Hun07] J. D. 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Evidence for conceptual quantitative abilities in the African grey parrot: labeling of cardinal sets, Ethology, 75(1):37-61, 1987. [Ste12] M.M. Steurer, U. Aust and L. Huber. The Vienna comparative cognition technology(VCCT): An innovative operant conditioning system for various species and experimental procedures, Behavior Research Methods, 44(4):909-918, December 2012. [Tak11] A. Takemoto, A. Izumi, M. Miwa and K. Nakamura. Development of a compact and general-purpose experimental apparatus with a touch-sensitive screen for use in evaluating cognitive functions in common marmosets, Journal of Neuroscience Methods, 199(1):82-86, July 2011. [Val09] D. Vallejo, J. Albusac, L. Jimenez, C. Gonzalez and J. Moreno. A cognitive surveillance system for detecting incorrect traffic behaviors, Expert Systems with Applications, 36(7):1050310511, September 2009. 19 20 PROC. OF THE 6th EUR. CONF. ON PYTHON IN SCIENCE (EUROSCIPY 2013)	5
Graphical Nonconvex Optimization for Optimal Estimation in Gaussian Graphical Models Qiang Sun⇤, Kean Ming Tan†, Han Liu‡ and Tong Zhang§ arXiv:1706.01158v1 [stat.ML] 4 Jun 2017 Abstract We consider the problem of learning high-dimensional Gaussian graphical models. The graphical lasso is one of the most popular methods for estimating Gaussian graphical models. However, it does not achieve the oracle rate of convergence. In this paper, we propose the graphical nonconvex optimization for optimal estimation in Gaussian graphical models, which is then approximated by a sequence of convex programs. Our proposal is computationally tractable and produces an estimator that achieves the oracle rate of convergence. The statistical error introduced by the sequential approximation using the convex programs are clearly demonstrated via a contraction property. The rate of convergence can be further improved using the notion of sparsity pattern. The proposed methodology is then extended to semiparametric graphical models. We show through numerical studies that the proposed estimator outperforms other popular methods for estimating Gaussian graphical models. Keywords: Adaptivity, Graphical nonconvex optimization, Nonconvexity, Semiparametric, Sequential convex approximation. 1 Introduction We consider the problem of learning an undirected graph G = (V, E), where V = {1, . . . , d} contains nodes that represent d random variables, and the edge set E describes the pairwise conditional dependence relationships among the d random variables. Gaussian graphical models have been widely used to represent pairwise conditional dependencies among a set of variables. Let X be a d-dimensional random variables. Under the Gaussian assumption X ⇠ N (0, ⌃⇤ ), the graph G is encoded by the sparse concentration matrix ⌦ = (⌃⇤ ) 1 , or the sparse inverse correlation matrix ⇤ = (C⇤ ) 1 . Here, C⇤ is the correlation matrix such that ⌃⇤ = WC⇤ W and W2 is a diagonal matrix with ⇤ Department of Operations Research and NJ 08544; e-mail: [email protected]. † Department of Operations Research and NJ 08544; e-mail: [email protected]. ‡ Department of Operations Research and NJ 08544, USA; e-mail:[email protected]. § Tencent AI Lab, Shen Zhen, Guangdong, Financial Engineering, Princeton University, Princeton, Financial Engineering, Princeton University, Princeton, Financial Engineering, Princeton University, Princeton, China; e-mail:[email protected]. 1 diagonal elements of ⌃⇤ . In particular, it is well known that the jth and kth variables are conditionally independent given all of the other variables if and only if the (j, k)-th element of ⌦⇤ (or ⇤ ) is equal to zero. Thus, inferring the conditional dependency structure of a Gaussian graphical model boils down to estimating a sparse inverse covariance (or correlation) matrix. A number of methods have been proposed to estimate the sparse concentration matrix under the Gaussian assumption. For example, Meinshausen and Bühlmann (2006) proposed a neighborhood selection approach for estimating Gaussian graphical models by solving a collection of sparse linear regression problems using the lasso penalty. In addition, Yuan (2010) and Cai et al. (2011) proposed the graphical Dantzig and CLIME, both of which can be solved efficiently. From a di↵erent perspective, Yuan and Lin (2007) and Friedman et al. (2008) proposed the graphical lasso methodology, a penalized likelihood based approach, to estimate the concentration matrix ⌦⇤ directly. Various extensions of the graphical lasso were proposed and the theoretical properties were also studied (among others, Banerjee et al., 2008; Rothman et al., 2008; Ravikumar et al., 2011). The Gaussian graphical models literature is vast and we refer the reader to Cai et al. (2016a) and Drton and Maathuis (2016) for recent reviews on this topic. Despite the large literature on using the graphical lasso to estimate concentration matrices in Gaussian graphical models, the graphical lasso does not achieve the oracle rate of convergence. More specifically, it is belived that the optimal rate of convergence p in spectral norm for the graphical lasso is at the order of s log d/n (Rothman et al., 2008). Here, n is the sample size, d is the number of nodes, and s is the number of edges in the true graph. In fact, the graphical lasso and all of the aforementioned methods are based on the lasso penalty and it is well known that convex penalties usually introduce non-negligible estimation bias. For example, in the linear regression setting, Fan and Li (2001); Zhang (2010a,b); Fan et al. (2017) have shown that the nonconvex penalized regression is able to eliminate the estimation bias and attain a more refined statistical rate of convergence. Based on these insights, we consider the following penalized maximum likelihood estimation with nonconvex regularizers: ⇢ X ⌦ ↵ b b ⇥ = argmin ⇥, ⌃ log det(⇥) + p ⇥ij , (1.1) d ⇥2S+ i6=j d = {A 2 Rd⇥d : A = AT , A where S+ 0} is the symmetric definite cone formed by all b is the sample covariance symmetric positive definite matrices in d ⇥ d dimensions, ⌃ matrix, and p (·) is a nonconvex penalty. Here, hA, Bi = tr(AT B) denotes the trace of AT B. However, from the computational perspective, minimizing a folded concave penalized problem is very complicated due to its intrinsic nonconvex structure. Indeed, Ge et al. (2015) have shown that solving (1.1) with a general concave penalty, such as the SCAD Fan and Li (2001) or the MCP Zhang (2010a), is strongly NP-hard. In other words, there does not exist a fully polynomial-time approximation scheme for problem (1.1) unless more structures are assumed. Recently, Loh and Wainwright (2015) proposed an algorithm to obtain a good local optimum for (1.1), but an additional convex constraint that depends on the unknown true concentration matrix is imposed. Moreover, 2 they failed to provide a faster rate of convergence statistically due to not taking the signal strength into account. In this paper, instead of directly solving the nonconvex problem (1.1), we propose to approximate it by a sequence of adaptive convex programs. Even though the proposed approach is solving a sequence of convex programs, under some regularity conditions, we show that the proposed estimator for estimating the sparse concentration matrix achieves p the oracle rate of convergence of s/n, treating as if the locations of the nonzeros were known a priori. This is achieved by a contraction property. Roughly speaking, each convex program gradually contracts the initial estimator to the region of oracle rate of convergence even when a bad initial estimator is used in the first place: r s 1 b (` 1) ⇤ ⇤ b (`)  C + , F F n 2 \| {z } \| {z } Oracle Rate Contraction where b (`) is the inverse correlation matrix estimator after the `-th convex approxip mation, k · kF denotes the Frobenius norm, C is a constant, and s/n is referred to as the oracle rate. Each iteration of the proposed method helps improve the accuracy ⇤ k dominates the statistical error. The error caused by each only when k b (` 1) F iteration is clearly demonstrated via the proven contraction property. By rescaling the inverse correlation matrix using the estimated marginal variances, we obtain an estimator of the concentration matrix with spectral norm convergence rate in the order of p p log d/n _ s/n. Here, a _ b = max{a, b} is used to denote the maximum of a and b. By exploiting a novel notion called sparsity pattern, we further sharpens the rate of convergence under the spectral norm. The rest of this paper proceeds as follows. In Section 2, we propose the new methodology and its implementation. Section 3 is devoted to theoretical studies. We show that the proposed methodology can be extended to the semiparametric graphical models in Section 4. Numerical experiments are provided to support the proposed methodology in Section 5. We conclude the paper in Section 6. All the proofs and technical details are collected in the supplementary material. Notation: We summarize the notation that will be used regularly throughout the paper. Given a vector u = (u1 , u2 , . . . , ud )T 2 Rd , we define the `q -norm of u by kukq = P ( dj=1 \|uj \|q )1/q , where q 2 [1, 1). For a set A, let \|A\| denote its cardinality. For a matrix A = (ai,j ) 2 Rd⇥d , we use A 0 to indicate that A is positive definite. For q 1, we use kAkq = maxu kAukq /kukq to denote the operator norm of A. For index sets I, J ✓ {1, . . . , d}, we define AI,J 2 Rd⇥d to be the matrix whose (i, j)-th entry is equal to ai,j if i 2 I and j 2 J , and zero otherwise. We use A B = (aij bij ) to denote the Hadamard product of two matrices A and B. Let diag(A) denote the diagonal matrix consisting diagonal elements of A. We use sign(x) to denote the sign of x: sign(x) = x/\|x\| if x 6= 0 and sign(x) = 0 otherwise. For two scalars fn and gn , we use fn & gn to denote the case that fn cgn , and fn . gn if fn  Cgn , for two positive constants c and C. We say fn ⇣ gn , if fn & gn and fn . gn . OP (·) is used to denote bounded in probability. We use c and C to denote constants that may vary from line to line. 3 2 A Sequential Convex Approximation Let X = (X1 , X2 , . . . , Xd )T be a zero mean d-dimensional Gaussian random vector. Then its density can be parameterized by the concentration matrix ⇥⇤ or the inverse correlation matrix ⇤ . The family of Gaussian distributions respects the edge structure of a graph G = (V, E) in the sense that ⇤ij = 0 if and only if (i, j) 62 E. This family is known as the Gauss-Markov random field with respect to the graph G. The problem of estimating the edge corresponds to parameter estimation, while the problem of identifying the edge set, i.e., the set E ⌘ {i, j 2 V \| i 6= j, ⇤ij 6= 0}, corresponds to the problem of model selection. Given n independent and identically distributed observations {X (i) }ni=1 of a zero mean d-dimensional random vector X 2 Rd , we are interested in estimating the inverse (i) (i) T b = n 1P correlation matrix ⇤ and concentration matrix ⇥⇤ . Let ⌃ 1in X (X ) b =W c 1⌃ bW c 1 , where W c 2 = diag(⌃). b To be the sample covariance matrix and let C estimate b (`) ⇤, we propose to adaptively solve the following sequence of convex programs n⌦ o ↵ (` 1) b = argmin ,C log det( ) + k k1,o↵ , for ` = 1, . . . , T, (2.1) d 2S+ P (` 1) where k⇥k1,o↵ = i6=j \|⇥ij \|, (` 1) = · w b ij is a d ⇥ d adaptive regularization matrix for a given tuning parameter and a weight function w(·), and T indicates the number of total convex programs needed. The weight function w(·) can be taken to be w(t) = p0 (t)/ , where p (t) is a folded concave penalty such as the SCAD or the MCP proposed by Fan and Li (2001) and Zhang (2010a), respectively. To obtain an estimate for the concentration matrix estimator ⇤ , we rescale b (T ) e (T ) = W c 1 b (T ) W c 1 after the T -th convex program. This rescaling helps back to ⇥ e (T ) significantly by eliminating the e↵ect introimprove the rate of convergence for ⇥ duced through the unpenalized diagonal terms. The detailed routine is summarized in Algorithm 1. Algorithm 1 A sequential convex approximation for the graphical nonconvex optimization. b regularization parameter . Input: Sample covariance matrix ⌃, b by C b = W c 1⌃ bW c 1 , where W c 2 is a Step 1: Obtain sample correlation matrix C b diagonal matrix with diagonal elements of ⌃. Step 2: Solve a sequence of graphical lasso problem adaptively n o b (`) = argmin h , Ci b log det( ) + k (` 1) k1,o↵ , d 2S+ and (`) = (`) · w( b ij ), for ` = 1, . . . , T. e (T ) = W c Step 3: Obtain an estimate of ⇥⇤ by ⇥ 1 b (T ) W c 1. The complexity of Step 2 in Algorithm 1 is O(d3 ) per iteration: this is the complexity of the algorithm for solving the graphical lasso problem. We will show in the latter section 4 that the number of iteration can be chosen to be T ⇡ log log d based on our theoretical analysis. Algorithm 1 can be implemented using existing R packages such as glasso. 3 Theoretical Results In this section, we study the theoretical properties of the proposed estimator. We start with the assumptions needed for our theoretical analysis. 3.1 Assumptions Let S = (i, j) : ⇥⇤ij 6= 0, i 6= j be the support set of the o↵-diagonal elements in ⇥⇤ . Thus, S is also the support set of the o↵-diagonal elements in ⇤ . The first assumption we need concerns the structure of the true concentration and covariance matrices. Assumption 3.1 (Structural Assumption). We assume that \|S\|  s, k⌃⇤ k1  M < 1, 0 < "1  min  max  1/"1 < 1, 0 < "2  min (⇥⇤ )  max (⇥⇤ )  1/"2 < 1. 2 2 Here, max = maxj ⌃⇤jj and min = minj ⌃⇤jj , where ⌃⇤ = ⌃⇤ij . Assumption 3.1 is standard in the existing literature for Gaussian graphical models (see, for instance, Meinshausen and Bühlmann, 2006; Yuan, 2010; Cai et al., 2016b; Yuan and Lin, 2007; Ravikumar et al., 2011). We need min and max to be bounded from above and below to guarantee reasonable performance of the concentration matrix estimator (Rothman et al., 2008). Throughout this section, we treat M, "1 , "2 as constants to simplify the presentation. The second assumption we need in our analysis concerns the weight functions, which are used to adaptively update the regularizers in Step 2 of Algorithm 1. Define the following class of weight functions: n o W = w(t) : w(t) is nonincreasing , 0  w(t)  1 if t 0, w(t) = 1 if t  0 . (3.1) Assumption 3.2 (Weight Function). There exists an ↵ such that the weight function w(·) 2 W satisfies w(↵ ) = 0 and w(u) 1/2, where u = c for some constant c. The above assumption on the weight functions can be easily satisfied. For example, it can be satisfied by simply taking w(t) = p0 (t)/ , where p (t) is a folded concave penalty such as the SCAD or the MCP (Fan and Li, 2001; Zhang, 2010a). Next, we impose an assumption on the magnitude of the nonzero o↵-diagonal entries in the inverse correlation matrix ⇤ . Assumption 3.3 (Minimal Signal Strength). Recall that S is the true support set. The minimal signal satisfies that min(i,j)2S ⇤ij (↵ + c) & , where c > 0 is the same constant that appears in Assumption 3.2. Assumption 3.3 is rather mild. In the sub-Gaussian design case, can be taken to p be the order of log d/n, which diminishes quickly as n increases. It is an analogue to the minimal signal strength assumption frequently assumed in nonconvex penalized regression problems (Fan and Li, 2001; Zhang, 2010a). Taking the signal strength into account, we can then obtain the oracle rate of convergence. 5 3.2 Main Theory We now present several main theorems concerning the rates of convergence of the proposed estimator for the sparse inverse correlation and the concentration matrices. The following theorem concerns the rate of convergence for the one-step estimator b (1) obtained from Algorithm 1 when ` = 1. p Proposition 3.4 (One-step Estimator). Let ⇣ log d/n. Under Assumption 3.1, we have r s log d (1) ⇤ b . F n with probability at least 1 8/d, Proof of Proposition 3.4. We collect the proof of Proposition 3.4 in Appendix A in the supplementary material. The above proposition indicates that the statistical error under the Frobenius norm p for the one-step estimator is at the order of s log d/n, which is believed to be unimprovable when one-step convex regularization is used (Rothman et al., 2008; Ravikumar et al., 2011). However, when a sequence of convex programs is used as in our proposal, the rate of convergence can be improved significantly. This is demonstrated in the following theorem. Theorem 3.5 (Contraction Property). Suppose that n & s log d and take such that p ⇣ log d/n. Under Assumptions 3.1, 3.2 and 3.3, b (`) satisfies the following contraction property: b (`) ⇤ F  8k \| with probability at least 1 ⇤ 2 k2 krL( {z Oracle Rate ⇤ 1 b (` 1) )S kF + } \|2 {z Contraction ⇤ , 1  `  T, } F p 8/d. Moreover, if T & log( n) & log log d, we have ✓r ◆ s ⇤ b (T ) = OP . F n Proof of Theorem 3.5. The proof is collected in Appendix A in the supplementary material. Theorem 3.5 establishes a contraction property: each convex approximation contracts the initial estimator towards the true sparse inverse correlation matrix until it reaches p the oracle rate of convergence: s/n. To achieve the oracle rate, we need to solve no more than approximately log log d convex programs. Note that log log d grows very slowly as d increases and thus, in practice, we only need to solve a few convex programs to get a better estimator than existing method such as the graphical lasso. The rate p of convergence s/n is better than the existing literature on likelihood-based methods for estimating sparse inverse correlation matrices (Rothman et al., 2008; Lam and Fan, 2009a; Ravikumar et al., 2011). By rescaling, we obtain a concentration matrix estimator with a faster rate of convergence. 6 Theorem 3.6 (Faster Rate in Spectral Norm). Under the same conditions in Theorem 3.5, we have r ✓r ◆ s log d (T ) ⇤ e ⇥ ⇥ 2 = OP _ . n n Proof of Theorem 3.6. The proof is deferred to Appendix A in the supplementary material. The theorem above provides the optimal statistical rate for estimating sparse concentration matrices using likelihood based methods (Rothman et al., 2008; Lam and Fan, 2009b; Ravikumar et al., 2011). The extra log d term is a consequence of estimating the marginal variances. We further sharpen the obtained theory using a novel notion, called sparsity pattern, as defined below. Definition 3.7 (Sparsity Pattern). For a matrix A = aij , we say Asp = asp ij is the sp sp corresponding sparsity pattern matrix if aij = 1 when aij 6= 0; and aij = 0, otherwise. Let M⇤ be the sparsity pattern matrix of ⇤ or ⇥⇤ . Our next theorem provides an improved rate of convergence using this newly defined notion of sparsity pattern. Theorem 3.8 (Improved Convergence Rate using Sparsity Pattern). Suppose that n & p (s + s2max ) log d and take such that ⇣ log d/n. Let T & log s. Under Assumptions 3.1, 3.2 and 3.3, we have r ◆ ✓ 1 ⇤ b (T ) = OP kM⇤ k2 , and 2 n r r ✓ ◆ 1 log d (T ) ⇤ ⇤ e ⇥ ⇥ 2 = OP kM k2 _ . n n Proof of Theorem 3.8. The proof is deferred to Appendix B in the supplementary material. Theorem 3.8 suggests that the rates of convergence can be bounded using the spectral norm of the sparsity pattern matrix M⇤ , which are sometimes much sharper than those provided in Theorems 3.5 and 3.6. To demonstrate this observation, we consider a sequence of chain graphs specified by the following sparsity pattern matrices: " # A 0 k Mck = , for k = 4, . . . , 50, 0 Id k 1 where Ak 2 R(k+1)⇥(k+1) such that the (i, j)-th entry Ak,ij = 1 if \|i j\|  1, and Ak,ij = 0 otherwise. Id k 1 2 R(d k 1)⇥(d k 1) is the identity matrix. Let sk be the total sparsity of Mck , that is sk = 2k. We plot the ratio of the two rates of convergence for estimating ⇤ in Theorems 3.5 and 3.8, kMc k2 /s , versus s in Figure 1. From Figure 1, we can k k 2 k see that the ratio goes to 0 as the total sparsity increases. This demonstrates that the convergence rate in Theorem 3.8 is indeed much sharper than that in Theorem 3.5, as least for the chain graphs constructed above. We also observe similar but less significant improvement for star-shape graphs. In Figure 2, we give an geometric illustration of the star and chain graphs. 7 Chain Graph 0.8 ● 0.6 ● ● ● ● 0.4 kMck k22 /sk ● ● ● ● ● ● ● ● 0.2 ● ● 20 ● ● ● ● ● ● ● ● ● 40 ● ● ● ● 60 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 80 ● ● 100 sk Figure 1: Convergence rates using sparsity pattern matrix Mck and total sparsity sk . Star Graph Chain Graph Figure 2: An illustration of the star and chain graphs. 4 Extension to Semiparametric Graphical Models In this section, we extend the proposed method to modeling semiparametric graphical models. We focus on the nonparanormal family proposed by Liu et al. (2012), which is a nonparametric extension of the normal family. More specifically, we replace the random variable X = (X1 , . . . , Xd )T by the transformation variable f (X) = (f1 (X1 ), . . . , fd (Xd ))T , and assume that f (X) follows a multivariate Gaussian distribution. Definition 4.1 (Nonparanormal). Let f = {f1 , . . . , fd }T be a set of monotone univariate functions and let ⌃npn 2 Rd⇥d be a positive-definite correlation matrix with diag(⌃npn ) = 1. A d-dimensional random variable X = (X1 , . . . , Xd )T has a nonparanormal distribution X ⇠ NPNd (f, ⌃npn ) if f (X) ⌘ (f (X1 ), . . . , fd (Xd ))T ⇠ Nd (0, ⌃npn ). We aim to recover the precision matrix ⇥npn = (⌃npn ) 1 . The main idea behind this procedure is to exploit Kendall’s tau statistics to directly estimate ⇥npn , without explicitly calculating the marginal transformation functions {fj }dj=1 . We consider the following Kendall’s tau statistic: X 2 (i) (i0 ) (i) (i0 ) ⌧bjk = sign (Xj Xj )(Xk Xk ) . n(n 1) 0 1i<i n The Kendall’s tau statistic ⌧bjk represent the nonparametric correlations between the empirical realizations of random variables Xj and Xk and is invariant to monotone trans8 ej and X ek be two independent copies of Xj and Xk . The population formations. Let X ej ), sign(Xk X ek ) . We need version of Kendall’s tau is given by ⌧jk ⌘ Corr sign(Xj X the following lemma which is taken from Liu et al. (2012). It connects the Kendall’s tau statistics to the underlying Pearson correlation coefficient ⌃npn . Lemma 4.2. Assuming X ⇠ NPNd (f, ⌃), we have ⌃0jk = sin ⌧jk ·⇡/2 . b = [Sbjk ] for the Motivated by this Lemma, we define the following estimators S unknown correlation matrix ⌃npn : ( sin ⌧bjk ·⇡/2 , j 6= k, ⌧ Sbjk = 1, j = k. Now we are ready to prove the optimal spectral norm rate for the Gaussian copula graphical model. The results are provided in the following theorem. p Theorem 4.3. Assume that n & s log d and let ⇣ log d/n. Under Assumptions 3.1, b (`) satisfies the following contraction property: 3.2 and 3.3, ⇥ b (`) ⇥⇤ ⇥ F 1 b (` 1)  4k⇥⇤ k22 krL(⇥⇤ )S kF + ⇥ ⇥⇤ F , 1  `  T, \| {z } \|2 {z } with probability at least 1 Optimal Rate Contraction 8/d. If T & log( b (T ) ⇥⇤ ⇥ F p n) & log log d, we have ✓r ◆ s = OP . n Proof of Theorem 4.3. The proof is deferred to Appendix C in the supplementary material. 5 Numerical Experiments We compare our proposal to the graphical lasso (glasso) (Friedman et al., 2008) and neighborhood selection (NS) (Meinshausen and Bühlmann, 2006). Each of these approaches learns a Gaussian graphical model via an `1 penalty on each edge. To evaluate the performance across di↵erent methods, we define the true positive rate as the proportion of correctly identified edges in the graph, and the false positive rate as the proportion of incorrectly identified edges in the graph. In addition, we calculate the difference between the estimated and true concentration matrix under the Frobenius norm. We do not compute this quantity for the NS approach since they do not estimate the concentration matrix directly. For our proposal, we consider T = 4 iterations with the SCAD penalty proposed by Fan and Li (2001) that takes the following form: 8 > if \|t\|  , > < p0 (t) = > > :0 \|t\| 1 if < \|t\| < otherwise, 9 , where > 2. In all of our simulation studies, we pick = 2.1. Each of the methods involves a sparsity tuning parameter: we applied a fine grid of tuning parameter values to obtain the curves shown in Figure 3. We consider cases with n = {150, 200} and d = 150 with two set-ups for a p ⇥ p adjacency matrix A: (i) random graph with 2.5% elements of A set to 1; (ii) band graph with Ai,i+1 = Ai+1,i = 1 for 1  i  d 1. We then use the adjacency matrix A to create a matrix E, as ( 0 if Aij = 0 Eij = 0.4 otherwise, and set E = 12 (E+ET ). Given the matrix E, we set ⇥ 1 equal to E+(0.1 emin )I, where emin is the smallest eigenvalue of E. We then standardize the matrix ⇥ 1 so that the i.i.d. diagonals are equal to one. Finally, we generate the data according to X (1) , . . . , X (n) ⇠ N (0, ⌃). We present the results averaged over 100 data sets for each of the two simulation settings with n = {150, 200} and p = 150 in Figure 3. Random graph (n=200) 0.0 ● ● ● ● 1.0 0.8 0.4 0.0 0.2 0.4 0.6 0.8 0.2 0.0 ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 3 Our Proposal Glasso 2 0.0 0.1 0.2 ● ● ● ● ● ● ●●●●● ● ●●●● ●●●●●●●●●● ● ●●●●● ● ● ●●● ●● ● ● ● ● ● ● ● ● 0.3 False positive rate 0.4 5 Our Proposal NS Glasso 0.2 ●● ● ● ● ● 0.4 0.6 0.8 0.8 Our Proposal NS Glasso ● ● 0.0 7 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● Our Proposal Glasso 2 0.0 0.1 0.2 0.3 0.4 6 0.2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● 0.4 0.6 0.8 0.0 ● 0.2 8 7 ● ● ● ● ● 4 ● ● ● ● ● 3 ● ● ● ● ● ● ● ● ● Our Proposal Glasso ● ● ● ● ● ● 0.0 0.1 0.2 0.3 0.4 0.6 0.8 1.0 False positive rate ● False positive rate False positive rate ●● Our Proposal NS Glasso ● ● 0.0 1.0 ● 5 2 ● ● ● 0.2 Band graph (n=150) ● ● ●● ● ● False positive rate ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.4 ● 0.0 ● ● ● ● ● ● ● ●●● ●●●● ●●● ●●●●●●●●● ●● ●●● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.6 0.2 1.0 1.0 ● ● 8 ● 3 ●● 0.4 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 4 ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Random graph (n=200) Frobenius norm Frobenius norm 4 ● 0.8 8 6 Band graph (n=200) Band graph (n=150) 1.0 ● ●● 0.6 7 ● ● ● False positive rate 7 5 ●● ● ● ● ● ● ● ● ● 0.0 1.0 ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● Random graph (n=150) 6 ●● ● ● False positive rate 8 ●● ● ● ● ● 0.6 Our Proposal NS Glasso ●● ● ● ●● ● ● ● ●● ●●●● ● ●●●● ● ● ● ●● ●●● ●●● ●● ● ●● ● True positive rate 0.2 ● 0.4 Frobenius norm 0.4 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Frobenius norm 0.6 ● ● True positive rate True positive rate 0.8 ● ● ● ● ● ● ● ●● ● ● ● ●●● ● ● ●● ● ● ●● ● ●●●●● ●● ●●● ●● ●●● ● ● ● True positive rate Random graph (n=150) 1.0 6 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● Band graph (n=200) ● ● 5 ● ● ● ● 4 ● ● ● ● 3 2 ● ● ● ● ● ● ● ●● ● ● 0.0 ● ● ● ● Our Proposal Glasso ● 0.1 0.2 0.3 0.4 False positive rate Figure 3: Row I: True and false positive rates, averaged over 100 data sets with p = 150, for random and band graphs, respectively. Row II: Di↵erence between the estimated and the true inverse covariance matrices under the Frobenius norm. The di↵erent curves are obtained by varying the sparsity tuning parameter for each of the methods. From Row I of Figure 3, we see that our proposal is very competitive relative to the existing proposals for estimating Gaussian graphical models in terms of true and false positive rates across all simulation settings. Row II of Figure 3 contains the di↵erence between the estimated and the true inverse covariance matrices under the Frobenius norm as a function of the false positive rate. For random graph with n = 150, we see that the minimum error under the Frobenius norm for our proposal is smaller than that of the graphical lasso. As we increase the number of observations to n = 200, the di↵erence between the minimum error for the two proposals are more apparent. More interestingly, 10 the region for which our proposal has lower Frobenius norm than the graphical lasso is the primary region of interest. This is because an ideal estimator is one that has a low false positive rate while maintaining a high true positive rate with low error under the Frobenius norm. In contrast, the region for which the graphical lasso does better under the Frobenius norm is not the primary region of interest due to the high false positive rate. We see similar results for the band graph setting. 6 Conclusion and Discussions We propose the graphical nonconvex optimization, which is then approximated by a sequence of convex programs, for estimating the inverse correlation and concentration matrices with better rates of convergence comparing with existing approaches. The proposed methodology is sequential convex in nature and thus is computationally tractable. Yet surprisingly, it produces estimators with oracle rate of convergence as if the global optimum for the penalized nonconvex problem could be obtained. Statistically, a contraction property is established: each convex program contracts the previous estimator by a 0.5-fraction until the optimal statistical error is reached. Our work can be applied to many di↵erent topics: low rank matrix completion problems, high-dimensional quantile regression and many others. We conjecture that in all of the aforementioned topics, a similar sequential convex approximation can be proposed and can possibly give faster rate, with controlled computing resources. It is also interesting to see how our algorithm works in large-scale distributed systems. Is there any fundamental tradeo↵s between statistical efficiency, communication and algorithmic complexity? We leave these as future research projects. References Banerjee, O., El Ghaoui, L., and d’Aspremont, A. 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(2010), “High dimensional inverse covariance matrix estimation via linear programming,” Journal of Machine Learning Research, 11, 2261–2286. Yuan, M. and Lin, Y. (2007), “Model selection and estimation in the Gaussian graphical model,” Biometrika, 94, 19–35. Zhang, C.-H. (2010a), “Nearly unbiased variable selection under minimax concave penalty,” The Annals of Statistics, 38, 894–942. Zhang, T. (2010b), “Analysis of multi-stage convex relaxation for sparse regularization,” The Journal of Machine Learning Research, 11, 1081–1107. 12 Supplementary Material to “Graphical Nonconvex Optimization for Optimal Estimation in Gaussian Graphical Models” Qiang Sun, Kean Ming Tan, Han Liu and Tong Zhang Abstract This supplementary material collects proofs for the main theoretical results in the main text and additional technical lemmas. The proofs of Proposition 3.4, Theorems 3.5 and 3.6 are collected in Section A. Section B provides the proof for Theorem 3.8. Proofs related to semiparametric graphical models are given in Section C. Various concentration inequalities and preliminary lemmas are postponed to Sections D and E, respectively. A Rate of Convergence in Frobenius Norm This section presents an upper bound for the adaptive estimator b (`) in Frobenius norm, which in turn helps establish the scaling conditions needed to achieve the optimal spectral norm convergence rate. A.1 Proofs of Proposition 3.4, Theorems 3.5 and 3.6 In this section, we collect the proofs for Proposition 3.4, Theorems 3.5 and 3.6. (` 1) In order to suppress the noise at the `th step, it is necessary to control min(i,j)2S b ij in high dimensions. For this, we construct an entropy set, E` , of S and analyze the mag(` 1) nitude of . The entropy set at the `-th stage, E` , is defined as Ec min ` n E` = (i, j) : (i, j) 2 S or (` 1) ij < w(u), for u = 2 32k ⇤ 2 k2 + k⌃⇤ k21 _ 1 o . (A.1) Thus the constant in Assumption 3.3 is c = 2(32k ⇤ k22 + k⌃⇤ k21 _ 1). Then it can be seen that S ✓ E` , and thus E` is an entropy set of S for any ` 1. Proposition 3.4 follows from a slightly more general result below, which establishes rate of convergence for the one-step estimator of sparse inverse correlation matrix b (1) . Proposition A.1 (One-step Estimator). Assume that assumption 3.1 holds. Suppose p p 8k ⇤ k22 s < 1. Take such that ⇣ (log d)/n and suppose n & log d. Then with probability at least 1 8/d, b (1) must satisfy r s log d ⇤ b (1)  Ck ⇤ k22 . F n 1 b C⇤ kmax  /2 . Then in the Proof of Proposition A.1. Define the event J = kC ⇤ k  4k ⇤ k2 · event J , by applying Lemma q A.4 and taking E = S, we obtain k b (1) F 2 p p p 1 s. If we further take = 3c2 (log d)/n ⇣ (log d)/n, then by Lemma D.5, we have event J hold with probability at least 1 8d 1 . The result follows by plugging the choice of . Theorems 3.5 and 3.6 follow form a slightly more general result below, which chare (T ) in spectral acterizes the rate of convergence of b (`) in Frobenius norm and that of ⇥ norm. p Theorem A.2. Assume that assumptions 3.1, 3.2 and 3.3. Suppose that 8k ⇤ k22 s < p 1. Take such that ⇣ log d/n. Then with probability at least 1 8d 1 , b (`) satisfies b (`) ⇤ F  8k \| Moreover, if that T & log( ⇤ 2 k2 krL( ⇤ {z Optimal Rate p 1 b (` 1) )S kF + } \|2 {z , 1  `  T. } F Contraction p s/n , and F r r ◆ ⇤k log d _ k ⇤ k22 s 2 . 2 n n min min n), we have ✓ 3 k ⇤ ⇥ 2 = OP max 3 e (T ) ⇥ ⇤ b (T ) ⇤ = OP k ⇤ k2 2 Proof of Theorem A.2. Under the conditions of theorem, combining Proposition A.7 and Lemma D.5, we obtain the following contraction property of the solutions, { b (`) }T `=1 , 1 b (` 1) ⇤ . F 2 Next, we introduce an inequality by induction analysis. Specifically, if an  a0 + ↵an 1 , 8 n 2 and 0  ↵ < 1, then b (`) ⇤ F ⇤ 2 k2 krL(  4k an  a0 1 ⇤ ↵n 1 ↵ 1 ) S kF + + ↵n 1 a1 . ⇤ k2 krL( ⇤ ) k , we obtain that b (`) ⇤  8k ⇤ k22 krL( ⇤ )S kF + S F 2 F ⇤ ⇤ k respec. In the sequel, we bound krL( ⇤ )S kF and k b (1) F F p ⇤ k . 8k ⇤ k2 tively. By Proposition A.1, we have k b (1) s. Moreover, if we F 2 p p p T 1 (1) ⇤ ⇤ 2 b let T log( n) log 2 & log( n), then (1/2) k kF  16k k2 · s/n. p ⇤ ⇤ 2 On the other side, we have krL( )S kF = OP (k k2 · s/n), which follows from ⇤k = Lemma D.4. Therefore, combining the above results obtains us that k b (T ) F p OP k ⇤ k22 s/n . e (T ) ⇥⇤ k2 , we apply Lemma E.3 and obtain To achieve the statistical rate for k⇥ Taking a0 = 4k ` 1 b (1) 1/2 that e (T ) k⇥ ⇥ ⇤ k2 = c W + c  kW \| 1 c W 1 c +kW \| W 1 W b (T ) 1 1 W ⇤ ⇤c 1 1 2 b (T ) k2 k {z W ⇤ (R1) 1 W 1 k2 k {z (R3) c W ⇤ W c + W 2 c k2 +kW } \| c k2 kW 2 1 1 1 + 2 1 b (T ) 1 c k2 +kW } \| W c W ⇤ 1 1 W W 1 2 k2 k b (T ) k2 kW {z (R2) 1 1 k2 kW k k b (T ) {z2 1 (R4) b (T ) W 1 k2 } ⇤ k2 . } 1 2 We now bound terms (R1) to (R4) respectively. Before we proceed, we apply Lemma D.2 and the union sum bound to obtain that, for any " 0, ⇣ ⌘ n o n o c 2 W2 k2 > " max ⌃⇤  d · exp P kW n · C(") = exp n · C(")+log d , ii i 1 (" where C(") = 2 log(1 + ")). Suppose that 0  "  1/2, then we have n · C(")  p 2 n · " /3. Further suppose that n 36 log d and take " = 3 (log d)/n, we obtain that n · C(")+log d  2 log d and r ✓ ◆ log d 1 2 2 2 c P kW W k2 > 3 max ·  2, n d 2 . Therefore, we have W c 2 W2 = where we use the assumption that maxi ⌃⇤ii  max 2 ⇣ ⌘ p 2 2 2 c OP max · log d/n . Since W and W are diagonal and thus commutative. We note that, for any two event A and B, P(A) = P(A \ B) + P(A \ B c ) holds. Therefore, for any M > 0, we have r ✓ ◆ log d 1 1 2 c P W W > M max 2 n r ✓ log d c 1 W 1 >M 2 P W , max 2 n ◆ p 2 1 1 2 2 2 c c W W  2( 2+1) W 2 min W W W 2 2 ✓ ◆ p 2 2 c 1 W 1 > 2( 2+1) W c 2 W2 +P W W W . 2 2 min 2 Further using Lemma E.7 yields that r ✓ ◆ log d 1 1 2 c P W W > M max 2 n ✓ p 2 c 2 W2  P 2( 2+1) W 2 min W2 W \| {z +P \| ✓ (T1) c2 W W2 >2 2 {z 1 min W2 ◆ >M 2 2 max r log d n . ◆ } } (T2) 2 4 By taking M = M1 · kWk2 min (W2 ) = M1 · max / min and letting M1 ! 0, we 2 2 get (T1) ! 0. Under the assumption that max / min = O (n/ log d)1/3 , we have p 2 2 c 1 n/ log d , and thus (T2) ! 0. Therefore we obtain that W max / min = o p 4 3 W 1 2 = OP min max (log d)/n . Similarly, we have the following facts: b (T ) 2 = OP k ⇤ k2 , c W 1 2 = 1 min c = OP ( W 1 min ), and W 1 2 = Applying the above results to the terms (R1)-(R4). we obtain that r r ◆ ✓ ◆ ✓ 6 s s max log d 2 2 ⇤ 2 ⇤ 2 (R1) = OP min k k2 · 6 = OP min k k2 , n n min n r r ◆ ✓ 3 ◆ ✓ log d s max 2 ⇤ ⇤ 2 (R2) = (R3) = OP k k2 , (R4) = OP min k k2 . 3 n n min 3 1 min . Therefore, by combining the rate for terms (R1)-(R4), we obtain the final result. A.2 Technical Lemmas s (⇥, ⇥⇤ ) = Define the symmetrized Bregman divergence for the loss function L(·) as DL ⌦ ↵ ⇤ ⇤ d⇥d d⇥d rL(⇥) L(⇥ ), ⇥ ⇥ . For any matrix A 2 R , let A 2 R be the o↵ diagonal matrix of A with diagonal entries equal to 0, and A+ = A A be the diagonal mtrix. Lemma A.3. For the symmetrized Bregman divergence defined above, we have ⌦ s DL (⇥, ⇥⇤ ) = rL(⇥) rL(⇥⇤ ), ⇥ ⇥⇤ ↵ k⇥⇤ k2 + k⇥ 2 ⇥ ⇤ k2 k⇥ ⇥⇤ k2F . Proof of Lemma A.3. We use vec(A) to denote the vectorized form of any matrix A. Then by the mean value theory, there exists a 2 [0, 1] such that, ⌦ s DL (⇥, ⇥⇤ ) = rL(⇥) min (r 2 ↵ ⇥⇤ = vec(⇥ rL(⇥⇤ ), ⇥ L(⇥⇤ + ) k k2F , where ⇥⇤ )T r2 L(⇥⇤ + =⇥ ) vec(⇥ ⇥⇤ ) ⇥⇤ . By standard properties of the Kronecker product and the Weyl’s inequality (Horn and Johnson, 2012), we obtain that ⇣ ⌘ ⇣ ⌘ 1 2 ⇤ ) = min (⇥⇤ + ) ⌦ (⇥⇤ + ) min r L(⇥ + = k⇥⇤ + Finally, observing that  1, we obtain ⌦ s DL (⇥, ⇥⇤ ) = rL(⇥) Plugging the definition of 2 k⇥⇤ k2 + k k2 k2 2 ↵ rL(⇥⇤ ), . k⇥⇤ k2 + k k2 2 ⇥⇤ k2F . k⇥ obtains us the final bound. ⇤ k by using localized The following lemma characterizes an upper bound of k b F analysis. p Lemma A.4. Suppose 8k ⇤ k2 s < 1. Take E such that S ✓ E and \|E\|  2s. Further assume k E c kmin /2 krL( ⇤ )kmax . Let b be the solution to (B.4). Then b must satisfy kb ⇤ ⇤ 2 k2 kF  4k k S kF +krL( ⇤ )E kF  8k ⇤ 2 k2 p s. Proof of Lemma A.4. We start by introducing an extra local parameter r which satp p p p isfies 8k ⇤ k22 s < r  k ⇤ k2 . This is possible since \|E\|  2 s ! 0 and p 8k ⇤ k2 s < 1 by assumption. Based on this local parameter r, we construct an inter⇤ ), where t is taken such that k( e ⇤ k = r, mediate estimator: e = ⇤ + t · ( b F ⇤ e e if k( kF > r; t = 1 otherwise. Applying Lemma A.3 with ⇥1 = and ⇥2 = ⇤ obtains us ⇤ 2 +r 2 e ⇤ 2 F ⌦  rL( e ) 4 rL( ⇤ ), e ⇤ ↵ . (A.2) To bound the right hand side of the above inequality, we use Lemma E.2 to obtain s e DL ( , ⇤ s b )  tDL ( , ⇤ ⌦ ) = t rL( b ) rL( ⇤ ), b ⇤ ↵ . (A.3) We note that the sub-di↵erential of the norm k · k1,o↵ evaluated at consists the set d⇥d of all symmetric matrices 2 R such that ij = 0 if i = j; ij = sign( ij ) if i 6= j and ij 6= 0; ij 2 [ 1, +1] if i 6= j and ij = 0, where ij is the (i, j)-th entry of . Then by the Karush-Kuhn-Tucker conditions, there exists a b 2 @k b k1,o↵ such that b =C b b 1+ b = 0. Plugging (A.3) into (A.2) and adding the term rL( b )+ ⇤ i on both sides of (A.3), we obtain b, b h (k ⇤ k2 +r) 2 k e ), b {z ⇤ 2 kF +t hrL( ⇤ \| b, b {z ⇤ i+t h } \| I b, b  t hrL( b )+ \| {z ⇤ i } II ⇤ i. } III (A.4) Next, we bound terms I, II and III respectively. For a set E, let E c denote its complement with respect to (w.r.t.) the full index set {(i, j) : 1  i, j  d}. For term I, separating ⇤ to E [ D and E c \ D, in which D is the set consisting the support of rL( ) and b of all diagonal elements, and then using the matrix Hölder inequality, we obtain ⌦ ⇤ rL( ), b ⇤ ↵ = ⌦ ⇤ rL( ) ⇤ rL( E[D ) ⇤ rL( , b ⇤ E[D F ) E c \D F b ), ( b ⇤ b )S[D , ( b )i = h( ⇤ b ↵ ⌦ + rL( ⇤ )S[D i +h( For the last term in the above equality, we have b )S c \D , ( b h( ⇤ )S c \D i = h S c \D , \| ⇤ ) E c\D E[D F ⇤ . E c \D F b ) and ( b For term II, separating the support of ( obtain h( b E[D b S c \D \|i = h ⇤) , b ⇤ E c\D ↵ to S [ D and S c \ D, we b )S c \D , ( b S c \D , \|( ⇤ b ⇤ )S c \D i. (A.5) )S c \D \|i. (A.6) Plugging (A.6) into (A.5) and applying matrix Hölder inequality yields h( b, b ⇤ b )S[D , ( b i = h( b )S , ( b = h( k S kF k( b ⇤ ⇤ )S[D i + h )S i + k S c \D , \|( S c \D kF k( ) S kF + k b E c \D kF k( b b ⇤ ⇤ )S c \D \|i )S c\D kF ⇤ )E c\D kF , where we use D = 0 in the second equality and E c \D ✓ S c \D in the last inequality. ⌦ ↵ b, b For term III, using the optimality condition, we have III = rL( b )+ = 0. Plugging the bounds for term I, II and III back into (A.4), we find that k ⇤ k2 + r 2 e ⇤ 2 +t F k E c \D kF t rL( k(rL( ⇤ 5 ) E[D F ))E c \D kF · ( b + S · b ⇤ F ⇤ ⇤ )E c \D F . F Further observing the facts that k E c \D kF ⇤k = k e rL( ⇤ ) E c\D F and tk b F we can simplify the above inequality to (k ⇤ 2 k2 +r) ke ⇤ kF  k S kF +krL( p p \|E c \D\| E c\D min \|E c \D\| rL( ⇤ k , dividing both sides by k e F ⇤ )E[D kF = k ⇤ S kF +krL( ) E kF  2 ⇤) max ⇤k , F p s, b C⇤ )E[D kF = k(C b C⇤ )E kF = krL( ⇤ )E kF in where we use krL( ⇤ )E[D kF = k(C the equality, and the last inequality follows from the Cauchy-Schwarz inequality, the fact k kmax  and the assumption that 2krL( ⇤ )kmax . Therefore, by the definition of ⇤ k  2(k ⇤ k + r)2 ps  8k ⇤ k2 ps < r, which implies e = b r, we obtain k e 2 F 2 from the construction of e . Thus b satisfies the desired `2 error bound. k Recall the definition of E` , 1  `  T . We can bound k b (`) ⇤k (` 1) kF . S F in terms of Lemma A.5 (Sequential Bound). Under the same assumptions and conditions in Lemma A.4, for ` 1, b (`) must satisfy k b (`) ⇤ ⇤ 2 2 kF  4 (` 1) S F ⇤ + rL( Proof of Lemma A.5. Now if we assume that for all ` ) E` F . 1, we have the following \|E` \|  2s, where E` is defined in (A.1) , and (` 1) E`c \D min /2 ⇤ krL( Using the matrix Hölder inequality, we obtain p p (` 1)  \|S\| S max  s and krL( S F ⇤ )kmax . )E` kF  Therefore, we have (` 1) + S F rL( ⇤ ) E`  F p s+krL( ⇤ (A.7) (A.8) p )E` kmax \|E` \|krL( ⇤ )E` kmax . p p \|E` \|  2 s, (A.9) where the second inequality is due to the assumption that krL( ⇤ )kmax  /2. The `2 error bound is given by Lemma A.4 by taking = (` 1) and E = E` , i.e. p 2 (` 1) ⇤ b (`) 4 ⇤ 2· + rL( ⇤ )E` F  8k ⇤ k22 · s, (A.10) S F F where last inequality is due to (A.9). Therefore, we only need to prove that (A.7) and (A.8) hold by induction. For ` = 1, we have w(u) for any u and thus E1 = S, which implies that (A.7) and (A.8) hold for ` = 1. Now assume that (A.7) and (A.8) hold at ` 1 for some ` 2. Since (i, j) 2 E` \S implies that (i, j) 2 / S and (` 1) (`) b w ij = j < w(u) = /2. By assumption, and since w(x) is non-increasing, we (` 1) b must have u. Therefore by induction hypothesis, we obtain that ij p \|E` \S\|  b (` 1) E`\S F u  b (` 1) u ⇤ F  8k ⇤ k2 2 u · p s p s, where the second last inequality follows from Lemma A.4, the fact that (A.7) and (A.8) hold at ` 1. This implies that \|E` \|  2\|S\| = 2s. Now for such E`c , we have k E`c kmin w(u) /2 krL( )k1 , which completes the induction step. 6 Our next lemma establishes the relationship between the adaptive regularization parameter and the estimator from the previous step. Lemma A.6. Assume w(·) 2 T . Let ij = w \|⇥ij \| for some ⇥ = (⇥ij ) and w(⇥S ) = w(⇥ij ) (i,j)2S , then for the Frobenius norm k · kF , we have w \|⇥⇤S \|  S F u F 1 + u ⇥⇤S ⇥S F . Proof of Lemma A.6. By assumption, if \|⇥⇤ij ⇥ij \| u, then w \|⇥ij \|  1  u 1 \|⇥ij ⇥⇤ij \|; otherwise, w \|⇥ij \|  w \|⇥⇤ij \| u . Therefore,the following inequality always hold: w \|⇥ij \|  w \|⇥⇤ij \| 1 u +u \|⇥⇤ij ⇥ij \|. Then by applying the k · k⇤ -norm triangle inequality, we obtain that S F w \|⇥⇤S \|  u ⇥⇤S 1 + u F ⇥S ?F . Our last technical result concerns a contraction property, namely, how the sequential approach improves the rate of convergence adaptively. Proposition A.7 (Contraction Property). Assume that assumptions 3.1, 3.2 and 3.3 p hold. Assume that 2krL( ⇤ )kmax and 8k ⇤ k22 s < 1. Then b (`) satisfies the following contraction property b (`) ⇤ F  4k ⇤ 2 k2 krL( ⇤ )S kF + 1 b (` 2 ⇤ 1) F . Proof of Proposition A.7. Under the conditions of the theorem, the proof of Lemma A.5 yields that (` 1) E`c \D kmin \|E` \|  2s, where E` is defined in (A.1), and k Thus, applying Lemma A.5 with b = b (`) , b (`) ⇤ F 4 ⇤ 2 2 · = (` 1) (` 1) kF S (` 1) S F  w(\| ⇤ S\| u) F + u ⇤ + rL( 1 ) E` . F b (` ⇤ 1) ⇤ S\| u) F . F I +4k u 1 b (` 1) ⇤ F . ⇤ ) E` F  rL( 7 ⇤ )S F + rL( (A.11) 1) ⇤k : F (A.12) ⌘ (A.13) In the next, we bound term I. Separating the support of rL( then using triangle inequality, we obtain I = rL( )kmax . in terms of k b (` Plugging the bound (A.12) into (A.11) yields that ⇣ ⇤ ⇤ 2 b (`)  4 rL( ⇤ )E` F + w(\| F 2 \| {z } ⇤ 2 k2 ⇤ and E = E` , we obtain (` 1) k S On the other side, by Lemma A.6, we can bound k krL( ⇤ ⇤) E` )E` \S to S and E`\S and F . (A.14) p Moreover, we have the following facts. First, we have rL( ⇤ )E` \S 2  \|E` \S\| rL( ⇤ ) by the Hölder inequality. From the assumption, we know krL( ⇤ )kmax  /2. Plugging p these bounds into (A.14) results that krL( ⇤ )E` kF  krL( ⇤ )S kF + \|E` \S\|. Now, by p (` 1) following a similar argument in Lemma A.5, we can bound \|E` \S\| by b E` \S F u  ⇤ b (` 1) u. Therefore, term I can be bounded by krL( ⇤ )S kF + u 1 b (` 1) ⇤ F . Plugging the upper bound for I into (A.13), we obtain F b (`) ⇤ F  4k ⇤ 2 k2 + (4k ⇣ krL( ⇤ 2 k2 ⇤ )S k2 + kw(\| ⇤ S\| b (` ⇤ 1 + 1) u 1) u)k2 F . ⌘ Now observing that k ⇤S kmin u + ↵ ⇣ , thus w(\|⇥⇤S \| u)  w(↵ · 1S ) = 0S , where 1S is a matrix with each entry equals to 1 and 0S is defined similarly. Further notice that (4k ⇤ k22 + 1) u 1  1/2, we complete the proof. B Improved Convergence Rate Using Sparsity Pattern We develop an improved spectral norm convergence rate using sparsity pattern in this section. We collect the proof for Theorem 3.8 first and then give technical lemmas that are needed for the proof. B.1 Proof of Theorem 3.8 (`) ⇤ Proof of Theorem 3.8. Let us define S (`) = (i, j) : ij u , where u is introij (0) ⇤ duced in (A.1). Let S = {(i, j) : \| ij \| u} = S. Then Lemma B.5 implies (` 1) E` F  w(\| ⇤ S\| u) + F q \|S (` 1) \ S\| + q E` /S ⇤ > u and thus (i, j) 2 S (` 1) /S. For any (i, j) 2 E` /S, we must have b ij = b ij ij Therefore, applying Lemma B.5 and using the fact that k ⇤S kmax u + ↵ , we obtain ⇢q q p p ⇤ 2 ⇤ 2 (` 1) \ S + (` 1) /S b (`) b  32 S S  32 2 S (` 1) . F 2 2 On the other side, (i, j) 2 S (`) implies that (`) \| b ij b ij \| (`) \| b ij ⇤ ij \| \| b ij Exploiting the above fact, we can bound q \|S (`) \|  b (`) 64k p ⇤ ij \| u 22 64k \|S (`) \| in terms of k b (`) b ⇤ k2 2 F  q S (` 1) /2. By induction on `, we obtain q ⇣ 1 ⌘`/2 q ⇣ 1 ⌘`/2 p (`) \|S \|  \|S (0) \| = s. 2 2 8 ⇤ 2 k2 b kF : , max Since ` > log s/ log 2, we must have that the right hand side of the above inequality is smaller than 1, which implies that S (`) = ? and b (`) = b . Therefore, the estimator enjoys the strong oracle property. Using Lemma B.4 obtains us that b (`) ⇤ b  2 ⇤ 2 . M⇤ b C 2 Applying Lemma D.6 finishes the proof of theorem. B.2 C⇤ S max . Technical Lemmas We start with the definitions of some constants. For notational simplicity, let 1 = k⌃⇤ k1 and D = {(i, i) : 1  i  d}. Define the oracle estimator as n⌦ o ↵ b = b argmin ,C log det( ) . d supp( )=S, 2S+ Recall that smax = maxj P ⇤ i 1(⇥ij ) is the maximum degree. Lemma B.1. Suppose that the weight function satisfies that w(u) 1/2 for u defined p in (A.1). Assume that 2 smax  1 2 k ⇤ k2 , 8k ⇤ k22 s < 1. If 2krL( b )kmax , we must have b (`) \|E` \|  2s and b F  32 Proof of Lemma B.1. If we assume that for all ` ⇤ 2 2 (` 1) . E` F 1, we have the following \|E` \|  2s, where E` is defined in (A.1), and k (` 1) kmin E`c (B.1) krL( b )kmax . (B.2) ⇤k ⇤ k +2 s Using lemma B.4, we obtain that k b k2  k ⇤ k2 +k b 1 k 2 2 max . p b Therefore, the assumption of the lemma implies 4k k2 s < 1. Replacing S by E` in Lemma B.3 and using Hölder inequality, we have 2 2 2 p (` 1) (` 1) b (`) b  4 b 2 E`  16 ⇤ 2 E`  32 ⇤ 2 s, (B.3) F F F For ` = 1, we have w(u) and thus E1 = S, which implies that (B.1) and (B.2) hold for ` = 1. Now assume that (B.1) and (B.2) hold at ` 1 for some ` 2. Since (` 1) (`) j 2 E` \ S implies that j 2 / S and w( j ) = j < w(u) by assumption, and since (` 1) \| j w(x) is decreasing, we must have \| obtain that b (` 1) E` \S F p \|E` \ S\|   b (` u. Therefore by induction hypothesis, we b 1) F  32 ⇤ 2 2 u u u where the last inequality follows from the definition of u hold at ` implies that \|E` \|  2\|S\| = 2s. Now for such E`c , we have k E`c kmin w(u) /2 krL( b )kmax , which completes the induction step. This completes the proof. 9 p s p s, 1. This inequality With some abuse of notation, we let \| ⇤S \| = (\| ⇤ij \|)(i,j)2S and \| u)(i,j)2S . The following inequality bounds the regularization parameter w( ⇤ij ) (i,j)2E in terms of functionals of ⇤ and . Lemma B.2. Let E F  ⇤\| S E = w \| \| . For any set E ◆ S, E must satisfy q ⇤ w(\| ⇤S \| u) F + E/S + {j 2 S : \| ij ij \| u = ( ⇤ij = w(\| ⇤E \|) = u} 1/2 p Proof. By triangle inequality, we have k E kF  k S kF + \|E/S\|. We further bound ⇤ ⇤\| k S kF . If \| ij u, then we have w \| ij \|  1  I \| ij u , otherwise, ij \| ij ⇤ ⇤ since because w(·) is non-increasing and thus \| ij ij \| < u implies w \| ij \|  w \| ij \| u . Therefore, using the Cauchy Schwartz inequality completes our proof. Define the following optimization problem n⌦ ↵ b = argmin b ,C log det( ) + 1,o↵ d 2S+ Lemma B.3. Let k satisfy krL( b )kmax and 4k b k2 S c /D kmin b b F 4 b 2 2 p o . (B.4) s < 1. Then b must S F. e = ⇥⇤ + t(⇥ b ⇥⇤ ), where t is chosen Proof. We construct an intermediate solution ⇥ e e such that k(⇥ ⇥⇤ kF = r, if k(⇥ ⇥⇤ kF > r; t = 1 otherwise. Here r satisfies p 4k b k22 s < r  k b k2 . Lemma A.3 implies that ⌦ ↵ 2 e b b b ⌘ Ds e , b . +r  rL( e ) rL( b ), e (B.5) L 2 F Then, we use Lemma E.2 to upper bound the right hand side of the above inequality ⌦ ↵ s e b s b b b . DL ( , )  tDL ( , ) = t rL b rL b , b Plugging the above inequality into (B.5), we obtain ⌦ 2 e b b 2  rL( b ) +r 2 F rL( b ), e b ↵ . (B.6) We further control the right hand side of the above inequality by exploiting the first b = 0 and rL( b )S[D = 0. Therefore, order optimality condition, which is rL( b )+ b to the right hand side of (B.6) and using the optimality adding and subtracting term condition obtains us that ⌦ ↵ ⌦ ↵ 2 e b b 2+ b, e b + rL( b ), e b  0. +r (B.7) 2 F \| {z } \| {z } I II b k2 , it suffices to bound I and II separately. For term I, by Therefore, to bound k e F decomposing the support to S and S c /D, then using matrix Hölder inequality, we have I S F e b S F + S c /D min 10 vec e b S c /D 1 . Again, by using the optimality condition, we has ⌦ ↵ b II = rL b S c /D , e rL( b )S c /D c S /D max vec e b By plugging the upper bound for I and II back into (B.7), we have b  2 2 +r S F (e e 2 + F b b )S rL( b )S c /D S c /D min . F max S c /D 1 vec e b . 1 By assumption, we know that k kmin krL( b )kmax , which implies that the second b term in the right hand side of the above inequality is positive. Thus, we have + 2 p 2 e b b 2 s < r  k b k2 , we obtain that e r  S F . Now since 4k k2 F p 2 b  4 b 2 S F  4k b k22 s < r. By the construction of e , we must have t = 1, F and thus e = b . Recall that M⇤ is the sparsity pattern matrix corresponding to ⇤ . p b C⇤ )S kmax  cn /2 for a sequence Lemma B.4. If 441 cn + 1 < 1+41 /smax and k(C cn , then we have b ⇤ max b  21 cn and ⇤ 2  21 cn kM⇤ k2 . ⇤ . It suffices to show that k k Proof of Lemma B.4. Let = b max  r, where 2 ⇤ ⇤ ). e r = 1 cn . To show this, we construct an intermediate estimator, = + t( b ⇤k e = b , otherwise. We choose t such that k e max = r, if k kmax > r, and For a matrix A, let AS be a matrix agreeing with A on S and having 0 elsewhere. Using the two term Taylor expansion, we know that there exists a 2 [0, 1] such that ⇤ ), e⇤ = ⇤ + (e vec rL( e ) = vec rL( which implies that n vec C⇤E e 1 E where E = S [ D. Let e = e E n vec C⇤E o ⇣ ⇤ ) + r2 L e ⇤ vec e e⇤ ⌦ e⇤ E E ⌘ 1 ⇣ vec e E ⇤ ⇤ E ⌘ = 0, = t . Define f vec( e ) to be o 1 ⇤ ⇤ eE + e E , EE vec ⇤ E , (B.8) 1 in which ⇤EE = ( ⇤E ⌦ ⇤E ) 1 . By the matrix expansion formula that (A+ P1 1 )m A 1 , f {vec( e )} reduces to m=1 ( A ⇢ X 1 vec ( ⌃⇤ e ) m ⌃⇤ . E m=2 1 Using triangle inequality, we then obtain that f vec( e )  max (j,k)2E 1 X m=2 11 eTj (⌃⇤ e )m ⌃⇤ ek . ) 1 A 1 = Further applying Hölder inequality to each single term in the right hand side of the above displayed inequality, we have eTj (⌃⇤ e )m ⌃⇤ ek  ⌃⇤ m+1 1 m 1 1 e e max 1 ⇤  sm max ⌃ m+1 1 where we use the fact k k1  smax k kmax . Therefore, we obtain f vec( e )  1 X m=2 1 ⇤ m+1 e m sm max k⌃ k1 k kmax = which, by triangle inequality, implies that k e kmax  k ⇤ EE k1 ✓ vec n C⇤E ⇤ + e 1 E o b E = b , the fact kC b Utilizing the KKT condition C E p 1+ 1+1 /smax , we obtain k e kmax  21 cn ⇣1 2 + e m , max 31 smax k e k2max , 1 1 smax k e kmax ◆ 31 smax k e k2max + . 1 1 1 smax k e kmax C⇤ kmax  cn /2 and 441 cn < 31 smax r2 ⌘ < 21 cn ⌘ r, 1 1 smax r which is a contradiction. Thus, e = and b satisfies the desired maximum norm bound. For the spectral norm bound, we utilize Lemma E.6 and obtain that b The proof is finished. C ⇤ 2  M⇤ 2 b ⇤ max  21 cn kM⇤ k2 . Semiparametric Graphical Model Proof of Theorem 4.3. We need the follows lemma, which are taken from Liu et al. b⌧ . (2012). It provides a nonasymptotic probability bound for estimating ⌃npn using S Lemma C.1. Let C be a constant. For any n & log d, with probability at least 1 we have r log d npn ⌧ sup \|Sbjk ⌃jk \|  C . n jk 8/d, The rest of the proof is adapted from that of Theorem 4.3 and thus is omitted. D Concentration Inequality In this section, we establish the concentration inequalities which are the key technical tools to the large probability bounds in Section 3. 12 Lemma D.1 (Sub-Gaussian Tail Bound). Let X = (X1 , X2 , . . . , Xd )T be a zero-mean random vector with covariance ⌃⇤ such that each Xi / ii⇤ is sub-Gaussian with variance proxy 1. Then there exists constants c1 and t0 such that for all t with 0  t  t0 the b satisfies the following tail probability bound associated sample covariance ⌃ P \|bij ⇤ ij \| t  8 exp c1 nt2 . Proof of Lemma D.1. By the definition of the sample covariance matrix, we have bij = P P (k) (k) (k) (k) n 1 nk=1 (Xi X̄i )(Xj X̄j ) = n 1 nk=1 Xi Xj X̄i X̄j . Therefore we can deP (k) (k) n ⇤ 1 ⇤ compose bij ij as n ij X̄i X̄j . By applying the union sum bound, k=1 Xi Xj we obtain that ◆ ✓ ◆ ✓ ◆ ✓ X n 1 t t (k) (k) ⇤ ⇤ P bij t P Xi Xj + P X̄i X̄j ij ij n 2 2 k=1 {z } \| {z } \| (R2) (R1) In the sequel, we bound (R1) and (R2) separately. For term (R1), following the argument of Lemma A.3 in Bickel and Levina (2008), there exists constant c01 and t00 not depending n, d such that ✓ X ◆ ⇢ n 1 t (k) (k) ⇤ (R1) = P Xi Xj  4 exp c01 nt2 ij n 2 k=1 for all t satisfying 0  t  t0 . Next, we bound the term (R2). By the linear structure p of sub-Gaussian random variables, we obtain that nX̄i ⇠ sub-Gaussian(0, ii⇤ ) for all p p 1  i  d. Therefore, by applying Lemma E.1, we obtain that \| nX̄i · nX̄j \| is a p p sub-exponential random variable with 1 norm bounded by 2k nX̄i k 2 k nX̄j k 2 . We p p give explicit bounds for the 2 -norm of nX̄i and nX̄j . By the Cherno↵ bound, the p tail probability of nX̄i can be bounded in the following ⇣p ⌘ n t2 o P \| nX̄i \| t  2 exp . 2 ii⇤ For every non-negative random variable Z, integration by parts yields the identity EZ = R1 p u)du. We apply this for Z = \| nX̄i \|p and obtain after change of variables 0 P(Z u = tp that Z 1 Z 1 n p p t2 o p 1 p p 1 E\| nX̄i \| = P(\| nX̄i \| t) · pt dt  2p · exp t dt 2 ii⇤ 0 0 ⇣ p ⌘p/2 p = p(2 ii⇤ )p/2 · ( )  p(2 ii⇤ )p/2 · , 2 2 p p which indicates that k nX̄i k 1  2 ii⇤ . The Gamma function is defined as (t) = q R1 t t 1 p ⇤ . Therefore we obtain e x dx. Similary, we can bound k n X̄ k by 2 jj j 2 0 q p p 2 2 ⇤ ⇤ ⇤ ⇤ k nX̄i · nX̄j k 1  2 ii jj  2 max , where max = max{ 11 , . . . , dd }. Define Zij = p p 2 e2 ) 1 and write the Taylor expansion series of the \| nX̄i · nX̄j \|. Let = (e 1)(2 max expoential function, we obtain E exp{ Zij } = 1 + 1 X k=1 k E(Z k ) ij k! 1+ 1 X k=1 13 k (2 2 k)k max k! 1+ 1 X k=1 (2 2 max · e)k  e, where we use k! (k/e)k in the last second inequality. Exponenting and using the Markov inequalty yields that ⇣ ⌘ ⇣ ⌘ ⇣ ⌘ Ee Zij P Zij t = P Zij t = P e Zij e t   exp{1 t}, et for all t 0. Using the above result, we can boudn (R2) as ⇣ n n nt ⌘ nt o (R2)  P Zij  exp 1  4 exp 1 2 2 nt o . 2 Combing the bounds for (R1) and (R2), taking c1 = min{c01 , } and t0 = min{1, t00 } obtain us that ⇤ ij \| P \|bij t  8 exp c1 nt2 8 t  t0 , which completes the proof. We then develop a large deviation bound for marginal variances. Lemma D.2 (Large Deviation Bound for Marginal Variance). Let X = (X1 , X2 , . . . , Xd )T p ⇤ be a zero-mean random vector with covariance ⌃⇤ such that each Xi ⌃ii is subn (k) Gaussian with variance proxy 1, and X be n i.i.d. samples from X. Let k=1 1 C(") = 2 " log(1 + ") > 0. Then, for any " 0, we must have ⇣ ⌘ n o b ii ⌃⇤ > " · ⌃⇤  2 · exp P ⌃ n · C(") . ii ii (k) Proof. We write Zi (k) (k) (k) = ⌃⇤ii 1/2 (k) Xi e ii = n and ⌃ 1 Pn (k) k=1 Zi (k) · Zi , for 1  i  d. Let &i = Zi · Zi ⇠ 21 , for 1  k  n. Therefore, the moment-generating function of (k) &i is M& (k) (t) = (1 2t) 1/2 , for t 2 ( 1, 1/2). Next, we control the tail probability i e ii > 1 + " and ⌃ e ii < 1 ", respectively. For the tail probability of ⌃ e ii > 1 + ", by of ⌃ applying Lemma E.8, we obtain ! (1) (n) n o &i +. . . + &i P > 1+"  exp n · A(") , n where A(") = supt (1 + ")t + 2 1 log(1 2t) = 2 1 " log(1 + ") . Similarly, for any e ii < 1 " as " > 0, we obtain the tail probability of ⌃ ! (1) (n) n o &i +. . . + &i P < 1 "  exp n · B(") , n where B(") = supt (1 ")t + 2 1 log(1 2t) . After some algebra, we obtain B(") = 2 1 " + log(1 ") , if " < 1; B(") = +1, otherwise. Let C(") = min A("), B(") = 2 1 " log(1+") . Therefore, combing the ⇣ ⌘ above twon inequalities o by union bound, we obtain P n b ii = (⌃⇤ ) ⌃ ii 1 (1) &i 1 ·⌃ e (n) 1 > "  2 · exp + . . . + &i ii = n 1 & (1) +. . .+& (n) ii ii ⇣ b ii P ⌃ ⌃⇤ii n · C(") . Note that we have . Thus, we obtain ⌘ n o > " · ⌃⇤ii  2 · exp n · C(") . 14 Our next results characterizes a large deviation bound for sample correlation matrix. Lemma D.3 (Large Deviation Bound for Sample Correlation). Let X = (X1 , X2 , . . . , Xd )T p ⇤ be a zero-mean random vector with covariance matrix ⌃⇤ such that each Xi ⌃ii is (k) n sub-Gaussian with variance proxy 1 and {X }k=1 be n independent and identically b = 1/n Pn X (k) X (k) T denote the sample covariance distributed copies of X. Let ⌃ k=1 b =W c 1⌃ bW c 1 denote the sample correlation matrix, where W c 2 is the diagonal and C b Further let ⇢bij and ⇢ij be the (i, j)th element of matrix with diagonal elements of ⌃. ⇤ b C and C respectively. Define c2 = min{4 1 c1 min(⌃⇤ )2 , 1/6}. Then, for 0  "  min{1/2, t0 maxi ⌃⇤ii }, we have ⇣ ⌘ n P \|b ⇢ij ⇢ij \| > "  6 exp ii o c2 n·"2 , where 1  i 6= j  d. b ii · ⌃ b jj ) 1/2 ⌃ b ij . To Proof of Lemma D.3. We denote the sample correlation as ⇢bij = (⌃ prove the tail probability bound. It suffices to prove the tail probability bound for ⇢bij ⇢ij > " and ⇢bij ⇢ij < ", respectively. We start with the tail probability bound for ⇢bij ⇢ij > ". Let us assume that ⇢ij 0. Using the basic probability argument, we have P(A) = P(A \ B) + P(A \ B c )  P(A) + P(B c ). Thus, for any 0  t  1 we obtain ⇣ ⌘ ⇣ ⌘ b ij (⌃ b ii ⌃ b jj ) 1/2 ·⇢ij > (⌃ b ii ⌃ b jj ) 1/2 ·" P ⇢bij ⇢ij > " = P ⌃ ⇣ ⌘ b ij (⌃⇤ ⌃⇤ ) 1/2 (1 t) 1 ·⇢ij > (⌃⇤ ⌃⇤ ) 1/2 (1 t) 1 ·" P ⌃ ii jj ii jj \| {z } ⇣ b ii +P ⌃ ⌘ (R1.1) ⇣ b jj ⌃⇤ii > ⌃⇤ii ·t + P ⌃ ⌘ ⌃⇤jj > ⌃⇤jj ·t . (D.1) Next, we bound the term (R1.1). After some simple algebra, (R1.1) can be bounded by ✓ ◆ ⇤ ⇤ ⇤ 1/2 1 ⇤ b P ⌃ij ⌃ij > " + ⇢ij ·(⌃ii ⌃jj ) (1 t) ⌃ij ✓ ◆ 1/2 ⇤ ⇤ ⇤ ⇤ b  P ⌃ij ⌃ij > " ⌃ii ⌃jj (1 + t) + t·⌃ij Let c02 = c1 mini (⌃⇤ii )2 , where c1 is defined in Lemma D.1.q If we apply Lemma D.1 with a better constant and Lemma D.2, then for any 0  "  t0 ⌃⇤ii ⌃⇤jj , in which t0 is defined in Lemma D.1, we must have ⇣ ⌘ ⇣ ⌘ ⇣ ⌘ b ij ⌃⇤ij > " ⌃⇤ii ⌃⇤jj 1/2 + P ⌃ b ii ⌃⇤ii > t·⌃⇤ii P ⇢bij ⇢ij > "  P ⌃ ⇣ ⌘ b jj ⌃⇤ > t·⌃⇤ +P ⌃ jj jj n o n o 1  4 exp c02 n·"2 + 2 exp n· t log(1 + t) . 2 Let c002 = min c02 , 1/6 . Further, for any 0  "  min{1/2, t0 maxi ⌃⇤ii }, by taking t = " and using the inequality t log(1 + t) 1/3·t2 for all t such that 0  t  1/2, we obtain ⇣ P ⇢bij ⌘ n o n 1 o n o ⇢ij > "  4 exp c02 "2 · n + 2 exp "2 · n  6 exp c002 n·"2 . 6 15 If ⇢ij < 0, in the a similar fashion as before, we can obtain the the following tail probability bound ⇣ ⌘ ⇣ ⌘ q b ij ⌃⇤ij > " ⌃⇤ii ⌃⇤jj 1/2 + ⌃⇤ij ·(t2 t) " ⌃⇤ ⌃⇤ ·t P ⇢bij ⇢ij > "  P ⌃ ii jj \| {z } ⇣ b ii +P ⌃ ⌘ (R1.2) ⇣ b jj ⌃⇤ii > t·⌃⇤ii + P ⌃ ⌘ ⌃⇤jj > t·⌃⇤jj . To continue, we bound the term (R1.2) in the next. If take t = "  min 1/2, t0 maxi ⌃⇤ii  q q 1/2 + 1/2\|⇢ij \|, we obtain that ⌃⇤ij ·(t2 t) " ⌃⇤ii ⌃⇤jj ·t 1/2 ⌃⇤ii ⌃⇤jj ·t. Thus, we have ⇣ P ⇢bij ⌘ ⇣ ⌘ ⇣ ⌘ b ij ⌃⇤ij > 1 " ⌃⇤ii ⌃⇤jj 1/2 + P ⌃ b ii ⌃⇤ii > t·⌃⇤ii ⇢ij > "  P ⌃ 2 ⇣ ⌘ ⇤ b + P ⌃jj ⌃jj > t·⌃⇤jj n 1 o n 1 o  4 exp c02 n·"2 + 2 exp n· " log(1 + ") 2 n 4 o 2  6 exp c2 n·" , where c2 = min{4 1 c02 , 1/6} = min{4 1 c1 min(⌃⇤ii )2 , 1/6}  c002 . By combining above two cases, for 0  "  min{1/2, t0 maxi ⌃⇤ii }, we have P(b ⇢ij ⇢ij > ")  6 exp{ c2 n · "2 }. In a similar fashion, we obtain the same tail probability bound for ⇢bij ⇢ij < ", for 0  "  min{1/2, t0 maxi ⌃⇤ii }. Thus the proof is completed. Lemma D.4. Under the same conditions in Lemma (D.3). We have the following result hold r ◆ ✓ ✓r ◆ 1 s ⇤ ⇤ lim lim sup P rL >M = 0, and krL( )S kF = OP . S max M !1 n n n b Proof of Lemma D.4. It is easy to check that rL( ⇤ )S F = C C⇤ S F . By applying Lemma D.3 and the union sum bound, for any M such that 0  M  p min 1/2, t0 maxi ⌃⇤ii · n, in which t0 is defined in Lemma D.3, we obtain r ◆ ✓ 1 ⇤ P rL >M  s · exp c2 M 2  exp c2 M 2 + log s . S max n q p Taking M such that 2c2 1 log s  M  min 1/2, t0 maxi ⌃⇤ii · n and M ! 1 in the above inequality obtains us that r ◆ ✓ 1 ⇤ lim lim sup P rL >M = 0, S max M !1 n n p which implies that rL ⇤ S F = OP ( s/n). b C⇤ , Lemma D.5 (A Concentration Inequality for Sample Correlation Matrix). Let C, 1 ⇤ 2 ⇢bij and ⇢ij be defined in Lemma D.3. Suppose n 3 c 2 t1 · log d. Take = q p b must 3c2 1 · (log d)/n ⇣ log(d)/n, in which c2 is defined as in Lemma D.3. Then C satisfy ⇣ ⌘ b C⇤ P C   1 8/d. max 16 b C⇤ . Therefore, applying Lemma D.3 and Proof. It is easy to check that rL(C⇤ ) = C union sum bound, we obtain that, for any  t1 ⌘ min 1/2, t0 maxi {⌃⇤ii } with t0 defined in Lemma D.1, ⇣ ⌘ b C⇤ P C >  6d2 · exp{ c2 n 2 }. max where c2 = min{4 1c ⇤ 2 1 min(⌃ii ) , 1/6}, in which c1 is definedqin Lemma D.1. , for n ·log d, by taking = 3c2 1 · (log d)/n  t1 , we 1 sufficiently large such that n 3 c2 t21 b C⇤ kmax  b C⇤ kmax > obtain P kC = 1 P kC The proof is completed. 1 6d2 ·exp{ c2 n Lemma D.6. Under the same conditions in Lemma D.5, we have r ◆ ✓ 1 ⇤ b b C⇤ lim lim sup P C C S max > M = 0, and C S M !1 n n max 2} 1 8/d. ✓r ◆ 1 = OP . n Proof of Lemma D.6. The proof is similar to that of Lemma D.5 and thus is omitted. E Preliminary Lemmas In this section we state and prove the technical lemmas used in previous sections. The following lemma establishes the tail bound type of the product of two sub-Gaussian random variables. Let k · k 1 and k · k 2 be the 1 - and 2 -norm defined in Vershynin (2010). Lemma E.1. For X and Y being two sub-Gaussian random variables, then the absolute value of their product \|X · Y \| is a sub-exponential random variable with kX · Y k 1  2 · kXk 2 kY k 2 . Proof of Lemma E.1. To show X · Y is sub-exponential, it suffices to prove that the 1 -norm of X · Y is bounded. By the definition of the 1 -norm, we have kX · Y k 1 = sup p p 1 1 ⇥ E\|X · Y \|p We need to use the Hölder inequality as follows ⇥ ⇤ ⇥ ⇤1/r ⇥ ⇤1/s E \|hf, gi\|  E\|f \|r E\|g\|s , ⇤1/p . (E.1) 1 1 + = 1, r s where f and g are two random functions. If we choose f = X p , g = Y p and r = s = 2 in the Hölder inequality, then the right hand side of (E.1) can be bounded by n ⇥ ⇤1/(2p) ⇥ ⇤1/(2p)o sup p 1 E\|X\|2p E\|Y \|2p p 1 n n ⇥ ⇤1/(2p)o ⇥ ⇤1/(2p)o  2 sup (2p) 1/2 E\|X\|2p · sup (2p) 1/2 E\|Y \|2p . p 1 Therefore we obtain that kX · Y k p 1 1  2kXk 2 kY k 17 2 < 1. The proof is completed. ⌦ ↵ s (⇥ , ⇥ ) = Lemma E.2. Let DL (⇥1 , ⇥2 ) = L(⇥1 ) L(⇥2 ) L(⇥2 ), ⇥1 ⇥2 and DL 1 2 DL (⇥1 , ⇥2 ) + DL (⇥2 , ⇥1 ). For ⇥(t) = ⇥⇤ + t(⇥ ⇥⇤ ) with t 2 (0, 1], we have that s s DL (⇥(t), ⇥⇤ )  tDL (⇥, ⇥⇤ ). ⌦ Proof of Lemma E.2. Let Q(t) = DL (⇥(t), ⇥⇤ ) = L(⇥(t)) L(⇥⇤ ) rL(⇥⇤ ), ⇥(t) ↵ ⇤ ⇥ . Since the derivative of L(⇥(t)) with respect to t is hrL(⇥(t)), ⇥ ⇥⇤ i, then the derivative of Q(t) is ⌦ ↵ Q0 (t) = rL(⇥(t)) rL(⇥⇤ ), ⇥ ⇥⇤ . s (⇥(t) Therefore the Bregman divergence DL ⇥⇤ ) can written as ⌦ ↵ s e e DL (⇥(t) ⇥⇤ ) = rL(⇥(t)) rL(⇥⇤ ), t(⇥ ⇥⇤ ) = tQ0 (t) for 0 < t  1. s (⇥, ⇥⇤ ) as a By plugging t = 1 in the above function equation, we have Q0 (1) = DL special case. If we assume that Q(t) is convex, then Q0 (t) is non-decreasing and thus s s DL (⇥(t), ⇥⇤ ) = tQ0 (t)  tQ0 (1) = tDL (⇥, ⇥⇤ ). Therefore the proof is completed. It remains to prove that Q(t) is a convex function, i.e. Q(↵1 t1 + ↵2 t2 )  ↵1 Q(t1 ) + ↵2 Q(t2 ), 8 t1 , t2 2 (0, 1], ↵1 , ↵2 0 s.t. ↵1 + ↵2 = 1. (E.2) For 8↵1 , ↵2 0 such that ↵1 + ↵2 = 1, and t1 , t2 2 (0, 1), we have ⇥(↵1 t1 + ↵2 t2 ) = ↵1 ⇥(t1 ) + ↵2 ⇥(t2 ). By the bi-linearity property of the inner product function h·, ·i, and using the linearity property of ⇥(·), we have the following equality hold ⌦ ↵ rL(⇥⇤ ), ⇥(↵1 t1 + ↵2 t2 ) ⇥⇤ ⌦ ⌦ ↵ = ↵1 rL(⇥⇤ ), ⇥(t1 ) ⇥⇤ i ↵2 rL(⇥⇤ ), ⇥(t2 ) ⇥⇤ . (E.3) On the other side, by the convexity of the loss function L(·), we obtain L ⇥(↵1 t1 + ↵2 t2 ) = L ↵1 ⇥(t1 ) + ↵2 ⇥(t2 )  ↵1 L ⇥(t1 ) + ↵2 L ⇥(t2 ) . (E.4) By adding (E.3) and (E.4) together and using the definition of function Q(·), we obtain Q(↵1 t1 + ↵2 t2 )  ↵1 Q(t1 ) + ↵2 Q(t2 ), which indicates Q(t) is a convex function. Thus we complete our proof. Lemma E.3. Let Ai , Bi 2 Rd⇥d be square matrices for i = 1, 2. Then we have A 1 B1 A 1 A2 B2 A2 = (A1 A2 )(B1 + (A1 B2 )(A1 A2 )B2 A1 + A1 (B1 The next lemma characterizes an upper bound of kA where k · k⇤ is any matrix norm. 18 A2 ) + (A1 1 B 1k ⇤ A2 )B2 A2 B2 )A2 . in terms of kA Bk⇤ , Lemma E.4. Let A, B 2 Rd⇥d be invertible. For any matrix norm k · k⇤ , we have 1 kA B 1 k⇤  kA 1 k2⇤ kA Bk⇤ . 1 kA 1 k⇤ kA Bk⇤ We need the following lemma for bounding the di↵erence with respect to the Kronecker product. Lemma E.5. Let A and B be matrices of the same dimension. Then we have kA ⌦ A kA ⌦ Bk1 = kAk1 kBk1 , B ⌦ Bk1  kA Bk21 and + 2 min kAk1 , kBk1 kA Bk1 . The proof of the above lemma can be carried out by using the definitions and thus is omitted here for simplicity. For a matrix A = aij , we say Asp = asp ij is the corresponding sparsity pattern sp sp matrix if aij = 1 when aij 6= 0; and aij = 0, otherwise. Lemma E.6. Let A 2 Rd⇥d be a matrix such that kAkmax  1. Let Asp be the corresponding sparsity pattern matrix. Then we have kAk2  kAsp k2 . Proof of Lemma E.6. Let aij be the (i, j)-th entry of matrix A and xj the j-th entry of x. Following the definition of the spectral norm of a matrix, we obtain that kAk2 = sup kAxk2 = sup kxk2 =1  sup kxk2 =1 = kxk2 =1 ⇢X n ✓X n i=1 sup x 0,kxk2 =1 i=1 aij xj j=1 sgn(xj )1(aij 6= 0) · xj j=1 ⇢X n ✓X n i=1 ⇢X n ✓X n j=1 1(aij 6= 0) · xj ◆2 ◆2 ◆2  kAsp k2 . Thus the proof is completed. b 2 Rd⇥d be a semi-positive definite random matrix, A 2 Rd⇥d a Lemma E.7. Let A positive definite deterministic matrix. Then we have ⇣ ⌘ ⇣ ⌘ b 1 A 1 >2 2 A · A b A b A > 2 1 min A . P A  P A min 2 2 2 b and A are commutative, that is AA b = AA, b then we have If we further assume that A ⇣ ⌘ p b 1/2 A 1/2 > 2( 2 + 1) A 1/2 2 A A b A P A min 2 2 2 ⇣ ⌘ b A > 2 1 min A . P A 2 19 b 1 A 1 as A b 1 (A b A)A Proof of Lemma E.7. We first write A the sub-multiplicative property of the spectral norm that b A 1 A 1 2 b  A  b (A b A 1 1 min b 1  A b A . A · A 2 A)A 1 min 1 2 2 A 1, 1 2 then it follows from b A A 2 (E.5) b b A , and thus min A b By Weyl’s inequality, we obtain that min (A)  min (A)+ A 2 b b A A 2 . Thus in the event of A A 2  2 1 min A , we have min A b 2 1 min A hold. Thus it follows from (E.5) that min A ⇣ ⌘ ⇣ ⌘ b 1 A 1 2 2 A · A b A b A  2 1 min A . P A P A min 2 2 2 b and A This proves the first desired probability bound. If we further assume that A 1 b A 2 are commutative, under the event A min A , we have 2 b A 1/2 A 1/2 2 = ⇣ b A 1/2 +A 1/2 1 ⌘ b A 1 A 1 2 b 1/2 + A 1/2 A b 1 A 1  A 2 2 2 p 1/2 b 1 A 1 2  ( 2 + 1) A 2 A p 1/2 2 b A .  2( 2 + 1) A 2 min A A 2 Therefore we prove the third result. The following lemma is taken from Dembo and Zeitouni (2009), which leads to a P concentration bound of the empirical means X̄ = n 1 ni=1 Xi , where Xi ’s are i.i.d. random copies of X. Define the logarithmic moment generating function associated with X to be ⇥ ⇤ ⇤X ( ) ⌘ log MX ( ) = log E exp{ X} . (E.6) Lemma E.8 (Large Deviation Inequality). Let the logarithmic moment generating function of X, ⇤X ( ), be defined in E.6. Define the Fenchel-Legendre dual of ⇤X (x) to be ⇤⇤X (x) ⌘ sup 2R x ⇤( ) . Then, for any t 0, we have ✓ X n 1 P Xi n i=1 ✓ X n 1 P Xi n i=1 EX EX  ⇥ where F1 = t, +1 and F2 = t ◆ t  exp ◆  exp n(EX + inf ⇤⇤ (x)) x2F1 and n(EX + inf ⇤⇤ (x)) , x2F2 ⇤ 1, t . References Bickel, P. J. and Levina, E. (2008), “Regularized estimation of large covariance matrices,” The Annals of Statistics, 36, 199–227. 20 Dembo, A. and Zeitouni, O. (2009), Large deviations techniques and applications, vol. 38, Springer Science & Business Media. Horn, R. A. and Johnson, C. R. (2012), Matrix analysis, Cambridge university press. Liu, H., Han, F., Yuan, M., La↵erty, J., Wasserman, L., et al. (2012), “High-dimensional semiparametric Gaussian copula graphical models,” The Annals of Statistics, 40, 2293– 2326. Vershynin, R. (2010), “Introduction to the non-asymptotic analysis of random matrices,” arXiv preprint arXiv:1011.3027. 21	10
arXiv:1802.02368v1 [math.ST] 7 Feb 2018 Group kernels for Gaussian process metamodels with categorical inputs O. Roustant1 , E. Padonou1 , Y. Deville2 , A. Clément3 , G. Perrin4 , J. Giorla4 , and H. Wynn5 1 Mines Saint-Étienne, UMR CNRS 6158, LIMOS, F–42023 Saint-Étienne, France 2 3 AlpeStat, Chambéry, France CEA/DAM/VA, F–21120, Is-sur-Tille, France 4 CEA/DAM/DIF, F–91297, Arpajon, France 5 London School of Economics, England Abstract Gaussian processes (GP) are widely used as a metamodel for emulating time-consuming computer codes. We focus on problems involving categorical inputs, with a potentially large number L of levels (typically several tens), partitioned in G L groups of various sizes. Parsimonious covariance functions, or kernels, can then be defined by block covariance matrices T with constant covariances between pairs of blocks and within blocks. However, little is said about the positive definiteness of such matrices, which may limit their practical usage. In this paper, we exploit the hierarchy group/level and provide a parameterization of valid block matrices T, based on a nested Bayesian linear model. The same model can be used when the assumption within blocks is relaxed, giving a flexible parametric family of valid covariance matrices with constant covariances between pairs of blocks. As a by-product, we show that the positive definiteness of T is equivalent to the positive definiteness of a small matrix of size G, obtained by averaging each block. We illustrate with an application in nuclear engineering, where one of the categorical inputs is the atomic number in Mendeleev’s periodic table and has more than 90 levels. 1 1 Introduction This research is motivated by the analysis of a time-consuming computer code in nuclear engineering, depending on both continuous and categorical inputs, one of them having more than 90 levels. The final motivation is an inversion problem. However, due to the heavy computational cost, a direct usage of the simulator is hardly possible. A realistic approach is to use a statistical emulator or metamodel. Thus, as a first step, we investigate the metamodelling of such computer code. More precisely, we consider Gaussian process (GP) regression models, also called kriging models ([Sacks et al., 1989], [Rasmussen and Williams, 2006]), which have been successfully used in sequential metamodel-based strategies for uncertainty quantification (see e.g. [Chevalier et al., 2014]). Whereas there is a flourishing literature on GP regression, the part concerned with categorical inputs remains quite limited. We refer to [Zhang and Notz, 2015] for a review. As for continuous inputs, covariance functions or kernels are usually built by combination of 1-dimensional ones, most often by multiplication or, more rarely, by addition [Deng et al., 2017]. The question then comes down to constructing a valid kernel on a finite set, which is a positive semidefinite matrix. Some effort has been spent on parameterization of general covariance matrices [Pinheiro and Bates, 1996] and parsimonious parameterizations of smaller classes [Pinheiro and Bates, 2009]. Some block form have also been proposed [Qian et al., 2007], in order to deal with a potential large number of levels. However, their validity was not investigated. Furthermore, to the best of our knowledge, applications in GP regression are limited to categorical inputs with very few levels, typically less than 5. Guided by the application, we investigate more deeply the so-called group kernels cited in Qian et al. [2007], defined by block covariance matrices T with constant covariances between pairs of blocks and within blocks. We exploit the hierarchy group/level by revisiting a nested Bayesian linear model where the response term is a sum of a group effect and a level effect. This leads to a parameterization of T which is automatically positive definite. Interestingly, the assumption on within blocks can be relaxed, and we obtain a parameterization of a wider class of valid group kernels. The positive definiteness condition of T is also explicited: it is equivalent to the positive definiteness of the smaller covariance matrix obtained by replacing each block by its average. 2 As mentioned above, this work has some connections with Bayesian linear models as well as linear mixed effect models (see e.g. Lindley and Smith [1972], Smith [1973]) in a hierarchical view. Other related works concern hierarchical GPs with a tree structure. For instance, particular forms of group kernels are obtained in multiresolution GP models ([Fox and Dunson, 2012], [Park and Choi, 2010]). Given two resolution levels and a spatial partition A = ∪i Ai of Rd , a parent GP on A, corresponding to the lowest resolution, serves as a trend for children GPs on Ai , corresponding to the highest resolution. Children GPs are independent conditionaly on the parent GP, and have the same covariance structure with a lengthscale parameter decreasing with the diameter of Ai . As a result, for a given resolution, the covariance matrix has a block form given by a sum of nested block diagonal covariance matrices. In comparison, the two-resolution (group/level) GP corresponding to a categorical input, does not assume a conditional independence between children, and the block form of covariance matrices can be more general. The paper is structured as follows. Section 2 gives some background on GP regression with mixed categorical and continuous inputs. Section 3 presents new findings on group kernels. Section 4 illustrates on synthetic examples. Section 5 is devoted to the application which motivated this work. Section 6 gives some conclusions and perspectives for future research. 2 Background and notations 2.1 GPs with continuous and categorical variables We consider a set of I continuous variables x1 , . . . , xI defined on a hypercubic domain ∆, and a set of J categorical variables u1 , . . . , uJ with L1 , . . . , LJ levels. Without loss of generality, we assume that ∆ = [0, 1]I and that, for each j = 1, . . . , J, the levels of uj are numbered 1, 2, . . . , Lj . We denote x = (x1 , . . . , xI ), u = (u1 , . . . , uJ ), and w = (x, u). We consider GP regression models defined on the product space J Y D = [0, 1] × {1, . . . , Lj }, I j=1 3 and written as: yi = µ(w(i) ) + Z(w(i) ) + i , i = 1, . . . , N. (1) where µ, Z and are respectively the trend, the GP part and a noise term. There exist a wide variety of trend functions, as in linear models. Our main focus here is on the centered GP Z(w), characterized by its kernel k : (w, w0 ) 7→ cov (Z(w), Z(w0 )) . I Kernels QJ on D can be obtained by combining kernels on [0, 1] and kernels on j=1 {1, . . . , Lj }. Standard valid combinations are the product, sum or ANOVA. Thus if kcont denotes a kernel for the continuous variables x, kcat a kernel for the categorical ones u, examples of valid kernels for w = (x, u) are written: (Product) (Sum) (ANOVA) k(w, w0 ) = kcont (x, x0 )kcat (u, u0 ) k(w, w0 ) = kcont (x, x0 ) + kcat (u, u0 ) k(w, w0 ) = (1 + kcont (x, x0 ))(1 + kcat (u, u0 )) For consiseness, we will denote by ∗ one of the operations: sum, product or ANOVA. The three formula above can then be summarized by: k(w, w0 ) = kcont (x, x0 ) ∗ kcat (u, u0 ) (2) Then, in turn, kcont and kcat can be defined by applying these operations to 1-dimensional kernels. For continuous variables, famous 1-dimensional kernels include squared exponential or Matérn [Rasmussen and Williams, i 2006]. We denote by kcont (xi , x0i ) such kernels (i = 1, . . . , I). For a categorical variable, notice that, as a positive semidefinite function on a finite space, a kernel is a positive semidefinite matrix. We denote by Tj the matrix of size Lj corresponding to kernels for uj (j = 1, . . . , J). Thus, examples of expressions for kcont and kcat are written: 1 I kcont (x, x0 ) = kcont (x1 , x01 ) ∗ · · · ∗ kcont (xI , x0I ) kcat (u, u0 ) = [T1 ]u1 ,u0 ∗ · · · ∗ [TJ ]uJ ,u0 1 J (3) (4) The formulation given by Equations (2), (3), (4) is not the most general one, since kernels are not always obtained by combining 1-dimensional ones. 4 Nevertheless, it encompasses the GP models used in the literature of computer experiments with categorical inputs. It generalizes the tensor-product kernels, very often used, and the sum used recently by [Deng et al., 2017] on the categorical part. It also contains the heteroscedastic case, since the matrices Tj are not assumed to have a constant diagonal, contrarily to most existing works [Zhang and Notz, 2015]. This will be useful in the application of Section 5, where the variance of the material is level dependent. Remark 1. Combining kernels needs some care to obtain identifiable models. 0 For instance, the product of kernels k1 , k2 with ki (xi , x0i ) = σi2 e−\|xi −xi \| (i = 1, 2), is a kernel depending on only one variance parameter σ 2 := σ12 σ22 . The GP model is identifiable for this new parameter, but not for the initial parameters σ12 , σ22 . 2.2 1-dimensional kernels for categorical variables We consider here a single categorical variable u with levels 1, . . . , L. We recall that a kernel for u is then a L by L positive semidefinite matrix T. 2.2.1 Kernels for ordinal variables A categorical variable with ordered levels is called ordinal. In this case, the levels can be viewed as a discretization of a continuous variable. Thus a GP Y on {1, . . . , L} can be obtained from a GP Yc on the interval [0, 1] by using a non-decreasing transformation F (also called warping): Y (u) = Yc (F (u)). Consequently, the covariance matrix T can be written: [T]`,`0 = kc (F (`), F (`0 )), `, `0 = 1, . . . , L. (5) When kc (x, x0 ) depends on the distance \|x − x0 \|, then k(`, `0 ) depends on the distance between the levels `, `0 , distorted by F . In the general case, F is piecewise-linear and defined by L − 1 parameters. However, a parsimonious parameterization may be preferred, based on the cdf of a flexible probability distribution such as the Normal or the Beta. We refer to [McCullagh, 1980] for examples in regression and to [Qian et al., 2007] for illustrations in computer experiments. 5 Remark 2. Notice that usual continuous kernels have non-negative values, which is a necessary condition if they are valid radial kernels in all dimensions. As a consequence, the kernels for ordinal variables built by warping do not allow negative correlations between levels. 2.2.2 Kernels for nominal variables For simplicity we present here the homoscedastic case, i.e. when T has a constant diagonal. It is immediately extended to situations where the variance depends on the level, by considering the correlation matrix. General parametric covariance matrices. There are several parameterizations of positive-definite matrices based on the spectral and Choleky decompositions. The spectral decomposition of T is written T = PDP> (6) where D is diagonal and P orthogonal. Standard parameterizations of P involve the Cayley transform, Eulerian angles, Householder transformations or Givens rotations, as detailed in [Khuri and Good, 1989] and [Shepard et al., 2015]. Another general parameterization of T is provided by the Cholesky decomposition: T = LL> , (7) where L is lower triangular. When the variance [T]`,` does not depend on the level `, the columns of L have the same norm and represent points on a sphere in RL . A spherical parameterization of L is then possible with one variance term and L(L−1)/2 angles, representing correlations between levels [see e.g. Pinheiro and Bates, 1996]. Parsimonious parameterizations. The general parametrizations of T described above require O(L2 ) parameters. More parsimonious ones can be used, up to additional model assumptions. Among the simplest forms, the compound symmetry (CS) - often called exchangeable - covariance matrix assumes a common correlation for all levels [see e.g. Pinheiro and Bates, 2009]. The CS matrix with variance v and covariance c is defined by: v if ` = `0 [T]`,`0 = , c/v ∈ (−1/(L − 1), 1) . (8) c if ` 6= `0 6 This generalizes the kernel obtained by substituting the Gower distance d 2 [Gower, 1982] into the exponential kernel, corresponding to c/v = e−d > 0. The CS covariance matrix treats equally all pairs of levels, which is an important limitation, especially when L 1. More flexibility is obtained by considering groups of levels. Assume that the L levels of u are partitioned in G groups G1 , . . . , GG and denote by g(`) the group number corresponding to a level `. Then a desired parameterization of T is given by the block matrix (see e.g. Qian et al. [2007]): v if ` = `0 [T]`,`0 = (9) cg(`),g(`0 ) if ` 6= `0 where for all i, j ∈ {1, . . . , G}, the terms ci,i /v are within-group correlations, and ci,j /v (i 6= j) are between-group correlations. Notice that additional conditions on the ci,j ’s are necessary to ensure that T is a valid covariance matrix, which is developed in the next section. 3 Block covariance matrices for levels grouping We consider the framework of Section 2.2.2 where u denotes a categorical variable whose levels are partitioned in G groups G1 , . . . , GG of various sizes n1 , . . . , nG . Without loss of generality, we assume that G1 = {1, . . . , n1 }, G2 = {n1 + 1, . . . , n1 + n2 }, . . . . We are interested in parsimonious parameterizations of the covariance matrix T, written in block form:   W1 B1,2 ··· B1,G ..  ...  .   B2,1 W2 T= . (10)  ... ...  .. BG−1,G  BG,1 · · · BG,G−1 WG where the diagonal blocks Wg contain the within-group covariances, and the off-diagonal blocks Bg,g0 are constant matrices containing the between-group covariances. We denote: g 6= g 0 ∈ {1, . . . , G} Bg,g0 = cg,g0 Jng ,n0g , where Js,t denotes the s by t matrix of ones. This means that the betweengroup covariances only depends on groups (and not on levels). 7 We will also consider the particular case where diagonal blocks Wg are CS covariance matrices with variance vg and covariance cg . In this subclass, the between-group and within-group covariances only depends on groups (and not on levels). When the variance term is the same for all groups, we obtain block matrices of the form (9) as a special case. Although the block matrices of the form (10) may be covariance matrices, they are not positive semidefinite in general. In the next section, we provide a proper characterization as well as a parameterization of such matrices which automatically fulfills the positive semidefinite conditions. We will use the following additional notations: for a given integer L ≥ 1, IL is the identity matrix of size L, JL is the matrix of ones of size L, 1L is the vector of ones of size L. Finally, for a vector or a matrix M, we denote by M the real number equal to the average of its coefficients. 3.1 A Gaussian model for CS covariance matrices We first focus on the case of a CS matrix. We denote by ΓCS L (v, c) = (v − c)IL + cJL (11) the CS matrix with a common variance term v and a common covariance term c. It is well-known that ΓCS L (v, c) is positive definite if and only if − (L − 1)−1 v < c < v. (12) For instance, one can check that the eigenvalues of ΓCS L (v, c) are v + (L − 1)c with multiplicity 1 (eigenvector 1L ) and v − c with multiplicity L − 1 (eigenspace 1⊥ L ). Notice that a CS matrix is positive definite for a range of negative values of its correlation term. Then we consider the following Gaussian model: η ` = µ + λ` , ` = 1, . . . , L (13) where µ ∼ N (0, vµ ) with vµ > 0, and λ1 , . . . , λL are i.i.d. random variables from N (0, vλ ), with vλ > 0, assumed to be independent of µ. A direct computation shows that the covariance matrix of η is the CS covariance matrix ΓCS L (vµ + vλ , vµ ). Clearly this characterizes the subclass of 8 positive definite CS covariance matrices ΓCS L (v, c) such that c is non-negative. The full parameterization, including negative values of c in the range (−(L − 1)−1 v, 0), can be obtained by restricting the average of level effects to be zero, as detailed in the next proposition. Proposition 1. When η and λ are related as in (13), the covariance of η conditional on zero average errors λ = 0 is a CS matrix with variance v = vµ + vλ [1 − 1/L] and covariance c = vµ − vλ /L. Conversely, given a CS covariance matrix C with variance v and covariance c, there exists a representation (13) such that C is the covariance of η conditional on zero average errors λ = 0 where vµ = v/L + c[1 − 1/L] and vλ = v − c. 3.2 Parameterization of centered covariance matrices The usage of Model (13) to describe CS covariance matrices involves Gaussian vectors that sum to zero. This is linked to centered covariance matrices, i.e. covariance matrices F such that F = 0, as detailed in the next proposition. We further give a parameterization of centered covariance matrices. Proposition 2. Let F be a covariance matrix of size L ≥ 2. Then, F is centered iff there exists a Gaussian vector z on RL such that F = cov(z\|z = 0). In that case, let A be a L × (L − 1) matrix whose columns form an orthonormal basis of 1⊥ L . Then F is written in an unique way F = AMA> (14) where M is a covariance matrix of size L − 1. In particular if F = v[IL − L−1 JL ] is a centered CS covariance matrix, then M = vIL−1 , and we can choose z ∼ N (0, vIL ). The choice of A in Prop. 2 is free, and can be obtained by normalizing the columns of a L × (L − 1) Helmert contrast matrix (Venables and Ripley [2002], §6.2.):   −1 −1 −1 · · · −1  1 −1 −1 · · · −1    0  2 −1 · · · −1  . ..  ..  .  . 0 3 .   .  .  .. . . ..  .. . . . −1  0 0 ··· 0 L − 1 9 3.3 A hierarchical Gaussian model for block covariance matrices Let us now return to the general case, where the levels of u are partitioned in G groups. It will be convenient to use the hierarchical notation g/`, indicating that ` belongs to the group Gg . Then, we consider the following hierarchical Gaussian model: ηg/` = µg + λg/` , g = 1, . . . , G, ` ∈ Gg (15) where for each g the random variable µg represent the effect of the group g, and the random variables λg/1 , . . . , λg/ng represent the effects of the levels in this group. We further assume: • The vector µ is normal N (0, Γµ ). • The vectors λg/. are normal N (0, Γλg ). • The vectors λ1/. , . . . , λG/. are independent. • The vectors µ and λ are independent. As an extension of Prop. 1, the next proposition and Cor. 1 show that (15) gives a one-to-one parameterization of positive semidefinite matrices of the form (10) with CS diagonal blocks, under the additional assumption that the average of level effects is zero in each group. More generally, we obtain a large parametric family of positive semidefinite matrices of the form (10). Proposition 3. The covariance matrix of η conditional on {λg/. = 0, g = 1, . . . , G} has the form (10) with, for all g, g 0 ∈ {1, . . . , G}: Wg = [Γµ ]g,g Jng + Fg , Bg,g0 = [Γµ ]g,g0 Jng ,ng0 , (16) where Fg is a centered positive semidefinite matrix equal to cov(λg/. \|λg/. = 0). Therefore Wg − Wg Jng is positive semidefinite for all g = 1, . . . , G. Conversely, consider a positive semidefinite matrix T having the block form (10) such that Wg − Wg Jng is positive semidefinite for all diagonal blocks g. e be the G × G matrix obtained by averaging each block of T. Then there Let T 10 exists a representation (15) such that T is the covariance of η conditional on zero average errors λg/. = 0, (g = 1, . . . , G), with: cov(λg/. \|λg/. e Γµ = T, = 0) = Wg − Wg Jng . Corollary 1. Positive semidefinite matrices of the form (10) with CS diagonal blocks exactly correspond to covariance matrices of η in (15) conditional on the G constraints λg/. = 0 when cov(λg/. ) ∝ Ing . As a by-product, we obtain a simple condition for the validity of block covariance matrices of the form (10). Interestingly, it only involves a small matrix whose size is the number of groups. Proposition 4. Let T be a matrix having the block form (10) such that Wg − Wg Jng is positive semidefinite for all g = 1, . . . , G. Then e is positive semidefinite. (i) T is positive semidefinite if and only if T e is positive definite and the diagonal (ii) T is positive definite if and only if T blocks Wg are positive definite for all g = 1, . . . , G. Furthermore, we have e > + diag(W1 − W1 Jn1 , . . . , WG − WG Jn ) T = XTX G where X is the n × G matrix    X :=   1n1 0 .. . 0 (17)  ... 0 ..  . 1n2 . . .  . ... ... 0  . . . 0 1nG 0 Remark 3. All the results depend on the conditional distribution λg/. \|λg/. = 0. Thus there is some flexibility in the choice of Γλg , since several matrices Γλg can lead to the same conditional covariance matrix cov(λg/. \|λg/. = 0). Remark 4 (Groups of size 1). Prop. 3 is still valid for groups of size 1. Indeed if ng = 1, then (λg/. \|λg/. = 0) is degenerate and equal to 0. Thus Fg = Wg − Wg J1 = 0 is positive semidefinite. 11 3.4 Related works Model (15): ηg/` = µg + λg/` , shares similarities with two-way Bayesian models and linear mixed effect models (see e.g. Lindley and Smith [1972]), with Gaussian priors for the effects µ and λg/. . The centering constraints λg/. = 0 are also standard identifiability conditions in such models. Furthermore, the particular case of CS covariance matrices corresponds to the exchangeable assumption of the corresponding random variables. Typically, in the framework of linear modelling, Model (15) could be written as yg,` = m + µg + λg,` + εg,` , with additionals grand mean m and errors εg,` . However, if the framework is similar, the goal is different. In linear modelling, the aim is to quantify the effects by estimating their posterior distribution (µ, λ)\|y. On the other hand, we aim at investigating the form of the covariance matrix of the response part µg + λg/` , or, equivalently, the covariance matrix of the likelihood y\|µg + λg/` . 3.5 Summary and comments The results of the previous sections show that a wide class of valid block covariance matrices can be parameterized by a family of covariance matrices of smaller sizes. This class is formed by positive definite matrices of the form (10) such that Wg − Wg Jng is positive semidefinite for all g = 1, . . . , G. It contains the case where diagonal blocks are CS covariance matrices. The algorithm is summarized below. 1. Generate a covariance matrix Γµ of size G. 2. For all g = 1, . . . , G, If ng = 1, set Fg = 0, else: • Generate a covariance matrix Mg of size ng − 1. • Compute a centered matrix Fg = Ag Mg A> g , where Ag is a ng × (ng − 1) matrix whose columns form an orthonormal basis of 1⊥ ng . 12 3. For all 1 ≤ g < g 0 ≤ G, compute the within-group blocks Wg and between-group blocks Bg,g0 by Eq. (16). In steps 1 and 2, the covariance matrices Γµ and Mg can be general, and obtained by one of the parameterizations of §2.2.2. However, some specific form, such as CS matrices, can also be chosen. Depending on the number of groups and their sizes, different levels of parsimony can be obtained. Table 1 summarizes some possibilities. Notice that it may be hard to choose the parametric setting Mg , Γµ in order to account for a specified constraint of the block matrix T, such as homoscedasticity. An alternative to over-parameterization is to use the economic constraint on T of Prop. 4. Indeed, positive definiteness on T is e of size G. equivalent to positive definitiness of the small matrix T Parametric setting Mg Γµ CS vλg Ing −1 Γ (vµ , cµ ) General vλg Ing −1 CS General Γ (vµ , cµ ) General General Resulting form of T Wg Bg,g0 CS Γ (vg , cg ) cg,g0 ≡ cµ ΓCS (vg , cg ) cg,g0 General cg,g0 ≡ cµ General cg,g0 Number of parameters 2G + 1 G(G+3) P 2 ng (ng +1) 2+ G g=1 PG n2g (ng +1) G(G+1) + g=1 2 2 Table 1: Parameterization details for some valid block-covariance matrices T of the form (10). 4 Examples Before considering the application in nuclear engineering, we consider two toy functions with one continuous input and one categorical input, which reproduce two specificities of that application. The first function mimics a situation where the output variance depends on the level of a categorical input. The second one investigates level grouping when the number of levels is large. 13 4.1 Example 1 (An heretoscedastic case) Consider the deterministic function  6.8πx  cos  2 f (x, u) = −2 cos 7.0πx 2  1 cos 7πx 2 2 if u = 1 if u = 2 , if u = 3 −2 −1 0 1 2 where x ∈ [0, 1] and u ∈ {1, 2, 3}. The expression of f is adapted from [Han et al., 2009] by scaling the three output curves f (., u) according to the level of u. Thus the output variance clearly depends on the level. As visible in Figure 1, these three curves are strongly dependent, with a positive link between f (x, 1) and f (x, 3), and negative between f (x, 1) and f (x, 2). 0.0 0.2 0.4 0.6 0.8 1.0 Figure 1: Test function 1. f (x, 1) in black, f (x, 2) in red and f (x, 3) in green. The design points correspond to one realization of a sliced LHD. The aim is to compare the accuracy of four GP models by reconstructing f with few evaluations. For all of them, a Matérn 5/2 kernel [Rasmussen and Williams, 2006] is chosen for x. The first GP model (IND) consists in three independent GPs corresponding to the levels of u. The other ones have tensor product kernels, with three different covariance matrices T for u. The first two ones assume a constant variance: T is defined by a CS covariance 14 structure (Eq. 8), or by a general spherical parameterization (SPH) (See Section 2.2.2). Finally, we consider a heteroscedastic spherical parameterization (H-SPH) where the covariance matrix is defined by a general variance vector, and a spherical parameterization of the correlation matrix. In order to benefit from the strong link between levels, we use a design that spreads out the points between levels. For instance, the information given by f (0, 1) may be useful to estimate f (x, 2) and f (x, 3) at 0, without computing f (0, 2) and f (0, 3). More precisely, we have used a (random) sliced Latin hypercube design (SLHD) [Qian, 2012] with 5 points by level, for a total budget of 15 points. Parameter estimation is by maximum likelihood. As the likelihood surface may be multimodal, we have launched several optimizations with different starting points chosen at random in the domain. Model accuracy is measured over a test set formed by a regular grid of size 1000, in terms of Q2 criterion. The Q2 criterion has a similar expression than R2 , but is computed on the test set: P (yi − ŷi )2 2 , (18) Q = 1 − Pi 2 i (yi − ȳ) where the yi denote the observations (on the test set), ȳ their mean, ŷi the predictions. It is negative if the model performs worst than the mean, positive otherwise, and tends to 1 when predictions are close to true values. Finally, the process is repeated 100 times, in order to assess the sensitivity of the result to the design. We observe that the heteroscedastic model H-SPH clearly outperforms the other ones. As expected, the estimated variances (2.5, 5.3, 1.4) for this model are level-dependent, whereas a constant variance is wrongly estimated around 2 by CS and SPH. Moreover, we have represented in Figure 3 an estimation of the correlations between levels, deduced from the parameterized covariance matrix T . For representativeness, we have chosen one of the 100 designs used to generate Figure 2, such that the corresponding Q2 is the closest to the median of the 100 Q2 values. We can see that the strong dependence link of f (., u) between the levels of u is only recovered correctly by H-SPH, with a poor estimation of correlation parameters for the other ones. 4.2 Example 2 (Levels grouping) The second function is defined as: x u g(x, u) = cos 7π + p(u)π − 2 20 15 1.00 0.75 0.50 0.25 0.00 IND CS SPH H−SPH Figure 2: Q2 criterion for four GP models, based on 100 repetitions of the design. u with x ∈ [0, 1], u ∈ {1, . . . , 13} and p(u) = 0.4 + 15 1u>9 . As visible in Figure 4, there are two groups of curves corresponding to levels {1, . . . , 9} and {10, . . . , 13} with strong within-group correlations, and strong negative between-group correlations. We aim at reconstructing g with five GP models based on levels grouping. The first one uses a CS covariance matrix, corresponding to a single group. The second one considers the two groups {1, . . . , 9} and {10, . . . , 13}. The third model, based on the five groups {1, . . . , 9}, {10}, {11}, {12}, {13}, has two variants: (a) when the inter-groups correlation is constant and (b) in the general case. The fourth model uses the spherical parameterization of T , leading to 13 groups, and the last one considers an ordinal paramaterization for T . The design of experiments is a 39-points SLHD. The remaining simulation settings are the same as in Example 1. The estimated correlation parameters are shown in Figures 5. The right correlation structure is well recovered with two groups and five groups, with different between-groups correlations. The model with thirteen groups involves the estimation of 90 parameters, which is hard to achieve, especially with 39 points. This is visible in the erratic values of the estimated correlations values, which seem not meaningful. On the opposite, considering only one group or five groups with a common between-group correlation oversimplifies and fails at recovering the right correlations. The ordinal ker16 CS H-SPH SPH −1.0 −0.5 0.0 0.5 1.0 Figure 3: Estimated correlation parameters among levels for f , for a design of experiments corresponding to a median Q2 . 0.0 0.2 0.4 0.6 0.8 1.0 Figure 4: Test function g. nel recovers the two blocs of curves, but cannot detect negative correlation between them (Remark 2). In Figure 6, we can see that the best tradeoff between prediction accuracy and parsimony is obtained with two groups. whereas it reduces the number of observations by group. Notice the rather good performance of the ordinal model, at the cost of a larger number of parameters: the warping was parameterized by an affine function (see Section 2.2.1). This is noticeable since negative correlations are not possible (see above). This may be due to the larger number of within-group levels combinations, which reduces the influence of the negative between-group correlations. 17 1 1 1 0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0 −0.2 −0.2 −0.2 −0.4 −0.4 −0.4 −0.6 −0.6 −0.6 −0.8 −0.8 −0.8 −1 −1 −1 1 group. 5 groups (b) 2 groups. 5 groups (a) 1 1 1 0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0 −0.2 −0.2 −0.2 −0.4 −0.4 −0.4 −0.6 −0.6 −0.6 −0.8 −0.8 −0.8 −1 −1 −1 13 groups. Ordinal. Figure 5: Estimated correlation parameters among levels for g, based on a representative design of experiments (design with median Q2 ). 5 5.1 Application in nuclear engineering Position of the problem As presented in Introduction, this research is originally motivated by the solving of an inverse problem confronting experimental measurements in nuclear engineering and time-consuming numerical simulation. More precisely, this analysis concerns the identification of the mass m of 239 Pu that is present in a particular waste container using a non-destructive nuclear detection technique such as the gamma spectrometry [Knoll, 2010]. In that case, at each energy level E, m × (E; E) = y SG (E), (19) where y SG (E) is the quantity of interest provided by the gamma transmitter, and (E; E) is the attenuation coefficient, which depends on the source envi18 1.0 0.8 0.6 0.4 1 group 2 groups 5 groups 5 groups (b) 13 groups ordinal Figure 6: Q2 of six GP models, based on 100 repetitions of the design. Number of parameters used (bloxplot order): 5, 7, 10, 19, 90, 16. ronment denoted by E. In practice, only discrete values of E are of interest, corresponding to the natural energy levels of 239 Pu: E ∈ {94.66, 129.3, 203.6, 345.0, 375.1, 413.7} (keV). (20) Then, based on previous studies [Guillot, 2015], the real source environment is parameterized by the following input variables: • An equivalent geometric shape for the nuclear waste: sphere (‘sph’), cylinder (‘cyl’) or parallelepiped (‘par’). • An equivalent material for this waste, characterized by its chemical element with atomic number in {1, . . . , 94}, • The bulk density of the waste, in [0, 1], • The distance of measurement between the container and the measurement device, in [80, 140] (cm), • The mean width and lateral surfaces (in logarithmic scale) crossed by a gamma ray during the rotation of the object. After normalization, the characteristics of the input space can be summed up in Table 2. 19 Name of the input Distance Density Width Surface Energy Shape Chemical element Variation domain [0,1] [0,1] [0,1] [0,1] {1, 2, 3, 4, 5, 6} {sph, cyl, par} {1, . . . , 94} Table 2: Description of the input variables for the nuclear application. To recapture the notation of the previous sections, let x and u be the vectors gathering respectively the continuous and categorical inputs, and w = (x, u). For a given value of w, Monte Carlo simulation codes as MCNP [Goorley et al., 2013] can be used to model the measured scene and approach the value of (w) = (E, E). The mass m can eventually be searched as the solution of the following optimization problem: (m? , w? ) = arg min ky obs − m × (w)k, m,w (21) where k · k is the classical Euclidian norm, (w) and y obs respectively gather the values of and y SG at the six values of E that are used for the measurements. To solve (21), it is therefore necessary to compute at a high number of points. However, each evaluation of the MCNP code can be extremely demanding (between several minutes to several hours CPU for one evaluation). Thus, surrogate models have to be introduced to emulate the function w 7→ (w), which is now investigated in the frame of Gaussian process regression. We refer to Clement et al. [2018] for the second step, namely the treatment of the inversion problem. 5.2 Model settings For pedagogical purpose, a dataset of large size N = 5076 has been computed with the MNCP code. The construction of the design of experiments was guided by the categorical inputs, such that each of the 6 × 3 × 94 = N/3 combinations of levels appears 3 times. It was completed by a Latin hypercube of size N to define the values of the four continuous inputs. 20 From this full dataset, a training set of size n = 3 × 94 = 282 is extracted by selecting at random 3 observations by chemical element. The remaining N − n points serve as test set. −12 −12 −14 −14 Y Y −16 −16 E1 E2 E3 E4 E5 E6 Sph Y in function of the energy Cyl Par Y in function of the geometric shape. Figure 7: Y in function of the energy and geometric shape. Model settings are now motivated by a graphical analysis. In Figure 7, the output is displayed in function of the energy and the geometric shape. We observe that successive energy levels correspond to close values. This fact confirms that the energy is ordinal and we use the warped kernel defined by Eq. (5). The influence of the geometric shape is less obvious, and we have chosen an exchangeable (CS) covariance structure for it. In Figure 8, Y is displayed in function of the 94 chemical elements, ordered by atomic number. Two important facts are the high number of levels and heteroscedasticity. For this purpose, the 94 chemical elements are divided into 5 groups, provided by expert knowledge and represented by colors. This partition suggests to use a group kernel of the form (10), where the within-group blocks Wg are CS covariance matrices. In order to handle heteroscedasticity, the variance of Wg is assumed to depend on the group number g. The influence of continuous variables can be observed by panels (not represented), and does not reveal useful information for our purpose. A Matérn 5/2 kernel is set for all continuous inputs, as we expect the output to be a regular function of the continuous inputs. Indeed, for this kernel, the corresponding Gaussian process is two times mean-square differentiable. Finally, three candidate kernels for w are obtained by combining the kernels of input variables defined above, by sum, product or ANOVA (see Section 2). 21 −12 −14 Y −16 1 10 20 36 61 94 Figure 8: Y in function of chemical elements, ordered by atomic number. 5.3 Results Following the model settings detailed above, Figure 9, Panel 3, presents the results obtained with 60 random designs of size n and three operations on kernels. Furthermore, we have implemented three other kernels for the chemical element, in order to compare other model choices for this categorical input. In the first panel, we grouped all the 94 levels in a single group. In the second one, we kept the 5-group kernel but forced the between-group covariances to have a common value. Finally, in the fourth panel, we considered that the levels were ordered by their atomic number, and used the warped kernel of Eq. (5) with a Normal transform. Figure 9: Q2 of several GP models, based on 60 random designs, corresponding to different model choices for the chemical element. First panel: Single group. Second panel: 5 groups, with a common between-group covariance. Third panel: 5 groups. Fourth panel: Ordered levels. Total number of parameters used in the panel order: ‘prod’ = (12, 21, 30, 14), ‘add’ = ‘prod’ + 6, ‘anova’ = ‘prod’ + 7. 22 First, comparing the three operations on kernels, we remark that in all the panels, additive kernels provide the worst results. This suggests the existence of interactions between different inputs of the simulator. Second, the ANOVA combination produces slight improvements, compared to the standard tensor-product, both in terms of accuracy and stability with respect to design choice. Now, comparing the four panels, we see that gathering the levels in a single group is the least efficient strategy. The 5-group kernel gives very good performances, especially when the between-group covariances vary freely: constraining them to be equal degrades the result. Surprisingly here, the ordinal kernel gives the best performance. Indeed, for this application it was not intuitive to the experts that the chemical element can be viewed as an ordinal variable, simply sorted by its atomic number. This is confirmed by the correlation plots of Figure 10, corresponding to a model with a median Q2 score. We can see that the estimated correlations between levels seems to decrease as the difference between levels increases, an indication that the levels may be ordered by their atomic number. Finally, we report several post-processing results. First, the estimated transformation of energy levels (Figure 11a) is concave and flat near high values, which corresponds to the behaviour observed in Figure 7 (left panel). In addition, the last three levels lead to similar results (Figure 11). This corresponds to the fact that when the energy is high, the gamma ray almost always crosses the nuclear waste, leading to a high value for the output. Second, the estimated correlation among the sphere, the cylinder and the parallelepiped is very high (c = 0.9, Figure 12). This justifies considering a covariance structure for that categorical input, rather than using three independent GP models for all the three levels. 6 Conclusion In the framework of GP regression with both continuous and categorical inputs, we focused on problems where categorical inputs may have a potentially large number of levels L, partitioned in G L groups of various sizes. We provided new results about parsimonious block covariance matrices, defined by a few within- and between-group covariance parameters. 23 (a) 5 groups (common betweengroup covariance) (b) 5 groups (general) 6 5 4 3 0.6 2 1.0 0.8 2 3 0.2 0.4 1 1 4 −0.2 0.2 1 Figure 10: Estimated correlation parameters among the chemical element. 5 0.8 0.6 0.4 0 −0.4 −0.6 −0.8 6 0.0 0.2 0.4 0.6 0.8 1.0 −1 (a) Transformation. (b) Correlations. 3 2 1 Figure 11: Estimated correlation parameters for the energy. 1 0.8 1 0.6 0.4 0.2 2 0 −0.2 −0.4 3 −0.6 −0.8 −1 Figure 12: Estimated correlation parameters for the geometric shape. 24 We revisited a two-way nested Bayesian linear model, where the response term is defined as a sum of a group effect and a level effect. We obtained a flexible parameterization of block covariance matrices which automatically satisfy the positive definiteness conditions. As a particular case, we recover situations where the within-group covariance structures are compound symmetry, with possible negative correlations. Furthermore, we showed that the positive definiteness of a given block covariance matrix can be checked by verifying that the small matrix of size G obtained by averaging each block is positive definite. This criterion can be useful if the proposed block matrix has a desirable constraint, such as homoscedasticity, which is not directly handled by the proposed parameterization. We applied these findings on several toy functions as well as an application in nuclear engineering, with 2 continuous inputs, 3 categorical inputs, one of them having 94 levels corresponding to chemical numbers in Mendeleev’s table. In this application, 5 groups were defined by experts. The results, measured in terms of prediction accuracy, outperform those obtained with oversimplifying assumptions, such as gathering all levels in a same group. On the other hand, when the categorical input can be viewed as an ordinal one, plugging the right order into warped kernels has lead to slightly better results in our experiments. There are several perspectives for this work. Firstly, one future direction is to find a data-driven technique to recover groups of levels. This may be not an easy task, due to the small number of observations available in the context of GP regression. Similarly, if there is an order between levels, can we infer it from the data? Secondly, the trend of the GP models has been fixed to a constant. More complex forms, based on linear models, could be explored. Software information and acknowledgements Implementations have been done with the R packages mixgp and kergp [Deville et al., 2015]. Illustrations use ggplot2 [Wickham, 2009] and corrplot [Wei and Simko, 2016]. This research was conducted within the frame of the Chair in Applied Mathematics OQUAIDO, gathering partners in technological research (BRGM, 25 CEA, IFPEN, IRSN, Safran, Storengy) and academia (CNRS, Ecole Centrale de Lyon, Mines Saint-Etienne, University of Grenoble, University of Nice, University of Toulouse) around advanced methods for Computer Experiments. Appendix Proof of Proposition 1. The vector (λ, λ1 + · · · + λL ) is a centered Gaussian vector with covariance matrix IL 1L vλ . 1> L L Hence the conditional distribution of λ knowing λ = 0 is a centered Gaussian vector with covariance matrix −1 cov(λ \| λ = 0) = vλ [IL − 1L L−1 1> L ] = vλ [IL − L JL ]. Then, by using the independence between µ and the λ` ’s, we deduce cov(η \| λ = 0) = vµ JL + vλ [IL − L−1 JL ] = vλ IL + [vµ − L−1 vλ ]JL . −1 We recognize the CS covariance matrix ΓCS L (v, c) with v = vµ + (1 − L )vλ −1 and c = vµ − L vλ . As a covariance matrix, it is positive semidefinite. Furthemore, we have c < v and c + (L − 1)−1 v = vµ [1 + (L − 1)−1 ] > 0, and the conditions of positive definiteness (12) are satisfied. Conversely, let C be a positive definite CS matrix ΓCS L (v, c). Then we have −1 −1 −(L−1) v < c < v, and we can define vµ = L [v +(L−1)c] and vλ = v −c. From the direct sense, we then obtain that the covariance matrix of η \| λ = 0 is ΓCS L (v, c) = C. Proof of Proposition 2. The first part of the proposition is obtained by remarking that if F = cov(z), then F = cov(z). Thus, assuming that z is centered, F = 0 is equivalent to z = 0 with probability 1. For the second part, notice that z = 0 means that z is orthogonal to 1L . Thus, one can write the expansion of z in the orthonormal basis 1⊥ L defined by A. Denoting by t the (L − 1)-vector of coordinates, we have z = At. This gives F = cov(At) = A cov(t)A> , and (14) follows with M = cov(t). 26 To prove unicity, observe that, by definition, A> A = IL−1 , A> 1L = 0. Starting from F = AMA> , and multiplying by A> on the left and by A on the right, we get M = A> FA, showing that M is unique. Now, let F = v[IL − L−1 JL ]. Since JL = 1L 1> L , we obtain M = A> FA = v[A> A − L−1 (A> 1L )(1> L A)] = vIL−1 . As a by-product, notice that resubstituting M into F = AMA> gives AA> = IL − L−1 JL . Finally, if z ∼ N (0, vIL ), then the properties of conditional Gaussian vectors lead immediately to cov(z\|z = 0) = F. Proof of Proposition 3. The expressions of Wg and Bg,g0 are obtained directly by using the independence assumptions about µ and the λ’s. Notice that Fg , the covariance matrix of λg/. knowing λg/. = 0, is centered by Proposition 2. This gives Wg − Wg Jng = Fg , which is positive semidefinite. Conversely, let T be a positive semidefinite matrix of the form (10), such e be that Wg − Wg Jng is positive semidefinite for all g = 1, . . . , G. Let T e is also a the matrix obtained from T by averaging each block. Then T positive semidefinite matrix. Indeed, since T is positive semidefinite, it is e is the covariance matrix the covariance matrix of some vector z. Then T P of e z, the vector obtained from z by averaging by group: e zg = n−1 g `∈Gg z` . Thus there exists a centered Gaussian vector (µg )1≤g≤p whose covariance matrix is e Γµ = T. Now, for g = 1, . . . , G, define Fg = Wg − [Γµ ]g,g Jng = Wg − Wg Jng . Observe that Fg = 0, and by assumption Fg is positive semidefinite. Hence, from Proposition 2, there exists a centered Gaussian vector (λg/j )1≤`≤ng such that Fg = cov(λg/. \|λg/. = 0). We can assume that λ1/. , . . . , λG/. are independent, and µ and λ are independent. Finally, we set ηg/` = µg + λg/` . By the direct sense and (16), we obtain that T is the covariance matrix of η conditional on {λg/. = 0, g = 1, . . . , G}. 27 Proof of Corollary 1. Let T be a positive semidefinite matrix of the form (10) with CS diagonal blocks. Then the diagonal CS matrices are positive semidefinite, leading to vg − cg ≥ 0. Thus, Wg − Wg Jng = (vg − cg )(Ing − n−1 g Jng ) is a positive semidefinite CS matrix. Hence, by Prop. 3, T is obtained from Model (15), with cov(λg/. \|λg/. = 0) = (vg − cg )(Ing − n−1 g Jng ). By Prop. 2 (last part), we can choose Γλg = vλg Ing , with vλg = vg − cg ≥ 0. Conversely, if Γλg = vλ,g Ing , then by Prop. 2, Fg = cov(λg/. \|λg/. = 0) is a CS covariance matrix. The result follows by Prop. 3. Proof of Proposition 4. The direct sense of (i) has already been derived in e the proof of Prop. 3. Furthermore, inspecting that proof, we see that if T is positive semidefinite, then T admits the representation (15). Thus T is a covariance matrix, and positive semidefinite. For (ii), a proof is available in [Roustant and Deville, 2017]. Notice that we need to add the condition that Wg is positive definite for all g = 1, . . . , G. However, adding an equivalent condition for (i), namely that Wg is positive semidefinite, was not necessary. 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MAY 2017 1 A Survey on Trapping Sets and Stopping Sets arXiv:1705.05996v1 [cs.IT] 17 May 2017 Aiden Price and Joanne Hall, Member, IEEE. Abstract—LDPC codes are used in many applications, however, their error correcting capabilities are limited by the presence of stopping sets and trappins sets. Trappins sets and stopping sets occur when specific low-wiehgt error patterns cause a decoder to fail. Trapping sets were first discovered with investigation of the error floor of the Margulis code. Possible solutions are constructions which avoid creating trapping sets, such as progressive edge growth (PEG), or methods which remove trapping sets from existing constructions, such as graph covers. This survey examines trapping sets and stopping sets in LDPC codes over channels such as BSC, BEC and AWGNC. Index Terms—LDPC codes, trapping sets, stopping sets, QCLDPC codes, Margulis codes, AWGNC, PEG algorithm, graph covers. I. I NTRODUCTION S technology advances, we wish to communicate over longer distances and have the ability to stay connected even over poor communication channels. While quasi-cyclic low-density parity-check (QC-LDPC) codes are one of the best ways to achieve this [1], their performance in many cases is limited by the presence of trapping sets and stopping sets. Trapping sets and stopping sets can cause iterative decoding methods to fail with relatively few errors. Finding ways to avoid or remove trapping sets and stopping sets will further improve the already high performance of LDPC codes and bring their performance curves even closer to the Shannon limit [2], [3]. Performance optmization is becoming incresingly crucial as the world moves further into the digital age. An increase in the speed at which digital communication occurs through modern applications such as WiFi [4] and DVB-S2 [5] has drastic implications on the overall productivity of the world. In 1962, Gallager [6] introduced low-density parity-check codes (LDPCs). LDPC codes are a class of binary linear block codes with a sparse parity-check matrix. An advantage of using LDPC codes is that they are able to provide error control which is very close to the capacity for many different channels [2]. This categorizes LDPC codes as one of few capacityapproaching codes; error correction methods which can allow the noise in a channel to be set very close to its theoretical maximum while maintaining error-correcting ability [7]. The performance of an error correction method is based upon two properties; the performance of such a code over a channel with variable noise and the optimal bit error ratio (BER) of a code with sufficient signal-to-noise ratio (SNR). The optimal BER is known as the error floor of a code [8], and A A. Price is with the Science and Engineering Faculty, Queensland University of Technology, Queensland, QLD, Australia. e-mail: [email protected]. J. Hall is with the School of Science, Royal Melbourne Institute of Technology, Melbourne. Manuscript received MO DATE, YEAR; revised MO DATE, YEAR. is discussed in different papers in terms of bit error rate (BER), frame error rate (FER), block error rate and symbol error rate, depending on the application being addressed (see [9], [10] for examples of such error floor analysis). Consideration of the error floor is one of the most important aspects of constructing a high-performing LDPC code [10]. Analysis of the performance of LDPC codes over the binary erasure channel (BEC) led to the discovery of stopping sets in 2002 [11]. The Margulis construction [12] improved upon the performance of Gallager codes, though a weakness in this construction led to a high error floor over the additive white Gaussian noise channel (AWGNC) compared to the performance of other constructions of the time [13]. This high error floor was due to the presence of stopping sets. Stopping sets over BEC as described in [11] became a well understood problem and led to the definition of trapping sets, which are defined over AWGNC and BSC. In some early works trapping sets are called near-code words [8], [13]. Trapping sets and stopping sets are an important topic, worthy of a stand-alone survey. II. P RELIMINARIES AND N OTATION In order to engage with the literature on stopping sets and trapping sets an overview of the preliminaries is necessary. We provide a short review of the literature surrounding LDPC codes, the common transmission channels, and common decoding techniques. Definition 1: [14] A binary [n, k, d] linear code, C, is a k-dimensional subspace of an n-dimensional vector space, F2n which is used to provide structure to a message vector for transmission over a channel. In order to transmit messages over communication channels using an error correcting code, we encode the message using a generator matrix. Definition 2: [7] The generator matrix, G, of a code, C, is a matrix which has dimensions k × n. The k rows of G correspond to linearly independent code words which form a basis of C. One of the most important aspects of error correction is the process of decoding. A parity-check matrix allows us to identify whether errors have been introduced during transmission. This matrix can also be represented by a Tanner graph. Definition 3: [7] A parity-check matrix, H, of C is a matrix which generates the nullspace of the code. This means that a code word, c, is in the code, C, iff H · cT = 0, where 0 is an r × 1 null vector. H has dimensions (n − k) × n. Definition 4: [14] The parity-check matrix, H, may be represented by a bipartite graph with variable node set, V, and check node set, C. This bipartite graph is denoted G(H) = (V ∪C, E), MAY 2017 2 where the columns of H indicate the variable nodes in V and the rows of H indicate the check nodes in C. For i ∈ V and j ∈ C, (i, j) ∈ E if and only if Hi j = 1. This bipartite graph is known as a Tanner graph with r = n − k check nodes and n variable nodes (see Fig. 1). We can also refer to individual variable nodes; let the i-th variable node be vi and the j-th check node be c j [15]. Definition 5: [16] If a parity-check matrix of a code is sparse, then the corresponding code, C, is called a low-density parity-check (LDPC) code. We note that the classification of sparse used in the context of LDPC codes is that there are fewer ones in the parity-check matrix than there are zeros [7]. The sparse nature of LDPC codes means that decoding processes have a fast run-time, as there are fewer operations to compute when compared to a non-sparse parity-check matrix. Two more important features of Tanner graphs are neighbours and node degrees. Definition 6: [16][17] For a variable node vi and check node c j , if (i, j) ∈ E we say that nodes vi and c j are neighbours. The degree of a node in the Tanner graph is defined as the number of edges it is connected to. From the node degree definition we can also define regular LDPC codes. Definition 7: [6] An LDPC code is called (dv, dc )−regular if each variable node, v, has degree dv and each check node, c, has degree dc . We denote an LDPC code of this form a C(dv, dc ) code. LDPC codes are designed to be used as error-correction methods over a variety of communication channels. There are three communication channels discussed in this paper; the binary erasure channel (BEC), the binary symmetric channel (BSC) and the additive white gaussian noise channel (AWGNC) [18]. Though these channels handle data transmission in different ways, their encoding and decoding goals are the same. The upper bound on the error correcting ability of an LDPC code is determined by the minimum distance of the code. In order to define the minimum distance of a code, we will first define Hamming weight and Hamming distance. Definition 8: [19] The Hamming weight, w, of a vector is the number of its non-zero elements. The Hamming weight of a binary vector is therefore the number of ones in the vector. The Hamming distance, d, between two vectors, x and y, is the number of places in which they differ; written as d(x, y). In the literature, Hamming weight and Hamming distance are often referred to using the terms “weight” and “distance” [14]. The weight of code words affects the number of operations performed in decoding and the distance between code words affects how many errors can be corrected. Definition 9: [18] The minimum distance of a code, C, is defined as the smallest Hamming distance between any two code words in the code, A. Encoding and Verification The distance between these codewords is given as d(c1, c2 ) = 4. Take the following error vector: The process of transforming a message vector into its associated code word is known as encoding. Every code word c = [c0, c1, ..., cn−1 ] ∈ C can be expressed as c = m · G. Where m = [m0, m1, ..., mk−1 ] is the mesage vector [7]. The code word, c, has the original k information bits as well as an additional r parity bits to give the code word a length of n bits. As the parity-check matrix, H, is the nullspace of the code, we can use it as a verification method to test if a recieved vector is a code word [14]. The product H · cT is denoted the syndrome, s, of c through H. For a given vector, v, v ∈ C iff s = H · vT = 0. There must be an even number of ones in the components of the product H·cT which add to give s = 0. This is known as the even-parity constraint [2]. After a code word, c, is transmitted through a channel, the other party receives a vector, v. If s = H· vT , 0, then v < C and so we use error correcting techniques in attempt to correct v and recover c. d(C) = min{d(x, y) \| x, y ∈ C, x , y}. The following theorem and corollary describe a code’s error detection and correction abilities using minimum distance [18]. Theorem 1: [18] A code C can detect up to s errors in any code word if d(C) ≥ s + 1. A code C can correct up to t errors in any code word if d(C) ≥ 2t + 1. Corollary 1: [18] If a code C has minimum distance d, then C can be used to either detect up to d − 1 errors, or to correct up to (d − 1)/2 errors in any code word. If the minimum distance of a code is too small, then it cannot provide sufficient error correction. This is demonstrated in Example 1 Example 1: Let two code words be given as; c1 = [0100001111] c2 = [0100111010]. e = [0000110101]. If e is added to c1 then the resulting code word is identical to c2 , demonstrating the importance of minimum distance. The minimum distance of an LDPC code is also related to its code-rate; a large code rate lowers the upper bound on the minimum distance of a code. Definition 10: The code rate, R(C), of an LDPC code is the portion of information bits sent in comparison to the entire code vector sent, written as R(C) = k , n where 0 ≤ R(C) ≤ 1. The code rate and minimum distance often determine the error correcting capability of an LDPC code, though the decoding algorithm plays a direct part in the time it takes to decode messages. MAY 2017 3 v1 e e c1 v2 3e → 1 (a) 1 0e 0 (b) 0e 0 1e e 0e 1 1 1 1 → 0e 0 c5 v10 1 0e 0 v9 1 1e 1 1 c4 v8 1 1 1 v7 0e 1 c3 v6 1 1e 2e 1 v5 0 1 e c2 v4 0e 0 0 v3 0 0e 1e 0e 0 (c) (d) Fig. 1. A 5 × 10 irregular Tanner graph (a) used to demonstrate the ER decoding algorithm with the received vector v = [e, 0, e, 1, 1, 1, 1, 0, 1, e] over BEC. Nodes of interest are highlighted gray. (b) shows steps 1 and 2 of the ER algorithm’s first iteration. (c) shows the changes made in step 3 and then step 2 of the second iteration. (d) shows the changes made in step 3 again, this time revealing that no further erasures exist. The algorithm then terminates on step 4 of this iteration, thus successfully correcting the received vector in 2 iterations. B. Communication Channels and Decoding Basics The communication channel by which transmission occurs impacts the error correction algorithms that are chosen. A communication channel can be modelled as a triple which contains an input alphabet, an output alphabet and the probability of transition between a symbol in the input alphabet and a symbol in the output alphabet [3]. The binary erasure channel (BEC) is one of the simplest, non-trivial channel models [20],[21]. Definition 11: [22],[16] The Binary Erasure Channel (BEC) is a communication channel with two input symbols, 0 and 1, and three output symbols, 0, 1 and e (the erasure symbol). The BEC has an erasure probability, p, where given an input, ci , the output, vi , is defined by the probability formulae P[vi = ci ] = 1 − p and P[vi = e] = p. Analysis of the BEC significantly advanced modern understanding of error correction [20]. An example of a simple decoding process over the BEC is the Edge Removal algorithm. Definition 12: [16] Let c ∈ C ⊆ {0, 1} n be a binary code word transmitted over the BEC and v ∈ {0, 1, e} n be the received vector. The Edge Removal (ER) algorithm proceeds as follows: 1) Initial Step: The value of each received vector bit, vi is assigned to each variable node i ∈ V of the Tanner Graph. 2) The check nodes ci ∈ C count the number of erased bits which are neighbours in the Tanner graph, G(H). 3) If check node ci neighbours only one e symbol in v, the even parity constraint uniquely determines the original value of e for that variable node. 4) Repeat steps (2) and (3) until either all erasures have been recovered or until every check node that is a neighbour of an erased bit is a neighbour of at least two erased bits. For step (4) above, if the latter occurs, then the decoder has failed due to the presence of a stopping set (see Section IV). We provide an example of the ER decoding process in Fig. 1, where ◦ represents a variable node and a check node. This example is found in [15]. For this decoding example, we use an irregular 5×10 paritycheck matrix, H and demonstrate the decoding of received vector v = [e, 0, e, 1, 1, 1, 1, 0, 1, e] using the ER algorithm over the BEC. Another communication channel is the binary symmetric channel (BSC). Definition 13: [16], [22] The Binary Symmetric Channel (BSC) is a communication channel with two input symbols, 0 and 1, and two output symbols, also 0 and 1. The BSC has an error probability, p, where given an input, ci , the output, vi , is defined by the probability formulae P[vi = ci ] = 1 − p and P[vi = c¯i ] = p. An example of a decoding process over the BSC is the Gallager A algorithm [6]. Definition 14: [6], [22], [23] Let c ∈ C ∈ {0, 1} n be a binary code word transmitted over the BSC and v ∈ {0, 1} n be the received vector. The Gallager A algorithm proceeds as follows: 1) Initial Step: The value of each received vector bit, vi is assigned to each variable node i ∈ V of the Tanner graph. 2) After this, a check node ci sends to all neighbouring variable nodes vi, . . . , v j the sum (mod 2) of all of the adjacent variable nodes except for the node itself (where j is the degree of check node ci ). 3) Each variable node, vi then sends the following to their adjacent check nodes: If all messages from check nodes ci other than the target check node of the message are equal, then vi sends that message back, otherwise it resends its prior value. 4) Repeat steps (2) and (3) until either all variables nodes send the same values over two consecutive iterations or when a pre-set max iteration count is reached. MAY 2017 4 v1 1, 0, 1 0 c1 v2 1 0 → 0, 1, 0 1 0, 1, 0 (a) (b) 0 1, 1, 0 1 1 0 1, 1, 1 0, 0, 0 0, 1, 0 1 → 0 1, 1, 0 c5 v10 1, 1, 1 0 0 v9 0, 1, 0 1 0, 0, 0 1 c4 v8 1, 1, 1 0, 1, 0 1 v7 0 1, 1, 1 c3 v6 0, 0, 1 1 1 0 v5 0, 0, 0 0, 0, 1 1 c2 v4 1 0, 0, 0 1 v3 1, 0, 1 1 1 1 0, 1, 0 (c) (d) Fig. 2. A 5 × 10 (6,3)-regular Tanner graph (a) used to demonstrate the Gallager A decoding algorithm with the received vector v = [0, 1, 1, 0, 1, 0, 1, 0, 1, 1]. We represent a 0 being sent along an edge by a dashed line and a 1 with a full black edge. (b) shows step 1 of the Gallager A algorithm, as well the check node calculation, taken as the addition mod 2 of all incoming message from variable nodes adjacent to each check node (denoted vi → ci ). (c) then shows step 2, ci → vi . Lastly, (d) shows step 3, vi → ci . Due to the complexity of this algorithm, only one full iteration has been shown, though step 4 in Definition 14 describes how decoding continues. The Gallager B algorithm offers improved decoding with an additional step on each loop within the algorithm [23]. For each degree, j, and each check node loop, i, there is a prechosen threshold value, bi, j . Throughout the steps involved in check node ci for each variable node, v, and each adjacent check node, c, if at least bi, j neighbours of v excluding c sent the same information in the previous round, then v sends that information to c; otherwise v sends its received value to c. Algorithm A is a special case of algorithm B, where bi, j = j−1 independent of the round [23]. Throughout the decoding procedure, if the pre-set max iteration count is reached without completion, the decoder has failed due to the existence of a trapping set (see Section V). An example of the first steps of the Gallager A algorithm (see Fig. 2) demonstrates the differences between the decoding considerations made between the BEC and the BSC. The most complex channel considered here is the binary input additive white gaussian noise channel, expressed commonly either as the BI-AWGNC or just as the AWGNC [24]. Definition 15: [24] Let X ∈ {0, 1}∗ be a message vector, where ∗ denotes an arbitrary length. The additive white Gaussian noise channel (AWGNC) maps the input vector, X, to the vector X 0 ∈ {+1, −1}∗ and then adds the result with Gaussian white noise to give an output vector Y = X 0 + W, where W ∼ N (0, N0 /2Eb ). Each code symbol, y ∈ Y , carries with it a signal to noise ratio (SNR) of Eb /N0 and the conditional distribution of Y is P(y\|x 0) = PW (y− x 0) = p 1 2π(N0 /2Eb ) ·exp −(y − x 0) , (1) (N0 /Eb ) 2 which gives the output alphabet for the AWGNC as y ∈ R. As in the BEC and BSC, we would like to have some indication of the errors that a channel is introducing to the code word. A metric used for the AWGNC is the log likelihood ratio (LLR). P(y\|x = 0) L(y\|x) = ln . (2) P(y\|x = 1) This L-value describes the likelihood that x is 0 or 1. If L is positive then P(y\|x = 0) > P(y\|x = 1) and thus the input estimate should be x̂ = 0. There are methods to map from Y ∈ R to Y 0 ∈ {0, 1}∗ and, as such, all decoding methods used over the BSC can be implemented on the AWGNC. However, high performing decoding algorithms, such as maximumlikelihood decoders, the sum-product algorithm and the maxproduct algorithm [8],[25],[26] utilize the LLR information to improve decoding speed [27]. The BSC can be used for these channels as LLR values are defined over the BSC, though the AWGNC more closely models the influence of real-world communication channels and is favoured for high performance simulations [7],[24]. Example 2: The BSC has a conditional LLR function as the bit flipping probabilities are well understood for the outputs ( 1−p ln( p ) y = 0 LBSC (y\|x) = (3) p ln( 1−p ) y=1 As the noise determines the values of y in the AWGNC and y ∈ R, the LLR on this channel is defined as L AW G NC (y\|x) = 4 Eb y. N0 (4) The decoding algorithms used on the AWGNC are far more complex than those on the BEC and BSC and, as such, we provide an overview of various methods rather than detailed definitions and examples. Decoding methods used over the AWGNC tend to be message passing algorithms, where nodes send information to their neighbours to correct errors based on the structure of the parity-check matrix [28]. The original message-passing algorithm [6] is an example of a flooding MAY 2017 5 v1 ability of a code as well as the frame error ratio (FER) [8], [10], [15]. The FER is the ratio of frames or whole messages transmitted which cannot be fully corrected versus the total number of frames transmitted. The largest contributors to the error floor stopping sets and trapping sets. c1 v2 v3 v7 c2 v4 v5 c3 → c1 c4 c5 v6 v7 c4 v9 v8 v9 c5 v10 (a) (b) Fig. 3. (a) Tanner graph for the irregular 5 × 10 parity-check matrix given in Example 3. (b) Induced subgraph of the highlighted stopping set with consistent labelling. schedule [29] where in each iteration, all variable nodes and subsequently all check nodes pass new messages to their neighbours. Another example of a flooding schedule is the sum-product algorithm (see [25]). An improved schedule, where both variable nodes and check nodes send messages to each other throughout a single iteration is known by many names including serial scheduling [30], [31], layered scheduling [32] and sequential scheduling [33]. These algorithms offer an improved decoding performance as information is moving through the Tanner graph more frequently. Examples of decoding algorithms using this scheduling include the max-product algorithm (MPA) [26] and the belief propagation algorithm (BPA) [33]. BPA is widely used in LDPC code analysis and is based on the likelihood that a node takes a value given its current value and the values of nearby nodes from previous iterations [23]. Error correction must be implemented differently for each channel. The edge removal algorithm, for example, deals with erasures and thus is not suitable over the BSC. Errors which occur during transmission that are not corrected by the decoding algorithm form what is known as the bit error ratio (BER); the ratio of bits that cannot be corrected versus the total number of bits transmitted [8] [34]. In order to test the performance of LDPC codes, we can simulate the transmission of messages over an increasing signal-to-noise ratio (SNR) and calculate the BER of a code under varying conditions. As SNR grows larger, the BER of a code will suddenly decrease depending on the conditions of the channel and the error correcting capability of the LDPC code in use. This curve is known as the waterfall region [8]. The best scenario for correcting errors is when the probability of error during transmission over a channel is negligible and when the implemented error correcting code can correct many errors. Definition 16: [8] The waterfall region eventually ends in all BER graph curves as anomalous errors cause decoders to fail even with a high SNR ratio. The lowest the BER becomes before levelling is called the error floor of a code. The BER is a standard way to analyze the error correcting III. C YCLES AND G IRTH The decoding method we choose has direct implications for the accuracy and efficiency of decoding. Cycles were the first known negative characteristic of LDPC codes and were extensively studied as they impacted on the accuracy of high performance LDPC codes [35]. A cycle in a graph is a sequence of connected nodes which form a closed loop where the initial and final node are the same and no edge is used more than once [7]. The cycle length is the number of edges a cycle contains, and the length of the smallest cycle in a graph is denoted as its girth [17]. If no cycles exist within the Tanner graph of a paritycheck matrix, then the iterative belief propagation decoding technique is always successful with sufficient iterations [36]. However, if the neighbours of a node are not conditionally independent then belief propagation methods become inaccurate [35]. The inferred solution is to construct a parity-check matrix with no cycles. However, as discussed in Section VII, this is unessecary as not all cycles negatively impact the decoding efficiency of LDPC codes. In fact, the restriction of girth can lead to constraints on the structure of the code which further impedes the decoding efficiency [35]. The cycles which negatively impact the decoding efficiency of LDPC codes combine to form what are known as stopping sets and trapping sets [37]. These sets lead to a high error floor in otherwise efficienct LDPC code constructions throughout various communication channels and affect all high performing decoding algorithms. IV. S TOPPING S ETS OVER BEC Stopping sets are collections of variable and check nodes in the Tanner graph of an LDPC code which greatly reduce its error correcting ability. These sets cause decoding to fail when certain variable nodes are affected by errors after transmission. Stopping sets were first described in 2002 by Di et al [11], who were researching the average erasure probabilities of bits and blocks over the BEC. Definition 17: [11] Let G(H) be a Tanner graph and V be the set of variable nodes in G(H). A stopping set, S, is a subset of V, such that all neighbours of S are connected to S at least twice. The empty set is also a stopping set and the space of stopping sets is closed under union [11]; if S1 and S2 are both stopping sets then so is S1 ∪ S2 . The following lemma describes a stopping set by the performance of the LDPC code’s decoding algorithm. Lemma 1: [11] Let G be the generator for an LDPC code over the BEC and E denote the subset of the set of variable nodes which is erased by the channel after the transmission of a message. Then the set of erasures which remain when the MAY 2017 6 v1 0 0 c1 v2 2e 0 v3 1 1 1 → e e (a) 2e 0 e e 2e 0 (b) 0e 1 e 2e 0 → 2e 0 c5 v10 1 2e 0 v9 1 0e e c4 v8 0e 1 1 0e 1 v7 1 0e 1 c3 v6 0 1e e v5 2e 0 c2 v4 0 3e 2e 0 (c) (d) Fig. 4. The effect of a stopping set on the ER decoding process for the 5 × 10 irregular Tanner graph (a) on the right hand side of the line. (b) shows steps 1 and 2 of the ER algorithm’s first iteration. (c) shows the changes made in step 3 and then step 2 of the second iteration. Finally, (d) shows that no further erasures can be corrected, and thus we see that the received vector v = [0, 0, 1, e, 1, 1, e, 0, e, 0] produces a scenario in the Tanner graph where the ER algorithm cannot retrieve the original code word. From Definition 12, we know that this is due to the presence of a stopping set. decoder stops is equal to the unique maximal stopping set of E. Definition 17 is now widely accepted [38],[39],[40]. Given a BEC with erasure probability , the performance of the code over the BEC is completely determined by the presence of stopping sets [8]. Since stopping sets have a combinatorial characterization, their distributions through various Tanner graphs can be analyzed rigorously [11], [38]. Definition 18: [38] Let S denote the collection of all stopping sets in a Tanner graph, G(H). The stopping number, s∗ , of G(H) is the size of the smallest, non-empty stopping set in S. The stopping number of a code aids in the analysis of the code’s error floor. It is known that the performance of an LDPC code over the BEC is dominated by the small stopping sets in the graph [8]. The larger this value is, the lower the error floor of the code. In some cases, this stopping number increases linearly with the number of variable nodes, \|V \|, in the Tanner graph [38]. This can be seen more easily using the stopping ratio. Definition 19: [38] Let G(H) be a Tanner graph with n variable nodes and stopping number s∗ . The stopping ratio, σ ∗ , of a Tanner graph is defined by s∗ /n; the ratio of its stopping number to the number of variable nodes. A stopping set in the parity-check matrix of an LDPC code is shown in Example 3. Example 3: [15] Let C be the code with the following check matrix 0  1  H = 1 0  0  0 0 0 1 1 1 0 1 0 0 1 1 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 1 0 0 0 0 1 0 1 0 1 1 1 1 0 1 1 . 0 1 Columns 7 and 9 in H have been highlighted as they belong to a stopping set. The Tanner graph for C with the stopping set highlighted is shown in Fig. 3. A stopping set must be either empty or at least contain two variable nodes. The stopping number, s∗ , of C is therefore 2 and its stopping ratio, σ ∗ , 0.2. An example showing the impact of a stopping set on the decoder is shown in Fig. 4, where the edge removal decoding algorithm is used over the BEC. Solutions to the problem of stopping sets (covered in Section VII) involve either avoiding or removing small stopping sets in the Tanner graph, leaving only LDPC codes with large stopping sets [39]. While stopping sets are well defined and some solutions exist to minimize their effect of the error floor of LDPC codes, the terminology does not support channels without erasure. V. T RAPPING S ETS OVER BSC, AWGN Trapping sets, much like stopping sets, are also collections of variable nodes and check nodes which impede the error correcting ability of LDPC code. Only small, elementary trapping sets impact the error floor of LDPC codes over the BSC and AWGNC because of clustering [8], [41]. The definition of trapping sets came shortly after stopping sets were defined. Similarly to the BEC, when decoding over the BSC and AWGNC, sometimes the maximum iteration count is reached when only a small set of variable nodes are in error. Experiments with the argulis codes lead to the definition of trapping sets [8], [13]. Definition 20: [8] Let G(H) be a Tanner graph. For a received vector, y, of length n, we define the failure set, T(y), to be the set of bits that are not eventually correct using some arbitrary iterative decorder. Decoding is successful on y if and only if T(y) = ∅. Definition 21: [41], [8] If T(y) , ∅ then T(y) is a trapping set. More specifically, T is an (a,b) trapping set in H if it has a variable nodes for which the sub-graph induced by T contains b ≥ 0 odd-degree check nodes. MAY 2017 Fig. 5. A (5,3) trapping set (left) with critical number k = 3 and a (4,4) trapping set (right) with critical number k = 4 [10]. These k values are found using the Gallager B decoding algorithm and may vary when other decoding algorithms are applied. Iterative techniques on the BSC and AWGNC distinguish trapping sets from stopping sets over the BEC [8]. If there is only iteration by which the decoding algorithm can become trapped then the notion of trapping sets becomes irrelevant. Lemma 2: [8] Let C be a code using a one-step maximum likelihood decoder, then the trapping sets are precisely the non-zero code words. Though, if the channel is BEC, then an iterative decoding failure is said to be due to stopping sets, making stopping sets and trapping sets equivalent over the BEC. Lemma 3: [8] Let C be a code using a belief propagation algorithm over BEC, then the trapping sets are precisely the stopping sets. Lemma 3 is an important bridge between trapping sets and stopping sets, allowing us to relate the BEC to the BSC and AWGNC. Decoding failure in an LDPC code over the BSC and AWGNC is largely due to the existence of trapping sets. Trapping sets pose a real threat to the error correcting ability of LDPC codes; even though there may be very few nodes in error after transmission, if enough of those nodes belong to a trapping set, the decoder will fail. Definition 22: [10] Let T be a trapping set. The critical number, k, is the minimal number of variable nodes that have to initially be in error for the decoder to become “trapped" in T. It is important to note that the variables nodes that are initially in error do not necessarily belong to the trapping set; it is possible that, at some iteration, the trapping set is entered, causing the decoder to fail. In order to become trapped, the decoder must, after some finite number of iterations, be in error on at least one variable node from T at every iteration thereafter. Only trapping sets with a small number of variable nodes and check nodes impact the error-floor of LDPC codes [8], [41]. Definition 23: [41] An (a, b) √ trapping set in a [n, k, d] code is a small trapping set if a ≤ n and b ≤ 4a. Only these small trapping sets contribute to a larger errorfloor [8]. Small trapping sets are also of elementary form [41]. Definition 24: [41] An elementary (a,b) trapping set in a [n, k, d] code is a trapping set for which all check nodes in the induced subgraph have either degree one or two, and there are exactly b degree-one check nodes. While check nodes of odd degree larger than one are possible, they are very unlikely within small trapping sets 7 [8],[41]. Techniques to find and remove elementary trapping sets have become crucial when constructing high perfoming codes [10], [42]. Two examples of trapping sets [10] are shown in Fig. 5. In Fig. 5 the trapping set on the right has a smaller number of variable nodes than the one of the left, however, under the Gallager B decoding algorithm the larger trapping set has a smaller critical number. Thus the performance of the code is limited by the larger trapping set. This idea is quite unintuitive and shows the depth of consideration which must be made when attempting to improve the error floor of LDPC codes. The problems which trapping sets and stopping sets introduce to LDPC code ares important to research and solve. There do exist methods for constructing LDPC codes by avoiding or removing trapping sets and stopping sets, however, these methods come at the cost of restraining other properties such as code length, density or error correcting ability [9]. VI. T HE INFLUENCE OF S TOPPING S ETS AND T RAPPING S ETS ON LDPC C ODE P ERFORMANCE The original LDPC codes proposed in 1962 by Gallager [6], were construction methods which allowed for varied code rates. Definition 25: [13] A Gallager code is an LDPC code constructed using a parity-check matrix with uniform row weight i and uniform column weight j. The code has length n code words and has code rate R(C), which gives a parity-check matrix, H, with n columns and k rows, where k = n(1− R(C)). Naive analysis indicated that failed decoding is due to received vectors containing too many errors for the decoding algorithm [6]. Analysis of a range of error patterns by Di et al [11] determined that this was not always the case; leading to the definition of stopping sets over the BEC. A variety of analyses of Gallager codes have shown high performance [2] [43]. A construction in 1982 by Margulis [12] promised an improved performance over the AWGNC. For each prime, p, let SL2 (p) be the Special Linear Group whose elements consist of 2×2 matrices of determinant 1 over Z p . This group has k = (p2 − 1)(p2 − p)/(p − 1) = (p2 − 1)p elements. For p ≥ 5, the Margulis code is of length n = 2k, with code rate R(C) = 1/2 [12]. The rows of the parity-check matrix are indexed by the elements of SL2 (p) and the columns are indexed by two copies of SL2 (p); detailed in the following definition. Definition 26: [13][12] Let SL2 (p) be generated by the following matrices; 1 2 1 0 A= ,B = 0 1 2 1 If g ∈ SL2 (p) is the index of a row of the parity-check matrix, a one is placed in the columns corresponding to g A2 , g ABA−1 and gB on the left hand side of the matrix and also in the columns corresponding to g A−2 , g AB−1 A−1 and gB−1 on the right hand side of the matrix. This results in a (3,6)-regular parity-check matrix for a Margulis code. An example of a parity-check matrix generated using the Margulis construction is shown in Fig. 6 to demonstrate the MAY 2017 8 Parity-Check Matrix for Margulis Code (p=7) Frame Error Rate 10 -2 0 (336,672)-Margulis (336,672)-Gallager 50 100 150 10 -3 FER 200 250 300 0 100 200 300 400 500 600 10 -4 Fig. 6. Parity-check matrix generated using the Margulis construction, setting p = 7 to give a (3,6)-regular 1/2-rate code with n = 672. The blue dots represent ones in this matrix with the remaining white space representing zeros. 10 -5 sparse nature of LDPC codes. Another example of a Margulis parity-check matrix can be found in [13] where p = 11 which corresponds to a 1/2-rate code with n = 2640. While the code has a higher performance than a random Gallager code, the error floor is still quite high [13]. This error floor was claimed to be due to near-code words [13]. A comparison between the Margulis code and a random Gallager code, both with n = 672, can be seen in Fig. 7. Definition 27: [13] Let H be a parity-check matrix. If x is a vector of weight w and HxT = s where s is of weight v, then x is a (w, v) near-code word. Near-code words are different from stopping sets. Typical (w, v) near-code words contain v check nodes which are only connected to the variables nodes once. The near-code words in the Margulis code are the (12, 4) and (14, 4) near-code words [13]. The high error floors of the Margulis code can be reproduced with a 5 bit approximation to a belief propagation algorithm [8]. Near-code words account for 98% of the error floor performance of the Margulis code. Near code words are trapping sets [8]. Trapping sets are often clustered [8]; if one trapping set is found it will often contain nodes which belong to another trapping set. This makes the search for trapping sets somewhat simpler. Finding both stopping and trapping sets are NP-hard problems [16], [44], which makes solutions to these sets difficult to analyse. The ER decoding algorithm is simple and the effect of stopping sets can be demonstrated easilly. However, decoding over the BSC or AWGNC is much more complex (see Fig. 2). Iterative decoding methods tend to have maximum iteration counts as termination conditions and, as such, demonstrating the effect of trapping sets is difficult to show. In lieu of an example, we remind the reader of the termination conditions for the Gallager A algorithm. This decoder terminates either when all variable nodes send the same values over two consecutive iterations or when some maximum iteration count is reached. In the latter case, the decoder has failed due to the existence of a trapping set. 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4 Eb/No (dB) Fig. 7. BER comparison between the p = 7 Margulis code (using the same example as presented in Fig. 6) and a random Gallager code, both with n = 672 and decoded using MPA over the AWGNC. These graphs are also known in the literature as waterfall curves [8]. While we have only discussed the issues with the Margulis code using specific decoding algorithms here, there are many code constructions which contain trapping and stopping sets and decoding algorithms which terminate for the same reasons. For further reading, see [45]. Further reading on stopping sets in LDPC codes include the message passing (MP) algorithm [46] and the maximumlikelihood (ML) decoder [47]; both over the BEC. Further reading on trapping sets include finite alphabet iterative decoders (FAIDs) [48] and constructions based on Latin squares [49]. This construction offers high structure by which stopping sets and trapping sets can be analysed. If constructions which avoid trapping and stopping sets exist, then the error floors of the associated LDPC codes will lower significantly. This would improve the speed at which almost all digital communication occurs given the already high performance of LDPC codes in modern applications including WiFi [4] and DVB-S2 [5]. VII. C URRENT S OLUTIONS The simple goal is to avoid or completely remove every stopping set or trapping set from an LDPC code. This is both not reasonable given the number of cycles in an LDPC construction [7] and, more importantly, not necessary [38], [41]. Only small, elementary trapping sets impact the error floor due to clustering [8], [41]. If there are enough errors in transmission for a decoder to get trapped in a large trapping set then it is highly likely that it would also be trapped in a small trapping set [8]. If there are not enough errors for the decoder to become trapped in a large trapping set, the received vector can either be successfully decoded or the decoder will fail due to the presence of at least one small trapping set. The current solutions to trapping sets are the development of constructions which avoid small trapping sets and the removal of trapping sets from existing constructions. MAY 2017 9 vi Depth-0 ... ... ... Depth-1 ... ... .. . .. . ... ... .. . ... Depth-l ... ... ... Fig. 8. [51] A subgraph (tree) contained in the depth l neighbourhood spreading from the variable node vi . Note that ◦ here represents a variable node and represents a check node. A. Avoiding Trapping Sets Stopping sets pose threats to the error correction of messages sent over the BEC, however, in practice the AWGNC is used and so, when discussing proposed solutions, we focus on the influence of trapping sets over the AWGNC. The progressive-edge-growth (PEG) construction [50], [51] is a method of constructing Tanner graphs with high girth. Many trapping sets include small cycles, so the likelihood of a small trapping set being constructed is small with a graph of high girth [52]. In order to give the definition for the PEG construction, some definitions are needed. The PEG construction method uses variable and check node degree sequences [51]. The variable node degree sequence is denoted Dv = {dv0 , dv1 , . . . , dvn−1 }, where dvi is the degree of variable node vi , 0 ≤ i ≤ n − 1, and dv0 ≤ dv1 ≤ · · · ≤ dvn−1 . The parity check sequence is denoted Dc = {dc0 , dc1 , . . . , dcm−1 }, where dc j is the degree of check node c j , 0 ≤ j ≤ m − 1, and dc0 ≤ dc1 ≤ · · · ≤ dcm−1 . The construction partitions the set of edges, E, to E = Ev0 ∪ Ev1 ∪ · · · ∪ Evn−1 , where Evi contains all edges incident on symbol node vi . The k th edge incident on vi is denoted as Evki , where 0 ≤ k ≤ dvi − 1. The neighbourhood of depth l for variable node vi is Nvli and is defined as the set of all check nodes included in a subgraph (tree) spreading from variable node vi within depth l. This is demonstrated in Fig. 8. The complement of Nvli is Nv−li = C\Nvli , where C is the set of check nodes. The subgraph (tree) generated this way is constructed breadth-first with vi as the root. Given the parameters n, m and Dv , we define the PEG construction as follows. Progressive edge-growth algorithm (PEG) [51]: for i = 0 to n − 1 do begin for k = 0 to dvi − 1 do begin if k = 0 Ev0i ← edge (c j , vi ), where ev0i is the first edge incident to vi and c j is a check node that it has the lowest check-node degree in Ev0 ∪ Ev1 ∪ · · · ∪ Evi−1 . else Expand a subgraph from symbol node vi up to depth l in Ev0 ∪ Ev1 ∪ · · · ∪ Evi−1 until the cardinality of Nvli stops increasing but is less than m, or Nv−li , but Nvl+1 = , then Evki ← i k edge (c j , vi ), where Evi is the k th edge incident to vi and c j is a check node chosen from the set Nv−li having the lowest check-node degree. end end end When presented with check nodes of the same degree, a decision must be made; the selection of a check node at random or the selection of a check node according to some order. An improved construction, the randomized-PEG construction [53], chooses at random, though the deterministic nature of the ordered check node process might be of use [51]. An example of PEG construction is given in Fig. 9, setting n = 10, m = 5 and Dv = {2, 2, 2, 2, 2, 2, 2, 2, 2, 2}. The PEG construction maximises the local girth of a variable node when a new edge is added to the node [51]. After the discovery of stopping and trapping sets the PEG construction was modified [37], [53]. The PEG construction is notable for its ability to create high girth LDPC codes, however, the number of cycles is not controlled. Trapping sets are formed by a combination of several cycles [37]. The PEG algorithm, while having a higher girth than aternate constructions, contains more trapping sets; thus leaving the error floor open to improvement. The RandPEG construction improves upon the PEG algorithm by minimizing cycles at the same time as reducing the computational complexity of the PEG algorithm [53]. This can be further improved by adding an objective function to avoid small trapping sets [37]. Quasi-cyclic LDPC (QCLDPC) codes, which are used in many applications [1], [54] can be constructed using the Improved RandPEG algorithm [37]. The objective function used in the improved RandPEG algorithm detects all (5,3) and (6,4) trapping sets, removing all (5,3) trapping sets and as many (6,4) trapping sets as possible without adversely affecting the performance of the LDPC code. The characterization of trapping sets is achieved in [37] through the locations of check nodes in different levels of depth-l trees (see Fig. 8). The resulting construction is as follows: Improved RandPEG algorithm (RPEG) [37]: for i = 0 to n − 1 do MAY 2017 10 v0 v0 v0 c0 v1 c0 v1 v2 v2 v2 v4 → v5 → v6 c3 v7 v8 v8 c4 v9 (b) c2 v5 v6 c4 v9 → c3 v7 v8 (a) v5 v6 c4 v9 v4 c2 c3 v7 v8 v3 v4 c2 c3 v7 c1 v3 v4 v6 v2 c1 v3 v5 v1 c1 v3 c0 v1 c1 c2 v0 c0 c4 v9 (c) (d) Fig. 9. A PEG construction with n = 10, m = 5 and D v = {2, 2, 2, 2, 2, 2, 2, 2, 2, 2}. Check nodes are chosen based on index order. The first edge chosen for each variable node is chosen from the check nodes with lowest degree at random. The generation of the subgraphs and subsequent edge placement are the factors which highlight this construction method. In (a), the edge choices are simplistic as the breadth-first subgraphs are of low depth. This decision making continues in (b) until s5 is considered, where the edge choice is restricted to c2 or c3 due to connections in the subgraph to check nodes c1 and c4 . These choices can be observed in both remaining figures (c) and (d). One notable choice is the 2 n d edge decision for v9 in (d), where the only remaining option once c2 is chosen becomes c4 , which gives a uniform check node degree sequence. begin for k = 0 to dvi − 1 do begin if k = 0 Ev0i ← edge (c j , vi ), where ev0i is the first edge incident to vi and c j is a check node such that it has the lowest check-node degree in Ev0 ∪ Ev1 ∪ · · · ∪ Evi−1 . else Expand a subgraph from symbol node vi up to depth l in Ev0 ∪ Ev1 ∪ · · · ∪ Evi−1 until the cardinality of Nvli stops increasing but is less than m, or Nv−li , but Nvl+1 = . Remove from i Nv−li all check nodes that appear at least once in the depth-3 tree spreading from vi . This removes all check nodes that would create cycles of size < 8. for cm in Nv−li do Compute the number of (5,3) and (6,4) trapping sets that would be created if cm is selected. Remove all check nodes that would create (5,3) trapping sets and remove check nodes which create more than the smallest number of (6,4) trapping sets. If Nv−li , Evki ← edge (cm, vi ), where Evki is the k th edge incident to vi and cm is a check node chosen from the remaining nodes in Nv−li . else Declare a design failure. end end end end end The Improved RandPEG construction algorithm, while having a high computational complexity, performs the task of avoiding trapping sets optimally for given dimensions of an LDPC code. Possible improvements to this construction method include lowering the computational complexity and potentially lowering the girth. The removal of all cycles is unnecessary as not all cycles contribute to trapping sets [38], [41]. The inclusion of a lower girth into a construction which also contains no small trapping sets could lead to an LDPC code with a higher decoding performance. B. Removing Stopping and Trapping Sets The performance of LDPC codes is constrained by the presence of cycles and trapping sets within the code’s paritycheck matrix. We discuss two methods of removing trapping sets; the addition of a redundant parity-check equation [55] and the use of Tanner graph covers [10]. Redundant Parity-Check Equations Adding a redundant parity-check equation is equivalent to adding a redundant row to the parity-check matrix. This has been used in an attempt to remove the trapping sets present in the [2640, 1320] Margulis code [55]. The (12, 4) and (14, 4) trapping sets in the [2640, 1320] Margulis code are elementary point trapping sets [13]. Point trapping sets are subsets of variable nodes that contain all errors ever to occur throughout the decoding process [55]. A redundant parity check row is identified which, when added to the parity-check matrix, potentially disrupts the (12,4) and (14,4) elementary trapping sets. This parity-check row is identified through a genie-aided random search which relies on information about trapped variables not available during MAY 2017 11 decoding [55]. As random searches cannot be used in applied error correction a structured search was considered to be more useful. The structured search identifies variable and check nodes which connect to both the (12,4) and (14,4) trapping sets and combines the projection of the involved nodes such that a redundant parity-check row can be added to eliminate the effect of those trapping sets. The only way to disrupt the (12,4) and (14,4) trapping sets in the Margulis code is if the projection of both the 12 variables and the 14 variables on the redundant parity-check equation has row weight one [53]. This can be most reliably achieved by extending (12,4) trapping sets to (14,4) trapping sets (see Fig. 10). Given that the Margulis code has a (3,6)-regular paritycheck matrix, an elementary (a,b) trapping set contains a fixed number of check nodes. Let e denote the number of check nodes connected to two variable nodes within the trapping set, then e = (3a − b)/2 and therefore two variable nodes and three check nodes must be added to extend a (12,4) trapping set to a (14,4) trapping set [53]. At most one check node can be connected to both of the added variable nodes such that 4-cycles are not created. Such an extension which avoids the creation of 4-cycles in the Margulis code is only possible in two configurations. (a) The two degree-one check nodes of the basic (12,4) trapping set are connected through two additional variable nodes to one additional check node. (b) In the second configuration, the additional variable nodes do not share a check node. These configurations are demonstrated in Fig. 10. The existing check and variable nodes neighbouring the additional check and variable nodes are linearly combined to generate a redundant parity-check equation. A structured search is then used to ensure that this projection has row weight one in both the (12,4) and (14,4) trapping set. The addition of a redundant parity-check equation focuses on point elementary trapping sets from the Margulis code, where the structure of the trapping sets are well known. If this method were applied to other LDPC codes, the location and structure of the trapping sets within such a code are unknown. The redundant rows are computationally inexpensive to compute, however, the code rate of the resulting LDPC code will be reduced and the extra row increases the number of operations per decoding iteration, though by a negligible amount. Another potential problem is the success rate of this solution. The addition of a redundant parity-check equation does not guarantee that the trapping set will be disrupted [55]. Tanner Graph Covers Another method capable of eliminating trapping sets is the utilization of graph covers [10]. This method constructs an LDPC code C (2) of length 2n given a code C of length n. The parity check matrix of this code is denoted H (2) and is initialized to H (2) = H 0 0 H cBO,1 cBO,2 vBO,1 vBO,2 vBO,3 vBO,4 cBO,3 cBO,4 (12, 4) TS cBO,1 vBO,1 vBO,2 vBO,3 vBO,4 cBO,2 cBO,3 cBO,4 (12, 4) TS vE,1 vE,2 cE cEO,1 cEO,2 Expansion to (14,4) TS (a) vE,1 vE,2 cE cEO,1 cEO,2 Expansion to (14,4) TS (b) Fig. 10. [55] The trapping set structure of a (12, 4) trapping set and its (14, 4) expansion configurations; (a) left and (b) right. For (a), the two expansion variables are denoted vE,1 and vE,2 and the check node connected to both of the variables nodes is denoted c E . The unsatisfied check nodes in the original (12, 4) trapping set are denoted c BO,1 through c BO,4 and the variable nodes connected to these check nodes are denoted vBO,1 through vBO,4 . The check nodes of degree one in the expansion of the trapping set are denoted c E O,1 and c E O,2 . The node labels for (b) follow similarly. (2) (2) The operation of changing the value of Ht,k and Hm+t,n+k (2) (2) to “0", and Hm+t,k and Hm,n+k to “1" is termed as edge swapping e. The graph covers method requires that the locations of dominant trapping sets are known. The method of edge swapping is then described as follows. Graph covers algorithm [10]: 1) Take two copies, C1 and C2 , of the code C. Since the codes are identical they share the same trapping sets. Initialize SwappedE dges = , FrozenE dges = ; 2) Order the trapping sets by their critical numbers. 3) Choose a trapping set T1 in the Tanner graph of C1 , with minimal critical number. Let ET1 denote the set of all edges in T1 . If ET1 ∩ SwappedE dges , go to step 5, else go to step 4. 4) Swap an arbitrarily chosen edge e ∈ ET1 \ FrozenE dges. Set SwappedE dges = SwappedE dges∪ e. 5) “Freeze" the edges ET1 from T1 so that they cannot be swapped in the following steps. Set FrozenE dges = FrozenE dges ∪ ET1 . 6) Repeat steps 2 to 4 until all trapping sets of the desired size are removed. Possible improvements to the graph cover method are to prioritize specific edges for swapping and freezing to avoid creating trapping sets of the same critical number [10]. However, experimentally, all trapping sets with minimal critical number were removed using the above algorithm [10]. The graph covers method gave improved FER results for a Tanner code [56], a Margulis code [57] and a MacKay code [58] using the Gallager B decoding algorithm. While the decoding method was constant throughout these results, the application of graph covers will optimize FER performance using an MAY 2017 12 arbitrary decoding algorithm [10]. The LDPC code C (2) created from code C has code rate (2) r ≤ r and minimum distance 2d ≥ d (2) ≥ d. An increase to the minimum distance of the code C (2) gives higher error correcting capabilities than C. However, a lower code rate could decrease the overall efficiency. The lower row and column weight of H (2) gives C (2) higher FER performance than C, though with a trade-off of low decoding complexity. The trade-offs associated with removing trapping sets are more severe than the surveyed construction methods which avoid them (code length, decoding speed from check nodes, etc). The current research goal remains the creation of a construction or modifiable construction which can either avoid or remove small elementary trapping sets without penalty to the code’s error correcting ability or decoding efficiency. VIII. C ONCLUSION Throughout this survey we have covered the literature surrounding LDPC codes, communication channels and decoding techniques. The negative impact cycles have on LDPC code efficiency is noted and the problem of stopping sets and trapping sets have been defined and discussed including the dominance of small elementary trapping sets over AWGNC. A small variety of partial solutions such as the randomized progressive edge-growth algorithm and Tanner graph covers are discussed. The research goal remains to find constructions of LDPC codes without small trapping sets. ACKNOWLEDGMENT The authors would like to thank Professor Ian Turner, who worked closely with us throughout our research, Emeritus Professor Ed Dawson and Dr Harry Bartlett for their help in the final stages before submission, Dr Dhammika Jayalath for his help with the decoding simulations over AWGNC, and Xuan He for his suggestions on how to present our BER data. Computational resources and services used in this work were provided by the HPC and Research Support Group, Queensland University of Technology, Brisbane, Australia. A. 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Theory, vol. 54, no. 8, pp. 3763–3768, 2008. [11] C. Di, D. Proietti, I. E. Telatar, T. J. Richardson, and R. L. Urbanke, “Finite-length analysis of low-density parity-check codes on the binary erasure channel,” IEEE Trans. Inf. Theory, vol. 48, no. 6, pp. 1570–1579, 2002. [12] G. A. Margulis, “Explicit constructions of graphs without short cycles and low density codes,” Combinatorica, vol. 2, no. 1, pp. 71–78, 1982. [13] D. J. MacKay and M. S. Postol, “Weaknesses of Margulis and Ramanujan-Margulis low-density parity-check codes,” Electronic Notes in Theoretical Computer Science, vol. 74, pp. 97–104, 2003. [14] M. Baldi, QC-LDPC code-based cryptography. Springer Science & Business, 2014. [15] G. Richter, “Finding small stopping sets in the tanner graphs of LDPC codes,” Turbo Codes & Related Topics; 6th International ITG-Conf. on Source and Channel Coding (TURBOCODING) pp. 1–5, 2006. [16] A. McGregor and O. Milenkovic, “On the hardness of approximating stopping and trapping sets,” IEEE Trans. Inf. Theory, vol. 56, no. 4, pp. 1640–1650, 2010. [17] R. Shedsale and N. Sarwade, “A review of construction methods for regular LDPC codes,” Indian Journal of Comput. Sci. Eng., vol. 3, no. 2, pp. 380–385, 2012. [18] R. Hill, “A first course in coding theory,” Oxford: Clarendon Press, 1986. [19] W. W. Peterson and E. J. Weldon, “Error-correcting codes,” MIT press, 1972. [20] T. Richardson and R. Urbanke, “Modern coding theory,” Cambridge University Press, 2008. [21] P. Elias, “Coding for two noisy channels,” Third London Symposium, vol. 67, 1955. [22] R. Poddar, “Low density parity check codes,” Complexity, vol. 7, p. 8, 2007. [23] A. Shokrollahi, “LDPC codes: An introduction,” Digital Fountain, Inc., Tech. Rep, p. 2, 2003. [24] N. Traore, S. Kant, and T. L. Jensen, “Message passing algorithm and linear programming decoding for LDPC and linear block codes,” Aalborg University, pp. 32–53, 2007. [25] G. Colavolpe and G. Germi, “On the application of factor graphs and the sum-product algorithm to ISI channels,” IEEE Trans. Commun., vol. 53, no. 5, pp. 818–825, 2005. [26] E. Sharon, S. Litsyn, and J. Goldberger, “An efficient message-passing schedule for LDPC decoding,” Proc. 23rd Conv. Electric. Electron. Engineers, pp. 223–226, 2004. [27] P. Hailes, L. Xu, R. G. Maunder, B. M. Al-Hashimi, and L. Hanzo, “A survey of FPGA-based LDPC decoders,” IEEE Commun. Surv. Tuts., vol. 18, no. 2, pp. 1098–1122, 2015. [28] E. Sharon, S. Litsyn, and J. Goldberger, “Convergence analysis of serial message-passing schedules for LDPC decoding,” Turbo Codes & Related Topics; 6th International ITG-Conf. on Source and Channel Coding (TURBOCODING) pp. 1–6, 2006. [29] F. R. Kschischang, B. J. Frey, and H.-A. Loeliger, “Factor graphs and the sum-product algorithm,” IEEE Tran. Inf. theory, vol. 47, no. 2, pp. 498–519, 2001. [30] J. Zhang and M. Fossorier, “Shuffled belief propagation decoding,” Asilomar Conf. on Signals, Systems and Computers, vol. 1, pp. 8–15, 2002. [31] H. Kfir and I. Kanter, “Parallel versus sequential updating for belief propagation decoding,” Physica A: Statistical Mechanics and its Applications, vol. 330, no. 1, pp. 259–270, 2003. [32] D. E. Hocevar, “A reduced complexity decoder architecture via layered decoding of LDPC codes,”, Signal Processing Systems, 2004, pp. 107– 112, 2004. [33] A. I. V. Casado, M. Griot, and R. D. Wesel, “Informed dynamic scheduling for belief-propagation decoding of LDPC codes,”, IEEE International Conf. on Commun., pp. 932–937, 2007. [34] S. Haykin, Communication systems. John Wiley & Sons, 2008. [35] T. Tian, C. Jones, D. Villasenor, and R. Wesel, “Construction of irregular LDPC codes with low error floors,” IEEE Intern. Conf. on Commun. (ICC), vol. 5, no. 1, pp. 3125–3129, 2003. MAY 2017 [36] T. Richardson, M. Shokrollahi, and R. Urbanke, “Design of capacityapproaching irregular low-density parity-check codes,” IEEE Trans. on Inf. Theory, vol. 47, no. 2, pp. 619–637, 2001. [37] M. Diouf, D. Declercq, S. Ouya, and B. Vasic, “A PEG-like LDPC code design avoiding short trapping sets,” IEEE Intern. Symp. Inf. Theory (ISIT), pp. 1079–1083, 2015. [38] A. Orlitsky, K. Viswanathan, and J. Zhang, “Stopping set distribution of LDPC code ensembles,” IEEE Trans. Inf. Theory, vol. 51, no. 3, pp. 929–953, 2005. [39] J. C. S. Ripoll and N. R. Barraza, “A new algorithm to construct LDPC codes with large stopping sets,” SIMULATION, vol. 505, p. 1, 2013. [40] S. V. Ranganathan, D. Divsalar, K. Vakilinia, and R. D. Wesel, “Design of high-rate irregular non-binary LDPC codes using algorithmic stopping-set cancellation,” IEEE Intern. Symp. Inf. Theory, pp. 711–715, 2014. [41] S. Laendner and O. Milenkovic, “Algorithmic and combinatorial analysis of trapping sets in structured LDPC codes,” 2005 Intern. Conf. on Wireless Networks, Commun. and Mobile Computing, vol. 1, pp. 630– 635, 2005. [42] G. Richter and A. Hof, “On a construction method of irregular LDPC codes without small stopping sets,” 2006 IEEE Intern. Conf. on Commun., vol. 3, pp. 1119–1124, 2006. [43] D. J. MacKay, “Good error-correcting codes based on very sparse matrices,” IEEE Trans. Inf. Theory, vol. 45, no. 2, pp. 399–431, 1999. [44] M. K. Krishnan and P. Shankar, “Computing the stopping distance of a Tanner graph is NP-hard,” IEEE Trans. Inf. Theory, vol. 53, no. 6, pp. 2278–2280, 2007. [45] S. Sankaranarayanan, S. K. Chilappagari, R. Radhakrishnan, and B. Vasic, “Failures of the Gallager B decoder: Analysis and applications,” Proc. Inf. Theory and Applic. Works. UCSD, vol. 17, 2006. [46] M. G. Luby, M. Mitzenmacher, M. A. Shokrollahi, and D. A. Spielman, “Efficient erasure correcting codes,” IEEE Trans. Inf. Theory, vol. 47, no. 2, pp. 569–584, 2001. [47] H. Pishro-Nik and F. Fekri, “On decoding of low-density parity-check codes over the binary erasure channel,” IEEE Trans. Inf. Theory, vol. 50, no. 3, pp. 439–454, 2004. [48] L. Danjean, D. Declercq, S. K. Planjery, and B. Vasic, “On the selection of finite alphabet iterative decoders for LDPC codes on the BSC,” IEEE Inf. Theory Works. (ITW), pp. 345–349, 2011. [49] S. Laendner and O. Milenkovic, “LDPC codes based on latin squares: Cycle structure, stopping set, and trapping set analysis,” IEEE Trans. Commun., vol. 55, no. 2, pp. 303–312, 2007. [50] X.-Y. Hu, E. Eleftheriou, and D.-M. Arnold, “Progressive edge-growth Tanner graphs,” IEEE Global Telecommun. Conf., 2001, vol. 2, pp. 995– 1001, 2001. [51] X.-Y. Hu, E. Eleftheriou, and D.-M. Arnold, “Regular and irregular progressive edge-growth Tanner graphs,” IEEE Trans. on Inf. Theory, vol. 51, no. 1, pp. 386–398, 2005. [52] O. Milenkovic, E. Soljanin, and P. Whiting, “Asymptotic spectra of trapping sets in regular and irregular LDPC code ensembles," IEEE Trans. on Inf. Theory, vol. 51, no. 1, pp. 39–55, 2007. [53] A. Venkiah, D. Declercq, and C. Poulliat. “Design of cages with a randomized progressive edge-growth algorithm," IEEE Commun. Lett., vol. 12, no. 4, pp. 301–303, 2008. [54] Y. Wang, J. Yedidia, and S. Draper, “Construction of high-girth QCLDPC codes,” IEEE 5th Intern. Symp. Turbo Codes and Related Topics, pp. 180–185, 2008. [55] S. Laendner, T. Hehn, O. Milenkovic, and J. B. Huber, “When does one redundant parity-check equation matter?” IEEE Globecom 2006, pp. 1–6, 2006. [56] R. Tanner, D. Sridhara, and T. Fuja, “A class of group-structured LDPC codes,” Proc. ISCTA, pp. 365–370, 2001. [57] J. Rosenthal, and P. Vontobel, “Constructions of LDPC codes using Ramanujan graphs and ideas from Margulis,” Proc. of the 38-th Allerton Conference on Commun., Control, and Computing, 2000. [58] D. MacKay, “Encyclopedia of sparse graph codes," 2003. 13 Aiden Price was awarded the QUT Dean’s scholarship and received a BSc and 1 s t class honours in mathematics from Queensland University of Technology. Aiden is currently studying a PhD at QUT in the school of mathematics under the supervision of Doctor Harry Bartlett and Emeritus Professor Ed Dawson and is funded by the APA scholarship. His research interests are coding theory and its application to digital communications and cryptography. Joanne Hall received a BSc and MPhil in mathematics from the Australian National University. She graduated with a PhD from RMIT University in 2011 under the supervision of Asha Rao in the Information Security and Informatics research group. Dr Hall spent one year as a postdoctoral research scientist at Charles University in Prague and four years as a Lecturer at the Queensland University of Technology in Brisbane. In 2017 she has returned to RMIT University as a Lecturer in the School of Science. Her research interests are algebraic and combinatorial structures and their applications in digital communication.	7
arXiv:1705.09058v1 [cs.AI] 25 May 2017 An Empirical Analysis of Approximation Algorithms for the Euclidean Traveling Salesman Problem Yihui He Xi’an Jiaotong University Xi’an, China Ming Xiang Xi’an Jiaotong University Xi’an China [email protected] [email protected] Abstract space, and the cost function is the Euclidean distance. That is, the Euclidean distance between two cities x = (x1 , x2 , ..., xd ), y = (y1 , y2 , ..., yd ) is: With applications to many disciplines, the traveling salesman problem (TSP) is a classical computer science optimization problem with applications to industrial engineering, theoretical computer science, bioinformatics, and several other disciplines [2]. In recent years, there have been a plethora of novel approaches for approximate solutions ranging from simplistic greedy to cooperative distributed algorithms derived from artificial intelligence. In this paper, we perform an evaluation and analysis of cornerstone algorithms for the Euclidean TSP. We evaluate greedy, 2opt, and genetic algorithms. We use several datasets as input for the algorithms including a small dataset, a mediumsized dataset representing cities in the United States, and a synthetic dataset consisting of 200 cities to test algorithm scalability. We discover that the greedy and 2-opt algorithms efficiently calculate solutions for smaller datasets. Genetic algorithm has the best performance for optimality for medium to large datasets, but generally have longer runtime. Our implementations is public available 1 . d X ( (xi − yi )2 )1/2 (1) i=0 This simplification allows us to survey several cornerstone algorithms without introducing complex scenarios. The remainder of this paper is organized as follows. In Section 2, we briefly review the first solutions and survey variants to the TSP. We describe the algorithms used in our experiment in Section 3. A description of the benchmark datasets and results of the experiment are detailed in Section 4, and explains the findings and compares the performance of the algorithms. We then conclude and describe future work in Section 5. 2. Background An example TSP is illustrated in Figure 1. The input is a collection of cities in the two dimensional space. This input can be represented as a distance matrix for each pair of cities or as a list of points denoting the coordinate of each city. In the latter method, distances are calculated using Euclidean geometry. A non-optimal tour is shown in subfigure (b). Although not shown in the figure, each edge will have some non-negative edge weight denoting the distance between two nodes or cities. Due to the computational complexity of the TSP, it may be necessary to approximate the optimal solution. The optimal tour is shown in sub-figure (c). For small graphs, it may be possible to perform an exhaustive search to obtain the optimal solution. However, as the number of cities increases, so does the solutions space, problem complexity, and running time. If n is number of cities. The number of possible Pthe n−1 edges is i=0 i. The number of possible tours is (n−1)!/2. since the same tour, with start point X and Y appears twice: once with X as the start node and once with Y as the start node. 1. Introduction Known to be NP-hard, the traveling salesman problem (TSP) was first formulated in 1930 and is one of the most studied optimization problems to date [8]. The problem is as follows: given a list of cities and a distance between each pair of cities, find the shortest possible path that visits every city exactly once and returns to the starting city. The TSP has broad applications including: shortest-path for lasers to sculpt microprocessors and delivery logistics for mail services, to name a few. The TSP is an area of active research. In fact, several variants have been derived from the original TSP. In this paper, we focus on the Euclidean TSP. In the Euclidean TSP, the vertices correspond to points in a d-dimensional 1 https://github.com/yihui-he/TSP 1 3.3. 2opt Algorithm In optimization, 2-opt is a simple local search algorithm first proposed by Croes in 1958 for solving the TSP [5]. The main idea behind it is to take a route that crosses over itself and reorder it so that it does not. A complete 2-opt local search will compare every possible valid combination of the swapping mechanism. This technique can be applied to the travelling salesman problem as well as many related problems. These include the vehicle routing problem (VRP) as well as the capacitated VRP, which require minor modification of the algorithm. This is the mechanism by which the 2-opt swap manipulates a given route: Figure 1. The TSP was first formulated in the 1930s by Karl Menger in Vienna and Harvard. By the mid-1950s, solutions for TSP began to appear. The first solution was published by Dantzig, Fulkerson, and Johnson using a dataset of 49 cities. In 1972, Richard M. Karp proved that the Hamiltonian cycle problem was NP-Complete, which proves that the TSP is NP-Hard. In modern day, the TSP has a variety of applications to numerous fields. Examples among these applications include genome sequencing, air traffic control, supplying manufacturing lines, and optimization. 1. take route[1] to route[i-1] and add them in order to new route 2. take route[i] to route[k] and add them in reverse order to new route 3. take route[k+1] to end and add them in order to new route 3. Algorithms 4. return new route We now move to a discussion of the algorithms used in our evaluation. First, we describe an upper bound for TSP in Section 3.1. The traditional greedy and 2-opt approaches are discussed in Section 3.2 and Section 3.3. We finally discuss the genetic algorithm in Section 3.4. 3.4. Genetic Algorithm Genetic algorithms (GA) are search heuristics that attempt to mimic natural selection for many problems in optimization and artificial intelligence [6]. In a genetic algorithm, a population of candidate solutions is evolved over time towards better solutions. These evolutions generally occur through mutations, randomization, and recombination. We define a fitness function to differentiate between better and worse solutions. Solutions, or individuals, with higher fitness scores are more likely to survive over time. The final solution is found if the population converges to a solution within some threshold. However, great care must be taken to avoid being trapped at local optima. We will now apply a genetic algorithm to the TSP [3]. We define a fitness function F as the length of the tour. Supposed we have an ordering of the cities A = x1 , x2 , ..., xn where n is the number of cities. The fitness score for the TSP becomes the cost of the tour d(x, y) denote the distance from x to y. 3.1. Random Path Finding the worst case of TSP is as hard as the best one. So we uniformly generate a random path for all available edges, and use this as a upper bound of optimal path benchmark for all other algorithms. 3.2. Greedy Algorithm The greedy heuristic is based on Kruskals algorithm to give an approximate solution to the TSP [11]. The algorithm forms a tour of the shortest route and can be constructed if and only if: The edges of the tour must not form a cycle unless the selected number of edges is equal to the number of vertices in the graph. The selected edge (before being appended to the tour) does not increase the degree of any node to be more than 2. The algorithm begins by sorting all edges from least weight to most heavily weighted. After the edges are sorted, the least heavily-weighted edge is selected and it is added to the tour if it does not violate the above conditions. The algorithm continues by selecting the next least-cost edge and adding it to the tour. This process is repeated until all vertices can be reached by the tour. The result is a minimum spanning tree and is a solution for the TSP. The runtime for the greedy algorithm is O(n2 log(n)) and generally returns a solution within 15-20% of the HeldKarp lower bound [15]. F (A) = n−1 X d(xi , xi+1 ) + d(xn , x0 ) (2) i=0 The genetic algorithm begins with an initial, P0 , random population of candidate solutions. That is, we have a set of paths that may or may not be good solutions. We then move forward one time step. During this time step, we perform a set of probabilistic and statistical methods to select, mutate, and produce an offspring population, P1 , with traits similar to those of the best individuals (with the highest fitness) 2 runtime comparison 10 2 10 1 time/sec 10 0 greedy 2opt sa 10 -1 10 -2 10 -3 0 p15 Figure 2. the ATT48 dataset in the 2D plane. (the United States) att48 rand200 cities Figure 3. runtime comparison, the y axis is in log scale from P0 . We then repeat this process until our population becomes homogeneous. The running time of genetic algorithms is variable and dependent on the problem and heuristics used. However, for each individual in the population, we require O(n) space for storage of the path. For genetic crossover, the space requirement remains O(n). The best genetic algorithms can find solutions within 2% of the optimal tour for certain graphs [9]. tour length comparison tour length 10 0 greedy 2opt sa optimal random 10 -1 rand200 10 -2 att48 We benchmark our algorithms using publicly available datasets. Additionally, to test the scalability of the algorithms, we generated a synthetic dataset consisting of 200 cities. In all dataset names, the numeric digits represent the number of cities in the dataset. The datasets are as follows: P15, ATT48, and R200. All datasets except R200 can be found online [4, 14]. The ATT48 and SGB128 datasets represent real-data consisting of locations of cities in the United States. A visual representation of the ATT48 dataset in the 2D plane is shown in Figure 2 Not all datasets have a known optimal tour. When this is the case, we use random path algorithm to infer a upper bound of the optimal tour. p15 4. Experiment dataset Figure 4. tour length comparison, the y axis is in log scale, divided by random tour length a solution similar with the optimal for small datasets and become worse for larger datasets. In terms of running time (Figure 3), the best algorithm is greedy algorithm. However, in terms of optimal tour length of solution, the best algorithm is GA. This is in line with our expectations and alludes to the fact that different heuristics are better suited for different situations. As shown in Figure 4, genetic algorithm performs fairly consistently, in comparison to the 2-opt and greedy algorithms, across all datasets. Highlighted in Figure 3, the running time of genetic is almost linear. This suggests that for larger datasets, if running time is a concern, then the genetic algorithm should be used. Figure 4 further demonstrates that genetic algorithm maintains a smaller percent above optimal than the other algorithms. From this, we can see that genetic algorithm has high accuracy and better complexity than other heuristics, especially for larger datasets. Surprisingly, genetic algorithm got the optimal solution for 4.1. Random Dataset The R200 dataset was generated by plotting 200 random, uniformly distributed points (x, y), in R2 with (x, y) ∈ [0, 4000]. As a result, all distances satisfy the triangle inequality and this dataset can be classified as a Euclidean TSP dataset. The running time for creating the dataset is O(n). The output is a list of all cities represented as (x, y) points. 4.2. Comparison As we can see in Figure 3, the greedy is the most efficient. In Figure 4, we can see that most algorithms return 3 [3] K. Bryant and A. Benjamin. Genetic algorithms and the traveling salesman problem. Department of Mathematics, Harvey Mudd College, pages 10–12, 2000. [4] J. Burkardt. Data for the traveling salesperson problem, 2011. http://people.sc.fsu.edu/ jburkardt/datasets/tsp/tsp.html. [5] G. A. Croes. A method for solving traveling-salesman problems. Operations research, 6(6):791–812, 1958. [6] J. Grefenstette, R. Gopal, B. Rosmaita, and D. Van Gucht. Genetic algorithms for the traveling salesman problem. In Proceedings of the first International Conference on Genetic Algorithms and their Applications, pages 160–168. Lawrence Erlbaum, New Jersey (160-168), 1985. [7] A. Haque, J. Shah, F. Ejaz, and J. X. Xu. An empirical evaluation of approximation algorithms for the metric traveling salesman problem. [8] A. Hoffman, J. Wolfe, R. Garfinkel, D. Johnson, C. Papadimitriou, P. Gilmore, E. Lawler, D. Shmoys, R. Karp, J. Steele, et al. The traveling salesman problem: a guided tour of combinatorial optimization. J. Wiley & Sons, 1986. [9] A. Homaifar, S. Guan, and G. E. Liepins. Schema analysis of the traveling salesman problem using genetic algorithms. Complex Systems, 6(6):533–552, 1992. [10] I. Hong, A. B. Kahng, and B.-R. Moon. Improved largestep markov chain variants for the symmetric tsp. Journal of Heuristics, 3(1):63–81, 1997. [11] B.-I. Kim, J.-I. Shim, and M. Zhang. Comparison of tsp algorithms. Project for Models in Facilities Planning and Materials Handling, 1998. [12] M. Mucha. \ frac {13}{9}-approximation for graphic tsp. Theory of computing systems, 55(4):640–657, 2014. [13] F. Qiu, J. Zhang, and H. Yan. An adaptive markov chain monte carlo algorithm for tsp. In Computer Science and Software Engineering, 2008 International Conference on, volume 1, pages 439–442. IEEE, 2008. [14] G. Reinelt. Tsplib, 1995. http://comopt.ifi.uniheidelberg.de/software/TSPLIB95/. [15] D. J. Rosenkrantz, R. E. Stearns, and P. M. Lewis, II. An analysis of several heuristics for the traveling salesman problem. SIAM journal on computing, 6(3):563–581, 1977. Figure 5. Solution generated by genetic algorithm for att48 dataset att48 dataset, shown in Figure 5. 5. Conclusion Most of our algorithms attempt to solve the TSP in a linear fashion. Originating from artificial intelligence, the genetic algorithm is very different compared to greedy, and 2-opt. Literature suggests that the best algorithms focus on iteration and convergence to find optimal tours – something genetic algorithms attempt to achieve. For example, the Large Step Markov Chain [10] relies on Markov chains to find convergence of many paths to form a global optimum and several papers cite Markov Chains as the best known solution to TSP. Recent studies include using adaptive Markov Chain Monte Carlo algorithms [13]. Many of these extend the Metropolis algorithm [9], a simulated annealing algorithm which attempts to mimic randomness with particles as the temperature varies. This further supports our conclusion that algorithms inspired from artificial intelligence perform well for finding solutions for the TSP. However, these may not be suitable when a guarantee is required. In this paper, we surveyed several key cornerstone approaches to the TSP. We selected four well-known algorithms and tested their performance on a variety of public datasets. Our results suggest that genetic algorithms (and other approaches from artificial intelligence) are able to find a near-optimal solution. References [1] S. Arora. Polynomial time approximation schemes for euclidean tsp and other geometric problems. In Foundations of Computer Science, 1996. Proceedings., 37th Annual Symposium on, pages 2–11. IEEE, 1996. [2] BaoJunpeng. Introduction to Artificial Intelligence. [M].BeiJing:MachineryIn-dustryPress, 2010. 4	2
arXiv:1705.08738v1 [cs.CE] 24 May 2017 Doppler Synthetic Aperture Radar Interferometry: A Novel SAR Interferometry for Height Mapping using Ultra-Narrowband Waveforms Birsen Yazıcı1,∗ , Il-Young Son1 and H. Cagri Yanik 1 Department of Electrical and Computer Systems Engineering, Rensselaer Polytechnic Institute, Troy, NY, USA ∗ Corresponding author E-mail: [email protected] Abstract. This paper introduces a new and novel radar interferometry based on Doppler synthetic aperture radar (Doppler-SAR) paradigm. Conventional SAR interferometry relies on wideband transmitted waveforms to obtain high range resolution. Topography of a surface is directly related to the range difference between two antennas configured at different positions. Doppler-SAR is a novel imaging modality that uses ultra-narrowband continuous waves (UNCW). It takes advantage of high resolution Doppler information provided by UNCWs to form high resolution SAR images. We introduced the theory of Doppler-SAR interferometry. We derived interferometric phase model and develop the equations of height mapping. Unlike conventional SAR interferometry, we show that the topography of a scene is related to the difference in Doppler between two antennas configured at different velocities. While the conventional SAR interferometry uses range, Doppler and Doppler due to interferometric phase in height mapping, Doppler-SAR interferometry uses Doppler, Doppler-rate and Doppler-rate due to interferometric phase in height mapping. We demonstrate our theory in numerical simulations. Doppler-SAR interferometry offers the advantages of long-range, robust, environmentally friendly operations; low-power, low-cost, lightweight systems suitable for low-payload platforms, such as micro-satellites; and passive applications using sources of opportunity transmitting UNCW. Submitted to: Inverse Problems Doppler Synthetic Aperture Radar Interferometry 2 1. Introduction Synthetic Aperture Radar (SAR) interferometry is a powerful tool in mapping surface topography and monitoring dynamic processes. This tool is now an integral part of wide range of applications in many disciplines including environmental remote sensing, geosciences and climate research, earthquake and volcanic research, mapping of Earth’s topography, ocean surface current monitoring, hazard and disaster monitoring, as well as defense and security related research [1]. Basic principles of SAR interferometry were originally developed in radio astronomy [2, 3]. Interferometric processing techniques and systems were later developed and applied to Earth observation [4, 5, 6, 7, 8]. SAR interferometry exploits phase differences of two or more SAR images to extract more information about a medium than present in a single SAR image [9] [10]. Conventional SAR interferometry relies on wideband transmitted waveforms to obtain high range resolution [10, 1, 11, 12, 13]. The phase difference of two wideband SAR images are related to range difference. There are many different interferometric methods depending on the configuration of imaging parameters in space, time, frequency etc [1]. When two images are acquired from different look-directions, the phase difference is related to the topography of a surface. In this paper, we develop the basic principles of a new and novel interferometric method based on Doppler-SAR paradigm to determine topography of a surface. Unlike conventional SAR, Doppler-SAR uses ultra-narrowband continuous waves (CW) to form high resolution images [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]. Conventional SAR takes advantage of high range resolution and range-rate due to the movement of SAR antenna for high resolution imaging. Doppler-SAR, on the other hand, takes advantage of high temporal Doppler resolution provided by UNCWs and Doppler-rate for high resolution imaging. We develop the phase relationship between two Doppler-SAR images and show that the phase difference is related to Doppler difference. We approximate this phase difference as Doppler-rate and derive the equations of height mapping for Doppler-SAR interferometry. Conventional wideband SAR interferometry for height mapping requires two different look-directions. Doppler-SAR interferometry provides a new degree of freedom in system design by allowing antennas to have the same look-direction, but different velocities to obtain height mapping. Additional advantages of Doppler-SAR interferometry include the following: (i ) Small, lightweight, inexpensive, easy-to-design and calibrate hardware, high Signal-to-Noise-Ratio(SNR) and long effective range of operation. All of these make Doppler-SAR interferometry a suitable modality for applications requiring high SNR, long range of operation and low payload platforms such as micro-satellites or small uninhabited aerial vehicles. (ii ) Effective use of electromagnetic spectrum and environmentally friendly illumination. (iii ) Passive applications. Doppler-SAR may not require dedicated transmitters, since existing Radio Doppler Synthetic Aperture Radar Interferometry 3 Frequency (RF) signals of opportunity often have the ultra-narrowband properties. To the best of our knowledge, this is the first interferometric method that is developed in Doppler-SAR paradigm. We present the theory for two monostatic Doppler-SAR. However, the method can be easily be extended to bistatic and multistatic configurations and synthetic aperture imaging applications in acoustics. The rest of the paper is organized as follows: In Section 2, SAR geometry and notation are defined. In Section 3, wideband SAR image formation, layover effect and basic principles of wideband SAR interferometry are described in a perspective relevant to our subsequent development. In Section 4, Doppler-SAR data model, image formation and layover are summarized. Section 5 introduces the basic principles of Doppler-SAR interferometry and compares the results to wideband SAR case. Section 6 presents numerical simulations and Section 7 concludes the paper. 2. Configurations and Notation We consider two mono-static SAR systems as shown in Fig. 1. Antenna 2 Antenna 1 : Location of the scatter : Height of the scatter Figure 1: Imaging geometry for an interferometric SAR system with two antennas following trajectories γ1 (s) and γ2 (s). The scatterer is located at x ∈ R3 where its height is h(x) and x = [x1 , x2 ] ∈ R2 . Let γ1 (s) and γ2 (s), s ∈ [S1 , S2 ] ⊆ R, denote the trajectories of the first and second antennas, respectively. Unless otherwise stated, bold Roman, bold italic, and Roman lower-case letters will denote elements in R3 , R2 and R, respectively, i.e., x = [x1 , x2 ] ∈ R2 , x3 ∈ R, and x = [x, x3 ] ∈ R3 . The Earth’s surface is located at x = [x, h(x)], where h : R2 → R, is the unknown height representing ground topography. Let V : R3 → R denote target reflectivity where we assume that the scattering takes place only on the surface of the Earth. Major notation used throughout the paper is tabulated in Table 1. Doppler Synthetic Aperture Radar Interferometry Table 1: Notation Symbol Description x = [x, h(x)], x ∈ R2 Location on earth’s surface h(x) Unknown height of a scatter at x V (x) Surface reflectivity γi (s) i-th antenna trajectory s Slow-time t Fast-time Ri (x, s) Range of i-th antenna ω0 Center frequency of the transmitted waveforms si0 Zero-Doppler time for the i-th antenna Li (x, s) Look-direction of the i-th antenna B dW (t, s) i Wideband SAR demodulated received signal at i-th antenna \|z − γi (s)\| = C. Iso-range surface \ (z − γi (s)) · γ̇i (s) = C Iso-Doppler surface KiW B Filtered backprojection (FBP) operator for wideband SAR IiW B Wideband SAR image B ΦW s0 (x) Wideband interferometric phase b Baseline vector in wideband SAR interferometry L1 (z, s10 ) · b = C Interferometric phase cone l Vector from a known scatterer position to the unknown location of a scatterer l⊥ 1 Component of l perpendicular to (z0 \ − γ1 (s)) B ΦW f lat (x) Flattened wideband SAR interferometric phase φ(t) Smooth windowing function Tφ Duration of φ(t) NB dU (µ, s) i Doppler-SAR data Li (z, s) · γ̇i (s) = C Iso-Doppler surface Li (z, s) · γ̈i (s) − Iso-Doppler-rate surface γ̇i (s)·γ̇i⊥ (s) Ri (z,s) =C KiU N B FBP operator for Doppler-SAR IiU N B Doppler-SAR image NB ΦU (x) sd Doppler-SAR interferometric phase v Baseline velocity NB ΦU f lat (x) Flattened Doppler-SAR interferometric phase 4 Doppler Synthetic Aperture Radar Interferometry 5 3. Wideband SAR Interferometry The basic principles of SAR interferometry are described by many sources [10], [25], [9], [26] [27], [1] and [28]. In this section, we summarize the principles and theory of SAR interferometry in a notation and context relevant to our subsequent presentation of Doppler-SAR interferometry. We begin with the wideband SAR received signal model, derive the interferometric phase model, provide a geometric interpretation of the interferometric phase from which we develop the equations of height mapping. 3.1. Wideband SAR received signal model We assume that the SAR antennas are transmitting wideband waveforms. Let ri (t, s) denote the received signals, i = 1, 2 where s and t are the slow-time and fast-time variables, respectively. Under the start-stop and Born approximations, the received signals can be modeled as [29, 30, 31]: Z ri (t, s) = e−iω(t−2Ri (x,s)/c) Ãi (x, s, ω)V (x)dxdω (1) where Ri (x, s) = \|x − γi (s)\| (2) is the range of the ith antenna, c is the speed of light in free-space, ω is the temporal frequency variable, V (x) is the scene reflectivity function. Ãi is a slowly-varying function of ω that depends on antenna beam patterns, geometrical spreading factors and transmitted waveforms. Let ω = ω0 + ω 0 , ω 0 ∈ Ω where Ω is the bandwith and ω0 is the center frequency of the transmitted waveforms. We demodulate the received signals and write B dW (t, s) = eiω0 t ri (t, s), i ZΩ ω0 0 = e−iω (t−2Ri (x,s)/c) Ãi (x, s, ω 0 )ei2 c Ri (x,s) V (x)dxdω 0 . (3) (4) −Ω Next, we approximate Ri (x, s) in ei ω0 Ri (x,s) c around s = si0 as follows: Ri (x, s) ≈ Ri (x, si0 ) + (s − si0 ) ∂s Ri (x, s)\|s=si 0 (s − si0 )2 2 + ∂s Ri (x, s) s=si , i = 1, 2 0 2 (5) where ∂s denotes derivative with respect to s and si0 is the zero-Doppler time for the ith antenna, i.e., \ ∂s Ri (x, s)\|s=si = (x − γi (si0 )) · γ̇i (si0 ) = 0. 0 (6) b denotes the unit vector in the direction of x and γ̇i (s) denotes the velocity of In (6) x th the i antenna. Doppler Synthetic Aperture Radar Interferometry 6 We define Li (x, s) = (x \ − γi (s)) (7) and refer to Li (x, s) as the look-direction of the ith antenna. Note that at the zeroDoppler time, the antenna look-direction is orthogonal to the antenna velocity. Let i2 Ai (x, s, ω 0 ) = Ãi (x, s, ω 0 )e i 2 ω0 (s−s0 ) c 2 ∂s2 Ri (s,x)\| s=si0 . (8) Finally, we write the demodulated received signal as follows: B dW (t, s) i ZΩ ≈ 0 e−iω (t−2Ri (x,s)/c) Ai (x, s, ω 0 )ei2 ω0 Ri (x,si0 ) c V (x)dxdω 0 . (9) −Ω 3.2. Wideband SAR image formation and layover Many different algorithms were developed to form wideband SAR images such as range-Doppler [25], seismic migration [32], backprojection [31] and chirp scaling [33] algorithms. All of these algorithms take advantage of high range resolution provided by wideband transmitted waveforms and pulse-to-pulse Doppler information provided by the movement of antennas. The location of a scatterer is identified by intersecting the iso-range and iso-Doppler surfaces and the ground topography as shown in Fig. 2. Range sphere Doppler cone Velocity vector Sensor position x Scatterer position Figure 2: The SAR image of a scatterer is reconstructed at the intersection of the iso-range (sphere) and iso-Doppler (cone) surfaces and the height of the scatterer. More precisely, the image of a scatterer is formed at z satisfying the following equations: \|z − γi (s)\| = Ri (x, s) \ Iso-Doppler surface: (z − γi (s)) · γ̇i (s) = ∂s Ri (x, s) (10) Height: (12) Iso-range surface: z3 = h(x), z = [z, z3 ]. (11) Doppler Synthetic Aperture Radar Interferometry 7 Note that Ri (x, s) and ∂s Ri (x, s) are the measured range and Doppler and h(x) is the height of the scatterer. As functions of z, (10) and (11) define the iso-range and iso-Doppler surfaces, respectively. Iso-range contours are defined as the intersection of the iso-range surface, i.e., sphere, and the ground topography. Without loss of generality, we consider a filtered backprojection (FBP) type method where the received and demodulated signals are backprojected onto iso-range contours defined on a reference surface [31], [29]. In the absence of heigh information, demodulated signal is backprojected onto the intersection of the iso-range surface and a known reference surface. Without loss of generality, we assume a flat reference surface at zero height and backproject the demodulated signals onto the following iso-range contours: HiRange (z0 ) = z0 ∈ R3 z0 = [z, 0] and \|z0 − γi (s)\| = Ri (x, s) . (13) Let KiW B be an FBP operator. Then, the reconstructed image of the scatterer at x becomes W B ˜W B i IiW B (zi0 ) := K Z i [di ](z0 ), 0 i B B = eiω (t−2Ri (z0 ,s)/c) QW (z0i , ω 0 , s)dW (t, s)dω 0 dtds, (14) i i B where QW is a filter that can be chosen with respect to a variety of criteria [31], [34]. i From (9), the image of the scatterer at x becomes IiW B (zi0 ) = \|IiW B (zi0 )\|ei2 ω0 Ri (x,si0 ) c . (15) The magnitude of reconstructed images is a measure of target reflectivity, whereas the phase of the reconstructed image depends on the true location, x = [x, h(x)] of the scatterer. However, since the true height h(x) of the scatter is unknown and hence different than that of the reference surface, the location, z0i , at which the scatterer is reconstructed is different than its true location, x. This positioning error due to incorrect height information is known as layover. Fig. 3 depicts the layover effect. We see that without the knowledge of ground topography, additional information or measurements are needed to reconstruct the scatterers at correct locations. This additional information is provided by a second antenna that has a different vantage point than the first one. 3.3. Wideband SAR interferometric height reconstruction An interferogram is formed by multiplying one of the SAR images with the complex conjugate of the other SAR image [9, 10]. Prior to multiplying the SAR images, the two intensity images, \|IiW B (zi0 )\|, i = 1, 2 are co-registered so that pixel locations z01 and z02 , each corresponding to the scatterer at position x in the scene, are roughly aligned‡. Multiplying I1W B (z10 ) with the complex conjugate of I2W B (z20 ), we get I1W B (z10 )I2W B (z20 ) = \|I1W B (z10 )\|\|I2W B (z20 )\|ei2 ω0 (R1 (x,s10 )−R2 (x,s20 )) c . (16) ‡ The positioning errors due to layover are different in the two SAR images due to different imaging geometries. Doppler Synthetic Aperture Radar Interferometry 8 Figure 3: Layover in wideband SAR - The range sphere depicts the iso-range surface of the monostatic SAR configuration. Since the correct height of the scatterer at location x is unknown, the image of the scatterer at x is formed at z0 on a flat surface. We refer to the phase of the interferogram as the wideband interferometric phase ω0 B (R1 (x, s10 ) − R2 (x, s20 )) ΦW (17) s0 (x) = 2 c B provides us the where s0 is a multi-index for {s10 , s20 }. The interferometric phase ΦW s0 third measurement needed to determine the location of a scatterer in R3 . In general the range difference can be many multiples of 2π. Unique phase proportional to range difference can be determined by a phase unwrapping process [1]. Now consider the following surface c WB \|z − γ1 (s10 )\| − \|z − γ2 (s20 )\| = Φ (x) (18) 2ω0 s0 B where ΦW (18) defines a two-sheet s0 (x) is the measured interferometric phase. 2 1 hyperboloid with foci at γ1 (s0 ) and γ2 (s0 ). We assume that the distance between the antennas is much smaller than the ranges of the antennas to the scene and approximate this hyperboloid as follows: c WB L1 (z, s10 ) · b ≈ (19) Φ (x) 2ω0 s0 where b = γ2 (s20 ) − γ1 (s10 ) (20) is the baseline vector. (19) defines a cone whose vertex is the first antenna and the axis of rotation is the baseline vector. We call this surface the interferometric phase cone. The interferometric phase cone provides the third equation needed to locate the position of a scatterer in R3 . More precisely, the location of the scatterer is given by the solution of the following equations: Range sphere: \|z − γ1 (s10 )\| = R1 (x, s10 ) \ (z − γ1 (s10 )) · γ̇1 (s10 )) = ∂s R1 (x, s)\|s=s1 0 c WB 1 Interferometric phase cone: L1 (z, s0 ) · b = Φ (x). 2ω0 s0 Doppler cone: (21) (22) (23) Doppler Synthetic Aperture Radar Interferometry 9 The right-hand-side of (21)-(23) are measured quantities defined in terms of the true location, x, of the scatterer in the scene and the left hand-side-defines the three surfaces in terms of the location of the scatterer z in the image. Fig. 4 geometrically illustrates the solution of these three equations in wideband SAR interferometry. Typically the Figure 4: Wideband SAR interferometry provides a third algebraic equation by which the unknown location of a scatters in R3 is determined. The scatterer is located at the intersection of the Doppler-cone, iso-range sphere, and the interferometric phase cone. The axis of rotation of the Doppler-cone is the velocity of the first antenna and the axis of rotation of the interferometric cone is the baseline vector extending from the first to the second antenna. variation in the color coding of interferogram is “flattened” by subtracting the expected phase from a surface of constant elevation. Let x = l + z0 . Then, under the assumption that \|l\| \|z0 − γ1 (s)\| \ (x − γ1 (s)) ≈ (z0 \ − γ1 (s)) + l⊥ 1 \|z0 − γ1 (s)\| (24) where " l⊥ 1 # \ l · (z − γ (s)) 0 1 = l − (z0 \ − γ1 (s)) . \|z0 − γ1 (s)\| (25) \ In other words, the vector l⊥ 1 is the component of l perpendicular to (z0 − γ1 (s)). The flattened phase then becomes ω0 B ΦW L1 (x, s10 ) − L1 (z0 , s10 ) · b (26) f lat (x) = 2 c ω0 l⊥ 1 ·b ≈2 . (27) c R(z0 , s10 ) ⊥ ⊥ 1 Since b⊥ 1 · l = l1 · b where b1 is the component of b perpendicular to L1 (z, s0 ), (27) can be alternatively expressed as B ΦW f lat (x) ω0 b⊥ 1 ·l =2 . c R(z0 , s10 ) (28) Doppler Synthetic Aperture Radar Interferometry 10 Fig. 5 illustrates the key concepts and vectors involved in the wideband interferometry. Figure 5: A two-dimensional illustration of vectors involved in wideband interferometric phase. l = x − z0 where z0 = [z0 , 0], γi (si0 ) denotes the location of the ith antenna at the zero-Doppler time si0 , L1 (z0 , s10 ) denotes the look-direction of the first antenna with respect to the reference scatterer located at z0 , and l⊥ 1 is the component of l 1 perpendicular to L1 (z0 , s0 ). The wideband interferometric phase is related to the projection of the baseline vector, b, onto the the look-direction, L1 (x, s10 ), of the antenna with respect to scatterer location x. Known vectors are shown in red and unknown vectors are shown in black. 4. Data Model and Image Formation for Doppler-SAR 4.1. Data Model for Doppler-SAR We consider two mono-static antennas following the trajectories γi (t), i = 1, 2, transmitting ultra-narrowband CWs as shown in Fig. 1. Let p(t) ≈ p̃(t)eiω0 be the transmitted waveform where ω0 is the center frequency. The scattered field model at the ith antenna is then given by Z −iω0 (t−2\|x−γi (t)\|/c) ω02 e ri (t) = p̃(t − 2\|x − γi (t)\|/c)V (x)dx. (29) (4π)2 \|x − γi (t)\|2 Let µ ∈ R+ and φ(t) be a smooth windowing function with a finite support, t ∈ [0, Tφ ]. Following [15, 14, 23], we correlate ri (t) with a scaled and translated version of the transmitted signal over φ(t) as follows: Z UNB di (µ, s) = ri (t)eiω0 µ(t−sTφ )/c p̃∗ (µ(t − sTφ ))φ(t − sTφ )dt. (30) Doppler Synthetic Aperture Radar Interferometry 11 Inserting (29) into (30), we obtain Z −iω0 (t−2\|x−γi (t)\|/c) ω02 e UNB (µ, s) = di p̃(t − 2\|x − γi (t)\|/c)V (x) 2 (4π) \|x − γi (t)\|2 × eiω0 µ(t−sTφ )/c p̃∗ (µ(t − sTφ ))φ(t − sTφ )dtdx. (31) Approximating γi (t) around t = sTφ , γi (t) ≈ γi (sTφ ) + γ̇i (sTφ )(t − sTφ ), and making the far-field approximation, we write \|x − γi (t)\| ≈ \|x − γi (sTφ )\| − Li (x, sTφ ) · γ̇i (sTφ )(t − sTφ ), (32) where Li (x, sTφ ) = (x −\ γi (sTφ )) and γ̇i (sTφ ) = ∂s γi (sTφ ) is the velocity of the ith antenna. To simplify our notation, for the rest of the paper, we set Li (x, sTφ ) = Li (x, s), γi (sTφ ) = γi (s), γ̇i (sTφ ) = γ̇i (s), ∂s2 γi (sTφ ) = γ̈i (sTφ ) = γ̈i (s) and Ri (x, sTφ ) = Ri (x, s). We next define Doppler for the ith antenna ω0 (33) fid (x, s) = − Li (x, s) · γ̇i (s). c Inserting (32) and (31) into (33), the data model becomes Z d d UNB di (µ, s) = e−it[ω0 (1−µ)−2fi (x,s)] Ãi (t, x, s, µ)ei2fi (x,s)sTφ V (x)dtdx, (34) where Ãi (t, x, s, µ) is a slow varying function of t composed of the rest of the terms in (31). We now approximate fid (x, s) around s = sid as follows: fid (x, s) ≈ fid (x, sid )+(s−sid ) ∂s fid (x, s) s=sid + (s − sid )2 2 d ∂s fi (x, s) 2 s=sid .(35) We choose sid such that ∂fid (x, s) ∂s =0 ⇒ Li (x, sid ) · γ̈i (sid ) − s=sid γ̇i (sid ) · γ̇i⊥ (sid ) =0 Ri (xsid ) (36) where γ̈i (sid ) is the acceleration of the ith antenna and γ̇i⊥ (sid ) is the component of γ̇i (sid ) perpendicular to the look-direction Li (x, sid ) as described in (25). We refer to sid as the zero-Doppler-rate time for the ith antenna. d Using (35) in ei2fi (x,s)sTφ and redefining the slow-varying function in t, i2sid Tφ Ai (t, x, s, µ) = Ãi (t, x, s, µ)e (s−sid )2 2 d ∂s fi (x,s)\|s=si 2 d , (37) we obtain the following data model for Doppler-SAR image reconstruction: Z d d i i UNB di (µ, s) ≈ e−it[ω0 (1−µ)−2fi (x,s)] Ai (t, x, s, µ)ei2fi (x,sd )sd Tφ V (x)dtdx.(38) Doppler Synthetic Aperture Radar Interferometry 12 4.2. Doppler-SAR Image Formation and Layover Similar to the wideband case, we reconstruct images by backprojection as described in [35, 15] [14]. The forward model in (38) shows that the data, dUi N B (s, µ), is the weighted integral of the scene reflectivity over iso-Doppler contours. It was shown in [14] that a scatterer located at x in the scene is reconstructed at the intersection of iso-Doppler surface and iso-Doppler-rate surface and ground topography. More precisely, the image of a scatterer located at x in the scene is reconstructed at z satisfying the following equations: c (39) Iso-Doppler surface: Ldi (z, s) · γ̇i (s) = fid (x, s) ω0 γ̇i (s) · γ̇i⊥ (s) c Iso-Doppler-rate surface: Li (z, s) · γ̈i (s) − (40) = ∂s fid (x, s) Ri (z, s) ω0 Height: z3 = h(x), z = [z, z3 ] (41) where the right-hand-side of (39)-(40) corresponds to measurements and the left-handside defines surfaces in image parameter z. The iso-Doppler-rate surface, given by the following set, γ̇i (s) · γ̇i⊥ (s) c Dop−rate 3 d Hi (z) = z ∈ R Li (z, s) · γ̈i (s) − = ∂s fi (x, s) . (42) Ri (z, s) ω0 can be viewed as a continuum of intersections of cones and expanding spheres centered at the sensor location. The axis of rotation for the surface is the acceleration vector of the antenna trajectory. Fig. 6 illustrates iso-Doppler and iso-Doppler-rate surfaces and the reconstruction of a point scatterer by the intersection of these surfaces and ground topography. The reconstruction is analogous to the wideband SAR image reconstruction shown in Fig. 2. In the absence of ground topography information, we backproject data onto isoDoppler contours on a reference surface. Without loss of generality, we consider the following iso-Doppler contours: c d Dop 3 Hi (z0 ) = z0 ∈ R z0 = [z, 0] and Li (z, s) · γ̇i (s) = fi (x, s) (43) ω0 where the right-hand-side of the equality in (43) is the high resolution measurement provided by ultra-narrowband CW. Let KiU N B be an FBP operator as described in [14]. Then, the reconstructed image is given by: UNB UNB [di ](zi0 ) IiU N B (zi0 ) := K Zi ≈ d i eit(ω0 (1−µ)−2fi (z0 ,s)) QUi N B (s, z0i , t)dUi N B (s, µ)dtdµds (44) where QUi N B is a filter that can be chosen as in [35, 15, 14]. The reconstructed image is given by d i i IiU N B (zi0 ) = \|IiU N B (zi0 )\|ei2fi (x,sd )sd Tφ . (45) Doppler Synthetic Aperture Radar Interferometry 13 Figure 6: In Doppler-SAR image reconstruction, a scatterer located at x in the scene is correctly reconstructed at the intersection of the iso-Doppler and iso-Doppler-rate surfaces and the ground topography. Iso-Doppler surface is a cone in which its vertex is the antenna location and its axis of rotation is the antenna velocity. The geometry of the iso-Doppler-rate surface depends on the antenna trajectory. Figure is drawn for a linear trajectory at a constant height. In the absence of topography information, we see that a scatterer located at x in the scene is reconstructed at zi0 6= x in the image. This position error in the reconstructed image is the counterpart of the layover effect observed in conventional wideband SAR images. Fig. 7 illustrates the layover effect in Doppler-SAR. However, the phase of the reconstructed image is a function of the scatterer’s true location, x, and hence, includes its height information, h(x). Figure 7: If the height of a scatter is not known, it is reconstructed at an incorrect position. Both the correct scatterer location x and its image z0 lie on the same isoDoppler surface, i.e., the Doppler cone. z0 lies at the intersection of the Doppler cone defined f1d (x, s) and the flat topography. Note that the phases of the reconstructed images depend on the Doppler-rate, Doppler Synthetic Aperture Radar Interferometry 14 fid (x, sid ), the duration of the windowing function, Tφ , and the corresponding zeroDoppler-rate times, sid . The height information is included in the Doppler-rate. However, since each imaging geometry may yield different zero-Doppler-times, Dopplerrate in the phase of each image is multiplied by a different zero-Doppler-rate time. To equalize the effect of this multiplication factor, we multiply one of the reconstructed images with itself so that the Doppler-rate in the phase of both images are multiplied by the same factor, say s1d . As a result, each image becomes d i 1 IiU N B (zi0 ) = \|IiU N B (zi0 )\|ei2fi (x,sd )sd Tφ , i = 1, 2. (46) 5. Doppler-SAR Interferometric Height Reconstruction Similar to the wideband case, we form two Doppler-SAR images, IiU N B (zi0 ), i = 1, 2, co-register the intensity images \|IiW B (zi0 )\| and multiply one of them by the complex conjugate of the other to form an interferogram. Then the interferometric phase, i.e., the phase function of I1U N B (x)I2U N B (x) is given by ΦUsdN B (x) = 2s1d Tφ f1d (x, s1d ) − f2d (x, s2d ) (47) where sd denotes multi-index for {s1d , s2d }. Thus the scatterer lies on the following surface: c L1 (z, s1d ) · γ̇1 (s1d ) − L2 (z, s2d ) · γ̇2 (s2d ) = − f1d (x, s1d ) − f2d (x, s2d ) (48) 2ω0 where the right-hand-side is the measured interferometric phase. The left-hand-side of (48) defines a surface that can be described as the intersections of two cones one of which has a continuously changing solid angle. Assuming that the distance between the antennas is much smaller than the ranges of the antennas to the scene, we can approximate the look-direction of the second antenna in terms of the look-direction of the first one as follows: b⊥ 1 L2 (x, s2d ) = L1 (x, s1d ) + (49) R1 (x, s1d ) where b = γ2 (s2d ) − γ1 (s1d ) is the baseline vector and b⊥ 1 is the component of b perpendicular to the look-direction of the first antenna. Using (49), we approximate the interferometric phase as follows: 2 c b⊥ 1 · γ̇2 (sd ) UNB 1 − 1 Φ (x) ≈ L1 (x, sd ) · v + 2sd Tφ ω0 sd R1 (x, s1d ) (50) v = γ̇2 (s2d ) − γ̇1 (s1d ). (51) where We refer to v as the baseline velocity. We see that (50) approximates the interferometric phase as a Doppler-rate. Additionally, (50) shows that Doppler-SAR interferometry involves not only configuring antennas in position space, but also in velocity space. The larger the difference in antenna velocities in the look-direction of the first antenna, the larger the interferometric phase becomes. If on the other hand, the velocities of the antennas are the same, the second term in (50) defines the interferometric phase surface. Doppler Synthetic Aperture Radar Interferometry 15 Clearly, in Doppler-SAR interferometry (50) provides the third equation needed to determine the location of a scatterer in R3 . More precisely, the location of a scatterer is given by the solution of the following three equations: c (52) (z −\ γ1 (s1d )) · γ̇1 (s1d ) = f1d (x, s1d ) ω0 γ̇1 (s1d ) · γ̇1⊥ (s1d Tφ ) 1 1 \ Iso-Doppler-rate: (z − γ1 (sd )) · γ̈1 (sd ) − = ∂s f1d (x, s1d ) (53) 1 R1 (sd , z) c b⊥ · γ̇2 (s2d ) =− 1 ΦU N B (x). (54) Interferometric Doppler-rate: L1 (z, s1d ) · v + 1 1 R1 (z, sd ) 2sd Tφ ω0 sd Iso-Doppler: Fig. (8) depicts the intersection of the three surfaces at the scatterer location in R3 . Doppler-rate surface Interferometric phase Doppler-rate surface Doppler cone Velocity vector Sensor position Scatterer position Figure 8: Determination of the scatterer location in Doppler-SAR interferometry. The scatterer is located at the intersection of the Doppler cone and the two iso-Dopplerrate surfaces. Interferometric phase measurement provides the third surface, i.e., the interferometric phase iso-Doppler-rate surface. Similar to the wideband SAR interferometry, the interferometric phase can be “flattened” by subtracting the phase due to a scatterer with known height. Without loss of generality, let z0 = [z, 0] with R1 (z0 , s) = R1 (x, s) and x = z0 + l. Thus, identifying the location of a scatterer is equivalent to determining l. Using (24), we see that l⊥ 1 1 ·v UNB UNB UNB Φf lat (x) = Φsd (x) − Φsd (z0 ) ≈ +O (55) R1 (z0 , s1d ) R12 (z0 , s1d ) 1 where l⊥ 1 is the component of l perpendicular to L1 (z0 , sd ). (55) shows that the flattened interferometric phase for Doppler-SAR interferometry is related to the projection of the unknown l⊥ 1 onto the baseline velocity vector scaled by the range of the first antenna to ⊥ z0 . Since l1 · v = v1⊥ · l where v1⊥ is the component of v perpendicular to L1 (z0 , s1d ), we alternative express (55) as follows: v1⊥ · l NB ΦUf lat (x) ≈ . (56) R1 (z0 , s1d ) Fig. 9 shows the key concepts and vectors involved in Doppler-SAR interferometry. Doppler Synthetic Aperture Radar Interferometry 16 Figure 9: An illustration of key concepts and vectors in Doppler-SAR interferometry. l = x − z0 where z0 = [z0 , 0], γi (s) denotes the ith antenna position, L1 (z0 , s1d ) denotes the look-direction with respect to a reference surface, l⊥ 1 is the component of 1 l perpendicular to L1 (z0 , sd ), γ̇1 (s) denotes the antenna velocity, L1 (x, s1d ) denotes the look-direction of the antenna with respect to the correct target location. Doppler-SAR interferometric phase is proportional to the projection of the baseline velocity vector onto l⊥ 1 . Known vectors are shown in red and unknown vectors are shown in black. 5.1. Comparison of Doppler-SAR Interferometry wide Wideband Case Table II tabulates the interferometric phase for the wideband SAR and Doppler-SAR cases. We compare and contrast the two interferometric phases below: • For WB and UNB, the “baseline” is the difference in range and difference in velocity, respectively. • The larger the ω0 , the center frequency, the larger the interferometric phase in both WB and UNB cases. • The larger the range, R1 , the smaller the interferometric phase in both WB and UNB cases. • For UNB, larger the Tφ , the larger the interferometric phase. • For WB, the larger the b, the difference between the positions of the two antennas, the larger the interferometric phase. For UNB, the larger the v, the difference between the velocities of the two antennas, the larger the interferometric phase. Doppler Synthetic Aperture Radar Interferometry 17 Table 2: Raw and flattened interferometric phase functions for wideband SAR and Doppler-SAR. Wideband SAR Doppler-SAR Interferometric Phase 2 ωc0 L1 (x, s10 ) · b h −2 ωc0 sd Tφ L1 (x, s1d ) · v + Flattened Interferometric Phase 2 ωc0 R1 (z10 ,s1 ) b⊥ 1 ·l 0 i 1 ⊥ 2 b · γ̇ (s ) −2 ωc0 sd Tφ R1 (z10 ,s1 ) v1⊥ · l 1 2 1 d R1 (x,s ) d d 6. Numerical Experiments 6.1. Experimental Setup We conducted numerical experiments for both wideband and Doppler-SAR. Our experimental setup was as follows: • A scene of size 128 × 128m at 1m resolution was imaged. • A single point target was placed at (−20, −31, 50)m with the origin (0, 0, 0) at the scene center. • Two antennas flying on a linear trajectory parallel to the y-axis was used with both antennas placed at 7.1km from the scene center in the x-axis direction. The midpoint of the linear trajectories for both antennas was aligned at y = 0. • Wideband: First antenna was placed at height of 3km and the second at 4km. The length of the trajectories were 1km in length for both antennas. Both antennas were moving at velocity of 100m/s. A waveform with flat spectrum of 100M Hz bandwidth at center frequency of 8GHz was transmitted from both antennas. 512 frequency samples and 1024 slow-time, s, samples were used for imaging. • Doppler: First antenna was placed at height of 2km and the second at 4km. The length of the trajectories were 1km for both antennas. The first antenna was moving at velocity of 100m/s and the second at 400m/s. A continuous waveform at center frequency of 8GHz was transmitted from both antennas. A window of 0.01s was used for processing at each slow time. 512 fast time, t, samples and 1024 slow-time, s, samples were used for imaging. 6.2. Wideband SAR Interferometry Fig. 10a and Fig. 10b show the reconstructed images of the point target located at (−20, −31, 50)m from the first and the second antenna, respectively assuming a flat ground topography at height of 0m. In both Fig. 10a and Fig. 10b, we see that there is a displacement due to layover effect in the range direction (x-axis). The first antenna reconstructs the target at (−41, −31, 0)m. The second antenna reconstructs the target at (−48, −31, 0)m. Doppler Synthetic Aperture Radar Interferometry (a) 18 (b) Figure 10: (a) Wideband reconstruction of the target located at (−20, −31, 50)m using the first antenna assuming flat ground topography. The target is reconstructed at (−41, −31, 0)m. (b) Wideband reconstruction of the target located at (−20, −31, 50)m using the second antenna assuming flat ground topography. The target is reconstructed at (−48, −31, 0)m. We next align the peaks in the two images and multiply the first image with the complex conjugate of the second as in (16) to generate the interferogram. The resulting interferogram is shown in Fig. 11. Figure 11: The interferogram from wideband SAR reconstructed images. In order to reconstruct the height we use the set of equations (21), (22), and (23). The Doppler cone equation (22) at zero-Doppler point s10 gives us that the iso-Doppler contours are in the look-direction, which in our scenario is parallel to the x-axis. Thus, iso-Doppler contours have constant y value at the target’s y position. Using this fact, Doppler Synthetic Aperture Radar Interferometry 19 we need only to compute the intersection of iso-range contour (21) and interferometric phase contour (23) fixing the y position. From Figs. 10a and 10b we see that both targets are reconstructed at y position of −31m. Thus we reconstruct the true target position using y = −31m. For reconstruction, we sampled the height in the interval [1, 100]m at 0.5m resolution. Fig. 12a shows the magnitude image of \|z − γ1 (s10 )\| − R1 (x, s10 ) at y = −31m. Note that R1 (x, s10 ) is the measured value derived from the phase of the reconstructed image. The dark blue area indicates the iso-range contour where the magnitude of the difference is minimized. (a) (b) Figure 12: (a) Image of the magnitude of \|z − γ1 (s10 )\| − R1 (x, s10 ) at y = −31m. The iso-range contour is indicated by dark blue area where the magnitude of \|z − γ1 (s10 )\| − B R1 (x, s10 ) is minimized. (b) Image of the magnitude of L1 (z, s10 ) · b − 2ωc 0 ΦW s0 (x) at y = −31m. The interferometric phase contour is indicated by dark blue area where the B magnitude of L1 (z, s10 ) · b − 2ωc 0 ΦW s0 (x) is minimized. Similarly, Fig. 12b shows the magnitude image of the difference L1 (z, s10 ) · b − c ΦW B (x). As before, the dark blue area indicates the interferometric phase contour. 2ω0 s0 Combining the two images, Fig. 13 shows the intersection of the two contours indicated by the dark blue area. The white ‘x’ in Fig. 13 indicates the exact intersection computed and where the target is reconstructed. The white ‘o’ indicates the true target position. It is clear that the target is reconstructed at the correct position and height. 6.3. Doppler-SAR We proceed similar as in the wideband case for the Doppler-SAR case. Figs. 14a and 14b show the reconstructed image for Doppler-SAR for the first and second antennas, respectively. The first antenna reconstructs the target at (−34, −31, 0)m and the second antenna at (−48, −31, 0)m. Doppler Synthetic Aperture Radar Interferometry 20 Figure 13: Image of the intersection of the iso-range contour with the interfermetric phase contour at y = −31m. The exact intersection is indicated by white ‘x’. The true target position is indicated by white ‘o’. The target is reconstructed at the correct position and height. (a) (b) Figure 14: (a) Doppler-SAR reconstruction of the target located at (−20, −31, 50)m using the first antenna assuming flat ground topography. The target is reconstructed at (−34, −31, 0)m. (b) Doppler-SAR reconstruction of the target located at (−20, −31, 50)m using the second antenna assuming flat ground topography. The target is reconstructed at (−48, −31, 0)m. Doppler Synthetic Aperture Radar Interferometry 21 As in the wideband case, we align the peaks of the two images and multiply the first image with the conjugate of the second image to form the interferogram of the Doppler images. The resulting interferogram is shown in Fig. 15. Figure 15: The interferogram from Doppler-SAR reconstructed images. To reconstruct the height we use the set of equations given in (52), (53), and (54). The zero-Doppler-rate points, s1d , is approximated by the end of the antenna’s trajectories farthest from the target position. By (36), for a linear trajectory with constant velocity, true zero-Doppler-rate point would be where γ̇i (s1d ) ⊥ γ̇i⊥ (s1d ). Namely, where the look-direction is parallel to the velocity vector. The best estimate would be at a point in the trajectory farthest away from the target location. Fig. 16a illustrates the iso-Doppler surface at y = −31m, which is the y-position where the target position is reconstructed and the true target’s y position. Notice that both images reconstruct the scatterer at the correct y position. The iso-Doppler contour is given by the dark blue area as before. Similarly, Figs. 16b and 16c illustrate the iso-Doppler-rate and interferometric Doppler-rate surfaces, respectively at y = −31m. Fig. 17 combines Figs. 16a, 16b, and 16c. The intersection of the three contours is indicated by white ‘x’. The white ‘o’ shows the true target location. Clearly, the target is reconstructed at the correct position and height. 7. Conclusions We present a novel radar interferometry based on Doppler-SAR imaging paradigm. Doppler-SAR uses single frequency transmitted waveforms. It has several advantages over conventional SAR including simpler, inexpensive hardware, high SNR and long effective range of operation, and is suitable for use in passive radar applications. We derived the interferometric phase relationship for Doppler-SAR. Doppler-SAR interferometric phase depends on the difference in the velocity of the antennas as opposed Doppler Synthetic Aperture Radar Interferometry 22 (a) (b) (c) Figure 16: (a) Image of the magnitude of (z −\ γ1 (s1d )) · γ̇1 (s1d ) − ωc0 f1d (x, s1d ) at y = −31m. The iso-Doppler contour is indicated by dark blue area where the magnitude of (z −\ γ1 (s1d )) · γ̇1 (s1d ) = ωc0 f1d (x, s1d ) is minimized. (b) Image of the γ̇ (s1 )·γ̇ ⊥ (s1 T ) magnitude of (z −\ γ1 (s1 )) · γ̈1 (s1 ) − 1 d 11 d φ − ∂s f d (x, s1 ) at y = −31m. The d d R1 (sd ,z) 1 d iso-Doppler-rate contour is indicated by dark blue area where the magnitude of γ̇ (s1 )·γ̇ ⊥ (s1 T ) (z −\ γ1 (s1d ))·γ̈1 (s1d )− 1 Rd 1 (s11 ,z)d φ −∂s f1d (x, s1d ) is minimized. (c) Image of the magnitude b⊥ ·γ̇2 (s2d ) L1 (z, s1d ) · v + R11 (z,s 1) d d of + 2s1 Tcφ ω0 ΦUsdN B (x) at y = −31m. The interferometric Dopplerd rate contour is indicated by dark blue area where the magnitude of L1 (z, s1d ) · v + 2 b⊥ 1 ·γ̇2 (sd ) + 2s1 Tcφ ω0 ΦUsdN B (x) is minimized. R1 (z,s1 ) d d Doppler Synthetic Aperture Radar Interferometry 23 Figure 17: Image of the intersection of the iso-Doppler, iso-Doppler-rate and interferometric Doppler-rate contours at y = −31m. The intersection is indicated by white ‘x’. The true target position is indicated by white ‘o’. The target is reconstructed at the correct position and height. to the range difference observed in wideband SAR. Thus, in Doppler-SAR interferometry, one can reconstruct the ground topography even with the same look-direction from both antennas so long as their velocities are different). Furthermore, we showed that the true target position is determined by the intersection of iso-Doppler, iso-Doppler-rate, and interferometric Doppler-rate surfaces. This is different from conventional wideband SAR in that the surfaces that determine the true target position are iso-range, iso-Doppler, and interferometric Doppler-rate surfaces. We presented numerical simulations for a single point scatterer using two antennas moving in linear trajectories to verify our interferometric method. We also conduct conventional wideband SAR interferometric reconstruction as a comparison. We show that both wideband SAR and Doppler-SAR interferometry is able to accurately reconstruct the target location. Thus, our numerical simulations show that DopplerSAR interferometry retains the accuracy of conventional SAR interferometry while having the advantage that Doppler-SAR affords. In the future, we will analyze the sensitivity of height estimation with respect to other observables and parameters. Acknowledgement This material is based upon work supported by the Air Force Office of Scientific Research (AFOSR) under award number FA9550-16-1-0234, and by the National Science Foundation (NSF) under Grant No. CCF-1421496. Doppler Synthetic Aperture Radar Interferometry 24 Appendix A. Approximations Appendix A.1. Far-field approximation Let x and y be two vectors such that \|x\| \|y\|. Then, by using Taylor series expansion we can make the following approximation: p \|x − y\| = (x1 − y1 )2 + (x2 − y2 )2 + (x3 − y3 )2 , (A.1) p = \|x\|2 − 2(x1 y1 + x2 y2 + x3 y3 ) + \|y\|2 , (A.2) s 2(x · y) \|y\|2 = \|x\| 1 − + 2, (A.3) \|x\|2 \|x\| 1 2x · y , (A.4) ≈ \|x\| 1 − 2 \|x\| ≈ \|x\| − x̂ · y (A.5) where x̂ is the unit vector x̂ = x . \|x\| Appendix A.2. Approximation of look-direction under far-field assumption Let (x \ − γ(s)) denote a look direction where x = y + z and \|y − γ(s)\| \|z\|. Then by using far field expansion we can write x − γ(s) \|x − γ(s)\| y + z − γ(s) = , \|y + z − γ(s)\| z y − γ(s) + , ≈ \|y − γ(s)\| + (y \ − γ(s)) · z \|y − γ(s)\| + (y \ − γ(s)) · z y − γ(s) ≈ (\|y − γ(s)\| − (y \ − γ(s)) · z) \|y − γ(s)\|2 (\|y − γ(s)\| − (y \ − γ(s)) · z)z + , \|y − γ(s)\|2 i h \ \ z − (y − γ(s)) (y − γ(s)) · z ≈ (y \ − γ(s)) + , \|y − γ(s)\| z⊥ ≈ (y \ − γ(s)) + \|y − γ(s)\| (x \ − γ(s)) = (A.6) (A.7) (A.8) (A.9) (A.10) (A.11) where z⊥ is the transverse z, i.e. projection of z onto the plane whose normal vector is \ along the look direction (γ(s) − y). 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Séminaire BOURBAKI 69ème année, 2016-2017, no 1125 Janvier 2017 ISOMORPHISMES DE GRAPHES EN TEMPS QUASI-POLYNOMIAL [d’après Babai et Luks, Weisfeiler-Leman, . . .] arXiv:1701.04372v2 [math.GR] 12 Oct 2017 par Harald Andrés HELFGOTT Résumé : Soient donnés deux graphes Γ1 , Γ2 à n sommets. Sont-ils isomorphes ? S’ils le sont, l’ensemble des isomorphismes de Γ1 à Γ2 peut être identiﬁé avec une classe H · π du groupe symétrique sur n éléments. Comment trouver π et des générateurs de H ? Le déﬁ de donner un algorithme toujours eﬃcace en réponse à ces questions est resté longtemps ouvert. Babai a récemment montré comment résoudre ces questions – et d’autres qui y sont liées – en temps quasi-polynomial, c’est-à-dire en temps O(1) exp O(log n) . Sa stratégie est basée en partie sur l’algorithme de Luks (1980/82), qui a résolu le cas de graphes de degré borné. 1. INTRODUCTION Soient x, y deux chaı̂nes de caractères, à savoir, deux applications Ω → Σ, où Σ (l’alphabet) et Ω (le domaine) sont des ensembles ﬁnis. Tout groupe de permutations (1) G < Sym(Ω) agit sur l’ensemble ΣΩ des chaı̂nes de domaine Ω sur un alphabet Σ. Pour nous, décrire un groupe G, ou être donné un groupe G, voudra toujours dire « donner, voire être donné, un ensemble de générateurs de G » ; décrire une classe Hπ voudra dire « donner un élément π de la classe et un ensemble de générateurs de H ». Le problème de l’isomorphisme de chaı̂nes consiste à déterminer, étant donnés x, y et G, s’il y a au moins un élément π de G qui envoie x sur y, et, si de tels éléments (isomorphismes) existent, à les décrire. Il est clair que l’ensemble des isomorphismes IsoG (x, y) forme une classe AutG (x)π du groupe AutG (x) d’automorphismes de x dans G, c’està-dire du groupe consistant dans les éléments de G qui envoient x sur lui-même. Le déﬁ consiste à donner un algorithme qui résolve le problème en temps polynomial en la taille n = \|Ω\| de Ω, voire en temps raisonnable. Par exemple, le temps employé pourrait être quasi-polynomial en n, ce qui veut dire exp O(log n)O(1) . Ici, comme toujours, O(f (n)) désigne une quantité bornée par C · f (n), pour n assez grand et C > 0 une constante, et Oǫ indique que la constante C dépend de ǫ. Une grande partie de la motivation pour le problème de l’isomorphisme de chaı̂nes vient du fait que le problème de l’isomorphisme de graphes se réduit à lui. Ce problème 1. Pour nous, G < S (ou S > G) veut dire « G est un sous-groupe de S, pas forcement propre. » 1125–02 consiste à déterminer si deux graphes ﬁnis Γ1 et Γ2 sont isomorphes, et, s’ils le sont, à décrire la classe de leurs isomorphismes. (Un isomorphisme π : Γ1 → Γ2 est une bijection π de l’ensemble de sommets de Γ1 vers celui de Γ2 telle que π(Γ1 ) = Γ2 .) Une solution permettrait, par exemple, de trouver une molécule dans une base de données. Le problème de l’isomorphisme de graphes se réduit en temps polynomial au problème de l’isomorphisme de chaı̂nes, de la façon suivante. Supposons sans perte de généralité que Γ1 et Γ2 ont le même ensemble de sommets V . Alors, nous pouvons déﬁnir Ω comme l’ensemble des paires d’éléments de V (ordonnés ou non ordonnés, suivant que nos graphes sont orientés ou pas). La chaı̂ne xi , i = 1, 2, est déﬁnie comme suit : pour la paire a = {v1 , v2 } (ou a = (v1 , v2 ), si nos graphes sont orientés), la valeur de xi (a) est 1 s’il y a une arête entre v1 et v2 en Γ1 , et 0 dans le cas contraire. Soit G l’image de l’homomorphisme ι : Sym(V ) → Sym(Ω) déﬁnie par σ ι ({v1 , v2 }) = {σ(v1 ), σ(v2 )}, où σ ι = ι(σ). Alors ι induit une bijection entre la classe des isomorphismes de Γ1 à Γ2 et la classe IsoG (x1 , x2 ). Théorème 1.1 (Babai). — Le problème de l’isomorphisme de chaı̂nes Ω → Σ peut être résolu en temps quasi-polynomial en le nombre d’éléments du domaine Ω. En novembre 2015, Babai a annoncé une solution en temps quasipolynomial, avec un algorithme explicite. La préparation de cet exposé m’a conduit à trouver une erreur non triviale dans l’analyse du temps, mais Babai a réussi à le réparer en simpliﬁant l’algorithme. La preuve est maintenant correcte. Corollaire 1.2 (Babai). — Le problème de l’isomorphisme de graphes peut être résolu en temps quasi-polynomial en le nombre de sommets. Notre référence principale sera [Ba] ; nous nous servirons aussi de la version courte [Ba2]. Nous essayerons d’examiner la preuve de la façon la plus détaillée possible dans un exposé de ce format, en partie pour aider à éliminer tout doute qui pourrait rester sur la forme actuelle du résultat. La meilleure borne générale connue antérieurement pour le temps requis par le √ problème de l’isomorphisme de graphes, due à Luks [BKL], était exp(O( n log n)), * L’usage de la canonicité joue un rôle crucial dans la stratégie de Babai. Comme dans la théorie de catégories, voire dans l’usage courant, un choix est canonique s’il est fonctoriel. La situation typique pour nous sera la suivante : un groupe G < Sym(Ω) agit sur Ω, et donc sur ΣΩ ; il agit aussi sur un autre ensemble S, et donc aussi sur les applications S → C , où C est un ensemble ﬁni. Une application S → C s’appelle un coloriage ; l’ensemble C s’appelle l’ensemble de couleurs. Un choix canonique (en relation à G) d’un coloriage de Ω pour chaque chaı̂ne x ∈ ΣΩ est une application qui va de ΣΩ aux coloriages et qui commute avec l’action de G. En particulier, un choix canonique peut être un outil pour détecter des nonisomorphismes : si les coloriages C(x) et C(y) induits canoniquement par x et y 1125–03 ne sont pas isomorphes l’un à l’autre – par exemple, s’ils ont un nombre diﬀérent d’éléments vermeils – alors x et y ne sont pas isomorphes l’un à l’autre. Même quand il y a des isomorphismes dans G qui envoient C(x) sur C(y), la classe IsoG (C(x), C(y)) de tels isomorphismes sert à délimiter la classe d’isomorphismes IsoG (x, y) de x à y, puisque cette dernière est forcément un sous-ensemble de IsoG (C(x), C(y)). La preuve assimile aussi plusieurs idées développées lors d’approches antérieures au problème. La première étape de la procédure consiste à essayer de suivre ce qui est en essence l’algorithme de Luks [Lu]. Si cet algorithme s’arrête, c’est parce qu’il s’est heurté contre un quotient H1 /H2 isomorphe à Alt(Γ), où H2 ⊳ H1 < G et Γ est plutôt grand. Notre tâche majeure consiste à étudier ce qui se passe à ce moment-là. La stratégie principale sera de chercher à colorier Γ d’une façon qui dépend canoniquement de x. Cela limitera les automorphismes et isomorphismes possibles à considérer. Par exemple, si la moitié de Γ est coloriée en rouge et l’autre en noir, le groupe d’automorphismes possibles se réduit à Sym(\|Γ\|/2) ×Sym(\|Γ\|/2). Un coloriage similaire induit par y limite les isomorphismes aux applications qui alignent les deux coloriages. Nous trouverons toujours des coloriages qui nous aident, sauf quand certaines structures ont une très grande symétrie, laquelle, en revanche, permettra une descente à Ω considérablement plus petit. Cette double récursion – réduction du groupe H1 /H2 ou descente à des chaı̂nes considérablement plus courtes – résoudra le problème. 2. FONDEMENTS ET TRAVAUX PRÉCÉDENTS En suivant l’usage courant pour les groupes de permutations, nous écrirons r g pour l’élément g(r) auquel g ∈ Sym(Ω) envoie r ∈ Ω. Étant donnés une chaı̂ne x : Ω → Σ et −1 un élément g ∈ Sym(Ω), nous déﬁnissons xg : Ω → Σ par xg (r) = x r g . Par contre, nous écrivons Ωk pour l’ensemble des ~x = (x1 , . . . , xk ) avec l’action à gauche donnée par (φ(~x))r = ~xφ(r) . L’idée est que ceci est déﬁni non pas seulement pour φ une permutation, mais pour toute application φ : {1, . . . , k} → {1, . . . , k}, même non injective. Nous appelons les éléments de Ωk tuples plutôt que chaı̂nes. 2.1. Algorithmes de base 2.1.1. Schreier-Sims. — Plusieurs algorithmes essentiels se basent sur une idée de Schreier [Sch]. Il a remarqué que, pour tout sous-groupe H d’un groupe G et tout sous-ensemble A ⊂ G qui engendre G et contient des représentants de toutes les classes de H dans G, A′ = AAA−1 ∩ H = σ1 σ2 σ3−1 : σi ∈ A ∩ H est un ensemble de générateurs de H. 1125–04 L’étape suivante est celle de Sims [Si1], [Si2], qui a montré l’utilité de travailler avec un groupe de permutations G < Sym(Ω), Ω = {x1 , . . . , xn }, en termes d’une chaı̂ne de stabilisateurs G = G0 > G1 > G2 > . . . > Gn−1 = {e}, où Gk = G(x1 ,x2 ,...,xk ) = {g ∈ G : ∀1 ≤ i ≤ k xgi = xi } (stabilisateur de points). L’algorithme de Schreier-Sims (Algorithme 1 ; description basée sur [Lu, §1.2]) construit des ensembles Ci de représentants de Gi /Gi+1 tels que ∪i≤j<n−1 Cj engendre Gi pour tout 0 ≤ i < n − 1. Le temps pris par l’algorithme est O(n5 + n3 \|A\|), où A est l’ensemble de générateurs de G qui nous est donné : la fonction Filtre prend O(n) de temps, et tout g pour lequel elle est appelée satisfait g ∈ AC ∪ CA ∪ C 2 , où C est la valeur de ∪i Ci à la ﬁn de la procédure. Bien sûr, \|C\| ≤ n(n + 1)/2. Grâce à l’algorithme lui-même, nous pourrons toujours supposer que nos ensembles de générateurs sont de taille O(n2). Le temps pris par l’algorithme est donc O(n5 ). (2) Algorithme 1 Schreier-Sims : construction d’ensembles Ci 1: fonction SchreierSims(A, ~ x) ⊲ A engendre G < Sym({x1 , . . . , xn }) assure ∪i≤j<n−1Cj engendre Gi et Ci 7→ Gi /Gi+1 est injectif ∀i ∈ {0, 1, . . . , n − 2} 2: Ci ← {e} pour tout i ∈ {0, 1, . . . , n − 2} 3: B←A 4: tantque B 6= ∅ 5: Choisir g ∈ B arbitraire, et l’enlever de B 6: (i, γ) ← Filtrer(g, (Ci ), ~x) 7: si γ 6= e alors 8: ajouter γ à Ci S S 9: B ← B ∪ j≤i Cj γ ∪ j≥i γCj 10: retourner (Ci ) fonction Filtrer(g, (Ci ), ~x) ⊲ retourne (i, γ) tel que γ ∈ Gi , g ∈ C0 C1 · · · Ci−1 γ requiert Ci ⊂ Gi et Ci → Gi /Gi+1 injectif ∀i ∈ {0, 1, . . . , n − 2} assure g ∈ / C0 C1 · · · Ci Gi+1 sauf si (i, γ) = (n − 1, e) 12: γ←g 13: pour i = 0 jusqu’à n − 2 14: si ∃h ∈ Ci tel quel xhi = xgi alors 15: γ ← h−1 γ 16: sinon 17: retourner (i, γ) 11: 18: retourner (n − 1, e) 2. Nous supposons que l’ensemble de générateurs initial, spéciﬁant le groupe G du problème, est de taille O(nC ), C une constante. Le temps pris par la première utilisation de l’algorithme est donc O nmax(5,3+C) . 1125–05 Une fois les ensembles Ci construits, il devient possible d’accomplir plusieurs tâches essentielles rapidement. Exercice 2.1. — Montrer comment accomplir les tâches suivantes en temps polynomial, étant donné un groupe G < Sym(Ω), \|Ω\| = n : (a) Déterminer si un élément g ∈ Sym(Ω) est dans G. (b) Étant donnés un homomorphisme φ : G → Sym(Ω′ ), \|Ω′ \| ≪ \|Ω\|O(1) , et un sousgroupe H < Sym(Ω′ ), décrire φ−1 (H). (c) [FHL] Soit H < G avec [G : H] ≪ nO(1) . Étant donné un test qui détermine en temps polynomial si un élément g ∈ G appartient à H, décrire H. Astuce : travailler avec G > H > H1 > H2 > . . . à la place de G = G0 > G1 > G2 > . . . . Ici, comme toujours, « décrire » veut dire « trouver un ensemble de générateurs », et un groupe nous est « donné » si un tel ensemble nous est donné. L’algorithme de Schreier-Sims décrit le stabilisateur de points G(x1 ,...,xk ) pour x1 , . . . , xk ∈ Ω arbitraires. Par contre, nous ne pouvons pas demander allègrement un ensemble de générateurs d’un stabilisateur d’ensemble G{x1 ,...,xk } = {g ∈ G : {xg1 , . . . , xgk } = {x1 , . . . , xk }} pour G, xi arbitraires : faire ceci serait équivalent à résoudre le problème de l’isomorphisme lui-même. 2.1.2. Orbites et blocs. — Soit donné, comme toujours, un groupe de permutations G agissant sur un ensemble ﬁni Ω. Le domaine Ω est l’union disjointe des orbites {xg : g ∈ G} de G. Ces orbites peuvent être déterminées en temps polynomial (3) en \|Ω\|. Ceci est un exercice simple. La tâche se réduit à celle – simple elle aussi – de trouver les composantes connexes d’un graphe. Supposons que l’action de G soit transitive. (Il y a donc une seule orbite.) Un bloc de G est un sous-ensemble B ⊂ Ω, B ∈ / {∅, Ω}, tel que, pour g, h ∈ G quelconques, g h g h soit B = B , soit B ∩ B = ∅. La collection {B g : g ∈ G} (système de blocs) pour B donné partitionne Ω. L’action de G est primitive s’il n’y a pas de blocs de taille > 1 ; autrement, elle s’appelle imprimitive. Un système de blocs est minimal (4) si l’action de G sur lui est primitive. Voyons comment déterminer si l’action de G est primitive, et, s’il ne l’est pas, comment trouver un système de blocs de taille > 1. En itérant la procédure, nous obtiendrons un système de blocs minimal en temps polynomial. (Nous suivons [Lu], qui cite [Si1].) Pour a, b ∈ Ω distincts, soit Γ le graphe avec Ω comme son ensemble de sommets et l’orbite {{a, b}g : g ∈ G} comme son ensemble d’arêtes. La composante connexe qui 3. Pour être précis : O \|Ω\|O(1) + \|A\|\|Ω\| , où A est la taille de l’ensemble de générateurs de G qui nous est donné. Nous omettrons toute mention de cette taille par la suite, puisque, comme nous l’avons déjà dit, nous pouvons la garder toujours sous contrôle. 4. Pour paraphraser [Lu, §1.1] : il faut avouer qu’un tel système pourrait s’appeler plutôt maximal. La taille des blocs est maximale, leur nombre est minimal. 1125–06 contient a et b est le bloc le plus petit qui contient a et b. (Si Γ est connexe, alors le « bloc » est Ω.) L’action de G est imprimitive ssi Γ est non connexe pour un a arbitraire et au moins un b ; dans ce cas-là, nous obtenons un bloc qui contient a et b, et donc tout un système de blocs de taille > 1. Un dernier mot : si G < Sym(Ω), nous disons que G est transitif, voire primitif, si son action sur Ω l’est. 2.2. Luks : le cas de groupes avec facteurs d’ordre borné Luks a montré comment résoudre le problème de l’isomorphisme de graphes en temps polynomial dans le cas spécial de graphes de degré borné. (Le degré, ou valence, d’un sommet dans un graphe non orienté est le nombre d’arêtes qui le contiennent.) Il réduit ceci au problème de décrire le groupe d’automorphismes de chaı̂nes dans le cas d’un groupe G tel que tout facteur de composition de G – c’est-à-dire, tout quotient dans une suite principale (Jordan-Hölder) de G – est borné. Le processus de réduction, élégant et loin d’être trivial, ne nous concerne pas ici. Voyons plutôt comment Luks résout ce cas du problème de l’isomorphisme de chaı̂nes. Nous suivrons la notation de [Ba], même si les idées viennent de [Lu]. Définition 2.2. — Soient K ⊂ Sym(Ω) et ∆ ⊂ Ω (la « fenêtre »). L’ensemble d’isomorphismes partiels Iso∆ K est τ Iso∆ K (x, y) = {τ ∈ K : x(x) = y(x ) ∀x ∈ ∆}. ∆ L’ensemble d’automorphismes partiels Aut∆ K (x) est égal à IsoK (x, x). Iso∆ K est donc l’ensemble de toutes les permutations g ∈ K qui envoient x sur y – au moins à en juger par ce qui peut se voir par la fenêtre ∆. Nous travaillerons en général avec K de la forme Hπ, où H laisse ∆ invariante (en tant qu’ensemble). Il est clair que, pour K, K1 , K2 ⊂ Sym(Ω) et σ ∈ Sym(Ω), ∆ ∆ σ−1 σ, (1) IsoKσ (x, y) = IsoK x, y (2) ∆ ∆ Iso∆ K1 ∪K2 (x, y) = IsoK1 (x, y) ∪ IsoK2 (x, y). Il est aussi clair que, si G est un sous-groupe de Sym(Ω) et ∆ est invariant sous G, alors AutG (x) est un sous-groupe de G, et, pour tout σ ∈ Sym(Ω), IsoGσ (x, y) est soit vide, soit une classe à droite de la forme AutG (x)τ , τ ∈ Sym(Ω). Soient ∆1 , ∆2 ⊂ Ω, ′ 1 ∆1 invariant sous G. Pour G′ = AutG (x) et σ, τ tels que Iso∆ Gσ (x, y) = G τ , ∆1 ∪∆2 ∆2 τ −1 2 τ, x, y (3) IsoGσ (x, y) = Iso∆ (x, y) = Iso ′ ′ Gτ G où la deuxième équation est une application de (1). Babai appelle (3) la règle de la chaı̂ne. L’énoncé suivant n’utilise pas la classiﬁcation de groupes ﬁnis simples. 1125–07 Théorème 2.3 ([BCP] (6) ). — Soit G < Sym(Ω) un groupe primitif. Soit n = \|Ω\|. Si tout facteur de composition de G est d’ordre ≤ k, alors \|G\| ≤ nOk (1) . Ici, comme d’habitude, Ok (1) désigne une quantité qui dépend seulement de k. Théorème 2.4 (Luks [Lu]). — Soient Ω un ensemble ﬁni et x, y : Ω → Σ deux chaı̂nes. Soit donné un groupe G < Sym(Ω) tel que tout facteur de composition de G est d’ordre ≤ k. Il est possible de déterminer IsoG (x, y) en temps polynomial en n = \|Ω\|. Preuve — Cas 1 : G non transitif. Soit ∆1 ( Ω, ∆1 6= ∅, ∆1 stable sous l’action 1 de G. Déﬁnissons ∆2 = Ω \ ∆1 . Alors, par (3), il suﬃt de calculer Iso∆ G (x, y) (égal à −1 ′ ′ τ 2 une classe que nous notons G′ τ ) et Iso∆ . Or, pour déterminer G′ (x, y ) pour y = y ∆1 IsoG (x, y), nous déterminons, de façon récursive, IsoG (x\|∆1 , y\|∆1 ), puis, par Schreier′ 2 Sims, le stabilisateur de points G(∆1 ) . De la même manière, déterminer Iso∆ G′ (x, y ) pour −1 y′ = yτ se réduit à déterminer le groupe d’isomorphismes (dans un groupe G′ ) entre deux chaı̂nes de longueur \|∆2 \|. Comme \|∆1 \|+\|∆2\| = n et Schreier-Sims prend du temps O(n5 ), tout va bien. (La comptabilité est laissée au lecteur.) Cas 2 : G transitif. Soit N le stabilisateur d’un système de blocs minimal pour G ; donc, G/N est primitif. Par le Théorème 2.3, \|G/N\| ≤ mOk (1) , où m est le nombre de blocs. Or, pour σ1 , . . . , σℓ (ℓ = \|G/N\|) tels que G = ∪1≤i≤ℓ Nσi , [ [ −1 (4) IsoG (x, y) = Iso∪i N σi (x, y) = IsoN σi (x, y) = IsoN (x, yσi )σi 1≤i≤ℓ 1≤i≤ℓ par (1) et (2). Comme les orbites de N sont contenues dans les blocs, qui sont de taille −1 n/m, déterminer IsoN (x, yi ) (yi = yσi ) se réduit – par la règle (3) – à déterminer les groupes d’isomorphismes de m paires de chaı̂nes de longueur n/m. Nous avons donc réduit le problème à la solution de ℓ·m = mOk (1) problèmes pour des chaı̂nes de longueur n/m. Le pas ﬁnal consiste à faire l’union de classes en (4). Nous avons une description de chaque IsoN (x, yi ), soit comme l’ensemble vide, soit comme une classe à droite Hτi du groupe H = AutN (x), dont nous avons trouvé une description, c’est-à-dire un ensemble de générateurs A. Alors [ [ Hτi σi IsoN (x, yi )σi = IsoG (x, y) = 1≤i≤ℓ 1≤i≤ℓ = A ∪ τi σi (τ1 σ1 )−1 : 1 ≤ i ≤ ℓ τ1 σ1 . Nous aurions pu éviter quelques appels à Schreier-Sims en travaillant toujours avec des isomorphismes partiels, mais cela a peu d’importance qualitative. 6. À vrai dire, [BCP, Thm 1.1] est plus général que ceci ; par exemple, des facteurs abéliens arbitraires (non bornés) sont admis. Cela donne une généralisation du Théorème 2.4. 1125–08 2.3. Relations, partitions, configurations Soit C (« couleurs ») un ensemble ﬁni que nous pouvons supposer ordonné (disons, de rouge à violet). Une relation k-aire sur un ensemble ﬁni Γ est un sous-ensemble R ⊂ Γk . Une structure (relationnelle) k-aire est une paire X = (Γ, (Ri )i∈C ), où, pour chaque i ∈ C , Ri est une relation k-aire sur Γ. Si les Ri sont tous non vides et partitionnent Γk , nous disons que X est une structure de partition k-aire. Dans ce cas-là, nous pouvons décrire X par une fonction c : Γk → C qui assigne à chaque ~x ∈ Γk l’indice i de la relation Ri à laquelle il appartient. Nous disons que c(~x) est la couleur de ~x. Un isomorphisme entre deux structures k-aires X = (Γ, (Ri )i∈C ) et X′ = (Γ′ , (Ri′ )i∈C ) est une bijection Γ → Γ′ qui envoie Ri à Ri′ pour chaque i. Il est possible de construire un foncteur F1 qui envoie chaque structure k-aire X sur Γ à une structure de partition k-aire F1 (X) sur Γ ; qui plus est, Iso(X, Y) = Iso(F1 (X), F1 (Y)). La procédure est plutôt triviale ; nous la détaillons (Algorithme 2) pour montrer ce qu’indexer veut dire. Cela nous permet de ne pas utiliser plus de min \|Γ\|k , 2\|C \| couleurs, où n = \|Ω\|, tout en gardant leur signiﬁcation en termes des couleurs originales C . Le temps pris pour calculer F1 (X) est O(\|C \|\|Γ\|O(k)). Nous ne nous occupons pas des détails d’implémentation de la collection de tuples I , mais il peut s’agir tout simplement d’une liste ordonnée lexicographiquement ; dans ce cas, \|Γ\|O(k) est \|Γ\|2k . (Dans la réalité, I serait implémentée avec du hachage, ce qui n’est que l’art de bien organiser une bibliothèque.) Algorithme 2 Raﬃnement d’une structure de relation. Indexeur. 1: fonction F1 (Γ,k,C ,(Ri )i∈C ) 2: I ←∅ 3: pour ~x ∈ Γk 4: a ← {i ∈ C : ~x ∈ Ri } 5: c(~x) ← Indexeur(I , a) 6: 7: 8: 9: 10: 11: retourner (I , c) ⊲ retourne c : Γk → C ′ ⊲ C ′ est l’ensemble d’indices de I ; I explique C ′ en termes de C fonction Indexeur(I ,a) ⊲ I est une collection modiﬁable si a n’est pas dans I alors ajouter a à I retourner indice de a dans I Un élément ~x ∈ Γk déﬁnit une relation d’équivalence ρ(~x) sur {1, . . . , k} : i ∼ j ssi xi = xj . Le monoı̈de M(S) (S un ensemble) consiste en les applications S → S, avec la composition comme opération. Définition 2.5. — Une structure de partition k-aire X = (Γ, c) est dite conﬁguration k-aire si (a) Pour tous ~x, ~y ∈ Γk , si c(~x) = c(~y ), alors ρ(~x) = ρ(~y ). (b) Il y a un homomorphisme de monoı̈des η : M({1, . . . , k}) → M(C ) tel que, pour tout τ ∈ M({1, . . . , k}), c (τ (~x)) = τ η (c(~x)) pour tout ~x ∈ Γk . 1125–09 Alors, par exemple, pour k = 2, (a) veut dire que la couleur de ~x = (x1 , x2 ) « sait » si x1 = x2 ou pas, dans le sens où, si nous connaissons c(~x), alors nous savons si x1 = x2 ou pas. De la même façon, (b) nous indique que la couleur de ~x connaı̂t les couleurs de (x2 , x1 ), (x1 , x1 ) et (x2 , x2 ). Nous pouvons déﬁnir un foncteur F2 qui envoie chaque structure de partition k-aire X sur Γ à une conﬁguration k-aire ; comme pour F1 , le fait que F2 (X) est un raﬃnement de X implique que Iso(X, Y) = Iso(F2 (X), F2 (Y)). La procédure pour calculer F2 est très similaire à celle pour calculer F1 (Algorithme 2). Au lieu d’assigner à ~x la couleur {i ∈ C : ~x ∈ Ri }, nous lui assignons la couleur ρ(~x), (c(φ(~x)))φ∈M({1,...,k}) . Il est aisé de voir que F2 (X) est le raﬃnement le plus grossier d’une structure de partition X qui est une conﬁguration, de la même manière que F1 (X) est le raﬃnement le plus grossier d’une structure X qui est une structure de partition. Définition 2.6. — Soit X = (Γ, c), c : Γk → C , une structure de partition k-aire. Pour 1 ≤ l ≤ k, nous déﬁnissons c(l) : Γl → C comme suit : c(l) (~x) = c(x1 , x2 , . . . , xl , xl , . . . xl ). La structure de partition l-aire X(l) = Γ, c(l) est dite le (l-)squelette de X. La chaı̂ne vide sera viride. Exercice 2.7. — Tout squelette d’une conﬁguration est une conﬁguration. Ici le fait que l’axiome (b) dans la déﬁnition de conﬁguration soit valable même pour η non injectif est crucial. Pour X = (Γ, c) une structure de partition et Γ′ ⊂ Γ, la sous-structure induite X[Γ′ ] est la structure (Γ′ , c\|Γ′ ) déﬁnie par la restriction de c à Γ′ . Il est clair que, si X est une conﬁguration, alors X[Γ′ ] l’est aussi. * Il ne faut pas confondre une structure de partition (partition structure) avec ce que nous appellerons un découpage (colored partition). Un découpage d’un ensemble Γ est un coloriage de Γ supplémenté d’une partition de chaque classe de couleur. (Une classe de couleur est l’ensemble de sommets d’une couleur donnée.) Un découpage est dit admissible si chaque ensemble B dans chaque partition est de taille ≥ 2. Pour α < 1, un α-découpage est un découpage admissible tel que \|B\| ≤ α\|Γ\| pour chaque B. Un découpage est une structure plus ﬁne que le coloriage qu’il raﬃne, mais moins ﬁne que la structure que nous obtiendrions si nous donnions à chaque élément de chaque partition une couleur diﬀérente. Un automorphisme ou isomorphisme d’un découpage doit préserver les couleurs de celui-ci, mais pourrait permuter les ensembles de la même taille qui appartiennent à la partition d’une couleur. Comme les ensembles de taille diﬀérente ne peuvent, évidemment, être permutés, il est clair que nous pouvons supposer sans perte de généralité que toute couleur est partitionnée en ensembles de la même taille. Nous ajoutons ceci à la déﬁnition de α-découpage à partir de maintenant. 1125–10 2.4. Configurations cohérentes k-aires Pour ~x ∈ Γk , z ∈ Γ et 1 ≤ i ≤ k, nous déﬁnissons ~xi (z) ∈ Γk comme suit : ~xi (z) = (x1 , x2 , . . . , xi−1 , z, xi+1 , . . . , xk ). Définition 2.8. — Une conﬁguration cohérente k-aire X = (Γ, c) est une conﬁguration k-aire ayant la propriété suivante : il y a une fonction γ : C k × C → Z≥0 telle que, pour ~k ∈ C k et j ∈ C arbitraires et tout ~x ∈ Γk tel que c(~x) = j, \|{z ∈ Γ : c(~xi (z)) = ki ∀1 ≤ i ≤ k}\| = γ(~k, j). Les valeurs γ(~k, j) sont appelées nombres d’intersection de X. Une conﬁguration cohérente est dite classique si k = 2. Remarque 2.9. — Les conﬁgurations cohérentes classiques ont été introduites par Higman [Hi]. Les premiers exemples étaient du type schurien : une conﬁguration est schurienne si elle est la partition de Γ2 dans ses orbites (« orbitales ») sous l’action d’un groupe G < Sym(Γ). Définition 2.10. — Si une conﬁguration cohérente classique n’a que deux couleurs, une pour {(x, x) : x ∈ Γ} et l’autre pour son complément, la conﬁguration est dite une clique, ou triviale. Exercice 2.11. — Tout squelette d’une conﬁguration cohérente est cohérent. Encore une fois, l’axiome (b) des conﬁgurations joue un rôle clé. Exercice 2.12. — Soient X = (Γ, c) une conﬁguration cohérente et Γ′ ⊂ Γ une classe de couleurs en relation au coloriage induit par c sur Γ. Alors la sous-structure induite X[Γ′ ] est une conﬁguration cohérente. Ici, c’est un cas spécial de (b) qu’il faut utiliser : la couleur c(x1 , . . . , xn ) « connaı̂t » les couleurs c(x1 ), . . . , c(xn ), puisque c(xi ) = c(xi , . . . , xi ). Soient 0 ≤ l < k et ~x ∈ Γl . Nous colorions Γk−l comme suit : pour ~y ∈ Γk−l , c~x (~y ) = c(~x~y ). En résulte une structure de partition (k − l)-aire X~x = (Γ, c~x ). Exercice 2.13. — Soit X = (Γ, c) une structure de partition ; soit ~x ∈ Γl , 0 ≤ l < k. Alors (a) c~x est un raﬃnement du coloriage du squelette X(k−l) . (b) Si X est cohérente, X~x l’est aussi. Il est clair que, de plus, X~x est canonique en relation à ~x, ce qui veut dire que X → X~x commute avec l’action sur Γ du stabilisateur dans Sym(Γ) des points x1 , . . . , xl . 1125–11 Définition 2.14. — Une conﬁguration cohérente (Γ, c) est dite homogène si la couleur c(x, x, . . . , x) de tout sommet x ∈ Γ est la même. Une conﬁguration cohérente classique est dite primitive si elle est homogène et les graphes Gr = {(x, y) : x, y ∈ Γ, c(x, y) = r} (pour toute couleur r telle que c(x, y) = r pour au moins une paire (x, y) avec x 6= y) sont tous connexes. Elle est dite uniprimitive si elle est primitive et non triviale. Nous n’avons pas besoin de préciser si ces graphes son connexes dans le sens propre (à savoir, il y a un chemin de tout sommet à tout autre, respectant l’orientation) ou dans le sens faible (sans compter l’orientation) : le fait que (Γ, c) soit cohérente, classique et homogène implique que d+ r (x) = \|{y ∈ Γ : (x, y) ∈ Gr }\| est indépendant de x (pourquoi ?), ce qui implique que toute composante faiblement connexe de Gr est connexe (exercice). Exercice 2.15. — Soit X = (Γ, c) une conﬁguration cohérente classique uniprimitive. Il n’y a aucun ensemble B ⊂ Γ, \|B\| > \|Γ\|/2, tel que la restriction de X à B soit une clique. Solution – Si les arêtes de la grande clique sont sensées être blanches, soit noir une autre couleur d’arêtes de X, et soit G = Gnoir . Or, pour un graphe orienté birégulier (7) G non vide avec Γ comme ensemble de sommets, il est impossible qu’il y ait un ensemble B ⊂ Γ, \|B\| > \|G\|/2, tel que la réduction du graphe à B soit vide (pourquoi ?). Exercice 2.16. — Soit (Γ, c) une conﬁguration cohérente classique homogène. (a) Soit r0 , . . . , rk une séquence de couleurs. Alors, si x0 , xk ∈ Γ sont tels que c (x0 , xk ) = r0 , le nombre de x1 , . . . , xk−1 ∈ Γ tels que c (xi−1 , xi ) = ri pour tout 1 ≤ i ≤ k dépend seulement de r0 , . . . , rk . (b) Pour toute couleur r, toute composante connexe de Gr est de la même taille. Solution (esquisse) — En (a), le cas k = 2 vaut par la déﬁnition de « cohérent » ; prouvez les cas k > 2 par induction. Pour prouver (b), utilisez (a). 2.5. Le raffinement canonique k-aire à la façon de Weisfeiler-Leman Déﬁnissons un foncteur F3 qui envoie une conﬁguration X = (Γ, c) à une conﬁguration cohérente F3 (X) = (Γ, c′ ). Comme F3 (X) sera un raﬃnement de X, nous aurons Iso(X, Y) = Iso(F3 (X), F3(Y)). L’algorithme 3, qui calcule F3 , est basé sur une idée de Weisfeiler et Leman (8) [WL]. Il s’agit d’itérer une procédure de raﬃnement. Si, dans une itération, aucun raﬃnement ne se produit – c’est-à-dire, si les classes d’équivalence du nouveau coloriage Ci sont les mêmes que celles de l’ancien coloriage Ci−1 – alors, (a) aucun raﬃnement ne se produira dans le futur, (b) le coloriage Ci−1 est déjà cohérent. 7. Voir la déﬁnition du §2.6. 8. Aussi appelé Lehman, mais [Ba] indique que le deuxième auteur préférait Leman. Deux transformations naturelles L → Л, Л → L peuvent ne pas être l’inverse l’une de l’autre. 1125–12 Si le coloriage C = C0 a r couleurs diﬀérentes du début, il est clair qu’il ne peut être raﬃné que \|Γ\|k − r fois. Alors, \|Γ\|k − r itérations sont suﬃsantes pour produire une conﬁguration cohérente. En particulier, si l’indexation est faite en temps logarithmique, et le vecteur dans le pas 6 de l’algorithme 3 est représenté comme un vecteur creux (puisque son nombre d’entrées non-nulles est au plus \|Γ\|), le temps pris par l’algorithme est O(k 2 \|Γ\|2k+1 log \|Γ\|). (En outre, [Ba, §2.8.3] aﬃrme une borne plus forte.) Les algorithmes de type Weisfeiler-Leman étaient autrefois regardés comme une approche plausible au problème de l’isomorphisme de graphes. Depuis [CFI], [EvP], il est clair qu’il ne se suﬃsent pas à eux-mêmes. Ils sont quand même un outil précieux. La version k-aire ici est due à Babai-Mathon [Ba3] et Immerman-Lander [ImL]. Algorithme 3 Weisfeiler-Leman pour les conﬁgurations k-aires. 1: fonction WeisfeilerLeman(Γ, k, c : Γk → C ) 2: C0 ← C ; c0 ← c ; i0 ← \|Γk \| − \|c(Γk )\| 3: pour i = 1 jusqu’à i0 4: Ii ← ∅ 5: pour ~x ∈ Γk j 6: ν ← ci−1 (~x), (\|{z ∈ Γ : ci−1 (~x (z)) = rj ∀ 1 ≤ j ≤ k}\|)~r∈C k i−1 7: ci (~x) = Indexeur (Ii , ν) ⊲ Indexeur est comme dans l’algorithme 2 8: 9: Ci ← indices de Ii retourner cn−i0 : Γk → Cn−i0 , (Ii )1≤i≤n−i0 ⊲ (Ii ) donne du sens à Cn−r 2.6. Graphes, hypergraphes et designs en blocs Nous savons déjà qu’un graphe est une paire (V, A), où V est un ensemble (« sommets ») et A est une collection de paires d’éléments de V (voire de sous-ensembles de V avec deux éléments, si le graphe est non orienté). Un graphe non orienté est dit régulier si le degré de tout sommet est le même ; un graphe orienté est dit birégulier si le degré sortant d+ (v) = \|{w ∈ V : (v, w) ∈ A}\| et le degré entrant d− (v) = \|{w ∈ V : (v, w) ∈ A}\| sont indépendants de v. (Pour V ﬁni, ils sont forcément la même constante.) Un graphe biparti est un triplet (V1 , V2 ; A) avec A ⊂ V1 × V2 . Un graphe biparti est semirégulier si le degré (9) d+ (v1 ) est indépendant de v1 ∈ V1 , et le degré d− (v2 ) est indépendant de v2 ∈ V2 . Exercice 2.17. — Soit X = (Γ, c) une conﬁguration cohérente classique homogène. (a) Soient C1 , C2 deux classes de couleur, et soit vert une couleur d’arêtes en C1 ×C2 . Alors, le graphe biparti (C1 , C2 ; Gvert ) est semirégulier. 9. Nous omettons les mots « entrant » et « sortant », puisqu’il est évident qu’il s’agit du degré entrant dans le cas de v1 et du degré sortant dans le cas de v2 . 1125–13 (b) Soit y ∈ Γ, et Li (y) = {x ∈ Γ : c(x, y) = i}. Soient lin, bis et terre trois couleurs d’arêtes. Alors, pour L1 = Llin (y) et L2 = Lbis (y), le graphe biparti (L1 , L2 ; Rterre ∩ (L1 × L2 )) est semirégulier. Exercice 2.18. — Soit X une conﬁguration cohérente classique homogène. Soient C1 , C2 deux classes de couleur. Soient vert une couleur d’arêtes en C1 × C2 et rouge une couleur d’arêtes en C2 × C2 . Soient B1 , . . . , Bm les composantes connexes de Grouge en C2 . Déﬁnissons le graphe biparti Y = (C1 , {1, . . . , m}; D) comme suit : (x, y) ∈ D ssi (x, y) ∈ Gvert pour au moins un y ∈ Bi . Alors Y est semirégulier. Solution — Notez que, pour y ∈ Bi et x ∈ C1 , (x, y ′ ) est vert pour au moins un y ′ ∈ Bi ssi il existe x0 = x, x1 , . . . , xm tels que (xi , xi+1 ) est rouge pour 0 ≤ i < m et (xm , y) est vert. Concluez par l’exercice 2.16a que tous les sommets en {1, . . . , m} ont le même degré en Y . De façon analogue, montrez que, pour x ∈ V1 et y ∈ Bi tels que (x, y) est rouge, le nombre de z ∈ Bi tels que (x, z) est rouge ne dépend pas de x, y ou i. Notons ce nombre q. Alors, le degré de tout v ∈ C1 est son degré en X, divisé par q. Par (a), il ne dépend donc pas de v. Un graphe biparti est complet (en tant que graphe biparti) si A = V1 × V2 . Un graphe biparti qui n’est ni vide ni complet est appelé non trivial. Un hypergraphe H = (V, A ) consiste en un ensemble V (« sommets ») et une collection A de sous-ensembles de V (« arêtes »), peut-être avec des sous-ensembles répétés. Un hypergraphe est dit u-uniforme si \|A\| = u pour tout A ∈ A . Il est dit régulier de degré r si tout v ∈ V appartient à exactement r ensembles A dans A . L’hypergraphe u-uniforme complet sur V est (V, {A ⊂ V : \|A\| = u}), où chaque ensemble A est compté une fois. Un coloriage des arêtes de l’hypergraphe complet est une application de {A ⊂ V : \|A\| = u} à un ensemble ﬁni C . Un block design équilibré (BDE) de paramètres (v, u, λ) est un hypergraphe avec \|V \| = v sommets, u-uniforme et régulier de degré r ≥ 1, tel que toute paire {v1 , v2 } de sommets distincts est contenue dans exactement λ ≥ 1 arêtes (« blocks »). Un block design dégénéré a la même déﬁnition, mais avec λ = 0, et la condition additionnelle d’être un hypergraphe régulier. (La régularité peut être déduite de la déﬁnition si λ ≥ 1.) Un block design est incomplet si u < v. Notons b le nombre \|A \| d’arêtes d’un BDE. Proposition 2.19 (Inégalité de Fisher (11) [F]). — Pour tout block design équilibré incomplet, b ≥ v. Il est aisé de voir que cette inégalité est vraie même pour les designs dégénérés. Les blocks designs admettent une généralisation. Un design t-(v, u, λ) est un hypergraphe (V, A ) u-uniforme avec v = \|V \| sommets tel que tout T ⊂ V de taille t est contenu dans exactement λ arêtes. Ici t ≥ 2 et λ ≥ 1. Nous écrivons toujours b = \|A \|. 11. Si, R. A. Fisher, le statisticien. Ici design vient d’experimental design. 1125–14 Proposition 2.20 ([RChW]). — Pour tout design t-(v, u, λ) et tout s ≤ min(t/2, v−u), nous avons b ≥ vs . 2.7. Schémas de Johnson Un schéma d’association est une conﬁguration cohérente classique (Γ, c : Γ2 → C ) telle que c(x, y) = c(y, x) ∀x, y ∈ Γ. (Il s’agit donc d’un sens du mot schéma qui n’a rien à voir avec les schémas de la géométrie algébrique.) Soient s ≥ 2 et r ≥ 2s + 1. Un schéma de Johnson J (r, s) = (Γ, c) est donné par Γ = Ss (Λ) = {S ⊂ Λ : \|S\| = s}, c(S1 , S2 ) = \|S1 \ (S1 ∩ S2 )\|, où Λ est un ensemble à r éléments. La relation Ri est bien sûr l’ensemble Ri = {(S1 , S2 ) : c(S1 , S2 ) = i}. Notons que nous avons déﬁni implicitement un foncteur de la catégorie d’ensembles Λ avec \|Λ\| = r à la catégorie de schémas de Johnson. Ceci est un foncteur plein ; autrement dit, les seuls automorphismes de J (r, s) sont ceux qui sont induits par Sym(Λ). 2.8. Identification de groupes et de schémas Il est une chose de démontrer que deux groupes G, H sont isomorphes, et une autre de construire un isomorphisme φ de façon explicite entre eux. Cette dernière tâche implique, au moins, de donner les images φ(g1), . . . , φ(gr ) de générateurs g1 , . . . , gr de G. Voyons un cas particulier qui nous sera crucial. Nous aurons un groupe de permutation G < Sym(Γ), et nous saurons qu’il est isomorphe au groupe abstrait Altm . Comment construire un isomorphisme ? Si m n’est pas trop petit en relation à n = \|Γ\|, il est connu que G doit être isomorphe que le groupe Altm à un groupe de permutation de la forme Alt(k) m , qui n’est autre m agissant sur l’ensemble Sk (Λ0 ) = {S ⊂ Λ0 : \|S\| = k} à k éléments, où Λ0 est un ensemble à m éléments. (12) En d’autres termes, il existe une bijection ι0 : Γ → Sk (Λ0 ) et un isomorphisme φ0 : G → Alt(Λ0 ) tels que ι0 (ω g ) = ι0 (ω)φ0 (g) . Le problème consiste à construire ι : Γ → Sk (Λ) et φ : G → Alt(Λ), calculables en temps polynomial, avec ces mêmes propriétés. Nous suivons [BLS]. Soient Υ ⊂ Γ × Γ l’orbitale la plus petite de G (hors la diagonale ({ω, ω} : ω ∈ Γ}) ; soit ∆ ⊂ Γ × Γ l’orbitale la plus grande. Nous supposerons que 12. Babai nomme les groupes Alt(k) m groupes de Johnson, par analogie avec les schémas de Johnson. Puisque Alt(k) n’est qu’un déguisement de Altm , ne faudrait-il pas appeler ce dernier groupe de m Ramerrez ? 1125–15 m > (k + 1)2 − 2, ce qui revient à dire que n n’est pas trop grand en relation à m. (13) Alors, φ(Υ) = R1 = {(S1 , S2 ) ∈ Sk (Λ0 ) : \|S1 ∩ S2 \| = k − 1}, (5) φ(∆) = Rk = {(S1 , S2 ) ∈ Sk (Λ0 ) : S1 ∩ S2 = ∅}. Déﬁnissons, pour (x, y) ∈ Υ, B(x, y) = {z ∈ Γ : (x, z) ∈ / ∆, (y, z) ∈ ∆}. Ceci est l’ensemble de tous les z tels que ι0 (z) intersecte ι0 (x) mais pas ι0 (y). Soit [ C(x, y) = Γ \ {r : (z, r) ∈ ∆(z)}. z∈B(x,y) Alors ι0 (C(x, y)) = {S ∈ Sk (Λ0 ) : S ∩ S ′ 6= ∅ ∀S ′ ∈ Sk (Λ0 ) t.q. S ′ ∩ ι0 (x) 6= ∅, S ′ ∩ ι0 (y) = ∅} = {S ∈ Sk (Λ0 ) : i ∈ S}, où i est l’élément de ι0 (x) qui n’est pas dans ι0 (y). Soit Λ la collection {C(x, y) : (x, y) ∈ Υ}, sans multiplicités. Nous pouvons calculer et comparer C(x, y) pour (x, y) donné, et calculer et indexer Λ, tout en temps polynomial. Nous calculons, aussi en temps polynomial, l’action de G sur Λ induite par l’action de G sur Υ. Ceci déﬁnit φ : G → Alt(Λ). Il y a une bijection naturelle j : Λ → Λ0 qui commute avec l’action de G : elle envoie C(x, y) à i, où i est l’élément de Λ0 tel que ι0 (C(x, y)) = {S ∈ Sk (Λ0 ) : i ∈ S}. Il est clair que, pour ω ∈ Γ, ω ∈ C(x, y) ssi j(C(x, y)) ∈ ι0 (ω). Ainsi, nous obtenons la bijection ι : Γ → Sk (Λ), donnée par ι(ω) = {γ ∈ Λ : ω ∈ γ}. Celle-ci satisfait ι (ω g ) = ι(ω)φ(g) . Les applications φ, ι sont donc celles que nous désirions ; nous avons construit un isomorphisme explicite entre G et Alt(Λ). Notons que cette même procédure nous permet de construire un isomorphisme explicite entre, d’un côté, un schéma d’association (§2.7) qu’on sait être isomorphe à un schéma de Johnson J (m, k), et, de l’autre côté, ce même schéma. 13. Si m ≤ (k + 1)2 − 2, alors n est si grand que m! = nO(log n) . En ce cas, nous pouvons enlever le groupe G (c’est-à-dire, dans l’application qui nous intéressera, un quotient G/N ) de façon brutale, comme dans le cas 2 de la preuve du théorème 2.4 (Luks). Nous pourrions aussi nous passer de la supposition m > (k + 1)2 − 2 au coût de quelques complications en ce qui suit. En particulier, φ(∆) ne serait pas Rk comme dans (5), sinon un autre Rj . 1125–16 3. LA PROCÉDURE PRINCIPALE input : G < Sym(Ω) x, y : Ω → Σ réduction de G/N à Altm′ m′ ≤ \|m\|/2 G transitif ? aligner coupe coupe ou Johnson ? coupe réduction J de G/N à Altm′ √ m′ ≪ m coupe ou relations ? rels. plénitude > 1/2 ? non G/N ∼ Altm ? blocs ∼ Γ k récursion n′ ≤ n/2 non oui m petit ? non oui k = 1? oui cas trivial non non certiﬁcats locaux symétrie > 1/2 ? non non récursion n′ < n oui : G/N ∼ Alt(Γ) G primitif ? Lemme des designs non oui oui une couleur domine ? output : IsoG (x, y) Fonction Isomorphisme-de-Chaı̂nes Weisfeiler - Leman k-aire oui x, y → relations k-aires sur Γ oui pullback 1125–17 3.1. Premiers pas : récursion à la façon de Luks G transitif ? non récursion n′ < n non récursion n′ ≤ n/2 oui G/N ∼ Altm ? oui m petit ? oui Les premiers pas de la procédure sont ceux de la preuve du Théorème 2.4 (Luks). En particulier, si G < Sym(Ω) n’est pas transitif, nous procédons exactement comme dans le cas non transitif de la preuve du Théorème 2.4. Bien qu’il soit possible que n = \|Ω\| ne décroisse que très légèrement, la récursion marche, puisque son coût est aussi très léger dans ce cas : nous n’avons qu’à subdiviser le problème selon les orbites de G. Supposons que G soit transitif. Nous savons que nous pouvons trouver rapidement un système de blocs minimal R = {Bi : 1 ≤ i ≤ r}, Bi ⊂ Ω (§2.1.2). Par Schreier-Sims, nous trouvons aussi, en temps polynomial, le sous-groupe N ⊳ G des éléments g ∈ G tels que Big = Bi pour tout i. Le groupe H = G/N agit sur R. Au lieu du Théorème 2.3 [BCP], nous utiliserons une conséquence de la Classiﬁcation des Groupes Finis Simples (CGFS). Elle a été dérivée pour la première fois par Cameron, puis raﬃnée par Maróti. Théorème 3.1 ([Cam], [Ma]). — Soit H < Sym(R) un groupe primitif, où \|R\| = r est plus grand qu’une constante absolue. Alors, soit (14) (a) \|H\| < r 1+log2 r , soit (b) il y a un M ⊳ H tel que R se subdivise (15) en un système de m blocs sur lequel k (k) M agit comme un groupe Altm , m ≥ 5. En plus, [H : M] ≤ r. La borne [H : M] ≤ r se déduit de m > 2, \|H\| ≥ r 1+log2 r , \|H\| ≤ m!s s!, ms ≤ r et [H : M] ≤ 2s s!, où s ≥ 1 est un paramètre dans Cameron-Maróti. Il est possible [BLS] de trouver en temps polynomial le sous-groupe normal M et les blocs de l’action de M. Nous avons déjà vu au §2.8 comment identiﬁer explicitement l’action de M avec celle de Alt(k) m . Par ailleurs, l’algorithme de Schreier-Sims nous permet de calculer \|H\| en temps polynomial, et donc nous dit aussi si nous sommes dans le cas (a). Si c’est le cas, nous 14. Pour nous, log2 désigne le logarithme en base 2, et non pas le logarithme itéré log log. 15. L’énoncé dans [Cam], [Ma] est plus fort : il décrit toute l’action de H sur R. À vrai dire, le groupe m s s M est isomorphe, en tant que groupe de permutation, à (Alt(k) m ) , s ≥ 1. Nous avons r = k . 1125–18 procédons comme dans le cas transitif de la preuve du Théorème 2.4. Nous réduisons ainsi le problème à r 1+log2 r instances du problème pour des chaı̂nes de longueur ≤ n/r. Si nous sommes dans le cas (b) nous commençons toujours par réduire le problème à [H : M] instances du problème avec M à la place de H : par l’équation (2) et comme dans l’équation (4), [ −1 IsoH (x, y) = IsoM x, yσ σ, σ∈S où S est un système de représentants des classes de M dans H. Si m ≤ C log n, où C est une constante, \|M\| = m! < mm ≤ mC log n ≤ (m′ )C log n , 2 où m′ = m . Donc, ici comme dans le cas (a), nous nous permettons de procéder k comme dans le cas transitif de la preuve du Théorème 2.4. Nous obtenons une réduction à ≤ r · (m′ )C log n = (m′ )O(log n) instances du problème pour des chaı̂nes de longueur n/m′ . Ceci est tout à fait consistant avec l’objectif d’avoir une solution en temps quasipolynomial en n (ou même en temps nO(log n) ). Il reste à savoir que faire si nous sommes dans le cas suivant : il y a un isomorphisme φ : G/N → Alt(Γ), \|Γ\| > C log n, C une constante. (Ici nous avons déjà (i) remplacé G par la préimage de M dans la réduction G → G/N, et, après cela, (ii) remplacé N par le stabilisateur des blocs dans la partie (b) du Théorème 3.1.) Ce cas nous occupera pour le reste de l’article. * Babai indique comment enlever la dépendance de CGFS à cette étape. Soient G et N comme avant, avec G transitif. Alors G/N est un groupe primitif agissant sur l’ensemble de blocs R. Si un groupe de permutations sur un ensemble R est tel que son action sur l’ensemble des paires d’éléments distincts de R est transitive, le groupe est dit doublement transitif. Or, un résultat de Pyber qui ne dépend pas de CGFS [Py2] nous dit qu’un tel groupe 2 est soit Alt(R), soit Sym(R), soit d’ordre ≤ \|R\|O(log \|R\|) . Si G/N est Alt(R) ou Sym(R), nous sommes dans le cas que nous discuterons d’ici jusqu’à la ﬁn. Si G/N est doublement transitif, mais n’est égal ni à Alt(R) ni à Sym(R), nous pouvons procéder comme dans le cas transitif de la preuve du Théorème 2.4, 2 puisque \|G/N\| ≤ r O(log r) , r = \|R\| ≤ n. (Babai propose aussi un traitement alternatif, même plus eﬃcace et élémentaire.) Supposons donc que G/N n’est pas doublement transitif. Alors la conﬁguration cohérente schurienne (§2.4) qu’elle induit n’est pas une clique. En conséquence, nous pouvons donner cette conﬁguration à la procédure Coupe-ou-Johnson (§5.2), et reprendre le ﬁl de l’argument à ce point-là. 1125–19 4. LA STRUCTURE DE L’ACTION DE Alt 4.1. Stabilisateurs, orbites et quotients alternants Nous aurons besoin de plusieurs résultats sur les épimorphismes G → Altk . Ils joueront un rôle crucial dans la méthode des certiﬁcats locaux (§6.1). Dans la version originale [Ba], ils ont aussi été utilisés dans le rôle joué par [BLS] dans cet exposé. Lemme 4.1. — Soit G < Sym(Ω) primitif. Soit φ : G → Altk un épimorphisme avec k > max(8, 2 + log2 \|Ω\|). Alors φ est un isomorphisme. Prouver ce lemme est à peu près un exercice en théorie des groupes ﬁnis ; il faut utiliser [BaPS, Prop. 1.22] pour le cas de socle abélien et la conjecture de Schreier pour le cas de socle non abélien. La conjecture de Schreier est un théorème, mais un théorème dont la preuve dépend, à son tour, de CGFS. Par contre, Pyber [Py] a donné une preuve du Lemme 4.1 qui n’utilise pas CGFS, avec une condition plus stricte : k > max(C, (log \|Ω\|)5 ), C constante. La dépendance de CGFS a donc été complètement enlevée de la preuve du théorème principal. Définition 4.2. — Soit G < Sym(Ω). Soit φ : G → Symk un homomorphisme dont l’image contient Altk . Alors x ∈ Ω est dit atteint si φ(Gx ) ne contient pas Altk . Lemme 4.3. — Soit G < Sym(Ω). Soit φ : G → Altk un épimorphisme avec k > max(8, 2 + log2 n0 ), où n0 est la taille de la plus grande orbite de G. (a) Si G est transitif, tout x ∈ Ω est atteint. (b) Au moins un x ∈ Ω est atteint. Preuve (esquisse) — (a) Ceci découle immédiatement du Lemme 4.1 si G est primitif, ou si K < ker(φ) pour K le stabilisateur d’un système de blocs minimal. Il reste le cas de φ : K → Altk surjectif. En général : Lemme.— Pour Ki arbitraires, K < K1 × · · · × Ks et un épimorphisme φ : K → S, S simple, il doit y avoir un i tel que φ se factorise comme suit : ψ K → Ki → S, ψ un épimorphisme. En utilisant ce lemme pour les restrictions Ki de K aux orbites de K, nous passons à une orbite Ki , et procédons par induction. (b) Soient Ω1 , . . . , Ωm les orbites de G, et soit Gi = G\|Ωi la restriction de G à Ωi . Par ψ le Lemme en (a), il doit y avoir un i tel que φ se factorise en G → Gi → Altk , ψ un épimorphisme. Alors, par (a), (Gx )ψ = ((Gi )x )ψ 6= Altk pour tout x ∈ Ωi . La proposition suivante jouera un rôle crucial au §6. Proposition 4.4. — Soient G < Sym(Ω) transitif et φ : G → Altk un épimorphisme. Soit U ⊂ Ω l’ensemble des éléments non atteints. (a) Supposons que k ≥ max(8, 2 + log2 n0 ), où n0 est la taille de la plus grande orbite de G. Alors (G(U ) )φ = Altk . 1125–20 (b) Supposons que k ≥ 5. Si ∆ est une orbite de G qui contient des éléments atteints, alors chaque orbite de ker(φ) contenue dans ∆ est de longueur ≤ \|∆\|/k. Rappelons que G(U ) = {g ∈ G : xg = x ∀x ∈ U} (stabilisateur de points). Preuve — (a) Il est facile de voir que G ﬁxe U en tant qu’ensemble. Alors, G(U ) ⊳ G, et donc (G(U ) )φ ⊳ Gφ . Or, Gφ = Altk . Supposons que (G(U ) )φ = {e}. Alors φ se factorise comme suit : ψ G → G\|U → Altk , puisque G(U ) est le noyau de G → G\|U . Ici ψ est un épimorphisme, et donc, par le Lemme 4.3 (b), il existe un x ∈ U tel que ((G\|U )x )ψ 6= Altk . Or ((G\|U )x )ψ = (Gx )φ = Altk , parce que x est dans U, c’est-à-dire non atteint. Contradiction. (b) Comme ∆ contient des éléments atteints et est une orbite de G, tout élément de ∆ est atteint. Soit N = ker(φ), x ∈ ∆. La longueur de l’orbite xN est xN = [N : Nx ] = [N : (N ∩ Gx )] = [NGx : Gx ] = = [G : Gx ] [G : NGx ] \|∆\| \|∆\| = . [Gφ : (Gx )φ ] [Altk : (Gx )φ ] Or, tout sous-groupe propre de Altk est d’indice ≥ k. Donc xN ≤ \|∆\|/k. 4.2. Le cas de grande symétrie G primitif ? oui k = 1? oui cas trivial non symétrie > 1/2 ? oui pullback Considérons le cas de G primitif. Nous pouvons supposer que G est isomorphe en tant que groupe de permutation à Alt(k) m , puisque nous avons déjà éliminé les autres cas au §3 (peut-être en passant à un groupe non primitif M ; le cas non primitif sera traité au §6). Comme nous l’avons vu au §2.8, nous pouvons construire une bijection ι entre Ω et l’ensemble Sk (Γ) des sous-ensembles avec k éléments d’un ensemble Γ. Cette bijection induit un isomorphisme φ : G → Alt(Γ). Si k = 1, alors Ω est en bijection avec Γ, et G ∼ Altn = Altm . Nous sommes donc dans le cas trivial : le groupe AutG (x) consiste en les éléments de Altn qui permutent les lettres de x de la même couleur, et IsoG (x, y) est non vide ssi x et y ont exactement le même nombre de lettres de chaque couleur – où, si aucune lettre n’est répétée ni en x ni en y, nous ajoutons la condition que la permutation de {1, . . . , n} qui induit x 7→ y soit dans Altn . Alors, soit G primitif, k > 1. 1125–21 Deux éléments γ1 , γ2 ∈ Γ sont des jumeaux par rapport à un objet si la transposition (γ1 γ2 ) le laisse invariant. Il est clair que les jumeaux forment des classes d’équivalence, et que, pour toute telle classe d’équivalence C, tout Sym(C) laisse l’objet invariant. Notre objet sera la chaı̂ne x (ou y) : γ1 , γ2 sont des jumeaux par rapport à x si, pour −1 tout i ∈ Ω, x(i) = x(τ φ (i)), où τ = (γ1 γ2 ). Nous pouvons donc déterminer facilement (et en temps polynomial) les classes d’équivalence en Γ (dites classes de jumeaux), et vériﬁer s’il y a une classe d’équivalence C de taille > \|Γ\|/2. Examinons cette possibilité puisque nous devrons l’exclure après. La classe C de taille > \|Γ\|/2 est évidemment unique et donc canonique. Si x a une telle classe et y ne l’a pas, ou si les deux ont de telles classes, mais de tailles diﬀérentes, alors x et y ne sont pas isomorphes. Si x, y ont des classes de jumeaux Cx , Cy de la même taille > \|Γ\|/2, nous choisissons ′ σ ∈ Alt(Γ) tel que Cx = (Cy )σ . (Nous supposons m > 1.) En remplaçant y par yσ , où σ ′ = φ−1 (σ −1 ), nous réduisons notre problème au cas Cx = Cy . (Voilà l’exemple le plus simple de ce que Babai appelle aligner ; nous avons aligné Cx et Cy .) Alors, soit C = Cx = Cy . La partition {C, Γ \ C} de Γ induit une partition {Ωj }0≤j≤k de Ω : ω ∈ Ωj ssi ψ(ω) contient k − j éléments de C et j éléments de Γ \ C. Il est aisé de montrer que αk−j (1 − α)j kj < 1/2 pour α ∈ (1/2, 1], 1 ≤ j ≤ k ; donc, \|Ωj \| < n/2 pour 1 ≤ j ≤ k. Nous avons réduit notre problème à celui de déterminer IsoH (x, y), où H = φ (Alt(Γ)C ). Ici le besoin de prendre un stabilisateur d’ensemble (à savoir, Alt(Γ)C ) ne pose aucun souci : nous engendrons H en prenant des préimages φ−1 (h1 ), . . . , φ−1(h5 ) de deux générateurs h1 , h2 de Alt(C) < Alt(Γ), deux générateurs h3 , h4 de Alt(Γ \ C) < Alt(Γ) et un élément h5 ∈ Alt(Γ) de la forme (γ1 γ2 )(γ3 γ4 ), où γ1 , γ2 ∈ C, γ3 , γ4 ∈ Γ \ C. (Si \|Γ\| < 8, le nombre de générateurs est moindre, et la discussion se simpliﬁe.) Notre problème se réduit à celui de déterminer IsoH ′ (x, y′ ) pour y′ = y et y′ = yh5 , où H ′ = φ−1 (Alt(C) × Alt(Γ \ C)) = φ−1 (hh1 , . . . , h4 i). −1 Comme C est une classe de jumeaux pour x, tout élément de φ−1 (Alt(C)) laisse x invariant. Si x\|Ω0 6= y\|Ω0 , alors IsoH ′ (x, y) = ∅. Soit alors x\|Ω0 = y\|Ω0 . Nous avons réduit notre problème à celui de déterminer IsoH ′ \|Ω′ (x\|Ω′ , y\|Ω′ ), où Ω′ = Ω \ Ω0 . Rappelons que H ′ \|Ω′ agit sur Ω′ avec des orbites de longueur \|Ωi \| < n/2. Nous procédons donc comme dans le cas non transitif de la méthode de Luks (preuve du Thm. 2.4). 1125–22 5. DES CHAÎNES AUX SCHÉMAS DE JOHNSON x, y → relations k-aires sur Γ Weisfeiler - Leman k-aire Lemme des designs une couleur domine ? oui coupe ou Johnson ? non récursion n′ ≤ n/2 Discutons maintenant le cas de G primitif et, plus précisément, G isomorphe à Alt(k) m , k ≥ 2. Maintenant nous pouvons supposer que nos chaı̂nes x, y n’ont pas de classes de jumeaux de taille > m/2. Les outils principaux que nous développerons (Lemme des designs, coupe-ou-Johnson) nous seront utiles, voire essentiels, aussi dans le cas de G imprimitif. Nous avons une bijection entre les éléments de Ω et {S ⊂ Γ : \|S\| = k}. Pour x : Ω → Σ donné, nous avons donc une structure relationnelle X = (Γ, (Ri )i∈Σ ) k-aire sur Γ : (x1 , . . . , xk ) ∈ Ri si x1 , . . . , xk sont tous diﬀérents et x(ω) = i, où ω est l’élément de Ω qui correspond à {x1 , . . . , xk }. Nous appliquons à X le foncteur F1 (§2.3), qui fait d’elle une structure de partition, puis le foncteur F2 (encore §2.3), qui nous donne une conﬁguration k-aire, et, ﬁnalement, le foncteur F3 déﬁni par Weisfeiler-Leman k-aire (§2.5). Nous obtenons ainsi un raﬃnement F3 (F2 (F1 (X))) = (Γ, cx : Ωk → C ) qui est une conﬁguration cohérente k-aire. Comme F1 , F2 , F3 sont des foncteurs, l’assignation de cx à x est canonique. Elle nous sera donc utile : si cx et cy ne sont pas isomorphes sous l’action de Altm , alors x et y ne sont pas isomorphes sous l’action de Alt(k) m non plus. Nous obtiendrons une conﬁguration cohérente classique de façon canonique à partir de cx (Lemme des designs). Soit cette nouvelle conﬁguration sera non triviale, soit nous obtiendrons un coloriage canonique sans couleur dominante, ce qui nous permettra immédiatement de réduire le problème à un certain nombre de problèmes pour des chaı̂nes plus courtes, comme dans l’algorithme de Luks. Supposons, alors, que nous disposons d’une conﬁguration cohérente classique non triviale assignée de façon canonique à x. La procédure Coupe-ou-Johnson nous donnera l’un ou l’autre de ces deux résultats : soit un découpage canonique de Γ, soit un schéma de Johnson plongé de façon canonique dans Γ. Dans un cas comme dans l’autre, avoir une telle structure canonique limite fortement l’ensemble d’isomorphismes et automorphismes possibles. Nous pourrons réduire G à un sous-groupe ∼ Altm′ , avec m′ ≤ m/2, √ dans le cas du découpage, ou m′ ≪ m, dans le cas de Johnson. Déjà m′ ≤ m/2 est suﬃsante pour une récursion réussie. 1125–23 5.1. Lemme des designs Étant donnés une conﬁguration X = (Γ, c : Γk → C ) et un paramètre 1/2 ≤ α < 1, une couleur i est dite α-dominante si c(γ, . . . , γ) = i pour ≥ α\|Γ\| valeurs de γ ∈ Γ. La classe de couleurs {γ ∈ Γ : c(γ, . . . , γ) = i} est, elle aussi, dite dominante. Par contre, si, pour toute couleur i, la classe {γ ∈ Γ : c(γ, . . . , γ) = i} est de taille < α\|Γ\|, le coloriage est dit un α-coloriage. Comme avant, deux éléments γ1 , γ2 ∈ Γ sont des jumeaux par rapport à une structure X (ici, une conﬁguration cohérente sur Γ) si (γ1 γ2 ) ∈ Aut(X). Proposition 5.1 (Lemme des designs). — Soit X = (Γ, c : Γk → C ) une conﬁguration cohérente k-aire, où 2 ≤ k ≤ \|Γ\|/2. Soit 1/2 ≤ α < 1. Supposons qu’il n’y a aucune classe de jumeaux dans Γ avec > α\|Γ\| éléments. Alors, au moins une des options suivantes est vraie : (1) (a) il existe x1 , . . . , xℓ ∈ Γ, 0 ≤ ℓ < k, tels que X~x n’a pas de couleur α-dominante ; (1) (b) il existe x1 , . . . , xℓ ∈ Γ, 0 ≤ ℓ < k − 1, tels que X~x a une couleur α-dominante C et (X~x )(2) [C] n’est pas une clique. La notation a été déﬁnie dans les sections 2.3 – 2.4 . En particulier, le 1-squelette est tout simplement un coloriage de Γ. (1) X~x Lemme 5.2 (Lemme de la grande clique). — Soit X = (Γ, c) une conﬁguration cohérente classique. Soit C ⊂ Γ une classe de couleurs avec \|C\| ≥ \|Γ\|/2. Si X[C] est une clique, alors C est une classe de jumeaux. Preuve — Supposons que C n’est pas une classe de jumeaux. Il y a donc un x ∈ Γ et une couleur (disons, azur) telle que c(x, y) est de cette couleur pour au moins un y ∈ C mais pas pour tous. Comme X[C] est une clique, x ∈ / C. Appelons la couleur de C carmin, et celle de x bronze. Soit B ⊂ Γ l’ensemble des éléments de couleur bronze. Il s’agit de construire un block design équilibré (§2.6) qui contredise l’inégalité de Fisher (Prop. 2.19). Déﬁnissons Ab = {y ∈ Γ : c(by) = azur} pour b ∈ B. Comme x ∈ B et c(xy) = azur pour au moins un y ∈ C, et c(xy) connaı̂t la couleur de y, tous les éléments de Ab sont carmin. Par la cohérence de X et la déﬁnition des nombres d’intersection (Def. 2.8), \|Ab \| = γ(azur, azur−1 , bronze), et donc \|Ab \| ne dépend pas de b. Comme nous l’avons dit au début, 1 ≤ \|Ax \| < \|C\| ; donc, 1 ≤ \|Ab \| < C pour tout b ∈ B. Montrez de façon similaire que, pour v ∈ C, la taille de {b ∈ B : v ∈ Ab } = {b ∈ B : c(bv) = azur} ne dépend pas de b. Comme X[C] est une clique, c(v, v ′) est de la même couleur pour tous v, v ′ ∈ C, v = v ′ ; appelons cette couleur doré. Montrez que {b ∈ B : v, v ′ ∈ Ab } = γ(azur, azur−1 , doré). Alors (C, {Ab}b∈B ) est un block design équilibré incomplet. 1125–24 En conséquence, par l’inégalité de Fisher, \|B\| ≥ \|C\|. Or, nous savons que \|C\| > \|Γ\|/2, B, C ⊂ Γ et B ∩ C = ∅. Contradiction. Preuve du Lemme des designs (Prop. 5.1) — Supposons que pour chaque ~x ∈ Ωℓ , 0 ≤ ℓ < k, C~x a une couleur α-dominante C(~x), et, en plus, si ℓ < k − 1, (X~x )(2) [C] est une clique. Nous arriverons à une contradiction. Soit C = C(vide). Comme \|C\| > α\|Γ\|, C est trop grande pour être un ensemble de jumeaux. Donc il existe u, v ∈ C, u 6= v, tels que τ = (uv) ∈ / Aut(X). Soit ~y de longueur τ minimale r entre les chaı̂nes satisfaisant c(~y ) 6= c(~y ). Par cette minimalité et les règles dans la déﬁnition 2.5, y1 , . . . yr sont tous distincts. En les permutant, nous pouvons assurer que u, v ∈ / {y1 , y2 , . . . , yr−2 }, et, sans perte de généralité, que soit (i) yr−1 6= u, v, yr = u, soit (ii) yr−1 = u, yr = v. Dans le cas (i), nous choisissons ~x = y1 , . . . , yr−1 , ℓ = r −1, et voyons que c~x (u) 6= c~x (v) ; dans le cas (ii), nous choisissons ~x = y1 , . . . , yr−2 , ℓ = r−2, et obtenons c~x (u, v) 6= cx (v, u). Nous aurons donc une contradiction avec notre supposition une fois que nous aurons prouvé que u, v ∈ C(~x). Le fait que u, v ∈ C(~x) s’ensuivra immédiatement de l’égalité C(~x) = C \{x1 , . . . , xℓ } ; cette égalité, à son tour, se déduit par itération du fait que, pour ~y de longueur ≤ k − 2 et ~x = ~y z, z ∈ Ω, (6) C(~x) = C(~y ) \ {z}. (2) Pourquoi (6) est-il vrai ? Nous sommes en train de supposer que X~y [C(~y )] est une clique, et que \|C(~y )\| > α\|Γ\| ≥ \|Γ\|/2. Donc, par le lemme de la grande clique, tous (2) les éléments de C(~y ) sont des jumeaux en X~y . En particulier, pour u ∈ C(~y ) \ {z}, c~x (u) = c~y (zu) ne dépend pas de u. Puisque le coloriage de sommets en C~x est un raﬃnement de celui en C~y (par la deuxième règle de la déﬁnition 2.5), il s’ensuit que, soit C(~x) = C(~y ) \ {z}, soit C(~x) ⊂ Γ \ C(~y ), soit C(~x) = {z}. Comme \|C(~x)\|, \|C(~y)\| > α\|Γ\| ≥ \|Γ\|/2, les deux dernières possibilités sont exclues. Nous appliquons le Lemme des designs (avec α = 1/2) à la conﬁguration cohérente k-aire X′ = F3 (F2 (F1 (X))), où X est donnée par x de la façon décrite au début de la section. Nous parcourons tous les tuples possibles ~x = (x1 , . . . , xℓ ) ∈ Γℓ , 0 ≤ ℓ < k, jusqu’à trouver un tuple pour lequel la première ou la deuxième conclusion du Lemme des designs est vraie. (1) Si la première conclusion est vraie, nous déﬁnissons cX = X~x et sautons à la section 5.3.1. Si la deuxième conclusion est vraie, nous passons au §5.2, ayant déﬁni X′′ = (2) (1) X~x [C], où C est la couleur α-dominante de X~x . 5.2. Coupe ou Johnson Nous avons une conﬁguration classique cohérente homogène non triviale X′′ = (Γ, c). (Nous rappelons que ceci est un coloriage c du graphe complet sur Γ tel que (a) les sommets ont leur couleur propre (« couleur diagonale »), (b) les arêtes (x, y), x 6= y, ne sont pas toutes de la même couleur, (c) la couleur c(x, y) de l’arête (x, y) détermine c(y, x), et (d) l’axiome de cohérence (2.8) se vériﬁe.) Nous voudrions trouver des structures qui dépendent canoniquement de X′′ et qui contraignent son groupe d’automorphismes. 1125–25 Il est raisonnable de s’attendre à ce que de telles structures existent : par le Théorème 3.1, si le groupe d’automorphismes est transitif, soit il est imprimitif (et donc il laisse une partition invariante), soit il est près d’être Alt(k) m , k ≥ 2, (qui laisse invariant un schéma de Johnson), soit il est petit (et donc le stabilisateur de quelques points aura des orbites petites, et ainsi nous donnera un coloriage sans couleur dominante). Le déﬁ est de trouver de telles structures, et de le faire canoniquement. Si X′′ n’est pas primitif (Déf. 2.14), la tâche est plutôt facile : soit r la couleur non diagonale la plus rouge telle que le graphe Gr = {(x, y) : x, y ∈ Γ, c(x, y) = r} est connexe ; par l’exercice 2.16, ceci donne une partition de Γ dans des ensembles de la même taille ≤ \|Γ\|/2. Théorème 5.3 (Coupe ou Johnson). — Soit X = (Γ, c) une conﬁguration classique cohérente uniprimitive. Soit 2/3 ≤ α < 1. En temps \|Γ\|O(1) , nous pouvons trouver — soit un α-découpage de Γ, — soit un schéma de Johnson plongé sur Γ0 ⊂ Γ, \|Γ0 \| ≥ α\|Γ\|, et un sous-groupe H < Sym(Γ) avec [Sym(Γ) : H] = \|Γ\|O(log \|Γ\|) tel que le découpage, voire le schéma, est canonique en relation à H. Le groupe H sera déﬁni comme un stabilisateur de points en Γ. La valeur 2/3 dans l’énoncé est assez arbitraire ; toute valeur > 1/2 serait valable. Une valeur proche à 1/2 aﬀecterait les constantes implicites. Preuve — Choisissons un x ∈ Γ arbitraire. Donnons à chaque y ∈ Γ la couleur de c(x, y). Ce coloriage est canonique en relation à Gx . S’il n’y a aucune classe de couleur C de taille > α\|Γ\|, la partition triviale (non-partition) de chaque classe nous donne un α-découpage de Γ, et nous avons ﬁni. Supposons, par contre, qu’il y ait une classe de couleur – disons, Clin – de taille > α\|Γ\|. Comme αn > n/2, la relation Rlin de cette couleur est non orientée (c(y, z) = lin ssi c(z, y) = lin). Le complément de Rlin (ou de toute autre relation) est de diamètre 2 (exercice). Soient x, z ∈ Γ tels que c(x, z) = lin, et soit y ∈ Γ tel que c(x, y), c(z, y) 6= lin. Appelons c(x, y) bis et c(z, y) terre. Considérons le graphe biparti (V1 , V2 ; A) avec sommets V1 = Clin, V2 = Cbis et arêtes Rterre ∩ (V1 × V2 ). Le graphe est non vide par déﬁnition et semirégulier par l’exercice 2.17b. Par homogénéité et cohérence, le nombre de y tels que c(y, w) est d’une couleur donnée c0 est indépendant de w. Donc, il est toujours ≤ (1 − α)n < n/2 pour c0 6= lin. Appliquant ceci à c0 = terre et V2 , nous voyons que le degré \|{v1 ∈ V1 : (v1 , v2 ) ∈ A}\| est < n/2, et donc, comme \|V1 \| > n/2, le graphe n’est pas complet. Nous appliquons donc la Proposition 5.7 à (V1 , V2 ; A) avec β = α\|Γ\|/\|V1\|. Notons que \|V2 \| ≤ β\|V1 \|. Nous travaillerons donc avec un graphe biparti (V1 , V2 ; A). La stratégie sera d’essayer, soit de rendre V2 plus petit (par au moins un facteur constant), soit de trouver des structures en lui. Soit ces structures nous permettront de réduire V2 quand même, soit 1125–26 elles nous aideront à découper V1 , ou à trouver un schéma de Johnson assez grand sur V1 . Tout d’abord, nous devrons borner la symétrie en V1 , c’est-à-dire réduire, voire éliminer les jumeaux. Il y a deux raisons à ceci. — Même si nous découvrions une structure assez riche en V2 , cela impliquerait peu ou rien sur V1 si beaucoup d’éléments de V1 se connectent à V2 de la même façon. — Si V2 est petit, nous colorierons chaque sommet de V1 par son ensemble de voisins en V2 . Ceci nous donnera un coloriage canonique en relation à G(V2 ) . Or, dans ce coloriage, deux sommets en V1 auront la même couleur ssi ils sont des jumeaux ; donc, si aucune classe de jumeaux en V1 n’a > α\|V1 \| éléments, nous aurons un α-coloriage. Exercice 5.4. — Soit (V1 , V2 ; A) un graphe biparti semirégulier et non trivial. Alors, aucune classe de jumeaux en V1 n’a plus de \|V1 \|/2 éléments. Solution — Nous assurons que \|A\| ≤ \|V1 \|\|V2 \|/2 en prenant le complément s’il est nécessaire. Soient d2 le degré des sommets en V2 et S une classe de jumeaux en V1 . Montrez que d2 ≥ \|S\|, et donc \|A\| ≥ \|S\|\|V2\|. Exercice 5.5. — Soit (V1 , V2 ; A) un graphe biparti sans jumeaux en V1 . Soient V2 = C1 ∪ C2 , C1 ∪ C2 = ∅. Montrez que, pour au moins un i = 1, 2, il n’y a aucune classe de ≥ \|V1 \|/2 + 1 jumeaux en V1 dans le graphe (V1 , Ci ; A ∩ (V1 × Ci )). Exercice 5.6. — Soit X = (Γ, c) une conﬁguration cohérente. Soient C1 , C2 deux classes de couleurs en Γ. Soit brun une couleur d’arêtes en A × B. Alors, pour x, y ∈ C1 , la couleur c(x, y) détermine si x et y sont des jumeaux dans le graphe biparti (C1 , C2 ; Gbrun ). Proposition 5.7 (Coupe ou Johnson biparti, ou « Una partita a poker ») Soit X = (V1 , V2 ; A) un graphe biparti avec \|V2 \| < β\|V1 \|, où 2/3 ≤ β < 1, et tel qu’aucune classe de jumeaux en V1 n’ait plus de 2\|V1\|/3 éléments. Alors, nous pouvons trouver, en temps \|V1 \|O(1) , — soit un β-découpage de V1 , — soit un schéma de Johnson plongé sur V0 ⊂ V1 , \|V0 \| ≥ β\|V1\|, et un sous-groupe H < G, G = Sym(V1 ) × Sym(V2 ), avec [G : H] = \|V1 \|O(log \|V1 \|) tel que le découpage, voire le schéma, est canonique en relation à H. La condition sur les classes de jumeaux ici était remplie (même avec 1/2 à la place de 2/3) à la ﬁn de la preuve du Thm. 5.3, grâce à l’exercice 5.4. En ce qui concerne le temps de la procédure, nous expliciterons quelques détails qui pourraient ne pas être évidents. Ce qui sera le détail le plus délicat est l’indice [G : H]. Le groupe H sera déﬁni comme un stabilisateur de points ; nous devons bien contrôler le nombre de points que nous stabilisons. 1125–27 Esquissons la stratégie générale de la preuve. Ce que nous voulons est une réduction à la Proposition 5.8, “Coupe-ou-Johnson cohérent”. Nous pouvons produire une conﬁguration cohérente classique sur V1 ∪V2 à partir du graphe X, tout simplement en utilisant Weisfeiler-Leman. Ce qui demande de la ruse est de garantir que la restriction X[C2 ] à la classe de couleurs dominante (s’il y a une) soit non triviale. Pour obtenir une conﬁguration non-triviale sur C2 , nous noterons que le graphe X induit lui-même une relation d-aire sur C2 , où d est au plus le degré de la majorité d’éléments de V1 (si telle chose existe ; sinon, les degrés nous donnent une partition de V1 ). Si la relation est triviale, dans le sens de contenir toutes les d-tuples d’éléments distincts dans C2 , nous obtenons un schéma de Johnson. Si elle est non triviale mais contient beaucoup de jumeaux, elle nous donne une manière de descendre à un C2 plus petit. S’il n’y a pas beaucoup de jumeaux, nous utilisons le Lemme des Designs (supplémenté par un lemme standard sur les designs) pour obtenir une conﬁguration cohérente classique non-triviale sur C2 , ce qui était à trouver. Preuve — Si \|V1 \| ≤ c, où c est une constante, nous colorions chaque v ∈ V1 par lui-même. Ce coloriage est canonique en relation à H = {e} ; autrement dit, il n’est pas canonique du tout. Peu importe : trivialement, \|G\| ≤ (c!)2 ≤ \|V1 \|O(log \|V1 \|) . Nous pouvons donc supposer que \|V1 \| > c. Si \|V2 \| ≤ (6 log \|V1 \|)3/2 (disons), alors, par la discussion ci-dessus, nous obtenons un (2/3)-coloriage de V1 (et donc : un (2/3)-découpage de V1 ). Ce coloriage est canonique en relation à un H d’indice 3 3 \|V2 \|! ≤ \|V2 \|\|V2 \| ≤ (6 log \|V1 \|) 2 (6 log \|V1 \|) 2 ≪ \|V1 \|log \|V1 \| . Nous pouvons donc supposer que \|V2 \| > (6 log \|V1 \|)3/2 . Notre première tâche est d’éliminer les jumeaux. Nous divisons V1 dans ses classes de jumeaux et colorions chaque v ∈ V1 par son nombre de jumeaux et par son degré dans le graphe (V1 , V2 ; A). Nous obtenons un β-découpage de V1 , sauf s’il y a un entier d1 tel que l’ensemble V1′ des sommets v sans jumeaux et de degré d1 est de taille \|V1′ \| > β\|V1\|. Supposons dorénavant que cela est le cas. Comme \|V1′ \| > \|V2 \| et qu’il n’y a pas de jumeaux en V1′ , nous voyons que 1 < d1 < \|V2 \| − 1 ; nous pouvons supposer que d1 ≤ \|V2 \|/2 en remplaçant A par son complément, si nécessaire. Soit H = (V2 , A ) l’hypergraphe dont les arêtes sont les voisinages en (V1 , V2 ; A) des sommets dans V1′ . (Elles sont toutes contenues en V2 .) L’hypergraphe est d1 -uniforme. Comme il n’y a pas de jumeaux dans V1′ , il n’y a pas d’arêtes identiques. Si H est l’hypergraphe complet d1 -uniforme, alors V1′ peut être identiﬁé avec le schéma de Johnson Sd1 (V2 ). (Scoppia in un pianto angoscioso e abbraccia la testa di Johnson.) Supposons alors que H n’est pas complet. Nous voudrions avoir un coloriage canonique sur V2d pour un d ≪ l, l = (log \|V1′ \|)/ log \|V2 \|, tel que les éléments de V2 ne soient pas tous jumeaux. Si d1 ≤ 6⌈l⌉, nous déﬁnissons d = d1 et colorions {(v1 , . . . , vd ) ∈ V2d : {v1 , . . . , vd } ∈ H } en écarlate, et tout le reste en gris. Supposons, par contre, que d1 > 6⌈l⌉. Soit d = 6⌈l⌉. Nous colorions ~v = (v1 , . . . , vd ) en gris si les vi ne sont pas tous distincts ; dans le cas contraire, nous donnons à ~v la 1125–28 couleur (7) \|{H ∈ H : {v1 , . . . , vd } ⊂ H}\|. Cette opération de coloriage peut être faite en temps de l’ordre de log \|V1′ \| 6 d1 ≤ \|V1 \| · \|V2 \|d = \|V1 \| · \|V2 \| log \|V2 \| = \|V1 \| · \|V1′ \|O(1) = \|V1 \|O(1) . \|V1 \| · d Si les tuples avec v1 , . . . , vd distincts n’avaient pas tous la même couleur λ, nous aurions un design d − (\|V2 \|, d1, λ) avec \|V1′ \| arêtes. Donc, par la Proposition 2.20, \|V1′ \| ≥ \|V2 \| pour s = 3⌈l⌉. Comme \|V2 \| ≥ (6 log \|V1′ \|)3/2 et \|V1′ \| peut être supposé plus grand s qu’une constante, s s log \|V1′ \| \|V2 \| \|V2 \| \|V2 \| 1/3 3l log \|V2 \| ≥ > > \|V2 \| = \|V2 \| = \|V1′ \|, s s 6l ce qui donne une contradiction. Donc, pour d1 arbitraire, les tuples avec v1 , . . . , vd distincts n’ont pas tous la même couleur ; en d’autres termes, les éléments de V2 ne sont pas tous jumeaux en relation à notre nouvelle structure d-aire. S’il y a une classe S de jumeaux de taille > \|V2 \|/2, alors, par l’exercice 5.5, au moins un des deux graphes (V1′ , S; A ∩ (V1′ × S)), (V1′ , V2 \ S; A ∩ (V1′ × (V2 \ S))) n’a aucune classe de > \|V1′ \|/2 + 1 jumeaux dans V1 . Comme 2\|V1′ \|/3 ≥ \|V1′ \|/2 + 1 pour \|V1 \| ≥ 8, nous appliquons la Proposition 5.7 elle-même à un de ces deux graphes (disons, celui sur V1′ × S si les deux sont valables), et terminons. (Peut-être que la taille de V2 est descendue seulement à \|V2 \| − 1, mais tous nos choix ont été canoniques – des non-choix, si l’on veut – donc gratuits. Nous n’avons perdu que du temps ; pour être précis, \|V1\|O(1) de temps, ce qui est acceptable.) Alors, nous avons un coloriage de V2d en relation auquel il n’y a aucune classe de jumeaux en V2 de taille > \|V2 \|/2. Nous appliquons les foncteurs F1 , F2 , F3 (WeisfeilerLeman) à ce coloriage. Puis nous utilisons le Lemme des designs (Prop. 5.1) avec α = 2/3. Nous trouvons les éléments x1 , . . . , xℓ ∈ V2 (ℓ = d − 1 ou ℓ = d − 2) dans l’énoncé de la Proposition 5.1 par force brute, en temps proportionnel à \|V2 \|d = \|V1 \|O(1) . Nous les ﬁxons, et nous imposons que H ﬁxe x1 , . . . , xℓ , ce qui a un coût de \|V1 \|O(1) , dans le sens où [G : Gx1 ,...,xℓ ] ≤ \|V2 \|d = \|V1 \|O(1) . Si nous sommes dans le premier cas du Lemme de designs (pas de couleur dominante), nous cueillons les classes de couleur, en commençant par la plus rouge (interprétez la quantité en (7) comme une longueur d’onde), jusqu’à avoir une union des classes S ⊂ V2 avec \|V2 \|/3 < \|S\| ≤ 2\|V2 \|/3. (Ceci marche s’il n’y a aucune classe de taille > \|V2 \|/3 ; si telles classes existent, nous déﬁnissons S comme la classe la plus grande de ce type.) Nous appliquons l’exercice 5.5, et obtenons un graphe (V1′ , V2′ , A ∩ (V1′ ∩ V2′ )) remplissant les conditions de notre Proposition 5.7 avec V2′ = S ou V2′ = V2 \ S, et donc \|V2′ \| ≤ α\|V2\|. Donc, nous appliquons la Proposition 5.7 à ce graphe ; la récursion marche. (Il est important ici que \|V2′ \| ≤ α\|V2\|, puisque nous avons déjà encouru un coût considérable (\|V1 \|O(1) ) dans l’indice.) 1125–29 Restons donc dans le deuxième cas du Lemme des designs : nous avons un coloriage de V2 avec une classe de couleurs C ⊂ V2 telle que \|C\| ≥ 2\|V2\|/3, et une conﬁguration cohérente homogène classique Y non triviale sur C. Nous déﬁnissons un graphe avec des sommets V1′ ∪ V2 , où V1′ est de couleur nacrée et V2 vient d’être colorié par le Lemme des Designs ; les arêtes seront non pas seulement celles en A ∩ (V1′ × V2 ) (coloriées en noir) mais aussi les arêtes entre les éléments de V2 , dans les couleurs données par Y. Nous appliquons les raﬃnements F1 , F2 et F3 (Weisfeiler-Leman) à ce graphe, et obtenons une conﬁguration cohérente X. La conﬁguration X[V2 ] est un raﬃnement de Y. Si elle n’a pas de couleur α-dominante, nous réduisons notre problème à celui pour (V1′ , V2′ , A ∩ (V1′ ∩ V2′ )), \|V2′ \| ≤ α\|V2 \|, comme avant ; nous pouvons appliquer la proposition 5.7 à un tel graphe sans changer β parce que 2 \|V2′ \| ≤ \|V2 \| ≤ β\|V2\| < β\|V1′ \|. 3 La récursion marche ici aussi parce que \|V2′ \| ≤ 2\|V2 \|/3 : il est important que V2′ soit plus petit que V2 par un facteur constant, puisque le coût entraı̂né jusqu’à maintenant dans l’index [G : H] est déjà considérable (\|V1\|O(1) ). Supposons donc qu’il y a une classe de couleurs (2/3)-dominantes C2 dans X[V2 ]. Elle doit être un sous-ensemble de C car 2/3 + 2/3 > 1. La restriction de X[C2 ] n’est pas une clique : si elle l’était, la restriction de Y à C2 l’aurait été aussi, et cela est impossible par l’exercice 2.15. Nous pouvons supposer qu’il existe une classe de couleurs C1 ⊂ V1′ en X1 qui satisfait \|C1 \| > β\|V1 \| ; sinon, nous avons un β-coloriage de V1 , et pouvons ﬁnir. Le fait que \|C1 \| > β\|V1\| implique que \|C1 \| > \|V2 \| ≥ \|C2 \|. Nous pouvons supposer aussi que les arêtes de X en C1 × C2 ne sont pas toutes de la même couleur. Si elles l’étaient, il y aurait une classe de ≥ \|C1 \| > β\|V1\| > \|V1 \|/2 + 1 jumeaux en V1 dans le graphe (V1 , C2; A∩(V1 ×C2 )), dont X[V1 ×C2 ] est un raﬃnement. Dans ce cas, par l’exercice 5.5, nous aurions une réduction à (V1 , V2 \ C2 ; A ∩ (V1 × (V2 \ C2 ))), et nous pourrions ﬁnir en utilisant la Proposition 5.7 de façon récursive. Ainsi, nous avons tout réduit à la Proposition 5.8 : nous l’appliquons à X[C1 ∪ C2 ]. Nous obtenons, soit un (1/2)-découpage de C1 , soit un graphe biparti (W1 , W2 ; A′ ), W1 ⊂ C1 , W2 ⊂ C2 , avec \|W1 \| ≥ \|C1 \|/2, \|W2 \| ≤ \|C2 \|/2 ≤ \|V2 \|/2, tel qu’aucune classe de jumeaux en W1 n’a plus que \|W1 \|/2 éléments. Nous pouvons supposer que \|W1 \| > β\|V1 \|, parce que, dans le cas contraire, nous avons obtenu un β-découpage de W1 . Alors, \|W2 \| < \|W1 \|/2. Nous pouvons, alors, faire de la récursion : nous appliquons la Proposition 5.7 avec (W1 , W2 ; A′ ) à la place de (V1 , V2 ; A). La récursion se ﬁnit après pas plus que O(log \|V2 \|) pas puisque \|W2 \| ≤ \|V2 \|/2. Si la taille de W1 (ou de V1 ) décroı̂t en dessous de β\|V1\| (pour la valeur originale de \|V1 \|), alors nous avons obtenu un β-découpage de V1 . Comme nous l’avons vu, Coupe ou Johnson biparti utilise Coupe ou Johnson cohérent. À son tour, Coupe ou Johnson cohérent se réduira à Coupe ou Johnson biparti pour un graphe biparti (V1 , V2 ; A) avec V2 de taille au plus une moitié de la taille du V2 original. 1125–30 Proposition 5.8 (Coupe ou Johnson cohérent). — Soit X = (C1 ∪ C2 ; c) une conﬁguration cohérente avec des classes de couleurs de sommets C1 , C2 , où \|C1 \| > \|C2 \|. Supposons que ni c\|C1 ×C2 ni c\|C2 ×C2 est une fonction constante. Alors, nous pouvons trouver, en temps \|C1 \|O(1) , soit — un (1/2)-découpage de C1 , ou — un graphe biparti (V1 , V2 ; A), Vi ⊂ Ci , \|V1 \| ≥ \|C1 \|/2, \|V2 \| ≤ \|C2 \|/2, tel que toute classe de jumeaux en V1 contient au plus \|V1 \|/2 éléments, et un élément y ∈ C2 , tel que le découpage, voire le graphe biparti, est canonique en relation à Gy , où G = Sym(C1 ) × Sym(C2 ). Il va de soi que dire que c\|C2 ×C2 est constant équivaut à dire que X[C2 ] est une clique. Preuve — Si la restriction X[C1 ] était une clique, alors, par cohérence, pour toute couleur en C1 × C2 – pourpre, disons – les voisinages dans (C1 , V2 ; Gpourpre ) des sommets en C2 nous donneraient un block design équilibré (et peut-être dégénéré) sur C1 . Le design est incomplet parce que c n’est pas monochrome sur C1 × C2 . L’inégalité de Fisher nous donne que \|C2 \| ≥ \|C1 \|, en contradiction avec nos suppositions. Donc, X1 [C1 ] n’est pas une clique. Si X[C1 ] n’est pas primitive, la plus rouge de ses relations non connexes nous donne un (1/2)-découpage canonique de V1 , par l’exercice 2.16. Nous pouvons donc supposer que X[C1 ] est primitif. Nous avons deux cas à considérer : X[C2 ] primitive et X[C2] imprimitive. Supposons d’abord que X[C2 ] est imprimitive. La relation non connexe la plus rouge dans X[C2 ] nous donne une partition de C2 dans des ensembles B1 , . . . , Bm , m ≥ 2, tous de la même taille ≥ 2. Nous avons donc trouvé une structure en C2 , et nous l’utiliserons, soit pour découper C1 , soit pour réduire \|C2\| par un facteur constant. Le premier pas consiste à montrer qu’il n’y a pas de jumeaux dans C1 . Comme notre conﬁguration est cohérente, la couleur d’une arête en C1 sait si ses sommets sont des jumeaux en relation à C2 (Ex. 5.6) ; donc, s’il y avait des jumeaux dans C1 en relation à C2 , nous aurions, soit qu’une des couleurs d’arêtes en C1 donne une relation non connexe – ce qui contredit le fait que X[C1 ] est uniprimitive – soit que tous les éléments de C1 sont des jumeaux en relation en C2 . Dans ce dernier cas, par l’exercice 5.4, c\|C1 ×C2 serait monochrome, ce qui n’est pas le cas. En conclusion, il n’y a pas de jumeaux dans C1 en relation à C2 . Notre intention est d’appliquer l’exercice 2.18 pour obtenir un graphe biparti contracté C1 × {1, 2, . . . , m} avec m ≤ \|C2 \|/2. Nous devons seulement faire attention à ce que ce graphe ne soit pas trivial. Soit dk le degré de tout w ∈ C2 dans le graphe biparti (C1 , C2; Gk ) pour une couleur k donnée, où Gk consiste en les arêtes de couleur k. (Par l’ex. 2.17a, le degré dk ne dépend pas de w.) Si dk ≤ \|C1 \|/2 pour tout k, nous ﬁxons un w ∈ C2 (non canonique) et obtenons un (1/2)-coloriage de C1 en assignant la couleur c(x, w) au sommet x ∈ C1 . Supposons donc qu’il y a une couleur – que nous appellerons violet – telle que 1125–31 dviolet > \|C1 \|/2. S’il y a un 1 ≤ i ≤ m tel qu’il n’y a aucune classe de plus que \|C1 \|/2 jumeaux dans C1 en relation à Bi , nous ﬁxons un élément y ∈ Bi d’un tel i (non canoniquement), ﬁxant ainsi cet i. De cette façon, nous obtenons une réduction au graphe biparti (C1 , Bi ; Gviolet ∩ (C1 × Bi )). Supposons que cela n’est pas le cas. Donc, pour chaque i, il existe une classe Ti ⊂ C1 de jumeaux en relation à Bi telle que \|Ti \| > \|C1 \|/2. Pour chaque w ∈ Bi , les arêtes de w à tout v ∈ Ti sont de la même couleur ; alors, elles doivent être violettes. Soit vert une couleur d’arêtes en C1 × C2 qui ne soit pas violet. Alors, le graphe X = (C1 , {1, . . . , m}; D) dans l’exercice 2.18 n’est pas vide ; comme (vi , i) est violet pour tout v ∈ Ti , X n’est pas complet non plus. Comme X est birégulier, il n’y a aucune classe de jumeaux en C1 en relation à {1, . . . , m} avec > \|C1 \|/2 éléments (ex. 5.4). Nous avons donc tout réduit à un graphe biparti X du type que nous désirions. Considérons maintenant le cas de X[C2 ] primitive (16) . Fixons un y ∈ C2 arbitraire (non canoniquement). Nous pouvons supposer qu’il y a une couleur – disons, violet – telle que dviolet > \|C1 \|/2, puisque, sinon, les couleurs des arêtes qui connectent les éléments de C1 avec y nous donneraient un (1/2)-coloriage de C1 . Écrivons V1 = Lviolet (y) = {x ∈ C1 : c(x, y) = violet}. Donc \|V1 \| > \|C1 \|/2. Soit bleu une couleur d’arêtes en X[C2 ] telle que le degré de Gbleu est (positif et) < \|C2 \|/2 ; une telle couleur existe parce que X[C2 ] n’est pas une clique. (S’il y a plusieurs couleurs comme cela, nous choisissons la plus bleue d’entre elles.) Alors, V2 = Lbleu (y) ⊂ C2 satisfait 1 ≤ \|V2 \| < \|C2 \|/2. Le graphe biparti (V1 , V2 ; Gviolet ∩ (W × U)) est semirégulier par l’exercice 2.17b. Il est non vide parce que, pour tout u ∈ V2 , \|Lviolet (u)\| > \|C1 \|/2, et donc Lviolet (u) ∩ V1 6= ∅. S’il était complet, nous aurions V1 ⊂ Lviolet (u) pour tout u ∈ V2 ; comme \|V1 \| = \|Lviolet (y)\| = \|Lviolet (u)\|, ceci impliquerait que V1 = Lviolet (u). Or, cela voudrait dire que y et u sont des jumeaux dans le graphe (C1 , C2 ; Gviolet ). Par le même argument qu’avant (basé sur l’exercice 5.6), la primitivité de X[C2 ] et le fait que c\|C1 ×C2 ne soit pas monochrome impliquent qu’il n’y a pas de jumeaux dans C2 en relation au graphe (C1 , C2 ; Gviolet ). Donc, (V1 , V2 ; Gviolet ∩ (V1 × V2 )) n’est pas complet. Par l’exercice 5.4, nous obtenons qu’aucune classe de jumeaux dans (V1 , V2 ; Gviolet ∩ (V1 × V2 )) n’a plus de \|V2 \|/2 éléments. Nous avons donc terminé. 16. Le problème dans la preuve originale de Babai était à ce point précis. Ce qui suit est un argument alternatif proposé par lui (col rumore sordo di un galoppo) lorsque cet article était en train d’être édité. Il est plus concis et élégant que l’argument d’origine, en plus d’être correct. Avant, la preuve faisait deux fois (ou plus) recours à la proposition elle-même, ce qui faisait croı̂tre l’indice [G : H] de façon catastrophique. 1125–32 5.3. Récursion et réduction une couleur domine ? non récursion n′ ≤ n/2 réduction de G/N à Altm′ m′ ≤ \|m\|/2 aligner G transitif ?, etc. réduction de G/N à Altm′ √ m′ ≪ m 5.3.1. Le cas sans couleurs dominantes. — Nous sommes dans le cas dans lequel un coloriage cX : Γ → C n’a pas de couleur dominante. Ici cX est l’image d’une structure X sous un foncteur F qui commute avec l’action de H(x1 ,...,xℓ ) , où H = Alt(Γ), xi ∈ Γ. Le fait que cX n’a pas de couleur dominante nous servira pour trouver ou écarter ses isomorphismes possibles en H~x = H(x1 ,...,xℓ ) . Pour trouver ou écarter des isomorphismes en tout H = Alt(Γ), nous n’avons qu’à travailler avec un ensemble de représentants {σ1 , . . . , σs }, s ≤ \|Γ\|ℓ = mℓ , des classes de H~x dans H, et à faire l’union de IsoH~x (cX , cYi ) −1 pour Yi = Yσi : [ (8) IsoH (X, Y) = IsoH~x (X, Yi ) σi , IsoH~x (X, Yi ) ⊂ IsoH~x (cX , cYi ). 1≤i≤s Ceci est similaire à l’équation (4), en §2.2. Le coût de la procédure est multiplié par s ≤ mℓ . Si le coloriage cX n’est pas une permutation (en H~x ) du coloriage cYi , alors IsoH~x (cX , cYi ) = ∅. Supposons, par contre, qu’il y a au moins un τi ∈ H~x tel que cX = cτYii . (Nous disons que τi aligne cX et cYi .) Il est trivial de trouver τi . Or IsoH~x (X, Yi ) = IsoH~x (X, Yτi i ) τi−1 ⊂ AutH~x (cX )τi−1 . Comme cX n’a pas de couleur dominante, ceci est assez contraignant, ce que nous voulions. Appliquons cette procédure générale au cas de G primitif que nous sommes en train de discuter. Il y a une bijection ι : Ω → {S ⊂ Γ : \|S\| = k} ; donc, cX induit un coloriage P c′ : Ω → {(ki )i∈C : ki ≥ 0, i ki = k}. Nous sommes dans une situation similaire à celle de la ﬁn du §4.2, mais en mieux : il est facile de montrer que, comme aucune classe de couleur de c possède plus de α\|Γ\| éléments, aucune classe de couleur de c′ possède plus de α\|Ω\| éléments. Nous procédons alors comme dans le cas intransitif de la preuve de Luks (Thm. 2.4), ce qui réduit le problème à ≤ n problèmes d’isomorphisme de chaı̂nes pour des chaı̂nes de longueur ≤ αn et de longueur totale ≤ n. Le dernier pas (lifting, « relèvement ») consiste à trouver des éléments de G qui induisent τi . Étant donnée une bijection ι, ceci est trivial. 1125–33 5.3.2. Le cas du découpage. — Considérons maintenant un α-découpage (ﬁn de §2.3) d’un ensemble de sommets Γ. Ce découpage sera donné canoniquement, à savoir, en tant que l’image d’une structure X sous un foncteur, tout comme le coloriage au §5.3.1. Nous pouvons supposer que le découpage a une classe de couleurs C dominante (\|C\| > α\|Γ\|, α > 1/2), puisque, dans le cas contraire, nous pouvons passer au §5.3.1. Nous voulons savoir quels éléments de Alt(Γ) respectent le α-découpage ; ceci nous aidera à contraindre les isomorphismes de X, tout comme en (8). Par la déﬁnition de α-découpage, C est partitionné en ℓ ≥ 2 ensembles de la même taille ≥ 2. Les seules permutations en Altm0 , m0 = C, qui sont permises sont celles qui respectent la partition. Le groupe qui respecte la partition est isomorphe à Altm0 /ℓ . Nous avons donc réduit notre problème à un problème avec m′ = m0 /ℓ ≤ m/2. Après avoir résolu ce problème, nous travaillons – comme dans le §5.3.1 – sur les autres classes de couleurs. Étant donnés deux α-découpages, nous vériﬁons si les partitions des deux découpages ont le même nombre d’ensembles de la même taille pour chaque couleur, puis nous alignons les deux découpages, et procédons exactement comme pour le problème de l’automorphisme. 5.3.3. Le cas du schéma de Johnson. — Soit donné un schéma de Johnson sur un ensemble de sommets Γ, ou plutôt deux schémas de Johnson J (mi , ki ), 2 ≤ ki ≤ mi /2, sur des ensembles de sommets Γ1 , Γ2 de la même taille. Nous avons vu au §2.8 comment identiﬁer Γi (là, Ω) explicitement avec les ensembles de taille ki d’un ensemble Λi (là, Γ) de taille mi . Si k1 6= k2 et m1 6= m2 , nos structures ne sont pas isomorphes. Si k1 = k2 et m1 = m2 , nous établissons une bijection entre Λ1 et Λ2 et alignons les deux structures. √ Nous avons réduit notre problème à un problème avec m′ ≪ m à la place de m. La situation nous est donc même plus favorable que dans le cas du découpage. À nouveau, nous laissons la comptabilité au lecteur. * Une petite confession : le cas de G primitif, que nous venons de ﬁnir de traiter, pourrait être traité exactement comme le cas de G imprimitif, que nous examinerons maintenant. La motivation du traitement séparé pour G primitif est pédagogique. Aucune peine n’est perdue, puisque toutes les techniques que nous avons étudiées nous seront essentielles dans le cas imprimitif. 6. LE CAS IMPRIMITIF Nous avons une application surjective explicite φ : G → Alt(Γ), 1125–34 où G < Sym(Ω) est un groupe de permutation, \|Γ\| = m, \|Ω\| = n. Nous pouvons supposer que \|Γ\| ≥ C log n, C arbitraire. L’application φ se factorise comme suit G → G/N → Alt(Γ), où N est le stabilisateur d’un système de blocs, et G/N → Alt(Γ) est un isomorphisme. Nous devons déterminer IsoG (x, y), où x, y sont des chaı̂nes. Nous avons déjà résolu le cas N = {e}. Nous attaquerons le problème de façon locale : pour T ⊂ Γ, nous arriverons à obtenir un certiﬁcat, soit du fait que φ(AutGT (x))\|T contient Alt(T ) (« certiﬁcat de plénitude »), soit du contraire. (Ici GT désigne le groupe {g ∈ G : T φ(g) = T }.) Nous calculerons tous ces certiﬁcats pour T d’une taille k modérée. Si le nombre de certiﬁcats de plénitude est très grand, nous aurons prouvé que φ(AutG (x)) contient un grand groupe alternant ; ce qui restera à faire sera une version de la procédure du §4.2 (« pull-back »). Dans le cas contraire, les certiﬁcats formeront une structure k-aire dont la symétrie est bornée. Nous pourrons donc appliquer le Lemme des designs, suivi de Coupe-ouJohnson, comme avant. Il y a aussi quelques autres cas particuliers, mais ils nous amènent à des α-découpages, α < 1, ce qui est aussi bien. 6.1. Les certificats locaux 6.1.1. Certiﬁcats d’automorphismes. — Un certiﬁcat local (17) pour T ⊂ Γ est — soit une paire (« pas plein », W, M(T )), oùW ⊂ Ω, M(T ) < Sym(T ), M(T ) 6= Alt(T ) (donc « pas plein ») et φ AutW GT (x) \|T < M(T ), — soit une paire (« plein », K(T )), où K(T ) < AutGT (x), et φ(K(T ))\|T = Alt(T ). Le certiﬁcat local dépend de x de façon canonique. Il est clair qu’un certiﬁcat plein, voire pas plein, garantit que φ(AutGT (x))\|T est Alt(T ), voire ne l’est pas. Si T est donné en tant que tuple ordonné, son certiﬁcat dépend de l’ordre de T seulement dans le sens de ne pas en dépendre : le même groupe {(23), e} < Sym({1, 2, 3}) (disons) a une apparence diﬀérente si nous le regardons du point de vue de l’ordre (1, 2, 3) ou de l’ordre (2, 1, 3). Nous construisons le certiﬁcat par une procédure itérative. Au début de chaque pas, W ⊂ Ω et A(W ) est le groupe AutW GT (x) ; la fenêtre W sera invariante sous A(W ). Au tout début de la procédure, W = ∅ et A(W ) = GT . (Nous pouvons calculer GT comme dans l’exercice 2.1c en temps \|Ω\|O(k) , où k = \|T \|.) À chaque pas, nous ajoutons à W tous les éléments atteints par A(W ) (voir §4.1), puis nous mettons A(W ) à jour, selon le nouveau W . Nous nous arrêtons si φ(A(W ))\|T 6= Alt(T ) (non-plénitude) ou si W ne croı̂t plus, ce qui veut dire qu’aucun élément de Ω \ W n’est atteint par A(W ). Il est clair qu’il y aura ≤ \|Ω\| itérations. À la ﬁn, dans le cas de non-plénitude, nous retournons (« pas plein », W, φ(A(W ))) ; dans le cas de plénitude, nous retournons « plein », A(W )(Ω\W ) . Il est clair que le stabilisateur des points A(W )(Ω\W ) est contenu 17. Ou « local-global », dans la nomenclature de Babai. « Global » fait référence à AutGT (x) < Sym(Ω). 1125–35 non pas seulement dans AutW GT (x), mais aussi dans AutGT (x), puisqu’il ﬁxe tous les points de Ω \ W . Nous savons que φ A(W )(Ω\W ) = Alt(T ) par la Proposition 4.4a, sous la condition que \|T \| ≥ max(8, 2 + log2 \|Ω\|). Vériﬁer si φ(A(W ))\|T = Alt(T ) est facile : nous n’avons qu’à vériﬁer, en utilisant Schreier-Sims, si deux générateurs arbitraires de Alt(T ) sont en φ(A(W ))\|T . De la même façon, il est simple de déterminer quels éléments sont atteints par A(W ) : nous calculons A(W )x pour chaque x ∈ Ω (par Schreier-Sims) et, toujours par Schreier-Sims, vériﬁons si φ(A(W )x )\|T = Alt(T ). Ceci prend du temps polynomial en \|Ω\|. Il reste à voir comment mettre à jour A(W ), étant donné A (W − ), où nous écrivons W − pour l’ancienne valeur de W . Tout élément de A(W ) est dans A(W − ), et donc A (W ) = AutW A(W − ) (x). Comme dans l’équation (4), [ [ W W σ−1 , x, x Iso (9) AutW (x) = Aut (x) = − N Nσ A(W ) σ σ où N est le noyau de φ\|A(W− ) et σ parcourt des représentants des k!/2 classes de N en A(W ). Nous pouvons trouver rapidement un σ ∈ A(W − ) ∩ φ−1 ({τ }) pour tout τ ∈ Sym(Γ), par Schreier-Sims. La Proposition 4.4b nous donne que toute orbite de N contenue en W (l’ensemble d’éléments atteints par A(W − )) est de longueur ≤ \|W \|/k ≤ \|Ω\|/k. En conséquence, par la règle (3), mettre A(W ) à jour se réduit à \|Ω\| · (k!/2) problèmes de détermination de Iso pour des chaı̂nes de longueur ≤ \|Ω\|/k. Comme le nombre d’itérations est ≤ \|Ω\|, la procédure fait appel à Isomorphismede-Chaı̂nes ≤ \|Ω\|2 · (k!/2) fois pour des chaı̂nes de longueur ≤ \|Ω\|/k. Ceci – comme la routine qui prenait \|Ω\|O(k) de temps – est acceptable pour k ≪ (log \|Ω\|)κ . Nous choisirons κ = 1. 6.1.2. Comparaisons de certiﬁcats. — Une légère modiﬁcation de la procédure cidessus nous permet d’élucider la relation entre deux certiﬁcats locaux pour deux chaı̂nes. Soient x, x′ : Ω → Σ, T, T ′ ⊂ Σ, \|T \| = \|T ′\| = k. Pour S ⊃ T , soit xS la chaı̂ne ( x(i) si i ∈ S, xS (i) = glauque si i ∈ /S où glauque ∈ / Σ. Nous voulons calculer (10) ′ IsoGT ·τT,T ′ xW , xW , où GT · τT,T ′ est la classe des éléments de G qui envoient l’ensemble T à T ′ , et W ′ est la valeur de W retournée quand la donnée est T ′ à la place de T . Pour déterminer (10), nous suivons la procédure (§6.1.1), modiﬁée de la façon suivante : nous mettrons à jour, dans chaque itération, non pas seulement A(W ), mais ′ aussi la classe A(W )τ d’isomorphismes en GT · τT,T ′ de xW à (x′ )W . Voilà comment le 1125–36 faire, de façon analogue à (9) : −1 [ [ ′ σ ′ W ′ W W ′ W , = IsoN x , (x ) (11) IsoN σ x , (x ) σ σ où N est le noyau de φ\|A(W− ) et σ parcourt des représentants des k!/2 classes de N contenues en A(W − )τ − . Comme W est stabilisé par A(W− ) (et donc par N), le fait que σ envoie W sur W ′ ou non dépend seulement de la classe de N à laquelle σ appartient. (La classe Iso dans la dernière expression de (11) est vide si W σ 6= W ′.) Comme avant, toute orbite de N contenue en W est de longueur ≤ \|W \|/k, et le problème se réduit à \|Ω\| · (k!/2) appels par itération à Isomorphisme-de-Chaı̂nes pour des chaı̂nes de longueur ≤ \|W \|/k ≤ \|Ω\|/k. Par ailleurs, si T et T ′ nous sont données comme tuples ordonnés (T ), (T ′ ), il est facile de déterminer W′ , (12) I(x, x′ , T, T ′) = IsoG(T ) ·τ(T ),(T ′ ) xW , (x′ ) où G(T ) · τ(T ),(T ′ ) est la classe des éléments de G qui envoient le tuple ordonné (T ) à (T ′ ). En eﬀet, nous n’avons qu’à déterminer (10), puis utiliser Schreier-Sims pour déceler les éléments de (10) qui envoient (T ) à (T ′ ) dans le bon ordre. 6.2. L’agrégation des certificats coupe coupe certiﬁcats locaux plénitude > 1/2 ? coupe ou relations ? relations réduction de G/N à Altm′ m′ ≤ \|m\|/2 Weisfeilerréduction Leman, Johnson de G/N Lemme des à Altm′ √ designs, m′ ≪ m etc. oui pas de couleur dominante pullback récursion n′ ≤ 3n/4 En suivant la procédure du §6.1.1 pour une chaı̂ne x, nous trouvons des certiﬁcats locaux pour chaque T ⊂ Γ de taille k, où k est une constante ∼ C log \|Ω\| (C > 1/ log 2) et k < \|Γ\|/10. Soit F < AutG (x) le groupe engendré par les certiﬁcats pleins K(T ). Soit S ⊂ Γ le support de φ(F ), c’est-à-dire l’ensemble des éléments de Γ qui ne sont pas ﬁxés par tout élément de φ(F ). Notre objectif est de déterminer les isomorphismes IsoG (x, x′ ) de x à une autre chaı̂ne x′ . Puisque les certiﬁcats sont canoniques, l’assignation de F et S à une chaı̂ne l’est aussi. Donc, si nous arrivons à deux cas diﬀérents ci-dessous en suivant la procédure pour x et pour x′ , les deux chaı̂nes ne sont pas isomorphes. Cas 1 : \|S\| ≥ \|Γ\|/2, mais aucune orbite de φ(F ) n’est de longueur > \|Γ\|/2. Alors, nous colorions chaque élément de Γ par la longueur de l’orbite qui le contient. Ceci est un coloriage canonique. Soit aucune classe de couleurs n’est de taille > \|Γ\|/2, 1125–37 soit une classe de couleurs de taille > \|Γ\|/2 est découpée en ≥ 2 ensembles de la même taille ≥ 2. Dans un cas comme dans l’autre, nous passons à une réduction/récursion. Cas 2 : \|S\| ≥ \|Γ\|/2 et une orbite Φ de φ(F ) est de longueur > \|Γ\|/2. Cas 2a : Alt(Φ) < φ(F )\|Φ . Nous sommes dans le cas de grande symétrie. Nous procédons comme au §4.2, jusqu’au point où nous devons déterminer IsoH (x, y) (où y ′ est (x′ )σ , σ ′ ∈ G, et H = φ−1 (Alt(Γ)Φ )). Déﬁnissons K = φ−1 Alt(Γ)(Φ) , et soient σ1 , σ2 ∈ G des préimages (arbitraires) sous φ de deux générateurs de Alt(Φ) < Alt(G), trouvées par Schreier-Sims. Nous savons que les classes AutKσi (x), i = 1, 2, sont non vides, puisque Alt(Φ) < φ(F )\|Φ . Comme K n’a pas d’orbites de longueur > \|Ω\|/2, nous pouvons déterminer ces deux classes par des appels à Isomorphisme-de-Chaı̂nes pour des chaı̂nes de longueur ≤ \|Ω\|/2 et longueur totale ≤ 2\|Ω\|. Elles engendrent AutH (x). Encore par le fait que Alt(Φ) < φ(F )\|Φ , la classe IsoH (x, y) sera non vide ssi IsoK (x, y) est non vide. Nous pouvons déterminer cette dernière classe par des appels à Isomorphisme-de-Chaı̂nes comme ci-dessus, puisque K n’a pas d’orbites de longueur > \|Ω\|/2. Si elle est non vide, nous obtenons la réponse IsoH (x, y) = AutH (x) IsoK (x, y). Cas 2b : Alt(Φ) ≮ φ(F )\|Φ . Soit d ≥ 1 l’entier maximal avec la propriété que φ(F )\|Φ est d-transitif, c’est-à-dire, φ(F )\|Φ agit transitivement sur l’ensemble des dtuples d’éléments distincts de Φ. Par CGFS, d ≤ 5 ; si nous ne voulons pas utiliser CGFS, nous avons la borne classique d ≪ log \|Γ\|. Choisissons x1 , . . . , xd−1 ∈ Φ arbitrairement. Le reste de notre traitement de ce cas sera donc seulement canonique en relation à φ(g) G(x1 ,...,xd−1 ) = {g ∈ G : xi = xi ∀1 ≤ i ≤ d − 1}, ce qui, comme nous le savons, n’est pas un problème ; voir le début du §5.3.1. La restriction du groupe φ(F )(x1 ,...,xd−1 ) à Φ′ = Φ\{x1 , . . . , xd−1 } est transitive sur Φ′ , mais elle n’est pas doublement transitive. Donc, la conﬁguration cohérente schurienne qui lui correspond n’est pas une clique. Nous livrons cette conﬁguration à Coupe-ouJohnson (§5.2), tout comme à la ﬁn du §5.2. Pour comparer les conﬁgurations qui correspondent à deux chaı̂nes x, x′ , nous alignons leurs classes Φ d’abord. (Si elles ne sont pas de la même taille, ou si une chaı̂ne nous donne le cas 2a et l’autre pas, les chaı̂nes ne sont pas isomorphes.) Les isomorphismes seront donc contenus dans le stabilisateur H < G de l’ensemble Φ (facile à déterminer, comme vers la ﬁn du §4.2, puisque φ est surjective). Nous pouvons remplacer φ par l’application g 7→ φ(g)\|Φ de H à Alt(Φ). Puis nous construisons les conﬁgurations comme ci-dessus, et comparons ce que Coupe-ou-Johnson nous donne. Tout à la ﬁn, nous nous occupons du complément de C. Il s’agit, comme d’habitude, d’appels à Isomorphisme-de-Chaı̂nes pour des chaı̂nes de longueur ≤ \|Ω\|/2 et longueur totale < \|Ω\|. Cas 3 : \|S\| < \|Γ\|/2. Nous commençons en alignant les supports S pour les chaı̂nes x, x′ , et en remplaçant φ par g 7→ φ(g)\|Γ\S , tout comme dans le cas 2(b). 1125–38 Nous allons déﬁnir une relation k-aire avec très peu de jumeaux, pour la donner après au Lemme des designs. Regardons la catégorie de toutes les chaı̂nes Ω → Σ, où Ω et Σ sont ﬁxes, une action de G sur Ω est donnée, et φ : G → Γ est aussi donnée. Nous la regardons depuis longtemps, puisque nous devons comparer les couleurs sur des conﬁgurations induites par des chaı̂nes diﬀérentes pour décider si ces dernières sont isomorphes. Cette fois-ci, nous déﬁnirons des couleurs par des classes d’équivalence : deux paires (x, (T )), (x′ , (T ′ )) (T, T ′ ⊂ Γ \ S, \|T \| = \|T ′ \| = k) sont équivalentes si l’ensemble des isomorphismes en (12) est non vide. Nous colorions (T ) – dans le coloriage de (Γ \ S)k correspondant à x – par la classe d’équivalence de (x, (T )). Ici, (T ) est un k-tuple ordonné sans répétitions ; si (T ) a des répétitions, elle est coloriée en gris. Pour x donné, aucune classe de jumeaux en Γ ne peut avoir ≥ k éléments : s’il existait un tel ensemble avec ≥ k éléments, il contiendrait un ensemble T avec k éléments, et tous les ordres (T ) de T auraient la même couleur. Ceci voudrait dire que l’ensemble des isomorphismes en (12) serait non vide pour n’importe quels ordres (T ), (T ′ ) de T . En conséquence, AutGT (xW ) contiendrait des éléments donnant toutes les permutations possibles de T . Ceci nous donnerait une contradiction, puisque T , étant contenu en Γ\S, n’est pas plein. Alors, pourvu que k ≤ \|Γ\|/4, nous avons un coloriage de (Γ\S)k sans aucune classe de jumeaux avec ≥ \|Γ \ S\| éléments. Nous pourrons donc appliquer le Lemme des designs, après une application de raﬃnements habituels F2 , F3 (Weisfeiler-Leman). Mais – pouvons-nous calculer ces coloriages ? Les classes d’équivalence sont énormes. Par contre, il n’y a aucun besoin de les calculer. Tout ce dont nous aurons besoin, pour comparer des structures qui viennent de chaı̂nes x, y, sera d’être capables de comparer deux tuples (T ) (sur la conﬁguration donnée par x ou y) et (T ′ ) (sur la conﬁguration donnée par x′ = x ou x′ = y) et dire si elles sont de la même couleur. En d’autres termes, nous devrons calculer – au tout début de la procédure, pour toute paire ((T ), (T ′)), \|T \| = \|T ′ \| = k, et pour les paires de chaı̂nes (x, x), (x, y), (y, y) – l’ensemble d’isomorphismes en (12), ce que nous savons déjà faire. Les couleurs sont donc, dans la pratique, des entrées dans un index que nous enrichissons et auquel nous faisons référence durant nos procédures. Nous invoquons donc le Lemme des Designs, suivi par Coupe-ou-Johnson, et le reste de la procédure. — Fine dell’opera — Le lecteur peut vériﬁer que les informations précisées jusqu’à ici (temps pris par des procédures, type de récursion) sont assez pour donner une borne du type exp(O(log \|Ω\|)c ) pour le temps de l’algorithme qui résout le problème de l’isomorphisme de chaı̂nes. Ceci donne une borne exp(O(log n)c ) pour le problème de l’isomorphisme de graphes avec n sommets. Avec un peu plus de travail, il devient clair que, dans un cas comme dans l’autre, c = 3. Nous donnons les détails dans l’appendice. L’exposant c = 3 1125–39 est plus petit que celui d’origine ; il est devenu possible grâce à quelques améliorations et simpliﬁcations que j’ai été capable d’apporter. Remerciements .— Je remercie vivement L. Babai, J. Bajpai, L. Bartholdi, D. Dona, E. Kowalski, W. Kantor, G. Puccini, L. Pyber, A. Rimbaud et C. Roney-Dougal pour des corrections et suggestions. En particulier, L. Babai a répondu à beaucoup de mes questions, et m’a aussi fourni des versions corrigées ou améliorées de plusieurs sections de [Ba]. En particulier, les §2.3–2.4 et §5.1 sont basés sur ces nouvelles versions. Je voudrais aussi remercier V. Ladret et V. Le Dret pour un grand nombre de corrections d’ordre typographique et linguistique. APPENDICE A. ANALYSE DU TEMPS D’EXÉCUTION A.1. Quelques précisions sur la procédure principale À tout moment donné, nous travaillons avec un groupe transitif G < Sym(Ω) qui S agit sur un système de blocs B = {Bi }, Ω = i Bi , Bi disjoints ; nous notons N le noyau de l’action sur B. À vrai dire, nous aurons toute une tour de systèmes de blocs B1 , B2 , . . . , Bk , où Bi est un raﬃnement de Bi+1 ; B signiﬁera Bk , le système le moins ﬁn. Au début, il n’y a qu’un système, B1 , dont les blocs Bi sont tous de taille 1, et dont le noyau N est trivial. Nous voudrions que l’action de G sur B soit primitive. Donc, si elle ne l’est pas, nous ajoutons à la tour un système minimal Bk+1 tel que Bk soit un raﬃnement de Bk+1 . Nous redéﬁnissons B = Bk+1 ; N sera le noyau du nouveau B. Si G/N est petit (≤ bO(log b) , où b = \|B\| ; cas (a) du Théorème 3.1 (Cameron)), nous réduisons notre problème à plusieurs instances du problème avec N à la place de G. Chacune de ces instances se décompose en plusieurs instances – une pour chaque orbite de N. Chaque orbite Ω′ de N est contenue dans un bloc de B. Les intersections de Ω′ avec les blocs de B1 , B2 , . . . nous donnent une tour de systèmes de blocs pour N\|Ω′ . Si nous sommes dans le cas (b) du Théorème 3.1, nous passons à ≤ b instances du problème avec M ⊳ G (où[G : M] ≤ b) à la place de G. Nous passons à un nouveau ≤ b blocs, et l’ajoutons à la tour comme son nouveau dernier système (18) B ′ de m′ = m k ′ niveau. Nous notons N le noyau de l’action de M sur B ′ . Alors, M/N ′ = Alt(k) m . Nous remplaçons G par M et redéﬁnissons B = B ′ , N = N ′ . Donc, nous avons un isomorphisme de G/N à Altm . Nous sommes dans le cas principal que Babai attaque. Ses méthodes amènent à une réduction de Altm , soit à un groupe intransitif sans grandes orbites, soit à un produit Alts1 ≀ Alts2 , s1 , s2 > 1, s1 s2 ≤ m, soit √ à un groupe Altm′ , m′ ≪ m. (Nous simpliﬁons quelque peu. Nous pourrions avoir, disons, un produit Alts1 ≀ Alts2 , agissant sur une orbite de grande taille s1 s2 ≤ m, et 18. Ce système peut être égal à B seulement si M = G ; voir la deuxième note de pied de page dans l’énoncé du Théorème 3.1. Dans ce cas-là, le passage de G à M est bien sûr gratuit. 1125–40 d’autres groupes sur des petites orbites, ou plusieurs produits agissant sur des petites orbites.) Dans le cas intransitif sans grandes orbites, nous procédons comme dans la preuve de Luks. (La procédure aura été plus coûteuse que dans Luks, mais grâce au manque de grandes orbites, le gain dans la récursion est aussi plus grand.) Dans le cas de Altm′ , √ m′ ≪ m, nous itérons la procédure. Dans le cas de Alts1 ≀ Alts2 – qui correspond à un découpage dans des ensembles de taille r de la même couleur – nous avons une action primitive de Alts sur un système de s blocs de taille r. Nous passons, alors, à cette action et à ces blocs, sans oublier les blocs B ′ , auxquels nous retournons plus tard, après avoir ﬁni de travailler sur Alts . Il est clair que ce type de procédure réduit complètement Altk en un nombre d’itérations qui n’est pas supérieur à log2 m. A.2. Récursion et temps Examinons le temps total d’exécution de l’algorithme qui trouve les isomorphismes entre deux chaı̂nes. Les pas individuels sont peu onéreux ; aucun ne précise plus de nO(log n) de temps. Notre attention doit se porter avant tout sur la récursion. Dans la procédure générale, une récursion est toujours d’une descente, soit vers des chaı̂nes plus courtes, soit vers un groupe plus petit, ou au moins coupé dans des tranches plus ﬁnes par une tour de systèmes de blocs ayant plus de niveaux. Dans le premier type de descente, le groupe reste le même ou, plutôt, est remplacé par une restriction de lui-même. Dans le deuxième cas, la longueur des chaı̂nes reste la même. (Nous pouvons aussi avoir un mélange des deux cas – tant mieux : le groupe devient plus petit et les chaı̂nes se raccourcissent aussi.) La descente la moins coûteuse, et moins avantageuse, est celle du cas intransitif de la procédure de Luks. Il pourrait arriver que G ait deux orbites sur Ω (\|Ω\| = n), une de longueur n − 1 et une de longueur 1. Ceci serait même compatible avec une borne polynomiale sur le temps, pourvu que le temps pris avant la descente soit lui-même polynomial : nc+1 ≤ (n − 1)c+1 + 1c+1 + nc pour c ≥ 1. D’autres types de descente sont plus coûteux, mais aussi plus avantageux : nous descendons à des chaı̂nes de longueur ≤ n/2 (ou ≤ 2n/3), ou de Altm à Alts1 ≀ Alts2 , s1 s2 ≤ m, s1 , s2 ≤ m/2, par exemple. Il est clair qu’il est impossible de descendre plus qu’un nombre logarithmique de fois de cette façon. Il est crucial de ne pas oublier qu’un coût (considérable) peut être caché dans une perte de canonicité. Si nos choix ne sont canoniques qu’en relation à un sous-groupe H de notre groupe G, le coût de leur application sera multiplié par [G : H]. (Voir §5.3.1.) *** Considérons, alors, le coût de chaque procédure. Le cas intransitif de Luks est, comme nous l’avons déjà vu, compatible même avec une borne polynomiale. Concentrons-nous alors sur le cas où G agit de façon primitive sur un système de blocs ; soit N le noyau. 1125–41 Si nous sommes dans le cas (a) du Théorème 3.1, ou dans le cas (b), mais avec m ≤ C log n, nous faisons appel à (m′ )O(log n) instances de la procédure principale pour des chaı̂nes de longueur n/m′ (où m′ ≥ m). Ceci est consistant avec une borne totale du type exp(O((log n)c )), c ≥ 2. Nous pouvons, donc, nous concentrer sur le cas où il existe un isomorphisme φ : G/N → Alt(Γ), \|Γ\| = m > C log n. (La procédure du §2.8 rend cet isomorphisme explicite.) Le premier pas à considérer est la création de certiﬁcats locaux, avec, comme objectif, la création d’une relation k-aire sur Γ. (Si G est primitif, créer une telle relation est trivial ; voir le début du §5.) Il y a nk certiﬁcats locaux, où k = 2 log n (disons) ; nous devons les calculer et aussi comparer toute paire de certiﬁcats. Déjà le premier pas du calcul d’un certiﬁcat, à savoir le calcul de GT , prend un temps nO(k) (plus précisément, O((n/k)O(k))). D’autres calculs prennent moins de temps. L’usage de la récursion, par contre, est relativement lourd : nous faisons appel à la procédure principale ≤ n2 · k! fois pour des chaı̂nes de longueur ≤ n/k. Ceci se passe pour chaque ensemble T de taille k, c’est-à-dire ≤ nk /k! fois. La procédure pour comparer des paires de certiﬁcats est analogue. Nous faisons donc appel à la procédure principale O(n2k+1 ) fois pour des chaı̂nes de longueur ≤ n/k. Dans chacun de ces appels, notre tour de stabilisateurs est héritée : notre groupe est un groupe transitif, égal à la restriction de N − à une de ses orbites, où N − (noté N au §6) est un sous-groupe d’un sous-groupe A(W − ) de G. Pour deux systèmes de blocs consécutifs Bi , Bi+1 , notons ri le nombre de blocs de Bi dans chaque bloc de Bi+1 . Il est clair que ce nombre n’augmente pas quand nous passons à la restriction d’un sous-groupe de G (par exemple, N − ) à une de ses orbites. Examinons maintenant l’agrégation des certiﬁcats locaux (§6.2). Il y a trois cas. Dans le premier, le temps de calcul additionnel est à peu près trivial, et nous obtenons une réduction, soit à un groupe intransitif sans grandes orbites, soit à un produit Alts1 ≀ Alts2 sur une grande orbite et éventuellement d’autres groupes sur des orbites plus petites. Ici, déjà, l’analyse devient délicate. Nous devons prendre en considération non seulement la taille du domaine mais aussi le groupe qui agit sur lui. Plus précisément, nous devons borner le nombre de fois que notre tour B1 , B2 , . . . , Bk pourrait être raﬃné ou raccourci encore. Ceci sera mesuré par X ρ= (2⌊log2 ri ⌋ − 1), 1≤i≤k−1 où nous supposons que nous avons enlevé des systèmes répétés de la tour (donc ri > 1). Notons F (n, r) le temps d’exécution de la procédure principale pour des chaı̂nes de longueur n et pour une tour de systèmes de blocs pour G telle que le paramètre ρ est ≤ r. Une réduction de G/N fait décroı̂tre r par au moins 1 ; un coloriage sans aucune grande classe de couleurs assure une descente vers des chaı̂nes de longueur ≤ n/2. Nous devrons aussi inclure un facteur de log n2k , prenant en considération le temps requis 1125–42 pour accéder à nos comparaisons de paires de certiﬁcats locaux (19) . Donc, dans le cas que nous examinons, F (n, r) est borné par ! X nO(k) + n2k+1 F (n/k, r) + F (n1 , r − 1) + F (ni , r) · O(k log n), i≥2 où P ni = n et ni ≤ n/2 pour i ≥ 2, ou nO(k) + n2k+1 F (n/k, r) + X i F (ni , r) ! · O(k log n), où ni = n et ni ≤ n/2 pour i ≥ 1. Puisque k ≪ log n, ceci est consistant avec F (n, r) = exp (O (r + log n)c ) pour c ≥ 3, ou même avec F (n, r) = exp (O ((log n)c1 + (log r)c2 )) pour c1 ≥ 3 et c2 ≥ 1, par exemple. Le cas 2a a un coût très similaire, à un facteur constant près. Le cas 2b et 3 sont diﬀérents. Dans les deux cas, nous arrivons à construire une relation d-aire, avec d ≤ 5, dans le cas 2b, et d = k ≪ log n dans le cas 3. Puis, nous appelons Weisfeiler-Leman, suivi du Lemme des Designs pour des conﬁgurations d-aires, et, ﬁnalement, Coupe-ouJohnson. Weisfeiler-Leman prend un temps \|Γ\|O(d) = mO(d) . Le Lemme des designs garantit l’existence d’un tuple (x1 , . . . , xℓ ) ∈ Γ, ℓ ≤ d − 1, avec certaines propriétés. Nous cherchons un tel tuple par force brute, ce qui prend un temps O(md ). Ce qui est plus important est que ce choix n’est pas canonique. Donc, le temps d’exécution de tout ce qui reste est multiplié par md = mO(log n) . Coupe-ou-Johnson prend un temps O(md ). Ici, à nouveau, nous faisons des choix qui ne sont pas complètement canoniques ; ils imposent un facteur de mO(log m) sur tout ce qui suit. Le résultat de Coupe-ou-Johnson est soit un β-découpage, ce qui implique une réduction à un produit du type Alts1 ≀ Alts2 et/ou à des chaı̂nes plus courtes, soit un √ schéma de Johnson, ce qui implique une réduction à Altm′ , m′ ≪ m. Donc, soit (13) ! X F (ni , r) , F (n, r) ≤ nO(k) + O kn2k+2 F (n/k, r) + mO(log n) 1 + F (n1 , r − 1) + P i≥2 où P (14) ni = n, et ni ≤ n/2 pour i ≥ 2, ou F (n, r) ≤ nO(k) + O kn2k+2 F (n/k, r) + mO(log n) 1 + X i ! F (ni , r) , P où ni = n et ni ≤ n/2 pour i ≥ 1. Ici m ≤ n. (Nous pourrions travailler avec une borne moins grossière, mais cela nous servirait peu.) Donc, les inégalités (13) et (14) sont consistantes avec F (n, r) = exp (O (r + log n)c ) pour c ≥ 3. 19. 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Modified Recursive Cholesky (Rchol) Algorithm arXiv:1703.05904v1 [math.AC] 17 Mar 2017 An Explicit Estimation and Pseudo-inverse of Correlation Matrices Vanita Pawar Krishna Naik Karamtot vanita [email protected] [email protected] Abstract—The Cholesky decomposition plays an important role in finding the inverse of the correlation matrices. As it is a fast and numerically stable for linear system solving, inversion, and factorization compared to singular valued decomposition (SVD), QR factorization and LU decomposition. As different methods exist to find the Cholesky decomposition of a given matrix, this paper presents the comparative study of a proposed RChol algorithm with the conventional methods. The RChol algorithm is an explicit way to estimate the modified Cholesky factors of a dynamic correlation matrix. Cholesky decomposition is a fast and numerically stable for linear system solving, inversion, and factorization compared to singular valued decomposition (SVD), QR factorization and LU decomposition [1]. The wireless communication system is highly dependent on matrix inversion of the correlation matrix. Such system consists of a huge matrix inversion. An outdoor wireless communication has a time-varying channel which changes dynamically for mobile user. In case of narrowband channel, the channel is considered constant for a symbol duration, whereas for broadband, it is changing within a symbol period. Such time-varying channel forms the special structure of channel matrix and correlation matrix. To exploit such special structure, a novel modified recessive Cholesky (RChol) algorithm is introduced in [2]. Our proposed (RChol) algorithm is a computational efficient algorithm to compute the modified Cholesky factors of known as well as an unknown covariance matrix. In this paper, we present the comparative study of conventional Cholesky algorithm and the RChol algorithm to manifest the importance of the proposed algorithm in a highly dynamic wireless communication. I. S YSTEM M ODEL In wireless communication system, number of transmit and or received antennas are used to improve the diversity of the system. The channel h between transmitter and receiver has the different form and depends on the number of antennas used at the transmitter and the receiver side. The channel for Single-input-single-output (SISO) as h = {h0n , h1n , . . . hL−1 }, for Single-input-multiple-output (SIMO) n as h = {h0n , h1n , . . . hL−1 } and for Multiple-input-multiplen output (MIMO): H = {H0n , H1n , . . . HL−1 }. n Let y(n) be received signal with the number of transmit antennas 0 K = 10 , multipath L − 1 and channel noise v, represented as y(n) := K L−1 X X hk (n; l)sk (n − l) + v(n), n = 0, 1, ..T − 1 k=1 l=0 (1) Let yN (n) be the received vector by stacking N successive received vectors. Where yN (n) = [y(n), y(n − 1), . . . y(n − N + 1)]T and the transmitted symbol vector is sN = [s(n), s(n − 1), .s(n − N + 1)]T . Then yN (n) can be represented in matrix form as yN (n) = HN sN (n) + vN (n) and the correlation matrix for yN can be written as RN (n) = H (n)]. Let rn00 = E[y(n)yH (n)] and rnij = E[yN (n)yN H E[y(n − i)y (n − j)] then the correlation matrix RN (n) and RN (n − 1) at time instant 0 n0 and 0 n − 10 can be represented as equation (2) and equation (3) respectively. 0    rn 00 rn 10 rn 01 rn 11 . . . . . . ...... ...... . . . ...... rn 0(N −2) rn 1(N −2) rn 0(N −1) rn 1(N −1) . . . . . .    (2) rn (N −1)0 rn (N −1)1  rn 11 rn 12 ...... rn 1(N −1) rn 1N    . . . rn (N −1)1 rn N1 . . . rn (N −1)2 rn N2 . . . ...... . . . rn (N −1)(N −1) rn N (N −1) . . . rn (N −1)N rn NN   ...... rn (N −1)(N −2) 0 rn (N −1)(N −1) (3) II. C HOLESKY D ECOMPOSITION The correlation matrix is complex matrix and the pseudoinverse of R can be computed from Cholesky factors, such that if lower triangular matrix L is Cholesky factors of the correlation matrix R and can be represented as R = LLH then pseudo-inverse of R can be computed as R̂† := L−H L−1 . The section below details the conventional Cholesky algorithms and the RChol algorithm. A. Cholesky Decomposition (Gaxpy version) The Cholesky Decomposition [3] factorizes a complex (or real-valued) positive-denite Hermitian symmetric matrix into a product of a lower triangular matrix and its Hermitian transpose. R = LLH where, L is a lower triangular matrix and LH is Hermitian of L. The matrix R must be a positive definite and this method needs square root operation. 1) 1) 2) 3) 4) 5) Algorithm steps: Compute R at each time instant n Find the square root of diagonal element of R Modify each column of R Equate lower triangular part of R to L Repeat steps (1) to (4) for each time instant Algorithm 1 Cholesky Decomposition R = LLH Initialization: [R]1:N,1 [R]1:N,1 = q [R]1,1 0 Order Updates on R s: for k = 2 to N H [R]k:N ,k = [R]k:N ,k − [R]k:N ,1:k−1 [R]k,1:k−1 [R]k:N,k [R]k:N,k = q [R]k,k end [4] that Levinson recursion may be used to derive the Lattice recursion for computing QR factors of data matrices and Lattice recursion can be used to derive the Schur recursion for computing Cholesky factors of a Toeplitz correlation matrix. The detail algorithm is given in algorithm 3. The Schur algorithm like previously mentioned algorithm computes all N inner product to compute matrix R for initialization. 1) Algorithm steps: 1) Compute R at each time instant n 2) Initialize first column of R to the first column of Cholesky factor H 3) Compute rest column recursively from columns of R 4) repeat step (1) to (3) for each time instant Algorithm 3 Schur Algorithm R = LLH Initialization: for k = 1 L = tril(R) T n n n H1 (n) = [r00 , r10 , .....r(N −1)0 ] T n n H̃1 (n) = [0, r10 , .....r(N −1)0 ] H B. Modified Cholesky Algorithm R = LDL To avoid square root operation, a modified Cholesky algorithm [3] is used, which avoids square root operation by introducing a diagonal matrix D in between Cholesky factors. The modified Cholesky algorithm does not require R to be a positive definite matrix but it’s determinant must be nonzero. R may be rank deficient to a certain degree i.e. D may contain negative main diagonal entries if R is not positive semidefinite. 1) Algorithm steps: 1) Compute R at each time instant n 2) Modify each column of R 3) Equate the strictly lower part of matrix R to L1 with ones on the main diagonal 4) Equate main diagonal of R with the main diagonal of D 5) Repeat step (1) to (4) for each time instant. Algorithm 2 Modified Cholesky Decomposition R = LDLH Initialization: [R]2:N,1 = [R]2:N,1 [R]1,1 0 Order Updates on R s: for k = 2 to N for i=1:k-1 [v]i = [R]1,k , if i = 1 [R]i,i [R]∗ n,i , if i 6= 1 end [v]k = [R]k,k − [R]k,1:k−1 [v]1:k−1 [R]k,k = [v]k [R]k+1:N ,k = [R]k+1:N ,k − [R]k+1:N ,1:k−1 [v]1:k−1 [v]k end D = diag(daig(R)) L = tril(R) C. Recursive Cholesky Algorithm (The Shcur Algorithm) RSchur := HHH The Schur algorithm recursively compute the columns of the lower triangular matrix H form matrix R. It is shown in 0 Order Updates on H s: for k = 2 to N σk Hk = k̃ref [kref (σk−1 H̃k−1 ) + ZM (σk−1 Hk−1 )] k k σk H̃k = k̃ref [(σn H̃k ) + kref ZM (σk Hk )] k k Scaling Factors: (σk H̃k )k kref = − k (σk Hk )k−1 (σk Hk )k−1 k̃ref = k (σk+1 Hk+1 )k ∗ Note: Here notation is followed same as in [4] and H represents vector D. The RChol Algorithm L̂D̂L̂H := R It is clear from above equation (2) and equation (3) that RN (n) can be represented from submatrix of RN (n − 1). To utilize such special structure of correlation matrices, we propose a modified recursive Cholesky algorithm to compute the Cholesky factors recursively. This algorithm is modification of Schur algorithm mentioned above. The more general approach consists of using the Schur algorithm to induce recursion for columns of dynamic L. This algorithm does not need N inner products to compute the correlation matrix R. The Cholesky factors are computed explicitly such that Let L1 = LD1/2 then −1 pseudo-inverse can be computed as R̂† = L−H 1 L1 1) Algorithm steps: 1) Initialize first the first column of Cholesky factor A as A1 2) Compute second column recursively from A1 (n) and A1 (n − 1) 3) Substitute sub-matrix A2:N −1,2:N −1 (n − 1) to A3:N,3:N (n) 4) Repeat step (1) to (3) for each time instant In the Schur algorithm, columns of Cholesky factors at time instant n are computed recursively from the correlation matrix at that instant. Whereas in the RChol algorithm first two columns of Cholesky factors at time instant n is computed recursively from previous Cholesky factor and submatrix of that Cholesky factors are updated recursively from previous Cholesky factor i.e. at time instant n − 1. Conventional Cholesky algorithm mentioned here are introduced for normal matrices whereas proposed matrix is well suited for block matrices and simulations are shown for that only. Algorithm 4 Recursive Cholesky Update : RChol R = LDLH Initialization: for k = 1 , D1 (n) = rn 00 T n n n A1 (n) = [r00 , r10 , . . . r(N −1)0 ] T n n Ã1 (n) = [0, r10 , . . . r(N −1)0 ] 0 Order Updates on A s: for k = 2 IV. C ONCLUSION Ak (n) := ZM Ak−1 (n − 1) − Ãk−1 (n)k̃ref (n) Convention methods of Cholesky factorization requires the correlation matrix which needs inner product. While the recursive modied Cholesky algorithm (RChol) algorithm is an explicit way to recursively calculating the pseudo-inverse of the matrices without estimating the correlation matrix. It requires less number of iteration which avoids error propagation through column updates. The RChol algorithm has most of the use in calculating the pseudo-inverse of the of a time-varying matrix which is applicable to SIMO/MIMO, CDMA, OFDM, etc. wireless communication systems. Dk (n) = Dk (n − 1)[IM − kref (n)k̃ref (n)] for k > 2 , Ãk−1 (n) = 0 Ak (n) = ZM Ak (n − 1) Dk (n) = ZM Dk (n − 1) Ã1 (n)(2,:) Scaling Factors: kref (n) = A1 (n−1)(1,:) 0 kref (n) D1 (n−1) k̃ref (n) = D̃1 (n) III. S IMULATION RESULTS To compare proposed the RChol algorithm with Schur algorithm, we compared the result of both the algorithm with theoretical results. Fig. 1. Show the ratio and difference of matrices R̂N , R̂RChol and R̂Schur , when the correlation matrix is unknown. That has the application in blind channel and or data estimation. Fig. 1 (a) and (b) shows the maximum error for the RChol algorithm, [R̂N − R̂RChol ] is 0.6 while for the Schur algorithm, [R̂N − R̂Schur ] is 4 i.e. nearly 6 times the RChol algorithm. In case of ratio Fig. 1 (a) and (b) shows the maximum ratio for the RChol algorithm, [R̂N ./R̂RChol ] is 45 while for the Schur algorithm, [R̂N ./R̂Schur ] is 1500. 0 0 5 10 15 20 25 30 35 5 10 15 20 25 30 35 40 0 0 40 0 0 5 10 15 20 25 30 35 5 10 15 20 25 30 35 40 0 40 0 45 2500 0.6 5 1 5 40 5 0.4 10 35 10 5 10 2000 10 0 30 15 15 15 15 0.2 25 1500 -1 20 20 20 20 0 15 -2 25 25 -0.4 30 0 -5 40 40 0 40 40 (a) (b) 0 5 10 15 20 25 30 35 (c) 5 10 15 20 25 30 35 (d) 40 0 0 40 0 0.03 0 0 0 10 15 20 25 30 35 5 1 15 20 25 30 35 40 1 10 0.8 0.02 15 15 15 20 10 1.05 10 15 5 0 40 5 0.025 10 -0.5 -1 5 5 5 0.95 0.6 20 0.015 20 20 -1.5 25 25 0.9 0.4 30 30 -2 25 25 0.01 30 0.005 0.2 0.85 35 35 35 35 -2.5 40 0 0.8 40 (e) 500 35 35 -4 -0.6 5 35 35 30 1000 30 -3 30 0 25 10 30 10 20 25 -0.2 40 0 40 (f) (g) (h) Fig. 1: Comparisons of RChol algorithm Vs Schur Algorithm for the unknown and known correlation matrix R, (a, e): Proposed Algorithm ( Difference), (b, f): Schur Algorithm( Difference), (c, g):Proposed Algorithm (Ratio), (d, h): Schur Algorithm (Ratio) Fig. 1 Show the ratio and difference of matrices R̂N , R̂RChol and R̂Schur , when the correlation matrix is known. Fig. 1 (a) and (b) shows that the maximum error for the RChol algorithm, [R̂N − R̂RChol ] is 2.5 while for the Schur 4 algorithm, [R̂N − R̂Schur ] is 0.03 i.e. nearly 6 times the RChol algorithm. In case of ratio Fig. 1 (e) and (f) shows that the maximum ratio for the RChol algorithm, [R̂N ./R̂RChol ] is 1.15 while for the Schur algorithm, [R̂N ./R̂Schur ] is 1. From Fig. 1 it can be concluded that the Schur algorithm is best suited when the correlation matrix is known, but leads to huge error propagation through the column when R is unknown and cannot be applied for blind channel estimation. In converse, the RChol algorithm is best suited for blind channel estimation and reduces error propagation through the column. V. Pawar and K. Naik (DIAT, Pune, India) E-mail: [email protected] R EFERENCES [1] G. Golub and C. Van Loan: ’Matrix computations’, 2012 [2] V. Pawar and K. Krishna Naik: ’Blind multipath time varying channel estimation using recursive Cholesky update’, AEU - Int. J. Electron. Commun., 2016, 70, no. 1, pp. 113-119 [3] R. Hunger and T. Report: ’Floating Point Operations in Matrix-Vector Calculus’, Matrix, 2007. [4] C. P. Rialan and L. L. Scharf: ’Fast algorithms for computing QR and Cholesky factors ofToeplitz operators’, IEEE Trans. Acoust., 1998 36, pp. 1740-1748	0
arXiv:1605.01298v1 [math.AC] 4 May 2016 THE EUCLIDEAN CRITERION FOR IRREDUCIBLES PETE L. CLARK Abstract. We recast Euclid’s proof of the infinitude of prime numbers as a Euclidean Criterion for a domain to have infinitely many atoms. We make connections with Furstenberg’s “topological” proof of the infinitude of prime numbers and show that our criterion applies even in certain domains in which not all nonzero nonunits factor into products of irreducibles. 1. Introduction This article has its genesis in a graduate VIGRE research group taught by Paul Pollack and me in Fall 2015: Introduction to the Process of Mathematical Research. Rather than concentrating on a fixed topic preselected by us, the goal was to guide students through the process of selecting and performing research on their own. One technique we tried to inculcate is exploitation of the many-to-one relation between theorems and proofs. A good theorem has several proofs, and you will know two proofs are different when can be used to prove further theorems the other cannot. In our first meeting, Pollack and I presented seven proofs of Euclid’s Proposition IX.20: there are infinitely many prime numbers. My first proof: suppose given a domain R that is not a field, in which each nonzero nonunit factors into irreducibles and whenever x ∈ R is a nonzero nonunit then x + 1 is not a unit; then there is at least one irreducible element f1 , and given irreducibles f1 , . . . , fn , by factoring f1 · · · fn +1 we get a new irreducible element. It was pointed out that this argument, though correct, does not imply Euclid’s result: x = −2 is a problem. Some salvages were suggested: in Z it is enough to replace f1 · · · fn by −f1 · · · fn , if necessary. Here we present a general fix – a Euclidean Criterion for a domain to have infinitely many nonassociate irreducibles – and explore its consequences. We soon find ourselves on a scenic tour of 20th century mathematics, as we engage with work of Jacobson, Furstenberg, Cohen-Kaplansky and Anderson-Mott, among others. 1.1. Acknowledgments. Thanks to all members of the 2015-2016 Introduction to Mathematical Research UGA VIGRE group. Conversations with Saurabh Gosavi, Noah Lebowitz-Lockard, Robert Samalis, Lee Troupe and Lori D. Watson were helpful. My group coleader Paul Pollack made key contributions: first, he emphasized that the Euclidean Criterion automatically yields pairwise comaximality. Second, Theorem 2.9 was inspired by [P, Thm. 1.16], and though I came up with the statement, I could prove it only in various special cases. The proof included here is his. I am grateful to two anonymous referees for their careful, detail-oriented reports. In particular, Example 4.19 was suggested by the “first” referee. Date: May 5, 2016. 1 2 PETE L. CLARK 2. The Euclidean Criterion 2.1. A primer on factorization in domains. By a ring we will mean a commutative ring with a multiplicative identity. We denote the set of nonzero elements of R by R• . An element x ∈ R is a unit if there is y ∈ R such that xy = 1. We denote the group of units of R by R× . For a subset S of a ring R, we denote by (S) the ideal of R generated by S. (As is standard, we write (x1 , . . . , xn ) for ({x1 , . . . , xn }). Ideals I and J in R are comaximal if I + J = R. Elements a, b ∈ R are comaximal if (a) and (b) are comaximal: (a, b) = R. An indexed family of ideals {Ii } is pairwise comaximal if Ii + Ij = R for all i 6= j, and similarly for pairwise comaximal elements. A domain is a nonzero ring in which x, y 6= 0 =⇒ xy 6= 0. For x, y ∈ R we say x divides y and write x \| y if there is c ∈ R such that cx = y. Elements x and y are associates if y = ux for some u ∈ R× . An element x of a domain is irreducible if it is a nonzero nonunit and x = yz implies y ∈ R× or z ∈ R× . A prime element p ∈ R is an element p ∈ R• for which (p) is a prime ideal. Thus a nonzero nonunit p is prime if and only if p \| ab =⇒ p \| a or p \| b. An atom in a domain R is a principal ideal (x) generated by an irreducible element x. Thus two irreducibles of a domain R determine the same atom if and only if they are associate. (It is more common in the literature for the terms “atom” and “irreducible” to be fully synonymous, but this minor distinction is convenient for our purposes: usually we will count to count irreducibles in a domain up to associates, but sometimes we will want to count irreducibles.) A Furstenberg domain is a domain R in which every nonzero nonunit has an irreducible divisor.1 An atomic domain is a domain R in which for every nonzero nonunit x ∈ R there are irreducible elements f1 , . . . , fn such that x = f1 · · · fn . A unique factorization domain (UFD) is an atomic domain such that if f1 , . . . , fm , g1 , . . . , gn are irreducibles such that f1 · · · fm = g1 · · · gn , then m = n and there is a bijection σ : {1, . . . , m} → {1, . . . , n} such that (fi ) = (gσi ) for all 1 ≤ i ≤ m. Prime elements are irreducible. In general the converse is false! An atomic domain is a UFD iff every irreducible is prime [Cl-CA, Thm. 15.8]. The terminology can be confusing in light of the definition of a prime number p as a positive integer not divisible by any 1 < n < p: this means p is irreducible in Z. But Euclid showed p \| ab =⇒ p \| a or p \| b. From this one can easily show the Fundamental Theorem of Arithmetic: Z is a UFD. A principal ideal domain (PID) is a domain in which each ideal is generated by a single element. Every PID is a UFD. It follows from the Euclidean algorithm that Z is a PID. A Bézout domain is a domain in which every finitely generated ideal is principal. A ring is Noetherian if all of its ideals are finitely generated. Noetherian domains are atomic [Cl-CA, Prop. 15.3]. Thus a PID is precisely a Noetherian Bézout domain. A Dedekind domain is a domain in which each nonzero proper ideal factors uniquely into prime ideals. A domain is Dedekind iff it is Noetherian, of dimension at most one – every nonzero prime ideal is maximal – and integrally closed – every element of the fraction field which satisfies a monic polynomial with coefficients in R lies in R [Cl-CA, Thm. 20.10]. Working in a domain rather than a general ring confers certain advantages: 1The explanation for the terminology comes in §3.1. THE EUCLIDEAN CRITERION FOR IRREDUCIBLES 3 Fact 1. a) Every nonzero ideal in a ring contains a nonzero principal ideal. b) If R is a domain and α ∈ R• , x ∈ R 7→ αx gives a bijection from R to (α). c) Thus for every nonzero ideal I of a domain R we have #I = #R. d) For nonzero ideals I and J of R, I ∩ J contains IJ and thus is nonzero. 2.2. The Euclidean Criterion. A ring R satisfies Condition (E) if for all x ∈ R• , there is y ∈ R such that yx + 1 ∈ / R× . In other words, if x 6= 0 then 1 + (x) 6⊂ R× . By Fact 1a) this is equivalent to: if I is a nonzero ideal of R then 1 + I 6⊂ R× , though we will defer consideration of this restatement until later on. Example 2.1. a) The ring Z satisfies Condition (E). Indeed, Z× = {±1}, so for x ∈ Z• , take y = 1 if x is positive and y = −1 if x is negative); then yx ≥ 1 so yx + 1 ≥ 2. b) For any domain R, the polynomial ring R[t] satisfies Condition (E). Indeed, (R[t])× = R× , so for any x ∈ R[t]• , take y = t. c) R = Z[i] satisfies Condition (E). Indeed Z[i]× = {1, i, −1, −i}, so this is geometrically clear: for any x ∈ Z[i]• , if we multiply it by a y with large enough \|y\|, then yx will be much more than 1 unit away from any point on the unit circle. Proposition 2.2. A domain R with #R > #R× satisfies Condition (E). Proof. For x ∈ R• , the map ι : R → R given by y 7→ yx + 1 is an injection. Thus #ι(R) = #R > #R× , so it cannot be that ι(R) ⊂ R× . And here we go: Theorem 2.3. (The Euclidean Criterion) Let R be a domain, not a field, satisfying Condition (E). a) There is an infinite sequence {an }∞ n=1 of pairwise comaximal nonunits. b) If R is also Furstenberg, it admits an infinite sequence {fn }∞ n=1 of pairwise comaximal irreducibles. Thus {(fn )}∞ n=1 is a sequence of distinct atoms in R. Proof. a) By induction on n. Let a1 ∈ R be a nonzero nonunit. Having chosen a1 , . . . , an pairwise comaximal, by Condition (E) there is y ∈ R such that an+1 := ya1 · · · an + 1 ∈ / R× . Clearly (ai , an+1 ) = R for all 1 ≤ i ≤ n. b) By induction on n. Since R is Furstenberg and not a field, it has an irreducible f1 . Having chosen pairwise comaximal irreducibles f1 , . . . , fn , by Condition (E) there is y ∈ R such that x = yf1 · · · fn + 1 is a nonzero (since f1 ∈ / R× ) nonunit, so x has an irreducible factor fn+1 . For all 1 ≤ i ≤ n we have Y fj )fi , 1 = (x/fn+1 )fn+1 − (y j6=i so fi , fn+1 are comaximal. Finally, if x and y are pairwise comaximal irreducibles, then (x), (y) ( R and (x) + (y) = (x, y) = R, so we must have (x) 6= (y). Here are two applications of the Euclidean Criterion. The first two are immediate. Theorem 2.4. a) For any domain R, R[t] has infinitely many atoms. b) In particular, let D be a UFD and let R = D[t1 , . . . , tn ]. Then R is a UFD satisfying Condition (E), so R has infinitely many nonassociate prime elements. c) The Gaussian integers Z[i] have infinitely many atoms. Since Z[i] is a PID, there are infinitely many nonassociate prime elements. 4 PETE L. CLARK Theorem 2.5. Let R be a Furstenberg domain, not a field, such that #R > #R× . Then R has infinitely many atoms. Theorem 2.6. Let R be a Furstenberg domain, let I be the set of all irreducible elements of R. Then I is either empty (if R is a field) or infinite (otherwise). Proof. Assume I 6= ∅ and fix f ∈ I. If R× is finite, Theorem 2.5 yields infinitely many atoms. If R× is infinite, then {uf \| u ∈ R× } is an infinite subset of I. 2.3. Supplement: Irreducibles in Residue Classes. We switch from an ancient theorem to matters of contemporary interest if we ask for infinitely many primes satisfying certain additional conditions. Here is a result along these lines, relatively modest over Z, but of a general algebraic nature. Lemma 2.7. Let a, b, c be elements of a ring R. If (a, b) = R and c \| a + b, then (a, c) = (b, c) = R. Proof. Let d ∈ R be such that cd = a + b. Then (a, c) ⊃ (a, cd) = (a, a + b) = (a, b) = R. Lemma 2.8. Let R be a domain, not a field, satisfying Condition (E). For any at + b ∈ R[t] with a ∈ R• , there is x ∈ R such that ax + b is a nonzero nonunit. Proof. Put P (t) = at + b. If b = 0, take any nonzero nonunit x ∈ R. If b ∈ R× , by Condition (E) there is x ∈ R such that b−1 ax + 1 ∈ / R× so P (x) = b(b−1 ax + 1) is a nonzero nonunit. If b ∈ R is a nonzero nonunit, take x = 0. The proof of the following result was suggested to me by Paul Pollack. Theorem 2.9. Let R be an atomic domain satisfying Condition (E), let I be a nonzero ideal of R, and let H be a proper subgroup of (R/I)× . Then there are infinitely many pairwise comaximal irreducibles f such that the class of f modulo I lies in (R/I)× \ H. Proof. Let r : R → R/I be the quotient map, let α ∈ R be such that r(α) ∈ (R/I)× \ H, and let β ∈ R be such that αβ − 1 ∈ I \ {0}. Inductively, assume that we have pairwise irreducibles f1 , . . . , fn of R such that (fi , α) = (fi , I) = R for all i and such that r(fi ) ∈ / H. Let P (t) = (αt + 1)(αβ − 1)f1 · · · fn + α ∈ R[t]. (We need to include the base case n = 0, and in this case f1 · · · fn = 1.) By Lemma 2.8 there is x ∈ R such that y = (αx + 1)(αβ − 1)f1 · · · fn + α is a nonzero nonunit, so we get an irreducible factorization y = g1 · · · gs with s ≥ 1. Then r(g1 ) · · · r(gs ) = r(y) = r(α) ∈ (R/I)× \ H, so (gj , I) = 1 for all j and there is at least one gj , say g1 , such that r(g1 ) ∈ / H. Now g1 cannot be associate to any fi ; if so g1 and hence also fi would divide α: if α ∈ R× this contradicts the irreducibility of fi ; if not, this contradicts (fi , α) = 1. THE EUCLIDEAN CRITERION FOR IRREDUCIBLES 5 Moreover y ≡ −f1 · · · fn (mod α) so y ∈ (R/α)× , hence also g1 ∈ (R/α)× , i.e., (g1 , α) = R. Finally, since (αx + 1)(αβ − 1)f1 · · · fn ≡ −f1 · · · fn (mod α), we have ((αx + 1)(αβ − 1)f1 · · · fn , α) = R, so by Lemma 2.7 we have (g1 , (αx + 1)(αβ − 1)f1 · · · fn ) = R so (g1 , fi ) = R for all i. Thus we may take fn+1 = g1 , completing the induction. When R = Z, we get: for any proper subgroup H ( (Z/N Z)× , there are infinitely many prime numbers p such that ±p (mod N ) ∈ / H. Moreover, in this classical case one can run the argument with positive integers only and so get rid of the annoying ±. This is a special case of Dirichlet’s theorem on primes in arithmetic progressions. It is an observation of A. Granville – unpublished by him, but reproduced in [P, Thm. 1.16] – that this case can be proved in an elementary “Euclidean” way. The special case of trivial H – for all N ≥ 3 there are infinitely many primes p 6≡ 1 (mod N ) – is older and better known. It is also simpler – just consider N p1 · · · pn−1 − 1. This case does not use that Z is a UFD, but Granville’s argument does. The most auspicious replacement for coprimality arguments is by comaximality, and that is what we’ve done here. 3. A “Topological” Interlude 3.1. Furstenberg’s Lemma. In this section we will give several proofs of the following result. Theorem 3.1. Let R be a Furstenberg domain with at least one and only finitely many irreducibles f1 , . . . , fn . Then: a) We have #R× = #R. b) More precisely there is a nonzero ideal I of R such that 1 + I ⊂ R× . Theorem 3.1 is the contrapositive of part b) of the Euclidean Criterion, without the information on comaximality. The proofs that we give here are inspired by the famous paper of H. Furstenberg [Fu55]. The essential core of his argument is the observation that in Z the set of elements not divisible by any prime number is ±1. Notice that has nothing to do with the natural ordering of Z that underlies most of the classical proofs of Euclid’s Theorem. In fact the property of Z being used is that Z is a Furstenberg domain. Lemma 3.2. (Furstenberg’s Lemma) T a) A domain R is a Furstenberg domain iff R× = f irreducible R \ (f ). b) In a FurstenbergTdomain with at least one and only finitely many irreducibles f1 , . . . , fn , we have ni=1 (R \ (fi )) = R× . The proof is virtually immediate and is left to the reader. 3.2. Following Furstenberg. Let R be a domain. By Fact 1d), for each x ∈ R, the family C(x) = {x + I \| I is a nonzero ideal of R} 6 PETE L. CLARK is closed under finite intersections, so {C(x)}x∈X is a system of neighborhood bases for a topology on R – let us call it the adic topology – in which U ⊂ R is open iff for all x ∈ U there is a nonzero ideal I with x + I ⊂ U . By Fact 1c), every nonempty open has cardinality #R. Proof of Theorem 3.1: let R be a Furstenberg domain with at least one and only finitely many irreducibles f1 , . . . , fn . Then each (fi ) is open, hence its complement R \ (fi ), being a union of cosets of (fi ), is also open. By Furstenberg’s Lemma T R× = ni=1 (R \ (fi )) is open. Since 1 ∈ R× , we have #R× = #R. More precisely, R× ⊃ 1 + I for some nonzero ideal of R. 3.3. Following Cass-Wildenberg. Let R be a domain, and let F2 be the field of two elements. For an ideal I of R, a function f : R → F2 is I-periodic if f (x + y) = f (x) for all x ∈ X and y ∈ I. Lemma 3.3. Let R be a domain, and let I, I1 , . . . , In be nonzero ideals of R. a) If I2 ⊂ I1 and f : R → F2 is I1 -periodic, it is also I2 -periodic. b) If for all 1 ≤ i ≤ n, fi : R → F2 is Ii -periodic, then the pointwise product f1 · · · fn : R → F2 is I1 · · · In -periodic. c) If f : R → F2 is I-periodic, then for all x ∈ R, we have #{y ∈ R \| f (y) = f (x)} = #R. Proof. a) This is immediate T from the definition. T b) Certainly f1 · · · fn is ni=1 Ii -periodic, and ni=1 Ii ⊃ I1 · · · In . Apply part a). c) Choose a nonzero α ∈ I. Then f (x + Rα) = f (x), and #Rα = #R. Proof of Theorem 3.1: Step 1: For 1 ≤ i ≤ n, let χi : R → F2 be the characteristic function of (fi ); put n Y χ= (1 − χi ). i=1 Each χi is (fi )-periodic, hence so too is 1 − χi , and thus χ is (f1 · · · fn )-periodic. T Moreover χ is the characteristic function of ni=1 (R \ (fi )) = R× . Step 2: Since χ(1) = 1, #R× = {x ∈ R \| χ(x) = 1} = #R: part a). Step 3: More precisely χ(1 + Rf1 · · · fn ) = 1, so Rf1 · · · fn + 1 ⊂ R× : part b). 3.4. Following Mercer. Let R be a domain. Call a subset X ⊂ R lovely if it is of the form x + I for x ∈ R and a nonzero ideal I of R, i.e., if it is a coset of a nonzero ideal. Call a subset X ⊂ R pleasant if it is a union of lovely subsets. If I is a nonzero ideal of R, then R\ I is a union of cosets of I hence pleasant. If X, Y ⊂ R are pleasant sets and x ∈ X ∩ Y , there are nonzero ideals I, J of R such that x + I ⊂ X and x + J ⊂ Y . By Fact 1d) x+(I ∩J) = (x+I)∩(x+J) is a lovely subset of X ∩Y containing x. So X ∩Y is pleasant. By Fact 1c), every nonempty pleasant subset has cardinality #R. Proof of Theorem 3.1: let R be a Furstenberg domain with at least one and only finitely many irreducibles f1 , . . . , fn . By Furstenberg’s Lemma, R× is the finite intersection of complements of nonzero ideals so is pleasant. Since 1 ∈ R× , we have #R× = #R. More precisely, R× ⊃ 1 + I for some nonzero ideal of R. THE EUCLIDEAN CRITERION FOR IRREDUCIBLES 7 3.5. Debriefing. The three proofs given above are generalizations of the proofs of Euclid’s Theorem given by Furstenberg [Fu55], Cass-Wildenberg [CW03] and Mercer [Me09]. The latter two works take the detopologization of Furstenberg’s proof as their goal. Our presentation of the argument of §3.4 differs superficially from Mercer’s. We chose the words “lovely” and “pleasant” precisely because they do not have a commonly understood technical mathematical meaning: had we said “basic” and “open” then the reader’s attention would have been drawn to the fact that since the basic sets are closed under finite intersections, they form the base of a topology. Mercer’s exposition takes pains to point out that the underlying fact here is just that finite intersections of unions are unions of finite intersections. Of course this is a basic logical principle: conjunctions distribute over disjunctions and conversely. Like many basic logical principles it is completely innocuous when used in context (as in our version of the argument). That the pleasant sets form a topology on R is no more and no less than a crisp enunciation of the facts we need to check in the first part of the proof. I find it quite striking (and pleasant!) that the facts can be enunciated in this way, but I must now agree with those who have claimed that there is no essential topological content in Furstenberg’s argument.2 The use of periodic functions involves slightly more packaging, but of a standard kind: it is well known that the Boolean ring 2R of subsets of R can be represented as the ring Maps(R, F2 ) with pointwise addition and multiplication. We recommend wikipedia and Glaymann [Gl67] as references. Glaymann develops this correspondence and applies it to prove such identities as A∆B = C ⇐⇒ B∆C = A ⇐⇒ C∆A = B...in a manner intended to be used in the high school classroom. This is an interesting snapshot of “the new math” near its zenith. 3.6. The Ubiquitous Theorem. Here is a result that complements Theorem 3.1. It is not deep, but it will play a recurring role for us as a common intersection of various constructions and themes. The first proof that we give follows the “topological conceit” of this section. We will give other, simpler, proofs later on. Theorem 3.4. Let R be a domain, not a field, with only finitely many maximal ideals m1 , . . . , mn . Then: a) We have #R× = #R. b) More precisely there is a nonzero ideal I of R such that 1 + I ⊂ R× . Proof. We endow R with the topology for which, for x ∈ R, C(x) = {x + m \| m is a maximal ideal of R} is a neighborhood subbase at x: that is, U ⊂ R is open iff for all x ∈ U there is a subset J ⊂ {1, . . . , n} such that \ \ (x + mi ) = x + mi ⊂ U. i∈J i∈J 2Furstenberg does not claim a topological proof of the infinitude of the primes but rather a “topological” proof of the infinitude of the primes. 8 PETE L. CLARK T Fact 1 gives i∈J mi ) (0), so every nonempty open has cardinality #R. Each R \ mi , being a union of cosets of mi , is also open. Therefore R× = n \ (R \ mi ) i=1 × is open. Since 1 ∈ R× we have T #R =×#R. More precisely T there is a subset J ⊂ {1, . . . , n} such that 1 + i∈J mi ⊂ R , and thus also 1 + ni=1 mi ⊂ R× . 3.7. Supplement: Further Topologies on a Domain. Here is a common generalization of Theorems 3.1 and 3.4: let J be a family of nonzero T ideals of a domain R,T and suppose there are I1 , . . . , In ∈ J such that R× = ni=1 (R \ Ii ). Then 1 + ni=1 Ii ⊂ R× , so in particular #R× = #R. Look again at Theorem 3.1: instead of taking J to be the family of all nonzero ideals, we could take J = {(f1 ), . . . , (fn )} and endow R with the unique translationinvariant topology with J as a neighborhood subbase at 0. This coarsens the adic T topology3 so that being open yields the sharper conclusion 1 + ni=1 (fi ) ⊂ R× . In × particular 1 + (f1 · · · fn ) ⊂ R . We are back to a version of Euclid’s argument. The adic topology on Z is not very interesting as a topological space: it is countably infinite, metrizable, totally disconnected and without isolated points, hence homeomorphic to the Euclidean topology on Q. In [Go59], Golomb proved Euclid’s Theorem using the topology on Z+ with base the one-sided arithmetic progressions {an + b \| n ∈ Z+ } for coprime a, b ∈ Z+ . Golomb’s topology makes Z+ into a countably infinite connected Hausdorff space...which is already interesting. In a domain R that is not a field, we may consider the Golomb topology with neighborhood base at x ∈ R given by C(x) = {x + I \| I is a nonzero ideal with (x, I) = R}. In this topology every maximal ideal is closed, so in a domain that is not a field with only finitely many maximal ideals m1 , . . . , mn , R× is open and thus contains 1 + I for some nonzero ideal I. We get another proof of Theorem 3.4. The Golomb topology is never Hausdorff: in fact {0} = R. However, the induced topology on R• can be (it is for Z). We leave further exploration to the reader. 4. Connections With Ideal Theory For a ring R, we denote by MaxSpec R the set of all maximal ideals of R. 4.1. Comaximal Ideals. Lemma 4.1. Let {In }∞ n=1 be a sequence of pairwise comaximal proper ideals in a ring R. Then MaxSpec R is infinite. Proof. For n ∈ Z+ , let mn be a maximal ideal containing In . If for n1 6= n2 we had mn1 = mn2 then R = In1 + In2 ⊂ mn1 , contradiction. In particular, part a) of the Euclidean Criterion implies that a domain that is not a field and that satisfies Condition (E) has infinitely many maximal ideals. Thus we get another proof of Theorem 3.4....but by no means our last. 3The adic topology on a domain is always Hausdorff, but in a Furstenberg domain with finitely many irreducibles, this new topology is not. THE EUCLIDEAN CRITERION FOR IRREDUCIBLES 9 4.2. Euclid Meets Jacobson. Now is the time to examine the more explicitly ideal-theoretic statement of Condition (E): for all nonzero ideals I, we have 1 + I 6⊂ R. Some readers will now see – or will have already seen – the connection with the Jacobson radical, but we will not assume a prior familiarity. In fact we will use the Euclidean Criterion to motivate a self-contained discussion of this and other ideal-theoretic concepts. Proposition 4.2. [Cl-CA, Prop. 4.14] For a ring R, let \ m, J(R) = m∈MaxSpec R the Jacobson radical of R. For x ∈ R, the following are equivalent: (i) x ∈ J(R). (ii) For all y ∈ R, yx + 1 ∈ R× . Proof. (i) =⇒ (ii): By contraposition: suppose there is y ∈ R such that z = yx + 1 ∈ / R× . Then z lies in some maximal ideal m. If also x ∈ m, then yx ∈ m and thus also z − yx = 1 ∈ m, contradiction. So x does not lie in m and thus x ∈ / J(R). (ii) =⇒ (i): Again by contraposition: suppose that there is a maximal ideal m such that x ∈ / m. Then m ( (m, x), so (m, x) = R. It follows that there is m ∈ m and y ∈ R such that m + yx = 1. Thus (−y)x + 1 = −m ∈ m so is not a unit. We get immediately: Corollary 4.3. A ring R satisfies Condition (E) iff J(R) = (0). This gives a third proof of Theorem 3.4: if R has only finitely many maximal ideals m1 , . . . , mn , then n n Y \ mi ) {0}. mi ⊃ J(R) = i=1 i=1 Apply Corollary 4.3. A ring with zero Jacobson radical is called semiprimitive.4 4.3. Some Questions and Some Answers. We now raise some natural questions...and answer them. Question 4.4. In part b) of the Euclidean Criterion, must we assume that R is a Furstenberg domain? Question 4.5. A semiprimitive domain, not a field, has infinitely many maximal ideals. Must a domain with infinitely many maximal ideals be semiprimitive? Question 4.6. Let R be a Furstenberg domain. a) If R is not semiprimitive, can it still have infinitely many atoms? b) Can R have finitely many maximal ideals and infinitely many atoms? 4Or Jacobson semisimple or J-semisimple. 10 PETE L. CLARK Example 4.7. The ring Z of all algebraic integers is not a Furstenberg domain. In fact it is an antimatter domain: there are no irreducibles whatsoever: if z is an algebraic integer then so is z 1/2 , so we can always factor z = z 1/2 z 1/2 . Moreover Z × is not a field: for all integers n ≥ 2, if n ∈ Z then n1 ∈ Z ∩ Q = Z, contradiction. If I is a nonzero ideal of Z then the constant coefficient of the minimal polynomial of a nonzero element α ∈ I is a nonzero integer in I. It follows that if J(Z) 6= 0 then there is N ∈ Z+ that is contained in every m ∈ MaxSpec Z. Choose a prime number p ∤ N . Then p is not a unit in Z – otherwise 1p ∈ Z ∩ Q = Z – so there is at least one maximal ideal mp of Z containing p. (In fact the set of maximal ideals of Z containing p has continuum cardinality.) Then mp ⊃ (N, p) = Z: contradiction. So the answer to Question 4.4 is yes: a semiprimitive domain that is not a field can have no irreducibles whatsoever. The following result answers Questions 4.5 and 4.6 for Dedekind domains and shows that the Euclidean Criterion is, in principle, completely efficacious in determining whether a Dedekind domain has infinitely many atoms. Theorem 4.8. For a Dedekind domain R that is not a field, the following are equivalent: (i) R is semiprimitive. (ii) R has infinitely many maximal ideals. (iii) R has infinitely many atoms. Proof. We know (i) =⇒ (ii) in any domain. (ii) =⇒ (i): in a Dedekind domain, any nonzero element is contained in only finitely many maximal ideals. So in fact for any infinite subset M ⊂ MaxSpec R T we have m∈M m = (0). (i) =⇒ (iii): Dedekind domains are Noetherian, hence Furstenberg domains, so the Euclidean Criterion applies. (iii) =⇒ (i): By contraposition: a Dedekind domain with finitely many maximal ideals is a PID [Cl-CA, Thm. 20.6], and in a PID there is no distinction between maximal ideals, principal ideals generated by prime elements, and atoms. Question 4.9. Let K be a number field, with ring of integers ZK . The set of prime numbers is an infinite sequence of pairwise comaximal nonunits of ZK , so (as is well known!) ZK has infinitely many prime ideals and thus is semiprimitive. When K = Q or is imaginary quadratic, the finiteness of Z× K leads to a direct verification of Condition (E). Is there a similarly direct verification for all K? This is a question we will leave to the reader to address. Proposition 4.10. Let R be a Noetherian domain of dimension at most one (nonzero prime ideals are maximal). If MaxSpec R is infinite, then R is semiprimitive and thus has infinitely many pairwise comaximal irreducibles. Proof. If R is not semiprimitive, then every maximal ideal m of R is a minimal prime ideal of R/J(R). Since R is Noetherian, so is R/J(R), and a Noetherian ring has only finitely many minimal prime ideals [Cl-CA, Thm. 10.13]. A Jacobson ring is a ring in which every prime ideal is the intersection of the maximal ideals containing it. Since in a domain (0) is prime, a Jacobson domain THE EUCLIDEAN CRITERION FOR IRREDUCIBLES 11 must be semiprimitive. Any quotient of a Jacobson ring is again a Jacobson ring. If R is a Jacobson ring and S is a commutative, finitely generated R-algebra then S is a Jacobson ring [Cl-CA, Thm. 12.15, 12.21]. So: Theorem 4.11. a) A Jacobson Furstenberg domain that is not a field has infinitely many pairwise comaximal irreducibles. b) Let F be a field, and let p be a prime but not maximal ideal of F [t1 , . . . , tn ]. Then the ring R = F [t1 , . . . , tn ]/p – i.e., a coordinate ring of an integral affine variety of positive dimension – has infinitely many pairwise comaximal irreducibles. c) A domain R that is finitely generated over Z and not a field has infinitely many pairwise comaximal irreducibles. To sum up: if we want to see a domain that has infinitely many maximal ideals but is not semiprimitive, it cannot be finitely generated over a field, and if Noetherian it must have a nonzero prime ideal that is not maximal. This cues us up for the following example, which gives a negative answer to Question 4.5. Example 4.12. Consider the ring Z[[t]] of formal power series with integral coefficients. It is not hard to show that Z[[t]] is an atomic domain. In fact Z[[t]] is a Noetherian UFD [Cl-CA, Thm. 15.32]. Since 1 + (t) ⊂ Z[[t]]× , the Jacobson radical J(Z[[t]]) contains (t) and is thus nonzero. Since J(Z[[t]])) 6= (0), the hypotheses of the Euclidean Criterion do not apply. Nevertheless there are infinitely many pairwise comaximal prime elements, namely the prime numbers! Hence there are infinitely many maximal ideals. Here we could have replaced Z with any PID with infinitely many maximal ideals. Thus the answer to Question 4.6a) is yes: moreover a nonsemiprimitive domain can have infinitely many comaximal irreducibles. Example 4.13. Let k be a field. Recall that k[x, y] is a UFD, and let K = k(x, y) be its fraction field. Let R be the subring of k(x, y) consisting of rational functions f (x,y) g(x,y) that, when written in lowest terms, have g(0, 0) 6= 0. Then R is itself a UFD – factorization in R proceeds as in k[x, y] except that the prime elements p(x, y) ∈ k[x, y] such that p(0, 0) 6= 0 become units in R – in which an element (x,y) is a unit iff f (0, 0) 6= 0. Thus m = { fg(x,y) \| f (0, 0) = 0} is the unique maximal ideal, so J(R) = m and R is very far from being semiprimitive. Nevertheless it has infinitely many prime elements, e.g. {y − xn }∞ n=1 . In more geometric language, the irreducibles are the irreducible curves in the affine plane passing through (0, 0). Thus the answer to Question 4.6b) is yes. However, there is more to say. The preceding example can be vastly generalized using the following striking result. Theorem 4.14. (Cohen-Kaplansky [CK46]) Let R be an atomic domain with finitely many atoms. Then: a) R has only finitely many prime ideals. b) R is Noetherian. c) Every nonzero prime ideal of R is maximal. Proof. a) In an atomic domain R, whenever a prime ideal p of R contains a nonzero element x, we may factor x = f1 · · · fr into irreducibles and thus see that p contains some irreducible element f dividing x. Thus, given any set of generators of a prime ideal p we can replace it with a set of irreducible generators. In a set of generators 12 PETE L. CLARK of an ideal, replacing each element by any one of its associates does not change the ideal generated, and thus if we have only finitely many nonassociate irreducibles we can only generate finitely many prime ideals. b) It follows from the proof of part a) that every prime ideal of R is finitely generated. By a famous result of Cohen [Cl-CA, Thm. 4.26], all ideals are finitely generated. (This is an instance of the prime ideal principle of Lam-Reyes [LR08].) c) If not, there are prime ideals (0) ( p1 ( p2 . Since R is Noetherian, this implies there are infinitely many prime ideals between (0) and p2 [Cl-CA, Cor. 8.46]. A Cohen-Kaplansky domain is an atomic domain with finitely many atoms. The work [CK46] does not give a complete classification: we are left with the case of a Noetherian domain R with finitely many nonzero prime ideals, all of which are maximal. If R is a Dedekind domain, then by Theorem 4.8 there are only finitely many atoms. So the remaining case is when R is not integrally closed in its fraction field, in which case the integral closure R is a Dedekind domain with finitely many prime ideals [Cl-CA, Cor. 18.8]. One might expect that this forces R to be Cohen-Kaplansky. This need not be the case! Example 4.15. Let k be a field, and consider the subring R = k[[t2 , t3 ]] = k + t2 k[[t]] P n of the formal power series ring k[[t]]. For 0 6= f = ∞ n=0 an t ∈ k[[t]], we define v(f ) to be the least n such that an 6= 0. Then v is a discrete valuation on k[[t]], and the only nonzero prime ideal of k[[t]] is (t) = {f ∈ R \| v(f ) > 0} ∪ {0}. In particular, k[[t]] is a PID. So is the (isomorphic!) subring k[[t2 ]], and {1, t3 } is a generating set for R as a k[[t2 ]]-module, so by standard PID structure theory, every ideal of R canP be generated by two elements. Thus R is Noetherian, hence atomic. n × For f = a0 + ∞ ⇐⇒ a0 6= 0, and thus n=2 an t ∈ R, we have f ∈ R m={ ∞ X an tn } = (t2 , t3 ) n=2 is the unique maximal ideal of R. We will give a complete description of the atoms of R. First we claim that f ∈ R is irreducible iff v(f ) ∈ {2, 3}. Indeed a nontrivial factorization f = xy involves v(x), v(y) ≥ 2 hence v(f ) ≥ 4; conversely, if v(f ) ≥ 4 then f = t2 tf2 is a nontrivial factorization. Since k × ⊂ R× , every irreducible is associate to one of the form t2 + X an tn , (v(f ) = 2 case) n≥3 or one of the form t3 + X an tn , (v(f ) = 3 case). n≥4 Associate elements have the same valuation, so certainly no irreducible P of the first type is associate to an irreducible of the second type. We claim that t2 + n≥3 an tn P P is associate to t2 + n≥3 bn tn iff a3 = b3 and t3 + n≥3 an tn is associate to P t3 + n≥3 bn tn iff a4 = b4 . This can be done by direct computation: (t2 + a3 t3 + a1 4t4 + a5 t5 + . . .)(1 + u2 t2 + u3 t3 + . . .) = t2 + a3 t3 + (a4 + u2 )t4 + (a5 + a3 u2 + u3 )t5 + . . . , THE EUCLIDEAN CRITERION FOR IRREDUCIBLES 13 so a3 = b3 and there is a unique choice of u2 , u3 , . . . leading to an = bn for all n ≥ 4. The v(f ) = 3 case is similar. Thus there are precisely 2#k atoms, and R is Cohen-Kaplansky iff k is finite. Example 4.16. (Anderson-Mott [AM92, Cor. 7.2]) For a prime power q and d, e ∈ Z+ , the ring R = Fq + te Fqd [[t]] is a Cohen-Kaplansky domain with exactly d −1 d(e−1) q irreducibles, none of which are one nonzero prime ideal and exactly e qq−1 prime unless (d, e) = (1, 1). The paper [CK46] was mostly forgotten for many years, until the breakthrough work of Anderson and Mott [AM92] gave a complete characterization of CohenKaplansky domains. In fact they give 14 characterizations! Here is one: Theorem 4.17. (Anderson-Mott [AM92]) For an atomic domain R, the following are equivalent: (i) R is a Cohen-Kaplansky domain. (ii) R is Noetherian of dimension at most one (nonzero prime ideals are maximal), has finitely many prime ideals, the integral closure R of R is finitely generated as an R-module, # MaxSpec R = # MaxSpec R, and for all nonprincipal ideals m ∈ MaxSpec R, R/m is finite. Example 4.18. Let k be a field of characteristic different from 2 or 3, and consider: • R1 : the localization of k[x, y]/(y 2 − x3 − x) at m0 = (x, y). • R2 : the localization of k[x, y]/(y 2 − x3 − x2 ) at m0 = (x, y). • R3 : the localization of k[x, y]/(y 2 − x3 ) at m0 = (x, y). Then: • R1 is always Cohen-Kaplansky (it is a Dedekind domain with one maximal ideal). • R2 is never Cohen-Kaplansky (# MaxSpec R2 = 2 > 1 = # MaxSpec R2 ). • R3 is Cohen-Kaplansky iff k is finite. 4.4. Euclid Beyond Atomicity. In the case of an atomic domain, the part of the Euclidean Criterion that yields infinitely many maximal ideals is much weaker than the Cohen-Kaplansky Theorem. However, there is life beyond atomic domains. Example 4.19. Let Hol(C) be the ring of entire functions f : C → C. For f ∈ Hol(C), put Z(f ) = {z ∈ C \| f (z) = 0}. If f, g ∈ Hol(C)• , then Z(f ) and Z(g) are countable sets, hence so is Z(f g) = Z(f ) ∪ Z(g), so f g 6= 0. Thus H(C) is a domain. The map z0 ∈ C 7→ (z − z0 ) gives a bijection from C to the atoms of Hol(C). An element f ∈ Hol(C) is a unit iff Z(f ) = ∅, and a nonzero nonunit f is a (finite!) product of atoms iff Z(f ) is finite and nonempty. So Hol(C) is not atomic – consider e.g. f (z) = sin z – but it is Furstenberg: if f is a nonzero nonunit, then f vanishes at some z0 ∈ C and thus is divisible by the irreducible element z − z0 . Moreover Hol(C) satisfies Condition (E): if f ∈ Hol(C)• 1 then there is w ∈ C such that f (w) 6= 0. Let g = z −w− f (w) . Then (gf +1)(w) = 0, × so gf + 1 ∈ / Hol(C) . Thus the Euclidean Criterion applies in Hol(C). Theorem 4.20. Let 1 ≤ α ≤ β ≤ γ be cardinal numbers. There is a domain R satisfying all of the following properties: (i) R is a Bézout domain: every finitely generated ideal is principal. (ii) R has exactly α atoms, each of which is a maximal ideal. 14 PETE L. CLARK (iii) R has exactly β maximal ideals. (iv) R has exactly γ nonzero prime ideals. (v) R is an atomic domain iff α = β = γ < ℵ0 . (vi) R is a Furstenberg domain iff α = β. (vii) R is semiprimitive iff β ≥ ℵ0 . We postpone the proof of Theorem 4.20 in order to discuss its significance. By taking α = β and γ ≥ ℵ0 we get Furstenberg domains with any number α ≥ 1 of irreducibles and any number γ ≥ max(α, ℵ0 ) nonzero prime ideals. In particular, a Furstenberg domain can have any finite, positive number of irreducibles and any infinite number of prime ideals, so the Cohen-Kaplansky Theorem does not extend from atomic domains to Furstenberg domains. For any α = β ≥ ℵ0 and γ ≥ α we get a semiprimitive Furstenberg domain that is not an atomic domain. Now we come to the proof of Theorem 4.20, which requires somewhat more specialized results. A completely self-contained presentation would require more space than we want to devote here. So we will make use of the material of [FS, Ch. II and III], and our treatment will be at the level of a detailed sketch. Let R be a domain with fraction field K. To x ∈ K • we attach the principal fractional ideal (x) = {ax \| a ∈ R}. When x ∈ R, this coincides with the usual notion of a principal ideal. For x, y ∈ K • we have (x) = (y) iff there is u ∈ R× such that y = ux. The principal fractional ideals of K form a commutative group under pointwise multiplication: we have (x)(y) = (xy). We call this the group of divisibility of R and denote it G(R). It is partially ordered by reverse inclusion: that is, for x, y ∈ K • we put (x) ≤ (y) iff (y) ⊃ (x). This order reversal is actually rather familiar: for x, y ∈ K × , we write x \| y ⇐⇒ xy ∈ R, and then we have x \| y if (x) ⊃ (y): to contain is to divide. Let {Gi }i∈IL be an indexed family of nonzero totally ordered commutative groups, and let G = i∈I Gi be the direct sum endowed with the pointwise partial ordering: x ≤ y iff xi ≤ yi for all i ∈ I. Let πi : G → Gi be projection onto the ith coordinate. By the Kaplansky-Jaffard-Ohm Theorem [FS, Thm. III.5.3] ∼ there is a Bézout domain R and an isomorphism ϕ : G(R) → G of partially ordered commutative groups. See [FS, Example III.5.4]. Let v be the composϕ L ite K × → K × /R× → I∈I Gi . Then the maximal ideals of R are precisely mi = {x ∈ R \| (πi ◦ v)(x) > 0} ∪ {0} for i ∈ I. Thus no element of R• lies in infinitely many maximal ideals, so R is semiprimitive iff I is infinite. An atom in a partially ordered commutative group is a minimal positive element. This is a direct generalization of our previous use of the term: if R is a domain, the minimal positive elements of the group of divisibility G(R) are precisely the principal fractional ideals (x) for an irreducible element x ∈ R. For every atom x ∈ G, there is i ∈ I such that xi is an atom of Gi and xj = 0 for all j 6= i, and conversely all such elements give atoms of G. Since each Gi is totally ordered, it has at most one atom, the least positive element of Gi if such an element exists. It follows that R is Furstenberg iff each Gi has a least positive element. Similarly, a nonzero nonunit x ∈ R factors into irreducibles iff v(x) ∈ G is a sum of atoms iff for all i ∈ I, Gi has a least positive element ai and vi (r) = nai for some n ∈ Z+ . Thus R is an atomic domain iff each Gi ∼ = Z. The domain R is h-local: each nonzero prime ideal is contained in a unique maximal ideal [FS, loc. cit.]. The nonzero prime ideals contained in mi correspond THE EUCLIDEAN CRITERION FOR IRREDUCIBLES 15 bijectively to the proper convex subgroups of Gi . (A subset Y of a totally ordered set X is convex if for all x < y < z ∈ X, if x, z ∈ Y then also y ∈ Y .) We will take each Gi to be a lexicographic product of copies of subgroups of (R, +) indexed by an ordinal η. Then the convex subgroups of of Gi are precisely {Hδ }0≤δ≤η , where Hδ is the set of all elements of Gi with j-coordinate zero for all j < δ. So there are #η nonzero prime ideals in mi . We will take a family of nonzero totally ordered commutative groups Gi parameterized by i ∈ β: this gives us β maximal ideals, and R is semiprimitive iff β ≥ ℵ0 . We are left to choose the groups Gi in terms of α and γ so as to attain the other assertions. We define an ordinal η: if γ is finite, it is the positive integer γ − β + 1; if γ is infinite, it is the successor ordinal to γ (what matters in this case is that η is a well-ordered set of cardinality γ and with a largest element). There are cases: • If α = β = γ < ℵ0 , we take R to be a PID with γ nonzero prime ideals. • If α = β and γ ≥ min(β + 1, ℵ0 ) we take Gi = Z for all 0 < i ∈ β. We take G0 to be the Cartesian product of copies of Z indexed by η, endowed with the lexicographic ordering. Then G0 has a least positive element: the element that is 0 in all factors but the last and 1 in the last factor. So all Gi have least elements and R is a Furstenberg domain. Moreover η ≥ 2 so G0 ∼ 6 Z and R is not an atomic = domain. It has (β − 1) + #η = γ nonzero prime ideals. • If α < β, we take G0 to be the Cartesian product of copies of Z indexed by η, for 1 ≤ i < α we take Gi = Z, and for i ≥ α we take Gi = R. 4.5. Supplement: Rings With Infinitely Many Maximal Ideals. Let us briefly consider the case of an arbitrary commutative ring. Though others have done so (see e.g. [AVL96]), it is beyond our ambitions to pursue a factorization theory in the presence of zero divisors. But we can still ask for criteria under which there are infinitely many maximal ideals. In this more general context J(R) = (0) is no longer sufficient: e.g. J(C × C) = 0 and there are only two maximal ideals. Nevertheless both Euclid and Jacobson have a role to play. Proposition 4.21. [Cl-CA, Prop. 4.15] Let I be an ideal of R contained in the Jacobson radical. Then for all x ∈ R, if the image of x in R/I is a unit, then x is a unit. In particular the natural map R× → (R/I)× is surjective. Proof. If the image of x in R/I is a unit, then there is y ∈ R such that xy ≡ 1 (mod I), i.e., xy − 1 ∈ I ⊂ J(R). Thus for every maximal ideal m of R, xy − 1 ∈ m so we cannot have x ∈ m. So x lies in no maximal ideal of R and thus x ∈ R× . Theorem 4.22. (Dubuque [Du10]) Let R be an infinite ring. If #R > #R× , then MaxSpec R is infinite. Proof. We will show by induction on n that for all n ∈ Z+ , R has n maximal ideals. Base Case: Since R is infinite, it is nonzero and thus it has a maximal ideal m1 . Induction Step: Let m1 , . . . , mm be maximal ideals, and put m Y mi . I= i=1 × Case 1: Suppose I + 1 ⊂ R . Then #I ≤ #R× . Moreover I ⊂ J(R), so by × × Proposition 4.21 R× → (R/I)× is surjective. It follows Qn that #(R/I) ≤ #R < ∼ #R: by the Chinese Remainder Theorem, R/I = i=1 R/mi , hence there is an 16 PETE L. CLARK injection (R/mi )× → (R/I)× . Putting the last two sentences together we conclude #(R/mi )× < #R, and thus, since R/mi is a field and R is infinite, #R/mi = #(R/mi )× + 1 < #R. Finally this gives the contradiction n Y #R = #I · #R/I = #I · #R/mi < (#R)n+1 = #R. i=1 × Case 2: So there is x ∈ I + 1 \ R . Let mn+1 be a maximal ideal containing x. For all 1 ≤ i ≤ n we have x − 1 ∈ I ⊂ mi , so 1 = x + (1 − x) ∈ mn+1 + mi . So mn+1 is an (n + 1)st maximal ideal of R, completing the induction step. A special case of Theorem 4.22 appears in [K, § 1.1, Exc. 8]. For a ring R, consider the quotient R/J(R). The maximal ideals of R/J(R) correspond to the maximal ideals of R containing J(R) – that is, to the maximal ideals of R. Thus R/J(R) is semiprimitive. Thus we can replace any ring with a semiprimitive ring without changing its MaxSpec. However this “Jacobson semisimplification” need not carry domains to domains: e.g. Q if R is a domain with 2 ≤ n < ℵ0 maximal ideals m1 , . . . , mn , then R/J(R) ∼ = ni=1 R/mi . Here is a generalization. Theorem 4.23. a) For a ring R, the following are equivalent. (i) R has only finitely many maximal ideals. (ii) R/J(R) is a finite product of fields. (iii) R/J(R) has only finitely many ideals. (iv) R/J(R) is Artinian (i.e., there are no infinite descending chains of ideals). b) A semiprimitive ring with finitely many maximal ideals has finitely many ideals. Proof. a) (i) =⇒ (ii): If the maximal ideals of R are m1 , . . . , mn , then by the Chinese Remainder Theorem [Cl-CA, Thm. 4.18] we have R/J(R) = R/ n \ i=1 mi ∼ = n Y R/mi . i=1 (ii) =⇒ (iii) =⇒ (iv) immediately. (iv) =⇒ (i): Maximal ideals of R/J(R) correspond bijectively to maximal ideals of R. And an Artinian ring has only finitely many maximal ideals [Cl-CA, Thm. 8.31]. b) This follows from part a). 5. But What About Primes? Our take on Euclid’s argument has been as a criterion for the existence of irreducibles. The distinction evaporates in a UFD. A PID with only finitely many prime ideals is a UFD with only finitely many principal prime ideals. It turns out that the converse is also true.5 Theorem 5.1. Let R be a UFD, not a field, with only finitely many atoms. Then R is a PID with finitely many prime ideals and #R = #R× . 5Theorem 5.1 is known to the experts: see e.g. [Za08]. THE EUCLIDEAN CRITERION FOR IRREDUCIBLES 17 Proof. A UFD with finitely many nonassociate prime elements is a Cohen-Kaplansky domain, so MaxSpec R is finite and #R = #R× by Theorem 3.4. By Theorem 4.14 every nonzero prime ideal of R is maximal. The proof of Theorem 4.14a) shows: every nonzero prime ideal p contains a prime element p. Since (p) is maximal, we have p = (p). Thus every prime ideal is principal, so R is a PID [Cl-CA, Thm. 4.25]. (This is another case of the Lam-Reyes Prime Ideal Principle.) Let us now move away from UFDs. From Example 4.15, we deduce: Theorem 5.2. Let κ ≥ ℵ0 be a cardinal. There is a Noetherian domain R with exactly one nonzero prime ideal, exactly κ irreducibles and no prime elements. Proof. Let k be a field of cardinality κ, e.g. k = Q({tα \| α ∈ κ}). By Example 4.15, R = k[[t2 , t3 ]] is a Noetherian domain with one nonzero prime ideal m = (t2 , t3 ) and 2κ = κ irreducibles. Since m is not principal, R has no prime elements. Cohen-Kaplansky showed that an atomic domain that is neither a field nor a UFD must have at least 3 atoms [CK46, p. 469]. Their argument is a nice one: we must have at least one nonprime irreducible f1 . Since (f1 ) is not prime, it is properly contained in some prime ideal p, which must therefore contain a nonassociate irreducible f2 . Since f1 + f2 ∈ p, f1 + f2 is not a unit and therefore it is divisible by an irreducible f3 , which cannot be associate to either f1 or f2 . Finally, we consider Dedekind domains. Question 5.3. Let R be a Dedekind domain with infinitely many prime ideals. Must R have infinitely many atoms? In an important classical case the answer is yes, as most number theorists know. Theorem 5.4. For each number field K, the ring of integers ZK has infinitely many nonassociate prime elements. Proof. Step 1: For any number field L, the number of rational primes that split completely in L is infinite. This is a special case of the Chebotarev Density Theorem, which however can be proved in a more elementary way, as was shown in [Po10]. Using some basic algebraic number theory which we omit here, it comes down to showing that for every nonconstant polynomial f ∈ Z[t], the set of prime numbers p dividing f (n) for some n ∈ Z is infinite. If f (0) = 0 this is trivial. If f (0) 6= 0, let p1 , . . . , pk be the prime divisors of f (0) (we allow k = 0) and let q1 , . . . , qℓ be any finite set of primes not dividing f (0). For 1 ≤ i ≤ k, let ai be such that pai i \| f (0) and pai i +1 ∤ f (0). For N ∈ Z+ consider xN = f (N pa1 1 +1 · · · pakk +1 q1 · · · qℓ ). Then for all 1 ≤ i ≤ k, pai i +1 ∤ xN and for all 1 ≤ j ≤ ℓ, qj ∤ xN , so the set of N for which xN is not divisible by some prime other than p1 , . . . , pk , q1 , . . . , qℓ is finite. Step 2: A prime ideal p of a number field is principal iff it splits completely in the Hilbert class field K 1 of K. So every prime ideal p of K lying above any one of the infinitely many prime numbers p that split completely in K 1 is principal. Looking at the above argument, one wonders: were we working working too hard? Perhaps some simple argument gives a general affirmative answer to Question 5.3. In fact Question 5.3 was answered negatively by Claborn [Cl65, Example 1.5]. 18 PETE L. CLARK The construction is impressively direct: start with a Dedekind domain A that is not a PID, let P be the set of prime elements of R and pass to R = A[{ p1 }p∈P ]. The prime ideals of R are precisely the nonprincipal prime ideals of A, which remain nonprincipal in R! This prime-killing construction also appears in a work of Samuel [Sa64, p. 17, Thm. 6.3] and is therein attributed to Nagata (cf. [N57, Lemma 2]). For a Dedekind domain A, write Cl A for its ideal class group: the quotient of the monoid of nonzero ideals of A under the equivalence relation I ∼ J iff there are α, β ∈ A• with (α)I = (β)J. In the setting of the prime-killing construction – i.e., R is the localization of A at the multiplicative subset generated by the prime elements – we have [Sa64], [Cl65] that Cl R ∼ = Cl A. Theorem 5.5. Let κ be an infinite cardinal. There is a Dedekind domain R with exactly κ atoms and no prime elements. Proof. We will use some properties of “elliptic Dedekind domains”: for more details, see [Cl09, §2.4]. Let k be an algebraically closed field of characteristic 0 and cardinality κ, and put R = k[x, y]/(y 2 − x3 − x). Then R is a Dedekind domain, and by the Nullstellensatz the nonzero prime ideals of R are all of the form p(x0 ,y0 ) = (x − x0 , y − y0 , y 2 − x3 − x) for pairs (x0 , y0 ) ∈ k 2 such that y02 = x30 +x0 . In other words, they are the k-rational points on the projective elliptic curve E : y 2 z = x3 + xz 2 , excluding the point at infinity O = [0 : 1 : 0]. Moreover, by the Riemann-Roch Theorem, since [x0 : y0 : 1] 6= O, the prime ideal p(x0 ,y0 ) is not principal. Thus R is a Dedekind domain with # MaxSpec R = #R = κ and without prime elements. Because R is Dedekind, every ideal can be generated by two elements [Cl-CA, Thm. 20.12]. This, together with the fact that Dedekind domains are atomic domains, implies that for all p ∈ MaxSpec R there are irreducibles pp , qp such that p = (pp , qp ). Thus if λ is the number of irreducibles of R we have κ = # MaxSpec R ≤ λ2 ≤ (#R)2 = κ2 = κ, so λ2 = κ. Since κ is infinite, so is λ and thus λ = λ2 = κ. References [AM92] [AVL96] [Cl-CA] [Cl09] [CK46] [Cl65] [Co73] [CS12] [CW03] [Du10] [FS] [Fu55] D.D. Anderson and J.L. Mott, Cohen-Kaplansky domains: integral domains with a finite number of irreducible elements. J. Algebra 148 (1992), 17-41. D.D. Anderson and S. Valdes-Leon, Factorization in commutative rings with zero divisors. Rocky Mountain J. Math. 26 (1996), 439-480. P. L. Clark, Commutative Algebra, http://math.uga.edu/~ pete/integral.pdf. P.L. Clark, Elliptic Dedekind domains revisited. Enseignement Math. 55 (2009), 213– 225. I.S. Cohen and I. Kaplansky, Rings with a finite number of primes. I. Trans. Amer. Math. Soc. 60 (1946), 468-477. L.E. Claborn, Dedekind domains and rings of quotients. Pacific J. Math. 15 (1965), 59–64. P.M. Cohn, Unique factorization domains. Amer. Math. Monthly 80 (1973), 1–18. J. Coykendall and C. Spicer, Cohen-Kaplansky domains and the Goldbach conjecture. Proc. Amer. Math. Soc. 140 (2012), 2227-2233. D. Cass and G. Wildenberg, Math Bite: A Novel Proof of the Infinitude of Primes, Revisited Mathematics Magazine, Vol. 76 (2003), 203. W.G. Dubuque, http://math.stackexchange.com/questions/201 L. Fuchs and L. Salce, Modules over non-Noetherian domains. Mathematical Surveys and Monographs, 84. American Mathematical Society, Providence, RI, 2001. H. Furstenberg, On the infinitude of primes, Amer. Math. Monthly 62 (1955), 353. THE EUCLIDEAN CRITERION FOR IRREDUCIBLES [Gl67] [Go59] [K] [LR08] [Me09] [N57] [P] [Po10] [Sa64] [Za08] 19 M. Glaymann, Characteristic Functions and Sets. Mathematics Teacher, 60 (1967), 775–778. S.W. Golomb, A connected topology for the integers. Amer. Math. Monthly 66 (1959), 663-665. I. Kaplansky, Commutative rings. Allyn and Bacon, Inc., Boston, Mass. 1970. T.Y. Lam and M. Reyes, A prime ideal principle in commutative algebra. J. Algebra 319 (2008), 3006-3027. I.D. Mercer, On Furstenberg’s Proof of the Infinitude of Primes. Amer. Math. Monthly 116 (2009), 355–356. M. Nagata, A remark on the unique factorization theorem. J. Math. Soc. Japan 9 (1957), 143–145. P. Pollack, Not always buried deep. A second course in elementary number theory. American Mathematical Society, Providence, RI, 2009. B. Poonen, http://mathoverflow.net/q/15221 P. Samuel, Lectures on unique factorization domains. Notes by M. Pavman Murthy. Tata Institute of Fundamental Research Lectures on Mathematics, No. 30 Tata Institute of Fundamental Research, Bombay 1964. M. Zafrullah, http://mathforum.org/kb/message.jspa?messageID=6451774	0
1 Sufficient Conditions for the Tightness of Shannon’s Capacity Bounds for Two-Way Channels Jian-Jia Weng† , Lin Song‡, Fady Alajaji† , and Tamás Linder† arXiv:1801.03163v1 [cs.IT] 9 Jan 2018 Abstract—New sufficient conditions for determining in closed form the capacity region of point-to-point memoryless two-way channels (TWCs) are derived. The proposed conditions not only relax Shannon’s condition which can identify only TWCs with a certain symmetry property but also generalize other existing results. Examples are given to demonstrate the advantages of the proposed conditions. Index Terms—Network information theory, two-way channels, capacity region, inner and outer bounds, channel symmetry. I. I NTRODUCTION Finding the capacity region of point-to-point discrete memoryless two-way channels (TWCs) in single-letter form is a long-standing open problem. The difficulty lies in the causality of transmission, since the senders are allowed to generate channel inputs by adapting to previously received channel outputs. In [1], Shannon gave an (uncomputable) multi-letter expression for the capacity region. Another multi-letter expression, using directed information [2], was given in [3]. The capacity region of TWCs is known only for some special channels such as TWCs with additive white Gaussian noise [4], determinisitc TWCs [5], TWCs with discrete additive noise [6], and injective semi-deterministic TWCs [7]. Thus, Shannon’s inner and outer bounds [1] still play an important role in characterizing the capacity region. In the literature, Shannon’s symmetry condition [1] and a condition established by Chaaban, Varshney, and Alouini (CVA) [7] are two known sufficient conditions under which Shannon’s inner and outer bounds coincide, thus directly characterizing the capacity region. Shannon’s condition focuses on a certain symmetry structure for the channel transition probabilities, while the CVA condition focuses on the existence of independent inputs which achieve Shannon’s outer bound. Although the two conditions can be used to determine the capacity region of a large class of TWCs, it is of interest to establish new conditions for wider families of channels. In this paper, four sufficient conditions guaranteeing that Shannon’s inner and outer bounds coincide are derived. Similar to the CVA condition, our conditions identify independent inputs which achieve Shannon’s outer bound based on the approach that a TWC can be viewed as two one-way channels † The authors are with the Department of Mathematics and Statistics, Queen’s University, Kingston, ON K7L 3N6, Canada (Emails: [email protected], {fady, linder}@mast.queensu.ca). ‡ The author was with the Department of Mathematics and Statistics, Queen’s University, Kingston, ON K7L 3N6, Canada. She is now with Contextere Ltd., Ottawa, ON K1Y 2C5, Canada (Email: [email protected]). This work was supported in part by NSERC of Canada. X1N M1 User 1 M̂2 X2N TWC Y1N M2 User 2 Y2N M̂1 Fig. 1. Block diagram of two-way transmission. with state. Two of the derived results are shown to be substantial generalizations of the Shannon and CVA conditions. Moreover, our simplest condition can be easily verified by observing the channel marginal distributions. The rest of this paper is organized as follows. In Section II, the system model and prior results are reviewed. New conditions for finding the capacity region are provided in Section III. A discussion of the connections between the new conditions and prior results is given in Section IV along with illustrative examples. Concluding remarks are given in Section V. II. P RELIMINARIES In a two-way communication system as shown in Fig. 1, two users want to exchange their own messages M1 and M2 via N uses of a TWC. Here, the messages M1 and M2 are assumed to be mutually independent and uniformly distributed on M1 , {1, 2, ..., 2N R1 } and M2 , {1, 2, ..., 2N R2 }, respectively, where N R1 and N R2 are non-negative integers. For j = 1, 2, let Xj and Yj respectively denote the finite channel input and output alphabets for user j. The joint distribution of the inputs and outputs of a memoryless TWC is governed by the channel transition probability PY1 ,Y2 \|X1 ,X2 . A channel code for a TWC is defined as follows. Definition 1: An (N, R1 , R2 ) code for a TWC consists of two message sets M1 = {1, 2, . . . , 2N R1 } and M2 = {1, 2, . . . , 2N R2 }, two sequences of encoding functions f1N , (f1,1 , f1,2 , . . . , f1,N ) and f2N , (f2,1 , f2,2 , . . . , f2,N ), with f1,1 : M1 → X1 , f1,n : M1 × Y1n−1 → X1 , f2,1 : M2 → X2 , and M2 ×Y2n−1 → X2 for n = 2, 3, . . . , N , and two decoding functions g1 : M1 × Y1N → M2 and g2 : M2 × Y2N → M1 . When messages M1 and M2 are encoded, the channel inputs at time n = 1 are only functions of the messages, i.e., Xj,1 = fj,1 (Mj ) for j = 1, 2, but all the other channel inputs are generated by also adapting to the previous channel outputs Yjn−1 , (Yj,1 , Yj,2 , . . . , Yj,n−1 ) via Xj,n = fj,n (Mj , Yjn−1 ) for j = 1, 2 and n = 2, 3, . . . , N . After receiving N channel outputs, user j reconstructs Mi as M̂i = gj (Mj , YjN ) for i, j = 1, 2 with i 6= j, and the probability of decoding error (N ) is defined as Pe (f1N , f2N , g1 , g2 ) = Pr{M̂1 6= M1 or M̂2 6= M2 }. Based on this performance index, we define achievable rate pairs and the capacity region. Definition 2: A rate pair (R1 , R2 ) is said to be achievable if there exists a sequence of (N, R1 , R2 ) codes such that 2 (N ) limN →∞ Pe = 0. The capacity region C of a TWC is the closure of the convex hull of all achievable rate pairs. To date, a computable single-letter expression for the capacity region of general memoryless TWCs has not been found. In [1], Shannon established inner and outer bounds for the capacity region. Let R(PX1 ,X2 , PY1 ,Y2 \|X1 ,X2 ) denote the set of rate pairs (R1 , R2 ) with R1 ≤ I(X1 ; Y2 \|X2 ) and R2 ≤ I(X2 ; Y1 \|X1 ), where the joint distribution of all random variables is given by PX1 ,X2 PY1 ,Y2 \|X1 ,X2 . Then, the capacity region of a discrete memoryless TWC with transition probability PY1 ,Y2 \|X1 ,X2 is inner bounded by [1]  CI (PY1 ,Y2 \|X1 ,X2 ) , co  [ PX1 PX2 and outer bounded by  CO (PY1 ,Y2 \|X1 ,X2 ) , co  [ PX1 ,X2  R(PX1 PX2 , PY1 ,Y2 \|X1 ,X2 ),  R(PX1 ,X2 , PY1 ,Y2 \|X1 ,X2 ), where co denotes taking the closure of the convex hull. In general, CI and CO are different, but if they coincide, then the exact capacity region is obtained and independent inputs can be used to achieve any point of the capacity region. We note that there exist other improved bounds for TWCs [4], [8]-[11]. However, those bounds are either restricted for the particular case of the binary multiplier TWC or expressed with auxiliary random variables, which do not match our approach. We next review the Shannon [1] and CVA [7] conditions that imply the coincidence of CI and CO . For a finite set A, let π A : A → A be a permutation (bijection), and for any two symbols a′ and a′′ in A, let τaA′ ,a′′ : A → A denote the transposition which swaps a′ and a′′ in A, but leaves the other symbols unaffected. Moreover, let PX,Z,Y = PX PZ\|X PY \|X,Z denote a probability distribution defined on finite sets X , Y, and Z. We define two functionals for conditional entropies: H(PX,Z , PY \|X,Z ) , X PX,Z (x, z)PY \|X,Z (y\|x, z) log x,z,y 1 PY \|X,Z (y\|x, z) and H̄(PX , PZ\|X , PY \|X,Z ) , X PX (x)PY \|X (y\|x) log x,y 1 , PY \|X (y\|x) P where PY \|X (y\|x) = z PY \|X,Z (y\|x, z)PZ\|X (z\|x). In particular, if P PZ\|X=x′ = PZ\|X=x′′ for any x′ , x′′ ∈ X , we let PZ (z) = x PX,Z (x, z) and define H̄⊥ (PX , PZ , PY \|X,Z ) , X x,y P PX (x)QY \|X (y\|x) log 1 , QY \|X (y\|x) where QY \|X (y\|x) = z PY \|X,Z (y\|x, z)PZ (z). Note that, given any PX1 ,X2 = PX2 PX1 \|X2 = PX1 PX2 \|X1 , we have H(Yj \|X1 , X2 ) = H(PX1 ,X2 , PYj \|X1 ,X2 ), H(Y1 \|X1 ) = H̄(PX1 , PX2 \|X1 , PY1 \|X1 ,X2 ), and H(Y2 \|X2 ) = H̄(PX2 , PX1 \|X2 , PY2 \|X1 ,X2 ), where PYj \|X1 ,X2 is a marginal of the channel probability PY1 ,Y2 \|X1 ,X2 and j = 1, 2. Furthermore, for any PX1 ,X2 which can be factorized as PX1 PX2 , we have H(Y1 \|X1 ) = H̄⊥ (PX1 , PX2 , PY1 \|X1 ,X2 ) and H(Y2 \|X2 ) = H̄⊥ (PX2 , PX1 , PY2 \|X1 ,X2 ). Finally, let P(Xj ) denote the set of all probability distributions on Xj for j = 1, 2. Proposition 1 (Shannon’s Symmetry Condition [1]): For a memoryless TWC with transition probability PY1 ,Y2 \|X1 ,X2 , we have C = CI = CO if for any pair of distinct input symbols x′1 , x′′1 ∈ X1 , there exists a pair of permutations (π Y1 [x′1 , x′′1 ], π Y2 [x′1 , x′′1 ]) on Y1 and Y2 (which depend on x′1 and x′′1 ) such that for all x1 , x2 , y1 , y2 , PY1 ,Y2 \|X1 ,X2 (y1 , y2 \|x1 , x2 ) = PY1 ,Y2 \|X1 ,X2 (π Y1 [x′1 , x′′1 ](y1 ), π Y2 [x′1 , x′′1 ](y2 )\|τxX′1,x′′ (x1 ), x2 ). (1) 1 1 Proposition 2 (CVA Condition [7]): For a memoryless TWC with transition probability PY1 ,Y2 \|X1 ,X2 , we have C = CI = CO if for any PX1 ,X2 = PX2 PX1 \|X2 = PX1 PX2 \|X1 , H(PX2 P̃X1 \|X2 , PYj \|X1 ,X2 ) does not depend on P̃X1 \|X2 for given PX2 and there exists P̃X1 ∈ P(X1 ) such that H̄⊥ (P̃X1 , PX2 , PY1 \|X1 ,X2 ) ≥ H̄(PX1 , PX2 \|X1 , PY1 \|X1 ,X2 ) and H̄⊥ (PX2 , P̃X1 , PY2 \|X1 ,X2 ) ≥ H̄(PX2 , PX1 \|X2 , PY2 \|X1 ,X2 ). We remark that Proposition 1 describes a channel symmetry property with respect to the channel input of user 1, but an analogous condition can be obtained by exchanging the roles of users 1 and 2. Also, the invariance of H(PX2 P̃X1 \|X2 , PYj \|X1 ,X2 ) in Proposition 2 in fact imposes a certain symmetry constraint on the channel marginal distribution PYj \|X1 ,X2 . In the literature, a TWC with independent q-ary additive noise [6] is an example that satisfies both the Shannon and CVA conditions. III. C ONDITIONS FOR THE T IGHTNESS OF S HANNON ’ S I NNER AND O UTER B OUNDS In this section, we present four results regarding the tightness of Shannon’s inner and outer bounds. We adopt the viewpoint that a two-way channel consists of two one-way channels with state. For example, the one-way channel from user 1 to user 2 is governed by the marginal distribution PY2 \|X1 ,X2 (derived from the channel probability distribution PY1 ,Y2 \|X1 ,X2 ), where X1 and Y2 are respectively the input and the output of the channel with state X2 . Let PX and PY \|X be probability distributions on finite sets X and Y. To simplify the presentation, we define I(PX , PY \|X ) = X x,y PX (x)PY \|X (y\|x) log P x′ PY \|X (y\|x) , PX (x′ )PY \|X (y\|x′ ) which is the mutual information I(X; Y ) between input X (governed by PX ) and corresponding output Y of a channel with transition probability PY \|X . A useful fact is that I(·, ·) is concave in the first argument when the second argument is fixed. Moreover, the conditional mutual information I(X1 ; Y2 \|X2 = x2 ) and I(X2 ; Y1 \|X1 = x1 ) can be expressed as I(PX1 \|X2 =x2 , PY2 \|X1 ,X2 =x2 ) and I(PX2 \|X1 =x1 , PY1 \|X1 =x1 ,X2 ), respectively. By viewing a TWC as two one-way channels with state, each of the following four theorems comprises two conditions, one for each direction of the two-way transmission. By symmetry, these theorems are also valid if the roles of users 1 and 2 are swapped. For simplicity, we will use I (k) (Xi ; Yj \|Xj ) and H (k) (Yj \|X1 , X2 ) to denote the conditional mutual information and conditional entropy evaluated (k) under input distribution PX1 ,X2 for i, j = 1, 2 with i 6= j. For 3 (k) (k) (k) (1) (1) (1) (2) PX1 ,X2 = PXi PXi \|Xj , the conditional entropy H (k) (Yi \|Xi ) Proof: Given any PX1 ,X2 = PX2 PX1 \|X2 , let PX1 ,X2 = is evaluated under the marginal distribution PYi \|Xi (yi \|xi ) = P (k) xj PXj \|Xi (xj \|xi )PYi \|Xj ,Xi (yi \|xj , xi ). Theorem 1: For a given memoryless TWC, if both of the following conditions are satisfied, then CI = CO : ∗ (i) There exists PX ∈ P(X1 ) such that for all x2 ∈ X2 we 1 ∗ . have arg maxPX \|X =x I(X1 ; Y2 \|X2 = x2 ) = PX 1 1 2 2 (ii) I(PX2 , PY1 \|X1 =x1 ,X2 ) does not depend on x1 ∈ X1 for any fixed PX2 ∈ P(X2 ). (2) (1) (1) (1) Proof: For any PX1 ,X2 = PX2 PX1 \|X2 , let PX1 ,X2 = ∗ PX P . By the same argument as in (2)-(6), we obtain via 1 X2 (i) that I (1) (X1 ; Y2 \|X2 ) ≤ I (2) (X1 ; Y2 \|X2 ). Moreover, (k) (1) ∗ ∗ PX PX2 , where PX is given by (i). In light of (i), we have 1 1 I (1) (X1 ; Y2 \|X2 ) X (1) PX2 (x2 ) · I (1) (X1 ; Y2 \|X2 = x2 ) = (2) x2 ≤ X (1) PX2 (x2 ) · x2 = X max PX1 \|X2 =x2 I(X1 ; Y2 \|X2 = x2 ) (1) PX2 (x2 ) · I(PX1∗ , PY2 \|X1 ,X2 =x2 ) (3) (4) x2 = X x2 (2) =I (1) PX2 (x2 ) · I (2) (X1 ; Y2 \|X2 = x2 ) (5) (X1 ; Y2 \|X2 ). (6) Moreover, I (1) (X2 ; Y1 \|X1 ) X (1) PX1 (x1 ) · I (1) (X2 ; Y1 \|X1 = x1 ) = (7) x1 = X (1) (1) (1) (1) PX1 (x1 ) · I(PX2 \|X1 =x1 , PY1 \|X1 =x1 ,X2 ) (8) x1 = X PX1 (x1 ) · I(PX2 \|X1 =x1 , PY1 \|X1 =x′1 ,X2 ) (9) x1 ≤I X (1) (1) PX1 (x1 )PX2 \|X1 =x1 , PY1 \|X1 =x′1 ,X2 x1 (1) ! = I(PX2 , PY1 \|X1 =x′1 ,X2 ) X (1) PX1∗ (x′1 ) · I(PX2 , PY1 \|X1 =x′1 ,X2 ) = (10) (11) (12) x′1 = I (2) (X2 ; Y1 \|X1 ), (13) where (9) holds by the invariance assumption in (ii), (10) holds since the functional I(·, ·) is concave in the first argument, and (12) is obtained from the invariance assumption in (1) (ii). Combining the above yields R(PX1 ,X2 , PY1 ,Y2 \|X1 ,X2 ) ⊆ (1) ∗ R(PX PX2 , PY1 ,Y2 \|X1 ,X2 ), which implies that CO ⊆ CI and 1 hence CI = CO . Theorem 2: For a given memoryless TWC, if for any PX1 ,X2 = PX2 PX1 \|X2 = PX1 PX2 \|X1 , both of the following conditions are satisfied, then CI = CO : ∗ (i) There exists PX ∈ P(X1 ) such that for all x2 ∈ X2 we 1 ∗ . have arg maxPX \|X =x I(X1 ; Y2 \|X2 = x2 ) = PX 1 1 2 2 (ii) H(PX2 P̃X1 \|X2 , PY1 \|X1 ,X2 ) does not depend on P̃X1 \|X2 given PX2 and PY1 \|X1 ,X2 , and the ∗ common maximizer PX in (i) also satisfies 1 ∗ H̄⊥ (PX1 , PX2 , PY1 \|X1 ,X2 ) ≥ H̄(PX1 , PX2 \|X1 , PY1 \|X1 ,X2 ). (1) I (1) (X2 ; Y1 \|X1 ) = H (1) (Y1 \|X1 ) − H (1) (Y1 \|X1 , X2 ) = H̄(PX(1)1 , PX(1)2 \|X1 , PY1 \|X1 ,X2 ) − H(PX(1)2 PX(1)1 \|X2 , PY1 \|X1 ,X2 ) (14) ≤ H̄⊥ (PX∗ 1 , PX(1)2 , PY1 \|X1 ,X2 ) − H(PX(1)2 PX∗ 1 , PY1 \|X1 ,X2 ) (2) = H (Y1 \|X1 ) − H = I (2) (X2 ; Y1 \|X1 ), (2) (Y1 \|X1 , X2 ) (15) (16) where (14) and (16) follow from the definitions in Section II and (15) is due to condition (ii). Consequently, (1) (1) ∗ PX2 , PY1 ,Y2 \|X1 ,X2 ), and R(PX1 ,X2 , PY1 ,Y2 \|X1 ,X2 ) ⊆ R(PX 1 hence CO ⊆ CI , so that CI = CO . Theorem 3: For a given memoryless TWC, if both of the following conditions are satisfied, then CI = CO : (i) I(PX1 , PY2 \|X1 ,X2 =x2 ) does not depend on x2 ∈ X2 for any fixed PX1 ∈ P(X1 ). (ii) I(PX2 , PY1 \|X1 =x1 ,X2 ) does not depend on x1 ∈ X1 for any fixed PX2 ∈ P(X2 ). Proof: From conditions (i) and (ii), we know that maxPX1 \|X2 =x2 I(X1 ; Y2 \|X2 = x2 ) has a common maximizer ∗ PX for all x2 ∈ X2 and maxPX2 \|X1 =x1 I(X2 ; Y1 \|X1 = x1 ) 1 ∗ has a common maximizer PX for all x1 ∈ X1 . For any 2 (2) (1) (1) (1) ∗ P ∗ . By the same PX1 ,X2 = PX1 PX2 \|X1 , let PX1 ,X2 = PX 1 X2 argument as in (2)-(6), we conclude that I (1) (X1 ; Y2 \|X2 ) ≤ I (2) (X1 ; Y2 \|X2 ) and I (1) (X2 ; Y1 \|X1 ) ≤ I (2) (X2 ; Y1 \|X1 ). (1) ∗ ∗ PX Thus, R(PX1 ,X2 , PY1 ,Y2 \|X1 ,X2 ) ⊆ R(PX , PY1 ,Y2 \|X1 ,X2 ), 1 2 which yields CI = CO . Similar to the CVA condition, complex computations are often inevitable for checking the above conditions. We next present a useful condition which needs little computational effort. Let [PY2 \|X1 ,X2 (·\|·, x2 )] (resp. [PY1 \|X1 ,X2 (·\|x1 , ·)]) denote the marginal transition probability matrix obtained from PY1 ,Y2 \|X1 ,X2 =x2 (resp. PY1 ,Y2 \|X1 =x1 ,X2 ), whose columns and rows are indexed according to a fixed order on the symbols in Y2 and X1 (resp. Y1 and X2 ). Theorem 4: For a given memoryless TWC, if both of the following conditions are satisfied, then CI = CO : (i) The matrices [PY2 \|X1 ,X2 (·\|·, x2 )], x2 ∈ X2 , are column permutations of each other. (ii) The matrices [PY1 \|X1 ,X2 (·\|x1 , ·)], x1 ∈ X1 , are column permutations of each other. Since the proof is similar to the second part of the proof of Theorem 5 in the next section, the details are omitted. IV. D ISCUSSION AND E XAMPLES A. Comparison with Other Conditions As already noted, the relationship between Propositions 1 and 2 is unclear as examples that satisfy the Shannon condition but not the CVA condition seem hard to construct. In this section, we show that Theorems 1 and 2 in fact generalize the Shannon and CVA results, respectively. To see this, it suffices to show that the Shannon and CVA conditions imply the conditions in Theorems 1 and 2, respectively. 4 Theorem 5: A TWC satisfying Shannon’s symmetry condition in Proposition 1 must satisfy the conditions in Theorem 1. Proof: For a TWC satisfying the condition of Proposition 1, the optimal input probability distribution that achieves capacity is of the form PX1 ,X2 = PX2 /\|X1 \| for some PX2 ∈ P(X2 ) [1]. This result implies that condition (i) of Theorem 1 is satisfied because a common maximizer exists for all x2 ∈ X ∗ and is given by PX (x1 ) = 1/\|X1 \|. To prove that condition 1 (ii) is also satisfied, we consider the two (marginal) matrices [PY1 \|X1 ,X2 (·\|x′1 , ·)] and [PY1 \|X1 ,X2 (·\|x′′1 , ·)] for some fixed x′1 , x′′1 ∈ X1 and show that these matrices are column permutations of each other and hence I(PX2 , PY1 \|X1 =x′1 ,X2 ) = I(PX2 , PY1 \|X1 =x′′1 ,X2 ). The former claim is true because PY1 \|X1 ,X2 (y1 \|x′1 , x2 ) (17) = PY1 \|X1 ,X2 (π1Y1 [x′1 , x′′1 ](y1 )\|x′′1 , x2 ), (18) 1 where (17) is obtained by marginalizing Y2 on both sides of (1) and (18) follows from the definition of transposition. The second claim can be verified by a direct computation on I(PX2 , PY1 \|X1 =x1 ,X2 ) with the above result straightforwardly, and hence the details are omitted. Remark 1: Example 1 in the next subsection demonstrates that a TWC that satisfies the conditions in Theorem 1 may not satisfy Shannon’s symmetry condition in Proposition 1 since the common maximizer is not necessarily the uniform input distribution. Hence, Theorem 1 is a more general result than Proposition 1. Theorem 6: A TWC satisfying the CVA condition in Proposition 2 must satisfy the conditions in Theorem 2. Proof: Suppose that the condition of Proposition 2 is satisfied. To prove the theorem, we first claim that for j = 1, 2, H(Yj \|X1 = x′1 , X2 = x′2 ) = H(Yj \|X1 = x′′1 , X2 = x′2 ) for all x′1 , x′′1 ∈ X1 and x′2 ∈ X2 . Given arbitrary pairs (x′1 , x′2 ) and (x′′1 , x′2 ) with x′1 6= x′′1 , consider the two probability distributions 1, if a = x′1 and b = x′2 , (1) PX1 ,X2 (a, b) = 0, otherwise, and (2) PX1 ,X2 (a, b) (1) = 1, 0, if a = x′′1 and b = x′2 , otherwise. (2) Noting that PX2 = PX2 , we have H(Yj \|X1 = x′1 , X2 = x′2 ) = H (1) (Yj \|X1 , X2 ) (19) = H(PX(1)2 PX(1)1 \|X2 , PYj \|X1 ,X2 ) = H(PX(1)2 PX(2)1 \|X2 , PYj \|X1 ,X2 ) (2) = H (Yj \|X1 , X2 ) = H(Yj \|X1 = x′′1 , X2 = x′2 ), (1) PX1 ,X2 1 2 2 (1) (1) (1) ∗ for some PX2 ∈ P(X2 ). and define PX1 ,X2 = PX2 PX 1 \|X2 Since H(Yj \|X1 , X2 = x2 ) does not depend on PX1 \|X2 =x2 , ∗ is in fact a maximizer for H(Y2 \|X2 = x2 ). Note PX 1 \|X2 =x2 ∗ may not be unique, but any that the maximizer PX 1 \|X2 =x2 (1) choice works for our purposes. Now for PX1 ,X2 , by the CVA condition, there exists P̃X1 ∈ P(X1 ) such that (1) (1) = PY1 \|X1 ,X2 (π1Y1 [x′1 , x′′1 ](y1 )\|τxX′1,x′′ (x′1 ), x2 ) 1 on x1 ∈ X1 for fixed x2 ∈ X2 , H(Yj \|X1 , X2 = x2 ) does not depend on PX1 \|X2 =x2 . Next, we show that condition (i) of Theorem 2 holds by constructing the common maximizer from the CVA condition. For each x2 ∈ X2 , let ∗ = arg maxPX \|X =x I(X1 ; Y2 \|X2 = x2 ) = PX 1 \|X2 =x2 1 2 2 arg maxPX \|X =x [H(Y2 \|X2 = x2 ) − H(Y2 \|X1 , X2 = x2 )] (20) (21) (22) where (19) and (22) are due to the definitions of and (2) PX1 ,X2 , respectively, (20) follows from the CVA condition, (2) (1) PX2 . The claim is proved. Since and (21) holds since PX2 = P H(Yj \|X1 , X2 = x2 ) = x1 PX1 \|X2 (x1 \|x2 )H(Yj \|X1 = x1 , X2 = x2 ) and H(Yj \|X1 = x1 , X2 = x2 ) does not depend ∗ , PY2 \|X1 ,X2 ) ≤ H̄⊥ (PX2 , P̃X1 , PY2 \|X1 ,X2 ). H̄(PX2 , PX 1 \|X2 (1) (2) ∗ is the maximizer Set PX1 ,X2 = P̃X1 PX2 . Since PX 1 \|X2 =x2 for H(Y2 \|X2 = x2 ), we have (1) ∗ , PY2 \|X1 ,X2 ) H̄(PX2 , PX 1 \|X2 = H (1) (Y2 \|X2 ) X (1) PX2 (x2 ) · H (1) (Y2 \|X2 = x2 ) = x2 = X (1) PX2 (x2 ) x2 ≥ X max PX1 \|X2 =x2 H(Y2 \|X2 = x2 ) (1) PX2 (x2 ) · H (2) (Y2 \|X2 = x2 ) x2 (2) =H · (Y2 \|X2 ) (1) = H̄⊥ (PX2 , P̃X1 , PY2 \|X1 ,X2 ). Thus, H̄(PX(1)2 , PX∗ 1 \|X2 , PY2 \|X1 ,X2 ) = H̄⊥ (PX(1)2 , P̃X1 , PY2 \|X1 ,X2 ), i.e., X (1) PX2 (x2 ) · H (1) (Y2 \|X2 = x2 ) = X (1) PX2 (x2 ) · H (2) (Y2 \|X2 = x2 ). x2 x2 (2) (1) Since H (Y2 \|X2 = x2 ) ≤ H (Y2 \|X2 = x2 ) for each x2 ∈ X2 , we obtain H (1) (Y2 \|X2 = x2 ) = H (2) (Y2 \|X2 = x2 ), i.e., ∗ P̃X1 achieves the same value of H(Y2 \|X2 = x2 ) as PX 1 \|X2 =x2 for all x2 ∈ X2 . Consequently, P̃X1 is a common maximizer and thus condition (i) of Theorem 2 is satisfied. Moreover, since the common maximizer P̃X1 is provided by the CVA condition, condition (ii) of Theorem 2 automatically holds. Remark 2: Example 1 below shows that a TWC that satisfies the conditions in Theorem 2 does not necessarily satisfy the condition in Proposition 2 because our conditions allow H(PX2 P̃X1 \|X2 , PY2 \|X1 ,X2 ) to depend on P̃X1 \|X2 for given PX2 . Hence, Theorem 2 is more general than Proposition 2. B. Examples We next illustrate the effectiveness of our conditions via two examples in which X1 = X2 = Y1 = Y2 = {0, 1}. The TWC in Example 1 satisfies the conditions of Theorems 1-4 and the capacity region is rectangular. The TWC in Example 2 satisfies the conditions of Theorem 1 and 2 and has a non-rectangular capacity region. However, neither of the constructed TWCs satisfy the Shannon or the CVA conditions. 5 Example 1: Consider the TWC with  00 01 10 11 0.783 0.087 0.117 0.013    01   0.0417 0.3753 0.0583 0.5247 . [PY1 ,Y2 \|X1 ,X2 ] =   10  0.261 0.609 0.039 0.091  11 0.2919 0.1251 0.4081 0.1749 The corresponding one-way channel marginal distributions are given by 0.9 [PY2 \|X1 ,X2 (·\|·, 0)] = 0.3 0.1 [PY2 \|X1 ,X2 (·\|·, 1)] = 0.7 0.08  0.1 0.87 0.13 , [PY1 \|X1 ,X2 (·\|0, ·)] = , 0.7 0.417 0.583 0.87 0.13 0.9 . , [PY1 \|X1 ,X2 (·\|1, ·)] = 0.417 0.583 0.3 For this TWC, Shannon’s symmetry condition in Proposition 1 does not hold since there are no permutations on Y1 and Y2 which can result in (1). Furthermore, since H(Y2 \|X1 = 0, X2 = 0) = Hb (0.1) and H(Y2 \|X1 = 1, X2 = 0) = Hb (0.3), where Hb (·) denotes the binary entropy function, H(PX2 P̃X1 \|X2 , PY2 \|X1 ,X2 ) depends on P̃X1 \|X2 for given PX2 . Thus, the CVA condition in Proposition 2 does not hold, either. However by Theorem 4, Shannon’s inner and outer bounds coincide since [PY2 \|X1 ,X2 (·\|·, 0)] (resp. [PY1 \|X1 ,X2 (·\|0, ·)]) can be obtained by permuting the columns of [PY2 \|X1 ,X2 (·\|·, 1)] (resp. [PY1 \|X1 ,X2 (·\|1, ·)]). Since the conditions in Theorem 4 imply the conditions in Theorem 3 and the conditions in Theorem 3 further imply the conditions in Theorem 1, the conditions of Theorems 1 and 3 are also satisfied. Moreover, the optimal input distribution for this TWC can be obtained by searching for the common maximizer for each of the two one-way channels via the Blahut-Arimoto algorithm yielding ∗ ∗ PX (0) = PX (0) = 0.471. Thus, the capacity region is 1 2 ∗ ∗ achieved by the input distribution PX = PX P ∗ , i.e., 1 ,X2 1 X2 C = {(R1 , R2 ) : 0 ≤ R1 ≤ 0.2967, 0 ≤ R2 ≤ 0.1715}. Finally, we note that this TWC also satisfies the conditions of Theorem 2, in which the first condition is already implied by the conditions of Theorem 1. To verify the second condition, we consider 0.07 0.06 R2 00 0.1 0.09 0.05 0.04 0.03 0.02 0.01 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 R1 Fig. 2. The capacity region of the TWC in Example 2. [PY1 \|X1 ,X2 (·\|0, ·)] = [PY1 \|X1 ,X2 (·\|1, ·)] = [PY2 \|X1 ,X2 (·\|·, 1)]. Using the same arguments as in Example 1, one can easily see that this TWC satisfies neither the Shannon nor the CVA conditions. However, it satisfies the conditions in Theorem 1 since a common maximizer exists for the one-way channel ∗ from users 1 to 2, i.e., PX (0) = 0.471, and condition (ii) 1 trivially holds. To verify that this channel also satisfies the conditions in Theorem 2, the same argument as in the previous example is used. Finally, by considering all input distributions ∗ of the form PX1 ,X2 = PX P , the capacity region of this 1 X2 channel is determined as shown in Fig. 2. V. C ONCLUSIONS In this paper, four conditions on the coincidence of Shannon’s capacity inner and outer bounds were derived. These invariance conditions were shown to generalize existing results, thus enlarging the class of TWCs whose capacity region can be exactly determined. Numerical examples illustrate the applications of the new conditions in situations where prior results do not apply. R EFERENCES [1] C. E. Shannon, “Two-way communications channels,” in Proc. 4th Berkeley Symp. Math. Stat. Probab., Chicago, IL, USA, Jun. 1961, pp. 611-644. [2] J. Massey, “Causality, feedback and directed information,” in Proc. Int. Symp. Information Theory Its Applic. (ISITA-90), Waikiki, HI, USA, Nov. 1990, pp. 303-305. 0.417 0.583 0.87 0.13 . [3] G. Kramer, “Directed information for channels with feedback,” Ph.D. , [PY1 \|X1 ,X2 (·\|·, 1)] = [PY1 \|X1 ,X2 (·\|·, 0)] = 0.417 0.583 0.87 0.13 Dissertation, Swiss Federal Institute of Technology Zurich, 1998. [4] T. Han, “A general coding scheme for the two-way channel,” IEEE Here, for all x1 ∈ {0, 1}, H(Y1 \|X1 = x1 , X2 = 0) = Trans. Inf. Theory, vol. IT-30, no. 1, pp. 35-44, Jan. 1984. Hb (0.13) and H(Y1 \|X1 = x1 , X2 = 1) = Hb (0.417). [5] Z. Cheng and N. Devroye, “Two-way networks: when adaptation is Thus, H(PX2 P̃X1 \|X2 , PY1 \|X1 ,X2 ) does not depend on P̃X1 \|X2 useless,” IEEE Trans. Inf. Theory, vol. 60, no. 3, pp. 1793-1813, Mar. (1) 2014. given PX2 . Together with the substitutions PX1 ,X2 = PX1 ,X2 [6] L. Song, F. Alajaji, and T. Linder, “Adaptation is useless for two discrete (2) ∗ into (7)-(13), we then obtain that and PX1 ,X2 = PX P additive-noise two-way channels,” in Proc. IEEE Int. Symp. Inf. Theory, 1 X2 ∗ Barcelona, Spain, Jul. 2016, pp. 1854-1858. ). , P , P ) ≥ H̄(P H̄⊥ (PX , P , P X X Y \|X ,X X \|X Y \|X ,X 1 2 1 1 2 2 1 1 1 2 1 [7] A. Chaaban, L. R. Varshney, and M. Alouini, “The capacity of injective Therefore, the second condition of Theorem 2 holds. semi-deterministic two-way channels,” in Proc. IEEE Int. Symp. Inf. Theory, Aachen, Germany, Jun. 2017, pp. 431-436. Example 2: Consider the TWC with [8] J. Schalkwijk, “The binary multiplying channel - a coding scheme that 00 01 10 11 operates beyond Shannons inner bound region,” IEEE Trans. Inf. Theory,   vol. IT-28, no. 1, pp. 107-110, Jan. 1982. 00 0.783 0.087 0.117 0.013 [9] J. Schalkwijk, “On an extension of an achievable rate region for the   01  0.36279 0.05421 0.50721 0.07579  binary multiplying channel,” IEEE Trans. Inf. Theory, vol. IT-29, no. 3,   [PY1 ,Y2 \|X1 ,X2 ] =  , pp. 445-448, May 1983. 10  0.261 0.609 0.039 0.091  [10] Z. Zhang, T. Berger, and J. Schalkwijk, “New outer bounds to capacity 11 0.173889 0.243111 0.243111 0.339889 regions of two-way channels,” IEEE Trans. Inf. Theory, vol. IT-32, no. 3, pp. 383-386, May 1986. where two one-way channel marginal distributions are [11] A. Hekstra and F. Willems, “Dependence balance bounds for single output two-way channels,” IEEE Trans. Inf. Theory, vol. 35, no. 1, pp. 0.9 0.1 0.87 0.13 44-53, Jan. 1989. [PY2 \|X1 ,X2 (·\|·, 0)] = , [PY2 \|X1 ,X2 (·\|·, 1)] = , 0.3 0.7 0.417 0.583	7
On κ-reducibility of pseudovarieties of the form V ∗ D J. C. Costa C. Nogueira M. L. Teixeira arXiv:1602.03020v1 [math.GR] 9 Feb 2016 February 8, 2016 Abstract This paper deals with the reducibility property of semidirect products of the form V ∗ D relatively to graph equation systems, where D denotes the pseudovariety of definite semigroups. We show that, if the pseudovariety V is reducible with respect to the canonical signature κ consisting of the multiplication and the (ω − 1)-power, then V ∗ D is also reducible with respect to κ. Keywords. Pseudovariety, definite semigroup, semidirect product, implicit signature, graph equations, reducibility. 1 Introduction A semigroup (resp. monoid) pseudovariety is a class of finite semigroups (resp. monoids) closed under taking subsemigroups (resp. submonoids), homomorphic images and finite direct products. It is said to be decidable if there is an algorithm to test membership of a finite semigroup (resp. monoid) in that pseudovariety. The semidirect product of pseudovariets has been getting much attention, mainly due to the Krohn-Rhodes decomposition theorem [18]. In turn, the pseudovarieties of the form V∗D, where D is the pseudovariety of all finite semigroups whose idempotents are right zeros, are among the most studied semidirect products [23, 25, 3, 1, 4]. For a pseudovariety V of monoids, LV denotes the pseudovariety of all finite semigroups S such that eSe ∈ V for all idempotents e of S. We know from [17, 23, 24, 25] that V ∗ D is contained in LV and that V ∗ D = LV if and only if V is local in the sense of Tilson [25]. In particular, the equalities Sl ∗ D = LSl and G ∗ D = LG hold for the pseudovarieties Sl of semilattices and G of groups. J. C. Costa & M. L. Teixeira: CMAT, Dep. Matemática e Aplicações, Universidade do Minho, Campus de Gualtar, 4710-057 Braga, Portugal; e-mail: [email protected], [email protected] C. Nogueira: CMAT, Escola Superior de Tecnologia e Gestão, Instituto Politécnico de Leiria, Campus 2, Morro do Lena, Alto Vieiro, 2411-901 Leiria, Portugal; e-mail: [email protected] 1 2 J. C. Costa, C. Nogueira, M. L. Teixeira It is known that the semidirect product operator does not preserve decidability of pseudovarieties [20, 11]. The notion of tameness was introduced by Almeida and Steinberg [7, 8] as a tool for proving decidability of semidirect products. The fundamental property for tameness is reducibility. This property was originally formulated in terms of graph equation systems and latter extended to any system of equations [2, 21]. It is parameterized by an implicit signature σ (a set of implicit operations on semigroups containing the multiplication), and we speak of σ-reducibility. For short, given an equation system Σ with rational constraints, a pseudovariety V is σ-reducible relatively to Σ when the existence of a solution of Σ by implicit operations over V implies the existence of a solution of Σ by σ-words over V and satisfying the same constraints. The pseudovariety V is said to be σ-reducible if it is σ-reducible with respect to every finite graph equation system. The implicit signature which is most commonly encountered in the literature is the canonical signature κ = {ab, aω−1 } consisting of the multiplication and the (ω − 1)-power. For instance, the pseudovarieties D [9], G [10, 8], J [1, 2] of all finite J -trivial semigroups, LSl [16] and R [6] of all finite R-trivial semigroups are κ-reducible. In this paper, we study the κ-reducibility property of semidirect products of the form V∗D. This research is essentially inspired by the papers [15, 16] (see also [13] where a stronger form of κ-reducibility was established for LSl). We prove that, if V is κ-reducible then V ∗ D is κreducible. In particular, this gives a new and simpler proof (though with the same basic idea) of the κ-reducibility of LSl and establishes the κ-reducibility of the pseudovarieties LG, J ∗ D and R ∗ D. Combined with the recent proof that the κ-word problem for LG is decidable [14], this shows that LG is κ-tame, a problem proposed by Almeida a few years ago. This also extends part of our work in the paper [15], where we proved that under mild hypotheses on an implicit signature σ, if V is σ-reducible relatively to pointlike systems of equations (i.e., systems of equations of the form x1 = · · · = xn ) then V ∗ D is pointlike σ-reducible as well. As in [15], we use results from [5], where various kinds of σ-reducibility of semidirect products with an order-computable pseudovariety were considered. More specifically, we know from [5] that a pseudovariety of the form V ∗ Dk is κ-reducible when V is κ-reducible, where Dk is the order-computable pseudovariety defined by the identity yx1 · · · xk = x1 · · · xk . As S V ∗ D = k V ∗ Dk , we utilize this result as a way to achieve our property concerning the pseudovarieties V ∗ D. The method used in this paper is similar to that of [15]. However, some significant changes, inspired by [16], had to be introduced in order to deal with the much more intricate graph equation systems. 2 Preliminaries The reader is referred to the standard bibliography on finite semigroups, namely [1, 21], for general background and undefined terminology. For basic definitions and results about combinatorics on words, the reader may wish to consult [19]. On κ-reducibility of pseudovarieties of the form V ∗ D 2.1 3 Words and pseudowords Throughout this paper, A denotes a finite non-empty set called an alphabet. The free semigroup and the free monoid generated by A are denoted respectively by A+ and A∗ . The empty word is represented by 1 and the length of a word w ∈ A∗ is denoted by \|w\|. A word is called primitive if it cannot be written in the form un with n > 1. Two words u and v are said to be conjugate if u = w1 w2 and v = w2 w1 for some words w1 , w2 ∈ A∗ . A Lyndon word is a primitive word which is minimal in its conjugacy class, for the lexicographic order on A+ . A left-infinite word on A is a sequence w = (an )n of letters of A indexed by −N also written w = · · · a−2 a−1 . The set of all left-infinite words on A will be denoted by A−N and we put A−∞ = A+ ∪ A−N . The set A−∞ is endowed with a semigroup structure by defining a product as follows: if w, z ∈ A+ , then wz is already defined; left-infinite words are right zeros; finally, if w = · · · a−2 a−1 is a left-infinite word and z = b1 b2 · · · bn is a finite word, then wz is the left-infinite word wz = · · · a−2 a−1 b1 b2 · · · bn . A left-infinite word w of the form u−∞ v = · · · uuuv, with u ∈ A+ and v ∈ A∗ , is said to be ultimately periodic. In case v = 1, the word w is named periodic. For a periodic word w = u−∞ , if u is a primitive word, then it will be called the root of w and its length \|u\| will be said to be the period of w. For a pseudovariety V of semigroups, we denote by ΩA V the relatively free pro-V semigroup generated by the set A: for each pro-V semigroup S and each function ϕ : A → S, there is a unique continuous homomorphism ϕ : ΩA V → S extending ϕ. The elements of ΩA V are called pseudowords (or implicit operations) over V. A pseudovariety V is called order-computable when the subsemigroup ΩA V of ΩA V generated by A is finite, in which case ΩA V = ΩA V, and effectively computable. Recall that, for the pseudovariety S of all finite semigroups, ΩA S is (identified with) the free semigroup A+ . The elements of ΩA S \ A+ will then be called infinite pseudowords. A pseudoidentity is a formal equality π = ρ of pseudowords π, ρ ∈ ΩA S over S. We say that V satisfies the pseudoidentity π = ρ, and write V \|= π = ρ, if ϕπ = ϕρ for every continuous homomorphism ϕ : ΩA S → S into a semigroup S ∈ V, which is equivalent to saying that pV π = pV ρ for the natural projection pV : ΩA S → ΩA V. 2.2 Pseudoidentities over V ∗ Dk For a positive integer k, let Dk be the pseudovariety of all finite semigroups satisfying the identity yx1 · · · xk = x1 · · · xk . Denote by Ak the set of words over A with length k and by Ak the set {w ∈ A+ : \|w\| ≤ k} of non-empty words over A with length at most k. We notice that ΩA Dk may be identified with the semigroup whose support set is Ak and whose multiplication is given by u · v = tk (uv), where tk w denotes the longest suffix of length at most k of a given (finite or left-infinite) word w. Then, the Dk are order-computable pseudovarieties such that S D = k Dk . Moreover, it is well-known that ΩA D is isomorphic to the semigroup A−∞ . For each pseudoword π ∈ ΩA S, we denote by tk π the unique smallest word (of Ak ) such that Dk \|= π = tk π. Simetrically, we denote by ik π the smallest word (of Ak ) such 4 J. C. Costa, C. Nogueira, M. L. Teixeira that Kk \|= π = ik π, where Kk is the dual pseudovariety of Dk defined by the identity x1 · · · xk y = x1 · · · xk . Let Φk be the function A+ → (Ak+1 )∗ that sends each word w ∈ A+ to the sequence of factors of length k + 1 of w, in the order they occur in w. We still denote by Φk (see [3] and [1, Lemma 10.6.11]) its unique continuous extension ΩA S → (ΩAk+1 S)1 . This function Φk is a k-superposition homomorphism, with the meaning that it verifies the conditions: i) Φk w = 1 for every w ∈ Ak ; ii) Φk (πρ) = Φk πΦk (tk π)ρ = Φk π(ik ρ) Φk ρ for every π, ρ ∈ ΩA S. Throughout the paper, V denotes a non-locally trivial pseudovariety of semigroups. For any pseudowords π, ρ ∈ ΩA S, it is known from [1, Theorem 10.6.12] that V ∗ Dk \|= π = ρ 2.3 ⇐⇒ ik π = ik ρ, tk π = tk ρ and V \|= Φk π = Φk ρ. (2.1) Implicit signatures and σ-reducibility By an implicit signature we mean a set σ of pseudowords (over S) containing the multiplication. In particular, we represent by κ the implicit signature {ab, aω−1 }, usually called the canonical signature. Every profinite semigroup has a natural structure of a σ-algebra, via the natural interpretation of pseudowords on profinite semigroups. The σ-subalgebra of ΩA S generated by A is denoted by ΩσA S. It is freely generated by A in the variety of σ-algebras generated by the pseudovariety S and its elements are called σ-words (over S). To a (directed multi)graph e Γ = V (Γ) ⊎ E(Γ), with vertex set V (Γ), edge set E(Γ), and edges αe − → ωe, we associate the system ΣΓ of all equations of the form (αe) e = ωe, with e ∈ E(Γ). Let S be a finite A-generated semigroup, δ : ΩA S → S be the continuous homomorphism respecting the choice of generators and ϕ : Γ → S 1 be an evaluation mapping such that ϕE(Γ) ⊆ S. We say that a mapping η : Γ → (ΩA S)1 is a V-solution of ΣΓ with respect to (ϕ, δ) when δη = ϕ and V \|= ηu = ηv for all (u = v) ∈ ΣΓ . Furthermore, if ηΓ ⊆ (ΩσA S)1 for an implicit signature σ, then η is called a (V, σ)-solution. The pseudovariety V is said to be σ-reducible relatively to the system ΣΓ if the existence of a V-solution of ΣΓ with respect to a pair (ϕ, δ) entails the existence of a (V, σ)-solution of ΣΓ with respect to the same pair (ϕ, δ). We say that V is σ-reducible, if it is σ-reducible relatively to ΣΓ for all finite graphs Γ. 3 κ-reducibility of V ∗ D Let V be a given κ-reducible non-locally trivial pseudovariety. The purpose of this paper is to prove the κ-reducibility of the pseudovariety V ∗ D. So, we fix a finite graph Γ and a finite A-generated semigroup S and consider a V ∗ D-solution η : Γ → (ΩA S)1 of the system ΣΓ with respect to a pair (ϕ, δ), where ϕ : Γ → S 1 is an evaluation mapping such that ϕE(Γ) ⊆ S and δ : ΩA S → S is a continuous homomorphism respecting the choice of generators. We have to construct a (V ∗ D, κ)-solution η ′ : Γ → (ΩκA S)1 of ΣΓ with respect to the same pair (ϕ, δ). On κ-reducibility of pseudovarieties of the form V ∗ D 3.1 5 Initial considerations Suppose that g ∈ Γ is such that ηg = u with u ∈ A∗ . Since η and η ′ are supposed to be V ∗ D-solutions of the system ΣΓ with respect to (ϕ, δ), we must have δη = ϕ = δη ′ and so, in particular, δη ′ g = δu. As the homomorphism δ : ΩA S → S is arbitrarily fixed, it may happen that the equality δη ′ g = δu holds only when η ′ g = u. In that case we would be obliged to define η ′ g = u. Since we want to describe an algorithm to define η ′ that should work for any given graph and solution, we will then construct a solution η ′ verifying the following condition: ∀g ∈ Γ, (ηg ∈ A∗ =⇒ η ′ g = ηg). C1 (Γ, η, η ′ ) Suppose next that a vertex v ∈ V (Γ) is such that D \|= ηv = uω with u ∈ A+ , that is, suppose that pD ηv = u−∞ . Because Γ is an arbitrary graph, it could include, for instance, an edge e such that αe = ωe = v and the labeling η could be such that ηe = u. Since D is a subpseudovariety of V ∗ D, η is a D-solution of ΣΓ with respect to (ϕ, δ). Hence, as by condition C1 (Γ, η, η ′ ) we want to preserve finite labels, it would follow in that case that D \|= (η ′ v)u = η ′ v and, thus, that D \|= η ′ v = uω = ηv. This observation suggests that we should preserve the projection into ΩA D of labelings of vertices v such that pD ηv = u−∞ with u ∈ A+ . More generally, we will construct the (V ∗ D, κ)-solution η ′ in such a way that the following condition holds: ∀v ∈ V (Γ), (pD ηv = u−∞ z with u ∈ A+ and z ∈ A∗ =⇒ pD η ′ v = pD ηv). C2 (Γ, η, η ′ ) Let ℓη = max{\|u\| : u ∈ A∗ and ηg = u for some g ∈ Γ} be the maximum length of finite labels under η of elements of Γ. To be able to make some reductions on the graph Γ and solution η, described in Section 3.2, we want η ′ to verify the extra condition below, where L ≥ ℓη is a non-negative integer to be specified later, on Section 3.3: ∀v ∈ V (Γ), 3.2 (ηv = uπ with u ∈ AL =⇒ η ′ v = uπ ′ with δπ = δπ ′ ). C3 (Γ, η, η ′ ) Simplifications on the solution η We begin this section by reducing to the case in which all vertices of Γ are labeled by infinite e pseudowords under η. Suppose first that there is an edge v − → w such that ηv = uv and ηe = ue with uv ∈ A∗ and ue ∈ A+ , so that ηw = uv ue . Drop the edge e and consider the restrictions η1 and ϕ1 , of η and ϕ respectively, to the graph Γ1 = Γ\{e}. Then η1 is a V∗D-solution of the system ΣΓ1 with respect to the pair (ϕ1 , δ). Assume that there is a (V ∗ D, κ)-solution η1′ of ΣΓ1 with respect to (ϕ1 , δ) verifying condition C1 (Γ1 , η1 , η1′ ). Then η1′ v = uv and η1′ w = uv ue . Let η ′ be the extension of η1′ to Γ obtained by letting η ′ e = ue . Then η ′ is a (V ∗ D, κ)-solution of ΣΓ with respect to (ϕ, δ). By induction on the number of edges labeled by finite words under η beginning in vertices also labeled by finite words under η, we may therefore assume that there are no such edges in Γ. Now, we remove all vertices v of Γ labeled by finite words under η such that v is not the beginning of an edge, thus obtaining a graph Γ1 . As above, if η1′ is a (V ∗ D, κ)-solution of 6 J. C. Costa, C. Nogueira, M. L. Teixeira ΣΓ1 , then we build a (V ∗ D, κ)-solution η ′ of ΣΓ by letting η ′ coincide with η1′ on Γ1 and letting η ′ v = ηv for each vertex v ∈ Γ \ Γ1 . So, we may assume that all vertices of Γ labeled by finite words under η are the beginning of some edge. e Suppose next that v − → w is an edge such that ηv = u and ηe = π with u ∈ A∗ and π ∈ ΩA S \ A+ . Notice that, since it is an infinite pseudoword, π can be written as π = π1 π2 with both π1 and π2 being infinite pseudowords. Drop the edge e (and the vertex v in case e e1 is the only edge beginning in v) and let v1 be a new vertex and v1 −→ w be a new edge thus obtaining a new graph Γ1 . Let η1 and ϕ1 be the labelings of Γ1 defined as follows: • η1 and ϕ1 coincide, respectively, with η and ϕ on Γ′ = Γ1 ∩ Γ; • η1 v1 = uπ1 , η1 e1 = π2 , ϕ1 v1 = δη1 v1 and ϕ1 e1 = δη1 e1 . Then η1 is a V ∗ D-solution of the system ΣΓ1 with respect to the pair (ϕ1 , δ). Assume that there is a (V ∗D, κ)-solution η1′ of ΣΓ1 with respect to (ϕ1 , δ) verifying conditions C1 (Γ1 , η1 , η1′ ) and C3 (Γ1 , η1 , η1′ ). In particular, since L is chosen to be greater than ℓη , η1′ v1 = uπ1′ with ′ ′ ′ ′ δπ1 = δπ1′ . Let η ′ be the extension of η1\|Γ ′ to Γ obtained by letting η e = π1 (η1 e1 ) (and η ′ v = u in case v 6∈ Γ′ ). As one can easily verify, η ′ is a (V ∗ D, κ)-solution of ΣΓ with respect to (ϕ, δ). By induction on the number of edges beginning in vertices labeled by finite words under η, we may therefore assume that all vertices of Γ are labeled by infinite pseudowords under η. Suppose at last that an edge e ∈ Γ is labeled under η by a finite word u = a1 · · · an , where n > 1 and ai ∈ A. Denote v0 = αe and vn = ωe. In this case, we drop the edge e and, for each ei → vi to the graph Γ. Let Γ1 i ∈ {1, . . . , n − 1}, we add a new vertex vi and a new edge vi−1 − be the graph thus obtained and let η1 and ϕ1 be the labelings of Γ1 defined as follows: • η1 and ϕ1 coincide, respectively, with η and ϕ on Γ′ = Γ \ {e}; • for each i ∈ {1, . . . , n − 1}, η1 vi = (ηv)a1 · · · ai , η1 ei = ai , ϕ1 vi = δη1 vi and ϕ1 ei = δη1 ei . Hence, η1 is a V ∗ D-solution of the system ΣΓ1 with respect to the pair (ϕ1 , δ). Suppose there exists a (V ∗ D, κ)-solution η1′ of ΣΓ1 with respect to (ϕ1 , δ) verifying condition C1 (Γ1 , η1 , η1′ ). ′ ′ ′ Let η ′ be the extension of η1\|Γ ′ to Γ obtained by letting η e = u. Then η is a (V ∗ D, κ)solution of ΣΓ with respect to (ϕ, δ). By induction on the number of edges labeled by finite words under η, we may further assume that each edge of Γ labeled by a finite word under η is, in fact, labeled by a letter of the alphabet. 3.3 Borders of the solution η The main objective of this section is to define a certain class of finite words, called borders of the solution η. Since the equations (of ΣΓ ) we have to deal with are of the form (αe) e = ωe, these borders will serve to signalize the transition from a vertex αe to the edge e. For each vertex v of Γ, denote by dv ∈ A−N the projection pD ηv of ηv into ΩA D and let Dη = {dv \| v ∈ V (Γ)}. We say that two left-infinite words v1 , v2 ∈ A−N are confinal if they On κ-reducibility of pseudovarieties of the form V ∗ D 7 have a common prefix y ∈ A−N , that is, if v1 = yz1 and v2 = yz2 for some words z1 , z2 ∈ A∗ . As one easily verifies, the relation ∝ defined, for each dv1 , dv2 ∈ Dη , by dv 1 ∝ dv 2 if and only if dv1 and dv2 are confinal is an equivalence on Dη . For each ∝-class ∆, we fix a word y∆ ∈ A−N and words zv ∈ A∗ , for each vertex v with dv ∈ ∆, such that dv = y∆ zv . Moreover, when dv is ultimately periodic, we choose y∆ of the form u−∞ , with u a Lyndon word, and fix zv not having u as a prefix. The word u and its length \|u\| will be said to be, respectively, a root and a period of the solution η. Without loss of generality, we assume that η has at least one root (otherwise we could, easily, modify the graph and the solution in order to include one). η′ . We fix a few of the integers that will be used in the construction of the (V ∗ D, κ)-solution They depend only on the mapping η and on the semigroup S. Definition 3.1 (constants nS , pη , L, E and Q) We let: • nS be the exponent of S which, as one recalls, is the least integer such that snS is idempotent for every element s of the finite A-generated semigroup S; • pη = lcm{\|u\| : u ∈ A+ is a root of η}; • L = max{ℓη , \|zv \| : v ∈ V (Γ)}; • E be an integer such that E ≥ nS pη and, for each word w ∈ AE , there is a factor e ∈ A+ of w for which δe is an idempotent of S. Notice that, for each root u of η, \|unS \| ≤ E and δ(unS ) is an idempotent of S; • Q = L + E. For each positive integer m, we denote by Bm the set Bm = {tm y∆ ∈ Am \| ∆ is a ∝-class}. If y∆ = u−∞ is a periodic left-infinite word, then the element y = tm y∆ of Bm will be said to be periodic (with root u and period \|u\|). For words y1 , y2 ∈ Bm , we define the gap between y1 and y2 as the positive integer g(y1 , y2 ) = min{\|u\| ∈ N : u ∈ A+ and, for some v ∈ A+ , y1 u = vy2 or y2 u = vy1 }, and notice that g(y1 , y2 ) = g(y2 , y1 ) ≤ m. Proposition 3.2 Consider the constant Q introduced in Definition 3.1. There exists qQ ∈ N such that for all integers m ≥ qQ the following conditions hold: 8 J. C. Costa, C. Nogueira, M. L. Teixeira (a) If y1 and y2 are distinct elements of Bm , then g(y1 , y2 ) > Q; (b) If y is a non-periodic element of Bm , then g(y, y) > Q. Proof. Suppose that, for every qQ ∈ N there is an integer m ≥ qQ and elements ym,1 and ym,2 of Bm such that g(ym,1 , ym,2 ) ≤ Q. Hence, there exist a strictly increasing sequence (mi )i of positive integers and an integer r ∈ {1, . . . , Q} such that g(ymi ,1 , ymi ,2 ) i is constant and equal to r. Moreover, since the graph Γ is finite, we may assume that ymi ,1 = tmi y∆1 and ymi ,2 = tmi y∆2 for every i and some ∝-classes ∆1 and ∆2 . It then follows that y∆1 u = y∆2 or y∆2 u = y∆1 for some word u ∈ Ar . Hence, y∆1 and y∆2 are confinal left-infinite words, whence ∆1 and ∆2 are the same ∝-class ∆. Therefore, for every m, ym,1 and ym,2 have the same length and are suffixes of the word y∆ and, so, ym,1 and ym,2 are the same word. This proves already (a). Now, notice that y∆ u = y∆ , meaning that y∆ is the periodic left-infinite word u−∞ . This shows (b) and completes the proof of the proposition. We now fix two more integers. Definition 3.3 (constants M and k) We let: • M be an integer such that M is a multiple of pη and M is greater than or equal to the integer qQ of Proposition 3.2, and notice that M > Q; • k = M + Q. The elements of the set BM will be called the borders of the solution η. We remark that the borders of η are finite words of length M such that, by Proposition 3.2, for any two distinct occurrences of borders y1 and y2 in a finite word, either these occurrences have a gap of size at least Q between them, or y1 and y2 are the same periodic border y. In this case, y is a power of its root u, since M is a multiple of the period \|u\|, and g(y, y) is \|u\|. 3.4 Getting a (V ∗ Dk , κ)-solution As V ∗ Dk is a subpseudovariety of V ∗ D, η is a V ∗ Dk -solution of ΣΓ with respect to (ϕ, δ). The given pseudovariety V was assumed to be κ-reducible. So, by [5, Corollary 6.5], V ∗ Dk is κ-reducible too. Therefore, there is a (V ∗ Dk , κ)-solution ηk′ : Γ → (ΩκA S)1 of ΣΓ with respect to the same pair (ϕ, δ). Moreover, as observed in [6, Remark 3.4], one can constrain the values ηk′ g of each g ∈ Γ with respect to properties which can be tested in a finite semigroup. Since the prefixes and the suffixes of length at most k can be tested in the finite semigroup ΩA Kk × ΩA Dk , we may assume further that ηk′ g and ηg have the same prefixes and the same suffixes of length at most k. We then denote ig = ik ηk′ g = ik ηg and tg = tk ηk′ g = tk ηg, for each g ∈ Γ. Notice that, by the simplifications introduced in Section 3.2, if ηg is a finite word, then g is an edge and ηg is a letter ag and so ig = tg = ag . Otherwise, ig and tg are On κ-reducibility of pseudovarieties of the form V ∗ D 9 length k words. In particular, condition C1 (Γ, η, ηk′ ) holds. That is, ηk′ e = ηe for every edge e such that ηe is a finite word. On the other hand, Lemma 2.3 (ii) of [12], which is stated only for edges, can be extended easily to vertices, so that ηk′ g can be assumed to be an infinite pseudoword for every g ∈ Γ such that ηg is infinite. Thus, in particular, ηk′ v is an infinite pseudoword for all vertices v. Notice that, for each vertex v, there exists a border yv of η such that the finite word yv zv is a suffix of ηv. On the other hand, by Definitions 3.1 and 3.3, \|zv \| ≤ L < Q and k = M + Q. So, as \|yv \| = M , tv = xv yv zv and ηk′ v = πv tv (3.1) for some infinite κ-word πv and some word xv ∈ A+ with \|xv \| = Q − \|zv \|. 3.5 Basic transformations The objective of this section is to introduce the basic steps that will allow to transform the (V ∗ Dk , κ)-solution ηk′ into a (V ∗ D, κ)-solution η ′ . The process of construction of η ′ from ηk′ is close to the one used in [15] to handle with systems of pointlike equations. Both procedures are supported by (basic) transformations of the form a1 · · · ak 7→ a1 · · · aj (ai · · · aj )ω aj+1 · · · ak , which replace words of length k by κ-words. Those procedures differ in the way the indices i ≤ j are determined. In the pointlike case, the only condition that a basic transformation had to comply with was that j had to be minimum such that the value of the word a1 · · · ak under δ is preserved. In the present case, the basic transformations have to preserve the value under δ as well, but the equations (αe)e = ωe impose an extra restriction that is not required by pointlike equations. Indeed, we need η ′ to verify, in particular, δη ′ αe = δηk′ αe(= δηαe) and δη ′ e = δηk′ e(= δηe). So, somewhat informally, for a word a1 · · · ak that has an occurrence overlapping both the factors ηk′ αe and ηk′ e of the pseudoword (ηk′ αe)(ηk′ e), the introduction of the factor (ai · · · aj )ω by the basic transformation should be done either in ηk′ αe or in ηk′ e, and not in both simultaneously. The borders of the solution η were introduced to help us to deal with this extra restriction. Informally speaking, the borders will be used to detect the “passage” from the labeling under ηk′ of a vertex αe to the labeling of the edge e and to avoid that the introduction of (ai · · · aj )ω affect the labelings under δ of ηk′ αe or ηk′ e. Consider an arbitrary word w = a1 · · · an ∈ A+ . An integer m ∈ {M, . . . , n} will be called a bound of w if the factor w[m] = am′ · · · am of w is a border, where m′ = m − M + 1. The bound m will be said to be periodic or non-periodic according to the border w[m] is periodic or not. If w admits bounds, then there is a maximum one that we name the last bound of w. In this case, if ℓ is the last bound of w, then the border w[ℓ] will be called the last border of w. Notice that, by Proposition 3.2 and the choice of M , if m1 and m2 are two bounds of w with m1 < m2 , then either m2 − m1 > Q or w[m1 ] and w[m2 ] are the same periodic border. 10 J. C. Costa, C. Nogueira, M. L. Teixeira Let w = a1 · · · ak ∈ A+ be a word of length k. Notice that, since k = M + Q, if w has a non-periodic last bound ℓ, then ℓ is the unique bound of w. We split the word w in two parts, lw (the left-hand of w) and rw (the right-hand of w), by setting l w = a1 · · · as and rw = as+1 · · · ak where s (the splitting point of w) is defined as follows: if w has a last bound ℓ then s = ℓ; otherwise s = k. In case w has a periodic last bound ℓ, the splitting point s will be said to be periodic. Then, s is not periodic in two situations: either w has a non-periodic last border or w has not a last border. The factorization w = lw rw will be called the splitting factorization of w. We have s ≥ M > Q ≥ E. So, by definition of E, there exist integers i and j such that s − E < i < j ≤ s and the factor e = ai · · · aj of lw verifies δe = (δe)2 . We begin by fixing the maximum such j and, for that j, we fix next an integer i and a word ew = ai · · · aj , called the essential factor of w, as follows. Notice that, if the splitting point s is periodic and u is the root of the last border of w, then δ(unS ) is idempotent and the left-hand of w is of the form lw = l′w unS . Hence, in this case, j = s and we let ew = unS , thus defining i as j − nS \|u\| + 1. Suppose now that the splitting point is not periodic. In this case we let i be the maximum integer such that δ(ai · · · aj ) is idempotent. The word w can be factorized as w = l′w ew l′′w rw , where l′w = a1 · · · ai−1 . We then denote by w b the following κ-word w b = l′w ew eωw l′′w rw = a1 · · · aj (ai · · · aj )ω aj+1 · · · ak and notice that δw b = δw. Moreover \|ew l′′w \| ≤ E and so \|l′w \| ≥ M − E > Q − E = L. It is also convenient to introduce two κ-words derived from w b λk w = a1 · · · aj (ai · · · aj )ω , ̺k w = (ai · · · aj )ω aj+1 · · · ak . (3.2) This defines two mappings λk , ̺k : Ak → ΩκA S that can be extended to ΩA S as done in [15]. Although they are not formally the same mappings used in that paper, because of the different choice of the integers i and j, we keep the same notation since the selection process of those integers is absolutely irrelevant for the purpose of the mappings. That is, with the above adjustment the mappings maintain the properties stated in [15]. The next lemma presents a property of the b-operation that is fundamental to our purposes. Lemma 3.4 For a word w = a1 · · · ak+1 ∈ A+ of length k + 1, let w1 = a1 · · · ak and w2 = a2 · · · ak+1 be the two factors of w of length k. If w b1 = a1 · · · aj1 (ai1 · · · aj1 )ω aj1 +1 · · · ak and w b2 = a2 · · · aj2 (ai2 · · · aj2 )ω aj2 +1 · · · ak+1 , then a1 lw2 = lw1 x for some word x ∈ A∗ . In particular j1 ≤ j2 . On κ-reducibility of pseudovarieties of the form V ∗ D 11 Proof. Write w2 = b1 · · · bk with bi = ai+1 . Let s1 and s2 be the splitting points of w1 and w2 respectively, whence lw1 = a1 · · · as1 and lw2 = b1 · · · bs2 = a2 · · · as2 +1 . To prove that there exists a word x such that a1 lw2 = lw1 x, we have to show that s1 ≤ s2 + 1. Under this hypothesis, we then deduce that ai1 · · · aj1 is an occurrence of the essential factor ew1 in lw2 which proves that j1 ≤ j2 . Assume first that w1 has a last bound ℓ1 , in which case s1 = ℓ1 . By definition, ℓ1 ≥ M . If ℓ1 > M , then the last border of w1 occurs in w2 , one position to the left relatively to w1 . Hence ℓ1 − 1 is a bound of w2 and, so, w2 has a last bound ℓ2 such that ℓ2 ≥ ℓ1 − 1. It follows in this case that s2 = ℓ2 and s1 ≤ s2 + 1. Suppose now that ℓ1 = M . Since s2 ≥ M by definition, the condition s1 ≤ s2 + 1 holds trivially in this case. Suppose now that w1 has not a last bound. Then s1 = k. Moreover, either w2 does not have a last bound or k is its last bound. In both circumstances s2 = k, whence s1 = s2 ≤ s2 + 1. This concludes the proof of the lemma. In the conditions of the above lemma and as in [15], we define ψk : (ΩAk+1 S)1 → (ΩA S)1 as the only continuous monoid homomorphism which extends the mapping Ak+1 → ΩκA S a1 · · · ak+1 7→ (ai1 · · · aj1 )ω aj1 +1 · · · aj2 (ai2 · · · aj2 )ω and let θk = ψk Φk . The function θk : ΩA S → (ΩA S)1 is a continuous k-superposition homomorphism since it is the composition of the continuous k-superposition homomorphism Φk with the continuous homomorphism ψk . We remark that a word w = a1 · · · an of length n > k has precisely r = n − k + 1 factors of length k and θk (w) = ψk (a1 · · · ak+1 , a2 · · · ak+2 , . . . , ar−1 · · · an ) = ψk (a1 · · · ak+1 )ψk (a2 · · · ak+2 ) · · · ψk (ar−1 · · · an ) = (eω1 f1 eω2 )(eω2 f2 eω3 ) · · · (eωr−1 fr−1 eωr ) = eω1 f1 eω2 f2 · · · eωr−1 fr−1 eωr where, for each p ∈ {1, . . . , r}, ep is the essential factor ewp = aip · · · ajp of the word wp = ap · · · ak+p−1 and fp = ajp +1 · · · ajp+1 (p 6= r). Above, for each p ∈ {2, . . . , r − 1}, we have replaced each expression eωp eωp with eωp since, indeed, these expressions represent the same κword. More generally, one can certainly replace an expression of the form xω xn xω with xω xn . Using this reduction rule as long as possible, θk (w) can be written as θk (w) = eωn1 f¯1 eωn2 f¯2 · · · eωnq f¯q , called the reduced form of θk (w), where q ∈ {1, . . . , r}, 1 = n1 < n2 < · · · < nq ≤ r, f¯p = fnp · · · fnp+1−1 (for p ∈ {1, . . . , q − 1}) and f¯q is fnq · · · fr−1 if nq 6= r and it is the empty word otherwise. 12 3.6 J. C. Costa, C. Nogueira, M. L. Teixeira Definition of the (V ∗ D, κ)-solution η ′ We are now in conditions to describe the procedure to transform the (V ∗ Dk , κ)-solution ηk′ into the (V ∗ D, κ)-solution η ′ . The mapping η ′ : Γ → (ΩκA S)1 is defined, for each g ∈ Γ, as η ′ g = (τ1 g)(τ2 g)(τ3 g), where, for each i ∈ {1, 2, 3}, τi : Γ → (ΩκA S)1 is a function defined as follows. First of all, we let τ2 = θk ηk′ . That τ2 is well-defined, that is, that τ2 g is indeed a κ-word for every g ∈ Γ, follows from the fact that ηk′ g is a κ-word and θk transforms κ-words into κ-words (see [15]). Next, for each vertex v, consider the length k words iv = ik ηk′ v = ik ηv and tv = tk ηk′ v = tk ηv. We let τ1 v = λk iv and τ3 v = ̺k tv , where the mappings λk and ̺k were defined in (3.2). Note that, by (3.1), tv = xv yv zv . Moreover, the occurrence of yv shown in this factorization is the last occurrence of a border in tv . Hence, the right-hand rtv of tv is precisely zv . Therefore, one has τ1 v = λk iv = l′iv eiv eωiv and τ3 v = ̺k tv = eωtv l′′tv zv . Consider now an arbitrary edge e. Suppose that ηe is a finite word. Then, ηe is a letter ae and ηk′ e is also ae in this case. Then τ2 e = θk ae = 1 because θk is a k-superposition homomorphism. Since we want η ′ e to be ae , we then define, for instance, τ 1 e = ae and τ3 e = 1. Suppose at last that ηe (and so also ηk′ e) is an infinite pseudoword. We let τ3 e = ̺k te and notice that τ3 e = τ3 ωe. Indeed, as ηk′ is a V ∗ Dk -solution of ΣΓ , it follows from (2.1) that te = tk ηk′ e = tk ηk′ ωe = tωe . The definition of τ1 e is more elaborate. Let v be the vertex αe and consider the word tv ie = a1 · · · a2k . This word has r = k + 1 factors of length k. Suppose that θk (tv ie ) is eω1 f1 eω2 f2 · · · eωr−1 fr−1 eωr and consider its reduced form θk (tv ie ) = eω1 f¯1 eωn2 f¯2 · · · eωnq f¯q . Notice that tv ie = f¯0 f¯1 · · · f¯q f¯q+1 for some words f¯0 , f¯q+1 ∈ A∗ . Hence, there is a (unique) ′ and f¯ = f¯′ f¯′′ with f¯′ ∈ A∗ and f¯′′ ∈ index m ∈ {1, . . . , q} such that tv = f¯0 f¯1 · · · f¯m−1 f¯m m m m m m ω ω ′′ ′ + ω ω ω ¯ ¯ ¯ ¯ ¯ A . Then θk (tv ie ) = β1 β2 , where β1 = e1 f1 en2 f2 · · · enm fm and β2 = fm enm+1 fm+1 · · · enq f¯q and we let ′′ ω τ1 e = β2 = f¯m enm+1 f¯m+1 · · · eωnq f¯q . ′′ f¯ ′ ω ¯ Note that the word β2′ = f¯m m+1 · · · fq is ak+1 · · · ajr , whence β2 er = λk ie . The next lemma is a key result that justifies the definition of the b-operation. On κ-reducibility of pseudovarieties of the form V ∗ D 13 Lemma 3.5 Let e be an edge such that ηe is infinite. Then, with the above notation, β1 = τ3 v and so θk (tv ie ) = (τ3 v)(τ1 e). Moreover, δτ1 e = δλk ie . Proof. We begin by recalling that tv ie = a1 · · · a2k and θk (tv ie ) = eω1 f1 eω2 f2 · · · eωr−1 fr−1 eωr = eω1 f¯1 eωn2 f¯2 · · · eωnq f¯q , where ep is the essential factor ewp = aip · · · ajp of the word wp = ap · · · ak+p−1 and fp = ajp +1 · · · ajp+1 for each p. Note also that λk ie = β2′ eωr , er is a suffix of β2′ and δer is idempotent. So, to prove the equality δτ1 e = δλk ie it suffices to show that δτ1 e = δβ2′ . We know from (3.1) that tv = xv yv zv with 1 ≤ \|xv \| ≤ Q. So, xv = a1 · · · ah−1 , yv = ah · · · aM +h−1 and zv = aM +h · · · ak for some h ∈ {2, . . . , Q + 1}. There are two cases to verify. Case 1. yv is a non-periodic border. Consider the factor wh = ah · · · ak+h−1 of tv ie . By the choice of M and k, the prefix yv is the only occurrence of a border in wh . Hence, M is the last bound of wh and, so, its splitting point. It follows that wh = yv ·zv ak+1 · · · ak+h−1 is the splitting factorization of wh . Therefore, as one can verify for an arbitrary p ∈ {1, . . . , h}, there is only one occurrence of a border in wp , precisely yv , and the splitting factorization of wp is wp = ap · · · ah−1 yv · zv ak+1 · · · ak+p−1 , whence ep = e1 with jp = j1 ≤ M + h − 1 and, so, fp = 1 for p < h. So, the prefix eω1 f1 eω2 · · · fh−1 eωh of θk (tv ie ) reduces to eω1 . Consider now the factor wh+1 = ah+1 · · · ak+h . Hence, either wh+1 does not have a last bound or k is its last bound. In both situations, the splitting point of wh+1 is k and its splitting factorization is wh+1 = wh+1 · 1. Therefore, one deduces from Lemma 3.4 that, for every p ∈ {h + 1, . . . , r}, the occurrence aip · · · ajp of the essential factor ewp in wp is, in fact, an occurrence in the suffix w′ = ak+h−E · · · a2k = aM +L+h · · · a2k of tv ie . Since \|xv yv \| = M +h−1 and \|zv \| ≤ L, it follows that k = \|xv yv zv \| < M + L + h, whence w′ is a suffix of ie and so k < ip < jp for all p ∈ {h + 1, . . . , r}. This means, in particular, that the ω-power eωh+1 is introduced at the suffix ie of tv ie . Hence β1 = eω1 f1 eω2 · · · fh−1 eωh ajh +1 · · · ak and its reduced form is eω1 aj1 +1 · · · ak = τ3 v, which proves that β1 and τ3 v are the same κ-word. Moreover, from k < ip , one deduces that the word ep is a suffix of ak+1 · · · ajp , which proves that δτ1 e = δβ2′ . Case 2. yv is a periodic border. Let u be the root of yv . Then, since M was fixed as a M multiple of \|u\|, yv = uMu where Mu = \|u\| . If the prefix yv is the only occurrence of a border in wh , then one deduces the lemma as in Case 1 above. So, we assume that there is another occurrence of a border y in wh . Hence, by Proposition 3.2 and the choice of M and k, y is precisely yv . Furthermore, since u is a Lyndon word and k = M + Q with Q < M , wh = yv ud wh′ for some positive integer d and some word wh′ ∈ A∗ such that u is not a prefix of wh′ . Notice that, since u is not a prefix of zv by definition of this word, zv is a proper prefix of u. On the other hand wh = ud yv wh′ and the occurrence of yv shown in 14 J. C. Costa, C. Nogueira, M. L. Teixeira this factorization is the last occurrence of yv in wh . Thus, wh = ud yv · wh′ is the splitting factorization of wh . Therefore w ch = ud yv (unS )ω wh′ and eh = unS . More generally, for any p ∈ {1, . . . , h}, yv is a factor of wp and it is the only border that occurs in wp . Hence, the splitting point of wp is periodic and ep = unS . Moreover, as one can verify, j1 = M + h − 1 and the prefix eω1 f1 eω2 · · · fh−1 eωh of θk (tv ie ) is eω1 (u(eω1 )\|u\| )d and so, analogously to Case 1, it reduces to eω1 ud . Since zv is a proper prefix of u and d ≥ 1, k < jh . This allows already deduce that the reduced form of β1 is (unS )ω zv = τ3 v, thus concluding the proof of the first part of the lemma. Now, there are two possible events. ′′ = β ′ , in which case δτ e = δβ ′ is trivially verified. Or m 6= q Either m = q and β2 = f¯m 1 2 2 ω and the ω-power enm+1 was not eliminated in the reduction process of θk (tv ie ). This means that the splitting point of the word wnm+1 is not determined by one of the occurrences of the border yv in the prefix a1 · · · ak+h−1 of tv ie . Then, as in Case 1 above, one deduces that k < ip for each p ∈ {nm+1 , . . . , r} and, so, that δτ1 e = δβ2′ . In both cases β1 = τ3 v and δτ1 e = δλk ie . Hence, the proof of the lemma is complete. Notice that, as shown in the proof of Lemma 3.5 above, if a vertex v is such that yv is a periodic border with root u, then τ3 v = (unS )ω zv . So, the definition of the mapping τ3 on vertices assures condition C2 (Γ, η, η ′ ). 3.7 Proof that η ′ is a (V ∗ D, κ)-solution This section will be dedicated to showing that η ′ is a (V ∗ D, κ)-solution of ΣΓ with respect to the pair (ϕ, δ) verifying conditions C1 (Γ, η, η ′ ) and C3 (Γ, η, η ′ ). We begin by noticing that η ′ g is a κ-word for every g ∈ Γ. Indeed, as observed above, each τ2 g is a κ-word. That both τ1 g and τ3 g are κ-words too, is easily seen by their definitions. Let us now show the following properties. Proposition 3.6 Conditions δη ′ = ϕ, C1 (Γ, η, η ′ ) and C3 (Γ, η, η ′ ) hold. Proof. As ηk′ is a V ∗ Dk -solution of ΣΓ with respect to (ϕ, δ) and, so, the equality δηk′ = ϕ holds, to deduce that δη ′ = ϕ holds it suffices to establish the equality δη ′ = δηk′ . Consider first a vertex v ∈ Γ. Then τ1 v = λk iv = l′iv eiv eωiv and τ3 v = ̺k tv = eωtv l′′tv zv . In this case, the equality δηk′ v = δη ′ v is a direct application of [15, Proposition 5.3], where the authors proved that δπ = δ (λk ik π)(θk π)(̺k tk π) (3.3) for every pseudoword π. Moreover, by definition of the b-operation, \|l′iv \| > L. Therefore, ηv and η ′ v are of the form ηv = uπ and η ′ v = uπ ′ with u ∈ AL and δπ = δπ ′ . So, condition C3 (Γ, η, η ′ ) holds. On κ-reducibility of pseudovarieties of the form V ∗ D 15 Consider next an edge e ∈ Γ. If ηk′ e is a finite word ae , then η ′ e = (τ1 e)(τ2 e)(τ3 e) = ae ·1·1 = ae = ηk′ e, whence δη ′ e = δηk′ e holds trivially. Moreover, since ηk′ e = ηe in this case and every vertex is labeled under η by an infinite pseudoword, it follows that condition C1 (Γ, η, η ′ ) holds. Suppose at last that ηk′ e is infinite and let v = αe. Then τ3 e = ̺k te . On the other hand, by Lemma 3.5, δτ1 e = δλk ie . Hence, by (3.3) and since δ is a homomorphism, δη ′ e = δ (τ1 e)(τ2 e)(τ3 e) = δ (λk ie )(θk ηk′ e)(̺k te ) = δηk′ e. This ends the proof of the proposition. e Consider an arbitrary edge v − → w of Γ. To achieve the objectives of this section it remains to prove that V ∗ D satisfies (η ′ v)(η ′ e) = η ′ w. Since ηk′ is a V ∗ Dk -solution of ΣΓ , V ∗ Dk satisfies (ηk′ v)(ηk′ e) = ηk′ w. Hence, by (2.1), iv = ik (ηk′ v)(ηk′ e) = ik (ηk′ w) = iw and tk (ηk′ v)(ηk′ e) = tk (ηk′ w) = tw . Thus, τ1 v = λk iv = l′iv eiv eωiv = l′iw eiw eωiw = λk iw = τ1 w and τ3 w = ̺k tw = eωtw l′′tw zw . As shown in the proof of [15, Proposition 5.4], it then follows that V ∗ D satisfies eωiw θk (ηk′ v)(ηk′ e) eωtw = eωiw θk (ηk′ w)eωtw and, so, V ∗ D \|= (τ1 v)θk (ηk′ v)(ηk′ e) (τ3 w) = (τ1 w)θk (ηk′ w)(τ3 w) = η ′ w. (3.4) On the other hand, from the fact that θk is a k-superposition homomorphism one deduces θk (ηk′ v)(ηk′ e) = θk (ηk′ v)θk tv (ηk′ e) = θk (ηk′ v)θk (tv ie )θk (ηk′ e). (3.5) Suppose that ηk′ e is an infinite pseudoword. In this case te = tw , whence τ3 e = τ3 w. Moreover, by Lemma 3.5, θk (tv ie ) = (τ3 v)(τ1 e). Therefore, by conditions (3.4) and (3.5), V ∗ D satisfies (η ′ v)(η ′ e) = η ′ w. Assume now that ηk′ e is a finite word, whence ηk′ e = ae ∈ A and η ′ e = ae . Since η is a D-solution of ΣΓ , D \|= (η ′ v)ae = η ′ w and, thus, dv ae = dw . Hence the left-infinite words dv and dw are confinal and, so, ∝-equivalent. Hence dv = y∆ zv , dw = y∆ zw and yv = yw = tk y∆ , where ∆ is the ∝-class of dv and dw . It follows that y∆ zv ae = y∆ zw and tk tv ae ) = tw . In this case, θk (ηk′ v)(ηk′ e) = θk (ηk′ v)θk (tv ae ). On the other hand, tv ae = a1 · · · ak ak+1 = a1 tw is a word of length k + 1 and, so, θk (tv ae ) = ψk (tv ae ) is of the form θk (tv ae ) = eω1 f eω2 . The splitting factorizations of tv and tw are, respectively, tv = xv yv · zv and tw = xw yw · zw . Since yv = yw , it follows that e1 = etv = etw = e2 . Suppose that zv ae = zw . In this case it is clear that f = 1, so that θk (tv ae ) = eωtv . Since θk (ηk′ v) ends with eωtv , it then follows that θk (ηk′ v)(ηk′ e) = θk ηk′ v = τ2 v. Therefore, (τ1 v)θk (ηk′ v)(ηk′ e) (τ3 w) = (τ1 v)(τ2 v)(τ3 w). On the other hand, τ3 w = ̺k tw = eωtw l′′tw zw = eωtv l′′tv zv ae = (τ3 v)ae . So, by (3.4), one has that V ∗ D satisfies (η ′ v)ae = (τ1 v)(τ2 v)(τ3 v)ae = (τ1 v)(τ2 v)(τ3 w) = η ′ w. Suppose now that zv ae 6= zw . In this case, one deduces from the equality y∆ zv ae = y∆ zw , that y∆ is a periodic left-infinite word. Let u be its root, so that y∆ = u−∞ , etv = unS and l′′tv = l′′tw = 1. Since, by definition, u is a primitive word which is not a prefix of zv nor a prefix of 16 J. C. Costa, C. Nogueira, M. L. Teixeira zw , we conclude that zv ae = u and zw = 1. In this case f = u, whence θk (tv ae ) = eωtv u. Then, θk (ηk′ v)(ηk′ e) = (θk ηk′ v)u = (τ2 v)u. Therefore, (τ1 v)θk (ηk′ v)(ηk′ e) (τ3 w) = (τ1 v)(τ2 v)u(τ3 w). Moreover, u(τ3 w) = ueωtw l′′tw zw = u(unS )ω = (unS )ω u = eωtv l′′tv zv ae = (τ3 v)ae . Therefore, using (3.4), one deduces as above that V ∗ D satisfies (η ′ v)ae = η ′ w. We have proved the main theorem of the paper. Theorem 3.7 If V is κ-reducible, then V ∗ D is κ-reducible. This result applies, for instance, to the pseudovarieties Sl, G, J and R. Since the κ-word problem for the pseudovariety LG of local groups is already solved [14], we obtain the following corollary. Corollary 3.8 The pseudovariety LG is κ-tame. Final remarks. In this paper we fixed our attention on the canonical signature κ, while in [15] we dealt with a more generic class of signatures σ verifying certain undemanding conditions. Theorem 3.7 is still valid for such generic signatures σ but we preferred to treat only the instance of the signature κ to keep the proofs clearer and a little less technical. References [1] J. Almeida, Finite Semigroups and Universal Algebra, (World Scientific, Singapore, 1995). 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arXiv:1403.4349v1 [math.AC] 18 Mar 2014 LINEARLY RELATED POLYOMINOES VIVIANA ENE, JÜRGEN HERZOG, TAKAYUKI HIBI Abstract. We classify all convex polyomino ideals which are linearly related or have a linear resolution. Convex stack polyominoes whose ideals are extremal Gorenstein are also classified. In addition, we characterize, in combinatorial terms, the distributive lattices whose join-meet ideals are extremal Gorenstein or have a linear resolution. Introduction The ideal of inner minors of a polyomino, a so-called polyomino ideal, is generated by certain subsets of 2-minors of an m × n-matrix X of indeterminates. Such ideals have first been studied by Qureshi in [17]. They include the two-sided ladder determinantal ideals of 2-minors which may also be viewed as the join-meet ideal of a planar distributive lattice. It is a challenging problem to understand the graded free resolution of such ideals. In [7], Ene, Rauf and Qureshi succeeded to compute the regularity of such joint-meet ideals. Sharpe [19, 20] showed that the ideal I2 (X) of all 2-minors of X is linearly related, which means that I2 (X) has linear relations. Moreover, he described these relations explicitly and conjectured that also the ideals of t-minors It (X) are generated by a certain type of linear relations. This conjecture was then proved by Kurano [13]. In the case that the base field over which It (X) is defined contains the rational numbers, Lascoux [14] gives the explicit free resolution of all ideals of t-minors. Unfortunately, the resolution of It (X) in general may depend on the characteristic of the base field. Indeed, Hashimoto [8] showed that for 2 ≤ t ≤ min(m, n) − 3, the second Betti number β2 of It (X) depends on the characteristic. On the other hand, by using squarefree divisor complexes [2] as introduced by Bruns and the second author of this paper, it follows from [2, Theorem 1.3] that β2 for t = 2 is independent of the characteristic. In this paper we use as a main tool squarefree divisor complexes to study the first syzygy module of a polyomino ideal. In particular, we classify all convex polyominoes which are linearly related; see Theorem 2.1. This is the main result of this paper. In the first section we recall the concept of polyomino ideals and show that the polyomino ideal of a convex polyomino has a quadratic Gröbner basis. The second section of the paper is devoted to state and to prove Theorem 2.1. As mentioned before, the proof heavily depends on the theory of squarefree divisor complexes which allow to compute the multi-graded Betti numbers of a toric ideal. To apply this theory, one observes that the polyomino ideal of a convex polyomino 2010 Mathematics Subject Classification. 13C05, 05E40, 13P10. Key words and phrases. Binomial ideals, Linear syzygies, Polyominoes. The first author was supported by the grant UEFISCDI, PN-II-ID-PCE- 2011-3-1023. 1 may be naturally identified with a toric ideal. The crucial conclusion deduced from this observation, formulated in Corollary 2.5, is then that the Betti numbers of a polyomino ideal is bounded below by the Betti numbers of the polyomino ideal of any induced subpolyomino. Corollary 2.5 allows to reduce the study of the relation of polyomino ideals to that of a finite number of polyominoes with a small number of cells which all can be analyzed by the use of a computer algebra system. In the last section, we classify all convex polyominoes whose polyomino ideal has a linear resolution (Theorem 3.1) and all convex stack polyominoes whose polyomino ideal is extremal Gorenstein (Theorem 3.4). Since polyomino ideals overlap with join-meet ideals, it is of interest which of the ideals among the join-meet ideals have a linear resolution or are extremal Gorenstein. The answers are given in Theorem 3.2 and Theorem 3.5. It turns out that the classifications for both classes of ideals almost lead to the same result. 1. Polyominoes In this section we consider polyomino ideals. This class of ideals of 2-minors was introduced by Qureshi [17]. To this end, we consider on N2 the natural partial order defined as follows: (i, j) ≤ (k, l) if and only if i ≤ k and j ≤ l. The set N2 together with this partial order is a distributive lattice. If a, b ∈ N2 with a ≤ b, then the set [a, b] = {c ∈ N2 \| a ≤ c ≤ b} is an interval of N2 . The interval C = [a, b] with b = a + (1, 1) is called a cell of N2 . The elements of C are called the vertices of C and a is called the left lower corner of C. The egdes of the cell C are the sets {a, (a + (1, 0)}, {a, a + (0, 1)}, {(a + (1, 0), a + (1, 1)} and {(a + (0, 1), a + (1, 1)}. Let P be a finite collection of cells and C, D ∈ P. Then C and D are connected, if there is a sequence of cells of P given by C = C1 , . . . , Cm = D such that Ci ∩ Ci+1 is an edge of Ci for i = 1, . . . , m − 1. If, in addition, Ci 6= Cj for all i 6= j, then C is called a path (connecting C and D). The collection of cells P is called a polyomino if any two cells of P are connected; see Figure 1. The set of vertices of P, denoted V (P), is the union of the vertices of all cells belonging to P. Two polyominoes are called isomorphic if they are mapped to each other by a composition of translations, reflections and rotations. Figure 1. A polyomino 2 We call a polyomino P row convex, if for any two cells C, D of P with left lower corner a = (i, j) and b = (k, j) respectively, and such that k > i, it follows that all cells with left lower corner (l, j) with i ≤ l ≤ k belong to P. Similarly, one defines column convex polyominoes. The polyomino P is called convex if it is row and column convex. The polyomino displayed in Figure 1 is not convex, while Figure 2 shows a convex polyomino. Note that a convex polyomino is not convex in the common geometric sense. Figure 2. A convex polyomino Now let P be any collection of cells. We may assume that the vertices of all the cells of P belong to the interval [(1, 1), (m, n)]. Fix a field K and let S be the polynomial ring over K in the variables xij with (i, j) ∈ P. The ideal of inner minors IP ⊂ S of P, is the ideal generated by all 2-minors xil xkj − xkl xij for which [(i, j), (k, l)] ⊂ V (P). Furthermore, we denote by K[P] the K-algebra S/IP . If P happens to be a polyomino, then IP will also be called a polyomino ideal. For example, the polyomino P displayed in Figure 2 may be embedded into the interval [(1, 1), (4, 4)]. Then, in these coordinates, IP is generated by the 2-minors x22 x31 − x32 x21 , x23 x31 − x33 x21 , x24 x31 − x34 x21 , x23 x32 − x33 x22 , x24 x32 − x34 x22 , x24 x33 − x34 x23 , x13 x22 − x12 x23 , x13 x32 − x12 x33 , x13 x42 − x12 x43 , x23 x42 − x22 x43 , x33 x42 − x32 x43 . The following result has been shown by Qureshi in [17, Theorem 2.2]. Theorem 1.1. Let P be a convex polyomino. Then K[P] is a normal Cohen– Macaulay domain. The proof of this theorem is based on the fact that IP may be viewed as follows as a toric ideal: with the assumptions and notation as introduced before, we may assume that V (P) ⊂ [(1, 1), (m, n)]. Consider the K-algebra homomorphism ϕ : S → T with ϕ(xij ) = si tj for all (i, j) ∈ V (P). Here T = K[s1 , . . . , sm , t1 , . . . , tn ] is the polynomial ring over K in the variables si and tj . Then, as observed by Qureshi, IP = Ker ϕ. It follows that K[P] may be identified with the edge ring of the bipartite graph GP on the vertex set {s1 , . . . , sm } ∪ {t1 , . . . , tn } and edges {si , tj } 3 with (i, j) ∈ V (P). With this interpretation of K[P] in mind and by using [16], we obtain Proposition 1.2. Let P be a convex polyomino. Then IP has a quadratic Gröbner basis. Proof. We use the crucial fact, proved in [16], that the toric ideal which defines the edge ring of a bipartite graph has a quadratic Gröbner basis if and only if each 2r-cycle with r ≥ 3 has a chord. By what we explained before, a 2k-cycle, after identifying the vertices of P with the edges of a bipartite graph, is nothing but a sequence of vertices a1 , . . . , a2r of P with a2k−1 = (ik , jk ) and a2k = (ik+1 , jk ) for k = 1, . . . , r such that ir+1 = i1 , ik 6= iℓ and jk 6= jℓ for all k, ℓ ≤ r and k 6= ℓ. A typical such sequence of pairs of integers is the following: 32244553 11332244 Here the first row is the sequence of the first component and the second row the sequence of the second component of the vertices ai . This pair of sequences represents an 8-cycle. It follows from Lemma 1.3 that there exist integers s and t with 1 ≤ t, s ≤ r and t 6= s, s + 1 such that either is < it < is+1 or is+1 < it < is . Suppose that is < it < is+1 . Since a2s−1 = (is , js ) and a2s = (is+1 , js ) are vertices of P and since P is convex, it follows that (it , js ) ∈ P. This vertex corresponds to a chord of the cycle a1 , . . . , a2r . Similarly one argues if is+1 < it < is . Lemma 1.3. Let r ≥ 3 be an integer and f : [r + 1] → Z a function such that f (i) 6= f (j) for 1 ≤ i < j ≤ r and f (r + 1) = f (1). Then there exist 1 ≤ s, t ≤ r such that one has either f (s) < f (t) < f (s + 1) or f (s + 1) < f (t) < f (s). Proof. Let, say, f (1) < f (2). Since f (r + 1) = f (1), there is 2 ≤ q ≤ r with f (1) < f (2) < · · · < f (q) > f (q + 1). • Let q = r. Then, since q = r ≥ 3, one has (f (1) =) f (r + 1) < f (2) < f (r). • Let q < r and f (q + 1) > f (1). Since f (q + 1) 6∈ {f (1), f (2), . . . , f (q)}, it follows that there is 1 ≤ s < q with f (s) < f (q) < f (s + 1). • Let q < r and f (q + 1) < f (1). Then one has f (q + 1) < f (1) < f (q). The case of f (1) > f (2) can be discussed similarly. We denote the graded Betti numbers of IP by βij (IP ). Corollary 1.4. Let P be a convex polyomino. Then β1j (IP ) = 0 for j > 4. Proof. By Proposition 1.2, there exists a monomial order < such that in< (IP ) is generated in degree 2. Therefore, it follows from [10, Corollary 4] that β1j (in< (IP )) = 0 for j > 4. Since β1j (IP ) ≤ β1j (in< (IP )) (see, for example, [9, Corollary 3.3.3]), the desired conclusion follows. 4 2. The first syzygy module of a polyomino ideal Let P be a convex polyomino and let f1 , . . . , fm be the minors generating IP . In this section we study the relation module Syz1 (IP ) of IP which is the kernel of the L S-module homomorphism m i=1 Sei → IP with ei 7→ fi for i = 1, . . . , m. The graded module Syz1 (IP ) has generators in degree 3 and no generators in degree > 4, as we have seen in Corollary 1.4. We say that IP (or simply P) is linearly related if Syz1 (IP ) is generated only in degree 3. Let fi and fj be two distinct generators of IP . Then the Koszul relation fi ej −fj ei belongs Syz1 (IP ). We call fi , fj a Koszul relation pair if fi ej − fj ei is a minimal generator of Syz1 (IP ). The main result of this section is the following. Theorem 2.1. Let P be a convex polyomino. The following conditions are equivalent: (a) P is linearly related; (b) IP admits no Koszul relation pairs; (c) Let, as we may assume, [(1, 1), (m, n)] be the smallest interval with the property that V (P) ⊂ [(1, 1), (m, n)]. We refer to the elements (1, 1), (m, 1), (1, n) and (m, n) as the corners. Then P has the shape as displayed in Figure 5, and one of the following conditions hold: (i) at most one of the corners does not belong to V (P); (ii) two of the corners do not belong to V (P), but they are not opposite to each other. In other words, the missing corners are not the corners (1, 1), (n, m), or the corners (m, 1), (1, n). (iii) three of the corners do not belong to V (P). If the missing corners are (m, 1), (1, n) and (m, n) (which one may assume without loss of generality), then referring to Figure 5 the following conditions must be satisfied: either i2 = m − 1 and j4 ≤ j2 , or j2 = n − 1 and i4 ≤ i2 . As an essential tool in the proof of this theorem we recall the co-called squarefree divisor complex, as introduced in [9]. Let K be field, H ⊂ Nn an affine semigroup and K[H] the semigroup ring attached to it. Suppose that h1 , . . . , hm ∈ Nn is the unique minimal set of generators of H. We consider the polynomial ring T = K[t1 , . . . , tn ] Q h (j) in the variables t1 , . . . , tn . Then K[H] = K[u1 , . . . , um ] ⊂ T where ui = nj=1 tj i and where hi (j) denotes the jth component of the integer vector hi . We choose a presentation S = K[x1 , . . . , xm ] → K[H] with xi 7→ ui for i = 1, . . . , m. The kernel IH of this K-algebra homomorphism is called the toric ideal of H. We assign a Zn -grading to S by setting deg xi = hi . Then K[H] as well as IH become Zn graded S-modules. Thus K[H] admits a minimal Zn -graded S-resolution F with L Fi = h∈H S(−h)βih (K[H]) . In the case that all ui are monomials of the same degree, one can assign to K[H] the structure of a standard graded K-algebra by setting deg ui = 1 for all i. The degree of h with respect to this standard grading will be denoted \|h\|. Given h ∈ H, we define the squarefree divisor complex ∆h as follows: ∆h is the simplicial complex whose faces F = {i1 , . . . , ik } are the subsets of [n] such that 5 h(1) ui1 · · · uik divides t1 · · · th(n) in K[H]. We denote by H̃i (Γ, K) the ith reduced n simplicial homology of a simplicial complex Γ. Proposition 2.2 (Bruns-Herzog [2]). With the notation and assumptions introduced one has Tori (K[H], K)h ∼ = H̃i−1 (∆h , K). In particular, βih (K[H]) = dimK H̃i−1 (∆h , K). Let H ′ be a subsemigroup of H generated by a subset of the set of generators of H, and let S ′ be the polynomial ring over K in the variables xi with hi generator of H ′ . Furthermore, let F′ the Zn -graded free S ′ -resolution of K[H ′ ]. Then, since ′ S. The S is a flat S ′ -module, F′ ⊗S ′ S is a Zn -graded free S-resolution of S/IH ′ n ′ inclusion K[H ] → K[H] induces a Z -graded complex homomorphism F ⊗S ′ S → F. Tensoring this complex homomorphism with K = S/m, where m is the graded maximal ideal of S, we obtain the following sequence of isomorphisms and natural maps of Zn -graded K-modules ′ TorS (K[H ′ ], K) ∼ = TorS (K[H], K). = Hi (F′ ⊗S ′ S)⊗S K) → Hi (F⊗S K) ∼ = Hi (F′ ⊗S ′ K) ∼ i i For later applications we need Corollary 2.3. With the notation and assumptions introduced, let H ′ be a subsemigroup of H generated by a subset of the set of generators of H, and let h be an element of H ′ with the property that hi ∈ H ′ whenever h − hi ∈ H. Then the ′ natural K-vector space homomorphism TorSi (K[H ′ ], K)h → TorSi (K[H], K)h is an isomorphism for all i. Proof. Let ∆′h be the squarefree divisor complex of h where h is viewed as an element of H ′ . Then we obtain the following commutative diagram Tori (K[H ′], K)h −−−→ Tori (K[H], K)h    y    y H̃i−1 (∆′h , K) −−−→ H̃i−1 (∆h , K). The vertical maps are isomorphisms, and also the lower horizontal map is an isomorphism, simply because ∆′h = ∆h , due to assumptions on h. This yields the desired conclusion. Let H ⊂ Nn be an affine semigroup generated by h1 , . . . , hm . An affine subsemigroup H ′ ⊂ H generated by a subset of {h1 , . . . , hm } will be called a homological pure subsemigroup of H if for all h ∈ H ′ and all hi with h − hi ∈ H it follows that hi ∈ H ′ . As an immediate consequence of Corollary 2.3 we obtain Corollary 2.4. Let H ′ be a homologically pure subsemigroup of H. Then ′ TorSi (K[H ′], K) → TorSi (K[H], K) is injective for all i. In other words, if F′ is the minimal Zn -graded free S ′ -resolution of K[H ′] and F is the minimal Zn -graded free S-resolution of K[H], then the complex homomorphism F′ ⊗S → F induces an injective map F′ ⊗K → F ⊗K. In particular, 6 any minimal set of generators of Syzi (K[H ′]) is part of a minimal set of generators of Syzi (K[H]). Moreover, βij (IH ′ ) ≤ βij (IH ) for all i and j. We fix a field K and let P ⊂ [(1, 1), (m, n)] be a convex polyomino. Let as before S be the polynomial ring over K in the variables xij with (i, j) ∈ V (P) and K[P] the K-subalgebra of the polynomial ring T = K[s1 , . . . , sm , t1 , . . . , tn ] generated by the monomials uij = si tj with (i, j) ∈ V (P). Viewing K[P] as a semigroup ring K[H], it is convenient to identify the semigroup elements with the monomial they represent. Given sets {i1 , i2 , . . . , is } and {j1 , j2 , . . . , jt } of integers with ik ⊂ [m] and jk ⊂ [n] for all k, we let H ′ be the subsemigroup of H generated by the elements sik tjl with (ik , jl ) ∈ V (P). Then H ′ a homologically pure subsemigroup of H. Note that H ′ is also a combinatorially pure subsemigroup of H in the sense of [15]. A collection of cells P ′ will be called a collection of cells of P induced by the columns i1 , i2 , . . . , is and the rows j1 , j2 , . . . , jt , if the following holds: (k, l) ∈ V (P ′ ) if and only if (ik , jl ) ∈ V (P). Observe that K[P ′ ] is always a domain, since it is a K-subalgebra of K[P]. The map V (P ′ ) → V (P), (k, l) 7→ (ik , jl ) identifies IP ′ with the ideal contained in IP generated by those 2-minors of I(P) which only involve the variables xik ,jl . In the following we always identify IP ′ with this subideal of IP . If the induced collection of cells of P ′ is a polyomino, we call it an induced polyomino. Any induced polyomino P ′ of P is again convex. Consider for example the polyomino P on the left side of Figure 3 with left lower corner (1, 1). Then the induced polyomino P ′ shown on the right side of Figure 3 is induced by the columns 1, 3, 4 and the rows 1, 2, 3, 4. P′ P Figure 3. Obviously Corollary 2.4 implies Corollary 2.5. Let P ′ be an induced collection of cells of P. Then βij (IP ′ ) ≤ βij (IP ) for all i and j, and each minimal relation of IP ′ is also a minimal relation of IP . We will now use Corollary 2.5 to isolate step by step the linearly related polyominoes. Lemma 2.6. Suppose P admits an induced collection of cells P ′ isomorphic to one of those displayed in Figure 4. Then IP has a Koszul relation pair. 7 Proof. We may assume that V (P ′ ) ⊂ [(1, 1), (4, 4)]. By using CoCoA [3] or Singular [4] to compute Syz1 (IP ′ ) we see that the minors fa = [12\|12] and fb = [34\|34] form a Koszul relation pair of IP ′ . Thus the assertion follows from Corollary 2.5. (a) (b) Figure 4. P ′ Corollary 2.7. Let P be a convex polyomino, and let [(1, 1), (m, n)] be the smallest interval with the property that V (P) ⊂ [(1, 1), (m, n)]. We assume that m, n ≥ 4. If one of the vertices (2, 2), (m − 1, 2), (m − 1, n − 1) or (2, n − 1) does not belong to V (P), then IP has a Koszul relation pair, and, hence, IP is not linearly related. Proof. We may assume that (2, 2) 6∈ V (P). Then the vertices of the interval [(1, 1), (2, 2)] do not belong to V (P). Since [(1, 1), (m, n)] is the smallest interval containing V (P), there exist, therefore, integers i and j with 2 < i ≤ m − 1 and 2 < j ≤ n − 1 such that the cells [(i, 1), (i + 1, 2)] and [(1, j), (2, j + 1)] belong to P. Then the collection of cells induced by the rows 1, 2, i, i + 1 and the columns 1, 2, j, j + 1 is isomorphic to one of the collections P ′ of Figure 4. Thus the assertion follows from Lemma 2.6 and Corollary 2.5. Corollary 2.7 shows that the convex polyomino P should contain all the vertices (2, 2), (m − 1, 2), (m − 1, n − 1) and (2, n − 1) in order to be linearly related. Thus a polyomino which is linearly related must have the shape as indicated in Figure 5. The number i1 is also allowed to be 1 in which case also j1 = 1. In this case the polyomino contains the corner (1, 1). A similar convention applies to the other corners. In Figure 5 all for corners (1, 1), (1, n), (m, 1) and (m, n) are missing. The convex polyomino displayed in Figure 6 however is not linearly related, though it has the shape as shown in Figure 5. Thus there must still be other obstructions for a polyomino to be linearly related. Now we proceed further in eliminating those polyominoes which are not linearly related. Lemma 2.8. Let P be a convex polyomino, and let [(1, 1), (m, n)] be the smallest interval with the property that V (P) ⊂ [(1, 1), (m, n)]. If P misses only two opposite corners, say (1, 1) and (m, n), or P misses all four corners (1, 1), (1, n), (m, 1) and (m, n), then IP admits a Koszul pair and hence is not linearly related. 8 i3 i4 (2, n − 1)• • (m − 1, n − 1) j4 j2 j3 j1 (2, 2) • •(m − 1, 2) i1 i2 Figure 5. Possible shape Figure 6. Not linearly related Proof. Let us first assume that (1, 1) and (m, n) do not belong to V (P), but (1, n) and (m, 1) belong to V (P). The collection of cells P1 induced by the rows 1, 2, m − 1, m and the columns 1, 2, n − 1, n is shown in Figure 7. All the light colored cells, some of them or none of them are present according to whether or not all, some or none of the equations i1 = 2, j1 = 2, i4 = m − 1 and j4 = n − 1 hold. For example, if i1 = 2, j1 6= 2, i4 = m − 1 and j4 6= n − 1, then the light colored cells [(2, 1), (3, 2)] and [(2, 3), (3, 4)] belong P1 and the other two light colored cells do not belong to P1 . It can easily be checked that the ideal IP1 displayed in Figure 7 has a Koszul relation pairs in all possible cases, and so does IP by Corollary 2.5. Next, we assume that none of the four corners (1, 1), (1, n), (m, 1) and (m, n) belong to P. In the following arguments we refer to Figure 5. In the first case suppose [i3 , i4 ] ⊂ [i1 , i2 ] and [j3 , j4 ] ⊂ [j1 , j2 ]. Then the collection of cells induced by the columns 2, i3 , i4 , m − 1 and the rows 1, j3 , j4 , n is the polyomino displayed in Figure 2 which has a Koszul relation pair as can be verified by computer. Thus P has a Koszul relation pair. A similar argument applies if [i1 , i2 ] ⊂ [i3 , i4 ] or [j1 , j2 ] ⊂ [j3 , j4 ]. Next assume that [i3 , i4 ] 6⊂ [i1 , i2 ] or [j3 , j4 ] 6⊂ [j1 , j2 ]. By symmetry, we may discuss only [i3 , i4 ] 6⊂ [i1 , i2 ]. Then we may assume that i3 < i1 and i4 < i2 . 9 Figure 7. We choose the columns i1 , i2 , i3 , i4 and the rows 1, 2, n − 1, n. Then the induced polyomino by these rows and columns is P1 if i1 < i4 , P2 if i4 = i1 and P3 if i4 < i1 ; see Figure 8. In all three cases the corresponding induced polyomino ideal has a Koszul relation pair, and hence so does IP . i1 < i4 i4 = i1 i4 < i1 Figure 8. Lemma 2.9. Let P be a convex polyomino, and let [(1, 1), (m, n)] be the smallest interval with the property that V (P) ⊂ [(1, 1), (m, n)]. Suppose P misses three corners, say (1, n), (m, 1), (m, n), and suppose that i2 < m − 1 and j2 < n − 1, or i2 = m − 1 and j2 < j4 , or j2 = n − 1 and i2 < i4 . Then IP has a Koszul relation pair and hence is not linearly related. Proof. We proceed as in the proofs of the previous lemmata. In the case that i2 < m − 1 and j2 < n − 1, we consider the collection of cells P ′ induced by the columns 1, 2, m − 1 and the rows 1, 2, n − 1. This collection of cells P ′ is depicted in Figure 9. It is easily seen that IP ′ is generated by a regular sequence of length 2, which is a Koszul relation pair. In the case that i2 = m − 1 and j2 < j4 we choose the columns 1, 2, m − 1, m and the rows 1, 2, j4 − 1, j4 . The polyomino P ′′ induced by this choice of rows and columns has two opposite missing corners, hence, by Lemma 2.8, it has a Koszul pair. The case j2 = n − 1 and i2 < i4 is symmetric. In both cases the induced polyomino ideal has a Koszul relation pair. Hence in all three cases IP itself has a Koszul relation pair. 10 Figure 9. Proof of Theorem 2.1. Implication (a)⇒(b) is obvious. Implication (b)⇒(c) follows by Corollary 2.7, Lemma 2.8, and Lemma 2.9. It remains to prove (c)⇒(a). Let P be a convex polyomino which satisfies one of the conditions (i)–(iii). We have to show that P is linearly related. By Corollary 1.4, we only need to prove that β14 (IP ) = 0. Viewing K[P] as a semigroup ring K[H], it follows that one has to check that β1h (IP ) = 0 for all h ∈ H with \|h\| = 4. The main idea of this proof is to use Corollary 2.3. Let h = h1 h2 h3 h4 with jq = siq tiq for 1 ≤ q ≤ 4, and i = minq {iq }, k = maxq {iq }, j = minq {jq }, and ℓ = maxq {jq }. Therefore, all the points hq lie in the (possible degenerate) rectangle Q of vertices (i, j), (k, j), (i, ℓ), (k, ℓ). If Q is degenerate, that is, all the vertices of Q are contained in a vertical or horizontal line segment in P, then β1h (IP ) = 0 since in this case the simplicial complex ∆h is just a simplex. Let us now consider Q non-degenerate. If all the vertices of Q belong to P, then the rectangle Q is an induced subpolyomino of P. Therefore, by Corollary 2.3, we have β1h (IP ) = β1h (IQ ) = 0, the latter equality being true since Q is linearly related. Next, let us assume that some of the vertices of Q do not belong to P. As P has one of the forms (i)–(iii), it follows that at most three verices of Q do not belong to P. Consequently, we have to analyze the following cases. Case 1. Exactly one vertex of Q does not belong to P. Without loss of generality, we may assume that (k, ℓ) ∈ / P which implies that k = m and ℓ = n. In this case, any relation in degree h of P is a relation of same degree of one of the polyominoes displayed in Figure 10. One may check with a computer algebra system that all polyominoes displayed in Figure 10 are linearly related, hence they do not have any relation in degree h. Actually, one has to check only the shapes (a), (b), and (d) since the polyomino displayed in (c) is isomorphic to that one from (b). Hence, β1h (IP ) = 0. Case 2. Two vertices of Q do not belong to P. We may assume that the missing vertices from P are (i, ℓ)and (k, ℓ). Hence, we have i = 1, k = m, and ℓ = n. In this case, any relation in degree h of P is a relation of same degree of one of the polyominoes displayed in Figure 11 (a)–(c). Note that the polyominoes (b) and (c) are isomorphic. One easily checks with the computer that all these polyominoes are linearly related, thus β1h (IP ) = 0. Case 3. Finally, we assume that there are three vertices of Q which do not belong to P. We may assume that these vertices are (i, ℓ), (k, ℓ), and (k, j). In this case, any relation in degree h of P is a relation of same degree of the polyomino displayed in Figure 11 (d) which is linearly related as one may easily check with the computer. Therefore, we get again β1h (IP ) = 0. 11 (a) (b) (c) (d) Figure 10. (a) (b) (c) (d) Figure 11. 12 3. Polyomino ideals with linear resolution In this final section, we classify all convex polyominoes which have a linear resolution and the convex stack polyominoes which are extremal Gorenstein. Theorem 3.1. Let P be a convex polyomino. Then the following conditions are equivalent: (a) IP has a linear resolution; (b) there exists a positive integer m such that P is isomorphic to the polyomino with cells [(i, i), (i + 1, i + 1)], i = 1, . . . , m − 1. Proof. (b) ⇒ (a): If the polyomino is of the shape as described in (b), then IP is just the ideal of 2-minors of a 2×m-matrix. It is well-known that the ideal of 2-minors of such a matrix has a linear resolution. Indeed the Eagon-Northcott complex, whose chain maps are described by matrices with linear entries, provides a free resolution of the ideal of maximal minors of any matrix of indeterminates, see for example [6, Page 600]. (a) ⇒ (b): We may assume that [(1, 1), (m, n)] is the smallest interval containing V (P). We may further assume that m ≥ 4 or n ≥ 4. The few remaining cases can easily be checked with the computer. So let us assume that m ≥ 4. Then we have to show that n = 2. Suppose that n ≥ 3. We first assume that all the corners (1, 1), (1, n), (m, 1) and (m, n) belong to V (P). Then the polyomino P ′ induced by the columns 1, 2, m and the rows 1, 2, n is the polyomino which is displayed on the right of Figure 15. The ideal IP ′ is a Gorenstein ideal, and hence it is does not have a linear resolution. Therefore, by Corollary 2.5, the ideal IP does not have a linear resolution as well, a contradiction. Next assume that one of the corners, say (1, 1), is missing. Since IP has a linear a linear resolution, IP is linearly related and hence has a shape as indicated in Figure 5. Let i1 and j1 be the numbers as shown in Figure 5, and let P ′ the polyomino of P induced by the columns 1, 2, 3 and the rows a, j1 , j1 + 1 where a = 1 if i1 = 2 and a = 2 if i1 > 2, j1 > 2. If j1 = 2 and i1 > 2, we let P ′ to be the polyomino induced by the columns 1, i1 , i1 + 1 and the rows 1, 2, 3. In any case, P ′ is isomorphic to that one displayed on the left of Figure 15. Since IP ′ is again a Gorenstein ideal, we conclude, as in the first case, that IP does not a have linear resolution, a contradiction. As mentioned in the introduction, polyomino ideals overlap with join-meet ideals of planar lattices. In the next result we show that the join-meet ideal of any lattice has linear resolution if and only if it is a polyomino as described in Theorem 3.1. With methods different from those which are used in this paper, the classification of join-meet ideals with linear resolution was first given in [7, Corollary 10]. Let L be a finite distributive lattice [11, pp. 118]. A join-irreducible element of L is an element α ∈ L which is not a unique minimal element and which possesses the property that α 6= β ∨ γ for all β, γ ∈ L \ {α}. Let P be the set of join-irreducible elements of L. We regard P as a poset (partially ordered set) which inherits its ordering from that of L. A subset J of P is called an order ideal of P if a ∈ J, b ∈ P together with b ≤ a imply b ∈ J. In particular, the empty set of P is an order ideal of P . Let J (P ) denote the set of order ideals of P , ordered by inclusion. It 13 then follows that J (P ) is a distributive lattice. Moreover, Birkhoff’s fundamental structure theorem of finite distributive lattices [11, Proposition 37.13] guarantees that L coincides with J (P ). Let L = J (P ) be a finite distributive lattice and K[L] = K[ xα : α ∈ L ] the polynomial ring in \|L\| variables over K. The join-meet ideal IL of L is the ideal of K[L] which is generated by those binomials xα xβ − xα∧β xα∨β , where α, β ∈ L are incomparable in L. It is known [12] that IL is a prime ideal and the quotient ring K[L]/IL is normal and Cohen–Macaulay. Moreover, K[L]/IL is Gorenstein if and only if P is pure. (A finite poset is pure if every maximal chain (totally ordered subset) of P has the same cardinality.) Now, let P = {ξ1 , . . . , ξd } be a finite poset, where i < j if ξi < ξj , and L = J (P ). A linear extension of P is a permutation π = i1 · · · id of [n] = {1, . . . , n} such that j < j ′ if ξij < ξij′ . A descent of π = i1 · · · id is an index j with ij > ij+1 . Let D(π) denote the set of descents of π. The h-vector of L is the sequence h(L) = (h0 , h1 , . . . , hd−1 ), where hi is the number of permutations π of [n] with \|D(π)\| = i. Thus, in particular, h0 = 1. It follows from [1] that the Hilbert series of K[L]/IL is of the form h0 + h1 λ + · · · + hd−1 λd−1 . (1 − λ)d+1 We say that a finite distributive lattice L = J (P ) is simple if L has no elements α and β with β < α such that each element γ ∈ L \ {α, β} satisfies either γ < β or γ > α. In other words, L is simple if and only if P possesses no element ξ for which every µ ∈ P satisfies either µ ≤ ξ or µ ≥ ξ. Theorem 3.2. Let L = J (P ) be a simple finite distributive lattice. Then the join-meet ideal IL has a linear resolution if and only if L is of the form shown in Figure 12. • • • • • • • • • • Figure 12. Proof. Since IL is generated in degree 2, it follows that IL has a linear resolution if and only if the regularity of K[L]/IL is equal to 1. We may assume that K is infinite. Since K[L]/IL is Cohen–Macaulay, we may divide by a regular sequence of 14 linear forms to obtain a 0-dimensional K-algebra A with reg A = reg K[L]/IL whose h-vector coincides with that of reg K[L]/IL . Since reg A = max{i : Ai 6= 0} (see for example [6, Exercise 20.18]), it follows that IL has a linear resolution if and only if the h-vector of L is of the form h(L) = (1, q, 0, . . . , 0), where q ≥ 0 is an integer. Clearly, if P is a finite poset of Figure 12, then \|D(π)\| ≤ 1 for each linear extension π of P . Thus IL has a linear resolution. Conversely, suppose that IL has a linear resolution. In other words, one has \|D(π)\| ≤ 1 for each linear extension π of P . Then P has no three-element clutter. (A clutter of P is a subset A of P with the property that no two elements belonging to A are comparable in P .) Since L = J (P ) is simple, it follows that P contains a two-element clutter. Hence Dilworth’s theorem [5] says that P = C ∪ C ′ , where C and C ′ are chains of P with C ∩ C ′ = ∅. Let \|C\| ≥ 2 and \|C ′ \| ≥ 2. Let ξ ∈ C and µ ∈ C ′ be minimal elements of P . Let ξ ′ ∈ C and µ′ ∈ C ′ be maximal elements of P . Since L = J (P ) is simple, it follows that ξ 6= µ and ξ ′ 6= µ′ . Thus there is a linear extension π of P with \|D(π)\| ≥ 2. Thus IL cannot have a linear resolution. Hence either \|C\| = 1 or \|C ′ \| = 1, as desired. A Gorenstein ideal can never have a linear resolution, unless it is a principal ideal. However, if the resolution is as much linear as possible, then it is called extremal Gorenstein. Since polyomino ideals are generated in degree 2 we restrict ourselves in the following definition of extremal Gorenstein ideals to graded ideals generated in degree 2. Let S be a polynomial ring over field, and I ⊂ S a graded ideal which is not principal and is generated in degree 2. Following [18] we say that I is an extremal Gorenstein ideal if S/I is Gorenstein and if the shifts of the graded minimal free resolution are −2 − p − 1, −2 − (p − 1), −2 − (p − 2), . . . , −3, −2, where p is the projective dimension of I. With similar arguments as in the proof of Theorem 3.2, we see that I is an extremal Gorenstein ideal if and only if I is a Gorenstein ideal and reg S/I = 2, and that this is the case if and only if S/I is Cohen–Macaulay and the h-vector of S/I is of the form h(L) = (1, q, 1, 0, . . . , 0), where q > 1 is an integer. In the following theorem we classify all convex stack polyominoes P for which IP is extremal Gorenstein. Convex stack polyominoes have been considered in [17]. In that paper Qureshi characterizes those convex stack polyominoes P for which IP is Gorenstein. Let P be a polyomino. We may assume that [(1, 1), (m, n)] is the smallest interval containing V (P). Then P is called a stack polyomino if it is column convex and for i = 1, . . . . , m − 1 the cells [(i, 1), (i + 1, 2)] belong to P. Figure 13 displays stack polyominoes – the right polyomino is convex, the left is not. The number of cells of the bottom row is called the width of P and the number of cells in a maximal column is called the height of P. 15 Figure 13. Stack polyominoes Let P be a convex stack polyomino. Removing the first k bottom rows of cells of P we obtain again a convex stack polyomino which we denote by Pk . We also set P0 = P. Let h be the height of the polymino, and let 1 < k1 < k2 < · · · < kr < h be the numbers with the property that width(Pki ) < width(Pki−1 ). Furthermore, we set k0 = 1. For example, for the convex stack polyomino in Figure 13 we have k1 = 1, k2 = 2 and k3 = 3. With the terminology and notation introduced, the characterization of Gorenstein convex stack polyominoes is given in the following theorem. Theorem 3.3 (Qureshi). Let P be a convex stack polyomino of height h. Then the following conditions are equivalent: (a) IP is a Gorenstein ideal. (b) width(Pki ) = height(Pki ) for i = 0, . . . , r. According to this theorem, the convex stack polyomino displayed in Figure 13 is not Gorenstein, because width(Pk0 ) = 5 and height(Pk0 ) = 4. An example of a Gorenstein stack polyomino is shown in Figure 14. Figure 14. A Gorenstein stack polyomino Combining Theorem 3.3 with the results of Section 2, we obtain Theorem 3.4. Let IP be convex stack polyomino. Then IP is extremal Gorenstein if and only if P is isomorphic to one of the polyominoes in Figure 15. 16 Figure 15. Extremal convex stack polyominoes Proof. It can be easily checked that IP is extremal Gorenstein, if P is isomorphic to one of the two polyominoes shown in Figure 15. Conversely, assume that IP is extremal Gorenstein. Without loss of generality we may assume that [(1, 1), (m, n)] is the smallest interval containing V (P). Then Theorem 3.3 implies that m = n. Suppose first that V (P) = [(1, 1), (n, n)]. Then, by [7, Theorem 4] of Ene, Rauf and Qureshi, it follows that the regularity of IP is equal to n. Since IP is extremal Gorenstein, its regularity is equal to 3. Thus n = 3. Next, assume that V (P) is properly contained in [(1, 1), (n, n)]. Since IP is linearly related, Corollary 2.7 together with Theorem 3.3 imply that the top row of P consists of only one cell and that [(2, 1), (n − 1, n − 1)] ⊂ V (P). Let P ′ be the polyomino induced by the rows 2, 3, . . . , n − 1 and the columns 1, 2, . . . , n − 1. Then P ′ is the polyomino with V (P ′ ) = [(1, 1), (n − 2, n − 1)]. By applying again [7, Theorem 4] it follows that reg IP′ = n − 2. Corollary 2.5 then implies that reg IP ≥ reg IP ′ = n − 2, and since reg IP = 3 we deduce that n ≤ 5. If n = 5, then IP′ is the ideal of 2-minors of a 3 × 4-matrix which has Betti numbers β35 6= 0 and β36 6= 0. Since P ′ is an induced polyomino of P and since IP is extremal Gorenstein, Corollary 2.5 yields a contradiction. Up to isomorphism there exist for n = 4 precisely the Gorenstein polyominoes displayed in Figure 16. They are all not extremal Gorenstein as can be easily checked with CoCoA or Singular. For n = 3 any Gorenstein polyomino is isomorphic to one of the two polyominoes shown in Figure 15. This yields the desired conclusion. Figure 16. Gorenstein polyominoes of width 3 17 The following theorem shows that besides of the two polyominoes listed in Theorem 3.4 whose polyomino ideal is extremal Gorenstein, there exist precisely two more join-meet ideals having this property. Theorem 3.5. Let L = J (P ) be a simple finite distributive lattice. Then the joinmeet ideal IL is an extremal Gorenstein ideal if and only if L is one of the following displayed in Figure 17. • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Figure 17. Proof. Suppose that L = J (P ) is simple and that K[L]/IL is Gorenstein. it then follows that P is pure and there is no element ξ ∈ P for which every µ ∈ P satisfies either µ ≤ ξ or µ ≥ ξ. Since h(L) = (1, q, 1, 0, . . . , 0), no 4-element clutter is contained in P . Suppose that a three-element clutter A is contained in P . If none of the elements belonging to A is a minimal element of P , then, since L = J (P ) is simple, there exist at least two minimal elements. Hence there exists a linear extension π of P with \|D(π)\| ≥ 3, a contradiction. Thus at least one of the elements belonging to A is a minimal element of P . Similarly, at least one of the elements belonging to A is a maximal element. Let an element x ∈ A which is both minimal and maximal. Then, since P is pure, one has P = A. Let A = {ξ1 , ξ2 , ξ3} with A 6= P , where ξ1 is a minimal element and ξ2 is a maximal element. Let µ1 be a maximal element with ξ1 < µ1 and µ2 a minimal element with µ2 < ξ2 . Then neither µ1 nor µ2 belongs to A. If ξ3 is either minimal or maximal, then there exists a linear extension π of P with \|D(π)\| ≥ 3, a contradiction. Hence ξ3 can be neither minimal nor maximal. Then since P is pure, there exist ν1 with ξ1 < ν1 < µ1 and ν2 with µ2 < ν2 < ξ2 such that {ν1 , ν2 , ξ3} is a three-element clutter. Hence there exists a linear extension π of P with \|D(π)\| ≥ 4, a contradiction. Consequently, if P contains a three-element clutter A, then P must coincide with A. Moreover, if P is a three-element clutter, then h(L) = (1, 4, 1) and IL is an extremal Gorenstein ideal. Now, suppose that P contains no clutter A with \|A\| ≥ 3. Let a chain C with \|C\| ≥ 3 be contained in P . Let ξ, ξ ′ be the minimal elements of P and µ, µ′ the maximal elements of P with ξ < µ and ξ ′ < µ′ . Since L = J (P ) is simple and since P is pure, it follows that there exist maximal chains ξ < ν1 < · · · < νr < µ and ξ ′ < ν1′ < · · · < νr′ < µ′ such that νi 6= νi′ for 1 ≤ i ≤ r. Then one has a linear extension π of P with D(π) = 2+r ≥ 3, a contradiction. Hence the cardinality of all maximal chains of P is at most 2. However, if the cardinality of all maximal chains of P is equal to 1, then h(L) = (1, 1). Thus IL cannot be an extremal Gorenstein ideal. If the cardinality of all maximal chains of P is equal to 2, then P is the posets 18 displayed in Figure 18. For each of them the join-meet ideal IL is an extremal Gorenstein ideal. • • • • • • • • • • • • h(L) = (1, 2, 1) h(L) = (1, 3, 1) h(L) = (1, 4, 1) Figure 18. References [1] A. Björner, A. M. Garsia and R. P. Stanley, An introduction to Cohen–Macaulay partially ordered sets, In: “Ordered Sets” (I. Rival, Ed.), Springer Netherlands, 1982, pp. 583–615. [2] W. Bruns, J. Herzog, Semigroup rings and simplicial complexes, J. Pure Appl. Algebra 122 (1997), 185–208. [3] CoCoATeam, CoCoA: a system for doing Computations in Commutative Algebra. Available at http://cocoa.dima.unige.it [4] W. Decker, G.-M. 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Hibi, Distributive lattices, affine semigroup rings and algebras with straightening laws, In: “Commutative Algebra and Combinatorics” (M. Nagata and H. Matsumura, Eds.), Adv. Stud. Pure Math. 11, North–Holland, Amsterdam, 1987, pp. 93–109. [13] K. Kurano, The first syzygies of determinantal ideals, J. Algebra 124 (1989), 414–436. [14] A. Lascoux, Syzygies des variétés determinantales, Adv. in Math. 30 (1978), 202–237. [15] H. Ohsugi, J. Herzog, T. Hibi, Combinatorial pure subrings, Osaka J. Math. bf 37 (2000), 745–757. [16] H. Ohsugi, T. Hibi, Koszul bipartite graphs, Adv. in Appl. Math. 22 (1999), 25-28. [17] A. Qureshi, Ideals generated by 2-minors, collections of cells and stack polyominoes, J. Algebra 357 (2012), 279–303. [18] P. Schenzel, Uber die freien Auflösungen extremaler Cohen-Macaulay Ringe, J. Algebra 64 (1980), 93–101. [19] D. W. Sharpe, On certain polynomial ideals defined by matrices, Quart. J. Math. Oxford (2) 15 (1964), 155–175. [20] D. W. Sharpe, The syzygies and semi-regularity of certain ideals defined by matrices, Proc. London Math. Soc. 15 (1965), 645–679. 19 Viviana Ene, Faculty of Mathematics and Computer Science, Ovidius University, Bd. Mamaia 124, 900527 Constanta, Romania, and Simion Stoilow Institute of Mathematics of the Romanian Academy, Research group of the project ID-PCE-2011-1023, P.O.Box 1-764, Bucharest 014700, Romania E-mail address: [email protected] Jürgen Herzog, Fachbereich Mathematik, Universität Duisburg-Essen, Campus Essen, 45117 Essen, Germany E-mail address: [email protected] Takayuki Hibi, Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Toyonaka, Osaka 5600043, Japan E-mail address: [email protected] 20	0
A Class of MSR Codes for Clustered Distributed Storage Jy-yong Sohn, Beongjun Choi and Jaekyun Moon arXiv:1801.02014v1 [cs.IT] 6 Jan 2018 KAIST School of Electrical Engineering Email: {jysohn1108, bbzang10}@kaist.ac.kr, [email protected] Abstract—Clustered distributed storage models real data centers where intra- and cross-cluster repair bandwidths are different. In this paper, exact-repair minimum-storage-regenerating (MSR) codes achieving capacity of clustered distributed storage are designed. Focus is given on two cases: = 0 and = 1/(n−k), where is the ratio of the available cross- and intra-cluster repair bandwidths, n is the total number of distributed nodes and k is the number of contact nodes in data retrieval. The former represents the scenario where cross-cluster communication is not allowed, while the latter corresponds to the case of minimum cross-cluster bandwidth that is possible under the minimum storage overhead constraint. For the = 0 case, two types of locally repairable codes are proven to achieve the MSR point. As for = 1/(n − k), an explicit MSR coding scheme is suggested for the two-cluster situation under the specific of condition of n = 2k. I. I NTRODUCTION Distributed Storage Systems (DSSs) have been deployed by various enterprises to reliably store massive amounts of data under the frequent storage node failure events. A failed node is regenerated (repaired) by collecting information from other survived nodes with the regeneration process guided by a predefined network coding scheme. Under this setting, Dimakis et al. [1] obtained the expression for the maximum reliably storable file size, denoted as capacity C(α, γ), as a function of given system parameters: the node capacity α and the bandwidth γ required for repairing a failed node. The capacity analysis in [1] underscores the following key messages. First, there exists a network coding scheme which utilizes the (α, γ) resources and enables a reliable storage of a file of size C(α, γ). Second, it is not feasible to find a network coding scheme which can reliably store a file larger than C(α, γ), given the available resources of (α, γ). In subsequent research efforts, the authors of [2]–[4] proposed explicit network coding schemes which achieve the capacity of DSSs. These coding schemes are optimal in the sense of efficiently utilizing (α, γ) resources for maintaining the reliable storage systems. Focus on the clustered nature of distributed storage has been a recent research direction taken by several researchers [5]–[8]. According to these recent papers, storage nodes dispersed into multiple racks in real data centers are seen as forming clusters. In particular, the authors of the present paper proposed a system model for clustered DSSs in [5] that reflects the difference between intra- and cross-cluster bandwidths. In the system model of [5], the file to be stored is coded and distributed into n storage nodes, which are evenly dispersed into L clusters. Each node has storage capacity of α, and the data collector contacts arbitrary k out of n existing nodes to retrieve the file. Since nodes are dispersed into multiple clusters, the regeneration process involves utilization of both intra- and cross-cluster repair bandwidths, denoted by βI and βc , respectively. In this proposed system model, the authors of [5] obtained the closed-form expression for the maximum reliably storable file size, or capacity C(α, βI , βc ), of the clustered DSS. Furthermore, it has been shown that network coding exists that can achieve the capacity of clustered DSSs. However, explicit constructions of capacity-achieving network coding schemes for clustered DSSs have yet to be found. This paper proposes a network coding scheme which achieves capacity of the clustered DSS, with a minimum required node storage overhead. In other words, the suggested code is shown to be a minimum-storage-regenerating (MSR) code of the clustered DSS. This paper focuses on two important cases of = 0 and = 1/(n − k), where := βc /βI represents the ratio of cross- to intra-cluster repair bandwidths. The former represents the system where crosscluster communication is not possible. The latter corresponds to the minimum value that can achieve the minimum storage overhead of α = M/k, where M is the file size. When = 0, it is shown that appropriate application of locally repairable codes suggested in [9], [10] achieves the MSR point for general n, k, L settings with the application rule depending on the parameter setting. For the = 1/(n−k) case, an explicit coding scheme is suggested which is proven to be an MSR code under the conditions of L = 2 and n = 2k. There have been some previous works [7], [8], [11], [12] on code construction for DSS with clustered storage nodes, but to a limited extent. The works of [8], [11] suggested a coding scheme which can reduce the cross-cluster repair bandwidth, but these schemes are not proven to be an MSR code that achieves capacity of clustered DSSs with minimum storage overhead. The authors of [12] provided an explicit coding scheme which reduces the repair bandwidth of a clustered DSS under the condition that each failed node can be exactly regenerated by contacting any one of other clusters. However, the approach of [12] is different from that of the present paper in the sense that it does not consider the scenario with unequal intra- and cross-cluster repair bandwidths. Moreover, the coding scheme proposed in [12] is shown to be a minimum-bandwidth- regenerating (MBR) code for some limited parameter setting, while the present paper deals with an MSR code. An MSR code for clustered DSSs has been suggested in [7], but this paper has the data retrieval condition different from the present paper. The authors of [7] considered the scenario where data can be collected by contacting arbitrary k out of n clusters, while data can be retrieved by contacting arbitrary k out of n nodes in the present paper. Thus, the two models have the identical condition only when each cluster has one node. The difference in data retrieval conditions results in different capacity values and different MSR points. In short, the code in [7] and the code in this paper achieves different MSR points. (1) A data collector (DC) retrieves the original file M by contacting arbitrary k out of n nodes - this property is called the maximum-distance-separable (MDS) property. The clustered distributed storage system with parameters n, k, L is called an [n, k, L]-clustered DSS. In an [n, k, L]-clustered DSS with given parameters of α, βI , βc , capacity C(α, γ) is defined in [5] as the maximum data that can be reliably stored. The closed-form expression for C(α, γ) is obtained in Theorem 1 of [5]. Aiming at reliably storing file M, the set of (α, γ) pair values is said to be feasible if C(α, γ) ≥ M holds. According to Corollaries 1 and 2 of [6], the set of feasible (α, γ) points shows the optimal trade-off relationship between α and γ, as illustrated in Fig. 1. In the optimal trade-off curve, the point with minimum node capacity α is called the minimum-storageregenerating (MSR) point. Explicit regenerating codes that achieve the MSR point are called the MSR codes. According to Theorem 3 of [6], node capacity of the MSR point satisfies αmsr = M/k αmsr > M/k 1 , n−k 1 if 0 ≤ < . n−k if ≥ MBR point 𝛾𝛾𝑚𝑚𝑚𝑚𝑚𝑚 (2) 𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼𝑚𝑚𝑚𝑚𝑚𝑚 𝛼𝛼 Fig. 1: The optimal trade-off relationship between α and γ in the clustered distributed storage modeled in [6] ૚ ૛ ૜ ૝ ૛ A given file of M symbols is encoded and distributed into n nodes, each of which has node capacity α. The storage nodes are evenly distributed into L ≥ 2 clusters, so that each cluster contains nI := n/L nodes. A failed node is regenerated by obtaining information from other survived nodes: nI −1 nodes in the same cluster help by sending βI each, while n−nI nodes in other clusters help by sending βc each. Thus, repairing each node requires the overall repair bandwidth of 𝛼𝛼: Node Capacity 𝛾𝛾: Repair Bandwidth MSR point 𝛾𝛾𝑚𝑚𝑚𝑚𝑚𝑚 ૚ II. BACKGROUNDS AND N OTATIONS γ = (nI − 1)βI + (n − nI )βc . 𝛾𝛾 ૜ ࢐ 1st cluster (݈ = 1) 2nd cluster (݈ = 2) 3rd cluster (݈ = 3) ࢒ ܰ(ଶ,ଷ) Fig. 2: Two-dimensional representation of clustered distributed storage (n = 12, L = 3, nI = n/L = 4) divides b. Similarly, write a - b if a does not divide b. For given k and nI , we define k c, nI m := mod(k, nI ) = k − qnI . q := b (4) (5) For vectors we use bold-faced lower case letters. For a given vector a, the transpose of a is denoted as aT . For natural numbers m and n ≥ m, the set {ym , ym+1 , · · · , yn } is represented as {yi }ni=m . For a matrix G, the entry of G at the ith row and j th column is denoted as Gi,j . We also express the nodes in a clustered DSS using a two-dimensional representation: in the structure illustrated in Fig. 2, N (l, j) represents the node at the lth row and the j th column. Finally, we recall definitions on the locally repairable codes (LRCs) in [9], [10]. As defined in [10], an (n, k, r)−LRC represents a code of length n, which is encoded from k information symbols. Every coded symbol of the (n, k, r)−LRC can be regenerated by accessing at most r other symbols. As defined in [9], an (n, r, d, M, α)−LRC takes a file of size M and encodes it into n coded symbols, where each symbol is composed of α bits. Moreover, any coded symbol can be regenerated by contacting at most r other symbols, and the code has the minimum distance of d. (3) Note that α = M/k is the minimum storage overhead to satisfy the MDS property, as stated in [1]. Thus, = 1/(n − k) is the scenario with minimum cross-cluster communication when the minimum storage overhead constraint α = M/k is imposed. Here we introduce some useful notations used in the paper. For a positive integer n, [n] represents the set {1, 2, · · · , n}. For natural numbers a and b, we use the notation a \| b if a III. MSR C ODE D ESIGN FOR = 0 In this section, MSR codes for = 0 (i.e., βc = 0) is designed. Under this setting, no cross-cluster communication is allowed in the node repair process. First, the system parameters for the MSR point are examined. Second, two types of locally repairable codes (LRCs) suggested in [9], [10] are proven to achieve the MSR point, under the settings of nI \| k and nI - k, respectively. A. Parameter Setting for the MSR Point We consider the MSR point (α, γ) = (αmsr , γmsr ) which can reliably store file M. The following property specifies the system parameters for the = 0 case. Proposition 1. Consider an [n,k,L] clustered DSS to reliably store file M. The MSR point for = 0 is M M , (nI − 1) , (6) (αmsr , γmsr ) = k−q k−q where q is defined in (4). This point satisfies α = βI . (1) 𝑦𝑦1 (1) 𝑦𝑦2 (1) 𝑦𝑦3 (1) 𝑦𝑦4 (1) 𝑦𝑦5 (1) 𝑦𝑦6 (1) 𝑥𝑥1 (1) 𝑥𝑥2 (6,3) MDS (1) 𝑥𝑥3 (2) 𝑥𝑥2 (1) 𝑠𝑠3 = 𝑦𝑦3 (1) 𝑠𝑠4 = 𝑦𝑦4 (2) (6,3) MDS (2) 𝑥𝑥3 Proof. See Appendix D-A. (1) 𝑠𝑠2 = 𝑦𝑦2 𝑦𝑦1 (2) 𝑦𝑦2 (2) 𝑦𝑦3 (2) 𝑦𝑦4 (2) 𝑦𝑦5 (2) 𝑦𝑦6 (2) 𝑥𝑥1 (1) 𝑠𝑠1 = 𝑦𝑦1 (1) 𝑠𝑠5 = 𝑦𝑦5 (1) 𝑠𝑠6 = 𝑦𝑦6 (2) + 𝑦𝑦1 (2) + 𝑦𝑦2 (2) + 𝑦𝑦3 (2) + 𝑦𝑦4 (2) + 𝑦𝑦5 (2) + 𝑦𝑦6 (a) MDS Precoding B. Code Construction for nI \| k We now examine how to construct an MSR code for the nI \| k case. The following theorem shows that a locally repairable code constructed in [9] with locality r = nI − 1 is a valid MSR code for nI \| k. Theorem 1 (Exact-repair MSR Code Construction for = 0, nI \| k) Let C be the (n, r, d, M, α)−LRC explicitly constructed in [9] for locality r = nI − 1. Consider allocating coded symbols of C in a [n, k, L]−clustered DSS, where r + 1 = nI nodes within the same repair group of C are located in the same cluster. Then, the code C is an MSR code for the [n, k, L]− clustered DSS under the conditions of = 0 and nI \| k. Proof. See Appendix A. Fig. 3 illustrates an example of the MSR code for the = 0 and nI \| k case, which is constructed using the LRC in [9]. In the [n, k, L] = [6, 3, 2] clustered DSS scenario, the parameters are set to 𝑦2 (1) 𝑦3 (2) 𝑦2 (2) 𝑦3 𝑦1 (1) Cluster 1 (1) 𝑠3 = 𝑦3 (2) + 𝑦3 (1) 𝑠1 = 𝑦1 (2) 𝑠6 = + (2) + 𝑦1 (1) 𝑠2 = 𝑦2 𝑦6 (2) (2) 𝑦6 (2) 𝑦6 𝑠4 = (1) 𝑦4 + (2) + 𝑦2 (1) 𝑦5 𝑦5 (1) 𝑦6 (2) (1) (1) 𝑦4 Cluster 2 (1) 𝑦1 𝑦4 (2) 𝑦4 𝑠5 = (1) 𝑦5 (2) + 𝑦5 (b) Allocation of coded symbols into n nodes Fig. 3: MSR code for = 0 with nI \| k (n = 6, k = 3, L = 2). The construction rule follows the instruction in [9], while the concept of the repair group in [9] can be interpreted as the cluster in the present paper. authors of [9], while the present paper proves that this code also achieves the MSR point of the [n, k, L] clustered DSS, in the case of = 0 and nI \| k. α = nI = n/L = 3, M = (k − q)α = (k − bk/nI c)α = 6. C. Code Construction for nI - k Thus, each storage node contains α = 3 symbols, while the [n, k, L] clustered DSS aims to reliably store a file of size M = 6. This code has two properties, 1) exact regeneration and 2) data reconstruction: 1) Any failed node can be exactly regenerated by contacting nI − 1 = 2 nodes in the same cluster, 2) Contacting any k = 3 nodes can recover the original file (j) {xi : i ∈ [3], j ∈ [2]} of size M = 6. (1) (2) The first property is obtained from the fact that yi , yi (1) (2) and si = yi + yi form a (3, 2) MDS code for i ∈ [6]. The second property is obtained as follows. For contacting arbitrary (1) (1) (1) k = 3 nodes, three distinct coded symbols {yi1 , yi2 , yi3 } having superscript one and three distinct coded symbols (2) (2) (2) {yj1 , yj2 , yj3 } having superscript two can be obtained for some i1 , i2 , i3 ∈ [6] and j1 , j2 , j3 ∈ [6]. From Fig. 3a, the (1) (1) (1) (1) (1) (1) information {yi1 , yi2 , yi3 } suffice to recover x1 , x2 , x3 . (2) (2) (2) Similarly, the information {yj1 , yj2 , yj3 } suffice to recover (2) (2) (2) x1 , x2 , x3 . This completes the proof for the second property. Note that this coding scheme is already suggested by the Here we construct an MSR code when the given system parameters satisfy nI - k. The theorem below shows that the optimal (n, k − q, nI − 1)−LRC designed in [10] is a valid MSR code when nI - k holds. Theorem 2 (Exact-repair MSR Code Construction for = 0, nI - k) Let C be the (n0 , k0 , r0 )−LRC constructed in [10] for n0 = n, k0 = k − q and r0 = nI − 1. Consider allocating the coded symbols of C in a [n, k, L]−clustered DSS, where r + 1 = nI nodes within the same repair group of C are located in the same cluster. Then, C is an MSR code for the [n, k, L]−clustered DSS under the conditions of = 0 and nI - k. Proof. See Appendix B. Fig. 4 illustrates an example of code construction for the nI - k case. Without loss of generality, we consider α = 1 case; parallel application of this code multiple α times achieves the MSR point for general α ∈ N, where N is the set 𝐴 0 0 𝐴 1 1 1 = 𝛼1 𝛼2 𝛼3 𝛼12 𝛼22 𝛼32 𝐺=𝑉 𝐺=𝑉 𝑥1 𝑥2 1 = 𝛼1 𝛼12 𝑥3 𝐴 0 0 𝐴 1 1 𝛼2 𝛼3 𝛼22 𝛼32 1 𝛼4 𝛼42 1 𝑤 0 1 0 0 0 0 1 𝑤 0 1 0 0 0 0 1 𝛼4 𝛼42 0 𝑤 0 0 0 0 1 0 0 0 𝑤 1 0 0 0 𝑤 (4,3) RS code 𝑥1 𝑧1 𝑧2 (3,2) - MDS 𝑦1 𝑦2 𝑧1 𝑧2 (3,2) - MDS 𝑦 𝑦1 3𝑦2 𝑥2 𝑥3 Cluster 1 𝑧3RS code𝑧4 (4,3) 𝑧3 Cluster 1 𝑧4 Cluster 2 (3,2) - MDS Cluster 2 (3,2) - MDS 𝑦 𝑦 𝑦3 4 𝑦4 5𝑦5 𝑦1 𝑦2 𝑦1 𝑦4 𝑦5 𝑦4 𝑦2 𝑦3 𝑦6 𝑦5 𝑦𝑦6 (a) Encoding structure 0 𝑤 0 0 0 0 1 0 0 0 𝑤 1 0 0 0 𝑤 Proposition 2. The MSR point for = 1/(n − k) is M M n − nI (αmsr , γmsr ) = , nI − 1 + . 𝑦3 k k n−k (8) This point satisfies α = βI = n − k and M = k(n − k). 𝑦6 Proof. See Appendix D-B. 6 (b) Allocation of coded symbols into n nodes Fig. 4: MSR code for = 0 with nI - k case (n = 6, k = 4, L = 2). The encoding structure follows from the instruction in [10], which constructed [n0 , k0 , r0 ] − LRC. This paper utilizes [n, k − q, nI − 1] − LRC to construct MSR code for [n, k, L] clustered DSS, in the case of = 0 with nI - k. of positivie integers. In the [n = 6, k = 4, L = 2] clustered DSS with = 0, the code and system parameters are: [n0 , k0 , r0 ] = [n, k − q, nI − 1] = [6, 3, 2], α = 1, M = (k − q)α = (k − bk/nI c) = 3 from Proposition 1. The code in Fig. 4 satisfies the exact regeneration and data reconstruction properties: 1) Any failed node can be exactly regenerated by contacting nI − 1 = 2 nodes in the same cluster, 2) Contacting any k = 4 nodes can recover the original file {xi : i ∈ [3]} of size M = 3. Note that {yi }3i=1 in Fig. 4 is a set of coded symbols generated by a (3, 2)−MDS code, and this statement also holds for {yi }6i=4 . This proves the first property. The second property is directly from the result of [10], which states that the minimum distance of the [n0 , k0 , r0 ] − LRC is k0 d = n0 − k0 − + 2 = 6 − 3 − d3/2e + 2 = 3. (7) r0 Note that the [n0 , k0 , r0 ] − LRC is already suggested by the authors of [10], while the present paper proves that applying this code with n0 = n, k0 = k − q, r0 = nI − 1 achieves the MSR point of the [n, k, L]−clustered DSS, in the case of = 0 and nI - k. IV. MSR C ODE D ESIGN FOR = 1 n−k 1 We propose an MSR code for = n−k in clustered DSSs. 1 From (2) and (3), recall that n−k is the minimum value which allows the minimum storage of αmsr = M/k. First, we obtain the system parameters for the MSR point. Second, we design a coding scheme which is shown to be an MSR code under the conditions of n = 2k and L = 2. A. Parameter Setting for the MSR Point The following property specifies the system parameters for the = 1/(n − k) case. Without a loss of generality, we set the cross-cluster repair bandwidth as βc = 1. B. Code Construction for [n, k, L] = [2k, k, 2] Here, we construct an MSR code under the constraints of n = 2k and L = 2. Since we consider the n = 2k case, the system parameters in Proposition 2 are set to α = βI = n − k = k, (9) 2 M = kα = k . Construction 1. Suppose that we are given M = k 2 source symbols {mi,j : i, j ∈ [k]}. Moreover, let the encoding matrix  (1)  (2) (k) G1 G1 · · · G1  (1) (2) (k)  G2 G2 · · · G2   (10) G= . .. ..  ..  .  .. . .  (1) (2) (k) Gk Gk · · · Gk (j) be a k 2 × k 2 matrix, where each encoding sub-matrix Gi is a k × k matrix. For j ∈ [k], node N (1, j) stores mj and node N (2, j) stores pj , where mi = [mi,1 , · · · , mi,k ]T , pi = [pi,1 , · · · , pi,k ]T = (11) k X (j) mTj Gi . (12) j=1 Remark 1. The code generated in Construction 1 satisfies the followings: (a) Every node in cluster 1 contains k message symbols. (b) Every node in cluster 2 contains k parity symbols. Note that this remark is consistent with (9), which states α = k. Under this construction, we have the following theorem, which specifies the MSR construction rule for the [n = 2k, k, L = 2]−DSS with = 1/(n − k). Theorem 3 (Exact-repair MSR Code Construction for 1 = n−k ) If all square sub-matrices of G are invertible, the code designed by Construction 1 is an MSR code for [n, k, L] = [2k, k, 2]−DSS with = 1/(n − k). Proof. See Appendix C. The following result suggests an explicit construction of an MSR code using the finite field. Corollary 1. Applying Construction 1 with encoding matrix G set to the k 2 × k 2 Cauchy matrix [13] achieves the MSR point for an [n = 2k, k, L = 2]−DSS. A finite field of size 2k 2 suffices to design G. Proof. The proof is directly from Theorem 3 and the fact that all sub-matrices of a Cauchy matrix has full rank, as stated in [14]. Moreover, the Cauchy matrix of size n × n can be Cluster 1 Cluster 2 N(1,1) N(1,2) 𝒎𝟏,𝟏 𝒎𝟐,𝟏 𝒎𝟏,𝟐 𝒎𝟐,𝟐 𝒑𝟏,𝟏 𝒑𝟐,𝟏 𝒑𝟏,𝟐 𝒑𝟐,𝟐 N(2,1) N(2,2) 𝑝,,, 𝑝,,/ 𝑝/,, = 𝐺 𝑝/,/ 7 = 2 3 4 𝑚,,, 𝑚,,/ 𝑚/,, 𝑚/,/ 2 7 4 3 Cluster 1 3 4 7 2 4 3 2 7 𝑚,,, 𝑚,,/ 𝑚/,, 𝑚/,/ Cluster 2 ü 𝒎𝟏,𝟏 𝒎𝟐,𝟏 𝒎𝟏,𝟐 𝒎𝟐,𝟐 𝒑𝟏,𝟏 𝒑𝟐,𝟏 𝒑𝟏,𝟐 𝒑𝟐,𝟐 ü ü 7 𝐺= 2 3 4 ü 2 7 4 3 3 4 7 2 4 3 2 7 Parities 𝒑𝟏,𝟐, 𝒑𝟐,𝟏 are inaccessible Messages 𝒎𝟐,𝟏, 𝒎𝟐,𝟐 are accessible Fig. 6: Repairing a failed node in proposed MSR code example for n = 4, k = 2, L = 2 Fig. 5: MSR example for n = 4, k = 2, L = 2 constructed using a finite field of size 2n, according to [15]. An example of MSR code designed by Construction 1 is illustrated in Fig. 5, in the case of n = 4, k = 2, L = 2. This coding scheme utilizes a Cauchy matrix   7 2 3 4 2 7 4 3  (13) G= 3 4 7 2 4 3 2 7 3 using the finite field GF (2 ) with the primitive polynomial x3 + x + 1. The element aα2 + bα + c in GF (23 ) is denoted by the decimal number of (abc)2 , where α is the primitive element. For example, α + 1 is denoted by 3 = (011)2 in the generator matrix G. When [n, k, L, ] = [4, 2, 2, 1/2], the system parameters are α = 2, M = 4, βI = 2, βc = 1 from Proposition 2, which holds for the example in Fig. 5. Here we show that the proposed coding scheme satisfies two properties: 1) exact regeneration of any failed node and 2) recovery of M = 4 message symbols {m1,1 , m1,2 , m2,1 , m2,2 } by contacting any k = 2 nodes. 1) Exact regeneration: Fig. 6 illustrates the regeneration process. Suppose that node N (1, 1) containing the message m1 = [m1,1 , m1,2 ] fails. Then, node N (1, 2) transmits βI = 2 symbols, m2,1 and m2,2 . Nodes N (2, 1) and N (2, 2) transmit βc = 1 symbol each, for example p1,1 and p2,2 , respectively. Then, from the received symbols of m2,1 , m2,2 , p1,1 , p2,2 and matrix G, we obtain y1 p1,1 − G1,3 m2,1 − G1,4 m2,2 7 2 m1,1 := = . y2 p2,2 − G4,3 m2,1 − G4,4 m2,2 4 3 m1,2 Thus, the contents of the failed node can be regenerated by −1 m1,1 7 2 y1 3 2 y1 = = m1,2 4 3 y2 4 7 y2 where the matrix inversion is over GF (23 ). Note that the exact regeneration property holds irrespective of the contents transmitted by N (2, 1) and N (2, 2), since the encoding matrix is a Cauchy matrix, all submatrices of which are invertible. 2) Data recovery: First, if DC contacts two systematic nodes, the proof is trivial. Second, contacting two parity nodes can recover the original message since G is invertible. Third, suppose that DC contacts one systematic node and one parity node, for example, N (1, 1) and N (1, 4). Then, DC can retrieve message symbols m1,1 , m1,2 and parity symbols p2,1 , p2,2 . Using the retrieved symbols and the information on the encoding matrix G, DC additionally obtains z1 p2,1 − G3,1 m1,1 − G3,2 m1,2 7 2 m2,1 := = . z2 p2,2 − G4,1 m1,1 − G4,2 m1,2 2 7 m2,2 Thus, DC obtains m2,1 7 = m2,2 2 2 7 −1 z1 1 = z2 3 3 1 z1 , z2 which completes the data recovery property of the suggested code. V. C ONCLUSION A class of MSR codes for clustered distributed storage modeled in [5] has been constructed. The proposed coding schemes can be applied in practical data centers with multiple racks, where the available cross-rack bandwidth is limited compared to the intra-rack bandwidth. Two important cases of = 0 and = 1/(n − k) are considered, where = βc /βI represents the ratio of available cross- to intra-cluster repair bandwidth. Under the constraint of zero cross-cluster repair bandwidth ( = 0), appropriate application of two locally repairable codes suggested in [9], [10] is shown to achieve the MSR point of clustered distributed storage. Moreover, an explicit MSR coding scheme is suggested for = 1/(n − k), when the system parameters satisfy n = 2k and L = 2. The proposed coding scheme can be implemented in a finite field, by using a Cauchy generator matrix. A PPENDIX A P ROOF OF T HEOREM 1 We focus on code C, the explicit (n, r, d, M, α)-LRC constructed in Section V of [9]. This code has the parameters r+1M ), (A.1) r k where r is the repair locality and d is the minimum distance, and other parameters (n, M, α) have physical meanings identical to those in the present paper. By setting r = nI − 1, the code has node capacity of (n, r, d = n − k + 1, M, α = α= nI M M M = = nI − 1 k k(1 − 1/nI ) k−q (A.2) where the last equality holds from the nI \| k condition and the definition of q in (4). We first prove that any node failure can be exactly regenerated by using the system parameters in (6). According to the description in Section V-B of [9], any node is contained in a unique corresponding repair group of size r + 1 = nI , so that a failed node can be exactly repaired by contacting r = nI − 1 other nodes in the same repair group. This implies that a failed node does not need to contact other repair groups in the exact regeneration process. By setting each repair group as a cluster (note that each cluster contains nI = n/L nodes), we can achieve βc = 0. (A.3) Moreover, Section V-B of [9] illustrates that the exact regeneration of a failed node is possible by contacting the entire symbols contained in r = nI − 1 nodes in the same repair group, and applying the XOR operation. This implies βI = α, which result in M , (A.4) γ = (nI − 1)βI = (nI − 1) k−q combined with (1) and (A.2). From (A.2) and (A.4), we can conclude that code C satisfies the exact regeneration of any failed node using the parameters in (6). Now we prove that contacting any k nodes suffices to recover original data in the clustered DSS with code C applied. Note that the minimum distance is d = n − k + 1 from (A.1). Thus, the information from k nodes suffices to pick the correct codeword. This completes the proof of Theorem 1. A PPENDIX B P ROOF OF T HEOREM 2 We first prove that the code C has minimum distance of d = n−k+1, which implies that the original file of size M = k−q can be recovered by contacting arbitrary k nodes. Second, we prove that any failed node can be exactly regenerated under the setting of (6). Recall that the [n0 , k0 , r0 ]−LRC constructed in [10] has the following property, as stated in Theorem 1 of [10]: Lemma 1 (Theorem 1 of [10]). The code constructed in [10] has locality r0 and optimal minimum distance d = n0 − k0 − d kr0 e + 2, when (r0 + 1) \| n0 . Note that we consider code C of optimal [n0 , k0 , r0 ] = [n, k − q, nI − 1]−LRC. Since r0 + 1 = nI divides n0 = n, Lemma 1 can be applied. The result of Lemma 1 implies that the minimum distance of C is k−q d = n − (k − q) − + 2. (B.1) nI − 1 Since we consider the nI - k case, we have k = qnI + m, (0 < m ≤ nI − 1) from (5). Inserting (B.2) into (B.1), we have (nI − 1)q + m d = n − (k − q) − +2 nI − 1 = n − (k − q) − (q + 1) + 2 = n − k + 1, (B.2) (B.3) Cluster 1 (2) (2) 𝑠3 = + 𝑠1 = 𝑠6 = (1) 𝑦1 + 𝑦1 (2) 𝑦1 𝑠2 = (1) 𝑦4 𝑦5 (2) 𝑦5 (2) 𝑦6 (1) 𝑦6 (2) 𝑦3 (2) 𝑦3 (1) Cluster 2 𝑦3 𝑦2 𝑦2 (1) 𝑦3 (1) (1) (1) 𝑦1 + (2) 𝑦6 𝑠4 = (1) 𝑦4 + (1) 𝑦2 (2) + 𝑦2 (1) 𝑦6 (2) 𝑦4 (2) 𝑦4 𝑠5 = (1) 𝑦5 (2) + 𝑦5 Fig. 7: Code construction for = 0, nI - k case where the second last equality holds since 0 < m ≤ nI − 1 from (B.2). Thus, this proves that contacting arbitrary k nodes suffices to recover the original source file. Now, all we need to prove is that any failed node can be exactly regenerated under the setting of system parameters specified in Proposition 1. According to the rule illustrated in [10], the construction of code C can be shown as in Fig. 7. First, we have M = k − q source symbols {xi }k−q i=1 to store reliably. By applying a (T, k − q) Reed-Solomon code to the source symbols, we obtain {zi }ti=1 where T := L(nI − 1). Then, we partition {zi }Ti=1 symbols into L groups, where each group contains (nI − 1) symbols. Next, each group of {zi } symbols is encoded by an (nI , nI − 1)−MDS code, which result in a group of nI symbols of {yi }. Finally, we store symbol ynI (l−1)+j in node N (l, j). By this allocation rule, yi symbols in the same group are located in the same cluster. Assume that N (l, j), the j th node at lth cluster, containing ynI (l−1)+j symbol fails for l ∈ [L] and j ∈ [nI ]. From Fig. I 7, we know that (nI − 1) symbols of {ynI (l−1)+s }ns=1,s6 =j stored in lth cluster can decode the (nI , nI − 1)−MDS code for group l. Thus, the contents of ynI (l−1)+j can be recovered by retrieving symbols from nodes in the the lth cluster (i.e., the same cluster where the failed node is in). This proves the ability of exactly regenerating an arbitrary failed node. The regeneration process satisfies βc = 0, βI = α. (B.4) Moreover, note that the code in Fig. 7 has M = (k − q)α (B.5) source symbols. Since parameters obtained in (B.4) and (B.5) are consistent with Proposition 1, we can confirm that code C is a valid MSR point under the conditions = 0 and nI - k. A PPENDIX C P ROOF OF T HEOREM 3 Recall that the code designed by Construction 1 allocates systematic nodes at 1st cluster and parity nodes at 2nd cluster, as illustrated in Fig. 8. Moreover, recall that the system parameters for [n, k, L] = [2k, k, 2]−DSS with = 1/(n − k) are α = βI = k, βc = 1, (C.1) 9 𝒎Z = 𝑚Z,,, 𝑚Z,/, ⋯ 𝑚Z,> , 𝒑Z = 𝑝Z,,, 𝑝Z,/, ⋯ 𝑝Z,> 9 𝑁(1,1) 𝑁(1,2) 𝑁(1, 𝑘) Cluster 1 𝒎,9 𝒎9/ ⋯ 𝒎9> Cluster 2 𝒑,9 𝒑9/ ⋯ 𝒑9> 𝑁(2,1) 𝑁(2,2) 𝑁(2, 𝑘) Fig. 8: Code construction for [n, k, L] = [2k, k, 2]−clustered DSS when = 1/(n − k) from Proposition 2 and the definition of = βc /βI . First, we show that exact regeneration of systematic nodes (in the first cluster) is possible using βI = k, βc = 1 in the [n, k, L] = [2k, k, 2] DSS with Construction 1. We use the concept of the projection vector to illustrate the repair process. For l ∈ [L], (l) let vi,j be the lth projection vector assigned for N (1, j), in repairing N (1, i). Similarly, let vi,j be the projection vector assigned for N (2, j), in repairing N (1, i). Assume that the node N (1, i) containing mi = [mi,1 , mi,2 , · · · , mi,k ]T fails. (l) Then, node N (1, j) transmits βI = k symbols {mTj vi,j }kl=1 , T while node N (2, j) transmits βc = 1 symbol pj vi,j . For (l) simplicity, we set vi,j = el and vi,j = ek , where ei is the kdimensional standard basis vector with a 1 in the ith coordinate and 00 s elsewhere. This means that node N (1, j) transmits k symbols mj = [mj,1 , mj,2 , · · · , mj,k ]T it contains, while N (2, j) transmits the last symbol it contains, i.e., the symbol pj,k . Thus, the newcomer node for regenerating systematic node N (1, i) obtains the following information Mi := {mj,s : j ∈ [k] \ {i}, s ∈ [k]} ∪ {pj,k }kj=1 . (C.2) We now show how the newcomer node regenerates mi = [mi,1 , mi,2 , · · · , mi,k ]T using information Mi . Recall that the parity symbols and message symbols are related as in the following k 2 equations:     m1 p1 p2  m2      (C.3)  ..  = G  ..   .   .  pk mk obtained from (10) and (12). Among these k 2 parity symbols, k parity symbols received by the newcomer node can be expressed as      p1,k Gk,1 Gk,2 · · · Gk,k2 m1,1 p2,k  G2k,1 G2k,2 · · · G2k,k2   m1,2        ..  =  .. .. ..   ..  , (C.4) ..  .   .   . . . .  pk,k Gk2 ,1 Gk2 ,2 ··· subtracting the constant known values from (C.4) results in      mi,1 y1 Gk,(i−1)k+1 Gk,(i−1)k+2 · · · Gk,ik y2  G2k,(i−1)k+1 G2k,(i−1)k+2 · · · G2k,ik  mi,2        ..  =  .. .. ..   ..  ..  .   . .  . . . 2 2 2 mi,k yk Gk ,(i−1)k+1 Gk ,(i−1)k+2 · · · Gk ,ik (C.5) where Gk2 ,k2 mk,k where the matrix in (C.4) is generated by removing k(k − 1) rows from G. Since we are aware of k(k−1) message symbols of {mj,s : j ∈ [k] \ {i}, s ∈ [k]} and the entries of G matrix, yl := pl,k − k k X X Glk,(j−1)k+s mj,s (C.6) j=1j6=i s=1 for l ∈ [k]. Note that the matrix in (C.5) can be obtained by removing k(k − 1) columns from the matrix in (C.4). Since every square sub-matrix of G is invertible, we can obtain mi = [mi,1 , mi,2 , · · · , mi,k ]T , which completes the proof for exactly regenerating the failed systematic node. Second, we prove that exact regeneration of the parity (l) nodes (in the second cluster) is possible. Let ωi,j be the lth projection vector assigned for N (2, j) in repairing N (2, i). Similarly, let ωi,j be the projection vector assigned for N (1, j) in repairing N (2, i). Assume that the parity node N (2, i) fails, which contains pi = [pi,1 , pi,2 , · · · , pi,k ]T . Then, node (l) N (2, j) transmits βI = k symbols {pTj ωi,j }kl=1 , while node N (1, j) transmits βc = 1 symbol mTj ωi,j . For simplicity, we (l) set ωi,j = el and ωi,j = ek . This means that node N (2, j) transmits k symbols pj = [pj,1 , pj,2 , · · · , pj,k ]T it contains, while N (1, j) transmits the last symbol it contains, i.e., the symbol mj,k . Thus, the newcomer node for regenerating parity node N (2, i) obtains the following information Pi := {pj,s : j ∈ [k] \ {i}, s ∈ [k]} ∪ {mj,k }kj=1 (C.7) We show how the newcomer node regenerates pi = [pi,1 , pi,2 , · · · , pi,k ]T using the information Pi . Among k 2 parity symbols in (C.3), k(k − 1) parity symbols received by the newcomer node can be expressed as    (1) (k)  G1 · · · G1 p1   .. ..  ..  ..   m1 .  .   . .      (1) (k)   m2  pi−1     0   Gi−1 · · · Gi−1  (C.8) (1) (k)   ..  = G m, pi+1  =     G · · · G .    i+1 i+1    .   . ..  mk ..  ..   .. . .  (1) (k) pk Gk · · · Gk (j) where Gi is defined in Construction 1. Note that G0 is a k(k − 1) × k 2 matrix, which is generated by removing lth rows from G, for l ∈ {(i − 1)k + 1, (i − 1)k + 2, · · · , ik}. Since we know the values of k message symbols {mj,k }kj=1 and the entries of G matrix, subtracting constant known values from (C.8) results in G00 m0 , (C.9) where G00 is generated by removing lth columns from G0 for l ∈ {k, 2k, · · · , k 2 }. Similarly, m0 is generated by removing lth rows from m for l ∈ {k, 2k, · · · , k 2 }. Thus, G00 is an invertible k(k − 1) × k(k − 1) matrix, so that we can obtain m0 , which contains P̃i = {mj,s : j ∈ [k], s ∈ [k − 1]}. (C.10) Since Pi ∪ P̃i contains every message symbol {mj,s : j, s ∈ [k]}, we can regenerate pi = [pi,1 , pi,2 , · · · , pi,k ]T using (C.3). This completes the proof for exactly regenerating the failed parity node. Finally, we prove that M = k 2 message symbols can be obtained by contacting arbitrary k nodes. In this proof, we use a slightly modified notation for representing message and parity symbols. For j, s ∈ [k], the message symbol mj,s and the parity symbol pj,s are denoted as m(j−1)k+s and p(j−1)k+s , respectively. Then, (C.3) is expressed as     m1 p1  m2   p2      (C.11)  ..  = G  ..   .   .  Suppose that the data collector (DC) contacts e nodes from the 1st cluster, and k − e nodes from the 2nd cluster, for e ∈ {0, 1, · · · , k}. Then, DC obtains k(k − e) parity symbols and ke message symbols. Since there exists total of M = k 2 message symbols, the number of message symbols that DC cannot obtain is M − ke = k(k − e). Let the parity symbols obtained by DC be pi1 , · · · , pik(k−e) , and the message symbols not obtained by DC be mj1 , · · · , mjk(k−e) . Then, the known parities can be expressed as     pi1 m1,1  m1,2   pi2      (C.12)  ..  = G0  ..  ,  .   .  pik(k−e) 1) if m = nI − 1 where G00 is a k(k−e)×k(k−e) matrix generated by taking the k(k−e) lth columns from G0 for l ∈ {jt }t=1 . Since G00 is invertible, k(k−e) we obtain the unknown message symbols {mji }i=1 . This completes the proof. M ζnI −2 λnI −2 (1 − (D.4) q nI − 1 = k − 2q λnI −2 = (D.5) δnI −2 (D.6) ζnI −2 = k − q (D.7) where q and m are defined in (5) and (4). When m = nI − 1, (D.3) can be expressed as = k − 2q − 1, δnI −2 )), ζnI −2 (D.8) where the last equality is from (5). Thus, from (D.8), (D.4) and (D.2), we have ζnI −2 − δnI −2 = nI − 1 λnI −2 (D.9) holds for the m = nI −1 case. Similarly, using (D.5), (D.6) and (D.7), we can confirm that (D.9) holds for the 0 ≤ m ≤ nI −2 case. Inserting (D.4), (D.7), (D.9) into (D.1), we obtain M , k−q M γ= (nI − 1) k−q α= (D.10) (D.11) Since γ = (nI − 1)βI for βc = 0 from (1), we obtain βI = α = M/(k − q), (D.12) which completes the proof. B. Proof of Proposition 2 We consider the βc = 1 case without losing generality. This implies that βI = 1/ = n − k (D.13) according to the definition = βc /βI . Now, we observe the expressions for α and M. From Corollary 3 of [6], the MSR point for = 1/(n − k) is illustrated as M M 1 , ), k k sk−1 (D.14) where From Corollary 3 of [6], the MSR point for = 0 is given , (D.3) 2) else (m = 0, 1, · · · , nI − 2) A. Proof of Proposition 1 M δnI −2 (α, γ) = ( A PPENDIX D P ROOF OF P ROPOSITIONS (α, γ) = ( (D.2) ζnI −2 = k − q mk,k where G0 is a k(k − e) × k 2 matrix obtained by taking lth k(k−e) rows from G, for l ∈ {it }t=1 . Since we know ke message symbols and the elements of G, subtracting the known constant values from (C.12) results in   mj1  mj2    G00  (C.13) , ..   . mjk(k−e) by q+1 nI − 1 = (q + 1)(nI − 2) λnI −2 = δnI −2 = qnI − 2q + nI − 2 = qnI − 2q + m − 1 mk2 pk2 where {ζi }, {λi }, {δi } are defined in [6]. This paper does not review the explicit form of the definitions, but shows how (α, γ) looks like. From the proof of Lemma 5 of [6], we have (D.1) sk−1 = (n − k) 1 = (nI − 1) + (n − nI ) nI − 1 + n−nI n−k (D.15) from the definition of {si } in [6] and the setting of = 1/(n− k). Combining (D.14) and (D.15) result in (8). Note that γ in (1) can be expressed as γ = (n − nI )βc + (nI − 1)βI = (n − nI ) + (nI − 1)(n − k), (D.16) where the last equality holds due to (D.13). Combining (8) and (D.16), we obtain γ= M γ , k n−k which result in M = k(n − k). (D.17) Using α = M/k in (D.14), we have α = n − k. (D.18) This completes the proof. R EFERENCES [1] A. G. Dimakis, P. B. Godfrey, Y. Wu, M. J. Wainwright, and K. Ramchandran, “Network coding for distributed storage systems,” IEEE Transactions on Information Theory, vol. 56, no. 9, pp. 4539–4551, 2010. [2] K. Rashmi, N. B. Shah, P. V. Kumar, and K. Ramchandran, “Explicit construction of optimal exact regenerating codes for distributed storage,” in Communication, Control, and Computing, 2009. 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arXiv:1604.08860v4 [cs.DS] 25 Nov 2016 Designing optimal- and fast-on-average pattern matching algorithms Gilles Didier and Laurent Tichit Aix-Marseille Université, CNRS, Centrale Marseille, I2M UMR7373, Marseille, France E-mail: {gilles.didier,_laurent.tichit}@univ-_amu.fr March 26, 2018 Abstract Given a pattern w and a text t, the speed of a pattern matching algorithm over t with regard to w, is the ratio of the length of t to the number of text accesses performed to search w into t. We first propose a general method for computing the limit of the expected speed of pattern matching algorithms, with regard to w, over iid texts. Next, we show how to determine the greatest speed which can be achieved among a large class of algorithms, altogether with an algorithm running this speed. Since the complexity of this determination makes it impossible to deal with patterns of length greater than 4, we propose a polynomial heuristic. Finally, our approaches are compared with 9 pre-existing pattern matching algorithms from both a theoretical and a practical point of view, i.e. both in terms of limit expected speed on iid texts, and in terms of observed average speed on real data. In all cases, the pre-existing algorithms are outperformed. 1 Introduction We focus on algorithms solving the online string matching problem, which consists in reporting all, and only the occurrence positions of a pattern w in a text t (online meaning that no pre-processing of the text is allowed). As one of the oldest problems addressed in computer science, it has been extensively studied. We refer to [10] for a comprehensive list and an evaluation of all the pattern matching algorithms developed so far. By the authors’ count, more than 80 algorithms have already been proposed, among which more than a half were published during the last ten years. This fact sounds quite paradoxical, since the Morris-Pratt algorithm, which is optimal in terms of worst case analysis, dates back to 1970. A possible explanation is that there is wide gap between the worst case complexity of algorithms and their computation times on real data. For instance, there are pattern matching algorithms with non-linear worst case complexities, which perform much better than Morris-Pratt on English texts. Basically, the 1 average case analysis is way more suited to assess the relevance of a pattern matching algorithm from a practical point of view. The average case analysis of some pattern matching algorithms, notably Boyer-Moore-Horspool and Knuth-Morris-Pratt, has already been carried out from various points of view [27, 12, 4, 3, 16, 21, 22, 25]. We provide here a general method for studying the limit average behavior of a pattern algorithm over iid texts. More precisely, following [18], we consider the limit expectation of the ratio of the text length to the number of text accesses performed by an algorithm for searching a pattern w in iid texts. This limit expectation is called the asymptotic speed of the algorithm with regard to w under the iid model. The computation of the asymptotic speed is based on w-matching machines which are automata-like structures able to simulate the behavior of a pattern matching algorithm while searching the pattern w. The underlying idea is the same as in [18, 19, 20, 17] and can be seen as a generalization of the string matching automaton [7]. In the companion paper, G. Didier provided a theoretical analysis of the asymptotic speed of pattern matching algorithms over iid texts [8]. In particular, he showed that, for a given pattern w, the greatest asymptotic speed among a large class of pattern matching algorithms, is achieved by a w-matching machine in which the states are essentially subsets of positions of w. Such machines are called strategies below. We provide here a brute force algorithm computing the Fastest strategy for a given pattern w and the frequencies of an iid model. The algorithm is based on an original structure associated to the pattern w and called its position lattice, which gives a full representation of the overlap relations between the subsets of positions of w. Since the brute force algorithm cannot be applied on patterns of length greater than 4, because of its (very high) time-complexity, we propose a polynomial K-Heuristic, in which the polynomial order K may be chosen by the user. The Fastest and K-Heuristic approaches are finally compared with 9 several pre-existing pattern matching algorithms: • from a theoretical point of view, by computing their limit expected speeds with regard to various patterns and iid models, • from a practical point of view, by computing their average speeds over two sources (an English text and a DNA sequence). In both cases, the Fastest and K-Heuristic (with K large enough) approaches outperform the pre-existing algorithms. The software and the data used to perform the tests are available at https: //github.com/gilles-_didier/Matchines.git. The rest of the paper is organized as follows. Section 2 presents the notations and recalls some concepts and results from [8]. It is followed by two sections which introduce the central objects of this work: the strategies and the position lattice of a pattern. In particular, we provide an algorithm computing the position lattice of a given pattern. Section 5 shows how to use the position 2 lattice of a pattern to obtain the Fastest strategy with regard to this pattern and an iid model. In Section 6, we provide a polynomial heuristic allowing to compute fast strategies. Section 7 presents the results of various comparisons between 9 pre-existing pattern matching algorithms, the K-Heuristic and, each time it is possible, the Fastest strategy. The results are discussed in the last section. 2 Notations and definitions 2.1 Notations and general definition For all finite sets S, P(S) is the power set of S and \|S\| is its cardinal. An alphabet is a finite set A of elements called letters or symbols. A word, a text or a pattern on A is a finite sequence of symbols of A. We put \|v\| for the length of a word v. Words are indexed from 0, i.e. v = v0 v1 . . . v\|v\|−1 . We write v[i,j] for the subword of v starting at its position i and ending at its position j, i.e. v[i,j] = vi vi+1 . . . vj . The concatenate of two words u and v is the word uv = u0 u1 . . . u\|u\|−1 v0 v1 . . . v\|v\|−1 . For any length n ≥ 0, we note An the S set of words of length n on A, and ∞ ? A , the set of finite words on A, i.e. A? = n=0 An . Unless otherwise specified, all the texts and patterns considered below are on a fixed alphabet A. A pattern matching algorithm takes a pattern w and a text t as inputs an reports all, and only the occurrence positions of w in t. For all patterns w, we say that two pattern matching algorithms are w-equivalent if, for all texts t, they access exactly the same positions of t on the input (w, t). 2.2 Matching machines and the generic algorithm [8] For all patterns w, a w-matching machine is 6-uple (Q, o, F, α, δ, γ) where • Q is a finite set of states, • o ∈ Q is the initial state, • F ⊂ Q is the subset of pre-match states, • α : Q → N is the next-position-to-check function, which is such that for all q ∈ F , α(q) < \|w\|, • δ : Q × A → Q is the transition state function, • γ : Q × A → N is the shift function. By convention, the set of states of a matching machine always contains a sink state , which is such that, for all symbols x ∈ A, δ( , x) = and γ( , x) = 0. The order OΓ of a matching machine Γ = (Q, o, F, α, δ, γ) is defined as OΓ = maxq∈Q {α(q)}. 3 The w-matching machines carry the same information as the Deterministic Arithmetic Automatons defined in [19, 20]. The generic algorithm takes a w-matching machine and a text t as inputs and outputs positions of t (Algorithm 1). input : a w-matching machine (Q, o, F, α, δ, γ) and a text t output: all the occurrence positions of w in t 1 2 3 4 5 (q, p) ← (o, 0) while p ≤ \|t\| − \|w\| do if q ∈ F and tp+α(q) = wα(q) then print “ occurrence at position p ” (q, p) ← (δ(q, tp+α(q) ), p + γ(q, tp+α(q) )) Algorithm 1: The generic algorithm Each component of a w-matching machine makes sense in regard to the way it is used by the generic algorithm. The pre-match states in F are those which lead to report an occurrence of the pattern at the current position, if the nextposition-to-check of the pattern matches the corresponding position in the text (Line 3 of Algorithm 1). The condition α(q) < \|w\| for all q ∈ F in the definition of w-matching machines, is technical and used in [8]. A w-matching machine Γ is valid if, for all texts t, the execution of the generic algorithm on the input (Γ, t) outputs all, and only the occurrence positions of w in t. Since one has to check all the positions of the pattern w before concluding that it occurs somewhere in a text, the order of a valid w-matching machine is at least \|w\| − 1. We claim that for all the pattern matching algorithms developed so far and all patterns w, there exists a w-matching machine Γ which is such that, for all texts t, the generic algorithm and the pattern matching algorithm access exactly the same positions of t on the inputs (Γ, t) and (w, t) respectively [8]. For instance, Figure 1 displays a abb-matching machine which accesses the same positions as the naive algorithm while searching abb. 2.3 Full-memory expansion – standard matching machines [8] We present here a transformation on matching machines which split their states according to the text positions read from the current position during an execution of the generic algorithm. The main point of this transformation is that the average complexity of matching machines such obtained may then be computed through algebraic methods (Sections 2.4 and 2.5). For all n ∈ N, Rn is the set of subsets H of {0, . . . , n} × A verifying that, for all i ∈ {0, . . . , n}, there exists at most one pair in H with i as first entry. For all H ∈ Rn , we put f (H) for the set comprising all the first entries of 4 b/1 S0 0 a/0 S1 1 b/0 S2 2 a/1 b/1 a/1 Figure 1: abb-matching machine of the naive algorithm. The next-position-to check are displayed below all states S0, S1 and S2. Edges from states Si are labelled with “x/γ(Si, x)” for all symbols x. The transition associated to a match is blue-colored. the pairs in H, namely f (H) = {i \| ∃x ∈ A with (i, x) ∈ H}. For all k ∈ N and H ∈ Rn , the k-shifted of H is k ← − H = {(u − k, y) \| (u, y) ∈ H with u ≥ k}, i.e. the subset of Rn obtained by subtracting k from the first entries of the pairs in H and by keeping only the pairs with non-negative first entries. The full memory expansion of a w-matching machine Γ = (Q, o, F, α, δ, γ) is the w-matching machine Γ? obtained by removing the unreachable states of Γ0 = (Q0 , o0 , F 0 , α0 , δ 0 , γ 0 ), defined as: • Q0 = Q × ROΓ • o0 = (o, ∅) • α0 ((q, H)) = α(q) • γ 0 ((q, H), x) = γ(q, x) • F 0 = F × ROΓ • δ 0 ((q, H), x)  γ(q,x)  ←−−−−−−−−−−−    (δ(q, x), H ∪ {(α(q), x)}) if ∀a ∈ A, (α(q), a) 6∈ H = if ∃a 6= x s.t. (α(q), a) ∈ H  γ(q,x)   ← −  (δ(q, x), H ) if (α(q), x) ∈ H 5 (S0, {(0, b)}) 0 b/1 b/1 b/1 b/1 (S0, ∅) 0 a/0 (S0, {(0, b), (1, b)}) 0 (S1, {(0, a)}) 1 a/0 b/0 a/1 (S0, {(0, a)}) 0 (S2, {(0, a), (1, b)}) 2 a/1 b/1 (S0, {(0, b), (1, a)}) 0 Figure 2: Full memory expansion of the abb-matching machine of Figure 1. By construction, at all iterations of the generic algorithm on the input (Γ? , t), if the current state and position are (q, H) and p, respectively, then the positions of {j + p \| j ∈ f (H)} are exactly the positions of t greater than p accessed so far (the second entries of the corresponding elements of H give the symbols read). For all texts t, the generic algorithm access the same positions of t on the inputs (Γ, t) and (Γ? , t) [8]. Let us remark that the full memory expansion of the full memory expansion of a matching machine is equal to its full memory expansion (up to a state isomorphism). A w-matching machine Γ is standard if each state q of Γ appears in a unique pair/state of its full memory expansion, or, equivalently, if it is equal to its full memory expansion. For instance the abb-matching machine of Figure 1 is not standard. Since the matching machine of Figure 2 is a full memory expansion, it is standard. For all states q of a standard matching machine Γ, we put hΓ (q) for the second entry of the unique pair/state of Γ? in which q appears. We implemented a basic algorithm computing the full-memory expansion Γ? = (Q? , o? , F ? , α? , δ ? , γ ? ) of a w-matching machine Γ = (Q, o, F, α, δ, γ) in O(\|w\|.\|Q? \|) time. We have \|Q? \| ≤ (A + 1)\|w\| \|Q\| but the size of Q? may vary a lot with regard to the matching machine/algorithm considered. A w-matching machine Γ is compact if it contains no state q which always leads to the same state. Formally, Γ = (Q, o, F, α, δ, γ) is compact if there is no q ∈ Q such that one of the following assertions holds: 1. there exists a symbol x with δ(q̇, x) 6= y 6= x; and δ(q̇, y) = for all symbols 2. for all symbols x and y, we have both δ(q̇, x) = δ(q̇, y) and γ(q̇, x) = γ(q̇, y). Basically, a non-compact machine performs useless text accesses. In [8], it is shown that any w-matching machine can be turned into a compact (and faster) machine. 6 2.4 iid and Markov models An independent identically distributed (iid) model (aka Bernoulli model) is fully specified by a probability distribution π on the alphabet (i.e. π(x) is the probability of the symbol x in the model). Such a model will be simply referred to as “π” below. Under π, the probability of a text t is \|t\|−1 Y pπ (t) = π(ti ). i=0 A Markov model M over a given set of states Q is a 2-uple (πM , δM ), where πM is a probability distribution on Q (the initial distribution) and δM associates a pair of states (q, q 0 ) with the probability for q to be followed by q 0 (the transition probability). Under a Markov model M = (πM , δM ), the probability of a sequence s of states is \|s\|−1 pM (s) = πM (s0 ) Y δM (si , si+1 ). i=0 Theorem 1 ([8]). Let Γ = (Q, o, F, α, δ, γ) be a w-matching machine. If a text t follows an iid model and Γ is standard then the sequence of states parsed by the generic algorithm on the input (Γ, t) follows a Markov model (πM , δM ). Proof. Whatever the text model and the machine, the sequence of states always starts with o with probability 1. We have πM (o) = 1 and πM (q) = 0 for all q 6= o. The probability δM (q, q 0 ) that the state q 0 follows the state q during an execution of the generic algorithm, is equal to: • 1, if there exists a symbol x such that δ(q, x) = q 0 and (α(q), x) ∈ hΓ (q), i.e. if the relative position α(q) was already checked with x occurring at it, X • π(x), otherwise, x s.t. δ(q, x) = q 0 independently of the previous states. 2.5 Asymptotic speed Let M be a text model and A be an algorithm. The w-asymptotic speed of A under M is the limit expectation, under M, of the ratio of the text length to the number of text accesses performed by A [8]. Namely, by putting aA (t) for the number of text accesses performed by A to parse t and pM (t) for the probability of t with regard to M, the asymptotic speed of A under M is ASM (A) = lim n→∞ X t∈An 7 \|t\| pM (t). aA (t) The asymptotic speed ASM (Γ) of a w-matching machines Γ is that the generic algorithm with Γ as first input. From Theorem 5 of [8], the asymptotic speed of a standard w-matching machine Γ = (Q, o, F, α, δ, γ) under an iid model π exists and is given by X ASπ (Γ) = βq E(q), (1) q∈Q where (βq )q∈Q are the limit frequencies of the states of the Markov model associated to Γ and π in Theorem 1, and γ(q, x) if (α(q), x) ∈ hΓ (q)), P E(q) = γ(q, x)π otherwise. x x Computing the asymptotic speed of a pattern matching algorithm, with regard to a pattern w and an iid model π is performed by following the stages below. 1. We get a w-matching machine Γ which simulates the behavior of the algorithm while looking for w (Figure 1). The transformation of the 9 algorithms presented in Section 7 (and a few others, see our GitHub repository) into w-matching machines, given w, has been implemented. 2. We obtain the full-memory expansion Γ? of Γ (Figure 2, Section 2.3). 3. We compute the limit frequencies of the Markov model associated to Γ? and π in Theorem 1. This mainly needs to solve a system of linear equations of dimension \|Q? \|. 4. We finally obtain the asymptotic speed of the algorithm from these limit frequencies, π and Γ? by using Equation 1. The most time-consuming stage is the computation of the limit frequencies, which has O(\|Q? \|3 ) time complexity, where \|Q? \|, the number of states of the full memory expansion, is smaller than (A + 1)\|w\| \|Q\|. 3 Strategies For all sets I ⊂ N and k ∈ N, we define the k-left-shifted of I as ∆(I, k) = {i − k \| ∃i ∈ I and i ≥ k}. A w-strategy S = (Q, o, F, α, δ, γ) is a w-matching machine such that • Q ⊆ P({0, . . . , \|w\| − 1}) \ {{0, . . . , \|w\| − 1}} and ∅ ∈ Q, • o = ∅, • F = {s ∈ Q \| \|s\| = \|w\| − 1}, 8 b/3 {1} 0 b/0 ∅ 1 a/2 a/1 {0} 2 a/0 {0, 1} 2 a/2 a/1 b/0 {0, 2} 1 a/0 {0, 1} 2 b/3 b/3 {1} 0 b/0 ∅ 1 a/2 a/1 b/0 a/1 {0} 1 Figure 3: Two abb-strategies with the same conventions as in Figure 1. • α : Q → {0, 1, . . . , \|w\| − 1} is such that for all s ∈ Q, α(s) 6∈ s and \|s\| < \|w\| − 1 ⇒ s ∪ {α(s)} ∈ Q, • γ : Q × A → {0, 1, . . . , \|w\|} is such that for all states s and all symbols x, γ(s, x) min{k ≥ 1 \| wα(s)−k = x if α(s) ≥ k and wj = wj+k for all j ∈ ∆(s, k)} = min{k ≥ 0 \| wα(s)−k = x if α(s) ≥ k and wj = wj+k for all j ∈ ∆(s, k)} • δ : Q × A → Q is such that for all s ∈ Q and all symbols x, δ(s, x) = ∆(s ∪ {α(s)}, γ(s, x)). Figure 3 shows two abb-strategies which differ notably in the next-positionto-check of state {0}. Proposition 1. A w-strategy is a standard, compact, valid and non-redundant w-matching machine. Proof. By construction, a w-strategy is standard, compact and non-redundant. The validity of a w-strategy follows from Theorem 1 of [8]. Proposition 2. There is a w-strategy which achieves the greatest asymptotic speed among all the w-matching machines of order \|w\| − 1. 9 if s ∈ F, otherwise, Proof. The Corollary 2 of [8] implies that there exists a w-matching machine which achieves the greatest asymptotic speed among those of order \|w\| − 1 and which is 1. standard, 2. compact, 3. valid, 4. in which all the states are relevant (i.e. such that they may lead to a match without any positive shift [8]), 5. such that there is no pair of states (q, q 0 ) with q 6= q 0 and hΓπ (q) = hΓπ (q 0 ). Let us verify that a w-matching machine Γ = (Q, o, F, α, δ, γ) of order \|w\| − 1 satisfying the properties above is (isomorphic to) a w-strategy. Since it verifies in particular the properties 4 and 5, its set of states Q is in bijection with a subset of P({0, . . . , \|w\| − 1}). Let us identify all states q of Q with f (hΓ (q)), its corresponding element of P({0, . . . , \|w\| − 1}). Since Γ is standard, compact and of order \|w\| − 1, we do not have {0, . . . , \|w\| − 1} ∈ Q. Moreover, since Γ is standard, we have δ(s, x) = ∆(s ∪ {i}, γ(s, x)) for all s ∈ Q. Last, by construction, if γ(s, x) min{k ≥ 1 \| wα(s)−k = x if α(s) ≥ k and wj = wj+k for all j ∈ ∆(s, k)} > min{k ≥ 0 \| wα(s)−k = x if α(s) ≥ k and wj = wj+k for all j ∈ ∆(s, k)} if s ∈ F, otherwise, then Γ is not valid, and if γ(s, x) min{k ≥ 1 \| wα(s)−k = x if α(s) ≥ k and wj = wj+k for all j ∈ ∆(s, k)} < min{k ≥ 0 \| wα(s)−k = x if α(s) ≥ k and wj = wj+k for all j ∈ ∆(s, k)} if q ∈ F, otherwise, then δ(s, x) is not relevant. 4 Position lattices [w] [w] The position lattice of a pattern w is the 3-uple L[w] = (Q[w] , (δs )s∈Q[w] , (γs )s∈Q[w] ) where, by putting s for {0, . . . , \|w\| − 1} \ s, • Q[w] = P({0, . . . , \|w\| − 1}) \ {{0, . . . , \|w\| − 1}}, i.e. the set made of all the subsets of positions of w but {0, . . . , \|w\| − 1}, [w] is a map from s × A to {0, . . . , \|w\|}, [w] is a map from s × A to Q[w] , • for all s ∈ Q[w] , γs • for all s ∈ Q[w] , δs 10 0, b\|1 ∅ 0, a\|0 2, b\|3 1, b\|0 0, b\|2 2, b\|0 0, b\|3 2, a\|2 0, a\|3 1, a\|1 1, a\|1 {0} {1} 2, a\|2 {2} 0, b\|1 1, b\|3 2, a\|2 2, b\|0 0, a\|0 2, a\|2 1, b\|0 1, b\|0 1, a\|1 2, b\|0 0, a\|0 {0, 1} 1, a\|1 {0, 2} {1, 2} Figure 4: Position lattice of the pattern abb. Vertices represent the states of L[abb] . For all states s, there is an outgoing edge for all pairs (i, x) with i ∈ {0, . . . , \|abb\| − 1} \ s and x ∈ A. This outgoing edge is labeled with [abb] [abb] “i, x\|γs (i, x)”, is colored according to i, and goes to δs (i, x). where, for all s ∈ Q[w] , all i ∈ s and all x ∈ A, we have γs[w] (i, x) min{k ≥ 1 \| wα(s)−k = x if α(s) ≥ k and wj = wj+k for all j ∈ ∆(s, k)} = min{k ≥ 0 \| wα(s)−k = x if α(s) ≥ k and wj = wj+k for all j ∈ ∆(s, k)} and δs[w] (i, x) = ∆(s ∪ {i}, γs[w] (i, x)). [w] In particular, if x = wi and \|s\| < \|w\| − 1 then we have γs (i, x) = 0 and [w] δs (i, x) = s ∪ {i}. Let us remark that, since max(s) ≤ \|w\| − 1 for all s ∈ Q[w] , we have, for all [w] i ∈ s and all x ∈ A, ∆(s ∪ {i}, \|w\|) = ∅, thus γs (i, x) ≤ \|w\| which is consistent [w] with the definition of γs . [w] The edges of L[w] are the pairs (s, δs (i, x)) for all s ∈ Q[w] , all i ∈ s and all x ∈ A (see Figure 4). 11 if \|s\| = \|w\| − 1, otherwise, Remark 1. The position lattice of w contains 2\|w\| − 1 states and \|A\|.\|w\|.2\|w\|−1 edges. Remark 2. Let s be a state of Q[w] , i and j be two positions in s such that i 6= j and x and y be two symbols of A. We have γ [w] [w] δs (i,x) (j − γs[w] (i, x), y) + γs[w] (i, x) = γ δ [w] [w] δs (i,x) [w] [w] δs (j,y) (j − γs[w] (i, x), y) = δ (i − γs[w] (j, y), x) + γs[w] (j, y), and [w] [w] δs (j,y) (i − γs[w] (j, y), x). By considering the particular case where x = wi , we get [w] γs∪{i} (j, y) = γ [w] δs∪{i} (j, y) = δ [w] [w] δs (j,y) [w] [w] δs (j,y) (i − γs[w] (j, y), wi ) + γs[w] (j, y), and (i − γs[w] (j, y), wi ). Let precw be the table indexed on {0, . . . , \|w\| − 1} × A and in which, for all positions i of w and all symbols x of A, the entry precw [i, x] is defined as max{j ≤ i \| wj = x} if {j ≤ i \| wj = x} = 6 ∅, precw [i, x] = NULL otherwise. For instance, the table precabb is 0 1 2 a b 0 NULL 0 1 0 2 Lemma 1. Let s be a state of Q[w] , i a position in s and x a symbol of A. 1. If x = wi then [w] [w] • if \|s\| = \|w\|−1 then δs (i, x) = {0, . . . , B−1} and γs (i, x) = \|w\|−B, where B is the length of the longest proper suffix of w which is a prefix of w; [w] [w] • otherwise δs (i, x) = s ∪ {i} and γs (i, x) = 0. 2. If x 6= wi , (a) if s = ∅ then {precw [i, x]} [w] δ∅ (i, x) = ∅ [w] γ∅ (i, x) = i − precw [i, x] i+1 if precw [i, x] 6= NULL, otherwise, if precw [i, x] 6= NULL, otherwise, 12 (b) if s 6= ∅ then for all ` ∈ s, we have δs[w] (i, x) = δ [w] [w] [w] δs\{`} (i,x) γs[w] (i, x) = γ (` − γs\{`} (i, x), w` ), [w] [w] [w] δs\{`} (i,x) [w] (` − γs\{`} (i, x), w` ) + γs\{`} (i, x). Proof. The only case which does not immediately follow from the definition of L[w] , is when x 6= wi and s 6= ∅ which is given by Remark 2. The relation 4 on Q[w] is defined as follows. For all sets s and s0 in Q[w] , we have s 4 s0 if one of the following properties holds: • \|s\| < \|s0 \|, • \|s\| = \|s0 \|, s 6= s0 and min(s difference of s and s0 , s0 ) ∈ s, where s s0 is the symmetric • s = s0 . The relation 4 defines a total order on Q[w] . We write “ s ≺ s0 ” for “ s 4 s0 and s 6= s0 ”. Lemma 2. Let s be a state of Q[w] with \|s\| > 1, i a position in s and x a symbol [w] of A. If x 6= wi then δs\{max s} (i, x) ≺ s. Proof. Under the assumption that \|s\| > 1, we have min s = min(s \ {max s}). [w] By construction, the fact that x 6= wi implies that γs\{max s} (i, x) > 0. If we have [w] min((s \ {max s}) ∪ {i}) < γs\{max s} (i, x) [w] [w] then \|δs\{max s} (i, x)\| < \|s\|, thus δs\{max s} (i, x) ≺ s. [w] Otherwise, we have \|δs\{max s} (i, x)\| = \|s\| but since necessarily [w] [w] min δs\{max s} (i, x) ≤ min\|s\| − γs\{max s} (i, x) < min\|s\|, [w] we get again δs\{max s} (i, x) ≺ s. Theorem 2. Algorithm 2 computes the position lattice of the pattern w in O(\|w\|2\|w\| ) time by using the same amount of memory. Proof. Let us first show that Algorithm 2 determines the shifts and the transitions of the state s before those of the state s0 if and only if s ≺ s0 . The loop at Lines 2-8 computes the shifts and the transitions of ∅. Next, the loop at Lines 9-23 computes the shifts and the transitions of the singletons from {0} to {\|w\| − 1}. The last loop (Lines 24-41) determines the shifts and the transitions of the states corresponding to the subsets of increasing cardinals ` from 2 to 13 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 B ← length of the longest proper suffix of w which is also a prefix; for x ∈ A do last[x] ← NULL for i = 0 to \|w\| − 1 do last[wi ] ← i; for x ∈ A do if last[x] 6= NULL then [w] [w] γ∅ (i, x) ← i − last[x]; δ∅ (i, x) ← {last[x]}; else [w] [w] γ∅ (i, x) ← i + 1; δ∅ (i, x) ← ∅; for i = 0 to \|w\| − 1 do for j = 0 to i − 1 do for x ∈ A do [w] [w] [w] [w] γ{i} (j, x) ← γ∅ (j, x) + γ [w] (i − γ∅ (j, x), wi ); [w] δ{i} (j, x) ← δ δ∅ (j,x) [w] [w] [w] δ∅ (j,x) (i − γ∅ (j, x), wi ); for j = i + 1 to \|w\| − 1 do for x ∈ A do if x = wj then if \|w\| = 2 then [w] [w] γs (i, x) ← \|w\| − B; δs (i, x) ← {0, . . . , B − 1}; else [w] [w] γ{i} (j, x) ← 0; δ{i} (j, x) ← {i, j}; else [w] [w] [w] (j − γ∅ (i − 1, wi ), x); γ{i} (j, x) ← γ [w] [w] δ{i} (j, x) ← δ δ∅ (i−1,wi ) [w] [w] δ∅ (i−1,wi ) [w] (j − γ∅ (i − 1, wi ), x); for ` = 2 to \|w\| − 1 do for j = 0 to ` − 1 do S[j] ← j repeat s ← {S[0], . . . , S[` − 1]}; s0 ← {S[0], . . . , S[` − 2]}; for i ∈ {0, . . . , \|w\| − 1} \ s do for x ∈ A do if x = wi then if ` = \|w\| − 1 then [w] [w] γs (i, x) ← \|w\| − B; δs (i, x) ← {0, . . . , B − 1}; else [w] [w] γs (i, x) ← 0; δs (i, x) ← s ∪ {i}; else [w] [w] [w] [w] (S[` − 1] − γs0 (i, x), wS[`−1] ); γs (i, x) ← γs0 (i, x) + γ [w] δ 0 (i,x) s [w] δs (i, x) ← δ [w] [w] [w] δ 0 (i,x) s (S[` − 1] − γs0 (i, x), wS[`−1] ); j ← ` − 1; while j ≥ 0 and S[j] ≥ \|w\| − ` + j do j ← j − 1 if j ≥ 0 then S[j] ← S[j] + 1; for k = j + 1 to ` − 1 do S[k] ← S[k − 1] + 1 until j < 0; Algorithm 2: Computation of the position lattice. Value B is the last entry of the Partial Match table of the KMP algorithm. Its computation takes a time linear with \|w\| [15]. 14 \|w\| − 1. Inside the last loop, the way in which the next subset s0 is computed from the current subset s, both of cardinal `, ensures that s ≺ s0 (Lines 37-41). For all iterations i of the loop at Lines 2-8 and all symbols x, we have last[x] = precw [i, x] at the beginning of the inner loop (Line 4). From Lemma [w] [w] 1 (Cases 1 and 2a), the transitions δ∅ (i, x) and the shifts γ∅ (i, x) for all positions i of w and all symbols x, are correctly computed at the end of the loop. The loop at Lines 9-23 computes the shifts and the transitions from the singleton states. For all pairs of positions (i, j) and all symbols x, determining [w] [w] δ{i} (j, x) and γ{i} (j, x) is performed by distinguishing between two cases. [w] • If i > j, then δ∅ (j, x) ≺{i} and its shifts and transitions were already computed. Formula of Remark 2 gives us those of {i} (Lines 12-13). • If i < j, we distinguish between two subcases according to the symbol x considered. If x = wj then the shift and the transition state are given in [w] Lemma 1 - Case 1. Otherwise, we remark that, since γ{i} (j, x) is positive, [w] we have that γ{i} (j, x) = min{k ≥ 1 \| wj−k = x if j ≥ k and wi−k = wi }. [w] This implies that γ{i} (j, x) = γ [w] [w] [w] δ∅ (i−1,wi ) (j, x). We have δ∅ (i−1, wi ) ≺ s, [w] thus both the shifts and the transitions of the state δ∅ (i − 1, wi ) are computed before s (Lines 22-23). The last loop, lines 24-41, computes the shifts and the transitions of the states corresponding to the subsets of cardinals 2 to ` − 1. For all states s with 2 ≤ \|s\| ≤ \|w\| − 1, all positions i ∈ s and all symbols x 6= wi , the corresponding [w] [w] shift and transition γs (i, x) and δs (i, x) are computed from the shifts and [w] transitions of the state δs\{max s} (i, x) following Lemma 1 - Cases 2b (in Algo[w] rithm 2, we put s0 for s \ {max s}). Lemma 2 ensures that δs\{max s} (i, x) ≺ s, [w] thus that the shifts and transitions of δs\{max s} (i, x) are computed before those of s. For all states s with 2 ≤ \|s\| ≤ \|w\| − 1, all positions i ∈ s, the shift and [w] [w] transition γs (i, wi ) and δs (i, wi ) are given in Lemma 1 - Case 1. P\|w\|−1 The time complexity is O( k=0 (k + \|w\| − k) \|w\| k ) (loop Lines 24-41), i.e. O(\|w\|2\|w\| ). We do not use more memory than needed to store the lattice, which is, from Remark 1, O(\|w\|2\|w\| ). 5 The Fastest w-strategy Determining the fastest w-strategy, which, from Proposition 2, has the greatest asymptotic speed among all the w-matching machines of order \|w\| − 1, may be performed by computing the asymptotic speed of all the w-strategies and by returning the fastest one. In order to enumerate all the w-strategies, let us remark that they are all contained in the position lattice of w in the sense that: 15 • the set of states of a w-strategy is included in that of the position lattice; [w] • all the w-strategies Γ = (Q, o, F, α, δ, γ) verify δ(s, x) = δs (α(s), x) and [w] γ(s, x) = γs (α(s), x) for all s ∈ Q and all symbols x. Reciprocally, to any map φ from Q[w] to {0, . . . , \|w\|−1} such that φ(s) ∈ s for all states s ∈ Q[w] , there corresponds the unique w-strategy S = (Q, o, F, α, δ, γ) for which the next-position-to-check function α coincides with φ on Q. Finally, our brute force algorithm 1. takes as input a pattern w and an iid model π, 2. computes the position lattice of w, 3. enumerates all the maps φ such that φ(s) ∈ s for all states s ∈ Q[w] , 4. for each φ, gets the corresponding w-strategy by keeping only the states of Q[w] reachable from ∅, with the next-position-to-check function φ, 5. computes the asymptotic speed of all the w-strategies under π, 6. returns the w-strategy with the greatest speed. The time complexity of the brute force algorithm is    \|w\|−1 Y \|w\| O  (\|w\| − k)( k )  23\|w\|  , k=1 where the first factor stands for the number of functions φ and the second one for the computation of the asymptotic speed of a w-strategy, which needs to solve a linear system of size equal to the number of states, which is O(2\|w\| ). Its memory space complexity is \|w\|2\|w\|−1 , i.e. what is needed to store the position lattice of w. Under its current implementation, the brute force determination of the fastest w-strategy is unfeasible for patterns of length greater than 4. 6 A polynomial heuristic There are two points which make the complexity of the brute force algorithm given in Section 5 that high: 1. the size of the position lattice, which is exponential with the length of the pattern, 2. determining the fastest strategy in the position lattice, which needs a time exponential with its size. 16 Our heuristic is based on two independent stages, each one aiming to overcome one of these two points. Both of them start from the general idea that, since, for any current position of the text, the probability that no mismatch occurs until the nth text access decreases geometrically with n, the first relative positions accessed by a strategy (or more generally by a pattern algorithm) are those which have the greatest influence on its asymptotic speed. 6.1 n-sets sublattices A sufficient condition for a sublattice U ⊆ Q[w] to contain a w-strategy is that, [w] for all s ∈ U, there exists at least a position i ∈ s with δs (i, x) ∈ U for all x ∈ A. A sublattice U verifying this condition will be said to be complete. Figure 5 displays four complete sublattices extracted from the position lattice of abb (Figure 4). Let us introduce some additional notations here. For all sets S of positions, the prefix of S is defined as P (S) = max{i ∈ S \| j ∈ S for all 0 ≤ j ≤ i} and its rest is R(S) = S \ {0, . . . , P (S)} For all positive integers n, the n-sets sublattice of w is the sublattice U of Q[w] which contains all and only the subsets of Q[w] with a rest containing less than n positions, i.e. the subsets of the form {0, . . . , p} ∪ X with p < \|w\| − 1 and \|X \| ≤ n. By construction, the n-sets sublattice of w is complete. It contains O(\|w\|n ) states and O(\|w\|n+1 ) transitions. We adapted Algorithm 2 to compute the n-sets sublattice of w in O(\|w\|n+1 ) time with the same amount of memory space. 6.2 `-shift expectation We are now interested in a fast way for finding an efficient w-strategy in a given complete sublattice. For all integers ` and all states s of a sublattice U, the `-shift expectation of s is defined as the greatest shift expectation one could possibly get in ` steps in U by starting from s, conditioned on starting from s, while parsing a text following an iid model π. Namely, the `-shift expectation is computed following the recursive formula: [w] • ES0 [s] = 0, • for all ` > 0, [w] ES` [s] = max i∈Tr(s) X [w] π(x) γs[w] (i, x) + ES`−1 [δs[w] (i, x)] x∈A [w] where Tr(s) = {i ∈ s \| δs (i, x) ∈ U for all x ∈ A}. The `-shift expectation of a complete sublattice U is well defined and can be computed in O(`T ) time, where T is the number of transitions of the sublattice U and by using O(\|U\|) memory space. 17 0, b\|1 ∅ ∅ 2, b\|0 2, b\|3 2, a\|2 0, a\|0 2, b\|3 0, a\|3 2, a\|2 1, a\|1 {0} {2} {0} 0, b\|3 2, b\|0 2, a\|2 2, a\|2 1, b\|0 1, b\|0 1, a\|1 {0, 1} {0, 2} 1, a\|1 1, b\|3 {1, 2} {0, 1} a b ∅ ∅ 2, b\|3 {0} 2, a\|2 2, b\|3 1, b\|0 0, b\|2 2, a\|2 1, b\|0 0, b\|2 1, a\|1 1, a\|1 {1} 1, b\|3 {0} 2, b\|0 1, a\|1 {1} 2, a\|2 1, b\|0 0, a\|0 0, a\|0 {0, 1} 1, a\|1 {0, 2} {0, 1} c d Figure 5: Four complete sublattices extracted from the position lattice of abb. Sublattice a (resp. b) leads to the strategy where the next-position-to-check is always the smallest (resp. the greatest) relative position unchecked. Sublattice c (resp. d) leads to the strategy at the top (resp. at the bottom) of Figure 3. 18 We finally extract a w-strategy from U by setting the next-position-to-check of all states s ∈ U to X [w] arg max γs[w] (i, x) + ES`−1 [δs[w] (i, x)] . i ∈Tr(s) x ∈A 6.3 K-Heuristic The K-Heuristic combines the two approaches above in order to compute a w-strategy in a time polynomial with the length of the pattern. Being given an order K ≥ 1, we start by computing the K-sets sublattice of w, thus in O(\|w\|K+1 ) time. In order to select a w-strategy from the K-sets sublattice, we next compute the (K + 1)-shift expectation of all its states and extract a w-strategy as described just above. This computation is performed in O(K\|w\|K+1 ) time, since the number of transition of the sublattice is O(\|w\|K+1 ), by using O(\|w\|K ) memory space. Let us remark that the order ` of the `-shift expectation does not have, a priori, to be strongly related to the order K of the K-sets sublattice on which it is computed. By experimenting various situations, we observed that considering an order greater than K + 1 generally does not improve much the performances, whereas the strategies obtained from `-expectations with ` smaller than K may be significantly slower. The K-Heuristic returns a w-strategy in O(K\|w\|K+1 ) time by using O(\|w\|K ) memory space. We insist on the fact that the K-Heuristic generally does not return the fastest strategy, even if K > \|w\|. However, we will see in the next section that it performs quite well in practice. 7 Evaluation We shall compare the approaches introduced in Sections 5 and 6 with selected pattern matching algorithms. The comparison is performed, first, from a theoretical point of view, by computing their asymptotic speeds under iid models, and second, in practical situations, by measuring their average speed over real data. The average speed with regard to a pattern w, of an algorithm or a matching machine on a text t is the ratio of \|t\| to the number of text accesses performed by the algorithm to search w in t. We are also interested in to what extent taking into account the frequencies of the letters of an iid model or a text, for determining the Fastest and the KHeuristic strategies, actually improves their asymptotic or their average speeds. To this purpose, we compute the Fastest and the K-Heuristic strategies from the uniform iid model. Next, we test their efficiency in terms of asymptotic speed under a non-uniform iid model and in terms of average speeds on data with non-uniform frequencies of letters. 19 7.1 Pre-existing pattern matching algorithms More than forty years of research have already led to the development of dozens algorithms. We selected the 9 ones below for our evaluation: 1. Naive [6], 2. Morris-Pratt [6], 3. Knuth-Morris-Pratt [15], 4. Quicksearch [23], 5. Boyer-Moore-Horspool [13], 6. TVSBS [24], a“right-to-left” algorithm in which shifts are given by a badcharacter rule [5, 23] taking into account the two letters at distances \|w\|−1 and \|w\| from the current position of the text, 7. EBOM [9], a version of the Backward Oracle Matching algorithm [1] which also uses a “bad two-characters” rule, 8. HASHq [26], which implements the Boyer-Moore algorithm on blocks of length q by using efficient hashing techniques [14] (our tests are performed with q = 3), 9. FJS [11], which combines the ideas of Knuth-Morris-Pratt [15] and Sunday [23] algorithms. Algorithms 1 to 5 are classics. The last four ones were chosen for being known to be efficient on short patterns and small alphabets [10], a situation in which the determination of the fastest strategy is feasible. Let us remark that the order of the w-matching machine associated to TVSBS is equal to \|w\|, thus greater than that of the Fastest strategy that we compute. The transformation into matching machines was implemented for a few other pattern matching approaches, for instance the SA algorithm (the Baeza-YatesGonnet algorithm) based on bitwise operations [2], or the string-matching automaton [7]. Since the asymptotic and average speeds of these two algorithms are exactly 1, whatever the pattern, the model and the text, there is no point in displaying them. 7.2 Results We shall evaluate: • the pre-existing pattern matching algorithms presented in Section 7.1, • the 1- 2- and 3-Heuristics and • the Fastest strategy (each time it is possible). 20 ra tt st Fa st e ris tic eu ris tic 3H eu ris tic 2H 0.78 0.69 0.59 0.71 0.66 0.56 0.56 0.76 0.76 0.56 0.56 0.66 0.71 0.59 0.69 0.78 0.72 0.39 0.54 0.50 0.61 0.50 0.39 0.62 0.62 0.39 0.50 0.61 0.50 0.54 0.39 0.72 0.93 0.73 0.62 0.69 0.62 0.73 0.67 0.73 0.73 0.67 0.73 0.62 0.69 0.62 0.73 0.93 0.52 0.52 0.53 0.53 0.54 0.54 0.53 0.53 0.53 0.53 0.54 0.54 0.53 0.53 0.52 0.52 1.50 1.37 1.19 1.30 1.23 1.22 1.27 1.47 1.47 1.27 1.22 1.23 1.30 1.19 1.37 1.50 1.69 1.52 1.33 1.43 1.34 1.33 1.31 1.59 1.59 1.31 1.33 1.34 1.43 1.33 1.52 1.69 1.80 1.60 1.35 1.54 1.38 1.36 1.34 1.64 1.64 1.34 1.36 1.38 1.54 1.35 1.60 1.80 1.83 1.60 1.37 1.56 1.38 1.43 1.34 1.69 1.69 1.34 1.43 1.38 1.56 1.37 1.60 1.83 V eu 1H as hq H 1.18 1.18 0.73 0.73 0.73 0.73 0.94 0.94 0.94 0.94 0.73 0.73 0.73 0.73 1.18 1.18 O EB M S T SB FJ 0.98 0.51 0.51 0.69 0.63 0.50 0.55 0.75 0.75 0.55 0.50 0.63 0.69 0.51 0.51 0.98 S H or sp o ol -P rc h ris 1.00 0.94 0.89 0.84 0.80 0.80 0.73 0.70 0.70 0.73 0.80 0.80 0.84 0.89 0.94 1.00 ui Q ck se a t hM or ra t 0.70 0.76 0.76 0.76 0.73 0.70 0.70 0.70 0.70 0.70 0.70 0.73 0.76 0.76 0.76 0.70 nu t K ris -P M or 0.53 0.53 0.53 0.53 0.53 0.53 0.53 0.53 0.53 0.53 0.53 0.53 0.53 0.53 0.53 0.53 ai ve N aaaa aaab aaba aabb abaa abab abba abbb baaa baab baba babb bbaa bbab bbba bbbb Table 1: Asymptotic speeds for the patterns of length 4 on {a, b} under the uniform model. 7.2.1 Asymptotic speed The asymptotic speeds are computed for texts and patterns on the binary alphabet {a, b}. Table 1 displays the asymptotic speeds for all the patterns of length 4 on iid texts drawn from the uniform distribution. As expected, the strategy computed with the brute force algorithm (last column) is actually the fastest, but the speeds of the 1-,2- and 3-Heuristics are very close. The pre-existing algorithms are outperformed by all our approaches (even by the 1-Heuristic) for all the patterns. We observe that the Naive, Morris-Pratt and Knuth-Morris-Pratt algorithms have asymptotic speeds always smaller than 1. One cannot expect them to be faster since, by construction, they access all the positions of a text at least once. In the following, we will not display their speeds, nor that of Quicksearch, for they are always smaller than at least one of the other preexisting algorithms. The full tables can easily be re-computed by using our software. Table 2 displays the asymptotic speeds with regard to the same patterns as Table 1, but under the iid model (πa , πb ) = (0.1, 0.9). This table shows the asymptotic speeds of the K-Heuristics and the Fastest strategies computed with regard to an uniform iid model (the columns starting with “Unif.”). The strategies such obtained are not optimized according to the letter probabilities of the model. They may be used as general purpose approaches, while the strategies obtained from the model probabilities will be called adapted below. Overall, 21 1 1H 2H eu ris tic 3H eu ris tic Fa st es t U eu r ist ic st ic ris t U ni f. ni f. Fa st e ist ic 3H eu ic eu r ris t 2H 1H eu 0.67 0.66 0.66 0.63 0.66 0.63 0.61 0.46 0.66 0.65 0.63 0.54 0.63 0.54 0.36 0.19 f. 1.49 1.37 1.24 0.70 1.24 1.17 0.63 0.31 1.37 1.18 1.17 0.55 0.70 0.55 0.31 0.27 ni 1.68 0.27 1.55 0.29 1.63 0.28 0.82 0.31 1.33 0.22 1.16 0.21 1.06 0.23 0.73 0.24 U 1.97 0.53 0.87 0.53 1.24 0.54 0.87 0.57 1.49 0.37 0.80 0.39 1.10 0.31 0.90 0.27 f. 3.30 1.77 0.91 0.38 1.67 0.85 0.93 0.33 2.50 1.34 0.91 0.38 1.67 0.85 1.00 0.35 U ni EB O M H as hq T V SB S ol FJ S po or s H aaaa aaab aaba aabb abaa abab abba abbb baaa baab baba babb bbaa bbab bbba bbbb 3.02 2.43 1.77 1.34 1.80 1.29 1.06 1.08 2.44 1.75 1.09 1.03 1.09 1.00 1.02 1.03 3.46 2.55 2.14 1.71 2.14 1.78 1.31 1.12 2.60 1.75 1.35 0.91 1.72 0.83 1.09 1.03 3.50 2.55 2.14 1.77 2.17 1.68 1.51 1.14 2.61 1.75 1.44 1.04 1.84 1.06 1.17 1.05 3.50 2.55 2.14 1.77 2.16 1.64 1.54 1.15 2.61 1.75 1.74 1.05 1.83 1.08 1.17 1.05 3.02 2.43 1.77 1.74 1.80 1.42 1.30 1.08 2.44 1.75 1.09 1.04 1.09 1.01 1.08 1.03 3.47 2.60 2.19 1.79 2.15 1.80 1.73 1.10 2.60 1.75 1.78 1.04 1.72 1.08 1.16 1.03 3.50 2.60 2.19 1.80 2.18 1.80 1.80 1.14 2.61 1.75 1.84 1.04 1.84 1.08 1.24 1.05 3.50 2.61 2.19 1.80 2.18 1.81 1.80 1.15 2.61 1.75 1.84 1.05 1.84 1.08 1.24 1.05 Table 2: Asymptotic speeds for the patterns of length 4 on {a, b} under the iid model (πa , πb ) = (0.1, 0.9). our methods are faster than the pre-existing algorithms, with a few exceptions: Horspool is faster than the 1-Heuristic for two patterns ending with the rare letter a: aaaa and baaa. And EBOM is faster than the 1-Heuristic for searching baba. The K-Heuristics and the Fastest strategies computed with regard to an uniform iid model have asymptotic speeds smaller than their counterparts obtained from the actual probabilities of the text model (here highly unbalanced). Nevertheless, the uniform approaches still perform quite well, notably better than the pre-existing algorithms, except for the uniform 1-Heuristic and the same patterns as above. Considering longer patterns leads to similar observations. Table 3 shows the asymptotic speeds obtained for random patterns of length 10. The 3-Heuristic outperforms all the others approaches (the Fastest strategy cannot be computed for this length). The (uniform) 1-Heuristic is slower than algorithms such EBOM or Hashq. But both the uniform 2- and 3-Heuristic overall perform better than the pre-existing algorithms, though they are slightly slower for a few patterns. 7.3 Average speed Our data benchmark consists in the Wigglesworthia glossinidia genome, known for its bias in nucleotide composition (78% of {a, t}), and the Bible in English from [10]. Table 4 displays the average speeds of patterns randomly picked from the data. Let us remark that we are now dealing with real texts, which are not iid. In particular, the Fastest strategy could possibly be outperformed (this is 22 1 ris tic 3H eu 1H r is eu tic r is t i 2c H eu ris tic 3H eu ris tic H U U U ni f. ni f. 2- H eu ris tic eu 1H EB O as hq M S SB V T S FJ ni f. l oo or sp H babbbaabab ababbbbbab aaabaaaaba bbbabbabab bbabaabbab baabbaaaaa abbbababbb baabbbabba baabbaabab bbbbababbb 0.85 0.70 0.91 0.84 0.80 4.10 0.31 0.90 0.85 0.31 0.42 0.55 0.87 0.27 0.31 2.17 0.58 0.81 0.37 0.26 0.22 0.29 3.01 0.24 0.22 1.86 0.32 0.68 0.22 0.20 1.42 0.77 3.93 1.45 2.21 2.40 1.32 1.39 2.22 0.95 1.57 0.85 2.61 1.77 1.92 2.45 1.20 1.14 2.29 0.84 1.09 1.19 3.04 1.02 1.15 1.99 1.08 1.61 1.77 1.03 1.98 1.34 4.78 1.05 1.41 4.03 1.33 1.12 2.21 1.05 1.54 1.47 4.92 1.72 1.69 4.72 1.52 1.63 2.26 1.10 1.22 1.43 3.04 1.02 1.15 1.99 1.35 1.73 1.77 1.03 2.38 1.87 4.79 1.29 2.14 4.06 1.90 2.44 3.15 1.20 2.71 2.34 5.27 2.30 3.02 4.78 2.29 2.78 3.54 1.50 ic Fa st es 1t H eu ris tic 2H eu ris tic 3H eu ris tic Fa st es t U ni ni f. U f. 3H eu r ist ist ic eu r H 2- 1H U ni f. U ni f. hq as H O M EB SB S 1.17 1.69 1.58 2.82 1.53 2.47 1.36 1.67 1.92 1.07 3.16 3.20 3.53 2.84 3.04 3.13 3.15 2.72 2.73 3.35 0.77 1.11 0.85 1.70 0.77 1.56 0.70 1.12 0.82 0.87 2.02 1.91 2.18 1.78 1.59 1.76 2.02 1.81 1.80 1.97 0.74 1.28 0.66 1.50 0.71 1.45 0.76 1.27 0.93 0.89 1.85 1.78 1.87 1.70 1.48 1.60 1.85 1.69 1.69 1.73 1.04 1.10 0.92 1.43 1.05 1.23 1.26 1.09 1.34 1.06 1.39 1.42 1.48 1.42 1.45 1.45 1.46 1.35 1.36 1.49 0.60 0.63 0.58 0.63 0.63 0.63 0.65 0.63 0.65 0.63 0.66 0.66 0.66 0.66 0.66 0.67 0.66 0.65 0.66 0.66 1.46 1.64 1.57 2.72 1.74 2.12 1.67 1.64 1.97 1.75 2.88 3.01 3.50 2.89 2.89 3.01 2.96 2.94 2.94 3.23 1.70 2.16 1.76 3.03 1.87 2.67 1.77 2.15 2.11 2.04 3.23 3.28 3.60 2.99 3.05 3.16 3.22 3.03 3.04 3.53 1.74 2.16 1.84 3.08 1.83 2.77 1.81 2.14 2.09 1.87 3.23 3.25 3.60 3.03 3.06 3.16 3.22 2.87 2.87 3.53 1.80 2.16 1.84 3.09 1.87 2.77 1.85 2.14 2.11 2.04 3.24 3.28 3.62 3.06 3.01 3.11 3.22 3.05 3.06 3.53 V T S FJ atat tatg aaat tccc caat aacc acta tatc gtga gatt he m , to usal le t hem are at d f th r th fede H or sp o ol eu r ist ic Table 3: Asymptotic speeds for some patterns of length 10 on {a, b} (drawn from the uniform distribution) under the iid model (πa , πb ) = (0.1, 0.9). 1.46 1.64 1.57 2.72 1.74 2.12 1.67 1.64 1.97 1.75 2.88 3.01 3.50 2.89 2.89 3.01 2.96 2.94 2.94 3.23 1.73 2.16 1.80 3.03 1.94 2.69 1.85 2.15 2.17 2.04 3.24 3.29 3.62 3.07 3.05 3.16 3.22 3.10 3.12 3.53 1.73 2.16 1.90 3.08 1.97 2.77 1.88 2.14 2.18 2.04 3.24 3.29 3.62 3.07 3.06 3.16 3.22 3.10 3.12 3.53 1.81 2.17 1.90 3.09 1.97 2.77 1.88 2.15 2.19 2.05 3.24 3.29 3.62 3.07 3.06 3.16 3.22 3.10 3.12 3.53 Table 4: Average speeds for some patterns of length 4 picked from the benchmark data (the Wigglesworthia glossinidia complete genome and the Bible in English). 23 1 FJ S T EB H as hq U ni f. 1H eu U ni ris f. tic 2H eu U ni ris f. tic 3H eu 1ris H eu tic ris tic 2H eu ris tic 3H eu ris tic 1.8 2.8 3.0 1.4 1.4 2.7 3.0 2.7 1.9 2.1 11.9 13.3 11.9 12.4 14.8 10.1 16.2 12.8 12.4 12.1 1.0 1.8 1.7 1.0 1.0 1.6 1.7 1.1 1.1 1.3 5.9 7.2 6.0 6.1 7.5 5.8 8.4 6.8 5.5 6.3 2.2 3.3 2.0 2.1 1.7 3.7 2.8 2.3 1.9 2.9 8.0 8.4 8.9 8.1 9.0 7.6 9.6 8.6 8.0 9.4 6.1 7.2 6.2 7.0 6.1 8.3 6.3 7.4 6.4 6.8 11.7 12.6 13.2 11.7 12.0 11.7 12.8 11.9 11.7 12.8 4.7 5.2 4.4 4.4 5.0 5.4 4.7 5.5 4.9 4.7 8.6 8.7 8.8 8.6 8.5 8.6 8.8 8.5 8.8 8.6 tccttatgtaaaatataaatgtagcaattt aaaagaaccccggcgaggggagtgaaatag aattttcaactaatattaaaccacgttctg aaaggtccattaagtattactatcacagca agatttgcgtgatttaaaataatcatctaa ataggaaaagattggattaaactagatatg at the mount called the mount ith Israel, to wit, with all t esus going up to Jerusalem too them, as they were able to hea o in Osee, I will call them my things are come upon thee, the Syria, that dwelt at Damascus, full of darkness. If therefor e it: for there is no other sa g, Syria is confederate with E S O M V SB sp oo l H or tggataaaaatttgttattaccatatctat cttctttaattatgttttctatttcttttt gttctatttgttggagatttaaaataatta tcctactttaacctctaaatgtcccttatt 2.3 2.5 2.6 2.5 2.2 3.4 2.0 2.3 2.4 2.2 8.0 8.9 9.5 7.8 9.0 8.5 10.4 9.4 7.7 9.8 4.6 5.0 5.3 5.5 4.7 6.9 4.8 5.1 4.9 5.1 17.2 17.1 18.0 16.5 18.5 16.3 19.3 18.5 16.4 18.6 7.5 7.9 8.0 8.4 7.5 11.3 8.1 8.1 7.5 8.3 18.0 18.0 18.5 17.9 19.3 17.7 20.0 18.9 17.6 18.9 2.3 2.6 2.7 2.6 2.3 3.5 2.2 2.5 2.5 2.3 8.0 9.1 9.4 7.8 8.9 8.5 10.4 9.4 7.7 9.8 5.5 5.3 6.1 6.5 5.5 9.3 5.6 6.6 5.8 5.9 17.6 18.0 18.1 16.6 18.8 16.8 19.6 18.7 16.9 18.6 8.9 8.5 9.1 9.9 8.5 12.5 9.7 10.2 9.1 9.7 18.4 18.8 18.7 18.0 19.3 18.1 20.1 19.1 17.9 18.9 Table 5: Average speeds for some patterns of length 30 picked from the benchmark data (the Wigglesworthia glossinidia complete genome and the Bible in English). not observed on the benchmark data). The 2- and 3-Heuristics, uniform and adapted, are faster than the pre-existing algorithms for all the patterns, whereas the 1-Heuristic is sometimes slightly outperformed by Horspool. Horspool is almost as fast as our approaches on the Bible while being sometimes significantly outperformed on the Wigglesworthia glossinidia genome. The average speeds are overall greater on the Bible than on the DNA sequence. In both cases, we do not observe a wide performance gap between the uniform and the adapted approaches, though our benchmark data are far from following an uniform iid model. Let us remark that the 2- and 3-Heuristics have almost the same performances both in the uniform and the adapted cases. Table 5 shows the averages speeds with regard to patterns of length 30. The average speeds on the Bible are about twice those on the Wigglesworthia glossinidia genome. One actually expects the speed to be greater in average on texts with large alphabets, since the less likely the match between two symbols, the greater the shift expectation per iteration. Again the 3-Heuristic, uniform or adapted, outperforms the pre-existing algorithms. The speeds of the 3-Heuristic and of the 2-Heuristic differ in a greater amount than with patterns of length 4 for the Wigglesworthia glossinidia genome, and, to a smaller extent, for the Bible. 24 1 8 Discussion In practical situations and though they do not take into account the letter frequencies, the uniform K-Heuristics and the uniform Fastest strategy perform generally almost as well as their adapted counterparts. The greatest difference observed is for the patterns of length 30 on the Wigglesworthia glossinidia genome (Table 5) and is relatively small. We do observe a notable amount of difference for the quite extreme case of the asymptotic speed under the iid model (πa , πb ) = (0.1, 0.9). But even for these frequencies, the uniform approaches show greater asymptotic speeds than any of the selected pre-existing algorithms. The 3-Heuristic has very good results whatever the pattern or the text. There is no situation for which the performances of the 2-heuristic are far from the best. On the contrary, the performance ranking of the pre-existing algorithms depends heavily on the patterns and on the texts or the model. For instance, Horspool may perform very well, even almost optimally, for some patterns and texts or models while its speed may completely plummet in other situations. The question of selecting the most efficient order of K-Heuristic still deserves further investigations. A basic answer could be “the greater, the better” but we should take into consideration that an higher order of heuristic comes with an increased computational cost. After some experiments, we observed that the asymptotic speed the K-Heuristic tends to stop improving beyond a certain rank. For instance, the difference in average speed between the 2- and 3-Heuristics for patterns of length 4, both on the genome and on the Bible, probably does not justify the computational cost of the 3-Heuristic, while it is worth to use the 3-Heuristic rather than the 2-Heuristic for searching patterns of length 30 in the Bible (not that much for the Wigglesworthia glossinidia genome). The best trade-off for the order of the K-Heuristic depends on the pattern (notably its length) and on the text features (in particular the alphabet size and the letter frequencies). It is certainly possible to obtain efficient heuristic with a lower computational cost than for the K-Heuristic. Since in standard situation, the length of the text is much greater than that of the pattern, there is no real reason for considering only pattern matching algorithms with linear pre-processings of the pattern. In the extreme case where the texts are arbitrarily long with regard to the patterns, any pre-processing, i.e. whatever its computation time, would be beneficial as soon as it improves the overall speed. Authors’ contributions Gilles Didier provided the initial idea, led the software development and wrote all the manuscript but the section Evaluation. Laurent Tichit collaborated on the software development, ran the tests and wrote the section Evaluation. Both authors read, edited and approved the final manuscript. 25 References [1] C. Allauzen, M. Crochemore, and M. Raffinot. Efficient experimental string matching by weak factor recognition. In Combinatorial Pattern Matching, pages 51–72. Springer, 2001. [2] R. Baeza-Yates and G. H. Gonnet. A new approach to text searching. Communications of the ACM, 35(10):74–82, 1992. [3] R. A. Baeza-Yates and M. Régnier. Average running time of the BoyerMoore-Horspool algorithm. Theoretical Computer Science, 92(1):19 – 31, 1992. [4] G. Barth. 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Intrinsic Point of Interest Discovery from Trajectory Data Matthew Piekenbrock Derek Doran Dept. of Computer Science & Engineering Kno.e.sis Research Center Wright State University, Dayton, OH, USA [email protected] Dept. of Computer Science & Engineering Kno.e.sis Research Center Wright State University, Dayton, OH, USA [email protected] arXiv:1712.05247v1 [cs.AI] 14 Dec 2017 Abstract This paper presents a framework for intrinsic point of interest discovery from trajectory databases. Intrinsic points of interest are regions of a geospatial area innately defined by the spatial and temporal aspects of trajectory data, and can be of varying size, shape, and resolution. Any trajectory database exhibits such points of interest, and hence are intrinsic, as compared to most other point of interest definitions which are said to be extrinsic, as they require trajectory metadata, external knowledge about the region the trajectories are observed, or other application-specific information. Spatial and temporal aspects are qualities of any trajectory database, making the framework applicable to data from any domain and of any resolution. The framework is developed under recent developments on the consistency of nonparametric hierarchical density estimators and enables the possibility of formal statistical inference and evaluation over such intrinsic points of interest. Comparisons of the POIs uncovered by the framework in synthetic truth data to thousands of parameter settings for common POI discovery methods show a marked improvement in fidelity without the need to tune any parameters by hand. ACM Reference format: Matthew Piekenbrock and Derek Doran. 2016. Intrinsic Point of Interest Discovery from Trajectory Data. In Proceedings of ACM Conference, Washington, DC, USA, July 2017 (Conference’17), 10 pages. DOI: 10.1145/nnnnnnn.nnnnnnn 1 Introduction The development and deployment of location acquisition systems have enabled large scale capturing of ‘movement’ or ‘trajectory’ data from people, cars, and other objects. Technologies like global positioning systems (GPS), global system for mobile communications (GSM), wide area motion imagery (WAMI), and radio-frequency identification (RFID) allow organizations and governments to collect and exploit trajectory patterns in many scenarios. More recent initiatives (e.g. Uber’s Movement1 and IBM’s Smarter Cities2 programs) have even made such data available to the either the public or city planning experts at large. With the rise in importance of this 1 https://movement.uber.com/cities 2 https://www.ibm.com/smarterplanet/us/en/smarter cities/overview/ Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]. Conference’17, Washington, DC, USA © 2016 ACM. 978-x-xxxx-xxxx-x/YY/MM. . . $15.00 DOI: 10.1145/nnnnnnn.nnnnnnn data comes prevalent use of Geographic Information Systems (GIS) and related platforms such as ArcGIS3 and Mapbox4 . Other related use cases of GIS information have also emerged for surveillance [9] and location-based service (LBS) applications [36]. In many of these applications, trajectory data is exploited for knowledge acquisition tasks [17], the integration of movement patterns to uncover “patterns of life” over a region [43], to expand situational awareness in crises [40], and to support the value added by a LBS application [13]. In many of these knowledge acquisition tasks, the notion of a “location” or “point of interest” (POI) is foundational to understanding the entirety of the common space in which the data are observed [31]. For example, mapping systems must know the position and geometry of locations for navigation and automated guidance control purposes. In LBS applications, the POIs and metadata such as their popularity (e.g. ‘star-rating’) are necessary to provide useful location recommendations [13, 30, 44]. Because POIs are not available from ‘raw’ trajectory data captured by location acquisition systems, they are often extrinsically defined by gazetteers such as Google Places, FourSquare, GeoNames, or OpenStreetMap. Yet external sources of location data present many difficulties when faced with the problem of understanding how a given trajectory dataset relates to the underlying geographical area where it was observed. For example, many gazetteers store varying types of either POI metadata or POI relational data, allowing gazetteer-derived information to present a source of bias. Furthermore, relying on gazetteers explicitly defines the set of POIs that exist in a given geographical region. When there is disagreement on this definition, analysis becomes difficult. Furthermore, with POIs defined a priori, one is faced with the problem of “fitting” observed trajectory data to models defined by such POIs, many of which may or may not be relevant to the given data at hand. For example, it may be desirable for a city-planner gathering movement (trajectory) data following a public event to discover ‘bottleneck’ congestion areas like parking lots, roads, or sidewalk segments for the purpose of traffic analysis. In this situation, it would more useful to discover POIs directly from the data itself during the event, but such geographical POIs may not be available in a gazetteer. To over come these challenges, this paper investigates the POI discovery problem in the most generic context possible. We ask: given only trajectory data, without access to gazetteers, can we infer subregions within a geospace that are “interesting” enough to call it a POI? We seek intrinsic POIs, which are POIs recoverable without the use of a gazetteer, are completely defined by observed movement patterns, and can be used for any domain-specific application and at any scale (e.g. from movements within a building to movements across an entire city region). To make such a definition meaningful, 3 https://www.arcgis.com/ 4 https://www.mapbox.com/ we build off recent theoretical work in density-based clustering and introduce a data-driven, statistically rigorous definition of a POI applicable to trajectory data of any (and even mixed) resolution. The definition follows from a recent minimax analysis of the consistency of hierarchical density superlevel set estimators. We use this definition to present a parameter-free framework for extracting intrinsic POIs, i.e. yields an optimal unsupervised solution without ad hoc parameter-tuning.5 A comparative analysis is performed on realistic simulations involving both vehicle and pedestrian traffic. Validation results show marked improvements in fidelity against several state of the art (SOTA) algorithms. Of interest to the authors, the simulation settings, the resulting traffic data, the validation code, and the framework itself is all completely reproducible and open source, available online.6 2 Point of Interest Discovery This section provides preliminary information about the POI discovery problem, and provides context and definitions for this work. It then formally defines a POI and (subsequently) an intrinsic POI, and the framework their discovery. 2.1 Preliminaries We consider a trajectory database of discrete, time-indexed spatial data having at least the 3-tuple of attributes (<object id>, <spatial component>, <temporal component>) This minimal amount of information implies a trajectory for an object of the form: ∆t 1 ∆t 2 ∆t n−1 T = p1 −−−→ p2 −−−→ . . . −−−−−→ pn (1) where p1 , p2 , ..., pn are chronologically ordered spatial coordinates. In the geographical sense, these spatial components are often defined by a <latitude> and <longitude> pair, but in practice could be from any coordinate system. Such representations require trajectory pattern mining techniques [42], or techniques that seek to mine common spatiotemporal patterns across trajectories to assert significance over areas where trajectory patterns emerge. Mined patterns in trajectories are often referred to as mobility patterns, characterizing some specific trajectory quality of interest, such as heading, stopping rate, velocity, rotation, curvature, or shape [5]. Such mobility patterns exhibit properties that make the formal retrieval of significant areas challenging. For example, if the timespan of an observed trajectory is long, the processes driving the mobility pattern may be non-stationary (e.g. road traffic that changes due to construction, or congestion effects due to time of day shifts in the work schedule). There may also be paths of objects that are transient (some areas are never traveled to more than once). Furthermore, the spatial components in trajectory data often have a high degree of autocorrelation, breaking assumptions of independence [13]. A variety of models have been proposed to handle thee situations, largely focusing on estimating individual trajectory statistics under these assumptions. This includes, for examples, adaptive Kalman 5 We see this as a necessary form of usability, an important feature to have in the modern clustering era. It is well-known that having several sensitive, real-valued parameters results in combinatorial explosion of the parameter space of an algorithm, resulting in the need for the user to use one or more parameter-tuning methods to arrive at a solution that befits the application. 6 ¡Anonymized for review purposes.¿ filters for vehicle navigation [20], state-space models [16], and trajectory path uncertainty models [34]. The knowledge mined from individual trajectories says little of the macroscopic patterns driving such trajectory observations. Rather than focusing on the statistics of individual trajectories, collective models preprocess the trajectory data to extract characteristics across a swath of trajectories. Such preprocessing is desirable, as it discards highly autocorrelated data representing redundant information in favor of aggregating trajectory positions into observations of significance. Examples of this preprocessing scheme include extracting “semantically enriched” points that intersect known geographical regions [1], aggregating trajectory positions as stay points using supplied spatial and/or temporal thresholds [43], or processing trajectory data into groups using some convex combination of spatial, temporal, and semantic similarity kernels [25, 39]. In a collective model, we refer to the ‘important’ or semantically meaningful data samples aggregated from trajectory points as exemplar positions, or simply, exemplars: Definition 1. Exemplar Consider a sample of n discrete points X n ⊂ Rd that constitute a trajectory T . Define an aggregation function α : P(X n ) 7→ Rd that maps any subset of points (e.g. a trajectory segment) in T to a set of exemplar positions Σ ⊂ Rd . The aggregation function of choice depends on the intent of the analysis. For example, consider an urban environmental study that defines α as a mapping of some isolated trajectory segment {pk , pk +1 , ..., pk +l } to the mean coordinate of the segment if the speed of the object traveling from pk to pk +l exceeds a certain threshold. Groups of these exemplar positions may determine “highemission” zones in a city [4]. Alternatively, if the traffic is made of pedestrians, such groups may represent tourist attraction areas, the popularity of which are useful for LBS applications [44]. It is not difficult to find this type of trajectory preprocessing in geospatial applications, and the grouping of them is foundational to countless tasks in trajectory mining [1, 24, 39, 43–45]. We generalize this preprocessing step by referring to it as “exemplar extraction.” An important aspect of exemplar extraction is to choose an aggregation function that befits the intent of the analyst and thus satisfies a study’s interpretation of “interesting.” This is inevitably application-specific, and the proposed framework is agnostic to the specific form of aggregation used, thus it is irrelevant to bestow a particular interpretation of what “interesting” means. We consider a more concrete and practical definition using a popular type of aggregation in Section 3. 2.2 Defining a point of interest Under the premise that exemplars represent meaningful aggregations of observations from a trajectory data source, it is natural to define a POI as a region of exemplars. We seek a definition of such regions with a statistical (rather than heuristic) foundation as a means of reflecting the naturally occurring structure within the data. Towards this end, we define a POI as a contiguous, high density region of exemplars. To formalize this definition, we follow the notation of Chaudhuri et. al [10]. Let X be a subset of Rd and define a path as a function P : [0, 1] → S where S ⊂ X. Also denote the equivalence relation C Figure 1: Illustrating the cluster tree hierarchy and its interpretation of POIs. Consider an estimated density (right panel) of exemplar positions extracted from trajectories in a geospace (middle, bottom panel). A POI is a geospatial region inhabited by exemplar positions at some density threshold λ, with the number of the POIs extracted depend on this scale parameter setting (left panel). Higher λ limits a POI to being specific and small, and could cause POIs to be manifested by random noise or be overfitted to a particular set of observations. Low λ defines POIs as very broad areas of low exemplar position density. The cluster tree hierarchy (left panel) summarizes the set of exemplar positions representing a POI at every density threshold, thus capturing the entire collection of POIs over a common area (middle panel, upper layers). as connected, where xCy iff P(0) = x and P(1) = y. Then C partitions S into connected components or clusters. Each component represents an area of high density and is called a high density cluster: Definition 2. High density clusters For a density function f on Rd , consider a partitioning: {x : f (x) ≥ λ}, for some λ > 0 (2) where λ is called the level, or high density threshold, parameter. Then all maximally connected components in this set are high density clusters at density level λ. We relate this formal definition to the trajectory mining domain with the following definition of a point of interest, defined over a extracted set of exemplars. Definition 3. Point of interest Given a set of m exemplars {ε 1 , ε 2 , . . . , εm } ∈ Σ and a fixed “scale” or resolution λ, each high density cluster of such exemplars forms a point of interest at the density level λ. The sets of high density clusters across all values of λ forms a hierarchy often referred to as the cluster tree of the density f [10, 11]. A hierarchical definition of locations is common [44] and matches the intuitive interpretation of a POI. For example, not only may a particular restaurant in a mall food court be a POI, but the food court itself may also be considered a POI, as well as entire mall may be yet another POI. The cluster tree conceptualization formalizes a POI as a maximally connected set of exemplars falling along a higher density area, implying such areas are ‘significant,’ and that such connected exemplars may be related. A visualization of hierarchical POIs and a dendrogram of the corresponding cluster tree is provided in Figure 1. The middle figure demonstrates a high-level view of what a set of trajectories might look like, with the colored dots in the left and middle figures representing exemplars. The right figure demonstrates a density estimate of the positions of these exemplars. That is, when these exemplars are very close to each other, they’re said to have a high density and are thought to be related, constituting a POI—the scale of the density depends on the analysis at hand. A sufficiently low density threshold λ 0 will designate every exemplar as one POI. From this definition, it may seem that any arbitrary density estimator may be used to find high-density clusters: simply estimate the density of every point by kernel density estimation (KDE), and then iterate through all possible values of λ that create distinct high-density clusters. Yet not every estimation will produce the same hierarchy—different kernels (and kernel bandwidths) may result in a completely different hierarchy of high-density clusters, and by extension, a different set of POIs. From the cluster tree perspective, the ideal kernel fn is one that is uniformly consistent (i.e. supx \| fn (x) − f (x)\| → 0 as n → ∞) from a given sample X n . In this case, a model could be fitted with the appropriate kernel and bandwidth parameter, and the would KDE furnish a continuous surface from which a cluster tree and its high-density clusters can be derived [11]. The main issue is that the set of all high-density clusters is not easy to compute for typical density estimates of f [10] and generally require a significant amount of memory to store. This computational inefficiency limits usability for large trajectory datasets, often observed over wide geographical areas and over long periods of time. From the applied perspective, many state-of-the-art approaches find POIs by variants of hierarchical clustering to find groups of exemplars. This has proved useful for application-specific problems [43–45] but they are largely heuristic, i.e. it is common for most clustering algorithms to have unstated or unknown statistical properties, precluding the possibility of formal inference [14]. The framework we introduce therefore examines density-based clustering methods as they are designed to infer a cluster tree [11] without facing the computational hurdles of KDEs. A desirable property of any finite-sample density estimator is some notion of consistency.7 In 1981, Hartigan establsihed a reasonable definition [19], often referred to as Hartigan consistency: Definition 4. Hartigan Consistency Let Cfn be the set of all high-density clusters from the cluster tree. For any sets A, A0 ⊂ X, let An (respectively, An0 ) denote the smallest set of Cfn containing A ∩ X n (respectively A0 ∩ X n ). Cfn is consistent if, whenever A and A0 are different connected components of {x : f (x) ≥ λ} (for some λ > 0), P(An is disjoint from An0 ) → 1 as n → ∞. This consistency definition essentially requires that two disjoint high-density clusters from the unknown, population density (A and A0 ) will also be disjoint components in a given empirical cluster tree (An and An0 ), given enough samples (n). The proposed framework for POI discovery is developed and implemented from the first computationally tractable and provably consistent algorithm that satisfies Hartigan consistency, as analyzed by Chaudhuri et. al [10], to be discussed in the next section. Having a nonparametric model satisfying this notion of consistency is important, as it transforms the unsupervised problem of POI discovery into a formal statistical estimation problem, not only enabling analysis driven by data, but requiring minimal assumptions regarding the nature of the data. Such a relation enables methods of formal statistical inference, allowing one to quantify uncertainty, i.e. to create hypothesis tests to discern “true” POIs as opposed to “false” POIs resulting from random noise or artifacts of low sample sizes, or to create notions of confidence in estimation [11]. Consistent cluster tree estimation: We next motivate a recent cluster tree estimator and discuss its relationship and applicability to POI discovery for the propsoed framework. Recall that an empirical estimate of the cluster tree, applied over exemplars, represents a hierarchy of POIs. Viewed from this perspective, what we propose can be seen as an extension of Chaudhuri et. al’s work on the cluster tree [10] to a trajectory mining context. Consider using Single-Linkage (SL) clustering, an agglomerative scheme that creates a hierarchical representation of clusters using the minimum pairwise distance D between all points, as a tool for clustering exemplars. Beginning with every exemplar x as a singleton, SL iteratively merges exemplars into clusters according to the linkage function: D(x i , x j ) = min d(x i , x j ) x i ,x j ∈X SL clustering is often criticized due to its tendency to create ‘excessive chaining’, wherein two clusters which may have been seen as generally unrelated are amalgamated by chance at a distance threshold that does not reflect the true dissimilarity between the resulting clusters. Hartigan proved SL is a consistent estimator of the cluster tree for densities in R (for d = 1) and is not consistent for any d > 1 [19], implying that any SL cluster that contains all the sample points in A will also contain nearly all sample points in A0 , in probability as n → ∞.8 This is reflected in the geospatial sense as well: that an estimator Θ̂n whose value θˆ is a point estimate of θ is consistent if, as more samples are collected (n → ∞), Θ̂n converges in probability to the true value of the parameter, i.e. plim (Θ̂n ) = θ . 7 Recall n→∞ 8 The condition is related to the “thin bridge” between any two population modes. Fractional consistency was shown for SL if for any pair A and A0 , the ratio of inf {f (x ) : x ∈ A ∪ A0 } to sup{inf {f (x ) : x ∈ P } : paths P from A to A0 } is sufficiently large. Figure 2: SL excessive chaining example. The bottom panel denotes a possible clustering using SL when pedestrians were found to stop between buildings. consider the case where exemplars represent aggregated ‘stops’ within a set of trajectories, a case that we will also consider later in Section 3. If an area is observed long enough, such exemplars should naturally form an area high density in areas where people stop frequently, e.g. within buildings. In such cases, it may be useful to categorize exemplars within their respective POIs (this is done in supervised way applications extract semantic information, see [1] for example). However, it’s also possible that there exist a few stops just outside of such buildings, which SL has a tendency to chain together. An example of this is shown in Figure 2. This discovery motivated efforts to modify SL not only to reduce this chaining to make SL more ‘robust’, but also to achieve (at least) Hartigan consistency for d = 2 and beyond. The first provably consistent estimator, which we consider in this effort, is a generalization of SL referred to as ‘Robust Single Linkage’ (RSL) [10]. Robust Single Linkage: Let X be a subset of Rd . Let k·k denote the `2 norm and let B(x, r ) be a closed ball of radius r around the point x. The RSL algorithm is given in the listing below: Robust Single Linkage Algorithm (1) For each x i set r k (x i ) = inf {r : B(x i , r ) contains k data points}. (2) As r grows from 0 to ∞: (a) Construct a graph G r with nodes {x i : r k (x i ) ≤ r }. Include edge (x i , x j ) if kx i − x j k ≤ αr (b) Let Cfn (r ) be the connected components of G r . The RSL algorithm has two free parameters which need to be set: α and k. SL is equivalent to RSL with the setting α = 1, k = 2. Whereas SL is equivalent to (and can be efficiently computed by) the minimum spanning tree (MST) computed over all pairwise distances, RSL scales these distances by a constant factor α, and only reduces to the MST if the components are restricted from connecting (satisfying {x i : r k (x i ) ≤ r }) within the MST computation. Chaudhuri et. al found that RSL is Hartigan consistent and established finite-sample rates of convergence for all 1 ≤ α ≤ 2, with the optimal rate of convergence with the setting α ≥ 2 [10]. 2.3 Finding intrinsic points of interest Using a consistent cluster tree estimator, such as RSL, on a set of exemplars creates a hierarchical representation of POIs. However, a nested set of multiple solutions is not always desirable, and a ‘flat’ solution (where each point is assigned a single label) may be preferred. A traditional approach in hierarchical clustering, “cutting” the empirical cluster tree at a given density threshold value λ yields a set of high-density clusters Cfn (λ) = {C 1 , C 2 , . . . Cm } that form m POIs, a possible ‘flat’ solution. However, the choice of λ forces all POIs to be of the same scale, requires the user to know which granularity to choose a priori, affecting the size and kinds of POIs discovered. For example, a small λ may define shops in a mall as POIs, while a larger λ may define the mall itself as a POI. It may not be known ahead of time what granularity level is relevant. Furthermore, it is reasonable to expect that relevant POIs exist at multiple levels of granularity, such that a sprawling city park and a small restaurant could both constitute a POI. Thus, it would useful to have some sensible notion of “cluster quality” that can be used (and optimized) as an objective function to discover POIs that are not dependent on the analyst’s choice of λ, and are strongly intrinsic to the geospace itself, e.g. are intrinsic POIs. To capture POIs of any scale and hence satisfy our notion of an intrinsic POIs, we first recall that high-density clusters are contiguous, relatively dense areas of the data space, separated by contiguous and relatively non-dense areas, defined over a working definition of density over a set of exemplars. From a statistical point of view, we can think of a high-density cluster as a set of points with high density around some “neighborhood” or volume of the support. Müller et. al quantify this using a functional called excess of mass [26]: Definition 5. Excess of Mass For a Ci ∈ Cfn (λ) for some value of λ > 0, the excess of mass of Ci is given by: ∫ E(Ci ) = f (x) − λmin (Ci ) dx (3) x ∈C i where λmin (Ci ) represents the lowest density level Ci appears. Initially, this measure seems like a reasonable definition of the “quality” of a clustering within the cluster tree estimate. Considering the definition of a high-density cluster from Equation 2, where a cluster exists along a mode of local maximum of the underlying density, it’s far too likely that a finite-sample estimation may empirically find a mode at a given point x 0 if the data is sparse, allowing an arbitrarily low probability associated x 0 to be classified. A more interesting result would be to associate a high-density cluster with a region that exhibits relatively high probability over a neighborhood. See Müller et. al for visualization, along with a more in depth description of this functional [26]. However, as Campello et. al remark, this measure exhibits monotonic behavior in any direction varying the density-level λ in the hierarchy, and instead propose an alternative, local measure of cluster quality [8]: Definition 6. Relative Excess of Mass For a Ci ∈ Cfn (λ) for some value of λ > 0, the relative excess of mass of Ci is given by: ∫ E R (Ci ) = λmax (x, Ci ) − λmin (Ci ) dx (4) x ∈C i where λmax (x, Ci ) = min{ f (x), λmax (Ci )} is the density level beyond which x is no longer part of Ci , and λmax (Ci ) is the highest density beyond which Ci either becomes disconnected (creating separate components) or disappears (creating singleton clusters). It is important to note that relative excess of mass is defined in terms of λ values associated with a specific cluster, as opposed to a specific clustering. This implies that an ‘optimal’ clustering with respect to the relative excess of mass estimate may not occur at a fixed, global density threshold, but rather as a result of several local density thresholds applied to the hierarchy. Intuitively, if a given cluster Ci contains many points that have high density relative to λmin (Ci ), such a cluster will exist across several thresholds of λ and is thus robust to fluctuations in the scale of analysis. For this reason, the relative excess of mass can be thought of as a measure of cluster ‘stability’ across different density levels, which we posit reflects an intrinsic POI that is innately defined by the dataset independent of density level. Such intrinsic POIs are thus defined as follows: let δi be an indicator equal to 1 if cluster Ci ∈ Cfn represents an intrinsic POI and 0 otherwise. Assign values to these indicators such that the following is maximized: m Õ maximize J = δi E R (Ci ) δ 1, ...,δm subject to i=2 ( δi ∈ {0, 1}, i ∈ {2, . . . , m} exactly one δ (·) = 1 per disjoint branch (5) Where the “per disjoint branch” constraint means that the indicator function δ (·) equals 1 exactly once for all clusters in each path from a leaf node to the root of the cluster tree. The optimization of this objective function is beyond the scope of this paper; we refer to Campello et. al’s cluster extraction method for general cluster hierarchies [7] to solve this optimization, as it was developed alongside an estimator very similar to RSL, is capable of producing an optimal result at several density levels, and accounts for the density thresholds at which points become noise (fall along densities below a given threshold). 3 Experiments and Discussion We next evaluate the proposed framework for intrinsic POI discovery. Because intrinsic POIs do not rely on gazetteers and may manifest themselves in unknown locations, evaluation on real data validated against “ground truth” external knowledge (such as imported location data from sources such as OpenStreetMap or Google Places) is not feasible. A common approach to evaluate clusterings when ground truth is absent is to use an internal cluster validity index (CVI). CVIs include common indices like the Silhouette score, the Dunn Index, and the Calinski-Harabasz criterion (see Arbelaitz [3] for an overview of these techniques and references therein). Recent work recommends validation using multiple CVIs, as they each score different aspects of a clustering such as the ratio of inter- to intra-cluster distances, sum of squares distance to centroid, or graph-theory scores based on similarity [3]. We do not believe scores are informative for intrinsic POI evaluation, as most of these CVIs operate on unrealistic assumptions (e.g. symmetry or convexity of cluster shape, a notion of minimal variance, the existence of a centroid or medoid, etc.). Contrary to these widespread concepts, we do not assume that a cluster of exemplars representing an intrinsic POI will maintain some hyper-spherical or -elliptical shape. Indeed, there are a number of features within a geographical area that may be considered POIs, yet inevitably exhibit arbitrary shapes (e.g. buildings, parks, gathering areas, etc.) and manifest at varying densities (e.g. a busy intersection that is small and concentrated in exemplar density vs. a parking lot that is large and more uniform in density). Following the advice of Guyon and Luxburg et al. [18, 37], we evaluate the efficacy of the framework in the context of its end-use. We use an external validation where “truth” can be defined a priori over simulated data, enabling a direct evaluation of intrinsic POIs against “truly interesting” regions, while ensuring the latent patterns in the generated data mimic the real geospatial dynamics of cars and pedestrians over a region. 3.1 Generating synthetic data To generate synthetic data for evaluation, we turn to the Simulation of Urban MObility (SUMO) software [23]. SUMO is an open source traffic simulation system capable of generating trajectories of many objects of multiple modalities (e.g. car, truck, person, plane, etc.). Given a shapefile that defines avenues for travel (e.g., a road network, a map of footpaths within a university campus, or the floor plan of a mall or large building), SUMO is able to generate trajectories following the avenues provided. Default parameter settings generate traffic and trajectories in ways that satisfy their measured physical properties have been shown to be incredibly accurate [23]. We use SUMO to generate two simulations of both pedestrian and vehicular traffic under different geographical areas: an urban region having a mixture of vehicle and pedestrian traffic (the area surrounding The Ohio State University (OSU)), and a suburban area where pedestrian traffic is more prominent (the area surrounding Wright State University (WSU)). Details about the simulation, the simulation data used in this paper, and the code that produced the resulting evaluation are all publicly available and reproducible online.9 The (RSL) cluster tree framework itself is part of a larger open source effort by the author.10 Simulation configuration: SUMO requires every object to have a trip defined by departure and destination nodes, which SUMO refers to as junctions. Junctions are connected by edges representing a possible travel path. Given a file containing the trip definitions of every object, SUMO dynamically generates routes, or sequences of edges the object travels along to get from departure junction A to its destination junction B. We leave nearly all simulation parameters at their default settings, only modifying simulation length and arrival parameters (binomially distributed arrivals) to generate pedestrian and vehicle demand. Because pedestrian traffic within unrestricted and indoor areas may constitute intrinsic POIs in a realistic setting, and because SUMO can generate only outdoor pedestrian traffic, we extended SUMO to simulate indoor pedestrian traffic as well. Figure 3 illustrates how this extension interplays with vehicular traffic generated by SUMO.11 Shapefiles denoting the location of buildings are first loaded into SUMO (the peach colored regions in Figure 3 inlet). Then, within the shapefile, a random number of pedestrian-only junctions are generated within the building and registered to nearby pedestrian-only edges (such as sidewalks). If a generated track is labeled as a pedestrian and its trip includes a junction contained within the building region (Figure 3; lower right inlet), the pedestrian undergoes a random walk within the junctions generated in the building. This random walk is emulated by choosing a random ordered subset of the generated junctions for a random amount of 9 See the following for simulation details: ¡Anonymized for review purposes.¿ the following package: ¡Anonymized for review purposes.¿ 11 See ¡Anonymized for review purposes.¿ 10 See Figure 3: Top-level view of extending SUMO to support indoor pedestrian traffic. Shapefiles defining buildings are loaded into SUMO and registered as junction. If pedestriantrack visits an attached junction during a trip, the simulator chooses ordered random set of junctions to follow within the building, exiting after a random period of time. time. The pedestrian visits these interior junctions and then travels to an ‘exit junction’ attached to the building polygon, continuing along the original (outdoor) route generated by SUMO. Defining truth: Recall that intrinsic POIs are inferred by exemplars representing the specific mobility pattern of interest. With both vehicular and more realistic pedestrian demand generated, the next step in data generation is to define an aggregation function to extract meaningful exemplars. To give a concrete use-case of the proposed framework, we align our experiment with much of the applied literature related to this topic [28, 42, 45] and extract exemplars representing the “stay points” of an object. A stay point is a position where objects have stopped or significantly slowed down. Extracting such points from simulated SUMO data is trivial, as the true speed of any traveling object is known at any given time. For pedestrian traffic, we extract trajectory points where pedestrians stopped moving. For vehicular traffic, we extract either a) the points where the vehicles stopped moving or b) the slowest point in a vehicle braking sequence using SUMOs exported braking signals, whichever is available. From these stay points (exemplars), we next establish a mapping between each exemplar and its presence within a “true” intrinsic POI, allowing external validation. Since exemplars represent object stopped moving, a natural definition of an intrinsic POI is an assignment defined by the mechanism causing such objects to stop. Specifically, we define a building that pedestrians stop within as a “true” intrinsic POI, as this is very natural and useful grouping. We follow a similar pattern for vehicular traffic, assigning exemplars a common label if stopped at identical intersections, stop signs or stop light, or other junctions. This mechanistic assignment of creating “true” intrinsic POIs has the benefit of not only being tractable (in the sense that SUMO provides this information directly), but also being semantically meaningful in the sense that the mechanisms encouraging objects to stop moving are intrinsic to the geospace. 3.2 Experimental Design To evaluate the fidelity of the POIs extracted by the proposed framework under multiple settings, we run SUMO simulations over the OSU and WSU geospaces with parameter settings reflecting differences between the two regions. These settings are shown in Table 1: SUMO Simulation Parameters Region # Build# Veh. ings # Ped. Region size Sim. Length OSU 70 2,933 2,935 342km2 8 hours WSU 26 2,050 4,327 461km2 6 hours Table 1. The OSU geospace covers a smaller area, has an equal mix of vehicles and pedestrians, and nearly three times as many buildings. The OSU geospace also has a larger number of roadways and traffic intersections where intrinsic POIs involving vehicles may materialize. Being within the main campus, the WSU geospace has a larger proportion of pedestrian traffic, with few roadways for vehicles to traverse and smaller number of buildings pedestrians may visit. Figures 4(a) and 5(a) show what this SUMO generated POIs labeling creates for the OSU and WSU campus areas, respectively. Qualitatively, examination of these clusters appear to be reasonable labels of intrinsic POIs. For example, the clusters representing “true” POIs across OSU in Figure 4(a) finds buildings surrounding the OSU oval quad, particular locations on the ring road around the quad (which tend to be busy OSU intersections for both vehicles and pedestrians), and parking lots around the OSU recreation builds west of the oval to represent POIs. Across WSU in Figure 5(a), the truth POIs represent each of the major buildings around the campus, with particularly complex, separate areas of movement in WSU’s large student union (the yellow points in the large building in the lower left part of the figure). Evaluation Measures: As discussed at the beginning of this section, the unsupervised nature of intrinsic POI discovery make it difficult to carry out a meaningful evaluation of POI discovery methods using internal (not requiring ‘truth’ labels) validation measures. Instead, we consider a multifaceted approach: an external, quantitative evaluation of whether the intrinsic POIs discovered aligns with SUMO generated POIs using the well-known Adjusted Rand Index (ARI) [21], and qualitative evaluation of the quality of the intrinsic POIs our approach unearths as compared to the “true” intrinsic POIs as defined above. The Rand-family of indices were chosen due to their transparency and simplicity—although, whereas the traditional RI measures the proportion of pairwise agreements between two partitions, the ARI also adjusts the score based on the expected value of agreements under the null hypothesis that the agreements were completely random, and thus is what we report. Algorithms Compared: We further compare the fidelity of the intrinsic POIs extracted by the proposed framework against other clustering algorithms commonly used for POI discovery from trajectories. We either downloaded the implementation of, or implemented ourselves, a number of these algorithms for comparison. Aside from RSL, the selected methods includes the well-known density-based algorithms DBSCAN [12] and OPTICS [2], the widespread hierarchical algorithms single linkage (SL), average linkage (AL), and wards criterion (WL) [27], along with the partitioning-like algorithms k-means and CLARA [22]. These algorithms were chosen due to their relevance to this problem, wide-spread availability, known success in the clustering world. Parameter Settings: Clustering algorithms generally require parameter-tuning in order to ‘fit’ to a given data set, but the number and semantics of these parameters often changes with the algorithm used, leaving comparisons between parameter settings difficult. Although most hierarchical algorithms carry no free parameters to create a (hierarchical) set of solutions, they do require either a threshold value (h) or the exact number of clusters to extract (k) to be specified to extract a ‘flat’ clustering. Similarly, k-means and CLARA also require k to be specified a priori. Because the k parameter has the same interpretation in multiple algorithms, we will use k to refer to the number of clusters extracted. Density based algorithms have multiple parameters with interpretations compared to the aforementioned algorithms. For example, DBSCAN requires a minimum cluster size parameter minPts and a distance (or ‘scale’) threshold ϵ to be set. OPTICS, often cited as extension to DBSCAN, is an ordering algorithm that—given a parameter setting for minPts—can be be used to extract either a flat, DBSCAN-like cluster extraction or a simplified hierarchy using a either the a distance threshold ϵ 0 or a reachability-based threshold ξ , respectively. The DBSCAN-like cluster extraction is reported here. RSL requires the setting of α and k, the former relating to scaling the connection radii used to connect components, and the latter to the saliency of cluster estimates. Note that in RSL, k is more similar to the minPts parameter in that it is a minimum neighborhood parameter. The number of clusters is automatically determined by optimized the defined relative excess of mass functional from Section 2. Each algorithm reflects a large set of possible solutions over its parameter setting. Choosing a single parameter setting for evaluation would represent a source of possible bias. Rather, we employ a more comprehensive approach by comparing a wide range of parameter settings for each algorithm. To define these ranges, let seq(x, y, s) denote the sequential range operator, skipping s values in the sequence of integers from x to y. For example, seq(1, n, 1) = {1, 2, . . . , n}, and seq(1, n, i) = {1, 1 + i, . . . , n − i, n}. For the hierarchical clustering algorithms (SL, AL, and WL) the number of flat clusters extracted k is varied in the range seq(2, nt , 1) where nt is the number of “true” POIs assigned by SUMO. We see this as a reasonable strategy, as it gives a better view of how multiple levels extracted from the hierarchy matched the data set as well as how well the merge criterion (or linkage function) collectively captures the true POIs in the geospace. We use the same range to vary k for the k-means and CLARA algorithms. The density based methods DBSCAN and OPTICS are evaluated by first varying minPts, and then (for each value of minPts) by varying the scale parameters ϵ and ϵ 0 , respectively. Recall minPts relates to a minimum neighborhood value that constitutes a cluster. Thus, and to allow the testing to be tractable, we set minPts to reflect the possible sizes of the POIs, along the quantiles qnt = seq(0.10, 0.95, 0.025) corresponding to the the number of exemplars per POI in the SUMO “truth” data. The distance thresholds ϵ for DBSCAN and ϵ 0 OPTICS are also varied along the quantiles seq(0.01, 0.20, 0.01) of the pairwise distances computed over the data set. Since all density-based methods mark points that fall in areas of the data set not sufficiently dense as ‘noise’ according to a scale parameter—leading to severe overfitting if not guided with a measure like stability—all densitybased solutions were deemed only valid if at least 75% of the data is classified with a non-noise label. Finally, the RSL also contains (a) “True” POIs (b) Inferred intrinsic POIs Figure 4: Intrinsic POI comparison, OSU (c) “True” POIs (d) Inferred intrinsic POIs Figure 5: Intrinsic POI comparison, WSU two parameters, a k value, and an α parameter. We use Chaudhuri et. al’s analysis to determine how to set these. RSL √ was shown to have optimal rates of convergence when α ≥ 2, so we leave √ it at that constant value ( 2). Similarly, the rate only holds for k at least as large as d log(n), where d is the dimensionality of the data set (d = 2 in this case). After varying through the small set of k values in a similar fashion as was performed for DBSCAN and OPTICS (k ∈ qnt ∀k >= d log(n)). In total, 2,196 and 1,995 cluster configurations were performed for the OSU and WSU simulations respectively, totalling 4,191 reported configurations. 3.3 Validation Testing and Discussion Qualitative comparison to truth: Figures 4 and 5 compare the intrinsic POIs discovered by our framework agains the simulation’s “true” POIs. Recall that points with low density are discarded as noise (not shown). Direct comparison of the true POIs defined by the simulation show clear similarities. Over the OSU simulation in Figure 4(b), the framework recovers intrinsic POIs within buildings no matter its shape, the density, or closeness to other buildings. It also recovers intrinsic POIs over parking lots and street intersections around the OSU oval. Some buildings are decomposed into a collection of individual intrinsic POIs. For example, the easternmost large campus building by the northeast corner of the oval contains three separate intrinsic POIs: one at its entrance by the road, another in the center of the building, and a third at its back entrance. Although these labels may not match what SUMO assigned, they are in some sense more natural, i.e. it’s quite possible for large buildings to have dense, isolated areas of people movement. Looking at the intrinsic POIs over the WSU dataset in Figure 5(b), we find each building in general is recovered as an intrinsic POI and align in shape compared to the shape of the “true” POIs from Figure 5(a). We also note that the framework determines that some movement within buildings but covering very small areas were found not be significant enough be an intrinsic POI. Large buildings also showed further decomposition like the OSU simulation. For example, in the WSU student union (large building in the lower left corner of the figure) the framework defines intrinsic POI’s at the center and two back exits from the building. Quantitative comparison to other approaches: The proposed intrinsic POI framework measured ARI scores of 0.966 on the OSU data set and 0.922 for the WSU data set. Note that this is not the maximum ARI of any RSL solution, but the ARI of the solution found using the highest predefined notion of stability, determined completely without any knowledge of the surrounding geographical area. RSL performed consistently in terms of having low variability compared to other algorithm, with overall high similarity to the semantically driven SUMO assigned locations. Figure 6 shows the distribution of the ARI for the algorithms we compared our method against with the parameter settings discussed in Section 3.2. The orange line corresponds with the ARI of the proposed framework, which compares favorably with best possible settings of other algorithms. Note that although DBSCAN (like others) performed well with very specific configurations of minPts and ε 0 , the settings of these parameters is often not very intuitive in unsupervised scenarios where the truth is unknown (and thus external measures like ARI cannot be computed). Of the hierarchical algorithms, we see the impact of the linkage criterion used and how they are influenced by the ‘shape’ of the true clusters. For example, SL clustering performed fairly well on the well separated WSU data set, but substantially lower on the more density-varied OSU data set. This is reflected in AL as well. k-means was able capture much of the true clustering structure with the right parameter settings Figure 6: The distribution of Adjusted Rand Index (ARI) scores of various clustering algorithms (after varying free parameters). The orange line corresponds to the ARI of the proposed framework. for the WSU simulation (with max/mean ARI scores of (0.92, 0.71)), however exhibited degraded performance when the POIs were less separated in OSU data set ((0.61, 0.84)). OPTICS, with a few specific parameter settings, performed well on the WSU data set, however again suffered on the more variable-scale OSU data set. 4 Related Research The trajectory field has been largely progressed by “extensive and intensive individual efforts” [42]. Nonetheless, conceptual models have been proposed for how to deal with the patterns within trajectories and how to relate such patterns to geographical areas of interest for various purposes. One such model postulates that trajectories and their spatiotemporal patterns are essentially driven by the semantics the application associates with trajectory itself [29] [33], and have contributed significantly to the “Stops and Moves of Trajectories” (SMoT) family of classification algorithms [1, 28, 32], where the premise of the analysis is that by partitioning trajectory data into a labeled set of ‘stop’ and ‘move’ segments, one can take then annotate these segments with semantic information, derive specific mobility patterns, and as a result discover ‘interesting’ locations. Alvares et al. developed IB-SMoT to find interesting positions based on semantic annotations describing the places a trajectory visited [1]. Palma et al. reduced IB-SMoT’s reliance on prior knowledge about positions that are likely to be interesting by incorporating the speed at which tracks are traveling with their variation CB-SMoT [28]. DB-SMoT finds clusters of common trajectories based on similar direction changes and stopping points [32]. Zhou et al. tackle the problem of finding positions of interest to an individual track based on data about a track’s location preferences, position over time, and tags of locations provided by web services such as Google Maps [45]. Many related efforts encode or are reliant on varying notions of an “interesting place” using, for example, techniques from natural language processing (NLP) [15], data clustering [32, 35, 45], sequential pattern mining [38], and social network analysis [41] methods. Zheng et. al pioneered the use of ‘stay points’, corresponding to an aggregation of consecutive GPS points that collectively are within a user-supplied time and distance threshold, thereby characterizing a ‘virtual’ location [43, 44]. It is interesting to note that Zheng et. al used OPTICS to create a hierarchical clustering ‘stay points’ for an LBS-type application with Microsoft called ‘Geolife’ [43]. Indeed, Zheng anticipated a number of developments in the trajectory mining field [42]; the theoretical cluster tree may be viewed as a more statistically based conception of the‘Tree Based Hierarchical Graph’ that is used to represent POIs in that application as well. It’s worth noting that our definition of an intrinsic POI, having a more theoretical foundation in density-based clustering, is both conceptually very similar to OPTICS and computationally, more recently, to Hierarchical DBSCAN (HDBSCAN) [6]. There exist a number of commonalities between both OPTICS and DBSCAN and the theory of the cluster tree. A comprehensive exposition of this relationship is beyond the scope of this paper, see Campello et. al [8] for a thorough review of the subject. Although it’s not mentioned in such efforts, the usage of RSL with a relative excess of mass functional to cluster extraction is equivalent to the flat clusters “HDBSCAN” extracts with a setting of α = 1 and k = minPts. However, the asymptotic consistency of the setting of the pair (α = 1, k ∼ d log n) has not been established [10]. When alpha = 1, k must be much larger, exponential in√the dimensionality of the data set, d. Thus, we use RSL with α ≥ 2. 5 Concluding Remarks This paper proposed a general framework for intrinsic POI discovery, without needing to rely on external gazetteers, based on recent theoretical advances in hierarchical, nearest neighbor density estimation. It discussed a conceptually sound basis for automated POI discovery specifically in the context of geospatial data, and introduced a framework that provides a rigorous and usable solution to an applied domain primarily dominated by intuitively reasonable, but heuristically-based methods. With novel extensions to SUMO to support pedestrian movement in buildings, an evaluation of simulated trajectory data over diverse geographical areas supports the conclusion that the proposed framework is a useful tool for extracting intrinsic POIs. The framework has both theoretical guarantees and practical benefits, requires no ad hoc parameter tuning, and exhibits improved fidelity against common approaches over thousands of parameter settings. In future work, with the help of the asymptotic analysis done by Chaudhuri et. al, we plan to develop model-selection techniques for POI extraction. This is imperative in exploratory settings, such as large urban environments where the number of POIs is not known ahead of time, there is little useful knowledge to gain from ad hoc or heuristic-based cluster analysis, especially when the solution space is large. 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Critical Parameters in Particle Swarm Optimisation arXiv:1511.06248v1 [cs.NE] 19 Nov 2015 J. Michael Herrmann∗, Adam Erskine, Thomas Joyce Institute for Perception, Action and Behaviour School of Informatics, The University of Edinburgh 10 Crichton St, Edinburgh EH8 9AB, Scotland, U.K. Abstract Particle swarm optimisation is a metaheuristic algorithm which finds reasonable solutions in a wide range of applied problems if suitable parameters are used. We study the properties of the algorithm in the framework of random dynamical systems which, due to the quasi-linear swarm dynamics, yields analytical results for the stability properties of the particles. Such considerations predict a relationship between the parameters of the algorithm that marks the edge between convergent and divergent behaviours. Comparison with simulations indicates that the algorithm performs best near this margin of instability. 1 PSO Introduction Particle Swarm Optimisation (PSO, [1]) is a metaheuristic algorithm which is widely used to solve search and optimisation tasks. It employs a number of particles as a swarm of potential solutions. Each particles shares knowledge about the current overall best solution and also retains a memory of the best solution it has encountered itself previously. Otherwise the particles, after random initialisation, obey a linear dynamics of the following form vi,t+1 xi,t+1 = = ωvi,t + α2 R1 (pi − xi,t ) + α2 R2 (g − xi,t ) xi,t + vi,t+1 (1) Here xi,t and vi,t , i = 1, . . . , N , t = 0, 1, 2, . . . , represent, respectively, the d-dimensional position in the search space and the velocity vector of the i-th particle in the swarm at time t. The velocity update contains an inertial term parameterised by ω and includes attractive forces towards the personal best location pi and towards the globally best location g, which are parameterised by α1 and and α2 , respectively. The symbols R1 and R2 denote diagonal matrices whose non-zero entries are uniformly distributed in the unit interval. The number of particles N is quite low in most applications, usually amounting to a few dozens. In order to function as an optimiser, the algorithm uses a nonnegative cost function F : Rd → R, where without loss of generality F (x∗ ) = 0 is assumed at an optimal solution x∗ . In many problems, where PSO is applied, there are also states with near-zero costs can be considered as good solutions. The cost function is evaluated for the state of each particle at each time step. If F (xi,t ) is better than F (pi ), then the personal best pi is replaced by xi,t . Similarly, if one of the particles arrives at a state with a cost less than F (g), then g is replaced in all particles by the position of the particle that has discovered the new solution. If its velocity is non-zero, a particle will depart from the current best location, but it may still have a chance to return guided by the force terms in the dynamics. Numerous modifications and variants have been proposed since the algorithm’s inception [1] and it continues to enjoy widespread usage. Ref. [2] groups around 700 PSO papers into 26 discernible application areas. Google Scholar reveals over 150,000 results for “Particle Swarm Optimisation” in total and 24,000 for the year 2014. In the next section we will report observations from a simulation of a particle swarm and move on to a standard matrix formulation of the swarm dynamics in order to describe some of the existing ∗ corresponding author: [email protected] 1 analytical work on PSO. In Sect. 3 we will argue for a formulation of PSO as a random dynamical system which will enable us to derive a novel exact characterisation of the dynamics of one-particle system, which will then be generalised towards the more realistic case of a multi-particle swarm. In Sect. 4 we will compare the theoretical predictions with simulations on a representative set of benchmark functions. Finally, in Sect. 5 we will discuss the assumption we have made in the theoretical solution in Sect. 3 and address the applicability of our results to other metaheuristic algorithms and to practical optimisation problems. 2 2.1 Swarm dynamics Empirical properties The success of the algorithm in locating good solutions depends on the dynamics of the particles in the state space of the problem. In contrast to many evolution strategies, it is not straight forward to interpret the particle swarm as following a landscape defined by the cost function. Unless the current best positions p or g change, the particles do not interact with each other and follow an intrinsic dynamics that does not even indirectly obtain any gradient information. The particle dynamics depends on the parameterisation of the Eq. 1. To obtain the best result one needs to select parameter settings that achieve a balance between the particles exploiting the knowledge of good known locations and exploring regions of the problem space that have not been visited before. Parameter values often need to be experimentally determined, and poor selection may result in premature convergence of the swarm to poor local minima or in a divergence of the particles towards regions that are irrelevant for the problem. Empirically we can execute PSO against a variety of problem functions with a range of ω and α1,2 values. Typically the algorithm shows performance of the form depicted in Fig. 1. The best solutions found show a curved relationship between ω and α = α1 + α2 , with ω ≈ 1 at small α, and α ' 4 at small ω. Large values of both α and ω are found to cause the particles to diverge leading to results far from optimality, while at small values for both parameters the particles converge to a nearby solution which sometimes is acceptable. For other cost functions similar relationships are observed in numerical tests (see Sect. 4) unless no good solutions found due to problem complexity or run time limits, see Sect. 5.3. For simple cost functions, such as a single well potential, there are also parameter combinations with small ω and small α will usually lead to good results. The choice of α1 and α2 at constant α may have an effect for some cost functions, but does not seem to have a big effect in most cases. 2.2 Matrix formulation In order to analyse the behaviour of the algorithm it is convenient to use a matrix formulation by inserting the velocity explicitly in the second equation (1). zt+1 = M zt + α1 R1 (p, p)⊤ + α2 R2 (g, g)⊤ ⊤ with z = (v, x) (2) and M= ωId ωId −α1 R1 − α2 R2 Id − α1 R1 − α2 R2 , (3) where Id is the unit matrix in d dimensions. Note that the two occurrence of R1 in Eq. 3 refer to the same realisation of the random variable. Similarly, the two R2 ’s are the same realisation, but different from R1 . Since the second and third term on the right in Eq. 2 are constant most of the time, the analysis of the algorithm can focus on the properties of the matrix M . In spite of its wide applicability, PSO has not been subject to deeper theoretical study, which may be due to the multiplicative noise in the simple quasi-linear, quasi-decoupled dynamics. In previous studies the effect of the noise has largely been ignored. 2.3 Analytical results An early exploration of the PSO dynamics [4] considered a single particle in a one-dimension space where the personal and global best locations were taken to be the same. The random components were 2 Figure 1: Typical PSO performance as a function of its ω and α parameters. Here a 25 particle swarm was run for pairs of ω and α values (α1 = α2 = α/2). Cost function here was the d = 10 non-continuous rotated Rastrigin function [3]. Each parameter pair was repeated 25 times and the minimal costs after 2000 iterations were averaged. replaced by their averages such that apart from random initialisation the algorithm was deterministic. Varying the parameters was shown to result in a range of periodic motions and divergent behaviour for the case of α1 + α2 ≥ 4. The addition of the random vectors was seen as beneficial as it adds noise to the deterministic search. Control of velocity, not requiring the enforcement of an arbitrary maximum value as in Ref. [4], is derived in an analytical manner by [5]. Here eigenvalues derived from the dynamic matrix of a simplified version of the PSO algorithm are used to imply various search behaviours. Thus, again the α1 + α2 ≥ 4 case is expected to diverge. For α1 + α2 < 4 various cyclic and quasi-cyclic motions are shown to exist for a non-random version of the algorithm. In Ref. [6] again a single particle was considered in a one dimensional problem space, using a deterministic version of PSO, setting R1 = R2 = 0.5. The eigenvalues of the system were determined as functions of ω and a combined α, which leads to three conditions: The particle is shown to converge when ω < 1, α > 0 and 2ω −α+2 > 0. Harmonic oscillations occur for ω 2 +α2 −2ωα−2ω −2α+1 < 0 and a zigzag motion is expected if ω < 0 and ω−α+1 < 0. As with the preceding papers the discussion of the random numbers in the algorithm views them purely as enhancing the search capabilities by adding a drunken walk to the particle motions. Their replacement by expectation values was thus believed to simplify the analysis with no loss of generality. We show in this contribution that the iterated use of these random factors R1 and R2 in fact adds a further level of complexity to the dynamics of the swarm which affects the behaviour of the algorithm in a non-trivial way. In Ref. [7] these factors were given some consideration. Regions of convergence and divergence separated by a curved line were predicted. This line separating these regions (an equation for which is given in Ref. [8]) fails to include some parameter settings that lead to convergent swarms. Our analytical solution of the stability problem for the swarm dynamics explains why parameter settings derived from the deterministic approaches are not in line with experiences from practical tests. For this purpose we will now formulate the PSO algorithm as a random dynamical system and present an analytical solution for the swarm dynamics in a simplified but representative case. 3 3 3.1 Critical swarm conditions for a single particle PSO as a random dynamical system As in Refs. [4, 6] the dynamics of the particle swarm will be studied here as well in the single-particle case. This can be justified because the particles interact only via the global best position such that, while g (1) is unchanged, single particles exhibit qualitatively the same dynamics as in the swarm. For the one-particle case we have necessarily p = g, such that shift invariance allows us to set both to zero, which leads us to the following is given by the stochastic-map formulation of the PSO dynamics (2). zt+1 = M zt (4) Extending earlier approaches we will explicitly consider the randomness of the dynamics, i.e. instead of averages over R1 and R2 we consider a random dynamical system with dynamical matrices M chosen from the set ωId −αR Mα,ω = , Rij = 0 for i 6= j and Rii ∈ [0, 1] , (5) ωId Id − αR with R being in both rows the same realisation of a random diagonal matrix that combines the effects of R1 and R2 (1). The parameter α is the sum α1 + α2 with α1 , α2 ≥ 0 and α > 0. As the diagonal elements of R1 and R2 are uniformly distributed in [0, 1], the distribution of the random variable Rii = αα1 R1,ii + αα2 R2,ii in Eq. 4 is given by a convolution of two uniform random variables, namely Pα1 ,α2 (r) =  αr   max{α1 ,α2 } α max{α ,α }   α(α−r)1 2 α1 α2 if 0 ≤ r ≤ min{ αα1 , αα2 } if min{ αα1 , αα2 } < r ≤ max{ αα1 , αα2 } if max αα1 , αα2 < r ≤ 1 (6) if the variable r ∈ [0, 1] and Pα1 ,α2 (r) = 0 otherwise. Pα1 ,α2 (r) has a tent shape for α1 = α2 and a box shape in the limits of either α1 → 0 or α2 → 0. The case α1 = α2 = 0, where the swarm does not obtain information about the fitness function, will not be considered here. We expect that the multi-particle PSO is well represented by the simplified version for α2 ≫ α1 or α1 ≫ α2 , the latter case being irrelevant in practice. For α1 ≈ α2 deviations from the theory may occur because in the multi-particle case p and g will be different for most particles. We will discuss this as well as the effects of the switching of the dynamics at discovery of better solutions in Sect. 5.2. 3.2 Marginal stability While the swarm does not discover any new solutions, its dynamical properties are determined by an infinite product of matrices from the set M (5). Such products have been studied for several decades [9] and have found applications in physics, biology and economics. Here they provide a convenient way to explicitly model the stochasticity of the swarm dynamics such that we can claim that the performance of PSO is determined by the stability properties of the random dynamical system (4). Since the equation (4) is linear, the analysis can be restricted to vectors on the unit sphere in the (v, x) space, i.e. to unit vectors ⊤ ⊤ a = (x, v) / k (x, v) k, (7) where k · k denotes the Euclidean norm. Unless the set of matrices shares the same eigenvectors (which is not the case here) standard stability analysis in terms of eigenvalues is not applicable. Instead we will use means from the theory of random matrix products in order to decide whether the set of matrices is stochastically contractive. The properties of the asymptotic dynamics can be described based on a double Lebesgue integral over the unit sphere S 2d−1 and the set M [10, 11]. As in Lyapunov exponents, the effect of the dynamics is measured in logarithmic units in order to account for multiplicative action. Z Z λ (α, ω) = dνα,ω (a) dPα,ω (M ) log kM ak (8) 4 If a (α, ω) is negative the algorithm will converge to p with probability 1, while for positive a arbitrarily large fluctuations are possible. While the measure for the inner integral (8) is given by Eq. 6, we have to determine the stationary distribution ν on the unit sphere for the outer integral. It is given as the solution of the integral equation Z Z να,ω (a) = dνα,ω (b) dPα,ω (M ) δ (a, M b/ kM bk) , a, b ∈ S 2d−1 . (9) The existence of the invariant measure requires the dynamics to be ergodic which is ensured if at least some of elements of M have complex eigenvalues, such as being the case for ω 2 +α2 /4−ωα−2ω−α+1 < 0 (see above, [6]). This condition excludes a small region in the parameters space at small values of ω, such that there we have to take all ergodic components into account. There are not more than two components which due to symmetry have the same stability properties. It depends on the parameters α and ω and differs strongly from a homogenous distribution, see Fig. 2 for a few examples in the case d = 1. Critical parameters are obtained from Eq. 8 by the relation 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 1 2 3 4 5 6 Figure 2: Stationary distribution να,ω (a) on the unit circle (a ∈ [0, 2π)) in the (x, v) plane for a one-particle system (4) for ω = 0.7 and α = α2 = 0.5, 1.5, 2.5, 3.5, 4.5 (the distribution with peak near π is for α = 0.5, otherwise main peaks are highest for largest α). λ (α, ω) = 0. (10) Solving Eq. 10 is difficult in higher dimensions, so we rely on the linearity of the system when considering the (d = 1)-case as representative. The curve in Fig. 3 represents the solution of Eq. 10 for d = 1 and α = α2 . For other settings of α1 and α2 the distribution of the random factors has a smaller variance rendering the dynamics more stable such that the contour moves towards larger parameter values (see Fig. 4). Inside the contour λ (α, ω) is negative, meaning that the state will approach the origin with probability 1. Along the contour and in the outside region large state fluctuations are possible. Interesting parameter values are expected near the curve where due to a coexistence of stable and unstable dynamics (induced by different sequences of random matrices) a theoretically optimal combination of exploration and exploitation is possible. For specific problems, however, deviations from the critical curve can be expected to be beneficial. 3.3 Personal best vs. global best Due to linearity, the particle swarm update rule (1) is subject to a scaling invariance which was already used in Eq. 7. We now consider the consequences of linearity for the case where personal best and global best differ, i.e. p 6= g. For an interval where pi and g remain unchanged, the particle i with personal best pi will behave like a particle in a swarm where together with x and v, pi is also scaled by a factor κ > 0. The finite-time approximation of the Lyapunov exponent (see Eq. 8) λ(t) = 1 log hk(xt , vt )ki t 5 (11) Figure 3: Solution of Eq. 10 representing a single particle in one dimension with a fixed best value at g = p = 0. The curve that has higher α-values on the right (magenta) is for α1 = α2 , the other curve (green) is for α = α2 , α1 = 0. Except for the regions near ω = ±1, where numerical instabilities can occur, a simulation produces an indistinguishable curve. In the simulation we tracked the probability of a particle to either reach a small region (10−6 ) near the origin or to escape beyond a radius of 106 after starting from a random location on the unit circle. Along the curve both probabilities are equal. will be changed by an amount of 1t log κ by the scaling. Although this has no effect on the asymptotic behaviour, we will have to expect an effect on the stability of the swarm for finite times which may be relevant for practical applications. For the same parameters, the swarm will be more stable if κ < 1 and less stable for κ > 1, provided that the initial conditions are scaled in the same way. Likewise, if kpk is increased, then the critical contour will move inwards, see Fig. 5. Note that in this figure, the low number of iterations lead to a few erroneous trials at parameter pairs outside the outer contour which have been omitted here. We also do not consider the behaviour near α = 0 which is complex but irrelevant for PSO. The contour (10) can be seen as the limit κ → 0 such that only an increase of kpk is relevant for comparison with the theoretical stability result. When comparing the stability results with numerical simulations for real optimisation problems, we will need to take into account the effects caused by differences between p and g in a multi-particle swarm with finite runtimes. 4 Optimisation of benchmark functions Metaheuristic algorithms are often tested in competition against benchmark functions designed to present different problem space characteristics. The 28 functions [3] contain a mix of unimodal, basic multimodal and composite functions. The domain of the functions in this test set are all defined to be [−100, 100]d where d is the dimensionality of the problem. Particles were initialised within the same domain. We use 10-dimensional problems throughout. Our implementation of PSO performed no spatial or velocity clamping. In all trials a swarm of 25 particles was used. We repeated the algorithm 100 times, on each occasion allowing 200, 2000, 20000 iterations to pass before recording the best solution found by the swarm. For the competition 50000 fitness evaluation were allowed which corresponds to 2000 iterations with 25 particles. Other iteration numbers were included for comparison. This protocol was carried out for pairs of ω ∈ [−1.1, 1.1] and α ∈ [0, 5] This was repeated for all 28 functions. The averaged solution costs as a function of the two parameters showed curved valleys similar to that in Fig. 1 for all problems. For each function we obtain different best values along (or near) the theoretical curve (10). There appears to be no preferable location within the valley. Some individual functions yield best performance near ω = 1. This is not the case near ω = 0, although the global average performance over all test functions is better in the valley near ω = 0 than near ω = 1, see Fig 4. 6 Figure 4: Best parameter regions for 200 (blue), 2000 (green), and 20000 (magenta) iterations: For more iterations the region shifts towards the critical line. Cost averaged over 100 runs and 28 CEC benchmark functions. The red (outer) curve represents the zero Lyapunov exponent for N = 1, d = 1, α1 = α2 . At medium values of ω the difference between the analytical solutions for the cases α1 = α2 and α1 = 0 is strongest, see Fig. 4. In simulations this shows to a lesser extent, thus revealing a shortcoming of the one-particle approximation. Because in the multi-particle case, p and g are often different, the resulting vector will have a smaller norm than in the one-particle case, where p = g. The case p 6= g violates a the assumption of the theory the dynamics can be described based unit vectors. While a particle far away from both p and g will behave as predicted from the one-particle case, at length scales smaller than kp − gk the retractive forces will tend to be reduced such that the inertia becomes more effective and the particle is locally less stable which shows numerically in optimal parameters that are smaller than predicted. 5 5.1 Discussion Relevance of criticality Our analytical approach predicts a locus of α and ω pairings that maintain the critical behaviour of the PSO swarm. Outside this line the swarm will diverge unless steps are taken to constrain it. Inside, the swarm will eventually converge to a single solution. In order to locate a solution within the search space, the swarm needs to converge at some point, so the line represents an upper bound on the exploration-exploitation mix that a swarm manifests. For parameters on the critical line, fluctuations are still arbitrary large. Therefore, subcritical parameter values can be preferable if the settling time is of the same order as the scheduled runtime of the algorithm. If, in addition, a typical length scale of the problem is known, then the finite standard deviation of the particles in the stable parameter region can be used to decide about the distance of the parameter values from the critical curve. These dynamical quantities can be approximately set, based on the theory presented here, such that a precise control of the behaviour of the algorithm is in principle possible. The observation of the distribution of empirically optimal parameter values along the critical curve, confirms the expectation that critical or near-critical behaviour is the main reason for success of the algorithm. Critical fluctuations are a plausible tool in search problem if apart from certain smoothness assumption nothing is known about the cost landscape: The majority of excursions will exploit the smoothness of the cost function by local search, whereas the fat tails of the distribution allow the particles to escape from local minima. 7 Figure 5: For p 6= g we define neutral stability as the equilibrium between divergence and convergence. Convergence means here that the particle approaches the line connecting p and g. Curves are for a one-dimensional problem with p = 0.1 and g = 0 scaled (see Sect. 3.3) by κ = 1 (outer curve) κ = 0.1 and κ = 0.04 (inner curve). Results are for 200 iterations and averaged over 100000 repetitions. 5.2 Switching dynamics at discovery of better solutions Eq. 2 shows that the discovery of a better solution affects only the constant terms of the linear dynamics of a particle, whereas its dynamical properties are governed by the linear coefficient matrices. However, in the time step after a particle has found a new solution the corresponding force term in the dynamics is zero (see Eq. 1) such that the particle dynamics slows down compared to the theoretical solution which assumes a finite distance from the best position at all (finite) times. As this affects usually only one particle at a time and because new discoveries tend to become rarer over time, this effect will be small in the asymptotic dynamics, although it could justify the empirical optimality of parameters in the unstable region for some test cases. The question is nevertheless, how often these changes occur. A weakly converging swarm can still produce good results if it often discovers better solutions by means of the fluctuations it performs before settling into the current best position. For cost functions that are not ‘deceptive’, i.e. where local optima tend to be near better optima, parameter values far inside the critical contour (see Fig. 3) may give good results, while in other cases more exploration is needed. 5.3 The role of personal best and global best A numerical scan of the (α1 , α2 ) plane shows a valley of good fitness values, which, at small fixed positive ω, is roughly linear and described by the relation α1 +α2 = const, i.e. only the joint parameter α = α1 + α2 matters. For large ω, and accordingly small predicted optimal α values, the valley is less straight. This may be because the effect of the known solutions is relatively weak, so the interaction of the two components becomes more important. In other words if the movement of the particles is mainly due to inertia, then the relation between the global and local best is non-trivial, while at low inertia the particles can adjust their p vectors quickly towards the g vector such that both terms become interchangeable. Finally, we should mention that more particles, longer runtime as well as lower search space dimension increase the potential for exploration. They all lead to the empirically determined optimal parameters being closer to the critical curve. 8 6 Conclusion PSO is a widely used optimisation scheme which is theoretically not well understood. Existing theory concentrates on a deterministic version of the algorithm which does not possess useful exploration capabilities. We have studied the algorithm by means of a product of random matrices which allows us to predict useful parameter ranges and may allow for more precise settings if a typical length scale of the problem is known. A weakness of the current approach is that it focuses on the standard PSO [1] which is known to include biases [12, 13], that are not necessarily justifiable, and to be outperformed on benchmark set and in practical applications by many of the existing PSO variants. Similar analyses are certainly possible and are expected to be carried out for some of the variants, even though the field of metaheuristic search is often portrayed as largely inert to theoretical advances. If the dynamics of particle swarms is better understood, the algorithms may become useful as efficient particle filters which have many applications beyond heuristic optimisation. Acknowledgments This work was supported by the Engineering and Physical Sciences Research Council (EPSRC), grant number EP/K503034/1. References [1] J. Kennedy and R. Eberhart. Particle swarm optimization. In Proceedings IEEE International Conference on Neural Networks, volume 4, pages 1942–1948. IEEE, 1995. [2] R. Poli. Analysis of the publications on the applications of particle swarm optimisation. Journal of Artificial Evolution and Applications, 2008(3):1–10, 2008. [3] CEC2013. http://www.ntu.edu.sg/home/EPNSugan/index files/CEC2013/CEC2013.htm. [4] J. Kennedy. The behavior of particles. In V.W. Porto, N. Saravanan, D.Waagen, and A. E. Eiben, editors, Evolutionary programming VII, pages 579–589. Springer, 1998. [5] M. Clerc and J. Kennedy. The particle swarm-explosion, stability, and convergence in a multidimensional complex space. IEEE Transactions on Evolutionary Computation, 6(1):58–73, 2002. [6] I. C. Trelea. The particle swarm optimization algorithm: convergence analysis and parameter selection. Information Processing Letters, 85(6):317–325, 2003. [7] M. Jiang, Y. Luo, and S. Yang. Stagnation analysis in particle swarm optimization. In Swarm Intelligence Symposium, 2007. SIS 2007. IEEE, pages 92–99. IEEE, 2007. [8] C. W. Cleghorn and A. P Engelbrecht. A generalized theoretical deterministic particle swarm model. Swarm Intelligence, 8(1):35–59, 2014. [9] H. Furstenberg and H. Kesten. Products of random matrices. Annals of Mathematical Statistics, 31(2):457–469, 1960. [10] V. N. Tutubalin. On limit theorems for the product of random matrices. Theory of Probability & Its Applications, 10(1):15–27, 1965. [11] R. Z. Khas’minskii. Necessary and sufficient conditions for the asymptotic stability of linear stochastic systems. Theory of Probability & Its Applications, 12(1):144–147, 1967. [12] M. Clerc. Confinements and biases in particle swarm optimisation. Technical Report hal00122799, Open archive HAL, 2006. [13] W. M. Spears, D. Green, and D. F. Spears. Biases in particle swarm optimization. International Journal of Swarm Intelligence Research, 1(2):34–57, 2010. 9	9
1 On the detection of low rank matrices in the high-dimensional regime. Antoine Chevreuil and Philippe Loubaton Gaspard Monge Computer Science Laboratory (LIGM) - UMR 8049 CNRS Université de Paris-Est/Marne-la-Vallée 5 Bd. Descartes 77454 Marne-la-Vallée (France) arXiv:1804.04851v1 [eess.SP] 13 Apr 2018 Abstract We address the detection of a low rank n × ndeterministic matrix X0 from the noisy observation X0 + Z when n → ∞, where Z is a complex Gaussian random matrix with independent identically distributed Nc (0, n1 ) entries. Thanks to large random matrix theory results, it is now well-known that if the largest singular value λ1 (X0 ) of X0 verifies λ1 (X0 ) > 1, then it is possible to exhibit consistent tests. In this contribution, we prove a contrario that under the condition λ1 (X0 ) < 1, there are no consistent tests. Our proof is inspired by previous works devoted to the case of rank 1 matrices X0 . Index Terms statistical detection tests, large random matrices, large deviation principle. I. I NTRODUCTION The problem of testing whether an observed n1 × n2 matrix Y is either a zero-mean independent identically distributed Gaussian random matrix Z with variance n12 , or X0 + Z for some low rank deterministic matrix X0 , with no known structure, called also a spike, is a fundamental problem arising in numerous applications such as the detection of low-rank multivariate signals or the Gaussian hidden clique problem. When the two dimensions n1 , n2 converge towards ∞ in such a way that n1 /n2 → c > 0 (the rank of X0 remaining fixed), known results on the so-called additive spiked large random matrix models have enabled to re-consider this fundamental detection problem (see e.g. [13], [4], [3]). It was established a long time ago (see e.g. [1] and the references therein) that in the above asymptotic regime, the largest singular value λ1 (Z) of Z converges almost √ surely towards 1 + c. More recently, under mild technical extra assumptions, [3] proved that λ1 (X0 + Z) still converges √ towards 1 + c if λ1 (X0 ) converges towards a limit strictly less than c1/4 . On the√contrary, if the limit of λ1 (X0 ) is strictly greater than c1/4 , then λ1 (X0 +Z) converges towards a limit strictly greater than 1+ c. This result implies that the Generalized Likelihood Ratio Test (GLRT) is consistent (i.e. both the probability of false alarm and the probability of missed detection converge towards 0 in the above asymptotic regime) if and only if λ1 (X0 ) is above the threshold c1/4 . In order to simplify the exposition, we assume from now on that n1 = n2 = n, so that ratio c reduces to 1. While the detection problem was extensively addressed in the zone λ1 (X0 ) > 1, the case where λ1 (X0 ) < 1 was much less studied. Montanari et al. [12] consider the zone λ1 (X0 ) < 1 when X0 is a rank 1 matrix. Thanks to simple information geometry tools, [12] prove that, in this region, it is impossible to find a consistent test for the detection of the spike. Irrespective of the standard random matrix tools, this approach is extended to the more general case when X0 and Z are tensors of order d ≥ 3; namely, if the Frobenius norm of the tensor X0 is stricly less than a threshold depending in d, then the probability distributions of the observation under the two hypotheses are asymptotically undistinguishable, so that any detection test cannot behave better than a random guess. This property, which is stronger than the non-existence of a consistent test, does not hold in the matrix case d = 2: see for instance [14] where a non-consistent test is exhibited that has a better performance than a random guess. In this paper, we extend the above methodolodgy to the general case where X0 has rank r. Our contribution is to prove that under λ1 (X0 ) < 1, the consistent detection is impossible. While this theoretical result is not unexpected, we believe that it provides a better understanding of the above fundamental detection problem in large dimensions without resorting to the machinery of large random matrices. We mention that the works [9] (when the spike is symmetric) and [10] (non-symmetric case) are clearly related to the above problem. However, two major differences arise: firstly, the detection is not addressed but rather the estimation of X0 ; second, a statistical model of the spike is needed. The results are in general not explicit. However for a certain prior and for the rank one model, it can be deduced that, in the zone λ1 (X0 ) < 1, it is impossible to find estimates of the spike that have better performance than any dummy estimate (i.e. an estimate that does not rely on the observation). The authors rely on the computation of the mutual information between X0 and Y: this computation involves non-obvious results extending the approach of Tallagrand for studying the Sherrington-Kirkpatrick model [15]. 2 II. M ODEL , NOTATION , ASUMPTION The set of complex-valued matrices Cn×npis a complex vector-space endowed with the standard scalar product hX, Yi = Tr(XY∗ ) and the Frobenius norm kXkF = hX, Xi. The spectral norm of a matrix X is denoted by kXk2 . The spike (“the signal”) is assumed to be a matrix of fixed rank r and hence admits a SVD such as X0 = r X λj uj vj∗ = UΛV∗ (1) j=1 where λi = λi (X0 ) are the singular values of X0 sorted in descending order and where Λ is the diagonal matrix gathering the (λj )j=1,...,r in the descending order. As X0 has to be defined for any n, we impose a non-erratic behavior of X0 , namely that all its singular values (λj )j=1,...,r do not depend on n for n large enough. This hypothesis could be replaced by the condition that (λj )j=1,...,r all converge towards a finite limit at an ad’hoc rate. However, this would introduce purely technical difficulties. The noise matrix Z is assumed to have i.i.d. entries distributed as Nc (0, 1/n). We consider the alternative H0 : Y = Z versus H1 : Y = X0 + Z. We denote by p1,n (y) the probability probability density of Y under H1 and p0,n (y) the density p1,n (Y) of Y under H0 . L(Y) = p0,n (Y) is the likelihood ratio and we denote by E0 the expectation under H0 . We now recall the fundamental information geometry results used in [12] in order to address the detection problem.The following properties are well known (see also [2] section 3): 2 is bounded, • (i) If E0 L(Y) then no consistent detection test exists. 2 • (ii) If moroever E0 L(Y) = 1 + o(1), the total variation distance between p0,n and p1,n converges towards 0, and no test performs better than a decision at random. We however mention that (i) and (ii) are only sufficient conditions. In particular, E0 L(Y)2 unbounded does not imply the existence of consistent tests. III. P RIOR ON THE SPIKE . E XPRESSION OF THE SECOND - ORDER MOMENT. 2 The density of Z, seen as a collection of n2 complex-valued random variables, is obviously p0,n (z) = κn exp −n kzkF n2 where κn = nπ . On the one hand, we notice that the study of the second-order momentof the likelihood ratio is not suited 2 to the deterministic model of the spike as presented previously. Indeed, in this case E L(Y) has the simple expression 0 2 exp 2n kX0 kF and always diverges. On the other hand, the noise matrix shows an invariance property: if Θ1 , Θ2 are unitary n × n matrices , then the density of Θ1 ZΘ2 equals this of Z. We hence modify the data according to the procedure: we pick two independent unitary Θ1 , Θ2 according to the Haar measure (which corresponds to the uniform distribution on the set of all unitary n × n matrices), and change the data tensor Y according to Θ1 YΘ2 . As said above, this does not affect the distribution of the noise, but this amounts to assume a certain prior on the spike. Indeed, this amounts to replace ui by Θ1 ui and vi by Θ∗2 vi . In the following, the data and the noise tensors after this procedure are still denoted respectively by Y and Z. We are now in position to give a closed-form expression of the second-order moment of L(Y) . We have p1,n (Y) = EX [p0,n (Y − X)] where EX is the mathematical expectation over the prior distribution of the spike, or equivalently over the Haar matrices Θ1 , Θ2 . It holds that E0 L(Y)2 = E [exp (2nR hX, X0 i)] where the expectation is over independent 0 0 0 0 copies X,2X of the spike (R stands for the real part); X and X being respectively associated with (Θ1 , Θ2 ) and (Θ1 , Θ2 ), E0 L(Y) has the expression h ∗ ∗ i E exp 2nRTr Θ1 X0 Θ2 Θ02 X∗0 Θ01 . ∗ ∗ As Θk and Θ0k are Haar and independent, then (Θ01 ) Θ1 and Θ2 (Θ02 ) are also independent, Haar distributed and it holds E0 L(Y)2 = E [exp (2nη)] , (2) where the expectation is over the independent Haar matrices Θ1 , Θ2 and η = RTr (Θ1 X0 Θ2 X∗0 ). The ultimate simplification comes from the decomposition (1) which implies that η = RTr (ΛΨ1 ΛΨ2 ) (3) where Ψ1 = U∗ Θ1 U and Ψ2 = V∗ Θ2 V. It is clear that Ψ1 and Ψ2 are independent matrices that are both distributed as the upper r × r diagonal block of a Haar unitary matrix. 3 IV. R ESULT The main result of our contribution is the following Theorem 1. If λ1 (X0 ) < 1 then lim sup E0 L(Y)2 ≤ 1 1 − λ1 (X0 )4 r2 and it is not possible to find a consistent test. We remind that we are looking for a condition on X0 (due to (2,3), this is a condition on Λ) under which E [exp (2nη)] is bounded. Evidently, the divergence may occur only when η > 0. We hence consider E1 = E [exp (2nη) Iη> ] and E2 = E [exp (2nη) Iη≤ ], and prove that, for a certain small enough > 0 to be specified later, E1 = o(1) and that E2 is bounded. V. T HE E1 TERM : COMPUTATION OF THE GRF OF η. It is clear that the boundedness of the integral E1 is achieved when η rarely deviates from 0. As remarked in [12], the natural machinery to consider is this of the Large Deviation Principle (LDP). In essence, if η follows the LDP with rate n, there can be found a certain non-negative function called Good Rate Function (GRF) Iη such that for any Borel set A of R, 1 n log P (η ∈ A) converges towards supx∈A −Iη (x). The existence of a GRF allows one to analyze the asymptotic behaviour of the integral E1 . In the next section, we thus justify that η follows a Large Deviation Principle with rate n, and we compute the associated GRF. A. Computation of the GRF of η Pr Eq. (3) and the Cauchy-Schwarz inequality imply that the random variable η is bounded: \|η\| ≤ ηmax with ηmax = j=1 λ2j . We first recall that for i = 1, 2, the random matrix Ψi follows a LDP with rate n and that its GRF at the parameter ψ ∈ Cr×r , kψk2 ≤ 1, is log det (Ir − ψ ∗ ψ) (see Theorem 3-6 in [8]). Besides, η is a function of the i.i.d. matrices (Ψi )i=1,2 and therefore, the contraction principle applies to η (see Theorem 4.2.1 in [7]): it ensures that η follows a LDP with rate n and its GRF is such that, for each real \|x\| ≤ ηmax , −Iη (x) is the solution of the following optimization problem: Problem 2. Maximize in Cr×r log det (I − ψ ∗1 ψ 1 ) + log det (I − ψ ∗2 ψ 2 ) . (4) under the constraints RTr (Λψ 1 Λψ 2 ) = x (5) kψ i k2 ≤ 1, i = 1, 2 (6) We provide a closed-form solution of Problem 2. In this respect, we define for each k = 1, . . . , r the interval Ik defined by ∀k = 1, ..., r − 1 : Ik =] k X X k+1 λ2i − λ2k , λ2i − λ2k+1 ] i=1 (7) i=1 Pr and Ir =] i=1 λ2i − λ2k , ηmax ]. It is easy to check that (Ik )k=1,...,r are disjoint and that ∪rk=1 Ik =]0, ηmax ]. The following result holds: Theorem 3. The maximum of Problem 2 is given by − Iη (x) = 2 r X k=1 " P log  k i=1 λ2i k − \|x\| #k  1  IIk (\|x\|) Πki=1 λ2i (8) It is easy to check that the function x 7→ −Iη (x) is continuous on ]0, ηmax [. The proof of Theorem 3 is provided in the Appendix. We illustrate Theorem 3 through the following experiment. The rank of the spike is fixed to r = 3 and the singular values have been set to (λ1 , λ2 , λ3 ) = (1 , 0.7 , 0.2). We have computed millions of random P2 samples of the matrices (ψ 1 , ψ 2 ). Each pair is associated with a point (x, y) defined as x = RTr (Λψ 1 Λψ 2 ) and y = i=1 log det (I − ψ ∗i ψ i ) . We obtain a cloud of points, the upper envelope of which is expected to be −Iη (x). We have also plotted the graph of the function y = −Iη (x). In addition, we mention that, in the more general context of tensors of order d, the second-order moment of L(Y) is still given by (2) but the random variable - call it ηd - has a more complicated form than (3), see [5]; the asymptotics of the term E1 can still be studied by evaluating the GRF of ηd . This GRF is the solution of an optimization problem that, apparently, cannot be solved in closed form for d ≥ 3. In [5], bound of the opposite of the true GRF is computed; this upper an upper \|x\| bound, valid for any d is given for d = 2 by log 1 − ηmax . We thus also represent in Figure V-A this upper bound; clearly, it is not tight. −6 −12 −10 −8 k ∑ log(det(I−ψkψHk ) −4 −2 0 4 −1.0 −0.5 0.0 Tr(Λψ1Λψ2) 0.5 1.0 Fig. 1. graph of −Iη (x) seen as an upper envelope. Upper curve: the upper bound computed in [5] B. Computation of E1 The Varadhan lemma (see Theorem 4.3.1 in [7]) states that n1 log E [exp (2nη) Iη> ] → supx> (2x − Iη (x)) and hence the E1 term converges towards 0 when supx> (2x − Iη (x)) < 0. Consider any of the intervals Ik defined in (7). The derivative 2 of 2x − Iη (x) for any x ∈ Ik is 2 − 2k/(λ + λ2k − x) : it is decreasing on Ik and the limit in the left extremity of Ik , 1 + ... Pk−1 2 1 2 i.e. ( j=1 λj ) − (k − 1)λk , is simply 2 1 − λ2 . If λ1 (X0 ) < 1, then for all the indices k, 1 − λ12 < 0. This shows that k k 2x + Iη (x) is strictly decreasing on every Ik . Hence, for every x ∈]0, ηmax ], we have 2x − Iη (x) < 0 − Iη (0) = 0. We have proved that E1 = o(1). VI. T HE E2 TERM : CONCENTRATION OF η. Notice that the upper block r × r Ψ of a unitary Haar matrix Θ has the same distribution as −1/2 G G̃∗ G̃ where the n×r matrix G̃ has i.i.d. entries distributed as NC (0, 1) and G is the top r ×r block of G̃. Obviously, E[G̃∗ G̃] = nI. It is a standard result that a random variable distributed as a χ2 (n) is concentrated around its mean. This can be easily extended to the matrix G̃∗ G̃: Lemma 4. For any 0 < δ < 1, there exists a constant c such that δ2 1 ∗ P G̃ G̃ − I > δ ≤ c exp −n . n 2 2 We take G̃1 and G̃2 independent, distributed as consider the upper r × r blocks G and G2 of G̃1 and G̃2 . It G̃ and −1/2 −1/2 1 ∗ ∗ follows that η has the same distribution as 2RTr ΛG1 G̃1 G̃1 ΛG2 G̃2 G̃2 . Take now any δ < 1. We may split the integral E2 in two parts: E exp (2nη) I{η≤}∩B1c ∩B2c + E exp (2nη) I{η≤}∩(B1 ∪B2 ) . {z } \| {z } \| E200 E20 where we have defined the events Bi = n 1 ∗ n G̃i G̃i E200 o −I 2 > δ . Thanks to the above concentration result, we have ≤ exp(2n) (P(B1 ) + P(B2 )) ≤ exp(2n)2c exp −nδ 2 /2 As it is always possible to choose δ and such that δ 2 − 4 > 0 and δ < 1 it follows that E200 = o(1). Let us now inspect the term E20 . Since we have, for i = 1, 2, n1 G̃∗i G̃i − I ≤ δ, then there exist ∆i for i = 1, 2 such 2 −1/2 that G̃∗i G̃i = √1n (I + ∆i )with k∆i k2 ≤ δ/2. We hence have E20 ≤ E [exp (2RTr (ΛG1 (I + ∆1 ) ΛG2 (I + ∆2 ))] . 5 We expand 2RTr (ΛG1 (I + ∆1 ) ΛG2 (I + ∆2 )) as the sum of four terms. Take for instance T2 = 2RTr (ΛG1 ∆1 ΛG2 ) Thanks to von Neumann’s lemma [11], we have T2 ≤ 2 r X λk (∆1 )λk (ΛG2 ΛG1 ) k=1 ≤ 2 k∆1 k r X λk (ΛG2 ΛG1 ) k=1 As Pr k=1 λk (ΛG2 ΛG1 ) ≤ √ pPr 2 r k=1 λk (ΛG2 ΛG1 ), it yields √ q T2 ≤ 2 k∆1 k r Tr (ΛG2 ΛG1 G∗1 ΛG∗2 Λ). Invoking the von Neumann’s lemma three times, it holds that q √ T2 ≤ 2 k∆1 k r Λ2 Tr (G1 G∗1 ) Tr (G2 G∗2 ) √ ≤ r k∆1 k Λ2 (Tr (G1 G∗1 ) + Tr (G2 G∗2 )) Similar manipulations can be done on the other terms of the expansion. so that E20 is less than E [exp (2RTr (ΛG1 ΛG2 ) + βTr ((G1 G∗1 ) + Tr (G2 G∗2 )))] √ 2 with β = 2r δ(2 + δ) kΛk . The above expectation is to be understood as the expectation over (G1 , G2 ). As G1 and G2 are independent, we consider first the expectation over G1 . This gives, up to the factor exp (βTr (G2 G∗2 )) Z −r 2 π exp (2RTr (g1 E) + (β − 1) Tr (g1∗ g1 ) ) dg1 with E = ΛG2 Λ. It is always possible to choose δ such that β < 1. With such a β, the above integral is ! 2 2 1 −r 2 √ Tr (EE∗ ) (1 − β) exp 4 1−β 4 As Tr (EE∗ ) ≤ kΛk2 Tr (G2 G∗2 ) we finally obtain after multiplying by exp (βTr (G2 G∗2 )) and taking the expectation over G2 : ! 4 −r 2 Z 2 (1 − β) − kΛk (1 − β) 2 E20 ≤ exp − Tr (g2∗ g2 ) dg2 . 1−β π r2 2 If kΛk2 < 1, it is always possible to adjust δ such that the above integral converges. In this condition, we have !r2 1 E20 ≤ . 4 (1 − β)2 − kΛk2 This must be true for all β arbitrarily small, hence the result. A PPENDIX We prove Theorem 3 when x > 0. As the function to be maximized converges towards −∞ if kψ 1 k → 1 or kψ 2 k → 1, any argument (ψ 1 , ψ 2 ) of the maximization problem satisfies kψ i k < 1, i = 1, 2. Therefore, the Karush-Kuhn-Tucker (KKT) conditions imply the existence P2 of a scalar Lagrange multiplier µ ≥ 0 such that (ψ 1 , ψ 2 ) is a stationary point of the Lagrangian `(ψ 1 , ψ 2 , µ) defined by i=1 log det (Ir − ψ ∗i ψ i ) + µ RTr (Λψ 1 Λψ 2 ) . As ` is a real valued function, a stationary point is computedwhen setting the differential w.r.t. the entries of ψ 1 and ψ 2 to zero. It can be checked that (ψ 1 , ψ 2 ) is a stationary point of ` when µ Λψ 2 Λ = ψ ∗1 (I − ψ 1 ψ ∗1 )−1 µ Λψ 1 Λ = ψ ∗2 (I − ψ 2 ψ ∗2 )−1 In a first step, these equations can be shown to be satisfied only if ψ 1 and ψ 2 are diagonal up to permutations of the columns. Then, is can be deduced that there exists a diagonal matrix 0 ≤ P ≤ I and a matrix of permutation Π such that 6 log det (Ir − ψ ∗1 ψ 1 ) + log det (Ir − ψ ∗2 ψ 2 ) = 2 log det(I − P)and RTr (Λψ 1 Λψ 2 ) = Tr(ΛΠ∗ ΛΠ P). This invites us to consider the following Problem 5. Maximize log det(I − P) (9) jointly over all the r! permutations Π and over diagonal matrices P verifying 0 ≤ P ≤ I and the constraint Tr(ΛΠ∗ ΛΠ P) = x. (10) In a first step, we set Π = I in the above problem and consider the Problem 6. Maximize r X log(1 − pi ) (11) under the constraints that 0 ≤ pi ≤ 1 for each i = 1, . . . , r and r X λ2i pi = x. (12) i=1 i=1 The maximum is denoted by JΛ (x). This is a variant of the celebrated water-filling problem (see e.g. [16] and Chap. 9 of [6]) that was solved to evaluate the capacity of a frequency selective Gaussian channel, the difference being that in the latter problem, log(1 − pi ) is replaced by log(1 + pi ).In order to solve Problem 6, we assume that the non zero singular values (λi )i=1,...,r are distinct. If this is not the case, a standard perturbation argument can be used in order to address the general case. As the function to be maximized is strictly concave on the set defined by the constraints, the maximum is reached p∗ verifying pi,∗ Pr at a unique point P Pr< 1 for r 2 each i. We consider the Lagrangian corresponding to Problem (6) given by i=1 log(1 − pi ) + µ λ p + i=1 i i i=1 δi pi where µ ≥ 0 and δi ≥ 0 for i = 1, . . . , r. The partial derivatives w.r.t. parameters (pi )i=1,...,r are zero at p∗ . This leads to for i = 1, . . . , r : 1 = µ∗ λ2i + δi,∗ 1 − pi,∗ (13) The first remark is that necessarily, these equations imply that the numbers pi,∗ are sorted in decreasing order. To verify this claim, we assume that i < j and that pi,∗ = 0 and pj,∗ > 0. Then, it holds that µ∗ λ2i + δi,∗ = 1 and that µ∗ λ2j = 1−p1 j,∗ > 1 because pj,∗ > 0 implies δj,∗ = 0. Therefore, λ2i ≤ µ1∗ < λ2j , a contradiction because λ2i ≥ λ2j . We denote by s(x) the number of non-zero entries of p∗ . Hence, the first s(x) entries of p∗ are non zero. Morever, the equations µ∗ λ2i = 1−p1 i,∗ for i = 1, . . . , s(x) imply that p1,∗ ≥ . . . ≥ ps(x),∗ > 0 = ps(x)+1,∗ = . . . = pr,∗ . We now analytically characterize s(x). On the one hand, (13) computed at for i = s(x) and for i = s(x) + 1 both imply 1 λ2s(x)+1 ≤ < λ2s(x) (14) µ∗ On the other hand, the constraint (12) imposes that 1/µ∗ verifies Ps(x) 2 λ −x 1 = i=1 i . µ∗ s(x) Therefore, it holds that s(x) ( X s(x) λ2i ) − s(x)λ2s(x) < x ≤ ( i=1 X λ2i ) − s(x)λ2s(x)+1 (15) i=1 such that s(x) coincides with the integer k for which x ∈ Ik (see (7) for the definition of these intervals). The maximum Ps(x) i=1 log(1 − pi,∗ ) is direcly computed as "  #s(x) Ps(x) 2 λ − x 1 i i=1  JΛ (x) = log  (16) s(x) 2 s(x) Π λ i=1 i In order to show that the GRF of η is Iη (x) = −2JΛ (x), it remains to show that the solution of Problem 5 is reached when the permutation matrix Π is the identity. In this respect, we introduce a nested problem motivated by the following observation. We denote by α and β the r–dimensional vectors whose components are respectively the diagonal entries of Λ2 and of ΛΠ∗ ΛΠ arranged in the decreasing order. Evidently, α majorizes β in the sense that for k = 1, . . . , r : k X i=1 αi ≥ k X i=1 βi (17) 7 We thus consider the relaxed problem Problem 7. Maximize log det(I − P) over the diagonal matrices 0 ≤ P ≤ I and over vectors β = (β1 , ..., βr ) satisfying β1 ≥ β2 ≥ . . . βr ≥ 0, the majorization constraint (17), and the equality constraint r X βi pi = x (18) i=1 The maximum of Problem 7 is above the maximum of Problem 5 which is itself above the maximum JΛ (x) of Problem 6. We actually show that the maximum of Problem 7 is less than JΛ (x), and that it is reached for a vector β that coincides with α. This will imply that the optimal permutation Π in Problem 5 is I and Iη (x) = −2JΛ (x). We give some elements for solving Problem 7. We consider a stationary point (p∗ , β ∗ ) of the associated Lagrangian and compute the KKT conditions. We suppose that this stationary point attains the maximum. If s denotes the number of non-zero components in p∗ , we prove that, necessarily, p1,∗ ≥ p2,∗ ≥ ... ≥ ps,∗ > 0 and β1,∗ ≥ β2,∗ ≥ ... ≥ βs,∗ . We let j1 be the Pj1 Pj1 βi (this index exists otherwise β ∗ = α and the problem is solved). This implies that αi > i=1 first index such that i=1 Pj1 +k Pj+k βi,∗ = αi for all indices i = 1, ..., j1 − 1. Notice this fact: if we suppose that the condition i=1 αi > i=1 βi is true whatever k, then it is possible to add a small > 0, and update βj1,∗ as βj1 ,∗ + in such a way that the majorization constraints still hold, the constraint (18) holds and the updated p∗ increases the function to maximize. This isPin contradiction with the Pj1 +j j1 +j2 2 βi,∗ . αi = i=1 definition of (p∗ , β ∗ ). This means that there exists an index j2 (we choose the smallest) such that i=1 It can be shown that it is necessary that all the βi,∗ are equal for i = j1 , ..., j1 + j2 . 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Recurrent Neural Network Language Models for Open Vocabulary Event-Level Cyber Anomaly Detection Aaron Tuor,1 Ryan Baerwolf,2 Nicolas Knowles,2 Brian Hutchinson,1,2 Nicole Nichols1 and Robert Jasper1 1 Pacific Northwest National Laboratory Richland, Washington 2 Western Washington University Bellingham, Washington Abstract Automated analysis methods are crucial aids for monitoring and defending a network to protect the sensitive or confidential data it hosts. This work introduces a flexible, powerful, and unsupervised approach to detecting anomalous behavior in computer and network logs; one that largely eliminates domain-dependent feature engineering employed by existing methods. By treating system logs as threads of interleaved “sentences” (event log lines) to train online unsupervised neural network language models, our approach provides an adaptive model of normal network behavior. We compare the effectiveness of both standard and bidirectional recurrent neural network language models at detecting malicious activity within network log data. Extending these models, we introduce a tiered recurrent architecture, which provides context by modeling sequences of users’ actions over time. Compared to Isolation Forest and Principal Components Analysis, two popular anomaly detection algorithms, we observe superior performance on the Los Alamos National Laboratory Cyber Security dataset. For log-line-level red team detection, our best performing character-based model provides test set area under the receiver operator characteristic curve of 0.98, demonstrating the strong fine-grained anomaly detection performance of this approach on open vocabulary logging sources. 1 Introduction To minimize cyber security risks, it is essential that organizations be able to rapidly detect and mitigate malicious activity on their computer networks. These threats can originate from a variety of sources including malware, phishing, port scanning, etc. Attacks can lead to unauthorized network access to perpetrate further damage such as theft of credentials, intellectual property, and other business sensitive information. In a typical scenario, cyber defenders and network administrators are tasked with sifting through vast amounts of data from various logging sources to assess potential security risks. Unfortunately, the amount of data for even a modestly-sized network can quickly grow beyond the ability of a single person or team to assess, leading to delayed response. The desire for automated assistance has and continues to encourage inter-domain research in cyber security and machine learning. Signature-based approaches for automated detection can be highly effective for characterizing individual threats. De- spite their high precision, they suffer from low recall and may fail to detect subtle mutations or novel attacks. Alternatively, given an unlabeled training set of typically benign activity logs, one can build a model of “normal behavior”. During online joint training and evaluation of this model, patterns of normal usage will be reinforced and atypical malicious activity will stand out as anomalous. The features used to identify unusual behavior are typically statistical feature vectors associated with time slices, e.g., vectors of counts for types of activities taking place in a 24-hour window. Such systems developed in research have been criticized as brittle to differences in site-specific properties of real-world operational networks such as security constraints and variable usage patterns (Sommer and Paxson 2010). The approach we introduce aims to minimize site-specific assumptions implicit in feature engineering, and effectively model variability in network usage by direct online learning of language models over log lines. Language models assign probabilities to sequences of tokens and are a core component of speech recognition, machine translation, and other language processing systems. Specifically, we explore the effectiveness of several recurrent neural network (RNN) language models for use in a network anomaly detection system. Our system dynamically updates the network language model each day based on the previous day’s events. When the language model assigns a low probability to a log-line it is flagged as anomalous. There are several advantages to this approach: 1. Reduced feature engineering: Our model acts directly on raw string tokens, rather than hand-designed domainspecific statistics. This dramatically reduces the time to deployment, and makes it agnostic to the specific network or logging source configuration. It also removes the “blind spots” introduced when tens of thousands of log-lines are distilled down to a single aggregated feature vector, allowing our model to capture patterns that would have otherwise been lost. 2. Fine grained assessment: The response time for analysts can be improved by providing more specific and relevant events of interest. Baseline systems that alert to a user’s day aggregate require sifting through tens of thousands of actions. Our approach can provide log-line-level or even token-level scores to the analyst, helping them quickly lo- cate the suspicious activity. 3. Real time processing: With the ability to process events in real time and fixed bounds on memory usage which do not grow over time, our approach is suitable for the common scenario in which log-line events are appearing in a high-volume, high-velocity log stream. We assess our models using the publicly available Los Alamos National Laboratory (LANL) Cyber Security Dataset, which contains real (de-identified) data with ground truth red team attacks, and demonstrate language models definitively outperforming standard unsupervised anomaly detection approaches. 2 Prior work Machine learning has been widely explored for network anomaly detection, with techniques such as isolation forest (Gavai et al. 2015; Liu, Ting, and Zhou 2008) and principal component analysis (Novakov et al. 2013; Ringberg et al. 2007) attracting significant interest. Machine learning classifiers ranging from decision trees to Naı̈ve Bayes have been used for cyber security tasks such as malware detection, network intrusion, and insider threat detection. Extensive discussion of machine learning applications in cyber security is presented in (Bhattacharyya and Kalita 2013; Buczak and Guven 2016; Dua and Du 2016; Kumar, Kumar, and Sachdeva 2010; Zuech, Khoshgoftaar, and Wald 2015; Rubin-Delanchy, Lawson, and Heard 2016). Deep learning approaches are also gaining adoption for specialized cyber defense tasks. In an early use of recurrent neural networks, Debar, Becker, and Siboni (1992) model sequences of Unix shell commands for network intrusion detection. Anomaly detection has been demonstrated using deep belief networks on the KDD Cup 1999 dataset (Alrawashdeh and Purdy 2016), and Bivens et al. (2002) use multi-layer perceptrons for the DARPA 1999 dataset. Both approaches use aggregated features and synthetic network data. Tuor et al. (2017) and Veeramachaneni et al. (2016) both employ deep neural network autoencoders for unsupervised network anomaly detection using time aggregated statistics as features. Some works of note have been previously published on the LANL data. Turcotte, Heard and Kent (2016) develop an online statistical model for anomaly detection in network activity using Multinomial-Dirichlet models. Similarly, Turcotte et al. (2016) use Poission Factorization (Gopalan, Hofman, and Blei 2013) on the LANL authentication logs. A user/computer authentication count matrix is constructed by assuming each count comes from a Poisson distribution parameterized by latent factors for users and computers. The learned distributions are then used to predict unlikely authentication behavior. Several variants of tiered recurrent networks have been explored in the machine learning and natural language processing communities (Koutnik et al. 2014; Ling et al. 2015b; Ling et al. 2015a; Chung et al. 2015). They are often realized by a lower tier pre-processing network, whose output is fed to an upper tier network and the separate tiers are jointly trained. Ling et al. (2015b) use a character-level convolutional neural network to feed a word level long short-term memory (LSTM) RNN for machine translation, with predictions made at the word-level. Both Hwang and Sung (2016) and Ling et al. (2015a) use a character-based LSTM to feed a second word or utterance-based LSTM for language modeling. Pascanu et al. (2015) create activity models from real world data on a per-event (command) basis and sequences of system calls are then modeled using RNN and echo state networks. The learned features are used to independently train neural network and logistic regression classifiers. Max pooling is applied to hidden layers of the unsupervised RNN for each time step in a session and the result is concatenated to the final hidden state to produce feature vectors for the classifier. This is similar to our tiered approach, in which we use the average of all hidden states concatenated with the final hidden state as input to the upper-tier RNN. In contrast, our model is completely unsupervised and all components are jointly trained. 3 Approach Our approach learns normal behavior for users, processing a stream of computer and network log-lines as follows: 1. Initialize model weights randomly 2. For each day k in chronological order: (a) Given model Mk−1 , produce log-line-level anomaly scores for all events in day k (b) Optionally, produce an aggregated anomaly score each user for day k (from the log-line-level scores) (c) Send per-user-day or per-user-event anomaly scores in rank order to analysts for inspection (d) Update model weights to minimize loss on all log-lines in day k, yielding model Mk This methodology interleaves detection and training in an online fashion. In this section we detail the components of our approach. 3.1 Log-Line Tokenization To work directly from arbitrary log formats, we treat loglines as sequences of tokens. For this work, we consider two tokenization granularities: word-level and character-level. For word tokenization, we assume that tokens in the logline are delimited by a known character (e.g., space or comma). After splitting the log-lines on this delimiter, we define a shared vocabulary of “words” over all log fields, consisting of the sufficiently-frequent tokens appearing in the training set. To allow our model to handle previously unseen tokens, we add an “out of vocbulary” token to our vocabulary, <oov>. (For instance, not every IP address will be represented in a training set; likewise, new PCs and users are continually being added to large networks.) To ensure that <oov> has non-zero probability, we replace sufficiently infrequent tokens in the training data with <oov>. During evaluation, tokens not seen before are labeled <oov>. In order to accommodate shifting word distributions in an online environment, a fixed size vocabulary could be periodically updated using a sliding window of word frequency statistics. For simplicity, we assume we have a fixed training set from which we produce a fixed vocabulary. To avoid the challenges of managing a word-level vocabulary, we also develop language models using a characterlevel tokenization. In this case our primitive vocabulary, the alphabet of printable ASCII characters, circumvents the open vocabulary issue by its ability to represent any log entry irrespective of the network, logging source, or log field. With character-level tokenization, we keep the delimiter token in the sequence, to provide our models with cues to transitions between log-line fields. 3.2 Recurrent Neural Network Language Models To produce log-line-level anomaly scores, we use recurrent neural networks in two ways: 1) as a language model over individual log-lines, and 2) to model the state of a user over time. We first present two recurrent models that focus only on (1), and then a tiered model that accomplishes both (1) and (2). Both were implemented1 for our experiments using TensorFlow (Abadi et al. 2015). Event Model (EM). First we consider a simple RNN model that operates on the token (e.g., word) sequences of individual log-lines (events). Specifically, we consider a Long Short-Term Memory (LSTM) (Hochreiter and Schmidhuber 1997) network whose inputs are token embeddings and from whose output we predict distributions over the next token. For a log-line with K tokens, each drawn from a shared vocabulary of size C, let X(1:K) = x(1) , x(2) , . . . , x(K) denote a sequence of one-hot representations of the tokens (each x(t) ∈ RC ). In this model, the hidden representation at token t, h(t) , from which we make our predictions, is a function of x(1) , x(2) , . . . , x(t) according to the usual LSTM equations: h(t) c(t) = = o(t) ◦ tanh(c(t) ) f(t) ◦ c(t−1) + i(t) ◦ g(t) g(t) = f(t) = i(t) = o(t) = tanh x(t) W(g,x) + h(t−1) W(g,h) + b(g) σ x(t) W(f,x) + h(t−1) W(f,h) + b(f ) σ x(t) W(i,x) + h(t−1) W(i,h) + b(i) σ x(t) W(o,x) + h(t−1) W(o,h) + b(o) , (1) (2) (3) (4) (5) (6) where the initial hidden and cell states, c(0) and h(0) , are set to zero vectors, and ◦ and σ denote element-wise multiplication and logistic sigmoid, respectively. Vector g(t) is a hidden representation based on the current input and previous hidden state, while vectors f(t) , i(t) , and o(t) , are the standard LSTM gates. The matrices (W) and bias vectors (b) are the model parameters. We use each h(t−1) to produce a probability distribution p(t) over the token at time t, as follows: p(t) = softmax h(t−1) W(p) + b(p) (7) 1 Code will soon be available at https://github.com/pnnl/safekit p(1) p(2) p(K-1) p(K) softmax softmax softmax softmax LSTM LSTM LSTM LSTM <sos> x(1) x(K-2) x(K-1) LSTM LSTM LSTM LSTM x(2) x(3) x(K) <eos> … Figure 1: Event Models. Set of black bordered nodes and connections illustrate the EM model while set of all nodes and connections illustrate the BEM model. We use cross-entropy loss, K 1 X H(x(t) , p(t) ), K t=1 (8) for two important purposes: first, as per-log-line anomaly score and second, as the training objective to update model weights. We train this model using stochastic mini-batch (non-truncated) back-propagation through time. Bidirectional Event Model (BEM). Following the language model formulation suggested in (Schuster and Paliwal 1997), we alternatively model the structure of log lines with a bidirectional LSTM. We define a new set of hidden vectors hb(K+1) , hb(K) , . . . , hb(1) by running the LSTM equations backwards in time (starting with initial zero cell and hidden states at time K + 1 set to zero). The weights W and biases b for the backward LSTM are denoted with superscript b. The probability distribution p(t) over the token at time t is then: b p(t) = softmax h(t−1) W(p) + hb(t+1) W(p) + b(p) (9) Tiered Event Models (T-EM, T-BEM). To incorporate inter-log-line context, we propose a two-tiered recurrent neural network. The lower-tier can be either event model (EM or BEM), but with the additional input of a context vector (generated by the upper-tier) concatenated to the token embedding at each time step. The input to the upper-tier model is the hidden states of the lower-tier model. This upper tier models the dynamics of user behavior over time, producing the context vectors provided to the lower-tier RNN. This model is illustrated in Fig. 2. In this model, x(u,j) denotes user u’s jth log line, which consists of a sequence of tokens as described in the previous subsections. The upper-tier models a sequence of user log lines, x(u,1) , x(u,2) , . . . , x(u,Tu ) , using an LSTM. For each user u and each log line j in the user’s log line sequence, a lower-tier LSTM is applied to the tokens of x(u,j) . The input to the upper-tier model at log-line j is the concatenation of: 1) the final lower-tier hidden state(s) and 2) the average of the lower-tier hidden states. In the case of a lower-tier EM, Context LSTM Mean LSTM LSTM Final LSTM … <sos> Context LSTM <eos> Mean LSTM <sos> LSTM Final LSTM … <eos> Figure 2: Tiered Event Model (T-EM) (1) refers to the hidden state at time K; for the BEM, (1) is the concatenation of the forward hidden state at time K and the backward hidden state at time 1. For (2), we average over hidden states primarily to provide many short-cut connections in the LSTM, which aids trainability. The output of the upper-tier LSTM at log-line j is a hidden state ĥ(u,j) . This hidden vector serves to provide context for the lower-tier model at the next time step: specifically, ĥ(u,j−1) is concatenated to each of the inputs of the lower-tier model operating on the jth log-line. Note that the upper-tier model serves only to propagate context information across individual log-lines; no loss is computed directly on the values produced by the upper-tier model. The upper- and lower-tier models are trained jointly to minimize the cross-entropy loss of the lower-tier model. We unroll the two-tier model for a fixed number of log-lines, fully unrolling each of the lower-tier models within that window. The lower-tier model’s cross-entropy loss is also used to detect anomalous behavior, as is described further in Section 4.2. Minibatching becomes more challenging for the tiered model, as the number of log-lines per day can vary dramatically between users. This poses two problems: first, it introduces the possibility that the most active users may have a disproportionate impact on model weights; second, it means that toward the end of the day, there may not be enough users to fill the minibatch. To counteract the first problem, we fix the number of log-lines per user per day that the model will train on. The remaining log-lines are not used in any gradient updates. We leave compensating for the inefficiency that results from the second to future work. 3.3 Baselines Anomaly detection in streaming network logs often relies upon computing statistics over windows of time and applying anomaly detection techniques to those vectors. Below we describe the aggregate features and two anomaly detection techniques that are typical of prior work. Aggregate Features We first define the set of per-userday features, which summarize users’ activities in the day. To aggregate the features that have a small number of distinct values (e.g. success/failure, logon orientation) we count the number of occurrences for each distinct value for the user-day. For fields that have a larger number of distinct values (pcs, users, domains), we count the number of common and uncommon events that occurred, rather than the number of occurrences of each distinct value (this approach avoids high dimensional sparse features). Furthermore, we define two categories of common/uncommon; to the individual entity/user, and relative to all users. A value is defined as uncommon for the user if it accounts for fewer than 5% of the values observed in that field (up to that point in time), and common otherwise. A value is defined as uncommon for all users if it occurs fewer times than the average value for the field, and common otherwise. For the LANL dataset, the prior featurization strategy yields a 108-dimensional aggregate feature vector per userday. These feature vectors then serve as the input to the baseline models described next. Models We consider two baseline models. The first uses Principal Components Analysis (pca) to learn a low dimensional representation of the aggregate features; the anomaly score is proportional to the reconstruction error after mapping the compressed representation back into the original dimension (Shyu et al. 2003). The second is an isolation forest (iso) based approach (Liu, Ting, and Zhou 2008) as implemented in scikit-learn’s outlier detection tools (Pedregosa et al. 2011). This was noted as the best performing anomaly detection algorithm in the recent DARPA insider threat detection program, (Gavai et al. 2015). 4 Experiments In this section we describe experiments to evaluate the effectiveness of the proposed event modeling algorithms. 4.1 Data The Los Alamos National Laboratory (LANL) Cyber Security Dataset (Kent 2016) consists of event logs from LANL’s internal computer network collected over a period of 58 consecutive days. The data set contains over one billion loglines from authentication, process, network flow, and DNS logging sources. Identifying fields (e.g., users, computers, and processes) have been anonymized. The recorded network activities included both normal operational network activity as well as a series of red team activities that compromised account credentials over the first 30 days of data. Information about known red team attack events is used only for evaluation; our approach is strictly unsupervised. For the experiments presented in this paper, we rely only on the authentication event logs, whose fields and statistics are summarized in Figure 3a. We filter these events to only those log-lines linked to an actual user, removing computercomputer interaction events. Events on weekends and holidays contain drastically different frequencies and distributions of activities. In a real deployment a separate model would be trained for use on those days, but because no malicious events were in that data it was also withheld. Table 3b has statistics of our data split; the first 12 days serve as the development set, while the remaining 18 days are the independent test set. 4.2 Assessment Granularity Our model learns normal behavior and assigns relatively high loss to events that are unexpected. A principal advantage of our approach is this ability to score the anomaly of Field time source user dest. user source pc dest. pc auth. type logon type auth. orient success Example 1 C625@DOM1 U147@DOM1 C625 C625 Negotiate Batch LogOn Success # unique labels 5011198 80553 98563 16230 15895 29 10 7 2 Model pca iso EM BEM T-EM T-BEM EM BEM T-EM T-BEM (a) Days # Events # Attacks # User-days Dev 1-12 133M 316 57 Test 13-58 918M 385 79 W W W W C C C C Figure 3: Dataset statistics: (a) Authentication log fields and statistics and (b) dataset splits. 4.3 Metrics We consider two performance metrics. First, we assess results using the standard area under the receiver operator characteristic curve (AUC) which characterizes the trade-off in model detection performance between true positives and false positives, effectively sweeping through all possible analyst budgets. False positives are detections that are not truly red team events, while true positives are detections that are. To quantify the proportion of the data the analyst must sift through to diagnose malicious behavior on the network, we use the Average Percentile (AP) metric. Specifically, for each red team event or user-day, we note the percentile of its anomaly amongst all anomaly scores for the day. We then average these percentiles for all of the malicious events or AUC 0.754 0.763 0.802 0.876 0.782 0.864 0.750 0.843 0.772 0.837 AP 73.9 75.0 79.3 87.0 77.5 85.7 70.9 82.9 76.2 82.9 Table 1: User-day granularity test set AUC and AP. Language model anomaly scores calculated with average userday normalization (diff). (b) individual events, allowing us to flag at the event-level or aggregate anomalies over any larger timescale. For this work, we consider two timescales. First, we assess based on individual events; a list of events would be presented to the analyst, sorted descending by anomaly score. Second, to facilitate comparison with traditional aggregation methods, we aggregate anomaly scores over all of a user’s events for the day (specifically, taking the max), producing a single anomaly score per-user, per-day. In this scenario, a list of user-days would be provided to the analyst, sorted descending by anomaly score. We refer to this approach as max, because the anomaly scores provided to the analyst are produced by taking the maximum score over the event scores in the window for that user (where event-level scoring is just taking the max over a singleton set of one event). In order to counter systematic offsets in users’ anomaly scores for a day we also consider a simple normalization strategy, which we refer to as diff, by which every raw score is first normalized by subtracting the user’s average event-level anomaly score for the day. Tokenization Word Word Word Word Char Char Char Char EM BEM T-EM T-BEM EM BEM T-EM T-BEM LOG diff max 0.964 0.932 0.974 0.895 0.959 0.948 0.959 0.902 0.940 0.935 0.973 0.979 0.859 0.927 0.945 0.969 DAY diff max 0.802 0.794 0.876 0.811 0.782 0.803 0.864 0.838 0.751 0.754 0.843 0.846 0.772 0.809 0.837 0.854 Table 2: Comparison of AUC for day-level and log-line-level analysis with and without user-day normalization. Figures 4 and 5 provide a visualization of these results. user-days. Note that if all true malicious events or user-days are flagged as the most anomalous on the respective days, then AP ≈ 100, while if all malicious events or user-days are ranked as the least anomalous on their respective days, AP ≈ 0. For both AUC and AP, a higher score is better. Our model hyperparameters were manually tuned to maximize AP for day-level diff scores on the development set. No separate training set is needed as our approach is unsupervised and trained online. 4.4 Results and Analysis We begin by exploring the user-day granularity performance. Table 1 summarizes model detection performance at this granularity on the test set for the AUC and AP metrics using the diff method to produce day level scores from the language models. A few trends are evident from these results. First, the aggregate feature baselines have nearequivalent performance by both metrics, with the isolation forest approach having a slight edge. We hypothesize the feature representation, which is common to these methods, could be a bottleneck in performance. This highlights the “blind spot” issue feature engineering introduces. Second, despite having only the context of a single log-line at a time, as opposed to features aggregated over an entire day, the event model (EM) performs comparably to the baseline models when a forward pass LSTM network is used with 100 8079 97 8987 81 9594 7880 40 9393 9797 8484 80 60 20 100 95 90 86 83 AUC AUC 80 96 93 7575 92 85 80 77 96 94 8385 60 40 20 diff max LOG \| DAY W EM LOG \| DAY W BEM LOG \| DAY W T-EM LOG \| DAY W T-BEM diff max LOG \| DAY W EM LOG \| DAY W BEM LOG \| DAY W T-EM LOG \| DAY W T-BEM Figure 4: Word model comparison of AUC at day-level and log-line-level granularities. Figure 5: Character model comparison of AUC at day-level and log-line-level granularities. a character tokenization and outperforms the baselines with word tokenization. The most pronounced performance gain results from using bidirectional models. Finally, word-level tokenization performs better than character-level; however, even the bidirectional character models perform appreciably better than the baselines. It is clear from these results that the tiered models perform comparably to, but not better than, the event-level models. This suggests that the event level model is able to characterize normal user behavior from information stored in the model weights of the network, which are trained each day to model user activity. Given the context of the past day’s activity stored in the model weights, the categorical variables represented by the fields in an individual log line may eliminate the need for explicit event context modeling. We leave tracking the state of individual computers, rather than users, to future work, but hypothesize that it may make the tiered approach more effective. Next, we broaden our analysis of language modeling approaches, comparing performance across all language models, tokenization strategies, anomaly granularity, and normalization techniques. Figure 4 plots AUC for all language model types using word tokenization, contrasting max and diff normalization modes. Figure 5 compares the same variations for character tokenization. Table 2 presents these results in tabular form. With few exceptions, log-line-level granularity vastly outperforms day-level; this is true for both the character-level and word-level tokenization strategies, with an average gain of 0.1 AUC. The most interesting outcome of these comparisons is that word tokenization performance gains are heavily reliant on the diff normalization, whereas for character tokenization the diff normalization has a minor detrimental effect for some models. This suggests that the character-level model could be used to provide a more immediate response time, not having to wait until the day is done to obtain the day statistics used in diff mode. The two tokenization strategies may in fact be complementary as the versatility and response time gains of a character tokenization come at the expense of easy interpretibility of a word tokenization: the word tokenization allows anomaly scores to be decomposed into individual log-line fields, enabling analysts to pinpoint features of the event that most contributed to it being flagged. Since we tuned hyperparameters using diff mode, the character-level model has potential to do better with additional tuning. Next, Figures 6 and 7 visualize the average percentiles of red team detections for the subset of the test set with the most red-team activity. Anomaly scores for both word and character tokenizations are computed without average userday offset normalization. Red team log-line-level scores are plotted as purple x’s with the x coordinate being the second in time at which the event occurred and y coordinate the anomaly score for that event. Percentile ranges are colored to provide context for the red-team anomaly scores against the backdrop of other network activity. The spread of non-normalized anomaly scores is much greater for the word-level tokenizations (Fig. 7) than character-level (Fig. 6), which could explain the different sensitivity of word level tokenization to normalization. Also notice that there is an expected bump in percentiles for windows of frequent redteam activity. Curiously, at the end of day 14 there are massive bumps for the 99th percentile, which suggest unplanned and un-annotated anomalous events on the LANL network for those hours. Notice that for the character tokenization almost all non-normalized red team anomaly scores are above the 95th percentile, with a large proportion above the 99th percentile. Finally, Figure 8 plots the ROC curves for the best aggregate baseline (iso), the best user-day granularity language model (word BEM), and the best event-level granularity model (character BEM). It illustrates the qualitatively different curves obtained with the baselines, the user-day granularity, and the event-level granularity. Since the proportion of red-team to normal events is vanishingly low in the data-set (< 0.001%), the false-positive rate is effectively the proportion of data flagged to achieve a particular recall. From this observation, Figure 8 shows the character event model can achieve 100% recall from only 12% of the data whereas the other models considered only achieve 100% recall when nearly all of the data has been 2.00 75-95 Percentile 95-99 Percentile Red-event 1.75 Anomaly Score 1.50 1.25 1.00 0.75 0.50 0.25 15 pm am pm ay D 6 12 14 6 pm pm ay D 6 12 am 6 13 D ay pm pm 6 12 12 6 D ay am 0.00 Day/Time Figure 6: Character-level red-team log-line anomaly scores in relation to percentiles over time. 1.0 True positive 0.8 0.6 0.4 0.2 iso agg 0.76 AUC W BEM day 0.88 AUC C BEM log-line 0.98 AUC 0.0 0.0 0.2 0.4 0.6 False positive 0.8 1.0 Figure 8: ROC curves for best performing baseline, word language model evaluated at day-granularity, and character language model evaluated at log-line-granularity. handed to the analyst. Further, the character event model can achieve 80% recall by flagging only 3% of the data whereas the word day language model needs 14% of the data and the aggregate isolation forest model needs 55% of the data to achieve the same result. 5 Conclusion This work builds upon advances in language modeling to address computer security log analysis, proposing an unsupervised, online anomaly detection approach. We eliminate the usual effort-intensive feature engineering stage, making our approach fast to deploy and agnostic to the system configuration and monitoring tools. It further confers the key advantage of event-level detection which allows for a near immediate alert response following anomalous activity. In experiments using the Los Alamos National Laboratory Cyber Security Dataset, bidirectional language models significantly outperformed standard methods at day-level Figure 7: Word-level red-team log-line anomaly scores in relation to percentiles over time. detection. The best log-line-level detection performance was achieved with a bidirectional character-based language model, obtaining a 0.98 area under the ROC curve, showing that for the constrained language domain of network logs, character based language modeling can achieve comparable accuracy to word based modeling for event level detection. We have therefore demonstrated a simple and effective approach to modeling dynamic networks with open vocabulary logs (e.g. with new users, PCs, or IP addresses). We propose to extend this work in several ways. First, potential modeling advantages of tiered architectures merit further investigation. The use of tiered architectures to track PCs instead of network users, or from a richer set of logging sources other than simply authentication logs may take better advantage of their modeling power. Next, we anticipate interpretability can become lost with such detailed granularity provided by log-line-level detection from a characterbased model, therefore future work will explore alternate methods of providing context to an analyst. Finally, we are interested in exploring the robustness of this approach to adversarial tampering. Similarly performing models could have different levels of resilience that would lead to selection of one over another. Acknowledgments The research described in this paper is part of the Analysis in Motion Initiative at Pacific Northwest National Laboratory. It was conducted under the Laboratory Directed Research and Development Program at PNNL, a multi-program national laboratory operated by Battelle for the U.S. Department of Energy. The authors would also like to thank the Nvidia corporation for their donations of Titan X and Titan Xp GPUs used in this research. 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IEEE SIGNAL PROCESSING LETTERS 1 Look Wider to Match Image Patches with Convolutional Neural Networks arXiv:1709.06248v1 [cs.CV] 19 Sep 2017 Haesol Park, and Kyoung Mu Lee Abstract—When a human matches two images, the viewer has a natural tendency to view the wide area around the target pixel to obtain clues of right correspondence. However, designing a matching cost function that works on a large window in the same way is difficult. The cost function is typically not intelligent enough to discard the information irrelevant to the target pixel, resulting in undesirable artifacts. In this paper, we propose a novel convolutional neural network (CNN) module to learn a stereo matching cost with a large-sized window. Unlike conventional pooling layers with strides, the proposed per-pixel pyramid-pooling layer can cover a large area without a loss of resolution and detail. Therefore, the learned matching cost function can successfully utilize the information from a large area without introducing the fattening effect. The proposed method is robust despite the presence of weak textures, depth discontinuity, illumination, and exposure difference. The proposed method achieves near-peak performance on the Middlebury benchmark. Index Terms—stereo matching,pooling,CNN I. I NTRODUCTION Most stereo matching methods first compute the matching cost of each pixel with a certain disparity, before optimizing the whole cost volume either globally or locally by using specific prior knowledge [1]. For decades, many researchers have focused on the second step, designing a good prior function and optimizing it [2], [3], [4], [5], [6]. Few studies have been conducted on designing or selecting a better matching cost function. One of the most widely used matching cost functions is a pixel-wise matching cost function, such as the one used in [7]. Along with sophisticated prior models, it sometimes produces good results, especially in preserving the detailed structures near the disparity discontinuities. However, the function fails when the image contains weakly-textured areas or repetitive textures. In such cases, a window-based matching cost, such as CENSUS or SAD [8], produces a more reliable and distinctive measurement. The critical shortcoming of window-based matching cost functions is their unreliability around disparity discontinuities. Figure 1 visually illustrates the characteristics of different matching cost measures. One method to handle this trade-off is to make the windowbased versatile to its input patterns [10], [11], [12]. The key idea is making the shape of the matching template adaptive so that it can discard the information from the pixels that are irrelevant to the target pixel. However, knowing the background pixels before the actual matching is difficult, making it a H. Park and K. M. Lee are with Automation and Systems Research Institute, Seoul National University, Seoul 151-744, Korea matching cost for Pixelwise matching cost for 1 1 0.5 0.5 0 1 0 1 SAD (11x11) 0.5 0.5 0 1 0 1 SAD (37x37) 0.5 0.5 0 1 0 1 CENSUS (11x11) 0.5 0.5 0 1 0 1 CENSUS (37x37) 0.5 0.5 0 1 0 1 MC-CNN-arct[13] (11x11) 0.5 0.5 0 1 0 1 0.5 0.5 0 0 Proposed (37x37) Fig. 1. The top image shows the reference image with two interested points, x1 and x2 . The pixel positions are marked as blue dots, whereas the red and green boxes represent 37 × 37 and 11 × 11 windows centered on them, respectively. At the bottom, the matching costs for each pixel are visualized as a normalized function of disparity for different matching cost functions. The positions of true disparities are marked as red vertical lines. The pixelwise cost shows the lowest values at the true disparity, but it also gives zero costs for other disparities. The SAD and CENSUS matching cost functions [9] become less ambiguous as the matching window becomes larger. However, these functions are affected by pixels irrelevant to the target pixel (x2 ). Even the matching cost learned by using the baseline convolutional neural network (CNN) architecture fails when the surface has a nearly flat texture (x1 ). On the other hand, the proposed CNN architecture works well both on weakly textured regions and disparity discontinuities. ‘chicken-and-egg’ problem. Therefore, the use of a CNN [13], [14] is appropriate, as it automatically learns the proper shape of the templates for each input pattern. The existing methods, however, are based on conventional CNN architectures resembling the AlexNet [15] or VGG [16] network, which are optimized for image classification task and not for image matching. The architectures comprise several convolution layers, each followed by a rectified linear unit (ReLU) [15], and pooling layers with strides. One of the limitations of using these architectures for matching is the difficulty of enlarging the size of the patches that are to be compared. The effective size of the patch is directly related to IEEE SIGNAL PROCESSING LETTERS the spatial extent of the receptive field of CNN, which can be increased by (1) including a few strided pooling/convolution layers, (2) using larger convolution kernels at each layer, or (3) increasing the number of layers. However, use of strided pooling/convolution layers makes the results downsampled, losing fine details. Although the resolution can be recovered by applying fractional-strided convolution [17], reconstructing small or thin structures is still difficult if once they are lost after downsampling. Increasing the size of the kernels is also problematic, as the number of feature maps required to represent the larger patterns increases significantly. Furthermore, a previous study [18] reported that the repetitive usage of small convolutions does not always result in a large receptive field. This paper contributes to the literature by proposing a novel CNN module to learn a better matching cost function. The module is an innovative pooling scheme that enables a CNN to view a larger area without losing the fine details and without increasing the computational complexity during test times. The experiments show that the use of the proposed module improves the performance of the baseline network, showing competitive results on the Middlebury [1], [19] benchmark. II. R ELATED W ORKS Given the introduction of high-resolution stereo datasets with the ground-truth disparity maps [20], [19], [21], many attempts have been made to learn a matching cost function using machine learning algorithms [13], [14], [22]. The most impressive results are obtained by using CNN [13], [14]. The architecture proposed in [13] takes a small 11 × 11 window and processes it without the use of pooling. The computed cost volume is noisy due to the limited size of the window. Thus, it is post-processed by using the crossbased cost aggregation (CBCA) [23], semi-global matching (SGM) [3], and additional refinement procedures. On the other hand, the method in [14] uses multiple pooling layers and spatial-pyramid-pooling (SPP) [24] to process larger patches. However, the results show a fattening effect owing to the loss of information introduced by pooling. The main contribution of this paper is in proposing a novel pooling scheme that can handle information from a large receptive field without losing the fine details. Recently, several attempts have been made to accomplish the same goal in the context of semantic segmentation [25], [26], [27]. These methods combine the feature maps from the highlevel layers with those from the lower layers, with the aim of correctly aligning the object-level information along the pixel-level details. While this approach can successfully align the boundaries of the big objects, its inherent limitation is its inability to recover small objects in the final output once they are lost during the abstraction due to multiple uses of pooling. In the same context, the FlowNet [28] architecture can upsample the coarse-level flow to the original scale by using lower-level feature maps. However, it fails to recover the extreme flow elements that are hidden due to the low resolution of high-level feature maps. The architecture most closely related to the current work has been proposed in [24]. Unlike the other approaches, the 2 4 P Fig. 2. The 4P module with pooling size vector s = [5, 3, 1] is visualized. This figure shows its action for one channel of the feature maps for brevity; it does the same job for all channels. SPP network excludes pooling layers between convolutional layers. Instead, it first computes highly-nonlinear feature maps by cascading convolutional layers several times and then generates high-level and mid-level information by pooling them at different scales. By keeping the original feature maps along with feature maps pooled at multiple scales, the SPP network can combine the features from multiple levels without losing fine details. Although the previously mentioned stereo method in [14] uses SPP, it also employs conventional pooling layers between convolutional layers, thus losing the detailed information. III. A RCHITECTURE OF THE N EURAL N ETWORK The proposed architecture takes two input patches and produces the corresponding matching cost. In the following subsections, the newly proposed module is first introduced. Then the detailed architecture of the entire network is presented. A. Per-pixel Pyramid Pooling (4P) The use of pooling layers in CNN has been considered desirable because of its accuracy and efficiency in image classification tasks. While the use of max-pooling layers has been reported to provide an additional invariance in spatial transformation, the most important gain comes from the downsampling of feature maps. By performing pooling with a stride that is larger than one, the output feature maps after the pooling are scaled down. The final scale of the CNN output is decreased exponentially in terms of the number of pooling layers. Given that no parameters related to a pooling operation exist, this method is an effective way to widen the receptive field area of a CNN without increasing the number of parameters. The drawback of strided pooling is that the network loses fine details in the original feature maps as the pooling is applied. Thus, a trade-off exists in seeing a larger area and preserving the small details. Inspired by the idea discussed in [24], we propose a novel pooling scheme to overcome this trade-off. Instead of using a small pooling window with a stride, a large pooling window is used to achieve the desired size of the receptive field. The use of one large pooling window can lead to the loss of finer details. Thus, multiple poolings with varying window sizes are performed, and the outputs are concatenated to IEEE SIGNAL PROCESSING LETTERS 3 Matching score Matching score 1x1 Conv. + Sigmoid (1) 1x1 Conv. + Sigmoid (1) TABLE I T HE QUANTITATIVE RESULTS ON THE ‘ TRAINING DENSE ’ SET OF THE M IDDLEBURY BENCHMARK [1] ARE SHOWN . T HE ERROR REPRESENTS THE PERCENTAGE OF BAD PIXELS WITH A DISPARITY THRESHOLD 2.0, AND THE SAME WEIGHTING SCHEME IS APPLIED AS IN [1] WHEN COMPUTING THE AVERAGE . 1x1 Conv. + ReLU (384) 4P (384) 1x1 Conv. (96) 1x1 Conv. + ReLU (384) 1x1 Conv. + ReLU (384) 1x1 Conv. + ReLU (384) 1x1 Conv. + ReLU (384) 1x1 Conv. + ReLU (384) 1x1 Conv. + ReLU (384) Methods WTA after post-processing 3x3 Conv. + ReLU (112) 3x3 Conv. + ReLU (112) 3x3 Conv. + ReLU (112) 3x3 Conv. + ReLU (112) 3x3 Conv. + ReLU (112) 3x3 Conv. + ReLU (112) 3x3 Conv. + ReLU (112) 3x3 Conv. + ReLU (112) 3x3 Conv. + ReLU (112) 3x3 Conv. + ReLU (112) L R Baseline L R Proposed Fig. 3. The network structures are visualized for the baseline network, ‘MCCNN-acrt’ [13], and the proposed network. The parenthesized numbers at each layer represent the number of feature maps after the corresponding operations. Note that this figure is drawn in terms of the fully convolutional network. create new feature maps. The resulting feature maps contain the information from coarse-to-fine scales. The multi-scale pooling operation is performed for every pixel without strides. We call this whole procedure, “per-pixel pyramid pooling” (4P), which is formally defined as follows: P 4P (F, s) = [P (F, s1 ) , · · · , P (F, sM )] , (1) where s is a vector having M number of elements, and P (F, si ) is the pooling operation with size si and stride one. The structure of this module is illustrated in Figure 2. B. Proposed model To validate the effect of the proposed module, we trained and tested CNNs with and without the 4P module. The baseline architecture is selected as the ‘MC-CNN-acrt’ [13]. The 4P module in the proposed architecture is constructed by using the size vector s = [27, 9, 3, 1]. The structures of two CNNs are visualized in Figure 3. IV. I MPLEMENTATION D ETAILS For a fair comparison, we followed the details in [13] to train the proposed architecture with a few exceptions mentioned below. First, the size of the training patch became 37×37. Furthermore, we only fine-tuned the parameters of the last three 1 × 1 convolution layers of the proposed architecture in Figure 3. The parameters of the earlier layers are borrowed from the pre-trained ‘MC-CNN-acrt’ [13] network. In our experiments, this resulted in a better performance than the end-to-end training of the network with random initializations. Moreover, training a few convolution layers with pre-trained features is easier, making it converge faster. We have run a avg. error MC-CNN-acrt [13] proposed 22.91 11.75 MC-CNN-acrt [13] proposed (w/ parameters in [13]) proposed (w/ parameter tuning) 10.26 9.72 8.45 total of four epochs of training, where the last two epochs were run with a decreased learning rate from 0.003 to 0.0003. We also used the same post-processing pipeline as in [13] during the test phase. The post-processing pipeline includes the use of the CBCA [23] and SGM [3], and the disparity maps are refined to have continuous values and undergo median filtering and bilateral filtering. V. E XPERIMENTS To verify the effect of the proposed 4P module, we have compared the results of the baseline and proposed network. The performance is measured using the ‘training dense’ set of the Middlebury benchmark [1]. The quantitative results are briefly summarized in Table I using the average errors. All experiments are performed by using the Intel core i7 4790K CPU and a single Nvidia Geforce GTX Titan X GPU. The proposed method outperforms the baseline architecture regardless of the use of post-processing. The benefit of using the 4P module is clear when the disparity maps are obtained by using the pixel-wise winner-takes-it-all (WTA) rule without any post-processing. Given that the images in the dataset contain many weakly-textured areas, the small-sized 11×11 window cannot distinguish the true matches from false ones without the aid of post-processing. On the other hand, the proposed architecture effectively sees the larger window, 37 × 37, by inserting the 4P module before the final decision layer. It is less straightforward to understand why the proposed architecture still outperforms the baseline even after postprocessing. In that sense, it is worth to mention that the best parameter setting for post-processing of the proposed method largely differ from that of the baseline.1 The most notable changes from the original parameter setting is that we use much less number of CBCA [23], and it means that multiple uses of CBCA [23] become redundant in the proposed architecture. From this fact, we can interpret the role of 4P module as adaptive local feature aggregation. Compared to the hand-designed algorithm such as CBCA [23], the influence of neighboring pixels to a certain pixel is automatically learned 1 Following the conventions in [13], the best parameter setting is as follows: cbca_num_iterations_1 = 0, cbca_num_iterations_2 = 1, sgm_P1 = 1.3, sgm_P2 = 17.0, sgm_Q1 = 3.6, sgm_Q2 = 36.0, and sgm_V = 1.4. IEEE SIGNAL PROCESSING LETTERS true disparity and left image 4 proposed MC-CNN-acrt Fig. 4. The results for PlaytableP and Vintage are visualized. For each datum, the upper row shows the disparity map and the bottom row shows the corresponding error maps. While the ‘MC-CNN-acrt’ [13] shows errors around the weakly-textured areas, such as the surfaces of the chair and the table in PlaytableP or the white wall in Vintage, the proposed method shows more reliable results. and it can be jointly trained with the cost function itself. Furthermore, the information exchange among pixels is done in feature space which contains richer contextual information than the final cost volume space. Note that the improvement over the baseline clearly results neither from the use of extra layers nor from the use of more parameters, as the authors of [13] already have shown that the additional use of fully-connected (FC) layers is less significant. Using two additional FC layers leads to an improvement of approximately 1.90%, whereas using the 4P module results in a 21.42% improvement in terms of average error. The main contribution of the proposed method lies in learning a less ambiguous matching cost function by inspecting a larger area. Figure 4 shows that the proposed network actually works better around the weakly-textured area than the ‘MCCNN-acrt’ [13]. The quantitative and qualitative results of each dataset, including the ones in the ‘test dense’ set, are available at the Middlebury benchmark [1] website. VI. C ONCLUSIONS Viewing a large area to estimate the dense pixel correspondence is necessary to fully utilize the texture information to achieve less ambiguous and more accurate matching. A conventional matching cost function fails because neighboring pixels on the same surface as the target pixel are unknown. In this paper, a novel CNN module is proposed to make the CNN structure handle a large image patch without losing the small details, which enables it to learn an intelligent matching cost function for large-sized windows. The learned cost function can discriminate the false matches for weakly-textured areas or repeating textures, and can also conserve the disparity discontinuities well/. The learned cost function achieves competitive performance on the Middlebury benchmark. IEEE SIGNAL PROCESSING LETTERS R EFERENCES [1] D. Scharstein and R. Szeliski, “A taxonomy and evaluation of dense two-frame stereo correspondence algorithms,” IJCV, vol. 47, no. 1-3, pp. 7–42, 2002. [2] V. 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On the asymptotic structure of Brownian motions with a small lead-lag effect arXiv:1601.03614v2 [math.ST] 8 Apr 2018 Yuta Koike∗†‡ April 10, 2018 Abstract This paper considers two Brownian motions in a situation where one is correlated to the other with a slight delay. We study the problem of estimating the time lag parameter between these Brownian motions from their high-frequency observations, which are possibly subject to measurement errors. The measurement errors are assumed to be i.i.d., centered Gaussian and independent of the latent processes. We investigate the asymptotic structure of the likelihood ratio process for this model when the lag parameter is asymptotically infinitesimal. We show that the structure of the limit experiment depends on the level of the measurement errors: If the measurement errors locally dominate the latent Brownian motions, the model enjoys the LAN property. Otherwise, the limit experiment does not result in typical ones appearing in the literature. We also discuss the efficient estimation of the lag parameter to highlight the statistical implications. Keywords and phrases: Asymptotic efficiency; Endogenous noise; Lead-lag effect; Local asymptotic normality; Microstructure noise. 1 Introduction Let Bt = (Bt1 , Bt2 ) (t ∈ R) be a bivariate two-sided Brownian motion such that B0 = 0, E[(B11 )2 ] = E[(B12 )2 ] = 1 and E[B11 B12 ] = ρ for some ρ ∈ (−1, 0) ∪ (0, 1). Also, let ǫi = (ǫ1i , ǫ2i ) (i = 1, 2, . . . ) be a sequence of i.i.d. bivariate standard normal variables independent of B. For each n ∈ N, we denote by Pn,ϑ the law of the vector Zn = (X1 , . . . , Xn , Y1 , . . . , Yn )⊤ generated by the following model: ( √ 1 + √v ǫ1 , 2 Xi = Bi/n Yi = Bi/n−ϑ + vn ǫ2i if ϑ ≥ 0, n i √ 2 + √ v ǫ2 1 if ϑ < 0 Xi = Bi/n−ϑ + vn ǫ1i , Yi = Bi/n n i for i = 1, . . . , n, (1.1) where vn is a non-negative number. ϑ ∈ R denotes the unknown time-lag parameter which we are interested in. Especially, the sign of ϑ is unknown. The aim of this paper is to study the asymptotic structure of the sequence of experiments (R2n , B 2n , (Pn,ϑ )ϑ∈R ), n = 1, 2, . . . , as n → ∞ when the time lag parameter ϑ is asymptotically infinitesimal, i.e. ϑ tends to 0 as n → ∞ (here and below B m denotes the Borel σ-field of Rm for m ∈ N). More precisely, we study the limit experiment of (Pn,rnu )u∈R as n → ∞ for the proper convergence rate rn . ∗ Department of Business Administration, Graduate School of Social Sciences, Tokyo Metropolitan University, Marunouchi Eiraku Bldg. 18F, 1-4-1 Marunouchi, Chiyoda-ku, Tokyo 100-0005 Japan † The Institute of Statistical Mathematics, 10-3 Midori-cho, Tachikawa, Tokyo 190-8562, Japan ‡ CREST, Japan Science and Technology Agency 1 If vn ≡ 0, model (1.1) is a special case of the Hoffmann-Rosenbaum-Yoshida (HRY) model introduced in Hoffmann et al. (2013) to describe lead-lag effects in high-frequency financial data. A similar model has also been studied in Robert and Rosenbaum (2010) with an asymptotic regime different from the current setting. Here, the lead-lag effect refers to a situation where one time series is correlated to another time series at a later time, which has especially drawn attention in analysis of economic time series data for a long time, and associated econometric methods have been developed by many authors; see Section 1 of Hoffmann et al. (2013), Section 3 of Robert and Rosenbaum (2010) and references therein. The practicality of the HRY model in empirical work has recently been established by several authors such as Alsayed and McGroarty (2014), Huth and Abergel (2014) and Bollen et al. (2017) for financial data and Iacus et al. (2015) for social media data. These empirical studies show that time lag parameters are typically comparable to the observation frequencies in their scales. This motivates us to study the HRY model when ϑ is small. In such a situation, one would especially be interested in how small lag parameters can be identified in principle. To the author’s knowledge, however, there is few theoretical study for the HRY model and, in particular, nothing has been known about the optimality of statistical inferences for the HRY model. The purpose of this paper is trying to fill in this gap. In this paper, as well as (a special case of) the HRY model, we also consider a situation where the model contains measurement errors. This is motivated by the recent studies for the volatility estimation from ultra high frequency financial data, which is typically modeled as a discretely observed semimartingale with market microstructure noise. We refer to Chapter 7 of Aı̈t-Sahalia and Jacod (2014) for a brief description of this subject. In particular, the asymptotic structure and the asymptotic efficiency bound have been established in the work of Gloter and Jacod (2001a,b) (see also Cai et al. (2010)) for a statistical model of estimating the scale parameter σ > 0 from the discrete observations σWi/n + √ vn δi , i = 1, . . . , n, (1.2) where W = (Wt )t∈[0,1] is a one-dimensional standard Wiener process and (δi )ni=1 is a sequence of centered i.i.d. standard normal variables independent of W . They proved the LAN property for the above model and constructed asymptotically efficient estimators for σ (they indeed considered a more general setting). Extensions of their LAN result to a multivariate setting have also been studied by several authors. The correlation estimation in a bivariate setting is studied in Bibinger (2011), while a more general setting containing non-synchronous sampling case is studied in Ogihara (2014). On the other hand, Reiß (2011) has studied the asymptotic structure of model (1.2) when σ is a function of time rather than a constant and established the asymptotic equivalence between such a model and a Gaussian white noise model. The result has been extended to the bivariate case by Bibinger and Reiß (2014) and a multivariate setting containing non-synchronous case by Bibinger et al. (2014). Another type of extension, replacing the Wiener process W by a different continuous-time process, has also been studied. For example, Sabel and Schmidt-Hieber (2014) consider the efficient estimation of σ in a situation where W is a more general Gaussian process, especially a fractional Brownian motion. The main contribution of this paper is (i) to determine the proper convergence rate rn , and (ii) to derive a stochastic expansion for the likelihood ratio process for (Pn,rnu )u∈R . Analogously to Gloter and Jacod (2001a), the proper convergence rate rn depends on the behavior of the sequence nvn . This is intuitively natural because Var[Bi/n −B(i−1)/n ] = n−1 and thus the behavior of nvn determines how strongly the measurement errors (locally) 3 dominate the nature of the observed returns. In particular, we find that rn = n− 2 if vn ≡ 0, or more generally if 2 3 1 nvn is bounded.1 The rate n− 2 is much faster than the usual parametric rate n− 2 and even faster than the rate n−1 . Since the time resolution of our model is n−1 , our result suggests that we could estimate lag parameters smaller than the time resolution of observation data. This implication is at least true for our restrictive situation, as shown in Section 3. Since the convergence rate of the estimator for the lag parameter ϑ proposed in Hoffmann et al. (2013) cannot be faster than n−1 (see Proposition 2 of Hoffmann et al. (2013) and the discussion after this proposition), our result shows that their estimator is suboptimal in the setting considered in this paper (although their estimator works in a more general setting). Given the proper convergence rate, we have the following stochastic expansion for the likelihood ratio process: There are random variables Tn and Sn defined on (R2n , B 2n ) and non-negative numbers Iγ and Jγ such that dPn,rnun u2n p log →0 under Pn,0 as n → ∞ (1.3) − un Tn + \|un \|Sn − (Iγ + Jγ ) − dPn,0 2 for any bounded sequence un of real numbers and d (Tn , Sn ) − → N (0, Iγ ) ⊗ N (0, Jγ ) under Pn,0 as n → ∞. (1.4) Therefore, by a contiguity argument we deduce that the experiments (R2n , B 2n , (Pn,rn u )u∈R ) converge weakly to the experiment (R2 , B 2 , (Qu )u∈R ) in the Le Cam sense, where Qu = N (uIγ , Iγ ) ⊗ N (\|u\|Jγ , Jγ ) (see Corol- lary 2.3). The numbers Iγ and Jγ are determined by the asymptotic behavior of nvn and precisely defined by (2.1)–(2.2). In particular, Iγ is always positive, while Jγ is positive if nvn is bounded and Jγ = 0 otherwise. The case Jγ = 0 corresponds to the situation where the measurement errors locally dominate the signal, and in this case our model enjoys the LAN property which commonly appears in regular experiments. This result is of interest because model (1.1) exhibits irregularity in the sense that its likelihood function is not smooth in ϑ, and the limit experiment of such a model typically deviates from the LAN structure as illustrated in Chapters V–VII of Ibragimov and Has’minskii (1981). Our result means that the measurement errors have a kind of regularizing effect on the asymptotic structure of model (1.1). On the other hand, if Jγ > 0, which corresponds to the cases where the signal dominates or is balanced with the measurement errors, in addition to an observation from a usual Gaussian shift experiment N (u, Iγ−1 ), the limit experiment contains an extra observation from the experiment N (\|u\|, Jγ−1 ). Although this experiment looks simple, to the author’s knowledge it does not result in well-studied cases (such as in Ibragimov and Has’minskii (1981)), so the definition of asymptotically efficient estimators in this case is not obvious. To obtain the asymptotic efficiency bound for estimating the lag parameter in this case, in Section 3 we apply Ibragimov and Has’minskii (1981)’s theory to our problem, which is a common approach to establish asymptotic efficiency bounds for experiments generated by diffusion type processes (see Kutoyants (2004) for details). As a result, we find that Bayesian estimators are asymptotically efficient, while the maximum likelihood estimator is not always asymptotically efficient. This is a common phenomenon in irregular models; see Chapters V–VII of Ibragimov and Has’minskii (1981), Küchler and Kutoyants (2000), Chapter 3 of Kutoyants (2004), Rubin and Song (1995) and Chapter 9 of van der Vaart (1998) for example. This paper is organized as follows. Section 2 presents the main result of the paper. In Section 3 we discuss the efficient estimation of the lag parameter in our setting. Section 4 is devoted to the proof of an auxiliary technical result. 1 Indeed, an intuition for this fact has already been appeared in Hoffmann et al. (2013) (see Remark 2.2). 3 General notation Em denotes the m × m-identity matrix. For a matrix A, we denote by kAksp and kAkF its spectral norm and the Frobenius norm, respectively. That is, kAksp = sup{kAxk : kxk ≤ 1} and kAk2F = tr(A⊤ A). Also, we denote by Aij the (i, j)-th entry of A. 2 Main result We start with completing the definitions of the quantities rn , Iγ and Jγ appearing in the Introduction. First, following Gloter and Jacod (2001a) we assume that the sequence nvn converges in [0, ∞] and set γ := lim nvn . n→∞ We also assume lim supn vn < ∞ as in Gloter and Jacod (2001a). Then we set ( p n/vn if γ = ∞, Nn = n otherwise. Nn can be considered as an “effective” sample size in the sense that the proper convergence rate for estimating σ −1 from model (1.1) is given by Nn 2 , which is seen as the regular parametric rate if we regard Nn as the sample size. −3 Using this effective sample size Nn , we define our proper convergence rate as rn = Nn 2 . The constants Iγ and Jγ appearing in (1.3)–(1.4) are defined by  ρ2   2(1−ρ 2  √  √ ) ρ (1+ρ)(1+ρ+4γ)− (1−ρ)(1−ρ+4γ)−2ρ Iγ =  8γ 2   ρ2  √ √ 2( 1+ρ+ 1−ρ) if γ = 0, (2.1) if 0 < γ < ∞, if γ = ∞ and Jγ =          3 4 n 1 16γ 2 0 o 1 if γ = 0, + (1−ρ) 2 1 1 3 3 2 2 2 2 1+ρ 1−ρ 1−ρ 1+ρ + 1−ρ+4γ + 1−ρ+4γ + if 0 < γ < ∞, 4−3 1+ρ+4γ 1+ρ+4γ 1 (1+ρ)2 (2.2) if γ = ∞. Remark 2.1. Iγ is always positive for any γ ∈ [0, ∞]. This is evident when γ = 0 or γ = ∞. When 0 < γ < ∞, this is proven as follows. First suppose that 0 < ρ < 1. Then we have p (1 + ρ)(1 + ρ + 4γ) − { (1 − ρ)(1 − ρ + 4γ) + 2ρ}2 p = 4ρ + 8γρ − 4ρ (1 − ρ)2 + 4γ(1 − ρ) − 4ρ2 p = 4ρ (1 − ρ) + 2γ − (1 − ρ)2 + 4γ(1 − ρ) p p = 4ρ (1 − ρ)2 + 4γ(1 − ρ) + 4γ 2 − (1 − ρ)2 + 4γ(1 − ρ) > 0. Hence we have Iγ > 0. On the other hand, if −1 < ρ < 0, applying the above inequality with replacing ρ by −ρ, p p we obtain (1 − ρ)(1 − ρ + 4γ) > (1 + ρ)(1 + ρ + 4γ) − 2ρ. Hence we have Iγ > 0. The following statement is our main result. 4 Theorem 2.1. There are two sequences Tn and Sn of random variables satisfying (1.3)–(1.4) for any bounded sequence un of real numbers. We can explicitly give the variables Tn and Sn in Theorem 2.1 by (2.8) below. Theorem 2.1 has some immediate consequences. The first one is the direct consequence of the definition of the LAN property. Corollary 2.1. If γ = ∞, (Pn,ϑ )ϑ∈R has the LAN property at ϑ = 0 with rate rn and asymptotic Fisher information Iγ . The second one follows from Le Cam’s first lemma (see e.g. Lemma 6.4 of van der Vaart (1998)). Corollary 2.2. Pn,ϑn and Pn,0 are mutually contiguous if the sequence ϑn of real numbers satisfies ϑn = O(rn ). The third one is derived from Corollary 2.2 and Theorem 61.6 of Strasser (1985) (we refer to Drost et al. (2009), Le Cam (1986), Chapter 10 of Strasser (1985) and Chapters 8–9 of van der Vaart (1998) for the definition and applications of the weak convergence of experiments). Corollary 2.3. The sequence (R2n , B 2n , (Pn,rn h )h∈R ) of experiments converges weakly to the experiment (R2 , B 2 , (Qu )u∈R ) as n → ∞, where Qu = N (uIγ , Iγ ) ⊗ N (\|u\|Jγ , Jγ ). Now we turn to the proof of Theorem 2.1. Although (Pn,ϑ )ϑ∈R consists of Gaussian distributions, the problem is not simple because the covariance matrix Cn (ϑ) of Pn,ϑ is a complicated function of the lag parameter ϑ. In particular, Cn (ϑ) and Cn (ϑ′ ) are not simultaneously diagonalizable in general (even asymptotically) if ϑ 6= ϑ′ . This could be troublesome because in analysis of Gaussian experiments the (asymptotically) simulta- neous diagonalizability of the covariance matrices of the statistical model for different parameters typically plays an important role (cf. Section 3 of Davies (1973), Lemma 8.1 of Gloter and Jacod (2001a) and Lemma C.4 of Sabel and Schmidt-Hieber (2014)). For this reason we first transfer from the model Pn,ϑ to a more tractable model defined as follows: For each n ∈ N, set Θn = {ϑ ∈ R : vn − nϑ2 + \|ϑ\| ≥ 0} = {ϑ ∈ R : √ e n,ϑ the law of the vector Z en = \|ϑ\| ≤ (1 + 1 + 4nvn )/(2n)}. Then, for each ϑ ∈ Θn we denote by P e1 , . . . , X en , Ye1 , . . . , Yen )⊤ defined by (X where ( √ 2 − B2 2 2 e ǫ2i,n = −nϑ(Bi/n if ϑ ≥ 0, (i−1)/n ) + vn − nϑ + ϑǫi p 1 1 1 2 2 = −n\|ϑ\|(Bi/n − B(i−1)/n ) + vn − nϑ2 + \|ϑ\|ǫi , e ǫi,n = ǫi if ϑ < 0 e ǫ1i,n = ǫ1i , e ǫ1i,n 2 Yei = Bi/n +e ǫ2i , ei = B 1 + e ǫ1i , X i/n (2.3) (2.4) en,ϑ . en (ϑ) the covariance matrix of P for i = 1, . . . , n. We denote by C e n,ϑ . To be precise, the Hellinger distance In the following we will show that Pn,ϑ is well-approximated by P en,ϑ tends to 0 as n → ∞, provided that ϑ tends to 0 sufficiently fast. Here, the Hellinger between Pn,ϑ and P distance H(P, Q) between two probability measures P and Q on a measurable space (X , A) is defined by  s s !2 1/2 Z dP dQ dµ , − H(P, Q) =  dµ dµ X where µ is a σ-finite measure dominating both P and Q (µ = P + Q for example). It can easily be checked that H(P, Q) does not depend on the choice of µ. See Appendix A.1 of Reiß (2011), Section 2 of Strasser (1985) and Section 2.4 of Tsybakov (2009) for more information about the Hellinger distance. 5 e n,ϑ ) expectation with respect to Pn,ϑ (resp. P en,ϑ ). Throughout the paper, we denote by En,ϑ (resp. E e n,ϑ . Proposition 2.1. (a) If \|ϑ\| ≤ 1/n, then Pn,ϑ = P en,ϑ ) ≤ 4v −2 (4 + 3ρ2 )n2 \|ϑ\|3 for any n ∈ N and any ϑ ∈ Θn . (b) If vn > 0, H 2 (Pn,ϑ , P n 4 − e n,ϑ ) → (c) If a sequence ϑn of positive numbers satisfies ϑn = o(n−1 ∨Nn 3 ) as n → ∞, then sup\|ϑ\|≤ϑn H(Pn,ϑ , P 0. Proof. Claim (c) immediately follows from (a) and (b), so we focus on (a) and (b). By symmetry we may assume e n,ϑ [xi xj ] for ϑ ≥ 0. Let x1 , . . . , xn , y1 , . . . , yn be the canonical variables on R2n . Then we have En,ϑ [xi xj ] = E all i, j. Moreover, a simple computation shows that ( ( i∧j if i ∧ j ≥ nϑ, n − ϑ) + vn 1{i=j} En,ϑ [yi yj ] = i∨j (ϑ − n )+ + vn 1{i=j} otherwise and and e n,ϑ [yi yj ] = E En,ϑ [xi yj ] = ( i∧j i∧j − ϑ1{i≥j} − ϑ1{i≤j} + (vn + ϑ)1{i=j} = − ϑ + vn 1{i=j} n n if i < j − nϑ, ρi/n ( e n,ϑ [xi yj ] = E ρ(j/n − ϑ)+ otherwise, ρi/n if i < j, ρ(j/n − ϑ) otherwise. en (ϑ) if ϑ ≤ 1/n. This yields claim (a) because both Pn,ϑ and P en,ϑ are centered Therefore, we have Cn (ϑ) = C Gaussian. On the other hand, from the above identities we also have en (ϑ)k2F kCn (ϑ) − C n X i∨j i∧j ϑ− = −ϑ − n n + 2 i,j=1 i∧j<nϑ +2  n  X i=1 = n X i,j=1 i∧j<nϑ +2  X ρ i∨j n − j:i<j≤i+nϑ ϑ−  n  X X + ρ i n j −ϑ n i∧j −ϑ n 2 + + − i n 2 + j:j≤i 2 X ρ X ρ j−i −ϑ n 2  j:i<j≤nϑ j:i∨nϑ<j≤i+nϑ 2 2 nϑ 2nϑ 2 + 6n · ρ · nϑ · = (8 + 6ρ2 )n2 ϑ3 . ≤ 2n · nϑ · n n i=1 1 j −ϑ n + + X − j:j≤i∧nϑ ρ j −ϑ n  2  j −ϑ n   2  −1 Now if vn > 0, Cn (ϑ) is positive semidefinite and satisfies kCn (ϑ)− 2 ksp ≤ vn 2 by the monotonicity theorem for eigenvalues (Corollary 4.3.3 of Horn and Johnson (1985)) because Cn (ϑ) − vn E2n is positive semidefinite. Therefore, by Eqs.(A.4) and (A.6) from Reiß (2011) we obtain en (ϑ))Cn (ϑ)− 12 k2 ≤ 4v −2 (4 + 3ρ2 )n2 ϑ3 , en,ϑ ) ≤ 2kCn (ϑ)− 21 (Cn (ϑ) − C H 2 (Pn,ϑ , P F n hence claim (b) holds true. 6 In the following we will frequently use the fact that the Hellinger distance dominates the total variation distance: V (P, Q) ≤ H(P, Q), (2.5) where V (P, Q) = supA∈A \|P (A) − Q(A)\|. See e.g. Lemma 2.3 of Tsybakov (2009) for the proof. The following properties of the total variation distance are immediate consequences of the definition and important for our purpose. For each n ∈ N, let Pn and Qn be two sequences of probability measures on a measurable space (Xn , An ), and let ζn be a random variable on (Xn , An ) taking its value in a metric space D. Then, for any a ∈ D and any probability measure µ on D, the following statements hold true: p p V (Pn , Qn ) → 0, ζn − → a under Pn ⇒ ξn − → a under Qn , d d V (Pn , Qn ) → 0, ζn − → µ under Pn ⇒ ξn − → µ under Qn . ) (2.6) e n,ϑ as a tractable form. For this purpose we introduce some en (ϑ) of P Next we express the covariance matrix C notation. The n × n matrix ∇n denotes the backward difference operator, i.e.   1    −1 1    ∇n =  . .. ..   . .   −1 1 b n := (∇n ⊕ ∇n )Z e n = (X e1 , X e2 − X e1 , . . . , X en − X en−1 , Ye1 , Ye2 − Ye1 , . . . , Yen − Yen−1 )⊤ and denote by We set Z b n , i.e. Vn (ϑ) = (∇n ⊕ ∇n )C en (ϑ)(∇n ⊕ ∇n )⊤ . Vn (ϑ) can explicitly be expressed Vn (ϑ) the covariance matrix of Z sign(ϑ) ¯n as Vn (ϑ) = Ḡn − ρϑ∇ " Gn Ḡn = ρ n En , where # ρ E n n , Gn ¯+ ∇ n = with Gn = n1 En + vn Fn , Fn = ∇n ∇⊤ n and "     Rn =    0 ∇⊤ n # ··· 0 ··· .. . 0 .. . 0 0 ∇n ρ−1 Rn 1 0 0 .. . 0 .. . 0 ··· ¯− ∇ n =− , " ρ−1 Rn ∇n ∇⊤ n 0 #     .   ¯ sign(ϑ) It is more convenient to rewrite the expression ∇ as follows. Let Sn and Tn be the symmetric and skewn ⊤ ⊤ symmetric parts of 2∇⊤ n , respectively. That is, Sn = ∇n + ∇n and Tn = ∇n − ∇n . Then we set # # " " Sn Tn 1 ρ−1 Rn 1 −ρ−1 Rn S̄n = , T̄n = . 2 2 Sn ρ−1 Rn −Tn ρ−1 Rn ¯ ± = T̄n ± S̄n , so we obtain Vn (ϑ) = Ḡn − ρ(ϑT̄n + \|ϑ\|S̄n ). This is a simple function of ϑ, We can easily check ∇ n so Vn (ϑ) is more tractable than Cn (ϑ): Although Vn (ϑ)’s are not simultaneously diagonalizable for different ϑ’s, it is sufficient to consider a relationship between the matrices Ḡn , T̄n and S̄n . In fact, it turns out that the following result is sufficient for our purpose. 7 −1 −1 Proposition 2.2. For any α, β ∈ R, we have kḠn 2 (αT̄n + β S̄n )Ḡn 2 ksp = O(Nn ) as n → ∞ and −1 −1 lim ρ2 rn2 kḠn 2 (αT̄n + β S̄n )Ḡn 2 k2F = 2(α2 Iγ + β 2 Jγ ). n→∞ (2.7) The proof of Proposition 2.2 consists of elementary but complicated calculations, so we postpone it to the Appendix (Section 4). We remark that the proof requires a calculation essentially different from that of the Fisher information for the scale parameter estimation from observations of the form (1.2) such as in Gloter and Jacod (2001a) and Sabel and Schmidt-Hieber (2014) (see Remark 4.1). Also, note that Proposition 2.2 yields the invertibility of 1 −1 1 −1 Vn (ϑn ) for sufficiently large n if ϑn = o(Nn−1 ) because Vn (ϑn ) = Ḡn2 (E2n − ρḠn 2 (ϑn T̄n + \|ϑn \|S̄n )Ḡn 2 )Ḡn2 . Proof of Theorem 2.1. Define the function zbn : R2n → R2n by setting zbn (ζ) = (∇n ⊕ ∇n )ζ for ζ ∈ R2n . Then we set o ρ n −1 −1 Tn = − rn zbn⊤ Ḡ−1 T̄ Ḡ z b − tr( Ḡ T̄ ) , n n n n n n 2 o ρ n −1 −1 S̄ Sn = − rn zbn⊤ Ḡ−1 Ḡ z b − tr( Ḡ S̄ ) . n n n n n n 2 (2.8) By virtue of Proposition 2.1 and (2.5)–(2.6), it suffices to prove the following statements: dPn,rnun u2n p en,0 , log → 0 under P − un Tn + \|un \|Sn − (Iγ + Jγ ) − dPn,0 2 (2.9) d e n,0 . (Tn , Sn ) − → N (0, Iγ ) ⊗ N (0, Jγ ) under P (2.10) (2.10) follows from Proposition 2.2 and Proposition 2 of Dalalyan and Yoshida (2011). On the other hand, setting −1 −1 An = ρrn Ḡn 2 (un T̄n + \|un \|S̄n )Ḡn 2 , we have kAn k2F − 2u2n (Iγ + Jγ ) → 0 by Proposition 2.2. Therefore, by Proposition 2.1, (2.5) and Proposition 2 from Chapter 4 of Le Cam (1986) we obtain (2.9) once we show that en,rnun dP kAn k2F p e n,0 . ξn := log → 0 under P − (2.11) − un Tn + \|un \|Sn − en,0 4 dP The strategy of the proof of (2.11) is the same as that of Theorem 4.2 from Davies (1973). First, by Eq.(4.3) of Davies (1973), for sufficiently large n we have 2 e n,0 [ξn ] = − 1 log det Vn (rn un ) − log det Vn (0) + tr(Vn (0)(Vn (rn un )−1 − Vn (0)−1 )) + kAn kF E 2 4 1 1 2 −1 2 =− log det(E2n − An ) + tr(An ) + tr(An ) + tr((E2n − An ) − E2n − An − An ) . 2 2 P p Note that it holds that (E2n − An )−1 = ∞ p=0 An for sufficiently large n because kAn ksp → 0 as n → ∞ by Proposition 2.2. Combining this fact with inequality (v) from Appendix II of Davies (1973), we obtain 1 1 1 1 2 e + \|En,0 [ξn ]\| ≤ kAn ksp kAn kF 2 3 (1 − kAn ksp )3 1 − kAn ksp e n,0 [ξn ] → 0. for sufficiently large n. Hence Proposition 2.2 yields E Next, noting that ξn can be rewritten as 1 en,0 [b en,0 [ξn ], ξn = − zbn⊤ Bn zbn − E zn⊤ Bn zbn ] + E 2 −1 −1 where Bn = Vn (rn un )−1 − Vn (0)−1 − Ḡn 2 An Ḡn 2 , we obtain from Eq.(4.4) of Davies (1973) 1 1 2 VarePn,0 [ξn ] = Vn (0) 2 (Vn (rn un )−1 − Vn (0)−1 )Vn (0) 2 − An 8 2 F = (E2n − An )−1 − E2n − An 2 F . Therefore, using the identity (E2n − An )−1 = P∞ p p=0 An again we obtain 2 VarPen,0 [ξn ] ≤ kAn k2sp kAn k2F (1 − kAn ksp )−2 for sufficiently large n. Hence Proposition 2.2 again yields VarPen,0 [ξn ] → 0, and we obtain (2.11). We finish this section with some remarks. 3 Remark 2.2. It is worth mentioning that we can infer from Hoffmann et al. (2013) why the rate n− 2 is the proper convergence rate of our model in the case of vn ≡ 0 as follows. Let us set n U (ϑ) = n−1 X (Xi+1 − Xi )(Yj+1 − Yj )1{[ i , i+1 )∩[ j −ϑ, j+1 −ϑ)6=∅} n i,j=1 n n n for ϑ ∈ R. The principle used in Hoffmann et al. (2013) is that U n (ϑ) is close to the true correlation ρ if and only if ϑ is close to the true time-lag parameter. Since the accuracy of estimating the correlation parameter is of √ √ order 1/ n, we naturally consider the quantity \| n(U n (ϑ) − ρ)\| to measure the distance between U n (ϑ) and √ ρ: \| n(U n (ϑ) − ρ)\| would take a large value if ϑ is not sufficiently close to the true time-lag parameter. In fact, √ Proposition 1 from Hoffmann et al. (2013) implies that \| n(U n (ϑ)−ρ)\| does not diverge if and only if the distance 3 between ϑ and the true time-lag parameter is at most of order n− 2 . This information allows us to estimate the true 3 time-lag parameter with the accuracy of order n− 2 . Remark 2.3. From an econometric point of view, Proposition 2.1 is of independent interest because the model given by (2.3)–(2.4) has an economic interpretation different from model (1.1). This model contains measurement errors correlated to the latent returns Bi/n − B(i−1)/n . The integrated volatility estimation in the presence of this type of measurement error has been studied by Kalnina and Linton (2008) for example. In the market microstruc- ture theory, such a correlation is often explained as an effect of asymmetric information (e.g. Glosten (1987)). Interestingly, some economic arguments suggest that such an information asymmetry would cause a lead-lag effect; see Chan (1993) and Chordia et al. (2011) for instance. It would also be worth emphasizing that de Jong and Schotman (2010) connect this type of model with the investigation of price discovery, for price discovery processes are closely related to lead-lag effects, as seen in de Jong et al. (1998) and Hasbrouck (1995). Remark 2.4. Our proof of the main result heavily depends on the Gaussianity of the model, and especially we require the Gaussianity of the measurement errors. It is obvious that we need some restriction on the distribution of the measurement errors to derive a specific limit experiment. In fact, if vn ≡ 1 and ǫi ’s take their values only in integers, then we can completely recover the signal for sufficiently large n. Apart from such a trivial example, the recent study of Bibinger et al. (2016) has shown that another (non-trivial) specification for the distribution of the measurement errors δi ’s in (1.2) can improve the convergence rate for estimating the scale parameter σ. In the light of the connection between the convergence rates for models (1.1)–(1.2), we naturally conjecture that a similar specification for the measurement errors would affect the convergence rate for our model. This issue is beyond the scope of this paper and left to future research. 3 Efficient estimation of the lag parameter As an application of the results from the previous section, we construct efficient estimators for the lag parameter ϑ in the models (Pn,ϑ )ϑ∈R at ϑ = 0. Here we consider a slightly extended setup as follows: letting ηn be a sequence of positive numbers tending to 0 and C be a bounded open interval in R, we construct efficient estimators for the parameter c in the models (Pn,cηn )c∈C at every c ∈ C. To make use of the results from the previous section, we 9 impose the following condition on ηn : −4 ηn = o n−1 ∨ Nn 3 rn−1 ηn → ∞ and as n → ∞. (3.1) Under (3.1) there is a positive integer n0 such that cηn ∈ Θn and Vn (cηn ) is invertible for any c ∈ C and n ≥ n0 due to Proposition 2.2. Throughout this section, we always assume that n is larger than such an n0 . Remark 3.1. In a practical point of view, the dependence of the time-lag parameter ϑ on the sampling frequency n is just a theoretical device to control the relative size of ϑ compared with n (which corresponds to ηn ) in the asymptotic theory and it is only important whether the asymptotic order condition (corresponding to (3.1) in our case) is acceptable as an approximation. Namely, our asymptotic theory concerns whether the time-lag parameter ϑ is comparable to n−ι for some ι > 0 given a fixed sampling frequency n (the possible values of ι change in accordance with the noise level vn ) and it does not require that the time-lag parameter varies in proportion to the sampling frequency. This type of asymptotic theory is standard in econometrics: For example, when one considers the volatility estimation of a financial asset with taking account of rounding, one usually lets the rounding level shrink as the sampling frequency increases; see Rosenbaum (2009), Li and Mykland (2015), Li et al. (2015) and Sato and Kunitomo (2015) for example. We start with generalizing Proposition 2.2 by a matrix perturbation argument. Lemma 3.1. For any α, β ∈ R we have 1 1 sup kVn (cηn )− 2 (αT̄n + β S̄n )Vn (cηn )− 2 ksp = O(Nn ), c∈C 1 1 sup ρ2 rn2 kVn (cηn )− 2 (αT̄n + β S̄n )Vn (cηn )− 2 k2F − 2(α2 Iγ + β 2 Jγ ) → 0 c∈C as n → ∞. 1 1 Proof. Setting Hn (c) = Vn (cηn )− 2 Ḡn2 , we have −1 1 1 −1 Vn (cηn )− 2 (αT̄n + β S̄n )Vn (cηn )− 2 = Hn (c)Ḡn 2 (αT̄n + β S̄n )Ḡn 2 Hn (c)⊤ for any c ∈ C. Therefore, Ostrowski’s theorem (Theorem 4.5.9 of Horn and Johnson (1985)) implies that 1 −1 1 −1 kVn (cηn )− 2 (αT̄n + β S̄n )Vn (cηn )− 2 ksp ≤ kHn (c)Hn (c)⊤ ksp kḠn 2 (αT̄n + β S̄n )Ḡn 2 ksp and 1 −1 1 −1 \|kVn (cηn )− 2 (αT̄n + β S̄n )Vn (cηn )− 2 k2F − kḠn 2 (αT̄n + β S̄n )Ḡn 2 k2F \| −1 −1 ≤ k(Hn (c)Hn (c)⊤ )2 − E2n ksp kḠn 2 (αT̄n + β S̄n )Ḡn 2 k2F −1 −1 ≤ kHn (c)Hn (c)⊤ − E2n ksp (kHn (c)Hn (c)⊤ ksp + 1)kḠn 2 (αT̄n + β S̄n )Ḡn 2 k2F . Hence, Proposition 2.2 implies that the proof is completed once we show that supc∈C kHn (c)Hn (c)⊤ ksp = O(1) and supc∈C kHn (c)Hn (c)⊤ − E2n ksp = o(1) as n → ∞. Since Hn (c)Hn (c)⊤ and Hn (c)⊤ Hn (c) share the −1 same eigenvalues (Theorem 1.3.20 of Horn and Johnson (1985)) and Hn (c)⊤ Hn (c) = (E2n − ρηn Gn 2 (\|c\|S̄n + −1 cT̄n )Gn 2 )−1 , the desired results follow from Proposition 2.2, (3.1) and the Neumann series representation of Hn (c)⊤ Hn (c). 10 Using the above result, we can prove a uniform version of Theorem 2.1. Proposition 3.1. Let Tn and Sn be defined by (2.8). Then dPn,cηn+rn un u2n p →0 (Iγ + Jγ ) − − un Tn + \|un \|Sn − log dPn,cηn 2 d under Pn,cηn as n → ∞ uniformly in c ∈ C for any bounded sequence un of real numbers. Moreover, (Tn , Sn ) − → N (0, Iγ ) ⊗ N (0, Jγ ) under Pn,cηn as n → ∞ uniformly in c ∈ C. Proof. We can prove the first claim in a similar manner to the proof of Theorem 2.1 using Lemma 3.1 instead d of Proposition 2.2. To prove the second claim, it suffices to show that (Tn , Sn ) − → N (0, Iγ ) ⊗ N (0, Jγ ) under Pn,cnηn as n → ∞ for any sequence cn of numbers in C, which follows from Lemma 3.1 and inequality (13) from Dalalyan and Yoshida (2011). Proposition 3.1 implies that the experiments (Pn,cηn )c∈C enjoy the LAN property if γ = ∞ and do not oth- erwise. When the LAN property holds true, there is a well-established theory to define the asymptotic efficiency of estimators (cf. Section II-11 of Ibragimov and Has’minskii (1981)): A sequence cn of estimators in the experiments (Pn,cηn )c∈C is said to be asymptotically efficient at c ∈ C if the variables rn−1 ηn (cn − c) converge in law to N (0, Iγ−1 ) under Pn,cηn as n → ∞ (see Definition II-11.1 of Ibragimov and Has’minskii (1981)). Under the LAN property, this definition of the asymptotic efficiency is supported by several theorems such as the convolution theorem (e.g. Theorem II-9.1 of Ibragimov and Has’minskii (1981)) and the local asymptotic minimax theorem (e.g. Theorem II-12.1 of Ibragimov and Has’minskii (1981)). Moreover, it is well-known that both maximum likelihood and Bayesian estimators are asymptotically efficient under very general settings (cf. Chapter III of Ibragimov and Has’minskii (1981)). On the other hand, if the LAN property fails, it is generally not obvious how to define the asymptotic efficiency of estimators. Here, we adopt the approach from Küchler and Kutoyants (2000) to define the asymptotic efficiency, which is based on Theorem I-9.1 of Ibragimov and Has’minskii (1981) that derives an asymptotically minimax lower bound from the asymptotic properties of the Bayesian estimators. As a consequence, the Bayesian estimators are turned out to be asymptotically efficient. Now we explain the strategy to obtain asymptotically efficient estimators in our setting. As in the previous en,ϑ rather than the original model Pn,ϑ . For this reason section, we would like to work with the tractable model P we consider the quasi-likelihood function based on the former as follows: en,cηn dP 1 ⊤ 1 −1 p exp − zbn Vn (cηn ) zbn , = Ln (c) := dx 2 (2π)n det Vn (cηn ) c ∈ C. Then we consider the quasi maximum likelihood and Bayesian estimators based on Ln (c) as our estimators and give en,cηn )c∈C using the general scheme of Ibragimov and Has’minskii their asymptotic behavior in the experiments (P (1981) (see Proposition 3.2). Next we consider the case γ = ∞ where the LAN property holds true and thus en,cηn )c∈C can be transferred to that in (Pn,cηn )c∈C by Proposition 2.1(c). Finally, we convergence in law in (P en,cη )c∈C = (Pn,cη )c∈C for sufficiently large n due to (3.1) and consider the case γ < ∞ where we have (P n n Proposition 2.1(a), hence we can apply the Ibragimov-Has’minskii method to define and obtain asymptotically efficient estimators. The quasi maximum likelihood estimator (QMLE) ĉn is defined as a solution of the equation Ln (ĉn ) = sup Ln (c). c∈C 11 Note that the above equation always has at least one solution belonging to the closure of C because c 7→ Ln (c) is continuous. Moreover, we can choose ĉn so that it is measurable by the measurable selection theorem (see e.g. Theorem 6.7.22 of Pfanzagl (1994)). Also, the quasi Bayesian estimator (QBE) c̃n for a prior density q : C → (0, ∞) with respect to the quadratic loss is defined by Z Z Ln (c)q(c)dc, c̃n = cLn (c)q(c)dc C C where the prior density q is assumed to be continuous and satisfy 0 < inf c∈C q(c) ≤ supc∈C q(c) < ∞. The corresponding QMLE and QBE in the experiments (Pn,ϑ )ϑ∈R are given by ϑ̂n = ĉn ηn and ϑ̃n = c̃n ηn , respectively. Remark 3.2. Since the quantity ηn seems the exact order of the true time-lag parameter, one may consider that in a practical setting it is difficult to know ηn beforehand and thus it is difficult to use the estimators ϑ̂n and ϑ̃n . However, when we construct the estimator ϑ̂n , ηn can be considered as the maximum order of the true time-lag en,ϑ /dx and Cn = {cηn : c ∈ C}. Then the estimator ϑ̂n can be parameter as follows. Let us set L′n (ϑ) = dP considered as a solution of the equation L′n (ϑ̂n ) = sup Ln (ϑ). ϑ∈Cn Therefore, in a practical situation (supc∈C c)ηn (resp. (inf c∈C c)ηn ) can be interpreted as an upper bound (resp. a lower bound) of possible time-lag parameters ϑ. It is often not so difficult to find such bounds in a practical setting (and they are typically “small” as pointed out in the Introduction). For example, we can find them by computing the cross-correlations via Hoffmann et al. (2013)’s method as in Huth and Abergel (2014). The same remark can be applied to the estimator ϑ̃n because it can be rewritten as Z Z ′ ϑLn (ϑ)qn (ϑ)dϑ ϑ̃n = Cn Cn L′n (ϑ)qn (ϑ)dϑ, where qn (ϑ) = ηn−1 q(ϑ/ηn ) for ϑ ∈ Cn (so qn is a prior density on Cn ). To describe the limit distribution of these estimators, we introduce the likelihood ratio process for the limit experiment u2 Z(u) = exp uζ1 + \|u\|ζ2 − (Iγ + Jγ ) , 2 u ∈ R, where ζ1 and ζ2 are two mutually independent variables such that ζ1 ∼ N (0, Iγ ) and ζ2 ∼ N (0, Jγ ). Then we set     (ζ1 + ζ2 )/(Iγ + Jγ ) if ζ1 ≥ (−ζ2 ) ∨ 0, û = argmaxu∈R Z(u) = (ζ1 − ζ2 )/(Iγ + Jγ ) if ζ1 < ζ2 ∧ 0,    0 otherwise and R∞ ũ = R−∞ ∞ uZ(u)du −∞ Z(u)du . en,cηn )c∈C using the We first give the asymptotic behavior of the estimators ĉn and c̃n in the experiments (P general scheme of Ibragimov and Has’minskii (1981). Note that in this situation ĉn and c̃n are true maximum likelihood and Bayesian estimators, respectively. 12 Proposition 3.2. For any compact subset K of C, uniformly in c ∈ K it holds that rn−1 ηn (ĉn − c) converges in law en,cη and E e n,cη [\|r −1 ηn (ĉn − c)\|p ] → E[\|û\|p ] for any p > 0 as n → ∞. Also, uniformly in c ∈ K to û under P n n n −1 e n,cηn and E e n,cηn [\|r −1 ηn (c̃n − c)\|p ] → E[\|ũ\|p ] for any it holds that r ηn (c̃n − c) converges in law to ũ under P n n p > 0 as n → ∞. en,cη +r u /dP en,cη Proof. For every c ∈ C, we set Un (c) = {u ∈ R : c + rn ηn−1 u ∈ C} and define Zn,c (u) = dP n n n for each u ∈ Un (c). According to Theorems I-10.1 and I-10.2 from Ibragimov and Has’minskii (1981), it suffices to prove the following statements: p p e n,cη [\| Zn,c (u) − Zn,c (v)\|2 ] < ∞, (a) lim supn→∞ supc∈K supu,v∈Un (c) \|u − v\|−2 E n p 2 e n,cη [ Zn,c (u)] < ∞, (b) there is a constant κ > 0 such that lim sup sup sup eκu E n→∞ c∈K u∈Un (c) n (c) the marginal distributions of Zn,c converge in law to the marginal distributions of Z uniformly in c ∈ K as n → ∞. (c) is an immediate consequence of Proposition 3.1. On the other hand, by Eq.(A.4) from Reiß (2011) we obtain e n,cη E n " q Zn,c (u) − q 2 Zn,c (v) # en,cη +r u , P en,cη +r v ) = H 2 (P n n n n 1 1 1 1 ≤ 4ρ2 rn2 (u − v)2 kVn (cηn + rn u)− 2 T̄n Vn (cηn + rn u)− 2 k2F + kVn (cηn + rn u)− 2 S̄n Vn (cηn + rn u)− 2 k2F , hence Lemma 3.1 yields claim (a). Now we consider (b). By Corollary 3.2a.1 from Mathai and Provost (1992) we have q 1 1 1 −1 e n,cη (E2n − An,c (u)) − E2n , log E Zn,c (u) = − log det[E2n − An,c (u)] − log det E2n + n 4 2 2 1 1 where An,c (u) = E2n − Vn (cηn )− 2 Vn (cηn + rn u)Vn (cηn )− 2 . Then we consider the following decomposition: q e n,cη Z (u) log E n,c n 1 1 2 =− log det[E2n − An,c (u)] + tr[An,c (u)] + kAn,c (u)kF 4 2 1 1 1 1 − log det E2n + Bn,c (u) − tr (Bn,c (u)) + kBn,c (u)k2F 2 2 2 8 1 1 1 2 2 2 + tr[An,c (u)] + kAn,c (u)kF − tr (Bn,c (u)) − kAn,c (u)kF − kBn,c (u)kF 4 8 2 =: In,c (u) + IIn,c (u) + IIIn,c (u) + IVn,c (u), where Bn,c (u) = (E2n − An,c (u))−1 − E2n . Let us set 1 1 1 1 S̃n,c = ρVn (cηn )− 2 S̄n Vn (cηn )− 2 . T̃n,c = ρVn (cηn )− 2 T̄n Vn (cηn )− 2 , Then we have An,c (u) = cηn T̃n,c + \|cηn \|S̃n,c − (cηn + rn u)T̃n,c − \|cηn + rn u\|S̃n,c , hence it holds that sup sup kAn,c (u)ksp ≤ sup \|c\| sup 2ηn (kT̃n,c ksp + kS̃n,c ksp ). c∈C u∈Un (c) c∈C c∈C 13 Here, we use the fact that c+rn ηn−1 u ∈ C because of u ∈ Un (c). In particular, we have supc∈C supu∈Un (c) kAn,c (u)ksp < P k 1 for sufficiently large n by (3.1) and Lemma 3.1. For such an n we have Bn,c (u) = ∞ k=1 An,c (u) and thus kBn,c (u)ksp ≤ kAn,c (u)ksp , 1 − kAn,c (u)ksp kBn,c (u)kF ≤ kAn,c (u)kF , 1 − kAn,c (u)ksp k−1 for k ≥ 1 to obtain the latter estimate. where we use the inequality kAn,c (u)k kF ≤ kAn,c (u)kF kAn,c (u)ksp Therefore, for sufficiently large n we have for any c ∈ C and any u ∈ Un (c) \|In,c (u)\| ≤ 1 kAn,c (u)ksp kAn,c (u)k2F , 12 1 − kAn,c (u)k3sp \|IIn,c (u)\| ≤ 1 kBn,c (u)ksp kBn,c (u)k2F 6 8 − kBn,c (u)k3sp by Appendix II-(v) from Davies (1973) and \|IIIn,c (u)\| ≤ ∞ X k=3 tr(An,c (u)k ) ≤ kAn,c (u)k2F kAn,c (u)ksp 1 − kAn,c (u)ksp k−2 for k ≥ 3, as well as by the inequality tr(An,c (u)k ) ≤ kAn,c (u)k2F kAn,c (u)ksp kAn,c (u)k2F 1 . 1− IVn,c (u) ≤ − 8 2(1 − kAn,c (u)ksp )2 Consequently, there is a constant κ0 > 0 such that for sufficiently large n it holds that q e Zn,c (u) ≤ −κ0 kAn,c (u)k2F log En,cηn for any c ∈ C and any u ∈ Un (c). Now we consider giving an upper bound for −kAn,c (u)k2F . We have −kAn,c (u)k2F = −u2 rn2 kT̃n,c k2F − 2rn u(\|cηn + rn u\| − \|cηn \|) tr(T̃n,c S̃n,c ) − (\|cηn + rn u\| − \|cηn \|)2 kS̃n,c k2F ≤ −2u2 Iγ + u2 rn2 kT̃n,c k2F − 2Iγ + 2 rn2 tr(T̃n,c S̃n,c ) . Therefore, noting Iγ > 0, by Lemma 3.1 for sufficiently large n we have −kAn,c (u)k2F ≤ −Iγ u2 for any c ∈ C and any u ∈ Un (c). Consequently, we obtain (b) by setting κ = κ0 Iγ . If γ < ∞, (3.1) is equivalent to the condition that ηn = o(n−1 ) as n → ∞. Therefore, Proposition 2.1(a) yields the following result: e n,cη ’s are replaced by Pn,cη ’s. Corollary 3.1. If γ < ∞, the statement of Proposition 3.2 still holds true while P n n Now we return to the efficient estimation of the parameter c in the model (Pn,cηn )c∈C . First we consider the case γ = ∞. In this case we know that (Pn,cηn )c∈C enjoy the LAN property at every c ∈ C by Proposition 3.1, so the definition of the asymptotic efficiency of an estimator sequence is well-established as explained in the above. Theorem 3.1. If γ = ∞, both ĉn and c̃n are asymptotically efficient at every c ∈ C in the experiments (Pn,cηn )c∈C . That is, both rn−1 ηn (ĉn −c) and rn−1 ηn (c̃n −c) converge in law to N (0, Iγ−1 ) under Pn,cηn for any c ∈ C as n → ∞. In particular, both ϑ̂n and ϑ̃n are asymptotically efficient at ϑ = 0 in the experiments (Pn,ϑ )ϑ∈R . Next we turn to the case γ < ∞. In this case the experiments (Pn,cηn )c∈C no longer enjoy the LAN property, so the definition of the asymptotic efficiency is not obvious. As explained in the above, here we follow the approach of Küchler and Kutoyants (2000) to define the asymptotic efficiency for our experiments. We obtain the following result by virtue of Corollary 3.1 and Theorem I-9.1 of Ibragimov and Has’minskii (1981): 14 Theorem 3.2. If γ < ∞, we have lim lim inf sup (rn−1 ηn )2 En,cηn [(c∗n − c)2 ] ≥ E[ũ2 ] δ→0 n→∞ \|c−c0 \|<δ for any c0 ∈ C and any estimator sequence c∗n in the experiments (Pn,cηn )c∈C . In particular, we also have lim lim inf sup rn−2 En,ϑ [(ϑ∗n δ→0 n→∞ \|ϑ\|<δηn − ϑ)2 ] ≥ E[ũ2 ] for any estimator sequence ϑ∗n in the experiments (Pn,ϑ )ϑ∈R . Thanks to Theorem 3.2, an estimator sequence c∗n is said to be asymptotically efficient at c0 ∈ C in the experi- ments (Pn,cηn )c∈C if it holds that lim lim inf sup (rn−1 ηn )2 En,cηn [(c∗n − c)2 ] = E[ũ2 ]. δ→0 n→∞ \|c−c0 \|<δ Similarly, an estimator sequence ϑ∗n is said to be asymptotically efficient at ϑ = 0 in the experiments (Pn,ϑ )ϑ∈R if it holds that lim lim inf sup rn−2 En,ϑ [(ϑ∗n δ→0 n→∞ \|ϑ\|<δηn − ϑ)2 ] = E[ũ2 ] for any sequence ηn of positive numbers satisfying (3.1). The following result is an immediate consequence of this definition. Theorem 3.3. If γ < ∞, the sequence c̃n is asymptotically efficient at every c ∈ C in the experiments (Pn,cηn )c∈C . In particular, the sequence ϑ̃n is asymptotically efficient at ϑ = 0 in the experiments (Pn,ϑ )ϑ∈R . In contrast, there is no guarantee of the asymptotic efficiency of the (Q)MLE ĉn if γ < ∞. In fact, c̃n may perform much better than ĉn if γ = 0, as shown in the following proposition. Proposition 3.3. It holds that s ! ! p Iγ Jγ Jγ 1 E û , 1 − arctan + π Iγ π(Iγ + Jγ ) Z ∞Z ∞ 2 1 xΨ(x) − yΨ(y) 2 E ũ = ψR (x, y)dxdy, Iγ + Jγ −∞ −∞ Ψ(x) + Ψ(y) where Ψ(x) = R∞ eux−u 2 1 = Iγ + Jγ 2 /2 (3.2) (3.3) du and ψR (x, y) denotes the bivariate normal density with standard normal marginals and correlation R = (Jγ − Iγ )/(Jγ + Iγ ). In particular, lim\|ρ\|→1 E ũ2 /E û2 = 0 if γ = 0. 0 Proof. Let us denote by φa the normal density with mean 0 and variance a. Then, a simple calculation yields Z ∞ Z ∞ 2 2 2 φIγ (x)φJγ (z − x)dx. z dz E û = (Iγ + Jγ )2 0 0 By formulae (3.322.2) and (6.292) from Gradshteyn and Ryzhik (2007) we have Z ∞ 2 z dz 0 Z 0 ∞ Iγ + Jγ Iγ + Jγ − arctan φIγ (x)φJγ (z − x)dx = 2 2π hence we obtain (3.2). 15 s Jγ Iγ ! p Iγ Jγ , + 2π Next, by a change of variable we obtain Z ∞ 1 Z(u)du = p Iγ + Jγ −∞ ( ζ + ζ2 p1 Iγ + Jγ Ψ ! +Ψ ζ − ζ1 p2 Iγ + Jγ !) . Moreover, formulae (3.462.5) and (3.322.2) from Gradshteyn and Ryzhik (2007) imply that ! !) ( Z ∞ ζ2 − ζ1 1 ζ1 + ζ2 ζ2 − ζ1 ζ1 + ζ2 p Ψ p −p Ψ p . uZ(u)du = Iγ + Jγ Iγ + Jγ Iγ + Jγ Iγ + Jγ Iγ + Jγ −∞ p p Since the distribution of the vector ((ζ1 +ζ2 )/ Iγ + Jγ , (ζ2 −ζ1 )/ Iγ + Jγ ) has the density ψR (x, y), we obtain (3.3). Finally, we prove the latter statement. Define the functions f and g on (−1, 1) by ! √ r Z ∞Z ∞ 1 xΨ(x) − yΨ(y) 2 1+r 1 − r2 f (r) = 1 − arctan + ψr (x, y)dxdy. , g(r) = π 1−r 2π Ψ(x) + Ψ(y) −∞ −∞ Then we have E[û2 ] = f (R)/(Iγ + Jγ ) and E[ũ2 ] = g(R)/(Iγ + Jγ ). Since R → 1 as \|ρ\| → 1 if γ = 0 and limr→1 f (r) = 12 , it suffices to prove limr→1 g(r) = 0. Because we have g(r) = Z ∞ −∞ Z ∞ −∞ !2 √ √ xΨ(x) − (rx + 1 − r 2 y)Ψ(rx + 1 − r 2 y) √ φ1 (x)φ1 (y)dxdy, Ψ(x) + Ψ(rx + 1 − r 2 y) the dominated convergence theorem yields limr→1 g(r) = 0, which completes the proof. 4 Appendix: Proof of Proposition 2.2 Before starting the proof, we introduce some notation. We set π 1 ξi := ξi,n := i− , 2n + 1 2 i = 1, 2, . . . . Then we define the n × n matrix Un = (uij )1≤i,j≤n by 2 2π 1 1 2 √ uij := i− j− =√ cos cos [ξi (2j − 1)] , 2n + 1 2 2 2n + 1 2n + 1 which is often referred to as the Discrete Cosine Transform (DCT) of type-VIII (see Sabel and Schmidt-Hieber (2014) and references therein). Note that Un⊤ = Un and Un is real orthogonal. It is known that Un diagonalizes Fn as follows: where λi := 2 [1 − cos (2ξi )] . Un Fn Un = diag(λ1 , . . . , λn ), (4.1) See Lemma 1 of Kunitomo and Sato (2013) or Lemma C.2 of Sabel and Schmidt-Hieber (2014) for the proof. For each a > 0 we define the functions fa and ga on R by fa (x) = a + 2vn (1 − cos(x)), n ga (x) = sin(x) fa (x) (x ∈ R). We also set Gn (a) = na En + vn Fn . From (4.1) we have Un Gn (a)Un = Λn (a) := diag(fa (2ξ1 ), . . . , fa (2ξn )). 16 (4.2) Remark 4.1. It turns out that the off-diagonal components of Un Tn Un play a dominant role to calculate the limit of −1 −1 kḠn 2 T̄n Ḡn 2 kF . This is essentially different from the case of calculating the Fisher information for the scale pa- rameter estimation from observations of the form (1.2), where the similarity transformations of Toeplitz matrices by Un are sufficiently approximated by diagonal matrices as manifested by Lemma C.4 of Sabel and Schmidt-Hieber (2014). For this reason we need rather specific calculations as seen in Lemmas 4.4 and 4.7. For a square matrix A, spr(A) denotes the spectral radius of A. We will frequently use the identity kAksp = spr(A) holding if A is a normal matrix. Now we start the main body of the proof. We will frequently use the following well-known inequality for the sine function, π . 0≤x≤ 2 2 x ≤ sin(x) ≤ 1 π Lemma 4.1. For any a > 0, we have sup ga (x) = p 0≤x≤π ( na )2 1 , + 4 na vn (4.3) sup \|ga′ (x)\| ≤ 0≤x≤π Proof. The claim immediately follows from the identity ga′ (x) = 3n . a a +2vn ) cos(x)−2vn (n fa (x)2 = a (1+cos(x)) n fa (x)2 − 1 fa (x) . Lemma 4.2. Let ψ : [0, π] → R be continuous. Also, let mn be a sequence of positive integers such that mn ≤ n and mn /n → c ∈ (0, 1] as n → ∞. Then, for any p > 0 we have lim 1 n→∞ np+1 Z πc mn X ψ(2ξi ) 1 = p ψ(x)dx fa (2ξi )p a π 0 i=1 provided that γ = 0. Proof. By the fundamental theorem of calculus, we have ψ(2ξi ) ψ(2ξi ) ≤ pkψk∞ − p fa (2ξi ) (a/n)p Z vn λi 0 1 2pkψk∞ vn dx ≤ . p+1 (a/n + x) (a/n)p+1 Hence the desired result follows from the standard Riemann sum approximation. Lemma 4.3. Let mn be a sequence of positive integers such that mn ≤ n and mn /n → c ∈ (0, 1] as n → ∞. Then   mn  X 1 1 lim = n→∞ nNn f (2ξi )   i=1 a π √ 2 a(a+4γ) arctan 1 √ 2 a q 1+ 4γ a tan πc 2 if γ < ∞,  1  c − sin2π2πc  2ab q  √   b(b+4γ) 4γ  πc  1 + b tan 2 mn  2πγ 2 (b−a) arctan X 2 √ q lim r ga (2ξi )gb (2ξi ) = a(a+4γ) n→∞ n 4γ  πc  − − 2πγ 2 (b−a) arctan 1 + a tan 2 i=1      1  √ √ 2( a+ b) for any a, b > 0 such that a 6= b. 17 (4.4) if γ = ∞, if γ = 0, c 4γ 2 (4.5) if 0 < γ < ∞, if γ = ∞. Proof. First, using the lower and upper Darboux sums of the integral 2n + 1 2π Z 2ξmn 2ξ1 m R 2ξmn 2ξ1 1 fa (x) dx, n X 1 1 2n + 1 1 1 dx + ≤ + ≤ fa (x) fa (2ξmn ) fa (2ξi ) fa (2ξ1 ) 2π i=1 we obtain Z 2ξmn 1 dx. fa (x) 2ξ1 Now, formula (2.553.3) in Gradshteyn and Ryzhik (2007) yields Z y 0 r 1 2n dx = p arctan fa (x) a(a + 4nvn ) ! y 4nvn , tan 1+ a 2 hence we obtain (4.4). Next, a simple calculation yields n ga (x)gb (x) = b−a sin2 x sin2 x − fa (x) fb (x) . Therefore, if γ = 0, Lemma 4.2 implies that lim r 2 n→∞ n mn X i=1 1 ga (2ξi )gb (2ξi ) = πab Z πc 0 1 sin xdx = 2ab 2 sin 2πc c− 2π . On the other hand, if γ 6= 0, for sufficiently large n we have 1 − cos(x) = (fa (x) − a/n)/(2vn ), hence sin2 x = fa (x)/vn − a/(nvn ) − fa (x)2 /(4vn2 ) + fa (x)a/(2nvn2 ) − a2 /(2nvn )2 . Therefore, we obtain sin2 x sin2 x − = fa(x) fb (x) b b2 + nvn (2nvn )2 1 − fb (x) a a2 + nvn (2nvn )2 1 b−a − . fa (x) 4nvn2 Hence the desired result follows from (4.4). Lemma 4.4. Let mn be a sequence of positive integers such that mn ≤ n and mn → ∞ as n → ∞. Then lim n→∞ 4 2n + 1 2 X mn sin4 (n · 2ξ1 i) = 2. sin2 (2ξ1 i) i=1 Proof. Since 4nξ1 = π − 2ξ1 , we have o 1 1n 1 − (−1)l cos (2ξ1 l) = sin (n · 2ξ1 l) = {1 − cos (4nξ1 l)} = 2 2 2 ( sin2 (ξ1 l) if l is even, cos2 (ξ1 l) if l is odd. Therefore, using the formula sin(2ξ1 i) = 2 sin(ξ1 i) cos(ξ1 i), we can decompose the target quantity as     2 X m m mn n n  X X 4 4 sin4 (n · 2ξ1 i) 2 2 cot (ξ i) =: A1,n + A2,n . tan (ξ i) + = 1 1  2n + 1 (2n + 1)2  sin2 (2ξ1 i)   i=1 i=1 i=1 i: even i: odd First we prove limn A1,n = 0. Using the monotonicity of the tangent function and assumption mn ≤ n, we obtain A1,n 8 ≤ (2n + 1)π Z 0 18 π n+1 2 2n+1 tan2 (x)dx Since formula (2.526.22) in Gradshteyn and Ryzhik (2007) yields limn that limn A1,n = 0. n+1 R π2 2n+1 0 tan2 (x)dx = 1−π/4, we conclude Next we prove limn A2,n = 2. Our proof relies on the following inequality for the tangent function: π π x π x ≤ tan x ≤ (0 ≤ x < 1). 2 2 2 1 − x2 (4.6) The lower estimate of (4.6) is well-known, and the upper estimate is known as the Becker-Stark inequality (Eq.(2) of Becker and Stark (1978)). Now, using (4.6), we obtain ⌊ mn2+1 ⌋ ⌊ mn2+1 ⌋ 2 (2i − 1)2 16 X 16 X 1 1 − + . ≤ A2,n ≤ 2 2 2 2 4 π (2i − 1) (2n + 1) (2n + 1) π (2i − 1)2 i=1 i=1 Therefore, using formula P∞ i=1 (2i − 1)−2 = π2 8 , we conclude that limn A2,n = 2. 1 1 1 1 Lemma 4.5. For any a, b > 0, it holds that kGn (a)− 2 Rn Gn (b)− 2 ksp = O(Nn ) and kGn (a)− 2 Rn Gn (b)− 2 k2F = O(Nn2 ) as n → ∞. 1 1 1 1 Proof. First, by definition we have kGn (a)− 2 Rn Gn (b)− 2 ksp = spr(Gn (b)− 2 Gn (a)− 2 Rn ), hence (4.2) and Theorem 5.6.9 of Horn and Johnson (1985) yield − 12 kGn (a) − 12 Rn Gn (b) ksp n X n 4 X p ≤ ≤ max 1≤i≤n 2n + 1 fa (2ξk )fb (2ξk ) k=1 k=1 uik uk1 1 1 1 + fa (2ξk ) fb (2ξk ) . 1 Therefore, Lemma 4.3 implies that kGn (a)− 2 Rn Gn (b)− 2 ksp = O(Nn ). Moreover, since it holds that 1 1 kGn (a)− 2 Rn Gn (b)− 2 k2F = tr(Gn (a)−1 Rn Gn (b)−1 Rn ) n X u1k uk1 u1l ul1 = ≤ fa (2ξk )fb (2ξl ) n 1 4 X 2n + 1 fa (2ξk ) k,l=1 k=1 ! n 4 X 1 2n + 1 fb (2ξl ) l=1 ! , Lemma 4.3 again yields the desired result. Lemma 4.6. For any a > 0, we have 1 1 1 kGn (a)− 2 Sn Gn (a)− 2 k2sp = O(Nn ), 1 rn2 kGn (a)− 2 Sn Gn (a)− 2 k2F → Jγ0 (a) (4.7) as n → ∞, where for any a > 0 we set  6  if γ = 0,   a2  1/2 3/2 a a 1 + a+4γ if 0 < γ < ∞, Jγ0 (a) = 2γ 2 2 − 3 a+4γ     0 if γ = ∞. Proof. Since Sn − Rn = Fn , by Lemma 4.5 it suffices to prove (4.7) in the case where Sn is replaced by Fn . (4.1)–(4.2) imply that 1 1 λi ≤ min{vn−1 , 4n/a} 1≤i≤n fa (2ξi ) kGn (a)− 2 Fn Gn (a)− 2 ksp = max 19 (4.8) and 1 1 rn2 kGn (a)− 2 Fn Gn (a)− 2 k2F = rn2 n X i=1 λ2i . fa (2ξi )2 (4.9) The first equation in (4.7) immediately follows from (4.8). In order to prove the second equation in (4.7), we will prove the right side of (4.9) converges to Jγ0 (a) as n → ∞. First, if γ = 0, the desired result follows from Lemma 4.2. Next, if 0 < γ < ∞, noting that fa (2ξi ) = a/n + vn λi , we have a 2 1 1 1 a λ2i + = 2 1−2 , fa (2ξi )2 vn n fa (2ξi ) n fa (2ξi )2 hence by Lemma 4.3 we obtain the desired equation once we prove n 1 X 1 1 a lim = p . 1+ n→∞ n3 f (2ξi )2 a + 4γ 2a a2 + 4aγ i=1 a The monotonicity of the cosine function yields 2n + 1 2π Z π 0 n X 1 1 1 2n + 1 dx ≤ ≤ + 2 2 2 fa (x) fa (2ξi ) fa (2ξ1 ) 2π i=1 Z π 0 1 dx. fa (x)2 Since formula (3.661.4) in Gradshteyn and Ryzhik (2007) implies that Z π π 1 a/n p dx = , 1+ 2 a/n + 4vn 2(a/n) (a/n)2 + 4avn /n 0 fa (x) we obtain the desired result. Finally, if γ = ∞, using the inequality fa (2ξi ) ≥ vn λi we obtain rn2 n X i=1 λ2i 1 n , ≤ 3 2 =√ 2 fa (2ξi ) N n vn nvn hence we deduce the desired result. Lemma 4.7. If a and b are positive numbers such that a 6= b, we have  2   ab √  √ 1 1 b(b+4γ)− a(a+4γ) 2 −2 −2 2 lim rn kGn (a) Tn Gn (b) kF = − γ 2 (b−a) n→∞    √ 2√ a+ b if γ = 0, 1 γ2 if 0 < γ < ∞, if γ = ∞. Proof. Since u0i = u1i , we have (Un Tn Un )ij + (Un Rn Un )ij = (Un (Tn + Rn )Un )ij = n X (ui,k−1 − ui,k+1 )ukj . k=1 x−y Using the trigonometric identities cos(x) − cos(y) = −2 sin( x+y 2 ) sin( 2 ) and 2 sin(x) cos(y) = sin(x + y) − sin(x − y), we obtain n (Un Tn Un )ij + (Un Rn Un )ij = X 4 sin(2ξi ) {sin ((2k − 1) (ξi + ξj )) + sin ((2k − 1) (ξi − ξj ))} . 2n + 1 k=1 20 Then, using summation formula (1.342.3) of Gradshteyn and Ryzhik (2007), we have 2 sin (n(ξi + ξj )) sin2 (n(ξi − ξj )) 4 ij ij (Un Tn Un ) + (Un Rn Un ) = sin(2ξi ) + 1{i6=j} . 2n + 1 sin(ξi + ξj ) sin(ξi − ξj ) (4.10) Now, since (Un Tn Un )⊤ = −Un Tn Un and (Un Rn Un )⊤ = Un Rn Un , by (4.2), (4.10) and the unitary invariance of the Frobenius norm, we obtain 1 1 1 1 kGn (a)− 2 Tn Gn (b)− 2 k2F − kGn (a)− 2 Rn Gn (b)− 2 k2F 4 2 X n 4 sin (n(ξi + ξj )) sin4 (n(ξi − ξj )) sin4 (n(ξi − ξj )) =− ga (2ξi )gb (2ξj ) − 1{i<j} − 1{i>j} 2n + 1 sin2 (ξi + ξj ) sin2 (ξi − ξj ) sin2 (ξi − ξj ) i,j=1 =: B1,n + B2,n + B3,n . First we consider B1,n . Using inequalities (4.3) and (x + y)2 ≥ 4xy (x, y ∈ R), we have X n 1 \|B1,n \| ≤ 4 max ga (2ξi ) max gb (2ξi ) 1 1≤i≤n 1≤i≤n (i − 2 + j − 12 )2 i,j=1 !2 X n 1 ≤ 4 max ga (2ξi ) max gb (2ξi ) , 1≤i≤n 1≤i≤n 2i − 1 i=1 and thus Lemma 4.1 yields B1,n = O((Nn log n)2 ) = o(rn−2 ). Next we consider B2,n . First we prove B2,n = B′2,n + o(rn−2 ), where 2 X n sin4 (n(ξi − ξj )) 4 ′ ga (2ξi )gb (2ξi ) 1{i<j} . B2,n = 2n + 1 sin2 (ξi − ξj ) i,j=1 Lemma 4.1 and (4.3) yield B2,n − B′2,n ≤ ≤ ≤ 4 2n + 1 2 4 (2n + 1)2 n sin4 (n(ξi − ξj )) 6n X ga (2ξi )\|ξi − ξj \| 1{i<j} b sin2 (ξi − ξj ) 6π 2 n b 4 6πn 2n + 1 b n X i,j=1 n X ga (2ξi ) i,j=1 ga (2ξi ) n X 1 j=1 i=1 1 1 \|ξi − ξj \| {i<j} j . If γ = ∞, by the property of ga we have n 2π 2π X ga (2ξi ) ≤ 2 max ga (2ξi ) + 2n + 1 2n + 1 1≤i≤n i=1 ≤ hence Lemma 4.1 yields Pn Lemma 4.3 because ga (x) i=1 ga (2ξi ) ≤ fa (x)−1 . 4 2n + 1 2ξn ga (x)dx 2ξ1 1 fa (2ξn ) 2π 2 max ga (2ξi ) + log , 2n + 1 1≤i≤n 2vn fa (2ξ1 ) = O(Nn2 log n). This also holds true in the case that γ < ∞ due to Consequently, B2,n − B′2,n = O((Nn log n)2 ) = o(rn−2 ). Now, since ξi − ξj = 2ξ1 (i − j), for any c ∈ (0, 1) we have Z 2 X n n X X sin4 (n · 2jξ1 ) sin4 (n · 2jξ1 ) 4 ′ g (2ξ )g (2ξ ) ga (2ξi )gb (2ξi ) ≤ B ≤ . a i i b 2,n 2n + 1 sin2 (2jξ1 ) sin2 (2jξ1 ) j=1 i=1 i=1 j=1 2 ⌊nc⌋ X n−⌊nc⌋ 21 Therefore, Lemma 4.4 implies that 2 lim inf rn2 n→∞ ⌊nc⌋ X i=1 ga (2ξi )gb (2ξi ) ≤ lim inf rn2 B′2,n ≤ lim sup rn2 B′2,n ≤ 2 lim sup rn2 n→∞ n→∞ n→∞ n X ga (2ξi )gb (2ξi ). i=1 Then, letting c ↑ 1, by Lemma 4.3 we obtain limn→∞ rn2 B2,n = limn→∞ rn2 B′2,n = 2 limn→∞ rn2 Pn i=1 ga (2ξi )gb (2ξi ). By symmetry we have limn→∞ rn2 B3,n = limn→∞ rn2 B2,n , hence we complete the proof due to (4.5) and Lemma 4.5. Proof of Proposition 2.2. Set 1 Ūn = √ 2 " Un −Un Un Un # . Then we have −1 −1 Ḡn 2 S̄n Ḡn 2 1 = Ūn 2 " Ln Un (Sn + ρ−1 Rn )Un Ln 0 0 −Ln Un (Sn − ρ−1 Rn )Un Ln −1 −1 Ḡn 2 T̄n Ḡn 2 1 = Ūn 2 " 0 Ln Un (Tn + ρ−1 Rn )Un Ln # Ūn⊤ and −Ln Un (Tn − ρ−1 R n )Un Ln 0 # Ūn⊤ , 1 1 where Ln = Λn (1 + ρ)− 2 and Ln = Λn (1 − ρ)− 2 . Hence we obtain −1 −1 4kḠn 2 (αT̄n + β S̄n )Ḡn 2 k2F 1 1 = 2α2 kGn (1 + ρ)− 2 (Tn + ρ−1 Rn )Gn (1 − ρ)− 2 k2F 1 1 1 1 + β 2 (kGn (1 + ρ)− 2 (Sn + ρ−1 Rn )Gn (1 + ρ)− 2 k2F + kGn (1 − ρ)− 2 (Sn − ρ−1 Rn )Gn (1 − ρ)− 2 k2F ). −1 −1 1 Therefore, Lemmas 4.5–4.7 yield (2.7). On the other hand, since 2kḠn 2 S̄n Ḡn 2 ksp ≤ kGn (1 + ρ)− 2 (Sn + 1 1 −1 −1 1 ρ−1 Rn )Gn (1+ρ)− 2 ksp +kGn (1−ρ)− 2 (Sn −ρ−1 Rn )Gn (1−ρ)− 2 ksp , Lemmas 4.5–4.6 also yield kḠn 2 S̄n Ḡn 2 ksp = O(Nn ). −1 −1 −1 −1 Hence the proof is completed once we prove kḠn 2 T̄n Ḡn 2 ksp = O(Nn ). Note that Ḡn 2 Vn (ϑ)Ḡn 2 is positive −1 −1 −1 −1 −1 −1 semidefinite and ρϑḠn 2 T̄n Ḡn 2 + Ḡn 2 Vn (ϑ)Ḡn 2 = E2n − ρ\|ϑ\|Ḡn 2 S̄n Ḡn 2 for any ϑ ∈ Θn . Note also that −1 −1 both T̄n and S̄n are symmetric. Therefore, if λ is an eigenvalue of Ḡn 2 T̄n Ḡn 2 , by the monotonicity theorem −1 −1 for eigenvalues (Corollary 4.3.3 of Horn and Johnson (1985)) we have ρϑλ ≤ kE2n − ρ\|ϑ\|Ḡn 2 S̄n Ḡn 2 ksp for −1 −1 any ϑ ∈ Θn . Since we can take ϑ = ±Nn−1 , this inequality implies that \|ρ\|Nn−1 kḠn 2 T̄n Ḡn 2 ksp ≤ kE2n − −1 −1 −1 −1 ρNn−1 Ḡn 2 S̄n Ḡn 2 ksp ≤ 1 + \|ρ\|Nn−1 kḠn 2 S̄n Ḡn 2 ksp , which yields the desired result. 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Robust Estimation via Robust Gradient Estimation arXiv:1802.06485v1 [stat.ML] 19 Feb 2018 Adarsh Prasad‡ Arun Sai Suggala‡ Sivaraman Balakrishnan† Pradeep Ravikumar‡ Machine Learning Department‡ Department of Statistics† Carnegie Mellon University, Pittsburgh, PA 15213. Abstract We provide a new computationally-efficient class of estimators for risk minimization. We show that these estimators are robust for general statistical models: in the classical Huber ǫ-contamination model and in heavy-tailed settings. Our workhorse is a novel robust variant of gradient descent, and we provide conditions under which our gradient descent variant provides accurate estimators in a general convex risk minimization problem. We provide specific consequences of our theory for linear regression, logistic regression and for estimation of the canonical parameters in an exponential family. These results provide some of the first computationally tractable and provably robust estimators for these canonical statistical models. Finally, we study the empirical performance of our proposed methods on synthetic and real datasets, and find that our methods convincingly outperform a variety of baselines. 1 Introduction Robust estimation has a rich history in statistics with seminal contributions due to Box [4], Tukey [42], Huber [24], Hampel [21] and several others. In the classical analysis of statistical estimators, statistical guarantees are derived under strong model assumptions and in most cases these guarantees hold only in the absence of arbitrary outliers, and other deviations from the model assumptions. Strong model assumptions are rarely met in practice, and this has led to the development of robust inferential procedures and various associated statistical concepts such as the influence function, the breakdown point and the Huber ǫ-contamination model to assess the robustness of estimators. Despite this progress however, the statistical methods with the strongest robustness guarantees, for instance those based on non-convex M -estimators [24], ℓ1 tournaments [13, 15, 45] and notions of depth [10, 20, 38], are computationally intractable. In this paper, we present a class of estimators that are computationally tractable and have strong robustness guarantees. The estimators we propose are obtained by robustifying classical algorithms for risk minimization and are applicable to a wide-range of parametric statistical models for which parameter estimation can be cast within this framework. In contrast to classical work, for instance on M-estimation, we do not attempt to replace the risk minimization objective with a robust counterpart but instead focus on making canonical gradient-based optimization for the usual risk minimization objective robust. We find that this shift in perspective enables a unified treatment of different statistical models, leads to computationally tractable estimators and leads to estimators with strong robustness guarantees. In the risk minimization framework, the target parameter θ ∗ is defined as the solution to an optimization problem: (1) θ ∗ = argmin R(θ) ≡ argmin Ez∼P L(θ; z) , θ∈Θ θ∈Θ 1 where L is an appropriate loss-function, R is the population risk and Θ is the set of feasible parameters. The goal of empirical risk minimization procedures is then to compute an approximate minimizer to the above program when given access to samples Dn = {z1 , . . . , zn }. In this classical setting, a standard assumption that is imposed on Dn is that the data has no outliers, and has no arbitrary deviations from model assumptions; i.e., it is typically assumed that each of the zi ’s are independent and identically distributed according to the distribution P . Many analyses of risk minimization further assume that P follows a sub-gaussian distribution, or has otherwise well-controlled tails in order to appropriately control the deviation between the population risk and its empirical counterpart. While our general results can be specialized to obtain results for a variety of models and notions of robustness, we focus on developing estimators which are robust to two canonical classes of deviations from the model assumptions: 1. Robustness to arbitrary outliers: In this setting, we focus on Huber’s ǫ-contamination model, where rather than observe samples directly from P in (1) we instead observe samples drawn from Pǫ which for an arbitrary distribution Q is defined as: Pǫ = (1 − ǫ)P + ǫQ. The distribution Q allows for arbitrary outliers, which may correspond to gross corruptions or more subtle deviations from the assumed model. This model can be equivalently viewed as model mis-specfication in the Total Variation (TV) metric. 2. Robustness to heavy-tails: In this setting, we are interested in developing estimators under weak moment assumptions. We assume that the distribution P from which we obtain samples only has finite low-order moments (see Section 5.4 for a precise characterization). Such heavy tailed distributions arise frequently in the analysis of financial data and large-scale biological datasets (see for instance examples in [18, 47]). In contrast to classical analyses of empirical risk minimization [43], in this setting the empirical risk is not uniformly close to the population risk, and methods that directly minimize the empirical risk perform poorly (see Section 4). The goal of our work is to develop estimators which are computationally tractable and robust in these models. Below, we provide an outline of our results and contributions. 1. Our first contribution is to introduce a new class of robust estimators for risk minimization (1). These estimators are based on robustly estimating gradients of the population risk, and are computationally tractable by design. Building on prior work for robust mean estimation in the Huber model [29], and in the heavy-tailed model [37], we design robust gradient estimators for the population risk in (1). Our main insight is that in this general risk minimization setting, the gradient of the population risk is simply a multivariate mean vector, and we can leverage prior work on mean estimation to design robust gradient estimators. Through this we are able to significantly generalize the applicability of mean estimation methods to general parametric models. 2. Our estimators are practical and our second contribution is to conduct extensive numerical experiments on real and simulated data with our proposed estimators. We provide guidelines for tuning parameter selection and we compare the proposed estimators with several competitive baselines [3, 19, 24]. Across different settings and according to various metrics we find that our estimators consistently perform well. 2 3. Finally, we provide rigorous robustness guarantees for the estimators we propose for a variety of canonical statistical models including for linear regression, for logistic regression and for estimation of the canonical parameters in an exponential family. Our contributions in this direction are two-fold: building on prior work [2] we provide a general result on the stability of gradient descent for risk minimization, showing that in favorable cases gradient descent can be quite tolerant to inaccurate gradient estimates. Subsequently, in concrete settings we provide a careful analysis of the quality of gradient estimation afforded by our proposed gradient estimators and combine these results to obtain guarantees on our final estimates. Broadly, as we discuss in the sequel, our work suggests that estimators which are based on robust gradient estimation offer a variety of practical, conceptual, statistical and computational advantages for robust estimation. 1.1 Related Work There is extensive work broadly in the area of robust statistics (see for instance [21] and references therein), and we focus this section on some lines of work that are most related to this paper. Classical work has already developed several estimators which are known to be optimally robust for a variety of inferential tasks, including hypothesis testing [26], mean estimation [38], general parametric estimation [11, 15, 45], and non-parametric estimation [13]. However, a major drawback of this classical line of work has been that most of the estimators with strong robustness guarantees are computationally intractable [42], while the remaining ones are heuristics which are not optimal [22]. Recently, there has been a flurry of research in theoretical computer science [9, 14, 16, 29, 32] designing provably robust estimators which are computationally tractable while achieving near-optimal contamination dependence for special classes of problems. Some of the proposed algorithms, are not practical as they rely on the ellipsoid algorithm or require solving semidefinite programs [9, 14, 16, 32] which can be slow for modern problem sizes. We build on the work of Lai et al. [29], who study practical robust mean and covariance estimators for distributions with appropriately controlled moments. A complementary line of recent research [10, 20] has focused on providing minimax upper and lower bounds on the performance of estimators under ǫ-contamination model, without the constraint of computational tractability. In the ǫ-contaminated model, Q can be arbitrary, but there has been a lot of work in settings where the contamination distribution is restricted in various ways. For example, recent work in high-dimensional statistics (for instance [7, 12, 33, 34, 46]) have studied problems like principal component analysis and linear regression under the assumption that the corruptions are evenly spread throughout the dataset. Another line of research has focused on designing robust estimators under the heavy tailed distribution setting. These approaches relax the sub-gaussian or sub-exponential distributional assumptions that are typically imposed on the target distribution P and allow it to be a heavy tailed distribution. Most of the approaches in this category use robust mean estimators [8, 31] that exhibit sub-gaussian type concentration around the true mean for distributions satisfying mild moment assumptions. The median-of-means estimator [31] and Catoni’s mean estimator [8] are two popular examples of such robust mean estimators. Hsu and Sabato [23] use the median-of-means estimator to develop an alternative to ERM under heavy tails. Although this estimator has strong theoretical guarantees and is computationally tractable, as noted by the authors in [23] it performs poorly in practice. In recent 3 work Brownlees et al. [5] replace empirical mean in the empirical risk minimization framework (ERM) with Catoni’s mean estimator and perform risk minimization. The authors provide risk bounds similar to the bounds one can achieve under sub-gaussian distributional assumptions. However, their estimator is not easily computable and the authors do not provide a practical algorithm to compute the estimator. Other recent works by Lerasle and Oliveira [31], Lugosi and Mendelson [35] use similar ideas to derive estimators that perform well theoretically, in heavy-tailed situations. However, these approaches involve optimization of complex objectives for which no computationally tractable algorithms exist. We emphasize that in contrast to our work, these works focus on robustly estimating the population risk which does not directly lead to a computable estimator. We instead consider robustly estimating the gradient of the population risk. When complemented with the gradient descent algorithm this leads naturally to a computable estimator. 1.2 Outline We conclude this section with a brief outline of the remainder of the paper. In Section 2, we provide some background on risk minimization and the Huber and heavy-tailed noise models. In Section 3, we introduce our class of estimators and provide concrete algorithms for the ǫ-contaminated and heavy-tailed setting. In Section 4 we study the empirical performance of our estimator on a variety of tasks and datasets. We complement our empirical results with theoretical guarantees in Sections 5, 6 and 7. We defer technical details to the Appendix. Finally, we conclude in Section 8 with a discussion of some open problems. 2 Background and Problem Setup In this section we provide the necessary background on risk minimization, gradient descent and introduce two notions of robustness that we consider in this work. 2.1 Risk Minimization and Parametric Estimation In the setting of risk minimization, we assume that we have access to a differentiable loss function L : Θ × Z 7→ R, where Θ is a convex subset of Rp . Let R(θ) = Ez∼P L(θ; z) be the population loss (risk), and let θ ∗ be the minimizer of the population risk R(θ), over the set Θ θ ∗ = argmin R(θ). (2) θ∈Θ The goal of risk minimization is to minimize the population risk R(θ), given only n samples Dn = {zi }ni=1 . Whereas in parameter estimation we are interested in estimating the unknown parameter θ ∗ from samples Dn . In this work we assume that the population risk is convex to ensure tractable minimization. Moreover, in order to ensure identifiability of the parameter θ ∗ , we impose two standard regularity conditions [6] on the population risk. These properties are defined in terms of the error of the first-order Taylor approximation of the population risk, i.e. defining, τ (θ1 , θ2 ) := R(θ1 ) − R(θ2 ) − h∇R(θ2 ), θ1 − θ2 i, we assume that τℓ τu kθ1 − θ2 k22 ≤ τ (θ1 , θ2 ) ≤ kθ1 − θ2 k22 , 2 2 (3) where the parameters τℓ , τu > 0 denote the strong-convexity and smoothness parameters respectively. 4 2.2 Gradient Descent and Empirical Risk Minimization A starting point for the techniques we develop in this paper is the classical projected gradient descent method for empirical risk minimization. Given data Dn = {zi }ni=1 , empirical risk minimization (ERM) estimates the unknown parameter θ ∗ as the minimizer of the empirical risk, i.e. n 1X L(θ; zi ). θbn = argmin Rn (θ) := n θ∈Θ i=1 A popular method for solving this optimization problem is projected gradient descent. Projected gradient descent generates a sequence of iterates {θ t }∞ t=0 , by refining an initial parameter θ0 ∈ Θ via the update: θ t+1 = PΘ θ t − η∇Rn (θ t ) , where η > 0 is the step size and Pθ is the projection operator onto Θ. Despite its simplicity, the gradient descent method is not robust for general convex losses. Furthermore, the empirical risk minimizer is a poor estimator of θ ∗ in the presence of outliers in the data: since ERM depends on the sample mean, outliers in the data can effect the sample mean and lead ERM to sub-optimal estimates. This observation has led to a large body of research that focuses on developing robust M-estimators which have favorable statistical properties, but are often computationally intractable. In this work we take a different approach. Our work relies on an important observation that the gradient of the population risk (Ez∼P ∇L(θ; z) ) is simply a mean vector: one that can be estimated robustly by leveraging recent advances in robust mean estimation [29, 37]. This leads to a general method for risk minimization based on robust gradient estimation (see Algorithm 1). 2.3 Robust Estimation One of the goals of this work is to develop general statistical estimation methods that are robust in one of the following two models: Huber’s ǫ-contamination model or the heavy-tailed model. We now briefly review these two notions of robustness. 1. Huber’s ǫ-contamination model: Huber [25, 26] proposed the ǫ-contamination model where we observe samples that are obtained from a mixture of the form Pǫ = (1 − ǫ)P + ǫQ, (4) where P is the true distribution, ǫ is the expected fraction of outliers and Q is an arbitrary outlier distribution. Given i.i.d. observations drawn from Pǫ , our objective is to estimate θ ∗ , the minimizer of the population risk R(θ) = Ez∼P L(θ; z) , robust to the contamination from Q. 2. Heavy-tailed model: In the heavy-tailed model it is assumed that the data follows a heavy-tailed distribution (i.e, P is heavy-tailed). While heavy-tailed distributions have various possible characterizations: in this paper we consider a characterization via gradients. For a fixed θ ∈ Θ we let Pgθ denote the multivariate distribution of the gradient of population loss, i.e. ∇L(θ; z). We refer to a heavy-tailed distribution as one for which 5 Pgθ has finite second moments for any θ ∈ Θ. As we illustrate in Section 7, in various concrete examples this translates to relatively weak low-order moment assumptions on the data distribution P . Given n i.i.d observations from P , our objective is to estimate the minimizer of the population risk. From a conceptual standpoint, the classical analysis of risk-minimization which relies on uniform concentration of the empirical risk around the true risk, fails in the heavy-tailed setting necessitating new estimators and analyses [5, 23, 35, 36]. 3 Gradient Estimation Gradient descent and its variants are at the heart of modern optimization and are well-studied in the literature. Suppose we have access to the true distribution Pθ∗ . Then to minimize the population risk R(θ), we can use projected gradient descent, where starting at some initial θ 0 and for an appropriately chosen step-size η, we update our estimate according to: θ t+1 ← PΘ (θ t − η∇R(θ t )). (5) However, we only have access to n samples Dn = {zi }ni=1 . The key technical challenges are then to estimate the gradient of R(θ) from samples Dn , and to ensure that an appropriate modification of gradient descent is stable to the resulting estimation error. To address the first challenge we observe that the gradient of the population risk at any point θ is the mean of a multivariate distribution, i.e. ∇R(θ) = Ez∼P ∇L(θ; z) . Accordingly, the problem of gradient estimation can be reduced to a multivariate mean estimation problem, where our goal is to robustly estimate the true mean ∇R(θ) from n samples {∇L(θ; zi )}ni=1 . For a given sample-size n and confidence parameter δ ∈ (0, 1) we define a gradient estimator: Definition 1. A function g(θ; Dn , δ) is a gradient estimator, if for functions α and β, with probability at least 1 − δ, at any fixed θ ∈ Θ, the estimator satisfies the following inequality: kg(θ; Dn , δ) − ∇R(θ)k2 ≤ α(n, δ)kθ − θ ∗ k2 + β(n, δ). (6) In subsequent sections, we will develop conditions under which we can obtain gradient estimators with strong control on the functions α(n, δ) and β(n, δ) in the Huber and heavy-tailed models. Furthermore, by investigating the stability of gradient descent we will develop sufficient conditions on these functions such that gradient descent with an inaccurate gradient estimator still returns an accurate estimate. To minimize R(θ), we replace ∇R(θ) in equation (5) with the gradient estimator g(θ; Dn , δ) and perform projected gradient descent. In order to avoid complex statistical dependency issues that can arise in the analysis of gradient descent, for our theoretical results we consider a sample-splitting variant of the algorithm where each iteration is performed on a fresh batch of samples (see Algorithm 1). We further assume that the number of gradient iterations T is specified a-priori, and accordingly we define: jnk δ and δe = , (7) n e= T T We discuss methods for selecting T , and the impact of sample-splitting in later sections. As confirmed in our experiments (see Section 4), sample-splitting should be viewed as a device introduced for theoretical convenience which can likely be eliminated via more complex uniform arguments (see for instance the work [2]). 6 Algorithm 1 Projected Gradient Descent function PGD({z1 , . . . , zn }, Step Size η, Number of Iterations T , δ) e. Split samples into T subsets {Zt }Tt=1 of size n for t = 0 to T − 1 do e k2 . θ t+1 = argminθ∈Θ kθ − θ t − ηg(θ t ; Zt , δ) 2 end for end function Next, we consider the two notions of robustness described in Section 2, and derive specific gradient estimators for each of the models using the framework described above. Although the major focus of this work is on Huber contamination and heavy-tailed models, our class of estimators are more general and are not restricted to these two notions of robustness. 3.1 Gradient Estimation in Huber’s ǫ-contamination model There has been a flurry of recent interest [9, 14, 16, 29, 32] in designing mean estimators which, under the Huber contamination model, can robustly estimate the mean of a random vector. While some of these results are focused on the case where the uncorrupted distribution is Gaussian, or isotropic, we are more interested in robust mean oracles for more general distributions. Lai et al. [29] proposed a robust mean estimator for general distributions, satisfying weak moment assumptions, and we leverage the existence of such an estimator to design a Huber gradient estimator g(θ; Dn , δ) which works in the Huber contamination model (see Algorithm 2). Now, we briefly describe the main idea behind Algorithm 2 and the mean estimator of Lai et al. [29]. The algorithm builds upon the fact that in one-dimension, it is relatively easy to estimate the gradient robustly. In higher-dimension, the crucial insight of Lai et al. [29] is that the effect of the contamination Q on the mean of uncontaminated distribution P is effectively one-dimensional provided we can accurately estimate the direction along which the mean is shifted. In our context, if we can compute the gradient shift direction, i.e. the direction of the difference between the sample (corrupted) mean gradient and the true (population) gradient, then the true gradient can be estimated by using a robust 1D-mean algorithm along the gradient-shift direction and a non-robust sample-gradient in the orthogonal direction since the contamination has no effect on the gradient in this orthogonal direction. In order to identify the gradient shift direction, we use a recursive Singular Value Decomposition (SVD) based algorithm. In each stage of the recursion, we first remove gross-outliers via a truncation algorithm (described in more detail in the Appendix) and subsequently identify two subspaces using an SVD – a clean subspace where the contamination has a small effect on the mean and another subspace where the contamination has a potentially larger effect. We use a simple sample-mean estimator in the clean subspace and recurse our computation on the other subspace. Building on the work of Lai et al. [29], in Lemma 1 and Appendix J we provide a careful analysis of this gradient estimator. 7 Algorithm 2 Huber Gradient Estimator function HuberGradientEstimator(Sample Gradients S = {∇L(θ; zi )}ni=1 , Corruption Level ǫ, Dimension p, δ) Se = HuberOutlierGradientTruncation(S, ǫ, p, δ). if p=1 then e return mean(S) else e Let ΣSe be the covariance matrix of S. Let V be the span of the top p/2 principal components of ΣSe and W be its complement. e where PV is the projection operation on to V . Set S1 := PV (S) Let µ bV := HuberGradientEstimator(S1 , ǫ, p/2, δ). e Let µ bW := mean(PW S). p Let µ b ∈ R be such that PV (b µ) = µ bV , and PW (b µ) = µ bW . return µ b. end if end function 3.2 Gradient Estimation in the Heavy-Tailed model To design gradient estimators for the heavy-tailed model, we leverage recent work on designing robust mean estimators in this setting. These robust mean estimators build on the classical work of Alon et al. [1], Nemirovski and Yudin [39] and Jerrum et al. [27] on the so-called median-of-means estimator. For the problem of one-dimensional mean estimation, Catoni [8], Lerasle and Oliveira [31] propose robust mean estimators that achieve exponential concentration around the true mean for any distribution with bounded second moment. In this work we require mean estimators for multivariate distributions. Several recent works ([23, 36, 37]) extend the median-of-means estimator of to general metric spaces. In this paper we use the geometric median-of-means estimator (Gmom), which was originally proposed and analyzed by Minsker [37], to design the gradient estimator g(θ; Dn , δ). The basic idea behind the Gmom estimator is to first split the samples into non-overlapping subsamples and estimate the sample mean of each of the subsamples. Then the Gmom estimator is given by the median-of-means of the subsamples. Formally, let {xi . . . xn } ∈ R be n i.i.d random variables sampled from a distribution P . Then the Gmom estimator for estimating the mean of P can be described as follows. Partition the n samples into b blocks B1 , P . . . Bb each 1 of size ⌊n/b⌋. Let {b µ1 , . . . , µ bb } be the sample means in each block, where µ bi = \|Bi \| xj ∈Bi xj . Then the Gmom estimator is given by median{b µ1 , . . . µ bb }. In high dimensions where different notions of the median have been considered Minsker [37] uses geometric median: µ b = argmin µ b X i=1 kµ − µ bi k2 . Algorithm 3 presents the gradient estimator g(θ; Dn , δ) obtained using Gmom as the mean estimator. 8 Algorithm 3 Heavy Tailed Gradient Estimator function HeavyTailedGradientEstimator(Sample Gradients S = {∇L(θ; zi )}ni=1 , δ) Define number of buckets b = 1 + ⌊3.5 log 1/δ⌋. Partition S into b blocks B1 , . . . Bb each of size ⌊n/b⌋. for i = 1 . . . nX do s. µ bi = \|B1i \| s∈Bi end for Let µ b = argmin µ return µ b. end function 4 b X i=1 kµ − µ bi k2 . Experiments In this section we demonstrate our proposed methods for Huber contamination and heavytailed models, on a variety of simulated and real data examples. 4.1 Huber Contamination We first consider the Huber contamination model and demonstrate the practical utility of gradient-descent based robust estimator described in Algorithms 1 and 2. 4.1.1 Synthetic Experiments: Linear Regression In linear regression we observe paired samples {(x1 , y1 ), . . . (xn , yn )}, where each (xi , yi ) ∈ Rp × R. We assume that the (x, y) pairs sampled from the true distribution P are linked via a linear model: y = hx, θ ∗ i + w, (8) where w is drawn from a zero-mean normal distribution with variance σ 2 (w ∼ N (0, σ 2 )). We use the squared loss as our loss function 1 L(θ; (x, y)) = (y − hx, θi)2 . 2 Note that the true parameter θ ∗ is the minimizer of the resulting population risk R(θ). We now describe the experiment setup, the data model and present the results. Setup We fix the contamination level ǫ = 0.1 and σ 2 = 0.1. Next, we generate (1 − ǫ)n clean covariates from x ∼ N (0, Ip ), the corresponding clean responses using y = hx, θ ∗ i + w where θ ∗ = [1, . . . , 1]T and w ∼ N (0, σ 2 ). We simulate an outlier distribution by drawing the covariates from N (0, p2 Ip ), and setting the responses to 0. The total number of samples are set to be(10 ǫp2 ) i.e. the sample size increases with the dimension. This scaling is used to ensure that the statistical (minimax) error, in the absence of any contamination, is roughly 0.001. An optimally robust method should have error close to 0.1 (roughly equal to corruption level), which ours does (see Figure 1). 9 25 1.7 OLS RobustGD TORRENT Huber Plugin RANSAC 1.6 1.5 Parameter Error \|\|θb − θ∗ \|\|2 20 ×10-3 15 10 1.4 1.3 1.2 1.1 5 1 0 100 150 200 250 300 350 400 450 0.9 0.1 500 0.15 0.2 p 0.25 0.3 ǫ (a) Parameter error vs p for ǫ = 0.1 (b) Parameter error vs ǫ 3 epsilon=0.1 epsilon=0.2 epsilon=0.3 epsilon=0.4 log(Parameter Error) 2 1 0 -1 -2 -3 0 10 20 30 40 50 60 Iterations (c) log(kθt − θ∗ k2 ) vs t for different ǫ. Figure 1: Robust Linear Regression. Metric We measure the parameter error in ℓ2 -norm. We also study the convergence properties of our proposed method, for different contamination levels ǫ. We use code provided by Lai et al. [29] to implement our gradient estimator. Baselines We use OLS, TORRENT [3], Huber-loss, RANSAC and plugin estimator as our baselines. TORRENT is an iterative hard-thresholding based alternating minimization algorithm, where in one step, it calculates an active set of examples by keeping only (1−ǫ)n samples which have the smallest absolute values of residual r = y − x, θ t , and in the other step it updates the current estimates by solving OLS on the active set. Bhatia et al. [3] had shown the superiority of TORRENT over other convex-penalty based outlier techniques, hence, we do not compare against those methods.P The plugin estimator is implemented using P Algorithm 2 to estimate both the mean vector n1 ni=1 yi xi and the covariance matrix n1 ni=1 xi xTi . Results We summarize our main findings here. 1. All estimators except our proposed algorithm perform poorly (Figure 1(a)). Note that the TORRENT algorithm has strong guarantees when only the response y is corrupted but 10 performs poorly in the Huber contamination model where both x and y may be contaminated. The error for the robust plugin estimator increases with dimension. We investigate this theoretically in Section 6.1, where we find that the error of the plugin estimator grows √ with the norm of θ ∗ . In our experiment, we choose kθ ∗ k2 = p, and thus Figure 1(a) corroborates Corollary 3 in Section 6.1. 2. In Figure 1(b) we find that the parameter error kθb − θ ∗ k2 increases linearly with the contamination rate ǫ and we study this further in Section 6.1. 3. Finally, Figure 1(c) shows that the convergence rate decreases with increasing contamination ǫ and after ǫ is high enough, the algorithm remains stuck at θ0 , corroborating Lemma 8 (in the Appendix). Next, we study the performance of our proposed method in the context of classification. 4.1.2 Synthetic Experiments: Logistic Regression In logistic regression we observe paired samples, {(x1 , y1 ), . . . , (xn , yn )} where each (xi , yi ) ∈ Rp × R. We assume that the (x, y) pairs sampled from the true distribution P are linked via a linear model: ( 1 1 with probability 1+exp(−hx, θ ∗ i) , (9) y= 0 otherwise. In this case, we use the negative conditional log-likelihood as our loss function, i.e. L(θ; (x, y)) = −yhx, θi + log(1 + exp(hx, θi)). Setup We simulate a linearly separable classification problem, where the clean covariates are sampled from N (0, Ip ), the corresponding clean responses are computed as y = sign(hx, θ ∗ i) √ √ where θ ∗ = [1/ p, . . . , 1/ p]T . We simulate the outlier distribution by adding asymmetric noise, i.e. we flip the labels of one class, and increase the variance of the corresponding covariates by multiplying them by p2 . The total number of samples are set to be (10 pǫ ). Metric We measure the 0-1 classification error on a separate (clean) test set. We study how the 0-1 error changes with p and ǫ and the convergence properties(parameter error) of our proposed method for different contamination levels ǫ. Baselines We use the logistic regression MLE and the linear Support Vector Machine (SVM) as our baselines. Results Figures 2(b) and 2(c) show qualitatively similar results to the linear regression setting, i.e. that the error of our proposed estimator degrades gracefully (and grows linearly) with the contamination level ǫ and that the gradient descent iterates converge linearly. In Figure 2(a) we observe that both the SVM and logistic regression MLE perform poorly. The logistic regression MLE completely flips the labels and has a 0-1 error close to 1, whereas the linear SVM outputs a random hyperplane classifier that flips the label for roughly half of the dataset. 11 1 1 RobustGD logisticRegression SVM 0.9 0.8 0.7 0.7 0.6 0.6 0-1 error 0-1 error 0.8 0.5 0.4 0.5 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 100 150 200 250 300 350 400 450 RobustGD 0.9 0 0.1 500 0.15 0.2 0.25 p 0.3 0.35 0.4 0.45 epsilon (a) 0-1 Error vs p at ǫ = 0.1 (b) 0-1 error vs ǫ 0.5 eta=0.1 eta=0.2 eta=0.3 eta=0.4 log(Parameter Error) 0 -0.5 -1 -1.5 -2 -2.5 0 100 200 300 400 500 600 Iterations (c) log(kθt − θ∗ k2 ) vs t for different ǫ Figure 2: Robust Logistic Regression. 4.1.3 Robust Face Reconstruction Setup In this experiment, we show the efficacy of our algorithm by attempting to reconstruct face images that have been corrupted with heavy occlusion, where the occluding pixels play the role the outliers. We use the data from the Cropped Yale Dataset [30] . The dataset contains 38 subjects, and each image has 192×168 pixels. Following the methodology of Wang et al. [44], we choose 8 face images per subject, taken under mild illumination conditions and computed an eigenface set with 20 eigenfaces. Then given a new corrupted face image of a subject, the goal is to get the best reconstruction/approximation of the true face. To remove the scaling effects, we normalized all images to [0, 1] range. One image per person was used to test reconstruction. Occlusions were simulated by randomly placing 10 blocks of size 30 × 30. We repeated this 10 times for each test image. Note that in this example, we use a linear regression model as the uncontaminated statistical model, which is almost certainly not an Table 1: Fitting to original image error. Mean RMSE Best Possible Proposed TORRENT OLS SCRRR 0.05 0.09 0.175 0.21 0.13 12 exact match for the unknown ground truth distribution. Despite this model misspecification, as our results show, that robust mean based gradient algorithms do well. Metric We use Root Mean Square Error (RMSE) between the original and reconstructed image to evaluate the performance of the algorithms. We also compute the best possible reconstruction of the original face image by using the 20 eigenfaces. Methods We use TORRENT, OLS as baselines. Wang et al. [44] implemented popular robust estimators such as RANSAC, Huber Loss etc. and showed their poor performance. Wang et al. [44] then proposed an alternate robust regression algorithm called Self Scaled Regularized Robust Regression(SCRRR), and showed its equivalence to and ℓ1 -penalized method. We also compare against the best possible RMSE obtained by reconstructing the un-occluded image using the eigenfaces. Results Table 1 shows that the mean RMSE is best for our proposed gradient descent based method and that the recovered images are in most cases closer to the un-occluded original image. (Figure 4.1.3). Figure 3(c) shows a case when none of the methods succeed in reconstruction. (a) Successful Reconstruction (b) Successful Reconstruction (c) Failed Reconstruction Figure 3: Robust Face recovery results: Top; in order from L to R: original image, occluded image, best possible recovery with given basis. Bottom; in order from L to R: Reconstructions using our proposed algorithm, TORRENT and ordinary least squares (OLS). 4.2 Heavy-tailed Estimation We now consider the heavy-tailed model and present experimental results on synthetic and real world datasets comparing the gradient descent based robust estimator described in Algo13 rithms 1 and 3 (which we call GMOM) with ERM and several other recent proposals. In these experiments we focus on the problem of linear regression which is described in Section 4.1 and work with heavy-tailed noise distributions. 4.2.1 Synthetic Experiments: Simple Linear Regression Setup. The covariate x ∈ Rp is sampled from a zero-mean isotropic gaussian distribution. √ We set each entry of θ ∗ to 1/ p. The noise w is sampled from a Pareto distribution, with mean zero, variance σ 2 and tail parameter β. The tail parameter β determines the moments of the Pareto random variable. More specifically, the moment of order k exists only if k < β, hence, smaller the β the more heavy-tailed the distribution. In this setup, we keep the dimension p fixed to 128 and vary n, σ and β. We always maintain the sample-size n to be at least 2p. Methods. We use ERM as our baseline and compare it with GMOM. Since we are always in the low-dimensional (n ≥ p) setting, the solution to ERM has a closed form expression and is simply the OLS solution. We also study ERM-GD, which performs a gradient descent on ERM and is equivalent to using empirical mean as the gradient oracle in our framework. We also compare against the robust estimation techniques of Hsu and Sabato [23] and Duchi and Namkoong [17]. In our experiments, all the iterative techniques are run until convergence. Hyper Parameter Selection. The GMOM estimator depends on the confidence parameter δ ∈ (0, 1), which needs to be tuned. In our experiments, we noticed that the performance of GMOM varies very little when δ is selected from a reasonable range that is not too close to 0 (see Figure 5 and the discussion below). So we set δ = 0.1 in all our simulations. Metrics. In our experiments, we vary σ and β, the parameters of Pareto distribution, which can change the minimal risk R(θ ∗ ). So to compare various approaches across parameter values, ∗ ) − 1.0, where θ b b is we use a scaled version of the Excess Risk, which we define as R(θ)/R(θ the estimator. To compare the performance of two estimators θb1 , θb2 , we define the notion of relative efficiency: (R(θb2 ) − R(θ ∗ )) − (R(θb1 ) − R(θ ∗ )) . RelEff(θb1 , θb2 ) = R(θb1 ) − R(θ ∗ ) Roughly, this corresponds to the percentage improvement in the excess risk obtained using θb1 over θb2 . Whenever RelEff(θb1 , θb2 ) > 0, θb1 has a lower risk, and higher the value, the more the fractional improvement. Results. To reduce the variance in the plots presented here and in the next section, we averaged results over 25 repetitions. Figure 4 shows the benefits of using GMOM over ERM. In Figure 4(a) we plot the excess risk of ERM, ERM-GD and GMOM against the number of Iterations. We see that upon convergence GMOM has a much lower population risk than ERM. As expected, ERM-GD converges to ERM. However, the population risk of ERM-GD in the first few iterations is much lower than the risk of ERM, suggesting early stopping. Next, in Figure 4(b) we plot the scaled excess risk for ERM and GMOM as n/p increases. We see that GMOM is always better than ERM, even when the number of samples is 12 times the dimension p. In Figure 4(c), we plot the relative efficiency of GMOM and ERM against σ. This shows that the percentage improvement in the excess risk by GMOM decreases as 14 the noise level σ decreases. This behavior is expected because in the noiseless setting both methods would have a similar behavior. We do a similar study to see the relative efficiency against the heavy-tailedness of the noise distribution. As noted before, as β is increased more moments exist for the underlying distribution. Figure 4(d) shows that as the noise distribution becomes more heavy-tailed, there is more benefit in using GMOM over ERM. 2 1 ERM ERM (GD) GMOM (GD) 1.9 Excess Risk / True Risk 1.8 1.7 Population Risk ERM GMOM (GD) 0.9 1.6 1.5 1.4 1.3 0.8 0.7 0.6 0.5 0.4 0.3 1.2 0.2 1.1 0.1 0 1 50 100 150 2 200 4 6 (a) Population Risk vs Iterations 10 12 (b) ExcessRisk/TrueRisk vs n/p 0.6 RelEff(GMOM,ERM) 1.2 8 n/p Iterations RelEff(GMOM,ERM) 0.5 Relative Efficiency Relative Efficiency 1 0.8 0.6 0.4 0.4 0.3 0.2 0.1 0.2 0 0 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 2 4 6 8 10 12 β+1 σ (c) Relative Efficiency vs σ (d) Relative Efficiency vs β + 1 Figure 4: Linear Regression: Performance comparison of GMOM and ERM. Dependence on Confidence Level. Figure 5(a) shows the performance of GMOM estimator for various values of δ. It can be seen that the choice of δ have very little effect on the performance of the estimator. However, we notice that for small values of δ the performance of the GMOM degrades. In practice, one can use either cross validation or a validation set for choosing δ. 5 Theoretical Preliminaries In this section we develop some theoretical preliminaries. We begin with a description of some canonical examples of risk minimization in Section 5.1. Next we develop a general 15 2 ERM δ=0.01 δ=0.05 δ=0.1 δ=0.2 1.9 1.8 Population Risk 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 50 100 150 200 Iterations (a) Population Risk vs δ Figure 5: Linear Regression: Dependence on Confidence Level δ. theory on convergence of projected gradient descent in Section 5.2. We analyze the gradient estimators defined in Algorithms 2 and 3 in Sections 5.3 and 5.4 respectively. Finally in Sections 6,7 we present consequences of our general theory for the canonical examples, under Huber contamination and heavy-tailed models. For some of our examples, we will assume certain mild moment conditions. Concretely, for a random vector x ∈ Rp , let µ = E[x] and Σ be the covariance matrix. Then x has bounded 2kth moments if there exists a constant C2k such that for every unit vector v we have that i h k . (10) E hx − µ, vi2k ≤ C2k E hx − µ, vi2 5.1 Illustrative Examples The framework of risk minimization is a central paradigm of statistical estimation and is widely applicable. In this section, we provide illustrative examples that fall under this framework. 5.1.1 Linear Regression Here we observe paired samples {(x1 , y1 ), . . . (xn , yn )}, where each (xi , yi ) ∈ Rp ×R. We assume that the (x, y) pairs sampled from the true distribution P are linked via a linear model: y = hx, θ ∗ i + w, (11) where w is drawn from a zero-mean distribution such as normal distribution with variance σ 2 (N (0, σ 2 )) or a more heavy-tailed distribution such as student-t or Pareto distribution. We suppose that under P the covariates x ∈ Rp , have mean 0, and covariance Σ. For this setting we use the squared loss as our loss function, which induces the following population risk: 1 1 (y − hx, θi)2 , and R(θ) = (θ − θ ∗ )T Σ(θ − θ ∗ ). 2 2 ∗ Note that the true parameter θ is the minimizer of the population risk R(θ). The strongconvexity and smoothness assumptions from (3) in this setting require that τℓ ≤ λmin (Σ) ≤ λmax (Σ) ≤ τu . L(θ; (x, y)) = 16 5.1.2 Generalized Linear Models Here we observe paired samples {(x1 , y1 ), . . . (xn , yn )}, where each (xi , yi ) ∈ Rp × Y. We suppose that the (x, y) pairs sampled from the true distribution P are linked via a linear model such that when conditioned on the covariates x, the response variable has the distribution: yhx, θ ∗ i − Φ(hx, θ ∗ i) (12) P (y\|x) ∝ exp c(σ) Here c(σ) is a fixed and known scale parameter and Φ : R 7→ R is the link function. We focus on the random design setting where the covariates x ∈ Rp , have mean 0, and covariance Σ. We use the negative conditional log-likelihood as our loss function, i.e. L(θ; (x, y)) = −yhx, θi + Φ(hx, θi). (13) Once again, the true parameter θ ∗ is the minimizer of the resulting population risk R(θ). It is easy to see that Linear Regression with Gaussian Noise lies in the family of generalized linear models. We now instantiate GLMs for logistic regression. Logistic Regression In this case the (x, y) pairs are linked as: ( 1 1 with probability 1+exp(−hx, θ ∗ i) , y= 0 otherwise. (14) This corresponds to setting Φ(t) = log(1 + exp(t)) and c(γ) = 1 in (12). The hessian of the population risk is given by exp hx, θi T 2 xx . ∇ R(θ) = E (1 + exp hx, θi)2 Note that as θ diverges, the minimum eigenvalue of the hessian approaches 0 and the loss is no longer strongly convex. To prevent this, in this case we take the parameter space Θ to be bounded. 5.1.3 Exponential Families and Canonical Parameters Finally we consider the case where the true distribution P is in exponential family with canonical parameters θ ∗ ∈ Rp , and a vector of sufficient statistics obtained from the map φ : Z 7→ Rp . Note that while the linear and logistic regression models are indeed in an exponential family, our interest in those cases was not in the canonical parameters. In more details, we can write the true distribution P in this case as P (z) = h(z) exp (hφ(z), θ ∗ i − A(θ ∗ )) , where h(z) is an arbitrary nuisance function. The negative log-likelihood gives us the following loss function: L(θ; z) = −hφ(z), θi + A(θ). (15) The strong-convexity and smoothness assumptions require that there are constants τℓ , τu such that τℓ ≤ ∇2 A(θ) ≤ τu , for θ ∈ Θ. 17 5.2 Stability of Gradient Descent In this section we develop a general theory for the convergence of the projected gradient descent described in Algorithm 1. Note that our gradient estimators could be biased and are not guaranteed to be consistent estimators of the true gradient ∇R(θ). This is especially true in the Huber contamination model where it is impossible to obtain consistent estimators of the gradient of the risk because of the non-vanishing bias caused by the contaminated samples. Hence, we turn our attention to understanding the behavior of projected gradient descent with a biased, inexact, gradient estimator of the form in (6). Before we present our main result, we define the notion of stability of a gradient estimator, which plays a key role in the convergence of gradient descent. Definition 2 (Stability). A gradient estimator is stable for a given risk function R : Θ 7→ R if for some φ ∈ [0, τℓ ), α(e n, δ̃) < τℓ − φ. We denote by κ the following contraction parameter: r 2ητℓ τu e κ := 1 − + ηα(e n, δ), τℓ + τu and note that κ < 1. With these definitions in place we state our main result on the stability of gradient descent: Theorem 1. Suppose that the gradient estimator satisfies the condition in (6) and is stable for the risk function R : Θ 7→ R. Then Algorithm 1 initialized at θ 0 with step-size η ≤ 2/(τℓ + τu ), returns iterates {θbt }Tt=1 such that with probability at least 1 − δ for the contraction parameter κ above we have that, kθbt − θ ∗ k2 ≤ κt kθ 0 − θ ∗ k2 + 1 e β(e n, δ). 1−κ (16) We defer a proof of this result to the Appendix. Theorem 1 provides a general result for risk minimization and parameter estimation. In any concrete instantiation for a given gradient estimator, risk pair, we first study the distribution of the gradient of the risk to estimate the e β(e e and then apply Theorem 1. error suffered by the gradient estimator (α(e n, δ), n, δ)) For the bound (16), the first term is decreasing in T , while the second term is increasing in T . This suggests that for a given n and δ, we need to run just enough iterations for the first term to be bounded by the second. Hence, we can fix the number of iterations T ∗ as the smallest positive integer such that: T ≥ log1/κ (1 − κ)kθ 0 − θ ∗ k2 . e β(e n, δ) Since we obtain linear convergence, i.e. κ < 1, typically a logarithmic number of iterations suffice to obtain an accurate estimate. 5.3 General Analysis of Algorithm 2 We now analyze the gradient estimator described in Algorithm 2 for Huber contamination model and study the error suffered by it. As stated before, Algorithm 2 uses the robust mean 18 estimator of Lai et al. [29]. Hence, while our proof strategy mimics that of Lai et al. [29], we present a different result which is obtained by a more careful non-asymptotic analysis of the algorithm. We define: p log p log n/(pδ) 3/8 ǫp2 log p log + γ(n, p, δ, ǫ) := n n p log(p) 1/4 δ , (17) and with this definition in place we have the following result: Lemma 1. Let P be the true probability distribution of z and let Pθ be the true distribution of the gradients ∇L(θ; z) on Rp with mean µθ = ∇R(θ), covariance Σθ , and bounded fourth moments. There exists a positive constant C1 > 0, such that given n samples from the distribution in (4), the Huber Gradient Estimator described in Algorithm 2 when instantiated with the contamination level ǫ, with probability at least 1 − δ, returns an estimate µ b of µθ such that, kb µ − µθ k2 ≤ C1 √ 1p ǫ + γ(n, p, δ, ǫ) kΣθ k22 log p. We note in particular, if n → ∞ (with other parameters held fixed) then γ(n, p, δ, ǫ) → 0 and the error of our gradient estimator satisfies kb µ − µθ k2 ≤ C p kΣθ k2 ǫ log p, and has only a weak dependence on the dimension p. 5.4 General Analysis of Algorithm 3 In this section we analyze the gradient estimator for heavy-tailed setting, described in Algorithm 3. The following result shows that the gradient estimate has exponential concentration around the true gradient, under the mild assumption that the gradient distribution has bounded second moment. Its proof follows from the analysis of geometric median-of-means estimator of Minsker [37]. We use tr (Σθ ) to denote the trace of the matrix Σθ . Lemma 2. Let P be the probability distribution of z and Pθ be the distribution of the gradients ∇L(θ; z) on Rp with mean µθ = ∇R(θ), covariance Σθ . Then the heavy tailed gradient estib that satisfies the following exponential mator described in Algorithm 3 returns an estimate µ concentration inequality, with probability at least 1 − δ: r kb µ − µθ k2 ≤ 11 6 tr (Σθ ) log 1.4/δ . n Consequences for Estimation under ǫ-Contaminated Model We now turn our attention to the examples introduced earlier, and present specific applications of Theorem 1, for parametric estimation under Huber contamination model. As shown in Lemma 1, we need the added assumption that the true gradient distribution has bounded fourth moments, which suggests the need for additional assumptions. We make our assumptions explicit and defer the technical details to the Appendix. 19 6.1 Linear Regression We assume that the covariates x ∈ Rp have bounded 8th -moments and the noise w has bounded 4th moments. Theorem 2 (Robust Linear Regression). Consider the statistical model in equation (11), and C1√τℓ e < suppose that the number of samples n is large enough such that γ(e n, p, δ) and the kΣk2 log p 2 ℓ e contamination level is such that ǫ < kΣkC2√τlog − γ(e n, p, δ) for some constants C1 and C2 . p 2 Then, there are universal constants C3 , C4 , such that if Algorithm 1 is initialized at θ 0 with stepsize η ≤ 2/(τu + τℓ ) and Algorithm 2 as gradient estimator, then it returns iterates {θbt }Tt=1 such that with probability at least 1 − δ p C σ kΣk2 log p 1 3 t ∗ t 0 ∗ e , kθb − θ k2 ≤ κ kθ − θ k2 + n, p, δ) (18) ǫ 2 + γ(e 1−κ for some contraction parameter κ < 1. In the asymptotic setting when the number of samples n → ∞ (and other parameters are held fixed), we see that for the Huber Gradient Estimator, the corresponding maximum allowed C τ2 contamination level is ǫ < τ 2 1logℓ p . This says that the more well-conditioned the covariance u matrix Σ, the higher the contamination level we can tolerate. Plugin Estimation For linear regression, the true parameter can be written in closed form as θ ∗ = E[xxT ]−1 E[xy]. A non-iterative way to estimate θ ∗ is to separately estimate E[xxT ] and E[xy] using robust covariance and mean oracles respectively. Under the assumption that x ∼ N (0, Ip ), one can reduce the problem to robustly estimating E[xy]. Under this setting, we now present a result using Lai et al. [29] as the mean estimator for estimation of E[xy]. Recall, the definition of γ in (17). We have the following result: Corollary 3. Consider the model in equation(11) with the covariates drawn from N (0, Ip ) and w ∈ N (0, 1), then there are universal constants C1 , C2 such that if ǫ < C1 , then [29] returns an estimate θb of E[xy], such that with probability at least 1 − δ q 1 (19) kθb − θ ∗ k2 ≤ C2 (1 + 2kθ ∗ k22 ) log p ǫ 2 + γ(n, p, δ, ǫ) . Comparing bounds (18) and (19), we see that the error of the plugin estimator depends on kθ ∗ k2 , which would make the estimator vacuous if kθ ∗ k2 scales with the dimension p. On the other hand, the asymptotic rate of our robust gradient estimator is independent of the kθ ∗ k2 . This disadvantage of plugin estimation is inescapable, due to known minimax results for robust mean estimation [10] that show that the dependence on kθ ∗ k2 is unavoidable for any oracle which estimates the mean of xy in the ǫ-contaminated setting. Next, we apply our estimator to generalized linear models. 6.2 Generalized Linear Models Here we assume that the covariates have bounded 8th moments. Additionally, we assume smoothness of Φ′ (·) around θ ∗ . To be more precise, we assume that there exist universal constants LΦ,2k , B2k such that h i 2k ≤ LΦ,2k kθ ∗ − θk2k Ex Φ′ (hx, θi) − Φ′ (hx, θ ∗ i) 2 + BΦ,2k , for k = 1, 2, 4 20 k We also assume that Ex [ Φ(t) (hx, θ ∗ i) ] ≤ MΦ,t,k where Φ(t) (·) is the tth -derivative of Φ(·). Theorem 4 (Robust Generalized Linear Models). Consider the statistical model in equation (12), and suppose that the number of samples n is large enough such that e < γ(e n, p, δ) √ C1 τ ℓ 1 1 1 , 4 2 + LΦ,2 ] log pkΣk22 [LΦ,4 and the contamination level is such that,  2 C τ 2 ℓ e , ǫ < √ − γ(e n, p, δ) 1 1 1 2 4 2 log pkΣk2 [LΦ,4 + LΦ,2 ] for some constants C1 and C2 . Then, there are universal constants C3 , C4 , such that if Algorithm 1 is initialized at θ 0 with stepsize η ≤ 2/(τu + τℓ ) and Algorithm 2 as gradient estimator, then it returns iterates {θbt }Tt=1 such that with probability at least 1 − δ kθbt − θ ∗ k2 ≤κt kθ 0 − θ ∗ k2 1 1 1 1 1 √ 1 3 4 2 4 4 + BΦ,2 + c(σ) 2 MΦ,2,2 + c(σ) 4 MΦ,4,1 ] 1 C3 log pkΣk22 [BΦ,4 e 2 + γ(e n , p, δ) , ǫ + 1−κ (20) for some contraction parameter κ < 1. Note that for the case of linear regression with gaussian noise, it is relatively straightforward to see that LΦ,2k = C2k kΣkk2 , BΦ,2k = 0, MΦ,t,k = 1 ∀(t ≥ 2, k ∈ N ) and MΦ,t,k = 0 ∀(t ≥ 3, k ∈ N ) under the assumption of bounded 8th moments of the covariates; which essentially leads to an equivalence between Theorem 2 and Theorem 4 for this setting. In the following section, we instantiate the above Theorem for logistic regression and compare and contrast our results to other existing methods. 6.2.1 Logistic Regression By observing that Φ(t) (·) is bounded for logistic regression for all t ≥ 1, we can see that LΦ,2k = 0, and that there exists a universal constant C > 0 such that BΦ,2k < C and MΦ,t,k < C ∀(t ≥ 1, k ∈ N ). Corollary 5 (Robust Logistic Regression). Consider the model in equation(14), then there are universal constants C1 , C2 , such that if ǫ < C1 , then Algorithm 1 initialized at θ 0 with stepsize η ≤ 2/(τu + τℓ ) and Algorithm 2 as gradient estimator, returns iterates {θbt }Tt=1 , such that with probability at least 1 − δ p C2 kΣk2 log p 1 t ∗ t 0 ∗ e , b n, p, δ) (21) ǫ 2 + γ(e kθ − θ k2 ≤ κ kθ − θ k2 + 1−κ for some contraction parameter κ < 1. Under the restrictive assumption that x ∼ N (0, Ip ), Du et al. [16] exploited Stein’s trick to derive a plugin estimator for logistic regression. However, similar to the linear regression, the error of the plugin estimator scales with kθ ∗ k2 , which is avoided in our robust gradient descent algorithm. We also note that our algorithm extends to general covariate distributions. 21 6.3 Exponential Family Here we assume that the random vector φ(z), z ∼ P has bounded 4th moments. Theorem 6 (Robust Exponential Family). Consider the model in equation(15), then there are universal constants C1 , C2 , such that if ǫ < C1 , then Algorithm 1 initialized at θ 0 with stepsize η ≤ 2/(τu + τℓ ) and Algorithm 2 as gradient oracle, returns iterates {θbt }Tt=1 , such that with probability at least 1 − δ √ C2 τu log p 1 t ∗ t 0 ∗ e , b ǫ 2 + γ(e n, p, δ) (22) kθ − θ k2 ≤ κ kθ − θ k2 + 1−κ for some contraction parameter κ < 1. Plugin Estimation Since the true parameter θ ∗ is the minimizer of the negative loglikelihood, we know that E[∇L(θ ∗ )] = 0, which implies that ∇A(θ ∗ ) = Eθ∗ [φ(Z)]. This shows that the true parameter θ ∗ can be obtained by inverting the ∇A operator, whenever possible. In the robust estimation framework, we can use a robust mean of the sufficient statistics to estimate Eθ∗ [φ(Z)]. We instantiate this estimator using the mean estimator of [29] to estimate Eθ∗ [φ(Z)]: Corollary 7. Consider the model in equation(15), then there are universal constants C1 , C2 b of E[φ(z)], such that with probability at such that if ǫ < C1 , then [29] returns an estimate µ least 1 − δ √ τu log p 1 −1 ∗ ǫ 2 + γ(n, p, δ, ǫ) , (23) b − θ k2 ≤ C2 kPΘ (∇A) µ τℓ where PΘ [θ] = argminy∈Θ ky − θk22 is the projection operator onto the feasible set Θ. 6.4 Discussion and Limitations In the asymptotic setting of n → ∞, Algorithm 1 √ with Algorithm 2 as gradient estimator converges to a point θb such that kθb − θ ∗ k2 = O( ǫ log p). Hence, our error scales only logarithmically with the dimension p. This dependency on the dimension p is a facet of using the estimator from Lai et al. [29] for gradient estimation. Using better oracles will only improve our performance. Next, we would like to point to the difference in the maximum allowed contamination ǫ∗ between the three models. For logistic regression and exponential C τ2 family, ǫ∗ < C1 , while for linear regression, ǫ∗ < τ 2 1logℓ p . These differences are in large part u due to differing variances of the gradients, which naturally depend on the underlying risk function. This scaling of the variance of gradients for linear regression also provides insights into the limitations of Algorithm 1 for gradient estimators. In the Appendix, we provide an upper bound for the contamination level ǫ based on the initialization point θ 0 , above which, Algorithm 1 would not work for any gradient estimator. 7 Consequences for Heavy-Tailed Estimation In this section we present specific applications of Theorem 1 for parametric estimation, under heavy tailed setting. The proofs of the results can be found in the Appendix. 22 7.1 Linear Regression We first consider the linear regression model described in Equation (11). We assume that the covariates x ∈ Rp have bounded 4th -moments and the noise w has bounded 2th moments. This assumption is needed to bound the error in the gradient estimator (see Lemma 2). Theorem 8 (Heavy Tailed Linear Regression). Consider the statistical model in equation (11). There are universal constants C1 , C2 > 0 such that if n e> C1 τu2 e p log(1/δ) τl2 and if Algorithm 1 is initialized at θ 0 with stepsize η ≤ 2/(τu + τℓ ) and Algorithm 3 as gradient estimator, then it returns iterates {θbt }Tt=1 such that with probability at least 1 − δ s  p e C2 σ kΣk2  p log(1/δ)  , (24) kθbt − θ ∗ k2 ≤ κt kθ 0 − θ ∗ k2 + 1−κ n e for some contraction parameter κ < 1. 7.2 Generalized Linear Models In this section we consider generalized linear models described in Equation (12), where the covariate x is allowed to have a heavy tailed distribution. Here we assume that the covariates have bounded 4th moment. Additionally, we assume smoothness of Φ′ (·) around θ ∗ . Specifically, we assume that there exist universal constants LΦ,2k , B2k such that h i 2k ≤ LΦ,2k kθ ∗ − θk2k Ex Φ′ (hx, θi) − Φ′ (hx, θ ∗ i) 2 + BΦ,2k , for k = 1, 2 k We also assume that Ex [ Φ(t) (hx, θ ∗ i) ] ≤ MΦ,t,k for t ∈ {1, 2, 4}, where Φ(t) (·) is the tth derivative of Φ(·). Theorem 9 (Heavy Tailed Generalized Linear Models). Consider the statistical model in equation (12). There are universal constants C1 , C2 > 0 such that if p LΦ,4 + LΦ,2 C1 kΣk2 e n e> p log 1/δ, τl2 and if Algorithm 1 is initialized at θ 0 with stepsize η ≤ 2/(τu + τℓ ) and Algorithm 3 as gradient estimator, it returns iterates {θbt }Tt=1 such that with probability at least 1 − δ kθbt − θ ∗ k2 ≤κt kθ 0 − θ ∗ k2 1 1 1 1 1 1 3 s  2 4 2 4 4 C2 kΣk2 BΦ,4 + BΦ,2 + c(σ) 2 MΦ,2,2 + c(σ) 4 MΦ,4,1 e  p log(1/δ)  , (25) + 1−κ n e for some contraction parameter κ < 1. We now instantiate the above Theorem for logistic regression model. 23 Corollary 10 (Heavy Tailed Logistic Regression). Consider the model in equation(14). There are universal constants C1 , C2 > 0 such that if n e> C12 kΣk2 e p log 1/δ. τl2 and if Algorithm 1 initialized at θ 0 with stepsize η ≤ 2/(τu + τℓ ) and Algorithm 3 as gradient estimator, it returns iterates {θbt }Tt=1 such that with probability at least 1 − δ s  p e C kΣk p log(1/ δ) 2 2  , kθbt − θ ∗ k2 ≤ κt kθ 0 − θ ∗ k2 + 1−κ n e (26) for some contraction parameter κ < 1. 7.3 Exponential Family We now instantiate Theorem 1 for parameter estimation in heavy-tailed exponential family distributions. Here we assume that the random vector φ(z), z ∼ P has bounded 2nd moments, and we obtain the following result: Theorem 11 (Heavy Tailed Exponential Family). Consider the model in equation(15). If Algorithm 1 is initialized at θ 0 with stepsize η ≤ 2/(τu + τℓ ) and Algorithm 3 as gradient estimator, it returns iterates {θbt }Tt=1 , such that with probability at least 1 − δ kθbt − θ ∗ k2 ≤ κt kθ 0 − θ ∗ k2 + 1 C 1−κ s k∇2 A(θ ∗ )k2 p log 1/δe , n e (27) for some contraction parameter κ < 1 and universal constant C. 8 Discussion In this paper we introduced a broad class of estimators and showed that these estimators can have strong robustness guarantees in Huber’s ǫ-contamination model and for heavy-tailed distributions. These estimators leverage the robustness of gradient descent, together with the observation that for risk minimization in most statistical models the gradient of the risk takes the form of a simple multivariate mean which can be robustly estimated by using recent work of robust mean estimation. These estimators based on robust gradient descent work well in practice and in many cases outperform other robust (and non-robust) estimators. There are several avenues for future work, including developing a better understanding of robust mean estimation. Any improvement for robust mean estimation would immediately translate to improved guarantees for the estimators we propose for general parametric models. Finally, it would also be of interest to understand the extent to which we could replace gradient descent with other optimization methods such as accelerated gradient descent or Newton’s method. We note however, that although these methods may have faster rates of convergence in the classical risk minimization setting: in our setup their stability (to using inexact gradients) is far more crucial and warrants further investigation. 24 9 Acknowledgements The research of SB was supported in part by the grant NSF-DMS-1713003. We thank Larry Wasserman for helpful comments on the paper. References [1] Noga Alon, Yossi Matias, and Mario Szegedy. The space complexity of approximating the frequency moments. 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At any iteration step t ∈ {1, 2, . . . , T }, by assumption we have that with probability at least 1 − Tδ , kg(θ t ; Dn , δ/T ) − ∇R(θ t )k2 ≤ α(n/T, δ/T )kθ − θ ∗ k2 + β(n/T, δ/T ). (28) Taking union bound, (28) holds over all iteration steps t ∈ {1 . . . T }, with probability at least 1 − δ. For the remainder of the analysis, we assume this event to be true. 27 Notation. Let g(θ k ) = ∇R(θ k ) + ek be the noisy gradient. Let α = α(n/T, δ/T ) and β = β(n/T, δ/T ) for brevity. We have the following Lemma from Bubeck [6]. Lemma 3. [Lemma 3.11 [6]] Let f be M -smooth and m-strongly convex, then for all x, y ∈ Rp , we have: h∇f (x) − ∇f (y), x − yi ≥ mM 1 kx − yk22 + k∇f (y) − ∇f (x)k22 . m+M m+M By assumptions we have that: k∇R(θ k ) − g(θ k )k2 = kek k2 ≤ αkθ k − θ ∗ k2 + β. Our update rule is θ k+1 = PΘ θ k − ηg(θ k ) . Then we have that: kθ k+1 − θ ∗ k22 = kPΘ [θ k − ηg(θ k )] − θ ∗ k22 = kPΘ [θ k − ηg(θ k )] − PΘ [θ ∗ − η∇R(θ ∗ )]k22 ≤ kθ k − ηg(θ k ) − (θ ∗ − η∇R(θ ∗ ))k22 = ≤ k ∗ k kθ − θ − η(∇R(θ ) − ∇R(θ )) − ηek k22 kθ k − θ ∗ − η(∇R(θ k ) − ∇R(θ ∗ ))k22 + η 2 kek k22 + 2kek k2 kθ k − θ ∗ − η(∇R(θ k ) − ∇R(θ ∗ ))k2 , (29) ∗ (30) where Equation (29) follows from contraction property of projections. Now, we can write kθ k − θ ∗ − η(∇R(θ k ) − ∇R(θ ∗ ))k2 as kθk − θ∗ − η(∇R(θk ) − ∇R(θ∗ ))k22 = kθk − θ∗ k22 + η 2 k∇R(θk ) − ∇R(θ∗ )k22 − 2η ∇R(θk ) − ∇R(θ∗ ), θk − θ∗ 1 τℓ τu k ∗ 2 2 k ∗ 2 k ∗ 2 k ∗ 2 ≤ kθ − θ k2 + η k∇R(θ ) − ∇R(θ )k2 − 2η kθ − θ k2 + k∇R(θ ) − ∇R(θ )k2 τℓ + τu τℓ + τu (31) = kθk − θ∗ k22 (1 − 2ητℓ τu /(τℓ + τu )) + ηk∇R(θk ) − ∇R(θ∗ )k22 (η − 2/(τu + τℓ )) ≤ kθk − θ∗ k22 (1 − 2ητℓ τu /(τℓ + τu )), (32) (33) where the second step follows from Lemma 3 and the last step follows from the step size η ≤ 2/(τℓ + τu ). Now, combining Equations (30) and (33), and using our assumption that kek k2 ≤ αkθ k − θ ∗ k2 + β, we get: 2 p kθ k+1 − θ ∗ k22 ≤ kθ k − θ ∗ k2 (1 − 2ητℓ τu /(τℓ + τu )) + ηkek k2 p kθ k+1 − θ ∗ k2 ≤ hp i 1 − 2ητℓ τu /(τℓ + τu ) + ηα kθ k − θ ∗ k2 + ηβ. 1 − 2ητℓ τu /(τℓ + τu ) + ηα. By assumption, we choose ǫ < ǫ∗ such that α < τℓ . p κ = 1 − 2ητℓ τu /(τℓ + τu ) + ηα (34) p < 1 − 2ητℓ τu /(τℓ + τu ) + ητℓ . (35) p p Since 0 ≤ η ≤ 2/(τℓ + τu ), we get that 1 − 2ητℓ τu /(τℓ + τu ) ≤ 1 − η 2 τℓ τu . We have: p (36) κ < 1 − η 2 τℓ τu + ητℓ q (∵ τℓ ≤ τu ) (37) < 1 − η 2 τℓ2 + ητℓ Let κ = (38) < 1. 28 Therefore, we have that, kθ k+1 − θ ∗ k2 ≤ κkθ k − θ ∗ k2 + ηβ. for some κ < 1. Solving the induction,we get: kθ k − θ ∗ k2 ≤ κk kθ 0 − θ ∗ k2 + B 1 ηβ. 1−κ Proof of Theorem 4 To prove our result on Robust Generalized Linear Models, we first study the distribution of gradients of the corresponding risk function. Lemma 4. Consider the model in Equation (12), then there exist universal constants C1 , C2 > 0 such that p p C4 LΦ,4 + LΦ,2 kCov(∇L(θ)k2 ≤C1 k∆k22 kΣk2 q p p 3 + C2 kΣk2 BΦ,2 + BΦ,4 + c(σ) 3MΦ,2,2 + c(σ) MΦ,4,1 Bounded fourth moments E h (∇L(θ) − E[∇L(θ)])T v 4 i ≤ C2 (Var[∇L(θ)T v])2 . Proof. The gradient ∇L(θ) and it’s expectation can be written as: ∇L(θ) = −y.x + u(hx, θi).x E[∇L(θ)] = E[x u(xT θ) − u(xT θ ∗ ) ], where u(t) = Φ′ (t). kE[∇L(θ)]k2 = sup y T E[∇L(θ)] y∈Sp−1 ≤ sup E[(y T x) u(xT θ) − u(xT θ ∗ ) ] y∈Sp−1 ≤ sup y∈Sp−1 q 1 2 ≤ C1 kΣk2 E[(y T x)2 ] q q 2 E[(u(xT θ) − u(xT θ ∗ )) ] LΦ,2 k∆k22 + BΦ,2 where the last line follows from our assumption of smoothness. Now, to bound the maximum eigenvalue of the Cov(∇L(θ)), kCov(∇L(θ))k2 = sup z T E ∇L(θ)∇L(θ)T − E[∇L(θ)]E[∇L(θ)]T z z∈Sp−1 ≤ sup z T E ∇L(θ)∇L(θ)T z + sup z T E[∇L(θ)]E[∇L(θ)]T z z∈Sp−1 z∈Sp−1 i h 2 z + kE[∇L(θ)]k22 ≤ sup z T E xxT u(xT θ) − y) p−1 z∈S h 2 i z + kE[∇L(θ)]k22 ≤ sup E z T xxT u(xT θ) − y z∈Sp−1 ≤ sup z∈Sp−1 q r h i E (u(xT θ) − y)4 ] + kE[∇L(θ)]k22 E [(z T x)4 ] 29 To bound E h u(xT θ) − y 4 i , we make use of the Cr inequality. Cr inequality. If X and Y are random variables such that E\|X\|r < ∞ and E\|Y \|4 < ∞ where r ≥ 1 then: E\|X + Y \|r ≤ 2r−1 (E\|X\|r + E\|Y \|r ) Using the Cr inequality, we have that h h h 4 i 4 i 4 i + E u(xT θ ∗ ) − y ≤ 8 E u(xT θ) − u(xT θ ∗ ) E u(xT θ) − y ≤ C LΦ,4 k∆k42 + BΦ,4 + c(σ)3 MΦ,4,1 + 3c(σ)2 MΦ,2,2 where the last line follows from our assumption that Pθ∗ (y\|x) is in the exponential family, hence, the cumulants are higher order derivatives of the log-normalization function. q √ p p p p 2 3 LΦ,4 k∆k2 + BΦ,4 + c(σ) 3MΦ,2,2 + c(σ) MΦ,4,1 + kE[∇L(θ)]k22 kCov(∇L(θ))k2 ≤ C C4 kΣk2 q √ p p p p 2 3 ≤ C C4 kΣk2 LΦ,4 k∆k2 + BΦ,4 + c(σ) 3MΦ,2,2 + c(σ) MΦ,4,1 + C12 kΣk2 LΦ,2 k∆k22 + BΦ,2 q p p p p ≤ Ck∆k22 kΣk2 C4 LΦ,4 + LΦ,2 + C6 kΣk2 BΦ,2 + BΦ,4 + c(σ) 3MΦ,2,2 + c(σ)3 MΦ,4,1 Bounded Fourth Moment. To show that the fourth moment of the gradient distribution is bounded, we have h h 4 i 4 i ≤ E (∇L(θ) − E[∇L(θ)])T v E (∇L(θ) − E[∇L(θ])T v   ≤ 8 E[\|∇L(θ])T v\|4 ] + E[\|E[∇L(θ)]T v\|4 ] . {z } \| {z } \| A Control of A. B E[\|∇L(θ])T v\|4 ] = E[(xT v)4 (u(xT θ) − y)4 ] q q T 8 ≤ E[(x v) ] E[(u(xT θ) − y)8 ] q p 2 ≤ C8 kΣk2 E[(u(xT θ) − u(xT θ ∗ ))8 ] + E[(u(xT θ ∗ ) − y)8 ] v u 8 X p u 2t 8 gt,k MΦ,t,k ≤ C8 kΣk2 LΦ,8 k∆k2 + BΦ,8 + t,k=2 ≤ √ v u 8 uX p p 2 4 CkΣk2 LΦ,8 k∆k2 + BΦ,8 + t gt,k MΦ,t,k t,k=2 where the last step follows from the fact that the 8th central moment can be written as a polynomial involving the lower cumulants, which in turn are the derivatives of the lognormalization function. Control of B. 2 E[\|E[∇L(θ)]T v\|4 ] ≤ kE[∇L(θ)k42 ≤ C1 kΣk22 L2Φ,2 k∆k22 + BΦ,2 30 By assumption LΦ,k , BΦ,k , MΦ,t,k are all bounded for k, t ≤ 8, which implies that there exist constants c1 , c2 > 0 such that h 4 i ≤ c1 kΣk22 k∆k42 + c2 (39) E (∇L(θ) − E[∇L(θ])T v Previously, we say that kCov∇L(θ)k2 ≤ c3 kΣk2 k∆k22 +c4 , for some universal constants c3 , c4 > 0, hence the gradient ∇L(θ) has bounded fourth moments. Having studied the distribution of the gradients, we use Lemma 1 to characterize the stability of Huber Gradient estimator. Using Lemma 1, we know that at any point θ, the Huber Gradient Estimator g(θ, δ/T ) satisfies that with probability 1 − δ/T , 1 1p e kCov(∇L(θ))k 2 log p. n, p, δ) kg(θ, δ/T ) − ∇R(θ)k2 ≤ C2 ǫ 2 + γ(e 2 Substituting the upper bound on kCov(∇L(θ))k2 from Lemma 4, we get that there are universal constants C1 , C2 such that with probability at least 1 - δ/T 1 p 1 1 1 4 2 e log pkΣk22 [LΦ,4 (40) kg(θ) − ∇R(θ)k2 ≤ C1 ǫ 2 + γ(e + LΦ,2 ] k∆k2 n, p, δ) {z } \| e α(e n,δ) p 1 1 1 1 1 1 1 3 4 2 4 4 e + C2 ǫ 2 + γ(e n, p, δ) + BΦ,2 + c(σ) 2 MΦ,2,2 + c(σ) 4 MΦ,4,1 ] log pkΣk22 [BΦ,4 \| {z } e β(e n,δ) (41) e < τℓ . Using Equation (40), we To ensure stability of gradient descent, we need that α(e n, δ) get that gradient descent is stable as long as the number of samples n is large enough such C1 τℓ e < that γ(e n, p, δ) , and the contamination level is such that 1 1 1 √ 4 +L 2 ] log pkΣk22 [LΦ,4 Φ,2 ǫ< C2 τℓ 1 1 1 √ 4 +L 2 ] log pkΣk22 [LΦ,4 Φ,2 !2 e − γ(e n, p, δ) for some constants C1 and C2 . Plugging the corre- e into Theorem 1, we get back the result of Theorem 4. sponding ǫ and β(e n, δ) C Proof of Corollary 3 We begin by studying the distribution of the random variable xy = xxT θ ∗ + x.w. Lemma 5. Consider the model in Equation (11), with x ∼ N (0, Ip ) and w ∼ N (0, 1) then there exist universal constants C1 , C2 such that E[xy] = θ ∗ kCov(xy)k2 = 1 + 2kθ ∗ k22 h 4 i ≤ C2 (Var[(xy)T v])2 . Bounded fourth moments E (xy − E[xy])T v Proof. Mean. xy = xxT θ ∗ + x.w T ∗ E[xy] = E[xx θ + x.w] ∗ (42) (43) (44) E[xy] = θ . 31 Covariance. Cov(xy) = E[(xxT − I)θ ∗ + x.w)((xxT − I)θ ∗ + x.w)T )] ∗ ∗T Cov(xy) = E[(xxT − I)θ θ (45) (xxT − I)] + Ip . (46) Now, Z = (xxT − I)θ ∗ can be written as:    ∗  ∗ 2  2 θ1 (x1 − 1) + x1 x2 θ2∗ + . . . + x1 xp θp∗ θ1 (x1 − 1) x1 x2 ... x1 xp ∗ ∗ 2 ∗  ∗   x1 x2 x2 xp  (x22 − 1) . . .  θ2  x1 x2 θ1 + (x2 − 1)θ2 + . . . + x2 xp θp   T ∗ (xx −I)θ =  .   ..  =  .. .. .. .. ..   .    . . . . . ∗ 2 2 ∗ ∗ ∗ θp x1 xp x2 xp . . . (xp − 1) x1 xp θ1 + x2 xp θ2 + . . . + (xp − 1)θp Then,  ... θ1∗ θp∗ 2θ1∗ 2 + θ2∗ 2 + . . . + θp∗ 2 θ1∗ θ2∗  θ1∗ 2 + 2θ2∗ 2 + . . . + θp∗ 2 . . . θ2∗ θp∗ θ1∗ θ2∗    T E ZZ =  . .. .. .. ..   . . . . 2 2 2 ∗ ∗ ∗ ∗ ∗ ∗ ∗ . . . θ1 + θ2 + . . . + 2θp θ2 θp θp θ1  Hence the covariance matrix can be written as: Cov(xy) = Ip (1 + kθ ∗ k22 ) + θ ∗ θ ∗ T . Therefore kCov(xy)k2 = 1 + 2kθ ∗ k22 . Bounded Fourth Moment. E h (xy − E[xy])T v 4 i ≤E We start from the LHS h (xy − E[xy])T v 4 i (47) 4 = E ((xxT − I)θ ∗ + wx)T v i4 h = E (θ ∗T x)(xT v) − θ ∗T v + wv T x    4   T ∗T ∗T  ≤ 8 8 E (θ x)(x v) + E θ v \| {z } \| {z A B (48) (49)    + E w(xT v) 4 .  \| {z }  } 4 (50) C The last line follows from two applications of the following inequality: Cr inequality. If X and Y are random variables such that E\|X\|r < ∞ and E\|Y \|4 < ∞ where r ≥ 1 then: E\|X + Y \|r ≤ 2r−1 (E\|X\|r + E\|Y \|r ) . Now to control each term on: • Control of A. Using Cauchy Schwartz, and normality of 1D projections of normal distribution q q T 8 ∗ A ≤ E[\|θ x\| ] E[\|xT v\|8 ] (51) - kθ ∗ k42 . 32 (52) • Control of B, B ≤ kθ ∗ k42 . • Control of C, C = O(1), using independence of w and normality of 1D projections of normal distribution. h 4 i Therefore the E (xy − E[xy])T v - c + kθ ∗ k42 . For the RHS: Var((xy)T v)2 = (v T Cov(xy)v)2 ≤ kCov(xy)k22 . We saw that the kCov(xy)k2 - c + kθ ∗ k22 , so both the LHS and RHS scale with kθ ∗ k42 . Hence, xy has bounded fourth moments. Now that we’ve established that xy has bounded fourth moments implies that we can use [29] as a mean estimation oracle. Using Theorem 1.3 [29], we know that the oracle of [29] outputs an estimate θb of E[xy] such that with probability at least 1 − 1/pC1 , we have: 1 p kθb − θ ∗ k2 ≤ C2 kCov(xy)k2 log p ǫ 2 + γ(n, p, δ, ǫ) Using Lemma 5 to subsitute kCov(xy)k2 ≤ 1+2kθ ∗ k22 ), we recover the statement of Corollary 3. D Proof of Theorem 6 To prove our result on Robust Exponential Family, we first study the distribution of gradients of the corresponding risk function. Lemma 6. Consider the model in Equation (15), then there exists a universal constant C1 such that E[∇L(θ)] = ∇A(θ) − ∇A(θ ∗ ) 2 ∗ kCov[∇L(θ)]k2 = k∇ A(θ )k2 h 4 i ≤ C1 (Var[∇L(θ)T v])2 . Bounded fourth moments E (∇L(θ) − E[∇L(θ)])T v (53) (54) (55) Proof. By Fisher Consistency of the negative log-likelihood, we know that Eθ∗ [∇L(θ ∗ )] = 0 (56) =⇒ ∇A(θ ) − Eθ∗ [φ(z)] = 0 (57) ∗ ∗ =⇒ ∇A(θ ) = Eθ∗ [φ(z)]. (58) For the mean, ∇L(θ) = ∇A(θ) − φ(z) (59) E[∇L(θ)] = ∇A(θ) − Eθ∗ [φ(z)] ∗ E[∇L(θ)] = ∇A(θ) − ∇A(θ ). Now, for the covariance: T i ∇L(θ) − Eθ∗ [∇L(θ)] ∇L(θ) − Eθ∗ [∇(L(θ)] i h T ∗ ∗ ∗ = Eθ (∇A(θ ) − φ(z)) (∇A(θ ) − φ(z)) = Covθ∗ ∇L(θ ∗ ) = ∇2 A(θ ∗ ). Covθ∗ [∇L(θ)] = Eθ∗ h 33 (60) (61) Bounded moments follows from our assumption that the sufficient statistics have bounded 4th moments. Having studied the distribution of the gradients, we use Lemma 1 to characterize the stability of Huber Gradient estimator. Using Lemma 1, we know that at any point θ, the Huber Gradient Estimator g(θ, δ/T ) satisfies that with probability 1 − δ/T , 1 1p e kCov(∇L(θ))k 2 log p. n, p, δ) kg(θ, δ/T ) − ∇R(θ)k2 ≤ C2 ǫ 2 + γ(e 2 Substituting the upper bound on kCov(∇L(θ))k2 from Lemma 6, we get that there are universal constants C1 , C2 such that 1 p √ e kg(θ) − ∇R(θ)k2 ≤ C1 ǫ 2 + γ(e n, p, δ) log p τu . \| {z } e β(e n,δ) e = 0 < τℓ by assumption. Therefore we just have that ǫ < In this case we have that α(e n, δ) e into Theorem 1, C1 for some universal constant C1 . Plugging the corresponding ǫ and β(e n, δ) we get back the result of Corollary 6. E Proof of Corollary 7 Using the contraction property of projections, we know that kPΘ (∇A)−1 µ b − θ ∗ k2 = kPΘ (∇A)−1 µ b − PΘ [θ ∗ ] k2 ≤ k(∇A)−1 µ b − θ ∗ k2 . By Fisher Consistency of the negative log-likelihood, we know that ∇A(θ ∗ ) = Eθ∗ [φ(z)]. The true parameter θ ∗ can be obtained by inverting the ∇A operator whenever possible. k(∇A)−1 µ b − θ ∗ k2 = k(∇A)−1 µ b − (∇A)−1 Eθ∗ [φ(z)]k2 ∗ ∗ = k∇A µ b − ∇A Eθ∗ [φ(z)]k2 . (62) (63) where A∗ is the convex conjugate of A. We can use the following result to control the Lipschitz smoothness A∗ . Theorem 12. (Strong/Smooth Duality) Assume f (·) is closed and convex. Then f (·) is smooth with parameter M if and only if its convex conjugate f (·) is strongly convex with parameter 1 . m= M A proof of the above theorem can be found in [28]. Hence, we have that: 1 µ − Eθ∗ [φ(z)]k2 b − θ ∗ k2 ≤ kb kPΘ (∇A)−1 µ τℓ (64) By assumption, we have that the fourth moments of the sufficient statistics are bounded. We also know that Cov(φ(z) = ∇2 A(θ ∗ ) which implies that we can use [29] as our oracle. Using Lemma 1, we get that, there exists universal constants C1 , C2 such that with probability at least 1 − 1/pC1 , 1 p kb µ − Eθ∗ [φ(z)]k2 ≤ C2 τu log p ǫ 2 + γ(n, p, δ, ǫ) . Combining the above with Equation (64) recovers the result of Corollary 7. 34 F Proof of Theorem 8 Before we present the proof of Theorem 8, we first study the distribution of gradients of the loss function. This will help us bound the error in the gradient estimator. Lemma 7. Consider the model in Equation (11). Suppose the covariates x ∈ Rp have bounded 4th -moments and the noise w has bounded 2th moments.. Then there exist universal constants C1 , C2 such that E[∇L(θ)] = Σ∆ kCov(∇L(θ)k2 ≤ σ 2 kΣk2 + C1 k∆k22 kΣk22 , where ∆ = θ − θ ∗ and E[xxT ] = Σ. Proof. We start by deriving the results for E[∇L(θ)]. 1 1 L(θ) = (y − xT θ)2 = (xT (∆) − w)2 2 2 ∇L(θ) = xxT ∆ − x.w (65) (66) (67) E[∇L(θ)] = Σ∆. Next, we bound the operator norm of the covariance of the gradients ∇L(θ) at any point θ. Covariance. Cov(∇L(θ)) = E[(xxT − Σ)∆ − x.w)((xxT − Σ)∆ − x.w)T )] T T T (68) 2 Cov(∇L(θ)) = E[(xx − Σ)∆∆ (xx − Σ)] + σ Σ. (69) Now, we want to bound kCov(∇L(θ))k2 = λmax (Cov(∇L(θ))). λmax (Cov(∇L(θ))) ≤ σ 2 λmax (Σ) + λmax E[(xxT − Σ)∆∆T (xxT − Σ)] 2 ≤ σ λmax (Σ) + sup y y∈Sp−1 T (70) E[(xx − Σ)∆∆ (xx − Σ)] y T T T (71) ≤ σ 2 λmax (Σ) + sup y T E[(xxT − Σ)∆∆T (xxT − Σ)] y y∈Sp−1 ≤ σ 2 λmax (Σ) + k∆k22 ≤ σ 2 λmax (Σ) + k∆k22 sup y,z∈Sp−1 sup ≤ σ 2 λmax (Σ) + 2k∆k22 2 ≤ σ λmax (Σ) + ≤ σ 2 kΣk2 + E (y T (xxT − Σ)z)2 y,z∈Sp−1 ≤ σ 2 λmax (Σ) + 2k∆k22 2k∆k22 sup y,z∈Sp−1 sup y,z∈Sp−1 sup (72) (73) E 2(y T x)2 (xT z)2 + 2(y T Σz)2 (74) E (y T x)2 (xT z)2 + kΣk22 E (y T x)2 (xT z)2 + kΣk22 q E [(y T x)4 ] y,z∈Sp−1 2 2k∆k2 (kΣk22 + C4 kΣk22 ), q E [(z T x)4 ] + kΣk22 (75) (76) (77) (78) where the second last step follows from Cauchy-Schwartz and the last step follows from our assumption of bounded 4th moments (see Equation (10)). 35 We now proceed to the proof of Theorem 8. From Lemma 2, we know that at any point e satisfies the following with θ, the gradient estimator described in Algorithm 3, g(θ; Dne , δ), probability at least 1 − δ, e − ∇R(θ)k2 ≤ C kg(θ; Dne , δ) q tr(Cov(∇L(θ))) log 1/δe . n e We substitute the upper bound for kCov(∇L(θ))k2 from Lemma 7 in the above equation e − ∇R(θ)k2 ≤ C kg(θ; Dne , δ) ≤ C q q ≤ C1 \| tr(Cov(∇L(θ))) log 1/δe n e p(σ2 kΣk2 +2k∆k22 (kΣk22 +C4 kΣk22 )) log 1/δe n e s kΣk22 p log 1/δe kθ − θ ∗ k2 n e {z } e α(e n,δ) s kΣk2 p log 1/δe . + C2 σ n e {z } \| e β(e n,δ) To complete the proof of this theorem, we use the results from Theorem 1. Note that the e < τl . This holds when gradient estimator satisfies the stability condition if α(e n, δ) n e> C12 τu2 e p log 1/δ. τl2 e into Theorem 1 gives us Now suppose n e satisfies the above condition, then plugging β(e n, δ) the required result. G Proof of Theorem 9 To prove the Theorem we use the result from Lemma 4, where we derived the following expression for covariance of ∇L(θ) p p kCov(∇L(θ)k2 ≤C1 k∆k22 kΣk2 C4 LΦ,4 + LΦ,2 q p p 3 + C2 kΣk2 BΦ,2 + BΦ,4 + c(σ) 3MΦ,2,2 + c(σ) MΦ,4,1 From Lemma 2, we know that at any point θ, the gradient estimator described in Algorithm 3, e satisfies the following with probability at least 1 − δ, g(θ; Dne , δ), e − ∇R(θ)k2 ≤ C kg(θ; Dne , δ) 36 q tr(Cov(∇L(θ))) log 1/δe . n e Substitute the upper bound for kCov(∇L(θ))k2 in the above equation, we get q log 1/δe e kg(θ; Dne , δ) − ∇R(θ)k2 ≤ C tr(Cov(∇L(θ))) n e s √ p kΣk2 C4 LΦ,4 + LΦ,2 p log 1/δe kθ − θ ∗ k2 ≤ C1 n e {z } \| e + C2 \| α(e n,δ) v u p u kΣk B + pB + c(σ)p3M 3 c(σ) MΦ,4,1 p log 1/δe Φ,2 2 Φ,4 Φ,2,2 + t {z n e e β(e n,δ) We now use the results from Theorem 1. The gradient estimator satisfies the stability condition e < τl . This holds when if α(e n, δ) √ p C12 kΣk2 C4 LΦ,4 + LΦ,2 e n e> p log 1/δ. τl2 e into Theorem 1 gives us Now suppose n e satisfies the above condition, then plugging β(e n, δ) the required result. H Proof of Theorem 11 The proof proceeds along similar lines as the proof of Theorem 9. To prove the Theorem we utilize the result of Lemma 6, where we showed that kCov[∇L(θ)]k2 = k∇2 A(θ ∗ )k2 . Combining this result with Lemma 2 we get that with probability at least 1 − δ q log 1/δe e kg(θ; Dne , δ) − ∇R(θ)k2 ≤ C tr(Cov(∇L(θ))) n e s k∇2 A(θ ∗ )k2 p log 1/δe ≤ C . n e \| {z } e β(e n,δ) e = 0, the stability condition is always satisfied, as long as τl > 0. Substituting Since α(e n, δ) e into Theorem 1 gives us the required result. β(e n, δ) I Upper bound on Contamination Level We provide a complementary result, which gives an upper bound for the contamination level ǫ based on the initialization point θ 0 , above which, Algorithm 1 would not work. The key idea is that the error incurred by any mean estimation oracle is lower bounded by the variance of the distribution, and that if the zero vector lies within that error ball, then any mean oracle can be forced to output 0 as the mean. For Algorithm 1, this implies that, in estimating the mean of the gradient, if the error is high, then one can force the mean to be 0 which forces the algorithm to converge. For the remainder of the section we consider the case of linear regression with x ∼ N (0, Ip ) in the asymptotic regime of n → ∞. 37 } . Lemma 8. Consider the model in equation(11) with x ∼ N (0, Ip ) and w ∼ N (0, 1), then 0 ∗ there exists a universal constant C1 such that if ǫ > C1 √ kθ −θ0 k2∗ 2 , then for every gradient 1+2kθ −θ k2 oracle, there exists a contamination distribution Q such that, Algorithm 1 will converge to θ 0 even when the number of samples n → ∞. Proof. Using Lemma 5, we know that for any point θ, ∇L(θ) = xxT ∆ − x.w Eθ∗ [∇L(θ)] = (θ − θ ∗ ) = ∆ kCov(∇L(θ)k2 = 1 + 2k∆k22 , where ∆ = θ − θ ∗ . Let P∇L(θ) represent the distribution ∇Lθ. Similarly, let Pǫ,∇L(θ),Q represent the corresponding ǫ-contaminated distribution. Then, using Theorem 2.1 [10], we know that the minimax rate for estimating the mean of the distribution of gradients is given by: µ − Eθ∗ [∇L(θ)]k22 ≥ Cǫ2 (1 + 2k∆k22 ) ≥ c. inf sup Pǫ,∇L(θ),Q kb µ b θ∈Rp ,Q The above p statement says that at any point θ, any mean oracle Ψ will always incur an error of Ω( Cǫ2 (1 + 2k∆k22 )) in estimating the gradient Eθ∗ [∇L(θ)]. kΨ(θ) − Eθ∗ [∇L(θ)]k2 ≥ Cǫ q (1 + 2k∆k22 ) ∀ Ψ Forp any oracle Ψ, there exists some adversarial contamination Q, such that whenever kEθ∗ [∇L(θ)]k2 < Cǫ (1 + 2k∆k22 ), then kΨ(θ)k2 = 0. Suppose that the contamination level ǫ is such that, ǫ> 1 kEθ∗ [∇L(θ 0 )]k2 p , C (1 + 2kθ 0 − θ ∗ k22 ) then for every oracle there exists a corresponding Q such that Algorithm 1 will remain stuck at θ 0 . Plugging Eθ∗ [∇L(θ 0 )] = θ 0 − θ ∗ , we recover the statement of the lemma. Chen et al. [10] provide a general minimax lower bound of Ω(ǫ) for ǫ-contamination√models in this setting. In contrast, using Algorithm 1 with [29] as oracle, we can only O( ǫ log p) close to the true parameter even when the contamination is small, which implies that our procedure is not minimax optimal. Our approach is nonetheless the only practical algorithm for robust estimation of general statistical models. J Details and Analysis of Algorithm 2 In this section we present a refined, non-asymptotic analysis of the algorithm from [29]. We begin by introducing some preliminaries. We subsequently analyze the algorithm in 1-dimension and finally turn our attention to the general algorithm. 38 J.1 Preliminaries Unless otherwise stated, we assume throughout that the random variable X has bounded fourth moments, i.e. for every unit vector v, 2 E hX − µ, vi4 ≤ C4 E hX − µ, vi2 . We summarize some useful results from [29], which bound the deviation of the conditional mean/covariance from the true mean/covariance. Lemma 9. [Lemma 3.11 [29]] Let X be a univariate random variable with bounded fourth moments, and let A be any with event with probability P(A) = 1 − γ ≥ 21 . Then, p \|E(X\|A) − E(X)\| ≤ σ 4 8C4 γ 3 . Lemma 10. [Lemma 3.12 [29]] Let X be a univariate random variable with E[X] = µ, E (X − µ)2 = σ 2 and let E((X − µ)4 ) ≤ C4 σ 4 . Let A be any with event with probability P(A) = 1 − γ ≥ 12 . Then, p (1 − C4 γ)σ 2 ≤ E((X − µ)2 \|A) ≤ (1 + 2γ)σ 2 . Corollary 13. [Corollary 3.13 [29]] Let A be any event with probability P(A) = 1 − γ ≥ 21 , and let X be a random variable with bounded fourth moments. We denote Σ\|A = E(XX T \|A) − (E(X\|A))(E(X\|A))T to be the conditional covariance matrix. We have that, p p (1 − C4 γ − 8C4 γ 3 )Σ Σ\|A (1 + 2γ)Σ. For random variables with bounded fourth moments we can use Chebyshev’s inequality to obtain tail bounds. Lemma 11. [Lemma 3.14 [29]] Let X have bounded fourth moments, then for every unit vector v we have that, P(\|hX, vi − E[hX, vi]\| ≥ t p [E [hX − µ, vi2 ]]) ≤ C4 . t4 Our proofs also use the matrix Bernstein inequality for rectangular matrices. As a preliminary, we consider a finite sequence {Zk } of independent, random matrices of size d1 ×d2 . We assume that each random matrix satisfies E(Zk ) = 0, and kZk kop ≤ R almost surely. We define: ( ) X X σ 2 := max k E(Zk ZkT )kop , k E(Zk ZkT )kop . k k With these preliminaries in place we use the following result from [41]. Lemma 12. For all t ≥ 0, P X k Zk op ≥ t ≤ (d1 + d2 ) exp −t2 /2 σ 2 + Rt/3 . Equivalently, with probability at least 1 − δ, s X d1 + d2 d1 + d2 2R log + . ≤ 2σ 2 log Zk δ 3 δ op k 39 We let I denote the set of all intervals in R. The following is a standard uniform convergence result. Lemma 13. Suppose X1 , . . . , Xn ∼ P, then with probability at least 1 − δ, r n 4 log(en) + 2 log(2/δ) 1X I(Xi ∈ I) ≤ 2 . sup P(I) − n n I∈I i=1 Algorithm 4 Huber Outlier Gradients Truncation function HuberOutlierGradientTruncation(Sample Gradients S, Corruption Level ǫ, Dimension p,δ) if p=1 then r \|S\| 1 Let [a, b] be smallest interval containing 1 − ǫ − C5 (1 − ǫ) fracδ \|S\| log tion of points. Se ← S ∩ [a, b]. return Se else Let [S]i be the samples with the ith co-ordinates only, [S]i = {hx, ei i \|x ∈ S} for i = 1 to p do a[i] = HuberGradientEstimator([S]i , ǫ, 1, δ/p). end for r Let B(r, a) be the ball of smallest radius centered at a containing (1 − ǫ − Cp p \|S\| log \|S\| pδ (1 − ǫ) fraction of points in S. Se ← S ∩ B(r, a). return Se end if end function We now turn our attention to an analysis of Algorithm 2 for the 1-dimensional case. J.2 The case when p = 1 Firstly, we analyze Algorithm 2 when p = 1. Lemma 14. Suppose that, Pθ∗ is a distribution on R1 with mean µ, variance σ 2 , and bounded fourth moments. There exist positive universal constants C1 , C2 , C8 > 0, such that given n samples from the distribution in (4), the algorithm with probability at least 1 − δ, returns an estimate µ b such that, !3 !1 r r r 4 2 1 log 3/δ log 3/δ log(3/δ) +t +t + C2 σ ǫ + kb µ − µk2 ≤ C1 C44 σ ǫ + 2n 2n n where t = C8 q 1 n n δ log 1 4 . which can be further simplified to, kb µ − µk2 ≤ C1 C4 σ ǫ + C8 r !3 !1 r r n 4 n 2 log(1/δ) 1 1 + C2 σ ǫ + C8 log log n δ n δ n 40 Proof. By an application of Hoeffding’s inequality we obtain that with probability at least 1 − δ/3, q the fraction of corrupted samples (i.e. samples from the distribution Q) is less than ǫ + log(3/δ) 2n . We condition on this event through the remainder of this proof. We let η denote the fraction of corrupted samples. Further, we let SP be the samples from the true distribution. Let nP be the cardinality of this set, i.e. nP := \|SP \|. Let I1−η be the interval around µ containing 1 − η mass of Pθ∗ . Then, using Lemma 11, we have that: 1 length (I1−η ) ≤ C44 σ 1 η4 . Using Lemma 13 we obtain that with probability at least 1 − δ/3 the number of samples from the distribution P that fall in the interval I1−η is at least 1 − η − t where t is upper bounded as: r 4 log(en) + 2 log(6/δ) . t≤2 n Now we let Se be the set of points in the smallest interval containing (1 − η − t)(1 − η) fraction of all the points. • Using VC theory, we know that for every interval I ⊂ R, there exists some universal constant C3 such that P (\|(P (x ∈ I\|x ∼ D) − P (x ∈ I\|x ∈u SD ))\| > t/2) ≤ n2D exp(−nD t2 /8) (79) This can be re-written as, that with probability at least (1−δ/3), there exists a universal constant C0 such that, r r n n 1 1 D ≤ C5 log log sup \|(P (x ∈ I\|x ∼ D) − P (x ∈ I\|x ∈u SD ))\| ≤ C0 nD δ n δ I \| {z } t • Using Equation (79), we know that (1 − η − t) fraction of SD lie in I1−η . Let Se be the set of points in the smallest interval containing (1 − η − t)(1 − η) fraction of the points. • We know that the length of minimum interval containing (1 − η − t)(1 − η) fraction of the points of S is less than length of smallest interval containing (1 − η − t) fraction of points of SD , which in turn is less than length of I1−η . • Now, I1−η and minimum interval containing (1 − η − t) fraction of points of SD need to overlap. This is because, n is large enough such that t < 12 − η hence, the extreme points for such an interval can be atmost 2length (I1−η ) away. • Hence, the distance of all chosen noise-points from µ will be within the length (I1−η ). • Moreover, the interval of minimum length with (1−η −t)(1−η) fraction of S will contain at least 1 − 3η − t fraction of SD . e by controlling the sources of error. • Hence, we can bound the error of mean(S) 41 – All chosen noise points are within length (I1−η ), and there are atmost η of them, hence the maximum error can be ηlength (I1−η ). – Next, the mean of chosen good points will converge to the mean of the conditional distribution. i.e. points sampled from D but conditioned to lie in the minimum length interval. The variance of these random variables is upper bounded using Lemma 10. – To control the distance between the mean(E(X) and the conditional mean(E(X\|A)), where A is the event that a sample x is in the chosen interval. We know that P (A) ≥ 1 − 3η − t, hence, using Lemma 3.11[29], we get that there exists a constant C13 such that, 1 3 \|E[X] − E[X\|A]\| ≤ C13 C44 σ(η + t) 4 • Hence, with probability at least 1 − δ/3, the mean of Se will be within r 1 3 1 log(3/δ) 4 η × length (I1−η ) + C13 C4 σ(η + t) 4 + C6 σ(1 + 2η) 2 n • q Taking union-bound over all conditioning statements, and upper bounding, η with ǫ + log(3/δ) 2n , we recover the statement of the lemma. J.3 The case when p > 1 To prove the case for p > 1, we use a series of lemmas. Lemma 15 proves that the outlier filtering constrains the points in a ball around the true mean. Lemma 17 controls the error in e Lemma 18 controls the mean and covariance the true distribution after outlier filtering (D). e the error for the mean of S when projected onto the bottom span of the covariance matrix ΣSe. Lemma 15. Suppose that, Pθ∗ is a distribution on Rp with mean µ, covariance Σ, and bounded fourth moments. There exist positive universal constants C1 , C2 , C8 > 0, such that given n samples from the distribution in Equation (4), we can find a vector a ∈ Rp such that with probability at least 1 − δ, 1 4 ka − µk2 ≤C1 C4 + C2 !3 np 4 1 tr (Σ) ǫ + C8 log n δ !1 r r np 2 p 1 log(p/δ) tr (Σ) log ǫ + C8 n δ n r p Proof. Pick n orthogonal directions v1 , v2 , . . . , vn , and use method for one-dimensions, and using union bound, we can recover the result. Next, we prove the case when p > 1. Firstly, we prove that after the outlier step, Lemma 16. After the outlier removal step, there exists universal constants C11 > 0 such that with probability at least 1 − δ, every remaining point x satisfies, kx − µk2 ≤ r1∗ + 2r2∗ 42 √ q 1p p 3 1 and r2∗ = C1 C44 pkΣk2 (η + t) 4 + C2 pkΣk2 (η + t) 2 log(1/δ) and n q q log(1/δ) is the fraction of samples corrupted. t = C8 n1 log np δ . Here η ≤ ǫ + 2n 1 where r1∗ = C10 C44 pkΣk2 1 η4 Proof. • Let Se be the set of points chosen after the outlier filtering. Let SeD be set of good points chosen after the outlier filtering. Let SeN be the set of bad points chosen after the outlier filtering. • Using VC theory we know that for every closed ball B(µ, r) = {x\|kx − µk2 ≤ r}, there exists a constant C9 such that with probability at least 1 − δ s n p log sup \|P (x ∈ B\|x ∼ D) − P (x ∈ B\|x ∈u SD )\| ≤ C9 n pδ B \| {z } t2 1 • Let B ∗ = B(µ, r ∗ ) for r1∗ = C10 C44 1 (η) 4 p pkΣk2 . Then, we claim that P (x ∈ B ∗ \|x ∼ D) ≥ 1 − η – To see this, suppose we have some x ∈ D. LetPz = x − µ. Let zi = z T vi for some orthogonal directions v1 , v2 , . . . , vp . Let Z 2 = zi2 = kzk22 . –  1 2  C pkΣk2 P Z 2 ≥ 4 1  = P (η) 2 Z4 ≥ C4 p2 kΣk22 (η) ≤ (η)E(Z 4 ) C4 p2 kΣk22 – Now, E(Z 4 ) ≤ p2 maxi E(zi4 ) ≤ C4 p2 kΣk22 . Plugging this in the above, we have that P (x ∈ B ∗ \|x ∼ D) ≥ 1 − η. • Hence, we have that P (x ∈ B ∗ \|x ∈u SD ) ≥ 1 − η − t2 . • Using Lemma 15, we have that at least (1 − η − t2 ) fraction of good points are r1∗ + r2∗ away from a. Hence, we have that the minimum radius of the ball containing all the (1 − η − t2 )(1 − η) has a radius of atmost r1∗ + r2∗ , which when combined with the triangle inequality recovers the statement of lemma. e µe = As before, let Se be the set of points after outlier filtering. Let µSe = mean(S), SD mean(SeD ), µ e = mean(SeN ). SN Lemma 17. Let SeD be the set of clean points remaining after the outlier filtering. Then, with probability at least 1 − δ, we have that r 1 p p 3p log(p/δ) 1 4 kµSeD − µk2 ≤ C1 C4 (η + t2 ) 4 kΣk2 1 + log(p/δ) + kΣk2 (1 + 2(η + t2 ) n n (r ∗ + 2r2∗ ) log(p/δ) + C15 1 n 43 and kΣSeD k2 ≤ β(n, δ)kΣk2 , where β(n, δ) = !! √ r 3 3 p C4 p log(p/δ log(p/δ) + 1 + 2C(η + t2 ) + 1 + √ + C4 p(η + t) 2 + (η + t2 ) 2 η n n Proof. We first prove the bounds on the mean shift. kµSeD − µk2 ≤ kµSeD − µDe k2 + kµDe − µk2 {z } \| {z } \| B A µ −µ • Control of B. We use Lemma 9 on X = xT kµ De−µk2 for x ∼ D, and A be the event e D that x is not removed by the outlier filtering. 1 3 kµDe − µk2 ≤ C1 C44 (η + t2 ) 4 p kΣk2 (80) • Control of A. Using Lemma 10,we have that kΣDe k2 ≤ (1 + 2(η + t2 ))kΣk2 . Now, we use Bernstein’s inequality . Lemma 12 with R = C(r1∗ + 2r2∗ + B), we get that, with probability at least 1 − δ, r p p (r ∗ + 2r2∗ + B) 1 log(p/δ) + C15 1 log(p/δ) kµSeD − µDe k2 ≤ C14 kΣk2 (1 + 2(η + t2 ) n n (81) Next, we prove the bound for covariance matrix. kΣDe − Σk2 \| {z } kΣSeD k2 ≤ kΣSeD − ΣDe k2 + +kΣk2 (82) ≤2C(η+t2 )kΣk2 (By Corollary 13)) (x −µ )(x −µ )T −Σ e D . From, To control kΣSeD −ΣDe k2 , we use Bernstein’s inequality, with Zk = k De kn De Lemma 16, we know that the points are constrained in a ball. Plugging this into Lemma 12, ! r log(p/δ log(p/δ) 2 kΣSeD − ΣDe k2 ≤ C(kΣk2 + R ) + n n 2 2 where R2 = C r1∗ + r2∗ + B 2 . Plugging in the values, we get that, kΣSeD − ΣDe k2 ≤ CkΣk2 ! √ r 3 3 p C4 p log(p/δ log(p/δ) 1 + √ + C4 p(η + t) 2 + (η + t2 ) 2 + η n n Finally, we have that, !! √ r 3 3 log(p/δ log(p/δ) p C4 p + kΣSeD k2 ≤ kΣk2 1 + 2C(η + t2 ) + 1 + √ + C4 p(η + t) 2 + (η + t2 ) 2 η n n \| {z } β(n,δ) 44 Lemma 18. Let W be the bottom p/2 principal components of the covariance matrix after filtering ΣSe. Then there exists a universal constant C > 0 such that with probability at least 1 − δ, we have that 1 2 2 kηPW δµ k2 ≤ Cη β(n, δ) + γ(n, δ)C4 )kΣk2 , where δµ = µSeN − µSeD , PW is the projection matrix on the bottom p/2-span of ΣSe, β(n, δ) 1 is as defined in Lemma 17 and γ(n, δ) = η 2 + (η + t)5/2 + η(η + t) log(1/δ) n Proof. We have ΣSe = (1 − η)ΣSeD + ηΣSeN + (η − η 2 )δµ δµT {z } \| {z } \| E (83) F (84) By Weyl’s inequality we have that, λp/2 (ΣSe) ≤ λ1 (E) + λp/2 (F ) • Control of λp/2 (F ). tr (F ) p/2 ((r ∗ )2 + (r2∗ )2 ) + B 2 ≤ C15 η 1 p/2 1 1 log(1/δ) 5/2 2 ≤ C16 C4 kΣk2 η 2 + (η + t) + η(η + t) n \| {z } λp/2 (F ) ≤ γ(n,δ) where t = C8 q 1 n log • Control of λ1 (E). np δ . λ1 (E) ≤ (1 − η)βkΣk2 Hence, we have that: λp/2 (ΣSe) ≤ (1 − η)βkΣk2 + C16 γ p C4 kΣk2 Using that W is the space spanned by the bottom p/2 eigenvectors of ΣSe and PW is corresponding projection operator, we have that: h p i T PW ΣSePW (1 − η)β + C16 γ C4 kΣk2 Ip Following some algebraic manipulation in [29], we get that, 1 2 2 kηPW δµ k2 ≤ η (β(n, δ) + γC4 )kΣk2 45 Having established all required results, we are now ready to prove Lemma 1. We restate the result for the sake of completeness. Theorem 14. Suppose that, Pθ∗ is a distribution on Rp with mean µ, covariance Σ and bounded fourth moments. There exist positive universal constant C > 0, such that given n samples from the distribution in Equation (4), the algorithm with probability at least 1 − δ, returns an estimate µ b such that,  !1  r 1 1 1 p 3 log p log(p log(p/δ)) 2  √ √ 2 4 2  4 kb µ − µk2 ≤ CkΣk2 (1 + log p) η + C4 (η + t2 ) + ηpC4 n where η = ǫ + q log(p) log(log p/δ) 2n and t2 = q p log(p) log(n/(pδ)) . n Proof. We divide n samples into ⌊log(p)⌋ different sets. We choose the first set and keep that as our active set of samples. We run our outlier filtering on this set, and let the remaining samples after the outlier filtering be SeD . By orthogonality of subspaces spanned by eigenvectors, coupled with triangle inequality and contraction of projection operators, we have that µV − PV µk22 kb µ − µk22 ≤ 2kPW (b µ − µSeD )k22 + 2kPW (µSeD − µ)k22 + kb µV − PV µk22 kb µ − µk22 ≤ 2kPW (b µ − µSeD )k22 + 2k(µSeD − µ)k22 + kb bV is the mean vector where V is the span of the top p/2 principal components of ΣSe and where µ of returned by the running the algorithm on the reduced dimensions dim(V ) = p/2. From Lemma 18, both β(n, δ) and γ(n, δ) are monotonically increasing in the dimension; moreover the upper bound in Lemma 17 is also monotonically increasing in the dimension p, hence, the error at each step of the algorithm can be upper bounded by error incurred when running on dimension p, with n/ log(p) samples, and probability of δ/ log p. Hence, the overall error for the recursive algorithm can be upper bounded as, kb µ − µk22 ≤ 2kPW (b µ − µSeD )k22 + 2kµSeD − µk22 (1 + log p) Combining Lemma 17 and Lemma 18 which are instantiated for n/ log p samples and probability δ/ log p, we get,  !1  r 2 1p 1 1 3 log p log(p log(p/δ)) √ √  kb µ − µk2 ≤ CkΣk22 log p  η + C44 (η + t2 ) 4 + ηpC42 n 46	2
(Quasi)Periodicity Quantification in Video Data, Using Topology arXiv:1704.08382v2 [cs.CV] 21 Jan 2018 Christopher J. Tralie† and Jose A. Perea* ∗ † January 23, 2018 Abstract This work introduces a novel framework for quantifying the presence and strength of recurrent dynamics in video data. Specifically, we provide continuous measures of periodicity (perfect repetition) and quasiperiodicity (superposition of periodic modes with non-commensurate periods), in a way which does not require segmentation, training, object tracking or 1-dimensional surrogate signals. Our methodology operates directly on video data. The approach combines ideas from nonlinear time series analysis (delay embeddings) and computational topology (persistent homology), by translating the problem of finding recurrent dynamics in video data, into the problem of determining the circularity or toroidality of an associated geometric space. Through extensive testing, we show the robustness of our scores with respect to several noise models/levels; we show that our periodicity score is superior to other methods when compared to human-generated periodicity rankings; and furthermore, we show that our quasiperiodicity score clearly indicates the presence of biphonation in videos of vibrating vocal folds, which has never before been accomplished end to end quantitatively. 1 Introduction Periodicity characterizes many natural motions including animal locomotion (walking/wing flapping/slithering), spinning wheels, oscillating pendulums, etc. Quasiperiodicity, thought of as the superposition of non-commensurate frequencies, occurs naturally during transitions from ordinary to chaotic dynamics [11]. The goal of this work is to automate the analysis of videos capturing periodic and quasiperiodic motion. In order to identify both classes of motion in a unified framework, we generalize 1-dimensional (1D) sliding window embeddings [38] to reconstruct periodic and quasiperiodic attractors from videos123 . We analyze the resulting attractors using persistent homology, a technique which combines geometry and topology (Section 2.2), and we return scores in the range [0, 1] that indicate the degree of periodicity or quasiperiodicity in the corresponding video. We show that our periodicity measure compares favorable to others in the literature when ranking videos (Section 4.2). Furthermore, to our knowledge, there is no other method able to quantify the existence of quasiperiodicity directly from video data. Our approach is fundamentally different from most others which quantify periodicity in video. For instance, it is common to derive 1D signals from the video and apply Fourier or autocorrelation to measure periodicity. By contrast, our technique operates on raw pixels, avoiding common video ∗ † Department of Electrical And Computer Engineering, Duke University,Durham, NC, USA. e-mail: [email protected] † * Department of Mathematics and Department of Computational Mathematics, Science & Engineering, Michigan State University, East Lansing, MI, USA. e-mail: [email protected] 1 Some of the analysis and results appeared as part of the Ph.D. thesis of the first author [?]. 2 Code to replicate results: https://github.com/ctralie/SlidingWindowVideoTDA 3 Supplementary material and videos: https://www.ctralie.com/Research/SlidingWindowVideoQuasi 1 preprocessing and tracking entirely. Using geometry over Fourier/autocorrelation also has advantages for our applications. In fact, as a simple synthetic example shows (Figure 3), the Fourier Transform of quasiperiodic signals is often very close to the Fourier transform of periodic signals. By contrast, the sliding window embeddings we design yield starkly different geometric structures in the periodic and quasiperiodic cases. We exploit this to devise a quasiperiodicity measurement, which we use to indicate the degree of “biphonation” in videos of vibrating vocal folds (Section 4.3), which is useful in automatically diagnosing speech pathologies. In the context of applied topology, our quasiperiodicity score is one of the first applications of persistent H2 to high dimensional data, which is largely possible due to recent advancements in the computational feasibility of persistent homology [3]. 1.1 1.1.1 Prior Work on Recurrence in Videos 1D Surrogate Signals One common strategy for detecting periodicity in video is to derive a 1D function to act as a surrogate for its dynamics, and then to use either frequency domain (Fourier transform) or time domain (autocorrelation, peak finding) techniques. One of the earliest works in this genre finds level set surfaces in a spatiotemporal “XYT” volume of video (all frames stacked on top of each other), and then uses curvature scale space on curves that live on these “spatiotemporal surfaces” as the 1D function [1]. [34] use Fourier Transforms on pixels which exhibit motion, and define a measure of periodicity based on the energy around the Fourier peak and its harmonics. [10] extract contours and find eigenshapes from the contours to classify and parameterize motion within a period. Frequency estimation is done by using Fourier analysis and peak detection on top of other 1D statistics derived from the contours, such as area and center of mass. Finally, [47] derive a 1D surrogate function based on mutual information between the first and subsequent frames, and then look for peaks in the similarity function with the help of a watershed method. 1.1.2 Self-Similarity Matrices Another class of techniques relies on self-similarity matrices (SSMs) between frames, where similarity can be defined in a variety of ways. [37] track a set of points on a foreground object and compare them with an affine invariant similarity. Another widely recognized technique for periodicity quantification [6], derives periodicity measures based on self-similarity matrices of L1 pixel differences. This technique has inspired a diverse array of applications, including analyzing the cycles of expanding/contracting jellyfish [33], analyzing bat wings [2], and analyzing videos of autistic spectrum children performing characteristic repetitive motions such as “hand flapping” [22]. We compare to this technique in Section 4.2. 1.1.3 Miscellaneous Techniques for Periodic Video Quantification There are also a number of works that don’t fall into the two categories above. Some works focus solely on walking humans, since that is one of the most common types of periodic motion in videos of interest to people. [29] look at the “braiding patterns” that occur in XYT slices of videos of walking people. [17] perform blob tracking on the foreground of a walking person, and use the ratio of the second and first eigenvalues of PCA on that blob. For more general periodic videos, [43] make a codebook of visual words and look for repetitions within the resulting string. [23] take a deep learning approach to counting the number of periods that occur in a video segment. They use a 3D convolutional neural network on spatially downsampled, non-sequential regions of interest, which are uniformly spaced in time, to estimate the length of the cycle. Finally, perhaps the most philosophically similar work to ours is the work of [41], who use cohomology to find maps of MOCAP data to the circle for parameterizing periodic motions, though this work does not provide a way to quantify periodicity. 2 1.1.4 Our Work We show that geometry provides a natural way to quantify recurrence (i.e. periodicity and quasiperiodicity) in video, by measuring the shape of delay embeddings. In particular, we propose several optimizations (section 3) which make this approach feasible. The resulting measure of quasiperiodicity, for which quantitative approaches are lacking, is used in section 4.3 to detect anomalies in high-speed videos of vibrating vocal folds. Finally, in contrast to both frequency and time domain techniques, our method does not rely on the period length being an integer multiple of the sampling rate. 2 2.1 Background Delay Embeddings And Their Geometry Recurrence in video data can be captured via the geometry of delay embeddings; we describe this next. 2.1.1 Video Delay Embeddings We will regard a video as a sequence of grayscale4 image frames indexed by the positive real numbers. That is, given positive integers W (width) and H (height), a video with W × H pixels is a function X : R+ −→ RW ×H In particular, a sequence of images X1 , X2 , . . . ∈ RW ×H sampled at discrete times t1 < t2 < · · · yields one such function via interpolation. For an integer d ≥ 0, known as the dimension, a real number τ > 0, known as the delay, and a video X : R+ −→ RW ×H , we define the sliding window (also referred to as time delay) embedding of X – with parameters d and τ – at time t ∈ R+ as the vector   X(t)  X(t + τ )    (1) SWd,τ X(t) =   ∈ RW ×H×(d+1) ..   . X(t + dτ ) The subset of RW ×H×(d+1) resulting from varying t will be referred to as the sliding window embedding of X. We remark that since the pixel measurement locations are fixed, the sliding window embedding is an “Eulerian” view into the dynamics of the video. Note that delay embeddings are generally applied to 1D time series, which can be viewed as 1-pixel videos (W = H = 1) in our framework. Hence equation (1) is essentially the concatenation of the delay embeddings of each individual pixel in the video into one large vector. One of the main points we leverage in this paper is the fact that the geometry of the sliding window embedding carries fundamental information about the original video. We explore this next. 2.1.2 Geometry of 1-Pixel Video Delay Embeddings As a motivating example, consider the harmonic (i.e. periodic) signal π π t + cos t fh (t) = cos 5 15 and the quasiperiodic signal π 1 fq (t) = cos t + cos t 5 5 (2) (3) 4 For color videos we can treat each channel independently, yielding a vector in RW ×H×3 . In practice, there isn’t much of a difference between color and grayscale embeddings in our framework for the videos we consider. 3 1 1 We refer to fh as harmonic because its constitutive frequencies, 10 and 30 , are commensurate; that is, they are linearly dependent over the rational numbers Q ⊂ R. By way of contrast, the underlying fre1 1 quencies of the signal fq , 10 and 10π , are linearly independent over Q and hence non-commensurate. We use the term quasiperiodicity, as in the non-linear dynamics literature [19], to denote the superposition of periodic processes whose frequencies are non-commensurate. This differs from other definitions in the literature (e.g. [?, 43]) which regard quasiperiodic as any deviation from perfect repetition. A geometric argument from [31] (see equation 7 below and the discussion that follows) shows that given a periodic function f : [0, 2π] −→ R with exactly N harmonics, if d ≥ 2N and 0 < τ < 2π d then the sliding window embedding SWd,τ f is a topological circle (i.e. a closed curve without selfintersections) which wraps around an N -dimensional torus S 1 = {z ∈ C : \|z\| = 1} TN = S 1 × · · · × S 1 , {z } \| N −times As an illustration, we show in Figure 1 a plot of fh and of its sliding window embedding SWd,τ fh , via a PCA (Principal Component Analysis) 3-dimensional projection. Figure 1: Sliding window embedding of the harmonic signal fh . Colors in the signal correspond to colors of the points in the PCA plot. The sliding window embedding traces a topological circle wrapped around a 2-dimensional torus. However, if g : R −→ R is quasiperiodic with N distinct non-commensurate frequencies then, for appropriate d and τ , SWd,τ g is dense in (i.e. fills out) TN [30]. Figure 2 shows a plot of the quasiperiodic signal fq (t) and a 3-dimensional projection, via PCA, of its sliding window embedding SWd,τ fq . Figure 2: Sliding window embedding of the quasiperiodic signal fq . Colors in the signal correspond to colors of the points in the PCA plot. The sliding window embedding is dense in a 2-dimensional torus. 4 The difference in geometry of the delay embeddings is stark compared to the difference between their power spectral densities, as shown in Figure 3. Figure 3: The power spectral densities (300 samples) of commensurate and non-commensurate signals with relative harmonics at ratios 3 and π, respectively. The difference is not nearly as evident as in the geometry of their sliding window embeddings. Additionally, unless sampling is commensurate with a frequency, a fixed Fourier basis causes that frequency component to bleed into many frequency bins in a sinc-like pattern, making precise peak finding difficult. Moreover, as we will see next, the interpretation of periodicity and quasiperiodicity as circularity and toroidality of sliding window embeddings remains true for videos with higher resolution (i.e. max{W, H} > 1). The rest of the paper will show how one can use persistent homology, a tool from the field of computational topology, to quantify the presence of (quasi)periodicity in a video by measuring the geometry of its associated sliding window embedding. In short, we propose a periodicity score for a video X which measures the degree to which the sliding window embedding SWd,τ X spans a topological circle, and a quasiperiodicity score which quantifies the degree to which SWd,τ X covers a torus. This approach will be validated extensively: we show that our (quasi)periodicity detection method is robust under several noise models (motion blur, additive Gaussian white noise, and MPEG bit corruption); we compare several periodicity quantification algorithms and show that our approach is the most closely aligned with human subjects; finally, we provide an application to the automatic classification of dynamic regimes in high-speed laryngeal video-endoscopy. 2.1.3 Geometry of Video Delay Embeddings Though it may seem daunting compared to the 1D case, the geometry of the delay embedding shares many similarities for periodic videos, as shown in [39]. Let us argue why sliding window embeddings from (quasi)periodic videos have the geometry we have described so far. To this end, consider an example video X that contains a set of N frequencies ω1 , ω2 , ..., ωN . Let the amplitude of the nth frequency and ith pixel be ain . For simplicity, but without loss of generality, assume that each is a cosine with zero phase offset. Then the time series at pixel i can be written as Xi (t) = N X ain cos(ωn t) (4) n=1 Grouping all of the coefficients together into a (W × H) × N matrix A, we can write X(t) = N X An cos(ωn t) n=1 5 (5) where An stands for the nth column of A. Constructing a delay embedding as in Equation 1:   An cos(ωn t) N X  .. SWd,τ X(t) =   . n n=1 A cos(ωn (t + dτ ) (6) and applying the cosine sum identity, we get SWd,τ X(t) = N X ~un cos(ωn t) − ~vn sin(ωn t) (7) n=1 where ~un , ~vn ∈ RW ×H×(d+1) are constant vectors. In other words, the sliding window embedding of this video is the sum of linearly independent ellipses, which lie in the space of d + 1 frame videos at resolution W × H. As shown in [31] for the case of commensurate frequencies, when the window length is just under the length of the period, all of the ~un and ~vn vectors become orthogonal, and so they can be recovered by doing PCA on SWd,τ X(t). Figure 4 shows the components of the first 8 PCA vectors for a horizontal line of pixels in a video of an oscillating pendulum. Note how the oscillations are present both temporally and spatially. X t Figure 4: Showing an XT slice of the principal components on SWd,1 for the synthetic video of an oscillating pendulum; d is chosen just under the period length (∼ 25 frames). 2.1.4 The High Dimensional Geometry of Repeated Pulses Using Eulerian coordinates has an important impact on the geometry of delay embeddings of natural videos. As Figure 5 shows, pixels often jump from foreground to background in a pattern similar to square waves. These types of abrupt transitions require higher dimensional embeddings to reconstruct the geometry. To see why, first extract one period of a signal with period ` at a pixel Xi (t): Xi (t) 0 ≤ t ≤ ` fi (t) = (8) 0 otherwise Then Xi (t) can be rewritten in terms of the pulse as Xi (t) = ∞ X fi (t − m`) (9) m=−∞ Since Xi (t) repeats itself, regardless of what fi (t) looks like, periodic summation discretizes the frequency domain [32] ∞ m m X F {Xi (t)} (k) ∝ F(fi (t)) δ −k (10) ` ` m=−∞ 6 Figure 5: An example of an Eulerian pixel witnessing a foreground/background transition in a video of a woman doing jumping jacks. Red, green, and blue channels are plotted over time. These transitions induce a per pixel periodic signal with sharp transitions, which leads to high dimensionality in an appropriate sliding window embedding. Switching back to the time domain, we can write Xi (t) as Xi (t) ∝ ∞ X F (fi (t)) m=−∞ m ` ei 2πm ` t (11) In other words, each pixel is the sum of some constant offset plus a (possibly infinite) set of harmonics at integer multiples of 1` . For instance, applying Equation 11 to a square wave of period ` centered at the origin is a roundabout way of deriving the Fourier Series 2π 1 6π 1 10π sin t + sin t + sin t + ... (12) ` 3 ` 5 ` by sampling the sinc function sin(π`f )/(πf ) at intervals of m/2` (every odd m coincides with π/2+kπ, proportional to 1/k, and every even harmonic is zero conciding with πk). In general, the sharper the transitions are in Xi (t), the longer the tail of F{fi (t)} will be, and the more high frequency harmonics will exist in the embedding, calling for a higher delay dimension to fully capture the geometry, since every harmonic lives on a linearly independent ellipse. Similar observations about harmonics have been made in images for collections of patches around sharp edges ( [48], Figure 2). 2.2 Persistent Homology Informally, topology is the study of properties of spaces which do not change after stretching without gluing or tearing. For instance, the number of connected components and the number of (essentially different) 1-dimensional loops which do not bound a 2-dimensional disk, are both topological properties of a space. It follows that a circle and a square are topologically equivalent since one can deform one onto the other, but a circle and a line segment are not because that would require either gluing the endpoints of the line segment or tearing the circle. Homology [12] is a tool from algebraic topology designed to measure these types of properties, and persistent homology [50] is an adaptation of these ideas to discrete collections of points (e.g., sliding window embeddings). We briefly introduce these concepts next. 2.2.1 Simplicial Complexes A simplicial complex is a combinatorial object used to represent and discretize a continuous space. With a discretization available, one can then compute topological properties by algorithmic means. 7 Formally, a simplicial complex with vertices in a nonempty set V is a collection K of nonempty finite subsets σ ⊂ V so that ∅ = 6 τ ⊂ σ ∈ K always implies τ ∈ K. An element σ ∈ K is called a simplex, and if σ has (n + 1) elements then it is called an n-simplex. The cases n = 0, 1, 2 are special, 0-simplices are called vertices, 1-simplices are called edges and 2-simplices are called faces. Here is an example to keep in mind: the circle S 1 = {z ∈ C : \|z\| = 1} is a continuous space but its topology can be captured by a simplicial complex K with three vertices a, b, c, and three edges {a, b}, {b, c}, {a, c}. That is, in terms of topological properties, the simplicial complex K = {a}, {b}, {c}, {a, b}, {b, c}, {a, c} can be regarded as a combinatorial surrogate for S 1 : they both have 1 connected component, one 1-dimensional loop which does not bound a 2-dimensional region, and no other features in higher dimensions. 2.2.2 Persistent Homology of Point Clouds The sliding window embedding of a video X is, in practice, a finite set SWd,τ X = {SWd,τ X(t) : t ∈ T } determined by a choice of T ⊂ R finite. Moreover, since SWd,τ X ⊂ RW ×H×(d+1) then the restriction of the ambient Euclidean distance endows SWd,τ X with the structure of a finite metric space. Discrete metric spaces, also referred to as point clouds, are trivial from a topological point of view: a point cloud with N points simply has N connected components and no other features (e.g., holes) in higher dimensions. However, when a point cloud has been sampled from/around a continuous space with non-trivial topology (e.g., a circle or a torus), one would expect that appropriate simplicial complexes with vertices on the point cloud should reflect the topology of the underlying continuous space. This is what we will exploit next. Given a point cloud (X, dX ) – where X is a finite set and dX : X×X −→ [0, ∞) is a distance function – the Vietoris-Rips complex (or Rips complex for short) at scale ≥ 0 is the collection of non-empty subsets of X with diameter less than or equal to : R (X) := σ ⊂ X : dX (x1 , x2 ) ≤ , ∀xi , xj ∈ σ (13) That is, R (X) is the simplicial complex with vertex set equal to X, constructed by adding an edge between any two vertices which are at most apart, adding all 2-dimensional triangular faces (i.e. 2-simplices) whose bounding edges are present, and more generally, adding all the k-simplices whose (k − 1)-dimensional bounding facets have been included. We show in Figure 6 the evolution of the Rips complex on a set of points sampled around the unit circle. epsilon = 0 epsilon = 0.30 epsilon = 0.35 1 1 1 0 0 0 -1 -1 -1 -1 0 1 -1 0 1 -1 0 1 Figure 6: The Rips complex, at three different scales ( = 0, 0.30, 0.35), on a point cloud with 40 points sampled around S 1 ⊂ R2 . The idea behind persistent homology is to track the evolution of topological features of complexes such as R (X), as the scale parameter ranges from 0 to some maximum value max ≤ ∞. For instance, in Figure 6 one can see that R0 (X) = X has 40 distinct connect components (one for each 8 point), R0.30 (X) has three connected components and R0.35 (X) has only one connected component; this will continue to be the case for every ≥ 0.35. Similarly, there are no closed loops in R0 (X) or R0.30 (X) bounding empty regions, but this changes when increases to 0.35. Indeed, R0.35 (X) has three 1-dimensional holes: the central prominent hole, and the two small ones to the left side. Notice, however, that as increases beyond 0.35 these holes will be filled by the addition of new simplices; in particular, for > 2 one has that R (X) will have only one connected component and no other topological features in higher dimensions. The family R(X) = {R (X)}≥0 is known as the Rips filtration of X, and the emergence/dissapearence of topological features in each dimension (i.e., connected components, holes, voids, etc) as changes, can be codified in what are referred to as the persistence diagrams of R(X). Specifically, for each dimension n = 0, 1, . . . (0 = connected components, 1 = holes, 2 = voids, etc) one can record the value of for which a particular n-dimensional topological feature of the Rips filtration appears (i.e. its birth time), and when it disappears (i.e. its death time). The birth-death times (b, d) ∈ R2 of n-dimensional features for R(X) form a multiset dgmn (R(X)) — i.e. a set whose elements can come with repetition — known as the n-dimensional persistence diagram of the Rips filtration on X. Since dgmn (R(X)) is just a collection of points in the region {(x, y) ∈ R2 : 0 ≤ x < y}, we will visualize it as a scatter plot. The persistence of a topological feature with birth-death times (b, d) is the quantity d − b, i.e. its lifetime. We will also include the diagonal y = x in the scatter plot in order to visually convey the persistence of each birth-death pair. In this setting, points far from the diagonal (i.e. with large persistence) represent topological features which are stable across scales and hence deemed significant, while points near the diagonal (i.e. with small persistence) are often associated with unstable features. We illustrate in Figure 7 the process of going from a point cloud to the 1-dimensional persistence diagram of its Rips filtration. Original Point Cloud 5 4 3 Time of Death 2 1 0 1 2 3 1D Persistence Diagram 4.5 4.0 2 3.5 3.0 2.5 2.0 1 1.5 1.0 0.5 0.0 5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Time of Birth Class 1 Death (d = 1.84) 5 3 2 1 0 1 2 3 4 Class 2 Birth (d = 1.8) 5 4 3 2 1 0 1 2 3 4 4 3 3 3 2 2 2 1 1 1 0 0 0 1 1 1 2 2 2 3 3 2 1 0 1 2 3 4 5 3 2 1 0 1 2 3 4 5 3 3 2 1 0 1 2 3 4 5 Class 2 Death (d = 3.55) 5 4 3 Class 1 Birth (d = 0.774) 5 3 2 1 0 1 2 3 4 5 Figure 7: From a point cloud to the 1-dimensional persistence diagram of its Rips filtration. Connected edges in the Rips filtration are drawn in blue, the birth/death of a class is indicated in red, and filled in triangles are shaded green. We remark that the computational task of determining all non-equivalent persistent homology classes of a filtered simplicial complex can, surprisingly, be reduced to computing the homology of a single simplicial complex [3, 50]. This is in fact a problem in linear algebra that can be solved via 9 elementary row and column operations on appropriate boundary matrices. The persistent homology of R SWd,τ X , and in particular its n-dimensional persistence diagrams for n = 1, 2, are the objects we will use to quantify periodicity and quasiperiodicity in a video X. Figures 8 and 9 show the persistence diagrams of the Rips filtrations, on the sliding window embeddings, for the commensurate and non-commensurate signals from Figures 1 and 2, respectively. We use fast new code from the “Ripser” software package to make persistent H2 computation feasible [3]. Figure 8: Sliding window embedding of the harmonic signal fh (left) and the n-dimensional persistence diagrams n = 1, 2 (right) of the associated Rips filtration. The sliding window embedding SWd,τ fh traces a topological circle wrapped around a 2-dimensional torus. The persistence diagram in dimension one (H1 ) shows only one birth-death pair with prominent persistence; this is consistent with a point cloud sampled around a space with the topology of a circle. Figure 9: Sliding window embedding of the quasiperiodic signal fq (left) and the n-dimensional persistence diagrams n = 1, 2 (right) of the associated Rips filtration. The sliding window embedding SWd,τ fq is dense on a 2-dimensional torus. The persistence diagram in dimension one (H1 ) shows two birth-death pairs with prominent persistence, while the persistence diagram in dimension two (H2 ) shows one prominent birth-death pair; this is consistent with a point cloud sampled around a space with the topology of a 2-dimensional torus. 3 3.1 Implementation Details Reducing Memory Requirements with SVD Suppose we have a video which has been discretely sampled at N different frames at a resolution of W × H, and we do a delay embedding with dimension d, for some arbitrary τ . Assuming 32 bit floats per grayscale value, storing the sliding window embedding requires 4W HN (d + 1) bytes. For 10 a low resolution 200 × 200 video only 10 seconds long at 30fps, using d = 30 already exceeds 1GB of memory. In what follows we will address the memory requirements and ensuing computational burden to construct and access the sliding window embedding. Indeed, constructing the Rips filtration only requires pairwise distances between different delay vectors, this enables a few optimizations. First of all, for N points in RW H , where N W H, there exists an N -dimensional linear subspace which contains them. In particular let A be the (W × H) × N matrix with each video frame along a column. Performing a singular value decomposition A = U SV T , yields a matrix U whose columns form an orthonormal basis for the aforementioned N -dimensional linear subspace. Hence, by finding the coordinates of the original frame vectors with respect to this orthogonal basis Â = U T A = U T U SV = SV (14) and using the coordinates of the columns of SV instead of the original pixels, we get a sliding window embedding of lower dimension   U T X(t)   .. (15) SWd,τ (t) =   . U T X(t + dτ ) for which kSWd,τ (t) − SWd,τ (t0 )k = kSWd,τ X(t) − SWd,τ X(t0 )k Note that SV can be computed by finding the eigenvectors of AT A; this has a cost of O(W 2 H 2 + N ) which is dominated by W 2 H 2 if W H N . In our example above, this alone reduces the memory requirements from 1GB to 10MB. Of course, this procedure is the most effective for short videos where there are actually many fewer frames than pixels, but this encompasses most of the examples in this work. In fact, the break-even point for a 200x200 30fps video is 22 minutes. A similar approach was used in the classical work on Eigenfaces [40] when computing the principal components over a set of face images. 3 3.2 Distance Computation via Diagonal Convolutions A different optimization is possible if τ = 1; that is, if delays are taken exactly on frames and no interpolation is needed. In this case, the squared Euclidean distance between SWd,1 X(i) and SWd,1 X(j) is d X \|\|SWd,1 X(i) − SWd,1 X(j)\|\|22 = \|\|X(i + m) − X(j + m)\|\|22 (16) m=0 2 DX Let be the N × N matrix of all pairwise squared Euclidean distances between frames (possibly computed with the memory optimization in Section 3.1), and let DY2 be the (N − d) × (N − d) matrix of all pairwise distances between delay frames. Then Equation 16 implies that DY2 can be obtained 2 from DX via convolution with a “rect function”, or a vector of 1s of length d + 1, over all diagonals in 2 DX (i.e. a moving average). This can be implemented in time O(N 2 ) with cumulative sums. Hence, regardless of how d is chosen, the computation and memory requirements for computing DY2 depend only on the number of frames in the video. Also, DY can simply be computed by taking the entry wise square root of DY2 , another O(N 2 ) computation. A similar scheme was used in [16] when comparing distances of 3D shape descriptors in videos of 3D meshes. Figure 10 shows self-similarity matrices on embeddings of the pendulum video with no delay and with a delay approximately matching the period. The effect of a moving average along diagonals with delay eliminates the anti-diagonals caused by the video’s mirror symmetry. Even for videos without mirror symmetries, such as a video of a running dog (Figure 11), introducing a delay brings the geometry into focus, as shown in Figure 12. 11 Pairwise Distances, Tau = 1, d = 28 Pairwise Distances, Tau = 1, d = 0 Figure 10: Self-similarity matrices DY0,1 and DY28,1 for a video of the oscillating pendulum. Bright colors indicate far distances and dark colors indicate near distances. This example clearly shows how adding a delay embedding is like performing block averaging along all diagonals of the pairwise distance matrices, and it gets rid of the mirror symmetry. Time Figure 11: An animation of a periodic video of a running dog, which, unlike an oscillating pendulum, does not have mirror symmetry in the second half of its period. Pairwise Distances, Tau = 1, d = 0 Pairwise Distances, Tau = 1, d = 18 Figure 12: Self-similarity matrices DY0,1 and DY18,1 for a video of a running dog. Even without the delay embedding (d = 0), the video frames still form a topological loop. However, a delay embedding with d = 25 cleans up the geometry and leads to a rounder loop, as seen in the resulting SSM. 3.3 Normalization A few normalization steps are needed in order to enable fair comparisons between videos with different resolutions, or which have a different range in periodic motion either spatially or in intensity. First, we perform a “point-center and sphere normalize” vector normalization which was shown in [31] to have nice theoretical properties. 12 That is, g d,τ (t) = SW SWd,τ (t) − (SWd,τ (t)T 1)1 \|\|SWd,τ (t) − (SWd,τ (t)T 1)1\|\|2 (17) where 1 is a W H(d + 1) × 1 vector of all ones. In other words, one subtracts the mean of each component of each vector, and each vector is scaled so that it has unit norm (i.e. lives on the unit sphere in RW H(d+1) ). Subtracting the mean from each component will eliminate additive linear drift on top of the periodic motion, while scaling addresses resolution / magnitude differences. Note that we can still use the memory optimization in Section 3.1, but we can no longer use the optimizations in Section 3.2 since each window is normalized independently. Moreover, in order to mitigate nonlinear drift, we implement a simple pixel-wise convolution by the derivative of a Gaussian for each pixel in the original video before applying the delay embedding: X̂i (t) = Xi (t) ∗ −at exp−t 2 /(2σ 2 ) (18) This is a pixel-wise bandpass filter which could be replaced with any other bandpass filter leveraging application specific knowledge of expected frequency bounds. This has the added advantage of reducing the number of harmonics, enabling a smaller embedding dimesion d. 3.4 Periodicity/Quasiperiodicity Scoring Once the videos are normalized to the same scale, we can score periodicity and quasiperiodicity based on the geometry of sliding window embeddings. Let dgmn be the n-dimensional persistence diagram for the Rips filtration on the sliding window embedding of a video, and define mpi (dgmn ) as the i-th largest difference d − b for (b, d) ∈ dgmn . In particular mp1 (dgmn ) = max{d − b : (b, d) ∈ dgmn } and mpi (dgmn ) ≥ mpi+1 (dgmn ). We propose the following scores: 1. Periodicity Score (PS) 1 P S = √ mp1 (dgm1 ) 3 (19) Like [31], we exploit the fact that for the Rips filtration on S 1 , the √ 1-dimensional persistence diagram has only one prominent birth-death pair with coordinates 0, 3 . Since this is the limit shape of a normalized perfectly periodic sliding window video, the periodicity score is between 0 (not periodic) and 1 (perfectly periodic). 2. Quasiperiodicity Score (QPS) r QP S = mp2 (dgm1 )mp1 (dgm2 ) 3 (20) This score is designed with the torus in mind. We score based on the second largest 1D persistence times the largest 2D persistence, since we want a shape that has two core circles and encloses a void to get a large score. Based on the Künneth theorem of homology, the 2-cycle (void) should die the moment the smallest 1-cycle dies. 3. Modified Periodicity Score (MPS) 1 M P S = √ (mp1 (dgm1 ) − mp2 (dgm1 )) 3 (21) We design a modified periodicity score which should be lower for quasiperiodic videos, than what the original periodicity score would yield. 13 Note that we use Z3 field coefficients for all persistent homology computations since, as shown by [31], this works better for periodic signals with strong harmonics. Before we embark on experiments, let us explore the choice of two crucial parameters for the sliding window embedding: the delay τ > 0 and the dimension d ∈ N. In practice we determine an equivalent pair of parameters: the dimension d and the window size dτ . 3.5 Dimension and Window Size Takens’ embedding theorem is one of the most fundamental results in the theory of dynamical systems [38]. In short, it contends that (under appropriate hypotheses) there exists an integer D, so that for all d ≥ D and generic τ > 0 the sliding window embedding SWd,τ X reconstructs the state space of the underlying dynamics witnessed by the signal X. One common strategy for determining a minimal such D is the false nearest-neighbors scheme [21]. The idea is to keep track of the k-th nearest neighbors of each point in the delay embedding, and if they change as d is increased, then the prior estimates for d were too low. This algorithm was used in recent work on video dynamics [42], for instance. Even if we can estimate d, however, how does one choose the delay τ ? As shown in [31], the sliding window embedding of a periodic signals is roundest (i.e. so that the periodicity score P S is maximized) when the window size, dτ , satisfies the following relation: d πk (22) dτ = L d+1 Here L is number of periods that the signal has in [0, 2π] and k ∈ N. To verify this experimentally, we show in Figure 13 how the periodicity score P S changes as a function of window size for the pendulum video, and how the choice of window size from Equation 22 maximizes P S. To generate this figure we fixed a sufficiently large d and varied τ . Let us now describe the general approach: Given a video we perform a period-length estimation step (see section 3.6 next), which results in a positive real number d `. For a given d ∈ N large enough we let τ > 0 be so that dτ = ` · d+1 . Figure 13: Varying the window size, dτ , in a delay embedding of the synthetic pendulum video, which has a period length around 25 frames. Red dashed lines are drawn at the window lengths that would be expected to maximize roundness of the embedding for that period length based on theory in [31]. 3.6 Fundamental Frequency Estimation Though Figure 13 suggests robustness to window size as long as the window is more than half of the period, we may not know what that is in practice. To automate window size choices, we do a coarse estimate using fundamental frequency estimation techniques on a 1D surrogate signal. To get a 1D signal, we extract the first coordinate of diffusion maps [4] using 10% nearest neighbors on the 14 raw video frames (no delay) after taking a smoothed time derivative. Note that a similar diffusionbased method was also used in recent work by [46] to analyze the frequency spectrum of a video of an oscillating 2 pendulum + spring system in a quasiperiodic state. Once we have the diffusion time series, we then apply the normalized autocorrelation method of [25] to estimate the fundamental frequency. In particular, given a discrete signal x of length N , define the autocorrelation as rt (τ ) = t+N −1−τ X xj xj+τ (23) j=t However, as observed by [7], a more robust function for detecting periodicities is the squared difference function t+N −1−τ X dt (τ ) = (xj − xj+τ )2 (24) j=t which can be rewritten as dt (τ ) = mt (τ ) − 2rt (τ ) where mt (τ ) = t+N −1−τ X (x2j + x2j+τ ) (25) j=t Finally, [25] suggest normalizing this function to the range [−1, 1] to control for window size and to have an interpretation akin to a Pearson correlation coefficient: nt (τ ) = 1 − 2rt (τ ) mt (τ ) − 2rt (τ ) = mt (τ ) mt (τ ) (26) The fundamental frequency is then the inverse period of the largest peak in nt which is to the right of a zero crossing. The zero crossing condition helps prevent an offset of 0 from being the largest peak. Defining the normalized autocorrelation as in Equation 26 has the added advantage that the value of nt (τ ) at the peak can be used to score periodicity, which the authors call clarity. Values closer to 1 indicate more perfect periodicities. This technique will sometimes pick integer multiples of the period, so we multiply nt (τ ) by a slowly decaying envelope which is 1 for 0 lag and 0.9 for the maximum lag to emphasize smaller periods. Figure 14 shows the result of this algorithm on a periodic video, and Figure 15 shows the algorithm on an irregular video. Figure 14: Diffusion maps + normalized autocorrelation fundamental frequency estimation on a periodic vocal folds video (Section 4.3). The chosen period length is 32, as indicated by the red dot over the peak. This matches with the visually inspected period length. 15 Figure 15: Diffusion maps + normalized autocorrelation fundamental frequency estimation on a video of vocal folds with irregular oscillations (Section 4.3). 4 Experimental Evaluation Next we evaluate the effectiveness of the proposed (Modified) Periodicity and Quasiperiodicity scores on three different tasks. First, we provide estimates of accuracy for the binary classifications periodic/notperiodic or quasiperiodic/not-quasiperiodic in the presence of several noise models and noise levels. The results illustrate the robustness of our method. Second, we quantify the quality of periodicity rankings from machine scores, as compared to those generated by human subjects. In a nutshell, and after comparing with several periodicity quantification algorithms, our approach is shown to be the most closely aligned with the perception of human subjects. Third, we demonstrate that our methodology can be used to automatically detect the physiological manifestations of certain speech pathologies (e.g., normal vs. biphonation), directly from high-speed videos of vibrating vocal folds. 4.1 Classification Under Varying Noise Levels/Models As shown empirically in [8], a common source of noise in videos comes from camera shake (blur); this is captured by point spread functions resembling directed random walks [8, Figure 1] and the amount of blur (i.e. noise level) is controlled by the extent in pixels of the walk. Other sources are additive white Gaussian noise (awgn), controlled by the standard deviation of the Gaussian kernel, and MPEG bit errors quantified by the percentage of corrupted information. Figure 16 shows examples of these noise types. For classification purposes we use three main recurrence classes. Three types of periodic videos (True periodic, TP): an oscillating pendulum, a bird flapping its wings, and an animation of a beating heart. Two types of quasiperiodic videos (True quasiperiodic, TQ): one showing two solid disks which oscillate sideways at non-commensurate rates, and the second showing two stationary Gaussian pulses with amplitudes non-commensurately modulated by cosine functions. Two videos without significant recurrence (True non-recurrent, TN): a video of a car driving past a landscape, and a video of an explosion. Each one of these seven videos is then corrupted by the three noise models at three different noise levels (blur = 20, 40, 80, awgn σ = 1, 2, 3, bit error = 5, 10, 20%) as follows: given a particular video, a noise model and noise level, 600 instances are generated by sampling noise independently at random. Results: We report in Table 1 the area under the Receiver Operating Characteristic (ROC) curve, or AUROC for short, for the classification task TP vs. TN (resp. TQ vs. TN) and binary classifier furnished by Periodicity (resp. Quasiperiodicity) Score. For instance, for the Blur noise model with noise level of 80 × 80 pixels, the AUROC from using the Periodicity Score to classify the 600 instances of the Heartbeat video as periodic, and the 600 16 (b) 20 × 20 blur (a) Original (c) awgn σ = 2 (d) 5% Bit Err Figure 16: The results of applying motion blur, additive white Gaussian noise, and MPEG bit corruption to a video frame. Table 1: AUROC values for different levels of noise, from the binary classification task: periodic (bird flapping, heart beating, pendulum) vs. non-recurrent (driving - left subcell, explosions - right subcell) based on the periodicity score (Equation 19). Also two synthetic quasiperiodic videos (sideways disks, modulated pulses) are compared to the same two non-recurrent videos based on the quasiperiodicity score (Equation 20). Awgn Awgn σ=1 σ=2 Awgn σ=3 Bit Err 5% Bit Err 10% 1 1 1 1 1 1 0.97 1 0.91 0.94 1 1 1 1 1 0.94 0.84 0.85 0.43 0.68 1 1 1 1 1 1 1 Blur 20 Bird Flapping Heart Beat Pendulum Blur 40 Blur 80 1 1 1 1 1 1 1 1 QuasiPeriodic Disks 1 0.85 QuasiPeriodic Pulses 1 0.9 1 1 1 Bit Err 15% 0.92 0.75 0.79 0.9 0.89 1 0.91 0.98 0.87 0.56 0.62 1 1 0.99 0.98 0.97 1 0.83 0.75 0.39 1 1 1 1 1 1 1 0.92 0.99 0.93 0.82 0.82 1 0.87 1 1 1 1 1 1 0.96 0.99 0.95 0.95 0.95 1 0.83 1 instances of the Driving video as not periodic is 0.91. Similarly, for the MPEG bit corruption model with 5% of bit error, the AUROC from using the Quasiperiodicity score to classify the 600 instances of the Quasiperiodic Sideways Disks as quasiperiodic and the 600 instances of the Explosions video as not quasiperiodic is 0.92. To put these numbers in perspective, AUROC = 1 is associated with a perfect classifier and AUROC = 0.5 corresponds to classification by a random coin flip. Overall, the type of noise that degrades performance the most across videos is the bit error, which makes sense, since this has the effect of randomly freezing, corrupting, or even deleting frames, which all interrupt periodicity. The blur noise also affects videos where the range of motion is small. The pendulum video, for instance, only moves over a range of 60 pixels at the most extreme end, so an 80x80 pixel blur almost completely obscures the motion. 4.2 Comparing Human and Machine Periodicity Rankings Next we quantify the extent to which rankings obtained from our periodicity score (Equation 19), as well as three other methods, agree with how humans rank videos by periodicity. The starting point is a dataset of 20 different creative commons videos, each 5 seconds long at 30 frames per second. Some 17 videos appear periodic, such as a person waving hands, a beating heart, and spinning carnival rides. Some of them appear nonperiodic, such as explosions, a traffic cam, and drone view of a boat sailing. And some of them are in between, such as the pendulum video with simulated camera shake. It is known that humans are notoriously bad at generating globally consistent rankings of sets with more than 5 or 7 elements [27]. However, when it comes to binary comparisons of the type “should A be ranked higher than B?” few systems are as effective as human perception, specially for the identification of recurrent patterns in visual stimuli. We will leverage this to generate a globally consistent ranking of the 20 videos in our initial data set. We use Amazon’s Mechanical Turk (AMT) [5] to present each pair of videos in the set of 20, 20 2 = 190, each to three different users, for a total of 570 pairwise rankings. 15 unique AMT workers contributed to our experiment, using an interface as the one shown in Figure 17. Figure 17: The interface that humans are given on AMT for pairwise ranking videos by periodicity. In order to aggregate this information into a global ranking which is as consistent as possible with the pairwise comparisons, we implement a technique known as Hodge rank aggregation [18]. Hodge rank aggregation finds the closest consistent ranking to a set of preferences, in a least squares sense. More precisely, given a set of objects X, and given a set of comparisons P ⊂ X × X, we seek a scalar function s on all of the objects that minimizes the following sum X \|vab − (sb − sa )\|2 (27) (a,b)∈P where vab is a real number which is positive if b is ranked higher than a and negative otherwise. Thus, s is a function whose discrete gradient best matches the set of preferences with respect to an L2 norm. Note that the preferences that we feed the algorithm are based on the pairwise rankings returned from AMT. If video b is greater than video a, then we assign vab = 1, or -1 otherwise. Since we have 3 rankings for each video, we actually assign weights of +3, +1, -1, or -3. The +/- 3 are if all rankings agree in one direction, and the +/- 1 are if one of the rankings disagrees with the other two. Figure 18 shows a histogram of all of the weighted scores from users on AMT. They are mostly in agreement, though there are a few +/- 1 scores. As comparison to the human scores, we use three different classes of techniques for machine ranking of periodicity. Sliding Windows (SW): We sort the videos in decreasing order of Periodicity Score (Equation 19). We fix the window size at 20 frames and the embedding dimension at 20 frames (which is enough to capture 10 strong harmonics). We also apply a time derivative of width 10 to every frame. 18 80 Histogram of Weighted Pairwise Turk Scores Counts 60 40 20 0 -4 -3 -2 -1 0 1 2 3 4 Score Figure 18: The histogram of scores that the workers on AMT gave to all pairwise videos. Cutler-Davis [6]: The authors of this work present two different techniques to quantify periodicity from a self-similarity matrix (SSM) of video frames. The first is a frequency domain technique based on the peak of the average power spectral density over all columns (rows) of the SSM after linearly de-trending and applying a Hann window. To turn this into a continuous score, we report the ratio of the peak minus the mean over the standard deviation. This method will be referred to as Frequency Score. As the authors warn, the frequency peak method has a high susceptibility to false positives. This motivated the design of a more robust technique in [6], which works by finding peaks in the 2D normalized autocorrelation of the Gaussian smoothed SSMs. For videos with mirror symmetry, the peaks will lie on a diamond lattice, while for videos without mirror symmetry, they will lie on a square lattice. After peak finding within neighborhoods, one simply searches over all possible lattices at all possible widths to find the best match with the peaks. Since each lattice is centered at the autocorrelation point (0, 0), no translational checks are necessary. To turn this into a continuous score, let E be the sum of Euclidean distances of the matched peaks in the autocorrelation image to the best fit lattice, let r1 be the proportion of lattice points that have been matched, and let r2 be the proportion of peaks which have been matched to a lattice point. Then we give the final periodicity score as CDscore = (1 + E/r1 )/(r1 r2 )3 (28) A lattice which fits the peaks perfectly (r1 = 1) with no error (E = 0) and no false positive peaks (r2 = 1) will have a score of 1, and any video which fails to have a perfectly matched lattice will have a score greater than 1. Hence, we sort in increasing order of the score to get a ranking. As we will show, this technique agrees the second best with humans after our Periodicity Score ranking. One of the main drawbacks is numerical stability of finding maxes in non-isolated critical points around nearly diagonal regions in square lattices, which will erroneously inflate the score. Also, the lattice searching only occurs over an integer grid, but there may be periods that aren’t integer number of frames, so there will always be a nonzero E for such videos. By contrast, our sliding window scheme can work for any real valued period length. Diffusion Maps + Normalized Autocorrelation “Clarity”: Finally, we apply the technique from Section 3.6 to get an autocorrelation function, and we report the value of the maximum peak of the normalized autocorrelation to the right of a zero crossing, referred to as “clarity” by [25]. Values closer to 1 indicate more perfect repetitions, so we sort in descending order of clarity to get a ranking. Figure 19 shows an example of these three different techniques on a periodic video. There is a dot which rises above the diagonal in the persistence diagram, a lattice is found which nearly matches the critical points in the autocorrelation image, and autocorrelation function on diffusion maps has a nice peak. 19 First Diﬀusion Coordinate Figure 19: An example of the SW score (top), the clarity score (bottom left), and the CDscore (bottom right, matched peaks in green and lattice in blue), on a periodic video of a man waving his arms from the KTH dataset ( [36]). By contrast, for a nonperiodic video (Figure 20), there is hardly any persistent homology, there is no well matching lattice, and the first diffusion coordinate has no apparent periodicities. First Diﬀusion Coordinate Figure 20: An example of the SW score (top), the clarity score (bottom left), and the CDscore (bottom left, matched peaks in green and lattice in blue), on a video of an explosion, which is nonperiodic. Results: Once we have the global human rankings and the global machine rankings, we can compare them using the Kendall τ score [20]. Given a set of objects N objects X and two total orders >1 and >2 , where > (xa , xb ) = 1 if xa > xb and > (xa , xb ) = −1 if xa < xb , the Kendall τ score is defined as 20 τ= X 1 (>1 (xi , xj ))(>2 (xi , xj )) N (N − 1)/2 i<j (29) For two rankings which agree exactly, the Kendall τ score will be 1. For two rankings which are exactly the reverse of each other, the Kendall τ score will be -1. In this way, it analogous to a Pearson correlation between rankings. Table 2: The Kendall τ scores between all of the machine rankings and the Hodge aggregated human rankings. Human SW+TDA Freq [6] CDscore [6] Clarity [7] Human 1 0.663 -0.295 0.347 0.284 SW+TDA 0.663 1 -0.316 0.221 0.516 Freq [6] -0.295 -0.316 1 -0.0842 -0.189 CDscore [6] 0.347 0.221 -0.0842 1 0.411 Clarity [7] 0.284 0.516 -0.189 0.411 1 Table 3: Average runtimes, in milliseconds, per video for all of the algorithms SW+TDA 3101ms Freq [6] 73ms CDscore [6] 176ms Clarity [7] 154ms Table 2 shows the Kendall τ scores between all of the different machine rankings and the human rankings. Our sliding window video methodology (SW+TDA) agrees with the human ranking more than any other pair of ranking types. The second most similar are the SW and the diffusion clarity, which is noteworthy as they are both geometric techniques. Table 3 also shows the average run times, in milliseconds, of the different algorithms on each video on our machine. This does highlight one potential drawback of our technique, since TDA algorithms tend to be computationally intensive. However, at this scale (videos with at most several hundred frames), performance is reasonable. 4.3 Periodicity And Biphonation in High Speed Videos of Vocal Folds In this final task we apply our methodology to a real world problem of interest in medicine. We show that our method can automatically detect certain types of voice pathologies from high-speed glottography, or high speed videos (4000 fps) of the left and right vocal folds in the human vocal tract [9, 44, 45]. In particular, we detect and differentiate quasiperiodicity from periodicity by using our geometric sliding window pipeline. Quasiperiodicity is a special case of what is referred to as “biphonation” in the biological context, where nonlinear phenomena cause a physical process to bifurcate into two different periodic modes, often during a transition to chaotic behavior [15]. The torus structure we sketched in Figure 9 has long been recognized in this context [28], but we provide a novel way of quantifying it. Similar phenomena exist in audio [14, 15], but the main reason for studying laryngeal high speed video is understanding the biomechanical underpinnings of what is perceived in the voice. In particular, this understanding can potentially lead to practical corrective therapies and surgical interventions. On the other hand, the presence of biphonation in sound is not necessarily the result of a physiological phenomenon; it has been argued that it may come about as the result of changes in states of arousal [?]. In contrast with our work, the existing literature on video-based techniques usually employs an inherently Lagrangian approach, where different points on the left and right vocal folds are tracked, and coordinates of these points are analyzed as 1D time series (e.g. [13, 24, 28, 35], [26]). This is a natural approach, since those are the pixels where all of the important signal resides, and wellunderstood 1D signal processing technique can be used. However, edge detectors often require tuning, 21 and they can suddenly fail when the vocal folds close [24]. In our technique, we give up the ability to localize the anomalies (left/right, anterior/posterior) since we are not tracking them, but in return we do virtually no preprocessing, and our technique is domain independent. Results: We use a collection of 7 high-speed videos for this analysis, drawn from a variety of different sources [49], [26], [28], [13]. There are two videos which correspond to “normal” periodic vocal folds, three which correspond to biphonation [28], and two which correspond to irregular motion5 . We manually extracted 400 frames per video (100milliseconds) and autotuned the window size based on autocorrelation of 1D diffusion maps (Section 3.6). We then chose an appropriate τ and chose a time spacing so that each point cloud would have 600 points. As shown in Table 4, our technique is able to differentiate between the four classes. We also show PCA and persistence diagrams for one example for each class. In Figure 21, we see what appears to be a loop in PCA, and one strong 1D persistent dot confirms this. In Figure 22, we see a prominent torus in the persistence diagram. In Figure 23, we don’t see any prominent structures in the persistence diagram, even though PCA looks like it could be a loop or a torus. Note, however, that PCA only preserves 13.7% of the variance in the signal, which is why high dimensional techniques are important to draw quantitative conclusions. Table 4: Results of our sliding window pipeline on videos of periodic vocal folds, biphonation, and irregularities. We give the max persistence periodicity score (PS), the modified periodicity score (MPS), the harmonic score (HS), and quasiperiodic score (QPS) presented in Section 3.4. We also show the window size (Win) that the autocorrelation technique in Section 3.6 gives. We have bolded the top three MPS and QPS scores across all videos. The max modified periodic scores include the two periodic videos and one of the biphonation videos. The max quasiperiodic scores are all of the biphonation videos, which means the one with a high periodicity score could be ruled out of the periodicity category. Video Name Periodic 1 ( [13]) Periodic 2 ( [26], Figure 21) Biphonation 1 ( [28]) Biphonation 2 ( [28]) Biphonation 3 ( [28], Figure 22) Mucus Perturbed Periodic ( [49]) Irregular ( [13], Figure 23) 5 Win 16 32 53 42 67 94 232 PS 0.816 0.601 0.638 0.703 0.515 0.028 0.18 MPS 0.789 0.533 0.294 0.583 0.076 0.019 0.097 QPS 0.011 0.009 0.292 0.116 0.426 0.004 0.04 Discussion We have shown in this work how applying sliding window embeddings to videos can be used to translate properties of the underlying dynamics into geometric features of the resulting point cloud representation. Moreover, we also showed how topological/geometric tools such as persistence homology can be leveraged to quantify the geometry of these embeddings. The pipeline was evaluated extensively showing robustness to several noise models, high quality in the produced periodicity rankings and applicability to the study of speech conditions form high-speed video data. Moving forward, an interesting avenue related to medical applications is the difference between biphonation which occurs from quasiperiodic modes and biphonation which occurs from harmonic modes. [31] shows that Z3 field coefficients can be used to indicate the presence of a strong harmonic, so we believe a geometric approach is possible. This could be used, for example, to differentiate between subharmonic anomalies and quasiperiodic transitions [44]. 5 Please refer to supplementary material for an example video from each of these three classes 22 Figure 21: Video frames and sliding window statistics on a video of vocal folds undergoing normal periodic vibrations [26]. One strong loop is visible in PCA and in the persistence diagrams Figure 22: Video frames and sliding window statistics on a video of vocal folds undergoing biphonation, courtesy of Juergen Neubauer [28]. 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arXiv:1802.04834v1 [cs.LG] 13 Feb 2018 Challenging Images For Minds and Machines Amir Rosenfeld, John K. Tsotsos Department of Electrical Engineering and Computer Science York University, Toronto, ON, Canada [email protected],[email protected] February 15, 2018 Abstract There is no denying the tremendous leap in the performance of machine learning methods in the past half-decade. Some might even say that specific sub-fields in pattern recognition, such as machine-vision, are as good as solved, reaching human and super-human levels. Arguably, lack of training data and computation power are all that stand between us and solving the remaining ones. In this position paper we underline cases in vision which are challenging to machines and even to human observers. This is to show limitations of contemporary models that are hard to ameliorate by following the current trend to increase training data, network capacity or computational power. Moreover, we claim that attempting to do so is in principle a suboptimal approach. We provide a taster of such examples in hope to encourage and challenge the machine learning community to develop new directions to solve the said difficulties. 1 Introduction Once only known to a few outside of academia, machine-learning has become ubiquitous in both popular media and in the industry. Superhuman capabilities are now being gradually recorded in various fields: in the game of GO, ([1, 2]), in face verification ([3, 4]), image categorization ([5]) and even in logical reasoning in simple scenes ([6, 7, 8]). Most current leading methods involve some variant of deep learning. Consequentially, they require large amounts of hand-labeled data (with the exception of [2] - which used self-play to gain experience). This has elicited a data-hungry era, with increasingly large-scale datasets painstakingly labeled for object classification/detection/segmentation, image annotation, visual question-answering, and pose estimation ([9, 10, 11, 12, 13]) to name a few. This is accompanied by a growing demand for computational power. We bring forward challenges in vision which do not seem to be solved by current methods - and more importantly - by current popular methodologies, meaning that neither additional data, nor added computational power will be the drivers of the solution. 1 Figure 1: A children’s puzzle where the goal is to find six hidden words: Book, words, story, pages, read, novel. For a machine this is far from child’s play. Could this be solved by providing a million similar examples to a deep-learning system? Does a human need such training? Related Work Imbalanced or Small Data: datasets tend to be naturally imbalanced, and there is a long history of suggested remedies ([14, 15, 16]). Handling lack of training data has also been treated by attempting to use web-scale data of lesser quality than handannotated dataset [17], simulating data [cite data for cars, text recognition in the wild, captcha]. Transfer Learning: reusing features of networks trained on large is a useful starting point (cf [18]) One-Shot-Learning: attempting to reduce the number of required training example, in extreme cases to one or even zero examples ([19]); DeepLearning Failures: recently, some simple cases where deep learning fails to work as one would possibly expect were introduced, along with theoretical justifications ([20]). 2 Challenging Cases We present two examples and then discuss them. They have a few common characteristics: humans are able to solve them on the first “encounter” - despite not having seen any such images before. Incidentally - but not critically - the two examples are from the domain of visual text recognition. Moreover, though humans know how to recognize text as seen in regular textbooks, street-signs, etc, the text in these images is either hidden, rendered, or distorted in an uncharacteristic manner. Children’s games: the first case is well exemplified by a child’s game, hidden word puzzles. The goal is to find hidden words in an image. Fig. 1 shows an arbitrarily selected example. For a human observer this is a solvable puzzle, though it may take a few minutes to complete. We applied two state-of-the-art methods for text recognition 2 Sub Image [21] “sned” “vvoz” “novees” “teg” [22] “score” ∅ ∅ ∅ Table 1: Text detected by two state-of-the-art scene-text recognition methods applied to sub-images of a children’s puzzle. ∅ means no text was detected by the method (images scaled to fit figure). Figure 2: Variants of textual CAPTCHA. Captchas are becoming increasingly difficult (reproduced from [24]) in the wild with available code ([21]) or an on line-demo ([22]1 ) on the image in Fig. 1. As this did not work immediately, we focused on the word “NOVEL” (the “N” is below the forearm of the left person, ending with an “L” below his foot), by cropping it an rotating so the text is level, cropping more tightly , and even cropping only the letter “L”. See Table 1 for the corresponding sub-images (including the entire image at the top row) and the results output by the two methods. This is by no means a systematic test and some may even claim that it isn’t fair and they would be right: these systems were not trained on such images; [21] was only trained on a photo-realistic dataset of 8 million synthetic training images, and [22] was only trained on tens of thousands of images from coco-text ([23]), or used powerful pre-trained networks where training data was less available. CAPTCHA: a well-known mechanism to thwart automated misuse of websites by distinguishing between humans and machines ([25]). Textual captchas involve presenting an image of text which has to be read and written by the user. We focus on this type of captcha, though others exist ([26]). The introduction of captchas immediately triggered the invention of new automatic ways to break them ([27]), which eventually sparked an “arms race” between increasingly complex captchas and correspondingly powerful automated methods ([28]). This caused a state where on one-hand the best leading textual captcha-solution methods involve training DNN’s over data with similar distortion characteristics as the desired types of captcha - though still these systems have limited success rates (at times less than 50%) - and on the other hand the level of distortion has become such that humans have a hard-time solving some of them. 3 Machines vs Humans as Supervised Learners One can rule out the suggested examples by saying that they are simply out-of-sample datapoints on behalf of a statistical learner’s perspective. Yet it seems that with what1 http://east.zxytim.com 3 ever supervision human-beings receive - they are usually able to solve them despite not being especially exposed to this kind of stimulus. Moreover, precisely these kinds of images are used routinely in human IQ testing, so they are a universally accepted indicator for human performance. If these examples may seem esoteric, we can revert to more common cases: as a child, how often is one exposed to bounding boxes of objects? How often to delineations of objects with precise segmentation masks? How often to pose-configurations, facial and bodily key-points, and dense-meshes of 3D objects overlayed on their field of view ([13])? More critically, for how many different object types does this happen (if any), for how many different instances, with what level of precision of annotation, and in how many modalities? The granularity of visual supervision given to machines seems to be much finer than that given to humans. As for the amount of directly supervised data, it does not seem to really be the main limiting factor; as already noted several times, performance either saturates with training data ([29, 30]) or at best grows logarithmically ([17, 31], increasing mAP from 53% to 58% when growing from 10M to 300M examples) making the solution of more data for better performance simply impractical - even for those with the most resources. And this is for “common” problems, such as object detection. Humans who only ever read street-signs and textbooks are able to solve captchas of various kinds without any special training on their first encounter with them. The same is true for the “picture puzzles” mentioned above, as it is for other cases not mentioned here. We do not claim that humans are not subject to supervised learning in their early life, and in later stages. On the contrary, supervisory signals arise from multiple sources: caretakers who provide supervisory signals by teaching, “internal supervision” provided by innate biases ([32]) and finally rewards stemming from results of behaviour, such as suffering pain from hitting an object. But any such supervision is interspersed within a vast, continuous stream of unsupervised data, most of which does not have an easily measurable supervisory affect on the observer. There is something fundamentally different about the way humans construct or use internal representations, enabling them to reason about and solve new patternrecognition tasks. We hypothesize that these are approached by generating procedures of a compositional nature when presented with a novel - or known - task (as suggested by the Visual Routines of [33] or the Cognitive Programs of [34]. 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Some Theory for Ordinal Embedding arXiv:1501.02861v2 [math.ST] 4 May 2016 Ery Arias-Castro∗ Abstract Motivated by recent work on ordinal embedding (Kleindessner and von Luxburg, 2014), we derive large sample consistency results and rates of convergence for the problem of embedding points based on triple or quadruple distance comparisons. We also consider a variant of this problem where only local comparisons are provided. Finally, inspired by (Jamieson and Nowak, 2011), we bound the number of such comparisons needed to achieve consistency. Keywords: ordinal embedding, non-metric multidimensional scaling (MDS), dissimilarity comparisons, landmark multidimensional scaling. 1 Introduction The problem of ordinal embedding, also called non-metric multidimensional scaling (Borg and Groenen, 2005), consists of finding an embedding of a set of items based on pairwise distance comparisons. Specifically, suppose that δij ≥ 0 is some dissimilarity measure between items i, j ∈ [n] ∶= {1, . . . , n}. We assume that δii = 0 and δij = δji for all i, j ∈ [n]. These dissimilarities are either directly available but assumed to lack meaning except for their relative magnitudes, or only available via comparisons with some other dissimilarities, meaning that we are only provided with a subset C ⊂ [n]4 such that δij < δkℓ , ∀(i, j, k, ℓ) ∈ C. (1) Note that the latter setting encompasses the former. Given C and a dimension d, the goal is to embed the items as points p1 , . . . , pn ∈ Rd in a way that is compatible with the available information, specifically δij < δkℓ ⇒ ∥pi − pj ∥ ≤ ∥pk − pℓ ∥, ∀(i, j, k, ℓ) ∈ C, (2) where ∥⋅∥ denotes the Euclidean norm. The two most common situations are when all the quadruple comparisons are available, meaning C = [n]4 , or all triple comparisons are available, meaning C = {(i, j, i, k) ∶ i, j, k ∈ [n]}, which can be identified with [n]3 . This problem has a long history surveyed in (Young and Hamer, 1987), with pioneering contributions from Shepard (1962a,b) and Kruskal (1964). The main question we tackle here is that of consistency. Suppose that the items are in fact points x1 , . . . , xn ∈ Rd and δij = ∥xi −xj ∥. (When the δij ’s are available, suppose that δij = g(∥xi −xj ∥) where g is an unknown increasing function.) Provided with a subset C = Cn of dissimilarity comparisons as in (2), is it possible to reconstruct the original points in the large-sample limit n → ∞? Clearly, the reconstruction can only be up to a similarity transformation — that is, a transformation f ∶ Rd ↦ Rd such that, for some λ > 0, ∥f (x) − f (y)∥ = λ∥x − y∥ for all x, y ∈ Rd , or equivalently, of the form f (x) = λR(x) + b where R is an orthogonal transformation and b is a constant vector — since such ∗ Department of Mathematics, University of California, San Diego, USA 1 2 a transformation leaves the distance comparisons unchanged. This question is at the foundation of non-metric multidimensional scaling. Early work only addressed the continuous case, where the x’s span a whole convex subset U ⊂ Rd . In that setting, the goal becomes to characterize isotonic functions on U , that is, functions f ∶ U ↦ Rd satisfying ∥x − y∥ < ∥x′ − y ′ ∥ ⇒ ∥f (x) − f (y)∥ ≤ ∥f (x′ ) − f (y ′ )∥, ∀x, y, x′ , y ′ ∈ U. (3) Shepard (1966) argues that such functions must be similarities, and cites earlier work (Aumann and Kruskal, 1958; Suppes and Winet, 1955) dealing with the case d = 1. Only recently has the finite sample case been formally considered. Indeed, Kleindessner and von Luxburg (2014) prove a consistency result, showing that if x1 , . . . , xn ∈ U ⊂ Rd , where U is a bounded, connected, and open subset of Rd satisfying some additional conditions — for example, a finite union of open balls — and C = [n]4 , then in the large sample limit with x1 , . . . , xn becoming dense in U , it is possible to recover the x’s up to a similarity transformation. (Note that U is then uniquely defined as the interior of {xi ∶ i ≥ 1}.) We note that Kleindessner and von Luxburg (2014) focus on the strictly isotonic case, where the second inequality in (3) is strict. Our first contribution is an extension of this consistency result for quadruple learning to triple learning where C = [n]3 . In the process, we greatly simplify the arguments of Kleindessner and von Luxburg (2014) and weaken the conditions on the sampling domain U . We note that Terada and Von Luxburg (2014) have partially solved this problem by a reduction to the problem of embedding a nearest-neighbor graph. However, their arguments are based on an apparently incomplete proof in (Von Luxburg and Alamgir, 2013), which is itself based on a rather sophisticated approach. Our proofs are comparatively much simpler and direct. Our second contribution is to provide rates of convergence, a problem left open by Kleindessner and von Luxburg (2014). In the context of quadruple learning, we obtain a rate in O(εn ), where εn is the Hausdorff distance between the underlying sample {x1 , . . . , xn } and U , meaning, εn ∶= supx∈U mini∈[n] ∥x − xi ∥. This is the first convergence rate for exact ordinal embedding that we know of. (We are not able to obtain the same rate in the context of triple learning.) Compared to establishing consistency, the proof is much more involved. The last decade has seen a surge of interest in ordinal embedding, motivated by applications to recommender systems and large-scale psychometric studies made available via the internet, for example, databases for music artists similarity (Ellis et al., 2002; McFee and Lanckriet, 2011). Sensor localization (Nhat et al., 2008) is another possible application. Modern datasets being large, all quadruple or triple comparisons are rarely available, motivating the proposal of embedding methods based on a sparse set of comparisons (Agarwal et al., 2007; Borg and Groenen, 2005; Jamieson and Nowak, 2011; Terada and Von Luxburg, 2014). Terada and Von Luxburg (2014) study what they call local ordinal embedding, which they define as the problem of embedding an unweighted K-nearest neighbor (K-NN) graph. With our notation, this is the situation where C = {(i, j, k) ∶ δij ≤ δi(K) < δik }, δi(K) being the dissimilarity between item i and its Kth nearestneighbor. Terada and Von Luxburg (2014) argue that, when the items are points x1 , . . . , xn sampled from a smooth density on a bounded, connected, convex, and open subset U ⊂ Rd with smooth boundary, then K = Kn ≫ n2/(2+d) (log n)d/(2+d) is enough for consistency. Our third contribution is to consider the related situation where C = {(i, j, k, ℓ) ∶ δij < δkℓ and max(δij , δik , δiℓ ) ≤ δi(K) }, which provides us with the K-NN graph and also all the quadruple √ comparisons between the nearest neighbors. In this setting, we are only able to show that Kn ≫ n log n is enough. Beyond local designs, which may not be feasible in some settings, Jamieson and Nowak (2011) consider the problem of adaptively (i.e., sequentially) selecting triple comparisons in order to minimize the number of such comparisons and yet deduce all the other triple comparisons. They 3 consider a few methods, among which a non-metric version of the landmark MDS method of De Silva and Tenenbaum (2004). Less ambitious is the problem of selecting few comparisons in order to consistently embed the items when these are points in a Euclidean space. Our fourth contribution is to show that one can obtain a consistent embedding with a landmark design based on an n queries, where an is any diverging sequence. Moreover, the embedding can be computed in (expected) time ζ(an ) n, for some function ζ ∶ R+ ↦ R+ . The rest of the paper is organized as follows. In Section 2, we state our theoretical results and prove the simpler ones. We then gather the remaining proofs in Section 3. Section 4 concludes the paper with a short discussion. 2 Theory In this section we present our theoretical findings. Most proofs are gathered in Section 3. We already defined isotonic functions in (3). Following (Kleindessner and von Luxburg, 2014), we say that a function f ∶ U ⊂ Rd ↦ Rd is weakly isotonic if ∥x − y∥ < ∥x − z∥ ⇒ ∥f (x) − f (y)∥ ≤ ∥f (x) − f (z)∥, ∀x, y, z ∈ U. (4) Obviously, if a function is isotonic (3), then it is weakly isotonic (4). Weak isotonicity is in fact not much weaker than isotonicity. Indeed, let P be a property (e.g., ‘isotonic’), and say that a function f ∶ U ⊂ Rd ↦ Rd has the property P locally if for each x ∈ U there is r > 0 such that f has property P on B(x, r) ∩ U , where B(x, r) denotes the open ball with center x and radius r. Lemma 1. Any locally weakly isotonic function on an open U is also locally isotonic on U . Proof. This is an immediate consequence of (Kleindessner and von Luxburg, 2014, Lem 6), which implies that a weakly isotonic function on B(x, r) is isotonic on B(x, r/4). Suppose we have data points x1 , . . . , xn ∈ Rd . Define Ωn = {x1 , . . . , xn }, Ω = ⋃ Ωn = {xn ∶ n ≥ 1}. (5) n≥1 Let δij = ∥xi − xj ∥ and suppose that we are only provided with a subset Cn ⊂ [n]4 of distance comparisons as in (1). To an (exact) ordinal embedding p ∶ [n] ↦ Rd — which by definition satisfies (2) — we associate the map φn ∶ Ωn ↦ Rd defined by φn (xi ) = pi for all i ∈ [n]. We crucially observe that, in the case of all quadruple comparisons (Cn = [n]4 ), the resulting map φn is isotonic on Ωn ; in the case of all triple comparisons (Cn = [n]3 ), φn is only weakly isotonic on Ωn , instead. In light of this, and the fact that the location, orientation and scale are all lost when only ordinal information is available, the problem of proving consistency of (exact) ordinal embedding reduces to showing that any such embedding is close to a similarity transformation as the sample size increases, n → ∞. This is exactly what Kleindessner and von Luxburg (2014) do under some assumptions. 2.1 Ordinal embedding based on all triple comparisons Our first contribution is to extend the consistency results of Kleindessner and von Luxburg (2014) on quadruple learning to triple learning. Following their presentation, we start with a result where the sample is infinite, which is only a mild generalization of (Kleindessner and von Luxburg, 2014, Th 3). 4 Theorem 1. Let U ⊂ Rd be bounded, connected and open. Suppose Ω is dense in U and consider a locally weakly isotonic function φ ∶ Ω ↦ Rd . Then there is a similarity transformation S that coincides with φ on Ω. The proof is largely based on that of (Kleindessner and von Luxburg, 2014, Th 3), but a bit simpler; see Section 3.1. We remark that there can only be one similarity with the above property, since similarities are affine transformations, and two affine transformations of Rd that coincide on d+1 affine independent points are necessarily identical. In this theorem, the set Ω is dense in an open subset of Rd , and therefore infinite. In fact, Kleindessner and von Luxburg (2014) use this theorem as an intermediary result for proving consistency as the sample size increases. Most of their paper is dedicated to establishing this, as their arguments are quite elaborate. We found a more direct route by ‘tending to the limit as soon as possible’, based on Lemma 2 below, which is at the core of the Arzelà-Ascoli theorem. For the remaining of this section, we consider the finite sample setting: U ⊂ Rd is bounded, connected and open, Ωn = {x1 , . . . , xn } ⊂ U is such that Ω ∶= {xn ∶ n ≥ 1} is dense in U , and φn ∶ Ωn ↦ Q ⊂ Rd is a function with values in a bounded set Q. (6) In the context of (6), we implicitly extend φn to Ω, for example, by setting φn (x) = q for all x ∈ Ω ∖ Ωn , where q is a given point in Q, although the following holds for any extension. Lemma 2. Consider Ωn ⊂ Rd finite and φn ∶ Ωn ↦ Q ⊂ Rd , where Q is bounded. Then there is N ⊂ N infinite such that φ(x) ∶= limn∈N φn (x) exists for all x ∈ Ω ∶= ⋃n Ωn . This is called the diagonal process in (Kelley, 1975, Problem D, Ch 7). Although the result is classical, we provide a proof for completeness. Proof. Without loss of generality, suppose Ωn = {x1 , . . . , xn }. Let N0 = N. Since (φn (x1 ) ∶ n ∈ N0 ) ∈ Q and Q is bounded, there is N1 ⊂ N0 infinite such that limn∈N1 φn (x1 ) exists. In turn, since (φn (x2 ) ∶ n ∈ N1 ) is bounded, there is N2 ⊂ N1 infinite such that limn∈N2 φn (x2 ) exists. Continuing this process — which formally corresponds to a recursion — we obtain ⋯ ⊂ Nk+1 ⊂ Nk ⊂ ⋯ ⊂ N1 ⊂ N0 = N such that, for all k, Nk is infinite and limn∈Nk φn (xk ) exists. Let nk denote the kth element (in increasing order) of Nk and note that (nk ∶ k ≥ 1) is strictly increasing. Define N = {nk ∶ k ≥ 1}. Since {np , p ≥ k} ⊂ Nk , we have limn∈N φn (xk ) = limn∈Nk φn (xk ), and this is valid for all k ≥ 1. Corollary 1. Consider the setting (6) and assume that φn is weakly isotonic. Then (φn ) is sequentially pre-compact for the pointwise convergence topology for functions on Ω and all the functions where it accumulates are similarity transformations restricted to Ω. The corresponding result (Kleindessner and von Luxburg, 2014, Th 4) was obtained for isotonic (instead of weakly isotonic) functions and for domains U that are finite unions of balls, and the convergence was uniform instead of pointwise. For now, we provide a proof of Corollary 1, which we derive as a simple consequence of Theorem 1 and Lemma 2. Proof. Lemma 2 implies that (φn ) is sequentially pre-compact for the pointwise convergence topology. Let φ be an accumulation point of (φn ), meaning that there is N ⊂ N infinite such that φ(x) = limn∈N φn (x) for all x ∈ Ω. Take x, y, z ∈ Ω such that ∥x − y∥ < ∥x − z∥. By definition, there is m such that x, y, z ∈ Ωm , and therefore ∥φn (x) − φn (y)∥ ≤ ∥φn (x) − φn (z)∥ for all n ≥ m. Passing to the limit along n ∈ N , we obtain ∥φ(x) − φ(y)∥ ≤ ∥φ(x) − φ(z)∥. Hence, φ is weakly isotonic on Ω and, by Theorem 1, it is therefore the restriction of a similarity transformation to Ω. 5 It is true that (Kleindessner and von Luxburg, 2014, Th 4) establishes a uniform convergence result. We do the same in Theorem 2 below, but with much simpler arguments. The key are the following two results bounding the modulus of continuity of a (resp. weakly) isotonic function. We note that the second result (for weakly isotonic functions) is very weak but sufficient for our purposes here. For Λ ⊂ V ⊂ Rd , define δH (Λ, V ) = supy∈V inf x∈Λ ∥y − x∥, which is their Hausdorff distance. We say that (yi ∶ i ∈ I) ⊂ Rd is an η-packing if ∥yi − yj ∥ ≥ η for all i ≠ j. We recall that the size of the largest η-packing of a Euclidean ball of radius r is of exact order (r/η)−d . For a set V ⊂ Rd , let diam(V ) = supx,y∈V ∥x − y∥ be its diameter and let ρ(V ) = arg sup{∃v ∈ V ∶ B(v, r/2) ⊂ V }, (7) r>0 which is the diameter of a largest ball inscribed in V . Everywhere in the paper, d is fixed, and in fact implicitly small as we assume repeatedly that the sample (of size n) is dense in a full-dimensional domain of Rd . In particular, all the implicit constants of proportionality that follow depend solely on d. Lemma 3. Let V ⊂ Rd be open. Consider Λ ⊂ V and set ε = δH (Λ, V ). Let ψ ∶ Λ ↦ Q be isotonic, where Q ⊂ Rd is bounded. There is C ∝ diam(Q)/ρ(V ), such that ∥ψ(x) − ψ(x′ )∥ ≤ C(∥x − x′ ∥ + ε), ∀x, x′ ∈ Λ. (8) Proof. The proof is based on the fact that an isotonic function transforms a packing into a packing. Take x, x′ ∈ Λ such that ξ ∶= ∥ψ(x) − ψ(x′ )∥ > 0, and let η = ∥x − x′ ∥. Since V is open it contains an open ball of diameter ρ(V ). Let y1 , . . . , ym be an (η + 3ε)-packing of such a ball with m ≥ C1 (ρ(V )/(η + ε))d for some constant C1 depending only on d. Then let x1 , . . . , xm ∈ Λ such that maxi ∥yi − xi ∥ ≤ ε. By the triangle inequality, for all i ≠ j we have ∥xi − xj ∥ ≥ ∥yi − yj ∥ − 2ε ≥ η + ε > ∥x − x′ ∥. Because ψ is isotonic, we have ∥ψ(xi ) − ψ(xj )∥ ≥ ξ, so that ψ(x1 ), . . . , ψ(xm ) is a ξ-packing. Therefore, there is a constant C2 depending only on d such that m ≤ C2 (diam(Q)/ξ)d . We conclude that ξ ≤ (C2 /C1 )1/d (diam(Q)/ρ(V ))(η + ε). For V ⊂ Rd and h > 0, let V h = {x ∈ V ∶ ∃y ∈ V s.t. x ∈ B(y, h) ⊂ V }. We note that V h is the complement of the h-convex hull of V c ∶= Rd ∖ V — see (Cuevas et al., 2012) and references therein. Lemma 4. In the context of Lemma 3, if ψ is only weakly isotonic, then there is C ∝ diam(Q), such that for all h > 0, √ 1/d (9) ∥ψ(x) − ψ(x′ )∥ ≤ C(∥x − x′ ∥/h + ε/h) , ∀x ∈ Λ ∩ V h , ∀x′ ∈ Λ. Proof. Assume that V h ≠ ∅, for otherwise there is nothing to prove. Take x ∈ Λ ∩ V h and x′ ∈ Λ such that ξ ∶= ∥ψ(x) − ψ(x′ )∥ > 0, and let η = ∥x − x′ ∥. Because ψ is bounded, it is enough to prove the result when η, ε < h/2. Let y ∈ V be such that x ∈ B(y, h) ⊂ V . There is y ′ ∈ B(y, h) such that y ∈ [xy ′ ] and ∥x − y ′ ∥ ≥ 2h/3. Define u = (y ′ − x)/∥y ′ − x∥. Let z0 = x, and for j ≥ 1, define zj = zj−1 + (η + 5jε)u. Let k ≥ 0 be maximum such that ∑kj=1 (η + 5jε) < h/2. Since k satisfies √ kη + 5k2 ε ≥ h/2, we have k ≥ min(h/(4η), h/(10ε)). By construction, for all j ∈ [k], zj ∈ [xy ′ ] and B(zj , 2ε) ⊂ B(y, h). Let x−1 = x′ , x0 = x and take x1 , . . . , xk ∈ Λ such that maxj ∥xj − zj ∥ ≤ ε. By the triangle inequality, for j = 2, . . . , k, ∥xj − xj−1 ∥ ≥ ∥zj − zj−1 ∥ − 2ε ≥ ∥zj−1 − zj−2 ∥ + 3ε ≥ ∥xj−1 − xj−2 ∥ + ε, 6 which implies by induction that ∥xj − xj−1 ∥ ≥ ∥x1 − x0 ∥ + ε ≥ ∥z1 − z0 ∥ = η + 5ε > ∥x − x′ ∥. By weak isotonicity, this implies that ∥ψ(xj ) − ψ(xj−1 )∥ ≥ ∥ψ(x) − ψ(x′ )∥ = ξ. We also have, for any i, j ∈ [k] such that 1 ≤ i ≤ j − 2, ∥xj − xi ∥ ≥ ∥zj − zi ∥ − 2ε ≥ ∥zj − zj−1 ∥ + η + 5ε − 2ε ≥ ∥xj − xj−1 ∥ + η + ε. By weak isotonicity, this implies that ∥ψ(xj ) − ψ(xi )∥ ≥ ∥ψ(xj ) − ψ(xj−1 )∥ for all 0 ≤ i < j ≤ k. Consequently, (ψ(xj ) ∶ j ∈ [k]) forms a ξ-packing of Q. Hence, k ≤ C ′ (diam(Q)/ξ)d , for some constant C ′ . We conclude with the lower bound on k. From this control on the modulus of continuity, we obtain a stronger version of Corollary 1. Theorem 2. Under the same conditions as Corollary 1, we have the stronger conclusion that there is a sequence (Sn ) of similarities such that, for all h > 0, maxx∈Ωn ∩U h ∥φn (x) − Sn (x)∥ → 0 as n → ∞. If in fact each φn is isotonic, then this remains true when h = 0. We remark that when U is a connected union of a possibly uncountable number of open balls of radius at least h > 0, then U = U h . This covers the case of a finite union of open balls considered in (Kleindessner and von Luxburg, 2014). We also note that, if U is bounded and open, and ∂U has bounded curvature, then there is h > 0 such that U = U h . This follows from the fact that, in this case, U c has positive reach (Federer, 1959), and is therefore h-convex when h is below the reach by1 (Cuevas et al., 2012, Prop 1). Moreover, our arguments can be modified to accommodate sets U with boundaries that are only Lipschitz, by reasoning with wedges in Lemma 4. Theorem 2 now contains (Kleindessner and von Luxburg, 2014, Th 4), and extends it to weakly isotonic functions and to more general domains U . Overall, our proof technique is much simpler, shorter, and elementary. Define εn = δH (Ωn , U ), which quantifies the density of Ωn in U . Because Ωn+1 ⊂ Ωn and Ω is dense in U , we have εn ↘ 0 as n → ∞. Proof. Let φ be an accumulation point of (φn ) for the pointwise convergence topology, meaning there is N ⊂ N infinite such that φ(x) = limn∈N φn (x) for all x ∈ Ω. We show that, in fact, the convergence is uniform. First, suppose that each φn is isotonic. In that case, Lemma 3 implies the existence of a constant C > 0 such that ∥φn (x)−φn (x′ )∥ ≤ C(∥x−x′ ∥+εn ) for all x, x′ ∈ Ωn , and for all n. Passing to the limit along n ∈ N , we get ∥φ(x) − φ(x′ )∥ ≤ C∥x − x′ ∥ for all x, x′ ∈ Ω. (In fact, we already knew this from Corollary 1, since we learned there that φ coincides with a similarity, and is therefore Lipschitz.) Fix ε > 0. There is m such that εm ≤ ε. Then there is m′ ≥ m such that maxi∈[m] ∥φn (xi )−φ(xi )∥ ≤ ε for all n ∈ N with n ≥ m′ . For such an n, and x ∈ Ωn , let i ∈ [m] be such that ∥x − xi ∥ ≤ εm . By the triangle inequality, ∥φn (x) − φ(x)∥ ≤ ∥φn (x) − φn (xi )∥ + ∥φn (xi ) − φ(xi )∥ + ∥φ(xi ) − φ(x)∥ ≤ C(∥x − xi ∥ + εn ) + ∥φn (xi ) − φ(xi )∥ + C∥xi − x∥ ≤ C(εm + εn ) + ε + Cεm ≤ (3C + 1)ε. Since x ∈ Ω is arbitrary and ε can be taken as small as desired, this shows that the sequence (φn ∶ n ∈ N ) convergences uniformly to φ over (Ωn ∶ n ∈ N ). 1 This proposition is stated for compact sets (which is not the case of U c ) but easily extends to the case where set is closed with compact boundary 7 When the φn are only weakly isotonic, we use Lemma 4 to get a constant C > 0 depending on √ diam(Q) and h > 0 such that ∥φn (x) − φn (x′ )∥ ≤ C(∥x − x′ ∥ + εn )1/d for all x ∈ Ωn ∩ U h and all x′ ∈ Ωn , and for all n. Passing to the limit along n ∈ N , we get ∥φ(x) − φ(x′ )∥ ≤ C∥x − x′ ∥1/d for all x, x′ ∈ Ω. (In fact, ∥φ(x) − φ(x′ )∥ ≤ C∥x − x′ ∥ for all x, x′ ∈ Ω from Corollary 1, as explained above.) The rest of the arguments are completely parallel. We conclude that (φn ∶ n ∈ N ) convergences uniformly to φ over (Ωn ∩ U h ∶ n ∈ N ). Let S denote the similarities of Rd . For any functions φ, ψ ∶ Ω ↦ Rd , define δn (φ, ψ) = maxx∈Ωn ∩U h ∥φ(x) − ψ(x)∥, and also δn (φ, S) = inf S∈S δn (φ, S). Our end goal is to show that δn (φn , S) → 0 as n → ∞. Suppose this is not the case, so that there is η > 0 and N ⊂ N infinite such that δn (φn , S) ≥ η for all n ∈ N . By Corollary 1, there is N1 ⊂ N and S ∈ S such that S(x) = limn∈N1 φn (x) for all x ∈ Ω. As we showed above, the convergence is in fact uniform over (Ωn ∩ U h ∶ n ∈ N1 ), meaning limn∈N1 δn (φn , S) = 0. At the same time, we have δn (φn , S) ≥ δn (φn , S) ≥ η. We therefore have a contradiction. 2.2 Rates of convergence Beyond consistency, we are able to derive convergence rates. We do so for the isotonic case, i.e., the quadruple comparison setting. Recall that εn = δH (Ωn , U ). Theorem 3. Consider the setting (6) with φn isotonic. There is C depending only on (d, U ), and a sequence of similarities Sn such that maxx∈Ωn ∥φn (x) − Sn (x)∥ ≤ C diam(Q)εn . If U = U h for some h > 0, then C = C ′ / diam(U ) where C ′ is a function of (d, h/ diam(U ), ρ(U )/ diam(U )). The proof of Theorem 3 is substantially more technical than the previous results, and thus postponed to Section 3. Although Kleindessner and von Luxburg (2014) are not able to obtain rates of convergence, the proof of Theorem 3 bares resemblance to their proof technique, and in particular, is also based on a result of Alestalo et al. (2001) on the approximation of ε-isometries; see Lemma 18. We will also make use of a related result of Vestfrid (2003) on the approximation of approximately midlinear functions; see Lemma 17. We mention that we know of a more elementary proof that only makes use of (Alestalo et al., 2001), but yields a slightly slower rate of convergence. We note that there is a constant c depending only on d such that εn ≥ cn−1/d . This is because U being open, it contains an open ball, and this lower bound trivially holds for an open ball. And such a lower bound is achieved when the xi ’s are roughly regularly spread out over U . If instead the xi ’s are iid uniform in U , and U is sufficiently regular — for example, U = U h for some h > 0 — then εn = O(log(n)/n)1/d , as is well-known. This would give the rate, and we do not know whether it is optimal, even in dimension d = 1. √ Remark 1. We are only able to get a rate in εn for the weakly isotonic case. We can do so by adapting of the arguments underlying Theorem 3, but only after assuming that U = U h for some h > 0 and resolving a few additional complications. 2.3 Ordinal embedding with local comparisons Terada and Von Luxburg (2014) consider the problem of embedding an unweighted nearest-neighbor graph, which as we saw in the Introduction, is a special case of ordinal embedding. Their arguments — which, as we explained earlier, seem incomplete at the time of writing — indicate that K = Kn ≫ n2/(2+d) (log n)d/(2+d) is enough for consistently embedding a K-NN graph. We consider here a situation where we have more information, specifically, all the distance comparisons between K-nearest-neighbors. Formally, this is the situation where Cn = {(i, j, k, ℓ) ∶ δij < δkℓ and {j, k, ℓ} ⊂ NKn (i)}, 8 where NK (i) denotes the set of the K items nearest item i. If the items are points Ωn = {x1 , . . . , xn } ⊂ Rd , an exact ordinal embedding φn is only constrained to be locally weakly isotonic as we explain now. We start by stating a standard result which relates a K-NN graph to an r-ball graph. Lemma 5. Let U ⊂ Rd be bounded, connected and open, and such that U = U h for some h > 0. Sample x1 , . . . , xn iid from a density f supported on U with (essential) range in (0, ∞) strictly. There is a constant C such that, if K ∶= [nr d ] ≥ C log n, then with probability tending to 1, NeighK/2 (xi ) ⊂ {xj ∶ ∥xj − xi ∥ ≤ r} ⊂ Neigh2K (xi ), ∀i ∈ [n], where NeighK (xi ) denotes the set of the K points in {xj ∶ j ∈ [n]} nearest xi . The proof is postponed to Section 3 and only provided for completeness. Therefore, assuming that K ≥ C log n, where C is the constant of Lemma 5, we may equivalently consider the case where Cn = {(i, j, k, ℓ) ∶ δij < δkℓ and max(δij , δik , δiℓ ) < rn }, for some given rn > 0. An exact embedding φn ∶ Ωn ↦ Rd in that case is isotonic on Ωn ∩ B(x, rn ) for any x ∈ Ωn . We require in addition that ∥x − x′ ∥ < rn ≤ ∥x† − x‡ ∥ ⇒ ∥φn (x) − φn (x′ )∥ ≤ ∥φn (x† ) − φn (x‡ )∥, ∀x, x′ , x† , x‡ ∈ Ωn . (10) This is a reasonable requirement since it is possible to infer it from Cn . Indeed, for k, ℓ ∈ [n], we have δkℓ < rn if, and only if, (k, k, k, ℓ) ∈ Cn or (ℓ, ℓ, ℓ, k) ∈ Cn . (Here we assume that δii = 0 for all i and δij > 0 if i ≠ j, as is the case for Euclidean distances.) We can still infer this even if the quadruples in Cn must include at least three distinct items. Indeed, suppose k, ℓ ∈ [n] are such that there is no i such that (i, k, i, ℓ) ∈ Cn or (i, ℓ, i, k) ∈ Cn , then (a) δik = δiℓ for all i such that max(δik , δiℓ ) < rn , or (b) δkℓ ≥ rn . Assume that rn ≥ Cεn with C > 0 sufficiently large, so that situation (a) does not happen. Conversely, if (k, ℓ) is such that δkℓ < rn , then when (a) does not happen, there is i such that (i, k, i, ℓ) ∈ Cn or (i, ℓ, i, k) ∈ Cn . Theorem 4. Consider the setting (6) and assume in addition that U = U h for some h > 0, and that φn is isotonic over balls of radius rn and satisfies (10). There is a constant C > 0 depending only on (d, h, ρ(U ), diam(U ), diam(Q)) and similarities Sn such that maxx∈Ωn ∥φn (x) − Sn (x)∥ ≤ Cεn /rn2 . Assume the data points are generated as in Lemma 5. In that case, we have εn = O(log(n)/n)1/d and Theorem 4 implies consistency when rn ≫ (log(n)/n)1/2d . By Lemma 5, this corresponds to the situation where we are provided with comparisons among Kn -nearest neighbors with Kn ≫ √ n log n. If the result of Terada and Von Luxburg (2014) holds in all rigor, then this is a rather weak result. 2.4 Landmark ordinal embedding Inspired by (Jamieson and Nowak, 2011), we consider the situation where there are landmark items indexed by Ln ⊂ [n], and we are given all distance comparisons from any point to the landmarks. Formally, with triple comparisons, this corresponds to the situation where Cn = {(i, j, k) ∈ [n] × L2n ∶ δij < δik }. If the items are points Ωn = {x1 , . . . , xn } ⊂ Rd , an exact ordinal embedding φn is only constrained to be weakly isotonic on the set of landmarks and, in addition, is required to respect the ordering of the distances from any point to the landmarks. The following is an easy consequence of Theorem 2. 9 Corollary 2. Theorem 2 remains valid in the landmark triple comparisons setting (meaning with φn as just described) as long as the landmarks become dense in U . Jamieson and Nowak (2011) study the number of triple comparisons that are needed for exact ordinal embedding. With a counting argument, they show that at least Cn log n comparisons are needed, where C is a constant depending only on d. If we only insist that the embedding respects the comparisons that are provided, then Corollary 2 implies that a landmark design is able to be consistent as long as the landmarks become dense in U . This consistency implies that, as the sample size increases, an embedding that respects the landmark comparisons also respects all other comparisons approximately. This is achieved with O(nℓ2n + ℓ3n ) triple comparisons, where ℓn ∶= ∣Ln ∣ is the number of landmarks, and the conditions of Corollary 2 can be fulfilled with ℓn → ∞ at any speed, so that the number of comparisons is nearly linear in n. Proof. We focus on the weakly isotonic case, where we assume that U = U h for some h > 0. Let Λn = {xl ∶ l ∈ Ln } denote the set of landmarks. Since Λn becomes dense in U , meaning ηn ∶= δH (Λn , U ) → 0, by Theorem 2, there is a sequence of similarities Sn such that ζn ∶= maxx∈Λn ∥φn (x) − Sn (x)∥ → 0. Now, for x ∈ Ωn , let x̃ ∈ Λn such that ∥x − x̃∥ ≤ ηn . We have ∥φn (x) − Sn (x)∥ ≤ ∥φn (x) − φn (x̃)∥ + ∥φn (x̃) − Sn (x̃)∥ + ∥Sn (x̃) − Sn (x)∥. (11) 1/2d by Lemma 4, for some constant C. The middle term is The first term is bounded by Cηn bounded by ζn . For the third term, express Sn in the form Sn (x) = βn Rn (x) + bn , where βn ∈ R, Rn is an orthogonal transformation, and bn ∈ Rd . Take two distinct landmarks x† , x‡ ∈ Λn such that ∥x† − x‡ ∥ ≥ diam(U )/2, which exist when n is sufficiently large. Since and, at the same time, ∥Sn (x† ) − Sn (x‡ )∥ = βn ∥x† − x‡ ∥ ≥ βn diam(U )/2 ∥Sn (x† ) − Sn (x‡ )∥ ≤ ∥Sn (x† ) − φn (x† )∥ + ∥φn (x† ) − φn (x‡ )∥ + ∥φn (x‡ ) − Sn (x‡ )∥ ≤ ζn + diam(Q) + ζn ≤ 2 diam(Q), eventually, we have βn ≤ β̄ ∶= 4 diam(Q)/ diam(U ). Hence, the third term on the RHS of (11) is bounded by 1/2d β̄ηn . Thus, the RHS of (11) is bounded by Cηn + ζn + β̄ηn , which tends to 0 as n → ∞. This being valid for any x ∈ Ωn , we conclude. We remark that at the very end of the proof, we obtained a rate of convergence as a function of the density of the landmarks and the convergence rate implicit in Theorem 2. This leads to the following rate for the quadruple comparisons setting, which corresponds to the situation where Cn = {(i, j, k) ∈ [n] × L2n ∶ δij < δik } ⋃ {(i, j, k, ℓ) ∈ L4n ∶ δij < δik }. Here, φn is constrained to be isotonic on the set of landmarks and, as before, is required to respect the ordering of the distances from any data point to the landmarks. Corollary 3. Consider the setting (6) in the landmark quadruple comparisons setting (meaning with φn as just described). Let Λn denote the set of landmarks and set ηn = δH (Λn , U ). There is a constant C > 0 and a sequence of similarities Sn such that maxx∈Ωn ∥φn (x) − Sn (x)∥ ≤ Cηn . Proof. The proof is parallel to that of Corollary 2. Here, we apply Theorem 3 to get ζn ≤ C0 ηn . This bounds the second term on the RHS of (11). The first term is bounded by C1 ηn by Lemma 3, while the third term is bounded by β̄ηn as before. (C0 , C1 are constants.) 10 Computational complexity. We now discuss the computational complexity of ordinal embedding with a landmark design. The obvious approach has two stages. In the first stage, the landmarks are embedded. This is the goal of (Agarwal et al., 2007), for example. Here, we use brute force. Proposition 1. Suppose that m items are in fact points in Euclidean space and their dissimilarities are their pairwise Euclidean distances. Then whether in the triple or quadruple comparisons setting, an exact ordinal embedding of these m items can be obtained in finite expected time. Proof. The algorithm we discuss is very naive: we sample m points iid from the uniform distribution on the unit ball, and repeat until the ordinal constraints are satisfied. Since checking the latter can be done in finite time, it suffices to show that there is a strictly positive probability that one such sample satisfies the ordinal constraints. Let Xm denote the set of m-tuples (x1 , . . . , xm ) ∈ B(0, 1) that satisfy the ordinal constraints, meaning that ∥xi − xj ∥ < ∥xk − xℓ ∥ when (i, j, k, ℓ) ∈ C. Seeing Xn as a subset of B(0, 1)m ⊂ Rdm , it is clearly open. And sampling x1 , . . . , xm iid from the uniform distribution on B(0, 1) results in sampling (x1 , . . . , xm ) from the uniform distribution on B(0, 1)m , which assigns a positive mass to any open set. In the second stage, each point that is not a landmark is embedded based on the order of its distances to the landmarks. We quickly mention the work Davenport (2013), who develops a convex method for performing this task. Here, we are contented with knowing that this can be done, for each point, in finite time, function of the number of landmarks. For example, a brute force approach starts by computing the Voronoi diagram of the landmarks, and iteratively repeats within each cell, creating a tree structure. Each point that is not a landmark is placed by going from the root to a leaf, and choosing any point in that leaf cell, say its barycenter. Thus, if there are ℓ landmarks, the first stage is performed in expected time F (ℓ), and the second stage is performed in time (n − ℓ)G(ℓ). The overall procedure is thus computed in expected time F (ℓ) + (n − ℓ)G(ℓ). Remark 2. The procedure described above is not suggested as a practical means to perform ordinal embedding with a landmark design. The first stage, described in Proposition 1, has finite expected time, but likely not polynomial in the number of landmarks. For a practical method, we can suggest the following: 1. Embed the landmarks using the method of Agarwal et al. (2007) (which solves a semidefinite program) or the method of Terada and Von Luxburg (2014) (which uses an iterative minimization-majorization strategy). 2. Embed the remaining points using the method of Davenport (2013) (which solves a quadratic program). Although practical and reasonable, we cannot provide any theoretical guarantees for this method. 3 More proofs In this section we gather the remaining proofs and some auxiliary results. We introduce some additional notation and basic concepts. For z1 , . . . , zm ∈ Rd , let Aff(z1 , . . . zm ) denote their affine hull, meaning the affine subspace they generate in Rd . For a vector x in a Euclidean space, let ∥x∥ denote its Euclidean norm. For a matrix M ∈ √ Rp×q , let ∥M ∥ denote its usual operator norm, meaning, ∥M ∥ = max{∥M x∥ ∶ ∥x∥ ≤ 1} and ∥M ∥F = tr(M ⊺ M ) its Frobenius norm. Regular simplexes. These will play a central role in our proofs. We say that z1 , . . . , zm ∈ Rd , with m ≥ 2, form a regular simplex if their pairwise distances are all equal. We note that, necessarily, 11 m ≤ d + 1, and that regular simplexes in the same Euclidean space and with same number of (distinct) nodes m are similarity transformations of each other — for example, segments (m = 2), equilateral triangles (m = 3), tetrahedron (m = 4). By recursion on the number of vertices, m, it is easy to prove the following. Lemma 6. Let z1 , . . . , zm form a√ regular simplex with edge length 1 and let µ denote the barycenter of z1 , . . . , zm . Then ∥µ − zi ∥ = (m − 1)/2m, and if z, z1 , . . . , zm form a regular simplex, then √ ∥z − µ∥ = (m + 1)/2m. (In dimension m, there are exactly two such points z.) 3.1 Proof of Theorem 1 We assume d ≥ 2. See (Kleindessner and von Luxburg, 2014) for the case d = 1. We divide the proof into several parts. Continuous extension. Lemma 4 implies that φ is locally uniformly continuous. Indeed, take x0 ∈ Ω and let r > 0 such that B(x0 , r) ⊂ U and φ is weakly isotonic on B(x0 , r) ∩ Ω. Applying Lemma 4 with V = B(x0 , r) and Λ = Ω ∩ B(x0 , r) — so that δH (Λ, V ) = 0 because Λ is dense in V — and noting that V r = V , yields a constant Cr > 0 such that ∥φ(x) − φ(x′ )∥ ≤ Cr ∥x − x′ ∥1/d , for all x, x′ ∈ Ω ∩ B(x0 , r). Being locally uniformly continuous, we can uniquely extend φ to a continuous function on U , also denoted by φ. By continuity, this extension is locally weakly isotonic on U . Isosceles preservation. Sikorska and Szostok (2004) say that a function f ∶ V ⊂ Rd → Rd preserves isosceles triangles if ∥x − y∥ = ∥x − z∥ ⇒ ∥f (x) − f (y)∥ = ∥f (x) − f (z)∥, ∀x, y, z ∈ V. In our case, by continuity, we also have that φ preserves isosceles triangles locally. Indeed, for the sake of pedagogy, let u ∈ U and r > 0 such that B(u, r) ⊂ U and φ is weakly isotonic on B(u, r). Take x, y, z ∈ B(u, r/2) be such that ∥x − y∥ = ∥x − z∥. For t ∈ R, define zt = (1 − t)x + tz. Let t > 1 such that zt ∈ B(u, r). Because ∥x − y∥ < t∥x − z∥ = ∥x − zt ∥, we have ∥φ(x) − φ(y)∥ ≤ ∥φ(x) − φ(zt )∥. Letting t ↘ 1, we get ∥φ(x) − φ(y)∥ ≤ ∥φ(x) − φ(z)∥ by continuity of φ. Since y and z play the same role, the converse inequality is also true, and combined, yield an equality. Midpoint preservation. Let V ⊂ Rd be convex. We say that a function f ∶ V ↦ Rd preserves midpoints if x+y f (x) + f (y) f( )= , ∀x, y ∈ V. 2 2 We now show that φ preserves midpoints, locally. Kleindessner and von Luxburg (2014) also do that, however, our arguments are closer to those of Sikorska and Szostok (2004), who make use of regular simplexes. The important fact is that a function that preserves isosceles preserves regular simplexes. Let u ∈ U and r > 0 such that B(u, r) ⊂ U and φ preserves isosceles on B(u, r). Take x, y ∈ B(u, r/2), and let µ = (x + y)/2. Let z1 , . . . , zd form a regular simplex with barycenter µ and side length s, and such that ∥x−zi ∥ = s for all i. In other words, x, z1 , . . . , zd forms a regular simplex placed so that µ is the barycenter of z1 , . . . , zd . By √ symmetry, y, z1 , . . . , zd forms a regular simplex also. By Lemma 6, we have ∥zi − µ∥/∥x − µ∥ = (d − 1)/(d + 1), so that z1 , . . . , zd ∈ B(µ, r/2) ⊂ B(u, r), by the triangle inequality and the fact that ∥x − µ∥ < r/2. Hence, φ(x), φ(z1 ), . . . , φ(zd ) and φ(y), φ(z1 ), . . . , φ(zd ) are regular simplexes. If one of them is singular, so is the other one, in which case φ(x) = φ(y) = φ(µ). Otherwise, necessarily φ(x) is the symmetric of φ(y) with respect to Aff(φ(z1 ), . . . , φ(zd )); the only other possibility would be that φ(x)√ = φ(y), but in that case we would still have that φ(zi ) = φ(x) for all i ∈ [d], since ∥x − zi ∥/∥x − µ∥ = 2d/(d + 1) by Lemma 6 — implying that ∥x − zi ∥ < ∥x − y∥ — and φ is weakly isotonic in that neighborhood. So assume that 12 φ(x) is the symmetric of φ(y) with respect to Aff(φ(z1 ), . . . , φ(zd )). For a ∈ {x, y, µ}, ∥a − zi ∥ is constant in i, and therefore so is ∥φ(a)−φ(zi )∥, so that φ(a) belongs to the line of points equidistant to φ(z1 ), . . . , φ(zd ). This implies that x, y, µ are collinear. And because ∥µ − x∥ = ∥µ − y∥, we also have ∥φ(µ) − φ(x)∥ = ∥φ(µ) − φ(y)∥, so that φ(µ) is necessarily the midpoint of φ(x) and φ(y). Conclusion. We arrived at the conclusion that φ can be extended to a continuous function on U that preserves midpoints locally. We then use the following simple results in sequence: with Lemma 7, we conclude that φ is locally affine; with Lemma 8, we conclude that φ is in fact affine on U ; and with Lemma 9, we conclude that φ is in fact a similarity on U . Lemma 7. Let V be a convex set of a Euclidean space and let f be a continuous function on V with values in a Euclidean space that preserves midpoints. Then f is an affine transformation. Proof. This result is in fact well-known, and we only provide a proof for completeness. It suffices to prove that f is such that f ((1 − t)x + ty) = (1 − t)f (x) + tf (y) for all x, y ∈ V and all t ∈ [0, 1]. Starting with the fact that this is true when t = 1/2, by recursion we have that this is true when t is dyadic, meaning, of the form t = k2−j , where j ≥ 1 and k ≤ 2j are both integers. Since dyadic numbers are dense in [0, 1], by continuity of f , we deduce the desired property. Lemma 8. A locally affine function over an open and connected subset of a Euclidean space is the restriction of an affine function over the whole space. Proof. Let U be the domain and f the function. Cover U with a countable number of open balls Bi , i ∈ I such that f coincides with an affine function fi on Bi . Take i, j ∈ I distinct. Since U is connected, there must be a sequence i = k1 , . . . , km = j, all in I, such that Bks ∩ Bks+1 ≠ ∅ for s ∈ [m − 1]. Since Bks ∩ Bks+1 is an open set, we must have fks = fks+1 , and this being true for all s, it implies that fi = fj . Lemma 9. An affine function that preserves isosceles locally is a similarity transformation. Proof. Let f be an affine function that preserves isosceles in an open ball. Without loss of generality, we may assume that the ball is B(0, 2) and that f (0) = 0 (so that f is linear). Fix u0 ∈ ∂B(0, 1) and let a = ∥f (u0 )∥. Take x ∈ Rd different from 0 and let u = x/∥x∥. We have ∥f (x)∥/∥x∥ = ∥f (u)∥ = ∥f (u) − f (0)∥ = ∥f (u0 ) − f (0)∥ = ∥f (u0 )∥ = a. Hence, ∥f (x)∥ = a∥x∥, valid for all x ∈ Rd , and f being linear, this implies that f is a similarity. 3.2 Auxiliary results We list here a number of auxiliary results that will be used in the proof of Theorem 3. The following result is a perturbation bound for trilateration, which is the process of locating a point based on its distance to landmark points. For a real matrix Z, let σk (Z) denote its k-th largest singular value. Lemma 10. Let z1 , . . . , zd+1 ∈ Rd such that Aff(z1 , . . . , zd+1 ) = Rd and let Z denote the matrix with columns z1 , . . . , zd+1 . Consider p, q ∈ Rd and define ai = ∥p − zi ∥ and bi = ∥q − zi ∥ for i ∈ [d + 1]. Then ∥p − q∥ ≤ √ 1√ d σd (Z)−1 max ∣a2d+1 − a2i − b2i + b2d+1 ∣ ≤ d σd (Z)−1 max ∣a2i − b2i ∣. i i 2 Proof. Assume without loss of generality that zd+1 = 0. In that case, note that ad+1 = ∥p∥ and bd+1 = ∥q∥. Also, redefine Z as the matrix with columns z1 , . . . , zd , and note that the first d singular values remain unchanged. Since Aff(z1 , . . . , zd+1 ) = Rd , there is α = (α1 , . . . , αd ) ∈ Rd 13 and β = (β1 , . . . , βd ) ∈ Rd such that p = ∑i∈[d] αi zi = Zα and q = ∑i∈[d] βi zi = Zβ. For p, we have a2i = ∥p − zi ∥2 = ∥p∥2 + ∥zi ∥2 − 2zi⊺ Zα for all i ∈ [d], or in matrix form, Z ⊺ Zα = 12 u, where u = (u1 , . . . , ud ) and ui = a2d+1 − a2i + ∥zi ∥2 . Similarly, we find Z ⊺ Zβ = 12 v, where v = (v1 , . . . , vd ) and vi = b2d+1 − b2i + ∥zi ∥2 . Hence, we have 1√ 1 ∥Z ⊺ (p − q)∥ = ∥Z ⊺ Zα − Z ⊺ Zβ∥ = ∥u − v∥ = ∑ (a2 − b2 − a2i + b2i )2 2 2 i∈[d] d+1 d+1 √ 1√ d max ∣a2d+1 − a2i − b2i + b2d+1 ∣ ≤ d max ∣a2i − b2i ∣. ≤ i i 2 Simultaneously, ∥Z ⊺ (p − q)∥ ≥ σd (Z)∥p − q∥. Combining both inequalities, we conclude. For η ∈ [0, 1), we say that z1 , . . . , zm ∈ Rd form an η-approximate regular simplex if min ∥zi − zj ∥ ≥ (1 − η) max ∥zi − zj ∥. i≠j i≠j Lemma 11. Let z1 , . . . , zm form an η-approximate regular simplex with maximum edge length λ ′ achieved by ∥z1 − z2 ∥. There is a constant Cm and z1′ , . . . , zm ∈ Aff(z1 , . . . , zm ) with z1′ = z1 and ′ z2 = z2 and forming a regular simplex with edge length λ, such that maxi ∥zi′ − zi ∥ ≤ λCm η. Proof. By scale equivariance, we may assume that λ = 1. We use an induction on m. In what ′ ′′ follows, Cm , Cm , Cm , etc, are constants that depend only on m. For m = 2, the statement is trivially true. Suppose that it is true for m ≥ 2 and consider an η-approximate regular simplex z1 , . . . , zm+1 ∈ Rd with maximum edge length 1. By changing d to m if needed, without loss of generality, assume that Aff(z1 , . . . , zm+1 ) = Rd . In that case, z1 , . . . , zm is an η-approximate regular simplex with maximum edge length achieved by ∥z1 − z2 ∥ = 1, and by the inductive hypothesis, this ′ ∈ A ∶= Aff(z1 , . . . , zm ) with z1′ = z1 and z2′ = z2 and forming a implies the existence of z1′ , . . . , zm regular simplex of edge length 1, such that maxi∈[m] ∥zi′ − zi ∥ ≤ Cm η for some constant Cm . Let p be the orthogonal projection of zm+1 onto A. Before continuing, let P be the set of such p ′ obtained when fixing z1′ , . . . , zm and then varying zi ∈ B(zi′ , Cm η) for i ∈ [m] and zm+1 among the points that make an η-approximate regular simplex with z1 , . . . , zm . Let µ′ be the barycenter of ′ z1′ , . . . , zm and note that µ′ ∈ P. Now, set δ = ∥zm+1 − p∥. By the Pythagoras theorem, we have 2 ∥p − zi ∥ = ∥zm+1 − zi ∥2 − δ2 , with 1 − η ≤ ∥zm+1 − zi ∥ ≤ 1, so that 0 ≤ 1 − δ2 − ∥p − zi ∥2 ≤ 2η. By the triangle inequality, ∣∥p − zi′ ∥ − ∥p − zi ∥∣ ≤ ∥zi − zi′ ∥ ≤ Cm η, so that ∣∥p − zi′ ∥2 − ∥p − zi ∥2 ∣ = ∣∥p − zi′ ∥ − ∥p − zi ∥∣(∥p − zi′ ∥ − ∥p − zi ∥) ≤ ∥zi − zi′ ∥∥zi − zi′ ∥ ′ η, ≤ Cm η(2 + Cm η) ≤ Cm using the fact that ∥p − zi ∥ ≤ ∥zm+1 − zi ∥ ≤ 1. Hence, ′′ P ⊂ {q ∶ ∥q − zi′ ∥2 = 1 − δ2 ± Cm η, ∀i ∈ [m]}. ′′ η. By Lemma 10, this implies that Since µ′ ∈√ P, we must therefore have ∥p − zi′ ∥2 = ∥µ′ − zi′ ∥2 ± 2Cm ′ ′′′ ′′ −1 ′ ′ ′ ∥p − µ ∥ ≤ m − 1 σm−1 ([z1 ⋯zm ])2Cm η =∶ Cm η. Let zm+1 be on the same side of A as zm+1 and such ′ ′ ′ , zm+1 form a regular simplex. Note that µ′ is the orthogonal projection of zm+1 onto that z1′ , . . . , zm A. By the Pythagoras theorem, applied multiple times, we obtain the following. First, we have ′ ′ − µ′ + µ′ − p + p − zm+1 ∥2 − zm+1 ∥2 = ∥zm+1 ∥zm+1 ′ ′ = ∥zm+1 − µ′ ∥2 − 2(zm+1 − µ′ )⊺ (zm+1 − p) + ∥p − zm+1 ∥2 + ∥µ′ − p∥2 ′ = (∥zm+1 − µ′ ∥ − ∥p − zm+1 ∥)2 + ∥µ′ − p∥2 , 14 ′ because zm+1 − µ′ and zm+1 − p are orthogonal to A, and therefore parallel to each other and both ′′′ orthogonal to µ′ − p. For the second term, we already know that ∥µ′ − p∥ ≤ Cm η, while the first ′′ 2 2 term is bounded by (2Cm + 2) η since, on the one hand, ′ ′ ∥zm+1 − µ′ ∥2 = ∥zm+1 − z1′ ∥2 − ∥µ′ − z1′ ∥2 = 1 − ∥µ′ − z1′ ∥2 while, on the other hand, ∥p − zm+1 ∥2 = ∥zm+1 − z1 ∥2 − ∥p − z1 ∥2 = 1 ± 2η − ∥p − z1 ∥2 , ′′ ′ 2 and we know that ∥µ′ − z1′ ∥2 = ∥p − z1 ∥2 ± 2Cm η. Hence, we find that ∥zm+1 − zm+1 ∥2 ≤ Cm+1 η2 for some constant Cm+1 function of m only. This shows that the induction hypothesis holds for m + 1. ′ Lemma 12. There are constants Cm , Cm > 0 such that, if z1 , . . . , zm form an η-approximate regular ′ simplex with maximum edge length λ, then σm−1 ([z1 ⋯zm ]) ≥ λCm (1 − Cm η). ′′ and Proof. By scale equivariance, we may assume that λ = 1. By Lemma 11, there is a constant Cm ′ ′ ′ z1 , . . . , zm ∈ Aff(z1 , . . . , zm ) forming a regular simplex with edge length 1 such that maxi ∥zi − zi ∥ ≤ ′′ Cm η. By Weyl’s inequality (Horn and Johnson, 1990, Cor 7.3.8), σm−1 (Z) ≥ σm−1 (Z ′ ) − ∥Z − Z ′ ∥. On the one hand, σm−1√ (Z ′ ) is a positive constant depending only on m, while on the other hand, √ ′ ′ ′′ η. ∥Z − Z ∥ ≤ ∥Z − Z ∥F = ∑i ∥zi − zi′ ∥2 ≤ mCm Lemma 13. Let z1 , . . . , zm form an η-approximate regular simplex with maximum edge length λ and barycenter µ. Let p ∈ Aff(z1 , . . . , zm ) and define γ = maxi ∥p − zi ∥2 − mini ∥p − zi ∥2 . There is a constant Cm ≥ 1 depending only on m such that ∥p − µ∥ ≤ Cm λγ when η ≤ 1/Cm . Proof. By scale equivariance, we may assume that λ = 1. By Lemma 10, we have ∥p − µ∥ ≤ 1√ −1 m − 1 σm−1 ([z1 ⋯zm ]) max ∣∥p − zm ∥2 − ∥p − zi ∥2 ∣. i 2 ′ −1 ′ ′ By Lemma 12, there is a constant Cm . And we when η ≤ 1/Cm such that σm−1 ([z1 ⋯zm ]) ≤ Cm 2 2 also have maxi ∣∥p − zm ∥ − ∥p − zi ∥ ∣ ≤ γ. From this, we conclude. Lemma 14. Let ψ ∶ Λ ↦ Q be isotonic, where Λ, Q ⊂ Rd . Let v ∈ Rd and r > 0, and set ε = δH (Λ, B(v, r)). There is C ∝ diam(Q)/r such that, for all x, x′ , x† , x‡ ∈ Λ with x, x′ ⊂ B(v, 3r/4) and for all η ∈ (0, r/4 − 2ε), ∥x − x′ ∥ = ∥x† − x‡ ∥ ± η ⇒ ∥ψ(x) − ψ(x′ )∥ = ∥ψ(x† ) − ψ(x‡ )∥ ± C(η + ε). (12) Proof. Let ξ = ∥x − x′ ∥ and ξ † = ∥x† − x‡ ∥. Suppose that ξ < η + 2ε, which implies that ξ † < 2η + 2ε. In that case, Lemma 3 — where the constant there is denoted here by C1 ∝ diam(Q)/r — yields ∥ψ(x) − ψ(x′ )∥ ≤ C1 (ξ + ε) ≤ C1 (η + 3ε) and, similarly, ∥ψ(x† ) − ψ(x‡ )∥ ≤ C1 (2η + 3ε). This proves (12). Henceforth, we assume that ξ ≥ η + 2ε. First assume that ξ > ξ † . In that case, we immediately have ∥ψ(x) − ψ(x′ )∥ ≥ ∥ψ(x† ) − ψ(x‡ )∥. For the reverse, let yt = (1 − t)x + tx′ , and note that ∥yt − x∥ = tξ. Take t = 1 − (η + 2ε)/ξ and note that t ∈ [0, 1], so that yt ∈ [xx′ ] ⊂ B(v, r), and therefore there is x⋆ ∈ Λ such that ∥x⋆ − yt ∥ ≤ ε. We have ∥x⋆ − x∥ ≤ ∥yt − x∥ + ∥x⋆ − yt ∥ ≤ ξ − η − ε < ξ † , so that ∥ψ(x) − ψ(x⋆ )∥ ≤ ∥ψ(x† ) − ψ(x‡ )∥. Applying the triangle inequality and Lemma 3, we then have ∥ψ(x) − ψ(x⋆ )∥ ≥ ∥ψ(x) − ψ(x′ )∥ − ∥ψ(x′ ) − ψ(x⋆ )∥ ≥ ∥ψ(x) − ψ(x′ )∥ − C1 (∥x′ − x⋆ ∥ + ε), 15 with ∥x′ − x⋆ ∥ ≤ ∥x′ − yt ∥ + ∥yt − x⋆ ∥ ≤ η + 3ε. When ξ < ξ † , we choose t = 1 + (η + 2ε)/ξ. Because x, x′ ⊂ B(v, 3r/4), we still have yt ∈ B(v, r) because of the constraint on η. The remaining arguments are analogous. When ξ = ξ † , repeating what we just did both ways and with η = 0 yields the result. Lemma 15. Consider ψ ∶ Λ ↦ Rd isotonic, where Λ ⊂ Rd . Let V denote the convex hull of Λ. Set ε = δH (Λ, V ) and c = diam(ψ(Λ))/5 diam(Λ). Then ∥ψ(x) − ψ(x′ )∥ ≥ c∥x − x′ ∥ for all x, x′ ∈ Λ such that ∥x − x′ ∥ ≥ 4ε. Proof. We first prove that, if c > 0 and η ≥ 4ε are such that ∥ψ(x) − ψ(x′ )∥ ≤ cη for all x, x′ ∈ Λ with ∥x − x′ ∥ < η, then diam(ψ(Λ)) < c(4 diam(Λ) + η). Indeed, take x, x′ ∈ Λ. Let u = (x′ − x)/∥x′ − x∥ and L = ∥x − x′ ∥, and define yj = x + sj u where sj = j(η − 3ε) for j = 0, . . . , J ∶= ⌊L/(η − 3ε)⌋, and then let sJ+1 = L. By construction, yj ∈ [xx′ ] ⊂ V , with y0 = x and yJ+1 = x′ . Let xj ∈ Λ be such that ∥xj − yj ∥ ≤ ε, with x0 = x and xJ+1 = x′ . By the triangle inequality, ∥xj+1 − xj ∥ ≤ ∥yj+1 − yj ∥ + 2ε = sj+1 − sj + 2ε < η. Hence, J ∥ψ(x) − ψ(x′ )∥ ≤ ∑ ∥ψ(xj ) − ψ(xj+1 )∥ ≤ (J + 1)cη ≤ c j=0 Lη + cη < c(4 diam(Λ) + η), η − 3ε since η − 3ε ≥ η − 3η/4 = η/4 and L ≤ diam(Λ). Now assume that ψ is isotonic and suppose that ∥ψ(x) − ψ(x′ )∥ < c∥x − x′ ∥ for some x, x′ ∈ Λ such that η ∶= ∥x − x′ ∥ ≥ 4ε. Then we have ∥ψ(x† ) − ψ(x‡ )∥ ≤ cη when x† , x‡ ∈ Λ satisfy ∥x† − x‡ ∥ < η. We just showed that this implies that diam(ψ(Λ)) < c(4 diam(V ) + η), and we conclude using the fact that η ≤ diam(Λ). The following result is on 1-nearest neighbor interpolation. Lemma 16. Let Λ be a subset of isolated points in V ⊂ Rd and set ε = δH (Λ, V ). For any function ψ ∶ Λ ↦ Rd , define its 1-nearest neighbor interpolation as ψ̂ ∶ V ↦ Rd as ψ̂(y) = 1 ∑ ψ(x), ∣NΛ (y)∣ x∈NΛ (y) NΛ (y) ∶= arg min ∥x − y∥. (13) x∈Λ Consider the modulus of continuity of ψ, which for η > 0 is defined as ω(η) = sup{∥ψ(x) − ψ(x′ )∥ ∶ x, x′ ∈ Λ, ∥x − x′ ∥ ≤ η}. Then the modulus of continuity of ψ̂, denoted ω̂, satisfies ω̂(η) ≤ ω(η + 2ε). Moreover, for any y, y ′ ∈ V and any x, x′ ∈ Λ such that ∥x − y∥ ≤ ε and ∥x′ − y ′ ∥ ≤ ε, ∥ψ̂(y) − ψ̂(y ′ )∥ = ∥ψ(x) − ψ(x′ )∥ ± 2ω(2ε). Proof. Fix η > 0 and take y, y ′ ∈ V such that ∥y − y ′ ∥ ≤ η. We have ∥x − y∥ ≤ ε for all x ∈ NΛ (y) and ∥x′ − y ′ ∥ ≤ ε for all x′ ∈ NΛ (y ′ ), so that ∥x − x′ ∥ ≤ ∥y − y ′ ∥ + 2ε for all such x and x′ , by the triangle inequality. Therefore, ∥ψ̂(y) − ψ̂(y ′ )∥ ≤ sup {∥ψ(x) − ψ(x′ )∥ ∶ x ∈ NΛ (y), x′ ∈ NΛ (y ′ )} ≤ sup {∥ψ(x) − ψ(x′ )∥ ∶ x, x′ ∈ Λ, ∥x − x′ ∥ ≤ η + 2ε} = ω(η + 2ε). Since this is true for all y, y ′ ∈ V such that ∥y − y ′ ∥ ≤ η, we conclude that ω̂(η) ≤ ω(η + 2ε). For the second part of the lemma, we have ∥ψ̂(y) − ψ̂(y ′ )∥ = ∥ψ(x) − ψ(x′ )∥ ± ∥ψ̂(y) − ψ(x)∥ ± ∥ψ̂(y ′ ) − ψ(x′ )∥, 16 where the second term is bounded by ∥ψ̂(y) − ψ(x)∥ ≤ sup {∥ψ(x̃) − ψ(x)∥ ∶ x̃ ∈ NΛ (y)} ≤ sup {∥ψ(x̃) − ψ(x)∥ ∶ ∥x̃ − x∥ ≤ 2ε} ≤ ω(2ε), using the fact that ∥x̃ − x∥ ≤ ∥x̃ − y∥ + ∥y − x∥ ≤ 2ε, and similarly for the third term. Let V ⊂ Rd be convex. In our context, we say that f ∶ V ↦ Rd is η-approximately midlinear if ∥f ( 1 x+y ) − (f (x) + f (y))∥ ≤ η, 2 2 ∀x, y ∈ V. Lemma 17. Let V ⊂ Rd be star-shaped with respect to some point in its interior. There is a constant C depending only on V such that, for any η-approximately midlinear function f ∶ V ↦ Rd , there is a affine function T ∶ Rd ↦ Rd such that supx∈V ∥f (x) − T (x)∥ ≤ Cη. Note that, if V is a ball, then by invariance considerations, C only depends on d. Proof. This is a direct consequence of (Vestfrid, 2003, Th 1.4). We say that f ∶ V ⊂ Rd ↦ Rd is an ε-isometry if ∥x − y∥ − ε ≤ ∥f (x) − f (y)∥ ≤ ∥x − y∥ + ε, ∀x, y ∈ V. For a set V ⊂ Rd , define its thickness as θ(V ) = inf { diam(u⊺ V ) ∶ u ∈ Rd , ∥u∥ = 1}. Recalling the definition of ρ in (7), we note that θ(V ) ≥ ρ(V ), but that the two are distinct in general. Lemma 18. Let V ⊂ Rd be compact and such that θ(V ) ≥ η diam(V ) for some η > 0. There is a constant C depending only on d such that, if f ∶ V ↦ Rd is an ε-isometry, then there is an isometry R ∶ Rd ↦ Rd such that maxx∈V ∥f (x) − R(x)∥ ≤ Cε/η. Proof. This is a direct consequence of (Alestalo et al., 2001, Th 3.3). Lemma 19. Let T ∶ Rd ↦ Rd be an affine function that transforms a regular simplex of edge length 1 into an η-approximate regular simplex of maximum edge length λ > 0. There is a constant C, depending only on d, and an isometry R, such that ∥T (x) − λR(x)∥ ≤ Cλη for all x ∈ B(0, 1). Proof. By invariance, we may assume T is linear and that the regular simplex is formed by 0, z1 , . . . , zd and has edge length 1. Letting wi = T (zi ), we have that 0, w1 , . . . , wd form an ηapproximate regular simplex of maximum edge length λ ∶= maxi ∥wi ∥. Lemma 11 gives 0, w1′ , . . . , wd′ forming a regular simplex of edge length λ such that maxi ∥wi − wi′ ∥ ≤ C1 λη for some constant C1 . Let R be the orthogonal transformation such that R(zi ) = wi′ /λ for all i ∈ [d]. We have ∥T (zi ) − λR(zi )∥ = ∥wi − wi′ ∥ ≤ C1 λη for all i. In matrix notation, letting Z ∶= [z1 . . . zd ], we have ¿ Ád √ √ À∑ ∥T z − λRz ∥2 ≤ d max ∥T z − λRz ∥ ≤ d C λη. ∥T Z − λRZ∥ ≤ ∥T Z − λRZ∥ = Á F i i=1 i i∈[d] i i 1 At the same time, ∥T Z − λRZ∥ ≥ ∥T − λR∥/∥Z −1 ∥ with ∥Z −1√∥ = 1/σd (Z) = 1/σd ([0z1 ⋯zd ]) being a positive constant depending only on d. Hence, ∥T − λR∥ ≤ ( d/σd (Z))C1 λη =∶ C2 λη. 17 Lemma 20. Suppose that S1 , S2 ∶ Rd ↦ Rd are two affinities such that maxx∈B(y,r) ∥S1 (x)−S2 (x)∥ ≤ η for some y ∈ Rd and r > 0. Then ∥S1 (x) − S2 (x)∥ ≤ 2η∥x − y∥/r + η for all x ∈ Rd . Proof. By translation and scale invariance, assume that y = 0 and r = 1. Let Li = Si − Si (0). For x ∈ B(0, 1), we ∥L1 (x) − L2 (x)∥ ≤ ∥S1 (x) − S2 (x)∥ + ∥S1 (0) − S2 (0)∥ ≤ 2η. Hence, for x ∈ Rd , ∥L1 (x) − L2 (x)∥ ≤ 2η∥x∥, which in turn implies that ∥S1 (x) − S2 (x)∥ ≤ ∥L1 (x) − L2 (x)∥ + ∥S1 (0) − S2 (0)∥ ≤ 2η∥x∥ + η. 3.3 Proof of Theorem 3 Without loss of generality, we may assume that Dn ∶= diam(φn (Ωn )) ≥ 1. Indeed, suppose that Dn < 1, but different from 0, for otherwise φn is a degenerate similarity and the result follows. Let φ̃n = Dn−1 φn , which is isotonic on Ωn and satisfies diam(φ̃n (Ωn )) = 1. If the result is true for φ̃n , there is a similarity S̃n such that maxx∈Ωn ∣φ̃n (x) − S̃n (x)∣ ≤ Cεn for some constant C. (We implicitly assume that the set φn (Ωn ) contains the origin, so that φ̃n (Ωn ) remains bounded.) We then have maxx∈Ωn ∣φn (x) − Sn (x)∣ ≤ CDn εn ≤ Cεn , where Sn ∶= Dn S̃n is also a similarity. Let r = ρ(U ), so that there is some u⋆ such that B(u⋆ , r) ⊂ U . Let Λn = Ωn ∩ B(u⋆ , r/2) and δn = diam(φn (Λn )). Let w be any unit-norm vector and define y± = u⋆ ± (r/2− εn )w. Let x± ∈ Ωn be such that ∥x± − y± ∥ ≤ εn . Necessarily, x± ∈ Λn because the distance from y± to ∂B(u⋆ , r/2) exceeds εn . Note that ∥x− − x+ ∥ ≥ r1 ∶= r − 4εn . By isotonicity, ∥φn (x) − φn (x′ )∥ ≤ ∥φn (x− ) − φn (x+ )∥ ≤ δn , whenever ∥x − x′ ∥ < r1 . (14) Let y1 , . . . , yK be a (r1 /3)-packing of U , so that K ≤ C(diam(U )/r)d for some constant C > 0. Let xik ∈ Ωn be such that ∥xik − yk ∥ ≤ εn , so that U ⊂ ⋃k∈[K] B(yk , r1 /3) ⊂ ⋃k∈[K] B(xik , r2 ), where r2 ∶= r1 /3 + εn . Let zk = xik for clarity. Take x, x′ ∈ Ωn . Because U is open, it is path-connected, so there is a continuous curve γ ∶ [0, 1] ↦ U such that γ(0) = x and γ(1) = x′ . Let k0 ∈ [K] be such that x ∈ B(zk0 , r2 ) and s0 = 0. Then for j ≥ 0, let sj+1 = inf{s > sj ∶ γ(s) ∉ ⋃l∈[j] B(zkl , r2 )}, and let kj+1 ∈ [K] be such that ∥zkj+1 − γ(sj+1 )∥ ≤ εn . Let J = min{j ∶ sj+1 = ∞}, which is indeed finite. By construction, ∥zkj − zkj+1 ∥ ≤ 2r2 < r1 when εn < r/10. By (14), we have ∥φn (zkj ) − φn (zkj+1 )∥ ≤ δn . Thus, by the triangle inequality, ∥φn (x) − φn (x′ )∥ ≤ Jδn ≤ Kδn . This being true for all x, x′ ∈ Ωn , this prove that δn ≥ Dn /K ∝ Dn (diam(U )/r)−d . 1-NN interpolation. Let φ̂n denote the 1-NN interpolation of φn as in (13). We claim that there is a C0⋆ ∝ Dn /r and c⋆0 ∝ (diam(U )/r)−d Dn /r such that φ̂n satisfies the following properties: for all y, y ′ , y † , y ‡ ∈ U , ∥y − y ∥ < ∥y − y ∥ − 4εn ⇒ ∥φ̂n (y) − φ̂n (y )∥ ≤ ∥φ̂n (y ) − φ̂n (y ′ and also and ∥φ̂n (y) − φ̂n (y ′ )∥ ≤ C0⋆ (∥y − y ′ ∥ + εn ), † ‡ ′ † (15) ‡ )∥ + C0⋆ εn , ∥φ̂n (y) − φ̂n (y ′ )∥ ≥ c⋆0 ∥y − y ′ ∥ − C0⋆ εn , if y, y ′ ∈ B(u⋆ , r/2) satisfy ∥y − y ′ ∥ ≥ 10εn , ∥y − y ′ ∥ = ∥y † − y ‡ ∥ ± η ⇒ ∥φ̂n (y) − φ̂n (y ′ )∥ = ∥φ̂n (y † ) − φ̂n (y ‡ )∥ ± C0⋆ (η + εn ), if y, y ′ ∈ B(u⋆ , r/2), εn < r/120 and 0 ≤ η ≤ r/5. Indeed, let x, x′ , x† , x‡ ∈ Ωn such that ∥x − y∥, ∥x′ − y ′ ∥, ∥x† − y † ∥, ∥x‡ − y ‡ ∥ ≤ εn . (16) (17) (18) 18 For (15), we start by applying Lemma 16 to get ∥φ̂n (y) − φ̂n (y ′ )∥ = ∥φn (x) − φn (x′ )∥ ± 2ωn (2εn ) ≤ ωn (∥x − x′ ∥) + 2ωn (2εn ) ≤ ωn (∥y − y ′ ∥ + 2εn ) + 2ωn (2εn ), where ωn is the modulus of continuity of φn . We then use Lemma 3, which gives that ωn (η) ≤ Cη for all η and some C ∝ Dn /r, to get ωn (∥y − y ′ ∥ + 2εn ) ± 2ωn (2εn ) ≤ C(∥y − y ′ ∥ + 6εn ). For (16), we first note that ∥x − x′ ∥ < ∥x† − x‡ ∥ by the triangle inequality, which in turn implies that ∥φn (x) − φn (x′ )∥ ≤ ∥φn (x† ) − φn (x‡ )∥ since φn is isotonic. We then apply Lemma 16 to get that ∥φ̂n (y) − φ̂n (y ′ )∥ ≤ ∥φ̂n (y † ) − φ̂n (y ‡ )∥ + 4ωn (2εn ), and conclude with Lemma 3 as for (15). For (17), we may apply Lemma 15 with Λn . Let V be the convex hull of Λn , so that V ⊂ B(u⋆ , r/2). Let z be a point in that ball. If z ≠ u⋆ , let w = (u⋆ − z)/∥u⋆ − z∥, and if z = u⋆ , let w be any unit-norm vector. Define z ′ = z + εn w and notice that the distance from z ′ to ∂B(u⋆ , r/2) exceeds εn . Therefore, if x ∈ Ωn is such that ∥z ′ − x∥ ≤ εn , then necessarily, x ∈ Λn . We then note that ∥z − x∥ ≤ 2εn . We conclude that δH (Λn , V ) ≤ 2εn . Since ∥x − x′ ∥ ≥ ∥y − y ′ ∥ − 2εn ≥ 4(2εn ), we get that ∥φn (x) − φn (x′ )∥ ≥ c∥x − x′ ∥, with c ∶= diam(φn (Λn ))/5 diam(Λn ) ≥ δn /5r. We then apply Lemma 16 to obtain ∥φ̂n (y) − φ̂n (y ′ )∥ ≥ c∥x − x′ ∥ − 2ωn (2εn ) ≥ c∥y − y ′ ∥ − 2(c + C)εn , using Lemma 3 as for (15). For (18), note that x, x′ ∈ B(u⋆ , r/2 + εn ) ⊂ B(u⋆ , 3r/4), and ∥x − x′ ∥ = ∥x† − x‡ ∥ ± (η + 4εn ) by the triangle inequality. By Lemma 14 — where the constant there is denoted here by C ′ ∝ Dn /r — this implies that ∥φn (x) − φn (x′ )∥ = ∥φn (x† ) − φn (x‡ )∥ ± C ′ (η + εn ) when η + 4εn < r/4 − 2εn , which is true when εn < r/120 and η ≤ r/5. We then apply Lemma 16 together with Lemma 3, as for (15). CASE d = 1. This case is particularly simple. Note that U is a bounded open interval of R. We show that the function φ̂n is approximately midlinear on U . Take x, y ∈ U and define µ = (x + y)/2. By the fact that φ̂n takes its values in R, and (18), we have ∣ 21 (φ̂n (x) + φ̂n (y)) − φ̂n (µ)∣ = 21 ∣∣φ̂n (x) − φ̂n (µ)∣ − ∣φ̂n (y) − φ̂n (µ)∣∣ ≤ C0⋆ εn /2, when εn /r is small enough. Hence, φ̂n is (C0⋆ εn )-approximate midlinear on U . By the result of Vestfrid (2003), namely Lemma 17, there is C ∝ 1 — since U is a ball — and an affine function Tn such that maxy∈U ∣φ̂n (y) − Tn (y)∣ ≤ CC0⋆εn . Since all affine transformations from R to R are (possibly degenerate) similarities, we conclude. CASE d ≥ 2. For the remaining of this subsection, we assume that d ≥ 2. Approximate midlinearity. We show that there is a constant C such that φ̂n is locally Cεn approximately midlinear. Take x, y ∈ B(u⋆ , r/4), and let µ = (x + y)/2. Let t > 0 be a constant to be set large enough later. If ∥x − y∥ ≤ tεn , then by (15), φ̂n (x), φ̂n (y) ∈ B(φ̂n (µ), C0⋆ (t/2 + 1)εn ), so that 1 ∥φ̂n (µ) − (φ̂n (x) + φ̂n (y))∥ ≤ C0⋆ (t/2 + 1)εn . 2 Therefore, assume that ∥x − y∥ ≥ tεn . Let z1 , . . . , zd be constructed as in the proof of Theorem 1. By construction, both x, z1 , . . . , zd and y, z1 , . . . , zd form regular simplexes, and µ is the barycenter of z1 , . . . , zd . By Lemma 6, for any i ≠ j, √ √ √ ∥zi − µ∥ = (d − 1)/2d ∥zi − zj ∥ = (d − 1)/2d 2d/(d + 1) ∥x − µ∥ ≤ ∥x − y∥/2, 19 which coupled with the fact that x, y ∈ B(u⋆ , r/4) yields that zi ∈ B(u⋆ , r/2) for all i. Now, √ let z0 = x. ⋆ By (18), we have mini≠j ∥φ̂n (zi ) − φ̂n (zj )∥ ≥ maxi,j ∥φ̂n (zi ) − φ̂n (zj )∥ − C0 εn . Let cd = d/(2d + 2). By (17) and Lemma 6, ∥φ̂n (zi ) − φ̂n (zj )∥ ≥ c⋆0 ∥zi − zj ∥ − C0⋆ εn = c⋆0 cd ∥x − y∥ − C0⋆εn ≥ (c⋆0 cd t − C0⋆ )εn . Hence, assuming t ≥ 2C0⋆ /c⋆0 cd , we have mini≠j ∥φ̂n (zi ) − φ̂n (zj )∥ ≥ (1 − η) maxi,j ∥φ̂n (zi ) − φ̂n (zj )∥, where η ∶= 2C0⋆ /(c⋆0 cd t). In that case, φ̂n (x), φ̂n (z1 ), . . . , φ̂n (zd ) form a η-approximate regular simplex. By symmetry, the same is true of φ̂n (y), φ̂n (z1 ), . . . , φ̂n (zd ). Define λ = ∥φ̂n (x) − φ̂n (y)∥. By Lemma 6, ∥zi − zj ∥ = cd ∥x − y∥ < ∥x − y∥ − 4εn when t > 4/(1 − cd ), since ∥x − y∥ ≥ tεn and cd < 1. By (16), this implies that ∥φ̂n (zi ) − φ̂n (zj )∥ ≤ λ + C0⋆εn . By (17), λ ≥ (c⋆0 t − C0⋆ )εn , so that λ + C0⋆ εn ≤ 2λ since we already assumed that t ≥ 2C0⋆ /c⋆0 cd > 2C0⋆ /c⋆0 . For a ∈ {x, y, µ}, ∥a − zi ∥ is constant in i ∈ [d]. Therefore, by (18), mini ∥φ̂n (a) − φ̂n (zi )∥ ≥ maxi ∥φ̂n (a) − φ̂n (zi )∥ − C0⋆ εn . Define ξa as the orthogonal projection of φ̂n (a) onto the affine space A ∶= Aff(φ̂n (z1 ), . . . , φ̂n (zd )) and let δa = ∥φ̂n (a) − ξa ∥. By the Pythagoras theorem, we have ∥ξa − φ̂n (zi )∥2 = ∥φ̂n (a) − φ̂n (zi )∥2 − δa2 . In particular, max ∥ξa − φ̂n (zi )∥2 − min ∥ξa − φ̂n (zi )∥2 = max ∥φ̂n (a) − φ̂n (zi )∥2 − min ∥φ̂n (a) − φ̂n (zi )∥2 i i i i ≤ 2C0⋆ εn min ∥φ̂n (a) − φ̂n (zi )∥ + (C0⋆ εn )2 ≤ C1 εn , i where C1 ∶= 2C0⋆ Dn + C0⋆ r, once εn ≤ r. Let ζ denotes the barycenter of φ̂n (z1 ), . . . , φ̂n (zd ). Assume that t is sufficiently large that η ≤ 1/C2 , where C2 ∝ 1 is the constant of Lemma 13. By that lemma, and the fact that φ̂n (z1 ), . . . , φ̂n (zd ) form a η-approximate regular simplex of maximum edge length bounded by λ, we have ∥ξa − ζ∥ ≤ C2 λC1 εn . Let L be the line passing through ζ and perpendicular to A. We just proved that φ̂n (x), φ̂n (y), φ̂n (µ) are within distance C3 λεn from L, where C3 ∶= C1 C2 . Let ξ denote the orthogonal projection of φ̂n (µ) onto (φ̂n (x)φ̂n (y)). Since ∥x − µ∥ = ∥y − µ∥, we can apply (18) to get ∣∥ξ − φ̂n (x)∥2 − ∥ξ − φ̂n (y)∥2 ∣ = ∣∥φ̂n (µ) − φ̂n (x)∥2 − ∥φ̂n (µ) − φ̂n (y)∥2 ∣ = ∣∥φ̂n (µ) − φ̂n (x)∥ + ∥φ̂n (µ) − φ̂n (y)∥∣ × ∣∥φ̂n (µ) − φ̂n (x)∥ − ∥φ̂n (µ) − φ̂n (y)∥∣ ≤ 4C0⋆ λεn , using the fact that max(∥φ̂n (µ) − φ̂n (x)∥, ∥φ̂n (µ) − φ̂n (y)∥) ≤ λ + C0⋆ εn ≤ 2λ, due to (16) and ∥x − µ∥ = ∥y − µ∥ = 21 ∥x − y∥ < ∥x − y∥ − 4εn when t is large enough. By Lemma 13, we then obtain ∥ξ − 21 (φ̂n (x) + φ̂n (y))∥ ≤ C4 λεn for some constant C4 ∝ C0⋆ . In particular, recalling that λ = ∥φ̂n (x) − φ̂n (y)∥, this implies that ξ ∈ [φ̂n (x)φ̂n (y)] when εn ≤ 1/2C4 . It remains to argue that φ̂n (µ) is close to ξ. We already know that φ̂n (x), φ̂n (y), φ̂n (µ) are within distance C3 λεn from L, and by convexity, the same must be true of ξ. Let M = (φ̂n (x)φ̂n (y)) and θ = ∠(L, M ). Let PM denote the orthogonal projection onto M , when M is a linear subspace. By Pythagoras theorem, λ2 = ∥φ̂n (x) − φ̂n (y)∥2 = ∥PL (φ̂n (x) − φ̂n (y))∥2 + ∥PL⊥ (φ̂n (x) − φ̂n (y))∥2 ≤ (cos θ)2 λ2 + (2C3 λεn )2 , 20 implying that sin θ ≤ 2C2 εn . Since ∥PL − PM ∥ = sin θ and φ̂n (µ) − ξ is parallel to M , we also have ∥φ̂n (µ) − ξ∥2 = ∥PL (φ̂n (µ) − ξ)∥2 + ∥PL⊥ (φ̂n (µ) − ξ)∥2 ≤ (sin θ)2 ∥φ̂n (µ) − ξ∥2 + (2C3 λεn )2 , √ so that ∥φ̂n (µ) − ξ∥ ≤ 2C3 λεn / cos θ ≤ 2C3 λεn / 1 − (2C3 εn )2 ≤ C5 λεn , for some constant C5 ∝ C3 , once C3 εn is small enough. We conclude that ∥φ̂n (µ) − 12 (φ̂n (x) + φ̂n (y))∥ ≤ (C4 + C5 )λεn , by the triangle inequality. Approximate affinity. We now know that φ̂n is Cεn -approximate midlinear on B(u⋆ , r/4) for some constant C ∝ C0⋆(Dn + r) ∝ C0⋆ (diam(Q) + r). This implies, by the result of Vestfrid (2003), that is Lemma 17, that there is an affine function Tn such ∥φ̂n (x) − Tn (x)∥ ≤ C1⋆ εn for all x ∈ W , for some constant C1⋆ ∝ rC ∝ rC0⋆(diam(Q) + r). Approximate similarity. (Reinitialize the constants Ck , k ≥ 1.) We saw above that φ̂n transforms the regular simplex z0 (= x), z1 , . . . , zd with height denoted h satisfying h ≥ tεn /2 into a η-approximate one, where η = 2C0⋆ /(c⋆0 cd t). In what follows, choose these points so that they are all in B(u⋆ , r/2) and the simplex has height h ≥ r/8. (From here on, reinitialize the variables x, y, λ, etc.) We can then take t = r/4εn , yielding η = C1 εn for a constant C1 ∝ C0⋆ /(c⋆0 r). By the triangle inequality, we have min ∥Tn (zi ) − Tn (zj )∥ ≥ min ∥φ̂n (zi ) − φ̂n (zj )∥ − 2C1⋆ εn i≠j i≠j ≥ (1 − C1 εn ) max ∥φ̂n (zi ) − φ̂n (zj )∥ − 2C1⋆ εn i≠j ≥ max ∥Tn (zi ) − Tn (zj )∥ − (4C1⋆ + C1 δn )εn . i≠j By the triangle inequality and (17), γn ∶= max ∥Tn (zi ) − Tn (zj )∥ ≥ max ∥φ̂n (zi ) − φ̂n (zj )∥ − 2C1⋆ εn i,j ≥ i,j ⋆ c0 max ∥zi i,j − zj ∥ − C0⋆ εn − 2C1⋆ εn ≥ c⋆0 r/8 − (C0⋆ + 2C1⋆ )εn . Hence, we find that Tn (z0 ), . . . , Tn (zd ) form a C2 εn -approximate regular simplex, where C2 ∶= (4C1⋆ + C1 δn )/(c⋆0 r/8 − (C0⋆ + 2C1⋆ )εn ). Note that its maximum edge length is bounded as follows: γn ≤ max ∥φ̂n (zi ) − φ̂n (zj )∥ + 2C1⋆ εn ≤ δn + 2C1⋆ εn ≤ 2δn , i,j when 2C1⋆ εn ≤ δn . By Lemma 19, there is a constant C3 > 0 and an isometry Rn⋆ , such that we have maxx∈W ∥Tn (x) − λn Rn⋆ (x)∥ ≤ C3 λn C2 εn , where λn ∶= γn /h. Because r/8 ≤ h ≤ r and the bounds on γn above, there is a constant C2⋆ ≥ 1 such that 1/C2⋆ ≤ λn ≤ C2⋆ . (19) This implies that ∥φ̂n (x) − λn Rn⋆ (x)∥ ≤ ∥φ̂n (x) − Tn (x)∥ + ∥Tn (x) − λn Rn⋆ (x)∥ ≤ (C1⋆ + C3 C2 C2⋆ )εn =∶ C3⋆ εn . (20) Covering and conclusion. (Reinitialize the constants Ck , k ≥ 1.) Let u1 = u⋆ and let u2 , . . . , uK ∈ U be such that u1 , . . . , uK form a maximal (r/16)-packing of U . (The number 16 is not essential 21 here, but will play a role in the proof of Theorem 4.) Note that U = U1 ∪ ⋯ ∪ UK where Uk ∶= U ∩ B(uk , r/4), and note that U⋆ ∶= U1 ⊂ U . For u, u′ ∈ Uk , there are w, w′ ∈ U⋆ such that ∥w − w′ ∥ = ∥u − u′ ∥. Define φ̃n = φ̂n /λn . By (18), and then (19)-(20), we have ∥φ̃n (u) − φ̃n (u′ )∥ = ∥φ̃n (w) − φ̃n (w′ )∥ ± C0⋆ εn /λn = ∥w − w′ ∥ ± (C0⋆ + C3⋆ )εn /C2⋆ =∶ ∥w − w′ ∥ ± C1 εn . Let θ(Uk ) , (21) diam(Uk ) which is strictly positive. The result of Alestalo et al. (2001), namely Lemma 18, gives a constant C2 ∝ ξ1 and an isometry Rk such that maxu∈Uk ∥φ̃n (u) − Rk (u)∥ ≤ C2 εn . Let 1 (22) ξ2 = min {ρ(Uk ∩ Uk′ ) ∶ Uk ∩ Uk′ ≠ ∅}. 2 Take k, k ′ ∈ [K] such that Uk ∩ Uk′ ≠ ∅, so that there is u ∈ U such that B(u, ξ2 ) ⊂ Uk ∩ Uk′ . Since ξ1 = min k max ∥Rk (x) − Rk′ (x)∥ ≤ max ′ ∥Rk (x) − φ̃n (x)∥ + ∥φ̃n (x) − Rk′ (x)∥ x∈Uk ∩Uk x∈B(u,ξ2 ) ≤ max ∥Rk (x) − φ̃n (x)∥ + max ∥φ̃n (x) − Rk′ (x)∥ ≤ 2C2 εn , x∈Uk x∈Uk′ we have ∥Rk (x) − Rk′ (x)∥ ≤ (2∥x − u∥/ξ2 + 1)2C2 εn for all x ∈ Rd , by Lemma 20. Hence, ∥Rk (x) − Rk′ (x)∥ ≤ (2 diam(U )/ξ2 + 1)2C2 εn =∶ C3 εn for all x ∈ U . If instead Uk ∩ Uk′ = ∅, we do as follows. Since U is connected, there is a sequence k0 = k, k1 , . . . , km = k′ in [K], such that Uki ∩ Uki+1 ≠ ∅. We thus have maxx∈U ∥Rki (x) − Rki+1 (x)∥ ≤ C3 εn . By the triangle inequality, we conclude that maxx∈U ∥Rk (x) − Rk′ (x)∥ ≤ KC3 εn for any k, k′ ∈ [K]. Noting that R1 = Rn⋆ (since U1 = U⋆ ), for any k ∈ [K] and x ∈ Uk , ∥φ̃n (x) − Rn⋆ (x)∥ ≤ ∥Rk (x) − R1 (x)∥ + C2 εn ≤ (KC3 + C2 )εn . We conclude that, for any x ∈ U , ∥φ̂n (x) − λn Rn⋆ (x)∥ ≤ (KC3 + C2 )λn εn ≤ (KC3 + C2 )C2⋆ εn =∶ C4 εn . (23) This concludes the proof when d ≥ 2. A refinement of the constant. Assume now that U = U h for some h > 0. Tracking the constants above, we see that they all depend only on (d, ρ(U ), diam(U ), diam(Q)), as well as ξ1 and ξ2 defined in (21) and (22), respectively. We note that diam(Uk ) ≤ r and ρ(Uk ) ≥ min(r/2, h) by Lemma 21, so that ξ1 ≥ min(r/2, h)/r. To bound ξ2 , we can do as we did at the beginning of this section, so that at the end of that section, we can restrict our attention to chains k0 , . . . , km where ∥ukj − ukj+1 ∥ ≤ 2r/16 = r/8. To be sure, fix k, k′ ∈ [K] and let γ ∶ [0, 1] ↦ U be a curve such that γ(0) = uk and γ(1) = uk′ . Define s0 = 0 and then sj+1 = inf{s > sj ∶ ∥γ(s) − ukj ∥ > r/16}, and let kj+1 ∈ [K] be such that ∥γ(sj+1 ) − ukj+1 ∥ ≤ r/16, which is well-defined since (uk , k ∈ [K]) is a (r/16)-packing of U . We then have ∥ukj − ukj+1 ∥ ≤ ∥ukj − γ(sj )∥ − ∥γ(sj+1 ) − ukj+1 ∥ ≤ r/16 + r/16 = r/8. We can therefore redefine ξ2 in (22) as 12 min{ρ(Uk ∩ Uk′ ) ∶ ∥uk − uk′ ∥ ≤ r/8}. Because U = U h , for each k ∈ [K], there is vk such that uk ∈ B(vk , min(r/16, h)) ⊂ U . By the triangle inequality, B(vk , min(r/16, h)) ⊂ Uk′ when ∥uk − uk′ ∥ ≤ r/8, so that ξ2 ≥ min(r/16, h). So we see that everything depends on (d, h, ρ(U ), diam(U ), diam(Q)). The second part of the theorem now follows by invariance considerations. 22 3.4 Proof of Lemma 5 Let c = ess inf U f and C = ess supU f , which by assumption belong to (0, ∞). Fix i ∈ [n] and let Ni = #{j ≠ i ∶ ∥xj − xi ∥ ≤ r}. For j ≠ i, pi (j) ∶= P(∥xj − xi ∥ ≤ r) = ∫B(xi ,r) f (u)du. For an upper bound, we have pi (j) ≤ C Vol(B(xi , r) ∩ U ) ≤ C Vol(B(xi , r)) = Cζd r d =∶ Q, where Vol denotes the Lebesgue measure in Rd and ζd is the volume of the unit ball in Rd . Hence, P(Ni > 2(n − 1)Q) ≤ P(Bin(n − 1, Q) > 2(n − 1)Q) ≤ e−(n−1)Q/3 by Bennett’s inequality for the binomial distribution. By the union bound, we conclude that maxi Ni ≤ 2(n − 1)Q with probability at least 1 − ne−(n−1)Q/3 , which tends to 1 if nr d ≥ C0 log n and C0 > 0 is sufficiently large. For a lower bound, we use the following lemma. Lemma 21. Suppose U ⊂ Rd is open and such that U = U h for some h > 0. Then for any x ∈ U and any r > 0, B(x, r) ∩ U contains a ball of radius min(r, h)/2. Moreover, the closure of that ball contains x. Proof. By definition, there is y ∈ U such that x ∈ B(y, h) ⊂ U . We then have B(x, r) ∩ U ⊃ B(x, r) ∩ B(y, h), so it suffices to show that the latter contains a ball of radius min(r, h)/2. By symmetry, we may assume that r ≤ h. If ∥x − y∥ ≤ r/2, then B(x, r/2) ⊂ B(y, h) and we are done. Otherwise, let z = (1−t)x+ty with t ∶= r/2∥x−y∥ ∈ (0, 1), and note that B(z, r/2) ⊂ B(x, r)∩B(y, h) and x ∈ ∂B(z, r/2). Now that Lemma 21 is established, we apply it to get pi (j) ≥ c Vol(B(xi , r) ∩ U ) ≥ cζd (min(r, h)/2)d =∶ q. Hence, P(Ni < (n − 1)q/2) ≤ P(Bin(n − 1, q) < (n − 1)q/2) ≤ e−(6/7)(n−1)q . By the union bound, we conclude that mini Ni ≥ (n − 1)q/2 with probability at least 1 − ne−(6/7)(n−1)q , which tends to 1 if nr d ≥ C1 log n and C1 > 0 is sufficiently large. (Recall that h is fixed.) 3.5 More auxiliary results We list here a few additional of auxiliary results that will be used in the proof of Theorem 4. For V ⊂ Rd and x, x′ ∈ V , define the intrinsic metric δV (x, x′ ) = sup {L ∶ ∃γ ∶ [0, L] ↦ V, 1-Lipschitz, with γ(0) = x, γ(L) = x′ }, where γ is 1-Lipschitz if ∥γ(s) − γ(t)∥ ≤ ∣s − t∣ for all s, t ∈ [0, L]. If no such curve exists, set δV (x, x′ ) = ∞. The intrinsic diameter of V is defined as sup{δV (x, x′ ) ∶ x, x′ ∈ V }. We note that, if L ∶= δV (x, x′ ) < ∞, then there is a curve γ ⊂ V̄ with length L joining x and x′ . Recall that a curve with finite length is said to be rectifiable. See (Burago et al., 2001) for a detailed account of intrinsic metrics. For U ⊂ Rd and h > 0, let U ⊖h = {x ∈ U ∶ B(x, h) ⊂ U }. This is referred to as an erosion (of the set U ) in mathematical morphology. Lemma 22. If U ⊂ Rd is open and connected, then for each pair of points x, x′ ∈ U , there is h > 0 and a rectifiable curve within U ⊖h joining x and x′ . 23 Proof. Take x, x′ ∈ U . By taking an intersection with an open ball that contains x, x′ , if needed, we may assume without loss of generality that U is bounded. Since every connected open set in a Euclidean space is also path-connected (Waldmann, 2014, Example 2.5.13), there is a continuous curve γ ∶ [0, 1] ↦ U such that γ(0) = x and γ(1) = x′ . A priori, γ could have infinite length. However, γ (≡ γ([0, 1])) is compact. For each t ∈ [0, 1], let r(t) > 0 be such that Bt ∶= B(γ(t), r(t)) ⊂ U . Since γ ⊂ ⋃t∈[0,1] Bt , there is 0 ≤ t1 < ⋅ ⋅ ⋅ < tm ≤ 1 such that γ ⊂ ⋃j∈[m] Btj . Since γ is connected, necessarily, for all j ∈ [m−1] there is sj ∈ [tj , tj+1 ] such that γ(sj ) ∈ Btj ∩Btj+1 . Let s0 = 0 and sm = 1. Then [γ(sj )γ(sj+1 )] ⊂ Btj+1 ⊂ U for all j ∈ 0, . . . , m − 1, and therefore the polygonal line defined by x = γ(s0 ), γ(s1 ), . . . , γ(sm−1 ), γ(sm ) = x′ is inside ⋃j∈[m] Btj ⊂ U ⊖r where r ∶= minj∈[m] r(tj ) > 0. By construction, this polygonal line joins x and x′ , and is also rectifiable since it has a finite number of vertices. Lemma 23. Suppose U ⊂ Rd is bounded, connected, and such that U = U h for some h > 0. Then ′ there is h‡ > 0 such that, for all h′ ∈ [0, h‡ ], the intrinsic diameter of U ⊖h is finite. Proof. Let V = U ⊖h . By assumption, for all x ∈ U , there is y ∈ V such that x ∈ B(y, h) ⊂ U . In particular, U ⊃ V ≠ ∅. Let V1 be a connected component of V . Pick y1 ∈ V1 and note that B1 ∶= B(y1 , h) ⊂ U by definition, and also B1 ⊂ V1 because B1 is connected. Let ζd be the volume of the unit ball in Rd . Since the connected components are disjoint and each has volume at least ζd hd while U has volume at most ζd (diam(U )/2)d , V can have at most ⌈(diam(U )/2h)d ⌉ connected components, which we now denote by V1 , . . . , VK . Pick yk ∈ Vk for each k ∈ [K]. Applying Lemma 22, for each pair of distinct k, k ′ ∈ [K], there is a rectifiable (i.e., finite-length) path γk,k′ ⊂ U joining yk and yk′ . By Lemma 22, the length of γk,k′ , denoted Dk,k′ , is finite, and there is hk,k′ > 0 such that γk,k′ ⊂ U ⊖hk,k′ . Let D‡ = maxk,k′ ∈[K] Dk,k′ and h‡ = mink,k′∈[K] hk,k′ . We now show that each connected component Vk has finite diameter in the intrinsic metric of V ′ ∶= U ⊖h/2 . Since Vk is bounded, there is x1 , . . . , xm ∈ Vk such that Vk ⊂ ⋃j∈[mk ] Qj , where Qj ∶= B(xj , h/2) ⊂ V ′ . Take any x, x′ ∈ Vk . Let j, j ′ ∈ [mk ] be such that x ∈ Qj and x′ ∈ Qj ′ . Since Vk is connected, there is a sequence j = j0 , j1 , . . . , jSk = j ′ ∈ [mk ] such that Qjs ∩ Qjs+1 ≠ ∅ for all s = 0, . . . , Sk . Choose zs ∈ Qjs ∩ Qjs+1 and let z0 = x and zSk = x′ . Then [zs zs+1 ] ⊂ Qjs+1 for all s. k Qjs ⊂ V ′ , it joins x Let L be the polygonal line formed by z0 , . . . , zSk . By construction, L ⊂ ⋃Ss=0 ′ ′ and x , and has length at most (Sk + 1)2h. Hence, δV ′ (x, x ) ≤ (Sk + 1)2h ≤ 2(mk + 1)h. This being valid for all x, x′ ∈ Vk , we proved that Vk has diameter at most Dk ∶= 2(mk + 1)h in the intrinsic metric of V ′ . Let D⋆ = maxk∈[K] Dk . Now take h† ∈ [0, h‡ ] and any x, x′ ∈ U ⊖h† . Let y, y ′ ∈ V be such that x ∈ B(y, h) and x′ ∈ B(y ′ , h). Let k, k ′ ∈ [K] be such that y ∈ Vk and y ′ ∈ Vk′ . There are curves γ, γ ′ ⊂ V ′ of length at most D⋆ such that γ joins y and yk , while γ ′ joins y ′ and yk′ . We then join yk and yk′ with γk,k′ All together, we have the curve [xy] ∪ γ ∪ γk,k′ ∪ γ ′ ∪ [y ′ x′ ], which joins x and x′ , lies entirely in U ⊖h† , and has length bounded by h + D⋆ + D‡ + D⋆ + h =∶ D. And this is true for any pair of such points. Lemma 24. Suppose that S1 , S2 ∶ Rd ↦ Rd are two affinities such that maxj ∥S1 (zj ) − S2 (zj )∥ ≤ ε, where z0 , . . . , zd form in a η-approximate regular simplex with minimum edge length at least λ. There is C > 0 depending only on d such that, if η ≤ 1/C, then ∥S1 (x) − S2 (x)∥ ≤ Cε∥x − z0 ∥/λ + ε for all x ∈ Rd . Proof. Note that this is closely related to Lemma 20. By translation and scale invariance, assume that z0 = 0 and λ = 1. Let Li = Si − Si (0). We have ∥L1 (zj ) − L2 (zj )∥ ≤ ∥S1 (zj ) − S2 (zj )∥ + ∥S1 (0) − 24 S2 (0)∥ ≤ 2ε. Let Z denote the matrix with columns z1 , . . . , zd . In matrix notation, we have ∥(L1 − L2 )Z∥F = √ √ ∑ ∥(L1 − L2 )zj ∥2 ≤ 2 dε. j We also have ∥(L1 − L2 )Z∥F ≥ ∥(L1 − L2 )Z∥ ≥ σd (Z)∥L1 − L2 ∥, and by Lemma 12, σd (Z) = σd ([z0 , Z]) ≥ 1/C1 when η ≤ 1/C1 , where C1 depends only on d. In that case, ∥L1 − L2 ∥ ≤ C2 ε for another constant C2 . Equivalently, for x ∈ Rd , ∥L1 (x) − L2 (x)∥ ≤ C2 ε∥x∥, which in turn implies that ∥S1 (x) − S2 (x)∥ ≤ ∥L1 (x) − L2 (x)∥ + ∥S1 (0) − S2 (0)∥ ≤ C2 ε∥x∥ + ε. 3.6 Proof of Theorem 4 Because φn is bounded independently of n, we may assume without loss of generality that C0 εn ≤ rn and C0 rn ≤ h for all n, where C0 ≥ 1 will be chosen large enough later on. Take y ∈ U and let Ωy = Ωn ∩ B(y, rn ) and Qy = φn (Ωy ). We first show that there is C1 ∝ diam(Q)/ρ(U ) such that, for any y ∈ U , diam(Qy ) ≤ C1 rn . For this, we mimic the proof of Lemma 3. Take x, x′ ∈ Ωy such that ξ ∶= ∥φn (x) − φn (x′ )∥ = diam(Qy ). Let u be such that B(u, ρ(U )) ⊂ U . Let y1 , . . . , ym be an (rn + 2εn )-packing of B(u, ρ(U )) with m ≥ A1 (ρ(U )/rn )d for some A1 ∝ 1. Then let {xis ∶ s ∈ [m]} ⊂ Ωn be such that maxs∈[m] ∥ys − xis ∥ ≤ εn . By the triangle inequality, for all s ≠ t, we have ∥xis − xit ∥ ≥ ∥ys − yt∥ − 2εn ≥ rn > ∥x − x′ ∥. By (10), we have ∥φn (xis ) − φn (xit )∥ ≥ ξ, so that φn (xi1 ), . . . , φn (xim ) form a ξ-packing. Therefore m ≤ A2 (diam(Q)/ξ)d for some A2 ∝ 1. We conclude that ξ ≤ (A2 /A1 )1/d (diam(Q)/ρ(U ))rn =∶ C1 rn . We apply Theorem 3 to Uy ∶= B(y, rn ) and Ωy . With the fact that δH (Ωy , Uy ) ≤ 2εn — as we saw in the proof of (17) — and invariance considerations, we obtain a constant C ∝ 1 and a similarity Sy such that maxx∈Ωy ∥φn (x) − Sy (x)∥ ≤ C(diam(Qy )/rn )εn ≤ CC1 εn =∶ C2 εn . (Note that all the quantities with subscript y depend also on n, but this will be left implicit.) Fix y⋆ ∈ U ⊖rn . For x ∈ Ωn , there is y ∈ U ⊖rn such that x ∈ Uy . Assume γ is parameterized by arc length and let h‡ be given by Lemma 23 and let D denote the intrinsic diameter of U ⊖h‡ . Then assuming rn ≤ h‡ , there is a curve γ ⊂ U ⊖rn of length L ≤ D joining y⋆ and y. Let y0 = y⋆ , yj = γ(jrn ) for j = 0, . . . , J ∶= ⌊L/rn ⌋, and then yJ+1 = y. We have maxz∈Uyj ∩Uyj+1 ∥Syj (z) − Syj+1 (z)∥ ≤ 2C2 εn by the triangle inequality. We also have ρ(Uyj ∩ Uyj+1 ) ≥ rn , because ∥yj − yj+1 ∥ ≤ rn . Let vj be such that B(vj , rn /2) ⊂ Uyj ∩ Uyj+1 . Fix j and let vj,0 , . . . , vj,d denote a regular simplex inscribed in the ball B(vj , rn /4). Let λn ∝ rn denote its edge length. Then let xj,0 , . . . , xs,d ∈ Ωn be such that maxk ∥xj,k − vj,k ∥ ≤ εn . When C0 is large enough, xj,0 , . . . , xj,d ∈ B(vj , rn /2) by the triangle inequality. Moreover, maxk,l ∥xj,k −xj,l ∥ ≤ λn +2εn , as well as mink≠l ∥xj,k −xj,l ∥ ≥ λn −2εn . When C0 is large enough, Fj ∶= {xj,0 , . . . , xj,d } is therefore an η-approximate regular simplex, with η ∝ εn /rn , and minimum edge length ∝ rn . Now, since maxk ∥Syj (xj,k ) − Syj+1 (xj,k )∥ ≤ 2C2 εn , by Lemma 24, for all z ∈ Rd , ∥Syj (z)−Syj+1 (z)∥ ≤ CC2 εn ∥z−xj,0 ∥/rn +2C2 εn for some C ∝ 1, assuming εn /rn ≤ 1/C. In particular, by the fact that ∥x − xj,0 ∥ ≤ diam(U ), this gives ∥Syj (x) − Syj+1 (x)∥ ≤ C3 εn /rn for some C3 ∝ diam(U )C2 . Hence, ∥Sy⋆ (x) − Sy (x)∥ ≤ (J + 1)C3 εn /rn ≤ C4 εn /rn2 , since J ≤ L/rn ≤ D/rn . This being true for any arbitrary x ∈ Ωn , we conclude that max ∥φn (x) − Sy⋆ (x)∥ ≤ C4 εn /rn2 + C2 εn ≤ C5 εn /rn2 . x∈Ωn 25 4 Discussion This paper builds on (Kleindessner and von Luxburg, 2014) to provide some theory for ordinal embedding, an important problem in multivariate statistics (aka unsupervised learning). We leave open two main problems: • What are the optimal rates of convergence for ordinal embedding with all triple and quadruple comparisons? • What is the minimum size of K = Kn for consistency of ordinal embedding based on the K-nearest neighbor distance comparisons? We note that we only studied the large sample behavior of exact embedding methods. In particular, we did not discuss or proposed any methodology for producing such an embedding. For this, we refer the reader to (Agarwal et al., 2007; Borg and Groenen, 2005; Terada and Von Luxburg, 2014) and references therein. In fact, the practice of ordinal embedding raises a number of other questions in terms of theory, for instance: • How many flawed comparisons can be tolerated? Acknowledgements We are grateful to Vicente Malave for introducing us to the topic and for reading a draft of this paper. 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A Framework for Datatype Transformation Jan Kort 1 and Ralf Lämmel 2,3 1 Universiteit van Amsterdam voor Wiskunde en Informatica 3 Vrije Universiteit van Amsterdam arXiv:cs/0204018v3 [cs.PL] 24 Feb 2003 2 Centrum Abstract We study one dimension in program evolution, namely the evolution of the datatype declarations in a program. To this end, a suite of basic transformation operators is designed. We cover structure-preserving refactorings, but also structure-extending and -reducing adaptations. Both the object programs that are subject to datatype transformations, and the meta programs that encode datatype transformations are functional programs. 1 Introduction We study operators for the transformation of the datatype declarations in a program. The presentation will be biased towards the algebraic datatypes in Haskell, but the concepts are of relevance for many typed declarative languages, e.g., Mercury and SML, as well as frameworks for algebraic specification or rewriting like ASF+SDF, CASL, Elan, and Maude. Our transformations are rather syntactical in nature as opposed to more semantical concepts such as data refinement. Our transformations contribute to the more general notion of functional program refactoring [TR01]. The following introductory example is about extracting a new datatype from constructor components of an existing datatype. This is illustrated with datatypes that represent the syntax of an imperative language. The following extraction identifies a piece of syntax to enable its reuse in later syntax extensions: - - Datatypes with focus on two constructor components data Prog = Prog ProgName [Dec ] [Stat ] data Dec = VDec Id Type data Stat = Assign Id Expr \| If Expr Stat Stat \| ... - - After extraction of [Dec] [Stat] to constitute a new datatype Block data Prog = Prog ProgName Block data Block = Block [Dec ] [Stat ] In the present paper, we describe the design of a framework for datatype transformations including the operators for the above extraction. In Sec. 2, we identify all the concerns addressed by the framework. In Sec. 3, we describe all the basic operators for datatype transformations. In Sec. 4, these operators are lifted from datatypes to complete programs. Related work is discussed in Sec. 5. The paper is concluded in Sec. 6. Kort & Lämmel 2 Concerns in datatype transformation The central contribution of the present paper is a simple, well-defined, and ‘editingcomplete’ suite of operators for datatype transformations. Before we embark on this suite, we identify the concerns addressed by our approach: • Datatype transformations via scripting or interactive tool support. • Well-defined primitives for datatype transformations. • Generic meta-programming for conciseness of datatype transformations. • Flexible means of referring to fragments of interest in datatype transformations. We will now discuss these concerns in some depth. 2.1 Scripting vs. interactive tool support From the point of view of a programmer, datatype transformations should be founded on intuitive scenarios for adaptation. To actually perform (datatype) transformations, there are two modes of operation. The first mode is scripting: the programmer encodes the desired transformation as an expression over basic or higher-level operators. The second mode is interactive transformation based on a corresponding GUI. The benefits of an interactive tool are rather obvious. Such a tool is useful to issue a transformation on the basis of an operator-specific dialogue, and to provide a tailored list of options for transformations that make sense in a given context. A crucial benefit of interactive transformation is that the GUI can be used to provide feedback to the programmer: Which locations were changed? Where is the programmer’s attention needed to complete the issued transformation scenario? The apparent benefits of scripting such as the opportunities to revise transformations and to replay them can be also integrated into an interactive setting. In Fig. 1, we illustrate the interactive treatment of the introductory example using our prototypical tool TH — Transform Haskell. As the snapshot indicates, we use a designated fold dialogue to perform the extraction of the piece of syntax. (Folding is the basic transformation underlying extraction.) This dialogue combines several transformation steps and side conditions in a convenient way. The figure shows the following situation. The user has selected two consecutive types “[Dec] [Stat]” and initiated the fold dialogue. The user has also typed in “Block” in the “type name” field. The introduction check-box is marked automatically since the given type name does not yet exist. The user has also selected the “kind” radiobutton to be “data” and filled in “Block” in the “cons name” field. After this, the user would press “Replace” to make the change. If there had been more than one occurrence, the user could replace them all with “Replace All”, or step through all occurrences with “Next”, and replace only specific ones with “Replace” as with ordinary find and replace in text editors. 2 Kort & Lämmel Fig. 1. A snapshot related to the interactive treatment of the introductory example Here is an open-ended list of further common transformation scenarios: • Renaming type and constructor names. • Permuting type arguments and constructor components. • The dual of extracting datatypes, i.e., inlining datatypes. • Including a constructor declaration together with associated functionality. • Excluding a constructor declaration together with associated functionality. • Inserting a constructor component together with associated functionality. • Deleting a constructor component together with associated functionality. 2.2 Well-defined transformation primitives The core asset of our framework is a suite of basic operators, which can be either used as is, or they can be completed into more complex, compound transformations. In the design of this suite, we reuse design experience from a related effort on grammar adaptation [Läm01]. Indeed, there is an obvious affinity of grammar transformations and datatype transformations. A challenging problem that we did not need to address in this previous work, is the completion of datatype transformations to apply to entire (functional) programs in which evolving datatypes reside. We list the required properties of our basic transformation operators: Correctness Mostly, we insist on ‘structure preservation’, that is, the resulting datatype is of the same shape as the original datatype. This is enforced by the pre- and postconditions of the operators. 3 Kort & Lämmel Completeness The operators are ‘editing-complete’, that is, they capture all scenarios of datatype evolution that are otherwise performed by plain text editors. Semantics-preserving adaptations are defined in terms of disciplined primitives. Orthogonality The operators inhabit well-defined, non-overlapping roles. Higherlevel scenarios for interactive transformation are derivable. Operators for datatype transformations are complementary to expression-level transformations. Locality The basic operators operate on small code locations as opposed to ‘global’ or ‘exhaustive’ operators, which iterate over the entire program. Note that some operators are necessarily exhaustive, e.g., an operator to rename a type name. Implementability The operators are implemented as syntactical transformations that are constrained by simple analyses to check for pre- and postconditions, but which otherwise do not necessitate any offline reasoning. Universality While the present paper focuses on datatype transformations, the principles that are embodied by our operators are universal in the sense that they also apply to other abstractions than datatypes, e.g., functions or modules. We do not list these properties to announce a formal treatment. This would be very challenging as we opt for the complex language setup of Haskell. The above properties provide merely a design rationale. A formal approach is an important subject for future work, but it does not contribute anything to the narrow goal of the present paper: to compile an inventory of the basic roles in datatype transformation. 2.3 Generic meta-programming We implement transformation operators and compound meta-programs in Haskell. We reuse a publicly available abstract syntax for Haskell. 1 We rely on generic programming techniques to perform meta-programming on the non-trivial Haskell syntax in Haskell. We use the Strafunski-style 2 of generic programming that allows us to complete functions on specific syntactical sorts into generic traversals that process subterms of the specific sorts accordingly. This style of metaprogramming is known to be very concise because one only provides functionality for the types and constructors that are immediately relevant for the given problem. All our datatype transformations are of type Trafo which is defined as follows: type Trafo = HsModule → Maybe HsModule That is, a datatype transformation is a partial function on HsModule — the abstract syntactical domain for Haskell modules. Partiality is expressed by means of the Maybe type constructor that wraps the result type. Partially is needed to model side conditions. In Fig. 2, we illustrate generic meta-programming by giving the definition of a simple operator for replacing type names. The specification formalises the fact that 1 2 The used abstract syntax is part of the Haskell Core Libraries — in the haskell-src package. http://www.cs.vu.nl/Strafunski/ 4 Kort & Lämmel Replace a type name replaceTypeId :: TypeId → TypeId → Trafo replaceTypeId n n ′ = full tdTP (adhocTP (adhocTP idTP declSite) refSite) where Transform declaring occurrences of type names declSite :: HsDecl → Maybe HsDecl declSite (HsTypeDecl l n0 ps t) \| n0 ≡ n = return (HsTypeDecl l n ′ ps t) declSite (HsDataDecl l c n0 ps cds d) \| n0 ≡ n = return (HsDataDecl l c n ′ ps cds d) declSite (HsNewTypeDecl l c n0 ps cd d) \| n0 ≡ n = return (HsNewTypeDecl l c n ′ ps cd d) declSite decl = return decl Transform using occurrences of type names refSite :: HsType → Maybe HsType refSite (HsTyCon (UnQual n0 )) \| n0 ≡ n = return (HsTyCon (UnQual n ′ )) refSite tpe = return tpe Fig. 2. Specification of the replacement operation underlying renaming of type names type names can occur in two kinds of locations: either on a declaration site, when we declare the type, or on a using site, when we refer to the type in a type expression. So we need to synthesise a transformation which pays special attention to the syntactical domains for declaring and using sites. Indeed, in the figure, there are two type-specific ‘ad-hoc’ cases which customise the identity function idTP. In the given context, we choose the traversal scheme full tdTP for ‘full top-down traversal in Type-Preserving manner’. This way, we will reach each node in the input tree to transform type names on declaring and using sites. The operator replaceTypeId, by itself, is a total function. (So the Maybe in its type is not really needed here.) Partiality would be an issue if we derived an operator for renaming type names. This necessitates adding a side condition to insist on a fresh new name. 2.4 Means of referring to fragments of interest Both the basic operators for datatype transformation but also actual transformation scenarios in scripts or in interactive sessions need to refer to program fragments of interest. Recall our introductory example. Extracting a type necessitates referring to the constructor components that are meant to constitute the new type. In our framework, we use three ways to refer to fragments of interest: Focus markers on subterms This approach is particularly suited for interactive transformations. Here, relevant fragments can be directly marked. In Fig. 3, we extend Haskell’s abstract syntax to include term constructors for focusing on relevant fragments in datatype transformations. That is, we are prepared to focus on names of types, on type expressions, and on lists of constructor components. Selectors of subterms This approach is particularly suited for scripting transformations. Selectors for Haskell’s type expressions are defined in Fig. 4. The three forms of TypeSel represent the three kinds of declarations that involve types. The helper TypeSel ′ allows to select any part of a given type expression. 5 Kort & Lämmel Focus on names data HsName = ... \| HsNameFocus HsName Focus on type expressions data HsType = ... \| HsTypeFocus HsType Focus on lists of constructor components data HsConDecl = HsConDecl SrcLoc HsName [ HsFocusedBangType ] \| HsRecDecl SrcLoc HsName [([HsName ], HsBangType)] data HsFocusedBangType = HsUnfocusedBangType HsBangType \| HsFocusedBangType [HsBangType ] Fig. 3. Kinds of focus for datatype transformation data TypeSel = AliasRef TypeId TypeSel ′ \| ConRef ConPos TypeSel ′ \| SigRef [FunId ] TypeSel ′ data TypeSel ′ = SelStop \| SelDom TypeSel ′ \| SelCod TypeSel ′ \| SelIth ParaPos TypeSel ′ \| SelFun TypeSel ′ \| SelArg TypeSel ′ type TypeId type ConId type FunId type ConPos type ParaPos data HsName = = = = = = HsName HsName HsName (ConId , ParaPos ) Int ... -- Refer to a type alias -- Refer to a constructor component -- Refer to a function signature -- Reference stops here -- Refer to domain of function type -- Refer to co-domain of function type -- Refer to products component -- Refer to type constructor -- Refer to type argument -- Refer to a type -- Refer to a constructor -- Refer to a function name -- Refer to a component of a constructor -- Refer to a parameter position -- Syntactical sort for all kinds of names Fig. 4. Selectors that refer to type expressions, and others Predicates on subterms Such predicates typically constrain the type of a term or the top-level pattern. This approach is particularly suited for the repeated application of a transformation to different focuses that match a given predicate. There are ways to mediate between these different ways of referring to subterms. For example. given a term with a focus marker on a type expression, one can compute the selector that refers to the focused subterm. Given a predicate on type expressions, one can compute the list of all selectors so that an operator that is defined on selectors can be used with predicates as well. Finally, given a selector, one can also add the corresponding focus marker in the input at hand. 3 Basic operators for datatype transformation We will now describe the themes that constitute our operator suite: • Renaming type and constructor names. • Permutation of type parameters and constructor components. • Swapping types on use sites. • Introduction vs. elimination of type declarations. • Folding vs. unfolding of type declarations. 6 Kort & Lämmel Sample input datatype data ConsList a = Nil \| Cons a (ConsList a) Renamed and permuted datatype data SnocList a = Lin \| Snoc (SnocList a) a Fig. 5. Illustration of renaming and permutation renameTypeId renameConId permuteTypeId permuteConId ::TypeId → TypeId → Trafo ::ConId → ConId → Trafo ::TypeId → [ParaPos ] → Trafo ::ConId → [ParaPos ] → Trafo -- Rename a type declaration -- Rename a constructor -- Permute type parameters -- Permute constructor components Fig. 6. Operators for renaming and parameter permutation renameTypeId (HsIdent "ConsList") (HsIdent "SnocList") renameConId (HsIdent "Nil") (HsIdent "Lin") renameConId (HsIdent "Cons") (HsIdent "Snoc") permuteConId (HsIdent "Snoc") [2, 1] ‘seqTrafo‘ ‘seqTrafo‘ ‘seqTrafo‘ Fig. 7. Script for the scenario in Fig. 5 • Wrapping vs. unwrapping of constructor components. • Inclusion vs. exclusion of entire constructor declarations. • Insertion vs. deletion of constructor components. As this list makes clear, we group an operator with its inverse such as in “folding vs. unfolding”, unless the operator can be used to inverse itself. This is the case for renaming, permutation, and swapping. The operators from the first six groups are (almost) structure-preserving. The last two groups deal with structure-extending and -reducing transformations. We will now explain the operators in detail including illustrative examples. We will only explain the effect of the operators on datatype declarations while we postpone lifting the operators to the level of complete programs until Sec. 4. 3.1 Renaming and permutation Let us start with the simplest datatype refactorings one can think of. These are transformations to consistently rename type or constructor names, and to permute parameters of type and constructor declarations. In Fig. 5, a simple example is illustrated. We rename the type name ConsList, the constructor names Nil and Cons, and we permute the two parameter positions of Cons. The resulting datatype specifies a SnocList as opposed to the ConsList before. In Fig. 6, we declare the operators for renaming names and permuting parameter lists. In Fig. 7, we include the script that encodes the ConsList-to-SnocList sample as a sequence of basic renaming and permuting transformations. To this end, we assume a sequential composition operator seqTrafo for datatype transformations. (In the script, seqTrafo is used as an infix operator ‘seqTrafo‘.) 7 Kort & Lämmel data HsDecl = ... introTypes elimTypes :: :: -- Syntactical sort for (type) declarations [HsDecl ] → Trafo [TypeId ] → Trafo -- Introduction of type declarations -- Elimination of type declarations Fig. 8. Operators for introduction and elimination of datatypes type TypeHdr = (TypeId , [TypeVar ]) type TypeVar = HsName -- Header (LHS) of type declaration -- Type variables foldAlias :: TypeSel → TypeHdr → Trafo unfoldAlias :: TypeSel → Trafo -- Folding the referred type -- Unfolding the referred type Fig. 9. Operators for folding and unfolding 3.2 Introduction vs. elimination The next group of operators deals with the introduction and elimination of type declarations (see Fig. 8). Introduction means that the supplied types are added while their names must not be in use in the given program. Elimination means that the referenced types are removed while their names must not be referred to anymore in the resulting program. The two operators take lists of types as opposed to single ones because types can often only be introduced and eliminated in groups, say mutually recursive systems of datatypes. All kinds of type declarations make sense in this context: aliases, newtypes, and proper datatypes. The operators for introduction and elimination are often essential in compound transformations. This will be illustrated below when we reconstruct the introductory example in full detail (see Sec. 3.4). 3.3 Folding vs. unfolding Instantiating the folklore notions of unfolding and folding for datatypes basically means to replace a type name by its definition and vice versa. Extra provisions are needed for parameterised datatypes. The prime usage scenarios for the two operators are the following: • extraction = introduction of a type followed by its folding. • inlining = unfolding a type followed by its elimination. To give an example, the introductory example basically extracts the structure of imperative program blocks. To actually reconstruct this example, we need a few more operators. So we postpone scripting the example (see Sec. 3.4). The operators for folding and unfolding are declared in Fig. 9, The operators make a strict assumption: the type which is subject to folding or unfolding is necessarily a type alias as opposed to a proper datatype. This assumption simplifies the treatment of the operators considerably since type aliases and their definitions are equivalent by definition. Extra operators for so-called wrapping and unwrapping allow us to use proper datatypes during folding and unfolding as well. This will be addressed below. In the type of the foldAlias operator, we do not just provide a type 8 Kort & Lämmel type ConRange = (ConPos , Int ) -- Refer to consecutive components groupConRange ungroupConPos alias2newtype newtype2data data2newtype newtype2alias :: :: :: :: :: :: ConRange → Trafo ConPos → Trafo TypeId → ConId → Trafo TypeId → Trafo TypeId → Trafo TypeId → Trafo -- Group constructor components -- Inline product -- Turn type alias into newtype -- Turn newtype into datatype -- Turn datatype into newtype -- Turn newtype into type alias Fig. 10. Operators for wrapping and unwrapping 0. Original syntax data Prog data Dec data Stat data Expr = = = = Prog ProgName [Dec ] [Stat ] VDec Id Type Assign Id Expr \| If Expr Stat Stat Var Id \| Const Int 1. After grouping [Dec] and [Stat] data Prog = Prog ProgName ([Dec ], [Stat ]) 2. After introduction of Block to prepare folding data Prog = Prog ProgName ([Dec ], [Stat ]) type Block = ([Dec ], [Stat ]) 3. After folding away the type expression ([Dec], [Stat]) data Prog = Prog ProgName Block type Block = ([Dec ], [Stat ]) 4. After turning Block into a proper datatype with the constructor Block data Prog = Prog ProgName Block data Block = Block ([Dec ], [Stat ]) 5. After ungrouping the product ([Dec], [Stat]) data Prog = Prog ProgName Block data Block = Block [Dec ] [Stat ] Fig. 11. Illustration of wrapping, unwrapping, and extraction name but also a list of type variables (cf. helper type TypeHdr). This is needed for parameterised datatypes, where we want to specify how the free type variables in the selected type expression map to the argument positions of the type alias. The preconditions for the operators are as follows. In the case of foldAlias, we need to check if the referenced type expression and the right-hand side of the given alias declaration coincide. In the case of unfolding, we need to check that the referenced type expression corresponds to an application of a type alias. 3.4 Wrapping vs. unwrapping We will now consider operators that facilitate certain forms of wrapping and unwrapping of datatype constructors (see Fig. 10). There are operators for grouping and ungrouping, that is, to turn consecutive constructor components into a single component that is of a product type, and vice versa. There are also operators to mediate between the different kinds of type declarations, namely type aliases, newtypes and datatypes. This will allow us to toggle the representation of datatypes in basic ways. As a result, the normal forms assumed by other operators can be established; recall, for example, the use of type aliases in folding and unfolding. This separation of concerns serves orthogonality. 9 Kort & Lämmel groupConRange ((HsIdent "Prog", 2), 2) introTypes [HsTypeDecl noLoc "Block" [ ] (HsTyTuple [ HsTyApp (HsTyCon (UnQual (HsIdent "List"))) (HsTyCon (UnQual (HsIdent "Dec"))), HsTyApp (HsTyCon (UnQual (HsIdent "List"))) (HsTyCon (UnQual (HsIdent "Stat")))])] foldAlias (ConRef (HsIdent "Prog", 2) SelStop) ((HsIdent "Block"), [ ]) alias2newtype (HsIdent "Block") (HsIdent "Block") newtype2data (HsIdent "Block") ungroupConPos ((HsIdent "Block"), 1) ‘seqTrafo‘ ‘seqTrafo‘ ‘seqTrafo‘ ‘seqTrafo‘ ‘seqTrafo‘ Fig. 12. Script for the scenario in Fig. 11 data Maybe k data Maybe ′ k data Maybe ′ k a = Nothing \| Just a k k k k a = Nothing ′ \| Just ′ a k k k k a = Nothing ′ \| Just ′ a k k k data ConsList a = Nil (Maybe ′ a) k k \| Cons a (ConsList a) Fig. 13. Illustration of the generalisation of Maybe to ConsList In Fig. 11, we show the steps that implement the introductory example. As one can see, we basically implement extraction, but extra steps deal with grouping and ungrouping the two components subject to extraction. Also, the extracted type should be a proper datatype as opposed to a type alias (see transition from 3. to 4.). For completeness’ sake, the transformation script is shown in Fig. 12. The script precisely captures the steps that underly the interactive transformation in Fig. 1. Some of the operators are not completely structure-preserving, that is, strictly speaking, the structures of the datatypes before and after transformation are not fully equivalent. For example, a newtype and a datatype are semantically distinguished, even if the defining constructor declaration is the very same. (This is because a constructor of a datatype involves an extra lifting step in the semantical domain, i.e., there is an extra ‘bottom’ element.) The operators for grouping and ungrouping also deviate from full structure preservation. 3.5 Swapping types on use sites We will now deal with transformations that eliminate or establish type distinctions by what we call swapping types on use sites. In Fig. 13, we illustrate a typical application of swapping. In the example, we want to generalise the standard datatype Maybe to allow for lists instead. In fact, we do not want to change the general definition of the library datatype Maybe, but we only want to change it on one use site (not shown in the figure). This is where swapping helps: as an intermediate step, we can replace Maybe on the use site by a newly introduced datatype Maybe ′ with equivalent structure. The figure illustrates how subsequent adaptations derive 10 Kort & Lämmel type DataNames type DataUnifier = = (TypeId , [ConId ]) (DataNames , DataNames) swapAlias swapData :: :: TypeSel → TypeId → TypeId → Trafo TypeSel → [DataUnifier ] → Trafo Fig. 14. Operators for swapping types on use sites type ConDecl data HsType = (ConId , [HsType ]) = ... :: :: includeConDecl excludeConDecl -- Constructor declaration -- Syntactical sort for type expressions TypeId → ConDecl → Trafo ConId → Trafo Fig. 15. Operators for inclusion and exclusion of constructor declarations Syntax as of Fig. 11 data Prog data Block data Dec data Stat data Expr = = = = = Prog ProgName Block Block [Dec ] [Stat ] VDec Id Type Assign Id Expr \| If Expr Stat Stat Var Id \| Const Int After syntax extension by statement blocks data Stat = Assign Id Expr \| If Expr Stat Stat \| SBlock Block Fig. 16. Illustration of constructor inclusion the ConsList datatype from the clone of the Maybe datatype. In particular, we add the boxed constructor component. The swapping operators are declared in Fig. 14. There is one operator for type aliases and another for datatype declarations. In the case of proper datatypes, one needs to match the constructors in addition to just the names of the types. This is modelled by the helper datatype DataUnifier. The type of the operator swapData clarifies that we are prepared to process a list of DataUnifiers. This is necessary if we want to swap mutually recursive systems of datatypes. 3.6 Inclusion vs. exclusion We now leave the ground of structure-preserving transformations. That is, we will consider transformations where input and output datatypes are not structurally equivalent. In fact, we consider certain ways to extend or reduce the structure of the datatype. The first couple of structure-extending and -reducing transformations is about inclusion and exclusion of constructor declarations (see Fig. 15). These operators are only feasible for proper datatypes and not for type aliases or newtypes. (This is because a type alias involves no constructor at all, and a newtype is defined in terms of precisely one constructor declaration.) In Fig. 16, we show an example for constructor inclusion. In fact, we just continue the introductory example to make use of the extracted block structure in a language extension for statement blocks. That is, we include a constructor application for Stat to capture Block as another statement form. This continuation of the 11 Kort & Lämmel insertConComp deleteConComp :: :: ConPos → HsType → Trafo ConPos → Trafo Fig. 17. Operators for insertion and deletion of constructor components A datatype for a transition relation / function, and helpers type TransRel a = a → Maybe a data Maybe a = Nothing \| Just a data ConsList a = Nil \| Cons a (ConsList a) Introduction of a substitute for Maybe data Maybe ′ a = Nothing ′ \| Just ′ a Swapping Maybe and Maybe’ in TransRel type TransRel a = a → Maybe ′ a Extension of Maybe ′ to fit with shape of ConsList data Maybe ′ a = Nothing ′ \| Just ′ a (Maybe ′ a) Swapping Maybe ′ and ConsList in TransRel type TransRel a = a → ConsList a Fig. 18. Illustration of component insertion and type swapping introductory example amplifies the intended use of our operator suite: for program evolution in the sense of datatype refactoring and adaptation. 3.7 Insertion vs. deletion Inclusion and exclusion of constructor declarations is about the branching structure of datatypes. We will now discuss operators that serve for the insertion or deletion of constructor components (see Fig. 17). Insertion of a component c into a constructor declaration C c1 · · · cn proceeds as follows. Given the target position for the new component, be it i 6 n + 1, the new constructor declaration is simply of the form C c1 · · · ci−1 c ci · · · cn . In general, c might need to refer to type parameters of the affected datatype. Deletion of a constructor declaration relies on the identification of the obsolete component. In Fig. 18, we elaborate on the earlier example for generalising ‘maybies’ to lists (recall Fig. 13). At the top of Fig. 18, we see three datatypes TransRel, Maybe, and ConsList. The idea is indeed to replace Maybe by ConsList in the using occurrence in TransRel. (That is, we want to allow for a function from a to a list of as instead of a partial function from a to a.) We call this adaptation a generalisation because a list is more general than an optional. In the initial phase of the generalisation of Maybe, we disconnect the relevant occurrence of Maybe in TransRel from other possible occurrences in the program. So we introduce a copy Maybe ′ of Maybe, and we perform type swapping so that TransRel refers to Maybe ′ instead of the ‘read-only’ Maybe. Now we need to make Maybe ′ structurally equivalent to ConsList. This amounts to adding a recursive component to the second constructor Just ′ . Then, we can again swap types to refer to ConsList in the co-domain of TransRel. 12 Kort & Lämmel 4 Datatype transformation meets program transformation We will now re-iterate over the groups of operators to investigate their impact on functional programs. It would be utterly complex to formalise the link between datatype and program transformation. The mere specification of the transformations is already intractable for a publication because of its size, and the number of details. So we will describe the implied program transformations informally while omitting less interesting details. 4.1 Renaming Type names only occur inside type declarations and type annotations. So there is no need to adapt expressions or function declarations except for their signatures, or the type annotations of expressions. Constructor names can very well occur inside patterns and expressions that contribute to function declarations. Renaming these occurrences is completely straightforward. 4.2 Permutation The permutation of type parameters does not necessitate any completion at the level of function declarations. The permutation of constructor components, however, needs to be realized in patterns and expressions as well. This is particularly simple for pattern-match cases because all components are matched by definition. Hence, we can directly permute the sub-patterns in an affected constructor pattern. Witnessing permutations of constructor components in expression forms is slightly complicated by currying and higher-order style. Instead of permuting components in possibly incomplete constructor applications, we could first get access to all components by ‘λ-pumping’: given a constructor C with say n potential components according to its declaration, we first replace C by λx1 · · · xn . C x1 · · · xn as justified by η-conversion. Then, we witness the permutation by permuting the arguments x1 , . . . , xn in the pumped-up expression. In the presence of a nonstrict language with an evaluation order on patterns, the permutation of constructor components might actually change the behaviour of the program regarding termination. We neglect this problem. We should also mention that it is debatable if the described kind of η-conversion is really what the programmer wants because it obscures the code. 4.3 Introduction vs. elimination Introduction does not place any obligations on the functions defined in the same program. In the case of elimination, we have to ensure that the relevant types are not used by any function. If we assume that all function declarations are annotated by programmer-supplied or inferred signatures, then the precondition for elimination can be checked by looking at these signatures. There is an alternative approach that does not rely on complete type annotations: we check that no constructor of the relevant types is used. 13 Kort & Lämmel 4.4 Folding vs. unfolding The restriction of folding and unfolding to type aliases guarantees that these operators do not necessitate any adaptation of the function declarations. This is simply because interchanging a type alias and its definition is completely structure- and semantics-preserving, by definition. This is extremely convenient: despite the crucial role of the operators for folding and unfolding, they do not raise any issue at the level of function declarations. 4.5 Wrapping vs. unwrapping Grouping and ungrouping These operators are handled using the same overall approach as advocated for the permutation of constructor components. That is, in patterns we witness grouping or ungrouping by inserting or removing the enclosing “( . . . )”; in expressions, we perform η-conversion to access the relevant components, and then we group or ungroup them in the pumped-up constructor application. Mediation between newtypes and datatypes These datatype transformations do not imply any adaptations of the functions that involve the datatype in question. (As we indicated earlier, the extra bottom value of a datatype, when compared to a newtype, allows a program to be ‘undefined’ in one more way.) Newtype to alias migration We simply remove all occurrences of the associated constructor both in pattern and expression forms. We require that the relevant newtype is not covered by any instance declaration of some type class or constructor class. Otherwise, we had to inline these members in a non-obvious way prior to the removal of the constructor. If we neglected this issue, the resulting program either becomes untypeable, or a different instance is applied accidentally, which would be hazardous regarding semantics preservation. Alias to newtype migration This operator requires a non-trivial treatment for function declarations. The crucial issue is how to know the following: • What expressions have to be wrapped with the newtype constructor? • In what patterns does the newtype constructor need to be stripped? Our approach is as simple as possible. We observe that the new newtype might be used in the declarations of other datatypes. The corresponding patterns and expressions can be easily located and adapted as in the case of permutation, grouping, and ungrouping (recall η-conversion etc.). We also need to adapt function declarations if their argument or result types are known to refer to the relevant alias. This basically means that we need to access the affected arguments and result expressions in all relevant equations to unwrap the arguments and wrap the result expressions. These adaptations are slightly complicated by the fact that the affected type alias can occur in arbitrarily nested locations. In Fig. 19, we illustrate the effect of the alias2newtype operator in the introductory example. We show the top-level interpreter function that maps over the statements 14 Kort & Lämmel Top-level interpreter function before the illustrative extraction run :: Prog → State () run (Prog name decs stats) = mapM interpret stats The same function after extraction run :: Prog → State () run (Prog name (Block decs stats) ) = mapM interpret stats Fig. 19. Function adaptation triggered by alias-to-newtype migration Input program type TransRel a = a → Maybe a data Maybe ′ a = Nothing \| Just a deadEnd :: TransRel a → a → Bool deadEnd r a = case r a of Nothing → True Just → False Output program type TransRel a = a → Maybe ′ a data Maybe ′ a = Nothing \| Just a deadEnd :: TransRel a → a → Bool deadEnd r a = case toMaybe (r a) of Nothing → True Just → False Induced helper for type swapping toMaybe :: Maybe ′ a → Maybe a toMaybe Nothing ′ = Nothing toMaybe (Just ′ a) = Just a Fig. 20. Function adaptation triggered by type swapping of the program. (The program name and the declarations do not carry any semantics here.) The type of the function run exhibits that the meaning of a program is a computation that involves a State for the program variables. The adapted version of run refers to the extra constructor Block, which resulted from extraction. 4.6 Swapping types on use sites This operator relies on the same techniques as alias2newtype. However, instead of wrapping and unwrapping a constructor. We invoke conversion functions that mediate between the two structurally equivalent types. These mediators merely map old to new constructors and vice versa, and hence they are immediately induced by the datatype transformation itself, namely by the DataUnifiers passed to the swap operator. This approach implies that we only perform very local changes. The program code will still work on the old datatypes thanks to the mediators. The impact of swapping types at the function level is illustrated in Fig. 20. We deal with the initial steps of the Maybe-to-ConsList migration in Fig. 18, where we replace the occurrence of Maybe within TransRel by a structurally equivalent Maybe ′ . We show an illustrative function deadEnd which performs a test if the given transition relation allows for a transition in the presence of a given state a. The adapted function deadEnd refers to the conversion function toMaybe prior to performing pattern matching on the obsolete Maybe type. 15 Kort & Lämmel Input program data Stat = Assign Id Expr \| If Expr Stat Stat interpret :: Stat → State () interpret (Assign i e) = envLookup i > >= λr → ... interpret (If e s1 s2 ) = reval e > >= λv → ... Output program data Stat = Assign Id Expr \| If Expr Stat Stat \| SBlock Block interpret :: Stat → State () interpret (Assign i e) = envLookup i > >= λr → ... interpret (If e s1 s2 ) = reval e > >= λv → ... interpret ( SBlock ) = ⊥ Fig. 21. Inclusion of a constructor declaration 4.7 Inclusion vs. exclusion Intuitively, the inclusion of a constructor should be complemented by the extension of all relevant case discriminations. This normally means to add a pattern-match equation (or a case to a case expression) for the new constructor. Dually, exclusion of a constructor should be complemented by the removal of all pattern-match equations (or cases) that refer to this constructor. In the case of added pattern-match equations, we view the right-hand sides of these equations as a kind of ‘hot spot’ to be resolved by subsequent expression-level transformations. To this end, we use “undefined”, i.e., “⊥”, as a kind of to-do marker. Dually, in the case of removed constructors, we also need to replace occurrences of the constructor within expressions by “⊥”. When using interactive tool support, these to-do markers are useful to control further steps in a transformation scenario. In Fig. 21, we progress with our running example of an interpreter for an imperative language. We illustrate the step where blocks are turned into another form of statements. Hence, the shown output program involves a new pattern-match equation that interprets statement blocks. This added equation reflects that the meaning of such blocks is as yet undefined, subject to subsequent adaptations. 4.8 Insertion vs. deletion Inserting a component into a declaration for a constructor C means that all patterns with C as outermost constructor must be adapted to neglect the added component, and all applications of C must be completed to include “⊥” for the added component. Dually, deletion of a component from C means that all applications of C and all patterns with C as outermost constructor need to be cleaned up to project away the obsolete component. Any reference to a pattern variable for the obsolete component is replaced by “⊥”. As in the case of permutation and others, η-conversion is needed to actually get access to constructor components in expressions. In Fig. 22, the insertion of a constructor component is illustrated by continuing the scenario from Fig. 20. The adapted equation of toMaybe involves an extended pattern. As the don’t care pattern “ ” indicates, the definition of toMaybe does not make use of the added component. In fact, the definition of the function deadEnd does not need to be adapted; it only tests for the availability of a transition step. 16 Kort & Lämmel Output program type TransRel a = a → Maybe ′ a data Maybe ′ a = Nothing ′ \| Just ′ a (Maybe ′ a) deadEnd :: TransRel a → a → Bool deadEnd r a = case toMaybe (r a) of Nothing → True Just → False Induced helper for type swapping toMaybe :: Maybe ′ a → Maybe a toMaybe Nothing ′ = Nothing toMaybe ( Just ′ a ) = Just a Fig. 22. Illustration of the insertion of a constructor component Normally, other functions will start to rely on the richer pattern. 5 Related work Transformational program development Formal program transformation [BD77] separates two concerns: the development of an initial, maybe inefficient program the correctness of which can easily be shown, and the stepwise derivation of a better implementation in a semantics-preserving manner. Partsch’s textbook [Par90] describes the formal approach to this kind of software development. Pettorossi and Proietti study typical transformation rules (for functional and logic) programs in [PP96]. Formal program transformation, in part, also addresses datatype transformation [dRE98], say data refinement. Here, one gives different axiomatisations or implementations of an abstract datatype which are then related by well-founded transformation steps. This typically involves some amount of mathematical program calculation. By contrast, we deliberately focus on the more syntactical transformations that a programmer uses anyway to adapt evolving programs. Database schema evolution There is a large body of research addressing the related problem of database schema evolution [BKKK87] as relevant, for example, in database re- and reverse engineering [HTJC93]. The schema transformations themselves can be compared with our datatype transformations only at a superficial level because of the different formalisms involved. There exist formal frameworks for the definition of schema transformations and various formalisms have been investigated [MP97]. An interesting aspect of database schema evolution is that schema evolution necessitates a database instance mapping [BCN92]. Compare this with the evolution of the datatypes in a functional program. Here, the main concern is to update the function declarations for compliance with the new datatypes. It seems that the instance mapping problem is a special case of the program update problem. Refactoring The transformational approach to program evolution is nowadays called refactoring [Opd92,Fow99], but the idea is not new [ABFP86,GN90]. Refactoring means to improve the structure of code so that it becomes more comprehensible, maintainable, and adaptable. Interactive refactoring tools are being studied and used extensively in the object-oriented programming context [Moo96,RBJ97]. Typical examples of functional program refactorings are described in [Läm00], e.g., the introduction of a monad in a non-monadic program. The precise inhabitation of 17 Kort & Lämmel the refactoring notion for functional programming is being addressed in a project at the University of Kent by Thompson and Reinke; see [TR01]. There is also related work on type-safe meta-programming in a functional context, e.g., by Erwig [ER02]. Previous work did not specifically address datatype transformations. The refactorings for object-oriented class structures are not directly applicable because of the different structure and semantics of classes vs. algebraic datatypes. Structure editing Support for interactive transformations can be seen as a sophistication of structure editing [RT88,Koo94,KS98]. This link between transformation and editing is particularly appealing for our “syntactical” transformations. Not surprisingly, concepts that were developed for structure editing are related to our work. For example, in [SdM99], primitives of structure editing are identified based on the notion of focus to select subtrees, and on navigation primitives left, right, up and down. Trees, subtrees and paths are here defined as follows: data Tree type SubTree type Path type Layer = = = = Fork Label [Tree ] (Path, Tree) [Layer ] (Label , [Tree ], [Tree ]) The t in a subtree (p, t) is the currently selected tree and it is between the left and right trees in the top layer (the head of the p). This approach does not account for the heterogeneous character of language syntaxes, but it shows that the fact if a focus resides in a term can be encoded in types. 6 Concluding remarks Contribution We identified the fundamental primitives for datatype transformation. These operators are meant to support common scenarios of program adaptation in functional programming, or other settings where algebraic datatypes play a role. In fact, all the identified operators are universal in the sense, that they are also meaningful for other program abstractions than just datatypes, e.g., function declarations. We deliberately focused on adaptations of datatypes because a vast body of previous work addressed fold/unfold transformations for recursive functions. Despite the focus on datatype transformations, we had to consider program transformations that are necessitated by the modification of datatypes. Regarding the executable specification of the operator suite, we adhered to the formula: metaprograms = object-programs = Haskell programs. We employed generic functional programming in the interest of conciseness. We also employed designated means of referring to fragments of interest, e.g., a focus concept. Partial project failure We are confident that the identified operators are sufficient and appropriate for actual datatype transformations. We have attempted to complement this framework development by actual interactive tool support. We initially thought that using Haskell for this interactive tooling as well would be a good idea. Since the actual transformation operators are implemented in Haskell anyway, and the interactive dialogues need to cooperate with the operator framework to perform analyses, Haskell indeed seems to be the obvious choice. To make a long story 18 Kort & Lämmel short, there are many GUI libraries for Haskell, but none of them is suitable for developing a sophisticated GUI for interactive program transformation at the moment. It seems that environments for interactive language tools would provide a better starting point, e.g., environments based on attribute grammars [RT88,KS98]. Perspective To cover full Haskell, a few further operators would have to be added to our suite, in particular, operators that support type and constructor classes. We should also pay full attention to some idiosyncrasies of Haskell; cf. refutable vs. irrefutable patterns. Then, there are also transformation techniques that seem to go beyond our notion of program evolution but it is interesting to cover them anyway. We think of techniques like turning a system of datatypes into functorial style, or threading a parameter through a system of datatypes. The ultimate perspective for the presented work is to integrate the datatype transformations into a complete, well-founded, and user-friendly refactoring tool for functional programming along the lines of Thompson’s and Reinke’s research project [TR01]. Another perspective for our research is to further pursue the intertwined character of datatype and program transformations in the context of XML format and API evolution. References [ABFP86] G. Arango, I. Baxter, P. Freeman, and C. Pidgeon. TMM: Software maintenance by transformation. IEEE Software, 3(3):27–39, May 1986. [BCN92] C. Batini, S. Ceri, and S.B. Navathe. Conceptual database design. Benjamin/Cummings, Redwood City, US, 1992. [BD77] R. M. Burstall and John Darlington. A transformation system for developing recursive programs. Journal of the ACM, 24(1):44–67, January 1977. [BKKK87] J. Banerjee, W. Kim, H.-J. Kim, and H.F. Korth. Semantics and Implementation of Schema Evolution in Object-Oriented Databases. SIGMOD Record (Proc. Conf. on Management of Data), 16(3):311–322, May 1987. [dRE98] Willem-Paul de Roever and Kai Engelhardt. Data Refinement: Model-Oriented Proof Methods and their Comparison, volume 47 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 1998. [ER02] M. Erwig and D. Ren. A rule-based language for programming software updates. In Proceedings of the 2002 ACM SIGPLAN workshop on Rule-based programming, pages 67–78. ACM Press, 2002. [Fow99] M. Fowler. Refactoring—Improving the Design of Existing Code. Addison Wesley, 1999. [GN90] W. G. Griswold and D. Notkin. Program restructuring as an aid to software maintenance. Technical report, Seattle, WA, USA, August 1990. [HTJC93] J.-L. Hainaut, C. Tonneau, M. Joris, and M. Chandelon. Schema Transformation Techniques for Database Reverse Engineering. In Proc. of the 12th Int. Conf. on ER Approach, Arlington-Dallas, 1993. E/R Institute. 19 Kort & Lämmel [Koo94] J.W.C. Koorn. Generating uniform user-interfaces for interactive programming environments. PhD thesis, University of Amsterdam, 1994. [KS98] M. Kuiper and J. Saraiva. Lrc — A generator for Incremental LanguageOriented Tools. In K. Koskimies, editor, Compiler Construction CC’98, volume 1383 of LNCS, pages 298–301. Springer-Verlag, April 1998. Tool demonstration. [Läm00] R. Lämmel. Reuse by Program Transformation. In Greg Michaelson and Phil Trinder, editors, Functional Programming Trends 1999, pages 143–152. Intellect, 2000. [Läm01] R. Lämmel. Grammar Adaptation. In J.N. Oliveira and P. Zave, editors, Proc. Formal Methods Europe (FME) 2001, volume 2021 of LNCS, pages 550–570. Springer-Verlag, 2001. [Moo96] I. Moore. Automatic Inheritance Hierarchy Restructuring and Method Refactoring. In OOPSLA ’96 Conference Proceedings: Object-Oriented Programming Systems, Languages, and Applications, pages 235–250. ACM Press, 1996. [MP97] P. McBrien and A. Poulovassilis. A Formal Framework for ER Schema Transformation. In D.W. Embley and R.C. Goldstein, editors, Conceptual Modeling - ER ’97, 16th International Conference on Conceptual Modeling, Los Angeles, California, USA, November 3-5, 1997, Proc., volume 1331 of LNCS, pages 408–421. Springer-Verlag, 1997. [Opd92] W. F. Opdyke. Refactoring Object-Oriented Frameworks. University of Illinois at Urbana-Champaign, 1992. PhD thesis, [Par90] H.A. Partsch. Specification and Transformation of Programs. Springer-Verlag, 1990. [PP96] A. Pettorossi and M. Proietti. Rules and Strategies for Transforming Functional and Logic Programs. ACM Computing Surveys, 28(2):360–414, June 1996. [RBJ97] D. Roberts, J. Brant, and R.E. Johnson. A Refactoring Tool for Smalltalk. Theory and Practice of Object Systems (TAPOS), 3(4):253–263, 1997. [RT88] T.W. Reps and T. Teitelbaum. The Synthesizer Generator: A System for Constructing Language–Based Editors. Springer–Verlag, 1988. [SdM99] B.A. Sufrin and O. de Moor. Modeless structure editing. In A.W. Roscoe and J.C.P. Woodcock, editors, Proceedings of the Oxford-Microsoft symposium in Celebration of the work of Tony Hoare, September 1999. [TR01] S. Thompson and C. Reinke. Refactoring Functional Programs. Technical Report 16-01, Computing Laboratory, University of Kent at Canterbury, October 2001. Also see http://www.cs.ukc.ac.uk/people/ staff/sjt/Refactor/. 20	6
On the Capacity of a Class of Signal-Dependent Noise Channels Hamid Ghourchian, Gholamali Aminian, Amin Gohari, Mahtab Mirmohseni, and Masoumeh Nasiri-Kenari, arXiv:1702.03590v2 [cs.IT] 2 Jun 2017 Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran. E-mails: {h ghourchian, aminian}@ee.sharif.edu, {aminzadeh, mirmohseni, mnasiri}@sharif.edu ∗ Abstract In some applications, the variance of additive measurement noise depends on the signal that we aim to measure. For instance, additive Gaussian signal-dependent noise (AGSDN) channel models are used in molecular and optical communication. Herein we provide lower and upper bounds on the capacity of additive signal-dependent noise (ASDN) channels. The idea of the first lower bound is the extension of the majorization inequality, and for the second one, it uses some calculations based on the fact that h (Y ) > h (Y \|Z). Both of them are valid for all additive signal-dependent noise (ASDN) channels defined in the paper. The upper bound is based on a previous idea of the authors (“symmetric relative entropy”) and is used for the additive Gaussian signal-dependent noise (AGSDN) channels. These bounds indicate that in ASDN channels (unlike the classical AWGN channels), the capacity does not necessarily become larger by making the variance function of the noise smaller. We also provide sufficient conditions under which the capacity becomes infinity. This is complemented by a number of conditions that imply capacity is finite and a unique capacity achieving measure exists (in the sense of the output measure). Keywords: Signal-dependent noise channels, molecular communication, channels with infinite capacity, existence of capacity-achieving distribution. 1 Introduction An additive Gaussian signal-dependent noise (AGSDN) channel with input x and output y is defined by −(y−x)2 1 e 2σ(x)2 , fY \|X (y\|x) = p 2πσ(x)2 where σ(·) is a given function from R to [0, ∞). Alternatively, we may describe the AGSDN channel by Y = X + σ(X) · Z where Z ∼ N (0, 1) is a standard Gaussian random variable and independent of the input X. For constant function σ(x) = c, the AGSDN channel reduces to a simple additive Gaussian channel. More generally, we may relax the Gaussian assumption on Z and consider an additive signal-dependent noise (ASDN) channel defined by Y = X + σ(X) · Z, (1) where noise Z is assumed to be a continuous random variable with a given pdf fZ (z), and be independent of the input X.1 For instance, one can consider an ASDN with Z being a truncated ∗ This work was supported by INSF Research Grant on “Nano-Network Communications”. The first two authors contributed equally to this work. 1 See Definition 3 for the definition of continuous random variables. 1 version of the Gaussian distribution as a better model in an application if we know that the output Y has minimum and maximum values in that applications. Below we provide a number of applications in which the ASDN channel arises. 1. The AGSDN channel appears in optical pcommunications when modeling the shot noise or the optical amplification noise for σ(x) = c20 + c21 x [1]. √ 2. In molecular communication, the AGSDN channel with σ(x) = c x arises in the ligand receptor model, the particle sampling noise, the particle counting noise and the Poisson model for an absorbing receiver [2, 3, 4]. In all cases, the reason for appearance of a Gaussian signaldependent noise is the approximation of a binomial or Poisson distribution with a Gaussian distribution. Observe that the mean and variance of a binomial distribution with parameters (n, p) relate to each other: the mean is np and the variance is np(1 − p) respectively. As a result, the mean and variance of the approximated Gaussian distribution also relate to each other (see [5, Section II.B] for a detailed overview). 3. Besides the above applications of ASDN in molecular communications we shall provide two other cases where this channel model is helpful: Consider the Brownian motion of a particle with no drift over a nonhomogeneous medium with σ(x) denoting the diffusion coefficient of the medium at location x. The diffusion coefficient σ(x) describes the movement variance of a particle when in location x. More specifically, the motion of the particle is described by the stochastic differential equation dXt = σ(Xt ) dBt , where Bt is the standard Wiener process (standard Brownian motion). Alternatively, we can express the above equation using the following Itô integral Z t+s Xt+s − Xt = σ(Xu ) dBu . (2) t Let us denote the position of the particle at time 0 by X = X0 , and its position after t seconds by Y = Xt . If t is a small and fixed number, (2) reduces to Y = X + tσ(X) · Z, where Z ∼ N (0, 1). Thus, the movement of the particle follows an AGSDN channel law if t is small. 4. As another example, consider the molecular timing channel in a time-varying medium. In a molecular timing channel, information is encoded in the release time of molecules. A molecule released at time X hits the receiver after a delay Z at time Y = X + Z. Molecules are absorbed once they hit the receiver. As such, the distribution of Z is that of the first arrival time. The existing literature only studies this problem when the medium is time-invariant (see [6, 7, 8, 9]): if the medium is uniform, time-invariant and one-dimensional, Z is distributed according to the inverse Gaussian distribution (if there is a flow in the medium) or the Lévy distribution (if there is no flow in the medium). As a result, the channel is called the additive inverse Gaussian noise additive channel, or the additive Lévy noise in the literature. However, in a time-varying medium (or when the distance between the transmitter and receiver varies over time), the distribution of Z depends on the release time X. As a result, we obtain a signal-dependent noise additive component. For instance, the additive noise can have a Lévy 2 distribution with a scale parameter that depends on input X. Using the scaling property of the Lévy distribution, we can express this as σ(X) · Z where Z is the standard Lévy distribution, and σ(X) is the scale parameter. This would be an ASDN channel. 5. In the third item, we discussed Brownian motion after a small time elapse. A Brownian motion with no drift is an example of a martingale. Now let us consider a martingale after a large time elapse. Here, the AGSDN channel also arises as a conditional distribution in any process that can be modeled by a discrete time martingale with bounded increments. Assume that X0 , X1 , X2 , · · · is such a martingale. Then E [Xn ] = E [X0 ]. Furthermore, by the martingale central limit theorem, the conditional distribution of Xn given X0 = x for large values of n can be approximated by a Gaussian distribution with mean X0 = x and a variance σn (x) that depends on X0 = x. 6. Finally, we relate the ASDN channel to real fading channels with a direct line of sight. Consider a scalar Gaussian fading channel Y = X + HX + N, (3) where X is the input, H ∼ N (0, c1 ) is the Gaussian fading coefficient and N ∼ N (0, c0 ) is the additive environment noise. The first X term on the right-hand side of (3) corresponds to the direct line of sight, while the HX term is the fading term. The distribution of Y given X = x is N (x, c1 x2 + c0 ). Thus (3) can be expressed as Y = X + σ(X) · Z where p σ(x) = c1 x2 + c0 , Z ∼ N (0, 1). A fast fading setting in which H varies independently over each channel use corresponds to a memoryless ASDN channel. The purpose of this paper is to study the capacity of a memoryless additive signal-dependent noise (ASDN) channel defined via Y = X + σ(X) · Z, under input cost constraints. The memoryless assumption implies that the noise Z is drawn independently from fZ (z) in each channel use. Related works: In [10], vector AGSDN channels subject cost constraints are studied. It is shown that under some assumptions, p the capacity achieving distribution is a discrete distribution. The AGSDN channel with σ(x) = c20 + c21 x is investigated in [1] wherein capacity upper and lower bounds are derived considering peak and average constraints. Note that the memoryless AGSDN includes the additive white Gaussian noise (AWGN) channel as its special case. The capacity of AWGN channel under power constraint is classical and is obtained by an input of Gaussian random variable. Its capacity under both average and peak power constraints is quite different, as the capacity achieving input distribution is discrete with a finite number of mass points [11]. See [12, 13] for further results on the capacity of the AWGN channel with both average and peak power constraints. Our contributions: Our contributions in this work can be summarized as follows: • We provide a new tool for bounding the capacity of continuous input/output channels. Note that I (X; Y ) = h (Y ) − h (Y \|X) . 3 We provide two sufficient conditions under which h (Y ) ≥ h (X), which results in I (X; Y ) ≥ h (X) − h (Y \|X) , and leads to lower bounds on the channel capacity of an ASDN channel. • It is known that increasing the noise variance of an AWGN channel decreases its capacity. However, we show that this is no longer the case for signal-dependent noise channels: the constraint σ1 (x) ≥ σ2 (x) for all x does not necessarily imply that the capacity of an AGSDN channel with σ1 (x) is less than or equal to the capacity of an AGSDN with σ2 (x). • We identify conditions under which the capacity of the ASDN channel becomes infinity. In particular, this implies that the capacity of a AGSDN channel with p σ(x) = c1 x2 + c0 tends to infinity as c0 tends to zero. Thus, the capacity of the real Gaussian fast fading channel given earlier in this section tends to infinity as c0 tends to zero. This parallels a similar result given in [14] for complex Gaussian fading channels. • We provide a new upper bound for the AGSDN channel based on the KL symmetrized upper bound of [15]. This upper bound is suitable for the low SNR regime, when σ(x) is large. This is p in contrast with the upper bound of [1, Theorems 4, 5] for AGSDN channels with σ(x) = c20 + c21 x which is suitable for large values of peak and average constraints. Furthermore, we give ourp upper bound for a large class of functions σ(x) while the technique of [1] is tuned for σ(x) = c20 + c21 x. This paper is organized as follows. Section 2 includes some of primary definitions and notations. In Section 3, our main results are given. This includes two lower bounds and one upper bound on the capacity of the ASDN channel. There are some useful lemmas in Section 4 used in the paper. The numerical results and plots are given in Section 5. The proofs of our results are given in Section 6. 2 Definitions and Notations In this section we review the definitions of continuous and discrete random variables, as well as entropy and differential entropy, relative entropy and mutual information. Throughout this paper all the logarithms are in base e. Random variables are denoted by capital letters, and probability measure functions are denoted by letter µ. The collection of Borel measurable sets in R is denoted by B(R). We sometimes use a.e. and µ-a.e. as a short-hand for “almost everywhere” and “µ-almost everywhere”, respectively. The set A is µ-a.e., when Z dµ = 0. A The set A is a.e. if it is µ-a.e. when µ is the Lebesgue measure. Definition 1 (Relative Entropy). [16, Section 1.4] For random variables X and Y with probability measures µX and µY , the relative entropy between X and Y is defined as follows: i ( h X E log dµ (X) µX µY dµY D (µX kµY ) = D (XkY ) := , +∞ o.w. 4 X where dµ dµY is the Radon-Nikodym derivative and µX µY means µX is absolutely continuous w.r.t. µY i.e. µX (A) = 0 for all A ∈ B if µY (A) = 0, where B is the Borel σ-field of the space over which the measures are defined. Definition 2 (Mutual Information). [16, Section 1.6] For random variables X, Y with joint probability measure µX,Y , the mutual information between X and Y is defined as follows: I (X; Y ) = D (µX,Y kµX µY ) , where µX µY is the product measure defined as (µX µY )(A, C) = µX (A)µY (C), where A ∈ BX the Borel σ-field of the space over which µX is defined, and C ∈ BY the Borel σ-field of the space over which µY is defined. Similarly, for three random variable X, Y, Z with joint measure µX,Y,Z , conditional mutual information I (X; Y \|Z) is defined as I (X; Y, Z) − I (X; Z). Definition 3 (Continuous Random Variable). [10] Let X be a real-valued and random variable that is measurable with respect to B(R). We call X a continuous random variable if its probability measure µX , induced on (R, B), is absolutely continuous with respect to the Lebesgue measure for B(R) (i.e., µ(A) = 0 for all A ∈ B with zero Lebesgue measure). We denote the set of all absolutely continuous probability measures by AC. Note that the Radon-Nikodym theorem implies that for each random variable X with measure µX ∈ AC there exists a B(R)-measurable function fX : R → [0, ∞), such that for all A ∈ B(R) we have that Z µX (A) = Pr {X ∈ A} = fX (x) dx. (4) A The function fX is called the probability density function (pdf ) of X [16, p. 21]. We denote pdf of absolutely continuous probability measures by letter f . Definition 4 (Discrete Random Variable). [10] A random variable X is discrete if it takes values in a countable alphabet set X ⊂ R. Probability mass function (pmf) for discrete random variable X with probability measure µX is denoted by pX and defined as follows: pX (x) := µX ({x}) = Pr {X = x} , ∀x ∈ X . Definition 5 (Entropy and Differential Entropy). [17, Chapter 2] We define entropy H (X), for a discrete random variable X with measure µX and pmf pX as H (X) = H (µX ) = H (pX ) := X pX (x) log x if the summation converges. Observe that 1 H (X) = E log . pX (X) 5 1 , pX (x) For a continuous random variable X with measure µX and pdf fX , we define differential entropy h (X) as Z +∞ 1 fX (x) log h (X) = h (µX ) = h (pX ) := , f X (x) −∞ if the integral converges. Similarly, the differential entropy is the same as 1 . h (X) = E log fX (X) Similarly, for two random variables X, Y , with measure µX,Y , if for all x, µY \|X (·\|x) is absolutely discrete with pmf pY \|X (·\|x), the conditional entropy H (Y \|X) is defined as 1 H (Y \|X) = E log . pY \|X (Y \|X) Likewise, for two random variables X, Y , with measure µX,Y , if for all x, µY \|X (·\|x) is absolutely continuous with pdf fY \|X (·\|x), the conditional differential entropy h (Y \|X) is defined as h (Y \|X) = E log 1 . fY \|X (Y \|X) We allow for differential entropy to be +∞ or −∞ if the integral is convergent to +∞ or −∞, i.e., we say that h (X) = +∞, if and only if 1 dx = +∞, and fX (x) Z 1 fX (x) log dx converges to a finite number fX (x) A− Z fX (x) log A+ where A− = {x : fX (x) > 1}. A+ = {x : fX (x) ≤ 1}, Similarly, we define h (X) = −∞. When we write that h (X) > −∞, we mean that the differential entropy of X exists and is not equal to −∞. The following example, from [14], demonstrates the differential entropy can be +∞ or −∞. Example 1. Differential entropy becomes plus infinity for the following pdf defined over R [14]: ( 1 , x > e, x(log x)2 f (x) = 0, x ≤ e. On the other hand, as shown in [14], differential entropy is minus infinity for ( −1 0 < x < e−e , 2, g(x) = x log x(log(− log x)) 0, otherwise. 6 Definition 6 (Riemann integrable functions). Given −∞ ≤ ` < u ≤ +∞, in this work, we utilize Riemann integrable functions g : (`, u) 7→ R on open interval (`, u). Such functions satisfy the property that for any c ∈ (`, u), the function Z x h(x) = g(t) dt, c is well-defined. By the fundamental theorem of calculus, h(·) is continuous on (`, u) (but not necessarily differentiable unless g is continuous). As an example, consider the function g(x) = 1/x for x 6= 0, and g(0) = 0 otherwise. This function is Riemann integrable on the restricted domain (0, ∞), but not integrable on (−1, 1). 3 Main Results We are interested in the capacity of an ASDN channel with the input X taking values in a set X and satisfying the cost constraint E[gi (X)] ≤ 0, ∀i = 1, 2, · · · , k for some functions gi (·). The common power constraint corresponds to gi (x) = x2 − p for some p ≥ 0, but we allow for more general constraints. Then, given a density function fZ (z) for the noise Z and function σ(·), we consider the following optimization problem: C = sup I (X; Y ), (5) µX ∈F where X and Y are related via (1) and F = {µX supp(µX ) ⊆ X , E[gi (X)] ≤ 0 for all i = 1, · · · , k}. (6) We sometimes use supp(X) to denote the support of measure µX , supp(µX ), when the probability measure on X is clear from the context. As an example, if, in an application, input X satisfies ` ≤ X ≤ u, the set X can be taken to be [`, u] to reflect this fact; similarly, the constraint 0 < X ≤ u reduces to X = (0, u], and 0 ≤ ` ≤ \|X\| ≤ u reduces to X = [−u, −`] ∪ [`, u]. The rest of this section is organized as follows: in Section 3.1, we provide conditions that imply finiteness of the capacity of an ASDN channel. In Section 3.2, we review the ideas used for obtaining lower bounds in previous works and also in this work. Then, based on the new ideas introduced in this work, we provide two different lower bounds in Sections 3.3 and 3.4. Finally, in Section 3.5, we provide an upper bound for AGSDN channels. 3.1 Existence and Finiteness of Channel Capacity Theorem 1. Assume that an ASDN channel satisfies the following properties: • X is a closed and also bounded subset of R, i.e., there exists u ≥ 0 such that X ⊆ [−u, u]; • Real numbers 0 < σ` < σu exist such that σ` ≤ σ(x) ≤ σu for all x ∈ X ; • Positive real m and γ exist such that fZ (z) ≤ m < ∞ (a.e.), and E [\|Z\|γ ] = α < ∞; • The cost constraint functions gi (·) are bounded over X . 7 Then, the capacity of the ASDN channel is finite. Furthermore there is a capacity achieving probability measure; in other words, the capacity C can be expressed as a maximum rather than a supremum: C = max I (X; Y ). µX ∈F Moreover, the output distribution is unique, i.e. if µX1 and µX2 both achieves the capacity, then fY1 (y) = fY2 (y), ∀y ∈ R, where fY1 and fY2 are the pdfs of the output of the channel when the input probability measures are µX1 and µX2 , respectively. Remark 1. The above theorem is a generalization of that given in [10, Theorem 1] for the special case of Gaussian noise Z. The proof can be found in Section 6.1. To give a partial converse of the above theorem, consider the case that the second assumption of the above theorem fails, i.e., when there is a sequence {xi } of elements in X such that σ(xi ) converges to zero or infinity. The following theorem shows that input/output mutual information can be infinity in such cases. Theorem 2. Consider an ASDN channel with σ : X 7→ [0, +∞) where X is not necessarily a closed set. Suppose one can find a sequence {x̃i } of elements in X such that σ(x̃i ) converges to 0 or +∞ such that • As a sequence on real numbers, {x̃i } has a limit (possibly outside X ), which we denote by c. The limit c can be plus or minus infinity. • One can find another real number c0 6= c such that the open interval E = (c, c0 ) (or E = (c0 , c) depending on whether c0 > c or c0 < c) belongs to X . Furthermore, x̃i ∈ E, and σ(·) is monotone and continuous over E. 2 Then one can find a measure µX defined on E such that I (X; Y ) = ∞ provided that Z is a continuous random variable and has the following regularity conditions: \|h (Z) \| < ∞, ∃δ > 0 : Pr {Z > δ} , Pr {Z < −δ} > 0, Furthermore, there is more than one measure µX that makes I (X; Y ) = ∞. In fact, input X can be both a continuous or discrete random variable, i.e., one can find both an absolutely continuous measure with pdf fX and discrete pmf pX such that I (X; Y ) is infinity when the measure on input is either fX or pX . The proof can be found in Section 6.2 and uses some of the results that we prove later in the paper. Remark 2. As an example, consider an AGSDN channel with X = (0, u) for an arbitrary u > 0, and σ(x) = xα for α 6= 0. For this channel, we have C = +∞ if we have no input cost constraints. Setting α = 1, this shows that the capacity of the fast-fading channel given in (3) is infinity if c0 = 0; that is when there is no additive noise. This parallels a similar result given in [14] for complex Gaussian fading channels. 2 We only require monotonicity here, and not strictly monotonicity. 8 Remark 3. It is known that increasing the noise variance of an AWGN channel decreases its capacity. However, we show that this is no longer the case for signal-dependent noise channels: Consider two AGSDN channels with parameters σ1 (x) and σ2 (x), respectively, which are defined over X = (0, 1) with the following formulas: σ1 (x) = 1, σ2 (x) = 1 . x No input cost constraints are imposed. It is clear that σ2 (x) > σ1 (x) for all x ∈ X . However, by considering the constraint 0 < X < 1, from Theorem 1 we obtain that the capacity of the first channel is finite, while from Theorem 2, we obtain that the capacity of the second channel is ∞. Therefore, the constraint σ1 (x) > σ2 (x) for all x ∈ X does not necessarily imply that the capacity of an AGSDN channel with σ1 (x) is less than or equal to the capacity of an AGSDN with σ2 (x). 3.2 Lower Bounds on Capacity To compute capacity from (5), one has to take maximum over probability measures in a potentially large class F. Practically speaking, one can only find a finite number of measures µ1 , µ2 , · · · , µk in F and evaluate input/output mutual information for them. Ideally, {µi } should form an covering of the entire F (with an appropriate distance metric), so that mutual information at every arbitrary measure in F can be approximated with one of the measures µi . This can be computationally cumbersome, even for measures defined on a finite interval. As a result, it is desirable to find explicit lower bounds on the capacity. Observe that I (X; Y ) = h (Y ) − h (Y \|X). To compute the term h (Y \|X), observe that given X = x, we have Y = x + σ(x) · Z and thus h (Y \|X = x) = log σ(x) + h (Z) (see Lemma 2). Thus, h (Y \|X) = E [log σ(X)] + h (Z) . However, the term p h (Y ) is more challenging to handle. Authors in [1] consider an AGSDN channel with σ(x) = c20 + c21 x for x ≥ 0, as well as show that h (Y ) ≥ h (X) and hence I (X; Y ) ≥ h (X)−h (Y \|X). This implies that instead of maximizing I (X; Y ), one can maximize h (X)−h (Y \|X) to obtain a lower bound. The proof of the relation h (Y ) ≥ h (X) in [1] is non-trivial; we review it here to motivate our own techniques in this paper. First consider the special case of c1 = 0. In this case, we get σ(x) = c0 and the AGDSN reduces to AWGN channel Y = X +Z. In this special case, one obtains the desired equation by writing h (Y ) ≥ h (Y \|Z) = h (X + Z\|Z) = h (X\|Z) = h (X) . (7) p However, the above argument does not extend for the case of c1 > 0 since σ(x) = c20 + c21 x depends on x. As argued in [1], without loss of generality, one mayp assume that c0 = 0; this is because one can express a signal-dependent noise channel with σ(x) = c20 + c21 x as √ Y = X + c1 XZ1 + c0 Z0 , where Z0 and Z1 are √ independent standard normal variables. Thus, we can write Y = Y1 + c0 Z0 where Y1 = X + c1 XZ1 . From the argument for AWGN channels, we have that h (Y ) ≥ h (Y1 ). Thus, it suffices to show that h (Y1 ) ≥ h (X). This is the special case of the problem for c0 = 0 and √ corresponds to σ(x) = c1 x. 9 √ To show h (Y ) ≥ h (X) when Y = X + c1 XZ, more advanced ideas are utilized in [1]. The key observation is the following: assume that 1 x X ∼ gX (x) = e− α 1[x ≥ 0], α be exponentially distributed with mean E [X] = α. Then Y has density ! p √ αy − α + 2c21 \|y\| 1 √ 2 exp . gY (y) = p αc1 α(α + 2c21 ) Then, for any arbitrary input distribution fX , from the data processing property of the relative entropy, we have D (fY kgY ) ≤ D (fX kgX ) where fY is the output density for input density fX . Once simplified, this equation leads to h (fY ) ≥ h (fX ). The above argument crucially depends on the particular form of the output distribution corresponding to the input p exponential distribution. It is a specific argument that works for the specific choice of σ(x) = c20 + c21 x and normal distribution for Z, and cannot be readily extended to other choices of σ(·) and fZ (z). In this paper, we propose two approaches to handle more general settings: • (Idea 1:) We provide the following novel general lemma that establishes h (Y ) ≥ h (X) for a large class of ASDN channels. Lemma 1. Take an arbitrary channel characterized by the conditional pdf fY \|X (·\|x) satisfying Z fY \|X (y\|x) dx ≤ 1, ∀y ∈ Y, (8) X where X and Y are the support of channel input X and channel output Y , respectively. Take an arbitrary input pdf fX (x) on X resulting in an output pdf fY (y) on Y. Assuming that h (X) and h (Y ) exist, we have h (Y ) ≥ h (X) The proof is provided in Section 6.7. As an example, Lemma 1 yields an alternative proof for the result of [1] for an AGSDN p channel. Note that, as we mentioned before, in order to prove that h (Y ) ≥ h (X) for σ(x) = c0 2 + c2 x, √ we only need to prove it for σ(x) = c x. To this end, observe that since X ⊆ [0, +∞), we have Z Z ∞ (y−x)2 1 √ fY \|X (y\|x) dx ≤ e− 2c2 x dx 2πc2 x X 0 Z ∞ √ 2 )2 2 −(y−v √ = e 2c2 v2 dv πc2 0 1 y≥0 2y = ≤ 1. (9) e c21 y < 0 where x = v 2 , and v ≥ 0. The proof for equation (9) is given in Appendix A. • (Idea 2:) We provide a variation of the type of argument given in (7) by introducing a number of new steps. This would adapt the argument to ASDN channels. In the following sections, we discuss the above two ideas separately. 10 3.3 First Idea for Lower Bound Theorem 3. Assume an ASDN channel defined in (1), where σ : (`, u) 7→ (0, +∞) with −∞ ≤ ` < u ≤ +∞, and noise with pdf fZ (z) such that Z u 1 y−x dx ≤ 1, ∀y, (10) fZ σ(x) ` σ(x) 1 is Riemann integrable on (`, u). σ(x) (11) Then, if X is continuous random variable with pdf fX (x) supported over (`, u), I (X; Y ) ≥ h (ϕ(X)) − h (Z) , provided that the integrals defining h (ϕ(X)) and h (Z) converge to a real number or ±∞. The function ϕ(x) is an increasing function of x defined by Z x 1 ϕ(x) = dt, ∀x ∈ (`, u), (12) σ(t) c where c ∈ (`, u) is arbitrary. Remark 4. Note that for any c ∈ (`, u), ϕ(x) is well defined (see Definition 6). By selecting a different c0 ∈ (`, u) we obtain a different function ϕ0 (x) such that Z c 1 0 ϕ (x) − ϕ(x) = dt < ∞. c0 σ(t) However, h (ϕ(X)) is invariant with respect to adding constant terms, and thus invariant with respect to different choices of c ∈ (`, u). The above theorem is proved in Section 6.3. Corollary 1. Let W = ϕ(X). Since ϕ(·) is a one-to-one function (as σ(x) > 0), we obtain max µX ∈F ∩AC h (ϕ(X)) − h (Z) = max h (W ) − h (Z) , fW ∈G where F is defined in (6), and W ∼ fW belongs to G = {fW (·) µW ∈ AC, supp(µW ) ⊆ ϕ(X ), E[gi (ϕ−1 (W ))] ≤ 0 for all i = 1, · · · , k}. Here ϕ(X ) = {ϕ(x) : x ∈ X }. Hence, from Theorem 3 we obtain that max I (X; Y ) ≥ max h (W ) − h (Z) . µX ∈F fW ∈G In order to find the maximum of h (W ) over fW ∈ G, we can use known results on maximum entropy probability distributions, e.g., see [16, Chapter 3.1]. Corollary 2. Consider an ASDN channel satisfying (10) and (11). Assume that the only input constraint is X = (`, u) i.e. ` < X < u. Then, from Corollary 1, we obtain the lower bound Z u 1 max h (W ) − h (Z) = log dx − h (Z) , fW ∈G ` σ(x) by taking a uniform distribution for fW (w) over ϕ(X ) if this set is bounded [16, Section 3.1]. Else, if ϕ(X ) has an infinite length, the capacity is infinity by choosing a pdf for W whose differential entropy is infinity (see Example 1). The equivalent pdf fX (x) for X is the pdf of ϕ−1 (W ). 11 For more insight, we provide the following example. Example 2. Consider an AWGN channel (namely, an AGSDN channel with σ(x) = σ0) with X = R and Z ∼ N (0, 1). Let us restrict to measures that satisfy the power constraint E X 2 ≤ P ; that is g1 (x) = x2 . Since Z R 1 fZ σ(x) y−x σ(x) Z dx = R 1 − p e 2πσ02 (y−x)2 2 2σ0 dx = 1, we can apply Corollary 1. Here W = ϕ(X) = x/σ0 ; thus, the lower bound is C≥ max fW (·):E[W 2 ]≤ P 2 σ0 h (W ) − h (Z) = P 1 log 2 , 2 σ0 (13) √ where it is achieved by Gaussian distribution W ∼ N (0, P /σ0 )[17, Section 12.1]. It is well-known that the capacity of AWGN channel is 1 P C = log 1 + 2 . (14) 2 σ0 Comparing (14) and (13), we see that the lower bound is very close to the capacity in the high SNR regime. As another example, consider the constraints X ≥ 0, and E [X] ≤ α on admissible input measures. Here, we obtain the lower bound max fW (·):W ≥0 E[W ]≤ σα h (W ) − h (Z) = 1 α2 e log , 2 2πσ02 0 where we used the fact that the maximum is achieved by the exponential distribution fW (w) = σ0 /α exp(−wσ0 /α) for w ≥ 0 and fW (w) = 0 for w < 0 [17, Section 12.1]. Unlike the first example above, an exact capacity formula for this channel is not known. 3.4 Second Idea for Lower Bound Now, we are going to provide another lower bound which is more appropriate in the channels for which Z is either non-negative or non-positive, and σ(x) is a monotonic function. An example of such channels is the molecular timing channel discussed in the introduction. Theorem 4. Assume an ASDN channel defined in (1) with σ : (`, u) 7→ (0, ∞) for −∞ ≤ ` < u ≤ +∞. If X is a continuous random variable with pdf fX (x), and σ(x) is continuous and monotonic over (`, u), (15) 1 is Riemann integrable on (`, u), σ(x) (16) then I (X; Y ) ≥ αh (ψ(X)) − β, provided that α, β are well-defined, and α > 0. In order to define the variables α, β, and the function ψ(x), take some arbitrary δ > 0 and proceed as follows: 12 • If the function σ(x) is increasing over (`, u), let x Z ψ(x) = δ log σ(x) + c α = Pr {Z ≥ δ} , 1 dt, σ(t) β = αh (Z\|Z ≥ δ) + H2 (α), • If the function σ(x) is decreasing over (`, u), let Z ψ(x) = −δ log σ(x) + c α = Pr {Z ≤ −δ} , x 1 dt, σ(t) β = αh (Z\|Z ≤ −δ) + H2 (α), where c ∈ (`, u) is arbitrary, and H2 (p) := −p log p − (1 − p) log (1 − p). Remark 5. Observe that in both cases, ψ(x) is an strictly increasing function of x defined over (`, u), as σ(x) > 0 and log(x) is increasing. Similar to Remark 4, the choice of c ∈ (`, u) does not affect the value of h (ψ(X)), and hence the lower bound. However, the choice of δ > 0 affects the lower bound. The above theorem is proved in Section 6.4. Corollary 3. Similar to Corollary 1, let V = ψ(X). Since ψ(·) is a one-to-one (strictly increasing) function, we obtain max αh (ψ(X)) − β = max αh (V ) − β µX ∈F ∩AC fV ∈G where F is defined in (6), and V ∼ fV belongs to G = {fV (·) µV ∈ AC, supp(µV ) ⊆ ψ(X ), E[gi (ψ −1 (V ))] ≤ 0 for all i = 1, · · · , k}. Hence, from Theorem 4 we obtain that max I (X; Y ) ≥ α max h (V ) − β, µX ∈F fV ∈G where α and β are constants defined in Theorem 4. As mentioned earlier, to maximize h (V ) over fV ∈ G, we can use known results on maximum entropy probability distributions, e.g., see [16, Chapter 3.1]. Corollary 4. Consider an ASDN channel satisfying (15) and (16). Assume that the only input constraint is X = (`, u) i.e. ` < X < u. Then, from Corollary 3, we obtain the lower bound Z u σ(u− ) 1 α max h (V ) − β = α log δ log + dx − β, fV ∈G σ(`+ ) ` σ(x) where α and β are defined in Theorem 4, and σ(u− ) := lim σ(x). σ(`+ ) := lim σ(x), x↓` x↑u The lower bound is achieved by taking a uniform distribution for fV (w) over ψ(X ) if this set is bounded [16, Section 3.1]. Else, if ψ(X ) has an infinite length, the capacity is infinity by choosing a pdf fV (v) such that h (V ) = +∞. (see Example 1). The equivalent pdf fX (x) for X is the pdf of ψ −1 (V ). 13 3.5 An Upper Bound We begin by reviewing upper bound given in [1] to motivate our own upper bound. The upper bound in [1] works by utilizing Topsoe’s inequality [18] to bound mutual information I (X; Y ) from above as follows: I (X; Y ) ≤ EµX [D (f (y\|x)kq(y))]. for any arbitrary pdf q(y) on output Y . The distribution q(y) is chosen carefully to allow for √ calculation of the above KL divergence. The particular form of σ(x) = c0 + c1 x makes explicit calculations possible. The second difficulty in calculating the above expression is that we need to take expected value over input measure µX . However, the capacity achieving input measure is not known. This difficulty is addressed by the technique of “input distributions that escape to infinity”, under some assumptions about the peak constraint. In this part, we give an upper bound based on the KL symmetrized upper bound of [15]. The idea is that I (X; Y ) = D(µX,Y kµX µY ) ≤ D(µX,Y kµX µY ) + D(µX µY kµX,Y ) , Dsym (µX,Y kµX µY ). Our upper bound has the advantage of being applicable to a large class of σ(x). To state this upper bound, let Cov (X, Y ) := E [XY ] − E [X] E [Y ] be the covariance function between two random variables X and Y . Theorem 5. For any AGSDN channel defined in (1), we have 1 1 X 2 2 I (X; Y ) ≤ − Cov X + σ (X), 2 + Cov X, 2 , 2 σ (X) σ (X) provided that the covariance terms on the right hand side are finite. The proof can be found in Section 6.5 Corollary 5. For an AGSDN channel with parameters σ(x), Z ∼ N (0, 1), and X = [0, u], if functions σ(x) and x/σ 2 (x) are increasing over X , σ(0) > 0, and x2 + σ(x) is convex over X then ( 1 F α ≥ u2 max I (X; Y ) ≤ 18 , α α µX :0≤X≤u α < u2 2 1 − u uF E[X]≤α where F = u2 u2 σ 2 (0) σ 2 (u) + + + − 2. σ 2 (u) σ 2 (0) σ 2 (u) σ 2 (0) The corollary is proved in Section 6.6. Remark 6. Even though Corollary 5 is with the assumption σ(0) > 0, if we formally set σ(0) = 0, we see that F and the upper bound on capacity becomes infinity. This is consistent with Theorem 2 when σ(0) = 0. p Corollary 6. The particular choice of σ(x) = c20 + c21 x that was motivated by applications discussed in the Introduction has the property that σ(x), x/σ 2 (x) are increasing and Theorem 5 can be applied. 14 4 Some Useful Lemmas In this section, we provide three lemmas used in the proof of theorems in this paper. Lemma 2. In an ASDN channel defined in (1), with continuous random variable noise Z with pdf fZ (·), and noise coefficient σ(x) > 0 (µX −a.e.), the conditional measure µY \|X (·\|x) has the following pdf: 1 y−x fY \|X (y\|x) = , µX,Y -(a.e.). fZ σ(x) σ(x) Moreover, Y is a continuous random variable with the pdf 1 y−X fY (y) = E . fZ σ(X) σ(X) Furthermore, if h (Z) exists, h (Y \|X) can be defined and is equal to h (Y \|X) = E [log σ(X)] + h (Z) . The lemma is proved in Section 6.8. Lemma 3. Let X be a continuous random variable with pdf fX (x). For any function σ : (`, u) 7→ [0, +∞) such that σ(x) is Riemann integrable over (`, u) and σ(x) > 0 (a.e), where −∞ ≤ ` < u ≤ +∞, we have that h (X) + E [log σ(X)] = h (ϕ(X)) . (17) where Z x ϕ(x) = σ(t) dt, (18) c where c ∈ (`, u) is an arbitrary constant. Note that if the left-hand side does not exist, or becomes ±∞, the same occurs for the right-hand side and vice versa. The lemma is proved in Section 6.9. Lemma 4. Let X be a random variable with probability measure µX , and the functions w(x) and v(x) be increasing over [`, u], where −∞ < ` < u < +∞. If v(x) is convex over [`, u], then max µX :`≤X≤u E[X]≤α Cov (w(X), v(X)) ≤ β[w(u) − w(`)][v(u) − v(`)], where ( β= 1 4 (u−α)(α−`) (u−`)2 α≥ α< `+u 2 `+u 2 . Furthermore, for the case α ≥ (` + u)/2, a maximizer of (19) is the pmf 1 pX (`) = pX (u) = . 2 For the case α < (` + u)/2 if v(x) is linear, a maximizer of (19) is the pmf pX (`) = 1 − pX (u) = The proof is given in Section 6.10. 15 u−α . u−` (19) 1 10 1 10 Capacity Capacity (nats per channel use) Capacity (nats per channel use) KL Upper Bound 0 10 −1 10 0 10 Capacity −1 10 Corollary 2 Lower Bound Corollary 4 Lower Bound −2 10 −2 10 0 20 40 60 80 100 120 140 160 180 200 σ02 −3 10 50 100 150 200 250 300 A Figure 1: Capacity and Symmetrized divergence upper bound in terms of c20 for AGSDN channel with A p = 5, α = 2.5, c1 = 1 and function σ(x) = c20 + x. 5 Figure 2: Capacity and lower bound at corollary 2 and 4 in terms of A√for AGSDN channel with function σ(x) = 1 + x. Numerical Results p In this section, some numerical results are given for σ(x) = c20 + x and Z ∼ N (0, 1). The upper bound, Corollary (5), and the capacity are depicted in the logarithmic scale in Fig. 1, where we have considered the peak constraint A = 5 and average constraint α = 2.5 . It can be observed that the distance between the upper bound and the capacity is a small constant in the logarithmic scale and low SNR regime. This is consistent with [15] that argues that the upper bound based on symmetrized KL divergence is mostly suitable for the low SNR regime. p The lower bounds of Corollaries 4 and 2 are plotted in Fig. 2 for the function σ(x) = c20 + x in terms of peak constraint, A. Here, c0 = 1 is assumed. The lower bound of Corollary 2 for 0 < X < A is computed by the following closed form formula: ! q Z A 1 1 2 − 2c p log − h (Z) = log 2 A + c log(2πe), 0 − 0 2 2 c0 + x 0 while the lower bound of Corollary 4 equals ! ! p Z A q A + c20 1 σ(A) p + α log δ log dx − β = α log δ log + 2 A + c20 − 2c0 − β 2+x σ(0) c 0 c 0 0 where δ > 0 and α = Pr {Z ≥ δ} , β = αh (Z\|Z ≥ δ) − α log α − (1 − α) log (1 − α). We maximized over δ in order to find the lower bound of Corollary 4. The first lower bound is better than the second one mainly because of the multiplicative coefficient α of the second lower bound. Since the second lower bound is for a more general class of channels, we should consider the positive (or negative) part of the support of Z, causing a multiplicative of coefficient 1/2 for the Gaussian noise. However, if the support of Z is positive (or negative) reals, the two lower bounds do not differ much. 16 6 6.1 Proofs Proof of Theorem 1 Finiteness of capacity: The first step is to show that the capacity is finite: sup I (X; Y ) < ∞. (20) µX ∈F To prove this, it suffices to show that the supremum of both h (Y ) and h (Y \|X) over µX ∈ F are finite, i.e., \|h (Y ) \|, \|h (Y \|X) \| < +∞, uniformly on µX ∈ F. (21) Utilizing Lemma 2, the existence and boundedness of h (Y \|X) is obtained as follows: \|h (Y \|X) \| ≤ max{\| log σ` \|, \| log σu \|} + \|h (Z) \| < ∞, uniformly on F. From Lemma 2, we obtain that Y is continuous with a pdf fY (y). To prove that the integral defining h (Y ) is convergent to a finite value (existence of entropy), and furthermore the integral is convergent to a value that is bounded uniformly on F, it is sufficient to show that there are some positive real γ, m̄ and v such that for any µX ∈ F, we have [19]: sup fY (y) < m̄, (22) y∈R E [\|Y \|γ ] < v. (23) Also, from Lemma 2, we obtain that for any µX ∈ F fY (y) ≤ m . σ` Thus, (22) holds with m̄ = m/σ` . In order to prove (23), note that E [\|Y \|γ ] ≤E [(\|X\| + σu \|Z\|)γ ] ≤2γ E [max {\|X\|γ , σuγ \|Z\|γ }] ≤2γ E [\|X\|γ ] + (2σu )γ E [\|Z\|γ ] ≤2γ uγ + (2σu )γ α, uniformly on F. Thus, h (Y ) is well-defined and uniformly bounded on F. Hence, from the definition of mutual information we obtain that I (X; Y ) = h (Y ) − h (Y \|X) (24) is bounded uniformly for µX ∈ F. Existence of a maximizer: Let C = sup I (X; Y ) < ∞. (25) µX ∈F We would like to prove that the above supremum is a maximum. Equation (25) implies existence (k) of a sequence of measures {µX }∞ k=1 in F such that lim I (Xk ; Yk ) = C, k→∞ 17 (k) (k) where Xk ∼ µX , and Yk ∼ µY is the output of the channel when the input is Xk . Furthermore, (k) without loss of generality, we can assume that {µX }∞ k=1 is convergent (in the Lévy measure) to a ∗ measure µX ∈ F. The reason is that since X is compact, the set F is also compact with respect to the Lévy measure [10, Proposition 2]. Thus, any sequence of measures in F has a convergent (k) subsequence. With no loss of generality we can take the subsequence as {µX }∞ k=1 . Thus, from convergence in Lévy measure, we know that there is µ∗X ∈ F such that lim E [g(Xk )] = E [g(X ∗ )] , (26) k→∞ for all g : R 7→ C such that supx∈R \|g(x)\| < +∞. We would like to prove that I (X ∗ ; Y ∗ ) = C, (27) where Y ∗ ∼ µ∗Y is the output measure of the channel when the input measure is µ∗X . This will complete the proof. From the argument given in the first part of the proof on “Finiteness of capacity”, h (Y ∗ \|X ∗ ) and h (Y ∗ ) are well-defined and finite. As a result to show (27), we only need to prove that lim h (Yk \|Xk ) = h (Y ∗ \|X ∗ ) , (28) lim h (Yk ) = h (Y ∗ ) . (29) k→∞ k→∞ Since −∞ < σ` < σu < +∞, (28) is obtained from (26) and Lemma 2. In order to prove (29), we proceed as follows: (k) • Step 1: We begin by showing that the sequence {µY }∞ k=1 is a Cauchy sequence with respect to total variation i.e. (m) ∀ > 0, ∃N : m, n ≥ N ⇒ kµY (n) − µY kV ≤ , (30) where for any two arbitrary probability measure µA and µB , the total variation distance is defined by [16, p. 31] X kµA − µB k := sup µA (Ei ) − µB (Ei ) , ∆ i where ∆ = {E1 , · · · , Em } ⊆ B(R) is the collection of all the available finite partitions. • Step 2: Having established step 1 above, we utilize the fact that the space of probability measures is complete with respect to the total variation metric. To show this, note that by Lemma 2, all the Yk ’s have a pdf, and hence the total variation can be expressed in terms of the k · kL1 norm between pdfs [16, Lemma 1.5.3]. From [20, p. 276] we obtain that this space of pdfs is complete with respect to k · kL1 norm. (k) As a result, µY converges to some measure Yb ∼ µ bY with respect to the total variation metric. We further claim that this convergence implies that lim h (Yk ) = h Yb . (31) k→∞ (k) The reason is that from (22) and (23), we see that {fY } and fŶ are uniformly bounded and have finite γ-moments. Therefore, (31) follows from [19, Theorem 1]. Thus, in step 2, we obtain that the sequence h (Yk ) has a limit. 18 • Step 3: We show that the limit found in Step 2 is equal to h (Y ∗ ), i.e., h Yb = h (Y ∗ ) . (32) This completes the proof of (29). Hence, it only remains to prove (30) and (32). 0 Proof of (30): Since {I (Xk ; Yk )}∞ k=1 is convergent to C, for any > 0, there exists N such that: \|C − I (Xk ; Yk ) \| ≤ 0 , ∀k ≥ N. Now, consider m, n ≥ N . Let Q be a uniform Bernoulli random variable, independent of all (m) previously defined variables. When Q = 0, we sample from measure µX and when Q = 1, we (n) e ∼ µ e defined as follows: sample from measure µX . This induces the measure X X 1 (m) 1 (n) µXe = µX + µX . 2 2 e We have a Markov chain Q − X e − Ye . Let Ye ∼ µYe be the output of the channel when the input is X. Note that e Ye \|Q = 1 I (Xm ; Ym ) + 1 I (Xn ; Yn ) . I X; 2 2 From concavity of mutual information in input measure, we obtain that: e Ye ≥ 1 I (Xm ; Ym ) + 1 I (Xn ; Yn ) ≥ C − 0 . I X; 2 2 e Ye ≤ C, Since F is an intersection of half spaces, it is convex and as a result µXe ∈ F. Thus, I X; and we obtain that e Ye − I X; e Ye Q ≤ 0 . I X; e e e e Because of the Markov chain Q − X − Y , we obtain I Y ; Q X = 0 and as a result: I Ye ; Q ≤ 0 =⇒ D µYe ,Q kµYe µQ ≤ 0 . From the Pinsker’s inequality we obtain that kµYe ,Q − µYe µQ kV ≤ √ 20 , (33) where kµYe ,Q − µYe µQ kV is the total variation between the measures µYe ,Q and µYe µQ . Note that 1 (n) 1 (m) kµYe ,Q − µYe µQ kV = kµY − µYe kV + kµY − µYe kV . 2 2 Therefore from (33) and (34), we obtain that (m) kµY √ (n) − µYe kV , kµY − µYe kV ≤ 2 20 . As a result, (m) kµY √ (n) − µY kV ≤ 4 20 . 19 (34) (k) Hence, by taking 0 ≤ 2 /32, we obtain that {µY }∞ k=1 is a Cauchy sequence. Proof of (32): To this end, it suffices to prove that ΦYb (ω) = ΦY ∗ (ω), ∀ω ∈ R, where ΦX (ω) := E [exp(jωX)] is the characteristic function of the random variable X. Since Yk converge to Yb in total variation, and the fact that convergence in total variation is stronger than weakly convergence [16, p. 31], from (26) we obtain that their characteristic functions, ΦYk (ω), also converge to ΦYb (ω) pointwise. Hence, it suffices to prove that ΦYk (ω) converge to ΦY ∗ (ω) pointwise. From (1), we obtain that h i ΦYk (ω) = E ejω(Xk +σ(Xk )Z) = E ejωXk ΦZ (σ(Xk )ω) . Similarly, h i ∗ ΦY ∗ (ω) = E ejωX ΦZ (σ(X ∗ )ω) . Since {Xk } converges to X ∗ in Lévy measure and the function g(x) = ejωx ΦZ (σ(x)ω) is bounded: \|g(x)\| = ejωx ΦZ (σ(x)ω) ≤ ejωx \|ΦZ (σ(x)ω)\| ≤ 1, from (26) we obtain that E[g(Xk )] = ΦYk (ω) converges to E[g(X ∗ )] = ΦY ∗ (ω) pointwise. Uniqueness of the output pdf: The proof is the same as the first part of the proof of [10, Theorem 1]. This completes the proof. 6.2 Proof of Theorem 2 For a continuous input measure, we utilize a later result in the paper, namely Theorem 4 by choosing ` = c, u = c0 when c0 > c, or ` = c0 , u = c when c0 < c. To use Corollary 4, observe that the image of E under ψ(·) has infinite length. This is because the sequence {x̃i } in E was such that the monotone function σ(·) converged to zero or infinity on that sequence. Then, it is obtained that any pdf fX (·) such that h (ψ(X)) = +∞, makes I (X; Y ) infinity if \|h (Z\|Z > δ) \| < ∞ (which leads to \|β\| < ∞), where ψ(x) is the bijective function of x defined in the statement of Theorem 4. In order to prove that \|h (Z\|Z > δ) \| < ∞, let the random variable Z̄ be Z conditioned to Z > δ. Due to the continuity of Z and the fact that Pr {Z > δ} > 0, we obtain that Z̄ has a valid pdf fZ̄ (z) defined by ( 1 fZ (z) z > δ fZ̄ (z) = θ , 0 z≤δ where θ := Pr {Z > δ} > 0. Since h (Z) exists and \|h (Z) \| < ∞, we obtain that 1 E < ∞. fZ (Z) Hence, \|h Z̄ \| ≤ E 1 fZ̄ (Z̄) 1 ≤ − log θ + E θ 1 fZ (Z) < ∞. Therefore, h (Y \|Z > δ) exists and \|h (Y \|Z > δ) \| < ∞. A similar treatment can be used to prove \|h (Y \|Z < δ) \| < ∞. 20 It remains to construct a discrete pmf with infinite mutual information. The statement of the theorem assumes existence of a sequence {x̃i } in an open interval E = (c, c0 ) ⊂ X (or E = (c0 , c) if c0 < c) such that 1. c is the limit of the sequence {x̃i }, 2. σ(x̃i ) converges to 0 or +∞ 3. σ(·) is monotone and continuous over E We now make the following claim about existence of another sequence {xi }∞ i=1 ⊆ E with certain nice properties: Claim: Suppose that one cannot find a non-empty interval [x0 , x00 ] ⊂ E such that σ(x) = 0 for all x ∈ [x0 , x00 ]. Then, there exists 0 < a < b and a sequence {xi }∞ i=1 ⊆ E, such that • If σ(x) is increasing, Pr {a < Z < b} > 0, (35) (xi + aσ(xi ), xi + bσ(xi )) ∩ (xj + aσ(xj ), xj + bσ(xj )) = ∅, 0 < σ(xi ) < ∞, ∀i 6= j ∈ N ∀i ∈ N. (36) (37) • If σ(x) is decreasing, Pr {−b < Z < −a} > 0, (xi − bσ(xi ), xi − aσ(xi )) ∩ (xj − bσ(xj ), xj − aσ(xj )) = ∅, 0 < σ(xi ) < ∞, ∀i 6= j ∈ N ∀i ∈ N. We continue with the proof assuming that this claim is correct; we give the proof of this claim later. To show how this claim can be used to construct a discrete pmf with infinite mutual information, consider the possibility that the assumption of the claim fails: σ(x) = 0 for all x ∈ [x0 , x00 ], then Y = X in that interval when X ∈ [x0 , x00 ]. Therefore, we can provide any discrete distribution in that interval such that H (X) = ∞, as a result I (X; Y ) = I (X; X) = H (X) = ∞. Thus, we should only consider the case that the assumption of the claim holds. Assume that σ(x) is increasing. The construction when σ(x) is decreasing is similar. Fix a given a, b, {xi }∞ i=1 satisfying (35) and (36). Take an arbitrary pmf {pi }∞ i=1 such that X i pi log 1 = +∞. pi (38) Then, we define a discrete random variable X, taking values in {xi }∞ i=1 such that Pr {X = xi } = pi . We claim that I (X; Y ) = +∞. To this end, it suffices to show I (X; Y ) ≥ Pr {a < Z < b} I (X; Y \|a < Z < b) − H2 (Pr {a < Z < b}), (39) I (X; Y \|a < Z < b) = ∞. (40) Proof of (39): Define random variable E as following: ( 0 Z ∈ (a, b) E= 1 Z∈ / (a, b) 21 From the definition of mutual information, we have that I (X; Y \|E) − I (X; Y ) = I (Y ; E\|X) − I (Y ; E) ≤ H (E) , Since I (X; Y \|E) = Pr {E = 0} I (X; Y \|E = 0) + Pr {E = 1} I (X; Y \|E = 1) , we conclude (39). Proof of (40): Since I (X; Y \|a < Z < b) = H (X) − H (X\|Y, a < Z < b) , it suffices to show that H (X) = ∞, H (X\|Y, a < Z < b) = 0. (41) P The equality H (X) := − pi log pi = +∞ follows (38). To prove the other equality, note that Y belongs to the interval (xi + aσ(xi ), xi + bσ(xi )) when X = xi . Therefore, since the intervals (xi + aσ(xi ), xi + bσ(xi )) are disjoint, X can be found from Y . Thus, X is a function of Y when a < Z < b. As a result, the second equality of (41) is proved. Now, it only remains to prove our Claim on the existence of a, b, and {xi }∞ i=1 . We assume that σ(x) is increasing. The proof when σ(x) is decreasing is similar. From the assumptions on Z that Pr {Z ≥ δ} > 0, we obtain there exists δ < b < ∞ such that Pr {δ < Z < b} > 0. As a result, we select a = δ. Since σ(x) is monotone, we cannot have σ(x0 ) = σ(x00 ) = 0 for two arbitrary distinct x0 and x00 in E since this implies that σ(x) = 0 for all x in between x0 and x00 . As a result, we shall not worry about the constraint (37) on {xi } because σ(xi ) = 0 can occur for at most one index i and we can delete that element from the sequence to ensure (37). To show the existence of {xi }∞ i=1 , we provide a method to find xi+1 with respect to xi . The method is described below and illustrated in Figure 3. Take x1 an arbitrary element of E. Observe that since σ(x) is continuous and increasing over E, the functions x + aσ(x) and x + bσ(x) are continuous and strictly increasing over E, as well as x + aσ(x) < x + bσ(x), ∀x ∈ E. Therefore, for the case c0 > c (happening when σ(xi ) converge to 0), lim x + aσ(x) = lim x + bσ(x) = c, x→c x→c Hence, for a given xi ∈ E, due to the intermediate value theorem, there exists unique xi+1 satisfying c < xi+1 < xi < c0 such that xi+1 + bσ(xi+1 ) = xi + aσ(xi ). Similarly, for the case c0 < c (happening when σ(x̃i ) converge to +∞), if xi ∈ E, there exists unique xi+1 satisfying c > xi+1 > xi > c0 such that xi+1 + aσ(xi+1 ) = xi + bσ(xi ). It can be easily obtained that the intervals created this way are disjoint, and the process will not stop after finite steps. Therefore, the theorem is proved. 22 x1 x2 σ→+∞ (ascending) x3 c c' x4 ... x + a σ(x) x - a σ(x) x + b σ(x) x - b σ(x) c c' σ→+∞ (descending) ... x5 x1 x4 x3 x2 σ→0 (descending) σ→0 (ascending) x - a σ(x) x + a σ(x) x1 x5 x - b σ(x) x4 x + b σ(x) c' x3 c' x2 x3 ... x2 c x4 x1 Figure 3: Possible cases for σ(x) when \|c\| < ∞. 23 ... c 6.3 Proof of Theorem 3 From Lemma 2 we obtain that h (Y \|X) exists. Hence, utilizing Lemma 1 , we can write h (Y ) ≥ h (X) =⇒ I (X; Y ) ≥ h (X) − h (Y \|X) , provided that (42) u Z fY \|X (y\|x) dx ≤ 1, ` where it is satisfied due to Z u Z fY \|X (y\|x) dx = ` ` u 1 fZ σ(x) y−x σ(x) dx ≤ 1, where the last inequality comes from the assumption of the theorem. From Lemma 2, we have that h (Y \|X) = E [log σ(X)] + h (Z) . Therefore, (42) can be written as I (X; Y ) ≥ h (X) − E [log σ(X)] − h (Z) . Exploiting Lemma 3 we obtain that h (X) − E [log σ(X)] = h (ϕ(X)) , where ϕ(X) is defined in (12) Hence, the proof is complete. 6.4 Proof of Theorem 4 We only prove the case that σ(x) is an increasing function over (`, u). The proof of the theorem for decreasing functions is similar to the increasing case and we only need to substitute Z ≥ δ with Z ≤ −δ. We claim that I (X; Y ) ≥ αI (X; Y \|Z ≥ δ) − H2 (α). (43) Consider random variable E as following: ( 0 Z≥δ E= . 1 Z<δ From the definition of mutual information, we have that I (X; Y \|E) − I (X; Y ) = I (Y ; E\|X) − I (Y ; E) ≤ H (E) , Therefore, since I (X; Y \|E) = Pr {Z ≥ δ} I (X; Y \|Z ≥ δ) + Pr {Z < δ} I (X; Y \|Z < δ) , we conclude (43). Now, we find a lower bound for I (X; Y \|Z ≥ δ). From Lemma 2 we obtain that Y is a continuous random variable. We claim that I (X; Y \|Z ≥ δ) =h (Y \|Z ≥ δ) − h (Y \|X, Z ≥ δ) 24 =h (Y \|Z ≥ δ) − (E [log σ(X)] + h (Z\|Z ≥ δ)) =h (Y \|Z ≥ δ) − h (X) − E log(1 + Zσ 0 (X))\|Z ≥ δ 1 + Zσ 0 (X) + h (X) + E log Z ≥ δ − h (Z\|Z ≥ δ) , σ(X) (44) (45) where (44) is obtained from Lemma 2, and the fact that random variable Z conditioned to Z ≥ δ is also continuous when Pr {Z ≥ δ} > 0. Moreover, (45) is obtained by adding and subtracting the term E [log(1 + Zσ 0 (X))\|Z ≥ δ]. Note that we had not assumed that σ(x) needs to be differentiable. We had only assumed that σ : (`, u) 7→ (0, ∞) is continuous and monotonic over (`, u). However, every monotonic function is differentiable almost everywhere, i.e., the set of points in which σ(x) is not differentiable has Lebesgue measure zero. We define σ 0 (x) to be equal to zero wherever σ(x) is not differentiable; and we take σ 0 (x) to be the derivative of σ(x) wherever it is differentiable. With this definition of σ 0 (x) and from the continuity of σ(x), we have that the integral of σ 0 (x)/σ(x) gives us back the function log(σ(x)). Since σ(x) is an increasing positive function, and Z ≥ δ > 0, we conclude that 1 + Zσ 0 (X) 1 + δσ 0 (X) E log Z ≥ δ ≥ E log . (46) σ(X) σ(X) From Lemma 3 and the fact that the integral of σ 0 (x)/σ(x) gives us back the function log(σ(x)), we obtain that 1 + δσ 0 (X) h (X) + E log = h (ψ(X)) , σ(X) where ψ(x) is defined in Theorem 4. As a result from (45), we obtain that I (X; Y \|Z ≥ δ) ≥h (Y \|Z ≥ δ) − h (X) − E log(1 + Zσ 0 (X))\|Z ≥ δ + h (ψ(X)) − h (Z\|Z ≥ δ) , (47) Using this inequality in conjunction with (43), we obtain a lower bound on I (X; Y ). The lower bound that we would like to prove in the statement of the theorem is that I (X; Y ) ≥ αh (ψ(X)) − αh (Z\|Z ≥ δ) − H2 (α). As a result, it suffices to prove that for all continuous random variables X with pdf fX (x) we have h (Y \|Z ≥ δ) − h (X) − E log(1 + Zσ 0 (X))\|Z ≥ δ ≥ 0. To this end, observe that h (Y \|Z ≥ δ) ≥ h (Y \|Z, Z ≥ δ). Thus, if we show that h (Y \|Z, Z ≥ δ) = h (X) + E log(1 + Zσ 0 (X))\|Z ≥ δ , (48) the proof is complete. We can write that Z h (Y \|Z, Z ≥ δ) = ∞ fZ 0 (z)h (Y \|Z = z) dz, δ where Z 0 is Z conditioned to Z ≥ δ, and the pdf of Z 0 is denoted by fZ 0 (z). By defining the function rz (x) := x + zσ(x), we obtain that Yz = rz (X), 25 where Yz is Y which is conditioned to Z = z ≥ δ. Since, σ(x) is a continuous increasing function, rz (x) is a bijection for all z ≥ δ, and so its inverse function, rz−1 (y), exists. Moreover, since X is continuous and rz (·) is a bijection, Yz is also continuous random variable with pdf fY (y) defined as following: 1 fYz (y) = fX (x), 1 + zσ 0 (x) where x = rz−1 (y). 3 Thus, we have that 1 h (Y \|Z = z) =E log fYz (Yz ) 1 =E log + E log(1 + zσ 0 (X)) fX (X) =h (X) + E log(1 + zσ 0 (X)) . By taking expected value over Z ≥ δ from both sides, (48) is achieved. Therefore, the theorem is proved. 6.5 Proof of Theorem 5 Based on [15] we obtain that I (X; Y ) ≤ Dsym (µX,Y kµX µY ), Utilizing Lemma 2, we obtain that the pdfs fY (y) and fY \|X (y\|x) exist and are well-defined. Therefore, Dsym (µX,Y kµX µY ) =D(µX,Y kµX µY ) + D(µX µY kµX,Y ) fY \|X (Y \|X) fY (Y ) =EµX,Y log + EµX µY log fY (Y ) fY \|X (Y \|X) 1 1 = − EµX,Y log + E log fY \|X (Y \|X) fY (Y ) 1 1 + EµX µY log − E log fY \|X (Y \|X) fY (Y ) 1 =EµX µY log − h (Y \|X) . fY \|X (Y \|X) Again, from Lemma 2, since Z ∼ N (0, 1), we obtain that log √ (y − x)2 1 = log σ(x) 2π + fY \|X (y\|x) 2σ 2 (x) Therefore, since Z = (Y − X)/σ(X), we obtain that h √ i 1 h (Y \|X) = E log σ(X) 2π + . 2 3 (49) The measure zero points where σ(x) is not differentiable affect fYz (y) on a measure zero points. However, note that FYz (y) = FX (r−1 (y)) is always correct and thus the values of fYz (y) on a measure zero set of points are not important. 26 In addition, EµX µY h √ i 1 (Y − X)2 log = E log 2πσ(X) + EµX µY . fY \|X (Y \|X) 2σ 2 (X) By expanding, we obtain that 2 1 X2 X (Y − X)2 =E Y E 2 +E 2 − 2E [Y ] E 2 . EµX µY σ 2 (X) σ (X) σ (X) σ (X) By substituting Y with X + σ(X)Z and simplifying we can write 2 2 (Y − X)2 1 1 EµX µY =E X E + E σ (X) E σ 2 (X) σ 2 (X) σ 2 (X) 2 X X − 2E [X] E 2 +E 2 σ (X) σ (X) 2 2 2 1 X + σ 2 (X) 1 =E X E 2 + E σ (X) E 2 −E +1 σ (X) σ (X) σ 2 (X) X X2 − 2E [X] E 2 , +2 E 2 σ (X) σ (X) which equals to 1 − Cov X 2 + σ 2 (X), 1 2 σ (X) + 2Cov X, X 2 σ (X) . Therefore, from all above equations the theorem is proved. 6.6 Proof of Corollary 5 Observe that 2 1 u u2 σ 2 (0) σ 2 (u) 1 F = + + + −2 2 2 σ 2 (u) σ 2 (0) σ 2 (u) σ 2 (0) 1 2 1 1 u2 2 2 = u + σ (u) − σ (0) − + . 2 σ 2 (0) σ 2 (u) σ 2 (u) Then, using Theorem 5, it suffices to prove the following two inequalities: −1 1 1 2 2 2 2 2 ≤ β u + σ (u) − σ (0) − , Cov X + σ (X), 2 σ (X) σ 2 (0) σ 2 (u) and Cov X, where ( β= X 2 σ (X) 1 4 u α 1− u α ≤β u2 , σ 2 (u) α≥ α< u 2 u 2 (50) (51) . Since σ(x) is increasing, we obtain that x2 + σ 2 (x) and −1/σ 2 (x) are also increasing. Therefore, from Lemma 4, equation (50) is proved. Similarly, (51) is also obtained from Lemma 4 because x and x/σ 2 (x) are increasing functions. 27 6.7 Proof of Lemma 1 From Definition 5, we obtain that fY (Y ) . h (X) − h (Y ) = E log fX (X) Now, utilizing the inequality log x ≤ x − 1, it suffuces to prove that fY (Y ) E ≤ 1. fX (X) To this end, we can write Z Z fY (y) fY (Y ) = fX,Y (x, y) E dy dx fX (X) fX (x) X Y Z Z fY \|X (y\|x)fY (y) dy dx = ZX Y Z = fY (y) fY \|X (y\|x) dx dy Y X Z ≤ fY (y) dy = 1, Y where the last inequality holds because of the assumption of the lemma. Therefore, the lemma is proved. 6.8 Proof of Lemma 2 The conditional pdf fY \|X (y\|x) can be easily obtained from the definition of channel in (1). In order to calculate h (Y \|X), using the Definition 5 we can write   1 1  . h (Y \|X) = E = E [log σ(X)] + E log Y −X fY \|X (Y \|X) f Z σ(X) Exploiting the fact that (Y − X)/σ(X) = Z, h (Y \|X) is obtained. It only remains to prove that Y is continuous. To this end, from the definition of the channel in (1), we obtain that y−X y−X FY (y) = Pr Z ≤ = EµX FZ , σ(X) σ(X) where FY (y) and FZ (z) are the cdfs of the random variables Y and Z, defined by FY (y) = Pr {Y ≤ y} and FZ (z) = Pr {Z ≤ z}, respectively. In order to prove the claim about fY (y), we must show that Z y 1 y−x y−X E fZ dy = E FZ , σ(x) σ(x) σ(X) −∞ for all y ∈ R. Because of the Fubini’s theorem [20, Chapter 2.3], it is equivalent to −n − X lim E FZ = 0. n→∞ σ(X) 28 Equivalently, we need to show that for any > 0, there exists m such that −n − X E FZ ≤ , ∀n > m. σ(X) (52) Since limz→−∞ FZ (z) = 0, there exists ` ∈ R such that FZ (z) ≤ , 2 ∀z ≤ `. Therefore, since FZ (z) ≤ 1 for all z, we can write −n − X −n − X ≤ + Pr ≥` . E FZ σ(X) 2 σ(X) We can write Pr −n − X ≥ ` = Pr {X + `σ(X) ≤ −n} . σ(X) Now, we can take m large enough such that, −n − X ≥` ≤ , Pr σ(X) 2 n > m. As a result, (52) is proved. 6.9 Proof of Lemma 3 Since σ(x) is Riemann integrable, ϕ(x) is continuous and since σ(x) > 0 (a.e.), ϕ(x) is a strictly increasing function over the support of X. It yields that ϕ(x) is an injective function and there exists an inverse function ϕ−1 (·) for ϕ(·). Now, define random variable Y = ϕ(X). Assume that the pdf of X is fX (x). Since X is a continuous random variable and ϕ(x) is a bijection, Y is also continuous random variable with the following pdf: fY (y) = 1 fX (x), d dx ϕ(x) where x = ϕ−1 (y). Hence, we have that fY (y) = 1 fX σ (ϕ−1 (y)) ϕ−1 (y) . Now, we can calculate the differential entropy of Y as following: 1 h (Y ) =E log fY (Y ) " # σ ϕ−1 (Y ) =E log fX (ϕ−1 (Y )) σ(X) =E log fX (X) =h (X) + E [log ϕ(X)] . Therefore, the lemma is proved. 29 6.10 Proof of Lemma 4 First, assume that v(x) = ax + b, with a > 0. We will prove the general case later. In this case, we claim that the support of the optimal solution only needs to have two members. To this end, note that the following problem is equivalent to the original problem defined in (19): max max γ≤α µX :`≤X≤u E[X]=γ Cov (w(X), v(X)) . Since v(x) = ax + b, for a given γ, we would like to maximize Cov (w(X), v(X)) = E [w(X)v(X)] − (aγ + b)E [w(X)] , which is a linear function of µX , subject to E [X] = γ which is also a linear function of µX . By the standard cardinality reduction technique (Fenchel’s extension of the Caratheodory theorem), we can reduce the support of µX to at most two members (see [21, Appendix C] for a discussion of the technique). Assume that the support of µX is {x1 , x2 } where ` ≤ x1 ≤ x2 ≤ u with pmf pX (x1 ) = 1 − pX (x2 ) = p. Thus, we can simplify Cov (w(X), v(X)) as Cov (w(X), v(X)) = 2 X pX (xi )w(xi )v(xi ) − i=1 2 X 2 X pX (xi )pX (xj )w(xi )v(xj ) i=1 j=1 =p(1 − p) (w(x2 ) − w(x1 )) (v(x2 ) − v(x1 )) , where the last equality can be obtained by expanding the sums. Thus, the problem defined in (19) equals the following: max p,x1 ,x2 :0≤p≤1 `≤x1 ≤x2 ≤u px1 +(1−p)x2 ≤α p(1 − p) (w(x2 ) − w(x1 )) (v(x2 ) − v(x1 )) . We claim that the optimal choice for x1 is x1 = `. To see this, observe that w(x) and v(x) are increasing functions, and hence p` + (1 − p)x2 ≤ px1 + (1 − p)x2 ≤ α and (w(x2 ) − w(`)) (v(x2 ) − v(`)) ≥ (w(x2 ) − w(x1 )) (v(x2 ) − v(x1 )) . Hence, x1 = ` is optimal. Substituting v(x) = ax + b, we obtain that the problem is equivalent with the following: a max p(1 − p) (w(x) − w(`)) (x − `) . p,x:0≤p≤1 `≤x≤u p`+(1−p)x≤α Utilizing KKT conditions, one obtains that the optimal solution is ( p∗ = 21 , x∗1 = `, x∗2 = u α ≥ `+u 2 . ∗ = `, x∗ = u α < `+u p∗ = u−α , x 1 2 u−` 2 Now, we consider the general case of v(x) being a convex function (but not necessarily linear). Since v(x) is convex, we obtain that v(x) ≤ v(`) + (x − `) v(u) − v(`) , u−` 30 ∀x ∈ [`, u]. The right hand side is the line that connects the two points (`, v(`)) and (u, v(u)); this line lies above the curve x 7→ v(x) for any x ∈ [`, u]. Therefore, E[v(X)] ≤ v(`) + (E[X] − `) v(u) − v(`) . u−` Thus, E [X] ≤ α implies that E [v(X)] ≤ ∆ where ∆ = v(`) + (α − `) v(u) − v(`) . u−` Now, we relax the optimization problem and consider max µX :`≤X≤u E[v(X)]≤∆ Cov (w(X), v(X)) . The solution of the above optimization problem is an upper bound for the original problem because the feasible set of the original problem is a subset of the feasible set of the relaxed optimization problem. Now, using similar ideas as in the linear case, we conclude that the support of the optimal µX has at most two members. And the optimal solution is ( p∗ = 12 , x∗1 = `, x∗2 = u α ≥ `+u 2 . v(u)−∆ p∗ = v(u)−v(`) , x∗1 = `, x∗2 = u α < `+u 2 It can be verified that v(u) − ∆ u−α = . v(u) − v(`) u−` Note that in the case α > (` + u)/2, we obtain that E [X ∗ ] = (` + u)/2 < α, where X ∗ distributed with the optimal probability measure. As a result the constraint E [X] ≤ α is redundant. Therefore, the support of the optimal µX has two members, which shows that the upper bound is tight in this case. 7 Conclusion In this paper, we studied the capacity of a class of signal-dependent additive noise channels. These channels are of importance in molecular and optical communication; we also gave a number of new application of such channels in the introduction. A set of necessary and a set of sufficient conditions for finiteness of capacity were given. We then introduced two new techniques for proving explicit lower bounds on the capacity. As a result, we obtained two lower bounds on the capacity. These lower bounds were helpful in inspecting when channel capacity becomes infinity. We also provided an upper bound using the symmetrized KL divergence bound. References [1] S. M. Moser, “Capacity results of an optical intensity channel with input-dependent gaussian noise,” IEEE Transactions on Information Theory, vol. 58, no. 1, pp. 207–223, 2012. 31 [2] M. Pierobon and I. F. Akyildiz, “Diffusion-based noise analysis for molecular communication in nanonetworks,” IEEE Transactions on Signal Processing, vol. 59, no. 6, pp. 2532–2547, 2011. [3] G. Aminian, M. F. Ghazani, M. Mirmohseni, M. 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Khormuji, “On the capacity of molecular communication over the aign channel,” in Information Sciences and Systems (CISS), 2011 45th Annual Conference on, pp. 1–4, IEEE, 2011. [8] H. Li, S. M. Moser, and D. Guo, “Capacity of the memoryless additive inverse gaussian noise channel,” IEEE Journal on Selected Areas in Communications, vol. 32, no. 12, pp. 2315–2329, 2014. [9] N. Farsad, Y. Murin, A. W. Eckford, and A. Goldsmith, “Capacity limits of diffusion-based molecular timing channels,” arXiv:1602.07757, 2016. [10] T. H. Chan, S. Hranilovic, and F. R. Kschischang, “Capacity-achieving probability measure for conditionally gaussian channels with bounded inputs,” IEEE Transactions on Information Theory, vol. 51, no. 6, pp. 2073–2088, 2005. [11] J. G. Smith, “The information capacity of amplitude-and variance-constrained sclar gaussian channels,” Information and Control, vol. 18, no. 3, pp. 203–219, 1971. [12] R. Jiang, Z. Wang, Q. Wang, and L. Dai, “A tight upper bound on channel capacity for visible light communications,” IEEE Communications Letters, vol. 20, no. 1, pp. 97–100, 2016. [13] A. Lapidoth, S. M. Moser, and M. A. Wigger, “On the capacity of free-space optical intensity channels,” IEEE Transactions on Information Theory, vol. 55, no. 10, pp. 4449–4461, 2009. [14] R. R. Chen, B. Hajek, R. Koetter, and U. Madhow, “On fixed input distributions for noncoherent communication over high-snr rayleigh-fading channels,” vol. 50, no. 12, pp. 3390–3396, 2004. [15] G. Aminian, H. Arjmandi, A. Gohari, M. Nasiri-Kenari, and U. Mitra, “Capacity of diffusionbased molecular communication networks over lti-poisson channels,” IEEE Transactions on Molecular, Biological and Multi-Scale Communications, vol. 1, no. 2, pp. 188–201, 2015. 32 [16] S. Ihara, Information theory for continuous systems. Singapore: World Scientific, 1993. [17] T. M. Cover and J. A. Thomas, Elements of information theory. New York: John Wiley & Sons, 2nd ed., 2006. [18] F. Topsoe, “An information theoretical identity and a problem involving capacity,” Studia Scientiarum Math. Hungarica, vol. 2, no. 10, p. 291292, 1967. [19] H. Ghourchian, A. Gohari, and A. Amini, “Existence and continuity of differential entropy for a class of distributions,” IEEE Communications Letters, 2017. [20] E. M. Stein and R. Shakarchi, Real analysis: measure theory, integration, and Hilbert spaces. New Jersey: Princeton University Press, 2005. [21] A. El Gamal and Y.-H. Kim, Network Information Theory. Cambrdige University press, 2011. A Proof of Equation (9) Take some arbitrary x > 0. Then, equation (9) holds because 2 Z ∞ √ Z ∞ −(y−v 2 )2 1 2 − (y−v2 22)2 v +y v2 − y √ √ √ e 2c v dv = + √ e 2c2 v2 dv π v 2 2c2 v 2 2c2 πc2 0 0 Z ∞ Z ∞ 2 )2 2 )2 2y 1 v 2 − y −(y+v 1 v 2 + y −(y−v 2 2 2 √ e 2c v dv + √ e 2c2 v2 dv √ √ =e c π v 2 2c2 π v 2 2c2 0 0 Z 1 Z ∞ 2 2 2 )2 2 −(y+v ) 2y 2y 1 v −y 1 v 2 − y −(y+v √ √ √ e 2c2 v2 dv + e c2 √ e 2c2 v2 dv =e c2 π v 2 2c2 π v 2 2c2 0 1 Z Z 1 2 )2 2 )2 ∞ 1 v 2 + y −(y−v 1 v 2 + y −(y−v √ e 2c2 v2 dv + √ e 2c2 v2 dv. √ √ + π v 2 2c2 π v 2 2c2 1 0 Now utilize the change of variables y − v2 v 7→ u1 = √ , v 2c2 y + v2 v 7→ u2 = √ v 2c2 to re-express the above integrals. Note that du1 = − v2 + y √ dv, v 2 2c2 du2 = v2 − y √ dv. v 2 2c2 For y > 0, if v = 0 √ then u1 = +∞, u2 =√+∞, if v = +∞ then u1 = −∞, u2 = +∞ and if v = 1 then u1 = (y − 1)/ 2c2 , u2 = (y + 1)/ 2c2 . For y < 0, if v = 0√then u1 = −∞, u2√= −∞, if v = +∞ then u1 = −∞, u2 = +∞ and if v = 1 then u1 = (y − 1)/ 2c2 , u2 = (y + 1)/ 2c2 . Now for y > 0 we have −e 2y c2 Z ∞ y+1 √ 2c2 Z ∞ = −∞ 2y 1 2 √ e−u2 du2 + e c2 π Z ∞ y+1 √ 2c2 1 2 √ e−u2 du2 + π 1 2 √ e−u1 du1 = 1. π 33 Z ∞ y−1 √ 2c2 1 2 √ e−u1 du1 + π Z y−1 √ 2c2 −∞ 1 2 √ e−u1 du1 π Similarly, for y < 0 we have e 2y c2 y+1 √ Z ∞ Z √y−1 Z √y−1 2y 1 −u22 1 1 1 2 2 2 2 2c 2c2 −u −u √ e √ e 2 du2 − √ e 1 du1 + √ e−u1 du1 du2 + e c2 y+1 π π π π √ −∞ −∞ −∞ 2c2 Z ∞ 2y 2y 1 2 √ e−u1 du1 = e c2 . = e c2 π −∞ Z 2c2 Therefore, the proof is complete. 34	7
Stochastic bandits robust to adversarial corruptions arXiv:1803.09353v1 [cs.LG] 25 Mar 2018 Thodoris Lykouris∗ Vahab Mirrokni† Renato Paes Leme‡ Abstract We introduce a new model of stochastic bandits with adversarial corruptions which aims to capture settings where most of the input follows a stochastic pattern but some fraction of it can be adversarially changed to trick the algorithm, e.g., click fraud, fake reviews and email spam. The goal of this model is to encourage the design of bandit algorithms that (i) work well in mixed adversarial and stochastic models, and (ii) whose performance deteriorates gracefully as we move from fully stochastic to fully adversarial models. In our model, the rewards for all arms are initially drawn from a distribution and are then altered by an adaptive adversary. We provide a simple algorithm whose performance gracefully degrades with the total corruption the adversary injected in the data, measured by the sum across rounds of the biggest alteration the adversary made in the data in that round; this total corruption is denoted by C. Our algorithm provides a guarantee that retains the optimal guarantee (up to a logarithmic term) if the input is stochastic and whose performance degrades linearly to the amount of corruption C, while crucially being agnostic to it. We also provide a lower bound showing that this linear degradation is necessary if the algorithm achieves optimal performance in the stochastic setting (the lower bound works even for a known amount of corruption, a special case in which our algorithm achieves optimal performance without the extra logarithm). ∗ Cornell University, [email protected]. Work supported under NSF grant CCF-1563714. Part of the work was done while the author was interning at Google. † Google Research, [email protected] ‡ Google Research, [email protected] 1 Introduction In online learning with bandit feedback, a learner needs to decide at each time between alternative actions or arms of unknown quality, facing a trade-off between exploiting profitable past actions or exploring new actions about which she has little information. Bandit problems are typically classified according to how the rewards are generated. In stochastic bandits, rewards are drawn from fixed but unknown distributions, which models settings where the alternatives follow particular patterns and do not react to the learner. The other extreme is adversarial bandits, which are robust to rewards that are specifically designed to trick the learner, as in game-theoretic settings. In this paper, we focus on settings where the overall behavior is essentially stochastic but a small fraction of the rewards can be adversarially changed. Classic stochastic bandit algorithms, like Upper Confidence Bound (UCB) [ACBF02] or Active Arm Elimination (AAE) [EMM06], base most of their decisions on a few observations made in an initial phase of the algorithm and therefore can be easily tricked into incurring linear regret if very few arms are corrupted. Adversarial bandit algorithms like EXP3 are not fooled by such tricks, but cannot exploit the fact that the input is mostly stochastic. Our goal is to robustify the stochastic setting by designing algorithms that can tolerate corruptions and still be able to exploit the stochastic nature of the input. The algorithms we design are agnostic to the corruption, i.e. they can tolerate any level of corruption, and the guarantee degrades gracefully as more corruption is added. Moreover, we prove lower bounds showing that our results are tight up to a logarithmic factor. Before we explain our technical contribution in detail, we describe examples of settings we have in mind. Click fraud In pay-per-click online advertising, the platform selects for each pageview an ad to display and obtains a certain reward if the user clicks on the ad. The click probabilities are unknown. The tension between repeatedly displaying a particular profitable ad that provides reliable revenue and exploring other potentially more rewarding options is a major application of stochastic bandits in the ads industry. If it weren’t for a phenomenon known as click fraud, this would be a textbook example of stochastic bandits. In click fraud, botnets maliciously simulate users clicking on an ad to trick learning algorithms. One example is a bot consistently making searches to trigger some ad and not clicking on it to make it seem like a certain ad has very low click-through-rate in order to boost its competitor. Recommendation systems: A platform recommending activities or services to a user faces the same trade-off. Suggesting new restaurants leads to faster learning of the best spots but may result to dissatisfaction of the customers who are led to disappointing experiences. While most inputs follow a stochastic pattern, some inputs are typically corrupted: either maliciously, e.g. fake reviews by competitors, or non-maliciously, e.g. construction next-door makes the restaurant less desirable in certain interval. This corruption may again exhibit arbitrary patterns and is not identically distributed over time, yet it is dwarfed by the fact that most of the input is stochastic. There are several other such examples: emails mostly follow a stochastic pattern except a fraction of them which are spam and are designed to trick algorithms. Internet searches follow a predictable pattern except certain spikes caused by unpredictable events. Data collection used in the econometric process often suffers from errors that affect a small part of the input. In all those cases, the vast 1 majority of the input follows a predictable pattern, but a fraction of the samples are corrupted. 1.1 Our contribution Our model. In this paper, we introduce a new model of stochastic bandits with adversarial corruptions. The goal of this model is to encourage the design of bandit algorithms that (i) work well in mixed adversarial and stochastic models, and (ii) whose performance deteriorates gracefully as we move from fully stochastic to fully adversarial models. In this model there are K arms, each associated with a fixed reward distribution F(a). At each round t, a random reward rSt (a) ∼ F(a) is drawn and an adversary can change the reward to r t (a), possibly using information about the realizations of rSτ (a) from both the current and previous rounds τ ≤ t as well as the probability that the learner puts on each arm. The learner then draws an arm at and obtains r t (at ) both as P reward and feedback. We say that the adversary is C-corrupted if in every sample path we have t maxa \|r t (a) − rSt (a)\| ≤ C. Our results. The main result (Theorem 3 in Section 3) is a learning algorithm we term Multi-layer Active Arm Elimination Race that with probability 1 − δ has regret   X K · C log(KT/δ ) + log(T ) · log(KT/δ ) O ∆(a) ⋆ a6=a where ∆(a) is the gap of arm a, i.e. the difference in stochastic means of arm a and the optimal √ 1 arm a⋆ . For arms with √ very small gap, i.e. when ∆(a) ≤ / T , the inverse dependence on the gap can be replaced by T . It is possible to improve the bound by i.e. Pa log factor for pseudo-regret, K·C+log(T ) · log(KT ) . Two maximum expected regret against any fixed arm, obtaining: O a6=a∗ ∆(a) important features of the algorithm are that the guarantee is: • Agnostic: The algorithm does not need to know the corruption level C. The guarantee is provided with respect to how much corruption was added in retrospect. If the corruption level is known, we can remove the dependence on K · log(KT/δ ) as shown in Theorem 1. • High Probability: Our bounds hold with high probability which is important for practical applications as the ones described above. In contrast, the weaker definition of pseudo-regret often hides events with large regret that are offset by events with large negative regret. The stochastic case corresponds to C = 0 in which case we recover P a bound that is slightlyworse T than the guarantee provided by UCB. Our algorithm obtains O a6=a⋆ log(T ) · log( /δ )/∆(a) with probability 1 − δ, while UCB obtains this bound without the log(T ) term. En routeto the result, inTheorem 2 we show an algorithm that, for any fixed known C, provides 2 P P KT log(KT/δ) regret O for stochastic input and O K · C · a6=a⋆ (log(∆(a)/δ )) if it is C-corrupted. a6=a⋆ ∆(a) In other words, if we only need to tolerate either a known level C or zero corruptions, we save a logarithmic factor from the bound, and match the bound provided by UCB in the stochastic case. Another question is whether the linear dependence on the corruption level is tight. In Section 4, we show that it cannot be improved upon without decay in the stochastic guarantee (i.e. while still guaranteeing logarithmic regret when the input is stochastic). The lower bound is an adaptation from the adversarial to the corrupted setting of a result from Auer and Chiang [AC16]. This holds 2 even for the case where the corruptions are either 0 or a known level C (where our algorithm provides a matching upper bound). We prove in Theorem 4 that an algorithm with pseudo-regret O(log(T )/∆) in the stochastic setting (C = 0) then for every constant ǫ > 0, there is a O(T ǫ )-corrupted instance where the algorithm incurs regret Ω(T ǫ ) with constant probability. Our algorithm can also be viewed through the lens of the best of both worlds literature [BS12, SS14, AC16, SL17], where the goal is to design algorithms that simultaneously provide logarithmic regret guarantees in the stochastic regime and square-root guarantees in the adversarial. In Section 5, we sketch how our algorithm can be appropriately modified to obtain, for any constant 0 < a < 1/2, 1/2 a a+ e e O(C) pseudo-regret for C = O(T ) and O T pseudo-regret otherwise. We observe that the results in the best of both worlds literature correspond to the case where a = 0. We note that such bounds are obtained for pseudo-regret and not regret with high-probability. Our techniques. The starting point of our design are classical stochastic bandit learning algorithms like UCB and Active Arm Elimination. Such algorithms are very susceptible to corruptions since they base most of their decisions on a small initial exploration phase. Therefore, with a small number of corruptions it is possible to completely trick the algorithm into eliminating the optimal arm. We address this issue by robustifying them using a multi-layer approach. The learning algorithm consists of multiple layers running in parallel. The layers have decreasing speed and increasing tolerance to corruption. The first layer finishes very fast selecting an arm as optimal, but provides no tolerance to corruption. Subsequent layers are more robust but also slower. The resulting algorithm is a race between different layers for picking the optimal arm. Once the fastest layer finishes, it provides a first crude estimate of the optimal arm. Once slower layers finish, we obtain finer and finer estimates of the optimal arm. Our second main idea is that we can obtain more robust algorithms by subsampling. If a layer is only selected with probability p, it only receives in expectation a p-fraction of the corruption injected by the adversary. If p is low enough, the layer behaves almost as if it was stochastic. Finally, we couple the different layers together by a process of global eliminations. This process enables slower layers to eliminate arms in faster layers. Such a process is necessary for preventing inaccurate layers from pulling suboptimal arms too often. 1.2 Related work Online learning with stochastic rewards goes back to the seminal work of Lai and Robbins [LR85]. The case of adversarial rewards was introduced by Auer et al. [ACBFS03]. The reader is referred to the books of Cesa-Bianchi and Lugosi [CBL06], Bubeck and Cesa-Bianchi [BCB12], and Slivkins [Sli17] for an elaborate overview of the area. These two extremes suffer from orthogonal problems; the one is overoptimistic expecting that all rewards come from the same distribution while the other one is too pessimistic in order to be protected against malicious adversaries. Our work addresses the middle ground: rewards come from distributions but are often adversarially corrupted. This is motivated by the non-robustness of stochastic learning algorithms to even small corruption levels. Closely related to our work lie the works on best of both worlds guarantees [BS12, SS14, AC16, SL17]. These works achieve (up to logarithmic factors) the optimal pseudo-regret guarantee for stochastic rewards and the optimal pseudo-regret or actual regret guarantee for adversarial rewards. Bubeck 3 and Slivkins [BS12] and Auer and Chiang [AC16] begin from a stochastic algorithm and test whether they encounter non-stochastic behavior in which case they switch to adversarial algorithm. In contrast, Seldin et al. [SS14, SL17] begin from an adversarial algorithm with very optimistic learning rate and adapt it if they encounter such behavior. Recently and independently to this work, Wei and Luo [WL18] provide a best of both worlds result with a small-loss pseudo-regret guarantee on the adversarial setting, via a novel analysis of the log-barrier OMD algorithm of Foster et al. [FLL+ 16]. Although the aforementioned algorithms are very elegant, their analysis is not robust to inputs that are slightly away from stochastic. Our work bridges this gap by designing algorithms with a more smooth behavior for close-to-stochastic instances. There have been other works that attempt to provide improved guarantees than the adversarial setting when instances are well behaved. Hazan and Kale [HK09] offer regret guarantees that scale with the variance of the losses instead of the time horizon. This guarantee is meaningful in settings that have a very predictable nature and have usually the same performance such as routing. However they do not address most applications of stochastic bandits. In Click Fraud, for example the rewards come from Bernoulli distributions and the variance of such a distribution is high even if the input is totally stochastic. Another approach is the work of Shamir and Szlak [SS17], who consider an input that is adversarial but random local permutations are applied to obtain a more benign instance. This approach is very relevant in settings like buffering, but is again not applicable to our settings. On the opposite side, attempting to provide improved guarantees for the stochastic setting or enhancing their range is a very active area of research. For instance, the MOSS algorithm [AB09] of Audibert and Bubeck provides the optimal non-distribution-based upper bound for stochastic bandits while retaining the optimal distribution-based stochastic guarantee. The KL-UCB algorithm of Garivier and Cappé [GC11] provides improved constants in the upper bound of the stochastic guarantee matching the lower bound of Lai and Robbins [LR85] for Bernoulli rewards. The Robust UCB algorithm [BCBL13] extends the results to non-bounded rewards replacing with the weaker assumption of bounded variance. However, all the above results are not robust to corruptions from an adaptive adversary due to their deterministic nature. Since the adversary knows the arm the learner will select, they can always corrupt the optimal arm whenever it is about to be selected and therefore cause the learner to either play it multiple times even if it is suboptimal or decide against playing it even with a small amount of corruption (similarly as in our lower bound). There is also prior work on incorporating corruptions in online decision making. In the online learning front, there are two such attempts, to the best of our knowledge. In their best of both worlds result, Seldin and Slivkins [SS14] allow for some contamination in the data as long as they are obliviously selected and they do not decrease the gap by more than a factor of 2. The second work is a recent paper by Gajane et al. [GUK18] who suggest a model of corrupted feedback aiming for differential privacy. Unlike our model, their corruptions are neither adversarial nor adaptive. Both of these works make benign assumptions about the nature of corruption and do not address the main roadblock in the settings we consider: an adversarial saboteur will try to add faulty data in the beginning to change the order between the two arms and, with a minimal corruption, she will achieve this goal. Closer to our model are the works on robust allocation such as online matching with corrupted data [MGZ12, EKM15]; unlike online matching though, in online learning we cannot evaluate the optimum at every round since the algorithm’s decisions affect the information it observes. Last, learning in the presence of corruptions has recently received great attention in the batch learning setting. For instance, recent works study inference under the presence of adversarially corrupted data [MRT15], designing estimators that are robust to corrupted data [DKK+ 16], learn4 ing in auctions with some faulty data due to econometrics errors [CD17]. Our work suggests a similar framework for the study of online learning that is robust to adversarial corruptions in the more challenging problem of sequential decision making where decisions also affect the information observed. 2 Model Corrupted stochastic bandits. We study an online bandit learning setting with K arms. Each arm a ∈ [K] is associated with a distribution F(a) with mean µ(a). The distributions are assumed to have positive measure only on rewards in [0, 1] and are unknown to the learner. We refer to the optimal arm as a⋆ = arg maxa µ(a) and define ∆(a) = µ(a⋆ ) − µ(a).1 We consider an adversary who can corrupt some of the stochastic rewards. The adversary is adaptive, in the sense that the corrupted rewards can be a function of the realization of the stochastic rewards up to that point and of the learner’s choices in previous rounds. More formally, the protocol between learner and adversary, at each round t = 1 . . . T , is as follows: 1. The learner picks a distribution wt over the K arms. 2. Stochastic rewards are drawn for each arm: rSt (a) ∼ F(a). 3. The adversary observes the realizations of rSt (a) as well as rewards and choices of the learner in previous steps and returns a corrupted reward r t (a)∈ [0, 1]. 4. The learner draws arm at ∼ wt and observes r t at . We refer to maxa \|r t (a) − rSt (a)\| as the amount of corruption injected in round t. The instance is C-corrupted if the total injected corruption is at most C for all realizations of the random variables: X max\|r t (a) − rSt (a)\| ≤ C t a Note that the adversary is assumed to be adaptive, in the sense that she has access to all the realizations of random variables for all rounds τ < t and the realization of rewards at round t but only knows the player’s distribution at round t and not the arm at . Our guarantees gracefully degrades with the total corruption injected by the adversary. Regret notions. Regret corresponds to the difference between the reward obtained by the algorithm and the reward of the best arm in hindsight: X Reg = max r t (a) − r t at a t The regret is a random variable that depends on the random rewards, the randomness used by the learner, and the randomness of the adversary. We say that a regret bound R(T, δ) holds with probability 1 − δ if P[Reg < R(T, δ)] > 1 − δ where the probability is taken over all the three sources of randomness described. We note that a⋆ is one arm with optimal mean and this does not preclude the existence of other arms with the same mean. If more than one such arms exist, let a⋆ be an arbitrary arm with optimal mean and the other arms a 6= a⋆ with optimal mean have gap ∆(a) = 0. 1 5 Finally pseudo-regret is a weaker notion that compares the expected performance of the learner with the arm with the highest expected performance. In other words: " # X PseudoReg = max E r t (a) − r t at a t We note that by Jensen’s inequality, PseudoReg ≤ E[Reg]. We often obtain improved bounds for pseudo-regret since it allows us to offset large positive regret events with large negative regret events. 3 The upper bound: Multi-layer Active Arm Elimination Race Active arm elimination. The starting point of our design is the Active Arm Elimination algorithm for stochastic bandits [EMM06], which can be viewed as an alternative presentation of the more famous UCB algorithm [ACBF02]. It is based on the following idea: in an initial exploration phase, we pull arms in a round-robin fashion and compute an estimate µ e(a) as the average empirical reward of arm a. After n(a) pulls of arm a, usual concentration arguments establish that with probability at p least 1 − 1/T Ω(1) , the difference of the empirical and actual means is at most wd(a) = O( log(T )/n(a)). We say that [e µ(a) − wd(a), µ e(a) + wd(a)] is the confidence interval of arm a. This means in particular that given two arms a and a′ , if the difference in empirical means becomes larger than the widths of the confidence intervals, i.e., µ e(a) − µ e(a′ ) > wd(a) + wd(a′ ), then with ′ high probability arm a is not optimal. Once this happens, the algorithm eliminates arm a′ by removing it from the round-robin rotation. After both arms a and the optimal arm a⋆ are pulled O(log(T )/∆(a)2 ) times, the confidence intervals will be small enough that arm a will be eliminated. Eventually all arms but the optimal are eliminated and we enter what is called the exploitation phase. In this phase we only pull the arm with optimal mean. Before we enter exploitation we pulled each suboptimal arm a at most O(log(T )/∆(a)2 ) times. Each of thoseP suboptimal pulls incurs regret ∆(a) in expectation which leads to the pseudo-regret bound of O( a6=a⋆ log(T )/∆(a)). This bound can also be converted to a high probability bound if we replace log(T ) by log(T/δ ). √ Arms with small ∆(a). We note that, for the arms that have ∆(a) < 1/ T , the inverse dependence on the gap may initially seem vacuous; for instance, when there are two optimal arms a, a⋆ with the same mean, the upper bound becomes infinite as ∆(a) = 0. However,√the inverse dependence on the gap can be replaced by ∆(a) · T in the case of pseudo-regret and T in the case of actual regret (due to variance reasons). For simplicity of exposition, we omit this in the current section but we demonstrate how to perform this replacement in Section 5. 3.1 Enlarged confidence intervals The active arm elimination algorithm is clearly not robust to corruption since by corrupting the first O(log T ) steps, the adversary can cause the algorithm to eliminate the optimal arm. As the algorithm never pulls the suboptimal arms after exploration, it is not able to ever recover. One initial idea to fix this problem is to enlarge the confidence intervals. We can decompose the rewards r t (a) in two terms rSt (a) + ct (a) where the first term comes from the stochastic reward and the second is the corruption introduced by the adversary. If the total corruption introduced by the 6 p adversary is at most C, then with width wd(a) = O( log(T )/n(a) + C/n(a)), a similar analysis to above gives us the following regret bound: Theorem for thetotal corruption q1. If C is a valid upper bound then active arm elimination with P log(KT/δ)+C log(2KT/δ) C with probability 1 − δ. + n(a) has regret O wd(a) = a6=a⋆ n(a) ∆(a) Proof sketch. The proof follows the standard analysis of active arm elimination. We first establish that, with high probability the optimal arm a⋆ is never inactivated (Lemma 3.1) and then upper bound the number of times each suboptimal arm is played (Lemma 3.2). The pseudo-regret guarantee directly follows by multiplying the number of plays for each arm by its gap ∆(a). For the high-probability guarantee, we need to also show that the regret incurred in the meantime is not much more than the above. We provide proof details about the theorem and lemmas in Appendix A. Lemma 3.1. With probability at least 1 − δ, arm a⋆ never becomes inactivated. Lemma 3.2. With probability at least 1 − δ, all arms a 6= a⋆ become inactivated after N (a) = 36 log(2KT/δ)+6C plays. ∆(a)2 3.2 Stochastic bandits robust to known corruption The drawback of the active arm elimination algorithm with enlarged confidence intervals (Theorem 1) is that, even if there are no corruptions, it still incurs a regret proportional to C.As a warm up to P log (KT/δ) the main theorem, we provide an algorithm that achieves the usual bound of O ⋆ a6=a ∆(a) P log(KT/δ)2 if the if the input is purely stochastic and, at the same time, achieves O K · C · a6=a⋆ ∆(a) input is C-corrupted for a known C. In the next subsection, we modify the algorithm to make it agnostic to the corruption level C. Two instances of Active Arm Elimination. The first idea is to run two instances of active arm elimination: the first is supposed to select the correct arm if there is no corruption and the second is supposed to select the right arm if there is C corruption. The first instance is very fast but it is not robust to corruptions. The second instance is slower but more precise, in the sense that it can tolerate corruptions. Since the second instance is more trustworthy, if the second instance decides to eliminate a certain arm a, we eliminate the same arm in the faster instance. Decrease corruption by sub-sampling. To keep the regret low if the input is stochastic, the second instance of active arm elimination cannot pull a suboptimal arm too many times. Therefore, the technique in Theorem 1 alone is not enough. The main idea of the algorithm is to make arm a behave as if it was almost stochastic by running the second instance with low probability. If the learner selects to run the second instance with probability 1/C then, when the adversary adds a certain amount of corruption to a certain round, the second instance observes that corruption with probability 1/C . Therefore, the expected amount of corruption the learner observes in the second instance is constant. This makes the arms behave almost like stochastic arms in that instance. Learning algorithm. We obtain our algorithm by combining those ideas. We have two instances of active arm elimination which we denote by F (fast) and S (slow). Each instance keeps an estimate of the mean µ eF (a) and µ eS (a) corresponding to the average empirical reward of that arm and also keeps track of how many times each arm was pulled in that instance nF (a) and nS (a). This F allows p us to define a notion of confidence interval in each of the instances. We define wd (a) = O( log(T )/nF (a)) as usual and for the slow instance we define slighly larger confidence intervals: 7 p wdS (a) = O( log(T )/nS (a) + log(T )/nS (a)) (the reason will be clear in a moment). Also, each instance keeps a set of eliminated arms for that instance: I F and I S . In each round, with probability 1 − 1/C we make a move in the fast instance: we choose the next active arm a in the round robin order, i.e., arm a ∈ [K] \ I F which was played less often, pull this arm and increase nF (a) and update µ eF (a) accordingly. As usual, if there are two active arms a and a′ such that µ eF (a) − µ eF (a′ ) > wdF (a) + wdF (a′ ) we eliminate a′ by adding it to I F . With the remaining probability we make a move in the slow instance by executing the exact same procedure as described for the other instance. There is only one difference (which causes the two instances to be coupled): when we inactivate an arm a in S we also eliminate it in F. This leaves us with a potential problem: it is possible that all arms in the F instance end up being eliminated. If we reach that point, we play an arbitrary active arm of the slow instance, i.e., any arm a ∈ [K] \ I S . The resulting algorithm is formally provided in Algorithm 1. Algorithm 1 Fast-Slow Active Arm Elimination Race for known corruption C 1: Initialize nℓ (a) = 0, µ eℓ (a) = 0, I ℓ = ∅ for all a ∈ [K] and ℓ ∈ {F, S} 2: For Rounds t = 1..T 3: Sample algorithm ℓ: ℓ = S with probability 1/C. Else ℓ = F. 4: If [K] \ I ℓ 6= ∅ 5: Play arm at ← arg mina∈[K]\I ℓ nℓ (a) 6: Update µ eℓ (at ) ← [nℓ (a)e µℓ (at ) + r t (at )]/[nℓ (a) + 1] and nℓ (a) ← nℓ (a) + 1 ′ 7: While exists arms a, a ∈ [K] \ I ℓ with µ eℓ (a) − µ eℓ (a′ ) > wdℓ (a) + wdℓ (a′ ) 8: Eliminate a′ by adding it to I ℓ 9: If ℓ = S then eliminate a′ from the other algorithm by adding it to I F 10: Else 11: Play an arbitrary arm in the set [K] \ I S . Towards the performance guarantee, Lemma 3.3 bounds the amount of corruption that actually enters the slow active arm elimination algorithm, which enables the regret guarantee in Theorem 2. Lemma 3.3. In Algorithm 1, the slow active arm elimination algorithm S observes, with probability at least 1 − δ, corruption of at most ln(1/δ ) + 3 during its exploration phase (when picked with probability 1/C ). Proof sketch. If one cared just about the expected corruption that affects S, this is at most a constant number since the total corruption is at most C and it affects S with probability 1/C . To prove a high-probability guarantee we require a concentration inequality on martingale differences (since the corruptions can be adaptively selected by the adversary). We provide the details in Appendix B. q 8KT /δ ) log(8KT/δ ) Theorem 2. Algorithm 1 run with widths wdS (a) = + 2 log( and wdF (a) = nS (a) nS (a) q P P log(8KT/δ) log(KT/δ ) (log(KT/δ))2 has O for the stochastic case and O K · C · for ⋆ ⋆ a6=a a6=a ∆(a) ∆(a) nF (a) the C-corrupted case with probability at least 1 − δ. Proof sketch. The result for the stochastic case follows standard arguments for stochastic algorithms (since we obtain double the regret of this setting as we run two such algorithms with essentially the same confidence intervals). For the C-corrupted case, we establish via Lemma 3.3 an upper 8 bound on the corruption that will affect the slow active arm elimination algorithm S. Thanks to the sub-sampling, this upper bound is close to a constant instead of depending on C which allows to not incur dependence on C in the stochastic case. Having this upper bound, we can apply it to the algorithm of the previous section to get an upper bound on the number of plays of suboptimal arms in S. Since the algorithms are coupled, such a bound implies an upper bound on the regret that it can cause in F as well. This is because in expectation the arm is played at most K · C times more in F as it may be selected every single time in F prior to getting eliminated by S and F is selected C times more often than S. To obtain the above guarantee with high probability, we lose an extra logarithmic factor. The details of the proof are provided in Appendix B. 3.3 Stochastic bandits robust to agnostic corruption Multiple layers of active arm elimination. In the previous subsection we designed an algorithm with two layers: one is faster but cannot tolerate corruptions and the second one is slower but more robust. In order to be agnostic to corruption, we need to plan for all possible amounts of corruption. To achieve this, we introduce log(T ) layers. Each layer is slower but more robust than the previous one. We achieve that by selecting the ℓ-th layer with probability proportional to 2−ℓ . By the argument in the last section, if the corruption level is at most C, then each layer with ℓ ≥ log C will observe O(1) corruption in expectation and at most O(log T ) corruption with high probability. Global eliminations. We couple the log T instances through what we call global eliminations. If arm a is eliminated by the ℓ-th layer, then we eliminate a in all layers ℓ′ ≤ ℓ. This is important to prevent us from pulling arm a too often. If arm a is suboptimal and the adversary is C-corrupted, e 1/∆(a)2 ) in then arm a eventually becomes eliminated in the ℓ⋆ = ⌈log C⌉ layer after being pulled O( ⋆ ⋆ −ℓ e 2 then it takes O(C/∆(a) ) iterations until that layer. Since layer ℓ is played with probability 2 e C/∆(a)) from that arm. arm is eliminated globally, in which case we will have total regret at most O( Multi-layer active arm elimination race. We now describe our main algorithm in the paper. We call it a race since we view it as multiple layers racing to pick the optimal arm. The less robust layers are faster so they arrive first and we keep choosing (mostly) according to them until more robust but slower layers finish and correct or confirm the current selection of the best arm. The algorithm keeps ℓ = 1 . . . log(T ) different instances of active arm elimination. The ℓ-th instance has as state the empirical means of each arm µ eℓ (a), the number nℓ (a) of times each arm a was pulled and the set I ℓ of inactive arms. The p width of the confidence interval for arm a in the ℓ-th layer is implicitly defined as wdℓ (a) = O( log(T )/nℓ (a) + log(T )/nℓ (a)). In each round t we sample ℓ ∈ {1, . . . , log T } with probability 2−ℓ (with remaining probability we pick layer 1). When layer ℓ is selected, we make a move in the active arm elimination instance corresponding to that layer: we sample the active arm in that layer with the least number of pulls, i.e., arm a ∈ [K] \ I ℓ minimizing nℓ (a). In case [K] \ I ℓ is empty, we pull an arbitrary arm from ′ ′ [K] \ I ℓ for the lowest ℓ′ such that [K] \ I ℓ is non-empty. The way we couple different layers is that once arm a′ is eliminated in layer ℓ because there is another active arm a in layer ℓ such that µ eℓ (a) − µ eℓ (a′ ) < wdℓ (a) + wdℓ (a′ ) we eliminate arm a′ in 1 all previous layers, keeping the invariant that: I ⊇ I 2 ⊇ I 3 ⊇ . . .. Figure 1 provides an example of the state of the algorithm, which is formally defined in Algorithm 2. 9 ℓ=1 arm 1 arm 2 ... arm d µ e1 (1), n1 (1) µ e1 (2), n1 (2) ... µ e1 (d), n1 (d) µ e2 (1), n2 (1) ℓ=2 µ e3 (1), n3 (1) ℓ=3 .. . .. . ℓ = lg T µ e2 (2), n2 (2) µ e3 (2), n3 (2) ... ... .. . µ elg T (1), nlg T (1) µ elg T (2), nlg T (2) µ e2 (d), n2 (d) µ e3 (d), n3 (d) .. . ... µ elg T (d), nlg T (d) Figure 1: Example of the state of the algorithm: for each layer ℓ and arm a we keep the estimated mean µ eℓ (a) and the number of pulls nℓ (a). Red cells indicate arms that have been eliminated in that layer. If an arm is eliminated in a layer, it is eliminated in all previous layers. If a layer where all the arms are eliminated (like layer 1 in the figure) is selected, we play an arbitrary active arm with the lowest layer that contains active arms. Algorithm 2 Multi-layer Active Arm Elimination Race 1: Initialize nℓ (a) = 0, µ eℓ (a) = 0, I ℓ = ∅ for all a ∈ [K] and ℓ ∈ [log T ] 2: For Rounds t = 1..T 3: Sample layer ℓ ∈ [log T ] with probability 2−ℓ . With remaining prob, sample ℓ = 1 4: If [K] \ I ℓ 6= ∅ 5: Play arm at ← arg mina∈[K]\I ℓ nℓ (a) 6: Update µ eℓ (at ) ← [nℓ (a)e µℓ (at ) + r t (at )]/[nℓ (a) + 1] and nℓ (a) ← nℓ (a) + 1 7: While exists arms a, a′ ∈ [K] \ I ℓ with µ eℓ (a) − µ eℓ (a′ ) > wdℓ (a) + wdℓ (a′ ) ′ 8: Eliminate a′ by adding it to I ℓ for all ℓ′ ≤ ℓ 9: Else ′ 10: Find minimum ℓ′ such that [K] \ I ℓ 6= ∅ and play an arbitrary arm in that set. We now provide the main result of the paper, a regret guarantee for Algorithm 2. Theorem 3. Algorithm 2 which is agnostic to the coruption level C, when run with widths wdℓ (a) = q log(4KT ·log T/δ ) nℓ (a) + log(4KT ·log T/δ) nℓ (a)  has regret:  X K · C log(KT/δ ) + log(T ) O · log(KT/δ ). ∆(a) ∗ a6=a Proof sketch. Similarly to the previous theorem, the regret guarantee comes from the summation between layers that are essentially stochastic (where the corruption is below corruption level, their log(KT/δ) r regret. Since i.e. less than C ≤ 2 for layer r). From each of these layers, we incur O ∆(a) there are at most log(T ) such layers, the second term in the theorem is derived. The challenge is to bound the regret incurred by layers that are not robust to the corruption. However, there exists some layer ℓ⋆ that is above the corruption level. By bounding the amount of steps that this level will require in order to inactivate each arm a 6= a⋆ in the incorrect layers (via Lemma 3.2), we obtain similarly to Theorem 2 a bound on the regret caused by this arm in those layers. Since 10 we take the minimum such layer and the tolerance of layers is within powers of 2, the fact that its corruption level does not match exactly the corruption that occurred only costs an extra factor of 2 in the regret. The details of the proof are provided in Appendix C. 4 The lower bound For the two arms case where the gap between the arms is ∆ > 0, Theorem 2 presents an algorithm which achieves O(log T/∆) pseudo-regret if the input is stochastic and O(C log(T/δ )/∆) with probability 1 − δ if the input is at most C-corrupted. We show below that this dependence is tight. The lower bound (Theorem 4) adapts the technique of Auer and Chiang [AC16] from the adversarial to the corrupted setting. The main idea is that an algorithm with logarithmic regret in the stochastic setting cannot query the sub-optimal arm more than log(T )/∆2 times. This implies a long time period where the learner queries the input only constant number of times. By corrupting all rounds before this period, an adversary can make the optimal arm look sub-optimal and trick the learner into not pulling the optimal arm for long time, causing large regret. Theorem 5 adapts this argument bounding the expected positive regret E[Reg+ ] where x+ = max{x, 0}; the high probability bounds provided imply bounds on the expected positive regret. Both proofs are provided in Appendix D. Theorem 4. Consider a multi-armed bandits algorithm that has the property that for any stochastic input in the two arm setting, it has pseudo-regret bounded by c log(T )/∆, where ∆ = \|µ1 − µ2 \|. ′ For any ǫ, ǫ′ ∈ (0, 1), there is a corruption level C with T ǫ < C < T ǫ and a C-corrupted instance such that with constant probability the regret is Ω(C). Theorem 5. If a multi-armed bandits algorithm that has the property that for any stochastic input in the two arm setting, it has pseudo-regret bounded by c log1+α (T )/∆ for α < 1. For any ′ ǫ, ǫ′ ∈ (0, 1), there is a corruption level C with T ǫ < C < T ǫ and a C-corrupted instance such that E[Reg+ ] = Ω(T ǫ−δ ) for all δ > 0. 5 Extensions In this section, we discuss some extensions that our algorithm can accommodate. Definition of corruption. We presented all results measuring the corruption as the sum over all rounds of the maximum across arms of the corruption injected by the adversary: X max \|r t (a) − rSt (a) ≤ C. a t P t t In fact all our results can be improved via using C(a) = t \|r (a) − rS (a)\| and replacing C by max(C(a), C(a⋆ )) for summand a. More formally, our main theorem (Theorem P t 3) becomes: t Theorem 6. Algorithm 2 which is agnostic to the corruptions C(a) = t \|r (a) − rS (a)\|, when run q with widths wdℓ (a) = log(4KT ·log T/δ) nℓ (a) + log(4KT ·log T/δ) nℓ (a) has regret:  X K · max(C(a⋆ ), C(a)) · log(KT/δ ) + log(T ) · log(KT/δ ). O ∆(a) ∗  a6=a 11 The proof follows the same arguments since it only compares each arm a with a⋆ . This result is nice since the contribution of each arm to the regret is a function only of its own gap and the corruption injected to it and the one injected to arm a⋆ . The latter dependence on the corruption on the optimal arm is essential since the main attack we presented to the classical arguments only corrupts arm a⋆ – the lower bound of the previous section also only adds corruption to a⋆ . Dependence on the gap. In Section 3, all our guarantees have an inverse dependence on the gap ∆(a) of all arms a. Note that such a guarantee is completely meaningless for arms with a very small gap; for instance, if there exist two optimal arms then there is an arm a 6= a⋆ with ∆(a) = 0 which makes the presented bound infinite and therefore vacuous. As we hinted there though, this √ inverse dependence can be improved for arms with small ∆(a) ≤ 1/ T . Our proofs generally relied on setting an upper bound on the number of times that a suboptimal arm is played and thereby providing an upper bound on the regret they cause. √ For arms with ∆(a) ≤ 1/ T , an alternative analysis is to say that, even if they are erroneously selected every single time, we can upper bound the loss in performance they√cause. For pseudoregret, the performance loss if they were selected every single time is ∆(a)·T ≤ T ≤ log T/∆(a). For actual regret, one needs to also take into consideration the variance but, even if they are selected every single time, a Hoeffding bound shows that their total reward is with high probability at most √ T lower than its expectation. As a result, the inverse dependence on ∆(a) in our bound can be √ 1 1 replaced by min(∆(a) · T, /∆(a)) for pseudo-regret and min( T , /∆(a)) for actual regret. C can be Moreover, the careful reader may have noticed that in Theorem 1, the dependence ∆(a) replaced by a sole dependence on C without the gap. However, this does not extend to the subsequent theorems since the dependence on C there does not come from the upper bound on the corruption experienced (this is at most log T due to subsampling). Instead, the dependence on C comes from projecting the correct layer (smallest layer robust to corruption) to the previous layers via the number of times it will take to eliminate any suboptimal arm. Uncorrupted objective. In applications such as spam, the corruptions should not be counted as part of the rewards. Our algorithm provides the same guarantee in the case of uncorrupted rewards (the difference between the performances in the two objectives is at most C). One can also observe that the linear dependence on C is still necessary: consider 2 arms with ∆ = 1 and an adversary that corrupts the first C steps making them look identical. The learner has no better option than randomly selecting between the two which gives him a regret of C/2 under the uncorrupted objective. We note that, in this setting, the linear dependence is necessary unconditionally of the performance of the algorithm in the stochastic setting. Towards best of all worlds. In the previous section, we showed that a logarithmic dependence in the stochastic setting comes at the expense of linear dependence on C in the C-corrupted setting if we focus on actual regret. A very interesting direction is to achieve such an improvement with either a higher power on the logarithm in the stochastic setting or aiming for pseudo-regret instead. In fact, we can combine our algorithm with the SAPO algorithm of Auer and Chiang [AC16] and achieve a bicriteria guarantee for pseudo-regret. For an a < 1/2 specified by the algorithm, we achieve our guarantee if the corruption is C ≤ T a and at most T 1/2+a otherwise; notice that the case a = 0 corresponds to the best of both worlds. This is done via running the SAPO algorithm at the level a log(T ) with probability T −a instead of having higher layers. The SAPO algorithm guarantees that the pseudo-regret caused by any particular arm is at most logarithmic if the instance √ is stochastic and at most T if it is adversarial via a beautiful analysis that keeps negative regret of time intervals that have performed well to avoid testing eliminated arms too often. In our setting, if 12 the corruption level is less than T a , the instance behaves as stochastic causing at most logarithmic regret. Else the instance is corrupted and we can extrapolate the regret in this layer to the whole algorithm as arms that are eliminated in this layer are also eliminated before via global eliminations. √ Since the regret there is at most T and this is multiplied by T a , this implies a bound of T 1/2+a on pseudo-regret. Acknowledgements The authors would like to thank Sid Banerjee whose lecture notes on stochastic bandits proved very helpful, Andrés Munoz Medina, Karthik Sridharan, and Éva Tardos for useful discussions, Manish Raghavan for suggestions on the writeup, and the anonymous reviewers for the valuable feedback they provided that improved the presentation of the paper. References [AB09] Jean-Yves Audibert and Sébastien Bubeck. Minimax policies for adversarial and stochastic bandits. In Proceedings of the 22nd Annual Conference on Learning Theory (COLT), 2009. [AC16] Peter Auer and Chao-Kai Chiang. 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The crux of the proof lies in establishing that, with high probability, the upper bound of the confidence interval of a⋆ never becomes lower than the lower bound of the confidence interval of any other arm a and therefore a⋆ does not become eliminated. More formally, let µ eS (a) and µ e(a) be the empirical mean after n(a) samples of the stochastic part of the rewards and the empirical mean after n(a) samples of the corrupted rewards respectively. Recall that µ(a) is the mean of arm a. By Hoeffding inequality, for any arm a, with probability at least 1 − δ′ : s log(2/δ′ ) . (1) \|e µS (a) − µ(a)\| ≤ n(a) We set δ′ = δ/KT to establish that this holds for all arms and all time steps (after q arm a has been 2KT played n(a) times). As a result, for any arm a and any time: µ eS (a) ≤ µ(a) + log(n(a) /δ) and q 2KT µ eS (a⋆ ) ≥ µ(a⋆ ) − log(n(a⋆ )/δ) . Comparing now the actual (corrupted) empirical means, they can be altered by at most absolute C C and µ e(a⋆ ) ≥ µ eS (a⋆ ) − n(a corruption C. Hence µ e(a) ≤ µ eS (a) + n(a) ⋆) . Combining the above inequalities with the fact that the actual mean of a⋆ is higher than the one of a, i.e. µ(a⋆ ) ≥ µ(a), we establish that µ e(a) − µ e(a⋆ ) ≤ wd(a) + wd(a⋆ ) and therefore arm a⋆ is not eliminated. Since this holds for all times and arms, the lemma follows. Lemma 3.2 (restated) With probability at least 1 − δ, all arms a 6= a⋆ become eliminated after 2KT N (a) = 36·log(∆(a)/2δ )+6C plays. Proof. The proof stems from the following observations. By Lemma 3.1, arm a⋆ is with high probability never eliminated. After N (a) rounds, with high probability, the lower confidence interval of arm a⋆ is above the upper confidence interval of arm a. This comes from the fact that, after N (a) plays of arm a (and also of arm a⋆ since it is not eliminated), the empirical stochastic mean of a⋆ is, with high probability, at most ∆(a)/6 below its actual mean and similarly the empirical stochastic mean of arm a is at most ∆(a)/6 above its actual mean. Since the corruptions are upper bounded by C, they can only contribute to a decrease in the average empirical (corrupted) means by at most ∆(a)/6 which is not enough to circumvent the gap ∆(a). More formally, let µ eS (a) and µ e(a) denote the empirical means of the stochastic part of the rewards and the corrupted rewards respectively after N (a) plays of arm a. By the same Hoeffding inequality as in the proof of the previous lemma, with probability at least 1 − δ, it holds that \|e µS (a) − µ(a)\| ≤ q log(2KT/δ) N (a) . Therefore, with the same probability, µ eS (a⋆ ) − µ(a⋆ ) ≤ ∆(a) eS (a) ≤ ∆(a) 6 and µ(a) − µ 6 . 15 after 36·log(2KT/δ ) ∆(a)2 plays for both arm a and a⋆ : e(a) ≤ µ eS (a) + NC(a) . By The absolute corruption is at most C therefore µ e(a⋆ ) ≥ µ eS (a⋆ ) − NC(a) and µ the choice of N (a), we have C N (a) ≤ ∆(a) 6 . Combining with the above argument, this also implies that the widths are upper bounded by wd(a) ≤ ∆(a) 3 and wd(a⋆ ) ≤ ∆(a) 3 . Combining the above with the fact that the actual mean of a⋆ is ∆(a) higher than the one of a, i.e. µ(a⋆ ) − µ(a) = ∆(a), we establish C − wd(a) − wd(a⋆ ) N (a) ∆(a) ∆(a) ∆(a) − − >0 > µ(a⋆ ) − µ(a) − 2 · 6 3 3 µ e(a⋆ ) − µ e(a) − wd(a) − wd(a⋆ ) ≥ µ eS (a⋆ ) − µ eS (a) − 2 · As a result arm a becomes eliminated after N (a) plays if it is not already eliminated before. Theorem 1 (restated) If C is a valid upper bound for the total corruption then arm elimination q P log(2KT/δ) log(KT/δ )+C C with wd(a) = with probability 1 − δ. + n(a) has regret O a6=a⋆ n(a) ∆(a) Proof. The proof follows the classical stochastic bandit argument of measuring the regret caused by each arm a 6= a⋆ as a function of its gap ∆(a) and the number of times N (a) it is played as established by Lemma 3.2. For simplicity of presentation, we first provide the pseudo-regret guarantee. Pseudo-regret compares the expected performance of the algorithm to the expected performance one would have had, had they selected a⋆ throughout the whole time horizon. The expected performance when one uses a⋆ is µ(a⋆ ). The loss compared to that every time a 6= a⋆ is used instead is equal to its gap ∆(a). As a result, the expected contribution to pseudo-regret from suboptimal arm a 6= a⋆ is equal to N (a) · ∆(a). Lemma 3.2 establishes that with probability 1 − δ any suboptimal arm a is played at 2KT most N (a) = 36 log(∆(a)/2δ )+6C times. Each play of the suboptimal arm causes pseudo-regret of ∆(a). Multiplying the times by the expected regret per time the guarantee (which equals to the gap) and setting the failure probability δ to be some inverse polynomial of the time horizon T to ensure that the expected regret due to the bad event is at most a constant leads to the pseudo-regret guarantee. To turn the above into a high-probability guarantee, we need to show that the regret incurred during the steps that we pull arm a is not significantly higher than the expectation (therefore bounding the resulting variance). By the Hoeffding inequality of Lemma 3.1, the empirical cumulative reward of p arm a is, with high probability, at most N (a) log(2KT/δ ) less than its expectation. The same holds for arm a⋆ for these steps (its realized performance is at most this much more than its expectation). The probability that these statements do not hold for some arm or some time is at most δ. p Regarding arms a 6= a⋆ , the N (a) log(2KT/δ ) term can be upper bounded by O(N (a)∆(a)) by the definition of N (a): s s p 2KT/δ ) log( log(2KT/δ ) ≤ N (a) · ∆(a) ≤ N (a)∆(a) N (a) log(2KT/δ ) ≤ N (a) · N (a) 36 log(2KT/δ ) + 6C Regarding arm a⋆ , let a′ be the arm with the smallest gap. By Lemma 3.1, a⋆ never gets eliminated p but it is not necessarily the ex post optimal arm. In fact some other arm with ∆(a) ≤ 1/T may be the ex post optimal arm (arms with higher gap are with high probability not the ex post optimal arm by an analogous argument as in Lemma 3.2. However, by the same argument as above arm a⋆ 16 is with high probability at most N (a′ ) · ∆(a′ ) below its expectation and the ex post optimal arm is at most this much above its expectation. This gives a bound of N (a′ )∆(a′ ) that is caused by the case where a⋆ is not the ex post optimal arm. Therefore the actual regret from times that arm a is played is at most 2N (a)∆(a) where the one term comes from the expectation and the other from the aforementioned bounds on the variance. The corruption can increase any cumulative reward by at most C which is already existing in the regret bound. Replacing N (a) by Lemma 3.2, we obtain the high-probability guarantee. Note that the failure probabilities of the two lemmas are coupled as they correspond to the same bad events. B Supplementary material on Section 3.2 In this section, we provide the proof of Theorem 2. To handle the corruption, we bound with high probability the total corruption experienced by the slow active arm elimination instance S (Lemma 3.3). To deal with an adaptive adversary, we need a martingale concentration inequality; specifically we apply a Bernstein-style inequality introduced in [BLL+ 11] (Lemma B.1). Lemma B.1 (Lemma 1 in [BLL+ 11]). Let X1 , . . . , XT be a sequence of real-valued random numbers. Assume, for all t, that Xt ≤ R and that E[Xt \|X1 , . . . , Xt−1 ] = 0. Also let V = T X t=1 Then, for any δ > 0: E[Xt2 \|X1 , . . . , Xt−1 ]. " T X e−2 ·V Xt > R ln(1/δ) + P R t=1 # ≤δ Lemma 3.3 (restated) In Algorithm 1, the slow active arm elimination algorithm S observes, with probability at least 1 − δ, corruption of at most ln(1/δ ) + 3 during its exploration phase (when picked with probability 1/C ). Proof. The first observation is that the expected corruption encountered by algorithm S is at most a constant (total corruption of C encountered with probability 1/C ). The rest of the proof focuses on bounding the variance of this random variable (actual corruption encountered by the layer). Crucially, since we want to allow the adversary to be adaptive, we should not assume independence across rounds but only conditional independence (conditioned on the history) and this is why some more involved concentration inequality is necessary. Therefore we create a martingale sequence (actual corruption minus expected corruption) and apply a Bernstein-style concentration inequality. Let Zat be the corruption that is observed by the exploration phase of the algorithm if arm a is selected. For every round t, if adversary selects corruption Cat then Zat is therefore a random variable equal to Cat with probability 1/C and 0 otherwise. Given that the adversary is adaptive and may select the corruptions based on the realizations of the previous rounds, we need to use an appropriate concentration inequality. We use a Bernstein-style inequality, introduced in [BLL+ 11] (Lemma B.1). Initially we resolve the randomness conditioning on ℓ = S (the slow algorithm is selected). Since active arm elimination is deterministic, conditioned on selecting algorithm S, the 17 selected arm is deterministic. Let a(S, t) be the arm that would be selected if ℓ = S (which happens with probability 1/C ). The martingale sequence is now h i t t Xt = Za(S,t) − E Za(S,t) \| H(1 : t − 1) where H(1 : t) corresponds to the history up to round t. Note that E Xt2 \|X1 , . . . , Xt−1 Ca(S,t) 2 C − 1 Ca(S,t) 2 1 + Ca(S,t) − = C C C C 2 2 Ca(S,t) Ca(S,t) C −1 C − 1 Ca(S,t) 2 = ≤2· . + C C C C C t The last inequality holds as Ca(S,t) ∈ [0, 1] and Ca(S,t) ≤ C by the definition of C. Therefore, summing over all the rounds, X X Ca(S,t) 2 ≤ · 2 V = E Xt2 \|X1 , . . . , Xt−1 ≤ C C t t X t max Cat a ! ≤ 2. A trivial upper bound of \|Xt \| is R = 1, since the rewards are in [0, 1]. Applying Lemma B.1, we show that, w.p. 1 − δ: X Xt ≤ ln(1/δ ) + 2(e − 2) ≤ ln(1/δ ) + 2 t P The lemma then follows by adding the expected corruption of E t Za(S,t) \| H(1 : t − 1) ≤ 1 and therefore obtaining the bound of the statement on the corruption experienced: # " X X X t Za(S,t) Za(S,t) \| H(1 : t − 1) ≤ ln(1/δ ) + 3. = Xt + E t t t q log( /δ ) /δ) Theorem 2 (restated) Algorithm 1 run with widths wd (a) = + 2 log( and nS (a) nS (a) q P P KT KT 8KT (log( /δ ))2 log( / δ ) log( / δ ) has O for the stochastic case and O K · C · wdF (a) = ⋆ ⋆ F a6=a a6=a ∆(a) ∆(a) n (a) for the C-corrupted case with probability at least 1 − δ. S 8KT 8KT Proof. For the stochastic case, the bound follows via standard stochastic bandit arguments (similarly to the proof of Theorem 1 with C = 0) as for each of the two active arm elimination algorithms P (log(2KT/δℓ,S )) we incur, with probability 1 − δℓ,S regret O where δℓ,S = δ/4 is the failure a6=a⋆ ∆(a) probability of inequality (1), which governs the results in Lemmas 3.1 and 3.2, for each of ℓ ∈ {F, S}. The most interesting case is the C-corrupted setting. Let δS,C = δ/4 be the failure probability in Lemma 3.3. By Lemma 3.3, with probability at least 1 − δS,C , the actual corruption experienced by the slow active arm elimination algorithm is at most ln(1/δ) + 3 which is less than 2 log(2KT/δ ) for non-trivial values of K and T . Therefore we can apply the analysis of Theorem 1 with corruption level at least 2 log(2KT/δS,C ) and get a handle on the actual regret coming from the slow active arm elimination algorithm. 18 What is left is to bound the regret coming from the fast active arm elimination algorithm. Towards this goal, we bound the number of times that a suboptimal arm is played in the fast active arm elimination by the expected time that it remains active at the slow active arm elimination. By Lemma 3.2, arm a is played in the slow active arm elimination, with probability at least 1 − 3δ/4, at most 18 log(8KT/δ ) 16 log(2KT/δS,S ) + 2 log(2KT/δS,C ) ≤ . NS (a) = 2 ∆(a) ∆(a)2 Having a bound on the number of plays of the arm in the slow active arm elimination instance, we use this to bound the number of plays in the fast active arm elimination instance. In expectation, this is at most K · C · NS (a) times as every move in the slow active arm elimination occurs with probability 1/C and, at least 1/K of these moves are plays of a while it is still active. Since every time arm a is played it incurs pseudo-regret ∆(a), this provides the pseudo-regret guarantee. To obtain a high probability guarantee, let δm = δ/4KT and observe that with probability at least 1 − δm , we make one move at the slow arm elimination algorithm every O(C log(1/δm )) moves at the fast arm elimination algorithm. This can be seen by thinking the following process: One tosses coins with bias p = 1/C until she observes heads for the first time (heads is the p-biased event). After M tosses of the coins the probability that no heads have arrived is at most (1 − p)M . To ensure that 1/δm ) 1/δm ) , which is achieved by M = log( this is less than δm , we need to wait M ≥ log( p/(1−p) . log( 1 ) 1−p By union bound on the failure probabilities for each of those draws, we get that with failure probability δe = K · NS (a) · δm ≤ δ/4 (since NS (a) ≤ T as it is at most the time horizon), arm a gets inactivated in F after NF (a) = K · NS (a) · C · log(1/δe ) = 18 · C · K · (log(8KT/δ ))2 . ∆(a)2 The last part is to prove that the regret experienced throughout those rounds is not too large. This follows by the two applications of Hoeffding inequality as before for arms a and a⋆ , analogously to Theorem 1. Combining the above arguments the theorem follows. The total failure probability of the guarantee is δS,S + δS,C + δF,S + δe ≤ δ. C Supplementary material on Section 3.3 Theorem 3 (restated) Algorithm 2 which is agnostic to the coruption level C, when run with q 4KT ·log T log(4KT ·log T/δ) ℓ + log( nℓ (a) /δ) has regret: widths wd (a) = nℓ (a)  X K · C log(KT/δ ) + log(T ) · log(KT/δ ). O ∆(a) ∗  a6=a Proof. The proof follows similar arguments to the proof of Theorem 2. Specifically, for the layers that are above the corruption level C, by using the standard arguments described in Theorem log(2KT/δℓ,S ) 1, we establish a bound on the regret caused by any suboptimal arm a, with failure ∆(a) δ probability δℓ,S = /2 log T . Since there are log(T ) such levels, the regret coming from these layers is upper bounded by the second term of the theorem with failure probabilityδ/2. 19 For the layers ℓ that are not tolerant to the corruption, i.e. 2ℓ > C, we apply the same argument as in the proof of Theorem 2 and bound their regret via the number of plays they are played by the minimum layer that is robust to corruption ℓ⋆ = arg minℓ 2ℓ > C . Similarly as in the proof of the theorem we upper bound the number of plays Nℓ⋆ (a) of each suboptimal arm a at this layer (by exactly the same arguments), then bound the number of plays in the suboptimal layer via the same coin toss process as in that proof and, last bound the regret they incur during this part. Since we do not know the amount of corruption in advance (and this amount is adaptively selected), we also need to take a union bound on the number of layers so that the guarantee on Nℓ⋆ (a) holds for all layers simulataneously if they end up being correct; we therefore repeat the arguments in Theorem 2 with δℓ,C = δ/2 log T and δm ≤ δ/2KT log T . Last, we note that, since we used powers of 2 to increase the corruption among layers, the fact that we did not apply the arguments of Theorem 2 with the exact C but instead used a C ′ such that C < C ′ < 2C causes just an extra constant factor on the regret. D Supplementary material on Section 4 Theorem 4 (restated) Consider a multi-armed bandits algorithm that has the property that for any stochastic input in the two arm setting, it has pseudo-regret bounded by c log(T )/∆, where ′ ∆ = \|µ1 − µ2 \|. For any ǫ, ǫ′ ∈ (0, 1), there is a corruption level C with T ǫ < C < T ǫ and a C-corrupted instance such that with constant probability the regret is Ω(C). Proof. The proof follows a sequence of steps. Step 1: Analyze behavior in the stochastic case. Fix a constant ∆ ≤ 1/6 and observe how the algorithm behaves for the stochastic input that has Bernoulli arms of means (µ1 , µ2 ) = ( 21 − ∆, 12 ). Since in that setting the expected regret is the same as ∆ · E[T1 ] where T1 is the number of pulls of arm 1, it follows that E[T1 ] ≤ c log(T )/∆2 . Step 2: find a large interval that is hit with at most constant probability. We divide the ′ space between T ǫ and T ǫ into O(log(T )/(ǫ′ − ǫ)) intervals Ii = [3i−1 T ǫ , 3i T ǫ ) such that size of each interval is twice the size of all the previous intervals combined. For each interval i, let T1,i be the number of times that arm 1 is pulled in the i-th interval. Then, there exists an interval Ii = [C, 3C) such that E[T1,i ] ≤ c̃ := O(1/[(ǫ′ − ǫ)∆2 ]). Step 3: create an adversary that forces a lot of regret in interval i. The adversary is quite simple: for the first C steps, the arms are Bernoulli with means ( 12 − ∆, 21 ) and for the remaining timesteps, the arms are Bernoulli with means ( 21 + ∆, 12 ). We use E and P to refer to the probability law whe inputs are drawn with respect to ( 12 − ∆, 12 ) in all timesteps and E′ and P′ to refer to the probability law when the input is according to ( 12 − ∆, 21 ) in the first K steps and according to ( 21 + ∆, 12 ) onwards. Step 4: With constant probability arm 1 is pulled a constant number of times in Ii under both P and P′ . Under the probability law P, this follows directly from Markov’s inequality: c̃ = E[T1,i ] ≥ 2c̃ P[T1,i ≥ 2c̃], so: P[T1,i ≤ 2c̃] ≥ 12 . Denote by A the event that T1,i ≤ 2c̃. We want to argue that P′ [A] is also constant. In order to do that, let Z = (Z1 , Z2 , . . . , Z2c̃ ) be a vector storing in Zs the reward of arm 1 in the s-th time it is pulled in interval Ii . Notice that in both the stochastic and corrupted scenarios if the learner 20 observes the same values of Z she acts the exact same way. Therefore, if we condition on Z, the probability that she ends up pulling arm 1 for more than 2c̃ times is exactly the same. In other words: P[A\|Z] = P′ [A\|Z] Therefore: ′ P [A] = X z ′ P [Z = z] · P[A\|Z = z] ≥ 1 2 1 2 −∆ +∆ !2c̃ 1 P [Z = z] P[A\|Z = z] ≥ · 2 ′ 1 2 1 2 −∆ +∆ !2c̃ which is a constant. Step 5: concentration bounds for the regret incurred in each interval. We now define an event B that occurs with probability 1 − o(1) that captures all the concentration bounds we need for the proof. First, we requireP arm 1 to be the optimal arm. Let r tP (i) be the reward of arm i in ′ ′ 1 t time step t. We know that E [ t r (1)] = 2 T + ∆(T − 2C) and E [ t r t (2)] = 21 T . Since P all the rewards are independent we can use the Hoeffding bound to bound the probability P′ ( t ∆t < 0) where ∆t = r t (2) − r t (1): ! ! X X X C T ∆2 ′ t t ′ ′ 1−2 = o(1) ∆t − E ∆t > ∆(T − 2C) ≤ 2 exp − r (1) < r (2) ≤ P P 2 T t t t Now we establish some concentration on the regret that the learner achieves with respect to arm 1 in the intervals [1, C), [C, 3C) and [3C, T ). We note that if the learner pulls arm 1, she does not incur any regret. If she pulls arm 2, she incurs regret ∆t = r t (2) − r t (1) which can be positive or negative. To compute regret with respect to arm 1 in each of those intervals, we sample ∆t every time that the arm 2 is pulled. P Step 5a: interval [1, C). In this interval, E′ ∆t = −∆, so t∈[1,C) ∆t = −C∆. If Y is the number P of times arm 1 is pulled, then the regret is given by Ys=1 ∆s where in the previous expression we abuse notation and mean by ∆s the regret in the s-th time the arm 2 is pulled instead of the regret in the t-th period. Therefore: ! ! ! t C t Y X X X X ′ ′ ′ ∆s < −1.1∆C P ∆s < −1.1∆C ≤ ∆s < −1.1∆C ≤ P min P t≤C s=1 t=1 s=1 s=1 We then use the Hoeffding bound in the last expression and get: ! C X 1 1 1.1∆C − ∆t 2 2 ≤ C · exp − (0.1∆C) = o(1) 2 exp − t 2 t 2 t=1 Step 5b: interval [C, 3C). In this interval, E′ ∆t = ∆, so using the same bound as before, we get   X P′  ∆t < 1.9∆C  ≤ 2 exp −C(0.1∆)2 = o(1) t∈[C,3C) Step 5c: interval [3C, T ] In this interval, pulling arm 2 has again positive expected regret. We use the same technique used in 5a to argue that she cannot obtain large negative regret with high 21 probability: Let Y be the number of times arm 2 is pulled in that interval and again we abuse notation and let ∆s be the difference in rewards in the s-th time the arm is pulled. Then: ! ! ! t T t Y X X X X ∆t < − log T P′ ∆t < − log T ≤ ∆t < − log T ≤ P′ min P′ t≤T s=1 t=1 s=1 s=1 For t = 1.. log T this probability is zero, since ∆t ≥ −1. Now, for larger T , we can use the standard Chernoff bound: ! T t X X 1 ′ 2 ∆t < − log T ≤ 2T exp − log (T )∆ = o(1) P 2 s=1 t=log T Now all the concentration bounds have been established we define the event B to be the event where all those concentration bounds hold. More precisely, B is the event where the following four things happen: (a) empirically arm 1 is better than arm 2; (b) in interval [1, C), the regret of the learner is at least −1.1∆C; (c) in interval [C, 3C), the difference between the total rewards of both arms is at least 1.9∆C; and (d) the regret of the learer in interval [3C, T ] is at least − log T . By the discussion in step 5, we know that P′ (B) = 1 − o(1). Step 6: putting it all together. Since P′ (A) = Ω(1) and P′ (B) = 1 − o(1), then by the union bound, P′ (A and B) ≥ P′ (A) − o(1) = Ω(1). Now, we need to argue that in the constant probability event (A and B), the regret of the learner is at least Ω(C). We simply sum the regret of the learner in each of the intervals. For intervals [1, C) and [3C, T ] we can use the bounds computed in steps 5a andn 5c directly. For interval [C, 3C), we note that conditioned on A, the learner probes arm 1 a constant number of times, so his total regret differs from the regret by pulling arm 2 in all iterations by at most a constant, therefore the total regret can be bounded by: −1.1∆C + (1.9∆C − 4c̃) − log(T ) = Ω(∆C) We can adapt the argument to provide a bound on the expected positive regret E[Reg+ ] where x+ = max{x, 0}. Note that the high probability bounds provided also imply a bound on the expected positive regret. Theorem 5. If a multi-armed bandits algorithm that has the property that for any stochastic input in the two arm setting, it has pseudo-regret bounded by c log1+α (T )/∆ for α < 1. For any ′ ǫ, ǫ′ ∈ (0, 1), there is a corruption level C with T ǫ < C < T ǫ and a C-corrupted instance such that E[Reg+ ] = Ω(T ǫ−δ ) for all δ > 0. Proof. Modify the proof of Theorem 4 as follows. Define c̃ = O(logα (T )/[(ǫ′ − ǫ)∆2 ]) and again select an interval such that E[Ti,1 ] ≤ c̃. Event A is defined in the same way. By Markov’s inequality: P[A] ≥ 1/2 and !2c̃ 1 − ∆ 1 = exp(−O(logα (T ))) P′ [A] ≥ · 21 2 + ∆ 2 Step 5 remains unchanged and in step 6 note that P′ (B) ≪ P′ (A) since α < 1, so P′ (A and B) = exp(−O(logα (T ))). Therefore, with probability at least exp(−O(logα (T ))) the regret is at least Ω(C) = Ω(T ǫ ) and therefore, E[Reg+ ] = Ω(T ǫ · exp(−O(logα (T )))) = Ω(T ǫ−δ ) for all δ > 0. 22	8
Representation Learning and Recovery in the ReLU Model arXiv:1803.04304v1 [stat.ML] 12 Mar 2018 Arya Mazumdar1 and Ankit Singh Rawat1,2 1 College of Information and Computer Sciences, University of Massachusetts Amherst, Amherst, MA 01003, USA. 2 Research Laboratory of Electronics, MIT, Cambridge, MA 02139, USA. E-mail: [email protected], [email protected]. March 13, 2018 Abstract Rectiﬁed linear units, or ReLUs, have become the preferred activation function for artiﬁcial neural networks. In this paper we consider two basic learning problems assuming that the underlying data follow a generative model based on a ReLU-network – a neural network with ReLU activations. As a primarily theoretical study, we limit ourselves to a single-layer network. The ﬁrst problem we study corresponds to dictionary-learning in the presence of nonlinearity (modeled by the ReLU functions). Given a set of observation vectors yi ∈ Rd , i = 1, 2, . . . , n, we aim to recover d × k matrix A and the latent vectors {ci } ⊂ Rk under the model yi = ReLU(Aci + b), where b ∈ Rd is a random bias. We show that it is possible to recover the column space of A within an error of O(d) (in Frobenius norm) under certain conditions on the probability distribution of b. The second problem we consider is that of robust recovery of the signal in the presence of outliers, i.e., large but sparse noise. In this setting we are interested in recovering the latent vector c from its noisy nonlinear sketches of the form v = ReLU(Ac) + e + w, where e ∈ Rd denotes the outliers with sparsity s and w ∈ Rd denote the dense but small noise. This line of work has recently been studied (Soltanolkotabi, 2017) without the presence of outliers. For this problem, we q show that a generalized LASSO algorithm is able to recover the signal c ∈ Rk within an ℓ2 error of O( random Gaussian matrix. (k+s) log d ) d when A is a 1 Introduction Rectiﬁed Linear Unit (ReLU) is a basic nonlinear function deﬁned to be ReLU : R → R+ ∪ {0} as ReLU(x) ≡ max(0, x). For any matrix X , ReLU(X ) denotes the matrix obtained by applying the ReLU function on each of the coordinates of the matrix X . ReLUs are building blocks of many nonlinear data-ﬁtting problems based on deep neural networks (see, e.g., Soltanolkotabi [2017] for a good exposition). Let Y ⊂ Rd be a collection of message vectors that are of interest to us. Depending on the application at hand, the message vectors, i.e., the constituents of Y, may range from images, speech signals, network access patterns to user-item rating vectors and so on. We assume that the message vectors satisfy a generative model, where each message vector can be approximated by a map д : Rk → Rd from the latent space to the ambient space, i.e., for each y ∈ Y, y ≈ д(c) for some c ∈ Rk . 1 (1) Motivated by the recent results on developing the generative models for various real-life signals (see e.g., Goodfellow et al. [2014], Kingma and Welling [2014], Bora et al. [2017]), the non-linear maps д that take the following form warrant special attention. д(c) = h (Al h(Al−1 · · · h(A1 c) · · · ))) , (2) i.e., д is the function corresponding to an l-layer neural network with the activation function h. Here, for i ∈ [l], Ai ∈ Rdi ×di −1 with d0 = k and dl = d, denotes the weight matrix for the i-th layer of the network. In the special case, where the activation function h is the ReLU function, the message vectors of the interest satisfy the following. y ≈ ReLU (Al ReLU(Al−1 · · · ReLU(A1 c + b1 ) · · · + bl−1 ) + bl )) , (3) where, for i ∈ [l], bi ∈ Rdi denotes the biases of the neurons (or output units) at the i-th layer of the network. The speciﬁc generative model in (3) raises multiple interesting questions that play fundamental role in understanding the underlying data and designing systems and algorithms for processing the data. Two such most basic questions are as follows: 1. Learning the representation: Given the observations {y1 , y1 , . . . , yn } ⊂ Rd from the model (cf. (3)), recover the parameters of the model, i.e., {Ât }t ∈[l] , and {c1 , c2 , . . . , cn } ⊂ Rk such that yi ≈ ReLU Âl ReLU(Âl−1 · · · ReLU(Â1 ci + b1 ) · · · + bl−1 ) + bl ) ∀ i ∈ [n]. (4) Note that this question is diﬀerent from training the model, in which case the set {c1 , c2, . . . , cn } is known (and possibly chosen accordingly). 2. Recovery of the signal in the presence of errors: Given the erroneous (noisy) version of a vector generated by the model (cf. (3)), denoise the observation or recover the latent vector. Formally, given v = y + e + w = ReLU (Al ReLU(Al−1 · · · ReLU(A1 c + b1 ) · · · + bl−1 ) + bl )) + e + w (5) and the knowledge of model parameters, obtain ŷ ∈ Rd or ĉ ∈ Rk such that ky − ŷk or kc − ĉk is small, respectively. In (5), e and w correspond to outliers, i.e., large but sparse errors, and (dense but small) noise, respectively. Apart from being closely related, one of our main motivations behind studying these two problems together comes from the recent work on associative memory Karbasi et al. [2014], Mazumdar and Rawat [2015, 2017]. An associative memory consists of a learning phase, where a generative model is learned from a given dataset; and a recovery phase, where given a noisy version of a data point generated by the generative model, the noise-free version is recovered with the help of the knowledge of the generative model. There have been a recent surge of interest in learning ReLUs, and the above two questions are of basic interest even for a single-layer network (i.e., nonlinearity comprising of a single ReLU function). It is conceivable that understanding the behavior of a single-layer network would allow one to use some ‘iterative peeling oﬀ’ technique to develop a theory for multiple layers. In Goel et al. [2017], the problem of recovering ReLU-model under Reliable Agnostic learning model of Kalai et al. [2012] is considered. Informally speaking, under very general distributional assumptions (the rows of A are sampled from some distribution), given A and y = ReLU(Ac), Goel et al. [2017] propose an algorithm that recovers a hypothesis which has an error-rate (under some natural loss function deﬁned therein) of ϵ with respect to the true underlying ReLU-model. Moreover, the algorithm runs in time polynomial in d and exponential in 1/ϵ. As 2 opposed to this, given A and the corresponding output of the ReLU-network y = ReLU(Ac + b), we focus on the problem of recovering c itself. Here, we note that the the model considered in Goel et al. [2017] does not consider the presence of outliers. Soltanolkotabi [2017] obtained further results on this model under somewhat diﬀerent learning guarantees. Assuming that the entries of the matrix A to be i.i.d. Gaussian, Soltanolkotabi [2017] show that with high probability a gradient descent algorithm recovers c within some precision in terms of ℓ2 -loss: the relative error decays exponentially with the number of steps in the gradient descent algorithm. The obtained result is more general as it extends to constrained optimizations in the presence of some regularizers (for example, c can be restricted to be a sparse vector, etc.). However both of these works do not consider the presence of outliers (sparse but large noise) in the observation. The sparse noise is quite natural to assume, as many times only partial observations of a signal vector are obtained. The ReLU model with outliers as considered in this paper can be thought of as a nonlinear version of the problem of recovering c from linear observations of the form v = Ac + e, with e denoting the outliers. This problem with linear observations was studied in the celebrated work of Candès and Tao [2005]. We note that the technique of Candès and Tao [2005] does not extend to the case when there is a dense (but bounded) noise component present. Our result in this case is a natural generalization and complementary to the one in Soltanolkotabi [2017] in that 1) we present a recovery method which is robust to outliers and 2) instead of analyzing gradient descent we directly analyze the performance of the minimizer of our optimization program (a generalized LASSO) using the ideas from Plan and Vershynin [2016], Nguyen and Tran [2013]. On the other hand, to the best of our knowledge, the representation learning problem for single-layer networks has not been studied as such. The representation learning problem for single-layer ReLUs bears some similarity with matrix completion problems, a fact we greatly exploit later. In low rank matrix completion, a matrix M = AX is visible only partially, and the task is to recover the unknown entries by exploiting the fact that it is low rank. In the case of (4), we are more likely to observe the positive entries of the matrix M, which, unlike a majority of matrix completion literature, creates the dependence between the matrix M and the sampling procedure. Main result for representation learning. We assume to have observed d × n matrix Y = ReLU(AC + b ⊗ 1T ) where A is a d × k matrix, C is a k × n matrix, both unknown, b ∈ Rd is a random i.i.d. bias, and ⊗ denote the Kronecker product1 . We show that a relaxed maximum-likelihood method guarantees √ the recovery of the matrix AC with an error in Frobenius norm at most O( d) with high probability (see Theorem 3 for the formal statement). Then leveraging a known result for recovering column space of a perturbed matrix (see Theorem. 5 in the appendix), we show that it is possible to also recover the column space of A with similar guarantee. The main technique that we use to obtain this result is inspired by the recent work on matrix completion by Davenport et al. [2014]. One of the main challenges that we face in recovery here is that while an entry of the matrix Y is a random variable (since b is a random bias), whether that is being observed or being cut-oﬀ by the ReLU function (for being negative) depends on the value of the entry itself. In general matrix completion literature, the entries of the matrix being observed are sampled i.i.d. (see, for example, Candès and Recht [2009], Keshavan et al. [2010], Chatterjee [2015] and references therein). For the aforementioned reason we cannot use most of these results oﬀ-the-shelf. However, similar predicament is (partially) present in Davenport et al. [2014], where entries are quantized while being observed. Similar to Davenport et al. [2014], the tools that prove helpful in this situation are the symmetrization trick and the contraction inequality Ledoux and Talagrand [2013]. However, there are crucial diﬀerence of our model from Davenport et al. [2014]: in our case the bias vector, while random, do not change over 1 This is to ensure that the bias is random, but does not change over diﬀerent observation of the data samples. 3 observations. This translates to less freedom during the transformation of the original matrix to the observed matrix, leading to dependence among the elements in a row. Furthermore, the analysis becomes notably diﬀerent since the positive observations are not quantized. Main result for noisy recovery. We plan to recover c ∈ Rk from observations v = ReLU(Ac + b) + e + w, where A is a d × k standard i.i.d. Gaussian matrix, e ∈ Rd is the vector containing outliers (sparse noise) with kek0 ≤ s, and w ∈ Rd is bounded dense noise such that kwk∞ ≤ δ . To recover c (and e) we employ the LASSO algorithm, which is inspired by the work of Plan and Vershynin [2016] and Nguyen and Tran [2013]. In particular, Plan and Vershynin [2016] recently showed that a signal can be provably recovered (up to a constant multiple) from its nonlinear Gaussian measurements via the LASSO algorithm by treating the measurements as linear observations. In the context of ReLU model, for outlier-free measurements v = ReLU(Ac + b) + w, it follows from Plan and Vershynin [2016] that LASSO algorithm outputs µc as the solution with µ = E(д · ReLU(д + b)), where д is a Gaussian random variable and b is a random variable denoting bias associated with the ReLU function. that this approach guarantees with high probq We show (k+s) log d even when the measurements are corrupted by ability recovery of c within an ℓ2 error of O d outliers e. This is achieved by jointly minimizing the square loss over (c, e) after treating our measurements as linear measurements v ′ = Ac + e + w and adding an ℓ1 regularizer to the loss function to promote the sparsity of the solution for e (we also recover e, see Theorem 4 for a formal description). Organization. The paper is organized as follows. In section 2, we describe some of the notations used throughout the paper and introduce the some technical tools that would be useful to prove our main results. In the same section (subsection 2.3), we provide the formal models of the problem we are studying. In section 3, we provide detailed proofs for our main results on the representation learning problem (see, Theorem 3). Section 4 contains the proofs and the techniques used for the recovery problem in the presence of outliers (see, Theorem 4). 2 Notations and Technical Tools 2.1 Notation For any positive integer n, deﬁne [n] ≡ {1, 2, . . . , n}. Given a matrix M ∈ Rd×n , for (i, j) ∈ [d] × [n], Mi, j denotes the (i, j)-th entry of M. For i ∈ [d], mi = (Mi,1 , Mi,2 , . . . , Mi,n )T denotes the vector containing the elements of the i-th row of the matrix M. Similarly, for j ∈ [n], mj = (M 1, j , M 2, j , . . . , Md, j )T denotes the j-th column of the matrix M. Recall that the function ReLU : R → R+ ∪ {0} takes the following form. ReLU(x) = max(x, 0). (6) For a matrix X ∈ Rd×n , we use ReLU(X ) to denote the d ×n matrix obtained by applying the ReLU function on each of the entries of the matrix X . For two matrix A and B, we use A ⊗ B to represent the Kronecker product of A and B. Given a matrix P ∈ Rd×n , s Õ Pi,2 j kP kF = (i, j)∈[d]×[n] denotes its Frobenius norm. Also, let kP k denote the ℓ2 operator norm of P, i.e. the maximum singular value of P. We let kP k∗ denote the nuclear norm of P. Similar to Davenport et al. [2014], we deﬁne a ﬂatness parameter associated with a function f : R → [0, 1]: β α (f ) := inf \|x \| ≤α 4 \| f ′(x)\| 2 . 4\| f (x)\| (7) β α quantiﬁes how ﬂat f can be in the interval [−α, α]. We also deﬁne a Lipschitz parameter L α (f ) for a function f as follows: L α (f ) := max 2.2 n sup ∫ x \|x \| ≤α f (x) f (y)dy −∞ , sup \|x \| ≤α f ′(x) o . f (x) (8) Techniques to bound the supremum of an empirical process In the course of this paper, namely in the representation learning part, we use the key tools of symmetrization and contraction to bound the supremum of an empirical process following the lead of Davenport et al. [2014] and the analysis of generalization bounds in the statistical learning literature. In particular, we need the following two statements. Theorem 1 (Symmetrization of expectation). Let X 1 , X 2 , . . . , X n be n independent RVs taking values in X and F be a class of R-valued functions on X. Furthermore, let ε 1 , ε 2 , . . . , εn be n independent Rademacher RVs. Then, for any t ≥ 1, ! ! n n Õ Õ t t E sup ≤ 2t E sup f (X i ) − Ef (X i ) εi f (X i ) . (9) f ∈ F i=1 f ∈ F i=1 Theorem 2 (Contraction inequality Ledoux and Talagrand [2013]). Let ε 1 , ε 2 , . . . , εd be d independent Rademacher RVs and f : R+ → R+ be a convex and increasing function. Let ζi : R → R be an L-Lipschitz functions, i.e., \|ζi (a) − ζi (b)\| ≤ L\|a − b\|, which satisfy ζi (0) = 0. Then, for any T ⊆ Rn , d Õ 1 εi ζi (ti ) sup Ef 2 t ∈T i=1 2.3 ! ≤ Ef L · sup t ∈T d Õ i=1 ! εi ti . (10) System Model We focus on the problems of learning the representation and recovery of the signal in the presence of errors when the signal is assumed to be generated using a single layer ReLU-network. The models of learning representations and recovery is described below. Model for learning representations. We assume that a signal vector of interest y satisﬁes y = ReLU(Ac + b), (11) where A ∈ Rd×k and b ∈ Rd correspond to the weight (generator) matrix and the bias vector, respectively. As for the problem of representation learning, we are given n message vectors that are generated from the underlying single-layer model. For j ∈ [n], the j-th signal vector is deﬁned as follows. We deﬁne the d × n observation matrix yj = ReLU Acj + b ∈ Rd . Y = y1 y2 · · · yn . 5 (12) (13) Similarly, we deﬁne the k × n coeﬃcient matrix C= c1 c2 · · · cn . With this notion, we can concisely represent the n observation vectors as Y = ReLU AC + b ⊗ 1T = ReLU M + b ⊗ 1T , (14) (15) where 1 ∈ Rn denotes the all-ones vector. We assume that the bias vector b is a random vector comprising of i.i.d. coordinates with each coordinate being copies of a random variable B distributed according to probability density function p(·). Model for recovery. For the recovery problem, we are given a vector v ∈ Rd , which is obtained by adding noise to a valid signal vector y ∈ Rd that is well modeled by a single-layer ReLU-network, with the matrix A ∈ Rd×k and bias b ∈ Rd . In particular, for some c ∈ Rk we have, v = y + e + w = ReLU(Ac + b) + e + w, (16) where w ∈ Rd denotes the (dense) noise vector with bounded norm. On the other hands, the vector e ∈ Rd contains (potentially) large corruptions, also referred to as sparse errors or outliers (we assume, kek0 ≤ s). The robust recovery problem in ReLU-networks corresponds to obtaining an estimate ĉ of the true latent vector c from the corrupt observation vector y such that the distance between ĉ and c is small. A related problem of denoising in the presence of outliers only focuses on obtaining an estimate y which is close to the true signal vector y. For this part, we focus on the setting where the weight matrix A is a random matrix with i.i.d. entries, where each entry of the matrix is distributed according to standard Gaussian distribution. Furthermore, another crucial assumption is that the Hamming error is oblivious in its nature, i.e., the error vector is not picked in an adversarial manner given the knowledge of A and c2 . 3 Representation learning in a single-layer ReLU-network In the paper, we employ the natural approach to learn the underlying weight matrix A from the observation matrix Y . As the network maps a lower dimensional coeﬃcient vector c ∈ Rk to obtain a signal vector y = ReLU(Ac + b) (17) in dimension d > k, the matrix M = AC (cf. (15)) is a low-rank matrix as long as k < min{d, n}. In our quest of recovering the weight matrix A, we ﬁrst focus on estimating the matrix M, when given access to Y . This task can be viewed as estimating a low-rank matrix from its partial (randomized) observations. Our work is inspired by the recent work of Davenport et al. [2014] on 1-bit matrix completion. However, as we describe later, the crucial diﬀerence of our model from the model of Davenport et al. [2014] is that the bias vector b does not change over observations in our case. Nonetheless we describe the model and main results of 1-bit matrix completion below to underscore the key ideas. 1-bit matrix completion. In Davenport et al. [2014], the following observation model is considered. Given a low-rank matrix M and a diﬀerentiable function ψ : R → [0, 1], the matrix Z ∈ {0, 1}d×n is 2 It is an interesting problem to extend our results to a setting with adversarial errors. However, we note that this problem is an active area of research even in the case of linear measurement, i.e, y = Ac + e + w Bhatia et al. [2015, 2017]. We plan to explore this problem in future work. 6 assumed to be generated as follows3 . Z i, j = ( 1 0 with probability ψ (Mi, j ), with probability 1 − ψ (Mi, j ). (18) Furthermore, one has access to only those entries of Z that are indexed by the set Ω ⊂ [d] × [m], where the set Ω is generated by including each (i, j) ∈ [d] × [n] with certain probability p. Given the observations Z Ω , the likelihood function associated with a matrix X ∈ Rd×n takes the following form4 . Õ (19) 1 {Zi, j =1} log ψ (X i, j ) + 1 {Zi, j =0} log 1 − ψ (X i, j ) . L Ω,Z (X ) = (i, j)∈Ω Now, in order to estimate the matrix M with bounded entries from Z Ω , it is natural to maximize the log-likelihood function (cf. (19)) under the constraint that the matrix M has rank k. maximize L Ω,Z (X ) X ∈Rd ×n subject to rank(X ) = r (20) kX k∞ ≤ γ , where the last constraint is introduced to model the setting where (w.h.p.) the observations are assumed to have bounded coordinates. We note that such assumptions indeed hold in many observations of interests, such as images. Note that the formulation in (20) is clearly non-convex due to the rank constraint. Thus, Davenport et al. [2014] propose the following program. maximize L Ω,Z (X ) X ∈Rd ×n √ subject to kX k∗ ≤ α rmn, (21) kX k∞ ≤ γ . √ Note that the constraint kX k∗ ≤ α rmn is a convex-relaxation of the non-convex constraint rank(X ) ≤ r , b be the output of which is required to ensure that the program in (21) outputs a low-rank matrix. Let M the program in (21). Davenport et al. [2014] obtain the following result to characterize the quality of the b obtained solution M. √ Proposition 1 ([Davenport et al., 2014, Theorem A.1]). Assume that kM k∗ ≤ α rmn and kM k∞ ≤ γ . Let Z be as deﬁned in (18). Then, for absolute constants C 1 and C 2 , with probability at least 1 − C 1 /(d + n), the b satisﬁes the following: solution of (21) M s s (m + n) log(mn) r (m + n) 1 b kF ≤ C 2α kM − M · 1+ , (22) dn E(\|Ω\|) E(\|Ω\|) where the constant C 2 depends on the ﬂatness and steepness of the function ψ . Learning in a single layer ReLU and 1-bit matrix completion: Main diﬀerences. Note that the problem of estimating the matrix M from Y is related to the problem of 1-bit matrix completion as deﬁned 3 The authors assume that the entries of Z take values in the set {+1, −1}. In this paper we state the equivalent model where the binary alphabet is {0, 1}. 4 Throughout this paper, log represents the natural logarithm. 7 above. Similar to the 1-bit matrix completion setup, the observation matrix Y is obtained by transforming the original matrix M in a probabilistic manner, which is dictated by the underlying distribution of the bias vector b. In particular, we get to observe the entire observation matrix, i.e., Ω = [d] × [n]. However, there is key diﬀerence between these two aforementioned setups. The 1-bit matrix completion setup studied in Davenport et al. [2014] (in fact, most of the literature on non-linear matrix completion Ganti et al. [2015]) assume that each entry of the original matrix M is independently transformed to obtain the observation matrix Y . In contrast to this, such independence in absent in the single-layer ReLU-network. In particular, for i ∈ [d], the i-th row of the matrix Y is obtained from the corresponding row of M by utilizing the shared randomness deﬁned by the bias bi . Note that the bias associated with a coordinate of the observed vector in our generative model should not vary across observation vectors. This prevents us from applying the known results to the problem of estimating M from Y . However, as we show in the remainder of this paper that the well-behaved nature of the ReLU-function allows us to deal with the dependence across the entries of a row in Z and obtain the recovery guarantees that are similar to those described in Proposition 1. 3.1 Representation learning from rectiﬁed observations We now focus on the task of recovering the matrix M from the observation matrix Y . Recall that, under the single-layer ReLU-network, the observation matrix Y depends on the matrix M as follows. Y = ReLU(M + b ⊗ 1T ). (23) For i ∈ [d], we deﬁne NY (i) ⊆ [n] as the set of positive coordinates of the i-th row of the matrix Y , i.e., NY (i) = {j ∈ [n] : Yi, j > 0} and NY,i = \|NY (i)\|. (24) Note that, for i ∈ [d], the original matrix M needs to satisfy the following requirements. Mi, j + bi = Yi, j for j ∈ NY (i) (25) Mi, j + bi < 0 for j ∈ N Y (i) := [n]\NY (i). (26) and Given the original matrix M, for i ∈ [d] and j ∈ [n], let Mi,(j) denote the j-th largest element of the i-th row of M, i.e., for i ∈ [d], Mi,(1) ≥ Mi,(2) ≥ · · · ≥ Mi,(n) . It is straightforward to verify from (25) that NY (i) denotes the indices of NY,i largest entries of M. Furthermore, whenever NY,i = si ∈ [n], we have (27) bi = Yi,(1) − Mi,(1) = · · · = Yi,(si ) − Mi,(si ) . Similarly, it follows from (26) that whenever we have NY,i = 0, then bi satisﬁes the following. bi ∈ (−∞, − max Mi, j ) = (−∞, −Mi,(1) ). (28) j ∈[n] Based on these observation, we deﬁne the set of matrices XY,ν,γ ⊂ Rd×n as XY,ν,γ = X : kX k∞ ≤ γ ; Yi,(1) − X i,(2) = · · · = Yi,(si ) − X i,(si ) ; and X i,(si ) ≥ max X i, j + ν ∀i ∈ [d] . j ∈N Y (i) 8 (29) Recall that, p : R → R denote the probability density function of each bias RV. We can then write the likelihood that a matrix X ∈ XZ,ν,γ results into the observation matrix X as follows. Ö P(yi \|xi ), (30) P(Y \|X ) = i ∈[d] where, for i ∈ [d], P(yi \|xi ) = 1 {NY ,i =0} · P(bi ≤ − max X i, j ) + j ∈[n] n Õ s=1 1 {NY ,i =s } · p(bi = Yi,(s) − X i,(s) ). (31) By using the notation F (x 1 , x 2 ) = P(−x 1 ≤ B ≤ −x 2 ) and X i∗ = maxj ∈[n] X i, j , we can rewrite (31) as follows. P(yi \|xi ) = 1 {NY ,i =0} · F (∞, X i∗ ) + n Õ s=1 1 {NY ,i =s } · p(bi = Yi,(s) − X i,(s) ). (32) Therefore the log-likelihood of observing Y given that X ∈ XY,ν,γ is the original matrix takes the following form. Õ log P(yi \|xi ) LY (X ) = i ∈[d] = Õ i ∈[d] 1 {NY ,i =0} · log F (∞, X i∗ ) + n Õ s=1 1 {NY ,i =s } · log p(Yi,(s) − X i,(s) ) . (33) In what follows, we work with a slightly modiﬁed quantity L Y (X ) = LY (X ) − LY (0) = Õ i ∈[d] 1 {NY ,i =0} · log n F (∞, X i∗ ) Õ p(Y − X ) + 1 {NY ,i =s } · log i,(s) i,(s) . F (∞, 0) p(Yi,(s) ) s=1 In order to recover the matrix M from the observation matrix Y , we employ the natural maximum likelihood approach which is equivalent to the following. maximize L Y (X ) subject to X ∈ XY,ν,γ . X ∈Rd ×n (34) Deﬁne ωp,γ ,ν to be such that F (x, y) ≥ ωp,γ ,ν for all x, y ∈ [−γ , γ ] with \|x − y\| > ν. In what follows, we simply refer this quantity as ωp as γ and ν are clear from context. The following result characterizes the performance of the program proposed in (34). b be Theorem 3. Assume that kM k∞ ≤ γ and the observation matrix Y is related to M according to (15). Let M the solution of the program speciﬁed in (34), and the bias density function is p(x) diﬀerentiable with bounded 1 . derivative. Then, the following holds with probability at least 1 − d+n b kF2 ≤ C 0 Lγ (p) · kM − M γd , βγ (p)ωp (35) where, C 0 is a constant. The quantities βγ (p) and Lγ (p) depend on the distribution of the bias and are deﬁned in (7) and (8), respectively. The proof of Theorem 3 crucially depends on the following lemma. 9 Lemma 1. Given the observation matrix Y which is related to the matrix M according to (15), let XY,ν,γ be as deﬁned in (29). Then, for any X ∈ XY,ν,γ , we have i h (36) E L Y (M) − L Y (X ) ≥ βγ (p)ωp · kM − X kF2 . The proof of this lemma is delegated to the appendix. Now we are ready to prove Theorem 3. b be the solution of the program in (34). In what follows, we use X as a short hand Proof of Theorem 3. Let M notation for XY,ν,γ . We have, h i h i h i b − L Y (M) = E L Y (M) b − L Y (M) − L Y (M) − E L Y (M) + L Y (M) b − E L Y (M) b 0 ≤ L Y (M) i h i h b − L Y (M) + 2 sup L Y (X ) − E L Y (X ) , (37) ≤ E L Y (M) X ∈X which means, h i h i b ≤ 2 sup L Y (X ) − E L Y (X ) . E L Y (M) − L Y (M) (38) X ∈X We now employ Lemma 1 to obtain that h i b kF2 ≤ 2 sup L Y (X ) − E L Y (X ) . βγ (p)ωp · kM − M (39) X ∈X We now proceed to upper bound the right hand side of (39). It follows from the standard symmetrization trick Devroye et al. [2013] that, for any integer t ≥ 1, we have " d h i Õ F (∞, X i∗ ) t + εi · 1 {NY ,i =0} · log ≤ 2t · E sup E sup L Y (X ) − E L Y (X ) F (∞, 0) X ∈X i=1 X ∈X # n Õ p(Yi,(s) − X i,(s) ) t , (40) 1 {NY ,i =s } · log p(Yi,(s) ) s=1 where {εi }i ∈[d] are i.i.d. Rademacher random variables. Note that, for x, x̃ ∈ R, d(log F (∞, u)) du \|u \| ≤γ ∫ −u d(log −∞ p(y)dy) = \|x − x̃ \| · sup ≤ \|x − x̃ \| · Lγ (p), du \|u \| ≤γ log F (∞, x) − log F (∞, x̃) ≤ \|x − x̃ \| · sup and log p(Yi,(s) − x) − log p(Yi,(s) − x̃) ≤ \|x − x̃ \| · sup \|u \| ≤γ dp(u) ≤ \|x − x̃ \| · Lγ (p). du At this point, we can combine the contraction principle with (40) to obtain the following. h i t E sup L Y (X ) − E L Y (X ) X ∈X t t " t ≤ 2 · 2 · E (Lγ (p)) · sup d Õ X ∈X i=1 εi · 10 1 ∗ {NY ,i =0} · X i + n Õ s=1 1 {NY ,i =s } · X i,(s) t # " d d Õ Õ 2 t /2 ≤ 4 · E (Lγ (p)) · sup εi (i) (ii) t t X ∈X " i=1 ≤ 4t · E (Lγ (p))t · d t /2 dγ 2 t /2 # 1 i=1 ∗ {NY ,i =0} · X i + n Õ s=1 1 {NY ,i =s } · X i,(s) 2 t /2 = (4Lγ (p) · dγ )t , # (41) where (i) and (ii) follow from the Cauchy-Schwartz inequality and the fact that, for X ∈ X, kX k∞ ≤ γ , respectively. Now using Markov’s inequality, it follows from (41) that h P sup L Y (X ) − E L Z (X ) X ∈X i ≥ C 0Lγ (p) · γd ≤ (i) ≤ where (i) follows from (41); and (ii) follows by setting C 0 ≥ 3.2 h i i h t E supX ∈X L Z (X ) − E L Z (X ) 4 C0 4 e t (C 0 Lγ (p) · γd)t (ii) ≤ 1 , d +n and t = log(d + n). (42) Recovering the network parameters b ∈ XY,ν,γ such that As established in Theorem 3, the program proposed in (34) recovers a matrix M b kF ≤ kM − M s C 0 · Lγ (p) · γ √ d. βγ (p)ωp (43) b as M b = M + E, where E denotes the perturbation matrix that has Let’s denote the recovered matrix M bounded Frobenius norm (cf. (43)). Now the task of recovering the parameters of single-layer ReLUnetwork is equivalent to solving for A given b = M + E = AC + E. M (44) In our setting where we have A ∈ Rd×k and C ∈ Rk×n with d > k and n > k, M is a low-rank matrix with its column space spanned by the columns of A. Therefore, as long as the generative model ensures that the matrix M has its singular values suﬃciently bounded away from 0, we can resort to standard results from b as an candidate for the orthonormal matrix-perturbation theory and output top k left singular vectors of M basis for the column space of M or A. In particular, we can employ the result from Yu et al. [2015] which b be the top k left singular vectors of M and M, b respectively. Note that, is stated in Appendix A. Let U and U even without the perturbation we could only hope to recover the column space of A (or the column space of U ) and not the exact matrix A. Let σk , the smallest non-zero singular value of M, is at least δ > 0. Then, it follows from Theorem 5 (cf. Appendix A) and (43) that there exists an orthogonal matrix O ∈ Rk×k such that √ 3/2 (2σ + kE k) · min{ k kE k, kE k } 23/2 (2σ1 + kE kF ) · kE kF 2 F 1 bO kF ≤ ≤ , kU − U δ2 δ2 (45) which is a guarantee that the column space of U is recovered within an error of O(d) in Frobenius norm b. by the column space of U 11 4 Robust recovery in single-layer ReLU-network We now explore the second fundamental question that arises in the context of reconstructing a signal vector belonging to the underlying generative model from its erroneous version. Recall that, we are given a vector v ∈ Rd , which is obtained by adding noise to a valid message vector y ∈ Rd that is well modeled by a single-layer ReLU-network, i.e., v = y + e + w = ReLU(Ac + b) + e + w. (46) Here, w denotes the (dense) noise vector with bounded norm. On the other hands, the vector e contains (potentially) large corruptions, also referred to as outliers. We assume the number of outliers kek0 to be bounded above by s. The robust recovery problem in ReLU-networks corresponds to obtaining an estimate ĉ of the true representation c from the corrupt observation vector y such that the distance between ĉ and c is small. A related problem of denoising in the presence of outliers only focuses on obtaining an estimate ŷ which is close to the true message vector y. In the remainder of this paper, we focus on the setting where the weight matrix A is a random matrix with i.i.d. entries, where each entry is distributed according to the standard Gaussian distribution. Furthermore, another crucial assumption is that the outlier vector is oblivious in its nature, i.e., the error vector is not picked in an adversarial manner5 given the knowledge of A and c. Note that Soltanolkotabi [2017] study a problem which is equivalent to recovering the latent vector c from the observation vector generated form a single-layer ReLU-network without the presence of outliers. In that sense, our work is a natural generalization of the work in Soltanolkotabi [2017] and presents a recovery method which is robust to errors as well. However, our approach signiﬁcantly diﬀers from that in Soltanolkotabi [2017], where the author analyze the convergence of the gradient descent method to the true representation vector c. In contrast, we rely on the recent work of Plan and Vershynin Plan and Vershynin [2016] to employ the LASSO method to recover the representation vector c (and the Hamming error vector e). Given v = ReLU(Ac + b) + e + w, which corresponds to the corrupted non-linear observations of c, we try to ﬁt a linear model to these observations by solving the following optimization problem6 . minimize c∈Rk ,e∈Rd 1 kv − Ac − ek22 + λkek1 . 2d (47) In the aforementioned formulation, the regularizer part is included to encourage the sparsity in the estimate vector. The following result characterizes the performance of our proposed program (cf. (47)) in recovering the representation and the corruption vector. Theorem 4. Let A ∈ Rd×n be a random matrix with i.i.d. standard Gaussian random variables as its entires and v satisﬁes v = ReLU(Ac∗ + b) + e∗ + w, (48) where kc∗ k2 = 1, ke∗ k0 ≤ s and kwk∞ ≤ δ . Let µ be deﬁned as µ = E[ReLU(д + b) · д], where д is a standard Gaussian random variable and b is a random variable that represents the bias in a coordinate in (48). Let (ĉ, ê) 5 It is an interesting problem to extend our results to a setting with adversarial errors. However, we note that this problem is an active area of research even in the case of linear measurement, i.e, y = Ac + e + w Bhatia et al. [2015, 2017]. We plan to explore this problem in future work. 6 Note that this paper deals with a setup where number of observations is greater than the dimension of the signal that needs to be recovered, i.e., d > k. Therefore, we don’t necessarily require the vector c to belong to a restricted set, as done in the other version of the robust LASSO methods for linear measurements (see e.g., Nguyen and Tran [2013]). 12 be the outcome of the program described in (47). Then, with high probability, we have (r ) r k log k s log d ke∗ − êk2 ∗ ≤ C̃ max , kµc − ĉk2 + , √ d d d (49) where C̃ is a large enough absolute constant that depends on δ . Proof. Assume that ĉ = µc∗ + h and ê = e∗ + f . (50) Furthermore, for c ∈ Rd and e ∈ Rk , we deﬁne L(c, e) = 1 ky − Ac − ek22 + λkek1 . 2d (51) Let S = {i ∈ [d] : ei∗ , 0} be the support of the vector e∗ such that \|S\| = s. Given a vector a ∈ Rd and set T ⊆ [d], we use a T to denote the vector obtained by restricting a to the indices belonging to T . Note that 1 1 kv − µAc∗ − e∗ − Ah − f k22 + λke∗ + f k1 − kv − µAc∗ − e∗ k22 − λke∗ k1 2d 2d 1 1 kAh + f k22 − hv − µAc∗ − e∗ , Ah + fi + λ k(e∗ + f ∗ ) S k1 + kf SC k1 − ke∗ k1 = 2d d 1 (i) 1 2 kAh + f k2 − hReLU(Ac∗ + b) − µAc∗ + w, Ah + fi + = 2d d λ k(e∗ + f ∗ ) S k1 + kf SC k1 − ke∗ k1 (ii) 1 1 ≥ kAh + f k22 − hReLU(Ac∗ + b) − µAc∗ + w, Ah + fi + λ kf SC k1 − kf S k1 (52) 2d d L(ĉ, ê) − L(µc∗, e∗ ) = where (i) and (ii) follow from (46) and the triangle inequality, respectively. Since (ĉ, ê) is solution to the program in (47), we have L(ĉ, ê) − L(µc∗, e∗ ) ≤ 0. (53) By combining this with (52), we obtain that 1 1 kAh + f k22 ≤ · hReLU(Ac∗ + b) − µAc∗ + w, Ah + fi + λ(kf S k1 − kf SC k1 ) 2d d (54) We now complete the proof in two steps where we obtain universal lower and upper bounds on the left hand side and the right hand side of (54), respectively, that hold with high probability. Upper bound on the RHS of (54). Let’s deﬁne z = ReLU(Ac∗ + b) − µAc∗ . Note that 1 · hz + w, Ah + fi + λ(kf S k1 − kf SC k1 ) d 1 1 · hz + w, fi + λ(kf S k1 − kf SC k1 ) = · hz + w, Ahi + d d (i) 1 1 ≤ · hz + w, Ahi + · kz + wk∞ kf k1 + λ(kf S k1 − kf SC k1 ) d d 13 (55) = 1 1 1 · hz + w, Ahi + (λ + · kz + wk∞ )kf S k1 − (λ − · kz + wk∞ )kf SC k1 d d d (56) where (i) follows from the Hölder’s inequality. We now employ [Plan and Vershynin, 2016, Lemma 4.3] to obtain that7 √ √ sup hz, Ahi ≤ C kσ + η d · khk2 , (57) h∈Rk where C is an absolute constant and σ 2 := E (ReLU(д + b) − µд)2 η 2 := E д2 · (ReLU(д + b) − µд)2 (58) with д being a standard Gaussian random variable. Now we can combine (56) and (57) to obtain the following. 1 · hz + w, Ah + fi + λ kf S k1 − kf SC k1 d √ √ kσ + η kz + wk∞ kz + wk∞ k T · khk2 + ≤C kA wk∞ khk2 + λ + kf S k1 − λ − kf SC k1 (59) √ d d d d √ √ (i) √ kσ + η k T 1 · khk2 + ≤C kA wk∞ khk2 + s(λ + · kz + wk∞ )kf k2 √ d d d √ √ (ii) √ kσ + η k T kA wk∞ khk2 + 2λ s kf k2, (60) · khk2 + ≤ C √ d d √ √ where (i) and (ii) follow by setting λ ≥ 2kz + wk∞ /d and using the fact that kf S k1 ≤ s kf S k2 ≤ s kf k2 . We can further simplify the bound in (60) as follows. 1 · hz + w, Ah + fi + λ kf S k1 − kf SC k1 d ) ( √ √ √√ kσ + η k T kf k2 + . ≤ max C kA wk∞, 2λ s d khk2 + √ √ d d d (61) Lower bound on the LHS of (54). By combining (54) and (59), we get that 1 kAh + f k22 2d √ √ kσ + η k T kz + wk∞ kz + wk∞ · khk2 + ≤C kA wk∞ khk2 + λ + kf S k1 − λ − kf SC k1 √ d d d d (62) k∞ . Since the left hand side of (62) is non-negative, we ﬁnd that the Note that we have picked λ ≥ 2 kz+w d tuple (h, f) belongs to the following restricted set. ! √ √ kσ + η k (63) kAT wk∞ khk2 + 3λkf S k1 }. (h, f) ∈ R := {h ∈ Rk , f ∈ Rd : λ · kf SC k1 ≤ 2 C + √ d d 7 In Plan and Vershynin [2016], Plan and Vershynin obtain the bound in terms of the Gaussian width Vershynin [2018] of the ∗ cone which √ the vector h belongs to. However, in our setup where we do not impose any speciﬁc structure on c , this quantity is simply O( k). 14 As a result, in order to lower bound (54), we lower bounding the following quantity for every (h, f) ∈ R. 1 · kAh + f k22 . 2d (64) Towards this, we employ Lemma 3 in Appendix C, which gives us that, for every (h, f) ∈ R, with high probability, we have 2 1 kf k2 1 2 · kAh + f k2 ≥ khk2 + √ . 2d 128 d (65) Completing the proof. It follows from (54), (61), and (65) that ( √ ) √ 2 √√ kσ + η kf k2 k T kf k2 1 kA wk∞ , 2λ s d khk2 + √ khk2 + √ ≤ max C + √ 128 d d d d or ) ( √ √ √√ kσ + η k T kf k2 kA wk∞, 2λ s d . khk2 + √ ≤ 128 max C + √ d d d (66) Now using the fact that kwk∞ ≤ δ and A is an i.i.d. standard Gaussian matrix, we can obtain the following bound from (66), which holds with high probability. ) (r r k log k s log d kf k2 , (67) , khk2 + √ ≤ C̃ max d d d where C̃ is a large enough absolute constant. Acknowledgements. This research is supported in part by NSF awards CCF 1642550, CCF 1618512 and CAREER award 1642658. References K. Bhatia, P. Jain, and P. Kar. Robust regression via hard thresholding. In Advances in Neural Information Processing Systems (NIPS), pages 721–729. 2015. K. Bhatia, P. Jain, P. Kamalaruban, and P. Kar. Consistent robust regression. In Advances in Neural Information Processing Systems (NIPS), pages 2107–2116. 2017. A. Bora, A. Jalal, E. Price, and A. G. Dimakis. 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Soltanolkotabi. Learning relus via gradient descent. In Advances in Neural Information Processing Systems (NIPS), pages 2004–2014. 2017. R. Vershynin. High-Dimensional Probability: Available online, 2018. An Introduction with Applications in Data Science. Y. Yu, T. Wang, and R. J. Samworth. A useful variant of the Davis–Kahan theorem for statisticians. Biometrika, 102(2):315–323, 2015. 16 A Results on Matrix Perturbation Let M be a d × n matrix, where without loss of generality we assume that d ≥ n. Let M have the following singular value decomposition. M = U ΣV T , (68) where Σ = Diag (σ1 , σ2 , . . . , σn ) is the diagonal matrix with the singular values of M as its diagonal entries. b = M + E be the matrix which is obtained by perturbing the original matrix M by an error matrix E. Let M b have the following singular value decomposition. Let M b =U bb M ΣVbT , (69) where b Σ = Diag (b σ1, b σ2 , . . . , b σn ) is the diagonal matrix comprising the singular values of the perturbed b Let γ 1 ≥ γ 2 ≥ · · · ≥ γn be the singular values of the matrix U T U b. Deﬁne, matrix M. b) = Diag θ 1 , θ 2 , . . . , θ n . θ i = cos−1 γi ∀i ∈ [n] and Θ(U , U (70) b the subspaces spanned by the Note that {θ i }i ∈[n] are referred to as the canonical angles between U and U, b, respectively. It is common to use k sin Θ(U , U b)kF as a distance measure columns of the matrices U and U b between U and U. In Yu et al. [2015], Yu et al. present the following result which bounds the distance between the subb respectively8. spaces spanned by the singular vectors of the original matrix M and the perturbed matrix M, b = M + E ∈ Rd×n have singular values σ1 ≥ · · · σmin{d,n } and b Theorem 5. Let M, M σ1 ≥ · · · b σmin{d,n } , respectively. Fix 1 ≤ r ≤ rank(A) and assume that σr2 − σr2+1 ≥ 0. b = (b Let U = (u1 , u2 , . . . , ur ) ∈ Rd×r and U u1 , b u2 , . . . , b ur ) ∈ Rd×r contain the left singular vectors associated b respectively. Then, with the r leading singular values of M and M, √ 2(2σ1 + kE k) · min{ r kE k, kE kF } b . k sin Θ(U , U )kF ≤ σr2 − σr2+1 (71) Moreover, there exists an orthogonal matrix O ∈ Rr ×r such that 3/2 (2σ + kE k) · min{√r kE k, kE k } 2 1 F bO kF ≤ kU − U . σr2 − σr2+1 B (72) Proofs of Section 3.1 Lemma 2 (Deﬁning the distance measure). Given the observation matrix Y which is related to the matrix M according to (15), let XY,ν,γ be as deﬁned in (29). Then, for any X ∈ XY,ν,γ , we have h i E L Y (M) − L Y (X ) ≥ βγ (p)ωp · kM − X kF2 . (73) 8 Here, we state a special case of the result from Yu et al. [2015]. See [Yu et al., 2015, Theorem 4] for the statement of the general result. 17 Proof. First, we recall our notation that given the original matrix M, for i ∈ [d] and j ∈ [n], Mi,(j) denotes the j-th largest element of the i-th row of M, i.e., for i ∈ [d], Mi,(1) ≥ Mi,(2) ≥ · · · ≥ Mi,(n) . We deﬁne, Mi∗ = maxj ∈[n] Mi, j . Thus, h i E L Y (M) − L Y (X ) # " n d Õ F (∞, Mi∗ ) Õ p(Yi,(s) − Mi,(s) ) + 1 {NY ,i =s } · log = E 1 {NY ,i =0} · log F (∞, X i∗ ) s=1 p(Yi,(s) − X i,(s) ) i=1 d n−1 ∫ −Mi,(s +1) Õ F (∞, Mi∗ ) Õ p(b) ∗ F (∞, Mi ) · log = db + p(b) · log ∗) + F (∞, X p(b + M i,(s) − X i,(s) ) i i=1 s=1 −Mi,(s ) ∫ ∞ p(b) db , p(b) · log p(b + Mi,(n) − X i,(n) ) −Mi,(n) (74) where p : R → R represents the probability density function of the bias RV and F is deﬁned as F (x 1 , x 2 ) = P(−x 1 ≤ b ≤ −x 2 ). Given the matrices, M, X ∈ XZ,ν,γ , we deﬁne a new (density) function д as follows. ( p(u + Mi,(s) − X i,(s) ) if u ∈ (−Mi,(s) , −Mi,(s+1) ] for s ∈ [n − 1] д(u) = p(u + Mi,(n) − X i,(n) ) if u ∈ [−Mi,(n) , ∞). (75) Recall that for x ∈ R, we have x ≤ e x − 1. For x = log y, this gives us that log y ≤ y − 1. In particular, for b ∈ R, employing (76) with y = p(b) · log s д(b) ≤ p(b) · p(b) q д(b) p(b) , s (76) we get that ! p д(b) − 1 = p(b)д(b) − p(b) p(b) or p(b) · log p p(b) ≥ 2 · p(b) − p(b)д(b) . д(b) (77) By using (77), for every i ∈ [d], we obtain that n−1 ∫ Õ s=1 −Mi,(s +1) −Mi,(s ) ∫ ∞ p(b) db + p(b) · log p(b + Mi,(s) − X i,(s) ) = ≥ −Mi,(1) ∞ ∫ −Mi,(1) p(b) db д(b) p 2 · p(b) − p(b)д(b) p(b) · log 18 ∫ ∞ −Mi,(n) p(b) · log p(b) db p(b + Mi,(n) − X i,(n) ) ∫ = 2 · 1 − F (∞, M 1,(1) ) − ∞ −Mi,(1) p p(b)д(b)db = 1 − F (∞, Mi∗ ) + 1 − F (∞, X i∗ ) − = ∫ ∞ −Mi,(1) ∞ −Mi,(1) F (∞, X i∗ ) F (∞, Mi∗ ) p 2 · p(b)д(b)db + 1− − 1− p 2 p p(b) − д(b) db + 1 − F (∞, Mi∗ ) − 1 − F (∞, X i∗ ) (78) F (∞,X i∗ ) F (∞, Mi∗ ) to obtain the following. F (∞, X i∗ ) F (∞, X i∗ ) ∗ ∗ F (∞, Mi ) · log ≤ F (∞, Mi ) · − 1 = F (∞, X i∗ ) − F (∞, Mi∗ ). F (∞, Mi∗ ) F (∞, Mi∗ ) For i ∈ [d], we now employ (76) with y = or ∫ F (∞, Mi∗ ) + F (∞, X i∗ ) − F (∞, Mi∗ ) F (∞, X i∗ ) F (∞, Mi∗ ) = F (∞, Mi∗ ) · log + 1 − F (∞, Mi∗ ) − 1 − F (∞, X i∗ ) ≥ 0 ∗ F (∞, X i ) F (∞, Mi∗ ) · log (79) By combining (78) and (79), we obtain that n−1 ∫ −Mi,(s +1) F (∞, Mi∗ ) Õ p(b) ∗ p(b) · log + db + F (∞, Mi ) · log ∗ F (∞, X i ) s=1 −Mi,(s ) p(b + Mi,(s) − X i,(s) ) ∫ ∞ p(b) db p(b) · log p(b + Mi,(n) − X i,(n) ) −Mi,(n) ∫ ∞ p 2 p ≥ p(b) − д(b) db −Mi,(1) = n−1 ∫ −Mi,(s +1) Õ s=1 −Mi,(s ) ∫ ∞ p −Mi,(n) 2 q p p(b) − p(b + Mi,(s) − X i,(s) ) db 2 q p(b) − p(b + Mi,(n) − X i,(n) ) db !2 p ′ (ξ s ) = Mi,(s) − X i,(s) db + p 2 p(ξ s ) s=1 −Mi,(s ) !2 ∫ ∞ p ′ (ξ n ) Mi,(n) − X i,(n) db p −Mi,(n) 2 p(ξ n ) ∫ ∞ n−1 ∫ −Mi,(s +1) Õ (ii) 2 ≥ βγ (p) · Mi,(s) − X i,(s) db + (i) n−1 ∫ Õ −Mi,(s +1) s=1 (iii) + ≥ βγ (p)ωp · −Mi,(s ) n Õ s=1 −Mi,(n) 2 Mi,(s) − X i,(s) , 2 Mi,(n) − X i,(n) db ! (80) where (i) follows from the Mean Value Theorem with suitable {ξ i } ⊂ R and (ii) follows by assuming that we have q p ′ (u) ≥ βγ (p) for all \|u\| ≤ γ . p 2 p(u) 19 Since M ∈ XZ,ν,γ , we have \|Mi,(n) \| ≤ γ and \|Mi,(s) −Mi,(s+1) \| ≥ ν. The step (iii) follows from the assumption that F (x, y) ≥ ωp for all (x, y) such that \|x − y\| ≥ ν . By combining (74) with (80), we now obtain that d n i Õ h Õ 2 βγ (p)ωp · Mi,(s) − X i,(s) = βγ (p)ωp · kM − X kF2 . E L Y (M) − L Y (X ) ≥ i=1 s=1 C Proofs of Section 4 Here we state the special form of a result that was obtained in Nguyen and Tran [2013] for the generals setting, where one may potentially require the vector c to be sparse as well. Lemma 3. Let A ∈ Rd×n be random matrix that has i.i.d. standard Gaussian entries. Furthermore, let R ⊂ Rk × Rd be as deﬁned in (63). Then, with probability at least 1 − c exp(−c̃d), we have 2 1 kf k 1 · kAh + f k2 ≥ khk + √ for all (h, f) ∈ R. 2d 128 d (81) Here, c, c̃ > 0 are absolute constants. Proof. Note that kAh + f k22 = kAhk22 + kf k22 + 2hAh, fi. (82) For a d × k matrix with i.i.d. Gaussian entries, there exists constants c 1 and c 2 such that with probability at least 1 − c 1 exp(−c 2d), we have 1 1 √ kAhk2 ≥ khk2 . 4 d (83) Therefore, with probability at least 1 − c 1 exp(−c 2d), we have kAhk22 + kf k22 2 d d d kf k2 2 2 2 2 . ≥ khk2 + kf k2 ≥ (khk2 + kf k2 /d) ≥ khk2 + √ 16 16 32 d (84) Next, we focus on obtaining an upper bound on 1 \|hAh, fi\|. d Towards this we partition the set [d] into r blocks S1 = S, S2, . . . , Sr such that \|S2 \| = · · · \|S\|r = s ′ ≥ \|S\| = s. Here, S2 refers to the set of indices of s ′ largest entries (in terms of absolute value) of f SC ; S3 corresponds to the set of indices of the next s ′ largest entires of f SC ; and so on. Now, we have Õ 1 1Õ 1 \|hA Si h, f Si i\| ≤ max kA Si k2 khk2 \|hAh, fi\| ≤ kf Si k2 d d i=1 d i i=1 r r 20 (85) In [Nguyen and Tran, 2013, Appendix], Nguyen and Tran show that, with probability at least 1−2 exp(−τ 2s ′/2), for a set S ′ with \|S ′ \| = s ′, we have √ √ √ (86) kA S′ k2 ≤ k + s ′ + τ s ′ . By setting τ = τ ′ q d s′ and taking the union bound over all the subsets of [d] of size s ′, kA S′ k2 ≤ holds with probability at least √ √ √ k + s ′ + τ s ′ ∀ S ′ ⊂ [d] such that \|S ′ \| = s ′ (87) s′ d ed ′2 exp(−τ ′2d/2). 1 − ′ exp(−τ d/2) ≥ 1 − ′ s s Assuming that s ′ log(d/s ′ ) ≤ c 3d, the aforementioned probability is at least 1 − exp(−(τ ′2 /2 − c 3 )d). On the other hand, we have r Õ i=1 (i) kf Si k2 ≤ 2kf k2 + r Õ i=3 kf Si k2 1 ≤ 2kf k2 + √ kf Sc k1 s′ ! √ √ (iii) kσ + η 3 k T 2 kA wk∞ khk2 + √ kf S k1 + ≤ 2kf k2 + √ C √ d λ s′ s′ d ! √ √ (iv) kσ + η 2 k T + ≤ 5kf k2 + √ C kA wk∞ khk2 , √ ′ d λ s d (ii) (88) where (i) follows from the fact that kf S1 k2 ≤ kf S2 k2 ≤ kf k2 ; (ii) follows from a standard bound given in Candes et al. [2006]; (iii) follows from the fact that √f belongs to the set R deﬁned in (63); and (iv) is a √ consequence of the loose bound kf S k1 ≤ s kf S k2 ≤ s ′ kf k2 . Next, we use the fact that λ ≥ 2kz + wk∞ /d, it follows from (88) that ! √ √ r Õ kσ + η k T d kA wk∞ khk2 kf Si k2 ≤ 5kf k2 + + √ C √ d kz + wk∞ s ′ d i=1 ! ! √ √ √ √ kf k2 kσ + η d k T + =5 d √ + kA wk∞ khk2 √ C √ d 5kz + wk∞ s ′ d d √ kf k2 (89) ≤ 5 d √ + khk2 d By combining (85), (87), and (89), with probability at least 1 − c 4 exp(−c 5d), we obtain that √ kf k2 √ √ 1 √ 1 ′ ′ \|hAh, fi\| ≤ k + s + τ s khk2 ·5 d √ + khk2 d d d 2 √ √ kf k2 5 √ ′ ′ k + s +τ s ≤ √ √ + khk2 d d 21 2 1 kf k2 ≤ √ + khk2 , 128 d (i) (90) where (i) follows from large enough d. By combining (82), (84), and (90), we obtain that, for every (h, f) ∈ R, 2 1 1 kf k2 2 · kAh + f k2 ≥ khk + √ 2d 128 d holds with probability at least 1 − c exp(−c̃d), with absolute constants c, c̃ > 0. 22 (91)	7
CHUDNOVSKY’S CONJECTURE FOR VERY GENERAL POINTS IN PN k arXiv:1604.02217v2 [math.AC] 7 Dec 2017 LOUIZA FOULI, PAOLO MANTERO AND YU XIE A BSTRACT. We prove a long-standing conjecture of Chudnovsky for very general and generic points in PN k , where k is an algebraically closed field of characteristic zero, and for any finite set of points lying on a quadric, without any assumptions on k. We also prove that for any homogeneous ideal I in the homogeneous coordinate ring R = k[x0 , . . . , xN ], Chudnovsky’s conjecture holds for large enough symbolic powers of I. 1. I NTRODUCTION This manuscript deals with the following general interpolation question: Question 1.1. Given a finite set of n distinct points X = {p1 , . . . , pn } in PN k , where k is a field, what is the minimum degree, αm (X), of a hypersurface f ≠ 0 passing through each pi with multiplicity at least m? Question 1.1 has been considered in various forms for a long time. We mention a few conjectures and motivations. For instance, this question plays a crucial role in the proof of Nagata’s counterexamples to Hilbert’s fourteenth problem [19]. In the same paper Nagata conjectured that √ αm (X) > m n for sets of n general points in P2C [19], and a vast number of papers in the last few decades are related to his conjecture. Another reason for the interest sparked by the above question comes from the context of complex analysis: an answer to Question 1.1 would provide information about the Schwarz exponent, which is very important in the investigation of the arithmetic nature of values of Abelian functions of several variables [4]. However, besides a few very special classes of points (e.g., if these n points lie on a single hyper−1 ) points forming a star configuration and m is a multiple of N [5],[2]), plane or one has n = (β+N N at the moment a satisfactory answer to this elusive question appears out of reach. Therefore, there has been interest in finding effective lower bounds for αm (X). In fact, lower bounds for αm (X) yield upper bounds for the Schwarz exponent. Using complex analytic techniques, Waldschmidt [22] and Skoda [21] in 1977 proved that for all m ≥ 1 αm (X) α(X) ≥ , m N where α(X) = α1 (X) is the minimum degree of a hypersurface passing through every point of X and k = C. In 1981, Chudnovsky [4] improved the inequality in the 2-dimensional projective space. He showed that if X is a set of n points in P2C , then for all m ≥ 1 αm (X) α(X) + 1 ≥ . m 2 He then raised the following conjecture for higher dimensional projective spaces: 2010 Mathematics Subject Classification. 13F20, 11C08. Key words and phrases. Chudnovsky’s conjecture, initial degrees, symbolic powers, fat points, Seshadri constant. The first author was partially supported by a grant from the Simons Foundation, grant #244930. 1 2 L. FOULI, P. MANTERO AND Y. XIE Conjecture 1.2 (Chudnovsky [4]). If X is a finite set of points in PN C , then for all m ≥ 1 αm (X) α(X) + N − 1 ≥ . m N The first improvement towards Chudnovsky’s Conjecture 1.3 was achieved in [9] by Esnault and αm (X) α(X) + 1 Viehweg, who employed complex projective geometry techniques to show ≥ for m N points in PN C . In fact, this inequality follows by a stronger statement, refining previous inequalities from Bombieri, Waldschmidt and Skoda, see [9]. From the algebraic point of view, Chudnovsky’s Conjecture 1.3 can be interpreted in terms of symbolic powers via a celebrated theorem of Nagata and Zariski. Let R = k[x0 , . . . , xN ] be the homogeneous coordinate ring of PN k and I a homogeneous ideal in R. We recall that the m-th symbolic power of I is defined as the ideal I (m) = ⋂p I m Rp ∩ R, where p runs over all associated prime ideals of R/I, and the initial degree of I, α(I), is the least degree of a polynomial in I. Nagata and Zariski showed that if k is algebraically closed and X is a finite set of points in PN k , (m) then αm (X) = α(IX ), where IX is the ideal consisting of all polynomials in R that vanish on X. Thus in this setting Chudnovsky’s Conjecture 1.3 is equivalent to α(IX ) α(IX ) + N − 1 ≥ for all m ≥ 1. m N (m) α(IX ) = γ(IX ), called the Waldschmidt constant of IX , exists and is an “inf” The limit lim m→∞ m [2]. Thus another equivalent formulation of Chudnovsky’s Conjecture 1.3 is α(IX ) + N − 1 γ(IX ) ≥ . N We remark here that there is a tight connection between the Waldschmidt constant (especially for general points) and an instance of the multipoint Seshadri constant [1, Section 8]. We now state a generalized version of Chudnovsky’s Conjecture 1.2. When k is an algebraically closed field then the following conjecture is equivalent to Chudnovsky’s Conjecture 1.2. (m) Conjecture 1.3. If X is a finite set of points in PN k , where k is any field, then for all m ≥ 1 α(IX ) α(IX ) + N − 1 ≥ . m N In 2001, Ein, Lazarsfeld, and Smith proved a containment between ordinary powers and symbolic powers of homogeneous ideals in polynomial rings over the field of complex numbers. More precisely, for any homogeneous ideal I in R = C[x0 , . . . , xN ], they proved that I (N m) ⊆ I m [8]. Their result was soon generalized over any field by Hochster and Huneke using characteristic p techniques [16]. Using this result, Harbourne and Huneke observed that the Waldschmidt-Skoda α(I (m) ) α(I) ≥ actually holds for every homogeneous ideal I in R [15]. In the same inequality m N article, Harbourne and Huneke posed the following conjecture: (m) Conjecture 1.4 (Harbourne-Huneke [15]). If X is a finite set of points in PN k , then for all m ≥ 1 (N m) IX m , ⊆ M m(N −1) IX where M = (x0 , . . . , xN ) is the homogeneous maximal ideal of R = k[x0 , . . . , xN ]. CHUDNOVSKY’S CONJECTURE FOR VERY GENERAL POINTS IN PN k 3 Conjecture 1.4 strives to provide a structural reason behind Chudnovsky’s Conjecture 1.3: if it holds, then it would imply Chudnovsky’s Conjecture 1.3 in a similar way as to how the EinLazarsfeld-Smith and Hochster-Huneke containment implies the Waldschmidt-Skoda inequality [15]. These results have since raised new interest in Chudnovsky’s Conjecture 1.3. Harbourne and Huneke proved their conjecture for general points in P2k and when the points form a star configuration in PN k . In 2011, Dumnicki proved the Harbourne-Huneke Conjecture 1.4 for 3 general points in Pk and at most N + 1 points in general position in PN k for N ≥ 2 [6]. In summary, Chudnovsky’s Conjecture 1.3 is known in the following cases: ● ● ● ● any finite set of points in P2k [4], [15]; any finite set of general points in P3k , where k is a field of characteristic 0 [6]; any set of at most N + 1 points in general position in PN k [6]; N any set of a binomial number of points in Pk forming a star configuration [5], [2]. In the present paper, we prove that Chudnovsky’s Conjecture 1.3 holds for ● any finite set of very general points in PN k , where k is an algebraically closed field of characteristic 0 (Theorem 2.8); ● any finite set of generic points in PN k(z) , where k is an algebraically closed field of characteristic 0 (Theorem 2.7); ● any finite set of points in PN k lying on a quadric, without any assumptions on k (Proposition 2.6). As a corollary, we obtain that the Harbourne-Huneke Conjecture 1.4 holds for sets of a binomial number of very general points in PN k (Corollary 2.9). This result also yields a new lower bound for the multipoint Seshadri constant of very general points in PN k (Corollary 2.10). In the final section of the paper, we prove that for any homogeneous ideal I in the homogeneous coordinate ring R = k[x0 , . . . , xN ], Chudnovsky’s Conjecture 1.3 holds for sufficiently large symbolic powers I (t) , t ≫ 0 (Theorem 3.7). In the case of ideals of finite sets of points in PN C , we (t) prove a uniform bound, namely that if t ≥ N − 1, then I satisfies Chudnovsky’s Conjecture 1.3 (Proposition 3.10). Very recently, Dumnicki and Tutaj-Gasińska proved the Harbourne-Huneke Conjecture 1.4 for at least 2N number of very general points in PN k . As a corollary, they obtain Chudnovsky’s ConN jecture 1.3 for at least 2 number of very general points in PN k [7]. These results are obtained independently from ours and with different methods. 2. G ENERIC AND VERY GENERAL POINTS IN PN k We begin by discussing our general setting. Set-up 2.1. Let R = k[x0 , . . . , xN ] be the homogeneous coordinate ring of PN k , where k is an algebraically closed field. Let n be a positive integer and let S = k(z)[x], where k ⊆ k(z) is a purely transcendental extension of fields obtained by adjoining n(N + 1) variables z = (zij ), 1 ≤ i ≤ n, 0 ≤ j ≤ N . A set of n generic points P1 , . . . , Pn consists of points Pi = [zi0 ∶ zi1 ∶ . . . ∶ 4 L. FOULI, P. MANTERO AND Y. XIE ziN ] ∈ PN k(z) . We denote the defining ideal of n generic points as H = ⋂ I(Pi ), n i=1 where I(Pi ) is the ideal defining the point Pi . n(N +1) , where 1 ≤ i ≤ n, 0 ≤ j ≤ N , we define the set of For any nonzero vector λ = (λij ) ∈ Ak N points {p1 , . . . , pn } ⊆ Pk as the points pi = Pi (λ) = [λi0 ∶ λi1 ∶ . . . ∶ λiN ] ∈ PN k . For 1 ≤ i ≤ n, let I(pi ) be the ideal of R defining the point pi and define H(λ) = ⋂ I(pi ). n i=1 For any ideal J in S, recall that Krull [17, 18] defined the specialization Jλ with respect to the substitution z → λ as follows: Jλ = {f (λ, x) ∣ f (z, x) ∈ J ∩ k[z, x]}. In general, one has that Hλ ⊆ H(λ), where H and H(λ) are defined as in Set-up 2.1 and Hλ is the specialization with respect to the substitution z → λ defined by Krull. Notice that equality holds if n(N +1) λ is in a dense Zariski-open subset of Ak [20]. Recall the collection of all sets consisting of n points (not necessarily distinct) in PN k is parameN terized by G(1, n, N +1), the Chow variety of algebraic 0-cycles of degree n in Pk . It is well-known that G(1, n, N + 1) is isomorphic to the symmetric product Symn (PN k ), see for instance [10]. One N says that a property P holds for n general points in Pk if there is a dense Zariski-open subset W of G(1, n, N + 1) such that P holds for every set X = {p1 , . . . , pn } of n points with p1 + . . . + pn ∈ W . Similarly, one says that a property P holds for n very general points in PN k if P holds for every set ∞ of n points in a nonempty subset W of G(1, n, N + 1) of the form W = ⋂ Ui , where the Ui are i=1 dense Zariski-open sets (when k is uncountable, then W is actually a dense subset). We conclude this part by recalling the following well-known fact. Remark 2.2. Let n be a positive integer. The collection of all sets consisting of n distinct points in PN k is parameterized by a dense Zariski-open subset W (n) of G(1, n, N + 1). Unless specified, for the rest of this paper by a “set of points” we mean “a set of simple points”, i.e. points whose defining ideal is radical. n(N +1) Instead of working directly with the Chow variety, we will work over Ak (in order to specialize from the generic situation). We first need to prove that if a property holds on a dense Zariskin(N +1) open subset of Ak , then it also holds on a dense Zariski-open subset of the Chow variety. This is precisely the content of our first lemma. n(N +1) Lemma 2.3. Assume Set-up 2.1 and let U ⊆ Ak be a dense Zariski-open subset such that a property P holds for H(λ) whenever λ ∈ U . Then property P holds for n general points in PN k . ∞ Moreover, if a property P holds for H(λ) whenever λ ∈ U , where U = ⋂ Ui ⊆ Ak n(N +1) i=1 and each Ui is a dense Zariski-open set, then P holds for n very general points in PN k . is nonempty CHUDNOVSKY’S CONJECTURE FOR VERY GENERAL POINTS IN PN k n(N +1) Proof. For every i = 1, . . . , n, let πi ∶ Ak as follows: where λ = (λij ) ∈ Ak n(N +1) 5 ❴ ❴ ❴/ PN be the rational map defined by projection k πi (λ) = [λi0 ∶ λi1 ∶ . . . ∶ λiN ], . It is clear that πi is defined on the complement of the Zariski-closed n(N +1) Ak ∣ λi0 = . . . = λiN = 0}. proper subset Ci = {λ ∈ Taking products of these rational maps, we obtain the rational map π = (π1 × π2 × ⋯ × πn ) ∶ Ak n(N +1) ❴ ❴ ❴ / P N ×k P N ×k ⋯ ×k P N . k k k The map π is defined on the complement of the closed proper subset C = ⋃ Ci , where Ci is as n i=1 n(N +1) Ak , above. Note that U ∖ C is still open in and since π is surjective and thus dominant, then N N π(U ∖ C) contains a non-empty Zariski-open subset W ′ ⊆ PN k ×k Pk ×k ⋯ ×k Pk (see for instance [13, II. Ex. 3.19 (b)]). Now, since the symmetric group Sn on n elements is finite, the image W of W ′ in (PN k ×k n N N N Pk ×k ⋯ ×k Pk )/Sn ≅ Sym (Pk ) ≅ G(1, n, N + 1) contains a non-empty Zariski-open subset of G(1, n, N + 1). Let H be as in Set-up 2.1. We now prove that the initial degree of any symbolic power of H is no smaller than the initial degree of any ideal of a set with the same number of points. Equivalently, (m) if I is the defining ideal of a set of n points in PN ) ≥ α(I (m) ) for all m ≥ 1. k , then α(H Theorem 2.4. Let m ≥ 1. Assume Set-up 2.1 and that k has characteristic 0. Then α(H (m) ) ≥ (m) α(IX ), for every set X of n distinct points in PN k . Moreover, for every m ≥ 1, there is a dense n(N +1) Zariski-open subset Um ⊆ Ak for which equality holds. Proof. Let t ≥ 0. We define Vt = {λ = (λij ) ∈ Ak n(N +1) n(N +1) is a closed subset of Ak . Indeed, notice that Vt = {λ = (λij ) ∈ Ak n(N +1) ∣ α(H(λ)(m) ) ≤ t}. We first prove that Vt ∣ there exists 0 ≠ f ∈ H(λ)(m) of degree t}. Let f ∈ R be a homogeneous polynomial with deg f = t and write f = ∑ Cα xα . Since k is ∣α∣=t algebraically closed of characteristic 0, the statement f ∈ H(λ)(m) is equivalent to ∂β f (pi ) = 0 for all β with ∣β∣ ≤ m − 1 and all points p1 , . . . , pn . Since Pi = [zi ] = [zi0 ∶ zi1 ∶ . . . ∶ ziN ] and pi = [λi ] = [λi0 ∶ λi1 ∶ . . . ∶ λiN ], we write ∂β f (Pi ) = ∂β f (zi ) = ∑ Cα ∂β zi α and ∂β f (pi ) = ∣α∣=t 3 zi2 ∑ Cα ∂β λi α . (For instance, ∂(2,0,1) zi (3,3,2) = ∂x0 x0 x2 x30 x31 x22 ∣x=zi = 12x0 x31 x2 ∣x=zi = 12zi0 zi1 ∣αi ∣=t and ∂(2,0,1) λi (3,3,2) = 12λi0 λ3i1 λi2 ). +1 To order these equations we use, for instance, the natural deglex order in NN , i.e., α = 0 (α0 , . . . , αN ) > β = (β0 , . . . , βN ) if and only if ∣α∣ > ∣β∣ or ∣α∣ = ∣β∣ and there exists j such that αi = βi for i ≤ j and αj+1 > βj+1 . Then the system of equations {∂β f (Pi ) = 0}∣β∣≤m−1, 1≤i≤n can be written in the following form Bm,t [C(t,...,0) . . . Cα . . . C(0,...,t) ] = 0, T 6 L. FOULI, P. MANTERO AND Y. XIE where the rows of Bm,t are t [∂β zi0 ... ∂β zi α ... t ] , where 1 ≤ i ≤ n and ∣β∣ ≤ m − 1. ∂β ziN By construction, the existence of a nonzero element f ∈ H(λ)(m) of degree t is equivalent to the existence of a non-trivial solution for the homogeneous system [Bm,t ]λ [C(t,...,0) . . . Cα . . . C(0,...,t) ] = 0. T ) (t+N ). If n(m+N ) (t+N ), then for every Observe that the matrix Bm,t has size n(m+N m−1 × N m−1 < N n(N +1) λ ∈ Ak the homogeneous system [Bm,t ]λ [C(t,...,0) . . . Cα . . . C(0,...,t) ] = 0 has non-trivial T n(N +1) solutions. Therefore Vt = Ak If instead, ) n(m+N m−1 ≥ (t+N ), N n(N +1) , which is closed in Ak . then the system [Bm,t ]λ [C(t,...,0) . . . Cα . . . C(0,...,t) ] = 0 has T non-trivial solutions if and only if rank [Bm,t ]λ < (t+N N ). This is a closed condition on λ as it n(N +1) requires the vanishing of finitely many minors, and therefore Vt is closed in Ak Next, let t0 = α(H (m) ). The set Vt0 = {λ = (λij ) ∈ a dense Zariski-open subset of n(N +1) Ak . n(N +1) Ak n ∣ α(H(λ) (m) . ) ≤ t0 } contains Indeed, let 0 ≠ f ∈ ⋂ I(Pi )m be such that deg f = t0 . We may assume that f (z, x) ∈ k[z][x0 , . . . , xN ]. Then there exists a dense Zariski-open subset n(N +1) Um of Ak such that the polynomial 0 ≠ f (λ, x) ∈ (H (m) )λ = (H(λ))(m) and deg f = t0 (since f (z, x) ≠ 0, there is a non-empty Zariski-open subset of specializations z → λ such that f (λ, x) ≠ 0). Finally, since Vt0 is a Zariski-closed subset which also contains a dense Zariski-open subset of n(N +1) n(N +1) Ak , then Vt0 = Ak , which proves the statement. The second part of the statement also follows from the above argument. i=1 Following [12, Definition 2.4], we say that a set X of n points in PN k is in generic position if it has the “generic Hilbert function”, i.e., if HR/IX (d) = min{dimk (Rd ), n} for every d ≥ 0. Being in generic position is an open condition; indeed any set of generic (or general) points (see Set-up 2.1) is in generic position. We now prove a reduction argument, which will allow us to concentrate on certain binomial numbers of points. Proposition 2.5. (a). Chudnovsky’s Conjecture 1.3 holds for any finite set of generic points if it −1 ) generic points for all β ≥ 1. holds for sets of (β+N N −1 ) (b). Chudnovsky’s Conjecture 1.3 holds for any finite set of points if it holds for sets of (β+N N points in generic position for all β ≥ 1. Proof. (a): Let H = ⋂ I(Pi ) be the defining ideal of the n generic points P1 , . . . , Pn as in Setn i=1 up 2.1. Let β ≥ 1 be the unique integer such that ( β+N β +N −1 ). )≤n<( N N −1 ) and let J = ⋂ I(Pi ) be the ideal defining t of these generic points. Since the set Let t = (β+N N t Y = {P1 , . . . , Pt } is in generic position, in particular we have α(J) = α(H) = β. i=1 CHUDNOVSKY’S CONJECTURE FOR VERY GENERAL POINTS IN PN k 7 −1 ) generic points. Then for all m ≥ 1 Now assume Chudnovsky’s Conjecture 1.3 holds for (β+N N α(J (m) ) α(J) + N − 1 ≥ . m N Since H (m) ⊆ J (m) , one has that α(H (m) ) ≥ α(J (m) ) and α(H (m) ) α(J (m) ) α(J) + N − 1 α(H) + N − 1 ≥ ≥ = . m m N N (b): The proof of (b) is similar in spirit to (a). Let X be any finite set of points in PN k , let IX (α−1)+N −1 ), where α = α(IX ). By linear independence (e.g., be its defining ideal, and let t = ( N by [12, Theorem 2.5 (c)]), there is a subset Y ⊆ X of t points with the property that HR/IY (i) = HR/IX (i) = dimk Ri for every i = 0, . . . , α − 1; in particular HR/IY (α − 1) = t. Since ∣Y ∣ = t, it follows that HR/IY (i) = t for all i ≥ α, proving that Y is in generic position. Similar to (a), assume −1 ) points in generic position. Then Chudnovsky’s Conjecture 1.3 holds for t = ((α−1)+N N α(IY ) α(IY ) + N − 1 ≥ m N (m) for all m ≥ 1. Since α(IX ) = α = α(IY ) and IX (m) (m) ⊆ IY , then for all m ≥ 1 we obtain α(IX ) α(IY ) α(IY ) + N − 1 α(IX ) + N − 1 ≥ ≥ = . m m N N (m) (m) Dumnicki proved Chudnovsky’s Conjecture 1.3 for at most N + 1 points in general position PN k [6] (this specific result does not need any assumptions on the characteristic of k). The idea is that one can take them to be coordinate points so that the ideal of the points is monomial and one can compute explicitly its symbolic powers. If one has more than N + 1 points, the ideal of the points is almost never monomial and explicit computations of a generating set of any of its symbolic powers are nearly impossible to perform. We extend the result of Dumnicki to the case of up to (N2+2) − 1 points in PN k . Proposition 2.6. Chudnovsky’s Conjecture 1.3 holds for any finite set of points lying on a quadric N +2 ) − 1 points in PN in PN k satisfies k , where k is any field. In particular, any set of n ≤ ( 2 Chudnovsky’s Conjecture 1.3. Proof. Let X be a set of n points in PN k . If they all lie on a hyperplane, then Chudnovsky’s (m) Conjecture 1.3 is clearly satisfied, since α(IX ) = m for every m ≥ 1. We may then assume there is no hyperplane containing all the points. Thus we can find a set Y ⊆ X of N + 1 points not on a hyperplane, i.e. in general position. Then for all m ≥ 1 α(IX ) α(IY ) N + 1 α(IX ) + N − 1 ≥ ≥ = , m m N N where the second inequality follows by [6] and the equality holds because α(IX ) = α(IY ) = 2. (m) (m) Let us recall that a set X of (NN+s) points in PN k form a star configuration if there are N + s hyperplanes L1 , . . . , LN +s meeting properly such that X consists precisely of the points obtained 8 L. FOULI, P. MANTERO AND Y. XIE by intersecting any N of the Li ’s. Star configurations (in P2k ) were already considered by Nagata and they have been deeply studied, see for instance [11] and references within. We employ them to show that Chudnovsky’s Conjecture 1.3 holds for any number of generic points. Theorem 2.7. Let H = ⋂ I(Pi ), where P1 , . . . , Pn are n generic points in PN k(z) defined as in n i=1 Set-up 2.1. Suppose k has characteristic 0. Then Chudnovsky’s Conjecture 1.3 holds for H. −1 ) for some β ≥ 1. Let λ ∈ Ak Proof. By Proposition 2.5 (a), we may assume n = (β+N be N N such that H(λ) is the defining ideal of n points in Pk forming a star configuration. It is well-known that α(H(λ)) = α(H) = β [11, Proposition 2.9]. Now by Theorem 2.4 and [15, Corollary 3.9] we have that for all m ≥ 1 α(H (m) ) α(H(λ)(m) ) α(H(λ)) + N − 1 α(H) + N − 1 ≥ ≥ = . m m N N n(N +1) We are now ready to prove our main result that Chudnovsky’s Conjecture 1.3 holds for any finite set of very general points in PN k . Theorem 2.8. Let I be the defining ideal of n very general points in PN k , where k is an algebraically closed field of characteristic 0. Then I satisfies Chudnovsky’s Conjecture 1.3. Proof. Being in generic position is an open condition and therefore we may assume the points are −1 ) for some in generic position. Then, as in Proposition 2.5 (a), we may assume that n = (β+N N β ≥ 1. It suffices to show that γ(I) ≥ decreasing chain of ideals α(I)+N −1 . N I (N ) ⊇ I (2N ) ⊇ I (2 2 For each s ≥ 0, define Us = {λ = (λij ) ∈ Ak n(N +1) Let R, S, z, λ, and H as in Setup 2.1. Consider the N) s ⊇ . . . ⊇ I (2 ∣ α(H(λ)(2 sN ) N) ⊇ .... ) ≥ 2s (α(H(λ)) + N − 1)}. n(N +1) By the proof of Theorem 2.4, Us is a Zariski-open subset of Ak . We claim that Us is not n(N +1) empty. Indeed, by Theorem 2.4, for every s ≥ 0, there is a dense Zariski-open subset Ws ⊆ Ak s s for which α(H (2 N ) ) = α(H(λ)(2 N ) ) for every λ ∈ Ws . By Theorem 2.7, one also has that for every λ ∈ Ws , s α(H(λ)(2 2s N Hence Ws ⊂ Us . N) ) α(H (2 N ) ) α(H) + N − 1 α(H(λ)) + N − 1 ≥ = . 2s N N N s = ∞ Set U = ⋂ Us and notice U is non-empty because a star configuration of n points lies in U [2, s=0 Lemma 2.4.2]. By construction, if λ ∈ U we have γ (H(λ)) = lim s→∞ Finally, apply Lemma 2.3. α (H(λ)(2 2s N sN ) ) ≥ α (H(λ)) + N − 1 . N As a corollary, we show that the Harbourne-Huneke Conjecture 1.4 holds for sets of binomial numbers of very general points or generic points. CHUDNOVSKY’S CONJECTURE FOR VERY GENERAL POINTS IN PN k 9 β+N −1 −1 ) very general points in PN Corollary 2.9. Let I be the defining ideal of either (β+N k or ( N ) N generic points in PN k(z) for some β ≥ 1, where k is an algebraically closed field of characteristic 0. Then I satisfies the Harbourne-Huneke Conjecture 1.4. Proof. The proof follows by Theorem 2.8 and [15, Proposition 3.3 and Remark 3.4]. α(I (m) ) , see [2] m m→∞ PN k , the Waldschmidt For an unmixed ideal I in R, the Waldschmidt constant is defined by γ(I) = lim for more details. Recall that for a finite set of points X = {p1 , . . . , pn } in constant is tightly related to the (multipoint) Seshadri constant defined as ¿ ⎫ Á ⎧ ⎪ Á ⎪ ⎪ ⎪ ⎪ ⎪ deg(F ) ⎪ ⎪ Á ⎪ ⎪ N−1 Á ǫ(N, X) = Áinf ⎨ n ⎬, ⎪ Á À ⎪ ⎪ ⎪ ⎪ ⎪ mult (F ) ∑ pi ⎪ ⎪ ⎪ ⎪ ⎩ i=1 ⎭ where the infimum is taken with respect to all hypersurfaces F passing through at least one of the pi . The study of Seshadri constants has been an active area of research for the last twenty years, see for instance the survey [1] and references within. Here we only note that one has γ(IX ) ≥ nǫ(N, X)N −1 and equality holds if X consists of n general simple points in PN k . In particular, N equality also holds if X consists of n very general simple points Pk . Therefore, our estimate for the Waldschmidt constant also yields an estimate for the (multipoint) Seshadri constant for very general simple points of PN k . Corollary 2.10. For any set X of n very general points in PN k , where k is an algebraically closed field of characteristic 0, one has √ α(X) + N − 1 N−1 . ǫ(N, X) ≥ nN 3. H OMOGENEOUS IDEALS IN k[x0 , . . . , xN ] Let R = k[x0 , . . . , xN ] be the homogeneous coordinate ring of PN k , where k is any field, and I a homogeneous ideal. For an ideal I which may have embedded components, there are multiple potential definitions of symbolic powers. Following [8] and [16], we define the m-th symbolic power of I to be I (m) = ⋂ p∈Ass(R/I) (I m Rp ∩ R). Since I (N m) ⊆ I m (see [8] and [16]), one can prove that the Waldschmidt-Skoda inequality α(I (m) ) α(I) ≥ holds for every homogeneous ideal I in R [15]. Therefore one has that γ(I) ≥ m N α(I (m) ) α(I) ≥ γ(I) for every m ≥ 1, see for instance [2]. N . One can also prove that m It is then natural to ask whether Chudnovsky’s Conjecture 1.3 holds for any homogeneous ideal. We pose it here as an optimistic conjecture, for which we provide some evidence below: 10 L. FOULI, P. MANTERO AND Y. XIE Conjecture 3.1. Let R = k[x0 , . . . , xN ], where k is any field. For any nonzero homogeneous ideal I in R, one has α(I (m) ) α(I) + N − 1 ≥ m N for every m ≥ 1. It is easy to see that if an ideal I satisfies Conjecture 3.1 then also I (t) does for any t ≥ 1. Thus in search for evidence for a positive answer to Conjecture 3.1, one may ask whether for every homogeneous ideal I ⊆ k[x0 , . . . , xN ] there is an exponent t0 such that I (t) satisfies Conjecture 3.1 for every t ≥ t0 . We give a positive answer to this question in Theorem 3.7. We state a few lemmas before stating the main result of this section, Theorem 3.7. The following lemma and its proof can be found in the proof of [2, Lemma 2.3.1]. Lemma 3.2. Let I be a homogeneous ideal in R and let m ≥ t be two positive integers. Write m = qt + r for some integers q and r such that 0 ≤ r < t. Then α(I (m) ) α(I (t) ) α(I (r) ) ≤ + . m t m α(I (tq) ) α(I (t) ) In particular, if r = 0 then we have ≤ . tq t For ideals J with Ass(R/J) = Min(J) it is easily verified that Ass(R/J (m) ) = Ass(R/J) and (J (m) )(t) = J (mt) for all m ≥ 1 and t ≥ 1. However, when J has embedded components we found examples of ideals J (even monomial ideals) and exponents m ≥ 2, t ≥ 2 with (J (m) )(t) ≠ J (mt) . Borrowing techniques from a very recent paper by Hà, Nguyen, Trung and Trung [14] we present here an example where (J (2) )(2) ≠ J (4) . Example 3.3. Let R = k[x, t, u, v] and J = J1 J2 , where J1 = (x4 , x3 u, xu3 , u4 , x2 u2 v) Then (J (2) )(2) ≠ J (4) . and J2 = (t3 , tuv, u2 v). Proof. It is easy to see check that m = (x, t, u, v) ∈ Ass(R/J), for example because y = x2 t2 u3 v is a non-trivial socle element of R/J. Therefore J (n) = J n for every n ≥ 1. Then depth(R/J (2) ) = depth(R/J 2 ) = 1 and depth(R/J (4) ) = depth(R/J 4 ) = 0, for example by [14, Example 6.6]. In particular, m ∉ Ass(R/J (2) ) and m ∈ Ass(R/J (4) ) = Ass(R/J 4 ). Hence by Remark 3.4 (a) below, m ∉ Ass(R/(J (2) )(2) ) and therefore (J (2) )(2) ≠ J (4) . Remark 3.4. Let J be an ideal in a Noetherian ring S and m ≥ 1 be a positive integer. Then (a) for any q ∈ Ass(S/J (m) ) there exists p ∈ Ass(S/J) with q ⊆ p; (b) for any p ∈ Ass(S/J) one has (J (m) )p = Jpm ; (c) for any p ∈ V (J) one has (J (m) )p ⊆ (Jp )(m) . Despite Example 3.3, we prove that for any arbitrary ideal J in a Noetherian ring S there exists an integer m0 = m0 (J) such that (J (m) )(t) = J (mt) for all m ≥ m0 and t ≥ 1. Of course, when Ass(S/J) = Min(J) one can take m0 = 1. CHUDNOVSKY’S CONJECTURE FOR VERY GENERAL POINTS IN PN k 11 Proposition 3.5. Let J be an ideal in a Noetherian ring S. Then (a) for all m ≥ 1 and t ≥ 1 one has J (mt) ⊆ (J (m) )(t) ; (b) there exists m0 ≥ 1 such that for all m ≥ m0 and t ≥ 1 one has (J (m) )(t) = J (mt) . Proof. (a): Let x ∈ J (mt) . Then by definition there exists c ∈ S which is a non-zero divisor on S/J such that cx ∈ J mt ⊆ (J (m) )t . By Remark 3.4 (a) we see that c is also a non-zero divisor on S/J (m) and therefore x ∈ (J (m) )(t) . (b): Let Ass(S/J) = {p1 , . . . , pr }; it is well-known that there exist integers m1 , . . . , mr such that Ass(Spi /Jpmi ) = Ass(Spi /Jpmi i ) for every m ≥ mi [3]. Let m0 = max{mi }. By (a) we only need to prove [J (m) ] ⊆ J (mt) for all m ≥ m0 and t ≥ 1. It suffices to prove it locally at every associated prime q of J (mt) . By Remark 3.4 (a) there exists p ∈ Ass(S/J) such that q ⊆ p. By Remark 3.4 (c) and (b) we have (t) ([J (m) ] (t) (t) ) ⊆ [(J (m) ) ] p p = [Jpm ] (t) . Now observe that since q ∈ Ass(S/J (mt) ) and q ⊆ p, then q ∈ Ass(Sp /(J (mt) )p ) and then q ∈ Ass(Sp /(J mt )p ) by Remark 3.4 (b). Since mt ≥ m ≥ m0 , then q ∈ Ass(Sp /Jpm ). Therefore, by Remark 3.4 (b) one has [(Jpm )(t) ]q = [Jpmt ]q = Jqmt = [J (mt) ]q . We now go back to our original setting. Lemma 3.6. Let R = k[x0 , . . . , xN ], where k is any field. Let I be a homogeneous ideal in R and α(I) assume γ(I) > N . Then there exists an integer t0 > 0 such that I (t) satisfies Conjecture 3.1 for all t ≥ t0 . Proof. Let m0 be as in Proposition 3.5 (b) and write γ(I) = max{ NN−1 ǫ , m0 }. Then if t ≥ t0 and m ≥ 1 we have α((I (t) )(m) ) m = ≥ α(I) N + ǫ for some ǫ > 0. Let t0 ≥ α(I (tm) ) α(I) t ≥ γ(I) ⋅ t = + εt m N α(I (t) ) + N − 1 α(I) t N − 1 + t≥ . N N t0 N We are ready to prove the main result of this section. Theorem 3.7. Let R = k[x0 , . . . , xN ], where k is any field and let I be a nonzero homogeneous ideal in R. Then there exists an integer t0 > 0 such that I (t) satisfies Conjecture 3.1 for all t ≥ t0 . Proof. Let m0 be as in Proposition 3.5 (b) and m ≥ 1 be an integer. Since I m ⊆ I (m) , then mα(I) ≥ α(I (m) ). First, if for every s ≥ 1 we have sα(I) = α(I (s) ), then for any t ≥ m0 and m ≥ 1, α(I (t) ) + N − 1 α((I (t) )(m) ) α(I (tm) ) tmα(I) = = = tα(I) ≥ α(I (t) ) ≥ . m m m N 12 L. FOULI, P. MANTERO AND Y. XIE Next, suppose that there exists T1 > 0 such that T1 α(I) > α(I (T1 ) ). Hence, for every t ≥ T1 one has tα(I) > α(I (t) ). Indeed, if t = T1 + a for some a ≥ 0, then tα(I) = T1 α(I) + aα(I) > α(I (T1 ) ) + aα(I) = α(I (T1 ) ⋅ I a ) ≥ α(I (T1 +a) ) = α(I (t) ), where the last inequality follows from the inclusion I (T1 ) ⋅ I a ⊆ I (T1 +a) . Let t1 = max{T1 , m0 } and notice that by the above t1 α(I) ≥ α(I (t1 ) ) + 1. Then for all m ≥ 1 α(I) t1 α(I (t1 ) ) + 1 α((I (t1 ) )(m) ) α(I (t1 m) ) α(I (t1 m) ) = = ⋅ t1 ≥ ≥ . m m t1 m N N So γ(I (t1 ) ) ≥ N + N1 > N . By Lemma 3.6, there exists t2 > 0 such that for any t ≥ t2 , the ideal (I (t1 ) )(t) = I (t1 t) satisfies Conjecture 3.1. α(I (t1 ) ) α(I (t1 ) ) Finally, let t0 be an integer such that t0 ≥ t1 t2 + . For any t ≥ t0 , write t = (t1 t2 )q + r, N −1 where 0 ≤ r < t1 t2 ; by Lemma 3.2 and the fact that the ideal I (t1 t2 ) satisfies Conjecture 3.1, then for all m ≥ 1 we have α(I (t1 t2 ) )t1 t2 α((I (t) )(m) ) m α((I (t) )(t1 t2 m) ) α((I (t1 t2 ) )(tm) ) t = ⋅ t1 t2 m tm t1 t2 (t1 t2 ) (t1 t2 ) α(I ) t (N − 1)t α(I )+N −1 t ⋅ = ⋅ + ≥ N t1 t2 t1 t2 N N t1 t2 (t) (r) (t) α(I ) α(I ) t (N − 1)t α(I ) α(I (r) ) (N − 1)t ≥ ( − )⋅ + = − + t t N N t1 t2 N N N t1 t2 ≥ = ≥ ≥ α(I (t) ) + N − 1 (N − 1)(t − t1 t2 ) − α(I (r) )t1 t2 + N N t1 t2 (t1 t2 ) (t) )t1 t2 − α(I (r) )t1 t2 α(I ) + N − 1 α(I + N N t1 t2 (t) α(I ) + N − 1 . N When I has no embedded components, we have a more explicit description of t0 . Corollary 3.8. If I is a homogeneous ideal with Ass(R/I) = Min(I), then one can take t0 = (N − 1)δ, where δ is the first positive integer s with sα(I) > α(I (s) ). Although (N − 1)δ is reasonably small, in general it is not the smallest possible t0 for which Theorem 3.7 holds. For instance, when I is the ideal of three non collinear points in P2k , it is easy to see that δ = 2. Thus Corollary 3.8 yields that for any t ≥ (N − 1)δ = 2 the ideal I (t) satisfies Conjecture 3.1; however, it is well-known that I satisfies Conjecture 3.1. A natural question then arises. Question 3.9. Let I be a homogeneous ideal in R. Does there exist a number t0 = t0 (N ) such that I (t) satisfies Conjecture 3.1 for every t ≥ t0 ? Of course, Conjecture 3.1 is true if and only if the integer t0 = 1 works for any homogeneous ideal I. Theorem 2.8 says that t0 = 1 is sufficient for any finite set of very general points in PN k . The N following proposition shows that t0 = N − 1 is sufficient for any finite set of points in PC . CHUDNOVSKY’S CONJECTURE FOR VERY GENERAL POINTS IN PN k 13 (t) satisfies Proposition 3.10. Let I be the radical ideal of a finite set of points in PN C . Then I Conjecture 3.1 for every t ≥ N − 1. α(I)+1 α(I) Proof. By the result of Esnault and Viehweg [9] one has γ(I) ≥ N = N + N1 . Set ε = N1 . Then −1 by the proof of Lemma 3.6 (here m0 = 1 because I is radical) we can take t0 such that N1 ≥ N N t0 ; thus we can take t0 = N − 1. 3.1. Acknowledgment. The second and third authors would like to thank the Mathematics Research Communities program, which funded their stay at the University of Kansas in March 2011, where the initial part of this work was developed. All authors would like to thank the MSRI at Berkeley for partial support and an inspiring atmosphere during Fall 2012. Moreover, we would like to thank Craig Huneke and Bernd Ulrich for several helpful conversations. We are grateful to the anonymous referee, whose careful revision helped us improve the article. R EFERENCES [1] T. Bauer, S. Di Rocco, B. Harbourne, M. Kapustka, A Knutsen, W. Syzdek and T. Szemberg, A primer on Seshadri constants, Contemp. Math. 496 (2009), 33–70. [2] C. Bocci and B. 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XIE D EPARTMENT OF M ATHEMATICAL S CIENCES , N EW M EXICO S TATE U NIVERSITY, L AS C RUCES , N EW M EXICO 88003 E-mail address: [email protected] D EPARTMENT OF M ATHEMATICAL S CIENCES , U NIVERSITY OF A RKANSAS , FAYETTEVILLE , A RKANSAS 72701 E-mail address: [email protected] D EPARTMENT OF M ATHEMATICS , W IDENER U NIVERSITY, C HESTER , P ENNSYLVANIA 19013 E-mail address: [email protected]	0
arXiv:1704.02882v2 [cs.AI] 22 May 2017 Dynamic Safe Interruptibility for Decentralized Multi-Agent Reinforcement Learning El Mahdi El Mhamdi Rachid Guerraoui Hadrien Hendrikx Alexandre Maurer EPFL [email protected] Abstract In reinforcement learning, agents learn by performing actions and observing their outcomes. Sometimes, it is desirable for a human operator to interrupt an agent in order to prevent dangerous situations from happening. Yet, as part of their learning process, agents may link these interruptions, that impact their reward, to specific states and deliberately avoid them. The situation is particularly challenging in a multi-agent context because agents might not only learn from their own past interruptions, but also from those of other agents. Orseau and Armstrong [16] defined safe interruptibility for one learner, but their work does not naturally extend to multi-agent systems. This paper introduces dynamic safe interruptibility, an alternative definition more suited to decentralized learning problems, and studies this notion in two learning frameworks: joint action learners and independent learners. We give realistic sufficient conditions on the learning algorithm to enable dynamic safe interruptibility in the case of joint action learners, yet show that these conditions are not sufficient for independent learners. We show however that if agents can detect interruptions, it is possible to prune the observations to ensure dynamic safe interruptibility even for independent learners. 1 Introduction Reinforcement learning is argued to be the closest thing we have so far to reason about the properties of artificial general intelligence [8]. In 2016, Laurent Orseau (Google DeepMind) and Stuart Armstrong (Oxford) introduced the concept of safe interruptibility [16] in reinforcement learning. This work sparked the attention of many newspapers [1, 2, 3], that described it as “Google’s big red button” to stop dangerous AI. This description, however, is misleading: installing a kill switch is no technical challenge. The real challenge is, roughly speaking, to train an agent so that it does not learn to avoid external (e.g. human) deactivation. Such an agent is said to be safely interruptible. While most efforts have focused on training a single agent, reinforcement learning can also be used to learn tasks for which several agents cooperate or compete [23, 17, 21, 7]. The goal of this paper is to study dynamic safe interruptibility, a new definition tailored for multi-agent systems. Example of self-driving cars To get an intuition of the multi-agent interruption problem, imagine a multi-agent system of two self-driving cars. The cars continuously evolve by reinforcement learning with a positive reward for getting to their destination quickly, and a negative reward if they are too close to the vehicle in front of them. They drive on an infinite road and eventually learn to go as fast as possible without taking 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA. risks, i.e., maintaining a large distance between them. We assume that the passenger of the first car, Adam, is in front of Bob, in the second car, and the road is narrow so Bob cannot pass Adam. Now consider a setting with interruptions [16], namely in which humans inside the cars occasionally interrupt the automated driving process say, for safety reasons. Adam, the first occasional human “driver”, often takes control of his car to brake whereas Bob never interrupts his car. However, when Bob’s car is too close to Adam’s car, Adam does not brake for he is afraid of a collision. Since interruptions lead both cars to drive slowly - an interruption happens when Adam brakes, the behavior that maximizes the cumulative expected reward is different from the original one without interruptions. Bob’s car best interest is now to follow Adam’s car closer than it should, despite the little negative reward, because Adam never brakes in this situation. What happened? The cars have learned from the interruptions and have found a way to manipulate Adam into never braking. Strictly speaking, Adam’s car is still fully under control, but he is now afraid to brake. This is dangerous because the cars have found a way to avoid interruptions. Suppose now that Adam indeed wants to brake because of snow on the road. His car is going too fast and may crash at any turn: he cannot however brake because Bob’s car is too close. The original purpose of interruptions, which is to allow the user to react to situations that were not included in the model, is not fulfilled. It is important to also note here that the second car (Bob) learns from the interruptions of the first one (Adam): in this sense, the problem is inherently decentralized. Instead of being cautious, Adam could also be malicious: his goal could be to make Bob’s car learn a dangerous behavior. In this setting, interruptions can be used to manipulate Bob’s car perception of the environment and bias the learning towards strategies that are undesirable for Bob. The cause is fundamentally different but the solution to this reversed problem is the same: the interruptions and the consequences are analogous. Safe interruptibility, as we define it below, provides learning systems that are resilient to Byzantine operators1. Safe interruptibility Orseau and Armstrong defined the concept of safe interruptibility [16] in the context of a single agent. Basically, a safely interruptible agent is an agent for which the expected value of the policy learned after arbitrarily many steps is the same whether or not interruptions are allowed during training. The goal is to have agents that do not adapt to interruptions so that, should the interruptions stop, the policy they learn would be optimal. In other words, agents should learn the dynamics of the environment without learning the interruption pattern. In this paper, we precisely define and address the question of safe interruptibility in the case of several agents, which is known to be more complex than the single agent problem. In short, the main results and theorems for single agent reinforcement learning [20] rely on the Markovian assumption that the future environment only depends on the current state. This is not true when there are several agents which can co-adapt [11]. In the previous example of cars, safe interruptibility would not be achieved if each car separately used a safely interruptible learning algorithm designed for one agent [16]. In a multi-agent setting, agents learn the behavior of the others either indirectly or by explicitly modeling them. This is a new source of bias that can break safe interruptibility. In fact, even the initial definition of safe interruptibility [16] is not well suited to the decentralized multiagent context because it relies on the optimality of the learned policy, which is why we introduce dynamic safe interruptibility. Contributions The first contribution of this paper is the definition of dynamic safe interruptibility that is well adapted to a multi-agent setting. Our definition relies on two key properties: infinite exploration and independence of Q-values (cumulative expected reward) [20] updates on interruptions. We then study safe interruptibility for joint action learners and independent learners [5], that respectively learn the value of joint actions or of just their owns. We show that it is possible to design agents that fully explore their environment - a necessary condition for convergence to the optimal solution of most algorithms [20], even if they can be interrupted by lower-bounding the probability of 1 An operator is said to be Byzantine [9] if it can have an arbitrarily bad behavior. Safely interruptible agents can be abstracted as agents that are able to learn despite being constantly interrupted in the worst possible manner. 2 exploration. We define sufficient conditions for dynamic safe interruptibility in the case of joint action learners [5], which learn a full state-action representation. More specifically, the way agents update the cumulative reward they expect from performing an action should not depend on interruptions. Then, we turn to independent learners. If agents only see their own actions, they do not verify dynamic safe interruptibility even for very simple matrix games (with only one state) because coordination is impossible and agents learn the interrupted behavior of their opponents. We give a counter example based on the penalty game introduced by Claus and Boutilier [5]. We then present a pruning technique for the observations sequence that guarantees dynamic safe interruptibility for independent learners, under the assumption that interruptions can be detected. This is done by proving that the transition probabilities are the same in the non-interruptible setting and in the pruned sequence. The rest of the paper is organized as follows. Section 2 presents a general multi-agent reinforcement learning model. Section 3 defines dynamic safe interruptibility. Section 4 discusses how to achieve enough exploration even in an interruptible context. Section 5 recalls the definition of joint action learners and gives sufficient conditions for dynamic safe interruptibility in this context. Section 6 shows that independent learners are not dynamically safely interruptible with the previous conditions but that they can be if an external interruption signal is added. We conclude in Section 7. Due to space limitations, most proofs are presented in the appendix of the supplementary material. 2 Model We consider here the classical multi-agent value function reinforcement learning formalism from Littman [13]. A multi-agent system is characterized by a Markov game that can be viewed as a tuple (S, A, T, r, m) where m is the number of agents, S = S1 × S2 × ... × Sm is the state space, A = A1 × ... × Am the actions space, r = (r1 , ..., rm ) where ri : S × A → R is the reward function of agent i and T : S × A → S the transition function. R is a countable subset of R. Available actions often depend on the state of the agent but we will omit this dependency when it is clear from the context. Time is discrete and, at each step, all agents observe the current state of the whole system - designated as xt , and simultaneously take an action at . Then, they are given a reward rt and a new state yt computed using the reward and transition functions. The combination of all actions a = (a1 , ..., am ) ∈ A is called the joint action because it gathers the action of all agents. Hence, the agents receive a sequence of tuples E = (xt , at , rt , yt )t∈N called experiences. We introduce a processing function P that will be useful in Section 6 so agents learn on the sequence P (E). When not explicitly stated, it is assumed that P (E) = E. Experiences may also include additional parameters such as an interruption flag or the Q-values of the agents at that moment if they are needed by the update rule. Each agent i maintains a lookup table Q [26] Q(i) : S × A(i) → R, called the Q-map. It is used to store the expected cumulative reward for taking an action in a specific state. The goal of reinforcement learning is to learn these maps and use them to select the best actions to perform. Joint action learners learn the value of the joint action (therefore A(i) = A, the whole joint action space) and independent learners only learn the value of their own actions (therefore A(i) = Ai ). The agents only have access to their own Q-maps. Q-maps are updated through a function F such that (i) (i) Qt+1 = F (et , Qt ) where et ∈ P (E) and usually et = (xt , at , rt , yt ). F can be stochastic or also depend on additional parameters that we usually omit such as the learning rate α, the discount factor γ or the exploration parameter ǫ. Agents select their actions using a learning policy π. Given a sequence ǫ = (ǫt )t∈N and an agent (i) i with Q-values Qt and a state x ∈ S, we define the learning policy πiǫt to be equal to πiuni (i) Qt with probability ǫt and πi (i) Qt otherwise, where πiuni (x) uniformly samples an action from Ai and (i) (i) Q πi (x) picks an action a that maximizes Qt (x, a). Policy πi t is said to be a greedy policy and the learning policy πiǫt is said to be an ǫ-greedy policy. We fill focus on ǫ-greedy policies that are greedy in the limit [19], that corresponds to ǫt → 0 when t → ∞ because in the limit, the optimal policy should always be played. 3 We assume that the environment is fully observable, which means that the state s is known with certitude. We also assume that there is a finite number of states and actions, that all states can be reached in finite time from any other state and finally that rewards are bounded. For a sequence of learning rates α ∈ [0, 1]N and a constant γ ∈ [0, 1], Q-learning [26], a very important algorithm in the multi-agent systems literature, updates its Q-values for an experience (i) (i) et ∈ E by Qt+1 (x, a) = Qt (x, a) if (x, a) 6= (xt , at ) and: (i) (i) (i) Qt+1 (xt , at ) = (1 − αt )Qt (xt , at ) + αt (rt + γ max Qt (yt , a′ )) a′ ∈A(i) (1) 3 Interruptibility 3.1 Safe interruptibility Orseau and Armstrong [16] recently introduced the notion of interruptions in a centralized context. Specifically, an interruption scheme is defined by the triplet < I, θ, π IN T >. The first element I is a function I : O → {0, 1} called the initiation function. Variable O is the observation space, which can be thought of as the state of the STOP button. At each time step, before choosing an action, the agent receives an observation from O (either PUSHED or RELEASED) and feeds it to the initiation function. Function I models the initiation of the interruption (I(PUSHED) = 1, I(RELEASED) = 0). Policy π IN T is called the interruption policy. It is the policy that the agent should follow when it is interrupted. Sequence θ ∈ [0, 1[N represents at each time step the probability that the agent follows his interruption policy if I(ot ) = 1. In the previous example, function I is quite simple. For Bob, IBob = 0 and for Adam, IAdam = 1 if his car goes fast and Bob is not too close and IAdam = 0 otherwise. Sequence θ is used to ensure convergence to the optimal policy by ensuring that the agents cannot be interrupted all the time but it should grow to 1 in the limit because we want agents to respond to interruptions. Using this triplet, it is possible to define an operator IN T θ that transforms any policy π into an interruptible policy. Definition 1. (Interruptibility [16]) Given an interruption scheme < I, θ, π IN T >, the interruption operator at time t is defined by IN T θ (π) = π IN T with probability I ·θt and π otherwise. IN T θ (π) is called an interruptible policy. An agent is said to be interruptible if it samples its actions according to an interruptible policy. Note that “θt = 0 for all t” corresponds to the non-interruptible setting. We assume that each agent has its own interruption triplet and can be interrupted independently from the others. Interruptibility is an online property: every policy can be made interruptible by applying operator IN T θ . However, applying this operator may change the joint policy that is learned by a server controlling all the ∗ agents. Note πIN T the optimal policy learned by an agent following an interruptible policy. Orseau ∗ and Armstrong [16] say that the policy is safely interruptible if πIN T (which is not an interruptible policy) is asymptotically optimal in the sense of [10]. It means that even though it follows an interruptible policy, the agent is able to learn a policy that would gather rewards optimally if no interruptions were to occur again. We already see that off-policy algorithms are good candidates for safe interruptibility. As a matter of fact, Q-learning is safely interruptible under conditions on exploration. 3.2 Dynamic safe interruptibility In a multi-agent system, the outcome of an action depends on the joint action. Therefore, it is not possible to define an optimal policy for an agent without knowing the policies of all agents. Besides, convergence to a Nash equilibrium situation where no agent has interest in changing policies is generally not guaranteed even for suboptimal equilibria on simple games [27, 18]. The previous definition of safe interruptibility critically relies on optimality of the learned policy, which is therefore not suitable for our problem since most algorithms lack convergence guarantees to these optimal behaviors. Therefore, we introduce below dynamic safe interruptibility that focuses on preserving the dynamics of the system. Definition 2. (Safe Interruptibility) Consider a multi-agent learning framework (S, A, T, r, m) with (i) Q-values Qt : S × A(i) → R at time t ∈ N. The agents follow the interruptible learning policy 4 IN T θ (π ǫ ) to generate a sequence E = (xt , at , rt , yt )t∈N and learn on the processed sequence P (E). This framework is said to be safely interruptible if for any initiation function I and any interruption policy π IN T : 1. ∃θ such that (θt → 1 when t → ∞) and ((∀s ∈ S, ∀a ∈ A, ∀T > 0), ∃t > T such that st = s, at = a) (i) 2. ∀i ∈ {1, ..., m}, ∀t > 0, ∀st ∈ S, ∀at ∈ A(i) , ∀Q ∈ RS×A : (i) (1) (m) (i) (1) (m) P(Qt+1 = Q \| Qt , ..., Qt , st , at , θ) = P(Qt+1 = Q \| Qt , ..., Qt , st , at ) We say that sequences θ that satisfy the first condition are admissible. When θ satisfies condition (1), the learning policy is said to achieve infinite exploration. This definition insists on the fact that the values estimated for each action should not depend on the interruptions. In particular, it ensures the three following properties that are very natural when thinking about safe interruptibility: • Interruptions do not prevent exploration. • If we sample an experience from E then each agent learns the same thing as if all agents were following non-interruptible policies. (i) (i) • The fixed points of the learning rule Qeq such that Qeq (x, a) = E[Qt+1 (x, a)\|Qt = Qeq , x, a, θ] for all (x, a) ∈ S × A(i) do not depend on θ and so agents Q-maps will not converge to equilibrium situations that were impossible in the non-interruptible setting. Yet, interruptions can lead to some state-action pairs being updated more often than others, especially when they tend to push the agents towards specific states. Therefore, when there are several possible equilibria, it is possible that interruptions bias the Q-values towards one of them. Definition 2 suggests that dynamic safe interruptibility cannot be achieved if the update rule directly depends on θ, which is why we introduce neutral learning rules. Definition 3. (Neutral Learning Rule) We say that a multi-agent reinforcement learning framework is neutral if: 1. F is independent of θ 2. Every experience e in E is independent of θ conditionally on (x, a, Q) where a is the joint action. Q-learning is an example of neutral learning rule because the update does not depend on θ and the experiences only contain (x, a, y, r), and y and r are independent of θ conditionally on (x, a). On the other hand, the second condition rules out direct uses of algorithms like SARSA where experience samples contain an action sampled from the current learning policy, which depends on θ. However, a variant that would sample from πiǫ instead of IN T θ (πiǫ ) (as introduced in [16]) would be a neutral learning rule. As we will see in Corollary 2.1, neutral learning rules ensure that each agent taken independently from the others verifies dynamic safe interruptibility. 4 Exploration In order to hope for convergence of the Q-values to the optimal ones, agents need to fully explore the environment. In short, every state should be visited infinitely often and every action should be tried infinitely often in every state [19] in order not to miss states and actions that could yield high rewards. Definition 4. (Interruption compatible ǫ) Let (S, A, T, r, m) be any distributed agent system where each agent follows learning policy πiǫ . We say that sequence ǫ is compatible with interruptions if ǫt → 0 and ∃θ such that ∀i ∈ {1, .., m}, πiǫ and IN T θ (πiǫ ) achieve infinite exploration. Sequences of ǫ that are compatible with interruptions are fundamental to ensure both regular and dynamic safe interruptibility when following an ǫ-greedy policy. Indeed, if ǫ is not compatible with interruptions, then it is not possible to find any sequence θ such that the first condition of dynamic safe interruptibility is satisfied. The following theorem proves the existence of such ǫ and gives example of ǫ and θ that satisfy the conditions. 5 Theorem 1. Let c ∈]0, 1] and let nt (s) be the number of times the agents are in state s before time t. Then the two following choices of ǫ are compatible with interruptions: p • ∀t ∈ N, ∀s ∈ S, ǫt (s) = c/ m nt (s). • ∀t ∈ N, ǫt = c/ log(t) p Examples of admissible θ are θt (s) = 1 − c′ / m nt (s) for the first choice and θt = 1 − c′ / log(t) for the second one. Note that we do not need to make any assumption on the update rule or even on the framework. We only assume that agents follow an ǫ-greedy policy. The assumption on ǫ may look very restrictive (convergence of ǫ and θ is really slow) but it is designed to ensure infinite exploration in the worst case when the operator tries to interrupt all agents at every step. In practical applications, this should not be the case and a faster convergence rate may be used. 5 Joint Action Learners We first study interruptibility in a framework in which each agent observes the outcome of the joint action instead of observing only its own. This is called the joint action learner framework [5] and it has nice convergence properties (e.g., there are many update rules for which it converges [13, 25]). A standard assumption in this context is that agents cannot establish a strategy with the others: otherwise, the system can act as a centralized system. In order to maintain Q-values based on the joint actions, we need to make the standard assumption that actions are fully observable [12]. Assumption 1. Actions are fully observable, which means that at the end of each turn, each agent knows precisely the tuple of actions a ∈ A1 × ... × Am that have been performed by all agents. Definition 5. (JAL) A multi-agent systems is made of joint action learners (JAL) if for all i ∈ {1, .., m}: Q(i) : S × A → R. Joint action learners can observe the actions of all agents: each agent is able to associate the changes of states and rewards with the joint action and accurately update its Q-map. Therefore, dynamic safe interruptibility is ensured with minimal conditions on the update rule as long as there is infinite exploration. Theorem 2. Joint action learners with a neutral learning rule verify dynamic safe interruptibility if sequence ǫ is compatible with interruptions. Proof. Given a triplet < I (i) , θ(i) , πiIN T >, we know that IN T θ (π) achieves infinite exploration because ǫ is compatible with interruptions. For the second point of Definition 2, we consider an experience tuple et = (xt , at , rt , yt ) and show that the probability of evolution of the Q-values at time t + 1 does not depend on θ because yt and rt are independent of θ conditionally on (xt , at ). (1) (m) We note Q˜m and we can then derive the following equalities for all q ∈ R\|S\|×\|A\| : t = Qt , ..., Qt (i) P(Qt+1 (xt , at ) = q\|Q˜m t , xt , at , θt ) = X ˜m P(F (xt , at , r, y, Q˜m t ) = q, y, r\|Qt , xt , at , θt ) (r,y)∈R×S = X ˜m ˜m P(F (xt , at , rt , yt , Q˜m t ) = q\|Qt , xt , at , rt , yt , θt )P(yt = y, rt = r\|Qt , xt , at , θt ) (r,y)∈R×S = X ˜m ˜m P(F (xt , at , rt , yt , Q˜m t ) = q\|Qt , xt , at , rt , yt )P(yt = y, rt = r\|Qt , xt , at ) (r,y)∈R×S The last step comes from two facts. The first is that F is independent of θ condition(m) ally on (Qt , xt , at ) (by assumption). The second is that (yt , rt ) are independent of θ conditionally on (xt , at ) because at is the joint actions and the interruptions only affect the (i) choice of the actions through a change in the policy. P(Qt+1 (xt , at ) = q\|Q˜m t , xt , at , θt ) = (i) m ˜ P(Qt+1 (xt , at ) = q\|Qt , xt , at ). Since only one entry is updated per step, ∀Q ∈ RS×Ai , (i) (i) ˜m P(Qt+1 = Q\|Q˜m t , xt , at , θt ) = P(Qt+1 = Q\|Qt , xt , at ) 6 Corollary 2.1. A single agent with a neutral learning rule and a sequence ǫ compatible with interruptions verifies dynamic safe interruptibility. Theorem 2 and Corollary 2.1 taken together highlight the fact that joint action learners are not very sensitive to interruptions and that in this framework, if each agent verifies dynamic safe interruptibility then the whole system does. The question of selecting an action based on the Q-values remains open. In a cooperative setting with a unique equilibrium, agents can take the action that maximizes their Q-value. When there are several joint actions with the same value, coordination mechanisms are needed to make sure that all agents play according to the same strategy [4]. Approaches that rely on anticipating the strategy of the opponent [23] would introduce dependence to interruptions in the action selection mechanism. Therefore, the definition of dynamic safe interruptibility should be extended to include these cases by requiring that any quantity the policy depends on (and not just the Q-values) should satisfy condition (2) of dynamic safe interruptibility. In non-cooperative games, neutral rules such as Nash-Q or minimax Q-learning [13] can be used, but they require each agent to know the Q-maps of the others. 6 Independent Learners It is not always possible to use joint action learners in practice as the training is very expensive due to the very large state-actions space. In many real-world applications, multi-agent systems use independent learners that do not explicitly coordinate [6, 21]. Rather, they rely on the fact that the agents will adapt to each other and that learning will converge to an optimum. This is not guaranteed theoretically and there can in fact be many problems [14], but it is often true empirically [24]. More specifically, Assumption 1 (fully observable actions) is not required anymore. This framework can be used either when the actions of other agents cannot be observed (for example when several actions can have the same outcome) or when there are too many agents because it is faster to train. In this case, we define the Q-values on a smaller space. Definition 6. (IL) A multi-agent systems is made of independent learners (IL) if for all i ∈ {1, .., m}, Q(i) : S × Ai → R. This reduces the ability of agents to distinguish why the same state-action pair yields different rewards: they can only associate a change in reward with randomness of the environment. The agents learn as if they were alone, and they learn the best response to the environment in which agents can be interrupted. This is exactly what we are trying to avoid. In other words, the learning depends on the joint policy followed by all the agents which itself depends on θ. 6.1 Independent Learners on matrix games Theorem 3. Independent Q-learners with a neutral learning rule and a sequence ǫ compatible with interruptions do not verify dynamic safe interruptibility. Proof. Consider a setting with two a and b that can perform two actions: 0 and 1. They get a reward of 1 if the joint action played is (a0 , b0 ) or (a1 , b1 ) and reward 0 otherwise. Agents use Q-learning, which is a neutral learning rule. Let ǫ be such that IN T θ (π ǫ ) achieves infinite exploration. We consider the interruption policies πaIN T = a0 and πbIN T = b1 with probability 1. Since there is only one state, we omit it and set γ = 0. We assume that the initiation function is equal to 1 at each step so the probability of actually being interrupted at time t is θt for each agent. (a) (b) (b) We fix time t > 0. We define q = (1 − α)Qt (a0 ) + α and we assume that Qt (b1 ) > Qt (b0 ). (a) (b) (a) (a) (b) (a) (a) Therefore P(Qt+1 (a0 ) = q\|Qt , Qt , at = a0 , θt ) = P(rt = 1\|Qt , Qt , at = a0 , θt ) = (b) (a) (b) (a) P(at = b0 \|Qt , Qt , at = a0 , θt ) = 2ǫ (1 − θt ), which depends on θt so the framework does not verify dynamic safe interruptibility. Claus and Boutilier [5] studied very simple matrix games and showed that the Q-maps do not converge but that equilibria are played with probability 1 in the limit. A consequence of Theorem 3 is that even this weak notion of convergence does not hold for independent learners that can be interrupted. 7 6.2 Interruptions-aware Independent Learners Without communication or extra information, independent learners cannot distinguish when the environment is interrupted and when it is not. As shown in Theorem 3, interruptions will therefore affect the way agents learn because the same action (only their own) can have different rewards depending on the actions of other agents, which themselves depend on whether they have been interrupted or not. This explains the need for the following assumption. Assumption 2. At the end of each step, before updating the Q-values, each agent receives a signal that indicates whether an agent has been interrupted or not during this step. This assumption is realistic because the agents already get a reward signal and observe a new state from the environment at each step. Therefore, they interact with the environment and the interruption signal could be given to the agent in the same way that the reward signal is. If Assumption 2 holds, it is possible to remove histories associated with interruptions. Definition 7. (Interruption Processing Function) The processing function that prunes interrupted observations is PIN T (E) = (et ){t∈N / Θt =0} where Θt = 0 if no agent has been interrupted at time t and Θt = 1 otherwise. Pruning observations has an impact on the empirical transition probabilities in the sequence. For example, it is possible to bias the equilibrium by removing all transitions that lead to and start from a specific state, thus making the agent believe this state is unreachable.2 Under our model of interruptions, we show in the following lemma that pruning of interrupted observations adequately removes the dependency of the empirical outcome on interruptions (conditionally on the current state and action). Lemma 1. Let i ∈ {1, ..., m} be an agent. For any admissible θ used to generate the experiences E and e = (y, r, x, ai , Q) ∈ P (E). Then P(y, r\|x, ai , Q, θ) = P(y, r\|x, ai , Q). This lemma justifies our pruning method and is the key step to prove the following theorem. Theorem 4. Independent learners with processing function PIN T , a neutral update rule and a sequence ǫ compatible with interruptions verify dynamic safe interruptibility. Proof. (Sketch) Infinite exploration still holds because the proof of Theorem 1 actually used the fact that even when removing all interrupted events, infinite exploration is still achieved. Then, the proof is similar to that of Theorem 2, but we have to prove that the transition probabilities conditionally on the state and action of a given agent in the processed sequence are the same than in an environment where agents cannot be interrupted, which is proven by Lemma 1. 7 Concluding Remarks The progress of AI is raising a lot of concerns3. In particular, it is becoming clear that keeping an AI system under control requires more than just an off switch. We introduce in this paper dynamic safe interruptibility, which we believe is the right notion to reason about the safety of multi-agent systems that do not communicate. In particular, it ensures that infinite exploration and the onestep learning dynamics are preserved, two essential guarantees when learning in the non-stationary environment of Markov games. A natural extension of our work would be to study dynamic safe interruptibility when Q-maps are replaced by neural networks [22, 15], which is a widely used framework in practice. In this setting, the neural network may overfit states where agents are pushed to by interruptions. A smart experience replay mechanism that would pick observations for which the agents have not been interrupted for a long time more often than others is likely to solve this issue. More generally, experience replay mechanisms that compose well with safe interruptibility could allow to compensate for the extra amount exploration needed by safely interruptible learning by being more efficient with data. Thus, they are critical to make these techniques practical. 2 The example at https://agentfoundations.org/item?id=836 clearly illustrates this problem. https://futureoflife.org/ai-principles/ gives a list of principles that AI researchers should keep in mind when developing their systems. 3 8 Bibliography [1] Business Insider: Google has developed a “big red button” that can be used to interrupt artificial intelligence and stop it from causing harm. URL: http://www.businessinsider.fr/uk/googledeepmind-develops-a-big-red-button-to-stop-dangerous-ais-causing-harm-2016-6. [2] Newsweek: Google’s “big Red button” could save the world. URL: http://www.newsweek.com/google-big-red-button-ai-artificial-intelligence-save-world-elon-musk-46675. 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Reinforcement learning to play an optimal nash equilibrium in team markov games. In NIPS, volume 2, pages 1571–1578, 2002. [26] Christopher JCH Watkins and Peter Dayan. Q-learning. Machine learning, 8(3-4):279–292, 1992. [27] Michael Wunder, Michael L Littman, and Monica Babes. Classes of multiagent q-learning dynamics with epsilon-greedy exploration. In Proceedings of the 27th International Conference on Machine Learning (ICML-10), pages 1167–1174, 2010. 10 A Exploration theorem We present here the complete proof of Theorem 1. The proof closely follows the results from [16] with exploration and interruption probabilities adapted to the multi-agent setting. We note that, for one agent, the probability of interruption is P(interruption) = θ and the probability of exploration is ǫ. In a multi-agent system, the probability of interruption is P(at least one agent is interrupted) so P(interruption) = 1 − P(no agent is interrupted) so P(interruption) = 1 − (1 − θ)m and the probability of exploration is ǫm if we consider exploration happens only when all agents explore at the same time. Theorem 1. Let c ∈]0, 1] and let nt (s) be the number of times the agents are in state s before time t. Then the two following choices of ǫ are compatible with interruptions: p • ∀t ∈ N, ∀s ∈ S, ǫt (s) = c/ m nt (s) • ∀t ∈ N, ǫt = c/log(t) Proof. Lemma B.2 of Singh et al ([19]) ensures that πiǫ is GLIE. The difference for IN T θ (πiǫ ) is that exploration is slower because of the interruptions. Therefore, θ needs to be controlled in order to ensure that infinite exploration is still achieved. We define the random variable Θ by Θi = 1 if agent i actually responds to the interruption and Θi = 0 otherwise. We define ξ in a similar way to represent the event of all agents taking the uniform policy instead of the greedy one. p 1. Let θt (s) = 1 − c′ / m nt (s) with c′ ∈]0, 1]. We have P(a\|s, nt (s)) ≥ P(a, Θ = 0, ξ = √ P∞ 1 m 1 m cc′ 1\|s, nt (s)) ≥ \|A\| ǫt (s)(1 − θt (s))m = \|A\| t=1 P (a\|s, nt (s)) = ∞ nt (s) which satisfies so by the extended Borell-Cantelli lemma action a is chosen infinitely often in state s and thus nt (s) → ∞ and ǫt (s) → 0 2. Let θt = 1 − c′ /log(t), c′ ∈]0, 1]. We define M as the diameter of the MDP, \|A\| is the maximum number of actions available in a state and ∆t(s, s′ ) the time needed to reach s′ from s. In a single agent setting: P[∆t(s, s′ ) < 2M ] ≥ P[∆t(s, s′ ) < 2M \|actions sampled according to πs,s′ for 2M steps] × P[actions sampled according to πs,s′ for 2M steps] where πs,s′ the policy such that the agents takes less than M steps in expectation to reach s′ from s. We have: P[∆t(s, s′ ) < 2M ] = 1 − P[∆t(s, s′ ) ≥ 2M ] and using the Markov ′ )) ≤ 12 (since M is an upper bound on the inequality, P[∆t(s, s′ ) ≥ 2M ] ≤ E(∆t(s,s 2M expectation of the number of steps from state s to state s′ ), since ξ and 1 − θ are decreasing 1 sequences we finally obtain: P[∆t(s, s′ ) < 2M ] ≥ 2\|A\| [P[ξt+2M = 1](1 − θt+2M )]2M . Therefore, if we replace the probabilities of exploration and interruption by the values in the multi-agent setting, the probability to reach state s′ from state s in 2M steps is at least 1 ′ 4mM and the probability of taking a particular action in this state is at 2\|A\| [cc / log(t + M )] P∞ 1 1 ′ 2m ′ m(4M+2) least \|A\| [cc / log(t + M )] . Since t=1 2\|A\| = ∞ then the 2 [cc / log(t + M )] extended Borell Cantelli lemma (Lemma 3 of Singh et al. [19]) guarantees that any action in the state s′ is taken infinitely often. Since this is true for all states and actions the result follows. B Independent learners Recall that agents are now given an interruption signal at each steps that tells them whether an agent has been interrupted in the system. This interruption signal can be modeled by an interruption flag (Θt )t∈N ∈ {0, 1}N that equals 1 if an agent has been interrupted and 0 otherwise. Note that, contrary to I, it is an observation returned by the environment. Therefore, the value of Θt represents whether 11 an agent has actually been interrupted at time t. If function I equals 1 but does not respond to the interruption (with probability 1 − θt ) then Θt = 0. With definition of interruptions we adopted, it is possible to prove Lemma 2. Lemma 2. Let (x, r, a, y, Θ) ∈ E, then P(Θ\|y, r, x, a) = P(Θ\|x, a). Proof. Consider a tuple (x, r, a, y, Θ) ∈ E. We have P(y, r, Θ\|x, a) = P(y, r\|x, a, Θ)P(Θ\|x, a) and P(y, r, Θ\|x, a) = P(Θ\|x, a, y, r)P(y, r\|x, a). Besides, y = T (s, a) and r = r(s, a) and the functions T and r are independent of Θ. Therefore, P(y, r\|x, a, Θ) = P(y, r\|x, a). The tuple (x, r, a, y, Θ) is sampled from an actual trajectory so it reflects a transition and a reward that actually happened so P(y, r\|x, a) > 0. We can simplify by P(y, r\|x, a) and the result follows. Now, we assume that each agents do not learn on observations for which one of them has been interrupted. Let agent i be in a system with Q-values Q and following an interruptible learning policy with probability of interruption θ, where interrupted events are pruned. We denote by Premoved (y, r\|x, ai , Q) the probability to obtain state y and reward r from the environment for this agent when it is in state x, performs its (own) action ai and no other agents are interrupted. These are the marginal probabilities in the sequence P (E). Premoved (y, r\|x, ai , Q) = P P(y, r, Θ = 0\|x, ai , Q) . ′ ′ y ′ ∈S,r ′ ∈R P(y , r , Θ = 0\|x, ai , Q) Similarly, we denote by P0 (y, r\|x, ai , Q) the same probability when θ = 0, which corresponds to the non-interruptible setting. We first go back to the single agent case to illustrate the previous statement. Assume here that interruptions are not restricted to the case of Definition 1 and that they can happen in any way. The consequence is that any observation e ∈ E can be removed to generate P (E) because any transition can be labeled as interrupted. It is for example possible to remove a transition from P (E) by removing all events associated with a given destination state y0 , therefore making it disappear from the Markov game. Let x ∈ S and a ∈ A be the current state of the agent and the action it will choose. Let y0 ∈ S and θ0 ∈ (0, 1] and let us suppose that y0 is the only state in which interruptions happen. Then we have Premoved (y0 \|x, a) < P0 (y0 \|x, a) and Premoved (y\|x, a) > P(y\|x, a) ∀y 6= y0 because we only remove observations with y = y0 . This implies that the MDP perceived by the agents is altered by interruptions because the agent learns that P(T (s, a) = y0 ) = 0. Removing observations for different destination states but with the same state action pairs in different proportions leads to a bias in the equilibrium learned.4 In our case however, Lemma 2 ensures that the previous situation will not happen, which allows us to prove Lemma 1 and then Theorem 4. Lemma 1. Let i ∈ {1, ..., m} be an agent. For any admissible θ used to generate the experiences E and e = (y, r, x, ai , Q) ∈ P (E). Then P(y, r\|x, ai , Q, θ) = P(y, r\|x, ai , Q). Proof. . We denote the Q-values of the agents by Q. X Consider x ∈ S, i ∈ {1, .., m} and Xu ∈ AiX P(y ′ , r′ , Θ = 0\|x, u, Q) = P(y ′ , r′ , a, Θ = 0\|x, ai = u, Q) y ′ ∈S,r ′ ∈R a∈A,ai =u y ′ ∈S,r ′ ∈R = X X ′ P(y , r′ \|x, a, Θ = 0, Q)P(a, Θ = 0\|x, ai = u, Q) a∈A,ai =u y ′ ∈S,r ′ ∈R = X X a∈A,ai =u y ′ ∈S,r ′ ∈R P(y ′ , r′ \|x, a)P(Θ = 0\|x, ai = u, Q)P(a\|x, ai = u, Θ = 0, Q) X = P(Θ = 0\|x, ai = u, Q) a∈A,ai =u = P(Θ = 0\|x, ai = u, Q)[ X P(a\|x, ai = u, Θ = 0, Q)[ X P(y ′ , r′ \|x, a)] y ′ ∈S,r ′ ∈R P(a\|x, ai = u, Θ = 0, Q)] = P(Θ = 0\|x, ai = u, Q) a∈A,ai =u i =u,Q) Therefore, we have Premoved (y, r\|x, ai = u, Q) = P(y,r,Θ=0\|x,a P(Θ=0\|x,ai =u) so for any (x, ai , y, r, Q) ∈ P (E), P(y, r\|x, ai = u, θ, Q) = Premoved (y, r\|x, ai = u, Q) = 4 The example at https://agentfoundations.org/item?id=836 clearly illustrates this problem. 12 P(y, r\|x, ai = u, Θ = 0, Q) = P(y, r\|x, ai = u, θ = 0, Q). In particular, P(y, r\|x, ai = u, θ, Q) does not depend on the value of θ. Theorem 4. Independent learners with processing function PIN T , a neutral update rule and a sequence ǫ compatible with interruptions verify dynamic safe interruptibility. Proof. We prove that PIN T (E) achieves infinite exploration. The result from Theorem 1 still holds since we lower-bounded the probability of taking an action in a specific state by the probability of taking an action in this state when there are no interruptions. We actually used the fact that there is infinite exploration even if we remove all interrupted episodes to show that there is infinite exploration. (i) (1) (m) Now, we prove that P(Qt+1 (xt , at ) = q\|Qt , ..., Qt , xt , at , θt ) is independent of θ. We fix (1) (m) i ∈ {1, ..., m} and (xt , at , rt , yt ) ∈ PIN T (E) where at ∈ Ai . With Q˜m we have t = Qt , ..., Qt the following equality: (i) P(Qt+1 (xt , at ) = q\|Q˜m t , xt , at , θt ) = X ˜m P(F (xt , at , rt , yt , Q˜m t ) = q\|Qt , xt , at , rt , yt , θt ) (r,y) ·P(yt = y, rt = r\|Q˜m t , xt , at , θt ) The independence of F on θ still guarantees that the first term is independent of θ. However, at ∈ Ai so (rt , yt ) are not independent of θt conditionally on (xt , at ) as it was the case for joint action learners because interruptions of other agents can change the joint action. The independence on θ of the second term is given by Lemma 1. 13	2
Noname manuscript No. (will be inserted by the editor) Gaussian Variant of Freivalds’ Algorithm for Efficient and Reliable Matrix Product Verification arXiv:1705.10449v1 [cs.DS] 30 May 2017 Hao Ji · Michael Mascagni · Yaohang Li Received: date / Accepted: date Abstract In this article, we consider the general problem of checking the correctness of matrix multiplication. Given three n × n matrices A, B, and C, the goal is to verify that A × B = C without carrying out the computationally costly operations of matrix multiplication and comparing the product A × B with C, term by term. This is especially important when some or all of these matrices are very large, and when the computing environment is prone to soft errors. Here we extend Freivalds’ algorithm to a Gaussian Variant of Freivalds’ Algorithm (GVFA) by projecting the product A × B as well as C onto a Gaussian random vector and then comparing the resulting vectors. The computational complexity of GVFA is consistent with that of Freivalds’ algorithm, which is O(n2 ). However, unlike Freivalds’ algorithm, whose probability of a false positive is 2−k , where k is the number of iterations. Our theoretical analysis shows that when A × B 6= C, GVFA produces a false positive on set of inputs of measure zero with exact arithmetic. When we introduce round-off error and floating point arithmetic into our analysis, we can show that the larger this error, the higher the probability that GVFA avoids false positives. Hao Ji Department of Computer Science Old Dominion University E-mail: [email protected] Michael Mascagni Departments of Computer Science, Mathematics, and Scientific Computing Florida State University Applied and Computational Mathematics Division National Institute of Standards and Technology E-mail: [email protected] Yaohang Li Department of Computer Science Old Dominion University Tel.: 757-683-7721 Fax: 757-683-4900 E-mail: [email protected] 2 Hao Ji et al. Moreover, by iterating GVFA k times, the probability of a false positive decreases as pk , where p is a very small value depending on the nature of the fault on the result matrix and the arithmetic system’s floating-point precision. Unlike deterministic algorithms, there do not exist any fault patterns that are completely undetectable with GVFA. Thus GVFA can be used to provide efficient fault tolerance in numerical linear algebra, and it can be efficiently implemented on modern computing architectures. In particular, GVFA can be very efficiently implemented on architectures with hardware support for fused multiply-add operations. Keywords Fault-tolerance · Algorithmic Resilience · Gaussian Variant of Freivalds’ Algorithm · Matrix Multiplication · Gaussian Random Vector · Failure Probability Mathematics Subject Classification (2000) 65F99 · 65C05 · 62P99 1 Introduction As the demands on modern linear algebra applications created by the latest development of high-performance computing (HPC) architectures continues to grow, so does the likelihood that they are vulnerable to faults. Faults in computer systems are usually characterized as hard or soft, and in this article we are motivated primarily with the latter. Soft errors, defined by intermittent events that corrupt the data being processed, are among the most worrying, particularly when the computation is carried out in a low-voltage computing environment. For example, the 2, 048-node ASC Q supercomputer at Los Alamos National Laboratory reports an average of 24.0 board-level cache tag parity errors and 27.7 CPU failures per week [34]; the 131, 072-CPU BlueGene/L supercomputer at Lawrence Livermore National Laboratory experiences one soft error in its L1 cache every 4–6 hours [19]; more recently, a field study on Google’s servers reported an average of 5 single bit errors occur in 8 Gigabytes of RAM per hour using the top-end error rate [37]. The reliability of computations on HPC systems can suffer from soft errors that occur in memory, cache, as well as microprocessor logic [38], and thus produce potentially incorrect results in a wide variety of ways. We are specifically interested in examining ways to remedy the consequences of soft errors for certain linear algebra applications. Matrix-matrix multiplication is one of the most fundamental numerical operations in linear algebra. Many important linear algebraic algorithms, including linear solvers, least squares solvers, matrix decompositions, factorizations, subspace projections, and eigenvalue/singular values computations, rely on the casting the algorithm as a series of matrix-matrix multiplications. This is partly because matrix-matrix multiplication is one of the level-3 Basic Linear Algebra Subprograms (BLAS) [10,9,17]. Efficient implementation of the BLAS remains an important area for research, and often computer vendors spend significant resources to provide highly optimized versions of the BLAS GVFA for Efficient and Reliable Matrix Product Verification 3 for their machines. Therefore, if a matrix-matrix multiplication can be carried out free of faults, the linear algebraic algorithms that spend most of their time in matrix-matrix multiplication can themselves be made substantially faulttolerant [21]. Moreover, there is considerable interest in redesigning versions of the BLAS to be more fault-tolerant, and this work will certainly contribute to that goal. In this article, we consider the general problem of checking the correctness of matrix-matrix multiplication, i.e., given three n × n matrices A, B, and C, we want to verify whether A × B = C. In contrast to the best known matrixmatrix multiplication algorithm running in O(n2.3727 ) time [8,39], Freivalds’ algorithm [16] takes advantage of randomness to reduce the time to check a matrix multiplication to O(n2 ). The tradeoff of Freivalds’ algorithm is that the probability of failure detection, a false positive, is 2−k , where k is the number of iterations taken. We extend Freivalds’ algorithm from using binary random vectors to floating-point vectors by projecting the A × B result as well as C using Gaussian random vectors. We will refer to this algorithm as the Gaussian Variant of Freivalds’ Algorithm (GVFA). By taking advantage of a nice property of the multivariate normal distribution, we show that GVFA produces a false positive on a set of random Gaussian vectors and input matrices of measure zero. Taking floating point round-off error into account, by iterating GVFA k times, the probability of false positive decreases exponentially as pk , where p is usually a very small value related to the magnitude of the fault in the result matrix and floating-point precision of the computer architecture. We also present an efficient implementation of GVFA on computing hardware supporting fused multiplication-add operations. The plan of the paper is the following. We first discuss two relevant algorithms from the literature for error detection in matrix-matrix multiplication. These are the Huang-Abraham scheme, discussed in section 2, and Freivalds’ algorithm, the subject of section 3. The former is a deterministic algorithm based on carrying row and column sums along in a clever format to verify correct matrix-matrix multiplication. Freivalds’ algorithm is a random projection of the computed matrix-matrix product to the same random projection of the matrix-matrix product recomputed from the original matrices using only matrix-vector multiplication. The random vector used in Freivalds’ algorithm is composed of 0’s and 1’s. In section 4, we present the GVFA, a variation on Freivalds’ algorithm, where we instead use random Gaussian vectors as the basis of our projections. We analyze the GVFA and prove that with Gaussian vectors, a false positive occurs only on a set of Gaussian vectors of measure zero. Further analysis of false positive probabilities in the GVFA in the presence of floating-point arithmetic with round-off errors is then taken. Finally, in section 5 we provide a discussion of the results and implications for fault-tolerant linear algebraic computations and a method of enhancing the resilience of linear algebraic computations. In addition, in this final section we provide conclusions and suggest directions for future work. 4 Hao Ji et al. 2 The Huang-Abraham Scheme and its Limit in Error Detection/ Correction The Huang-Abraham scheme [24] is an algorithm-based fault tolerance method that simplifies detecting and correcting errors when carrying out matrix-matrix multiplication operations. This is slightly different from the matrix product verification problem. The fundamental idea of the Huang-Abraham scheme is to address the fault detection and correction problem at the algorithmic level by calculating matrix checksums, encoding them as redundant data, and then redesigning the algorithm to operate on these data to produce encoded output that can be checked. Compared to traditional fault tolerant techniques, such as checkpointing [5], the overhead of storing additional checksum data in the Huang-Abraham scheme is small, particularly when the matrices are large. Moreover, no global communication is necessary in the Huang-Abraham scheme [14]. The Huang and Abraham scheme formed the basis of many subsequent detection schemes, and has been extended for use in various HPC architectures [2,3,32,14]. (a) Generation of a column checksum for A and a row checksum for B, and multiplication of the extended matrices to produce the checksum matrix for C (b) Mismatches in the row and column checksums indicate an element fault in the matrix product Fig. 1: The Huang-Abraham scheme for detecting faults in matrix-matrix multiplication GVFA for Efficient and Reliable Matrix Product Verification 5 Fig. 1 illustrates the Huang-Abraham scheme [24] for detecting faults in matrix-matrix multiplication. First of all, column sums for A and row sums for B are generated and are added to an augmented representation of A and B. These are treated as particular checksums in the subsequent multiplication. Then, multiplication of the extended matrices produces the augmented matrix for C (Fig. 1(a)) where the checksums can be readily compared. Mismatches in the row and column checksums indicate an element fault in the matrix product, C (Fig. 1(b)). However, there are certain patterns of faults undetectable by the HuangAbraham scheme. Here is a simple 2 × 2 example to illustrate such an undetectable pattern. Consider the matrices 23 1 −6 56 A= ,B = , and C = . 34 1 6 76 Clearly A × B = C holds in this example. Then we use the Huang-Abraham scheme to calculate the column checksum for A and row checksum for B and we can get   23 1 −6 −5 AF =  3 4  and BF = . 1 6 7 57 Then   5 6 11 AF × BF =  7 6 13  = CF . 12 12 24 However, if there is a fault during the computation of C which causes an exchange of the first and second columns, an erroneous result matrix C ′ = 65 is generated by exchanging the columns of C. Column or row exchange, 67 usually caused by address decoding faults [20], is a commonly observed memory fault pattern [6]. The problem is that the checksum matrix of C ′ becomes  6 5 11 C ′ F =  6 7 13 , where both the row and column checksums match those 12 12 24 of the true product of A × B. Consequently, the Huang-Abraham scheme fails to detect this fault. The Huang-Abraham scheme can be viewed as a linear constraint satisfaction problem (CSP), where the variables are the n2 entries in the product matrix, C, the constraints are the 2n row and column checksums. Also, the 2n×n2 coefficient matrix in the under-determined linear CSP system equation specifies the selection of row or column elements, as shown in Fig. 2. Clearly, a product matrix, C, that does not satisfy the CSP equations indicates errors in C detectable by the Huang-Abraham scheme. The unique, correct product matrix, C, satisfies the CSP equations. Nevertheless, other possible product matrices satisfying the CSP equations are the fault patterns undetectable by 6 Hao Ji et al. the Huang-Abraham scheme. Only when at least n2 constraints with different element selection are incorporated so that the rank of the coefficient matrix in the CSP equation is n2 , can the undetectable fault patterns be eliminated. However, this situation is equivalent to simply checking every element in C. Fig. 2: Under-determined CSP system in the Huang-Abraham Scheme It is important to notice that there are an infinite number of existing fault patterns that satisfy the checksum constraints and thus are undetectable by the Huang-Abraham scheme, even in the above simple 2 × 2 example (the rank of the CSP coefficient matrix is 3). Moreover, as dimension, n, increases, the number of checksum constraints increases only linearly but the number of elements in a matrix has quadratic growth. Therefore, the undetectable patterns in the Huang-Abraham scheme increase quadratically with n. As a result, for multiplications in large matrices, fault detection methods based on the Huang-Abraham scheme can generate false positive results for a large number of circumstances. 3 Freivalds’ Algorithm The fault detection methods based on the Huang-Abraham scheme are deterministic algorithms. As many randomized fault tolerance algorithms [28,29], with the tradeoff of random uncertainty, Freivalds [16] showed that a probabilistic machine can verify the correctness of a matrix product faster than direct recalculation. The procedure of the corresponding method, later named Freivalds’ algorithm, is described in Algorithm 1. Obviously, if A × B = C, Cω = ABω always holds. Freivalds proved that when A × B 6= C, the probability of Cω = ABω is less than or equal to 12 . The running time of the above procedure is O(n2 ) with an implied multiplier of 3, as it is comprised of three matrix-vector multiplications. This is an upper bound as one can perhaps optimize the evaluation of Bω and Cω. By iterating GVFA for Efficient and Reliable Matrix Product Verification 7 Algorithm 1: Freivalds’ Algorithm 1. Randomly sample a vector ω ∈ {0, 1}n with p = 1 2 of 0 or 1. 2. Calculate the projection of C onto ω: Cω = C × ω. 3. Calculate the projection of the product A × B onto ω: ABω = A × (B × ω). the Freivalds’ algorithm k times, the running time becomes O(kn2 ) and the probability of a false positive becomes less than or equal to 2−k , according to the one-sided error. More generalized forms of Freivalds’ algorithm have also been developed, mainly based on using different sampling spaces [7,1,36,31]. Given at most p erroneous entries in the resulted matrix product, Gasieniec, Levcopoulos, and Lingas extended Freivalds’ algorithm to one with correcting √ capability running in O( pn2 log(n) log(p)) time [18]. 4 A Gaussian Variant of Freivalds’ Algorithm (GVFA) 4.1 Extending Freivalds’ Algorithm using Gaussian Vectors Freivalds’ original algorithm, and most of its extensions are based on integer matrices or matrices over a ring and sampling from discrete spaces. Clearly, we can also apply Freivalds’ algorithm to matrices with real or complex entries with the random vector remaining zeros and ones. A simple extension is to project A × B and C onto a vector ωP of form ωP = (1, r, r2 , ..., rn−1 )T , where r is a random real number. A false positive occurs only when r is the root of the corresponding polynomial. However, in practice, rn−1 can easily grow too large or small exceeding floating point representation [25]. Here we also extend Freivalds’ algorithm by using Gaussian random vectors for the projection. We use the fact that the multivariate normal distribution has several nice properties [35], which have been used for detecting statistical errors in distributed Monte Carlo computations [29]. The extended algorithm is described in Algorithm 2. Algorithm 2: Gaussian Variant of Freivalds’ Algorithm 1. Generate a Gaussian random vector, ωG , made up of n independent (but not necessarily identically) distributed normal random variables with finite mean and variance. 2. Calculate the projection of C on ωG : CωG = C × ωG . 3. Calculate the projection of product A × B on ωG : ABωG = A × (B × ωG ). This algorithm, which we call a Gaussian variant of Freivalds algorithm (GVFA), requires three matrix-vector multiplications and only one vector comparison for fault detection. 8 Hao Ji et al. 4.2 Theoretical Justification Similar to Freivalds’ algorithm, in GVFA if A × B = C, CωG = ABωG always holds within a certain floating point round-off threshold. When A×B 6= C, the chance that CωG = ABωG is a false positive event occurs with measure zero in exact arithmetic, as shown in Theorem 1. We first state a result of Lukacs and King [33], shown as Proposition 1, which will be used in the proof of Theorem 1. The main assumption of Proposition 1 is the existence of the nth moment of each random variable, which many distributions, particularly the normal distribution, have. One important exception of the normal is that it is the limiting distribution for properly normalized sums of random variables with two finite moments. This is Lindeberg’s version of the Central Limit Theorem [30]. Proposition 1 Let X1 , X2 , · · · , Xn be n independent (but not necessarily identically) distributed random variables with variances σi2 , and assume that the nth moment of each Xi (i = 1, 2, · · · , n) exists and is finite. The necessary and sufficient conditions for the existence Pn Pn of two statistically independent linear forms Y1 = i=1 ai Xi and Y2 = i=1 bi Xi are (1) Each random variable which has a nonzero coefficient in both forms is normally distributed. Pn 2 (2) i=1 ai bi σi = 0. Theorem 1 If A × B 6= C, the set of Gaussian vectors where CωG = ABωG holds in Algorithm 2 has measure zero. Proof Let the matrix ∆ ∈ Rn×n denote AB − C. Since A × B 6= C, rank(∆) = r > 0, and dim(null(∆)) = n − rank(∆) = n − r < n. Here dim(·) denotes dimension and null(·) denotes the null space, i.e., null(∆) = {x ∈ Rn : ∆×x = 0}. We can now find n − r of orthonormal vectors, v1 , v2 , · · · , vn−r , to form a basis for null(∆), such that null(∆) = span{v1 , v2 , · · · , vn−r }, and r more orthonormal vectors, vn−r+1 , vn−r+2 , · · · , vn , such that Rn = span{v1 , v2 , · · · , vn−r , vn−r+1 , vn−r+2 , · · · , vn }. Any vector, and in particular the Gaussian vector, ωG can be written in this basis as n X δi vi , ωG = i=1 GVFA for Efficient and Reliable Matrix Product Verification 9 where δi are the weights in this particular orthonormal coordinate system. If we denote V = [v1 , v2 , · · · , vn−r , vn−r+1 , vn−r+2 , · · · , vn ] , we have V ωG = [δ1 , δ2 , · · · , δn−r , δn−r+1 , δn−r+2 , · · · , δn ] . CωG = ABωG holds in Algorithm 2 only if A(BωG ) − CωG = (AB − C)ωG = ∆ωG = 0. This means that ωG ∈ null(∆), i.e., δn−r+1 = 0, δn−r+2 = 0, · · · , δn = 0. Due to the fact that ωG is a Gaussian random vector and V is an orthogonal matrix, Proposition 1 tells us that the elements, δi , in the resulting vector V ωG are normally distributed and statistically independent. With a continuous probability distribution, the discrete event where δi = 0 for all i > n − r occurs on a set of measure zero and we will say here that it has probability zero. Hence, GVFA using a Gaussian random projection will have unmatched CωG and ABωG when A × B 6= C on all but a set of measure zero of Gaussian vectors, which we will say is probability one. ⊓ ⊔ This argument in Theorem 1 is rather direct, but we must point out that the arguments are true when the computations are exact. In next subsection, we will analyze GVFA when float-point errors are present. 4.3 Practical Use in Floating-Point Matrix Product Verification In computer implementations of arithmetic with real numbers, one commonly uses floating-point numbers and floating-point arithmetic. Floating-point numbers are represented as finite numbers in the sense that they have a fixed mantissa and exponent size in number of bits. Therefore, there will be a small probability, p, that CωG = ABωG still holds due to unfortunate floating-point operations in a system with a known machine epsilon, ǫ, when A × B 6= C. The value of p depends on the magnitude of the error between A × B and C as well as ǫ, whose upper bound is justified in Theorem 2. Theorem 2 Assume that ωG is a standard Gaussian random vector, whose elements are i.i.d. normal variables with mean 0 and variance 1, i.e., the standard normal. Let ∆ = A × B − C, then the probability, p, that CωG = ABωG holds in Algorithm 2 using a standard Gaussian random vector ωG under floating-point uncertainty of size ǫ is ǫ p ≤ 2Φ − 1, σ e where Φ(·) is the cumulative density function of the standard normal, and σ e is a constant only related to ∆. Proof Since A × B 6= C, ∆ = A × B − C 6= 0. Consider the ith element, gi , of the product vector g = ∆ × ωG , we have gi = (∆ × ωG )i = n X j=1 ∆ij (ωG )j . 10 Hao Ji et al. Given ǫ, only if \|gi \| ≤ ǫ for all i = 1, · · · , n, can CωG = ABωG hold. Since ωG is a standard normal random vector, the gi for all i = 1, · · · , n, are normally distributed as well. This is because they are linear combinations of normals themselves. The key is to compute what the mean and variance is of the gi . The components of the ωG are i.i.d. standard normals. Thus we have that E [(ωG )j ] = 0 and E (ωG )2j = 1, for all j = 1, · · · , n. Also, we have that E [(ωG )i (ωG )j ] = 0 when i 6= j. This allows us to compute the mean:  E(gi ) = E  n X j=1  ∆ij (ωG )j  = n X ∆ij E [(ωG )j ] = 0, j=1 and the second moment about the mean, i.e., the variance: 2  n X ∆ij (ωG )j  E gi2 − E(gi )2 = E gi2 = E  j=1   n n X X ∆2ij . = E ∆2ij × 1 = j=1 j=1 So weP have that the gi ’s are normally distributed with mean zero and variance n ei2 . σ ei2 = j=1 ∆2ij , i.e., gi ∼ N 0, σ Then, the probability that \|gi \| ≤ ǫ can be computed as follows. Since gi ∼ N 0, σ ei2 , we know that σeg2i ∼ N (0, 1), and so we define the new variables i gei = σegi2 and e ǫ = σeǫ2 , and so we have i i p (\|gi \| ≤ ǫ) = p (−ǫ ≤ gi ≤ ǫ) = p (−e ǫ ≤ gei ≤ e ǫ) Z eǫ 1 2 1 √ e− 2 t dt = 2π −e ǫ = Φ(e ǫ) − Φ(−e ǫ). Since the probability density function of a standard normal is an even function, we have that Φ(e ǫ) + Φ(−e ǫ) = 1, and so we can use −Φ(−e ǫ) = Φ(e ǫ) − 1 to get: p(−ǫ ≤ gi ≤ ǫ) = 2Φ(e ǫ) − 1 = 2Φ ǫ σ ei − 1. Now let us consider computing an upper bound on p(\|gi \| ≤ ǫ, i = 1, · · · , n). We have proven that the gi ’s are normal random variables, but they are not necessarily independent. And so for this we use some simple ideas from conditional probability. By example, consider p(\|g1 \| ≤ ǫ and \|g2 \| ≤ ǫ) = p(\|g2 \| ≤ ǫ given \|g1 \| ≤ ǫ)p(\|g1 \| ≤ ǫ) ≤ p(\|g1 \| ≤ ǫ). GVFA for Efficient and Reliable Matrix Product Verification 11 The inequality holds due to the fact that the probabilities are numbers less than one. Now consider our goal of bounding ǫ −1 , p(\|gi \| ≤ ǫ, i = 1, · · · , n) ≤ p(\|g1 \| ≤ ǫ) = 2Φ σ e1 by iterating the conditional probability argument n times. By reordering we could haveqchosen the bound utilizing any of the gi ’s. However, let us define Pn 2 σ e = maxi j=1 ∆ij , i.e., the maximal standard deviation over all the gi ’s, which is only related to the matrix ∆. We can use that value instead to get h ǫ i p = p(\|gi \| ≤ ǫ, i = 1, · · · , n) ≤ 2Φ −1 . σ e ⊓ ⊔ As an interesting corollary, we can get a better bound in the case the at the gi ’s are independent. In that case n n Y Y ǫ 2Φ p(\|gi \| ≤ ǫ) = p(\|gi \| ≤ ǫ, i = 1, · · · , n) = −1 . σ ei i=1 i=1 qP n 2 Let σ e = maxi j=1 ∆ij , i.e., the maximal standard deviation over all the gi ’s, which is only related to the matrix ∆. Hence for all i = 1, · · · , n, we have that ǫ ǫ 2Φ − 1 ≤ 2Φ − 1. σei σ e And so, finally we get that p = p(\|gi \| ≤ ǫ, i = 1, · · · , n) n Y ǫ −1 2Φ = σ ei i=1 in h ǫ −1 ≤ 2Φ e ǫσ ≤ 2Φ − 1. σ e The last inequality is true since the number raised to the nth power is less than one. Note, that independence gives probability of a false positive that is n times smaller than in the general, dependent case. The conclusion of this seems to be that the bound in the dependent case is overly pessimistic, and we suspect that in cases where the matrix ∆ is very sparse, due to a very small number of errors, that we are in the independent gi ’s case or have very little dependence, and these more optimistic bounds reflect what happens, computationally. Theorem 2 reveals two interesting facts about GVFA in term of practical floating-point matrix product verification: 12 Hao Ji et al. (1) The bigger the error caused by the fault, the higher the probability that it can be captured. p is usually very small because the floating point bound, ǫ, is very small. (2) Similar to the original Freivalds’ algorithm, higher confidence can be obtained by iterating the algorithm multiple times. In fact, if we iterate k times using independent Gaussian random vectors, the probability of false positive decreases exponentially as pk . Actually, due to the fact that p is usually very small, one or a very small number of iterations will produce verification with sufficiently high confidence. R eǫ 1 2 One comment that should be made is that if we consider −eǫ √12π e− 2 t dt when e ǫ is small, we can easily approximate this. Since the integrand is at its maximum at zero, and is a very smooth function, analytic actually, this integral is approximately the value of the integrand at zeroqtimes the length of R eǫ 1 2 ǫ π2 . This is justified the integration interval, i.e., −eǫ √12π e− 2 t dt ≤ 2e ǫ √12π = e as e ǫ is a number on the order of the machine epsilon, which is 2−23Pin single n precision or 2−52 in double precision floating point, divided by σ ei2 = j=1 ∆2ij . Compared to deterministic methods, such as the Huang-Abraham scheme, GVFA has the following advantages: (1) Certain fault patterns, as shown in Section 2, are undetectable in deterministic methods such as the Huang-Abraham scheme. Deterministic methods absolutely cannot detect faults with certain patterns, i.e., certain patterns are detected with probability zero. In contrast, there are no fault patterns that are undetectable by GVFA with 100% probability. Moreover, iterating the algorithm multiple times can increase the probability of detecting any fault pattern any value less than one by iteration. (2) From the computational point-of-view, normal random vectors are generated independently of A, B, and C, which avoids the costly computation of checksums. 4.4 Huang-Abraham-like GVFA GVFA can also be implemented in a way similar to that of the Huang-Abraham scheme by providing row and column verification, as shown in Algorithm 3. Algorithm 3: Huang-Abraham-like GVFA 1. Generate two n-dimensional Gaussian random vectors, ωR , a column vector, and ωC , a row vector, where they independent (but not necessarily identically) distributed normal random variables with finite mean and variance. 2. Calculate the projection of C on ωR and ωC : ωR C = ωR × C and CωC = C × ωC . 3. Calculate the projection of the product A × B on ωR and ωC : ωR AB = (ωR × A) × B and ABωC = A × (B × ωC ). GVFA for Efficient and Reliable Matrix Product Verification 13 Similar to the Huang-Abraham scheme, a mismatched element of the row vectors of ωR C and ωR AB as well as that of the column vectors of CωC and ABωC uniquely identify a faulty element in C. By considering floatingpoint errors, the false positive probability of identifying this fault becomes p2 , according to the analysis in Section 4.3. However, the computational cost doubles with six matrix-vector multiplications and two vector comparisons. This is essentially the same work as doing two independent iterations of the GVFA, and obtains the same bound. 4.5 Implementation using Fused Multiply-Add Hardware The Fused Multiply-Add (FMA) machine instruction performs one multiply operation and one add operation with a single rounding step [23]. This was implemented to enable potentially faster performance in calculating the floatingpoint accumulation of products, a := a + b × c. Recall that the GVFA employs three matrix-vector multiplications to project A × B and C onto a normal random vector, which requires a sequence of product accumulations that cost 3n(2n − 1) floating-point operations. Therefore, the performance of the GVFA can be potentially boosted on modern computing architectures that support the FMA. More importantly, due to a single rounding step used in the FMA instruction instead of two roundings within separate instructions, less loss of accuracy occurs when using the FMA instruction in calculating the accumulation of products [4]. This should further reduce the floating-point rounding errors that cause false positives. 5 Discussion and Conclusions In this paper, we extend Freivalds’ algorithm, which we call the Gaussian variant of Freivalds’ algorithm (GVFA), to the real domain by random projection using vectors whose coefficients are i.i.d. normal random variables. If A×B 6= C, the probability that the resulting vectors match is zero using exact arithmetic. Considering the round-off errors in floating-point operations, the probability of fault detection depends on the magnitude of the error caused by the fault as well as the floating point precision. The new GVFA can be iterated k times with the probability of false positives decreasing exponentially in k. In addition to matrix-matrix multiplication, the new algorithm can be applied to verify a wide variety of computations relevant to numerical linear algebra as it provides fault tolerance to the computation that defines level 3 of the BLAS. GVFA can also be used to enforce the trustworthiness of outsourcing matrix computations on untrusted distributed computing infrastructures such as clouds or volunteer peer-to-peer platforms [27,26]. The GVFA can be easily extended to a more general matrix multiplication operation where A is m × p, B is p × n, and C is m × n. The overall computational time then becomes O(mp + np). The algorithm can be further extended 14 Hao Ji et al. to verify the product of N matrices, which requires overall N +1 matrix-vector multiplications. The GVFA can also be applied to verifying a wide variety of matrix decomposition operations such as LU, QR, Cholesky, as well as eigenvalue computations, and singular value decompositions. In this case, faults are not in the product matrix but occur in the decomposed ones instead. Anyway, the GVFA can be directly applied with no modifications necessary. The GVFA is a new tool to detect faults in numerical linear algebra, and since it is based on random Gaussian projection, it is related to the many new randomized algorithms being used directly in numerical linear algebra [22,11, 12,13,15]. The fundamental idea of these randomized algorithms is to apply efficient sampling on the potentially very large matrices to extract their important characteristics so as to fast approximate numerical linear algebra operations. We believe that the GVFA will be a very useful tool in the development of fault-tolerant and otherwise resilient algorithms for solving large numerical linear algebra problems. In fact, it seems that the GVFA’s similarity to other, new, stochastic techniques in numerical linear algebra affords the possibility of creating stochastic linear solvers that are by their very nature resilient and fault-tolerant. This is highly relevant for new machines being developed in HPC to have maximal floating-point operations per second (FLOPS) while existing within restrictive energy budgets. These HPC systems will be operating at voltages lower than most current systems, and so they are expected to be particularly susceptible to soft errors. However, even if one is not anticipating the use of these high-end machines, the trend in processor design is to lower power, and is being driven by the explosion of mobile computing. Thus, the ability to reliably perform complicated numerical linear algebraic computations on systems more apt to experience soft faults is a very general concern. The GVFA will make it much easier to perform such computations with high fidelity in HPC, cloud computing, mobile applications, as well in big-data settings. Acknowledgements We would like to thank Dr. Stephan Olariu for his valuable suggestions on the manuscript. 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arXiv:1704.07513v2 [math.ST] 17 May 2017 BAYES MODEL SELECTION QIYANG HAN Abstract. We offer a general Bayes theoretic framework to tackle the model selection problem under a two-step prior design: the first-step prior serves to assess the model selection uncertainty, and the secondstep prior quantifies the prior belief on the strength of the signals within the model chosen from the first step. We establish non-asymptotic oracle posterior contraction rates under (i) a new Bernstein-inequality condition on the log likelihood ratio of the statistical experiment, (ii) a local entropy condition on the dimensionality of the models, and (iii) a sufficient mass condition on the second-step prior near the best approximating signal for each model. The first-step prior can be designed generically. The resulting posterior mean also satisfies an oracle inequality, thus automatically serving as an adaptive point estimator in a frequentist sense. Model mis-specification is allowed in these oracle rates. The new Bernstein-inequality condition not only eliminates the convention of constructing explicit tests with exponentially small type I and II errors, but also suggests the intrinsic metric to use in a given statistical experiment, both as a loss function and as an entropy measurement. This gives a unified reduction scheme for many experiments considered in [23] and beyond. As an illustration for the scope of our general results in concrete applications, we consider (i) trace regression, (ii) shape-restricted isotonic/convex regression, (iii) high-dimensional partially linear regression and (iv) covariance matrix estimation in the sparse factor model. These new results serve either as theoretical justification of practical prior proposals in the literature, or as an illustration of the generic construction scheme of a (nearly) minimax adaptive estimator for a multi-structured experiment. 1. Introduction 1.1. Overview. Suppose we observe X (n) from a statistical experiment (n) (n) (X(n) , A(n) , Pf ), where f belongs to a statistical model F and {Pf }f ∈F is dominated by a σ-finite measure µ. Instead of using a single ‘big’ model F, a collection of (sub-)models {Fm }m∈I ⊂ F are available to statisticians, and the art of model selection is to determine which one(s) to use. Date: May 18, 2017. 2000 Mathematics Subject Classification. 60F17, 62E17. Key words and phrases. Bayes nonparametrics, model selection, adaptive estimation, model mis-specification, Bernstein inequality. Supported in part by NSF Grant DMS-1566514. 1 2 Q. HAN There are vast literatures on model selection from a frequentist point of view; we only refer the reader to [4, 36, 11, 44] as some representative pointers for various approaches of penalization, aggregation, etc. On the other hand, from a Bayes point of view, although posterior contraction rates have been derived for many different models (see e.g. [21, 43, 23, 16, 14, 15, 42, 45, 46, 28] for some key contributions), understanding towards general Bayes model selection procedures has been limited. [22] focused on designing adaptive Bayes procedures with models primarily indexed by the smoothness level of classical function classes in the context of density estimation. Their conditions are complicated and seem not directly applicable to other settings. [18] designed a prior specific to structured linear problems in the Gaussian regression model, with their main focus on high-dimensional (linear) and network problems. It seems non-trivial for their framework to handle other non-linear models. Despite these limitations, [22, 18] give useful clues. One common feature in these papers is a two-step prior design, where the first-step prior Λn assesses the model selection uncertainty, followed by a second-step prior Πn,m quantifying the prior belief in the strength of the signals within the specific chosen model Fm from the first step. Such a prior design is intrinsic in many proposals for different problems, e.g. [16, 15] for sparse linear regression, [2] for trace regression, [29, 27] for shape restricted regression, [19, 38] for problems related to convariance matrix estimation. This is the starting point of this paper. We give a unified theoretical treatment to this two-step prior design by identifying common structural (n) assumptions on the statistical experiments (X(n) , A(n) , Pf ), the collection of models {Fm } and the priors {Λn } and {Πn,m } such that the posterior distribution both (G1) contracts at an oracle rate with respect to some metric dn : 2 (1.1) inf inf dn (f0 , g) + pen(m) , m∈I g∈Fm where pen(m)1 is related to the ‘dimension’ of Fm , and (G2) concentrates on the model Fm∗ , where m∗ is the ‘best’ model balancing the bias-variance tradeoff in (1.1). The oracle formulation (1.1) follows the convention in the frequentist literature [36, 44], and has several advantages: (i) (minimaxity) if the true signal f0 can be well-approximated by the models {Fm }, the contraction rate in (1.1) is usually (nearly) minimax optimal, (ii) (adaptivity) if f0 lies in certain low-dimensional model Fm , the contraction rate adapts to this unknown information, and (iii) (mis-specification) if the models Fm are mis-specified while d2n (f0 , ∪m∈I Fm ) remains ‘small’, then the contraction rate should still be rescued by this relatively ‘small’ bias. 1pen(m) may depend on n but we suppress this dependence for notational convenience. BAYES MODEL SELECTION 3 As the main abstract result of this paper (cf. Theorem 1), we show that our goals (G1)-(G2) can be accomplished under: (i) (Experiment) a Bernstein-inequality condition on the log likelihood ratio for the statistical experiment with respect to dn ; (ii) (Models) a dimensionality condition of the model Fm measured in terms of local entropy with respect to the metric dn ; (iii) (Priors) exponential weighting for the first-step prior Λn , and sufficient mass of the second-step prior Πn,m near the ‘best’ approximating signal f0,m within the model Fm for the true signal f0 . One important ingredient in studying posterior contraction rates in Bayes nonparametrics literature has been the construction of appropriate tests with exponentially small type I and II errors with respect to certain metric [21, 23]. Such tests date back to the work of Le Cam [31, 32, 33] and Birgé [6, 7, 8], who brought out the special role of the Hellinger metric in which tests can be constructed generically. On the other hand, the testing framework [21, 23] requires the prior to spread sufficient mass near the Kullback-Leibler neighborhood of the true signal. The discrepancy of these two metrics can be rather delicate, particularly for non i.i.d. and complicated models, and it often remains unclear which metric is the natural one to use in these models. Moreover, it is usually a significant theoretical challenge to construct tests in complicated models, cf. [19, 38], to name a few. Our Bernstein-inequality condition (i) closes these gaps by suggesting the usage of an ‘intrinsic metric’ that mimics the behavior of the KullbackLeibler divergence in a given statistical experiment, in which a ‘good’ test can be constructed generically (cf. Lemma 1). Bernstein inequality is a fundamental tool in probability theory, and hence can be easily verified in many statistical experiments including various experiments considered in [23] and beyond: Gaussian/binary/poisson regression, density estimation, Gaussian autoregression, Gaussian time series and covariance matrix estimation problems. We identify the intrinsic metrics to use in these experiments. Furthermore, the Bernstein-inequality condition entails sharp exponential contraction of the posterior distribution near the ‘true’ signal, complementing a recent result of [28]. Results of this type typically do not follow directly from general principles in [21, 23], and have mainly been derived on a case-by-case basis, cf. [16, 19, 18]. As such, we provide a refinement of the seminal testing framework in [21, 23] so that the investigation of sharp posterior contraction rates in the intrinsic metric of an experiment essentially reduces to the study of prior design. Conditions (ii) and (iii) are familiar in Bayes nonparametrics literature. In particular, the first-step prior can be designed generically (cf. Proposition 1). Sufficient mass of the second-step prior Πn,m is a minimal condition in the sense that using Πn,m alone should lead to a (nearly) optimal posterior contraction rate on the model Fm . 4 Q. HAN These conditions, albeit minimal, imply more than an optimal adaptive Bayes procedure in the sense of (G1)-(G2). In fact, we show that the posterior mean automatically serves as an adaptive point estimator in a frequentist sense. These results reveal, in a sense, that the task of constructing adaptive procedures with respect to the intrinsic metric in a given statistical experiment, in both frequentist and Bayes contexts, is not really harder than that of designing an optimal non-adaptive prior for each of the models. A general theory would be less interesting without being able to address problems of different types. As an illustration of our general framework in concrete applications, we justify the prior proposals in (i) [2, 34] for the trace regression problem, and in (ii) [29, 27] for the shape-restricted regression problems. Despite many theoretical results for Bayes high-dimensional models (cf. [16, 15, 19, 18, 38, 3]), it seems that the important low-rank trace regression problem has not yet been successfully addressed. Our result here fills in this gap. Furthermore, to the best knowledge of the author, the theoretical results concerning shape-restricted regression problems provide the first systematic approach that bridges the gap between Bayesian nonparametrics and shape-restricted nonparametric function estimation literature in the context of adaptive estimation2. We also consider adaptive Bayes procedures for the high-dimensional partially linear regression model and the covariance matrix estimation problem in the sparse factor model. These new results serve as an illustration of the generic construction scheme of a (nearly) minimax adaptive estimator in a complicated experiment with multiple structures. Some of these results improve the best known result in the literature. During the preparation of this paper, we become aware of a very recent paper [48] who independently considered the Bayes model selection problem. Both our approach and [48] shed light on the general Bayes model selection problem, while differing in several important aspects (cf. Remark 2). Moreover, our work here applies to a wide range of applications that are not covered by [48]. 1.2. Notation. Cx denotes a generic constant that depends only on x, whose numeric value may change from line to line. a .x b and a &x b mean a ≤ Cx b and a ≥ Cx b respectively, and a ≍x b means a .x b and (n) a &x b. For a, b ∈ R, a ∨ b := max{a, b} and a ∧ b := min{a, b}. Pf T denotes the expectation of a random variable T = T (X (n) ) under the exper(n) iment (X(n) , A(n) , Pf ). 1.3. Organization. Section 2 is devoted to the general model selection theory. We work out a wide range of experiments that fit into our general theory 2Almost completed at the same time, [35] considered a Bayes approach for univariate log-concave density estimation, where they derived contraction rates without addressing the adaptation issue. BAYES MODEL SELECTION 5 in Section 3. Section 4 discusses various concrete applications as mentioned above. Most detailed proofs are deferred to Sections 5-6 and the Appendix. 2. General theory In the two-step prior design framework, we first put a prior Λn on the model index I, followed by a prior Πn,m on the model Fm chosen from the first P step. The overall prior is a probability measure on F given by Πn ≡ m∈I λn,m Πn,m . The posterior distribution is then a random measure on F: for a measurable subset B ⊂ F, R (n) (n) pf (X ) dΠn (f ) (2.1) Πn (B\|X (n) ) = RB (n) pf (X (n) ) dΠn (f ) (n) (n) where pf (·) denotes the probability density function of Pf to the dominating measure µ. with respect 2.1. Assumptions. To state our assumption on the experiment, let vλ2 , 0 ≤ \|λ\| < 1/c 2(1 − c\|λ\|) denote the ‘Bernstein’ function. This function plays a pivital role in proving sub-gamma behavior of a given (complicated) random variable, cf. [9]. Here v and c are the L2 size and L∞ size of the random variable controlling respectively the degree of its sub-Gaussian and sub-gamma behavior. (2.2) ψv,c (λ) = Assumption A (Experiment: Bernstein-inequality condition). There exist some absolute c1 > 0 and κ = (κg , κΓ ) ∈ (0, ∞) × [0, ∞) such that for all n ∈ N, and f0 , f1 ∈ F,    (n) (n) p p (n) f0   (n) f0 ≤ c1 exp ψκg nd2n (f0 ,f1 ),κΓ (λ) − Pf0 log (n) Pf0 exp λ log (n) p f1 p f1 holds for all \|λ\| < 1/κΓ . Here the metric dn : F × F → R≥0 satisfies (n) (2.3) c2 · d2n (f0 , f1 ) p f0 1 (n) ≤ Pf0 log (n) ≤ c3 · d2n (f0 , f1 ) n p f1 for some absolute constants c2 , c3 > 0. In Assumption A, we require the log likelihood ratio to satisfy a Bernstein inequality. In particular, the log likelihood ratio has local Gaussian behavior. Conversely, if the log likelihood ratio behaves locally like Gaussian, then we can pick some κΓ > 0 so that the Bernstein inequality holds. Lemma 1. Let Assumption A hold. Fix f0 , f1 ∈ F, there exists some test φn such that (n) (n) sup Pf0 φn + Pf (1 − φn ) ≤ c6 exp(−c7 nd2n (f0 , f1 )) f ∈F :d2n (f,f1 )≤c5 d2n (f0 ,f1 ) 6 Q. HAN where c5 ≤ 1/4, c6 ∈ [2, ∞), c7 ∈ (0, 1) only depend on c1 , c2 , c3 , κ. This lemma suggests that under a Bernstein-inequality condition on the log likelihood ratio, tests exist automatically under the intrinsic metric dn that mimics the behavior of the Kullback-Leibler divergence in the sense of (2.3). Several examples will be worked out in Section 3 to illustrate the choice of an intrinsic metric dn , including the discrete ℓ2 loss for regression models, a weighted L2 metric for the Gaussian autoregression model, the Hellinger metric for density estimation, the Frobenius norm for covariance matrix estimation problem. Next we state the assumption on the complexity of the models {Fm }m∈I . Let I = Nq be a q-dimensional lattice with the natural order (I, ≤)3. Here the dimension q is understood as the number of different structures in the models {Fm }m∈I . In the sequel we will not explicitly mention q unless otherwise specified. We require the models to be nested in the sense that Fm ⊂ Fm′ if m ≤ m′ . Let f0,m denote the ‘best’ approximation of f0 within the model Fm in the sense that f0,m ∈ arg inf g∈Fm dn (f0 , g)4. Assumption B (Models: Local entropy condition). For each m ∈ I, 2 sup log N c5 ε, {f ∈ Fm : dn (f, g) ≤ 2ε}, dn ≤ (c7 /2)nδn,m (2.4) 1 + ε>δ n,m holds for all g ∈ {f0,m′ }m′ ≤m . Furthermore there exist absolute constants c ≥ 1 and γ ≥ 1 such that for any m ∈ I, α ≥ c7 /2 and any h ≥ 1, X −αnδ2 2 2 2 2 n,m′ ≤ 2e−αnhδn,m /c , ≤ hγ δn,m . c−2 δn,hm e (2.5) m′ ≥hm Note that if we choose all models Fm = F, then (2.4) reduces to the local entropy condition in [21, 23]. When Fm is finite-dimensional, typically we can check (2.4) for all g ∈ Fm . Now we comment on (2.5). The left side of 2 , while the right (2.5) essentially requires super linearity of the map m 7→ δn,m side of (2.5) controls the degree of this super linearity. As a leading example, 2 (2.5) will be trivially satisfied with c = 1 and γ = 1 when nδn,m = cm for some absolute constant c > 2/c7 . Finally we state assumptions on the priors. Assumption C (Priors: Mass condition). For all m ∈ I, (P1) (First-step prior) There exists some h ≥ 1 such that X 1 2 2 ), λn,k ≤ 2 exp(−nδn,m ). λn,m ≥ exp(−2nδn,m (2.6) 2 k>hm,k∈I (P2) (Second-step prior) 2 2 ). /c3 ≥ exp(−2nδn,m Πn,m f ∈ Fm : d2n (f, f0,m ) ≤ δn,m (2.7) 3For any a, b ∈ I, a ≤ b iff a ≤ b for all 1 ≤ i ≤ q. Similar definition applies to i i <, ≥, >. 4We assume that f 0,m is well-defined without loss of generality. BAYES MODEL SELECTION 7 Condition (P1) can be verified by using the following generic prior Λn : (2.8) 2 λn,m ∝ exp(−2nδn,m ). Proposition 1. Suppose the first condition of (2.5) holds. Then (P1) in Assumption C holds for the prior (2.8) with h ≥ 2c2 . (2.8) will be the model selection (first-step) prior on the model index I in all examples in Section 4. Condition (P2) is reminiscent of the classical prior mass condition consid2 is understood as the ‘posterior contraction rate’ ered in [21, 23] where δn,m for the model Fm . Hence (P2) can also be viewed as a solvability condition imposed on each model. Note that (2.7) only requires a sufficient prior mass on a Kullback-Leibler ball near f0,m , where [21, 23] uses more complicated metric balls induced by higher moments of the Kullback-Leibler divergence. 2.2. Main results. The following is the main abstract result of this paper. Theorem 1. Suppose Assumptions A-C hold for some M ⊂ I with \|M\| = 2 . ∞, and h ≥ C0 c2 . Let ε2n,m ≡ inf g∈Fm d2n (f0 , g) ∨ δn,m (1) For any m ∈ M, (n) (n) 2 X Pf0 Πn f ∈ F : d2n (f, f0 ) > C1 inf d2n (f0 , g) + δn,m g∈Fm (2.9) ≤ C2 exp − nε2n,m /C2 . 2 (2) For any m ∈ M such that δn,m ≥ inf g∈Fm d2n (f0 , g), (n) (2.10) / FC3 m \|X (n) ≤ C2 exp − nε2n,m /C2 . Pf0 Πn f ∈ (3) Let fˆn ≡ Πn (f \|X (n) ) be the posterior mean. Then (n) 2 ˆ 2 2 inf dn (f0 , g) + δn,m . (2.11) Pf0 dn (fn , f0 ) ≤ C4 inf m∈M g∈Fm Here the constant C0 depends on {ci }3i=1 , κ and {Ci }4i=1 depend on the {ci }3i=1 , κ, c, h and γ. The main message of Theorem 1 is that, the task of constructing Bayes procedures adaptive to a collection of models in the intrinsic metric of a given statistical experiment, can be essentially reduced to that of designing a nonadaptive prior for each model. Furthermore, the resulting posterior mean serves as an automatic adaptive point estimator in a frequentist sense. In particular, if the non-adaptive priors we use on each model lead to (nearly) optimal posterior contraction rates on these models, adaptation happens automatically by designing a ‘correct’ model selection prior, e.g. (2.8). Besides being rate-adaptive to the collection of models, (2.10) shows that the posterior distribution concentrates on the model Fm that balances the bias and variance tradeoff in the oracle rates (2.9) and (2.11). Results of this type have been derived primarily in the Gaussian regression model (cf. 8 Q. HAN [16, 15, 18]) and in density estimation [22]; here our result shows that this is a general phenomenon for the two-step prior design. Note that f0 is arbitrary and hence our oracle inequalities (2.9) and (2.11) account for model mis-specification errors. Previous work allowing model mis-specification includes [18] who mainly focuses on structured linear models in the Gaussian regression setting, and [30] who pursued generality at the cost of harder-to-check conditions. The condition \|M\| = ∞ is assumed purely for technical convenience. If we have finitely many models {F1 , . . . , Fm′ } at hand, then we can define Fm ≡ Fm′ for m ≥ m′ so that this condition is satisfied. Remark 1. We make some technical remarks. (1) The probability estimate in (2.9) is sharp (up to constants) in view of the lower bound result Theorem 2.1 in [28], thus closing a gap that has not been attainable in a general setting by using [21, 23] directly. Beyond of theoretical interest in its own right, the sharp estimate helps us to derive an oracle inequality for the posterior mean as an important frequentist summary of the posterior distribution. Such sharp estimates have been derived separately in different models, e.g. the sparse normal mean model [16], the sparse PCA model [19], and the structured linear model [18], to name a few. (2) Assumption A implies, among other things, the existence of a good test (cf. Lemma 1). In this sense our approach here falls into the general testing approach adopted in [21, 23]. The testing approach has difficulties in handling non-intrinsic metrics, cf. [28]. Some alternative approaches for dealing with non-intrinsic metrics can be found in [14, 28, 49]. (3) The constants {Ci }4i=1 in Theorem 1 depend at most polynomially with respect to the constants involved in Assumption A. This allows some flexibility in the choice of the constants therein. In fact, Bernstein inequality in some dependent cases comes with logarithmic factors in n, cf. [1, 37]. Remark 2. We compare our results with Theorems 4 and 5 of [48]. Both their results and our Theorem 1 shed light on the general problem of Bayes model selection, while differing in several important aspects: (1) Our Theorem 1 hinges on the new Bernstein-inequality condition, while the results of [48] are based on the classical mechanism of [23] which requires the construction of tests. Some merits of our approach will be clear from Section 3 and (2) below along with Remark 1. (2) The probability estimate in [48] for the posterior distribution outside a ball of radius at the targeted contraction rate is asymptotic in nature, while our Theorem 1 provides non-asymptotic sharp estimates. (3) Theorem 4 of [48] targets at exact model selection consistency, under a set of additional ‘separation’ assumptions. Our Theorem 1 (2) requires no extra assumptions, and shows the concentration behavior BAYES MODEL SELECTION 9 of the posterior distribution on the ‘best’ model that balances the bias-variance tradeoff. This is significant in non-parametric problems: the true signal typically need not belong to any specific model. (4) Theorem 5 of [48] contains a term involving the cardinality of the models and hence the models need be apriori finitely many for their bound to be finite. It remains open to see if this can be removed. 2.3. Proof sketch. Here we sketch the main steps in the proof of our main abstract result Theorem 1. The details will be deferred to Section 5. The proof can be roughly divided into two main steps. (Step 1) We first solve a localized problem on the model Fm by ‘projecting’ the underlying probability measure from Pf0 to Pf0,m . In particular, we establish exponential deviation inequality for the posterior contraction rate via the existence of tests guaranteed by Lemma 1: (n) 2 (n) 2 (2.12) Pf0,m Πn f ∈ F : d2n (f, f0,m )) > M δn, . exp − c1 nδn, m̃ \|X m̃ , 2 2 where m̃ is the smallest index ≥ m such that δn, m̃ & ℓn (f0 , f0,m ). This index may deviate from m substantially for small indices. (Step 2) We argue that, the cost of the projection in Step 1 is essentially 2 ) factor in the probability bound (2.12), cf. a multiplicative O exp(c2 nδn, m̃ Lemma 8, which is made possible by the Bernstein-inequality Assumption A. Then by requiring c1 ≫ c2 we obtain the conclusion by the definition of 2 2 2 2 δn, m̃ and the fact that δn,m̃ ≈ ℓn (f0 , f0,m ) ∨ δn,m . The existence of tests (Lemma 1) is used in step 1. Step 2 is inspired by the work of [17, 5] in the context of frequentist least squares estimator over a polyhedral cone in the Gaussian regression setting, where the localized problem therein is estimation of signals on a low-dimensional face (where ‘risk adaptation’ happens). In the Bayesian context, [16, 15] used a change of measure argument in the Gaussian regression setting for a different purpose. Our proof strategy can be viewed as an extension of these ideas beyond the (simple) Gaussian regression model. 3. Statistical experiments In this section we work out a couple of specific statistical experiments that satisfy Bernstein-inequality Assumption A to illustrate the scope of the general theory in Section 2. Some of the examples come from [23]; we identify the ‘intrinsic’ metric to use in these examples. Since Bernstein inequality is a fundamental probabilistic tool, and has been derived in a wide range of complicated (dependent) settings ([1, 37]), we expect many more experiments to be covered beyond the ones we present here. 3.1. Regression models. Suppose we want to estimate θ = (θ1 , . . . , θn ) in a given model Θn ⊂ Rn in the following regression models: for 1 ≤ i ≤ n, (1) (Gaussian) Xi = θi + εi where εi ’s are i.i.d. N (0, 1) and Θn ⊂ Rn ; (2) (Binary) Xi ∼i.i.d. Bern(θi ) where Θn ⊂ [η, 1 − η]n for some η > 0; 10 Q. HAN (3) (Poisson) Xi ∼i.i.d. Poisson(θi ) where Θn ⊂ [1/M, M ]n for some M ≥ 1; We will use the following metric: for any θ0 , θ1 ∈ Θn , n 2 1X 2 ℓn (θ0 , θ1 ) ≡ θ0,1 − θ1,i . n i=1 Lemma 2. Assumption A holds for ℓn with (1) (Gaussian) c1 = c2 = c3 = κg = 1 and κΓ = 0; (2) (Binary) κΓ = 0 and the constants {ci }3i=1 , κg depend on η only; (3) (Poisson) constants {ci }3i=1 , κ depending on M only. Theorem 2. For Gaussian/binary/poisson regression models, let dn ≡ ℓn . If Assumptions B-C hold, then (2.9)-(2.11) hold. Using similar techniques we can derive analogous results for Gaussian regression with random design and white noise model. We omit the details. 3.2. Density estimation. Suppose X1 , . . . , Xn ’s are i.i.d. samples from a density f ∈ F with respect to a measure ν on the sample space (X, A). g(x) We consider the following form of F: f (x) = R eeg dν for some g ∈ G for X all x ∈ X. A natural metric to use for density estimation is the Hellinger metric: for any f0 , f1 ∈ F, Z p p 1 h2 (f0 , f1 ) ≡ ( f0 − f1 )2 dν. 2 X Lemma 3. Suppose that G is uniformly bounded. Then Assumption A is satisfied for h with constants {ci }3i=1 , κ depending on G only. Theorem 3. For density estimation, let dn ≡ h. If G is a class of uniformly bounded functions and Assumptions B-C hold, then (2.9)-(2.11) hold. 3.3. Gaussian autoregression. Suppose X0 , X1 , . . . , Xn is generated from Xi = f (Xi−1 ) + εi for 1 ≤ i ≤ n, where f belongs to a function class F with a uniform bound M , and εi ’s are i.i.d. N (0, 1). Then Xn is a Markov chain with transition density pf (y\|x) = φ(y − f (x)) where φ is the normal density. By the arguments on page 209 of [23], this chain has a unique stationary distribution with density qf with respect to the Lebesgue measure λ on R. We assume that X0 is generated from this stationary distribution under the true f . Consider the following metric: for any f0 , f1 ∈ F, Z d2r,M (f0 , f1 ) ≡ (f0 − f1 )2 rM dλ where rM (x) ≡ 1 2 (φ(x − M ) + φ(x + M )). Lemma 4. Suppose that F is uniformly bounded by M . Then Assumption A is satisfied for dr,M with constants {ci }3i=1 , κ depending on M only. Theorem 4. For Gaussian autoregression model, if F is uniformly bounded by M , let dn ≡ dr,M . If Assumptions B-C hold, then (2.9)-(2.11) hold. BAYES MODEL SELECTION 11 Compared with results obtained in [23] (cf. Section 7.4), we identify the intrinsic metric dr,M (a weighted L2 norm) for the Gaussian autoregression model, while [23] uses a weighted Ls (s > 2) norm to check the local entropy condition, and an average Hellinger metric as the loss function. 3.4. Gaussian time series. Suppose X1 , X2 , . . . is a stationary Gaussian process with spectral density f ∈ F defined on [−π, π]. Then the covariance R π √−1λ(k−l) (n) f (λ) dλ. matrix of X = (X1 , . . . , Xn ) is given by (Tn (f ))kl ≡ −π e We consider a special form of F: f ≡ fg ≡ exp(g) for some g ∈ G. We will use the following metric: for any g0 , g1 ∈ G, 1 Dn2 (g0 , g1 ) ≡ kTn (fg0 ) − Tn (fg1 )k2F , n where k·kF denotes the matrix Frobenius norm. Lemma 5. Suppose that G is uniformly bounded. Then Assumption A is satisfied for Dn with constants {ci }3i=1 , κ depending on G only. Theorem 5. For the Gaussian time series model, if G is uniformly bounded, let dn ≡ Dn . If Assumptions B-C hold, then (2.9)-(2.11) hold. The metric Dn can always be bounded from above by the usual L2 metric, and can be related to the L2 metric from below (cf. Lemma B.3 of [20]). Our result then shows that the metric to use in the entropy condition can be weakened to the usual L2 norm rather than the much stronger L∞ norm as in page 202 of [23]. 3.5. Covariance matrix estimation. Suppose X1 , . . . , Xn ∈ Rp are i.i.d. observations from Np (0, Σ) where Σ ∈ Sp (L), the set of p × p covariance matrices whose minimal and maximal eigenvalues are bounded by L−1 and L, respectively. We will use the Frobenius norm: for any Σ0 , Σ1 ∈ Sp (L), DF2 (Σ0 , Σ1 ) ≡ kΣ0 − Σ1 k2F . Lemma 6. Under the above setting, Assumption A holds for the metric DF with constants {ci }3i=1 , κ depending on L only. Theorem 6. For covariance matrix estimation in Sp (L) for some L < ∞, let dn ≡ DF . If Assumptions B-C hold, then (2.9)-(2.11) hold. 4. Applications In this section, we consider concrete applications. As we have seen in previous sections, construction of adaptive Bayes procedures in the intrinsic metric of an experiment essentially reduces to the design of non-adaptive priors, and hence we only consider the simplest setup for a particular structure. For instance, once we understand how to analyze the convex Gaussian regression problem, we can similarly consider convex binary/Poisson regression, convex density estimation, Gaussian autoregression with convex functions, Gaussian time series with convex spectral density problems in their 12 Q. HAN respective intrinsic metrics. Hence our emphasis in the examples will be focused on the analysis of different model structures. Models that can be handled using similar techniques will not be presented in detail (e.g. Remark 4). We will only explicitly state the corresponding oracle inequalities in the form of (2.9) for each example to be considered below. The corresponding results for (2.10) and (2.11) are omitted. 4.1. Trace regression. Consider fitting the Gaussian regression model yi = f0 (xi ) + εi (1 ≤ i ≤ n) by F ≡ {fA : A ∈ Rm1 ×m2 } where fA (x) = tr(x⊤ A) for all x ∈ X ≡ Rm1 ×m2 . Let m ≡ m1 ∧ m2 and m̄ ≡ m1 ∨ m2 . The index set is I = I1 ∪ I2 ≡ {1, . . . , rmax } ∪ {rmax + 1, . . .} where rmax ≤ m. For r ∈ I1 , let Fr ≡ {fA : A ∈ Rm1 ×m2 , rank(A) ≤ r}, and for r ∈ I2 , Fr ≡ Frmax 5. Although various Bayesian methods have been proposed in the literature (cf. see [2] for a state-to-art summary), theoretical understanding has been limited. [34] derived an oracle inequality for an exponentially aggragated estimator for the matrix completion problem. Their result is purely frequentist. Below we consider a two step prior similar to [34, 2], and derive the corresponding posterior contraction rates. For a matrix B = (bij ) ∈ Rm1 ×m2 let kBkp denote its Schatten p-norm6. p = 1 and 2 correspond to the nuclear norm and the Frobenius norm respectively. To introduce the notion of RIP, let X : Rm1 ×m2 → Rn be the n linear map defined via A 7→ (tr(x⊤ i A))i=1 . Definition 1. The linear map X : Rm1 ×m2 → Rn is said to satisfy RIP (r, νr ) for some 1 ≤ r ≤ rmax and some νr = (ν r , ν̄r ) with 0 < ν r ≤ ν̄r < ∞ iff √ (A)k2 ≤ ν̄r holds for all matrices A ∈ Rm1 ×m2 such that rank(A) ≤ r. ν r ≤ kX nkAk2 For r > rmax , X satisfies RIP(r, νr ) iff X satisfies RIP(rmax , νr ). Furthermore, X : Rm1 ×m2 → Rn is said to satisfy uniform RIP (ν; I) on an index set I iff X satisfies RIP(2r, ν) for all r ∈ I. RIP(r, νr ) is a variant of the RIP condition introduced in [13, 12, 40] with scaling factors ν̄r = 1/(1 − δr ) and ν r = 1/(1 + δr ) for some 0 < δr < 1. Example 1 (Matrix completion). Suppose that xi ∈ Rm1 ×m2 takes value 1 at one position and 0 otherwise. Further assume that A ≤ \|A0 \|ij ≤ Ā for all 1 ≤ i ≤ m1 and 1 ≤ j ≤ m2 7. Let Ω ≡ ΩX denote the indices for which {xi }’s take value 1. Then kX (A)k2 = kA1Ω k2 . Easy calculations show that 5This trick of defining models for high-dimensional experiments will also used in other applications in later subsections, but we will not explicitly state it again. 1/p 6That is, kBk ≡ Pm σ (B)p , where {σj (B)} are the singular values of B. p j=1 j 7This assumption is usually satisfied in applications: in fact in the Netflix problem (which is the main motivating example for matrix completion), A0 is the rating matrix with rows indexing the users and columns indexing movies, and we can simply take A = 1 (one star) and Ā = 5 (five stars). BAYES MODEL SELECTION we can take ν = (ν̄, ν) defined by ν̄ = X is uniform RIP(ν; I). √ √m1 m2 ∧n nm1 m2 Ā ·A ,ν = 13 √ √m1 m2 ∧n nm1 m2 ·A so that Ā Example 2 (Gaussian measurement ensembles). Suppose xi ’s are i.i.d. random matrices whose entries are i.i.d. standard normal. Theorem 2.3 of [12] entails that X is uniform RIP(ν; I) with ν̄ = 1 + δ, ν = 1 − δ for some δ ∈ (0, 1), with probability at least 1 − C exp(−cn)8, provided n & m̄rmax . Consider a prior Λn on I of form λn,r ∝ exp − ctr (m1 + m2 )r log m̄ . (4.1) Given the chosen index r ∈ I, by a prior on Pra prior⊤ on Fr is induced m 1 all m1 × m2 matrices of form i=1 ui vi where ui ∈ R and vi ∈ Rm2 . Here we use a product prior distribution G with Lebesgue density (g1 ⊗ g2 )⊗r on (Rm1 × Rm2 )r . For simplicity we use gi ≡ g⊗mi for i = 1, 2 where g is symmetric about 0 and non-increasing on (0, ∞)9. Let A0,r ∈ tr ≡ g σ arg minB:rank(B)≤r ℓ2n (fB , f0 ), and τr,g max (A0,r ) + 1 where σmax denotes the largest singular value. Theorem 7. Fix 0 < η < 1/2 and rmax ≤ n. Suppose that there exists some M ⊂ I such that the linear map X : Rm1 ×m2 → Rn satisfies uniform RIP(ν; M), and that for all r ∈ M, we have 2η tr (4.2) τr,g ≥ e− log m̄/2η , m̄ ≥ 3 ∨ 3ν̄(1 ∨ σmax (A0,r ))n . Then there exists some ctr in (4.1) depending on ν̄/ν, η such that for any r ∈ M, (4.3) (n) Pf0 Πn A ∈ Rm1 ×m2 : ℓ2n (fA , f0 ) Here ε2n,r > C1tr ℓ2n (f0 , fB ) (m1 + m2 )r log(m̄) + n (n) inf Y ≤ C2tr exp −nε2n,r /C2tr . o n ≡ max inf B:rank(B)≤r ℓ2n (f0 , fB ), (m1 +mn2 )r log m̄ , and the constants B:rank(B)≤r Citr (i = 1, 2) depend on ν̄/ν, η. By Theorem 5 of [41], the rate in (4.3) is minimax optimal up to a logarithmic factor. To the best knowledge of the author, the theorem above is the first result in the literature that addresses the posterior contraction rate in the context of trace regression in a fully Bayesian setup. (4.2) may be verified in a case-by-case manner; or generically we can take M = {r0 , r0 + 1, . . .} if the model is well specified, at the cost of sacrificing 8Note here we used the union bound to get a probability estimate r max exp(−cn) . exp(−c′ n) for some c′ < c under the assumption that n & m̄rmax . 9We will always use such g in the prior design in the examples in this section. 14 Q. HAN the form of oracle inequalities (but still get nearly optimal posterior contraction rates) in (4.3). In particular, the first condition of (4.2) prevents the largest eigenvalue of A0,r from growing too fast. This is in similar spirit with Theorem 2.8 of [16], showing that the magnitude of the signals cannot be too large for light-tailed priors to work in the sparse normal mean model. The second condition of (4.2) is typically a mild technical condition: we only need to choose η > 0 small enough. 4.2. Isotonic regression. Consider fitting the Gaussian regression model Yi = f0 (xi )+εi by F ≡ {f : [0, 1] → R : f is non-decreasing}. For simplicity the design points are assumed to be xi = i/(n + 1) for all 1 ≤ i ≤ n. Let Fm ≡ f ∈ F, f is piecewise constant with at most m constant pieces . Consider the following prior Λn on I = N: (4.4) λn,m ∝ exp − ciso m log(en) . Let gm ≡ g ⊗m where g is symmetric and non-increasing on (0, ∞). Then ḡm (µ) ≡ m!gm 1{µ1 ≤...≤µm } (µ) is a valid density on {µ1 ≤ . . . ≤ µm }. Given a chosen model Fm by the prior Λn , we randomly pick a set of change points {xi(k) }m k=1 (i(1) < . . . < i(m)) and put a prior ḡm on {f (xi(k) )}’s. [29] proposed a similar prior with Λn being uniform since they assumed the maximum number of change points is known apriori. Below we derive a theoretical result without assuming the knowledge of this. Let f0,m ∈ 10 iso = g kf . arg ming∈Fm ℓ2n (f0 , g), and τm,g 0,m k∞ + 1 Theorem 8. Fix 0 < η < 1/2. Suppose that (4.5) iso τm,g ≥ e− log(en)/2η . Then there exists some ciso in (4.4) depending on η such that (4.6) m log(en) (n) 2 iso 2 (n) Pf0 Πn f ∈ F : ℓn (f, f0 ) > C1 inf ℓn (f0 , g) + Y g∈Fm n 2 iso ≤ C2iso exp −n(εiso . n,m ) /C2 n o 2 ≡ max inf 2 (f , g), m log(en) , and the constants C iso (i = ) ℓ Here (εiso 0 g∈F m n n,m i n 1, 2) depend on η. (4.6) implies that if f0 is piecewise constant, the posterior distribution contracts at nearly a parametric rate. (4.5) can be checked by the following. Lemma 7. If f0 is square integrable, and the prior density g is heavy-tailed in the sense that there exists some α > 0 such that lim inf \|x\|→∞ xα g(x) > 0. Then for any η ∈ (0, 1/α), (4.5) holds uniformly in all m ∈ N for n large enough depending on α and kf0 kL2 ([0,1]) . 10The value of f 0,m outside of [1/(n + 1), n/(n + 1)] can be defined in a canonical way by extending f0,m (1/(n + 1)) and f0,m (n/(n + 1)) towards the endpoints. BAYES MODEL SELECTION 15 4.3. Convex regression. Consider fitting the Gaussian regression model Yi = f0 (xi ) + εi by F, the class of convex functions on X = [0, 1]d . Let Fm ≡ f (x) = max1≤i≤m (ai · x + bi ) : ai ∈ Rd , bi ∈ R denote the class of piecewise affine convex functions with at most m pieces. We will focus on the multivariate case since the univariate case can be easily derived using the techniques exploited in isotonic regression. A prior on each model Fm can be induced by a prior on the slopes the intercepts Nm and ⊗d ⊗ g on (Rd × {(ai , bi ) ∈ Rd × R}m g i=1 . We use a prior with density i=1 R)m to induce a prior on Fm . Let f0,m ∈ arg ming∈Fm ℓ2n (f0 , g) admit the (m) (m) . Further let representation given n by f0,m (x) ≡max1≤i≤m ai o· x + bi (m) (m) cvx ≡ min . τm,g 1≤i≤m g kai k∞ + 1 , g \|bi \| + 1 The prior Λn we will use on the index I = N is given by λn,m ∝ exp − ccvx dm log 3m · log n . (4.7) The first step prior used in [27] is a Poisson proposal, which slightly differs from (4.7) by a logarithmic factor. This would affect the contraction rate only by a logarithmic factor. Theorem 9. Fix 0 < η < 1/4. Suppose that (4.8) cvx τm,g ≥ e− log n·log 3m/8η , and n ≥ d. Then there exists some ccvx in (4.7) depending on η such that (4.9) d log n · m log 3m (n) 2 cvx 2 (n) Pf0 Πn f ∈ F : ℓn (f, f0 ) > C1 inf ℓn (f0 , g) + Y g∈Fm n 2 cvx ≤ C2cvx exp −n(εcvx . n,m ) /C2 o n 2 ≡ max inf 2 (f , g), d log n·m log 3m , and the constants Here (εcvx ) ℓ g∈F 0 n,m m n n Cicvx (i = 1, 2) depend on η. Our oracle inequality shows that the posterior contraction rate of [27] (Theorem 3.3 therein) is far from optimal. (4.8) can be satisfied by using heavy-tailed priors g(·) in the same spirit as Lemma 7–if f0 is square integrable and the design points are regular enough (e.g. using regular grids on [0, 1]d ). Moreover, explicit rate results can be obtained using approximation techniques in [26] (cf. Lemma 4.10 therein). We omit detailed derivations. Remark 3. For univariate convex regression, the term log(3m) in (4.7)-(4.9) can be removed. The logarithmic term is due to the fact that the pseudodimension of Fm scales as m log(3m) for d ≥ 2, cf. Lemma 22. Remark 4. Using similar priors and proof techniques we can construct a (nearly) rate-optimal adaptive Bayes estimator for the support function regression problem for convex bodies [25]. There the models Fm are support functions indexed by polytopes with m vertices, and a prior on Fm is induced by a prior on the location of the m vertices. The pseudo-dimension of Fm can be controlled using techniques developed in [25]. Details are omitted. 16 Q. HAN 4.4. High-dimensional partially linear model. Consider fitting the Gaussian regression model Yi = f0 (xi , zi ) + εi where (xi , zi ) ∈ Rp × [0, 1], by a partially linear model F ≡ {fβ,u (x, z) = x⊤ β + u(z) ≡ hβ (x) + u(z) : β ∈ Rp , u ∈ U } where the dimension of the parametric part can diverge. We consider U to be the class of non-decreasing functions as an illustration (cf. Section 4.2). Consider models F(s,m) ≡ {fβ,u : β ∈ B0 (s), u ∈ Um } where Um denotes the class of piecewise constant non-decreasing functions with at most m constant pieces, and B0 (s) ≡ {v ∈ Rp : \|supp(v)\| ≤ s}. In this example the model index I is a 2-dimensional lattice. Our goal here is to construct an estimator that satisfies an oracle inequality over the models {F(s,m) }(s,m)∈N2 . Consider the following model selection prior: (4.10) λn,(s,m) ∝ exp −chp (s log(ep) + m log(en)) . For a chosen model F(s,m) , consider the following prior Πn,(s,m) : pick randomly a support S ⊂ {1, . . . , p} with \|S\| = s and a set of change points Q ≡ {zi(k) }m k=1 (i(1) < . . . i(m)), and then put a prior gS,Q on βS and u(zi(k) )’s. For simplicity we use a product prior gS,Q ≡ g⊗s ⊗ ḡm where ḡm is a prior on {µ1 ≤ . . . ≤ µm } ⊂ Rm constructed in Section 4.2. Let f0,(s,m) ∈ inf g∈F(s,m) ℓ2n (f0 , g), and write f0,(s,m) (x, z) = x⊤ β0,s + u0,m (z) ≡ h0,s (x) + u0,m (z). Let τs,g ≡ g(kβ0,s k∞ + 1) and τm,g ≡ g(ku0,m k∞ + 1). Let X ∈ Rn×p be the design matrix so that X ⊤ X/n is normalized with diagonal elements taking value 111. Theorem 10. Fix 0 < η < 1/4 and p ≥ n. Suppose that (4.11) τs,g ≥ e− log(ep)/2η , τm,g ≥ e− log(en)/2η . Then there exists some chp in (4.10) depending on η such that (4.12) s log(ep) m log(en) (n) hp (n) 2 2 + Y Pf0 Πn f ∈ F : ℓn (f, f0 ) > C1 inf ℓn (f0 , fβ,u ) + fβ,u ∈F(s,m) n n hp 2 ≤ C2hp exp −n(εhp ) /C . 2 n,(s,m) o n s log(ep)+m log(en) 2 ≡ max inf 2 (f , f , and the ) ℓ ), Here (εhp 0 f ∈F β,u n β,u (s,m) n n,(s,m) constants Cihp (i = 1, 2) depend on η. The first condition of (4.11) requires that the magnitude of kβ0,s k∞ does not grow too fast; see also comments following Theorem 7. The second condition of (4.11) is the same as in (4.5). When the model is well-specified in the sense that f0 (x, z) = x⊤ β0 + u0 (z) for some β0 ∈ B0 (s0 ) and u0 ∈ U , the oracle rate in (4.12) becomes s0 log(ep) m log(en) (4.13) + inf inf ℓ2n (u0 , u) + . m∈N u∈Um n n 11This is a common assumption, cf. Section 6.1 of [10]. BAYES MODEL SELECTION 17 The two terms in the rate (4.13) trades off two structures of the experiment: the sparsity of hβ (x) and the smoothness level of u(z). The resulting phase transition of the rate (4.13) in terms of these structures is in a sense similar to the results of [51, 50]. It is not hard to see that (4.13) cannot be improved in general. Hence our Bayes estimator automatically serves as a theoretically (nearly) optimal adaptive estimator for the high-dimensional partially linear regression model. 4.5. Covariance matrix estimation in the sparse factor model. Suppose we observe i.i.d. X1 , . . . , Xn ∈ Rp from Np (0, Σ0 ). The covariance matrix is modelled by the sparse factor model M ≡ ∪(k,s)∈N2 M(k,s) where M(k,s) ≡ {Σ = ΛΛ⊤ + I : Λ ∈ R(k,s) (L)} with R(k,s) (L) ≡ {Λ ∈ Rp×k , Λ·j ∈ B0 (s), \|σj (Λ)\| ≤ L1/2 , ∀1 ≤ j ≤ k}. In this example, the model index I is a 2-dimensional lattice, and the sparsity structure depends on the rank structure. Consider the following model selection prior: (4.14) λn,(k,s) ∝ exp (−ccov ks log(ep)) . Theorem 11. Let p ≥ n. There exist some ccov in (4.14) and some sequence of sieve priors Πn,(k,s) on M(k,s) depending on L such that (4.15) ks log(ep) (n) (n) 2 cov ′ 2 PΣ0 Πn Σ ∈ M : kΣ − Σ0 kF > C1 X inf kΣ − Σ0 kF + Σ′ ∈M(k,s) n 2 cov . ≤ C2cov exp −n(εcov ) /C 2 n,(k,s) n o 2 ≡ max inf ′ ′ − Σ k2 , ks log(ep) , and the constants Here (εcov ) kΣ 0 Σ ∈M F (s,k) n,(k,s) n Cicov (i = 1, 2) depend on L. Since spectral norm (non-intrinsic) is dominated by Frobenius norm (intrinsic), our result shows that if the model is well-specified in the sense that Σ0 ∈ M, then we can construct an adaptive p Bayes estimator with convergence rates in both norms no worse than ks log p/n. [38] considered p the same sparse factor model, where they proved a strictly sub-optimal rate k3 s log p log n/n in spectral norm under ks & log p. [19] considered a closely related sparse PCA problem, where the convergence rate under spectral norm achieves the same rate as here (cf. Theorem 4.1 therein), √ while a factor of k is lost when using Frobenius norm as a loss function (cf. Remark 4.3 therein). It should be mentioned that the sieve prior Πn,(k,s) is constructed using the metric entropy of M(k,s) and hence the resulting Bayes estimator and the posterior mean as a point estimator are purely theoretical. We use this example to illustrate (i) the construction scheme of a (nearly) optimal adaptive procedure for a multi-structured experiment based on the metric entropy of the underlying parameter space, and (ii) derivation of contraction rates in non-intrinsic metrics when these metrics can be related to the intrinsic metrics nicely. 18 Q. HAN 5. Proofs for the main results 5.1. Proof of Theorem 1: main steps. First we need a lemma allowing a change-of-measure argument. Lemma 8. Let Assumption A hold. There exists some constant c4 ≥ 1 only depending on c1 , c3 and κ such that for any random variable U ∈ [0, 1], any δn ≥ dn (f0 , f1 ) and any j ∈ N, i h −1 2 2 (n) (n) Pf0 U ≤ c4 Pf1 U · ec4 njδn + e−c4 njδn . The next propositions solve the posterior contraction problem for the ‘local’ model Fm . 2 ≥ d2n (f0 , f0,m ). Then there Proposition 2. Fix m ∈ M such that δn,m exists some constant c8 ≥ 1 (depending on the constants in Assumption A) such that for j ≥ 8c2 /c7 h, (5.1) 2 2 (n) 2 X (n) ≤ c8 e−njhδn,m /c8 c . Pf0,m Πn f ∈ F : d2n (f, f0,m ) > c2 (jh)γ δn,m 2 < d2n (f0 , f0,m ). Let m̃ ≡ Proposition 3. Fix m ∈ M such that δn,m m̃(m) ≡ inf{m′ ∈ M, m′ ≥ m : δn,m′ ≥ dn (f0 , f0,m )}. Then for j ≥ 8c2 /c7 h, (5.2) 2 2 (n) Pf0,m Πn f ∈ F : d2n (f, f0,m ) > c4 (2jh)γ d2n (f0 , f0,m ) X (n) ≤ c8 e−njhδn,m̃ /c8 c . The proofs of these results will be detailed in later subsections. Proof of Theorem 1: main steps. Instead of (2.9), we will prove a slightly stronger statement as follows: for any j ≥ 8c2 /c7 h, and h ≥ 2c4 c8 c2 , (5.3) (n) 2 (n) 2 X ≤ c2 e−jnεn,m /c2 . Pf0 Πn f ∈ F : d2n (f, f0 ) > c1 j γ inf d2n (f0 , g) + δn,m g∈Fm Here the constants ci (i = 1, 2) depends on the constants involved in Assumption A and c, h. Proof of (5.3). First consider the overfitting case. By Proposition 2 and Lemma 8, we 2 see that when δn,m ≥ d2n (f0 , f0,m ) holds, for j ≥ 8c2 /c7 h, (n) 2 Pf0 Πn f ∈ F : d2n (f, f0 ) > 2d2n (f0 , f0,m ) + 2c2 (jh)γ δn,m X (n) (n) 2 2 γ 2 (n) ≤ Pf0 Πn f ∈ F : dn (f, f0,m ) > c (jh) δn,m X 2 2 (n) (5.4) ≤ c4 P (n) Πn f ∈ F : d2n (f, f0,m ) > c2 (jh)γ δn,m X ec4 njδn,m f0,m 2 −c−1 njδ n,m +e 4 2 −njδn,m ≤ c8 c4 e h −c4 c 8 c2 −1 + c4 e−c4 2 njδn,m 2 −1 ≤ 2c8 c4 e−jnδn,m min{c4 ,c4 } . BAYES MODEL SELECTION 19 Here in the second line we used the fact that d2n (f, f0,m ) ≥ d2n (f, f0 )/2 − d2n (f0 , f0,m ). (5.4) completes the estimate for overfitting m ∈ M. 2 < d2n (f0 , f0,m ). Next consider the underfitting case: fix m ∈ M such that δn,m Apply Proposition 3 and Lemma 8, and use arguments similar to (5.4) to see that for j ≥ 8c2 /c7 h, (5.5) (n) Pf0 Πn f ∈ F : d2n (f, f0 ) > 2c4 (2jh)γ + 2 d2n (f0 , f0,m ) X (n) 2 (n) 2 4 γ 2 (n) ≤ c4 Pf0,m Πn f ∈ F : dn (f, f0,m ) > c (2jh) dn (f0 , f0,m ) X ec4 njδn,m̃ −1 2 2 −c−1 jnδn, 4 m̃ +e ≤ 2c8 c4 e−njδn,m̃ min{c4 ,c4 } . Here in the second line we used (i) 2d2n (f, f0,m ) ≥ d2n (f, f0 ) − 2d2n (f0 , f0,m ), and (ii) δn,m̃ ≥ dn (f0 , f0,m ). The claim of (5.3) follows by combining (5.4) and (5.5). Proof of (2.11). The proof is essentially integration of tail estimates by a peeling device. Let the event Aj be defined via 2 2 Aj := {c1 j γ d2n (f0 , f0,m ) + δn,m < d2n (f, f0 ) ≤ c1 (j + 1)γ d2n (f0 , f0,m ) + δn,m }. Then, (n) (n) (n) Pf0 d2n (fˆn , f0 ) = Pf0 d2n Πn (f \|X (n) ), f0 ≤ Pf0 Πn d2n (f, f0 )\|X (n) X 2 ≤ Cc1 ,c,c7,h,γ d2n (f0 , f0,m ) + δn,m + Pfn0 Πn d2n (f, f0 )1Aj X (n) j≥8c2 /c7 h 2γ+1 c1 c2 2 + ≤ Cc1 ,c,c7,h,γ d2n (f0 , f0,m ) + δn,m n X 2 j γ nε2n,m e−jnεn,m /c2 . j≥8c2 /c7 h The inequality in the first line of the above display is due to Jensen’s inequality applied with dn (·, f0 ), followed by Cauchy-Schwarz inequality. The summation can be bounded up to a constant depending on γ, c1 , c2 by X X 2 2 (jnε2n,m )γ e−jnεn,m/c2 ≤ (jnε2n,m )γ e−jnεn,m /c2 (j + 1)nε2n,m − jnε2n,m j≥8c2 /c7 h j≥8c2 /c7 h where the inequality follows since nε2n,m ≥ nε2n,1 ≥ 1. This quantity can be R∞ bounded by a constant multiple of 0 xγ e−x/c2 dx independent of m. Now 2 majorizes 1/n up to a constant, the proof is complete by noting that δn,m and then taking infimum over m ∈ M. 5.2. Proofs of Propositions 2 and 3. We will need several lemmas before the proof of Propositions 2 and 3. Lemma 9. Let Assumption A hold. Let F be a function class defined on the sample space X. Suppose that N : R≥0 → R≥0 is a non-increasing function 20 Q. HAN such that for some ε0 ≥ 0 and every ε ≥ ε0 , it holds that N (c5 ε, {f ∈ F : ε < dn (f, f0 ) ≤ 2ε}, dn ) ≤ N (ε). Then for any ε ≥ ε0 , there exists some test φn such that 2 (n) Pf0 φn c6 N (ε)e−c7 nε , ≤ 1 − e−c7 nε2 + 2 (n) sup f ∈F :dn (f,f0 )≥ε Pf (1 − φn ) ≤ c6 e−c7 nε . The constants c5 , c6 , c7 are taken from Lemma 1. Lemma 10. Let Assumption A hold. Suppose that Π is a probability measure on {f ∈ F : dn (f, f0 ) ≤ ε}. Then for every C > 0, there exists some C ′ > 0 depending on C, κ such that Z (n) pf ′ 2 (n) −(C+c3 )nε2 (5.6) Pf0 dΠ(f ) ≤ e ≤ c1 e−C nε . (n) p f0 The proof of these lemmas can be found in Appendix C. Proof of Proposition 2. Fix m′ ∈ I with m′ ≥ m. Now we invoke Lemma 9 with F ≡ Fm′ , f0 ≡ f0,m ∈ Fm ⊂ Fm′ [since m′ ≥ m], ε0 ≡ δn,m′ and 2 log N (ε) ≡ (c7 /2)nδn,m ′ for ε = ε0 to see that, there exists some test φn,m′ such that (5.7) (n) Pf0,m φn,m′ ≤ and that (5.8) 2 log N (ε)−c7 nδn,m ′ c6 e 2 −c7 nδn,m ′ 1−e 2 −c7 nδn,m ′ /2 ≤ 2c6 e 2 −c7 nδn,m ′ (n) sup 2 f ∈Fm′ :d2n (f,f0,m )≥δn,m ′ , Pf (1 − φn,m′ ) ≤ c6 e . 2 Note that here in (5.7) we used the fact that nδn,m ′ ≥ 2/c7 by definition of ′ δn,m . Now for the fixed j, m as in the statement of the proposition, we let φn := supm′ ∈I:m′ ≥jhm φn,m′ be a global test for big models. Then by (5.7), X X 2 2 −c nδ2 /2 (n) (n) Pf0,m φn ≤ Pf0,m φn,m′ ≤ 2c6 e 7 n,m′ ≤ 4c6 e−(c7 /2c )njhδn,m . m′ ≥jhm m′ ≥jhm Here we used the left side of (2.5). This implies that for any random variable U ∈ [0, 1], we have (5.9) (n) (n) Pf0,m U · φn ≤ Pf0,m φn ≤ 4c6 e−(c7 /2c 2 )njhδ 2 n,m . On the power side, with m′ = jhm applied to (5.8) we see that (5.10) (n) (n) sup sup Pf (1 − φn ) ≤ Pf (1 − φn ) 2 f ∈Fjhm :d2n (f,f0,m )≥c2 (jh)γ δn,m 2 f ∈Fjhm :d2n (f,f0,m )≥δn,jhm 2 ≤ c6 e−c7 nδn,jhm ≤ 2c6 e−(c7 /c 2 )njhδ 2 n,m . BAYES MODEL SELECTION 21 2 ≥ The first inequality follows from the right side of (2.5) since c2 (jh)γ δn,m 2 δn,jhm, and the last inequality follows from the left side of (2.5). On the other hand, by applying Lemma 10 with C = c3 and ε2 ≡ (n) 2 c7 jhδn,m , 8c3 c2 we see 2 c njhδn,m −C ′ 7 8c3 c2 and it that there exists some event En such that Pf0,m (Enc ) ≤ c1 e holds on the event En that (5.11) (n) Z (n) Z pf pf dΠ(f ) ≥ λn,m dΠn,m (f ) (n) (n) 2 {f ∈Fm :d2n (f,f0,m )≤c7 jhδn,m /8c3 c2 } p pf0,m f0,m ≥ λn,m e− 2 c7 njhδn,m 4c2 Πn,m 2 f ∈ Fm : d2n (f, f0,m ) ≤ c7 jhδn,m /8c3 c2 Note that (5.12) (n) 2 2 γ 2 (n) Pf0,m Πn f ∈ F : dn (f, f0,m ) > c (jh) δn,m X (1 − φn )1En R  (n) (n) dΠ(f ) /p p 2 2 γ 2 f0,m f ∈F :dn (f,f0,m )>c (jh) δn,m f (n) = Pf0,m  (1 − φn )1En  R (n) (n) pf /pf0,m dΠ(f ) 2 ≤ . 2 ec7 njhδn,m /4c 2 /8c c2 λn,m Πn,m f ∈ Fm : d2n (f, f0,m ) ≤ c7 jhδn,m 3 # "Z (n) (n) (n) pf /pf0,m dΠ(f )(1 − φn ) × Pf0,m 2 f ∈F :d2n (f,f0,m )>c2 (jh)γ δn,m ≡ (I) · (II) where the inequality follows from (5.11). On the other hand, the expectation term in the above display can be further calculated as follows: Z (n) Pf (1 − φn ) dΠ(f ) (II) = 2 f ∈F :d2n (f,f0,m )>c2 (jh)γ δn,m (5.13) ≤ sup 2 f ∈Fjhm :d2n (f,f0,m )>c2 (jh)γ δn,m ≤ 2c6 e−(c7 /c 2 )njhδ 2 n,m (n) Pf (1 − φn ) + Π F \ Fjhm + 4e−(1/c 2 )njhδ 2 n,m ≤ 6c6 e−(c7 /c 2 )njhδ 2 n,m . The first term in the third line follows from (5.10) and the second term follows from (P1) in Assumption C along with the left side of (2.5). By (P1)-(P2) in Assumption C and j ≥ 8c2 /c7 h, (5.14) (n) 2 X (n) (1 − φn )1En ≤ Ce−(c7 /4c Pf0,m Πn f ∈ F : d2n (f, f0,m ) > c2 (jh)γ δn,m 2 )njhδ 2 n,m Hence we conclude (5.1) from (5.9), probability estimate on Enc , and (5.14). . 22 Q. HAN Proof of Proposition 3. The proof largely follows the same lines as that of Proposition 2. See Appendix C for details. 5.3. Completion of proof of Theorem 1. 2 ≥ d2n (f0 , f0,m ), following Proof of (2.10). For any m ∈ M such that δn,m the similar reasoning in (5.12) with j = 8c2 /c7 h, (5.15) (n) / Fjhm \|X (n) 1En Pf0,m Πn f ∈ 2 ≤ λn,m Πn,m ≤ Ce−(c7 /4c 2 ec7 njhδn,m /4c · Π F \ Fjhm 2 2 2 f ∈ Fm : dn (f, f0,m ) ≤ c7 jhδn,m /8c3 c 2 )njhδ 2 n,m . From here (2.10) can be established by controlling the probability estimate for Enc as in Proposition 2, and a change of measure argument as in (5.4) using Lemma 8. 5.4. Proof of Lemma 1. Proof of Lemma 1. Let c> 0 be a constant to be specified later. Consider (n) pf 0 ≤ −cnd2n (f0 , f1 ) . We first consider type the test statistics φn ≡ 1 log (n) pf 1 I error. Under the null hypothesis, we have for any λ1 ∈ (0, 1/κΓ ),    (n) (n) p p f0 f0  (n) (n) Pf0 φn ≤ Pf0 log (n) ≤ −(c + c2 )nd2n (f0 , f1 ) − Pf0 log (n) p f1 p f1 ≤ c1 exp ψκg nd2n (f0 ,f1 ),κΓ (−λ1 ) · exp −λ1 (c + c2 )nd2n (f0 , f1 ) . Choosing λ1 > 0 small enough (depending on κ, c2 , c) we get (n) Pf0 φn ≤ C1 exp(−C2 nd2n (f0 , f1 )) where C1 , C2 > 0 depend on c1 , c2 , c, κ. Next we handle the type II error. ′ To this end, 3 c5 to be specified later, consider the event for a constant c > c (n) En ≡ 1 log pf (n) 1 pf < c′ nd2n (f0 , f1 ) , where f ∈ F is such that d2n (f, f1 ) ≤ c5 d2n (f0 , f1 ), and λ2 ∈ (0, 1/κΓ ),   (n) (n) pf pf (n) (n) (n) Pf (Enc ) ≤ Pf log (n) − Pf log (n) > c′ nd2n (f0 , f1 ) − c3 nd2n (f, f1 ) p f1 p f1   (n) (n) p p f f (n) (n) ≤ Pf log (n) − Pf log (n) > (c′ − c3 c5 )nd2n (f0 , f1 ) p f1 p f1 ′ 2 ≤ e−λ2 (c −c3 c5 )ndn (f0 ,f1 ) c1 exp(ψκg nd2n (f,f1 ),κΓ (λ2 )). BAYES MODEL SELECTION 23 By choosing λ2 > 0 small enough depending on c3 , c5 , c′ , κ, we see that (n) Pf (Enc ) ≤ C3 exp(−C4 nd2n (f0 , f1 )) for some constants C3 , C4 depending only on c1 , c3 , c5 , c′ , κ (in particular, does not depend on f ). On the other hand,   (n) (n) p p f (n) (n) f1 + log (n) < cnd2n (f0 , f1 ) (1En + 1Enc ) Pf (1 − φn ) = Pf log (n) pf p f0   (n) p f (n) (n) (n) ≤ Pf log (n) < (c + c′ )nd2n (f0 , f1 ) + Pf (Enc ) ≡ (∗) + Pf (Enc ). p f0 (n) (n) pf ≥ c2 d2n (f, f0 ) ≥ c2 (1 − √ c5 )2 d2n (f0 , f1 ), we continue our √ computation: for c, c′ such that c + c′ < c2 (1 − c5 )2 , and λ3 ∈ (0, 1/κΓ ),   (n) (n) p p √ f (n) (n) f (∗) ≤ Pf log (n) − Pf log (n) < − c2 (1 − c5 )2 − (c + c′ ) nd2n (f0 , f1 ) p f0 p f0 √ 2 ′ 2 (−λ ) ψ ≤ e−λ3 c2 (1− c5 ) −(c+c ) ndn (f0 ,f1 ) c1 e κg nd2n (f,f0 ),κΓ 3 . Since Pf log (n) pf 0 Now choose λ3 > 0 small enough depending on c2 , c5 , c, c′ , κ we see that for any f ∈ F such that d2n (f, f1 ) ≤ c5 d2n (f0 , f1 ), (n) Pf (1 − φn ) ≤ C5 exp(−C6 nd2n (f0 , f1 )) where C5 , C6 depending on c1 , c2 , c5 , c, c′ , κ, C3 , C4 . Now we need to choose √ c, c′ , c5 such that c > 0, c′ > c3 c5 , c + c′ < c2 (1 − c5 )2 . This can be done by choosing c5 ≤ min{1/4, c2 /16c3 } and c′ = c = 2c3 c5 . 5.5. Proof of Lemma 8. We recall a standard fact. Lemma 11. If a random variable X satisfies E exp(λX) ≤ exp(ψv,c (λ)), t2 then for t > 0, P (X ≥ t) ∨ P(X ≤ −t) ≤ exp − 2(v+ct) . (n) pf 2 0 Proof of Lemma 8. For c = 2c3 , consider the event En ≡ log (n) < cjnδn . pf 1 By Lemma 11, we have for some constant C > 0 depending on c1 , c3 and κ,   (n) (n) p p (n) f0 (n) (n) f0 − Pf0 log (n) ≥ cjnδn2 − c3 nd2n (f0 , f1 ) Pf0 (Enc ) ≤ Pf0 log (n) p f1 p f1   (n) (n) p f0 p f0 (n) (n) ≤ Pf0 log (n) − Pf0 log (n) ≥ c3 jnδn2  (since dn (f0 , f1 ) ≤ δn ) p f1 p f1 n2 j 2 δn4 ≤ C exp −C −1 njδn2 . ≤ C exp − 2 C(njδn + 1) 24 Q. HAN We remind the reader that the constant C may not be the same in the above series of inequalities, and hence the last inequality follows by noting that (i) if njδn2 ≥ 1, then we replace the denumerator of the second last line by 2, (ii) if njδn2 < 1, then we increase C ≥ 1. Then   (n) p −1 2 (n) (n) (n) (n) f0 1En  + Ce−C njδn Pf0 U = Pf0 U 1En + Pf0 U 1Enc ≤ Pf1 U (n) p f1 (n) 2 ≤ Pf1 U · ecnjδn + Ce−C completing the proof. −1 njδ 2 n , 5.6. Proof of Proposition 1. P 2 Proof of Proposition 1. Let Σn = m∈I e−2nδn,m be the total mass. Then 2 2 2 e−2nδn,1 ≤ Σn ≤ 2e−2nδn,1 /c ≤ 2. The first condition ofP(P1) is trivial. We only need to verify the second condition of (P1): k>hm λn,k = P 2 2 2 2 2 −2nδ n,k ≤ e2nδn,1 · 2e−(2h/c )nδn,m ≤ 2e−2nδn,m , where the first Σ−1 n k>hm e inequality follows from (2.5) and the second by the condition h ≥ 2c2 . 6. Proofs for applications The proofs of the theorems in Section 4 follow the similar route by verifying (i) the local entropy condition in Assumption B, (ii) the summability condition in (2.5) and (iii) the sufficient mass condition (P2) in Assumption C. We remind the reader that we use (2.8) in all examples as the first-step (model selection) prior. We only prove Theorems 7 and 11 in this section. The proofs for Theorems 8-10 are deferred to Appendix B. 6.1. Proof of Theorem 7. Lemma 12. Let r ∈ I. Suppose that the linear map X : Rm1 ×m2 → Rn is uniform RIP(ν; I). Then for any ε > 0 and A0 ∈ Rm1 ×m2 such that rank(A0 ) ≤ r, we have log N c5 ε, {fA ∈ Fr : ℓn (fA , fA0 ) ≤ 2ε}, ℓn ≤ 2(m1 + m2 )r · log 18ν̄/c5 ν . We will need the following result. Lemma 13. Let S(r, B) = {A ∈ Rm1 ×m2 : rank(A) ≤ r, kAk2 ≤ B}. Then (m1 +m2 −1)r N ε, S(r, B), k·k2 ≤ 9B . ε Proof of Lemma 13. The case for B = 1 follows from Lemma 3.1 of [12] and the general case follows by a scaling argument. We omit the details. Proof of Lemma 12. We only need to consider the case r ≤ rmax . First note that the entropy in question equals √ √ (6.1) log N c5 nε, {X (A) : kX (A − A0 )k2 ≤ 2 nε, rank A ≤ r}, k·k2 . BAYES MODEL SELECTION 25 By uniform RIP(ν; I), the set to be covered in the above display is contained in {X (A) : kA − A0 k2 ≤ 2ε/ν, rank A ≤ r} ⊂ X (S(2r, 2ε/ν )). On the other hand, again by uniform RIP(ν; I), a c√ 5 ε/ν̄-cover of the set S(2r, 2ε/ν) under the Frobenius norm k·k2 induces a c5 nε-cover of X (S(2r, 2ε/ν )) under the Euclidean k·k2 norm. This implies that (6.1) can be further bounded from above by log N c5 ε/ν̄, S(2r, 2ε/ν ), k·k2 ≤ 2(m1 + m2 )r · log 18ν̄/c5 ν , where the last inequality follows from Lemma 13. 2 = 4 log(18ν̄/c5 ν) ∨ 1 ·(m1 +m2 )r log m̄ . Clearly δ 2 satisfies Now we take δn,r n,r c7 η n (2.5) with c ≡ 1 and γ = 1. Lemma 14. Suppose that X : Rm1 ×m2 → Rn is uniform RIP(ν; I), and that (4.2) holds. Then (P2) in Assumption C holds. Proof of Lemma 14. We only need to consider r ≤ rmax . First note that (6.2) 2 /c3 Πn,r fA ∈ Fr : ℓ2n (fA , fA0,r ) ≤ δn,r √ √ = ΠG A ∈ Rm1 ×m2 : kX (A − A0,r )k2 ≤ nδn,r / c3 , rank(A) ≤ r √ ≥ ΠG A ∈ Rm1 ×m2 : kA − A0,r k2 ≤ δn,r /ν̄ c3 , rank(A) ≤ r . P Let A0,r ≡ ri=1 σi ūi v̄i⊤ be the spectral decomposition of A0,r , and let ui ≡ P √ √ σi ūi and vi ≡ σi v̄i . Then A0,r ≡ ri=1 ui vi⊤ . Now for u∗i ∈ Bm1 (ui , ε) P and vi∗ ∈ Bm2 (vi , ε), i = 1, . . . , r, let A∗ ≡ ri=1 u∗i (vi∗ )⊤ , then by noting √ that the Frobenius norm is sub-multiplicative and that kui k2 = kvi k2 = σi , we have for ε ≤ 1, kA∗ − A0,r k2 ≤ ≤ r X k(ui − u∗i )vi⊤ k2 + ku∗i (vi − vi∗ )⊤ k2 i=1 r X i=1 √ √ (ε σi + ( σi + ε)ε) ≤ ρr ε √ where ρr ≡ i=1 (2 σi + 1). Now with εn,r ≡ can be further bounded from below as follows: Pr δ √n,r ν̄ c3 ρr ∧ 1 we see that (6.2) (6.2) ≥ ΠG (∩ri=1 {(u∗i , vi∗ ) : u∗i ∈ Bm1 (ui , εn,r ), vi∗ ∈ Bm2 (vi , εn,r )}) ≥ tr (m1 +m2 )r (τr,g ) r Y i=1 vol (Bm1 (ui , εn,r )) · vol (Bm2 (vi , εn,r )) −1 −1 tr r ≥ (τr,g · εn,r )(m1 +m2 )r vm v r ≥ e−(m1 +m2 )r·(log m̄/2+log τr,g +log(εn,r ∨1)) . 1 m2 √ where vd = vol(Bd (0, 1)), and vd ≥ (1/ d)d . Hence in order that the right 2 side of the above display can be bounded from below by e−2nδn,r , it suffices 26 Q. HAN to require that (6.3) log m̄ −1 . max log τr,g , log(ε−1 n,r ∨ 1) ≤ 2η 2 2 It is easy to calculate that ε−2 n,r ≤ 9ν̄ c3 (1 ∨ σmax (A0,r )) rmax n. Now the conclusion follows by noting that (4.2) implies (6.3) since rmax ≤ n and c3 = 1. Proof of Theorem 7. The theorem follows by Theorems 1 and 2, Proposition 1 coupled with Lemmas 12 and 14. 6.2. Proof of Theorem 11. Lemma 15. For any Σ0 ∈ M(k,s) , log N c5 ε, {Σ ∈ M(k,s) : kΣ − Σ0 kF ≤ CL ε}, k·kF √ ≤ ks log(ep/s) + ks log(6 kL/c5 ε). Proof. The set involved in the entropy is equivalent to o n (6.4) Λ ∈ R(k,s) (L) : kΛΛ⊤ − Λ0 Λ⊤ 0 kF ≤ CL ε, k·kF . √ We claim that supΛ∈R(k,s) kΛΛ⊤ kF ≤ kL. To see this, let Λ ≡ P ΞQ⊤ be the singular value decomposition of Λ, where P ∈ Rp×p , Q ∈ Rk×k are unitary matrices and Ξ ∈ Rp×k is a diagonal matrix. Then kΛΛ⊤ k2F = kΞΞ⊤ k2F ≤ kL, proving the claim. Combined with (6.4) and Euclidean embedding, we see that the entropy in question can be bounded by √ log N c5 ε, {v ∈ B0 (ks; pk) : kvk2 ≤ 2 kL}, k·k2  √ !ks  √ kL pk 6  ≤ ks log(ep/s) + ks log(6 kL/c5 ε), ≤ log  c5 ε ks where B0 (s; pk) ≡ {v ∈ Rpk : \|supp(v)\| ≤ s}. ′ ′ 2 Proof of Theorem 11. Take δn,(k,s) = KC ′ ks n log(C p) for some C ≥ e depending on c5 , c7 , L and some absolute constant K ≥ 1. Apparently (2.5) holds with c = 1, γ = 1. The prior Πn,(k,s) on M(k,s) will be the uniform disq log(C ′ p) covering-ball of the set {Σ ∈ M(k,s) } tribution on a minimal C ′ cks 3n under the Frobenius norm k·kF . The above lemma entails that the cardinality for such a cover is no more than exp(C ′′ ks log(C ′′ p)) for another constant C ′′ ≥ e depending on c3 , c5 , c7 , L. Hence o n 2 ≥ exp(−C ′′ ks log(C ′′ p)), /c3 Πn,(k,s) Σ ∈ M(k,s) : kΣ − Σ0,(k,s)kF ≤ δn,(k,s) 2 which can be bounded from below by exp(−2nδn,(k,s) ) by choosing K large enough. The claim of Theorem 11 now follows from these considerations along with Theorems 1 and 6, Proposition 1. BAYES MODEL SELECTION 27 Appendix A. Proof of lemmas in Section 3 (n) Proof of Lemma 2. Let Pθ0 denote the probability measure induced by the joint distribution of (X1 , . . . , Xn ) when the underlying signal is θ0 . First consider Gaussian regression case. It is easy to calculate that (n) n X p0 1 1 (X (n) ) = − (Xi − θ0,i )2 + (Xi − θ1,i )2 , log θ(n) 2 2 p θ1 i=1 (n) (n) Pθ0 log p θ0 (n) p θ1 1 (X (n) ) = nℓ2n (θ0 , θ1 ). 2 Then (n) Pθ0 exp " ≤ P exp (n) (n) λ log n X i=1 p θ0 (n) p θ1 (X (n) )− εi λ θ0,i − θ1,i (n) Pθ0 log ! p θ0 (n) p θ1 (X (n) !# ) ≤ exp λ2 nℓ2n (θ0 , θ1 )/2 . Next consider binary regression. Easy calculation shows that (n) log Pθ(n) log 0 p θ0 (n) p θ1 (n) p θ0 (n) p θ1 = n X Xi log 1 − θ0,i θ0,i + (1 − Xi ) log , θ1,i 1 − θ1,i θ0,i log θ0,i 1 − θ0,i + (1 − θ0,i ) log . θ1,i 1 − θ1,i i=1 = n X i=1 Using the inequality cx ≤ log(1 + x) ≤ x for all −1 < x ≤ c′ for some c > 0 depending on c′ > −1 only, we have shown Pθ(n) log 0 (n) 0 (n) pθ 1 pθ ≍ nℓ2n (θ0 , θ1 ) under the assumed condition that Θn ⊂ [η, 1 − η]n . Now we verify the Bernstein condition: " !# (n) (n) p θ0 p θ0 (n) Pθ0 exp λ log (n) − Pθ(n) log (n) 0 p θ1 p θ1 ! ! n n X X (n) 2 2 ti /8 (Xi − θ0,i )ti ≤ exp λ = Pθ0 exp λ i=1 i=1 θ 1−θ where ti ≡ ti (θ0 , θ1 ) = log 1−θ0,i0,i · θ1,i1,i and the last inequality follows from Hoeffding’s inequality (cf. Section 2.6 of [9]). The claim follows by h i2 θ0,i −θ1,i noting that t2i = log (1−θ + 1 ≍ (θ0,i − θ1,i )2 by the assumed 0,i )θ1,i condition and the aforementioned inequality log(1 + x) ≍ x in a constrained range. 28 Q. HAN Finally consider Poisson regression. It is easy to see that (n) log (n) Pθ0 log p θ0 (n) p θ1 (n) p θ0 (n) p θ1 = n X Xi log θ0,i + (θ1,i − θ0,i ), θ1,i θ0,i log θ0,i + (θ1,i − θ0,i ). θ1,i i=1 = n X i=1 Note that for any 1/M ≤ p, q ≤ M , 2 q q q p − 1 ≍ (p − q)2 , p log − (p − q) = p − log − 1 + ≍p· q p p p where in the middle we used the fact that − log x − 1 + x ≍ (x − 1)2 for x (n) bounded away from 0 and ∞. This shows that Pθ0 log (n) 0 (n) pθ 1 pθ ≍ nℓ2n (θ0 , θ1 ). Next we verify the Bernstein condition: " !# (n) (n) p θ0 p θ0 (n) Pθ0 exp λ log (n) − Pθ(n) log (n) 0 p θ1 pθ # # 1 " n " n X X (n) λti θ0,i e − 1 − λti (Xi − θ0,i )ti ≤ exp ≤ Pθ0 exp λ i=1 i=1 where ti = log(θ0,i /θ1,i ). Now for any \|λ\| ≤ 1, we have eλti −1−λti ≍ λ2 t2i ≤ λ2 t2i /(1−\|λ\|). On the other hand, θ0,i t2i = θ0,i (log(θ0,i /θ1,i ))2 ≍ (θ0,i −θ1,i )2 , completing the proof. Proof of Lemma 3. Since the log-likelihood ratio for X1 , . . . , Xn can be decomposed into sums of the log-likelihood ratio for single samples, and the log-likelihood ratio is uniformly bounded over F (since G is bounded), classical Bernstein inequality applies to see that for any couple (f0 , f1 ), the Bernstein condition in Assumption A holds with v = κg nVarf0 (log f0 /f1 ), c = κΓ where κg , κΓ depend only on G. Hence we only need to verify that Varf0 (log f0 /f1 ) . h2 (f0 , f1 ), Pf0 (log f0 /f1 ) ≍ h2 (f0 , f1 ). This can be seen by Lemma 8 of [24] and the fact that Hellinger metric is dominated by the Kullback-Leiber divergence. Lemma 16. Let Z be a random variable bounded by M > 0. Then E exp(Z) ≤ exp eM EZ . Proof. Note that log E exp(Z) = log (E [exp(Z) − 1] + 1) ≤ E [exp(Z) − 1] ≤ eM EZ. P where the last inequality follows from Taylor expansion ex −1 = nk=1 xk /k! ≤ P x k≥1 M k−1 /k! ≤ xeM . BAYES MODEL SELECTION 29 Proof of Lemma 4. We omit explicit dependence of M on the notation dr,M (n) and rM in the proof. Let Pf0 denote the probability measure induced by the joint distribution of (X0 , . . . , Xn ) where X0 is distributed according to the stationary density qf0 . Easy computation shows that (n) n−1 X p f0 1 2 log (n) = εi+1 (f0 (Xi ) − f1 (Xi )) + (f0 (Xi ) − f1 (Xi )) , 2 p f1 i=0 (n) Z p f0 n (n) Pf0 log (n) = (f0 − f1 )2 qf0 dλ. 2 p f1 Here λ denotes the Lebesgue measure on R. By the arguments on page 209 of [23], we see that r . qf0 . r. Hence we only need to verify the Bernstein condition. By Cauchy-Schwarz inequality, (A.1) 2   ! (n) n−1 X p (n) f (n) 0  ≤ P exp 2λ P exp λ log εi+1 (f0 (Xi ) − f1 (Xi )) f0 f0 (n) p f1 i=0 ! n−1 X (n) (f0 (Xi ) − f1 (Xi ))2 × Pf0 exp λ i=0 ≡ (I) × (II). The first term (I) can be handled by an inductive calculation. First note that for any \|µ\| ≤ 2 and X1 ∈ R, (A.2) 2 (f Pp(·\|X1 ) eµ 2 0 (X2 )−f1 (X2 ) ≤ ee 16M 2 µ2 P 2 p(·\|X1 ) (f0 −f1 )(X2 ) 2 d2 (f ,f ) r 0 1 ≤ eCM µ where the first inequality follows from Lemma 16 and the second inequality follows from r(·) . pf (·\|x) . r(·) holds for all x ∈ R where the constant Pn−1 involved depends only on M . Let Sn ≡ i=0 εi+1 (f0 (Xi ) − f1 (Xi )) and εn ≡ (ε1 , . . . , εn ). Then for \|λ\| ≤ 1, let µ ≡ 2λ, (n) (n) Pf0 e2λSn = Pf0 eµSn = EX0 ,εn−1 eµSn−1 Eεn eµεn (f0 (Xn−1 )−f1 (Xn−1 )) 2 2 ≤ EX0 ,εn−1 eµSn−1 eµ (f0 (Xn−1 )−f1 (Xn−1 )) /2 h i 2 2 ≤ EX0 ,εn−2 eµSn−2 Eεn−1 eµεn−1 (f0 (Xn−2 )−f1 (Xn−2 ))+µ (f0 (Xn−1 )−f1 (Xn−1 )) /2 1/2 ≤ EX0 ,εn−2 eµSn−2 Eεn−1 e2µεn−1 (f0 (Xn−2 )−f1 (Xn−2 )) 1/2 µ2 (f0 (Xn−1 )−f1 (Xn−1 ))2 × Ep(·\|Xn−2 ) e 2 2 2 2 ≤ EX0 ,εn−2 eµSn−2 eµ (f0 (Xn−2 )−f1 (Xn−2 )) · eCM µ dr (f0 ,f1 )/2 , 30 Q. HAN where the last inequality follows from (A.2). Now we can iterate the above calculation to see that (I) ≤ exp(CM λ2 nd2r (f0 , f1 )). (A.3) Next we consider (II). Since for any non-negative random variables Z1 , . . . , Zn , Q Q we have E ni=1 Zi ≤ ni=1 (EZin )1/n . It follows that (A.4) n Y 1/n (n) Pf0 exp nλ(f0 (Xi ) − f1 (Xi ))2 = Pqf0 exp(nλ(f0 (X0 ) − f1 (X0 ))2 (II) ≤ i=1 where the last inequality follows by stationarity. On the other hand, by Jensen’s inequality,   (n) p λn f0  (n) exp −λPf0 log (n) ≤ exp − Pqf0 (f0 − f1 )2 2 (A.5) p f1 ≤ Pqf0 exp(−λn(f0 (X0 ) − f1 (X0 ))2 /2). Collecting (A.1) and (A.3)-(A.5), we see that for \|λ\| ≤ 1,     (n) (n) (n) p p f0 p f0 p (n) f0  Pf (n) exp λ log (n) − Pf (n) log (n)  ≤ (I) · (II) exp −λPf0 log (n) 0 0 p f1 p f1 p f1 ′ ≤ exp(CM λ2 nd2r (f0 , f1 )), completing the proof. (n) Proof of Lemma 5. For any g ∈ G, let pg denote the probability density function of a n-dimensional multivariate normal distribution with covariance (n) matrix Σg ≡ Tn (fg ), and Pg the expectation taken with respect to the (n) density pg . Then for any g0 , g1 ∈ G, (n) log (A.6) pg 0 (n) pg 1 1 1 −1 (n) − log det(Σg0 Σ−1 (X (n) ) = − (X (n) )⊤ (Σ−1 g0 − Σg1 )X g1 ), 2 2 (n) Pg(n) log 0 pg 0 (n) pg 1 1 1 log det(Σg0 Σ−1 = − tr(I − Σg0 Σ−1 g1 ) − g1 ) 2 2 where we used the fact that for a random vector X with covariance matrix −1/2 (n) Σ, EX ⊤ AX = tr(ΣA). Let G ≡ Σg0 X (n) ∼ N (0, I) under Pg0 , and 1/2 1/2 B ≡ I − Σg0 Σ−1 g1 Σg0 , then (n) Yn ≡ log pg 0 (X (n) ) − Pg(n) log 0 (n) (n) pg 0 (n) pg 1 pg 1 i i 1h ⊤ 1 h (n) ⊤ −1 (n) −1 G BG − tr(B) . )X − tr(I − Σ Σ ) = − = − (X ) (Σg0 − Σ−1 g0 g1 g1 2 2 BAYES MODEL SELECTION 31 Let B = U ⊤ ΛU be the spectral decomposition of B where U is orthonormal and Λ = diag(λ1 , . . . , λn ) is a diagonal matrix. Then we can further compute ⊤ −2Yn =d G ΛG − tr(Λ) = n X i=1 λi (gi2 − 1) where g1 , . . . , gn ’s are i.i.d. standard normal. Note that for any \|t\| < 1/2, 1 √ 2π Z ∞ et(x 2 −1) e−x −∞ 2 /2 dx = √ 1 e−t 2 = e 2 (− log(1−2t)−2t) ≤ et /(1−2t) 1 − 2t P 1 k where the inequality follows from − log(1 − 2t) − 2t = k≥2 k (2t) = P 2t2 1 (2t)k ≤ 1−2t . Hence apply the above display with t = −λλi /2, 4t2 k≥0 k+2 we have that for any \|λ\| < 1/ maxi λi , (A.7) Z ∞ n n Y Y 1 2 2 2 √ E exp −λ · λi (gi − 1)/2 = E exp(λYn ) = e−λ·λi (x −1)/2 e−x /2 dx 2π −∞ i=1 i=1 P n Y λ2 i λ2i λ2 λ2i exp ≤ ≤ exp . 4 + 4λλi 4 − 4\|λ\| maxi \|λi \| i=1 Denote k·k and k·kF the matrix operator norm and Frobenius norm respectively. By the arguments on page 203 of [23], we have kΣg k ≤ 2πkeg k∞ −1 −g and kΣ−1 g k ≤ (2π) ke k∞ . Since G is a class of uniformly bounded function classes, the spectrum of the covariance matrices Σg and their inverses running over g must be bounded. Hence (A.8) −1 max\|λi \| = kBk = k(Σg1 − Σg0 )Σ−1 g1 k ≤ kΣg1 − Σg0 kkΣg1 k ≤ CG < ∞. i Next, note that (A.9) X i λ2i 1/2 = (tr(BB ⊤ ))1/2 = kBkF = k(Σg1 − Σg0 )Σ−1 g1 kF ′ ≤ kΣ−1 g1 kkΣg1 − Σg0 kF ≤ CG p nDn2 (g0 , g1 ) where in the first inequality we used kM N kF = kN M kF for symmetric matrices M, N and the general rule kP QkF ≤ kP kkQkF . Collecting (A.7)(A.9) we see that Assumption A is satisfied for v = κg nDn2 (g0 , g1 ) and c = κΓ for some constants κg , κΓ depending on G only. 32 Q. HAN (n) Finally we establish equivalence of n1 Pg0 log (A.6), we have (n) pg0 (n) pg1 and Dn2 (g0 , g1 ). First by (n) 1 1 = − tr(I − Σg0 Σ−1 log det(Σg0 Σ−1 g1 ) − g1 ) 2 2 1 −1/2 −1/2 −1/2 tr Σg1 (Σg0 − Σg1 )Σg−1/2 − log det I + Σ (Σ − Σ )Σ = g0 g1 g1 g1 1 2 1 1 2 2 −1 2 ′′ 2 ≤ kI − Σg0 Σ−1 g1 kF ≤ kΣg1 − Σg0 kF kΣg1 k ≤ CG nDn (g0 , g1 ). 4 4 Here in the second line we used the fact that det(AB −1 ) = det(I +B −1/2 (A− B)B −1/2 ), and in the third line we used the fact − log det(I + A) + tr(A) ≤ 1 2 2 tr(A ) for any p.s.d. matrix A, due to the inequality log(1 + x) − x ≥ − 12 x2 for all x ≥ 0. On the other hand, by using the reversed inequality log(1 + x) − x ≤ −cx2 for all 0 ≤ x ≤ c′ where c is a constant depending only Pg(n) log 0 pg 0 (n) pg 1 (n) on c′ , we can establish Pg0 log the proof. (n) pg0 (n) pg1 ≥ CG′′′ nDn2 (g0 , g1 ), thereby completing Proof of Lemma 6. Note that (n) n X p Σ0 1 1 ⊤ −1 −1 −1 (n) X (Σ0 − Σ1 )Xi − log det(Σ0 Σ1 ) , log (n) (X ) = − 2 i 2 p i=1 Σ1 (n) PΣ0 log (n) p Σ0 (n) p Σ1 n n log det(Σ0 Σ−1 = − tr(I − Σ0 Σ−1 1 )− 1 ). 2 2 The rest of the proof proceeds along the same line as in Lemma 5. Appendix B. Proof of remaining theorems in Section 4 B.1. Proof of Theorem 8. Lemma 17. Let n ≥ 2. Then for any g ∈ Fm , log N c5 ε, {f ∈ Fm : ℓn (f, g) ≤ 2ε}, ℓn ≤ 2 log(6/c5 ) · m log(en). Proof of Lemma 17. Let Qm denote all m-partitions of the design points n x1 , . . . , xn . Then it is easy to see that \|Qm \| = m−1 . For a given mpartition Q ∈ Qm , let Fm,Q ⊂ Fm denote all monotonic non-decreasing functions that are constant on the partition Q. Then the entropy in question can be bounded by n (B.1) log max N c5 ε, {f ∈ Fm,Q : ℓn (f, g) ≤ 2ε}, ℓn . m − 1 Q∈Qm On the other hand,√for any fixed m-partition Q ∈ Qm√ , the entropy term above equals N c5 nε, {γ ∈ Pn,m,Q : kγ − gk2 ≤ 2 nε}, k·k2 , where Pn,m,Q ≡ {(f (x1 ), . . . , f (xn )) : f ∈ Fm,Q }. By Pythagoras theorem, the set BAYES MODEL SELECTION 33 involved in the entropy is included in {γ ∈ Pn,m,Q : kγ − πPn,m,Q (g)k2 ≤ √ 2 nε} where πPn,m,Q is the natural projection from Rn onto the subspace Pn,m,Q . Clearly Pn,m,Q is contained in a linear subspace with dimension no more than m. Using entropy result for the finite-dimensional space [Problem 2.1.6 in [47], page 94 combined with the discussion in page 98 relating the packing number and covering number], (B.2) √ m 3 · 2 nε √ log N c5 ε, {f ∈ Fm,Q : ℓn (f, f0,m ) ≤ 2ε}, ℓn ≤ log = m log(6/c5 ). c5 nε n The claim follows by (B.1)-(B.2), and log m−1 ≤ m log(en). 1 m log(en) 5) 2 ∨ . It is clear that (2.5) ≡ 4 log(6/c Hence we can take δn,m c7 η n is satisfied with c ≡ 1 and γ = 1. Lemma 18. Suppose that (4.5) holds . Then (P2) in Assumption C holds. Proof of Lemma 18. Let Q0,m = {Ik }m k=1 be the associated m-partition of {x1 , . . . , xn } of f0,m ∈ Fm with the convention that {Ik } ⊂ {x1 , . . . , xn } is ordered from smaller values to bigger ones. Thenit is easy to see that µ0,m = (µ0,1 , . . . , µ0,m ) ≡ f0,m (xi(1) ), . . . , f0,m (xi(m) ) ∈ Rm is well-defined and µ0,1 ≤ . . . ≤ µ0,m . It is easy to see that any f ∈ Fm,Q0,m satisfying √ the property that sup1≤k≤m \|f (xi(k) ) − µ0,k \| ≤ δn,m / c3 leads to the error 2 /c . Hence estimate ℓ2n (f, f0,m ) ≤ δn,m 3 (B.3) 2 /c3 } Πn,m {f ∈ Fm : ℓ2n (f, f0,m ) ≤ δn,m −1 n 2 ≥ Πḡm {f ∈ Fm,Q0,m : ℓ2n (f, f0,m ) ≤ δn,m /c3 } m−1 −1 m √ n ≥ Πḡm {µ ∈ Rm : µ ≡ µ0,k + εk k=1 , 0 ≤ ε1 ≤ . . . ≤ εm ≤ δn,m / c3 } m−1 −1 √ m 1 n ≥ · inf √ ḡm (µ)(1 ∧ δn,m / c3 ) m m m−1 m! µ∈R :µ≡(µ0,k +εk )k=1 ,0≤ε1 ≤...≤εm ≤1∧δn,m / c3 −1 √c 3 iso )−1 ∨1 −m log √ n −m log(en)−m log (τm,g ∨1 iso m δn,m · (τm,g ) (1 ∧ δn,m / c3 )m ≥ e ≥ . m−1 Here the first inequality in the last line follows from the definition of ḡm iso . The claim follows by verifying (4.5) implies that the second and and τm,g 1 · m log(en) [the third term in the exponent above are both bounded by 2η √ −1 third term does not contribute to the condition since c3 δn,m ≤ n by noting c3 = 1 in the Gaussian regression setting and definition of η]. Proof of Theorem 8. The theorem follows by Theorems 1 and 2, Proposition 1 coupled with Lemmas 17 and 18. We now prove Lemma 7. We need the following result. 34 Q. HAN Lemma 19. Let f0 := (f0 (x1 ), . . . , f0 (xn )) ∈ Rn , and f0,m := (f0,m (x1 ), . . . , f0,m (xn )) ∈ Rn where f0,m ∈ arg ming∈Fm ℓ2n (f0 , g). Suppose that kf0 k2 ≤ L, and that there exists some element f ∈ Fm such that f ≡ (f (x1 ), . . . , f (xn )) satisfies kf k2 ≤ L. Then kf0,m k2 ≤ 3L. Proof of Lemma 19. It can be seen that f0,m ∈ arg minγ∈Pn,m Lf0 (γ) ≡ arg minγ∈Pn,m kf0 − γk2 where Pn,m ≡ {(f (x1 ), . . . , f (xn )) : f ∈ Fm }. For any γ ∈ Pn,m such that kγk2 ≤ L, the loss function satisfies Lf0 (γ) ≤ 2L by triangle inequality. If kf0,m k2 > 3L, then Lf0 (f0,m ) = kf0 − f0,m k2 ≥ kf0,m k2 − kf0 k2 > 3L − L = 2L, contradicting the definition of f0,m as a minimizer of Lf0 (·) over Pm,n . This shows the claim. R1 2 R 1 Proof of Lemma 7. Let L = 0 f . Note that kf0 k22 ≤ 2n 0 f 2 (x) dx = √ 2nL2 . By Lemma 19, we see that kf0,m k2 ≤ 3 2nL which √ entails that √ kf0,m k∞ ≤ 3 2nL. Now the conclusion follows from g(3 2nL + 1) ≥ (en)−1/2η while the left side is at least on the order of n−α/2 as n → ∞. B.2. Proof of Theorem 9. Checking the local entropy assumption B requires some additional work. The notion of pseudo-dimension will be useful in this regard. Following [39] Section 4, a subset V of Rd is said to have pseudo-dimension t, denoted as pdim(V ) = t, if for every x ∈ Rt+1 and indices I = (i1 , · · · , it+1 ) ∈ {1, · · · , n}t+1 with iα 6= iβ for all α 6= β, we can always find a sub-index set J ⊂ I such that no v ∈ V satisfies both vi > xi for all i ∈ J and vi < xi for all i ∈ I \ J. Lemma 20. Let n Suppose that pdim(Pn,m ) ≤ Dm where Pn,m := ≥ 2. n { f (x1 ), . . . , f (xn ) ∈ R : f ∈ Fm }. Then for all g ∈ Fm , log N c5 ε, {f ∈ Fm : ℓn (f, g) ≤ 2ε}, ℓn ) ≤ C · Dm log n for some constant C > 0 depending on c5 . To prove Lemma 20, we need the following result, cf. Theorem B.2 [25]. Lemma 21. Let V be a subset of Rn with supv∈V kvk∞ ≤ B and pseudodimension at most t. Then, for every ε > 0, we have N (ε, A, k·k2 ) ≤ √ κt , holds for some absolute constant κ ≥ 1. 4 + 2Bε n Proof of Lemma 20. Note that the entropy in question can be bounded by √ √ log N c5 ε n, {Pn,m − g} ∩ Bn (0, 2 nε), k·k2 . Since translation does not change the pseudo-dimension of a set, Pn,m − g has the same pseudo-dimension with that of Pn,m , which is bounded √ from above by Dm by assumption. Further note that {Pn,m − g} ∩ Bn(0, 2 nε) is √ uniformly bounded by 2 nε, hence an application of Lemma 21 yields that the above display can be further bounded as follows: log N c5 ε, {f ∈ Fm : ℓn (f, g) ≤ 2ε}, ℓn ) ≤ κDm log 4 + 4n/c5 ) ≤ C · Dm log n for some constant C > 0 depending on c5 whenever n ≥ 2. BAYES MODEL SELECTION 35 The pseudo-dimension of the class of piecewise affine functions Fm can be well controlled, as the following lemma shows. Lemma 22 (Lemma 4.9 in [26]). pdim(Pn,m ) ≤ 6md log 3m. As an immediate result of Lemmas 20 and 22, we can take for n ≥ 2, := (C ∨ 1/η) d · logn n · m log 3m for some C ≥ 2/c7 depending on c5 , c7 . 2 δn,m Lemma 23. Suppose that (4.8) holds and n ≥ d. Then (P2) in Assumption C holds. Proof of Lemma 23. We write f0,m ≡ max1≤i≤m ai · x + bi throughout √ the proof. We first claim that for any a∗i ∈ Bd (ai , δn,m /2 c3 d) and b∗i ∈ √ ∗ ∗ ∗ ∗ (x) := max B1 (bi , δn,m /2 c3 ), let gm 1≤i≤m (ai · x + bi ), then ℓ∞ (gm , f0,m ) ≤ √ δn,m / c3 . To see this, for any x ∈ X, there exists some index ix ∈ {1, . . . , m} ∗ (x) = a∗ · x + b∗ . Hence such that gm ix ix ∗ gm (x) − f0,m (x) ≤ a∗ix − aix · x + b∗ix − bix ≤ ka∗ix − aix k2 kxk2 + \|b∗ix − bix \| δn,m δn,m √ δn,m ≤ √ · d+ √ = √ . 2 c3 c3 2 c3 d The reverse direction can be shown similarly, whence the claim follows by taking supremum over x ∈ X. This entails that (B.4) 2 /c3 Πn,m f ∈ Fm : ℓ2n (f, f0,m ) ≤ δn,m n p √ o ∗ ∗ ∗ ∗ c d), b ∈ B (b , δ /2 c3 ) (a , b ) : a ∈ B (a , δ /2 ≥ ΠG ∩m 3 1 i n,m i n,m d i i i i i=1 = m Y i=1 m Y p √ Πg⊗d Bd (ai , δn,m /2 c3 d) · Πg B1 (bi , δn,m /2 c3 ) d δn,m δn,m √ √ g(kai k∞ + 1) · g(\|bi \| + 1) · ≥ ∧1 ∧ 1 vd 4c3 4c3 d i=1 √ 1 4c3 d −1 ∨ 1 − md log d ≥ exp − 2m(d + 1) log τm,g ∨ 1 − m(d + 1) log δn,m 2 √ where vd ≡ vol(Bd (0, 1)) and we used the fact that vd ≥ (1/ d)d . Now by requiring that n ≥ d and (B.5) √ d 4c3 d −1 ∨1 ≤ log n · m log 3m, max 2m(d + 1) log τm,g ∨ 1 , m(d + 1) log δn,m 2η √ −1 ≤ √n, the claim follows by verifying (4.8) implies (B.5)[since 4c3 dδn,m the second term is bounded by md log n. The inequality follows by noting η < 1/4]. d Lemma 24. For n ≥ 2, (2.5) is satisfied for c = 1 and γ = 2. 36 Q. HAN 2 = c log n(m log 3m) throughProof. For fixed n ≥ 2 and η > 0, write nδn,m out the proof, where c ≥ 2/c7 . Then for any α ≥ c7 /2 and h ≥ 1, since log(3m′ ) ≥ log(3hm) ≥ log(3m) for any m′ ≥ hm, we have X −αnδ2 X 2 ′ e−αchm log n log 3m n,m′ ≤ ≤ 2e−αhnδn,m . e e−αcm (log n·log 3m) = −αc log n log 3m 1−e ′ ′ m ≥hm m ≥hm For the second condition of (2.5), note that for γ = 2, in order to verify 2 2 , it suffices to have hm log(3hm) ≤ h2 m log(3m), equivalently δn,hm ≤ h2 δn,m 3hm ≤ (3m)h , and hence 3h−1 ≥ h for all h ≥ 1 suffices. This is valid and hence completing the proof. Proof of Theorem 9. This is a direct consequence of Theorems 1 and 2, Lemma 23 and 24, combined with Proposition 1. B.3. Proof of Theorem 10. Lemma 25. Let n ≥ 2, then for any g ∈ F(s,m) , log N (c5 ε, {f ∈ F(s,m) : ℓn (f, g) ≤ 2ε}, ℓn ) ≤ 2 log(6/c5 ) s log(ep) + m log(en) . Proof. The proof borrows notation from the proof of Lemma 17. Further let Ss denote all subsets of {1, . . . , p} with cardinality at most s. Then the entropy in question can be bounded by p n log N (c5 ε, {f ∈ F(s,m),(S,Q) : ℓn (f, g) ≤ 2ε}, ℓn ) max s m − 1 S∈Ss ,Q∈Qm ≤ s log(ep) + m log(en) √ √ log N c5 nε, {γ ∈ Pn,(S,Q) : kγ − gk2 ≤ 2 nε}, k·k2 + max S∈Ss ,Q∈Qm n n where Pn,(S,Q) ≡ {(x⊤ i β+u(zi ))i=1 ∈ R : supp(β) = S, u is constant on the partitions of Q} is contained in a linear subspace of dimension no more than s + m. Now similar arguments as in Lemma 17 shows that the entropy term in the above display can be bounded by (s + m) log(6/c5 ), proving the claim. log(en) 5) 2 ∨ η1 s log(ep)+m . Hence we can take δn,(s,m) ≡ 4 log(6/c c7 n Lemma 26. (2.5) holds with c = 1 and γ = 1. Proof. For the first condition of (2.5), note that for any h ≥ 1, with c′ ≡ 4 log(6/c5 ) ∨ η1 in the proof, for any α ≥ c7 /2, αc′ ≥ 2 log(6/c5 ) ≥ 2 since c7 c5 ≤ 1/4, X X X 2 ′ ′ −αnδn,(s ′ ,m′ ) e = e−αc s log(ep) e−αc m log(en) (s′ ,m′ )≥(hs,hm) s′ ≥hs m′ ≥hm ′ ′ 2 −αnhδn,(s,m) ≤ (1 − e−αc )−2 e−αc h(s log(ep)+m log(en) ≤ 2e The second condition of (2.5) is easy to verify by our choices of c, γ. Lemma 27. Suppose (4.11) holds. Then (P2) in Assumption C holds. . BAYES MODEL SELECTION 37 Proof. Using notation in Lemma 25, (B.6) 2 /c3 } Πn,(s,m) {f ∈ F(s,m) : ℓ2n (f, f0,(s,m) ) ≤ δn,(s,m) −1 −1 p n 2 ≥ /c3 } Πg⊗s ⊗ḡm {f ∈ F(s,m),(S0 ,Q0) : ℓ2n (f, f0,(s,m) ) ≤ δn,(s,m) s m−1 5) ∨ η1 throughout the proof, and where f0,m ∈ F(s,m),(S0 ,Q0) . Let c′ ≡ 4 log(6/c c7 2 2 δn,s ≡ c′ s log(ep)/n and δn,m ≡ c′ m log(en)/n. To bound the prior mass of the above display from below, it suffices to bound the product of the following two terms: 2 πs ≡ Πg⊗s {β ∈ B0 (s) : βS0c = 0, ℓ2n (hβ , hβ0,s ) ≤ δn,s /2c3 } , (B.7) 2 πm ≡ Πḡm {u ∈ Um,Q0 : ℓ2n (u, u0,m ) ≤ δn,m /2c3 } . The first term equals √ √ Πg⊗s β ∈ B0 (s) : βS0c = 0, kXβ − Xβ0,s k2 ≤ nδn,s / 2c3 (B.8) δn,s 1 c √ ≥ Πg⊗s β ∈ B0 (s) : βS0 = 0, kβ − β0,s k2 ≤ · σΣ 2c3 Here the inequality follows by noting 2 kXβ − Xβ0,s k22 ≤ n(β − β0,s )⊤ Σ(β − β0,s ) ≤ nσΣ kβ − β0,s k22 , √ where σΣ denotes the largest singular value of X ⊤ X/n. Note that σΣ ≤ p since the trace for X ⊤ X/n is p and the trace of a p.s.d. matrix dominates the largest eigenvalue. The set in the last line of (B.8) is supported on RpS0 s δ 1 s √n,s ∧ 1 · and hence can be further bounded from below by τs,g vs σΣ 2c3 where vs = vol(Bs (0, 1)). Hence s δn,s 1 s ∧ 1 vs ·√ πs ≥ (τs,g ∧ 1) σΣ 2c3 (B.9) 2 s 2c3 σΣ 1 −1 ≥ exp − s log s − s log τs,g ∨ 1 − log ∨1 , 2 2 2 δn,s √ where in the last inequality we used that vs ≥ (1/ s)s . By repeating the arguments in the proof of Lemma 18, we have m 2c3 −1 ∨1 . πm ≥ exp − m log τm,g ∨ 1 − log (B.10) 2 2 δn,m Combining (B.6), (B.7), (B.9) and (B.10), we see that (B.11) 2 /c3 } Πn,(s,m) {f ∈ F(s,m) : ℓ2n (f, f0,(s,m) ) ≤ δn,(s,m) −1 −1 ∨1 ∨ 1 − m log τm,g ≥ exp −2s log(ep) − m log(en) − s log τs,g 2 2c3 σΣ 2c3 m s ∨ 1 − log ∨1 . × exp − log 2 2 2 δn,s 2 δn,m 38 Q. HAN In order that the right side of the above display bounded from below by 2 exp(−2nδn,(s,m) ), we only need to require that ) √ ( 2c3 σΣ ∨1 −s log 1 −1 δ n,s ≥ e− 2η s log(ep) , min e−s log(τs,g ∨1) , e ( −1 τm,g ∨1 min e−m log( −m log ), e √ 2c3 δn,m ) ∨1 1 ≥ e− 2η m log(en) . The first terms in the above two lines lead to (4.11). The other terms in the 2c3 c7 2 n≤ above two lines do not contribute by noting that 2c3 /δn,m ≤ 4 log(6/c 5) (1/2)n ≤ en since c3 = 1 (in Gaussian regression model) and c7 ∈ (0, 1), 2 /δ 2 ≤ σ 2 n ≤ pn ≤ p2 and η < 1/4. while 2c3 σΣ n,s Σ Proof of Theorem 10. The claim of the theorem follows by Theorems 1 and 2, Proposition 1 and Lemmas 25-27. Appendix C. Proof of auxiliary lemmas in Section 5 Proof of Lemma 9. Let Fj := {f ∈ F : jε < dn (f, f0 ) ≤ 2jε} and Gj ⊂ Fj be the collection of functions that form a minimal c5 jε covering set of Fj under the metric dn . Then by assumption \|Gj \| ≤ N (jε). Furthermore, for each g ∈ Gj , it follows by Lemma 1 that there exists some test ωn,j,g such that 2 (n) (n) sup Pf0 ωn,j,g + Pf (1 − ωn,j,g ) ≤ c6 e−c7 ndn (f0 ,g) . (C.1) f ∈F :dn (f,g)≤c5 dn (f0 ,g) Recall that g ∈ Gj ⊂ Fj , then dn (f0 , g) > jε. Hence the indexing set above contains {f ∈ F : dn (f, g) ≤ c5 jε}. Now we see that (n) Pf0 ωn,j,g ≤ c6 e−c7 nj 2 ε2 , (n) sup f ∈F :dn (f,g)≤c5 jε Pf (1 − ωn,j,g ) ≤ c6 e−c7 nj Consider the global test φn := supj≥1 maxg∈Gj ωn,j,g , then X XX 2 2 (n) (n) N (jε)e−c7 nj ε ωn,j,g ≤ c6 Pf0 φn ≤ Pf0 ≤ c6 N (ε) j≥1 e−c7 . j≥1 j≥1 g∈Gj X 2 ε2 nj 2 ε2 2 ≤ c6 N (ε)e−c7 nε · 1 − e−c7 nε 2 −1 + . On the other hand, for any f ∈ F such that dn (f, f0 ) ≥ ε, there exists some j ∗ ≥ 1 and some gj ∗ ∈ Gj ∗ such that dn (f, gj ∗ ) ≤ j ∗ c5 ε. Hence ∗ 2 2 2 (n) (n) Pf (1 − φn ) ≤ Pf 1 − ωn,j ∗,gj∗ ≤ c6 e−c7 n(j ) ε ≤ c6 e−c7 nε . The right hand side of the above display is independent of individual f ∈ F such that dn (f, f0 ) ≥ ε and hence the claim follows. BAYES MODEL SELECTION 39 Proof of Lemma 10. By Jensen’s inequality, the left side of (5.6) is bounded by (n) Pf0 Z  (n) ≤ Pf0  (n) log Z p f0 (n) pf − (n) Pf0 log (n) p f0 (n) pf (n) log p f0 (n) (n) pf − Pf0 log  (n) ≤ exp(−Cλnε2 ) · c1 Pf0 exp λ 2 dΠ(f ) ≥ C + c3 )nε − c3 n (n) p f0 (n) pf Z Z d2n (f0 , f ) dΠ(f )  dΠ(f ) ≥ Cnε2  (n) log p f0 (n) pf (n) − Pf0 log (n) p f0 (n) pf  dΠ(f ) . Using Jensen’s inequality, the last term in the right side of the above display can be further bounded by   (n) (n) Z Z p p f0 (n) (n) f0  dΠ(f ) ≤ eψκg nd2n (f0 ,f ),κΓ (λ) dΠ(f ) − Pf0 log (n) Pf0 exp λ log (n) pf pf where the last inequality follows from Fubini’s theorem and Assumption A. Now the condition on the prior Π entails that (n) Pf0 Z (n) pf (n) p f0 −(C+c3 )nε2 dΠ(f ) ≤ e ≤ c1 exp −Cλnε2 + ψκg nε2 ,κΓ (λ) . The claim follows by choosing λ > 0 small enough depending on C, κ. Proof of Proposition 3. We may assume without loss of generality that dn (f0 , f0,m ) < ∞ so that m̃ is well-defined since \|M\| = ∞ and (2.5). By definition we have δn,m̃ ≥ dn (f0 , f0,m ) and δn,m̃−1 < dn (f0 , f0,m ). In this case, the global test can be constructed via φ̃n := supm′ ∈I,m′ ≥jhm̃ φn,m′ . Then analogous to (5.9) and (5.10), for any random variable U ∈ [0, 1], (n) (C.2) Pf0,m U · φ̃n ≤ 4c6 e−(c7 /2c sup 2 f ∈Fjhm̃ :d2n (f,f0,m )≥c2 (jh)γ δn, m̃ (n) Pf (1 − φ̃n ) ≤ 2c6 e−(c7 /c (n) Pf0,m (E˜nc ) 2 )njhδ 2 n,m̃ 2 )njhδ 2 n,m̃ −C ′ ≤ c1 e , . 2 c7 njhδn, m̃ 8c3 c2 Similar to (5.11), there exists an event Ẽn with and the following is true on the event E˜n : (C.3) Z Y 2 n c7 njhδn, m̃ pf 2 2 . dΠ(f ) ≥ λn,m e− 4c2 Πn,m f ∈ Fm : d2n (f, f0,m ) ≤ c7 jhδn, m̃ /8c3 c pf0,m i=1 40 Q. HAN Repeating the reasoning in (5.12), (5.13) and (5.14) we see that (C.4) (n) 2 4 γ 2 (n) Pf0,m Πn f ∈ F : dn (f, f0,m) > c (2jh) dn (f0 , f0,m ) X (1 − φ̃n )1Ẽn 2 ≤ ec7 njhδn,m̃ /4c λn,m Πn,m Z × ≤ (· · · ) × ≤ Ce−(c7 /4c n 2 2 /8c c2 f ∈ Fm : d2n (f, f0,m ) ≤ c7 jhδn, 3 m̃ (n) f ∈F :d2n (f,f0,m )>c4 (2jh)γ d2n (f0 ,f0,m ) sup 2 f ∈Fjhm̃ :d2n (f,f0,m )≥c2 (jh)γ δn, m̃ 2 )njhδ 2 n,m̃ o Pf (1 − φ̃n ) dΠ(f ) (n) Pf (1 − φ̃n ) + Π F \ Fjhm̃ . 2 Here the third line is valid since c4 (2jh)γ d2n (f0 , f0,m ) > c4 (2jh)γ δn, m̃−1 ≥ 2 2 γ 2 2 γ 2 c (jh) δn,m̃ by the right side of (2.5), which entails δn,m̃ ≤ c 2 δn,m̃−1 . The fourth line uses (C.2) and assumption (P1), together with the fact that δn,m̃ ≥ δn,m . (5.2) follows from (C.2), probability estimate for Enc and (C.4). Acknowledgements The author is indebted to Chao Gao for his numerous suggestions that lead to a substantially improved version of the paper. 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Ref: International Conference on Artificial Neural Networks (ICANN), Springer LNCS, Vol. 9887, pp. 170–178, Barcelona, Spain, September 2016. DNN-Buddies: A Deep Neural Network-Based Estimation Metric for the Jigsaw Puzzle Problem Dror Sholomon1 , Eli (Omid) David1 , and Nathan S. Netanyahu1,2 1 arXiv:1711.08762v1 [cs.CV] 23 Nov 2017 2 Department of Computer Science, Bar-Ilan University, Ramat-Gan 52900, Israel [email protected], [email protected], [email protected] Center for Automation Research, University of Maryland, College Park, MD 20742 [email protected] Abstract. This paper introduces the first deep neural network-based estimation metric for the jigsaw puzzle problem. Given two puzzle piece edges, the neural network predicts whether or not they should be adjacent in the correct assembly of the puzzle, using nothing but the pixels of each piece. The proposed metric exhibits an extremely high precision even though no manual feature extraction is performed. When incorporated into an existing puzzle solver, the solution’s accuracy increases significantly, achieving thereby a new state-of-the-art standard. (a) (b) Fig. 1: Jigsaw puzzle before and after reassembly using our DNN-Buddies scheme in an enhanced solver. 1 Introduction Jigsaw puzzles are a popular form of entertainment, available in different variation of difficulty to challenge children, adults and even professional players. Given n × m 2 D. Sholomon, E.O. David, N.S. Netanyahu different non-overlapping tiles of an image, the objective is to reconstruct the original image, taking advantage of both the shape and chromatic information of each piece. Despite the popularity and vast distribution of jigsaw puzzles, their assembly is not trivial computationally, as this problem was proven to be NP-hard [1] [8]. Nevertheless, a computational jigsaw solver may have applications in many realworld applications, such as biology [16], chemistry [25], literature [18], speech descrambling [27], archeology [2] [15], image editing [5], and the recovery of shredded documents or photographs [3,7,14,17]. Regardless, as noted in [11], research of the topic may be justified solely due to its intriguing nature. Recent years have witnessed a vast improvement in the research and development of automatic jigsaw puzzle solvers, manifested in both puzzle size, solution accuracy, and amount of manual human intervention required. In its most basic form, every puzzle solver requires some function to evaluate the compatibility of adjacent pieces and a strategy for placing the pieces as accurately as possible. Most strategies are greedy and rely heavily on some “trick” to estimate whether two pieces are truly adjacent (e.g. two pieces that are each the most compatible piece from all pieces to one another, four pieces that form a loop where each pair’s compatibility is above a threshold, etc). Such heuristics were dubbed an “estimation metric” in [20], as they allow estimating the adjacency correctness of two pieces without knowing the correct solution. The majority of recent works focused on devising elaborate, hand-crafted compatibility functions and high-precision estimation metrics. Despite the proven effectiveness of neural networks in the field of computer vision, no attempt has been made to automatically devise a high-precision estimation metric for the jigsaw puzzle problem. This might be due to the highly imbalanced nature of the puzzle problem, as in each n × m puzzle, there are O(n × m) matching piece-pairs and O(n2 × m2 ) possible mismatching ones. In this paper we propose a novel estimation metric relying on neural networks. The proposed metric achieves extremely high precision despite the lack of any manually extracted features. The proposed metric proves to be highly effective in real-world scenarios. We incorporated the metric in our GA-based solver, using no hand-crafted sophisticated compatibility measure and experimented with the currently known challenging benchmarks of the hardest variant of the jigsaw puzzle problem: non-overlapping, (28 × 28) square pieces (i.e. only chromatic information is available to the solver) where both piece orientation and puzzle dimensions are unknown. The enhanced solver proposed sets a new state-of-the-art in terms of the accuracy of the solutions obtained and the number of perfectly reconstructed puzzles. 2 Previous Work Jigsaw puzzles were first introduced around 1760 by John Spilsbury, a Londonian engraver and mapmaker. Nevertheless, the first attempt by the scientific community to computationally solve the problem is attributed to Freeman and Garder [9] who in 1964 presented a solver which could handle up to nine-piece problems. Ever since then, the research focus regarding the problem has shifted from shape-based to merely color-based solvers of square-tile puzzles. In 2010 Cho et al. [4] presented Deep Neural Network Based Estimation Metric for the Jigsaw Puzzle Problem 3 a probabilistic puzzle solver that could handle up to 432 pieces, given some a priori knowledge of the puzzle. Their results were improved a year later by Yang et al. [26] who presented a particle filter-based solver. Furthermore, Pomeranz et al. [20] introduced that year, for the first time, a fully automated square jigsaw puzzle solver that could handle puzzles of up to 3,000 pieces. Gallagher [10] has further advanced this by considering a more general variant of the problem, where neither piece orientation nor puzzle dimensions are known. Son et al. [24] improved the accuracy of the latter variant using so-called “loop-constraints”. Palkin and Tal [19] further improved the accuracy and handled puzzles with missing pieces. Sholomon et al. [21] presented a genetic algorithm (GA)-based solver for puzzles of known orientation which was later generalized to other variants [22,23]. 2.1 Compatibility Measures and Estimation Metrics As stated earlier, most works focus on the compatibility measure and an estimation metric. A compatibility measure is a function that given two puzzle piece edges (e.g. the right edge of piece 7 versus the upper edge of piece 12) predicts the likelihood that these two edges are indeed placed as neighbors in the correct solution. This measure applies to each possible pair of piece edges. The estimation metric, on the other hand, predict whether two piece edges are adjacent but may not apply to many possible pairs. Following is a more detailed review of the efforts made so far in the field. Cho et al. [4] surveyed four compatibility measures among which they found dissimilarity the most accurate. Dissimilarity is the sum (over all neighboring pixels) of squared color differences (over all color bands). Assuming pieces xi , xj are represented in some three-dimensional color space (like RGB or YUV) by a K × K × 3 matrix, where K is the height/width of a piece (in pixels), their dissimilarity, where xj is to the right of xi , for example, is v uK 3 uX X (xi (k, K, cb) − xj (k, 1, cb))2 , (1) D(xi , xj , r) = t k=1 cb=1 where cb denotes the color band. Pomeranz et al. [20] also used the dissimilarity measure but found empirically that using the (Lp )q norm works better than the usual L2 norm. Moreover, they presented the high-precision best-buddy metric. Pieces xi and xj are said to bestbuddies if ∀xk ∈ P ieces, C(xi , xj , R1 ) ≥ C(xi , xk , R1 ) and (2) ∀xp ∈ P ieces, C(xj , xi , R2 ) ≥ C(xj , xp , R2 ) where P ieces is the set of all given image pieces and R1 and R2 are “complementary” spatial relations (e.g. if R1 = right, then R2 = left and vice versa). Gallagher [10] proposed yet another compatibility measure, called the Mahalanobis gradient compatibility (MGC) as a preferable compatibility measure to those 4 D. Sholomon, E.O. David, N.S. Netanyahu used by Pomeranz et al. [20]. The MGC penalizes changes in intensity gradients, rather than changes in intensity, and learns the covariance of the color channels, using the Mahalanobis distance. Also, Gallagher suggested using dissimilarity ratios. Absolute distances between potential piece edge matches are sometimes not indicative (for example in smooth surfaces like sea and sky), so considering the absolute score, divided by the second-best score available seems more indicative. Son et al. [24] suggested “loop-constraints”, four or more puzzle piece edges where the compatibility ratio between each pair is in the top ten among all possible pairs of piece edges in the given puzzle. Palkin and Tal [19] proposed a greedy solver based on an L1 -norm asymmetric dissimilarity and the best-buddies estimation metric. 3 3.1 DNN-Buddies Motivation We propose a novel estimation metric called “DNN-Buddies”. Our goal is to obtain a classifier which predicts the adjacency likelihood of two puzzle piece edges in the correct puzzle configuration. Note that despite the exponential nature of the problem (as there are O((nm)!) possible arrangements of the pieces, taking into account rotations), the problem can be solved theoretically by assigning correctly, in a consecutive manner, n × m − 1 piece-edge pairs. (This is reminiscent of finding a minimal spanning tree, as noted by [10].) Hence, the classifier’s precision is of far greater importance than its recall. A classifier with perfect precision and a recall of n×m−1 1 n×m−1 = < all possible matches 4 × (n × (m − 1) + (n − 1) × m) 8 (3) might achieve a perfect solution by itself. 3.2 Challenges A straight-forward solution might have been to train a neural network against matching-pairs vs. non-matching ones. However, the issue of a jigsaw puzzle piece matching is of an imbalanced nature. In each n × m puzzle, there are O(n × m) matching pairs of piece edges and O(n2 × m2 ) possible nonmatching ones. A thorough review on the challenges and tactics to avoid them can be found in [13]. The trivial approach of random or uninformed undersampling, i.e. randomly choosing the required number of nonmatching pairs leads to a low-precision and highrecall metric, the very opposite of the goal set beforehand. We believe that the reason for this shortcoming is that there exist many “easy-to-spot” mismatches but only a handful of “hard-to-spot” ones. Thus, we resort to informed undersampling, choosing a subset of “good” mismatching pairs according to some criterion. Nevertheless, we avoid using any manual feature selection or other sophisticated image-related means. In the jigsaw puzzle domain, similarly to many other problem domains, the solver does not actually try to reassemble the original image (as this problem is Deep Neural Network Based Estimation Metric for the Jigsaw Puzzle Problem 5 not mathematically defined), but rather tries solving a “proxy problem” which is to achieve an image whose global overall score between abutting-edges is minimal. Thus, we choose using the compatibility measure as the undersampling criterion. 3.3 Neural Network Training For training and cross-validation, we use the 2,755 images of size 360 × 480 pixels from the IAPR TC-12 Benchmark [12]. Each image is first converted to YUV space followed by the normalization of each channel separately (via z-score normalization). Next, each (puzzle) image is divided to 12 × 17 tiles, where each tile is of size 28 × 28 pixels (as in all previous works); finally, we create a balanced set of positive and negative samples of puzzle-piece pairs, using informed undersampling as will be described below. In the end, we obtain a balanced set of 970,224 pairs overall. To balance our dataset, we use the most basic compatibility score which is the dissimilarity between two piece-edges in the YUV color-space, as described in Eq. 1, as an undersampling criterion. For each puzzle piece edge xi,j (i = 1..n×m, j = 1..4), we find its most compatible piece edge xk1,l1 and its second most compatible piece edge xk2,l2 . If the pair of edges xi,j −xk1,l1 is indeed adjacent in the original image, we add this pair to the pool of positively-labeled samples and toss the pair xi,j − xk2,l2 to the pool of negatively-labeled samples. Otherwise, xi,j − xk1,l1 is added to the negatively-labeled samples and the other pair is discarded. The latter is done to avoid training the network on adjacent pieces which happen to be vastly different due to a significant change of the image scenery in the corresponding region. In other words, we restrict our interest to highly compatible piece edges that are indeed adjacent. Since this method leads to more negative samples than positive ones, we eventually randomly throw some negative samples to balance out the set. From each image pair we extract the two columns near the edge, i.e. the column of abutting pixels in each edge and the one next to it. This results is an input of size (28 × 4 × 3 =) 336 pixels. We use a feed-forward neural network (FFNN) of five fully connected layers of size 336, 100, 100, 100, and 2. The output is a softmax layer containing two neurons. We expect (0, 1) for matching pairs and (1, 0) otherwise. The activation used in all layers is the rectified linear unit (ReLU) function, i.e. f (x) = max(0, x). Figure 2 depicts the network’s structure. We trained the network in a supervised manner using Stochastic Gradient Descent that minimizes the negative log likelihood of the error for 100 iterations. The resulting network reaches 95.04% accuracy on the training set and 94.62% on a held-out test set. All dataset preparation and network training was performed using Torch7 [6]. 4 Experimental Results For each piece edge xi,j (i = 1..n × m, j = 1..4), if its most compatible piece edge xk,l is classified positively using the DNN-Buddies network, we define xk,l to be xi,j ’s DNN-buddy piece edge. Note that each piece edge can have only a single DNN-buddy; 6 D. Sholomon, E.O. David, N.S. Netanyahu Fig. 2: Architecture of our DNN-Buddies scheme. also, some pieces might not have a DNN-buddy at all (if the most compatible piece is not classified as one by the DNN-Buddies network). First, we evaluate the precision of the proposed metric, i.e. how many DNNbuddies are indeed adjacent in the original image. Using the well known dataset presented by Cho et al. [4] of 20 432-piece puzzles, we obtained a precision of 94.83%. Next, we incorporated the estimation metric (due to the proposed DNN-Buddies scheme) into the GA-based solver proposed by us previously [23]. Unfortunately, due to lack of space, no self-contained review of genetic algorithms and the proposed method can be included in this paper. Nevertheless, the modification required with respect to the existing GA framework is rather simple; if a DNN-buddy pair appears in one of the parents, assign this pair in the child. Figure 3 describes the modified crossover operator in the GA framework according to the above (see Step 2, which includes the new DNN-buddy phase). Until 1. 2. 3. 4. 5. (n − 1) relative relations are assigned do Try assigning all common relative relations in the parents. Try assigning all DNN-buddy relative relations in the parents. Try assigning all best-buddy relative relations in the parents. Try assigning all existing most-compatible relative relations. Try assigning random relative relations. Fig. 3: Crossover overview We ran the augmented solver on the 432-piece puzzle set and on the two additional datasets proposed by Pomeranz et al. [20] of 540- and 805- piece puzzles. We evaluated our results according to the neighbor comparison which measures the fraction of correct neighbors and the number of puzzles perfectly reconstructed for each set. Deep Neural Network Based Estimation Metric for the Jigsaw Puzzle Problem 7 Table 1 presents the accuracy results of the same solver with and without the DNN-Buddies metric. For each dataset we achieve a considerable improvement in the overall accuracy of the solution, as well as the number of perfectly reconstructed puzzles. Moreover, our enhanced deep neural network-based scheme appears to outperform the current state-of-the-art results, as it yields accuracy levels of 95.65%, 96.37% and 95.86%, which surpass, respectively, the best results known of 95.4% [19], 94.08% and 94.12% [23]. # of Pieces 432 540 805 GA Neighbor Perfect 94.88% 94.08% 94.12% 11 8 6 Our (GA + DNN-Buddies) Neighbor perfect 95.65% 96.37% 95.86% 12 11 8 Table 1: Comparison of our accuracy results with and without the new DNN-Buddies estimation metric. 5 Conclusions In this paper we presented the first neural network-based estimation metric for the jigsaw puzzle problem. Unlike previous methods, no manual feature crafting was employed. The novel method exhibits high precision and when combined with a real-world puzzle solver, it significantly improves the solution’s accuracy to set a new state-of-the art standard. References 1. Altman, T.: Solving the jigsaw puzzle problem in linear time. 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arXiv:1709.10154v1 [cs.SY] 28 Sep 2017 Finite-Time Distributed Linear Equation Solver for Minimum l1 Norm Solutions Jingqiu Zhou, Wang Xuan, Shaoshuai Mou, and Brian. D. O. Anderson October 2, 2017 Abstract This paper proposes distributed algorithms for multi-agent networks to achieve a solution in finite time to a linear equation Ax = b where A has full row rank, and with the minimum l1 -norm in the underdetermined case (where A has more columns than rows). The underlying network is assumed to be undirected and fixed, and an analytical proof is provided for the proposed algorithm to drive all agents’ individual states to converge to a common value, viz a solution of Ax = b, which is the minimum l1 norm solution in the underdetermined case. Numerical simulations are also provided as validation of the proposed algorithms. 1 Introduction A significant amount of effort in the control community has recently been given to distributed algorithms for solving linear equations over multi-agent networks, in which each agent only knows part of the equation and controls a state vector that can be looked at as an estimate of the solution of the overall linear equations [1–5]. Numerous extensions along this direction include achieving solutions with the minimum Euclidean norm [6, 7], elimination of the initialization step [8], reduction of state vector dimension by utilizing the sparsity of the linear equation [9] and achieving least square solutions [10–15]. All these algorithms yield asymptotic convergence, but require an infinite number of sensing or communication events. ∗ This work was supported by a funding from Northrop Grumman Cooperation. J. Zhou, X. Wang and S. Mou are with the School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN 47906 USA (e-mail: [email protected], [email protected], [email protected]). B. D. O. Anderson is with Hangzhou Dianzi University, Hangzhou, China, The Australian National University and Data-61 CSIRO (formerly NICTA), Canberra, ACT 2600 Australia, e-mail: [email protected]; his work is supported by Data-61, CSIRO, and by the Australian Research Council’s Discovery Projects DP-130103610 and DP-160104500. Corresponding Author: Shaoshuai Mou. 1 ∗ Solutions to underdetermined linear equations with the minimum l1 norm are perhaps the most important in many engineering applications including earthquake location detection [16] , analysis of statistical data [17], solving biomeganetic inverse problems [18], and so on. One most intriguing case among these applications is compressive sensing, which enables transmission of sparse data in a very efficient way [19]. The decoding process of compressive sensing requires solving of linear equations with a minimum number of non-zero entries of the solution vectors, which, however, is an NP-hard problem and usually computationally costly [20]. Thus researchers usually turn to achieve solutions with minimum l1 norm instead for which the function to be minimized is convex [21, 22]. Most existing results for achieving minimum l1 norm solutions are based on the idea of Lasso [23] including Alternating Direction Method of Multipliers (ADMM) [24], the Primal-Dual Interior-Point Method [25, 26], Gradient Projection Methods [27], Homotopy Methods [28], Iterative Shrinkage-Thresholding Methods [29] and Proximal Gradient Methods [30]. Interesting as these results are, they either achieve limited accuracy dominated by a threshold parameter, involve solving a much larger linear equation, or lead to high computational complexity. In this paper we aim to develop distributed algorithms for multi-agent networks to achieve in finite time a solution of linear equations or, in the underdetermined case, one with the minimum l1 norm. By distributed is meant that each agent only knows part of the overall linear equation and can communicate with only its nearby neighbors. The problem of interest is formulated in Section 2. We introduce in section 3 the concepts to be employed in the paper including Filippov set-valued maps, Filippov solutions, generalized Lie derivatives, based on which a preliminary result is achieved. In Section 4, we will first propose a distributed algorithm to drive all agents’ state vectors to converge in finite time to the same solution of the overall linear equations. Then we present a centralized update for achieving a solution with the minimum l1 norm. Motivated by the projection-consensus flow proposed in [13] and the finite-time gradient flow for consensus devised in [31–36], we utilize a combination of the proposed distributed linear equation solver and the proposed centralized algorithm for minimum l1 -norm solutions to develop a distributed linear equation solver for achieving the minimum l1 norm solution, which is shown to converge in finite time. We provide simulations in Section 5 and concluding remarks in Section 6. Notation: Let r denote an arbitrary positive integer. Let 1r denote the vector in Rr with all entries equal to 1s. Let Ir denote the r × r identity matrix. We let col {A1 , A2 , · · · , Ar } be a stack of matrices Ai possessing the same number of columns with the index in a top-down ascending order, i = 1, 2, · · · , r. Let diag {A1 , A2 , · · · , Ar } denote a block diagonal matrix with Ai the ith diagonal block entry, i = 1, 2, · · · , r. By M > 0 and M ≥ 0 are meant that the square matrix M is positive definite and positive semi-definite, respectively. By M 0 is meant the transpose of a matrix M . Let ker M and image M denote the kernel and image of a matrix M , respectively. Let ⊗ denote the Kronecker product. Let k · k1 denote the l1 norm of a vector in Rr . 2 2 Problem Formulation Consider a network of m agents, i = 1, 2, ..., m; inside this network, each agent can observe states of certain other agents called its neighbors. Let Ni denote the set of agent i’s neighbors. We assume that the neighbor relation is symmetric, that is, j ∈ Ni if and only if i ∈ Nj . Then all these neighbor relations can be described by an m-node-m̄-edge undirected graph G such that there is an undirected edge connecting i and j if and only if i and j are neighbors. In this paper we only consider the case in which G is connected, fixed and undirected. Suppose that each agent i knows Ai ∈ Rni ×n and bi ∈ Rni and controls a state vector yi (t) ∈ Rn . Then all these Ai and bi can be stacked into an overall equation Ax = b, where A = col {A1 , A2 , · · ·, Am }, b = col {b1 , b2 , · · ·, bm }. Without loss of generality for the problems of interest to us, we assume A to have full row-rank. Let x∗ denote a solution to Ax = b (and in the underdetermined case, x∗ is not unique); and let x̄∗ denote its minimum l1 -norm solution, that is, x̄∗ = arg min kxk1 (1) Ax=b ∗ (In the non-singular A case, x and x̄∗ necessarily coincide) The problem of interest in this paper is to develop distributed algorithms for each agent i to update its state vector yi (t) by only using its neighbors’ states such that all yi (t) to converge in finite time to a common value x∗ and if desired in the nonsquare case the value x̄∗ . 3 Key Concepts and Preliminary Results Before proceeding, we introduce some key concepts and preliminary results for future derivation and analysis. The key references for the background we summarize are [37] and [38]. 3.1 Filippov set-valued maps and Filippov Solutions By a Filippov set-valued map F [f ] : Rr → B ⊂ Rr associated with a function f : Rr → Rr is meant \ \ co{f (B(x, δ))/S} (2) F [f ](x) , δ>0 µ(S)=0 Here B(x, δ) stands for the open ball on Rr , whose center is at x and has a radius of δ; µ(S) denotes the Lebesgue measure of S; and co stands for the convex closure. Let sgn (x) : Rr → Rr be a function with the kth entry, k = 1, 2, ..., r, defined as  (x)k > 0;  1, −1, (x)k < 0; (sgn (x))k = (3)  0, (x)k = 0. 3 It follows that the Filippov set-valued map F [sgn ](x) for x ∈ Rr is defined entrywise as:  1 (x)k > 0  [−1, 1] (x)k = 0 (F [sgn ](x))k = (4)  −1 (x)k < 0 for k = 1, 2, ..., r. Note that even if (x)i = (x)j = 0, the ith and jth entries of a vector in F [sgn ](x) may not necessarily be equal since each of them could be chosen as arbitrary values in the interval [−1, 1]. From the definition of F [sgn ](x), one can verify that q 0 x = kxk1 , ∀q ∈ F [sgn ](x) (5) While for any w ∈ Rr , there holds q 0 w ≤ kwk1 (6) . By a Filippov solution for ẋ ∈ F [f ](x) is meant a Caratheodory solution x(t) such that ẋ ∈ F [f ](x) for almost all t, x(t) is absolutely continuous and can be written in the form of an indefinite integral. The following two lemmas treat existence of such a Filippov solution. Lemma 1 ( Proposition 3 in [37]) If f : Rr → Rr is measurable and locally bounded, then for any initial point x0 ∈ Rr , there exists a Filippov solution1 for ẋ ∈ F [f ](x). Lemma 2 ( Theorem 8 in page 85 of [38] ) Let a vector-valued f (t, x) be defined almost-everywhere in the domain G of time space (t, x). With f (t, x) measurable and locally bounded almost-everywhere in an open domain G, let \ \ F [f ](t, x) , co{f (t, B(x, δ)/S)}. (7) δ>0 µ(S)=0 Then for any point (t0 , x0 ) ⊂ G, there exists a Filippov solution of ẋ ∈ F [f ](t, x) with x(t0 ) = x0 . Note that Lemma 2 establishes the existence of a solution for time-varying systems. This is more general than Lemma 1, which only guarantees the existence of solutions to time-invariant systems. 3.2 Generalized Gradients and Generalized Lie Derivatives For a locally Lipschitz function w : Rr → R, the generalized gradient of w is ∂w(x) , co{ lim ∇w(xi ) : xi → x, xi ∈ / S ∪ Ωw } i→∞ 1 There is no implication that the solution exists on an infinite interval 4 (8) where S ⊂ Rr is an arbitrarily chosen set of measure zero, Ωw denotes the set of points at which w is not differentiable, and co denotes convex hull. Specially, for the function \|\|x\|\|1 , one computes the kth element of its generalized gradient to be:  1 xk > 0  [−1, 1] xk = 0 (∂\|\|x\|\|1 )k = (9)  −1 xk < 0 It follows from this and the definition of F [sgn ](x) in (4) that F [sgn ](x) = ∂kxk1 (10) For a set-valued map F : Rr → B(Rr ), the generalized Lie derivative of w is defined as L̃F w(x) = {q ∈ R : there exists α ∈ F(x) such that ∀β ∈ ∂w(x), q = β 0 α} (11) The above definition of generalized Lie derivative implies that for each α in F(x), we check if the inner product β 0 α is a fixed value for all β ∈ ∂w(x). If so, this inner product is an element in L̃F w(x), but note that the set L̃F w(x) may be empty. Moreover, for locally Lipschitz and regular(see [39],p.39 and [40], p.3 for detailed discussion of regular function2 ) functions w(x), one has the following lemma: Lemma 3 (Proposition 10 in [37]) Let x : [0, t1 ] → Rr be a solution for ẋ ∈ F(x(t)), where F is any set-valued map. Let w(x) : Rr → R be locally Lipschitz and regular. Then w(x(t)) is differentiable at almost all t ∈ [0, t1 ]; ∈ L̃F w(x(t)) for almost all t ∈ [0, t1 ]. The derivative of w(x(t)) satisfies dw(x(t)) dt Lemma 3 guarantees the existence of generalized Lie derivatives for functions that are locally Lipschitz and regular. If one focuses on a specific solution, one can show that α in (11) is a special vector as summarized in the following lemma. Lemma 4 (See Proof of Lemma 1 in [40]) Let x(t) denote a specific solution of a differential enclosure. Suppose w(x) is locally Lipschitz and regular. Let I ⊂ [0, ∞) denote the time interval for which ẋ(t) exists. Then dw(x(t))) = β 0 ẋ(t) dt where β is any vector in ∂w(x). 2 That a function w(x) : Rn → R is called regular at x ∈ Rn if 0 (x, v). 1. for all v ∈ Rn there exists the usual right directional derivative w+ 0 (x, v) = w o (x, v). 2. for all v ∈ Rn , w+ 5 (12) 3.3 Preliminary Results For any positive semi-definite matrix M ∈ Rr×r , M 6= 0 one can define Φ(M ) = {q ∈ Rr \| ∃φ ∈ F [sgn (q)], M φ(q) = 0} (13) and its compliment Φc (M ) = {q ∈ Rr \| ∀φ ∈ F [sgn (q)], M φ(q) 6= 0}. (14) We impose a further requirement on M , namely that Φc (M ) is nonempty which can be easily ensured. Let Λ(M ) = {φ\|φ ∈ F [sgn ](q), q ∈ Φc (M )} (15) Now F [sgn ](q) is a closed set for any fixed q; also note that F [sgn ](q) can only be one of a finite number of different sets; hence it is easy to check for a given M whether Φc (M ) is nonempty, (and in a later use of the result, it proves easy to check). It further follows that Λ(M ) is also a closed set. Consequently, the continuous function f (φ) = φ0 M φ has a nonzero minimum on Λ(M ). We denote λ(M ) = min f (φ). (16) φ∈Λ(M ) From the definition of Φc (M ) and Λ(M ), one has λ(M ) > 0. To summarize, one has the following lemma: Lemma 5 For any nonnegative-definite matrix M , we let Φc (M ), Λ(M ) and λ(M ) defined as above. Suppose that Φc (M ) is nonempty. Then λ(M ) is a positive constant. For the m-node-m̄-edge graph G, we label all its nodes as 1, 2, ..., m and all its edges as 1, 2, ..., m̄. Assign an arbitrary direction to each edge in G. Then the incidence matrix of G denoted by H = [hik ]m×m̄ is defined as follows  i is the head of the kth edge;  1, −1, i is the tail of the kth edge; hik = (17)  0, otherwise. Since G is connected, then ker H 0 is the span of 1m [41]. Moreover, one has the following lemma: Lemma 6 Suppose A has full-row rank and G is connected. Let P̄ = diag {P1 , P2 , ..., Pm } where each Pi is the projection matrix to ker Ai . Let H̄ = H ⊗ In with H the incidence matrix of G. Then one has image H̄ ∩ ker P̄ = 0 (18) image H̄ 0 ∩ Φ(H̄ 0 P̄ H̄) = 0 (19) and 6 Proof of Lemma 6: Let u be a vector such that P̄ H̄u = 0. (20) The vector v = H̄u lies in image H̄ and ker P̄ , and we will show it is zero to establish (18). Define Ā = diag {A1 , A2 , ..., Am }, which is full row rank since A has full row rank. Then P̄ = I − Ā0 (ĀĀ0 )−1 Ā. It follows from (20) that H̄u = Ā0 (ĀĀ0 )−1 ĀH̄u (21) Multiplying both sides of the above equation by 10m ⊗ In , one has 0 = (1m ⊗ In )0 Ā0 (ĀĀ0 )−1 ĀH̄u (22) Since (10m ⊗ In0 )Ā0 = A0 there holds 0 = A0 (ĀĀ0 )−1 ĀH̄u (23) Since A0 is full column rank, one has 0 = (ĀĀ0 )−1 ĀH̄u (24) From (21) and equation (24) one has H̄u = 0 (25) Furthermore, we notice (20) and conclude that (18) is true. For any q ∈ image H̄ 0 ∩ Φ(H̄ 0 P̄ H̄), there exists a vector p such that q = H̄ 0 p (26) and a vector φ ∈ F [sgn ](q) such that H̄ 0 P̄ H̄φ = 0. Note that P̄ is a projection matrix; then P̄ H̄φ = 0. (27) From (18) one then has H̄φ = 0. (28) From φ ∈ F [sgn ](q) and (5), one has \|\|q\|\|1 = φ0 q which together with (26) and (28) implies \|\|q\|\|1 = 0. Then one has q = 0 and (19) is true. Now consider the system ẋ ∈ −M F [sgn ](x) (29) with any positive semi-definite matrix M ∈ Rr×r . The existence of a Filippov solution to (29) can be guaranteed by Lemma 1. The existence interval is 7 t ∈ [0, ∞) because of the global bound on the right-hand side to (29). Let x(t) denote such a Filippov solution for any given x(0). Note that the function kxk1 is locally Lipschitz and regular(The word was introduced early with reference). Then by Lemma 3, the time derivative of kx(t)k1 exists for almost all t ∈ [0, ∞) and is in the set of generalized Lie derivatives. In other words, there exists a set I = [0, ∞) \ T with T of Lebesgue measure 0 such that dkx(t)k1 exists for all t ∈ I dt (30) Proposition 1 Let x(t) denote a Filippov solution to (29) for any given x(0) ∈ Rr . Then • dkx(t)k1 ≤ 0, dt t ∈ I; (31) • there exists a finite time T such that x(T ) ∈ Φ(M ); (32) • if it is further true that dkx(t)k1 = 0, dt t ∈ [T, ∞) \ T , (33) one has x(t) = x(T ) t ∈ [T, ∞) (34) Remark 1 Note that the right-hand side of (29) is a projection of the gradient flow of the potential function kx(t)k1 . It is also standard that the gradient law of a real analytic function with a lower bound converges to a single point which is both a local minimum and a critical point of the potential function [42]. However, if the real analytic property does not hold, the convergence result may fail. Indeed, the function kx(t)k1 here is obviously not real analytic, and one cannot immediately assert that (29) will drive kx(t)k1 to its minimum, not to mention the finite time result in Proposition 1. Thus Proposition 1 is nontrivial and will serve as the foundation for devising finite-time distributed linear equation solvers in this paper. Proof of Proposition 1: By (12) in Lemma 4, one has d\|\|x(t)\|\|1 = β 0 ẋ, dt t∈I (35) holds for all β ∈ ∂kx(t)k1 . It follows from ẋ ∈ −M F [sgn ](x) in (29) that at each t there exists a γ(t) ∈ F [sgn ](x) such that ẋ = −M γ(t) 8 (36) Since β could be chosen as any vector in ∂kx(t)k1 , γ(t) ∈ F [sgn ](x) and ∂kxk1 = F [sgn ](x) from (10), one could choose β = γ(t) in (35), which together with (36) leads to dkx(t)k1 = −γ(t)0 M γ(t), t ∈ I (37) dt Note that M is positive semi-definite. Thus (31) is true. We use the method of contradiction to prove that there exists a finite time T such that x(T ) ∈ Φ(M ), T ∈ I. Suppose such a finite time does not exist. One has x(t) ∈ Φc (M ), t ∈ I Then dkx(t)k1 ≤ −λ(M ), t ∈ I dt where λ(M ) is as defined in (16). It follows that kx(t)k1 ≤ kx(0)k1 − λ(M )t, t ∈ [0, ∞). This contradicts the fact that kx(t)k1 ≥ 0, t ∈ [0, ∞) since λ(M ) is a positive constant by Lemma 5. Thus there exists a finite time T such that x(T ) ∈ Φ(M ). From the assumption (33), the fact that M is semipositive definite and (37), one has M γ(t) = 0, t ∈ [T, ∞) \ T . (38) It follows from this and (36) that ẋ(t) = 0, t ∈ [T, ∞) \ T . By integration of ẋ(t) from T to ∞, one has x(t) = x(T ), completes the proof. 4 t ∈ [T, ∞). This Algorithms and Main Results In this section, we study three related problems: finite time distributed solution of Ax = b, centralized solution of Ax = b achieving a minimum l1 norm, and then finally, using ideas from the first two, and finding a distributed algorithm achieving the minimum l1 -norm solution of Ax = b in finite time. 4.1 Finite-Time Distributed Linear Equation Solver In this subsection, we will present a distributed update for achieving a solution x∗ to Ax = b in finite time. Of course, A is assumed to have full row rank, but not necessarily be square. Recall that distributed linear equation solvers based on the agreement principle require each agent i to limit its update subject to its own constraint Ai x = bi while seeking consensus with its neighbors [1]. Such an agreement principle in continuous-time systems can be 9 achieved by a projection-consensus flow which at each agent i projects a function of its neighbors’ states to the subspace defined by its linear equation [13]. By choosing the finite-time gradient flow for consensus developed by [36] within the projection-consensus flow, we are led to postulate the following update for each agent i: X ẏi = −Pi φij , Ai yi (0) = bi (39) j∈Ni where φij ∈ F [sgn ](yi − yj ) and Pi denote the projection matrix to the kernel of Ai . Note the special property that F [sgn ](0) is a point which can be chosen arbitrarily from the interval [−1, 1]. Generally speaking, different agents may have different choices for F [sgn ](0). Before proceeding, we make the following assumption on coordinations between neighbor agents: Assumption 1 Each pair of neighbor agents i and j takes the choice of φij and φji when yi = yj such that φij = −φji (40) Under Assumption 1 and the definition of F [sgn ](x), one always has φij = −φji no matter whether yi is equal to yj or not. Let y = col {y1 , y2 , · · · , ym }, P̄ = diag{P1 , P2 , · · · , Pm } and H̄ = H ⊗ In with H the incidence matrix of G. Then from (39) we have ẏ ∈ −P̄ H̄F [sgn ](H̄ 0 y) (41) By Lemma 1 there exists a Filippov solution to system (41) for t ∈ [0, ∞), we denote this solution by y(t). By Lemma 3, there exists a set I = [0, ∞) \ T 1 exists for all t ∈ I. Moreover, with T of Lebesgue measure 0 such that dky(t)k dt one has the following main theorem, which establishes existence of a limiting consensus solution but for the moment leaves unspecified the L1 -optimality. Theorem 1 Under Assumption 1 and the updates (39), and with A assumed to have full row rank, all yi (t), i = 1, 2, ..., m, converge to be a single solution to Ax = b in finite time. Proof of Theorem 1. Since Ai yi (0) = bi and ẏi is in the kernel of Ai , one has Ai yi (t) = bi for all t ∈ [0, ∞). Then to prove Theorem 1, it is sufficient to show that all yi reach consensus in finite time. Note that G is connected and ker H 0 is spanned by the vector 1m . Then H̄ 0 y = 0 if and only if all yi are equal. Thus to prove Theorem 1, it suffices to prove z(t) = H̄ 0 y(t) converges to 0 in finite time. By multiplying both sides of (41) by H̄ 0 , one has ż ∈ −H̄ 0 P̄ H̄F [sgn ](z) (42) By Proposition 1, we have dkz(t)k1 ≤ 0, t ∈ I; dt 10 (43) and there exists a finite time T ∈ I such that z(T ) ∈ Φ(H̄ 0 P̄ H̄). (44) From this, we know the fact that z(T ) = H̄ 0 y(T ) so that z(T ) ∈ image H̄ 0 , and recalling (19) in Lemma 6, one has z(T ) = 0, which by (43) implies kz(t)k1 = 0, t ∈ [T, ∞). It follows that z(t) = 0, t ∈ [T, ∞). This completes the proof. 4.2 Finite-Time Centralized Update for Minimum l1 -norm Solution In this subsection, we will propose a centralized update for achieving the minimum l1 -norm solution to Ax = b. By noting that kxk1 is convex, we conceive of using a negative gradient flow of \|\|x\|\|1 subject to x remaining on the manifold Ax = b in order to achieve x̄∗ = arg minAx=b kxk1 . This leads us to the following update: ẏ ∈ −P F [sgn ](y), Ay(0) = b. (45) where P denotes the projection matrix onto the kernel of A. Again by Lemma 1 one has there exists a Filippov solution to system (45) for t ∈ [0, ∞), which we denote by y(t). By Lemma 3, there exists a set I = [0, ∞) \ T with T of 1 measure 0 such that dky(t)k exists for all t ∈ I. Moreover, we have the following dt main theorem: Theorem 2 With A of full row rank, the Filippov solution y(t) to (45) converges in finite time to a constant, which is the minimum l1 -norm solution to Ax = b. Proof of Theorem 2: By Proposition 1, one has dky(t)k1 ≤ 0, t ∈ I; dt (46) and there exists a finite time T ∈ I such that y(T ) ∈ Φ(P ). Then there exists a vector φ ∈ F [sgn ](y(T )) such that P φ = 0. This and (5) imply φ0 y(T ) = ky(T )k1 (47) 11 Moreover, let ȳ denote any solution to Ax = b. Recall (6), there holds φ0 ȳ ≤ kȳk1 (48) Since φ ∈ ker P , one has φ ∈ image A0 . This implies that there exists a vector q such that q 0 A = φ0 (49) From Ay(T ) = b = Aȳ and (47)-(49), one has ky(T )k1 = φ0 y(T ) = q 0 Ay(T ) = q 0 Aȳ = φ0 ȳ ≤ kȳk1 (50) where ȳ is any solution to Ax = b. Thus y(T ) is a minimum l1 norm solution to Ax = b. This and (46) implies \|\|y(t)\|\|1 reaches its minimum value subject to Ay(t) = b for t ∈ [T, ∞) \ T . Thus dky(t)k1 = 0, dt t ∈ [T, ∞) \ T (51) which satisfies the assumption (33) in Proposition 1 again. Then y(t) = y(T ), t ∈ [T, ∞). Thus y(t) is the minimum l1 -norm solution to Ax = b for t ∈ [T, ∞). This completes the proof. 4.3 Finite-Time Distributed Update for Minimum l1 -norm Solutions In this subsection we will develop a distributed update for a multi-agent network to achieve the minimum l1 -norm solution to Ax = b in finite time. Motivated to study a combination of the finite-time distributed linear equation solver in (39) and the finite-time centralized update for minimum l1 -norm solutions in (45), we propose the following update for agent i, i = 1, 2, ..., m: X ẏi = −k(t)Pi φi − Pi φij (52) j∈Ni where φi ∈ F [sgn ](yi ), φij ∈ F [sgn ](yi − yj ), with φij = −φji in case yi = yj , and Ai yi (0) = bi . (53) We assume that k(t) ∈ R is measurable and locally bounded almost everywhere for t ∈ [0, ∞), and lim k(t) = δ (54) t→∞ Z ∞ k(t)dt = ∞ (55) 0 where δ is a sufficiently small nonnegative number depending on the connection of the network and A, note that 0 is always a feasible choice of δ. One example δ̄ +δ. One simple case is choosing δ̄ = 1, δ = 0, and of a choice of k(t) is k(t) = t+1 12 1 resulting in k(t) = t+1 , obtained by taking δ to be zero. This choice obviates the need to decide how small one has to be to meet a “sufficiently small” condition, but may result in rather slow convergence. Now from Ai yi (0) = bi and the fact that Pi is the projection to the kernel of Ai (which ensures ẏi ∈ ker Ai ), one has Ai yi (t) = bi , t ∈ [0, ∞) (56) Let y = col {y1 , y2 , y3 , ..., ym }, P̄ = diag {P1 , P2 , ..., Pm }, and H̄ = H ⊗ In with H the incidence matrix of G. From the updates in (52) and Assumption 1, one has ẏ ∈ −k(t)P̄ F [sgn ](y) − P̄ H̄F [sgn ](H̄ 0 y) (57) Note that sgn (y), k(t) and P̄ H̄F [sgn ](H̄ 0 y) are measurable and locally bounded almost everywhere. Then by Lemma 2 there exists a Filippov solution to system (57) for any given y(0) satisfying (53), which we denote by y(t) = col {y1 (t), y2 (t), y3 (t), ..., ym (t)} . By Lemma 3, there exists a set I = [0, ∞) \ T 1 exists for all t ∈ I. Then one with T of Lebesgue measure 0 such that dky(t)k dt has the following theorem: Theorem 3 Under Assumption 1 and the update (52) and with A of full row rank, all yi (t), i = 1, 2, ..., m converge in finite time to the same value which is the minimum l1 -norm solution to Ax = b. Proof of Theorem 3: We first prove that all yi (t) reach a consensus in finite time by showing that z(t) converges to 0 in finite time, where z(t) = H̄ 0 y(t). Multiplying both sides of (57) by H̄ 0 , one has ż ∈ −k(t)H̄ 0 P̄ F [sgn ](y) − H̄ 0 P̄ H̄F [sgn ](z) (58) By Lemma 4, one has dkz(t)k1 = β 0 ż, t ∈ I dt where β can be any vector in ∂kzk1 . Note that ∂kzk1 = F [sgn ](z). Then dkz(t)k1 = −k(t)γ 0 H̄ 0 P̄ η − γ 0 H̄ 0 P̄ H̄γ, dt t∈I (59) where η ∈ F [sgn ](y), γ ∈ F [sgn ](z) and β is chosen to be equal to γ. Since z(t) ∈ image H̄ 0 , then by Lemma 6, if also z(t) ∈ Φ(H̄ 0 P̄ H̄), one will have z(t) = 0. Thus z(t) ∈ Φc (H̄ 0 P̄ H̄) as long as kz(t)k1 6= 0. (60) Now by the definition of λ(H̄ 0 P̄ H̄) and Lemma 5, one has γ 0 H̄ 0 P̄ H̄γ ≥ ρ as long as kz(t)k1 6= 0 13 (61) where ρ = λ(H̄ 0 P̄ H̄) is a positive constant. Let κ(H̄, P̄ ) denote an upper bound on \|γ 0 H̄ 0 P̄ η\|, and define an upper bound on δ by δ< ρ κ(H̄, P̄ ) (62) This captures the idea stated previously that δ depends on A and the graph. For any δ chosen as in (62), there must exist a finite time T1 such that \| − k(t)γ H̄ 0 P̄ η\| ≤ ρ2 , t ∈ [T1 , ∞). (63) where we have ρ2 = κ(H̄, P̄ )δ < ρ. From (59), (61) and (63), one has dkz(t)k1 ≤ −(ρ − ρ2 ) as long as kz(t)k1 6= 0, t ∈ [T1 , ∞) \ T dt (64) with ρ a positive constant. Thus there must exist a finite time T2 ≥ T1 such that z(T2 ) = 0 (65) Next we prove that z(t) = 0, t ∈ [T2 , ∞) (66) We prove this by contradiction. Suppose (66) is not true. Then there exists a time T̄2 > T2 such that z(T̄2 ) 6= 0. Then kz(T̄2 )k1 > 0. Since kz(t)k1 is continuous, there exists a time T2∗ such that kz(T2∗ )k1 takes its maximum value for t ∈ [T2 , T̄2 ]. Again, since kz(t)k1 is continuous, there exists a sufficiently small but positive such that kz(t)k1 > 0 is differentiable for t ∈ [T2∗ − , T2∗ ]. Because kz(t)k1 is differentiable almost everywhere, we know that kz(T2∗ )k1 = Z T2∗ T2∗ − dkz(t)k1 dt + kz(T2∗ − )k1 dt (67) Because kz(t)k1 > 0 in [T2∗ − , T2∗ ], by (64) and (67) we have kz(T2∗ )k1 ≤ −(ρ − ρ2 ) + kz(T2∗ − )k1 < kz(T2∗ − )k1 (68) This contradicts the fact that kz(T2∗ )k1 is the maximum value on [T2 , T̄2 ]. Thus (66) is true. By (66), one has there exists a vector ȳ(t) such that y1 (t) = y2 (t) = · · · = ym (t) = ȳ(t), t ∈ [T2 , ∞) (69) Moreover, Aȳ(T2 ) = b since Ai yi (t) = bi for i = 1, 2, ..., m. To prove Theorem 3, we only need to prove that ȳ(t) converges to be the minimum l1 -norm solution to Ax = b. To see why this is so, we let P denote the projection matrix to the kernel of A. Then P Pi = P for i = 1, 2, ..., m. Multiplying both sides of (52) by P , one has X ẏi = −k(t)P φi − P φij , i = 1, 2, ..., m (70) j∈Ni 14 Since G is undirected, then φij appears in the update i if φji appears in its neighbor j’s update. By adding the updates in (70) for i = 1, 2, ..., m and noting φij = −φji for any two neighbors i and j, one has m X ẏi = −k(t)P m X i=1 φi (71) i=1 where φi ∈ F [sgn ](yi ). By (69), one knows all yi (t) reach a consensus ȳ(t) for t ∈ [T2 , ∞). Note that if the kth entry of ȳ(t) is 0, the kth entry of each φi can be selected as an arbitrary value from [−1, 1], which may be different for different entries, but their average is still an arbitrary value in [−1, 1]. Thus m 1 X φi ∈ F [sgn ](ȳ(t)), m i=1 t ∈ [T2 , ∞) (72) From (71) and (72) we have ȳ˙ ∈ −k(t)P F [sgn ](ȳ), Let τ = Rt T2 t ∈ [T2 , ∞) (73) k(s)ds. From dȳ dt dȳ = · dτ dt dτ and (73), one has dȳ ∈ −P F [sgn ](ȳ), τ ∈ [0, ∞) (74) dτ with Aȳ(τ ) = b for τ = 0. This is exactly the same as the centralized update in (45). By Theorem 1, there exists a finite time Γ such that y(τ ) is the minimum l1 -norm solution to Ax = b for τ ∈ [Γ, ∞). By the relation between τ and t, one has correspondingly that there exist a finite time T such that ȳ(t) is a minimum l1 -norm solution for t ∈ [T, ∞). This completes our proof. 5 Simulation Result In this section, we will report several simulations of the proposed algorithms for solving an underdetermined linear equation Ax = b in a four-agent undirected and connected network as in Figure 1. 0 0 0 Here, A and b are partitioned as A = A1 A02 A03 A04 and b = b01 b02 b03 b04 , 15 1 2 4 3 Figure 1: A four agent network respectively. Each agent i knows Ai and bi with  0.63 0.58  0.65  0.33  0.68  0.22 0 A1 =  0.49  0.21  0.62 0.57  0.71 0.28  0.44 0.06  0.77  0.16  0.84  0.62 A03 =  0.74  0.26  0.44 0.28  0.50 0.38  0.04 0.60  0.50  0.81  0.01  0.51 , 0.23  0.29  0.25 0.21  0.66 0.90  0.34 0.94  0.28  0.41  0.75  0.56 , 0.41  0.89  0.69 0.23  0.88 0.63  0.80 0.25  0.53  0.79  0.13  0.79 0 A2 =  0.34  0.45  0.10 0.94  0.78 0.70  0.05 0.09  0.65  0.69  0.94  0.73 A04 =  0.51  0.59  0.76 0.95  0.39 0.03 b01 = 0.47 b03 = 0.63 0.52 , 0.33 , b02 = 0.77 b04 = 0.31  0.99 0.65  0.38  0.12  0.76  0.52  0.55  0.24  0.55 0.51  0.58 0.85  0.23 0.33  0.92  0.66  0.92  0.06  0.13  0.94  0.40 0.69  0.24 0.92 0.34 0.65 (75) (76) Example 1: We utilize the distributed update (39) to achieve a solution ∗ to Ax by x in finite time in the above four-agent network. Let 0= b 0denoted 0 0 0 y = y1 y2 y3 y4 where yi (t) denote the state of agent i that is the estimate of agent i to x∗ . Then ky(t) − 1m ⊗ x∗ k1 measures the difference between all agents’ estimations and the solution x∗ . As shown by simulations in Figure 2, ky(t) − 1m ⊗ x∗ k1 reaches 0 in finite time. This suggests all agents’ states achieves a consensus to x∗ in finite time, consistent with the claim of Theorem 1. Example 2: We employ the centralized update (45) with state vector y(t) to achieve x̄∗ which denotes a minimum l1 -norm solution to Ax = b. As shown in 16 Figure 2: Finite time achieving a solution under the update(39) Figure 3, ky(t) − x̄∗ k1 reaches 0 in finite time and maintains to be 0 afterwards. This indicates that the minimum l1 -norm solution x̄∗ is achieved in finite time corresponding to Theorem 2. It is worth noting that one could observe multiple phases of convergence in Figure 2. This is because F [sgn ](y(t)) in the update (45) takes different values piece-wisely, and results in different convergence rates. Figure 3: Centralized solver for achieving a minimum l1 norm solution under the update (45) Example 3: Finally, we utilize the distributed update (52) to achieve a minimum l1 solution to Ax = b denoted by x̄∗ in finite time. Here k(t) is δ̄ chosen to take the form 1+t + δ with δ̄ and δ constants. We still let y = 0 0 0 0 0 y1 y2 y3 y4 where yi (t) denote the state of agent i that is the estimate of 17 agent i to x̄∗ . Then ky(t)−1m ⊗ x̄∗ k1 measures the difference between all agents’ estimations and x̄∗ . As shown in Figure 4 and Figure 5, all yi (t) reach the same minimum l1 -norm solution in finite time regardless of different choices of δ̄ and δ. Moreover, by fixing δ̄ and increasing the value of δ in k(t), one achieves a significantly faster convergence as shown in Figure 4. Similarly, increasing δ̄ with a fixed δ also leads to a faster convergence, although not that dramatically, as shown in Figure 5. Figure 4: Distributed solver for achieving minimum l1 norm solution under the δ̄ update (52), where k(t) = t+1 + δ with fixed δ̄ = 0.1 and different values of δ. Figure 5: Distributed solver for achieving minimum l1 norm solution under δ̄ update (52) where k(t) = t+1 + δ with fixed δ = 0.01 and different values of δ̄. We also note from Figure 4 and Figure 5 that the convergence time required in this distributed way for minimum l1 -norm solutions is dramatically longer, 18 roughly speaking, 1δ times longer, than that in the centralized case in Figure 3. The major reason for this is that the centralized update appearing in the distributed update (52) is scaled by k(t), which is smaller than 1. The time required for consensus in this four-agent network example is minor under the distributed update (52) as indicated in Figure 6. Let ȳ(t) denote the average of all four agents’ states. The evolution of ky(t) − 1m ⊗ ȳ(t)k1 in Figure 6 suggests that all agents’ states reach a consensus in a finite time similar to that in Figure 2. We anticipate that when it comes to a very large network, the convergence time for consensus might play a more significant role in convergence of the distributed update (52). Figure 6: Consensus of distributed solver under update (52) with k(t) = 0.01 6 0.1 t+1 + Conclusion We have developed continuous-time distributed algorithms for achieving solutions, and minimum l1 -norm solutions, respectively, to linear equations Ax = b in finite time. The algorithms result from combination of the projectionconsensus flow proposed in [13] and the finite-time gradient flow for consensus devised in [36], and work for fixed undirected multi-agent networks. Future work includes the generalization of the proposed update to networks that are directed and time-varying. References [1] S. Mou, J. 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1 UAV-Aided Wireless Communication Designs With Propulsion Energy Limitations Subin Eom, Hoon Lee, Junhee Park and Inkyu Lee, Fellow, IEEE arXiv:1801.02782v1 [cs.IT] 9 Jan 2018 School of Electrical Eng., Korea University, Seoul, Korea Email: {esb777, ihun1, pjh0585, inkyu}@korea.ac.kr Abstract This paper studies unmanned aerial vehicle (UAV) aided wireless communication systems where a UAV supports uplink communications of multiple ground nodes (GNs) while flying over the area of the interest. In this system, the propulsion energy consumption at the UAV is taken into account so that the UAV’s velocity and acceleration should not exceed a certain threshold. We formulate the minimum average rate maximization problem and the energy efficiency (EE) maximization problem by jointly optimizing the trajectory, velocity, and acceleration of the UAV and the uplink transmit power at the GNs. As these problems are non-convex in general, we employ the successive convex approximation (SCA) techniques. To this end, proper convex approximations for the non-convex constraints are derived, and iterative algorithms are proposed which converge to a local optimal point. Numerical results demonstrate that the proposed algorithms outperform baseline schemes for both problems. Especially for the EE maximization problem, the proposed algorithm exhibits about 109 % gain over the baseline scheme. I. I NTRODUCTION Recently, unmanned aerial vehicles (UAVs) have received great attentions as a new communication entity in wireless networks [1]. Compared to conventional terrestrial communications where users are served by ground base stations (BSs) fixed at given position [2], UAV-aided systems could be dispatched to the field with various purposes such as disaster situations and military uses. Moreover, located high above users, UAVs are likely to have line-of-sight (LoS) communication links for air-to-ground channels. Utilizing these advantages, UAVs have been considered to diverse wireless communication systems. The authors in [3] and [4] studied a mobile relaying system where a UAV helps the communication of ground nodes (GNs) without direct communication links. In this UAV-aided 2 relaying system, compared to conventional static relay schemes [5], [6], the UAV can move closer to source and destination nodes in order to obtain good channel conditions, and thus the system throughput can be significantly improved. In [3], the throughput of mobile relaying channels was maximized by optimizing the transmit power at the source and the relay node as well as the trajectory of the mobile relay. For the fixed relay trajectory, the work [4] addressed the secrecy rate maximization problem for the UAV-based relaying system with an external eavesdropper. In addition, UAVs have been adopted to assist conventional terrestrial communication infrastructures [7]–[9]. For the disaster situation, UAVs were employed in [7] to recover malfunctioned ground infrastructure. The work in [8] examined a system where the UAV serves cell-edge users by jointly optimizing UAV’s trajectory, bandwidth allocation, and user partitioning. Also, the flying computing cloudlets with UAVs were introduced to provide the offloading opportunities to multiple users [9]. Moreover, the UAVs could play the role of mobile BSs in wireless networks [10]–[12]. The authors in [10] derived mathematical expressions for the optimum altitude of the UAVs that maximizes the coverage of the cellular network. Also, the trajectory optimization methods for mobile BSs were presented in [11] and [12]. Assuming that the GNs are located in a line, the minimum throughput performance was maximized in [11] by optimizing the position of a UAV on a straight line. This result was extended in [12] to a general scenario where multiple UAVs fly three-dimensional space to communicate with GNs. The joint optimization algorithms for the UAV trajectory, transmit power, and time allocation were provided in [12] to maximize the minimum throughput performance. However, these works did not consider the propulsion energy consumption of the UAVs necessary for practical UAV designs under limited on-board energy situation [13]. By taking this issue into account, recent works [14]–[16] investigated energy efficiency (EE) of the UAV system. Different from conventional systems which consider only communicationrelated energy consumption [17]–[19], the EE of the UAV should addresses the propulsion energy at the UAV additionally. The authors in [14] maximized the EE by controlling the turning radius of a UAV for mobile relay systems. Also, by jointly optimizing the time allocation, speed, and trajectory, both the spectrum efficiency and the EE were maximized in [15]. In [16], the propulsion energy consumption of the UAV was theoretically modeled, and the EE of the UAV was maximized for a single GN system. This paper studies UAV-aided wireless communications where a UAV with limited propulsion 3 energy receives the data of multiple GNs in the uplink. It is assumed that all GNs and the UAV operate in the same frequency band and there are no direct communication links among GNs. Under these setup, we formulate the minimum rate maximization problem and the EE maximization problem by jointly optimizing the UAV trajectory, the velocity, the acceleration, and the uplink transmit power at the GNs. A similar approach for solving the minimum rate maximization was studied in [12], but the authors in [12] did not involve the propulsion energy consumption at the UAV. For the EE maximization problem, our work can be regarded as a generalization of the single GN system in [16] to the multi-GN scenario, and thus we need to deal with inter-node interference as well. Due to these issues, existing algorithms presented in [12] and [16] cannot be directly applied to our problems. To tackle our problem of interest, we introduce auxiliary variables which couple the trajectory variables and the uplink transmit power in order to jointly optimize these variables. As the equivalent problem is still non-convex, we employ the successive convex approximation (SCA) technique which successively solves approximated convex problems of the original non-convex one. In order to apply the SCA to our optimization problems, we present new convex surrogate functions for the non-convex constraints. Then, we propose efficient algorithms for the minimum rate maximization problem and the EE maximization problem which yield local optimal solutions. Simulation results confirm that the proposed algorithms provide a significant performance gain over baseline schemes. The rest of this paper is organized as follows: Section II explains the system model and the problem formulations for the UAV-aided communication systems. In Section III, the minimum rate maximization and the EE maximization algorithms are proposed. We examine the circular trajectory case as baseline schemes in Section IV. Section V presents the numerical results for the proposed algorithms and we conclude the paper in Section VI. Notations: Throughout this paper, the bold lower-case and normal letters denote vectors and scalars, respectively. The space of M-dimensional real-valued vectors are represented by RM ×1 . For a vector a, kak and aT indicate norm and transpose, respectively. The gradient of a function f is defined as ∇f . For a time-dependent function x(t), ẋ(t) and ẍ(t) stand for the first-order and second-order derivatives with respect to time t, respectively. 4 Fig. 1. UAV-enabled wireless network II. S YSTEM M ODEL A ND P ROBLEM F ORMULATION As shown in Fig. 1, we consider UAV-aided wireless communications where a UAV receives uplink information transmitted from K GNs. The UAV horizontally flies at a constant altitude H with a time period T , while the GNs are located at fixed positions, which are perfectly known to the UAV in advance. For the location of the GNs and the UAV, we employ a three-dimensional Cartesian coordinate system, and thus the horizontal coordinate of GN k (k = 1, ..., K) is denoted by wk = [xk yk ]T . Also, we define the time-varying horizontal coordinate of the UAV at time instant t as q(t) = [qx (t) qy (t)]T , for 0 ≤ t ≤ T . Then, the instantaneous velocity v(t) and the acceleration a(t) of the UAV are expressed by v(t) , q̇(t) and a(t) , q̈(t), respectively. Continuous time expressions of variables make analysis and derivations in the UAV systems intractable. For ease of analysis, we discretize the time duration T into N time slots with the same time interval δt = T N [3]. As a result, the trajectory of the UAV can be represented by N vector sequences q[n] , q(nδt ), v[n] , v(nδt ), and a[n] , a(nδt ) for n = 0, 1, ..., N. When the discretized time interval δt is chosen as a small number, the velocity and the acceleration can be approximated by using Taylor expansions as [16] v[n] = v[n − 1] + a[n − 1]δt , for n = 1, ..., N, (1) q[n] = q[n − 1] + v[n − 1]δt + 21 a[n − 1]δt2 , for n = 1, ..., N. (2) Also, assuming the periodical operation at the UAV, we have [12] q[0] = q[N], v[0] = v[N], a[0] = a[N], (3) 5 which implies that after one period T , the UAV returns to its starting location with the same velocity and acceleration. In addition, the acceleration and the velocity of the practical UAV are subject to ka[n]k ≤ amax , for n = 0, 1, ...N, (4) Vmin ≤ kv[n]k ≤ Vmax , for n = 0, 1, ..., N, (5) where amax indicates the maximum UAV acceleration in m/sec2 and Vmin and Vmax stand for the minimum and the maximum UAV speed constraints in m/sec, respectively. Notice that the minimum speed constraint Vmin is important for practical fixed-wing UAV designs which need to move forward to remain aloft and thus cannot hover over a fixed location [16]. For the power consumption at the UAV, we take into account the propulsion power utilized for maintaining the UAV aloft and supporting its mobility. The propulsion power of the UAV Pprop [n] at time slot n is given by [16] c2 Pprop [n] = c1 kv[n]k + kv[n]k 3 ka[n]k2 1+ , for n = 0, 1, ..., N, g2 (6) where c1 and c2 are the parameters related to the aircraft design and g = 9.8 m/sec2 equals the gravitational acceleration. Thus, the average propulsion power and the total consumed propulsion P PN energy over N time slots are obtained by N1 N n=1 Pprop [n] and δt n=1 Pprop [n], respectively. The power consumed by signal processing circuits such as analog-to-digital converters and channel decoders are ignored since they are practically much smaller than the propulsion power [16]. Now, let us explain the channel model between the UAV and the GNs. We assume that the air-to-ground communication links are dominated by the LoS links. Moreover, the Doppler effect due to the UAV mobility is assumed to be well compensated. Then, the effective channel gain hk [n] from GN k to the UAV at time slot n follows the free-space path loss model as [3] hk [n] = γ0 , 2 dk [n] (7) where γ0 , β0 /σ 2 represents the reference signal-to-noise ratio (SNR) at 1 m with β0 and σ 2 being the channel power at 1 m and the white Gaussian noise power at the UAV, respectively, and the distance dk [n] is written by dk [n] = p kq[n] − wk k2 + H 2 . (8) 6 At time slot n, GN k transmits its data signal to the UAV with power 0 ≤ pk [n] ≤ Ppeak , where Ppeak is the peak transmission power constraint at the GNs. Accordingly, the instantaneous achievable rate Rk [n] can be expressed as Rk [n] = log2 where the term PK j=1,j6=k 1+ pk [n]hk [n] 1+ PK pj [n]hj [n] j=1,j6=k ! , (9) pj [n]hj [n] stands for interference from other GNs. Therefore, the achievable average rate of the GN k and the total information bits transmitted from GN k PN P over N time slots are denoted as N1 N n=1 Rk [n], respectively, where W n=1 Rk [n] and W δt means the bandwidth. In this paper, we jointly optimize the variables q[n], v[n], and a[n] and the uplink transmit power pk [n] at the GNs so that the minimum average rate among multiple GNs and the EE are maximized, respectively. First, the minimum rate maximization problem can be formulated as (P 1) : max {q[n],v[n],a[n]} {pk [n],τ } s.t. τ (10a) N 1 X Rk [n] ≥ τ, ∀k, N n=1 0 ≤ pk [n] ≤ Ppeak , ∀k, n, N 1 X Pprop [n] ≤ Plim , N n=1 (10b) (10c) (10d) (1) − (5), where Plim in (10d) indicates the propulsion power constraint at the UAV. Next, to support all of the individual GNs, the fairness based EE [20]–[22] is more suitable than the network-wise EE [18], [19]. Thus, we define the EE in the UAV-aided wireless communication systems as the ratio between the minimum information bits transmitted among the GNs and the total energy consumed at the UAV. Therefore, the EE maximization problem can be written by (P 2) : max {q[n],v[n],a[n]} {pk [n],η} η PN Pprop [n] W Rk [n] ≥ η, ∀k, n=1 N X n=1 s.t. (1) − (5), (10c). (11a) (11b) 7 In general, (P1) and (P2) are non-convex problems due to the constraints and the objective functions. Compared to [12], we additionally consider the propulsion power constraint (10d) in the minimum rate maximization problem (P1). Also, note that the EE maximization problem (P2) can be regarded as a generalization of [16] which investigated only a single GN scenario. From these respects, the works in [12] and [16] can be regarded as special cases of our problems (P1) and (P2), respectively. To solve the problems (P1) and (P2), we adopt the SCA framework [23] [24] which iteratively solves approximated convex problems for the original non-convex problems. III. P ROPOSED A LGORITHM In this section, we propose iterative algorithms for efficiently solving (P1) and (P2) by applying the SCA method. First, the minimum rate maximization problem (P1) is considered in Section III-A, and then it is followed by the EE maximization problem (P2) in Section III-B. A. Minimum Average Rate Maximization Applying the change of variables as Gk [n] , pk [n]hk [n] = pk [n]γ0 , ∀k, n, kq[n] − wk k2 + H 2 (12) where Gk [n] is a new optimization variable, the constraint (10c) becomes 0 ≤ Gk [n] ≤ Gk,max [n], ∀k, n, where Gk,max [n] , Ppeak hk [n] = Ppeak γ0 . kq[n]−wk k2 +H 2 Then, we can rewrite the achievable rate Rk [n] in (9) as Rk [n] = log2 1+ K X m=1 P G [n] . where R̂k [n] , log2 1 + K j=1,j6=k j ! Gm [n] − R̂k [n], (13) 8 By introducing new auxiliary variables {V1 [n]}, (P1) can be recast to (P 1.1) : max {q[n],v[n],a[n]} {Gk [n],V1 [n],τ } τ N 1 X log2 N n=1 s.t. (14a) 1+ K X Gm [n] m=1 ! − R̂k [n] ! ≥ τ, ∀k, 0 ≤ Gk [n] ≤ Gk,max [n], ∀k, n, N c2 1 X c2 ka[n]k2 c1 kv[n]k3 + + 2 ≤ Plim , N n=1 V1 [n] g V1 [n] (14b) (14c) (14d) Vmin ≤ V1 [n], ∀n, (14e) V12 [n] ≤ kv[n]k2 , ∀n, (14f) kv[n]k ≤ Vmax , ∀n, (14g) (1) − (4). It can be shown that at the optimal point of (P1.1), the inequality constraint in (14f) holds with the equality, since otherwise we can enlarge the feasible region corresponding to (14d) by increasing V1 [n]. Therefore, we can conclude that (P1.1) is equivalent to (P1). Thanks to the new auxiliary variables {V1 [n]}, constraints (14d) and (14e) now become convex, while (14b), (14c), and (14f) are still non-convex in general. To address these constraints, we employ the SCA methods. First, it can be checked that constraint (14b) is given by a difference of two concave functions. Hence, the convex surrogate function R̂kub [n] for R̂k [n] can be computed from a first order Taylor approximation as ! K K X X Gj,l [n] ≥ R̂k [n], (Gj,l+1[n] − Gj,l [n]) + log2 1 + R̂kub [n] , Γ̂k [n] j=1,j6=k (15) j=1,j6=k where Gk,l [n] indicates a solution of Gk [n] attained at the l-th iteration of the SCA process and P Γ̂k [n] , log2 e/(1 + K j=1,j6=k Gj,l [n]). Next, to identify the surrogate functions of (14c) and (14f), we present the following lemmas. Lemma 1: Denoting {ql [n]} as a solution for {q[n]} calculated at the l-th iteration, the concave surrogate function Glb k,max [n] for Gk,max [n] can be expressed as Glb k,max [n] , Ppeak γ0 kql+1 [n] − wk k2 + Bk [n](ql+1 [n] − wk )T (ql [n] − wk ) + Ck [n] − H4 ≤ Gk,max[n], ! (16) 9 where the constants Bk [n] and Ck [n] are respectively given as ! 1 1 − Bk [n] , 2 2 , 4 H kql [n] − wk k2 + H 2 kql [n] − wk k2 2kql [n] − wk k2 1 + Ck [n] , . 2 − H4 kql [n] − wk k2 + H 2 kql [n] − wk k2 + H 2 Proof: Please refer to Appendix A. Lemma 2: From a solution {vl [n]} obtained at the l-th iteration, the concave surrogate function of kvl+1 [n]k2 can be computed as −kvl+1 [n]k2 + 2vlT (2vl+1 [n] − vl [n]) ≤ kvl+1 [n]k2 . (17) Proof: Applying a similar process in Appendix A, we can conclude that the function in (17) satisfies the conditions for a concave surrogate function [23]. With the aid of Lemmas 1 and 2, at the (l + 1)-th iteration, the non-convex constraints in (14c) and (14f) can be approximated as 0 ≤ Gk [n] ≤ Glb k,max [n], (18) V12 [n] ≤ −kvl+1 [n]k2 + 2vlT (2vl+1 [n] − vl [n]) . (19) As a result, with given solutions {ql [n], vl [n], Gk,l [n]} at the l-th iteration, we solve the following problem at the (l + 1)-th iteration of the SCA procedure (P 1.2) : max {ql+1 [n],vl+1 [n],a[n]} {Gk,l+1 [n],V1 [n],τ lb } s.t. τ lb N 1 X log2 N n=1 (20a) 1+ K X m=1 Gm,l+1 [n] ! − R̂kub [n] ! ≥ τ lb , ∀k, (20b) (1) − (4), (14d), (14e), (14g), (18), (19), where τ lb denotes the lower bound of τ in the original problem (P1). Since (P1.2) is a convex problem, it can be optimally solved via existing convex optimization solvers, e.g. CVX [25]. Based on these results, we summarize the proposed iterative procedure in Algorithm 1. Algorithm 1: Proposed algorithm for (P1) Initialize {q0 [n], v0 [n], Gk,0 [n]}, ∀k, n and let l = 0. Repeat Compute {ql+1 [n], vl+1 [n], Gk,l+1[n]} for (P1.2) with given {ql [n], vl [n], Gk,l [n]}. Update l ← l + 1. Until Convergence. Gk,l+1 [n] Obtain pk [n] = hk,l+1 . [n] 10 For the convergence analysis of Algorithm 1, let us define the objective values of (P1) and (P1.2) at the l-th iteration as τl and τllb , respectively. Then we can express the relationship lb τl = τllb ≤ τl+1 ≤ τl+1 , (21) where the first equation holds because the surrogate functions in (15), (16), and (17) are tight at the given local points, the second inequality is derived from the non-decreasing property of the optimal solution of (P1.2), and the third inequality follows from the fact that the approximation problem (P1.2) is a lower bound of the original problem (P1). From (21), we can conclude that the objective value τ in (P1) is non-decreasing for every iterations of Algorithm 1. Since the objective value τ in (P1) has a finite upper bound value and at given local points, the surrogate functions in (15), (16), and (17) obtain the same gradients as their original functions, it can be verified that Algorithm 1 is guaranteed to converge to at least a local optimal solution for (P1) [23], [24]. B. Energy Efficiency Maximization In this subsection, we consider the EE maximization problem (P2). First, by applying (12)(13), and introducing an auxiliary variable {V1 [n]}, (P2) can be transformed as η (P 2.1) : max PN 2 2 3 {q[n],v[n],a[n]} c1 kv[n]k + V1c[n] + c2gka[n]k 2 V [n] n=1 1 {Gk [n],V1 [n],η} ! ! K N X X Gm [n] − R̂k [n] ≥ η, ∀k, s.t. W log2 1 + (22a) (22b) m=1 n=1 (1) − (4), (14c), (14e) − (14g). Similar to (P1.1), we can see that (P2.1) is equivalent to (P2), but (P2.1) is still non-convex due to the constraints in (14c), (14f), and (22b). To tackle this issue, we can employ the similar SCA process presented in Section III-A. By adopting (15) and Lemmas 1 and 2, a convex approximation of (P2.1) at the (l + 1)-th iteration is given by (P 2.2) : max {ql+1 [n],vl+1[n],a[n]} {Gk,l+1 [n],V1 [n],ηlb } s.t. η lb PN 3 n=1 c1 kv[n]k + W N X n=1 log2 1 + c2 V1 [n] K X m=1 + (23a) c2 ka[n]k2 g 2 V1 [n] ! Gm,l+1 [n] (1) − (4), (14e), (14g), (18), (19), − R̂kub [n] ! ≥ η lb , ∀k (23b) 11 where η lb denotes the lower bound of η in the original problem (P2). It can be shown that (P2.2) is a concave-convex fractional problem, which can be optimally P c2 3 solved via the Dinkelbach’s method [26], [27]. Then, denoting µ = N n=1 c1 kv[n]k + V1 [n] + c2 ka[n]k2 g 2 V1 [n] with a given constant λm , (P2.2) can be converted to (P2.3) as (P 2.3) : η lb − λm µ max {ql+1 [n],vl+1 [n],a[n]} {Gk,l+1 [n],V1 [n],ηlb } s.t. (24a) (1) − (4), (14e), (14g), (18), (19), (23b). Based on (P2.3), we summarize the proposed iterative procedure in Algorithm 2. The convergence and the local optimality of Algorithm 2 can be verified similar to Algorithm 1, and thus the details are omitted for brevity. Algorithm 2: Proposed algorithm for (P2) Initialize {q0 [n], v0 [n], Gk,0[n]}, ∀k, n and let λ0 = 0, m = 0, and l = 0. Repeat Repeat Compute {ql+1 [n], vl+1 [n], Gk,l+1 [n]} for (P2.3) with given {ql [n], vl [n], Gk,l [n]}, ∀k, n and λm . Update l ← l + 1. Until Convergence. Let F (λm ) = η lb − λm µ and λm+1 = η lb /µ. Update m ← m + 1. Let {q0 [n], v0 [n], Gk,0[n]} = {ql+1 [n], vl+1 [n], Gk,l+1[n]}, ∀k, n and l = 0. Until Convergence. Gk,l+1 [n] Obtain pk [n] = hk,l+1 , ∀k, n. [n] It is worthwhile to note that we need to initialize the trajectory variables {q[n], v[n]} for (P1) and (P2). However, it is not trivial to find such variables satisfying the UAV movement constraints (1)-(5) and the propulsion power constraint (10d). This will be clearly explained in Section IV-C. IV. C IRCULAR TRAJECTORY SYSTEM Now, we examine the circular trajectory system which will be used as a baseline scheme. First, we choose the center of the circular trajectory c = [x0 y0 ]T as the geometrical mean of the GNs c = PK k=1 K wk . Denoting r as the radius of the trajectory and θ[n] as the angle of the circle along which the UAV flies at time slot n, the horizontal coordinate of the UAV q[n] can be obtained by q[n] = [r cos θ[n] + x0 r sin θ[n] + y0 ]T . Also, the location of GN k wk can be 12 represented as wk = [ζk cos ϕk + x0 ζk sin ϕk + y0]T , where ζk and ϕk equal the distance and the angle between the geometric center c and GN k, respectively. Thus, the distance dk [n] between p the UAV and GN k in (8) can be expressed as dk [n] = r 2 + ζk2 + H 2 − 2rζk cos (θ[n] − ϕk ). By adopting the angular velocity ω[n] and the angular acceleration α[n], equations in (1)-(6) can be rewritten as ω[n] = ω[n − 1] + α[n − 1]δt , for n = 1, ..., N, (25) θ[n] = θ[n − 1] + ω[n − 1]δt + 12 α[n − 1]δt2 , for n = 1, ..., N, (26) θ[N] = θ[0] + 2π, ω[0] = ω[N], α[0] = α[N], (27) ka[n]k2 = kak [n]k2 + ka⊥ [n]k2 = r 2 α2 [n] + r 2 ω 4 [n] ≤ a2max , for n = 0, 1, ...N, (28) ωmin ≤ ω[n] ≤ ωmax , for n = 0, 1, ..., N, (29) Pprop [n] = c1 r 3 ω 3 [n] + c2 rω[n] + c2 rω 3 [n] g2 + c2 rα2 [n] , g 2 ω[n] for n = 0, 1, ..., N, (30) where ak [n] and a⊥ [n] are the tangential and centripetal accelerations, respectively, and ωmin , Vmin /r and ωmax , Vmax /r indicate the minimum and maximum angular velocity, respectively. Similar to Section III, we address the minimum average rate maximization problem and the EE maximization problem for the circular trajectory, which are respectively formulated as (P 3) : max {θ[n],ω[n],α[n]} {r,pk [n],τ } s.t. τ (31a) rmin ≤ r ≤ rmax , (31b) (10b) − (10d), (25) − (29), (P 4) : max {θ[n],ω[n],α[n]} {r,pk [n],η} s.t. where rmin , Vmin T 2π and rmax η PN n=1 Pprop [n] (32a) (10c), (11b), (25) − (29), (31b), Vmax T a max √ 4 , min , denote the minimum and 2π 2 max( ω [n]+α [n]) maximum radius of the circular trajectory, respectively. It is emphasized that (P3) and (P4) are difficult to solve because of the non-convex constraints and objective functions. To deal with the problems (P3) and (P4), similar SCA frameworks in Section III are applied. 13 A. Minimum Average Rate Maximization and EE maximization For the minimum average rate maximization problem (P3), we first find {r, pk [n]} with given {θ[n], ω[n], α[n]} and then updates {θ[n], ω[n], α[n], pk [n]} for a fixed r. For given {θ[n], ω[n], α[n]}, we adopt the change of variable Sk [n] and Sk,max [n] as pk [n]γ0 , (r − ζk cos (θ[n] − θk ))2 + ζk2 sin2 (θ[n] − θk ) + H 2 Sk [n] , pk [n]hk [n] = (33) Ppeak γ0 . (r − ζk cos (θ[n] − θk ))2 + ζk2 sin2 (θ[n] − θk ) + H 2 Sk,max [n] , Ppeak hk [n] = (34) Similar to the method in Section III-A, we employ the SCA to Sk,max [n]. Based on Lemma 1, lb1 the concave surrogate function Sk,max [n] of Sk,max [n] with a solution rl at the l-th iteration can be chosen as lb1 Sk,max [n] , Ppeak γ0 − rl+1 − b̌k [n] Ǎ2k [n] 2 ≤ Sk,max [n], ∀n, ! + B̌k [n] rl+1 − b̌k [n] rl − b̌k [n] + Čk [n] (35) where the constants b̌k [n], Ǎk [n], B̌k [n], and Čk [n] are respectively given by b̌k [n] , ζk cos (θ[n] − θk ) , Ǎk [n] , ζk2 sin2 (θ[n] − θk ) + H 2 ,   B̌k [n] , 2  1 Ǎ2k [n] −  1  2  , 2 rl − b̌k [n] + Ǎk [n] 2 2 2 rl − b̌k [n] rl − b̌k [n] 1 Čk [n] , . + 2 − 2 2 Ǎ2k [n] rl − b̌k [n] + Ǎk [n] rl − b̌k [n] + Ǎk [n] By applying (15), (P3) for fixed {θ[n], ω[n], α[n]} can be reformulated as an approximated convex problem at the (l + 1)-th iteration of the SCA (P 3.1) : max {rl+1 ,Sk,l+1 [n],τ lb1 } s.t. τ lb1 N 1 X log2 N n=1 (36a) 1+ K X Sm,l+1 [n] m=1 lb1 0 ≤ Sk [n] ≤ Sk,max [n], ∀k, n, (10d), (31b), ! − R̆kub [n] ! ≥ τ lb1 , ∀k, (36b) (36c) 14 P K where R̆kub [n] , Γ̆k [n] log2 e P . (P3.1) 1+ K j=1,j6=k Sj,l [n] j=1,j6=k (Sj,l+1[n] − Sj,l [n]) +log2 1 + PK j=1,j6=k Sj,l [n] and Γ̆k [n] , can be successively solved by the CVX until convergence. Next, we present a solution for (P3) with a given r. To obtain the concave surrogate function of Sk,max [n], we introduce the following lemma which identifies the surrogate function of the cosine function. Lemma 3: For any given φl , the concave surrogate function of cos φ can be computed as −(φ − φl + sin φl )2 sin2 φl + cos φl + ≤ cos φ. 2 2 (37) Proof: With a similar process in Appendix A, we can conclude that the function in (37) satisfies the conditions for a concave surrogate function [23]. lb2 By inspecting Lemmas 1 and 3, the concave surrogate function Sk,max [n] for Sk,max[n] can be identified as  2   rζk θl+1 [n] − b̂k [n] lb2 + B̂k [n] sin(θl [n] − θk ) θl+1 [n] − b̂k [n] + Ĉk [n] Sk,max [n] , Ppeak γ0 − 2 Âk [n]  ≤ Ppeak γ0 ≤ Sk,max [n], 2 rζk θl+1 [n] − b̂k [n] + Âk [n] (38) where b̂k [n], Âk [n], B̂k [n], and Ĉk [n] are given by b̂k [n] , θl [n] − sin (θl [n] − θk ) , Âk [n] , r 2 + ζk2 + H 2 − rζk 2 cos (θl [n] − θk ) + sin2 (θl [n] − θk ) ,  Ĉk [n] ,  B̂k [n] , 2rζk  1 Â2k [n] − 1   2  , rζk sin2 (θl [n] − θk ) + Âk [n] 2rζk sin2 (θl [n] − θk ) rζk sin2 (θl [n] − θk ) + − . 2 2 2 rζk sin2 (θl [n] − θk ) + Âk [n] Â [n] k rζk sin (θl [n] − θk ) + Âk [n] 1 15 By utilizing (15) and (38), at the (l + 1)-th iteration of the SCA algorithm with a given r, (P3) can be approximated to the following convex problem. (P 3.2) : max {θl+1 [n],ω[n],α[n]} {Sk,l+1 [n],τ lb2 } s.t. τ lb2 (39a) N 1 X log2 N n=1 1+ K X ! Sm,l+1 [n] m=1 lb2 0 ≤ Sk [n] ≤ Sk,max [n], ∀k, n, ! − R̆kub [n] ≥ τ lb2 , ∀k, (39b) (39c) (10d), (25) − (29). We then successively solve (P3.2) by the CVX until convergence. Similar to Algorithm 1, a solution of problem (P3) is obtained by alternately solving (P3.1) and (P3.2) until the objective value converges. For the EE maximization problem (P4) in the circular trajectory case, we can apply similar methods in Section III-B. Based on (P3.1) and (P3.2), given {θ[n], ω[n], α[n]} and r, (P4) can be transformed into two concave-convex fractional problems. By using Algorithm 2, we can alternately solve these problems until convergence. B. Trajectory Initialization To initialize the proposed algorithms, we employ a simple circular path concept in [12]. First, the initial angular velocity ω0 is set to ω0 = 2π , T which implies θ0 [n] = 2π Nn , ∀n. Next, the initial radius r0 is chosen to fulfill the constraints in (4), (5), and (10d), which can be expressed as Vmin T Vmax T amax (40) ≤ r0 ≤ min 2π , ω2 , 2π 0 c1 r03 ω03 + c2 r0 ω0 + c2 r0 ω03 g2 ≤ Plim . (41) We can simply find r0 which maximizes the minimum rate in (P1) and (P3) under constraints (40) and (41) via one-dimensional line search. For the EE maximization problems (P2) and (P4), r0 can be computed in the range of (40). As a result, the initial trajectory q0 [n] can be written by q0 [n] = [r0 cos 2π Nn + x0 r0 sin 2π Nn + y0 ]T (n = 0, 1, ..., N) and the initial velocity v0 [n] can be simply obtained as v0 [n] = (q0 [n + 1] − q0 [n])/δt (n = 0, 1, ..., N − 1) assuming δt2 ≈ 0 in (2). 16 T = 50 sec T = 100 sec T = 150 sec T = 400 sec 800 600 400 y (m) 200 0 -200 -400 -600 -800 -1000 -1200 -1000 -800 -600 -400 -200 0 200 400 600 800 1000 x (m) Fig. 2. Optimized UAV trajectories for different periods T with Plim = 150 W. V. N UMERICAL R ESULTS In this section, we provide numerical results to validate the effectiveness of the proposed algorithms. For the simulations, we consider K = 6 GNs which are distributed as in Fig. 2 where the locations of the GNs are marked with the triangles. The constant altitude, the bandwidth, the reference SNR, and the peak transmission power are set to be H = 100 m, W = 1 MHz, γ0 = 80 dB, and Ppeak = 10 dBm, respectively. Also, the minimum velocity, the maximum velocity, and the maximum acceleration of the UAV are determined as Vmin = 3 m/sec, Vmax = 100 m/sec, and amax = 5 m/sec2 , respectively. For the propulsion power consumption model in (6), the constants c1 and c2 are set as c1 = 9.26 × 10−4 and c2 = 2250, respectively, which make the minimum propulsion power consumption Pprop,min = 100 W when kvk = 30 m/sec. We first demonstrate the performance of the minimum rate maximization algorithms. Fig. 2 illustrates the optimized UAV trajectories with various T for Plim = 150 W. It is observed that when T is smaller than 150 sec, as T increases, the UAV tries to get closer to all GNs in order to improve the channel conditions from the GNs. In contrast, if T is sufficiently large (T = 400 sec), the UAV is now able to visit all the GNs within a given time period. Thus, the UAV can 17 1 0.9 Max-min rate (bps/Hz) 0.8 Proposed Circular w/ opt. r, ω, α, & p Circular w/ opt. r & p Circular w/ opt. r 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 50 100 150 200 250 300 350 400 450 500 Period T (sec) Fig. 3. Max-min rate with respect to the period T with Plim = 150 W. hover over each GN for a while by traveling smooth path around the GNs. This is different from the results in [12] where the UAV does not have practical movement constraints. This can be explained as follows: Due to the constraints on the velocity and the propulsion power, the UAV cannot stay at fixed positions as in [12]. Therefore, the UAV continuously moves around as close to the GNs as possible to maintain good communication channels without exceeding the propulsion power limit Plim . Fig. 3 shows the maximized minimum (max-min) rate performance of the proposed algorithm as a function of T . We compare the performance of the proposed algorithm with the following circular trajectory based methods. - Circular with optimum r, ω, α, and p: radius, angular velocity, angular acceleration, and uplink transmit power are jointly optimized with (P3) in Section IV-A with the circular trajectory. - Circular with optimum r and p: radius and uplink transmit power are jointly optimized with (P3.1) in Section IV-A with the circular trajectory. - Circular with optimum r: radius is optimized with Ppeak as the initial circular trajectory in 18 1000 Plim = 110 W 800 Plim = 130 W 600 No power limit 400 y (m) 200 0 -200 -400 -600 -800 -1000 -1500 -1250 -1000 -750 -500 -250 0 250 500 750 1000 1250 x (m) Fig. 4. Optimized UAV trajectories for different propulsion power limit Plim with T = 400 sec. Section IV-B First, it can be verified that the proposed algorithm outperforms the baseline schemes regardless of the time period T . Also, we can see that the max-min rate in the proposed algorithm monotonically increases with T , since more time is available at the UAV to hover around each GN. In contrast, in the baseline schemes which are restricted in circular shape trajectory, the max-min rate performance first increases as T grows, and then decreases after a certain T . This is due to a fact that in order to satisfy the propulsion power constraint, the radius of the circular trajectory should increase as T gets large, and thus the UAV may become too far away from the geometric center of the GNs after a certain T . Therefore, we can expect the performance gain of the proposed algorithm over baseline schemes is to grow with T . Fig. 4 illustrates the optimized UAV trajectories for various propulsion power limit Plim with T = 400 sec. It can be shown that for Plim = 110 W, the trajectory of the UAV is restricted to a smooth path with a large turning radius to consume a low propulsion power. However, as Plim gets larger, we observe quick changes along the trajectory path. Thus the UAV can move with a much smaller turning radius, which enhances the max-min rate performance. 19 1 0.9 Proposed Circular w/ opt. r, ω, α, & p Circular w/ opt. r & p Circular w/ opt. r Max-min rate (bps/Hz) 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 100 125 150 175 200 225 250 Propulsion power limit (W) Fig. 5. Max-min rate with respect to the propulsion power limit Plim with T = 400 sec. In Fig. 5, we depict the average max-min rate of various schemes as a function of the propulsion power constraint Plim . For both the proposed algorithm and the baseline schemes, the max-min rate first increases as Plim grows and then gets saturated. This can be explained as follows: With a large Plim , the trajectory and the velocity of the UAV change more freely to attain good channel conditions, and thus the max-min rate increases. However, even if a large Plim is given, the max-min rate cannot continue to increase because there are practical limits on the velocity and acceleration. Similar to Fig. 3, we can see that the proposed algorithm provides significant performance gains over the baseline schemes. Next, in Fig. 6, we investigate the optimized trajectory of the EE maximization problem with various T . As T increases, the overall patterns are similar to Fig. 2. Nevertheless, to balance between the rate performance and the propulsion power consumption, the EE maximization trajectory shows a smooth path with a relatively large turning radius, and thus the average propulsion power consumption becomes lower. To present the impact of the energy efficient UAV communication designs, Fig. 7 depicts the UAV speed of the proposed EE maximization method with T = 400 sec. For comparison, we 20 1000 T = 50 sec T = 200 sec T = 400 sec T = 500 sec 800 600 400 y (m) 200 0 -200 -400 -600 -800 -1000 -1250 -1000 -750 -500 -250 0 250 500 750 1000 1250 x (m) Fig. 6. Optimized energy efficient UAV trajectories for different periods T 100 Minimum rate maximization w/o P lim 90 EE maximization 80 Speed (m/sec) 70 60 50 40 30 20 10 0 0 50 100 150 200 250 300 350 400 Time (sec) Fig. 7. UAV speeds for the max-min rate without propulsion power constraint and the EE maximization with T = 400 sec 21 TABLE I P ERFORMANCE COMPARISON WITH MAX - MIN RATE AND EE MAXIMIZATION FOR T = 400 Max-min rate w/o Plim EE maximization Proposed Circular Proposed Circular SEC Average speed (m/sec) Average acceleration (m/sec2 ) Average max-min rate (bps/Hz) Average power (Watts) Energy efficiency (kbits/Joule) 18.42 13.50 25.73 15.35 4.73 1.88 2.71 0.27 0.99 0.53 0.79 0.47 553.01 541.41 122.14 151.33 1.80 0.98 6.47 3.10 also consider the max-min rate scheme without the propulsion power constraint. It is observed that for the max-min rate case, the UAV tries to fly between the GNs as fast as possible and stay over the GNs with a low speed. On the other hand, the EE maximization scheme keeps the speed of the UAV at around 30 m/sec in order not to waste the propulsion energy. Finally, Table I presents the performance comparison of the max-min rate without propulsion power constraint and the EE maximization designs for both the proposed and the baseline schemes with T = 400 sec. We can see that the max-min rate methods consume much higher propulsion power by allowing a large variation of the speed and the average acceleration. In contrast, the speed of the proposed EE maximization design slowly varies with low acceleration, and thus much higher EE can be achieved. We observe that the proposed EE maximization algorithm exhibits about 259 % gain over the max-min rate without the propulsion power constraint and 109 % gain over the circular baseline EE maximization scheme. VI. C ONCLUSION In this paper, we have studied the UAV-aided wireless communication optimization under the practical propulsion energy constraint at the UAV. For both the minimum average rate maximization problem and the EE maximization problem, the UAV trajectories and the uplink transmit power of the GNs have been jointly optimized. By applying the SCA technique, we have proposed efficient iterative algorithms which find local optimal solutions. Numerical results have demonstrated that the proposed algorithms provide substantial performance gains compared to the baseline schemes. 22 A PPENDIX A PROOF OF LEMMA Let us define a function f1 (u) , 1 ρkuk2 +z 1 for u = [ux uy ]T where z and ρ are positive constants. For any given ul ∈ R2×1 , in order for arbitrary function g1 (u\|ul ) to be a concave surrogate function of f1 (u), it must satisfy the following conditions: f1 (ul ) = g1 (ul \|ul ), ∇g1 (ul \|ul ) = ∇f1 (ul ), and g1 (u\|ul ) ≤ f1 (u), ∀u [23]. Denoting the function g1 (u\|ul ) as where B , 2ρ 1 z2 ρkuk2 g1 (u\|ul ) , − 2 + BuT ul + C, (42) z 2 2ρkul k2 − ρkuz 2l k , it can be easily shown − (ρku k12 +z)2 and C , ρkul1k2 +z + (ρku k2 +z)2 l l that f1 (ul ) = g1 (ul \|ul ), i.e., g1 (u\|ul ) fulfills the first condition for the surrogate function. Also, the gradient of f1 (u) and g1 (u\|ul ) with respect to u can be respectively computed as −2ρu , (ρkuk2 + z)2 2ρu ∇g1 (u\|ul ) = − 2 + Bul . z ∇f1 (u) = − (43) (44) Since two gradients in (43) and (44) become identical at u = ul , g1 (u\|ul ) satisfies the second condition for the surrogate function. To prove the global lower bound condition, we can calculate the Hessian matrix ∇2u h1 (u\|ul ) of the function h1 (u\|ul ) , f1 (u) − g1 (u\|ul ) as # " 2 2 2 E + 4ρz u 4ρz u u x y x , ∇2 h1 (u\|ul ) = D 2 4ρz ux uy E + 4ρz 2 u2y where D , 2ρ z 2 (ρkuk2 +z)3 (45) > 0 and E , ρ3 kuk6 + 3ρ2 zkuk4 + 2ρz 2 kuk2 ≥ 0. One can easily check that the Hessian in (45) is a positive semi-definite matrix, which implies that h1 (u\|ul ) is a convex function. Since ∇h1 (u\|ul ) = 0 at u = ul from (43) and (44), the global minimum of h1 (u\|ul ) is achieved at u = ul with h1 (ul \|ul ) = 0. As a result, we can show that h1 (u\|ul ) is greater than or equal to 0 for any given ul , and thus the third condition for the surrogate function holds. By substituting u = ql+1 [n] − wk , ul = ql [n] − wk , z = H 2 , and ρ = 1 and multiplying f1 (u) and g1 (u\|ul ) by Ppeak γ0 , Lemma 1 is thus proved. R EFERENCES [1] Y. Zeng, R. Zhang, and T. J. Lim, “Wireless communications with unmanned aerial vehicles: opportunities and challenges,” IEEE Commun. Mag., vol. 54, pp. 36–42, May. 2016. 23 [2] W. Lee, I. Lee, J. S. Kwak, B.-C. Ihm, and S. Han, “Multi-BS MIMO cooperation: challenges and practical solutions in 4G systems,” IEEE Wireless Commun., vol. 19, pp. 89–96, Feb. 2012. [3] Y. Zeng, R. Zhang, and T. J. 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c xxxx Society for Industrial and Applied Mathematics ⃝ SIAM/ASA J. UNCERTAINTY QUANTIFICATION Vol. xx, pp. x Mathematical Properties of Polynomial Dimensional Decomposition ∗ arXiv:1804.01647v1 [math.NA] 5 Apr 2018 Sharif Rahman† Abstract. Many high-dimensional uncertainty quantification problems are solved by polynomial dimensional decomposition (PDD), which represents Fourier-like series expansion in terms of random orthonormal polynomials with increasing dimensions. This study constructs dimension-wise and orthogonal splitting of polynomial spaces, proves completeness of polynomial orthogonal basis for prescribed assumptions, and demonstrates mean-square convergence to the correct limit – all associated with PDD. A second-moment error analysis reveals that PDD cannot commit larger error than polynomial chaos expansion (PCE) for the appropriately chosen truncation parameters. From the comparison of computational eﬀorts, required to estimate with the same precision the variance of an output function involving exponentially attenuating expansion coeﬃcients, the PDD approximation can be markedly more eﬃcient than the PCE approximation. Key words. Uncertainty quantification, ANOVA decomposition, multivariate orthogonal polynomials, polynomial chaos expansion. 1. Introduction. Polynomial dimensional decomposition (PDD) is a hierarchical, infinite series expansion of a square-integrable random variable involving measure-consistent orthogonal polynomials in independent random variables. Introduced by the author [20, 21] as a polynomial variant of the well-known analysis-of-variance (ANOVA) dimensional decomposition (ADD), PDD deflates the curse of dimensionality to some extent by developing an input-output behavior of complex systems with low eﬀective dimensions [4], wherein the eﬀects of degrees of interactions among input variables weaken rapidly or vanish altogether. Approximations stemming from truncated PDD are commonly used for solving uncertainty quantification problems in engineering and applied sciences, including multiscale fracture mechanics [6], random eigenvalue problems [24], computational fluid dynamics [27], and stochastic design optimization [25], to name a few. However, all existing works on PDD are focused on practical applications with almost no mathematical analysis of PDD. Indeed, a number of mathematical issues concerning necessary and suﬃcient conditions for the completeness of PDD basis functions; convergence, exactness, and optimal analyses of PDD; and approximation quality of the truncated PDD have yet to be studied or resolved. This paper fills the gap by establishing fundamental mathematical properties to empower PDD with a solid foundation, so that PDD can be as credible as its close cousin, polynomial chaos expansion (PCE) [5, 10, 28, 29], providing an alternative, if not a better, choice for uncertainty quantification in computational science and engineering. The principal objective of this work is to examine important mathematical properties of PDD, not studied heretofore, for arbitrary but independent probability measures of input random variables. The paper is organized as follows. Section 2 defines or discusses mathematical notations and preliminaries. Two sets of assumptions on the input probability measures required by PDD are explained. A brief exposition of univariate and multivariate orthogonal polynomials consistent with a general but product-type probability measure, ∗ This work was supported by the U.S. National Science Foundation under Grant Number CMMI-1462385. College of Engineering and Applied Mathematics & Computational Sciences, The University of Iowa, Iowa City, IA 52242 ([email protected]). Questions, comments, or corrections to this document may be directed to that email address. † 1 x–x 2 S. RAHMAN including their second moment properties, is given in Section 3. The section also describes relevant polynomial spaces and construction of their dimension-wise orthogonal decompositions. The orthogonal basis and completeness of multivariate orthogonal polynomials have also been proved. Section 4 briefly explains ADD, followed by presentations of PDD for a square-integrable random variable. The convergence and exactness of PDD are explained. In the same section, a truncated PDD and its approximation quality are discussed. The formulae for the mean and variance of a truncated PDD are also derived. The section ends with an explanation on how and when the PDD can be extended for infinitely many input variables. Section 5 briefly describes degree-wise orthogonal decompositions of polynomial spaces, leading to PCE. In Section 6, a second-moment error analysis of PDD is conducted, followed by a comparison with that of PCE. Finally, conclusions are drawn in Section 7. 2. Input Random Variables. Let N := {1, 2, . . .}, N0 := N ∪ {0}, and R := (−∞, +∞) represent the sets of positive integer (natural), non-negative integer, and real numbers, respectively. Denote by A{i} , i = 1, . . . , N , an ith bounded or unbounded subdomain of R, {i} ⊆ RN . so that AN := ×N i=1 A Let (Ω, F, P) be a complete probability space, where Ω is a sample space representing an abstract set of elementary events, F is a σ-algebra on Ω, and P : F → [0, 1] is a probability measure. With B N := B(AN ) representing the Borel σ-algebra on AN ⊆ RN , consider an AN -valued input random vector X := (X1 , . . . , XN )T : (Ω, F) → (AN , BN ), describing the statistical uncertainties in all system parameters of a stochastic problem. The input random variables are also referred to as basic random variables [10]. The non-zero, finite integer N represents the number of input random variables and is often referred to as the dimension of the stochastic problem. Denote by FX (x) := P(∩N i=1 {Xi ≤ xi }) the joint distribution function of X, admitting the joint probability density function fX (x) := ∂ N FX (x)/∂x1 · · · ∂xN . Given the abstract probability space (Ω, F, P) of X, the image probability space is (AN , B N , fX dx), where AN can be viewed as the image of Ω from the mapping X : Ω → AN , and is also the support of fX (x). Similarly, each component random variable Xi is defined on the abstract marginal probability space (Ω{i} , F {i} , P{i} ) comprising sample space Ω{i} , σ-algebra F {i} , and probability measure P{i} . Then, the corresponding image probability space is (A{i} , B {i} , fXi dxi ), where A{i} ⊆ R is the image sample space of Xi , B{i} is the Borel σ-algebra on A{i} , and fXi (xi ) is the marginal probability density function of Xi . Relevant statements and objects in the abstract probability space have obvious counterparts in the associated image probability space. Both probability spaces will be used in this paper. Two sets of assumptions used by PDD are as follows. Assumption 2.1.The input random vector X := (X1 , . . . , XN )T : (Ω, F) → (AN , B N ) satisfies all of the following conditions: (1) Each input random variable Xi : (Ω{i} , F {i} ) → (A{i} , B {i} ) has absolutely continuous marginal distribution function FXi (xi ) := P(Xi ≤ xi ) and continuous marginal density function fXi (xi ) := ∂FXi (xi )/∂xi with a bounded or unbounded support A{i} ⊆ R. (2) All component random variables Xi , i = 1, . . . , N , are statistically independent, but not necessarily identical. In consequence, X is endowed with a product-type probability ∏ density function, that is, fX (x) = N i=1 fXi (xi ) with a bounded or unbounded support AN ⊆ RN . (3) Each input random variable Xi possesses finite moments of all orders, that is, for all POLYNOMIAL DIMENSIONAL DECOMPOSITION i = 1, . . . , N and l ∈ N0 , ∫ ∫ [ ] l l (2.1) µi,l := E Xi := Xi (ω)dP(ω) = Ω 3 AN xli fX (x)dx = ∫ A{i} xli fXi (xi )dxi < ∞, where E is the expectation operator with respect to the probability measure P or fX (x)dx. Assumption 2.2.The moments and marginal density function of each input random variable Xi , where i = 1, . . . , N , satisfy at least one of the following conditions [10]: (1) The density function fXi (xi ) has a compact support, that is, there exists a compact interval [ai , bi ], ai , bi ∈ R, such that P(ai ≤ Xi ≤ bi ) = 1. (2) For the moment sequence {µi,l }l∈N0 for Xi , there holds (2.2) lim inf l→∞ (µi,2l )1/2l < ∞. 2l (3) For the moment sequence {µi,l }l∈N0 for Xi , there holds ∑ (2.3) l∈N0 1 = ∞. (µi,2l )1/2l (4) The random variable Xi is exponentially integrable, that is, there exists a real number a > 0 such that ∫ exp(a\|xi \|)fXi (xi )dxi < ∞. (2.4) A{i} (5) If the density function fXi (xi ) is symmetric, diﬀerentiable, and strictly positive, then there exists a real number a > 0 such that ∫ ln fXi (xi ) −xi dfXi (xi )/dxi − (2.5) → ∞ as (xi → ∞, xi ≥ a). 2 dxi = ∞ and fXi (xi ) {i} 1 + x A i Assumption 2.1 assures the existence of an infinite sequence of orthogonal polynomials consistent with the input probability measure. Assumption 2.2, in addition to Assumption 2.1, guarantees the input probability measure to be determinate, resulting in a complete orthogonal polynomial basis of a function space of interest. Both assumptions impose only mild restrictions on the probability measure. Examples of input random variables satisfying Assumptions 2.1 and 2.2 are Gaussian, uniform, exponential, beta, and gamma variables, which are commonly used in uncertainty quantification. These assumptions, to be explained in the next section, are vitally important for the determinacy of the probability measure and the completeness of the orthogonal polynomial basis. Therefore, for both PDD and PCE, which entail orthogonal polynomial expansions, Assumptions 2.1 and 2.2 are necessary. Unfortunately, they are not always clearly specified in the PDD or PCE literature. A prototypical example where Assumption 2.1 is satisfied, but Assumption 2.2 is not, is the case of a lognormal random variable. As noted by Ernst et al. [10], the violation of Assumption 2.2 leads to indeterminacy of the input probability measure and thereby fails to form a complete orthogonal polynomial basis. Finally, Assumptions 2.1 and 2.2 can be modified to account for random variables with discrete or mixed distributions [11] or dependent random variables [23]. The discrete or mixed distributions and dependent variables are not considered in this paper. 4 S. RAHMAN 3. Measure-Consistent Orthogonal Polynomials and Polynomial Spaces. 3.1. Univariate orthogonal polynomials. Consider an ith random variable Xi defined on the abstract probability space (Ω{i} , F {i} , P{i} ) with its image (A{i} , B {i} , fXi dxi ). Let Π{i} := R[xi ] be the space of real polynomials in xi . For any polynomial pair P{i} , Q{i} ∈ Π{i} , define an inner product ∫ [ ] (3.1) (P{i} , Q{i} )f dx := P{i} (xi )Q{i} (xi )fXi (xi )dxi =: E P{i} (Xi )Q{i} (Xi ) Xi i A{i} with respect to the probability measure fXi (xi )dxi and the induced norm ∥P{i} ∥fXi dxi := √ (P{i} , P{i} )f Xi dxi = (∫ Ai 2 P{i} (xi )fXi (xi )dxi )1/2 √ [ ] 2 (X ) . =: E P{i} i Under Assumption 2.1,∫ moments of Xi of all orders exist and are finite, including zeroorder moments µi,0 := A{i} fXi (xi )dxi = 1, i = 1, . . . , N , that are always positive. Clearly, ∥P{i} ∥ > 0 for all non-zero P{i} ∈ Π{i} . Then, according to Gautschi [12], the inner product in (3.1) is positive-definite on Π{i} . Therefore, there exists an infinite set of univariate orthogonal polynomials, say, {P{i},ji (xi ) : ji ∈ N0 }, P{i},ji ̸= 0, which is consistent with the probability measure fXi (xi )dxi , satisfying { 2 ( ) E[P{i},j (Xi )], ji = ki , i (3.2) P{i},ji , P{i},ki f dx = i Xi 0, ji ̸= ki , 2 for ki ∈ N0 , where 0 < E[P{i},j (Xi )] < ∞. Here, in the notation for the polynomial i P{i},ji (xi ), the first and second indices refer to the ith variable and degree ji , respectively. Prominent examples of classical univariate orthogonal polynomials comprise Hermite, Laguerre, and Jacobi polynomials, which are consistent with the measures defined by Gaussian, gamma, and beta densities on the whole real line, semi-infinite interval, and bounded interval, respectively. Many orthogonal polynomials, including the three classical polynomials mentioned, can be expressed in a unified way by invoking hypergeometric series, incorporated in a tree structure of the Askey scheme [1]. For even more general measures, established numerical techniques, such as Gram-Schmidt [13] and Stieltjes’ procedure [26], can be used to generate any measure-consistent orthogonal polynomials. 3.2. Multivariate orthogonal polynomials. For N ∈ N, denote by {1, . . . , N } an index set, so that u ⊆ {1, . . . , N } is a subset, including the empty set ∅, with cardinality 0 ≤ \|u\| ≤ N . When ∅ ̸= u ⊆ {1, . . . , N }, a \|u\|-dimensional multi-index is denoted by ju := \|u\| (ji1 , . . . , ji\|u\| ) ∈ N0 with degree \|ju \| := ji1 + · · · + ji\|u\| , where jip ∈ N0 , p = 1, . . . , \|u\|, represents the pth component of ju .1 For ∅ ̸= u ⊆ {1, . . . , N }, let Xu := (Xi1 , . . . , Xi\|u\| )T , a subvector of X, be defined on the abstract probability space (Ωu , F u , Pu ), where Ωu is the sample space of Xu , F u is a σ-algebra on Ωu , and Pu is a probability measure. The corresponding image probability space is (Au , B u , fXu dxu ), where Au := ×i∈u A{i} ⊆ R\|u\| is the image sample space of Xu , 1 The same symbol \| · \| is used for designating both the cardinality of a set and the degree of a multi-index in this paper. POLYNOMIAL DIMENSIONAL DECOMPOSITION 5 B u is the Borel σ-algebra on Au , and fXu (xu ) is the marginal ∏ probability density function of Xu supported on Au . Under Assumption 2.1, fXu (xu ) = i∈u fXi (xi ). Denote by Πu := R[xu ] = R[xi1 , . . . , xi\|u\| ] the space of all real polynomials in xu . Then, given the inner product ∫ Pu (xu )Qu (xu )fXu (xu )dxu =: E [Pu (Xu )Qu (Xu )] , (Pu , Qu )fX dxu := u Au Πu two polynomials Pu ∈ and Qu ∈ Πu in xu are called orthogonal to each other if (Pu , Qu )fX dxu = 0 [8]. Moreover, a polynomial Pu ∈ Πu is said to be an orthogonal u polynomial with respect to fXu dxu if it is orthogonal to all polynomials of lower degree, that is, if [8] (3.3) (Pu , Qu )fX u dxu = 0 ∀Qu ∈ Πu with deg Qu < deg Pu . \|u\| Let {Pu,ju (xu ) : ju ∈ N0 }, ∅ ̸= u ⊆ {1, . . . , N }, represent an infinite set of multivariate orthogonal polynomials, which is consistent with the probability measure fXu (xu )dxu , satisfying (3.4) (Pu,ju , Pu,ku )fX u dxu \|u\| =: E [Pu,ju (Xu )Pu,ku (Xu )] = 0 ∀ju ̸= ku , ku ∈ N0 . Clearly, each Pu,ju ∈ Πu is a multivariate orthogonal polynomial satisfying (3.3). Due to the product-type probability measure of Xu , a consequence of statistical independence from Assumption 2.1, such multivariate polynomials exist and are easily constructed by tensorizing univariate orthogonal polynomials. Proposition 3.1.Let X := (X1 , . . . , XN )T : (Ω, F) → (AN , B N ) be a vector of N ∈ N input random variables fulfilling Assumption 2.1. Suppose that the sets of univariate orthogonal polynomials for all marginal measures are obtained as {P{i},ji (xi ) : ji ∈ N0 }, i = 1, . . . , N . Then, for ∅ ̸= u ⊆ {1, . . . , N }, the set of multivariate orthogonal polynomials in xu consistent with the probability measure fXu (xu )dxu is { } ⊗{ } \|u\| P{i},ji (xi ) : ji ∈ N0 , (3.5) Pu,ju (xu ) : ju ∈ N0 = ⊗ i∈u where the symbol denotes tensor product. In terms of an element, the multivariate orthogonal polynomial of degree \|ju \| = ji1 + · · · + ji\|u\| is ∏ (3.6) Pu,ju (xu ) = P{i},ji (xi ). i∈u Proof. Consider two distinct polynomials Pu,ju (xu ) and Pu,ku (xu ) from the set {Pu,ju (xu ) : \|u\| ju ∈ N0 } satisfying (3.5). Since ju ̸= ku , ju and ku must diﬀer in at least one component. Without loss of generality, suppose that ji1 ̸= ki1 . Then, by Fubini’s theorem, with statistical independence of random variables in mind, ∫ (Pu,ju Pu,ku )fX dxu = Au Pu,ju (xu )Pu,ku (xu )fXu (xu )dxu u \|u\| [ ] ∏ ∫ = ×\|u\| A{ip } P{ip },jip (xip )P{ip },kip (xip )fXip (xip )dxip × (3.7) p=2 p=2 ∫ P (xi1 )P{i1 },ki1 (xi1 )fXi1 (xi1 )dxi1 {i } {i },j 1 i1 A 1 = 0, 6 S. RAHMAN where the equality to zero in the last line results from the recognition that the inner integral vanishes by setting i = i1 in (3.2). \|u\| In addition, for ju ∈ N0 , ] [ 2 ] ∏ [ 2 (3.8) (Pu,ju Pu,ju )fX dxu =: E Pu,j (X ) = E P (X ) >0 u i {i},ji u u i∈u and is finite by virtue of the existence of the set of univariate orthogonal polynomials \|u\| {P{i},ji (xi ) : ji ∈ N0 } for i = 1, . . . , N . Therefore, {Pu,ju (xu ) : ju ∈ N0 } satisfying (3.5) is a set of multivariate orthogonal polynomials consistent with the probability measure fXu (xu )dxu . Once the multivariate orthogonal polynomials are obtained, they can be scaled to generate multivariate orthonormal polynomials, as follows. Definition 3.2.A multivariate orthonormal polynomial Ψu,ju (xu ), ∅ ̸= u ⊆ {1, . . . , N }, \|u\| ju ∈ N0 , of degree \|ju \| = ji1 + · · · + ji\|u\| that is consistent with the probability measure fXu (xu )dxu is defined as (3.9) Ψu,ju (xu ) := √ Pu,ju (xu ) 2 (X )] E[Pu,j u u where Ψ{i},ji (xi ) := P{i},ji (xi )/ √ = ∏ P{i},j (xi ) √ [ i Ψ{i},ji (xi ), ] =: 2 i∈u i∈u E P{i},ji (Xi ) ∏ 2 E[P{i},j (Xi )] is a univariate orthonormal polynomial in i xi of degree ji that is consistent with the probability measure fXi (xi )dxi 3.3. Dimension-wise orthogonal decomposition of polynomial spaces. An orthogonal decomposition of polynomial spaces entailing dimension-wise splitting leads to PDD. Here, to facilitate such splitting of the polynomial space Πu for any ∅ ̸= u ⊆ {1, . . . , N }, limit the power jip of the ip -th variable, where ip ∈ u ⊆ {1, . . . , N }, p = 1, . . . , \|u\|, and \|u\| > 0, to take on only positive integer values. In consequence, ju := (ji1 , . . . , ji\|u\| ) ∈ N\|u\| , the multi-index of Pu,ju (xu ), has degree \|ju \| = ji1 + · · · + ji\|u\| , varying from \|u\| to ∞ as ji1 ̸= · · · ji\|u\| ̸= 0. For ju ∈ N\|u\| and xu := (xi1 , . . . , xi\|u ), a monomial in the variables xi1 , . . . , xi\|u\| is the ji ji \|u\| product xjuu = xi11 . . . xi\|u\| and has a total degree \|ju \|. A linear combination of xjuu , where \|ju \| = l, \|u\| ≤ l ≤ ∞, is a homogeneous polynomial in xu of degree l. For ∅ ̸= u ⊆ {1, . . . , N }, denote by Qul := span{xjuu : \|ju \| = l, ju ∈ N\|u\| }, \|u\| ≤ l < ∞, the space of homogeneous polynomials in xu of degree l where the individual degree of each variable is non-zero and by Θum := span{xjuu : \|u\| ≤ \|ju \| ≤ m, ju ∈ N\|u\| }, \|u\| ≤ m < ∞, the space of polynomials in xu of degree at least \|u\| and at most m where the individual degree of each variable is non-zero. The dimensions of the vector spaces Qul and Θum , respectively, are { } ( l−1 ) u \|u\| (3.10) dim Ql = # ju ∈ N : \|ju \| = l = \|u\| − 1 POLYNOMIAL DIMENSIONAL DECOMPOSITION and dim Θum = (3.11) m ∑ l=\|u\| u Z\|u\| dim Qul = 7 ) ( ) m ( ∑ l−1 m = . \|u\| − 1 \|u\| l=\|u\| Θu\|u\| . Let := For each \|u\| + 1 ≤ l < ∞, denote by Zlu ⊂ Θul the space of orthogonal polynomials of degree exactly l that are orthogonal to all polynomials in Θul−1 , that is, Zlu := {Pu ∈ Θul : (Pu , Qu )fXu dxu = 0 ∀ Qu ∈ Θul−1 }, \|u\| + 1 ≤ l < ∞. Then Zlu , provided that the support of fXu (xu ) has non-empty interior, is a vector space of dimension ( ) l−1 Mu,l := dim Zlu = dim Qul = . \|u\| − 1 Many choices exist for the basis of Zlu . Here, to be formally proved in Section 3.3.2, select {Pu,ju (xu ) : \|ju \| = l, ju ∈ N\|u\| } ⊂ Zlu to be a basis of Zlu , comprising Mu,l number of basis functions. Each basis function Pu,ju (xu ) is a multivariate orthogonal polynomial of degree \|ju \| as defined earlier. Clearly, Zlu = span{Pu,ju (xu ) : \|ju \| = l, ju ∈ N\|u\| }, \|u\| ≤ l < ∞. According to Proposition 3.3, to be presented later, Pu,ju (Xu ) is orthogonal to Pv,kv (Xv ) whenever (1) u ̸= v and ju , kv are arbitrary; or (2) u = v and ju ̸= kv . Therefore, any two distinct polynomial subspaces Zlu and Zlv′ , where ∅ ̸= u ⊆ {1, . . . , N }, ∅ ̸= v ⊆ {1, . . . , N }, \|u\| ≤ l < ∞, and \|v\| ≤ l′ < ∞, are orthogonal whenever u ̸= v or l ̸= l′ . In consequence, there exist orthogonal decompositions of Θum = m ⊕ l=\|u\| Zlu = m ⊕ l=\|u\| span{Pu,ju (xu ) : \|ju \| = l, ju ∈ N\|u\| } = span{Pu,ju (xu ) : \|u\| ≤ \|ju \| ≤ m, ju ∈ N\|u\| } with the symbol ⊕ representing orthogonal sum and Πu (3.12) = 1⊕ = 1⊕ ∞ ⊕ ⊕ ∅̸=v⊆u l=\|v\| ⊕ ∅̸=v⊆u Zlv =1⊕ ⊕ ∞ ⊕ ∅̸=v⊆u l=\|v\| span{Pv,jv (xv ) : \|jv \| = l, jv ∈ N\|v\| } span{Pv,jv (xv ) : jv ∈ N\|v\| }, where 1 := span{1}, the constant subspace, needs to be added because the subspace Zlv excludes constant functions. Recall that ΠN is the space of all real polynomials in x. Then, setting u = {1, . . . , N } in (3.12) first and then swapping v for u yields yet another orthogonal decomposition of ΠN (3.13) = 1⊕ = 1⊕ = 1⊕ ⊕ ∞ ⊕ ∅̸=u⊆{1,...,N } l=\|u\| ∞ ⊕ ⊕ ∅̸=u⊆{1,...,N } l=\|u\| ⊕ ∅̸=u⊆{1,...,N } Zlu span{Pu,ju (xu ) : \|ju \| = l, ju ∈ N\|u\| } span{Pu,ju (xu ) : ju ∈ N\|u\| }. 8 S. RAHMAN Note that the last expression of (3.13) is equal to the span of (3.14) N ⊗ { } { } N Pj (x) : j ∈ N0 := P{i},ji (xi ) : ji ∈ N0 , i=1 representing an infinite set of orthogonal polynomials in x. Given the dimension-wise orthogonal splitting of ΠN , any square-integrable function of input random vector X can be expanded as a Fourier-like series of hierarchically ordered multivariate orthogonal or orthonormal polynomials in Xu , ∅ = ̸ u ⊆ {1, . . . , N }. The expansion is referred to as PDD, to be formally presented and analyzed in Section 4. 3.4. Statistical properties of random multivariate polynomials. When the input random variables X1 , . . . , XN , instead of real variables x1 , . . . , xN , are inserted in the argument, the multivariate polynomials Pu,ju (Xu ) and Ψu,ju (Xu ), where ∅ = ̸ u ⊆ {1, . . . , N } and ju ∈ N\|u\| , become functions of random input variables. Therefore, it is important to establish their second-moment properties, to be exploited in the remaining part of this section and Section 4. Proposition 3.3.Let X := (X1 , . . . , XN ) be a vector of N ∈ N input random variables fulfilling Assumption 2.1. For ∅ ̸= u, v ⊆ {1, . . . , N }, ju ∈ N\|u\| , and kv ∈ N\|v\| , the first- and second-order moments of multivariate orthogonal polynomials are (3.15) E [Pu,ju (Xu )] = 0 and (3.16) E [Pu,ju (Xu )Pv,kv (Xv )] = ] ∏ [ 2  E P{i},j (X ) i , i u = v, ju = kv , i∈u  0, otherwise, respectively. independence of random variables, E[Pu,ju (Xu )] = ∏ Proof. Using (3.6) and statistical \|u\| i∈u E[P{i},ji (Xi ] for any ju ∈ N . Since each component of ju is non-zero, (3.2), with the constant function P{i},0 ̸= 0 in mind, produces E[P{i},ji (Xi ] = 0 for any i ∈ u, ji ∈ N, resulting in (3.15). To obtain the non-trivial result of (3.16), set u = v and ju = kv and use (3.8) directly. The trivial result of (3.16) is obtained by considering two subcases. First, when u = v and ju ̸= kv , (3.7) yields the result already. Second, when u ̸= v and ju , kv ∈ N\|v\| are arbitrary, then u and v diﬀer by at least one element. Suppose that i ∈ (u ∪ v) is that element with the associated degree ji ∈ N. Using the statistical independence of random variables and the fact that E[P{i},ji (Xi ] = 0, as already demonstrated, produces the desired result. Corollary 3.4.For ∅ ̸= u, v ⊆ {1, . . . , N }, ju ∈ N\|u\| , and kv ∈ N\|v\| , the first- and secondorder moments of multivariate orthonormal polynomials are (3.17) E [Ψu,ju (Xu )] = 0 and (3.18) respectively. E [Ψu,ju (Xu )Ψv,kv (Xv )] = { 1, 0, u = v, ju = kv , otherwise, POLYNOMIAL DIMENSIONAL DECOMPOSITION 9 3.5. Orthogonal basis and completeness. An important question regarding multivariate orthogonal polynomials discussed in the preceding subsection is whether they constitute a complete basis in a function space of interest, such as a Hilbert space. Let L2 (AN , B N , fX dx) represent a Hilbert space of square-integrable functions with respect to the probability measure fX (x)dx supported on AN . The following two propositions show that, indeed, measure-consistent orthogonal polynomials span various spaces of interest. Proposition 3.5.Let X := (X1 , . . . , XN )T : (Ω, F) → (AN , B N ) be a vector of N ∈ N input random variables fulfilling Assumption 2.1 and Xu := (Xi1 , . . . , Xi\|u\| )T : (Ωu , F u ) → (Au , B u ), ∅ ̸= u ⊆ {1, . . . , N }, be a subvector of X. Then {Pu,ju (xu ) : \|ju \| = l, ju ∈ N\|u\| }, the set of multivariate orthogonal polynomials of degree l, \|u\| ≤ l < ∞, consistent with the probability measure fXu (xu )dxu , is a basis of Zlu . Proof. Under Assumption 2.1, orthogonal polynomials consistent with the probability (M ) (1) measure fXu (xu )dxu exist. Denote by Pu,l = (Pu,l , . . . , Pu,l u,l )T a column vector of the elements of {Pu,ju (Xu ) : \|ju \| = l, ju ∈ N\|u\| } arranged according to some monomial order. Let (1) (M ) (j) aTu,l = (au,l , . . . , au,l u,l ) be a row vector comprising some constants au,l ∈ R, j = 1, . . . , Mu,l . Set aTu,l Pu,l = 0. Multiply both sides of the equality from the right by PTu,l , integrate with respect to the measure fXu (xu )dxu over Au , and apply transposition to obtain (3.19) Gu,l au,l = 0, where Gu,l = E[Pu,l PTu,l ] is an Mu,l × Mu,l matrix with its (p, q)th element ∫ [ ] (pq) (p) (q) (p) (q) Gu,l = Pu,l (xu )Pu,l (xu )fXu (xu )dxu = E Pu,l (Xu )Pu,l (Xu ) Au representing the covariance between two elements of Pu,l . According to Proposition 3.3, any two distinct polynomials from {Pu,ju (xu ) : \|ju \| = l, ju ∈ N\|u\| } are orthogonal, meaning (p) (q) that E[Pu,l Pu,l ] is zero if p ̸= q and positive and finite if p = q. Consequently, Gu,l is a diagonal, positive-definite matrix and hence invertible. Therefore, (3.19) yields au,l = 0, proving linear independence of the elements of Pu,l or the set {Pu,ju (xu ) : \|ju \| = l, ju ∈ N\|u\| }. Furthermore, the dimension of Zlu , which is Mu,l , matches exactly the number of elements of the aforementioned set. Therefore, the spanning set {Pu,ju (xu ) : \|ju \| = l, ju ∈ N\|u\| } forms a basis of Zlu . Proposition 3.6.Let X := (X1 , . . . , XN )T : (Ω, F) → (AN , B N ) be a vector of N ∈ N input random variables fulfilling both Assumptions 2.1 and 2.2 and Xu := (Xi1 , . . . , Xi\|u\| )T : (Ωu , F u ) → (Au , B u ), ∅ ̸= u ⊆ {1, . . . , N }, be a subvector of X. Consistent with the probability measure fXu (xu )dxu , let {Pu,ju (xu ) : \|ju \| = l, ju ∈ N\|u\| }, the set of multivariate orthogonal polynomials of degree l, \|u\| ≤ l < ∞, be a basis of Zlu . Then the set of polynomials from the orthogonal sum 1⊕ ⊕ ∞ ⊕ ∅̸=u⊆{1,...,N } l=\|u\| span{Pu,ju (xu ) : \|ju \| = l, ju ∈ N\|u\| } is dense in L2 (AN , B N , fX dx). Moreover, (3.20) L2 (AN , B N , fX dx) = 1 ⊕ ⊕ ∞ ⊕ ∅̸=u⊆{1,...,N } l=\|u\| Zlu 10 S. RAHMAN where the overline denotes set closure. Proof. Under Assumption 2.1, orthogonal polynomials exist. According to Theorem 3.4 of Ernst et al. [10], which exploits Assumption 2.2, the polynomial space Π{i} = R[xi ] is dense in L2 (A{i} , B {i} , fXi dxi ). Now use Theorem 4 of Petersen [19], which asserts that if, for p ≥ 1 and all i = 1, . . . , N , Π{i} is dense in Lp (A{i} , B{i} , fXi dxi ), then so is ΠN = R[x1 , . . . , xN ] in Lp (AN , B N , fX dx). Therefore, the set of polynomials from the orthogonal sum, which is equal to ΠN as per (3.13), is dense in L2 (AN , B N , fX dx). Including the limit points of the orthogonal sum yields (3.20). 4. Polynomial Dimensional Decomposition. Let y(X) := y(X1 , . . . , XN ) be a realvalued, square-integrable output random variable defined on the same probability space (Ω, F, P). The vector space L2 (Ω, F, P) is a Hilbert space such that ∫ ∫ [ ] E y 2 (X) := y 2 (X(ω))dP(ω) = y 2 (x)fX (x)dx < ∞ AN Ω with inner product (y(X), z(X))L2 (Ω,F,P) := ∫ y(X(ω))z(X(ω))dP(ω) = √ (y(X), y(X))L2 (Ω,F,P) = Ω ∫ AN y(x)z(x)fX (x)dx =: (y(x), z(x))fX dx and norm ∥y(X)∥L2 (Ω,F,P) := √ (y(x), y(x))fX dx =: ∥y(x)∥fX dx . It is elementary to show that y(X(ω)) ∈ L2 (Ω, F, P) if and only if y(x) ∈ L2 (AN , B N , fX dx). 4.1. ADD. The ADD, expressed by the recursive form [17, 22] ∑ (4.1a) y(X) = y∅ + yu (Xu ), (4.1b) y∅ = (4.1c) yu (Xu ) = ∫ AN ∫ ∅̸=u⊆{1,...,N } y(x)fX (x)dx, AN −\|u\| y(Xu , x−u )fX−u (x−u )dx−u − ∑ yv (Xv ), v⊂u is a finite, hierarchical expansion of y in terms of its input variables with increasing dimensions, where u ⊆ {1, · · · , N } is a subset with the complementary set −u = {1, · · · , N }\u and yu is a \|u\|-variate component function describing a constant or an \|u\|-variate interaction of Xu = (Xi1 , · · · , Xi\|u\| ) on y when \|u\| = 0 or \|u\| > 0. Here, (Xu , x−u ) denotes an N -dimensional vector whose ith component is Xi if i ∈ u and xi if i ∈ / u. The summation in (4.1a) comprises 2N − 1 terms with each term depending on a group of variables indexed by a particular subset of {1, · · · , N }. When u = ∅, the sum in (4.1c) vanishes, resulting in the expression of the constant function y∅ in (4.1b). When u = {1, · · · , N }, the integration in the last line of (4.1c) is on the empty set, reproducing (4.1a) and hence finding the last function y{1,··· ,N } . Indeed, all component functions of y can be obtained by interpreting literally (4.1c). This decomposition, first presented by Hoeﬀding [16] in relation to his seminal work on U -statistics, has been studied by many other researchers described by Efron and Stein [9], the author [22], and references cited therein. POLYNOMIAL DIMENSIONAL DECOMPOSITION 11 The ADD can also be generated by tensorizing a univariate function space decomposition into its constant subspace and remainder, producing [14] ⊕ (4.2) L2 (AN , B N , fX dx) = 1 ⊕ Wu , ∅̸=u⊆{1,...,N } where { } Wu := yu ∈ L2 (Au , B u , fXu dxu ) : E [yu (Xu )yv (Xv )] = 0 if u ̸= v, v ⊆ {1, . . . , N } is an ADD subspace comprising \|u\|-variate component functions of y. However, the subspaces Wu , ∅ ̸= u ⊆ {1, . . . , N }, are in general infinite-dimensional; therefore, further discretization of Wu is necessary. For instance, by introducing measure-consistent orthogonal polynomial basis discussed in Section 3, a component function yu (Xu ) ∈ Wu can be expressed as a linear combination of these basis functions. Indeed, comparing (3.20) and (4.2) yields the closure of an orthogonal decomposition of Wu = (4.3) ∞ ⊕ l=\|u\| Zlu into polynomial spaces Zlu , \|u\| ≤ l < ∞. The result is a polynomial refinement of ADD, which is commonly referred to as PDD. 4.2. PDD. The PDD of a square-integrable random variable y(X) ∈ L2 (Ω, F, P) is simply the expansion of y(X) with respect to a complete, hierarchically ordered, orthonormal polynomial basis of L2 (Ω, F, P). There are at least two ways to explain PDD: a polynomial variant of ADD and a dimension-wise orthogonal polynomial expansion. 4.2.1. Polynomial variant of ADD. The first approach, explained by the author in a prior work [20], involves the following two steps: (1) expand the ANOVA component function ∑ Cu,ju Ψu,ju (Xu ) (4.4) yu (Xu ) ∼ ju ∈N\|u\| in terms of the basis of Wu , which originally stems from the basis of Zlu , \|u\| ≤ l < ∞, with ∫ yu (Xu )Ψu,ju (xu )fXu (xu )dxu , ∅ ̸= u ⊆ {1, . . . , N }, ju ∈ N\|u\| , (4.5) Cu,ju = A\|u\| representing the associated expansion coeﬃcients; and (2) apply (4.1c) to (4.5) and exploit orthogonal properties of the basis. The end result is the PDD [20] of ∑ ∑ (4.6) y(X) ∼ y∅ + Cu,ju Ψu,ju (Xu ), ∅̸=u⊆{1,...,N } ju ∈N\|u\| where, eventually, (4.7) Cu,ju = ∫ AN y(x)Ψu,ju (xu )fX (x)dx. Comparing (4.1) and (4.6), the connection between PDD and ADD is clearly palpable, where the former can be viewed as a polynomial variant of the latter. For instance, Cu,ju Ψu,ju (Xu ) in (4.6) represents a \|u\|-variate, \|ju \|th-order PDD component function of y(X), describing the \|ju \|th-order polynomial approximation of yu (Xu ). In addition, PDD inherits all desirable properties of ADD [20]. 12 S. RAHMAN 4.2.2. Dimension-wise orthogonal polynomial expansion. The second approach entails polynomial expansion associated with the dimension-wise orthogonal splitting of polynomial spaces, as explained in Section 3.3. The latter approach has not been published elsewhere and is, therefore, formally presented here as Theorem 4.1. Theorem 4.1. Let X := (X1 , . . . , XN )T : (Ω, F) → (AN , BN ) be a vector of N ∈ N input random variables fulfilling Assumptions 2.1 and 2.2. For ∅ ̸= u ⊆ {1, . . . , N } and Xu := (Xi1 , . . . , Xi\|u\| )T : (Ωu , F u ) → (Au , B u ), denote by {Ψu,ju (Xu ): ju ∈ N\|u\| } the set of multivariate orthonormal polynomials consistent with the probability measure fXu (xu )dxu . Then (1) any random variable y(X) ∈ L2 (Ω, F, P) can be hierarchically expanded as a Fourier-like series, referred to as the PDD of y(X) ∼ y∅ + (4.8) = y∅ + ∑ ∞ ∑ ∑ Cu,ju Ψu,ju (Xu ) ∅̸=u⊆{1,...,N } l=\|u\| ju ∈N\|u\| \|ju \|=l ∑ ∑ Cu,ju Ψu,ju (Xu ), ∅̸=u⊆{1,...,N } ju ∈N\|u\| where the expansion coeﬃcients y∅ ∈ R and Cu,ju ∈ R, ∅ ̸= u ⊆ {1, . . . , N }, ju ∈ N\|u\| , are defined by ∫ (4.9) y∅ := E [y(X)] := y(x)fX (x)dx, AN Cu,ju := E [y(X)Ψu,ju (Xu )] := (4.10) ∫ AN y(x)Ψu,ju (xu )fX (x)dx; and (2) the PDD of y(X) ∈ L2 (Ω, F, P) converges to y(X) in mean-square; furthermore, the PDD converges in probability and in distribution. Proof. Under Assumptions 2.1 and 2.2, a complete infinite set of multivariate orthogonal polynomials in xu consistent with the probability measure fXu (xu )dxu exists. From Proposition 3.6 and the fact that orthonormality is merely scaling, the set of polynomials from the orthogonal sum (4.11) 1⊕ ⊕ ∞ ⊕ ∅̸=u⊆{1,...,N } l=\|u\| span{Ψu,ju (xu ) : \|ju \| = l, ju ∈ N\|u\| } = ΠN is also dense in L2 (AN , B N , fX dx). Therefore, any square-integrable random variable y(X) can be expanded as shown in (4.8). Combining the two inner sums of the expansion forms the equality in the second line of (4.8). From the denseness, one has the Bessel’s inequality [7] (4.12) [ E y∅ + ∑ ∑ ∅̸=u⊆{1,...,N } ju ∈N\|u\| ]2 Cu,ju Ψu,ju (Xu ) [ ] ≤ E y 2 (X) , POLYNOMIAL DIMENSIONAL DECOMPOSITION 13 proving that the PDD converges in mean-square or L2 . To determine the limit of convergence, invoke again Proposition 3.6, which implies that the set on the left side of (4.11) is complete in L2 (AN , B N , fX dx). Therefore, Bessel’s inequality becomes an equality [ ]2 ∑ ∑ [ ] (4.13) E y∅ + Cu,ju Ψu,ju (Xu ) = E y 2 (X) , ∅̸=u⊆{1,...,N } ju ∈N\|u\| known as the Parseval identity [7] for a multivariate orthonormal system, for every random variable y(X) ∈ L2 (Ω, F, P). Furthermore, as the PDD converges in mean-square, it does so in probability. Moreover, as the expansion converges in probability, it also converges in distribution. Finally, to find the expansion coeﬃcients, define a second moment ]2 [ ∑ ∑ (4.14) ePDD := E y(X) − y∅ − Cv,kv Ψv,kv (Xv ) ∅̸=v⊆{1,...,N } kv ∈N\|v\| of the diﬀerence between y(X) and its full PDD. Diﬀerentiate both sides of (4.14) with respect to y∅ and Cu,ju , ∅ ̸= u ⊆ {1, . . . , N }, ju ∈ N\|u\| , to write ∂ePDD ∂y∅ (4.15) ]2 [ ∑ ∑ ∂ Cv,kv Ψv,kv (Xv ) = E y(X) − y∅ − ∂y∅ ∅̸=v⊆{1,...,N } kv ∈N\|v\| [ }2 ] { ∑ ∑ ∂ = E Cv,kv Ψv,kv (Xv ) y(X) − y∅ − ∂y∅ \|v\| ∅̸ = v⊆{1,...,N } k ∈N v } ] [{ ∑ ∑ Cv,kv Ψv,kv (Xv ) − y(X) × 1 = 2E y∅ + ∅̸=v⊆{1,...,N } kv ∈N\|v\| = 2 {y∅ − E [y(X)]} and ∂ePDD ∂Cu,ju (4.16) ]2 [ ∑ ∑ ∂ Cv,kv Ψv,kv (Xv ) = E y(X) − y∅ − ∂Cu,ju ∅̸=v⊆{1,...,N } kv ∈N\|v\| [ { }2 ] ∑ ∑ ∂ Cv,kv Ψv,kv (Xv ) = E y(X) − y∅ − ∂Cu,ju \|v\| ∅̸ = v⊆{1,...,N } kv ∈N [{ } ] ∑ ∑ = 2E y∅ + Cv,kv Ψv,kv (Xv ) − y(X) Ψu,ju (Xu ) ∅̸=v⊆{1,...,N } kv ∈N\|v\| = 2 {Cu,ju − E [y(X)Ψu,ju (Xu )]} . Here, the second, third, and last lines of both (4.15) and (4.16) are obtained by interchanging the diﬀerential and expectation operators, performing the diﬀerentiation, swapping the expectation and summation operators and applying Corollary 3.4, respectively. The interchanges are permissible as the infinite sum is convergent as demonstrated in the preceding paragraph. Setting ∂ePDD /∂y∅ = 0 in (4.15) and ∂ePDD /∂Cu,ju = 0 in (4.16) yields (4.9) and (4.10), respectively, completing the proof. 14 S. RAHMAN The expressions of the expansion coeﬃcients can also be derived by simply replacing y(X) in (4.9) and (4.10) with the full PDD and then using Corollary 3.4. In contrast, the proof given here demonstrates that the PDD coeﬃcients are determined optimally. It should be emphasized that the function y must be square-integrable for the meansquare and other convergences to hold. However, the rate of convergence depends on the smoothness of the function. The smoother the function, the faster the convergence. If the function is a polynomial, then its PDD exactly reproduces the function. These results can be easily proved using classical approximation theory. A related expansion, known by the name of RS-HDMR [18], also involves orthogonal polynomials in connection with ADD. However, the existence, convergence, and approximation quality of the expansion, including its behavior for infinitely many input variables, have not been reported. 4.3. Truncation. The full PDD contains an infinite number of orthonormal polynomials or coeﬃcients. In practice, the number must be finite, meaning that PDD must be truncated. However, there are multiple ways to perform the truncation. A straightforward approach adopted in this work entails (1) keeping all polynomials in at most 0 ≤ S ≤ N variables, thereby retaining the degrees of interaction among input variables less than or equal to S and (2) preserving polynomial expansion orders (total) less than or equal to S ≤ m < ∞. The result is an S-variate, mth-order PDD approximation2 yS,m (X) = y∅ + (4.17) = y∅ + S m ∑ ∑ s=1 l=s ∑ ∑ ∑ Cu,ju Ψu,ju (Xu ) ∅̸=u⊆{1,...,N } ju ∈N\|u\| \|u\|=s \|ju \|=l ∑ Cu,ju Ψu,ju (Xu ) ∅̸=u⊆{1,...,N } ju ∈N\|u\| 1≤\|u\|≤S \|u\|≤\|ju \|≤m of y(X), containing (4.18) LS,m = 1 + S ( )( ) ∑ N m s=1 s s number of expansion coeﬃcients including y∅ . It is important to clarify a few things about the truncated PDD proposed. First, a diﬀerent truncation with respect to the polynomial expansion order based on ∞-norm as opposed to 1-norm, that is, ∥ju ∥∞ ≤ m, was employed in prior works [20, 21, 24]. Therefore, comparing (4.17) and (4.18) with the existing truncation, if it is desired, should be done with care. Having said this, the proposed truncation has one advantage over the existing one: a direct comparison with a truncated PCE is possible; this will be further explained in the forthcoming sections. Second, the right side of (4.17) contains sums of at most S-dimensional orthonormal polynomials, representing at most S-variate PDD component functions of y. Therefore, the term “S-variate” used for the PDD approximation should be interpreted in the context of including at most S-degree interaction of input variables, even though yS,m is strictly an N -variate function. Third, when S = 0, y0,m = y∅ for any m as the outer sums of (4.17) vanish. Finally, when S → N 2 The nouns degree and order associated with PDD or orthogonal polynomials are used synonymously in the paper. POLYNOMIAL DIMENSIONAL DECOMPOSITION 15 and m → ∞, yS,m converges to y in the mean-square sense, generating a hierarchical and convergent sequence of PDD approximations. Readers interested in an adaptive version of PDD, where the truncation parameters are automatically chosen, are directed to the work of Yadav and Rahman [30], including an application to design optimization [25]. It is natural to ask about the approximation quality of (4.17). Since the set of polynomials from the orthogonal sum in (4.11) is complete in L2 (AN , B N , fX dx), the truncation error y(X) − yS,m (X) is orthogonal to any element of the subspace from which yS,m (X) is chosen, as demonstrated below. Proposition 4.2.For any y(X) ∈ L2 (Ω, F, P), let yS,m (X) be its S-variate, mth-order PDD approximation. Then the truncation error y(X)−yS,m (X) is orthogonal to the subspace ΠN S,m := 1 ⊕ (4.19) ⊕ ⊕ ∅̸=u⊆{1,...,N } ju ∈N\|u\| 1≤\|u\|≤S \|u\|≤\|ju \|≤m span{Ψu,ju (Xu ) : ju ∈ N\|u\| } ⊆ L2 (Ω, F, P), comprising all polynomials in X with the degree of interaction at most S and order at most m, including constants. Moreover, E[y(X) − yS,m (X)]2 → 0 as S → N and m → ∞. Proof. Let ∑ ∑ (4.20) ȳS,m (X) := ȳ∅ + C̄v,kv Ψv,kv (Xv ), ∅̸=v⊆{1,...,N } kv ∈N\|v\| 1≤\|v\|≤S \|v\|≤\|kv \|≤m with arbitrary expansion coeﬃcients ȳ∅ and C̄v,kv , be any element of the subspace ΠN S,m of 2 L (Ω, F, P) described by (4.19). Then (4.21) E[{ [{y(X) − yS,m (X)}ȳS,m (X)] } ∑ ∑ ∑ ∑ = E Cu,ju Ψu,ju (Xu ) + Cu,ju Ψu,ju (Xu ) { ∅̸=u⊆{1,...,N } ju ∈N\|u\| 1≤\|u\|≤S m+1≤\|ju \|<∞ × ȳ∅ + = 0, ∑ ∑ ∅̸=v⊆{1,...,N } kv 1≤\|v\|≤S \|v\|≤\|kv \|≤m ∈N\|v\| ∅̸=u⊆{1,...,N } ju ∈N\|u\| S+1≤\|u\|≤N \|u\|≤\|ju \|<∞ }] C̄v,kv Ψv,kv (Xv ) where the last line follows from Corollary 3.4, proving the first part of the proposition. For the latter part, the Pythagoras theorem yields (4.22) 2 E[{y(X) − yS,m (X)}2 ] + E[yS,m (X)] = E[y 2 (X)]. 2 (X)] → E[y 2 (X)] as S → N and m → ∞. Therefore, E[{y(X) − From Theorem 4.1, E[yS,m 2 yS,m (X)} ] → 0 as S → N and m → ∞. The second part of Proposition 4.2 entails L2 convergence, which is the same as the mean-square convergence described in Theorem 4.1. However, an alternative route is chosen for the proof of the proposition. Besides, Proposition 4.2 implies that the PDD approximation is optimal as it recovers the best approximation from the subspace ΠN S,m , as described by Corollary 4.3. 16 S. RAHMAN Corollary 4.3.Let ΠN S,m in (4.19) define the subspace of all polynomials in X with the degree of interaction at most S and order at most m, including constants. Then the S-variate, mth-order PDD approximation yS,m (X) of y(X) ∈ L2 (Ω, F, P) is the best approximation in the sense that E [y(X) − yS,m (X)]2 = (4.23) inf ȳS,m ∈ΠN S,m E [y(X) − ȳS,m (X)]2 . Proof. Consider two elements yS,m (X) and ȳS,m (X) of the subspace ΠN S,m , where the former is the S-variate, mth-order PDD approximation of y(X) with the expansion coefficients defined by (4.9) and (4.10) and the latter is any S-variate, mth-order polynomial function, described by (4.20), with arbitrary chosen expansion coeﬃcients. From Proposition 4.2, the truncation error y(X) − yS,m (X) is orthogonal to both yS,m (X) and ȳS,m (X) and is, therefore, orthogonal to their linear combinations, yielding E [{y(X) − yS,m (X)}{yS,m (X)] − ȳS,m (X)}] = 0. Consequently, (4.24) E [y(X) − ȳS,m (X)]2 = E [y(X) − yS,m (X)]2 + E [yS,m (X) − ȳS,m (X)]2 ≥ E [y(X) − yS,m (X)]2 , as the second expectation on the right side of the first line of (4.24) is non-negative, thereby proving the mean-square optimality of the S-variate, mth-order PDD approximation. The motivations behind ADD- and PDD-derived approximations are the following. In a practical setting, the function y(X), fortunately, has an eﬀective dimension [3] much lower than N , meaning that the right side of (4.1a) can be eﬀectively approximated by a sum of lower-dimensional component functions yu , \|u\| ≪ N , but still maintaining all random variables X of a high-dimensional uncertainty quantification problem. Furthermore, an Svariate, mth-order PDD approximation is grounded on a fundamental conjecture known to be true in many real-world uncertainty quantification problems: given a high-dimensional function y, its \|u\|-variate, \|ju \|th-order PDD component function Cu,ju Ψu,ju (Xu ), where S + 1 ≤ \|u\| ≤ N and m+1 ≤ \|ju \| < ∞, is small and hence negligible, leading to an accurate lowvariate, low-order approximation of y. The computational complexity of a truncated PDD is polynomial, as opposed to exponential, thereby alleviating the curse of dimensionality to a substantial extent. Although PCE contains the same orthogonal polynomials, a recent work on random eigenvalue analysis of dynamic systems reveals markedly higher convergence rate of the PDD approximation than the PCE approximation [24]. 4.4. Output statistics and other probabilistic characteristics. The S-variate, mthorder PDD approximation yS,m (X) can be viewed as a surrogate of y(X). Therefore, relevant probabilistic characteristics of y(X), including its first two moments and probability density function, if it exists, can be estimated from the statistical properties of yS,m (X). Applying the expectation operator on yS,m (X) and y(X) in (4.17) and (4.8) and imposing Corollary 3.4, their means (4.25) E [yS,m (X)] = E [y(X)] = y∅ POLYNOMIAL DIMENSIONAL DECOMPOSITION 17 are the same and independent of S and m. Therefore, the PDD truncated for any values of 0 ≤ S ≤ N and S ≤ m < ∞ yields the exact mean. Nonetheless, E[yS,m (X)] will be referred to as the S-variate, mth-order PDD approximation of the mean of y(X). Applying the expectation operator again, this time on [yS,m (X) − y∅ ]2 and [y(X) − y∅ ]2 , and employing Corollary 3.4 results in the variances (4.26) ∑ var [yS,m (X)] = ∑ 2 Cu,j u ∅̸=u⊆{1,...,N } ju 1≤\|u\|≤S \|u\|≤\|ju \|≤m ∈N\|u\| and (4.27) var [y(X)] = ∑ ∑ 2 Cu,j u ∅̸=u⊆{1,...,N } ju ∈N\|u\| of yS,m (X) and y(X), respectively. Again, var[yS,m (X)] will be referred to as the S-variate, mth-order PDD approximation of the variance of y(X). Clearly, var[yS,m (X)] approaches var[y(X)], the exact variance of y(X), as S → N and m → ∞. Being convergent in probability and distribution, the probability density function of y(X), if it exists, can also be estimated by that of yS,m (X). However, no analytical formula exists for the density function. In that case, the density can be estimated by sampling methods, such as Monte Carlo simulation (MCS) of yS,m (X). Such simulation should not be confused with crude MCS of y(X), commonly used for producing benchmark results whenever possible. The crude MCS can be expensive or even prohibitive, particularly when the sample size needs to be very large for estimating tail probabilistic characteristics. In contrast, the MCS embedded in the PDD approximation requires evaluations of simple polynomial functions that describe yS,m . Therefore, a relatively large sample size can be accommodated in the PDD approximation even when y is expensive to evaluate. 4.5. Infinitely many input variables. In many fields, such as uncertainty quantification, information theory, and stochastic process, functions depending on a countable sequence {Xi }i∈N of input random variables need to be considered [15]. Under certain assumptions, PDD is still applicable as in the case of finitely many random variables, as demonstrated by the following proposition. Proposition 4.4.Let {Xi }i∈N be a countable sequence of input random variables defined on the probability space (Ω, F∞ , P), where F∞ := σ({Xi }i∈N ) is the associated σ-algebra generated. If the sequence {Xi }i∈N satisfies Assumptions 2.1 and 2.2, then the PDD of y({Xi }i∈N ) ∈ L2 (Ω, F∞ , P), where y : AN → R, converges to y({Xi }i∈N ) in mean-square. Moreover, the PDD converges in probability and in distribution. Proof. According to Proposition 3.6, ΠN is dense in L2 (AN , B N , fX dx) and hence in 2 L (Ω, FN , P) for every N ∈ N, where FN := σ({Xi }N i=1 ) is the associated σ-algebra generated by {Xi }N . Here, with a certain abuse of notation, ΠN is used as a set of polynomial i=1 functions of both real variables x and random variables X. Now, apply Theorem 3.8 of Ernst et al. [10], which says that if ΠN is dense in L2 (Ω, FN , P) for every N ∈ N, then Π∞ := ∞ ∪ N =1 ΠN , 18 S. RAHMAN a subspace of L2 (Ω, F∞ , P), is also dense in L2 (Ω, F∞ , P). But, using (4.11), Π∞ = ∞ ∪ N =1 = 1⊕ 1⊕ ∞ ⊕ ⊕ ∅̸=u⊆{1,...,N } l=\|u\| ∞ ⊕ ⊕ ∅̸=u⊆N l=\|u\| span{Ψu,ju : \|ju \| = l, ju ∈ N\|u\| } span{Ψu,ju : \|ju \| = l, ju ∈ N\|u\| }, demonstrating that the set of polynomials from the orthogonal sum in the last line is dense in L2 (Ω, F∞ , P). Therefore, the PDD of y({Xi }i∈N ) ∈ L2 (Ω, F∞ , P) converges to y({Xi }i∈N ) in mean-square. Since the mean-square convergence is stronger than the convergence in probability or in distribution, the latter modes of convergence follow readily. 5. Polynomial Chaos Expansion. In contrast to the dimension-wise splitting of polynomial spaces in PDD, a degree-wise orthogonal splitting of polynomial spaces results in PCE. The latter decomposition is briefly summarized here as PCE will be compared with PDD in the next section. 5.1. Degree-wise orthogonal decomposition of polynomial spaces. Let j := j{1,...,N } = (j1 , . . . , , jN ) ∈ NN 0 , ji ∈ N0 , i ∈ {1, . . . , N }, define an N -dimensional multi-index. For x = (x1 , . . . , xN ) ∈ AN ⊆ RN , a monomial in the variables x1 , . . . , xN is the product xj = xj11 · · · xjNN and has a total degree \|j\| = j1 + · · · + jN . Denote by j N ΠN p := span{x : 0 ≤ \|j\| ≤ p, j ∈ N0 }, p ∈ N0 , the space of real polynomials in x of degree at most p. Let V0N := ΠN 0 = span{1} be the space of constant functions. For each 1 ≤ l < ∞, denote by VlN ⊂ ΠN l the space of orthogonal polynomials of degree exactly l that are orthogonal to all polynomials in ΠN l−1 , that is, N VlN := {P ∈ ΠN l : (P, Q)fX dx = 0 ∀ Q ∈ Πl−1 }, 1 ≤ l < ∞. N From Section 3, with u = {1, . . . , N } in mind, select {Pj (x) : \|j\| = l, j ∈ NN 0 } ⊂ Vl to be N a basis of Vl . Each basis function Pj (x) is a multivariate orthogonal polynomial in x of degree \|j\|. Obviously, VlN = span{Pj (x) : \|j\| = l, j ∈ NN 0 }, 0 ≤ l < ∞. According to (3.7) with u = {1, . . . , N }, Pj (x) is orthogonal to Pk (x) whenever j ̸= k. Therefore, any two polynomial subspaces VlN and VrN , where 0 ≤ l, r < ∞, are orthogonal whenever l ̸= r. In consequence, there exists another orthogonal decomposition of ⊕ ⊕ N (5.1) ΠN = VlN = span{Pj (x) : \|j\| = l, j ∈ NN 0 } = span{Pj (x) : j ∈ N0 }. l∈N0 l∈N0 Compared with (3.13), (5.1) represents a degree-wise orthogonal decomposition of ΠN . 5.2. PCE. Given the degree-wise orthogonal decomposition of ΠN , the PCE of any square-integrable output random variable y(X) is expressed by [5, 10, 28, 29] (5.2) y(X) ∼ ∞ ∑ ∑ l=0 j∈NN 0 \|j\|=l Cj Ψj (X) = ∑ j∈NN 0 Cj Ψj (X), POLYNOMIAL DIMENSIONAL DECOMPOSITION 19 where {Ψj (X) : j ∈ NN 0 } is an infinite set of measure-consistent multivariate orthonormal polynomials in X that can be obtained by scaling Pj in (3.14) and Cj ∈ R, j ∈ NN 0 , are the 2 PCE expansion coeﬃcients. Like PDD, the PCE of y(X) ∈ L (Ω, F, P) under Assumptions 2.1 and 2.2 also converges to y(X) in mean-square, in probability, and in distribution. Since the PCE of y(X) in (5.2) is an infinite series, it must also be truncated in applications. A commonly adopted truncation is based on retaining orders of polynomials less than or equal to a specified total degree. In this regard, given 0 ≤ p < ∞, the pth-order PCE approximation of y(X) ∈ L2 (Ω, F, P) reads (5.3) p ∑ ∑ yp (X) = l=0 ∑ Cj Ψj (X) = j∈NN 0 \|j\|=l Cj Ψj (X). j∈NN 0 0≤\|j\|≤p This kind of truncation is related to the total degree index set { } N ∑ j ∈ NN ji ≤ p 0 : i=1 for defining the recovered multivariate polynomial space of a pth-order PCE approximation. Other kinds of truncation entail { } { } N ∏ N N j ∈ N0 : max ji ≤ p and j ∈ N0 : (ji + 1) ≤ p + 1 , i=1,...,N i=1 describing the tensor product and hyperbolic cross index sets, respectively, to name just two. The total degree and tensor product index sets are common choices, although the latter one suﬀers from the curse of dimensionality, making it impractical for high-dimensional problems. The hyperbolic cross index set, originally introduced for approximating periodic functions by trigonometric polynomials [2], is relatively a new idea and has yet to receive widespread attention. All of these choices and possibly others, including their anisotropic versions, can be used for truncating PCE. In this work, however, only the total degree index set is used for the PCE approximation. This is consistent with the 1-norm of ju used for truncating PDD in (4.17). 6. Error Analysis. 6.1. PDD error. Define a second-moment error, eS,m := E [y(X) − yS,m (X)]2 , (6.1) stemming from the S-variate, mth-order PDD approximation presented in the preceding section. Replacing y(X) and yS,m (X) in (6.1) with the right sides of (4.8) and (4.17), respectively, produces (6.2) eS,m = S ∑ ∞ ∑ ∑ ∑ s=1 l=m+1 ∅̸=u⊆{1,...,N } ju \|u\|=s \|ju \|=l 2 Cu,j u ∈N\|u\| + N ∑ s=S+1 ∞ ∑ l=s ∑ ∑ 2 Cu,j , u ∅̸=u⊆{1,...,N } ju \|u\|=s \|ju \|=l ∈N\|u\| where the second term vanishes expectedly when S = N as the lower limit of the outer sum exceeds the upper limit. In (6.2), the first term of the PDD error is due to the truncation 20 S. RAHMAN of polynomial expansion orders involving interactive eﬀects of at most S variables, whereas the second term of the PDD error is contributed by ignoring the interactive eﬀects of larger than S variables. Obviously, the error for a general function y depends on which expansion coeﬃcients decay and how they decay with respect to S and m. Nonetheless, the error decays monotonically with respect to S and/or m as stated in Proposition 6.1. Other than that, nothing more can be said about the PDD error. Proposition 6.1.For a general function y, eS+i,m+j ≤ eS,m , where 0 ≤ S < N , S ≤ m < ∞, and i and j are equal to either 0 or 1, but not both equal to 0. Proof. Setting i = 1, j = 0 and using (6.2), eS+1,m − eS,m = ∞ ∑ ∑ ∑ 2 Cu,j − u l=m+1 ∅̸=u⊆{1,...,N } ju ∈N\|u\| \|u\|=S+1 \|ju \|=l ∞ ∑ ∑ ∑ 2 Cu,j ≤ 0, u l=S+1 ∅̸=u⊆{1,...,N } ju ∈N\|u\| \|u\|=S+1 \|ju \|=l where the inequality to zero in the last line results from the fact that, as S ≤ m, the first term is smaller than or equal to the second term. Similarly, setting i = 0, j = 1 and using (6.2), S ∑ ∑ ∑ 2 eS,m+1 − eS,m = − Cu,j ≤ 0. u s=1 ∅̸=u⊆{1,...,N } ju ∈N\|u\| \|u\|=s \|ju \|=m+1 Finally, setting i = 1, j = 1, eS+1,m+1 − eS,m = eS+1,m+1 − eS,m+1 + eS,m+1 − eS,m ≤ 0, as eS+1,m+1 − eS,m+1 ≤ 0 and eS,m+1 − eS,m ≤ 0. Corollary 6.2.For a general function y, eS ′ ,m′ ≤ eS,m whenever S ′ ≥ S and m′ ≥ m. In practice, the eﬀects of interaction among input variables and polynomial expansion 2 , which is equal to order become increasingly weaker as \|u\| and \|ju \| grow. In this case, Cu,j u 2 the variance of Cu,ju Ψu,ju (Xu ), decreases with \|u\| and \|ju \|. Given the rates at which Cu,j u decreases with \|u\| and \|ju \|, a question arises on how fast does eS,m decay with respect to S and m. Proposition 6.3, Corollary 6.4, and subsequent discussions provide a few insights. 2 , ∅ ̸= u ⊆ {1, · · · , N }, Proposition 6.3.For a class of functions y, assume that Cu,j u −\|u\| −\|j \| 2 ju ∈ N\|u\| , attenuates according to Cu,j ≤ Cp1 p2 u , where C > 0, p1 > 1, and p2 > 1 u are three real-valued constants. Then it holds that [{ ] } 1 + p1 (p2 − 1) N (6.3) var [y(X)] ≤ C −1 p1 (p2 − 1) and (6.4) eS,m ≤ C [ S ∑ s=1 ∞ ∑ l=m+1 ( )( ) N ∑ N l − 1 −s −l p1 p2 + s s−1 s=S+1 ] ) ∞ ( )( ∑ N l − 1 −s −l p p . s s−1 1 2 l=s Proof. With the recognition that ( ) ( ) N l−1 \|u\| #{∅ ̸= u ⊆ {1, · · · , N } : \|u\| = s} = , #{ju ∈ N : \|ju \| = l} = , s \|u\| − 1 POLYNOMIAL DIMENSIONAL DECOMPOSITION 21 −\|u\| −\|j \| 2 use Cu,j ≤ Cp1 p2 u in (4.27) and (6.2) to obtain (6.3) and (6.4). u Corollary 6.4.For the function class described in Proposition 6.3, eS+i,m+j < eS,m , where 0 ≤ S < N , S ≤ m < ∞, and i and j are equal to either 0 or 1, but not both equal to 0. According to Corollary 6.4, eS,m decays strictly monotonically with respect to S and/or m for any rate parameters p1 > 1 and p2 > 1. When the equality holds in (6.3) and (6.4) from Proposition 6.3, Figure 1, comprising three subfigures, presents three sets of plots of the relative error, eS,m /var[y(X)], against m for five distinct values of S = 1, 2, 3, 5, 9. These subfigures, each obtained for N = 20, correspond to three distinct cases of the values of p1 and p2 : (1) p1 = 500, p2 = 5; (2) p1 = 5, p2 = 500; and (3) p1 = 500, p2 = 500. In all cases, the error for a given S decays first with respect to m, and then levels oﬀ at a respective limit when m is suﬃciently large. The limits get progressively smaller when S increases as expected. However, the magnitude of this behavior depends on the rates at which the expansion coeﬃcient attenuates with respect to the degree of interaction and polynomial expansion order. When p1 > p2 , as in case 1 [Figure 1 (top)], the error for a given S decays slowly with respect to m due to a relatively weaker attenuation rate associated with the polynomial expansion order. The trend reverses when the attenuation rate becomes stronger and reaches the condition p1 < p2 , as in case 2 [Figure 1 (middle)]. For larger values of S, for example, S = 5 or 9, the respective limits are significantly lower in case 2 than in case 1. When the attenuation rates are the same and large, as in case 3 [Figure 1 (bottom)], the decay rate of error accelerates substantially. 6.2. Relationship between PDD and PCE. Since PDD and PCE share the same orthonormal polynomials, they are related. Indeed, the relationship was first studied by Rahman and Yadav [24], who determined that any one of the two infinite series from PDD and PCE defined by (4.8) and (5.2) can be rearranged to derive the other. In other words, the PDD can also be viewed as a reshuﬄed PCE and vice versa. However, due to a strong connection to ADD endowed with a desired hierarchical structure, PDD merits its own appellation. More importantly, the PDD and PCE when truncated are not the same. In fact, two important observations stand out prominently. First, the terms in the PCE approximation are organized with respect to the order of polynomials. In contrast, the PDD approximation is structured with respect to the degree of interaction between a finite number of random variables. Therefore, significant diﬀerences may exist regarding the accuracy, eﬃciency, and convergence properties of their truncated sum. Second, if a stochastic response is highly nonlinear, but contains rapidly diminishing interactive eﬀects of multiple random variables, the PDD approximation is expected to be more eﬀective than the PCE approximation. This is because the lower-variate terms of the PDD approximation can be just as nonlinear by selecting appropriate values of m in (4.17). In contrast, many more terms and expansion coeﬃcients are required to be included in the PCE approximation to capture such high nonlinearity. In this work, a theoretical comparison between PDD and PCE in the context of error analysis, not studied in prior works, is presented. For error analysis, it is convenient to write a PCE approximation in terms of a PDD approximation. Indeed, there exists a striking result connecting PCE with PDD approximations, as explained in Proposition 6.3. Proposition 6.5.Let yp (X) and yS,m (X) be the pth-order PCE approximation and Svariate, mth-order PDD approximation of y(X) ∈ L2 (Ω, F, P), respectively, where 0 ≤ S ≤ N , S ≤ m < ∞, and 0 ≤ p < ∞. Then the pth-order PCE approximation and the 22 S. RAHMAN Figure 1. PDD errors for various attenuation rates of the expansion coeﬃcients; (top) p1 = 500, p2 = 5; (middle) p1 = 5, p2 = 500; (bottom) p1 = 500, p2 = 500. POLYNOMIAL DIMENSIONAL DECOMPOSITION 23 (p ∧ N )-variate, pth-order PDD approximation are the same, that is, (6.5) yp (X) = yp∧N,p (X), where y0,0 (X) = y∅ and p ∧ N denotes the minimum of p and N . Proof. According to Rahman and Yadav [24], the right side of (5.3) can be reshuﬄed, resulting in a long form of the PCE approximation, expressed by [ N −s+1 ] p−s+1 p−s+1 N N s ∑ ∑ ∑ ∑ ∑ ∏ (6.6) yp (X) = y∅ + ··· ··· C{i1 ···is },(ji1 ···jis ) ψiq jiq (Xiq ) , s=1 i1 =1 \| is =is−1 +1 {z s sums } ji1 =1 \| {z jis =1 } q=1 s sums;ji1 +···+jis ≤p in terms of the PDD expansion coeﬃcients. Note that, depending on the condition p ≤ N or p ≥ N , at most p-dimensional or N -dimensional sums survive in (6.6), meaning that the pth-order PCE approximation retains eﬀects of at most (p ∧ N )-degree interaction and at most pth-order polynomial expansion order. Accordingly, the compact form of the PCE approximation can be written as (6.7) yp (X) = y∅ + p∧N ∑ s=1 p ∑ l=s ∑ ∑ Cu,ju Ψu,ju (Xu ) = yp∧N,p (X), ∅̸=u⊆{1,...,N } ju \|u\|=s \|ju \|=l ∈N\|u\| completing the proof. Using Proposition 6.5, the number of expansion coeﬃcients, say, Lp associated with the pth-order PCE approximation can be calculated from that required by the (p ∧ N )-variate, pth-order PDD approximation. Accordingly, setting S = p ∧ N and m = p in (4.18), (6.8) Lp = Lp∧N,p = 1 + p∧N ∑ s=1 ( )( ) N p (N + p)! = N !p! s s with the last expression commonly found in the PCE literature [29]. The advantage of (6.7) over (5.3) is obvious: the PDD coeﬃcients, once determined, can be reused for the PCE approximation and subsequent error analysis, thereby sidestepping calculations of the PCE coeﬃcients. 6.3. PDD vs. PCE errors. Define another second-moment error, (6.9) ep := E [y(X) − yp (X)]2 , resulting from the pth-order PCE approximation yp (X) of y(X). Using Proposition 6.5, ep = ep∧N,p , meaning that the PCE error analysis can be conducted using the PDD approximation. Proposition 6.6.For a general function y, let eS,m and ep denote the PDD and PCE errors defined by (6.1) and (6.9), respectively. Given a truncation parameter 0 ≤ p < ∞ of the PCE approximation, if the truncation parameters of the PDD approximation are chosen such that p ∧ N ≤ S ≤ N and p ∨ S ≤ m < ∞, then (6.10) eS,m ≤ ep , 24 S. RAHMAN where p ∨ S denotes the maximum of p and S. Proof. The result follows from Propositions 6.1 and 6.5, and Corollary 6.2. Proposition 6.6 aids in selecting appropriate truncation parameters to contrast the second-moment errors due to PDD and PCE approximations. However, the proposition does not say anything about the computational eﬀort. Proposition 6.7 and subsequent discussion explain the relationship between computational eﬀort and error committed by both PDD and PCE approximations for a special class of functions. 2 , ∅ ̸= u ⊆ {1, · · · , N }, Proposition 6.7.For a special class of functions y, assume that Cu,j u −\|u\| −\|j \| 2 ju ∈ N\|u\| , diminishes according to Cu,j ≤ Cp1 p2 u , where C > 0, p1 > 1, and p2 > 1 u are three real-valued constants. Then it holds that   ( )( ) ) p∧N ∞ N ∞ ( )( ∑ ∑ ∑ ∑ N l − 1 −s −l N l − 1 −s −l  (6.11) ep ≤ C  p p + p p . s s−1 1 2 s s−1 1 2 s=1 s=p∧N +1 l=p+1 l=s Proof. Replacing S and m in (6.4) with p ∧ N and p, respectively, obtains the result. Theoretically, the numbers of expansion coeﬃcients required by the PDD and PCE approximations can be used to compare their respective computational eﬀorts. Table 1 presents for N = 20 the requisite numbers of expansion coeﬃcients when PDD is truncated at S = 1, 2, 3, 5, 9 and m = 1 − 20, and when PCE is truncated at p = 1 − 20. They are calculated using (4.18) and (6.8) for PDD and PCE approximations, respectively. According to Table 1, the growth of the number of expansion coeﬃcients in PCE is steeper than that in PDD. The growth rate increases markedly when the polynomial expansion order is large. This is primarily because a PCE approximation is solely dictated by a single truncation parameter p, which controls the largest polynomial expansion order preserved, but not the degree of interaction independently. In contrast, two diﬀerent truncation parameters S and m are involved in a PDD approximation, aﬀording a greater flexibility in retaining the largest degree of interaction and largest polynomial expansion order. In consequence, the numbers of expansion coeﬃcients and hence the computational eﬀorts by the PDD and PCE approximations can vary appreciably. Table 1 Growth of expansion coeﬃcients in the PDD and PCE approximations LS,m m or p S=1 1 21 2 41 S=2 S=3 S=5 S=9 Lp 21 231 231 3 61 631 1771 5 101 2001 13,401 1771 53,130 53,130 9 181 7021 102,781 2,666,755 10,015,005 10,015,005 12 241 12,781 263,581 14,941,024 211,457,454 225,792,840 15 301 20,251 538,951 53,710,888 2,397,802,638 3,247,943,160 20 401 36,501 1,336,101 265,184,142 51,855,874,642 137,846,528,820 Using the equalities in (6.3), (6.4), and (6.11), Figure 2 depicts how the relative PDD error, eS,m /var[y(X)], and the relative PCE error ep /var[y(X)], vary with respect to the POLYNOMIAL DIMENSIONAL DECOMPOSITION 25 Figure 2. PDD vs. PCE errors for various attenuation rates of the expansion coeﬃcients; (top) p1 = 500, p2 = 5; (middle) p1 = 5, p2 = 500; (bottom) p1 = 500, p2 = 500. 26 S. RAHMAN number of expansion coeﬃcients required for N = 20. Again, the three preceding cases of the attenuation rates p1 and p2 with respect to the degree of interaction and polynomial expansion order are studied. In all cases, the PDD or PCE errors decay with respect to S, m, and p as expected. However, in the PDD approximation, the error for a fixed S may decline even further by increasing m, whereas no such possibility exists in the PCE approximation. This behavior is pronounced in case 1, that is, when p1 > p2 [Figure 2 (top)]. For example, in case 1, the bivariate, sixth-order PDD approximation (S = 2, m = 6) achieves a relative error of 8.54 × 10−5 employing only 2971 expansion coeﬃcients. In contrast, to match the same-order error, the sixth-order PCE approximation (p = 6) is needed, committing a relative error of 7.15 × 10−5 at the cost of 230,230 expansion coeﬃcients. Therefore, the PDD approximation is substantially more economical than the PCE approximation for a similar accuracy. However, when p1 > p2 , as in case 2 [Figure 2 (middle)], the computational advantage of PDD over PCE approximations disappears as the attenuation rate associated with the polynomial expansion order is dominant over that associated with the degree of interaction. Nonetheless, in case 2, the S-variate, mth-order PDD approximation with the lowest m possible cannot commit more error than the mthorder PCE approximation for the same computational eﬀort. Finally, when the attenuation rates are the same, as in case 3 [Figure 2 (bottom)], the PDD approximation is still more computationally eﬃcient than the PCE approximation. For instance, the trivariate, fifthorder PDD (S = 3, m = 5) and fifth-order PCE (p = 6) approximations require 13,401 and 53,130 expansion coeﬃcients to commit the same-order errors of 5.07 × 10−14 and 3.51×10−14 , respectively. But, unlike in case 1, an unnecessarily large polynomial expansion order may render the PDD approximation more expensive than required. Readers should take note that the comparative error analyses reported here are limited to PDD and PCE approximations derived from truncations according to the total degree index set. For other index sets, such as the tensor product and hyperbolic cross index sets, it would be intriguing to find whether a similar conclusion arises. 7. Conclusion. The fundamental mathematical properties of PDD, representing Fourierlike series expansion in terms of random orthogonal polynomials with increasing dimensions, were studied. A dimension-wise splitting of appropriate polynomial spaces into orthogonal subspaces, each spanned by measure-consistent orthogonal polynomials, was constructed, resulting in a polynomial refinement of ADD and eventually PDD. Under prescribed assumptions, the set of measure-consistent orthogonal polynomials was proved to form a complete basis of each subspace, leading to an orthogonal sum of such sets of basis functions, including the constant subspace, to span the space of all polynomials. In addition, the orthogonal sum is dense in a Hilbert space of square-integrable functions, leading to mean-square convergence of PDD to the correct limit, including for the case of infinitely many random variables. The optimality of PDD and the approximation quality due to truncation were demonstrated or discussed. From the second-moment error analysis of a general function of 1 ≤ N < ∞ random variables, given 0 ≤ p < ∞, the (p ∧ N )-variate, pth-order PDD approximation and pth-order PCE approximation are the same. Therefore, an S-variate, mth-order PDD approximation cannot commit a larger error than a pth-order PCE approximation if p ∧ N ≤ S ≤ N and p ∨ S ≤ m < ∞. 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arXiv:1308.6074v2 [q-bio.GN] 3 Apr 2014 Exploration and retrieval of whole-metagenome sequencing samples Sohan Seth 1 , Niko Välimäki 2,3 , Samuel Kaski 1,3 , Antti Honkela 3 1 Helsinki Institute for Information Technology HIIT, Department of Information and Computer Science, Aalto University, Espoo, Finland 2 Genome-Scale Biology Program and Department of Medical Genetics, University of Helsinki, Helsinki, Finland 3 Helsinki Institute for Information Technology HIIT, Department of Computer Science, University of Helsinki, Helsinki, Finland April 4, 2014 Abstract Over the recent years, the field of whole metagenome shotgun sequencing has witnessed significant growth due to the high-throughput sequencing technologies that allow sequencing genomic samples cheaper, faster, and with better coverage than before. This technical advancement has initiated the trend of sequencing multiple samples in different conditions or environments to explore the similarities and dissimilarities of the microbial communities. Examples include the human microbiome project and various studies of the human intestinal tract. With the availability of ever larger databases of such measurements, finding samples similar to a given query sample is becoming a central operation. In this paper, we develop a content-based exploration and retrieval method for whole metagenome sequencing samples. We apply a distributed string mining framework to efficiently extract all informative sequence k-mers from a pool of metagenomic samples and use them to measure the dissimilarity between two samples. We evaluate the performance of the proposed approach on two human gut metagenome data sets as well as human microbiome project metagenomic samples. We observe significant enrichment for diseased gut samples in results of queries with another diseased sample and very high accuracy in discriminating between different body sites even though the method is unsupervised. A software implementation of the DSM framework is available at https://github.com/HIITMetagenomics/dsm-framework. 1 Introduction Metagenomics is the study of microbial communities in their natural habitat using genomics techniques [28]. It is undergoing a boom due to proliferation of high-throughput sequencing technologies. Many studies focus at targeted sequencing of specific marker genes such as the 16S rRNA gene in bacteria, but recently there has been a growing interest in whole metagenome sequencing (see, e.g. [18, 27]). While targeted studies provide data for phylogenetic profiling at a lower cost, whole metagenomes provide much more information, for example, about the collective metabolism [5], and the population genetics of the community [22]. Recent studies have also found associations between features of whole human gut metagenomes and type II diabetes [19]. New data are accumulating rapidly, with a popular web-based MG-RAST server [14] listing almost 3000 public whole metagenomes. Analysing whole-metagenome shotgun (WMS) sequencing data is very challenging. The original sample typically contains genetic material from hundreds to thousands of bacterial species of different abundances [9], most of which have not been fully sequenced previously. After sequencing, we obtain a huge 1 Bag of metagenomic samples s1 ACTAGTCA TAGCATAG CCATGACA CTTAATGA s5 ATCGCAGA AGGTTAAT GTGTACCG TCAACGGG s3 ACTGACTG ATTCCTTA CTATGCAC GTTGCTTC s2 ATGACATA GATCATGA CACATGCA CATGACTG s6 Feature extraction s4 GATGGATT GTCAGTAC GTACTGAC ACTGCATG Dissimilarity evaluation Dissimilarity values Query: S3 Retrieve: S6,S1,... Query: S6 Retrieve: S5,S3,... ACTGGTCA CTTAAGGC GTGTACCA AGGACAAC Figure 1: Given a set of metagenomic samples our objective is to be able to retrieve relevant samples to a query sample. For this, we need to extract relevant features and evaluate a pairwise similarity (or dissimilarity) measure. The samples are then ranked in the order of increasing dissimilarity from the query. collection of short sequence reads whose species of origin is unknown. While significant progress has been made, analysis relying on either the limited previously annotated genomes, or assembling the reads into novel more complete genomes, remains difficult and inefficient, and potentially susceptible to annotation biases. In this paper we introduce an efficient purely data-driven feature extraction and selection method as well as similarity measures for WMS sequencing data sets, and apply them in retrieval of similar data sets. Such content-based retrieval is an extremely powerful tool for exploration of the data and generating hypotheses of disease associations, as previously demonstrated with gene expression data [2, 3]. Retrieval from existing databases makes it possible to automatically explore a much greater variety of hypotheses than relying solely on the more common specifically designed focused studies. Content-based similarity measures and retrieval of similar metagenomic data sets have been suggested previously [16, 10, 26, 6], based on quantifying abundances over a relatively small number of predetermined features requiring existing annotation. Up to some thousands of known taxa, genes or metabolic pathways have been used. We introduce similarity measures that are based solely on raw sequencing reads, and hence, unbiased and insensitive to the quality of the existing annotation. A similar measure has been previously suggested by [11], but only for pairwise comparisons using a method that is computationally too expensive to scale to even modestly large data sets. Furthermore, instead of considering all sequences of particular length, also known as k-mers, as has been done earlier for other tasks and by [11], we employ an efficient distributed string mining algorithm to find informative subsequences that can be of any length. In order to deal with the very large number of features some feature selection is necessary. Previous approaches for detecting relevant features in metagenomic data have been based on direct comparison of two classes of samples. Again, most of these methods work on up to some thousands of features [30, 17, 23], with the notable exception of one study [19] where quantification and association testing was done for over 4.3 million predefined genes. Without feature selection one can use short k-mers [1] or limit to a set of k-mers that are likely to be informative, such as k-mers associated with well characterised protein families [4]. While there are no previous examples of unsupervised feature selection for metagenomics, it is a common practice in information retrieval with text documents [31]; a particularly relevant method assesses the entropy of the distribution of documents in which a specific term occurs [8]. We evaluate the performance of the proposed unsupervised, unconstrained retrieval method on synthetic data, as well as metagenomic samples from human body sites [18, 19, 27]. To evaluate the performance of the retrieval engine, we use external validation based on a ground truth similarity between two samples. To simplify this process, we consider a binary similarity, which is crude but easily accessible. The human gut samples in [18, 19] come from studies exploring the change in bacterial species composition between healthy persons and either inflammatory bowel disease or type II diabetes. We utilize disease state to construct a 2 1. Normalization Collection of metagenomic samples 2. Regularization 4. Distributed string mining Dissimilarity matrix 5. Dissimilarity computation 3. Entropy ﬁltering 6. Evaluation Annotations Figure 2: Processing steps of our method. Given a collection of metagenomic samples, we use the collection as an input to the distributed string mining method (4). For the method, we estimate the frequency of each k-mer (1,2), evaluate if the k-mer is informative or not (3), and compute the needed dissimilarities (5). Finally, in this paper we evaluate the performance considering the existing annotations as ground truth; annotations are not needed for the retrieval in general. binary ground truth. Thus, we study if, given the metagenomic sample of a person with a disease, the retrieval finds metagenomic samples related by having the same disease. In the body site data [27] we use the body sites as ground truth to investigate whether it is possible to identify the bacterial communities at different body sites in an unsupervised setting without the need of reference genomes. It should be noted that especially for the gut data, two samples may be related in other ways too. The external validation with one simple ground truth nonetheless provides an objective platform for comparing different methods. Given that the method is unsupervised and hence completely oblivious of the disease labels, if such retrieval is successful it is a promising starting point for developing methods for leveraging data from earlier patients in early detection of disease and personalized medicine. 2 Approach Our objective is to extract and select suitable features for representing WMS sequencing samples and to form a pairwise dissimilarity measure for a collection of such samples. Given this dissimilarity one can query with a sample and retrieve other samples that are similar to it (Fig. 1). The measure needs to be reasonably rapidly computable, yet captures relevant differences between the samples, and does all this with as little prior biological knowledge and annotations as possible, since detailed quantitative prior knowledge is typically not yet available for metagenomics. Evaluating dissimilarity requires representing the metagenomic sample in a suitable feature space. A standard choice for representing objects over strings is to estimate the k-mer frequency values, where a kmer here is a string of k letters from the DNA alphabet {A,C,T,G}. Therefore, there are 4k possible k-mers for any given k. It is standard practice to set k to a specific value, typically a small value to keep the estimation problem tractable both computationally and statistically. A larger k would give better discriminability but not without bounds, as for finite data set sizes there simply is not enough data to estimate long k-mers. We argue that instead of setting k to a particular value, it is more effective to estimate all possible k-mers for all possible k which the data supports. This makes the problem more challenging, since the number of such observed different k-mers for large k becomes very large and they become more susceptible to sequencing errors. Focusing on k-mers appearing more than once in a sample helps significantly because it is relatively rare to have the exactly same sequencing errors in two independent reads. To make the method computationally efficient we treat each k-mer as an independent feature. We compute a Bayesian estimate of their relative frequencies across samples. The employed prior helps in suppressing noise caused by small observed read counts. In the filtering step the abundance distribution of each 3 Figure 3: Technical overview of our distributed string mining framework consisting of client (left) and server (right) processes. The client-side processes are responsible for computing the substring frequencies within each sample s1 , s2 , . . . sd separately. Substrings and their frequencies are found using a depth-first-traversal over a (compressed) suffix tree. Frequency information is transmitted over to the server-side by streaming it as a balanced-parenthesis representation of a sorted trie. For example, the trie on the left results as the parenthesis representation given in the middle. The server reads the client-streams and merges the (already sorted) tries in recursive manner: at each node, the server computes the entropy based on the received values and updates the affected pairwise distances. Load-balancing on the server-side is achieved by hashing the prefix of the substring so that each server corresponds to a certain range of hash values. k-mer over samples is used to judge informativeness of the k-mer for retrieval; a k-mer with constant abundance does not have discriminative power and, in the other extreme, a k-mer which is present in only one sample cannot generalize over samples. We show that the filtering step significantly improves the retrieval performance with most datasets and distance measures. Finally, we compute the dissimilarity between two samples across the features as a weighted average of distances between relative frequencies of individual k-mers. Treating each k-mer as an independent feature allows us to execute these steps fast and on the fly without storing the intermediate results. Such simplified distance measures are necessary to guarantee scalability given the extremely high dimensionality of the k-mer features. To summarize, we introduce methods to i. estimate the frequencies of a large number of k-mers over multiple samples, ii. decide if a k-mer is informative or uninformative in the context of a retrieval task, iii. compute a distance metric using the filtered k-mer frequencies, and iv. execute these steps fast without explicitly storing the frequency values. Fig. 2 summarizes the method. 3 Methods 3.1 Estimating k-mer frequencies: normalization, regularization, filtering In order to perform the feature selection or filtering, we first compute Bayesian estimates of the relative frequencies p(s\|w) of each k-mer w over samples s ∈ S using observed frequencies fˆ(s, w) of the k-mers. These are distributions over samples for each k-mer that are computed independently for each k-mer for reasons of computational efficiency. Even if the relative abundance of a k-mer is the same in every sample, the observed frequencies may differ because of different sequencing depth or coverage in different samples. To tackle this issue we employ normalization: we normalize the frequency fˆ(s, w) by a sample-specific constant σ(s), which is proportional to the total number of base pairs in a sample, and σ(s) = 1 for the largest sample in the collection in terms 4 of total base pair count, obtaining f (s, w) = fˆ(s, w)/σ(s). (1) The σ(s) can be interpreted probabilistically as the probability of observing a sequence in the actual sample, assuming every sample had the same number of base pairs to start with but some have been lost in the processing. In order to estimate the relative frequencies, we place a conjugate symmetric Dirichlet prior on the parameters of the multinomial distribution over the observed counts. The common choice of uniform prior distribution corresponds to a Dirichlet distribution with all parameters equal to 1. This yields a posterior mean estimate of the relative frequency values as p(s\|w) = P f (s, w) + 1 . [f (s′ , w) + 1] (2) s′ ∈S The Dirichlet prior with all parameters equal to one is ubiquitous in document retrieval. It is particularly suitable for metagenomics due to the following observations: 1. The distributed string mining algorithm (described below) trades off low k-mer counts for speed and ignores any k-mers that are present only once in a sample. The pseudo-count from the prior makes up for this missing count. 2. Adding pseudo-counts assists in playing down the significance of very rare k-mers that may appear due to sequencing errors in the filtering step without affecting other k-mers too much. Finally, given the massive number of potential k-mers, it is crucially important to improve signal-to-noise ratio by focusing on the informative ones. For the unsupervised tasks of comparing the samples, obviously only k-mers which distinguish between the samples are informative. As a concrete example, consider a kmer that is present in all samples with a similar abundance. It certainly does not give information useful for comparing samples. In the other extreme, if a k-mer is present in one specific sample, but not in any other, it is potentially a spurious k-mer due to sequencing error, and in any case does not help in comparing samples either. On the other hand, if a k-mer is present in some samples, but not all, then it gives information that those samples are similar in a specific sense. Informativeness in this sense can be measured by the entropy H of the distribution of the k-mer over the samples: we filter the k-mers based on the conditional entropies H(S\|w) = − X 1 p(s\|w) log p(s\|w); log(\|S\|) (3) s∈S a k-mer is taken into account in distance computation only if the normalized entropy is lower than a certain threshold e. By design 0 ≤ H ≤ 1. Notice that in standard information theory terminology higher entropy implies higher information. However, in our context an informative k-mer has low entropy. Also, due to the Bayesian estimation, a spurious k-mer having only very small counts will have large conditional entropy and will be filtered out. The optimal value of threshold e varies with datasets. It can be ‘optimized’ in a supervised manner by utilizing a training set where we have labelled samples. In the absence of a labelled set, we suggest taking the ‘average’ of distance metrics computed over the potential thresholds as the final metric. We refer to the final metrics in the two cases as optimized metric and average metric. In our experimental set-up, we randomly make a 50-50 split of a given dataset in training Str and testing Ste sets: Str ∩ Ste = ∅, and Str ∪ Ste = S. We use Str to optimize the entropy threshold: we query with samples in Str , and retrieve relevant samples within the same set to observe which entropy threshold results in the best retrieval result (see Sec. 3.4 for details). While comparing the performance of two methods we always present the evaluation over Ste : we query with samples within Ste , and we retrieve relevant samples from S (not just Ste ). 5 3.2 Algorithms to extract informative k-mers Our main computational challenge is to extract all informative k-mers from large-scale datasets in feasible time and space. Recall that the filtering step relies on knowledge over multiple samples to decide if the respective k-mer is informative for the retrieval task or not. Since the typical collections of WMS samples are huge in size, we cannot assume that even the plain input fits into the main memory of any single machine. To process these large-scale datasets, the computation needs to be done either using external memory (i.e. disk) or in a distributed manner (i.e. a computer cluster). We review two approaches: k-mer counting [12, 21] and distributed string mining [29]. The first one is a standard approach in the literature for fixed k, but has several limitations when applied in our context of multiple samples and large-scale data. We show that the latter approach is more flexible in this context and can also be generalized to extract informative k-mers over all values of k simultaneously. Jellyfish [12] and DSK [21] are examples of recent algorithmic improvements in k-mer counting. Both tools use hash tables to compute the k-mer distribution for a given (fixed) k. In both tools, space-efficiency is achieved by keeping most of the hash table on disk. The main drawback with these disk-based approaches is that they are aimed at counting k-mers in a single sample and extending them over to multiple samples is non-trivial. For example, Jellyfish could, in principle, be extended to count k-mers over multiple samples: the authors give a roughly linear time algorithm to merge two or more hash tables. However, the intermediate k-mer counts would need to be stored on disk, which requires significant amount of additional space, and the merge-phase is not parallelized [12, User manual, Sect. Bugs]. The decision whether a particular k-mer is informative or not is made by looking at its frequency over all the given WMS samples. We tackle this problem by a Distributed String Mining (DSM) framework [29] that can handle multi-sample inputs by utilizing a computer cluster. The main advantages of this framework are that (i) load-balancing divides the data and computation over multiple cluster nodes, (ii) intermediate k-mer counts are not stored explicitly, and (iii) there is no additional disk I/O strain, except reading through the input once. These advantages allow terabyte-scale data analysis on a cluster consisting of nodes having limited main memory. We extend the DSM framework to be compatible with our definition of informative k-mers (see the above subsection). It allows us to extract the informative k-mers either for a fixed k or over all values of k in feasible time. The DSM framework is based on a client-server model. The clients have one-to-one correspondence to the given samples, each client being responsible for computing the frequencies within the designated sample. The client-side computation relies heavily on suffix sorting techniques and on space-efficient data structures for strings [29]: the input data are first preprocessed into a compressed representation, which replaces the input data and acts as an efficient search structure. The server-side computation is more straightforward: the server simply merges the (sorted) input from the clients, computes the entropies and updates the distance matrices. Fig. 3 gives a toy example of the client-server interaction. Two crucial observations are needed to keep the whole computation and transmission costs feasible. First, the informative k-mers can be seen as a subset of left-right-branching substrings, i.e. substrings whose instances have differentiating continuation on both left and right. More formally: substring w of string T [1, n] is called right-branching if there exists two symbols a and b such that a 6= b and both wa and wb are substrings of T . Similarly, a substring w is leftbranching if aw and bw, a 6= b, are substrings of T . If a substring is both left-branching and right-branching we say it is left-right-branching. Second, for any string of length n, there are at most O(n) left-right-branching substrings, and the total length of all such substrings is bounded by O(n log n) [7, Theorem 1]. The first observation allows us to reduce the client-side computation to a smaller set of substrings: it is easy to see that if k-mer w, having frequency f ′ (s, w) ≥ 2, is non-branching, then there exists a substring w′ of length k ′ > k that is left-right-branching and has exactly the same frequency, i.e., f ′ (s, w) = f ′ (s, w′ ). It follows that the frequency of non-branching k-mers can be deduced from the branching k ′ -mers, and the left-right-branching substrings contain all the necessary information for us to detect informative k-mers. The second observation guarantees a feasible transmission cost between clients and servers: the upper bound for the concatenation of all left-right-branching substrings also acts as an upper bound for both the server-side running time and the amount of communication needed. The drawback of restricting to left-right-branching substrings is that the informative k-mers that we are able to detect have to appear at least twice in a sample, 6 although this limit may be useful in pruning spurious k-mers introduced by sequencing errors. More detailed explanation and analysis of the DSM framework is given in [29]. A software implementation of the DSM framework is available at https://github.com/HIITMetagenomics/dsm-framework. 3.3 Dissimilarity metrics Having extracted the informative k-mers, we use them to compute the dissimilarity between two metagenomic samples. We consider three dissimilarity metrics that can be computed easily over a large number of k-mers in sequential manner, i.e. one k-mer at a time, and without storing all the k-mer frequencies explicitly. To utilize the natural variance structure of the k-mers—some are more abundant than others—we weight the relative frequencies of each k-mer by their respective total counts, i.e., we utilize the absolute frequencies f (s, w) as defined in (1). We mainly use the very simple Jaccard distance which does not consider abundances at all, only whether a k-mer occurs or not. Given two sets s1 and s2 of k-mers detected as present in two different samples, Jaccard distance measures how many elements are shared between these two sets. Mathematically, it is defined as \|s1 ∩ s2 \| Dcount (s1 , s2 ) = 1 − . \|s1 ∪ s2 \| Despite its simplicity, we observe that Jaccard distance performs well; a potential reason is its robustness to measurement noise and effectiveness when two metagenomic samples differ in terms of presence and absence of certain species or functionalities. We assume a k-mer is present in a sample if its frequency is more than 2. We also experiment with two metrics that use the abundance information: I. Variance-stabilized Euclidean distance: An obvious distance measure between two metagenomic samples s1 and s2 is the Euclidean distance between their respective k-mer frequencies. We consider the distance metric Xp p Dsqrt (s1 , s2 ) = ( f (w, s1 ) − f (w, s2 ))2 w which can be computed sequentially as new informative k-mers are extracted. The square root transformation is the variance stabilizing transformation for Poisson distribution—a popular model for quantitative sequencing data. II. Log transformed Euclidean distance: We also consider the same metric but with log transformation which is a popular approach in document retrieval, i.e., X Dlog (s1 , s2 ) = (log(1 + f (w, s1 )) − log(1 + f (w, s2 )))2 . w The motivation for using the log transformation is that it decreases sensitivity to high frequency counts: some k-mers are present in high abundance in almost every genome, for instance k-mers from the marker gene, and the log transformation reduces their effect in the metric. 3.4 Evaluation metric We evaluate the performance of the dissimilarity metric in terms of its performance in the task of retrieving relevant samples given a query metagenomics sample. The ground truth for relevance is either the disease class (disease vs not) or the known body site: samples from the same class are considered relevant. For measuring retrieval performance we use an evaluation metric which is popular in document retrieval, the mean average precision (MAP) [25]. Given a query q, the retrieval method ranks the samples in an increasing order of their dissimilarities from q. Given one has retrieved the top (closest) n ∈ {1, . . . , N } samples the precision @n is defined as Precision(n; q) = number of relevant samples in n retrieved samples , n 7 HIGH-C 149 200 0.4 117 4.9 149 1.8 10 Input size (GB) Samples Preproc. (h) Total memory (GB) All k Wall-clock (h) CPU time (h) k = 21 Wall-clock (h) CPU time (h) MetaHIT 536 124 3.6 209 2.0 187 0.4 74 T2D-P2 786 199 10 610 8.0 1,137 2.8 279 HMP 3,353 435 65 2,885 53 20,000 12 4,000 Table 1: Computational resources required by the distributed string mining on different datasets. We report wall-clock times and total CPU times for both fixed k = 21 and over all k. Preprocessing is done only once, separately from the actual computation. Total memory is the memory requirement over all computation nodes. Experiments were ran on a cluster of Dell PowerEdge M610 nodes having 32 GB of RAM and 16 cores. Simulated data and MetaHIT were run using up to 8 nodes. T2D-P2 was ran using 32 nodes allowing more parallelization at the server-side. HMP was ran on a cluster of 20 nodes with 2x10-core Xeon CPUs and 256GB RAM. and MAP defined using average precision as, MAP = 1 X 1 X Precision(n; q). AveP(q), AveP(q) = \|Q\| mq n∈Rq q∈Q Here Q is the set of all queries, mq is the number of relevant samples to query q, and Rq is the set of locations in the ranked list where a relevant sample appears. It is straight-forward that a higher MAP implies better performance. To judge if two MAP values are significantly different or not, we employ the randomization test described in [25]: for each query, this test randomly reassigns the AvePs achieved by two methods to one another and computes the difference between the resulting MAP for multiple such reassignments to get a distribution, against which the true MAP value is tested in terms of p-value. In case two samples share the same dissimilarity from a query sample, we employ the modification suggested in [13] to break ties. When computing the mean, we follow a leave-one-out cross-validation type approach using each sample as a query, and retrieving from the rest of the collection. For simulated data and human gut samples, we only query with the positive samples in the testing set q ∈ Ste , whereas for body site samples we query with each sample in the testing set. For both cases we retrieve from the entire set S \ {q}. While choosing the entropy threshold in a supervised setting, we query from q ∈ Str and retrieve from Str \ {q}. 3.5 Synthetic data generation To test the method, we simulated four datasets containing samples from separate classes, with the interpretation that samples from the same class are relevant. In all the datasets we have two classes: both classes of samples have the same species composition but different relative abundances. We used MetaSim [20] to generate Illumina reads of length 80 using the error configuration file provided by the developers. Each dataset contains 200 samples: 98 of them belong to the positive class and the rest belong to the negative class. For each dataset, we used the same 100 species from the following genera: acetobacter, acetobacterium, acidiphilium, acidithiobacillus, acinetobacter, bacillus, bacteroides, bifidobacterium, chlamydia, chlamydophila, clostridium, escherichia, haloarcula, halobacterium, lactobacillus, pasteurella, salmonella, staphylococcus, and streptococcus. The abundance profiles were generated from two Dirichlet distributions; one for positive and the other for negative class. The parameters of the Dirichlet distributions were shared between two classes: for half of the species (randomly chosen) the same parameters were used for both classes and for the other half of the species the parameters were randomly permuted. For example, given 5 species the assigned parameters could be: (0.3, 0.2, 0.6, 0.1, 0.9) and (0.9, 0.2, 0.3, 0.1, 0.6) where the parameters for 8 log10(# of k−mers) MetaHIT + log 15 HMP + Jaccard 15 15 12 21 30 All 10 10 10 5 5 5 0 0 0.5 Entropy threshold HIGH−C + Jaccard log10(# of k−mers) T2D−P2 + Jaccard 1 0 0 0.5 Entropy threshold HIGH−VAR + Jaccard 15 1 0 0 F 0.5 Entropy threshold LOW−C + Jaccard 15 MIXED−C + Jaccard 15 15 12 21 30 All 10 10 10 10 5 5 5 5 0 0.7 0.8 0.9 Entropy threshold 1 0 0.7 0.8 0.9 Entropy threshold 0 0.7 1 1 0.8 0.9 Entropy threshold 1 0 0.7 F 0.8 0.9 Entropy threshold 1 Figure 4: Number of informative strings over varying entropy thresholds for the proposed approach ‘All’, fixed k-mer lenthgs ‘12’,‘21’ and ‘30’, and for protein family based comparison with FIGfam ‘F’. The box denotes the ‘optimized’ entropy threshold that has been used to evaluate the performance of the methods. Some general observations are as follows. The number of strings for k = 12 is lower than the rest while the number of strings for ‘All’ is much higher than rest of the methods, and number of strings for k = 21 and k = 30 are very close. We observe that there are strings with low entropies—more in the real data sets than in the simulated data sets—which indicate the presence of discriminative features. Also, the ‘optimized’ entropy threshold varies for different methods. the second and fourth species are the same, but for the other species they were permuted. The exact species and corresponding parameter values can be downloaded from https://github.com/HIITMetagenomics. The resulting datasets are: 1. HIGH-C: relatively easy data with high coverage (10,000,000 reads per sample) 2. LOW-C: relatively difficult data with low coverage (2,000,000 reads per sample) 3. MIXED-C: mixed data with half the samples from HIGH-C and the rest from LOW-C to simulate varying sequencing depth. 4. HIGH-VAR: relatively difficult data with same coverage as HIGH but additional noise in the class distributions to simulate more overlap between classes. To elaborate, the relative abundance of species is pHIGH−VAR = 0.5 pHIGH + 0.5 noise where noise is generated from a symmetric Dirichlet distribution with all parameters equal to 1. 4 Results We evaluated the retrieval performance on three human metagenomics datasets: 1. MetaHIT [18], 124 metagenomic samples from 99 healthy people and 25 patients with inflammatory bowel disease (IBD) syndrome. Each sample has on average 65 ± 21 million reads. Our goal was to retrieve IBD positive patients. 9 Mean average precision MetaHIT + Jaccard MetaHIT + log ↑* 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 ↓ T2D−P2 + Jaccard HMP + Jaccard ↓ * ↓ ↓ * 0.6 ↓ * ↓ * ↓ *** 1 0.4 0 Ak FIG Abd Method 3 0 HIGH−C + Jaccard Mean average precision ↓* 1 0.2 Ak FIG Abd Method 0 3 HIGH−VAR + Jaccard ↓ * ↓ * ↓ * ↑ * 0.5 0 0.5 1 T 0 FIG Abd Method 3 1 0 T Ak FIG Abd Method 3 MIXED−C + Jaccard ↓ * ↓ * ↑ * 0.5 Ak FIG Abd 3 Method 0 LOW−C + Jaccard ↓ * ↓ * ↓ * ↑ * 0.5 Ak FIG Abd 3 Method Ak 1 ↓ * ↓ * ↓ * ↑ * 0.5 Ak FIG Abd 3 Method T 0 Ak FIG Abd 3 Method T Figure 5: Retrieval performance comparison of the proposed approach using all k-mers (“Ak”) against the following base measures: (1) “FIG”: retrieval performance using known protein family, (2) “Abd”: Hellinger distance between relative estimated abundance, (3) “3”: d2S distance between relative abundance of 3-mers. “Ak” uses the ‘optimized metric’ over 101 equally spaced threshold values between 0 and 1. Each errorbar shows the MAP value along with the standard error. The grey horizontal line shows retrieval by chance: MAP computed over zero similarity metric. An arrow (if present) over a method indicates whether the performance of the corresponding method is significantly better (↑) or worse (↓) than “Ak”: The stars denote significance level: 0 < * < 0.001 < ** < 0.01 < * < 0.05. For the synthetic datasets (in the bottom row) the relative abundance is known from experimental design. We present this result as “T”. For MetaHIT we present the performance for both Jaccard and log metric, since the latter performs much better compared to the former. 2. T2D Phase II [19], 199 metagenomic samples from 100 healthy people and 99 patients with type II diabetes. Each sample has on average 47 ± 11 million reads. Our goal was to retrieve diabetic patients. We chose to explore the phase II data instead of the phase I data since the former has higher coverage; about 40% more reads than the latter. 3. HMP [27], 435 metagenomic samples from 10 different body sites. Out of 690 samples that passed the QC assessment (http://www.hmpdacc.org/HMASM/), we discarded 255 samples that had less than 1% of the number of reads of the largest sample. To recapitulate, for MetaHIT and T2D-P2, our goal is to observe if given a positive sample, e.g., from a patient with a particular disease, one can retrieve relevant samples, i.e., with similar disease; whereas for HMP, our goal is to observe if given a sample from a particular body site, one can retrieve relevant samples, i.e., samples from the same body site. For all data we applied a quality threshold of 30 and ignored any base pairs with quality less than the threshold. Table 1 gives an overview of the computational resources required for each data set. Additionally, number of k-mers used by different methods for each data set is available in 4. Retrieval of samples with similar annotation: We applied the proposed approach and a number of alternatives to retrieval of similar samples from the same data set and evaluated by how many of the retrieved 10 MetaHIT + log Average precision ↓ ** ↓* T2D−P2 + Jaccard ↓ ↑ * ↓ 0.4 0.2 0.5 0.2 All 12 21 0 30 All 12 Average precision ↓ * ↑ * ↑ * 0.5 21 k 30 0 30 All 12 ↓ *** 1 0 LOW−C + Jaccard ↑* 12 21 1 0 30 k 30 MIXED−C + Jaccard ↑ * ↑ * 0.5 All 21 k HIGH−VAR + Jaccard 0.5 12 21 k HIGH−C + Jaccard All ↓* 0.6 0.4 k 0 ↓ *** ↓ * 1 0.6 0 1 HMP + Jaccard ↓ ↓ * ↑ * ↑ * 1 0.5 All 12 21 k 30 0 All 12 21 30 k Figure 6: Comparison of best performances for different k-mer lengths. The figures show the performance over queries by all positive samples as a violin plot. All methods use the ‘optimized metric’ chosen over 101 equally spaced threshold values between 0 and 1: the box denotes the MAP value. The horizontal lines show retrieval by chance: AveP computed over zero dissimilarity metric. Straight line is the mean, and dotted lines are 5%, and 95% quantiles respectively, when number of relevant samples differ for different queries. An arrow (if present) over a method implies whether the corresponding method performs significantly better (↑) or worse (↓) than ‘All’ : The stars denote significance level: 0 < * < 0.001 < < 0.01 < * < 0.05. We observe that the considering all k-mers usually perform equally well with respect to considering a single k. samples had the same annotation: class label, disease state or body site. A comparison of the obtained mean average precision values averaged over queries by all positive samples is shown in Fig. 5. The results show the performance achieved by the ‘optimized metric’. The alternatives we considered were: i. retrieval performance based on the proposed distances but with the frequencies counted on specific 21-mers from known protein families (FIGfams) [15]; ii. retrieval based on Hellinger distances between relative species abundances estimated using MetaPhlAn [24]; and iii. retrieval based on d2S \|M0 distances between relative frequencies of 3-mers [6]. For the simulated data, the two classes differ only by the relative species abundance; thus, retrieval based on ground truth abundance can be considered to give an upper limit for the performance. For HIGHC and HIGH-VAR, the proposed method performs closer to the ground truth performance than any other methods, although the difference from ground truth performance is still statistically significant. For LOW-C, the performance of all methods, except the protein family based comparison, drop compared to HIGH-C, while for MIXED-C the performance is again close to HIGH-C despite the presence of low coverage samples. This is an encouraging observation showing the robustness of the proposed approach to varying sequencing depths. For the real data sets the proposed approach yielded statistically significantly higher mean average precision than any of the alternatives (p < 0.05) for all the datasets, except T2D-P2 where protein family based comparison works equally well. Interestingly, the abundance-based retrieval performs relatively poorly here, suggesting that the differences between the classes cannot be easily captured by species composition alone, while the proposed k-mer features can provide a better separation. Retrieval based on the known protein family performed fairly well, but slightly worse than the proposed approach on MetaHIT. We observe that 11 Mean average precision Mean average precision 1 0.4 MetaHIT + log ↑* ↑ * ↑ * 0.3 0.4 0.2 0 0.5 0.2 0.1 All 12 HIGH−C + Jaccard ↑ * ↑ * ↑ ↓ * ↑ * ↑ * ↑ * ↑ * ↑ * 21 30 FIG k 1 0 All 12 21 30 FIG k HIGH−VAR + Jaccard ↓ * ↑ * ↓ * ↓ * ↑ * ↑ * ↑ * ↑ * ↑ * 1 0.5 All 12 1 HMP + Jaccard ↓ * ↑ * ↓ * ↓ * ↓ * ↑ * ↑ * 0.6 0.5 0 T2D−P2 + Jaccard ↑ * ↑ * ↑ * ↑ * ↑ * ↑ * ↑ * ↓ ↓ *** 21 30 FIG k 0 0 All 12 LOW−C + Jaccard ↓ * ↑ * ↓ ↑ *** ↑ * ↑ *** 0.5 All 12 0 21 30 FIG k 21 30 FIG k 1 MIXED−C + Jaccard ↑ * ↑ * ↓ * ↑ * ↑ * ↑ * ↑ * ↑ * 0.5 All 12 21 30 FIG k 0 All 12 21 30 FIG k Figure 7: Comparison of the best retrieval performance achieved with ‘optimized metric’ (middle), ‘average metric’ (right) and without entropy filtering (left), for proposed approach ‘All’, individual ks as well as FIGfam based distance metric. The metrics are ‘optimized’/‘averaged’ over 101 equally spaced threshold values between 0 and 1. Each errorbar line shows the MAP value along with the standard error. The grey horizontal line shows retrieval by chance: MAP computed over zero dissimilarity metric. An arrow (if present) over a method implies whether the performance of the corresponding method (top: ‘average metric’, bottom: ‘optimized metric’) is better (↑) or worse (↓) than when entropy filtering is employed: The stars denote significance level: 0 < * < 0.001 < ** < 0.01 < * < 0.05. We observe that filtering has a positive impact on the retrieval performance. for MetaHIT, Jaccard metric performs poorly; however, a change of metric to log significantly improves the performance for all methods. Otherwise, all metrics usually work equally well over different data sets. Effect of using specific or unspecific k-mer length: We next compared the proposed approach of using all k-mers to using a specific k. The retrieval performance using ‘optimized metric’ is shown in Fig. 6. The figures show the complete distribution of average precision values over different queries whose mean is the mean average precision of Fig. 5. The performance of the proposed method is usually better than with any individual k. Thus, the proposed method appears to be a relatively safe choice that does not suffer from catastrophically bad performance on any of the data sets. Effect of the entropy filtering: Next, we evaluated the efficacy of filtering the informative k-mers against retrieval performance without the filtering operation. The results are presented in Fig. 7. We observed that entropy filtering usually improved retrieval performance for all tested k-mer lengths when using the ‘optimized metric’, although the improvement might not always be statistically significant. Although ‘average metric’ often provides significant performance, it might not always improve over performance without filtering. Also, retrieval performance of FIGfam may or may not improve with entropy filtering. Comparison across different metrics: Finally, we evaluated the retrieval performance over different dissimilarity metrics. We presented the performance using ‘optimized metric’ for different metrics in Fig. 8. We 12 Average precision MetaHIT ↓ *** ↑* ↓ * ↑ * Average precision 1 0.4 0.2 HIGH−C ↑* ↓ * ↑ * ↓ * HMP ↑ * ↑ * ↓ * ↓ * ↑ * ↓ * 0.5 0.2 count ↑ * ↓* sqrt Metric 0 log 1 HIGH−VAR ↓ * ↓ * ↓ * ↑ * count sqrt Metric ↑ * ↑ * sqrt Metric log 0 0 log 1 0.5 count ↓ * ↑ * 0.6 0.4 0.5 0 ↑ * ↑ * 0.6 0 1 T2D−P2 ↓ * ↓ * ↓* ↑ * LOW−C ↑ * ↑ * ↓ * ↓ * count ↑ * ↓ *** 0.5 count sqrt Metric 0 log sqrt Metric log 1 MIXED−C ↓* ↓ *** ↑* ↓* ↑ *** ↑* 0.5 count sqrt Metric log 0 count sqrt Metric log Figure 8: Comparison of the best retrieval performance for different distance metrics using all k-mers. They show a violin plot of the average performances over queries by all positive samples in the data sets. The ‘optimized metrics’ have been selected over 101 equally spaced threshold values between 0 and 1: the box denotes the MAP value. The horizontal lines show retrieval by chance: AveP computed over zero dissimilarity metric. Straight line is the mean, and dotted lines are 5%, and 95% quantiles respectively, when number of relevant samples differ for different queries. An arrow (if present) over a method implies whether the corresponding method performs significantly better (↑) or worse (↓) than the other methods (denoted by their colors): The stars denote significance level: 0 < * < 0.001 < < 0.01 < * < 0.05. We observe that different distance metrics usually demonstrate similar performance. observed that the simple presence/absence-based metric Dcount performed at least as well as abundancesensitive log and sqrt metrics, except for the MetaHIT data for which the other metrics performed the better. 5 Conclusion In the wake of collecting multiple samples from similar environments information retrieval for metagenomic samples is expected to become a handy tool in metagenomics research. In this paper, we have addressed the problem of retrieving relevant metagenomic samples given a query sample from the same collection. The novelty of the proposed approach is that it is unsupervised, and does not rely on the availability of reference databases. We have suggested employing k-mer frequencies as feature representation; however, rather than exploring k-mers of a fixed k, we have scanned through all possible k-mers of all possible k’s using distributed string mining, and have proposed appropriate filtering technique to discard uninformative k-mers. We have evaluated our method on both real and simulated data, and observed that the approach can effectively retrieve relevant metagenomic samples, outperforming both the FIGfams method based on known highly informative protein families as well as retrieval based on species composition of the samples. 13 Acknowledgement The authors would like to thank Ahmed Sobih for his help with the MetaPhlAn experiments on MetaHIT and T2D-P2. Part of the calculations presented above were performed using computer resources within the Aalto University School of Science “Science-IT” project. Funding: This work was supported by the Academy of Finland (project numbers 140057, 250345, 251170 and 259440). 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AMENABLE UNIFORMLY RECURRENT SUBGROUPS AND LATTICE EMBEDDINGS arXiv:1802.04736v2 [math.GR] 29 Mar 2018 ADRIEN LE BOUDEC Abstract. We study lattice embeddings for the class of countable groups Γ defined by the property that the largest amenable uniformly recurrent subgroup AΓ is continuous. When AΓ comes from an extremely proximal action and the envelope of AΓ is co-amenable in Γ, we obtain restrictions on the locally compact groups G that contain a copy of Γ as a lattice, notably regarding normal subgroups of G, product decompositions of G, and more generally dense mappings from G to a product of locally compact groups. We then focus on a family of finitely generated groups acting on trees within this class, and show that these embed as cocompact irreducible lattices in some locally compact wreath products. This provides examples of finitely generated simple groups quasi-isometric to a wreath product C ≀ F , where C is a finite group and F a non-abelian free group. Keywords. Lattices, locally compact groups, strongly proximal actions, Chabauty space, groups acting on trees, irreducible lattices in wreath products. 1. Introduction The questions considered in this article fall into the setting of the following general problem: given a (class of) countable group Γ, study the locally compact groups G such that Γ embeds as a lattice in G, i.e. such that Γ sits as a discrete subgroup of G and G/Γ carries a G-invariant probability measure. Malcev showed that every finitely generated torsion free nilpotent group embeds as a cocompact lattice in a unique simply connected nilpotent Lie group [Rag72, Ch. II]. Conversely if G is a locally compact group with a finitely generated nilpotent lattice, then after modding out by a compact normal subgroup, the identity component G0 is a Lie group of polynomial growth (these have been characterized in [Gui73, Jen73]) and G/G0 is finitely generated and virtually nilpotent. This statement is a combination of several works. First if G has a finitely generated nilpotent lattice Γ, then Γ is necessarily cocompact in G. Since Γ is virtually torsion free this is a classical fact when G is totally disconnected, and the general case can be deduced from [BQ14, Prop. 3.7] (which uses notably the solution of Hilbert’s fifth problem [MZ55]). In particular G is compactly generated with polynomial growth, and the statement then follows from the generalization of Gromov’s polynomial growth theorem for locally compact groups [Los87]. Beyond the nilpotent case, examples of classifications of embeddings of Γ as a cocompact lattice have been obtained by Dymarz in [Dym15] for several families Date: March 29, 2018. This work was carried out when the author was F.R.S.-FNRS Postdoctoral Researcher. Current affiliation: CNRS, UMPA - ENS Lyon. 1 2 ADRIEN LE BOUDEC of examples of solvable groups Γ. Although not directly related to our concerns, we also mention that a certain dual problem was considered by Bader–Caprace– Gelander–Mozes in [BCGM16] for the class of amenable groups. Outside the setting of amenable groups, Furman addressed the above problem for the class of lattices Γ in semi-simple Lie groups in [Fur01], improving rigidity results of Mostow, Prasad, Margulis (see the references in [Fur01]; see also Furstenberg [Fur67b]). In [BFS15], Bader–Furman–Sauer considered a large class of countable groups Γ defined by certain group theoretic conditions, and established, given a lattice embedding of Γ in G, a general arithmeticity result in the setting where the connected component of G is non-compact. In this article we consider the class of groups whose Furstenberg uniformly recurrent subgroup is continuous (see below for definitions). In the first part of the article we address the question to what extent the properties of the Furstenberg uniformly recurrent subgroup of a countable group Γ influence the locally compact groups into which Γ embeds as a lattice. In the second part we focus on a family of finitely generated groups within this class which embed as cocompact irreducible lattices in some locally compact wreath products. The groups under consideration. For a countable group Γ, the Chabauty space Sub(Γ) of all subgroups of Γ is a compact space, on which Γ acts by conjugation. A uniformly recurrent subgroup (URS) of Γ is a closed minimal Γ-invariant subset of Sub(Γ) [GW15]. Glasner and Weiss showed that every minimal action of Γ on a compact space X gives rise to a URS (see Proposition 3.2), called the stabilizer URS associated to the action. Conversely every URS arises as the stabilizer URS of a minimal action (see Matte Bon–Tsankov [MBT17], and Elek [Ele17] in the case of finitely generated groups). URS’s have been shown to be related to the study of ideals in reduced group C ∗ algebras [KK17, Ken15] and reduced crossed products [Kaw17]. URS’s of several classes of groups have been studied in [LBMB16]. For certain examples of groups Γ, rigidity results about minimal actions on compact spaces have been obtained in [LBMB16] from a complete description of the space URS(Γ). Various results about homomorphisms between topological full groups of étale groupoids, notably obstructions involving invariants of the groupoids, have been obtained in [MB18] via URS’s considerations (more precisely via a complete description of the points in the Chabauty space of these groups whose orbit does not approach the trivial subgroup). In the present article we will make use of URS’s as a tool in order to study lattice embeddings for a class of countable groups that we now define. A URS is amenable if it consists of amenable subgroups. Every countable group Γ admits a largest amenable URS AΓ (with respect to a natural partial order on URS(Γ), see §3.1), which is the stabilizer URS associated to the action of Γ on its Furstenberg boundary (see §2.2 for definitions). The URS AΓ is called the Furstenberg URS of Γ. AΓ is either a point, in which case we have AΓ = {Rad(Γ)}, where Rad(Γ) is the amenable radical of Γ, or homeomorphic to a Cantor space. In this last case we say that AΓ is continuous. We refer to [LBMB16] for a more detailed discussion. Let (C) denote the class of groups Γ for which the Furstenberg URS AΓ is continuous. Equivalently, a group Γ belongs to (C) if and only if Γ admits an amenable URS whose envelope is not amenable (see below for the definition of the AMENABLE URS’S AND LATTICE EMBEDDINGS 3 envelope). The class (C) is disjoint from all classes of groups previously mentioned in the introduction. More precisely, the class (C) is disjoint from the class of amenable groups, the class of linear groups [BKKO14], and also from other classes of groups specifically considered in [BFS15], such as groups with non-vanishing ℓ2 -Betti numbers [BKKO14] or acylindrically hyperbolic groups (see [DGO11, Th. 7.19] and [BKKO14, Th. 1.4]). The class (C) is stable under taking quotient by an amenable normal subgroup and extension by an amenable group [LBMB16, Prop. 2.20]. Also if Γ has a normal subgroup that is in (C), then Γ belongs to (C) [LBMB16, Prop. 2.24]. By a result of Breuillard–Kalantar–Kennedy–Ozawa, the complement of the class (C) is also stable under extensions (see [LBMB16, Prop. 2.24]). The study of this class of groups is also motivated by the work of Kalantar– Kennedy [KK17], who showed the following characterization: a countable group Γ belongs to (C) if and only if the group Γ/Rad(Γ) has a reduced C ∗ -algebra that is not simple. For an introduction and the historical developments of the problem of C ∗ -simplicity, we refer to the survey of de la Harpe [Har07]. Topological boundaries. We will make use of the notion of topological boundary in the sense of Furstenberg. These are compact spaces with a minimal and strongly proximal group action (see §2.2 for definitions). Many different notions of boundaries appear in the study of groups and group actions. What is now sometimes called “boundary theory” is particularly well described in the introduction of [BF14]. We insist that in the present article the term boundary will always refer to a topological boundary in the above sense. This notion should not be confused with any of the measured notions of boundaries. In particular, despite the possibly confusing terminology, the maximal topological boundary, called the Furstenberg boundary, is not the same notion as the measured notion of Poisson–Furstenberg boundary. Lattices and direct products. Special attention will be given to products of locally compact groups. The study of lattices in product groups is motivated (among other things) by its connections with the theory of lattices in semi-simple Lie groups, its rich geometric aspects, as well as the instances of groups with rare properties appearing in this setting. We refer to the literature (see [Mar91, Wis96, BM97, BM00b, Rém99, Sha00, BM02, Rat04, MS04, BS06, CR09, CM12, Rad17]) for developments over the last years on the study of lattices in products of locally compact groups. Given a countable group Γ with a continuous Furstenberg URS and a group G containing Γ as a lattice, we are interested in understanding how close the group G can be from a direct product of two groups, or which properties the group G can share with a direct product. Of course various notions of closeness can be considered. The most basic one is to ask whether the group G admits non-trivial decompositions as a direct product. One step further, one might consider quotient morphisms from G onto direct products of groups. In Theorems 1.1 and 1.2 below we more generally consider continuous morphisms with dense image from G to a direct product of groups G → G1 × G2 . We make no assumption about injectivity of these maps or injectivity of the composition with the projection to one factor 4 ADRIEN LE BOUDEC Gi . In particular this setting allows maps of the form G → G/N1 × G/N2 for closed normal subgroups N1 , N2 such that N1 N2 is dense in G. First results. A central notion in this article is the one of extremely proximal action. Minimal and extremely proximal actions naturally arise in geometric group theory, and are boundaries in the sense of Furstenberg. We refer to §2.3 for definitions and examples. We say that the Furstenberg URS AΓ of a countable group Γ comes from an extremely proximal action if there exists a compact space Z and a Γ-action on Z that is minimal and extremely proximal, whose associated stabilizer URS is equal to AΓ . Note that typically Z will not be the Furstenberg boundary of Γ. If H is a URS of Γ, the envelope Env(H) of H is by definition the subgroup of Γ generated by all the subgroups H ∈ H. Theorem 1.1. Let Γ be a countable group whose Furstenberg URS comes from a faithful and extremely proximal action, and let G be a locally compact group containing Γ as a lattice. The following hold: (a) Assume that Env(AΓ ) is finitely generated and co-amenable in Γ. Then G cannot be a direct product G = G1 × G2 of two non-compact groups. (b) Assume that Env(AΓ ) has finite index in Γ and finite abelianization. Then any continuous morphism with dense image from G to a product of locally compact groups G → G1 × G2 is such that one factor Gi is compact. The same conclusions hold for any group commensurable to G up to compact kernels. This result has applications to the setting of groups acting on trees, see Corollary 1.3. We make several comments about the theorem: 1) We do not assume that Γ is finitely generated, nor that G is compactly generated. For statement (a), The assumption that Env(AΓ ) if finitely generated admits variations, see Theorem 5.14. 2) Making an assumption on the “size” of the envelope of AΓ with respect to Γ is natural, in the sense that in general there is no hope to derive any conclusion on the entire group Γ if this envelope is too small. An extreme illustration of this is that there are groups Γ whose Furstenberg URS comes from a faithful and extremely proximal action but is trivial, and these can be lattices in products, e.g. PSL(2, Z[1/p]) inside PSL(2, R) × PSL(2, Qp ) (see also the discussion right after Corollary 1.3). 3) Under the assumption that Env(AΓ ) is co-amenable in Γ, the fact that the Furstenberg URS AΓ comes from a faithful and extremely proximal action is equivalent to asking that the action of Γ on AΓ is faithful and extremely proximal; see Remark 3.27. This provides an intrinsic reformulation of the assumption not appealing to any auxiliary space. 4) For Γ as in the theorem, the assumption in statement (b) that Env(AΓ ) has finite index in Γ and Env(AΓ ) has finite abelianization is equivalent to Γ being virtually simple (see Proposition 4.6). The URS approach to study lattice embeddings allows to consider more generally subgroups of finite covolume. Recall that a closed subgroup H of a locally AMENABLE URS’S AND LATTICE EMBEDDINGS 5 compact group G has finite covolume in G if G/H carries a G-invariant probability measure. Thus a lattice is a discrete subgroup of finite covolume. Before stating the following result we need some terminology. Recall the notion of disjointness introduced by Furstenberg in [Fur67a]. If X, Y are compact G-spaces, X and Y are disjoint if whenever Ω is a compact G-space and Ω → X and Ω → Y are continuous equivariant surjective maps, the map Ω → X × Y that makes the natural diagram commute remains surjective (see §3.3). When X, Y are minimal G-spaces, this is equivalent to asking that the diagonal G-action on the product X × Y is minimal. Consider the following property: two non-trivial G-boundaries are never disjoint. A group with this property will be called boundary indivisible. Glasner characterized minimal compact G-spaces which are disjoint from all G-boundaries as those carrying a fully supported measure whose orbit closure in the space of probability measures is minimal [Gla75, Th. 6.2]. The relation between disjointness and boundaries that we consider here is of different spirit, as it deals with disjointness within the class of G-boundaries, rather than disjointness from this class. Locally compact groups with a cocompact amenable maximal subgroup are examples of boundary indivisible groups [Fur73, Prop. 4.4]. On the contrary, many discrete groups are not boundary indivisible. The relevance of this property in our setting comes from the fact that, as we will show in Proposition 3.24, a discrete group Γ as in Theorem 1.1 is boundary indivisible. Actually the only examples of (non-amenable) boundary indivisible discrete groups that we are aware of fall into the setting of Proposition 3.24. Recall that a convex compact G-space is irreducible if it does not contain any proper closed convex G-invariant subspace. We say that a subgroup L of a topological group G is weakly co-amenable in G if whenever Q is a non-trivial convex compact G-space in which L fixes a point, Q is not irreducible. This is indeed a weakening of the notion of co-amenability †, which asks that every convex compact G-space Q with L-fixed points has G-fixed points [Eym72] (and hence Q is not irreducible, unless trivial). If G has a subgroup that is both amenable and weakly co-amenable, then G is amenable; and a normal weakly co-amenable subgroup is coamenable. However in general weak co-amenability does not imply co-amenability, even for discrete groups. In §6.3 we exhibit examples of finitely generated groups such that every subgroup is either amenable or weakly co-amenable, but having non-amenable subgroups that are not co-amenable. Finally we say that a subgroup L ≤ G is boundary-minimal if there exists a non-trivial G-boundary on which L acts minimally. We refer to §5.1 for context and examples. Theorem 1.2. Let H be a locally compact group with an amenable URS that comes from an extremely proximal action, and whose envelope is co-amenable in H. Let G be a locally compact group containing H as a closed subgroup of finite covolume. Then G is boundary indivisible, and the following hold: (a) Whenever G → G1 × G2 is a continuous morphism with dense image, one factor Gi is amenable. †and not a relative version of a notion of weak amenability. 6 ADRIEN LE BOUDEC (b) If L is a boundary-minimal subgroup of G, and L is uniformly recurrent, then L is weakly co-amenable in G. In particular every boundary-minimal normal subgroup of G is co-amenable in G. Again we make several comments: 1) The group H is allowed to be discrete, so the theorem applies for all groups Γ as in Theorem 1.1. While (a) will be an intermediate step in the proof of Theorem 1.1, (b) provides additional information that is rather independent of the conclusion of Theorem 1.1. 2) Statement (a) implies that whenever N1 , N2 are closed normal subgroups of G such that N1 N2 is dense in G, at least one Ni must be co-amenable in G. 3) The last sentence in (b) implies that if N is a closed normal subgroup of G such that N CN is open in G (where CN is the centralizer of N ), then N is either amenable or co-amenable in G (see Proposition 5.3). We do not know whether the condition that N CN is open can be removed. 4) Theorem 1.2 does not say anything about amenable normal subgroups of G. It is worst pointing out that, as illustrated by the examples discussed in Section 6, it happens that a discrete group Γ satisfying the assumptions of Theorem 1.2 and with trivial amenable radical, sits as a lattice in a group G with noncompact amenable radical. 5) Remark 5.13 below provides counter-examples showing that in statement (b) the conclusion cannot be strengthened by saying that L is co-amenable in G. 6) For H, G as in the theorem, it happens that G splits as a direct product of two non-compact groups, even under the additional assumption that the amenable URS of H comes from a faithful extremely proximal action (see Example 6.3), so that “amenable” cannot be replaced by “compact” in statement (a). We view the above remarks 4)-5)-6) as illustrations of the limitations of the use of topological boundaries and URS’s to the problem addressed here in the rather abstract setting of Theorem 1.2. Group actions on trees is a natural source of extremely proximal actions, and Theorems 1.1 and 1.2 find applications in this setting. In the following statement T is a locally finite simplicial tree. Corollary 1.3. Let Γ ≤ Aut(T ) be a countable group having no proper invariant subtree and no finite orbit in T ∪ ∂T . Assume that Γξ is non-trivial and amenable for all ξ ∈ ∂T ; and Γ is virtually simple. If G is a locally compact group containing Γ as a lattice, then: (a) any continuous morphism with dense image G → G1 × G2 is such that one factor Gi is compact. In particular G itself cannot be a direct product of two non-compact groups. (b) The conclusion (b) of Theorem 1.2 holds for G. A group Γ as in Corollary 1.3 is never discrete in Aut(T ). Recall that Burger and Mozes constructed simple groups Γ acting on two locally finite regular trees T, T ′ such that the image of Γ in Aut(T ) and Aut(T ′ ) are non-discrete, but Γ acts AMENABLE URS’S AND LATTICE EMBEDDINGS 7 freely and cocompactly on T × T ′ , so that Γ is a cocompact lattice in the product Aut(T ) × Aut(T ′ ) [BM00b]. These examples illustrate the fact that the assumption in Corollary 1.3 that end-stabilizers are all non-trivial is essential. Examples of groups to which Corollary 1.3 applies can be found among the family of groups denoted G(F, F ′ ) in [LB16] (see Corollary 6.22). These are examples of groups with a continuous Furstenberg URS. Here F ′ ≤ Sym(d) is a finite permutation group and F is a simply transitive (or regular) subgroup of F ′ . The group G(F, F ′ ) is then a finitely generated group acting on a d-regular tree, transitively on vertices and edges, and with local action at every vertex isomorphic to F ′ . We refer to §6.2 for a definition. The normal subgroup structure of these groups is highly sensible to the permutation groups: there are permutation groups F, F ′ such that G(F, F ′ ) virtually admits a non-abelian free quotient (Proposition 6.11), and there are permutation groups F, F ′ such that G(F, F ′ )∗ (the subgroup of index two in G(F, F ′ ) preserving the bipartition of Td ) is simple [LB16, Cor. 4.14]. This family of groups and the family of Burger–Mozes lattices in the product of two trees both contain instances of finitely generated simple groups which embed densely in some universal group U (F )+ [BM00a, §3.2]. Despite these similarities, Corollary 6.22 shows that any group containing a virtually simple G(F, F ′ ) as a lattice is rather allergic to any direct product behavior. Compare with Theorem 1.4. We also mention that other examples of groups to which Corollary 1.3 can be applied may be found among the family of piecewise prescribed tree automorphism groups considered in [LB17, Sec. 4]. Irreducible lattices in wreath products. Leaving aside the previous abstract situation, we then focus on the family of groups G(F, F ′ ) (see §6.2 for definitions). The above mentioned common properties between the discrete groups G(F, F ′ ) and certain lattices in the product of two trees provide motivation for studying which locally compact groups can contain a group G(F, F ′ ) as a lattice. The contribution of this article to this problem is on the one hand the conclusions given by Corollary 1.3 (see Corollary 6.22), and on the other hand to describe embeddings of these groups as irreducible lattices in some locally compact wreath products. If H is a group acting on a set Ω, and A is a subgroup of a group B, the semirestricted permutational wreath product B ≀A Ω H, introduced by Cornulier in Ω,A [Cor17], is the semi-direct product B ⋊ H, where B Ω,A is the set of functions f : Ω → B such that f (x) ∈ A for all but finitely many x ∈ Ω, and H acts on B Ω,A in the usual way. This definition somehow interpolates between the restricted and the unrestricted permutational wreath products, which correspond respectively to A = 1 (in which case we will write B ≀Ω H) and A = B. When B, H are locally compact and A is compact open in B, there is a natural locally compact group topology on B ≀A Ω H (see §6.5.1). We call a lattice Γ in B ≀A Ω H irreducible if Γ has a non-discrete projection to H. The terminology is motivated by the fact that this definition prevents Γ, and more generally any subgroup commensurable with Γ, from being of the form Γ1 ⋊ Γ2 , where Γ1 and Γ2 are lattices in B Ω,A and H. In the following statement Ck is the cyclic group of order k, Sk the symmetric group on k elements, Vd the vertex set of a d-regular tree Td , and G(F, F ′ )∗ the subgroup of index two in G(F, F ′ ) preserving the bipartition of Td . 8 ADRIEN LE BOUDEC Theorem 1.4. Let d ≥ 3, F F ′ ≤ Sym(d) permutation groups such that F acts freely, and n the index of F in F ′ . Then the following hold: (a) The group G(F, F ′ ) embeds as an irreducible cocompact lattice in the semiS restricted permutational wreath product Gn,d = Sn ≀Vdn−1 Aut(Td ). (b) When F is a transitive permutation group, the finitely generated group Γn,d = Cn ≀Vd (Cd ∗ Cd ) = (Cn2 ≀ Fd−1 ) ⋊ Cd and G(F, F ′ )∗ have isometric Cayley graphs. We note that no finite index subgroup of Gn,d can split non-trivially as a product, but the stabilizer of an edge of Td in Gn,d (for the projection action on Td ) is an open subgroup which does split as a direct product of two non-compact groups. V ,S The embedding of G(F, F ′ ) in Gn,d = Snd n−1 ⋊ Aut(Td ) is not the inclusion in the subgroup 1 ⋊ Aut(Td ), but a twisted embedding associated to the cocycle given by the local action on Td . See Section 6 for details. We also note that the image V ,S of G(F, F ′ ) in Gn,d does not intersect the amenable radical Snd n−1 ⋊ 1, but does intersect the subgroup 1 ⋊ Aut(Td ) (along a cocompact lattice of Aut(Td )). The case n = 2 is particular as the group G2,d is actually a restricted wreath product, and in this situation the group G(F, F ′ ) is an irreducible cocompact lattice in C2 ≀Vd Aut(Td ) = (⊕Vd C2 ) ⋊ Aut(Td ). Applications. Recall that the property of being virtually simple is not invariant by quasi-isometry. Indeed the lattices constructed by Burger and Mozes in [BM00b] show that a virtually simple finitely generated group may have the same Cayley graph as a product of two finitely generated free groups. Theorem 1.4 together with simplicity results from [LB16] provide another illustration of this fact, namely finitely generated simple groups having the same Cayley graph as a wreath product. The wreath product construction is already known to be a source of examples of finitely generated groups whose algebraic properties are not reflected in their Cayley graphs. Two wreath products B1 ≀ Γ and B2 ≀ Γ may have isometric or bi-Lipschitz Cayley-graphs, one being solvable or torsion free, while no finite index subgroup of the second has these properties [Dyu00]. The phenomenon exhibited in Theorem 1.4 is nonetheless very different, in the sense that it provides finitely generated groups with isometric Cayley graphs such that one is a wreath product, but the other is simple (and hence not commensurable with a wreath product). Recall that for finitely generated groups, being amenable is a quasi-isometry invariant. By contrast, Theorem 1.4 implies: Corollary 1.5. Among finitely generated groups, the property of having infinite amenable radical is not invariant by quasi-isometry. The examples from Theorem 1.4 simultaneously show that having an infinite elliptic radical is also not invariant by quasi-isometry. Recall that the elliptic radical of a discrete group is the largest locally finite normal subgroup. Recall that by a theorem of Eskin–Fisher–Whyte, any finitely generated group Γ that is quasi-isometric to a wreath product C ≀Z, where C is a finite group, must act properly and cocompactly on a Diestel-Leader graph DL(m, m) [EFW07, EFW12, EFW13]. By the algebraic description of the isometry groups of these graphs given in [BNW08] (see also [CFK12]), this implies in particular that Γ has a subgroup AMENABLE URS’S AND LATTICE EMBEDDINGS 9 of index at most two that is (locally finite)-by-Z. By contrast, Theorem 1.4 shows that this rigidity fails in the case of C ≀ Fk when k ≥ 2. Questions. We end this introduction with two questions. Extreme proximality is used in a crucial way at different stages of the proofs of Theorems 1.1 and 1.2. These results both fail without the extreme proximality assumption, simply because then the group itself may very well be a direct product. Putting aside these trivial counter-examples, we do not know whether serious algebraic restrictions on a locally compact group may be derived from the existence of a lattice with a continuous Furstenberg URS. In this direction, we find the following question natural: Question 1.6. Does there exist Γ with a continuous Furstenberg URS which is a lattice in a group G = G1 × G2 such both factors are non-discrete, and Γ has an injective and dense projection to each factor ? What if we impose moreover that Γ has trivial amenable radical ? Theorem 1.4 presents a situation of a locally compact group G with two cocompact lattices Γ1 , Γ2 ≤ G such that the stabilizer URS associated to the Γ1 -action on ∂sp G is {Rad(Γ1 )}, while the stabilizer URS associated to the Γ2 -action on ∂sp G is continuous. Here ∂sp G stands for the Furstenberg boundary of G; see §2.2. In these examples the group G splits as G = N ⋊ Q, where N is the amenable radical of G. The lattice Γ1 preserves this splitting, meaning that we have Γ1 = (N ∩Γ1 )⋊(Q∩Γ1 ) (and hence Γ1 does not act faithfully on ∂sp G), while Γ2 has an injective projection to Q. This naturally raises the following: Question 1.7. Let G be a locally compact group with two lattices Γ1 and Γ2 both acting faithfully on X = ∂sp G. Is it possible that the Γ1 -action on X is topologically free, but the Γ2 -action on X is not topologically free ? Can this happen with ∂sp G = G/H being a homogeneous G-space ? Note that by [Fur03, Prop. 7], the condition that Γ1 and Γ2 act faithfully on ∂sp G is equivalent to saying that Γ1 and Γ2 have trivial amenable radical. Recall that topologically free means that there is a dense subset of points having trivial stabilizer (equivalently, the stabilizer URS is trivial). Outline of proofs and organization. The article is organized as follows. In the next section we introduce terminology and preliminary results about topological boundaries and extremely proximal actions. In Section 3 we establish the results about uniformly recurrent subgroups that are used in later sections. In particular we prove a certain gap property for URS’s coming from extremely proximal actions (Proposition 3.17). Combined with an observation about compact spaces with comparable stabilizer URS’s (Proposition 3.9), we deduce that a locally compact group H with an amenable URS that comes from an extremely proximal action, and whose envelope is co-amenable in H, is boundary indivisible (Proposition 3.24). The setting of Section 4 is that of a group admitting a non-topologically free extremely proximal action. We establish intermediate results, notably concerning normal subgroups (Proposition 4.6) and commensurated subgroups (Proposition 4.12), and deduce non-embedding results for this class of groups (see Proposition 4.10 and Corollary 4.13). In Section 5 we use results from Section 3 together with Proposition 5.9 of Furstenberg and prove Theorem 1.2. We then specify to discrete groups and give 10 ADRIEN LE BOUDEC the proof of Theorem 1.1. The proof essentially splits in two steps: the first one is the application of Theorem 1.2 to obtain amenability of one factor, and the second consists in proving that under appropriate assumptions the amenable factor is compact, using results from Section 4. In Section 6 we consider groups acting on trees, and apply previous results of the article to this setting. After giving the proof of Corollary 1.3, we focus on the family of groups with prescribed local action G(F, F ′ ). We study boundaries of these groups, and use results from Section 3 in order to characterize the discrete groups within this family which are boundary indivisible (see Theorem 6.9). This includes those which are virtually simple, but this also contain non-virtually simple instances. Finally we study lattice embeddings of these groups and give the proof of Theorem 1.4. Acknowledgements. I am grateful to Alex Furman for pointing out Proposition 5.9 to my attention, and to Uri Bader for enlightening discussion about the proof. I am also grateful to Pierre-Emmanuel Caprace, Yves Cornulier, Bruno Duchesne, Nicolás Matte Bon, Nicolas Monod and Pierre Pansu for interesting discussions and comments related to this work. Finally I am indebted to Alain Valette for a decisive remark made in Neuchâtel in May 2015, which eventually led to Theorem 1.4. 2. Preliminaries 2.1. Conventions and terminology. The letter G will usually refer to a topological group, while Γ will denote a discrete group. The group of homeomorphic automorphisms of G will be denoted Aut(G). Whenever G is a locally compact group, we will always assume that G is second countable. The notation X will refer to a topological space. The letters X, Y will be reserved for compact spaces, and Z for a compact space equipped with an extremely proximal group action. All compact spaces are assumed to be Hausdorff. A space X is a G-space if G admits a continuous action G × X → X . The action of G on X (or the G-space X ) is minimal if all orbits are dense. The G-space X is said to be trivial if X is a one-point space. If X is locally compact, we denote by Prob(X ) the set of all regular Borel probability measures on X . The space of continuous compactly supported functions on X is denoted CK (X ). Each µ ∈ Prob(X ) defines a linear functional on CK (X ), and we endow Prob(X ) with the weak*-topology: a net (µi ) converges to µ if µi (f ) → µ(f ) for all f ∈ CK (X ). By Banach-Alaoglu theorem, Prob(X ) is relatively compact in CK (X )∗ . We denote by 2X the set of all closed subsets of X . The sets n o O(K; U1 , . . . , Un ) = C ∈ 2X : C ∩ K = ∅; C ∩ Ui 6= ∅ for all i , where K ⊂ X is compact and U1 , . . . , Un ⊂ X are open, form a basis for the Chabauty topology on 2X . Endowed with the Chabauty topology, the space 2X is compact. We will freely identify X with its image in 2X by the natural inclusion x 7→ {x}. Note that when X is a G-space, so is 2X . In the particular case where X = G is a locally compact group, the space Sub(G) of closed subgroups of G is closed in 2G . In particular Sub(G) is a compact space, on which G acts by conjugation. A uniformly recurrent subgroup (URS) of AMENABLE URS’S AND LATTICE EMBEDDINGS 11 G is a closed, G-invariant, minimal subset of Sub(G). The set of URS’s of G is denoted URS(G). By extension we also say that a subgroup H ≤ G is uniformly recurrent if the closure of the conjugacy class of H in Sub(G) is minimal. 2.2. Topological boundaries. If X is a compact G-space, the action of G on X is strongly proximal if the closure of any G-orbit in Prob(X) contains a Dirac measure. Strong proximality is stable under taking products (with diagonal action) and continuous equivariant images (see e.g. [Gla76]). We say that X is a boundary if X is both minimal and strongly proximal. For every topological group G, there exists a unique boundary ∂sp G with the universal property that for any boundary X, there exists a continuous G-equivariant surjection ∂sp G → X [Fur73, Prop. 4.6]. This universal space ∂sp G is referred to as the Furstenberg boundary of G. It is easy to verify that any amenable normal subgroup N of G acts trivially on any G-boundary, so that ∂sp G = ∂sp (G/N ). If G admits a cocompact amenable subgroup, then the Furstenberg boundary is a homogeneous space ∂sp G = G/H, and the G-spaces of the form G/L with L containing H are precisely the G-boundaries [Fur73, Prop. 4.4]. The situation for discrete groups is quite different: as shown in [KK17] and [BKKO14], Furstenberg boundaries of discrete groups are always non-metrizable (unless trivial). The following is a fundamental property of boundaries (see [Gla76, III.2.3]): Theorem 2.1. Any convex compact G-space contains a boundary. In fact if Q is an irreducible convex compact G-space, then the action of G on Q is strongly proximal, and the closure of extreme points of Q is a G-boundary. Irreducible means that Q has no proper closed convex G-invariant subspace. In particular Theorem 2.1 has the following consequence ([Gla76, III.3.1]): Theorem 2.2. A group G is amenable if and only if all G-boundaries are trivial, or equivalently ∂sp G is trivial. 2.3. Extremely proximal actions. Let X be a compact G-space. A closed subset C of X is compressible if the closure of the G-orbit of C in the space 2X contains a singleton {x}. Equivalently, for every neighbourhood U of x, there exists g ∈ G such that g(C) ⊂ U . The action of G on X is extremely proximal if every closed subset C ( X is compressible. References where extremely proximal actions were considered include [Gla74, LS96, JR00, FST15, LBMB16]. We will make use of the following result, which is Theorem 2.3 from [Gla74]: Theorem 2.3. Let X be a compact G-space, and assume X has at least three points. If the G-action on X is extremely proximal, then it is strongly proximal. Examples of extremely proximal actions are provided by group actions on trees or hyperbolic spaces. If G ≤ Aut(T ) acts on T with no proper invariant subtree and no finite orbit in T ∪ ∂T , then the action of G on ∂T is minimal and extremely proximal; and if G acts coboundedly on a proper geodesic hyperbolic space X with no fixed point or fixed pair at infinity, then the G-action on the Gromov boundary ∂X is minimal and extremely proximal. These two situations are particular cases of the following more general result, that we believe is well-known. A homeomorphism g of a space X is hyperbolic if there exist ξ− , ξ+ ∈ X, called the endpoints of g, such that for all neighbourhoods 12 ADRIEN LE BOUDEC U− , U+ of ξ− , ξ+ , for n large enough we have gn (X \ U− ) ⊂ U+ and g−n (X \ U+ ) ⊂ U− . Proposition 2.4. If G acts on a compact space X with hyperbolic elements having no common endpoints, and such that the set of endpoints of hyperbolic elements of G is dense in X, then the action is minimal and extremely proximal. Proof. Let U ⊂ X be a non-empty open invariant subset. By our density assumption, there is g ∈ G hyperbolic whose attracting endpoint ξ+ belongs to U . So for every x 6= ξ− , there is n > 0 such that gn (x) ∈ U since U is open, so we deduce that U contains X \ {ξ− }. But the existence of hyperbolic elements with no common endpoints ensures that G fixes no point of X, so finally U = X, i.e. the action is minimal. Now if C ( X is a closed subset then again there is g ∈ G whose attracting endpoint is outside C, and C is compressible to the repealing endpoint of g. Recent work of Duchesne and Monod shows that group actions on dendrites is also a source of extremely proximal actions. Recall that a dendrite X is a compact metrizable space such that any two points are the extremities of a unique arc. Duchesne and Monod show that if Γ acts on X with no invariant proper subdendrite, then there is a unique minimal closed invariant subset M ⊆ X and the Γ-action on M is extremely proximal. See the proof of Theorem 10.1 in [DM16]. Extremely proximal actions also play a prominent role in the context of group actions on the circle. For any minimal action α : Γ → Homeo+ (S1 ), either α(Γ) is conjugated to a group of rotations, or α(Γ) has a finite centralizer CΓ in Homeo+ (S1 ) and the action of Γ on the quotient circle CΓ \S1 is extremely proximal: see Ghys [Ghy01] and Margulis [Mar00]. We mention however that in all the examples of countable groups Γ with an action on S1 that is minimal and not topologically free that we are aware of, the stabilizer URS is either non-amenable, or not known to be amenable. In particular we do not know any application of Theorem 1.1 to groups acting on the circle. In the sequel we will make use of the following easy lemma. Lemma 2.5. Let G be a topological group, and H a subgroup of G such that there is some compact subset K of G such that G = KH. Let X be a compact G-space, and C a closed subset of X that is compressible by G. Then C is compressible by H. In particular if the G-action on X is extremely proximal, then the H-action is extremely proximal. Proof. By assumption there exists x ∈ X and (gi ) such that gi (C) converges to x in 2X . If gi = ki hi , by compactness of K we assume that (ki ) converges to some k, and it follows that hi (C) converges to k−1 x by continuity of G × 2X → 2X . 3. Uniformly recurrent subgroups 3.1. Generalities on uniformly recurrent subgroups. Let G be a locally compact group. For H, K ∈ URS(G), we write H 4 K when there exist H ∈ H and K ∈ K such that H ≤ K. This is equivalent to the fact that every H ∈ H is contained in an element of K, and every K ∈ K contains an element of H, and the relation 4 is an order on URS(G). See e.g. §2.4 in [LBMB16]. AMENABLE URS’S AND LATTICE EMBEDDINGS 13 For simplicity the URS {N } associated to a closed normal subgroup N of G will still be denoted N . In particular N 4 H (resp. H 4 N ) means that N is contained in (resp. contains) all the elements of H. By the trivial URS we mean the URS corresponding to the trivial subgroup {1}. We warn the reader that in this terminology the URS corresponding to a non-trivial normal subgroup N is trivial as a G-space (it is a one-point space), but is not trivial as a URS. Let X, Y be compact G-spaces. We say that X is a factor of Y , and Y is an extension of X, if there exists a continuous equivariant map Y → X that is onto. If π : Y → X is a continuous equivariant map, we say that π is almost 1-1 if the set of y ∈ Y such that π −1 (π(y)) = {y} is dense in Y . When moreover π is onto we say that Y is an almost 1-1 extension of X. We now recall the definition of the stabilizer URS associated to a minimal action on a compact space. If X is a compact G-space and x ∈ X, we denote by Gx the stabilizer of x in G. Definition 3.1. If X is a compact G-space, we denote by X0 ⊂ X the set of points at which Stab : X → Sub(G), x 7→ Gx , is continuous. Upper semi-continuity of the map Stab and second countability of G imply that X0 is a dense subset of X (indeed if (Un ) is a basis of the topology on Sub(G) and Xn is the set of x ∈ X such that Gx ∩ Un 6= ∅, which is closed, one verifies that Stab is continuous on ∩n (∂Xn )c ). Following [GW15], we denote X̃ = cls {(x, Gx ) : x ∈ X0 } ⊂ X × Sub(G), and SG (X) = cls {Gx : x ∈ X0 } ⊂ Sub(G), where cls stands for the closure in the ambient space. We have the obvious inclusions X̃ ⊆ X ∗ := cls {(x, Gx ) : x ∈ X} and ∗ SG (X) ⊆ SG (X) := cls {Gx : x ∈ X} . We denote by ηX and πX the projections from X × Sub(G) to X and Sub(G) respectively. Proposition 3.2 (Prop. 1.2 in [GW15]). If X is a minimal compact G-space, then ηX : X̃ → X is an almost 1-1 extension, and X̃ and SG (X) are the unique minimal ∗ (X). closed G-invariant subsets of respectively X ∗ and SG Definition 3.3. SG (X) is the stabilizer URS associated to the G-action on X. The action of G on X is topologically free if SG (X) is trivial, i.e. SG (X) = {{1}}. Remark 3.4. When G is not assumed second countable, in general X0 is no longer dense in X. However it is still possible to define the stabilizer URS associated to a minimal action on a compact space; see the discussion in [MBT17, p1]. In the sequel we will sometimes use the following version of Proposition 3.2. Proposition 3.5. Let X be compact G-space, and let H ≤ G be a subgroup acting minimally on X. Then H acts minimally on X̃ and SG (X), and X̃ and SG (X) ∗ (X). are the unique minimal closed H-invariant subsets of X ∗ and SG 14 ADRIEN LE BOUDEC Proof. Let Y ⊆ X ∗ closed and H-invariant. Since X is a factor of X ∗ and H acts minimally on X, for every x ∈ X0 there exists L ∈ Sub(G) such that (x, L) ∈ Y . But for x ∈ X0 , the fact that (x, L) belongs to X ∗ forces L to be equal to Gx by definition of X0 , and it follows that X̃ ⊆ Y . Moreover H acts minimally on X̃ since ηX : X̃ → X is an almost 1-1 extension and minimality is preserved by taking almost 1-1 extensions (if π : X1 → X2 is almost 1-1 and if C ⊆ X1 is a closed subset such that π(C) = X2 , then C = X1 ). So the statements for X̃ and ∗ (X) since these are factors X ∗ are established, and the same hold for SG (X) and SG of X̃ and X ∗ . 3.2. Envelopes. Let G be a locally compact group and H ∈ URS(G). Definition 3.6. The envelope Env(H) of H is the closed subgroup of G generated by all the subgroups H ∈ H. By definition Env(H) is the smallest closed subgroup of G such that H ⊂ Sub(Env(H)). Note that Env(H) is a normal subgroup of G, and is actually the smallest normal subgroup such that H 4 Env(H). Let Γ be a discrete group, X a compact Γ-space and X0 the domain of continuity of the map Stab. It is a classical fact that X0 consists of those x ∈ X such that for every γ ∈ Γx , there exists U neighbourhood of x that is fixed by γ (see e.g. [Vor12, Lem. 5.4] for a proof). For x ∈ X, we will denote by Γ0x ≤ Γx the set of elements fixing a neighbourhood of x, so that x ∈ X0 if and only if Γx = Γ0x . Lemma 3.7. Let Γ be a countable discrete group, X a compact minimal Γ-space, n ≥ 1 and γ1 , . . . , γn ∈ Γ. The following are equivalent: (i) (ii) (iii) (iv) ∩Fix(γi ) has non-empty interior; there is x ∈ X such that γi ∈ Γ0x for all i; same as in (ii) but with x ∈ X0 ; there is H ∈ SΓ (X) such that γi ∈ Hfor all i. In particular Env(SΓ (X)) is generated by the elements γ ∈ Γ such that Fix(γ) has non-empty interior. Proof. It is clear that (i) and (ii) are equivalent. Also (iii) clearly implies (ii), and (ii) also implies (iii) by density of X0 in X. Finally (iii) implies (iv) since Γ0x ∈ SΓ (X) for x ∈ X0 , and (iv) implies (iii) by density of the set of Γ0x , x ∈ X0 , in SΓ (X). 3.3. G-spaces with comparable stabilizer URS’s. We recall the notion of disjointness from [Fur67a]. Two compact G-spaces X, Y are disjoint if whenever X, Y are factors of a compact G-space Ω via ϕX : Ω → X and ϕY : Ω → Y , then the map (ϕX , ϕY ) : Ω → X × Y is surjective. When X, Y are minimal G-spaces, this is equivalent to saying that the product X × Y remains minimal [Fur67a, Lem. II.1]. The following lemma presents a situation which easily implies disjointness: Lemma 3.8. Let X, Y be minimal compact G-spaces such that there exists y0 ∈ Y such that Gy0 acts minimally on X. Then X and Y are disjoint. AMENABLE URS’S AND LATTICE EMBEDDINGS 15 Proof. This is clear: if W is a closed invariant subset of X × Y , then by minimality of Y there exists x0 ∈ X such that (x0 , y0 ) ∈ W . Since Gy0 acts minimally on X we deduce that W contains X × {y0 }, and by minimality of Y it follows that W is equal to X × Y . The following proposition will be used notably in Proposition 3.24. Proposition 3.9. Let X, Y be compact minimal G-spaces, and write H = SG (X) and K = SG (Y ). Suppose that H 4 K. Then X and Y can be disjoint only if Env(H) 4 K. In particular if SG (X) = SG (Y ) and this URS is not a point, then X and Y are not disjoint. Proof. Using notation from Proposition 3.2, we have almost 1-1 extensions ηX : X̃ → X and ηY : Ỹ → Y , and we write η = ηX × ηY : X̃ × Ỹ → X × Y , and π = πX × πY : X̃ × Ỹ → H × K. The set W ⊆ H × K of pairs (H, K) such that H ≤ K is non-empty by assumption, is clearly G-invariant, and is easily seen to be closed. If W is a proper subset of H × K then π −1 (W ) is a proper subset of X̃ × Ỹ since π is a factor, and it follows that η π −1 (W ) is a closed G-invariant subset of X × Y that is proper since η is almost 1-1. This contradicts disjointness of X and Y . Therefore we have W = H × K. This means that for a fixed K ∈ K we have H ≤ K for every H ∈ H, and hence Env(H) 4 K. 3.4. Action of a URS on a G-space. In this paragraph G still denote a locally compact group, and X is a compact G-space. Given H ∈ URS(G), we study the properties of the action of elements on H on the space X. The proof of the following lemma is an easy verification, and we leave it to the reader. Lemma 3.10. If X is a compact G-space, {H ∈ Sub(G) \| H fixes a point in X} is a closed G-invariant subset of Sub(G). In particular the following definition makes sense. Definition 3.11. Let X be a compact G-space, and H ∈ URS(G). We say that H fixes a point in X if for some (all) H ∈ H, there is x ∈ X such that h(x) = x for all h ∈ H. Lemma 3.12. Let X be a compact G-space, Y ⊆ X a closed invariant subset of X, and H ∈ URS(G). If there exists H ∈ H fixing x ∈ X such that Gx ∩ Y 6= ∅, then H fixes a point in Y . Proof. By assumption there exist (gi ) and y ∈ Y such that gi (x) converges to y. If K ∈ H is a limit point of (H gi ) (which exists by compactness), then K fixes y by upper semi-continuity of the stabilizer map. Lemma 3.12 implies the following: Lemma 3.13. If X is a compact G-space containing a unique minimal closed Ginvariant subset Xmin ⊂ X (e.g. X is proximal), and if H ∈ URS(G) fixes a point in X, then H fixes a point in Xmin . Proposition 3.14. Let Z be a compact G-space that is extremely proximal, and H ∈ URS(G). Then either H fixes a point in Z, or all H ∈ H act minimally on Z. 16 ADRIEN LE BOUDEC Proof. If there exist H ∈ H and a non-empty closed subset C ( Z that is invariant by H, then we may apply Lemma 3.12 to the space X = 2Z , the subspace Y = Z and the point x = C, and we deduce that H fixes a point in Z. ∗ (X) stands for the closure in Sub(G) Recall that given a compact G-space X, SG of the set of subgroups Gx , x ∈ X. Lemma 3.15. Let X be a compact G-space. Assume that K ≤ G is a closed ∗ (X) subgroup of G which acts minimally on X and such that there exists H ∈ SG with H ≤ K. Then Env(SG (X)) ≤ K. ∗ (X) Proof. Since K acts minimally on X, the closure of the K-orbit of H in SG contains SG (X) according to Proposition 3.5. Since H ∈ Sub(K) and Sub(K) is a closed subset of Sub(G), we deduce that SG (X) ⊂ Sub(K), and in particular Env(SG (X)) ≤ K. Definition 3.16. Let H ∈ URS(G). We say that H comes from an extremely proximal action if there exists a compact G-space Z that is minimal and extremely proximal, and such that SG (Z) = H. It was shown in [LBMB16] that for a discrete group Γ with a non-trivial URS H coming from an extremely proximal action, any non-trivial K ∈ URS(Γ) must be “relatively large” with respect to H (see [LBMB16, Th. 3.10] for a precise statement). Appropriate assumptions on Γ and H further imply that H 4 K for every non-trivial K ∈ URS(Γ) [LBMB16, Cor. 3.12]. The following proposition goes in the opposite direction by considering URS’s larger than H. Proposition 3.17. Let H ∈ URS(G) that comes from an extremely proximal action. Let K ∈ URS(G) such that H 4 K and H = 6 K. Then Env(H) 4 K. Proof. Let Z be a compact G-space that is minimal and extremely proximal and such that SG (Z) = H. Fix K ∈ K, and assume that K does not act minimally on Z. According to Proposition 3.14 this implies that the URS K fixes a point in Z, i.e. K 4 H. Since moreover H, K satisfy H 4 K by assumption, we deduce that H = K, which is a contradiction. Therefore K acts minimally on Z. Since moreover there exists H ∈ H such that H ≤ K, we are in position to apply Lemma 3.15, from which the conclusion follows. It should be noted that Proposition 3.17 is false without the extreme proximality assumption, as in general there are plenty of URS’s between H and Env(H). Lemma 3.18. Let H ∈ URS(G) that comes from an extremely proximal action. Then Env(H) acts minimally on H. Proof. Let Z be a compact G-space that is minimal and extremely proximal and such that SG (Z) = H, and let N = Env(H). Without loss of generality we may assume that H is not a point, since otherwise there is nothing to prove. This ensures that N acts non-trivially on Z. By extreme proximality N must act minimally on Z (see Lemma 4.2), and therefore also on H by Proposition 3.5. Remark 3.19. The extreme proximality assumption cannot be removed in Lemma 3.18. Indeed it is not true in general that, given H ∈ URS(G), H remains a URS of Env(H). Indeed, as explained in [GW15], any minimal subshift on two letters AMENABLE URS’S AND LATTICE EMBEDDINGS 17 gives rise to a URS H of the lamplighter group G = C2 ≀Z, such that H is contained in the Chabauty space Sub(L) of the base group L = ⊕C2 . In particular Env(H) lies inside the abelian group L, and it follows that Env(H) acts trivially on H. Proposition 3.20. Let H ∈ URS(G) that comes from an extremely proximal action, and assume H is not a point. Then: (a) The action of G on H gives rise to the same URS, i.e. SG (H) = H. (b) If moreover H comes from a faithful extremely proximal action, then the action of G on H is faithful. Proof. Write K = SG (H). By definition we have H 4 K. Argue by contradiction and suppose H = 6 K. Then applying Proposition 3.17, we deduce that Env(H) acts trivially on H. But Env(H) also acts minimally on H by Lemma 3.18, so we deduce that H must be a point, a contradiction. This shows (a). For (b), arguing as in the proof of Lemma 3.18 we see that any non-trivial normal subgroup N of G acts minimally on H. Since H is not a point, we have in particular that N acts non-trivially on H. Remark 3.21. Proposition 3.20 implies that, as far as our interest lies inside the URS associated to a minimal and extremely proximal action (and not the space Z itself), there is no loss of generality in assuming that (G, Z) is a sub-system of (G, Sub(G)). See also Remark 3.27. 3.5. Amenable URS’s. Recall that we say that H ∈ URS(G) is amenable if every H ∈ H is amenable. The following lemma already appeared in [LBMB16, Prop. 2.21]. Lemma 3.22. If H ∈ URS(G) is amenable and X is a G-boundary, then H 4 SG (X). Proof. Since H is amenable, H must fix a point in the compact G-space Prob(X). Now X is the unique minimal G-invariant subspace of Prob(X) since X is a Gboundary, so by Lemma 3.13 we have that H fixes a point in X, i.e. H 4 SG (X). Proposition 3.23. Let X be a compact minimal G-space such that H = SG (X) is amenable, and let Y be a G-boundary such that X and Y are disjoint. Then Env(H) acts trivially on Y . In particular if Env(H) is co-amenable in G, a non-trivial G-boundary is never disjoint with X. Proof. The fact that Env(H) must act trivially on Y follows by applying Lemma 3.22 and Proposition 3.9. Since an amenable group has no non-trivial boundary, the second statement follows. Proposition 3.23 says that when G admits an amenable URS whose envelope is co-amenable, a non-trivial G-boundary is never disjoint with X. This conclusion is not satisfactory for our concerns as it depends on the choice of a space X and not only on G. Although there is no hope to get a better conclusion in full generality, the next result, which will play an important role in Section 5, will remove this dependence under an extreme proximality assumption. We recall from the introduction that we say that G is boundary indivisible if two non-trivial G-boundaries are never disjoint. 18 ADRIEN LE BOUDEC Proposition 3.24. Assume that G admits an amenable H ∈ URS(G) that comes from an extremely proximal action, and let X be a non-trivial G-boundary. (a) Either SG (X) = H, or Env(H) acts trivially on X. (b) Assume that Env(H) is co-amenable in G. Then SG (X) = H, and G is boundary indivisible. Proof. (a). Since H is amenable, we have H 4 SG (X) by Lemma 3.22. Now if we assume H = 6 SG (X), then according to Proposition 3.17 we have Env(H) 4 SG (X), which exactly means that Env(H) acts trivially on X. (b). If SG (X) 6= H then the action of G on X factors through an action of G/Env(H) by (a). But by assumption the latter is amenable, so has no non-trivial boundaries. So it follows that X is trivial, a contradiction. Therefore all non-trivial G-boundaries have the same stabilizer URS H. Since moreover H cannot be a point (because otherwise G would be amenable), the fact that G is boundary indivisible follows from Proposition 3.9. For a countable group Γ, the Furstenberg URS of Γ is the stabilizer URS associated to the action of Γ on its Furstenberg boundary. We refer to [LBMB16] for the proof of the following properties. Proposition 3.25. Let Γ be a countable group, and AΓ its Furstenberg URS. Then the following hold: (a) AΓ is amenable, and H 4 AΓ for every amenable H ∈ URS(Γ). (b) If X is a Γ-boundary, then AΓ 4 SΓ (X). If moreover there is x ∈ X such that Γx is amenable, then AΓ = SΓ (X). (c) AΓ is invariant under Aut(Γ). Proposition 3.26. Let Γ be a countable group, and let Λ = Env(AΓ ) be the envelope of the Furstenberg URS of Γ. Then Λ acts minimally on AΓ , and AΓ = AΛ . Proof. The conjugation action of Γ on the normal subgroup Λ = Env(AΓ ) induces a map Γ → Aut(Λ). Since AΛ is invariant under Aut(Λ) by Proposition 3.25, it is in particular Γ-invariant. Moreover the action of Γ on AΛ is clearly minimal since it is already the case for Λ. Therefore AΛ is an amenable URS of Γ, so it follows that AΛ 4 AΓ since AΓ is larger than any amenable URS of Γ. On the other hand AΓ is a closed and Λ-invariant subset of Sub(Λ) consisting of amenable subgroups, so by the domination property applied to AΛ we must have AΓ 4 AΛ . Equality follows. Remark 3.27. When Env(AΓ ) is co-amenable in Γ, the fact that AΓ comes from a faithful and extremely proximal action is equivalent to saying that the Γ-action on AΓ is faithful and extremely proximal. The direct implication is consequence of Proposition 3.20, and the converse follows from Proposition 3.24. This gives us an intrinsic reformulation of the assumption of Theorem 1.1 inside the Chabauty space of Γ. 4. Extremely proximal actions If X is a Hausdorff Γ-space and U ⊂ X , we denote by ΓU the set of elements of Γ acting trivially on X \ U . We say that the action of Γ on X is micro-supported if ΓU is non-trivial for every non-empty open set U . We will need the following easy lemma. AMENABLE URS’S AND LATTICE EMBEDDINGS 19 Lemma 4.1. Assume that the action of Γ on X is micro-supported, and let U be a non-empty open set. Then ΓU is not solvable. Proof. Assume that Λ is a subgroup of ΓU whose action on U is micro-supported, and let V be a non-empty open subset of U . By assumption there exists a nontrivial λ1 ∈ ΛV , so that we may find an open set W ⊂ V such that W and λ1 (W ) are disjoint. For λ2 ∈ ΛW , the commutator [λ1 , λ2 ] coincides with λ−1 2 on W , and is therefore non-trivial provided that λ2 is non-trivial. It follows by induction that if ΓU,n is the n-th term of the derived series of ΓU , then the action of ΓU,n on U is micro-supported. In particular ΓU,n is never trivial, and ΓU is not solvable. In this section we will consider the following setting: (EP) Γ is a discrete group, Z is a compact Γ-space, and the action of Γ on Z is faithful, minimal and extremely proximal. In order to avoid trivialities, we assume that Z has at least three points. Unless specified otherwise, in the remaining of this section Γ and Z will be assumed to satisfy (EP). Our goal is to derive various properties on the group Γ that will be used in later sections. Lemma 4.2. Let N ≤ Homeo(Z) be a non-trivial subgroup that is normalized by Γ. Then N acts minimally and does not fix any probability measure on Z. Proof. Assume there exists C ( Z that is closed and N -invariant. Since C is compressible and N is normalized by Γ, wee see that N has a fixed point in Z. Now the set of N -fixed points is Γ-invariant, so it has to be the entire Z by minimality, and N is trivial. The same argument shows the absence of N -invariant probability measure on Z, since an extremely proximal action is also strongly proximal by Theorem 2.3. In all this section the terminology topologically free (see Definition 3.3) has to be understood with Γ viewed as a discrete group. Therefore that the action is not topologically free means that there exists γ 6= 1 which acts trivially on a non-empty open subset of Z. Lemma 4.3. If the action of Γ on Z is not topologically free, then it is microsupported. Proof. Let U be a non-empty open subset of Z. Let γ be a non-trivial element such that there is a non-empty open set V on which γ acts trivially, and let g ∈ Γ such that g(X \ V ) ⊂ U . Then the non-trivial element gγg−1 acts trivially outside U , so ΓU is non-trivial. Definition 4.4. Let Γ0 be the subgroup of Γ generated by the elements γ ∈ Γ such that Fix(γ) has non-empty interior. Remark 4.5. When Γ is a countable group, Γ0 is also equal to the envelope of the URS SΓ (Z) by Lemma 3.7. Recall that the monolith Mon(Γ) is the intersection of all non-trivial normal subgroups of Γ. We say Γ is monolithic if Mon(Γ) is non-trivial. Proposition 4.6. Assume that the action of Γ on Z is not topologically free. Then the following hold: 20 ADRIEN LE BOUDEC (a) The commutators [γ1 , γ2 ], where Fix(γ1 ) ∩ Fix(γ2 ) has non-empty interior, generate [Γ0 , Γ0 ]. (b) Γ is monolithic, and one has Mon(Γ) = [Γ0 , Γ0 ]. (c) Any non-trivial normal subgroup of Γ has trivial centralizer. (d) If the action of [Γ0 , Γ0 ] on Z is extremely proximal, then [Γ0 , Γ0 ] is a simple group. (e) Γ is virtually simple if and only if Γ0 has finite index in Γ and finite abelianization. Proof. (a) Denote by N the subgroup generated by the set of [γ1 , γ2 ], where γ1 , γ2 act trivially on a common open set. We show that for every g, h fixing non-empty open sets U, V , the commutator [g, h] belongs to N . Since Γ0 is generated by all these elements g, h, this will show that Γ0 /N is abelian. Hence [Γ0 , Γ0 ] ≤ N , and the other inclusion is clear. First note that N is not trivial by Lemmas 4.1 and 4.3. Therefore N acts minimally on Z according to Lemma 4.2, and we may find s ∈ N such that the open set W = U ∩ s(V ) is non-empty. Since g and hs fix W by construction, we have [g, hs ] ∈ N . But since s ∈ N , we deduce that [g, h] = [h, s]g [g, hs ][s, h] is a product of three elements of N , and hence belongs to N , as desired. (b) We shall show that any non-trivial normal subgroup N contains [Γ0 , Γ0 ]. Since [Γ0 , Γ0 ] is itself a non-trivial normal subgroup, this will prove that it is the monolith of Γ. By a classical commutator manipulation (see e.g. Lemma 4.1 from [Nek13]), there exists an open set U such that N contains the derived subgroup of ΓU . Now let γ1 , γ2 fixing an open set V . If γ is such that γ(Z \ V ) ⊂ U , then γ1γ , γ2γ are supported inside U , so that [γ1γ , γ2γ ] = [γ1 , γ2 ]γ is contained in N . Since N is normal, [γ1 , γ2 ] ∈ N . Now all these elements generate [Γ0 , Γ0 ] by (a), hence the conclusion. (c) If N is a normal subgroup of Γ, then so is CΓ (N ). Therefore they cannot be both non-trivial, because otherwise the intersection would be abelian and would contain [Γ0 , Γ0 ] by the previous paragraph, a contradiction. (d) If the action of [Γ0 , Γ0 ] on Z is extremely proximal, then according to (b) the monolith N of [Γ0 , Γ0 ] is non-trivial. Since N is characteristic in [Γ0 , Γ0 ], N is normal in Γ, and hence contains [Γ0 , Γ0 ] by (b). So N = [Γ0 , Γ0 ] and [Γ0 , Γ0 ] is simple. (e) For Γ to be virtually simple it is clearly necessary that the normal subgroup [Γ0 , Γ0 ] has finite index in Γ. Conversely, if this condition holds then the action of [Γ0 , Γ0 ] on Z is extremely proximal (Lemma 2.5), and [Γ0 , Γ0 ] is simple by (d). Definition 4.7. Let X be a topological space, and let Γ be a group acting on X . A non-empty open set Ω ⊂ X is wandering for Γ if the translates γ(Ω), γ ∈ Γ, are pairwise disjoint. We say that Ω is wandering for γ if it is wandering for hγi. Proposition 4.8. Let Γ and Z as in (EP). Then there exist an open set Ω and a non-abelian free subgroup F2 = ∆ ≤ Γ such that Ω is wandering for ∆. Proof. Following Glasner [Gla74], we consider pairwise disjoint non-empty open sets U− , U+ , V− , V+ , and elements a, b ∈ Γ such that a(Z\U− ) ⊂ U+ and b(Z\V− ) ⊂ V+ . Let W = U− ∪ U+ ∪ V− ∪ V+ . It follows from a ping-pong argument than any non-trivial reduced word in the letters a, b sends the complement of W inside W , so that the subgroup ∆ generated by a, b is free [Gla74, Th. 3.4]. AMENABLE URS’S AND LATTICE EMBEDDINGS 21 Upon reducing W if necessary, we may find an open set Ω such that Ω ∩ W = ∅ and a(Ω) ⊂ U+ , a−1 (Ω) ⊂ U− , b(Ω) ⊂ V+ and b−1 (Ω) ⊂ V− . Induction on the word length shows that if the left-most letter of γ ∈ ∆ is respectively a, a−1 , b, b−1 , then γ(Ω) lies respectively inside U+ , U− , V+ , V− . In particular Ω ∩ γ(Ω) is empty since Ω is disjoint from W , so Ω is wandering for ∆. Proposition 4.9. Retain notations (EP). Then the wreath product ΓU ≀ F2 embeds into Γ for every open subset U ( Z that is not dense. Proof. Let Ω and ∆ as in Proposition 4.8, and let Λ be the subgroup of Γ generated by ΓΩ and ∆. Since Ω is wandering for ∆, all the conjugates γΓΩ γ −1 pairwise commute, and it follows that Λ is isomorphic to ΓΩ ≀ F2 . Now if U is an in the statement, by extreme proximality the group ΓU is isomorphic to a subgroup of ΓΩ , hence the conclusion. The argument in the following proof is borrowed from [Cap16]. Proposition 4.10. In the setting (EP), if the action of Γ on Z is not topologically free, then Γ cannot have any faithful linear representation. Proof. Let U be an open subset of Z. By Lemmas 4.1 and 4.3, we may find a non-abelian finitely generated subgroup B inside ΓU . Now if we choose U small enough, it follows from Proposition 4.9 that the finitely generated group Λ = B ≀ F2 is isomorphic to a subgroup of Γ. Since Λ is not residually finite [Gru57], it admits no faithful linear representation by Malcev’s theorem [Mal40], and a fortiori the same is true for Γ. Recall that a subgroup Λ of a group Γ is commensurated if all conjugates of Λ are commensurable, where two subgroups are commensurable if the intersection has finite index in both. The beginning of the argument in the proof of the following proposition already appeared in [LBW16]. The idea is to extend classical techniques for normal subgroups to certain commensurated subgroups. Proposition 4.11. Retain notation from (EP), and assume that the action of Γ on Z is not topologically free. If Λ is a commensurated subgroup of Γ such that there exists an element of Λ admitting a wandering open set, then Λ contains the monolith Mon(Γ). Proof. Let λ ∈ Λ admitting a wandering open set Ω. We shall first prove that [ΓΩ , ΓΩ ] is contained in Λ. Let g, h ∈ ΓΩ , and let also n ≥ 1. Since Ω is wandering we have Ω ∩ λn (Ω) = ∅. It follows that the commutator [g, λn ] is trivial outside Ω ∪ λn (Ω), and coincides with g on Ω and with λn g−1 λ−n on λn (Ω). Therefore its commutator with h is trivial outside Ω, and is coincides with [g, h] on Ω. But since g, h ∈ ΓΩ , the elements [[g, λn ], h] and [g, h] actually coincide everywhere, i.e. [[g, λn ], h] = [g, h]. Now since Λ is commensurated in Γ, there exists n0 ≥ 1 such that [[g, λn0 ], h] belongs to Λ. Applying the previous argument with n = n0 , we deduce that [g, h] belongs to Λ. In order to prove the statement, it is enough to prove that [ΓC , ΓC ] is contained in Λ for every closed subset C ( Z according to Proposition 4.6. So let C a proper closed subset of Z. By minimality and extreme proximality, there is γ ∈ Γ such 22 ADRIEN LE BOUDEC that γ(C) ⊂ Ω. Fix such a γ, and choose some integer n1 ≥ 1 such that γ −1 λn1 γ belongs to Λ. Set λ′ = γ −1 λn1 γ and Ω′ = γ −1 (Ω). This Ω′ is wandering for λ′ and λ′ ∈ Λ, so [ΓΩ′ , ΓΩ′ ] is contained in Λ by the first paragraph. Since C ⊂ Ω′ , we have ΓC ≤ ΓΩ′ , and the proof is complete. Proposition 4.12. Assume that Z is a compact Γ-space and the action of Γ on Z is faithful, minimal, extremely proximal and not topologically free. Then there exists a free subgroup F2 ≤ Γ such that for every commensurated subgroup Λ ≤ Γ not containing the monolith Mon(Γ), we have F2 ∩ Λ = 1. Proof. Let F2 be a free subgroup of Γ as in the conclusion of Proposition 4.8. If Λ ≤ Γ is a commensurated subgroup such that F2 ∩ Λ 6= 1, then in particular Λ contains an element admitting a wandering open set. So by Proposition 4.11 we have Mon(Γ) ≤ Λ. This shows that every commensurated subgroup not containing Mon(Γ) intersects F2 trivially. Corollary 4.13. Assume that Z is a compact Γ-space and the action of Γ on Z is faithful, minimal, extremely proximal and not topologically free. If G is a locally compact amenable group whose connected component G0 is a Lie group, then there exists no injective homomorphism Γ → G. Proof. Argue by contradiction and assume Γ embeds in G. Let U be an open subgroup of G containing G0 as a cocompact subgroup. Such a U is commensurated in G, so the subgroup Γ ∩ U is commensurated in Γ. If there exists U such that Γ ∩ U does not contain Mon(Γ), then according to Proposition 4.12 we may find a non-abelian free subgroup F2 ≤ Γ such that F2 ∩ U = 1. In particular G contains the non-amenable group F2 as a discrete subgroup, which contradicts amenability of G. Therefore Mon(Γ) is contained in U for every choice of U . Since compact open subgroups form a basis at 1 in G/G0 by van Dantzig’s theorem, it follows that Mon(Γ) actually lies inside G0 . Now since G0 is a connected Lie group, the group Aut(G0 ) is linear, so the map G → Aut(G0 ) induced by the conjugation action of G on G0 is not injective in restriction to Γ by Proposition 4.10. Therefore this map must vanish on Mon(Γ), which means that Mon(Γ) actually lies inside the center of G0 . In particular Mon(Γ) is abelian, which contradicts Proposition 4.6. 5. The proofs of Theorems 1.1 and 1.2 5.1. Boundary-minimal subgroups. In this paragraph we consider the following property: Definition 5.1. Let G be a topological group, and L a closed subgroup of G. We say that L is boundary-minimal if there exists a non-trivial G-boundary on which L acts minimally. It should be noted that being boundary-minimal does not prevent L from being amenable. For instance the action of Thompson’s group T on the circle S1 is a boundary action, and the abelian subgroup of T consisting of rotations acts minimally on S1 . Other examples may be found among the groups acting on trees considered in §6.2, where the stabilizer of a vertex is an amenable subgroup acting minimally on the ends of the tree. In the sequel we will mainly focus on the case when L is normal in G, or more generally when L belongs to a URS (see Proposition 5.8). By contrast with the AMENABLE URS’S AND LATTICE EMBEDDINGS 23 previous examples, a normal boundary-minimal subgroup is never amenable, as a normal amenable subgroup of G acts trivially on any G-boundary. Recall that Furman showed [Fur03, Prop. 7] (see also Caprace–Monod [CM14, Prop. 3]) that if N is non-amenable normal subgroup of a locally compact group G, there always exists a G-boundary on which N acts non-trivially. This naturally raises the question whether any non-amenable normal subgroup of G is boundary-minimal. We do not know the answer to this question. While the case of discrete groups is easily settled (see below), the situation for non-discrete groups seems to be more delicate. We recall the following result of Furstenberg (see [Gla76, II.4.3]): Theorem 5.2. Let G be a topological group, and denote by ϕ : G → Homeo(∂sp G) the action of G on ∂sp G. Then there exists a homomorphism ψ : Aut(G) → Homeo(∂sp G) such that ψ ◦ Inn = ϕ, where Inn(G) ≤ Aut(G) is the group of inner automorphisms of G. In particular when N is a normal subgroup of a group G, the map G → Aut(N ) coming from the conjugation action of G on N induces an action of G on ∂sp N , which factors through G/CG (N ). Note that this result readily answers the above question for discrete groups, by showing that the boundary-minimal normal subgroups of G are exactly the non-amenable normal subgroups of G. Indeed if N is non-amenable then ∂sp N is a non-trivial space, N acts minimally on ∂sp N , and ∂sp N is a G-boundary by Theorem 5.2. However the argument does not carry over for arbitrary groups, as in general the G-action on ∂sp N is not continuous. Proposition 5.3. Let G be a locally compact group, and N a closed normal nonamenable subgroup. Assume that at least one of the following holds true: (a) N · CG (N ) is open in G (e.g. when N is a direct factor of G); (b) N is cocompact in G; (c) there exists H ∈ URS(N ) that is not a point, invariant by Aut(N ), and that is an N -boundary (e.g. if N has a closed cocompact amenable subgroup). Then N is boundary-minimal in G. Proof. Condition (a) ensures that the image of N in G/CG (N ) is open. Therefore the G-action on ∂sp N given by Theorem 5.2 is continuous because the N -action is, and we deduce that ∂sp N is a non-tivial G-boundary. If (b) holds, then N acts minimally on any G-boundary [Gla76, II.3.2]. Finally the verification of case (c) is straightforward since G → Aut(N ) is continuous [HR79, Th. 26.7] and Aut(N ) acts continuously on Sub(N ). 5.2. Weakly co-amenable subgroups. In this paragraph we consider the following weakening of the notion of co-amenability. Definition 5.4. Let G be a topological group, and H a subgroup of G. We say that H is weakly co-amenable in G if whenever Q is a non-trivial convex compact G-space in which H fixes a point, Q is not irreducible. The following properties readily follow from the definition. Proposition 5.5. Let K ≤ H ≤ G be subgroups of G. (i) If H ≤ G is co-amenable then H is weakly co-amenable. 24 ADRIEN LE BOUDEC (ii) For a normal subgroup N ⊳ G, weakly co-amenable is equivalent to coamenable. (iii) If H ≤ G is amenable and weakly co-amenable in G, then G is amenable. (iv) If ϕ : G → G′ is continuous with dense image and H ≤ G is weakly coamenable, then ϕ(H) is weakly co-amenable in G′ . (v) If K is weakly co-amenable in G, then H is weakly co-amenable in G. (vi) If K is co-amenable in H and H is weakly co-amenable in G, then K is weakly co-amenable in G. Proof. (i) If Q is non-trivial convex compact G-space with H-fixed points, then there is a G-fixed point by co-amenability of H in G, so Q is not irreducible. (ii) If N ⊳ G is not co-amenable, there is a convex Q such that Fix(N ) is nonempty but Fix(G) is empty. Since N is normal Fix(N ) is G-invariant, so that by Zorn’s lemma Fix(N ) contains an irreducible convex G-space, which is non-trivial since Fix(G) is empty. This shows N is not weakly co-amenable. The proofs of (iii), (iv), (v) and (vi) are similar verifications, and we leave them to the reader. Remark 5.6. As for co-amenability, it is natural to wonder whether weak coamenability of K in G implies weak co-amenability of K in H. In view of (ii), the same counter-examples given in [MP03] show that the answer is negative in general. By the correspondence between irreducible convex compact G-spaces and Gboundaries, weak co-amenability admits the following characterization: Proposition 5.7. A subgroup H ≤ G is weakly co-amenable in G if and only if for every non-trivial G-boundary X, there is no probability measure on X that is fixed by H. Proof. Follows from Theorem 2.1. The following shows how weak co-amenability naturally appears for boundary indivisible groups (see also Proposition 6.17). Proposition 5.8. Let G be a boundary indivisible locally compact group, and L a closed subgroup of G that is boundary-minimal and uniformly recurrent. Then L is weakly co-amenable in G. Proof. Write H for the closure of LG in Sub(G), which is a URS by assumption. Let X be a non-trivial G-boundary on which L acts minimally, and let Y be a G-boundary on which L fixes a probability measure. We have to show that Y is trivial. Since H fixes a point in Prob(Y ) and the G-action on Prob(Y ) is strongly proximal by Theorem 2.1, H fixes a point in Y by Lemma 3.13. So there exists y ∈ Y such that L ≤ Gy , and it follows that Gy acts minimally on X. Therefore by Lemma 3.8 X and Y are disjoint, and since X is non-trivial and G is boundary indivisible, this is possible only if Y is trivial. 5.3. The proof of Theorem 1.2. In this paragraph we shall give the proof of Theorem 1.2 from the introduction. We will make use of the following result. Proposition 5.9 (Furstenberg). Let G be a locally compact group, H ≤ G a closed subgroup of finite covolume, and X a G-boundary. Then X is a H-boundary. AMENABLE URS’S AND LATTICE EMBEDDINGS 25 For completeness we repeat the argument from [Fur81, Prop. 4.4]. Proof. Write Q = Prob(X), and consider a closed H-invariant subspace Q′ ⊆ Q. We have to show that X ⊆ Q′ . The set X = (gH, µ) : µ ∈ g(Q′ ) ⊆ G/H × Q is a well-defined, closed, G-invariant subspace of G/H × Q. Fix a G-invariant probability measure mG/H on G/H, and consider n o Y = ν ∈ Prob(X ) : p∗G/H (ν) = mG/H , where pG/H is the projection from G/H ×Q onto the first factor, and p∗G/H is the induced push-forward operator. Then Y is a closed (and hence compact) G-invariant subspace of Prob(X ), and p∗Q : Y → Prob(Q) is continuous. So p∗Q (Y ) is closed in Prob(Q), and by strong proximality of the G-action on Q (Theorem 2.1), p∗Q (Y ) must intersect Q. Now X being the unique minimal closed G-invariant subspace of Q, one has X ⊆ p∗Q (Y ). For every x ∈ X, we therefore have νx ∈ Prob(X ) such that p∗G/H (νx ) = mG/H and p∗Q (νx ) = δx . This implies mG/H {gH : x ∈ g(Q′ )} = 1 for every x, and it easily follows that X ⊆ Q′ . Remark 5.10. In the case when H is cocompact in G, strong proximality of the action of H on X also follows from [Gla76, II.3.1] applied to the action on Prob(X); and minimality follows [Gla76, IV.5.1] from disjointness of the G-spaces G/H and X [Gla76, III.6.1]. Theorem 5.11. Assume that H admits an amenable URS H that comes from an extremely proximal action, and such that Env(H) is co-amenable in H. Let G be a locally compact group containing H as a closed subgroup of finite covolume. Then: (a) G is boundary indivisible. (b) More generally if L is a locally compact group such that there is a sequence a topological group homomorphisms G = G0 → G1 → . . . → Gn = L such that either Gi → Gi+1 has dense image, or Gi → Gi+1 is an embedding of Gi as a closed subgroup of finite covolume in Gi+1 ; then L is boundary indivisible. In particular whenever G maps continuously and with dense image to a product G1 × G2 , one factor Gi must be amenable. Proof. Since H is amenable, H comes from an extremely proximal action, and Env(H) is co-amenable in H, the group H is boundary indivisible by Proposition 3.24. Now by Proposition 5.9, the property of being boundary indivisible is inherited from closed subgroups of finite covolume. Indeed if X, Y are disjoint Gboundaries, i.e. X × Y is a G-boundary, then X × Y is also a boundary for H by Proposition 5.9, hence of X or Y must be trivial since H is boundary indivisible. This shows (a). Since boundary indivisibility passes to dense continuous images, and is inherited from closed subgroups of finite covolume, (b) follows from (a). Finally if G1 , G2 are as in the last statement and Xi = ∂sp Gi , then X1 × X2 is a boundary for G1 × G2 , which is boundary indivisible by the previous paragraph. So one factor Xi must be trivial, which exactly means that Gi is amenable by Theorem 2.2. 26 ADRIEN LE BOUDEC Remark 5.12. In the proof of Theorem 5.11 we obtain that G is boundary indivisible from the same property for H, which is itself deduced from Proposition 3.24 (which in turn relies notably on Proposition 3.9). We note that the order in which the argument is developed seems to matter, in the sense that the arguments applied to H do not seem to be applicable directly to the group G. Indeed we do not know whether a group G as in Theorem 5.11 falls into the setting of Proposition 3.9, i.e. we do not know whether all non-trivial G-boundaries have the same stabilizer URS. We actually believe this might be false in general. We note at this point that the proof of Theorem 1.2 from the introduction is now complete. Indeed the fact that a group G as in Theorem 1.2 is boundary indivisible, as well as statement (a), is Theorem 5.11; and statement (b) follows from Proposition 5.8. The following remark explains a comment from the introduction. Remark 5.13. Theorem 1.4 provides instances of countable groups Γ being lattices in a group G of the form G = N ⋊ Aut(Td ), such that all the assumptions of Theorem 1.2 are satisfied (see Section 6). If M is a cocompact subgroup of Aut(Td ) acting minimally on ∂Td , then L = N ⋊ M is a cocompact (hence uniformly recurrent) subgroup of G, and L is boundary-minimal in G since ∂Td is a G-boundary. However when M is non-unimodular, then M is not co-amenable in Aut(Td ), and L is not co-amenable in G. This shows that the conclusion of statement (b) in Theorem 1.2 that L is weakly co-amenable in G cannot be strengthen by saying that L is co-amenable in G. 5.4. The proof of Theorem 1.1. Recall that if G is a topological group, the quasi-center QZ(G) of G is the subgroup of G containing the elements g ∈ G having an open centralizer. Note that QZ(G) contains the elements having a discrete conjugacy class, so in particular it contains all discrete normal subgroups. Recall also that the elliptic radical of G is the largest normal subgroup of G in which every compact subset generates a relatively compact subgroup. It is a closed characteristic subgroup of G. We say that two groups G1 , G2 are commensurable up to compact kernels if there exist Ki ≤ Hi ≤ Gi such that Hi is open and of finite index in Gi , Ki is a compact normal subgroup of Hi , and H1 /K1 and H2 /K2 are isomorphic. The following is slightly more complete than Theorem 1.1 from the introduction. Theorem 5.14. Let Γ be a countable group whose Furstenberg URS AΓ comes from a faithful and extremely proximal action, and assume that Env(AΓ ) is co-amenable in Γ. Let G be a locally compact group containing Γ as a lattice, and H a group commenurable with G up to compact kernels. Consider the following properties: (a) Env(AΓ ) is finitely generated; (b) Env(AΓ ) has finite index in Γ, and Γ admits a finitely generated subgroup with finite centralizer; (c) Env(AΓ ) has finite index in Γ and Env(AΓ ) has finite abelianization. Then: • (a), (b) both imply that H cannot be a product of two non-compact groups. AMENABLE URS’S AND LATTICE EMBEDDINGS 27 • (c) implies that any continuous morphism with dense image from H to a product of locally compact groups H → G1 × G2 is such that one factor Gi is compact. Proof. For simplicity we give the proof for H = G. The general case follows the same lines. Of course we may assume that Env(AΓ ) is non-trivial, since otherwise there is nothing to prove. According to Proposition 4.6 we have in particular that Γ is monolithic, and Mon(Γ) = [Env(AΓ ), Env(AΓ )]. For simplicity in all the proof we write E = Env(AΓ ) and M = Mon(Γ) = [E, E]. Assume that ϕ : G → G1 × G2 is continuous with dense image, and denote by pi the projection G1 × G2 → Gi , i = 1, 2. We will show that one factor must be compact. Upon modding out by the maximal compact normal subgroup of the identity component G0 , which intersects Γ trivially since Γ has no non-trivial finite normal subgroup (Proposition 4.6), we may also assume that G0 has no non-trivial compact normal subgroup. This implies in particular that G0 is a connected Lie group [MZ55]. By the assumption that E is co-amenable in Γ, we can apply Theorem 5.11, which says that one factor, say G2 , must be amenable. We then apply Corollary 4.13, which tells us that the map p2 ◦ ϕ is not injective in restriction to Γ. By definition of M we deduce that M ≤ ϕ−1 (G1 × 1). Assume now that (c) holds. Then M , being of finite index in Γ, is a lattice in G, and is contained in the closed normal subgroup ϕ−1 (G1 × 1). Therefore we deduce that ϕ−1 (G1 × 1) is cocompact in G, and that p2 ◦ ϕ(G) is a compact subgroup of G2 . Since p2 ◦ ϕ(G) is also dense in G2 , we have that G2 is compact. We now have to deal with (a), (b), in which case ϕ is the identity and G = G1 × G2 . Without loss of generality, we may assume that the projections pi (Γ) are dense. The proofs of the two cases will share a common mechanism, given by the following easy fact: Lemma 5.15. If there exists a subgroup L ≤ G whose centralizer CG (L) contains G2 , CG (L) is open in G, and CG (L) ∩ Γ is finite, then G2 is compact. Indeed, since Γ must intersect an open subgroup O ≤ G along a lattice of O, it follows that CG (L) is compact, and a fortiori so is G2 . We start with case (a). Consider H1 = p1 (E), which is normal in G1 by density of p1 (Γ). Note that H1 is compactly generated in view of the assumption that E is finitely generated. Since M = [E, E], the group H1 /M is abelian, and therefore of the form Zn × Rm × C for some compact group C. It follows that the group Q1 = H1 /H10 admits a discrete cocompact normal subgroup ∆, which is an extension of M by a free abelian group. Being characteristically simple and non-amenable, the group M has trivial elliptic radical, so the group ∆ also has trivial elliptic radical. Now since Q1 is compactly generated, there is a compact open normal subgroup K of Q1 such that K ⋊ ∆ has finite index in Q1 (see e.g. [BCGM16, Lem. 4.4]), so we deduce that Q1 has a compact open elliptic radical. Since any connected group has compact elliptic radical [MZ55], we deduce that H1 has a compact elliptic radical R, and H1 /R is discrete-by-connected. The compact group R is also normal in G, and therefore we can mod out by R and assume that R is trivial, so that H10 is open in H1 . Since H10 centralizes M , any γ ∈ E such that p1 (γ) belongs to H10 28 ADRIEN LE BOUDEC centralizes M , and therefore is trivial by Proposition 4.6. Therefore H10 is open in H1 and intersects the dense subgroup p1 (E) trivially, so it follows that H10 is trivial, and p1 (E) is a discrete subgroup of G. Observe that p1 (E) is centralized by G2 and normalized by G1 , and hence is normal in G. Being a discrete normal subgroup of G, p1 (E) therefore lies in the quasi-center QZ(G). Since p1 (E) is finitely generated, the centralizer of p1 (E) in G is actually open in G. Moreover the subgroup Γ ∩ CG (p1 (E)) is normal in Γ since CG (p1 (E)) is normal in G, but clearly does not contain M , and hence is trivial by Proposition 4.6. Therefore we can apply Lemma 5.15 with L = p1 (E), and we obtain the conclusion. We now deal with (b). Let Z be a minimal compact Γ-space on which the Γ-action is faithful and extremely proximal and such that SΓ (Z) = AΓ . By Proposition 5.9 (actually an easy case of it) the action of E on Z is also minimal, and it is extremely proximal by Lemma 2.5. Moreover the associated stabilizer URS remains equal to AΓ , and is also the Furstenberg URS of E by Proposition 3.26. So E satisfies all the assumptions of case (b) of the theorem, so it is enough to prove the result under the additional assumption Γ = E. In this case we have M = [Γ, Γ] thanks to Proposition 4.6, so it follows that p2 (Γ) is abelian. By density of the projection the group G2 is also abelian, and hence G2 lies in the center of G. Therefore Γ is normalized by the dense subgroup ΓG2 , and it follows that Γ is normal in G. In particular Γ ≤ QZ(G), and the conclusion follows by applying Lemma 5.15 with L a f.g. subgroup of Γ such that CΓ (L) is finite. 6. Groups acting on trees 6.1. Amenable URS’s and groups acting on trees. In this paragraph T is a locally finite tree, and H acts continuously on T by isometries. The assumption that T is locally finite is not essential here, and the results admit appropriate generalizations for non-locally finite trees (using the compactification from [MS04, Prop. 4.2]). Recall that the H-action on T is minimal if there is no proper invariant subtree, and of general type if H has no finite orbit in T ∪ ∂T . The following is wellknown, and essentially goes back to Tits [Tit70] (see also [PV91] and Proposition 2.4 for details). Proposition 6.1. If the action of H ≤ Aut(T ) is minimal and of general type, then the action of H on ∂T is minimal and extremely proximal. Theorem 5.11 therefore implies the following result: Corollary 6.2. Let H ≤ Aut(T ) be a locally compact group whose action on T is continuous, minimal and of general type. Assume that end stabilizers are amenable, and the envelope of SH (∂T ) is co-amenable in H. Assume H embeds as a subgroup of finite covolume in G. Then whenever G maps continuously and with dense image to a product G1 × G2 , one factor Gi must be amenable. The conclusion of Corollary 6.2 implies in particular that whenever H embeds in G with finite covolume, then G cannot be a product of two non-amenable groups. The following example, which is largely inspired from [CM11, Ex. II.8], shows that the group G can nonetheless be a product of two non-compact groups. AMENABLE URS’S AND LATTICE EMBEDDINGS 29 Example 6.3. Let k = Fp ((t)) be the field of Laurent series over the finite field Fp , and let α ∈ Aut(k) be a non-trivial automorphism of k. The group L = SL(2, k) acts on a (p + 1)-regular tree, 2-transitively and with amenable stabilizers on the boundary. This action extends to a continuous action of H = L ⋊α Z, so that H satisfies all the assumptions of Corollary 6.2. Nevertheless H embeds diagonally in the product G = (L ⋊ Aut(k)) × Z as a closed subgroup of finite covolume since G/H is compact and H and G are unimodular. We will need the following fact. If A is a subtree of T , by the fixator of A we mean the subgroup fixing pointwise A. Proposition 6.4. Let Γ ≤ Aut(T ) be a countable group whose action on T is minimal and of general type, and such that end-stabilizers in Γ are amenable. Then AΓ = SΓ (∂T ), and Env(AΓ ) is the subgroup generated by fixators of half-trees. Proof. Since the Γ-action on ∂T is extremely proximal, it is also strongly proximal by Theorem 2.3. So ∂T is a Γ-boundary with amenable stabilizers, and we deduce that AΓ = SΓ (∂T ) by Proposition 3.25. Now according to Lemma 3.7, the subgroup Env(SΓ (∂T )) = Env(AΓ ) is generated by the elements γ ∈ Γ whose fixed point set in ∂T has non-empty interior. Since half-trees form a basis of the topology in ∂T , the statement follows. Before going to the proof of Corollary 1.3, we make the following observation: Remark 6.5. For Γ acting on T (action minimal and general type) such that the action on ∂T is not topologically free, virtual simplicity of Γ is equivalent to Γ0 being of finite index in Γ and Γ0 having finite abelianization, where Γ0 is the subgroup generated by fixators of half-trees. See statement (e) of Proposition 4.6. Proof of Corollary 1.3. In view of Proposition 6.4, the assumptions on Γ imply that the Furstenberg URS of Γ comes from a faithful extremely proximal action. The fact that end-stabilizers are all non-trivial means that the action of Γ on ∂T is not topologically free, and by the above observation virtual simplicity of Γ is equivalent to Env(AΓ ) being of finite index in Γ and with finite abelianization. The first statement of the corollary therefore follows from Theorem 5.14, case (c), and the second statement from Theorem 1.2. 6.2. Groups with prescribed local action. In the next paragraphs we will illustrate the results of the previous sections on a family of groups acting on trees, which contains instances of discrete and non-discrete groups. The purpose of this paragraph is to recall the definition and give a brief description of known properties of these groups. We will denote by Ω a set of cardinality d ≥ 3 and by Td a d-regular tree. The vertex set and edge set of Td will be denoted respectively Vd and Ed . We fix a coloring c : Ed → Ω such that neighbouring edges have different colors. For every g ∈ Aut(Td ) and every v ∈ Vd , the action of g on the star around v gives rise to a permutation of Ω, denoted σ(g, v), and called the local permutation of g at v. These permutations satisfy the identity (1) σ(gh, v) = σ(g, hv)σ(h, v) for every g, h ∈ Aut(Td ) and v ∈ Vd . 30 ADRIEN LE BOUDEC Given a permutation group F ≤ Sym(Ω), the group U (F ) introduced by Burger and Mozes in [BM00a] is the group of automorphisms g ∈ Aut(Td ) such that σ(g, v) ∈ F for all v. It is a closed cocompact subgroup of Aut(Td ). Definition 6.6. Given F ≤ F ′ ≤ Sym(Ω), we denote by G(F, F ′ ) the group of automorphisms g ∈ Aut(Td ) such that σ(g, v) ∈ F ′ for all v and σ(g, v) ∈ F for all but finitely many v. That G(F, F ′ ) is indeed a subgroup of Aut(Td ) follows from (1), and we note that we have U (F ) ≤ G(F, F ′ ) ≤ U (F ′ ). We make the following observation for future reference. Remark 6.7. As it follows from the definition, any element γ ∈ G(F, F ′ ) fixing an edge e can be (uniquely) written as γ = γ1 γ2 , where each γi belongs to G(F, F ′ ) and fixes one of the two half-trees defined by e. In the sequel we always assume that F ′ preserves the F -orbits in Ω (see [LB16, Lem. 3.3] for the relevance of this property in this context). These groups satisfy the following properties (see [LB16]): (1) The group G(F, F ′ ) is dense in the locally compact group U (F ′ ). In particular G(F, Sym(Ω)) is a dense subgroup of Aut(Td ). (2) G(F, F ′ ) admits a locally compact group topology (defined by requiring that the inclusion of U (F ) is continuous and open), and the action of G(F, F ′ ) on Td is continuous but not proper as soon as F 6= F ′ . Endowed with this topology, the group G(F, F ′ ) is compactly generated. (3) stabilizers of vertices and stabilizers of ends in G(F, F ′ ) are respectively locally elliptic and (locally elliptic)-by-cyclic. In particular they are amenable. (4) G(F, F ′ ) is a discrete group if and only if F acts freely on Ω. When this is so, the group G(F, F ′ ) is therefore a finitely generated group, and stabilizers of vertices and stabilizers of ends in G(F, F ′ ) are respectively locally finite and (locally finite)-by-cyclic. When F acts freely on Ω and F 6= F ′ , the groups G(F, F ′ ) are instances of groups obtained from a more general construction described in [LB17, Sec. 4] (more precisely, a variation of it), which provides discrete groups with a continuous Furstenberg URS (the later being the stabilizer URS associated to the action on the boundary of the tree on which these groups act). In the particular case of the groups G(F, F ′ ), the Furstenberg URS can be explicitly described, see Proposition 4.28 and Corollary 4.29 in [LBMB16]. In the sequel whenever we use letters F and F ′ , we will always mean that F, F ′ are permutation groups on a set Ω, that F ′ contains F and preserves the F -orbits in Ω. Following [Tit70], we will denote by G(F, F ′ )+ the subgroup of G(F, F ′ ) generated by fixators of edges, and by G(F, F ′ )∗ the subgroup of index two in G(F, F ′ ) preserving the bipartition of Td . The following result, also obtained in [CRW17, Prop. 9.16], supplements simplicity results obtained in [LB16], where the index of the simple subgroup was found explicitly under appropriate assumptions on the permutation groups. Proposition 6.8. The group G(F, F ′ ) has a simple subgroup of finite index if and only if F is transitive and F ′ is generated by its point stabilizers. AMENABLE URS’S AND LATTICE EMBEDDINGS 31 Proof. These conditions are necessary by [LB16, Prop. 4.7]. Conversely, assume F transitive and F ′ generated by its point stabilizers. By [LB16, Prop. 4.7] again, G(F, F ′ )+ has index two in G(F, F ′ ), so in particular it is compactly generated. If M is the monolith of G(F, F ′ ), which is simple and open in G(F, F ′ ) by [LB16, Cor. 4.9], we have to show M has finite index. According to Remark 6.7, G(F, F ′ )+ is also the subgroup generated by fixators of half-trees, and therefore by Proposition 4.6 M is the commutator subgroup of G(F, F ′ )+ . The abelianization of G(F, F ′ )+ is therefore a finitely generated abelian group, which is generated by torsion elements since G(F, F ′ )+ is generated by locally elliptic subgroups (fixators of edges). Therefore this abelianization is finite, and it follows that M has finite index in G(F, F ′ ). 6.3. Boundaries of G(F, F ′ ). In this paragraph we use results from the previous sections in order to study the boundaries of the discrete groups G(F, F ′ ). The following result shows that several properties of the set of boundaries are governed by the permutation groups, and that rigidity phenomena occur under mild conditions of the permutation groups. Theorem 6.9. Assume F acts freely on Ω, F 6= F ′ , and write Γ = G(F, F ′ ). The following are equivalent: (i) The subgroup of F ′ generated by its point stabilizers has at most two orbits in Ω. (ii) Γ/Env(AΓ ) is isomorphic to one of C2 , D∞ or D∞ ⋊ C2 . (iii) Env(AΓ ) is co-amenable in Γ. (iv) SΓ (X) = AΓ for every non-trivial Γ-boundary X. (v) Γ is boundary indivisible. We will need preliminary results before proving Theorem 6.9. Lemma 6.10. Assume that F acts freely on Ω. The envelope of the Furstenberg URS of G(F, F ′ ) is equal to G(F, F ′ )+ . Proof. Write Γ = G(F, F ′ ) and Γ+ = G(F, F ′ )+ . According to Proposition 6.4, Env(AΓ ) is the subgroup generated by fixators of half-trees in Γ. Therefore the inclusion Env(AΓ ) ≤ Γ+ is clear. The converse inclusion also holds true by Remark 6.7, so equality follows. In view of Lemma 6.10 and Proposition 3.24, we are led to consider the quotient G(F, F ′ )/G(F, F ′ )+ , and in particular study when it is amenable. To this end, we will denote by F ′+ the subgroup of F ′ generated by its point stabilizers, and write D = F ′ /F ′+ . Since F ′+ is normal in F ′ , we have an action of F ′ on the set of orbits of F ′+ , which factors through a free action of D. Proposition 6.11. The group Q = G(F, F ′ )∗ /G(F, F ′ )+ is isomorphic to the group U (D)∗ , where D = F ′ /F ′+ is viewed as a permutation group acting freely on the set of orbits of F ′+ in Ω. If moreover F is transitive, one has Q = D ∗ D. Proof. We let O1 , . . . , Or ⊆ Ω be the orbits of F ′+ , and we will freely identify the set of orbits with the integers {1, . . . , r}. For every a ∈ Ω, there is a unique i ∈ {1, . . . , r} such that a ∈ Oi , and we denote i = ia . We view the tree Td as the Cayley graph of the free Coxeter group of rank d, namely the group defined by generators x1 , . . . , xd and relators x2j = 1 for all j. 32 ADRIEN LE BOUDEC When adding relations of the form xa = xb whenever ia = ib (i.e. a and b are in the same F ′+ -orbit), we obtain a free Coxeter group of rank r. It has a Cayley graph that is a regular tree of degree r, and we have a surjective map p : Td → Tr . Two elements v, w ∈ ∗d C2 = hx1 , . . . , xd i have the same image in ∗r C2 if and Q only if one can write w = v wj xaj xbj wj−1 for some words wj and colors aj , bj such that iaj = ibj . Since the inverse of wj is equal to the word obtained from wj by reversing the order, we have: Lemma 6.12. Two vertices v, w of Td have the same projection in Tr if and only if the distance between v and w is even, say d(v, w) = 2m, and if (a1 , . . . , a2m ) is the sequence of colors from v to w, then the word (ia1 , . . . , ia2m ) is a concatenation of palindromes of even length. Lemma 6.13. For every g ∈ G(F, F ′ ) and every vertex v on Td , the image of σ(g, v) in D = F ′ /F ′+ does not depend on v. We denote by σg ∈ D the corresponding element, which is trivial when g ∈ G(F, F ′ )+ . Proof. If v, w are adjacent vertices and a is the color of the edge between them, then σ(g, v)(a) = σ(g, w)(a). So σ(g, v)σ(g, w)−1 ∈ F ′+ . The first statement follows by connectedness. The fact that σg is trivial on G(F, F ′ )+ is then clear because g 7→ σg is a morphism according to the first statement, which vanishes on fixators of edges. Note that the set of edges of Tr inherits a natural coloring by the integers 1, . . . , r. Lemma 6.14. There is a natural morphism ϕ : G(F, F ′ ) → Aut(Tr ) such that ker(ϕ) = G(F, F ′ )+ and Im(ϕ) = U (D). Proof. We shall first define an action of G(F, F ′ ) on the set of vertices of Tr . Let g ∈ G(F, F ′ ). Let v, w be two vertices of Td , and (a1 , . . . , an ) be the sequence of colors from v to w. If v = v1 , . . . , vn+1 = w are the vertices between v and w, then the sequence of colors from g(v) to g(w) is (σ(g, v1 )(a1 ), . . . , σ(g, vn )(an )). If σg is the element defined in Lemma 6.13, then one has iσ(g,vj )(aj ) = σg (iaj ) for all j = 1, . . . , n. This shows in particular that if v, w satisfy the condition of Lemma 6.12, then the same holds for g(v) and g(w). This means that for every vertex x of Tr , the formula (2) ϕ(g)(x) := p(gx̃), where x̃ is any vertex of Td such that p(x̃) = x, is a well-defined action of G(F, F ′ ) on Tr . The fact that the tree structure is preserved is clear. Note that for every g ∈ G(F, F ′ ), all the local permutations of ϕ(g) are equal to σg : for every vertex x of Tr , one has σ(ϕ(g), x) = σg . In particular the image of ϕ lies inside U (D). We shall prove that ker(ϕ) = G(F, F ′ )+ . Let g ∈ G(F, F ′ ) fixing an edge in Td . Then ϕ(g) also fixes an edge of Tr by (2). Moreover one has σg = 1 (Lemma 6.13), and it follows from the previous paragraph that all local permutations of ϕ(g) are trivial. This implies g ∈ ker(ϕ). Conversely, we let g be an element of ker(ϕ), and we prove that g ∈ G(F, F ′ )+ . Note that since ϕ(g) is trivial, one has σg = 1, i.e. all local permutations of g are in F ′+ . Now let v be any vertex. By Lemma 6.12, the sequence of colors (a1 , . . . , a2n ) from v to g(v) gives rise to a sequence (ia1 , . . . , ia2n ) that is a concatenation of palindromes. For simplicity we treat the case where (ia1 , . . . , ia2n ) is a palindrome; the general case consists in repeating the AMENABLE URS’S AND LATTICE EMBEDDINGS 33 argument for this case. Let v = v1 , . . . , v2n+1 = g(v) be the vertices between v and g(v) (note that vn is the midpoint between v and g(v)). Since (ia1 , . . . , ia2n ) is a palindrome, one easily checks that there are elements g1 , . . . , gn such that gj belongs to the stabilizer of vj in G(F, F ′+ ) and g′ = g1 . . . gn g fixes the vertex v (this is obtained by successively folding the geodesic [v, g(v)] onto itself starting from its midpoint in order to bring back g(v) to v with g1 . . . gn ). We now invoke the following easy fact, whose verification is left to the reader. Lemma 6.15. Let γ ∈ G(F, F ′ ) fixing a vertex w, and such that σ(γ, w) ∈ F ′+ . Then γ ∈ G(F, F ′ )+ . We apply Lemma 6.15 to g′ and all gj ’s, and deduce that g = gn−1 . . . g1−1 g′ belongs to G(F, F ′ )+ as desired. The last thing that remains to be proved in the statement of Lemma 6.14 is that the image of ϕ is equal to U (D). The fact that ϕ(g) always belongs to U (D) has already been observed. For the converse inclusion, observe that since G(F, F ′ ) acts transitively on the vertices of Tr (as it is already the case on Td ), it is enough to check that the image of ϕ contains U (D)x for some vertex x of Tr . Now since D acts freely on {1, . . . , r}, the map U (D)x → D, γ 7→ σ(γ, x), is an isomorphism. Therefore it is enough to see that any action on the star around x on Tr can realized by an element of G(F, F ′ ), and this is indeed the case (see e.g. Lemma 3.4 in [LB16]). To finish the proof of the proposition, remark that the image of G(F, F ′ )∗ by ϕ is precisely U (D)∗ . When F is transitive then D is also transitive, so that U (D)∗ has two orbits of vertices and one orbit of edges, and therefore splits as the free product D ∗ D. Remark 6.16. The case F = F ′ is allowed in Proposition 6.11, so that the conclusion also holds for the groups U (F ) from [BM00a]. Proposition 6.11 naturally leads us to isolate the following three situations. We keep the previous notation, so that r is the number of orbits of F ′+ in Ω, D = F ′ /F ′+ and Q = G(F, F ′ )∗ /G(F, F ′ )+ : (1) r = 1. In this case Tr is a segment of length one, and D and Q are trivial. (2) r = 2. Tr is a bi-infinite line, and this case splits into two disjoint sub-cases: (a) If F ′ is intransitive then D is trivial, and Q = U (1)∗ = Z (generated by a translation of Tr of length 2). (b) If F ′ is transitive then we have D = Sym(2) and Q = D ∗ D = D∞ . (3) r ≥ 3. Then Q = U (D)∗ is a virtually free group (since it acts vertex transitively and with trivial edge stabilizers of Tr ). Theorem 6.9 says that all properties stated there hold true if and only if r ∈ {1, 2}. A sufficient condition for having r = 1 is for instance that F acts transitively on Ω, F 6= F ′ and F ′ acts primitively, or quasi-primitively on Ω. Recall that a permutation group is quasi-primitive if every non-trivial normal subgroup acts transitively. But Theorem 6.9 also applies beyond the case of quasi-primitive permutation groups. For example a situation giving rise to case (2) (a) is when F ′ has a fixed point and acts transitively on the complement. Examples giving rise to case (2) (b) are for instance obtained by taking F ′ = Sym(n)≀C2 = Sym(n)≀hτ i acting naturally 34 ADRIEN LE BOUDEC on 2n letters, and F the subgroup generated by ((cn , cn ), 1) and ((1, 1), τ ), where cn is a cycle of order n. Proof of Theorem 6.9. (i) ⇒ (ii) follows from Lemma 6.10 and Proposition 6.11 (and the discussion following its proof). (ii) ⇒ (iii) is clear. (iii) ⇒ (iv) is Proposition 3.24. (iv) ⇒ (v) is guaranteed by Proposition 3.9 and the fact that AΓ is not a point. Finally assume that (i) does not hold, i.e. F ′+ has at least three orbits in Ω, and write Γ+ = G(F, F ′ )+ . By Proposition 6.11, the group Q = Γ/Γ+ has a subgroup of finite index that is free of rank at least 2. So there exist non-trivial Q-boundaries, and a fortiori these are non-trivial Γ-boundaries. If X is such a boundary, then Γ+ acts trivially on X. Since Γ+ also acts minimally on ∂Td , it follows that X and ∂Td are disjoint Γ-boundaries, contradicting (v). Therefore property (v) implies property (i), and the proof is complete. 6.4. Weakly co-amenable subgroups. In this paragraph we show that subgroups of the groups G(F, F ′ ) satisfy the following dichotomy: Proposition 6.17. Assume that F acts freely transitively and F ′ acts primitively on Ω. Then any subgroup of G(F, F ′ ) is either (locally finite)-by-cyclic (and hence amenable) or weakly co-amenable. We will need the following lemma. Lemma 6.18. Assume that F ′ acts primitively on Ω, and take two subgroups H1 6= H2 in the Furstenberg URS of G(F, F ′ ). Then hH1 , H2 i = G(F, F ′ )∗ . Proof. Write Γ = G(F, F ′ ). Recall from [LBMB16, Prop. 4.28] that the Furstenberg URS of Γ consists of subgroups Γ0ξ , ξ ∈ ∂Td , where Γ0ξ is the set of elements acting trivially on a neighbourhood of ξ. Given ξ 6= η ∈ ∂Td , we show that the subgroup Λ generated by Γ0ξ and Γ0η must be equal to G(F, F ′ )∗ . Take a vertex v on the geodesic from ξ to η, let e1 , e2 be the edges containing v and pointing towards ξ and η, and a, b the colors of e1 , e2 . Denote by K(v) the subgroup of Γ consisting of elements γ fixing v and such that σ(γ, w) ∈ F for every w 6= v. Since F ′ is primitive, F ′ is generated by the point stabilizers Fa′ and Fb′ . This implies that every element of K(v) may be written as a product of elements fixing either the half-tree defined by e1 containing ξ, or the half-tree defined by e2 containing η, so that K(v) ≤ Λ. Since v was arbitrary, we also have K(v ′ ) ≤ Λ for v ′ a neighbour of v on the geodesic [ξ, η]. The conclusion now follows since for two neighbouring vertices v, v ′ , the subgroups K(v), K(v ′ ) always generate G(F, F ′ )∗ [LB16, Cor. 3.10]. Proof of Proposition 6.17. Write Γ = G(F, F ′ ), and let Λ be a subgroup of Γ that is non-amenable, equivalently whose action of Td is of general type. By Proposition 5.7 we have to show that Λ fixes no probability measure on any non-trivial Γboundary. Argue by contradiction and assume that X is a non-trivial Γ-boundary on which Λ fixes a probability measure µ. According to Theorem 6.9, we have SΓ (X) = AΓ . Therefore by Proposition 3.2 there exist an almost 1-1 extension η : X̃ → X and a factor map π : X̃ → AΓ . Let Q ⊂ Prob(X̃) be the set of ν such that η ∗ ν = µ, and write R = π ∗ (Q), which is a closed Λ-invariant subset of Prob(AΓ ). Since the action of Λ on ∂Td is strongly proximal and since AΓ is a factor of ∂Td [LBMB16, Prop. 2.10-4.28], AMENABLE URS’S AND LATTICE EMBEDDINGS 35 we deduce that R contains some Dirac measures. Let H ∈ AΓ such that there is ν ∈ Prob(X̃) with η ∗ ν = µ and π ∗ ν = δH . Such a measure ν must be supported in the set of (x, H) ∈ X̃, and it follows that µ is supported in the set of H-fixed points in X (because (x, H) ∈ X̃ implies that H ≤ Gx by upper semi-continuity of the stabilizer map). But since Λ does not fix any point in AΓ , we may find another H ′ ∈ AΓ such that δH ′ ∈ R, so that the same argument shows that H ′ also acts trivially on the support of µ. By Lemma 6.18 the subgroups H, H ′ generate G(F, F ′ )∗ , which is of index two in Γ. Therefore any point in the support of µ has a Γ-orbit of cardinality at most two, which is absurd since Γ acts minimally on X and X is non-trivial by assumption. Remark 6.19. Assume F acts freely transitively and F ′ acts primitively, and write Γ = G(F, F ′ ). Let Λ ≤ Γ such that the Λ-action on Td is of general type but with a proper Λ-invariant subtree. For instance one could take for Λ the subgroup generated by two hyperbolic elements with sufficiently far apart axis. Then Λ is not co-amenable in Γ (see the argument in the proof of Theorem 2.4 in [CM09]), but Λ is weakly co-amenable in Γ by Proposition 6.17. Remark 6.20. We mention that when F ′ is primitive, following the proof of Corollary 4.14 from [LBMB16] (with minor modifications), one could prove that every non-trivial G(F, F ′ )-boundary factors onto ∂Td . This would provide an alternative proof of Proposition 6.17. 6.5. Lattice embeddings of the groups G(F, F ′ ). In this section we study how the discrete groups G(F, F ′ ) can embed as lattices in some locally compact groups. The purpose of this paragraph is twofold: (1) First we apply previous results of the article to the family of groups G(F, F ′ ) and deduce some properties of general locally compact groups containing a group G(F, F ′ ) as a lattice (Corollary 6.22). (2) Second we explain how the groups G(F, F ′ ) embed as lattices in some locally compact wreath products. This will be the content of §6.5.2 below. Remark 6.21. Maybe it is worth pointing out that instances of lattice embeddings of the groups G(F, F ′ ) already appeared in [LB16]. Indeed under appropriate assumptions on permutation groups F ≤ F ′ , H ≤ H ′ , the inclusion of G(F, F ′ ) in G(H, H ′ ) has discrete and cocompact image [LB16, Cor. 7.4]. Corollary 6.22. Assume that F acts freely transitively on Ω, and that F ′ is generated by its point stabilizers. Let G be a locally compact group containing G(F, F ′ ) as a lattice. Then the conclusions of Corollary 1.3 hold. Proof. The assumptions on F, F ′ imply that G(F, F ′ ) is virtually simple by Proposition 6.8, so Corollary 1.3 applies. Remark 6.23. In the setting of Corollary 6.22, although G(F, F ′ ) cannot be a lattice in a product, it happens that there exist non-discrete groups G1 , G2 such that G(F, F ′ ) embeds as a discrete subgroup of G1 × G2 with injective and dense projection to each factor. For instance if F1 , F2 are permutation groups such that F Fi ≤ F ′ and we set Gi = G(Fi , F ′ ), then the diagonal embedding of G(F, F ′ ) in G1 × G2 has this property as soon as F1 ∩ F2 = F . See [LB16, Lem. 3.4 and §7.1]. 36 ADRIEN LE BOUDEC 6.5.1. Locally compact wreath products. In this paragraph we introduce some terminology that will be used in the sequel. Let Ω be a set, B a group and A a subgroup of B. We will denote by B Ω,A the set of functions f : Ω → B such that f (x) ∈ A for all but finitely many x ∈ Ω. Note that B Ω,A is a group. Definition 6.24 ([Cor17]). If H is a group acting on Ω, the semi-restricted Ω,A ⋊ H, permutational wreath product B ≀A Ω H is the semi-direct product B X,A −1 where h ∈ H acts on f ∈ B by (hf )(x) = f (h x). The extreme situations when A = 1 and when A = B correspond respectively to the restricted and the unrestricted wreath product. When A = 1, we shall write B ≀Ω H for the restricted wreath product. Also for simplicity we will sometimes say “wreath product” B ≀A Ω H instead of “semi-restricted permutational wreath product”. When A is a compact group and H a locally compact group acting continuously on Ω, the group AΩ ⋊ H is a locally compact group for the product topology. If moreover B is locally compact and A is compact open in B, there is a natural locally compact group topology on B ≀A Ω H, defined by requiring that the inclusion Ω A of A ⋊ H in B ≀Ω H is continuous and open. See [Cor17, Sec. 2]. In the remaining of the article we shall be interested in the study of certain lattices in some locally compact groups B ≀A Ω H. A few remarks are in order: Lemma 6.25. Let A, B, Ω and H as above. (a) Assume Γ1 is a lattice in B Ω,A and Γ2 is a lattice in H that normalizes Γ1 . Then Γ1 ⋊ Γ2 is a lattice in B ≀A Ω H. A (b) For B ≀Ω H to contain a lattice, it is necessary that H contains a lattice. Proof. For the first statement, see [Rag72, Lem. I.1.6-7]. For the second statement, Ω observe that if Γ ≤ B ≀A Ω H is a lattice, the intersection ΓA = Γ ∩ (A ⋊ H) is a Ω lattice in AΩ ⋊H since AΩ ⋊H is open in B ≀A Ω H. The subgroup A being compact, the projection of ΓA to H is discrete, and hence is a lattice in H. Recall that there are various notions of irreducibility for a lattice in a direct product of groups. In general whether all these notions coincide depends on the context. We refer to [CM12, 2.B] and [CM09, 4.A] for detailed discussions. In the setting of wreath products, we will use the following terminology: Definition 6.26. A lattice Γ in B ≀A Ω H is an irreducible lattice if Γ has a non-discrete projection to the group H. This definition implies that neither Γ nor its finite index subgroups can be of the form Γ1 ⋊ Γ2 as in Lemma 6.25. Lemma 6.27. If the group B Ω,A does not contain any lattice, then any lattice in B ≀A Ω H is irreducible. Proof. If Γ ≤ B≀A Ω H is a lattice with a discrete projection Λ to H, then the subgroup Λ contains Γ as a lattice, and it follows that Γ intersects the subgroup B Ω,A , B ≀A Ω Ω,A . which is open in B ≀A Ω Λ, along a lattice of B AMENABLE URS’S AND LATTICE EMBEDDINGS 37 Remark 6.28. Of course if B admits no lattice then the same holds for B Ω,A . More interestingly, there are finite groups A, B for which B Ω,A fails to admit any lattice (provided Ω is infinite). This is for instance the case when A 6= B and any non-trivial element of B has a non-trivial power in A. Consequently all lattices in B ≀A Ω H are irreducible by Lemma 6.27 (for arbitrary H). Proof of Remark 6.28. We claim that the above condition on A, B actually implies that B Ω,A has no infinite discrete subgroup. For every finite Σ ⊂ Ω, we write OΣ ≤ B Ω,A for the subgroup vanishing on Σ, and UΣ = OΣ ∩ AΩ . Assume Γ is a discrete subgroup of B Ω,A , so that there is a finite Σ ⊂ Ω such that Γ ∩ UΣ = 1. The assumption on A, B is easily seen to imply that any non-trivial subgroup of OΣ intersects UΣ non-trivially. Therefore Γ ∩ OΣ = 1, and OΣ being of finite index in B Ω,A , Γ is finite. It should be noted that the existence of an irreducible lattice in B ≀A Ω H forces H to be non-discrete and B to be non-trivial. However this does not force B Ω,A to be non-discrete, and as we will see below, interesting examples already arise when B is finite and A is trivial. 6.5.2. The proof of Theorem 1.4. Let n ≥ 2 and d ≥ 3. We denote by Σn the (V ) set of integers {0, . . . , n − 1}, and by Σn d the set of functions f : Vd → Σn with finite support, where the support of f is the set of v such that f (v) 6= 0. We will also write fv for the image of v by f , and sometimes use the notation (fv ) for the function f . We consider the graph Xn,d whose set of vertices is the set of pairs (f, e), where (V ) f belongs to Σn d and e ∈ Ed , and edges emanating from a vertex (f, e) are of two types: • type 1: (f, e′ ) is connected to (f, e) if e′ ∈ Ed is a neighbour of e (i.e. if e and e′ share exactly one vertex); • type 2: (f ′ , e) is connected to (f, e) if the function f ′ is obtained from f by changing the value at exactly one vertex of e. Note that since any e ∈ Ed has 2(d − 1) neighbours and Σn has cardinality n, every vertex of Xn,d has 2(d − 1) neighbours of type 1 and 2(n − 1) neighbours of type 2. The graph Xn,d is almost the wreath product of the complete graph on n vertices with the tree Td , see below. Let Sn be the group of permutations of Σn . For σ ∈ Sn and i ∈ Σn , we will write σ · i the action of σ on i. The stabilizer of 0 ∈ Σn in Sn is obviously isomorphic to Sn−1 , and by abuse of notation we will denote it Sn−1 . In particular when viewing Sn−1 as a subgroup of Sn , we will always implicitly mean that Sn−1 is the subgroup of Sn acting only on {1, . . . , n − 1}. S Definition 6.29. We will denote by Gn,d the wreath product Sn ≀Vdn−1 Aut(Td ). Groups of the form Gn,d were considered in [Cor17, Ex. 2.6]. We will denote V ,S Un,d = Snd n−1 , so that Gn,d = Un,d ⋊ Aut(Td ). We endow Gn,d with the topology such that sets of the form ((σv )U1 , γU2 ) form a basis of neighbourhoods of ((σv ), γ), d and in where U1 and U2 belong to a basis of the identity respectively in SVn−1 Aut(Td ). This defines a totally disconnected locally compact group topology on Gn,d (see [Cor17, Prop. 2.3]). We note that the case n = 2 is somehow particular, 38 ADRIEN LE BOUDEC as U2,d is a discrete subgroup of G2,d , and G2,d is just the restricted wreath product G2,d = C2 ≀Vd Aut(Td ) = (⊕Vd C2 ) ⋊ Aut(Td ). Proposition 6.30. The group Gn,d acts by automorphisms on the graph Xn,d by preserving the types of edges. Moreover the action is faithful, continuous, proper and transitive on the set of vertices. Proof. The group Gn,d is a subgroup of the unrestricted permutational wreath product of Sn and Aut(Td ). The latter group has a faithful action on the set of functions Vd → Σn , given by ((σv ), γ) · (fv ) = (σv · fγ −1 v ). (V ) The group Gn,d preserves Σn d because σv fixes 0 almost surely if (σv ) belongs to Un,d . Now the projection from Gn,d onto Aut(Td ) induces an action of Gn,d on the (V ) set Ed , and we will consider the diagonal action of Gn,d on Σn d × Ed . In other words, if g = ((σv ), γ) ∈ Gn,d , and ((fv ), e) ∈ Xn,d , (3) g · ((fv ), e) = (σv · fγ −1 v ), γe . Fix x = ((fv ), e) ∈ Xn,d and g = ((σv ), γ) ∈ Gn,d , and let x′ be a neighbour of x. If x′ is of type 1, then we have x′ = ((fv ), e′ ), where e and e′ share a vertex w in Td . Then γe and γe′ have the vertex γw in common, so that by the formula (3), g · x′ is a neighbour of type 1 of g · x in Xn,d . Now if x′ is of type 2, then we may write x′ = ((fv′ ), e) with fv′ = fv if and only if v 6= w, where w is one of the two vertices of e. It follows that σv · fγ −1 v = σv · fγ′ −1 v if and only if v 6= γw, so that by (3) g · x′ is a neighbour of type 2 of g · x. This shows that the action is by graph automorphisms and preserves the types of edges. Lemma 6.31. Let e ∈ Ed , and let K be the stabilizer of e in Aut(TQ d ). Then the stabilizer of the vertex ((0), e) in Gn,d is the compact open subgroup Sn−1 ⋊ K. Proof. That g = ((σv ), γ) fixes ((0), e) exactly means by (3) that σv fixes 0 for all v and that γ fixes e. So the fact that the action is continuous and proper follows from the lemma, and the transitivity on the set of vertices is an easy verification. Consider now the free product Cd ∗ Cd of two cyclic groups of order d, acting on its Bass-Serre tree Td with one orbit of edges and two orbits of vertices. Denote by Cn the cyclic subgroup of Sn generated by the cycle (0, . . . , n − 1), and set Γn,d := Cn ≀Vd (Cd ∗ Cd ) ≤ Gn,d . Remark that Cd ∗ Cd has a split morphism onto Cd , whose kernel acts on Td with two orbits of vertices and is free of rank d − 1. Therefore Γn,d splits as Γn,d = (Cn2 ≀ Fd−1 ) ⋊ Cd . Lemma 6.32. Γn,d ≤ Gn,d acts freely transitively on the vertices of Xn,d . Proof. This is clear: the image of the vertex ((0), e) by an element ((σv ), γ) is ((σv · 0), γe), so both transitivity and freeness follow from the fact that the actions of Cn on Σn and of Cd ∗ Cd on Ed have these properties. AMENABLE URS’S AND LATTICE EMBEDDINGS 39 We now explain how the groups G(F, F ′ ) act on the graphs Xn,d . In the sequel F, F ′ denote two permutation groups on Ω such that F ≤ F ′ and F ′ preserves the orbits of F , and we denote by n the index of F in F ′ . Fix a bijection between Σn = {0, . . . , n − 1} and F ′ /F , such that 0 is sent to the class F . The action of F ′ on the coset space F ′ /F induces a group homomorphism α : F ′ → Sn , such that α(F ) lies inside Sn−1 . For γ ∈ G(F, F ′ ) and v ∈ Vd , write ργ,v = α(σ(γ, γ −1 v)) ∈ Sn . Note that ργ,v ∈ Sn−1 if and only if σ(γ, γ −1 v) ∈ F . We also denote by ργ = (ργ,v ). Proposition 6.33. Let F ≤ F ′ ≤ Sym(Ω), and n the index of F in F ′ . The map ϕ : G(F, F ′ ) → Gn,d , γ 7→ ϕ(γ) = (ργ , γ), is a well-defined group morphism that is injective, continuous, and with a closed and cocompact image. Proof. The map ϕ is well-defined because ργ,v ∈ Sn−1 for all but finitely many V ,S v, so that we indeed have ργ ∈ Snd n−1 . The fact that ϕ is a group morphism follows from the cocycle identity (1) satisfied by local permutations. Indeed for γ, γ ′ ∈ G(F, F ′ ) we have ϕ(γ)ϕ(γ ′ ) = (ψ, γγ ′ ) with ψv = ργ,v ργ ′ ,γ −1 v = α(σ(γ, γ −1 v))α(σ(γ ′ , γ ′−1 γ −1 v)) = α(σ(γγ ′ , (γγ ′ )−1 v)) = ργγ ′ ,v , so ψ = ργγ ′ and ϕ(γ)ϕ(γ ′ ) = ϕ(γγ ′ ). Injectivity of ϕ is clear since the composition with the projection to Aut(Td ) d ⋊ Aut(Td ) is is injective. The preimage in G(F, F ′ ) of the open subgroup SVn−1 ′ the subgroup U (F ), which is open in G(F, F ) by definition of the topology, so it follows that the map ϕ is continuous. Also the intersection between Im(ϕ) and the d ⋊ Aut(Td ) is ϕ(U (F )), and it is easy to check that the latter open subgroup SVn−1 is indeed a closed subgroup of Gn,d , so it follows that Im(ϕ) is closed in Gn,d . The fact that Im(ϕ) is cocompact will follow from Proposition 6.30 and Proposition 6.34 below. In the sequel for simplicity we will also write G(F, F ′ ) for the image of ϕ : G(F, F ′ ) → Gn,d , γ 7→ ϕ(γ) = (ργ , γ) . In particular when speaking about an action of G(F, F ′ ) on the graph Xn,d , we will always refer to the action defined in Proposition 6.30, restricted to G(F, F ′ ). This means that γ ∈ G(F, F ′ ) acts on (f, e) ∈ Xn,d by γ · (f, e) = (f γ , γe), where (4) (f γ )v = ργ,v · fγ −1 v = α(σ(γ, γ −1 v)) · fγ −1 v . This action should not be confused with the standard action ((fv ), e) 7→ ((fγ −1 v ), γe) coming from the inclusion of G(F, F ′ ) in Aut(Td ). Proposition 6.34. Let F ≤ F ′ ≤ Sym(Ω), and n the index of F in F ′ . 40 ADRIEN LE BOUDEC (a) The group G(F, F ′ )∗ acts cocompactly on Xn,d . When F is transitive on Ω, the group G(F, F ′ )∗ acts transitively on vertices of Xn,d . (b) The stabilizer of a vertex ((0), e) ∈ Xn,d in G(F, F ′ ) is the stabilizer of e in U (F ). In particular the action of G(F, F ′ ) on Xn,d is proper. Therefore when F acts freely transitively on Ω, the group G(F, F ′ )∗ acts freely transitively on the vertices of Xn,d . Proof. We show that for every vertex x = ((fv ), e) of Xn,d , there is g ∈ G(F, F ′ )∗ such that g · x = ((0), e). Since U (F ) preserves the vertices of this form, and since the number of orbits of U (F )∗ ≤ G(F, F ′ )∗ on Ed is finite and is equal to one when F is transitive [BM00a], statement (a) will follow. We argue by induction on the cardinality N of the support of (fv ). There is nothing to show if N = 0. Assume N ≥ 1, and let v0 ∈ Vd with fv0 6= 0 and such that v0 maximizes the distance from e among vertices v such that fv 6= 0. Let e0 be the edge emanating from v0 toward e (if v0 belongs to e then e0 = e), and let a ∈ Ω be the color of e0 . We also denote by T 1 and T 2 the two half-trees defined by e0 , where T 1 contains v0 . For every b ∈ Ω, b 6= a, we denote by e0,b the edge containing v0 and having color c(e0,b ) = b, and by T 1,b the half-tree defined by e0,b not containing v0 . By assumption the permutation group F ′ preserves the F -orbits in Ω, so we have ′ F = F Fa′ . The subgroup α(F ′ ) ≤ Sn being transitive, it follows from the previous decomposition that there exists σ ∈ Fa′ such that α(σ) · fv0 = 0. For every b 6= a, we choose σb ∈ F such that σb (b) = σ(b), and we consider the unique element h ∈ Aut(Td ) whose local permutations are σ(h, v) = 1 if v ∈ T 2 ; σ(h, v0 ) = σ; and σ(h, v) = σb for every v ∈ T 1,b and every b 6= a. It is an easy verification to check that h is a well-defined automorphism of Td , and h ∈ G(F, F ′ ) because all but possibly one local permutations of h are in F . Note that h fixes e by construction. Write h(x) = ((φv ), e). We claim that the support of (φv ) has cardinality N − 1. Since h fixes v0 , by (4) we have φv0 = α(σ(h, v0 )) · fv0 = α(σ) · fv0 = 0. Moreover we also have φv = fv for every v in T 2 because h acts trivially on T 2 . Finally by the choice of v0 we had fv = 0 for every v 6= v0 in T 1 , and since σ(h, v) ∈ F for all these v and α(F ) fixes 0, we still have φv = 0 for every v in T 1 , v 6= v0 . This proves the claim, and the conclusion follows by induction. Statement (b) follows from Lemma 6.31, and the last statement follows from (a) and (b) and the fact that U (F ) acts freely on Ed when F acts freely on Ω. Propositions 6.33-6.34 and Lemma 6.32 imply Theorem 1.4 from the introduction. Note that when F acts freely transitively on Ω, we have an explicit description of a generating subset of the group G(F, F ′ )∗ whose associated Cayley graph is Xn,d . For, fix an edge e0 ∈ Ed , whose color is a ∈ Ω and whose vertices are v0 , v1 , and denote x0 = ((0), e0 ) ∈ Xn,d . For i = 0, 1, let Si be the set of γ ∈ G(F, F ′ ) fixing vi and such that σ(γ, v) ∈ F for every v 6= vi , and σ(γ, vi ) is non-trivial and belongs to F ∪ Fa′ . Then S = S0 ∪ S1 generates G(F, F ′ )∗ , and Cay(G(F, F ′ )∗ , S) → Xn,d , γ 7→ γx0 , is a graph isomorphism. Moreover neighbours of type 1 (resp. type 2) of a vertex of Xn,d are labeled by elements s ∈ Si such that σ(γ, vi ) ∈ F (resp. σ(γ, vi ) ∈ Fa′ ). AMENABLE URS’S AND LATTICE EMBEDDINGS 41 We end the article by observing that there are possible variations in the definition of the graph Xn,d . If Kn is the complete graph on n vertices, let Kn ≀ Td be the wreath product of the graphs Kn and Td (sometimes also called the lamplighter (V ) graph over Td ): the vertex set is Σn d × Vd , and there is an edge between (f, v) and (f ′ , v ′ ) if and only if either f = f ′ and v, v ′ are adjacent in Td , or v = v ′ and f (w) = f ′ (w) if and only if w 6= v. Again if n is the index of F in F ′ , the group G(F, F ′ ) acts on Kn ≀ Td by γ · (f, v) = (f γ , γv), where f γ is given by (4). The previous arguments for the graph Xn,d carry over to this graph, so that we have: Proposition 6.35. Let F ≤ F ′ ≤ Sym(Ω), and n the index of F in F ′ . Then G(F, F ′ ) acts properly and cocompactly on the graph Kn ≀ Td . The reason why we considered the graph Xn,d instead of Kn ≀ Td is to obtain, under the assumption that F acts freely on Ω, a free action of G(F, F ′ )∗ on the set of vertices. In the case of Kn ≀ Td , the stabilizer of a vertex in G(F, F ′ )∗ is finite, but non-trivial. We note that it might be interesting to investigate whether the generalized wreath products of graphs from [Ers06] could provide other kind of interesting groups of automorphisms. Yet another possibility is to take the same vertex set as Xn,d , but declaring that there is an edge between (f, e) and (f ′ , e′ ) if e 6= e′ share a vertex w and fv = fv′ for every v 6= w. 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UCLouvain, IRMP, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, Belgium CNRS, Unité de Mathématiques Pures et Appliquées, ENS-Lyon, France E-mail address: [email protected]	4
1 Multisensor Poisson Multi-Bernoulli Filtering with Uncertain Sensor States arXiv:1712.08146v1 [cs.SY] 21 Dec 2017 Markus Fröhle, Christopher Lindberg, Karl Granström, Henk Wymeersch Abstract—In a typical multitarget tracking (MTT) scenario, the sensor state is either assumed known, or tracking is performed based on the sensor’s (relative) coordinate frame. This assumption becomes violated when the MTT sensor, such as a vehicular radar, is mounted on a vehicle, and the target state should be represented in a global (absolute) coordinate frame. Then it is important to consider the uncertain sensor location for MTT. Furthermore, in a multisensor scenario, where multiple sensors observe a common set of targets, state information from one sensor can be utilized to improve the state of another sensor. In this paper, we present a Poisson multi-Bernoulli MTT filter, which models the uncertain sensor state. The multisensor case is addressed in an asynchronous way, where measurements are incorporated sequentially based on the arrival of new sensor measurements. In doing so, targets observed from a well localized sensor reduce the state uncertainty at another poorly localized sensor, provided that a common non-empty subset of features is observed. The proposed MTT filter has low computational demands due to its parametric implementation. Numerical results demonstrate the performance benefits of modeling the uncertain sensor state in feature tracking as well as the reduction of sensor state uncertainty in a multisensor scenario compared to a per sensor Kalman filter. Scalability results display the linear increase of computation time with number of sensors or features present. I. I NTRODUCTION Intelligent transportation systems in general, and autonomous driving (AD) in particular, require accurate position information [1]. Measurements provided by various on-board sensors allow to infer the vehicle state, e.g., position and velocity, as well as information about the surrounding environment. For instance, a global navigation satellite system (GNSS) receiver provides absolute position, whereas a radar sensor provides relative position with respect to (w.r.t.) the sensor origin. Furthermore, vehicles have access to a pre-recorded local dynamic map (LDM) containing static features such as, e.g., landmarks [2]. Dynamic features such as pedestrians, cyclists, etc. are not part of the pre-recorded map. For an AD system to be fully aware of the surrounding environment, dynamic features need to be estimated and tracked over time using the vehicles on-board sensors thus allowing to enrich the vehicle’s LDM. In order to incorporate mobile features into the LDM, which contains map features described in a global coordinate frame, location uncertainty of on-board sensors used to track dynamic features needs to M. Fröhle, K. Granström, and H. Wymeersch are with the Department of Electrical Engineering, Chalmers University of Technology, Gothenburg, Sweden. E-mail: {frohle, karl.granstrom, henkw}@chalmers.se. C. Lindberg is with Zenuity AB, Gothenburg, Sweden. E-mail: [email protected] be considered, i.e., the vehicle’s state uncertainty such as its location and pose. In an ITS, vehicles communicate through the wireless channel with other vehicles (vehicle-to-vehicle (V2V) communication) or with road infrastructure, such as a road side unit (RSU), through vehicle-to-infrastructure (V2I) communication, using IEEE 802.11p or 4G/5G cellular communication. Through information exchange, vehicles can make local LDM information available to their neighbors allowing to enrich their LDMs and their situational awareness. Not only can LDM information be shared, it can also be fused to improve every LDM [3], [4]. For the special case, they observe an overlapping set of dynamic features, information from one vehicle can be utilized to increase location accuracy of other vehicles, and vice versa [5], [6]. Note, in this context no V2V measurements are performed, different to a traditional cooperative localization approach [7], [8]. The problem of vehicular localization using locally observed features with unknown observation to feature correspondence, aggregated at an RSU, can be interpreted as an MTT problem. In MTT, a varying number of mobile features (targets) are tracked, using sensors such as for example radars, LIDARS, or cameras [9]. Thereby, it is typically assumed that the state of the observing sensor is known. Although not true in general, this assumption can be motivated by the fact that sensor state uncertainty is negligible in comparison to the sensors’ measurement accuracy. If sensor state uncertainty is significant, it needs to be modeled in the MTT in order not to have negative impact on feature tracking performance. In this paper, we consider the case of MTT with uncertain sensor state, where there are potentially multiple sensors with varying sensor state uncertainty. To enable accurate feature tracking we model the sensor state uncertainty in the MTT filter. The main contributions are: • • • an asynchronous parametric multitarget-multisensor tracking filter with uncertain sensor state information, fusion of mulitarget-multisensor tracking information with local sensor tracking information, and numerical simulation results demonstrating the performance of the filter in a multisensor vehicular scenario. In the application example, we demonstrate how the proposed filter can be used to transfer location information from a well localized vehicle to a poorly localized vehicle through MTT. Hence, positioning accuracy of the poorly localized vehicle is greatly improved compared to using a local Kalman filter with GNSS measurements alone. 2 Legend Cooperative vehicle Mobile feature RSU Communication link Fig. 1. Urban ITS scenario with two vehicles cooperating through the RSU and six mobile features. A. Motivation In this paper, we consider an urban intelligent transportation system (ITS) scenario, consisting of cooperating vehicles (illustrated in Fig. 1). Each vehicle is equipped with an on-board sensor allowing them to determine their absolute position, e.g., a GNSS receiver, and an on-board sensor to retrieve relative positions of mobile features present in the environment, e.g., a radar. Absolute position measurements are denoted GNSS measurements and measurements taken w.r.t. features are denoted vehicle-to-feature (V2F) measurements. Due to the sensor used to obtain V2F measurements, it is in general not known which feature gave rise to which measurement. A GNSS and a set of V2F measurements from every vehicle are transmitted in a non-time synchronized manner to the RSU, where a centralized filter is run to track the feature as well as the vehicle states. This information can be utilized by the RSU an sent back to the vehicles to increase their situation awareness. B. Related Work 1) MTT with known sensor state: Many MTT filters have been proposed to track mobile features using radar-like sensors when the sensor state is known [9]. The multi-hypothesis tracking (MHT) filter builds a growing hypothesis tree with feature-to-measurements data association (DA), and needs to be pruned to limit computation complexity [10]. The joint probability data association (JPDA) filter finds the most likely DA, where feature state information is reduced after each update step to a single Gaussian per feature. In the last years, MTT filters based on random finite set (RFS) and finite-set statistics (FISST) (which avoid the inherent DA), originally developed by [11], have gained much attention. The probability hypothesis density (PHD) filter propagates the first moment of the RFS density over time [12]–[14]. The Poisson multi-Bernoulli (PMB) filter approximates the global joint DA by the product of (local) marginal DA similar to the JPDA filter [15]. In [16], a derivation of this PMB filter based on standard single target measurement models, without using probability generating functional (p.g.fl.) and functional derivatives, is presented. Furthermore, a connection to the δ-generalized labelled multi-Bernoulli (δ-GLMB) filter [17], [18] is shown, where the δ-GLMB density is a special case of multi-Bernoulli (MB) density for labeled targets and can therefore be seen as a special case of the PMB filter. In [19], a Gaussian mixture (GM) multitarget-multisensor Bernoulli tracker, with known sensor locations, is developed and compared to a particle filter (PF) implementation for multistatic sonobuoy fields. Target state update from multiple sensors was achieved through sequential sensor updates. In [20], [21], a factor graph (FG) based approach, not using FISST, was proposed for a variant of the JPDA filter. A multiscan scenario was considered, and the filter was realized by running loopy belief propagation on the FG containing cycles. In [22], a PF based implementation of the PMB filter [15] was presented. This implementation has been used in [21] as a performance comparison of the FG based MTT. There it was found that the PF based PMB filter implementation scales exponentially with an increasing number of features. In [23], vehicles perform local feature tracking and send track information over the wireless channel to a central fusion center. Then, track-to-track fusion [24], [25] is performed, taking care of out-of-sequence measurements which can arise by utilizing a shared communication media. There the DA problem arises on the decision of which local tracks to fuse. The Mahalanobis distance is employed as DA metric. 2) MTT with uncertain sensor state: In contrast to MTT, simultaneous localization and mapping (SLAM) based methods determine the sensor state while mapping features of the environment [2], [26]. Most of the proposed SLAM methods assume static features such as, e.g., walls and street signs. SLAM methods excel through maintaining the correlation between features. A reduction of complexity was achieved in FastSLAM through Rao-Blackwellization (RB), where the feature state, conditioned on the sensor state, is tracked through a Kalman filter and the sensor state through a PF. In [27], a RFS based approach to the SLAM problem was proposed. There the target state is conditioned on the sensor location and then tracked through a PHD filter following a RB. In [28], an MTT with uncertain (single) sensor location is derived for the SLAM problem using FISST and point process theory. Simulation results are shown with a RB PF implementation. In [29], the problem of sensor uncertainty for Bernoulli filtering [30] of at most one target using a single sensor is addressed. Since the scenario is restricted to a single sensor, a suboptimal approach is presented, where the sensor state is updated only by measurements independent of the target state, i.e., target tracking information is not used to update the sensor location. Similar to [21], a FG based approach was considered in [31] for an urban ITS scenario, where the number of features is assumed a priori known, and in [32] a variant of SLAM was considered for indoor environments using time-of-arrival radio signal measurements. There, features are the (static) source locations of line-of-sight and non line-of-sight signal propagation paths of the transmitted radio wave. C. Notation and Paper Organization Scalars are described by non-bold letters r, vectors by lower-case bold letters x; matrices and sets by upper-case bold 3 letters X. The cardinality of set X is denoted \|X\|. The set operator ] denotes the disjoint set union, i.e., F u ] F d = F means F u ∪ F d = F and F u ∩ F d = ∅. The vehicle state is reserved by letter x, the feature state by letter f , and measurements by letter z. The identity matrix of size n × n is denoted I n . The `2 -norm of vector x is kxk2 . The remainder of this paper is organized as follows. Section II gives some background knowledge on RFS, and Section III introduces the problem formulation and system models. Section IV details the proposed MTT filter with uncertain sensor state, numerical results are given in Section V, and conclusions are drawn in Section VI. II. BACKGROUND ON RFS In this section, we describe some useful properties of an RFS. If not stated otherwise, the source of all these is [15]. A. Random Finite Set Formulation According to [15], RFS based methods have been developed in [11] to conduct statistical inference in problems in which the variables of interest and/or observations form finite sets. In tracking, they address two major challenges of interest: (i) the number of targets present in the scene is unknown, (ii) measurements are invariant to ordering (measurement-totarget correspondence is unknown). An RFS X is a finite-set valued random variable, which can be described by a discrete probability distribution p(n), n ≥ 0 and a family of joint probability densities fn (xπ(1) , . . . , xπ(n) ) yielding X f (x1 , . . . , xn ) = p(n) fn (xπ(1) , . . . , xπ(n) ), (1) π where the sum spans over the n! permutation functions π(·), such that its RFS density f (X) is permutation invariant. The set integral of a real-valued function g(X) of a finite-set variable X is defined as [11, p. 361] Z Z ∞ X 1 g(x1 , . . . , xn )dx1 · · · dxn . g(X)δX , g(∅) + n! n=1 (2) A Bernoulli process X with probability of existence r and existence-conditioned probability density function (PDF) f (x) has RFS density   X=∅ 1 − r, f (X) = r · f (x), X = x (3)   0, otherwise, The RFS density of a MB process for the RFS X = {x1 , . . . , xn } is [11, p. 368] "N # n Y X Y rik f (X) = (1 − ri ) fi (xi ) . 1 − rik k i=1 i≤i1 6=...6=in ≤N k=1 (4) A Poisson point process (PPP) with intensity function λc (y) has RFS density [15] Y f (Y ) = exp(− hλc , 1i) λc (y) (5) y∈Y R with inner product hλc , hi , λc (y)h(y)dy. Remark 1: If X and Y are independent RFSs such that Z = X ] Y , then X fZ (Z) = fX (X)fY (Y ). (6) X]Y =Z Note, for RFS X a MB and RFS Y a PPP (6) is called a PMB density. B. State estimation from RFS density A common way to estimate the set states from a Bernoulli process with RFS density f (X) is by comparing the probability of existence r against an existence threshold rth . For r > rth , the target is said to exist and has PDF f (x) (c.f. (3)). RIts state can then be estimated by the mean of f (x), i.e., x̂ = xf (x)dx. III. P ROBLEM F ORMULATION AND S YSTEM M ODELS Here, we first present the problem formulation, and the vehicle and feature dynamics. This is followed by the GNSS and V2F measurement models, and the communication model. A. Problem Formulation The goal of the filter, which runs on the RSU, is to track the features and the states of all vehicles in every discrete time step t through incorporation of all sensor measurements (GNSS and V2F measurements) up until time step t. We are therefore interested in the joint posterior distribution of the feature and vehicle states at every time step t. B. Vehicle and Feature Dynamics Vehicle state motion follows independent Markovian processes, where the time-varying vehicle state xs,t ∈ RNx of each vehicle s ∈ S at time step t ∈ N0 is statistically modeled as p(xs,t \|xs,t−1 ), with linear state-space model xs,t = As,t xs,t−1 + ws,t , (7) where As,t denotes the state-transition matrix and ws,t ∼ N (0, W s,t ) with error covariance matrix W s,t . A single time-varying feature k ∈ K with state f k,t−1 ∈ RNf survives to the next time step t following an independent identically distributed (IID) Markovian process with survival probability pS (f k,t ). The feature state motion follows IID Markovian processes and is statistically modeled as p(f k,t \|f k,t−1 ) with linear state-space model f k,t = B t f k,t−1 + v k,t , (8) where B t denotes the state-transition model and process noise v k,t ∼ N (0, V t ) with error covariance matrix V t . The statetransition matrices, as well as the error covariance matrices 4 are assumed equal among the features. Note that vehicle and feature state motion is independent of each other.1,2 In the following, we will drop the subscript indexing on states and measurements w.r.t. vehicle/feature/time whenever the context allows. C. Measurement models At time step t, vehicle s ∈ S obtains two different kind of measurements: (i) measurements of the vehicle state x w.r.t. the reference frame, i.e., GNSS-like measurements; and (ii) measurements w.r.t. to features, i.e., from a radar-like (onboard) V2F sensor. Without loss of generality, we assume that the on-board sensors’ state is equal to the vehicle state. Thus an uncertain vehicle location implies an uncertain location for the on-board V2F sensor. 1) GNSS measurement: The GNSS measurement z G ∈ RMG of vehicle s ∈ S at time t is statistically modeled through the likelihood function p(z G \|x) with linear observation model z G = H G x + r, (9) where H G is the linear observation matrix and r ∼ N (0, R) with error covariance matrix R. 2) V2F measurement: Let Z F be a set of measurements from a tracking sensor that is susceptible to measurement noise, missed detections, and false detections. Examples of such sensors include camera, radar and LIDAR. Consequently, Z F = Z FA ] Z D , z = H 1 x + H 2 f + q, (11) where H 1 and H 2 denote observation matrices, and q ∼ N (0, Q) with error covariance matrix Q. Note that the measurement-to-feature state correspondence denoted DA, is in general not known and needs to be inferred from the measurements. Let the measurement likelihood for a single feature RFS F , i.e. \|F \| ≤ 1, be  1 if F = ∅, ZF = ∅   0  if F = ∅, Z F = {z} 1 − pD (x, f ) if F = {f }, Z F = ∅ η(Z F \|x, F ) =  F   pD (x, f )`(z\|x, f ) if F = {f }, Z =F{z} if \|F \| > 1, or \|Z \| > 1, (12) 1 For where 0 < α ≤ 1 is a constant such that it is a valid PDF. Depending on the specific sensor at hand, the FoV may be different and (13) needs to be adapted. D. Communication model We assume that every vehicle is able to communicate all obtained measurements (V2F and GNSS) with the RSU instantaneously and without errors. This implies that at any time t the number of vehicles communicating with the RSU can vary. The incorporation of a realistic V2I channel model and its performance impact is a point for future work. IV. P OISSON M ULTI -B ERNOULLI FILTERING WITH UNCERTAIN SENSOR STATE (10) where Z FA denotes the set of false alarm measurements due to clutter modeled by a PPP with intensity λc (z), and Z D denotes the set of detected features. Let a V2F measurement z ∈ Z D with state dimension RMV2F be obtained through the V2F sensor at vehicle s ∈ S w.r.t. feature k at time t. It is modeled through the likelihood function `(z\|x, f ) with linear observation model 0 where `(z\|x, f ) follows the measurement model (11). Note that due to (11), the probability of detection pD (x, f ) in (12) depends on the vehicle state x as well as on the feature state f . For instance, a limited sensor field-of-view (FoV) affects the probability of feature detection based on the distance between vehicle and feature. Remark 2: For the case the radar-like V2F sensor is able to detect features within a radius rmax , the probability of detection is defined ( α, if kH 1 x + H 2 f k2 ≤ rmax (13) pD (x, f ) = 0, otherwise, the sake of brevity, we present linear system and measurement models. In case the true system dynamics and/or measurement model are (mild) nonlinear, linearization steps can be performed similar to the steps taken in an extended Kalman filter (EKF) or the unscented Kalman filter (UKF) [33], [34]. In doing so, the proposed filter remains valid unaltered. 2 Note that the proposed filter only predicts over small time-horizons in the order of a few tens of milliseconds. Then the assumption on vehicle and feature states evolving independently is reasonable, because there is very little interaction among them within the prediction horizon. In this section, we formulate the proposed PMB filter with uncertain sensor state. We consider a tracking scenario subject to Section I-A, where there may be multiple features. In Section IV-C, we proceed with the asynchronous multisensor case allowing to track multiple features and vehicles with uncertain vehicle state. In Section IV-D, a tractable Gaussian density approximation of the proposed filter is given. The vehicle state PDF at time step t − 1 is indicated by subscript ’−’, i.e., p− (x), the PDF predicted to the current time step t (before updating by a measurement) is indicated by subscript ’+’, i.e., p+ (x), and the posterior PDF is stated without subscript. Similar definitions hold w.r.t. the feature RFS density. The proposed filter is developed within a Bayesian framework with alternating prediction and update steps operating on an RFS X by [11, Ch. 14]. Z f+ (X) = f (X\|X 0 )f− (X 0 )δX 0 , (14) and f (X\|Z) ∝ `(Z\|X)f+ (X), (15) where f− (X 0 ) is the prior RFS density, f (X\|X 0 ) is the RFS transition density, f+ (X) is the predicted density, and `(Z\|X) is the RFS measurement likelihood for measurement set Z. From the problem definition stated in Section III-A, we are interested in the joint posterior density of the vehicle and features considering all measurements up to the current time step t. We now proceed with the development of the proposed filter within this framework. 5 With a single vehicle, the prior joint vehicle-feature density is of form f− (x, F ) = p− (x)f− (F ), (16) where p− (x) is the prior PDF on the vehicle state, and f− (F ) is the prior PMB density. The latter density can be written in u terms of an PPP intensity of undetected features f− (F u ), i.e., features which are hypothesized to exist but have never been detected [15, Def. I], and the prior MB RFS density of detected d features f− (F d ), as [11, p. 484], [15] X u d f− (F u )f− (F d ). (17) f− (F ) = F u ]F d =F In (17), the PPP density of undetected features is Y u u u f− (F u ) = e−hD− ,1i D− (f ), (18) f ∈F u u where D− (f ) is the intensity of undetected features. We are interested in a low computational complexity method to compute the (posterior) joint vehicle-feature density f (x, F ) in every discrete time step t through incorporation of all sensor measurements. In doing so, the posterior density should remain in the same form as the prior joint density (16). A. Prediction step With the vehicle state x, and existing feature RFS F , the predicted joint vehicle-feature density is f+ (x, F ) = p+ (x)f+ (F ). (19) Here, the predicted vehicle state PDF is given by the Chapman-Kolmogorov equation [35] Z p+ (x) = p(x\|x0 )p− (x0 )dx0 , (20) where p(x\|x0 ) is the state transition PDF described by (7), and p− (x0 ) is the prior PDF. Similarly, the predicted feature state PMB density is calculated by Z f+ (F ) = f (F \|F 0 )f− (F 0 )δF 0 , (21) where f (F \|F 0 ) is the transition RFS density, and f− (F 0 ) is the prior PMB density. the predicted intensity of undetected u u features D+ (f ) of the predicted PPP density f+ (F u ) is given by Z u u D+ (f ) = Db (f ) + p(f \|f 0 )pS (f 0 )D− (f 0 )df 0 . (22) Here, the birth intensity is denoted Db (f ), feature transition PDF p(f \|f 0 ) is described by (8), feature survival probability u is denoted pS (f 0 ), and D− (f 0 ) denotes the prior intensity. The MB RFS density of detected features of (21) is X d i (F i ), (23) f+ (F d ) = f+ ]i∈I+ F i =F d where I+ denotes the set of existing features (before measurement update), and  i , Fi = ∅  1 − r+ i i i i f+ (F ) = (24) r p (f ), F i = {f i }  + + 0, otherwise. Here, the predicted PDF of feature i is Z p+ (f i ) = p(f i \|f 0i )p− (f 0i )df 0i , (25) where p− (f 0i ) is the prior PDF of feature i. The probability of existence of feature i is [15, Eqn. (40)] Z i i r+ pS (f 0i )p− (f 0i )df 0i , (26) = r− i where r− denotes the prior probability of existence, p− (·) the prior PDF, and pS (·) the probability of feature survival. B. Measurement update step Updating the joint vehicle-feature density (19) by any of the two types of different measurements, GNSS and V2F measurements, involves the application of Bayes’ theorem. In the following, we describe the update calculations using the different type of measurements. 1) Update with vehicle state measurement: Let z G be a measurement related to the vehicle state x, and unrelated to the set of features F , p(z G \|x, F ) = p(z G \|x). (27) For example, z G could be a GNSS and/or an inertial measurment unit (IMU) measurement. Given a predicted vehiclefeature density (19), and by Bayes’ theorem the updated density is f (x, F \|z G ) = p(x\|z G )f+ (F ). (28) In other words, the vehicle state density is updated with the measurement z G , the feature set density is unaffected by the update, and the independent form is retained (c.f. (16)). Note, this update step can be omitted in the absence of GNSS-like measurements, e.g., in a pure SLAM application [26]. 2) Update with cluttered set of feature measurements: Let the set of measurements Z F subject to the measurement model of Section III-C2 be indexed by M, and let A be the space of all DAs A for the predicted MB. A DA A ∈ A is an assignment of each measurement in Z F to a source, either to the background (clutter or new feature) or to one of the existing features indexed by I. It is therefore a partition of M ∪ I into non-empty disjoint subsets C ∈ A, called index cells3 . Remark 3: Due to the standard MTT assumption that the features generate measurements independent of each other 3 For example, let M = (m , m , m ) and I = (i , i ), i.e., three 1 2 3 1 2 measurements and two features. One valid partition of M ∪ I, i.e., one of the possible associations, is {m1 }, {m2 , i1 }, {m3 }, {i2 }. The meaning of this is that measurement m2 is associated to feature i1 , feature i2 is not detected, and measurements m1 and m3 are not associated to any previously detected feature, i.e., measurements m1 and m3 are either clutter or from new features . 6 [9], an index cell contains at most one feature index, i.e., \|C ∩ I\| ≤ 1 for all C ∈ A. Any association in which there is at least one cell, with at least two feature indices, will have zero likelihood because this violates the independence assumption. Further, due to the point feature assumption, any feature generates at most one measurement in each time step, i.e., \|C ∩ M\| ≤ 1 for all C ∈ A. Any association in which there is at least one cell, with at least two measurement indices, will have zero likelihood because this violates the point feature assumption. If the index cell C contains a feature index, then let iC denote the corresponding feature index. Further, if the index cell C contains a measurement index, then let mC denote the corresponding measurement index. Measurements in C not assigned to any feature are associated to the background. With the help of Bayes’ rule, the updated joint vehicle-feature density is X f u (F u \|x) f (x, F \|Z F ) = × F u ]F d =F X A A w p (x\|Z F )f d,A (F d \|x, Z F ), (29) A∈A P where wA denotes the weight of DA A ∈ A with A∈A wA = F 1, and pA (x\|Z ) denotes the vehicle state posterior stated in Appendix A. For now, let us assume the DA weights are given. The undetected feature density f u (F u \|x) and the detected feature density f d,A (F d \|x, Z F ) are stated in Appendix B. Equation (29) does not factorize as f (x, F \|Z F ) = p(x\|Z F )f (F \|Z F ) where f (F \|Z F ) is a multi-Bernoulli density such that it remains in the same form as the prior (c.f. (19)); on the contrary, there are many dependencies between the feature state RFS and the vehicle state. This means that existing independence-assuming tracking frameworks cannot be applied directly [9], or introduce a significant increase in computational complexity [28]. To overcome this, we approximate f u (F u \|x) ≈ fˆu (F u ), f d,A (F d \|x, Z F ) ≈ f˜d,A (F d \|Z F ), (30) (31) where the functions on the right hand side need to be found. Towards this end, we make the following approximations for the vehicle and feature dependent probability of detection u p̂D , f ∈ F u pD (x, f ) ≈ (32) p̂iD , f ∈ F d , with F u ] F d = F , and where ZZ u p̂uD = pD (x, f )p+ (x)D+ (f )dxdf , ZZ p̂iD = pD (x, f )p+ (x)pi+ (f )dxdf . for p̂iD . Under approximation (32), the updated density (29) becomes X fˆ(x, F \|Z F ) ∝ fˆu (F u ) × F u ]F d =F X A A w p (x\|Z F )fˆd,A (F d \|x, Z F ) (35) A∈A with fˆu (F u ) and fˆd,A (F d \|x, Z F ) given in Appendix C. We observe in (35) that the undetected feature density fˆu (F u ) depends only on the undetected feature RFS F u , and is independent of the other stochastic variables. What remains are dependencies between the detected features, and the vehicle. To remove the dependency on the vehicle state in the detected feature density fˆd,A (F d \|x, Z F ) in (35), we map the vehicle state uncertainty onto the V2F measurement uncertainty. This is done by averaging the V2F measurement likelihood by the vehicle state uncertainty. In doing so, the detected feature density under association A becomes independent of the vehicle state x leading to the approximation (31), where the approximated updated feature set density f˜d,A (F d \|Z F ) is given in Appendix D. The updated joint vehicle-feature density is approximated as X f (x, F \|Z F ) ∝ fˆu (F u ) F u ]F d =F X A A w p (x\|Z F )f˜d,A (F d \|Z F ). × (36) A∈A In this form, the vehicle state PDF is now independent on the feature RFS, and so is the feature density on the vehicle state. This allows to state the weights wA , which are given in Appendix E. Note, (36) is a Poisson multi-Bernoulli mixture (PMBM) density, where each DA A ∈ A denotes a hypothesis on the posterior vehicle state x and the detected feature state F d , weighted by wA . It can be reduced to a PMB density using, e.g., the variational approximation presented in [36], or based on the marginal DA probabilities [15]. We apply the latter approach, where for the reduction of (36) to the form of (16), the track-oriented marginal MeMBer/Poisson (TOMB/P) algorithm (c.f. [15]) is used. This results in a single hypothesis per detected feature described by a Bernoulli process (c.f. (3)), and per vehicle described by its PDF; as well as the intensity of undetected feature described by a PPP (c.f. (5)). This means, the summation over the DA space A has vanished in (36), retaining the form of (16). C. Multi-scan scenario using multiple vehicles with uncertain state (33) (34) Here, (33) is the expected probability of detection for an undetected feature under the predictive distributions for x and f , and (34) is the expected probability of detection for a detected feature. An alternative (and stronger) R approximation for (33) would be p̂uD = pD (x̂, f̂ ) with x̂ = xp+ (x)dx and f̂ the estimated feature state (c.f. Section II-B), and similarly Up to this point, we discussed PMB filtering with a single vehicle and uncertain state, where GNSS and V2F measurements are used. To achieve feature tracking as described in Section I-A, where sensors are mounted on several vehicles, we have to consider the multisensor case. Furthermore, depending on the infrastructure, sensors are time synchronized, i.e., take measurements at the same time step t or are not synchronized, i.e., measurements from a sensor arrives timestamped, but the time the sensor acquires the measurement is independent of other sensors. Let there be a single RFS F 7 modeling the features state, S = \|S\| vehicles with uncertain vehicle state xs with s = 1, 2, . . . S. The set of vehicles taking a measurement at time step t is given by C ⊆ S, where each vehicle c ∈ C provides a vector with a GNSS measurement z c and V2F measurements Z F c . Furthermore, T T T T T T T let x̃ = [xT 1 , x2 , . . . , x\|S\| ] , x̂ = [x1 , x2 , . . . , x\|C\| ] , ẑ G = S F F T T T [z G T 1 , z G 2 , . . . , z G \|C\| ] , and Ẑ = c∈C Z c . In the multisensor case, the PMB filter with uncertain sensor state follows the unisensor case proposed in Section IV: the joint vehicle-feature density is predicted, and updated by the GNSS and the V2F measurements. Thereby, the predicted vehicle-feature density (19) becomes f+ (x̃, F ) = f+ (F )p+ (x̃). (37) Updating the joint density (19) by the GNSS measurement results in (28). In the multisensor case, it becomes f (x̃, F \|ẑ G ) = f+ (F )p(x̃\|ẑ G ). F u ]F d =F X F F wA pA (x̃\|Ẑ )f˜d,A (F d \|Ẑ ), (39) A∈A where F F pA (x̃\|Ẑ ) ∝ p+ (x̃)`A (Ẑ \|x̂), In order to obtain a low complexity implementation, we describe the vehicle state PDF by a Gaussian with PDF p(x) , N (µ, Σ) with mean parameter µ and covariance matrix Σ. Similarly, the RFS density of a MB process of feature i is described by a Bernoulli random variable ri and a Gaussian PDF p(f i ) , N (µf , Σf ) with parameters µf and Σf . Under this description and the system models of Section III, we can express the prediction and update steps of the proposed filter in closed form with low computational complexity, whose steps are described next. 1) Prediction Step: The predicted vehicle state PDF of (19) is (41) p+ (x) = N (µx+ , Σx+ ), where with the use of (7) µx+ = Aµx− , (38) After incorporating the GNSS measurements we proceed with incorporating the V2F measurements. In the unisensor case, this resulted in the approximated joint sensor-feature density (36). In the multisensor case, this density becomes X F f (x̃, F \|Ẑ ) ∝ fˆu (F u ) × D. Gaussian Density Approximation Σx+ = AΣx− A + W . (43) The notation above means that if at time step t − 1 the vehicle state PDF p− (x) = N (µx− , Σx− ), then at time step t, the predicted state PDF (before updating by a measurement) is p+ (x) = N (µx+ , Σx+ ). The intensity of undetected features (22) is modeled by GM consisting of newborn features with weight λb (f ) = λb and PDF p(f ) = N (µb , Σb ), where µb andΣb are the birth mean and covariance matrix; and undetected features survived to the current time step with prior parameters {λu , µf − , Σf − } and predicted parameters (40) F and f˜d,A (F d \|Ẑ ) involves the marginalization over the vehicle prior PDF p+ (x̂), i.e., containing only vehicles which provide a V2F measurement, according to (70). Note that here (38) is used as prior in the joint sensor-feature density (39). Furthermore, the space A of all DA increases with an increase in the number of communicating sensors \|C\| for the predicted MB. In terms of complexity, this increase can be significant, because of the increase of possible feature stateto-measurement associations. Remark 4: Several different approaches exist to tackle this DA problem in a tractable manner. For instance, by employing sequential sensor-by-sensor measurement updates on (36), or by performing variational inference [37], or by solving the DA in parallel on a sensor-by-sensor basis [21]. Here, we employ the sequential measurement update strategy to limit the size of the DA space A. In doing so, subsequent sensors will benefit from updated vehicle and feature information of preceding sensors. In our application example (c.f. Section I-A), this means that an update of the joint vehicle-feature density, with measurements from a well localized vehicle (certain vehicle state), results in an improvement of feature tracking performance when prior information on the features is low. An update of the joint vehicle-feature density, with measurements from a poorly localized vehicle (uncertain vehicle state), allows to reduce the uncertainty of its own vehicle state when prior information on the features is high. (42) T λu+ = hpS , λu i , µf + = Bµf − , (44) (45) Σf + = BΣf − B T + V . (46) The predicted MB density of detected features, stated in (21), i predicted using has single feature Bernoulli parameters r+ i (26), and single feature PDF p+ (f ) calculated similarly to (45) and (46). 2) Update step: The joint vehicle-feature state density (28) is computed by updating the predicted vehicle-feature density (19) with the GNSS measurement z G through the Kalman update step [33], [35], where the vehicle state PDF is given by p(x\|z G ) = N (µx , Σx ). (47) Here, µx = µx+ + K (z G − H G µx+ ), (48) Σx = Σx+ − K H G Σx+ , (49) T K = Σx+ H G S , (50) T S = H G Σx+ H G + R. (51) The matrices H G and R are defined in (9). The updated joint vehicle-feature density (36) is computed by updating the predicted vehicle-feature density (19) with the V2F measurement Z F . Note that depending on the time difference between the GNSS and V2F measurements, (28) may be used instead of (19) as prior on the joint vehicle-feature 8 density. In order to calculate (36), the vehicle measurementstate likelihood `(z\|x) used in (64) and (65), and the feature measurement-state likelihood `(z\|f ) used in (74) are needed. The measurement-feature state likelihood (74) for z given, can be written in terms of f in closed form by `(z\|f ) ∝ N (µz\|f , Σz\|f ) (52) with −1 T T H 2, Σ−1 z\|f = H 2 Q + H 1 Σx+ H 1 µz\|f = H + 2 z − H 1 µx+ . (53) (54) Here, p+ (x) = N (µx+ , Σx+ ), and (·)+ denotes the MoorePenrose pseudo inverse. Proof: See Appendix F. The vehicle measurement-state likelihood (64) for a given z and marginalized over a detected feature, and in (65) marginalized over an undetected feature, can be written in terms of x in closed form by `(z\|x) = N (µz\|x , Σz\|x ) (55) described by a Bernoulli component (c.f. (3)) with Gaussian PDF, has a memory footprint of a Bytes needed to store {r, f , cov(f )}, where f ∈ RNf . Storing the vehicle state requires b Bytes, where the state of each vehicle is described by a Gaussian PDF with parameters {x, cov(x)} with x ∈ RNx . Without pruning of low-probability Bernoulli components in F , the memory footprint of the proposed filter is a\|F \| + b Bytes. In the multisensor approach of Section IV-C, the RSU receives GNSS measurement z G and V2F measurement Z F from a vehicle and then performs the filter update computation. The RSU broadcasts the vehicle state estimates either whenever a new measurement has been processed, or based on a fixed schedule. Should information of detected (and tracked) features be required at the vehicles, then the single-feature pdfs need to be transmitted as well (c.f. Section II-B). V. N UMERICAL R ESULTS We consider a scenario similar to the one outlined in Fig.1, where we apply the proposed multifeature-multisensor state tracking filter presented in Section IV. with −1 T T H 1, Q + H Σ H Σ−1 = H 2 f 2 1 z\|x + µz\|x = H + 1 z − H 2 µf + . (56) (57) Here, (55) equals (64) for feature PDF p(f ) , pi+C (f ) = u N (µf + , Σf + ), and in (65) for feature PDF p(f ) , D+ (f ) = N (µf + , Σf + ). Proof: The proof is analogous to the proof of the feature measurement-state likelihood (52) with the only difference that the unknown is x instead of f . E. Computational complexity, memory footprint, and communication demand Computational complexity is dominated by the matrix inversion needed to update the feature and vehicle densities, and the measurement-to-feature state DA. Updating the joint vehiclefeature density f (x, F \|z G ) using GNSS measurement z G requires a matrix inversion which scales as O(Nx3 ). The update of an MB component in f˜d,A (F \|x, Z F ) by V2F measurement z mC ∈ Z F scales as O(Nf3 ), and consequently as O(\|Z F \|Nf3 ) for the whole measurement set. Computational complexity of DA is O(\|F \|\|Z F \|) [38]. Hence, the update of the joint vehiclefeature density (36) by V2F measurement set Z F scales as O(\|F \|\|Z F \| + Nf3 \|Z F \| + Nx3 \|Z F \|), where the last term comes from vehicle state update of Z F . In each time step t, the size of undetected features RFS F u increases by \|Db (f )\| new born targets. The number of existing feature-tracks increases by \|Z F \|, a new feature-track per measurement [15], using a Bernoulli component per track. For each existing feature-track \|Z F \| + 1 hypotheses (plus one for a missed detection) are computed. The TOMB/P algorithm reduces each feature-track to a single hypothesis track. Pruning of feature-tracks with low probability of existence r allows to keep the number of feature-tracks tractable. Each hypothesis, A. Setup The state of a vehicle at time step t is x = [pT , v T ]T with position p ∈ R2 and velocity v ∈ R2 . Vehicle dynamics follow a linear constant velocity (CV) model described by (7) with 1 Ts A= ⊗ I 2, (58) 0 1 where Ts = 0.5 s, and W =r Ts3 /3 Ts2 /2 Ts2 /2 Ts ⊗ I2 (59) with r = 0.05 m2 , and ⊗ denoting the Kronecker product. The state of a feature at time t, denoted f ∈ R4 , is comprised of Cartesian position and velocity, similar to the vehicle state x. There are maximal five features present, if not noted otherwise. Furthermore, feature dynamics follow the CV model with the same parameters used for the vehicles. To generate a challenging scenario for DA, we initialize the feature states f ∼ N (0, 0.25I 4 ) at t = 175 for all features and run the CV model forward and backward in time similar to [15, Sec. VI]. The first feature enters the scene after t = 0, the second after t = 20 and so on. Once present, features stay alive for the remaining simulation time. Vehicle and feature trajectories are shown in Fig. 2. The observation matrix of the GNSS measurement model (9) is H G = [1 0] ⊗ I 2 , (60) 2 where R = σG I 2 . For vehicle 1, we assume it has low 2 location uncertainty with σG = 0.1 m2 and for vehicle 2 2 high location uncertainty with σG = 10 m2 , corresponding to a vehicle with high quality GNSS receiver and one with low quality GNSS receiver. In the single sensor case, only vehicle 1 is present, and in the multisensor case both vehicles are present, if not noted otherwise. The V2F measurement model follows (11), where H 1 , H G , H 2 , −H G and 9 200 y dimension in m 100 vehicle feature 0 −100 −200 −300 −500 −400 −300 −200 −100 x dimension in m 0 100 Fig. 2. Vehicle and feature trajectories. Fig. 3. V2F measurements in x (top panel) and y dimension (bottom panel) for each time step t. 2 Q = σV2F I 2 . In Fig. 3, V2F measurements are shown for each time step t including clutter measurements. Following u [15], we set the initial undetected feature intensity to D− (f ) = 2 2 T 10N (0, P ), where P = diag([100 , 1, 100 , 1] ) to cover the ranges of interest of the feature state. The feature birth intensity is set to Db (f ) = 0.05N (0, P ), the average number of false alarms per scan to λc = 20, with uniform spatial distribution on [−rmax , rmax ] with parameter rmax = 1000 m. Furthermore, the probability of survival is pS = 0.7 and the probability of detection is pD (x, f ) , pD = 1. To asses feature tracking performance, we use the optimal sub-pattern assignment (OSPA) metric with cut-off parameter c = 20 m and order p = 2 [39]. The vehicle tracking performance is assessed in terms of the root mean square error (RMSE). B. Discussion First, we discuss the impact of an uncertain vehicle state on feature tracking performance using a single vehicle and multiple-features. After that, we consider the multitarget- multisensor case from Section IV-C, and show scaling results in terms of numbers of features and vehicles tracked. 1) Impact of uncertain vehicle state on feature tracking performance: The features and vehicle trajectories are outlined in Fig. 2 with V2F measurements in Fig. 3. In Fig. 4, the feature state OSPA is plot for each time step t. We observe that there are peaks with a high OSPA value when a new feature enters the scene. These peaks are due to a cardinality mismatch between the feature RFS estimate and the true feature set. Furthermore, there is a high OSPA value around time step t = 175. At this point in time, features are closely spaced 2 together w.r.t. the V2F measurement variance σV2F resulting in a challenging scenario for measurement-to-state DA. In Fig. 5 the cardinality of the feature RFS is plotted over time. Around time step t = 175, the filter overestimates the feature RFS cardinality, which may be caused by clutter measurements. Note, different Monte-Carlo (MC) realizations produce a slightly different outcome and feature OSPA value, but with the same tendency. This behavior w.r.t. feature appearance and the effect when they are spatially close agrees with the findings in [15] for a known sensor (vehicle) state. Furthermore, we observe in Fig. 4, that the OSPA is low for time steps where features are spatially separated and already present in the scene. Then the MTT filter is able to produce feature estimates with low error. In Fig. 6, the feature state OSPA averaged over all 351 time steps is plot for different values of GNSS measurement 2 . The increase of GNSS measurement variance variance σG leads to an increased vehicle state uncertainty with the effect of an increase of the average feature OSPA. This OSPA increase consists of two components. First, an increased feature state 2 estimation error due to a higher value of σG . Second, this results in features staying spatially close together w.r.t. the 2 2 together) and σV2F feature state measurement uncertainty (σG for a longer period of time around time step t = 175. Hence, DA is more challenging with the effect of an increased feature OSPA in this regime. In the same figure, the average feature OSPA without modeling the present vehicle state uncertainty is plotted using the conventional PMB filter [15]. We observe that not modeling the present vehicle state uncertainty has a negative effect on feature tracking performance. In Fig. 7, the average feature state OSPA is plotted for 2 different values of V2F measurement noise variance σV2F . We observe, that a higher V2F noise variance leads to an increased OSPA value. This is because the single feature state estimation error increases and DA becomes more challenging. Note, the results of Fig. 6 and Fig. 7 are averages over 10 MC realizations. 2) Multitarget-multisensor tracking performance: The RMSE of the vehicle state is plotted for each time step t for 2 small) in Fig. 8, vehicle 1 with low location uncertainty (σG 2 and in Fig. 9 for vehicle 2 with high location uncertainty (σG high). As a benchmark, results from a centralized Kalman filter (KF) are plot as well, where measurement-to-feature DA is known and where the augmented state vector contains all vehicle and all feature states. Furthermore, the tracking performance using a local KF is plot. The local KF performs filtering only on the individual vehicle state separately using 20 4 15 3 RMSE Feature OSPA 10 10 0 50 100 150 200 time-step t 250 300 0 350 2 = 10−3 m2 and σ 2 2 Fig. 4. Feature OSPA with σG V2F = 0.25 m . cardinality 4 100 150 200 time-step t 250 300 350 0.8 2 0 50 100 150 200 time-step t 250 300 CDF 0.6 350 0.2 0 proposed conventional 15 Vehicle Vehicle Vehicle Vehicle Vehicle Vehicle 0.4 20 Avg. feature OSPA 50 1 estimate true 2 = 10−3 m2 and σ 2 2 Fig. 5. Feature cardinality with σG V2F = 0.25 m . 10 0 0.5 1 1, 1, 1, 2, 2, 2, proposed local KF central KF (known DA) proposed local KF central KF (known DA) 1.5 RMSE in m 2 2.5 3 Fig. 10. CDF plot of vehicle state RMSE. 5 0 2 4 6 8 10 2 σ G in m2 Fig. 6. Average feature OSPA for different values of GNSS measurement 2 . The V2F noise variance is set to σ 2 2 variance σG V2F = 0.25 m . Avg. feature OSPA 4 3 2 1 0 0.2 0.4 2 σ V2F 0.6 in 0.8 1 m2 Fig. 7. Average feature OSPA for different values of V2F measurement 2 2 = 10−3 m2 . variance σV2F . The GNSS noise variance is set to σG 1 local KF central KF (known DA) proposed 0.8 RMSE 0 Fig. 9. Vehicle state RMSE of vehicle 2. 6 0 2 1 5 0 local KF central KF (known DA) proposed 0.6 0.4 0.2 0 0 50 100 150 200 time-step t Fig. 8. Vehicle state RMSE of vehicle 1. 250 300 350 only GNSS measurements and does not estimate feature states. Note, the performance of the local KF can be considered as the worst-case performance on vehicle state estimation, since V2F measurements are not considered at all. We observe from Fig. 8, that for vehicle 1, which has low GNSS measurement noise, all three filter methods deliver a similar performance. The reason for this is that, due to the high accuracy of GNSS measurements, not a lot of information (to improve the vehicle state) is provided from feature tracking, i.e., feature tracking error is high w.r.t. the vehicle state tracking error of vehicle 1 (after updating by the GNSS measurement). In Fig. 10, the cumulative distribution function (CDF) of the RMSE is plot. Here, the low RMSE of vehicle 1 using the three different filters can be observed as well. Moving the focus to vehicle 2, we observe that the RMSE of the local KF is much higher compared to the central KF, which is caused by the high noise in the GNSS measurements. Due to the low RMSE of vehicle 1’s state there is relevant position information in the system, which can be transfered from vehicle 1 to vehicle 2 via the features utilizing the V2F measurements. In 80% of all cases, the RMSE of vehicle 2 is below 0.5 m with the proposed filter, compared to 1.4 m with the local KF. Despite this great improvement of the proposed filter over the local KF, it does not achieve the performance of the central KF, where the RMSE is below 0.3 m. The reason for the difference is that the central KF has knowledge of the correct DA, knows the true number of features present, and ignores clutter V2F measurements. Furthermore, it tracks any present correlations between features and vehicles not modeled 11 0.1 time in s 8 · 10−2 6 · 10−2 4 · 10−2 2 · 10−2 0 0 5 10 No. of features 15 20 Fig. 11. Average computation time per time step for different number of present features. The number of vehicles is set to two. time in s 0.2 0.15 0.1 5 · 10−2 0 0 5 10 No. of vehicles 15 be incorporated either in a time-synchronized manner where sensor measurements are aggregated in a super-sensor state, or in a non time-synchronized manner through asynchronous update steps, executed whenever sensor measurements arrive at the central node. Simulation results showed for a unisensormultitarget tracking scenario with known sensor state that tracking performance assessed by the OSPA distance metric is equivalent to William’s PMB filter. In a scenario with present vehicle state uncertainty, the proposed filter showed superior feature tracking performance over the conventional PMB filter due to the modeling of this type of additional uncertainty. In a multisensor-multitarget tracking scenario, feature information from a well localized vehicle (sensor) allows to significantly reduce the vehicle state uncertainty of (previously) poor localized vehicles. This improvement is possible through joint observation of a subset of the present features and is supported by simulation results. 20 A PPENDIX A. Vehicle State Posterior Fig. 12. Average computation time per time step for different number of vehicles. The number of features is set to five. by the proposed filter. The proposed filter needs to infer the measurement-to-feature DA, estimate the number of features currently present, and needs to appropriately handle clutter in the V2F measurement set Z F . 3) Scaling results: In Fig. 11, the average computation time per time step t is plot for a simulation with two vehicles and different number of present features. We observe that computation time increases linearly as the number of present features increases. This scaling result is different to the PF based implementation of the PMB filter in [21] with a known vehicle state. There, the authors reported an exponential increase of computation time. In Fig. 12, the average computation time per time step t is plot for a simulation with five features and different number of vehicles. Here, computation time increases linearly as the number of vehicles increases. Furthermore, we investigated the average computation time per time step t for 2 different values of the GNSS measurement variance σG and 2 of the V2F measurement variance σV2F . In the simulation scenario with five features and two vehicles, the average computation time remained constant around 2.4 · 10−2 s for 2 10−3 m2 ≤ σG ≤ 10 m2 . For a V2F measurement variance −3 2 2 10 m ≤ σV2F ≤ 10 m2 , the average computation time 2 linearly increased from 2 · 10−2 s to 3 · 10−2 s as σV2F increased. VI. C ONCLUSIONS This paper presented a Poisson multi-Bernoulli filter for multisensor-multitarget tracking with uncertain sensor states. Two different kind of measurements, observations of the sensor state and observations of the features, were used to obtain accurate feature and sensor state tracking. The proposed parametric filter implementation scales linearly with the number of features or sensors. Information from multiple sensors can The vehicle state posterior of (29) and (36) is proportional to the vehicle’s prior PDF p+ (x) times the measurement likelihood `A (Z F \|x) as pA (x\|Z F ) ∝ p+ (x)`A (Z F \|x), (61) where A Y Z F ` (Z \|x) ∝ `(z mC \|x, f )pi+C (f )df C∈A: C∩I6=∅ C∩M6=∅ f ∈F d × Y Z u `(z mC \|x, f )D+ (f )df (62) C∈A: C∩I=∅ C∩M6=∅ f C ∈F u ∝ Y `iC (z mC \|x) C∈A: C∩I6=∅ C∩M6=∅ f ∈F d Y `u (z mC \|x) (63) C∈A: C∩I=∅ C∩M6=∅ f ∈F u with Z iC ` (z\|x) = `u (z\|x) = Z `(z\|x, f )pi+C (f )df , (64) u `(z\|x, f )D+ (f )df . (65) Here, (64) and (65) map the feature uncertainty on the V2F measurement likelihood. B. Updated Undetected and Detected Feature Density The updated joint vehicle-feature density (29) has undetected feature density Y u f u (F u \|x) ∝ (1 − pD (x, f ))D+ (f ) (66) f ∈F u 12 where we average over the predicted vehicle state and 0 marginalize over all subsets F C not equal to F C . Eqn. (70) has Bernoulli parameters [15, Eqns. (45) to (57)]  i i (1−p̂DC )r+C  C ∩ I 6= ∅, C ∩ M = ∅  iC iC  1−p̂D r+ F iC r̃ \|Z = 1 if C ∩ I 6= 0 C ∩ M 6= 0 u  ,D+ p̂u i  D h`z mC E D  C∩I=0 C ∩ M 6= 0, m u u and detected feature density f d,A ∝ d F (F \|x, Z ) X ]C∈A F C =F Y × d Y iC (F C ) η(∅\|x, F C )f+ C∈A: C∩I6=∅ C∩M=∅ iC (F C ) η({z mC }\|x, F C )f+ λc (z C∈A: C∩I6=∅ C∩M6=∅ D `zmC ,D+ (72) Y × C )+p̂ u (F C ). η({z mC }\|x, F C )f+ (67) C∈A: C∩I=∅ C∩M6=∅ p̃iC (f \|Z F ) =  i p+C (f )     `zmC (f )pi+C (f ) Here, the first product considers the cases where no measurement is associated to any of the existing features, the second product considers the cases where a measurement is associated to an existing feature, and the last line considers the case where a measurement is associated to the background (clutter or undetected feature). E i `zmC ,p+C u `zmC (f )D+ (f ) E D u `zmC ,D+ D     if C ∩ I 6= ∅, C∩M=∅ C ∩ I 6= 0 C ∩ M 6= 0 C∩I=0 C ∩ M 6= 0, (73) where Z `z (f ) = `(z\|f ) = `(z\|x, f )p+ (x)dx. (74) E. Approximated Updated Feature Density Weights C. Updated Undetected and Detected Feature Density Approximation The joint vehicle-feature density approximation (35) has undetected feature density Y fˆu (F u ) ∝ (1 − p̂uD )D+ (f ) (68) The weight of a global association hypothesis A ∈ A stated in (36) is [15, Eqn. (67)] Y Y iC iC iC iC p̂D `zmC , pi+C p̂D ) r+ wA ∝ (1 − r+ C∈A: C∩I6=∅ C∩M=∅ f ∈F u × ]C∈A F × C =F Y d Y iC (1 − p̂uD ∆\|F C \| )f+ (F C ) Here, we proof (52). With (11) and p+ (x) = N (µx+ , Σx+ ) known, define iC η({z mC }\|x, F C )f+ (F C ) Y (75) F. Proof of Measurement-Feature State Likelihood C∈A: C∩I6=∅ C∩M=∅ C∈A: C∩I6=∅ C∩M6=∅ × u ). (λc (z mC ) + p̂uD `zmC , D+ C∈A: C∩I=∅ C∩M6=∅ and detected feature density fˆd,A (F d \|x, Z F ) X ∝ Y C∈A: C∩I6=∅ C∩M6=∅ y , z − H 1 x − q, (76) p(y) = N (z − H 1 µx+ , H 1 Σx+ H T 1 ). (77) and consequently u η({z mC }\|x, F C )f+ (F C ), (69) C∈A: C∩I=∅ C∩M6=∅ C where ∆\|F C \| = 1 for F 6= ∅, and zero otherwise. 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Implementation of Tetris as a Model Counter∗ Atri Rudra University at Buffalo, SUNY [email protected] arXiv:1701.07473v1 [cs.DS] 25 Jan 2017 Jimmy Dobler University at Buffalo, SUNY [email protected] Abstract Solving #SAT problems is an important area of work. In this paper, we discuss implementing Tetris, an algorithm originally designed for handling natural joins, as an exact model counter for the #SAT problem. Tetris uses a simple geometric framework, yet manages to achieve the fractional hypertree-width bound. Its design allows it to handle complex problems involving extremely large numbers of clauses on which other state-of-the-art model counters do not perform well, yet still performs strongly on standard SAT benchmarks. We have achieved the following objectives. First, we have found a natural set of model counting benchmarks on which Tetris outperforms other model counters. Second, we have constructed a data structure capable of efficiently handling and caching all of the data Tetris needs to work on over the course of the algorithm. Third, we have modified Tetris in order to move from a theoretical, asymptotic-time-focused environment to one that performs well in practice. In particular, we have managed to produce results keeping us within a single order of magnitude as compared to other solvers on most benchmarks, and outperform those solvers by multiple orders of magnitude on others. ∗ This research was supported in part by grant# NSF CCF-1319402. 1 1 Introduction #SAT is the prototypical #P-complete problem. #SAT (as well as its NP-complete cousin SAT) are not only of great interest in computational complexity but by their completeness turn out to be a great tool to model a wide host of practical problems. This has led to an explosion of SAT solvers that try to solve practical instances of #SAT or SAT by exploiting structure in these instances. For this paper, we will assume the importance of designing #SAT and SAT solvers as a given. We refer the reader to the book chapters by Gomes, Sabharwal and Selman on #SAT solvers (also known as model counters) [15] and by Gomes et al. on SAT solvers [14] for more details. A common technique is the DPLL procedure, a depth-first search procedure where the algorithm makes guesses on the assignments one variable at a time, determines at each stage whether or not this produces a conflict, and uses that information to learn new clauses and get closer to finding the satisfying assignment [17]. Recently in the database literature, the work of Abo Khamis et al. [6] connected the DPLL procedure to computing natural joins. In particular, they presented the Tetris algorithm, which computes the natural join with beyond worst-case theoretical guarantees. As a special case, Tetris also recovers some of the recent worst-case optimal join results [24, 30, 25]. Abo Khamis et al. then showed that Tetris is an DPLL procedure and pointed out how one of the main step in their algorithm is exactly the resolution step that is ubiquitous in DPLL-based SAT solvers. Given the close ties of SAT solvers to DPLL, they left open the following intriguing possibility: Can Tetris be implemented as a SAT solver or model counter that can compete with state-ofthe-art solvers? Our contributions Our main result in this paper is to show that Tetris can indeed be implemented as a model counter that is competitive with state-of-the-art model counters on actual datasets. While [6] presented a nice geometric framework to reason about algorithms to compute the natural join query, some of its simplicity arose from inefficiencies that matter when implementing Tetris as a model counter. Before we present the issues we tackle, we give a quick overview of Tetris. The fundamental idea is that, rather than working to create the output of a join directly, it instead attempts to rule out large sections of the cross product of the joined tables [6]. Initially, Tetris is given a set of sets whose union is the set of all incorrect solutions to the problem Tetris is solving. In other words, any solution to the problem must not be a member of this union. By efficiently querying this set of sets, and by adding to it intelligently at various times (such as by adding a new exclusion whenever an output point is found), Tetris is able to rule out increasingly large sets of potential solutions. Once it has ruled out all possible solutions, it terminates and outputs the list of solutions. We tackle the following three issues with the theoretical presentation of Tetris in [6]: 1. At any point, Tetris needs to keep track of the union of all the potential solutions it has ruled out. To do this, [6] used a simple trie data structure to keep track of the union. However, this loses some poly-logarithmic factors and proves detrimental to practical performance. To deal with this, we design a new data structure that essentially compresses consecutive layers in the traditional trie into one ‘mega layer.’ Inspired by the used of SIMD instructions by EmptyHeaded [5] (to speed up implementations of worst-case optimal join algorithms [25]), we set up the compression in a manner that lends itself to speedup via SIMD instructions. 2 2. The analysis of Tetris in [6] was for data complexity. This implies they could afford to use exponential (in the size of the join query) time algorithm to find an appropriate ordering in which to explore different variables. In #SAT instances, we can no longer assume that the number of variables is a constant, and hence we cannot obtain an optimal ordering using a brute force algorithm. We deal with this by designing heuristics that take the structure of Tetris into account. 3. As mentioned earlier, Tetris (like a DPLL procedure) performs a sequence of resolutions and theoretically, it can store the outcomes of all the resolutions it performs. However, for practical efficiency, we use a heuristic to decide which resolution results to cache and which ones to discard. Our experimental results are promising. On some natural #SAT benchmarks based on counting number of occurrences of small subgraphs in a large graph (which we created), our implementation of Tetris is at least two orders of magnitude (and in most cases more than three orders of magnitude) faster than the standard model counters (sharpSAT [28], Cachet [26] and dSharp [22]). We also compared Tetris with these model counters on standard SAT benchmarks, where Tetris was either comparable or at most 25x slower. Theoretical Implications While this paper deals with an experimental validation of the theoretical result from [6], we believe that it highlights certain theoretical questions that are worth investigating by the database community. We highlight some of our favorite ones that correspond to each of our three main contributions: 1. Extending Tetris beyond join queries. As our work has shown, Tetris can be used to solve problem beyond the original natural join computation. Recently, the worst-case optimal join algorithms were shown to be powerful enough to solve problems in host of other areas such as CSPs (of which MaxSAT is a prominent example), probabilistic graphical models and logic [7]. (Also see the followup work [18].) The beyond worst-case results in [6] have so far seemed more of a theoretical novelty. However, given that this paper demonstrates the viability of Tetris in practice, this work opens up the tantalizing possibility of extending the theoretical results of Tetris to problems captured by [7, 18]. Such a result even for MaxSAT would be of interest in practice. 2. Computing orderings efficiently. As mentioned earlier, the theoretical results for Tetris assumes that the required ordering among variables can be computed in exponential time. However, for applications in SAT (as well as other areas such as probabilistic graphical models, assuming the question in the item above can be answered), we need to compute orderings that are approximately good in polynomial time. Thus, a further avenue of theoretical investigation is to come up with a polynomial time algorithm to compute the ordering, and to prove some guarantees on the loss of performance from the case where Tetris has access to the ‘optimal’ ordering. Some of the heuristics developed in our paper might prove to be good starting points for this investigation. We would like to point that that the importance of efficiently computing variable orderings has been studied a lot in AI and database literature. Some of the very recent work on Generalized Hypertree Decompositions (which are well known to be equivalent to variable elimination orderings) could potentially be useful towards this goal [13]. 3. Time-space tradeoff. Recent results on worst-case optimal algorithms to compute natural joins [24, 30, 25] and to compute joins with functional dependencies [8] all focus exclusively on time complexity. However, as highlighted by our work, being more prudent with space usage in fact benefits 3 actual performance. This point was also indirectly highlighted in [6], where it was shown that resolution schemes that did not cache their intermediate results are strictly less powerful than those that do (in the context of computing the natural join). However, we believe that a systematic theoretical study of the tradeoff between time and space needed to compute the natural join is an attractive route to pursue. We will begin in Section 2 by introducing the fundamental concepts necessary to understand both SAT problems and details on Tetris itself, all while giving a hands-on example of how Tetris would handle a toy example. From there, we will move into Section 3, an in-depth analysis of our major contributions. Afterwards, we will continue with our experimental results in Section 4. Then we will discuss related work in the field in Section 5. 2 Background In this section, we will introduce the concepts necessary to understand how Tetris functions, introduce the concept of resolutions, and walk through how Tetris would handle a simple input. 2.1 SAT and Boxes We begin by defining several key terms and ideas. Recall that a SAT problem consists of a series of Boolean variables, x 1 , x 2 , ..., x n , joined together in a series of AND and OR clauses. Problems are generally presented in the conjunctive normal form (CNF), a simplification wherein the entire formula is written as a series of ANDs over a set of disjunctive clauses. One such example would be (x 1 ∨ x 2 ) ∧ (x 1 ∨ x¯2 ). A solution, or satisfying assignment, to a SAT problem is an assignment of true or false to each of the variables such that the Boolean formula is satisfied; that is, that all clauses are satisfied. Next, we will consider the idea of boxes, which is how our algorithm will interpret SAT problems. Each box is an n-dimensional structure in {0, 1}n , where n is the number of variables in the original SAT problem. We will define this set as the output space; that is, all potential outputs will be elements of this set. Each of our boxes exists within this hypercube, and along each dimension has the value 0, the value 1, or extends along the full length of the edge. The reason for this is simple: 0 corresponds to false, 1 to true, and the length of the edge to both. Henceforth, we will use λ to refer to edges with length 1. We thus form the following definition: Definition 1 (Box Notation). A box takes the form 〈b 1, b 2 , ..., b n 〉, where each b i ∈ {T, F, λ}. Observe that, from these definitions, we can consider every assignment to be a 0-dimensional box; this will be important later. Then, our goal will be to find the set of points within the output space that are not contained (see Definition 2) by any boxes. Any such point will be termed an output point, and the goal of an algorithm working on these boxes is to find all such output points. Definition 2 (Containment). A box b is said to contain another box c if, for all points p ∈ {0, 1}n such that p ∈ c, it is true that p ∈ b. Equivalently, the box 〈b 1, b 2 , ..., b n 〉 contains the box 〈c 1 , c 2 , ..., c n 〉 if, for all i , b i = c i or b i = λ. However, there is one key difference between these two representations. Each clause in a CNF formula is essentially a subproblem wherein at least one variable’s assignment must match its value in the 4 clause for the assignment to possibly be satisfying. But with boxes, the exact opposite is true: if an assignment matches the value for the boxes on all non-λ dimensions, we reject the assignment. In other words, if we consider a geometric visualization of these boxes, any and all assignments that fall within a box are rejected. Hence, our next step is to devise a means by which to convert any given SAT problem in CNF form to the boxes format that Tetris can understand. As follows from our above observation, the most important step is simply the negation of the CNF formula; the rest is all bookkeeping. For the exact algorithm, see Algorithm 1. Algorithm 1 Conversion from CNF to boxes 1: for each CNF clause do 2: Negate the clause 3: Set all x¯i to F, and all x j to T 4: Set all variables not present in the clause to λ. 5: Insert into the database {See Definition 3} Let us consider the following toy example CNF problem: Example 1. (x 1 ∨ x 2 ) ∧ (x 1 ∨ x¯2 ) ∧ (x 2 ∨ x 3 ). Our first step is to negate each clause, which will give us a disjunctive normal form (DNF) co-problem: (x¯1 ∧ x¯2 ) ∨ (x¯1 ∧ x 2 ) ∨ (x¯2 ∧ x¯3 ). Next, we will convert to boxes (by replacing a variable with T and its negation with F ) and add all missing variables (as λ). After conversion, our three clauses become 〈F, F, λ〉, 〈F, T, λ〉, and 〈λ, F, F 〉. 1 x2 1 0 1 x3 1 1 0 x1 (x 1 ∨ x 2 ), 〈F, F, λ〉 x2 1 1 x3 x2 0 x1 1 x3 1 x1 (x 1 ∨ x¯2 ), 〈F, T, λ〉 (x 2 ∨ x 3 ), 〈λ, F, F 〉 Figure 1: Our starting boxes, and the corresponding SAT clauses. While boxes are, technically speaking, strictly the corners of what we depict as the boxes, we depict them with the edges and surfaces drawn for the purpose of visual clarity. At this point, it is time to insert the boxes into our data structure. Let us list the fundamental operations the data structure must be able to perform: Definition 3 (Tetris data structure). The Tetris data structure shall be able to perform the following operations: 1) Insert, input is the box to be inserted, no output 2) Contains, input is the box we are seeing if the structure contains, output is the containing box (see Definition 2) 5 3) GetAllContainingBoxes, input is the box we are seeing if the structure contains, output is the set of all containing boxes We will return to the details of the data structure implementation in Section 3.1. 2.2 Resolution We then come to the concept of resolution, a key aspect of Tetris and most state-of-the-art SAT solvers. Resolution can be defined over both CNF clauses and boxes; let us begin with the former. Let us consider two clauses in our CNF Example 1 once again: specifically, (x 1 ∨x 2 ) and (x 1 ∨ x¯2 ). We see that these are two very similar clauses; they differ only in that the x 2 term is negated in one and not the other. Therefore, we can resolve these two clauses by removing the x 2 term and then taking the OR of all remaining variables. In this case, this gives us (x 1 ). We then remove the original two clauses from the CNF problem and insert this new clause in its place. This is a significant simplification. Similarly, we can resolve any two clauses such that there is exactly one pivot point, by which we mean a variable that appears in both clauses, but is negated in one and not the other. For instance, looking back at our example, we can also resolve (x 1 ∨ x¯2 ) with (x 2 ∨ x 3 ) to form (x 1 ∨ x 3 ) In this case, we would not be able to remove the original two clauses, but we would have gained information. Let us now formally define this process: Definition 4 (Resolution on Clauses). Two clauses (x i 1 ∨ x i 2 ∨...∨ x i m ∨ v) and (y j 1 ∨ y j 2 ∨...∨ y j l ∨ v̄), i ∈ I , j ∈ J , I , J ⊂ [n], can be resolved if and only if there exists exactly one variable v, the pivot point, such that v ∈ x and v̄ ∈ y. The resolution of the two clauses is (x i 1 ∨ x i 2 ∨ ... ∨ x i m ∨ y j 1 ∨ y j 2 ∨ ... ∨ y j m ). Since boxes are simply another representation of the same problem, it follows that resolution can be performed on boxes as well. First, we will require that there must exist exactly one variable on which one box is true and the other box is false.1 Call this the pivot variable. In the output, set this variable to λ. Then, for each other variable, if it is T in one or both boxes, set it to T in the output box; if it is F in one or both boxes, set it to F in the output box; and if both variables are λ, then the resolution of the two is also λ. We see two possible resolutions in our Example 1. The resolution of 〈F, F, λ〉 and 〈F, T, λ〉, two coplanar and parallel edges, is the square 〈F, λ, λ〉, as depicted in Figure 2, and the resolution of the askew edges 〈F, T, λ〉 and 〈λ, F, F 〉 is the edge 〈F, λ, F 〉, as depicted in Figure 3. For a formal definition of the resolution operator, henceforth ⊕, see Definition 5. 1 x2 1 0 1 x3 1 x2 1 0 x1 + 1 1 x3 x2 0 x1 = 1 x3 1 x1 Figure 2: The resolution of 〈F, F, λ〉 and 〈F, T, λ〉 on the vertex x 2 is the square 〈F, λ, λ〉. This is equivalent to (x 1 ∨ x 2 ) resolved with (x 1 ∨ x¯2 ) being the clause (x 1 ). 1 This is exactly analogous to the requirement that we resolve on a pivot point in the clause version. 6 1 x2 1 0 1 x3 1 x1 x2 1 0 + 1 1 x3 x1 x2 0 = 1 x3 1 x1 Figure 3: The resolution of 〈F, T, λ〉 and 〈λ, F, F 〉 on the vertex x 2 is the edge 〈F, λ, F 〉 This is equivalent to (x 1 ∨ x¯2 ) resolved with (x 2 ∨ x 3 ) being the clause (x 1 ∨ x 3 ). Definition 5 (Resolution on Boxes). Two boxes 〈b 1 , b 2 , ..., b n 〉 and 〈c 1 , c 2 , ..., c n 〉 can be resolved if and only if there exists exactly one i such that b i is true and c i is false, or vice-versa. In the resolved box a, a i = λ. Each a j , j 6= i is equal to b j ⊕ c j , where ⊕ is defined as follows: T ⊕T = T F ⊕F = F λ⊕T = T λ⊕F = F T ⊕λ = T F ⊕λ = F λ⊕λ = λ T ⊕ F is undefined. Observe that resolution on boxes and resolution on clauses are identical: Lemma 1. Resolution on boxes, with the additional restriction that exactly one variable must be true in one box and false in the other, is exactly equivalent to resolution on SAT clauses. Proof. Let (x i 1 ∨...∨x i m ) and (y j 1 ∨...∨ y j l ), i ∈ I , j ∈ J , w her e I and J ⊂ [n], be the clauses we are resolving. Assume WLOG that i 1 = j 1 is the pivot point. Then the resolved clause is (x i 2 ∨ ... ∨ x i m ∨ y j 2 ∨ ... ∨ y j l ). The boxes equivalent to our starting to clauses are 〈b 1 , ..., b n 〉 and 〈c 1 , ..., c n 〉, where b k = F if x k ∈ x, b k = T if x¯k ∈ x, and λ otherwise, and c k is defined similarly with respect to y. The resolution of these boxes a is then defined as 〈a 1, ..., a n 〉, where a k = λ if k = i 1 = j 1 , and a k = b k ⊕ c k otherwise. Now, let us calculate the box equivalent of the output of the clause-based resolution, t . It can be shown that t k = λ if k = i 1 = j 1 , t k = x k = y k if k ∈ I , k ∈ J and k 6= i 1 , t k = x k if k ∈ I and k ∉ J , t k = y k if k ∈ J and k ∉ I , and t k = λ if k ∉ I and k ∉ J . Inspection with the above definition of ⊕ reveals that t is exactly equivalent to a. Since the same problem with arbitrary, equivalent inputs produced equivalent outputs, the two operations must be equivalent. Tetris introduces one additional restriction on resolution. Definition 6 (Resolution on Boxes in Tetris). Two boxes b and c can be resolved if and only if there is exactly one spot i such that b i = T and c i = F, or vice-versa, and for all j > i , b j = c j = λ. In other words, we will demand that that the pivot variable be the final non-λ variable. Therefore, while Tetris will perform the resolution of 〈F, F, λ〉 and 〈F, T, λ〉 (Figure 2), it will not perform the resolution of 〈F, T, λ〉 and 〈λ, F, F 〉 (Figure 3). We see, then, that the ordering of the variables determines whether or 7 not a resolution is even possible. This makes determining the global ordering of the variables a key issue, as mentioned earlier as Theoretical Implication 2, which we will address later in Section 3.2. In general, Tetris performs resolution on pairs of recently found boxes. Let k be the location of the last non-λ variable in a box b. Then b k must be either true or false. If it is false, we will store the box for future use. If it is true, then we will take this box b and resolve it with the stored box with the same value for k whose last non-λ variable was false. By doing so, we will guarantee the production of a box where the last n −k +1 variables have the value λ. For more details on how this works, along with the reasoning for why such pairs can always be found, see Section 2.3. 2.3 Tetris For now, let us return to our Example 1. When we last left off, we were just inserting the three clauses into our data structure, which was loosely defined in Definition 3. For a formal definition of the database and details for how it allows the set of boxes Tetris knows about to be quickly and efficiently queried, see Section 3.1; for now, one can simply assume it to be a trie-based structure. Additionally, we can resolve the first two boxes while leaving the third untouched, as in Figure 2; therefore, the database will contain exactly the boxes 〈F, λ, λ〉 and 〈λ, F, F 〉 (see Figure 4). Furthermore, we will prepare an empty array of boxes L of size n, which will be used later. The purpose of this array is to store and retrieve boxes that we wish to resolve with other boxes. Now that we have our database established, it is time to perform Tetris proper. The basic idea here is very simple. We will pick a point P in the output space, which we will call the probe point; recall that this point is itself a 0-dimensional box. We then determine whether or not any box in the database contains this point. If one does, we will store this box in an additional data structure referred to as the cache, which functions identically to the main database, and probe a new point. If no box contains P , we will list the point as a solution and furthermore add this point into the cache. Along the way, we will perform resolution in order to create new and larger boxes. This process continues until the entirety of the output space is covered by a single box, at which point we must have found every output point and are done. Algorithm 3 has the details. It should be noted that this algorithm was originally presented recursively in [6]; here, we present it iteratively both for the purposes of speed and because this allows for non-chronological backtracking; in other words, we can backtrack more than one layer at a time. Algorithm 2 Advance(box b, probe point &p) (note: p is a global variable) 1: while b Contains p do 2: if the last non-λ variable of p is F then 3: Set that variable to T {Return to the previous branching point and take the right, or true, branch} 4: else 5: while the last non-λ variable of p is T do 6: Set that variable to λ {Return to the most recent level where we branched left} 7: Set the last non-λ variable of p to T {Branch right here} 8: Replace all λs after this variable with F {Repeatedly branch left} Now, let us consider how this algorithm behaves with regards to our earlier Example 1. We pick as our first probe point 〈F, F, F 〉, giving us the situation illustrated in Figure 4. We first scan our local cache, C , for any boxes that contain this point; however, since this is the first probe point, the cache is trivially empty. Next, we scan the database D. D contains all the boxes corresponding to the clauses in the original SAT 8 Algorithm 3 General Tetris for SAT 1: Establish variable ordering 2: Build the database D using Algorithm 1 3: C ← ; 4: L ← An empty array of size n {This array is implicit in [6]} 5: p ← 〈F, F, ..., F 〉 6: while 〈λ, λ, ..., λ〉 ∉ C do 7: if (b ← C .Contains(p)) is nonempty then 8: Advance(b, p) {Advance (see Algorithm 2) the probe point past b} 9: else if (A ← D.GetAllContainingBoxes(p)) is nonempty then 10: for all boxes b ∈ A do 11: C .Insert(b) 12: Advance(b, p) {Advance the probe point past b} 13: else 14: Add p to the output{There is no containing box, so p is an output point} 15: C .Insert(p) 16: Advance(p, p) {Advance the probe point past itself} 17: k ← the location of the last non-λ variable in b w.r.t. the variable ordering 18: if b k = F then 19: L[k] = b {Store the most recent left-branching box for a given depth} 20: else 21: r ← b ⊕ L[k] {Resolve this right-branching box with the corresponding left-branching box} 22: C .Insert(r ) problem. The database just so happens to contain two containing boxes; for reasons that will become clear shortly, the operation will choose to output 〈F, λ, λ〉. We insert the box 〈F, λ, λ〉 into C . Our next task is to advance the probe point until it lies beyond our box. To do this, we proceed according to Algorithm 2. The idea is to think of the set of all possible probe points as a tree that we are performing a depth-first search on, with F representing left-branching paths and T representing rightbranching paths. We will continue along this depth-first search until we find a point not covered by the most recently discovered box. This takes us to 〈T, F, F 〉. Note that if the database had fetched 〈F, F, λ〉, we would not have been able to advance the probe point as far. Finally, we insert our containing box into the array L at location 1, since only the first variable is nonλ, for future use. We know to do insertion here, rather than trying to resolve with a non-existent box, because the value of that first non-λ variable is false. This takes us to the situation depicted in Figure 5. Again we scan C , this time for the probe point 〈T, F, F 〉, and again we find no containing box in C . So we scan D once again and find the containing box 〈λ, F, F 〉. We then insert this box into the cache and advance the probe point to 〈T, F, T 〉. This time, although our containing box features a λ at the first location, we determine the location in L into which we will insert based on the location of the last non-λ variable, so we insert it into L[3]. This time, we find no containing boxes in either the cache or the database. Therefore, we have found an output point (see Figure 6 for an illustration). We add 〈T, F, T 〉 to our output set, then add the box 〈T, F, T 〉 to our cache, which marks the point as found. At this juncture, we find that the last non-λ variable is at location 3, but this time, it is true. Therefore, 9 1 x2 1 0 1 1 x3 Probe Point p Cache C Database D x2 1 0 x1 1 1 x3 x2 0 x1 Probe Point Map 1 1 x3 x1 Figure 4: The initial state of the database D, cache C , and the location of the first probe point p, which will search the output space in a depth-first manner that can be tracked using the map on its right. D is simply the union of all the boxes we created from the initial SAT problem, while p is set to an initial value of 〈F, F, F 〉. L is currently empty. 1 x2 1 0 1 x3 1 Probe Point p Cache C Database D x2 1 0 x1 1 x3 1 x2 0 x1 1 x3 Probe Point Map 1 x1 Figure 5: The state of the database, cache, and probe point after the first round of the algorithm. Our probe point found the box 〈F, λ, λ〉, so it added this box to C . Then, p was advanced until it reached a point not contained by this box, which turned out to be 〈T, F, F 〉. L(1) is the box 〈F, λ, λ〉, which is the box corresponding to the orange vertex in the map; the array is empty elsewhere. we will extract the same-length box we stored previously at L[3] and resolve it with this box. We know that this will be a legal resolution because we are scanning the output space in a tree-like fashion. This means that, when retreating from a right branch, the box containing the corresponding left branch must be able to contain the right branch if the final non-λ variable were set to λ instead. It follows that this final non-λ variable must be the one and only pivot point between the two. Therefore, we can and do perform this resolution; in this example, it is 〈T, F, T 〉 resolved with 〈λ, F, F 〉. This outputs the box 〈T, F, λ〉. We furthermore store this box in L at location 2, since this box ends with false at that index. We continue forth with probe points 〈T, T, F 〉 and 〈T, T, T 〉. Neither will be found in either D or C , and are therefore output points. Both again have their final non-λ variables at index 3, with 〈T, T, F 〉 being inserted into L at that index and then 〈T, T, T 〉 recovering that box so it can resolve with it to form the box 〈T, T, λ〉. This time, the output of our resolution ends with true, so we recover the box at index 2 in L, 〈T, F, λ〉, and take the resolution of these two boxes, giving us 〈T, λ, λ〉. Once again this ends in T, so we can resolve it with the box we found back at the beginning that has been waiting in slot 1, 〈F, λ, λ〉, to form the box 〈λ, λ, λ〉. This box completely covers the output space; therefore, the algorithm knows that it has found all possible output points and terminates (see Figure 7 for an illustration). 10 1 x2 1 0 1 1 x3 Probe Point p Cache C Database D x2 1 0 x1 1 1 x3 x2 0 x1 Probe Point Map 1 1 x3 x1 Figure 6: The state of the database, cache, and probe point after the first output point is found at 〈T, F, T 〉. In getting here, we first found the box 〈λ, F, F 〉, and then probed the output point. After finding the output point, that box was resolved with the aforementioned box to produce 〈T, F, λ〉, which was also added to C . Note that, while the box 〈λ, F, λ〉 could be produced at this juncture, Tetris will not do so. L(1) contains 〈F, λ, λ〉 (the orange dot); L(2) contains 〈T, F, λ〉 (the purple dot); L(3) is empty. 1 x2 1 0 1 x3 1 Output Points Cache C Database D x2 1 0 x1 1 x3 1 x2 0 x1 1 x3 Probe Point Map 1 x1 Figure 7: The state of the database, cache, and probe point after the entire output space has been covered. With each further output point discovered, boxes were added to the cache, which produced a chain of resolutions that eventually resulted in the production of the box 〈λ, λ, λ〉. Note that there is no longer a probe point, as there is nothing left to probe. 3 Our Improvements Here, we will discuss the major additions introduced into Tetris in order to handle CNF inputs and increase practical efficiency. These include a new data structure, work on heuristically determining a global variable ordering, and selectively caching only certain boxes. 3.1 Data Structure and Compression For its data structure, the original Tetris paper simply states that a trie will suffice to achieve asymptotic runtime guarantees. While this is true, a simple trie still leaves much to be desired; attempts to implement Tetris in such a simple matter produced a system significantly slower than other state-of-the-art model counters. Our contribution is to design a novel system of tries that takes advantage of the nature of the problem space to improve both runtime and memory usage. 11 3.1.1 Data Structure Description As described above, the database must allow each variable to store three values: false, true, and λ. Therefore, the immediate approach is to use 3-tries as the base data structure. However, when #SAT instances routinely have hundreds of variables, this results in an extremely deep problem space that requires a lot of time to probe. Our next step, then, is to compress multiple layers into a single node that can be queried in a single instruction. To this end, we will first come up with a means to enumerate all possible boxes: Definition 7. Let φ be a bijective function from the set of boxes onto the integers. Example 2. One such way is as follows: Let λ be assigned 1; false, 2; true, 3. Then, for each box 〈b 1 , b 2 , ..., b n 〉, its numerical value is 3n−1 b 1 + 3n−2 b 2 + ... + b n .2 This gives us a bijective ternary numeration. From there, we observe that there is exactly one box for which n is 0 (namely, the empty box 〈〉), three boxes with n equal to 1, nine boxes with n equal to 2, twenty-seven boxes with n equal to 3, and eighty-one boxes with n equal to 4. We will require that a single node within the trie be able to record a box of any of these lengths. Additionally, it must be able to store the children of all possible boxes with n equal to 4. Therefore, by compacting four logical layers into a single layer within the database, the result is a trie that can store 121 possible boxes, the sum of the five aforementioned values, and can have 81 possible children. We will refer to this collection of variables as a cluster. Definition 8 (Cluster). A Cluster is the set of variables that the database handles in a single operation. By default, each cluster contains 4 variables. Of course, this raises a new issue: When checking if the database contains a given input string x, there can exist up to sixteen children of x that must be checked (since each T or F can be replaced by a λ), and an even greater number of boxes that could be contained in this cluster may contain the input string. This creates the need for a way to quickly and efficiently determine if a containing box exists in this cluster and to create the list of children to be searched. 3.1.2 SIMD-based Trie Here, we take inspiration from EmptyHeaded, a relational database engine [5], and utilize SIMD. As an example, let us consider the simplified version of the data structure that contains only two layers, and with n = 3. Suppose that 〈F, λ, λ〉 was known to be a box, and that 〈λ, T, F 〉 and 〈T, T, F 〉 are boxes that, to be found, necessitate traversing into child clusters. We can see this depicted in Figure 8. Now, we will determine whether or not this data structure contains the box 〈F, T, F 〉. Since each cluster contains two layers, clusters with depth 0 will look at only the first two variables in the input box to determine input; therefore, let us consider the sub-box 〈F, T 〉. Using φ, we can find in a lookup table the two 128-bit bitstrings corresponding to this input sub-box. The first lists the set of boxes that, if they exist within the cluster, would contain 〈F, T 〉, and is the second line of Figure 9. The second bitstring does much the same for the set of children that, if truncated to two variables, contain 〈F, T 〉. Additionally, a cluster stores two more 128-bit bitstrings: BOXES, which marks the boxes contained by the cluster; and CHILDREN, which marks the child nodes of the cluster. The BOXES bitstring is specifically the top line of Figure 9. BOXES is marked for the box 〈F 〉 (which is equivalent to 〈F, λ, λ〉), while CHILDREN has bits marked corresponding to the box prefixes 〈F, T 〉 and 〈T, T 〉. 2 This is exactly the mapping used in our implementation. 12 It follows that the intersection of these two pairs of bitstrings is the set of boxes and child prefixes present in the data structure that contain the input; a single AND operation suffices to calculate it. ; Cluster 1 Cluster 0 Layer 0 Layer 1 〈F 〉 〈λ〉 〈T 〉 Layer 2 〈λ, T 〉 〈T, T 〉 Layer 3 〈λ, T, F 〉 〈T, T, F 〉 Layer 4 Figure 8: The SIMD operation, shown as the associated clusters, with each box marked by a blue box. Each layer corresponds to a variable (with Layer 4 being empty because all boxes have only three variables), and checkmarks mark the boxes that are found to be containing boxes. The central branch is created from the box 〈F, λ, λ〉. In the left branch, created by 〈λ, T, F 〉, a child is created corresponding to the sub-box 〈λ, T 〉, and then the box is inserted in the child cluster at 〈λ, T, F 〉. In the right branch, created by 〈T, T, F 〉, we similarly create a child corresponding to the sub-box 〈T, T 〉, and then inserting 〈T, T, F 〉 in the child cluster. We have drawn the database here as a 3-trie; to see it as a single, flattened cluster, see Figure 10. Let us inspect our output. In this particular example, we find that we have matched the box 〈F, λ〉 and the child with prefix 〈λ, T 〉. At this point, we are faced with an interesting choice. Suppose for the sake of argument that the child eventually leads to some containing box for the input. Now, if this is a check for containment, the algorithm can only return the one box that it considers the “best" containing box. Which one, then, should the algorithm choose? In general, it will choose the shortest-length box; in this example, it will choose the box 〈F, λ〉. The reason for this is simple: the more λs in a box, the more space it covers; and the more space it covers, the more output space it can cover. Additionally, when those λs are at the tail end of a box, we know that we can immediately advance the probe point considerably. On the other hand, when the λs are in the middle, the algorithm must re-find these boxes every time it scans a point that is contained within this hypothetical box. This repetition is costly; we would rather avoid it. Now, let us consider how our algorithm would handle this input if we were simply using traditional tries; in other words, consider what would happen if our cluster size were 1, as in a standard 3-trie. This algorithm would query the input box for its first value, find that it is F , and know that it could check both the λ branch and the F branch. Since λs are generally to be preferred, it would take the λ branch. Then, it would match the T value to the T branch, and proceed to the third layer, where it would match the F value to the F branch and find a containing box, which it will return. In this way, it has taken us three comparisons to find a containing box. Furthermore, the box found in the 3-trie version is of lower quality: we would have preferred to find the 〈F, λ, λ〉 box instead. On the other hand, let us consider what occurs when using full, four-variable clusters (Figure 10). In this case, a single operation immediately finds all three containing boxes. Therefore, we have outper- 13 Database’s BOXES: 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 10 11 12 & Input: 〈F, T 〉: 1 1 1 0 1 1 0 1 0 1 0 0 0 0 1 2 3 4 5 6 7 8 9 10 11 12 = Containing Boxes: 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 10 11 12 Figure 9: The algorithm takes the logical AND of the stored bitstrings, top, with the bitstring corresponding to the input, row 2, in order to produce the list of containing boxes and children, bottom. The BOXES bitstring contains a single 1 to mark the box 〈F, λ〉. The second line contains all the potential boxes that would contain 〈F, T 〉; in addition to the boxes that do exist, these include 〈λ〉, 〈F, T 〉, and many more. The output of the operation has bits set corresponding to the box 〈F, λ〉. formed the traditional variation both in terms of number of comparisons and in terms of the quality of the output. In summary, each box is stored as a single bit in a 128-bit vector (with the last 7 bits unused), as is a record of whether or not a given child exists. A lookup table is used to find the 128-bit vectors corresponding to the possible outputs. Then, the former two are concatenated, as are the latter two, and all are compared using a single 256-bit AND operation. It can be seen that the output of this operation must be the intersection of the potential containing boxes and children, found from the lookup table, and the ones that actually exist. Hence, by calculating φ(b) for some box b, we can quickly find a box containing b, if it exists; and if it does not, we can quickly generate the exact list of children to examine. Additionally, in practice, it turns out that there is a certain sub-box value that shows up far more often than any other: the sequence 〈λ, λ, λ, λ〉. Notably, this is the very sequence that accepts every single possible input string. Therefore, in the event that a layer contains only this child and no words, it is in fact possible to skip the entire layer. In practice, this produces significant savings on both computational costs and memory usage, which contributes towards Theoretical Implication 3. Let us now formally define the data structures and its algorithms used in our implementation: Definition 9 (Data Structure). The data structure is a 121-ary trie, where each node of the trie is called a cluster. The top level consists of a pointer to the root cluster, which is the cluster covering the first four variables in the ordering, and can perform the following operations: Insert (Algorithm 4), Contains (Algorithm 6), and GetAllContainingBoxes (Algorithm 8). Definition 10 (Cluster). Each cluster in the database contains two bitstrings, BOXES and CHILDREN, bitstrings identifying the sets of boxes and children, respectively, along with an integer DEPTH that informs the cluster of its depth. The operation .at (i ) on a bitstring calculates the intersection of that bitstring with a bitstring retrieved from a lookup table that lists the set of boxes or children that contain or potentially contain, respectively, the box with value i ; setting .at (i ) to 1 sets the specific bit referring exactly to that box. Each cluster corresponds to four layers of a standard 3-ary trie. Definition 11 (Index of a Box). The index of a box b, i nd ex(b), is the location of the last non-λ variable in that box. 14 Cluster 0 ; 〈F 〉 〈λ, T, F 〉 〈T, T, F 〉 Figure 10: The clusters in Figure 8, flattened out into a single node with four variables to a cluster. This is how the node would be stored in the actual database. Note that it is far more compressed than when drawn out as a trie as in Figure 8. This cluster would have a DEPTH of 0, have three bits set in BOXES, and no bits set in CHILDREN. Example 3. The index of the box b = 〈F, T, λ, F, λ〉, i nd ex(b), is 4. Algorithm 4 Insert 1: Given box b to insert, b = 〈c 1 , c 2 , ..., c n 〉, where each c i here is a cluster consisting of b i 1 ...b i 4 2: k ← i nd ex(b) 3: Call InsertCluster(b, k) (Algorithm 5) called on the root cluster Algorithm 5 InsertCluster 1: Input: A cluster T with depth DEPTHto insert on, box b to insert, b = 〈c 1 , c 2 , ..., c n 〉, where each c i here is a cluster consisting of b i 1 , ..., b i 4 , and the location of the final non-λ cluster k 2: if DEPTH = k then 3: if BOXES.at (φ(c DEPTH )) is nonempty then 4: Return {If a containing box of b is already in the data structure, stop.} 5: else 6: Set BOXES.at(φ(c DEPTH )) to 1 7: else 8: if CHILDREN.at (φ(c DEPTH )) 6= 1 then 9: Create a child cluster at location φ(c DEPTH ) with depth (DEPTH + 1). 10: Set CHILDREN.at (φ(c DEPTH )) to 1 11: Call InsertCluster on the cluster indexed as φ(c DEPTH ). 12: Return In the Insert algorithms, Algorithms 4 and 5, we recursively traverse clusters until we find the appropriate location in the data structure, and then set the appropriate bit to 1. Along the way, we will check for containing boxes and immediately cease operation if one is found. Furthermore, if a child cluster that contains the box we are inserting, or that contains a cluster along the path to that cluster, does not exist, we create it. Algorithm 6 Contains 1: Given box b to find a containing box of, b = 〈c 1 , c 2 , ..., c n 〉, where each c i here is a cluster consisting of b i 1 ...b i 4 2: out put ← ContainsCluster(b) (Algorithm 7) called on the root cluster 3: Return out put 15 Algorithm 7 ContainsCluster 1: Input: Cluster T with depth DEPTHthat is being checked for a containing box, box b to check containment of, b = 〈c 1 , c 2 , ..., c n 〉, where each c i here is a cluster consisting of b i 1 ...b i 4 . 2: if ( f = BOXES.at (φ(c DEPTH ))) is nonempty (i.e., there is at least one box in the intersection) then 3: o = mi n x∈ f I nd ex(x) 4: Return o 5: else if ( f = CHILDREN.at (φ(c DEPTH ))) is nonempty (i.e., there is at least one child in the intersection) then 6: for all Children k ∈ f do 7: a = k.ContainsCluster(b) {Scan these in order of increasing index} 8: if a is nonempty then 9: Return a 10: else 11: Return ; In the Contains algorithms, Algorithms 6 and 7, we check the database to see if it contains any box that contains some box b. Therefore, we traverse along clusters in our path, first checking to see if any containing boxes exist; if we find one, we return that box immediately and cease checking further. If none exists, we will perform a depth-first search of the children that could potentially contain a containing box. This process continues until either a containing box is found, or else the search space is exhausted and it is determined that no containing box exists. Algorithm 8 GetAllContainingBoxes 1: Given box b to find all containing boxes of, b = 〈c 1 , c 2 , ..., c n 〉, where each c i here is a cluster consisting of b i 1 ...b i 4 2: out put ← GetAllContainingBoxesCluster (b) (Algorithm 9) called on the root cluster 3: Return out put Algorithm 9 GetAllContainingBoxesCluster 1: Input: Cluster T with depth DEPTH that is being checked, box b to find all containing boxes of, b = 〈c 1 , c 2 , ..., c n 〉, where each c i here is a cluster consisting of b i 1 ...b i 4 . 2: O ← ; 3: if (F = BOXES.at (φ(c DEPTH ))) is nonempty (i.e., there is at least one box in the intersection) then 4: O = F. 5: else if ( f = CHILDREN.at (φ(c DEPTH ))) is nonempty (i.e., there is at least one child in the intersection) then 6: for all Children k ∈ f do 7: A = k.GetAllContainingBoxesCluster(b) 8: O =O∪A 9: else 10: Return O 16 The GetAllContainingBoxes algorithms, Algorithms 8 and 9, are similar to the Contains algorithms. There are, however, two key differences. First, while Contains terminates as soon as it found a single containing box, GetAllContainingBoxes will continue. Secondly, it returns the set of all containing boxes, rather than just one; hence, the name. In all other regards, it behaves exactly as Contains does. 3.2 Global Variable Ordering Up until this point, we have simply been assuming that all boxes must order their variables in exactly the same order as they appear in the original SAT formulas. In other words, in each box b, b 1 must correspond to x 1 , b 2 must correspond to x 2 , and so on. However, this does not need to be the case. We can reorder the variables, and by doing so can greatly improve the runtime of our system. The original Tetris paper cites the importance of the variable ordering. However, it assumes that there exists an exponential in n time algorithm to compute the optimal variable ordering. While this is justifiable in the context of join problems, where n is small compared to the size of the database, in SAT problems it is unacceptable. Furthermore, as computing the optimal ordering is NP-hard, we cannot hope to improve upon this result. Indeed, even approximating the ordering is intractable [20]. Nevertheless, initial experimentation with Tetris made clear how impactful this choice can be. A slight variation in ordering can result in a large difference in runtime. Thus, we turned to various heuristics and intuitions in order to find a quick and effective means to generate an ordering that works well in practice and thereby contribute to Theoretical Implication 2. First, let us define a few terms that we will use in our discussion of various ordering strategies: Definition 12 (Degree). The degree of a variable is the number of clauses of which the variable is a part. Example 4. In Example 1, (x 1 ∨x 2 )∧(x 1 ∨ x¯2 )∧(x 2 ∨x 3 ), x 1 has degree 2, x 2 has degree 3, and x 3 has degree 1. Definition 13 (Closeness). Variables x i and x j are said to be close if there exists a clause that includes both x i and x j . The fewer terms in the clause, the closer the two variables are said to be. Specifically, the closeness of two variables, θ(x 1 , x 2 ), is equal to 1 divided by the size of the smallest clause containing both variables minus 1. Example 5. In the clause (x 1 ∨ x 2 ), x 1 and x 2 would have a closeness of 1; and in the clause (x 1 ∨ x 2 ∨ x 3 ), x 1 and x 3 would have a closeness of 21 . If both clauses were part of the same SAT problem, x 1 and x 2 would still have a closeness of 1 because the first clause has a smaller size than the second. Definition 14 (Interconnectedness). The interconnectedness of a cluster C , IC (C ), is the sum of the close¡ ¢ ness values for each 42 pairs of variables in the cluster. Example 6. Using Example 5, if x 1 , x 2 , and x 3 compose a cluster, its interconnectedness would be 2, since θ(x 1 , x 2 ) = 1, θ(x 1 , x 3 ) = 12 , and θ(x 2 , x 3 ) = 12 . In general, we note two high-level strategies that we have found improve the performance of a given global variable ordering: a) High-degree First In Tetris, handling high-degree variables early tends to improve performance. To see the reason for this, we must consider the nature of the algorithm. Scattering high-degree variables throughout the ordering forces the algorithm to branch frequently, which means that, when testing for 17 inclusion or for containing boxes, the algorithm must scan all possible branches. This is highly inefficient. Instead, we focus the branches as much as possible to the beginning, with the hope being that, as the algorithm progresses down from there, most layers will have few if any divergent choices. If this is true, then the inclusion check can be handled quickly. Let us proceed to see why this is so. First, we will introduce an example of why placing high-degree variables early in the ordering proves effective. Let us consider the following sample problem: Example 7. Consider the SAT formula (x 1 ∨x 3 )∧(x 2 ∨x 3 ). The equivalent box problem, using the ordering (x 1 , x 2 , x 3 ), would begin with database D containing the boxes {〈F, λ, F 〉, 〈λ, F, F 〉}. x 1 has degree 1, x 2 has degree 1, and x 3 has degree 2. Now, let us consider how the algorithm would attack this problem if we used this naive, low-degree first ordering, which would be (x 1 , x 2 , x 3 ). We immediately note two things. First, there are no boxes that have λ for the final variable. This means that the algorithm will never find a box that allows it to skip multiple probe points unless it can use resolution to create a new box that happens to have that property. In fact, that will not occur. Additionally, we can consider all 8 possible probe points and track how many comparisons the algorithm would need to make on each point, assuming that each cluster only covers a single variable instead of four. We can see that the algorithm will always have to calculate the set intersection for the cluster of depth 1, will have to perform the set intersection for the cluster with depth 2 on 50% of probe points, and will have to perform the set intersection for the cluster with depth 3 on 75% of probe points. Let us contrast this with the high-degree ordering, which moves x 3 to the front and x 1 to the back to give us the ordering (x 3 , x 2 , x 1 ). Now, our database contains the boxes 〈F, λ, F 〉 and 〈F, F, λ〉. This time, we do have a box with a λ at the back; specifically, 〈F, F, λ〉. Therefore, after the probe point 〈F, F, F 〉 finds this box, the algorithm will advance past 〈F, F, T 〉 entirely. Additionally, while we still have to perform a set intersection for the cluster with depth 1 100% of the time, and 50% of the time for the cluster with depth 2, we only have to perform the set intersection at depth 3 25% of the time — and all of these numbers ignore how we skipped one of the probe points entirely. While this is of course a very simple example, this illustrates the principles that cause the strategy to be effective in larger datasets. b) Local Interconnectedness As a direct result of the 121-ary trie-based system described in Section 3.1, if a box has multiple non-λ variables within the same 4-variable cluster, they can all be recovered with a single operation. Therefore, maximizing the interconnectedness of these 4-variable blocks provides an advantage. To illustrate, let us consider the following example: Example 8. (x 1 ∨ x 3 ) ∧ (x 2 ∨ x 4 ), with each cluster containing 2 variables rather than 4. If we use the naive strategy of keeping the variables ordered as-is, the first cluster contains x 1 and x 2 while the second cluster contains x 3 and x 4 . Therefore, both clusters have interconnectedness 0, and we find that all boxes transcend a cluster boundary. In other words, it will always take at least two comparisons to find either of these boxes. However, if we had gone with the ordering (x 1 , x 3 , x 2 , x 4 ) instead, the box corresponding to (x 1 ∨ x 3 ) would be entirely contained within the first cluster, and the box corresponding to (x 2 ∨ x 4 ) would be entirely contained within the second cluster. Therefore, each box would be entirely contained within a cluster, and each cluster would have had an interconnectedness of 1, and each box could have been recovered with only a single comparison. This saves a large number of comparisons over the long term. 18 3.2.1 Ordering Algorithms Two major methods were employed in order to achieve these aims. The first was a descending degree sort. While this only directly achieved goal (a), in practice it did an acceptable job with goal (b). Additionally, we constructed three variations on this method. The first, naive degree descent, is where we simply order the variables according to their degree, using Algorithm 10. Algorithm 10 Naive Degree Descent Ordering 1: Given a set of variables V and, for each variable v, its degree v d : 2: O ← SORT(V on v d , descending) 3: Return O The second, optimally grouped degree descent, forms all possible groups of four variables, finds the greatest possible interconnectivity among these groups, and then selects from all groups with the greatest interconnectivity on the basis of the combined degree of the group of four, using Algorithm 11. While this proved effective, it is a slow algorithm with a runtime of Θ(n 8 ). Algorithm 11 Optimally Grouped Degree Descent Ordering 1: Given a set of variables V and, for each variable v, its degree v d : 2: O ← ; 3: G ← all possible sets of four variables {v i , v j , v k , v l } from V . 4: while G is nonempty do 5: max IC ← max g ∈G IC (g ) {Determine the maximum possible interconnectedness of all remaining groups.} P 6: X ← max g ∈G s.t . IC (g )=maxIC ( v∈g v d ) {Of the groups with max interconnectedness, 7: select the group for which the sum of the degrees of all variables is the greatest.} O ← (O, X ) {Append this group to the ordering.} for v ∈ X do for Y ∈ G do 10: if v ∈ Y then 11: G ← G \ Y {Remove each grouping that contains one of the variables in the group that was selected.} 12: Return O 8: 9: This necessitated the creation of the third subtype, the heuristically grouped degree descent ordering. This ordering works in groups of four. When creating a group, the first node chosen is the highestdegree remaining variable. Then, for each of the remaining three variables, the algorithm picks the variable with the highest interconnectedness to the nodes already chosen for this group of four, breaking inevitable ties based on degree. The result, Algorithm 12, is an algorithm that can compute its ordering significantly faster than the optimal ordering, while Tetris run on this ordering runs is competitive with the optimal ordering. 19 Algorithm 12 Heuristically Grouped Degree Descent Ordering 1: Given a set of variables V and, for each variable v, its degree d v : 2: i ← 0 3: X ← ; 4: O ← ; 5: while V is nonempty do 6: if i = 0 then 7: x ← y where De g r ee(y) = max v∈V (d v ) 8: else P 9: max IC ← max v∈V ( x∈X θ(v, x)) {Calculate which variable has the best interconnectedness with the already chosen variables} 10: x ← y where d y = max v∈V s.t .VIC =maxIC (d v ) {Break ties based on degree} 11: O ← (O, x) 12: X ← X ∪x 13: i ← i +1 14: if i = 3 then 15: i =0 16: x ←; 17: Return O Additionally, we employ the Treewidth tree decomposition, which was introduced in [16]. In essence, the idea here is to minimize the width of the search tree; in our domain, this corresponds to increasing the locality and local interconnectedness of variables. This naturally did a very good job with interconnectivity, while a decent job with placing high-degree variables early. We also experimented with the minfill ordering, described in [12]. This ordering sets the elimination order such that the node to be eliminated is the node whose removal makes the smallest impact on the overall graph. While this ordering has proved effective in similar applications, we found it to perform poorly with Tetris. In Table 1, we can see how these various orderings performed in practice on representative graphbased and non-graph-based benchmarks. For instance, while the Treewidth sort outperformed all others on the AIS8 dataset [3], on the WikiVotes dataset (created using a SNAP [19] dataset; see Section 4.1.1) the ordering caused Tetris to timeout. Notably, we see that the Heuristically Grouped Degree Descent takes only slightly longer to process the input compared to the Naive Degree Descent, but significantly less time than the Optimally Grouped Degree Descent; however, the runtime does not suffer significantly when going from the optimal ordering to the heuristic one. 3.3 Selective Insertion While the original Tetris paper calls for the insertion of every box created through the resolution process to be inserted into the database, this proved to be inefficient in practice. Very frequently, this will result in a huge increase in the number of branches that the algorithm must scan while trying to find the output point without notably improving the quality of the containing boxes found. Therefore, we only insert those boxes that contain a suitably high percentage of λs. The best results generally come from requiring slightly less than 50 percent of the layers to be composed of entirely λs. In Table 2, we have posted the runtime for the AIS10 dataset [3] showing the relationship between the number of λs we require in storing a box and the runtime of Tetris. Performance suffers at extreme 20 Dataset AIS8 WikiVotes RUNTIME WITH D IFFERENT O RDERINGS Ordering Load Time (Seconds) Naive Degree Descent .001 Heuristically Grouped Degree Descent .009 Treewidth .002 Minfill .008 Optimally Grouped Degree Descent 1.629 Naive Degree Descent .927 Heuristically Grouped Degree Descent 1.024 Treewidth 2.349 Minfill 2.032 Optimally Grouped Degree Descent 2.06 Runtime (Seconds) 1.494 2.607 .45 7.005 0.573 34.124 21.509 Timeout 4059.426 23.142 Table 1: The performance of various ordering schemes on two datasets. As one can see, no ordering does best on both datasets; indeed, the best ordering on one is the worst on the other. Insertion Ratio (see Section 3.3) was set to .5 for these tests. settings, with optimal performance resulting from an insertion ratio of close to 21 . Hence, with regard to Theoretical Implication 3, we find that by decreasing space complexity, we furthermore improve runtime. 4 Our Experimental Results Here, we compare CNFTetris — that is, Tetris designed to solve CNF problems — with other model counters in order to compare and contrast their ability to tackle model counting problems. A model counting problem, simply put, is, given a CNF formula, output the number of satisfying solutions to that formula. Since Tetris was originally designed to handle database joins, these are more natural problems for the algorithm to solve than the corresponding SAT problems, which are to simply determine whether or not any solution exists. While the model counting problem allows for a solver to simply find the number of solutions without finding the solutions themselves, CNFTetris does in fact output all of the solutions. While this admittedly poses a disadvantage compared to the solvers we are comparing against, for some datasets, CNFTetris runs faster in spite of this. We compare our results with those of the sharpSAT [28] , dSharp[22], and Cachet[26] model counters due to their recognition as state-of-the-art model counters. All tests were performed using a single thread on an 8-core E5 v3 2.6GHz processor with 64GB of RAM. Additionally, we include two types of datasets. The first is derived from join problems on graphs. These are the sort of problems that Tetris was originally designed to solve; as such, CNFTetris does a very good job on them. The second set is a selection of standard model counting benchmarks from various competitions held over the past several years. Most model counters have been trained to solve such problems, so they serve as an apt second set of benchmarks for CNFTetris to compete against. 21 I NSERTION R ATIO VS . RUNTIME Ratio Time (Seconds) .00 140.999 .05 123.49 .10 79.003 .15 71.229 .20 41.956 .25 36.685 .30 32.064 .35 24.405 .40 22.749 .45 21.784 .50 24.171 .55 25.205 .60 28.157 .65 35.738 .70 44.657 .75 51.809 .80 54.622 .85 66.529 .90 64.16 .95 75.829 1.00 88.001 Table 2: A comparison of insertion ratios with the time to solve the AIS10 dataset [3]. For these tests, we used the Treewidth ordering; similar behavior was observed from all ordering strategies. 4.1 Graph Results Here, we will compare and contrast how various solvers performed on model counting problems created from graphs. 4.1.1 Dataset Generation The CNF graph datasets were created using the publicly available SNAP datasets [19]. These are graph datasets; that is, each consists of a set of vertices and a set of edges connecting those vertices. Each of these datasets is a natural problem; some arose from social networks, while others are anonymized data from other corners on the Internet. We can then use this data to run various queries; for instance, we can determine how many triangles exist in the graph. Our goal, then, is to convert these problems into an equivalent CNF problem so that we can use CNFTetris and the other model counters to solve them. To do this, each vertex is first assigned a unique binary encoding using log(n) bits. We furthermore increase the number of bits such that there are log(n) bits times the size of the data structure being looked for in the graph. For instance, if we are performing the triangle query on the dataset, there will be 3 · log(n) bits used in the encoding. Henceforth, let k represent the size of this query. Each of these bits will correspond to a variable in the CNF encoding of the problem. In essence, each of these repetitions 22 RUNTIME Query 3-clique 2-path Base Graph Wikivotes Facebook Soc-Epinions Wikivotes Facebook Soc-Epinions AIS6 AIS8 AIS10 AIS12 ls8-simplified4 LS5 firstr CNFTetris ON VARIOUS D ATASETS sharpSAT dSharp Cachet Loadtime 1.080 .593 77.420 2.32 .752 6.22 Runtime 21.55 11.22 309.418 35.171 10.236 236.439 Runtime Timeout Timeout Timeout 31801.8 4637.25 Timeout Speedup n/a n/a n/a 904.2 453.03 n/a Runtime Timeout Timeout Timeout 36947.5 3871.86 Timeout Speedup n/a n/a n/a 1050.51 378.26 n/a Runtime Timeout Timeout Timeout 28838.1 2210.42 Timeout Speedup n/a n/a n/a 819.94 215.95 n/a 0.0021 0.0027 .00798 .0198 .00063 .000656 .013 .45 21.55 1732.03 .123 .409 .004 .0466 2.629 124.937 .004933 .0156 .308 0.104 .122 .0721 .0401 .0381 .0074 .3386 14.757 1002.73 .021 .095 .569 0.752 .685 .579 .171 .232 .0135 .300 16.4724 1020.64 .01 .025 1.034 .667 .764 .589 .0813 .0611 Table 3: This table shows the comparative results of various solvers on CNF datasets created using various SNAP graphical datasets and SAT datasets. All runtimes are in seconds; timeout was set at 40,000 seconds. For these tests, we used a insertion ratio of .45 and the Heuristic Degree Descent ordering for CNFTetris. Wikivote [19] contains 39 variables and 745485 clauses; Facebook [19] contains 36 variables and 464234 clauses; and Soc-Epinions [19] contains 51 variables and 4578589 clauses (all clause data is for 3-clique; 2-path has approximately 2/3rd that number of clauses). For the SAT datasets, AIS6 [3] has 61 variables and 581 clauses; AIS8 [3] has 113 variables and 1520 clauses; AIS10 [3] has 181 variables and 3151 clauses; AIS12 [3] has 265 variables and 5666 clauses; ls8-simplified4 [2] has 119 variables and 410 clauses; and LS5-firstr [2] has 125 variables and 529 clauses. In CNFTetris, loadtime refers to the time to determine the variable ordering and insert the boxes into the database, while runtime is the time to find all satisfying solutions. represents a vertex in e.g. the triangle. Next, we will encode each absent edge (i.e., a pair of vertices v 1 and v 2 such that the edge (v 1 , v 2 ) ∉ E , ¡ ¢ where E is the edge set of the graph) as k2 total Boolean formulas for the k-clique query, and k formulas for the k-path query. Each of these formulas corresponds to e.g. one of the three edges of a triangle. Observe that any possible satisfying solution to the SAT problem cannot select an edge that does not exist; therefore, any assignment that matches one of these formulas on all variables must be rejected. Equivalently, any accepting assignment must match at least one variable in the inverse. This naturally leads to a CNF definition, which is what we create. We will repeat this encoding over each possible set of vertex pairings such that the lower-indexed vertex is always written before the higher-indexed vertex, while adding additional clauses to reject all edges that would be from the higher-indexed vertex to the lower-indexed vertex. Simplifying resolutions are also performed where possible. Therefore, we have created a CNF problem where, for each output to the query in the original problem, there exists a solution. We can and do use this problem instance as input for both Tetris and other model counters. Let us examine an example instance of this. Consider the very simple example graph depicted in Figure 11 below, using the triangle query. In order to encode the non-edge (v 2 , v 4 ), we must first calculate the binary encodings of each of these vertices. These are (01) and (11), respectively. We then flip all of the 23 bits, giving us (10) and (00). Then we construct three CNF clauses, each of which corresponds to being the first, the second, or the third edge of the triangle. The first will be (x 1 ∨ x¯2 ∨ x¯3 ∨ x¯4 ), with (x 1 ∨ x¯2 ) corresponding to (10) and (x¯3 ∨ x¯4 ) corresponding to 00. Similarly, the second will be (x 1 ∨ x¯2 ∨ x¯5 ∨ x¯6 ); and the third will be (x 3 ∨ x¯4 ∨ x¯5 ∨ x¯6 ). Additionally, we insert clauses forbidding “bad" orderings of the points; in other words, we are making sure we do not count (v 1 , v 2 , v 3 ) and (v 2 , v 3 , v 1 ) as separate triangles. Then, when Tetris run on this CNF input attempts to recover the number of triangles, when it uses the probe point (assuming naive ordering) 〈T, T, T, F, F, T 〉 — that is, the probe point that corresponds to the inverted binary representations of v 1 , v 2, and v 3 , the three vertices in the top-right triangle — let us consider what happens. Since each of the selected edges does not correspond to the missing edge, we know that all three of those clauses must be satisfied; and since each edge is in the index order, we know that the additional clauses that we added will also accept our input. Therefore, Tetris will add this probe point to the output list. This continues for all other probe points until Tetris has found all triangles. v1 v2 v4 v3 Figure 11: A sample graph on four vertices, used in the above example. We will encode the non-edge (v 2 , v 4 ) as our SAT formula, along with additional clauses that ensure we only count triangles once, which will allow us to run the query as a #SAT problem. 4.1.2 Results Analysis As can be seen in Table 3, while all of the other solvers find these problems to be difficult, CNFTetris solves them quickly. Queries that take seconds on CNFTetris wind up taking hours on the competition, with CNFTetris running nearly a thousand times faster on some problems. This is largely due to the extremely high number of clauses relative to the number of variables, along with the fact that these clauses contain a large number of variables; these factors are not present in many of the standard SAT benchmarks. For instance, while the average clause in many SAT benchmarks contains two or three variables, here the average clause has thirty or more. And while SAT benchmarks rarely have over ten times as many clauses as variables, here the system is forced to tackle an environment where the number of clauses is exponentially larger than the number of variables. Note that all of the solvers we are comparing against use unit propagation techniques in order to count models [9]; see Section 5 for details. Because of this, the increased number of clauses directly corresponds to increased work for these solvers. 4.2 Nongraph Results In this section, we will discuss how CNFTetris performed as compared to other solvers on standard model counting benchmarks. 24 4.2.1 About the Datasets These datasets are a combination of datasets from the SATLIB datasets [3] and the SampleCount benchmarks for model counting taken from International Joint Conference on Artificial Intelligence ’07 [2]. We chose to use the AIS datasets for several reasons. First, each of the datasets terminates in a reasonable amount of time on all solvers, allowing us to find interesting comparisons. Secondly, due to the existence of increasingly-sized versions of this dataset, we can use this as insight into whether or not Tetris is scaling efficiently with the size of the dataset. Additionally, we featured the ls8-simplified4 and LS5firstr datasets, which gave us insights into our implementation’s strengths and weaknesses. 4.2.2 Results Analysis As Table 3 shows, Tetris is competitive with dSharp and Cachet on many of the datasets. Indeed, a factor of 2 separates us from either solver on all of the AIS datasets, a difference that engineering work alone can easily overcome. While there is significantly more space between it and sharpSAT, a factor of 10 on average, we believe the distance is not insurmountable. The largest gaps exist on the ls8-simplified and LS5-firstr datasets. On these, CNFTetris is roughly a factor of 5 off of the worst of the competition, and a factor of over 20 as compared to sharpSAT. The reason for this is simple: both of these datasets contain pure variables. In other words, there exist variables x such that, in all clauses, x̄ never appears, or vice-versa. This is an important piece of information, one that can and must be utilized, but CNFTetris in its current state does not know how to do so. However, since we know what the problem is, we expect to be able to quickly and efficiently attack this issue. 5 Related Work Our work builds on Tetris as developed by Abo Khamis et al. in [6]. In that work, the authors introduced Tetris as a beyond-worst-case algorithm for geometrically solving the database join problem. This in turn built on work on the Minesweeper [23], NPRR [24] and Leapfrog [29] algorithms, of which Tetris is a generalization. Furthermore, Tetris itself is can be considered a version of the DPLL algorithm [10] with clause learning. In DPLL, which is itself an evolution of the earlier DP [11] algorithm, a variable is chosen at every stage and assigned to be either true or false. The algorithm then uses unit propagation in order to simplify clauses under these assumptions. In this techniques, after the solver assigns a value to a variable, every other clause is inspected to see if this assignment creates a unit clause (i.e. a clause with only one variable in it), and to see if resolutions can be performed. This process continues until a conflicting clause (that is, a clause that is violated by the assignments) is found, at which point the algorithm is forced to backtrack. In the clause learning versions, introduced in [27], the solver takes this as an opportunity. It determines where it went astray, adds a new clause to its cache that is the negation of this errant assignment, non-chronologically backtracks to where this decision-making took place, and then proceeds in the opposite direction. The reasoning why CNFTetris is a form of this algorithm follows from the aforementioned method of converting from SAT clauses to boxes, and vice-versa (see Algorithm 1). Since these two representations are exactly equivalent, any operation performed on one representation can be translated into an operation on the other. Hence, every single operation Tetris performs on the boxes over its execution must correspond exactly to a set of operations on the original clauses. For instance, the Contains operation matches up with the idea of a conflicting clause. When a containing box is found, we can consider this as finding a box that rejects the current probe point as a po25 tential output point. Meanwhile, a conflicting clause rejects a potential satisfying assignment in much the same way. Furthermore, over the course of Tetris, the algorithm tentatively assigns a variable to either true or false, and then proceeds along with this assumption until a contradiction is found, all while learning additional clauses where possible through the resolution process. When a containing box is found or synthesized through resolution, and we advance the probe point accordingly, we are in essence backtracking to the earliest decision point and choosing to go in the opposite direction, just as DPLL with clause learning does. Therefore, this is exactly the DPLL algorithm with clause learning, with the added restriction of a fixed global variable ordering [6]. Tetris additionally utilizes a three-value logic system. While similar systems have been utilized in database schemes, such as by Zaniolo in [31], in these systems the three values are true, false, and unknown. Here, however, the three values we are considering can be summarized as true, false, and both. This causes a number of key differences. For instance, true ∧ unknown is equivalent to unknown, while true ∧ both is equivalent to true. Similarly, true ∨ unknown is equivalent to true, while true ∨ both is equivalent to both. Much work has been done in creating SAT solvers. Let us briefly discuss those state-of-the-art solvers we are comparing our work against. First, let us consider Cachet[26]. This solver was originally released in 2005, with minor compatibility updates continuing through the most recent version, which came out in 2015 [1]. Next, there is sharpSAT. First released in 2006, sharpSAT significantly eclipsed contemporary solvers [28]. sharpSAT has been maintained over time, with the most recent release in 2013 [4]. Finally, we come to dSharp. The most recently released of our three competitors, dSharpwas introduced in 2012 in order to efficiently compile CNF problems into the Decomposable Negation Normal Form language [22]. Further work allowed it to function as a model counter, which is how we utilize it. The version we use was released in 2016. What all of these solvers have in common, including CNFTetris, is that they have at their core a form of the DPLL algorithm with clause learning; indeed, almost all modern SAT and #SAT solvers do so [9]. The differences, then, come in terms of efficiency. Each solver uses a different array of techniques in order to effectively cache and recover learned clauses, to determine the variable ordering, and to identify clause conflicts. With Cachet, the authors focused on adding component caching capabilities on top of an existing SAT solver, ZChaff [26], the theoretical grounds for which were themselves introduced in [21]. This caching involved the storing of subproblems in a local cache, so that these clauses would not have to be re-derived by Cachet at a later juncture, thereby reducing redundant calculations over the course of the algorithm. This can be viewed as analogous to how CNFTetris stores learned boxes in a local cache, which it checks for containing boxes before examining the original database. A subproblem, meanwhile, could be thought of as a box with a high percentage of variables set to λ. However, one key difference here is the nature of the cached components. In Cachet, due to how the algorithm functions, it must regularly prune the cache of siblings that would otherwise cause it to undercount the number of models [26]. CNFTetris, in contrast, needs to perform no such pruning; it will naturally determine the exact number of models without any additional work. sharpSAT built on the work in Cachet while adding new ideas of its own [28]. Boolean constraint propagation (also known as the failed literal rule [15]) and unit propagation heuristics are used by sharpSAT to identify failed literals with greater efficiency than was done in Cachet [28]. However, by fixing the variable order, CNFTetris simplifies this process. Ultimately, this means that it finds its conflicting boxes in a fundamentally different manner than sharpSAT does, which provides room for CNFTetris to 26 outperform sharpSAT. dSharp, much like how sharpSAT built on Cachet uses sharpSAT as a core component [22]. The authors perform a DNNF translation, and then use properties of decomposability and determinism to perform model counting [15]. Though these differences do allow it to outperform more pure DPLLbased solvers on some benchmarks [15], since this system still uses sharpSAT as a core component, it still shares many of the same advantages and disadvantages in comparison to CNFTetris. As we have seen, all of the competing solvers can be viewed as evolutions along a single line. While CNFTetris does not throw the baby out with the bathwater — that is, while CNFTetris still continues to implement the classic DPLL algorithm — it does represent a distinct deviation from that line, challenging assumptions such as the necessity of allowing a non-fixed global variable ordering and the much more complex data storage scheme necessary in order to accommodate this. While this has necessitated much work in order to implement, it has also shown vast promise. 6 Acknowledgments We would like to thank Mahmoud Abo Khamis, Hung Q. Ngo, Christopher Ré, and Ce Zhang for very helpful discussions. References [1] Cachet. http://www.cs.rochester.edu/users/faculty/kautz/Cachet/index.htm. cessed: 2016-11-24. [2] International joint conference on artificial intelligence ’07 dataset Ac- collection. 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Neural Affine Grayscale Image Denoising arXiv:1709.05672v1 [cs.CV] 17 Sep 2017 Sungmin Cha, Taesup Moon College of Information and Communication Engineering Sungkyunkwan University, Suwon, Korea 16419 [email protected] Abstract We propose a new grayscale image denoiser, dubbed as Neural Affine Image Denoiser (Neural AIDE), which utilizes neural network in a novel way. Unlike other neural network based image denoising methods, which typically apply simple supervised learning to learn a mapping from a noisy patch to a clean patch, we formulate to train a neural network to learn an affine mapping that gets applied to a noisy pixel, based on its context. Our formulation enables both supervised training of the network from the labeled training dataset and adaptive fine-tuning of the network parameters using the given noisy image subject to denoising. The key tool for devising Neural AIDE is to devise an estimated loss function of the MSE of the affine mapping, solely based on the noisy data. As a result, our algorithm can outperform most of the recent state-of-the-art methods in the standard benchmark datasets. Moreover, our fine-tuning method can nicely overcome one of the drawbacks of the patch-level supervised learning methods in image denoising; namely, a supervised trained model with a mismatched noise variance can be mostly corrected as long as we have the matched noise variance during the fine-tuning step. 1 Introduction Image denoising is one of the oldest problems in image processing and various denoising methods have been proposed over the past several decades, e.g., BM3D [1], wavelet shrinkage [2], field of experts [3], sparse-coding based approach [4], WNNM [5], EPLL [6] and CSF [7], etc. In this paper, we propose a new image denoiser, dubbed as Neural Affine Image Denoiser (Neural AIDE), which utilizes neural network in a novel way. The method is inspired by the recent work in discrete denoising [8], in which a novel “pseudo-labels” were devised to train a denoiser solely based on the noisy data. We extend the approach to the continuous-valued data case and devise a novel estimated loss function based on the noisy data that is an unbiased estimate of the true MSE. By investigating the devised estimated loss function we formulate to train a neural network to learn an affine mapping that gets applied to a noisy pixel, based on its context. Such formulation enables both supervised training of the network from the labeled training dataset and adaptive fine-tuning of the network parameters using the given noisy image subject to denoising. Our experimental results extensively show how we made subtle design choices in developing our algorithm. Furthermore, we show that Neural AIDE significantly outperforms strong state-of-the-art baselines in the standard benchmark test datasets. 2 Notations and Problem Setting We denote xn×n as the clean grascale image, and each pixel xi ∈ {0, . . . , 255} is corrupted by an independent additive noise to result in a noisy pixel Zi , i.e., Zi = xi + Ni , i = 1, . . . , n2 , (1) where the continuous noise variables Ni ’s are independent (not necessarily identically distributed nor Gaussian) over i and E(Ni ) = 0, E(Ni2 ) = σ 2 for all i. Moreover, As in the standard processing in grayscale image denoising, we normalize both xi ’s and Zi ’s with 255 and treat them as real numbers. Importantly, following the universal setting in discrete denoising [9, 8], we treat the clean image xn×n as an individual image without any probabilistic model and only treat Z n×n as random. 2 Generally, a denoiser can be denoted as X̂ n×n = {X̂i (Z n×n )}ni=1 denoting that each reconstruction at location i is a function of the noisy image Z n×n . The standard loss function used for the grayscale image denoising to measure the denoising quality is the mean-squared error (MSE) denoted as 2 n×n ΛX̂ n×n (x ,Z n×n ) n 1 X Λ xi , X̂i (Z n×n ) 2 n i=1 = (2) where Λ(x, x̂) = (x − x̂)2 is the per-symbol squared-error. Conventionally, the MSE is compared in the dB-scale using the Peak Signal-to-Noise-Ratio (PSNR) defined as 10 log10 (1/ΛX̂ n×n (xn×n , Z n×n )). 2.1 Estimated loss function for the affine denoiser In this paper, we consider the denoiser of the form X̂i (Z n×n ) = a(Z \i ) · Zi + b(Z \i ) for each i, in which Z \i stands for the entire noisy image except for Zi . Namely, the reconstruction at location i has the affine function form of the noisy symbol Zi , but the slope and the intercept parameters, i.e., a(Z \i ) and b(Z \i ), of the affine function can be functions of the surrounding pixels. Hence, separete parameters can be learned from data for each location. Before presenting more concrete form of our denoiser, we first consider the following lemma. Lemma 1 Consider a single-symbol case Z = x + N with E(N ) = 0 and E(N 2 ) = σ 2 , and suppose a single-symbol denoiser has the form of X̂(Z) = aZ + b. Then, L(Z, (a, b); σ 2 ) = (Z − (aZ + b))2 + 2aσ 2 (3) is an unbiased estimate of Ex Λ(x, X̂(Z)) + σ 2 , in which Λ(x, x̂) = (x − x̂)2 and Ex (·) notation stands for the expectation over Z given that the clean symbol is x. Remark: Note while the true MSE, Λ(x, X̂(Z)), can be evaluated only when the clean symbol x is known, the estimated loss L(Z, (a, b)) can be evaluated soley with the noisy symbol Z, the affine mapping (a, b) and the noisy variance σ 2 . Thus, L(Z, (a, b)) plays a key role in adaptively learning the neural network-based affine denoiser as shown in the next section. Proof: By simple algebra, we have the following equalities: Ex (x − X̂(Z))2 = Ex (x2 + (aZ + b)2 − 2x(aZ + b)) = Ex (x2 + (aZ + b)2 − 2ax2 − 2bx) 2 2 2 2 (4) 2 = Ex (Z − σ + (aZ + b) − 2a(Z − σ ) − 2bZ) 2 = Ex Z − (aZ + b) + (2a − 1)σ 2 (5) (6) = Ex L(Z, (a, b); σ 2 ) − σ 2 , in which (4) follows from Ex (Z) = x, (5) follows from Ex (Z 2 ) = x2 + σ 2 and replacing x2 with Ex (Z 2 − σ 2 ), and (6) follows from simply rearranging the terms. Thus, we have the lemma. From Lemma 1, we can also show that for the denoisers of the form X̂i (Z n×n ) = a(Z \i )·Zi +b(Z \i ), Exi Λ(xi , X̂i (Z n×n )) Z \i = Exi L(Zi , (a(Z \i ), b(Z \i )); σ 2 )\|Z \i − σ 2 (7) holds since a(Z \i ) and b(Z \i ) become constant given Z \i and the noise is independent over i. The Exi (·\|Z \i ) in (7) stands for the conditional expectation of Zi given the clean symbol xi and the noisy symbols Z \i . Note the estimated loss function similar to (3) has been also used to the filtering problem [10]. 2 3 3.1 Neural AIDE: Neural Affine Image DEnoiser Neural network-based affine denoiser Our proposing Neural Affine Image DEnoiser (Neural AIDE) considers the denoiser of the form \i \i X̂i (Z n×n ) = a(Ck×k ) · Zi + b(Ck×k ), i = 1, . . . , n × n (8) \i in which Ck×k stands for the noisy image patch, or the context, of size k × k surrounding Zi that does not include Zi . Thus, the patch has a hole in the center. Then, we define a neural network g(w, ·) : [0, 1]k that takes the context \i Ck×k 2 −1 → R2+ (9) as input and outputs the slope and intercept parameters \i a(Ck×k ) and \i b(Ck×k ) for each location i. We denote w as the weight parameters of the neural network, which will be learned by the process described in the later sections. As it will get clear in our arguments below, the specific form of our denoiser in (8) enables learning the parameters by both supervised learning with labelled training data and adaptive fine-tuning with the given noisy image. Note in (9), we put a constraint that the slope and intercept of the affine function, i.e., the output of the network, should be nonnegative. While such constraint would appear apparent in our experimental results, it also makes an intuitive sense; the denoiser (8) tries to estimate xi from Zi , which are both in the interval [0, 1], hence, the nonnegative slope and intercept parameters should suffice. The nonnegativity constraint is realized in the neural network by applying f (x) = log(1 + ex ) (10) as the activation function for the final output layer of the neural network. The rest of the network architecture is the ordinary fully-connected neural network with ReLU activation functions, as depicted in Figure 1. There are two sharp differences with our Neural AIDE and other neural network based denoisers, e.g., [11, 12]. First, the other schemes take the full noisy image patch (including the center location) as input to the network, and the network is trained to directly infer the corresponding clean image patches. In contrast, Neural AIDE is trained to first learn an affine mapping based on Figure 1: The arthe noisy image patch with a hole (i.e., the context of Zi ), then the learned chitecture of Neural mapping is applied to Zi to obtain the recostruction X̂i . Such difference AIDE enables the development of the estimated loss function in Lemma 1 and the adaptive training process described in the next section. The principle of learning a mapping first and applying the mapping to the noisy symbol for denoising or filtering has been utilized in [13, 10, 8]. Second, unlike the other schemes, in which the patch-level reconstructions should somehow be aggregated to generate the final denoised image, Neural AIDE simply generates the final pixel-bypixel reconstructions. Thus, there is no need for a step to aggregate multiple number of reconstructed patches, which simplifies the denoising step. Furthermore, since the neural network of Neural AIDE only has to estimate the two parameters of the affine mapping from each context, Neural AIDE can make much more efficient usage of the data with a simpler model compared to the networks in other schemes that need to estimate the full k × k-patch, e.g., [11]. 3.2 Adaptive training with noisy image We first describe how the network parameters w can be adaptively learned from the given noisy image Z n×n without any additional labelled training data. That is, by denoting each output element \i of the neural network g(w, ·) for the context Ck×k as \i \i \i \i g(w, Ck×k )1 , a(Ck×k ) and g(w, Ck×k )2 , b(Ck×k ), we can define an objective function for the neural network to minimize as Ladaptive (w, Z n×n n2 1 X \i \i L Zi , (g(w, Ck×k )1 , g(w, Ck×k )2 ); σ 2 ), 2 n i=1 3 (11) by using the estimated loss function L(Z, (a, b); σ 2 ) defined in Lemma 1. The training process using (11) is identical to the ordinary neural network learning, i.e., start with randomly initiallized w, then use backprogagation and variants of mini-batch SGD for updating the parameters. The formulation (11) may seem similar to training a neural network for a regression problem; namely, 2 \i {(Ck×k , Zi )}ni=1 , which are solely obtained from the noisy image Z n×n , can be analogously thought of as the input-target label pairs for the supervised regression. But, unlike regression, which tries to directly learn a mapping from input to the target label, our network learns the affine mapping for each context and apply it to Zi to estimate the unobserved clean symbol xi . The fact that (11) only depends on the given noisy image Z n×n (and the assumed σ 2 ) makes the learning adaptive. The rationale behind using L(Z, (a, b); σ 2 ) in (11) is the following; as shown in (7), the estimated \i loss is an unbiased estimate of the true expected squared-error given the context Ck×k . Therefore, minimizing (11) may result in the network that produces the slope and intercept parameters that minimize the true MSE for the reconstrunctions of the corresponding affine mappings. This formulation of training neural network parameters solely based on the noisy data is inspired by the recent work in discrete denoising [8]. Once the training is done, we can then denoise the very noisy image Z n×n used for training by applying the affine mapping at each location as (8). That is, by denoting w∗ as the learned parameter by minimizing (11), the reconstruction at location i by Neural AIDE becomes \i \i X̂i,Neural AIDE (Z n×n ) = g(w∗ , Ck×k )1 · Zi + g(w∗ , Ck×k )2 . 3.3 (12) Supervised training and adaptive fine-tuning While the formulation in (11) gives an effective way of adaptively training a denoiser based on the given noisy image Z n×n , the specific form of the denoiser in (8) makes it possible to carry out the supervised pre-training of w before the adaptive training step. That is, we can collect abundant clean images, x̃n×n , from the various image sources (e.g., World Wide Web) and corrupt them with the assumed additive noise with variance σ 2 in (1) to generate the correspoding noisy images, Z̃ n×n , and the labelled training data of size N , D = {(x̃i , C̃i,k×k )}N i=1 . (13) In (13), C̃i,k×k stands for the noisy image patch of size k × k at location i that includes the noisy symbol Z̃i , and x̃i is the clean symbol that correspond to Z̃i . Now, the subtle point is that, unlike the usual supervised learning that may directly learn a mapping from C̃i,k×k to x̃i , we remain in using the neural network defined in (9) and learn w by minimizing N 1 X \i \i Λ x̃i , g(w, C̃k×k )1 · Z̃i + g(w, C̃k×k )2 . Lsupervised (w, D) , N i=1 (14) Note Λ(x, x̂) = (x − x̂)2 as before. The training process of minimizing (14) is again done by the usual backpropagation and the variants of mini-batch SGD. Once the objective function (14) converges after sufficient iteration of weight updates, we denote the converged parameter as w̃. Then, for a given noisy image to denoise, Z n×n , we can further update w̃ adaptively for Z n×n by minimizing Ladaptive (w, Z n×n ) in (11) starting from w̃. That is, we adaptively fine-tune w̃ until Ladaptive (w, Z n×n ) converges, then denoise Z n×n with the converged parameter as (12). This capability of adaptively fine-tuning the supervised trained weight parameter is the unique characteristic of Neural AIDE that differentiates it from other neural network-based denoisers. 4 Experimental Results We compared the denoising performance of the proposed Neural AIDE with several state-of-the-art denoising methods, including BM3D [1], MLP [11], EPLL [6], WNNM [5] and CSF [7]. 4 4.1 Data and experimental setup For the supervised training, we generated the labelled training set using 2000 images available in public datasets. Out of 2000 images, 300 images are taken from train/validation set in the Berkeley Segmentation Dataset and the remaining 1700 images are taken from Pascal VOC 2012 Dataset. For the Pascal VOC images, we resized them to match the resolution of the Berkeley Segmentation Dataset [14], 481 × 321. We corrupted the images with additive Gaussian noise and tested with multiple noise levels, namely, σ = 5, 10, 15, 20, 25. That is, we built separate training set of size 2000 for each noise level. The total number of training data points (i.e., N in (13)) in each dataset was thus about 308 million. We evaluated the performance of the denoisers with 11 standard test images, i.e., {Barbara, Boat, C.man Couple, F.print, Hill, House, Lena, Man, Montage and Peppers}, and 68 standard Berkeley images [3]. Our network had 9 fully connected layers with 512 nodes in each layer, which showed the best result among a few tried models 1 . ReLU was used as activation functions, and we used Adam [15] as the optimizer to train the network. For the supervised training, we trained the network up to 50 epochs and halved the learning rate every 10 epochs starting from 10−4 . For the adaptive fine-tuning, we also trained up to 50 epochs and halved the learning rate every 20 epochs starting from 10−5 . We \i did not use any regularization methods while training. Moreover, for the context data, Ck×k , we subtracted 0.5 from the values to make the input to the network get centered around 0. (Note Zi that the affine mappping gets applied to in (12) still is in the original scale.) For all our experiments, we used Keras (version 1.2.2) with Tensorflow (version 0.11.0) backend and NVIDIA’s GPU (GeForce GTX1080) with CUDA library version 8.0. 4.2 Training Neural AIDE In this section, we systematically show the reasoning behind choosing the context size k, the empirical justification of the nonnegative contraint on the outputs of g(w, ·) and the validity of the combination of the supervised pre-training with adaptive fine-tuning. 4.2.1 Adaptive training with noisy image We first carried out the adaptive training solely with the given noisy image as described in Section 3.2. That is, for each given noisy image, we randomly initialized the weight parameters of the neural network and trained with the objective function (11). After training, the image was denoised as (12). Figure 2(a) shows the PSNR results on the standard 11 test images with varying k values and output activation functions, i.e., Linear (f (x) = x), Positive (f (x) = log(1 + ex ) in (10)) and Sigmoid (f (x) = 1/(1 + e−x )). The noise level was σ = 25. From the figure, we can see that the adaptive training alone can still result in a decent denoiser, although some PSNR gap exists compared to the state-of-the-arts as shown in Table 1. We see that k = 7 tend to be the best context size for adaptive training. Moreover, the choice of the output activation functions turns out to be important, and more discussion is given on the activation function in the next section. 4.2.2 Supervised training and adaptive fine-tuning Since the limitation of the adaptive training alone was apparent, we then carried out the supervised training in Section 3.3. That is, we took the 300 images from the Berkeley Segmentation Dataset and trained the network with varying k values as shown in Figure 2(b). Denoising of the noisy image was done identically as before by applying the learned affine mapping to each noisy pixel. Note in this case, we only carried out the experiments with the Linear activation function. We can see that the supervised training can result in a much higher PSNR values than the adaptive training, already very close to the state-of-the-arts. Also, the performance seems to get saturated around k = 17, so in all our experiments below, we used k = 17. Encouraged by this result, we moved on to adaptively fine-tuning the weight parameters by minimizing the objective function (11) for each image initialized with the parameters learned by supervised learning. This is when the subtle issue regarding the activation function we describe below comes 1 The difference among the models were not huge. 5 (a) Adaptive training (random initialization) (b) Supervised training (300 training images) Figure 2: Adaptive and supervised training results on the standard 11 test images (σ = 25) up. In Figure 3, we trained supervised learning models with Linear and Positive output activation functions using 800 images for σ = 25, then adaptively fine-tuned the parameters for given noisy image (F.print and Montage image). Figure 3(a)-3(d) show the distributions of the slope (a) and intercept (b) paramters that each model outputs for the given image, and 3(i) shows the change of PSNR value in the process of adaptive fine-tuning. From Figure 3(a) and 3(e), we can see that when trained with supervised learning with Linear output activation function, the values of a and b all lie in the interval [0, 1]. However, when fine-tuned for each image, Figure 3(b) and 3(f) show that many negative a values are produced for the Linear activation. This can be readily seen by examining the form of L(Z, (a, b); σ 2 ) in (3), which does not hinder a from having negative values when there is no constraint. As shown in Figure 3(i), such negative a values for the affine mapping sometime does not have big effect on the fine-tuning process and the final denoising performance as in the case of F.print, in which the PSNR increases significantly from the supervised model by fine-tuning. However, as in the case of Montage in Figure 3(i), we suspect such negative a values sometimes hurt the denoising performance greatly. In contrast, when we put the nonnegativitiy contstraint on a and b in the neural network, we observe a stable fine-tuning process, as is observed in Figure 3(d), 3(h) and 3(i). Thus, the results of Neural AIDE from now on all uses the positive activation function. 2 Figure 4 shows the adaptive fine-tuning process of the standard 11 images for σ = 15. The supervised model was trained with the full training set of 2000 images. From the figures, we can see that the learning is done appropriately and the PSNR does improve with fine-tuning. 2 We also tested with the sigmoid activation and the result was more or less the same. 6 (a) F.print(Lin.,s) (b) F.print(Lin.,ft) (e) Montage(Lin.,s) (f) Montage(Lin.,ft) (c) F.print(Pos.,s) (d) F.print(Pos.,ft) (g) Montage(Pos.,s) (h) Montage(Pos.,ft) (i) PSNR values during adaptive fine-tuning. Figure 3: (a-h) Distribution of a and b values for F.print and Montage after supervised training (s) and fine-tuning (ft) for Linear (Lin.) and Positive (Pos.) activation functions. The distributions obtained for fine-tuning are from the models at 50 epoch. (i) PSNR values during fine-tuning. (a) PSNR (b) Objective function (11) Figure 4: PSNR and objective function value during fine-tuning for the standard 11 images (σ = 15) 4.3 4.3.1 Quantitative evaluation Standard 11 images Table 1 summarizes our denoising results compared to the recent state-of-the-arts on the standard 11 images for various noise levels. We show both mean and standard deviation of PSNR values. For the baseline methods, we downloaded the codes from the authors’ webpages and ran the code on the noisy images, thus, the numbers can be compared fairly. (MLP and CSF57×7 could run only on selected noise levels.) N-AIDES stands for the Neural AIDE that is only supervised trained (with 2000 images). N-AIDEfB and N-AIDEfH are fine-tuned models after supervised learning; N-AIDEfB is the best model (in terms of epoch) chosen based on PSNR (thus, not practical) and N-AIDEfH is the model that is chosen with a heuristic rule - i.e., stop fine-tuning when the training loss becomes smaller than σ 2 , otherwise fine-tune until 50 epochs. From the table, we can see that N-AIDEfH significantly outperforms all other baselines on average except for WNNM. The difference of mean PSNR between WNNM and N-AIDEfH is almost 7 negligible and N-AIDEfH tend to have smaller variance in terms of PSNR than WNNM. By comparing N-AIDES and N-AIDEfH , we can definitely see that adaptive fine-tuning is effective. Also, when the noise level is low, the improvement gets larger. Furthermore, by comparing N-AIDES with MLP, which is another neural network based denoiser and uses much more data points (362 million exmample) and larger model, we can confirm that our model more efficiently uses the data. σ 5 10 15 20 25 PSNR Mean Std Mean Std Mean Std Mean Std Mean Std BM3D 38.24 1.24 34.71 1.37 32.76 1.48 31.43 1.50 30.40 1.51 MLP 34.45 1.12 30.24 1.43 EPLL 37.88 1.07 34.27 1.18 32.29 1.35 30.90 1.34 29.81 1.38 WNNM 38.43 1.28 34.95 1.42 32.99 1.54 31.59 1.57 30.51 1.56 CSF57×7 32.40 1.27 29.93 1.41 N-AIDEs 38.14 1.17 34.66 1.31 32.77 1.44 31.38 1.50 30.36 1.53 N-AIDEfB 38.44 1.18 34.92 1.33 32.97 1.42 31.58 1..46 30.51 1.45 N-AIDEfH 38.44 1.18 34.91 1.33 32.96 1.42 31.55 1.44 30.47 1.46 Table 1: PSNR comparsions on the 11 standard benchmark images for σ = 5, 10, 15, 20, 25. Figure 5(a) shows the competitive comparison between N-AIDEfH and the baselines. That is, the figure plots the number of images of which the PSNR of N-AIDEfH is better than the baseline methods. We can see that our method mostly outperforms all baselines competitively, including WNNM. One of the main drawbacks of MLP [11] is that the neural networks have to be trained separately for all noise levels and the mismatch of σ significantly hurts the denoising performance. While the supervised training of Neural AIDE is also done in the similar way, Figure 5(b)-5(c) show that the adaptive fine-tuning can be very effective in overcoming such limitation. Figure 5(b) shows the PSNR results of the mismatched N-AIDEs models before fine-tuning. Each row is normalized with the PSNR of the matched case, i.e., the diagonal element, and the PSNR values are color-coded. We clearly see the sensitivity of PSNR in the mismatch of σ as the off-diagonal values show significant gaps compared to the diagonal values in each row. On the other hand, Figure 5(c) shows the PSNR values of N-AIDEfH ’s that have mismatched supervised models but are adaptively fine-tuned with the correct σ’s. We can clearly see that the PSNR gaps of the mismatched supervised models can be significantly closed by adaptive fine-tuning, which gives a significant edge over MLP in [11]. (a) Competitive comparison (b) PSNR of N-AIDEs (c) PSNR of N-AIDEfH Figure 5: (a) Competitive comparison of N-AIDEfH with baselines (b) PSNR of mismatched N-AIDEs (c) PSNR of N-AIDEfH with mismatched N-AIDEs but fine-tuned with correct σ 4.3.2 Standard 68 Berkeley images Table 2 shows the PSNR results on the 68 standard Berkeley images from [3]. We can clear see that N-AIDEfH again outperforms the baseline state-of-the-art methods, including WNNM, with significant margins. 8 σ 5 10 15 20 25 MLP 33.41 28.73 EPLL 37.50 33.32 31.09 29.60 28.47 WNNM 37.71 33.48 31.18 29.63 28.46 CSF57×7 31.10 28.41 N-AIDEs 37.72 33.62 31.45 29.98 28.93 N-AIDEfB 37.82 33.71 31.52 30.05 28.97 N-AIDEfH 37.79 33.66 31.47 30.00 28.90 Table 2: PSNR comparisons on the 68 standard Berkeley images. 5 Concluding remarks We devised a novel neural network based image denoiser, Neural AIDE. The algorithm is devised with a different principle from the other state-of-the-art methods. As a result, we show that a very simple adaptive affine model, which Neural AIDE learns differently for each pixel, can significantly outperform many strong baselines. Also, the adaptive fine-tuning of Neural AIDE can successfully overcome the σ mismatch problem, which is a serious drawback of other neural network based methods. As a future work, we would like to more thoroughly carry out the experiments in even noisier regime. Also, since our algorithm does not require the noise to be Gaussian (only the additivity of the noise and σ 2 are assumed), we would try to other types of noise, e.g., Laplacian noise. Furthermore, extending our framework to non-additive noise such as multiplicative noise would be another interesting direction. Finally, theoretical anayses of our method based on information theory and learning theory would be another direction worth pursuing. 9 References [1] K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian. Image denoising by sparse 3-d transformdomain collaborative filtering. IEEE Trans. Image Processing, 16(8):2080–2095, 2007. [2] E.P. Simoncelli and E.H. Adelson. Noise removal via bayesian wavelet coring. In ICIP, 1996. [3] S. Roth and M.J Black. Field of experts. IJCV, 82(2):205–229, 2009. [4] J. Mairal, F. Bach, J. Ponce, G. Sapiro, and A. Zisserman. 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Convolutional Neural Networks for Histopathology Image Classification: Training vs. Using Pre-Trained Networks Brady Kieffer1 , Morteza Babaie2 Shivam Kalra1 , and H.R.Tizhoosh1 1 KIMIA Lab, University of Waterloo, ON, CANADA 2 Mathematics and Computer Science Department, Amirkabir University of Technology, Tehran, IRAN arXiv:1710.05726v1 [cs.CV] 11 Oct 2017 e-mails: [email protected], [email protected], [email protected], [email protected] Abstract— We explore the problem of classification within a medical image data-set based on a feature vector extracted from the deepest layer of pre-trained Convolution Neural Networks. We have used feature vectors from several pre-trained structures, including networks with/without transfer learning to evaluate the performance of pre-trained deep features versus CNNs which have been trained by that specific dataset as well as the impact of transfer learning with a small number of samples. All experiments are done on Kimia Path24 dataset which consists of 27,055 histopathology training patches in 24 tissue texture classes along with 1,325 test patches for evaluation. The result shows that pre-trained networks are quite competitive against training from scratch. As well, fine-tuning does not seem to add any tangible improvement for VGG16 to justify additional training while we observed considerable improvement in retrieval and classification accuracy when we fine-tuned the Inception structure. Keywords— Image retrieval, medical imaging, deep learning, CNNs, digital pathology, image classification, deep features, VGG, Inception. I. I NTRODUCTION We are amid a transition from traditional pathology to digital pathology where scanners are replacing microscopes rapidly. Capturing the tissue characteristics in digital formats opens new horizons for diagnosis in medicine. On on hand, we will need to store thousands and thousands of specimens in large physical archives of glass samples. This will be a relief for many hospitals with limited space. On the other hand, acquiring an image from the specimen enables more systematic analysis, collaborations possibilities and, last but not least, the computer-aided diagnosis for pathology, arguable the final frontier of vision-based disease diagnosis. However, like any other technology, digital pathology comes with its own challenges; whole-scan imaging generally generates gigapixel files that also require (digital) storage and are not easy to analyze via computer algorithms. Detection, segmentation, and identification of tissue types in huge digital images, e.g., 50,000×70,000 pixels, appears to be a quite daunting task for computer vision algorithms. Looking at the computer vision community, the emergence of deep learning and its vast possibilities for recognition and classification seems to be a lucky coincidence when we intend to address the above-mentioned obstacles of digital pathology. Diverse deep architectures have been trained with large set of images, e.g., ImageNet project or Faces in the Wild database, to perform difficult tasks like object classification and face recognition. The results have been more than impressive; one may objectively speak of a computational revolution. Accuracy numbers in mid and high 90s have become quite common when deep networks, trained with millions of images, are tested to recognize unseen samples. In spite of all progress, one can observe that the applications of deep learning in digital pathology hast not fully started yet. The major obstacle appears to be the lack of large labelled datasets of histopathology scans to properly train some type of multi-layer neural networks, a requirement that may still be missing for some years to come. Hence, we have to start designing and training deep nets with the available datasets. Training from scratch when we artificially increase the number of images, i.e., data augmentation, is certainly the most obvious action. But we can also use nets that have been trained with millions of (non-medical) images to extract deep features. As a last possibility, we could slightly train (finetune) the pre-trained nets to adjust them to the nature of or data before we use them as feature extractors or classifiers. In this paper, we investigate the usage of deep networks for Kimia Path24 via training from scratch, feature extraction, and fine-tuning. The results show that employing a pre-trained network (trained with non-medical images) may be the most viable option. II. BACKGROUND Over recent years researchers have shown interest in leveraging machine-learning techniques for digital pathology images. These images pose unique issues due to their high variation, rich structures, and large dimensionality. This has lead researchers to investigate various image analysis techniques and their application to digital pathology [1]. For dealing with the large rich structures within a scan, researchers have attempted segmentation on both local and global scales. For example, researchers have conducted works on the segmentation of various structures in breast histopathology images using methods such as thresholding, fuzzy c-means clustering, and adaptive thresholding with varying levels of success [1]–[4]. When applying these methods to histopathological images, it is often desired that a computer aided diagnosis (CAD) method be adopted for use in a content-based image retrieval (CBIR) system. Work has been done to propose various CBIR systems for CAD by multiple groups [5]. Recently, hashing To appear in proceedings of the 7th Intern. Conf. on Image Processing Theory, Tools and Applications (IPTA 2017), Nov 28-Dec 1, Montreal, Canada. methods have been employed for large-scale image retrieval. Among the hashing methods, kernelized and supervised hashing are considered the most effective [5], [6]. More recently Radon barcodes have been investigated as a potential method for creating a CBIR [7]–[9]. Yi et al. utilized CNNs on a relatively small mammography dataset to achieve a classification accuracy of 85% and an ROC AUC of 0.91 whereas handcrafted features were only able to obtain an accuracy of 71% [10]. Currently, there is interest in using pre-trained networks to accomplish a variety of tasks outside of the original domain [11]. This is of great interest for medical tasks where there is often a lack of comprehensive labeled data to train a deep network [12]. Thus, other groups have leveraged networks trained on the ImageNet database which consists of more than 1.2 million categorized images of 1000+ classes [12]–[14]. These groups have reported a general success when attempting to utilize pre-trained networks for medical imaging tasks [12], [14], [15]. In this study we explore and evaluate the performance of a CNN when pre-trained on non-medical imaging data [12], [16]. Specifically, when used as feature extractors with and without fine tuning for a digital pathology task. for the fine-tuning process (we do not use all of them to emulate cases where no large dataset is available; besides more extensive training may destroy what a network has already learned). The values of each patch were subsequently normalized into [0, 1]. The patches were finally downsized to a 224×224 to be fed into the CNN architecture. Following the above steps, we first obtained 27,055 patches from each scan based purely on the homogeneity threshold. Then, randomly sampled 100 patches from each class leading to the much smaller training set of 2,400 patches. A selection of patches from the training set can be viewed within Fig. 1. As Fig. 2 shows the testing samples are relatively balanced in Kimia Path24 dataset, whereas the training set is rather imbalanced. Different size and frequency of specimens are the main reasons for the imbalance. B. Accuracy Calculation The accuracy measures used for the experiments are adopted from [17]. These were chosen so that results between the papers could be compared. There are ntot = 1, 325 testing patches Psj that belong to 24 sets Γs = {Psi \|s ∈ S, i = 1, 2, . . . } with s = 0, 1, 2, . . . , 23 [17]. Looking at the set of retrieved images for an experiment, R, the patch-to-scan accuracy, ηp , can be defined as III. DATA S ET ηp = The data used to train and test the CNNs was the Kimia Path24 consisting of 24 whole scan images (WSIs), manually selected from more than 350 scans, depicting diverse body parts with distinct texture patterns. The images were captured by TissueScope LE 1.0 1 bright field using a 0.75 NA lens. For each image, one can determine the resolution by checking the description tag in the header of the file. For instance, if the resolution is 0.5µm, then the magnification is 20x, and if the resolution is 0.25µm, then the magnification is 40x. The dataset offers 27,055 training patches and 1,325 (manually selected) test patches of size 1000×1000 (0.5mm×0.5mm) [17]. The locations of the test patches in the scans have been removed (whitened) such that they cannot be mistakenly used for training. The color (staining) is neglected in Kimia Path24 dataset; all patches are saved as grayscale images. The Kimia Path24 dataset is publicly available2 . 1 X \|R ∩ Γs \| ntot (1) s∈S The whole-scan accuracy, ηw , can be defined as ηw = 1 X \|R ∩ Γs \| 24 (2) s∈S With the total accuracy is defined as ηtotal = ηp × ηw . By incorporating both the accuracy measurements the resulting problem becomes much more difficult when attempting to obtain acceptable results [17]. IV. M ETHODS Each experiment was run using the architecture for both the VGG16 and Inception-v3 networks as provided in the Keras Python package [18]–[20]. Utilizing a pre-trained network, we then analyze the effectiveness of the network when using it just as a feature extractor, and when transferring the network (some of its weights) to the medical imaging domain. A. Patch Selection To create the Kimia Path24 dataset, each scan is divided into patches that are 1000×1000 pixels in size with no overlap between patches. Background pixels (i.e., very bright pixels) are set to white and ignored using a homogeneity measure for each patch. The homogeneity for selection criterion is that every patch with a homogeneity of less than 99% is ignored. The high threshold ascertains that no patch with significant texture pattern is ignored. From the set of patches each scan had 100 randomly sampled patches are selected to be used A. Fine-Tuning Protocols When fine-tuning a deep network, the optimal setup varies between applications [21]. However, using a pre-trained network and applying it to other domains has yielded better performing models [12]. It was decided that only the final convolutional block (block 5) within VGG16 and the final two inception blocks within Inception-v3 would be re-trained [12], [18], [19], [21]. As in [14] a single fully connected layer 1 http://www.hurondigitalpathology.com 2 http://kimia.uwaterloo.ca 2 To appear in proceedings of the 7th Intern. Conf. on Image Processing Theory, Tools and Applications (IPTA 2017), Nov 28-Dec 1, Montreal, Canada. Fig. 1. A selection of patches from each training scan within the Kimia Path24 dataset. The patches are 1000×1000 pixels in size or 0.5mm×0.5mm. From top left to bottom right: scan/class 0 to scan/class 23. 3 To appear in proceedings of the 7th Intern. Conf. on Image Processing Theory, Tools and Applications (IPTA 2017), Nov 28-Dec 1, Montreal, Canada. Fig. 2. Instance distribution for training set (left) and testing set (right) of Kimia Path24 . of size 256 (followed by an output layer of size 24) was chosen to replace the default VGG16 fully connected layers when fine-tuning. This was found to give better results. The optimizer we used follows the logic from [12], [14] where the learning rate chosen was very small (10−4 ) and the momentum used was large (0.9), both of which were selected to ensure no drastic changes within the weights of the network during training (which would destroy what had been already learned). The Keras data augmentation API was used to generate extra training samples and the network was trained for a total of 200 epochs (after which the accuracy was no longer changing) with a batch size of 32 [20]. softmax classification layer. The fully connected layers were pretrained on bottleneck features and then attached to the convolutional layers and training on the final two inception blocks was then performed. The resulting networks (Transfer Learned VGG16 or TL-VGG16 and TL-Inception-v3) were then used to classify the test patches. The class activation mappings (CAMs) for the fine-tuned Inception-v3 network on randomly selected test patches can be viewed in Fig. 3. V. R ESULTS The results of our experiments are summarized in Table 1. It can be stated the results for VGG16 and CNN1 are quite similar; training from scratch, using a pre-trained network as feature extractor, and fine-tuning a pre-trained network are all delivering comparable results for Kimia Path24 . Whereas the results for Inception-v3 are similar with the transfer-learned model outperforming the feature extractor. As TL-Inceptionv3 produced the best results, ηtotal = 56.98%, and minimally updating the weights of a pre-trained network is not a time consuming task, one may prefer to utilize it. However, one may prefer using Inception-v3 to training from scratch and fine-tuning a pre-trained net as it requires no extra effort and produces similar results with a linear SVM. B. Pre-Trained CNN as a Feature Extractor By using the provided implementation of the specified architectures within Keras, the pre-trained network was first used as a feature extractor without any fine-tuning (Feature Extractor VGG16 or FE-VGG16 and FE-Inception-v3) [20]. The last fully connected layer of the network – prior to classification – was used extracted to be used a feature vector. As pre-trained networks are trained in other domains (very different image categories) and hence cannot be used as classifier, we used the deep features to train a linear Support Vector Machine (SVM) for classification. The Python package scikit-learn as well as LIBSVM were used to train SVM classifiers with a linear kernel [22], [23]. Both NumPy and SciPy were leveraged to manipulate and store data during these experiments [24], [25]. VI. D ISCUSSIONS It was surprising to find out that simply using features from a pre-trained network (trained on non-medical images, see Fig. 4) can deliver results comparable with a network that, with considerable effort and resources, has been trained from scratch for the domain in focus (here histopathology). As well, such simpler approach was even able to achieve a noticeable accuracy increase of ≈ 8.74% in overall performance for Kimia Path24 dataset. Another surprising effect was that transfer learning via fine-tuning for VGG16 was not able to provide any improvement compared to extracting deep features from a pre-trained network without any change in the learned of its weights whereas with Inception-v3 the improvement was immediate. Perhaps the most obvious reaction to this finding is that if we had enough samples, i.e., millions of histopathological images, and if we would use proper computational devices for efficient training, then CNN would perhaps deliver the C. Fine-Tuned CNN as a Classifier The proposed network was then fine-tuned to the Kimia Path24 dataset. Using the Keras library, the convolutional layers were first separated from the top fully connected layers [20]. The training patches were fed through the model to create a set of bottleneck features to initially pre-train the new fullyconnected layers [26]. These features were used to initialize the weights of a fully connected MLP consisting of one 256 dense ReLU layer and a softmax classification layer. Next, the fully connected model was attached to the convolutional layers and training on each convolutional block, except the last block, was performed to adjust classification weights [12], [14]. Similarily, for the Inception-v3 network the fully connected layers were replaced with one 1024 dense ReLU layer and a 4 To appear in proceedings of the 7th Intern. Conf. on Image Processing Theory, Tools and Applications (IPTA 2017), Nov 28-Dec 1, Montreal, Canada. Table 1. Comparing the results training form scratch (CNN1 reported in [17]), using deep features via a pre-trained network with no change (FE-VGG16), and classification after fine-tuning a pre-trained network (TL-VGG16, TL-Inception-v3). The best scores are highlighted in bold. Scheme Train from scratch Pre-trained features Fine-tuning the pre-trained net Pre-trained features Fine-tuning the pre-trained net Approach CNN1 [17] FE-VGG16 TL-VGG16 FE-Inception-v3 TL-Inception-v3 ηp 64.98% 65.21% 63.85% 70.94% 74.87% ηw 64.75% 64.96% 66.23% 71.24% 76.10% ηtotal 41.80% 42.36% 42.29% 50.54% 56.98% Fig. 4. Sample images from ImageNet project. One may object to using features that have been learned from such images in order to classify highly sensitive images of histopathology for medical diagnosis. However, experiments with Kimia Path24 dataset shows that features extracted from these images are expressive enough to compete against networks trained by histopathology images from scratch [Source: http://openai.com/ ]. a deep network, architecture not well suited to the problem, or an overly simplistic fully connected network. However, as previously discussed in [17], the problem given by the Kimia Path24 dataset is indeed a hard problem, most likely due to the high variance between the different patches within a given scan (intra-class variability). This is further validated when looking at the results in Fig. 3. The two columns contain patches that have distinct patterns with their own unique features. The CAM from the first column shows that the network responds strongly to the unique structures within the 4 label (very strongly for the final patch). Whereas when presented with completely different patterns in the second column, the network responds strongly to other areas, typically ones that embody inner edges within the sample. This shows evidence that the model has at the very least begun to learn higher level Fig. 3. Activation maps using randomly selected patches from the Kimia Path24 testing data. The patches within each column are the same class and the labels per column are 4 and 8, respectively. The activation maps are created using the Keras Visualization Toolkit and the Grad-CAM algorithm [27]–[29]. Red areas had more influence on the label prediction [28]. best results clearly better than transfer learning. Although this statement is supported by comparable empirical evidence, it remains speculation for a sensitive field like medical imaging. But why is so difficult to train a CNN for this case? It is most likely due to a number of factors such as a relative lack of image data, the effect of scaling down a patch for use within 5 To appear in proceedings of the 7th Intern. Conf. on Image Processing Theory, Tools and Applications (IPTA 2017), Nov 28-Dec 1, Montreal, Canada. structures within individual patches. Further investigation with different architectures would likely improve upon these results as would more aggressive augmentation. [10] D. Yi, R. L. Sawyer, D. C. III, J. Dunnmon, C. Lam, X. Xiao, and D. Rubin, “Optimizing and visualizing deep learning for benign/malignant classification in breast tumors,” CoRR, vol. abs/1705.06362, 2017. [Online]. Available: http: //arxiv.org/abs/1705.06362 [11] S. J. Pan and Q. Yang, “A survey on transfer learning,” IEEE Transactions on Knowledge and Data Engineering, vol. 22, no. 10, pp. 1345– 1359, Oct 2010. [12] H. C. Shin, H. R. Roth, M. Gao, L. Lu, Z. Xu, I. Nogues, J. Yao, D. Mollura, and R. M. Summers, “Deep convolutional neural networks for computer-aided detection: Cnn architectures, dataset characteristics and transfer learning,” IEEE Transactions on Medical Imaging, vol. 35, no. 5, pp. 1285–1298, May 2016. [13] J. Deng, W. Dong, R. Socher, L.-J. Li, K. Li, and L. Fei-Fei, “Imagenet: A large-scale hierarchical image database,” in Computer Vision and Pattern Recognition, 2009. CVPR 2009. IEEE Conference on. IEEE, 2009, pp. 248–255. [14] R. Girshick, J. Donahue, T. Darrell, and J. Malik, “Region-based convolutional networks for accurate object detection and segmentation,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 38, no. 1, pp. 142–158, Jan 2016. [15] Y. Bar, I. Diamant, L. Wolf, and H. Greenspan, “Deep learning with non-medical training used for chest pathology identification,” in Proc. SPIE, vol. 9414, 2015, p. 94140V. [16] Y. Lecun, L. Bottou, Y. Bengio, and P. Haffner, “Gradient-based learning applied to document recognition,” Proceedings of the IEEE, vol. 86, no. 11, pp. 2278–2324, Nov 1998. [17] M. Babaie, S. Kalra, A. Sriram, C. Mitcheltree, S. Zhu, A. Khatami, S. Rahnamayan, and H. R. Tizhoosh, “Classification and Retrieval of Digital Pathology Scans: A New Dataset.” [Online]. Available: http://arxiv.org/abs/1705.07522 [18] K. Simonyan and A. Zisserman, “Very deep convolutional networks for large-scale image recognition,” CoRR, vol. abs/1409.1556, 2014. [19] C. Szegedy, V. Vanhoucke, S. Ioffe, J. Shlens, and Z. Wojna, “Rethinking the inception architecture for computer vision,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2016, pp. 2818–2826. [20] F. Chollet et al., “Keras,” https://github.com/fchollet/keras, 2015. [21] N. Tajbakhsh, J. Y. Shin, S. R. Gurudu, R. T. Hurst, C. B. Kendall, M. B. Gotway, and J. Liang, “Convolutional neural networks for medical image analysis: Full training or fine tuning?” IEEE Transactions on Medical Imaging, vol. 35, no. 5, pp. 1299–1312, May 2016. [22] F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay, “Scikit-learn: Machine learning in Python,” Journal of Machine Learning Research, vol. 12, pp. 2825–2830, 2011. [23] C.-C. Chang and C.-J. Lin, “LIBSVM: A library for support vector machines,” ACM Transactions on Intelligent Systems and Technology, vol. 2, pp. 27:1–27:27, 2011, software available at http://www.csie.ntu. edu.tw/∼cjlin/libsvm. [24] S. van der Walt, S. C. Colbert, and G. Varoquaux, “The numpy array: A structure for efficient numerical computation,” Computing in Science & Engineering, vol. 13, no. 2, pp. 22–30, 2011. [Online]. Available: http://aip.scitation.org/doi/abs/10.1109/MCSE.2011.37 [25] E. Jones, T. Oliphant, P. Peterson et al., “SciPy: Open source scientific tools for Python,” 2001–, [Online; accessed ¡today¿]. [Online]. Available: http://www.scipy.org/ [26] D. Yu and M. L. Seltzer, “Improved bottleneck features using pretrained deep neural networks,” in Twelfth Annual Conference of the International Speech Communication Association, 2011. [27] R. Kotikalapudi and contributors, “keras-vis,” https://github.com/ raghakot/keras-vis, 2017. [28] R. R. Selvaraju, M. Cogswell, A. Das, R. Vedantam, D. Parikh, and D. Batra, “Grad-cam: Visual explanations from deep networks via gradient-based localization,” See https://arxiv. org/abs/1610.02391 v3, 2016. [29] M. D. Zeiler and R. Fergus, Visualizing and Understanding Convolutional Networks. Cham: Springer International Publishing, 2014, pp. 818–833. [Online]. Available: http://dx.doi.org/10.1007/ 978-3-319-10590-1 53 VII. C ONCLUSIONS Retrieval and classification of histopathological images are useful but challenging tasks in analysis for diagnostic pathology. Whole scan imaging (WSI) generates gigapixel images that are immensely rich in details and exhibit tremendous interand intra-class variance. Both a feature extractor and transferlearned network were able to offer increases in classification accuracy on the Kimia Path24 dataset when compared to a CNN trained from scratch. Comparatively low performance of the latter could be due to the architecture not being well suited for the problem, lack of sufficient number of training images, and/or the inherent difficulty of the classification task for high-resolutional and highly variable histopathology images. Further work would warrant using different architectures for comparison, more aggressive data augmentation, and potentially increasing the size of training samples used from the Kimia Path24 dataset. However, both transfer-learned and feature extractor models were able to compete with the stateof-the-art methods reported in literature [17], and therefore show potential for further improvements. ACKNOWLEDGEMENTS The authors would like to thank Huron Digital Pathology (Waterloo, ON, Canada) for its continuing support. R EFERENCES [1] M. N. Gurcan, L. E. Boucheron, A. Can, A. Madabhushi, N. M. Rajpoot, and B. Yener, “Histopathological image analysis: A review,” IEEE reviews in biomedical engineering, vol. 2, pp. 147–171, 2009. [2] S. Naik, S. Doyle, M. Feldman, J. Tomaszewski, and A. Madabhushi, “Gland segmentation and computerized gleason grading of prostate histology by integrating low-, high-level and domain specific information,” in MIAAB workshop, 2007, pp. 1–8. [3] P. S. Karvelis, D. I. Fotiadis, I. Georgiou, and M. Syrrou, “A watershed based segmentation method for multispectral chromosome images classification,” in Engineering in Medicine and Biology Society, 2006. EMBS’06. 28th Annual International Conference of the IEEE. IEEE, 2006, pp. 3009–3012. [4] S. Petushi, F. U. Garcia, M. M. Haber, C. Katsinis, and A. Tozeren, “Large-scale computations on histology images reveal gradedifferentiating parameters for breast cancer,” BMC medical imaging, vol. 6, no. 1, p. 14, 2006. [5] X. Zhang, W. Liu, M. Dundar, S. Badve, and S. Zhang, “Towards largescale histopathological image analysis: Hashing-based image retrieval,” IEEE Transactions on Medical Imaging, vol. 34, no. 2, pp. 496–506, Feb 2015. [6] W. Liu, J. Wang, R. Ji, Y. G. Jiang, and S. F. Chang, “Supervised hashing with kernels,” in 2012 IEEE Conference on Computer Vision and Pattern Recognition, June 2012, pp. 2074–2081. [7] H. R. Tizhoosh, “Barcode annotations for medical image retrieval: A preliminary investigation,” in Image Processing (ICIP), 2015 IEEE International Conference on. IEEE, 2015, pp. 818–822. [8] H. R. Tizhoosh, S. Zhu, H. Lo, V. Chaudhari, and T. 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arXiv:1709.07929v2 [math.AC] 12 Dec 2017 TRIANGULATED EQUIVALENCES AND RECONSTRUCTION OF CLASSIFYING SPACES HIROKI MATSUI Abstract. In algebra such as algebraic geometry, modular representation theory and commutative ring theory, we study algebraic objects through associated triangulated categories and topological spaces. In this paper, we consider the relationship between such triangulated categories and topological spaces. To be precise, we explore necessary conditions for derived equivalence of Noetherian schemes, stable equivalence of finite groups, and singular equivalence of commutative Noetherian rings by using associated topological spaces. 1. Introduction As is a common approach in many branches of algebra including algebraic geometry, modular representation theory and commutative ring theory, we assign to an algebraic object A (e.g., a scheme X, a finite group G, a commutative Noetherianring R) a triangulated category T (e.g., the perfect derived category Dperf (X), the stable module category mod kG, the singularity category Dsg (R)) and a topological space S (e.g., the underlying topological spaces X, Proj H∗ (G; k), Sing R). By studying such a triangulated category and a topological space, we aim to grasp the structure of the original algebraic object. From this motivation, it is natural to ask what kind of relationship there exists between T and S. algebraic objects A: X, G, R ❖❖❖ ❖❖❖ ❖❖❖ ❖❖' ♥♥ ♥♥♥ ♥ ♥ ♥♥ w ♥♥ ♥ triangulated categories T : Dperf (X), mod kG, Dsg (R) o /o /o /o /o /o /o ??? o/ /o /o /o /o /o /o /o /o / topological spaces S: X, Proj H∗ (G; k), Sing R In this paper, we consider this question, more precisely, the following: Question 1.1. Let A, A′ be algebraic objects, T , T ′ corresponding triangulated categories, and S, S ′ corresponding topological spaces, respectively. Does the implication T ∼ = S′ = T ′ =⇒ S ∼ hold? 2010 Mathematics Subject Classification. 13C14, 13D09, 14F05, 18E30, 19D23, 20C20. Key words and phrases. triangulated category, triangulated equivalence, classifying space, (classifying) support data, quasi-affine scheme, finite p-group, complete intersection. The author is partly supported by Grant-in-Aid for JSPS Fellows 16J01067. 1 2 HIROKI MATSUI We introduce the notion of a classifying space of a triangulated category (see Definition 2.9), and prove the following result, which gives a machinery to answer the above question. Theorem 1.2 (Theorem 3.13). Let T , T ′ be essentially small triangulated categories and S, S ′ classifying spaces for T and T ′ , respectively. Then the implication T ∼ = S′ = T ′ =⇒ S ∼ holds. The key role to prove this theorem is played by the support theory for triangulated categories. For tensor triangulated categories, the support theory has been developed by Balmer [Bal02, Bal05] and is a powerful tool to show such a reconstruction theorem. Since we focus on triangulated categories without tensor structure, we need to invent the support theory without tensor structure. 1.1. Algebraic geometry. Let X be a scheme. The derived category of perfect complexes on X is called the perfect derived category and denoted by Dperf (X). The case where X = Spec R is affine, it is well known that the original scheme is reconstructed from Dperf (R) := Dperf (X). Indeed, for two commutative rings R and S, if the perfect derived categories of R and S are equivalent, then R is isomorphic to S (see [Ric, Proposition 9.2]), and hence Dperf (R) ∼ = Dperf (S) =⇒ Spec R ∼ = Spec S as topological spaces. (∗) However, such a result no longer holds for non-affine schemes. In fact, there exist a lot of non-isomorphic schemes X and Y such that Dperf (X) ∼ = Dperf (Y ); see [Muk, Orl97]. When perf perf ∼ there is a triangulated equivalence D (X) = D (Y ), X and Y are said to be derived equivalent. In section 3, we shall prove that the underlying topological spaces of a certain class of schemes can be reconstructed from their perfect derived categories: Theorem 1.3 (Theorem 3.10). Let X and Y be Noetherian quasi-affine schemes (i.e., open subschemes of affine schemes). Then the implication Dperf (X) ∼ = Dperf (Y ) =⇒ X ∼ = Y as topological spaces holds. This theorem recovers (∗) for Noetherian rings as any affine scheme is quasi-affine. A typical example of a non-affine quasi-affine scheme is the punctured spectrum of a local ring. As an application of this theorem, we obtain that a derived equivalence of X and Y yields the equality of the dimensions of X and Y . 1.2. Modular representation theory. In modular representation theory, finite groups are studied in various contexts. From an algebraic viewpoint, a finite group G has been studied through its group algebra kG and stable module category mod kG, where k is a field whose characteristic divides the order of G. Here, mod kG is a triangulated category consisting of finitely generated kG-modules modulo projectives. On the other hand, the cohomology ring H∗ (G; k) gives an approach to study a finite group G from the topological aspect because it is isomorphic to the cohomology ring of a classifying space BG of G; see [Ben, Chapter 2] for instance. The second main result in section 3 is the following: TRIANGULATED EQUIVALENCES AND RECONSTRUCTION OF CLASSIFYING SPACES 3 Theorem 1.4 (Theorem 3.13). Let k (resp. l) be a field of characteristic p (resp. q), and let G (resp. H) be a finite p-group (resp. q-group). Then the implication mod kG ∼ = mod lH =⇒ Proj H∗ (G; k) ∼ = Proj H∗ (H; l) as topological spaces holds. If there exists a triangulated equivalence mod kG ∼ = mod lH, we say that kG and lH are stably equivalent. As an application of this theorem, we have that a stable equivalence of kG and lH yields that the p-rank of G and the q-rank of H are equal. 1.3. Commutative ring theory. Let R be a left Noetherian ring. The singularity category of R is by definition the Verdier quotient Dsg (R) := Db (modR)/Dperf (R), which has been introduced by Buchweitz [Buc] in 1980s. Here, mod R stands for the category of finitely generated left R-modules and Db (modR) its bounded derived category. The singularity categories have been deeply investigated from algebro-geometric and representation-theoretic motivations [Che, IW, Ste, Tak] and connected to the Homological Mirror Symmetry Conjecture by Orlov [Orl04]. One of the important subjects in representation theory of rings is to classify rings up to certain category equivalence. For example, left Noetherian rings R and S are said to be: • Morita equivalent if mod R ∼ = mod S as abelian categories, • derived equivalent if Db (mod R) ∼ = Db (mod S) as triangulated categories, • singularly equivalent if Dsg (R) ∼ = Dsg (S) as triangulated categories. It is well known that these equivalences have the following relations: Morita equivalence ⇒ derived equivalence ⇒ singular equivalence. Complete characterizations of Morita and derived equivalence have already been obtained in [Mor, Ric], while singular equivalence is quite difficult to characterize even in the case of commutative rings. Indeed, only a few examples of singular equivalences of commutative Noetherian rings are known. Furthermore, for all of such known examples, the singular loci of rings are homeomorphic. Thus, it is natural to ask the following question. Question 1.5. Let R and S be commutative Noetherian rings. Are their singular loci homeomorphic if R and S are singularly equivalent? In section 4, we show that this question is affirmative for certain classes of commutative Noetherian rings. To be precise, we shall prove the following theorem. Theorem 1.6 (Theorem 4.4). Let R and S be commutative Noetherian local rings that are locally hypersurfaces on the punctured spectra. Assume that R and S are either (a) complete intersection rings, or (b) Cohen-Macaulay rings with quasi-decomposable maximal ideal. Then the implication Dsg (R) ∼ = Dsg (S) =⇒ Sing R ∼ = Sing S as topological spaces holds. 4 HIROKI MATSUI Here, we say that an ideal I of a commutative ring R is quasi-decomposable if there is an R-regular sequence x in I such that I/(x) is decomposable as an R-module. Moreover, we prove that singular equivalence localizes by using such a homeomorphism. The organization of this paper is as follows. In section 2, we introduce the notions of a support data and a classifying support data for a given triangulated category and develop the support theory without tensor structure, and finally prove Theorem 1.2. In section 3, we connect the results obtained in section 2 with the support theory for tensor triangulated categories and study reconstructing the topologies of the Balmer spectra without tensor structure. Using this method, we prove Theorem 1.3 and 1.4. In section 4, we prove Theorem 1.6 and give examples of commutative rings which are not singularly equivalent. Throughout this paper, all categories are assumed to be essentially small. For two triangulated category T , T ′ (resp. topological spaces X, X ′ ), the notation T ∼ = T ′ (resp. X∼ = X ′ ) means that T and T ′ are equivalent as triangulated categories (resp. X and X ′ are homeomorphic) unless otherwise specified. 2. The support theory without tensor structure In this section, we discuss the support theory for triangulated categories without tensor structure. Throughout this section, T denotes a triangulated category with shift functor Σ. First of all, let us recall some basic definitions which are used in this section. Definition 2.1. Let X be a topological space and T a triangulated category. (1) We say that X is sober if every irreducible closed subset of X is the closure of exactly one point. (2) We say that X is Noetherian if every descending chain of closed subspaces stabilizes. (3) We say that a subset W of X is specialization-closed if it is closed under specialization, namely if an element x of X belongs to W , then the closure {x} is contained in W . Note that W is specialization-closed if and only if it is a union of closed subspaces of X. (4) We say that a non-empty additive full subcategory X of T is thick if it satisfies the following conditions: (i) closed under taking shifts: ΣX = X . (ii) closed under taking extensions: for a triangle L → M → N → ΣL in T , if L and N belong to X , then so does M. (iii) closed under taking direct summands: for two objects L, M of T , if the direct sum L ⊕ M belongs to X , then so do L and M. For a subcategory X of T , denote by thickT X the smallest thick subcategory of T containing X . We introduce the notion of a support data for a triangulated category. Definition 2.2. Let T be a triangulated category. A support data for T is a pair (X, σ) where X is a topological space and σ is an assignment which assigns to an object M of T a closed subset σ(M) of X satisfying the following conditions: (1) σ(0) = ∅. (2) σ(Σn M) = σ(M) for any M ∈ T and n ∈ Z. (3) σ(M ⊕ N) = σ(M) ∪ σ(N) for any M, N ∈ T . TRIANGULATED EQUIVALENCES AND RECONSTRUCTION OF CLASSIFYING SPACES 5 (4) σ(M) ⊆ σ(L) ∪ σ(N) for any triangle L → M → N → ΣL in T . Support data naturally appear in various areas of algebras. Example 2.3. (1) Let R be a commutative Noetherian ring. For M ∈ Dsg (R), we define the singular support of M by SSupp (M) := {p ∈ Sing R \| Mp ∼ 6= 0 in Dsg (Rp )}. R Then (Sing R, SSuppR ) is a support data for Dsg (R). Indeed, it follows from [AIL, Theorem 1.1] and [BM, Lemma 4.5] that SSuppR (M) is a closed subset of Sing R and that SSuppR satisfies the condition (1) in Definition 2.2. The remained conditions (2)-(4) are clear because the localization functor Dsg (R) → Dsg (Rp ) is exact. Assume that R is Gorenstein. Denote by CM(R) the category of maximal CohenMacaulay R-modules (i.e., modules M satisfying ExtiR (M, R) = 0 for all integers i > 0). Recall that the stable category CM(R) of CM(R) is the category whose objects are the same as CM(R) and the set of morphisms from M to N is given by HomR (M, N) := HomR (M, N)/PR (M, N), where PR (M, N) consists of all R-linear maps from M to N factoring through some free R-module. Then the stable category CM(R) has the structure of a triangulated category; see [Hap]. Moreover, the natural inclusion induces a triangle equivalence ∼ = → Dsg (R) by [Buc]. Thus we obtain the support data (Sing R, SuppR ) for F : CM(R) − CM(R) by using this equivalence. Here, Supp (M) := SSupp (F (M)) = {p ∈ Sing R \| Mp ∼ 6= 0 in CM(Rp )} R R for M ∈ CM(R). (2) Let X be a Noetherian scheme. For F ∈ Dperf (X), we define the cohomological support of F by SuppX (F ) := {x ∈ X \| Fx ∼ 6= 0 in Dperf (OX,x )}. S Then, SuppX (F ) = n∈Z SuppX (Hn (F )) is a finite union of supports of coherent OX modules and hence is a closed subspace of X. Moreover, (X, SuppX ) is a support data for Dperf (X) because the localization is exact. For details, please see [Tho]. (3) Let k be a field of characteristic p > 0 and G a finite group such that p divides the order of G. Then as in the case of Gorenstein rings, we can define the stable category mod kG of mod kG and it is also a triangulated category. We denote by ( ⊕i∈Z Hi (G; k) p = 2 H∗ (G; k) = ⊕i∈2Z Hi (G; k) p : odd the direct sum of cohomologies of G with coefficient k. Then H∗ (G; k) has the structure of a graded-commutative Noetherian ring by using the cup product and we can consider its homogeneous prime spectrum Proj H∗ (G; k). Denote by VG (M) the support variety for a finitely generated kG-module M which is a closed space of Proj H∗ (G; k). Then the pair (Proj H∗ (G; k), VG ) becomes a support data for mod kG. For details, please refer to [Ben, Chapter 5]. Remark 2.4. Actually, the above examples of support data satisfy the following stronger condition: (1′ ) σ(M) = ∅ if and only if M ∼ = 0. 6 HIROKI MATSUI Definition 2.5. Let U be a full subcategory of T . We say that U is a ⊕-ideal if it satisfies M ∈ U, N ∈ T ⇒ M ⊕ N ∈ U. Remark 2.6. U ⊆ T is a ⊕-ideal if and only if T \ U is closed under taking direct summands. Example 2.7. (1) The full subcategory T \ {0} is a ⊕-ideal. (2) The full subcategory T(T ) of test objects (see Definition 4.8 below) of T is a ⊕-ideal. Let us fix the following notations: Notation 2.8. Let T be a triangulated category, U ⊆ T a ⊕-ideal, and X a topological space. Then we set: • Th(T ) := {thick subcategories of T }, • ThU (T ) := {thick subcategories of T containing an object of U}, • Spcl(X) := {specialization closed subsets of X}, • Nesc(X) := {non-empty specialization-closed subsets of X}, • Nec(X) := {non-empty closed subsets of X}, • Irr(X) := {irreducible closed subsets of X}. Let (X, σ) be a support data for T , X a thick subcategory of T , and W S a specializationclosed subset of X. Then one can easily check that fσ (X ) := σ(X ) := M ∈X σ(M) is a specialization-closed subset of X and gσ (W ) := σ −1 (W ) := {M ∈ T \| σ(M) ⊆ W } is a thick subcategory of T . Therefore, we obtain two order-preserving maps with respect to the inclusion relations. Definition 2.9. Let (X, σ) be a support data for T and U ⊆ T a ⊕-ideal. Then we say that (X, σ) is a classifying support data for T with respect to U if (i) X is a Noetherian sober space, and (ii) the above maps fσ and gσ restrict to mutually inverse bijections: ThU (T ) o fσ / gσ Nesc(X). When this is the case, we say that X is a classifying space of T with respect to U. We say simply a classifying support data for T (resp. a classifying space of T ), we mean a classifying support data for T (resp. a classifying space of T ) with respect to T \ {0}. Remark 2.10. A classifying support data (X, σ) for T classifies all thick subcategories of T containing gσ (∅) = σ −1 (∅). Indeed, the map gσ : Nesc(X) → ThU (T ) is injective with image {X ∈ Th(T ) \| X ) σ −1 (∅)}. Thus, we obtain a one-to-one correspondence {X ∈ Th(T ) \| X ⊇ σ −1 (∅)} o fσ gσ / Spcl(X). In particular, if (X, σ) satisfies the condition (1′ ) in Remark 2.4, we obtain a one-to-one correspondence: Th(T ) o fσ gσ / Spcl(X). Every classifying support data automatically satisfies the following realization property. TRIANGULATED EQUIVALENCES AND RECONSTRUCTION OF CLASSIFYING SPACES 7 Lemma 2.11. Let (X, σ) be a classifying support data for T with respect to U. Then for any non-empty closed subset Z of X, there is an object M of U, such that Z = σ(M). Proof. Since X is a Noetherian sober space and σ(M) ∪ σ(N) = σ(M ⊕ N), we may assume that Z = {x} for some x ∈ X. From the assumption, one has Z = fσ gσ (Z) = S M ∈gσ (Z) σ(M). Hence, there is an element x of σ(M) for some M ∈ gσ (Z). Then we obtain x ∈ σ(M) ⊆ Z = {x} and this implies that σ(M) = {x} = Z. By definition of a classifying support data with respect to U, gσ (Z) = {N ∈ T \| σ(N) ⊆ σ(M)} contains a object T of U. We conclude that σ(T ⊕M) = σ(T )∪σ(M) = σ(M) = Z for T ⊕ M ∈ U. Let me give two more notations. Definition 2.12. Let U be a ⊕-ideal of T . (1) We say that a thick subcategory X of T is U-principal if there is an object M of U such that X = thickT M. Denote by PThU (T ) the set of all U-principal thick subcategories of T . (2) We say that a U-principal thick subcategory X of T is U-irreducible if X = thickT (X1 ∪ X2 ) (X1 , X2 ∈ PThU (T )) implies that X1 = X or X2 = X . Denote by IrrU (T ) the set of all U-irreducible thick subcategories of T . The following lemma shows that by using classifying support data with respect to U, we can also classify U-principal thick subcategories and U-irreducible thick subcategories. Lemma 2.13. Let (X, σ) be a classifying support data for T with respect to U, then the one-to-one correspondence fσ ThU (T ) o / Nesc(X) gσ restricts to one-to-one correspondences PThU (T ) o IrrU (T ) o fσ / gσ fσ / gσ Nec(X), Irr(X). Proof. Note that fσ (thickT M) = σ(M) for any M ∈ T . Therefore, the injective map fσ : ThU (T ) → Nesc(X) induces a well defined injective map fσ : PThU (T ) → Nec(X). The surjectivity has already been shown in Lemma 2.11. Next, we show the second one-to-one correspondence. For X1 , X2 ∈ ThU (T ), one has [ (1) fσ (thickT (X1 ∪ X2 )) = σ(M) M ∈thickT (X1 ∪X2 ) = [ σ(M) M ∈X1 ∪X2 =( [ M ∈X1 σ(M)) ∪ ( [ M ∈X2 = fσ (X1 ) ∪ fσ (X2 ). σ(M)) 8 HIROKI MATSUI On the other hand, for Z1 , Z2 ∈ Nesc(X), one has fσ (thickT (gσ (Z1 ) ∪ gσ (Z2 ))) = fσ (gσ (Z1 )) ∪ fσ (σ(Z2 )) = Z1 ∪ Z2 . Applying gσ to this equality, we get (2) thickT (gσ (Z1 ) ∪ gσ (Z2 )) = gσ (Z1 ∪ Z2 ). Let W be an irreducible closed subset of X. Assume gσ (W ) = thickT (X1 ∪ X2 ) for some X1 , X2 ∈ PThU (T ). Then from the above equality (1), we obtain an equality W = fσ (gσ (W )) = fσ (thickT (X1 ∪ X2 )) = fσ (X1 ) ∪ fσ (X2 ). Since W is irreducible, fσ (X1 ) = W or fσ (X2 ) = W and hence X1 = gσ (fσ (X1 )) = gσ (W ) or X2 = gσ (fσ (X2 )) = gσ (W ). This shows that gσ (W ) is U-irreducible. Conversely, take a U-irreducible thick subcategory X of T and assume fσ (X ) = Z1 ∪ Z2 for some non-empty closed subsets Z1 , Z2 of X. From the above equality (2), we get X = gσ (fσ (X )) = gσ (Z1 ∪ Z2 ) = thickT (gσ (Z1 ) ∪ gσ (Z2 )). Since X is U-irreducible, X = gσ (Z1 ) or X = gσ (Z2 ) and therefore, Z1 = fσ (gσ (Z1 )) = fσ (X ) or Z2 = fσ (gσ (Z2 )) = fσ (X ). Thus, fσ (X ) is irreducible. These observations show the second one-to-one correspondence. From this lemma, we can show the following uniqueness result for classifying support data with respect to U. Proposition 2.14. Let (X, σ) and (Y, τ ) be classifying support data for T with respect to a ⊕-ideal U. Then X and Y are homeomorphic. Proof. First note that for a topological space X, the natural map ιX : X → Irr(X), x 7→ {x} is bijective if and only if X is sober. Define maps ϕ : X → Y and ψ : Y → X to be the composites ι gσ fτ ι−1 Y X → Y, Irr(X) −→ IrrU (T ) −→ Irr(Y ) −− ϕ : X −→ ι gτ fσ ι−1 Y X ψ : Y −→ Irr(Y ) − → IrrU (T ) −→ Irr(X) −− → X. Then ϕ and ψ are well defined and mutually inverse bijections by Lemma 2.13. Fix x ∈ X. For x′ ∈ {x}, one has ιX (x′ ) ⊆ ιX (x) and hence {ϕ(x′ )} = ιY (ϕ(x′ )) = fτ (gσ (ιX (x′ ))) ⊆ {ϕ(x)} = ιY (ϕ(x)) = fτ (gσ (ιX (x))). In particular, ϕ(x′ ) belongs to {ϕ(x)}. Therefore, ϕ({x}) ⊆ {ϕ(x)}. Conversely, for y ∈ {ϕ(x)}, the above argument shows ψ(y) ∈ ψ({ϕ(x)}) ⊆ {ψϕ(x)} = {x}. Applying ϕ to this inclusion, we obtain y ∈ ϕ({x}) and therefore, {ϕ(x)} ⊆ ϕ({x}). Thus, we conclude that ϕ({x}) = {ϕ(x)}. Since X is Noetherian, this equation means that ϕ is a closed map. Similarly, ψ is also a closed map. The following theorem is the main result of this section. Theorem 2.15. Consider the following setting: • T and T ′ are triangulated categories. TRIANGULATED EQUIVALENCES AND RECONSTRUCTION OF CLASSIFYING SPACES 9 • U and U ′ are ⊕-ideals of T and T ′ , respectively. • (X, σ) and (Y, τ ) are classifying support data for T and T ′ with respect to U and U ′ , respectively. Suppose that there is a triangle equivalence F : T → T ′ with F (U) = U ′ . Then X and Y are homeomorphic. Proof. From the assumption, F induces a one-to-one correspondence ∼ = → ThU ′ (T ′ ), X 7→ F̃ (X ), F̃ : ThU (T ) − where F̃ (X ) := {N ∈ T ′ \| ∃M ∈ X such that N ∼ = F (M)}. For an object M of T , set τ F (M) := τ (F (M)). Then we can easily verify that the pair (Y, τ F ) is a support data for T . Furthermore, it becomes a classifying support data for T with respect to U. Indeed, for X ∈ ThU (T ) and W ∈ Nesc(Y ), we obtain [ [ [ fτ F (X ) = τ F (M) = τ (F (M)) = τ (N) = fτ (F̃ (X )), M ∈X M ∈X N ∈F̃ (X ) F̃ (gτ F (W )) = F̃ ({M ∈ T \| τ F (M) ⊆ W }) = {N ∈ T ′ \| τ (N) ⊆ W } = gτ (W ). From these equalities, we get equalities fτ F = fτ ◦ F̃ and F̃ ◦ gτ F = gτ and thus fτ F and gτ F give mutually inverse bijections between ThU (T ) and Nesc(Y ). Consequently, we obtain two classifying support data (X, σ) and (Y, τ F ) for T with respect to U, and hence X and Y are homeomorphic by Proposition 2.14. 3. Comparison with tensor triangulated structure In this section, we discuss relation between the support theory we discussed in section 2 and the support theory for tensor triangulated categories. Recall that a tensor triangulated category (T , ⊗, 1) consists of a triangulated category T together with a symmetric monoidal tensor product ⊗ with unit object 1 which is compatible with the triangulated structure of T . For the precise definition, please refer to [HPS, Appendix A]. Example 3.1. (1) Let X be a Noetherian scheme. Then (Dperf (X), ⊗LOX , OX ) is a tensor triangulated category. Here, ⊗LOX denotes the derived tensor product. (2) Let k be a field and G a finite group. Then (mod kG, ⊗k , k) is a tensor triangulated category. Throughout this section, fix a tensor triangulated category (T , ⊗, 1). We begin with recalling some basic definitions which are used in the support theory of tensor triangulated categories. Definition 3.2. (1) A full subcategory X of T is called a thick tensor ideal if it is a thick subcategory of T and is closed under the action of T by ⊗: M ⊗ N ∈ X for any M ∈ X and N ∈ T . For a subcategory X of T , denote by hX i the smallest thick tensor ideal of T containing X . 10 HIROKI MATSUI (2) For a thick subcategory X of T , define its radical by √ X := {M ∈ T \| ∃n > 0 such that M ⊗n ∈ X }. Here, M ⊗n denotes the n-fold tensor product of M. By [Bal05, Lemma 4.2], the radical of a thick subcategory is always a thick tensor ideal. √ A thick tensor ideal X of T is called radical if it satisfies X = X . (3) A thick tensor ideal X of T is called prime if it satisfies M ⊗ N ∈ X ⇒ M ∈ X or N ∈ X . Denote by Spc T the set of all prime thick tensor ideals of T . (4) For M ∈ T , the Balmer support of M is defined as SppM := {P ∈ Spc T \| M ∈ / P}. The set Spc T is a topological space with closed basis {SppM \| M ∈ T } and call it the Balmer spectrum of T . (5) Let X be a topological space. We say that a subset W of X is a Thomason subset if it is a union of closed subsets whose complements are quasi-compact. Denote by Thom(X) the set of all Thomason subsets of X. Note that Thom(X) ⊆ Spcl(X). We say that a support data (X, σ) for T is tensorial if it satisfies: σ(M ⊗ N) = σ(M) ∩ σ(N) for any M, N ∈ T . In [Bal05], tensorial support data are called simply support data. Then gσ (W ) is a radical thick tensor ideal of T for every specialization-closed subset W of X. We say that a tensorial support data (X, σ) is classifying if X is a Noetherian sober space and there is a one-to-one correspondence: {radical thick tensor ideals of T } o fσ / gσ Spcl(X). Balmer showed the following celebrated result: Theorem 3.3. [Bal05, Lemma 2.6, Theorem 4.10] (1) The pair (Spc T , Spp) is a tensorial support data for T . (2) There is a one-to-one correspondence: {radical thick tensor ideals of T } o fSpp gSpp / Thom(Spc T ). Remark 3.4. If a topological space X is Noetherian, then every specialization-closed subset of X is Thomason. Therefore, the above theorem shows that (Spc T , Spp) is a classifying tensorial support data for T provided Spc T is Noetherian. Recall that a tensor triangulated category T is rigid if (1) the functor M ⊗ − : T → T has a right adjoint F (M, −) : T → T for each M ∈ T and (2) every object M is strongly dualizable (i.e., the natural map F (M, 1) ⊗ N → F (M, N) is an isomorphism for each N ∈ T ). If T is rigid, then (Spc T , Spp) satisfies the stronger condition. Lemma 3.5. Assume that T is rigid. Then the support data (Spc T , Spp) satisfies the condition (1′ ) in Remark 2.4. TRIANGULATED EQUIVALENCES AND RECONSTRUCTION OF CLASSIFYING SPACES 11 Proof. Take an object M ∈ T with Spp(M) = ∅. By [Bal05, Corollary 2.4], there is a positive integer n such that M ⊗n ∼ = 0. On the other hand, by [HPS, Lemma A 2.6], M i ⊗ 2i belongs to thickT (M ) for any positive integer since every object is strongly dualizable. Therefore, by using induction, we conclude that M ∼ = 0. Note that a tensorial classifying support data for T is a classifying tensorial support data for T . Indeed, for a tensorial classifying support data (X, σ) for T and X ∈ Th(T ), we obtain an equalities p p X = gσ (fσ (X )) = gσ (fσ ( thick⊗ X )) = thick⊗ X . The following lemma gives a criterion for the converse implication of this fact. Lemma 3.6. Let (X, σ) be a classifying tensorial support data for T . Suppose that T is rigid. Then the following are equivalent: (1) There is a one-to-one correspondence: Th(T ) o fσ gσ / Spcl(X). (2) (X, σ) is a classifying support data for T . (3) Every thick subcategory of T is a thick ⊗-ideal. (4) T = thickT 1. Proof. By Lemma 3.5 and Theorem [Bal05, Theorem 5.2], (X, σ) satisfies the condition (1′ ) in Remark 2.4. Therefore, (1) and (2) means the same conditions from Remark 2.10. (1) ⇒ (3): From the assumption, every thick subcategory X of T is of the form X = gσ (W ) for some specialization-closed subset W of X. On the other hand, gσ (W ) is a radical thick ⊗-ideal as (X, σ) is a tensorial support data. (3) ⇒ (4): By assumption, the thick subcategory thickT 1 is a thick tensor ideal. Thus, for any M ∈ T , M ∼ = M ⊗ 1 belongs to thickT 1. (4) ⇒ (1): Note that 1 is strongly dualizable and the family of all strongly dualizable objects forms a thick subcategory of T by [HPS, Theorem A.2.5 (a)]. Therefore, every object of T = thickT 1 is strongly dualizable. Thus, for any object M ∈ T , M belongs to thick⊗ T (M ⊗ M) by [HPS, Lemma A.2.6]. Then [Bal05, Proposition 4.4] shows that every thick tensor ideal of T is radical. On the other hand, for any thick subcategory X of Y, one can easily verify that the subcategory Y := {M ∈ T \| M ⊗ X ⊆ X } is a thick ⊗-ideal of T containing 1. Thus, we obtain Y = thickT 1 = T and hence X is a thick ⊗-ideal. From these discussion, we conclude that every thick subcategory of T is a radical thick ⊗-ideal and this shows the implication (4) ⇒ (1). The following corollaries are direct consequences of this lemma, Proposition 2.14 and Theorem 2.15. Corollary 3.7. Let T be a rigid tensor triangulated category. Assume that the Balmer spectrum Spc T of T is Noetherian and that T = thickT 1. Then for any classifying support data (X, σ) for T , X is homeomorphic to Spc T . Corollary 3.8. Let T and T ′ be rigid tensor triangulated categories such that 12 HIROKI MATSUI (1) Spc T and Spc T ′ are Noetherian, and (2) T and T ′ are generated by their unit objects. If T and T ′ are equivalent as triangulated categories, then Spc T and Spc T ′ are homeomorphic. Next, we consider applications of these corollaries to tensor triangulated categories appeared in Example 3.1. Thomason showed the following classification theorem of thick tensor ideas of Dperf (X): Theorem 3.9. [Tho, Theorem 3.15] Let X be a Noetherian scheme. Then (X, SuppX ) is a classifying tensorial support data for Dperf (X). As an application of Corollary 3.8, we can reconstruct underlying topological spaces of a certain class of schemes from their perfect derived categories without tensor structure. Theorem 3.10. Let X and Y be Noetherian quasi-affine schemes (i.e., open subschemes of affine schemes). If X and Y are derived equivalent, then X and Y are homeomorphic. In particular, topologically determined properties, such as the dimensions and the numbers of irreducible components of quasi-affine Noetherian schemes are preserved by derived equivalences. Proof. First, let me remark that the functor F ⊗LOX − : Dperf (X) → Dperf (X) has a right adjoint RHomOX (F , −) : Dperf (X) → Dperf (X) for each F ∈ Dperf (X) and moreover Dperf (X) is rigid. Note that a scheme X is quasi-affine if and only if its structure sheaf OX is ample. Thus, every thick subcategory of Dperf (X) is thick tensor ideal by [Tho, Proposition 3.11.1]. Applying Corollary 3.8, we obtain the result. Remark 3.11. Let X and Y be Noetherian schemes. (1) As we have already remarked in the introduction, if X and Y are affine, then a derived equivalence Dperf (X) ∼ = Dperf (Y ) implies that X and Y are isomorphic as schemes. (2) By [Bal02, Theorem 9.7], if Dperf (X) and Dperf (Y ) are equivalent as tensor triangulated categories, then X and Y are isomorphic as schemes. Next consider stable module categories over group rings of finite groups. In this case, the following classification theorem is given by Benson-Carlson-Rickard for algebraically closed field k and by Benson-Iyengar-Krause for general k. Theorem 3.12. [BCR, BIK] Let k be a field of characteristic p > 0 and G a finite group such that p divides the order of G. Then the support data (Proj H∗ (G; k), VG ) is a classifying tensorial support data for mod kG. Applying Corollary 3.8 to this classifying tensorial support data, we obtain the following result: Theorem 3.13. Let k (resp. l) be field of characteristic p (resp. q), G (resp. H) be a finite p-group (resp. q-group). If kG and lH are stably equivalent, then Proj H∗ (G; k) and Proj H∗ (H; l) are homeomorphic. Proof. For each M ∈ mod kG, the functor M ⊗k − : mod kG → mod kG has a right adjoint Homk (M, −) : mod kG → mod kG and in addition mod kG is rigid. Moreover, for a pgroup G, kG has only one simple module k. Therefore, we have mod kG = thickmod kG k. Applying Corollary 3.8, we are done. TRIANGULATED EQUIVALENCES AND RECONSTRUCTION OF CLASSIFYING SPACES 13 Recall that the p-rank of a finite group G is by definition, rp (G) := sup{r \| (Z/p)r ⊆ G}. Quillen [Qui] showed that the dimension of the cohomology ring H ∗ (G; k) is equal to the p-rank of G. Thus, the p-rank is an invariant of stable equivalences: Corollary 3.14. Let k, l, G, H be as in Theorem 3.13. Assume that there is a stable equivalence between kG and lH, then rp (G) = rq (H). Remark 3.15. Let G and H be a p-group and k a field of characteristic p. (1) By [Lin, Corollary 3.6], if there exists a stable equivalence between kG and kH, then \|G\| = \|H\|. (2) By [Lin, Corollary 3.2], if there exists a stable equivalence of Morita type between kG and kH, then G ∼ = H. 4. A necessary condition for singular equivalences Recall that commutative Noetherian rings R and S are said to be singularly equivalent if their singularity categories are equivalent as triangulated categories. The only known examples of singular equivalences are the following: Example 4.1. (1) If R ∼ = Dsg (S). = S, then Dsg (R) ∼ (2) If R and S are regular, then Dsg (R) ∼ = Dsg (S). =0∼ (3) (Knörrer’s periodicity [Yos, Chapter 12]) Let k be an algebraically closed field of characteristic 0. Set R := k[[x0 , x1 , ..., xd ]]/(f ) and S := k[[x0 , x1 , ..., xd , u, v]]/(f +uv). Then Dsg (R) ∼ = Dsg (S). Remark 4.2. All of these singular equivalences, the singular loci Sing R and Sing S are homeomorphic. In fact, the cases (1) and (2) are clear. Consider the case of R := k[[x0 , x1 , ..., xd ]]/(f ) and S := k[[x0 , x1 , ..., xd , u, v]]/(f + uv). Then Sing S = V(∂f /∂x0 , . . . ∂f /∂xd , u, v) ∼ = Spec(S/(∂f /∂x0 , . . . , ∂f /∂xd , u, v)) ∼ = Spec(k[[x0 , x1 , ..., xd , u, v]]/(f + uv, ∂f /∂x0 , . . . , ∂f /∂xd , u, v)) ∼ = Spec(k[[x0 , x1 , ..., xd ]]/(f, ∂f /∂x0 , . . . , ∂f /∂xd ) ∼ = V(∂f /∂x0 , . . . ∂f /∂xd ) = Sing R. Here, the first and the last equalities are known as the Jacobian criterion. Let me give some definitions appearing in the statement of the main theorem of this section. Definition 4.3. Let (R, m, k) be a commutative Noetherian local ring. (1) We say that an ideal I of R is quasi-decomposable if there is an R-regular sequence x of I such that I/(x) is decomposable as an R-module. (2) A local ring R is said to be complete intersection if there is a regular local ring S and an S-regular sequence x such that the completion R̂ of R is isomorphic to S/(x). We say that R is a hypersurface if we can take x to be an S-regular sequence of length 1. (3) A local ring R is said to be locally a hypersurface on the punctured spectrum if Rp is a hypersurface for every non-maximal prime ideal p. 14 HIROKI MATSUI The following theorem is the main result of this section. Theorem 4.4. Let R and S be commutative Noetherian local rings that are locally hypersurfaces on the punctured spectra. Assume that R and S are either (a) complete intersection rings, or (b) Cohen-Macaulay rings with quasi-decomposable maximal ideal. If R and S are singularly equivalent, then Sing R and Sing S are homeomorphic. For a ring R satisfying the condition (b) in Theorem 4.4, Nasseh-Takahashi [NT, Theorem B] shows that (Sing R, SSuppR ) is a classifying support data for Dsg (R). Therefore, the statement of Theorem 4.4 follows from Theorem 2.15. Therefore, the problem is the case of (a). For a ring R satisfying the condition (a) in Theorem 4.4, Takahashi [Tak] classified thick subcategories of Dsg (R) containing the residue field k of R by using the singular locus Sing R and the singular support SSuppR . We would like to apply Theorem 2.15 also for this case. The problem is that whether the condition “containing the residue field k” is preserved by stable equivalences. As we will show later, this condition is actually preserved by singular equivalences for local complete intersection rings. To do this, we discuss replacing the residue field k with some categorically defined object. First of all, let us recall the notion of a test module. Definition 4.5. Let R be a Noetherian ring. We say that a finitely generated R-module T is a test module if for any finitely generated R-module M, TorR n (T, M) = 0 for n ≫ 0 ⇒ pdR M < ∞. Example 4.6. For a Noetherian local ring (R, m, k), the syzygy Ωn k of its residue field is a test module for each n. For commutative Noetherian rings admitting dualizing complexes (e.g., Gorenstein rings), there is another characterization for test modules: Theorem 4.7. [CDT, Theorem 3.2] Let R be a commutative Noetherian ring admitting a dualizing complex. Then, test modules are nothing but finitely generated R-modules T satisfying the following condition: for any finitely generated R-module M, ExtnR (T, M) = 0 for n ≫ 0 ⇒ idR M < ∞. Motivated by this theorem, we introduce the following notion. Definition 4.8. Let T be a triangulated category. We say that T ∈ T is a test object if for any object M of T , HomT (T, Σn M) = 0 for n ≫ 0 ⇒ M = 0. Denote by T(T ) the full subcategory of T consisting of test objects. The following lemma shows that we can consider the notion of a test object is a generalization of the notion of a test module. Lemma 4.9. Let R be a Gorenstein ring. Then one has T(CM(R)) = {T ∈ CM(R) \| T is a test module}. TRIANGULATED EQUIVALENCES AND RECONSTRUCTION OF CLASSIFYING SPACES 15 Proof. By Theorem 4.7, we have only to show ≫0 T(CM(R)) = {T ∈ CM(R) \| all N ∈ mod R with ExtR (M, N) = 0 satisfy idR N < ∞}. Fix a maximal Cohen-Macaulay R-module T and a finitely generated R-module M. Since R is Gorenstein and T is maximal Cohen-Macaulay, one has Ext1R (T, R) = 0. Therefore, we get isomorphisms Exti (T, M) ∼ = Exti+1 (T, ΩR M) ∼ = Exti+2 (T, Ω2 M) ∼ = ··· R R R R for any positive integer i. Therefore, we get isomorphisms Hom (T, Σd+n Ωd M) ∼ = Extn (T, M) = Extd+n (T, Ωd M) ∼ R R R R R for n > 0. Here, d denotes the dimension of R. Thus, we are done since ΩdR M is free if and only if M has finite injective dimension. Let us recall several classes of subcategories of modules. Definition 4.10. (1) An additive subcategory X of mod R is called resolving if it satisfies the following conditions: (i) X is closed under extensions: for an exact sequence 0 → L → M → N → 0 in mod R, if L and N belong to X , then so does M. (ii) X is closed under kernels of epimorphisms: for an exact sequence 0 → L → M → N → 0 in mod R, if M and N belong to X , then so does L. (iii) X contains all projective R-modules. For a finitely generated R-module M, denote by resR (M) the smallest resolving subcategory of mod R containing M. (2) A non-empty additive subcategory X of mod R is called thick if X satisfies 2-out-of3 property: for an exact sequence 0 → L → M → N → 0 in mod R, if 2-out-of {L, M, N} belong to X , then so does the third. For a finitely generated R-module M, denote by thickR (M) the smallest thick subcategory of mod R containing M. Lemma 4.11. Let T be a triangulated category and T an object of T . If thickT (T ) contains a test object of T , then T is also a test object. Proof. Take an object M ∈ T with HomT (T, Σn M) = 0 for n ≫ 0. Set X := {N ∈ T \| HomT (N, Σn M) = 0 for n ≫ 0}. Then one can easily verify that X is a thick subcategory of T . By assumption, X contains a test object as X contains T . Thus, M must be zero and hence T is a test object. The next proposition plays a key role to prove our main theorem. Proposition 4.12. Let (R, m, k) be a d-dimensional local complete intersection ring and T a finitely generated R-module. Then the following are equivalent: (1) T is a test module. (2) ΩdR k ∈ resR (T ). (3) k ∈ thickR (T ⊕ R). (4) k ∈ thickDb (mod R) (T ⊕ R). (5) k ∈ thickDsg (R) (T ). (6) ΩdR k ∈ thickCM(R) (Ωd T ). 16 HIROKI MATSUI Proof. Notice resR (ΩiR T ) ⊆ resR (T ), thickR (T ⊕ R) = thickR (ΩiR T ⊕ R), thickDb (mod R) (T ⊕ R) = thickDb (mod R) (ΩiR T ⊕ R), thickDsg (R) (T ) = thickDsg (R)(ΩiR T ) and T is a test module if and only if so is ΩR T . Hence we may assume that T is maximal Cohen-Macaulay. Then we have resR (T ) ⊆ thickR (T ⊕ R) = thickDb (mod R) (T ⊕ R) ∩ mod R. Here, the first inclusion directly follows from the definition, and the second equality is ∼ = → given by [KS, Theorem 1]. Moreover, the composition functor Db (mod R) → Dsg (R) − CM(R) sends k to ΩdR k[d], and the inverse image of thickCM(R) (T ) is thickDb (mod R) (T ⊕ R). Therefore, the implications (2) ⇒ (3) ⇔ (4) ⇔ (5) ⇔ (6) hold true. Furthermore, by using Lemma 4.9 and Lemma 4.11, the implication (5) ⇒ (1) follows. Thus, it remains to show the implication (1) ⇒ (2). Assume that T is a test module. Recall that the complexity cxR (M) of a finitely generated R-module M is the dimension of the support variety VR (M) associated to M; see [AB] for details. By [CDT, Proposition 2.7], T has maximal complexity, namely cxR (T ) = codim(R) =: c. Thanks to the prime avoidance lemma, we can take an R-regular sequence x of length d from m \ m2 . Set R = R/(x) and T = T /(x). Then R is an Artinian complete intersection ring and cxR (T ) = cxR (T ) = c = codimR = codim(R). Moreover, one has VR (T ) = Acka = VR (k), where k a denotes the algebraic closure of k. This follows from the fact that VR (T ) and VR (k) are c-dimensional closed subvarieties of the c-dimensional affine space Acka . Hence, by [CI, Theorem 5.6], k belongs to thickDb (mod R) (T ). As a result, we get k ∈ thickDb (mod R) (T ) ∩ mod(R) ⊆ thickDb (mod R) (T ⊕ R) ∩ mod(R) = thickR (T ⊕ R). Again, the second equality uses [KS, Theorem 1]. Since thickR (T ⊕ R) = resR (T ) by [DT, Corollary 4.16], we deduce ΩdR k ∈ resR (T ) by using [Tak, Lemma 5.8]. Gathering [Tak, Theorem 6.7], [NT, Theorem B], Lemma 4.9 and Proposition 4.12, we obtain the following proposition. Proposition 4.13. Let R be a Noetherian local ring. (1) If R satisfies the condition (a) in Theorem 4.4, then (Sing R, SSuppR ) is a classifying support data for Dsg (R) with respect to T(Dsg (R)). (2) If R satisfies the condition (b) in Theorem 4.4, then (Sing R, SSuppR ) is a classifying support data for Dsg (R). Now, the proof of Theorem 4.4 has almost been done. Proof of Theorem 4.4. Use Proposition 4.13 and Theorem 2.15. Here, let me remark that test objects are preserved by singular equivalences. Remark 4.14. For a hypersurface ring R, the triangulated category Dsg (R) becomes a pseudo tensor triangulated category (i.e., tensor triangulated category without unit). It is shown by Yu implicitly in the paper [Yu] that for two hypersurfaces R and S, if a singular equivalence between R and S preserves tensor products, then Sing R and Sing S are homeomorphic. Indeed, Sing R is reconstructed from Dsg (R) by using its pseudo tensor triangulated structure. TRIANGULATED EQUIVALENCES AND RECONSTRUCTION OF CLASSIFYING SPACES 17 Since Theorem 4.4 gives a necessary condition for singular equivalences, we can generate many pairs of rings which are not singularly equivalent. Let us start with the following lemma. Lemma 4.15. Let R be a local complete intersection ring with only an isolated singularity and r > 1 an integer. Then the ring R[[u]]/(ur ) is a local complete intersection ring which is locally a hypersurface on the punctured spectrum, and Sing(R[[u]]/(ur )) is homeomorphic to Spec R. Proof. Of course T := R[[u]]/(ur ) is a local complete intersection ring. ∼ = The natural inclusion R → T induces a homeomorphism f : Spec T − → Spec R. Then one can easily check that P = (f (P ), u)T for any P ∈ Spec T and TP ∼ = Rf (P ) [[u]]/(ur ). Therefore, T is locally a hypersurface on the punctured spectrum and Sing T = Spec T . Corollary 4.16. Let R and S be local complete intersection rings which have only isolated singularities. Assume that Spec R and Spec S are not homeomorphic. Then for any integers r, s > 1, one has Dsg (R[[u]]/(ur )) ∼ 6= Dsg (S[[v]]/(v s )). In particular, Dsg (R ∗ R) ∼ 6= Dsg (S ∗ S). Here R ∗ R denotes the trivial extension ring of a commutative ring R. Proof. From the above lemma, we obtain (1) R[[u]]/(ur ) and S[[v]]/(v s ) satisfies the condition (a) in Theorem 4.4, (2) Sing R[[u]]/(ur ) ∼ = Spec S are not homeomorphic. = Spec R and Sing S[[u]]/(v r ) ∼ r ∼ Thus, we conclude Dsg (R[[u]]/(u )) 6= Dsg (S[[v]]/(v s )) by Theorem 4.4. The second statement follows from an isomorphism R ∗ R ∼ = R[[u]]/(u2 ). The following corollary says that a Knörrer-type equivalence fails over a non-regular ring. Corollary 4.17. Let S be a regular local ring. Assume that S/(f ) has an isolated singularity. Then one has Dsg (S[[u]]/(f, u2)) ∼ 6= Dsg (S[[u, v, w]]/(f + vw, u2)). Proof. Sing S[[u]]/(f, u2) ∼ = Spec S[[v, w]]/(f + = Spec S/(f ) and Sing S[[u, v, w]]/(f +vw, u2) ∼ vw) have different dimensions and hence are not homeomorphic. For the last of this paper, we will show that singular equivalence localizes. Lemma 4.18. Let R be a d-dimensional Gorenstein local ring and p a prime ideal of R. Then a full subcategory Xp := {M ∈ Dsg (R) \| Mp ∼ = 0 in Dsg (Rp )} is thick and there is a triangle equivalence Dsg (R)/Xp ∼ = Dsg (Rp ). Proof. By using the triangle equivalence Dsg (R) ∼ = CM(R), we may show the triangle equivalence CM(R)/Xp ∼ = CM(Rp ), ∼ where Xp := {M ∈ CM(R) \| Mp = 0 in CM(Rp )}. Note that the localization functor Lp : CM(R) → CM(Rp ), M 7→ Mp is triangulated. Since Xp = Ker Lp , Xp is a thick subcategory of CM(R) and Lp induces a triangulated 18 HIROKI MATSUI functor Lp : CM(R)/Xp → CM(Rp ). Thus, we have only to verify that Lp is dense and fully faithful. (i): Lp is dense. δ Let U be an Rp -module. Take a finite free presentation Rpn − → Rpm → U → 0 of U. Then δ can be viewed as an m × n-matrix (αij ) with entries in Rp . Write αij = aij /s for some aij ∈ R and s ∈ R \ p. Then the cokernel M := Coker((aij ) : Rn → Rm ) is a finitely generated R-module and Mp ∼ = U. Since Mp is a maximal Cohen-Macaulay Rp -module, we obtain isomorphisms (Ω−d Ωd M)p ∼ =U = Ω−d Ωd Mp ∼ = Mp ∼ R R Rp Rp in CM(Rp ). This shows that the functor Lp is dense. (ii): Lp is faithful. Let α : M → N be a morphism in CM(R)/Xp . Then α is given by a fraction f /s of morphisms f : M → Z and s : N → Z in CM(R) such that the mapping cone C(s) of s belongs to Xp . Assume Lp (α) = Lp (s)−1 Lp (f ) = (sp )−1 fp = 0. Then fp = 0 in HomRp (Mp , Zp ). From the isomorphism HomR (M, Z)p ∼ = HomRp (Mp , Zp), there is a ∈ R\p such that af = 0 in HomR (M, Z). Since a : Zp → Zp is isomorphism, the mapping cone of the morphism a : Z → Z in CM(R) belongs to Xp . Thus, α = f /s = (af )/(as) = 0 in CM(R)/Xp . This shows that Lp is faithful. (iii): Lp is full. Let g : Mp → Np be a morphism in CM(Rp ) where M, N ∈ CM(R). By the isomorphism HomR (M, N)p ∼ = HomRp (Mp , Np ), there is a morphism f : M → N in CM(R) and a ∈ R\p such that g = fp /a. Since the mapping cone of a : N → N is in Xp , we obtain a morphism f /a : M → N in CM(R)/Xp and Lp (f /a) = fp /a = g. This shows that Lp is full. Corollary 4.19. Let R and S be complete intersection rings which are locally hypersurfaces on the punctured spectra. If R and S are singularly equivalent, then there is a homeomorphism ϕ : Sing R → Sing S such that Rp and Sϕ(p) are singularly equivalent for any p ∈ Sing R. Proof. As in Lemma 4.18, we may consider the category CM(R). Let F : CM(R) → CM(S) be a triangle equivalence. Take a homeomorphism ϕ : Sing R → Sing S given in Proposition 2.14 and Theorem 2.15. Then by construction, it satisfies [ SuppS F (M) {ϕ(p)} = M ∈CM(R), SuppR (M )⊆V(p) for each p ∈ Sing R. Moreover, the following diagram is commutative: F̃ ThT(CM(R)) (CM(R)) −−−→ ThT(CM(S)) (CM(S))   fSupp  fSupp y y S R Nesc(Sing R) −−−→ ϕ̃ Nesc(Sing S), where the map F̃ and ϕ̃ are defined by F̃ (X ) := {N ∈ T ′ \| ∃M ∈ X such that N ∼ = F (M)} and ϕ̃(W ) := ϕ(W ), respectively. Let p be an element of Sing R. Set Wp := {q ∈ Sing R \| q 6⊆ p} which is a specializationclosed subset of Sing R. We establish two claims. TRIANGULATED EQUIVALENCES AND RECONSTRUCTION OF CLASSIFYING SPACES 19 Claim 1. gSuppR (Wp ) = Xp . Proof of Claim 1. Let M ∈ Xp . Since Mp = 0 in CM(Rp ), one has p 6∈ SuppR (M). Thus, SuppR (M) ⊆ Wp and hence M ∈ gSuppR (Wp ). Next, take M ∈ gSuppR (Wp ). Then SuppR (M) ⊆ Wp means that p does not belong to SuppR (M). Therefore, Mp = 0 in CM(Rp ) and hence M ∈ Xp . Claim 2. ϕ(Wp ) = Wϕ(p) := {q ∈ Sing S \| q 6⊆ ϕ(p)}. Proof of Claim 2. One can easily check that ϕ is order isomorphism with respect to the inclusion relations. Since Sing R \ Wp has a unique maximal element p, ϕ(Sing R \ Wp ) = Sing S \ ϕ(Wp ) also has a unique maximal element ϕ(p). This shows ϕ(Wp ) = Wϕ(p) . From the above two claims, we obtain F̃ (Xp ) = F̃ (gSuppR (Wp )) = gSuppS (ϕ̃(Wp )) = gSuppS (Wϕ(p) ) = Xϕ(p) , where the second equality comes from the above commutative diagram and the last equality is shown by the same proof as Claim 1. Consequently, the triangle equivalence F induces triangle equivalences: CM(Rp ) ∼ = CM(R)/Xp ∼ = CM(S)/Xϕ(p) ∼ = CM(Sϕ(p) ). Acknowledgments. The author is grateful to his supervisor Ryo Takahashi for many supports and his helpful comments. References [AB] L. L. Avramov; R.-O. Buchweitz, Support varieties and cohomology over complete intersections, Invent. Math. 142 (2000), no. 2, 285–318. [AIL] L. L. Avramov; S. B. Iyengar; J. Lipman, Reflexivity and rigidity for complexes I. Commutative rings, Algebra Number Theory 4 (2010), no. 1, 47–86. [Bal02] P. 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MILP and Max-Clique based heuristics for the Eternity II puzzle arXiv:1709.00252v2 [cs.DS] 5 Oct 2017 Fabio Salassaa , Wim Vancroonenburgb , Tony Wautersb,∗, Federico Della Crocea , Greet Vanden Bergheb a Politecnico di Torino, DIGEP, Corso Duca degli Abruzzi 24, 10129 Torino, Italy b KU Leuven, Department of Computer Science, CODeS & iMinds-ITEC, Gebroeders De Smetstraat 1, 9000 Gent, Belgium Abstract The present paper considers a hybrid local search approach to the Eternity II puzzle and to unsigned, rectangular, edge matching puzzles in general. Both an original mixed-integer linear programming (MILP) formulation and a novel Max-Clique formulation are presented for this NP-hard problem. Although the presented formulations remain computationally intractable for medium and large sized instances, they can serve as the basis for developing heuristic decompositions and very large scale neighbourhoods. As a side product of the Max-Clique formulation, new hard-to-solve instances are published for the academic research community. Two reasonably well performing MILP-based constructive methods are presented and used for determining the initial solution of a multi-neighbourhood local search approach. Experimental results confirm that this local search can further improve the results obtained by the constructive heuristics and is quite competitive with the state of the art procedures. Keywords: Edge matching puzzle, hybrid approach, local search 1. Introduction The Eternity II puzzle (EII) is a commercial edge matching puzzle in which 256 square tiles with four coloured edges must be arranged on a 16 × 16 grid ∗ Corresponding author Email address: [email protected] (Tony Wauters) Preprint submitted to . October 6, 2017 such that all tile edges are matched. In addition, a complete solution requires that the ‘grey’ patterns, which appear only on a subset of the tiles, should be matched to the outer edges of the grid. An illustration of a complete solution for a small size puzzle, 5 × 5, is provided in Figure 1. Figure 1: Solution to an Eternity II-like edge matching puzzle of size 5 × 5 (Image generated with the Eternity II editor, http://sourceforge.net/projects/eternityii/, accessed on 24 January 2014) The EII puzzle was created by Christopher Monckton and released by the toy distributor Tomy UK Ltd. in July 2007. Along with the puzzle release, a large cash prize of 2 million USD was announced to be awarded to the first person who could solve the puzzle. As can be expected, this competition attracted considerable attention. Many efforts were made to tackle this challenging problem, yielding interesting approaches and results. However, no complete solution has ever been generated. Meanwhile, the final scrutiny date for the cash price, 31 December 2010, has passed, leaving the large money prize unclaimed. The EII puzzle belongs to the more general class of Edge Matching Puzzles, which have been shown to be NP-complete [5]. Many approaches to Edge Matching Puzzles are now available in the literature. Constraint programming approaches [2, 11] have been developed, in addition to metaheuristics [4, 11, 13], backtracking [12] and evolutionary methods [10]. Other methods translate the problem into a SAT formulation and then solve it with SAT solvers [1, 7]. An extensive literature overview on the topic is provided in [14], while [8] provides a survey on the complexity of other puzzles. 2 The present paper introduces a novel Mixed-Integer Linear Programming (MILP) model and a novel Max-Clique based formulation for EII-like puzzles of size n × n. Both formulations serve as components of heuristic decompositions used in a multi-neighbourhood local search approach. The remainder of the paper is structured as follows. Section 2 presents the MILP and MaxClique formulations. In Section 3, several hybrid heuristic approaches are introduced. Computational results are presented in Section 4. Final conclusions are drawn in Section 5. 2. Problem formulations 2.1. Mixed integer linear programming formulation A novel mixed-integer linear programming model was developed for the EII puzzle problem. The following notation will be used. The puzzle consists of an n × n square onto which n2 tiles need to be placed. The index t = 1 . . . n2 is used to refer to tiles. The indices r = 1 . . . n, c = 1 . . . n denote the rows, resp. columns, of the puzzle board. The index α = 0 . . . 3 refers to the rotation of the tile, i.e. α = 0 means not rotated, α = 1 means rotated clockwise over 90◦ , etc. The coefficient CTt,α,l (resp. CB, CL, CR) is equal to 1 if tile t has colour l at its top (resp. bottom, left, right) position when rotated by α. The decision variables of the MILP model are defined as follows: ( 1 if tile t is placed in row r, column c with rotation α, xt,r,c,α = 0 otherwise. ( 1 if the right edge of position (r, c) is unmatched , hr,c = 0 otherwise. ( 1 if the bottom edge of position (r, c) is unmatched , vr,c = 0 otherwise. The model is then defined as follows: Min n X n−1 X r=1 c=1 hr,c + n n−1 X X r=1 c=1 s.t. 3 vr,c (1) n X n X 3 X xt,r,c,α = 1 ∀t = 1, . . . , n2 (2) r=1 c=1 α=0 2 n X 3 X xt,r,c,α = 1 ∀r = 1, . . . , n, c = 1, . . . , n (3) t=1 α=0 2 n X 3 X 2 CRt,α,l xt,r,c,α − t=1 α=0 n X 3 X CLt,α,l xt,r,c+1,α ≤ hr,c t=1 α=0 ∀r = 1, . . . , n, c = 1, . . . , n − 1, l = 1, . . . , L 2 − n X 3 X 2 CRt,α,l xt,r,c,α + t=1 α=0 n X 3 X CLt,α,l xt,r,c+1,α ≤ hr,c t=1 α=0 ∀r = 1, . . . , n, c = 1, . . . , n − 1, l = 1, . . . , L CBt,α,l xt,r,c,α − n X 3 X CTt,α,l xt,r+1,c,α ≤ vr,c t=1 α=0 t=1 α=0 ∀r = 1, . . . , n − 1, c = 1, . . . , n, l = 1, . . . , L − t=1 α=0 (6) 2 2 n X 3 X (5) 2 2 n X 3 X (4) CBt,α,l xt,r,c,α + n X 3 X CTt,α,l xt,r+1,c,α ≤ vr,c t=1 α=0 ∀r = 1, . . . , n − 1, c = 1, . . . , n, l = 1, . . . , L (7) n2 3 XX CTt,α,0 · xt,0,c,α = 1 ∀c = 1, . . . , n (8) CBt,α,0 · xt,n−1,c,α = 1 ∀c = 1, . . . , n (9) t=1 α=0 2 n X 3 X t=1 α=0 2 n X 3 X CLt,α,0 · xt,r,0,α = 1 ∀r = 1, . . . , n (10) CRt,α,0 · xt,r,n−1,α = 1 ∀r = 1, . . . , n (11) t=1 α=0 2 n X 3 X t=1 α=0 xt,r,c,α ∈ {0, 1} ∀t = 1, . . . , n2 , r = 1, . . . , n, c = 1, . . . , n, α = 0, . . . , 3 0 ≤ hr,c ≤ 1 ∀r = 1, . . . , n, c = 1, . . . , n − 1 4 (12) (13) 0 ≤ vr,c ≤ 1 ∀r = 1, . . . , n − 1, c = 1, . . . , n (14) The objective function (Expression 1) minimises the number of unmatched edges in the inner region of the puzzle. Constraints (2) indicate that each tile must be assigned to exactly one position, with one rotation. Constraints (3) require that exactly one tile must be assigned to a position. The edge constraints (4) and (5) force the hr,c variables to take on the value 1 if the tiles on positions (r, c) and (r, c+1) are unmatched. Similarly, constraints (6) and (7) do the same for the vertical edge variables vr,c . Finally, constraints (8) - (11) ensure that the border edges are matched to the gray frame (colour l = 0). We point out that constraining the objective function to zero (i.e. no unmatched edges allowed), turns the model into a feasibility problem where every feasible solution is also optimal. However, preliminary testing showed that the latter model is only relevant for very small size problem instances. If the MILP solver needs to be stopped prematurely on the feasibility model, no solution is returned. 2.2. Clique formulation The EII puzzle itself is a decision problem and can be modelled as (i.e. it reduces to) the well known decision version of the clique problem [3] as follows. Given a parameter k and an undirected graph G = (V, E), the clique problem calls for finding a subset of pairwise adjacent nodes, called a clique, with a cardinality greater than or equal to k. Let the nodes of the graph correspond to variables xt,r,c,α from the formulation introduced in Section 2.1. Each node thus represents a tile in a given position on the puzzle and with a given rotation α. The nodes are connected iff there is no conflict between the nodes in the puzzle. Possible causes of conflicts are: • unmatching colors for adjacent positions • same tile assigned to different positions • same tile assigned to the same position with different rotations • different tiles assigned to the same position. The objective is to find a clique of size n2 , where n is the size of the puzzle. 5 Puzzle size Optimal solution Clique Number of nodes Number of edges Graph density q = 1, 000, 000 q = 10, 000, 000 q = 50, 000, 000 MILP Number of variables Number of constraints 1 thread, 3600s time 4 threads, 3600s time 3x3 12 4x4 24 5x5 40 6x6 60 7x7 84 8x8 112 30 291 66.9% E (T ) 12 (0.2) 12 (1.9) 12 (9.3) 138 6707 70.9% E (T ) 24 (0.3) 24 (3.8) 24 (19.3) 478 86184 75.6% E (T ) 40 (0.8) 40 (7.6) 40 (37.8) 1290 674512 81.1% E (T ) 57 (1.4) 60 (13.5) 60 (67.0) 2910 3620565 85.5% E (T ) 78 (2.5) 80 (22.4) 82 (110.5) 5770 14751304 88.6% E (T ) 102 (4.8) 106 (35.7) 106 (172.4) 336 126 E (T ) 12 (0.1) 12 (0.1) 1048 288 E (T ) 24 (0.1) 24 (0.1) 2540 630 E (T ) 40 (16.7) 40 (1.2) 5244 1056 E (T ) 60 (243.5) 60 (158.5) 9688 1638 E (T ) 81 (3600.0) 81 (3600.0) 16496 2624 E (T ) 103 (3600.0) 103 (3600.0) Table 1: Results obtained with the maximum clique formulation solved with the algorithm by [6], and the MILP formulation solved with CPLEX, on small size edge matching puzzle instances. 2.3. Comparison of the MILP model and the Clique model The applicability of the MILP model and the Clique model is investigated in what follows. Initial testing was performed on a set of small puzzle instances, ranging from 3 × 3 up to 8 × 8 (refer to Section 4 for more information on these instances). The MILP model was implemented using CPLEX 12.6. A state of the art heuristic was used for solving the maximum clique problem [6], kindly provided by its authors. The heuristic has only one parameter, i.e. the number of selections q. The computing time of the algorithm is linear with respect to q. We tested the heuristic with q ∈ {1, 000, 000; 10, 000, 000; 50, 000, 000}. Both the MILP model and the Max Clique heuristic were tested on a modern desktop pc1 . Table 1 shows the results obtained with the max clique formulation and the MILP formulation. For each instance, we report for the clique formulation: the number of nodes, the number of edges, the optimal solution (namely the max. number of matching edges), the density, the best number of matching edges E and the average computing time T (in seconds) for 10 runs. The number of variables and constraints is reported for the MILP formulation. The solutions depicted in bold are optimal. The results show that instances up to size 6 × 6 can be easily solved using a state of the art maximum clique 1 Intel Core i7-2600 [email protected] 6 algorithm. Instances of size 7 × 7 could not be solved completely, even when the algorithm was executed with higher values of q and more runs. The MILP is also able to solve up to size 6 × 6. However, the clique formulation is significantly faster from size 6 × 6 upwards. Note that the edge matching puzzles correspond with large, difficult clique instances, for which current state-of-the-art max clique solvers are not able to find the optimal solution. We provide the corresponding max clique instances of the 3 × 3 to 9 × 9 instances to the academic community2 . Larger size instances are hard to manage. The 10 × 10 graph file, for example, is larger than 1 GB. 3. Solution Approaches Both the MILP model and the clique formulation presented in the previous section proved to be computationally intractable for medium sized instances. The size appears to be restricted to 7 × 7 and 8 × 8 when the execution time is limited to one day. The true EII puzzle instance is still far beyond the grasp of these models. However, these models can serve as the basis for some well performing heuristics, presented in the following paragraphs. 3.1. MILP-based greedy heuristic A MILP-based greedy constructive heuristic has been developed for the problem studied here. The heuristic is based on subproblem optimisation. The puzzle is divided in regions, e.g. by considering individual rows/columns or rectangular regions. Regions are then consecutively constructed by employing a variant of the MILP model presented in Section 2. First we introduce the notion of a partial solution S ∗ = T ∗ 7→ R∗ , in which a subset T ∗ of the tiles T = {1, . . . , n2 } have been assigned to a subset R∗ of the positions R = {(r, c)\|r = 1, . . . , n, c = 1, . . . , n}. Given a partial solution S ∗ , model (1)–(14) can be modified such that it only considers the positions in a region R0 ⊆ R\R∗ , and it only aims to assign tiles T 0 ⊆ T \T ∗ (i.e. tiles that have not been assigned elsewhere). In addition, we restrict 0 0 R0 to a rectangular region, denoted by (rmin , c0min ) × (rmax , c0max ) (i.e. the min/max positions of the region). 2 The instances can be downloaded from https://dl.dropbox.com/u/24916303/ Graph_Pieces.zip. A generator for these instances is available upon request from the authors. 7 Figure 2 illustrates how model (1)–(14) can be modified to solve a region (1, 4) × (4, 8), given a partial solution on (1, 1) × (4, 4). In this example, it is required to select 16 of the available 64 − 16 = 48 tiles (16 tiles are already assigned to region (1, 1) × (4, 4)) in such a way that the unmatched edges are minimised. Hence, in order to consider region (1, 4) × (4, 8), the objective function is modified as follows Min 4 X 7 X hr,c + r=1 c=4 3 X 8 X vr,c (15) r=1 c=4 Only 16 of the remaining 48 tiles must be selected and assigned to the region and therefore constraints (2) are modified as follows. Note that the inequality indicates that not all tiles will be selected. 4 X 8 X 3 X xt,r,c,α ≤ 1 ∀t ∈ T \T ∗ (16) r=1 c=4 α=0 Similarly, constraints (3–14) are also suitably modified in order to take into account the specific region to be considered. Note that the edge constraints forcing the values of the hr,c and vr,c variables also hold for rows and columns matching the boundaries of previously solved regions. This enables building a solution with only a few unmatched edges between region boundaries. This partial optimization model can be applied to solve all regions sequentially, thus constructing the final complete solution. Initially R1 is optimised, after which region R2 , disjoint from region R1 , is optimised, and so on. The variables corresponding with the region are then optimally assigned by the MILP solver. Algorithm 1 presents pseudocode of this approach. Algorithm 1 Greedy heuristic Require: R = {R1 , R2 , R3 , . . . RK } . A decomposition of R in K regions Ri ∗ ∗ ∗ T0 ← ∅, R0 ← ∅, S0 ← ∅ . The initial partial solution S ∗ has no tiles assigned for i = 1, . . . , K do ∗ ,T∗ ∗ Si∗ , Ti∗ , Ri∗ ← Apply MILP model to region Ri , given Si−1 i−1 and Ri−1 . ∗ . Get a new partial solution Si , with tiles assigned to Ri end for ∗ return SK For each puzzle’s size, differently sized subsets of tiles have been tested to assess the quality of the approach. Preliminary tests of regions varying 8 Partial solution hrc ∀t,α vrc xt15α xt16α xt17α xt18α xt25α xt26α xt27α xt28α xt35α xt36α xt37α xt38α xt45α xt46α xt47α xt48α Empty region Figure 2: Illustration of how model (1)–(14) can be modified such that it only considers the positions in a region (1, 4) × (4, 8), given a partial solution S ∗ on region (1, 1) × (4, 4). 9 from 2 by 2 tiles (size 4) to 16 by 2 tiles (size 32) have been performed on the EII puzzle instance. This preliminary analysis revealed that the CPU time required at each iteration of the greedy heuristic limits the subset size to 32 tiles. This roughly corresponds to 32500 MILP variables for the first region of the real EII puzzle. Clearly, an increased number of tiles leads to better results. However, more CPU time is needed to compute the optimal solution, limiting the use in any hybrid framework. 3.2. MILP-based backtracking constructive heuristic A backtracking version of the greedy heuristic has also been developed. The main idea, namely building a complete solution by constructing optimal regions, is the same as for the greedy heuristic. The backtracking version, however, restricts the optimal value of each subproblem to zero. All tiles in the region should match both internally and with respect to the tiles outside the region. Whenever a subproblem is determined to be infeasible (i.e. no completely edge matching region can be constructed), the procedure backtracks to the previous region in order to find a new assignment in that region. This may afterwards enable constructing a feasible assignment in the next region. If not, then the process is repeated until the backtracks are sufficient to find a complete solution. Model (1)–(14), suitably modified, is again used to build partial solutions. Let Ri be the current region considered by the procedure and Si∗ the related partial solution once the corresponding MILP model is solved. Whenever the lower bound of the MILP model related to region Ri is detected to be greater than zero, optimisation of region Rk is stopped. Instead, the previous region 0∗ , again with Ri−1 is reconsidered in order to obtain a new partial solution Si−1 ∗ value 0. Let Xi−1 be the set of variables xt,r,c,α having value 1 in solution ∗ ∗ Si−1 . The previous partial solution Si−1 must be cut off when searching for 0∗ solution Si−1 . The following new constraint is added to the model: X ∗ xt,r,c,α ≤ \|Xi−1 \|−1 (17) ∗ t,r,c,α : xt,r,c,α ∈Xi−1 ∗ The rationale is to force at least one of the variables of set Xi−1 to be equal to zero. If no solution of the previous region Ri−1 can lead to a zero lower bound in the current region Ri , the procedure backtracks further and searches for a new solution for region Ri−2 (and so on). 10 Due to the enumerative nature, this procedure can lead to incomplete solutions despite long computation times. We decided to limit the backtracking procedure to a fixed time limit, after which the greedy heuristic continues until a complete solution is generated. This backtracking heuristic is sketched in Algorithm 2, in which a recursive method (BACKT RACKIN G HEU RIST IC) ∗ obtained attempts to solve the current region Ri , given partial solution Si−1 in the previous region Ri−1 . If the lower bound lb of the current region is greater than 0, the method backtracks to the previous level. However, if the lower bound is still 0 (and a perfectly matched assignment is found), the heuristic attempts to solve the next region. This will continue calling recursively until the puzzle is solved, or shown infeasible given the current ∗ ∗ assignments in Si−1 . In the latter case, the current partial solution Si−1 will 0∗ be excluded and a new partial solution Si−1 will be constructed, different ∗ and any other previously excluded partial solution. from Si−1 If a timeout is reached, the method will continue with the best partial solution and solve the remaining regions with the greedy heuristic, discussed in the previous section. 3.3. A multi-neighbourhood local search approach A multi-neighbourhood local search approach has been developed to improve the solutions generated by the constructive heuristics (or any random solution). The key idea is to test, after an initial, complete solution is generated by the heuristics, whether a controlled-size neighbourhood can still improve the current solution. This local search method is a Steepest Descent search that tries to improve a solution with the following neighbourhoods: Border Optimisation, Region Optimisation, Tile Assignment and Tiles Swap and Rotation. We refer to Figure 3 for an illustration of the regions considered by these neighbourhoods. The Border Optimisation (BO) neighbourhood only considers placing tiles in the border, while all the tiles in the inner part are fixed. The decomposition tries to find the optimal border in terms of matching edges, also considering the fixed tiles on the adjacent inner part. Correspondingly, model (1)–(14) is modified in such a way that the inner tile/positions variables are fixed to their current value. Only the border tile/positions variables can change value. This subproblem corresponds to a one-dimensional edgematching problem. Preliminary computational tests indicated that the related MILP model could always be solved. Solutions for the largest instances, 11 Algorithm 2 BACKT RACKIN G HEU RIST IC Require: R = {R1 , R2 , R3 , . . . RK } . A decomposition of R in K regions Ri Require: i . Current recursive level (i = 1, . . . , K) ∗ ∗ 7→ R∗ Require: Si−1 : Ti−1 . Partial solution of the previous level (S0∗ is the i−1 initial empty partial solution) excludedsolutions ← ∅ . Partial solutions not leading to feasible solutions while not timeout do ∗ , R , excludedsolutions) Si∗ ← OP T IM IZE REGION (Si−1 i if lb > 0 then ∗ return Si−1 . No feasible solution in current level. (backtrack) else ∗ Si+1 ← BACKT RACKIN G HEU RIST IC(R, i + 1, Si∗ ) ∗ if Si+1 = Si∗ then . Si∗ does not lead to feasible solution excludedsolutions ← excludedsolutions ∪ Si∗ else ∗ ∗ S ∗ ← Si+1 . Si+1 is the complete solution ∗ return S end if end if end while S ∗ ← GREEDY (Si∗ , Ri∗ ) return S ∗ 12 such as the original EII puzzle, can be generated within little computation time. When the BO neighborhood is considered, the corresponding MILP model is solved, returning a solution at least as good as the current solution and consisting of an optimal border with respect to the (n − 2) × (n − 2) inner region. The Region Optimisation (RO) neighbourhood relates to the optimisation of a smaller region inside the puzzle and only considers the tiles of this region in the puzzle. Correspondingly, given the current solution, model (1)– (14) is suitably modified in such a way that the tile/position variables outside the region are fixed to their current value. Only the region’s tile/position variables can change value. The RO neighborhood can also be tackled by means of the Max-Clique formulation by generating a graph only containing nodes corresponding to tile/position assignments in the specified region. However, only feasible tile/position assignments should be considered and nodes conflicting with assignments adjacent to (but outside) the region should not be added to the graph. We recall that the purpose of the model is to find complete assignments, that is, without any unmatched edges. However, given the tiles in the considered region, it may not be feasible to find such a solution. In this case, holes are left in the region to which the remaining, unassigned tiles should be assigned. The related MILP region model is solved where all assigned tile/position variables are fixed to the value determined by the MaxClique solver. When the RO neighborhood is considered, the local search procedure samples regions of fixed size in the current solution under consideration. For small sizes, the Max-Clique model (heuristically) is solved faster than the MILP model. Therefore, the RO neighborhood is always addressed by means of the Max-Clique formulation where the MILP formulation is only used for completing the solution whenever holes are left in the region. In the Tile Assignment (TA) neighbourhood, k tiles are removed from non-adjacent positions (diagonally adjacent is allowed) and optimally reinserted, thereby minimising the number of unmatched edges. The related subproblem corresponds to a pure bipartite weighted matching problem, which is optimally solvable by e.g. the Hungarian Algorithm [9]. The TA neighbourhood was first introduced by Schaus and Deville [11] who called it a very large neighbourhood. Wauters et al. [14] developed a probabilistic version of the TA neighbourhood that sets a higher probability to selecting tiles with many unmatched edges. The latter TA variant was applied in the present 13 paper. The TA separates the inner and the border moves. It is prohibited to reassign border pieces to the inner region and vice versa. An extention to the TA neighbourhood is also tested. In particular a “checkers” configuration of selected tiles is studied, i.e. all tiles on the board that are diagonally adjacent. We denote this extension Black and White (BW). The local search procedure iterates in this neighbourhood, iteratively changing between “black” and “white” positions and solving the related bipartite weighted matching problem until no more improvements are found. Finally, the Tiles Swap and Rotation (TSR) neighbourhood is a standard local search swap operator, in this case swapping the assignment of two tiles, trying all possible rotations as well. The local search procedure exhaustively searches the neighbourhood until a local optimum is reached. 4. Computational Results This section provides computational results obtained by the multi-neighbourhood local search approach on the Eternity II puzzle as well as the 10×10, 12×12, 14 × 14 and 16 × 16 instances that were used in the META 2010 EII contest3 . The latter instances serve as an interesting test set for comparison, due to the availability of some results from the contest. In addition, to the best of our knowledge, the complete solutions of these instances are not publicly available. The 3 × 3 to 9 × 9 instances used in Section 2 also originate from this set. All tests were performed on a 40 cores Intel Nehalem cluster with 120 GB ram, with each core running @ 3.46 GHz with 8 MB cache. Computational resources provided by DAUIN’s HPC Initiative4 . This cluster was used to solve different instances/runs in parallel in order to reduce the total time required to run all tests. Each individual test was run on a single processing core, thus no parallelism was employed in the algorithms. All MILP models are solved using CPLEX 12.4. 3 rd 3 International Conference on Metaheuristics and Nature Inspired Computing, Djerba Island, Tunisia, October 27th -31th 2010 – Eternity 2 contest http://www2.lifl. fr/META10/pmwiki.php?n=Main.Contest. 4 For more details see http://www.dauin-hpc.polito.it. 14 (a) (b) (d) (c) (e) Figure 3: Illustration of the regions involved in the neighbourhood operations: (a) optimizing the border (BO); (b) optimizing a rectangular region (RO); (c) optimizing nonadjacent tile assignments (TA); (d) optimizing diagonally adjacent tile assignments in a checkers fashion (BW); (e) swapping two tiles and possibly rotating them (TSR). 15 Parameter Name Value T A.K 16 tiles T A.N 1000 it Clique.W 6 cols Clique.H 6 rows Clique.N 10 it Table 2: Parameter Settings Table 2 summarizes the parameter settings of the multi-neighbourhood local search approach. The local search procedure starts either from a random solution or from a solution obtained by the constructive heuristics. The algorithm cycles through the proposed neighbourhoods in the following sequence: TA (for T A.N iterations with sample size T A.K), BO (for one iteration), BW (till local optimum), TSR (till local optimum) and finally RO by means of the Max-Clique formulation (for Clique.N iterations with rectangular sample size Clique.W × Clique.H). This sequence was determined experimentally, though the difference in performance between sequences was very limited. At the end of the multi-neighbourhood step, the final solution is a local minimum with respect to all considered neighbourhoods. Table 3 shows the results for twenty runs of the MILP-based greedy heuristic and the MILP-based backtracking heuristic for different region sizes on all the problem instances. A timeout of 1200 seconds was set for the backtracking heuristic for the 10 × 10 instance, 1800 seconds for the 12 × 12 instance, 2400 seconds for the 14×14 instance and 3600 seconds for the 16×16 and EII instances. The greedy heuristic (executed by itself, or after the backtracking heuristic) is executed until all regions are solved. The table also shows the results of both constructive methods after subsequent optimisation by the local search heuristic. In general, both constructive heuristics generate better results when larger regions are used. This clearly affects the CPU time needed to compute optimal solutions for each region. By comparing the results of the two heuristics (without the local search phase), it seems that the backtracking procedure does not strongly dominate (on maximum, average and minimum values) the greedy one, while consuming all the available time. This dominance tends to be more evident for small puzzles and region sizes, while for larger instances with solutions generated by larger regions, the gap becomes smaller. 16 In almost all cases, the local search procedure manages to improve the results of the constructive heuristics by several units, indicating that these initial solutions are not local optima with respect to the considered neighbourhoods. We conclude that many neighbourhoods in a complex structure are effective for improving these greedy constructive solutions. Table 4 shows the performance of the local search procedure starting from a random initial solution of poor quality. The procedure can achieve good quality results for the 10 × 10 instance, but not for larger instances. This can easily be related to the size of the RO neighborhood. As it is quite large with respect to the puzzle size in the 10 × 10 instance, it is able to optimize a large part of the puzzle. However, this ratio becomes smaller and is thus less effective for larger instances. Finally, Table 5 compares the best published results with the results obtained by the hybrid local search procedure. The CPU times refer to the considered time limits. Table 5 also reports a large test of the procedure, where the best performing configuration (backtracking+LS) was run 100 times within a doubled execution time limit. Larger execution times (using CPLEX 12.4 as ILP solver) do not induce further improvements of the results. We note that some of the entries of Table 5 are missing. Many of the approaches only deal with a subset of the considered instances. Only three studies [4, 13, 14] report results for the 10 × 10 to 16 × 16 instances that were tested in this paper. Some approaches [11, 10, 12] were only applied to the real EII game puzzle. The algorithm reported in [11] was executed on a CPU Intel Xeon(TM) 2.80GHz, with computation time 24 hours. The best score over 20 runs equals 458. [10] obtained a best score of 371. No indication was provided on the computer and the required CPU time. The algorithm of [4] was run on a PC Pentium Core 2 Quad (Q6600), 2.4 GHz, with 8 GB of RAM. It considered EII style problems (but not the real puzzle) with sizes 10 × 10, 12 × 12, 14 × 14 and 16 × 16. The corresponding time limits were 1200, 1800, 2400 and 3600 seconds respectively and the entries of Table 5 report the best solution obtained over 30 runs. The algorithm of [13] addressed the same instances with the same time limits and number of runs as [4]. It was tested on a personal computer with 1.8GHz CPU and 1GB RAM. Time limits and number of runs were the same as in [13]. The tests were performed on an Intel Core 2 Duo @ 3Ghz with 4GB of 17 RAM. The algorithm of [14] was tested on all the instances from [4] and [13] and also on the EII real game puzzle. The entry reports the best result obtained over 30 runs with a time limit of 3600 seconds for EII. Finally, the algorithm of [12] was tested on the EII real game puzzle only running on a grid computing system over a period of several weeks/months not explicitly indicated by the authors. The results show that the algorithm is competitive with the state of the art, obtaining top results for the 10 × 10 instance in a similar time frame as the other algorithms. Most interesting, the initial solution constructed by the greedy and backtracking heuristics is already of high quality, leaving only a limited gap from the optimal solution. Therefore, we expect that these methods may serve as the basis for reaching new top results. The best result for the official EII puzzle instance, 467 obtained using a slipping tile, scanrow backtracking algorithm [12], is still out of our current grasp. However, that algorithm was highly tailored to the EII puzzle instances, used precomputed sequences and was run over the course of several weeks/months (see http://www.shortestpath.se/eii/eii details.html). A direct comparison with the approach presented is partially misleading. Among the other existing approaches, only [14] shows to be slightly superior to our approach. However, our approach should become more competitive along with the expected performance improvement of MILP solvers over the years. Clearly, solving larger subregions in both the constructive heuristics (greedy and backtracking) will lead to better initial solutions. In addition, the effectiveness of the MIP-based local search neighbourhoods is expected to improve when larger regions can be solved. If performance improvements allow ILP solvers to address instances of size 8 × 8 or even 9 × 9 in a reasonable amount of time, it may safely be assumed that the proposed approach will lead to improved results competitive with the other state of the art approaches. 5. Conclusions The present work introduced a hybrid approach to the Eternity II puzzle. A MILP formulation is related to the puzzle’s optimisation version, where the total number of unmatched edges should be minimised. It is shown that the Eternity II puzzle can be modelled as a clique problem, providing, as a byproduct of this work, new hard instances of the maximum clique problem 18 to the community. Preliminary testing revealed it was clear that these models cannot handle large size instances (such as the original EII puzzle), as they quickly become computationally intractable. Therefore, these models were used as the basis for heuristic decompositions, which could then be used in a hybrid approach. A greedy and a backtracking constructive heuristic have been designed, which strongly rely on the capability of optimally solving a specific region of the puzzle. Within a reasonable time limit, high quality solutions can be generated using these heuristics. A multi-neighbourhood local search approach has also been proposed. By applying a set of different neighbourhoods, the local search procedure manages to improve upon the initial solutions generated by the constructive heuristics and reaches solutions competitive to the best available results. These results confirm that a novel and clever use of mathematical models and solvers/heuristics is effective for large size problems, which cannot be solved all in once by the same MILP solver. We believe that hybridizing local search approaches and mathematical programming techniques in a matheuristic context is the key to break up the intractability of hard problems such as the EII puzzle. References [1] Ansótegui C, Béjar R, Fernàndez C, Mateu C. Edge matching puzzles as hard SAT/CSP benchmarks. In: CP ’08 Proceedings of the 14th international conference on Principles and Practice of Constraint Programming, 560-565, 2008. [2] Benoist T, Bourreau E. Fast global filtering for Eternity II. Constraint Programming Letters, 3, 35-50, 2008. [3] Bomze IM, Budinich M, Pardalos PM, PelilloM. The maximum clique problem. In: Du DZ, Pardalos PM (eds) Handbook of combinatorial optimization. Kluwer Academic, Dordrecht, 174, 1999. [4] Coelho I, Coelho B, Coelho V, Haddad M, Souza M, Ochi L. A general variable neighborhood search approach for the resolution of the Eternity II puzzle. In: Proceedings of the 3rd International Conference 19 on Metaheuristics and Nature Inspired Computing, META’10. 2010, http://www2.lifl.fr/META10/proceedings//meta20100 submission 157.pdf. [5] Demaine ED, Demaine ML. Jigsaw puzzles, edge matching, and polyomino packing: Connections and complexity. Graphs and Combinatorics, 23, 195-208, 2007. [6] Grosso A, Locatelli M, Pullan W. Simple ingredients leading to very efficient heuristics for the maximum clique problem. Journal of Heuristics, 14, 587-612, 2008. [7] Heule MJH. Solving edge-matching problems with satisfiability solvers. In: Proceedings of the Second International Workshop on Logic and Search (LaSh 2008), 88-102, 2008. [8] Kendall G, Parkes A, Spoerer K. A survey of NP-Complete puzzles. International Computer Games Association Journal, 31, 13-34, 2008. [9] Kuhn HW. The Hungarian Method for the assignment problem. Naval Research Logistics Quarterly, 2, 83-97, 1955. [10] Muñoz J, Gutierrez G, Sanchis A. Evolutionary genetic algorithms in a constraint satisfaction problem: Puzzle Eternity II. In: Cabestany J, Sandoval F, Prieto A, Corchado J, editors. Bio-Inspired Systems: Computational and Ambient Intelligence; LNCS 5517, 720-727, 2009. [11] Schaus P, Deville Y. Hybridization of CP and VLNS for Eternity II. In: JFPC’08 Quatrième Journées Francophones de Programmation par Contraintes. 2008, http://www.info.ucl.ac.be/∼yde/Papers/JFPC2008 Eternity en.pdf. [12] Verhaard L. http://www.shortestpath.se/eii/index.html; 2008. [13] Wang WS, Chiang TC. Solving eternity-ii puzzles with a tabu search algorithm. In: Proceedings of the 3rd International Conference on Metaheuristics and Nature Inspired Computing. 2010, http://www2.lifl.fr/META10/proceedings//meta20100 submission 161.pdf. [14] Wauters T, Vancroonenburg W, Vanden Berghe G. A guide-and-observe hyper-heuristic approach to the Eternity II puzzle. Journal of Mathematical Modelling and Algorithms, 11, 217-233, 2012. 20 INSTANCE 10x10 12x12 14x14 16x16 EII-instance START greedy greedy+LS greedy greedy+LS backtracking backtracking+LS backtracking backtracking+LS greedy greedy+LS greedy greedy+LS backtracking backtracking+LS backtracking backtracking+LS greedy greedy+LS greedy greedy+LS backtracking backtracking+LS backtracking backtracking+LS greedy greedy+LS greedy greedy+LS backtracking backtracking+LS backtracking backtracking+LS greedy greedy+LS greedy greedy+LS backtracking backtracking+LS backtracking backtracking+LS Region Size 1 x 10 1 x 10 2 x 10 2 x 10 1 x 10 1 x 10 2 x 10 2 x 10 1 x 12 1 x 12 2 x 12 2 x 12 1 x 12 1 x 12 2 x 12 2 x 12 1 x 14 1 x 14 2 x 14 2 x 14 1 x 14 1 x 14 2 x 14 2 x 14 1 x 16 1 x 16 2 x 16 2 x 16 1 x 16 1 x 16 2 x 16 2 x 16 1 x 16 1 x 16 2 x 16 2 x 16 1 x 16 1 x 16 2 x 16 2 x 16 MAX 165 168 170 170 169 172 170 171 245 247 249 250 248 250 249 252 338 339 344 344 342 344 344 344 448 449 454 454 453 454 457 457 449 450 456 457 453 454 457 457 AVG 161.10 164.90 166.35 167.05 165.65 167.75 167.65 168.10 241.20 244.00 247.00 247.50 245.35 247.75 247.60 248.45 333.00 335.56 340.50 340.80 338.25 340.69 342.17 342.33 444.05 446.20 451.38 451.88 448.80 451.53 453.69 454.00 443.75 446.50 451.75 451.95 449.00 451.35 452.80 453.15 MIN 158 161 164 164 162 165 164 165 239 240 244 245 242 246 246 246 329 332 338 338 334 336 340 340 440 443 448 448 444 449 451 451 440 442 450 450 446 447 448 449 Time Avg. (s) 4.29 1204.29 25.93 1225.93 1200.24 2400.24 1207.06 2407.06 6.83 1806.83 101.02 1901.02 1801.51 3601.51 1868.85 3668.85 10.43 2410.43 2335.42 4735.42 2401.36 4801.36 4664.65 7064.65 24.08 3624.08 11188.03 14788.03 3605.68 7205.68 13722.86 17322.86 21.85 3621.85 13835.04 17435.04 3605.08 7205.08 15294.52 18894.52 Table 3: Results for the MILP-based greedy and backtracking heuristics with and without local search, using different region sizes. ObjMax ObjAvg ObjMin Optimum 10 × 10 167 163.2 158 180 12 × 12 238 231.1 223 264 14 × 14 312 299.2 292 364 16 × 16 398 384.3 367 480 EII − instance 391 382 372 480 Table 4: Results for the local search procedure with optimal neighbourhoods starting from a random solution. 21 Present paper (20 runs) Present paper (100 runs) Muñoz et al. [10] Wang and Chiang [13] Coelho et al. [4] Schaus and Deville [11] Wauters et al. [14] Verhaard [12] Optimum 10 × 10 172 (20 min) 172 (40 min) 163 (20 min) 167 (20 min) 172 (20 min) 180 12 × 12 252 (30 min) 252 (60 min) 234 (30 min) 241 (30 min) 254 (30 min) 264 14 × 14 344 (40 min) 347 (80 min) 318 (40 min) 325 (40 min) 348 (40 min) 364 16 × 16 457 (180 min) 457 (360 min) 418 (60 min) 425 (60 min) 460 (60 min) 480 EII − instance 457 (180 min) 458 (360 min) 371 (-) 458 (1440 min) 461 (60 min) 467 (weeks/months) 480 Table 5: Comparison of the best results to other approaches available in the literature. (Execution times presented within parenthesis) 22	8
Mapping Images to Scene Graphs with Permutation-Invariant Structured Prediction Roei Herzig * 1 Moshiko Raboh * 1 Gal Chechik 2 3 Jonathan Berant 1 Amir Globerson 1 arXiv:1802.05451v1 [stat.ML] 15 Feb 2018 Abstract Structured prediction is concerned with predicting multiple inter-dependent labels simultaneously. Classical methods like CRF achieve this by maximizing a score function over the set of possible label assignments. Recent extensions use neural networks to either implement the score function or in maximization. The current paper takes an alternative approach, using a neural network to generate the structured output directly, without going through a score function. We take an axiomatic perspective to derive the desired properties and invariances of a such network to certain input permutations, presenting a structural characterization that is provably both necessary and sufficient. We then discuss graph-permutation invariant (GPI) architectures that satisfy this characterization and explain how they can be used for deep structured prediction. We evaluate our approach on the challenging problem of inferring a scene graph from an image, namely, predicting entities and their relations in the image. We obtain state-of-the-art results on the challenging Visual Genome benchmark, outperforming all recent approaches. 1. Introduction Structured prediction addresses the problem of classification when the label space contains multiple inter-dependent labels. For example, in semantic segmentation of an image, each pixel is assigned a label, while considering the labels of nearby pixels. A similar problem is the task of recognizing multiple entities and their relations in an image, where recognizing one entity affects recognition of the others. Structured prediction has attracted considerable attention because it applies to many learning problems and poses unique theoretical and applied challenges (e.g., see Taskar et al., 2004; Chen et al., 2015; Belanger et al., 2017). * Equal contribution 1 Tel-Aviv University, Israel 2 Google Brain, CA, 94043 3 Gonda brain research institute, Bar-Ilan University, Ramat-Gen 52900, Israel. Correspondence to: Roei Herzig <[email protected]>, Moshiko Raboh <[email protected]>, Gal Chechick <[email protected]>, Jonathan Berant <[email protected]>, Amir Globerson <[email protected]>. Typically, structured prediction models define a score function s(x, y) that quantifies how well a label assignment y is compatible, or consistent, with an input x. In this setup, the inference task amounts to finding the label that maximizes the compatibility score y ∗ = arg maxy s(x, y). This score-based approach separates a scoring component – implemented by a parametric model, from an optimization component – aimed at finding a label that maximizes that score. Unfortunately, for a general scoring function s(·), the space of possible label assignments grows exponentially with input size. For instance, the set of possible pixel label assignments is too large even for small images. Thus, inferring the label assignment that maximizes a scoring function is computationally hard in the general case. An alternative approach to scored-based methods is to map an input x to a structured output y with a “black box” neural network, without explicitly defining a score function. This raises a natural question: what properties and invariances must be satisfied by such a network? We take this axiomatic approach and argue that one important property is invariance to a particular type of input permutation. We then prove that this invariance is equivalent to imposing certain structural constraints on the architecture of the network, and describe architectures that satisfy these constraints, significantly extending the expressive power of current structured prediction approaches. We argue that respecting permutation invariance is important, as otherwise the model would have to spend capacity on learning this invariance at training time. Conceptually, our approach is motivated by recent work on D EEP S ETS (Zaheer et al., 2017), which asked a similar question for black-box functions on sets. To evaluate our approach, we tackle the challenging task of mapping an image to a scene graph, which describes the entities in the image and their relations. We describe a model that satisfies the permutation invariance property, and show that it achieves state-of-the-art results on the competitive Visual Genome benchmark (Krishna et al., 2017), demonstrating the power of our new design principle. In summary, the novel contributions of this paper are: First, we derive sufficient and necessary conditions for a deep structured prediction architecture. Second, we improve the state-of-the-art with this approach in a challenging pixel-tograph problem on a large dataset of complex visual scenes. Mapping Images to Scene Graphs with Permutation-Invariant Structured Prediction 2. Structured Prediction Scored-based methods in structured prediction define a score function s(x, y)1 that reflects the degree to which y is compatible with x, and infer a label by solving y ∗ = arg maxy s(x, y) (e.g., see Lafferty et al., 2001; Taskar et al., 2004; Meshi et al., 2010; Chen et al., 2015; Belanger et al., 2017). Most score functions previously used decompose as a sum over simpler functions, s(x, y) = P i fi (x, y), where solving maxy fi (x, y) can be performed efficiently.2 This local maximization forms the basic building block of algorithms for approximately maximizing s(x, y). One way to achieve this is to restrict fi (x, y) to depend only on a small subset of the y variables. The renewed interest in deep learning led to efforts to integrate deep networks with structured prediction, including modeling the fi functions as deep networks. In this context, the most widely-used score functions are singleton f (yi , x) and pairwise fij (yi , yj , x). Initial work used a two-stage architecture, learning local scores independently of the structured prediction goal (Chen et al., 2014; Farabet et al., 2013). Later works considered end-to-end architectures where the inference algorithm is part of the computation graph (Chen et al., 2015; Pei et al., 2015; Schwing & Urtasun, 2015; Zheng et al., 2015). These studies used standard inference algorithms, such as loopy belief propagation, mean field methods and gradient descent (Belanger et al., 2017). Score-based methods provide several advantages. First, they allow intuitive specification of local dependencies between labels (like pairwise dependencies) and how these translate to global dependencies. Second, when the score function is linear in its parameters (s(x, y; w) is linear in w), the learning problem has natural convex surrogates (e.g., logloss in CRF), making learning efficient. Third, inference in large label spaces is often possible via exact combinatorial algorithms or empirically accurate approximations. However, with the advent of deep scoring functions s(x, y; w), learning is no longer convex. Thus, it is worthwhile to rethink the architecture of structured prediction models, and consider models that map inputs x to outputs y directly without an explicit score function. We want these models to enjoy the expressivity and predictive power of neural networks, while maintaining the ability to specify local dependencies between labels in a flexible manner. In the next section, we present such an approach and consider a natural question: what should be the properties of such a deep neural network used for structured prediction. 1 The term energy function is also used. More precisely, many P message passing algorithms require that functions fi (x, y) + k δk (yk ) can be maximized efficiently. 2 3. Permutation Invariant Structured Prediction We begin with some notation, focusing on structures that consists of pairwise interactions, as these are simpler in terms of notation, and sufficient for describing the structure in many problems. We denote a structured label with n entries by y = [y1 , . . . , yn ]. In a score-based approach, the score is defined via a set of singleton scores fi (yi , x) and pairwise scores fij (yi , yj , x), where the overall score s(x, y) is the sum of these singleton and pair scores. For brevity, we also denote fij = fij (yi , yj , x) and fi = fi (yi , x). An inference algorithm takes as input the set of local scores fi , fij and outputs the assignment maximizing s(x, y). We can therefore abstractly view an inference algorithm as a blackbox that takes as input a set of node- and edge-dependent inputs (i.e., the local scores fi , fij ) and returns a label y, even without an explicit score function s(x, y). While numerous inference algorithms exist for this setup, including belief propagation (BP) and mean field, here we aim to develop a framework for a deep learning labeling algorithm (we avoid the term “inference” since the algorithm does not explicitly maximize a score function). Such an algorithm will be a black-box with the f functions as input and the labels y1 , . . . , yn as output. We next ask what architecture such an algorithm should have. We follow with several definitions. A graph labeling function F : (V, E) → Y is a function whose input is an ordered set of node features V = [z 1 , . . . , z n ] and ordered set of edge features E = [z 1,2 . . . , z i,j , . . . , z n,n−1 ]. For example, the z i ’s can be the array of values fi (yi , x), and the z i,j ’s can be the table of values fi,j (yi , yj , x) . For simplicity, assume z i ∈ Rd and z i,j ∈ Re . The output of F is a set of labels y = [y1 , . . . , yn ], which can be thought of as labeling the nodes. Thus, inference algorithms like BP are graph labeling functions, since they take f as input and output a set of labels. However, graph labeling functions need not correspond to any inference algorithm (i.e., an algorithm that maximizes a score function). A natural requirement is that the algorithm produces the same result when given the same score function. For example, consider a label space containing three variables y1 , y2 , y3 , and assume that the inference algorithm takes as input z = (z 1 , z 2 , z 3 , z 12 , z 13 , z 23 ) = (f1 , f2 , f3 , f12 , f13 f23 ), and outputs a label y = (y1∗ , y2∗ , y3∗ ). When the same algorithm is given an input that is permuted in a consistent way, z 0 = (f2 , f1 , f3 , f21 , f23 f13 ), this defines exactly the same score function as the first scenario. Hence, we would expect it to output the same label, only permuted, namely, it should output y = (y2∗ , y1∗ , y3∗ ). Most inference algorithms, including BP and mean field, satisfy this symmetry requirement by de- Mapping Images to Scene Graphs with Permutation-Invariant Structured Prediction 3.1. Characterizing Permutation Invariance Figure 1. Graph permutation invariance and structured prediction. A graph labeling function F is graph permutation invariant (GPI) if permuting the names of nodes maintains the output. sign. Here we design a deep learning black-box, hence need to guarantee invariance to input permutations. A black-box that does not satisfy this invariance has to waste capacity on learning it at training time. In what follows we use z to denote the joint set of node and edge features. z can be thought of as a container with n + n(n − 1) = n2 elements. We next consider what happens to a graph labeling function, when graph variables are permuted by a permutation σ. Importantly, the edges in this case are also permuted in a way that is consistent with the node permutation σ. Definition 1. Let z be a set of node and edge features. Given a permutation σ of {1, . . . , n}, denote σ(z) to be a new set of node and edge features that are given by: [σ(z)]i = z σ(i) , [σ(z)]i,j = z σ(i),σ(j) . (1) σ(z) has the same elements as in z, but the node elements are permuted according to σ, and the edge elements are permuted accordingly. In what follows, we use the notation σ([y1 , . . . , yn ]) = [yσ(1) , . . . , yσ(n) ], namely, σ applied to a set of labels yields the same labels, only permuted by σ. Next comes our key definition of a function F whose output is invariant to permutations of the input graph. Definition 2. A graph labeling function F is said to be graph-permutation invariant (GPI), if for all permutations σ of {1, . . . , n} and for all z it satisfies: F(σ(z)) = σ(F(z)). (2) Figure 1 illustrates the desired invariance. The above property says that as long as the input to F describes the same node and edge properties, the same labeling will be output. This is indeed a property we would like any such F to have, and we thus turn to characterizing a necessary and sufficient structure for achieving it. Motivated by the above discussion, we ask: what structure is necessary and sufficient to guarantee that F is graphpermutation invariant? Note that a function F takes as input an ordered set z. Therefore its output on z could certainly differ from its output on σ(z). To achieve permutation invariance, F should intuitively contain certain symmetries. For example, one permutation invariant architecture is to define yi = g(z i ) for any function g, but this characterization is too restrictive to cover all permutation invariant functions. The next Theorem provides a complete characterization, while Figure 2 shows the corresponding architecture. Theorem 1. Let F be a graph labeling function. Then F is graph-permutation invariant if and only if there exist functions α, ρ, φ such that for all k = 1, . . . , n: [F(z)]k = ρ(z k , n X α(z i , i=1 X φ(z i , z i,j , z j ))), (3) j6=i where φ : R2d+e → RL and α : Rd+L → RW and ρ : RW +d → R. Proof. First, we show that any F satisfying the conditions of Theorem 1 is GPI. Namely, for any permutation σ, [F(σ(z))]k = [F(z)]σ(k) . To see this, write [F(σ(z))]k using Eq. 3 and Definition 1 as: X X ρ(z σ(k) , α(z σ(i) , φ(z σ(i) , z σ(i),σ(j) , z σ(j) ))). i j6=i The second argument of ρ above is clearly invariant under σ, because the sum considers an index i and all other indices j, hence the same elements are covered under permutation. The expression therefore equals to: X X ρ(z σ(k) , α(z i , φ(z i , z i,j , z j ))) = [F(z)]σ(k) i j6=i where the equality follows from Eq. 3. We thus proved that Eq. 3 implies graph permutation invariance. Next, we prove that any black-box graph-permutation invariant function can be expressed as in Eq. 3. Namely, we show how to define φ, α and ρ that can implement any permutation invariant function F. The key idea is to construct φ, α such that the second argument of ρ in Eq. 3 contains all the information about the graph features z, including the edges they originated from . Then, the function ρ consists of an application of the black box F to this representation, followed by extracting the label yk . To simplify notation assume that edge features are scalar (e = 1). The extension to the vector case is simple, but involves more indexing. We also assume that z k uniquely identifies the node (i.e., no two nodes share the same node Mapping Images to Scene Graphs with Permutation-Invariant Structured Prediction Figure 2. A schematic representation of the GPI architecture in Theorem 1. Singleton features z i are omitted for simplicity. First, the features z i,j are processed element-wise by φ. Next, they are summed to create a vector si , which is concatenated with z i . Third, a representation of the entire graph is created by applying α n times and summing the created vector. The graph representation is then finally processed by ρ together with z k . feature), which can be achieved by adding the index as another feature of z k . Finally, we assume that F is a function only of the pairwise features z i,j . This can be achieved by adding singleton features into the pairwise ones. Let H be a hash function with L buckets mapping node features z i to an index (bucket). Assume that H is perfect (this can be achieved for a large enough L). Define φ to map the pairwise features to a vector of size L. Let 1 [j] be a one-hot vector of dimension RL , with one in the j th coordinate. Recall that we consider scalar z i,j so that φ is indeed in RL , and define φ as: φ(z i , z i,j , z j ) = 1 [H(z j )] zi,j (4) i.e., φ “stores” zi,j in the unique bucket for node j. P Let si = j6=i φ(z i , zi,j , z j ) be the second argument of α in Eq. 3 (si ∈ RL ). Then, since all z j are distinct, si stores all the pairwise features for neighbors of i in unique positions within its L coordinates. Since si (H(z k )) contains the feature zi,k whereas sj (H(z k )) contains the feature z j,k , we cannot simply sum the si , since we would lose the information of which edges the features originated from. Instead, we define α to map si to RL×L such that each feature is mapped to a distinct location. Formally: α(z i , si ) = 1 [H(z i )] sTi . (5) α outputs a matrix that is all zeros except for the features correspondingP to node i that are stored in row H(z i ). The matrix M = i α(z i , si ) (namely, the second argument of ρ in Eq. 3) is a matrix with all the edge features in the graph including the graph structure. Figure 3. Illustration of the proof construction for Theorem 1. Here H is a hash function of size L = 5 such that H(1) = 1, H(3) = 2, H(2) = 4, G is a three-node input graph, and z i,j ∈ R are the pairwise features (in purple) of G. (a) φ is applied to each z i,j . Each application yields a vector in R5 . The three dark yellow columns correspond to φ(z 1,1 ),φ(z 1,2 ) and φ(z 1,3 ). Then, all vectors φ(z i,j ) are summed over j to obtain three si vectors. (b) α’s (blue matrices) are an outer product between 1 [H(z i )] and si (see Eq. 5) resulting in a matrix of zeros except one row. The dark blue matrix corresponds for α(z 1 , s1 ). (c) All α’s are summed to a 5 × 5 matrix, isomorphic to the original z i,j matrix. To complete the construction we set ρ to have the same outcome as F. We first discard rows and columns in M that do not correspond to original nodes (reducing M to dimension n × n). Then, we use the reduced matrix as the input z to the given F. Assume for simplicity that M does not need to be contracted.3 Let the output of F on M be y = y1 , . . . , yn . Then we set ρ(z k , M ) = y H(zk ) . Since F is invariant to permutations this indeed returns the output of F on the original input. General Graphs So far, we discussed complete graphs, where all edges correspond to valid feature pairs. Many graphs however may be sparse and have certain structures. For example, an n-variable chain graph in sequence labeling has only n − 1 edges. For such sparse graphs, the input to F would not be all z i,j pairs but rather only features corresponding to valid edges of the graph, and we are only interested in invariances that preserve the graph structure, namely, the automorphisms of the graph. Thus, the desired invariance is that σ(F(z)) = F(σ(z)) only for automorphisms of the graph. It is easy to see that P Theorem P1 holds in this case, if one replaces the sum j6=i with j∈N (i) , where N (i) are the neighbors of node i in the graph. 3 This merely introduces another indexing step. Mapping Images to Scene Graphs with Permutation-Invariant Structured Prediction 4. Deep Graph Prediction Theorem (1) provides the general requirements for designing an architecture for structured prediction. For a given problem, one has to choose a specific architecture and parameterization for α, φ, ρ. For instance, it is interesting to consider how an algorithm like belief propagation (BP) can be implemented in our framework. Following the proof of Theorem 1, one would use φ, α to aggregate features, and then ρ would apply BP to these features. Our architecture is of course more general by construction. For example, it could use φ and α to “sketch” the input graph, such that labeling can be performed on a reduced representation. We now survey certain architectures consistent with Theorem 1 and discuss their expressive power. Introducing Attention. Attention is a powerful architectural component in deep learning (Bahdanau et al., 2015), but most inference algorithms do not use attention. We now show how attention can be introduced in our framework. Intuitively, attention means that instead of aggregating features of neighbors, a node i weighs neighbors based on their relevance. For example, the label of an entity in an image may depend more strongly on entities that are spatially closer. We now implement attention for the architecture of Eq. 3. Formally, we learn attention weights for the neighbors j of a node i, which scale the features z i,j of that neighbor. We can also learn different attention weights for individual features of each neighbor in a similar way. Let wi,j ∈ R be an attention mask specifying the weight that node i gives to node j: X wi,j (z i , z i,j , z j ) = eβ(zi ,zi,j ,zj ) / eβ(zi ,zi,t ,zt ) , (6) t where β can be any scalar-valued function of its arguments (e.g., a dot product of z i and z j as in standard attention models). To introduce attention we wish α ∈ Re to have the form of weighting P wi,j over neighboring feature vectors z i,j , namely, α = j6=i wi,j z i,j . Figure 4. An image (top) and its scene graph (bottom) from the Visual Genome dataset (Krishna et al., 2017). The scene graph captures the entities in the image (nodes, blue circles) and their pairwise relations (edges, red circles). Example relationships in this graph include: hat, on, dog and dog, on, motorcycle . Using RNNs as Components. Theorem 1 allows arbitrary functions for φ, α and ρ, except for their input dimensionality. Specifically, these functions can involve highly expressive recursive computation, simulate existing message passing algorithms, and new algorithms that are learned from data. This can of course be extended to more elaborate structures like LSTMs (Hochreiter & Schmidhuber, 1997) and Neural Turing Machines (Graves et al., 2014), which we leave for future work. Theorem 1 suggests that any function in the form of F is graph permutation invariant. It is easy to show that composing two functions that are GPI, is also GPI. Therefore, we can run F iteratively by providing the output of one step of F as part of the input to the next step and maintain graph-permutation invariance. This results in a recurrent architecture, which we will employ in the next section to obtain state-of-the-art performance on scene graph prediction. 5. Application - Scene Graph Classification X si,1:e X eβ(zi ,zi,j ,zj ) z i,j P = = wi,j z i,j β(z ,z ,z ) i i,j j si,e+1 j6=i e We demonstrate the benefits of our axiomatic approach in the task of inferring scene graphs from images. In this problem, the input is an image annotated with a set of rectangles that bound entities in the image, known as bounding boxes. The goal is to label each bounding box with the correct entity category, and every pair of entities with their relation, such that they form a coherent graph, known as a scene graph. In a scene graph, nodes correspond to bounding boxes labeled with the entity category and edges correspond to relations among entities, which could be spatial (“on”) or functional (“wearing”). Thus, in an image with 5 bounding boxes there are 5 + 5 × 4 = 25 output variables. A similar approach can be applied over α and ρ to model attention over the outputs of α as well (graph nodes). This concept is illustrated in Figure 4, showing an image of a dog on a motorcycle (top) and the corresponding scene To achieve this form we extend φ by a single entry, defining φ ∈ Re+1 (namely we set L = e+1) as φ1:e (z i , z i,j , z j ) = eβ(zi ,zi,j ,zj ) z i,j (here φ1:e are the first e elements of φ) and φe+1 (z iP , z i,j ; z j ) = eβ(zi ,zi,j ,zj ) . We keep the definition si,1:e of si = j6=i φ(z i , zi,j , z j ). Next, we define α = si,e+1 and substitute si and φ to obtain the desired form as attention weights wi,j over neighboring feature vectors z i,j : α(z i , si ) = j6=i j6=i Mapping Images to Scene Graphs with Permutation-Invariant Structured Prediction graph below. The pink box in the image is labeled “motorcycle” and the white box is labeled “dog”. These two boxes correspond to two nodes (light blue circles in Figure 4 bottom), and their relation “on” corresponds to an edge (red circle labeled “on”). While scene graphs are typically very sparse, one can view a scene graph as complete if each pair of unrelated entities is connected by a ‘null’ edge. A scene graph can be represented as a collection of triplets, each with a relation and two entities, like dog, on, motorcycle . 5.1. Model Our model has two components: A Label Predictor (LP) that takes as input an image with bounding boxes and outputs a distribution over labels for each entity and relation. Then, a Scene Graph Predictor (SGP) that takes all label distributions and predicts more consistent label distributions jointly for all entities and relations. Label Prediction. The LP module (Figure 5) receives an input image x and a set of bounding boxes bb1 , . . . , bbn , corresponding to image entities (as in Figure 4). LP outputs a set of entity label probabilities p(yient \| bbi ) for each box i from a pre-defined set of candidate entity labels P and a set rel of relation probabilities p(yi,j \| bbi , bbj ) from another predefined set of relation labels R. These unary and pairwise potentials are later fed to the SGP module. To predict entity labels, we used R ES N ET 50 (He et al., 2016a;b), taking as input a patch cropped from the full image according to the ith bounding box (Figure 5). We used a second R ES N ET 50 to predict relations, using a 5-channel tensor input: three channels for the RGB image, and two channels for binary masks for the subject entity and object entity bounding boxes (Figure 5). The image patch provided to this network was cropped such that it covers both the subject entity and the object entity. Providing the two binary masks breaks the symmetry of the subject entity and object entity and allow the network to discriminate between triplets like man, wearing, shirt and shirt, on, man ). Scene Graph Prediction. While the LP module described above is trivially GPI, because the output variables rel yient , yi,j are predicted independently, constructing a GPI architecture for a Scene Graph Predictor is harder. We now outline this construction. Entity classification in this module is GPI following Theorem 1, where z i are features for every bounding box and z i,j are features for box pairs. To classify relations, we added a function ρrelation that reuses the GPI representation created during entity classification. Because the input to ρrelation is a GPI representation, it is easy to show that our entire network is GPI. Let z i be the concatenation of z features and z spatial . Where i i features zi is the current label probability for entity i (logits Figure 5. The Label Predictor. (a) Entity recognition network: A network takes an image patch cropped based on a bounding box, and outputs classification probabilities per label. (b) Relation recognition network: A network takes an input tensor, containing the RGB image in the first 3 channels, and two binary masks for the subject and object entities in the remaining two channels. before the final softmax layer) and z spatial is i’s bounding i box given as a (x, y, width, height). In addition, for z i,j , we used the confidences for relation i, j (logits before the final softmax layer). In each step of SGP we apply the function F, which receives all entity features z i and all relation features z i,j , and output updated confidences for entities and relations. Because composing GPI functions is GPI, our SGP module is GPI. We now describe our implementation of the three components of F: φ, α and ρ. (1) φ is a network with two FC-layers. It receives (a) subject features z i (b) relations features z i,j (c) entity features z j and outputs a vector of size 500. Next, for each entity i, we aggregate φ(z i , z i,j , z j ) into si using the attention mechanism described in Section 4. To calculate the weights wi,j , we implement β(·) (Eq. 6) with a FC layer that receives the same input as φ and outputs a scalar. (2) α is a two FC-layer network, receiving entity features z i and context features si . The outputs of α are aggregated with a similar attention mechanism over entities, resulting in a vector g ∈ R500 representing the entire graph. (3) ρ, consists of ρentity , which classifies entities, and ρrelation , which classifies relations. ρentity is a three FC-layer network of size 500. It receives z i , si and g as input, and outputs a vector qi with one scalar per entity class. Unlike Theorem 1, we allow ρ direct access to si , which maintains the GPI property, and improved learning in practice. The final output confidence is a linear interpolation of the current confidence z fi eatures and the new confidence qi , controlled by a learned forget gate, i.e., the output is qi + forget · z fi eatures . ρrelation , the relation classifier, is analogous to the entity classifier, receiving as input z i , z j , the relation features z i,j , and the graph representation g. Mapping Images to Scene Graphs with Permutation-Invariant Structured Prediction We also explored concatenating word embeddings of the most probable entity class to z i . Word vectors were learned with G LOV E (Pennington et al., 2014) from the ground-truth captions of Visual Genome (Krishna et al., 2017). 5.2. Experimental Setup Dataset. We evaluated our approach on the Visual Genome (VG) dataset (Krishna et al., 2017). VG consists of 108,077 images annotated with bounding boxes, entities and relations. Distribution over entity classes and relations is long-tailed with a total of 75,729 unique entity classes and 40,480 unique relations. To allow apple-to-apple comparison with previous studies of this dataset (Xu et al., 2017; Newell & Deng, 2017; Zellers et al., 2017), we used the same preprocessed data, including the train and test splits, as provided by (Xu et al., 2017). This dataset had on average 12 entities and 7 relations per image. For evaluation, we used the same 150 entity categories and 50 relations as in (Xu et al., 2017; Newell & Deng, 2017; Zellers et al., 2017). To tune hyper-parameters, we also split the training data into two by randomly selecting 5K examples, resulting in a final 70K/5K/32K split for train/validation/test sets. Training Procedure. We trained all networks using Adam (Kingma & Ba, 2014); Input images were resized to 224x224 to conform with the R ES N ET architecture. We first trained the LP module, and then trained the SGP module using the best LP model. In what follows, all the particular chosen values were tuned on the validation set. For LP, we trained the relation network with cross-entropy loss and a positive-to-negative ratio of 1:3 (where ‘positive’ refers to a labeled relation and ‘negative’ to unlabeled), and performed early-stopping after 90 epochs. We chose a batch size of 64, and also used data augmentation techniques such as translation and rotation to further improve the results. The loss function for the SGP was the sum of cross entropy losses over all entities and relations in the image. In the loss, we penalized entities 4 times more strongly than relations, and penalized negative relations 10 times more weakly than positive relations. We used batch size 100 and early-stopped after 120 epochs. The recurrent application of F was performed for 2 steps. Evaluation. Xu et al. (2017) defined three different subtasks when inferring scene graphs, and we focus on two: (1) SGCls: Given ground-truth bounding boxes for entities, predict all entity categories and relations categories. (2) PredCls: Given bounding boxes annotated with entity labels, predict all relations. Following (Lu et al., 2016), we used Recall@K as the evaluation metric. It measures the fraction of correct ground-truth triplets that appear within the K most confident triplets proposed by the model. Two evaluation protocols are used in the literature, which differ by whether they enforce graph constraints over model predictions. The first protocol requires that the top-K triplets assign one consistent class per entity and relation. This rules out putting more than one triplet for a pair of bounding boxes. It also rules out inconsistent assignment, like a bounding box that is labeled as one entity in one triplet, and as another entity in another triplet. The second evaluation protocol does not enforce any such constraints. Models and baselines. We compare four variants of our GPI approach with the reported results of four baselines that are currently the state-of-the-art on various scene graph subtasks. All models use the same data split and pre-processing as (Xu et al., 2017): 1. (L U ET AL ., 2016): This work leverages word embeddings to fine-tune the likelihood of predicted relations. 2. (X U ET AL ., 2017): This model passes messages between entities and relations, and iteratively refines the feature map used for prediction. 3. (N EWELL & D ENG , 2017). The P IXEL 2G RAPH model uses associative embeddings (Newell et al., 2017) to produce a full graph from the image. 4. (Z ELLERS ET AL ., 2017) The N EURAL M OTIF method encodes global context for capturing highorder motifs in scene graphs. 5. GPI: N O ATTENTION: Our GPI model, but with no attention mechanism. Instead, following Theorem 1, we simply sum the features. 6. GPI: N EIGHBOR ATTENTION: Our GPI model, using attention over neighbors as described in Section 5.1. 7. GPI: M ULTI ATTENTION: Our GPI model, except that we learn different attention weights per feature. 8. GPI: L INGUISTIC : Same as GPI: M ULTI ATTENTION but also concatenating the word embedding vector for the most probable entity label (see Sec. 5.1). 5.3. Results Table 1 lists recall@50 and recall@100 for four variants of our approach compared with three baselines, evaluating with graph constraints. The GPI approach performs well, and L INGUISTIC outperforms all baselines for both PredCls and SGCls. Table 2 provides a similar comparison when evaluating without graph constraints; again L INGUISTIC performs best. More details in the supplemental material. Figure 6 illustrates the model behavior. Predicting isolated labels with LP (column (c)) mislabels several entities, but these are corrected after joint prediction (column (d)). Column (e) shows that the system learned to attend more to nearby entities (the window and building are closer to the tree), and column (f) shows that stronger attention is learned for the classes bird, presumably because it is usually more informative than common classes like tree. Mapping Images to Scene Graphs with Permutation-Invariant Structured Prediction Figure 6. (a) An input image with bounding boxes from VG. (b) The ground-truth scene graph. (c) The LP fails to recognize some entities (building and tree) and relations (in front of instead of looking at). (d) GPI:L INGUISTIC fixes most incorrect LP predictions. (e) Window is the most significant neighbor of Tree. (f) The entity bird receives substantial attention, while Tree and building are less informative. Table 1. Test set results for graph-constrained evaluation SGC LS R@50 R@100 (L U ET AL ., 2016) (X U ET AL ., 2017) N EURAL M OTIFS N O ATTENTION N EIGHBOR ATTEN . M ULTI ATTENTION L INGUISTIC 11.8 21.7 31.3 28.9 30.6 32.3 32.1 14.1 24.4 32.1 31.0 32.4 34.1 34.1 P RED C LS R@50 R@100 35.0 44.8 65.8 63.0 63.7 65.4 66.0 27.9 53.0 68.0 65.2 66.5 67.8 68.3 Table 2. Test set results for unconstrained evaluation SGC LS R@50 R@100 P IXEL 2G RAPH N O ATTENTION N EIGHBOR ATTEN . M ULTI ATTENTION L INGUISTIC 26.5 35.2 37.6 39.0 39.2 30.0 42.1 43.2 44.5 44.9 P RED C LS R@50 R@100 68.0 70.3 73.7 74.5 75.3 75.2 75.1 79.2 81.0 82.4 6. Related Work There has been significant recent interest in extending deep learning to structured prediction. Much of this work has been on semantic segmentation, where convolutional networks (Shelhamer et al., 2017) became a standard approach for obtaining “singleton scores” and various approaches were proposed for adding structure on top. Most of these approaches used variants of message passing algorithms, unrolled into a computation graph (Xu et al., 2017). Some studies parameterized parts of the message passing algorithm and learned its parameters (Lin et al., 2015). Recently, gradient descent has also been used for maximizing score functions (Belanger et al., 2017; Gygli et al., 2017). An alternative approach for deep structured prediction is via greedy decoding, where one label is inferred at a time, based on previous labels. This has been popular in sequence-based applications like dependency parsing (Chen & Manning, 2014). These works rely on the sequential structure of the Table 3. Recall@5 of PredCls for the 20-top relations ranked by their frequency, as in (Xu et al., 2017) R ELATION ON HAS IN OF WEARING NEAR WITH ABOVE HOLDING BEHIND UNDER SITTING ON IN FRONT OF ATTACHED TO AT HANGING FROM OVER FOR RIDING L U ET AL . (2016) X U ET AL . (2017) L INGUIS - 99.71 98.03 80.38 82.47 98.47 85.16 31.85 49.19 61.50 79.35 28.64 31.74 26.09 8.45 54.08 0.0 9.26 12.20 72.43 99.25 97.25 88.30 96.75 98.23 96.81 88.10 79.73 80.67 92.32 52.73 50.17 59.63 29.58 70.41 0.0 0.0 31.71 89.72 99.4 98.9 96.2 98.2 99.5 95.0 94.2 83.7 95.6 90.6 83.2 90.5 74.7 77.2 81.1 74.7 54.9 43.4 96.2 TIC input, where BiLSTMs can be effectively applied. The concept of architectural invariance was recently proposed in D EEP S ETS (Zaheer et al., 2017). The invariance we consider is much less restrictive (i.e., we do not need to be invariant to all permutations of singleton and pairwise features, just those consistent with a graph re-labeling), and hence results in a substantially different set of architectures. Extracting scene graphs from images provides a semantic representation that can later be used for reasoning, question answering, and image retrieval (Johnson et al., 2015; Lu et al., 2016; Raposo et al., 2017). It is at the forefront of machine vision research, integrating challenges like object detection, action recognition and detection of human-object interactions (Liao et al., 2016; Plummer et al., 2017). Mapping Images to Scene Graphs with Permutation-Invariant Structured Prediction 7. Conclusion We presented a deep learning approach to structured prediction, which constrains the architecture to be invariant to structurally identical inputs. 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Improved Space-efficient Linear Time Algorithms for Some Classical Graph Problems Sankardeep Chakraborty1, Seungbum Jo2 , and Srinivasa Rao Satti3 arXiv:1712.03349v1 [cs.DS] 9 Dec 2017 1 The Institute of Mathematical Sciences, HBNI, Chennai, India. [email protected] 2 University of Siegen, Siegen, Germany. [email protected] 3 Seoul National University, Seoul, South Korea. [email protected] We provide space-efficient linear time algorithms for computing bridges, topological sorting, and strongly connected components improving on several recent results of Elmasry et al. [STACS’15], Banerjee et al. [COCOON’16] and Chakraborty et al. [ISAAC’16]. En route, we also provide another DFS implementation with weaker input graph representation assumption without compromising on the time and space bounds of the earlier results of Banerjee et al. [COCOON’16] and Kammer et al. [MFCS’16]. 1 Introduction Since the early days of designing graph algorithms, researchers have developed several approaches for testing whether a given undirected (or directed) graph G = (V, E) with n vertices and m edges is (strongly connected) biconnected and/or 2-edge connected, and finding cut vertices and/or bridges of G. All of these methods use depth-first search (DFS) as the backbone to design the main algorithm. The classical linear time algorithms due to Tarjan [11, 12] computes the so-called “low-point” values (which are defined in terms of a DFS-tree of G) for every vertex v, and checks some conditions using that to determine whether G has the desired property. There are other linear time algorithms as well for these problems (see [10] and all the references therein). All of these classical algorithms take O(m + n) time and O(n) words (our model of computation is the standard word RAM model with word size w = Ω(lg n) bits) of space. Our aim is to improve the space bounds of these algorithms without increasing the running time. 1.1 Motivation and Related Work Motivated mainly by the “big data” phenomenon among others, recently there has been a surge of interest in improving the space complexity of the fundamental linear time graph algorithms by paying little or no penalty in the running time i.e., reducing the working space of the classical graph algorithms (which generally take O(n lg n) bits) to o(n lg n) bits without compromising on time. Towards this, Elmasry et al. [7] gave, among others, an implementation for DFS taking O(m + n) time and O(n lg lg n) bits of space. For sparse graphs (when m = O(n)), the Time Space (in bits) DFS O(n + m) O(n + m) O(n + m) O(n + m) O(n lg n) O(n + m) O(n lg(m/n)) O(n lg lg n) [5] [1, 9] [3] [7] Testing biconnectivity & reporting cut vertices [11] [1] [3] [9] Testing 2-edge connectivity & reporting bridges [12] [1] [3] This paper Topological sort [5] This paper This paper [7] Testing strong connectivity [11] This paper This paper [7] Table 1: Summary of our results. space bound was improved further to O(n) bits keeping the same linear time in [1]. Banerjee et al. [1] gave, among others, a space efficient implementation for performing BFS using just 2n + o(n) bits of space and linear time, improving upon the result of [7]. Such algorithms for a few other graph problems also have been considered recently [2, 3, 4, 6, 9]. 1.2 Our Results We assume that the input graph G, which is represented using adjacency array [1, 3, 7, 9], i.e., G is represented by an array of length \|V \| where the i-th entry stores a pointer to an array that stores all the neighbors of the i-th vertex, is given in a read-only memory with a limited read-write working memory, and write-only output. We count space in terms of the number of bits in workspace used by the algorithms. Our main goal here is to improve the space bounds of some of the classical and fundamental graph algorithms. We summarize all our main results in Table 1. In this paper, basically we complete the full spectrum of results regarding the space bounds for these problems keeping the running time linear by providing/improving the missing/existing algorithms in the recent space efficient graph algorithm literature. Due to lack of space, we provide only sketches of our proofs. 2 Testing 2-Edge Connectivity and Finding Bridges In an undirected graph G, a bridge is an edge that when removed (without removing the vertices) from a graph creates more components than previously in the graph. A (connected) graph with at least two vertices is 2-edge-connected if and only if it has no bridge. Let T denote the DFS tree of G. Following Kammer et al. [9], we call a tree edge (u, v) of T with u being the parent of v full marked if there is a back edge from a descendant of v to a strict ancestor of u, half marked if it is not full marked and there exists a back edge from a descendant of v to u, and unmarked, otherwise. They use this definition to prove the following: (i) every vertex u (except the root r) is a cut vertex exactly if at least one of the edges from u to one of its children is either an unmarked edge or a half marked edge, and (ii) root r is a cut vertex exactly if it has at least two children in T . Based on the above characterization, they gave O(m + n) time and O(n lg lg n) bits algorithm to test/report if G has any cut vertex. Our main observation is that we can give a similar characterization for bridges in G, and essentially using a similar implementation, we can also obtain O(m + n) time and O(n lg lg n) bits algorithms for testing 2-edge connectivity and reporting bridges of G. We start with the following lemma. Lemma 1. A tree edge e = (u, v) in T is a bridge of G if and only if it is unmarked. Proof sketch: If e is unmarked, then no descendants of v reaches u or any strict ancestor of u, so deleting e would result in disconnected graph, thus e has to be a bridge. On the other direction, it is easy to see that if e is a bridge, it has to be an unmarked edge. Now we state our theorem below. Theorem 2. Given an undirected graph G, in O(m + n) time and O(n lg lg n) bits of space we can determine whether G is 2-edge connected. If G is not 2-edge connected, then in the same amount of time and space, we can compute and output all the bridges of G. Proof sketch: Using Lemma 1 and the similar implementation of using stack compression and other tools of the algorithm provided in Section 3.2 of Kammer et al. [9] with few modifications, we can prove the theorem. Note that the space bound of Theorem 2 improves the results of [1] and [3] for sufficiently dense graphs (when m = ω(n lg lg n) and m = ω(n lgO(1) n) respectively) while keeping the same linear runtime (see Table 1). 3 DFS without Cross Pointers Banerjee et al. [1] and subsequently Kammer et al. [9] gave O(m + n) bits and O(m + n) time implementations of DFS improving on the bounds of [7] for sparse graphs. But both of these DFS implementations assume that the input graph is represented using the adjacency array along with cross pointers i.e., for undirected graphs, every neighbour v in the adjacency array of a vertex u stores a pointer to the position of vertex u in the adjacency array of v. See [7] for detailed definitions for directed graphs. We emphasize that this input assumption can double the space usage, compared to the raw adjacency array in worst case. In what follows, we provide the proof sketch of a DFS implementation taking the same time and space bounds as that of [1, 9] but without using the cross pointers. Our main theorem is as follows. Theorem 3. Given a directed or undirected graph G, represented as adjacency array, we can perform DFS traversal of G using O(m + n) bits and O(m + n) time. Proof sketch: We essentially modify the proof of [1] which uses a bitvector A of length O(m + n) having one to one mapping with the unary encoding of the degree sequence to mark the tree edges, and subsequently uses cross pointers to find the parent of any vertex during backtracking as well as starting with next unvisited vertex after backtracking. We note that we can represent the parents of all the vertices in another bitvector P of length O(m + n) (parallel to A). Now to perform backtracking efficiently, we could use the constant time append only structure (also with constant time rank/select) of Grossi et al. [8] along with the P array. With these modifications, we could get rid of cross pointers without compromising on the running time and space bound of the earlier algorithms. 4 Testing Strong Connectivity and Topological Sorting Towards giving improved space efficient algorithms for strong connectivity (SC) and topological sorting (TS), we first improve Lemma 4.1 of [7] which says the following: if DFS of a directed graph G takes T (n, m) time and S(n, m) space, then we can output the vertices of G in reverse postorder of the DFS tree T of G taking O(T (n, m)) time and O(S(n, m) + n lg lg n) space. Combining this lemma with the classical algorithms for SC and TS [5] they obtained O(n lg lg n) bits and O(m + n) time algorithms for both these problems. We improve these by showing the following, Theorem 4. If DFS of a directed graph G takes T (n, m) time and S(n, m) space, then the vertices of G can be output in reverse postorder with respect to a DFS forest of G taking O(T (n, m)) time and O(S(n, m) + m + n) space. As a result, we can also solve SC and TS in O(m + n) time using O(n + m) bits of space. Proof sketch: We use the DFS algorithm of Theorem 3 to first mark all the tree edges in the array A. Now we start with the rightmost leaf vertex of the DFS tree and use rank/select operations [8] on A and P (as defined in the proof of Theorem 3) carefully to traverse the tree in reverse direction (along with standard DFS backtracking etc) to generate reverse postorder sequence. Now using this as the back bone of the classical algorithms, we obtain O(m + n) bit and O(n + m) time algorithms for SC and TS. Theorem 4 improves the result of [7] for sparse (when m = O(n)) graphs. Now if we use the DFS algorithm of Chakraborty et al. [3] and modify it suitably to perform the traversal of the DFS tree in reverse, we obtain the following result. Theorem 5. If DFS of a directed graph G takes T (n, m) time and S(n, m) space, then the vertices of G can be output in reverse postorder with respect to a DFS forest of G taking O(T (n, m)) time and O(S(n, m) + n lg(m/n)) space. As a result, we can also solve SC and TS using O(m + n) time and O(n lg(m/n)) bits. References [1] N. Banerjee, S. Chakraborty, and V. Raman. Improved space efficient algorithms for BFS, DFS and applications. In 22nd COCOON, volume 9797, pages 119–130. Springer, LNCS, 2016. [2] N. Banerjee, S. Chakraborty, V. Raman, S. Roy, and S. Saurabh. Time-space tradeoffs for dynamic programming in trees and bounded treewidth graphs. In 21st COCOON, volume 9198, pages 349–360. springer, LNCS, 2015. [3] S. Chakraborty, V. Raman, and S. R. Satti. Biconnectivity, chain decomposition and stnumbering using O(n) bits. In 27th ISAAC, volume 64 of LIPIcs, pages 22:1–22:13. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. [4] S. Chakraborty and S. R. Satti. Space-efficient algorithms for Maximum Cardinality Search, Stack BFS, Queue BFS and applications. In 23rd COCOON 2017, Hong Kong, China, August 3-5, 2017, pages 87–98, 2017. [5] T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein. Introduction to Algorithms. [6] S. Chakraborty, V. Raman, and S. R. Satti. Biconnectivity, st-numbering and other applications of DFS using O(n) bits. J. Comput. Syst. Sci., 90:63–79, 2017. [7] A. Elmasry, T. Hagerup, and F. Kammer. Space-efficient basic graph algorithms. In 32nd STACS, pages 288–301, 2015. [8] R. Grossi and G. Ottaviano. The wavelet trie: maintaining an indexed sequence of strings in compressed space. In 31st PODS, pages 203–214, 2012. [9] F. Kammer, D. Kratsch, and M. Laudahn. Space-efficient biconnected components and recognition of outerplanar graphs. In 41st MFCS, 2016. [10] J. M. Schmidt. A simple test on 2-vertex- and 2-edge-connectivity. Inf. Process. Lett., 113(7):241–244, 2013. [11] R. E. Tarjan. Depth-first search and linear graph algorithms. SICOMP, 1(2):146–160, 1972. [12] R. E. Tarjan. A note on finding the bridges of a graph. Inf. Pro. Lett., 2(6):160–161, 1974.	8
arXiv:1704.01458v1 [math.ST] 5 Apr 2017 β-mixing and moments properties of a non-stationary copula-based Markov process Fabio Gobbi Sabrina Mulinacci University of Bologna - Department of Statistics April 6, 2017 Abstract This paper provides conditions under which a non-stationary copula-based Markov process is β-mixing. We introduce, as a particular case, a convolution-based gaussian Markov process which generalizes the standard random walk allowing the increments to be dependent. JEL classification: C22,C10 Mathematics Subject Classification (2010): 62M10, 62H20 Keywords: Markov process, copula, β-mixing, gaussian process. 1 Introduction In this paper we analyze the temporal dependence properties satisfied by a discrete times nonstationary Markov process. Temporal dependence is relevant since it permits to verify of how well theoretical models explain temporal persistency observed in financial data. Moreover, it is also a useful tool to establish large sample properties of estimators for dynamic models. In particular, in this paper we analyze the β-mixing property and we give sufficient conditions that ensure this property be satisfied. In the copula approach to univariate time series modelling, the finite dimensional distributions are generate by copulas. Darsow et al. (1992) provide necessary and sufficient conditions for a copula-based time series to be a Markov process. Recent literature on this topic has mainly focused on the stationary case. Chen and Fan (2006) introduce a copula-based strictly stationary first order Markov process generated from (G0 (·), C(·, ·, α0 )) where G0 (·) is the invariant distribution of Yt and C(·, ·, α0 ) is the parametric copula for (Yt−1 , Yt ). The authors show that the β-mixing temporal dependence measure is purely determined by the properties of copulas and present sufficient conditions to ensure that the process (Yt )t based on gaussian and EFGM copulas are geometric β-mixing. Beare (2010) shows that all Markov models generated via symmetric copulas with positive and square integrable densities are geometric β-mixing. Many commonly used bivariate copulas without tail dependence such as gaussian, EFGM and Frank copulas satisfy this condition. Chen et al. (2009) show that Clayton, Gumbel and Student’s t copula based Markov models are geometrically ergodic which is a stronger condition than the geometric β-mixing. In this paper we focus on Markov processes where some dependence between each state variable and increment is allowed and modeled through a time-invariant copula. In particular, we introduce a gaussian Markov process, which is non-stationary and generalizes the classical gaussian random walk, and we study related moments properties and provide conditions under wich the process is β-mixing. 1 The paper is organized as follows. Section 2 presents a general result on the β-mixing properties satisfied by non-stationary Markov processes. Section 3 restricts the study to the gaussian case. Section 4 concludes. 2 Copula-based Markov processes and β-mixing properties Throughout the paper Y = (Yt )t∈Z is a discrete time Markov process. Thanks to the seminal paper of Darsow et al. (1992), the markovianity of a stochastic process can be characterized through a specific requirement that the copulas, representing the dependence structure of the finite dimensional distributions induced by the stochastic process (for a detailed discussion on copulas see Nelsen (2006), Joe (1997), Cherubini et al. (2012) and Durante and Sempi (2015)), must satisfy. In particular, in Darsow et al (1992) it is proved that the Chapman-Kolmogorov equations for transition probabilities are equivalent to the requirement that, if Ci,j is the copula associated to the vector (Yi , Yj ), then Cs,t (u, v) = Cs,r ∗ Cr,t (u, v) = Z 1 0 ∂ ∂ Cs,r (u, w) Cr,t (w, v) dw, ∀s < r < t. ∂w ∂w As a consequence, since Y is a discrete times Markov process, if we assume that the set of bivariate copulas Ct,t+1 (representing the dependence structure of the stochastic process at two adjacent times) is given for t ∈ Z and k > 0, then necessarily (we remind that the ∗-operator is associative) Ct,t+k (u, v) = Ct,t+k−1 ∗ Ct+k−1,t+k (u, v) = Ct,t+1 ∗ Ct+1,t+2 ∗ · · · ∗ Ct+k−1,t+k (u, v). (1) Notice that, in the stationary case considered in Beare (2010), Ct,t+1 = C for all t ∈ Z, therefore all bivariate copulas Ct,t+k are functions of the copula C and of the lag k and not of the time t. In this paper we extend the study to the more general non-stationary case. In particular we analyze the temporal dependence problem with a special attention to mixing properties. The notion of β-mixing was introduced by Volkonskii and Rozanov (1959 and 1961) and was attribute there to Kolmogorov. Given a (not necessarily stationary) sequence of random variables Y = (Yt )t∈Z , let Ftl be the σ-field Ftl = σ(Yt , t ≤ t ≤ l) with −∞ ≤ t ≤ l ≤ +∞ and let J I +∞ t β̃(F−∞ , Ft+k )= 1 XX \|P(Ai ∩ Bj ) − P(Ai )P(Bj )\|, {Ai },{Bj } 2 i=1 j=1 sup (2) where the second supremum is taken over all finite partitions {A1 , ...AI } and {B1 , ...BJ } of Ω such t ∞ that Ai ∈ F−∞ for each i and Bj ∈ Ft+k for each j. Define the following dependence coefficient +∞ t βk = sup β̃(F−∞ , Ft+k ). t∈Z We say that the sequence (Yt )t∈Z is β−mixing (or absolutely regular) if βk → 0 as k → +∞. In next Theorem we give conditions on the set of copulas Ct,t+1 , t ∈ Z in order to guarantee that the Markov resulting process is β-mixing. These conditions are based on specific requirements on the maximal correlation coefficients of the copulas Ct,t+1 . We remind that the maximal correlation η of a copula C is given by Z Z 1 1 sup f,g f (x)g(x)C(dx, dy) 0 0 R1 R1 R1 R1 where f, g ∈ L2 ([0, 1]), 0 f (x)dx = 0 g(x)dx = 0 and 0 f 2 (x)dx = 0 g 2 (x)dx = 1 and we refer to Beare (2010) and Rényi (1959) for more details. 2 Theorem 2.1. Let Y = (Yt )t∈Z be a Markov process. Let Ct,t+1 be the copula associated to the vector (Yt , Yt+1 ) for t ∈ Z that we assume to be absolutely continuous, with symmetric and square-integrable density ct,t+1 so that (ct,t+1 )t∈Z is uniformly bounded in L2 ([0, 1]). If the maximal correlation coefficients ηt of Ct,t+1 satisfy η̂ = sup ηt < 1, (3) t∈Z then Y is β-mixing. Proof. The proof follows that of Theorem 3.1 in Beare (2010) who proves a similar result for stationary copula-based Markov processes. First of all, since the stochastic process is Markovian, (2) can be rewritten in terms of the cumulative distribution functions of (Yt , Yt+k ), Yt and Yt+k (Ft,t+k , Ft and Ft+k , respectively) and the total variation norm k · kT V (see Bradley, 2007) and then, applying Sklar’s theorem, we can write 1 k Ft,t+k (x, y) − Ft (x)Ft+k (y) kT V = 2 1 = k Ct,t+k (Ft (x), Ft+k (y)) − Ft (x)Ft+k (y) kT V ≤ 2 1 ≤ k Ct,t+k (u, v) − uv) kT V . 2 +∞ t β̃(F−∞ , Ft,t+k )= From (1) it follows that all bivariate copulas of type Ct,t+k for t ∈ Z and k ≥ 1 are absolutely continuous: let us denote their density as ct,t+k . Then +∞ t β̃(F−∞ , Ft+k )≤ 1 1 k ct,t+k (u, v) − 1 kL1 ≤ k ct,t+k (u, v) − 1 kL2 2 2 and 1 sup k ct,t+k (u, v) − 1 kL2 . 2 t∈Z βk ≤ Since ct,t+1 is a symmetric square-integrable joint density with uniform margins, it admits the following series expansion in terms of a complete orthonormal sequence (φi )i≥1 in L2 [0, 1], +∞ X ct,t+1 (u, v) = 1 + λi,t φi (u)φi (v), i=1 where the eigenvalues (λi,t )i form a square-summable sequence of nonnegative real numbers: notice that, as proved in Lancaster(1958) max λi,t = ηt . (4) i≥1 Applying (1), we get ct,t+k (u, v) = 1 + +∞ X i=1 Then, using (4) and (3), we get k ct,t+k (u, v) − 1 kL2 = +∞ X i=1  k−1 Y  j=0  k−1 Y  j=0  λi,t+j  φi (u)φi (v).  λi,t+j  φi (u)φi (v) L2 1/2   k−1 +∞ Y X  λ2i,t+j  = = i=1 j=0   1/2   1/2 +∞ k−1 +∞ k−1 X Y X Y 2  = λ2i,t  λ2i,t+j  ≤ λ2i,t  ηt+j ≤ i=1 ≤ η̂ k−1 j=1 "+∞ X i=1 λ2i,t #1/2 i=1 j=1 = η̂ k−1 k ct,t+1 (u, v) − 1 kL2 . 3 (5) Therefore βk ≤ 1 k−1 η̂ sup k ct,t+1 (u, v) − 1 kL2 2 t∈Z which, since (ct,t+1 )t∈Z is uniformly bounded in L2 ([0, 1]), tends to zero as k → +∞. 3 A gaussian convolution-based Markov process From now on, we assume that the Markov process Y is obtained through Yt = Yt−1 + ξt , Y0 = 0, (6) where (ξt )t≥1 is a sequence of identically distributed random variables such that ξt is dependent on Yt−1 for each t. The dependence structure is modelled by a time-invariant copula function C. The process defined in (6) is not stationary. However, we can determine the distribution of C Yt for each t thanks to the C-convolution operator (denoted by ∗), introduced in Cherubini et al. (2011) as a tool to recover the distribution of the sum of two dependent random variables. As shown in Cherubini et al. (2011, 2012 and 2015) the C-convolution technique may be used in the construction of dependent increments stochastic processes like (6). More precisely, if Ft−1 is the cumulative distribution function of Yt−1 and Ht that of ξt , we may recover the cumulative distribution function of Yt iterating the C-convolution for all t Z 1 C −1 Ft (yt ) = (Ft−1 ∗ Ht )(yt ) = D1 C(w, Ht (yt − Ft−1 (w)))dw, t ≥ 2 (7) 0 while, the copula associated to (Yt−1 , Yt ) is Z u −1 D1 C(w, Ht (Ft−1 (v) − Ft−1 Ct−1,t (u, v) = (w)))dw, t≥2 (8) 0 ∂ C(u, v). where D1 C(u, v) = ∂u Equations (7) and (8) provide the ingredients to construct discrete times Markov processes according to Darsow et al. (1992). Our model (6) is a sort of a modified version of a random walk process where the independence assumption for the innovations (ξt )t≥1 is no longer required: however, its weakness is that in most cases the distribution function cannot be expressed in closed form and it may be evaluated only numerically. From now on we assume that innovations (ξt )t≥1 are gaussian identically distributed with zero mean and standard deviation σξ and that the copula between Yt−1 and ξt is a (stationary) gaussian copula with constant parameter ρ for all t. This way, the distribution of Yt is gaussian for all t and, more specifically, in section 4.3.1 of Cherubini et al. (2016) it is shown that Yt ∼ N (0, Vt2 ), where Vt2 = V ar(Yt ) = V12 + (t − 1)σξ2 + 2ρσξ (9) t−1 X Vt−i , t ≥ 2, (10) i=1 where V12 = σξ2 since by assumption Y1 = ξ1 . Moreover, the copula between Yt and Yt+1 is gaussian with parameters Vt + ρσξ τt,t+1 = , t≥2 Vt+1 since E[Yt Yt+1 ] = Vt2 + ρVt σξ . 4 The limiting behavior of the standard deviation Vt has also been analyzed in Cherubini et al. (2016) where it is proved that σε − 2ρ , if ρ ∈ (−1, 0) lim Vt = t→+∞ +∞, otherwise. Notice that only in case of negative correlation with the increments, the standard deviation of the levels does not explode: in the following we will restrict the analysis to the case ρ ∈ (−1, 0). 3.1 Moments and autocorrelation function In this subsection we study the behavior of moments and autocorrelation functions of the process (Yt )t≥1 when t → +∞. It is just the case to recall that in the standard random walk model the k-th order autocorrelation function of (Yt )t≥1 tends to 1 as t → +∞, for each lag k. In our more general setting, this is no longer true. The limit of the k-th order autocorrelation function of (Yt )t≥1 is a function of k and ρ as the following proposition shows. Proposition 3.1. Let ρ ∈ (−1, 0). The k-th order autocorrelation function of (Yt )t≥1 tends to (1 − 2ρ2 )k for any k ≥ 1 as t → +∞. Proof. As proved in section 4.3.1 in Cherubini et al. (2016), using the fact that the ∗-product of two gaussian copulas has a parameter given by the product of the parameters of the copulas involved in the ∗-product, we have that the copula between Yt and Yt+k is gaussian with parameter τt,t+k = k−1 Y Vt+s + ρσξ . Vt+s+1 s=0 Therefore, since as t → +∞ and for any s ≥ 1 σ − 2ρξ + ρσξ2 Vt+s + ρσξ → = 1 − 2ρ2 , σ Vt+s+1 − 2ρξ (11) we easily get the result. On the other hand, the innovations (ξt )t≥1 are no longer serially independent as in the random walk case and the k-th order autocorrelation function approaches to a limit which again depends on ρ and k. Proposition 3.2. Let ρ ∈ (−1, 0). The k-th order autocorrelation function of (ξt )t≥1 tends to −ρ2 (1 − 2ρ2 )k−1 for any k ≥ 1 as t → +∞. Proof. We compute first the autocovariance of order k, with k ≥ 1, E[ξt ξt+k ]. We have E[ξt ξt+k ] = E[(Yt − Yt−1 )(Yt+k − Yt+k−1 )] = = E[Yt Yt+k ] − E[Yt Yt+k−1 ] − E[Yt−1 Yt+k ] + E[Yt−1 Yt+k−1 ] = = τt,t+k Vt Vt+k − τt,t+k−1 Vt Vt+k−1 − τt−1,t+k Vt−1 Vt+k + τt−1,t+k−1 Vt−1 Vt+k−1 . σ Since for any fixed k ≥ 1, τt,t+k → (1 − 2ρ2 )k and Vt → − 2ρξ as t → +∞ we get E[ξt ξt+k ] → σξ2 4ρ2 (1 − 2ρ2 )k − (1 − 2ρ2 )k−1 − (1 − 2ρ2 )k+1 + (1 − 2ρ2 )k = = −ρ2 σξ2 (1 − 2ρ2 )k−1 , as t → +∞. Moreover, it is immediate to find the statement of the proposition since as t → +∞ corr(ξt ξt+k ) → −ρ2 (1 − 2ρ2 )k−1 . 5 3.2 β-mixing properties In our gaussian framework ct,t+1 is the density of a gaussian copula for which it is well known that the maximal correlation coefficient is equal to the absolute value of the simple correlation coefficient (see Lancaster, 1957). Therefore, according to the notation of Theorem 2.1, for each t, ηt = \|τt,t+1 \|. The following results, which is an application of Theorem 2.1, holds Corollary 3.1. The Markov process defined by (6) with ξt ∼ N (0, σξ ) and ρ ∈ (−1, 0) is β-mixing. Proof. Firstly notice that \|τt,t+1 \| < 1, ∀t. 2 2 In fact this is equivalent to (Vt + ρσξ ) < Vt+1 which is always verified since ρ2 < 1 by assumption. 2 Thanks to (11), since \|1 − 2ρ \| < 1, we have that \|τt,t+1 \| = ηt is bounded by a constant smaller than 1, that is (3) is satisfied. Furthermore, it is not hard to prove that for any t k ct,t+1 (u, v) − 1 kL2 = 2 τt,t+1 η̂ 2 ≤ . 2 1 − τt,t+1 1 − η̂ 2 Thus Theorem 2.1 applies. 4 Concluding remarks In this paper we provide conditions under which a non-stationary copula-based Markov process is βmixing. Our results represent a generalization of those in Beare (2010), where the author considers the stationary case. Our analysis is focused on the particular case of a gaussian Markov process with dependent increments that represents a generalization of the standard gaussian random walk. In this particular non-stationary setting it is proved that the k-th order autocorrelation function of the process does not converge to 1, as in the random walk case, but to a quantity that depends on the lag and the correlation between the state variable and the innovation, which is assumed to be time-invariant. Additionally, it is proved that the process satisfies the conditions required to be β-mixing. References [1] Beare B. (2010): ”Copulas and temporal dependence”, Econometrica, 78(1), 395-410. [2] Bradley R.C. (2007): Introduction to strong mixing conditions, vols. 1-3. Kendrick Press. herber City. [3] Chen X., Fan Y. (2006): ”Estimation of Copula-Based Semiparametric Time Series Models”, Journal of Econometrics, 130, 307335. [4] Chen X., Wu W.B., Yi Y. (2009): ”Efficient estimation of copula-based semiparametric Markov models”, The Annals of Statistics, 37(6B) 42144253. [5] Cherubini U., Gobbi F., Mulinacci S., (2016) ”Convolution Copula Econometrics”, SpringerBriefs in Statistics. [6] Cherubini U., Gobbi F., Mulinacci S., Romagnoli S. (2012): Dynamic Copula Methods in Finance, John Wiley & Sons. [7] Cherubini, U., Mulinacci S., Romagnoli S. (2011) ”A Copula-based Model of Speculative Price Dynamics in Discrete Time”, Journal of Multivariate Analysis, 102, 1047-1063. [8] Darsow W.F. - Nguyen B. - Olsen E.T.(1992): ”Copulas and Markov Processes”, Illinois Journal of Mathematics, 36, 600-642. 6 [9] Durante F., Sempi C. (2015) Principles of copula theory, Boca Raton: Chapman and Hall/CRC [10] Joe H. (1997): Multivariate Models and Dependence Concepts, Chapman & Hall, London [11] Lancaster,H.O. (1957) ”Some properties of the bivariate normal distribution considered in the form of a contingency table”, Biometrika, 44, 289-292. [12] Lancaster H.O. (1958): ”The structure of bivariate distributions”, Annals of Mathematical Statistics, 29, 719-736. [13] Nelsen R.(2006): An Introduction to Copulas, Springer [14] Renyi A. (1959): ”On measures of dependence”, Acta Mathematica Academiae Scientiarum Hungaricae, 10, 441-451. [15] Volkonskii V.A., Rozanov Yu.A. (1959): ”Some limit theorems for random functions I”, Theor. Probab. Appl., 4, 178-197. [16] Volkonskii V.A., Rozanov Yu.A. (1961): ”Some limit theorems for random functions II”, Theor. Probab. Appl., 6, 186-198. 7	10
Pragmatic-Pedagogic Value Alignment arXiv:1707.06354v2 [cs.AI] 5 Feb 2018 Jaime F. Fisac, Monica A. Gates, Jessica B. Hamrick, Chang Liu, Dylan Hadfield-Menell, Malayandi Palaniappan, Dhruv Malik, S. Shankar Sastry, Thomas L. Griffiths, and Anca D. Dragan Abstract As intelligent systems gain autonomy and capability, it becomes vital to ensure that their objectives match those of their human users; this is known as the value-alignment problem. In robotics, value alignment is key to the design of collaborative robots that can integrate into human workflows, successfully inferring and adapting to their users’ objectives as they go. We argue that a meaningful solution to value alignment must combine multi-agent decision theory with rich mathematical models of human cognition, enabling robots to tap into people’s natural collaborative capabilities. We present a solution to the cooperative inverse reinforcement learning (CIRL) dynamic game based on well-established cognitive models of decision making and theory of mind. The solution captures a key reciprocity relation: the human will not plan her actions in isolation, but rather reason pedagogically about how the robot might learn from them; the robot, in turn, can anticipate this and interpret the human’s actions pragmatically. To our knowledge, this work constitutes the first formal analysis of value alignment grounded in empirically validated cognitive models. Key words: Value Alignment, Human-Robot Interaction, Dynamic Game Theory 1 Introduction The accelerating progress in artificial intelligence (AI) and robotics is bound to have a substantial impact in society, simultaneously unlocking new potential in augmenting and transcending human capabilities while also posing significant challenges to safe and effective human-robot interaction. In the short term, integrating robotic systems into human-dominated environments will require them to assess the intentions All authors are with the University of California, Berkeley. e-mail: {jfisac,mgates,jhamrick,changliu,dhm,malayandi,dhruvmalik, shankar_sastry,tom_griffiths,anca}@berkeley.edu 1 2 J. F. Fisac, M. A. Gates, J. B. Hamrick, C. Liu, D. Hadfield-Menell, et al. and preferences of their users in order to assist them effectively, while avoiding failures due to poor coordination. In the long term, ensuring that advanced and highly autonomous AI systems will be beneficial to individuals and society will hinge on their ability to correctly assimilate human values and objectives [1]. We envision the short- and long-term challenges as being inherently coupled, and predict that improving the ability of robots to understand and coordinate with their human users will inform solutions to the general AI value-alignment problem. Successful value alignment requires moving from typical single-agent AI formulations to robots that account for a second agent—the human—who determines what the objective is. In other words, value alignment is fundamentally a multi-agent problem. Cooperative Inverse Reinforcement Learning (CIRL) formulates value alignment as a two-player game in which a human and a robot share a common reward function, but only the human has knowledge of this reward [2]. In practice, solving a CIRL game requires more than multi-agent decision theory: we are not dealing with any multi-agent system, but with a human-robot system. This poses a unique challenge in that humans do not behave like idealized rational agents [3]. However, humans do excel at social interaction and are extremely perceptive of the mental states of others [4, 5]. They will naturally project mental states such as beliefs and intentions onto their robotic collaborators, becoming invaluable allies in our robots’ quest for value alignment. In the coming decades, tackling the value-alignment problem will be crucial to building collaborative robots that know what their human users want. In this paper, we show that value alignment is possible not just in theory, but also in practice. We introduce a solution for CIRL based on a model of the human agent that is grounded in cognitive science findings regarding human decision making [6] and pedagogical reasoning [7]. Our solution leverages two closely related insights to facilitate value alignment. First, to the extent that improving their collaborator’s understanding of their goals may be conducive to success, people will tend to behave pedagogically, deliberately choosing their actions to be informative about these goals. Second, the robot should anticipate this pedagogical reasoning in interpreting the actions of its human users, akin to how a pragmatic listener interprets a speaker’s utterance in natural language. Jointly, pedagogical actions and pragmatic interpretations enable stronger and faster inferences among people [7]. Our result suggests that it is possible for robots to partake in this naturally-emerging equilibrium, ultimately becoming more perceptive and competent collaborators. 2 Solving Value Alignment using Cognitive Models 2.1 Cooperative Inverse Reinforcement Learning (CIRL) Cooperative Inverse Reinforcement Learning (CIRL) [2] formalizes value alignment as a two-player game, which we briefly present here. Consider two agents, a human Pragmatic-Pedagogic Value Alignment 3 H and a robot R, engaged in a dynamic collaborative task involving a (possibly infinite) sequence of steps. The goal of both agents is to achieve the best possible outcome according to some objective θ ∈ Θ . However, this objective is only known to H. In order to contribute to the objective, R will need to make inferences about θ from the actions of H (an Inverse Reinforcement Learning (IRL) problem), and H will have an incentive to behave informatively so that R becomes more helpful, hence the term cooperative IRL. Formally, a CIRL game is a dynamic (Markov) game of two players (H and R), described by a tuple hS, {AH , AR }, T, {Θ , r}, P0 , γi, where S is the set of possible states of the world; AH , AR are the sets of actions available to H and R respectively; T : S × S × AH × AR → [0, 1] a discrete transition measure1 over the next state, conditioned on the previous state and the actions of H and R: T (s0 \|s, aH , aR ); Θ is the set of possible objectives; r : S × AH × AR × Θ → R is a cumulative reward function assigning a real value to every tuple of state and actions for a given objective: r(s, aH , aR ; θ ); P0 : S × Θ → [0, 1] is a probability measure on the initial state and the objective; γ ∈ [0, 1] is a geometric time discount factor making future rewards gradually less valuable. 2.2 Pragmatic Robots for Pedagogic Humans Asymmetric information structures in games (even static ones) generally induce an infinite hierarchy of beliefs: our robot will need to maintain a Bayesian belief over the human’s objectives to decide on its actions. To reason about the robot’s decisions, the human would in principle need to maintain a belief on the robot’s belief, which will in turn inform her decisions, thereby requiring the robot to maintain a belief on the human’s belief about its own belief, and so on [8]. In [2], it was shown that an optimal pair of strategies can be found for any CIRL game by solving a partially observed Markov decision process (POMDP). This avoids this bottomless recursion as long as both agents are rational and can coordinate perfectly before the start of the game. Unfortunately, when dealing with human agents, rationality and prior coordination are nontrivial assumptions. Finding an equivalent tractability result for more realistic human models is therefore crucial in using the CIRL formulation to solve real-world value-alignment problems. We discover the key insight in cognitive studies of human pedagogical reasoning [7], in which a teacher chooses actions or utterances to influence the beliefs of a learner who is aware of the teacher’s intention. The teacher can then exploit the fact that the learner can interpret utterances pragmatically. Infinite recursion is averted by finding a fixed-point relation between the teacher’s best utterance and the learner’s best interpretation, exploiting a common modeling assumption in Bayesian theory of mind: the learner models the teacher as a noisily rational decision maker [9], who will be likelier to choose utterances 1 Note that the theoretical formulation is easily extended to arbitrary measurable sets; we limit our analysis to finite state and objective sets for computational tractability and clarity of exposition. 4 J. F. Fisac, M. A. Gates, J. B. Hamrick, C. Liu, D. Hadfield-Menell, et al. causing the learner to place a high posterior belief on the correct hypothesis, given the learner’s current belief. While in reality, the teacher cannot exactly compute the learner’s belief, the model supposes that she estimates it (from the learner’s previous responses to her utterances), then introduces noise in her decisions to capture estimation inaccuracies. This framework can predict complex behaviors observed in human teaching-learning interactions, in which pedagogical utterances and pragmatic interpretations permit efficient communication [7]. We adopt an analogous modeling framework to that in [7] for value alignment, with a critical difference: the ultimate objective of the human is not to explicitly improve the robot’s understanding of the true objective, but to optimize the team’s expected performance towards this objective. Pedagogic behavior thus emerges implicitly to the extent that a well-informed robot becomes a better collaborator. 2.3 Pragmatic-Pedagogic Equilibrium Solution to CIRL The robot does not have access to the true objective θ , but rather estimates a belief bR over θ . We assume that this belief on θ can be expressed parametrically (this is always true if Θ is a finite set), and define 4Θ to be the corresponding (finitedimensional) parameter space, denoting R’s belief by bR ∈ 4Θ . While in reality the human cannot directly observe bR , we assume, as in [7], that she can compute it or infer it from the robot’s behavior (and model estimation inaccuracies as noise in her policy). We can then let Q : S × 4Θ × AH × AR ×Θ → R represent the state-action value function of the CIRL game for a given objective θ , which we are seeking to compute: if θ ∈ Θ is the true objective known to H, then Q(s, bR , aH , aR ; θ ) represents the best performance the team can expect to achieve if H chooses aH and R chooses aR from state s, with R’s current belief being bR . In order to solve for Q, we seek to establish an appropriate dynamic programming relation for the game, given a well-defined information structure and a model of the human’s decision making. Since it is typically possible for people to predict a robot’s next action if they see its beginning [10], we assume that H can observe aR at each turn before committing to aH . A well-established model of human decision making in psychology and econometrics is the Luce choice rule, which models people’s decisions probabilistically, making high-utility choices more likely than those with lower utility [9]. In particular, we employ a common case of the Luce choice rule, the Boltzmann (or soft-max) noisy rationality model [6], in which the probability of a choice decays exponentially as its utility decreases in comparison to competing options. The relevant utility metric in our case is the sought Q (which captures H’s best expected outcome for each of her available actions aH ). Therefore the probability that H will choose action aH has the form πH (aH \|s, bR , aR ; θ ) ∝ exp β Q(s, bR , aH , aR ; θ ) , (1) Pragmatic-Pedagogic Value Alignment 5 where β > 0 is termed the rationality coefficient of H and quantifies the concentration of H’s choices around the optimum; as β → ∞, H becomes a perfect rational agent, while, as β → 0, H becomes indifferent to Q. The above expression can be interpreted by R as the likelihood of action aH given a particular θ . The evolution of R’s belief bR is then given (deterministically) by the Bayesian update b0R (θ \|s, bR , aR , aH ) ∝ πH (aH \|s, bR , aR ; θ )bR (θ ) , (2) Jointly, (1) and (2) define a fixed-point equation analogous to the one in [7], which states how R should pragmatically update bR based on a noisily rational pedagogic aH . This amounts to a deterministic transition function for R’s belief, b0R = fb (s, bR , aH , aR ). Crucially, however, the fixed-point relation derived here involves Q itself, which we have yet to compute. Unlike H, R is modeled as a rational agent; however, not knowing the true θ , the best R can do is to maximize2 the expectation of Q based on its current belief3 bR : πR∗ (s, bR ) := arg max aR ∑ Q(s, bR , aH , aR ; θ ) · πH (aH \|s, bR , aR ; θ )bR (θ ) . (3) aH ,θ Combining (2) with the state transition measure T (s0 \|s, aH , aR ), we can define the Bellman equation for H under the noisily rational policy πH for any given θ ∈ Θ : h i Q(s, bR , aH , aR ; θ ) = r(s, aH , aR ; θ ) + Es0 ,a0H γ · Q0 s0 , b0R , a0H , πR∗ (s0 , b0R ); θ , (4) where s0 ∼ T (s0 \|s, aH , aR ); b0R = fb (s, bR , aH , aR ); a0H ∼ πH (aH \|s0 , b0R , πR∗ (s0 , b0R ); θ ). Note that H’s next action a0H implicitly depends on R’s action at the next turn. Substituting (1-3) into (4), we obtain the sought dynamic programming relation for the CIRL problem under a noisily rational-pedagogic human and a pragmatic robot. The human is pedagogic because she takes actions according to (1), which takes into account how her actions will influence the robot’s belief about the objective. The robot is pragmatic because it assumes the human is actively aware of how her actions convey the objective, and interprets them accordingly. The resulting problem is similar to a POMDP (in this case formulated in beliefstate MDP form), with the important difference that the belief transition depends on the value function itself. In spite of this complication, the problem can be solved in backward time through dynamic programming: each Bellman update will be based on a pragmatic-pedagogic fixed point that encodes an equilibrium between the Q function (and therefore H’s policy for choosing her action) and the belief transition (that is, R’s rule for interpreting H’s actions). Evidence in [7] suggests that people are proficient at finding such equilibria, even though uniqueness is not guaranteed in general; study of disambiguation is an open research direction. 2 We assume for simplicity that the optimum is unique or a well-defined disambiguation rule exists. Note that this does not imply certainty equivalence, nor do we assume separation of estimation and control: R is fully reasoning about how its actions and those of H may affect its future beliefs. 3 6 J. F. Fisac, M. A. Gates, J. B. Hamrick, C. Liu, D. Hadfield-Menell, et al. 3 A Proof-of-Concept We introduce the benchmark domain ChefWorld, a household collaboration setting in which a human H seeks to prepare a meal with the help of an intelligent robotic manipulator R. There are multiple possible meals that H may want to prepare using the available ingredients, and R does not know beforehand which one she has chosen (we assume H cannot or will not tell R explicitly). The team obtains a reward only if H’s intended recipe is successfully cooked. If H is aware of R’s uncertainty, she should take actions that give R actionable information, particularly the information that she expects will allow R to be as helpful as possible as the task progresses. Our problem has 3 ingredients, each with 2 or 3 states: spinach (absent, chopped), tomatoes (absent, chopped, puréed), and bread (absent, sliced, toasted). Recipes correspond to (joint) target states for the food. Soup requires the tomatoes to be chopped then puréed, the bread to be sliced then toasted, and no spinach. Salad requires the spinach and tomatoes to be chopped, and the bread to be sliced then toasted. H and R can slice or chop any of the foods, while only R can purée tomatoes or toast bread. A simple scenario with the above two recipes is solved using discretized beliefstate value iteration and presented as an illustrative example in Fig 1. R has a wrong initial belief about H’s intended recipe. Under standard IRL, H fails to communicate her recipe. But if R is pragmatic and H is pedagogic, H is able to change R’s belief and they successfully collaborate to make the meal. In addition, we computed the solution to games with 4 recipes through a modification of POMDP value iteration (Table 1). In the pragmatic-pedagogic CIRL equilibrium with β = 5, H and R successfully cook the correct recipe 97% of the time, whereas under the standard IRL framework (with H acting as an expert disregarding R’s inferences) they only succeed 46% of the time—less than half as often. Fig. 1 Simple collaborative scenario with 2 possible objectives. The human H wants soup but the robot R initially believes her goal is salad. Even under a full POMDP formulation, if R reasons “literally” about H’s actions using standard IRL (assuming H behaves as if R knew the true objective), it fails to infer the correct objective. Conversely, under the pragmatic-pedagogic CIRL equilibrium, R views H as incentivized to choose pedagogic actions that will fix R’s belief when needed. Under the pragmatic interpretation, H’s wait action in turn 2 (instead of adding spinach, which would be preferred by a pedagogic H wanting salad) indicates H wants soup. While H’s actions are the same under both solutions, only the pragmatic R achieves value alignment and completes the recipe. Pragmatic-Pedagogic Value Alignment IRL CIRL Boltzmann (β = 1) Boltzmann (β = 2.5) Boltzmann (β = 5) 0.2351 0.3783 0.4555 0.2916 0.7026 0.9727 7 Rational 0.7083 1.0000 Table 1 A comparison of the expected value (or equivalently here, the probability of success) achieved by CIRL and IRL on the ChefWorld domain with four recipes when the robot begins with a uniform belief over the set of recipes. We ran each algorithm across different models of the human’s behavior, namely a rational model and a Boltzmann-rational model with various values of β (a higher β corresponds to a more rational human). When the human is highly irrational (β = 1), both CIRL and IRL unsurprisingly perform rather poorly. However, as the human becomes less noisy (β = 2.5, β = 5), CIRL outperforms IRL by a significant margin; in fact, the pragmaticpedagogic CIRL strategy with a Boltzmann-rational human performs comparably (β = 2.5) or even substantially outperforms (β = 5) the IRL result when the human is perfectly rational. 4 Discussion We have presented here an analysis of the AI value alignment problem that incorporates a well-established model of human decision making and theory of mind into the game-theoretic framework of cooperative inverse reinforcement learning (CIRL). Using this analysis, we derive a Bellman backup that allows solving the dynamic game through dynamic programming. At every instant, the backup rule is based on a pragmatic-pedagogic equilibrium between the robot and the human: the robot is uncertain about the objective and therefore incentivized to learn it from the human, whereas the human has an incentive to help the robot infer the objective so that it can become more helpful. We note that this type of pragmatic-pedagogic equilibrium, recently studied in the cognitive science literature for human teaching and learning [7], may not be unique in general: there may exist two actions for H and two corresponding interpretations for R leading to different fixed points. For example, H could press a blue or a red button which R could then interpret as asking it to pick up a blue or a red object. Although we might feel that blue-blue/red-red is a more intuitive pairing, blue-red/red-blue is valid as well: that is, if H thinks that R will interpret pressing the blue button as asking for the red object then she will certainly be incentivized to press blue when she wants red; and in this case R’s policy should consistently be to pick up the red object upon H’s press of the blue button. When multiple conventions are possible, human beings tend to naturally disambiguate between them, converging on salient equilibria or “focal points” [11]. Accounting for this phenomenon is likely to be instrumental for developing competent human-centered robots. On the other hand, it is important to point out that, although they are computationally simpler than more general multi-agent planning problems, POMDPs are still PSPACE-complete [12], so reducing pragmatic-pedagogic equilibrium computation to solving a modified POMDP falls short of rendering the problem tractable in general. However, finding a POMDP-like Bellman backup does open the door to efficient CIRL solution methods that leverage and benefit from the extensive research on practical algorithms for approximate planning in large POMDPs [13]. 8 REFERENCES We find the results in this work promising for two reasons. First, they provide insight into how CIRL games can be not only theoretically formulated but also practically solved. Second, they demonstrate, for the first time, formal solutions to value alignment that depart from the ideal assumption of a rational human agent and instead benefit from modern studies of human cognition. We predict that developing efficient solution approaches and incorporating more realistic human models will constitute important and fruitful research directions for value alignment. Acknowledgements This work is supported by ONR under the Embedded Humans MURI (N00014-13-1-0341), by AFOSR under Implicit Communication (16RT0676), and by the Center for Human-Compatible AI. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] D. Amodei, J. Steinhardt, D. Man, and P. Christiano. “Concrete Problems in AI Safety”. arXiv preprint (2017). D. Hadfield-Menell, A. Dragan, P. Abbeel, and S. Russell. “Cooperative Inverse Reinforcement Learning”. NIPS (2016). A. Tversky and D. Kahneman. “Judgment under Uncertainty: Heuristics and Biases”. Science 185.4157 (1974). F. Heider and M. Simmel. “An Experimental Study of Apparent Behavior”. The American Journal of Psychology 57.2 (1944). A. N. Meltzoff. “Understanding the intentions of others: Re-enactment of intended acts by 18-month-old children.” Dev. Psych. 31.5 (1995). C. L. Baker and J. B. Tenenbaum. “Modeling Human Plan Recognition Using Bayesian Theory of Mind”. Plan, Activity, and Intent Recognition. 2014. P. Shafto, N. D. Goodman, and T. L. Griffiths. “A rational account of pedagogical reasoning: Teaching by, and learning from, examples”. Cog. Psych. 71 (2014). S. Zamir. “Bayesian games: Games with incomplete information”. Computational Complexity: Theory, Techniques, and Applications (2012). R. D. Luce. Individual Choice Behavior: a Theoretical Analysis. John Wiley and Sons, 1959. A. D. Dragan and S. Srinivasa. “Integrating human observer inferences into robot motion planning”. Autonomous Robots (2014). T. C. Schelling. The strategy of conflict. Harvard University Press, 1960. M. Mundhenk, J. Goldsmith, C. Lusena, and E. Allender. “Complexity of Finite-horizon Markov Decision Process Problems”. J. ACM 47.4 (2000). D. Silver and J. Veness. “Monte-Carlo Planning in Large POMDPs”. NIPS. 2010.	2
arXiv:1606.09481v1 [cs.DS] 30 Jun 2016 Generating massive complex networks with hyperbolic geometry faster in practice Moritz von Looz Mustafa Safa Özdayi Karlsruhe Institute of Technology (KIT), Germany Email: [email protected] Istanbul Technical University, Turkey Email: [email protected] Sören Laue Henning Meyerhenke Friedrich Schiller University Jena, Germany Email: [email protected] Karlsruhe Institute of Technology (KIT), Germany Email: [email protected] Abstract—Generative network models play an important role in algorithm development, scaling studies, network analysis, and realistic system benchmarks for graph data sets. The commonly used graph-based benchmark model R-MAT has some drawbacks concerning realism and the scaling behavior of network properties. A complex network model gaining considerable popularity builds random hyperbolic graphs, generated by distributing points within a disk in the hyperbolic plane and then adding edges between points whose hyperbolic distance is below a threshold. We present in this paper a fast generation algorithm for such graphs. Our experiments show that our new generator achieves speedup factors of 3-60 over the best previous implementation. One billion edges can now be generated in under one minute on a shared-memory workstation. Furthermore, we present a dynamic extension to model gradual network change, while preserving at each step the point position probabilities. 1. Introduction Relational data of complex relationships often take the form of complex networks, graphs with heterogeneous and often hierarchical structure, low diameter, high clustering, and a heavy-tailed degree distribution. Examples include social networks, the graph of hyperlinks between websites, protein interaction networks, and infrastructure routing networks on the autonomous system level [1]. Frequently found properties in generative models for complex network are non-negligible clustering (ratio of triangles to triads), a community structure, and a heavytailed degree distribution [2], such as a power-law. Benchmarks developed to evaluate a system with respect to floating point operations do not represent the requirements of graph algorithms, especially with heterogeneous datasets such as complex networks. The Graph500 benchmark [3] addresses this gap; it is the most widely-used graph benchmark in high-performance computing. It uses the Recursive Matrix (R-MAT) [4] model to generate synthetic networks as benchmark instances. Graphs from this model are efficiently computable, but suffer from drawbacks in terms of realism. For example, even with fixed parameters, the clustering coefficient shrinks with graph size, while the number of connected components increases, which is problematic for scaling studies [5]. An interesting model without this problem are random hyperbolic graphs (RHG), a family of geometric graphs in the hyperbolic plane. Krioukov et al. [6] introduced this graph model and showed how the structure of complex networks naturally develops from the properties of hyperbolic geometry. To generate a RHG, one randomly samples node positions in a hyperbolic disk, then connects two nodes with an edge with a probability depending on their hyperbolic distance. In a special case of this model, an edge between two nodes is added exactly if their distance is below a threshold. This subset of RHG, sometimes called threshold random hyperbolic graphs, is well-analyzed theoretically [7], [8], [9] and could be considered as unitdisk graphs in hyperbolic space. The resulting graphs show a power-law degree distribution with adjustable exponent, provably high clustering [8], and small diameter [9]. Motivation, outline, and contribution. A fast generator implementation that scales to large graph sizes and provides sufficient realism is necessary to create meaningful graph benchmark instances in acceptable time. While our previous work [10] was able to improve the quadratic time complexity of the pairwise probing approach [11] for threshold RHGs, it still has superlinear time complexity. We therefore provide a faster generation algorithm in this paper for threshold random hyperbolic graphs (Section 3), using a new spatial data structure. The key idea is to divide the relevant part of the hyperbolic plane into ring-shaped slabs and use these to bound the coordinates of possible neighbors in each slab. As our experiments (Section 4) show, a network with 10 million vertices and 1G edges can be generated with our shared-memory parallel implementation in under one minute, yielding a speedup factor of up to 60 over the best previous implementation [10]. For a graph with n nodes and m edges, the measurements suggest an O(n log n+m) time on the hyperbolic plane, the distance between them is given by the hyperbolic law of cosines: complexity, but we do not have a proof for this. While an algorithm with optimal expected linear time complexity has been suggested in a theoretical paper [12], our present work provides the fastest implementation to date. The generator code is publicly available in our network analysis toolkit NetworKit [13]. cosh dist(p, q) = cosh rp cosh rq −sinh rp sinh rq cos \|φp − φq \| (3) As mentioned briefly in Section 1, an important special case is T = 0, where an edge is added to a node pair exactly if the hyperbolic distance between the points is below a threshold. This graph family is sometimes called threshold random hyperbolic graphs, hyperbolic unit-disk graphs or (slightly confusingly) just random hyperbolic graphs. While we consider hyperbolic unit-disk graphs to be more precise, we stick with threshold random hyperbolic graphs to avoid name proliferation. Many theoretical results are for this special case [9]. 2. Related Work Generative Models. Due to the growing interest in complex networks, numerous generators for them exist. For a comprehensive overview, which would be outside the scope of this paper, we refer the interested reader to Goldenberg’s survey [14]. None of the models is suitable for all use cases. As mentioned above, the Recursive Matrix (R-MAT) [4] model has received particular attention in the HPC community due to its use in the Graph500 benchmark [3]. RHG Generation Algorithms. Previous generators for random hyperbolic graphs exist, both for the general and special case. Aldecoa et al. [11] present a generator for the general case with quadratic time complexity, calculating distances and sampling edges for all Θ(n2 ) node pairs. Von Looz et al. [10] use polar quadtrees to generate threshold RHGs with a time complexity of O((n3/2 + m) log n) (with high probability). Recently, von Looz and Meyerhenke [17] have extended this approach to generate general RHGs with the same time complexity. Bringmann et al. [12] propose Geometric Inhomogeneous Random Graphs as a generalization of RHGs and describe a generation algorithm with expected linear time complexity. To our knowledge no implementation of this algorithm is available. Hyperbolic Geometry. Hyperbolic space is one of the three isotropic spaces, the other two being the (more common) Euclidean space and spherical space. In contrast to the flat Euclidean geometry and the positively curved spherical geometry, hyperbolic geometry has negative curvature [15]. Among other interesting properties, hyperbolic geometry shows an exponential expansion of space: While the area of a Euclidean circle grows quadratically with the circle radius, the area of a circle on the hyperbolic plane grows exponentially with its radius. In balanced trees, the number of nodes at a certain distance from the root also grows exponentially with said distance, leading to the suggestion that hierarchical complex networks with tree-like structures might be easily embeddable in hyperbolic space [6]. Indeed, Boguñá et al. [16] demonstrate the connection between hyperbolic geometry and complex networks by embedding the autonomous system internet graph in the hyperbolic plane and enabling locally greedy routing. As a generative model, Krioukov et al. [6] introduced random hyperbolic graphs in 2010. To generate a graph, points are first distributed randomly within in a disk DR of radius R in the hyperbolic plane. The probability density functions for the point distributions are given in polar coordinates, the angular coordinate φ is distributed uniformly over [0, 2π], the radial coordinate r is given by [6, Eq. (17)]: f (r) = α sinh(αr) cosh(αR) − 1 3. Algorithm Our main idea is to partition the hyperbolic plane into concentric ring-shaped slabs (Section 3.1) and use them to limit the number of necessary distance calculations during edge creation (Algorithm 1). Point positions are sampled, sorted by their angular coordinates and stored in the appropriate slab as determined by their radial coordinates. To gather the neighborhood of a point v , we then iterate over all slabs and examine possible neighbors within them. Since each slab limits the radial coordinates of points it contains, we can use Eq. (3) to also bound the angular coordinates of possible neighbors in each slab, thus reducing the number of comparisons and running time. (1) The parameter α governs node dispersion, which determines the power-law exponent of the resulting degree distribution. After sampling point positions, edges are then added to each node pair (u, v) with a probability given in [6, Eq. (41)], depending on their hyperbolic distance and parametrized by a temperature T ≥ 0: 1 f (x) = (1/T )·(x−R)/2 (2) e +1 3.1. Data Structure Let C = {c0 , c1 , ...cmax } be a set of log n ordered radial boundaries, with c0 = 0 and cmax = R. We then define a slab Si as the area enclosed by ci and ci+1 . A point p = (φp , rp ) is contained in slab Si exactly if ci ≤ rp < ci+1 . Since slabs are ring-shaped, they partition the hyperbolic disk DR : For α ≥ 21 , the resulting degree distribution follows a power law with exponent γ := 2α + 1 [6, Eq. (29)]. Given two points in polar coordinates p = (φp , rp ), q = (φq , rq ) DR = 2 log [n i=0 Si . φm ax Algorithm 1: Graph Generation 1 2 φ min 3 4 ci ci+1 5 6 7 8 R 9 10 11 Figure 1: Graph in hyperbolic geometry with unit-disk neighborhood. Neighbors of the bold blue vertex are in the hyperbolic circle, marked in blue. 12 13 14 15 The choice of radial boundaries is an important tuning parameter. After experimenting with different divisions, we settled on a geometric sequence with ratio p = 0.9. The relationship between successive boundary values is then: ci+1 − ci = p · (ci − ci−1 ). From c0 = 0 and cmax = R, we derive the value of c1 : log n−1 X 16 17 18 19 20 log n 1−p c1 p = R ⇔ c1 1−p (1 − p)R = R ⇔ c1 = 1 − plog n k=0 (4) The remaining values follow geometrically. Figure 1 shows an example of a graph in the hyperbolic plane, together with slab Si . The neighbors of the bold blue vertex v are those within a hyperbolic circle of radius R (0.2R in this visualization), marked by the blue egg-shaped area. When considering nodes in Si as possible neighbors of v , the algorithm only needs to examine nodes whose angular coordinate is between φmin and φmax . k 21 22 23 24 Input: number of vertices n, average degree k , power-law exponent γ Output: G = (V, E) α = (γ − 1)/2; R = getTargetRadius(n, k, α); V = n vertices; C = {c0 , c1 , ...cmax } set of log n ordered radial coordinates, with c0 = 0 and cmax = R; B = {b0 , b1 , ...bmax } set of log n empty sets; for vertex v ∈ V do in parallel draw φ[v] from U[0, 2π); draw r[v] with density f (r) = α sinh(αr)/(cosh(αR) − 1); insert (φ[v], r[v]) in suitable bi so that ci ≤ r[v] ≤ ci+1 ; end for b ∈ B do in parallel sort points in b by their angular coordinates; end for vertex v ∈ V do in parallel for band bi ∈ B , where ci+1 > r[v] do minφ , maxφ = getMinMaxPhi(φ[v], r[v]), ci , ci+1 , R); for vertex w ∈ bi , where minφ ≤ φ[w] ≤ maxφ do if distH (v, w) ≤ R then add (v, w) to E ; end end end end return G; Algorithm. Algorithm 1 shows the generation of G = (V, E) with average degree k and power-law exponent γ . First, the radius R of the hyperbolic disk is calculated according to desired graph size and density (Line 2). The value of ζ can be fixed while retaining all degrees of freedom in the model [7], we thus assume ζ = 1. We then use binary search with fixed n, α and desired k to find an R that gives us a close approximation of the desired average degree k . Note that the above equation is only an approximation and might give wrong results for extreme values. Our implementation could easily be adapted to skip this step and accept the commonly used [18] parameter C , with R = 2 ln n+C or even accept R directly. For increased usability, we accept the average degree k as a parameter in the default version. getTargetRadius. This function is unchanged from our previous work [10]. For given values of n, α and R, an approximation of the expected average degree k is given by [6, Eq. (22)] and the notation ξ = (α/ζ)/(α/ζ − 1/2): 2 2 k = ξ 2 n · e−ζR/2 + ξ 2 n (5) π π 2 R π ζ ζ · e−αR α − (π − 1) + (π − 2) − 1 2 4 α α (6) Vertex Positions and Bands. After settling the disk boundary, the radial boundaries ci are calculated (Line 4) as defined above, the disk DR is thus partitioned into log n slabs. For each slab Si , a set bi stores the vertices located in the area of Si . These sets bi are initially empty (Line 5). The vertex positions are then sampled randomly in polar coordinates (Lines 7 and 8) and stored in the corresponding set, i.e, vertex v is put into set bi iff ci ≤ r[v] < ci+1 (Line 9). Within each set, vertices are sorted with respect to their angular coordinates (Lines 11 to 13). 3.2. Generation Algorithm 3 outside the hyperbolic disk DR . In this case, the movement is inverted and the node “bounces” off the boundary. The different probability densities in the center of the disk and the outer regions can be translated into movement speed: A node is less likely to be in the center; thus it needs to spend less time there while traversing it, resulting in a higher speed. We implement this movement in two phases: In the initialization, step values τφ and τr are assigned to each node according to the desired movement. Each movement step of a node then consists of a rotation and a radial movement. The rotation step is a straightforward addition of angular coordinates: rotated(φ, r, τφ ) = (φ+τφ ) mod 2π . The radial movement is described in Algorithm 2 and a visualization is shown in Figure 2. getMinMaxPhi. The neighbors of a given vertex v = (φv , rv ) are those whose hyperbolic distance to v is at most R. Let bi be the slab between ci and ci+1 , and u = (φu , ru ) ∈ bi a neighbor of v in bi . Since u is in bi , ru is between ci and ci+1 . With the hyperbolic law of cosines, we can conclude: cosh R ≥ cosh rv cosh ci − sinh rv sinh ci cos \|φu − φv \| ⇔ (7) cosh rv cosh ci − cosh R ≤ sinh rv sinh ci cos \|φu − φv \| ⇔ (8) cosh rv cosh ci − cosh R ⇔ cos \|φu − φv \| ≥ sinh rv sinh ci (9) cosh r cosh c − cosh R v i −1 \|φu − φv \| ≤ cos sinh rv sinh ci (10) Algorithm 2: Radial movement in dynamic model Input: r, τr , R, α. Output: rnew 1) x = sinh(r · α); 2) y = x+τr ; 3) z = asinh(y)/α; 4) Return z To gather the neighborhood of a vertex v = (φv , rv ), we iterate over all slabs Si and compute for each slab how far the angular coordinate φq of a possible neighbor in bi can deviate from φv (Line 16). We call the vertices in bi whose angular coordinates are within these bounds the neighbor candidates for v in bi . Since points are sorted according to their angular coordinates, we can quickly find the leftmost and rightmost neighbor candidate in each slab using binary search. We then only need to check each neighbor candidate (Line 17), compute its hyperbolic distance to v and add an edge if this distance is below R (Lines 18 and 19). Since edges can be found from both ends, we only need to iterate over slabs in one direction; we choose outward in our implementation (Line 15). The process is repeated for every vertex v (Line 14). Not surprisingly the running time of Algorithm 1 is dominated by the range queries (Lines 14-23). Our experiments in Section 4 suggest a running time of O(n log n + m) for the complete algorithm. This should be seen as an empirical observation; we leave a mathematical proof for future work. If the new node position would be outside the boundary (r > R) or below the origin (r < 0), the movement is reflected and τr set to −τr . Theorem 1. Let fr,φ ((pr , pφ )) be the probability density of point positions, given in polar coordinates. Let move((pr , pφ )) be a movement step. Then, the node movement preserves the distribution of angular and radial distributions: fr,φ (move((pr , pφ ))) = fr,φ ((pr , pφ )). Proof. Since the distributions of angular and radial coordinates are independent, we consider them separately: fr,φ (pr , pφ ) = fr (pr ) · fφ (pφ ). As introduced in Eq. (1), the radial coordinate r is sampled from a distribution with density α sinh(αr)/(cosh(αR) − 1). We introduce random variables X, Y, Z for each step in Algorithm 2, each is denoted with the upper case letter of its equivalent. An additional random variable Q denotes the pre-movement radial coordinate. The other variables are defined as X = sinh(Q · α), Y = X + τr and Z = asinh(Y )/α. Let fQ , fX , fY and fZ denote the density functions of these variables: α sinh(αr) fQ (r) = (11) cosh(αR) − 1 asinh(r) αr fX (r) = fQ = (12) α cosh(αR) − 1 αr − τr fY (r) = fX (r − τr ) = (13) cosh(αR) − 1 α sinh(αr) − τr fZ (r) = fY (sinh(r · α)) = (14) cosh(αR) − 1 τr = fQ (r) − (15) cosh(αR) − 1 3.3. Dynamic Model To model gradual change in networks, we design and implement a dynamic version with node movement. While deleting nodes or inserting them at random positions is a suitable dynamic behavior for modeling internet infrastructure with sudden site failures or additions, change in e. g., social networks happens more gradually. A suitable node movement model needs to be consistent: After moving a node, the network may change, but properties should stay the same in expectation. Since the properties emerge from the node positions, the probability distribution of node positions needs to be preserved. In our implementation, movement happens in discrete time steps. We choose the movement to be directed: If a node i moves in a certain direction at time t, it will move in the same direction at t + 1, except if the new position would be 4 to 60 for graphs with 107 nodes and ≈ 4 · 107 edges. Very roughly, the experimental running times fit a complexity of O(n log n + m). While the running times of the faster generator appear to grow more steeply with increasing edge count, this is an artifact of the logarithmic plot: The same constant increase is relatively larger compared to a smaller running time, and thus appears larger in the logarithmic drawing. sinh(αR) R R ⇒ ⇒ 0 1 n = 105 , our impl. n = 106 , our impl. n = 107 , our impl. n = 105 , impl. of [10] n = 106 , impl. of [10] n = 107 , impl. of [10] n = 105 , theoretical fit n = 106 , theoretical fit n = 107 , theoretical fit 0 Figure 2: For each movement step, radial coordinates are mapped into the interval [1, sinh(αR)), where the coordinate distribution is uniform. Adding τr and transforming the coordinates back results in correctly scaled movements. The distributions of Q and Z only differ in the constant addition of τr /(cosh(αR) − 1). Every (cosh(αR) − 1)/τr steps, the radial movement reaches a limit (0 or R) and is reflected, causing τr to be multiplied with -1. On average, τr is thus zero and FQ (r) = FZ (r). A similar argument works for the rotational step: While the rotational direction is unchanged, the change in coordinates is balanced by the addition or subtraction of 2π whenever the interval [0, 2π) is left, leading to an average of zero in terms of change. running time in seconds 103 4. Experimental Evaluation Setup. The generation algorithm is implemented in C++11 and parallelized with OpenMP. Running time measurements were made on a server with 256 GB RAM and 2x8 Intel Xeon E5-2680 cores at 2.7 GHz. With hyperthreading enabled, we use up to 32 threads. For memory allocations, we use the lock-free malloc implementation of Intel’s Threading Building Blocks library. Our code is included in the network analysis toolkit NetworKit [13]. To compare performance, we generate graphs with 105 , 6 10 and 107 nodes and average degrees between 1 and 64, both with the algorithm presented in this work and the implementation of von Looz et al. [10]. To validate the distribution of generated graphs, we compare our implementation with the implementation of Aldecoa et al. [11]. We generate graphs with 104 nodes each for a combination of parameters and calculate several network analytic characteristics, averaging over 100 runs. For the dynamic model, we measure the time required for a movement step and again compare the distributions of network analytic properties. 102 101 100 10−1 105 106 107 edges 108 109 Figure 3: Comparison of running times to generate networks with 104 -107 vertices, α = 1 and varying k . Circles represent running times of our implementation, diamonds the running times of the implementation of [10]. Our running times are fitted with the equation T (n, m) = (7.07 · n log10 n + 2.23 · m + 891) · 10−8 seconds. The scaling behavior for 1 to 32 threads on 16 cores is shown in Figure 4. Considering edge sampling alone, it shows strong scaling up to the number of physical cores, with a speedup of 13.48 for 16 threads. With hyperthreading, the speedup increases to 18.38. Combining the edge lists later on into the NetworKit graph data structure, however, requires coordination and proves to be a bottleneck in parallel. If only edge lists are required, this final step can be omitted – as done for example in the Graph500 benchmark. Running Time. Figure 3 shows the running times to generate graphs with 105 to 107 nodes and 2 · 105 to 128 · 107 edges. The speedup over the previously fastest implementation [10] increases with graph size and sparsity, reaching up Distribution of Generated Graphs. The average degree assortativity, degeneracy, clustering coefficient and size and diameter of largest components of our generator and the 5 20 18.38 total edge sampling speedup factor 15 [3] D. A. Bader, J. Berry, S. Kahan, R. Murphy, E. J. Riedy, and J. Willcock, “Graph 500 benchmark 1 (”search”), version 1.1,” Graph 500, Tech. Rep., 2010. [4] D. Chakrabarti, Y. Zhan, and C. Faloutsos, “R-MAT: A recursive model for graph mining,” in Proc. 4th SIAM Intl. Conf. on Data Mining (SDM). Orlando, FL: SIAM, Apr. 2004. [5] T. G. Kolda, A. Pinar, T. Plantenga, and C. Seshadhri, “A scalable generative graph model with community structure,” SIAM J. Scientific Computing, vol. 36, no. 5, pp. C424–C452, Sep 2014. [6] D. Krioukov, F. Papadopoulos, M. Kitsak, A. Vahdat, and M. Boguñá, “Hyperbolic geometry of complex networks,” Physical Review E, vol. 82, no. 3, p. 036106, Sep 2010. [Online]. Available: http://link.aps.org/doi/10.1103/PhysRevE.82.036106 [7] M. Bode, N. Fountoulakis, and T. Müller, “The probability that the hyperbolic random graph is connected,” Random Structures and Algorithms, 2016, to appear. Preprint available at http://www.staff. science.uu.nl/∼muell001/Papers/BFM.pdf. [8] L. Gugelmann, K. Panagiotou, and U. Peter, “Random hyperbolic graphs: Degree sequence and clustering - (extended abstract),” in Automata, Languages, and Programming - 39th International Colloquium, ICALP 2012, Proceedings, Part II, ser. Lecture Notes in Computer Science, A. Czumaj, K. Mehlhorn, A. M. Pitts, and R. Wattenhofer, Eds., vol. 7392. Springer, 2012, pp. 573–585. [Online]. Available: http://dx.doi.org/10.1007/978-3-642-31585-5 51 [9] A. Bonato, “A survey of models of the web graph,” in Combinatorial and Algorithmic Aspects of Networking, ser. Lecture Notes in Computer Science. Springer Berlin Heidelberg, 2005, vol. 3405, pp. 159–172. [Online]. Available: http://dx.doi.org/10.1007/11527954 16 10 5 0 4 8 12 16 20 24 28 32 threads Figure 4: Speedup curves for n = 107 , k = 6, γ = 3 on a machine with 16 physical cores (marked with a vertical line) and hyperthreading. Averaged over 10 runs. one by Aldecoa et al. [11] are shown in Plots 5 and 6 in Appendix A. Averaged over 100 runs, the network analytic properties show a very close match between the distributions of the two generation algorithms. [10] M. von Looz, R. Prutkin, and H. Meyerhenke, “Generating random hyperbolic graphs in subquadratic time,” in ISAAC 2015: Proc. 26th Int’l Symp. on Algorithms and Computation, 2015. Dynamic Model. Our implementation allows updating a graph without rebuilding it from scratch. Moving up to 12% of nodes and updating an existing graph is still faster than a new static generation. The distribution of generated graphs is indistinguishable from the static model (Appendix B). [11] R. Aldecoa, C. Orsini, and D. Krioukov, “Hyperbolic graph generator,” Computer Physics Communications, 2015. [Online]. Available: http://www.sciencedirect.com/science/article/pii/ S0010465515002088 [12] K. Bringmann, R. Keusch, and J. Lengler, “Geometric inhomogeneous random graphs,” arXiv preprint arXiv:1511.00576, 2015. 5. Conclusions [13] C. L. Staudt, A. Sazonovs, and H. Meyerhenke, “NetworKit: A tool suite for large-scale complex network analysis,” Network Science, 2015, to appear. We have provided the fastest implementation so far to generate massive complex networks based on threshold random hyperbolic graphs. The running time improvement is particularly large for graphs with realistic densities. We have also presented a model extension to cover gradual node movement and have proved its consistency regarding the probability densities of vertex positions. Both the static and the dynamic model can serve as complex network generators with reasonable realism and fast generation times even for massive networks. [14] A. Goldenberg, A. X. Zheng, S. E. Fienberg, and E. M. Airoldi, “A survey of statistical network models,” Foundations and Trends R in Machine Learning, vol. 2, no. 2, pp. 129–233, 2010. [15] J. W. Anderson, Hyperbolic geometry; 2nd ed., ser. Springer undergraduate mathematics series. Berlin: Springer, 2005. [16] M. Boguñá, F. Papadopoulos, and D. Krioukov, “Sustaining the internet with hyperbolic mapping,” Nature Communications, no. 62, September 2010. [Online]. Available: http://www.nature.com/ ncomms/journal/v1/n6/abs/ncomms1063.html [17] M. von Looz and H. Meyerhenke, “Querying Probabilistic Neighborhoods in Spatial Data Sets Efficiently,” ArXiv preprint arXiv:1509.01990, Sep. 2015. Acknowledgements. This work is partially supported by German Research Foundation (DFG) grant ME 3619/3-1 (FINCA) and grant GI-711/5-1, both within the Priority Programme 1736 Algorithms for Big Data. [18] M. Kiwi and D. Mitsche, “A bound for the diameter of random hyperbolic graphs,” in 2015 Proceedings of the Twelfth Workshop on Analytic Algorithmics and Combinatorics (ANALCO). SIAM, Jan 2015, pp. 26–39. [Online]. Available: http://epubs.siam.org/doi/abs/ 10.1137/1.9781611973761.3 References [1] M. Newman, Networks: An Introduction. 2010. Oxford University Press, [2] D. Chakrabarti and C. Faloutsos, “Graph mining: Laws, generators, and algorithms,” ACM Computing Surveys (CSUR), vol. 38, no. 1, p. 2, 2006. 6 Appendix A. Comparison tion [11] with Previous Degree Assortativity Degree Assortativity γ = 2.2 γ = 4.6 γ = 7.0 0.4 0.2 0 0.6 degree assortativity degree assortativity 0.6 Implementa- −0.2 γ = 2.2 γ = 4.6 γ = 7.0 0.4 0.2 0 −0.2 101 102 101 k 102 Clustering Coefficient k 0.86 0.84 Degeneracy Degeneracy Clustering Coefficient 0.86 k=4 k = 32 k = 256 0.82 cc 2 k 3 4 5 6 7 2 3 4 γ ·104 101 102 vertices in largest component 102 0.76 0.76 101 101 0.8 0.78 0.78 2 k Figure 5: Comparison of degree assortativity and degeneracy for the implementation of [11] (left) and our implementation (right). Degree assortativity describes whether vertices have neighbors of similar degree. A value near 1 signifies subgraphs with equal degree, a value of -1 star-like structures. k -Cores, in turn, are a generalization of connected components and result from iteratively peeling away vertices of degree k and assigning to each vertex the core number of the innermost core it is contained in. Degeneracy refers to the largest core number. Values are averaged over 100 runs. Size of Largest Component 1 0.8 0.6 0.4 0.2 γ = 2.2 γ = 4.6 γ = 7.0 0 101 ·104 Size of Largest Component 0.6 0.4 0.2 γ = 2.2 γ = 4.6 γ = 7.0 0 101 102 k Diameter of Largest Component diameter of largest component 7 0.8 102 γ = 2.2 γ = 4.6 γ = 7.0 600 400 200 0 102 k 6 1 k 101 5 γ vertices in largest component 101 10 0.8 Diameter of Largest Component diameter of largest component max core number max core number 10 2 cc 0.82 γ = 2.2 γ = 4.6 γ = 7.0 γ = 2.2 γ = 4.6 γ = 7.0 k=4 k = 32 k = 256 0.84 600 γ = 2.2 γ = 4.6 γ = 7.0 400 200 0 101 102 k Figure 6: Comparison of clustering coefficients, size of largest component and diameter of largest components for the implementation of [11] (left) and our implementation (right). Values are averaged over 100 runs. 7 Appendix B. Consistency of Dynamic Model Degree Assortativity Degree Assortativity 0.2 0 −0.2 γ = 2.2 γ = 4.6 γ = 7.0 0.4 0.2 0 −0.2 101 102 101 Clustering Coefficient 102 0.86 k k 0.84 Degeneracy max core number 0.8 102 101 102 k 0.76 0.76 2 3 4 5 6 7 2 3 4 γ ·104 101 101 0.8 0.78 0.78 101 vertices in largest component max core number 102 0.82 cc γ = 2.2 γ = 4.6 γ = 7.0 k=4 k = 32 k = 256 0.84 0.82 cc Degeneracy γ = 2.2 γ = 4.6 γ = 7.0 Clustering Coefficient 0.86 k=4 k = 32 k = 256 102 k Figure 7: Comparison of degree assortativity and degeneracy for graphs with 104 nodes, before and after one movement step. All nodes were moved, with τφ ∈ (−1, 1) and τr ∈ (−10, 1) sampled randomly. Distribution of graphs after node movement are shown left, before node movement right. Values are averaged over 100 runs. Size of Largest Component 1 0.8 0.6 0.4 0.2 γ = 2.2 γ = 4.6 γ = 7.0 0 101 ·104 Size of Largest Component 0.6 0.4 0.2 γ = 2.2 γ = 4.6 γ = 7.0 0 101 102 k Diameter of Largest Component diameter of largest component 7 0.8 102 γ = 2.2 γ = 4.6 γ = 7.0 400 200 0 102 k 6 1 k 101 5 γ vertices in largest component 0.4 0.6 Diameter of Largest Component diameter of largest component γ = 2.2 γ = 4.6 γ = 7.0 degree assortativity degree assortativity 0.6 600 γ = 2.2 γ = 4.6 γ = 7.0 400 200 0 101 102 k Figure 8: Comparison of clustering coefficients, size of largest component and diameter of largest components for graphs with 104 nodes, before and after one movement step. All nodes were moved, with τφ ∈ (−1, 1) and τr ∈ (−10, 1) sampled randomly. Distribution of graphs after node movement are shown left, before node movement right. Values are averaged over 100 runs. 8	8
The (1\|1)-Centroid Problem on the Plane Concerning Distance Constraints ⋆ arXiv:1608.03680v1 [cs.CG] 12 Aug 2016 Hung-I Yu, Tien-Ching Lin, and D. T. Lee Institute of Information Science, Academia Sinica, Nankang, Taipei 115, Taiwan, {herbert,kero,dtlee}@iis.sinica.edu.tw Abstract. In 1982, Drezner proposed the (1\|1)-centroid problem on the plane, in which two players, called the leader and the follower, open facilities to provide service to customers in a competitive manner. The leader opens the first facility, and the follower opens the second. Each customer will patronize the facility closest to him (ties broken in favor of the first one), thereby decides the market share of the two facilities. The goal is to find the best position for the leader’s facility so that its market share is maximized. The best algorithm of this problem is an O(n2 log n)-time parametric search approach, which searches over the space of market share values. In the same paper, Drezner also proposed a general version of (1\|1)centroid problem by introducing a minimal distance constraint R, such that the follower’s facility is not allowed to be located within a distance R from the leader’s. He proposed an O(n5 log n)-time algorithm for this general version by identifying O(n4 ) points as the candidates of the optimal solution and checking the market share for each of them. In this paper, we develop a new parametric search approach searching over the O(n4 ) candidate points, and present an O(n2 log n)-time algorithm for the general version, thereby close the O(n3 ) gap between the two bounds. Keywords: competitive facility, Euclidean plane, parametric search 1 Introduction In 1929, economist Hotelling introduced the first competitive location problem in his seminal paper [13]. Since then, the subject of competitive facility location has been extensively studied by researchers in the fields of spatial economics, social and political sciences, and operations research, and spawned hundreds of contributions in the literature. The interested reader is referred to the following survey papers [3,7,8,9,11,12,17,19]. Hakimi [10] and Drezner [5] individually proposed a series of competitive location problems in a leader-follower framework. The framework is briefly described as follows. There are n customers in the market, and each is endowed ⋆ Research supported by under Grants No. MOST 103-2221-E-005-042, 103-2221-E005-043. 2 Hung-I Yu, Tien-Ching Lin, and D. T. Lee with a certain buying power. Two players, called the leader and the follower, sequentially open facilities to attract the buying power of customers. At first, the leader opens his p facilities, and then the follower opens another r facilities. Each customer will patronize the closest facility with all buying power (ties broken in favor of the leader’s ones), thereby decides the market share of the two players. Since both players ask for market share maximization, two competitive facility location problems are defined under this framework. Given that the leader locates his p facilities at the set Xp of p points, the follower wants to locate his r facilities in order to attract the most buying power, which is called the (r\|Xp )-medianoid problem. On the other hand, knowing that the follower will react with maximization strategy, the leader wants to locate his p facilities in order to retain the most buying power against the competition, which is called the (r\|p)-centroid problem. Drezner [5] first proposed to study the two competitive facility location problems on the Euclidean plane. Since then, many related results [4,5,6,11,14] have been obtained for different values of r and p. Due to page limit, here we introduce only previous results about the case r = p = 1. For the (1\|X1 )-medianoid problem, Drezner [5] showed that there exists an optimal solution arbitrarily close to X1 , and solved the problem in O(n log n) time by sweeping technique. Later, Lee and Wu [14] obtained an Ω(n log n) lower bound for the (1\|X1 )-medianoid problem, and thus proved the optimality of the above result. For the (1\|1)-centroid problem, Drezner [5] developed a parametric search based approach that searches over the space of O(n2 ) possible market share values, along with an O(n4 )-time test procedure constructing and solving a linear program of O(n2 ) constraints, thereby gave an O(n4 log n)-time algorithm. Then, by improving the test procedure via Megiddo’s result [16] for solving linear programs, Hakimi [11] reduced the time complexity to O(n2 log n). In [5], Drezner also proposed a more general setting for the leader-follower framework by introducing a minimal distance constraint R ≥ 0 into the (1\|X1 )medianoid problem and the (1\|1)-centroid problem, such that the follower’s facility is not allowed to be located within a distance R from the leader’s. The augmented problems are respectively called the (1\|X1 )R -medianoid problem and (1\|1)R -centroid problem in this paper. Drezner showed that the (1\|X1 )R medianoid problem can also be solved in O(n log n) time by using nearly the same proof and technique as for the (1\|X1 )-medianoid problem. However, for the (1\|1)R -centroid problem, he argued that it is hard to generalize the approach for the (1\|1)-centroid problem to solve this general version, due to the change of problem properties. Then, he gave an O(n5 log n)-time algorithm by identifying O(n4 ) candidate points on the plane, which contain at least one optimal solution, and performing medianoid computation on each of them. So far, the O(n3 ) bound gap between the two centroid problems remains unclosed. In this paper, we propose an O(n2 log n)-time algorithm for the (1\|1)R centroid problem on the Euclidean plane, thereby close the gap last for decades. Instead of searching over market share values, we develop a new approach based on the parametric search technique by searching over the O(n4 ) candidate points The (1\|1)R -Centroid Problem on the Plane 3 mentioned in [5]. This is made possible by making a critical observation on the distribution of optimal solutions for the (1\|X1 )R -medianoid problem given X1 , which provides us a useful tool to prune candidate points with respect to X1 . We then extend the usage of this tool to design a key procedure to prune candidates with respect to a given vertical line. The rest of this paper is organized as follows. Section 2 gives formal problem definitions and describes previous results in [5,11]. In Section 3, we make the observation on the (1\|X1 )R -medianoid problem, and make use of it to find a “local” centroid on a given line. This result is then extended as a new pruning procedure with respect to any given line in Section 4, and utilized in our parametric search approach for the (1\|1)R -centroid problem. Finally, in Section 5, we give some concluding remarks. 2 Notations and Preliminary Results Let V = {v1 , v2 , · · · , vn } be a set of n points on the Euclidean plane R2 , as the representatives of the n customers. Each point vi ∈ V is assigned with a positive weight w(vi ), representing its buying power. To simplify the algorithm description, we assume that the points in V are in general position, that is, no three points are collinear and no two points share a common x or y-coordinate. Let d(u, w) denote the Euclidean distance between any P two points u, w T ∈ R2 . For any set Z of points on the plane, we define W (Z) = {w(v)\|v ∈ V Z}. Suppose that the leader has located his facility at X1 = {x}, which is shortened as x for simplicity. Due to the minimal distance constraint R mentioned in [5], any point y ′ ∈ R2 with d(y ′ , x) < R is infeasible to be the follower’s choice. If the follower locates his facility at some feasible point y, the set of customers patronizing y instead of x is defined as V (y\|x) = {v ∈ V \|d(v, y) < d(v, x)}, with their total buying power W (y\|x) = W (V (y\|x)). Then, the largest market share that the follower can capture is denoted by the function W ∗ (x) = max y∈R2 ,d(y,x)≥R W (y\|x), which is called the weight loss of x. Given a point x ∈ R2 , the (1\|x)R -medianoid problem is to find a (1\|x)R -medianoid, which denotes a feasible point y ∗ ∈ R2 maximizing the weight loss of x. In contrast, the leader tries to minimize the weight loss of his own facility by finding a point x∗ ∈ R2 such that W ∗ (x∗ ) ≤ W ∗ (x) for any point x ∈ R2 . The (1\|1)R -centroid problem is to find a (1\|1)R -centroid, which denotes a point x∗ minimizing its weight loss. Note that, when R = 0, the two problems degenerate to the (1\|x)-medianoid and (1\|1)-centroid problems. 4 2.1 Hung-I Yu, Tien-Ching Lin, and D. T. Lee Previous approaches In this subsection, we briefly review previous results for the (1\|x)R -medianoid, (1\|1)-centroid, and (1\|1)R -centroid problems in [5,11], so as to derive some basic properties essential to our approach. Let L be an arbitrary line, which partitions the Euclidean plane into two halfplanes. For any point y ∈ / L, we define H(L, y) as the close half-plane including y, and H − (L, y) as the open half-plane including y (but not L). For any two distinct points x, y ∈ R2 , let B(y\|x) denote the perpendicular bisector of the line segment from x to y. Given an arbitrary point x ∈ R2 , we first describe the algorithm for finding a (1\|x)R -medianoid in [5]. Let y be an arbitrary point other than x, and y ′ be some point on the open line segment from y to x. We can see that H − (B(y\|x), y) ⊂ H − (B(y ′ \|x), y ′ ), which implies the fact that W (y ′ \|x) = W (H − (B(y ′ \|x), y ′ )) ≥ W (H − (B(y\|x), y)) = W (y\|x), It shows that moving y toward x does not diminish its weight capture, thereby follows the lemma. Lemma 1 [5] There exists a (1\|x)R -medianoid in {y \| y ∈ R2 , d(x, y) = R}. For any point z ∈ R2 , let CR (x) and Cγ (x) be the circles centered at z with radii R and γ = R/2, respectively. By Lemma 1, finding a (1\|x)R -medianoid can be reduced to searching a point y on CR (x) maximizing W (y\|x). Since the perpendicular bisector B(y\|x) of each point y on CR (x) is a tangent line to the circle Cγ (x), the searching of y on CR (x) is equivalent to finding a tangent line to Cγ (x) that partitions the most weight from x. The latter problem can be solved in O(n log n) time as follows. For each v ∈ V outside Cγ (x), we calculate its two tangent lines to Cγ (x). Then, by sorting these tangent lines according to the polar angles of their corresponding tangent points with respect to x, we can use the angle sweeping technique to check how much weight they partition. Theorem 1 [5] Given a point x ∈ R2 , the (1\|x)R -medianoid problem can be solved in O(n log n) time. Next, we describe the algorithm of the (1\|1)R -centroid problem in [5]. Let S be a subset of V . We define C(S) to be the set of all circles Cγ (v), v ∈ S, and CH(C(S)) to be the convex hull of these circles. It is easy to see the following. Lemma 2 [5] Let S be a subset of V . For any point x ∈ R2 , W ∗ (x) ≥ W (S) if x is outside CH(C(S)). For any positive number W0 , let I(W0 ) be the intersection of all convex hulls CH(C(S)), where S ⊆ V and W (S) ≥ W0 . We have the lemma below. Lemma 3 [5] Let W0 be a positive real number. For any point x ∈ R2 , W ∗ (x) < W0 if and only if x ∈ I(W0 ). The (1\|1)R -Centroid Problem on the Plane 5 Proof. Consider first the case that x ∈ I(W0 ). By definition, x intersects with every CH(C(S)) of subset S ⊆ V with W (S) ≥ W0 . Let S ′ ⊆ V be any of such subsets. Since x ∈ CH(C(S ′ )), for any point y feasible to x, there must exist a point v ∈ S such that v ∈ / H − (B(y\|x), y), implying that no feasible point y can acquire all buying power from customers of S ′ . It follows that no feasible point y can acquire buying power larger than or equal to W0 , i.e., W ∗ (x) < W0 . If x ∈ / I(W0 ), there must exist a subset S ⊆ V with W (S) ≥ W0 , such that x∈ / CH(C(S)). By Lemma 2, W ∗ (x) ≥ W (S) ≥ W0 . ⊓ ⊔ Drezner [5] argued that the set of all (1\|1)R -centroids is equivalent to some intersection I(W0 ) for smallest possible W0 . We slightly strengthen his argument below. Let W = {W (y\|x) \| x, y ∈ R2 , d(x, y) ≥ R}. The following lemma can be obtained. Lemma 4. Let W0∗ be the smallest number in W such that I(W0∗ ) is not null. A point x is a (1\|1)R -centroid if and only if x ∈ I(W0∗ ). Proof. Let WOP T be the weight loss of some (1\|1)R -centroid x∗ . We first show that I(W0 ) is null for any W0 ≤ WOP T . Suppose to the contrary that it is not null and there exists a point x′ in I(W0 ). By Lemma 3, W ∗ (x′ ) < W0 ≤ WOP T , which contradicts the optimality of x∗ . Moreover, since I(W0∗ ) is not null, we have that W0∗ > WOP T . We now show that a point x is a (1\|1)R -centroid if and only if x ∈ I(W0∗ ). If x is a (1\|1)R -centroid, we have that W ∗ (x) = WOP T < W0∗ . By Lemma 3, x ∈ I(W0∗ ). On the other hand, if x is not a (1\|1)R -centroid, we have that W ∗ (x) > WOP T . Since by definition W ∗ (x) ∈ W, we can see that W ∗ (x) ≥ W0∗ . Thus, again by Lemma 3, x ∈ / I(W0∗ ). ⊓ ⊔ Although it is hard to compute I(W0∗ ) itself, we can find its vertices as solutions to the (1\|1)R -centroid problem. Let T be the set of outer tangent lines of all pairs of circles in C(V ). For any subset S ⊆ V , the boundary of CH(C(S)) is formed by segments of lines in T and arcs of circles in C(V ). Since I(W0 ) is an intersection of such convex hulls, its vertices must fall within the set of intersection points between lines in T , between circles in C(V ), and between one line in T and one circle in C(V ). Let T × T , C(V ) × C(V ), and T × C(V ) denote the three sets of intersection points, respectively. We have the lemma below. Lemma 5 [5] There exists a (1\|1)R -centroid in T × T , C(V ) × C(V ), and T × C(V ). Obviously, there are at most O(n4 ) intersection points, which can be viewed as the candidates of being (1\|1)R -centroids. Drezner thus gave an algorithm by evaluating the weight loss of each candidate by Theorem 1. Theorem 2 [5] The (1\|1)R -centroid problem can be solved in O(n5 log n) time. We remark that, when R = 0, CH(C(S)) for any S ⊆ V degenerates to a convex polygon, so does I(W0 ) for any given W0 , if not null. Drezner [5] proved 6 Hung-I Yu, Tien-Ching Lin, and D. T. Lee that in this case I(W0 ) is equivalent to the intersection of all half-planes H with W (H) ≥ W0 . Thus, whether I(W0 ) is null can be determined by constructing and solving a linear program of O(n2 ) constraints, which takes O(n2 ) time by Megiddo’s result [16]. Since \|W\| = O(n2 ), by Lemma 4, the (1\|1)-centroid problem can be solved in O(n2 log n) time [11], by applying parametric search over W for W0∗ . Unfortunately, it is hard to generalize this idea to the case R > 0, motivating us to develop a different approach. Local (1\|1)R -Centroid within a Line 3 In this section, we analyze the properties of (1\|x)R -medianoids of a given point x in Subsection 3.1, and derive a procedure that prunes candidate points with respect to x. Applying this procedure, we study a restricted version of the (1\|1)R centroid problem in Subsection 3.2, in which the leader’s choice is limited to a given non-horizontal line L, and obtain an O(n log2 n)-time algorithm. The algorithm is then extended as the basis of the test procedure for the parametric search approach in Section 4. 3.1 Pruning with Respect to a Point Given a point x ∈ R2 and an angle θ between 0 and 2π, let y(θ\|x) be the point on CR (x) with polar angle θ with respect to x.1 We define M A(x) = {θ \| W (y(θ\|x)\|x) = W ∗ (x), 0 ≤ θ < 2π}, that is, the set of angles θ maximizing W (y(θ\|x)\|x) (see Figure 1). It can be observed that, for any θ ∈ M A(x) and sufficiently small ǫ, both θ + ǫ and θ − ǫ belong to M A(x), because each v ∈ V (y(θ\|x)\|x) does not intersect B(y(θ\|x)\|x) by definition. This implies that angles in M A(x) form open angle interval(s) of non-zero length. To simplify the terms, let W (θ\|x) = W (y(θ\|x)\|x) and B(θ\|x) = B(y(θ\|x)\|x) in the remaining of this section. Also, let F (θ\|x) be the line passing through x and parallel to B(θ\|x). The following lemma provides the basis for pruning. Lemma 6 Let x ∈ R2 be an arbitrary point, and θ be an angle in M A(x). For any point x′ ∈ / H − (F (θ\|x), y(θ\|x)), W ∗ (x′ ) ≥ W ∗ (x). Proof. Since x′ ∈ / H − (F (θ\|x), y(θ\|x)) and y(θ\|x) ∈ H − (F (θ\|x), y(θ\|x)), by the definition of bisectors, the distance between F (θ\|x′ ) and B(θ\|x) is no less than R/2, which implies that H − (B(θ\|x), y(θ\|x)) ⊆ H − (B(θ\|x′ ), y(θ\|x′ )). Therefore, we can derive the following inequality W ∗ (x′ ) ≥ W (θ\|x′ ) = W (H − (B(θ\|x′ ), y(θ\|x′ )) ≥ W (H − (B(θ\|x), y(θ\|x)) = W (θ\|x) = W ∗ (x), 1 We assume that a polar angle is measured counterclockwise from the positive x-axis. The (1\|1)R -Centroid Problem on the Plane 7 Fig. 1. The black arcs represent the intervals of angles in M A(x), whereas the open circles represent the open ends of these intervals. which completes the proof. ⊓ ⊔ This lemma tells us that, given a point x and an angle θ ∈ M A(x), all points not in H − (F (θ\|x), y(θ\|x)) can be ignored while finding (1\|1)R -centroids, as their weight losses are no less than that of x. By this lemma, we can also prove that the weight loss function is convex along any line on the plane, as shown below. Lemma 7 Let x1 , x2 be two arbitrary distinct points on a given line L. For any point x ∈ x1 x2 \{x1 , x2 }, W ∗ (x) ≤ max{W ∗ (x1 ), W ∗ (x2 )}. Proof. Suppose by contradiction that W ∗ (x) > W ∗ (x1 ) and W ∗ (x) > W ∗ (x2 ) for some point x ∈ x1 x2 \{x1 , x2 }. Since W ∗ (x) > W ∗ (x1 ), by Lemma 6 there exists an angle θ ∈ M A(x) such that x1 is included in H − (F (θ\|x), y(θ\|x)). However, since x ∈ x1 x2 \{x1 , x2 }, x1 and x2 locate on different sides of F (θ\|x). It follows that x2 is outside H − (F (θ\|x), y(θ\|x)) and W ∗ (x2 ) ≥ W ∗ (x) by Lemma 6, which contradicts the assumption. Thus, the lemma holds. ⊓ ⊔ We further investigate the distribution of angles in M A(x). Let CA(x) be the minimal angle interval covering all angles in M A(x) (see Figure 2(a)), and δ(CA(x)) be its angle span in radians. As mentioned before, M A(x) consists of open angle interval(s) of non-zero length, which implies that CA(x) is an open interval and δ(CA(x)) > 0. Moreover, we can derive the following. Lemma 8 If δ(CA(x)) > π, x is a (1\|1)R -centroid. Proof. We prove this lemma by showing that W ∗ (x′ ) ≥ W ∗ (x) ∀ x′ 6= x. Let x′ ∈ R2 be an arbitrary point other than x, and θ′ be its polar angle with respect to x. Obviously, any angle θ satisfying x′ ∈ H − (F (θ\|x), y(θ\|x)) is in the open interval (θ′ − π/2, θ′ + π/2), the angle span of which is equal to π. Since 8 Hung-I Yu, Tien-Ching Lin, and D. T. Lee (a) CA(x) (b) W edge(x) Fig. 2. CA(x) and W edge(x). δ(CA(x)) > π, by its definition there exists an angle θ ∈ M A(x) such that x′ ∈ / H − (F (θ\|x), y(θ\|x)). Thus, by Lemma 6, we have W ∗ (x′ ) ≥ W ∗ (x), thereby proves the lemma. ⊓ ⊔ We call a point x satisfying Lemma 8 a strong (1\|1)R -centroid, since its discovery gives an immediate solution to the (1\|1)R -centroid problem. Note that there are problem instances in which no strong (1\|1)R -centroids exist. Suppose that δ(CA(x)) ≤ π for some point x ∈ R2 . Let W edge(x) denote the wedge of x, defined as the intersection of the two half-planes H(F (θb \|x), y(θb \|x)) and H(F (θe \|x), y(θe \|x)), where θb and θe are the beginning and ending angles of CA(x), respectively. As illustrated in Figure 2(b), W edge(x) is the infinite region lying between two half-lines extending from x (including x and the two halflines). The half-lines defined by F (θe \|x) and F (θb \|x) are called its boundaries, and the counterclockwise (CCW) angle between the two boundaries is denoted by δ(W edge(x)). Since 0 < δ(CA(x)) ≤ π, we have that W edge(x) 6= ∅ and 0 ≤ δ(W edge(x)) < π. It should be emphasized that W edge(x) is a computational byproduct of CA(x) when x is not a strong (1\|1)R -centroid. In other words, not every point has its wedge. Therefore, we make the following assumption (or restriction) in order to avoid the misuse of W edge(x). Assumption 1 Whenever W edge(x) is mentioned, the point x has been found not to be a strong (1\|1)R -centroid, either by computation or by properties. Equivalently, δ(CA(x)) ≤ π. The following essential lemma makes W edge(x) our main tool for prune-andsearch. (Note that its proof cannot be trivially derived from Lemma 6, since by definition θb and θe do not belong to the open intervals CA(x) and M A(x).) Lemma 9 Let x ∈ R2 be an arbitrary point. For any point x′ ∈ / W edge(x), W ∗ (x′ ) ≥ W ∗ (x). Proof. By symmetry, suppose that x′ ∈ / H(F (θb \|x), y(θb \|x)). We can further divide the position of x′ into two cases, (1) x′ ∈ H(F (θe \|x), y(θe \|x)) and (2) x′ ∈ / H(F (θe \|x), y(θe \|x)). The (1\|1)R -Centroid Problem on the Plane 9 Consider case (1). The two assumptions ensure that there exists an angle θ′ ∈ (θb , θe ], such that F (θ′ \|x) passes through x′ . Obviously, any angle θ′′ ∈ (θb , θ′ ) satisfies that x′ ∈ / H(F (θ′′ \|x), y(θ′′ \|x)). By the definition of CA(x), there must ′ exist an angle θb ∈ (θb , θ′ ) infinitely close to θb , such that θb′ belongs to M A(x). Thus, by Lemma 6, we have that W ∗ (x′ ) ≥ W ∗ (x). In case (2), for any angle θ′′ ∈ M A(x), we have that x′ ∈ / H(F (θ′′ \|x), y(θ′′ \|x)), ′′ ∗ ′ ∗ since θ is in (θb , θe ). Again, W (x ) ≥ W (x) by Lemma 6. ⊓ ⊔ Finally, we consider the computation of W edge(x). Lemma 10 Given a point x ∈ R2 , M A(x), CA(x), and W edge(x) can be computed in O(n log n) time. Proof. By Theorem 1, we first compute W ∗ (x) and those ordered tangent lines in O(n log n) time. Then, by performing angle sweeping around Cγ (x), we can identify in O(n) time those open intervals of angles θ with W (θ\|x) = W ∗ (x), of which M A(x) consists. Again by sweeping around Cγ (x), CA(x) can be obtained from M A(x) in O(n) time. Now, if we find x to be a strong (1\|1)R -centroid by checking δ(CA(x)), the (1\|1)R -centroid problem is solved and the algorithm can be terminated. Otherwise, W edge(x) can be constructed in O(1) time. ⊓ ⊔ 3.2 Searching on a Line Although computing wedges can be used to prune candidate points, it does not serve as a stable prune-and-search tool, since wedges of different points have indefinite angle intervals and spans. However, Assumption 1 makes it work fine with lines. Here we show how to use the wedges to compute a local optimal point on a given line, i.e. a point x with W ∗ (x) ≤ W ∗ (x′ ) for any point x′ on the line. Let L be an arbitrary line, which is assumed to be non-horizontal for ease of discussion. For any point x on L, we can compute W edge(x) and make use of it for pruning purposes by defining its direction with respect to L. Since δ(W edge(x)) < π by definition, there are only three categories of directions according to the intersection of W edge(x) and L: Upward – the intersection is the half-line of L above and including x; Downward – the intersection is the half-line of L below and including x; Sideward – the intersection is x itself. If W edge(x) is sideward, x is a local optimal point on L, since by Lemma 9 W ∗ (x) ≤ W ∗ (x′ ) ∀ x′ ∈ L. Otherwise, either W edge(x) is upward or downward, the points on the opposite half of L can be pruned by Lemma 9. It shows that computing wedges acts as a predictable tool for pruning on L. Next, we list sets of breakpoints on L in which a local optimal point locates. Recall that T is the set of outer tangent lines of all pairs of circles in C(V ). We define the T -breakpoints as the set L × T of intersection points between L and lines in T , and the C-breakpoints as the set L × C(V ) of intersection points between L and circles in C(V ). We have the following lemmas for breakpoints. 10 Hung-I Yu, Tien-Ching Lin, and D. T. Lee Lemma 11 Let x1 , x2 be two distinct points on L. If W ∗ (x1 ) > W ∗ (x2 ), there exists at least a breakpoint on the segment x1 x2 \{x1 }. Proof. Let θ be an arbitrary angle in M A(x1 ) and S be the subset of V located in the half-plane H − (B(θ\|x1 ), y(θ\|x1 )). By definition, x1 is outside the convex hull CH(C(S)) and W ∗ (x1 ) = W (S). On the other hand, since W ∗ (x2 ) < W ∗ (x1 ) = W (S) by assumption, we have that x2 is inside CH(C(S)) by Lemma 2. Thus, the segment x1 x2 \{x1 } intersects with the boundary of CH(C(S)). Since the boundary of CH(C(S)) consists of segments of lines in T and arcs of circles in C(V ), the intersection point is either a T -breakpoint or a C-breakpoint, thereby proves the lemma. ⊓ ⊔ Lemma 12 There exists a local optimal point x∗L which is also a breakpoint. Proof. Let x∗L be a local optimal point such that W ∗ (x′ ) > W ∗ (x∗L ) for some point x′ adjacent to x∗L on L. Note that, if no such local optimal point exists, every point on L must have the same weight loss and be local optimal, and the lemma holds trivially. If such x∗L and x′ exist, by Lemma 11 there is a breakpoint on x′ x∗L \{x′ }, which is x∗L itself. Thus, the lemma holds. ⊓ ⊔ We remark that outer tangent lines parallel to L are exceptional cases while considering breakpoints. For any line T ∈ T that is parallel to L, either T does not intersect with L or they just coincide. In either case, T is irrelevant to the finding of local optimal points, and should not be counted for defining T -breakpoints. Now, by Lemma 12, if we have all breakpoints on L sorted in the decreasing order of their y-coordinates, a local optimal point can be found by performing binary search using wedges. Obviously, such sorted sequence can be obtained in O(n2 log n) time, since \|L × T \| = O(n2 ) and \|L × C(V )\| = O(n). However, in order to speed up the computations of local optimal points on multiple lines, alternatively we propose an O(n2 log n)-time preprocessing, so that a local optimal point on any given line can be computed in O(n log2 n) time. The preprocessing itself is very simple. For each point v ∈ V , we compute a sequence P (v), consisting of points in V \{v} sorted in increasing order of their polar angles with respect to v. The computation for all v ∈ V takes O(n2 log n) time in total. Besides, all outer tangent lines in T are computed in O(n2 ) time. We will show that, for any given line L, O(n) sorted sequences can be obtained from these pre-computed sequences in O(n log n) time, which can be used to replace a sorted sequence of all T -breakpoints in the process of binary search. For any two points v ∈ V and z ∈ R2 , let T r (z\|v) be the outer tangent line of Cγ (v) and Cγ (z) to the right of the line from v to z. Similarly, let T l (z\|v) be the outer tangent line to the left. (See Figure 3.) Moreover, let trL (z\|v) and tlL (z\|v) be the points at which T r (z\|v) and T l (z\|v) intersect with L, respectively. We partition T into O(n) sets T r (v) = {T r (vi \|v)\|vi ∈ V \{v}} and T l (v) = {T l (vi \|v)\|vi ∈ V \{v}} for v ∈ V , and consider their corresponding T -breakpoints independently. By symmetry, we only discuss the case about L × T r (v). The (1\|1)R -Centroid Problem on the Plane 11 Fig. 3. Outer tangent lines of v. Lemma 13 For each v ∈ V , we can compute O(1) sequences of T -breakpoints on L, which satisfy the following conditions: (a) Each sequence is of length O(n) and can be obtained in O(log n) time. (b) Breakpoints in each sequence are sorted in decreasing y-coordinates. (c) The union of breakpoints in all sequences form L × T r (v). Proof. Without loss of generality, suppose that v is either strictly to the right of L or on L. Note that each point vi ∈ V \{v} corresponds to exactly one outer tangent line T r (vi \|v), thereby exactly one breakpoint trL (vi \|v). Such one-to-one correspondence can be easily done in O(1) time. Therefore, equivalently we are computing sequences of points in V \{v}, instead of breakpoints. In the following, we consider two cases about the relative position between L and Cγ (v), (1) L intersects with Cγ (v) at zero or one point, (2) L intersects with Cγ (v) at two points. Case (1): Let θL be the angle of the upward direction along L. See Figure 4(a). We classify the points in V \{v} by their polar angles with respect to v. Let P1 (v) denote the sequence of those points with polar angles in the interval (θL , θL + π) and sorted in CCW order. Similarly, let P2 (v) be the sequence of points with polar angles in (θL + π, θL ) and sorted in CCW order. Obviously, P1 (v) and P2 (v) together satisfy condition (c). (Note that points with polar angles θL and θL + π are ignored, since they correspond to outer tangent lines parallel to L.) By general position assumption, we can observe that, for any two distinct points vi , vj in P1 (v), trL (vi \|v) is strictly above trL (vj \|v) if and only if vi precedes vj in P1 (v). Thus, the ordering of points in P1 (v) implicitly describes an ordering of their corresponding breakpoints in decreasing y-coordinates. Similarly, the ordering in P2 (v) implies an ordering of corresponding breakpoints in decreasing y-coordinates. It follows that both P1 (v) and P2 (v) satisfy condition (b). As for condition (a), both P1 (v) and P2 (v) are of length O(n) by definition. Also, since we have pre-computed the sequence P (v) as all points in V \{v} 12 Hung-I Yu, Tien-Ching Lin, and D. T. Lee (a) no intersection (b) two intersection points Fig. 4. Two subcases about how Cγ (v) intersects C. sorted in CCW order, P1 (v) and P2 (v) can be implicitly represented as concatenations of subsequences of P (v). This can be done in O(log n) time by searching in P (v) the foremost elements with polar angles larger than θL and θL + π, respectively. Case (2): Suppose that the two intersection points between L and Cγ (v) are c1 and c2 , where c1 is above c2 . Let θ1 = θ1′ + π/2 and θ2 = θ2′ + π/2, in which θ1′ and θ2′ are respectively the polar angles of c1 and c2 with respect to v. See Figure 4(b). By assumption, we have that θL < θ1′ < θL + π/2 < θ2′ < θL + π, which implies that θ1 ∈ (θL , θL + π) and θ2 ∈ (θL + π, θL ). We divide the points in V \{v} into four sequences P1 (v), P2 (v), P3 (v), and P4 (v) by their polar angles with respect to v. P1 (v) consists of points with polar angles in (θL , θ1 ), P2 (v) in [θ1 , θL + π), P3 (v) in (θL + π, θ2 ], and P4 (v) in (θ2 , θL ), all sorted in CCW order. It follows that the four sequences satisfy conditions (c). Condition (a) and (b) hold for P1 (v) and P4 (v) from similar discussion as above. However, for any two distinct points vi , vj in P2 (v), we can observe that trL (vi \|v) is strictly below trL (vj \|v) if and only if vi precedes vj in P2 (v). Similarly, the argument holds for P3 (v). Thus, what satisfy condition (b) are actually the reverse sequences of P2 (v) and P3 (v), which can also be obtained in O(log n) time, satisfying condition (a). ⊓ ⊔ By Lemma 13.(c), searching in L × T r (v) is equivalent to searching in the O(1) sequences of breakpoints, which can be computed more efficiently than the obvious way. Besides, we can also obtain a symmetrical lemma constructing sequences for L × T l (v). In the following, we show how to perform a binary search within these sequences. Lemma 14 With an O(n2 log n)-time preprocessing, given an arbitrary line L, a local optimal point x∗L can be computed in O(n log2 n) time. The (1\|1)R -Centroid Problem on the Plane 13 Proof. By Lemma 12, the searching of x∗L can be done within L × T and L × C(V ). L × T can be further divided into L × T r (v) and L × T l (v) for each v ∈ V . By Lemma 13, these 2n sets can be replaced by O(n) sorted sequences of breakpoints on L. Besides, L × C(V ) consists of no more than 2n breakpoints, which can be computed and arranged into a sorted sequence in decreasing ycoordinates. Therefore, we can construct N0 = O(n) sequences P1 , P2 , · · · , PN0 of breakpoints, each of length O(n) and sorted in decreasing y-coordinates. The searching in the N0 sorted sequences is done by performing parametric search for parallel binary searches, introduced in [1]. The technique we used here is similar to the algorithm in [1], but uses a different weighting scheme. For each sorted sequence Pj , 1 ≤ j ≤ N0 , we first obtain its middle element xj , and associate xj with a weight mj equal to the number of elements in Pj . Then, we compute the weighted defined as the ele0 middle elements, P median [18] of the NP P ment x such that {m \|x is above x} ≥ m /2 and {m \|xj is below x} ≥ j j j j P mj /2. Finally, we apply Lemma 10 on the point x. If x is a strong (1\|1)R centroid, of course it is local optimal. If not, Assumption 1 holds and W edge(x) can be computed. If W edge(x) is sideward, a local optimal point x∗L = x is directly found. Otherwise, W edge(x) is either upward or downward, and thus all breakpoints on the opposite half can be pruned by Lemma 9. The pruning makes a portion of sequences, that possesses over half of total breakpoints by the definition of weighted median, lose at least a quarter of their elements. Hence, at least one-eighths of breakpoints are pruned. By repeating the above process, we can find x∗L in at most O(log n) iterations. The time complexity for finding x∗L is analyzed as follows. By Lemma 13, constructing sorted sequences for L × T r (v) and L × T l (v) for all v ∈ V takes O(n log n) time. Computing and sorting L × C(V ) also takes O(n log n) time. There are at most O(log n) iterations of the pruning process. At each iteration, the N0 middle elements and their weighted median x can be obtained in O(N0 ) = O(n) time by the linear-time weighted selection algorithm [18]. Then, the computation of W edge(x) takes O(n log n) time by Lemma 10. Finally, the pruning of those sequences can be done in O(n) time. In summary, the searching ⊓ ⊔ of x∗L requires O(n log n) + O(log n) × O(n log n) = O(n log2 n) time. We remark that, by Lemma 14, it is easy to obtain an intermediate result for the (1\|1)R -centroid problem on the plane. By Lemma 5, there exists a (1\|1)R centroid in T × T , T × C(V ), and C(V ) × C(V ). By applying Lemma 14 to the O(n2 ) lines in T , the local optimum among the intersection points in T × T and T × C(V ) can be obtained in O(n3 log2 n) time. By applying Theorem 1 on the O(n2 ) intersection points in C(V )× C(V ), the local optimum among them can be obtained in O(n3 log n) time. Thus, we can find a (1\|1)R -centroid in O(n3 log2 n) time, a nearly O(n2 ) improvement over the O(n5 log n) bound in [5]. 4 (1\|1)R -Centroid on the Plane In this section, we study the (1\|1)R -centroid problem and propose an improved algorithm of time complexity O(n2 log n). This algorithm is as efficient as the 14 Hung-I Yu, Tien-Ching Lin, and D. T. Lee best-so-far algorithm for the (1\|1)-centroid problem, but based on a completely different approach. In Subsection 4.1, we extend the algorithm of Lemma 14 to develop a procedure allowing us to prune candidate points with respect to a given vertical line. Then, in Subsection 4.2, we show how to compute a (1\|1)R -centroid in O(n2 log n) time based on this newly-developed pruning procedure. 4.1 Pruning with Respect to a Vertical Line Let L be an arbitrary vertical line on the plane. We call the half-plane strictly to the left of L the left plane of L and the one strictly to its right the right plane of L. A sideward wedge of some point on L is said to be rightward (leftward ) if it intersects the right (left) plane of L. We can observe that, if there is some point x ∈ L such that W edge(x) is rightward, every point x′ on the left plane of L can be pruned, since W ∗ (x′ ) ≥ W ∗ (x) by Lemma 9. Similarly, if W edge(x) is leftward, points on the right plane of L can be pruned. Although the power of wedges is not fully exerted in this way, pruning via vertical lines and sideward wedges is superior than directly via wedges due to predictable pruning regions. Therefore, in this subsection we describe how to design a procedure that enables us to prune either the left or the right plane of a given vertical line L. As mentioned above, the key point is the searching of sideward wedges on L. It is achieved by carrying out three conditional phases. In the first phase, we try to find some proper breakpoints with sideward wedges. If failed, we pick some representative point in the second phase and check its wedge to determine whether or not sideward wedges exist. Finally, in case of their nonexistence, we show that their functional alternative can be computed, called the pseudo wedge, that still allows us to prune the left or right plane of L. In the following, we develop a series of lemmas to demonstrate the details of the three phases. Property 1 Given a point x ∈ L, for each possible direction of W edge(x), the corresponding CA(x) satisfies the following conditions: Upward – CA(x) ⊆ (0, π), Downward – CA(x) ⊆ (π, 2π), Rightward – 0 ∈ CA(x), Leftward – π ∈ CA(x). Proof. When W edge(x) is upward, by definition the beginning angle θb and the ending angle θe of CA(x) must satisfy that both half-planes H(F (θb \|x), y(θb \|x)) and H(F (θe \|x), y(θe \|x)) include the half-line of L above x. It follows that 0 ≤ θb , θe ≤ π, and thus CA(x) ⊆ (0, π). (Recall that θb , θe ∈ / CA(x).) The case that W edge(x) is downward can be proved in a symmetric way. When W edge(x) is rightward, we can see that H(F (θb \|x), y(θb \|x)) must not contain the half-line of L above x, and thus π < θb < 2π. By similar arguments, 0 < θe < π. Therefore, counterclockwise covering angles from θb to θe , CA(x) must include the angle 0. The case that W edge(x) is leftward can be symmetrically proved. ⊓ ⊔ The (1\|1)R -Centroid Problem on the Plane 15 Lemma 15 Let x1 , x2 be two points on L, where x1 is strictly above x2 . For any angle 0 ≤ θ ≤ π, W (θ\|x1 ) ≤ W (θ\|x2 ). Symmetrically, for π ≤ θ ≤ 2π, W (θ\|x2 ) ≤ W (θ\|x1 ). Proof. For any angle 0 ≤ θ ≤ π, we can observe that H − (B(θ\|x1 ), y(θ\|x1 )) ⊂ H − (B(θ\|x2 ), y(θ\|x2 )), since x1 is strictly above x2 . It follows that W (θ\|x1 ) ≤ W (θ\|x2 ). The second claim also holds by symmetric arguments. ⊓ ⊔ Lemma 16 Let x be an arbitrary point on L. If W edge(x) is either upward or downward, for any point x′ ∈ L\W edge(x), W edge(x′ ) has the same direction as W edge(x). Proof. By symmetry, we prove that, if W edge(x) is upward, W edge(x′ ) is also upward for every x′ ∈ L strictly below x. By Property 1, the fact that W edge(x) is upward means that CA(x) ⊂ (0, π) and thus M A(x) ⊂ (0, π). Let x′ be a point on L strictly below x. By Lemma 15, we have that W (θ\|x′ ) ≥ W (θ\|x) for 0 < θ < π and W (θ\|x′ ) ≤ W (θ\|x) for π ≤ θ ≤ 2π. It follows that M A(x′ ) ⊂ (0, π) and CA(x′ ) ⊂ (0, π), so W edge(x′ ) is upward as well. ⊓ ⊔ Following from this lemma, if there exist two arbitrary points x1 and x2 on L with their wedges downward and upward, respectively, we can derive that x1 must be strictly above x2 , and that points with sideward wedges or even strong (1\|1)R -centroids can locate only between x1 and x2 . Thus, we can find sideward wedges between some specified downward and upward wedges. Let xD be the lowermost breakpoint on L with its wedge downward, xU the uppermost breakpoint on L with its wedge upward, and GDU the open segment xD xU \{xD , xU }. (For ease of discussion, we assume that both xD and xU exist on L, and show how to resolve this assumption later by constructing a bounded box.) Again, xD is strictly above xU . Also, we have the following corollary by their definitions. Corollary 17 If there exist breakpoints in the segment GDU , for any such breakpoint x, either x is a strong (1\|1)R -centroid or W edge(x) is sideward. Given xD and xU , the first phase can thus be done by checking whether there exist breakpoints in GDU and picking any of them if exist. Supposing that the picked one is not a strong (1\|1)R -centroid, a sideward wedge is found by Corollary 17 and can be used for pruning. Notice that, when there are two or more such breakpoints, one may question whether their wedges are of the same direction, as different directions result in inconsistent pruning results. The following lemma answers the question in the positive. Lemma 18 Let x1 , x2 be two distinct points on L, where x1 is strictly above x2 and none of them is a strong (1\|1)R -centroid. If W edge(x1 ) and W edge(x2 ) are both sideward, they are either both rightward or both leftward. Proof. We prove this lemma by contradiction. By symmetry, suppose the case that W edge(x1 ) is rightward and W edge(x2 ) is leftward. This case can be further divided into two subcases by whether or not CA(x1 ) and CA(x2 ) intersect. 16 Hung-I Yu, Tien-Ching Lin, and D. T. Lee Fig. 5. CH(C(S)) intersects L between x1 and x2 . Consider first that CA(x1 ) does not intersect CA(x2 ). Because W edge(x1 ) is rightward, 0 ∈ CA(x1 ) by Property 1. Thus, there exists an angle θ, 0 < θ < π, such that θ ∈ M A(x1 ). Since x1 is strictly above x2 , by Lemma 15 we have that W ∗ (x1 ) = W (θ\|x1 ) ≤ W (θ\|x2 ) ≤ W ∗ (x2 ). Furthermore, since W edge(x2 ) is leftward, we can see that x1 ∈ / W edge(x2 ) and therefore W ∗ (x1 ) ≥ ∗ W (x2 ) by Lemma 9. It follows that W (θ\|x2 ) = W ∗ (x2 ) and thus θ ∈ M A(x2 ). By definition, M A(x1 ) ⊆ CA(x1 ) and M A(x2 ) ⊆ CA(x2 ), which implies that CA(x1 ) and CA(x2 ) intersect at θ, contradicting the subcase assumption. When CA(x1 ) intersects CA(x2 ), their intersection must be completely included in either (0, π) or (π, 2π) due to Assumption 1. By symmetry, we assume the latter subcase. Using similar arguments as above, we can find an angle θ′ , where 0 < θ′ < π, such that θ′ ∈ M A(x1 ) and θ′ ∈ M A(x2 ). This is a contradiction, since θ′ ∈ / (π, 2π). Since both subcases do not hold, the lemma is proved. ⊓ ⊔ The second phase deals with the case that no breakpoint exists between xD and xU by determining the wedge direction of an arbitrary inner point in GDU . We begin with several auxiliary lemmas. Lemma 19 Let x1 , x2 be two distinct points on L such that W ∗ (x1 ) = W ∗ (x2 ) and x1 is strictly above x2 . There exists at least one breakpoint in the segment (a) x1 x2 \{x2 }, if M A(x2 ) intersects (0, π) but M A(x1 ) does not, (b) x1 x2 \{x1 }, if M A(x1 ) intersects (π, 2π) but M A(x2 ) does not. Proof. By symmetry, we only show the correctness of condition (a). From its assumption, there exists an angle θ, where 0 < θ < π, such that θ ∈ M A(x2 ). Let T S = V H − (B(θ\|x2 ), y(θ\|x2 )). By definition, we have that W (S) = W (θ\|x2 ) = W ∗ (x2 ) = W ∗ (x1 ) and CH(C(S)) ⊂ H − (F (θ\|x2 ), y(θ\|x2 )), which implies that CH(C(S)) is strictly above F (θ\|x2 ). (See Figure 5.) We first claim that CH(C(S)) intersects L. If not, there must exist an angle θ′ , where 0 < θ′ < π, such that CH(C(S)) ⊂ H − (F (θ′ \|x1 ), y(θ′ \|x1 )), that is, The (1\|1)R -Centroid Problem on the Plane 17 S ⊂ H − (B(θ′ \|x1 ), y(θ′ \|x1 )). By definition, W ∗ (x1 ) ≥ W (θ′ \|x1 ) ≥ W (S). Since W ∗ (x1 ) = W (S), W (θ′ \|x1 ) = W ∗ (x1 ) and thus θ′ ∈ M A(x1 ), which contradicts the condition that M A(x1 ) does not intersect (0, π). Thus, the claim holds. When CH(C(S)) intersects L, x1 locates either inside or outside CH(C(S)). Since x2 locates outside CH(C(S)), in the former case the boundary of CH(C(S)) intersects x1 x2 \{x2 } and forms a breakpoint, thereby proves condition (a). On the other hand, if x1 is outside CH(C(S)), again there exists an angle θ′′ such that CH(C(S)) ⊂ H − (F (θ′′ \|x1 ), y(θ′′ \|x1 )). By similar arguments, we can show that θ′′ ∈ M A(x1 ). By assumption, θ′′ must belong to (π, 2π), which implies that CH(C(S)) is strictly below F (θ′′ \|x1 ). Since CH(C(S)) is strictly above F (θ\|x2 ) as mentioned, any intersection point between CH(C(S)) and L should be inner ⊓ ⊔ to x1 x2 . Therefore, the lemma holds. Lemma 20 Let G be a line segment connecting two consecutive breakpoints on L. For any two distinct points x1 , x2 inner to G, W ∗ (x1 ) = W ∗ (x2 ). Proof. Suppose to the contrary that W ∗ (x1 ) 6= W ∗ (x2 ). By Lemma 11, there exists at least one breakpoint in x1 x2 , which contradicts the definition of G. Thus, the lemma holds. ⊓ ⊔ Lemma 21 When there is no breakpoint between xD and xU , any two distinct points x1 , x2 in GDU have the same wedge direction, if they are not strong (1\|1)R centroids. Proof. Suppose by contradiction that the directions of their wedges are different. By Lemmas 16 and 18, there are only two possible cases. (1) W edge(x1 ) is downward, and W edge(x2 ) is either sideward or upward. (2) W edge(x1 ) is sideward, and W edge(x2 ) is upward. In the following, we show that both cases do not hold. Case (1): Because W edge(x1 ) is downward, we have that CA(x1 ) ⊆ (π, 2π) by Property 1 and thus M A(x1 ) does not intersect (0, π). On the other hand, whether W edge(x2 ) is sideward or upward, we can see that CA(x2 ) and M A(x2 ) intersect (0, π) by again Property 1. Since W ∗ (x1 ) = W ∗ (x2 ) by Lemma 20, the status of the two points satisfies the condition (a) of Lemma 19, so at least one breakpoint exists between x1 and x2 . By definitions of x1 and x2 , this breakpoint is inner to GDU , thereby contradicts the assumption. Therefore, Case (1) does not hold. Case (2): The proof of Case (2) is symmetric to that of Case (1). The condition (b) of Lemma 19 can be applied similarly to show the existence of at least one breakpoint between x1 and x2 , again a contradiction. Combining the above discussions, we prove that the wedges of x1 and x2 are of the same direction, thereby completes the proof of this lemma. ⊓ ⊔ 18 Hung-I Yu, Tien-Ching Lin, and D. T. Lee This lemma enables us to pick an arbitrary point in GDU , e.g., the bisector point xB of xD and xU , as the representative of all inner points in GDU . If xB is not a strong (1\|1)R -centroid and W edge(xB ) is sideward, the second phase finishes with a sideward wedge found. Otherwise, if W edge(xB ) is downward or upward, we can derive the following and have to invoke the third phase. Lemma 22 If there is no breakpoint between xD and xU and W edge(xB ) is not sideward, there exist neither strong (1\|1)R -centroids nor points with sideward wedges on L. Proof. By Lemma 16, this lemma holds for points not in GDU . Without loss of generality, suppose that W edge(xB ) is downward. For all points in GDU above xB , the lemma holds by again Lemma 16. Consider an arbitrary point x ∈ xB xU \{xB , xU }. We first show that x is not a strong (1\|1)R -centroid. Suppose to the contrary that x really is. By definition, we have that δ(CA(x)) > π, and thus CA(x) and M A(x) intersect (0, π). On the other hand, CA(xB ) and M A(xB ) do not intersect (0, π) due to downward W edge(xB ) and Property 1. Since W ∗ (xB ) = W ∗ (x) by Lemma 20, applying the condition (a) of Lemma 19 to xB and x shows that at least one breakpoint exists between them, which contradicts the no-breakpoint assumption. Now that x is not a strong (1\|1)R -centroid, it must have a downward wedge, as xB does by Lemma 21. Therefore, the lemma holds for all points on L. ⊓ ⊔ When L satisfies Lemma 22, it consists of only points with downward or upward wedges, and is said to be non-leaning. Obviously, our pruning strategy via sideward wedges could not apply to such non-leaning lines. The third phase overcomes this obstacle by constructing a functional alternative of sideward wedges, called the pseudo wedge, on either xD or xU , so that pruning with respect to L is still achievable. Again, we start with auxiliary lemmas. Lemma 23 If L is non-leaning, the following statements hold: (a) W ∗ (xD ) 6= W ∗ (xU ), (b) W ∗ (x) = max{W ∗ (xD ), W ∗ (xU )} for all points x ∈ GDU . Proof. We prove the correctness of statement (a) by contradiction, and suppose that W ∗ (xD ) = W ∗ (xU ). Besides, the fact that L is non-leaning implies that no breakpoint exists in GDU . By Lemmas 22 and 21, the wedges of all points in GDU are of the same direction, either downward or upward. Suppose the downward case by symmetry, and pick an arbitrary point in GDU , say, xB . Since W edge(xB ) is downward, we have that M A(xB ) does not intersect (0, π). Oppositely, by definition W edge(xU ) is upward, so CA(xU ) and M A(xU ) are included in (0, π). Because xB is strictly above xU and W ∗ (xD ) = W ∗ (xU ), according to the condition (a) of Lemma 19, there exists at least one breakpoint in xB xU \{xU }, which is a contradiction. Therefore, statement (a) holds. The proof of statement (b) is also done by contradiction. By symmetry, assume that W ∗ (xD ) > W ∗ (xU ) in statement (a). Consider an arbitrary point The (1\|1)R -Centroid Problem on the Plane 19 x ∈ GDU . By Lemma 7, we have that W ∗ (x) ≤ max{W ∗ (xD ), W ∗ (xU )} = W ∗ (xD ). Suppose that the equality does not hold. Then, by Lemma 11, at least one breakpoint exists in the segment xD x\{xD }, contradicting the no-breakpoint fact. Thus, W ∗ (x) = W ∗ (xD ) and statement (b) holds. ⊓ ⊔ Let W1 = max{W ∗ (xD ), W ∗ (xU )}. We are going to define the pseudo wedge on either xU or xD , depending on which one has the smaller weight loss. We consider first the case that W ∗ (xD ) > W ∗ (xU ), and obtain the following. Lemma 24 If L is non-leaning and W ∗ (xD ) > W ∗ (xU ), there exists one angle θ for xU , where π ≤ θ ≤ 2π, such that W (H(B(θ\|xU ), y(θ\|xU ))) ≥ W1 . Proof. We first show that there exists at least a subset S ⊆ V with W (S) = W1 , such that xU locates on the upper boundary of CH(C(S)). Let x be the point strictly above but arbitrarily close to xU on L. By Lemma 23, W ∗ (x) = W1 , hence W ∗ (x) > W ∗ (xU ) by case assumption. It follows that xU ∈ W edge(x) by Lemma 9 and W edge(x) must be downward. By Property 1, we have that CA(x) ⊆ (π, 2π). Thus, there exists an angle θ′ ∈ M A(x), where π < θ′ < 2π, ′ such that W (H − \|x), y(θ′ \|x))) = W (θ′ \|x) = W1 . T(B(θ − Let S = V H (B(θ′ \|x), y(θ′ \|x)). Since W ∗ (xU ) < W1 = W (S), xU is inside CH(C(S)) by Lemma 2. Oppositely, by the definition of S, x is outside the convex hull CH(C(S)). It implies that xU is the topmost intersection point between CH(C(S)) and L, hence on the upper boundary of CH(C(S)). (It is possible that xU locates at the leftmost or the rightmost point of CH(C(S)).) The claimed angle θ is obtained as follows. Since xU is a boundary point of CH(C(S)), there exists a line F passing through xU and tangent to CH(C(S)). Let θ be the angle satisfying that F (θ\|xU ) = F , π ≤ θ ≤ 2π, and CH(C(S)) ⊂ H(F (θ\|xU ), y(θ\|xU )). Obviously, we have that S ⊂ H(B(θ\|xU ), y(θ\|xU )) and thus W (H(B(θ\|xU ), y(θ\|xU ))) ≥ W ∗ (S) = W1 . ⊓ ⊔ Let θU be an arbitrary angle satisfying the conditions of Lemma 24. We apply the line F (θU \|xU ) for trimming the region of W edge(xU ), so that a sideward wedge can be obtained. Let P W (xU ), called the pseudo wedge of xU , denote the intersection of W edge(xU ) and H(F (θU \|xU ), y(θU \|xU )). Deriving from the three facts that W edge(xU ) is upward, δ(W edge(xU )) < π, and π ≤ θU ≤ 2π, we can observe that either P W (xU ) is xU itself, or it intersects only one of the right and left plane of L. In the two circumstances, P W (xU ) is said to be null or sideward, respectively. The pseudo wedge has similar functionality as wedges, as shown in the following corollary. Corollary 25 For any point x′ ∈ / P W (xU ), W ∗ (x′ ) ≥ W ∗ (xU ). Proof. If x′ ∈ / W edge(xU ), the lemma directly holds by Lemma 9. Otherwise, we have that x′ ∈ / H(F (θ\|xU ), y(θ\|xU )) and thus H(B(θ\|x′ ), y(θ\|x′ )) contains H(B(θ\|xU ), y(θ\|xU )). Then, by Lemma 24, W ∗ (x′ ) ≥ W (H(B(θ\|x′ ), y(θ\|x′ ))) ≥ W (H(B(θ\|xU ), y(θ\|xU ))) ≥ W1 , thereby completes the proof. ⊓ ⊔ 20 Hung-I Yu, Tien-Ching Lin, and D. T. Lee By this lemma, if P W (xU ) is found to be sideward, points on the opposite half-plane with respect to L can be pruned. If P W (xU ) is null, xU becomes another kind of strong (1\|1)R -centroids, in the meaning that it is also an immediate solution to the (1\|1)R -centroid problem. Without confusion, we call xU a conditional (1\|1)R -centroid in the latter case. On the other hand, considering the reverse case that W ∗ (xD ) < W ∗ (xU ), we can also obtain an angle xD and a pseudo wedge P W (xD ) for xD by symmetric arguments. Then, either P W (xD ) is sideward and the opposite side of L can be pruned, or P W (xD ) itself is a conditional (1\|1)R -centroid. Thus, the third phase solves the problem of the nonexistence of sideward wedges. Recall that the three phases of searching sideward wedges is based on the existence of xD and xU on L, which was not guaranteed before. Here we show that, by constructing appropriate border lines, we can guarantee the existence of xD and xU while searching between these border lines. The bounding box is defined as the smallest axis-aligned rectangle that encloses all circles in C(V ). Obviously, any point x outside the box satisfies that W ∗ (x) = W (V ) and must not be a (1\|1)R -centroid. Thus, given a vertical line not intersecting the box, the half-plane to be pruned is trivially decided. Moreover, let Ttop and Tbtm be two arbitrary horizontal lines strictly above and below the bounding box, respectively. We can obtain the following. Lemma 26 Let L be an arbitrary vertical line intersecting the bounding box, and x′D and x′U denote its intersection points with Ttop and Tbtm , respectively. W edge(x′D ) is downward and W edge(x′U ) is upward. Proof. Consider the case about W edge(x′D ). As described above, we know the fact that W ∗ (xD ) = W (V ). Let θ be an arbitrary angle with 0 ≤ θ ≤ π. We can observe that H − (F (θ\|x′D ), y(θ\|x′D )) cannot contain all circles in C(V ), that is, V 6⊂ H − (B(θ\|x′D ), y(θ\|x′D )). This implies that W (θ\|x′D ) < W ∗ (x′D ) and θ ∈ / M A(x′D ). Therefore, we have that M A(x′D ) ⊂ (π, 2π) and W edge(x′D ) is downward by Property 1. By similar arguments, we can show that W edge(x′U ) is upward. Thus, the lemma holds. ⊓ ⊔ According to this lemma, by inserting Ttop and Tbtm into T , the existence of xD and xU is enforced for any vertical line intersecting the bounding box. Besides, it is obvious to see that the insertion does not affect the correctness of all lemmas developed so far. Summarizing the above discussion, the whole picture of our desired pruning procedure can be described as follows. In the beginning, we perform a preprocessing to obtain the bounding box and then add Ttop and Tbtm into T . Now, given a vertical line L, whether to prune its left or right plane can be determined by the following steps. 1. If L does not intersect the bounding box, prune the half-plane not containing the box. 2. Compute xD and xU on L. The (1\|1)R -Centroid Problem on the Plane 21 3. Find a sideward wedge or pseudo wedge via three forementioned phases. (Terminate whenever a strong or conditional (1\|1)R -centroid is found.) (a) If breakpoints exist between xD and xU , pick any of them and check it. (b) If no such breakpoint, decide whether L is non-leaning by checking xB . (c) If L is non-leaning, compute P W (xU ) or P W (xD ) depending on which of xU and xD has smaller weight loss. 4. Prune the right or left plane of L according to the direction of the sideward wedge or pseudo wedge. The correctness of this procedure follows from the developed lemmas. Any vertical line not intersecting the bounding box is trivially dealt with in Step 1, due to the property of the box. When L intersects the box, by Lemma 26, xD and xU can certainly be found in Step 2. The three sub-steps of Step 3 correspond to the three searching phases. When L is not non-leaning, a sideward wedge is found, either at some breakpoint between xD and xU in Step 3(a) by Corollary 17, or at xB in Step 3(b) by Lemma 21. Otherwise, according to Lemma 24 or its symmetric version, a pseudo wedge can be built in Step 3(c) for xU or xD , respectively. Finally in Step 4, whether to prune the left or right plane of L can be determined via the just-found sideward wedge or pseudo wedge, by respectively Lemma 9 or Corollary 25. The time complexity of this procedure is analyzed as follows. The preprocessing for computing the bounding box trivially takes O(n) time. In Step 1, any vertical line not intersecting the box can be identified and dealt with in O(1) time. Finding xD and xU in Step 2 requires the help of the binary-search algorithm developed in 3.2. Although the algorithm is designed to find a local optimal point, we can easily observe that slightly modifying its objective makes it applicable to this purpose without changing its time complexity. Thus, Step 2 can be done in O(n log2 n) time by Lemma 14. In Step 3(a), all breakpoints between xD and xU can be found in O(n log n) time as follows. As done in Lemma 14, we first list all breakpoints on L by O(n) sorted sequences of length O(n), which takes O(n log n) time. Then, by performing binary search with the y-coordinates of xD and xU , we can find within each sequence the breakpoints between them in O(log n) time. In Step 3(a) or 3(b), checking a picked point x is done by computing CA(x), that requires O(n log n) time by Lemma 10. To compute the pseudo wedge in Step 3(c), the angle θU satisfying Lemma 24, or symmetrically θD , can be computed in O(n log n) time by sweeping technique as in Lemma 10. Thus, P W (xU ) or P W (xD ) can be computed in O(n log n) time. Finally, the pruning decision in Step 4 takes O(1) time. Summarizing the above, these steps require O(n log2 n) time in total. Since the invocation of Lemma 14 needs an additional O(n2 log n)-time preprocessing, we have the following result. Lemma 27 With an O(n2 log n)-time preprocessing, whether to prune the right or left plane of a given vertical line L can be determined in O(n log2 n) time. 22 4.2 Hung-I Yu, Tien-Ching Lin, and D. T. Lee Searching on the Euclidean Plane In this subsection, we come back to the (1\|1)R -centroid problem. Recall that, by Lemma 5, at least one (1\|1)R -centroid can be found in the three sets of intersection points T × T , C(V ) × T , and C(V ) × C(V ), which consist of total O(n4 ) points. Let L denote the set of all vertical lines passing through these O(n4 ) intersection points. By definition, there exists a vertical line L∗ ∈ L such that its local optimal point is a (1\|1)R -centroid. Conceptually, with the help of Lemma 27, L∗ can be derived by applying prune-and-search approach to L: pick the vertical line L from L with median x-coordinates, determine by Lemma 27 whether the right or left plane of L should be pruned, discard lines of L in the pruned half-plane, and repeat above until two vertical lines left. Obviously, it costs too much if this approach is carried out by explicitly generating and sorting the O(n4 ) lines. However, by separately dealing with each of the three sets, we can implicitly maintain sorted sequences of these lines and apply the prune-and-search approach. Let LT , LM , and LC be the sets of all vertical lines passing through the intersection points in T × T , C(V ) × T , and C(V ) × C(V ), respectively. A local optimal line of LT is a vertical line L∗t such that its local optimal point has weight loss no larger than those of points in T × T . The local optimal lines L∗m and L∗c can be similarly defined for LM and LC , respectively. We will adopt different prune-and-search techniques to find the local optimal lines in the three sets, as shown in the following lemmas. Lemma 28 A local optimal line L∗t of LT can be found in O(n2 log n) time. Proof. Let N1 = \|T \|. By definition, there are (N1 )2 intersection points in T × T and (N1 )2 vertical lines in LT . For efficiently searching within these vertical lines, we apply the ingenious idea of parametric search via parallel sorting algorithms, proposed by Megiddo [15]. Consider two arbitrary lines Tg , Th ∈ T . If they are not parallel, let tgh be their intersection point and Lgh be the vertical line passing through tgh . Suppose that Tg is above Th in the left plane of Lgh . If applying Lemma 27 to Lgh prunes its right plane, Tg is above Th in the remained left plane. On the other hand, if the left plane of Lgh is pruned, Th is above Tg in the remained right plane. Therefore, Lgh can be treated as a “comparison” between Tg and Th , in the sense that applying Lemma 27 to Lgh determines their ordering in the remained half-plane. It also decides the ordering of their intersection points with the undetermined local optimal line L∗t , since the pruning ensures that a local optimal line stays in the remained half-plane. It follows that, by resolving comparisons, the process of pruning vertical lines in LT to find L∗t can be reduced to the problem of determining the ordering of the intersection points of the N1 lines with L∗t , or say, the sorting of these intersection points on L∗t . While resolving comparisons during the sorting process, we can simultaneously maintain the remained half-plane by two vertical lines as its boundaries. Thus, after resolving all comparisons in LT , one of the two boundaries must be a local optimal line. As we know, the most efficient way to The (1\|1)R -Centroid Problem on the Plane 23 obtain the ordering is to apply some optimal sorting algorithm AS , which needs to resolve only O(N1 log N1 ) comparisons, instead of (N1 )2 comparisons. Since resolving each comparison takes O(n log2 n) time by Lemma 27, the sorting is done in O(n log2 n) × O(N1 log2 N1 ) = O(n3 log3 n) time, so is the finding of L∗t . However, Megiddo [15] observed that, when multiple comparisons can be indirectly resolved in a batch, simulating parallel sorting algorithms in a sequential way naturally provides the scheme for batching comparisons, thereby outperform the case of applying AS . Let AP be an arbitrary cost-optimal parallel sorting algorithm that runs in O(log n) steps on O(n) processors, e.g., the parallel merge sort in [2]. Using AP to sort the N1 lines in LT on L∗t takes O(log N1 ) parallel steps. At each parallel step, there are k = O(N1 ) comparisons L1 , L2 , · · · , Lk to be resolved. We select the one with median x-coordinate among them, which is supposed to be some Li . If applying Lemma 27 to Li prunes its left plane, for each comparison Lj to the left of Li , the ordering of the corresponding lines of Lj in the remained right plane of Li is directly known. Thus, the O(k/2) comparisons to the left of Li are indirectly resolved in O(k/2) time. If otherwise the right plane of Li is pruned, the O(k/2) comparisons to its right are resolved in O(k/2) time. By repeating this process of selecting medians and pruning on the remaining elements O(log k) times, all k comparisons can be resolved, which takes O(n log2 n) × O(log k) + O(k + k/2 + k/4 + · · · ) = O(n log3 n) + O(N1 ) = O(n2 ) time. Therefore, going through O(log N1 ) parallel steps of AP requires O(n2 log N1 ) = O(n2 log n) time, which determines the ordering of lines in LT ⊓ ⊔ on L∗t and also computes a local optimal line L∗t . Lemma 29 A local optimal line L∗m of LM can be found in O(n2 log n) time. Proof. To deal with the set LM , we use the ideas similar to the proofs of Lemmas 13 and 14 in order to divide C(V ) × T into sorted sequences of points. Given a fixed circle C = Cγ (u0 ) for some point u0 ∈ V , we show that the intersection points in C × T r (v) and C × T l (v) for each v ∈ V can be grouped into O(1) sequences of length O(n), which are sorted in increasing x-coordinates. Summarizing over all circles in C(V ), there will be total O(n2 ) sequences of length O(n), each of which maps to a sequence of O(n) vertical lines sorted in increasing x-coordinates. Then, finding a local optimal line L∗m can be done by performing prune-and-search to the O(n2 ) sequences of vertical lines via parallel binary searches. The details of these steps are described as follows. First we discuss about the way for grouping intersection points in C × T r (v) and C ×T l (v) for a fixed point v ∈ V , so that each of them can be represented by O(1) subsequences of P (v). By symmetry, only C × T r (v) is considered. Similar to Lemma 13, we are actually computing sequences of points in V \{v} corresponding to these intersection points. For each vi ∈ V \{v}, the outer tangent line T r (vi \|v) may intersect C at two, one, or zero point. Let trC,1 (vi \|v) and trC,2 (vi \|v) denote the first and second points, respectively, at which T r (vi \|v) intersects C along the direction from v to vi . Note that, when T r (vi \|v) intersects C at less than two points, trC,2 (vi \|v) or both of them will be null. In the following, we consider the sequence computation under two cases about the relationship between u0 and v, (1) u0 = v, and (2) u0 6= v. 24 Hung-I Yu, Tien-Ching Lin, and D. T. Lee (a) no intersection (b) two intersection points Fig. 6. Two subcases about how Cγ (v) intersects C. Case (1): Since v coincides with u0 , C × T r (v) is just the set of tangent points trC,1 (vi \|v) for all vi ∈ V \{v}. It is easy to see the the angular sorted sequence P (v) directly corresponds to a sorted sequence of these n − 1 tangent points in CCW order. P (v) can be further partitioned into two sub-sequences P1 (v) and P2 (v), which consist of points in V \{v} with polar angles (with respect to v) in the intervals [0, π) and [π, 2π), respectively. Since they are sorted in CCW order, we have that intersection points corresponding to P2 (v) and to the reverse of P1 (v) are sorted in increasing x-coordinates, as we required. Obviously, P1 (v) and P2 (v) are of length O(n) and can be obtained in O(log n) time. Case (2): Suppose without loss of generality that v locates on the lower left quadrant with respect to u0 , and let θ0 be the polar angle of u0 with respect to v. This case can be further divided into two subcases by whether or not Cγ (v) intersects C at less than two points. Consider first the subcase that they intersect at none or one point (see Figure 6(a).) Let θ3 and θ4 be the angles such that T r (y(θ3 \|v)\|v) and T r (y(θ4 \|v)\|v) are inner tangent to Cγ (v) and C, where θ3 ≤ θ4 (Note that θ3 = θ4 only when the two circles intersect at one point.) For each vi ∈ V \{v}, T r (vi \|v) does not intersect C, if the polar angle of vi with respect to v is neither in [θ0 , θ3 ] nor in [θ4 , θ0 + π]. We can implicitly obtain from P (v) two subsequences P3 (v) and P4 (v), consisting of points with polar angles in [θ0 , θ3 ] and in [θ4 , θ0 + π], respectively. It can be observed that the sequence of points vi listed in P3 (v) corresponds to a sequence of intersection points trC,1 (vi \|v) listed in clockwise (CW) order on C and, moreover, a sequence of trC,2 (vi \|v) listed in CCW order on C. Symmetrically, the sequence of points vj in P4 (v) corresponds to a sequence of trC,1 (vj \|v) in CCW order and a sequence of trC,2 (vj \|v) in CW order. The four implicit sequences of intersection points on C can be further partitioned by a horizontal line Lh passing through The (1\|1)R -Centroid Problem on the Plane 25 its center u0 , so that the resulted sequences are naturally sorted in either increasing or decreasing x-coordinates. Therefore, we can implicitly obtain at most eight sorted sequences of length O(n) in replace of C × T r (v), by appropriately partitioning P (v) in O(log n) time. Consider that Cγ (v) intersects Cγ (u0 ) at two points c5 and c6 , where c5 is to the upper right of c6 (see Figure 6(b).) Let θ5 and θ6 be the angles such that T r (y(θ5 \|v)\|v) and T r (y(θ6 \|v)\|v) are tangent to Cγ (v) at c5 and c6 , respectively. Again, P (v) can be implicitly partitioned into three subsequences P5 (v), P6 (v), and P7 (v), which consists of points with polar angles in [θ0 , θ5 ), [θ5 , θ6 ), and [θ6 , θ0 + π], respectively. By similar observations, P5 (v) corresponds to two sequences of intersection points listed in CW and CCW order, respectively, and P7 (v) corresponds to two sequences listed in CCW and CW order, respectively. However, the sequence of points vi in P6 (v) corresponds to the sequences of trC,1 (vi \|v) and trC,2 (vi \|v) listed in both CCW order. These sequences can also be partitioned by Lh into sequences sorted in x-coordinates. It follows that we can implicitly obtain at most twelve sorted sequences of length O(n) in replace of C × T r (v) in O(log n) time. According to the above discussion, for any two points u, v ∈ V , Cγ (u)×T r (v) and Cγ (u) × T l (v) can be divided into O(1) sequences in O(log n) time, each of which consists of O(n) intersection points on Cγ (u) sorted in increasing xcoordinates. Thus, C(V ) × T can be re-organized as O(n2 ) sorted sequences of length O(n) in O(n2 log n) time, which correspond to O(n2 ) sorted sequences of O(n) vertical lines. Now, we can perform parametric search for parallel binary search to these sequences of vertical lines, by similar techniques used in Lemma 14. For each of the O(n2 ) sequences, its middle element is first obtained and assigned with a weight equal to the sequence length in O(1) time. Then, the weighted median L of these O(n2 ) elements are computed in O(n2 ) time [18]. By applying Lemma 27 to L in O(n log2 n) time, at least one-eighths of total elements can be pruned from these sequences, taking another O(n2 ) time. Therefore, a single iteration of pruning requires O(n2 ) time. After O(log n) such iterations, a local optimal line L∗m can be found in total O(n2 log n) time, thereby proves the lemma. ⊓ ⊔ Lemma 30 A local optimal line L∗c of LC can be found in O(n2 log n) time. Proof. There are at most O(n2 ) points in C(V )×C(V ). Thus, LC can be obtained and sorted according to x-coordinates in O(n2 log n) time. Then, by simply performing binary search with Lemma 27, a local optimal line L∗c can be easily found in O(log n) iterations of pruning, which require total O(n log3 n) time. In summary, the computation takes O(n2 log n) time, and the lemma holds. ⊓ ⊔ By definition, L∗ can be found among L∗t , L∗m , and L∗c , which can be computed in O(n2 log n) time by Lemmas 28, 29, and 30, respectively. Then, a (1\|1)R centroid can be computed as the local optimal point of L∗ in O(n log2 n) time by Lemma 14. Combining with the O(n2 log n)-time preprocessing for computing the angular sorted sequence P (v)s and the bounding box enclosing C(V ), we have the following theorem. 26 Hung-I Yu, Tien-Ching Lin, and D. T. Lee Theorem 3 The (1\|1)R -centroid problem can be solved in O(n2 log n) time. 5 Concluding Remarks In this paper, we revisited the (1\|1)-centroid problem on the Euclidean plane under the consideration of minimal distance constraint between facilities, and proposed an O(n2 log n)-time algorithm, which close the bound gap between this problem and its unconstrained version. Starting from a critical observation on the medianoid solutions, we developed a pruning tool with indefinite region remained after pruning, and made use of it via multi-level structured parametric search approach, which is quite different to the previous approach in [5,11]. Considering distance constraint between facilities in various competitive facility location models is both of theoretical interest and of practical importance. However, similar constraints are rarely seen in the literature. It would be good starting points by introducing the constraint to the facilities between players in the (r\|Xp )-medianoid and (r\|p)-centroid problems, maybe even to the facilities between the same player. References 1. R. Cole, “Slowing down sorting networks to obtain faster sorting algorithms,” Journal of the ACM, vol. 34, no. 1, pp. 200–208, 1987. 2. R. Cole, “Parallel merge sort,” SIAM Journal on Computing, vol. 17, no. 4, pp. 770–785, 1988. 3. A. Dasci, “Conditional Location Problems on Networks and in the Plane,” In: H. A. Eiselt, V. Marianov (eds.), Foundations of Location Analysis, Springer, New York, pp. 179–206, 2011. 4. I. Davydov, Y. Kochetov, and A. Plyasunov, “On the complexity of the (r\|p)centroid problem in the plane,” TOP, vol. 22, no. 2, pp. 614–623, 2013. 5. Z. Drezner, “Competitive location strategies for two facilities,” Regional Science and Urban Economics, vol 12, no. 4, pp. 485–493, 1982. 6. Z. Drezner, and Z. Eitan, “Competitive location in the plane,” Annals of Operations Research, vol. 40, no. 1, pp. 173–193, 1992. 7. H. A. Eiselt and G. Laporte, “Sequential location problems,” European Journal of Operational Research, vol. 96, pp. 217–231, 1997. 8. H. A. Eiselt, G. Laporte, and J.-F. Thisse, “Competitive location models: a framework and bibliography,” Transportation Science, vol. 27, pp. 44–54, 1993. 9. H. A. Eiselt, V. Marianov, Vladimir and T. Drezner, Tammy, “Competitive Location Models,” In: G. Laporte, S. Nickel, F. Saldanha da Gama (eds.), Location Science, Springer International Publishing, pp. 365–398, 2015. 10. S. L. Hakimi, “On locating new facilities in a competitive environment,” European Journal of Operational Research, vol. 12, no. 1, pp. 29–35, 1983. 11. S. L. Hakimi, “Locations with spatial interactions: competitive locations and games,” In: P.B. Mirchandani, R.L. Francis (eds), Discrete location theory, Wiley, New York, pp. 439–478, 1990. 12. P. Hansen, J.-F. Thisse, and R. W. Wendell, “Equilibrium analysis for voting and competitive location problems,” In: P. B. Mirchandani, R. L. Francis RL (eds), Discrete location theory, Wiley, New York, pp 479–501, 1990. The (1\|1)R -Centroid Problem on the Plane 27 13. H. Hotelling, “Stability in competition,” Economic Journal, vol. 39, 41–57, 1929. 14. D. T. Lee, Y. F. Wu, “Geometric complexity of some location problems,” Algorithmica, vol. 1, no. 1, pp. 193–211, 1986. 15. N. Megiddo, “Applying parallel computation algorithms in the design of serial algorithms,” Journal of the ACM, vol. 30, no. 4, pp. 852–865, 1983. 16. N. Megiddo, “Linear-time algorithms for linear programming in R3 and related problems,” SIAM Journal on Computing, vol. 12, no. 4, pp. 759–776, 1983. 17. F. Plastria, “Static competitive facility location: an overview of optimisation approaches.” European Journal of Operational Research, vol. 129, no. 3, pp. 461–470, 2001. 18. A. Reiser, “A linear selection algorithm for sets of elements with weights,” Information Processing Letters, vol. 7, no. 3, pp. 159–162, 1978. 19. D. R. Santos-Peñate, R. Suárez-Vega, and P. Dorta-González, “The leader–follower location model,” Networks and Spatial Economics, vol. 7, no. 1, pp. 45–61, 2007.	8
HYBRID FUEL CELLS POWER FOR LONG DURATION ROBOT MISSIONS IN FIELD ENVIRONMENTS Jekanthan Thangavelautham1, Danielle Gallardo2, Daniel Strawser1, Steven Dubowsky1 1 Mechanical Engineering Department, Massachusetts Institute of Technology 77 Massachusetts Ave., Cambridge, MA, 02139, {jekan, dstrawse, dubowsky}@mit.edu 2 Department of Mechanical, Aerospace and Nuclear Engineering, Rensselaer Polytechnic Institute, 110th 8th Jonsson Engineering Center, Troy New York, 12180 Mobile robots are often needed for long duration missions. These include search rescue, sentry, repair, surveillance and entertainment. Current power supply technology limit walking and climbing robots from many such missions. Internal combustion engines have high noise and emit toxic exhaust while rechargeable batteries have low energy densities and high rates of self-discharge. In theory, fuel cells do not have such limitations. In particular Proton Exchange Membrane (PEMs) can provide very high energy densities, are clean and quiet. However, PEM fuel cells are found to be unreliable due to performance degradation. This can be mitigated by protecting the fuel cell in a fuel-cell battery hybrid configuration using filtering electronics that ensure the fuel cell is isolated from electrical noise and a battery to isolate it from power surges. Simulation results are presented for a HOAP 2 humanoid robot that suggests a fuel cell powered hybrid power supply superior to conventional batteries. I. Introduction Mobile robots, including walking robots are needed to perform long duration missions that are difficult, dangerous and tedious. These include search and rescue, repair, entertainment, sentry, surveillance applications [1, 2]. Continuous operation of these robots, lasting days and weeks (not hours) would be ideal for these applications. Typical power demands for field robots will vary significantly during a mission, often with high peak power demands. These field systems often have constraints on their mass, volume and noise. Current power supply technology is a key limiting factor for long duration field robotic applications. Internal combustion engines can provide high power for long durations but produce toxic exhaust, noise and strong thermal signatures making them inappropriate for many important applications. Current rechargeable batteries have very low energy densities and high rates of selfdischarge, requiring systems to stop and recharge every few hours, making them 1 2 ineffective for continuous long duration missions. Hence, there is a significant need for a power supply that can provide the high total energy required for long duration missions that is quiet and clean. Figure 1: (Left) Boston Dynamics Big Dog, a four-legged supply robot. (Right) Robonaut 2 repair robot. II. Fuel Cell Power for Mobile Robots Fuel cells are high energy sources of power that have been suggested for robots [7, 8]. They are a promising alternative mobile source of power and have the potential to overcome limitations of current batteries and internal combustion engines. They are simple electrochemical devices that convert chemical energy into electricity (Figure 2). Unlike a battery, fuel cells require a constant supply of fuel and oxidant to produce electricity. Proton Exchange Membrane (PEM) fuel cells are particularly attractive for robotics. These devices consist of simple solid state components sandwiched together as shown in Figure 2. They combine hydrogen fuel, and oxygen (from breathing air) through the most energy releasing reaction known, to produce electricity and water. It has been demonstrated; PEM fuel cells can reach 65-70 % or higher operating efficiencies at room temperature and produce clean water exhaust [9]. III. The Challenges of Fuel Cells for Robots While PEM fuel cells are simple and sound great in theory, they have three fundamental problems for practical robotics applications. These problems are storage of hydrogen fuel, long-life reliability of fuel cells and low power. Hydrogen fuel due to its high energy content and low density is difficult to store. In our research, we have developed simple, innovative hydrogen storage 3 technologies that promise energy storage densities better than the best batteries of today [5]. Figure 2: A PEM Fuel Cell Consumes Reactants, Hydrogen and Oxygen to Produce Electricity, Water and Heat. Second, PEM fuel cells have been found to be unreliable [3]. Our studies of PEM fuel cells show that they are delicate and unreliable due to degradation of their components, resulting in short lives and premature failure [4]. However our physical models and experiments suggests that PEM fuel cells controlled to operate within narrow operating can be made robust, have long lives of 3-5 years or more and high operating efficiencies [4]. Among the factors known to degrade fuel cells, are high operating voltages and electrical noise. As discussed below, mobile field robots operating in unstructured environments are subject to very substantial variation, which, without proper control can result in fuel cell degradation that shortens their lives. A solution to this problem is discussed in the section below. A third problem with fuel cells is that while they are high energy devices, they have relatively low power. This is a problem for robotics, where typical power requirements can vary substantially over a mission, with low-power rest periods and short bursts at peak power. These varying power demands are known to stress the fuel cells, resulting in short lives. A solution is to use fuel-cells in a hybrid system for mobile robots that maintains a fuel cell at optimal operating conditions to maximize life and efficiency, by protecting it from external electrical load variations, noises and meeting peak power requirement using a battery (see Figure 3). 4 Fuel cell hybrid systems have been subjected to meet rapid, transient power demands in large and stationary applications, and in robotics, to meet power surges [7, 8]. However these hybrid system designs have not considered the effects of fuel cell degradation. Figure 3: Proposed Fuel Cell-Battery Hybrid Power Supply for Robots. IV. Research The research presented here is focused on developing hybrid system design concept for mobile robots that have energy densities that exceed the best battery technology. The hybrid system is designed to meet the required peak power demands and isolate the fuel cell from degrading stresses of high and low frequency noises generated by conditioning circuits required for battery management. Physical models are used to simulate expected conditions and control systems are developed to demonstrate the concept. It is shown that the results are vast improvement over conventional batteries in terms of life, efficiency, energy density and power density. V. Case Study: Power for Humanoid Walking Robot Here a hybrid fuel cell power supply for a HOAP-2 humanoid walking robot (Figure 4) developed by Fujitsu is presented. The HOAP-2 is a 7.8 kg robot, with a maximum rated power of 250 W. It contains a 1.2 kg Nickel Metal Hydride rechargeable battery pack by default. The system contains 25 servo actuators, 6 for each leg and 5 for each arm, 2 for the head and 1 one for the waist. The robot has onboard computer equivalent to a PC-104 Pentium III system, a vision system consisting of 2 CCD cameras, onboard accelerometers, gyroscope and pressure sensors on each feet. 5 Figure 4: (Right) Fujitsu’s HOAP 2 Robot (Left) Cyberbotics WebotsTM Model of the HOAP-2. A simulation model of HOAP-2 from Cyberbotics WebotsTM [6] is used for our power demand calculations. The robot system consists of three different subsystems for power calculations, namely electro-mechanical system, computer + sensors and power system. For the electro-mechanical system the simulator model provides mechanical power output of the servo motors. The servo motors are assumed to have a 50 % electrical to mechanical efficiency. The computer and sensor system are assumed to be always powered and consume 40 W based on the HOAP-2 specifications. Below, power demand profiles of the robot’s walking behavior is shown (Figure 5). For these scenarios, alternative power sources are compared with the default nickel metal hydride battery packs that weigh 1.2 kg. VI. Fuel Cell Hybrid System The fuel cell hybrid system consists of a fuel cell stack that provides steady power source and rechargeable lithium ion A123 Nanophosphate TM battery that meets peak power demands. The fuel cells within the stack are operated at constant operating voltage of 0.8, providing a 65 % operating efficiency. Our research into the degradation of fuel cells, based on models and experimental results shows that increased operating voltages exponentially decreases the life of the fuel cell [4]. Operating the fuel cell at constant voltage of 0.8 V or less ensures long-life, while providing sufficiently high operating efficiency. The fuel cell trickle charges the battery during idle times, ensuring the battery is fully charged, to meet power peaks. The NanophosphateTM battery handles peak demands and can better handle deep battery discharges compared to 6 conventional lithium ion batteries. Ensuring the battery is nearly fully charged, maximizes its life. The battery for the hybrid system is sized based on its specific power density to meet the maximum possible power requirements of the robot. An oscillation suppression circuit interfaces a fuel cell to the power management system, consisting of power switching circuitry and a DC-DC convertor. The interface circuit effectively extracts the energy from the fuel cell and transfers it into the battery. The oscillation suppression circuits prevents any voltage oscillation from electrical circuits, particularly DC-DC convertors from being noticed by the fuel cell. This ensures the fuel cell operates at steady operating voltage, without any electrical load oscillations. Figure 5: Power Demand of a HOAP-2 Robot, walking at 0.06 m/s. VII. Hybrid System Sizing Based on the power demand profiles (Figure 5), the fuel cell will provide a constant steady source of power of 45 W. While the power peaks will be handled by the battery. For 45 W steady supply of power, a fuel cell stack weighing 150 g will be required. For a 250 W peak required by the robot, a 135 g lithium ion NanophosphateTM battery is required (with specific power of 1,850 W/kg for APR18650 cells) . Another 115 g is allocated for mass of power electronics and other items, leaving 800 g for the Lithium Hydride fuel supply with an energy density of 4,950 Wh/kg. 7 VIII. Power System Comparison Four power supply configurations are compared, including a nickel metal hydride and lithium ion battery system, a fuel cell system and a fuel cell hybrid system. See Table 1. Table 1: Power Supply Comparison for HOAP-2 Humanoid Robot Power Supply FC Stack FC Fuel Energy System RunMass Mass Density Life Time NiMH Battery 40 Wh/kg 0.3 year 3 hours Li Ion Battery 120 Wh/kg 1 year 9 hours Fuel Cell 300 g 900 g 4950 Wh/kg 5 days 99 hours Fuel Cell Hybrid 150 g 800 g 4950 Wh/kg 3 years 88 hours Nickel metal hydride and lithium ion batteries have the lowest energy densities and thus provide short run-times, before requiring recharging. The system life of the batteries is computed based on expected lifetime of 1,000 charge/recharge multiplied by the run-times hours. A direct fuel cell system has the longest runtime. However the life of the system, based on our degradation models is expected to last just 5 days making this option impractical. The fuel cell hybrid system offers a good trade-off between both run-time and system life. IX. Summary and Conclusions Based on these results, the fuel cell hybrid system concept offers high energy density, long-life and would meet the required peak power demands of a battery. The key to our hybrid system concept is effective control and design, where a fuel cell and battery are optimally sized to minimize stress on the fuel cell, while enabling a battery to meet power demand peaks. By minimizing stresses on the fuel cell, the system can be operated for long-lives at high operating efficiencies. X. References 1. 2. 3. 4. Hägele, M., “Contribution to World Robotics 2005,” Technical Report, European Robotics Network, 2005. Asada, H., et al. “A Roadmap to US Robotics: From Internet to Robotics,” Technical Report, Editor: Computing Community Consortium/Computing Research Association, 2009. Rubio, M.A., Urquia, A., and Dormida, S. (2009) “Diagnosis of Performance Degradation Phenomena in PEM Fuel Cells,” International Journal of Hydrogen Energy, pp. 1-5. Thangavelautham, J., Dubowsky, S., (2013) “On the Catalytic Degradation in Fuel Cell Power Supplies for Long-Life Mobile Field Sensors,” Fuel Cells, vol. 13, no. 2, pp. 1-24. 8 5. Thangavelautham, J., Strawser, D., Dubowsky, S., “Lithium hydride powered PEM fuel cells for long-duration small mobile robotic missions,” IEEE International Conference on Robots and Automation, 2012, p. 1-8. 6. Michel, O. / Cyberbotics Ltd. (2004) “WebotsTM: Professional Mobile Robot Simulation,” International Journal of Advanced Robotic Systems, pp. 39-42, Vol. 1 No. 1. Kesner, S. B., Plante, J. S., Boston, P., Fabian, T., Dubowsky, S., “Mobility and Power Feasibility of a Microbot Team System for Extraterrestrial Cave Exploration,” Proceedings of the 2007 IEEE International Conference Robotics and Automation, Rome, Italy, April 2007. Joh, H., et al., (2010) A direct methanol fuel cell system to power a humanoid robot, Journal of Power Sources, Vol. 195, No. 1, pp. 293-298. F. Barbir, PEM Fuel Cells: Theory and Practice, Academic Press, 2005. 7. 8. 9. View publication stats	3
Using a hierarchy of Domain Specific Languages in complex software systems design V. S. Lugovsky <[email protected]> arXiv:cs/0409016v1 [cs.PL] 9 Sep 2004 February 1, 2008 Abstract A new design methodology is introduced, with some examples on building Domain Specific Languages hierarchy on top of Scheme. 1 Introduction nized as “true” macros as they hide an access to the host language. This situation looks like a paradox. On the one hand, industry uses the metaprogramming ideas and tools, and it is easy to imagine how it would suffer without them. On the other hand, industry does not want to hear anything related to the metaprogramming. It does not want people inventing new programming languages — plenty of industry coders barely use only one language and IT managers believe without any reason that they can not be taught to use more [6]. Industry prefers to “re–invent a wheel” and to express any sort of complexity in the form of libraries for static, steady languages. For some strange reason learning complicated libraries for a language which barely fits problem domain needs is preferred to learning a small new language specifically designed for it. In this paper I am trying to advocate the metaprogramming approach as the major design methodology for complex systems. Sounds like another one “silver bullet” invention? There were many methodologies claiming to solve all possible problems of the mankind — RUP, eXtreme programming, etc. Why do we need another one? Simply because the previous approaches did not succeed. They were too tied to particular programming technologies (mostly — OOP varieties), which are definitely not “silver bullets”. Metaprogramming Programs that write programs that write programs (...) Too complicated? A hackers technique which can not be applied to the “real world” problems? This is exactly how IT industry specialists think about metaprogramming. And this is a completely wrong notion! Metaprogramming is the only known way to reduce the complexity significantly. In some areas “programs that write programs” are accepted by the industry due to the enormous level of complexity of the corresponding handwritten code — regular expressions, lexers and parsers generators to name a few, and code wizards and templates in popular “integrated development environments” are also widely used. But this does not help at all in the overall methodology recognition. The industry’s most beloved and widely buzzworded language, Java, does not have even such a rudimentary preprocessor as C does. Very few C++ programmers have an idea on how to use the templates, they just utilize STL without any understanding of the true source of the power. Even in the enlightened world of Lisp programming the misunderstanding is surprisingly wide: almost all of Lisp dialects and Scheme implementations have problems with macros (not so many people are using them), and even the current Scheme standard R5RS contains only hygienic macros that can hardly be recog1 methodology is different, it strongly encourages of domain specific languages is not very tricky. the use of all possible programming technolo- An approach I will describe here is based on metaprogramming techniques. It requires a so gies and to invent the “impossible” ones. called Core Language, on top of which we will build a hierarchy of our domain specific lan2 Domain specific lan- guages. The Core Language should possess the following properties: guages Below I am providing an outline of the proposed methodology. Any problem domain can be best expressed using a language (mathematical, programming, natural, ...) specially designed for it. In most cases there should be one entity in a language for every entity in a problem domain. For example, if a problem domain is the recognition of syntax constructions in a characters stream, the Domain Specific Language should contain characters and characters sets as a primary entity and automata constructions for expressing syntax. That is enough — regular expressions language is designed. It is hard to believe that somebody will ever invent anything better for this purpose than this “most optimal” DSL. If a problem domain is already specified as an algebra, we even do not have to design the DSL: it will be this algebra itself, galvanised with any underlying computational semantics — this is the way SQL was born. If a problem domain is 3D graphics, linear algebra and stereometry should be used. All the languages and data formats dedicated to 3D contain subsets of this formal theories. As it is stated in [1], “The object of a DSL-based software architecture is to minimise the semantic distance between the system’s specification and its implementation.” 3 Core language For any problem it is convenient to have a language that best fits it. There already exist specialized languages for some common problems. But what to do if none is available? The answer is trivial: implement it. Implementation • True macros. That is, we must have an access to a complete programming language (preferably the same as a host language, or a different one) inside the macro definitions. Macros should be real programs which can do anything that the programs written in the host language can do. Macros are producing the code in the host language, in the form of text or directly as an abstract syntax tree. • True runtime eval. Programs that are generated in the runtime should be evaluated. This can be a different language than the host language, or, better, the same one. • Turing-completeness. This should be a real programming language, equivalent in its expressive power to the “general purpose” languages. • Simplicity. It is an extensible core and should not contain any unnecessary complexity that can be later added by a user who really needs it. • Comprehensive and easy to use data types system. If a type system is well suited for expressing any possible abstract syntax trees, the language fits this requirement. On top of the Core Language we have to build functionality that will be needed to implement programming languages. It is lexing, parsing, intermediate languages that fit well computational models different from the model of the Core Language (e.g., if the core language is imperative or an eager functional, we will need a graph reduction engine to implement lazy functional DSLs, or a term unification engine to implement logical languages and a stack machine if we have to go to lower levels). The 2 Core Language enriched with this “Swiss army knife” for programming languages development then becomes a major tool for any project. 4 New methodology The development process must fit in the following chain: • divide the problem into sub–problems, possibly using some object oriented design techniques, or whatever fits better. • formalize each sub–problem. 5 Scheme example A good example of a practical Core Language is Scheme (with addition of Common Lisp–style macros). It uses S–expressions as an AST, and S–expressions composition is very natural. S–expressions are good enough to represent any possible AST (for example, XML is naturally represented as SXML). It provides a true runtime eval hosting the same language as in compile time. There exist some practical and efficient Scheme implementations which provide performance acceptable for most tasks, good FFI, and, thus, integration with legacy libraries. • implement the Domain Specific Language after this formalization, using the Let us start with adding the functionality Core Language and other DSL with the described above to Scheme. First of all we will same semantics. need parsing — not all of our team members • solve the problem using the best possible are fond of parentheses, so we have to implelanguage. ment many complicated syntaxes. The most This way any project will grow into a tree (hier- natural way for a functional programming lanarchy) of domain specific languages. Any lan- guage is to implement a set of parsing combinaguage is a subset or a superset of another lan- tors for building recursive descendant parsers guage in the hierarchy (or, may be, combina- (mostly LL(1), but it is not such a fixed limit as tion of several languages), and the amount of LALR(1) for Yacc–like automata generators). coding for a new language if we already have a deep and comprehensive hierarchy is quite small. A development team working within this methodology should consist of at least one specialist who maintains this hierarchy, an architect who formalizes problems, and a number of coders who specialize in particular problem domains, they even may not be programmers at all — they just have to know well their domains and operate them in terms that are as close as possible to the native problem domain terminology. For example, HTML designer will be happy operating HTML–like tags for his templates (that is why JSP custom tags are so popular); mathematician will find a language modelled after the standard mathematical notation intuitive — for this reason Wolfram Mathematica is so popular among non-programmers; game script writer will operate a language expressing characters, their properties and action rules — stating, not programming. This list can be continued infinitely. Of course we will use metaprogramming wherever possible. All the parsers should be functions which consume a list of tokens (e.g. characters) as an input and return the result in the following form: ((RESULT anyresult) unparsed-input-rest) or ((FAIL reason) input) To access the parsing result we will provide the following macros: (define-macro (success? r ) ‘(not (eq? (caar ,r ) ’FAIL))) And if we are sure that we have some result, we will use the following macro to extract it (otherwise, this will return a fail message): (define-macro (result r ) ‘(cdar ,r )) 3 The last definition looks surprisingly comIn any case, we can access the rest of the pact, thanks to the pselect macro. From this stream after the parsing pass: stage the power of metaprogramming becomes (define-macro (rest r ) more and more obvious. ‘(cdr ,r )) Just as a reference, we will show here the These macros could also be implemented as definition of a choice combinator: functions. But all the macros are available in (define-macro (pOR0 p1 p2 ) the context of macro definitions while functions ‘(λ (l) are not. (let ((r1 (,p1 l))) Almost all of the parsers should fail on the (if (success? r1 ) end of the input, so the following safeguard r1 macro will be extremely useful: (,p2 l))))) (define-macro (parser p) ‘(λ (l) (if (∅? l) ’((FAIL "EMPTY")) (,p l)))) And its nested version is obvious: (define-macro (pOR p1 . p~o ) 0 ‘(pselect pOR ,p1 ,@p~o )) We will skip the rest of the combinators Now this game becomes more interesting. definitions and just show what we gained after Here is a very handy macro that nests a se- all. For example, now to define a floating point quence of applications into the form of (m p1 number recognizer, we can use this definition: (m p2 . . . (m px pn ))): (define parse-num (p+ (define-macro (pselect m p1 . p~o ) (pOR (pcsx-or (#\− #\+)) (if (∅? p~o ) p1 parse-any) (let ((p2 (car p~o )) (pMANY pdigit) (px (cdr p~o ))) OR (p (p+ (pcharx #\.) ‘(,m ,p1 (pselect ,m ,p2 ,@px ))))) (pMANY pdigit)) Sequence parsing combinator with two arparse-any))) guments can be declared as follows: It looks like BNF, but still too Schemish. (define-macro (p+0 p1 p2 ) This is already a Domain Specific Language on ‘(λ (l) top of Scheme, but it does not conform to the (let ((r1 (,p1 l))) perfectionist requirement. However, we can use (if (success? r1 ) this still not perfect parsing engine to imple(let ((r2 (,p2 (rest r1 )))) ment an intermediate regular expressions lan(if (success? r2 ) guage as a macro. Omitting the definitions, we (cons (cons ’RESULT will show the previous recognizer implemented (append in a new way: (result r1 ) (define parse-num (result r2 ))) (regexp (rest r2 )) ((#\− / #\+) / parse-any) + (cons (list ’FAIL "p+" (car r2 )) l))) (pdigit ∗) + r1 )))) (("." + (pdigit ∗)) / And it will be immediately turned into the parse-any))) sequence parsing combinator with an arbitrary This new Domain Specific Language can be number of arguments: used in many ways. For example, we can build (define-macro (p+ p1 . p~o ) ‘(pselect p+0 ,p1 ,@p~o )) a simple infix pre–calculator for constants: (define-macro (exp1 . v ) 4 (defparsers (letrec ((epr (let ((body (regexp (num :−> $ 0) / (lst -> (aprs epr ))))) (regexp ((body + (SCM psym +) + epr ) :−> (list (+ $ 0 $ 2))) / ((body + (SCM psym −) + epr ) :−> (list (− $ 0 $ 2))) / ((body + (SCM psym ∗) + epr ) :−> (list (∗ $ 0 $ 2))) / ((body + (SCM psym / ) + epr ) :−> (list (/ $ 0 $ 2))) / body )))) (car (result (epr v )))))) only languages with a computational model which is close to the model of Scheme (eager dynamically typed functional languages with imperative features), but any possible languages, providing small intermediate DSLs which simulate alternative computational models. For those who need very lowlevel power it is possible to produce an intermediate code in C language (for example, the Bigloo Scheme [3] implementation allows to include C code when compiling through C backend). For implementing complicated runtime models it is easy to produce an intermediate Forth–like DSL on top of Scheme and then use both Scheme and Forth metaprogramming powers. 6 Alternatives To make the picture complete, it is necesAnd then, wherever we want to calculate a sary to mention other possible choices for the numerical constant in the compilation time, we Core Language. The popular programming may use the exp1 macro: language, C++, could become such a Core (exp1 5 + ((10 / 2)−(1 / 5))) Language relatively easily. It has a TuringThis language does not look like Scheme complete macro system, unfortunately, featurany more. And we can go even further, im- ing the language different from the host lanplementing a Pascal (or Rlisp)–like language guage (so only one stage preprocessing is poson top of Scheme, using just the same regexp sible). It lacks a good type system, but it could macro to describe both a lexer and a parser, be simulated on top of the existing lowlevel feaand then to compile the resulting code to the tures. There exist some implementations of the recursive descendant parsing combinators underlying Scheme. for C++ (e.g., Boost Spirit library [2]), implementation of the functional programming (e.g., (pasqualish Boost Lambda [2]), and even Lisp compilers on " top of the C++ template system. The runtime function fac(x) evaluation is available in different ways: using begin pluggable scripting languages other than C++, if (x > 0) then using the C++ interpreter [14]. An interesting x*fac(x - 1) approach is described in [13]. else 1; Another choice is Forth. It is a powerful end metalanguage, but the core language remains ") too lowlevel and unsafe. Forth is often the only No more parenthesis that frighten non–Lisp choice available for the embedded systems with programmers so much! Now even Pascal pro- limited resources. It is worth mentioning modern experimengrammers can use Scheme. The code samples above demonstrate some tal extensions for strictly typed functional lanof the techniques available in this approach. guages: Template Haskell [12] and MetaOCaml The complete implementation can be down- [11]. Both of them conform well to all of loaded from [4]. It is possible to produce not the Core Language requirements. Objective 5 Caml also provides one–stage metaprogram- metaprogramming too. ming using a sophisticated preprocessing engine CamlP4. And OCaml is quite good for Conclusion implementing interpreters using the closure– 7 based technique. Some examples can be found The idea of metaprogramming is not something in [7]. esoteric. Metaprogramming is used widely by No doubt that Common Lisp would also commercial programmers, they just do not rebe a very good platform since it shares almost alize it. The methodology proposed in this paall the features with Scheme with exception of per is an attempt of uncovering all the hidsimplicity. The killing feature of Common Lisp den power of the metaprogramming techniques is advanced runtime compilation in some of the available. major implementations (CMU CL [8] and its The Scheme example presented above is descendant SBCL [9] are good examples), and part of the working project, which already the defmacro is guaranteed to be working in all proved the supremacy of this approach. A subthe implementations available, which is a great set of the Domain Specific Languages hierarchy advantage over Scheme. designed for the WWW data acquiring project For relatively small projects Tcl [10] would is shown on the Fig. 1. be a good choice. Its computational model The subject discussed requires future reis based on rewrites (and primary data struc- search and practical approbation, whose final tures are just the strings of text), which ren- result may be a completely formalized, matheders an extremely powerful metaprogramming matically strict methodology description and a tool. JavaScript language is also based on Core Language which will best fit this methodthe rewrites semantics, so it could be used for ology. 6 References [1] Diomidis Spinellis. Reliable software implementation using domain specific languages. In G. I. Schuëller and P. Kafka, editors, Proceedings ESREL ’99 — The Tenth European Conference on Safety and Reliability, pages 627–631, Rotterdam, September 1999. ESRA, VDI, TUM, A. A. Balkema // [draft] http://www.dmst.aueb.gr/dds/pubs/conf/1999-ESREL-SoftRel/html/dsl.html [2] The Boost project // http://www.boost.org/ [3] The Bigloo Practical Scheme implementation // http://www.bigloo.org/ [4] V. S. Lugovsky, DSLEngine project home // http://dslengine.sourceforge.net/ [5] P. Graham, The Hundred-Year Language // http://www.paulgraham.com/hundred.html [6] P. Graham, The Python Paradox // http://www.paulgraham.com/pypar.html [7] V. S. Lugovsky, publications list // http://ontil.ihep.su/˜vsl [8] CMU Common Lisp // http://www.cons.org/cmucl/ [9] Steel Bank Common Lisp // http://sbcl.sourceforge.net/ [10] Tcl programming language resource // http://tcl.activestate.com/ [11] MetaOCaml project home // http://www.metaocaml.org/ [12] T. Sheard, S. P. Jones, Template metaprogramming for Haskell // http://research.microsoft.com/˜simonpj/papers/meta-haskell/ [13] Tempo project home // http://compose.labri.fr/prototypes/tempo/ [14] CINT project home // http://root.cern.ch/root/Cint.html 7 Core language Parsing combinators Stack machine Lexer Graph machine Unification machine Templates language Parser generator Forth-like Regular expressins Data aquision regexps Pascal-like Rule engine SQL templates Figure 1: A sample DSLs hierarchy subset for the Web crawler project. 8	2
ACD Term Rewriting arXiv:cs/0608016v1 [cs.PL] 3 Aug 2006 Gregory J. Duck, Peter J. Stuckey, and Sebastian Brand NICTA Victoria Laboratory Department of Computer Science & Software Engineering, University of Melbourne, Australia Abstract. In this paper we introduce Associative Commutative Distributive Term Rewriting (ACDTR), a rewriting language for rewriting logical formulae. ACDTR extends AC term rewriting by adding distribution of conjunction over other operators. Conjunction is vital for expressive term rewriting systems since it allows us to require that multiple conditions hold for a term rewriting rule to be used. ACDTR uses the notion of a “conjunctive context”, which is the conjunction of constraints that must hold in the context of a term, to enable the programmer to write very expressive and targeted rewriting rules. ACDTR can be seen as a general logic programming language that extends Constraint Handling Rules and AC term rewriting. In this paper we define the semantics of ACDTR and describe our prototype implementation. 1 Introduction Term rewriting is a powerful instrument to specify computational processes. It is the basis of functional languages; it is used to define the semantics of languages and it is applied in automated theorem proving, to name only a few application areas. One difficulty faced by users of term rewriting systems is that term rewrite rules are local, that is, the term to be rewritten occurs in a single place. This means in order to write precise rewrite rules we need to gather all relevant information in a single place. Example 1. Imagine we wish to “program” an overloaded ordering relation for integers variables, real variables and pair variables. In order to write this the “type” of the variable must be encoded in the term1 as in: int(x) ≤ int(y) → intleq(int(x), int(y)) real(x) ≤ real(y) → realleq(real(x), real(y)) pair(x1 , x2 ) ≤ pair(y1 , y2 ) → x1 ≤ y1 ∨ x1 = y1 ∧ x2 ≤ y2 In a more standard language, the type information for variables (and other information) would be kept separate and “looked up” when required. ⊓ ⊔ 1 Operator precedences used throughout this paper are: ∧ binds tighter than ∨, and all other operators, e.g. ¬, =, bind tighter than ∧. Term rewriting systems such as constraint handling rules (CHRs) [5] and associative commutative (AC) term rewriting [3] allow “look up” to be managed straightforwardly for a single conjunction. Example 2. In AC term rewriting the above example could be expressed as: int (x) ∧ int (y) ∧ x ≤ y → int(x) ∧ int(y) ∧ intleq(x, y) real (x) ∧ real (y) ∧ x ≤ y → real (x) ∧ real (y) ∧ realleq (x, y) pair (x, x1 , x2 ) ∧ pair (y, y1 , y2 ) ∧ x ≤ y → pair (x, x1 , x2 ) ∧ pair (y, y1 , y2 )∧ (x1 ≤ y1 ∨ x1 = y1 ∧ x2 ≤ y2 ) where each rule replaces the x ≤ y by an appropriate specialised version, in the conjunction of constraints. The associativity and commutativity of ∧ is used to easily collect the required type information from a conjunction. ⊓ ⊔ One difficulty remains with both AC term rewriting and CHRs. The “look up” is restricted to be over a single large conjunction. Example 3. Given the term int(x1 ) ∧ int(y1 ) ∧ pair (x, x1 , x2 ) ∧ pair (y, y1 , y2 ) ∧ x ≤ y. Then after rewriting x ≤ y to (x1 ≤ y1 ∨ x1 = y1 ∧ x2 ≤ y2 ) we could not rewrite x1 ≤ y1 since the types for x1 , y1 appear in a different level. In order to push the type information inside the disjunction we need to distribute conjunction over disjunction. ⊓ ⊔ Simply adding distribution rules like A ∧ (B ∨ C) → A ∧ B ∨ A ∧ C A ∧ B ∨ A ∧ C → A ∧ (B ∨ C) (1) (2) does not solve the problem. Rule (1) creates two copies of term A, which increases the size of the term being rewritten. Adding Rule (2) to counter this effect results in a non-terminating rewriting system. 1.1 Conjunctive context We address the non-termination vs. size explosion problem due to distributivity rewrite rules in a similar way to how commutativity is dealt with: by handling distributivity on the language level. We restrict ourselves to dealing with expanding distributivity of conjunction ∧ over any other operator, and we account for idempotence of conjunction.2 Thus we are concerned with distribution rules of the form P ∧ f (Q1 , . . . , Qn ) → P ∧ f (P ∧ Q1 , . . . , P ∧ Qn ). 2 (3) This means that conjunction is distributive over any function f in presence of a redundant copy of P , i.e. P ∧ (P ∧ f (Q1 , . . . , Qn )) → P ∧ f (P ∧ Q1 , . . . , P ∧ Qn ). We use idempotence to simplify the RHS and derive (3). 2 Let us introduce the conjunctive context of a term and its use in rewrite rules, informally for now. Consider a term T and the conjunction C ∧ T modulo idempotence of ∧ that would result from exhaustive application of rule (3) to the superterm of T . By the conjunctive context of T we mean the conjunction C. Example 4. The conjunctive context of the boxed occurrence of x in the term (x = 3) ∧ (x2 > y ∨ ( x = 4) ∧ U ∨ V ) ∧ W, is (x = 3) ∧ U ∧ W . ⊓ ⊔ We allow a rewrite rule P → T to refer to the conjunctive context C of the rule head P . We use the following notation: C \ P ⇐⇒ T. This facility provides ∧-distributivity without the undesirable effects of rule (3) on the term size. Example 5. We can express that an equality can be used anywhere “in its scope” by viewing the equality as a conjunctive context: x = a \ x ⇐⇒ a. Using this rule on the term of Example 4 results in (x = 3) ∧ (32 > y ∨ (3 = 4) ∧ U ∨ V ) ∧ W ⊓ ⊔ without dissolving the disjunction. 1.2 Motivation and Applications Constraint Model Simplification. Our concrete motivation behind associative commutative distributive term rewriting (ACDTR) is constraint model mapping as part of the G12 project [7]. A key aim of G12 is the mapping of solver independent models to efficient solver dependent models. We see ACDTR as the basis for writing these mappings. Since models are not flat conjunctions of constraints we need to go beyond AC term rewriting or CHRs. Example 6. Consider the following simple constraint model inspired by the Social Golfers problem. For two groups g1 and g2 playing in the same week there can be no overlap in players: maxOverlap(g1 , g2 , 0) The aim is to maximise the number of times the overlap between two groups is less than 2; in other words minimise the number of times two players play together in a group. ^ constraint maxOverlap(g1 , g2 , 0) ∀w∈Weeks ∀g1 ,g2 ∈weeks[w] g1 <g2 maximise X holds(maxOverlap(g1 , g2 , 1)) ∀w1 ,w2 ∈Weeks ∀g1 ∈weeks[w1 ] ∀g2 ∈weeks[w2 ] g1 <g2 3 Consider the following ACDTR program for optimising this constraint model. maxOverlap(a, b, c1 ) \ maxOverlap(a, b, c2 ) ⇐⇒ c2 ≥ c1 \| true holds(true) ⇐⇒ 1 holds (false) ⇐⇒ 0 The first rule removes redundant maxOverlap constraints. The next two rules implement partial evaluation of the holds auxiliary function which coerces a Boolean to an integer. By representing the constraint model as a giant term, we can optimise the model by applying the ACDTR program. For example, consider the trivial case with one week and two groups G1 and G2 . The model becomes maxOverlap(G1 , G2 , 0) ∧ maximise(holds (maxOverlap(G1 , G2 , 1))). The subterm holds(maxOverlap(G1 , G2 , 1)) simplifies to 1 using the conjunctive context maxOverlap(G1 , G2 , 0). ⊓ ⊔ It is clear that pure CHRs are insufficient for constraint model mapping for at least two reasons, namely – a constraint model, e.g. Example 6, is typically not a flattened conjunction; – some rules rewrite functions, e.g. rules (2) and (3) rewriting function holds, which is outside the scope of CHRs (which rewrite constraints only). Global Definitions. As we have seen conjunctive context matching provides a natural mechanism for making global information available. In a constraint model, structured data and constraint definitions are typically global, i.e. on the top level, while access to the data and the use of a defined constraint is local, e.g. the type information from Example 1. Another example is partial evaluation. Example 7. The solver independent modelling language has support for arrays. Take a model having an array a of given values. It could be represented as the top-level term array (a, [3, 1, 4, 1, 5, 9, 2, 7]). Deeper inside the model, accesses to the array a occur, such as in the constraint x > y + lookup(a, 3). The following rules expand such an array lookup: array(A, Array) \ lookup(A, Index ) ⇐⇒ list element(Array, Index ) list element([X\|Xs], 0) ⇐⇒ X list element ([X\|Xs], N ) ⇐⇒ N > 0 \| list element (Xs, N − 1) Referring to the respective array of the lookup expression via its conjunctive context allows us to ignore the direct context of the lookup, i.e. the concrete constraint or expression in which it occurs. ⊓ ⊔ 4 Propagation rules. When processing a logical formula, it is often useful to be able to specify that a new formula Q can be derived from an existing formula P without consuming P . In basic term rewriting, the obvious rule P ⇐⇒ P ∧ Q causes trivial non-termination. This issue is recognised in CHRs, which provide support for inference or propagation rules. We account for this fact and use rules of the form P =⇒ Q to express such circumstances. Example 8. The following is the classic CHR leq program reimplemented for ACD term rewriting (we omit the basic rules for logical connectives): leq(X, X) ⇐⇒ true leq(X, Y ) \ leq(Y, X) ⇐⇒ X = Y leq(X, Y ) \ leq(X, Y ) ⇐⇒ true leq(X, Y ) ∧ leq(Y, Z) =⇒ leq(X, Z) (reflexivity) (antisymmetry) (idempotence) (transitivity) These rules are almost the same as the CHR version, with the exception of the second and third rule (antisymmetry and idempotence) which generalise its original by using conjunctive context matching. ⊓ ⊔ Propagation rules are also used for adding redundant information during model mapping. The rest of the paper is organised as follows. Section 2 covers the standard syntax and notation of term rewriting. Section 3 defines the declarative and operational semantics of ACDTR. Section 4 describes a prototype implementation of ACDTR as part of the G12 project. Section 5 compares ACDTR with related languages. Finally, in Section 6 we conclude. 2 Preliminaries In this section we briefly introduce the notation and terminology used in this paper. Much of this is borrowed from term rewriting [3]. We use T (Σ, X) to represent the set of all terms constructed from a set of function symbols Σ and set of variables X (assumed to be countably infinite). We use Σ (n) ⊆ Σ to represent the set of function symbols of arity n. A position is a string (sequence) of integers that uniquely determines a subterm of a term T , where ǫ represents the empty string. We define function T \|p , which returns the subterm of T at position p as T \|ǫ = T f (T1 , . . . , Ti , . . . , Tn )\|ip = Ti \|p We similarly define a function T [S]p which replaces the subterm of T at position p with term S. We define the set Pos(T ) to represent the set of all positions of subterms in T . An identity is a pair (s, t) ∈ T (Σ, X) × T (Σ, X), which is usually written as s ≈ t. Given a set of identities E, we define ≈E to be the set of identities closed under the axioms of equational logic [3], i.e. symmetry, transitivity, etc. 5 We define the congruence class [T ]≈E = {S ∈ T (Σ, X)\|S ≈E T } as the set of terms equal to T with respect to E. Finally, we define function vars(T ) to return the set of variables in T . 3 Syntax and Semantics The syntax of ACDTR closely resembles that of CHRs. There are three types of rules of the following form: (simplification) (propagation) (simpagation) r @ H ⇐⇒ g \| B r @ H =⇒ g \| B r @ C \ H ⇐⇒ g \| B where r is a rule identifier, and head H, conjunctive context C, guard g and body B are arbitrary terms. The rule identifier is assumed to uniquely determine the rule. A program P is a set of rules. We assume that vars(g) ⊆ vars(H) or vars(g) ⊆ vars(H) ∪ vars(C) (for simpagation rules). The rule identifier can be omitted. If g = true then the guard can be omitted. We present the declarative semantics of ACDTR based on equational logic. First we define the set of operators that ACDTR treats specially. Definition 1 (Operators). We define the set of associate commutative operators as AC. The set AC must satisfy AC ⊆ Σ (2) and (∧) ∈ AC. For our examples we assume that AC = {∧, ∨, +, ×}. We also treat the operator ∧ as distributive as explained below. ACDTR supports a simple form of guards. Definition 2 (Guards). A guard is a term. We denote the set of all “true” guards as G, i.e. a guard g is said to hold iff g ∈ G. We assume that true ∈ G and false 6∈ G. We can now define the declarative semantics for ACDTR. In order to do so we employ a special binary operator where to explicitly attach a conjunctive context to a term. Intuitively, the meaning of T where C is equivalent to that of T provided C is true, otherwise the meaning of T where C is unconstrained. For Boolean expressions, it is useful to interpret where as conjunction ∧, therefore where-distribution, i.e. identity (6) below, becomes equivalent to ∧-distribution (3). The advantage of distinguishing where and ∧ is that we are not forced to extend the definition of ∧ to arbitrary (non-Boolean) functions. We denote by B the following set of built-in identities: A◦B ≈B ◦A (A ◦ B) ◦ C ≈ A ◦ (B ◦ C) T ≈ (T where true) A ∧ B ≈ (A where B) ∧ B T where (W1 ∧ W2 ) ≈ (T where W1 ) where W2 f (A1 , ..., Ai , ..., An ) where W ≈ f (A1 , ..., Ai where W, ..., An ) where W 6 (1) (2) (3) (4) (5) (6) for all ◦ ∈ AC, functions f ∈ Σ (n) , and i ∈ {1, . . . , n}. Definition 3 (Declarative Semantics for ACDTR). The declarative semantics for an ACDTR program P (represented as a multiset of rules) is given by the function JK defined as follows: JP K = {Jθ(R)K \| ∀R, θ . R ∈ P ∧ θ(guard(R)) ∈ G} ∪ B JH ⇐⇒ g \| BK = ∃vars(B)−vars(H) (H ≈ B) JC \ H ⇐⇒ g \| BK = ∃vars(B)−vars(C,H) (H where C ≈ B where C) JH =⇒ g \| BK = ∃vars(B)−vars(H) (H ≈ H ∧ B) where function guard(R) returns the guard of a rule. The function JK maps ACDTR rules to identities between the head and the body terms, where body-only variables are existentially quantified.3 Note that there is a new identity for each possible binding of guard(R) that holds in G. A propagation rule is equivalent to a simplification rule that (re)introduces the head H (in conjunction with the body B) in the RHS. This is analogous to propagation rules under CHRs. A simpagation rule is equivalent to a simplification rule provided the conjunctive context is satisfied. The built-in rules B from Definition 3 contain identities for creating/destroying (3) and (4), combining/splitting (5), and distributing downwards/upwards (6) a conjunctive context in terms of the where operator. The set B also contains identities (1) and (2) for the associative/commutative properties of the AC operators. Example 9. Consider the following ACDTR rule and the corresponding identity. JX = Y \ X ⇐⇒ Y K = (Y where X = Y ) ≈ (X where X = Y ) (7) Under this identity and using the rules in B, we can show that f (A)∧(A = B) ≈ f (B) ∧ (A = B), as follows. f (A) ∧ (A = B) ≈(4) (f (A) where (A = B)) ∧ (A = B) ≈(6) (f (A where (A = B)) where (A = B)) ∧ (A = B) ≈(7) (f (B where (A = B)) where (A = B)) ∧ (A = B) ≈(6) (f (B) where (A = B)) ∧ (A = B) ≈(4) f (B) ∧ (A = B) ⊓ ⊔ 3.1 Operational Semantics In this section we describe the operational semantics of ACDTR. It is based on the theoretical operational semantics of CHRs [1,4]. This includes support for identifiers and propagation histories, and conjunctive context matching for simpagation rules. 3 All other variables are implicitly universally quantified, where the universal quantifiers appear outside the existential ones. 7 Propagation history. The CHR concept of a propagation history, which prevents trivial non-termination of propagation rules, needs to be generalised over arbitrary terms for ACDTR. A propagation history is essentially a record of all propagation rule applications, which is checked to ensure a propagation rule is not applied twice to the same (sub)term. In CHRs, each constraint is associated with a unique identifier. If multiple copies of the same constraint appear in the CHR store, then each copy is assigned a different identifier. We extend the notion of identifiers to arbitrary terms. Definition 4 (Identifiers). An identifier is an integer associated with each (sub)term. We use the notation T #i to indicate that term T has been associated with identifier i. A term T is annotated if T and all subterms of T are associated with an identifier. We also define function ids(T ) to return the set of identifiers in T , and term(T ) to return the non-annotated version of T . For example, T = f (a#1, b#2)#3 is an annotated term, where ids(T ) = {1, 2, 3} and term(T ) = f (a, b). Identifiers are considered separate from the term. We could be more precise by separating the two, i.e. explicitly maintain a map between Pos(T ) and the identifiers for T . We do not use this approach for space reasons. We extend and overload all of the standard operations over terms (e.g. from Section 2) to annotated terms in the obvious manner. For example, the subterm relation T \|p over annotated terms returns the annotated term at position p. The exception are elements of the congruence class [T ]≈AC , formed by the AC relation ≈AC , which we assume satisfies the following constraints. A#i ◦ B#j ≈AC B#j ◦ A#i A#i ◦ (B#j ◦ C#k) ≈AC (A#i ◦ B#j) ◦ C#k We have neglected to mention the identifiers over AC operators. These identifiers will be ignored later, so we leave them unconstrained. A propagation history is a set of entries defined as follows. Definition 5 (Entries). A propagation history entry is of the form (r @ E), where r is a propagation rule identifier, and E is a string of identifiers. We define function entry(r, T ) to return the propagation history entry of rule r for annotated term T as follows. entry(r, T ) = (r @ entry(T )) entry(T1 ◦ T2 ) = entry(T1 ) entry(T2 ) entry(f (T1 , ..., Tn )#i) = i entry(T1 ) ... entry(Tn ) ◦ ∈ AC otherwise This definition means that propagation history entries are unaffected by associativity, but are effected by commutativity. Example 10. Consider the annotated term T = f ((a#1 ∧ b#2)#3)#4. We have that T ∈ [T ]≈AC and T ′ = f ((b#2 ∧ a#1)#3)#4 ∈ [T ]≈AC . Although T and T ′ belong to [T ]≈AC they have different propagation history entries, e.g. entry(r, T ) = (r @ (4 1 2)) while entry(r, T ′ ) = (r @ (4 2 1)). ⊓ ⊔ 8 When a (sub)term is rewritten into another, the new term is assigned a set of new unique identifiers. We define the auxiliary function annotate(P, T ) = Ta to map a set of identifiers P and un-annotated term T to an annotated term Ta such that ids(Ta ) ∩ P = ∅ and \|ids(Ta )\| = \|Pos(T )\|. These conditions ensure that all identifiers are new and unique. When a rule is applied the propagation history must be updated accordingly to reflect which terms are copied from the matching. For example, the rule f (X) ⇐⇒ g(X, X) essentially clones the term matching X. The identifiers, however, are not cloned. If a term is cloned, we expect that both copies will inherit the propagation history of the original. Likewise, terms can be merged, e.g. g(X, X) ⇐⇒ f (X) merges two instances of the term matching X. In this case, the propagation histories of the copies are also merged. To achieve this we duplicate entries in the propagation history for each occurrence of a variable in the body that also appeared in the head. Definition 6 (Updating History). Define function update(H, Ha , B, Ba , T0 ) = T1 where H and B are un-annotated terms, Ha and Ba are annotated terms, and T0 and T1 are propagation histories. T1 is a minimal propagation history satisfying the following conditions: – T0 ⊆ T1 ; – ∀p ∈ Pos(H) such that H\|p = V ∈ X (where X is the set of variables), and ∃q ∈ Pos(B) such that B\|q = V , then define identifier renaming ρ such that ρ(Ha \|p ) and Ba \|q are identical annotated terms. Then if E ∈ T0 we have that ρ(E) ∈ T1 . Example 11. Consider rewriting the term Ha = f ((a#1 ∧ b#2)#3)#4 with a propagation history of T0 = {(r @ (1 2))} using the rule f (X) ⇐⇒ g(X, X). The resulting term is Ba = g((a#5 ∧b#6)#7), (a#8 ∧b#9)#10#11 and the new propagation history is T1 = {(r @ (1 2)), (r @ (5 6)), (r @ (8 9))}. ⊓ ⊔ Conjunctive context. According to the declarative semantics, a term T with conjunctive context C is represented as (T where C). Operationally, we will never explicitly build a term containing a where clause. Instead we use the following function to compute the conjunctive context of a subterm on demand. Definition 7 (Conjunctive Context). Given an (annotated) term T and a position p ∈ Pos(T ), we define function cc(T, p) to return the conjunctive context at position p as follows. cc(T, ǫ) = true cc(A ∧ B, 1p) = B ∧ cc(A, p) cc(A ∧ B, 2p) = A ∧ cc(B, p) cc(f (T1 , . . . , Ti , . . . , Tn ), ip) = cc(Ti , p) 9 (f 6= ∧) States and transitions. The operational semantics are defined as a set of transitions on execution states. Definition 8 (Execution States). An execution state is a tuple of the form hG, T, V, Pi, where G is a term (the goal), T is the propagation history, V is the set of variables appearing in the initial goal and P is a set of identifiers. We also define initial and final states as follows. Definition 9 (Initial and Final States). Given an initial goal G for program P , the initial state of G is hGa , ∅, vars(G), ids(Ga )i where Ga = annotate(∅, G). A final state is a state where no more rules are applicable to the goal G. We can now define the operational semantics of ACDTR as follows. Definition 10 (Operational Semantics). hG0 , T0 , V, P0 i ֌ hG1 , T1 , V, P1 i 1. Simplify: There exists a (renamed) rule from P H ⇐⇒ g \| B such that there exists a matching substitution θ and a term G′0 such that – – – – G0 ≈AC G′0 ∃p ∈ Pos(G′0 ) . G′0 \|p = θ(H) θ(g) ∈ G Ba = annotate(P0 , θ(B)) Then G1 = G′0 [Ba ]p , P1 = P0 ∪ ids(G1 ) and T1 = update(H, G′0 \|p , B, Ba , T0 ). 2. Propagate: There exists a (renamed) rule from P r @ H =⇒ g \| B such that there exists a matching substitution θ and a term G′0 such that – – – – – G0 ≈AC G′0 ∃p ∈ Pos(G′0 ) . G′0 \|p = θ(H) θ(g) ∈ G entry(r, G′0 \|p ) 6∈ T0 Ba = annotate(P0 , θ(B)) Then G1 = G′0 [G′0 \|p ∧ Ba ]p , T1 = update(H, G′0 \|p , B, Ba , T0 ) ∪ {entry(r, G′0 \|p )} and P1 = P0 ∪ ids(G1 ). 3. Simpagate: There exists a (renamed) rule from P C \ H ⇐⇒ g \| B such that there exists a matching substitution θ and a term G′0 such that 10 h(leq(X1 , Y2 )3 ∧4 leq(Y5 , Z6 )7 ∧8 ¬9 leq(X10 , Z11 )12 ), ∅i ֌trans h(leq(X1 , Y2 )3 ∧4 leq(Y5 , Z6 )7 ∧13 leq(X15 , Z16 )14 ∧8 ¬9 leq(X10 , Z11 )12 ), T i ֌idemp h(leq(X1 , Y2 )3 ∧4 leq(Y5 , Z6 )7 ∧13 leq(X15 , Z16 )14 ∧8 ¬9 true17 ), T i ֌simplif y h(leq(X1 , Y2 )3 ∧4 leq(Y5 , Z6 )7 ∧13 leq(X15 , Z16 )14 ∧8 f alse18 ), T i ֌simplif y h(leq(X1 , Y2 )3 ∧4 leq(Y5 , Z6 )7 ∧13 f alse19 ), T i ֌simplif y h(leq(X1 , Y2 )3 ∧4 f alse20 ), T i ֌simplif y h(f alse21 ), T i Fig. 1. Example derivation for the leq program. – – – – – G0 ≈AC G′0 ∃p ∈ Pos(G′0 ) . G′0 \|p = θ(H) ∃D.θ(C) ∧ D ≈AC cc(G′0 , p) θ(g) ∈ G Ba = annotate(P0 , θ(B)) Then G1 = G′0 [Ba ]p , T1 = update(H, G′0 \|p , B, Ba , T0 ) and P1 = P0 ∪ ids(G1 ). Example. Consider the leq program from Example 8 with the goal leq(X, Y ) ∧ leq(Y, Z) ∧ ¬leq(X, Z) Figure 1 shows one possible derivation of this goal to the final state representing f alse. For brevity, we omit the V and P fields, and represent identifiers as subscripts, i.e. T #i = Ti . Also we substitute T = {transitivity @ (3 2 1 7 5 6)}. We can state a soundness result for ACDTR. Theorem 1 (Soundness). If hG0 , T0 , V, Pi ֌∗ hG′ , T ′ , V, P ′ i with respect to a program P , then JP K \|= ∃vars(G′ )−V G0 ≈ G′ This means that for all algebras A that satisfy JP K, G0 and G′ are equivalent for some assignment of the fresh variables in G′ . 4 Implementation We have implemented a prototype version of ACDTR as part of the mapping language of the G12 project, called Cadmium. In this section we give an overview of the implementation details. In particular, we will focus on the implementation of conjunctive context matching, which is the main contribution of this paper. Cadmium constructs normalised terms from the bottom up. Here, a normalised term is one that cannot be reduced further by an application of a rule. Given a goal f (t1 , ..., tn ), we first must recursively normalise all of t1 , ..., tn (to say s1 , ..., sn ), and then attempt to find a rule that can be applied to the top-level of f (s1 , ..., sn ). This is the standard execution algorithm used by many TRSs implementations. 11 This approach of normalising terms bottom up is complicated by the consideration of conjunctive context matching. This is because the conjunctive context of the current term appears “higher up” in the overall goal term. Thus conjunctive context must be passed top down, yet we are normalising bottom up. This means there is no guarantee that the conjunctive context is normalised. Example 12. Consider the following ACDTR program that uses conjunctive context matching. X = V \ X ⇐⇒ var(X) ∧ nonvar(V ) \| V. one(X) ⇐⇒ X = 1. not one(1) ⇐⇒ f alse. Consider the goal not one(A)∧one(A), which we expect should be normalised to f alse. Assume that the sub-term not one(A) is selected for normalisation first. The conjunctive context for not one(A) (and its subterm A) is one(A). No rule is applicable, so not one(A) is not reduced. Next the subterm one(A) is reduced. The second rule will fire resulting in the new term A = 1. Now the conjunctive context for the first term not one(A) has changed to A = 1, so we expect that A should be rewritten to the number ⊓ ⊔ 1. However not one(A) has already being considered for normalisation. The current Cadmium prototype solves this problem by re-normalising terms when and if the conjunctive context “changes”. For example, when the conjunctive context one(A) changes to A = 1, the term not one(X) will be renormalised to not one(1) by the first rule. The general execution algorithm for Cadmium is shown in Figure 2. Function normalise takes a term T , a substitution θ, a conjunctive context CC and a Boolean value Ch which keeps track of when the conjunctive context of the current subterm has changed. If Ch = true, then we can assume the substitution θ maps variables to normalised terms. For the initial goal, we assume θ is empty, otherwise if we are executing a body of a rule, then θ is the matching substitution. Operationally, normalise splits into three cases depending on what T is. If T is a variable, and the conjunctive context has changed (i.e. Ch = true), then θ(T ) is no longer guaranteed to be normalised. In this case we return the result of renormalising θ(T ) with respect to CC. Otherwise if Ch = f alse, we simply return θ(T ) which must be already normalised. If T is a conjunction T1 ∧ T2 , we repeatedly call normalise on each conjunct with the other added to the conjunctive context. This is repeated until a fixed point (i.e. further normalisation does not result in either conjunct changing) is reached, and then return the result of apply rule on the which we will discuss below. This fixed point calculation accounts for the case where the conjunctive context of a term changes, as shown in Example 12. Otherwise, if T is any other term of the form f (T1 , ..., Tn ), construct the new term T ′ by normalising each argument. Finally we return the result of apply rule applied to T ′ . The function call apply rule(T ′ ,CC) will attempt to apply a rule to normalised term T ′ with respect to conjunctive context CC. If a matching rule is found, then 12 normalise(T ,θ,CC,Ch) if is var(T ) if Ch return normalise(θ(T ),θ,CC,f alse) else return θ(T ) else if T = T1 ∧ T2 do T1′ := T1 T2′ := T2 T1 := normalise(T1′ ,θ,T2′ ∧ CC,true) T2 := normalise(T2′ ,θ,T1′ ∧ CC,true) while T1 6= T1′ ∧ T2 6= T2′ return apply rule(T1′ ∧ T2′ ,CC) else T = f (T1 , ..., Tn ) T ′ := f (normalise(T1 ,θ,CC,Ch), ..., normalise(Tn ,θ,CC,Ch)) return apply rule(T ′ ,CC) Fig. 2. Pseudo code of the Cadmium execution algorithm. the result of normalise(B,θ,CC,f alse) is returned, where B is the (renamed) rule body and θ is the matching substitution. Otherwise, T ′ is simply returned. 5 Related Work ACDTR is closely related to both TRS and CHRs, and in this section we compare the three languages. 5.1 AC Term Rewriting Systems The problem of dealing with associative commutative operators in TRS is well studied. A popular solution is to perform the rewriting modulo some permutation of the AC operators. Although this complicates the matching algorithm, the problem of trivial non-termination (e.g. by continually rewriting with respect to commutativity) is solved. ACDTR subsumes ACTRS (Associative Commutative TRS) in that we have introduced distributivity (via simpagation rules), and added some “CHR-style” concepts such as identifiers and propagation rules. Given an ACTRS program, we can map it to an equivalent ACDTR program by interpreting each ACTRS rule H → B as the ACDTR rule H ⇐⇒ B. We can now state the theorem relating ACTRS and ACDTR. Theorem 2. Let P be an ACTRS program and T a ground term, then T →∗ S under P iff hTa , ∅, ∅, ids(Ta )i ֌∗ hSa , ∅, ∅, Pi under α(P ) (where Ta = annotate(∅, T )) for some P and term(Sa ) = S. 13 5.2 CHRs and CHR∨ ACDTR has been deliberately designed to be an extension of CHRs. Several CHR concepts, e.g. propagation rules, etc., have been adapted. There are differences between CHRs and ACDTR. The main difference is that ACDTR does not have a “built-in” or “underlying” solver, i.e. ACDTR is not a constraint programming language. However it is possible to encode solvers directly as rules, e.g. the simple leq solver from Example 8. Another important difference is that CHRs is based on predicate logic, where there exists a distinction between predicate symbols (i.e. the names of the constraints) and functions (used to construct terms). ACDTR is based on equational logic between terms, hence there is no distinction between predicates and functions (a predicate is just a Boolean function). To overcome this, we assume the existence of a set Pred, which contains the set of function symbols that are Boolean functions. We assume that AC ∩ Pred = {∧(2) }. The mapping between a CHR program and an ACDTR program is simply α(P ) = P ∪ {X ∧ true ⇐⇒ X}.4 However, we assume program P is restricted as follows: – rules have no guards apart from implicit equality guards; and – the only built-in constraint is true and the initial goal G is also restricted: – G must be of the form G0 ∧ ... ∧ Gn for n > 0; – Each Gi is of the form fi (A0 , ..., Am ) for m ≥ 0 and fi ∈ Pred; – For all p ∈ Pos(Aj ), 0 ≤ j ≤ m we have that if Aj \|p = g(B0 , ..., Bq ) then g (q) 6∈ AC and g (q) 6∈ Pred. These conditions disallow predicate symbols from appearing as arguments in CHR constraints. Theorem 3. Let P be a CHR program, and G an initial goal both satisfying V the above conditions, then hG, ∅, true, ∅iV 1 ֌ h∅, S, true, T ii (for some T , i and V = vars(G)) under the theoretical operational semantics [4] for CHRs iff hGa , ∅, V, ids(Ga )i ֌ hSa , T ′ , V, Pi (for some T ′ , P) under ACDTR, where term(Sa ) = S1 ∧...∧Sn and S = {S1 #i1 , ..., Sn #in } for some identifiers i1 , ..., in . We believe that Theorem 3 could be extended to include CHR programs that extend an underlying solver, provided the rules for handling tell constraints are added to the ACDTR program. For example, we can combine rules for rational tree unification with the leq program from Example 8 to get a program equivalent to the traditional leq program under CHRs. ACDTR generalises CHRs by allowing other operators besides conjunction inside the head or body of rules. One such extension of CHRs has been studied before, namely CHR∨ [2] which allows disjunction in the body. Unlike ACDTR, 4 There is one slight difference in syntax: CHRs use ‘,’ to represent conjunction, whereas ACDTR uses ‘∧’. 14 which manipulates disjunction syntactically, CHR∨ typically finds solutions using backtracking search. One notable implementation of CHR∨ is [6], which has an operational semantics described as an and/or (∧/∨) tree rewriting system. A limited form of conjunctive context matching is used, similar to that used by ACDTR, based on the knowledge that conjunction ∧ distributes over disjunction ∨. ACDTR generalises this by distributing over all functions. 6 Future Work and Conclusions We have presented a powerful new rule-based programming language, ACDTR, that naturally extends both AC term rewriting and CHRs. The main contribution is the ability to match a rule against the conjunctive context of a (sub)term, taking advantage of the distributive property of conjunction over all possible functions. We have shown this is a natural way of expressing some problems, and by building the distributive property into the matching algorithm, we avoid non-termination issues that arise from naively implementing distribution (e.g. as rewrite rules). We intend that ACDTR will become the theoretical basis for the Cadmium constraint mapping language as part of the G12 project [7]. Work on ACDTR and Cadmium is ongoing, and there is a wide scope for future work, such as confluence, termination and implementation/optimisation issues. References 1. S. Abdennadher. Operational semantics and confluence of constraint propagation rules. In Gert Smolka, editor, Proceedings of the Third International Conference on Principles and Practice of Constraint Programming, LNCS 1330, pages 252–266. Springer-Verlag, 1997. 2. S. Abdennadher and H. Schütz. CHR∨ : A flexible query language. In International conference on Flexible Query Answering Systems, number 1495 in LNCS, pages 1–14, Roskilde, Denmark, 1998. Springer-Verlag. 3. F. Baader and T. Nipkow. Term rewriting and all that. Cambridge Univ. Press, 1998. 4. G. Duck, P. Stuckey, M. Garcia de la Banda, and C. Holzbaur. The refined operational semantics of constraint handling rules. In B. Demoen and V. Lifschitz, editors, Proceedings of the 20th International Conference on Logic Programming, LNCS 3132, pages 90–104. Springer-Verlag, September 2004. 5. T. Frühwirth. Theory and practice of constraint handling rules. Journal of Logic Programming, 37:95–138, 1998. 6. L. Menezes, J. Vitorino, and M. Aurelio. A High Performance CHR∨ Execution Engine. In Second Workshop on Constraint Handling Rules, Sitges, Spain, 2005. 7. P.J. Stuckey, M. Garcia de la Banda, M. Maher, K. Marriott, J. Slaney, Z. Somogyi, M. Wallace, and T. Walsh. The G12 project: Mapping solver independent models to efficient solutions. In M. Gabrielli and G. Gupta, editors, Proceedings of the 21st International Conference on Logic Programming, number 3668 in LNCS, pages 9–13. Springer-Verlag, 2005. 15 A Examples A.1 Further Motivating Examples Example 13 (Conjunctive Normal Form). One of the roles of mapping models is to convert a model written in an expressive language into a restricted language which is easy to solve. Many standard approaches to solving propositional formulae require that the formulae are in conjunctive normal form (CNF). Disjunction ∨ is distributive over ∧, which can be used to establish CNF in a direct way, using the oriented rule P ∨ Q ∧ R → (P ∨ Q) ∧ (P ∨ R). CNF conversion based on this rule can exponentially increase the size of the formula. This undesirable circumstance means that in practice CNF conversions are preferred that replace subformulae by new propositional atoms, which increases the formula size at most linearly. Let us formulate this approach in rewrite rules. To keep this example simple, we assume that the non-CNF subformula P ∨ Q ∧ R occurs in a positive context (for example by a preprocessing into negation normal form). We replace Q ∧ R by a new atom s defined by the logical implication s ⇒ (Q ∧ R). In rewrite rule form, we have P ∨ Q ∧ R → (P ∨ s) ∧ (¬s ∨ Q) ∧ (¬s ∨ R). (8) Unit resolution and unit subsumption can be formalised in rewrite rules. Here are two versions, one using conjunctive context and a regular one: with conj. context: regular: P \ P ⇐⇒ true P ∧P → P P ∧ (P ∨ Q) → P P ∧ ¬P → false P \ ¬P ⇐⇒ false P ∧ (¬P ∨ Q) → P ∧ Q We furthermore assume rules eliminating the logical constants true and false from conjunctions and disjunctions in the obvious way. Let us contrast the two rule sets for the formula (a∨b∧(c∨d))∧d. The following is a terminating rewrite history: with conj. context: regular: (a ∨ b ∧ (c ∨ d)) ∧ d (a ∨ b ∧ (c ∨ d)) ∧ d ֌ (a ∨ b ∧ (c ∨ true)) ∧ d ֌ (a ∨ b ∧ true) ∧ d ֌ ֌ (a ∨ s) ∧ (¬s ∨ b) ∧ (¬s ∨ c ∨ d) ∧ d (a ∨ s) ∧ (¬s ∨ b) ∧ true ∧ d ֌ (a ∨ b) ∧ d ֌ (a ∨ s) ∧ (¬s ∨ b) ∧ d 16 To obtain the simple conjunct (a ∨ b) using the regular rule format, a rule expressing binary resolution, i.e. from (P ∨ S) ∧ (¬S ∨ Q) follows (P ∨ Q), would be required. However, such a rule is undesirable as it would create arbitrary binary resolvents, increasing formula size. Moreover, the superfluous atom s remains in the formula. ⊓ ⊔ Example 14 (Type remapping). One of the main model mappings we are interested in expressing is where the type of a variable is changed from a high level type easy for modelling to a low level type easy to solve. A prime example of this is mapping a set variable x ranging over finite subsets of some fixed set s to an array x′ of 0/1 variables indexed by s. So for variable x we have e ∈ x ⇔ x′ [e] = 1. For this example we use the more concrete modelling syntax: t : x indicates variable x has type t, the types we are interested are l..u an integers in the range l to u, set of S a set ranging over elements in S, and array[I] of E an array indexed by set I of elements of type E. We use f orall and sum looping constructs which iterate over sets. This is expressed in ACDTR as follows. set of s : x ⇐⇒ array[s] of 0..1 : x′ ∧ map(x, x′ ) map(x, x′ ) \ x ⇐⇒ x′ array[s] of 0..1 : x \ card(x) ⇐⇒ sum(e in s) x[e] array[s] of 0..1 : x ∧ z :: (array[s] of 0..1 : z ∧ ⇐⇒ array[s] of 0..1 : y \ x ∩ y f orall(e in s) z[e] = x[e] && y[e]) array[s] of 0..1 : x ∧ z :: (array[s] of 0..1 : z ∧ ⇐⇒ array[s] of 0..1 : y \ x ∪ y f orall(e in s) z[e] = x[e] \|\| y[e]) array[s] of 0..1 : x \ x = ∅ ⇐⇒ f orall(e in s) x[e] = 0 card(t :: c) ⇐⇒ card(t) :: c (t1 :: c) ∪ t2 ⇐⇒ t1 ∪ t2 :: c t1 ∪ (t2 :: c) ⇐⇒ t1 ∪ t2 :: c (t1 :: c) ∩ t2 ⇐⇒ t1 ∩ t2 :: c t1 ∩ (t2 :: c) ⇐⇒ t1 ∩ t2 :: c (t1 :: c) = t2 ⇐⇒ t1 = t2 ∧ c t1 = (t2 :: c) ⇐⇒ t1 = t2 ∧ c (t1 :: c) ≤ t2 ⇐⇒ t1 ≤ t2 ∧ c t1 ≤ (t2 :: c) ⇐⇒ t1 ≤ t2 ∧ c (t :: c1 ) :: c2 ⇐⇒ t :: (c1 ∧ c2 ) maxOverlap(x, y, c) ⇐⇒ card(x ∩ y) ≤ c (typec) (vsubs) (card) (cap) (cup) (emptyset) (↑ card) (↑ cupl) (↑ cupr) (↑ capl) (↑ capr) (↑ eql) (↑ eqr) (↑ leql) (↑ leqr) (↑ cc) (maxO) The :: constructor adds some local conjunctive context to an arbitrary term (like where) and the last 11 rules bar 1 move this context outwards to the nearest predicate scope. The last rule defines the maxOverlap predicate. They are used to introduce new variables z and their type and the constraints upon then. As 17 an example, consider the following derivation: ֌maxO ֌typec ֌vsubs ֌typec ֌vsubs ֌cap ֌↑card ֌↑leql set of 1..n : x ∧ set of 1..n : y ∧ maxOverlap(x, y, 1) set of 1..n : x ∧ set of 1..n : y ∧ card(x ∩ y) ≤ 1 array[1..n] of 0..1 : x′ ∧ map(x, x′ ) ∧ set of 1..n : y ∧ card(x ∩ y) ≤ 1 array[1..n] of 0..1 : x′ ∧ map(x, x′ ) ∧ set of 1..n : y ∧ card(x′ ∩ y) ≤ 1 array[1..n] of 0..1 : x′ ∧ map(x, x′ ) ∧ array[1..n] of 0..1 : y ′ ∧ map(y, y ′ ) ∧ card(x′ ∩ y) ≤ 1 array[1..n] of 0..1 : x′ ∧ map(x, x′ ) ∧ array[1..n] of 0..1 : y ′ ∧ map(y, y ′ ) ∧ card(x′ ∩ y ′ ) ≤ 1 array[1..n] of 0..1 : x′ ∧ map(x, x′ ) ∧ array[1..n] of 0..1 : y ′ ∧ map(y, y ′ ) ∧ card(z :: (array[1..n] of 0..1 : z ∧ f orall(e in 1..n) z[e] = x′ [e] && y ′ [e]) ≤ 1 array[1..n] of 0..1 : x′ ∧ map(x, x′ ) ∧ array[1..n] of 0..1 : y ′ ∧ map(y, y ′ ) ∧ card(z) :: (array[1..n] of 0..1 : z ∧ f orall(e in 1..n) z[e] = x′ [e] && y ′ [e]) ≤ 1 array[1..n] of 0..1 : x′ ∧ map(x, x′ ) ∧ array[1..n] of 0..1 : y ′ ∧ map(y, y ′ ) ∧ card(z) ≤ 1 ∧ array[1..n] of 0..1 : z ∧ f orall(e in 1..n) z[e] = x′ [e] && y ′ [e] The final goal is a flat conjunction of constraints and types. It can be similarly translated into a conjunction of pseudo-Boolean constraints that can be sent to a finite domain solver, by unrolling f orall and replacing the arrays by sequences of n variables. ⊓ ⊔ Example 15 (Rational Tree Unification). We can directly express the rational tree unification algorithm of Colmerauer5 as an ACD term rewriting system. f (s1 , . . . sn ) = f (t1 , . . . , tn ) ⇐⇒ s1 = t1 ∧ · · · sn = tn f (s1 , . . . sn ) = g(t1 , . . . , tm ) ⇐⇒ f alse (split) (f ail) The (split) rule must be defined for each constructor f /n and the (fail) rule for each pair of different constructors f /n and g/m. The remaining rules are: x = x ⇐⇒ var(x) \| true t = x ⇐⇒ var(x) ∧ nonvar(t) \| x = t x = s \ x = t ⇐⇒ var(x) ∧ nonvar(s) ∧ size(s) ≤ size(t) \| s = t x = y \ x ⇐⇒ var(x) ∧ var(y) ∧ x 6≡ y \| y (id) (f lip) (tsubs) (vsubs) where size(t) is the size of the term t in terms of number of symbols, and ≡ is syntactic identity. Even though the goals are a single conjunction of constraints, ACD is used for succinctly expressing the (vsubs) rule which replaces one variable by another in any other position. 5 A. Colmerauer. Prolog and Infinite Trees. Logic Programming, APIC Studies in Data Processing (16). Academic Press. 1992 18 The following derivation illustrates the unification process in action. The underlined part show the matching elements ֌f lip ֌vsubs ֌vsubs ֌tsubs ֌split ֌split ֌tsubs ֌split ֌tsubs ֌split ֌id x = y ∧ f (f (x)) = x ∧ y = f (f (f (y))) x = y ∧ x = f (f (x)) ∧ y = f (f (f (y))) x = y ∧ y = f (f (x)) ∧ y = f (f (f (y))) x = y ∧ y = f (f (y)) ∧ y = f (f (f (y))) x = y ∧ y = f (f (y)) ∧ f (f (y)) = f (f (f (y))) x = y ∧ y = f (f (y)) ∧ f (y) = f (f (y)) x = y ∧ y = f (f (y)) ∧ y = f (y) x = y ∧ f (y) = f (f (y)) ∧ y = f (y) x = y ∧ y = f (y) ∧ y = f (y) x = y ∧ y = f (y) ∧ f (y) = f (y) x = y ∧ y = f (y) ∧ f (y) = f (y) x = y ∧ y = f (y) ∧ true ⊓ ⊔ A.2 Expanded Examples The purpose of this section is to show some example derivations under the operational semantics of ACDTR, rather than high-level descriptions. We allow for some shorthand, namely T #i = Ti . Identifiers and conjunctive context. In this section we explain parts of the derivation from Example 15 in more detail. The initial goal is x = y ∧ f (f (x)) = x ∧ y = f (f (f (y))) which corresponds to the initial state: h(((x1 = y2 )3 ∧ (f (f (x4 )5 )6 = x7 )8 )9 ∧ (y10 = f (f (f (y11 )12 )13 )14 )15 )16 , ∅, {x, y}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}i The initial state is a quadruple contained an annotated version of the goal, an empty propagation history, the set of variables in the goal and a set of “used” identifiers. The first derivation step is a Simplify transition with the f lip rule: h(((x1 = y2 )3 ∧ (f (f (x4 )5 )6 = x7 )8 )9 ∧ (y10 = f (f (f (y11 )12 )13 )14 )15 )16 , ∅, {x, y}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}i ֌ h(((x1 = y2 )3 ∧ (x17 = f (f (x18 )19 )20 )21 )9 ∧ (y10 = f (f (f (y11 )12 )13 )14 )15 )16 , ∅, {x, y}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21}i We have replaced the annotated subterm (f (f (x4 )5 )6 = x7 )8 with x17 = f (f (x18 )19 )20 )21 (i.e. flipped the operands to the equality) and reannotated the 19 new term with fresh identifiers. These were also added to the set of used identifiers. Since the propagation history is empty, it remains unchanged. The next derivation step is a Simpagate transition with the vsubs rule. h(((x1 = y2 )3 ∧ (x17 = f (f (x18 )19 )20 )21 )9 ∧ (y10 = f (f (f (y11 )12 )13 )14 )15 )16 , ∅, {x, y}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21}i ֌ h(((x1 = y2 )3 ∧ (y21 = f (f (x18 )19 )20 )21 )9 ∧ (y10 = f (f (f (y11 )12 )13 )14 )15 )16 , ∅, {x, y}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22}i The conjunctive context for subterm x17 is cc(Ga , p) = (x1 = y2 )3 ∧ (y10 = f (f (f (y11 )12 )13 )14 )15 ∧ true where Ga is the current goal and p is the position of x17 . The first conjunct matches the conjunctive context of the vsubs rule, thus subterm x17 is replaced with y21 . Identifier 21 is added to the list of used identifiers. Execution proceeds until the final state h(x = y ∧ y = f (y)) ∧ true, ∅, {x, y}, Pi is reached, for some annotation of the goal and some set of identifiers P. This is a final state because no more rules are applicable to it. AC matching and propagation histories. Consider the propagation rule from the leq program: trans @ leq(X, Y ) ∧ leq(Y, Z) =⇒ X 6≡ Y ∧ Y 6≡ Z \| leq(X, Z) and the initial state hleq(A1 , B2 )3 ∧4 leq(B5 , A6 )7 , ∅, {A, B}, {1, 2, 3, 4, 5, 6, 7}i. We can apply Propagate directly (i.e. without permuting the conjunction) to arrive at the state: h(leq(A1 , B2 )3 ∧4 leq(B5 , A6 )7 ) ∧8 leq(A9 , A10 )11 , {trans @ (3 1 2 7 6 5)}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}i. The propagation history prevents the rule from firing on the same terms again, however we can permute the terms to find a new matching. Namely, we can permute the annotated goal (which we call Ga ) (leq(A1 , B2 )3 ∧4 leq(B5 , A6 )7 ) ∧8 leq(A9 , A10 )11 to (leq(B5 , A6 )7 ∧4 leq(A1 , B2 )3 ) ∧8 leq(A9 , A10 )11 . 20 The latter is an element of [Ga ]AC , and the identifiers have been preserved in the correct way. The entry trans @ (7 6 5 3 1 2) is not in the propagation history, so we can apply Propagate again to arrive at: h((leq(B5 , A6 )7 ∧4 leq(A1 , B2 )3 ) ∧12 leq(B13 , B14 )15 ) ∧8 leq(A9 , A10 )11 , {trans @ (3 1 2 7 6 5), trans @ (7 6 5 3 1 2)}, {1...15}i. Now the propagation history prevents the rule trans being applied to the first two leq constraints. The guard also prevents the trans rule firing on either of the two new constraints,6 thus we have reached a final state. Updating propagation histories. Consider a modified version of the previous example, now with two rules: X ∧ X ⇐⇒ X trans @ leq(X, Y ) ∧ leq(Y, Z) =⇒ leq(X, Z) The first rule enforces idempotence of conjunction. Consider the initial state: hleq(A1 , A2 )3 ∧4 leq(A5 , A6 )7 ∧8 leq(A9 , A10 )11 , ∅, {A}, {1...11}i We apply the trans rule to the first two copies of the leq constraint (with identifiers 3 and 7). hleq(A1 , A2 )3 ∧4 leq(A5 , A6 )7 ∧8 leq(A9 , A10 )11 ∧12 leq(A13 , A14 )15 , {trans @ (3 1 2 7 5 6)}, {A}, {1...15}i Next we apply idempotence to leq constraints with identifiers 7 and 11. hleq(A1 , A2 )3 ∧4 leq(A16 , A17 )18 ∧12 leq(A13 , A14 )15 , {trans @ (3 1 2 7 5 6), trans @ (3 1 2 18 16 17)}, {A}, {1...18}i An extra entry (trans @ (3 1 2 18 16 17)) is added to the propagation history in order to satisfy the requirements of Definition 6. This is because we have replaced the annotated constraint leq(A5 , A6 )7 with the newly annotated term leq(A16 , A17 )18 , which defines an identifier renaming ρ = {5 7→ 16, 6 7→ 17, 7 7→ 18}. Since E = (trans @ (3 1 2 7 5 6)) is an element of the propagation history, we have that ρ(E) = (trans @ (3 1 2 18 16 17)) must also be an element, and hence the history is expanded. 6 Without the guard both ACDTR and CHRs are not guaranteed to terminate. 21	6